# Properties

 Label 6336.2.a.cm 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 99) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{5} + 2q^{7} + O(q^{10})$$ $$q + 4q^{5} + 2q^{7} - q^{11} + 2q^{13} + 2q^{17} - 6q^{19} - 4q^{23} + 11q^{25} + 6q^{29} - 4q^{31} + 8q^{35} + 6q^{37} - 10q^{41} + 6q^{43} + 8q^{47} - 3q^{49} - 4q^{55} + 4q^{59} + 6q^{61} + 8q^{65} + 8q^{67} - 2q^{73} - 2q^{77} + 10q^{79} + 12q^{83} + 8q^{85} + 4q^{91} - 24q^{95} + 2q^{97} + O(q^{100})$$

## 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 4.00000 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.cm 1
3.b odd 2 1 6336.2.a.f 1
4.b odd 2 1 6336.2.a.cl 1
8.b even 2 1 1584.2.a.b 1
8.d odd 2 1 99.2.a.a 1
12.b even 2 1 6336.2.a.b 1
24.f even 2 1 99.2.a.c yes 1
24.h odd 2 1 1584.2.a.r 1
40.e odd 2 1 2475.2.a.j 1
40.k even 4 2 2475.2.c.b 2
56.e even 2 1 4851.2.a.g 1
72.l even 6 2 891.2.e.c 2
72.p odd 6 2 891.2.e.j 2
88.g even 2 1 1089.2.a.h 1
120.m even 2 1 2475.2.a.c 1
120.q odd 4 2 2475.2.c.g 2
168.e odd 2 1 4851.2.a.o 1
264.p odd 2 1 1089.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.a.a 1 8.d odd 2 1
99.2.a.c yes 1 24.f even 2 1
891.2.e.c 2 72.l even 6 2
891.2.e.j 2 72.p odd 6 2
1089.2.a.d 1 264.p odd 2 1
1089.2.a.h 1 88.g even 2 1
1584.2.a.b 1 8.b even 2 1
1584.2.a.r 1 24.h odd 2 1
2475.2.a.c 1 120.m even 2 1
2475.2.a.j 1 40.e odd 2 1
2475.2.c.b 2 40.k even 4 2
2475.2.c.g 2 120.q odd 4 2
4851.2.a.g 1 56.e even 2 1
4851.2.a.o 1 168.e odd 2 1
6336.2.a.b 1 12.b even 2 1
6336.2.a.f 1 3.b odd 2 1
6336.2.a.cl 1 4.b odd 2 1
6336.2.a.cm 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5} - 4$$ $$T_{7} - 2$$ $$T_{13} - 2$$ $$T_{17} - 2$$ $$T_{19} + 6$$ $$T_{23} + 4$$ $$T_{47} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$-2 + T$$
$11$ $$1 + T$$
$13$ $$-2 + T$$
$17$ $$-2 + T$$
$19$ $$6 + T$$
$23$ $$4 + T$$
$29$ $$-6 + T$$
$31$ $$4 + T$$
$37$ $$-6 + T$$
$41$ $$10 + T$$
$43$ $$-6 + T$$
$47$ $$-8 + T$$
$53$ $$T$$
$59$ $$-4 + T$$
$61$ $$-6 + T$$
$67$ $$-8 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$-10 + T$$
$83$ $$-12 + T$$
$89$ $$T$$
$97$ $$-2 + T$$