Properties

Label 637.2.bb.b
Level $637$
Weight $2$
Character orbit 637.bb
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{2} - 16q^{8} + 8q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 4q^{2} - 16q^{8} + 8q^{9} - 16q^{11} - 8q^{15} - 24q^{16} + 68q^{18} + 4q^{22} + 4q^{29} - 12q^{30} - 68q^{32} - 8q^{37} - 48q^{43} + 60q^{44} + 24q^{46} - 44q^{50} - 12q^{51} - 36q^{53} - 92q^{57} - 28q^{58} - 104q^{60} - 32q^{65} - 8q^{67} + 84q^{71} + 124q^{72} + 48q^{74} + 148q^{78} + 40q^{79} + 28q^{81} + 36q^{85} - 60q^{86} + 228q^{88} + 24q^{92} - 84q^{93} - 60q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1 −1.65980 + 1.65980i −2.39046 + 1.38013i 3.50987i −0.412430 + 1.53921i 1.67694 6.25841i 0 2.50608 + 2.50608i 2.30952 4.00020i −1.87023 3.23933i
227.2 −1.65980 + 1.65980i 2.39046 1.38013i 3.50987i 0.412430 1.53921i −1.67694 + 6.25841i 0 2.50608 + 2.50608i 2.30952 4.00020i 1.87023 + 3.23933i
227.3 −0.976208 + 0.976208i −0.928437 + 0.536034i 0.0940343i 0.742756 2.77200i 0.383068 1.42963i 0 −2.04421 2.04421i −0.925336 + 1.60273i 1.98097 + 3.43114i
227.4 −0.976208 + 0.976208i 0.928437 0.536034i 0.0940343i −0.742756 + 2.77200i −0.383068 + 1.42963i 0 −2.04421 2.04421i −0.925336 + 1.60273i −1.98097 3.43114i
227.5 0.595687 0.595687i −1.24122 + 0.716618i 1.29031i −0.370805 + 1.38386i −0.312498 + 1.16626i 0 1.96000 + 1.96000i −0.472916 + 0.819114i 0.603466 + 1.04523i
227.6 0.595687 0.595687i 1.24122 0.716618i 1.29031i 0.370805 1.38386i 0.312498 1.16626i 0 1.96000 + 1.96000i −0.472916 + 0.819114i −0.603466 1.04523i
227.7 1.67430 1.67430i −1.04118 + 0.601128i 3.60653i −0.825486 + 3.08075i −0.736784 + 2.74971i 0 −2.68981 2.68981i −0.777291 + 1.34631i 3.77599 + 6.54020i
227.8 1.67430 1.67430i 1.04118 0.601128i 3.60653i 0.825486 3.08075i 0.736784 2.74971i 0 −2.68981 2.68981i −0.777291 + 1.34631i −3.77599 6.54020i
362.1 −1.65980 1.65980i −2.39046 1.38013i 3.50987i −0.412430 1.53921i 1.67694 + 6.25841i 0 2.50608 2.50608i 2.30952 + 4.00020i −1.87023 + 3.23933i
362.2 −1.65980 1.65980i 2.39046 + 1.38013i 3.50987i 0.412430 + 1.53921i −1.67694 6.25841i 0 2.50608 2.50608i 2.30952 + 4.00020i 1.87023 3.23933i
362.3 −0.976208 0.976208i −0.928437 0.536034i 0.0940343i 0.742756 + 2.77200i 0.383068 + 1.42963i 0 −2.04421 + 2.04421i −0.925336 1.60273i 1.98097 3.43114i
362.4 −0.976208 0.976208i 0.928437 + 0.536034i 0.0940343i −0.742756 2.77200i −0.383068 1.42963i 0 −2.04421 + 2.04421i −0.925336 1.60273i −1.98097 + 3.43114i
362.5 0.595687 + 0.595687i −1.24122 0.716618i 1.29031i −0.370805 1.38386i −0.312498 1.16626i 0 1.96000 1.96000i −0.472916 0.819114i 0.603466 1.04523i
362.6 0.595687 + 0.595687i 1.24122 + 0.716618i 1.29031i 0.370805 + 1.38386i 0.312498 + 1.16626i 0 1.96000 1.96000i −0.472916 0.819114i −0.603466 + 1.04523i
362.7 1.67430 + 1.67430i −1.04118 0.601128i 3.60653i −0.825486 3.08075i −0.736784 2.74971i 0 −2.68981 + 2.68981i −0.777291 1.34631i 3.77599 6.54020i
362.8 1.67430 + 1.67430i 1.04118 + 0.601128i 3.60653i 0.825486 + 3.08075i 0.736784 + 2.74971i 0 −2.68981 + 2.68981i −0.777291 1.34631i −3.77599 + 6.54020i
423.1 −1.18517 1.18517i −0.552306 + 0.318874i 0.809250i 1.94174 + 0.520288i 1.03250 + 0.276656i 0 −1.41124 + 1.41124i −1.29664 + 2.24584i −1.68466 2.91792i
423.2 −1.18517 1.18517i 0.552306 0.318874i 0.809250i −1.94174 0.520288i −1.03250 0.276656i 0 −1.41124 + 1.41124i −1.29664 + 2.24584i 1.68466 + 2.91792i
423.3 0.0825572 + 0.0825572i −2.25055 + 1.29935i 1.98637i −1.70537 0.456951i −0.293070 0.0785278i 0 0.329103 0.329103i 1.87664 3.25044i −0.103066 0.178515i
423.4 0.0825572 + 0.0825572i 2.25055 1.29935i 1.98637i 1.70537 + 0.456951i 0.293070 + 0.0785278i 0 0.329103 0.329103i 1.87664 3.25044i 0.103066 + 0.178515i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.x odd 12 1 inner
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bb.b 32
7.b odd 2 1 inner 637.2.bb.b 32
7.c even 3 1 91.2.bc.a 32
7.c even 3 1 637.2.x.b 32
7.d odd 6 1 91.2.bc.a 32
7.d odd 6 1 637.2.x.b 32
13.f odd 12 1 637.2.x.b 32
21.g even 6 1 819.2.fm.g 32
21.h odd 6 1 819.2.fm.g 32
91.w even 12 1 91.2.bc.a 32
91.x odd 12 1 inner 637.2.bb.b 32
91.ba even 12 1 inner 637.2.bb.b 32
91.bc even 12 1 637.2.x.b 32
91.bd odd 12 1 91.2.bc.a 32
273.bw even 12 1 819.2.fm.g 32
273.ch odd 12 1 819.2.fm.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bc.a 32 7.c even 3 1
91.2.bc.a 32 7.d odd 6 1
91.2.bc.a 32 91.w even 12 1
91.2.bc.a 32 91.bd odd 12 1
637.2.x.b 32 7.c even 3 1
637.2.x.b 32 7.d odd 6 1
637.2.x.b 32 13.f odd 12 1
637.2.x.b 32 91.bc even 12 1
637.2.bb.b 32 1.a even 1 1 trivial
637.2.bb.b 32 7.b odd 2 1 inner
637.2.bb.b 32 91.x odd 12 1 inner
637.2.bb.b 32 91.ba even 12 1 inner
819.2.fm.g 32 21.g even 6 1
819.2.fm.g 32 21.h odd 6 1
819.2.fm.g 32 273.bw even 12 1
819.2.fm.g 32 273.ch odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\).