# Properties

 Label 429.2.bg.a Level $429$ Weight $2$ Character orbit 429.bg 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.bg (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$112q - 14q^{3} + 20q^{4} + 6q^{5} + 24q^{8} + 14q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q - 14q^{3} + 20q^{4} + 6q^{5} + 24q^{8} + 14q^{9} + 3q^{11} - 120q^{12} + q^{13} + 6q^{14} - 2q^{15} - 16q^{16} - 20q^{17} - 40q^{20} + 15q^{22} + 4q^{23} - 18q^{24} - 26q^{25} + 49q^{26} + 28q^{27} - 6q^{28} - 10q^{29} - 10q^{30} - 4q^{31} - 34q^{32} - 8q^{33} + 96q^{34} - 12q^{35} + 20q^{36} + 20q^{37} - 62q^{38} - 4q^{39} - 20q^{40} - 50q^{41} + 8q^{42} + 44q^{43} - 12q^{44} + 2q^{45} + 42q^{46} + 18q^{47} - 34q^{48} + 48q^{49} - 37q^{50} + 10q^{51} - 19q^{52} - 46q^{53} + 48q^{55} - 12q^{56} - 40q^{58} - 18q^{59} - 80q^{60} + 2q^{61} - 49q^{62} + 68q^{64} + 24q^{65} - 20q^{66} - 52q^{67} - 56q^{68} + 11q^{69} + 220q^{70} - 54q^{71} + 18q^{72} - 50q^{73} - 64q^{74} - 28q^{75} + 28q^{76} + 84q^{77} + 2q^{78} - 68q^{79} + 34q^{80} + 14q^{81} + 51q^{82} - 72q^{83} - 9q^{84} + 14q^{85} - 10q^{86} - 60q^{87} - 22q^{88} + 120q^{89} + 20q^{90} - 43q^{91} + 122q^{92} - 2q^{93} + 30q^{94} - 16q^{95} - 68q^{96} + 51q^{97} - 24q^{98} + 14q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −1.80037 + 1.99951i 0.104528 0.994522i −0.547661 5.21064i −0.759776 + 2.33835i 1.80037 + 1.99951i 0.0209204 + 0.199044i 7.05123 + 5.12302i −0.978148 0.207912i −3.30768 5.72907i
16.2 −1.68093 + 1.86686i 0.104528 0.994522i −0.450589 4.28707i 1.24954 3.84568i 1.68093 + 1.86686i 0.0277297 + 0.263830i 4.69608 + 3.41190i −0.978148 0.207912i 5.07897 + 8.79703i
16.3 −1.46128 + 1.62292i 0.104528 0.994522i −0.289460 2.75403i −0.299446 + 0.921600i 1.46128 + 1.62292i −0.235463 2.24028i 1.35900 + 0.987368i −0.978148 0.207912i −1.05810 1.83269i
16.4 −0.990763 + 1.10035i 0.104528 0.994522i −0.0201101 0.191335i −0.983732 + 3.02762i 0.990763 + 1.10035i 0.365713 + 3.47953i −2.16532 1.57319i −0.978148 0.207912i −2.35680 4.08210i
16.5 −0.946617 + 1.05132i 0.104528 0.994522i −0.000142744 0.00135811i 0.776847 2.39089i 0.946617 + 1.05132i −0.131132 1.24764i −2.28746 1.66194i −0.978148 0.207912i 1.77822 + 3.07997i
16.6 −0.491663 + 0.546047i 0.104528 0.994522i 0.152622 + 1.45210i −0.00147074 + 0.00452647i 0.491663 + 0.546047i −0.428655 4.07838i −2.05685 1.49439i −0.978148 0.207912i −0.00174856 0.00302859i
16.7 −0.0978317 + 0.108653i 0.104528 0.994522i 0.206822 + 1.96778i −0.621977 + 1.91425i 0.0978317 + 0.108653i 0.00250914 + 0.0238729i −0.470608 0.341917i −0.978148 0.207912i −0.147140 0.254854i
16.8 0.244102 0.271103i 0.104528 0.994522i 0.195146 + 1.85669i 0.211790 0.651823i −0.244102 0.271103i 0.465929 + 4.43302i 1.14126 + 0.829171i −0.978148 0.207912i −0.125013 0.216528i
16.9 0.502391 0.557961i 0.104528 0.994522i 0.150132 + 1.42841i −0.0412359 + 0.126911i −0.502391 0.557961i −0.139017 1.32266i 2.08726 + 1.51649i −0.978148 0.207912i 0.0500950 + 0.0867670i
16.10 0.844998 0.938465i 0.104528 0.994522i 0.0423614 + 0.403042i 1.13050 3.47932i −0.844998 0.938465i −0.100820 0.959240i 2.45734 + 1.78536i −0.978148 0.207912i −2.30995 4.00095i
16.11 1.06574 1.18363i 0.104528 0.994522i −0.0561096 0.533847i −1.11978 + 3.44633i −1.06574 1.18363i 0.311034 + 2.95930i 1.88542 + 1.36983i −0.978148 0.207912i 2.88578 + 4.99832i
16.12 1.44079 1.60016i 0.104528 0.994522i −0.275575 2.62192i 0.0727615 0.223937i −1.44079 1.60016i −0.0981000 0.933359i −1.10854 0.805401i −0.978148 0.207912i −0.253500 0.439075i
16.13 1.65656 1.83979i 0.104528 0.994522i −0.431601 4.10641i −0.543715 + 1.67338i −1.65656 1.83979i −0.506656 4.82051i −4.26418 3.09811i −0.978148 0.207912i 2.17798 + 3.77238i
16.14 1.71487 1.90455i 0.104528 0.994522i −0.477496 4.54307i 1.12068 3.44910i −1.71487 1.90455i 0.446007 + 4.24348i −5.32461 3.86856i −0.978148 0.207912i −4.64718 8.04915i
256.1 −0.273007 2.59749i 0.978148 + 0.207912i −4.71612 + 1.00244i 0.822674 + 0.597708i 0.273007 2.59749i 3.34488 0.710976i 2.27719 + 7.00848i 0.913545 + 0.406737i 1.32794 2.30007i
256.2 −0.258546 2.45990i 0.978148 + 0.207912i −4.02797 + 0.856172i −0.872789 0.634118i 0.258546 2.45990i −3.11026 + 0.661107i 1.61884 + 4.98227i 0.913545 + 0.406737i −1.33421 + 2.31092i
256.3 −0.220383 2.09680i 0.978148 + 0.207912i −2.39171 + 0.508373i 3.27922 + 2.38249i 0.220383 2.09680i −0.946033 + 0.201086i 0.290016 + 0.892577i 0.913545 + 0.406737i 4.27293 7.40092i
256.4 −0.140589 1.33762i 0.978148 + 0.207912i 0.186846 0.0397154i −0.768940 0.558668i 0.140589 1.33762i 3.61727 0.768874i −0.910638 2.80265i 0.913545 + 0.406737i −0.639178 + 1.10709i
256.5 −0.134448 1.27919i 0.978148 + 0.207912i 0.338055 0.0718558i −1.25069 0.908681i 0.134448 1.27919i 1.33154 0.283029i −0.932303 2.86933i 0.913545 + 0.406737i −0.994219 + 1.72204i
256.6 −0.0790719 0.752319i 0.978148 + 0.207912i 1.39656 0.296849i 2.62007 + 1.90359i 0.0790719 0.752319i −0.683673 + 0.145319i −0.801274 2.46607i 0.913545 + 0.406737i 1.22494 2.12165i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 412.14 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.c even 3 1 inner
143.q even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bg.a 112
11.c even 5 1 inner 429.2.bg.a 112
13.c even 3 1 inner 429.2.bg.a 112
143.q even 15 1 inner 429.2.bg.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bg.a 112 1.a even 1 1 trivial
429.2.bg.a 112 11.c even 5 1 inner
429.2.bg.a 112 13.c even 3 1 inner
429.2.bg.a 112 143.q even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!92$$$$T_{2}^{84} - 93223698381 T_{2}^{83} +$$$$32\!\cdots\!96$$$$T_{2}^{82} + 850388036538 T_{2}^{81} +$$$$29\!\cdots\!09$$$$T_{2}^{80} +$$$$37\!\cdots\!33$$$$T_{2}^{79} -$$$$24\!\cdots\!94$$$$T_{2}^{78} -$$$$40\!\cdots\!93$$$$T_{2}^{77} +$$$$84\!\cdots\!89$$$$T_{2}^{76} +$$$$42\!\cdots\!82$$$$T_{2}^{75} -$$$$17\!\cdots\!31$$$$T_{2}^{74} +$$$$80\!\cdots\!80$$$$T_{2}^{73} -$$$$14\!\cdots\!12$$$$T_{2}^{72} -$$$$55\!\cdots\!68$$$$T_{2}^{71} +$$$$79\!\cdots\!73$$$$T_{2}^{70} +$$$$21\!\cdots\!46$$$$T_{2}^{69} -$$$$16\!\cdots\!40$$$$T_{2}^{68} -$$$$12\!\cdots\!35$$$$T_{2}^{67} -$$$$34\!\cdots\!31$$$$T_{2}^{66} -$$$$41\!\cdots\!42$$$$T_{2}^{65} +$$$$31\!\cdots\!41$$$$T_{2}^{64} +$$$$24\!\cdots\!63$$$$T_{2}^{63} -$$$$79\!\cdots\!27$$$$T_{2}^{62} -$$$$48\!\cdots\!63$$$$T_{2}^{61} +$$$$50\!\cdots\!18$$$$T_{2}^{60} -$$$$26\!\cdots\!96$$$$T_{2}^{59} +$$$$32\!\cdots\!22$$$$T_{2}^{58} +$$$$46\!\cdots\!81$$$$T_{2}^{57} -$$$$14\!\cdots\!16$$$$T_{2}^{56} -$$$$17\!\cdots\!93$$$$T_{2}^{55} +$$$$25\!\cdots\!23$$$$T_{2}^{54} +$$$$36\!\cdots\!79$$$$T_{2}^{53} +$$$$36\!\cdots\!95$$$$T_{2}^{52} -$$$$21\!\cdots\!78$$$$T_{2}^{51} -$$$$13\!\cdots\!84$$$$T_{2}^{50} -$$$$11\!\cdots\!57$$$$T_{2}^{49} +$$$$30\!\cdots\!39$$$$T_{2}^{48} +$$$$34\!\cdots\!20$$$$T_{2}^{47} -$$$$27\!\cdots\!97$$$$T_{2}^{46} -$$$$39\!\cdots\!59$$$$T_{2}^{45} -$$$$16\!\cdots\!23$$$$T_{2}^{44} -$$$$77\!\cdots\!37$$$$T_{2}^{43} +$$$$10\!\cdots\!90$$$$T_{2}^{42} +$$$$10\!\cdots\!45$$$$T_{2}^{41} -$$$$18\!\cdots\!99$$$$T_{2}^{40} -$$$$17\!\cdots\!44$$$$T_{2}^{39} +$$$$13\!\cdots\!11$$$$T_{2}^{38} +$$$$83\!\cdots\!55$$$$T_{2}^{37} +$$$$16\!\cdots\!50$$$$T_{2}^{36} +$$$$21\!\cdots\!66$$$$T_{2}^{35} -$$$$45\!\cdots\!19$$$$T_{2}^{34} -$$$$43\!\cdots\!53$$$$T_{2}^{33} +$$$$34\!\cdots\!20$$$$T_{2}^{32} +$$$$26\!\cdots\!74$$$$T_{2}^{31} +$$$$63\!\cdots\!87$$$$T_{2}^{30} +$$$$17\!\cdots\!08$$$$T_{2}^{29} -$$$$15\!\cdots\!07$$$$T_{2}^{28} -$$$$25\!\cdots\!91$$$$T_{2}^{27} +$$$$96\!\cdots\!73$$$$T_{2}^{26} +$$$$13\!\cdots\!94$$$$T_{2}^{25} -$$$$14\!\cdots\!41$$$$T_{2}^{24} -$$$$34\!\cdots\!68$$$$T_{2}^{23} -$$$$19\!\cdots\!23$$$$T_{2}^{22} -$$$$69\!\cdots\!88$$$$T_{2}^{21} +$$$$26\!\cdots\!95$$$$T_{2}^{20} +$$$$20\!\cdots\!96$$$$T_{2}^{19} -$$$$13\!\cdots\!72$$$$T_{2}^{18} -$$$$10\!\cdots\!63$$$$T_{2}^{17} +$$$$30\!\cdots\!31$$$$T_{2}^{16} +$$$$14\!\cdots\!25$$$$T_{2}^{15} +$$$$65\!\cdots\!65$$$$T_{2}^{14} +$$$$67\!\cdots\!25$$$$T_{2}^{13} +$$$$15\!\cdots\!25$$$$T_{2}^{12} -$$$$79\!\cdots\!25$$$$T_{2}^{11} +$$$$24\!\cdots\!00$$$$T_{2}^{10} +$$$$16\!\cdots\!25$$$$T_{2}^{9} -$$$$22\!\cdots\!50$$$$T_{2}^{8} -$$$$65\!\cdots\!00$$$$T_{2}^{7} +$$$$74\!\cdots\!50$$$$T_{2}^{6} -$$$$82\!\cdots\!00$$$$T_{2}^{5} +$$$$46\!\cdots\!75$$$$T_{2}^{4} -$$$$36\!\cdots\!50$$$$T_{2}^{3} +$$$$15\!\cdots\!75$$$$T_{2}^{2} -$$$$14\!\cdots\!75$$$$T_{2} + 152587890625$$">$$T_{2}^{112} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.