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 $4$

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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:

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 \(S_{2}^{\mathrm{new}}(322, [\chi])\).