# Properties

 Label 8100.2.d.c Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

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: $$1$$ Twist minimal: no (minimal twist has level 36) 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.

 $$f(q)$$ $$=$$ $$q + i q^{7}+O(q^{10})$$ q + i * q^7 $$q + i q^{7} - 3 q^{11} - i q^{13} + 6 i q^{17} + 4 q^{19} + 3 i q^{23} + 3 q^{29} + 5 q^{31} - 2 i q^{37} - 3 q^{41} - i q^{43} - 9 i q^{47} + 6 q^{49} + 6 i q^{53} - 3 q^{59} - 13 q^{61} + 7 i q^{67} + 12 q^{71} - 10 i q^{73} - 3 i q^{77} - 11 q^{79} + 9 i q^{83} + 6 q^{89} + q^{91} - 11 i q^{97} +O(q^{100})$$ q + i * q^7 - 3 * q^11 - i * q^13 + 6*i * q^17 + 4 * q^19 + 3*i * q^23 + 3 * q^29 + 5 * q^31 - 2*i * q^37 - 3 * q^41 - i * q^43 - 9*i * q^47 + 6 * q^49 + 6*i * q^53 - 3 * q^59 - 13 * q^61 + 7*i * q^67 + 12 * q^71 - 10*i * q^73 - 3*i * q^77 - 11 * q^79 + 9*i * q^83 + 6 * q^89 + q^91 - 11*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 6 q^{11} + 8 q^{19} + 6 q^{29} + 10 q^{31} - 6 q^{41} + 12 q^{49} - 6 q^{59} - 26 q^{61} + 24 q^{71} - 22 q^{79} + 12 q^{89} + 2 q^{91}+O(q^{100})$$ 2 * q - 6 * q^11 + 8 * q^19 + 6 * q^29 + 10 * q^31 - 6 * q^41 + 12 * q^49 - 6 * q^59 - 26 * q^61 + 24 * q^71 - 22 * q^79 + 12 * q^89 + 2 * 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 1.00000i 0 0 0
649.2 0 0 0 0 0 1.00000i 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.c 2
3.b odd 2 1 8100.2.d.h 2
5.b even 2 1 inner 8100.2.d.c 2
5.c odd 4 1 324.2.a.a 1
5.c odd 4 1 8100.2.a.g 1
9.c even 3 2 2700.2.s.b 4
9.d odd 6 2 900.2.s.b 4
15.d odd 2 1 8100.2.d.h 2
15.e even 4 1 324.2.a.c 1
15.e even 4 1 8100.2.a.j 1
20.e even 4 1 1296.2.a.b 1
40.i odd 4 1 5184.2.a.ba 1
40.k even 4 1 5184.2.a.bb 1
45.h odd 6 2 900.2.s.b 4
45.j even 6 2 2700.2.s.b 4
45.k odd 12 2 108.2.e.a 2
45.k odd 12 2 2700.2.i.b 2
45.l even 12 2 36.2.e.a 2
45.l even 12 2 900.2.i.b 2
60.l odd 4 1 1296.2.a.k 1
120.q odd 4 1 5184.2.a.f 1
120.w even 4 1 5184.2.a.e 1
180.v odd 12 2 144.2.i.a 2
180.x even 12 2 432.2.i.c 2
315.bs even 12 2 5292.2.i.a 2
315.bt odd 12 2 5292.2.i.c 2
315.bu odd 12 2 1764.2.i.c 2
315.bv even 12 2 1764.2.i.a 2
315.bw odd 12 2 1764.2.l.a 2
315.bx even 12 2 1764.2.l.c 2
315.cb even 12 2 5292.2.j.a 2
315.cf odd 12 2 1764.2.j.b 2
315.cg even 12 2 5292.2.l.c 2
315.ch odd 12 2 5292.2.l.a 2
360.bo even 12 2 1728.2.i.c 2
360.br even 12 2 576.2.i.f 2
360.bt odd 12 2 576.2.i.e 2
360.bu odd 12 2 1728.2.i.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 45.l even 12 2
108.2.e.a 2 45.k odd 12 2
144.2.i.a 2 180.v odd 12 2
324.2.a.a 1 5.c odd 4 1
324.2.a.c 1 15.e even 4 1
432.2.i.c 2 180.x even 12 2
576.2.i.e 2 360.bt odd 12 2
576.2.i.f 2 360.br even 12 2
900.2.i.b 2 45.l even 12 2
900.2.s.b 4 9.d odd 6 2
900.2.s.b 4 45.h odd 6 2
1296.2.a.b 1 20.e even 4 1
1296.2.a.k 1 60.l odd 4 1
1728.2.i.c 2 360.bo even 12 2
1728.2.i.d 2 360.bu odd 12 2
1764.2.i.a 2 315.bv even 12 2
1764.2.i.c 2 315.bu odd 12 2
1764.2.j.b 2 315.cf odd 12 2
1764.2.l.a 2 315.bw odd 12 2
1764.2.l.c 2 315.bx even 12 2
2700.2.i.b 2 45.k odd 12 2
2700.2.s.b 4 9.c even 3 2
2700.2.s.b 4 45.j even 6 2
5184.2.a.e 1 120.w even 4 1
5184.2.a.f 1 120.q odd 4 1
5184.2.a.ba 1 40.i odd 4 1
5184.2.a.bb 1 40.k even 4 1
5292.2.i.a 2 315.bs even 12 2
5292.2.i.c 2 315.bt odd 12 2
5292.2.j.a 2 315.cb even 12 2
5292.2.l.a 2 315.ch odd 12 2
5292.2.l.c 2 315.cg even 12 2
8100.2.a.g 1 5.c odd 4 1
8100.2.a.j 1 15.e even 4 1
8100.2.d.c 2 1.a even 1 1 trivial
8100.2.d.c 2 5.b even 2 1 inner
8100.2.d.h 2 3.b odd 2 1
8100.2.d.h 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 $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 3$$ T11 + 3 $$T_{29} - 3$$ T29 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T + 13)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 11)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 121$$
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