# Properties

 Label 2736.2.f.g Level $2736$ Weight $2$ Character orbit 2736.f 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(1025,2736)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.f (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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4$$ x^8 + 18*x^6 + 37*x^4 + 22*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1368) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{5} - \beta_{4} q^{7}+O(q^{10})$$ q + b7 * q^5 - b4 * q^7 $$q + \beta_{7} q^{5} - \beta_{4} q^{7} + ( - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{3}) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + (\beta_{6} - \beta_{4} - \beta_1 - 1) q^{25} + ( - \beta_{6} + \beta_1 - 4) q^{29} + ( - \beta_{5} - 3 \beta_{3} + \beta_{2}) q^{31} + ( - \beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{2}) q^{35} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_{3} - \beta_{2}) q^{37} + ( - 2 \beta_{6} + 2 \beta_{4} + 4) q^{41} + (\beta_{4} + \beta_1 + 3) q^{43} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{47} + (2 \beta_{6} + \beta_{4} - \beta_1) q^{49} + ( - 2 \beta_{4} - 2 \beta_1) q^{53} + (\beta_{6} + \beta_{4} + 2 \beta_1 - 3) q^{55} + (\beta_{6} - 2 \beta_{4} + \beta_1) q^{59} + ( - 3 \beta_{4} - 2 \beta_1) q^{61} + ( - 3 \beta_{6} - 2 \beta_{4} + \beta_1 + 4) q^{65} + (2 \beta_{7} - 2 \beta_{3}) q^{67} + (\beta_{6} + 2 \beta_{4} + \beta_1 - 4) q^{71} + (2 \beta_{6} + 3 \beta_{4} + \beta_1 - 1) q^{73} - \beta_{3} q^{77} + ( - 2 \beta_{7} + 3 \beta_{5} - \beta_{3} + \beta_{2}) q^{79} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{83} + (\beta_{6} + \beta_{4} - 1) q^{85} + (\beta_{6} + 3 \beta_1 - 4) q^{89} + ( - 4 \beta_{7} + 6 \beta_{5} - 2 \beta_{3} - 2 \beta_{2}) q^{91} + (2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{95} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{2}) q^{97}+O(q^{100})$$ q + b7 * q^5 - b4 * q^7 + (-b5 - b3) * q^11 + (-b5 + b3 + b2) * q^13 + (b5 - b3) * q^17 + (-b7 - b6 - b3) * q^19 + (-b3 + b2) * q^23 + (b6 - b4 - b1 - 1) * q^25 + (-b6 + b1 - 4) * q^29 + (-b5 - 3*b3 + b2) * q^31 + (-b7 + 2*b5 - b3 - b2) * q^35 + (-2*b7 - 3*b5 + b3 - b2) * q^37 + (-2*b6 + 2*b4 + 4) * q^41 + (b4 + b1 + 3) * q^43 + (-b7 + 2*b5 - 2*b3) * q^47 + (2*b6 + b4 - b1) * q^49 + (-2*b4 - 2*b1) * q^53 + (b6 + b4 + 2*b1 - 3) * q^55 + (b6 - 2*b4 + b1) * q^59 + (-3*b4 - 2*b1) * q^61 + (-3*b6 - 2*b4 + b1 + 4) * q^65 + (2*b7 - 2*b3) * q^67 + (b6 + 2*b4 + b1 - 4) * q^71 + (2*b6 + 3*b4 + b1 - 1) * q^73 - b3 * q^77 + (-2*b7 + 3*b5 - b3 + b2) * q^79 + (-2*b7 + b5 - 2*b3 - 2*b2) * q^83 + (b6 + b4 - 1) * q^85 + (b6 + 3*b1 - 4) * q^89 + (-4*b7 + 6*b5 - 2*b3 - 2*b2) * q^91 + (2*b4 - 2*b3 - b2 + 2*b1 + 4) * q^95 + (-2*b7 + 2*b5 - 2*b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 4 q^{19} - 8 q^{25} - 32 q^{29} + 24 q^{41} + 28 q^{43} + 4 q^{49} - 8 q^{53} - 12 q^{55} + 8 q^{59} - 8 q^{61} + 24 q^{65} - 24 q^{71} + 4 q^{73} - 4 q^{85} - 16 q^{89} + 40 q^{95}+O(q^{100})$$ 8 * q - 4 * q^19 - 8 * q^25 - 32 * q^29 + 24 * q^41 + 28 * q^43 + 4 * q^49 - 8 * q^53 - 12 * q^55 + 8 * q^59 - 8 * q^61 + 24 * q^65 - 24 * q^71 + 4 * q^73 - 4 * q^85 - 16 * q^89 + 40 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18x^{6} + 37x^{4} + 22x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{6} + 17\nu^{4} + 20\nu^{2} + 3$$ v^6 + 17*v^4 + 20*v^2 + 3 $$\beta_{2}$$ $$=$$ $$( \nu^{7} + 18\nu^{5} + 37\nu^{3} + 24\nu ) / 2$$ (v^7 + 18*v^5 + 37*v^3 + 24*v) / 2 $$\beta_{3}$$ $$=$$ $$-4\nu^{7} - 70\nu^{5} - 113\nu^{3} - 32\nu$$ -4*v^7 - 70*v^5 - 113*v^3 - 32*v $$\beta_{4}$$ $$=$$ $$5\nu^{6} + 88\nu^{4} + 150\nu^{2} + 50$$ 5*v^6 + 88*v^4 + 150*v^2 + 50 $$\beta_{5}$$ $$=$$ $$( -9\nu^{7} - 158\nu^{5} - 263\nu^{3} - 84\nu ) / 2$$ (-9*v^7 - 158*v^5 - 263*v^3 - 84*v) / 2 $$\beta_{6}$$ $$=$$ $$7\nu^{6} + 123\nu^{4} + 206\nu^{2} + 65$$ 7*v^6 + 123*v^4 + 206*v^2 + 65 $$\beta_{7}$$ $$=$$ $$( 15\nu^{7} + 264\nu^{5} + 449\nu^{3} + 144\nu ) / 2$$ (15*v^7 + 264*v^5 + 449*v^3 + 144*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{3} + \beta_{2} ) / 2$$ (b5 - b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( -3\beta_{6} + 4\beta_{4} + \beta _1 - 8 ) / 2$$ (-3*b6 + 4*b4 + b1 - 8) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{7} - 12\beta_{5} + 9\beta_{3} - 6\beta_{2}$$ -2*b7 - 12*b5 + 9*b3 - 6*b2 $$\nu^{4}$$ $$=$$ $$25\beta_{6} - 33\beta_{4} - 10\beta _1 + 55$$ 25*b6 - 33*b4 - 10*b1 + 55 $$\nu^{5}$$ $$=$$ $$35\beta_{7} + 194\beta_{5} - 141\beta_{3} + 93\beta_{2}$$ 35*b7 + 194*b5 - 141*b3 + 93*b2 $$\nu^{6}$$ $$=$$ $$-395\beta_{6} + 521\beta_{4} + 161\beta _1 - 858$$ -395*b6 + 521*b4 + 161*b1 - 858 $$\nu^{7}$$ $$=$$ $$-556\beta_{7} - 3060\beta_{5} + 2217\beta_{3} - 1462\beta_{2}$$ -556*b7 - 3060*b5 + 2217*b3 - 1462*b2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 1.18847i 0.676777i − 0.626815i − 3.96694i 3.96694i 0.626815i − 0.676777i − 1.18847i
0 0 0 3.42865i 0 0.394100 0 0 0
1025.2 0 0 0 2.60938i 0 0.723074 0 0 0
1025.3 0 0 0 2.32945i 0 −4.34658 0 0 0
1025.4 0 0 0 0.0959660i 0 3.22941 0 0 0
1025.5 0 0 0 0.0959660i 0 3.22941 0 0 0
1025.6 0 0 0 2.32945i 0 −4.34658 0 0 0
1025.7 0 0 0 2.60938i 0 0.723074 0 0 0
1025.8 0 0 0 3.42865i 0 0.394100 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.f.g 8
3.b odd 2 1 2736.2.f.h 8
4.b odd 2 1 1368.2.f.c 8
12.b even 2 1 1368.2.f.d yes 8
19.b odd 2 1 2736.2.f.h 8
57.d even 2 1 inner 2736.2.f.g 8
76.d even 2 1 1368.2.f.d yes 8
228.b odd 2 1 1368.2.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.f.c 8 4.b odd 2 1
1368.2.f.c 8 228.b odd 2 1
1368.2.f.d yes 8 12.b even 2 1
1368.2.f.d yes 8 76.d even 2 1
2736.2.f.g 8 1.a even 1 1 trivial
2736.2.f.g 8 57.d even 2 1 inner
2736.2.f.h 8 3.b odd 2 1
2736.2.f.h 8 19.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{8} + 24T_{5}^{6} + 181T_{5}^{4} + 436T_{5}^{2} + 4$$ T5^8 + 24*T5^6 + 181*T5^4 + 436*T5^2 + 4 $$T_{7}^{4} - 15T_{7}^{2} + 16T_{7} - 4$$ T7^4 - 15*T7^2 + 16*T7 - 4 $$T_{11}^{8} + 32T_{11}^{6} + 157T_{11}^{4} + 68T_{11}^{2} + 4$$ T11^8 + 32*T11^6 + 157*T11^4 + 68*T11^2 + 4 $$T_{29}^{4} + 16T_{29}^{3} + 60T_{29}^{2} - 64T_{29} - 320$$ T29^4 + 16*T29^3 + 60*T29^2 - 64*T29 - 320

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 24 T^{6} + 181 T^{4} + 436 T^{2} + \cdots + 4$$
$7$ $$(T^{4} - 15 T^{2} + 16 T - 4)^{2}$$
$11$ $$T^{8} + 32 T^{6} + 157 T^{4} + 68 T^{2} + \cdots + 4$$
$13$ $$T^{8} + 88 T^{6} + 2368 T^{4} + \cdots + 16384$$
$17$ $$T^{8} + 24 T^{6} + 181 T^{4} + \cdots + 100$$
$19$ $$T^{8} + 4 T^{7} + 4 T^{6} + \cdots + 130321$$
$23$ $$T^{8} + 64 T^{6} + 1016 T^{4} + \cdots + 16$$
$29$ $$(T^{4} + 16 T^{3} + 60 T^{2} - 64 T - 320)^{2}$$
$31$ $$T^{8} + 216 T^{6} + 15424 T^{4} + \cdots + 1048576$$
$37$ $$T^{8} + 184 T^{6} + 8256 T^{4} + \cdots + 65536$$
$41$ $$(T^{4} - 12 T^{3} - 76 T^{2} + 1248 T - 3200)^{2}$$
$43$ $$(T^{4} - 14 T^{3} + 49 T^{2} - 16 T - 16)^{2}$$
$47$ $$T^{8} + 112 T^{6} + 3613 T^{4} + \cdots + 58564$$
$53$ $$(T^{4} + 4 T^{3} - 92 T^{2} - 64 T + 1280)^{2}$$
$59$ $$(T^{4} - 4 T^{3} - 128 T^{2} + 192 T + 3328)^{2}$$
$61$ $$(T^{4} + 4 T^{3} - 139 T^{2} - 268 T + 4976)^{2}$$
$67$ $$T^{8} + 208 T^{6} + 8512 T^{4} + \cdots + 409600$$
$71$ $$(T^{4} + 12 T^{3} - 32 T^{2} - 320 T + 256)^{2}$$
$73$ $$(T^{4} - 2 T^{3} - 219 T^{2} + 284 T + 1156)^{2}$$
$79$ $$T^{8} + 280 T^{6} + \cdots + 15745024$$
$83$ $$T^{8} + 336 T^{6} + 21464 T^{4} + \cdots + 512656$$
$89$ $$(T^{4} + 8 T^{3} - 228 T^{2} - 1152 T + 5888)^{2}$$
$97$ $$T^{8} + 272 T^{6} + 10048 T^{4} + \cdots + 16384$$