# Properties

 Label 4410.2.a.c 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 $$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 3 q^{11} - 5 q^{13} + q^{16} + 6 q^{17} + q^{19} - q^{20} + 3 q^{22} - 3 q^{23} + q^{25} + 5 q^{26} + 6 q^{29} + 4 q^{31} - q^{32} - 6 q^{34} + 11 q^{37} - q^{38} + q^{40} + 3 q^{41} - 10 q^{43} - 3 q^{44} + 3 q^{46} + 3 q^{47} - q^{50} - 5 q^{52} - 3 q^{53} + 3 q^{55} - 6 q^{58} + 4 q^{61} - 4 q^{62} + q^{64} + 5 q^{65} - 4 q^{67} + 6 q^{68} - 12 q^{71} + 4 q^{73} - 11 q^{74} + q^{76} - 10 q^{79} - q^{80} - 3 q^{82} - 12 q^{83} - 6 q^{85} + 10 q^{86} + 3 q^{88} + 6 q^{89} - 3 q^{92} - 3 q^{94} - q^{95} - 14 q^{97}+O(q^{100})$$ q - q^2 + q^4 - q^5 - q^8 + q^10 - 3 * q^11 - 5 * q^13 + q^16 + 6 * q^17 + q^19 - q^20 + 3 * q^22 - 3 * q^23 + q^25 + 5 * q^26 + 6 * q^29 + 4 * q^31 - q^32 - 6 * q^34 + 11 * q^37 - q^38 + q^40 + 3 * q^41 - 10 * q^43 - 3 * q^44 + 3 * q^46 + 3 * q^47 - q^50 - 5 * q^52 - 3 * q^53 + 3 * q^55 - 6 * q^58 + 4 * q^61 - 4 * q^62 + q^64 + 5 * q^65 - 4 * q^67 + 6 * q^68 - 12 * q^71 + 4 * q^73 - 11 * q^74 + q^76 - 10 * q^79 - q^80 - 3 * q^82 - 12 * q^83 - 6 * q^85 + 10 * q^86 + 3 * q^88 + 6 * q^89 - 3 * q^92 - 3 * q^94 - q^95 - 14 * 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.c 1
3.b odd 2 1 490.2.a.j 1
7.b odd 2 1 4410.2.a.m 1
7.d odd 6 2 630.2.k.e 2
12.b even 2 1 3920.2.a.g 1
15.d odd 2 1 2450.2.a.f 1
15.e even 4 2 2450.2.c.p 2
21.c even 2 1 490.2.a.g 1
21.g even 6 2 70.2.e.b 2
21.h odd 6 2 490.2.e.a 2
84.h odd 2 1 3920.2.a.be 1
84.j odd 6 2 560.2.q.d 2
105.g even 2 1 2450.2.a.p 1
105.k odd 4 2 2450.2.c.f 2
105.p even 6 2 350.2.e.h 2
105.w odd 12 4 350.2.j.a 4

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

## Hecke kernels

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

 $$T_{11} + 3$$ T11 + 3 $$T_{13} + 5$$ T13 + 5 $$T_{17} - 6$$ T17 - 6 $$T_{19} - 1$$ T19 - 1 $$T_{29} - 6$$ T29 - 6 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

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