Properties

Label 462.2.w.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 48q + 12q^{2} - 4q^{3} - 12q^{4} + 4q^{6} + 12q^{8} + 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 12q^{2} - 4q^{3} - 12q^{4} + 4q^{6} + 12q^{8} + 10q^{9} + 8q^{11} + 6q^{12} - 24q^{15} - 12q^{16} + 24q^{17} + 10q^{18} + 30q^{19} - 8q^{22} - 6q^{24} + 18q^{25} + 20q^{26} + 38q^{27} - 8q^{29} - 36q^{30} - 32q^{31} - 48q^{32} + 24q^{33} - 4q^{34} - 6q^{35} - 20q^{37} - 20q^{38} - 34q^{39} - 22q^{44} + 12q^{45} + 20q^{46} - 20q^{47} - 4q^{48} + 12q^{49} - 28q^{50} + 2q^{51} + 20q^{52} - 20q^{53} - 18q^{54} + 16q^{55} + 12q^{57} + 8q^{58} - 30q^{59} - 4q^{60} - 20q^{61} - 8q^{62} - 4q^{63} - 12q^{64} + 46q^{66} + 36q^{67} + 24q^{68} + 30q^{69} - 4q^{70} - 10q^{72} - 20q^{73} + 20q^{74} + 40q^{75} + 16q^{77} - 16q^{78} - 20q^{79} - 54q^{81} - 10q^{82} + 46q^{83} - 10q^{84} + 10q^{85} + 30q^{86} - 8q^{87} - 8q^{88} - 62q^{90} - 36q^{91} - 10q^{92} + 44q^{93} - 50q^{95} + 4q^{96} - 2q^{97} + 48q^{98} - 26q^{99} + O(q^{100}) \)

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 −1.67366 0.445927i 0.309017 + 0.951057i 1.77599 + 2.44445i −1.09191 1.34452i −0.951057 + 0.309017i −0.309017 + 0.951057i 2.60230 + 1.49266i 3.02150i
29.2 0.809017 + 0.587785i −1.57790 + 0.714295i 0.309017 + 0.951057i −1.72358 2.37230i −1.69640 0.349593i 0.951057 0.309017i −0.309017 + 0.951057i 1.97957 2.25418i 2.93233i
29.3 0.809017 + 0.587785i −1.35082 + 1.08410i 0.309017 + 0.951057i 1.96876 + 2.70976i −1.73006 + 0.0830614i 0.951057 0.309017i −0.309017 + 0.951057i 0.649451 2.92886i 3.34945i
29.4 0.809017 + 0.587785i −1.32326 1.11758i 0.309017 + 0.951057i −2.46323 3.39035i −0.413643 1.68193i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.502035 + 2.95770i 4.19070i
29.5 0.809017 + 0.587785i −0.979414 1.42855i 0.309017 + 0.951057i 0.571630 + 0.786781i 0.0473160 1.73140i 0.951057 0.309017i −0.309017 + 0.951057i −1.08149 + 2.79828i 0.972514i
29.6 0.809017 + 0.587785i 0.407282 + 1.68348i 0.309017 + 0.951057i 1.88925 + 2.60033i −0.660029 + 1.60136i −0.951057 + 0.309017i −0.309017 + 0.951057i −2.66824 + 1.37131i 3.21418i
29.7 0.809017 + 0.587785i 0.677330 + 1.59412i 0.309017 + 0.951057i −0.456676 0.628560i −0.389030 + 1.68780i 0.951057 0.309017i −0.309017 + 0.951057i −2.08245 + 2.15949i 0.776943i
29.8 0.809017 + 0.587785i 0.794507 1.53908i 0.309017 + 0.951057i −1.43689 1.97771i 1.54742 0.778140i −0.951057 + 0.309017i −0.309017 + 0.951057i −1.73752 2.44562i 2.44458i
29.9 0.809017 + 0.587785i 1.39186 1.03088i 0.309017 + 0.951057i 1.54008 + 2.11973i 1.73198 0.0158825i −0.951057 + 0.309017i −0.309017 + 0.951057i 0.874572 2.86969i 2.62013i
29.10 0.809017 + 0.587785i 1.60305 0.655932i 0.309017 + 0.951057i −2.04373 2.81296i 1.68244 + 0.411586i 0.951057 0.309017i −0.309017 + 0.951057i 2.13951 2.10298i 3.47700i
29.11 0.809017 + 0.587785i 1.60909 + 0.640962i 0.309017 + 0.951057i −0.354145 0.487439i 0.925033 + 1.46435i −0.951057 + 0.309017i −0.309017 + 0.951057i 2.17834 + 2.06273i 0.602508i
29.12 0.809017 + 0.587785i 1.65802 + 0.500979i 0.309017 + 0.951057i 0.732542 + 1.00826i 1.04690 + 1.37986i 0.951057 0.309017i −0.309017 + 0.951057i 2.49804 + 1.66126i 1.24628i
239.1 0.809017 0.587785i −1.67366 + 0.445927i 0.309017 0.951057i 1.77599 2.44445i −1.09191 + 1.34452i −0.951057 0.309017i −0.309017 0.951057i 2.60230 1.49266i 3.02150i
239.2 0.809017 0.587785i −1.57790 0.714295i 0.309017 0.951057i −1.72358 + 2.37230i −1.69640 + 0.349593i 0.951057 + 0.309017i −0.309017 0.951057i 1.97957 + 2.25418i 2.93233i
239.3 0.809017 0.587785i −1.35082 1.08410i 0.309017 0.951057i 1.96876 2.70976i −1.73006 0.0830614i 0.951057 + 0.309017i −0.309017 0.951057i 0.649451 + 2.92886i 3.34945i
239.4 0.809017 0.587785i −1.32326 + 1.11758i 0.309017 0.951057i −2.46323 + 3.39035i −0.413643 + 1.68193i −0.951057 0.309017i −0.309017 0.951057i 0.502035 2.95770i 4.19070i
239.5 0.809017 0.587785i −0.979414 + 1.42855i 0.309017 0.951057i 0.571630 0.786781i 0.0473160 + 1.73140i 0.951057 + 0.309017i −0.309017 0.951057i −1.08149 2.79828i 0.972514i
239.6 0.809017 0.587785i 0.407282 1.68348i 0.309017 0.951057i 1.88925 2.60033i −0.660029 1.60136i −0.951057 0.309017i −0.309017 0.951057i −2.66824 1.37131i 3.21418i
239.7 0.809017 0.587785i 0.677330 1.59412i 0.309017 0.951057i −0.456676 + 0.628560i −0.389030 1.68780i 0.951057 + 0.309017i −0.309017 0.951057i −2.08245 2.15949i 0.776943i
239.8 0.809017 0.587785i 0.794507 + 1.53908i 0.309017 0.951057i −1.43689 + 1.97771i 1.54742 + 0.778140i −0.951057 0.309017i −0.309017 0.951057i −1.73752 + 2.44562i 2.44458i
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.b yes 48
3.b odd 2 1 462.2.w.a 48
11.d odd 10 1 462.2.w.a 48
33.f even 10 1 inner 462.2.w.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.w.a 48 3.b odd 2 1
462.2.w.a 48 11.d odd 10 1
462.2.w.b yes 48 1.a even 1 1 trivial
462.2.w.b yes 48 33.f even 10 1 inner

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 \(S_{2}^{\mathrm{new}}(462, [\chi])\).