# Properties

 Label 1089.1.s.a Level $1089$ Weight $1$ Character orbit 1089.s Analytic conductor $0.543$ Analytic rank $0$ Dimension $8$ Projective image $D_{3}$ CM discriminant -11 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,1,Mod(40,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([10, 21]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1089.40");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1089.s (of order $$30$$, degree $$8$$, 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: $$0.543481798757$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1$$ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 99) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.891.1 Artin image: $S_3\times C_{15}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{30} + \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + \zeta_{30} q^{5} + \zeta_{30}^{14} q^{9} +O(q^{10})$$ q - z^7 * q^3 - z^13 * q^4 + z * q^5 + z^14 * q^9 $$q - \zeta_{30}^{7} q^{3} - \zeta_{30}^{13} q^{4} + \zeta_{30} q^{5} + \zeta_{30}^{14} q^{9} - \zeta_{30}^{5} q^{12} - \zeta_{30}^{8} q^{15} - \zeta_{30}^{11} q^{16} - \zeta_{30}^{14} q^{20} + \zeta_{30}^{10} q^{23} + \zeta_{30}^{6} q^{27} - \zeta_{30}^{4} q^{31} + \zeta_{30}^{12} q^{36} + \zeta_{30}^{3} q^{37} - q^{45} - \zeta_{30}^{2} q^{47} - \zeta_{30}^{3} q^{48} - \zeta_{30} q^{49} + \zeta_{30}^{9} q^{53} + \zeta_{30}^{13} q^{59} - \zeta_{30}^{6} q^{60} - \zeta_{30}^{9} q^{64} - \zeta_{30}^{10} q^{67} + 2 \zeta_{30}^{2} q^{69} - \zeta_{30}^{6} q^{71} - \zeta_{30}^{12} q^{80} - \zeta_{30}^{13} q^{81} + q^{89} + 2 \zeta_{30}^{8} q^{92} + \zeta_{30}^{11} q^{93} - \zeta_{30}^{14} q^{97} +O(q^{100})$$ q - z^7 * q^3 - z^13 * q^4 + z * q^5 + z^14 * q^9 - z^5 * q^12 - z^8 * q^15 - z^11 * q^16 - z^14 * q^20 + z^10 * q^23 + z^6 * q^27 - z^4 * q^31 + z^12 * q^36 + z^3 * q^37 - q^45 - z^2 * q^47 - z^3 * q^48 - z * q^49 + z^9 * q^53 + z^13 * q^59 - z^6 * q^60 - z^9 * q^64 - z^10 * q^67 + 2*z^2 * q^69 - z^6 * q^71 - z^12 * q^80 - z^13 * q^81 + q^89 + 2*z^8 * q^92 + z^11 * q^93 - z^14 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{3} + q^{4} - q^{5} + q^{9}+O(q^{10})$$ 8 * q + q^3 + q^4 - q^5 + q^9 $$8 q + q^{3} + q^{4} - q^{5} + q^{9} - 4 q^{12} - q^{15} + q^{16} - q^{20} - 8 q^{23} - 2 q^{27} - q^{31} - 2 q^{36} + 2 q^{37} - 8 q^{45} - q^{47} - 2 q^{48} + q^{49} + 2 q^{53} - q^{59} + 2 q^{60} - 2 q^{64} + 4 q^{67} + 2 q^{69} + 2 q^{71} + 2 q^{80} + q^{81} + 16 q^{89} + 2 q^{92} - q^{93} - q^{97}+O(q^{100})$$ 8 * q + q^3 + q^4 - q^5 + q^9 - 4 * q^12 - q^15 + q^16 - q^20 - 8 * q^23 - 2 * q^27 - q^31 - 2 * q^36 + 2 * q^37 - 8 * q^45 - q^47 - 2 * q^48 + q^49 + 2 * q^53 - q^59 + 2 * q^60 - 2 * q^64 + 4 * q^67 + 2 * q^69 + 2 * q^71 + 2 * q^80 + q^81 + 16 * q^89 + 2 * q^92 - q^93 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$\zeta_{30}^{3}$$ $$\zeta_{30}^{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
40.1
 −0.104528 − 0.994522i 0.669131 + 0.743145i −0.978148 − 0.207912i 0.913545 + 0.406737i −0.978148 + 0.207912i 0.669131 − 0.743145i 0.913545 − 0.406737i −0.104528 + 0.994522i
0 0.669131 + 0.743145i −0.978148 0.207912i 0.104528 + 0.994522i 0 0 0 −0.104528 + 0.994522i 0
94.1 0 0.913545 0.406737i −0.104528 0.994522i −0.669131 0.743145i 0 0 0 0.669131 0.743145i 0
112.1 0 −0.104528 0.994522i 0.913545 0.406737i 0.978148 + 0.207912i 0 0 0 −0.978148 + 0.207912i 0
403.1 0 −0.978148 + 0.207912i 0.669131 0.743145i −0.913545 0.406737i 0 0 0 0.913545 0.406737i 0
457.1 0 −0.104528 + 0.994522i 0.913545 + 0.406737i 0.978148 0.207912i 0 0 0 −0.978148 0.207912i 0
475.1 0 0.913545 + 0.406737i −0.104528 + 0.994522i −0.669131 + 0.743145i 0 0 0 0.669131 + 0.743145i 0
481.1 0 −0.978148 0.207912i 0.669131 + 0.743145i −0.913545 + 0.406737i 0 0 0 0.913545 + 0.406737i 0
844.1 0 0.669131 0.743145i −0.978148 + 0.207912i 0.104528 0.994522i 0 0 0 −0.104528 0.994522i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
9.c even 3 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
99.h odd 6 1 inner
99.m even 15 3 inner
99.o odd 30 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.1.s.a 8
3.b odd 2 1 3267.1.w.a 8
9.c even 3 1 inner 1089.1.s.a 8
9.d odd 6 1 3267.1.w.a 8
11.b odd 2 1 CM 1089.1.s.a 8
11.c even 5 1 99.1.h.a 2
11.c even 5 3 inner 1089.1.s.a 8
11.d odd 10 1 99.1.h.a 2
11.d odd 10 3 inner 1089.1.s.a 8
33.d even 2 1 3267.1.w.a 8
33.f even 10 1 297.1.h.a 2
33.f even 10 3 3267.1.w.a 8
33.h odd 10 1 297.1.h.a 2
33.h odd 10 3 3267.1.w.a 8
44.g even 10 1 1584.1.bf.b 2
44.h odd 10 1 1584.1.bf.b 2
55.h odd 10 1 2475.1.y.a 2
55.j even 10 1 2475.1.y.a 2
55.k odd 20 2 2475.1.t.a 4
55.l even 20 2 2475.1.t.a 4
99.g even 6 1 3267.1.w.a 8
99.h odd 6 1 inner 1089.1.s.a 8
99.m even 15 1 99.1.h.a 2
99.m even 15 1 891.1.c.a 1
99.m even 15 3 inner 1089.1.s.a 8
99.n odd 30 1 297.1.h.a 2
99.n odd 30 1 891.1.c.b 1
99.n odd 30 3 3267.1.w.a 8
99.o odd 30 1 99.1.h.a 2
99.o odd 30 1 891.1.c.a 1
99.o odd 30 3 inner 1089.1.s.a 8
99.p even 30 1 297.1.h.a 2
99.p even 30 1 891.1.c.b 1
99.p even 30 3 3267.1.w.a 8
396.be odd 30 1 1584.1.bf.b 2
396.bf even 30 1 1584.1.bf.b 2
495.bl even 30 1 2475.1.y.a 2
495.br odd 30 1 2475.1.y.a 2
495.bs even 60 2 2475.1.t.a 4
495.bt odd 60 2 2475.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 11.c even 5 1
99.1.h.a 2 11.d odd 10 1
99.1.h.a 2 99.m even 15 1
99.1.h.a 2 99.o odd 30 1
297.1.h.a 2 33.f even 10 1
297.1.h.a 2 33.h odd 10 1
297.1.h.a 2 99.n odd 30 1
297.1.h.a 2 99.p even 30 1
891.1.c.a 1 99.m even 15 1
891.1.c.a 1 99.o odd 30 1
891.1.c.b 1 99.n odd 30 1
891.1.c.b 1 99.p even 30 1
1089.1.s.a 8 1.a even 1 1 trivial
1089.1.s.a 8 9.c even 3 1 inner
1089.1.s.a 8 11.b odd 2 1 CM
1089.1.s.a 8 11.c even 5 3 inner
1089.1.s.a 8 11.d odd 10 3 inner
1089.1.s.a 8 99.h odd 6 1 inner
1089.1.s.a 8 99.m even 15 3 inner
1089.1.s.a 8 99.o odd 30 3 inner
1584.1.bf.b 2 44.g even 10 1
1584.1.bf.b 2 44.h odd 10 1
1584.1.bf.b 2 396.be odd 30 1
1584.1.bf.b 2 396.bf even 30 1
2475.1.t.a 4 55.k odd 20 2
2475.1.t.a 4 55.l even 20 2
2475.1.t.a 4 495.bs even 60 2
2475.1.t.a 4 495.bt odd 60 2
2475.1.y.a 2 55.h odd 10 1
2475.1.y.a 2 55.j even 10 1
2475.1.y.a 2 495.bl even 30 1
2475.1.y.a 2 495.br odd 30 1
3267.1.w.a 8 3.b odd 2 1
3267.1.w.a 8 9.d odd 6 1
3267.1.w.a 8 33.d even 2 1
3267.1.w.a 8 33.f even 10 3
3267.1.w.a 8 33.h odd 10 3
3267.1.w.a 8 99.g even 6 1
3267.1.w.a 8 99.n odd 30 3
3267.1.w.a 8 99.p even 30 3

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{1}^{\mathrm{new}}(1089, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1$$
$5$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$(T^{2} + 2 T + 4)^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$37$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$53$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$59$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$
$61$ $$T^{8}$$
$67$ $$(T^{2} - T + 1)^{4}$$
$71$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$(T - 2)^{8}$$
$97$ $$T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1$$