Properties

Label 462.2.w.a
Level $462$
Weight $2$
Character orbit 462.w
Analytic conductor $3.689$
Analytic rank $0$
Dimension $48$
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.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 12q^{2} - 4q^{3} - 12q^{4} + 6q^{6} - 12q^{8} + 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 12q^{2} - 4q^{3} - 12q^{4} + 6q^{6} - 12q^{8} + 10q^{9} - 8q^{11} + 6q^{12} + 36q^{15} - 12q^{16} - 24q^{17} + 30q^{19} - 8q^{22} - 4q^{24} + 18q^{25} - 20q^{26} - 22q^{27} + 8q^{29} - 24q^{30} - 32q^{31} + 48q^{32} - 14q^{33} - 4q^{34} + 6q^{35} - 10q^{36} - 20q^{37} + 20q^{38} - 6q^{39} + 22q^{44} + 12q^{45} + 20q^{46} + 20q^{47} - 4q^{48} + 12q^{49} + 28q^{50} + 8q^{51} + 20q^{52} + 20q^{53} + 18q^{54} + 16q^{55} - 2q^{57} + 8q^{58} + 30q^{59} - 4q^{60} - 20q^{61} + 8q^{62} + 4q^{63} - 12q^{64} - 14q^{66} + 36q^{67} - 24q^{68} - 70q^{69} - 4q^{70} + 10q^{72} - 20q^{73} - 20q^{74} - 30q^{75} - 16q^{77} - 16q^{78} - 20q^{79} + 26q^{81} - 10q^{82} - 46q^{83} - 10q^{84} + 10q^{85} - 30q^{86} + 8q^{87} - 8q^{88} - 38q^{90} - 36q^{91} + 10q^{92} - 36q^{93} + 50q^{95} - 4q^{96} - 2q^{97} - 48q^{98} + 70q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.809017 0.587785i −1.67853 0.427250i 0.309017 + 0.951057i 0.354145 + 0.487439i 1.10683 + 1.33227i −0.951057 + 0.309017i 0.309017 0.951057i 2.63492 + 1.43430i 0.602508i
29.2 −0.809017 0.587785i −1.63583 0.569257i 0.309017 + 0.951057i −0.732542 1.00826i 0.988815 + 1.42206i 0.951057 0.309017i 0.309017 0.951057i 2.35189 + 1.86242i 1.24628i
29.3 −0.809017 0.587785i −1.48497 + 0.891547i 0.309017 + 0.951057i 0.456676 + 0.628560i 1.72541 + 0.151568i 0.951057 0.309017i 0.309017 0.951057i 1.41029 2.64785i 0.776943i
29.4 −0.809017 0.587785i −1.31903 + 1.12257i 0.309017 + 0.951057i −1.88925 2.60033i 1.72695 0.132877i −0.951057 + 0.309017i 0.309017 0.951057i 0.479659 2.96141i 3.21418i
29.5 −0.809017 0.587785i −0.911344 1.47291i 0.309017 + 0.951057i 2.04373 + 2.81296i −0.128460 + 1.72728i 0.951057 0.309017i 0.309017 0.951057i −1.33891 + 2.68465i 3.47700i
29.6 −0.809017 0.587785i −0.520106 1.65212i 0.309017 + 0.951057i −1.54008 2.11973i −0.550316 + 1.64230i −0.951057 + 0.309017i 0.309017 0.951057i −2.45898 + 1.71855i 2.62013i
29.7 −0.809017 0.587785i 0.261877 1.71214i 0.309017 + 0.951057i 1.43689 + 1.97771i −1.21823 + 1.23122i −0.951057 + 0.309017i 0.309017 0.951057i −2.86284 0.896741i 2.44458i
29.8 −0.809017 0.587785i 0.455621 + 1.67105i 0.309017 + 0.951057i −1.96876 2.70976i 0.613613 1.61972i 0.951057 0.309017i 0.309017 0.951057i −2.58482 + 1.52273i 3.34945i
29.9 −0.809017 0.587785i 0.856700 + 1.50535i 0.309017 + 0.951057i 1.72358 + 2.37230i 0.191735 1.72141i 0.951057 0.309017i 0.309017 0.951057i −1.53213 + 2.57926i 2.93233i
29.10 −0.809017 0.587785i 1.61613 + 0.622992i 0.309017 + 0.951057i −1.77599 2.44445i −0.941292 1.45395i −0.951057 + 0.309017i 0.309017 0.951057i 2.22376 + 2.01367i 3.02150i
29.11 −0.809017 0.587785i 1.63204 0.580034i 0.309017 + 0.951057i −0.571630 0.786781i −1.66128 0.490033i 0.951057 0.309017i 0.309017 0.951057i 2.32712 1.89328i 0.972514i
29.12 −0.809017 0.587785i 1.72744 0.126348i 0.309017 + 0.951057i 2.46323 + 3.39035i −1.47179 0.913144i −0.951057 + 0.309017i 0.309017 0.951057i 2.96807 0.436515i 4.19070i
239.1 −0.809017 + 0.587785i −1.67853 + 0.427250i 0.309017 0.951057i 0.354145 0.487439i 1.10683 1.33227i −0.951057 0.309017i 0.309017 + 0.951057i 2.63492 1.43430i 0.602508i
239.2 −0.809017 + 0.587785i −1.63583 + 0.569257i 0.309017 0.951057i −0.732542 + 1.00826i 0.988815 1.42206i 0.951057 + 0.309017i 0.309017 + 0.951057i 2.35189 1.86242i 1.24628i
239.3 −0.809017 + 0.587785i −1.48497 0.891547i 0.309017 0.951057i 0.456676 0.628560i 1.72541 0.151568i 0.951057 + 0.309017i 0.309017 + 0.951057i 1.41029 + 2.64785i 0.776943i
239.4 −0.809017 + 0.587785i −1.31903 1.12257i 0.309017 0.951057i −1.88925 + 2.60033i 1.72695 + 0.132877i −0.951057 0.309017i 0.309017 + 0.951057i 0.479659 + 2.96141i 3.21418i
239.5 −0.809017 + 0.587785i −0.911344 + 1.47291i 0.309017 0.951057i 2.04373 2.81296i −0.128460 1.72728i 0.951057 + 0.309017i 0.309017 + 0.951057i −1.33891 2.68465i 3.47700i
239.6 −0.809017 + 0.587785i −0.520106 + 1.65212i 0.309017 0.951057i −1.54008 + 2.11973i −0.550316 1.64230i −0.951057 0.309017i 0.309017 + 0.951057i −2.45898 1.71855i 2.62013i
239.7 −0.809017 + 0.587785i 0.261877 + 1.71214i 0.309017 0.951057i 1.43689 1.97771i −1.21823 1.23122i −0.951057 0.309017i 0.309017 + 0.951057i −2.86284 + 0.896741i 2.44458i
239.8 −0.809017 + 0.587785i 0.455621 1.67105i 0.309017 0.951057i −1.96876 + 2.70976i 0.613613 + 1.61972i 0.951057 + 0.309017i 0.309017 + 0.951057i −2.58482 1.52273i 3.34945i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 365.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.w.a 48
3.b odd 2 1 462.2.w.b yes 48
11.d odd 10 1 462.2.w.b yes 48
33.f even 10 1 inner 462.2.w.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.w.a 48 1.a even 1 1 trivial
462.2.w.a 48 33.f even 10 1 inner
462.2.w.b yes 48 3.b odd 2 1
462.2.w.b yes 48 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(34\!\cdots\!30\)\( T_{5}^{26} - \)\(37\!\cdots\!00\)\( T_{5}^{25} + \)\(18\!\cdots\!66\)\( T_{5}^{24} + \)\(24\!\cdots\!40\)\( T_{5}^{23} - \)\(81\!\cdots\!48\)\( T_{5}^{22} - \)\(11\!\cdots\!40\)\( T_{5}^{21} + \)\(37\!\cdots\!22\)\( T_{5}^{20} + \)\(62\!\cdots\!40\)\( T_{5}^{19} - \)\(83\!\cdots\!20\)\( T_{5}^{18} - \)\(20\!\cdots\!80\)\( T_{5}^{17} + \)\(58\!\cdots\!61\)\( T_{5}^{16} + \)\(38\!\cdots\!40\)\( T_{5}^{15} + \)\(12\!\cdots\!21\)\( T_{5}^{14} - \)\(41\!\cdots\!80\)\( T_{5}^{13} - \)\(24\!\cdots\!10\)\( T_{5}^{12} + \)\(25\!\cdots\!00\)\( T_{5}^{11} + \)\(18\!\cdots\!88\)\( T_{5}^{10} - \)\(15\!\cdots\!00\)\( T_{5}^{9} - \)\(83\!\cdots\!75\)\( T_{5}^{8} + \)\(16\!\cdots\!60\)\( T_{5}^{7} + \)\(10\!\cdots\!24\)\( T_{5}^{6} - \)\(30\!\cdots\!40\)\( T_{5}^{5} + \)\(37\!\cdots\!68\)\( T_{5}^{4} + \)\(26\!\cdots\!60\)\( T_{5}^{3} + \)\(32\!\cdots\!92\)\( T_{5}^{2} - \)\(27\!\cdots\!40\)\( T_{5} + \)\(29\!\cdots\!36\)\( \)">\(T_{5}^{48} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).