# Properties

 Label 462.2.ba.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{4} - 2q^{5} + 16q^{6} + 16q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{4} - 2q^{5} + 16q^{6} + 16q^{7} - 8q^{9} - 2q^{10} + 4q^{11} + 2q^{14} - 6q^{15} + 8q^{16} + 30q^{17} - 10q^{19} - 20q^{20} + 4q^{21} - 2q^{22} + 4q^{23} + 8q^{24} - 12q^{26} - 20q^{29} - 18q^{30} + 34q^{31} + 8q^{33} - 2q^{35} - 16q^{36} - 14q^{37} + 12q^{38} - 18q^{39} + 12q^{40} + 28q^{41} + 4q^{42} + 6q^{44} - 12q^{45} + 42q^{46} + 24q^{47} - 44q^{49} + 14q^{51} - 32q^{54} + 14q^{55} - 4q^{56} - 10q^{58} - 30q^{59} + 2q^{60} - 28q^{61} + 8q^{62} + 16q^{63} + 16q^{64} - 12q^{65} - 4q^{66} + 16q^{67} - 30q^{68} - 30q^{70} - 24q^{71} - 116q^{73} - 44q^{74} + 12q^{75} - 32q^{77} - 18q^{80} + 8q^{81} - 28q^{82} - 8q^{83} - 2q^{84} - 80q^{85} - 18q^{86} - 10q^{87} - 14q^{88} - 24q^{89} - 4q^{90} + 48q^{91} + 8q^{92} + 76q^{93} + 6q^{94} + 98q^{95} - 8q^{96} - 120q^{97} - 40q^{98} + 8q^{99} + O(q^{100})$$

## 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 −1.44711 3.25025i 0.809017 + 0.587785i 1.04897 + 2.42892i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.77892 + 3.08119i
19.2 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i −1.26939 2.85109i 0.809017 + 0.587785i −0.613482 2.57364i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.56045 + 2.70278i
19.3 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.0572849 + 0.128664i 0.809017 + 0.587785i −2.64421 + 0.0903816i −0.951057 + 0.309017i 0.104528 + 0.994522i −0.0704201 0.121971i
19.4 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.949965 + 2.13366i 0.809017 + 0.587785i 2.43658 + 1.03106i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.16779 2.02267i
19.5 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −1.01671 2.28357i 0.809017 + 0.587785i −0.151373 2.64142i 0.951057 0.309017i 0.104528 + 0.994522i −1.24984 2.16479i
19.6 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.748568 1.68131i 0.809017 + 0.587785i 2.42705 + 1.05329i 0.951057 0.309017i 0.104528 + 0.994522i −0.920212 1.59385i
19.7 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 0.316797 + 0.711538i 0.809017 + 0.587785i −0.370234 + 2.61972i 0.951057 0.309017i 0.104528 + 0.994522i 0.389438 + 0.674526i
19.8 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 1.33064 + 2.98866i 0.809017 + 0.587785i 1.90285 1.83825i 0.951057 0.309017i 0.104528 + 0.994522i 1.63575 + 2.83319i
61.1 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −3.24579 + 2.92252i −0.309017 + 0.951057i −0.190680 2.63887i 0.587785 0.809017i 0.978148 + 0.207912i −2.18382 3.78249i
61.2 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −0.468951 + 0.422245i −0.309017 + 0.951057i −0.177812 + 2.63977i 0.587785 0.809017i 0.978148 + 0.207912i −0.315518 0.546493i
61.3 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 0.131864 0.118731i −0.309017 + 0.951057i −1.17519 2.37043i 0.587785 0.809017i 0.978148 + 0.207912i 0.0887206 + 0.153669i
61.4 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 1.55595 1.40099i −0.309017 + 0.951057i 2.62525 0.328695i 0.587785 0.809017i 0.978148 + 0.207912i 1.04687 + 1.81324i
61.5 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i −1.40387 + 1.26405i −0.309017 + 0.951057i 1.39244 2.24969i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.944546 + 1.63600i
61.6 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i −0.716223 + 0.644890i −0.309017 + 0.951057i 2.17783 + 1.50235i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.481886 + 0.834652i
61.7 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i 0.192170 0.173031i −0.309017 + 0.951057i −1.35536 + 2.27222i −0.587785 + 0.809017i 0.978148 + 0.207912i −0.129295 0.223946i
61.8 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i 2.61658 2.35598i −0.309017 + 0.951057i 1.99807 1.73428i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.76048 3.04924i
73.1 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.44711 + 3.25025i 0.809017 0.587785i 1.04897 2.42892i −0.951057 0.309017i 0.104528 0.994522i 1.77892 3.08119i
73.2 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.26939 + 2.85109i 0.809017 0.587785i −0.613482 + 2.57364i −0.951057 0.309017i 0.104528 0.994522i 1.56045 2.70278i
73.3 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.0572849 0.128664i 0.809017 0.587785i −2.64421 0.0903816i −0.951057 0.309017i 0.104528 0.994522i −0.0704201 + 0.121971i
73.4 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.949965 2.13366i 0.809017 0.587785i 2.43658 1.03106i −0.951057 0.309017i 0.104528 0.994522i −1.16779 + 2.02267i
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.b yes 64
7.d odd 6 1 462.2.ba.a 64
11.d odd 10 1 462.2.ba.a 64
77.n even 30 1 inner 462.2.ba.b yes 64

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$34\!\cdots\!86$$$$T_{5}^{42} -$$$$28\!\cdots\!44$$$$T_{5}^{41} -$$$$19\!\cdots\!45$$$$T_{5}^{40} +$$$$48\!\cdots\!70$$$$T_{5}^{39} +$$$$22\!\cdots\!03$$$$T_{5}^{38} +$$$$73\!\cdots\!44$$$$T_{5}^{37} +$$$$22\!\cdots\!54$$$$T_{5}^{36} +$$$$51\!\cdots\!82$$$$T_{5}^{35} +$$$$12\!\cdots\!50$$$$T_{5}^{34} +$$$$23\!\cdots\!88$$$$T_{5}^{33} +$$$$41\!\cdots\!95$$$$T_{5}^{32} +$$$$61\!\cdots\!06$$$$T_{5}^{31} +$$$$69\!\cdots\!03$$$$T_{5}^{30} +$$$$28\!\cdots\!24$$$$T_{5}^{29} -$$$$18\!\cdots\!78$$$$T_{5}^{28} -$$$$66\!\cdots\!52$$$$T_{5}^{27} -$$$$17\!\cdots\!76$$$$T_{5}^{26} -$$$$32\!\cdots\!52$$$$T_{5}^{25} -$$$$51\!\cdots\!00$$$$T_{5}^{24} -$$$$57\!\cdots\!28$$$$T_{5}^{23} -$$$$19\!\cdots\!83$$$$T_{5}^{22} +$$$$75\!\cdots\!84$$$$T_{5}^{21} +$$$$28\!\cdots\!24$$$$T_{5}^{20} +$$$$54\!\cdots\!50$$$$T_{5}^{19} +$$$$83\!\cdots\!11$$$$T_{5}^{18} +$$$$10\!\cdots\!50$$$$T_{5}^{17} +$$$$10\!\cdots\!70$$$$T_{5}^{16} +$$$$97\!\cdots\!04$$$$T_{5}^{15} +$$$$77\!\cdots\!62$$$$T_{5}^{14} +$$$$52\!\cdots\!06$$$$T_{5}^{13} +$$$$29\!\cdots\!43$$$$T_{5}^{12} +$$$$14\!\cdots\!70$$$$T_{5}^{11} +$$$$57\!\cdots\!74$$$$T_{5}^{10} +$$$$12\!\cdots\!24$$$$T_{5}^{9} -$$$$58\!\cdots\!02$$$$T_{5}^{8} +$$$$11\!\cdots\!54$$$$T_{5}^{7} +$$$$69\!\cdots\!27$$$$T_{5}^{6} +$$$$18\!\cdots\!80$$$$T_{5}^{5} -$$$$14\!\cdots\!42$$$$T_{5}^{4} +$$$$50\!\cdots\!32$$$$T_{5}^{3} +$$$$13\!\cdots\!74$$$$T_{5}^{2} -$$$$50\!\cdots\!18$$$$T_{5} +$$$$12\!\cdots\!41$$">$$T_{5}^{64} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.