# Properties

 Label 3600.1.e.a Level $3600$ Weight $1$ Character orbit 3600.e Self dual yes Analytic conductor $1.797$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -20, 5 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(3151,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3600.3151");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.e (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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.18000.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + 2 q^{29} + 2 q^{41} + q^{49} - 2 q^{61} + 2 q^{89}+O(q^{100})$$ q + 2 * q^29 + 2 * q^41 + q^49 - 2 * q^61 + 2 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3151.1
 0
0 0 0 0 0 0 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.e.a 1
3.b odd 2 1 400.1.b.a 1
4.b odd 2 1 CM 3600.1.e.a 1
5.b even 2 1 RM 3600.1.e.a 1
5.c odd 4 2 720.1.j.a 1
12.b even 2 1 400.1.b.a 1
15.d odd 2 1 400.1.b.a 1
15.e even 4 2 80.1.h.a 1
20.d odd 2 1 CM 3600.1.e.a 1
20.e even 4 2 720.1.j.a 1
24.f even 2 1 1600.1.b.a 1
24.h odd 2 1 1600.1.b.a 1
40.i odd 4 2 2880.1.j.a 1
40.k even 4 2 2880.1.j.a 1
60.h even 2 1 400.1.b.a 1
60.l odd 4 2 80.1.h.a 1
105.k odd 4 2 3920.1.j.a 1
105.w odd 12 4 3920.1.bt.a 2
105.x even 12 4 3920.1.bt.b 2
120.i odd 2 1 1600.1.b.a 1
120.m even 2 1 1600.1.b.a 1
120.q odd 4 2 320.1.h.a 1
120.w even 4 2 320.1.h.a 1
240.z odd 4 2 1280.1.e.a 2
240.bb even 4 2 1280.1.e.a 2
240.bd odd 4 2 1280.1.e.a 2
240.bf even 4 2 1280.1.e.a 2
420.w even 4 2 3920.1.j.a 1
420.bp odd 12 4 3920.1.bt.b 2
420.br even 12 4 3920.1.bt.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 15.e even 4 2
80.1.h.a 1 60.l odd 4 2
320.1.h.a 1 120.q odd 4 2
320.1.h.a 1 120.w even 4 2
400.1.b.a 1 3.b odd 2 1
400.1.b.a 1 12.b even 2 1
400.1.b.a 1 15.d odd 2 1
400.1.b.a 1 60.h even 2 1
720.1.j.a 1 5.c odd 4 2
720.1.j.a 1 20.e even 4 2
1280.1.e.a 2 240.z odd 4 2
1280.1.e.a 2 240.bb even 4 2
1280.1.e.a 2 240.bd odd 4 2
1280.1.e.a 2 240.bf even 4 2
1600.1.b.a 1 24.f even 2 1
1600.1.b.a 1 24.h odd 2 1
1600.1.b.a 1 120.i odd 2 1
1600.1.b.a 1 120.m even 2 1
2880.1.j.a 1 40.i odd 4 2
2880.1.j.a 1 40.k even 4 2
3600.1.e.a 1 1.a even 1 1 trivial
3600.1.e.a 1 4.b odd 2 1 CM
3600.1.e.a 1 5.b even 2 1 RM
3600.1.e.a 1 20.d odd 2 1 CM
3920.1.j.a 1 105.k odd 4 2
3920.1.j.a 1 420.w even 4 2
3920.1.bt.a 2 105.w odd 12 4
3920.1.bt.a 2 420.br even 12 4
3920.1.bt.b 2 105.x even 12 4
3920.1.bt.b 2 420.bp odd 12 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}$$ T7 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 2$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 2$$
$97$ $$T$$
This newform has the most leading zero Fourier coefficients among newforms with $Nk^2$ $\le 4000$.