Properties

 Label 462.2.ba.a Level $462$ Weight $2$ Character orbit 462.ba Analytic conductor $3.689$ Analytic rank $0$ Dimension $64$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.ba (of order $$30$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{4} - 22q^{5} - 16q^{6} + 4q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{4} - 22q^{5} - 16q^{6} + 4q^{7} - 8q^{9} + 2q^{10} + 4q^{11} + 2q^{14} - 6q^{15} + 8q^{16} + 30q^{17} + 10q^{19} + 20q^{20} - 4q^{21} - 2q^{22} + 4q^{23} - 8q^{24} - 12q^{26} - 10q^{28} - 20q^{29} + 18q^{30} - 16q^{31} - 14q^{33} + 42q^{35} - 16q^{36} - 14q^{37} + 12q^{38} + 18q^{39} + 18q^{40} - 28q^{41} - 6q^{42} + 6q^{44} - 12q^{45} - 42q^{46} + 24q^{47} + 116q^{49} + 26q^{51} + 32q^{54} - 14q^{55} - 4q^{56} + 20q^{58} + 30q^{59} + 2q^{60} - 32q^{61} - 8q^{62} + 4q^{63} + 16q^{64} + 12q^{65} + 4q^{66} + 16q^{67} - 30q^{68} - 20q^{70} - 24q^{71} - 64q^{73} + 4q^{74} + 12q^{75} - 48q^{77} - 60q^{79} - 18q^{80} + 8q^{81} - 68q^{82} + 8q^{83} + 2q^{84} - 80q^{85} - 18q^{86} + 10q^{87} - 8q^{88} - 24q^{89} + 4q^{90} - 172q^{91} + 8q^{92} - 104q^{93} - 6q^{94} - 118q^{95} + 8q^{96} + 120q^{97} + 40q^{98} + 8q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.993461 2.23135i −0.809017 0.587785i 2.46197 0.968858i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.22126 + 2.11528i
19.2 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.716911 1.61021i −0.809017 0.587785i 1.87226 + 1.86939i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.881296 + 1.52645i
19.3 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.281694 0.632696i −0.809017 0.587785i −2.57173 0.621464i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.346286 + 0.599785i
19.4 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 1.49050 + 3.34771i −0.809017 0.587785i −2.49930 0.868052i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.83226 3.17357i
19.5 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i −1.50643 3.38350i −0.809017 0.587785i 2.60972 0.435184i 0.951057 0.309017i 0.104528 + 0.994522i −1.85185 3.20750i
19.6 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i −0.0779848 0.175157i −0.809017 0.587785i −1.57120 2.12869i 0.951057 0.309017i 0.104528 + 0.994522i −0.0958664 0.166046i
19.7 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.688924 + 1.54735i −0.809017 0.587785i 0.640384 + 2.56708i 0.951057 0.309017i 0.104528 + 0.994522i 0.846892 + 1.46686i
19.8 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i 1.48256 + 3.32989i −0.809017 0.587785i 1.16475 2.37557i 0.951057 0.309017i 0.104528 + 0.994522i 1.82251 + 3.15668i
61.1 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −2.80028 + 2.52138i 0.309017 0.951057i −2.64322 + 0.115723i 0.587785 0.809017i 0.978148 + 0.207912i −1.88407 3.26331i
61.2 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −1.32335 + 1.19155i 0.309017 0.951057i 2.59833 0.498689i 0.587785 0.809017i 0.978148 + 0.207912i −0.890371 1.54217i
61.3 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −0.624935 + 0.562694i 0.309017 0.951057i 1.46102 2.20577i 0.587785 0.809017i 0.978148 + 0.207912i −0.420466 0.728269i
61.4 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i 1.09698 0.987728i 0.309017 0.951057i −2.51719 + 0.814704i 0.587785 0.809017i 0.978148 + 0.207912i 0.738068 + 1.27837i
61.5 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −2.17695 + 1.96014i 0.309017 0.951057i −2.29320 + 1.31956i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.46469 + 2.53692i
61.6 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −2.02809 + 1.82610i 0.309017 0.951057i 2.53777 + 0.748146i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.36453 + 2.36343i
61.7 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −1.26898 + 1.14260i 0.309017 0.951057i −0.554537 2.58698i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.853791 + 1.47881i
61.8 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 2.13316 1.92070i 0.309017 0.951057i 2.34031 + 1.23407i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.43522 2.48588i
73.1 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.993461 + 2.23135i −0.809017 + 0.587785i 2.46197 + 0.968858i −0.951057 0.309017i 0.104528 0.994522i 1.22126 2.11528i
73.2 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.716911 + 1.61021i −0.809017 + 0.587785i 1.87226 1.86939i −0.951057 0.309017i 0.104528 0.994522i 0.881296 1.52645i
73.3 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.281694 + 0.632696i −0.809017 + 0.587785i −2.57173 + 0.621464i −0.951057 0.309017i 0.104528 0.994522i 0.346286 0.599785i
73.4 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i 1.49050 3.34771i −0.809017 + 0.587785i −2.49930 + 0.868052i −0.951057 0.309017i 0.104528 0.994522i −1.83226 + 3.17357i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.n even 30 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.ba.a 64
7.d odd 6 1 462.2.ba.b yes 64
11.d odd 10 1 462.2.ba.b yes 64
77.n even 30 1 inner 462.2.ba.a 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.ba.a 64 1.a even 1 1 trivial
462.2.ba.a 64 77.n even 30 1 inner
462.2.ba.b yes 64 7.d odd 6 1
462.2.ba.b yes 64 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$22\!\cdots\!78$$$$T_{5}^{45} +$$$$87\!\cdots\!86$$$$T_{5}^{44} +$$$$27\!\cdots\!10$$$$T_{5}^{43} +$$$$78\!\cdots\!79$$$$T_{5}^{42} +$$$$19\!\cdots\!96$$$$T_{5}^{41} +$$$$39\!\cdots\!35$$$$T_{5}^{40} +$$$$48\!\cdots\!70$$$$T_{5}^{39} -$$$$59\!\cdots\!22$$$$T_{5}^{38} -$$$$59\!\cdots\!56$$$$T_{5}^{37} -$$$$21\!\cdots\!76$$$$T_{5}^{36} -$$$$54\!\cdots\!18$$$$T_{5}^{35} -$$$$10\!\cdots\!40$$$$T_{5}^{34} -$$$$13\!\cdots\!72$$$$T_{5}^{33} -$$$$28\!\cdots\!70$$$$T_{5}^{32} +$$$$43\!\cdots\!56$$$$T_{5}^{31} +$$$$16\!\cdots\!43$$$$T_{5}^{30} +$$$$40\!\cdots\!34$$$$T_{5}^{29} +$$$$83\!\cdots\!37$$$$T_{5}^{28} +$$$$14\!\cdots\!48$$$$T_{5}^{27} +$$$$24\!\cdots\!49$$$$T_{5}^{26} +$$$$36\!\cdots\!28$$$$T_{5}^{25} +$$$$50\!\cdots\!35$$$$T_{5}^{24} +$$$$65\!\cdots\!42$$$$T_{5}^{23} +$$$$77\!\cdots\!82$$$$T_{5}^{22} +$$$$78\!\cdots\!64$$$$T_{5}^{21} +$$$$62\!\cdots\!84$$$$T_{5}^{20} +$$$$29\!\cdots\!00$$$$T_{5}^{19} -$$$$94\!\cdots\!29$$$$T_{5}^{18} -$$$$38\!\cdots\!90$$$$T_{5}^{17} -$$$$49\!\cdots\!65$$$$T_{5}^{16} -$$$$41\!\cdots\!66$$$$T_{5}^{15} -$$$$20\!\cdots\!13$$$$T_{5}^{14} -$$$$27\!\cdots\!24$$$$T_{5}^{13} +$$$$77\!\cdots\!03$$$$T_{5}^{12} +$$$$86\!\cdots\!50$$$$T_{5}^{11} +$$$$68\!\cdots\!94$$$$T_{5}^{10} +$$$$32\!\cdots\!14$$$$T_{5}^{9} +$$$$14\!\cdots\!53$$$$T_{5}^{8} +$$$$21\!\cdots\!84$$$$T_{5}^{7} +$$$$64\!\cdots\!72$$$$T_{5}^{6} -$$$$81\!\cdots\!60$$$$T_{5}^{5} +$$$$19\!\cdots\!98$$$$T_{5}^{4} -$$$$60\!\cdots\!78$$$$T_{5}^{3} +$$$$26\!\cdots\!79$$$$T_{5}^{2} -$$$$42\!\cdots\!58$$$$T_{5} +$$$$12\!\cdots\!41$$">$$T_{5}^{64} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.