# Properties

 Label 6336.2.a.j Level $6336$ Weight $2$ Character orbit 6336.a Self dual yes Analytic conductor $50.593$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{5} + 2 q^{7}+O(q^{10})$$ q - 3 * q^5 + 2 * q^7 $$q - 3 q^{5} + 2 q^{7} - q^{11} + 4 q^{13} - 6 q^{17} - 8 q^{19} + 3 q^{23} + 4 q^{25} + 5 q^{31} - 6 q^{35} + q^{37} + 10 q^{43} - 3 q^{49} - 6 q^{53} + 3 q^{55} + 3 q^{59} + 4 q^{61} - 12 q^{65} + q^{67} - 15 q^{71} - 4 q^{73} - 2 q^{77} + 2 q^{79} + 6 q^{83} + 18 q^{85} + 9 q^{89} + 8 q^{91} + 24 q^{95} - 7 q^{97}+O(q^{100})$$ q - 3 * q^5 + 2 * q^7 - q^11 + 4 * q^13 - 6 * q^17 - 8 * q^19 + 3 * q^23 + 4 * q^25 + 5 * q^31 - 6 * q^35 + q^37 + 10 * q^43 - 3 * q^49 - 6 * q^53 + 3 * q^55 + 3 * q^59 + 4 * q^61 - 12 * q^65 + q^67 - 15 * q^71 - 4 * q^73 - 2 * q^77 + 2 * q^79 + 6 * q^83 + 18 * q^85 + 9 * q^89 + 8 * q^91 + 24 * q^95 - 7 * 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 −3.00000 0 2.00000 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.j 1
3.b odd 2 1 704.2.a.f 1
4.b odd 2 1 6336.2.a.i 1
8.b even 2 1 396.2.a.c 1
8.d odd 2 1 1584.2.a.p 1
12.b even 2 1 704.2.a.i 1
24.f even 2 1 176.2.a.a 1
24.h odd 2 1 44.2.a.a 1
33.d even 2 1 7744.2.a.m 1
40.f even 2 1 9900.2.a.h 1
40.i odd 4 2 9900.2.c.g 2
48.i odd 4 2 2816.2.c.e 2
48.k even 4 2 2816.2.c.k 2
72.j odd 6 2 3564.2.i.j 2
72.n even 6 2 3564.2.i.a 2
88.b odd 2 1 4356.2.a.j 1
120.i odd 2 1 1100.2.a.b 1
120.m even 2 1 4400.2.a.v 1
120.q odd 4 2 4400.2.b.k 2
120.w even 4 2 1100.2.b.c 2
132.d odd 2 1 7744.2.a.bc 1
168.e odd 2 1 8624.2.a.w 1
168.i even 2 1 2156.2.a.a 1
168.s odd 6 2 2156.2.i.b 2
168.ba even 6 2 2156.2.i.c 2
264.m even 2 1 484.2.a.a 1
264.p odd 2 1 1936.2.a.c 1
264.t odd 10 4 484.2.e.a 4
264.u even 10 4 484.2.e.b 4
312.b odd 2 1 7436.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 24.h odd 2 1
176.2.a.a 1 24.f even 2 1
396.2.a.c 1 8.b even 2 1
484.2.a.a 1 264.m even 2 1
484.2.e.a 4 264.t odd 10 4
484.2.e.b 4 264.u even 10 4
704.2.a.f 1 3.b odd 2 1
704.2.a.i 1 12.b even 2 1
1100.2.a.b 1 120.i odd 2 1
1100.2.b.c 2 120.w even 4 2
1584.2.a.p 1 8.d odd 2 1
1936.2.a.c 1 264.p odd 2 1
2156.2.a.a 1 168.i even 2 1
2156.2.i.b 2 168.s odd 6 2
2156.2.i.c 2 168.ba even 6 2
2816.2.c.e 2 48.i odd 4 2
2816.2.c.k 2 48.k even 4 2
3564.2.i.a 2 72.n even 6 2
3564.2.i.j 2 72.j odd 6 2
4356.2.a.j 1 88.b odd 2 1
4400.2.a.v 1 120.m even 2 1
4400.2.b.k 2 120.q odd 4 2
6336.2.a.i 1 4.b odd 2 1
6336.2.a.j 1 1.a even 1 1 trivial
7436.2.a.d 1 312.b odd 2 1
7744.2.a.m 1 33.d even 2 1
7744.2.a.bc 1 132.d odd 2 1
8624.2.a.w 1 168.e odd 2 1
9900.2.a.h 1 40.f even 2 1
9900.2.c.g 2 40.i odd 4 2

## Hecke kernels

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

 $$T_{5} + 3$$ T5 + 3 $$T_{7} - 2$$ T7 - 2 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 8$$ T19 + 8 $$T_{23} - 3$$ T23 - 3 $$T_{47}$$ T47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T - 2$$
$11$ $$T + 1$$
$13$ $$T - 4$$
$17$ $$T + 6$$
$19$ $$T + 8$$
$23$ $$T - 3$$
$29$ $$T$$
$31$ $$T - 5$$
$37$ $$T - 1$$
$41$ $$T$$
$43$ $$T - 10$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 3$$
$61$ $$T - 4$$
$67$ $$T - 1$$
$71$ $$T + 15$$
$73$ $$T + 4$$
$79$ $$T - 2$$
$83$ $$T - 6$$
$89$ $$T - 9$$
$97$ $$T + 7$$