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Properties

Label 637.2.bb.b
Level $637$
Weight $2$
Character orbit 637.bb
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(227,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.227");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bb (of order \(12\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{2} - 16 q^{8} + 8 q^{9} - 16 q^{11} - 8 q^{15} - 24 q^{16} + 68 q^{18} + 4 q^{22} + 4 q^{29} - 12 q^{30} - 68 q^{32} - 8 q^{37} - 48 q^{43} + 60 q^{44} + 24 q^{46} - 44 q^{50} - 12 q^{51} - 36 q^{53}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 227.8
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 \( T_{2}^{16} - 2 T_{2}^{15} + 2 T_{2}^{14} + 4 T_{2}^{13} + 49 T_{2}^{12} - 80 T_{2}^{11} + 70 T_{2}^{10} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display