Properties

Label 462.2.ba.a
Level $462$
Weight $2$
Character orbit 462.ba
Analytic conductor $3.689$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.ba (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64q - 8q^{4} - 22q^{5} - 16q^{6} + 4q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q - 8q^{4} - 22q^{5} - 16q^{6} + 4q^{7} - 8q^{9} + 2q^{10} + 4q^{11} + 2q^{14} - 6q^{15} + 8q^{16} + 30q^{17} + 10q^{19} + 20q^{20} - 4q^{21} - 2q^{22} + 4q^{23} - 8q^{24} - 12q^{26} - 10q^{28} - 20q^{29} + 18q^{30} - 16q^{31} - 14q^{33} + 42q^{35} - 16q^{36} - 14q^{37} + 12q^{38} + 18q^{39} + 18q^{40} - 28q^{41} - 6q^{42} + 6q^{44} - 12q^{45} - 42q^{46} + 24q^{47} + 116q^{49} + 26q^{51} + 32q^{54} - 14q^{55} - 4q^{56} + 20q^{58} + 30q^{59} + 2q^{60} - 32q^{61} - 8q^{62} + 4q^{63} + 16q^{64} + 12q^{65} + 4q^{66} + 16q^{67} - 30q^{68} - 20q^{70} - 24q^{71} - 64q^{73} + 4q^{74} + 12q^{75} - 48q^{77} - 60q^{79} - 18q^{80} + 8q^{81} - 68q^{82} + 8q^{83} + 2q^{84} - 80q^{85} - 18q^{86} + 10q^{87} - 8q^{88} - 24q^{89} + 4q^{90} - 172q^{91} + 8q^{92} - 104q^{93} - 6q^{94} - 118q^{95} + 8q^{96} + 120q^{97} + 40q^{98} + 8q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.993461 2.23135i −0.809017 0.587785i 2.46197 0.968858i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.22126 + 2.11528i
19.2 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.716911 1.61021i −0.809017 0.587785i 1.87226 + 1.86939i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.881296 + 1.52645i
19.3 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.281694 0.632696i −0.809017 0.587785i −2.57173 0.621464i −0.951057 + 0.309017i 0.104528 + 0.994522i 0.346286 + 0.599785i
19.4 −0.994522 + 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 1.49050 + 3.34771i −0.809017 0.587785i −2.49930 0.868052i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.83226 3.17357i
19.5 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i −1.50643 3.38350i −0.809017 0.587785i 2.60972 0.435184i 0.951057 0.309017i 0.104528 + 0.994522i −1.85185 3.20750i
19.6 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i −0.0779848 0.175157i −0.809017 0.587785i −1.57120 2.12869i 0.951057 0.309017i 0.104528 + 0.994522i −0.0958664 0.166046i
19.7 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.688924 + 1.54735i −0.809017 0.587785i 0.640384 + 2.56708i 0.951057 0.309017i 0.104528 + 0.994522i 0.846892 + 1.46686i
19.8 0.994522 0.104528i −0.743145 0.669131i 0.978148 0.207912i 1.48256 + 3.32989i −0.809017 0.587785i 1.16475 2.37557i 0.951057 0.309017i 0.104528 + 0.994522i 1.82251 + 3.15668i
61.1 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −2.80028 + 2.52138i 0.309017 0.951057i −2.64322 + 0.115723i 0.587785 0.809017i 0.978148 + 0.207912i −1.88407 3.26331i
61.2 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −1.32335 + 1.19155i 0.309017 0.951057i 2.59833 0.498689i 0.587785 0.809017i 0.978148 + 0.207912i −0.890371 1.54217i
61.3 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i −0.624935 + 0.562694i 0.309017 0.951057i 1.46102 2.20577i 0.587785 0.809017i 0.978148 + 0.207912i −0.420466 0.728269i
61.4 −0.207912 + 0.978148i −0.994522 0.104528i −0.913545 0.406737i 1.09698 0.987728i 0.309017 0.951057i −2.51719 + 0.814704i 0.587785 0.809017i 0.978148 + 0.207912i 0.738068 + 1.27837i
61.5 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −2.17695 + 1.96014i 0.309017 0.951057i −2.29320 + 1.31956i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.46469 + 2.53692i
61.6 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −2.02809 + 1.82610i 0.309017 0.951057i 2.53777 + 0.748146i −0.587785 + 0.809017i 0.978148 + 0.207912i 1.36453 + 2.36343i
61.7 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −1.26898 + 1.14260i 0.309017 0.951057i −0.554537 2.58698i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.853791 + 1.47881i
61.8 0.207912 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 2.13316 1.92070i 0.309017 0.951057i 2.34031 + 1.23407i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.43522 2.48588i
73.1 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.993461 + 2.23135i −0.809017 + 0.587785i 2.46197 + 0.968858i −0.951057 0.309017i 0.104528 0.994522i 1.22126 2.11528i
73.2 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.716911 + 1.61021i −0.809017 + 0.587785i 1.87226 1.86939i −0.951057 0.309017i 0.104528 0.994522i 0.881296 1.52645i
73.3 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i −0.281694 + 0.632696i −0.809017 + 0.587785i −2.57173 + 0.621464i −0.951057 0.309017i 0.104528 0.994522i 0.346286 0.599785i
73.4 −0.994522 0.104528i 0.743145 0.669131i 0.978148 + 0.207912i 1.49050 3.34771i −0.809017 + 0.587785i −2.49930 + 0.868052i −0.951057 0.309017i 0.104528 0.994522i −1.83226 + 3.17357i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 409.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.ba.a 64
7.d odd 6 1 462.2.ba.b yes 64
11.d odd 10 1 462.2.ba.b yes 64
77.n even 30 1 inner 462.2.ba.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.ba.a 64 1.a even 1 1 trivial
462.2.ba.a 64 77.n even 30 1 inner
462.2.ba.b yes 64 7.d odd 6 1
462.2.ba.b yes 64 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(22\!\cdots\!78\)\( T_{5}^{45} + \)\(87\!\cdots\!86\)\( T_{5}^{44} + \)\(27\!\cdots\!10\)\( T_{5}^{43} + \)\(78\!\cdots\!79\)\( T_{5}^{42} + \)\(19\!\cdots\!96\)\( T_{5}^{41} + \)\(39\!\cdots\!35\)\( T_{5}^{40} + \)\(48\!\cdots\!70\)\( T_{5}^{39} - \)\(59\!\cdots\!22\)\( T_{5}^{38} - \)\(59\!\cdots\!56\)\( T_{5}^{37} - \)\(21\!\cdots\!76\)\( T_{5}^{36} - \)\(54\!\cdots\!18\)\( T_{5}^{35} - \)\(10\!\cdots\!40\)\( T_{5}^{34} - \)\(13\!\cdots\!72\)\( T_{5}^{33} - \)\(28\!\cdots\!70\)\( T_{5}^{32} + \)\(43\!\cdots\!56\)\( T_{5}^{31} + \)\(16\!\cdots\!43\)\( T_{5}^{30} + \)\(40\!\cdots\!34\)\( T_{5}^{29} + \)\(83\!\cdots\!37\)\( T_{5}^{28} + \)\(14\!\cdots\!48\)\( T_{5}^{27} + \)\(24\!\cdots\!49\)\( T_{5}^{26} + \)\(36\!\cdots\!28\)\( T_{5}^{25} + \)\(50\!\cdots\!35\)\( T_{5}^{24} + \)\(65\!\cdots\!42\)\( T_{5}^{23} + \)\(77\!\cdots\!82\)\( T_{5}^{22} + \)\(78\!\cdots\!64\)\( T_{5}^{21} + \)\(62\!\cdots\!84\)\( T_{5}^{20} + \)\(29\!\cdots\!00\)\( T_{5}^{19} - \)\(94\!\cdots\!29\)\( T_{5}^{18} - \)\(38\!\cdots\!90\)\( T_{5}^{17} - \)\(49\!\cdots\!65\)\( T_{5}^{16} - \)\(41\!\cdots\!66\)\( T_{5}^{15} - \)\(20\!\cdots\!13\)\( T_{5}^{14} - \)\(27\!\cdots\!24\)\( T_{5}^{13} + \)\(77\!\cdots\!03\)\( T_{5}^{12} + \)\(86\!\cdots\!50\)\( T_{5}^{11} + \)\(68\!\cdots\!94\)\( T_{5}^{10} + \)\(32\!\cdots\!14\)\( T_{5}^{9} + \)\(14\!\cdots\!53\)\( T_{5}^{8} + \)\(21\!\cdots\!84\)\( T_{5}^{7} + \)\(64\!\cdots\!72\)\( T_{5}^{6} - \)\(81\!\cdots\!60\)\( T_{5}^{5} + \)\(19\!\cdots\!98\)\( T_{5}^{4} - \)\(60\!\cdots\!78\)\( T_{5}^{3} + \)\(26\!\cdots\!79\)\( T_{5}^{2} - \)\(42\!\cdots\!58\)\( T_{5} + \)\(12\!\cdots\!41\)\( \)">\(T_{5}^{64} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).