Properties

 Label 637.2.q.a Level $637$ Weight $2$ Character orbit 637.q Analytic conductor $5.086$ Analytic rank $0$ 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(491,637)]

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([0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("637.491");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.q (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: $$5.08647060876$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z - 1) * q^2 + (-2*z + 2) * q^3 + z * q^4 + (2*z - 1) * q^5 + (2*z - 4) * q^6 + (2*z - 1) * q^8 - z * q^9 $$q + ( - \zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + (2 \zeta_{6} - 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (2 \zeta_{6} - 1) q^{8} - \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + 2 q^{12} + (3 \zeta_{6} + 1) q^{13} + (2 \zeta_{6} + 2) q^{15} + ( - 5 \zeta_{6} + 5) q^{16} - 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 1) q^{18} + ( - 2 \zeta_{6} + 4) q^{19} + (\zeta_{6} - 2) q^{20} + ( - 6 \zeta_{6} + 6) q^{23} + (2 \zeta_{6} + 2) q^{24} + 2 q^{25} + ( - 7 \zeta_{6} + 2) q^{26} + 4 q^{27} + (3 \zeta_{6} - 3) q^{29} - 6 \zeta_{6} q^{30} + ( - 4 \zeta_{6} + 2) q^{31} + (3 \zeta_{6} - 6) q^{32} + (6 \zeta_{6} - 3) q^{34} + ( - \zeta_{6} + 1) q^{36} + (5 \zeta_{6} + 5) q^{37} - 6 q^{38} + ( - 2 \zeta_{6} + 8) q^{39} - 3 q^{40} + (3 \zeta_{6} + 3) q^{41} - 8 \zeta_{6} q^{43} + ( - \zeta_{6} + 2) q^{45} + (6 \zeta_{6} - 12) q^{46} + (4 \zeta_{6} - 2) q^{47} - 10 \zeta_{6} q^{48} + ( - 2 \zeta_{6} - 2) q^{50} - 6 q^{51} + (4 \zeta_{6} - 3) q^{52} - 3 q^{53} + ( - 4 \zeta_{6} - 4) q^{54} + ( - 8 \zeta_{6} + 4) q^{57} + ( - 3 \zeta_{6} + 6) q^{58} + (4 \zeta_{6} - 8) q^{59} + (4 \zeta_{6} - 2) q^{60} + \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} - q^{64} + (5 \zeta_{6} - 7) q^{65} + (2 \zeta_{6} + 2) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} - 12 \zeta_{6} q^{69} + ( - 2 \zeta_{6} + 4) q^{71} + ( - \zeta_{6} + 2) q^{72} + ( - 2 \zeta_{6} + 1) q^{73} - 15 \zeta_{6} q^{74} + ( - 4 \zeta_{6} + 4) q^{75} + (2 \zeta_{6} + 2) q^{76} + ( - 4 \zeta_{6} - 10) q^{78} + 4 q^{79} + (5 \zeta_{6} + 5) q^{80} + ( - 11 \zeta_{6} + 11) q^{81} - 9 \zeta_{6} q^{82} + ( - 16 \zeta_{6} + 8) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + (16 \zeta_{6} - 8) q^{86} + 6 \zeta_{6} q^{87} + (4 \zeta_{6} + 4) q^{89} - 3 q^{90} + 6 q^{92} + ( - 4 \zeta_{6} - 4) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} + 6 \zeta_{6} q^{95} + (12 \zeta_{6} - 6) q^{96} + (4 \zeta_{6} - 8) q^{97} +O(q^{100})$$ q + (-z - 1) * q^2 + (-2*z + 2) * q^3 + z * q^4 + (2*z - 1) * q^5 + (2*z - 4) * q^6 + (2*z - 1) * q^8 - z * q^9 + (-3*z + 3) * q^10 + 2 * q^12 + (3*z + 1) * q^13 + (2*z + 2) * q^15 + (-5*z + 5) * q^16 - 3*z * q^17 + (2*z - 1) * q^18 + (-2*z + 4) * q^19 + (z - 2) * q^20 + (-6*z + 6) * q^23 + (2*z + 2) * q^24 + 2 * q^25 + (-7*z + 2) * q^26 + 4 * q^27 + (3*z - 3) * q^29 - 6*z * q^30 + (-4*z + 2) * q^31 + (3*z - 6) * q^32 + (6*z - 3) * q^34 + (-z + 1) * q^36 + (5*z + 5) * q^37 - 6 * q^38 + (-2*z + 8) * q^39 - 3 * q^40 + (3*z + 3) * q^41 - 8*z * q^43 + (-z + 2) * q^45 + (6*z - 12) * q^46 + (4*z - 2) * q^47 - 10*z * q^48 + (-2*z - 2) * q^50 - 6 * q^51 + (4*z - 3) * q^52 - 3 * q^53 + (-4*z - 4) * q^54 + (-8*z + 4) * q^57 + (-3*z + 6) * q^58 + (4*z - 8) * q^59 + (4*z - 2) * q^60 + z * q^61 + (6*z - 6) * q^62 - q^64 + (5*z - 7) * q^65 + (2*z + 2) * q^67 + (-3*z + 3) * q^68 - 12*z * q^69 + (-2*z + 4) * q^71 + (-z + 2) * q^72 + (-2*z + 1) * q^73 - 15*z * q^74 + (-4*z + 4) * q^75 + (2*z + 2) * q^76 + (-4*z - 10) * q^78 + 4 * q^79 + (5*z + 5) * q^80 + (-11*z + 11) * q^81 - 9*z * q^82 + (-16*z + 8) * q^83 + (-3*z + 6) * q^85 + (16*z - 8) * q^86 + 6*z * q^87 + (4*z + 4) * q^89 - 3 * q^90 + 6 * q^92 + (-4*z - 4) * q^93 + (-6*z + 6) * q^94 + 6*z * q^95 + (12*z - 6) * q^96 + (4*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} + q^{4} - 6 q^{6} - q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 + q^4 - 6 * q^6 - q^9 $$2 q - 3 q^{2} + 2 q^{3} + q^{4} - 6 q^{6} - q^{9} + 3 q^{10} + 4 q^{12} + 5 q^{13} + 6 q^{15} + 5 q^{16} - 3 q^{17} + 6 q^{19} - 3 q^{20} + 6 q^{23} + 6 q^{24} + 4 q^{25} - 3 q^{26} + 8 q^{27} - 3 q^{29} - 6 q^{30} - 9 q^{32} + q^{36} + 15 q^{37} - 12 q^{38} + 14 q^{39} - 6 q^{40} + 9 q^{41} - 8 q^{43} + 3 q^{45} - 18 q^{46} - 10 q^{48} - 6 q^{50} - 12 q^{51} - 2 q^{52} - 6 q^{53} - 12 q^{54} + 9 q^{58} - 12 q^{59} + q^{61} - 6 q^{62} - 2 q^{64} - 9 q^{65} + 6 q^{67} + 3 q^{68} - 12 q^{69} + 6 q^{71} + 3 q^{72} - 15 q^{74} + 4 q^{75} + 6 q^{76} - 24 q^{78} + 8 q^{79} + 15 q^{80} + 11 q^{81} - 9 q^{82} + 9 q^{85} + 6 q^{87} + 12 q^{89} - 6 q^{90} + 12 q^{92} - 12 q^{93} + 6 q^{94} + 6 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 + q^4 - 6 * q^6 - q^9 + 3 * q^10 + 4 * q^12 + 5 * q^13 + 6 * q^15 + 5 * q^16 - 3 * q^17 + 6 * q^19 - 3 * q^20 + 6 * q^23 + 6 * q^24 + 4 * q^25 - 3 * q^26 + 8 * q^27 - 3 * q^29 - 6 * q^30 - 9 * q^32 + q^36 + 15 * q^37 - 12 * q^38 + 14 * q^39 - 6 * q^40 + 9 * q^41 - 8 * q^43 + 3 * q^45 - 18 * q^46 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 2 * q^52 - 6 * q^53 - 12 * q^54 + 9 * q^58 - 12 * q^59 + q^61 - 6 * q^62 - 2 * q^64 - 9 * q^65 + 6 * q^67 + 3 * q^68 - 12 * q^69 + 6 * q^71 + 3 * q^72 - 15 * q^74 + 4 * q^75 + 6 * q^76 - 24 * q^78 + 8 * q^79 + 15 * q^80 + 11 * q^81 - 9 * q^82 + 9 * q^85 + 6 * q^87 + 12 * q^89 - 6 * q^90 + 12 * q^92 - 12 * q^93 + 6 * q^94 + 6 * q^95 - 12 * q^97

Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.50000 + 0.866025i 1.00000 + 1.73205i 0.500000 0.866025i 1.73205i −3.00000 1.73205i 0 1.73205i −0.500000 + 0.866025i 1.50000 + 2.59808i
589.1 −1.50000 0.866025i 1.00000 1.73205i 0.500000 + 0.866025i 1.73205i −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
13.e 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.q.a 2
7.b odd 2 1 13.2.e.a 2
7.c even 3 1 637.2.k.c 2
7.c even 3 1 637.2.u.b 2
7.d odd 6 1 637.2.k.a 2
7.d odd 6 1 637.2.u.c 2
13.e even 6 1 inner 637.2.q.a 2
13.f odd 12 2 8281.2.a.q 2
21.c even 2 1 117.2.q.c 2
28.d even 2 1 208.2.w.b 2
35.c odd 2 1 325.2.n.a 2
35.f even 4 2 325.2.m.a 4
56.e even 2 1 832.2.w.a 2
56.h odd 2 1 832.2.w.d 2
84.h odd 2 1 1872.2.by.d 2
91.b odd 2 1 169.2.e.a 2
91.i even 4 2 169.2.c.a 4
91.k even 6 1 637.2.u.b 2
91.l odd 6 1 637.2.u.c 2
91.n odd 6 1 169.2.b.a 2
91.n odd 6 1 169.2.e.a 2
91.p odd 6 1 637.2.k.a 2
91.t odd 6 1 13.2.e.a 2
91.t odd 6 1 169.2.b.a 2
91.u even 6 1 637.2.k.c 2
91.bc even 12 2 169.2.a.a 2
91.bc even 12 2 169.2.c.a 4
273.u even 6 1 117.2.q.c 2
273.u even 6 1 1521.2.b.a 2
273.bn even 6 1 1521.2.b.a 2
273.ca odd 12 2 1521.2.a.k 2
364.v even 6 1 2704.2.f.b 2
364.bc even 6 1 208.2.w.b 2
364.bc even 6 1 2704.2.f.b 2
364.bv odd 12 2 2704.2.a.o 2
455.be odd 6 1 325.2.n.a 2
455.cn even 12 2 4225.2.a.v 2
455.cz even 12 2 325.2.m.a 4
728.bl odd 6 1 832.2.w.d 2
728.ci even 6 1 832.2.w.a 2
1092.bh odd 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.b odd 2 1
13.2.e.a 2 91.t odd 6 1
117.2.q.c 2 21.c even 2 1
117.2.q.c 2 273.u even 6 1
169.2.a.a 2 91.bc even 12 2
169.2.b.a 2 91.n odd 6 1
169.2.b.a 2 91.t odd 6 1
169.2.c.a 4 91.i even 4 2
169.2.c.a 4 91.bc even 12 2
169.2.e.a 2 91.b odd 2 1
169.2.e.a 2 91.n odd 6 1
208.2.w.b 2 28.d even 2 1
208.2.w.b 2 364.bc even 6 1
325.2.m.a 4 35.f even 4 2
325.2.m.a 4 455.cz even 12 2
325.2.n.a 2 35.c odd 2 1
325.2.n.a 2 455.be odd 6 1
637.2.k.a 2 7.d odd 6 1
637.2.k.a 2 91.p odd 6 1
637.2.k.c 2 7.c even 3 1
637.2.k.c 2 91.u even 6 1
637.2.q.a 2 1.a even 1 1 trivial
637.2.q.a 2 13.e even 6 1 inner
637.2.u.b 2 7.c even 3 1
637.2.u.b 2 91.k even 6 1
637.2.u.c 2 7.d odd 6 1
637.2.u.c 2 91.l odd 6 1
832.2.w.a 2 56.e even 2 1
832.2.w.a 2 728.ci even 6 1
832.2.w.d 2 56.h odd 2 1
832.2.w.d 2 728.bl odd 6 1
1521.2.a.k 2 273.ca odd 12 2
1521.2.b.a 2 273.u even 6 1
1521.2.b.a 2 273.bn even 6 1
1872.2.by.d 2 84.h odd 2 1
1872.2.by.d 2 1092.bh odd 6 1
2704.2.a.o 2 364.bv odd 12 2
2704.2.f.b 2 364.v even 6 1
2704.2.f.b 2 364.bc even 6 1
4225.2.a.v 2 455.cn even 12 2
8281.2.a.q 2 13.f odd 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 3$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 6T + 12$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} - 15T + 75$$
$41$ $$T^{2} - 9T + 27$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} + 12T + 48$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 6T + 12$$
$71$ $$T^{2} - 6T + 12$$
$73$ $$T^{2} + 3$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 192$$
$89$ $$T^{2} - 12T + 48$$
$97$ $$T^{2} + 12T + 48$$