Properties

Label 930.2.bj.a
Level $930$
Weight $2$
Character orbit 930.bj
Analytic conductor $7.426$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128q - 16q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q - 16q^{7} - 4q^{10} + 4q^{15} + 32q^{16} - 12q^{17} - 40q^{19} + 40q^{21} + 16q^{22} - 32q^{24} - 8q^{25} - 4q^{28} + 8q^{29} + 20q^{31} + 4q^{33} + 24q^{35} - 128q^{36} + 64q^{37} - 16q^{38} - 24q^{41} + 4q^{42} + 24q^{43} - 8q^{44} + 20q^{46} + 108q^{47} - 100q^{49} - 24q^{50} + 16q^{53} - 32q^{54} + 12q^{55} + 16q^{57} - 40q^{58} - 16q^{62} - 4q^{63} + 36q^{65} - 12q^{66} - 32q^{67} + 8q^{68} - 32q^{70} + 24q^{71} + 60q^{73} - 16q^{74} + 24q^{75} - 24q^{76} - 20q^{77} - 56q^{79} + 32q^{81} - 16q^{82} - 8q^{83} - 132q^{85} - 20q^{87} - 4q^{88} + 136q^{89} - 40q^{91} - 64q^{93} + 108q^{95} + 64q^{97} - 16q^{98} - 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −2.22990 + 0.166033i 1.00000i 0.154498 0.303220i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.512821 + 2.17647i
277.2 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −1.27713 + 1.83547i 1.00000i 2.18789 4.29397i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.01266 + 0.974271i
277.3 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.620092 2.14837i 1.00000i −0.961148 + 1.88636i 0.453990 + 0.891007i 0.951057 + 0.309017i −2.02491 + 0.948537i
277.4 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.426558 + 2.19501i 1.00000i −0.653972 + 1.28349i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.23471 + 0.0779318i
277.5 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0.591821 2.15633i 1.00000i −0.754948 + 1.48167i 0.453990 + 0.891007i 0.951057 + 0.309017i −2.22236 0.247211i
277.6 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 1.90676 1.16802i 1.00000i 0.193795 0.380343i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.45193 1.70056i
277.7 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.03470 + 0.927363i 1.00000i −1.56419 + 3.06990i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.597648 2.15472i
277.8 −0.156434 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.08697 + 0.802838i 1.00000i 2.05326 4.02976i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.466480 2.18687i
277.9 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.90323 1.17377i 1.00000i 0.771267 1.51370i −0.453990 0.891007i 0.951057 + 0.309017i 0.861584 2.06341i
277.10 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.17179 + 1.90444i 1.00000i 1.50670 2.95707i −0.453990 0.891007i 0.951057 + 0.309017i −2.06431 0.859444i
277.11 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.980551 2.00961i 1.00000i 1.28197 2.51601i −0.453990 0.891007i 0.951057 + 0.309017i 1.83147 1.28285i
277.12 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.683297 2.12911i 1.00000i −2.06274 + 4.04835i −0.453990 0.891007i 0.951057 + 0.309017i 1.99600 1.00795i
277.13 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −0.284153 + 2.21794i 1.00000i −0.874752 + 1.71680i −0.453990 0.891007i 0.951057 + 0.309017i −2.23508 + 0.0663075i
277.14 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.11441 + 1.93858i 1.00000i −1.22295 + 2.40017i −0.453990 0.891007i 0.951057 + 0.309017i −1.74038 + 1.40395i
277.15 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.96074 1.07493i 1.00000i −0.462930 + 0.908551i −0.453990 0.891007i 0.951057 + 0.309017i 1.36843 + 1.76845i
277.16 0.156434 + 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 2.23243 0.127537i 1.00000i −0.416181 + 0.816801i −0.453990 0.891007i 0.951057 + 0.309017i 0.475195 + 2.18499i
337.1 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −2.23208 0.133471i 1.00000i 0.703148 + 0.111368i 0.987688 0.156434i 0.587785 0.809017i 1.13227 1.92820i
337.2 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −1.80883 1.31459i 1.00000i 2.31722 + 0.367011i 0.987688 0.156434i 0.587785 0.809017i 1.99250 1.01486i
337.3 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −1.30319 + 1.81705i 1.00000i −1.71397 0.271467i 0.987688 0.156434i 0.587785 0.809017i −1.02737 1.98608i
337.4 −0.453990 + 0.891007i −0.891007 + 0.453990i −0.587785 0.809017i −0.789819 2.09193i 1.00000i −4.01115 0.635304i 0.987688 0.156434i 0.587785 0.809017i 2.22250 + 0.245984i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.r even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bj.a 128
5.c odd 4 1 930.2.bj.b yes 128
31.f odd 10 1 930.2.bj.b yes 128
155.r even 20 1 inner 930.2.bj.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bj.a 128 1.a even 1 1 trivial
930.2.bj.a 128 155.r even 20 1 inner
930.2.bj.b yes 128 5.c odd 4 1
930.2.bj.b yes 128 31.f odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(23\!\cdots\!22\)\( T_{7}^{112} + \)\(62\!\cdots\!48\)\( T_{7}^{111} + \)\(10\!\cdots\!08\)\( T_{7}^{110} + \)\(50\!\cdots\!96\)\( T_{7}^{109} + \)\(86\!\cdots\!49\)\( T_{7}^{108} + \)\(59\!\cdots\!08\)\( T_{7}^{107} + \)\(56\!\cdots\!32\)\( T_{7}^{106} + \)\(29\!\cdots\!32\)\( T_{7}^{105} + \)\(96\!\cdots\!35\)\( T_{7}^{104} + \)\(70\!\cdots\!84\)\( T_{7}^{103} - \)\(13\!\cdots\!98\)\( T_{7}^{102} - \)\(10\!\cdots\!80\)\( T_{7}^{101} - \)\(37\!\cdots\!08\)\( T_{7}^{100} - \)\(30\!\cdots\!80\)\( T_{7}^{99} + \)\(67\!\cdots\!60\)\( T_{7}^{98} + \)\(61\!\cdots\!36\)\( T_{7}^{97} + \)\(34\!\cdots\!83\)\( T_{7}^{96} + \)\(15\!\cdots\!12\)\( T_{7}^{95} + \)\(53\!\cdots\!32\)\( T_{7}^{94} + \)\(15\!\cdots\!76\)\( T_{7}^{93} + \)\(33\!\cdots\!72\)\( T_{7}^{92} + \)\(44\!\cdots\!24\)\( T_{7}^{91} - \)\(34\!\cdots\!04\)\( T_{7}^{90} - \)\(37\!\cdots\!60\)\( T_{7}^{89} - \)\(91\!\cdots\!43\)\( T_{7}^{88} + \)\(62\!\cdots\!68\)\( T_{7}^{87} + \)\(14\!\cdots\!50\)\( T_{7}^{86} + \)\(64\!\cdots\!32\)\( T_{7}^{85} + \)\(16\!\cdots\!63\)\( T_{7}^{84} + \)\(11\!\cdots\!04\)\( T_{7}^{83} - \)\(11\!\cdots\!80\)\( T_{7}^{82} - \)\(68\!\cdots\!40\)\( T_{7}^{81} - \)\(22\!\cdots\!91\)\( T_{7}^{80} - \)\(47\!\cdots\!36\)\( T_{7}^{79} - \)\(22\!\cdots\!84\)\( T_{7}^{78} + \)\(29\!\cdots\!28\)\( T_{7}^{77} + \)\(14\!\cdots\!65\)\( T_{7}^{76} + \)\(43\!\cdots\!12\)\( T_{7}^{75} + \)\(76\!\cdots\!68\)\( T_{7}^{74} + \)\(17\!\cdots\!68\)\( T_{7}^{73} - \)\(43\!\cdots\!66\)\( T_{7}^{72} - \)\(18\!\cdots\!84\)\( T_{7}^{71} - \)\(41\!\cdots\!92\)\( T_{7}^{70} - \)\(43\!\cdots\!32\)\( T_{7}^{69} + \)\(94\!\cdots\!99\)\( T_{7}^{68} + \)\(60\!\cdots\!88\)\( T_{7}^{67} + \)\(16\!\cdots\!32\)\( T_{7}^{66} + \)\(21\!\cdots\!56\)\( T_{7}^{65} - \)\(17\!\cdots\!19\)\( T_{7}^{64} - \)\(18\!\cdots\!32\)\( T_{7}^{63} - \)\(55\!\cdots\!82\)\( T_{7}^{62} - \)\(90\!\cdots\!96\)\( T_{7}^{61} - \)\(10\!\cdots\!04\)\( T_{7}^{60} + \)\(44\!\cdots\!00\)\( T_{7}^{59} + \)\(15\!\cdots\!80\)\( T_{7}^{58} + \)\(31\!\cdots\!32\)\( T_{7}^{57} + \)\(29\!\cdots\!81\)\( T_{7}^{56} - \)\(44\!\cdots\!56\)\( T_{7}^{55} - \)\(26\!\cdots\!36\)\( T_{7}^{54} - \)\(63\!\cdots\!76\)\( T_{7}^{53} - \)\(89\!\cdots\!52\)\( T_{7}^{52} - \)\(32\!\cdots\!16\)\( T_{7}^{51} + \)\(21\!\cdots\!04\)\( T_{7}^{50} + \)\(70\!\cdots\!48\)\( T_{7}^{49} + \)\(12\!\cdots\!79\)\( T_{7}^{48} + \)\(12\!\cdots\!76\)\( T_{7}^{47} - \)\(29\!\cdots\!74\)\( T_{7}^{46} - \)\(36\!\cdots\!96\)\( T_{7}^{45} - \)\(72\!\cdots\!27\)\( T_{7}^{44} - \)\(66\!\cdots\!92\)\( T_{7}^{43} + \)\(39\!\cdots\!04\)\( T_{7}^{42} + \)\(26\!\cdots\!04\)\( T_{7}^{41} + \)\(49\!\cdots\!51\)\( T_{7}^{40} + \)\(45\!\cdots\!12\)\( T_{7}^{39} - \)\(18\!\cdots\!04\)\( T_{7}^{38} - \)\(14\!\cdots\!60\)\( T_{7}^{37} - \)\(25\!\cdots\!33\)\( T_{7}^{36} - \)\(21\!\cdots\!92\)\( T_{7}^{35} + \)\(10\!\cdots\!60\)\( T_{7}^{34} + \)\(61\!\cdots\!08\)\( T_{7}^{33} + \)\(89\!\cdots\!54\)\( T_{7}^{32} + \)\(46\!\cdots\!76\)\( T_{7}^{31} - \)\(71\!\cdots\!64\)\( T_{7}^{30} - \)\(18\!\cdots\!88\)\( T_{7}^{29} - \)\(15\!\cdots\!45\)\( T_{7}^{28} + \)\(53\!\cdots\!08\)\( T_{7}^{27} + \)\(28\!\cdots\!48\)\( T_{7}^{26} + \)\(28\!\cdots\!48\)\( T_{7}^{25} + \)\(13\!\cdots\!73\)\( T_{7}^{24} - \)\(29\!\cdots\!24\)\( T_{7}^{23} - \)\(32\!\cdots\!74\)\( T_{7}^{22} - \)\(43\!\cdots\!32\)\( T_{7}^{21} + \)\(22\!\cdots\!78\)\( T_{7}^{20} + \)\(22\!\cdots\!40\)\( T_{7}^{19} + \)\(13\!\cdots\!04\)\( T_{7}^{18} - \)\(12\!\cdots\!64\)\( T_{7}^{17} - \)\(87\!\cdots\!19\)\( T_{7}^{16} + \)\(10\!\cdots\!20\)\( T_{7}^{15} + \)\(42\!\cdots\!60\)\( T_{7}^{14} + \)\(16\!\cdots\!92\)\( T_{7}^{13} - \)\(54\!\cdots\!88\)\( T_{7}^{12} - \)\(53\!\cdots\!52\)\( T_{7}^{11} - \)\(58\!\cdots\!68\)\( T_{7}^{10} - \)\(86\!\cdots\!36\)\( T_{7}^{9} - \)\(38\!\cdots\!76\)\( T_{7}^{8} + \)\(56\!\cdots\!48\)\( T_{7}^{7} + \)\(15\!\cdots\!28\)\( T_{7}^{6} + \)\(68\!\cdots\!00\)\( T_{7}^{5} + \)\(17\!\cdots\!60\)\( T_{7}^{4} + \)\(75\!\cdots\!36\)\( T_{7}^{3} + \)\(27\!\cdots\!88\)\( T_{7}^{2} + \)\(50\!\cdots\!44\)\( T_{7} + \)\(34\!\cdots\!76\)\( \)">\(T_{7}^{128} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).