Properties

 Label 8100.2.d.d Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ 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] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8100 = 2^{2} \cdot 3^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8100.d (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: $$64.6788256372$$ 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 1620) 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 $$q + \beta q^{7} - 3 q^{11} + 2 \beta q^{13} - 3 \beta q^{17} + 7 q^{19} + 3 \beta q^{23} + 3 q^{29} + 5 q^{31} - 2 \beta q^{37} - 3 q^{41} - 4 \beta q^{43} + 3 q^{49} - 3 \beta q^{53} - 3 q^{59} + 14 q^{61} + \beta q^{67} - 15 q^{71} + 5 \beta q^{73} - 3 \beta q^{77} - 8 q^{79} + 15 q^{89} - 8 q^{91} + 4 \beta q^{97} +O(q^{100})$$ q + b * q^7 - 3 * q^11 + 2*b * q^13 - 3*b * q^17 + 7 * q^19 + 3*b * q^23 + 3 * q^29 + 5 * q^31 - 2*b * q^37 - 3 * q^41 - 4*b * q^43 + 3 * q^49 - 3*b * q^53 - 3 * q^59 + 14 * q^61 + b * q^67 - 15 * q^71 + 5*b * q^73 - 3*b * q^77 - 8 * q^79 + 15 * q^89 - 8 * q^91 + 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 6 q^{11} + 14 q^{19} + 6 q^{29} + 10 q^{31} - 6 q^{41} + 6 q^{49} - 6 q^{59} + 28 q^{61} - 30 q^{71} - 16 q^{79} + 30 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q - 6 * q^11 + 14 * q^19 + 6 * q^29 + 10 * q^31 - 6 * q^41 + 6 * q^49 - 6 * q^59 + 28 * q^61 - 30 * q^71 - 16 * q^79 + 30 * q^89 - 16 * q^91

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$$.

 $$n$$ $$4051$$ $$6401$$ $$7777$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 0 0 2.00000i 0 0 0
649.2 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 8100.2.d.d 2
3.b odd 2 1 8100.2.d.i 2
5.b even 2 1 inner 8100.2.d.d 2
5.c odd 4 1 1620.2.a.f yes 1
5.c odd 4 1 8100.2.a.b 1
15.d odd 2 1 8100.2.d.i 2
15.e even 4 1 1620.2.a.c 1
15.e even 4 1 8100.2.a.e 1
20.e even 4 1 6480.2.a.p 1
45.k odd 12 2 1620.2.i.c 2
45.l even 12 2 1620.2.i.g 2
60.l odd 4 1 6480.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.c 1 15.e even 4 1
1620.2.a.f yes 1 5.c odd 4 1
1620.2.i.c 2 45.k odd 12 2
1620.2.i.g 2 45.l even 12 2
6480.2.a.b 1 60.l odd 4 1
6480.2.a.p 1 20.e even 4 1
8100.2.a.b 1 5.c odd 4 1
8100.2.a.e 1 15.e even 4 1
8100.2.d.d 2 1.a even 1 1 trivial
8100.2.d.d 2 5.b even 2 1 inner
8100.2.d.i 2 3.b odd 2 1
8100.2.d.i 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 3$$ T11 + 3 $$T_{29} - 3$$ T29 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 15)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 64$$