# Properties

 Label 2736.2.bm.r Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} - 30x^{5} - 5x^{4} + 114x^{3} + 300x^{2} + 116x + 19$$ x^8 - 2*x^7 - 30*x^5 - 5*x^4 + 114*x^3 + 300*x^2 + 116*x + 19 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{5} + ( - 2 \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{7}+O(q^{10})$$ q + (-b7 + b6 + b3) * q^5 + (-2*b6 - b4 + b2 - 1) * q^7 $$q + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{5} + ( - 2 \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{7} + ( - \beta_{6} + \beta_{5} - 1) q^{11} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1) q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - \beta_1) q^{19} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{23} + (2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1) q^{25} + ( - 2 \beta_{5} + 2 \beta_1 + 2) q^{29} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{31} + (\beta_{7} - 3 \beta_{4} + \beta_{3} - \beta_1 - 3) q^{35} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 1) q^{37} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_1 + 1) q^{41} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_1 - 2) q^{43} + (\beta_{7} - \beta_{6} - 2 \beta_{3} - \beta_{2} - 1) q^{47} + (\beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{49} + (\beta_{2} + 1) q^{53} + (\beta_{7} - 2 \beta_{6} + \beta_{3} + \beta_1) q^{55} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - \beta_1 - 2) q^{59} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{61} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{2} - 2) q^{65} + (4 \beta_{7} - 5 \beta_{6} - 2 \beta_{4} + \beta_{2} - 1) q^{67} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 1) q^{71} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{3} - 1) q^{73} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2) q^{77} + ( - 3 \beta_{7} + 7 \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{79}+ \cdots + ( - 5 \beta_{6} - \beta_{4} - \beta_1 + 5) q^{97}+O(q^{100})$$ q + (-b7 + b6 + b3) * q^5 + (-2*b6 - b4 + b2 - 1) * q^7 + (-b6 + b5 - 1) * q^11 + (b6 + b5 - b2 - b1) * q^13 + (-b7 + b6 + b4 + b3 - 2*b2 - 1) * q^17 + (-b7 + b6 + b5 + b4 - b2 - b1) * q^19 + (-b7 + b6 + b5 + 2*b3 - b2 - b1 - 2) * q^23 + (2*b7 + b6 - b5 - 2*b4 + b2 - b1) * q^25 + (-2*b5 + 2*b1 + 2) * q^29 + (b6 + b5 - b4 + b3 - b2 - 2*b1 - 4) * q^31 + (b7 - 3*b4 + b3 - b1 - 3) * q^35 + (-2*b6 - 2*b4 + 2*b2 - 1) * q^37 + (-b7 - b6 + b4 - b3 + 2*b1 + 1) * q^41 + (-b7 + b6 - b4 - b3 + b1 - 2) * q^43 + (b7 - b6 - 2*b3 - b2 - 1) * q^47 + (b6 + b5 + b4 - 2*b3 + b2 - 2*b1 - 1) * q^49 + (b2 + 1) * q^53 + (b7 - 2*b6 + b3 + b1) * q^55 + (2*b7 - 3*b6 + 2*b5 - 2*b3 - b1 - 2) * q^59 + (-b6 + b5 - 2*b4 + b2 + b1 - 2) * q^61 + (-2*b6 + 2*b5 + 3*b4 - 3*b2 - 2) * q^65 + (4*b7 - 5*b6 - 2*b4 + b2 - 1) * q^67 + (-b7 + b6 - b4 + b3 + 2*b2 + 1) * q^71 + (2*b7 - 3*b6 - 2*b3 - 1) * q^73 + (2*b4 - b3 + 2*b2 - 2) * q^77 + (-3*b7 + 7*b6 + 2*b5 + b4 + 3*b3 - 2*b2 - b1 + 2) * q^79 + (2*b7 + 8*b6 + 2*b5 + b4 - b3 - b2 + 4) * q^83 + (2*b7 + 4*b6 + 4*b4 - 2*b2 + 2) * q^85 + (-b7 - 5*b6 + 2*b3 + 2*b2 - 10) * q^89 + (b7 - 6*b6 + b5 + 2*b4 - b2 + b1) * q^91 + (b7 - 2*b6 + 2*b5 + 2*b4 + 2*b3 - b2 - b1 - 7) * q^95 + (-5*b6 - b4 - b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{5}+O(q^{10})$$ 8 * q + 2 * q^5 $$8 q + 2 q^{5} - 4 q^{17} - 6 q^{23} - 12 q^{25} + 12 q^{29} - 28 q^{31} - 18 q^{35} + 12 q^{41} - 18 q^{43} - 12 q^{47} - 24 q^{49} + 6 q^{53} + 12 q^{55} - 10 q^{59} - 4 q^{61} + 6 q^{67} + 8 q^{71} - 8 q^{73} - 28 q^{77} + 14 q^{79} - 8 q^{85} - 54 q^{89} + 26 q^{91} - 38 q^{95} + 60 q^{97}+O(q^{100})$$ 8 * q + 2 * q^5 - 4 * q^17 - 6 * q^23 - 12 * q^25 + 12 * q^29 - 28 * q^31 - 18 * q^35 + 12 * q^41 - 18 * q^43 - 12 * q^47 - 24 * q^49 + 6 * q^53 + 12 * q^55 - 10 * q^59 - 4 * q^61 + 6 * q^67 + 8 * q^71 - 8 * q^73 - 28 * q^77 + 14 * q^79 - 8 * q^85 - 54 * q^89 + 26 * q^91 - 38 * q^95 + 60 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} - 30x^{5} - 5x^{4} + 114x^{3} + 300x^{2} + 116x + 19$$ :

 $$\beta_{1}$$ $$=$$ $$( 44\nu^{7} - 2059\nu^{6} + 6647\nu^{5} - 15007\nu^{4} + 50938\nu^{3} + 80360\nu^{2} - 278981\nu + 154035 ) / 307713$$ (44*v^7 - 2059*v^6 + 6647*v^5 - 15007*v^4 + 50938*v^3 + 80360*v^2 - 278981*v + 154035) / 307713 $$\beta_{2}$$ $$=$$ $$( - 88 \nu^{7} + 4118 \nu^{6} - 13294 \nu^{5} + 30014 \nu^{4} - 101876 \nu^{3} + 146993 \nu^{2} + 250249 \nu - 308070 ) / 307713$$ (-88*v^7 + 4118*v^6 - 13294*v^5 + 30014*v^4 - 101876*v^3 + 146993*v^2 + 250249*v - 308070) / 307713 $$\beta_{3}$$ $$=$$ $$( 3500 \nu^{7} - 15922 \nu^{6} + 21212 \nu^{5} - 114745 \nu^{4} + 251431 \nu^{3} + 373886 \nu^{2} + 131509 \nu - 1214655 ) / 307713$$ (3500*v^7 - 15922*v^6 + 21212*v^5 - 114745*v^4 + 251431*v^3 + 373886*v^2 + 131509*v - 1214655) / 307713 $$\beta_{4}$$ $$=$$ $$( - 3772 \nu^{7} + 8669 \nu^{6} - 10351 \nu^{5} + 111605 \nu^{4} - 2846 \nu^{3} - 315175 \nu^{2} - 309125 \nu - 420924 ) / 307713$$ (-3772*v^7 + 8669*v^6 - 10351*v^5 + 111605*v^4 - 2846*v^3 - 315175*v^2 - 309125*v - 420924) / 307713 $$\beta_{5}$$ $$=$$ $$( 5106 \nu^{7} - 10484 \nu^{6} - 7253 \nu^{5} - 142319 \nu^{4} - 28670 \nu^{3} + 830669 \nu^{2} + 1282798 \nu + 414680 ) / 307713$$ (5106*v^7 - 10484*v^6 - 7253*v^5 - 142319*v^4 - 28670*v^3 + 830669*v^2 + 1282798*v + 414680) / 307713 $$\beta_{6}$$ $$=$$ $$( -1002\nu^{7} + 2264\nu^{6} - 178\nu^{5} + 29375\nu^{4} + 254\nu^{3} - 124940\nu^{2} - 273328\nu - 78329 ) / 43959$$ (-1002*v^7 + 2264*v^6 - 178*v^5 + 29375*v^4 + 254*v^3 - 124940*v^2 - 273328*v - 78329) / 43959 $$\beta_{7}$$ $$=$$ $$( - 32081 \nu^{7} + 65584 \nu^{6} - 5933 \nu^{5} + 958120 \nu^{4} + 99677 \nu^{3} - 3586874 \nu^{2} - 9255799 \nu - 2690259 ) / 307713$$ (-32081*v^7 + 65584*v^6 - 5933*v^5 + 958120*v^4 + 99677*v^3 - 3586874*v^2 - 9255799*v - 2690259) / 307713
 $$\nu$$ $$=$$ $$( -2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 3$$ (-2*b6 - 2*b5 + b4 + b2 + b1 + 1) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{6} - 2\beta_{5} + \beta_{4} + 4\beta_{2} + 7\beta _1 + 1 ) / 3$$ (-2*b6 - 2*b5 + b4 + 4*b2 + 7*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( -6\beta_{7} + 35\beta_{6} + 8\beta_{5} + 5\beta_{4} + 9\beta_{3} + 5\beta_{2} - \beta _1 + 47 ) / 3$$ (-6*b7 + 35*b6 + 8*b5 + 5*b4 + 9*b3 + 5*b2 - b1 + 47) / 3 $$\nu^{4}$$ $$=$$ $$2\beta_{7} - 15\beta_{6} - 12\beta_{5} - 2\beta_{4} + 4\beta_{3} + 14\beta_{2} - 2\beta _1 + 35$$ 2*b7 - 15*b6 - 12*b5 - 2*b4 + 4*b3 + 14*b2 - 2*b1 + 35 $$\nu^{5}$$ $$=$$ $$( -133\beta_{6} - 187\beta_{5} - 7\beta_{4} + 146\beta_{2} + 191\beta _1 + 56 ) / 3$$ (-133*b6 - 187*b5 - 7*b4 + 146*b2 + 191*b1 + 56) / 3 $$\nu^{6}$$ $$=$$ $$( -198\beta_{7} + 992\beta_{6} + 47\beta_{5} + 38\beta_{4} + 150\beta_{3} + 335\beta_{2} + 311\beta _1 + 797 ) / 3$$ (-198*b7 + 992*b6 + 47*b5 + 38*b4 + 150*b3 + 335*b2 + 311*b1 + 797) / 3 $$\nu^{7}$$ $$=$$ $$( -273\beta_{7} + 1618\beta_{6} - 119\beta_{5} - 485\beta_{4} + 693\beta_{3} + 1192\beta_{2} - 653\beta _1 + 4249 ) / 3$$ (-273*b7 + 1618*b6 - 119*b5 - 485*b4 + 693*b3 + 1192*b2 - 653*b1 + 4249) / 3

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{6}$$ $$1$$ $$-1$$ $$1$$

## 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
 −0.213988 + 0.172868i −0.654220 + 2.95767i −1.27736 − 1.04884i 3.14556 − 0.349646i −0.213988 − 0.172868i −0.654220 − 2.95767i −1.27736 + 1.04884i 3.14556 + 0.349646i
0 0 0 −2.00488 + 3.47255i 0 1.92982i 0 0 0
559.2 0 0 0 0.353597 0.612447i 0 0.0924751i 0 0 0
559.3 0 0 0 0.912850 1.58110i 0 4.99333i 0 0 0
559.4 0 0 0 1.73843 3.01105i 0 3.36658i 0 0 0
1855.1 0 0 0 −2.00488 3.47255i 0 1.92982i 0 0 0
1855.2 0 0 0 0.353597 + 0.612447i 0 0.0924751i 0 0 0
1855.3 0 0 0 0.912850 + 1.58110i 0 4.99333i 0 0 0
1855.4 0 0 0 1.73843 + 3.01105i 0 3.36658i 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
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.r 8
3.b odd 2 1 912.2.bb.h yes 8
4.b odd 2 1 2736.2.bm.s 8
12.b even 2 1 912.2.bb.g 8
19.d odd 6 1 2736.2.bm.s 8
57.f even 6 1 912.2.bb.g 8
76.f even 6 1 inner 2736.2.bm.r 8
228.n odd 6 1 912.2.bb.h yes 8

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

## Hecke kernels

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

 $$T_{5}^{8} - 2T_{5}^{7} + 18T_{5}^{6} - 44T_{5}^{5} + 286T_{5}^{4} - 576T_{5}^{3} + 1044T_{5}^{2} - 648T_{5} + 324$$ T5^8 - 2*T5^7 + 18*T5^6 - 44*T5^5 + 286*T5^4 - 576*T5^3 + 1044*T5^2 - 648*T5 + 324 $$T_{7}^{8} + 40T_{7}^{6} + 418T_{7}^{4} + 1056T_{7}^{2} + 9$$ T7^8 + 40*T7^6 + 418*T7^4 + 1056*T7^2 + 9 $$T_{11}^{8} + 40T_{11}^{6} + 520T_{11}^{4} + 2376T_{11}^{2} + 2916$$ T11^8 + 40*T11^6 + 520*T11^4 + 2376*T11^2 + 2916 $$T_{23}^{8} + 6T_{23}^{7} - 26T_{23}^{6} - 228T_{23}^{5} + 1378T_{23}^{4} - 912T_{23}^{3} - 4140T_{23}^{2} + 2736T_{23} + 12996$$ T23^8 + 6*T23^7 - 26*T23^6 - 228*T23^5 + 1378*T23^4 - 912*T23^3 - 4140*T23^2 + 2736*T23 + 12996 $$T_{31}^{4} + 14T_{31}^{3} + 10T_{31}^{2} - 282T_{31} - 45$$ T31^4 + 14*T31^3 + 10*T31^2 - 282*T31 - 45

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{7} + 18 T^{6} - 44 T^{5} + \cdots + 324$$
$7$ $$T^{8} + 40 T^{6} + 418 T^{4} + 1056 T^{2} + \cdots + 9$$
$11$ $$T^{8} + 40 T^{6} + 520 T^{4} + \cdots + 2916$$
$13$ $$T^{8} - 26 T^{6} + 691 T^{4} + \cdots + 225$$
$17$ $$T^{8} + 4 T^{7} + 60 T^{6} + \cdots + 129600$$
$19$ $$T^{8} - 4 T^{6} - 72 T^{5} + \cdots + 130321$$
$23$ $$T^{8} + 6 T^{7} - 26 T^{6} + \cdots + 12996$$
$29$ $$T^{8} - 12 T^{7} - 8 T^{6} + \cdots + 746496$$
$31$ $$(T^{4} + 14 T^{3} + 10 T^{2} - 282 T - 45)^{2}$$
$37$ $$T^{8} + 148 T^{6} + 6046 T^{4} + \cdots + 81225$$
$41$ $$T^{8} - 12 T^{7} - 12 T^{6} + \cdots + 129600$$
$43$ $$T^{8} + 18 T^{7} + 42 T^{6} + \cdots + 1946025$$
$47$ $$T^{8} + 12 T^{7} - 20 T^{6} + \cdots + 576$$
$53$ $$T^{8} - 6 T^{7} - 2 T^{6} + 84 T^{5} + \cdots + 36$$
$59$ $$T^{8} + 10 T^{7} + 198 T^{6} + \cdots + 14152644$$
$61$ $$T^{8} + 4 T^{7} + 154 T^{6} + \cdots + 1590121$$
$67$ $$T^{8} - 6 T^{7} + 242 T^{6} + \cdots + 693889$$
$71$ $$T^{8} - 8 T^{7} + 132 T^{6} + \cdots + 627264$$
$73$ $$T^{8} + 8 T^{7} + 102 T^{6} + \cdots + 393129$$
$79$ $$T^{8} - 14 T^{7} + 250 T^{6} + \cdots + 5331481$$
$83$ $$T^{8} + 504 T^{6} + \cdots + 13483584$$
$89$ $$T^{8} + 54 T^{7} + 1174 T^{6} + \cdots + 61496964$$
$97$ $$T^{8} - 60 T^{7} + 1624 T^{6} + \cdots + 10890000$$