Properties

Label 912.2.q.k
Level $912$
Weight $2$
Character orbit 912.q
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
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.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{18}^{3} q^{3} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{5} + ( -1 + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{3} ) q^{9} +O(q^{10})\) \( q -\zeta_{18}^{3} q^{3} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{5} + ( -1 + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{7} + ( -1 + \zeta_{18}^{3} ) q^{9} + ( 2 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{11} + ( -1 + \zeta_{18}^{3} ) q^{13} + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} ) q^{15} + ( 4 \zeta_{18}^{2} + 4 \zeta_{18}^{4} ) q^{17} + ( -1 - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{19} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{21} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} ) q^{23} + ( -3 + 4 \zeta_{18} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{25} + q^{27} + ( -4 - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{29} + ( -5 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{31} + ( 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{33} + ( -2 \zeta_{18} + 6 \zeta_{18}^{2} - 8 \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{35} + ( -1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{5} ) q^{37} + q^{39} + ( 4 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{41} + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{43} + ( 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{45} + ( -6 + 6 \zeta_{18}^{3} ) q^{47} + ( 2 - 4 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{49} + ( 4 \zeta_{18} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{51} + ( 2 \zeta_{18} - 6 \zeta_{18}^{2} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{53} + ( 12 \zeta_{18} - 8 \zeta_{18}^{2} - 8 \zeta_{18}^{4} + 12 \zeta_{18}^{5} ) q^{55} + ( 2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{57} + ( 6 \zeta_{18} - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{59} + ( 1 - 4 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{61} + ( 1 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{63} + ( 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{65} + ( -1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{67} + ( -2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{69} + ( -4 \zeta_{18} + 8 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{71} + ( -4 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{73} + ( 3 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{5} ) q^{75} + ( -2 + 10 \zeta_{18}^{4} - 10 \zeta_{18}^{5} ) q^{77} + ( 6 \zeta_{18} - 2 \zeta_{18}^{2} + 9 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{79} -\zeta_{18}^{3} q^{81} + ( 4 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 8 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{83} + ( 8 - 8 \zeta_{18} + 8 \zeta_{18}^{2} - 8 \zeta_{18}^{3} ) q^{85} + ( 4 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} ) q^{87} + ( -6 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 6 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{89} + ( 1 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{91} + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{93} + ( -8 + 10 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{95} + ( 4 \zeta_{18} - 8 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{97} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{3} - 6q^{7} - 3q^{9} + O(q^{10}) \) \( 6q - 3q^{3} - 6q^{7} - 3q^{9} + 12q^{11} - 3q^{13} + 3q^{21} - 9q^{25} + 6q^{27} - 12q^{29} - 30q^{31} - 6q^{33} - 24q^{35} - 6q^{37} + 6q^{39} - 12q^{41} + 9q^{43} - 18q^{47} + 12q^{49} + 9q^{57} - 6q^{59} + 3q^{61} + 3q^{63} - 3q^{67} + 6q^{71} + 9q^{73} + 18q^{75} - 12q^{77} + 27q^{79} - 3q^{81} + 24q^{83} + 24q^{85} + 24q^{87} - 18q^{89} + 3q^{91} + 15q^{93} - 48q^{95} - 6q^{97} - 6q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1 + \zeta_{18}^{2}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
0 −0.500000 + 0.866025i 0 −1.53209 + 2.65366i 0 2.06418 0 −0.500000 0.866025i 0
49.2 0 −0.500000 + 0.866025i 0 −0.347296 + 0.601535i 0 −0.305407 0 −0.500000 0.866025i 0
49.3 0 −0.500000 + 0.866025i 0 1.87939 3.25519i 0 −4.75877 0 −0.500000 0.866025i 0
577.1 0 −0.500000 0.866025i 0 −1.53209 2.65366i 0 2.06418 0 −0.500000 + 0.866025i 0
577.2 0 −0.500000 0.866025i 0 −0.347296 0.601535i 0 −0.305407 0 −0.500000 + 0.866025i 0
577.3 0 −0.500000 0.866025i 0 1.87939 + 3.25519i 0 −4.75877 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.q.k 6
3.b odd 2 1 2736.2.s.x 6
4.b odd 2 1 456.2.q.f 6
12.b even 2 1 1368.2.s.j 6
19.c even 3 1 inner 912.2.q.k 6
57.h odd 6 1 2736.2.s.x 6
76.f even 6 1 8664.2.a.z 3
76.g odd 6 1 456.2.q.f 6
76.g odd 6 1 8664.2.a.x 3
228.m even 6 1 1368.2.s.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.f 6 4.b odd 2 1
456.2.q.f 6 76.g odd 6 1
912.2.q.k 6 1.a even 1 1 trivial
912.2.q.k 6 19.c even 3 1 inner
1368.2.s.j 6 12.b even 2 1
1368.2.s.j 6 228.m even 6 1
2736.2.s.x 6 3.b odd 2 1
2736.2.s.x 6 57.h odd 6 1
8664.2.a.x 3 76.g odd 6 1
8664.2.a.z 3 76.f even 6 1

Hecke kernels

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

\( T_{5}^{6} + 12 T_{5}^{4} + 16 T_{5}^{3} + 144 T_{5}^{2} + 96 T_{5} + 64 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 9 T_{7} - 3 \)
\( T_{13}^{2} + T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( ( 1 + T + T^{2} )^{3} \)
$5$ \( 64 + 96 T + 144 T^{2} + 16 T^{3} + 12 T^{4} + T^{6} \)
$7$ \( ( -3 - 9 T + 3 T^{2} + T^{3} )^{2} \)
$11$ \( ( 136 - 24 T - 6 T^{2} + T^{3} )^{2} \)
$13$ \( ( 1 + T + T^{2} )^{3} \)
$17$ \( 4096 + 3072 T + 2304 T^{2} + 128 T^{3} + 48 T^{4} + T^{6} \)
$19$ \( 6859 + 171 T^{2} - 64 T^{3} + 9 T^{4} + T^{6} \)
$23$ \( 64 - 96 T + 144 T^{2} - 16 T^{3} + 12 T^{4} + T^{6} \)
$29$ \( 36864 + 2304 T^{2} + 384 T^{3} + 144 T^{4} + 12 T^{5} + T^{6} \)
$31$ \( ( -127 + 39 T + 15 T^{2} + T^{3} )^{2} \)
$37$ \( ( -111 - 45 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 36864 + 2304 T^{2} + 384 T^{3} + 144 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( 289 + 255 T + 378 T^{2} - 169 T^{3} + 66 T^{4} - 9 T^{5} + T^{6} \)
$47$ \( ( 36 + 6 T + T^{2} )^{3} \)
$53$ \( 87616 + 24864 T + 7056 T^{2} + 592 T^{3} + 84 T^{4} + T^{6} \)
$59$ \( 179776 + 40704 T + 11760 T^{2} + 272 T^{3} + 132 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( 289 + 765 T + 1974 T^{2} + 169 T^{3} + 54 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 1369 - 1221 T + 978 T^{2} - 173 T^{3} + 42 T^{4} + 3 T^{5} + T^{6} \)
$71$ \( 732736 - 112992 T + 22560 T^{2} - 920 T^{3} + 168 T^{4} - 6 T^{5} + T^{6} \)
$73$ \( 32761 - 3801 T + 2070 T^{2} - 173 T^{3} + 102 T^{4} - 9 T^{5} + T^{6} \)
$79$ \( 104329 + 51357 T + 34002 T^{2} - 4939 T^{3} + 570 T^{4} - 27 T^{5} + T^{6} \)
$83$ \( ( -64 - 96 T - 12 T^{2} + T^{3} )^{2} \)
$89$ \( 5184 - 5184 T + 6480 T^{2} + 1440 T^{3} + 252 T^{4} + 18 T^{5} + T^{6} \)
$97$ \( 732736 + 112992 T + 22560 T^{2} + 920 T^{3} + 168 T^{4} + 6 T^{5} + T^{6} \)
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