Properties

Label 930.2.bn.b
Level $930$
Weight $2$
Character orbit 930.bn
Analytic conductor $7.426$
Analytic rank $0$
Dimension $144$
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.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144q + 36q^{4} + 2q^{5} - 72q^{6} - 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 144q + 36q^{4} + 2q^{5} - 72q^{6} - 18q^{9} - 18q^{11} - 8q^{14} - 36q^{16} - 24q^{19} - 2q^{20} + 28q^{21} - 18q^{24} + 10q^{25} - 12q^{26} - 4q^{30} - 4q^{31} + 10q^{34} - 2q^{35} - 72q^{36} + 16q^{39} + 4q^{41} - 2q^{44} - 2q^{45} - 2q^{46} - 78q^{49} + 32q^{50} + 10q^{51} + 36q^{54} - 50q^{55} - 12q^{56} + 28q^{59} + 88q^{61} + 36q^{64} - 124q^{65} + 6q^{66} - 46q^{69} - 10q^{70} + 140q^{71} + 34q^{74} - 32q^{75} + 24q^{76} + 16q^{79} + 12q^{80} + 18q^{81} - 8q^{84} + 74q^{85} - 98q^{86} + 148q^{89} + 44q^{91} - 108q^{94} - 80q^{95} + 18q^{96} - 2q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.89205 1.19170i −0.500000 0.866025i −1.93366 1.74107i 0.951057 + 0.309017i −0.669131 0.743145i 2.07622 0.830237i
19.2 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.39553 + 1.74714i −0.500000 0.866025i 1.99915 + 1.80004i 0.951057 + 0.309017i −0.669131 0.743145i −0.593198 2.15595i
19.3 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −1.16157 1.91070i −0.500000 0.866025i 2.15311 + 1.93867i 0.951057 + 0.309017i −0.669131 0.743145i 2.22854 + 0.183353i
19.4 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −0.978641 + 2.01054i −0.500000 0.866025i −2.38970 2.15169i 0.951057 + 0.309017i −0.669131 0.743145i −1.05133 1.97350i
19.5 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i −0.293219 2.21676i −0.500000 0.866025i 1.44492 + 1.30102i 0.951057 + 0.309017i −0.669131 0.743145i 1.96575 + 1.06576i
19.6 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 0.991216 + 2.00437i −0.500000 0.866025i 0.807100 + 0.726716i 0.951057 + 0.309017i −0.669131 0.743145i −2.20419 0.376228i
19.7 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 1.47304 1.68231i −0.500000 0.866025i −2.98158 2.68462i 0.951057 + 0.309017i −0.669131 0.743145i 0.495182 + 2.18055i
19.8 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 2.02048 0.957953i −0.500000 0.866025i 1.95080 + 1.75650i 0.951057 + 0.309017i −0.669131 0.743145i −0.412606 + 2.19767i
19.9 −0.587785 + 0.809017i −0.406737 + 0.913545i −0.309017 0.951057i 2.14981 + 0.615060i −0.500000 0.866025i −2.84716 2.56360i 0.951057 + 0.309017i −0.669131 0.743145i −1.76122 + 1.37771i
19.10 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −2.23144 + 0.143766i −0.500000 0.866025i −0.807100 0.726716i −0.951057 0.309017i −0.669131 0.743145i −1.19530 + 1.88978i
19.11 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −1.60756 1.55426i −0.500000 0.866025i 2.84716 + 2.56360i −0.951057 0.309017i −0.669131 0.743145i −2.20233 + 0.386974i
19.12 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −1.25186 + 1.85280i −0.500000 0.866025i 2.38970 + 2.15169i −0.951057 0.309017i −0.669131 0.743145i 0.763121 + 2.10182i
19.13 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.815306 + 2.08213i −0.500000 0.866025i −1.99915 1.80004i −0.951057 0.309017i −0.669131 0.743145i 1.20526 + 1.88344i
19.14 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i −0.180627 2.22876i −0.500000 0.866025i −1.95080 1.75650i −0.951057 0.309017i −0.669131 0.743145i −1.90927 1.16390i
19.15 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 0.720399 2.11684i −0.500000 0.866025i 2.98158 + 2.68462i −0.951057 0.309017i −0.669131 0.743145i −1.28912 1.82706i
19.16 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 1.97807 + 1.04271i −0.500000 0.866025i 1.93366 + 1.74107i −0.951057 0.309017i −0.669131 0.743145i 2.00625 0.987398i
19.17 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 2.06638 0.854445i −0.500000 0.866025i −1.44492 1.30102i −0.951057 0.309017i −0.669131 0.743145i 0.523327 2.17397i
19.18 0.587785 0.809017i 0.406737 0.913545i −0.309017 0.951057i 2.23550 + 0.0505968i −0.500000 0.866025i −2.15311 1.93867i −0.951057 0.309017i −0.669131 0.743145i 1.35492 1.77881i
49.1 −0.587785 0.809017i −0.406737 0.913545i −0.309017 + 0.951057i −1.89205 + 1.19170i −0.500000 + 0.866025i −1.93366 + 1.74107i 0.951057 0.309017i −0.669131 + 0.743145i 2.07622 + 0.830237i
49.2 −0.587785 0.809017i −0.406737 0.913545i −0.309017 + 0.951057i −1.39553 1.74714i −0.500000 + 0.866025i 1.99915 1.80004i 0.951057 0.309017i −0.669131 + 0.743145i −0.593198 + 2.15595i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.g even 15 1 inner
155.u 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.bn.b 144
5.b even 2 1 inner 930.2.bn.b 144
31.g even 15 1 inner 930.2.bn.b 144
155.u even 30 1 inner 930.2.bn.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bn.b 144 1.a even 1 1 trivial
930.2.bn.b 144 5.b even 2 1 inner
930.2.bn.b 144 31.g even 15 1 inner
930.2.bn.b 144 155.u even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(18\!\cdots\!14\)\( T_{7}^{126} + \)\(14\!\cdots\!31\)\( T_{7}^{124} - \)\(28\!\cdots\!52\)\( T_{7}^{122} - \)\(11\!\cdots\!95\)\( T_{7}^{120} - \)\(16\!\cdots\!10\)\( T_{7}^{118} - \)\(50\!\cdots\!44\)\( T_{7}^{116} + \)\(28\!\cdots\!08\)\( T_{7}^{114} + \)\(70\!\cdots\!30\)\( T_{7}^{112} + \)\(79\!\cdots\!82\)\( T_{7}^{110} + \)\(21\!\cdots\!10\)\( T_{7}^{108} - \)\(11\!\cdots\!98\)\( T_{7}^{106} - \)\(21\!\cdots\!67\)\( T_{7}^{104} - \)\(18\!\cdots\!36\)\( T_{7}^{102} - \)\(10\!\cdots\!20\)\( T_{7}^{100} + \)\(21\!\cdots\!30\)\( T_{7}^{98} + \)\(35\!\cdots\!83\)\( T_{7}^{96} + \)\(27\!\cdots\!58\)\( T_{7}^{94} + \)\(18\!\cdots\!33\)\( T_{7}^{92} - \)\(24\!\cdots\!72\)\( T_{7}^{90} - \)\(32\!\cdots\!00\)\( T_{7}^{88} - \)\(24\!\cdots\!86\)\( T_{7}^{86} - \)\(33\!\cdots\!37\)\( T_{7}^{84} + \)\(13\!\cdots\!72\)\( T_{7}^{82} + \)\(19\!\cdots\!46\)\( T_{7}^{80} + \)\(13\!\cdots\!60\)\( T_{7}^{78} + \)\(45\!\cdots\!55\)\( T_{7}^{76} - \)\(30\!\cdots\!22\)\( T_{7}^{74} - \)\(52\!\cdots\!09\)\( T_{7}^{72} - \)\(40\!\cdots\!86\)\( T_{7}^{70} - \)\(14\!\cdots\!60\)\( T_{7}^{68} + \)\(36\!\cdots\!66\)\( T_{7}^{66} + \)\(82\!\cdots\!03\)\( T_{7}^{64} + \)\(65\!\cdots\!00\)\( T_{7}^{62} + \)\(31\!\cdots\!67\)\( T_{7}^{60} + \)\(83\!\cdots\!20\)\( T_{7}^{58} + \)\(11\!\cdots\!22\)\( T_{7}^{56} - \)\(27\!\cdots\!30\)\( T_{7}^{54} - \)\(76\!\cdots\!62\)\( T_{7}^{52} - \)\(25\!\cdots\!74\)\( T_{7}^{50} + \)\(63\!\cdots\!25\)\( T_{7}^{48} + \)\(21\!\cdots\!72\)\( T_{7}^{46} + \)\(17\!\cdots\!25\)\( T_{7}^{44} + \)\(25\!\cdots\!48\)\( T_{7}^{42} - \)\(30\!\cdots\!04\)\( T_{7}^{40} - \)\(28\!\cdots\!30\)\( T_{7}^{38} + \)\(13\!\cdots\!67\)\( T_{7}^{36} + \)\(12\!\cdots\!50\)\( T_{7}^{34} + \)\(88\!\cdots\!18\)\( T_{7}^{32} + \)\(11\!\cdots\!36\)\( T_{7}^{30} + \)\(10\!\cdots\!40\)\( T_{7}^{28} - \)\(73\!\cdots\!98\)\( T_{7}^{26} - \)\(33\!\cdots\!47\)\( T_{7}^{24} + \)\(47\!\cdots\!64\)\( T_{7}^{22} + \)\(67\!\cdots\!30\)\( T_{7}^{20} + \)\(19\!\cdots\!60\)\( T_{7}^{18} + \)\(26\!\cdots\!53\)\( T_{7}^{16} + \)\(17\!\cdots\!28\)\( T_{7}^{14} + \)\(60\!\cdots\!68\)\( T_{7}^{12} + \)\(10\!\cdots\!68\)\( T_{7}^{10} + \)\(85\!\cdots\!80\)\( T_{7}^{8} + \)\(34\!\cdots\!64\)\( T_{7}^{6} + \)\(45\!\cdots\!28\)\( T_{7}^{4} - \)\(52\!\cdots\!68\)\( T_{7}^{2} + \)\(11\!\cdots\!56\)\( \)">\(T_{7}^{144} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).