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Properties

Label 684.2.ce.a
Level $684$
Weight $2$
Character orbit 684.ce
Analytic conductor $5.462$
Analytic rank $0$
Dimension $240$
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] = [684,2,Mod(35,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.ce (of order \(18\), degree \(6\), 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: \(5.46176749826\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 12 q^{4} + 12 q^{10} + 24 q^{13} - 12 q^{16} - 12 q^{34} + 120 q^{49} - 48 q^{52} - 144 q^{58} + 48 q^{61} - 12 q^{64} - 72 q^{70} + 72 q^{73} - 144 q^{76} - 72 q^{82} + 240 q^{85} - 48 q^{88} - 48 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −1.41066 + 0.100136i 0 1.97995 − 0.282516i 2.62986 + 0.463716i 0 −0.0486289 + 0.0280759i −2.76475 + 0.596799i 0 −3.75629 − 0.390804i
35.2 −1.40902 − 0.121139i 0 1.97065 + 0.341373i −1.60652 − 0.283272i 0 2.17699 − 1.25689i −2.73532 − 0.719723i 0 2.22929 + 0.593747i
35.3 −1.38949 − 0.263288i 0 1.86136 + 0.731671i 3.08849 + 0.544584i 0 −3.20468 + 1.85022i −2.39370 − 1.50672i 0 −4.14804 − 1.56986i
35.4 −1.34626 + 0.433097i 0 1.62485 − 1.16613i −3.51342 − 0.619510i 0 −0.312440 + 0.180387i −1.68244 + 2.27363i 0 4.99830 − 0.687626i
35.5 −1.32804 + 0.486117i 0 1.52738 − 1.29117i −1.52467 − 0.268841i 0 −1.25899 + 0.726880i −1.40076 + 2.45721i 0 2.15551 − 0.384138i
35.6 −1.30310 − 0.549476i 0 1.39615 + 1.43205i −1.67868 − 0.295997i 0 2.09389 − 1.20891i −1.03245 − 2.63326i 0 2.02485 + 1.30811i
35.7 −1.14528 + 0.829654i 0 0.623347 − 1.90038i 1.58595 + 0.279646i 0 −0.149816 + 0.0864964i 0.862749 + 2.69363i 0 −2.04837 + 0.995517i
35.8 −1.12762 − 0.853506i 0 0.543054 + 1.92486i 4.17116 + 0.735488i 0 2.05168 − 1.18454i 1.03052 − 2.63401i 0 −4.07574 − 4.38947i
35.9 −1.10560 − 0.881847i 0 0.444691 + 1.94994i −0.968845 − 0.170834i 0 −4.03846 + 2.33161i 1.22790 − 2.54799i 0 0.920504 + 1.04325i
35.10 −1.06043 − 0.935670i 0 0.249045 + 1.98443i 0.968845 + 0.170834i 0 4.03846 − 2.33161i 1.59268 − 2.33739i 0 −0.867554 − 1.08768i
35.11 −1.05911 + 0.937173i 0 0.243412 − 1.98513i 0.903171 + 0.159253i 0 3.38909 − 1.95669i 1.60261 + 2.33059i 0 −1.10580 + 0.677762i
35.12 −1.03635 − 0.962279i 0 0.148038 + 1.99451i −4.17116 − 0.735488i 0 −2.05168 + 1.18454i 1.76586 − 2.20947i 0 3.61503 + 4.77605i
35.13 −0.767410 − 1.18789i 0 −0.822164 + 1.82320i 1.67868 + 0.295997i 0 −2.09389 + 1.20891i 2.79669 − 0.422500i 0 −0.936627 − 2.22124i
35.14 −0.739024 + 1.20575i 0 −0.907688 − 1.78216i 0.903171 + 0.159253i 0 −3.38909 + 1.95669i 2.81965 + 0.222612i 0 −0.859486 + 0.971311i
35.15 −0.618174 + 1.27195i 0 −1.23572 − 1.57257i 1.58595 + 0.279646i 0 0.149816 − 0.0864964i 2.76413 − 0.599655i 0 −1.33609 + 1.84438i
35.16 −0.500570 − 1.32266i 0 −1.49886 + 1.32417i −3.08849 − 0.544584i 0 3.20468 − 1.85022i 2.50171 + 1.31964i 0 0.825706 + 4.35763i
35.17 −0.363972 − 1.36657i 0 −1.73505 + 0.994788i 1.60652 + 0.283272i 0 −2.17699 + 1.25689i 1.99096 + 2.00900i 0 −0.197614 − 2.29853i
35.18 −0.248120 + 1.39228i 0 −1.87687 − 0.690904i −1.52467 − 0.268841i 0 1.25899 − 0.726880i 1.42762 − 2.44170i 0 0.752604 − 2.05606i
35.19 −0.192741 + 1.40102i 0 −1.92570 − 0.540067i −3.51342 − 0.619510i 0 0.312440 − 0.180387i 1.12780 − 2.59385i 0 1.54512 − 4.80296i
35.20 −0.146345 − 1.40662i 0 −1.95717 + 0.411703i −2.62986 − 0.463716i 0 0.0486289 − 0.0280759i 0.865531 + 2.69274i 0 −0.267406 + 3.76708i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.e even 9 1 inner
57.l odd 18 1 inner
76.l odd 18 1 inner
228.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.2.ce.a ✓ 240
3.b odd 2 1 inner 684.2.ce.a ✓ 240
4.b odd 2 1 inner 684.2.ce.a ✓ 240
12.b even 2 1 inner 684.2.ce.a ✓ 240
19.e even 9 1 inner 684.2.ce.a ✓ 240
57.l odd 18 1 inner 684.2.ce.a ✓ 240
76.l odd 18 1 inner 684.2.ce.a ✓ 240
228.v even 18 1 inner 684.2.ce.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.ce.a ✓ 240 1.a even 1 1 trivial
684.2.ce.a ✓ 240 3.b odd 2 1 inner
684.2.ce.a ✓ 240 4.b odd 2 1 inner
684.2.ce.a ✓ 240 12.b even 2 1 inner
684.2.ce.a ✓ 240 19.e even 9 1 inner
684.2.ce.a ✓ 240 57.l odd 18 1 inner
684.2.ce.a ✓ 240 76.l odd 18 1 inner
684.2.ce.a ✓ 240 228.v even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).