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Properties

Label 1260.2.cj.c
Level $1260$
Weight $2$
Character orbit 1260.cj
Analytic conductor $10.061$
Analytic rank $0$
Dimension $80$
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] = [1260,2,Mod(209,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.cj (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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{9} + 24 q^{11} + 12 q^{15} + 2 q^{21} - 40 q^{25} + 12 q^{29} + 32 q^{39} + 62 q^{49} - 32 q^{51} - 36 q^{65} + 16 q^{79} - 28 q^{81} + 10 q^{85} + 28 q^{91} + 60 q^{95} + 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −1.70512 − 0.304226i 0 2.21491 + 0.306896i 0 −0.889115 − 2.49188i 0 2.81489 + 1.03749i 0
209.2 0 −1.70512 − 0.304226i 0 0.841674 − 2.07161i 0 2.60259 − 0.475945i 0 2.81489 + 1.03749i 0
209.3 0 −1.69802 − 0.341647i 0 −2.12845 − 0.685343i 0 −2.64558 − 0.0305430i 0 2.76656 + 1.16025i 0
209.4 0 −1.69802 − 0.341647i 0 −0.470701 + 2.18596i 0 1.34924 + 2.27586i 0 2.76656 + 1.16025i 0
209.5 0 −1.67967 + 0.422727i 0 1.76262 + 1.37593i 0 2.56403 − 0.652477i 0 2.64260 − 1.42009i 0
209.6 0 −1.67967 + 0.422727i 0 −0.310282 − 2.21444i 0 −0.716955 − 2.54676i 0 2.64260 − 1.42009i 0
209.7 0 −1.42972 + 0.977700i 0 −1.20000 + 1.88680i 0 −1.83608 − 1.90495i 0 1.08821 − 2.79568i 0
209.8 0 −1.42972 + 0.977700i 0 −2.23401 + 0.0958296i 0 2.56777 + 0.637615i 0 1.08821 − 2.79568i 0
209.9 0 −1.22755 + 1.22193i 0 −1.00572 − 1.99713i 0 −0.0863616 + 2.64434i 0 0.0137791 − 2.99997i 0
209.10 0 −1.22755 + 1.22193i 0 1.22671 + 1.86954i 0 −2.24689 + 1.39696i 0 0.0137791 − 2.99997i 0
209.11 0 −1.13681 − 1.30677i 0 2.01816 + 0.962818i 0 −2.58349 + 0.570607i 0 −0.415306 + 2.97111i 0
209.12 0 −1.13681 − 1.30677i 0 0.175257 − 2.22919i 0 0.797583 + 2.52267i 0 −0.415306 + 2.97111i 0
209.13 0 −0.923807 − 1.46512i 0 −0.517749 + 2.17530i 0 −0.976914 − 2.45879i 0 −1.29316 + 2.70698i 0
209.14 0 −0.923807 − 1.46512i 0 −2.14274 − 0.639267i 0 2.61783 − 0.383362i 0 −1.29316 + 2.70698i 0
209.15 0 −0.800568 − 1.53593i 0 −0.661313 − 2.13604i 0 −2.41752 − 1.07500i 0 −1.71818 + 2.45924i 0
209.16 0 −0.800568 − 1.53593i 0 1.51921 + 1.64073i 0 2.13973 + 1.55613i 0 −1.71818 + 2.45924i 0
209.17 0 −0.527467 + 1.64978i 0 −0.200494 − 2.22706i 0 0.735084 − 2.54158i 0 −2.44356 − 1.74041i 0
209.18 0 −0.527467 + 1.64978i 0 1.82844 + 1.28716i 0 1.83353 − 1.90739i 0 −2.44356 − 1.74041i 0
209.19 0 −0.148608 + 1.72566i 0 1.75104 − 1.39063i 0 1.76230 + 1.97340i 0 −2.95583 − 0.512895i 0
209.20 0 −0.148608 + 1.72566i 0 2.07984 − 0.821129i 0 −2.59016 − 0.539492i 0 −2.95583 − 0.512895i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
9.d odd 6 1 inner
35.c odd 2 1 inner
45.h odd 6 1 inner
63.o even 6 1 inner
315.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.cj.c ✓ 80
3.b odd 2 1 3780.2.cj.c 80
5.b even 2 1 inner 1260.2.cj.c ✓ 80
7.b odd 2 1 inner 1260.2.cj.c ✓ 80
9.c even 3 1 3780.2.cj.c 80
9.d odd 6 1 inner 1260.2.cj.c ✓ 80
15.d odd 2 1 3780.2.cj.c 80
21.c even 2 1 3780.2.cj.c 80
35.c odd 2 1 inner 1260.2.cj.c ✓ 80
45.h odd 6 1 inner 1260.2.cj.c ✓ 80
45.j even 6 1 3780.2.cj.c 80
63.l odd 6 1 3780.2.cj.c 80
63.o even 6 1 inner 1260.2.cj.c ✓ 80
105.g even 2 1 3780.2.cj.c 80
315.z even 6 1 inner 1260.2.cj.c ✓ 80
315.bg odd 6 1 3780.2.cj.c 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.cj.c ✓ 80 1.a even 1 1 trivial
1260.2.cj.c ✓ 80 5.b even 2 1 inner
1260.2.cj.c ✓ 80 7.b odd 2 1 inner
1260.2.cj.c ✓ 80 9.d odd 6 1 inner
1260.2.cj.c ✓ 80 35.c odd 2 1 inner
1260.2.cj.c ✓ 80 45.h odd 6 1 inner
1260.2.cj.c ✓ 80 63.o even 6 1 inner
1260.2.cj.c ✓ 80 315.z even 6 1 inner
3780.2.cj.c 80 3.b odd 2 1
3780.2.cj.c 80 9.c even 3 1
3780.2.cj.c 80 15.d odd 2 1
3780.2.cj.c 80 21.c even 2 1
3780.2.cj.c 80 45.j even 6 1
3780.2.cj.c 80 63.l odd 6 1
3780.2.cj.c 80 105.g even 2 1
3780.2.cj.c 80 315.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{20} - 6 T_{11}^{19} - 28 T_{11}^{18} + 240 T_{11}^{17} + 750 T_{11}^{16} - 8046 T_{11}^{15} + \cdots + 1444 \) acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\). Copy content Toggle raw display