Properties

Label 304.2.a.c
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 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{7} - 2q^{9} + O(q^{10}) \) \( q - q^{3} + q^{7} - 2q^{9} + 6q^{11} + 5q^{13} + 3q^{17} - q^{19} - q^{21} - 3q^{23} - 5q^{25} + 5q^{27} + 9q^{29} + 4q^{31} - 6q^{33} + 2q^{37} - 5q^{39} - 8q^{43} - 6q^{49} - 3q^{51} - 3q^{53} + q^{57} - 9q^{59} - 10q^{61} - 2q^{63} - 5q^{67} + 3q^{69} + 6q^{71} - 7q^{73} + 5q^{75} + 6q^{77} + 10q^{79} + q^{81} + 6q^{83} - 9q^{87} - 12q^{89} + 5q^{91} - 4q^{93} - 10q^{97} - 12q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 1.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.c 1
3.b odd 2 1 2736.2.a.n 1
4.b odd 2 1 38.2.a.a 1
5.b even 2 1 7600.2.a.n 1
8.b even 2 1 1216.2.a.m 1
8.d odd 2 1 1216.2.a.e 1
12.b even 2 1 342.2.a.e 1
19.b odd 2 1 5776.2.a.m 1
20.d odd 2 1 950.2.a.d 1
20.e even 4 2 950.2.b.b 2
28.d even 2 1 1862.2.a.b 1
44.c even 2 1 4598.2.a.p 1
52.b odd 2 1 6422.2.a.h 1
60.h even 2 1 8550.2.a.m 1
76.d even 2 1 722.2.a.e 1
76.f even 6 2 722.2.c.c 2
76.g odd 6 2 722.2.c.e 2
76.k even 18 6 722.2.e.e 6
76.l odd 18 6 722.2.e.f 6
228.b odd 2 1 6498.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 4.b odd 2 1
304.2.a.c 1 1.a even 1 1 trivial
342.2.a.e 1 12.b even 2 1
722.2.a.e 1 76.d even 2 1
722.2.c.c 2 76.f even 6 2
722.2.c.e 2 76.g odd 6 2
722.2.e.e 6 76.k even 18 6
722.2.e.f 6 76.l odd 18 6
950.2.a.d 1 20.d odd 2 1
950.2.b.b 2 20.e even 4 2
1216.2.a.e 1 8.d odd 2 1
1216.2.a.m 1 8.b even 2 1
1862.2.a.b 1 28.d even 2 1
2736.2.a.n 1 3.b odd 2 1
4598.2.a.p 1 44.c even 2 1
5776.2.a.m 1 19.b odd 2 1
6422.2.a.h 1 52.b odd 2 1
6498.2.a.f 1 228.b odd 2 1
7600.2.a.n 1 5.b even 2 1
8550.2.a.m 1 60.h even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{5} \)
\( T_{7} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 6 T + 11 T^{2} \)
$13$ \( 1 - 5 T + 13 T^{2} \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ \( 1 + T \)
$23$ \( 1 + 3 T + 23 T^{2} \)
$29$ \( 1 - 9 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 3 T + 53 T^{2} \)
$59$ \( 1 + 9 T + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 5 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 + 7 T + 73 T^{2} \)
$79$ \( 1 - 10 T + 79 T^{2} \)
$83$ \( 1 - 6 T + 83 T^{2} \)
$89$ \( 1 + 12 T + 89 T^{2} \)
$97$ \( 1 + 10 T + 97 T^{2} \)
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