Properties

Label 930.2.s.b
Level $930$
Weight $2$
Character orbit 930.s
Analytic conductor $7.426$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{2} q^{3} - q^{4} + ( 2 \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} -\beta_{3} q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{2} q^{3} - q^{4} + ( 2 \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} -\beta_{3} q^{9} + ( -\beta_{3} + 2 \beta_{4} ) q^{10} + ( 2 \beta_{3} - \beta_{5} ) q^{11} + \beta_{2} q^{12} + 2 \beta_{1} q^{13} + \beta_{7} q^{14} + ( 1 + 2 \beta_{2} + 2 \beta_{4} ) q^{15} + q^{16} + 6 \beta_{2} q^{17} -\beta_{4} q^{18} -2 \beta_{7} q^{19} + ( -2 \beta_{3} - \beta_{4} ) q^{20} + \beta_{5} q^{21} + ( \beta_{1} + 2 \beta_{4} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{3} ) q^{24} + ( -3 + 4 \beta_{2} - 3 \beta_{3} ) q^{25} + 2 \beta_{5} q^{26} + ( -\beta_{2} - \beta_{4} ) q^{27} + \beta_{6} q^{28} + ( \beta_{5} - \beta_{7} ) q^{29} + ( 2 - \beta_{2} - \beta_{4} ) q^{30} + ( -1 + \beta_{3} + 2 \beta_{7} ) q^{31} + ( -\beta_{2} - \beta_{4} ) q^{32} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{33} + ( 6 + 6 \beta_{3} ) q^{34} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{35} + \beta_{3} q^{36} -4 \beta_{2} q^{37} -2 \beta_{6} q^{38} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{39} + ( \beta_{3} - 2 \beta_{4} ) q^{40} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{41} -\beta_{1} q^{42} + ( 6 \beta_{2} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{3} + \beta_{5} ) q^{44} + ( 2 - \beta_{2} + 2 \beta_{3} ) q^{45} + ( -4 + 2 \beta_{5} - 2 \beta_{7} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{47} -\beta_{2} q^{48} + ( 4 + 3 \beta_{2} + 4 \beta_{3} ) q^{50} + 6 \beta_{3} q^{51} -2 \beta_{1} q^{52} + ( 2 \beta_{1} + 3 \beta_{4} ) q^{53} - q^{54} + ( -4 + 2 \beta_{2} - 4 \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{55} -\beta_{7} q^{56} + 2 \beta_{1} q^{57} + ( -\beta_{1} - \beta_{6} ) q^{58} + ( 2 + 2 \beta_{3} - \beta_{7} ) q^{59} + ( -1 - 2 \beta_{2} - 2 \beta_{4} ) q^{60} + 10 q^{61} + ( \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{62} + ( -\beta_{1} - \beta_{6} ) q^{63} - q^{64} + ( 4 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 2 + \beta_{5} - \beta_{7} ) q^{66} + ( -2 \beta_{1} + 6 \beta_{4} ) q^{67} -6 \beta_{2} q^{68} + ( -4 - 4 \beta_{3} - 2 \beta_{7} ) q^{69} + ( -\beta_{1} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{70} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{71} + \beta_{4} q^{72} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{73} + ( -4 - 4 \beta_{3} ) q^{74} + ( 4 \beta_{3} - 3 \beta_{4} ) q^{75} + 2 \beta_{7} q^{76} + ( 2 \beta_{1} + 7 \beta_{2} + 7 \beta_{4} + 2 \beta_{6} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{6} ) q^{78} + ( 4 + 4 \beta_{3} + 4 \beta_{7} ) q^{79} + ( 2 \beta_{3} + \beta_{4} ) q^{80} + ( -1 - \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{82} + ( -4 \beta_{1} - \beta_{4} ) q^{83} -\beta_{5} q^{84} + ( -6 - 12 \beta_{2} - 12 \beta_{4} ) q^{85} + ( 6 + 6 \beta_{3} + 2 \beta_{7} ) q^{86} -\beta_{6} q^{87} + ( -\beta_{1} - 2 \beta_{4} ) q^{88} + ( -2 - 2 \beta_{5} + 2 \beta_{7} ) q^{89} + ( -1 - 2 \beta_{2} - \beta_{3} ) q^{90} + 14 q^{91} + ( -2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{92} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{93} + ( -4 + 2 \beta_{5} - 2 \beta_{7} ) q^{94} + ( 2 \beta_{1} - 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -1 - \beta_{3} ) q^{96} + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 5 \beta_{6} ) q^{97} + ( 2 + 2 \beta_{3} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} - 8q^{5} - 4q^{6} + 4q^{9} + O(q^{10}) \) \( 8q - 8q^{4} - 8q^{5} - 4q^{6} + 4q^{9} + 4q^{10} - 8q^{11} + 8q^{15} + 8q^{16} + 8q^{20} + 4q^{24} - 12q^{25} + 16q^{30} - 12q^{31} + 24q^{34} - 4q^{36} - 4q^{40} + 8q^{41} + 8q^{44} + 8q^{45} - 32q^{46} + 16q^{50} - 24q^{51} - 8q^{54} - 16q^{55} + 8q^{59} - 8q^{60} + 80q^{61} - 8q^{64} + 16q^{66} - 16q^{69} + 16q^{71} - 16q^{74} - 16q^{75} + 16q^{79} - 8q^{80} - 4q^{81} - 48q^{85} + 24q^{86} - 16q^{89} - 4q^{90} + 112q^{91} - 32q^{94} - 4q^{96} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/40\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 7 \nu \)\()/10\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 13 \nu \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( -9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48 \)\()/20\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu \)\()/40\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 3 \beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - 11 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{6} - 5 \beta_{1} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{5} - 13 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
1.09445 + 0.895644i
−0.228425 1.39564i
−1.09445 0.895644i
0.228425 + 1.39564i
−1.09445 + 0.895644i
0.228425 1.39564i
1.09445 0.895644i
−0.228425 + 1.39564i
1.00000i −0.866025 0.500000i −1.00000 −1.86603 1.23205i −0.500000 + 0.866025i −2.29129 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.2 1.00000i −0.866025 0.500000i −1.00000 −1.86603 1.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.3 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 2.23205i −0.500000 + 0.866025i −2.29129 1.32288i 1.00000i 0.500000 + 0.866025i 2.23205 0.133975i
439.4 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 2.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i 1.00000i 0.500000 + 0.866025i 2.23205 0.133975i
769.1 1.00000i 0.866025 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 0.866025i −2.29129 + 1.32288i 1.00000i 0.500000 0.866025i 2.23205 + 0.133975i
769.2 1.00000i 0.866025 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 0.866025i 2.29129 1.32288i 1.00000i 0.500000 0.866025i 2.23205 + 0.133975i
769.3 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 0.866025i −2.29129 + 1.32288i 1.00000i 0.500000 0.866025i −1.23205 1.86603i
769.4 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 0.866025i 2.29129 1.32288i 1.00000i 0.500000 0.866025i −1.23205 1.86603i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.b 8
5.b even 2 1 inner 930.2.s.b 8
31.c even 3 1 inner 930.2.s.b 8
155.j even 6 1 inner 930.2.s.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.b 8 1.a even 1 1 trivial
930.2.s.b 8 5.b even 2 1 inner
930.2.s.b 8 31.c even 3 1 inner
930.2.s.b 8 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 7 T_{7}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( ( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$7$ \( ( 49 - 7 T^{2} + T^{4} )^{2} \)
$11$ \( ( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$13$ \( ( 784 - 28 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1296 - 36 T^{2} + T^{4} )^{2} \)
$19$ \( ( 784 + 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 144 + 88 T^{2} + T^{4} )^{2} \)
$29$ \( ( -7 + T^{2} )^{4} \)
$31$ \( ( 961 + 186 T + 43 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$37$ \( ( 256 - 16 T^{2} + T^{4} )^{2} \)
$41$ \( ( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$43$ \( 4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8} \)
$47$ \( ( 144 + 88 T^{2} + T^{4} )^{2} \)
$53$ \( 130321 - 26714 T^{2} + 5115 T^{4} - 74 T^{6} + T^{8} \)
$59$ \( ( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( ( -10 + T )^{8} \)
$67$ \( 4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8} \)
$71$ \( ( 9216 + 768 T + 160 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$73$ \( 84934656 - 2359296 T^{2} + 56320 T^{4} - 256 T^{6} + T^{8} \)
$79$ \( ( 9216 + 768 T + 160 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( 151807041 - 2784546 T^{2} + 38755 T^{4} - 226 T^{6} + T^{8} \)
$89$ \( ( -24 + 4 T + T^{2} )^{4} \)
$97$ \( ( 29241 + 358 T^{2} + T^{4} )^{2} \)
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