# Properties

 Label 930.2.bg.h 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.

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

## 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 −4.34565 + 0.923697i −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 0.105382 0.0223997i −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 0.744050 0.158153i −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 −0.220951 2.10221i 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 0.0963276 + 0.916496i 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 0.167373 + 1.59245i 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 −2.71613 + 3.01657i −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 0.334019 0.370966i −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 2.56932 2.85352i −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 −2.71613 3.01657i −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 0.334019 + 0.370966i −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 2.56932 + 2.85352i −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 −0.220951 + 2.10221i 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 0.0963276 0.916496i 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 0.167373 1.59245i 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 −4.34565 0.923697i −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 0.105382 + 0.0223997i −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 0.744050 + 0.158153i −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 −2.37522 + 1.05752i 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 −2.21753 + 0.987309i 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.h 24
31.g even 15 1 inner 930.2.bg.h 24

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