# Properties

 Label 1080.2.f.g Level $1080$ Weight $2$ Character orbit 1080.f Analytic conductor $8.624$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(649,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1080.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.f (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: $$8.62384341830$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2702336256.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ x^8 + 9*x^6 + 56*x^4 + 225*x^2 + 625 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} - \beta_1 q^{7}+O(q^{10})$$ q + b3 * q^5 - b1 * q^7 $$q + \beta_{3} q^{5} - \beta_1 q^{7} + (2 \beta_{7} - \beta_{6} - 2 \beta_{3}) q^{11} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{13} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{17} + (\beta_{5} + \beta_{2} - 1) q^{19} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{23} + ( - 2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{25} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{29} + 3 q^{31} + (\beta_{7} + 2 \beta_{6} + \beta_{4}) q^{35} + (2 \beta_{5} - 2 \beta_{2}) q^{37} + (3 \beta_{7} + \beta_{6} - 3 \beta_{3}) q^{41} + (3 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{43} + (2 \beta_{7} - 4 \beta_{4} + 2 \beta_{3}) q^{47} + ( - 3 \beta_{5} - 3 \beta_{2} - 3) q^{49} + (2 \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{53} + (5 \beta_{5} - 2 \beta_{2} - 5) q^{55} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{59} + ( - \beta_{5} - \beta_{2} + 7) q^{61} + (\beta_{7} - 3 \beta_{6} + \cdots - \beta_{3}) q^{65}+ \cdots + ( - 5 \beta_{5} + 5 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100})$$ q + b3 * q^5 - b1 * q^7 + (2*b7 - b6 - 2*b3) * q^11 + (-b5 + b2 + b1) * q^13 + (b7 + b4 + b3) * q^17 + (b5 + b2 - 1) * q^19 + (b7 - b4 + b3) * q^23 + (-2*b5 + b2 - b1 - 2) * q^25 + (b7 + b6 - b3) * q^29 + 3 * q^31 + (b7 + 2*b6 + b4) * q^35 + (2*b5 - 2*b2) * q^37 + (3*b7 + b6 - 3*b3) * q^41 + (3*b5 - 3*b2 + 3*b1) * q^43 + (2*b7 - 4*b4 + 2*b3) * q^47 + (-3*b5 - 3*b2 - 3) * q^49 + (2*b7 + b4 + 2*b3) * q^53 + (5*b5 - 2*b2 - 5) * q^55 + (b7 - b6 - b3) * q^59 + (-b5 - b2 + 7) * q^61 + (b7 - 3*b6 - 4*b4 - b3) * q^65 + (-2*b5 + 2*b2) * q^67 + (-b7 - b6 + b3) * q^71 + (-2*b5 + 2*b2 - b1) * q^73 + (b7 + 6*b4 + b3) * q^77 + (b5 + b2 - 3) * q^79 + (2*b7 + 7*b4 + 2*b3) * q^83 + (-2*b5 - b1 - 7) * q^85 + (3*b7 - 3*b6 - 3*b3) * q^89 + (4*b5 + 4*b2 + 12) * q^91 + (2*b7 - b6 + 7*b4 - 2*b3) * q^95 + (-5*b5 + 5*b2 + 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 4 q^{19} - 18 q^{25} + 24 q^{31} - 36 q^{49} - 34 q^{55} + 52 q^{61} - 20 q^{79} - 60 q^{85} + 112 q^{91}+O(q^{100})$$ 8 * q - 4 * q^19 - 18 * q^25 + 24 * q^31 - 36 * q^49 - 34 * q^55 + 52 * q^61 - 20 * q^79 - 60 * q^85 + 112 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} - 84\nu^{4} - 756\nu^{2} - 2325 ) / 700$$ (-v^6 - 84*v^4 - 756*v^2 - 2325) / 700 $$\beta_{2}$$ $$=$$ $$( -11\nu^{6} - 224\nu^{4} - 616\nu^{2} - 2475 ) / 1400$$ (-11*v^6 - 224*v^4 - 616*v^2 - 2475) / 1400 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 279\nu ) / 280$$ (-v^7 + 279*v) / 280 $$\beta_{4}$$ $$=$$ $$( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750$$ (-9*v^7 - 56*v^5 - 154*v^3 - 625*v) / 1750 $$\beta_{5}$$ $$=$$ $$( -9\nu^{6} - 56\nu^{4} - 224\nu^{2} - 625 ) / 280$$ (-9*v^6 - 56*v^4 - 224*v^2 - 625) / 280 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} + 52\nu^{5} + 268\nu^{3} + 575\nu ) / 500$$ (3*v^7 + 52*v^5 + 268*v^3 + 575*v) / 500 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} + 56\nu^{5} + 504\nu^{3} + 2025\nu ) / 1400$$ (9*v^7 + 56*v^5 + 504*v^3 + 2025*v) / 1400
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} - 2\beta_{4} + 3\beta_{3} ) / 4$$ (b7 - b6 - 2*b4 + 3*b3) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} + 5\beta_{2} - 5\beta _1 - 10 ) / 4$$ (-b5 + 5*b2 - 5*b1 - 10) / 4 $$\nu^{3}$$ $$=$$ $$3\beta_{7} + \beta_{6} + 7\beta_{4} - 3\beta_{3}$$ 3*b7 + b6 + 7*b4 - 3*b3 $$\nu^{4}$$ $$=$$ $$( 11\beta_{5} - 47\beta_{2} + 11\beta _1 - 22 ) / 4$$ (11*b5 - 47*b2 + 11*b1 - 22) / 4 $$\nu^{5}$$ $$=$$ $$( -89\beta_{7} + 45\beta_{6} - 90\beta_{4} + 45\beta_{3} ) / 4$$ (-89*b7 + 45*b6 - 90*b4 + 45*b3) / 4 $$\nu^{6}$$ $$=$$ $$-42\beta_{5} + 42\beta_{2} + 14\beta _1 + 27$$ -42*b5 + 42*b2 + 14*b1 + 27 $$\nu^{7}$$ $$=$$ $$( 279\beta_{7} - 279\beta_{6} - 558\beta_{4} - 283\beta_{3} ) / 4$$ (279*b7 - 279*b6 - 558*b4 - 283*b3) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 −0.656712 − 2.13746i −0.656712 + 2.13746i −1.52274 − 1.63746i −1.52274 + 1.63746i 1.52274 − 1.63746i 1.52274 + 1.63746i 0.656712 − 2.13746i 0.656712 + 2.13746i
0 0 0 −1.52274 1.63746i 0 0.418627i 0 0 0
649.2 0 0 0 −1.52274 + 1.63746i 0 0.418627i 0 0 0
649.3 0 0 0 −0.656712 2.13746i 0 4.77753i 0 0 0
649.4 0 0 0 −0.656712 + 2.13746i 0 4.77753i 0 0 0
649.5 0 0 0 0.656712 2.13746i 0 4.77753i 0 0 0
649.6 0 0 0 0.656712 + 2.13746i 0 4.77753i 0 0 0
649.7 0 0 0 1.52274 1.63746i 0 0.418627i 0 0 0
649.8 0 0 0 1.52274 + 1.63746i 0 0.418627i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.f.g 8
3.b odd 2 1 inner 1080.2.f.g 8
4.b odd 2 1 2160.2.f.o 8
5.b even 2 1 inner 1080.2.f.g 8
5.c odd 4 1 5400.2.a.cg 4
5.c odd 4 1 5400.2.a.ch 4
12.b even 2 1 2160.2.f.o 8
15.d odd 2 1 inner 1080.2.f.g 8
15.e even 4 1 5400.2.a.cg 4
15.e even 4 1 5400.2.a.ch 4
20.d odd 2 1 2160.2.f.o 8
60.h even 2 1 2160.2.f.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.f.g 8 1.a even 1 1 trivial
1080.2.f.g 8 3.b odd 2 1 inner
1080.2.f.g 8 5.b even 2 1 inner
1080.2.f.g 8 15.d odd 2 1 inner
2160.2.f.o 8 4.b odd 2 1
2160.2.f.o 8 12.b even 2 1
2160.2.f.o 8 20.d odd 2 1
2160.2.f.o 8 60.h even 2 1
5400.2.a.cg 4 5.c odd 4 1
5400.2.a.cg 4 15.e even 4 1
5400.2.a.ch 4 5.c odd 4 1
5400.2.a.ch 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 $$S_{2}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}^{4} + 23T_{7}^{2} + 4$$ T7^4 + 23*T7^2 + 4 $$T_{11}^{4} - 47T_{11}^{2} + 196$$ T11^4 - 47*T11^2 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 9 T^{6} + \cdots + 625$$
$7$ $$(T^{4} + 23 T^{2} + 4)^{2}$$
$11$ $$(T^{4} - 47 T^{2} + 196)^{2}$$
$13$ $$(T^{4} + 44 T^{2} + 256)^{2}$$
$17$ $$(T^{4} + 33 T^{2} + 144)^{2}$$
$19$ $$(T^{2} + T - 14)^{4}$$
$23$ $$(T^{4} + 29 T^{2} + 196)^{2}$$
$29$ $$(T^{4} - 44 T^{2} + 256)^{2}$$
$31$ $$(T - 3)^{8}$$
$37$ $$(T^{4} + 44 T^{2} + 256)^{2}$$
$41$ $$(T^{2} - 76)^{4}$$
$43$ $$(T^{2} + 108)^{4}$$
$47$ $$(T^{4} + 132 T^{2} + 2304)^{2}$$
$53$ $$(T^{4} + 122 T^{2} + 2809)^{2}$$
$59$ $$(T^{2} - 12)^{4}$$
$61$ $$(T^{2} - 13 T + 28)^{4}$$
$67$ $$(T^{4} + 44 T^{2} + 256)^{2}$$
$71$ $$(T^{4} - 44 T^{2} + 256)^{2}$$
$73$ $$(T^{4} + 47 T^{2} + 196)^{2}$$
$79$ $$(T^{2} + 5 T - 8)^{4}$$
$83$ $$(T^{4} + 242 T^{2} + 49)^{2}$$
$89$ $$(T^{2} - 108)^{4}$$
$97$ $$(T^{4} + 467 T^{2} + 53824)^{2}$$