# Properties

 Label 429.2.bj.b Level $429$ Weight $2$ Character orbit 429.bj 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.bj (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112q + 28q^{3} + 6q^{5} - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q + 28q^{3} + 6q^{5} - 28q^{9} - 14q^{11} + 10q^{13} + 12q^{14} + 4q^{15} - 48q^{16} - 2q^{20} + 32q^{22} + 30q^{24} - 46q^{26} + 28q^{27} - 20q^{29} - 24q^{31} - 16q^{33} - 16q^{34} - 20q^{35} + 12q^{37} + 10q^{39} - 40q^{40} - 10q^{41} + 28q^{42} + 24q^{44} - 4q^{45} - 20q^{46} - 62q^{47} - 92q^{48} - 90q^{50} + 110q^{52} + 68q^{53} + 32q^{55} - 30q^{57} - 56q^{58} + 16q^{59} + 2q^{60} + 20q^{61} + 8q^{66} + 12q^{67} + 60q^{68} - 196q^{70} - 52q^{71} + 30q^{72} - 10q^{73} - 120q^{74} - 84q^{78} + 40q^{79} - 56q^{80} - 28q^{81} + 110q^{83} - 30q^{84} - 40q^{85} + 18q^{86} - 96q^{89} - 44q^{91} + 80q^{92} + 24q^{93} - 20q^{94} + 100q^{96} + 18q^{97} + 6q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1 −2.50252 + 0.396361i −0.309017 + 0.951057i 4.20341 1.36577i −1.89481 0.300108i 0.396361 2.50252i 0.290317 0.569780i −5.46267 + 2.78337i −0.809017 0.587785i 4.86076
73.2 −2.34881 + 0.372015i −0.309017 + 0.951057i 3.47641 1.12955i 3.36131 + 0.532379i 0.372015 2.34881i −2.10675 + 4.13473i −3.50742 + 1.78712i −0.809017 0.587785i −8.09314
73.3 −1.79125 + 0.283706i −0.309017 + 0.951057i 1.22598 0.398345i −3.18553 0.504538i 0.283706 1.79125i 0.465361 0.913322i 1.14880 0.585343i −0.809017 0.587785i 5.84923
73.4 −1.73374 + 0.274598i −0.309017 + 0.951057i 1.02835 0.334131i 2.33112 + 0.369213i 0.274598 1.73374i 1.54772 3.03757i 1.43692 0.732150i −0.809017 0.587785i −4.14295
73.5 −1.43832 + 0.227807i −0.309017 + 0.951057i 0.114751 0.0372850i −0.828408 0.131207i 0.227807 1.43832i −1.40334 + 2.75420i 2.43850 1.24248i −0.809017 0.587785i 1.22140
73.6 −0.778309 + 0.123272i −0.309017 + 0.951057i −1.31154 + 0.426146i 3.15337 + 0.499445i 0.123272 0.778309i −0.271920 + 0.533674i 2.37250 1.20885i −0.809017 0.587785i −2.51586
73.7 0.143806 0.0227766i −0.309017 + 0.951057i −1.88195 + 0.611483i −1.10538 0.175075i −0.0227766 + 0.143806i −0.0900281 + 0.176690i −0.516165 + 0.262999i −0.809017 0.587785i −0.162947
73.8 0.227816 0.0360826i −0.309017 + 0.951057i −1.85151 + 0.601594i −1.68185 0.266378i −0.0360826 + 0.227816i 2.29608 4.50631i −0.811131 + 0.413292i −0.809017 0.587785i −0.392764
73.9 1.10738 0.175392i −0.309017 + 0.951057i −0.706578 + 0.229581i 2.85202 + 0.451715i −0.175392 + 1.10738i −1.47375 + 2.89241i −2.74016 + 1.39618i −0.809017 0.587785i 3.23751
73.10 1.19422 0.189145i −0.309017 + 0.951057i −0.511738 + 0.166274i −0.797331 0.126285i −0.189145 + 1.19422i −1.27410 + 2.50056i −2.73431 + 1.39320i −0.809017 0.587785i −0.976071
73.11 1.48140 0.234631i −0.309017 + 0.951057i 0.237388 0.0771321i 3.64808 + 0.577798i −0.234631 + 1.48140i 1.93361 3.79492i −2.33921 + 1.19189i −0.809017 0.587785i 5.53984
73.12 1.66516 0.263735i −0.309017 + 0.951057i 0.801076 0.260285i −3.99771 0.633176i −0.263735 + 1.66516i −0.308156 + 0.604790i −1.73905 + 0.886089i −0.809017 0.587785i −6.82381
73.13 2.30315 0.364783i −0.309017 + 0.951057i 3.26933 1.06227i 0.468964 + 0.0742767i −0.364783 + 2.30315i −1.08156 + 2.12269i 2.98686 1.52188i −0.809017 0.587785i 1.10719
73.14 2.47003 0.391214i −0.309017 + 0.951057i 4.04586 1.31458i −0.0637683 0.0100999i −0.391214 + 2.47003i 1.47652 2.89783i 5.02262 2.55915i −0.809017 0.587785i −0.161460
112.1 −1.24336 2.44023i 0.809017 0.587785i −3.23319 + 4.45011i −1.70076 + 3.33794i −2.44023 1.24336i −0.416032 2.62673i 9.46926 + 1.49978i 0.309017 0.951057i 10.2600
112.2 −1.24128 2.43615i 0.809017 0.587785i −3.21847 + 4.42985i 1.65945 3.25685i −2.43615 1.24128i 0.168914 + 1.06648i 9.38582 + 1.48657i 0.309017 0.951057i −9.99400
112.3 −0.943260 1.85125i 0.809017 0.587785i −1.36183 + 1.87439i −0.530412 + 1.04099i −1.85125 0.943260i 0.785270 + 4.95800i 0.650274 + 0.102993i 0.309017 0.951057i 2.42745
112.4 −0.608810 1.19486i 0.809017 0.587785i 0.118537 0.163152i 0.900222 1.76679i −1.19486 0.608810i −0.329220 2.07862i −2.91613 0.461869i 0.309017 0.951057i −2.65912
112.5 −0.558611 1.09633i 0.809017 0.587785i 0.285666 0.393186i −1.49041 + 2.92509i −1.09633 0.558611i −0.495831 3.13056i −3.02123 0.478516i 0.309017 0.951057i 4.03944
112.6 −0.390599 0.766593i 0.809017 0.587785i 0.740473 1.01917i −0.202131 + 0.396704i −0.766593 0.390599i 0.246699 + 1.55759i −2.77007 0.438736i 0.309017 0.951057i 0.383063
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.14 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.d odd 10 1 inner
13.d odd 4 1 inner
143.s even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bj.b 112
11.d odd 10 1 inner 429.2.bj.b 112
13.d odd 4 1 inner 429.2.bj.b 112
143.s even 20 1 inner 429.2.bj.b 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bj.b 112 1.a even 1 1 trivial
429.2.bj.b 112 11.d odd 10 1 inner
429.2.bj.b 112 13.d odd 4 1 inner
429.2.bj.b 112 143.s even 20 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$16\!\cdots\!62$$$$T_{2}^{84} +$$$$20\!\cdots\!10$$$$T_{2}^{83} + 48230440900 T_{2}^{82} -$$$$10\!\cdots\!10$$$$T_{2}^{81} +$$$$43\!\cdots\!57$$$$T_{2}^{80} -$$$$52\!\cdots\!60$$$$T_{2}^{79} + 926562554050 T_{2}^{78} +$$$$23\!\cdots\!00$$$$T_{2}^{77} -$$$$86\!\cdots\!51$$$$T_{2}^{76} +$$$$10\!\cdots\!70$$$$T_{2}^{75} -$$$$50\!\cdots\!50$$$$T_{2}^{74} -$$$$42\!\cdots\!50$$$$T_{2}^{73} +$$$$15\!\cdots\!34$$$$T_{2}^{72} -$$$$18\!\cdots\!30$$$$T_{2}^{71} +$$$$16\!\cdots\!00$$$$T_{2}^{70} +$$$$67\!\cdots\!30$$$$T_{2}^{69} -$$$$23\!\cdots\!18$$$$T_{2}^{68} +$$$$26\!\cdots\!90$$$$T_{2}^{67} -$$$$28\!\cdots\!50$$$$T_{2}^{66} -$$$$78\!\cdots\!20$$$$T_{2}^{65} +$$$$26\!\cdots\!37$$$$T_{2}^{64} -$$$$27\!\cdots\!40$$$$T_{2}^{63} +$$$$63\!\cdots\!00$$$$T_{2}^{62} +$$$$55\!\cdots\!90$$$$T_{2}^{61} -$$$$20\!\cdots\!11$$$$T_{2}^{60} +$$$$19\!\cdots\!20$$$$T_{2}^{59} -$$$$78\!\cdots\!00$$$$T_{2}^{58} -$$$$20\!\cdots\!10$$$$T_{2}^{57} +$$$$10\!\cdots\!75$$$$T_{2}^{56} -$$$$89\!\cdots\!90$$$$T_{2}^{55} +$$$$52\!\cdots\!50$$$$T_{2}^{54} +$$$$17\!\cdots\!60$$$$T_{2}^{53} -$$$$45\!\cdots\!29$$$$T_{2}^{52} +$$$$29\!\cdots\!30$$$$T_{2}^{51} -$$$$14\!\cdots\!00$$$$T_{2}^{50} +$$$$10\!\cdots\!80$$$$T_{2}^{49} +$$$$16\!\cdots\!80$$$$T_{2}^{48} -$$$$10\!\cdots\!40$$$$T_{2}^{47} +$$$$31\!\cdots\!00$$$$T_{2}^{46} -$$$$45\!\cdots\!60$$$$T_{2}^{45} -$$$$44\!\cdots\!81$$$$T_{2}^{44} +$$$$40\!\cdots\!90$$$$T_{2}^{43} -$$$$11\!\cdots\!50$$$$T_{2}^{42} +$$$$10\!\cdots\!70$$$$T_{2}^{41} +$$$$75\!\cdots\!64$$$$T_{2}^{40} -$$$$10\!\cdots\!90$$$$T_{2}^{39} +$$$$56\!\cdots\!00$$$$T_{2}^{38} -$$$$18\!\cdots\!40$$$$T_{2}^{37} -$$$$87\!\cdots\!09$$$$T_{2}^{36} +$$$$11\!\cdots\!70$$$$T_{2}^{35} -$$$$57\!\cdots\!00$$$$T_{2}^{34} -$$$$58\!\cdots\!20$$$$T_{2}^{33} +$$$$27\!\cdots\!55$$$$T_{2}^{32} -$$$$36\!\cdots\!70$$$$T_{2}^{31} +$$$$29\!\cdots\!50$$$$T_{2}^{30} -$$$$17\!\cdots\!20$$$$T_{2}^{29} -$$$$26\!\cdots\!72$$$$T_{2}^{28} +$$$$14\!\cdots\!50$$$$T_{2}^{27} -$$$$12\!\cdots\!50$$$$T_{2}^{26} +$$$$29\!\cdots\!60$$$$T_{2}^{25} +$$$$72\!\cdots\!32$$$$T_{2}^{24} -$$$$94\!\cdots\!70$$$$T_{2}^{23} +$$$$65\!\cdots\!50$$$$T_{2}^{22} -$$$$30\!\cdots\!70$$$$T_{2}^{21} -$$$$17\!\cdots\!83$$$$T_{2}^{20} +$$$$49\!\cdots\!40$$$$T_{2}^{19} -$$$$44\!\cdots\!50$$$$T_{2}^{18} +$$$$23\!\cdots\!30$$$$T_{2}^{17} +$$$$68\!\cdots\!04$$$$T_{2}^{16} -$$$$18\!\cdots\!10$$$$T_{2}^{15} +$$$$13\!\cdots\!00$$$$T_{2}^{14} -$$$$53\!\cdots\!60$$$$T_{2}^{13} +$$$$13\!\cdots\!47$$$$T_{2}^{12} -$$$$23\!\cdots\!50$$$$T_{2}^{11} +$$$$93\!\cdots\!50$$$$T_{2}^{10} -$$$$16\!\cdots\!10$$$$T_{2}^{9} +$$$$97\!\cdots\!51$$$$T_{2}^{8} -$$$$12\!\cdots\!20$$$$T_{2}^{7} +$$$$29\!\cdots\!50$$$$T_{2}^{6} -$$$$47\!\cdots\!00$$$$T_{2}^{5} +$$$$20\!\cdots\!55$$$$T_{2}^{4} -$$$$36\!\cdots\!70$$$$T_{2}^{3} +$$$$40\!\cdots\!00$$$$T_{2}^{2} -$$$$63\!\cdots\!20$$$$T_{2} + 495504774241$$">$$T_{2}^{112} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.