Properties

Label 460.2.o.b
Level $460$
Weight $2$
Character orbit 460.o
Analytic conductor $3.673$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 460.o (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.67311849298\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(64\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640q - 6q^{4} - 22q^{5} + 2q^{6} - 100q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 640q - 6q^{4} - 22q^{5} + 2q^{6} - 100q^{9} - 11q^{10} - 22q^{14} + 2q^{16} - 11q^{20} - 44q^{21} - 8q^{24} - 38q^{25} - 46q^{26} - 52q^{29} - 11q^{30} - 44q^{34} + 18q^{36} - 11q^{40} - 76q^{41} - 154q^{44} - 30q^{46} - 100q^{49} + 21q^{50} + 178q^{54} + 242q^{56} - 11q^{60} - 44q^{61} + 6q^{64} - 22q^{65} + 44q^{66} - 12q^{69} + 22q^{70} - 22q^{74} + 88q^{76} - 110q^{80} - 252q^{81} - 374q^{84} + 94q^{85} - 198q^{86} - 44q^{89} - 242q^{90} - 62q^{94} + 372q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41365 0.0397971i 2.67003 0.783991i 1.99683 + 0.112519i 1.97695 + 1.04482i −3.80569 + 1.00203i 0.790909 0.113716i −2.81835 0.238531i 3.99064 2.56463i −2.75315 1.55569i
19.2 −1.41287 + 0.0615925i 2.08651 0.612653i 1.99241 0.174045i −0.546640 2.16822i −2.91023 + 0.994114i −1.11477 + 0.160279i −2.80430 + 0.368620i 1.45440 0.934688i 0.905878 + 3.02975i
19.3 −1.41242 0.0712068i −1.44577 + 0.424517i 1.98986 + 0.201148i −0.632143 2.14485i 2.07226 0.496647i 4.69086 0.674443i −2.79619 0.425797i −0.613721 + 0.394415i 0.740123 + 3.07445i
19.4 −1.40349 + 0.173829i −0.723511 + 0.212442i 1.93957 0.487934i 1.78262 1.34991i 0.978512 0.423927i −1.71228 + 0.246189i −2.63735 + 1.02196i −2.04542 + 1.31451i −2.26724 + 2.20446i
19.5 −1.40029 + 0.197940i 1.48939 0.437323i 1.92164 0.554348i −2.20194 + 0.389205i −1.99901 + 0.907190i −0.230848 + 0.0331909i −2.58113 + 1.15662i −0.496740 + 0.319236i 3.00632 0.980851i
19.6 −1.39440 + 0.235922i −2.94595 + 0.865009i 1.88868 0.657939i −1.40285 + 1.74127i 3.90375 1.90118i 1.84920 0.265875i −2.47835 + 1.36301i 5.40662 3.47462i 1.54533 2.75898i
19.7 −1.34217 0.445613i −0.610817 + 0.179352i 1.60286 + 1.19618i −1.58117 + 1.58110i 0.899744 + 0.0314662i 0.165059 0.0237318i −1.61828 2.31973i −2.18283 + 1.40282i 2.82677 1.41752i
19.8 −1.33042 + 0.479557i −0.820997 + 0.241066i 1.54005 1.27603i 0.912951 + 2.04121i 0.976668 0.714435i −3.61150 + 0.519256i −1.43699 + 2.43620i −1.90784 + 1.22609i −2.19348 2.27786i
19.9 −1.27975 0.601871i −2.51556 + 0.738636i 1.27550 + 1.54049i 2.08492 + 0.808149i 3.66385 + 0.568779i −0.0121874 + 0.00175228i −0.705145 2.73912i 3.25872 2.09425i −2.18177 2.28908i
19.10 −1.27640 0.608942i 1.81640 0.533344i 1.25838 + 1.55450i 0.889872 + 2.05137i −2.64323 0.425324i −4.44634 + 0.639287i −0.659592 2.75044i 0.491100 0.315611i 0.113335 3.16025i
19.11 −1.23367 + 0.691424i 0.820997 0.241066i 1.04387 1.70597i 0.912951 + 2.04121i −0.846157 + 0.865052i 3.61150 0.519256i −0.108234 + 2.82636i −1.90784 + 1.22609i −2.53761 1.88693i
19.12 −1.21441 0.724711i −0.381027 + 0.111880i 0.949588 + 1.76019i −1.73128 1.41516i 0.543804 + 0.140267i −2.86375 + 0.411745i 0.122443 2.82578i −2.39110 + 1.53666i 1.07690 + 2.97326i
19.13 −1.19551 0.755479i 1.52020 0.446372i 0.858502 + 1.80637i 1.91991 1.14628i −2.15465 0.614838i 4.13344 0.594300i 0.338324 2.80812i −0.411991 + 0.264771i −3.16126 0.0800530i
19.14 −1.09143 + 0.899318i 2.94595 0.865009i 0.382455 1.96309i −1.40285 + 1.74127i −2.43739 + 3.59345i −1.84920 + 0.265875i 1.34802 + 2.48653i 5.40662 3.47462i −0.0348340 3.16209i
19.15 −1.09031 0.900680i 2.64511 0.776673i 0.377553 + 1.96404i −2.22046 + 0.263774i −3.58352 1.53558i 2.67954 0.385260i 1.35732 2.48147i 3.86960 2.48684i 2.65856 + 1.71232i
19.16 −1.06659 + 0.928648i −1.48939 + 0.437323i 0.275227 1.98097i −2.20194 + 0.389205i 1.18245 1.84956i 0.230848 0.0331909i 1.54607 + 2.36847i −0.496740 + 0.319236i 1.98713 2.45994i
19.17 −1.05046 + 0.946853i 0.723511 0.212442i 0.206938 1.98927i 1.78262 1.34991i −0.558869 + 0.908221i 1.71228 0.246189i 1.66616 + 2.28559i −2.04542 + 1.31451i −0.594407 + 3.10591i
19.18 −0.971783 + 1.02744i −2.08651 + 0.612653i −0.111277 1.99690i −0.546640 2.16822i 1.39816 2.73913i 1.11477 0.160279i 2.15984 + 1.82622i 1.45440 0.934688i 2.75894 + 1.54540i
19.19 −0.925935 1.06895i −0.832655 + 0.244489i −0.285289 + 1.97955i 0.527840 + 2.17287i 1.03233 + 0.663681i 3.68197 0.529388i 2.38019 1.52797i −1.89022 + 1.21477i 1.83394 2.57617i
19.20 −0.921514 1.07276i −2.25339 + 0.661656i −0.301623 + 1.97713i 0.592624 2.15611i 2.78633 + 1.80762i −0.146802 + 0.0211070i 2.39893 1.49838i 2.11623 1.36002i −2.85909 + 1.35114i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner
115.i odd 22 1 inner
460.o even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.o.b 640
4.b odd 2 1 inner 460.2.o.b 640
5.b even 2 1 inner 460.2.o.b 640
20.d odd 2 1 inner 460.2.o.b 640
23.d odd 22 1 inner 460.2.o.b 640
92.h even 22 1 inner 460.2.o.b 640
115.i odd 22 1 inner 460.2.o.b 640
460.o even 22 1 inner 460.2.o.b 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.o.b 640 1.a even 1 1 trivial
460.2.o.b 640 4.b odd 2 1 inner
460.2.o.b 640 5.b even 2 1 inner
460.2.o.b 640 20.d odd 2 1 inner
460.2.o.b 640 23.d odd 22 1 inner
460.2.o.b 640 92.h even 22 1 inner
460.2.o.b 640 115.i odd 22 1 inner
460.2.o.b 640 460.o even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(19\!\cdots\!09\)\( T_{3}^{302} + \)\(24\!\cdots\!17\)\( T_{3}^{300} + \)\(29\!\cdots\!78\)\( T_{3}^{298} + \)\(34\!\cdots\!37\)\( T_{3}^{296} + \)\(39\!\cdots\!43\)\( T_{3}^{294} + \)\(42\!\cdots\!60\)\( T_{3}^{292} + \)\(45\!\cdots\!88\)\( T_{3}^{290} + \)\(47\!\cdots\!67\)\( T_{3}^{288} + \)\(47\!\cdots\!96\)\( T_{3}^{286} + \)\(46\!\cdots\!32\)\( T_{3}^{284} + \)\(44\!\cdots\!35\)\( T_{3}^{282} + \)\(41\!\cdots\!23\)\( T_{3}^{280} + \)\(38\!\cdots\!88\)\( T_{3}^{278} + \)\(34\!\cdots\!24\)\( T_{3}^{276} + \)\(29\!\cdots\!50\)\( T_{3}^{274} + \)\(25\!\cdots\!09\)\( T_{3}^{272} + \)\(21\!\cdots\!20\)\( T_{3}^{270} + \)\(17\!\cdots\!57\)\( T_{3}^{268} + \)\(14\!\cdots\!15\)\( T_{3}^{266} + \)\(11\!\cdots\!83\)\( T_{3}^{264} + \)\(89\!\cdots\!51\)\( T_{3}^{262} + \)\(68\!\cdots\!58\)\( T_{3}^{260} + \)\(51\!\cdots\!12\)\( T_{3}^{258} + \)\(38\!\cdots\!98\)\( T_{3}^{256} + \)\(27\!\cdots\!45\)\( T_{3}^{254} + \)\(19\!\cdots\!95\)\( T_{3}^{252} + \)\(13\!\cdots\!58\)\( T_{3}^{250} + \)\(94\!\cdots\!75\)\( T_{3}^{248} + \)\(63\!\cdots\!03\)\( T_{3}^{246} + \)\(42\!\cdots\!20\)\( T_{3}^{244} + \)\(27\!\cdots\!53\)\( T_{3}^{242} + \)\(17\!\cdots\!66\)\( T_{3}^{240} + \)\(10\!\cdots\!38\)\( T_{3}^{238} + \)\(67\!\cdots\!71\)\( T_{3}^{236} + \)\(40\!\cdots\!93\)\( T_{3}^{234} + \)\(24\!\cdots\!89\)\( T_{3}^{232} + \)\(13\!\cdots\!13\)\( T_{3}^{230} + \)\(79\!\cdots\!95\)\( T_{3}^{228} + \)\(44\!\cdots\!77\)\( T_{3}^{226} + \)\(24\!\cdots\!72\)\( T_{3}^{224} + \)\(12\!\cdots\!33\)\( T_{3}^{222} + \)\(68\!\cdots\!69\)\( T_{3}^{220} + \)\(35\!\cdots\!40\)\( T_{3}^{218} + \)\(17\!\cdots\!53\)\( T_{3}^{216} + \)\(88\!\cdots\!35\)\( T_{3}^{214} + \)\(42\!\cdots\!02\)\( T_{3}^{212} + \)\(20\!\cdots\!77\)\( T_{3}^{210} + \)\(94\!\cdots\!39\)\( T_{3}^{208} + \)\(42\!\cdots\!78\)\( T_{3}^{206} + \)\(19\!\cdots\!72\)\( T_{3}^{204} + \)\(83\!\cdots\!13\)\( T_{3}^{202} + \)\(35\!\cdots\!47\)\( T_{3}^{200} + \)\(14\!\cdots\!58\)\( T_{3}^{198} + \)\(60\!\cdots\!24\)\( T_{3}^{196} + \)\(23\!\cdots\!99\)\( T_{3}^{194} + \)\(92\!\cdots\!73\)\( T_{3}^{192} + \)\(35\!\cdots\!42\)\( T_{3}^{190} + \)\(13\!\cdots\!45\)\( T_{3}^{188} + \)\(47\!\cdots\!37\)\( T_{3}^{186} + \)\(16\!\cdots\!04\)\( T_{3}^{184} + \)\(58\!\cdots\!64\)\( T_{3}^{182} + \)\(19\!\cdots\!60\)\( T_{3}^{180} + \)\(65\!\cdots\!59\)\( T_{3}^{178} + \)\(20\!\cdots\!58\)\( T_{3}^{176} + \)\(66\!\cdots\!75\)\( T_{3}^{174} + \)\(20\!\cdots\!10\)\( T_{3}^{172} + \)\(61\!\cdots\!11\)\( T_{3}^{170} + \)\(17\!\cdots\!25\)\( T_{3}^{168} + \)\(51\!\cdots\!41\)\( T_{3}^{166} + \)\(14\!\cdots\!60\)\( T_{3}^{164} + \)\(39\!\cdots\!37\)\( T_{3}^{162} + \)\(10\!\cdots\!23\)\( T_{3}^{160} + \)\(28\!\cdots\!70\)\( T_{3}^{158} + \)\(72\!\cdots\!48\)\( T_{3}^{156} + \)\(18\!\cdots\!31\)\( T_{3}^{154} + \)\(45\!\cdots\!65\)\( T_{3}^{152} + \)\(11\!\cdots\!39\)\( T_{3}^{150} + \)\(26\!\cdots\!28\)\( T_{3}^{148} + \)\(60\!\cdots\!90\)\( T_{3}^{146} + \)\(13\!\cdots\!92\)\( T_{3}^{144} + \)\(30\!\cdots\!14\)\( T_{3}^{142} + \)\(66\!\cdots\!97\)\( T_{3}^{140} + \)\(14\!\cdots\!42\)\( T_{3}^{138} + \)\(29\!\cdots\!21\)\( T_{3}^{136} + \)\(58\!\cdots\!95\)\( T_{3}^{134} + \)\(11\!\cdots\!77\)\( T_{3}^{132} + \)\(22\!\cdots\!60\)\( T_{3}^{130} + \)\(40\!\cdots\!16\)\( T_{3}^{128} + \)\(73\!\cdots\!57\)\( T_{3}^{126} + \)\(12\!\cdots\!20\)\( T_{3}^{124} + \)\(21\!\cdots\!64\)\( T_{3}^{122} + \)\(33\!\cdots\!09\)\( T_{3}^{120} + \)\(50\!\cdots\!93\)\( T_{3}^{118} + \)\(71\!\cdots\!03\)\( T_{3}^{116} + \)\(92\!\cdots\!27\)\( T_{3}^{114} + \)\(10\!\cdots\!39\)\( T_{3}^{112} + \)\(11\!\cdots\!74\)\( T_{3}^{110} + \)\(11\!\cdots\!65\)\( T_{3}^{108} + \)\(97\!\cdots\!09\)\( T_{3}^{106} + \)\(81\!\cdots\!56\)\( T_{3}^{104} + \)\(67\!\cdots\!82\)\( T_{3}^{102} + \)\(58\!\cdots\!08\)\( T_{3}^{100} + \)\(50\!\cdots\!37\)\( T_{3}^{98} + \)\(41\!\cdots\!79\)\( T_{3}^{96} + \)\(30\!\cdots\!18\)\( T_{3}^{94} + \)\(20\!\cdots\!52\)\( T_{3}^{92} + \)\(13\!\cdots\!39\)\( T_{3}^{90} + \)\(94\!\cdots\!44\)\( T_{3}^{88} + \)\(70\!\cdots\!64\)\( T_{3}^{86} + \)\(49\!\cdots\!30\)\( T_{3}^{84} + \)\(31\!\cdots\!01\)\( T_{3}^{82} + \)\(17\!\cdots\!44\)\( T_{3}^{80} + \)\(90\!\cdots\!47\)\( T_{3}^{78} + \)\(52\!\cdots\!36\)\( T_{3}^{76} + \)\(31\!\cdots\!40\)\( T_{3}^{74} + \)\(18\!\cdots\!37\)\( T_{3}^{72} + \)\(88\!\cdots\!98\)\( T_{3}^{70} + \)\(44\!\cdots\!62\)\( T_{3}^{68} + \)\(20\!\cdots\!06\)\( T_{3}^{66} + \)\(11\!\cdots\!07\)\( T_{3}^{64} + \)\(36\!\cdots\!41\)\( T_{3}^{62} + \)\(16\!\cdots\!31\)\( T_{3}^{60} + \)\(94\!\cdots\!78\)\( T_{3}^{58} + \)\(25\!\cdots\!97\)\( T_{3}^{56} + \)\(17\!\cdots\!81\)\( T_{3}^{54} + \)\(60\!\cdots\!93\)\( T_{3}^{52} - \)\(43\!\cdots\!87\)\( T_{3}^{50} + \)\(12\!\cdots\!14\)\( T_{3}^{48} + \)\(32\!\cdots\!46\)\( T_{3}^{46} - \)\(59\!\cdots\!32\)\( T_{3}^{44} + \)\(49\!\cdots\!40\)\( T_{3}^{42} + \)\(14\!\cdots\!80\)\( T_{3}^{40} - \)\(97\!\cdots\!50\)\( T_{3}^{38} + \)\(86\!\cdots\!14\)\( T_{3}^{36} - \)\(70\!\cdots\!03\)\( T_{3}^{34} + \)\(23\!\cdots\!72\)\( T_{3}^{32} - \)\(10\!\cdots\!68\)\( T_{3}^{30} + \)\(63\!\cdots\!04\)\( T_{3}^{28} - \)\(21\!\cdots\!72\)\( T_{3}^{26} + \)\(10\!\cdots\!01\)\( T_{3}^{24} + \)\(26\!\cdots\!05\)\( T_{3}^{22} + \)\(17\!\cdots\!85\)\( T_{3}^{20} + \)\(79\!\cdots\!16\)\( T_{3}^{18} + \)\(14\!\cdots\!68\)\( T_{3}^{16} + \)\(42\!\cdots\!96\)\( T_{3}^{14} + \)\(29\!\cdots\!44\)\( T_{3}^{12} + \)\(47\!\cdots\!32\)\( T_{3}^{10} + \)\(88\!\cdots\!96\)\( T_{3}^{8} + \)\(18\!\cdots\!84\)\( T_{3}^{6} + \)\(24\!\cdots\!52\)\( T_{3}^{4} + \)\(18\!\cdots\!80\)\( T_{3}^{2} + \)\(58\!\cdots\!04\)\( \)">\(T_{3}^{320} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).