# Properties

 Label 4400.2.b.m Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 440) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{7} + 3 q^{9} +O(q^{10})$$ q - b * q^7 + 3 * q^9 $$q - \beta q^{7} + 3 q^{9} + q^{11} - 8 q^{19} + 4 \beta q^{23} - 10 q^{29} - 8 q^{31} + 5 \beta q^{37} - 2 q^{41} + 3 \beta q^{43} - 4 \beta q^{47} + 3 q^{49} + 7 \beta q^{53} - 4 q^{59} + 10 q^{61} - 3 \beta q^{63} + 2 \beta q^{67} - 4 \beta q^{73} - \beta q^{77} - 4 q^{79} + 9 q^{81} - 5 \beta q^{83} - 6 q^{89} + 5 \beta q^{97} + 3 q^{99} +O(q^{100})$$ q - b * q^7 + 3 * q^9 + q^11 - 8 * q^19 + 4*b * q^23 - 10 * q^29 - 8 * q^31 + 5*b * q^37 - 2 * q^41 + 3*b * q^43 - 4*b * q^47 + 3 * q^49 + 7*b * q^53 - 4 * q^59 + 10 * q^61 - 3*b * q^63 + 2*b * q^67 - 4*b * q^73 - b * q^77 - 4 * q^79 + 9 * q^81 - 5*b * q^83 - 6 * q^89 + 5*b * q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{9}+O(q^{10})$$ 2 * q + 6 * q^9 $$2 q + 6 q^{9} + 2 q^{11} - 16 q^{19} - 20 q^{29} - 16 q^{31} - 4 q^{41} + 6 q^{49} - 8 q^{59} + 20 q^{61} - 8 q^{79} + 18 q^{81} - 12 q^{89} + 6 q^{99}+O(q^{100})$$ 2 * q + 6 * q^9 + 2 * q^11 - 16 * q^19 - 20 * q^29 - 16 * q^31 - 4 * q^41 + 6 * q^49 - 8 * q^59 + 20 * q^61 - 8 * q^79 + 18 * q^81 - 12 * q^89 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 1.00000i − 1.00000i
0 0 0 0 0 2.00000i 0 3.00000 0
4049.2 0 0 0 0 0 2.00000i 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4400.2.b.m 2
4.b odd 2 1 2200.2.b.d 2
5.b even 2 1 inner 4400.2.b.m 2
5.c odd 4 1 880.2.a.f 1
5.c odd 4 1 4400.2.a.o 1
15.e even 4 1 7920.2.a.bg 1
20.d odd 2 1 2200.2.b.d 2
20.e even 4 1 440.2.a.a 1
20.e even 4 1 2200.2.a.f 1
40.i odd 4 1 3520.2.a.u 1
40.k even 4 1 3520.2.a.t 1
55.e even 4 1 9680.2.a.n 1
60.l odd 4 1 3960.2.a.p 1
220.i odd 4 1 4840.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.a 1 20.e even 4 1
880.2.a.f 1 5.c odd 4 1
2200.2.a.f 1 20.e even 4 1
2200.2.b.d 2 4.b odd 2 1
2200.2.b.d 2 20.d odd 2 1
3520.2.a.t 1 40.k even 4 1
3520.2.a.u 1 40.i odd 4 1
3960.2.a.p 1 60.l odd 4 1
4400.2.a.o 1 5.c odd 4 1
4400.2.b.m 2 1.a even 1 1 trivial
4400.2.b.m 2 5.b even 2 1 inner
4840.2.a.c 1 220.i odd 4 1
7920.2.a.bg 1 15.e even 4 1
9680.2.a.n 1 55.e even 4 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{13}$$ T13 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 64$$
$29$ $$(T + 10)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 196$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 100$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 100$$
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