# Properties

 Label 396.2.a.c Level $396$ Weight $2$ Character orbit 396.a Self dual yes Analytic conductor $3.162$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$396 = 2^{2} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 396.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.16207592004$$ 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 396.2.a.c 1
3.b odd 2 1 44.2.a.a 1
4.b odd 2 1 1584.2.a.p 1
5.b even 2 1 9900.2.a.h 1
5.c odd 4 2 9900.2.c.g 2
8.b even 2 1 6336.2.a.j 1
8.d odd 2 1 6336.2.a.i 1
9.c even 3 2 3564.2.i.a 2
9.d odd 6 2 3564.2.i.j 2
11.b odd 2 1 4356.2.a.j 1
12.b even 2 1 176.2.a.a 1
15.d odd 2 1 1100.2.a.b 1
15.e even 4 2 1100.2.b.c 2
21.c even 2 1 2156.2.a.a 1
21.g even 6 2 2156.2.i.c 2
21.h odd 6 2 2156.2.i.b 2
24.f even 2 1 704.2.a.i 1
24.h odd 2 1 704.2.a.f 1
33.d even 2 1 484.2.a.a 1
33.f even 10 4 484.2.e.b 4
33.h odd 10 4 484.2.e.a 4
39.d odd 2 1 7436.2.a.d 1
48.i odd 4 2 2816.2.c.e 2
48.k even 4 2 2816.2.c.k 2
60.h even 2 1 4400.2.a.v 1
60.l odd 4 2 4400.2.b.k 2
84.h odd 2 1 8624.2.a.w 1
132.d odd 2 1 1936.2.a.c 1
264.m even 2 1 7744.2.a.m 1
264.p odd 2 1 7744.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 3.b odd 2 1
176.2.a.a 1 12.b even 2 1
396.2.a.c 1 1.a even 1 1 trivial
484.2.a.a 1 33.d even 2 1
484.2.e.a 4 33.h odd 10 4
484.2.e.b 4 33.f even 10 4
704.2.a.f 1 24.h odd 2 1
704.2.a.i 1 24.f even 2 1
1100.2.a.b 1 15.d odd 2 1
1100.2.b.c 2 15.e even 4 2
1584.2.a.p 1 4.b odd 2 1
1936.2.a.c 1 132.d odd 2 1
2156.2.a.a 1 21.c even 2 1
2156.2.i.b 2 21.h odd 6 2
2156.2.i.c 2 21.g 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 9.c even 3 2
3564.2.i.j 2 9.d odd 6 2
4356.2.a.j 1 11.b odd 2 1
4400.2.a.v 1 60.h even 2 1
4400.2.b.k 2 60.l odd 4 2
6336.2.a.i 1 8.d odd 2 1
6336.2.a.j 1 8.b even 2 1
7436.2.a.d 1 39.d odd 2 1
7744.2.a.m 1 264.m even 2 1
7744.2.a.bc 1 264.p odd 2 1
8624.2.a.w 1 84.h odd 2 1
9900.2.a.h 1 5.b even 2 1
9900.2.c.g 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(396))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

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