Properties

Label 462.2.n.e
Level $462$
Weight $2$
Character orbit 462.n
Analytic conductor $3.689$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q - 12q^{2} - 12q^{4} + 24q^{8} - 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q - 12q^{2} - 12q^{4} + 24q^{8} - 8q^{9} - 4q^{11} + 4q^{15} - 12q^{16} + 4q^{17} - 8q^{18} + 8q^{22} + 20q^{25} - 12q^{27} + 40q^{29} - 2q^{30} - 28q^{31} - 12q^{32} + 32q^{33} - 8q^{34} - 32q^{35} + 16q^{36} - 16q^{37} - 8q^{41} + 6q^{42} - 4q^{44} - 28q^{45} - 8q^{49} - 40q^{50} - 34q^{51} + 6q^{54} + 8q^{55} + 40q^{57} - 20q^{58} - 2q^{60} + 56q^{62} - 14q^{63} + 24q^{64} - 44q^{65} + 32q^{66} + 4q^{68} - 20q^{69} + 64q^{70} - 8q^{72} - 16q^{74} - 26q^{75} + 24q^{81} + 4q^{82} + 144q^{83} - 6q^{84} + 8q^{87} - 4q^{88} + 56q^{90} - 32q^{91} - 22q^{93} - 44q^{95} + 64q^{97} - 8q^{98} - 52q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 −0.500000 + 0.866025i −1.72951 0.0938734i −0.500000 0.866025i −3.23623 1.86844i 0.946049 1.45086i −0.830341 + 2.51208i 1.00000 2.98238 + 0.324709i 3.23623 1.86844i
65.2 −0.500000 + 0.866025i −1.64448 0.543759i −0.500000 0.866025i 1.63087 + 0.941585i 1.29315 1.15228i −1.76745 1.96879i 1.00000 2.40865 + 1.78841i −1.63087 + 0.941585i
65.3 −0.500000 + 0.866025i −1.20836 1.24091i −0.500000 0.866025i −1.00577 0.580683i 1.67884 0.426020i 2.19957 1.47034i 1.00000 −0.0797091 + 2.99894i 1.00577 0.580683i
65.4 −0.500000 + 0.866025i −0.993776 + 1.41859i −0.500000 0.866025i 0.246102 + 0.142087i −0.731651 1.56993i 2.09441 + 1.61662i 1.00000 −1.02482 2.81953i −0.246102 + 0.142087i
65.5 −0.500000 + 0.866025i −0.470476 1.66693i −0.500000 0.866025i 1.00577 + 0.580683i 1.67884 + 0.426020i −2.19957 + 1.47034i 1.00000 −2.55730 + 1.56850i −1.00577 + 0.580683i
65.6 −0.500000 + 0.866025i 0.125098 + 1.72753i −0.500000 0.866025i 3.72427 + 2.15021i −1.55863 0.755426i −2.62857 0.301041i 1.00000 −2.96870 + 0.432220i −3.72427 + 2.15021i
65.7 −0.500000 + 0.866025i 0.301239 + 1.70565i −0.500000 0.866025i −1.38736 0.800992i −1.62776 0.591946i 0.229478 2.63578i 1.00000 −2.81851 + 1.02762i 1.38736 0.800992i
65.8 −0.500000 + 0.866025i 0.351333 1.69604i −0.500000 0.866025i −1.63087 0.941585i 1.29315 + 1.15228i 1.76745 + 1.96879i 1.00000 −2.75313 1.19175i 1.63087 0.941585i
65.9 −0.500000 + 0.866025i 0.783456 1.54473i −0.500000 0.866025i 3.23623 + 1.86844i 0.946049 + 1.45086i 0.830341 2.51208i 1.00000 −1.77239 2.42046i −3.23623 + 1.86844i
65.10 −0.500000 + 0.866025i 1.32652 + 1.11371i −0.500000 0.866025i 1.38736 + 0.800992i −1.62776 + 0.591946i −0.229478 + 2.63578i 1.00000 0.519310 + 2.95471i −1.38736 + 0.800992i
65.11 −0.500000 + 0.866025i 1.43353 + 0.972102i −0.500000 0.866025i −3.72427 2.15021i −1.55863 + 0.755426i 2.62857 + 0.301041i 1.00000 1.11004 + 2.78708i 3.72427 2.15021i
65.12 −0.500000 + 0.866025i 1.72543 0.151338i −0.500000 0.866025i −0.246102 0.142087i −0.731651 + 1.56993i −2.09441 1.61662i 1.00000 2.95419 0.522246i 0.246102 0.142087i
263.1 −0.500000 0.866025i −1.72951 + 0.0938734i −0.500000 + 0.866025i −3.23623 + 1.86844i 0.946049 + 1.45086i −0.830341 2.51208i 1.00000 2.98238 0.324709i 3.23623 + 1.86844i
263.2 −0.500000 0.866025i −1.64448 + 0.543759i −0.500000 + 0.866025i 1.63087 0.941585i 1.29315 + 1.15228i −1.76745 + 1.96879i 1.00000 2.40865 1.78841i −1.63087 0.941585i
263.3 −0.500000 0.866025i −1.20836 + 1.24091i −0.500000 + 0.866025i −1.00577 + 0.580683i 1.67884 + 0.426020i 2.19957 + 1.47034i 1.00000 −0.0797091 2.99894i 1.00577 + 0.580683i
263.4 −0.500000 0.866025i −0.993776 1.41859i −0.500000 + 0.866025i 0.246102 0.142087i −0.731651 + 1.56993i 2.09441 1.61662i 1.00000 −1.02482 + 2.81953i −0.246102 0.142087i
263.5 −0.500000 0.866025i −0.470476 + 1.66693i −0.500000 + 0.866025i 1.00577 0.580683i 1.67884 0.426020i −2.19957 1.47034i 1.00000 −2.55730 1.56850i −1.00577 0.580683i
263.6 −0.500000 0.866025i 0.125098 1.72753i −0.500000 + 0.866025i 3.72427 2.15021i −1.55863 + 0.755426i −2.62857 + 0.301041i 1.00000 −2.96870 0.432220i −3.72427 2.15021i
263.7 −0.500000 0.866025i 0.301239 1.70565i −0.500000 + 0.866025i −1.38736 + 0.800992i −1.62776 + 0.591946i 0.229478 + 2.63578i 1.00000 −2.81851 1.02762i 1.38736 + 0.800992i
263.8 −0.500000 0.866025i 0.351333 + 1.69604i −0.500000 + 0.866025i −1.63087 + 0.941585i 1.29315 1.15228i 1.76745 1.96879i 1.00000 −2.75313 + 1.19175i 1.63087 + 0.941585i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.d even 2 1 inner
231.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.n.e 24
3.b odd 2 1 462.2.n.f yes 24
7.c even 3 1 inner 462.2.n.e 24
11.b odd 2 1 462.2.n.f yes 24
21.h odd 6 1 462.2.n.f yes 24
33.d even 2 1 inner 462.2.n.e 24
77.h odd 6 1 462.2.n.f yes 24
231.l even 6 1 inner 462.2.n.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.n.e 24 1.a even 1 1 trivial
462.2.n.e 24 7.c even 3 1 inner
462.2.n.e 24 33.d even 2 1 inner
462.2.n.e 24 231.l even 6 1 inner
462.2.n.f yes 24 3.b odd 2 1
462.2.n.f yes 24 11.b odd 2 1
462.2.n.f yes 24 21.h odd 6 1
462.2.n.f yes 24 77.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\(T_{5}^{24} - \cdots\)
\(T_{17}^{12} - \cdots\)