Properties

Label 304.3.e.a
Level $304$
Weight $3$
Character orbit 304.e
Self dual yes
Analytic conductor $8.283$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(8.28340003655\)
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)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 9 q^{5} + 5 q^{7} + 9 q^{9} + O(q^{10}) \) \( q - 9 q^{5} + 5 q^{7} + 9 q^{9} - 3 q^{11} + 15 q^{17} + 19 q^{19} + 30 q^{23} + 56 q^{25} - 45 q^{35} + 85 q^{43} - 81 q^{45} - 75 q^{47} - 24 q^{49} + 27 q^{55} + 103 q^{61} + 45 q^{63} - 25 q^{73} - 15 q^{77} + 81 q^{81} - 90 q^{83} - 135 q^{85} - 171 q^{95} - 27 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0
0 0 0 −9.00000 0 5.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.e.a 1
3.b odd 2 1 2736.3.o.a 1
4.b odd 2 1 19.3.b.a 1
8.b even 2 1 1216.3.e.b 1
8.d odd 2 1 1216.3.e.a 1
12.b even 2 1 171.3.c.a 1
19.b odd 2 1 CM 304.3.e.a 1
20.d odd 2 1 475.3.c.a 1
20.e even 4 2 475.3.d.a 2
57.d even 2 1 2736.3.o.a 1
76.d even 2 1 19.3.b.a 1
76.f even 6 2 361.3.d.a 2
76.g odd 6 2 361.3.d.a 2
76.k even 18 6 361.3.f.a 6
76.l odd 18 6 361.3.f.a 6
152.b even 2 1 1216.3.e.a 1
152.g odd 2 1 1216.3.e.b 1
228.b odd 2 1 171.3.c.a 1
380.d even 2 1 475.3.c.a 1
380.j odd 4 2 475.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 4.b odd 2 1
19.3.b.a 1 76.d even 2 1
171.3.c.a 1 12.b even 2 1
171.3.c.a 1 228.b odd 2 1
304.3.e.a 1 1.a even 1 1 trivial
304.3.e.a 1 19.b odd 2 1 CM
361.3.d.a 2 76.f even 6 2
361.3.d.a 2 76.g odd 6 2
361.3.f.a 6 76.k even 18 6
361.3.f.a 6 76.l odd 18 6
475.3.c.a 1 20.d odd 2 1
475.3.c.a 1 380.d even 2 1
475.3.d.a 2 20.e even 4 2
475.3.d.a 2 380.j odd 4 2
1216.3.e.a 1 8.d odd 2 1
1216.3.e.a 1 152.b even 2 1
1216.3.e.b 1 8.b even 2 1
1216.3.e.b 1 152.g odd 2 1
2736.3.o.a 1 3.b odd 2 1
2736.3.o.a 1 57.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 \(S_{3}^{\mathrm{new}}(304, [\chi])\):

\( T_{3} \)
\( T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 9 + T \)
$7$ \( -5 + T \)
$11$ \( 3 + T \)
$13$ \( T \)
$17$ \( -15 + T \)
$19$ \( -19 + T \)
$23$ \( -30 + T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( -85 + T \)
$47$ \( 75 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -103 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 25 + T \)
$79$ \( T \)
$83$ \( 90 + T \)
$89$ \( T \)
$97$ \( T \)
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