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Properties

Label 930.2.bn.a
Level $930$
Weight $2$
Character orbit 930.bn
Analytic conductor $7.426$
Analytic rank $0$
Dimension $112$
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: \(112\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112q + 28q^{4} - 2q^{5} + 56q^{6} - 14q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 112q + 28q^{4} - 2q^{5} + 56q^{6} - 14q^{9} - 4q^{10} + 18q^{11} + 8q^{14} + 8q^{15} - 28q^{16} + 16q^{19} + 2q^{20} + 28q^{21} + 14q^{24} + 14q^{25} + 12q^{26} + 16q^{29} - 4q^{30} + 10q^{34} - 38q^{35} - 56q^{36} + 16q^{39} - 6q^{40} + 20q^{41} + 2q^{44} + 2q^{45} - 2q^{46} + 38q^{49} + 8q^{50} - 10q^{51} - 28q^{54} - 46q^{55} + 12q^{56} + 60q^{59} - 8q^{60} + 88q^{61} + 28q^{64} - 28q^{65} + 6q^{66} + 46q^{69} + 26q^{70} + 116q^{71} - 34q^{74} + 8q^{75} + 24q^{76} - 40q^{79} - 12q^{80} + 14q^{81} - 8q^{84} + 18q^{85} - 38q^{86} - 60q^{89} + 4q^{90} - 92q^{91} + 132q^{94} + 132q^{95} - 14q^{96} + 2q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 −2.15286 − 0.604297i 0.500000 + 0.866025i 2.63601 + 2.37347i 0.951057 + 0.309017i −0.669131 − 0.743145i 1.75431 − 1.38651i
19.2 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i −1.44692 − 1.70482i 0.500000 + 0.866025i −3.41731 − 3.07696i 0.951057 + 0.309017i −0.669131 − 0.743145i 2.22971 − 0.168512i
19.3 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i −0.509764 + 2.17719i 0.500000 + 0.866025i 0.159624 + 0.143726i 0.951057 + 0.309017i −0.669131 − 0.743145i −1.46175 − 1.69213i
19.4 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i −0.181057 + 2.22873i 0.500000 + 0.866025i 3.89646 + 3.50839i 0.951057 + 0.309017i −0.669131 − 0.743145i −1.69665 − 1.45649i
19.5 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i 0.230445 − 2.22416i 0.500000 + 0.866025i 0.872484 + 0.785588i 0.951057 + 0.309017i −0.669131 − 0.743145i 1.66393 + 1.49376i
19.6 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i 1.47095 + 1.68413i 0.500000 + 0.866025i −2.07600 − 1.86924i 0.951057 + 0.309017i −0.669131 − 0.743145i −2.22710 + 0.200114i
19.7 −0.587785 + 0.809017i 0.406737 − 0.913545i −0.309017 − 0.951057i 2.21090 + 0.334560i 0.500000 + 0.866025i −0.274259 − 0.246944i 0.951057 + 0.309017i −0.669131 − 0.743145i −1.57020 + 1.59200i
19.8 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i −2.19398 − 0.431813i 0.500000 + 0.866025i 2.07600 + 1.86924i −0.951057 − 0.309017i −0.669131 − 0.743145i −1.63893 + 1.52115i
19.9 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i −1.83960 + 1.27116i 0.500000 + 0.866025i −3.89646 − 3.50839i −0.951057 − 0.309017i −0.669131 − 0.743145i −0.0529000 + 2.23544i
19.10 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i −1.63062 + 1.53006i 0.500000 + 0.866025i −0.159624 − 0.143726i −0.951057 − 0.309017i −0.669131 − 0.743145i 0.279393 + 2.21854i
19.11 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i −1.39519 − 1.74741i 0.500000 + 0.866025i 0.274259 + 0.246944i −0.951057 − 0.309017i −0.669131 − 0.743145i −2.23376 + 0.101626i
19.12 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i 1.59977 + 1.56229i 0.500000 + 0.866025i −2.63601 − 2.37347i −0.951057 − 0.309017i −0.669131 − 0.743145i 2.20424 − 0.375951i
19.13 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i 1.81096 − 1.31165i 0.500000 + 0.866025i −0.872484 − 0.785588i −0.951057 − 0.309017i −0.669131 − 0.743145i 0.00330584 − 2.23607i
19.14 0.587785 − 0.809017i −0.406737 + 0.913545i −0.309017 − 0.951057i 2.19988 + 0.400657i 0.500000 + 0.866025i 3.41731 + 3.07696i −0.951057 − 0.309017i −0.669131 − 0.743145i 1.61720 − 1.54424i
49.1 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −2.15286 + 0.604297i 0.500000 − 0.866025i 2.63601 − 2.37347i 0.951057 − 0.309017i −0.669131 + 0.743145i 1.75431 + 1.38651i
49.2 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −1.44692 + 1.70482i 0.500000 − 0.866025i −3.41731 + 3.07696i 0.951057 − 0.309017i −0.669131 + 0.743145i 2.22971 + 0.168512i
49.3 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −0.509764 − 2.17719i 0.500000 − 0.866025i 0.159624 − 0.143726i 0.951057 − 0.309017i −0.669131 + 0.743145i −1.46175 + 1.69213i
49.4 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i −0.181057 − 2.22873i 0.500000 − 0.866025i 3.89646 − 3.50839i 0.951057 − 0.309017i −0.669131 + 0.743145i −1.69665 + 1.45649i
49.5 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i 0.230445 + 2.22416i 0.500000 − 0.866025i 0.872484 − 0.785588i 0.951057 − 0.309017i −0.669131 + 0.743145i 1.66393 − 1.49376i
49.6 −0.587785 − 0.809017i 0.406737 + 0.913545i −0.309017 + 0.951057i 1.47095 − 1.68413i 0.500000 − 0.866025i −2.07600 + 1.86924i 0.951057 − 0.309017i −0.669131 + 0.743145i −2.22710 − 0.200114i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.14
Significant digits:
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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.a ✓ 112
5.b even 2 1 inner 930.2.bn.a ✓ 112
31.g even 15 1 inner 930.2.bn.a ✓ 112
155.u even 30 1 inner 930.2.bn.a ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bn.a ✓ 112 1.a even 1 1 trivial
930.2.bn.a ✓ 112 5.b even 2 1 inner
930.2.bn.a ✓ 112 31.g even 15 1 inner
930.2.bn.a ✓ 112 155.u even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!30\)\( T_{7}^{94} + \)\(36\!\cdots\!28\)\( T_{7}^{92} + \)\(36\!\cdots\!88\)\( T_{7}^{90} - \)\(11\!\cdots\!79\)\( T_{7}^{88} - \)\(92\!\cdots\!06\)\( T_{7}^{86} - \)\(15\!\cdots\!71\)\( T_{7}^{84} - \)\(11\!\cdots\!68\)\( T_{7}^{82} + \)\(52\!\cdots\!42\)\( T_{7}^{80} + \)\(25\!\cdots\!64\)\( T_{7}^{78} + \)\(32\!\cdots\!51\)\( T_{7}^{76} + \)\(19\!\cdots\!92\)\( T_{7}^{74} - \)\(69\!\cdots\!43\)\( T_{7}^{72} - \)\(14\!\cdots\!96\)\( T_{7}^{70} - \)\(15\!\cdots\!59\)\( T_{7}^{68} - \)\(75\!\cdots\!22\)\( T_{7}^{66} + \)\(75\!\cdots\!31\)\( T_{7}^{64} + \)\(43\!\cdots\!52\)\( T_{7}^{62} + \)\(35\!\cdots\!38\)\( T_{7}^{60} + \)\(12\!\cdots\!24\)\( T_{7}^{58} - \)\(11\!\cdots\!15\)\( T_{7}^{56} - \)\(36\!\cdots\!22\)\( T_{7}^{54} - \)\(18\!\cdots\!04\)\( T_{7}^{52} - \)\(27\!\cdots\!96\)\( T_{7}^{50} + \)\(14\!\cdots\!87\)\( T_{7}^{48} + \)\(98\!\cdots\!46\)\( T_{7}^{46} + \)\(26\!\cdots\!52\)\( T_{7}^{44} + \)\(19\!\cdots\!00\)\( T_{7}^{42} - \)\(98\!\cdots\!31\)\( T_{7}^{40} - \)\(37\!\cdots\!86\)\( T_{7}^{38} - \)\(55\!\cdots\!68\)\( T_{7}^{36} - \)\(32\!\cdots\!46\)\( T_{7}^{34} + \)\(16\!\cdots\!68\)\( T_{7}^{32} + \)\(43\!\cdots\!70\)\( T_{7}^{30} + \)\(68\!\cdots\!54\)\( T_{7}^{28} + \)\(75\!\cdots\!40\)\( T_{7}^{26} + \)\(54\!\cdots\!41\)\( T_{7}^{24} + \)\(20\!\cdots\!22\)\( T_{7}^{22} + \)\(35\!\cdots\!79\)\( T_{7}^{20} + \)\(30\!\cdots\!70\)\( T_{7}^{18} + \)\(63\!\cdots\!73\)\( T_{7}^{16} - \)\(59\!\cdots\!96\)\( T_{7}^{14} + \)\(10\!\cdots\!48\)\( T_{7}^{12} - \)\(22\!\cdots\!24\)\( T_{7}^{10} + \)\(62\!\cdots\!24\)\( T_{7}^{8} - \)\(63\!\cdots\!00\)\( T_{7}^{6} + \)\(18\!\cdots\!92\)\( T_{7}^{4} - \)\(20\!\cdots\!96\)\( T_{7}^{2} + \)\(81\!\cdots\!36\)\( \)">\(T_{7}^{112} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).