# Properties

 Label 2736.2.bm.l 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: 6.0.31726512.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9$$ x^6 - x^5 + 10*x^4 + 3*x^3 + 84*x^2 - 27*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 304) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4}) q^{7}+O(q^{10})$$ q + (-b3 + b1 - 1) * q^5 + (b5 - b4) * q^7 $$q + ( - \beta_{3} + \beta_1 - 1) q^{5} + (\beta_{5} - \beta_{4}) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 1) q^{11} + (2 \beta_{3} + 4) q^{13} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{17} + ( - \beta_{5} + \beta_{4} + \beta_1 - 3) q^{19} + \beta_{4} q^{23} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{25} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{29} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{31} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{35} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{37} + ( - 3 \beta_{5} - 2 \beta_{2} + \beta_1) q^{41} + (2 \beta_{3} - 2) q^{43} + ( - \beta_{4} - 2 \beta_{3} - 4) q^{47} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{49} + (4 \beta_{3} + 8) q^{53} + (\beta_{5} - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{55} + (\beta_{5} - 2 \beta_{4} + 9 \beta_{3} + 9) q^{59} + (3 \beta_{3} - \beta_{2} + \beta_1) q^{61} + ( - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 2) q^{65} + ( - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{67} + (6 \beta_{3} + 6) q^{71} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{73} + ( - \beta_{2} + 7) q^{77} + (6 \beta_{3} - 2 \beta_1 + 6) q^{79} + (2 \beta_{5} - 2 \beta_{4} + \beta_{2} - 2 \beta_1) q^{83} + (4 \beta_{5} - 2 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{85} + (3 \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 + 6) q^{89} + (4 \beta_{5} - 2 \beta_{4}) q^{91} + ( - \beta_{5} + \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{95} + ( - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 3) q^{97}+O(q^{100})$$ q + (-b3 + b1 - 1) * q^5 + (b5 - b4) * q^7 + (-b5 + b4 + 2*b3 + 1) * q^11 + (2*b3 + 4) * q^13 + (-b5 + 2*b4 + b3 - b1 + 1) * q^17 + (-b5 + b4 + b1 - 3) * q^19 + b4 * q^23 + (-2*b5 + b4 + 2*b3 + b2 - b1) * q^25 + (b3 + b2 + b1 + 2) * q^29 + (-b5 - b4 - 2*b2) * q^31 + (-b5 + b3 + 2*b2 - b1 - 1) * q^35 + (-2*b3 + b2 - 2*b1 - 1) * q^37 + (-3*b5 - 2*b2 + b1) * q^41 + (2*b3 - 2) * q^43 + (-b4 - 2*b3 - 4) * q^47 + (b5 + b4 + b2) * q^49 + (4*b3 + 8) * q^53 + (b5 - 2*b3 - 4*b2 + 2*b1 + 2) * q^55 + (b5 - 2*b4 + 9*b3 + 9) * q^59 + (3*b3 - b2 + b1) * q^61 + (-4*b3 - 2*b2 + 4*b1 - 2) * q^65 + (-2*b5 + b4 - 3*b3 + 2*b2 - 2*b1) * q^67 + (6*b3 + 6) * q^71 + (-b5 + 2*b4 - 2*b3 + b1 - 2) * q^73 + (-b2 + 7) * q^77 + (6*b3 - 2*b1 + 6) * q^79 + (2*b5 - 2*b4 + b2 - 2*b1) * q^83 + (4*b5 - 2*b4 - 10*b3 - 4*b2 + 4*b1) * q^85 + (3*b4 + 3*b3 - b2 - b1 + 6) * q^89 + (4*b5 - 2*b4) * q^91 + (-b5 + b4 + 8*b3 - 2*b2 - 2*b1 + 4) * q^95 + (-3*b3 - 4*b2 + 2*b1 + 3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{5}+O(q^{10})$$ 6 * q - 2 * q^5 $$6 q - 2 q^{5} + 18 q^{13} + 2 q^{17} - 17 q^{19} - 5 q^{25} + 12 q^{29} - 4 q^{31} - 6 q^{35} - 3 q^{41} - 18 q^{43} - 18 q^{47} + 2 q^{49} + 36 q^{53} + 12 q^{55} + 27 q^{59} - 10 q^{61} + 11 q^{67} + 18 q^{71} - 5 q^{73} + 40 q^{77} + 16 q^{79} + 26 q^{85} + 24 q^{89} - 6 q^{95} + 21 q^{97}+O(q^{100})$$ 6 * q - 2 * q^5 + 18 * q^13 + 2 * q^17 - 17 * q^19 - 5 * q^25 + 12 * q^29 - 4 * q^31 - 6 * q^35 - 3 * q^41 - 18 * q^43 - 18 * q^47 + 2 * q^49 + 36 * q^53 + 12 * q^55 + 27 * q^59 - 10 * q^61 + 11 * q^67 + 18 * q^71 - 5 * q^73 + 40 * q^77 + 16 * q^79 + 26 * q^85 + 24 * q^89 - 6 * q^95 + 21 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 10x^{4} + 3x^{3} + 84x^{2} - 27x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 10\nu^{4} - 100\nu^{3} + 84\nu^{2} - 27\nu + 270 ) / 813$$ (-v^5 + 10*v^4 - 100*v^3 + 84*v^2 - 27*v + 270) / 813 $$\beta_{3}$$ $$=$$ $$( 30\nu^{5} - 29\nu^{4} + 290\nu^{3} + 190\nu^{2} + 2436\nu - 783 ) / 813$$ (30*v^5 - 29*v^4 + 290*v^3 + 190*v^2 + 2436*v - 783) / 813 $$\beta_{4}$$ $$=$$ $$( -161\nu^{5} - 16\nu^{4} - 1466\nu^{3} - 1923\nu^{2} - 14916\nu - 5310 ) / 2439$$ (-161*v^5 - 16*v^4 - 1466*v^3 - 1923*v^2 - 14916*v - 5310) / 2439 $$\beta_{5}$$ $$=$$ $$( 191\nu^{5} - 284\nu^{4} + 2027\nu^{3} - 597\nu^{2} + 15726\nu - 10107 ) / 2439$$ (191*v^5 - 284*v^4 + 2027*v^3 - 597*v^2 + 15726*v - 10107) / 2439
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-2\beta_{5} + \beta_{4} + 6\beta_{3} - \beta_{2} + \beta_1$$ -2*b5 + b4 + 6*b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{5} - \beta_{4} - 10\beta_{2} - 3$$ -b5 - b4 - 10*b2 - 3 $$\nu^{4}$$ $$=$$ $$10\beta_{5} - 20\beta_{4} - 57\beta_{3} - 16\beta _1 - 57$$ 10*b5 - 20*b4 - 57*b3 - 16*b1 - 57 $$\nu^{5}$$ $$=$$ $$32\beta_{5} - 16\beta_{4} - 66\beta_{3} + 103\beta_{2} - 103\beta_1$$ 32*b5 - 16*b4 - 66*b3 + 103*b2 - 103*b1

## 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
 −1.35887 + 2.35363i 0.162698 − 0.281802i 1.69617 − 2.93786i −1.35887 − 2.35363i 0.162698 + 0.281802i 1.69617 + 2.93786i
0 0 0 −1.85887 + 3.21966i 0 2.36936i 0 0 0
559.2 0 0 0 −0.337302 + 0.584224i 0 3.59084i 0 0 0
559.3 0 0 0 1.19617 2.07183i 0 1.22147i 0 0 0
1855.1 0 0 0 −1.85887 3.21966i 0 2.36936i 0 0 0
1855.2 0 0 0 −0.337302 0.584224i 0 3.59084i 0 0 0
1855.3 0 0 0 1.19617 + 2.07183i 0 1.22147i 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.l 6
3.b odd 2 1 304.2.n.d 6
4.b odd 2 1 2736.2.bm.m 6
12.b even 2 1 304.2.n.e yes 6
19.d odd 6 1 2736.2.bm.m 6
24.f even 2 1 1216.2.n.d 6
24.h odd 2 1 1216.2.n.e 6
57.f even 6 1 304.2.n.e yes 6
76.f even 6 1 inner 2736.2.bm.l 6
228.n odd 6 1 304.2.n.d 6
456.s odd 6 1 1216.2.n.e 6
456.v even 6 1 1216.2.n.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.d 6 3.b odd 2 1
304.2.n.d 6 228.n odd 6 1
304.2.n.e yes 6 12.b even 2 1
304.2.n.e yes 6 57.f even 6 1
1216.2.n.d 6 24.f even 2 1
1216.2.n.d 6 456.v even 6 1
1216.2.n.e 6 24.h odd 2 1
1216.2.n.e 6 456.s odd 6 1
2736.2.bm.l 6 1.a even 1 1 trivial
2736.2.bm.l 6 76.f even 6 1 inner
2736.2.bm.m 6 4.b odd 2 1
2736.2.bm.m 6 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{6} + 2T_{5}^{5} + 12T_{5}^{4} - 4T_{5}^{3} + 76T_{5}^{2} + 48T_{5} + 36$$ T5^6 + 2*T5^5 + 12*T5^4 - 4*T5^3 + 76*T5^2 + 48*T5 + 36 $$T_{7}^{6} + 20T_{7}^{4} + 100T_{7}^{2} + 108$$ T7^6 + 20*T7^4 + 100*T7^2 + 108 $$T_{11}^{6} + 29T_{11}^{4} + 235T_{11}^{2} + 507$$ T11^6 + 29*T11^4 + 235*T11^2 + 507 $$T_{23}^{6} - 10T_{23}^{4} + 100T_{23}^{2} - 180T_{23} + 108$$ T23^6 - 10*T23^4 + 100*T23^2 - 180*T23 + 108 $$T_{31}^{3} + 2T_{31}^{2} - 54T_{31} + 66$$ T31^3 + 2*T31^2 - 54*T31 + 66

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 2 T^{5} + 12 T^{4} - 4 T^{3} + \cdots + 36$$
$7$ $$T^{6} + 20 T^{4} + 100 T^{2} + \cdots + 108$$
$11$ $$T^{6} + 29 T^{4} + 235 T^{2} + \cdots + 507$$
$13$ $$(T^{2} - 6 T + 12)^{3}$$
$17$ $$T^{6} - 2 T^{5} + 48 T^{4} + 112 T^{3} + \cdots + 144$$
$19$ $$T^{6} + 17 T^{5} + 144 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 10 T^{4} + 100 T^{2} + \cdots + 108$$
$29$ $$T^{6} - 12 T^{5} + 36 T^{4} + \cdots + 2700$$
$31$ $$(T^{3} + 2 T^{2} - 54 T + 66)^{2}$$
$37$ $$T^{6} + 72 T^{4} + 864 T^{2} + \cdots + 2700$$
$41$ $$T^{6} + 3 T^{5} - 96 T^{4} + \cdots + 54675$$
$43$ $$(T^{2} + 6 T + 12)^{3}$$
$47$ $$T^{6} + 18 T^{5} + 134 T^{4} + \cdots + 12$$
$53$ $$(T^{2} - 12 T + 48)^{3}$$
$59$ $$T^{6} - 27 T^{5} + 516 T^{4} + \cdots + 263169$$
$61$ $$T^{6} + 10 T^{5} + 76 T^{4} + 228 T^{3} + \cdots + 36$$
$67$ $$T^{6} - 11 T^{5} + 160 T^{4} + \cdots + 245025$$
$71$ $$(T^{2} - 6 T + 36)^{3}$$
$73$ $$T^{6} + 5 T^{5} + 50 T^{4} - 91 T^{3} + \cdots + 289$$
$79$ $$T^{6} - 16 T^{5} + 208 T^{4} + \cdots + 2304$$
$83$ $$T^{6} + 113 T^{4} + 3031 T^{2} + \cdots + 3$$
$89$ $$T^{6} - 24 T^{5} + 120 T^{4} + \cdots + 97200$$
$97$ $$T^{6} - 21 T^{5} + 84 T^{4} + \cdots + 151875$$