# Properties

 Label 1089.1.s.b Level $1089$ Weight $1$ Character orbit 1089.s Analytic conductor $0.543$ Analytic rank $0$ Dimension $16$ Projective image $S_{4}$ CM/RM no Inner twists $16$

# Learn more

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: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{30})$$ Coefficient field: 16.0.26873856000000000000.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256$$ x^16 + 2*x^14 - 8*x^10 - 16*x^8 - 32*x^6 + 128*x^2 + 256 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.107811.1

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{3}) q^{5} - \beta_{8} q^{6} + \beta_{6} q^{7} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b7 * q^3 + b2 * q^4 + (b15 + b3) * q^5 - b8 * q^6 + b6 * q^7 + (-b15 + b9 + b7 + b5 - b3 - b2 - 1) * q^9 $$q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{15} + \beta_{3}) q^{5} - \beta_{8} q^{6} + \beta_{6} q^{7} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{9} + \beta_{13} q^{10} - \beta_{9} q^{12} + 2 \beta_{7} q^{14} + ( - \beta_{11} + \beta_{2}) q^{15} - \beta_{3} q^{16} + ( - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_1) q^{18} + ( - \beta_{15} + \beta_{9} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{20} - \beta_{13} q^{21} - \beta_{15} q^{27} + \beta_{8} q^{28} - \beta_{6} q^{29} + \beta_{14} q^{30} + (\beta_{15} - \beta_{9} - \beta_{7} + \beta_{3} + \beta_{2} + 1) q^{31} - \beta_{4} q^{32} + (\beta_{10} - \beta_1) q^{35} + (\beta_{15} + \beta_{11} - \beta_{5} + 1) q^{36} - \beta_{11} q^{37} + (2 \beta_{15} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{42}+ \cdots + \beta_{13} q^{98}+O(q^{100})$$ q + b1 * q^2 - b7 * q^3 + b2 * q^4 + (b15 + b3) * q^5 - b8 * q^6 + b6 * q^7 + (-b15 + b9 + b7 + b5 - b3 - b2 - 1) * q^9 + b13 * q^10 - b9 * q^12 + 2*b7 * q^14 + (-b11 + b2) * q^15 - b3 * q^16 + (-b14 - b13 - b12 + b10 + b8 + b6 - b1) * q^18 + (-b15 + b9 + b7 + b5 - b3 - b2 - 1) * q^20 - b13 * q^21 - b15 * q^27 + b8 * q^28 - b6 * q^29 + b14 * q^30 + (b15 - b9 - b7 + b3 + b2 + 1) * q^31 - b4 * q^32 + (b10 - b1) * q^35 + (b15 + b11 - b5 + 1) * q^36 - b11 * q^37 + (2*b15 - 2*b9 - 2*b7 - 2*b5 + 2*b3 + 2*b2 + 2) * q^42 - q^45 + (-b15 - b11 + b7 + b5 - 1) * q^47 + b11 * q^48 + (b15 + b3) * q^49 - b5 * q^53 + (-b13 + b4) * q^54 - 2*b7 * q^58 - b2 * q^59 - b15 * q^60 + (b13 + b12 - b6 - b4 + b1) * q^61 + (b14 + b13 + b12 - b10 - b8 + b1) * q^62 - b14 * q^63 - b5 * q^64 + (-b9 + 1) * q^67 + (2*b11 - 2*b2) * q^70 - b15 * q^71 + (-b14 - b13 - b12 + b10 + b8 - b1) * q^73 - b12 * q^74 + (-b15 - b11 + b5 - 1) * q^80 - b2 * q^81 + (b13 + b12 - b6 - b4 + b1) * q^83 + (b14 + b13 + b12 - b10 - b8 - b6 + b1) * q^84 + b13 * q^87 - b1 * q^90 - b3 * q^93 + (-b13 - b12 + b8 + b6 + b4 - b1) * q^94 + b12 * q^96 + (-b15 + b9 + b7 + b5 - b3 - b2 - 1) * q^97 + b13 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 16 * q - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^9 $$16 q - 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9} - 8 q^{12} + 4 q^{14} + 2 q^{15} - 2 q^{16} + 2 q^{20} + 4 q^{27} + 2 q^{31} + 4 q^{36} + 4 q^{37} - 4 q^{42} - 16 q^{45} - 2 q^{47} - 4 q^{48} - 2 q^{49} - 4 q^{53} - 4 q^{58} + 2 q^{59} + 4 q^{60} - 4 q^{64} + 8 q^{67} - 4 q^{70} + 4 q^{71} - 4 q^{80} + 2 q^{81} - 2 q^{93} + 2 q^{97}+O(q^{100})$$ 16 * q - 2 * q^3 - 2 * q^4 - 2 * q^5 + 2 * q^9 - 8 * q^12 + 4 * q^14 + 2 * q^15 - 2 * q^16 + 2 * q^20 + 4 * q^27 + 2 * q^31 + 4 * q^36 + 4 * q^37 - 4 * q^42 - 16 * q^45 - 2 * q^47 - 4 * q^48 - 2 * q^49 - 4 * q^53 - 4 * q^58 + 2 * q^59 + 4 * q^60 - 4 * q^64 + 8 * q^67 - 4 * q^70 + 4 * q^71 - 4 * q^80 + 2 * q^81 - 2 * q^93 + 2 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 2x^{14} - 8x^{10} - 16x^{8} - 32x^{6} + 128x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{5} ) / 4$$ (v^5) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{7} ) / 8$$ (v^7) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{8}$$ $$=$$ $$( \nu^{9} ) / 16$$ (v^9) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{10}$$ $$=$$ $$( \nu^{11} ) / 32$$ (v^11) / 32 $$\beta_{11}$$ $$=$$ $$( \nu^{12} ) / 64$$ (v^12) / 64 $$\beta_{12}$$ $$=$$ $$( \nu^{13} ) / 64$$ (v^13) / 64 $$\beta_{13}$$ $$=$$ $$( \nu^{15} ) / 128$$ (v^15) / 128 $$\beta_{14}$$ $$=$$ $$( -\nu^{13} + 32\nu^{3} ) / 64$$ (-v^13 + 32*v^3) / 64 $$\beta_{15}$$ $$=$$ $$( \nu^{14} - 32\nu^{4} ) / 128$$ (v^14 - 32*v^4) / 128
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{14} + 2\beta_{12}$$ 2*b14 + 2*b12 $$\nu^{4}$$ $$=$$ $$4\beta_{3}$$ 4*b3 $$\nu^{5}$$ $$=$$ $$4\beta_{4}$$ 4*b4 $$\nu^{6}$$ $$=$$ $$8\beta_{5}$$ 8*b5 $$\nu^{7}$$ $$=$$ $$8\beta_{6}$$ 8*b6 $$\nu^{8}$$ $$=$$ $$16\beta_{7}$$ 16*b7 $$\nu^{9}$$ $$=$$ $$16\beta_{8}$$ 16*b8 $$\nu^{10}$$ $$=$$ $$32\beta_{9}$$ 32*b9 $$\nu^{11}$$ $$=$$ $$32\beta_{10}$$ 32*b10 $$\nu^{12}$$ $$=$$ $$64\beta_{11}$$ 64*b11 $$\nu^{13}$$ $$=$$ $$64\beta_{12}$$ 64*b12 $$\nu^{14}$$ $$=$$ $$128\beta_{15} + 128\beta_{3}$$ 128*b15 + 128*b3 $$\nu^{15}$$ $$=$$ $$128\beta_{13}$$ 128*b13

## Character values

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

 $$n$$ $$244$$ $$848$$ $$\chi(n)$$ $$-\beta_{11}$$ $$-1 + \beta_{9}$$

## 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
 −1.40647 − 0.147826i 1.40647 + 0.147826i −1.05097 − 0.946294i 1.05097 + 0.946294i −0.294032 − 1.38331i 0.294032 + 1.38331i −0.575212 − 1.29195i 0.575212 + 1.29195i −0.294032 + 1.38331i 0.294032 − 1.38331i −1.05097 + 0.946294i 1.05097 − 0.946294i −0.575212 + 1.29195i 0.575212 − 1.29195i −1.40647 + 0.147826i 1.40647 − 0.147826i
−1.40647 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i 0.831254 + 1.14412i −1.05097 0.946294i 0 −0.104528 + 0.994522i 1.41421i
40.2 1.40647 + 0.147826i −0.669131 0.743145i 0.978148 + 0.207912i 0.104528 + 0.994522i −0.831254 1.14412i 1.05097 + 0.946294i 0 −0.104528 + 0.994522i 1.41421i
94.1 −1.05097 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i 1.34500 + 0.437016i −0.575212 + 1.29195i 0 0.669131 0.743145i 1.41421i
94.2 1.05097 + 0.946294i −0.913545 + 0.406737i 0.104528 + 0.994522i −0.669131 0.743145i −1.34500 0.437016i 0.575212 1.29195i 0 0.669131 0.743145i 1.41421i
112.1 −0.294032 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i 1.34500 0.437016i 1.40647 + 0.147826i 0 −0.978148 + 0.207912i 1.41421i
112.2 0.294032 + 1.38331i 0.104528 + 0.994522i −0.913545 + 0.406737i 0.978148 + 0.207912i −1.34500 + 0.437016i −1.40647 0.147826i 0 −0.978148 + 0.207912i 1.41421i
403.1 −0.575212 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i −0.831254 1.14412i 0.294032 1.38331i 0 0.913545 0.406737i 1.41421i
403.2 0.575212 + 1.29195i 0.978148 0.207912i −0.669131 + 0.743145i −0.913545 0.406737i 0.831254 + 1.14412i −0.294032 + 1.38331i 0 0.913545 0.406737i 1.41421i
457.1 −0.294032 + 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i 1.34500 + 0.437016i 1.40647 0.147826i 0 −0.978148 0.207912i 1.41421i
457.2 0.294032 1.38331i 0.104528 0.994522i −0.913545 0.406737i 0.978148 0.207912i −1.34500 0.437016i −1.40647 + 0.147826i 0 −0.978148 0.207912i 1.41421i
475.1 −1.05097 + 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i 1.34500 0.437016i −0.575212 1.29195i 0 0.669131 + 0.743145i 1.41421i
475.2 1.05097 0.946294i −0.913545 0.406737i 0.104528 0.994522i −0.669131 + 0.743145i −1.34500 + 0.437016i 0.575212 + 1.29195i 0 0.669131 + 0.743145i 1.41421i
481.1 −0.575212 + 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i −0.831254 + 1.14412i 0.294032 + 1.38331i 0 0.913545 + 0.406737i 1.41421i
481.2 0.575212 1.29195i 0.978148 + 0.207912i −0.669131 0.743145i −0.913545 + 0.406737i 0.831254 1.14412i −0.294032 1.38331i 0 0.913545 + 0.406737i 1.41421i
844.1 −1.40647 + 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i 0.831254 1.14412i −1.05097 + 0.946294i 0 −0.104528 0.994522i 1.41421i
844.2 1.40647 0.147826i −0.669131 + 0.743145i 0.978148 0.207912i 0.104528 0.994522i −0.831254 + 1.14412i 1.05097 0.946294i 0 −0.104528 0.994522i 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.2 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
11.b odd 2 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.b 16
3.b odd 2 1 3267.1.w.b 16
9.c even 3 1 inner 1089.1.s.b 16
9.d odd 6 1 3267.1.w.b 16
11.b odd 2 1 inner 1089.1.s.b 16
11.c even 5 1 1089.1.h.a 4
11.c even 5 3 inner 1089.1.s.b 16
11.d odd 10 1 1089.1.h.a 4
11.d odd 10 3 inner 1089.1.s.b 16
33.d even 2 1 3267.1.w.b 16
33.f even 10 1 3267.1.h.a 4
33.f even 10 3 3267.1.w.b 16
33.h odd 10 1 3267.1.h.a 4
33.h odd 10 3 3267.1.w.b 16
99.g even 6 1 3267.1.w.b 16
99.h odd 6 1 inner 1089.1.s.b 16
99.m even 15 1 1089.1.h.a 4
99.m even 15 3 inner 1089.1.s.b 16
99.n odd 30 1 3267.1.h.a 4
99.n odd 30 3 3267.1.w.b 16
99.o odd 30 1 1089.1.h.a 4
99.o odd 30 3 inner 1089.1.s.b 16
99.p even 30 1 3267.1.h.a 4
99.p even 30 3 3267.1.w.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1089.1.h.a 4 11.c even 5 1
1089.1.h.a 4 11.d odd 10 1
1089.1.h.a 4 99.m even 15 1
1089.1.h.a 4 99.o odd 30 1
1089.1.s.b 16 1.a even 1 1 trivial
1089.1.s.b 16 9.c even 3 1 inner
1089.1.s.b 16 11.b odd 2 1 inner
1089.1.s.b 16 11.c even 5 3 inner
1089.1.s.b 16 11.d odd 10 3 inner
1089.1.s.b 16 99.h odd 6 1 inner
1089.1.s.b 16 99.m even 15 3 inner
1089.1.s.b 16 99.o odd 30 3 inner
3267.1.h.a 4 33.f even 10 1
3267.1.h.a 4 33.h odd 10 1
3267.1.h.a 4 99.n odd 30 1
3267.1.h.a 4 99.p even 30 1
3267.1.w.b 16 3.b odd 2 1
3267.1.w.b 16 9.d odd 6 1
3267.1.w.b 16 33.d even 2 1
3267.1.w.b 16 33.f even 10 3
3267.1.w.b 16 33.h odd 10 3
3267.1.w.b 16 99.g even 6 1
3267.1.w.b 16 99.n odd 30 3
3267.1.w.b 16 99.p even 30 3

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 2T_{2}^{14} - 8T_{2}^{10} - 16T_{2}^{8} - 32T_{2}^{6} + 128T_{2}^{2} + 256$$ acting on $$S_{1}^{\mathrm{new}}(1089, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256$$
$3$ $$(T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2}$$
$5$ $$(T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2}$$
$7$ $$T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256$$
$11$ $$T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256$$
$31$ $$(T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2}$$
$37$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$(T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1)^{2}$$
$53$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{4}$$
$59$ $$(T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2}$$
$61$ $$T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256$$
$67$ $$(T^{2} - T + 1)^{8}$$
$71$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{4}$$
$73$ $$(T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16} + 2 T^{14} - 8 T^{10} - 16 T^{8} + \cdots + 256$$
$89$ $$T^{16}$$
$97$ $$(T^{8} - T^{7} + T^{5} - T^{4} + T^{3} - T + 1)^{2}$$
show more
show less