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Properties

Label 637.2.bc.c
Level $637$
Weight $2$
Character orbit 637.bc
Analytic conductor $5.086$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(31,637)]
 
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, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bc (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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 56 q^{9} + 16 q^{11} + 96 q^{15} + 56 q^{16} + 32 q^{18} - 96 q^{29} + 32 q^{39} + 64 q^{44} - 32 q^{46} - 80 q^{50} + 16 q^{53} - 192 q^{57} - 72 q^{58} + 64 q^{60} - 32 q^{65} + 64 q^{71} - 208 q^{72} - 208 q^{74} + 320 q^{78} + 80 q^{79} - 152 q^{81} + 48 q^{85} - 64 q^{86} - 256 q^{92} - 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.45320 − 0.657332i −2.10856 − 1.21738i 3.85404 + 2.22513i −0.923684 + 3.44724i 4.37249 + 4.37249i 0 −4.40035 − 4.40035i 1.46401 + 2.53574i 4.53196 − 7.84958i
31.2 −2.45320 − 0.657332i 2.10856 + 1.21738i 3.85404 + 2.22513i 0.923684 − 3.44724i −4.37249 − 4.37249i 0 −4.40035 − 4.40035i 1.46401 + 2.53574i −4.53196 + 7.84958i
31.3 −2.23197 − 0.598054i −2.73021 − 1.57629i 2.89197 + 1.66968i 0.0878004 − 0.327676i 5.15104 + 5.15104i 0 −2.18840 − 2.18840i 3.46936 + 6.00911i −0.391936 + 0.678853i
31.4 −2.23197 − 0.598054i 2.73021 + 1.57629i 2.89197 + 1.66968i −0.0878004 + 0.327676i −5.15104 − 5.15104i 0 −2.18840 − 2.18840i 3.46936 + 6.00911i 0.391936 − 0.678853i
31.5 −2.16854 − 0.581058i −1.07317 − 0.619597i 2.63287 + 1.52009i 0.348911 − 1.30215i 1.96719 + 1.96719i 0 −1.65126 − 1.65126i −0.732198 − 1.26820i −1.51325 + 2.62103i
31.6 −2.16854 − 0.581058i 1.07317 + 0.619597i 2.63287 + 1.52009i −0.348911 + 1.30215i −1.96719 − 1.96719i 0 −1.65126 − 1.65126i −0.732198 − 1.26820i 1.51325 − 2.62103i
31.7 −1.69959 − 0.455403i −0.328893 − 0.189887i 0.949157 + 0.547996i −0.376466 + 1.40499i 0.472508 + 0.472508i 0 1.12475 + 1.12475i −1.42789 − 2.47317i 1.27967 − 2.21646i
31.8 −1.69959 − 0.455403i 0.328893 + 0.189887i 0.949157 + 0.547996i 0.376466 − 1.40499i −0.472508 − 0.472508i 0 1.12475 + 1.12475i −1.42789 − 2.47317i −1.27967 + 2.21646i
31.9 −0.652006 − 0.174705i −0.603144 − 0.348225i −1.33746 − 0.772183i 0.969150 − 3.61692i 0.332417 + 0.332417i 0 1.69173 + 1.69173i −1.25748 − 2.17802i −1.26378 + 2.18894i
31.10 −0.652006 − 0.174705i 0.603144 + 0.348225i −1.33746 − 0.772183i −0.969150 + 3.61692i −0.332417 − 0.332417i 0 1.69173 + 1.69173i −1.25748 − 2.17802i 1.26378 − 2.18894i
31.11 −0.487625 − 0.130659i −1.64366 − 0.948970i −1.51134 − 0.872575i 0.231171 − 0.862742i 0.677500 + 0.677500i 0 1.33689 + 1.33689i 0.301088 + 0.521500i −0.225449 + 0.390490i
31.12 −0.487625 − 0.130659i 1.64366 + 0.948970i −1.51134 − 0.872575i −0.231171 + 0.862742i −0.677500 − 0.677500i 0 1.33689 + 1.33689i 0.301088 + 0.521500i 0.225449 − 0.390490i
31.13 −0.436103 − 0.116853i −2.75647 − 1.59145i −1.55552 − 0.898080i 0.00916361 − 0.0341991i 1.01614 + 1.01614i 0 1.21192 + 1.21192i 3.56540 + 6.17545i −0.00799256 + 0.0138435i
31.14 −0.436103 − 0.116853i 2.75647 + 1.59145i −1.55552 − 0.898080i −0.00916361 + 0.0341991i −1.01614 − 1.01614i 0 1.21192 + 1.21192i 3.56540 + 6.17545i 0.00799256 − 0.0138435i
31.15 0.0430431 + 0.0115334i −0.412583 − 0.238205i −1.73033 − 0.999007i −0.855060 + 3.19113i −0.0150115 − 0.0150115i 0 −0.125976 − 0.125976i −1.38652 − 2.40152i −0.0736088 + 0.127494i
31.16 0.0430431 + 0.0115334i 0.412583 + 0.238205i −1.73033 − 0.999007i 0.855060 − 3.19113i 0.0150115 + 0.0150115i 0 −0.125976 − 0.125976i −1.38652 − 2.40152i 0.0736088 − 0.127494i
31.17 0.428489 + 0.114813i −1.86327 − 1.07576i −1.56163 − 0.901608i −0.501910 + 1.87315i −0.674879 − 0.674879i 0 −1.19298 − 1.19298i 0.814518 + 1.41079i −0.430126 + 0.744999i
31.18 0.428489 + 0.114813i 1.86327 + 1.07576i −1.56163 − 0.901608i 0.501910 − 1.87315i 0.674879 + 0.674879i 0 −1.19298 − 1.19298i 0.814518 + 1.41079i 0.430126 − 0.744999i
31.19 1.43251 + 0.383841i −0.765995 − 0.442247i 0.172712 + 0.0997153i −1.01546 + 3.78973i −0.927546 − 0.927546i 0 −1.88821 − 1.88821i −1.10883 − 1.92056i −2.90931 + 5.03907i
31.20 1.43251 + 0.383841i 0.765995 + 0.442247i 0.172712 + 0.0997153i 1.01546 − 3.78973i 0.927546 + 0.927546i 0 −1.88821 − 1.88821i −1.10883 − 1.92056i 2.90931 − 5.03907i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.28
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.d odd 4 1 inner
91.i even 4 1 inner
91.z odd 12 1 inner
91.bb 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.bc.c 112
7.b odd 2 1 inner 637.2.bc.c 112
7.c even 3 1 637.2.i.b ✓ 56
7.c even 3 1 inner 637.2.bc.c 112
7.d odd 6 1 637.2.i.b ✓ 56
7.d odd 6 1 inner 637.2.bc.c 112
13.d odd 4 1 inner 637.2.bc.c 112
91.i even 4 1 inner 637.2.bc.c 112
91.z odd 12 1 637.2.i.b ✓ 56
91.z odd 12 1 inner 637.2.bc.c 112
91.bb even 12 1 637.2.i.b ✓ 56
91.bb even 12 1 inner 637.2.bc.c 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.i.b ✓ 56 7.c even 3 1
637.2.i.b ✓ 56 7.d odd 6 1
637.2.i.b ✓ 56 91.z odd 12 1
637.2.i.b ✓ 56 91.bb even 12 1
637.2.bc.c 112 1.a even 1 1 trivial
637.2.bc.c 112 7.b odd 2 1 inner
637.2.bc.c 112 7.c even 3 1 inner
637.2.bc.c 112 7.d odd 6 1 inner
637.2.bc.c 112 13.d odd 4 1 inner
637.2.bc.c 112 91.i even 4 1 inner
637.2.bc.c 112 91.z odd 12 1 inner
637.2.bc.c 112 91.bb even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 105 T_{2}^{52} - 16 T_{2}^{49} + 7237 T_{2}^{48} + 384 T_{2}^{47} + 3336 T_{2}^{45} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display