# Properties

 Label 930.2.be.a Level $930$ Weight $2$ Character orbit 930.be Analytic conductor $7.426$ Analytic rank $0$ Dimension $64$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.be (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$16$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{7} - 4q^{10} + 24q^{14} + 8q^{15} - 64q^{16} + 4q^{17} - 12q^{20} - 24q^{21} - 8q^{22} + 32q^{24} - 20q^{25} - 4q^{28} + 16q^{29} + 8q^{31} + 4q^{33} + 24q^{35} + 32q^{36} + 56q^{37} - 4q^{38} - 8q^{41} + 8q^{42} - 56q^{43} - 4q^{44} + 12q^{45} + 8q^{47} - 60q^{49} - 8q^{50} + 24q^{53} - 64q^{54} + 16q^{55} - 28q^{57} + 52q^{58} + 24q^{59} - 4q^{62} - 4q^{63} + 100q^{65} + 8q^{66} + 76q^{67} + 4q^{68} - 12q^{69} - 44q^{70} + 8q^{71} - 52q^{73} + 12q^{74} - 8q^{75} - 8q^{76} - 104q^{77} + 56q^{79} + 32q^{81} + 32q^{82} + 24q^{83} + 32q^{85} - 24q^{86} - 20q^{87} + 4q^{88} - 176q^{89} + 8q^{93} + 64q^{95} - 68q^{97} - 64q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i −2.04325 0.908360i 0.866025 0.500000i −0.239418 + 0.893522i 0.707107 0.707107i −0.866025 0.500000i 0.802490 + 2.08711i
37.2 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i −1.98218 + 1.03487i 0.866025 0.500000i 0.211801 0.790453i 0.707107 0.707107i −0.866025 0.500000i 2.13338 + 0.669853i
37.3 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i −1.08971 1.95257i 0.866025 0.500000i 0.294543 1.09925i 0.707107 0.707107i −0.866025 0.500000i −0.610134 + 2.15122i
37.4 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i −0.786015 + 2.09337i 0.866025 0.500000i −1.17109 + 4.37058i 0.707107 0.707107i −0.866025 0.500000i 2.03603 0.924436i
37.5 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i 0.303557 2.21537i 0.866025 0.500000i −1.35687 + 5.06392i 0.707107 0.707107i −0.866025 0.500000i −1.78115 + 1.35185i
37.6 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i 0.871485 + 2.05925i 0.866025 0.500000i 1.08173 4.03706i 0.707107 0.707107i −0.866025 0.500000i 0.839878 2.07234i
37.7 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i 1.64800 + 1.51133i 0.866025 0.500000i −0.191313 + 0.713991i 0.707107 0.707107i −0.866025 0.500000i −0.0966398 2.23398i
37.8 −0.707107 0.707107i −0.258819 + 0.965926i 1.00000i 1.95328 1.08844i 0.866025 0.500000i 0.340075 1.26918i 0.707107 0.707107i −0.866025 0.500000i −2.15082 0.611534i
37.9 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i −2.17687 + 0.511134i 0.866025 0.500000i 0.572203 2.13549i −0.707107 + 0.707107i −0.866025 0.500000i −1.90070 1.17785i
37.10 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i −1.66175 + 1.49620i 0.866025 0.500000i 1.33377 4.97771i −0.707107 + 0.707107i −0.866025 0.500000i −2.23300 0.117061i
37.11 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i −1.62734 1.53354i 0.866025 0.500000i −0.664038 + 2.47822i −0.707107 + 0.707107i −0.866025 0.500000i −0.0663294 2.23508i
37.12 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i 0.247692 + 2.22231i 0.866025 0.500000i −1.08066 + 4.03306i −0.707107 + 0.707107i −0.866025 0.500000i −1.39626 + 1.74655i
37.13 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i 0.795067 + 2.08994i 0.866025 0.500000i −0.0433914 + 0.161939i −0.707107 + 0.707107i −0.866025 0.500000i −0.915617 + 2.04001i
37.14 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i 1.01126 1.99433i 0.866025 0.500000i −0.754227 + 2.81481i −0.707107 + 0.707107i −0.866025 0.500000i 2.12527 0.695132i
37.15 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i 1.20229 1.88534i 0.866025 0.500000i 0.614903 2.29485i −0.707107 + 0.707107i −0.866025 0.500000i 2.18328 0.482987i
37.16 0.707107 + 0.707107i 0.258819 0.965926i 1.00000i 1.60244 + 1.55955i 0.866025 0.500000i 0.784034 2.92605i −0.707107 + 0.707107i −0.866025 0.500000i 0.0303243 + 2.23586i
223.1 −0.707107 + 0.707107i 0.965926 + 0.258819i 1.00000i −2.07384 0.836183i −0.866025 + 0.500000i −4.78156 1.28122i 0.707107 + 0.707107i 0.866025 + 0.500000i 2.05769 0.875153i
223.2 −0.707107 + 0.707107i 0.965926 + 0.258819i 1.00000i −1.98088 + 1.03736i −0.866025 + 0.500000i −0.414720 0.111124i 0.707107 + 0.707107i 0.866025 + 0.500000i 0.667172 2.13422i
223.3 −0.707107 + 0.707107i 0.965926 + 0.258819i 1.00000i −0.440662 2.19222i −0.866025 + 0.500000i −1.90738 0.511082i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.86173 + 1.23854i
223.4 −0.707107 + 0.707107i 0.965926 + 0.258819i 1.00000i −0.395276 2.20085i −0.866025 + 0.500000i 3.88090 + 1.03988i 0.707107 + 0.707107i 0.866025 + 0.500000i 1.83574 + 1.27674i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 553.16 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.p even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.be.a 64
5.c odd 4 1 930.2.be.b yes 64
31.e odd 6 1 930.2.be.b yes 64
155.p even 12 1 inner 930.2.be.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.be.a 64 1.a even 1 1 trivial
930.2.be.a 64 155.p even 12 1 inner
930.2.be.b yes 64 5.c odd 4 1
930.2.be.b yes 64 31.e odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$25\!\cdots\!52$$$$T_{7}^{45} - 432728185118 T_{7}^{44} +$$$$17\!\cdots\!36$$$$T_{7}^{43} -$$$$30\!\cdots\!68$$$$T_{7}^{42} -$$$$24\!\cdots\!00$$$$T_{7}^{41} +$$$$23\!\cdots\!57$$$$T_{7}^{40} -$$$$16\!\cdots\!88$$$$T_{7}^{39} +$$$$29\!\cdots\!98$$$$T_{7}^{38} +$$$$12\!\cdots\!24$$$$T_{7}^{37} -$$$$18\!\cdots\!01$$$$T_{7}^{36} +$$$$76\!\cdots\!44$$$$T_{7}^{35} -$$$$86\!\cdots\!50$$$$T_{7}^{34} -$$$$47\!\cdots\!08$$$$T_{7}^{33} +$$$$75\!\cdots\!57$$$$T_{7}^{32} -$$$$20\!\cdots\!12$$$$T_{7}^{31} +$$$$16\!\cdots\!80$$$$T_{7}^{30} +$$$$12\!\cdots\!08$$$$T_{7}^{29} -$$$$20\!\cdots\!74$$$$T_{7}^{28} +$$$$27\!\cdots\!28$$$$T_{7}^{27} +$$$$26\!\cdots\!48$$$$T_{7}^{26} -$$$$21\!\cdots\!40$$$$T_{7}^{25} +$$$$18\!\cdots\!93$$$$T_{7}^{24} +$$$$14\!\cdots\!76$$$$T_{7}^{23} -$$$$53\!\cdots\!10$$$$T_{7}^{22} +$$$$82\!\cdots\!88$$$$T_{7}^{21} +$$$$98\!\cdots\!75$$$$T_{7}^{20} -$$$$12\!\cdots\!04$$$$T_{7}^{19} +$$$$11\!\cdots\!18$$$$T_{7}^{18} -$$$$94\!\cdots\!96$$$$T_{7}^{17} -$$$$62\!\cdots\!32$$$$T_{7}^{16} +$$$$18\!\cdots\!56$$$$T_{7}^{15} -$$$$11\!\cdots\!70$$$$T_{7}^{14} +$$$$12\!\cdots\!96$$$$T_{7}^{13} +$$$$47\!\cdots\!61$$$$T_{7}^{12} -$$$$17\!\cdots\!40$$$$T_{7}^{11} +$$$$10\!\cdots\!06$$$$T_{7}^{10} -$$$$11\!\cdots\!24$$$$T_{7}^{9} +$$$$39\!\cdots\!93$$$$T_{7}^{8} +$$$$14\!\cdots\!60$$$$T_{7}^{7} +$$$$12\!\cdots\!28$$$$T_{7}^{6} +$$$$21\!\cdots\!96$$$$T_{7}^{5} +$$$$19\!\cdots\!16$$$$T_{7}^{4} -$$$$16\!\cdots\!76$$$$T_{7}^{3} +$$$$11\!\cdots\!20$$$$T_{7}^{2} -$$$$36\!\cdots\!96$$$$T_{7} +$$$$86\!\cdots\!76$$">$$T_{7}^{64} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.