# Properties

 Label 930.2.br.a Level $930$ Weight $2$ Character orbit 930.br Analytic conductor $7.426$ Analytic rank $0$ Dimension $176$ 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.br (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$176$$ Relative dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$176q + 44q^{4} + 4q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$176q + 44q^{4} + 4q^{7} - 4q^{9} - 22q^{10} - 38q^{13} - 44q^{16} + 12q^{18} + 8q^{19} - 18q^{21} - 4q^{22} + 88q^{25} - 90q^{27} + 36q^{28} + 24q^{31} + 18q^{33} + 14q^{34} + 4q^{36} - 42q^{37} - 42q^{39} + 22q^{40} - 12q^{42} - 34q^{43} - 8q^{45} + 10q^{46} + 22q^{49} + 26q^{51} - 2q^{52} + 4q^{55} + 114q^{57} + 32q^{63} + 44q^{64} - 42q^{66} + 20q^{67} + 16q^{69} + 8q^{70} - 12q^{72} - 28q^{73} + 12q^{76} - 92q^{78} - 56q^{79} - 124q^{81} - 32q^{82} - 12q^{84} - 36q^{87} - 6q^{88} + 24q^{90} - 140q^{91} - 104q^{93} - 36q^{94} + 88q^{97} - 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −0.951057 + 0.309017i −1.68681 + 0.393302i 0.809017 0.587785i 0.866025 0.500000i 1.48271 0.895304i 0.503857 + 4.79388i −0.587785 + 0.809017i 2.69063 1.32685i −0.669131 + 0.743145i
11.2 −0.951057 + 0.309017i −1.68179 + 0.414242i 0.809017 0.587785i 0.866025 0.500000i 1.47147 0.913668i −0.309626 2.94590i −0.587785 + 0.809017i 2.65681 1.39333i −0.669131 + 0.743145i
11.3 −0.951057 + 0.309017i −1.35745 1.07579i 0.809017 0.587785i 0.866025 0.500000i 1.62345 + 0.603659i 0.0557772 + 0.530685i −0.587785 + 0.809017i 0.685360 + 2.92066i −0.669131 + 0.743145i
11.4 −0.951057 + 0.309017i −1.00587 + 1.41004i 0.809017 0.587785i 0.866025 0.500000i 0.520913 1.65186i −0.0129897 0.123589i −0.587785 + 0.809017i −0.976447 2.83664i −0.669131 + 0.743145i
11.5 −0.951057 + 0.309017i −0.361945 1.69381i 0.809017 0.587785i 0.866025 0.500000i 0.867646 + 1.49906i 0.291242 + 2.77098i −0.587785 + 0.809017i −2.73799 + 1.22613i −0.669131 + 0.743145i
11.6 −0.951057 + 0.309017i 0.164188 + 1.72425i 0.809017 0.587785i 0.866025 0.500000i −0.688975 1.58912i −0.487372 4.63703i −0.587785 + 0.809017i −2.94608 + 0.566203i −0.669131 + 0.743145i
11.7 −0.951057 + 0.309017i 0.821805 + 1.52468i 0.809017 0.587785i 0.866025 0.500000i −1.25273 1.19610i 0.122441 + 1.16495i −0.587785 + 0.809017i −1.64927 + 2.50597i −0.669131 + 0.743145i
11.8 −0.951057 + 0.309017i 0.991446 1.42022i 0.809017 0.587785i 0.866025 0.500000i −0.504048 + 1.65709i −0.310184 2.95121i −0.587785 + 0.809017i −1.03407 2.81615i −0.669131 + 0.743145i
11.9 −0.951057 + 0.309017i 1.11822 1.32272i 0.809017 0.587785i 0.866025 0.500000i −0.654747 + 1.60353i 0.396236 + 3.76994i −0.587785 + 0.809017i −0.499172 2.95818i −0.669131 + 0.743145i
11.10 −0.951057 + 0.309017i 1.48579 + 0.890187i 0.809017 0.587785i 0.866025 0.500000i −1.68815 0.387484i 0.0245947 + 0.234003i −0.587785 + 0.809017i 1.41514 + 2.64526i −0.669131 + 0.743145i
11.11 −0.951057 + 0.309017i 1.51241 0.844159i 0.809017 0.587785i 0.866025 0.500000i −1.17753 + 1.27020i 0.00579634 + 0.0551485i −0.587785 + 0.809017i 1.57479 2.55344i −0.669131 + 0.743145i
11.12 0.951057 0.309017i −1.69764 0.343522i 0.809017 0.587785i −0.866025 + 0.500000i −1.72071 + 0.197892i −0.487372 4.63703i 0.587785 0.809017i 2.76399 + 1.16636i −0.669131 + 0.743145i
11.13 0.951057 0.309017i −1.50746 + 0.852971i 0.809017 0.587785i −0.866025 + 0.500000i −1.17010 + 1.27705i −0.0129897 0.123589i 0.587785 0.809017i 1.54488 2.57164i −0.669131 + 0.743145i
11.14 0.951057 0.309017i −1.43042 0.976675i 0.809017 0.587785i −0.866025 + 0.500000i −1.66222 0.486849i 0.122441 + 1.16495i 0.587785 0.809017i 1.09221 + 2.79411i −0.669131 + 0.743145i
11.15 0.951057 0.309017i −0.730003 1.57070i 0.809017 0.587785i −0.866025 + 0.500000i −1.17965 1.26824i 0.0245947 + 0.234003i 0.587785 0.809017i −1.93419 + 2.29323i −0.669131 + 0.743145i
11.16 0.951057 0.309017i −0.587767 + 1.62927i 0.809017 0.587785i −0.866025 + 0.500000i −0.0555271 + 1.73116i −0.309626 2.94590i 0.587785 0.809017i −2.30906 1.91527i −0.669131 + 0.743145i
11.17 0.951057 0.309017i −0.567467 + 1.63645i 0.809017 0.587785i −0.866025 + 0.500000i −0.0340008 + 1.73172i 0.503857 + 4.79388i 0.587785 0.809017i −2.35596 1.85727i −0.669131 + 0.743145i
11.18 0.951057 0.309017i 0.928002 + 1.46247i 0.809017 0.587785i −0.866025 + 0.500000i 1.33451 + 1.10412i 0.0557772 + 0.530685i 0.587785 0.809017i −1.27762 + 2.71435i −0.669131 + 0.743145i
11.19 0.951057 0.309017i 0.997625 1.41589i 0.809017 0.587785i −0.866025 + 0.500000i 0.511264 1.65487i 0.00579634 + 0.0551485i 0.587785 0.809017i −1.00949 2.82505i −0.669131 + 0.743145i
11.20 0.951057 0.309017i 1.43236 0.973832i 0.809017 0.587785i −0.866025 + 0.500000i 1.06132 1.36879i 0.396236 + 3.76994i 0.587785 0.809017i 1.10330 2.78975i −0.669131 + 0.743145i
See next 80 embeddings (of 176 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 911.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p 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.br.a 176
3.b odd 2 1 inner 930.2.br.a 176
31.h odd 30 1 inner 930.2.br.a 176
93.p even 30 1 inner 930.2.br.a 176

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.br.a 176 1.a even 1 1 trivial
930.2.br.a 176 3.b odd 2 1 inner
930.2.br.a 176 31.h odd 30 1 inner
930.2.br.a 176 93.p even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!14$$$$T_{7}^{70} +$$$$33\!\cdots\!72$$$$T_{7}^{69} +$$$$83\!\cdots\!27$$$$T_{7}^{68} -$$$$45\!\cdots\!74$$$$T_{7}^{67} +$$$$22\!\cdots\!45$$$$T_{7}^{66} +$$$$40\!\cdots\!16$$$$T_{7}^{65} -$$$$60\!\cdots\!11$$$$T_{7}^{64} +$$$$28\!\cdots\!90$$$$T_{7}^{63} +$$$$74\!\cdots\!05$$$$T_{7}^{62} -$$$$11\!\cdots\!78$$$$T_{7}^{61} -$$$$27\!\cdots\!59$$$$T_{7}^{60} +$$$$96\!\cdots\!98$$$$T_{7}^{59} -$$$$63\!\cdots\!31$$$$T_{7}^{58} +$$$$45\!\cdots\!52$$$$T_{7}^{57} +$$$$10\!\cdots\!15$$$$T_{7}^{56} -$$$$18\!\cdots\!94$$$$T_{7}^{55} -$$$$49\!\cdots\!30$$$$T_{7}^{54} +$$$$15\!\cdots\!18$$$$T_{7}^{53} -$$$$29\!\cdots\!60$$$$T_{7}^{52} +$$$$48\!\cdots\!28$$$$T_{7}^{51} +$$$$50\!\cdots\!98$$$$T_{7}^{50} -$$$$11\!\cdots\!58$$$$T_{7}^{49} -$$$$21\!\cdots\!79$$$$T_{7}^{48} +$$$$86\!\cdots\!58$$$$T_{7}^{47} -$$$$37\!\cdots\!94$$$$T_{7}^{46} -$$$$10\!\cdots\!82$$$$T_{7}^{45} +$$$$79\!\cdots\!66$$$$T_{7}^{44} -$$$$22\!\cdots\!44$$$$T_{7}^{43} -$$$$42\!\cdots\!35$$$$T_{7}^{42} +$$$$17\!\cdots\!54$$$$T_{7}^{41} +$$$$19\!\cdots\!43$$$$T_{7}^{40} -$$$$74\!\cdots\!38$$$$T_{7}^{39} -$$$$80\!\cdots\!88$$$$T_{7}^{38} +$$$$26\!\cdots\!98$$$$T_{7}^{37} +$$$$13\!\cdots\!56$$$$T_{7}^{36} -$$$$54\!\cdots\!30$$$$T_{7}^{35} +$$$$30\!\cdots\!77$$$$T_{7}^{34} +$$$$14\!\cdots\!58$$$$T_{7}^{33} +$$$$11\!\cdots\!07$$$$T_{7}^{32} -$$$$26\!\cdots\!40$$$$T_{7}^{31} +$$$$35\!\cdots\!86$$$$T_{7}^{30} -$$$$41\!\cdots\!88$$$$T_{7}^{29} +$$$$47\!\cdots\!66$$$$T_{7}^{28} -$$$$76\!\cdots\!02$$$$T_{7}^{27} +$$$$11\!\cdots\!17$$$$T_{7}^{26} -$$$$11\!\cdots\!90$$$$T_{7}^{25} +$$$$86\!\cdots\!86$$$$T_{7}^{24} -$$$$95\!\cdots\!72$$$$T_{7}^{23} +$$$$12\!\cdots\!67$$$$T_{7}^{22} -$$$$10\!\cdots\!20$$$$T_{7}^{21} +$$$$59\!\cdots\!38$$$$T_{7}^{20} -$$$$36\!\cdots\!56$$$$T_{7}^{19} +$$$$33\!\cdots\!85$$$$T_{7}^{18} -$$$$25\!\cdots\!82$$$$T_{7}^{17} +$$$$11\!\cdots\!90$$$$T_{7}^{16} -$$$$24\!\cdots\!20$$$$T_{7}^{15} -$$$$38\!\cdots\!17$$$$T_{7}^{14} +$$$$81\!\cdots\!76$$$$T_{7}^{13} +$$$$55\!\cdots\!08$$$$T_{7}^{12} -$$$$53\!\cdots\!26$$$$T_{7}^{11} +$$$$25\!\cdots\!07$$$$T_{7}^{10} -$$$$77\!\cdots\!92$$$$T_{7}^{9} +$$$$17\!\cdots\!81$$$$T_{7}^{8} -$$$$29\!\cdots\!92$$$$T_{7}^{7} +$$$$32\!\cdots\!88$$$$T_{7}^{6} -$$$$30\!\cdots\!40$$$$T_{7}^{5} +$$$$25\!\cdots\!44$$$$T_{7}^{4} +$$$$81\!\cdots\!24$$$$T_{7}^{3} +$$$$48\!\cdots\!96$$$$T_{7}^{2} +$$$$25\!\cdots\!80$$$$T_{7} +$$$$35\!\cdots\!16$$">$$T_{7}^{88} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.