# Properties

 Label 8100.2.d.l Level $8100$ Weight $2$ Character orbit 8100.d Analytic conductor $64.679$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{7}+O(q^{10})$$ q - b3 * q^7 $$q - \beta_{3} q^{7} + \beta_1 q^{11} - 2 \beta_{3} q^{13} + (\beta_{3} - \beta_{2}) q^{17} + ( - 2 \beta_1 + 1) q^{19} + (2 \beta_{3} + \beta_{2}) q^{23} + (\beta_1 - 6) q^{29} + ( - 4 \beta_1 - 1) q^{31} + ( - 3 \beta_{3} - \beta_{2}) q^{37} + ( - 3 \beta_1 + 6) q^{41} + (\beta_{3} + 3 \beta_{2}) q^{43} + ( - \beta_{3} - 2 \beta_{2}) q^{47} + (2 \beta_1 + 3) q^{49} + (\beta_{3} + 5 \beta_{2}) q^{53} + (\beta_1 - 6) q^{59} - 4 q^{61} + (6 \beta_{3} + 5 \beta_{2}) q^{67} + ( - 3 \beta_1 + 6) q^{71} + (3 \beta_{3} + 4 \beta_{2}) q^{73} + (\beta_{3} - \beta_{2}) q^{77} + ( - 6 \beta_1 + 4) q^{79} + ( - 7 \beta_{3} - 2 \beta_{2}) q^{83} - 3 \beta_1 q^{89} + (4 \beta_1 - 8) q^{91} + (\beta_{3} + \beta_{2}) q^{97}+O(q^{100})$$ q - b3 * q^7 + b1 * q^11 - 2*b3 * q^13 + (b3 - b2) * q^17 + (-2*b1 + 1) * q^19 + (2*b3 + b2) * q^23 + (b1 - 6) * q^29 + (-4*b1 - 1) * q^31 + (-3*b3 - b2) * q^37 + (-3*b1 + 6) * q^41 + (b3 + 3*b2) * q^43 + (-b3 - 2*b2) * q^47 + (2*b1 + 3) * q^49 + (b3 + 5*b2) * q^53 + (b1 - 6) * q^59 - 4 * q^61 + (6*b3 + 5*b2) * q^67 + (-3*b1 + 6) * q^71 + (3*b3 + 4*b2) * q^73 + (b3 - b2) * q^77 + (-6*b1 + 4) * q^79 + (-7*b3 - 2*b2) * q^83 - 3*b1 * q^89 + (4*b1 - 8) * q^91 + (b3 + b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100})$$ 4 * q + 4 * q^19 - 24 * q^29 - 4 * q^31 + 24 * q^41 + 12 * q^49 - 24 * q^59 - 16 * q^61 + 24 * q^71 + 16 * q^79 - 32 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1$$ -v^3 + 2*v^2 - 1
 $$\zeta_{12}$$ $$=$$ $$( \beta_{2} + 2\beta_1 ) / 4$$ (b2 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} + 2 ) / 4$$ (2*b3 + b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{2} ) / 2$$ (b2) / 2

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 0 0 2.73205i 0 0 0
649.2 0 0 0 0 0 0.732051i 0 0 0
649.3 0 0 0 0 0 0.732051i 0 0 0
649.4 0 0 0 0 0 2.73205i 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.l 4
3.b odd 2 1 8100.2.d.m 4
5.b even 2 1 inner 8100.2.d.l 4
5.c odd 4 1 1620.2.a.h yes 2
5.c odd 4 1 8100.2.a.s 2
15.d odd 2 1 8100.2.d.m 4
15.e even 4 1 1620.2.a.g 2
15.e even 4 1 8100.2.a.t 2
20.e even 4 1 6480.2.a.bp 2
45.k odd 12 2 1620.2.i.m 4
45.l even 12 2 1620.2.i.n 4
60.l odd 4 1 6480.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 15.e even 4 1
1620.2.a.h yes 2 5.c odd 4 1
1620.2.i.m 4 45.k odd 12 2
1620.2.i.n 4 45.l even 12 2
6480.2.a.bh 2 60.l odd 4 1
6480.2.a.bp 2 20.e even 4 1
8100.2.a.s 2 5.c odd 4 1
8100.2.a.t 2 15.e even 4 1
8100.2.d.l 4 1.a even 1 1 trivial
8100.2.d.l 4 5.b even 2 1 inner
8100.2.d.m 4 3.b odd 2 1
8100.2.d.m 4 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 $$S_{2}^{\mathrm{new}}(8100, [\chi])$$:

 $$T_{7}^{4} + 8T_{7}^{2} + 4$$ T7^4 + 8*T7^2 + 4 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{29}^{2} + 12T_{29} + 33$$ T29^2 + 12*T29 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 8T^{2} + 4$$
$11$ $$(T^{2} - 3)^{2}$$
$13$ $$T^{4} + 32T^{2} + 64$$
$17$ $$T^{4} + 24T^{2} + 36$$
$19$ $$(T^{2} - 2 T - 11)^{2}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$(T^{2} + 12 T + 33)^{2}$$
$31$ $$(T^{2} + 2 T - 47)^{2}$$
$37$ $$T^{4} + 56T^{2} + 676$$
$41$ $$(T^{2} - 12 T + 9)^{2}$$
$43$ $$T^{4} + 56T^{2} + 484$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$T^{4} + 168T^{2} + 6084$$
$59$ $$(T^{2} + 12 T + 33)^{2}$$
$61$ $$(T + 4)^{4}$$
$67$ $$T^{4} + 248T^{2} + 8464$$
$71$ $$(T^{2} - 12 T + 9)^{2}$$
$73$ $$T^{4} + 104T^{2} + 4$$
$79$ $$(T^{2} - 8 T - 92)^{2}$$
$83$ $$T^{4} + 312 T^{2} + 19044$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$T^{4} + 8T^{2} + 4$$