# Properties

 Label 2700.2.d.a Level $2700$ Weight $2$ Character orbit 2700.d Analytic conductor $21.560$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(649,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, 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("2700.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2700.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: $$21.5596085457$$ 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 540) 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 $$q + i q^{7} - 6 q^{11} - i q^{13} + q^{19} - 6 i q^{23} + 6 q^{29} + 8 q^{31} + 7 i q^{37} + 6 q^{41} - 4 i q^{43} + 12 i q^{47} + 6 q^{49} + 6 i q^{53} + 11 q^{61} + 7 i q^{67} + 6 q^{71} + 11 i q^{73} - 6 i q^{77} + q^{79} - 6 i q^{83} - 12 q^{89} + q^{91} + 13 i q^{97} +O(q^{100})$$ q + i * q^7 - 6 * q^11 - i * q^13 + q^19 - 6*i * q^23 + 6 * q^29 + 8 * q^31 + 7*i * q^37 + 6 * q^41 - 4*i * q^43 + 12*i * q^47 + 6 * q^49 + 6*i * q^53 + 11 * q^61 + 7*i * q^67 + 6 * q^71 + 11*i * q^73 - 6*i * q^77 + q^79 - 6*i * q^83 - 12 * q^89 + q^91 + 13*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 12 q^{11} + 2 q^{19} + 12 q^{29} + 16 q^{31} + 12 q^{41} + 12 q^{49} + 22 q^{61} + 12 q^{71} + 2 q^{79} - 24 q^{89} + 2 q^{91}+O(q^{100})$$ 2 * q - 12 * q^11 + 2 * q^19 + 12 * q^29 + 16 * q^31 + 12 * q^41 + 12 * q^49 + 22 * q^61 + 12 * q^71 + 2 * q^79 - 24 * q^89 + 2 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1351$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 2700.2.d.a 2
3.b odd 2 1 2700.2.d.k 2
5.b even 2 1 inner 2700.2.d.a 2
5.c odd 4 1 540.2.a.b 1
5.c odd 4 1 2700.2.a.k 1
15.d odd 2 1 2700.2.d.k 2
15.e even 4 1 540.2.a.e yes 1
15.e even 4 1 2700.2.a.m 1
20.e even 4 1 2160.2.a.h 1
40.i odd 4 1 8640.2.a.br 1
40.k even 4 1 8640.2.a.bu 1
45.k odd 12 2 1620.2.i.j 2
45.l even 12 2 1620.2.i.d 2
60.l odd 4 1 2160.2.a.s 1
120.q odd 4 1 8640.2.a.s 1
120.w even 4 1 8640.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 5.c odd 4 1
540.2.a.e yes 1 15.e even 4 1
1620.2.i.d 2 45.l even 12 2
1620.2.i.j 2 45.k odd 12 2
2160.2.a.h 1 20.e even 4 1
2160.2.a.s 1 60.l odd 4 1
2700.2.a.k 1 5.c odd 4 1
2700.2.a.m 1 15.e even 4 1
2700.2.d.a 2 1.a even 1 1 trivial
2700.2.d.a 2 5.b even 2 1 inner
2700.2.d.k 2 3.b odd 2 1
2700.2.d.k 2 15.d odd 2 1
8640.2.a.l 1 120.w even 4 1
8640.2.a.s 1 120.q odd 4 1
8640.2.a.br 1 40.i odd 4 1
8640.2.a.bu 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2700, [\chi])$$:

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} + 6$$ T11 + 6 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$(T - 11)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 121$$
$79$ $$(T - 1)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2} + 169$$