# Properties

 Label 2736.3.o.c Level $2736$ Weight $3$ Character orbit 2736.o Self dual yes Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM discriminant -19 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: yes Analytic conductor: $$74.5506003290$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 4) q^{5} + (3 \beta - 4) q^{7}+O(q^{10})$$ q + (-b - 4) * q^5 + (3*b - 4) * q^7 $$q + ( - \beta - 4) q^{5} + (3 \beta - 4) q^{7} + (5 \beta - 4) q^{11} + (7 \beta + 4) q^{17} + 19 q^{19} - 30 q^{23} + (9 \beta + 5) q^{25} + ( - 11 \beta - 26) q^{35} + (3 \beta - 44) q^{43} + (13 \beta - 44) q^{47} + ( - 15 \beta + 93) q^{49} + ( - 21 \beta - 54) q^{55} + ( - 15 \beta - 44) q^{61} + (33 \beta - 4) q^{73} + ( - 17 \beta + 226) q^{77} + 90 q^{83} + ( - 39 \beta - 114) q^{85} + ( - 19 \beta - 76) q^{95}+O(q^{100})$$ q + (-b - 4) * q^5 + (3*b - 4) * q^7 + (5*b - 4) * q^11 + (7*b + 4) * q^17 + 19 * q^19 - 30 * q^23 + (9*b + 5) * q^25 + (-11*b - 26) * q^35 + (3*b - 44) * q^43 + (13*b - 44) * q^47 + (-15*b + 93) * q^49 + (-21*b - 54) * q^55 + (-15*b - 44) * q^61 + (33*b - 4) * q^73 + (-17*b + 226) * q^77 + 90 * q^83 + (-39*b - 114) * q^85 + (-19*b - 76) * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{5} - 5 q^{7}+O(q^{10})$$ 2 * q - 9 * q^5 - 5 * q^7 $$2 q - 9 q^{5} - 5 q^{7} - 3 q^{11} + 15 q^{17} + 38 q^{19} - 60 q^{23} + 19 q^{25} - 63 q^{35} - 85 q^{43} - 75 q^{47} + 171 q^{49} - 129 q^{55} - 103 q^{61} + 25 q^{73} + 435 q^{77} + 180 q^{83} - 267 q^{85} - 171 q^{95}+O(q^{100})$$ 2 * q - 9 * q^5 - 5 * q^7 - 3 * q^11 + 15 * q^17 + 38 * q^19 - 60 * q^23 + 19 * q^25 - 63 * q^35 - 85 * q^43 - 75 * q^47 + 171 * q^49 - 129 * q^55 - 103 * q^61 + 25 * q^73 + 435 * q^77 + 180 * q^83 - 267 * q^85 - 171 * 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
 4.27492 −3.27492
0 0 0 −8.27492 0 8.82475 0 0 0
721.2 0 0 0 −0.725083 0 −13.8248 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 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.c 2
3.b odd 2 1 304.3.e.e 2
4.b odd 2 1 684.3.h.a 2
12.b even 2 1 76.3.c.b 2
19.b odd 2 1 CM 2736.3.o.c 2
24.f even 2 1 1216.3.e.f 2
24.h odd 2 1 1216.3.e.e 2
57.d even 2 1 304.3.e.e 2
60.h even 2 1 1900.3.e.a 2
60.l odd 4 2 1900.3.g.a 4
76.d even 2 1 684.3.h.a 2
228.b odd 2 1 76.3.c.b 2
456.l odd 2 1 1216.3.e.f 2
456.p even 2 1 1216.3.e.e 2
1140.p odd 2 1 1900.3.e.a 2
1140.w even 4 2 1900.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 12.b even 2 1
76.3.c.b 2 228.b odd 2 1
304.3.e.e 2 3.b odd 2 1
304.3.e.e 2 57.d even 2 1
684.3.h.a 2 4.b odd 2 1
684.3.h.a 2 76.d even 2 1
1216.3.e.e 2 24.h odd 2 1
1216.3.e.e 2 456.p even 2 1
1216.3.e.f 2 24.f even 2 1
1216.3.e.f 2 456.l odd 2 1
1900.3.e.a 2 60.h even 2 1
1900.3.e.a 2 1140.p odd 2 1
1900.3.g.a 4 60.l odd 4 2
1900.3.g.a 4 1140.w even 4 2
2736.3.o.c 2 1.a even 1 1 trivial
2736.3.o.c 2 19.b odd 2 1 CM

## Hecke kernels

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

 $$T_{5}^{2} + 9T_{5} + 6$$ T5^2 + 9*T5 + 6 $$T_{7}^{2} + 5T_{7} - 122$$ T7^2 + 5*T7 - 122

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9T + 6$$
$7$ $$T^{2} + 5T - 122$$
$11$ $$T^{2} + 3T - 354$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 15T - 642$$
$19$ $$(T - 19)^{2}$$
$23$ $$(T + 30)^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 85T + 1678$$
$47$ $$T^{2} + 75T - 1002$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 103T - 554$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 25T - 15362$$
$79$ $$T^{2}$$
$83$ $$(T - 90)^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$