# Properties

 Label 429.2.bn.b Level $429$ Weight $2$ Character orbit 429.bn Analytic conductor $3.426$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bn (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 14q^{3} - 8q^{4} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 14q^{3} - 8q^{4} + 14q^{9} - 40q^{10} + 15q^{11} - 104q^{12} + q^{13} - 6q^{14} - 6q^{15} + 32q^{16} + 8q^{17} - 12q^{19} + 42q^{20} - 9q^{22} + 8q^{23} + 30q^{25} - 57q^{26} - 28q^{27} + 18q^{28} - 10q^{29} + 10q^{30} + 30q^{32} + 30q^{33} - 12q^{35} - 8q^{36} + 30q^{38} + 4q^{39} + 20q^{40} + 72q^{41} - 12q^{42} - 108q^{43} + 6q^{45} - 18q^{46} + 2q^{48} - 40q^{49} + 111q^{50} - 26q^{51} + 13q^{52} - 46q^{53} - 38q^{55} - 100q^{56} - 12q^{58} - 18q^{59} - 46q^{61} - 9q^{62} + 52q^{64} + 24q^{65} - 32q^{66} + 48q^{67} - 8q^{68} - 7q^{69} + 18q^{71} + 32q^{74} - 216q^{76} + 4q^{77} - 26q^{78} + 108q^{79} - 66q^{80} + 14q^{81} + 39q^{82} + 27q^{84} + 6q^{85} + 60q^{87} + 28q^{88} - 120q^{89} - 20q^{90} + 47q^{91} + 78q^{92} - 6q^{93} - 50q^{94} + 60q^{95} - 69q^{97} + 72q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −1.08659 2.44051i 0.669131 + 0.743145i −3.43718 + 3.81737i −1.96001 2.69773i 1.08659 2.44051i −2.63121 2.36915i 7.96970 + 2.58951i −0.104528 + 0.994522i −4.45412 + 7.71476i
4.2 −0.976080 2.19231i 0.669131 + 0.743145i −2.51524 + 2.79345i 0.603494 + 0.830639i 0.976080 2.19231i 0.422137 + 0.380094i 4.01454 + 1.30440i −0.104528 + 0.994522i 1.23196 2.13382i
4.3 −0.819494 1.84061i 0.669131 + 0.743145i −1.37803 + 1.53046i −1.15996 1.59655i 0.819494 1.84061i 3.61282 + 3.25299i 0.113886 + 0.0370037i −0.104528 + 0.994522i −1.98805 + 3.44340i
4.4 −0.727584 1.63418i 0.669131 + 0.743145i −0.802903 + 0.891714i −0.105484 0.145187i 0.727584 1.63418i −2.03695 1.83408i −1.36116 0.442268i −0.104528 + 0.994522i −0.160512 + 0.278015i
4.5 −0.425286 0.955208i 0.669131 + 0.743145i 0.606707 0.673817i 2.35366 + 3.23953i 0.425286 0.955208i −1.51333 1.36261i −2.89052 0.939186i −0.104528 + 0.994522i 2.09345 3.62596i
4.6 −0.288687 0.648403i 0.669131 + 0.743145i 1.00118 1.11192i −2.40424 3.30915i 0.288687 0.648403i 0.504508 + 0.454261i −2.36005 0.766827i −0.104528 + 0.994522i −1.45159 + 2.51422i
4.7 0.0203023 + 0.0455997i 0.669131 + 0.743145i 1.33659 1.48444i 1.77101 + 2.43759i −0.0203023 + 0.0455997i 0.890300 + 0.801630i 0.189770 + 0.0616600i −0.104528 + 0.994522i −0.0751978 + 0.130246i
4.8 0.0299019 + 0.0671608i 0.669131 + 0.743145i 1.33464 1.48227i 0.0495558 + 0.0682077i −0.0299019 + 0.0671608i 1.42900 + 1.28668i 0.279296 + 0.0907488i −0.104528 + 0.994522i −0.00309907 + 0.00536775i
4.9 0.293907 + 0.660125i 0.669131 + 0.743145i 0.988877 1.09826i −1.15395 1.58828i −0.293907 + 0.660125i −3.62548 3.26440i 2.39009 + 0.776586i −0.104528 + 0.994522i 0.709310 1.22856i
4.10 0.434661 + 0.976265i 0.669131 + 0.743145i 0.574098 0.637600i 0.194734 + 0.268028i −0.434661 + 0.976265i 1.78328 + 1.60567i 2.90471 + 0.943797i −0.104528 + 0.994522i −0.177023 + 0.306614i
4.11 0.644170 + 1.44683i 0.669131 + 0.743145i −0.340099 + 0.377718i −1.31195 1.80574i −0.644170 + 1.44683i 1.36339 + 1.22760i 2.24690 + 0.730062i −0.104528 + 0.994522i 1.76748 3.06137i
4.12 0.772619 + 1.73533i 0.669131 + 0.743145i −1.07617 + 1.19521i 1.38712 + 1.90920i −0.772619 + 1.73533i −2.05909 1.85401i 0.707612 + 0.229917i −0.104528 + 0.994522i −2.24139 + 3.88220i
4.13 1.04016 + 2.33623i 0.669131 + 0.743145i −3.03778 + 3.37380i −0.492512 0.677885i −1.04016 + 2.33623i 3.64547 + 3.28240i −6.17741 2.00716i −0.104528 + 0.994522i 1.07140 1.85573i
4.14 1.08800 + 2.44369i 0.669131 + 0.743145i −3.44962 + 3.83119i 0.581262 + 0.800038i −1.08800 + 2.44369i −1.78483 1.60707i −8.02736 2.60825i −0.104528 + 0.994522i −1.32263 + 2.29087i
49.1 −1.91017 1.71993i −0.104528 + 0.994522i 0.481555 + 4.58169i 3.19249 + 1.03730i 1.91017 1.71993i −2.98254 + 0.313478i 3.93864 5.42108i −0.978148 0.207912i −4.31412 7.47228i
49.2 −1.64800 1.48387i −0.104528 + 0.994522i 0.304992 + 2.90181i −1.57487 0.511707i 1.64800 1.48387i 1.08496 0.114034i 1.19632 1.64659i −0.978148 0.207912i 1.83609 + 3.18020i
49.3 −1.60557 1.44566i −0.104528 + 0.994522i 0.278861 + 2.65319i −0.655963 0.213135i 1.60557 1.44566i 0.714969 0.0751462i 0.848055 1.16725i −0.978148 0.207912i 0.745073 + 1.29051i
49.4 −0.977278 0.879945i −0.104528 + 0.994522i −0.0282879 0.269141i 1.70562 + 0.554188i 0.977278 0.879945i 2.04084 0.214501i −1.75513 + 2.41573i −0.978148 0.207912i −1.17921 2.04245i
49.5 −0.646484 0.582097i −0.104528 + 0.994522i −0.129952 1.23641i −2.84399 0.924069i 0.646484 0.582097i 0.352201 0.0370178i −1.65836 + 2.28254i −0.978148 0.207912i 1.30070 + 2.25288i
49.6 −0.429748 0.386946i −0.104528 + 0.994522i −0.174102 1.65647i 0.866536 + 0.281555i 0.429748 0.386946i −3.42237 + 0.359706i −1.24596 + 1.71491i −0.978148 0.207912i −0.263445 0.456300i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 400.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
13.e even 6 1 inner
143.u even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bn.b 112
11.c even 5 1 inner 429.2.bn.b 112
13.e even 6 1 inner 429.2.bn.b 112
143.u even 30 1 inner 429.2.bn.b 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bn.b 112 1.a even 1 1 trivial
429.2.bn.b 112 11.c even 5 1 inner
429.2.bn.b 112 13.e even 6 1 inner
429.2.bn.b 112 143.u even 30 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$96\!\cdots\!80$$$$T_{2}^{82} +$$$$46\!\cdots\!16$$$$T_{2}^{81} +$$$$44\!\cdots\!17$$$$T_{2}^{80} +$$$$94\!\cdots\!05$$$$T_{2}^{79} +$$$$48\!\cdots\!42$$$$T_{2}^{78} -$$$$39\!\cdots\!65$$$$T_{2}^{77} -$$$$56\!\cdots\!63$$$$T_{2}^{76} -$$$$22\!\cdots\!10$$$$T_{2}^{75} -$$$$35\!\cdots\!71$$$$T_{2}^{74} +$$$$36\!\cdots\!44$$$$T_{2}^{73} -$$$$85\!\cdots\!30$$$$T_{2}^{72} +$$$$45\!\cdots\!56$$$$T_{2}^{71} +$$$$48\!\cdots\!45$$$$T_{2}^{70} +$$$$22\!\cdots\!20$$$$T_{2}^{69} +$$$$12\!\cdots\!24$$$$T_{2}^{68} -$$$$10\!\cdots\!99$$$$T_{2}^{67} +$$$$52\!\cdots\!69$$$$T_{2}^{66} -$$$$36\!\cdots\!10$$$$T_{2}^{65} +$$$$82\!\cdots\!85$$$$T_{2}^{64} +$$$$66\!\cdots\!27$$$$T_{2}^{63} -$$$$19\!\cdots\!49$$$$T_{2}^{62} +$$$$44\!\cdots\!77$$$$T_{2}^{61} -$$$$13\!\cdots\!54$$$$T_{2}^{60} +$$$$14\!\cdots\!12$$$$T_{2}^{59} -$$$$27\!\cdots\!34$$$$T_{2}^{58} +$$$$11\!\cdots\!79$$$$T_{2}^{57} -$$$$59\!\cdots\!42$$$$T_{2}^{56} -$$$$48\!\cdots\!61$$$$T_{2}^{55} +$$$$11\!\cdots\!41$$$$T_{2}^{54} -$$$$19\!\cdots\!23$$$$T_{2}^{53} +$$$$36\!\cdots\!41$$$$T_{2}^{52} -$$$$37\!\cdots\!26$$$$T_{2}^{51} +$$$$66\!\cdots\!76$$$$T_{2}^{50} -$$$$43\!\cdots\!31$$$$T_{2}^{49} +$$$$70\!\cdots\!05$$$$T_{2}^{48} -$$$$89\!\cdots\!42$$$$T_{2}^{47} +$$$$82\!\cdots\!71$$$$T_{2}^{46} +$$$$70\!\cdots\!75$$$$T_{2}^{45} -$$$$10\!\cdots\!31$$$$T_{2}^{44} +$$$$12\!\cdots\!15$$$$T_{2}^{43} -$$$$16\!\cdots\!70$$$$T_{2}^{42} +$$$$65\!\cdots\!37$$$$T_{2}^{41} -$$$$11\!\cdots\!31$$$$T_{2}^{40} -$$$$59\!\cdots\!00$$$$T_{2}^{39} +$$$$10\!\cdots\!17$$$$T_{2}^{38} -$$$$12\!\cdots\!11$$$$T_{2}^{37} +$$$$11\!\cdots\!72$$$$T_{2}^{36} -$$$$82\!\cdots\!20$$$$T_{2}^{35} +$$$$14\!\cdots\!67$$$$T_{2}^{34} -$$$$15\!\cdots\!91$$$$T_{2}^{33} +$$$$11\!\cdots\!26$$$$T_{2}^{32} +$$$$33\!\cdots\!72$$$$T_{2}^{31} +$$$$55\!\cdots\!59$$$$T_{2}^{30} +$$$$42\!\cdots\!60$$$$T_{2}^{29} +$$$$23\!\cdots\!99$$$$T_{2}^{28} +$$$$16\!\cdots\!57$$$$T_{2}^{27} -$$$$20\!\cdots\!21$$$$T_{2}^{26} -$$$$88\!\cdots\!56$$$$T_{2}^{25} -$$$$16\!\cdots\!01$$$$T_{2}^{24} -$$$$90\!\cdots\!16$$$$T_{2}^{23} -$$$$45\!\cdots\!71$$$$T_{2}^{22} -$$$$19\!\cdots\!74$$$$T_{2}^{21} +$$$$13\!\cdots\!05$$$$T_{2}^{20} +$$$$10\!\cdots\!50$$$$T_{2}^{19} +$$$$15\!\cdots\!46$$$$T_{2}^{18} +$$$$57\!\cdots\!09$$$$T_{2}^{17} +$$$$50\!\cdots\!19$$$$T_{2}^{16} +$$$$10\!\cdots\!45$$$$T_{2}^{15} +$$$$65\!\cdots\!51$$$$T_{2}^{14} +$$$$11\!\cdots\!35$$$$T_{2}^{13} +$$$$50\!\cdots\!45$$$$T_{2}^{12} -$$$$22\!\cdots\!27$$$$T_{2}^{11} +$$$$40\!\cdots\!36$$$$T_{2}^{10} -$$$$43\!\cdots\!37$$$$T_{2}^{9} +$$$$31\!\cdots\!22$$$$T_{2}^{8} -$$$$15\!\cdots\!30$$$$T_{2}^{7} + 50623041572 T_{2}^{6} - 1012957494 T_{2}^{5} + 8921117 T_{2}^{4} - 70728 T_{2}^{3} + 3515 T_{2}^{2} - 21 T_{2} + 1$$">$$T_{2}^{112} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.