Properties

Label 912.4.a.b
Level $912$
Weight $4$
Character orbit 912.a
Self dual yes
Analytic conductor $53.810$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.8097419252\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} - 7q^{5} + 15q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} - 7q^{5} + 15q^{7} + 9q^{9} + 49q^{11} + 14q^{13} + 21q^{15} - 33q^{17} + 19q^{19} - 45q^{21} + 148q^{23} - 76q^{25} - 27q^{27} - 278q^{29} - 94q^{31} - 147q^{33} - 105q^{35} + 160q^{37} - 42q^{39} + 400q^{41} - 73q^{43} - 63q^{45} - 173q^{47} - 118q^{49} + 99q^{51} + 170q^{53} - 343q^{55} - 57q^{57} + 12q^{59} + 419q^{61} + 135q^{63} - 98q^{65} - 444q^{67} - 444q^{69} + 952q^{71} - 27q^{73} + 228q^{75} + 735q^{77} + 556q^{79} + 81q^{81} + 276q^{83} + 231q^{85} + 834q^{87} + 1386q^{89} + 210q^{91} + 282q^{93} - 133q^{95} + 130q^{97} + 441q^{99} + O(q^{100}) \)

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 −3.00000 0 −7.00000 0 15.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)

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 912.4.a.b 1
4.b odd 2 1 114.4.a.b 1
12.b even 2 1 342.4.a.c 1
76.d even 2 1 2166.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.a.b 1 4.b odd 2 1
342.4.a.c 1 12.b even 2 1
912.4.a.b 1 1.a even 1 1 trivial
2166.4.a.d 1 76.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(912))\):

\( T_{5} + 7 \)
\( T_{7} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 7 + T \)
$7$ \( -15 + T \)
$11$ \( -49 + T \)
$13$ \( -14 + T \)
$17$ \( 33 + T \)
$19$ \( -19 + T \)
$23$ \( -148 + T \)
$29$ \( 278 + T \)
$31$ \( 94 + T \)
$37$ \( -160 + T \)
$41$ \( -400 + T \)
$43$ \( 73 + T \)
$47$ \( 173 + T \)
$53$ \( -170 + T \)
$59$ \( -12 + T \)
$61$ \( -419 + T \)
$67$ \( 444 + T \)
$71$ \( -952 + T \)
$73$ \( 27 + T \)
$79$ \( -556 + T \)
$83$ \( -276 + T \)
$89$ \( -1386 + T \)
$97$ \( -130 + T \)
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