# Properties

 Label 630.2.k.e Level $630$ Weight $2$ Character orbit 630.k Analytic conductor $5.031$ 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] = [630,2,Mod(361,630)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("630.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$630 = 2 \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 630.k (of order $$3$$, 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: $$5.03057532734$$ 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 70) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} - 1) q^{7} - q^{8} +O(q^{10})$$ q + z * q^2 + (z - 1) * q^4 - z * q^5 + (-2*z - 1) * q^7 - q^8 $$q + \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + ( - 2 \zeta_{6} - 1) q^{7} - q^{8} + ( - \zeta_{6} + 1) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + 5 q^{13} + ( - 3 \zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + \zeta_{6} q^{19} + q^{20} + 3 q^{22} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 \zeta_{6} q^{26} + ( - \zeta_{6} + 3) q^{28} + 6 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 q^{34} + (3 \zeta_{6} - 2) q^{35} - 11 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} + \zeta_{6} q^{40} - 3 q^{41} - 10 q^{43} + 3 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + 3 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} - q^{50} + (5 \zeta_{6} - 5) q^{52} + ( - 3 \zeta_{6} + 3) q^{53} - 3 q^{55} + (2 \zeta_{6} + 1) q^{56} + 6 \zeta_{6} q^{58} + 4 \zeta_{6} q^{61} + 4 q^{62} + q^{64} - 5 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} + 6 \zeta_{6} q^{68} + (\zeta_{6} - 3) q^{70} - 12 q^{71} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 11 \zeta_{6} + 11) q^{74} - q^{76} + (3 \zeta_{6} - 9) q^{77} + 10 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} - 3 \zeta_{6} q^{82} + 12 q^{83} - 6 q^{85} - 10 \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} + 6 \zeta_{6} q^{89} + ( - 10 \zeta_{6} - 5) q^{91} - 3 q^{92} + (3 \zeta_{6} - 3) q^{94} + ( - \zeta_{6} + 1) q^{95} + 14 q^{97} + (5 \zeta_{6} - 8) q^{98} +O(q^{100})$$ q + z * q^2 + (z - 1) * q^4 - z * q^5 + (-2*z - 1) * q^7 - q^8 + (-z + 1) * q^10 + (-3*z + 3) * q^11 + 5 * q^13 + (-3*z + 2) * q^14 - z * q^16 + (-6*z + 6) * q^17 + z * q^19 + q^20 + 3 * q^22 + 3*z * q^23 + (z - 1) * q^25 + 5*z * q^26 + (-z + 3) * q^28 + 6 * q^29 + (-4*z + 4) * q^31 + (-z + 1) * q^32 + 6 * q^34 + (3*z - 2) * q^35 - 11*z * q^37 + (z - 1) * q^38 + z * q^40 - 3 * q^41 - 10 * q^43 + 3*z * q^44 + (3*z - 3) * q^46 + 3*z * q^47 + (8*z - 3) * q^49 - q^50 + (5*z - 5) * q^52 + (-3*z + 3) * q^53 - 3 * q^55 + (2*z + 1) * q^56 + 6*z * q^58 + 4*z * q^61 + 4 * q^62 + q^64 - 5*z * q^65 + (-4*z + 4) * q^67 + 6*z * q^68 + (z - 3) * q^70 - 12 * q^71 + (-4*z + 4) * q^73 + (-11*z + 11) * q^74 - q^76 + (3*z - 9) * q^77 + 10*z * q^79 + (z - 1) * q^80 - 3*z * q^82 + 12 * q^83 - 6 * q^85 - 10*z * q^86 + (3*z - 3) * q^88 + 6*z * q^89 + (-10*z - 5) * q^91 - 3 * q^92 + (3*z - 3) * q^94 + (-z + 1) * q^95 + 14 * q^97 + (5*z - 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{5} - 4 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - q^5 - 4 * q^7 - 2 * q^8 $$2 q + q^{2} - q^{4} - q^{5} - 4 q^{7} - 2 q^{8} + q^{10} + 3 q^{11} + 10 q^{13} + q^{14} - q^{16} + 6 q^{17} + q^{19} + 2 q^{20} + 6 q^{22} + 3 q^{23} - q^{25} + 5 q^{26} + 5 q^{28} + 12 q^{29} + 4 q^{31} + q^{32} + 12 q^{34} - q^{35} - 11 q^{37} - q^{38} + q^{40} - 6 q^{41} - 20 q^{43} + 3 q^{44} - 3 q^{46} + 3 q^{47} + 2 q^{49} - 2 q^{50} - 5 q^{52} + 3 q^{53} - 6 q^{55} + 4 q^{56} + 6 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 5 q^{65} + 4 q^{67} + 6 q^{68} - 5 q^{70} - 24 q^{71} + 4 q^{73} + 11 q^{74} - 2 q^{76} - 15 q^{77} + 10 q^{79} - q^{80} - 3 q^{82} + 24 q^{83} - 12 q^{85} - 10 q^{86} - 3 q^{88} + 6 q^{89} - 20 q^{91} - 6 q^{92} - 3 q^{94} + q^{95} + 28 q^{97} - 11 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - q^5 - 4 * q^7 - 2 * q^8 + q^10 + 3 * q^11 + 10 * q^13 + q^14 - q^16 + 6 * q^17 + q^19 + 2 * q^20 + 6 * q^22 + 3 * q^23 - q^25 + 5 * q^26 + 5 * q^28 + 12 * q^29 + 4 * q^31 + q^32 + 12 * q^34 - q^35 - 11 * q^37 - q^38 + q^40 - 6 * q^41 - 20 * q^43 + 3 * q^44 - 3 * q^46 + 3 * q^47 + 2 * q^49 - 2 * q^50 - 5 * q^52 + 3 * q^53 - 6 * q^55 + 4 * q^56 + 6 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 5 * q^65 + 4 * q^67 + 6 * q^68 - 5 * q^70 - 24 * q^71 + 4 * q^73 + 11 * q^74 - 2 * q^76 - 15 * q^77 + 10 * q^79 - q^80 - 3 * q^82 + 24 * q^83 - 12 * q^85 - 10 * q^86 - 3 * q^88 + 6 * q^89 - 20 * q^91 - 6 * q^92 - 3 * q^94 + q^95 + 28 * q^97 - 11 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 0.866025i 0 −2.00000 1.73205i −1.00000 0 0.500000 0.866025i
541.1 0.500000 0.866025i 0 −0.500000 0.866025i −0.500000 + 0.866025i 0 −2.00000 + 1.73205i −1.00000 0 0.500000 + 0.866025i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.k.e 2
3.b odd 2 1 70.2.e.b 2
7.c even 3 1 inner 630.2.k.e 2
7.c even 3 1 4410.2.a.m 1
7.d odd 6 1 4410.2.a.c 1
12.b even 2 1 560.2.q.d 2
15.d odd 2 1 350.2.e.h 2
15.e even 4 2 350.2.j.a 4
21.c even 2 1 490.2.e.a 2
21.g even 6 1 490.2.a.j 1
21.g even 6 1 490.2.e.a 2
21.h odd 6 1 70.2.e.b 2
21.h odd 6 1 490.2.a.g 1
84.j odd 6 1 3920.2.a.g 1
84.n even 6 1 560.2.q.d 2
84.n even 6 1 3920.2.a.be 1
105.o odd 6 1 350.2.e.h 2
105.o odd 6 1 2450.2.a.p 1
105.p even 6 1 2450.2.a.f 1
105.w odd 12 2 2450.2.c.p 2
105.x even 12 2 350.2.j.a 4
105.x even 12 2 2450.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 3.b odd 2 1
70.2.e.b 2 21.h odd 6 1
350.2.e.h 2 15.d odd 2 1
350.2.e.h 2 105.o odd 6 1
350.2.j.a 4 15.e even 4 2
350.2.j.a 4 105.x even 12 2
490.2.a.g 1 21.h odd 6 1
490.2.a.j 1 21.g even 6 1
490.2.e.a 2 21.c even 2 1
490.2.e.a 2 21.g even 6 1
560.2.q.d 2 12.b even 2 1
560.2.q.d 2 84.n even 6 1
630.2.k.e 2 1.a even 1 1 trivial
630.2.k.e 2 7.c even 3 1 inner
2450.2.a.f 1 105.p even 6 1
2450.2.a.p 1 105.o odd 6 1
2450.2.c.f 2 105.x even 12 2
2450.2.c.p 2 105.w odd 12 2
3920.2.a.g 1 84.j odd 6 1
3920.2.a.be 1 84.n even 6 1
4410.2.a.c 1 7.d odd 6 1
4410.2.a.m 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(630, [\chi])$$:

 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{13} - 5$$ T13 - 5 $$T_{17}^{2} - 6T_{17} + 36$$ T17^2 - 6*T17 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} - T + 1$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 14)^{2}$$