# Properties

 Label 1089.4.a.u Level $1089$ Weight $4$ Character orbit 1089.a Self dual yes Analytic conductor $64.253$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)

## 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: yes Analytic conductor: $$64.2530799963$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{97})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + (4 \beta - 14) q^{7} + (9 \beta + 24) q^{8}+O(q^{10})$$ q + b * q^2 + (b + 16) * q^4 + (2*b + 6) * q^5 + (4*b - 14) * q^7 + (9*b + 24) * q^8 $$q + \beta q^{2} + (\beta + 16) q^{4} + (2 \beta + 6) q^{5} + (4 \beta - 14) q^{7} + (9 \beta + 24) q^{8} + (8 \beta + 48) q^{10} + ( - 2 \beta - 14) q^{13} + ( - 10 \beta + 96) q^{14} + (25 \beta + 88) q^{16} + ( - 14 \beta + 60) q^{17} + (2 \beta - 26) q^{19} + (40 \beta + 144) q^{20} + (10 \beta - 72) q^{23} + (28 \beta + 7) q^{25} + ( - 16 \beta - 48) q^{26} + (54 \beta - 128) q^{28} + ( - 6 \beta - 96) q^{29} + (8 \beta + 176) q^{31} + (41 \beta + 408) q^{32} + (46 \beta - 336) q^{34} + (4 \beta + 108) q^{35} + (52 \beta - 190) q^{37} + ( - 24 \beta + 48) q^{38} + (120 \beta + 576) q^{40} + ( - 14 \beta - 384) q^{41} + (26 \beta - 206) q^{43} + ( - 62 \beta + 240) q^{46} + ( - 74 \beta - 96) q^{47} + ( - 96 \beta + 237) q^{49} + (35 \beta + 672) q^{50} + ( - 48 \beta - 272) q^{52} + (54 \beta + 234) q^{53} + (6 \beta + 528) q^{56} + ( - 102 \beta - 144) q^{58} + (100 \beta + 36) q^{59} + ( - 34 \beta + 406) q^{61} + (184 \beta + 192) q^{62} + (249 \beta + 280) q^{64} + ( - 44 \beta - 180) q^{65} + ( - 96 \beta - 340) q^{67} + ( - 178 \beta + 624) q^{68} + (112 \beta + 96) q^{70} + ( - 54 \beta - 288) q^{71} + (28 \beta - 662) q^{73} + ( - 138 \beta + 1248) q^{74} + (8 \beta - 368) q^{76} + ( - 144 \beta - 254) q^{79} + (376 \beta + 1728) q^{80} + ( - 398 \beta - 336) q^{82} + (156 \beta - 240) q^{83} + (8 \beta - 312) q^{85} + ( - 180 \beta + 624) q^{86} + ( - 72 \beta + 414) q^{89} + ( - 36 \beta + 4) q^{91} + (98 \beta - 912) q^{92} + ( - 170 \beta - 1776) q^{94} + ( - 36 \beta - 60) q^{95} + (192 \beta - 322) q^{97} + (141 \beta - 2304) q^{98}+O(q^{100})$$ q + b * q^2 + (b + 16) * q^4 + (2*b + 6) * q^5 + (4*b - 14) * q^7 + (9*b + 24) * q^8 + (8*b + 48) * q^10 + (-2*b - 14) * q^13 + (-10*b + 96) * q^14 + (25*b + 88) * q^16 + (-14*b + 60) * q^17 + (2*b - 26) * q^19 + (40*b + 144) * q^20 + (10*b - 72) * q^23 + (28*b + 7) * q^25 + (-16*b - 48) * q^26 + (54*b - 128) * q^28 + (-6*b - 96) * q^29 + (8*b + 176) * q^31 + (41*b + 408) * q^32 + (46*b - 336) * q^34 + (4*b + 108) * q^35 + (52*b - 190) * q^37 + (-24*b + 48) * q^38 + (120*b + 576) * q^40 + (-14*b - 384) * q^41 + (26*b - 206) * q^43 + (-62*b + 240) * q^46 + (-74*b - 96) * q^47 + (-96*b + 237) * q^49 + (35*b + 672) * q^50 + (-48*b - 272) * q^52 + (54*b + 234) * q^53 + (6*b + 528) * q^56 + (-102*b - 144) * q^58 + (100*b + 36) * q^59 + (-34*b + 406) * q^61 + (184*b + 192) * q^62 + (249*b + 280) * q^64 + (-44*b - 180) * q^65 + (-96*b - 340) * q^67 + (-178*b + 624) * q^68 + (112*b + 96) * q^70 + (-54*b - 288) * q^71 + (28*b - 662) * q^73 + (-138*b + 1248) * q^74 + (8*b - 368) * q^76 + (-144*b - 254) * q^79 + (376*b + 1728) * q^80 + (-398*b - 336) * q^82 + (156*b - 240) * q^83 + (8*b - 312) * q^85 + (-180*b + 624) * q^86 + (-72*b + 414) * q^89 + (-36*b + 4) * q^91 + (98*b - 912) * q^92 + (-170*b - 1776) * q^94 + (-36*b - 60) * q^95 + (192*b - 322) * q^97 + (141*b - 2304) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8}+O(q^{10})$$ 2 * q + q^2 + 33 * q^4 + 14 * q^5 - 24 * q^7 + 57 * q^8 $$2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8} + 104 q^{10} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} - 50 q^{19} + 328 q^{20} - 134 q^{23} + 42 q^{25} - 112 q^{26} - 202 q^{28} - 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 220 q^{35} - 328 q^{37} + 72 q^{38} + 1272 q^{40} - 782 q^{41} - 386 q^{43} + 418 q^{46} - 266 q^{47} + 378 q^{49} + 1379 q^{50} - 592 q^{52} + 522 q^{53} + 1062 q^{56} - 390 q^{58} + 172 q^{59} + 778 q^{61} + 568 q^{62} + 809 q^{64} - 404 q^{65} - 776 q^{67} + 1070 q^{68} + 304 q^{70} - 630 q^{71} - 1296 q^{73} + 2358 q^{74} - 728 q^{76} - 652 q^{79} + 3832 q^{80} - 1070 q^{82} - 324 q^{83} - 616 q^{85} + 1068 q^{86} + 756 q^{89} - 28 q^{91} - 1726 q^{92} - 3722 q^{94} - 156 q^{95} - 452 q^{97} - 4467 q^{98}+O(q^{100})$$ 2 * q + q^2 + 33 * q^4 + 14 * q^5 - 24 * q^7 + 57 * q^8 + 104 * q^10 - 30 * q^13 + 182 * q^14 + 201 * q^16 + 106 * q^17 - 50 * q^19 + 328 * q^20 - 134 * q^23 + 42 * q^25 - 112 * q^26 - 202 * q^28 - 198 * q^29 + 360 * q^31 + 857 * q^32 - 626 * q^34 + 220 * q^35 - 328 * q^37 + 72 * q^38 + 1272 * q^40 - 782 * q^41 - 386 * q^43 + 418 * q^46 - 266 * q^47 + 378 * q^49 + 1379 * q^50 - 592 * q^52 + 522 * q^53 + 1062 * q^56 - 390 * q^58 + 172 * q^59 + 778 * q^61 + 568 * q^62 + 809 * q^64 - 404 * q^65 - 776 * q^67 + 1070 * q^68 + 304 * q^70 - 630 * q^71 - 1296 * q^73 + 2358 * q^74 - 728 * q^76 - 652 * q^79 + 3832 * q^80 - 1070 * q^82 - 324 * q^83 - 616 * q^85 + 1068 * q^86 + 756 * q^89 - 28 * q^91 - 1726 * q^92 - 3722 * q^94 - 156 * q^95 - 452 * q^97 - 4467 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.42443 5.42443
−4.42443 0 11.5756 −2.84886 0 −31.6977 −15.8199 0 12.6046
1.2 5.42443 0 21.4244 16.8489 0 7.69772 72.8199 0 91.3954
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1089.4.a.u 2
3.b odd 2 1 363.4.a.i 2
11.b odd 2 1 99.4.a.f 2
33.d even 2 1 33.4.a.c 2
44.c even 2 1 1584.4.a.bj 2
55.d odd 2 1 2475.4.a.p 2
132.d odd 2 1 528.4.a.p 2
165.d even 2 1 825.4.a.l 2
165.l odd 4 2 825.4.c.h 4
231.h odd 2 1 1617.4.a.k 2
264.m even 2 1 2112.4.a.bn 2
264.p odd 2 1 2112.4.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 33.d even 2 1
99.4.a.f 2 11.b odd 2 1
363.4.a.i 2 3.b odd 2 1
528.4.a.p 2 132.d odd 2 1
825.4.a.l 2 165.d even 2 1
825.4.c.h 4 165.l odd 4 2
1089.4.a.u 2 1.a even 1 1 trivial
1584.4.a.bj 2 44.c even 2 1
1617.4.a.k 2 231.h odd 2 1
2112.4.a.bg 2 264.p odd 2 1
2112.4.a.bn 2 264.m even 2 1
2475.4.a.p 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - T_{2} - 24$$ T2^2 - T2 - 24 $$T_{5}^{2} - 14T_{5} - 48$$ T5^2 - 14*T5 - 48 $$T_{7}^{2} + 24T_{7} - 244$$ T7^2 + 24*T7 - 244

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 24$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 14T - 48$$
$7$ $$T^{2} + 24T - 244$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 30T + 128$$
$17$ $$T^{2} - 106T - 1944$$
$19$ $$T^{2} + 50T + 528$$
$23$ $$T^{2} + 134T + 2064$$
$29$ $$T^{2} + 198T + 8928$$
$31$ $$T^{2} - 360T + 30848$$
$37$ $$T^{2} + 328T - 38676$$
$41$ $$T^{2} + 782T + 148128$$
$43$ $$T^{2} + 386T + 20856$$
$47$ $$T^{2} + 266T - 115104$$
$53$ $$T^{2} - 522T - 2592$$
$59$ $$T^{2} - 172T - 235104$$
$61$ $$T^{2} - 778T + 123288$$
$67$ $$T^{2} + 776T - 72944$$
$71$ $$T^{2} + 630T + 28512$$
$73$ $$T^{2} + 1296 T + 400892$$
$79$ $$T^{2} + 652T - 396572$$
$83$ $$T^{2} + 324T - 563904$$
$89$ $$T^{2} - 756T + 17172$$
$97$ $$T^{2} + 452T - 842876$$