# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$.