Properties

Label 637.2.k.j
Level $637$
Weight $2$
Character orbit 637.k
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(459,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.459");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.k (of order \(6\), degree \(2\), not 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 40 q^{4} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 40 q^{4} - 28 q^{9} - 12 q^{15} + 56 q^{16} + 24 q^{18} + 8 q^{22} - 48 q^{23} + 20 q^{25} - 24 q^{29} + 24 q^{30} + 92 q^{36} - 20 q^{39} + 12 q^{43} + 12 q^{50} - 36 q^{53} + 132 q^{58} + 276 q^{60} + 32 q^{64} - 12 q^{65} - 48 q^{67} - 48 q^{71} - 72 q^{72} - 48 q^{74} - 156 q^{78} - 48 q^{79} - 64 q^{81} + 12 q^{85} - 48 q^{86} + 56 q^{88} + 168 q^{92} - 168 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1 2.66170i −1.59481 2.76230i −5.08467 0.285259 0.164694i −7.35241 + 4.24492i 0 8.21047i −3.58685 + 6.21261i −0.438368 0.759275i
459.2 2.66170i 1.59481 + 2.76230i −5.08467 −0.285259 + 0.164694i 7.35241 4.24492i 0 8.21047i −3.58685 + 6.21261i 0.438368 + 0.759275i
459.3 2.09120i −0.663994 1.15007i −2.37313 3.02157 1.74450i −2.40503 + 1.38855i 0 0.780297i 0.618224 1.07080i −3.64811 6.31871i
459.4 2.09120i 0.663994 + 1.15007i −2.37313 −3.02157 + 1.74450i 2.40503 1.38855i 0 0.780297i 0.618224 1.07080i 3.64811 + 6.31871i
459.5 1.04004i −0.384681 0.666288i 0.918323 1.45206 0.838345i −0.692964 + 0.400083i 0 3.03516i 1.20404 2.08546i −0.871910 1.51019i
459.6 1.04004i 0.384681 + 0.666288i 0.918323 −1.45206 + 0.838345i 0.692964 0.400083i 0 3.03516i 1.20404 2.08546i 0.871910 + 1.51019i
459.7 0.565505i −1.54556 2.67698i 1.68020 −1.56330 + 0.902570i −1.51385 + 0.874021i 0 2.08118i −3.27749 + 5.67679i 0.510408 + 0.884053i
459.8 0.565505i 1.54556 + 2.67698i 1.68020 1.56330 0.902570i 1.51385 0.874021i 0 2.08118i −3.27749 + 5.67679i −0.510408 0.884053i
459.9 0.288856i −0.969921 1.67995i 1.91656 −3.71818 + 2.14669i 0.485264 0.280167i 0 1.13132i −0.381493 + 0.660765i −0.620085 1.07402i
459.10 0.288856i 0.969921 + 1.67995i 1.91656 3.71818 2.14669i −0.485264 + 0.280167i 0 1.13132i −0.381493 + 0.660765i 0.620085 + 1.07402i
459.11 1.30368i −0.134969 0.233774i 0.300429 −1.35808 + 0.784090i 0.304765 0.175956i 0 2.99901i 1.46357 2.53497i −1.02220 1.77050i
459.12 1.30368i 0.134969 + 0.233774i 0.300429 1.35808 0.784090i −0.304765 + 0.175956i 0 2.99901i 1.46357 2.53497i 1.02220 + 1.77050i
459.13 2.36389i −1.49347 2.58676i −3.58797 2.77356 1.60131i 6.11481 3.53039i 0 3.75379i −2.96088 + 5.12839i 3.78533 + 6.55639i
459.14 2.36389i 1.49347 + 2.58676i −3.58797 −2.77356 + 1.60131i −6.11481 + 3.53039i 0 3.75379i −2.96088 + 5.12839i −3.78533 6.55639i
459.15 2.40203i −0.888571 1.53905i −3.76975 0.611970 0.353321i 3.69684 2.13437i 0 4.25098i −0.0791164 + 0.137034i 0.848687 + 1.46997i
459.16 2.40203i 0.888571 + 1.53905i −3.76975 −0.611970 + 0.353321i −3.69684 + 2.13437i 0 4.25098i −0.0791164 + 0.137034i −0.848687 1.46997i
569.1 2.40203i −0.888571 + 1.53905i −3.76975 0.611970 + 0.353321i 3.69684 + 2.13437i 0 4.25098i −0.0791164 0.137034i 0.848687 1.46997i
569.2 2.40203i 0.888571 1.53905i −3.76975 −0.611970 0.353321i −3.69684 2.13437i 0 4.25098i −0.0791164 0.137034i −0.848687 + 1.46997i
569.3 2.36389i −1.49347 + 2.58676i −3.58797 2.77356 + 1.60131i 6.11481 + 3.53039i 0 3.75379i −2.96088 5.12839i 3.78533 6.55639i
569.4 2.36389i 1.49347 2.58676i −3.58797 −2.77356 1.60131i −6.11481 3.53039i 0 3.75379i −2.96088 5.12839i −3.78533 + 6.55639i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.k even 6 1 inner
91.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.k.j 32
7.b odd 2 1 inner 637.2.k.j 32
7.c even 3 1 637.2.q.j 32
7.c even 3 1 637.2.u.j 32
7.d odd 6 1 637.2.q.j 32
7.d odd 6 1 637.2.u.j 32
13.e even 6 1 637.2.u.j 32
91.k even 6 1 inner 637.2.k.j 32
91.l odd 6 1 inner 637.2.k.j 32
91.p odd 6 1 637.2.q.j 32
91.t odd 6 1 637.2.u.j 32
91.u even 6 1 637.2.q.j 32
91.x odd 12 2 8281.2.a.cx 32
91.ba even 12 2 8281.2.a.cx 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.j 32 1.a even 1 1 trivial
637.2.k.j 32 7.b odd 2 1 inner
637.2.k.j 32 91.k even 6 1 inner
637.2.k.j 32 91.l odd 6 1 inner
637.2.q.j 32 7.c even 3 1
637.2.q.j 32 7.d odd 6 1
637.2.q.j 32 91.p odd 6 1
637.2.q.j 32 91.u even 6 1
637.2.u.j 32 7.c even 3 1
637.2.u.j 32 7.d odd 6 1
637.2.u.j 32 13.e even 6 1
637.2.u.j 32 91.t odd 6 1
8281.2.a.cx 32 91.x odd 12 2
8281.2.a.cx 32 91.ba even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{16} + 26T_{2}^{14} + 269T_{2}^{12} + 1406T_{2}^{10} + 3892T_{2}^{8} + 5494T_{2}^{6} + 3581T_{2}^{4} + 850T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{3}^{32} + 38 T_{3}^{30} + 873 T_{3}^{28} + 13066 T_{3}^{26} + 144668 T_{3}^{24} + 1181268 T_{3}^{22} + 7395721 T_{3}^{20} + 34582652 T_{3}^{18} + 123857253 T_{3}^{16} + 327520988 T_{3}^{14} + \cdots + 614656 \) Copy content Toggle raw display