# Properties

 Label 4400.2.b.a Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - i q^{7} - 6 q^{9} +O(q^{10})$$ q + 3*i * q^3 - i * q^7 - 6 * q^9 $$q + 3 i q^{3} - i q^{7} - 6 q^{9} + q^{11} + 6 i q^{13} + 3 i q^{17} - 5 q^{19} + 3 q^{21} - 2 i q^{23} - 9 i q^{27} + 5 q^{29} - 5 q^{31} + 3 i q^{33} - i q^{37} - 18 q^{39} - 2 q^{41} + 12 i q^{43} + 2 i q^{47} + 6 q^{49} - 9 q^{51} + 13 i q^{53} - 15 i q^{57} + 2 q^{59} + q^{61} + 6 i q^{63} - 16 i q^{67} + 6 q^{69} - 15 q^{71} - 10 i q^{73} - i q^{77} + 2 q^{79} + 9 q^{81} - 14 i q^{83} + 15 i q^{87} - 9 q^{89} + 6 q^{91} - 15 i q^{93} - 16 i q^{97} - 6 q^{99} +O(q^{100})$$ q + 3*i * q^3 - i * q^7 - 6 * q^9 + q^11 + 6*i * q^13 + 3*i * q^17 - 5 * q^19 + 3 * q^21 - 2*i * q^23 - 9*i * q^27 + 5 * q^29 - 5 * q^31 + 3*i * q^33 - i * q^37 - 18 * q^39 - 2 * q^41 + 12*i * q^43 + 2*i * q^47 + 6 * q^49 - 9 * q^51 + 13*i * q^53 - 15*i * q^57 + 2 * q^59 + q^61 + 6*i * q^63 - 16*i * q^67 + 6 * q^69 - 15 * q^71 - 10*i * q^73 - i * q^77 + 2 * q^79 + 9 * q^81 - 14*i * q^83 + 15*i * q^87 - 9 * q^89 + 6 * q^91 - 15*i * q^93 - 16*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9}+O(q^{10})$$ 2 * q - 12 * q^9 $$2 q - 12 q^{9} + 2 q^{11} - 10 q^{19} + 6 q^{21} + 10 q^{29} - 10 q^{31} - 36 q^{39} - 4 q^{41} + 12 q^{49} - 18 q^{51} + 4 q^{59} + 2 q^{61} + 12 q^{69} - 30 q^{71} + 4 q^{79} + 18 q^{81} - 18 q^{89} + 12 q^{91} - 12 q^{99}+O(q^{100})$$ 2 * q - 12 * q^9 + 2 * q^11 - 10 * q^19 + 6 * q^21 + 10 * q^29 - 10 * q^31 - 36 * q^39 - 4 * q^41 + 12 * q^49 - 18 * q^51 + 4 * q^59 + 2 * q^61 + 12 * q^69 - 30 * q^71 + 4 * q^79 + 18 * q^81 - 18 * q^89 + 12 * q^91 - 12 * 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 3.00000i 0 0 0 1.00000i 0 −6.00000 0
4049.2 0 3.00000i 0 0 0 1.00000i 0 −6.00000 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.a 2
4.b odd 2 1 2200.2.b.b 2
5.b even 2 1 inner 4400.2.b.a 2
5.c odd 4 1 880.2.a.a 1
5.c odd 4 1 4400.2.a.be 1
15.e even 4 1 7920.2.a.e 1
20.d odd 2 1 2200.2.b.b 2
20.e even 4 1 440.2.a.d 1
20.e even 4 1 2200.2.a.a 1
40.i odd 4 1 3520.2.a.bh 1
40.k even 4 1 3520.2.a.a 1
55.e even 4 1 9680.2.a.a 1
60.l odd 4 1 3960.2.a.f 1
220.i odd 4 1 4840.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.d 1 20.e even 4 1
880.2.a.a 1 5.c odd 4 1
2200.2.a.a 1 20.e even 4 1
2200.2.b.b 2 4.b odd 2 1
2200.2.b.b 2 20.d odd 2 1
3520.2.a.a 1 40.k even 4 1
3520.2.a.bh 1 40.i odd 4 1
3960.2.a.f 1 60.l odd 4 1
4400.2.a.be 1 5.c odd 4 1
4400.2.b.a 2 1.a even 1 1 trivial
4400.2.b.a 2 5.b even 2 1 inner
4840.2.a.i 1 220.i odd 4 1
7920.2.a.e 1 15.e even 4 1
9680.2.a.a 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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{2} + 9$$ T3^2 + 9 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 9$$ T17^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 169$$
$59$ $$(T - 2)^{2}$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$(T + 15)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$(T + 9)^{2}$$
$97$ $$T^{2} + 256$$