# Properties

 Label 4410.2.a.b Level $4410$ Weight $2$ Character orbit 4410.a Self dual yes Analytic conductor $35.214$ 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] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.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: $$35.2140272914$$ 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)

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

## 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 0 −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 4410.2.a.b 1
3.b odd 2 1 490.2.a.h 1
7.b odd 2 1 630.2.a.d 1
12.b even 2 1 3920.2.a.t 1
15.d odd 2 1 2450.2.a.l 1
15.e even 4 2 2450.2.c.k 2
21.c even 2 1 70.2.a.a 1
21.g even 6 2 490.2.e.d 2
21.h odd 6 2 490.2.e.c 2
28.d even 2 1 5040.2.a.bm 1
35.c odd 2 1 3150.2.a.bj 1
35.f even 4 2 3150.2.g.c 2
84.h odd 2 1 560.2.a.d 1
105.g even 2 1 350.2.a.b 1
105.k odd 4 2 350.2.c.b 2
168.e odd 2 1 2240.2.a.q 1
168.i even 2 1 2240.2.a.n 1
231.h odd 2 1 8470.2.a.j 1
420.o odd 2 1 2800.2.a.m 1
420.w even 4 2 2800.2.g.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 21.c even 2 1
350.2.a.b 1 105.g even 2 1
350.2.c.b 2 105.k odd 4 2
490.2.a.h 1 3.b odd 2 1
490.2.e.c 2 21.h odd 6 2
490.2.e.d 2 21.g even 6 2
560.2.a.d 1 84.h odd 2 1
630.2.a.d 1 7.b odd 2 1
2240.2.a.n 1 168.i even 2 1
2240.2.a.q 1 168.e odd 2 1
2450.2.a.l 1 15.d odd 2 1
2450.2.c.k 2 15.e even 4 2
2800.2.a.m 1 420.o odd 2 1
2800.2.g.n 2 420.w even 4 2
3150.2.a.bj 1 35.c odd 2 1
3150.2.g.c 2 35.f even 4 2
3920.2.a.t 1 12.b even 2 1
4410.2.a.b 1 1.a even 1 1 trivial
5040.2.a.bm 1 28.d even 2 1
8470.2.a.j 1 231.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4410))$$:

 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 2$$ T17 - 2 $$T_{19}$$ T19 $$T_{29} + 6$$ T29 + 6 $$T_{31} + 8$$ T31 + 8

## Hecke characteristic polynomials

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