# Properties

 Label 462.2.bc.a Level $462$ Weight $2$ Character orbit 462.bc Analytic conductor $3.689$ Analytic rank $0$ Dimension $128$ CM no Inner twists $4$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 4 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128 q - 16 q^{2} + 16 q^{4} + 32 q^{8} - 4 q^{9} + 6 q^{11} - 12 q^{15} + 16 q^{16} + 2 q^{17} + 4 q^{18} + 2 q^{22} - 12 q^{25} - 18 q^{27} - 5 q^{28} - 38 q^{29} - 6 q^{30} - 3 q^{31} + 64 q^{32} - 4 q^{33} - 16 q^{34} + 31 q^{35} + 8 q^{36} + 2 q^{37} + 22 q^{39} + 5 q^{40} - 16 q^{41} + 17 q^{42} + q^{44} + 28 q^{45} + 38 q^{49} - 34 q^{50} + 16 q^{51} - 25 q^{53} + 6 q^{54} - 42 q^{55} + 20 q^{57} - 19 q^{58} - 40 q^{59} - 4 q^{60} + 40 q^{61} + 4 q^{62} + 6 q^{63} - 32 q^{64} - 20 q^{65} - 41 q^{66} + 16 q^{67} + 2 q^{68} - 68 q^{69} - 21 q^{70} - 80 q^{71} - 16 q^{72} + 10 q^{73} - 2 q^{74} - 14 q^{75} - q^{77} - 16 q^{78} + 5 q^{80} - 88 q^{81} - 8 q^{82} + 92 q^{83} - 48 q^{84} - 100 q^{85} + 40 q^{86} + 38 q^{87} - q^{88} - 164 q^{90} + 12 q^{91} + 20 q^{92} + 47 q^{93} + 40 q^{94} - 38 q^{95} - 16 q^{97} - 18 q^{98} - 138 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1 0.104528 0.994522i −1.64414 + 0.544800i −0.978148 0.207912i 1.25353 2.81547i 0.369957 + 1.69208i 1.28350 + 2.31358i −0.309017 + 0.951057i 2.40638 1.79146i −2.66902 1.54096i
95.2 0.104528 0.994522i −1.63004 + 0.585637i −0.978148 0.207912i 0.112013 0.251585i 0.412043 + 1.68233i −2.57741 0.597448i −0.309017 + 0.951057i 2.31406 1.90922i −0.238498 0.137697i
95.3 0.104528 0.994522i −1.45000 0.947367i −0.978148 0.207912i 0.625478 1.40485i −1.09374 + 1.34303i −1.00131 2.44895i −0.309017 + 0.951057i 1.20499 + 2.74736i −1.33177 0.768898i
95.4 0.104528 0.994522i −1.41798 0.994648i −0.978148 0.207912i −1.12093 + 2.51765i −1.13742 + 1.30625i 2.40939 1.09308i −0.309017 + 0.951057i 1.02135 + 2.82079i 2.38669 + 1.37795i
95.5 0.104528 0.994522i −1.18732 1.26106i −0.978148 0.207912i −0.157991 + 0.354854i −1.37826 + 1.04900i 0.106147 + 2.64362i −0.309017 + 0.951057i −0.180521 + 2.99456i 0.336396 + 0.194218i
95.6 0.104528 0.994522i −0.612442 + 1.62016i −0.978148 0.207912i −1.31655 + 2.95701i 1.54727 + 0.778440i −1.86309 1.87854i −0.309017 + 0.951057i −2.24983 1.98451i 2.80319 + 1.61843i
95.7 0.104528 0.994522i −0.611589 + 1.62048i −0.978148 0.207912i 0.684905 1.53832i 1.54768 + 0.777625i 1.54934 2.14465i −0.309017 + 0.951057i −2.25192 1.98214i −1.45830 0.841952i
95.8 0.104528 0.994522i 0.0618530 1.73095i −0.978148 0.207912i 1.30191 2.92414i −1.71500 0.242447i −2.04556 + 1.67800i −0.309017 + 0.951057i −2.99235 0.214128i −2.77204 1.60044i
95.9 0.104528 0.994522i 0.333518 + 1.69964i −0.978148 0.207912i 0.0756668 0.169950i 1.72519 0.154030i 1.63703 + 2.07849i −0.309017 + 0.951057i −2.77753 + 1.13372i −0.161110 0.0930169i
95.10 0.104528 0.994522i 0.398305 1.68563i −0.978148 0.207912i −1.53261 + 3.44230i −1.63476 0.572320i −2.01818 + 1.71083i −0.309017 + 0.951057i −2.68271 1.34279i 3.26324 + 1.88403i
95.11 0.104528 0.994522i 0.812842 1.52947i −0.978148 0.207912i 0.565716 1.27062i −1.43613 0.968262i 0.339893 2.62383i −0.309017 + 0.951057i −1.67858 2.48644i −1.20453 0.695433i
95.12 0.104528 0.994522i 0.891795 + 1.48482i −0.978148 0.207912i −0.333330 + 0.748672i 1.56991 0.731703i −2.01015 + 1.72026i −0.309017 + 0.951057i −1.40940 + 2.64832i 0.709728 + 0.409762i
95.13 0.104528 0.994522i 1.28109 1.16568i −0.978148 0.207912i −0.572012 + 1.28476i −1.02538 1.39592i 2.51605 + 0.818231i −0.309017 + 0.951057i 0.282395 2.98668i 1.21793 + 0.703172i
95.14 0.104528 0.994522i 1.39020 + 1.03313i −0.978148 0.207912i −1.35789 + 3.04988i 1.17278 1.27459i 1.07443 2.41777i −0.309017 + 0.951057i 0.865302 + 2.87250i 2.89123 + 1.66925i
95.15 0.104528 0.994522i 1.67644 + 0.435353i −0.978148 0.207912i 0.789465 1.77317i 0.608205 1.62175i −2.28945 + 1.32606i −0.309017 + 0.951057i 2.62093 + 1.45969i −1.68093 0.970486i
95.16 0.104528 0.994522i 1.70747 + 0.290781i −0.978148 0.207912i 1.75629 3.94468i 0.467667 1.66772i 2.64495 + 0.0650804i −0.309017 + 0.951057i 2.83089 + 0.992999i −3.73949 2.15900i
107.1 0.104528 + 0.994522i −1.64414 0.544800i −0.978148 + 0.207912i 1.25353 + 2.81547i 0.369957 1.69208i 1.28350 2.31358i −0.309017 0.951057i 2.40638 + 1.79146i −2.66902 + 1.54096i
107.2 0.104528 + 0.994522i −1.63004 0.585637i −0.978148 + 0.207912i 0.112013 + 0.251585i 0.412043 1.68233i −2.57741 + 0.597448i −0.309017 0.951057i 2.31406 + 1.90922i −0.238498 + 0.137697i
107.3 0.104528 + 0.994522i −1.45000 + 0.947367i −0.978148 + 0.207912i 0.625478 + 1.40485i −1.09374 1.34303i −1.00131 + 2.44895i −0.309017 0.951057i 1.20499 2.74736i −1.33177 + 0.768898i
107.4 0.104528 + 0.994522i −1.41798 + 0.994648i −0.978148 + 0.207912i −1.12093 2.51765i −1.13742 1.30625i 2.40939 + 1.09308i −0.309017 0.951057i 1.02135 2.82079i 2.38669 1.37795i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 431.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.f even 10 1 inner
231.be 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.bc.a 128
3.b odd 2 1 462.2.bc.b yes 128
7.c even 3 1 inner 462.2.bc.a 128
11.d odd 10 1 462.2.bc.b yes 128
21.h odd 6 1 462.2.bc.b yes 128
33.f even 10 1 inner 462.2.bc.a 128
77.o odd 30 1 462.2.bc.b yes 128
231.be even 30 1 inner 462.2.bc.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bc.a 128 1.a even 1 1 trivial
462.2.bc.a 128 7.c even 3 1 inner
462.2.bc.a 128 33.f even 10 1 inner
462.2.bc.a 128 231.be even 30 1 inner
462.2.bc.b yes 128 3.b odd 2 1
462.2.bc.b yes 128 11.d odd 10 1
462.2.bc.b yes 128 21.h odd 6 1
462.2.bc.b yes 128 77.o odd 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$15\!\cdots\!00$$$$T_{5}^{107} -$$$$35\!\cdots\!25$$$$T_{5}^{106} +$$$$13\!\cdots\!10$$$$T_{5}^{105} -$$$$58\!\cdots\!78$$$$T_{5}^{104} -$$$$48\!\cdots\!90$$$$T_{5}^{103} -$$$$40\!\cdots\!78$$$$T_{5}^{102} -$$$$25\!\cdots\!85$$$$T_{5}^{101} -$$$$27\!\cdots\!30$$$$T_{5}^{100} -$$$$27\!\cdots\!15$$$$T_{5}^{99} +$$$$28\!\cdots\!29$$$$T_{5}^{98} -$$$$66\!\cdots\!15$$$$T_{5}^{97} +$$$$48\!\cdots\!86$$$$T_{5}^{96} +$$$$20\!\cdots\!00$$$$T_{5}^{95} +$$$$56\!\cdots\!69$$$$T_{5}^{94} +$$$$32\!\cdots\!15$$$$T_{5}^{93} +$$$$42\!\cdots\!52$$$$T_{5}^{92} +$$$$23\!\cdots\!65$$$$T_{5}^{91} +$$$$49\!\cdots\!01$$$$T_{5}^{90} +$$$$95\!\cdots\!70$$$$T_{5}^{89} -$$$$38\!\cdots\!97$$$$T_{5}^{88} -$$$$19\!\cdots\!45$$$$T_{5}^{87} -$$$$62\!\cdots\!19$$$$T_{5}^{86} -$$$$24\!\cdots\!60$$$$T_{5}^{85} -$$$$53\!\cdots\!21$$$$T_{5}^{84} -$$$$15\!\cdots\!75$$$$T_{5}^{83} -$$$$18\!\cdots\!71$$$$T_{5}^{82} -$$$$90\!\cdots\!10$$$$T_{5}^{81} +$$$$19\!\cdots\!36$$$$T_{5}^{80} +$$$$98\!\cdots\!65$$$$T_{5}^{79} +$$$$38\!\cdots\!63$$$$T_{5}^{78} +$$$$12\!\cdots\!10$$$$T_{5}^{77} +$$$$33\!\cdots\!44$$$$T_{5}^{76} +$$$$81\!\cdots\!05$$$$T_{5}^{75} +$$$$15\!\cdots\!88$$$$T_{5}^{74} +$$$$19\!\cdots\!65$$$$T_{5}^{73} -$$$$16\!\cdots\!32$$$$T_{5}^{72} -$$$$21\!\cdots\!55$$$$T_{5}^{71} -$$$$96\!\cdots\!83$$$$T_{5}^{70} -$$$$32\!\cdots\!80$$$$T_{5}^{69} -$$$$91\!\cdots\!02$$$$T_{5}^{68} -$$$$22\!\cdots\!35$$$$T_{5}^{67} -$$$$48\!\cdots\!68$$$$T_{5}^{66} -$$$$83\!\cdots\!20$$$$T_{5}^{65} -$$$$89\!\cdots\!42$$$$T_{5}^{64} +$$$$98\!\cdots\!05$$$$T_{5}^{63} +$$$$10\!\cdots\!03$$$$T_{5}^{62} +$$$$40\!\cdots\!55$$$$T_{5}^{61} +$$$$12\!\cdots\!33$$$$T_{5}^{60} +$$$$33\!\cdots\!70$$$$T_{5}^{59} +$$$$82\!\cdots\!63$$$$T_{5}^{58} +$$$$18\!\cdots\!60$$$$T_{5}^{57} +$$$$38\!\cdots\!58$$$$T_{5}^{56} +$$$$74\!\cdots\!25$$$$T_{5}^{55} +$$$$13\!\cdots\!53$$$$T_{5}^{54} +$$$$22\!\cdots\!70$$$$T_{5}^{53} +$$$$35\!\cdots\!18$$$$T_{5}^{52} +$$$$48\!\cdots\!60$$$$T_{5}^{51} +$$$$58\!\cdots\!98$$$$T_{5}^{50} +$$$$53\!\cdots\!65$$$$T_{5}^{49} +$$$$14\!\cdots\!78$$$$T_{5}^{48} -$$$$86\!\cdots\!80$$$$T_{5}^{47} -$$$$27\!\cdots\!54$$$$T_{5}^{46} -$$$$57\!\cdots\!25$$$$T_{5}^{45} -$$$$96\!\cdots\!74$$$$T_{5}^{44} -$$$$13\!\cdots\!15$$$$T_{5}^{43} -$$$$15\!\cdots\!11$$$$T_{5}^{42} -$$$$13\!\cdots\!20$$$$T_{5}^{41} -$$$$43\!\cdots\!31$$$$T_{5}^{40} +$$$$15\!\cdots\!20$$$$T_{5}^{39} +$$$$45\!\cdots\!89$$$$T_{5}^{38} +$$$$83\!\cdots\!00$$$$T_{5}^{37} +$$$$12\!\cdots\!39$$$$T_{5}^{36} +$$$$15\!\cdots\!70$$$$T_{5}^{35} +$$$$15\!\cdots\!02$$$$T_{5}^{34} +$$$$13\!\cdots\!95$$$$T_{5}^{33} +$$$$96\!\cdots\!67$$$$T_{5}^{32} +$$$$39\!\cdots\!15$$$$T_{5}^{31} -$$$$17\!\cdots\!98$$$$T_{5}^{30} -$$$$58\!\cdots\!70$$$$T_{5}^{29} -$$$$73\!\cdots\!37$$$$T_{5}^{28} -$$$$66\!\cdots\!70$$$$T_{5}^{27} -$$$$46\!\cdots\!67$$$$T_{5}^{26} -$$$$23\!\cdots\!25$$$$T_{5}^{25} -$$$$64\!\cdots\!24$$$$T_{5}^{24} +$$$$31\!\cdots\!05$$$$T_{5}^{23} +$$$$74\!\cdots\!06$$$$T_{5}^{22} +$$$$83\!\cdots\!80$$$$T_{5}^{21} +$$$$79\!\cdots\!66$$$$T_{5}^{20} +$$$$64\!\cdots\!15$$$$T_{5}^{19} +$$$$46\!\cdots\!98$$$$T_{5}^{18} +$$$$28\!\cdots\!60$$$$T_{5}^{17} +$$$$13\!\cdots\!26$$$$T_{5}^{16} +$$$$44\!\cdots\!10$$$$T_{5}^{15} +$$$$33\!\cdots\!36$$$$T_{5}^{14} -$$$$76\!\cdots\!75$$$$T_{5}^{13} -$$$$57\!\cdots\!97$$$$T_{5}^{12} -$$$$24\!\cdots\!65$$$$T_{5}^{11} -$$$$58\!\cdots\!80$$$$T_{5}^{10} -$$$$29\!\cdots\!15$$$$T_{5}^{9} +$$$$40\!\cdots\!60$$$$T_{5}^{8} +$$$$23\!\cdots\!70$$$$T_{5}^{7} +$$$$87\!\cdots\!14$$$$T_{5}^{6} +$$$$28\!\cdots\!50$$$$T_{5}^{5} +$$$$68\!\cdots\!35$$$$T_{5}^{4} +$$$$10\!\cdots\!40$$$$T_{5}^{3} +$$$$17\!\cdots\!45$$$$T_{5}^{2} +$$$$28\!\cdots\!65$$$$T_{5} +$$$$21\!\cdots\!81$$">$$T_{5}^{128} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$.