# Properties

 Label 4400.2.a.x Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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 440) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - 2 * q^7 + q^9 $$q + 2 q^{3} - 2 q^{7} + q^{9} + q^{11} - 4 q^{17} - 4 q^{19} - 4 q^{21} + 6 q^{23} - 4 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} - 4 q^{37} - 6 q^{41} + 6 q^{43} + 2 q^{47} - 3 q^{49} - 8 q^{51} - 12 q^{53} - 8 q^{57} - 4 q^{59} + 14 q^{61} - 2 q^{63} - 10 q^{67} + 12 q^{69} - 8 q^{71} + 4 q^{73} - 2 q^{77} + 8 q^{79} - 11 q^{81} - 2 q^{83} + 4 q^{87} - 10 q^{89} - 16 q^{93} - 8 q^{97} + q^{99}+O(q^{100})$$ q + 2 * q^3 - 2 * q^7 + q^9 + q^11 - 4 * q^17 - 4 * q^19 - 4 * q^21 + 6 * q^23 - 4 * q^27 + 2 * q^29 - 8 * q^31 + 2 * q^33 - 4 * q^37 - 6 * q^41 + 6 * q^43 + 2 * q^47 - 3 * q^49 - 8 * q^51 - 12 * q^53 - 8 * q^57 - 4 * q^59 + 14 * q^61 - 2 * q^63 - 10 * q^67 + 12 * q^69 - 8 * q^71 + 4 * q^73 - 2 * q^77 + 8 * q^79 - 11 * q^81 - 2 * q^83 + 4 * q^87 - 10 * q^89 - 16 * q^93 - 8 * q^97 + 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 2.00000 0 0 0 −2.00000 0 1.00000 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.x 1
4.b odd 2 1 2200.2.a.c 1
5.b even 2 1 4400.2.a.h 1
5.c odd 4 2 880.2.b.b 2
20.d odd 2 1 2200.2.a.i 1
20.e even 4 2 440.2.b.a 2
60.l odd 4 2 3960.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.a 2 20.e even 4 2
880.2.b.b 2 5.c odd 4 2
2200.2.a.c 1 4.b odd 2 1
2200.2.a.i 1 20.d odd 2 1
3960.2.d.c 2 60.l odd 4 2
4400.2.a.h 1 5.b even 2 1
4400.2.a.x 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{13}$$ T13

## Hecke characteristic polynomials

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