# Properties

 Label 322.2.o.b Level $322$ Weight $2$ Character orbit 322.o Analytic conductor $2.571$ Analytic rank $0$ Dimension $160$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.o (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$8$$ over $$\Q(\zeta_{66})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 8q^{2} - 6q^{3} + 8q^{4} + 11q^{7} - 16q^{8} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 8q^{2} - 6q^{3} + 8q^{4} + 11q^{7} - 16q^{8} + 12q^{9} - 27q^{12} - 11q^{14} + 8q^{16} + 66q^{17} + 12q^{18} - 66q^{21} - 18q^{23} - 6q^{24} - 2q^{25} + 6q^{26} - 22q^{28} - 16q^{29} - 22q^{30} + 24q^{31} + 8q^{32} + 73q^{35} + 20q^{36} + 22q^{37} - 33q^{38} + 22q^{42} - 110q^{43} + 4q^{46} - 162q^{47} + 25q^{49} + 4q^{50} - 11q^{51} + 60q^{52} + 22q^{53} - 54q^{54} - 11q^{56} - 44q^{57} - 14q^{58} - 36q^{59} - 121q^{63} - 16q^{64} - 77q^{65} + 32q^{70} + 144q^{71} - 32q^{72} - 108q^{73} - 22q^{74} + 96q^{75} - 17q^{77} - 22q^{78} - 44q^{79} + 14q^{81} - 27q^{82} + 11q^{84} - 2q^{85} - 66q^{86} - 108q^{87} + 11q^{88} + 198q^{89} - 8q^{92} - 50q^{93} + 30q^{94} - 28q^{95} + 6q^{96} - 45q^{98} - 220q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 0.723734 0.690079i −0.618301 + 3.20805i 0.0475819 0.998867i 2.71652 + 2.13629i 1.76632 + 2.74845i 2.21316 1.44980i −0.654861 0.755750i −7.12418 2.85209i 3.44025 0.328504i
5.2 0.723734 0.690079i −0.290209 + 1.50575i 0.0475819 0.998867i −0.209478 0.164735i 0.829051 + 1.29003i 0.478644 + 2.60210i −0.654861 0.755750i 0.602045 + 0.241023i −0.265286 + 0.0253318i
5.3 0.723734 0.690079i −0.219533 + 1.13904i 0.0475819 0.998867i −0.936223 0.736254i 0.627147 + 0.975860i 0.155114 2.64120i −0.654861 0.755750i 1.53588 + 0.614873i −1.18565 + 0.113216i
5.4 0.723734 0.690079i 0.141475 0.734043i 0.0475819 0.998867i 1.50572 + 1.18411i −0.404157 0.628881i 2.62746 + 0.310594i −0.654861 0.755750i 2.26630 + 0.907289i 1.90687 0.182084i
5.5 0.723734 0.690079i 0.223563 1.15995i 0.0475819 0.998867i −2.96167 2.32908i −0.638660 0.993774i −1.92017 + 1.82015i −0.654861 0.755750i 1.48959 + 0.596343i −3.75071 + 0.358150i
5.6 0.723734 0.690079i 0.340605 1.76722i 0.0475819 0.998867i −0.793626 0.624115i −0.973017 1.51404i −0.911188 2.48390i −0.654861 0.755750i −0.221966 0.0888616i −1.00506 + 0.0959718i
5.7 0.723734 0.690079i 0.410973 2.13233i 0.0475819 0.998867i 3.20038 + 2.51681i −1.17404 1.82684i −2.40619 1.10010i −0.654861 0.755750i −1.59282 0.637669i 4.05302 0.387017i
5.8 0.723734 0.690079i 0.640458 3.32301i 0.0475819 0.998867i −0.463735 0.364685i −1.82962 2.84694i 1.94583 + 1.79269i −0.654861 0.755750i −7.84710 3.14151i −0.587283 + 0.0560787i
17.1 −0.995472 + 0.0950560i −2.26151 2.37180i 0.981929 0.189251i 2.68625 + 1.38486i 2.47672 + 2.14609i 2.03190 1.69451i −0.959493 + 0.281733i −0.368277 + 7.73108i −2.80573 1.12324i
17.2 −0.995472 + 0.0950560i −1.67171 1.75324i 0.981929 0.189251i −3.49473 1.80166i 1.83079 + 1.58639i 2.30654 1.29610i −0.959493 + 0.281733i −0.136486 + 2.86520i 3.65017 + 1.46131i
17.3 −0.995472 + 0.0950560i −1.02117 1.07097i 0.981929 0.189251i 0.591499 + 0.304939i 1.11835 + 0.969052i −2.51796 0.812315i −0.959493 + 0.281733i 0.0385527 0.809321i −0.617807 0.247332i
17.4 −0.995472 + 0.0950560i −0.855603 0.897331i 0.981929 0.189251i −2.48995 1.28366i 0.937026 + 0.811937i −1.18187 + 2.36710i −0.959493 + 0.281733i 0.0696000 1.46108i 2.60070 + 1.04116i
17.5 −0.995472 + 0.0950560i 0.322555 + 0.338286i 0.981929 0.189251i 0.680578 + 0.350862i −0.353250 0.306093i −0.141341 2.64197i −0.959493 + 0.281733i 0.132350 2.77837i −0.710848 0.284581i
17.6 −0.995472 + 0.0950560i 0.519459 + 0.544793i 0.981929 0.189251i 2.76848 + 1.42725i −0.568893 0.492948i 2.09567 + 1.61498i −0.959493 + 0.281733i 0.115784 2.43061i −2.89162 1.15763i
17.7 −0.995472 + 0.0950560i 1.77845 + 1.86518i 0.981929 0.189251i −1.66354 0.857616i −1.94769 1.68768i 2.61594 0.396066i −0.959493 + 0.281733i −0.173283 + 3.63765i 1.73753 + 0.695603i
17.8 −0.995472 + 0.0950560i 2.19647 + 2.30360i 0.981929 0.189251i 1.78887 + 0.922229i −2.40550 2.08438i −2.58627 + 0.557860i −0.959493 + 0.281733i −0.339309 + 7.12298i −1.86844 0.748010i
19.1 −0.995472 0.0950560i −2.26151 + 2.37180i 0.981929 + 0.189251i 2.68625 1.38486i 2.47672 2.14609i 2.03190 + 1.69451i −0.959493 0.281733i −0.368277 7.73108i −2.80573 + 1.12324i
19.2 −0.995472 0.0950560i −1.67171 + 1.75324i 0.981929 + 0.189251i −3.49473 + 1.80166i 1.83079 1.58639i 2.30654 + 1.29610i −0.959493 0.281733i −0.136486 2.86520i 3.65017 1.46131i
19.3 −0.995472 0.0950560i −1.02117 + 1.07097i 0.981929 + 0.189251i 0.591499 0.304939i 1.11835 0.969052i −2.51796 + 0.812315i −0.959493 0.281733i 0.0385527 + 0.809321i −0.617807 + 0.247332i
19.4 −0.995472 0.0950560i −0.855603 + 0.897331i 0.981929 + 0.189251i −2.48995 + 1.28366i 0.937026 0.811937i −1.18187 2.36710i −0.959493 0.281733i 0.0696000 + 1.46108i 2.60070 1.04116i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 313.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.o.b 160
7.d odd 6 1 inner 322.2.o.b 160
23.d odd 22 1 inner 322.2.o.b 160
161.o even 66 1 inner 322.2.o.b 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.o.b 160 1.a even 1 1 trivial
322.2.o.b 160 7.d odd 6 1 inner
322.2.o.b 160 23.d odd 22 1 inner
322.2.o.b 160 161.o even 66 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$30\!\cdots\!41$$$$T_{3}^{136} +$$$$93\!\cdots\!57$$$$T_{3}^{135} +$$$$22\!\cdots\!03$$$$T_{3}^{134} +$$$$70\!\cdots\!38$$$$T_{3}^{133} +$$$$67\!\cdots\!64$$$$T_{3}^{132} +$$$$55\!\cdots\!86$$$$T_{3}^{131} -$$$$50\!\cdots\!54$$$$T_{3}^{130} -$$$$22\!\cdots\!79$$$$T_{3}^{129} -$$$$65\!\cdots\!97$$$$T_{3}^{128} -$$$$29\!\cdots\!37$$$$T_{3}^{127} -$$$$54\!\cdots\!99$$$$T_{3}^{126} -$$$$18\!\cdots\!18$$$$T_{3}^{125} +$$$$15\!\cdots\!81$$$$T_{3}^{124} +$$$$22\!\cdots\!71$$$$T_{3}^{123} +$$$$43\!\cdots\!25$$$$T_{3}^{122} +$$$$12\!\cdots\!65$$$$T_{3}^{121} +$$$$30\!\cdots\!38$$$$T_{3}^{120} +$$$$99\!\cdots\!84$$$$T_{3}^{119} +$$$$13\!\cdots\!55$$$$T_{3}^{118} +$$$$29\!\cdots\!03$$$$T_{3}^{117} -$$$$15\!\cdots\!00$$$$T_{3}^{116} -$$$$31\!\cdots\!05$$$$T_{3}^{115} -$$$$10\!\cdots\!11$$$$T_{3}^{114} -$$$$27\!\cdots\!44$$$$T_{3}^{113} -$$$$17\!\cdots\!19$$$$T_{3}^{112} -$$$$27\!\cdots\!22$$$$T_{3}^{111} +$$$$29\!\cdots\!28$$$$T_{3}^{110} +$$$$55\!\cdots\!83$$$$T_{3}^{109} +$$$$25\!\cdots\!89$$$$T_{3}^{108} +$$$$42\!\cdots\!54$$$$T_{3}^{107} +$$$$71\!\cdots\!44$$$$T_{3}^{106} -$$$$14\!\cdots\!74$$$$T_{3}^{105} -$$$$52\!\cdots\!28$$$$T_{3}^{104} -$$$$10\!\cdots\!88$$$$T_{3}^{103} -$$$$28\!\cdots\!55$$$$T_{3}^{102} -$$$$94\!\cdots\!72$$$$T_{3}^{101} +$$$$46\!\cdots\!59$$$$T_{3}^{100} +$$$$13\!\cdots\!94$$$$T_{3}^{99} +$$$$80\!\cdots\!35$$$$T_{3}^{98} +$$$$52\!\cdots\!06$$$$T_{3}^{97} +$$$$19\!\cdots\!31$$$$T_{3}^{96} -$$$$34\!\cdots\!00$$$$T_{3}^{95} +$$$$29\!\cdots\!64$$$$T_{3}^{94} -$$$$29\!\cdots\!52$$$$T_{3}^{93} -$$$$64\!\cdots\!03$$$$T_{3}^{92} -$$$$63\!\cdots\!81$$$$T_{3}^{91} +$$$$64\!\cdots\!28$$$$T_{3}^{90} +$$$$64\!\cdots\!72$$$$T_{3}^{89} +$$$$97\!\cdots\!44$$$$T_{3}^{88} +$$$$33\!\cdots\!45$$$$T_{3}^{87} +$$$$51\!\cdots\!35$$$$T_{3}^{86} -$$$$33\!\cdots\!79$$$$T_{3}^{85} +$$$$10\!\cdots\!75$$$$T_{3}^{84} -$$$$17\!\cdots\!51$$$$T_{3}^{83} -$$$$54\!\cdots\!88$$$$T_{3}^{82} +$$$$15\!\cdots\!80$$$$T_{3}^{81} +$$$$97\!\cdots\!96$$$$T_{3}^{80} +$$$$75\!\cdots\!90$$$$T_{3}^{79} +$$$$23\!\cdots\!86$$$$T_{3}^{78} +$$$$13\!\cdots\!62$$$$T_{3}^{77} -$$$$11\!\cdots\!55$$$$T_{3}^{76} +$$$$15\!\cdots\!58$$$$T_{3}^{75} +$$$$71\!\cdots\!47$$$$T_{3}^{74} +$$$$17\!\cdots\!81$$$$T_{3}^{73} +$$$$34\!\cdots\!08$$$$T_{3}^{72} +$$$$44\!\cdots\!44$$$$T_{3}^{71} +$$$$35\!\cdots\!17$$$$T_{3}^{70} -$$$$34\!\cdots\!15$$$$T_{3}^{69} -$$$$59\!\cdots\!11$$$$T_{3}^{68} -$$$$12\!\cdots\!81$$$$T_{3}^{67} -$$$$11\!\cdots\!98$$$$T_{3}^{66} -$$$$19\!\cdots\!42$$$$T_{3}^{65} +$$$$12\!\cdots\!60$$$$T_{3}^{64} +$$$$27\!\cdots\!47$$$$T_{3}^{63} +$$$$34\!\cdots\!22$$$$T_{3}^{62} +$$$$19\!\cdots\!29$$$$T_{3}^{61} -$$$$10\!\cdots\!16$$$$T_{3}^{60} -$$$$34\!\cdots\!07$$$$T_{3}^{59} -$$$$44\!\cdots\!92$$$$T_{3}^{58} -$$$$43\!\cdots\!30$$$$T_{3}^{57} -$$$$52\!\cdots\!48$$$$T_{3}^{56} +$$$$10\!\cdots\!10$$$$T_{3}^{55} +$$$$26\!\cdots\!16$$$$T_{3}^{54} +$$$$29\!\cdots\!67$$$$T_{3}^{53} +$$$$11\!\cdots\!31$$$$T_{3}^{52} -$$$$11\!\cdots\!39$$$$T_{3}^{51} -$$$$10\!\cdots\!13$$$$T_{3}^{50} +$$$$45\!\cdots\!29$$$$T_{3}^{49} -$$$$19\!\cdots\!79$$$$T_{3}^{48} -$$$$49\!\cdots\!07$$$$T_{3}^{47} +$$$$37\!\cdots\!07$$$$T_{3}^{46} -$$$$30\!\cdots\!21$$$$T_{3}^{45} +$$$$18\!\cdots\!07$$$$T_{3}^{44} -$$$$11\!\cdots\!99$$$$T_{3}^{43} -$$$$11\!\cdots\!37$$$$T_{3}^{42} +$$$$43\!\cdots\!57$$$$T_{3}^{41} -$$$$52\!\cdots\!55$$$$T_{3}^{40} +$$$$40\!\cdots\!14$$$$T_{3}^{39} -$$$$14\!\cdots\!15$$$$T_{3}^{38} -$$$$13\!\cdots\!34$$$$T_{3}^{37} +$$$$29\!\cdots\!86$$$$T_{3}^{36} -$$$$33\!\cdots\!51$$$$T_{3}^{35} +$$$$26\!\cdots\!98$$$$T_{3}^{34} -$$$$13\!\cdots\!75$$$$T_{3}^{33} -$$$$35\!\cdots\!00$$$$T_{3}^{32} +$$$$87\!\cdots\!27$$$$T_{3}^{31} -$$$$10\!\cdots\!99$$$$T_{3}^{30} +$$$$86\!\cdots\!05$$$$T_{3}^{29} -$$$$47\!\cdots\!52$$$$T_{3}^{28} +$$$$15\!\cdots\!61$$$$T_{3}^{27} +$$$$20\!\cdots\!54$$$$T_{3}^{26} -$$$$83\!\cdots\!21$$$$T_{3}^{25} +$$$$84\!\cdots\!27$$$$T_{3}^{24} -$$$$64\!\cdots\!22$$$$T_{3}^{23} +$$$$41\!\cdots\!21$$$$T_{3}^{22} -$$$$23\!\cdots\!22$$$$T_{3}^{21} +$$$$11\!\cdots\!51$$$$T_{3}^{20} -$$$$52\!\cdots\!84$$$$T_{3}^{19} +$$$$21\!\cdots\!48$$$$T_{3}^{18} -$$$$79\!\cdots\!47$$$$T_{3}^{17} +$$$$27\!\cdots\!39$$$$T_{3}^{16} -$$$$85\!\cdots\!82$$$$T_{3}^{15} +$$$$24\!\cdots\!34$$$$T_{3}^{14} -$$$$65\!\cdots\!89$$$$T_{3}^{13} +$$$$15\!\cdots\!65$$$$T_{3}^{12} -$$$$34\!\cdots\!81$$$$T_{3}^{11} +$$$$66\!\cdots\!98$$$$T_{3}^{10} -$$$$11\!\cdots\!84$$$$T_{3}^{9} +$$$$17\!\cdots\!21$$$$T_{3}^{8} -$$$$21\!\cdots\!48$$$$T_{3}^{7} +$$$$21\!\cdots\!46$$$$T_{3}^{6} -$$$$17\!\cdots\!90$$$$T_{3}^{5} +$$$$95\!\cdots\!66$$$$T_{3}^{4} -$$$$19\!\cdots\!95$$$$T_{3}^{3} +$$$$20\!\cdots\!68$$$$T_{3}^{2} -$$$$13\!\cdots\!15$$$$T_{3} +$$$$39\!\cdots\!01$$">$$T_{3}^{160} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$.