# Properties

 Label 6336.2.a.x Level $6336$ Weight $2$ Character orbit 6336.a Self dual yes Analytic conductor $50.593$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$6336 = 2^{6} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6336.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$50.5932147207$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} + 4 q^{7}+O(q^{10})$$ q - 2 * q^5 + 4 * q^7 $$q - 2 q^{5} + 4 q^{7} + q^{11} + 2 q^{13} + 2 q^{17} - 8 q^{23} - q^{25} - 6 q^{29} - 8 q^{31} - 8 q^{35} - 6 q^{37} + 2 q^{41} - 8 q^{47} + 9 q^{49} + 6 q^{53} - 2 q^{55} - 4 q^{59} - 6 q^{61} - 4 q^{65} + 4 q^{67} - 14 q^{73} + 4 q^{77} - 4 q^{79} + 12 q^{83} - 4 q^{85} + 6 q^{89} + 8 q^{91} + 2 q^{97}+O(q^{100})$$ q - 2 * q^5 + 4 * q^7 + q^11 + 2 * q^13 + 2 * q^17 - 8 * q^23 - q^25 - 6 * q^29 - 8 * q^31 - 8 * q^35 - 6 * q^37 + 2 * q^41 - 8 * q^47 + 9 * q^49 + 6 * q^53 - 2 * q^55 - 4 * q^59 - 6 * q^61 - 4 * q^65 + 4 * q^67 - 14 * q^73 + 4 * q^77 - 4 * q^79 + 12 * q^83 - 4 * q^85 + 6 * q^89 + 8 * q^91 + 2 * q^97

## 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 0 0 −2.00000 0 4.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 6336.2.a.x 1
3.b odd 2 1 2112.2.a.bb 1
4.b odd 2 1 6336.2.a.n 1
8.b even 2 1 99.2.a.b 1
8.d odd 2 1 1584.2.a.o 1
12.b even 2 1 2112.2.a.j 1
24.f even 2 1 528.2.a.g 1
24.h odd 2 1 33.2.a.a 1
40.f even 2 1 2475.2.a.g 1
40.i odd 4 2 2475.2.c.d 2
56.h odd 2 1 4851.2.a.b 1
72.j odd 6 2 891.2.e.e 2
72.n even 6 2 891.2.e.g 2
88.b odd 2 1 1089.2.a.j 1
120.i odd 2 1 825.2.a.a 1
120.w even 4 2 825.2.c.a 2
168.i even 2 1 1617.2.a.j 1
264.m even 2 1 363.2.a.b 1
264.p odd 2 1 5808.2.a.t 1
264.t odd 10 4 363.2.e.e 4
264.u even 10 4 363.2.e.g 4
312.b odd 2 1 5577.2.a.a 1
408.b odd 2 1 9537.2.a.m 1
1320.u even 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 24.h odd 2 1
99.2.a.b 1 8.b even 2 1
363.2.a.b 1 264.m even 2 1
363.2.e.e 4 264.t odd 10 4
363.2.e.g 4 264.u even 10 4
528.2.a.g 1 24.f even 2 1
825.2.a.a 1 120.i odd 2 1
825.2.c.a 2 120.w even 4 2
891.2.e.e 2 72.j odd 6 2
891.2.e.g 2 72.n even 6 2
1089.2.a.j 1 88.b odd 2 1
1584.2.a.o 1 8.d odd 2 1
1617.2.a.j 1 168.i even 2 1
2112.2.a.j 1 12.b even 2 1
2112.2.a.bb 1 3.b odd 2 1
2475.2.a.g 1 40.f even 2 1
2475.2.c.d 2 40.i odd 4 2
4851.2.a.b 1 56.h odd 2 1
5577.2.a.a 1 312.b odd 2 1
5808.2.a.t 1 264.p odd 2 1
6336.2.a.n 1 4.b odd 2 1
6336.2.a.x 1 1.a even 1 1 trivial
9075.2.a.q 1 1320.u even 2 1
9537.2.a.m 1 408.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6336))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 4$$ T7 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2 $$T_{19}$$ T19 $$T_{23} + 8$$ T23 + 8 $$T_{47} + 8$$ T47 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T - 4$$
$11$ $$T - 1$$
$13$ $$T - 2$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T + 6$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 4$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T - 2$$
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