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Properties

Label 63.10.e.d
Level $63$
Weight $10$
Character orbit 63.e
Analytic conductor $32.447$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,10,Mod(37,63)]
 
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([0, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.37");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), 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: \(32.4472576783\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2902 q^{4} + 7436 q^{7} + 83446 q^{10} + 41812 q^{13} - 218794 q^{16} + 436858 q^{19} + 471004 q^{22} - 3902506 q^{25} - 7936878 q^{28} + 5798636 q^{31} + 26797032 q^{34} - 27318982 q^{37} + 52610058 q^{40}+ \cdots + 412446860 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −19.4807 + 33.7415i 0 −502.994 − 871.211i 1238.29 − 2144.78i 0 6002.22 − 2080.12i 19246.4 0 48245.5 + 83563.7i
37.2 −18.8691 + 32.6823i 0 −456.088 − 789.967i −419.379 + 726.386i 0 −6230.19 − 1240.29i 15101.9 0 −15826.6 − 27412.5i
37.3 −15.9183 + 27.5713i 0 −250.784 − 434.371i −502.573 + 870.481i 0 3102.35 + 5543.38i −332.099 0 −16000.2 − 27713.2i
37.4 −8.00111 + 13.8583i 0 127.964 + 221.641i 413.923 − 716.935i 0 −4764.21 − 4201.89i −12288.6 0 6623.68 + 11472.6i
37.5 −7.52085 + 13.0265i 0 142.874 + 247.464i −693.972 + 1202.00i 0 5243.90 − 3585.40i −11999.5 0 −10438.5 − 18080.1i
37.6 −4.60827 + 7.98176i 0 213.528 + 369.841i 895.962 − 1551.85i 0 −1495.06 + 6174.01i −8654.85 0 8257.68 + 14302.7i
37.7 4.60827 − 7.98176i 0 213.528 + 369.841i −895.962 + 1551.85i 0 −1495.06 + 6174.01i 8654.85 0 8257.68 + 14302.7i
37.8 7.52085 − 13.0265i 0 142.874 + 247.464i 693.972 − 1202.00i 0 5243.90 − 3585.40i 11999.5 0 −10438.5 − 18080.1i
37.9 8.00111 − 13.8583i 0 127.964 + 221.641i −413.923 + 716.935i 0 −4764.21 − 4201.89i 12288.6 0 6623.68 + 11472.6i
37.10 15.9183 − 27.5713i 0 −250.784 − 434.371i 502.573 − 870.481i 0 3102.35 + 5543.38i 332.099 0 −16000.2 − 27713.2i
37.11 18.8691 − 32.6823i 0 −456.088 − 789.967i 419.379 − 726.386i 0 −6230.19 − 1240.29i −15101.9 0 −15826.6 − 27412.5i
37.12 19.4807 − 33.7415i 0 −502.994 − 871.211i −1238.29 + 2144.78i 0 6002.22 − 2080.12i −19246.4 0 48245.5 + 83563.7i
46.1 −19.4807 − 33.7415i 0 −502.994 + 871.211i 1238.29 + 2144.78i 0 6002.22 + 2080.12i 19246.4 0 48245.5 − 83563.7i
46.2 −18.8691 − 32.6823i 0 −456.088 + 789.967i −419.379 − 726.386i 0 −6230.19 + 1240.29i 15101.9 0 −15826.6 + 27412.5i
46.3 −15.9183 − 27.5713i 0 −250.784 + 434.371i −502.573 − 870.481i 0 3102.35 − 5543.38i −332.099 0 −16000.2 + 27713.2i
46.4 −8.00111 − 13.8583i 0 127.964 − 221.641i 413.923 + 716.935i 0 −4764.21 + 4201.89i −12288.6 0 6623.68 − 11472.6i
46.5 −7.52085 − 13.0265i 0 142.874 − 247.464i −693.972 − 1202.00i 0 5243.90 + 3585.40i −11999.5 0 −10438.5 + 18080.1i
46.6 −4.60827 − 7.98176i 0 213.528 − 369.841i 895.962 + 1551.85i 0 −1495.06 − 6174.01i −8654.85 0 8257.68 − 14302.7i
46.7 4.60827 + 7.98176i 0 213.528 − 369.841i −895.962 − 1551.85i 0 −1495.06 − 6174.01i 8654.85 0 8257.68 − 14302.7i
46.8 7.52085 + 13.0265i 0 142.874 − 247.464i 693.972 + 1202.00i 0 5243.90 + 3585.40i 11999.5 0 −10438.5 + 18080.1i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.10.e.d ✓ 24
3.b odd 2 1 inner 63.10.e.d ✓ 24
7.c even 3 1 inner 63.10.e.d ✓ 24
21.h odd 6 1 inner 63.10.e.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.10.e.d ✓ 24 1.a even 1 1 trivial
63.10.e.d ✓ 24 3.b odd 2 1 inner
63.10.e.d ✓ 24 7.c even 3 1 inner
63.10.e.d ✓ 24 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 4523 T_{2}^{22} + 12970695 T_{2}^{20} + 22852156502 T_{2}^{18} + 29380521307660 T_{2}^{16} + \cdots + 11\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display