# Properties

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

# Related objects

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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + i * q^7 + 2 * q^9 $$q + i q^{3} + i q^{7} + 2 q^{9} + q^{11} - 2 i q^{13} - 3 i q^{17} - q^{19} - q^{21} + 6 i q^{23} + 5 i q^{27} + 9 q^{29} - 5 q^{31} + i q^{33} + 5 i q^{37} + 2 q^{39} - 6 q^{41} + 8 i q^{43} - 6 i q^{47} + 6 q^{49} + 3 q^{51} - 9 i q^{53} - i q^{57} + 6 q^{59} + 5 q^{61} + 2 i q^{63} - 8 i q^{67} - 6 q^{69} + 9 q^{71} + 10 i q^{73} + i q^{77} + 14 q^{79} + q^{81} - 6 i q^{83} + 9 i q^{87} + 15 q^{89} + 2 q^{91} - 5 i q^{93} + 8 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 + i * q^7 + 2 * q^9 + q^11 - 2*i * q^13 - 3*i * q^17 - q^19 - q^21 + 6*i * q^23 + 5*i * q^27 + 9 * q^29 - 5 * q^31 + i * q^33 + 5*i * q^37 + 2 * q^39 - 6 * q^41 + 8*i * q^43 - 6*i * q^47 + 6 * q^49 + 3 * q^51 - 9*i * q^53 - i * q^57 + 6 * q^59 + 5 * q^61 + 2*i * q^63 - 8*i * q^67 - 6 * q^69 + 9 * q^71 + 10*i * q^73 + i * q^77 + 14 * q^79 + q^81 - 6*i * q^83 + 9*i * q^87 + 15 * q^89 + 2 * q^91 - 5*i * q^93 + 8*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 2 q^{11} - 2 q^{19} - 2 q^{21} + 18 q^{29} - 10 q^{31} + 4 q^{39} - 12 q^{41} + 12 q^{49} + 6 q^{51} + 12 q^{59} + 10 q^{61} - 12 q^{69} + 18 q^{71} + 28 q^{79} + 2 q^{81} + 30 q^{89} + 4 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 2 * q^11 - 2 * q^19 - 2 * q^21 + 18 * q^29 - 10 * q^31 + 4 * q^39 - 12 * q^41 + 12 * q^49 + 6 * q^51 + 12 * q^59 + 10 * q^61 - 12 * q^69 + 18 * q^71 + 28 * q^79 + 2 * q^81 + 30 * q^89 + 4 * q^91 + 4 * 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 1.00000i 0 0 0 1.00000i 0 2.00000 0
4049.2 0 1.00000i 0 0 0 1.00000i 0 2.00000 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.j 2
4.b odd 2 1 550.2.b.c 2
5.b even 2 1 inner 4400.2.b.j 2
5.c odd 4 1 880.2.a.d 1
5.c odd 4 1 4400.2.a.t 1
12.b even 2 1 4950.2.c.s 2
15.e even 4 1 7920.2.a.bc 1
20.d odd 2 1 550.2.b.c 2
20.e even 4 1 110.2.a.c 1
20.e even 4 1 550.2.a.d 1
40.i odd 4 1 3520.2.a.ba 1
40.k even 4 1 3520.2.a.k 1
55.e even 4 1 9680.2.a.g 1
60.h even 2 1 4950.2.c.s 2
60.l odd 4 1 990.2.a.f 1
60.l odd 4 1 4950.2.a.bm 1
140.j odd 4 1 5390.2.a.x 1
220.i odd 4 1 1210.2.a.e 1
220.i odd 4 1 6050.2.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.c 1 20.e even 4 1
550.2.a.d 1 20.e even 4 1
550.2.b.c 2 4.b odd 2 1
550.2.b.c 2 20.d odd 2 1
880.2.a.d 1 5.c odd 4 1
990.2.a.f 1 60.l odd 4 1
1210.2.a.e 1 220.i odd 4 1
3520.2.a.k 1 40.k even 4 1
3520.2.a.ba 1 40.i odd 4 1
4400.2.a.t 1 5.c odd 4 1
4400.2.b.j 2 1.a even 1 1 trivial
4400.2.b.j 2 5.b even 2 1 inner
4950.2.a.bm 1 60.l odd 4 1
4950.2.c.s 2 12.b even 2 1
4950.2.c.s 2 60.h even 2 1
5390.2.a.x 1 140.j odd 4 1
6050.2.a.bc 1 220.i odd 4 1
7920.2.a.bc 1 15.e even 4 1
9680.2.a.g 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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{2} + 9$$ T17^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 9)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 25$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 9)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 14)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 64$$