Properties

Label 63.8.p.a
Level $63$
Weight $8$
Character orbit 63.p
Analytic conductor $19.680$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,8,Mod(17,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.17");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 63.p (of order \(6\), degree \(2\), 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: \(19.6802566055\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 1024 q^{4} - 2074 q^{7} + 1248 q^{10} - 59708 q^{16} - 105330 q^{19} + 544 q^{22} - 56250 q^{25} + 556220 q^{28} + 86862 q^{31} + 591034 q^{37} - 3036324 q^{40} - 837332 q^{43} + 3896752 q^{46} + 6626454 q^{49}+ \cdots + 96093708 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −18.8163 + 10.8636i 0 172.035 297.973i 38.2128 + 66.1865i 0 884.855 + 201.432i 4694.58i 0 −1438.04 830.255i
17.2 −16.6651 + 9.62158i 0 121.150 209.837i 115.111 + 199.378i 0 −904.118 + 78.1943i 2199.48i 0 −3836.67 2215.10i
17.3 −13.9069 + 8.02916i 0 64.9347 112.470i −95.0205 164.580i 0 874.061 + 244.050i 30.0190i 0 2642.88 + 1525.87i
17.4 −13.0229 + 7.51875i 0 49.0633 84.9800i −217.264 376.313i 0 −593.116 686.845i 449.223i 0 5658.81 + 3267.11i
17.5 −11.8878 + 6.86345i 0 30.2140 52.3321i 99.9401 + 173.101i 0 −671.322 + 610.631i 927.556i 0 −2376.15 1371.87i
17.6 −6.95719 + 4.01673i 0 −31.7317 + 54.9609i 79.8320 + 138.273i 0 726.865 543.333i 1538.12i 0 −1110.81 641.328i
17.7 −6.63630 + 3.83147i 0 −34.6397 + 59.9977i 101.591 + 175.960i 0 −81.3062 903.843i 1511.74i 0 −1348.37 778.484i
17.8 −4.38065 + 2.52917i 0 −51.2066 + 88.6924i −258.225 447.259i 0 −900.609 111.567i 1165.51i 0 2262.39 + 1306.19i
17.9 −0.522037 + 0.301398i 0 −63.8183 + 110.537i 136.040 + 235.628i 0 146.190 + 895.640i 154.097i 0 −142.035 82.0042i
17.10 0.522037 0.301398i 0 −63.8183 + 110.537i −136.040 235.628i 0 146.190 + 895.640i 154.097i 0 −142.035 82.0042i
17.11 4.38065 2.52917i 0 −51.2066 + 88.6924i 258.225 + 447.259i 0 −900.609 111.567i 1165.51i 0 2262.39 + 1306.19i
17.12 6.63630 3.83147i 0 −34.6397 + 59.9977i −101.591 175.960i 0 −81.3062 903.843i 1511.74i 0 −1348.37 778.484i
17.13 6.95719 4.01673i 0 −31.7317 + 54.9609i −79.8320 138.273i 0 726.865 543.333i 1538.12i 0 −1110.81 641.328i
17.14 11.8878 6.86345i 0 30.2140 52.3321i −99.9401 173.101i 0 −671.322 + 610.631i 927.556i 0 −2376.15 1371.87i
17.15 13.0229 7.51875i 0 49.0633 84.9800i 217.264 + 376.313i 0 −593.116 686.845i 449.223i 0 5658.81 + 3267.11i
17.16 13.9069 8.02916i 0 64.9347 112.470i 95.0205 + 164.580i 0 874.061 + 244.050i 30.0190i 0 2642.88 + 1525.87i
17.17 16.6651 9.62158i 0 121.150 209.837i −115.111 199.378i 0 −904.118 + 78.1943i 2199.48i 0 −3836.67 2215.10i
17.18 18.8163 10.8636i 0 172.035 297.973i −38.2128 66.1865i 0 884.855 + 201.432i 4694.58i 0 −1438.04 830.255i
26.1 −18.8163 10.8636i 0 172.035 + 297.973i 38.2128 66.1865i 0 884.855 201.432i 4694.58i 0 −1438.04 + 830.255i
26.2 −16.6651 9.62158i 0 121.150 + 209.837i 115.111 199.378i 0 −904.118 78.1943i 2199.48i 0 −3836.67 + 2215.10i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.8.p.a 36
3.b odd 2 1 inner 63.8.p.a 36
7.c even 3 1 441.8.c.a 36
7.d odd 6 1 inner 63.8.p.a 36
7.d odd 6 1 441.8.c.a 36
21.g even 6 1 inner 63.8.p.a 36
21.g even 6 1 441.8.c.a 36
21.h odd 6 1 441.8.c.a 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.8.p.a 36 1.a even 1 1 trivial
63.8.p.a 36 3.b odd 2 1 inner
63.8.p.a 36 7.d odd 6 1 inner
63.8.p.a 36 21.g even 6 1 inner
441.8.c.a 36 7.c even 3 1
441.8.c.a 36 7.d odd 6 1
441.8.c.a 36 21.g even 6 1
441.8.c.a 36 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(63, [\chi])\).