# Properties

 Label 3600.1.c.a Level $3600$ Weight $1$ Character orbit 3600.c Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(449,3600)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("3600.449");

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.c (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: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1800) Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.10800.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{2} q^{7} +O(q^{10})$$ q - z^2 * q^7 $$q - \zeta_{8}^{2} q^{7} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{13} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{17} + q^{19} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} + (\zeta_{8}^{3} + \zeta_{8}) q^{29} - q^{31} - \zeta_{8}^{2} q^{43} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{47} + (\zeta_{8}^{3} + \zeta_{8}) q^{59} + q^{61} - \zeta_{8}^{2} q^{67} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{77} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{83} + q^{91} - \zeta_{8}^{2} q^{97} +O(q^{100})$$ q - z^2 * q^7 + (z^3 + z) * q^11 + z^2 * q^13 + (-z^3 + z) * q^17 + q^19 + (z^3 - z) * q^23 + (z^3 + z) * q^29 - q^31 - z^2 * q^43 + (-z^3 + z) * q^47 + (z^3 + z) * q^59 + q^61 - z^2 * q^67 + (-z^3 + z) * q^77 + (-z^3 + z) * q^83 + q^91 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{19} - 4 q^{31} + 4 q^{61} + 4 q^{91}+O(q^{100})$$ 4 * q + 4 * q^19 - 4 * q^31 + 4 * q^61 + 4 * q^91

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.c.a 4
3.b odd 2 1 inner 3600.1.c.a 4
4.b odd 2 1 1800.1.c.a 4
5.b even 2 1 inner 3600.1.c.a 4
5.c odd 4 1 3600.1.l.a 2
5.c odd 4 1 3600.1.l.b 2
12.b even 2 1 1800.1.c.a 4
15.d odd 2 1 inner 3600.1.c.a 4
15.e even 4 1 3600.1.l.a 2
15.e even 4 1 3600.1.l.b 2
20.d odd 2 1 1800.1.c.a 4
20.e even 4 1 1800.1.l.a 2
20.e even 4 1 1800.1.l.b yes 2
60.h even 2 1 1800.1.c.a 4
60.l odd 4 1 1800.1.l.a 2
60.l odd 4 1 1800.1.l.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.c.a 4 4.b odd 2 1
1800.1.c.a 4 12.b even 2 1
1800.1.c.a 4 20.d odd 2 1
1800.1.c.a 4 60.h even 2 1
1800.1.l.a 2 20.e even 4 1
1800.1.l.a 2 60.l odd 4 1
1800.1.l.b yes 2 20.e even 4 1
1800.1.l.b yes 2 60.l odd 4 1
3600.1.c.a 4 1.a even 1 1 trivial
3600.1.c.a 4 3.b odd 2 1 inner
3600.1.c.a 4 5.b even 2 1 inner
3600.1.c.a 4 15.d odd 2 1 inner
3600.1.l.a 2 5.c odd 4 1
3600.1.l.a 2 15.e even 4 1
3600.1.l.b 2 5.c odd 4 1
3600.1.l.b 2 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

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