Properties

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

Related objects

Downloads

Learn more

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.

\(\operatorname{Tr}(f)(q) = \) \( 128q - 16q^{2} + 16q^{4} + 32q^{8} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q - 16q^{2} + 16q^{4} + 32q^{8} - 4q^{9} + 6q^{11} - 12q^{15} + 16q^{16} + 2q^{17} + 4q^{18} + 2q^{22} - 12q^{25} - 18q^{27} - 5q^{28} - 38q^{29} - 6q^{30} - 3q^{31} + 64q^{32} - 4q^{33} - 16q^{34} + 31q^{35} + 8q^{36} + 2q^{37} + 22q^{39} + 5q^{40} - 16q^{41} + 17q^{42} + q^{44} + 28q^{45} + 38q^{49} - 34q^{50} + 16q^{51} - 25q^{53} + 6q^{54} - 42q^{55} + 20q^{57} - 19q^{58} - 40q^{59} - 4q^{60} + 40q^{61} + 4q^{62} + 6q^{63} - 32q^{64} - 20q^{65} - 41q^{66} + 16q^{67} + 2q^{68} - 68q^{69} - 21q^{70} - 80q^{71} - 16q^{72} + 10q^{73} - 2q^{74} - 14q^{75} - q^{77} - 16q^{78} + 5q^{80} - 88q^{81} - 8q^{82} + 92q^{83} - 48q^{84} - 100q^{85} + 40q^{86} + 38q^{87} - q^{88} - 164q^{90} + 12q^{91} + 20q^{92} + 47q^{93} + 40q^{94} - 38q^{95} - 16q^{97} - 18q^{98} - 138q^{99} + O(q^{100}) \)

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 −1.64414 + 0.544800i −0.978148 0.207912i 1.25353 2.81547i 0.369957 + 1.69208i 1.28350 + 2.31358i −0.309017 + 0.951057i 2.40638 1.79146i −2.66902 1.54096i
95.2 0.104528 0.994522i −1.63004 + 0.585637i −0.978148 0.207912i 0.112013 0.251585i 0.412043 + 1.68233i −2.57741 0.597448i −0.309017 + 0.951057i 2.31406 1.90922i −0.238498 0.137697i
95.3 0.104528 0.994522i −1.45000 0.947367i −0.978148 0.207912i 0.625478 1.40485i −1.09374 + 1.34303i −1.00131 2.44895i −0.309017 + 0.951057i 1.20499 + 2.74736i −1.33177 0.768898i
95.4 0.104528 0.994522i −1.41798 0.994648i −0.978148 0.207912i −1.12093 + 2.51765i −1.13742 + 1.30625i 2.40939 1.09308i −0.309017 + 0.951057i 1.02135 + 2.82079i 2.38669 + 1.37795i
95.5 0.104528 0.994522i −1.18732 1.26106i −0.978148 0.207912i −0.157991 + 0.354854i −1.37826 + 1.04900i 0.106147 + 2.64362i −0.309017 + 0.951057i −0.180521 + 2.99456i 0.336396 + 0.194218i
95.6 0.104528 0.994522i −0.612442 + 1.62016i −0.978148 0.207912i −1.31655 + 2.95701i 1.54727 + 0.778440i −1.86309 1.87854i −0.309017 + 0.951057i −2.24983 1.98451i 2.80319 + 1.61843i
95.7 0.104528 0.994522i −0.611589 + 1.62048i −0.978148 0.207912i 0.684905 1.53832i 1.54768 + 0.777625i 1.54934 2.14465i −0.309017 + 0.951057i −2.25192 1.98214i −1.45830 0.841952i
95.8 0.104528 0.994522i 0.0618530 1.73095i −0.978148 0.207912i 1.30191 2.92414i −1.71500 0.242447i −2.04556 + 1.67800i −0.309017 + 0.951057i −2.99235 0.214128i −2.77204 1.60044i
95.9 0.104528 0.994522i 0.333518 + 1.69964i −0.978148 0.207912i 0.0756668 0.169950i 1.72519 0.154030i 1.63703 + 2.07849i −0.309017 + 0.951057i −2.77753 + 1.13372i −0.161110 0.0930169i
95.10 0.104528 0.994522i 0.398305 1.68563i −0.978148 0.207912i −1.53261 + 3.44230i −1.63476 0.572320i −2.01818 + 1.71083i −0.309017 + 0.951057i −2.68271 1.34279i 3.26324 + 1.88403i
95.11 0.104528 0.994522i 0.812842 1.52947i −0.978148 0.207912i 0.565716 1.27062i −1.43613 0.968262i 0.339893 2.62383i −0.309017 + 0.951057i −1.67858 2.48644i −1.20453 0.695433i
95.12 0.104528 0.994522i 0.891795 + 1.48482i −0.978148 0.207912i −0.333330 + 0.748672i 1.56991 0.731703i −2.01015 + 1.72026i −0.309017 + 0.951057i −1.40940 + 2.64832i 0.709728 + 0.409762i
95.13 0.104528 0.994522i 1.28109 1.16568i −0.978148 0.207912i −0.572012 + 1.28476i −1.02538 1.39592i 2.51605 + 0.818231i −0.309017 + 0.951057i 0.282395 2.98668i 1.21793 + 0.703172i
95.14 0.104528 0.994522i 1.39020 + 1.03313i −0.978148 0.207912i −1.35789 + 3.04988i 1.17278 1.27459i 1.07443 2.41777i −0.309017 + 0.951057i 0.865302 + 2.87250i 2.89123 + 1.66925i
95.15 0.104528 0.994522i 1.67644 + 0.435353i −0.978148 0.207912i 0.789465 1.77317i 0.608205 1.62175i −2.28945 + 1.32606i −0.309017 + 0.951057i 2.62093 + 1.45969i −1.68093 0.970486i
95.16 0.104528 0.994522i 1.70747 + 0.290781i −0.978148 0.207912i 1.75629 3.94468i 0.467667 1.66772i 2.64495 + 0.0650804i −0.309017 + 0.951057i 2.83089 + 0.992999i −3.73949 2.15900i
107.1 0.104528 + 0.994522i −1.64414 0.544800i −0.978148 + 0.207912i 1.25353 + 2.81547i 0.369957 1.69208i 1.28350 2.31358i −0.309017 0.951057i 2.40638 + 1.79146i −2.66902 + 1.54096i
107.2 0.104528 + 0.994522i −1.63004 0.585637i −0.978148 + 0.207912i 0.112013 + 0.251585i 0.412043 1.68233i −2.57741 + 0.597448i −0.309017 0.951057i 2.31406 + 1.90922i −0.238498 + 0.137697i
107.3 0.104528 + 0.994522i −1.45000 + 0.947367i −0.978148 + 0.207912i 0.625478 + 1.40485i −1.09374 1.34303i −1.00131 + 2.44895i −0.309017 0.951057i 1.20499 2.74736i −1.33177 + 0.768898i
107.4 0.104528 + 0.994522i −1.41798 + 0.994648i −0.978148 + 0.207912i −1.12093 2.51765i −1.13742 1.30625i 2.40939 + 1.09308i −0.309017 0.951057i 1.02135 2.82079i 2.38669 1.37795i
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.a 128
3.b odd 2 1 462.2.bc.b yes 128
7.c even 3 1 inner 462.2.bc.a 128
11.d odd 10 1 462.2.bc.b yes 128
21.h odd 6 1 462.2.bc.b yes 128
33.f even 10 1 inner 462.2.bc.a 128
77.o odd 30 1 462.2.bc.b yes 128
231.be even 30 1 inner 462.2.bc.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.bc.a 128 1.a even 1 1 trivial
462.2.bc.a 128 7.c even 3 1 inner
462.2.bc.a 128 33.f even 10 1 inner
462.2.bc.a 128 231.be even 30 1 inner
462.2.bc.b yes 128 3.b odd 2 1
462.2.bc.b yes 128 11.d odd 10 1
462.2.bc.b yes 128 21.h odd 6 1
462.2.bc.b yes 128 77.o odd 30 1

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 \(S_{2}^{\mathrm{new}}(462, [\chi])\).