# Properties

 Label 2736.2.k.i Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ 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] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, 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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 912) 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + 2 * q^5 - 2*b * q^7 $$q + 2 q^{5} - 2 \beta q^{7} - \beta q^{11} - 2 \beta q^{13} + 2 q^{17} + ( - 3 \beta + 1) q^{19} - \beta q^{23} - q^{25} + 5 \beta q^{29} - 6 q^{31} - 4 \beta q^{35} - 8 \beta q^{37} + 3 \beta q^{41} - 5 \beta q^{47} - q^{49} + 7 \beta q^{53} - 2 \beta q^{55} - 8 q^{59} - 8 q^{61} - 4 \beta q^{65} + 2 q^{67} - 8 q^{71} - 4 q^{77} + 8 q^{79} - 9 \beta q^{83} + 4 q^{85} - 7 \beta q^{89} - 8 q^{91} + ( - 6 \beta + 2) q^{95} + 2 \beta q^{97} +O(q^{100})$$ q + 2 * q^5 - 2*b * q^7 - b * q^11 - 2*b * q^13 + 2 * q^17 + (-3*b + 1) * q^19 - b * q^23 - q^25 + 5*b * q^29 - 6 * q^31 - 4*b * q^35 - 8*b * q^37 + 3*b * q^41 - 5*b * q^47 - q^49 + 7*b * q^53 - 2*b * q^55 - 8 * q^59 - 8 * q^61 - 4*b * q^65 + 2 * q^67 - 8 * q^71 - 4 * q^77 + 8 * q^79 - 9*b * q^83 + 4 * q^85 - 7*b * q^89 - 8 * q^91 + (-6*b + 2) * q^95 + 2*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5}+O(q^{10})$$ 2 * q + 4 * q^5 $$2 q + 4 q^{5} + 4 q^{17} + 2 q^{19} - 2 q^{25} - 12 q^{31} - 2 q^{49} - 16 q^{59} - 16 q^{61} + 4 q^{67} - 16 q^{71} - 8 q^{77} + 16 q^{79} + 8 q^{85} - 16 q^{91} + 4 q^{95}+O(q^{100})$$ 2 * q + 4 * q^5 + 4 * q^17 + 2 * q^19 - 2 * q^25 - 12 * q^31 - 2 * q^49 - 16 * q^59 - 16 * q^61 + 4 * q^67 - 16 * q^71 - 8 * q^77 + 16 * q^79 + 8 * q^85 - 16 * q^91 + 4 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 1.41421i − 1.41421i
0 0 0 2.00000 0 2.82843i 0 0 0
2431.2 0 0 0 2.00000 0 2.82843i 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
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.i 2
3.b odd 2 1 912.2.k.d yes 2
4.b odd 2 1 2736.2.k.h 2
12.b even 2 1 912.2.k.a 2
19.b odd 2 1 2736.2.k.h 2
24.f even 2 1 3648.2.k.f 2
24.h odd 2 1 3648.2.k.c 2
57.d even 2 1 912.2.k.a 2
76.d even 2 1 inner 2736.2.k.i 2
228.b odd 2 1 912.2.k.d yes 2
456.l odd 2 1 3648.2.k.c 2
456.p even 2 1 3648.2.k.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.a 2 12.b even 2 1
912.2.k.a 2 57.d even 2 1
912.2.k.d yes 2 3.b odd 2 1
912.2.k.d yes 2 228.b odd 2 1
2736.2.k.h 2 4.b odd 2 1
2736.2.k.h 2 19.b odd 2 1
2736.2.k.i 2 1.a even 1 1 trivial
2736.2.k.i 2 76.d even 2 1 inner
3648.2.k.c 2 24.h odd 2 1
3648.2.k.c 2 456.l odd 2 1
3648.2.k.f 2 24.f even 2 1
3648.2.k.f 2 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 2$$ T5 - 2 $$T_{7}^{2} + 8$$ T7^2 + 8 $$T_{11}^{2} + 2$$ T11^2 + 2 $$T_{31} + 6$$ T31 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2} + 8$$
$11$ $$T^{2} + 2$$
$13$ $$T^{2} + 8$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - 2T + 19$$
$23$ $$T^{2} + 2$$
$29$ $$T^{2} + 50$$
$31$ $$(T + 6)^{2}$$
$37$ $$T^{2} + 128$$
$41$ $$T^{2} + 18$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 50$$
$53$ $$T^{2} + 98$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 162$$
$89$ $$T^{2} + 98$$
$97$ $$T^{2} + 8$$