# Properties

 Label 7200.2.f.bg Level $7200$ Weight $2$ Character orbit 7200.f Analytic conductor $57.492$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(6049,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, 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("7200.6049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7200 = 2^{5} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7200.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: $$57.4922894553$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 800) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{7}+O(q^{10})$$ q + 2*b2 * q^7 $$q + 2 \beta_{2} q^{7} + \beta_{3} q^{11} + 4 \beta_1 q^{13} - 7 \beta_1 q^{17} - 3 \beta_{3} q^{19} - 2 \beta_{2} q^{23} - 2 \beta_{3} q^{31} - 2 \beta_1 q^{37} - 5 q^{41} + 4 \beta_{2} q^{47} - 13 q^{49} - 6 \beta_1 q^{53} - 4 \beta_{3} q^{59} + 10 q^{61} - \beta_{2} q^{67} - 4 \beta_{3} q^{71} - 9 \beta_1 q^{73} + 10 \beta_1 q^{77} - 2 \beta_{3} q^{79} - 5 \beta_{2} q^{83} - 5 q^{89} - 8 \beta_{3} q^{91} - 2 \beta_1 q^{97}+O(q^{100})$$ q + 2*b2 * q^7 + b3 * q^11 + 4*b1 * q^13 - 7*b1 * q^17 - 3*b3 * q^19 - 2*b2 * q^23 - 2*b3 * q^31 - 2*b1 * q^37 - 5 * q^41 + 4*b2 * q^47 - 13 * q^49 - 6*b1 * q^53 - 4*b3 * q^59 + 10 * q^61 - b2 * q^67 - 4*b3 * q^71 - 9*b1 * q^73 + 10*b1 * q^77 - 2*b3 * q^79 - 5*b2 * q^83 - 5 * q^89 - 8*b3 * q^91 - 2*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 20 q^{41} - 52 q^{49} + 40 q^{61} - 20 q^{89}+O(q^{100})$$ 4 * q - 20 * q^41 - 52 * q^49 + 40 * q^61 - 20 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$6401$$ $$6751$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6049.1
 − 1.61803i − 0.618034i 1.61803i 0.618034i
0 0 0 0 0 4.47214i 0 0 0
6049.2 0 0 0 0 0 4.47214i 0 0 0
6049.3 0 0 0 0 0 4.47214i 0 0 0
6049.4 0 0 0 0 0 4.47214i 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 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.f.bg 4
3.b odd 2 1 800.2.c.g 4
4.b odd 2 1 inner 7200.2.f.bg 4
5.b even 2 1 inner 7200.2.f.bg 4
5.c odd 4 1 7200.2.a.cf 2
5.c odd 4 1 7200.2.a.cn 2
12.b even 2 1 800.2.c.g 4
15.d odd 2 1 800.2.c.g 4
15.e even 4 1 800.2.a.k 2
15.e even 4 1 800.2.a.l yes 2
20.d odd 2 1 inner 7200.2.f.bg 4
20.e even 4 1 7200.2.a.cf 2
20.e even 4 1 7200.2.a.cn 2
24.f even 2 1 1600.2.c.o 4
24.h odd 2 1 1600.2.c.o 4
60.h even 2 1 800.2.c.g 4
60.l odd 4 1 800.2.a.k 2
60.l odd 4 1 800.2.a.l yes 2
120.i odd 2 1 1600.2.c.o 4
120.m even 2 1 1600.2.c.o 4
120.q odd 4 1 1600.2.a.ba 2
120.q odd 4 1 1600.2.a.bb 2
120.w even 4 1 1600.2.a.ba 2
120.w even 4 1 1600.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 15.e even 4 1
800.2.a.k 2 60.l odd 4 1
800.2.a.l yes 2 15.e even 4 1
800.2.a.l yes 2 60.l odd 4 1
800.2.c.g 4 3.b odd 2 1
800.2.c.g 4 12.b even 2 1
800.2.c.g 4 15.d odd 2 1
800.2.c.g 4 60.h even 2 1
1600.2.a.ba 2 120.q odd 4 1
1600.2.a.ba 2 120.w even 4 1
1600.2.a.bb 2 120.q odd 4 1
1600.2.a.bb 2 120.w even 4 1
1600.2.c.o 4 24.f even 2 1
1600.2.c.o 4 24.h odd 2 1
1600.2.c.o 4 120.i odd 2 1
1600.2.c.o 4 120.m even 2 1
7200.2.a.cf 2 5.c odd 4 1
7200.2.a.cf 2 20.e even 4 1
7200.2.a.cn 2 5.c odd 4 1
7200.2.a.cn 2 20.e even 4 1
7200.2.f.bg 4 1.a even 1 1 trivial
7200.2.f.bg 4 4.b odd 2 1 inner
7200.2.f.bg 4 5.b even 2 1 inner
7200.2.f.bg 4 20.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}^{2} + 20$$ T7^2 + 20 $$T_{11}^{2} - 5$$ T11^2 - 5 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 49$$ T17^2 + 49 $$T_{19}^{2} - 45$$ T19^2 - 45 $$T_{29}$$ T29 $$T_{31}^{2} - 20$$ T31^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 20)^{2}$$
$11$ $$(T^{2} - 5)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$(T^{2} + 49)^{2}$$
$19$ $$(T^{2} - 45)^{2}$$
$23$ $$(T^{2} + 20)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 20)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T + 5)^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 80)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 80)^{2}$$
$61$ $$(T - 10)^{4}$$
$67$ $$(T^{2} + 5)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$(T^{2} + 81)^{2}$$
$79$ $$(T^{2} - 20)^{2}$$
$83$ $$(T^{2} + 125)^{2}$$
$89$ $$(T + 5)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$