Properties

Label 322.2.m.a
Level $322$
Weight $2$
Character orbit 322.m
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.m (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(8\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160q - 8q^{2} + 2q^{3} + 8q^{4} - 2q^{5} + 4q^{6} + 15q^{7} + 16q^{8} + 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 160q - 8q^{2} + 2q^{3} + 8q^{4} - 2q^{5} + 4q^{6} + 15q^{7} + 16q^{8} + 28q^{9} + 2q^{10} - 8q^{11} - 9q^{12} - 12q^{13} + 7q^{14} + 4q^{15} + 8q^{16} - 14q^{17} - 28q^{18} - 2q^{19} - 18q^{20} + 72q^{21} - 16q^{22} - 22q^{23} - 2q^{24} - 30q^{25} - 6q^{26} - 70q^{27} + 14q^{28} - 20q^{30} + 4q^{31} - 8q^{32} + 40q^{33} - 28q^{34} - 79q^{35} - 12q^{36} - 14q^{37} - 9q^{38} + 16q^{39} + 2q^{40} + 12q^{41} + 16q^{42} + 106q^{43} - 8q^{44} + 40q^{45} - 50q^{47} + 18q^{48} - 55q^{49} + 28q^{50} - 47q^{51} - 16q^{52} - 4q^{53} - 24q^{54} - 15q^{56} + 8q^{57} + 22q^{58} - 42q^{59} - 2q^{60} - 80q^{61} + 8q^{62} + 5q^{63} - 16q^{64} + 71q^{65} + 4q^{66} + 12q^{67} + 8q^{68} - 72q^{70} - 112q^{71} + 16q^{72} - 64q^{73} + 14q^{74} - 126q^{75} + 4q^{76} - 69q^{77} + 54q^{78} + 52q^{79} - 2q^{80} - 30q^{81} - 49q^{82} + 26q^{83} - q^{84} - 22q^{85} - 46q^{86} + 4q^{87} - 3q^{88} - 72q^{89} - 8q^{90} + 44q^{91} + 54q^{93} + 6q^{94} - 58q^{95} - 2q^{96} + 20q^{97} - 19q^{98} - 264q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.235759 + 0.971812i −0.899762 + 2.59969i −0.888835 0.458227i −0.684642 0.274089i −2.31428 1.48730i −2.64563 0.0257032i 0.654861 0.755750i −3.59067 2.82373i 0.427774 0.600724i
9.2 −0.235759 + 0.971812i −0.457151 + 1.32085i −0.888835 0.458227i −4.01090 1.60572i −1.17584 0.755667i 1.65184 2.06674i 0.654861 0.755750i 0.822498 + 0.646819i 2.50607 3.51928i
9.3 −0.235759 + 0.971812i −0.289551 + 0.836604i −0.888835 0.458227i 2.56790 + 1.02803i −0.744757 0.478626i 2.64344 0.110492i 0.654861 0.755750i 1.74209 + 1.37000i −1.60446 + 2.25314i
9.4 −0.235759 + 0.971812i −0.245677 + 0.709838i −0.888835 0.458227i 1.32131 + 0.528973i −0.631908 0.406103i −0.478358 2.60215i 0.654861 0.755750i 1.91465 + 1.50569i −0.825572 + 1.15935i
9.5 −0.235759 + 0.971812i 0.207322 0.599019i −0.888835 0.458227i −0.965092 0.386365i 0.533255 + 0.342702i −1.55248 + 2.14239i 0.654861 0.755750i 2.04232 + 1.60610i 0.603003 0.846799i
9.6 −0.235759 + 0.971812i 0.635615 1.83649i −0.888835 0.458227i 0.639202 + 0.255898i 1.63487 + 1.05067i −0.520133 2.59412i 0.654861 0.755750i −0.610530 0.480126i −0.399382 + 0.560854i
9.7 −0.235759 + 0.971812i 0.774374 2.23741i −0.888835 0.458227i 3.37590 + 1.35151i 1.99177 + 1.28003i −0.224436 + 2.63621i 0.654861 0.755750i −2.04818 1.61070i −2.10931 + 2.96211i
9.8 −0.235759 + 0.971812i 0.902469 2.60751i −0.888835 0.458227i −1.02777 0.411456i 2.32125 + 1.49177i 2.58489 + 0.564210i 0.654861 0.755750i −3.62651 2.85192i 0.642162 0.901791i
25.1 −0.0475819 + 0.998867i −3.01871 1.20851i −0.995472 0.0950560i −0.530176 2.18541i 1.35077 2.95778i −2.53173 0.768337i 0.142315 0.989821i 5.48089 + 5.22602i 2.20817 0.425589i
25.2 −0.0475819 + 0.998867i −1.62223 0.649442i −0.995472 0.0950560i 0.0716832 + 0.295482i 0.725895 1.58949i 2.51052 + 0.835042i 0.142315 0.989821i 0.0386478 + 0.0368506i −0.298558 + 0.0575424i
25.3 −0.0475819 + 0.998867i −0.846875 0.339038i −0.995472 0.0950560i 0.0344685 + 0.142081i 0.378950 0.829784i 0.462244 2.60506i 0.142315 0.989821i −1.56895 1.49599i −0.143560 + 0.0276690i
25.4 −0.0475819 + 0.998867i −0.473143 0.189418i −0.995472 0.0950560i 0.994090 + 4.09770i 0.211716 0.463594i −2.53885 + 0.744474i 0.142315 0.989821i −1.98322 1.89099i −4.14036 + 0.797988i
25.5 −0.0475819 + 0.998867i 0.421611 + 0.168788i −0.995472 0.0950560i −0.751910 3.09942i −0.188658 + 0.413103i −2.62384 + 0.339832i 0.142315 0.989821i −2.02194 1.92791i 3.13168 0.603582i
25.6 −0.0475819 + 0.998867i 2.32492 + 0.930757i −0.995472 0.0950560i 0.378372 + 1.55967i −1.04033 + 2.27800i −1.47479 2.19659i 0.142315 0.989821i 2.36774 + 2.25763i −1.57591 + 0.303731i
25.7 −0.0475819 + 0.998867i 2.33543 + 0.934966i −0.995472 0.0950560i −0.786581 3.24233i −1.04503 + 2.28830i 2.61565 0.397974i 0.142315 0.989821i 2.40888 + 2.29686i 3.27608 0.631414i
25.8 −0.0475819 + 0.998867i 2.44097 + 0.977219i −0.995472 0.0950560i 0.657157 + 2.70884i −1.09226 + 2.39171i 2.36993 + 1.17620i 0.142315 0.989821i 2.83220 + 2.70050i −2.73704 + 0.527521i
39.1 −0.981929 0.189251i −0.150806 3.16580i 0.928368 + 0.371662i −0.753896 + 1.05870i −0.451052 + 3.13713i −0.989718 + 2.45366i −0.841254 0.540641i −7.01315 + 0.669675i 0.940632 0.896891i
39.2 −0.981929 0.189251i −0.121146 2.54317i 0.928368 + 0.371662i 1.79650 2.52283i −0.362342 + 2.52014i −0.443953 2.60824i −0.841254 0.540641i −3.46663 + 0.331023i −2.24148 + 2.13725i
39.3 −0.981929 0.189251i −0.0679970 1.42743i 0.928368 + 0.371662i −1.39707 + 1.96191i −0.203375 + 1.41451i 2.62689 0.315333i −0.841254 0.540641i 0.953474 0.0910457i 1.74312 1.66206i
39.4 −0.981929 0.189251i −0.0525591 1.10335i 0.928368 + 0.371662i −1.42783 + 2.00510i −0.157201 + 1.09336i −2.61494 0.402583i −0.841254 0.540641i 1.77179 0.169186i 1.78149 1.69865i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 317.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.c even 11 1 inner
161.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.m.a 160
7.c even 3 1 inner 322.2.m.a 160
23.c even 11 1 inner 322.2.m.a 160
161.m even 33 1 inner 322.2.m.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.m.a 160 1.a even 1 1 trivial
322.2.m.a 160 7.c even 3 1 inner
322.2.m.a 160 23.c even 11 1 inner
322.2.m.a 160 161.m even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(26\!\cdots\!65\)\( T_{3}^{134} + \)\(39\!\cdots\!64\)\( T_{3}^{133} + \)\(12\!\cdots\!54\)\( T_{3}^{132} - \)\(80\!\cdots\!64\)\( T_{3}^{131} - \)\(17\!\cdots\!40\)\( T_{3}^{130} + \)\(75\!\cdots\!25\)\( T_{3}^{129} + \)\(85\!\cdots\!39\)\( T_{3}^{128} - \)\(40\!\cdots\!87\)\( T_{3}^{127} + \)\(85\!\cdots\!35\)\( T_{3}^{126} + \)\(66\!\cdots\!14\)\( T_{3}^{125} - \)\(16\!\cdots\!85\)\( T_{3}^{124} + \)\(41\!\cdots\!73\)\( T_{3}^{123} + \)\(12\!\cdots\!35\)\( T_{3}^{122} - \)\(67\!\cdots\!19\)\( T_{3}^{121} - \)\(37\!\cdots\!52\)\( T_{3}^{120} + \)\(91\!\cdots\!68\)\( T_{3}^{119} - \)\(16\!\cdots\!09\)\( T_{3}^{118} - \)\(81\!\cdots\!33\)\( T_{3}^{117} + \)\(23\!\cdots\!76\)\( T_{3}^{116} + \)\(33\!\cdots\!97\)\( T_{3}^{115} - \)\(15\!\cdots\!75\)\( T_{3}^{114} + \)\(98\!\cdots\!00\)\( T_{3}^{113} + \)\(77\!\cdots\!85\)\( T_{3}^{112} - \)\(15\!\cdots\!04\)\( T_{3}^{111} - \)\(12\!\cdots\!32\)\( T_{3}^{110} + \)\(78\!\cdots\!85\)\( T_{3}^{109} - \)\(14\!\cdots\!29\)\( T_{3}^{108} - \)\(37\!\cdots\!00\)\( T_{3}^{107} + \)\(99\!\cdots\!50\)\( T_{3}^{106} + \)\(96\!\cdots\!30\)\( T_{3}^{105} + \)\(11\!\cdots\!52\)\( T_{3}^{104} + \)\(68\!\cdots\!24\)\( T_{3}^{103} - \)\(32\!\cdots\!49\)\( T_{3}^{102} - \)\(68\!\cdots\!18\)\( T_{3}^{101} + \)\(10\!\cdots\!95\)\( T_{3}^{100} + \)\(13\!\cdots\!60\)\( T_{3}^{99} - \)\(77\!\cdots\!85\)\( T_{3}^{98} + \)\(13\!\cdots\!46\)\( T_{3}^{97} - \)\(83\!\cdots\!33\)\( T_{3}^{96} - \)\(11\!\cdots\!74\)\( T_{3}^{95} + \)\(54\!\cdots\!08\)\( T_{3}^{94} + \)\(44\!\cdots\!28\)\( T_{3}^{93} + \)\(13\!\cdots\!83\)\( T_{3}^{92} + \)\(46\!\cdots\!11\)\( T_{3}^{91} - \)\(85\!\cdots\!92\)\( T_{3}^{90} - \)\(10\!\cdots\!62\)\( T_{3}^{89} - \)\(89\!\cdots\!96\)\( T_{3}^{88} + \)\(34\!\cdots\!67\)\( T_{3}^{87} + \)\(66\!\cdots\!29\)\( T_{3}^{86} + \)\(74\!\cdots\!55\)\( T_{3}^{85} + \)\(73\!\cdots\!97\)\( T_{3}^{84} - \)\(41\!\cdots\!11\)\( T_{3}^{83} - \)\(79\!\cdots\!48\)\( T_{3}^{82} + \)\(21\!\cdots\!56\)\( T_{3}^{81} - \)\(19\!\cdots\!28\)\( T_{3}^{80} - \)\(19\!\cdots\!94\)\( T_{3}^{79} + \)\(10\!\cdots\!84\)\( T_{3}^{78} - \)\(68\!\cdots\!86\)\( T_{3}^{77} - \)\(36\!\cdots\!31\)\( T_{3}^{76} + \)\(39\!\cdots\!12\)\( T_{3}^{75} + \)\(11\!\cdots\!67\)\( T_{3}^{74} + \)\(15\!\cdots\!49\)\( T_{3}^{73} + \)\(16\!\cdots\!08\)\( T_{3}^{72} + \)\(93\!\cdots\!24\)\( T_{3}^{71} + \)\(10\!\cdots\!47\)\( T_{3}^{70} + \)\(25\!\cdots\!35\)\( T_{3}^{69} + \)\(76\!\cdots\!37\)\( T_{3}^{68} - \)\(38\!\cdots\!47\)\( T_{3}^{67} - \)\(15\!\cdots\!12\)\( T_{3}^{66} - \)\(70\!\cdots\!90\)\( T_{3}^{65} + \)\(10\!\cdots\!68\)\( T_{3}^{64} + \)\(32\!\cdots\!59\)\( T_{3}^{63} - \)\(45\!\cdots\!76\)\( T_{3}^{62} + \)\(33\!\cdots\!03\)\( T_{3}^{61} - \)\(53\!\cdots\!92\)\( T_{3}^{60} - \)\(45\!\cdots\!17\)\( T_{3}^{59} + \)\(36\!\cdots\!84\)\( T_{3}^{58} + \)\(83\!\cdots\!36\)\( T_{3}^{57} + \)\(12\!\cdots\!90\)\( T_{3}^{56} - \)\(95\!\cdots\!64\)\( T_{3}^{55} + \)\(27\!\cdots\!58\)\( T_{3}^{54} - \)\(50\!\cdots\!91\)\( T_{3}^{53} + \)\(74\!\cdots\!87\)\( T_{3}^{52} - \)\(15\!\cdots\!39\)\( T_{3}^{51} + \)\(60\!\cdots\!39\)\( T_{3}^{50} - \)\(12\!\cdots\!83\)\( T_{3}^{49} + \)\(56\!\cdots\!83\)\( T_{3}^{48} - \)\(11\!\cdots\!85\)\( T_{3}^{47} + \)\(18\!\cdots\!59\)\( T_{3}^{46} - \)\(11\!\cdots\!07\)\( T_{3}^{45} + \)\(16\!\cdots\!39\)\( T_{3}^{44} - \)\(44\!\cdots\!77\)\( T_{3}^{43} + \)\(25\!\cdots\!21\)\( T_{3}^{42} - \)\(98\!\cdots\!81\)\( T_{3}^{41} + \)\(12\!\cdots\!03\)\( T_{3}^{40} - \)\(12\!\cdots\!84\)\( T_{3}^{39} + \)\(15\!\cdots\!39\)\( T_{3}^{38} - \)\(10\!\cdots\!68\)\( T_{3}^{37} + \)\(22\!\cdots\!54\)\( T_{3}^{36} - \)\(53\!\cdots\!89\)\( T_{3}^{35} + \)\(25\!\cdots\!38\)\( T_{3}^{34} - \)\(14\!\cdots\!39\)\( T_{3}^{33} + \)\(71\!\cdots\!72\)\( T_{3}^{32} - \)\(29\!\cdots\!11\)\( T_{3}^{31} + \)\(19\!\cdots\!27\)\( T_{3}^{30} - \)\(24\!\cdots\!93\)\( T_{3}^{29} + \)\(89\!\cdots\!52\)\( T_{3}^{28} + \)\(26\!\cdots\!71\)\( T_{3}^{27} + \)\(15\!\cdots\!32\)\( T_{3}^{26} - \)\(13\!\cdots\!17\)\( T_{3}^{25} - \)\(58\!\cdots\!35\)\( T_{3}^{24} - \)\(48\!\cdots\!40\)\( T_{3}^{23} - \)\(42\!\cdots\!65\)\( T_{3}^{22} + \)\(29\!\cdots\!78\)\( T_{3}^{21} + \)\(18\!\cdots\!17\)\( T_{3}^{20} + \)\(26\!\cdots\!00\)\( T_{3}^{19} + \)\(53\!\cdots\!78\)\( T_{3}^{18} - \)\(31\!\cdots\!33\)\( T_{3}^{17} + \)\(17\!\cdots\!13\)\( T_{3}^{16} - \)\(10\!\cdots\!10\)\( T_{3}^{15} - \)\(59\!\cdots\!70\)\( T_{3}^{14} + \)\(48\!\cdots\!87\)\( T_{3}^{13} + \)\(47\!\cdots\!91\)\( T_{3}^{12} - \)\(19\!\cdots\!13\)\( T_{3}^{11} - \)\(13\!\cdots\!18\)\( T_{3}^{10} + \)\(25\!\cdots\!12\)\( T_{3}^{9} + \)\(28\!\cdots\!27\)\( T_{3}^{8} - \)\(15\!\cdots\!54\)\( T_{3}^{7} - \)\(23\!\cdots\!14\)\( T_{3}^{6} - \)\(50\!\cdots\!12\)\( T_{3}^{5} + \)\(18\!\cdots\!50\)\( T_{3}^{4} + \)\(11\!\cdots\!93\)\( T_{3}^{3} - \)\(38\!\cdots\!88\)\( T_{3}^{2} + \)\(38\!\cdots\!23\)\( T_{3} + \)\(35\!\cdots\!41\)\( \)">\(T_{3}^{160} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).