Properties

Label 930.2.bo.a
Level $930$
Weight $2$
Character orbit 930.bo
Analytic conductor $7.426$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bo (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(256\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 256q - 64q^{2} - 64q^{4} - 2q^{5} - 64q^{8} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 256q - 64q^{2} - 64q^{4} - 2q^{5} - 64q^{8} + 4q^{9} - 2q^{10} + 20q^{15} - 64q^{16} - 6q^{17} - 6q^{18} - 4q^{19} + 3q^{20} + 20q^{23} - 2q^{25} + 42q^{31} + 256q^{32} + 8q^{33} + 14q^{34} - 16q^{35} + 4q^{36} + 36q^{38} + 8q^{39} + 3q^{40} - 79q^{45} - 10q^{46} - 6q^{47} - 40q^{49} - 7q^{50} + 68q^{51} - 34q^{53} - 6q^{57} - 20q^{60} + 2q^{62} - 72q^{63} - 64q^{64} + 8q^{66} - 6q^{68} + 10q^{69} - 16q^{70} - 6q^{72} - 2q^{75} - 24q^{76} + 100q^{77} + 8q^{78} + 40q^{79} - 2q^{80} + 12q^{81} - 26q^{83} - 30q^{85} + 16q^{87} - 49q^{90} - 20q^{91} - 22q^{93} + 4q^{94} + 56q^{95} + 130q^{98} + 102q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 0.309017 + 0.951057i −1.73157 0.0407018i −0.809017 + 0.587785i 0.493029 + 2.18104i −0.496376 1.65940i −0.863747 + 1.94001i −0.809017 0.587785i 2.99669 + 0.140956i −1.92193 + 1.14288i
179.2 0.309017 + 0.951057i −1.69787 0.342394i −0.809017 + 0.587785i −1.92997 1.12925i −0.199035 1.72058i −0.934663 + 2.09929i −0.809017 0.587785i 2.76553 + 1.16268i 0.477584 2.18447i
179.3 0.309017 + 0.951057i −1.66354 0.482315i −0.809017 + 0.587785i 0.575623 2.16071i −0.0553539 1.73117i −1.66082 + 3.73026i −0.809017 0.587785i 2.53474 + 1.60470i 2.23283 0.120245i
179.4 0.309017 + 0.951057i −1.64568 + 0.540140i −0.809017 + 0.587785i 2.22856 0.183115i −1.02225 1.39822i 0.318388 0.715111i −0.809017 0.587785i 2.41650 1.77779i 0.862815 + 2.06290i
179.5 0.309017 + 0.951057i −1.59415 0.677259i −0.809017 + 0.587785i −2.09420 + 0.783790i 0.151491 1.72541i 1.79370 4.02872i −0.809017 0.587785i 2.08264 + 2.15931i −1.39257 1.74950i
179.6 0.309017 + 0.951057i −1.46052 0.931061i −0.809017 + 0.587785i 1.43804 + 1.71232i 0.434165 1.67675i 0.429946 0.965674i −0.809017 0.587785i 1.26625 + 2.71967i −1.18414 + 1.89679i
179.7 0.309017 + 0.951057i −1.41936 + 0.992677i −0.809017 + 0.587785i −1.50750 + 1.65150i −1.38270 1.04314i 0.732525 1.64528i −0.809017 0.587785i 1.02919 2.81794i −2.03651 0.923374i
179.8 0.309017 + 0.951057i −1.39110 + 1.03191i −0.809017 + 0.587785i −1.53101 1.62973i −1.41128 1.00414i 0.325961 0.732121i −0.809017 0.587785i 0.870333 2.87098i 1.07685 1.95969i
179.9 0.309017 + 0.951057i −1.21541 1.23401i −0.809017 + 0.587785i 1.34040 1.78978i 0.798025 1.53726i 1.34591 3.02296i −0.809017 0.587785i −0.0455387 + 2.99965i 2.11639 + 0.721724i
179.10 0.309017 + 0.951057i −1.03655 + 1.38764i −0.809017 + 0.587785i −1.97712 + 1.04451i −1.64004 0.557016i −1.83029 + 4.11091i −0.809017 0.587785i −0.851111 2.87674i −1.60435 1.55758i
179.11 0.309017 + 0.951057i −0.835073 1.51745i −0.809017 + 0.587785i 2.23544 + 0.0527903i 1.18513 1.26312i −0.915394 + 2.05601i −0.809017 0.587785i −1.60531 + 2.53436i 0.640584 + 2.14235i
179.12 0.309017 + 0.951057i −0.732396 + 1.56958i −0.809017 + 0.587785i 0.719639 + 2.11710i −1.71909 0.211521i 1.13296 2.54466i −0.809017 0.587785i −1.92719 2.29911i −1.79110 + 1.33864i
179.13 0.309017 + 0.951057i −0.696767 + 1.58572i −0.809017 + 0.587785i 1.79380 1.33503i −1.72342 0.172650i 0.308507 0.692917i −0.809017 0.587785i −2.02903 2.20976i 1.82400 + 1.29346i
179.14 0.309017 + 0.951057i −0.584760 1.63035i −0.809017 + 0.587785i −2.12123 + 0.707392i 1.36986 1.05995i −1.37162 + 3.08072i −0.809017 0.587785i −2.31611 + 1.90673i −1.32826 1.79881i
179.15 0.309017 + 0.951057i −0.128921 1.72725i −0.809017 + 0.587785i 0.447993 + 2.19073i 1.60287 0.656359i 1.37162 3.08072i −0.809017 0.587785i −2.96676 + 0.445355i −1.94507 + 1.10304i
179.16 0.309017 + 0.951057i 0.0869307 + 1.72987i −0.809017 + 0.587785i −2.09670 0.777090i −1.61834 + 0.617235i 1.84852 4.15185i −0.809017 0.587785i −2.98489 + 0.300757i 0.0911422 2.23421i
179.17 0.309017 + 0.951057i 0.145674 1.72591i −0.809017 + 0.587785i −1.16344 1.90956i 1.68646 0.394792i 0.915394 2.05601i −0.809017 0.587785i −2.95756 0.502843i 1.45657 1.69658i
179.18 0.309017 + 0.951057i 0.198005 + 1.72070i −0.809017 + 0.587785i 2.18990 0.452039i −1.57529 + 0.720038i 0.478693 1.07516i −0.809017 0.587785i −2.92159 + 0.681412i 1.10663 + 1.94303i
179.19 0.309017 + 0.951057i 0.518984 + 1.65247i −0.809017 + 0.587785i −0.703473 2.12253i −1.41122 + 1.00422i −0.478693 + 1.07516i −0.809017 0.587785i −2.46131 + 1.71521i 1.80126 1.32494i
179.20 0.309017 + 0.951057i 0.608421 1.62167i −0.809017 + 0.587785i 0.879797 2.05571i 1.73032 + 0.0775177i −1.34591 + 3.02296i −0.809017 0.587785i −2.25965 1.97332i 2.22697 + 0.201486i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 809.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.h odd 30 1 inner
465.bm even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bo.a 256
3.b odd 2 1 930.2.bo.b yes 256
5.b even 2 1 930.2.bo.b yes 256
15.d odd 2 1 inner 930.2.bo.a 256
31.h odd 30 1 inner 930.2.bo.a 256
93.p even 30 1 930.2.bo.b yes 256
155.v odd 30 1 930.2.bo.b yes 256
465.bm even 30 1 inner 930.2.bo.a 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bo.a 256 1.a even 1 1 trivial
930.2.bo.a 256 15.d odd 2 1 inner
930.2.bo.a 256 31.h odd 30 1 inner
930.2.bo.a 256 465.bm even 30 1 inner
930.2.bo.b yes 256 3.b odd 2 1
930.2.bo.b yes 256 5.b even 2 1
930.2.bo.b yes 256 93.p even 30 1
930.2.bo.b yes 256 155.v odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(13\!\cdots\!09\)\( T_{17}^{114} + \)\(42\!\cdots\!81\)\( T_{17}^{113} - \)\(49\!\cdots\!13\)\( T_{17}^{112} + \)\(26\!\cdots\!39\)\( T_{17}^{111} - \)\(14\!\cdots\!41\)\( T_{17}^{110} + \)\(62\!\cdots\!96\)\( T_{17}^{109} + \)\(87\!\cdots\!13\)\( T_{17}^{108} - \)\(11\!\cdots\!70\)\( T_{17}^{107} + \)\(51\!\cdots\!21\)\( T_{17}^{106} - \)\(14\!\cdots\!49\)\( T_{17}^{105} + \)\(14\!\cdots\!33\)\( T_{17}^{104} - \)\(50\!\cdots\!37\)\( T_{17}^{103} + \)\(14\!\cdots\!78\)\( T_{17}^{102} - \)\(62\!\cdots\!55\)\( T_{17}^{101} - \)\(16\!\cdots\!76\)\( T_{17}^{100} + \)\(24\!\cdots\!15\)\( T_{17}^{99} - \)\(81\!\cdots\!12\)\( T_{17}^{98} + \)\(15\!\cdots\!31\)\( T_{17}^{97} - \)\(19\!\cdots\!44\)\( T_{17}^{96} + \)\(36\!\cdots\!45\)\( T_{17}^{95} - \)\(10\!\cdots\!15\)\( T_{17}^{94} + \)\(10\!\cdots\!02\)\( T_{17}^{93} + \)\(13\!\cdots\!48\)\( T_{17}^{92} - \)\(20\!\cdots\!99\)\( T_{17}^{91} + \)\(67\!\cdots\!18\)\( T_{17}^{90} - \)\(52\!\cdots\!14\)\( T_{17}^{89} + \)\(18\!\cdots\!85\)\( T_{17}^{88} + \)\(46\!\cdots\!12\)\( T_{17}^{87} + \)\(31\!\cdots\!69\)\( T_{17}^{86} + \)\(75\!\cdots\!19\)\( T_{17}^{85} + \)\(24\!\cdots\!36\)\( T_{17}^{84} + \)\(29\!\cdots\!58\)\( T_{17}^{83} - \)\(47\!\cdots\!63\)\( T_{17}^{82} + \)\(68\!\cdots\!16\)\( T_{17}^{81} - \)\(20\!\cdots\!18\)\( T_{17}^{80} + \)\(97\!\cdots\!48\)\( T_{17}^{79} - \)\(31\!\cdots\!78\)\( T_{17}^{78} + \)\(26\!\cdots\!22\)\( T_{17}^{77} - \)\(70\!\cdots\!34\)\( T_{17}^{76} - \)\(25\!\cdots\!50\)\( T_{17}^{75} + \)\(90\!\cdots\!29\)\( T_{17}^{74} - \)\(76\!\cdots\!97\)\( T_{17}^{73} + \)\(25\!\cdots\!52\)\( T_{17}^{72} - \)\(12\!\cdots\!94\)\( T_{17}^{71} + \)\(41\!\cdots\!42\)\( T_{17}^{70} - \)\(13\!\cdots\!61\)\( T_{17}^{69} + \)\(42\!\cdots\!34\)\( T_{17}^{68} - \)\(53\!\cdots\!38\)\( T_{17}^{67} + \)\(15\!\cdots\!38\)\( T_{17}^{66} + \)\(10\!\cdots\!79\)\( T_{17}^{65} - \)\(34\!\cdots\!18\)\( T_{17}^{64} + \)\(26\!\cdots\!98\)\( T_{17}^{63} - \)\(75\!\cdots\!79\)\( T_{17}^{62} + \)\(28\!\cdots\!26\)\( T_{17}^{61} - \)\(81\!\cdots\!45\)\( T_{17}^{60} + \)\(14\!\cdots\!95\)\( T_{17}^{59} - \)\(41\!\cdots\!25\)\( T_{17}^{58} - \)\(91\!\cdots\!39\)\( T_{17}^{57} + \)\(28\!\cdots\!58\)\( T_{17}^{56} - \)\(29\!\cdots\!71\)\( T_{17}^{55} + \)\(86\!\cdots\!31\)\( T_{17}^{54} - \)\(31\!\cdots\!82\)\( T_{17}^{53} + \)\(84\!\cdots\!04\)\( T_{17}^{52} - \)\(14\!\cdots\!13\)\( T_{17}^{51} + \)\(37\!\cdots\!44\)\( T_{17}^{50} + \)\(51\!\cdots\!74\)\( T_{17}^{49} - \)\(13\!\cdots\!88\)\( T_{17}^{48} + \)\(16\!\cdots\!89\)\( T_{17}^{47} - \)\(51\!\cdots\!06\)\( T_{17}^{46} + \)\(17\!\cdots\!14\)\( T_{17}^{45} - \)\(46\!\cdots\!57\)\( T_{17}^{44} + \)\(64\!\cdots\!66\)\( T_{17}^{43} - \)\(58\!\cdots\!92\)\( T_{17}^{42} - \)\(21\!\cdots\!22\)\( T_{17}^{41} + \)\(12\!\cdots\!59\)\( T_{17}^{40} - \)\(59\!\cdots\!57\)\( T_{17}^{39} + \)\(18\!\cdots\!15\)\( T_{17}^{38} - \)\(43\!\cdots\!33\)\( T_{17}^{37} + \)\(90\!\cdots\!41\)\( T_{17}^{36} - \)\(15\!\cdots\!30\)\( T_{17}^{35} + \)\(10\!\cdots\!46\)\( T_{17}^{34} + \)\(10\!\cdots\!03\)\( T_{17}^{33} - \)\(51\!\cdots\!04\)\( T_{17}^{32} + \)\(13\!\cdots\!48\)\( T_{17}^{31} - \)\(24\!\cdots\!88\)\( T_{17}^{30} + \)\(23\!\cdots\!73\)\( T_{17}^{29} + \)\(17\!\cdots\!02\)\( T_{17}^{28} - \)\(14\!\cdots\!38\)\( T_{17}^{27} + \)\(42\!\cdots\!18\)\( T_{17}^{26} - \)\(86\!\cdots\!61\)\( T_{17}^{25} + \)\(14\!\cdots\!17\)\( T_{17}^{24} - \)\(19\!\cdots\!91\)\( T_{17}^{23} + \)\(20\!\cdots\!90\)\( T_{17}^{22} - \)\(19\!\cdots\!99\)\( T_{17}^{21} + \)\(15\!\cdots\!10\)\( T_{17}^{20} - \)\(91\!\cdots\!67\)\( T_{17}^{19} + \)\(44\!\cdots\!56\)\( T_{17}^{18} - \)\(19\!\cdots\!91\)\( T_{17}^{17} + \)\(55\!\cdots\!49\)\( T_{17}^{16} - \)\(68\!\cdots\!04\)\( T_{17}^{15} + \)\(30\!\cdots\!42\)\( T_{17}^{14} - \)\(49\!\cdots\!41\)\( T_{17}^{13} - \)\(36\!\cdots\!71\)\( T_{17}^{12} - \)\(25\!\cdots\!55\)\( T_{17}^{11} - \)\(29\!\cdots\!55\)\( T_{17}^{10} + \)\(41\!\cdots\!02\)\( T_{17}^{9} + \)\(15\!\cdots\!81\)\( T_{17}^{8} + \)\(25\!\cdots\!36\)\( T_{17}^{7} + \)\(28\!\cdots\!52\)\( T_{17}^{6} + \)\(23\!\cdots\!12\)\( T_{17}^{5} + \)\(13\!\cdots\!32\)\( T_{17}^{4} + \)\(50\!\cdots\!88\)\( T_{17}^{3} + \)\(12\!\cdots\!84\)\( T_{17}^{2} + \)\(18\!\cdots\!64\)\( T_{17} + \)\(12\!\cdots\!36\)\( \)">\(T_{17}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).