Properties

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

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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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{5} - 2q^{7} + O(q^{10}) \) \( q - 2q^{5} - 2q^{7} - q^{11} - 2q^{13} - 4q^{17} - 6q^{19} - q^{25} + 8q^{29} - 8q^{31} + 4q^{35} + 10q^{37} - 8q^{41} - 2q^{43} + 8q^{47} - 3q^{49} + 2q^{53} + 2q^{55} - 12q^{59} + 10q^{61} + 4q^{65} + 12q^{67} - 8q^{71} + 6q^{73} + 2q^{77} - 2q^{79} - 16q^{83} + 8q^{85} + 14q^{89} + 4q^{91} + 12q^{95} - 2q^{97} + O(q^{100}) \)

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 −2.00000 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.a 1
3.b odd 2 1 132.2.a.b 1
4.b odd 2 1 1584.2.a.e 1
5.b even 2 1 9900.2.a.w 1
5.c odd 4 2 9900.2.c.f 2
8.b even 2 1 6336.2.a.ca 1
8.d odd 2 1 6336.2.a.cg 1
9.c even 3 2 3564.2.i.i 2
9.d odd 6 2 3564.2.i.d 2
11.b odd 2 1 4356.2.a.d 1
12.b even 2 1 528.2.a.e 1
15.d odd 2 1 3300.2.a.f 1
15.e even 4 2 3300.2.c.j 2
21.c even 2 1 6468.2.a.b 1
24.f even 2 1 2112.2.a.u 1
24.h odd 2 1 2112.2.a.c 1
33.d even 2 1 1452.2.a.f 1
33.f even 10 4 1452.2.i.d 4
33.h odd 10 4 1452.2.i.e 4
132.d odd 2 1 5808.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.2.a.b 1 3.b odd 2 1
396.2.a.a 1 1.a even 1 1 trivial
528.2.a.e 1 12.b even 2 1
1452.2.a.f 1 33.d even 2 1
1452.2.i.d 4 33.f even 10 4
1452.2.i.e 4 33.h odd 10 4
1584.2.a.e 1 4.b odd 2 1
2112.2.a.c 1 24.h odd 2 1
2112.2.a.u 1 24.f even 2 1
3300.2.a.f 1 15.d odd 2 1
3300.2.c.j 2 15.e even 4 2
3564.2.i.d 2 9.d odd 6 2
3564.2.i.i 2 9.c even 3 2
4356.2.a.d 1 11.b odd 2 1
5808.2.a.m 1 132.d odd 2 1
6336.2.a.ca 1 8.b even 2 1
6336.2.a.cg 1 8.d odd 2 1
6468.2.a.b 1 21.c even 2 1
9900.2.a.w 1 5.b even 2 1
9900.2.c.f 2 5.c odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)

Hecke kernels

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

\( T_{5} + 2 \)
\( T_{7} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + 2 T + 7 T^{2} \)
$11$ \( 1 + T \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 4 T + 17 T^{2} \)
$19$ \( 1 + 6 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 8 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 10 T + 37 T^{2} \)
$41$ \( 1 + 8 T + 41 T^{2} \)
$43$ \( 1 + 2 T + 43 T^{2} \)
$47$ \( 1 - 8 T + 47 T^{2} \)
$53$ \( 1 - 2 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 10 T + 61 T^{2} \)
$67$ \( 1 - 12 T + 67 T^{2} \)
$71$ \( 1 + 8 T + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 + 2 T + 79 T^{2} \)
$83$ \( 1 + 16 T + 83 T^{2} \)
$89$ \( 1 - 14 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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