Properties

 Label 2736.3.o.a Level $2736$ Weight $3$ Character orbit 2736.o Self dual yes Analytic conductor $74.551$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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)

 $$f(q)$$ $$=$$ $$q + 9 q^{5} + 5 q^{7}+O(q^{10})$$ q + 9 * q^5 + 5 * q^7 $$q + 9 q^{5} + 5 q^{7} + 3 q^{11} - 15 q^{17} + 19 q^{19} - 30 q^{23} + 56 q^{25} + 45 q^{35} + 85 q^{43} + 75 q^{47} - 24 q^{49} + 27 q^{55} + 103 q^{61} - 25 q^{73} + 15 q^{77} + 90 q^{83} - 135 q^{85} + 171 q^{95}+O(q^{100})$$ q + 9 * q^5 + 5 * q^7 + 3 * q^11 - 15 * q^17 + 19 * q^19 - 30 * q^23 + 56 * q^25 + 45 * q^35 + 85 * q^43 + 75 * q^47 - 24 * q^49 + 27 * q^55 + 103 * q^61 - 25 * q^73 + 15 * q^77 + 90 * q^83 - 135 * 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
 0
0 0 0 9.00000 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 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.a 1
3.b odd 2 1 304.3.e.a 1
4.b odd 2 1 171.3.c.a 1
12.b even 2 1 19.3.b.a 1
19.b odd 2 1 CM 2736.3.o.a 1
24.f even 2 1 1216.3.e.a 1
24.h odd 2 1 1216.3.e.b 1
57.d even 2 1 304.3.e.a 1
60.h even 2 1 475.3.c.a 1
60.l odd 4 2 475.3.d.a 2
76.d even 2 1 171.3.c.a 1
228.b odd 2 1 19.3.b.a 1
228.m even 6 2 361.3.d.a 2
228.n odd 6 2 361.3.d.a 2
228.u odd 18 6 361.3.f.a 6
228.v even 18 6 361.3.f.a 6
456.l odd 2 1 1216.3.e.a 1
456.p even 2 1 1216.3.e.b 1
1140.p odd 2 1 475.3.c.a 1
1140.w even 4 2 475.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 12.b even 2 1
19.3.b.a 1 228.b odd 2 1
171.3.c.a 1 4.b odd 2 1
171.3.c.a 1 76.d even 2 1
304.3.e.a 1 3.b odd 2 1
304.3.e.a 1 57.d even 2 1
361.3.d.a 2 228.m even 6 2
361.3.d.a 2 228.n odd 6 2
361.3.f.a 6 228.u odd 18 6
361.3.f.a 6 228.v even 18 6
475.3.c.a 1 60.h even 2 1
475.3.c.a 1 1140.p odd 2 1
475.3.d.a 2 60.l odd 4 2
475.3.d.a 2 1140.w even 4 2
1216.3.e.a 1 24.f even 2 1
1216.3.e.a 1 456.l odd 2 1
1216.3.e.b 1 24.h odd 2 1
1216.3.e.b 1 456.p even 2 1
2736.3.o.a 1 1.a even 1 1 trivial
2736.3.o.a 1 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 $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 9$$ T5 - 9 $$T_{7} - 5$$ T7 - 5

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 9$$
$7$ $$T - 5$$
$11$ $$T - 3$$
$13$ $$T$$
$17$ $$T + 15$$
$19$ $$T - 19$$
$23$ $$T + 30$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 85$$
$47$ $$T - 75$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 103$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 25$$
$79$ $$T$$
$83$ $$T - 90$$
$89$ $$T$$
$97$ $$T$$