Properties

Label 462.2.s.a
Level $462$
Weight $2$
Character orbit 462.s
Analytic conductor $3.689$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.s (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128q + 32q^{4} - 4q^{7} + 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 128q + 32q^{4} - 4q^{7} + 12q^{9} + 12q^{15} - 32q^{16} - 16q^{18} - 20q^{21} - 36q^{25} - 6q^{28} + 8q^{36} - 16q^{37} + 60q^{39} + 4q^{42} - 64q^{43} - 8q^{46} - 40q^{49} - 28q^{51} + 88q^{57} + 92q^{58} + 8q^{60} - 42q^{63} + 32q^{64} - 64q^{67} - 26q^{70} - 24q^{72} - 64q^{78} + 136q^{79} - 116q^{81} - 28q^{85} - 40q^{91} - 76q^{93} - 160q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −0.587785 + 0.809017i −1.72594 + 0.145412i −0.309017 0.951057i 0.826181 0.600255i 0.896839 1.48178i −1.44871 + 2.21388i 0.951057 + 0.309017i 2.95771 0.501942i 1.02122i
125.2 −0.587785 + 0.809017i −1.70765 + 0.289739i −0.309017 0.951057i 0.820027 0.595785i 0.769325 1.55182i −1.08488 2.41309i 0.951057 + 0.309017i 2.83210 0.989542i 1.01361i
125.3 −0.587785 + 0.809017i −1.63638 0.567690i −0.309017 0.951057i −3.26960 + 2.37551i 1.42111 0.990177i 1.78115 + 1.95640i 0.951057 + 0.309017i 2.35546 + 1.85791i 4.04145i
125.4 −0.587785 + 0.809017i −1.24874 1.20027i −0.309017 0.951057i 2.87147 2.08625i 1.70503 0.304749i 1.88621 1.85532i 0.951057 + 0.309017i 0.118698 + 2.99765i 3.54933i
125.5 −0.587785 + 0.809017i −1.03869 + 1.38604i −0.309017 0.951057i −1.46380 + 1.06351i −0.510803 1.65502i 1.28572 2.31234i 0.951057 + 0.309017i −0.842230 2.87935i 1.80935i
125.6 −0.587785 + 0.809017i −0.582733 1.63108i −0.309017 0.951057i 0.793236 0.576320i 1.66209 + 0.487284i 1.17577 + 2.37014i 0.951057 + 0.309017i −2.32084 + 1.90097i 0.980494i
125.7 −0.587785 + 0.809017i −0.389332 1.68773i −0.309017 0.951057i 0.0969774 0.0704582i 1.59424 + 0.677045i −2.45348 + 0.990165i 0.951057 + 0.309017i −2.69684 + 1.31417i 0.119871i
125.8 −0.587785 + 0.809017i −0.0377610 + 1.73164i −0.309017 0.951057i 2.99982 2.17950i −1.37873 1.04838i −2.21103 1.45305i 0.951057 + 0.309017i −2.99715 0.130777i 3.70798i
125.9 −0.587785 + 0.809017i 0.0377610 1.73164i −0.309017 0.951057i −2.99982 + 2.17950i 1.37873 + 1.04838i 0.934677 2.47515i 0.951057 + 0.309017i −2.99715 0.130777i 3.70798i
125.10 −0.587785 + 0.809017i 0.389332 + 1.68773i −0.309017 0.951057i −0.0969774 + 0.0704582i −1.59424 0.677045i 2.56691 0.641060i 0.951057 + 0.309017i −2.69684 + 1.31417i 0.119871i
125.11 −0.587785 + 0.809017i 0.582733 + 1.63108i −0.309017 0.951057i −0.793236 + 0.576320i −1.66209 0.487284i 0.441911 + 2.60858i 0.951057 + 0.309017i −2.32084 + 1.90097i 0.980494i
125.12 −0.587785 + 0.809017i 1.03869 1.38604i −0.309017 0.951057i 1.46380 1.06351i 0.510803 + 1.65502i −2.39933 1.11499i 0.951057 + 0.309017i −0.842230 2.87935i 1.80935i
125.13 −0.587785 + 0.809017i 1.24874 + 1.20027i −0.309017 0.951057i −2.87147 + 2.08625i −1.70503 + 0.304749i −2.61650 0.392306i 0.951057 + 0.309017i 0.118698 + 2.99765i 3.54933i
125.14 −0.587785 + 0.809017i 1.63638 + 0.567690i −0.309017 0.951057i 3.26960 2.37551i −1.42111 + 0.990177i −0.291040 + 2.62970i 0.951057 + 0.309017i 2.35546 + 1.85791i 4.04145i
125.15 −0.587785 + 0.809017i 1.70765 0.289739i −0.309017 0.951057i −0.820027 + 0.595785i −0.769325 + 1.55182i −0.540692 2.58991i 0.951057 + 0.309017i 2.83210 0.989542i 1.01361i
125.16 −0.587785 + 0.809017i 1.72594 0.145412i −0.309017 0.951057i −0.826181 + 0.600255i −0.896839 + 1.48178i 2.47331 + 0.939534i 0.951057 + 0.309017i 2.95771 0.501942i 1.02122i
125.17 0.587785 0.809017i −1.71575 + 0.237049i −0.309017 0.951057i −2.87147 + 2.08625i −0.816717 + 1.52741i 1.88621 1.85532i −0.951057 0.309017i 2.88762 0.813436i 3.54933i
125.18 0.587785 0.809017i −1.65754 0.502567i −0.309017 0.951057i 3.26960 2.37551i −1.38086 + 1.04557i 1.78115 + 1.95640i −0.951057 0.309017i 2.49485 + 1.66605i 4.04145i
125.19 0.587785 0.809017i −1.43017 + 0.977050i −0.309017 0.951057i −0.793236 + 0.576320i −0.0501805 + 1.73132i 1.17577 + 2.37014i −0.951057 0.309017i 1.09075 2.79469i 0.980494i
125.20 0.587785 0.809017i −1.31084 1.13212i −0.309017 0.951057i −0.826181 + 0.600255i −1.68640 + 0.395049i −1.44871 + 2.21388i −0.951057 0.309017i 0.436607 + 2.96806i 1.02122i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
11.c even 5 1 inner
21.c even 2 1 inner
33.h odd 10 1 inner
77.j odd 10 1 inner
231.u 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.s.a 128
3.b odd 2 1 inner 462.2.s.a 128
7.b odd 2 1 inner 462.2.s.a 128
11.c even 5 1 inner 462.2.s.a 128
21.c even 2 1 inner 462.2.s.a 128
33.h odd 10 1 inner 462.2.s.a 128
77.j odd 10 1 inner 462.2.s.a 128
231.u even 10 1 inner 462.2.s.a 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.s.a 128 1.a even 1 1 trivial
462.2.s.a 128 3.b odd 2 1 inner
462.2.s.a 128 7.b odd 2 1 inner
462.2.s.a 128 11.c even 5 1 inner
462.2.s.a 128 21.c even 2 1 inner
462.2.s.a 128 33.h odd 10 1 inner
462.2.s.a 128 77.j odd 10 1 inner
462.2.s.a 128 231.u even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(462, [\chi])\).