Properties

Label 462.2.bc.b
Level $462$
Weight $2$
Character orbit 462.bc
Analytic conductor $3.689$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.bc (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128q + 16q^{2} + 16q^{4} - 32q^{8} + 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 16q^{2} + 16q^{4} - 32q^{8} + 16q^{9} - 6q^{11} - 12q^{15} + 16q^{16} - 2q^{17} - 4q^{18} + 2q^{22} - 12q^{25} - 18q^{27} - 5q^{28} + 38q^{29} + 6q^{30} - 3q^{31} - 64q^{32} + 28q^{33} - 16q^{34} - 31q^{35} + 8q^{36} + 2q^{37} - 2q^{39} + 5q^{40} + 16q^{41} - 13q^{42} - q^{44} + 28q^{45} + 38q^{49} + 34q^{50} + 4q^{51} + 25q^{53} - 6q^{54} - 42q^{55} - 100q^{57} - 19q^{58} + 40q^{59} - 4q^{60} + 40q^{61} - 4q^{62} - 106q^{63} - 32q^{64} + 20q^{65} - 7q^{66} + 16q^{67} - 2q^{68} - 68q^{69} - 21q^{70} + 80q^{71} - 4q^{72} + 10q^{73} + 2q^{74} - 14q^{75} + q^{77} - 16q^{78} - 5q^{80} + 32q^{81} - 8q^{82} - 92q^{83} + 8q^{84} - 100q^{85} - 40q^{86} - 38q^{87} - q^{88} + 4q^{90} + 12q^{91} - 20q^{92} - 33q^{93} + 40q^{94} + 38q^{95} - 16q^{97} + 18q^{98} + 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 −0.104528 + 0.994522i −1.72793 0.119419i −0.978148 0.207912i −1.30191 + 2.92414i 0.299383 1.70598i −2.04556 + 1.67800i 0.309017 0.951057i 2.97148 + 0.412695i −2.77204 1.60044i
95.2 −0.104528 + 0.994522i −1.71803 + 0.219927i −0.978148 0.207912i 1.53261 3.44230i −0.0391388 1.73161i −2.01818 + 1.71083i 0.309017 0.951057i 2.90326 0.755682i 3.26324 + 1.88403i
95.3 −0.104528 + 0.994522i −1.60606 + 0.648515i −0.978148 0.207912i −0.565716 + 1.27062i −0.477084 1.66505i 0.339893 2.62383i 0.309017 0.951057i 2.15886 2.08311i −1.20453 0.695433i
95.4 −0.104528 + 0.994522i −1.29320 + 1.15223i −0.978148 0.207912i 0.572012 1.28476i −1.01074 1.40656i 2.51605 + 0.818231i 0.309017 0.951057i 0.344742 2.98013i 1.21793 + 0.703172i
95.5 −0.104528 + 0.994522i −1.13004 1.31264i −0.978148 0.207912i 0.157991 0.354854i 1.42357 0.986639i 0.106147 + 2.64362i 0.309017 0.951057i −0.446029 + 2.96666i 0.336396 + 0.194218i
95.6 −0.104528 + 0.994522i −0.840980 1.51418i −0.978148 0.207912i 1.12093 2.51765i 1.59380 0.678098i 2.40939 1.09308i 0.309017 0.951057i −1.58551 + 2.54680i 2.38669 + 1.37795i
95.7 −0.104528 + 0.994522i −0.790611 1.54108i −0.978148 0.207912i −0.625478 + 1.40485i 1.61528 0.625193i −1.00131 2.44895i 0.309017 0.951057i −1.74987 + 2.43679i −1.33177 0.768898i
95.8 −0.104528 + 0.994522i 0.110709 + 1.72851i −0.978148 0.207912i −1.75629 + 3.94468i −1.73061 0.0705756i 2.64495 + 0.0650804i 0.309017 0.951057i −2.97549 + 0.382724i −3.73949 2.15900i
95.9 −0.104528 + 0.994522i 0.257732 + 1.71277i −0.978148 0.207912i −0.789465 + 1.77317i −1.73033 + 0.0772875i −2.28945 + 1.32606i 0.309017 0.951057i −2.86715 + 0.882871i −1.68093 0.970486i
95.10 −0.104528 + 0.994522i 0.713675 1.57818i −0.978148 0.207912i −1.25353 + 2.81547i 1.49494 + 0.874731i 1.28350 + 2.31358i 0.309017 0.951057i −1.98134 2.25262i −2.66902 1.54096i
95.11 −0.104528 + 0.994522i 0.752814 1.55989i −0.978148 0.207912i −0.112013 + 0.251585i 1.47266 + 0.911743i −2.57741 0.597448i 0.309017 0.951057i −1.86654 2.34862i −0.238498 0.137697i
95.12 −0.104528 + 0.994522i 0.882151 + 1.49057i −0.978148 0.207912i 1.35789 3.04988i −1.57462 + 0.721511i 1.07443 2.41777i 0.309017 0.951057i −1.44362 + 2.62982i 2.89123 + 1.66925i
95.13 −0.104528 + 0.994522i 1.38347 + 1.04212i −0.978148 0.207912i 0.333330 0.748672i −1.18102 + 1.26696i −2.01015 + 1.72026i 0.309017 0.951057i 0.827988 + 2.88348i 0.709728 + 0.409762i
95.14 −0.104528 + 0.994522i 1.65546 + 0.509351i −0.978148 0.207912i −0.0756668 + 0.169950i −0.679604 + 1.59315i 1.63703 + 2.07849i 0.309017 0.951057i 2.48112 + 1.68643i −0.161110 0.0930169i
95.15 −0.104528 + 0.994522i 1.67530 0.439734i −0.978148 0.207912i 1.31655 2.95701i 0.262209 + 1.71209i −1.86309 1.87854i 0.309017 0.951057i 2.61327 1.47338i 2.80319 + 1.61843i
95.16 −0.104528 + 0.994522i 1.67553 0.438853i −0.978148 0.207912i −0.684905 + 1.53832i 0.261308 + 1.71223i 1.54934 2.14465i 0.309017 0.951057i 2.61482 1.47062i −1.45830 0.841952i
107.1 −0.104528 0.994522i −1.72793 + 0.119419i −0.978148 + 0.207912i −1.30191 2.92414i 0.299383 + 1.70598i −2.04556 1.67800i 0.309017 + 0.951057i 2.97148 0.412695i −2.77204 + 1.60044i
107.2 −0.104528 0.994522i −1.71803 0.219927i −0.978148 + 0.207912i 1.53261 + 3.44230i −0.0391388 + 1.73161i −2.01818 1.71083i 0.309017 + 0.951057i 2.90326 + 0.755682i 3.26324 1.88403i
107.3 −0.104528 0.994522i −1.60606 0.648515i −0.978148 + 0.207912i −0.565716 1.27062i −0.477084 + 1.66505i 0.339893 + 2.62383i 0.309017 + 0.951057i 2.15886 + 2.08311i −1.20453 + 0.695433i
107.4 −0.104528 0.994522i −1.29320 1.15223i −0.978148 + 0.207912i 0.572012 + 1.28476i −1.01074 + 1.40656i 2.51605 0.818231i 0.309017 + 0.951057i 0.344742 + 2.98013i 1.21793 0.703172i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.f even 10 1 inner
231.be even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.bc.b yes 128
3.b odd 2 1 462.2.bc.a 128
7.c even 3 1 inner 462.2.bc.b yes 128
11.d odd 10 1 462.2.bc.a 128
21.h odd 6 1 462.2.bc.a 128
33.f even 10 1 inner 462.2.bc.b yes 128
77.o odd 30 1 462.2.bc.a 128
231.be even 30 1 inner 462.2.bc.b yes 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bc.a 128 3.b odd 2 1
462.2.bc.a 128 11.d odd 10 1
462.2.bc.a 128 21.h odd 6 1
462.2.bc.a 128 77.o odd 30 1
462.2.bc.b yes 128 1.a even 1 1 trivial
462.2.bc.b yes 128 7.c even 3 1 inner
462.2.bc.b yes 128 33.f even 10 1 inner
462.2.bc.b yes 128 231.be even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(15\!\cdots\!00\)\( T_{5}^{107} - \)\(35\!\cdots\!25\)\( T_{5}^{106} - \)\(13\!\cdots\!10\)\( T_{5}^{105} - \)\(58\!\cdots\!78\)\( T_{5}^{104} + \)\(48\!\cdots\!90\)\( T_{5}^{103} - \)\(40\!\cdots\!78\)\( T_{5}^{102} + \)\(25\!\cdots\!85\)\( T_{5}^{101} - \)\(27\!\cdots\!30\)\( T_{5}^{100} + \)\(27\!\cdots\!15\)\( T_{5}^{99} + \)\(28\!\cdots\!29\)\( T_{5}^{98} + \)\(66\!\cdots\!15\)\( T_{5}^{97} + \)\(48\!\cdots\!86\)\( T_{5}^{96} - \)\(20\!\cdots\!00\)\( T_{5}^{95} + \)\(56\!\cdots\!69\)\( T_{5}^{94} - \)\(32\!\cdots\!15\)\( T_{5}^{93} + \)\(42\!\cdots\!52\)\( T_{5}^{92} - \)\(23\!\cdots\!65\)\( T_{5}^{91} + \)\(49\!\cdots\!01\)\( T_{5}^{90} - \)\(95\!\cdots\!70\)\( T_{5}^{89} - \)\(38\!\cdots\!97\)\( T_{5}^{88} + \)\(19\!\cdots\!45\)\( T_{5}^{87} - \)\(62\!\cdots\!19\)\( T_{5}^{86} + \)\(24\!\cdots\!60\)\( T_{5}^{85} - \)\(53\!\cdots\!21\)\( T_{5}^{84} + \)\(15\!\cdots\!75\)\( T_{5}^{83} - \)\(18\!\cdots\!71\)\( T_{5}^{82} + \)\(90\!\cdots\!10\)\( T_{5}^{81} + \)\(19\!\cdots\!36\)\( T_{5}^{80} - \)\(98\!\cdots\!65\)\( T_{5}^{79} + \)\(38\!\cdots\!63\)\( T_{5}^{78} - \)\(12\!\cdots\!10\)\( T_{5}^{77} + \)\(33\!\cdots\!44\)\( T_{5}^{76} - \)\(81\!\cdots\!05\)\( T_{5}^{75} + \)\(15\!\cdots\!88\)\( T_{5}^{74} - \)\(19\!\cdots\!65\)\( T_{5}^{73} - \)\(16\!\cdots\!32\)\( T_{5}^{72} + \)\(21\!\cdots\!55\)\( T_{5}^{71} - \)\(96\!\cdots\!83\)\( T_{5}^{70} + \)\(32\!\cdots\!80\)\( T_{5}^{69} - \)\(91\!\cdots\!02\)\( T_{5}^{68} + \)\(22\!\cdots\!35\)\( T_{5}^{67} - \)\(48\!\cdots\!68\)\( T_{5}^{66} + \)\(83\!\cdots\!20\)\( T_{5}^{65} - \)\(89\!\cdots\!42\)\( T_{5}^{64} - \)\(98\!\cdots\!05\)\( T_{5}^{63} + \)\(10\!\cdots\!03\)\( T_{5}^{62} - \)\(40\!\cdots\!55\)\( T_{5}^{61} + \)\(12\!\cdots\!33\)\( T_{5}^{60} - \)\(33\!\cdots\!70\)\( T_{5}^{59} + \)\(82\!\cdots\!63\)\( T_{5}^{58} - \)\(18\!\cdots\!60\)\( T_{5}^{57} + \)\(38\!\cdots\!58\)\( T_{5}^{56} - \)\(74\!\cdots\!25\)\( T_{5}^{55} + \)\(13\!\cdots\!53\)\( T_{5}^{54} - \)\(22\!\cdots\!70\)\( T_{5}^{53} + \)\(35\!\cdots\!18\)\( T_{5}^{52} - \)\(48\!\cdots\!60\)\( T_{5}^{51} + \)\(58\!\cdots\!98\)\( T_{5}^{50} - \)\(53\!\cdots\!65\)\( T_{5}^{49} + \)\(14\!\cdots\!78\)\( T_{5}^{48} + \)\(86\!\cdots\!80\)\( T_{5}^{47} - \)\(27\!\cdots\!54\)\( T_{5}^{46} + \)\(57\!\cdots\!25\)\( T_{5}^{45} - \)\(96\!\cdots\!74\)\( T_{5}^{44} + \)\(13\!\cdots\!15\)\( T_{5}^{43} - \)\(15\!\cdots\!11\)\( T_{5}^{42} + \)\(13\!\cdots\!20\)\( T_{5}^{41} - \)\(43\!\cdots\!31\)\( T_{5}^{40} - \)\(15\!\cdots\!20\)\( T_{5}^{39} + \)\(45\!\cdots\!89\)\( T_{5}^{38} - \)\(83\!\cdots\!00\)\( T_{5}^{37} + \)\(12\!\cdots\!39\)\( T_{5}^{36} - \)\(15\!\cdots\!70\)\( T_{5}^{35} + \)\(15\!\cdots\!02\)\( T_{5}^{34} - \)\(13\!\cdots\!95\)\( T_{5}^{33} + \)\(96\!\cdots\!67\)\( T_{5}^{32} - \)\(39\!\cdots\!15\)\( T_{5}^{31} - \)\(17\!\cdots\!98\)\( T_{5}^{30} + \)\(58\!\cdots\!70\)\( T_{5}^{29} - \)\(73\!\cdots\!37\)\( T_{5}^{28} + \)\(66\!\cdots\!70\)\( T_{5}^{27} - \)\(46\!\cdots\!67\)\( T_{5}^{26} + \)\(23\!\cdots\!25\)\( T_{5}^{25} - \)\(64\!\cdots\!24\)\( T_{5}^{24} - \)\(31\!\cdots\!05\)\( T_{5}^{23} + \)\(74\!\cdots\!06\)\( T_{5}^{22} - \)\(83\!\cdots\!80\)\( T_{5}^{21} + \)\(79\!\cdots\!66\)\( T_{5}^{20} - \)\(64\!\cdots\!15\)\( T_{5}^{19} + \)\(46\!\cdots\!98\)\( T_{5}^{18} - \)\(28\!\cdots\!60\)\( T_{5}^{17} + \)\(13\!\cdots\!26\)\( T_{5}^{16} - \)\(44\!\cdots\!10\)\( T_{5}^{15} + \)\(33\!\cdots\!36\)\( T_{5}^{14} + \)\(76\!\cdots\!75\)\( T_{5}^{13} - \)\(57\!\cdots\!97\)\( T_{5}^{12} + \)\(24\!\cdots\!65\)\( T_{5}^{11} - \)\(58\!\cdots\!80\)\( T_{5}^{10} + \)\(29\!\cdots\!15\)\( T_{5}^{9} + \)\(40\!\cdots\!60\)\( T_{5}^{8} - \)\(23\!\cdots\!70\)\( T_{5}^{7} + \)\(87\!\cdots\!14\)\( T_{5}^{6} - \)\(28\!\cdots\!50\)\( T_{5}^{5} + \)\(68\!\cdots\!35\)\( T_{5}^{4} - \)\(10\!\cdots\!40\)\( T_{5}^{3} + \)\(17\!\cdots\!45\)\( T_{5}^{2} - \)\(28\!\cdots\!65\)\( T_{5} + \)\(21\!\cdots\!81\)\( \)">\(T_{5}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).