# Properties

 Label 930.2.bn.b Level $930$ Weight $2$ Character orbit 930.bn Analytic conductor $7.426$ Analytic rank $0$ Dimension $144$ 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: $$144$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 36q^{4} + 2q^{5} - 72q^{6} - 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 36q^{4} + 2q^{5} - 72q^{6} - 18q^{9} - 18q^{11} - 8q^{14} - 36q^{16} - 24q^{19} - 2q^{20} + 28q^{21} - 18q^{24} + 10q^{25} - 12q^{26} - 4q^{30} - 4q^{31} + 10q^{34} - 2q^{35} - 72q^{36} + 16q^{39} + 4q^{41} - 2q^{44} - 2q^{45} - 2q^{46} - 78q^{49} + 32q^{50} + 10q^{51} + 36q^{54} - 50q^{55} - 12q^{56} + 28q^{59} + 88q^{61} + 36q^{64} - 124q^{65} + 6q^{66} - 46q^{69} - 10q^{70} + 140q^{71} + 34q^{74} - 32q^{75} + 24q^{76} + 16q^{79} + 12q^{80} + 18q^{81} - 8q^{84} + 74q^{85} - 98q^{86} + 148q^{89} + 44q^{91} - 108q^{94} - 80q^{95} + 18q^{96} - 2q^{99} + O(q^{100})$$

## 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 −1.89205 1.19170i −0.500000 0.866025i −1.93366 1.74107i 0.951057 + 0.309017i −0.669131 0.743145i 2.07622 0.830237i
19.2 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.39553 + 1.74714i −0.500000 0.866025i 1.99915 + 1.80004i 0.951057 + 0.309017i −0.669131 0.743145i −0.593198 2.15595i
19.3 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.16157 1.91070i −0.500000 0.866025i 2.15311 + 1.93867i 0.951057 + 0.309017i −0.669131 0.743145i 2.22854 + 0.183353i
19.4 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −0.978641 + 2.01054i −0.500000 0.866025i −2.38970 2.15169i 0.951057 + 0.309017i −0.669131 0.743145i −1.05133 1.97350i
19.5 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −0.293219 2.21676i −0.500000 0.866025i 1.44492 + 1.30102i 0.951057 + 0.309017i −0.669131 0.743145i 1.96575 + 1.06576i
19.6 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 0.991216 + 2.00437i −0.500000 0.866025i 0.807100 + 0.726716i 0.951057 + 0.309017i −0.669131 0.743145i −2.20419 0.376228i
19.7 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 1.47304 1.68231i −0.500000 0.866025i −2.98158 2.68462i 0.951057 + 0.309017i −0.669131 0.743145i 0.495182 + 2.18055i
19.8 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 2.02048 0.957953i −0.500000 0.866025i 1.95080 + 1.75650i 0.951057 + 0.309017i −0.669131 0.743145i −0.412606 + 2.19767i
19.9 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 2.14981 + 0.615060i −0.500000 0.866025i −2.84716 2.56360i 0.951057 + 0.309017i −0.669131 0.743145i −1.76122 + 1.37771i
19.10 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −2.23144 + 0.143766i −0.500000 0.866025i −0.807100 0.726716i −0.951057 0.309017i −0.669131 0.743145i −1.19530 + 1.88978i
19.11 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −1.60756 1.55426i −0.500000 0.866025i 2.84716 + 2.56360i −0.951057 0.309017i −0.669131 0.743145i −2.20233 + 0.386974i
19.12 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −1.25186 + 1.85280i −0.500000 0.866025i 2.38970 + 2.15169i −0.951057 0.309017i −0.669131 0.743145i 0.763121 + 2.10182i
19.13 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.815306 + 2.08213i −0.500000 0.866025i −1.99915 1.80004i −0.951057 0.309017i −0.669131 0.743145i 1.20526 + 1.88344i
19.14 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.180627 2.22876i −0.500000 0.866025i −1.95080 1.75650i −0.951057 0.309017i −0.669131 0.743145i −1.90927 1.16390i
19.15 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 0.720399 2.11684i −0.500000 0.866025i 2.98158 + 2.68462i −0.951057 0.309017i −0.669131 0.743145i −1.28912 1.82706i
19.16 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 1.97807 + 1.04271i −0.500000 0.866025i 1.93366 + 1.74107i −0.951057 0.309017i −0.669131 0.743145i 2.00625 0.987398i
19.17 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 2.06638 0.854445i −0.500000 0.866025i −1.44492 1.30102i −0.951057 0.309017i −0.669131 0.743145i 0.523327 2.17397i
19.18 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 2.23550 + 0.0505968i −0.500000 0.866025i −2.15311 1.93867i −0.951057 0.309017i −0.669131 0.743145i 1.35492 1.77881i
49.1 −0.587785 0.809017i −0.406737 0.913545i −0.309017 + 0.951057i −1.89205 + 1.19170i −0.500000 + 0.866025i −1.93366 + 1.74107i 0.951057 0.309017i −0.669131 + 0.743145i 2.07622 + 0.830237i
49.2 −0.587785 0.809017i −0.406737 0.913545i −0.309017 + 0.951057i −1.39553 1.74714i −0.500000 + 0.866025i 1.99915 1.80004i 0.951057 0.309017i −0.669131 + 0.743145i −0.593198 + 2.15595i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 919.18 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.b 144
5.b even 2 1 inner 930.2.bn.b 144
31.g even 15 1 inner 930.2.bn.b 144
155.u even 30 1 inner 930.2.bn.b 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bn.b 144 1.a even 1 1 trivial
930.2.bn.b 144 5.b even 2 1 inner
930.2.bn.b 144 31.g even 15 1 inner
930.2.bn.b 144 155.u even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!14$$$$T_{7}^{126} +$$$$14\!\cdots\!31$$$$T_{7}^{124} -$$$$28\!\cdots\!52$$$$T_{7}^{122} -$$$$11\!\cdots\!95$$$$T_{7}^{120} -$$$$16\!\cdots\!10$$$$T_{7}^{118} -$$$$50\!\cdots\!44$$$$T_{7}^{116} +$$$$28\!\cdots\!08$$$$T_{7}^{114} +$$$$70\!\cdots\!30$$$$T_{7}^{112} +$$$$79\!\cdots\!82$$$$T_{7}^{110} +$$$$21\!\cdots\!10$$$$T_{7}^{108} -$$$$11\!\cdots\!98$$$$T_{7}^{106} -$$$$21\!\cdots\!67$$$$T_{7}^{104} -$$$$18\!\cdots\!36$$$$T_{7}^{102} -$$$$10\!\cdots\!20$$$$T_{7}^{100} +$$$$21\!\cdots\!30$$$$T_{7}^{98} +$$$$35\!\cdots\!83$$$$T_{7}^{96} +$$$$27\!\cdots\!58$$$$T_{7}^{94} +$$$$18\!\cdots\!33$$$$T_{7}^{92} -$$$$24\!\cdots\!72$$$$T_{7}^{90} -$$$$32\!\cdots\!00$$$$T_{7}^{88} -$$$$24\!\cdots\!86$$$$T_{7}^{86} -$$$$33\!\cdots\!37$$$$T_{7}^{84} +$$$$13\!\cdots\!72$$$$T_{7}^{82} +$$$$19\!\cdots\!46$$$$T_{7}^{80} +$$$$13\!\cdots\!60$$$$T_{7}^{78} +$$$$45\!\cdots\!55$$$$T_{7}^{76} -$$$$30\!\cdots\!22$$$$T_{7}^{74} -$$$$52\!\cdots\!09$$$$T_{7}^{72} -$$$$40\!\cdots\!86$$$$T_{7}^{70} -$$$$14\!\cdots\!60$$$$T_{7}^{68} +$$$$36\!\cdots\!66$$$$T_{7}^{66} +$$$$82\!\cdots\!03$$$$T_{7}^{64} +$$$$65\!\cdots\!00$$$$T_{7}^{62} +$$$$31\!\cdots\!67$$$$T_{7}^{60} +$$$$83\!\cdots\!20$$$$T_{7}^{58} +$$$$11\!\cdots\!22$$$$T_{7}^{56} -$$$$27\!\cdots\!30$$$$T_{7}^{54} -$$$$76\!\cdots\!62$$$$T_{7}^{52} -$$$$25\!\cdots\!74$$$$T_{7}^{50} +$$$$63\!\cdots\!25$$$$T_{7}^{48} +$$$$21\!\cdots\!72$$$$T_{7}^{46} +$$$$17\!\cdots\!25$$$$T_{7}^{44} +$$$$25\!\cdots\!48$$$$T_{7}^{42} -$$$$30\!\cdots\!04$$$$T_{7}^{40} -$$$$28\!\cdots\!30$$$$T_{7}^{38} +$$$$13\!\cdots\!67$$$$T_{7}^{36} +$$$$12\!\cdots\!50$$$$T_{7}^{34} +$$$$88\!\cdots\!18$$$$T_{7}^{32} +$$$$11\!\cdots\!36$$$$T_{7}^{30} +$$$$10\!\cdots\!40$$$$T_{7}^{28} -$$$$73\!\cdots\!98$$$$T_{7}^{26} -$$$$33\!\cdots\!47$$$$T_{7}^{24} +$$$$47\!\cdots\!64$$$$T_{7}^{22} +$$$$67\!\cdots\!30$$$$T_{7}^{20} +$$$$19\!\cdots\!60$$$$T_{7}^{18} +$$$$26\!\cdots\!53$$$$T_{7}^{16} +$$$$17\!\cdots\!28$$$$T_{7}^{14} +$$$$60\!\cdots\!68$$$$T_{7}^{12} +$$$$10\!\cdots\!68$$$$T_{7}^{10} +$$$$85\!\cdots\!80$$$$T_{7}^{8} +$$$$34\!\cdots\!64$$$$T_{7}^{6} +$$$$45\!\cdots\!28$$$$T_{7}^{4} -$$$$52\!\cdots\!68$$$$T_{7}^{2} +$$$$11\!\cdots\!56$$">$$T_{7}^{144} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.