Properties

Label 304.2.a.f
Level $304$
Weight $2$
Character orbit 304.a
Self dual yes
Analytic conductor $2.427$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 3.00000 0 1.00000 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.f 1
3.b odd 2 1 2736.2.a.c 1
4.b odd 2 1 19.2.a.a 1
5.b even 2 1 7600.2.a.c 1
8.b even 2 1 1216.2.a.b 1
8.d odd 2 1 1216.2.a.o 1
12.b even 2 1 171.2.a.b 1
19.b odd 2 1 5776.2.a.c 1
20.d odd 2 1 475.2.a.b 1
20.e even 4 2 475.2.b.a 2
28.d even 2 1 931.2.a.a 1
28.f even 6 2 931.2.f.b 2
28.g odd 6 2 931.2.f.c 2
44.c even 2 1 2299.2.a.b 1
52.b odd 2 1 3211.2.a.a 1
60.h even 2 1 4275.2.a.i 1
68.d odd 2 1 5491.2.a.b 1
76.d even 2 1 361.2.a.b 1
76.f even 6 2 361.2.c.a 2
76.g odd 6 2 361.2.c.c 2
76.k even 18 6 361.2.e.e 6
76.l odd 18 6 361.2.e.d 6
84.h odd 2 1 8379.2.a.j 1
228.b odd 2 1 3249.2.a.d 1
380.d even 2 1 9025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 4.b odd 2 1
171.2.a.b 1 12.b even 2 1
304.2.a.f 1 1.a even 1 1 trivial
361.2.a.b 1 76.d even 2 1
361.2.c.a 2 76.f even 6 2
361.2.c.c 2 76.g odd 6 2
361.2.e.d 6 76.l odd 18 6
361.2.e.e 6 76.k even 18 6
475.2.a.b 1 20.d odd 2 1
475.2.b.a 2 20.e even 4 2
931.2.a.a 1 28.d even 2 1
931.2.f.b 2 28.f even 6 2
931.2.f.c 2 28.g odd 6 2
1216.2.a.b 1 8.b even 2 1
1216.2.a.o 1 8.d odd 2 1
2299.2.a.b 1 44.c even 2 1
2736.2.a.c 1 3.b odd 2 1
3211.2.a.a 1 52.b odd 2 1
3249.2.a.d 1 228.b odd 2 1
4275.2.a.i 1 60.h even 2 1
5491.2.a.b 1 68.d odd 2 1
5776.2.a.c 1 19.b odd 2 1
7600.2.a.c 1 5.b even 2 1
8379.2.a.j 1 84.h odd 2 1
9025.2.a.d 1 380.d even 2 1

Hecke kernels

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

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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