# Properties

 Label 3600.2.f.j Level $3600$ Weight $2$ Character orbit 3600.f Analytic conductor $28.746$ 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] = [3600,2,Mod(2449,3600)]

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, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3600.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.f (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: $$28.7461447277$$ 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: $$2$$ Twist minimal: no (minimal twist has level 20) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{7}+O(q^{10})$$ q + b * q^7 $$q + \beta q^{7} + \beta q^{13} - 3 \beta q^{17} - 4 q^{19} + 3 \beta q^{23} + 6 q^{29} + 4 q^{31} - \beta q^{37} - 6 q^{41} + 5 \beta q^{43} + 3 \beta q^{47} + 3 q^{49} + 3 \beta q^{53} - 12 q^{59} + 2 q^{61} + \beta q^{67} - 12 q^{71} + \beta q^{73} + 8 q^{79} + 3 \beta q^{83} - 6 q^{89} - 4 q^{91} - \beta q^{97} +O(q^{100})$$ q + b * q^7 + b * q^13 - 3*b * q^17 - 4 * q^19 + 3*b * q^23 + 6 * q^29 + 4 * q^31 - b * q^37 - 6 * q^41 + 5*b * q^43 + 3*b * q^47 + 3 * q^49 + 3*b * q^53 - 12 * q^59 + 2 * q^61 + b * q^67 - 12 * q^71 + b * q^73 + 8 * q^79 + 3*b * q^83 - 6 * q^89 - 4 * q^91 - b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{19} + 12 q^{29} + 8 q^{31} - 12 q^{41} + 6 q^{49} - 24 q^{59} + 4 q^{61} - 24 q^{71} + 16 q^{79} - 12 q^{89} - 8 q^{91}+O(q^{100})$$ 2 * q - 8 * q^19 + 12 * q^29 + 8 * q^31 - 12 * q^41 + 6 * q^49 - 24 * q^59 + 4 * q^61 - 24 * q^71 + 16 * q^79 - 12 * q^89 - 8 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2449.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
2449.2 0 0 0 0 0 2.00000i 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
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 3600.2.f.j 2
3.b odd 2 1 400.2.c.b 2
4.b odd 2 1 900.2.d.c 2
5.b even 2 1 inner 3600.2.f.j 2
5.c odd 4 1 720.2.a.h 1
5.c odd 4 1 3600.2.a.be 1
12.b even 2 1 100.2.c.a 2
15.d odd 2 1 400.2.c.b 2
15.e even 4 1 80.2.a.b 1
15.e even 4 1 400.2.a.c 1
20.d odd 2 1 900.2.d.c 2
20.e even 4 1 180.2.a.a 1
20.e even 4 1 900.2.a.b 1
24.f even 2 1 1600.2.c.d 2
24.h odd 2 1 1600.2.c.e 2
40.i odd 4 1 2880.2.a.f 1
40.k even 4 1 2880.2.a.m 1
60.h even 2 1 100.2.c.a 2
60.l odd 4 1 20.2.a.a 1
60.l odd 4 1 100.2.a.a 1
84.h odd 2 1 4900.2.e.f 2
105.k odd 4 1 3920.2.a.h 1
120.i odd 2 1 1600.2.c.e 2
120.m even 2 1 1600.2.c.d 2
120.q odd 4 1 320.2.a.f 1
120.q odd 4 1 1600.2.a.c 1
120.w even 4 1 320.2.a.a 1
120.w even 4 1 1600.2.a.w 1
140.j odd 4 1 8820.2.a.g 1
165.l odd 4 1 9680.2.a.ba 1
180.v odd 12 2 1620.2.i.h 2
180.x even 12 2 1620.2.i.b 2
240.z odd 4 1 1280.2.d.c 2
240.bb even 4 1 1280.2.d.g 2
240.bd odd 4 1 1280.2.d.c 2
240.bf even 4 1 1280.2.d.g 2
420.o odd 2 1 4900.2.e.f 2
420.w even 4 1 980.2.a.h 1
420.w even 4 1 4900.2.a.e 1
420.bp odd 12 2 980.2.i.i 2
420.br even 12 2 980.2.i.c 2
660.q even 4 1 2420.2.a.a 1
780.u even 4 1 3380.2.f.b 2
780.w odd 4 1 3380.2.a.c 1
780.bn even 4 1 3380.2.f.b 2
1020.q odd 4 1 5780.2.c.a 2
1020.x odd 4 1 5780.2.a.f 1
1020.bl odd 4 1 5780.2.c.a 2
1140.w even 4 1 7220.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 60.l odd 4 1
80.2.a.b 1 15.e even 4 1
100.2.a.a 1 60.l odd 4 1
100.2.c.a 2 12.b even 2 1
100.2.c.a 2 60.h even 2 1
180.2.a.a 1 20.e even 4 1
320.2.a.a 1 120.w even 4 1
320.2.a.f 1 120.q odd 4 1
400.2.a.c 1 15.e even 4 1
400.2.c.b 2 3.b odd 2 1
400.2.c.b 2 15.d odd 2 1
720.2.a.h 1 5.c odd 4 1
900.2.a.b 1 20.e even 4 1
900.2.d.c 2 4.b odd 2 1
900.2.d.c 2 20.d odd 2 1
980.2.a.h 1 420.w even 4 1
980.2.i.c 2 420.br even 12 2
980.2.i.i 2 420.bp odd 12 2
1280.2.d.c 2 240.z odd 4 1
1280.2.d.c 2 240.bd odd 4 1
1280.2.d.g 2 240.bb even 4 1
1280.2.d.g 2 240.bf even 4 1
1600.2.a.c 1 120.q odd 4 1
1600.2.a.w 1 120.w even 4 1
1600.2.c.d 2 24.f even 2 1
1600.2.c.d 2 120.m even 2 1
1600.2.c.e 2 24.h odd 2 1
1600.2.c.e 2 120.i odd 2 1
1620.2.i.b 2 180.x even 12 2
1620.2.i.h 2 180.v odd 12 2
2420.2.a.a 1 660.q even 4 1
2880.2.a.f 1 40.i odd 4 1
2880.2.a.m 1 40.k even 4 1
3380.2.a.c 1 780.w odd 4 1
3380.2.f.b 2 780.u even 4 1
3380.2.f.b 2 780.bn even 4 1
3600.2.a.be 1 5.c odd 4 1
3600.2.f.j 2 1.a even 1 1 trivial
3600.2.f.j 2 5.b even 2 1 inner
3920.2.a.h 1 105.k odd 4 1
4900.2.a.e 1 420.w even 4 1
4900.2.e.f 2 84.h odd 2 1
4900.2.e.f 2 420.o odd 2 1
5780.2.a.f 1 1020.x odd 4 1
5780.2.c.a 2 1020.q odd 4 1
5780.2.c.a 2 1020.bl odd 4 1
7220.2.a.f 1 1140.w even 4 1
8820.2.a.g 1 140.j odd 4 1
9680.2.a.ba 1 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{2} + 36$$ T17^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 4$$