# Properties

 Label 2736.3.o.l Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 57) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 3) q^{5} + ( - \beta_1 - 9) q^{7}+O(q^{10})$$ q + (b1 + 3) * q^5 + (-b1 - 9) * q^7 $$q + (\beta_1 + 3) q^{5} + ( - \beta_1 - 9) q^{7} + (\beta_1 - 3) q^{11} + (2 \beta_{3} - \beta_{2}) q^{13} + ( - 5 \beta_1 - 1) q^{17} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{19} + (2 \beta_1 + 14) q^{23} + (5 \beta_1 - 2) q^{25} + (4 \beta_{3} - 2 \beta_{2}) q^{29} + ( - 4 \beta_{3} - \beta_{2}) q^{31} + ( - 11 \beta_1 - 41) q^{35} + ( - 10 \beta_{3} + 7 \beta_{2}) q^{37} + (\beta_{3} + 3 \beta_{2}) q^{41} + (15 \beta_1 - 1) q^{43} + ( - 9 \beta_1 + 17) q^{47} + (17 \beta_1 + 46) q^{49} + ( - 3 \beta_{3} + 9 \beta_{2}) q^{53} + ( - \beta_1 + 5) q^{55} + ( - \beta_{3} - 7 \beta_{2}) q^{59} + ( - 3 \beta_1 - 19) q^{61} + (13 \beta_{3} - 3 \beta_{2}) q^{65} + ( - 18 \beta_{3} - 8 \beta_{2}) q^{67} + ( - 17 \beta_{3} + 5 \beta_{2}) q^{71} + ( - 3 \beta_1 + 21) q^{73} + ( - 5 \beta_1 + 13) q^{77} + ( - 3 \beta_{3} + 10 \beta_{2}) q^{79} + ( - 28 \beta_1 + 2) q^{83} + ( - 11 \beta_1 - 73) q^{85} + ( - 23 \beta_{3} - \beta_{2}) q^{89} + ( - 25 \beta_{3} + 9 \beta_{2}) q^{91} + (17 \beta_{3} - 7 \beta_{2} - 5 \beta_1 - 31) q^{95} + (5 \beta_{3} - 13 \beta_{2}) q^{97}+O(q^{100})$$ q + (b1 + 3) * q^5 + (-b1 - 9) * q^7 + (b1 - 3) * q^11 + (2*b3 - b2) * q^13 + (-5*b1 - 1) * q^17 + (3*b3 + b2 - 2*b1 - 1) * q^19 + (2*b1 + 14) * q^23 + (5*b1 - 2) * q^25 + (4*b3 - 2*b2) * q^29 + (-4*b3 - b2) * q^31 + (-11*b1 - 41) * q^35 + (-10*b3 + 7*b2) * q^37 + (b3 + 3*b2) * q^41 + (15*b1 - 1) * q^43 + (-9*b1 + 17) * q^47 + (17*b1 + 46) * q^49 + (-3*b3 + 9*b2) * q^53 + (-b1 + 5) * q^55 + (-b3 - 7*b2) * q^59 + (-3*b1 - 19) * q^61 + (13*b3 - 3*b2) * q^65 + (-18*b3 - 8*b2) * q^67 + (-17*b3 + 5*b2) * q^71 + (-3*b1 + 21) * q^73 + (-5*b1 + 13) * q^77 + (-3*b3 + 10*b2) * q^79 + (-28*b1 + 2) * q^83 + (-11*b1 - 73) * q^85 + (-23*b3 - b2) * q^89 + (-25*b3 + 9*b2) * q^91 + (17*b3 - 7*b2 - 5*b1 - 31) * q^95 + (5*b3 - 13*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 10 q^{5} - 34 q^{7}+O(q^{10})$$ 4 * q + 10 * q^5 - 34 * q^7 $$4 q + 10 q^{5} - 34 q^{7} - 14 q^{11} + 6 q^{17} + 52 q^{23} - 18 q^{25} - 142 q^{35} - 34 q^{43} + 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 90 q^{73} + 62 q^{77} + 64 q^{83} - 270 q^{85} - 114 q^{95}+O(q^{100})$$ 4 * q + 10 * q^5 - 34 * q^7 - 14 * q^11 + 6 * q^17 + 52 * q^23 - 18 * q^25 - 142 * q^35 - 34 * q^43 + 86 * q^47 + 150 * q^49 + 22 * q^55 - 70 * q^61 + 90 * q^73 + 62 * q^77 + 64 * q^83 - 270 * q^85 - 114 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu ) / 5$$ (-v^3 + v^2 + 9*v) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 6\nu^{2} + 6\nu + 5 ) / 5$$ (v^3 - 6*v^2 + 6*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( 3\nu^{3} + 2\nu^{2} - 2\nu - 25 ) / 5$$ (3*v^3 + 2*v^2 - 2*v - 25) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 4\beta _1 + 4 ) / 8$$ (b3 + b2 + 4*b1 + 4) / 8 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} - 5\beta_{2} + 4\beta _1 + 20 ) / 8$$ (3*b3 - 5*b2 + 4*b1 + 20) / 8 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + \beta_{2} + 14 ) / 2$$ (3*b3 + b2 + 14) / 2

## 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
 −1.63746 + 1.52274i −1.63746 − 1.52274i 2.13746 − 0.656712i 2.13746 + 0.656712i
0 0 0 −1.27492 0 −4.72508 0 0 0
721.2 0 0 0 −1.27492 0 −4.72508 0 0 0
721.3 0 0 0 6.27492 0 −12.2749 0 0 0
721.4 0 0 0 6.27492 0 −12.2749 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.l 4
3.b odd 2 1 912.3.o.b 4
4.b odd 2 1 171.3.c.f 4
12.b even 2 1 57.3.c.b 4
19.b odd 2 1 inner 2736.3.o.l 4
57.d even 2 1 912.3.o.b 4
76.d even 2 1 171.3.c.f 4
228.b odd 2 1 57.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.b 4 12.b even 2 1
57.3.c.b 4 228.b odd 2 1
171.3.c.f 4 4.b odd 2 1
171.3.c.f 4 76.d even 2 1
912.3.o.b 4 3.b odd 2 1
912.3.o.b 4 57.d even 2 1
2736.3.o.l 4 1.a even 1 1 trivial
2736.3.o.l 4 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 $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{2} - 5T_{5} - 8$$ T5^2 - 5*T5 - 8 $$T_{7}^{2} + 17T_{7} + 58$$ T7^2 + 17*T7 + 58

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 5 T - 8)^{2}$$
$7$ $$(T^{2} + 17 T + 58)^{2}$$
$11$ $$(T^{2} + 7 T - 2)^{2}$$
$13$ $$T^{4} + 188T^{2} + 3136$$
$17$ $$(T^{2} - 3 T - 354)^{2}$$
$19$ $$T^{4} + 494 T^{2} + 130321$$
$23$ $$(T^{2} - 26 T + 112)^{2}$$
$29$ $$T^{4} + 752 T^{2} + 50176$$
$31$ $$T^{4} + 956 T^{2} + 222784$$
$37$ $$T^{4} + 6108 T^{2} + \cdots + 7354944$$
$41$ $$T^{4} + 992 T^{2} + 12544$$
$43$ $$(T^{2} + 17 T - 3134)^{2}$$
$47$ $$(T^{2} - 43 T - 692)^{2}$$
$53$ $$T^{4} + 6768 T^{2} + \cdots + 4064256$$
$59$ $$T^{4} + 4832T^{2} + 256$$
$61$ $$(T^{2} + 35 T + 178)^{2}$$
$67$ $$T^{4} + 25904 T^{2} + \cdots + 162205696$$
$71$ $$T^{4} + 11616 T^{2} + \cdots + 112896$$
$73$ $$(T^{2} - 45 T + 378)^{2}$$
$79$ $$T^{4} + 8396 T^{2} + \cdots + 5456896$$
$83$ $$(T^{2} - 32 T - 10916)^{2}$$
$89$ $$T^{4} + 24288 T^{2} + \cdots + 94945536$$
$97$ $$T^{4} + 14048 T^{2} + \cdots + 21086464$$