# Properties

 Label 637.2.k.a Level $637$ Weight $2$ Character orbit 637.k Analytic conductor $5.086$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 1) q^{2} - 2 \zeta_{6} q^{3} - q^{4} + (\zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + (2 \zeta_{6} - 1) q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q + (2*z - 1) * q^2 - 2*z * q^3 - q^4 + (z - 2) * q^5 + (-2*z + 4) * q^6 + (2*z - 1) * q^8 + (z - 1) * q^9 $$q + (2 \zeta_{6} - 1) q^{2} - 2 \zeta_{6} q^{3} - q^{4} + (\zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 4) q^{6} + (2 \zeta_{6} - 1) q^{8} + (\zeta_{6} - 1) q^{9} - 3 \zeta_{6} q^{10} + 2 \zeta_{6} q^{12} + ( - 3 \zeta_{6} - 1) q^{13} + (2 \zeta_{6} + 2) q^{15} - 5 q^{16} - 3 q^{17} + ( - \zeta_{6} - 1) q^{18} + (2 \zeta_{6} + 2) q^{19} + ( - \zeta_{6} + 2) q^{20} - 6 q^{23} + ( - 2 \zeta_{6} + 4) q^{24} + (2 \zeta_{6} - 2) q^{25} + ( - 5 \zeta_{6} + 7) q^{26} - 4 q^{27} + (3 \zeta_{6} - 3) q^{29} + (6 \zeta_{6} - 6) q^{30} + ( - 2 \zeta_{6} - 2) q^{31} + ( - 6 \zeta_{6} + 3) q^{32} + ( - 6 \zeta_{6} + 3) q^{34} + ( - \zeta_{6} + 1) q^{36} + ( - 10 \zeta_{6} + 5) q^{37} + (6 \zeta_{6} - 6) q^{38} + (8 \zeta_{6} - 6) q^{39} - 3 \zeta_{6} q^{40} + ( - 3 \zeta_{6} - 3) q^{41} - 8 \zeta_{6} q^{43} + ( - 2 \zeta_{6} + 1) q^{45} + ( - 12 \zeta_{6} + 6) q^{46} + (2 \zeta_{6} - 4) q^{47} + 10 \zeta_{6} q^{48} + ( - 2 \zeta_{6} - 2) q^{50} + 6 \zeta_{6} q^{51} + (3 \zeta_{6} + 1) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} + ( - 8 \zeta_{6} + 4) q^{54} + ( - 8 \zeta_{6} + 4) q^{57} + ( - 3 \zeta_{6} - 3) q^{58} + (8 \zeta_{6} - 4) q^{59} + ( - 2 \zeta_{6} - 2) q^{60} + (\zeta_{6} - 1) q^{61} + ( - 6 \zeta_{6} + 6) q^{62} - q^{64} + (2 \zeta_{6} + 5) q^{65} + (2 \zeta_{6} - 4) q^{67} + 3 q^{68} + 12 \zeta_{6} q^{69} + ( - 2 \zeta_{6} + 4) q^{71} + ( - \zeta_{6} - 1) q^{72} + ( - \zeta_{6} - 1) q^{73} + 15 q^{74} + 4 q^{75} + ( - 2 \zeta_{6} - 2) q^{76} + ( - 4 \zeta_{6} - 10) q^{78} - 4 \zeta_{6} q^{79} + ( - 5 \zeta_{6} + 10) q^{80} + 11 \zeta_{6} q^{81} + ( - 9 \zeta_{6} + 9) q^{82} + (16 \zeta_{6} - 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + ( - 8 \zeta_{6} + 16) q^{86} + 6 q^{87} + (8 \zeta_{6} - 4) q^{89} + 3 q^{90} + 6 q^{92} + (8 \zeta_{6} - 4) q^{93} - 6 \zeta_{6} q^{94} - 6 q^{95} + (6 \zeta_{6} - 12) q^{96} + ( - 4 \zeta_{6} + 8) q^{97} +O(q^{100})$$ q + (2*z - 1) * q^2 - 2*z * q^3 - q^4 + (z - 2) * q^5 + (-2*z + 4) * q^6 + (2*z - 1) * q^8 + (z - 1) * q^9 - 3*z * q^10 + 2*z * q^12 + (-3*z - 1) * q^13 + (2*z + 2) * q^15 - 5 * q^16 - 3 * q^17 + (-z - 1) * q^18 + (2*z + 2) * q^19 + (-z + 2) * q^20 - 6 * q^23 + (-2*z + 4) * q^24 + (2*z - 2) * q^25 + (-5*z + 7) * q^26 - 4 * q^27 + (3*z - 3) * q^29 + (6*z - 6) * q^30 + (-2*z - 2) * q^31 + (-6*z + 3) * q^32 + (-6*z + 3) * q^34 + (-z + 1) * q^36 + (-10*z + 5) * q^37 + (6*z - 6) * q^38 + (8*z - 6) * q^39 - 3*z * q^40 + (-3*z - 3) * q^41 - 8*z * q^43 + (-2*z + 1) * q^45 + (-12*z + 6) * q^46 + (2*z - 4) * q^47 + 10*z * q^48 + (-2*z - 2) * q^50 + 6*z * q^51 + (3*z + 1) * q^52 + (-3*z + 3) * q^53 + (-8*z + 4) * q^54 + (-8*z + 4) * q^57 + (-3*z - 3) * q^58 + (8*z - 4) * q^59 + (-2*z - 2) * q^60 + (z - 1) * q^61 + (-6*z + 6) * q^62 - q^64 + (2*z + 5) * q^65 + (2*z - 4) * q^67 + 3 * q^68 + 12*z * q^69 + (-2*z + 4) * q^71 + (-z - 1) * q^72 + (-z - 1) * q^73 + 15 * q^74 + 4 * q^75 + (-2*z - 2) * q^76 + (-4*z - 10) * q^78 - 4*z * q^79 + (-5*z + 10) * q^80 + 11*z * q^81 + (-9*z + 9) * q^82 + (16*z - 8) * q^83 + (-3*z + 6) * q^85 + (-8*z + 16) * q^86 + 6 * q^87 + (8*z - 4) * q^89 + 3 * q^90 + 6 * q^92 + (8*z - 4) * q^93 - 6*z * q^94 - 6 * q^95 + (6*z - 12) * q^96 + (-4*z + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} - 3 q^{5} + 6 q^{6} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 - 3 * q^5 + 6 * q^6 - q^9 $$2 q - 2 q^{3} - 2 q^{4} - 3 q^{5} + 6 q^{6} - q^{9} - 3 q^{10} + 2 q^{12} - 5 q^{13} + 6 q^{15} - 10 q^{16} - 6 q^{17} - 3 q^{18} + 6 q^{19} + 3 q^{20} - 12 q^{23} + 6 q^{24} - 2 q^{25} + 9 q^{26} - 8 q^{27} - 3 q^{29} - 6 q^{30} - 6 q^{31} + q^{36} - 6 q^{38} - 4 q^{39} - 3 q^{40} - 9 q^{41} - 8 q^{43} - 6 q^{47} + 10 q^{48} - 6 q^{50} + 6 q^{51} + 5 q^{52} + 3 q^{53} - 9 q^{58} - 6 q^{60} - q^{61} + 6 q^{62} - 2 q^{64} + 12 q^{65} - 6 q^{67} + 6 q^{68} + 12 q^{69} + 6 q^{71} - 3 q^{72} - 3 q^{73} + 30 q^{74} + 8 q^{75} - 6 q^{76} - 24 q^{78} - 4 q^{79} + 15 q^{80} + 11 q^{81} + 9 q^{82} + 9 q^{85} + 24 q^{86} + 12 q^{87} + 6 q^{90} + 12 q^{92} - 6 q^{94} - 12 q^{95} - 18 q^{96} + 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 - 3 * q^5 + 6 * q^6 - q^9 - 3 * q^10 + 2 * q^12 - 5 * q^13 + 6 * q^15 - 10 * q^16 - 6 * q^17 - 3 * q^18 + 6 * q^19 + 3 * q^20 - 12 * q^23 + 6 * q^24 - 2 * q^25 + 9 * q^26 - 8 * q^27 - 3 * q^29 - 6 * q^30 - 6 * q^31 + q^36 - 6 * q^38 - 4 * q^39 - 3 * q^40 - 9 * q^41 - 8 * q^43 - 6 * q^47 + 10 * q^48 - 6 * q^50 + 6 * q^51 + 5 * q^52 + 3 * q^53 - 9 * q^58 - 6 * q^60 - q^61 + 6 * q^62 - 2 * q^64 + 12 * q^65 - 6 * q^67 + 6 * q^68 + 12 * q^69 + 6 * q^71 - 3 * q^72 - 3 * q^73 + 30 * q^74 + 8 * q^75 - 6 * q^76 - 24 * q^78 - 4 * q^79 + 15 * q^80 + 11 * q^81 + 9 * q^82 + 9 * q^85 + 24 * q^86 + 12 * q^87 + 6 * q^90 + 12 * q^92 - 6 * q^94 - 12 * q^95 - 18 * q^96 + 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
459.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i −1.00000 1.73205i −1.00000 −1.50000 + 0.866025i 3.00000 1.73205i 0 1.73205i −0.500000 + 0.866025i −1.50000 2.59808i
569.1 1.73205i −1.00000 + 1.73205i −1.00000 −1.50000 0.866025i 3.00000 + 1.73205i 0 1.73205i −0.500000 0.866025i −1.50000 + 2.59808i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 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.a 2
7.b odd 2 1 637.2.k.c 2
7.c even 3 1 13.2.e.a 2
7.c even 3 1 637.2.u.c 2
7.d odd 6 1 637.2.q.a 2
7.d odd 6 1 637.2.u.b 2
13.e even 6 1 637.2.u.c 2
21.h odd 6 1 117.2.q.c 2
28.g odd 6 1 208.2.w.b 2
35.j even 6 1 325.2.n.a 2
35.l odd 12 2 325.2.m.a 4
56.k odd 6 1 832.2.w.a 2
56.p even 6 1 832.2.w.d 2
84.n even 6 1 1872.2.by.d 2
91.g even 3 1 169.2.e.a 2
91.h even 3 1 169.2.b.a 2
91.k even 6 1 169.2.b.a 2
91.k even 6 1 inner 637.2.k.a 2
91.l odd 6 1 637.2.k.c 2
91.p odd 6 1 637.2.q.a 2
91.r even 6 1 169.2.e.a 2
91.t odd 6 1 637.2.u.b 2
91.u even 6 1 13.2.e.a 2
91.x odd 12 2 169.2.a.a 2
91.z odd 12 2 169.2.c.a 4
91.ba even 12 2 8281.2.a.q 2
91.bd odd 12 2 169.2.c.a 4
273.s odd 6 1 1521.2.b.a 2
273.x odd 6 1 117.2.q.c 2
273.bp odd 6 1 1521.2.b.a 2
273.bv even 12 2 1521.2.a.k 2
364.s odd 6 1 208.2.w.b 2
364.bi odd 6 1 2704.2.f.b 2
364.bk odd 6 1 2704.2.f.b 2
364.ca even 12 2 2704.2.a.o 2
455.bc even 6 1 325.2.n.a 2
455.da odd 12 2 325.2.m.a 4
455.dh odd 12 2 4225.2.a.v 2
728.be even 6 1 832.2.w.d 2
728.dd odd 6 1 832.2.w.a 2
1092.bn even 6 1 1872.2.by.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 7.c even 3 1
13.2.e.a 2 91.u even 6 1
117.2.q.c 2 21.h odd 6 1
117.2.q.c 2 273.x odd 6 1
169.2.a.a 2 91.x odd 12 2
169.2.b.a 2 91.h even 3 1
169.2.b.a 2 91.k even 6 1
169.2.c.a 4 91.z odd 12 2
169.2.c.a 4 91.bd odd 12 2
169.2.e.a 2 91.g even 3 1
169.2.e.a 2 91.r even 6 1
208.2.w.b 2 28.g odd 6 1
208.2.w.b 2 364.s odd 6 1
325.2.m.a 4 35.l odd 12 2
325.2.m.a 4 455.da odd 12 2
325.2.n.a 2 35.j even 6 1
325.2.n.a 2 455.bc even 6 1
637.2.k.a 2 1.a even 1 1 trivial
637.2.k.a 2 91.k even 6 1 inner
637.2.k.c 2 7.b odd 2 1
637.2.k.c 2 91.l odd 6 1
637.2.q.a 2 7.d odd 6 1
637.2.q.a 2 91.p odd 6 1
637.2.u.b 2 7.d odd 6 1
637.2.u.b 2 91.t odd 6 1
637.2.u.c 2 7.c even 3 1
637.2.u.c 2 13.e even 6 1
832.2.w.a 2 56.k odd 6 1
832.2.w.a 2 728.dd odd 6 1
832.2.w.d 2 56.p even 6 1
832.2.w.d 2 728.be even 6 1
1521.2.a.k 2 273.bv even 12 2
1521.2.b.a 2 273.s odd 6 1
1521.2.b.a 2 273.bp odd 6 1
1872.2.by.d 2 84.n even 6 1
1872.2.by.d 2 1092.bn even 6 1
2704.2.a.o 2 364.ca even 12 2
2704.2.f.b 2 364.bi odd 6 1
2704.2.f.b 2 364.bk odd 6 1
4225.2.a.v 2 455.dh odd 12 2
8281.2.a.q 2 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{2} + 3$$ T2^2 + 3 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 3T + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 6T + 12$$
$37$ $$T^{2} + 75$$
$41$ $$T^{2} + 9T + 27$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 6T + 12$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} + 48$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 6T + 12$$
$71$ $$T^{2} - 6T + 12$$
$73$ $$T^{2} + 3T + 3$$
$79$ $$T^{2} + 4T + 16$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} - 12T + 48$$