# Properties

 Label 630.2.a.d Level $630$ Weight $2$ Character orbit 630.a Self dual yes Analytic conductor $5.031$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(1,630)]

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, 0, 0]))

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

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

chi := DirichletCharacter("630.1");

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.a (trivial)

## 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: yes Analytic conductor: $$5.03057532734$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + q^5 - q^7 - q^8 $$q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} - 4 q^{11} - 6 q^{13} + q^{14} + q^{16} - 2 q^{17} + q^{20} + 4 q^{22} + q^{25} + 6 q^{26} - q^{28} - 6 q^{29} + 8 q^{31} - q^{32} + 2 q^{34} - q^{35} - 10 q^{37} - q^{40} - 2 q^{41} + 4 q^{43} - 4 q^{44} - 8 q^{47} + q^{49} - q^{50} - 6 q^{52} + 2 q^{53} - 4 q^{55} + q^{56} + 6 q^{58} + 8 q^{59} - 14 q^{61} - 8 q^{62} + q^{64} - 6 q^{65} - 12 q^{67} - 2 q^{68} + q^{70} + 16 q^{71} + 2 q^{73} + 10 q^{74} + 4 q^{77} - 8 q^{79} + q^{80} + 2 q^{82} - 8 q^{83} - 2 q^{85} - 4 q^{86} + 4 q^{88} - 10 q^{89} + 6 q^{91} + 8 q^{94} + 2 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 + q^5 - q^7 - q^8 - q^10 - 4 * q^11 - 6 * q^13 + q^14 + q^16 - 2 * q^17 + q^20 + 4 * q^22 + q^25 + 6 * q^26 - q^28 - 6 * q^29 + 8 * q^31 - q^32 + 2 * q^34 - q^35 - 10 * q^37 - q^40 - 2 * q^41 + 4 * q^43 - 4 * q^44 - 8 * q^47 + q^49 - q^50 - 6 * q^52 + 2 * q^53 - 4 * q^55 + q^56 + 6 * q^58 + 8 * q^59 - 14 * q^61 - 8 * q^62 + q^64 - 6 * q^65 - 12 * q^67 - 2 * q^68 + q^70 + 16 * q^71 + 2 * q^73 + 10 * q^74 + 4 * q^77 - 8 * q^79 + q^80 + 2 * q^82 - 8 * q^83 - 2 * q^85 - 4 * q^86 + 4 * q^88 - 10 * q^89 + 6 * q^91 + 8 * q^94 + 2 * q^97 - q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 1.00000 1.00000 0 −1.00000 −1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.a.d 1
3.b odd 2 1 70.2.a.a 1
4.b odd 2 1 5040.2.a.bm 1
5.b even 2 1 3150.2.a.bj 1
5.c odd 4 2 3150.2.g.c 2
7.b odd 2 1 4410.2.a.b 1
12.b even 2 1 560.2.a.d 1
15.d odd 2 1 350.2.a.b 1
15.e even 4 2 350.2.c.b 2
21.c even 2 1 490.2.a.h 1
21.g even 6 2 490.2.e.c 2
21.h odd 6 2 490.2.e.d 2
24.f even 2 1 2240.2.a.q 1
24.h odd 2 1 2240.2.a.n 1
33.d even 2 1 8470.2.a.j 1
60.h even 2 1 2800.2.a.m 1
60.l odd 4 2 2800.2.g.n 2
84.h odd 2 1 3920.2.a.t 1
105.g even 2 1 2450.2.a.l 1
105.k odd 4 2 2450.2.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 3.b odd 2 1
350.2.a.b 1 15.d odd 2 1
350.2.c.b 2 15.e even 4 2
490.2.a.h 1 21.c even 2 1
490.2.e.c 2 21.g even 6 2
490.2.e.d 2 21.h odd 6 2
560.2.a.d 1 12.b even 2 1
630.2.a.d 1 1.a even 1 1 trivial
2240.2.a.n 1 24.h odd 2 1
2240.2.a.q 1 24.f even 2 1
2450.2.a.l 1 105.g even 2 1
2450.2.c.k 2 105.k odd 4 2
2800.2.a.m 1 60.h even 2 1
2800.2.g.n 2 60.l odd 4 2
3150.2.a.bj 1 5.b even 2 1
3150.2.g.c 2 5.c odd 4 2
3920.2.a.t 1 84.h odd 2 1
4410.2.a.b 1 7.b odd 2 1
5040.2.a.bm 1 4.b odd 2 1
8470.2.a.j 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(630))$$:

 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 6$$ T13 + 6 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T + 6$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T + 10$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T + 8$$
$53$ $$T - 2$$
$59$ $$T - 8$$
$61$ $$T + 14$$
$67$ $$T + 12$$
$71$ $$T - 16$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 8$$
$89$ $$T + 10$$
$97$ $$T - 2$$