Properties

Label 462.2.ba.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 64q - 8q^{4} - 2q^{5} + 16q^{6} + 16q^{7} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q - 8q^{4} - 2q^{5} + 16q^{6} + 16q^{7} - 8q^{9} - 2q^{10} + 4q^{11} + 2q^{14} - 6q^{15} + 8q^{16} + 30q^{17} - 10q^{19} - 20q^{20} + 4q^{21} - 2q^{22} + 4q^{23} + 8q^{24} - 12q^{26} - 20q^{29} - 18q^{30} + 34q^{31} + 8q^{33} - 2q^{35} - 16q^{36} - 14q^{37} + 12q^{38} - 18q^{39} + 12q^{40} + 28q^{41} + 4q^{42} + 6q^{44} - 12q^{45} + 42q^{46} + 24q^{47} - 44q^{49} + 14q^{51} - 32q^{54} + 14q^{55} - 4q^{56} - 10q^{58} - 30q^{59} + 2q^{60} - 28q^{61} + 8q^{62} + 16q^{63} + 16q^{64} - 12q^{65} - 4q^{66} + 16q^{67} - 30q^{68} - 30q^{70} - 24q^{71} - 116q^{73} - 44q^{74} + 12q^{75} - 32q^{77} - 18q^{80} + 8q^{81} - 28q^{82} - 8q^{83} - 2q^{84} - 80q^{85} - 18q^{86} - 10q^{87} - 14q^{88} - 24q^{89} - 4q^{90} + 48q^{91} + 8q^{92} + 76q^{93} + 6q^{94} + 98q^{95} - 8q^{96} - 120q^{97} - 40q^{98} + 8q^{99} + O(q^{100}) \)

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 −1.44711 3.25025i 0.809017 + 0.587785i 1.04897 + 2.42892i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.77892 + 3.08119i
19.2 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i −1.26939 2.85109i 0.809017 + 0.587785i −0.613482 2.57364i −0.951057 + 0.309017i 0.104528 + 0.994522i 1.56045 + 2.70278i
19.3 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.0572849 + 0.128664i 0.809017 + 0.587785i −2.64421 + 0.0903816i −0.951057 + 0.309017i 0.104528 + 0.994522i −0.0704201 0.121971i
19.4 −0.994522 + 0.104528i −0.743145 0.669131i 0.978148 0.207912i 0.949965 + 2.13366i 0.809017 + 0.587785i 2.43658 + 1.03106i −0.951057 + 0.309017i 0.104528 + 0.994522i −1.16779 2.02267i
19.5 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −1.01671 2.28357i 0.809017 + 0.587785i −0.151373 2.64142i 0.951057 0.309017i 0.104528 + 0.994522i −1.24984 2.16479i
19.6 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i −0.748568 1.68131i 0.809017 + 0.587785i 2.42705 + 1.05329i 0.951057 0.309017i 0.104528 + 0.994522i −0.920212 1.59385i
19.7 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 0.316797 + 0.711538i 0.809017 + 0.587785i −0.370234 + 2.61972i 0.951057 0.309017i 0.104528 + 0.994522i 0.389438 + 0.674526i
19.8 0.994522 0.104528i 0.743145 + 0.669131i 0.978148 0.207912i 1.33064 + 2.98866i 0.809017 + 0.587785i 1.90285 1.83825i 0.951057 0.309017i 0.104528 + 0.994522i 1.63575 + 2.83319i
61.1 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −3.24579 + 2.92252i −0.309017 + 0.951057i −0.190680 2.63887i 0.587785 0.809017i 0.978148 + 0.207912i −2.18382 3.78249i
61.2 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i −0.468951 + 0.422245i −0.309017 + 0.951057i −0.177812 + 2.63977i 0.587785 0.809017i 0.978148 + 0.207912i −0.315518 0.546493i
61.3 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 0.131864 0.118731i −0.309017 + 0.951057i −1.17519 2.37043i 0.587785 0.809017i 0.978148 + 0.207912i 0.0887206 + 0.153669i
61.4 −0.207912 + 0.978148i 0.994522 + 0.104528i −0.913545 0.406737i 1.55595 1.40099i −0.309017 + 0.951057i 2.62525 0.328695i 0.587785 0.809017i 0.978148 + 0.207912i 1.04687 + 1.81324i
61.5 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i −1.40387 + 1.26405i −0.309017 + 0.951057i 1.39244 2.24969i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.944546 + 1.63600i
61.6 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i −0.716223 + 0.644890i −0.309017 + 0.951057i 2.17783 + 1.50235i −0.587785 + 0.809017i 0.978148 + 0.207912i 0.481886 + 0.834652i
61.7 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i 0.192170 0.173031i −0.309017 + 0.951057i −1.35536 + 2.27222i −0.587785 + 0.809017i 0.978148 + 0.207912i −0.129295 0.223946i
61.8 0.207912 0.978148i −0.994522 0.104528i −0.913545 0.406737i 2.61658 2.35598i −0.309017 + 0.951057i 1.99807 1.73428i −0.587785 + 0.809017i 0.978148 + 0.207912i −1.76048 3.04924i
73.1 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.44711 + 3.25025i 0.809017 0.587785i 1.04897 2.42892i −0.951057 0.309017i 0.104528 0.994522i 1.77892 3.08119i
73.2 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i −1.26939 + 2.85109i 0.809017 0.587785i −0.613482 + 2.57364i −0.951057 0.309017i 0.104528 0.994522i 1.56045 2.70278i
73.3 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.0572849 0.128664i 0.809017 0.587785i −2.64421 0.0903816i −0.951057 0.309017i 0.104528 0.994522i −0.0704201 + 0.121971i
73.4 −0.994522 0.104528i −0.743145 + 0.669131i 0.978148 + 0.207912i 0.949965 2.13366i 0.809017 0.587785i 2.43658 1.03106i −0.951057 0.309017i 0.104528 0.994522i −1.16779 + 2.02267i
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.b yes 64
7.d odd 6 1 462.2.ba.a 64
11.d odd 10 1 462.2.ba.a 64
77.n even 30 1 inner 462.2.ba.b yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.ba.a 64 7.d odd 6 1
462.2.ba.a 64 11.d odd 10 1
462.2.ba.b yes 64 1.a even 1 1 trivial
462.2.ba.b yes 64 77.n even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(34\!\cdots\!86\)\( T_{5}^{42} - \)\(28\!\cdots\!44\)\( T_{5}^{41} - \)\(19\!\cdots\!45\)\( T_{5}^{40} + \)\(48\!\cdots\!70\)\( T_{5}^{39} + \)\(22\!\cdots\!03\)\( T_{5}^{38} + \)\(73\!\cdots\!44\)\( T_{5}^{37} + \)\(22\!\cdots\!54\)\( T_{5}^{36} + \)\(51\!\cdots\!82\)\( T_{5}^{35} + \)\(12\!\cdots\!50\)\( T_{5}^{34} + \)\(23\!\cdots\!88\)\( T_{5}^{33} + \)\(41\!\cdots\!95\)\( T_{5}^{32} + \)\(61\!\cdots\!06\)\( T_{5}^{31} + \)\(69\!\cdots\!03\)\( T_{5}^{30} + \)\(28\!\cdots\!24\)\( T_{5}^{29} - \)\(18\!\cdots\!78\)\( T_{5}^{28} - \)\(66\!\cdots\!52\)\( T_{5}^{27} - \)\(17\!\cdots\!76\)\( T_{5}^{26} - \)\(32\!\cdots\!52\)\( T_{5}^{25} - \)\(51\!\cdots\!00\)\( T_{5}^{24} - \)\(57\!\cdots\!28\)\( T_{5}^{23} - \)\(19\!\cdots\!83\)\( T_{5}^{22} + \)\(75\!\cdots\!84\)\( T_{5}^{21} + \)\(28\!\cdots\!24\)\( T_{5}^{20} + \)\(54\!\cdots\!50\)\( T_{5}^{19} + \)\(83\!\cdots\!11\)\( T_{5}^{18} + \)\(10\!\cdots\!50\)\( T_{5}^{17} + \)\(10\!\cdots\!70\)\( T_{5}^{16} + \)\(97\!\cdots\!04\)\( T_{5}^{15} + \)\(77\!\cdots\!62\)\( T_{5}^{14} + \)\(52\!\cdots\!06\)\( T_{5}^{13} + \)\(29\!\cdots\!43\)\( T_{5}^{12} + \)\(14\!\cdots\!70\)\( T_{5}^{11} + \)\(57\!\cdots\!74\)\( T_{5}^{10} + \)\(12\!\cdots\!24\)\( T_{5}^{9} - \)\(58\!\cdots\!02\)\( T_{5}^{8} + \)\(11\!\cdots\!54\)\( T_{5}^{7} + \)\(69\!\cdots\!27\)\( T_{5}^{6} + \)\(18\!\cdots\!80\)\( T_{5}^{5} - \)\(14\!\cdots\!42\)\( T_{5}^{4} + \)\(50\!\cdots\!32\)\( T_{5}^{3} + \)\(13\!\cdots\!74\)\( T_{5}^{2} - \)\(50\!\cdots\!18\)\( T_{5} + \)\(12\!\cdots\!41\)\( \)">\(T_{5}^{64} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).