# Properties

 Label 930.2.bj.b Level $930$ Weight $2$ Character orbit 930.bj Analytic conductor $7.426$ Analytic rank $0$ Dimension $128$ 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.bj (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$16$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$128q + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q + 4q^{7} - 4q^{10} - 4q^{15} + 32q^{16} + 12q^{17} + 40q^{19} + 40q^{21} + 4q^{22} + 32q^{24} - 8q^{25} - 4q^{28} - 8q^{29} + 20q^{31} + 4q^{33} + 24q^{35} - 128q^{36} - 64q^{37} - 16q^{38} - 24q^{41} + 16q^{42} - 24q^{43} + 8q^{44} + 20q^{46} - 12q^{47} + 100q^{49} - 24q^{50} + 64q^{53} + 32q^{54} + 68q^{55} - 16q^{57} + 40q^{58} + 8q^{62} - 4q^{63} + 84q^{65} - 12q^{66} - 32q^{67} - 8q^{68} + 88q^{70} + 24q^{71} + 20q^{73} + 16q^{74} - 24q^{75} - 24q^{76} + 60q^{77} + 56q^{79} + 32q^{81} - 16q^{82} + 8q^{83} - 68q^{85} - 20q^{87} + 4q^{88} - 136q^{89} - 40q^{91} + 48q^{93} - 92q^{95} + 64q^{97} - 16q^{98} + 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.90277 + 1.17451i 1.00000i 1.72531 3.38612i 0.453990 + 0.891007i 0.951057 + 0.309017i 1.45771 + 1.69561i
277.2 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.82843 + 1.28719i 1.00000i −2.04975 + 4.02286i 0.453990 + 0.891007i 0.951057 + 0.309017i 1.55737 + 1.60455i
277.3 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.74178 1.40221i 1.00000i −0.335285 + 0.658034i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.11247 + 1.93969i
277.4 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.67223 1.48447i 1.00000i 1.10706 2.17272i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.20460 + 1.88386i
277.5 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 0.244623 + 2.22265i 1.00000i 0.795365 1.56099i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.15702 0.589310i
277.6 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.48798 1.66911i 1.00000i −0.506200 + 0.993473i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.88133 1.20856i
277.7 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.61098 1.55073i 1.00000i −1.36477 + 2.67852i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.78365 1.34855i
277.8 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 2.17207 + 0.531157i 1.00000i 0.369895 0.725959i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.184831 2.22842i
277.9 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −2.23524 0.0607951i 1.00000i 0.773249 1.51759i −0.453990 0.891007i 0.951057 + 0.309017i −0.289622 2.21723i
277.10 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −2.20784 + 0.354206i 1.00000i 0.622042 1.22083i −0.453990 0.891007i 0.951057 + 0.309017i −0.695226 2.12524i
277.11 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −1.01777 1.99102i 1.00000i −0.681739 + 1.33799i −0.453990 0.891007i 0.951057 + 0.309017i 1.80729 1.31670i
277.12 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.869928 + 2.05991i 1.00000i −2.25496 + 4.42561i −0.453990 0.891007i 0.951057 + 0.309017i −2.17063 0.536977i
277.13 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0.145083 + 2.23136i 1.00000i 0.996252 1.95526i −0.453990 0.891007i 0.951057 + 0.309017i −2.18119 + 0.492358i
277.14 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 1.00799 1.99598i 1.00000i 1.85184 3.63444i −0.453990 0.891007i 0.951057 + 0.309017i 2.12909 + 0.683342i
277.15 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.22022 + 0.265769i 1.00000i 1.65375 3.24566i −0.453990 0.891007i 0.951057 + 0.309017i 0.0848219 + 2.23446i
277.16 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.23590 + 0.0275636i 1.00000i −1.08401 + 2.12750i −0.453990 0.891007i 0.951057 + 0.309017i 0.322547 + 2.21268i
337.1 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −2.20246 + 0.386226i 1.00000i 3.54459 + 0.561408i 0.987688 0.156434i 0.587785 0.809017i 0.655766 2.13775i
337.2 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −1.81236 1.30972i 1.00000i −2.09145 0.331253i 0.987688 0.156434i 0.587785 0.809017i 1.98976 1.02022i
337.3 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −1.39059 + 1.75108i 1.00000i −2.23420 0.353863i 0.987688 0.156434i 0.587785 0.809017i −0.928908 2.03399i
337.4 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −0.354402 2.20780i 1.00000i 5.01200 + 0.793822i 0.987688 0.156434i 0.587785 0.809017i 2.12806 + 0.686548i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.r even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bj.b yes 128
5.c odd 4 1 930.2.bj.a 128
31.f odd 10 1 930.2.bj.a 128
155.r even 20 1 inner 930.2.bj.b yes 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bj.a 128 5.c odd 4 1
930.2.bj.a 128 31.f odd 10 1
930.2.bj.b yes 128 1.a even 1 1 trivial
930.2.bj.b yes 128 155.r even 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$23\!\cdots\!92$$$$T_{7}^{110} +$$$$13\!\cdots\!36$$$$T_{7}^{109} -$$$$14\!\cdots\!71$$$$T_{7}^{108} -$$$$70\!\cdots\!72$$$$T_{7}^{107} +$$$$90\!\cdots\!12$$$$T_{7}^{106} -$$$$51\!\cdots\!88$$$$T_{7}^{105} +$$$$73\!\cdots\!75$$$$T_{7}^{104} +$$$$39\!\cdots\!44$$$$T_{7}^{103} -$$$$37\!\cdots\!18$$$$T_{7}^{102} +$$$$14\!\cdots\!80$$$$T_{7}^{101} -$$$$10\!\cdots\!08$$$$T_{7}^{100} -$$$$15\!\cdots\!60$$$$T_{7}^{99} +$$$$11\!\cdots\!80$$$$T_{7}^{98} -$$$$33\!\cdots\!24$$$$T_{7}^{97} -$$$$33\!\cdots\!57$$$$T_{7}^{96} +$$$$57\!\cdots\!92$$$$T_{7}^{95} -$$$$20\!\cdots\!88$$$$T_{7}^{94} +$$$$19\!\cdots\!56$$$$T_{7}^{93} +$$$$16\!\cdots\!52$$$$T_{7}^{92} -$$$$92\!\cdots\!96$$$$T_{7}^{91} +$$$$20\!\cdots\!36$$$$T_{7}^{90} +$$$$19\!\cdots\!80$$$$T_{7}^{89} -$$$$32\!\cdots\!23$$$$T_{7}^{88} +$$$$91\!\cdots\!08$$$$T_{7}^{87} -$$$$19\!\cdots\!10$$$$T_{7}^{86} -$$$$54\!\cdots\!68$$$$T_{7}^{85} +$$$$21\!\cdots\!23$$$$T_{7}^{84} -$$$$47\!\cdots\!16$$$$T_{7}^{83} +$$$$28\!\cdots\!40$$$$T_{7}^{82} +$$$$30\!\cdots\!20$$$$T_{7}^{81} -$$$$19\!\cdots\!31$$$$T_{7}^{80} +$$$$53\!\cdots\!64$$$$T_{7}^{79} +$$$$16\!\cdots\!36$$$$T_{7}^{78} -$$$$50\!\cdots\!12$$$$T_{7}^{77} +$$$$17\!\cdots\!05$$$$T_{7}^{76} -$$$$14\!\cdots\!08$$$$T_{7}^{75} -$$$$80\!\cdots\!32$$$$T_{7}^{74} +$$$$33\!\cdots\!48$$$$T_{7}^{73} -$$$$48\!\cdots\!26$$$$T_{7}^{72} -$$$$43\!\cdots\!04$$$$T_{7}^{71} +$$$$40\!\cdots\!88$$$$T_{7}^{70} -$$$$10\!\cdots\!32$$$$T_{7}^{69} +$$$$11\!\cdots\!79$$$$T_{7}^{68} +$$$$17\!\cdots\!88$$$$T_{7}^{67} -$$$$13\!\cdots\!88$$$$T_{7}^{66} +$$$$36\!\cdots\!36$$$$T_{7}^{65} -$$$$40\!\cdots\!59$$$$T_{7}^{64} -$$$$89\!\cdots\!12$$$$T_{7}^{63} +$$$$52\!\cdots\!78$$$$T_{7}^{62} -$$$$10\!\cdots\!36$$$$T_{7}^{61} +$$$$36\!\cdots\!36$$$$T_{7}^{60} +$$$$43\!\cdots\!60$$$$T_{7}^{59} -$$$$13\!\cdots\!00$$$$T_{7}^{58} +$$$$17\!\cdots\!72$$$$T_{7}^{57} +$$$$15\!\cdots\!01$$$$T_{7}^{56} -$$$$11\!\cdots\!56$$$$T_{7}^{55} +$$$$21\!\cdots\!44$$$$T_{7}^{54} -$$$$11\!\cdots\!36$$$$T_{7}^{53} -$$$$51\!\cdots\!92$$$$T_{7}^{52} +$$$$16\!\cdots\!44$$$$T_{7}^{51} -$$$$20\!\cdots\!56$$$$T_{7}^{50} -$$$$31\!\cdots\!12$$$$T_{7}^{49} +$$$$70\!\cdots\!59$$$$T_{7}^{48} -$$$$14\!\cdots\!04$$$$T_{7}^{47} +$$$$13\!\cdots\!46$$$$T_{7}^{46} +$$$$10\!\cdots\!44$$$$T_{7}^{45} -$$$$59\!\cdots\!07$$$$T_{7}^{44} +$$$$10\!\cdots\!88$$$$T_{7}^{43} -$$$$79\!\cdots\!96$$$$T_{7}^{42} -$$$$79\!\cdots\!96$$$$T_{7}^{41} +$$$$37\!\cdots\!11$$$$T_{7}^{40} -$$$$63\!\cdots\!48$$$$T_{7}^{39} +$$$$52\!\cdots\!76$$$$T_{7}^{38} +$$$$36\!\cdots\!20$$$$T_{7}^{37} -$$$$20\!\cdots\!33$$$$T_{7}^{36} +$$$$35\!\cdots\!68$$$$T_{7}^{35} -$$$$28\!\cdots\!80$$$$T_{7}^{34} -$$$$11\!\cdots\!12$$$$T_{7}^{33} +$$$$82\!\cdots\!54$$$$T_{7}^{32} -$$$$14\!\cdots\!04$$$$T_{7}^{31} +$$$$14\!\cdots\!56$$$$T_{7}^{30} -$$$$34\!\cdots\!68$$$$T_{7}^{29} -$$$$16\!\cdots\!25$$$$T_{7}^{28} +$$$$39\!\cdots\!68$$$$T_{7}^{27} -$$$$52\!\cdots\!52$$$$T_{7}^{26} +$$$$47\!\cdots\!88$$$$T_{7}^{25} -$$$$13\!\cdots\!47$$$$T_{7}^{24} -$$$$32\!\cdots\!44$$$$T_{7}^{23} +$$$$88\!\cdots\!86$$$$T_{7}^{22} -$$$$12\!\cdots\!92$$$$T_{7}^{21} +$$$$14\!\cdots\!78$$$$T_{7}^{20} -$$$$12\!\cdots\!80$$$$T_{7}^{19} +$$$$10\!\cdots\!24$$$$T_{7}^{18} -$$$$64\!\cdots\!04$$$$T_{7}^{17} +$$$$40\!\cdots\!21$$$$T_{7}^{16} -$$$$17\!\cdots\!20$$$$T_{7}^{15} +$$$$85\!\cdots\!00$$$$T_{7}^{14} -$$$$13\!\cdots\!08$$$$T_{7}^{13} +$$$$14\!\cdots\!92$$$$T_{7}^{12} +$$$$48\!\cdots\!48$$$$T_{7}^{11} +$$$$10\!\cdots\!12$$$$T_{7}^{10} +$$$$55\!\cdots\!84$$$$T_{7}^{9} +$$$$59\!\cdots\!24$$$$T_{7}^{8} +$$$$34\!\cdots\!88$$$$T_{7}^{7} +$$$$20\!\cdots\!48$$$$T_{7}^{6} +$$$$87\!\cdots\!60$$$$T_{7}^{5} +$$$$31\!\cdots\!60$$$$T_{7}^{4} +$$$$85\!\cdots\!36$$$$T_{7}^{3} +$$$$19\!\cdots\!28$$$$T_{7}^{2} +$$$$26\!\cdots\!64$$$$T_{7} +$$$$34\!\cdots\!76$$">$$T_{7}^{128} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.