# Properties

 Label 4400.2.a.k Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ 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] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.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: $$35.1341768894$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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)

 $$f(q)$$ $$=$$ $$q - q^{3} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + 3 * q^7 - 2 * q^9 $$q - q^{3} + 3 q^{7} - 2 q^{9} - q^{11} - 4 q^{13} - 3 q^{17} + 5 q^{19} - 3 q^{21} + 4 q^{23} + 5 q^{27} + 5 q^{29} - 7 q^{31} + q^{33} + 7 q^{37} + 4 q^{39} - 8 q^{41} - 6 q^{43} + 8 q^{47} + 2 q^{49} + 3 q^{51} - 9 q^{53} - 5 q^{57} - 13 q^{61} - 6 q^{63} - 12 q^{67} - 4 q^{69} + 3 q^{71} + 6 q^{73} - 3 q^{77} + q^{81} + 4 q^{83} - 5 q^{87} - 15 q^{89} - 12 q^{91} + 7 q^{93} + 12 q^{97} + 2 q^{99}+O(q^{100})$$ q - q^3 + 3 * q^7 - 2 * q^9 - q^11 - 4 * q^13 - 3 * q^17 + 5 * q^19 - 3 * q^21 + 4 * q^23 + 5 * q^27 + 5 * q^29 - 7 * q^31 + q^33 + 7 * q^37 + 4 * q^39 - 8 * q^41 - 6 * q^43 + 8 * q^47 + 2 * q^49 + 3 * q^51 - 9 * q^53 - 5 * q^57 - 13 * q^61 - 6 * q^63 - 12 * q^67 - 4 * q^69 + 3 * q^71 + 6 * q^73 - 3 * q^77 + q^81 + 4 * q^83 - 5 * q^87 - 15 * q^89 - 12 * q^91 + 7 * q^93 + 12 * q^97 + 2 * q^99

## 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
0 −1.00000 0 0 0 3.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$+1$$

## 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 4400.2.a.k 1
4.b odd 2 1 550.2.a.e 1
5.b even 2 1 4400.2.a.s 1
5.c odd 4 2 880.2.b.a 2
12.b even 2 1 4950.2.a.ba 1
20.d odd 2 1 550.2.a.j 1
20.e even 4 2 110.2.b.a 2
44.c even 2 1 6050.2.a.bk 1
60.h even 2 1 4950.2.a.q 1
60.l odd 4 2 990.2.c.d 2
220.g even 2 1 6050.2.a.f 1
220.i odd 4 2 1210.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.b.a 2 20.e even 4 2
550.2.a.e 1 4.b odd 2 1
550.2.a.j 1 20.d odd 2 1
880.2.b.a 2 5.c odd 4 2
990.2.c.d 2 60.l odd 4 2
1210.2.b.a 2 220.i odd 4 2
4400.2.a.k 1 1.a even 1 1 trivial
4400.2.a.s 1 5.b even 2 1
4950.2.a.q 1 60.h even 2 1
4950.2.a.ba 1 12.b even 2 1
6050.2.a.f 1 220.g even 2 1
6050.2.a.bk 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 3$$ T7 - 3 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T + 3$$
$19$ $$T - 5$$
$23$ $$T - 4$$
$29$ $$T - 5$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T + 8$$
$43$ $$T + 6$$
$47$ $$T - 8$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T + 13$$
$67$ $$T + 12$$
$71$ $$T - 3$$
$73$ $$T - 6$$
$79$ $$T$$
$83$ $$T - 4$$
$89$ $$T + 15$$
$97$ $$T - 12$$