Properties

Label 912.2.a.k
Level $912$
Weight $2$
Character orbit 912.a
Self dual yes
Analytic conductor $7.282$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.28235666434\)
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

\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} + q^{9} + 4 q^{11} + 2 q^{13} + 2 q^{15} - 6 q^{17} + q^{19} + 4 q^{23} - q^{25} + q^{27} - 2 q^{29} - 4 q^{31} + 4 q^{33} + 10 q^{37} + 2 q^{39} + 10 q^{41} - 4 q^{43} + 2 q^{45} + 4 q^{47} - 7 q^{49} - 6 q^{51} - 10 q^{53} + 8 q^{55} + q^{57} - 12 q^{59} + 14 q^{61} + 4 q^{65} + 12 q^{67} + 4 q^{69} - 8 q^{71} - 6 q^{73} - q^{75} + 4 q^{79} + q^{81} - 12 q^{83} - 12 q^{85} - 2 q^{87} - 6 q^{89} - 4 q^{93} + 2 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 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 2.00000 0 0 0 1.00000 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 912.2.a.k 1
3.b odd 2 1 2736.2.a.d 1
4.b odd 2 1 114.2.a.b 1
8.b even 2 1 3648.2.a.c 1
8.d odd 2 1 3648.2.a.x 1
12.b even 2 1 342.2.a.b 1
20.d odd 2 1 2850.2.a.j 1
20.e even 4 2 2850.2.d.b 2
28.d even 2 1 5586.2.a.y 1
60.h even 2 1 8550.2.a.ba 1
76.d even 2 1 2166.2.a.d 1
228.b odd 2 1 6498.2.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.b 1 4.b odd 2 1
342.2.a.b 1 12.b even 2 1
912.2.a.k 1 1.a even 1 1 trivial
2166.2.a.d 1 76.d even 2 1
2736.2.a.d 1 3.b odd 2 1
2850.2.a.j 1 20.d odd 2 1
2850.2.d.b 2 20.e even 4 2
3648.2.a.c 1 8.b even 2 1
3648.2.a.x 1 8.d odd 2 1
5586.2.a.y 1 28.d even 2 1
6498.2.a.p 1 228.b odd 2 1
8550.2.a.ba 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(912))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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