# Properties

 Label 3600.2.f.b 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 600) 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.

 $$f(q)$$ $$=$$ $$q + 5 i q^{7}+O(q^{10})$$ q + 5*i * q^7 $$q + 5 i q^{7} - 6 q^{11} + 3 i q^{13} + 2 i q^{17} + q^{19} + 2 i q^{23} + 6 q^{29} - 3 q^{31} - 6 i q^{37} - 4 q^{41} + 11 i q^{43} - 10 i q^{47} - 18 q^{49} - 8 i q^{53} + 6 q^{59} + 3 q^{61} + i q^{67} - 12 q^{71} - 10 i q^{73} - 30 i q^{77} - 8 q^{79} + 6 i q^{83} - 16 q^{89} - 15 q^{91} - 7 i q^{97} +O(q^{100})$$ q + 5*i * q^7 - 6 * q^11 + 3*i * q^13 + 2*i * q^17 + q^19 + 2*i * q^23 + 6 * q^29 - 3 * q^31 - 6*i * q^37 - 4 * q^41 + 11*i * q^43 - 10*i * q^47 - 18 * q^49 - 8*i * q^53 + 6 * q^59 + 3 * q^61 + i * q^67 - 12 * q^71 - 10*i * q^73 - 30*i * q^77 - 8 * q^79 + 6*i * q^83 - 16 * q^89 - 15 * q^91 - 7*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 12 q^{11} + 2 q^{19} + 12 q^{29} - 6 q^{31} - 8 q^{41} - 36 q^{49} + 12 q^{59} + 6 q^{61} - 24 q^{71} - 16 q^{79} - 32 q^{89} - 30 q^{91}+O(q^{100})$$ 2 * q - 12 * q^11 + 2 * q^19 + 12 * q^29 - 6 * q^31 - 8 * q^41 - 36 * q^49 + 12 * q^59 + 6 * q^61 - 24 * q^71 - 16 * q^79 - 32 * q^89 - 30 * 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 5.00000i 0 0 0
2449.2 0 0 0 0 0 5.00000i 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.b 2
3.b odd 2 1 1200.2.f.i 2
4.b odd 2 1 1800.2.f.k 2
5.b even 2 1 inner 3600.2.f.b 2
5.c odd 4 1 3600.2.a.a 1
5.c odd 4 1 3600.2.a.bq 1
12.b even 2 1 600.2.f.a 2
15.d odd 2 1 1200.2.f.i 2
15.e even 4 1 1200.2.a.i 1
15.e even 4 1 1200.2.a.j 1
20.d odd 2 1 1800.2.f.k 2
20.e even 4 1 1800.2.a.a 1
20.e even 4 1 1800.2.a.x 1
24.f even 2 1 4800.2.f.bj 2
24.h odd 2 1 4800.2.f.a 2
60.h even 2 1 600.2.f.a 2
60.l odd 4 1 600.2.a.e 1
60.l odd 4 1 600.2.a.f yes 1
120.i odd 2 1 4800.2.f.a 2
120.m even 2 1 4800.2.f.bj 2
120.q odd 4 1 4800.2.a.b 1
120.q odd 4 1 4800.2.a.ct 1
120.w even 4 1 4800.2.a.a 1
120.w even 4 1 4800.2.a.cs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.e 1 60.l odd 4 1
600.2.a.f yes 1 60.l odd 4 1
600.2.f.a 2 12.b even 2 1
600.2.f.a 2 60.h even 2 1
1200.2.a.i 1 15.e even 4 1
1200.2.a.j 1 15.e even 4 1
1200.2.f.i 2 3.b odd 2 1
1200.2.f.i 2 15.d odd 2 1
1800.2.a.a 1 20.e even 4 1
1800.2.a.x 1 20.e even 4 1
1800.2.f.k 2 4.b odd 2 1
1800.2.f.k 2 20.d odd 2 1
3600.2.a.a 1 5.c odd 4 1
3600.2.a.bq 1 5.c odd 4 1
3600.2.f.b 2 1.a even 1 1 trivial
3600.2.f.b 2 5.b even 2 1 inner
4800.2.a.a 1 120.w even 4 1
4800.2.a.b 1 120.q odd 4 1
4800.2.a.cs 1 120.w even 4 1
4800.2.a.ct 1 120.q odd 4 1
4800.2.f.a 2 24.h odd 2 1
4800.2.f.a 2 120.i odd 2 1
4800.2.f.bj 2 24.f even 2 1
4800.2.f.bj 2 120.m even 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 25$$ T7^2 + 25 $$T_{11} + 6$$ T11 + 6 $$T_{13}^{2} + 9$$ T13^2 + 9 $$T_{17}^{2} + 4$$ T17^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 25$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 9$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 64$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 3)^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 16)^{2}$$
$97$ $$T^{2} + 49$$