# Properties

 Label 2736.2.bm.o Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bm (of order $$6$$, degree $$2$$, 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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.954288.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7}+O(q^{10})$$ q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2*b3 + b1 + 1) * q^7 $$q + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{5} + ( - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{5} - \beta_1) q^{11} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{13} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{23} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{25} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2) q^{29} - \beta_{2} q^{31} + (2 \beta_{4} + 2 \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{37} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{41} + ( - \beta_{4} - 4 \beta_{3} - \beta_{2} + 4) q^{43} + (4 \beta_{5} - 2 \beta_{3} - 4) q^{47} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{49} + ( - \beta_{5} - 4 \beta_{3} - 8) q^{53} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{59} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{61} + ( - 4 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} - \beta_{2} + 4 \beta_1 + 5) q^{65} + ( - \beta_{5} + 2 \beta_{4} + 7 \beta_{3} + 2 \beta_1) q^{67} + (2 \beta_{5} + \beta_{4} - 5 \beta_{3} - \beta_{2} - \beta_1 - 5) q^{71} + (4 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{73} + (\beta_{2} + 5) q^{77} + ( - 3 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 4) q^{79} + (3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{83} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 4 \beta_1) q^{85} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{2}) q^{89} + (5 \beta_{4} + 2 \beta_{3}) q^{91} + (4 \beta_{4} - 2 \beta_{3} + 3 \beta_1 - 6) q^{95} + (\beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{97}+O(q^{100})$$ q + (b4 + b3 - b2 + 1) * q^5 + (-b5 + 2*b3 + b1 + 1) * q^7 + (b5 - b1) * q^11 + (-b5 + b4 - 2*b2) * q^13 + (-2*b5 + b4 + b3 - b2 + b1 + 1) * q^17 + (-2*b5 + b4 + b3 + b1 + 1) * q^19 + (2*b5 - b4 + b3 + 2*b2 + 2) * q^23 + (-b5 + b4 + 2*b3 + 2*b1) * q^25 + (b5 - b4 - b3 + 2*b2 - 2) * q^29 - b2 * q^31 + (2*b4 + 2*b2 + b1) * q^35 + (-2*b5 + 4*b4 - 2*b3 - 2*b2 + 2*b1 - 1) * q^37 + (-2*b4 - 2*b3 - 2*b2 + 2) * q^41 + (-b4 - 4*b3 - b2 + 4) * q^43 + (4*b5 - 2*b3 - 4) * q^47 + (b5 - b2 + b1 - 1) * q^49 + (-b5 - 4*b3 - 8) * q^53 + (-b4 + b3 - b2 - b1 - 1) * q^55 + (b4 + b3 - b2 + 1) * q^59 + (-b5 - 3*b4 - 2*b3 + 2*b1) * q^61 + (-4*b5 + 2*b4 + 10*b3 - b2 + 4*b1 + 5) * q^65 + (-b5 + 2*b4 + 7*b3 + 2*b1) * q^67 + (2*b5 + b4 - 5*b3 - b2 - b1 - 5) * q^71 + (4*b5 - 2*b4 + 3*b3 + 2*b2 - 2*b1 + 3) * q^73 + (b2 + 5) * q^77 + (-3*b4 - 4*b3 + 3*b2 - 4) * q^79 + (3*b5 - 2*b4 - 2*b3 + b2 - 3*b1 - 1) * q^83 + (-2*b5 + 4*b4 + 4*b3 + 4*b1) * q^85 + (b5 + 2*b4 - 4*b2) * q^89 + (5*b4 + 2*b3) * q^91 + (4*b4 - 2*b3 + 3*b1 - 6) * q^95 + (b4 + 3*b3 + b2 - b1 - 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{5}+O(q^{10})$$ 6 * q + 2 * q^5 $$6 q + 2 q^{5} - 3 q^{13} + 2 q^{17} + 4 q^{19} + 12 q^{23} - 5 q^{25} - 6 q^{29} - 2 q^{31} + 6 q^{35} + 12 q^{41} + 33 q^{43} - 18 q^{47} - 8 q^{49} - 36 q^{53} - 12 q^{55} + 2 q^{59} + 3 q^{61} - 19 q^{67} - 16 q^{71} + 11 q^{73} + 32 q^{77} - 9 q^{79} - 8 q^{85} - 6 q^{89} - q^{91} - 26 q^{95} - 24 q^{97}+O(q^{100})$$ 6 * q + 2 * q^5 - 3 * q^13 + 2 * q^17 + 4 * q^19 + 12 * q^23 - 5 * q^25 - 6 * q^29 - 2 * q^31 + 6 * q^35 + 12 * q^41 + 33 * q^43 - 18 * q^47 - 8 * q^49 - 36 * q^53 - 12 * q^55 + 2 * q^59 + 3 * q^61 - 19 * q^67 - 16 * q^71 + 11 * q^73 + 32 * q^77 - 9 * q^79 - 8 * q^85 - 6 * q^89 - q^91 - 26 * q^95 - 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} + 21\nu - 45 ) / 27$$ (v^5 + 5*v^4 + v^3 + 9*v^2 + 21*v - 45) / 27 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 4\nu^{4} + \nu^{3} + 18\nu^{2} - 33\nu + 9 ) / 27$$ (v^5 - 4*v^4 + v^3 + 18*v^2 - 33*v + 9) / 27 $$\beta_{3}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27$$ (-2*v^5 - v^4 - 2*v^3 + 12*v + 9) / 27 $$\beta_{4}$$ $$=$$ $$( 2\nu^{5} - 8\nu^{4} + 2\nu^{3} + 9\nu^{2} - 12\nu + 18 ) / 27$$ (2*v^5 - 8*v^4 + 2*v^3 + 9*v^2 - 12*v + 18) / 27 $$\beta_{5}$$ $$=$$ $$( -2\nu^{5} - \nu^{4} + 4\nu^{3} - 3\nu^{2} + 12\nu + 27 ) / 9$$ (-2*v^5 - v^4 + 4*v^3 - 3*v^2 + 12*v + 27) / 9
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2$$ (b4 + b3 - b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 + 1$$ b3 + b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$( 3\beta_{5} - 8\beta_{3} + \beta_{2} + \beta _1 - 5 ) / 2$$ (3*b5 - 8*b3 + b2 + b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$-3\beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 4$$ -3*b4 - 2*b3 + b2 + b1 + 4 $$\nu^{5}$$ $$=$$ $$( -3\beta_{5} + 9\beta_{4} - 11\beta_{3} - 8\beta_{2} + 4\beta _1 + 16 ) / 2$$ (-3*b5 + 9*b4 - 11*b3 - 8*b2 + 4*b1 + 16) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −1.62241 + 0.606458i 1.71903 + 0.211943i 0.403374 − 1.68443i −1.62241 − 0.606458i 1.71903 − 0.211943i 0.403374 + 1.68443i
0 0 0 −1.33641 + 2.31473i 0 3.93569i 0 0 0
559.2 0 0 0 0.675970 1.17081i 0 1.45735i 0 0 0
559.3 0 0 0 1.66044 2.87597i 0 2.71781i 0 0 0
1855.1 0 0 0 −1.33641 2.31473i 0 3.93569i 0 0 0
1855.2 0 0 0 0.675970 + 1.17081i 0 1.45735i 0 0 0
1855.3 0 0 0 1.66044 + 2.87597i 0 2.71781i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.o 6
3.b odd 2 1 912.2.bb.f yes 6
4.b odd 2 1 2736.2.bm.n 6
12.b even 2 1 912.2.bb.e 6
19.d odd 6 1 2736.2.bm.n 6
57.f even 6 1 912.2.bb.e 6
76.f even 6 1 inner 2736.2.bm.o 6
228.n odd 6 1 912.2.bb.f yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.e 6 12.b even 2 1
912.2.bb.e 6 57.f even 6 1
912.2.bb.f yes 6 3.b odd 2 1
912.2.bb.f yes 6 228.n odd 6 1
2736.2.bm.n 6 4.b odd 2 1
2736.2.bm.n 6 19.d odd 6 1
2736.2.bm.o 6 1.a even 1 1 trivial
2736.2.bm.o 6 76.f even 6 1 inner

## Hecke kernels

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

 $$T_{5}^{6} - 2T_{5}^{5} + 12T_{5}^{4} - 8T_{5}^{3} + 88T_{5}^{2} - 96T_{5} + 144$$ T5^6 - 2*T5^5 + 12*T5^4 - 8*T5^3 + 88*T5^2 - 96*T5 + 144 $$T_{7}^{6} + 25T_{7}^{4} + 163T_{7}^{2} + 243$$ T7^6 + 25*T7^4 + 163*T7^2 + 243 $$T_{11}^{6} + 16T_{11}^{4} + 64T_{11}^{2} + 48$$ T11^6 + 16*T11^4 + 64*T11^2 + 48 $$T_{23}^{6} - 12T_{23}^{5} + 28T_{23}^{4} + 240T_{23}^{3} - 416T_{23}^{2} - 4080T_{23} + 13872$$ T23^6 - 12*T23^5 + 28*T23^4 + 240*T23^3 - 416*T23^2 - 4080*T23 + 13872 $$T_{31}^{3} + T_{31}^{2} - 9T_{31} + 3$$ T31^3 + T31^2 - 9*T31 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144$$
$7$ $$T^{6} + 25 T^{4} + 163 T^{2} + \cdots + 243$$
$11$ $$T^{6} + 16 T^{4} + 64 T^{2} + 48$$
$13$ $$T^{6} + 3 T^{5} - 20 T^{4} + \cdots + 2883$$
$17$ $$T^{6} - 2 T^{5} + 24 T^{4} - 8 T^{3} + \cdots + 576$$
$19$ $$T^{6} - 4 T^{5} + 17 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 12 T^{5} + 28 T^{4} + \cdots + 13872$$
$29$ $$T^{6} + 6 T^{5} - 8 T^{4} - 120 T^{3} + \cdots + 192$$
$31$ $$(T^{3} + T^{2} - 9 T + 3)^{2}$$
$37$ $$T^{6} + 217 T^{4} + 14971 T^{2} + \cdots + 328683$$
$41$ $$T^{6} - 12 T^{5} - 48 T^{4} + \cdots + 248832$$
$43$ $$T^{6} - 33 T^{5} + 456 T^{4} + \cdots + 2187$$
$47$ $$T^{6} + 18 T^{5} + 16 T^{4} + \cdots + 714432$$
$53$ $$T^{6} + 36 T^{5} + 568 T^{4} + \cdots + 80688$$
$59$ $$T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144$$
$61$ $$T^{6} - 3 T^{5} + 150 T^{4} + \cdots + 638401$$
$67$ $$T^{6} + 19 T^{5} + 278 T^{4} + \cdots + 10201$$
$71$ $$T^{6} + 16 T^{5} + 216 T^{4} + \cdots + 2304$$
$73$ $$T^{6} - 11 T^{5} + 166 T^{4} + \cdots + 33489$$
$79$ $$T^{6} + 9 T^{5} + 138 T^{4} + \cdots + 151321$$
$83$ $$T^{6} + 132 T^{4} + 4464 T^{2} + \cdots + 15552$$
$89$ $$T^{6} + 6 T^{5} - 128 T^{4} + \cdots + 215472$$
$97$ $$T^{6} + 24 T^{5} + 208 T^{4} + \cdots + 62208$$