Properties

Label 304.2.a.a
Level $304$
Weight $2$
Character orbit 304.a
Self dual yes
Analytic conductor $2.427$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.42745222145\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} - q^{5} + 3 q^{7} + q^{9} + O(q^{10}) \) \( q - 2 q^{3} - q^{5} + 3 q^{7} + q^{9} - 5 q^{11} - 4 q^{13} + 2 q^{15} - 3 q^{17} + q^{19} - 6 q^{21} - 8 q^{23} - 4 q^{25} + 4 q^{27} - 2 q^{29} - 4 q^{31} + 10 q^{33} - 3 q^{35} + 10 q^{37} + 8 q^{39} + 10 q^{41} - q^{43} - q^{45} + q^{47} + 2 q^{49} + 6 q^{51} - 4 q^{53} + 5 q^{55} - 2 q^{57} - 6 q^{59} - 13 q^{61} + 3 q^{63} + 4 q^{65} + 12 q^{67} + 16 q^{69} - 2 q^{71} + 9 q^{73} + 8 q^{75} - 15 q^{77} - 8 q^{79} - 11 q^{81} + 12 q^{83} + 3 q^{85} + 4 q^{87} + 12 q^{89} - 12 q^{91} + 8 q^{93} - q^{95} - 8 q^{97} - 5 q^{99} + O(q^{100}) \)

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 −2.00000 0 −1.00000 0 3.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 304.2.a.a 1
3.b odd 2 1 2736.2.a.q 1
4.b odd 2 1 76.2.a.a 1
5.b even 2 1 7600.2.a.p 1
8.b even 2 1 1216.2.a.q 1
8.d odd 2 1 1216.2.a.c 1
12.b even 2 1 684.2.a.b 1
19.b odd 2 1 5776.2.a.p 1
20.d odd 2 1 1900.2.a.b 1
20.e even 4 2 1900.2.c.b 2
28.d even 2 1 3724.2.a.a 1
44.c even 2 1 9196.2.a.f 1
76.d even 2 1 1444.2.a.a 1
76.f even 6 2 1444.2.e.c 2
76.g odd 6 2 1444.2.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.a.a 1 4.b odd 2 1
304.2.a.a 1 1.a even 1 1 trivial
684.2.a.b 1 12.b even 2 1
1216.2.a.c 1 8.d odd 2 1
1216.2.a.q 1 8.b even 2 1
1444.2.a.a 1 76.d even 2 1
1444.2.e.a 2 76.g odd 6 2
1444.2.e.c 2 76.f even 6 2
1900.2.a.b 1 20.d odd 2 1
1900.2.c.b 2 20.e even 4 2
2736.2.a.q 1 3.b odd 2 1
3724.2.a.a 1 28.d even 2 1
5776.2.a.p 1 19.b odd 2 1
7600.2.a.p 1 5.b even 2 1
9196.2.a.f 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(304))\):

\( T_{3} + 2 \)
\( T_{5} + 1 \)
\( T_{7} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( 1 + T \)
$7$ \( -3 + T \)
$11$ \( 5 + T \)
$13$ \( 4 + T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( 8 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( -10 + T \)
$41$ \( -10 + T \)
$43$ \( 1 + T \)
$47$ \( -1 + T \)
$53$ \( 4 + T \)
$59$ \( 6 + T \)
$61$ \( 13 + T \)
$67$ \( -12 + T \)
$71$ \( 2 + T \)
$73$ \( -9 + T \)
$79$ \( 8 + T \)
$83$ \( -12 + T \)
$89$ \( -12 + T \)
$97$ \( 8 + T \)
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