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# Properties

 Label 1856.2.a.p Level $1856$ Weight $2$ Character orbit 1856.a Self dual yes Analytic conductor $14.820$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8202346151$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ q + 3 * q^3 + 3 * q^5 - 2 * q^7 + 6 * q^9 $$q + 3 q^{3} + 3 q^{5} - 2 q^{7} + 6 q^{9} + q^{11} - 3 q^{13} + 9 q^{15} - 4 q^{17} + 8 q^{19} - 6 q^{21} + 4 q^{25} + 9 q^{27} + q^{29} + 3 q^{31} + 3 q^{33} - 6 q^{35} + 8 q^{37} - 9 q^{39} - 2 q^{41} - 7 q^{43} + 18 q^{45} + 11 q^{47} - 3 q^{49} - 12 q^{51} - q^{53} + 3 q^{55} + 24 q^{57} + 4 q^{59} - 4 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} - 2 q^{71} - 12 q^{73} + 12 q^{75} - 2 q^{77} - 7 q^{79} + 9 q^{81} - 12 q^{85} + 3 q^{87} - 6 q^{89} + 6 q^{91} + 9 q^{93} + 24 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100})$$ q + 3 * q^3 + 3 * q^5 - 2 * q^7 + 6 * q^9 + q^11 - 3 * q^13 + 9 * q^15 - 4 * q^17 + 8 * q^19 - 6 * q^21 + 4 * q^25 + 9 * q^27 + q^29 + 3 * q^31 + 3 * q^33 - 6 * q^35 + 8 * q^37 - 9 * q^39 - 2 * q^41 - 7 * q^43 + 18 * q^45 + 11 * q^47 - 3 * q^49 - 12 * q^51 - q^53 + 3 * q^55 + 24 * q^57 + 4 * q^59 - 4 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 - 2 * q^71 - 12 * q^73 + 12 * q^75 - 2 * q^77 - 7 * q^79 + 9 * q^81 - 12 * q^85 + 3 * q^87 - 6 * q^89 + 6 * q^91 + 9 * q^93 + 24 * q^95 - 6 * q^97 + 6 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 3.00000 0 3.00000 0 −2.00000 0 6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.a.p 1
4.b odd 2 1 1856.2.a.b 1
8.b even 2 1 58.2.a.a 1
8.d odd 2 1 464.2.a.f 1
24.f even 2 1 4176.2.a.bh 1
24.h odd 2 1 522.2.a.k 1
40.f even 2 1 1450.2.a.i 1
40.i odd 4 2 1450.2.b.f 2
56.h odd 2 1 2842.2.a.d 1
88.b odd 2 1 7018.2.a.c 1
104.e even 2 1 9802.2.a.d 1
232.g even 2 1 1682.2.a.j 1
232.l odd 4 2 1682.2.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.a 1 8.b even 2 1
464.2.a.f 1 8.d odd 2 1
522.2.a.k 1 24.h odd 2 1
1450.2.a.i 1 40.f even 2 1
1450.2.b.f 2 40.i odd 4 2
1682.2.a.j 1 232.g even 2 1
1682.2.b.e 2 232.l odd 4 2
1856.2.a.b 1 4.b odd 2 1
1856.2.a.p 1 1.a even 1 1 trivial
2842.2.a.d 1 56.h odd 2 1
4176.2.a.bh 1 24.f even 2 1
7018.2.a.c 1 88.b odd 2 1
9802.2.a.d 1 104.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3} - 3$$ T3 - 3 $$T_{5} - 3$$ T5 - 3 $$T_{17} + 4$$ T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 3$$
$7$ $$T + 2$$
$11$ $$T - 1$$
$13$ $$T + 3$$
$17$ $$T + 4$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T - 1$$
$31$ $$T - 3$$
$37$ $$T - 8$$
$41$ $$T + 2$$
$43$ $$T + 7$$
$47$ $$T - 11$$
$53$ $$T + 1$$
$59$ $$T - 4$$
$61$ $$T + 4$$
$67$ $$T - 4$$
$71$ $$T + 2$$
$73$ $$T + 12$$
$79$ $$T + 7$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 6$$


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# Reinforcement Learning

Modified 2019-04-24 by Liam Paull

This section describes the basic procedure for making a submission with a model trained in simulation using reinforcement learning with PyTorch. It can be used as a starting point for any of the LF, LFV, and LFVI challenges.

That you have made a submission with the PyTorch template.

You win the AI-DO!

No questions found. You can ask a question on the website.

format: plain


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# what is the definition of cosine , sine [duplicate]

I know that sine is the ratio of the perpendicular to the hypotenuse of an acute angle. Similarly cosine is the ratio of the base and hypotenuse .

But now I found that there is sine and cosine of an angle over 90 degrees or negative degrees . I do not understand how they can determine the value of these sine and cosine . I have heard about the unit circle , and I find it is not convincing to determined the value of sine and cosine just by looking at the picture . I mean , is there any others method like using equations or theorems to calculate the cosine of an over 90 degrees?

## marked as duplicate by wythagoras, N. F. Taussig, Tim Raczkowski, muaddib, user147263 Jul 27 '15 at 21:46

Suppose that the black dot on the rear wheel (where the wheel connects with the coupling rod) was a distance of $1$ from the center of the wheel, and its initial position was directly to the right of the center of the wheel.

Given that the wheel had turned and angle of $\theta$ (from its initial position), Vertically, how far above the center of the wheel would the black dot be? Horizontally, how far to the right of the the center of the wheel would the black do be (a negative value would mean that it was to the left)?

Try drawing some diagrams of the wheel in different positions (drawing reference triangles would be helpful here).

Also try drawing a triangle with a hypotenuse of $1$. If $\theta$ is known, what would the length of the opposite side be? What would the length of the adjacent side be?

There are many definitions, but geometrically I believe the unit-circle definition is the best. It allows you to understand easly many trigonometric relations.

When you press the cos button on your calculator it probably takes use of the following definition:

cos(x) = $\sum_{0}^{\infty}\frac{(-1)^n x^{2n}}{2n!}$

• Depends on the calculator. Older calculators use the CORDIC algorithm. – Omnomnomnom Jul 27 '15 at 18:42
• This is extremely unhelpful based on the level of knowledge of the question. – Aloizio Macedo Jul 27 '15 at 19:27
• While this a possible definition of cosine, it is not very helpful intuitively for someone beginning off. I think it's better to derive this result assuming the unit curcle definition through Taylor Series. – Cataline Jul 27 '15 at 19:36