# Properties

 Label 3240.2.q.bg Level $3240$ Weight $2$ Character orbit 3240.q Analytic conductor $25.872$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.q (of order $$3$$, degree $$2$$, 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: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.3887771904.9 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576$$ x^8 + 26*x^6 + 217*x^4 + 672*x^2 + 576 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{7}+O(q^{10})$$ q - b1 * q^5 + (-b7 + b5 - b4 - b2 + b1) * q^7 $$q - \beta_1 q^{5} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 4) q^{97}+O(q^{100})$$ q - b1 * q^5 + (-b7 + b5 - b4 - b2 + b1) * q^7 + (b7 - b6 - b5 - b3 - 2) * q^11 + (-b5 + b4 + b3) * q^13 + (b6 + 2*b2) * q^17 + (2*b7 - 1) * q^19 + (b5 - 2*b1) * q^23 + (b1 - 1) * q^25 + (-b7 + b6 + b5 + 2*b4 + b3 + 2*b2 + 2) * q^29 + (b5 - 3*b1) * q^31 + (b7 + b2) * q^35 + (-2*b7 - b6 + 4) * q^37 + (-3*b4 - 2*b3 - 2*b1) * q^41 + (2*b7 + b6 - 2*b5 - 2*b4 + b3 - 2*b2 + 5*b1 - 6) * q^43 + (-b7 + b6 + b5 + 4*b4 + b3 + 4*b2 + b1 + 1) * q^47 + (b5 - 2*b4 - 2*b3 - 3*b1) * q^49 + (b7 - b6 + 4*b2 - 3) * q^53 + (-b7 + b6 + 2) * q^55 - b4 * q^59 + (-2*b6 + 2*b4 - 2*b3 + 2*b2 - 2) * q^61 + (-b7 - b6 + b5 - b4 - b3 - b2) * q^65 + (-2*b5 - 4*b4 - 2*b3 - 8*b1) * q^67 + (-3*b7 - b6 + 2*b2 + 8) * q^71 + (2*b7 + b6 + 2*b2) * q^73 + (3*b3 + 9*b1) * q^77 + (-2*b4 - 2*b2 + 4*b1 - 4) * q^79 + (-2*b7 + b6 + 2*b5 + 2*b4 + b3 + 2*b2 + 3*b1) * q^83 + (2*b4 + b3 + b1) * q^85 + (-b7 - b6 + 6*b2 - 2) * q^89 + (-3*b7 - 6*b2 + 9) * q^91 + (-2*b5 - b1) * q^95 + (-2*b7 + b6 + 2*b5 + 4*b4 + b3 + 4*b2 - b1 + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4 q^{5} - 2 q^{7}+O(q^{10})$$ 8 * q - 4 * q^5 - 2 * q^7 $$8 q - 4 q^{5} - 2 q^{7} - 4 q^{11} - 4 q^{17} - 10 q^{23} - 4 q^{25} + 4 q^{29} - 14 q^{31} + 4 q^{35} + 28 q^{37} - 4 q^{41} - 22 q^{43} - 10 q^{49} - 16 q^{53} + 8 q^{55} - 4 q^{61} - 24 q^{67} + 56 q^{71} + 4 q^{73} + 30 q^{77} - 16 q^{79} - 6 q^{83} + 2 q^{85} - 16 q^{89} + 60 q^{91} + 10 q^{97}+O(q^{100})$$ 8 * q - 4 * q^5 - 2 * q^7 - 4 * q^11 - 4 * q^17 - 10 * q^23 - 4 * q^25 + 4 * q^29 - 14 * q^31 + 4 * q^35 + 28 * q^37 - 4 * q^41 - 22 * q^43 - 10 * q^49 - 16 * q^53 + 8 * q^55 - 4 * q^61 - 24 * q^67 + 56 * q^71 + 4 * q^73 + 30 * q^77 - 16 * q^79 - 6 * q^83 + 2 * q^85 - 16 * q^89 + 60 * q^91 + 10 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 26x^{6} + 217x^{4} + 672x^{2} + 576$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 26\nu^{5} + 193\nu^{3} + 360\nu + 96 ) / 192$$ (v^7 + 26*v^5 + 193*v^3 + 360*v + 96) / 192 $$\beta_{2}$$ $$=$$ $$( -\nu^{6} - 20\nu^{4} - 91\nu^{2} - 48 ) / 24$$ (-v^6 - 20*v^4 - 91*v^2 - 48) / 24 $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 4\nu^{6} + 26\nu^{5} + 104\nu^{4} + 241\nu^{3} + 772\nu^{2} + 984\nu + 1344 ) / 192$$ (v^7 + 4*v^6 + 26*v^5 + 104*v^4 + 241*v^3 + 772*v^2 + 984*v + 1344) / 192 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 4\nu^{6} - 106\nu^{5} + 80\nu^{4} - 605\nu^{3} + 364\nu^{2} - 888\nu + 192 ) / 192$$ (-5*v^7 + 4*v^6 - 106*v^5 + 80*v^4 - 605*v^3 + 364*v^2 - 888*v + 192) / 192 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 20\nu^{5} + 40\nu^{4} - 91\nu^{3} + 206\nu^{2} - 48\nu + 240 ) / 48$$ (-v^7 + 2*v^6 - 20*v^5 + 40*v^4 - 91*v^3 + 206*v^2 - 48*v + 240) / 48 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 26\nu^{4} - 193\nu^{2} - 360 ) / 24$$ (-v^6 - 26*v^4 - 193*v^2 - 360) / 24 $$\beta_{7}$$ $$=$$ $$( \nu^{6} + 20\nu^{4} + 103\nu^{2} + 132 ) / 12$$ (v^6 + 20*v^4 + 103*v^2 + 132) / 12
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} ) / 3$$ (b7 + b6 - 2*b5 + 2*b4 + 2*b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{7} + 2\beta_{2} - 7$$ b7 + 2*b2 - 7 $$\nu^{3}$$ $$=$$ $$( -13\beta_{7} - 7\beta_{6} + 26\beta_{5} - 26\beta_{4} - 14\beta_{3} - 13\beta_{2} - 12\beta _1 + 12 ) / 3$$ (-13*b7 - 7*b6 + 26*b5 - 26*b4 - 14*b3 - 13*b2 - 12*b1 + 12) / 3 $$\nu^{4}$$ $$=$$ $$-17\beta_{7} - 4\beta_{6} - 30\beta_{2} + 67$$ -17*b7 - 4*b6 - 30*b2 + 67 $$\nu^{5}$$ $$=$$ $$( 157\beta_{7} + 67\beta_{6} - 314\beta_{5} + 338\beta_{4} + 134\beta_{3} + 169\beta_{2} + 300\beta _1 - 240 ) / 3$$ (157*b7 + 67*b6 - 314*b5 + 338*b4 + 134*b3 + 169*b2 + 300*b1 - 240) / 3 $$\nu^{6}$$ $$=$$ $$249\beta_{7} + 80\beta_{6} + 394\beta_{2} - 751$$ 249*b7 + 80*b6 + 394*b2 - 751 $$\nu^{7}$$ $$=$$ $$( - 1933 \beta_{7} - 751 \beta_{6} + 3866 \beta_{5} - 4490 \beta_{4} - 1502 \beta_{3} - 2245 \beta_{2} + \cdots + 3636 ) / 3$$ (-1933*b7 - 751*b6 + 3866*b5 - 4490*b4 - 1502*b3 - 2245*b2 - 4908*b1 + 3636) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$$.

 $$n$$ $$1297$$ $$1621$$ $$2431$$ $$3161$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1081.1
 − 1.16910i 2.61675i 2.16910i − 3.61675i 1.16910i − 2.61675i − 2.16910i 3.61675i
0 0 0 −0.500000 + 0.866025i 0 −1.95058 3.37850i 0 0 0
1081.2 0 0 0 −0.500000 + 0.866025i 0 −0.942347 1.63219i 0 0 0
1081.3 0 0 0 −0.500000 + 0.866025i 0 −0.281475 0.487529i 0 0 0
1081.4 0 0 0 −0.500000 + 0.866025i 0 2.17440 + 3.76617i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −1.95058 + 3.37850i 0 0 0
2161.2 0 0 0 −0.500000 0.866025i 0 −0.942347 + 1.63219i 0 0 0
2161.3 0 0 0 −0.500000 0.866025i 0 −0.281475 + 0.487529i 0 0 0
2161.4 0 0 0 −0.500000 0.866025i 0 2.17440 3.76617i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1081.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3240.2.q.bg 8
3.b odd 2 1 3240.2.q.bh 8
9.c even 3 1 3240.2.a.v yes 4
9.c even 3 1 inner 3240.2.q.bg 8
9.d odd 6 1 3240.2.a.t 4
9.d odd 6 1 3240.2.q.bh 8
36.f odd 6 1 6480.2.a.ca 4
36.h even 6 1 6480.2.a.by 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3240.2.a.t 4 9.d odd 6 1
3240.2.a.v yes 4 9.c even 3 1
3240.2.q.bg 8 1.a even 1 1 trivial
3240.2.q.bg 8 9.c even 3 1 inner
3240.2.q.bh 8 3.b odd 2 1
3240.2.q.bh 8 9.d odd 6 1
6480.2.a.by 4 36.h even 6 1
6480.2.a.ca 4 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3240, [\chi])$$:

 $$T_{7}^{8} + 2T_{7}^{7} + 21T_{7}^{6} + 50T_{7}^{5} + 391T_{7}^{4} + 786T_{7}^{3} + 1458T_{7}^{2} + 756T_{7} + 324$$ T7^8 + 2*T7^7 + 21*T7^6 + 50*T7^5 + 391*T7^4 + 786*T7^3 + 1458*T7^2 + 756*T7 + 324 $$T_{11}^{8} + 4T_{11}^{7} + 51T_{11}^{6} + 148T_{11}^{5} + 1909T_{11}^{4} + 5904T_{11}^{3} + 16956T_{11}^{2} + 15552T_{11} + 11664$$ T11^8 + 4*T11^7 + 51*T11^6 + 148*T11^5 + 1909*T11^4 + 5904*T11^3 + 16956*T11^2 + 15552*T11 + 11664 $$T_{17}^{4} + 2T_{17}^{3} - 38T_{17}^{2} - 120T_{17} - 72$$ T17^4 + 2*T17^3 - 38*T17^2 - 120*T17 - 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{2} + T + 1)^{4}$$
$7$ $$T^{8} + 2 T^{7} + \cdots + 324$$
$11$ $$T^{8} + 4 T^{7} + \cdots + 11664$$
$13$ $$T^{8} + 39 T^{6} + \cdots + 46656$$
$17$ $$(T^{4} + 2 T^{3} - 38 T^{2} + \cdots - 72)^{2}$$
$19$ $$(T^{4} - 50 T^{2} + 433)^{2}$$
$23$ $$T^{8} + 10 T^{7} + \cdots + 144$$
$29$ $$T^{8} - 4 T^{7} + \cdots + 2304$$
$31$ $$T^{8} + 14 T^{7} + \cdots + 576$$
$37$ $$(T^{4} - 14 T^{3} + \cdots - 1152)^{2}$$
$41$ $$T^{8} + 4 T^{7} + \cdots + 4761$$
$43$ $$T^{8} + 22 T^{7} + \cdots + 45805824$$
$47$ $$T^{8} + 113 T^{6} + \cdots + 5476$$
$53$ $$(T^{4} + 8 T^{3} + \cdots - 498)^{2}$$
$59$ $$(T^{4} + 3 T^{2} + 9)^{2}$$
$61$ $$T^{8} + 4 T^{7} + \cdots + 1327104$$
$67$ $$T^{8} + 24 T^{7} + \cdots + 358875136$$
$71$ $$(T^{4} - 28 T^{3} + \cdots - 16428)^{2}$$
$73$ $$(T^{4} - 2 T^{3} + \cdots - 552)^{2}$$
$79$ $$(T^{4} + 8 T^{3} + 60 T^{2} + \cdots + 16)^{2}$$
$83$ $$T^{8} + 6 T^{7} + \cdots + 5184$$
$89$ $$(T^{4} + 8 T^{3} + \cdots + 11344)^{2}$$
$97$ $$T^{8} - 10 T^{7} + \cdots + 1024$$