Properties

Label 912.2.bn.m
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{3} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{7} + ( -1 - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{3} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{5} + ( 2 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{7} + ( -1 - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} + ( -\zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( 1 - 2 \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{13} + ( 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{15} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( -2 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{19} + ( -2 - 3 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{21} + ( -2 \zeta_{24} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{23} + ( -3 + 3 \zeta_{24}^{4} ) q^{25} + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{27} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{31} + ( 4 + \zeta_{24} + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{33} + ( -\zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{35} + ( -1 + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{37} + ( -2 - 3 \zeta_{24} + 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{39} + ( 4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{41} + ( \zeta_{24}^{2} + 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{43} + ( 4 + \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{45} + ( 4 \zeta_{24} - 2 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{49} + ( 4 - 4 \zeta_{24} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{51} + ( 2 \zeta_{24} + 5 \zeta_{24}^{3} - 7 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{53} + ( -2 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{55} + ( -2 - 5 \zeta_{24} + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{57} + ( -3 \zeta_{24} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{59} + ( 7 + 2 \zeta_{24}^{2} - 7 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{61} + ( -2 + 4 \zeta_{24} + \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{63} + ( \zeta_{24} + \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{65} + \zeta_{24}^{2} q^{67} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{69} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{71} + 3 \zeta_{24}^{4} q^{73} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{75} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{77} + ( -4 - 7 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 7 \zeta_{24}^{6} ) q^{79} + ( 4 \zeta_{24} + 7 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{81} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{83} + ( 4 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{85} + ( -4 + 4 \zeta_{24} + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{87} + ( -6 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{89} + ( 4 - 7 \zeta_{24}^{2} + 4 \zeta_{24}^{4} ) q^{91} + ( \zeta_{24} + 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{93} + ( 5 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{95} + ( 4 - 4 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{97} + ( 8 - 2 \zeta_{24} + 4 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - \zeta_{24}^{5} - 8 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 4 q^{9} + O(q^{10}) \) \( 8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1 - \zeta_{24}^{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0 −1.57313 + 0.724745i 0 1.22474 + 0.707107i 0 3.73205 0 1.94949 2.28024i 0
65.2 0 −0.158919 1.72474i 0 −1.22474 0.707107i 0 3.73205 0 −2.94949 + 0.548188i 0
65.3 0 0.158919 + 1.72474i 0 −1.22474 0.707107i 0 0.267949 0 −2.94949 + 0.548188i 0
65.4 0 1.57313 0.724745i 0 1.22474 + 0.707107i 0 0.267949 0 1.94949 2.28024i 0
449.1 0 −1.57313 0.724745i 0 1.22474 0.707107i 0 3.73205 0 1.94949 + 2.28024i 0
449.2 0 −0.158919 + 1.72474i 0 −1.22474 + 0.707107i 0 3.73205 0 −2.94949 0.548188i 0
449.3 0 0.158919 1.72474i 0 −1.22474 + 0.707107i 0 0.267949 0 −2.94949 0.548188i 0
449.4 0 1.57313 + 0.724745i 0 1.22474 0.707107i 0 0.267949 0 1.94949 + 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.m 8
3.b odd 2 1 inner 912.2.bn.m 8
4.b odd 2 1 57.2.f.a 8
12.b even 2 1 57.2.f.a 8
19.d odd 6 1 inner 912.2.bn.m 8
57.f even 6 1 inner 912.2.bn.m 8
76.f even 6 1 57.2.f.a 8
76.f even 6 1 1083.2.d.b 8
76.g odd 6 1 1083.2.d.b 8
228.m even 6 1 1083.2.d.b 8
228.n odd 6 1 57.2.f.a 8
228.n odd 6 1 1083.2.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 4.b odd 2 1
57.2.f.a 8 12.b even 2 1
57.2.f.a 8 76.f even 6 1
57.2.f.a 8 228.n odd 6 1
912.2.bn.m 8 1.a even 1 1 trivial
912.2.bn.m 8 3.b odd 2 1 inner
912.2.bn.m 8 19.d odd 6 1 inner
912.2.bn.m 8 57.f even 6 1 inner
1083.2.d.b 8 76.f even 6 1
1083.2.d.b 8 76.g odd 6 1
1083.2.d.b 8 228.m even 6 1
1083.2.d.b 8 228.n odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2 T_{5}^{2} + 4 \)
\( T_{7}^{2} - 4 T_{7} + 1 \)
\( T_{17}^{4} - 24 T_{17}^{2} + 576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 81 + 18 T^{2} - 5 T^{4} + 2 T^{6} + T^{8} \)
$5$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( 1 - 4 T + T^{2} )^{4} \)
$11$ \( ( 4 + 28 T^{2} + T^{4} )^{2} \)
$13$ \( ( 1 + 6 T + 11 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$17$ \( ( 576 - 24 T^{2} + T^{4} )^{2} \)
$19$ \( ( 361 + 26 T^{2} + T^{4} )^{2} \)
$23$ \( 16 - 112 T^{2} + 780 T^{4} - 28 T^{6} + T^{8} \)
$29$ \( 65536 + 16384 T^{2} + 3840 T^{4} + 64 T^{6} + T^{8} \)
$31$ \( ( 121 + 26 T^{2} + T^{4} )^{2} \)
$37$ \( ( 1089 + 78 T^{2} + T^{4} )^{2} \)
$41$ \( ( 1024 + 32 T^{2} + T^{4} )^{2} \)
$43$ \( ( 169 - 104 T + 51 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$47$ \( 4096 - 7168 T^{2} + 12480 T^{4} - 112 T^{6} + T^{8} \)
$53$ \( 18974736 + 679536 T^{2} + 19980 T^{4} + 156 T^{6} + T^{8} \)
$59$ \( 78074896 + 1731856 T^{2} + 29580 T^{4} + 196 T^{6} + T^{8} \)
$61$ \( ( 1369 - 518 T + 159 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$67$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$71$ \( 5308416 + 442368 T^{2} + 34560 T^{4} + 192 T^{6} + T^{8} \)
$73$ \( ( 9 - 3 T + T^{2} )^{4} \)
$79$ \( ( 1369 - 444 T + 11 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$83$ \( ( 256 + 64 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2916 + 54 T^{2} + T^{4} )^{2} \)
$97$ \( ( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} )^{2} \)
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