Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \(x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{15}^{3} q^{2} -\zeta_{15}^{7} q^{3} + \zeta_{15}^{6} q^{4} -\zeta_{15}^{5} q^{5} + ( 1 + \zeta_{15}^{5} ) q^{6} + ( 4 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{7} + ( -1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{8} + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{9} +O(q^{10})\) \( q + \zeta_{15}^{3} q^{2} -\zeta_{15}^{7} q^{3} + \zeta_{15}^{6} q^{4} -\zeta_{15}^{5} q^{5} + ( 1 + \zeta_{15}^{5} ) q^{6} + ( 4 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{7} + ( -1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{8} + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{9} + ( 1 - \zeta_{15} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{10} + ( -3 - 2 \zeta_{15} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} - 2 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{11} + ( -1 + \zeta_{15} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{12} + ( -3 + 2 \zeta_{15} + \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{13} + ( 2 \zeta_{15} + 2 \zeta_{15}^{2} + 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{14} + ( -\zeta_{15}^{2} - \zeta_{15}^{7} ) q^{15} + ( -\zeta_{15}^{2} - \zeta_{15}^{7} ) q^{16} + ( 5 \zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{4} + \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{17} + \zeta_{15}^{2} q^{18} + ( 4 - 2 \zeta_{15} - 4 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{19} + ( \zeta_{15} + \zeta_{15}^{6} ) q^{20} + ( 2 + 2 \zeta_{15} ) q^{21} + ( 5 - 2 \zeta_{15}^{2} - \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{22} + ( \zeta_{15} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{23} -\zeta_{15} q^{24} + ( -1 - \zeta_{15}^{5} ) q^{25} + ( 1 - \zeta_{15} - \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} + \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{26} -\zeta_{15}^{6} q^{27} + ( -4 + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{28} + ( 3 + 2 \zeta_{15} - 2 \zeta_{15}^{2} - \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 3 \zeta_{15}^{6} ) q^{29} + q^{30} + ( 6 - 2 \zeta_{15} - 2 \zeta_{15}^{2} - 4 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{31} + q^{32} + ( 1 - 2 \zeta_{15} - 3 \zeta_{15}^{2} + \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 2 \zeta_{15}^{5} + \zeta_{15}^{6} ) q^{33} + ( -6 + \zeta_{15}^{2} - \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{34} + ( 4 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{35} + \zeta_{15}^{5} q^{36} + ( 3 - 7 \zeta_{15} + 5 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 4 \zeta_{15}^{5} - 2 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{37} + ( 2 + 2 \zeta_{15} + 2 \zeta_{15}^{2} ) q^{38} + ( 4 - 4 \zeta_{15} - \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 4 \zeta_{15}^{5} - \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{39} + ( -1 + \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{40} + ( -6 - 2 \zeta_{15} - 4 \zeta_{15}^{5} + 4 \zeta_{15}^{6} ) q^{41} + ( 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} ) q^{42} + ( 1 - \zeta_{15} - \zeta_{15}^{4} + 5 \zeta_{15}^{5} + 4 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{43} + ( 1 + 3 \zeta_{15} + 2 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{44} -\zeta_{15}^{4} q^{45} + ( \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{46} + ( -4 + 8 \zeta_{15} + \zeta_{15}^{2} - 5 \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - \zeta_{15}^{6} + 5 \zeta_{15}^{7} ) q^{47} -\zeta_{15}^{4} q^{48} + ( 4 \zeta_{15} + \zeta_{15}^{2} + 4 \zeta_{15}^{3} ) q^{49} + ( 1 - \zeta_{15} - \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{50} + ( -2 \zeta_{15} + 4 \zeta_{15}^{2} + \zeta_{15}^{3} + \zeta_{15}^{4} + 4 \zeta_{15}^{5} - 2 \zeta_{15}^{6} ) q^{51} + ( \zeta_{15} + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{52} + ( 4 \zeta_{15} - 2 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{53} + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{54} + ( -4 - \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - 2 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{55} + ( -2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} ) q^{56} + ( -2 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{6} ) q^{57} + ( -5 + 2 \zeta_{15} + 3 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{58} + ( 5 - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - \zeta_{15}^{4} - 3 \zeta_{15}^{5} + 4 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{59} + \zeta_{15}^{3} q^{60} + ( 6 + 2 \zeta_{15} + 4 \zeta_{15}^{3} - 10 \zeta_{15}^{4} + 4 \zeta_{15}^{5} + 6 \zeta_{15}^{6} - 8 \zeta_{15}^{7} ) q^{61} + ( -1 + 6 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{62} + ( 2 - 2 \zeta_{15} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{63} + \zeta_{15}^{3} q^{64} + ( -1 - 2 \zeta_{15} + 2 \zeta_{15}^{2} - \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 4 \zeta_{15}^{6} ) q^{65} + ( 1 - 2 \zeta_{15} + \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 4 \zeta_{15}^{4} - \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{66} + ( -8 + 8 \zeta_{15} + 8 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 4 \zeta_{15}^{4} + \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 12 \zeta_{15}^{7} ) q^{67} + ( 2 - 4 \zeta_{15} - \zeta_{15}^{2} - \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} ) q^{68} + ( 1 - \zeta_{15}^{2} + 2 \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{69} + ( 2 \zeta_{15} + 2 \zeta_{15}^{2} ) q^{70} + ( 12 - 6 \zeta_{15} - 2 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 8 \zeta_{15}^{4} + 4 \zeta_{15}^{5} + 4 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{71} + ( -1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{72} + ( 2 + 4 \zeta_{15} + 2 \zeta_{15}^{2} + 4 \zeta_{15}^{3} + 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 4 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{73} + ( -3 + 4 \zeta_{15} - 2 \zeta_{15}^{2} + \zeta_{15}^{3} - 3 \zeta_{15}^{4} + 5 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{74} -\zeta_{15}^{2} q^{75} + ( 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} ) q^{76} + ( -2 + 4 \zeta_{15} - 2 \zeta_{15}^{2} - 12 \zeta_{15}^{3} - 2 \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 10 \zeta_{15}^{6} ) q^{77} + ( -2 + 4 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - 4 \zeta_{15}^{5} + 3 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{78} + ( 4 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 8 \zeta_{15}^{4} + 5 \zeta_{15}^{5} + 4 \zeta_{15}^{6} ) q^{79} -\zeta_{15}^{2} q^{80} + ( 1 - \zeta_{15} - \zeta_{15}^{4} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{81} + ( -4 \zeta_{15} + 4 \zeta_{15}^{2} - 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{6} ) q^{82} + ( -4 + 4 \zeta_{15} + 2 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 6 \zeta_{15}^{7} ) q^{83} + ( 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{84} + ( 2 + \zeta_{15} + 3 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{85} + ( -8 + 5 \zeta_{15} + 4 \zeta_{15}^{2} - 8 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 8 \zeta_{15}^{7} ) q^{86} + ( -4 - \zeta_{15} + 4 \zeta_{15}^{2} + \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{87} + ( 1 - \zeta_{15} - \zeta_{15}^{2} + 3 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 3 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{88} + ( 4 + 2 \zeta_{15} - 6 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 6 \zeta_{15}^{6} ) q^{89} -\zeta_{15}^{7} q^{90} + ( 6 + 4 \zeta_{15} - 2 \zeta_{15}^{2} - 4 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 4 \zeta_{15}^{5} + 6 \zeta_{15}^{6} ) q^{91} + ( -1 + \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{92} + ( 1 - 3 \zeta_{15} + 4 \zeta_{15}^{2} - 3 \zeta_{15}^{4} - \zeta_{15}^{5} - 2 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{93} + ( -1 - 3 \zeta_{15} - \zeta_{15}^{2} + 5 \zeta_{15}^{4} - \zeta_{15}^{5} - 4 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{94} + ( -2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} ) q^{95} -\zeta_{15}^{7} q^{96} + ( 6 - 6 \zeta_{15}^{2} - 4 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{97} + ( 4 \zeta_{15}^{4} + \zeta_{15}^{5} + 4 \zeta_{15}^{6} ) q^{98} + ( -5 + \zeta_{15}^{2} - 5 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 6 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - q^{3} - 2q^{4} + 4q^{5} + 4q^{6} + 2q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - q^{3} - 2q^{4} + 4q^{5} + 4q^{6} + 2q^{7} - 2q^{8} + q^{9} - q^{10} - 23q^{11} - q^{12} - 17q^{13} + 2q^{14} - 2q^{15} - 2q^{16} + 11q^{17} + q^{18} - q^{20} + 18q^{21} + 22q^{22} - 2q^{23} - q^{24} - 4q^{25} - 2q^{26} + 2q^{27} - 18q^{28} + 10q^{29} + 8q^{30} + 14q^{31} + 8q^{32} + 4q^{33} - 19q^{34} + 4q^{35} - 4q^{36} - q^{37} + 20q^{38} + 6q^{39} - q^{40} - 42q^{41} - 2q^{42} - 23q^{43} + 17q^{44} - q^{45} + 3q^{46} + 11q^{47} - q^{48} - 3q^{49} + q^{50} - 11q^{51} + 13q^{52} + 2q^{53} + 2q^{54} - 22q^{55} + 2q^{56} + 10q^{57} + 28q^{59} - 2q^{60} - 4q^{61} + 14q^{62} - 4q^{63} - 2q^{64} - 13q^{65} - q^{66} - 20q^{67} + q^{68} - q^{69} + 4q^{70} + 42q^{71} + q^{72} + 4q^{73} - 41q^{74} - q^{75} - 10q^{76} + 52q^{77} - 4q^{78} - 42q^{79} - q^{80} + q^{81} - 2q^{82} + 6q^{83} - 2q^{84} + 22q^{85} - 3q^{86} - 10q^{87} - 3q^{88} + 16q^{89} - q^{90} + 28q^{91} - 2q^{92} + 13q^{93} + 6q^{94} - q^{96} + 44q^{97} - 8q^{98} - 3q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.978148 + 0.207912i
−0.104528 0.994522i
0.669131 0.743145i
0.669131 + 0.743145i
−0.104528 + 0.994522i
−0.978148 0.207912i
0.913545 0.406737i
0.913545 + 0.406737i
−0.809017 + 0.587785i 0.104528 0.994522i 0.309017 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i −0.408977 + 0.0869308i 0.309017 + 0.951057i −0.978148 0.207912i 0.104528 + 0.994522i
361.1 0.309017 + 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i 0.500000 + 0.866025i 0.500000 0.866025i 0.279773 + 2.66186i −0.809017 0.587785i −0.104528 + 0.994522i −0.669131 + 0.743145i
391.1 −0.809017 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i −2.44512 + 2.71559i 0.309017 0.951057i 0.669131 + 0.743145i −0.913545 + 0.406737i
421.1 −0.809017 + 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i −2.44512 2.71559i 0.309017 + 0.951057i 0.669131 0.743145i −0.913545 0.406737i
541.1 0.309017 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i 0.500000 0.866025i 0.500000 + 0.866025i 0.279773 2.66186i −0.809017 + 0.587785i −0.104528 0.994522i −0.669131 0.743145i
661.1 −0.809017 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i −0.408977 0.0869308i 0.309017 0.951057i −0.978148 + 0.207912i 0.104528 0.994522i
691.1 0.309017 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i 0.500000 + 0.866025i 0.500000 0.866025i 3.57433 1.59139i −0.809017 + 0.587785i 0.913545 + 0.406737i 0.978148 0.207912i
751.1 0.309017 + 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i 0.500000 0.866025i 0.500000 + 0.866025i 3.57433 + 1.59139i −0.809017 0.587785i 0.913545 0.406737i 0.978148 + 0.207912i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bg.a 8
31.g even 15 1 inner 930.2.bg.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.a 8 1.a even 1 1 trivial
930.2.bg.a 8 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 2 T_{7}^{7} - 32 T_{7}^{5} + 144 T_{7}^{4} - 128 T_{7}^{3} + 1280 T_{7}^{2} + 1152 T_{7} + 256 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \)
$5$ \( ( 1 - T + T^{2} )^{4} \)
$7$ \( 256 + 1152 T + 1280 T^{2} - 128 T^{3} + 144 T^{4} - 32 T^{5} - 2 T^{7} + T^{8} \)
$11$ \( 73441 + 119782 T + 97300 T^{2} + 43292 T^{3} + 11589 T^{4} + 2068 T^{5} + 265 T^{6} + 23 T^{7} + T^{8} \)
$13$ \( 841 - 2262 T + 3650 T^{2} + 5408 T^{3} + 2559 T^{4} + 692 T^{5} + 135 T^{6} + 17 T^{7} + T^{8} \)
$17$ \( 292681 - 160136 T + 25740 T^{2} - 2959 T^{3} + 2054 T^{4} - 454 T^{5} + 60 T^{6} - 11 T^{7} + T^{8} \)
$19$ \( 6400 + 12800 T + 9600 T^{2} + 3200 T^{3} + 480 T^{4} + 80 T^{5} + 20 T^{6} + T^{8} \)
$23$ \( 1 - 3 T + 23 T^{2} - T^{3} - T^{5} + 3 T^{6} + 2 T^{7} + T^{8} \)
$29$ \( 25 + 150 T + 1775 T^{2} + 1700 T^{3} + 840 T^{4} + 170 T^{5} + 30 T^{6} - 10 T^{7} + T^{8} \)
$31$ \( 923521 - 417074 T + 158565 T^{2} - 40486 T^{3} + 8279 T^{4} - 1306 T^{5} + 165 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( 961 - 2294 T + 9010 T^{2} + 8374 T^{3} + 13039 T^{4} + 34 T^{5} + 115 T^{6} + T^{7} + T^{8} \)
$41$ \( 11397376 + 8804608 T + 3098880 T^{2} + 665728 T^{3} + 100944 T^{4} + 11152 T^{5} + 860 T^{6} + 42 T^{7} + T^{8} \)
$43$ \( 3481 - 36993 T + 109700 T^{2} + 63557 T^{3} + 16539 T^{4} + 2693 T^{5} + 300 T^{6} + 23 T^{7} + T^{8} \)
$47$ \( 292681 + 135791 T + 75502 T^{2} + 8893 T^{3} - 225 T^{4} - 283 T^{5} + 112 T^{6} - 11 T^{7} + T^{8} \)
$53$ \( 246016 + 333312 T + 150080 T^{2} + 23392 T^{3} + 1344 T^{4} + 328 T^{5} + 60 T^{6} - 2 T^{7} + T^{8} \)
$59$ \( 436921 - 493767 T + 258455 T^{2} - 84682 T^{3} + 19779 T^{4} - 3358 T^{5} + 390 T^{6} - 28 T^{7} + T^{8} \)
$61$ \( ( 5296 + 88 T - 196 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$67$ \( 89397025 + 19477300 T + 4149050 T^{2} + 398800 T^{3} + 50755 T^{4} + 3920 T^{5} + 410 T^{6} + 20 T^{7} + T^{8} \)
$71$ \( 19927296 - 11570688 T + 3265920 T^{2} - 628128 T^{3} + 93024 T^{4} - 10512 T^{5} + 840 T^{6} - 42 T^{7} + T^{8} \)
$73$ \( 256 + 896 T + 960 T^{2} + 64 T^{3} - 496 T^{4} - 176 T^{5} + 240 T^{6} - 4 T^{7} + T^{8} \)
$79$ \( 6046681 - 3892597 T + 184305 T^{2} + 81458 T^{3} + 36729 T^{4} + 6692 T^{5} + 740 T^{6} + 42 T^{7} + T^{8} \)
$83$ \( 256 - 896 T + 1920 T^{2} + 256 T^{3} + 1104 T^{4} + 256 T^{5} - 40 T^{6} - 6 T^{7} + T^{8} \)
$89$ \( 18524416 - 1618304 T + 404032 T^{2} + 21728 T^{3} + 2880 T^{4} - 1448 T^{5} + 292 T^{6} - 16 T^{7} + T^{8} \)
$97$ \( ( 16 + 32 T + 184 T^{2} - 22 T^{3} + T^{4} )^{2} \)
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