Properties

Label 304.4.a.b
Level $304$
Weight $4$
Character orbit 304.a
Self dual yes
Analytic conductor $17.937$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(17.9365806417\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + 5q^{3} - 12q^{5} - 11q^{7} - 2q^{9} + O(q^{10}) \) \( q + 5q^{3} - 12q^{5} - 11q^{7} - 2q^{9} + 54q^{11} + 11q^{13} - 60q^{15} - 93q^{17} - 19q^{19} - 55q^{21} - 183q^{23} + 19q^{25} - 145q^{27} - 249q^{29} - 56q^{31} + 270q^{33} + 132q^{35} - 250q^{37} + 55q^{39} + 240q^{41} + 196q^{43} + 24q^{45} + 168q^{47} - 222q^{49} - 465q^{51} + 435q^{53} - 648q^{55} - 95q^{57} - 195q^{59} - 358q^{61} + 22q^{63} - 132q^{65} + 961q^{67} - 915q^{69} + 246q^{71} + 353q^{73} + 95q^{75} - 594q^{77} + 34q^{79} - 671q^{81} - 234q^{83} + 1116q^{85} - 1245q^{87} - 168q^{89} - 121q^{91} - 280q^{93} + 228q^{95} + 758q^{97} - 108q^{99} + O(q^{100}) \)

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 5.00000 0 −12.0000 0 −11.0000 0 −2.00000 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 304.4.a.b 1
4.b odd 2 1 19.4.a.a 1
8.b even 2 1 1216.4.a.a 1
8.d odd 2 1 1216.4.a.f 1
12.b even 2 1 171.4.a.d 1
20.d odd 2 1 475.4.a.e 1
20.e even 4 2 475.4.b.c 2
28.d even 2 1 931.4.a.a 1
44.c even 2 1 2299.4.a.b 1
76.d even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 4.b odd 2 1
171.4.a.d 1 12.b even 2 1
304.4.a.b 1 1.a even 1 1 trivial
361.4.a.b 1 76.d even 2 1
475.4.a.e 1 20.d odd 2 1
475.4.b.c 2 20.e even 4 2
931.4.a.a 1 28.d even 2 1
1216.4.a.a 1 8.b even 2 1
1216.4.a.f 1 8.d odd 2 1
2299.4.a.b 1 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -5 + T \)
$5$ \( 12 + T \)
$7$ \( 11 + T \)
$11$ \( -54 + T \)
$13$ \( -11 + T \)
$17$ \( 93 + T \)
$19$ \( 19 + T \)
$23$ \( 183 + T \)
$29$ \( 249 + T \)
$31$ \( 56 + T \)
$37$ \( 250 + T \)
$41$ \( -240 + T \)
$43$ \( -196 + T \)
$47$ \( -168 + T \)
$53$ \( -435 + T \)
$59$ \( 195 + T \)
$61$ \( 358 + T \)
$67$ \( -961 + T \)
$71$ \( -246 + T \)
$73$ \( -353 + T \)
$79$ \( -34 + T \)
$83$ \( 234 + T \)
$89$ \( 168 + T \)
$97$ \( -758 + T \)
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