# Properties

 Label 4680.2.l.g Level $4680$ Weight $2$ Character orbit 4680.l Analytic conductor $37.370$ 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] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.l (of order $$2$$, degree $$1$$, 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: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.57815240704.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16$$ x^8 - 2*x^7 + 2*x^6 + 89*x^4 - 170*x^3 + 162*x^2 - 72*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1560) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{5} + (2 \beta_{2} - \beta_1) q^{7}+O(q^{10})$$ q + b7 * q^5 + (2*b2 - b1) * q^7 $$q + \beta_{7} q^{5} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + \beta_{2} q^{13} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{23} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2}) q^{25} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{29} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_1 + 2) q^{35} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_1) q^{37} + (\beta_{7} + \beta_{6} + \beta_{3} - 2) q^{41} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{43} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{47} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} - 3) q^{49} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_1) q^{53} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{55} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2) q^{59} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2) q^{61} - \beta_{5} q^{65} + ( - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_{2}) q^{67} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} - 4) q^{71} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{73} + ( - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{77} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{79} + (\beta_{7} - \beta_{6} + 6 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 2) q^{85} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2) q^{89} + ( - \beta_{3} - 2) q^{91} + ( - 2 \beta_{7} + 2 \beta_{6} + 5 \beta_1) q^{97}+O(q^{100})$$ q + b7 * q^5 + (2*b2 - b1) * q^7 + (-b5 - b4 - b3) * q^11 + b2 * q^13 + (-b7 + b6 + b5 - b4 + b1) * q^17 + (-b7 + b6 + b5 - b4 + 2*b2 + b1) * q^23 + (-b6 - b5 - 2*b3 - b2) * q^25 + (b7 + b6 - b5 - b4 + 2) * q^29 + (-2*b5 - 2*b4 - b1 + 2) * q^35 + (b7 - b6 - b5 + b4 + b1) * q^37 + (b7 + b6 + b3 - 2) * q^41 + (-b7 + b6 - b5 + b4 - 2*b2) * q^43 + (-b7 + b6 - 2*b2) * q^47 + (-b7 - b6 + b5 + b4 - 3*b3 - 3) * q^49 + (-b7 + b6 + b5 - b4 - b1) * q^53 + (-b7 + 2*b6 - b4 - b3 - 3*b2 + 2*b1 + 1) * q^55 + (-b7 - b6 - 2*b5 - 2*b4 + 2) * q^59 + (2*b5 + 2*b4 + b3 + 2) * q^61 - b5 * q^65 + (-2*b7 + 2*b6 + 4*b2) * q^67 + (-b7 - b6 - 3*b3 - 4) * q^71 + (b7 - b6 - b5 + b4 + 2*b2 - 2*b1) * q^73 + (-3*b7 + 3*b6 + b5 - b4 - 2*b2 + b1) * q^77 + (2*b7 + 2*b6 - 2*b5 - 2*b4 - b3) * q^79 + (b7 - b6 + 6*b2 - 4*b1) * q^83 + (b7 + b6 + b5 + 3*b4 + 2*b3 - 4*b2 - b1 + 2) * q^85 + (b7 + b6 - 2*b5 - 2*b4 - 3*b3 + 2) * q^89 + (-b3 - 2) * q^91 + (-2*b7 + 2*b6 + 5*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{5}+O(q^{10})$$ 8 * q + 2 * q^5 $$8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100})$$ 8 * q + 2 * q^5 - 2 * q^11 + 16 * q^29 + 8 * q^35 - 14 * q^41 - 18 * q^49 + 10 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 - 30 * q^71 + 2 * q^79 + 24 * q^85 + 18 * q^89 - 14 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( 72\nu^{7} + 747\nu^{6} + 76\nu^{5} + 32\nu^{4} + 6804\nu^{3} + 63783\nu^{2} + 9448\nu - 2736 ) / 18170$$ (72*v^7 + 747*v^6 + 76*v^5 + 32*v^4 + 6804*v^3 + 63783*v^2 + 9448*v - 2736) / 18170 $$\beta_{2}$$ $$=$$ $$( -1881\nu^{7} + 2970\nu^{6} - 2894\nu^{5} - 836\nu^{4} - 167761\nu^{3} + 244926\nu^{2} - 234110\nu + 67844 ) / 36340$$ (-1881*v^7 + 2970*v^6 - 2894*v^5 - 836*v^4 - 167761*v^3 + 244926*v^2 - 234110*v + 67844) / 36340 $$\beta_{3}$$ $$=$$ $$( -1611\nu^{7} + 1683\nu^{6} - 792\nu^{5} - 716\nu^{4} - 144063\nu^{3} + 136611\nu^{2} - 60588\nu + 28512 ) / 18170$$ (-1611*v^7 + 1683*v^6 - 792*v^5 - 716*v^4 - 144063*v^3 + 136611*v^2 - 60588*v + 28512) / 18170 $$\beta_{4}$$ $$=$$ $$( 358\nu^{7} - 374\nu^{6} + 176\nu^{5} + 361\nu^{4} + 32014\nu^{3} - 30358\nu^{2} + 11647\nu + 2749 ) / 1817$$ (358*v^7 - 374*v^6 + 176*v^5 + 361*v^4 + 32014*v^3 - 30358*v^2 + 11647*v + 2749) / 1817 $$\beta_{5}$$ $$=$$ $$( 1988\nu^{7} - 2087\nu^{6} + 1089\nu^{5} + 1893\nu^{4} + 178781\nu^{3} - 169443\nu^{2} + 84217\nu + 6221 ) / 9085$$ (1988*v^7 - 2087*v^6 + 1089*v^5 + 1893*v^4 + 178781*v^3 - 169443*v^2 + 84217*v + 6221) / 9085 $$\beta_{6}$$ $$=$$ $$( 1665\nu^{7} - 3394\nu^{6} + 2666\nu^{5} + 740\nu^{4} + 147349\nu^{3} - 289098\nu^{2} + 213034\nu - 59636 ) / 7268$$ (1665*v^7 - 3394*v^6 + 2666*v^5 + 740*v^4 + 147349*v^3 - 289098*v^2 + 213034*v - 59636) / 7268 $$\beta_{7}$$ $$=$$ $$( - 9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276 ) / 36340$$ (-9117*v^7 + 17838*v^6 - 14166*v^5 - 4052*v^4 - 811589*v^3 + 1516102*v^2 - 1096418*v + 328276) / 36340
 $$\nu$$ $$=$$ $$( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 2$$ (b7 + b6 + b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 10\beta_{2} - 4\beta_1 ) / 2$$ (b7 - b6 - b5 + b4 - 10*b2 - 4*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -9\beta_{7} - 9\beta_{6} + 9\beta_{5} - 9\beta_{4} + 2\beta_{3} + 4\beta_{2} - 2\beta _1 + 4 ) / 2$$ (-9*b7 - 9*b6 + 9*b5 - 9*b4 + 2*b3 + 4*b2 - 2*b1 + 4) / 2 $$\nu^{4}$$ $$=$$ $$( 9\beta_{7} + 9\beta_{6} + 9\beta_{5} + 9\beta_{4} + 40\beta_{3} - 90 ) / 2$$ (9*b7 + 9*b6 + 9*b5 + 9*b4 + 40*b3 - 90) / 2 $$\nu^{5}$$ $$=$$ $$( -85\beta_{7} - 85\beta_{6} - 85\beta_{5} + 85\beta_{4} + 22\beta_{3} - 36\beta_{2} + 22\beta _1 + 36 ) / 2$$ (-85*b7 - 85*b6 - 85*b5 + 85*b4 + 22*b3 - 36*b2 + 22*b1 + 36) / 2 $$\nu^{6}$$ $$=$$ $$( -85\beta_{7} + 85\beta_{6} + 77\beta_{5} - 77\beta_{4} + 850\beta_{2} + 384\beta_1 ) / 2$$ (-85*b7 + 85*b6 + 77*b5 - 77*b4 + 850*b2 + 384*b1) / 2 $$\nu^{7}$$ $$=$$ $$( 801\beta_{7} + 809\beta_{6} - 809\beta_{5} + 801\beta_{4} - 230\beta_{3} - 300\beta_{2} + 230\beta _1 - 300 ) / 2$$ (801*b7 + 809*b6 - 809*b5 + 801*b4 - 230*b3 - 300*b2 + 230*b1 - 300) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$$.

 $$n$$ $$937$$ $$1081$$ $$2081$$ $$2341$$ $$3511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2809.1
 −2.15569 − 2.15569i −2.15569 + 2.15569i 0.353624 + 0.353624i 0.353624 − 0.353624i 0.594137 − 0.594137i 0.594137 + 0.594137i 2.20793 + 2.20793i 2.20793 − 2.20793i
0 0 0 −2.15569 0.594137i 0 0.633776i 0 0 0
2809.2 0 0 0 −2.15569 + 0.594137i 0 0.633776i 0 0 0
2809.3 0 0 0 0.353624 2.20793i 0 3.09417i 0 0 0
2809.4 0 0 0 0.353624 + 2.20793i 0 3.09417i 0 0 0
2809.5 0 0 0 0.594137 2.15569i 0 4.92778i 0 0 0
2809.6 0 0 0 0.594137 + 2.15569i 0 4.92778i 0 0 0
2809.7 0 0 0 2.20793 0.353624i 0 1.65573i 0 0 0
2809.8 0 0 0 2.20793 + 0.353624i 0 1.65573i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2809.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4680.2.l.g 8
3.b odd 2 1 1560.2.l.d 8
5.b even 2 1 inner 4680.2.l.g 8
12.b even 2 1 3120.2.l.n 8
15.d odd 2 1 1560.2.l.d 8
15.e even 4 1 7800.2.a.bt 4
15.e even 4 1 7800.2.a.by 4
60.h even 2 1 3120.2.l.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.d 8 3.b odd 2 1
1560.2.l.d 8 15.d odd 2 1
3120.2.l.n 8 12.b even 2 1
3120.2.l.n 8 60.h even 2 1
4680.2.l.g 8 1.a even 1 1 trivial
4680.2.l.g 8 5.b even 2 1 inner
7800.2.a.bt 4 15.e even 4 1
7800.2.a.by 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4680, [\chi])$$:

 $$T_{7}^{8} + 37T_{7}^{6} + 340T_{7}^{4} + 768T_{7}^{2} + 256$$ T7^8 + 37*T7^6 + 340*T7^4 + 768*T7^2 + 256 $$T_{11}^{4} + T_{11}^{3} - 20T_{11}^{2} + 26T_{11} - 4$$ T11^4 + T11^3 - 20*T11^2 + 26*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 2 T^{7} + 2 T^{6} + 6 T^{5} + \cdots + 625$$
$7$ $$T^{8} + 37 T^{6} + 340 T^{4} + \cdots + 256$$
$11$ $$(T^{4} + T^{3} - 20 T^{2} + 26 T - 4)^{2}$$
$13$ $$(T^{2} + 1)^{4}$$
$17$ $$T^{8} + 61 T^{6} + 908 T^{4} + \cdots + 64$$
$19$ $$T^{8}$$
$23$ $$T^{8} + 65 T^{6} + 1056 T^{4} + \cdots + 1024$$
$29$ $$(T^{4} - 8 T^{3} - 20 T^{2} + 144 T + 256)^{2}$$
$31$ $$T^{8}$$
$37$ $$T^{8} + 133 T^{6} + 5068 T^{4} + \cdots + 141376$$
$41$ $$(T^{4} + 7 T^{3} - 8 T^{2} - 22 T - 4)^{2}$$
$43$ $$T^{8} + 104 T^{6} + 3216 T^{4} + \cdots + 65536$$
$47$ $$T^{8} + 64 T^{6} + 1108 T^{4} + \cdots + 16384$$
$53$ $$T^{8} + 133 T^{6} + 5068 T^{4} + \cdots + 141376$$
$59$ $$(T^{4} - 2 T^{3} - 86 T^{2} + 36 T + 1072)^{2}$$
$61$ $$(T^{4} - 11 T^{3} - 22 T^{2} + 128 T + 32)^{2}$$
$67$ $$T^{8} + 192 T^{6} + 5440 T^{4} + \cdots + 65536$$
$71$ $$(T^{4} + 15 T^{3} - 30 T^{2} - 530 T + 1256)^{2}$$
$73$ $$T^{8} + 124 T^{6} + 3568 T^{4} + \cdots + 1024$$
$79$ $$(T^{4} - T^{3} - 176 T^{2} + 52 T + 7328)^{2}$$
$83$ $$T^{8} + 464 T^{6} + 65620 T^{4} + \cdots + 7311616$$
$89$ $$(T^{4} - 9 T^{3} - 136 T^{2} + 378 T + 268)^{2}$$
$97$ $$T^{8} + 545 T^{6} + \cdots + 41783296$$