# Properties

 Label 2736.2.a.n Level $2736$ Weight $2$ Character orbit 2736.a Self dual yes Analytic conductor $21.847$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.a (trivial)

## 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: $$21.8470699930$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{7}+O(q^{10})$$ q + q^7 $$q + q^{7} - 6 q^{11} + 5 q^{13} - 3 q^{17} - q^{19} + 3 q^{23} - 5 q^{25} - 9 q^{29} + 4 q^{31} + 2 q^{37} - 8 q^{43} - 6 q^{49} + 3 q^{53} + 9 q^{59} - 10 q^{61} - 5 q^{67} - 6 q^{71} - 7 q^{73} - 6 q^{77} + 10 q^{79} - 6 q^{83} + 12 q^{89} + 5 q^{91} - 10 q^{97}+O(q^{100})$$ q + q^7 - 6 * q^11 + 5 * q^13 - 3 * q^17 - q^19 + 3 * q^23 - 5 * q^25 - 9 * q^29 + 4 * q^31 + 2 * q^37 - 8 * q^43 - 6 * q^49 + 3 * q^53 + 9 * q^59 - 10 * q^61 - 5 * q^67 - 6 * q^71 - 7 * q^73 - 6 * q^77 + 10 * q^79 - 6 * q^83 + 12 * q^89 + 5 * q^91 - 10 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.a.n 1
3.b odd 2 1 304.2.a.c 1
4.b odd 2 1 342.2.a.e 1
12.b even 2 1 38.2.a.a 1
15.d odd 2 1 7600.2.a.n 1
20.d odd 2 1 8550.2.a.m 1
24.f even 2 1 1216.2.a.e 1
24.h odd 2 1 1216.2.a.m 1
57.d even 2 1 5776.2.a.m 1
60.h even 2 1 950.2.a.d 1
60.l odd 4 2 950.2.b.b 2
76.d even 2 1 6498.2.a.f 1
84.h odd 2 1 1862.2.a.b 1
132.d odd 2 1 4598.2.a.p 1
156.h even 2 1 6422.2.a.h 1
228.b odd 2 1 722.2.a.e 1
228.m even 6 2 722.2.c.e 2
228.n odd 6 2 722.2.c.c 2
228.u odd 18 6 722.2.e.e 6
228.v even 18 6 722.2.e.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.a.a 1 12.b even 2 1
304.2.a.c 1 3.b odd 2 1
342.2.a.e 1 4.b odd 2 1
722.2.a.e 1 228.b odd 2 1
722.2.c.c 2 228.n odd 6 2
722.2.c.e 2 228.m even 6 2
722.2.e.e 6 228.u odd 18 6
722.2.e.f 6 228.v even 18 6
950.2.a.d 1 60.h even 2 1
950.2.b.b 2 60.l odd 4 2
1216.2.a.e 1 24.f even 2 1
1216.2.a.m 1 24.h odd 2 1
1862.2.a.b 1 84.h odd 2 1
2736.2.a.n 1 1.a even 1 1 trivial
4598.2.a.p 1 132.d odd 2 1
5776.2.a.m 1 57.d even 2 1
6422.2.a.h 1 156.h even 2 1
6498.2.a.f 1 76.d even 2 1
7600.2.a.n 1 15.d odd 2 1
8550.2.a.m 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2736))$$:

 $$T_{5}$$ T5 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 6$$ T11 + 6 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 6$$
$13$ $$T - 5$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T - 3$$
$29$ $$T + 9$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T - 3$$
$59$ $$T - 9$$
$61$ $$T + 10$$
$67$ $$T + 5$$
$71$ $$T + 6$$
$73$ $$T + 7$$
$79$ $$T - 10$$
$83$ $$T + 6$$
$89$ $$T - 12$$
$97$ $$T + 10$$