Properties

Label 912.2.k.h
Level $912$
Weight $2$
Character orbit 912.k
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -\beta_{3} q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{3} q^{5} + \beta_{1} q^{7} + q^{9} + \beta_{1} q^{11} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{13} -\beta_{3} q^{15} + ( 2 + \beta_{3} ) q^{17} + ( 4 + \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 3 + \beta_{3} ) q^{25} + q^{27} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{33} + ( \beta_{1} + 2 \beta_{2} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{39} -4 \beta_{2} q^{41} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{43} -\beta_{3} q^{45} + ( -\beta_{1} + 2 \beta_{2} ) q^{47} + ( 3 + \beta_{3} ) q^{49} + ( 2 + \beta_{3} ) q^{51} -4 \beta_{1} q^{53} + ( \beta_{1} + 2 \beta_{2} ) q^{55} + ( 4 + \beta_{2} ) q^{57} -4 q^{59} + ( 2 + \beta_{3} ) q^{61} + \beta_{1} q^{63} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{65} + ( 4 - 4 \beta_{3} ) q^{67} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{69} -4 q^{71} + ( 6 - 3 \beta_{3} ) q^{73} + ( 3 + \beta_{3} ) q^{75} + ( -4 + \beta_{3} ) q^{77} + ( -2 + 2 \beta_{3} ) q^{79} + q^{81} + 2 \beta_{2} q^{83} + ( -8 - 3 \beta_{3} ) q^{85} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{89} + 4 q^{91} + ( 2 + 2 \beta_{3} ) q^{93} + ( 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{97} + \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 2q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 2q^{5} + 4q^{9} - 2q^{15} + 10q^{17} + 16q^{19} + 14q^{25} + 4q^{27} + 12q^{31} - 2q^{45} + 14q^{49} + 10q^{51} + 16q^{57} - 16q^{59} + 10q^{61} + 8q^{67} - 16q^{71} + 18q^{73} + 14q^{75} - 14q^{77} - 4q^{79} + 4q^{81} - 38q^{85} + 16q^{91} + 12q^{93} - 8q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} - \nu + 3 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu + 3 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + \beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} - 2 \beta_{1} + 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 + 1.26217i
−1.18614 1.26217i
0 1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.2 0 1.00000 0 −3.37228 0 0.792287i 0 1.00000 0
607.3 0 1.00000 0 2.37228 0 2.52434i 0 1.00000 0
607.4 0 1.00000 0 2.37228 0 2.52434i 0 1.00000 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
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.k.h yes 4
3.b odd 2 1 2736.2.k.o 4
4.b odd 2 1 912.2.k.g 4
8.b even 2 1 3648.2.k.g 4
8.d odd 2 1 3648.2.k.h 4
12.b even 2 1 2736.2.k.n 4
19.b odd 2 1 912.2.k.g 4
57.d even 2 1 2736.2.k.n 4
76.d even 2 1 inner 912.2.k.h yes 4
152.b even 2 1 3648.2.k.g 4
152.g odd 2 1 3648.2.k.h 4
228.b odd 2 1 2736.2.k.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.g 4 4.b odd 2 1
912.2.k.g 4 19.b odd 2 1
912.2.k.h yes 4 1.a even 1 1 trivial
912.2.k.h yes 4 76.d even 2 1 inner
2736.2.k.n 4 12.b even 2 1
2736.2.k.n 4 57.d even 2 1
2736.2.k.o 4 3.b odd 2 1
2736.2.k.o 4 228.b odd 2 1
3648.2.k.g 4 8.b even 2 1
3648.2.k.g 4 152.b even 2 1
3648.2.k.h 4 8.d odd 2 1
3648.2.k.h 4 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{2} + T_{5} - 8 \)
\( T_{31}^{2} - 6 T_{31} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( ( -8 + T + T^{2} )^{2} \)
$7$ \( 4 + 7 T^{2} + T^{4} \)
$11$ \( 4 + 7 T^{2} + T^{4} \)
$13$ \( 64 + 28 T^{2} + T^{4} \)
$17$ \( ( -2 - 5 T + T^{2} )^{2} \)
$19$ \( ( 19 - 8 T + T^{2} )^{2} \)
$23$ \( 256 + 76 T^{2} + T^{4} \)
$29$ \( 1024 + 112 T^{2} + T^{4} \)
$31$ \( ( -24 - 6 T + T^{2} )^{2} \)
$37$ \( 64 + 28 T^{2} + T^{4} \)
$41$ \( ( 48 + T^{2} )^{2} \)
$43$ \( 36 + 87 T^{2} + T^{4} \)
$47$ \( 16 + 19 T^{2} + T^{4} \)
$53$ \( 1024 + 112 T^{2} + T^{4} \)
$59$ \( ( 4 + T )^{4} \)
$61$ \( ( -2 - 5 T + T^{2} )^{2} \)
$67$ \( ( -128 - 4 T + T^{2} )^{2} \)
$71$ \( ( 4 + T )^{4} \)
$73$ \( ( -54 - 9 T + T^{2} )^{2} \)
$79$ \( ( -32 + 2 T + T^{2} )^{2} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( ( 176 + T^{2} )^{2} \)
$97$ \( ( 176 + T^{2} )^{2} \)
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