Properties

Label 304.7.e.a
Level $304$
Weight $7$
Character orbit 304.e
Self dual yes
Analytic conductor $69.936$
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: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(69.9364414204\)
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

\(f(q)\) \(=\) \( q - 54q^{5} - 610q^{7} + 729q^{9} + O(q^{10}) \) \( q - 54q^{5} - 610q^{7} + 729q^{9} + 1062q^{11} - 9630q^{17} + 6859q^{19} - 20610q^{23} - 12709q^{25} + 32940q^{35} + 142630q^{43} - 39366q^{45} + 75150q^{47} + 254451q^{49} - 57348q^{55} - 57062q^{61} - 444690q^{63} + 384050q^{73} - 647820q^{77} + 531441q^{81} + 1131030q^{83} + 520020q^{85} - 370386q^{95} + 774198q^{99} + O(q^{100}) \)

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 −54.0000 0 −610.000 0 729.000 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 304.7.e.a 1
4.b odd 2 1 19.7.b.a 1
12.b even 2 1 171.7.c.a 1
19.b odd 2 1 CM 304.7.e.a 1
76.d even 2 1 19.7.b.a 1
228.b odd 2 1 171.7.c.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.b.a 1 4.b odd 2 1
19.7.b.a 1 76.d even 2 1
171.7.c.a 1 12.b even 2 1
171.7.c.a 1 228.b odd 2 1
304.7.e.a 1 1.a even 1 1 trivial
304.7.e.a 1 19.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(304, [\chi])\):

\( T_{3} \)
\( T_{5} + 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 54 + T \)
$7$ \( 610 + T \)
$11$ \( -1062 + T \)
$13$ \( T \)
$17$ \( 9630 + T \)
$19$ \( -6859 + T \)
$23$ \( 20610 + T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( -142630 + T \)
$47$ \( -75150 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( 57062 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -384050 + T \)
$79$ \( T \)
$83$ \( -1131030 + T \)
$89$ \( T \)
$97$ \( T \)
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