Properties

Label 304.7.e.b
Level $304$
Weight $7$
Character orbit 304.e
Self dual yes
Analytic conductor $69.936$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{57}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 27 + 7 \beta ) q^{5} + ( 305 - 9 \beta ) q^{7} + 729 q^{9} +O(q^{10})\) \( q + ( 27 + 7 \beta ) q^{5} + ( 305 - 9 \beta ) q^{7} + 729 q^{9} + ( -531 + 70 \beta ) q^{11} + ( 4815 + 56 \beta ) q^{17} + 6859 q^{19} -20610 q^{23} + ( 29792 + 378 \beta ) q^{25} + ( -49221 + 1892 \beta ) q^{35} + ( -71315 + 2016 \beta ) q^{43} + ( 19683 + 5103 \beta ) q^{45} + ( -37575 - 5551 \beta ) q^{47} + ( 49248 - 5490 \beta ) q^{49} + ( 432543 - 1827 \beta ) q^{55} + ( 28531 - 12915 \beta ) q^{61} + ( 222345 - 6561 \beta ) q^{63} + ( -192025 - 19404 \beta ) q^{73} + ( -736515 + 26129 \beta ) q^{77} + 531441 q^{81} + 1131030 q^{83} + ( 487509 + 35217 \beta ) q^{85} + ( 185193 + 48013 \beta ) q^{95} + ( -387099 + 51030 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 54q^{5} + 610q^{7} + 1458q^{9} + O(q^{10}) \) \( 2q + 54q^{5} + 610q^{7} + 1458q^{9} - 1062q^{11} + 9630q^{17} + 13718q^{19} - 41220q^{23} + 59584q^{25} - 98442q^{35} - 142630q^{43} + 39366q^{45} - 75150q^{47} + 98496q^{49} + 865086q^{55} + 57062q^{61} + 444690q^{63} - 384050q^{73} - 1473030q^{77} + 1062882q^{81} + 2262060q^{83} + 975018q^{85} + 370386q^{95} - 774198q^{99} + O(q^{100}) \)

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
−3.27492
4.27492
0 0 0 −184.395 0 576.794 0 729.000 0
113.2 0 0 0 238.395 0 33.2060 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.b 2
4.b odd 2 1 76.7.c.a 2
12.b even 2 1 684.7.h.a 2
19.b odd 2 1 CM 304.7.e.b 2
76.d even 2 1 76.7.c.a 2
228.b odd 2 1 684.7.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.c.a 2 4.b odd 2 1
76.7.c.a 2 76.d even 2 1
304.7.e.b 2 1.a even 1 1 trivial
304.7.e.b 2 19.b odd 2 1 CM
684.7.h.a 2 12.b even 2 1
684.7.h.a 2 228.b odd 2 1

Hecke kernels

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

\( T_{3} \)
\( T_{5}^{2} - 54 T_{5} - 43959 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -43959 - 54 T + T^{2} \)
$7$ \( 19153 - 610 T + T^{2} \)
$11$ \( -4186839 + 1062 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 20324193 - 9630 T + T^{2} \)
$19$ \( ( -6859 + T )^{2} \)
$23$ \( ( 20610 + T )^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1379227753 + 142630 T + T^{2} \)
$47$ \( -26690123487 + 75150 T + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( -151305051239 - 57062 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( -306508276367 + 384050 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( ( -1131030 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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