Properties

Label 304.2.a.d
Level $304$
Weight $2$
Character orbit 304.a
Self dual yes
Analytic conductor $2.427$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(2.42745222145\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{3} - 4 q^{5} - 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 4 q^{5} - 3 q^{7} - 2 q^{9} - 2 q^{11} - q^{13} - 4 q^{15} + 3 q^{17} + q^{19} - 3 q^{21} + q^{23} + 11 q^{25} - 5 q^{27} - 5 q^{29} + 8 q^{31} - 2 q^{33} + 12 q^{35} - 2 q^{37} - q^{39} - 8 q^{41} - 4 q^{43} + 8 q^{45} - 8 q^{47} + 2 q^{49} + 3 q^{51} - q^{53} + 8 q^{55} + q^{57} - 15 q^{59} + 2 q^{61} + 6 q^{63} + 4 q^{65} - 3 q^{67} + q^{69} - 2 q^{71} + 9 q^{73} + 11 q^{75} + 6 q^{77} + 10 q^{79} + q^{81} + 6 q^{83} - 12 q^{85} - 5 q^{87} + 3 q^{91} + 8 q^{93} - 4 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −4.00000 0 −3.00000 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 304.2.a.d 1
3.b odd 2 1 2736.2.a.w 1
4.b odd 2 1 38.2.a.b 1
5.b even 2 1 7600.2.a.h 1
8.b even 2 1 1216.2.a.g 1
8.d odd 2 1 1216.2.a.n 1
12.b even 2 1 342.2.a.d 1
19.b odd 2 1 5776.2.a.d 1
20.d odd 2 1 950.2.a.b 1
20.e even 4 2 950.2.b.c 2
28.d even 2 1 1862.2.a.f 1
44.c even 2 1 4598.2.a.a 1
52.b odd 2 1 6422.2.a.b 1
60.h even 2 1 8550.2.a.u 1
76.d even 2 1 722.2.a.b 1
76.f even 6 2 722.2.c.f 2
76.g odd 6 2 722.2.c.d 2
76.k even 18 6 722.2.e.d 6
76.l odd 18 6 722.2.e.c 6
228.b odd 2 1 6498.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.b 1 4.b odd 2 1
304.2.a.d 1 1.a even 1 1 trivial
342.2.a.d 1 12.b even 2 1
722.2.a.b 1 76.d even 2 1
722.2.c.d 2 76.g odd 6 2
722.2.c.f 2 76.f even 6 2
722.2.e.c 6 76.l odd 18 6
722.2.e.d 6 76.k even 18 6
950.2.a.b 1 20.d odd 2 1
950.2.b.c 2 20.e even 4 2
1216.2.a.g 1 8.b even 2 1
1216.2.a.n 1 8.d odd 2 1
1862.2.a.f 1 28.d even 2 1
2736.2.a.w 1 3.b odd 2 1
4598.2.a.a 1 44.c even 2 1
5776.2.a.d 1 19.b odd 2 1
6422.2.a.b 1 52.b odd 2 1
6498.2.a.y 1 228.b odd 2 1
7600.2.a.h 1 5.b even 2 1
8550.2.a.u 1 60.h even 2 1

Hecke kernels

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

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 4 \) Copy content Toggle raw display
\( T_{7} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 4 \) Copy content Toggle raw display
$7$ \( T + 3 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 3 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T - 1 \) Copy content Toggle raw display
$29$ \( T + 5 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 8 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T + 15 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T - 9 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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