# Properties

 Label 930.2.bo.a Level $930$ Weight $2$ Character orbit 930.bo Analytic conductor $7.426$ Analytic rank $0$ Dimension $256$ 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.bo (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$256$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$256q - 64q^{2} - 64q^{4} - 2q^{5} - 64q^{8} + 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$256q - 64q^{2} - 64q^{4} - 2q^{5} - 64q^{8} + 4q^{9} - 2q^{10} + 20q^{15} - 64q^{16} - 6q^{17} - 6q^{18} - 4q^{19} + 3q^{20} + 20q^{23} - 2q^{25} + 42q^{31} + 256q^{32} + 8q^{33} + 14q^{34} - 16q^{35} + 4q^{36} + 36q^{38} + 8q^{39} + 3q^{40} - 79q^{45} - 10q^{46} - 6q^{47} - 40q^{49} - 7q^{50} + 68q^{51} - 34q^{53} - 6q^{57} - 20q^{60} + 2q^{62} - 72q^{63} - 64q^{64} + 8q^{66} - 6q^{68} + 10q^{69} - 16q^{70} - 6q^{72} - 2q^{75} - 24q^{76} + 100q^{77} + 8q^{78} + 40q^{79} - 2q^{80} + 12q^{81} - 26q^{83} - 30q^{85} + 16q^{87} - 49q^{90} - 20q^{91} - 22q^{93} + 4q^{94} + 56q^{95} + 130q^{98} + 102q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1 0.309017 + 0.951057i −1.73157 0.0407018i −0.809017 + 0.587785i 0.493029 + 2.18104i −0.496376 1.65940i −0.863747 + 1.94001i −0.809017 0.587785i 2.99669 + 0.140956i −1.92193 + 1.14288i
179.2 0.309017 + 0.951057i −1.69787 0.342394i −0.809017 + 0.587785i −1.92997 1.12925i −0.199035 1.72058i −0.934663 + 2.09929i −0.809017 0.587785i 2.76553 + 1.16268i 0.477584 2.18447i
179.3 0.309017 + 0.951057i −1.66354 0.482315i −0.809017 + 0.587785i 0.575623 2.16071i −0.0553539 1.73117i −1.66082 + 3.73026i −0.809017 0.587785i 2.53474 + 1.60470i 2.23283 0.120245i
179.4 0.309017 + 0.951057i −1.64568 + 0.540140i −0.809017 + 0.587785i 2.22856 0.183115i −1.02225 1.39822i 0.318388 0.715111i −0.809017 0.587785i 2.41650 1.77779i 0.862815 + 2.06290i
179.5 0.309017 + 0.951057i −1.59415 0.677259i −0.809017 + 0.587785i −2.09420 + 0.783790i 0.151491 1.72541i 1.79370 4.02872i −0.809017 0.587785i 2.08264 + 2.15931i −1.39257 1.74950i
179.6 0.309017 + 0.951057i −1.46052 0.931061i −0.809017 + 0.587785i 1.43804 + 1.71232i 0.434165 1.67675i 0.429946 0.965674i −0.809017 0.587785i 1.26625 + 2.71967i −1.18414 + 1.89679i
179.7 0.309017 + 0.951057i −1.41936 + 0.992677i −0.809017 + 0.587785i −1.50750 + 1.65150i −1.38270 1.04314i 0.732525 1.64528i −0.809017 0.587785i 1.02919 2.81794i −2.03651 0.923374i
179.8 0.309017 + 0.951057i −1.39110 + 1.03191i −0.809017 + 0.587785i −1.53101 1.62973i −1.41128 1.00414i 0.325961 0.732121i −0.809017 0.587785i 0.870333 2.87098i 1.07685 1.95969i
179.9 0.309017 + 0.951057i −1.21541 1.23401i −0.809017 + 0.587785i 1.34040 1.78978i 0.798025 1.53726i 1.34591 3.02296i −0.809017 0.587785i −0.0455387 + 2.99965i 2.11639 + 0.721724i
179.10 0.309017 + 0.951057i −1.03655 + 1.38764i −0.809017 + 0.587785i −1.97712 + 1.04451i −1.64004 0.557016i −1.83029 + 4.11091i −0.809017 0.587785i −0.851111 2.87674i −1.60435 1.55758i
179.11 0.309017 + 0.951057i −0.835073 1.51745i −0.809017 + 0.587785i 2.23544 + 0.0527903i 1.18513 1.26312i −0.915394 + 2.05601i −0.809017 0.587785i −1.60531 + 2.53436i 0.640584 + 2.14235i
179.12 0.309017 + 0.951057i −0.732396 + 1.56958i −0.809017 + 0.587785i 0.719639 + 2.11710i −1.71909 0.211521i 1.13296 2.54466i −0.809017 0.587785i −1.92719 2.29911i −1.79110 + 1.33864i
179.13 0.309017 + 0.951057i −0.696767 + 1.58572i −0.809017 + 0.587785i 1.79380 1.33503i −1.72342 0.172650i 0.308507 0.692917i −0.809017 0.587785i −2.02903 2.20976i 1.82400 + 1.29346i
179.14 0.309017 + 0.951057i −0.584760 1.63035i −0.809017 + 0.587785i −2.12123 + 0.707392i 1.36986 1.05995i −1.37162 + 3.08072i −0.809017 0.587785i −2.31611 + 1.90673i −1.32826 1.79881i
179.15 0.309017 + 0.951057i −0.128921 1.72725i −0.809017 + 0.587785i 0.447993 + 2.19073i 1.60287 0.656359i 1.37162 3.08072i −0.809017 0.587785i −2.96676 + 0.445355i −1.94507 + 1.10304i
179.16 0.309017 + 0.951057i 0.0869307 + 1.72987i −0.809017 + 0.587785i −2.09670 0.777090i −1.61834 + 0.617235i 1.84852 4.15185i −0.809017 0.587785i −2.98489 + 0.300757i 0.0911422 2.23421i
179.17 0.309017 + 0.951057i 0.145674 1.72591i −0.809017 + 0.587785i −1.16344 1.90956i 1.68646 0.394792i 0.915394 2.05601i −0.809017 0.587785i −2.95756 0.502843i 1.45657 1.69658i
179.18 0.309017 + 0.951057i 0.198005 + 1.72070i −0.809017 + 0.587785i 2.18990 0.452039i −1.57529 + 0.720038i 0.478693 1.07516i −0.809017 0.587785i −2.92159 + 0.681412i 1.10663 + 1.94303i
179.19 0.309017 + 0.951057i 0.518984 + 1.65247i −0.809017 + 0.587785i −0.703473 2.12253i −1.41122 + 1.00422i −0.478693 + 1.07516i −0.809017 0.587785i −2.46131 + 1.71521i 1.80126 1.32494i
179.20 0.309017 + 0.951057i 0.608421 1.62167i −0.809017 + 0.587785i 0.879797 2.05571i 1.73032 + 0.0775177i −1.34591 + 3.02296i −0.809017 0.587785i −2.25965 1.97332i 2.22697 + 0.201486i
See next 80 embeddings (of 256 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 809.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.h odd 30 1 inner
465.bm 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.bo.a 256
3.b odd 2 1 930.2.bo.b yes 256
5.b even 2 1 930.2.bo.b yes 256
15.d odd 2 1 inner 930.2.bo.a 256
31.h odd 30 1 inner 930.2.bo.a 256
93.p even 30 1 930.2.bo.b yes 256
155.v odd 30 1 930.2.bo.b yes 256
465.bm even 30 1 inner 930.2.bo.a 256

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bo.a 256 1.a even 1 1 trivial
930.2.bo.a 256 15.d odd 2 1 inner
930.2.bo.a 256 31.h odd 30 1 inner
930.2.bo.a 256 465.bm even 30 1 inner
930.2.bo.b yes 256 3.b odd 2 1
930.2.bo.b yes 256 5.b even 2 1
930.2.bo.b yes 256 93.p even 30 1
930.2.bo.b yes 256 155.v odd 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$13\!\cdots\!09$$$$T_{17}^{114} +$$$$42\!\cdots\!81$$$$T_{17}^{113} -$$$$49\!\cdots\!13$$$$T_{17}^{112} +$$$$26\!\cdots\!39$$$$T_{17}^{111} -$$$$14\!\cdots\!41$$$$T_{17}^{110} +$$$$62\!\cdots\!96$$$$T_{17}^{109} +$$$$87\!\cdots\!13$$$$T_{17}^{108} -$$$$11\!\cdots\!70$$$$T_{17}^{107} +$$$$51\!\cdots\!21$$$$T_{17}^{106} -$$$$14\!\cdots\!49$$$$T_{17}^{105} +$$$$14\!\cdots\!33$$$$T_{17}^{104} -$$$$50\!\cdots\!37$$$$T_{17}^{103} +$$$$14\!\cdots\!78$$$$T_{17}^{102} -$$$$62\!\cdots\!55$$$$T_{17}^{101} -$$$$16\!\cdots\!76$$$$T_{17}^{100} +$$$$24\!\cdots\!15$$$$T_{17}^{99} -$$$$81\!\cdots\!12$$$$T_{17}^{98} +$$$$15\!\cdots\!31$$$$T_{17}^{97} -$$$$19\!\cdots\!44$$$$T_{17}^{96} +$$$$36\!\cdots\!45$$$$T_{17}^{95} -$$$$10\!\cdots\!15$$$$T_{17}^{94} +$$$$10\!\cdots\!02$$$$T_{17}^{93} +$$$$13\!\cdots\!48$$$$T_{17}^{92} -$$$$20\!\cdots\!99$$$$T_{17}^{91} +$$$$67\!\cdots\!18$$$$T_{17}^{90} -$$$$52\!\cdots\!14$$$$T_{17}^{89} +$$$$18\!\cdots\!85$$$$T_{17}^{88} +$$$$46\!\cdots\!12$$$$T_{17}^{87} +$$$$31\!\cdots\!69$$$$T_{17}^{86} +$$$$75\!\cdots\!19$$$$T_{17}^{85} +$$$$24\!\cdots\!36$$$$T_{17}^{84} +$$$$29\!\cdots\!58$$$$T_{17}^{83} -$$$$47\!\cdots\!63$$$$T_{17}^{82} +$$$$68\!\cdots\!16$$$$T_{17}^{81} -$$$$20\!\cdots\!18$$$$T_{17}^{80} +$$$$97\!\cdots\!48$$$$T_{17}^{79} -$$$$31\!\cdots\!78$$$$T_{17}^{78} +$$$$26\!\cdots\!22$$$$T_{17}^{77} -$$$$70\!\cdots\!34$$$$T_{17}^{76} -$$$$25\!\cdots\!50$$$$T_{17}^{75} +$$$$90\!\cdots\!29$$$$T_{17}^{74} -$$$$76\!\cdots\!97$$$$T_{17}^{73} +$$$$25\!\cdots\!52$$$$T_{17}^{72} -$$$$12\!\cdots\!94$$$$T_{17}^{71} +$$$$41\!\cdots\!42$$$$T_{17}^{70} -$$$$13\!\cdots\!61$$$$T_{17}^{69} +$$$$42\!\cdots\!34$$$$T_{17}^{68} -$$$$53\!\cdots\!38$$$$T_{17}^{67} +$$$$15\!\cdots\!38$$$$T_{17}^{66} +$$$$10\!\cdots\!79$$$$T_{17}^{65} -$$$$34\!\cdots\!18$$$$T_{17}^{64} +$$$$26\!\cdots\!98$$$$T_{17}^{63} -$$$$75\!\cdots\!79$$$$T_{17}^{62} +$$$$28\!\cdots\!26$$$$T_{17}^{61} -$$$$81\!\cdots\!45$$$$T_{17}^{60} +$$$$14\!\cdots\!95$$$$T_{17}^{59} -$$$$41\!\cdots\!25$$$$T_{17}^{58} -$$$$91\!\cdots\!39$$$$T_{17}^{57} +$$$$28\!\cdots\!58$$$$T_{17}^{56} -$$$$29\!\cdots\!71$$$$T_{17}^{55} +$$$$86\!\cdots\!31$$$$T_{17}^{54} -$$$$31\!\cdots\!82$$$$T_{17}^{53} +$$$$84\!\cdots\!04$$$$T_{17}^{52} -$$$$14\!\cdots\!13$$$$T_{17}^{51} +$$$$37\!\cdots\!44$$$$T_{17}^{50} +$$$$51\!\cdots\!74$$$$T_{17}^{49} -$$$$13\!\cdots\!88$$$$T_{17}^{48} +$$$$16\!\cdots\!89$$$$T_{17}^{47} -$$$$51\!\cdots\!06$$$$T_{17}^{46} +$$$$17\!\cdots\!14$$$$T_{17}^{45} -$$$$46\!\cdots\!57$$$$T_{17}^{44} +$$$$64\!\cdots\!66$$$$T_{17}^{43} -$$$$58\!\cdots\!92$$$$T_{17}^{42} -$$$$21\!\cdots\!22$$$$T_{17}^{41} +$$$$12\!\cdots\!59$$$$T_{17}^{40} -$$$$59\!\cdots\!57$$$$T_{17}^{39} +$$$$18\!\cdots\!15$$$$T_{17}^{38} -$$$$43\!\cdots\!33$$$$T_{17}^{37} +$$$$90\!\cdots\!41$$$$T_{17}^{36} -$$$$15\!\cdots\!30$$$$T_{17}^{35} +$$$$10\!\cdots\!46$$$$T_{17}^{34} +$$$$10\!\cdots\!03$$$$T_{17}^{33} -$$$$51\!\cdots\!04$$$$T_{17}^{32} +$$$$13\!\cdots\!48$$$$T_{17}^{31} -$$$$24\!\cdots\!88$$$$T_{17}^{30} +$$$$23\!\cdots\!73$$$$T_{17}^{29} +$$$$17\!\cdots\!02$$$$T_{17}^{28} -$$$$14\!\cdots\!38$$$$T_{17}^{27} +$$$$42\!\cdots\!18$$$$T_{17}^{26} -$$$$86\!\cdots\!61$$$$T_{17}^{25} +$$$$14\!\cdots\!17$$$$T_{17}^{24} -$$$$19\!\cdots\!91$$$$T_{17}^{23} +$$$$20\!\cdots\!90$$$$T_{17}^{22} -$$$$19\!\cdots\!99$$$$T_{17}^{21} +$$$$15\!\cdots\!10$$$$T_{17}^{20} -$$$$91\!\cdots\!67$$$$T_{17}^{19} +$$$$44\!\cdots\!56$$$$T_{17}^{18} -$$$$19\!\cdots\!91$$$$T_{17}^{17} +$$$$55\!\cdots\!49$$$$T_{17}^{16} -$$$$68\!\cdots\!04$$$$T_{17}^{15} +$$$$30\!\cdots\!42$$$$T_{17}^{14} -$$$$49\!\cdots\!41$$$$T_{17}^{13} -$$$$36\!\cdots\!71$$$$T_{17}^{12} -$$$$25\!\cdots\!55$$$$T_{17}^{11} -$$$$29\!\cdots\!55$$$$T_{17}^{10} +$$$$41\!\cdots\!02$$$$T_{17}^{9} +$$$$15\!\cdots\!81$$$$T_{17}^{8} +$$$$25\!\cdots\!36$$$$T_{17}^{7} +$$$$28\!\cdots\!52$$$$T_{17}^{6} +$$$$23\!\cdots\!12$$$$T_{17}^{5} +$$$$13\!\cdots\!32$$$$T_{17}^{4} +$$$$50\!\cdots\!88$$$$T_{17}^{3} +$$$$12\!\cdots\!84$$$$T_{17}^{2} +$$$$18\!\cdots\!64$$$$T_{17} +$$$$12\!\cdots\!36$$">$$T_{17}^{128} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.