# Properties

 Label 930.2.bn.a Level $930$ Weight $2$ Character orbit 930.bn Analytic conductor $7.426$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.bn (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 28q^{4} - 2q^{5} + 56q^{6} - 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 28q^{4} - 2q^{5} + 56q^{6} - 14q^{9} - 4q^{10} + 18q^{11} + 8q^{14} + 8q^{15} - 28q^{16} + 16q^{19} + 2q^{20} + 28q^{21} + 14q^{24} + 14q^{25} + 12q^{26} + 16q^{29} - 4q^{30} + 10q^{34} - 38q^{35} - 56q^{36} + 16q^{39} - 6q^{40} + 20q^{41} + 2q^{44} + 2q^{45} - 2q^{46} + 38q^{49} + 8q^{50} - 10q^{51} - 28q^{54} - 46q^{55} + 12q^{56} + 60q^{59} - 8q^{60} + 88q^{61} + 28q^{64} - 28q^{65} + 6q^{66} + 46q^{69} + 26q^{70} + 116q^{71} - 34q^{74} + 8q^{75} + 24q^{76} - 40q^{79} - 12q^{80} + 14q^{81} - 8q^{84} + 18q^{85} - 38q^{86} - 60q^{89} + 4q^{90} - 92q^{91} + 132q^{94} + 132q^{95} - 14q^{96} + 2q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i −2.15286 0.604297i 0.500000 + 0.866025i 2.63601 + 2.37347i 0.951057 + 0.309017i −0.669131 0.743145i 1.75431 1.38651i
19.2 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i −1.44692 1.70482i 0.500000 + 0.866025i −3.41731 3.07696i 0.951057 + 0.309017i −0.669131 0.743145i 2.22971 0.168512i
19.3 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.509764 + 2.17719i 0.500000 + 0.866025i 0.159624 + 0.143726i 0.951057 + 0.309017i −0.669131 0.743145i −1.46175 1.69213i
19.4 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.181057 + 2.22873i 0.500000 + 0.866025i 3.89646 + 3.50839i 0.951057 + 0.309017i −0.669131 0.743145i −1.69665 1.45649i
19.5 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i 0.230445 2.22416i 0.500000 + 0.866025i 0.872484 + 0.785588i 0.951057 + 0.309017i −0.669131 0.743145i 1.66393 + 1.49376i
19.6 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i 1.47095 + 1.68413i 0.500000 + 0.866025i −2.07600 1.86924i 0.951057 + 0.309017i −0.669131 0.743145i −2.22710 + 0.200114i
19.7 −0.587785 + 0.809017i 0.406737 0.913545i −0.309017 0.951057i 2.21090 + 0.334560i 0.500000 + 0.866025i −0.274259 0.246944i 0.951057 + 0.309017i −0.669131 0.743145i −1.57020 + 1.59200i
19.8 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −2.19398 0.431813i 0.500000 + 0.866025i 2.07600 + 1.86924i −0.951057 0.309017i −0.669131 0.743145i −1.63893 + 1.52115i
19.9 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.83960 + 1.27116i 0.500000 + 0.866025i −3.89646 3.50839i −0.951057 0.309017i −0.669131 0.743145i −0.0529000 + 2.23544i
19.10 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.63062 + 1.53006i 0.500000 + 0.866025i −0.159624 0.143726i −0.951057 0.309017i −0.669131 0.743145i 0.279393 + 2.21854i
19.11 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.39519 1.74741i 0.500000 + 0.866025i 0.274259 + 0.246944i −0.951057 0.309017i −0.669131 0.743145i −2.23376 + 0.101626i
19.12 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 1.59977 + 1.56229i 0.500000 + 0.866025i −2.63601 2.37347i −0.951057 0.309017i −0.669131 0.743145i 2.20424 0.375951i
19.13 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 1.81096 1.31165i 0.500000 + 0.866025i −0.872484 0.785588i −0.951057 0.309017i −0.669131 0.743145i 0.00330584 2.23607i
19.14 0.587785 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 2.19988 + 0.400657i 0.500000 + 0.866025i 3.41731 + 3.07696i −0.951057 0.309017i −0.669131 0.743145i 1.61720 1.54424i
49.1 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −2.15286 + 0.604297i 0.500000 0.866025i 2.63601 2.37347i 0.951057 0.309017i −0.669131 + 0.743145i 1.75431 + 1.38651i
49.2 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −1.44692 + 1.70482i 0.500000 0.866025i −3.41731 + 3.07696i 0.951057 0.309017i −0.669131 + 0.743145i 2.22971 + 0.168512i
49.3 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −0.509764 2.17719i 0.500000 0.866025i 0.159624 0.143726i 0.951057 0.309017i −0.669131 + 0.743145i −1.46175 + 1.69213i
49.4 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −0.181057 2.22873i 0.500000 0.866025i 3.89646 3.50839i 0.951057 0.309017i −0.669131 + 0.743145i −1.69665 + 1.45649i
49.5 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i 0.230445 + 2.22416i 0.500000 0.866025i 0.872484 0.785588i 0.951057 0.309017i −0.669131 + 0.743145i 1.66393 1.49376i
49.6 −0.587785 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i 1.47095 1.68413i 0.500000 0.866025i −2.07600 + 1.86924i 0.951057 0.309017i −0.669131 + 0.743145i −2.22710 0.200114i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 919.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.g even 15 1 inner
155.u even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bn.a 112
5.b even 2 1 inner 930.2.bn.a 112
31.g even 15 1 inner 930.2.bn.a 112
155.u even 30 1 inner 930.2.bn.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bn.a 112 1.a even 1 1 trivial
930.2.bn.a 112 5.b even 2 1 inner
930.2.bn.a 112 31.g even 15 1 inner
930.2.bn.a 112 155.u even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$14\!\cdots\!30$$$$T_{7}^{94} +$$$$36\!\cdots\!28$$$$T_{7}^{92} +$$$$36\!\cdots\!88$$$$T_{7}^{90} -$$$$11\!\cdots\!79$$$$T_{7}^{88} -$$$$92\!\cdots\!06$$$$T_{7}^{86} -$$$$15\!\cdots\!71$$$$T_{7}^{84} -$$$$11\!\cdots\!68$$$$T_{7}^{82} +$$$$52\!\cdots\!42$$$$T_{7}^{80} +$$$$25\!\cdots\!64$$$$T_{7}^{78} +$$$$32\!\cdots\!51$$$$T_{7}^{76} +$$$$19\!\cdots\!92$$$$T_{7}^{74} -$$$$69\!\cdots\!43$$$$T_{7}^{72} -$$$$14\!\cdots\!96$$$$T_{7}^{70} -$$$$15\!\cdots\!59$$$$T_{7}^{68} -$$$$75\!\cdots\!22$$$$T_{7}^{66} +$$$$75\!\cdots\!31$$$$T_{7}^{64} +$$$$43\!\cdots\!52$$$$T_{7}^{62} +$$$$35\!\cdots\!38$$$$T_{7}^{60} +$$$$12\!\cdots\!24$$$$T_{7}^{58} -$$$$11\!\cdots\!15$$$$T_{7}^{56} -$$$$36\!\cdots\!22$$$$T_{7}^{54} -$$$$18\!\cdots\!04$$$$T_{7}^{52} -$$$$27\!\cdots\!96$$$$T_{7}^{50} +$$$$14\!\cdots\!87$$$$T_{7}^{48} +$$$$98\!\cdots\!46$$$$T_{7}^{46} +$$$$26\!\cdots\!52$$$$T_{7}^{44} +$$$$19\!\cdots\!00$$$$T_{7}^{42} -$$$$98\!\cdots\!31$$$$T_{7}^{40} -$$$$37\!\cdots\!86$$$$T_{7}^{38} -$$$$55\!\cdots\!68$$$$T_{7}^{36} -$$$$32\!\cdots\!46$$$$T_{7}^{34} +$$$$16\!\cdots\!68$$$$T_{7}^{32} +$$$$43\!\cdots\!70$$$$T_{7}^{30} +$$$$68\!\cdots\!54$$$$T_{7}^{28} +$$$$75\!\cdots\!40$$$$T_{7}^{26} +$$$$54\!\cdots\!41$$$$T_{7}^{24} +$$$$20\!\cdots\!22$$$$T_{7}^{22} +$$$$35\!\cdots\!79$$$$T_{7}^{20} +$$$$30\!\cdots\!70$$$$T_{7}^{18} +$$$$63\!\cdots\!73$$$$T_{7}^{16} -$$$$59\!\cdots\!96$$$$T_{7}^{14} +$$$$10\!\cdots\!48$$$$T_{7}^{12} -$$$$22\!\cdots\!24$$$$T_{7}^{10} +$$$$62\!\cdots\!24$$$$T_{7}^{8} -$$$$63\!\cdots\!00$$$$T_{7}^{6} +$$$$18\!\cdots\!92$$$$T_{7}^{4} -$$$$20\!\cdots\!96$$$$T_{7}^{2} +$$$$81\!\cdots\!36$$">$$T_{7}^{112} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.