Properties

 Label 7200.2.f.q Level $7200$ Weight $2$ Character orbit 7200.f Analytic conductor $57.492$ Analytic rank $0$ Dimension $2$ CM discriminant -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: $$2$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - 3 \beta q^{13} - 4 \beta q^{17} + 4 q^{29} + \beta q^{37} - 8 q^{41} + 7 q^{49} - 2 \beta q^{53} - 10 q^{61} + 3 \beta q^{73} - 16 q^{89} + 9 \beta q^{97} +O(q^{100})$$ q - 3*b * q^13 - 4*b * q^17 + 4 * q^29 + b * q^37 - 8 * q^41 + 7 * q^49 - 2*b * q^53 - 10 * q^61 + 3*b * q^73 - 16 * q^89 + 9*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 8 q^{29} - 16 q^{41} + 14 q^{49} - 20 q^{61} - 32 q^{89}+O(q^{100})$$ 2 * q + 8 * q^29 - 16 * q^41 + 14 * q^49 - 20 * q^61 - 32 * q^89

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
 1.00000i − 1.00000i
0 0 0 0 0 0 0 0 0
6049.2 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 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.q 2
3.b odd 2 1 7200.2.f.n 2
4.b odd 2 1 CM 7200.2.f.q 2
5.b even 2 1 inner 7200.2.f.q 2
5.c odd 4 1 288.2.a.e yes 1
5.c odd 4 1 7200.2.a.be 1
12.b even 2 1 7200.2.f.n 2
15.d odd 2 1 7200.2.f.n 2
15.e even 4 1 288.2.a.a 1
15.e even 4 1 7200.2.a.bf 1
20.d odd 2 1 inner 7200.2.f.q 2
20.e even 4 1 288.2.a.e yes 1
20.e even 4 1 7200.2.a.be 1
40.i odd 4 1 576.2.a.a 1
40.k even 4 1 576.2.a.a 1
45.k odd 12 2 2592.2.i.a 2
45.l even 12 2 2592.2.i.x 2
60.h even 2 1 7200.2.f.n 2
60.l odd 4 1 288.2.a.a 1
60.l odd 4 1 7200.2.a.bf 1
80.i odd 4 1 2304.2.d.l 2
80.j even 4 1 2304.2.d.l 2
80.s even 4 1 2304.2.d.l 2
80.t odd 4 1 2304.2.d.l 2
120.q odd 4 1 576.2.a.i 1
120.w even 4 1 576.2.a.i 1
180.v odd 12 2 2592.2.i.x 2
180.x even 12 2 2592.2.i.a 2
240.z odd 4 1 2304.2.d.h 2
240.bb even 4 1 2304.2.d.h 2
240.bd odd 4 1 2304.2.d.h 2
240.bf even 4 1 2304.2.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 15.e even 4 1
288.2.a.a 1 60.l odd 4 1
288.2.a.e yes 1 5.c odd 4 1
288.2.a.e yes 1 20.e even 4 1
576.2.a.a 1 40.i odd 4 1
576.2.a.a 1 40.k even 4 1
576.2.a.i 1 120.q odd 4 1
576.2.a.i 1 120.w even 4 1
2304.2.d.h 2 240.z odd 4 1
2304.2.d.h 2 240.bb even 4 1
2304.2.d.h 2 240.bd odd 4 1
2304.2.d.h 2 240.bf even 4 1
2304.2.d.l 2 80.i odd 4 1
2304.2.d.l 2 80.j even 4 1
2304.2.d.l 2 80.s even 4 1
2304.2.d.l 2 80.t odd 4 1
2592.2.i.a 2 45.k odd 12 2
2592.2.i.a 2 180.x even 12 2
2592.2.i.x 2 45.l even 12 2
2592.2.i.x 2 180.v odd 12 2
7200.2.a.be 1 5.c odd 4 1
7200.2.a.be 1 20.e even 4 1
7200.2.a.bf 1 15.e even 4 1
7200.2.a.bf 1 60.l odd 4 1
7200.2.f.n 2 3.b odd 2 1
7200.2.f.n 2 12.b even 2 1
7200.2.f.n 2 15.d odd 2 1
7200.2.f.n 2 60.h even 2 1
7200.2.f.q 2 1.a even 1 1 trivial
7200.2.f.q 2 4.b odd 2 1 CM
7200.2.f.q 2 5.b even 2 1 inner
7200.2.f.q 2 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 $$S_{2}^{\mathrm{new}}(7200, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 64$$ T17^2 + 64 $$T_{19}$$ T19 $$T_{29} - 4$$ T29 - 4 $$T_{31}$$ T31

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 64$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 16$$
$59$ $$T^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 16)^{2}$$
$97$ $$T^{2} + 324$$