Properties

Label 930.2.bg.i
Level $930$
Weight $2$
Character orbit 930.bg
Analytic conductor $7.426$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(3\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 0.809017 0.587785i −0.104528 + 0.994522i 0.309017 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i −2.78894 + 0.592807i −0.309017 0.951057i −0.978148 0.207912i −0.104528 0.994522i
121.2 0.809017 0.587785i −0.104528 + 0.994522i 0.309017 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i −1.97429 + 0.419648i −0.309017 0.951057i −0.978148 0.207912i −0.104528 0.994522i
121.3 0.809017 0.587785i −0.104528 + 0.994522i 0.309017 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i 4.43236 0.942127i −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 −0.540069 5.13841i 0.809017 + 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
361.2 −0.309017 0.951057i 0.669131 + 0.743145i −0.809017 + 0.587785i 0.500000 + 0.866025i 0.500000 0.866025i 0.0785623 + 0.747471i 0.809017 + 0.587785i −0.104528 + 0.994522i 0.669131 0.743145i
361.3 −0.309017 0.951057i 0.669131 + 0.743145i −0.809017 + 0.587785i 0.500000 + 0.866025i 0.500000 0.866025i 0.375052 + 3.56838i 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 −3.49940 + 3.88648i −0.309017 + 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
391.2 0.809017 + 0.587785i 0.913545 + 0.406737i 0.309017 + 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i 0.346545 0.384877i −0.309017 + 0.951057i 0.669131 + 0.743145i 0.913545 0.406737i
391.3 0.809017 + 0.587785i 0.913545 + 0.406737i 0.309017 + 0.951057i 0.500000 0.866025i 0.500000 + 0.866025i 1.17471 1.30465i −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 −3.49940 3.88648i −0.309017 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
421.2 0.809017 0.587785i 0.913545 0.406737i 0.309017 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i 0.346545 + 0.384877i −0.309017 0.951057i 0.669131 0.743145i 0.913545 + 0.406737i
421.3 0.809017 0.587785i 0.913545 0.406737i 0.309017 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i 1.17471 + 1.30465i −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 −0.540069 + 5.13841i 0.809017 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
541.2 −0.309017 + 0.951057i 0.669131 0.743145i −0.809017 0.587785i 0.500000 0.866025i 0.500000 + 0.866025i 0.0785623 0.747471i 0.809017 0.587785i −0.104528 0.994522i 0.669131 + 0.743145i
541.3 −0.309017 + 0.951057i 0.669131 0.743145i −0.809017 0.587785i 0.500000 0.866025i 0.500000 + 0.866025i 0.375052 3.56838i 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 −2.78894 0.592807i −0.309017 + 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
661.2 0.809017 + 0.587785i −0.104528 0.994522i 0.309017 + 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i −1.97429 0.419648i −0.309017 + 0.951057i −0.978148 + 0.207912i −0.104528 + 0.994522i
661.3 0.809017 + 0.587785i −0.104528 0.994522i 0.309017 + 0.951057i 0.500000 + 0.866025i 0.500000 0.866025i 4.43236 + 0.942127i −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 −4.59456 + 2.04563i 0.809017 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
691.2 −0.309017 + 0.951057i −0.978148 0.207912i −0.809017 0.587785i 0.500000 + 0.866025i 0.500000 0.866025i −0.381451 + 0.169833i 0.809017 0.587785i 0.913545 + 0.406737i −0.978148 + 0.207912i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.3
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.i 24
31.g even 15 1 inner 930.2.bg.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.i 24 1.a even 1 1 trivial
930.2.bg.i 24 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{24} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).