# Properties

 Label 930.2.bj.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128q - 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q - 16q^{7} - 4q^{10} + 4q^{15} + 32q^{16} - 12q^{17} - 40q^{19} + 40q^{21} + 16q^{22} - 32q^{24} - 8q^{25} - 4q^{28} + 8q^{29} + 20q^{31} + 4q^{33} + 24q^{35} - 128q^{36} + 64q^{37} - 16q^{38} - 24q^{41} + 4q^{42} + 24q^{43} - 8q^{44} + 20q^{46} + 108q^{47} - 100q^{49} - 24q^{50} + 16q^{53} - 32q^{54} + 12q^{55} + 16q^{57} - 40q^{58} - 16q^{62} - 4q^{63} + 36q^{65} - 12q^{66} - 32q^{67} + 8q^{68} - 32q^{70} + 24q^{71} + 60q^{73} - 16q^{74} + 24q^{75} - 24q^{76} - 20q^{77} - 56q^{79} + 32q^{81} - 16q^{82} - 8q^{83} - 132q^{85} - 20q^{87} - 4q^{88} + 136q^{89} - 40q^{91} - 64q^{93} + 108q^{95} + 64q^{97} - 16q^{98} - 8q^{99} + O(q^{100})$$

## 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 −2.22990 + 0.166033i 1.00000i 0.154498 0.303220i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.512821 + 2.17647i
277.2 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −1.27713 + 1.83547i 1.00000i 2.18789 4.29397i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.01266 + 0.974271i
277.3 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.620092 2.14837i 1.00000i −0.961148 + 1.88636i 0.453990 + 0.891007i 0.951057 + 0.309017i −2.02491 + 0.948537i
277.4 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.426558 + 2.19501i 1.00000i −0.653972 + 1.28349i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.23471 + 0.0779318i
277.5 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0.591821 2.15633i 1.00000i −0.754948 + 1.48167i 0.453990 + 0.891007i 0.951057 + 0.309017i −2.22236 0.247211i
277.6 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 1.90676 1.16802i 1.00000i 0.193795 0.380343i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.45193 1.70056i
277.7 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.03470 + 0.927363i 1.00000i −1.56419 + 3.06990i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.597648 2.15472i
277.8 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.08697 + 0.802838i 1.00000i 2.05326 4.02976i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.466480 2.18687i
277.9 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.90323 1.17377i 1.00000i 0.771267 1.51370i −0.453990 0.891007i 0.951057 + 0.309017i 0.861584 2.06341i
277.10 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.17179 + 1.90444i 1.00000i 1.50670 2.95707i −0.453990 0.891007i 0.951057 + 0.309017i −2.06431 0.859444i
277.11 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.980551 2.00961i 1.00000i 1.28197 2.51601i −0.453990 0.891007i 0.951057 + 0.309017i 1.83147 1.28285i
277.12 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.683297 2.12911i 1.00000i −2.06274 + 4.04835i −0.453990 0.891007i 0.951057 + 0.309017i 1.99600 1.00795i
277.13 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.284153 + 2.21794i 1.00000i −0.874752 + 1.71680i −0.453990 0.891007i 0.951057 + 0.309017i −2.23508 + 0.0663075i
277.14 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.11441 + 1.93858i 1.00000i −1.22295 + 2.40017i −0.453990 0.891007i 0.951057 + 0.309017i −1.74038 + 1.40395i
277.15 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.96074 1.07493i 1.00000i −0.462930 + 0.908551i −0.453990 0.891007i 0.951057 + 0.309017i 1.36843 + 1.76845i
277.16 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 2.23243 0.127537i 1.00000i −0.416181 + 0.816801i −0.453990 0.891007i 0.951057 + 0.309017i 0.475195 + 2.18499i
337.1 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −2.23208 0.133471i 1.00000i 0.703148 + 0.111368i 0.987688 0.156434i 0.587785 0.809017i 1.13227 1.92820i
337.2 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −1.80883 1.31459i 1.00000i 2.31722 + 0.367011i 0.987688 0.156434i 0.587785 0.809017i 1.99250 1.01486i
337.3 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −1.30319 + 1.81705i 1.00000i −1.71397 0.271467i 0.987688 0.156434i 0.587785 0.809017i −1.02737 1.98608i
337.4 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −0.789819 2.09193i 1.00000i −4.01115 0.635304i 0.987688 0.156434i 0.587785 0.809017i 2.22250 + 0.245984i
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.a 128
5.c odd 4 1 930.2.bj.b yes 128
31.f odd 10 1 930.2.bj.b yes 128
155.r even 20 1 inner 930.2.bj.a 128

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$23\!\cdots\!22$$$$T_{7}^{112} +$$$$62\!\cdots\!48$$$$T_{7}^{111} +$$$$10\!\cdots\!08$$$$T_{7}^{110} +$$$$50\!\cdots\!96$$$$T_{7}^{109} +$$$$86\!\cdots\!49$$$$T_{7}^{108} +$$$$59\!\cdots\!08$$$$T_{7}^{107} +$$$$56\!\cdots\!32$$$$T_{7}^{106} +$$$$29\!\cdots\!32$$$$T_{7}^{105} +$$$$96\!\cdots\!35$$$$T_{7}^{104} +$$$$70\!\cdots\!84$$$$T_{7}^{103} -$$$$13\!\cdots\!98$$$$T_{7}^{102} -$$$$10\!\cdots\!80$$$$T_{7}^{101} -$$$$37\!\cdots\!08$$$$T_{7}^{100} -$$$$30\!\cdots\!80$$$$T_{7}^{99} +$$$$67\!\cdots\!60$$$$T_{7}^{98} +$$$$61\!\cdots\!36$$$$T_{7}^{97} +$$$$34\!\cdots\!83$$$$T_{7}^{96} +$$$$15\!\cdots\!12$$$$T_{7}^{95} +$$$$53\!\cdots\!32$$$$T_{7}^{94} +$$$$15\!\cdots\!76$$$$T_{7}^{93} +$$$$33\!\cdots\!72$$$$T_{7}^{92} +$$$$44\!\cdots\!24$$$$T_{7}^{91} -$$$$34\!\cdots\!04$$$$T_{7}^{90} -$$$$37\!\cdots\!60$$$$T_{7}^{89} -$$$$91\!\cdots\!43$$$$T_{7}^{88} +$$$$62\!\cdots\!68$$$$T_{7}^{87} +$$$$14\!\cdots\!50$$$$T_{7}^{86} +$$$$64\!\cdots\!32$$$$T_{7}^{85} +$$$$16\!\cdots\!63$$$$T_{7}^{84} +$$$$11\!\cdots\!04$$$$T_{7}^{83} -$$$$11\!\cdots\!80$$$$T_{7}^{82} -$$$$68\!\cdots\!40$$$$T_{7}^{81} -$$$$22\!\cdots\!91$$$$T_{7}^{80} -$$$$47\!\cdots\!36$$$$T_{7}^{79} -$$$$22\!\cdots\!84$$$$T_{7}^{78} +$$$$29\!\cdots\!28$$$$T_{7}^{77} +$$$$14\!\cdots\!65$$$$T_{7}^{76} +$$$$43\!\cdots\!12$$$$T_{7}^{75} +$$$$76\!\cdots\!68$$$$T_{7}^{74} +$$$$17\!\cdots\!68$$$$T_{7}^{73} -$$$$43\!\cdots\!66$$$$T_{7}^{72} -$$$$18\!\cdots\!84$$$$T_{7}^{71} -$$$$41\!\cdots\!92$$$$T_{7}^{70} -$$$$43\!\cdots\!32$$$$T_{7}^{69} +$$$$94\!\cdots\!99$$$$T_{7}^{68} +$$$$60\!\cdots\!88$$$$T_{7}^{67} +$$$$16\!\cdots\!32$$$$T_{7}^{66} +$$$$21\!\cdots\!56$$$$T_{7}^{65} -$$$$17\!\cdots\!19$$$$T_{7}^{64} -$$$$18\!\cdots\!32$$$$T_{7}^{63} -$$$$55\!\cdots\!82$$$$T_{7}^{62} -$$$$90\!\cdots\!96$$$$T_{7}^{61} -$$$$10\!\cdots\!04$$$$T_{7}^{60} +$$$$44\!\cdots\!00$$$$T_{7}^{59} +$$$$15\!\cdots\!80$$$$T_{7}^{58} +$$$$31\!\cdots\!32$$$$T_{7}^{57} +$$$$29\!\cdots\!81$$$$T_{7}^{56} -$$$$44\!\cdots\!56$$$$T_{7}^{55} -$$$$26\!\cdots\!36$$$$T_{7}^{54} -$$$$63\!\cdots\!76$$$$T_{7}^{53} -$$$$89\!\cdots\!52$$$$T_{7}^{52} -$$$$32\!\cdots\!16$$$$T_{7}^{51} +$$$$21\!\cdots\!04$$$$T_{7}^{50} +$$$$70\!\cdots\!48$$$$T_{7}^{49} +$$$$12\!\cdots\!79$$$$T_{7}^{48} +$$$$12\!\cdots\!76$$$$T_{7}^{47} -$$$$29\!\cdots\!74$$$$T_{7}^{46} -$$$$36\!\cdots\!96$$$$T_{7}^{45} -$$$$72\!\cdots\!27$$$$T_{7}^{44} -$$$$66\!\cdots\!92$$$$T_{7}^{43} +$$$$39\!\cdots\!04$$$$T_{7}^{42} +$$$$26\!\cdots\!04$$$$T_{7}^{41} +$$$$49\!\cdots\!51$$$$T_{7}^{40} +$$$$45\!\cdots\!12$$$$T_{7}^{39} -$$$$18\!\cdots\!04$$$$T_{7}^{38} -$$$$14\!\cdots\!60$$$$T_{7}^{37} -$$$$25\!\cdots\!33$$$$T_{7}^{36} -$$$$21\!\cdots\!92$$$$T_{7}^{35} +$$$$10\!\cdots\!60$$$$T_{7}^{34} +$$$$61\!\cdots\!08$$$$T_{7}^{33} +$$$$89\!\cdots\!54$$$$T_{7}^{32} +$$$$46\!\cdots\!76$$$$T_{7}^{31} -$$$$71\!\cdots\!64$$$$T_{7}^{30} -$$$$18\!\cdots\!88$$$$T_{7}^{29} -$$$$15\!\cdots\!45$$$$T_{7}^{28} +$$$$53\!\cdots\!08$$$$T_{7}^{27} +$$$$28\!\cdots\!48$$$$T_{7}^{26} +$$$$28\!\cdots\!48$$$$T_{7}^{25} +$$$$13\!\cdots\!73$$$$T_{7}^{24} -$$$$29\!\cdots\!24$$$$T_{7}^{23} -$$$$32\!\cdots\!74$$$$T_{7}^{22} -$$$$43\!\cdots\!32$$$$T_{7}^{21} +$$$$22\!\cdots\!78$$$$T_{7}^{20} +$$$$22\!\cdots\!40$$$$T_{7}^{19} +$$$$13\!\cdots\!04$$$$T_{7}^{18} -$$$$12\!\cdots\!64$$$$T_{7}^{17} -$$$$87\!\cdots\!19$$$$T_{7}^{16} +$$$$10\!\cdots\!20$$$$T_{7}^{15} +$$$$42\!\cdots\!60$$$$T_{7}^{14} +$$$$16\!\cdots\!92$$$$T_{7}^{13} -$$$$54\!\cdots\!88$$$$T_{7}^{12} -$$$$53\!\cdots\!52$$$$T_{7}^{11} -$$$$58\!\cdots\!68$$$$T_{7}^{10} -$$$$86\!\cdots\!36$$$$T_{7}^{9} -$$$$38\!\cdots\!76$$$$T_{7}^{8} +$$$$56\!\cdots\!48$$$$T_{7}^{7} +$$$$15\!\cdots\!28$$$$T_{7}^{6} +$$$$68\!\cdots\!00$$$$T_{7}^{5} +$$$$17\!\cdots\!60$$$$T_{7}^{4} +$$$$75\!\cdots\!36$$$$T_{7}^{3} +$$$$27\!\cdots\!88$$$$T_{7}^{2} +$$$$50\!\cdots\!44$$$$T_{7} +$$$$34\!\cdots\!76$$">$$T_{7}^{128} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.