# Properties

 Label 2736.2.bm.p Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.897122304.10 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169$$ x^8 - 8*x^6 + 51*x^4 - 104*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_1) q^{7}+O(q^{10})$$ q + (-b7 - b6 + b3 + b2) * q^5 + (b5 - b1) * q^7 $$q + ( - \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{5} - \beta_1) q^{7} + (\beta_{7} + \beta_{6} - \beta_{2}) q^{11} + (\beta_{4} - 1) q^{13} + ( - \beta_{7} - \beta_{6}) q^{17} + ( - 3 \beta_{5} + 5 \beta_1) q^{19} + ( - \beta_{6} - \beta_{2}) q^{23} + ( - 8 \beta_{5} - 5 \beta_{4} + 4 \beta_1 - 5) q^{25} + ( - \beta_{7} - 2 \beta_{6} + \beta_{3} - \beta_{2}) q^{29} + (\beta_{5} + \beta_1 - 6) q^{31} + (\beta_{3} - \beta_{2}) q^{35} + ( - 6 \beta_{5} + 2 \beta_{4} + 6 \beta_1 + 1) q^{37} + (3 \beta_{5} + 2 \beta_{4} + 4) q^{43} + ( - 2 \beta_{7} + 2 \beta_{3} + 2 \beta_{2}) q^{47} + 6 q^{49} + ( - \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2}) q^{53} + (10 \beta_{5} + 4 \beta_{4} + 8) q^{55} + (\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2}) q^{59} + ( - 8 \beta_{5} + 3 \beta_{4} + 4 \beta_1 + 3) q^{61} + (\beta_{7} + 2 \beta_{6} - \beta_{3} - 2 \beta_{2}) q^{65} + ( - 6 \beta_{5} - 6 \beta_{4} + 3 \beta_1 - 6) q^{67} + ( - 3 \beta_{7} - 3 \beta_{6}) q^{71} + ( - 6 \beta_{5} + 5 \beta_{4} + 12 \beta_1) q^{73} + (\beta_{7} - \beta_{3}) q^{77} + ( - 5 \beta_{5} - 6 \beta_{4} + 10 \beta_1) q^{79} + (2 \beta_{7} + 3 \beta_{6} - \beta_{3} - 3 \beta_{2}) q^{83} + ( - 12 \beta_{5} - 2 \beta_{4} + 6 \beta_1 - 2) q^{85} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_{3}) q^{89} + ( - 2 \beta_{5} + \beta_1) q^{91} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{3} + 5 \beta_{2}) q^{95} - 6 \beta_{5} q^{97}+O(q^{100})$$ q + (-b7 - b6 + b3 + b2) * q^5 + (b5 - b1) * q^7 + (b7 + b6 - b2) * q^11 + (b4 - 1) * q^13 + (-b7 - b6) * q^17 + (-3*b5 + 5*b1) * q^19 + (-b6 - b2) * q^23 + (-8*b5 - 5*b4 + 4*b1 - 5) * q^25 + (-b7 - 2*b6 + b3 - b2) * q^29 + (b5 + b1 - 6) * q^31 + (b3 - b2) * q^35 + (-6*b5 + 2*b4 + 6*b1 + 1) * q^37 + (3*b5 + 2*b4 + 4) * q^43 + (-2*b7 + 2*b3 + 2*b2) * q^47 + 6 * q^49 + (-b7 + b6 + b3 + 2*b2) * q^53 + (10*b5 + 4*b4 + 8) * q^55 + (b7 + b6 + b3 + b2) * q^59 + (-8*b5 + 3*b4 + 4*b1 + 3) * q^61 + (b7 + 2*b6 - b3 - 2*b2) * q^65 + (-6*b5 - 6*b4 + 3*b1 - 6) * q^67 + (-3*b7 - 3*b6) * q^71 + (-6*b5 + 5*b4 + 12*b1) * q^73 + (b7 - b3) * q^77 + (-5*b5 - 6*b4 + 10*b1) * q^79 + (2*b7 + 3*b6 - b3 - 3*b2) * q^83 + (-12*b5 - 2*b4 + 6*b1 - 2) * q^85 + (3*b7 + 3*b6 - 3*b3) * q^89 + (-2*b5 + b1) * q^91 + (-2*b7 - 2*b6 - 3*b3 + 5*b2) * q^95 - 6*b5 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 12 q^{13} - 20 q^{25} - 48 q^{31} + 24 q^{43} + 48 q^{49} + 48 q^{55} + 12 q^{61} - 24 q^{67} - 20 q^{73} + 24 q^{79} - 8 q^{85}+O(q^{100})$$ 8 * q - 12 * q^13 - 20 * q^25 - 48 * q^31 + 24 * q^43 + 48 * q^49 + 48 * q^55 + 12 * q^61 - 24 * q^67 - 20 * q^73 + 24 * q^79 - 8 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{6} + 68\nu^{4} - 323\nu^{2} + 1092 ) / 663$$ (-2*v^6 + 68*v^4 - 323*v^2 + 1092) / 663 $$\beta_{2}$$ $$=$$ $$( 6\nu^{7} + 17\nu^{5} + 85\nu^{3} + 923\nu ) / 663$$ (6*v^7 + 17*v^5 + 85*v^3 + 923*v) / 663 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 17\nu^{5} + 85\nu^{3} - 1040\nu ) / 663$$ (-7*v^7 + 17*v^5 + 85*v^3 - 1040*v) / 663 $$\beta_{4}$$ $$=$$ $$( 8\nu^{6} - 51\nu^{4} + 408\nu^{2} - 832 ) / 663$$ (8*v^6 - 51*v^4 + 408*v^2 - 832) / 663 $$\beta_{5}$$ $$=$$ $$( 15\nu^{6} - 68\nu^{4} + 323\nu^{2} + 208 ) / 663$$ (15*v^6 - 68*v^4 + 323*v^2 + 208) / 663 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 8\nu^{5} - 38\nu^{3} + 13\nu ) / 39$$ (-v^7 + 8*v^5 - 38*v^3 + 13*v) / 39 $$\beta_{7}$$ $$=$$ $$( -\nu^{7} + 8\nu^{5} - 38\nu^{3} + 91\nu ) / 39$$ (-v^7 + 8*v^5 - 38*v^3 + 91*v) / 39
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} ) / 2$$ (b7 - b6) / 2 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + 4\beta_{4} + \beta _1 + 4$$ -2*b5 + 4*b4 + b1 + 4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 3\beta_{6} + 5\beta_{3} + 3\beta_{2} ) / 2$$ (2*b7 - 3*b6 + 5*b3 + 3*b2) / 2 $$\nu^{4}$$ $$=$$ $$-8\beta_{5} + 19\beta_{4} + 16\beta_1$$ -8*b5 + 19*b4 + 16*b1 $$\nu^{5}$$ $$=$$ $$( -11\beta_{7} + 16\beta_{6} + 11\beta_{3} + 27\beta_{2} ) / 2$$ (-11*b7 + 16*b6 + 11*b3 + 27*b2) / 2 $$\nu^{6}$$ $$=$$ $$51\beta_{5} + 51\beta _1 - 100$$ 51*b5 + 51*b1 - 100 $$\nu^{7}$$ $$=$$ $$( -151\beta_{7} + 151\beta_{6} - 102\beta_{3} + 102\beta_{2} ) / 2$$ (-151*b7 + 151*b6 - 102*b3 + 102*b2) / 2

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$1 + \beta_{4}$$ $$1$$ $$-1$$ $$1$$

## 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
 1.30421 + 0.752986i 2.07341 + 1.19709i −2.07341 − 1.19709i −1.30421 − 0.752986i 1.30421 − 0.752986i 2.07341 − 1.19709i −2.07341 + 1.19709i −1.30421 + 0.752986i
0 0 0 −2.05719 + 3.56317i 0 1.00000i 0 0 0
559.2 0 0 0 −0.876327 + 1.51784i 0 1.00000i 0 0 0
559.3 0 0 0 0.876327 1.51784i 0 1.00000i 0 0 0
559.4 0 0 0 2.05719 3.56317i 0 1.00000i 0 0 0
1855.1 0 0 0 −2.05719 3.56317i 0 1.00000i 0 0 0
1855.2 0 0 0 −0.876327 1.51784i 0 1.00000i 0 0 0
1855.3 0 0 0 0.876327 + 1.51784i 0 1.00000i 0 0 0
1855.4 0 0 0 2.05719 + 3.56317i 0 1.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
76.f even 6 1 inner
228.n odd 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.p 8
3.b odd 2 1 inner 2736.2.bm.p 8
4.b odd 2 1 2736.2.bm.q yes 8
12.b even 2 1 2736.2.bm.q yes 8
19.d odd 6 1 2736.2.bm.q yes 8
57.f even 6 1 2736.2.bm.q yes 8
76.f even 6 1 inner 2736.2.bm.p 8
228.n odd 6 1 inner 2736.2.bm.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.p 8 1.a even 1 1 trivial
2736.2.bm.p 8 3.b odd 2 1 inner
2736.2.bm.p 8 76.f even 6 1 inner
2736.2.bm.p 8 228.n odd 6 1 inner
2736.2.bm.q yes 8 4.b odd 2 1
2736.2.bm.q yes 8 12.b even 2 1
2736.2.bm.q yes 8 19.d odd 6 1
2736.2.bm.q yes 8 57.f even 6 1

## Hecke kernels

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

 $$T_{5}^{8} + 20T_{5}^{6} + 348T_{5}^{4} + 1040T_{5}^{2} + 2704$$ T5^8 + 20*T5^6 + 348*T5^4 + 1040*T5^2 + 2704 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}^{4} + 20T_{11}^{2} + 52$$ T11^4 + 20*T11^2 + 52 $$T_{23}^{8} - 44T_{23}^{6} + 1884T_{23}^{4} - 2288T_{23}^{2} + 2704$$ T23^8 - 44*T23^6 + 1884*T23^4 - 2288*T23^2 + 2704 $$T_{31}^{2} + 12T_{31} + 33$$ T31^2 + 12*T31 + 33

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 20 T^{6} + 348 T^{4} + \cdots + 2704$$
$7$ $$(T^{2} + 1)^{4}$$
$11$ $$(T^{4} + 20 T^{2} + 52)^{2}$$
$13$ $$(T^{2} + 3 T + 3)^{4}$$
$17$ $$T^{8} + 32 T^{6} + 816 T^{4} + \cdots + 43264$$
$19$ $$(T^{4} + 26 T^{2} + 361)^{2}$$
$23$ $$T^{8} - 44 T^{6} + 1884 T^{4} + \cdots + 2704$$
$29$ $$T^{8} - 96 T^{6} + 7344 T^{4} + \cdots + 3504384$$
$31$ $$(T^{2} + 12 T + 33)^{4}$$
$37$ $$(T^{4} + 78 T^{2} + 1089)^{2}$$
$41$ $$T^{8}$$
$43$ $$(T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9)^{2}$$
$47$ $$T^{8} - 128 T^{6} + \cdots + 11075584$$
$53$ $$T^{8} - 132 T^{6} + 16956 T^{4} + \cdots + 219024$$
$59$ $$T^{8} + 132 T^{6} + 16956 T^{4} + \cdots + 219024$$
$61$ $$(T^{4} - 6 T^{3} + 75 T^{2} + 234 T + 1521)^{2}$$
$67$ $$(T^{4} + 12 T^{3} + 135 T^{2} + 108 T + 81)^{2}$$
$71$ $$T^{8} + 288 T^{6} + \cdots + 283855104$$
$73$ $$(T^{4} + 10 T^{3} + 183 T^{2} - 830 T + 6889)^{2}$$
$79$ $$(T^{4} - 12 T^{3} + 183 T^{2} + 468 T + 1521)^{2}$$
$83$ $$(T^{4} + 128 T^{2} + 208)^{2}$$
$89$ $$T^{8} - 180 T^{6} + \cdots + 17740944$$
$97$ $$(T^{4} - 36 T^{2} + 1296)^{2}$$