# Properties

 Label 2736.3.o.h Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ 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] = [2736,3,Mod(721,2736)]

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([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (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: $$74.5506003290$$ 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: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 38) 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 $$\beta = 6\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} - 5 q^{7}+O(q^{10})$$ q + q^5 - 5 * q^7 $$q + q^{5} - 5 q^{7} + 5 q^{11} - 2 \beta q^{13} + 25 q^{17} - 19 q^{19} - 10 q^{23} - 24 q^{25} - 5 \beta q^{29} + 5 \beta q^{31} - 5 q^{35} - 3 \beta q^{37} + 5 \beta q^{41} - 5 q^{43} + 5 q^{47} - 24 q^{49} + 3 \beta q^{53} + 5 q^{55} + 10 \beta q^{59} + 95 q^{61} - 2 \beta q^{65} - 13 \beta q^{67} - 25 q^{73} - 25 q^{77} - 5 \beta q^{79} - 130 q^{83} + 25 q^{85} + 15 \beta q^{89} + 10 \beta q^{91} - 19 q^{95} + 2 \beta q^{97} +O(q^{100})$$ q + q^5 - 5 * q^7 + 5 * q^11 - 2*b * q^13 + 25 * q^17 - 19 * q^19 - 10 * q^23 - 24 * q^25 - 5*b * q^29 + 5*b * q^31 - 5 * q^35 - 3*b * q^37 + 5*b * q^41 - 5 * q^43 + 5 * q^47 - 24 * q^49 + 3*b * q^53 + 5 * q^55 + 10*b * q^59 + 95 * q^61 - 2*b * q^65 - 13*b * q^67 - 25 * q^73 - 25 * q^77 - 5*b * q^79 - 130 * q^83 + 25 * q^85 + 15*b * q^89 + 10*b * q^91 - 19 * q^95 + 2*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 10 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 10 * q^7 $$2 q + 2 q^{5} - 10 q^{7} + 10 q^{11} + 50 q^{17} - 38 q^{19} - 20 q^{23} - 48 q^{25} - 10 q^{35} - 10 q^{43} + 10 q^{47} - 48 q^{49} + 10 q^{55} + 190 q^{61} - 50 q^{73} - 50 q^{77} - 260 q^{83} + 50 q^{85} - 38 q^{95}+O(q^{100})$$ 2 * q + 2 * q^5 - 10 * q^7 + 10 * q^11 + 50 * q^17 - 38 * q^19 - 20 * q^23 - 48 * q^25 - 10 * q^35 - 10 * q^43 + 10 * q^47 - 48 * q^49 + 10 * q^55 + 190 * q^61 - 50 * q^73 - 50 * q^77 - 260 * q^83 + 50 * q^85 - 38 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 1.41421i − 1.41421i
0 0 0 1.00000 0 −5.00000 0 0 0
721.2 0 0 0 1.00000 0 −5.00000 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.h 2
3.b odd 2 1 304.3.e.c 2
4.b odd 2 1 342.3.d.a 2
12.b even 2 1 38.3.b.a 2
19.b odd 2 1 inner 2736.3.o.h 2
24.f even 2 1 1216.3.e.j 2
24.h odd 2 1 1216.3.e.i 2
57.d even 2 1 304.3.e.c 2
60.h even 2 1 950.3.c.a 2
60.l odd 4 2 950.3.d.a 4
76.d even 2 1 342.3.d.a 2
228.b odd 2 1 38.3.b.a 2
456.l odd 2 1 1216.3.e.j 2
456.p even 2 1 1216.3.e.i 2
1140.p odd 2 1 950.3.c.a 2
1140.w even 4 2 950.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 12.b even 2 1
38.3.b.a 2 228.b odd 2 1
304.3.e.c 2 3.b odd 2 1
304.3.e.c 2 57.d even 2 1
342.3.d.a 2 4.b odd 2 1
342.3.d.a 2 76.d even 2 1
950.3.c.a 2 60.h even 2 1
950.3.c.a 2 1140.p odd 2 1
950.3.d.a 4 60.l odd 4 2
950.3.d.a 4 1140.w even 4 2
1216.3.e.i 2 24.h odd 2 1
1216.3.e.i 2 456.p even 2 1
1216.3.e.j 2 24.f even 2 1
1216.3.e.j 2 456.l odd 2 1
2736.3.o.h 2 1.a even 1 1 trivial
2736.3.o.h 2 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 1$$ T5 - 1 $$T_{7} + 5$$ T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 5)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 288$$
$17$ $$(T - 25)^{2}$$
$19$ $$(T + 19)^{2}$$
$23$ $$(T + 10)^{2}$$
$29$ $$T^{2} + 1800$$
$31$ $$T^{2} + 1800$$
$37$ $$T^{2} + 648$$
$41$ $$T^{2} + 1800$$
$43$ $$(T + 5)^{2}$$
$47$ $$(T - 5)^{2}$$
$53$ $$T^{2} + 648$$
$59$ $$T^{2} + 7200$$
$61$ $$(T - 95)^{2}$$
$67$ $$T^{2} + 12168$$
$71$ $$T^{2}$$
$73$ $$(T + 25)^{2}$$
$79$ $$T^{2} + 1800$$
$83$ $$(T + 130)^{2}$$
$89$ $$T^{2} + 16200$$
$97$ $$T^{2} + 288$$