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Properties

Label 630.2.ce.c
Level $630$
Weight $2$
Character orbit 630.ce
Analytic conductor $5.031$
Analytic rank $0$
Dimension $32$
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] = [630,2,Mod(53,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.ce (of order \(12\), degree \(4\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 12 q^{7} + 8 q^{13} + 16 q^{16} + 32 q^{22} - 16 q^{25} - 56 q^{31} + 20 q^{37} + 4 q^{40} + 24 q^{46} + 4 q^{52} + 12 q^{58} - 48 q^{61} + 8 q^{67} + 24 q^{70} + 48 q^{73} - 36 q^{82} - 136 q^{85} - 16 q^{88} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −2.16447 + 0.561299i 0 2.06214 − 1.65758i 0.707107 − 0.707107i 0 0.0180339 − 2.23600i
53.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −1.24269 − 1.85895i 0 −1.31709 + 2.29462i 0.707107 − 0.707107i 0 2.11725 − 0.719217i
53.3 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.923861 + 2.03629i 0 −2.56861 − 0.634242i 0.707107 − 0.707107i 0 −2.20602 + 0.365351i
53.4 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 2.22449 + 0.227291i 0 1.18959 + 2.36324i 0.707107 − 0.707107i 0 −0.795286 + 2.08986i
53.5 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −2.22449 − 0.227291i 0 1.18959 + 2.36324i −0.707107 + 0.707107i 0 −0.795286 + 2.08986i
53.6 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.923861 − 2.03629i 0 −2.56861 − 0.634242i −0.707107 + 0.707107i 0 −2.20602 + 0.365351i
53.7 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 1.24269 + 1.85895i 0 −1.31709 + 2.29462i −0.707107 + 0.707107i 0 2.11725 − 0.719217i
53.8 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 2.16447 − 0.561299i 0 2.06214 − 1.65758i −0.707107 + 0.707107i 0 0.0180339 − 2.23600i
107.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −2.16447 − 0.561299i 0 2.06214 + 1.65758i 0.707107 + 0.707107i 0 0.0180339 + 2.23600i
107.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −1.24269 + 1.85895i 0 −1.31709 − 2.29462i 0.707107 + 0.707107i 0 2.11725 + 0.719217i
107.3 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.923861 − 2.03629i 0 −2.56861 + 0.634242i 0.707107 + 0.707107i 0 −2.20602 − 0.365351i
107.4 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 2.22449 − 0.227291i 0 1.18959 − 2.36324i 0.707107 + 0.707107i 0 −0.795286 − 2.08986i
107.5 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −2.22449 + 0.227291i 0 1.18959 − 2.36324i −0.707107 − 0.707107i 0 −0.795286 − 2.08986i
107.6 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.923861 + 2.03629i 0 −2.56861 + 0.634242i −0.707107 − 0.707107i 0 −2.20602 − 0.365351i
107.7 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 1.24269 − 1.85895i 0 −1.31709 − 2.29462i −0.707107 − 0.707107i 0 2.11725 + 0.719217i
107.8 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 2.16447 + 0.561299i 0 2.06214 + 1.65758i −0.707107 − 0.707107i 0 0.0180339 + 2.23600i
233.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −1.56834 − 1.59384i 0 1.65758 − 2.06214i −0.707107 + 0.707107i 0 1.92741 + 1.13362i
233.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −1.30155 + 1.81823i 0 0.634242 + 2.56861i −0.707107 + 0.707107i 0 0.786607 − 2.09314i
233.3 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.915403 + 2.04011i 0 −2.36324 − 1.18959i −0.707107 + 0.707107i 0 −1.41223 − 1.73367i
233.4 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.988555 − 2.00568i 0 −2.29462 + 1.31709i −0.707107 + 0.707107i 0 −0.435763 + 2.19320i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.ce.c ✓ 32
3.b odd 2 1 inner 630.2.ce.c ✓ 32
5.c odd 4 1 inner 630.2.ce.c ✓ 32
7.c even 3 1 inner 630.2.ce.c ✓ 32
15.e even 4 1 inner 630.2.ce.c ✓ 32
21.h odd 6 1 inner 630.2.ce.c ✓ 32
35.l odd 12 1 inner 630.2.ce.c ✓ 32
105.x even 12 1 inner 630.2.ce.c ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.ce.c ✓ 32 1.a even 1 1 trivial
630.2.ce.c ✓ 32 3.b odd 2 1 inner
630.2.ce.c ✓ 32 5.c odd 4 1 inner
630.2.ce.c ✓ 32 7.c even 3 1 inner
630.2.ce.c ✓ 32 15.e even 4 1 inner
630.2.ce.c ✓ 32 21.h odd 6 1 inner
630.2.ce.c ✓ 32 35.l odd 12 1 inner
630.2.ce.c ✓ 32 105.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} - 28 T_{11}^{14} + 694 T_{11}^{12} - 2336 T_{11}^{10} + 5499 T_{11}^{8} - 6880 T_{11}^{6} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display