Properties

Label 637.2.r.g
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 20q^{4} - 16q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 20q^{4} - 16q^{9} - 28q^{16} - 16q^{22} + 36q^{23} + 44q^{25} + 72q^{29} + 104q^{36} - 32q^{39} - 72q^{43} + 72q^{51} - 12q^{53} - 328q^{64} + 24q^{65} - 96q^{74} + 48q^{78} - 36q^{79} - 16q^{81} - 136q^{88} + 48q^{92} + 84q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −2.36789 + 1.36710i −0.577975 + 1.00108i 2.73793 4.74223i −1.62576 + 0.938635i 3.16060i 0 9.50370i 0.831889 + 1.44087i 2.56642 4.44516i
116.2 −2.36789 + 1.36710i 0.577975 1.00108i 2.73793 4.74223i 1.62576 0.938635i 3.16060i 0 9.50370i 0.831889 + 1.44087i −2.56642 + 4.44516i
116.3 −1.43372 + 0.827759i −0.247488 + 0.428663i 0.370370 0.641500i −2.48886 + 1.43695i 0.819443i 0 2.08473i 1.37750 + 2.38590i 2.37889 4.12036i
116.4 −1.43372 + 0.827759i 0.247488 0.428663i 0.370370 0.641500i 2.48886 1.43695i 0.819443i 0 2.08473i 1.37750 + 2.38590i −2.37889 + 4.12036i
116.5 −1.31930 + 0.761698i −1.49755 + 2.59383i 0.160367 0.277763i 2.55136 1.47303i 4.56272i 0 2.55819i −2.98531 5.17071i −2.24401 + 3.88673i
116.6 −1.31930 + 0.761698i 1.49755 2.59383i 0.160367 0.277763i −2.55136 + 1.47303i 4.56272i 0 2.55819i −2.98531 5.17071i 2.24401 3.88673i
116.7 −0.589067 + 0.340098i −1.16706 + 2.02141i −0.768667 + 1.33137i −2.81123 + 1.62306i 1.58766i 0 2.40608i −1.22407 2.12016i 1.10400 1.91219i
116.8 −0.589067 + 0.340098i 1.16706 2.02141i −0.768667 + 1.33137i 2.81123 1.62306i 1.58766i 0 2.40608i −1.22407 2.12016i −1.10400 + 1.91219i
116.9 0.589067 0.340098i −1.16706 + 2.02141i −0.768667 + 1.33137i 2.81123 1.62306i 1.58766i 0 2.40608i −1.22407 2.12016i 1.10400 1.91219i
116.10 0.589067 0.340098i 1.16706 2.02141i −0.768667 + 1.33137i −2.81123 + 1.62306i 1.58766i 0 2.40608i −1.22407 2.12016i −1.10400 + 1.91219i
116.11 1.31930 0.761698i −1.49755 + 2.59383i 0.160367 0.277763i −2.55136 + 1.47303i 4.56272i 0 2.55819i −2.98531 5.17071i −2.24401 + 3.88673i
116.12 1.31930 0.761698i 1.49755 2.59383i 0.160367 0.277763i 2.55136 1.47303i 4.56272i 0 2.55819i −2.98531 5.17071i 2.24401 3.88673i
116.13 1.43372 0.827759i −0.247488 + 0.428663i 0.370370 0.641500i 2.48886 1.43695i 0.819443i 0 2.08473i 1.37750 + 2.38590i 2.37889 4.12036i
116.14 1.43372 0.827759i 0.247488 0.428663i 0.370370 0.641500i −2.48886 + 1.43695i 0.819443i 0 2.08473i 1.37750 + 2.38590i −2.37889 + 4.12036i
116.15 2.36789 1.36710i −0.577975 + 1.00108i 2.73793 4.74223i 1.62576 0.938635i 3.16060i 0 9.50370i 0.831889 + 1.44087i 2.56642 4.44516i
116.16 2.36789 1.36710i 0.577975 1.00108i 2.73793 4.74223i −1.62576 + 0.938635i 3.16060i 0 9.50370i 0.831889 + 1.44087i −2.56642 + 4.44516i
324.1 −2.36789 1.36710i −0.577975 1.00108i 2.73793 + 4.74223i −1.62576 0.938635i 3.16060i 0 9.50370i 0.831889 1.44087i 2.56642 + 4.44516i
324.2 −2.36789 1.36710i 0.577975 + 1.00108i 2.73793 + 4.74223i 1.62576 + 0.938635i 3.16060i 0 9.50370i 0.831889 1.44087i −2.56642 4.44516i
324.3 −1.43372 0.827759i −0.247488 0.428663i 0.370370 + 0.641500i −2.48886 1.43695i 0.819443i 0 2.08473i 1.37750 2.38590i 2.37889 + 4.12036i
324.4 −1.43372 0.827759i 0.247488 + 0.428663i 0.370370 + 0.641500i 2.48886 + 1.43695i 0.819443i 0 2.08473i 1.37750 2.38590i −2.37889 4.12036i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
13.b even 2 1 inner
91.b odd 2 1 inner
91.r even 6 1 inner
91.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.g 32
7.b odd 2 1 inner 637.2.r.g 32
7.c even 3 1 637.2.c.g 16
7.c even 3 1 inner 637.2.r.g 32
7.d odd 6 1 637.2.c.g 16
7.d odd 6 1 inner 637.2.r.g 32
13.b even 2 1 inner 637.2.r.g 32
91.b odd 2 1 inner 637.2.r.g 32
91.r even 6 1 637.2.c.g 16
91.r even 6 1 inner 637.2.r.g 32
91.s odd 6 1 637.2.c.g 16
91.s odd 6 1 inner 637.2.r.g 32
91.z odd 12 2 8281.2.a.cs 16
91.bb even 12 2 8281.2.a.cs 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.g 16 7.c even 3 1
637.2.c.g 16 7.d odd 6 1
637.2.c.g 16 91.r even 6 1
637.2.c.g 16 91.s odd 6 1
637.2.r.g 32 1.a even 1 1 trivial
637.2.r.g 32 7.b odd 2 1 inner
637.2.r.g 32 7.c even 3 1 inner
637.2.r.g 32 7.d odd 6 1 inner
637.2.r.g 32 13.b even 2 1 inner
637.2.r.g 32 91.b odd 2 1 inner
637.2.r.g 32 91.r even 6 1 inner
637.2.r.g 32 91.s odd 6 1 inner
8281.2.a.cs 16 91.z odd 12 2
8281.2.a.cs 16 91.bb even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{3}^{16} + \cdots\)