# Properties

 Label 2736.3.o.i Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

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: $$\Q(\sqrt{-29})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 29$$ x^2 + 29 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 76) 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 $$\beta = 3\sqrt{-29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + q^{7}+O(q^{10})$$ q + 4 * q^5 + q^7 $$q + 4 q^{5} + q^{7} + 14 q^{11} + \beta q^{13} - 23 q^{17} + ( - \beta - 10) q^{19} - q^{23} - 9 q^{25} + 3 \beta q^{29} + 2 \beta q^{31} + 4 q^{35} + 2 \beta q^{37} - 2 \beta q^{41} - 68 q^{43} + 26 q^{47} - 48 q^{49} + 5 \beta q^{53} + 56 q^{55} + \beta q^{59} - 40 q^{61} + 4 \beta q^{65} + \beta q^{67} - 2 \beta q^{71} - 7 q^{73} + 14 q^{77} - 6 \beta q^{79} + 32 q^{83} - 92 q^{85} + 8 \beta q^{89} + \beta q^{91} + ( - 4 \beta - 40) q^{95} - 6 \beta q^{97} +O(q^{100})$$ q + 4 * q^5 + q^7 + 14 * q^11 + b * q^13 - 23 * q^17 + (-b - 10) * q^19 - q^23 - 9 * q^25 + 3*b * q^29 + 2*b * q^31 + 4 * q^35 + 2*b * q^37 - 2*b * q^41 - 68 * q^43 + 26 * q^47 - 48 * q^49 + 5*b * q^53 + 56 * q^55 + b * q^59 - 40 * q^61 + 4*b * q^65 + b * q^67 - 2*b * q^71 - 7 * q^73 + 14 * q^77 - 6*b * q^79 + 32 * q^83 - 92 * q^85 + 8*b * q^89 + b * q^91 + (-4*b - 40) * q^95 - 6*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 8 * q^5 + 2 * q^7 $$2 q + 8 q^{5} + 2 q^{7} + 28 q^{11} - 46 q^{17} - 20 q^{19} - 2 q^{23} - 18 q^{25} + 8 q^{35} - 136 q^{43} + 52 q^{47} - 96 q^{49} + 112 q^{55} - 80 q^{61} - 14 q^{73} + 28 q^{77} + 64 q^{83} - 184 q^{85} - 80 q^{95}+O(q^{100})$$ 2 * q + 8 * q^5 + 2 * q^7 + 28 * q^11 - 46 * q^17 - 20 * q^19 - 2 * q^23 - 18 * q^25 + 8 * q^35 - 136 * q^43 + 52 * q^47 - 96 * q^49 + 112 * q^55 - 80 * q^61 - 14 * q^73 + 28 * q^77 + 64 * q^83 - 184 * q^85 - 80 * 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
 − 5.38516i 5.38516i
0 0 0 4.00000 0 1.00000 0 0 0
721.2 0 0 0 4.00000 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

## 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.i 2
3.b odd 2 1 304.3.e.b 2
4.b odd 2 1 684.3.h.c 2
12.b even 2 1 76.3.c.a 2
19.b odd 2 1 inner 2736.3.o.i 2
24.f even 2 1 1216.3.e.k 2
24.h odd 2 1 1216.3.e.l 2
57.d even 2 1 304.3.e.b 2
60.h even 2 1 1900.3.e.b 2
60.l odd 4 2 1900.3.g.b 4
76.d even 2 1 684.3.h.c 2
228.b odd 2 1 76.3.c.a 2
456.l odd 2 1 1216.3.e.k 2
456.p even 2 1 1216.3.e.l 2
1140.p odd 2 1 1900.3.e.b 2
1140.w even 4 2 1900.3.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 12.b even 2 1
76.3.c.a 2 228.b odd 2 1
304.3.e.b 2 3.b odd 2 1
304.3.e.b 2 57.d even 2 1
684.3.h.c 2 4.b odd 2 1
684.3.h.c 2 76.d even 2 1
1216.3.e.k 2 24.f even 2 1
1216.3.e.k 2 456.l odd 2 1
1216.3.e.l 2 24.h odd 2 1
1216.3.e.l 2 456.p even 2 1
1900.3.e.b 2 60.h even 2 1
1900.3.e.b 2 1140.p odd 2 1
1900.3.g.b 4 60.l odd 4 2
1900.3.g.b 4 1140.w even 4 2
2736.3.o.i 2 1.a even 1 1 trivial
2736.3.o.i 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 $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 4)^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 14)^{2}$$
$13$ $$T^{2} + 261$$
$17$ $$(T + 23)^{2}$$
$19$ $$T^{2} + 20T + 361$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 2349$$
$31$ $$T^{2} + 1044$$
$37$ $$T^{2} + 1044$$
$41$ $$T^{2} + 1044$$
$43$ $$(T + 68)^{2}$$
$47$ $$(T - 26)^{2}$$
$53$ $$T^{2} + 6525$$
$59$ $$T^{2} + 261$$
$61$ $$(T + 40)^{2}$$
$67$ $$T^{2} + 261$$
$71$ $$T^{2} + 1044$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} + 9396$$
$83$ $$(T - 32)^{2}$$
$89$ $$T^{2} + 16704$$
$97$ $$T^{2} + 9396$$
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