Properties

Label 930.2.bj.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 4q^{7} - 4q^{10} - 4q^{15} + 32q^{16} + 12q^{17} + 40q^{19} + 40q^{21} + 4q^{22} + 32q^{24} - 8q^{25} - 4q^{28} - 8q^{29} + 20q^{31} + 4q^{33} + 24q^{35} - 128q^{36} - 64q^{37} - 16q^{38} - 24q^{41} + 16q^{42} - 24q^{43} + 8q^{44} + 20q^{46} - 12q^{47} + 100q^{49} - 24q^{50} + 64q^{53} + 32q^{54} + 68q^{55} - 16q^{57} + 40q^{58} + 8q^{62} - 4q^{63} + 84q^{65} - 12q^{66} - 32q^{67} - 8q^{68} + 88q^{70} + 24q^{71} + 20q^{73} + 16q^{74} - 24q^{75} - 24q^{76} + 60q^{77} + 56q^{79} + 32q^{81} - 16q^{82} + 8q^{83} - 68q^{85} - 20q^{87} + 4q^{88} - 136q^{89} - 40q^{91} + 48q^{93} - 92q^{95} + 64q^{97} - 16q^{98} + 8q^{99} + O(q^{100}) \)

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 −1.90277 + 1.17451i 1.00000i 1.72531 3.38612i 0.453990 + 0.891007i 0.951057 + 0.309017i 1.45771 + 1.69561i
277.2 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.82843 + 1.28719i 1.00000i −2.04975 + 4.02286i 0.453990 + 0.891007i 0.951057 + 0.309017i 1.55737 + 1.60455i
277.3 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.74178 1.40221i 1.00000i −0.335285 + 0.658034i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.11247 + 1.93969i
277.4 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i −1.67223 1.48447i 1.00000i 1.10706 2.17272i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.20460 + 1.88386i
277.5 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 0.244623 + 2.22265i 1.00000i 0.795365 1.56099i 0.453990 + 0.891007i 0.951057 + 0.309017i 2.15702 0.589310i
277.6 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.48798 1.66911i 1.00000i −0.506200 + 0.993473i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.88133 1.20856i
277.7 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 1.61098 1.55073i 1.00000i −1.36477 + 2.67852i 0.453990 + 0.891007i 0.951057 + 0.309017i −1.78365 1.34855i
277.8 −0.156434 0.987688i −0.987688 0.156434i −0.951057 + 0.309017i 2.17207 + 0.531157i 1.00000i 0.369895 0.725959i 0.453990 + 0.891007i 0.951057 + 0.309017i 0.184831 2.22842i
277.9 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −2.23524 0.0607951i 1.00000i 0.773249 1.51759i −0.453990 0.891007i 0.951057 + 0.309017i −0.289622 2.21723i
277.10 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −2.20784 + 0.354206i 1.00000i 0.622042 1.22083i −0.453990 0.891007i 0.951057 + 0.309017i −0.695226 2.12524i
277.11 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −1.01777 1.99102i 1.00000i −0.681739 + 1.33799i −0.453990 0.891007i 0.951057 + 0.309017i 1.80729 1.31670i
277.12 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i −0.869928 + 2.05991i 1.00000i −2.25496 + 4.42561i −0.453990 0.891007i 0.951057 + 0.309017i −2.17063 0.536977i
277.13 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 0.145083 + 2.23136i 1.00000i 0.996252 1.95526i −0.453990 0.891007i 0.951057 + 0.309017i −2.18119 + 0.492358i
277.14 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 1.00799 1.99598i 1.00000i 1.85184 3.63444i −0.453990 0.891007i 0.951057 + 0.309017i 2.12909 + 0.683342i
277.15 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.22022 + 0.265769i 1.00000i 1.65375 3.24566i −0.453990 0.891007i 0.951057 + 0.309017i 0.0848219 + 2.23446i
277.16 0.156434 + 0.987688i 0.987688 + 0.156434i −0.951057 + 0.309017i 2.23590 + 0.0275636i 1.00000i −1.08401 + 2.12750i −0.453990 0.891007i 0.951057 + 0.309017i 0.322547 + 2.21268i
337.1 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −2.20246 + 0.386226i 1.00000i 3.54459 + 0.561408i 0.987688 0.156434i 0.587785 0.809017i 0.655766 2.13775i
337.2 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −1.81236 1.30972i 1.00000i −2.09145 0.331253i 0.987688 0.156434i 0.587785 0.809017i 1.98976 1.02022i
337.3 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −1.39059 + 1.75108i 1.00000i −2.23420 0.353863i 0.987688 0.156434i 0.587785 0.809017i −0.928908 2.03399i
337.4 −0.453990 + 0.891007i 0.891007 0.453990i −0.587785 0.809017i −0.354402 2.20780i 1.00000i 5.01200 + 0.793822i 0.987688 0.156434i 0.587785 0.809017i 2.12806 + 0.686548i
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.b yes 128
5.c odd 4 1 930.2.bj.a 128
31.f odd 10 1 930.2.bj.a 128
155.r even 20 1 inner 930.2.bj.b yes 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bj.a 128 5.c odd 4 1
930.2.bj.a 128 31.f odd 10 1
930.2.bj.b yes 128 1.a even 1 1 trivial
930.2.bj.b yes 128 155.r even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(23\!\cdots\!92\)\( T_{7}^{110} + \)\(13\!\cdots\!36\)\( T_{7}^{109} - \)\(14\!\cdots\!71\)\( T_{7}^{108} - \)\(70\!\cdots\!72\)\( T_{7}^{107} + \)\(90\!\cdots\!12\)\( T_{7}^{106} - \)\(51\!\cdots\!88\)\( T_{7}^{105} + \)\(73\!\cdots\!75\)\( T_{7}^{104} + \)\(39\!\cdots\!44\)\( T_{7}^{103} - \)\(37\!\cdots\!18\)\( T_{7}^{102} + \)\(14\!\cdots\!80\)\( T_{7}^{101} - \)\(10\!\cdots\!08\)\( T_{7}^{100} - \)\(15\!\cdots\!60\)\( T_{7}^{99} + \)\(11\!\cdots\!80\)\( T_{7}^{98} - \)\(33\!\cdots\!24\)\( T_{7}^{97} - \)\(33\!\cdots\!57\)\( T_{7}^{96} + \)\(57\!\cdots\!92\)\( T_{7}^{95} - \)\(20\!\cdots\!88\)\( T_{7}^{94} + \)\(19\!\cdots\!56\)\( T_{7}^{93} + \)\(16\!\cdots\!52\)\( T_{7}^{92} - \)\(92\!\cdots\!96\)\( T_{7}^{91} + \)\(20\!\cdots\!36\)\( T_{7}^{90} + \)\(19\!\cdots\!80\)\( T_{7}^{89} - \)\(32\!\cdots\!23\)\( T_{7}^{88} + \)\(91\!\cdots\!08\)\( T_{7}^{87} - \)\(19\!\cdots\!10\)\( T_{7}^{86} - \)\(54\!\cdots\!68\)\( T_{7}^{85} + \)\(21\!\cdots\!23\)\( T_{7}^{84} - \)\(47\!\cdots\!16\)\( T_{7}^{83} + \)\(28\!\cdots\!40\)\( T_{7}^{82} + \)\(30\!\cdots\!20\)\( T_{7}^{81} - \)\(19\!\cdots\!31\)\( T_{7}^{80} + \)\(53\!\cdots\!64\)\( T_{7}^{79} + \)\(16\!\cdots\!36\)\( T_{7}^{78} - \)\(50\!\cdots\!12\)\( T_{7}^{77} + \)\(17\!\cdots\!05\)\( T_{7}^{76} - \)\(14\!\cdots\!08\)\( T_{7}^{75} - \)\(80\!\cdots\!32\)\( T_{7}^{74} + \)\(33\!\cdots\!48\)\( T_{7}^{73} - \)\(48\!\cdots\!26\)\( T_{7}^{72} - \)\(43\!\cdots\!04\)\( T_{7}^{71} + \)\(40\!\cdots\!88\)\( T_{7}^{70} - \)\(10\!\cdots\!32\)\( T_{7}^{69} + \)\(11\!\cdots\!79\)\( T_{7}^{68} + \)\(17\!\cdots\!88\)\( T_{7}^{67} - \)\(13\!\cdots\!88\)\( T_{7}^{66} + \)\(36\!\cdots\!36\)\( T_{7}^{65} - \)\(40\!\cdots\!59\)\( T_{7}^{64} - \)\(89\!\cdots\!12\)\( T_{7}^{63} + \)\(52\!\cdots\!78\)\( T_{7}^{62} - \)\(10\!\cdots\!36\)\( T_{7}^{61} + \)\(36\!\cdots\!36\)\( T_{7}^{60} + \)\(43\!\cdots\!60\)\( T_{7}^{59} - \)\(13\!\cdots\!00\)\( T_{7}^{58} + \)\(17\!\cdots\!72\)\( T_{7}^{57} + \)\(15\!\cdots\!01\)\( T_{7}^{56} - \)\(11\!\cdots\!56\)\( T_{7}^{55} + \)\(21\!\cdots\!44\)\( T_{7}^{54} - \)\(11\!\cdots\!36\)\( T_{7}^{53} - \)\(51\!\cdots\!92\)\( T_{7}^{52} + \)\(16\!\cdots\!44\)\( T_{7}^{51} - \)\(20\!\cdots\!56\)\( T_{7}^{50} - \)\(31\!\cdots\!12\)\( T_{7}^{49} + \)\(70\!\cdots\!59\)\( T_{7}^{48} - \)\(14\!\cdots\!04\)\( T_{7}^{47} + \)\(13\!\cdots\!46\)\( T_{7}^{46} + \)\(10\!\cdots\!44\)\( T_{7}^{45} - \)\(59\!\cdots\!07\)\( T_{7}^{44} + \)\(10\!\cdots\!88\)\( T_{7}^{43} - \)\(79\!\cdots\!96\)\( T_{7}^{42} - \)\(79\!\cdots\!96\)\( T_{7}^{41} + \)\(37\!\cdots\!11\)\( T_{7}^{40} - \)\(63\!\cdots\!48\)\( T_{7}^{39} + \)\(52\!\cdots\!76\)\( T_{7}^{38} + \)\(36\!\cdots\!20\)\( T_{7}^{37} - \)\(20\!\cdots\!33\)\( T_{7}^{36} + \)\(35\!\cdots\!68\)\( T_{7}^{35} - \)\(28\!\cdots\!80\)\( T_{7}^{34} - \)\(11\!\cdots\!12\)\( T_{7}^{33} + \)\(82\!\cdots\!54\)\( T_{7}^{32} - \)\(14\!\cdots\!04\)\( T_{7}^{31} + \)\(14\!\cdots\!56\)\( T_{7}^{30} - \)\(34\!\cdots\!68\)\( T_{7}^{29} - \)\(16\!\cdots\!25\)\( T_{7}^{28} + \)\(39\!\cdots\!68\)\( T_{7}^{27} - \)\(52\!\cdots\!52\)\( T_{7}^{26} + \)\(47\!\cdots\!88\)\( T_{7}^{25} - \)\(13\!\cdots\!47\)\( T_{7}^{24} - \)\(32\!\cdots\!44\)\( T_{7}^{23} + \)\(88\!\cdots\!86\)\( T_{7}^{22} - \)\(12\!\cdots\!92\)\( T_{7}^{21} + \)\(14\!\cdots\!78\)\( T_{7}^{20} - \)\(12\!\cdots\!80\)\( T_{7}^{19} + \)\(10\!\cdots\!24\)\( T_{7}^{18} - \)\(64\!\cdots\!04\)\( T_{7}^{17} + \)\(40\!\cdots\!21\)\( T_{7}^{16} - \)\(17\!\cdots\!20\)\( T_{7}^{15} + \)\(85\!\cdots\!00\)\( T_{7}^{14} - \)\(13\!\cdots\!08\)\( T_{7}^{13} + \)\(14\!\cdots\!92\)\( T_{7}^{12} + \)\(48\!\cdots\!48\)\( T_{7}^{11} + \)\(10\!\cdots\!12\)\( T_{7}^{10} + \)\(55\!\cdots\!84\)\( T_{7}^{9} + \)\(59\!\cdots\!24\)\( T_{7}^{8} + \)\(34\!\cdots\!88\)\( T_{7}^{7} + \)\(20\!\cdots\!48\)\( T_{7}^{6} + \)\(87\!\cdots\!60\)\( T_{7}^{5} + \)\(31\!\cdots\!60\)\( T_{7}^{4} + \)\(85\!\cdots\!36\)\( T_{7}^{3} + \)\(19\!\cdots\!28\)\( T_{7}^{2} + \)\(26\!\cdots\!64\)\( T_{7} + \)\(34\!\cdots\!76\)\( \)">\(T_{7}^{128} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).