# Properties

 Label 930.2.bg.g 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.

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

## 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 −3.47616 + 0.738880i 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 −0.813060 + 0.172821i 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 2.66379 0.566206i 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 −0.272387 2.59159i −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 −0.258263 2.45720i −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 0.408047 + 3.88231i −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 −0.348368 + 0.386902i 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 0.955933 1.06167i 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 2.70884 3.00848i 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 −0.348368 0.386902i 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 0.955933 + 1.06167i 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 2.70884 + 3.00848i 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 −0.272387 + 2.59159i −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 −0.258263 + 2.45720i −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 0.408047 3.88231i −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 −3.47616 0.738880i 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 −0.813060 0.172821i 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 2.66379 + 0.566206i 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 −2.05288 + 0.914002i −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 0.302260 0.134575i −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.g 24
31.g even 15 1 inner 930.2.bg.g 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bg.g 24 1.a even 1 1 trivial
930.2.bg.g 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$.