# Properties

 Label 630.2.g.g Level $630$ Weight $2$ Character orbit 630.g Analytic conductor $5.031$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(379,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("630.379");

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.g (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8}+O(q^{10})$$ q - b2 * q^2 - q^4 + (b2 - b1 - 1) * q^5 + b2 * q^7 + b2 * q^8 $$q - \beta_{2} q^{2} - q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{8} + (\beta_{3} + \beta_{2} + 1) q^{10} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + q^{14} + q^{16} + 2 \beta_{2} q^{17} + ( - \beta_{3} + \beta_1 - 4) q^{19} + ( - \beta_{2} + \beta_1 + 1) q^{20} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 + 2) q^{26} - \beta_{2} q^{28} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{31} - \beta_{2} q^{32} + 2 q^{34} + ( - \beta_{3} - \beta_{2} - 1) q^{35} - 2 \beta_{2} q^{37} + ( - \beta_{3} + 4 \beta_{2} - \beta_1) q^{38} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + (2 \beta_{3} - 2 \beta_1 + 6) q^{41} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} + (2 \beta_{3} - 2 \beta_1 + 2) q^{46} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{50} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{52} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{53} + (4 \beta_{3} - 6 \beta_{2} - 6) q^{55} - q^{56} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{58} + ( - \beta_{3} + \beta_1 - 4) q^{59} + (\beta_{3} - \beta_1 + 6) q^{61} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{62} - q^{64} + ( - 2 \beta_{3} - 5 \beta_{2} + \cdots + 1) q^{65}+ \cdots + \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 - q^4 + (b2 - b1 - 1) * q^5 + b2 * q^7 + b2 * q^8 + (b3 + b2 + 1) * q^10 + (-2*b3 + 2*b1) * q^11 + (b3 + 2*b2 + b1) * q^13 + q^14 + q^16 + 2*b2 * q^17 + (-b3 + b1 - 4) * q^19 + (-b2 + b1 + 1) * q^20 + (-2*b3 - 2*b1) * q^22 + (-2*b3 + 2*b2 - 2*b1) * q^23 + (-2*b3 + b2 + 2*b1) * q^25 + (-b3 + b1 + 2) * q^26 - b2 * q^28 + (-2*b3 + 2*b1 + 2) * q^29 + (-2*b3 + 2*b1 + 4) * q^31 - b2 * q^32 + 2 * q^34 + (-b3 - b2 - 1) * q^35 - 2*b2 * q^37 + (-b3 + 4*b2 - b1) * q^38 + (-b3 - b2 - 1) * q^40 + (2*b3 - 2*b1 + 6) * q^41 + (-2*b3 + 4*b2 - 2*b1) * q^43 + (2*b3 - 2*b1) * q^44 + (2*b3 - 2*b1 + 2) * q^46 + (2*b3 + 4*b2 + 2*b1) * q^47 - q^49 + (-2*b3 - 2*b1 + 1) * q^50 + (-b3 - 2*b2 - b1) * q^52 + (2*b3 + 6*b2 + 2*b1) * q^53 + (4*b3 - 6*b2 - 6) * q^55 - q^56 + (-2*b3 - 2*b2 - 2*b1) * q^58 + (-b3 + b1 - 4) * q^59 + (b3 - b1 + 6) * q^61 + (-2*b3 - 4*b2 - 2*b1) * q^62 - q^64 + (-2*b3 - 5*b2 - 2*b1 + 1) * q^65 + 8*b2 * q^67 - 2*b2 * q^68 + (b2 - b1 - 1) * q^70 + (2*b3 - 2*b1 + 6) * q^71 + (2*b3 - 2*b2 + 2*b1) * q^73 - 2 * q^74 + (b3 - b1 + 4) * q^76 + (2*b3 + 2*b1) * q^77 + (2*b3 - 2*b1 - 2) * q^79 + (b2 - b1 - 1) * q^80 + (2*b3 - 6*b2 + 2*b1) * q^82 + (b3 + b1) * q^83 + (-2*b3 - 2*b2 - 2) * q^85 + (2*b3 - 2*b1 + 4) * q^86 + (2*b3 + 2*b1) * q^88 - 10 * q^89 + (b3 - b1 - 2) * q^91 + (2*b3 - 2*b2 + 2*b1) * q^92 + (-2*b3 + 2*b1 + 4) * q^94 + (2*b3 - 7*b2 + 4*b1 + 1) * q^95 + (-4*b3 - 6*b2 - 4*b1) * q^97 + b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{5}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^5 $$4 q - 4 q^{4} - 4 q^{5} + 4 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{19} + 4 q^{20} + 8 q^{26} + 8 q^{29} + 16 q^{31} + 8 q^{34} - 4 q^{35} - 4 q^{40} + 24 q^{41} + 8 q^{46} - 4 q^{49} + 4 q^{50} - 24 q^{55} - 4 q^{56} - 16 q^{59} + 24 q^{61} - 4 q^{64} + 4 q^{65} - 4 q^{70} + 24 q^{71} - 8 q^{74} + 16 q^{76} - 8 q^{79} - 4 q^{80} - 8 q^{85} + 16 q^{86} - 40 q^{89} - 8 q^{91} + 16 q^{94} + 4 q^{95}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^5 + 4 * q^10 + 4 * q^14 + 4 * q^16 - 16 * q^19 + 4 * q^20 + 8 * q^26 + 8 * q^29 + 16 * q^31 + 8 * q^34 - 4 * q^35 - 4 * q^40 + 24 * q^41 + 8 * q^46 - 4 * q^49 + 4 * q^50 - 24 * q^55 - 4 * q^56 - 16 * q^59 + 24 * q^61 - 4 * q^64 + 4 * q^65 - 4 * q^70 + 24 * q^71 - 8 * q^74 + 16 * q^76 - 8 * q^79 - 4 * q^80 - 8 * q^85 + 16 * q^86 - 40 * q^89 - 8 * q^91 + 16 * q^94 + 4 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$127$$ $$281$$ $$451$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
1.00000i 0 −1.00000 −2.22474 0.224745i 0 1.00000i 1.00000i 0 −0.224745 + 2.22474i
379.2 1.00000i 0 −1.00000 0.224745 + 2.22474i 0 1.00000i 1.00000i 0 2.22474 0.224745i
379.3 1.00000i 0 −1.00000 −2.22474 + 0.224745i 0 1.00000i 1.00000i 0 −0.224745 2.22474i
379.4 1.00000i 0 −1.00000 0.224745 2.22474i 0 1.00000i 1.00000i 0 2.22474 + 0.224745i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.g.g 4
3.b odd 2 1 70.2.c.a 4
4.b odd 2 1 5040.2.t.t 4
5.b even 2 1 inner 630.2.g.g 4
5.c odd 4 1 3150.2.a.bs 2
5.c odd 4 1 3150.2.a.bt 2
12.b even 2 1 560.2.g.e 4
15.d odd 2 1 70.2.c.a 4
15.e even 4 1 350.2.a.g 2
15.e even 4 1 350.2.a.h 2
20.d odd 2 1 5040.2.t.t 4
21.c even 2 1 490.2.c.e 4
21.g even 6 2 490.2.i.f 8
21.h odd 6 2 490.2.i.c 8
24.f even 2 1 2240.2.g.i 4
24.h odd 2 1 2240.2.g.j 4
60.h even 2 1 560.2.g.e 4
60.l odd 4 1 2800.2.a.bl 2
60.l odd 4 1 2800.2.a.bm 2
105.g even 2 1 490.2.c.e 4
105.k odd 4 1 2450.2.a.bl 2
105.k odd 4 1 2450.2.a.bq 2
105.o odd 6 2 490.2.i.c 8
105.p even 6 2 490.2.i.f 8
120.i odd 2 1 2240.2.g.j 4
120.m even 2 1 2240.2.g.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 3.b odd 2 1
70.2.c.a 4 15.d odd 2 1
350.2.a.g 2 15.e even 4 1
350.2.a.h 2 15.e even 4 1
490.2.c.e 4 21.c even 2 1
490.2.c.e 4 105.g even 2 1
490.2.i.c 8 21.h odd 6 2
490.2.i.c 8 105.o odd 6 2
490.2.i.f 8 21.g even 6 2
490.2.i.f 8 105.p even 6 2
560.2.g.e 4 12.b even 2 1
560.2.g.e 4 60.h even 2 1
630.2.g.g 4 1.a even 1 1 trivial
630.2.g.g 4 5.b even 2 1 inner
2240.2.g.i 4 24.f even 2 1
2240.2.g.i 4 120.m even 2 1
2240.2.g.j 4 24.h odd 2 1
2240.2.g.j 4 120.i odd 2 1
2450.2.a.bl 2 105.k odd 4 1
2450.2.a.bq 2 105.k odd 4 1
2800.2.a.bl 2 60.l odd 4 1
2800.2.a.bm 2 60.l odd 4 1
3150.2.a.bs 2 5.c odd 4 1
3150.2.a.bt 2 5.c odd 4 1
5040.2.t.t 4 4.b odd 2 1
5040.2.t.t 4 20.d odd 2 1

## Hecke kernels

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

 $$T_{11}^{2} - 24$$ T11^2 - 24 $$T_{29}^{2} - 4T_{29} - 20$$ T29^2 - 4*T29 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$T^{4} + 20T^{2} + 4$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$(T^{2} + 8 T + 10)^{2}$$
$23$ $$T^{4} + 56T^{2} + 400$$
$29$ $$(T^{2} - 4 T - 20)^{2}$$
$31$ $$(T^{2} - 8 T - 8)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} - 12 T + 12)^{2}$$
$43$ $$T^{4} + 80T^{2} + 64$$
$47$ $$T^{4} + 80T^{2} + 64$$
$53$ $$T^{4} + 120T^{2} + 144$$
$59$ $$(T^{2} + 8 T + 10)^{2}$$
$61$ $$(T^{2} - 12 T + 30)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 12 T + 12)^{2}$$
$73$ $$T^{4} + 56T^{2} + 400$$
$79$ $$(T^{2} + 4 T - 20)^{2}$$
$83$ $$(T^{2} + 6)^{2}$$
$89$ $$(T + 10)^{4}$$
$97$ $$T^{4} + 264T^{2} + 3600$$