# Properties

 Label 637.2.k.j Level $637$ Weight $2$ Character orbit 637.k Analytic conductor $5.086$ Analytic rank $0$ Dimension $32$ Inner twists $4$

# Related objects

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 algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 40 q^{4} - 28 q^{9}+O(q^{10})$$ 32 * q - 40 * q^4 - 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 40 q^{4} - 28 q^{9} - 12 q^{15} + 56 q^{16} + 24 q^{18} + 8 q^{22} - 48 q^{23} + 20 q^{25} - 24 q^{29} + 24 q^{30} + 92 q^{36} - 20 q^{39} + 12 q^{43} + 12 q^{50} - 36 q^{53} + 132 q^{58} + 276 q^{60} + 32 q^{64} - 12 q^{65} - 48 q^{67} - 48 q^{71} - 72 q^{72} - 48 q^{74} - 156 q^{78} - 48 q^{79} - 64 q^{81} + 12 q^{85} - 48 q^{86} + 56 q^{88} + 168 q^{92} - 168 q^{95}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{16} + 26T_{2}^{14} + 269T_{2}^{12} + 1406T_{2}^{10} + 3892T_{2}^{8} + 5494T_{2}^{6} + 3581T_{2}^{4} + 850T_{2}^{2} + 49$$ T2^16 + 26*T2^14 + 269*T2^12 + 1406*T2^10 + 3892*T2^8 + 5494*T2^6 + 3581*T2^4 + 850*T2^2 + 49 $$T_{3}^{32} + 38 T_{3}^{30} + 873 T_{3}^{28} + 13066 T_{3}^{26} + 144668 T_{3}^{24} + 1181268 T_{3}^{22} + \cdots + 614656$$ T3^32 + 38*T3^30 + 873*T3^28 + 13066*T3^26 + 144668*T3^24 + 1181268*T3^22 + 7395721*T3^20 + 34582652*T3^18 + 123857253*T3^16 + 327520988*T3^14 + 647335780*T3^12 + 879006960*T3^10 + 835299200*T3^8 + 425728960*T3^6 + 146463552*T3^4 + 10373888*T3^2 + 614656