# Properties

 Label 2200.1.d.c Level $2200$ Weight $1$ Character orbit 2200.d Self dual yes Analytic conductor $1.098$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -88 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,1,Mod(901,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2200.901");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2200.d (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.09794302779$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.2200.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.193600000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^4 + q^8 + q^9 $$q + q^{2} + q^{4} + q^{8} + q^{9} - q^{11} - q^{13} + q^{16} + q^{18} + q^{19} - q^{22} + q^{23} - q^{26} + q^{29} - q^{31} + q^{32} + q^{36} + q^{38} - q^{43} - q^{44} + q^{46} - 2 q^{47} + q^{49} - q^{52} + q^{58} - 2 q^{61} - q^{62} + q^{64} - q^{71} + q^{72} + q^{76} + q^{81} - q^{83} - q^{86} - q^{88} - q^{89} + q^{92} - 2 q^{94} + q^{97} + q^{98} - q^{99}+O(q^{100})$$ q + q^2 + q^4 + q^8 + q^9 - q^11 - q^13 + q^16 + q^18 + q^19 - q^22 + q^23 - q^26 + q^29 - q^31 + q^32 + q^36 + q^38 - q^43 - q^44 + q^46 - 2 * q^47 + q^49 - q^52 + q^58 - 2 * q^61 - q^62 + q^64 - q^71 + q^72 + q^76 + q^81 - q^83 - q^86 - q^88 - q^89 + q^92 - 2 * q^94 + q^97 + q^98 - q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$551$$ $$1101$$ $$1201$$ $$\chi(n)$$ $$0$$ $$0$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
901.1
 0
1.00000 0 1.00000 0 0 0 1.00000 1.00000 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
88.b odd 2 1 CM by $$\Q(\sqrt{-22})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2200.1.d.c yes 1
5.b even 2 1 2200.1.d.a 1
5.c odd 4 2 2200.1.o.a 2
8.b even 2 1 2200.1.d.b yes 1
11.b odd 2 1 2200.1.d.b yes 1
40.f even 2 1 2200.1.d.d yes 1
40.i odd 4 2 2200.1.o.b 2
55.d odd 2 1 2200.1.d.d yes 1
55.e even 4 2 2200.1.o.b 2
88.b odd 2 1 CM 2200.1.d.c yes 1
440.o odd 2 1 2200.1.d.a 1
440.t even 4 2 2200.1.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.1.d.a 1 5.b even 2 1
2200.1.d.a 1 440.o odd 2 1
2200.1.d.b yes 1 8.b even 2 1
2200.1.d.b yes 1 11.b odd 2 1
2200.1.d.c yes 1 1.a even 1 1 trivial
2200.1.d.c yes 1 88.b odd 2 1 CM
2200.1.d.d yes 1 40.f even 2 1
2200.1.d.d yes 1 55.d odd 2 1
2200.1.o.a 2 5.c odd 4 2
2200.1.o.a 2 440.t even 4 2
2200.1.o.b 2 40.i odd 4 2
2200.1.o.b 2 55.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2200, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}$$ T7 $$T_{13} + 1$$ T13 + 1 $$T_{19} - 1$$ T19 - 1

## Hecke characteristic polynomials

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