# Properties

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

# Learn more

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: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + (2 \beta - 2) q^{7} + ( - 7 \beta + 8) q^{8}+O(q^{10})$$ q + b * q^2 + b * q^4 + (4*b - 10) * q^5 + (2*b - 2) * q^7 + (-7*b + 8) * q^8 $$q + \beta q^{2} + \beta q^{4} + (4 \beta - 10) q^{5} + (2 \beta - 2) q^{7} + ( - 7 \beta + 8) q^{8} + ( - 6 \beta + 32) q^{10} + ( - 8 \beta + 42) q^{13} + 16 q^{14} + ( - 7 \beta - 56) q^{16} + (30 \beta - 28) q^{17} + (18 \beta + 18) q^{19} + ( - 6 \beta + 32) q^{20} - 112 q^{23} + ( - 64 \beta + 103) q^{25} + (34 \beta - 64) q^{26} + 16 q^{28} + (46 \beta + 88) q^{29} + (104 \beta - 72) q^{31} + ( - 7 \beta - 120) q^{32} + (2 \beta + 240) q^{34} + ( - 20 \beta + 84) q^{35} + (44 \beta - 46) q^{37} + (36 \beta + 144) q^{38} + (74 \beta - 304) q^{40} + (2 \beta - 248) q^{41} + (86 \beta - 10) q^{43} - 112 \beta q^{46} + (48 \beta + 8) q^{47} + ( - 4 \beta - 307) q^{49} + (39 \beta - 512) q^{50} + (34 \beta - 64) q^{52} + (128 \beta - 22) q^{53} + (16 \beta - 128) q^{56} + (134 \beta + 368) q^{58} - 196 q^{59} + (52 \beta + 526) q^{61} + (32 \beta + 832) q^{62} + ( - 71 \beta + 392) q^{64} + (216 \beta - 676) q^{65} + (152 \beta + 388) q^{67} + (2 \beta + 240) q^{68} + (64 \beta - 160) q^{70} + ( - 184 \beta - 136) q^{71} + (252 \beta + 170) q^{73} + ( - 2 \beta + 352) q^{74} + (36 \beta + 144) q^{76} + (74 \beta + 78) q^{79} + ( - 182 \beta + 336) q^{80} + ( - 246 \beta + 16) q^{82} + ( - 324 \beta + 336) q^{83} + ( - 292 \beta + 1240) q^{85} + (76 \beta + 688) q^{86} + ( - 8 \beta - 482) q^{89} + (84 \beta - 212) q^{91} - 112 \beta q^{92} + (56 \beta + 384) q^{94} + ( - 36 \beta + 396) q^{95} + (420 \beta - 802) q^{97} + ( - 311 \beta - 32) q^{98} +O(q^{100})$$ q + b * q^2 + b * q^4 + (4*b - 10) * q^5 + (2*b - 2) * q^7 + (-7*b + 8) * q^8 + (-6*b + 32) * q^10 + (-8*b + 42) * q^13 + 16 * q^14 + (-7*b - 56) * q^16 + (30*b - 28) * q^17 + (18*b + 18) * q^19 + (-6*b + 32) * q^20 - 112 * q^23 + (-64*b + 103) * q^25 + (34*b - 64) * q^26 + 16 * q^28 + (46*b + 88) * q^29 + (104*b - 72) * q^31 + (-7*b - 120) * q^32 + (2*b + 240) * q^34 + (-20*b + 84) * q^35 + (44*b - 46) * q^37 + (36*b + 144) * q^38 + (74*b - 304) * q^40 + (2*b - 248) * q^41 + (86*b - 10) * q^43 - 112*b * q^46 + (48*b + 8) * q^47 + (-4*b - 307) * q^49 + (39*b - 512) * q^50 + (34*b - 64) * q^52 + (128*b - 22) * q^53 + (16*b - 128) * q^56 + (134*b + 368) * q^58 - 196 * q^59 + (52*b + 526) * q^61 + (32*b + 832) * q^62 + (-71*b + 392) * q^64 + (216*b - 676) * q^65 + (152*b + 388) * q^67 + (2*b + 240) * q^68 + (64*b - 160) * q^70 + (-184*b - 136) * q^71 + (252*b + 170) * q^73 + (-2*b + 352) * q^74 + (36*b + 144) * q^76 + (74*b + 78) * q^79 + (-182*b + 336) * q^80 + (-246*b + 16) * q^82 + (-324*b + 336) * q^83 + (-292*b + 1240) * q^85 + (76*b + 688) * q^86 + (-8*b - 482) * q^89 + (84*b - 212) * q^91 - 112*b * q^92 + (56*b + 384) * q^94 + (-36*b + 396) * q^95 + (420*b - 802) * q^97 + (-311*b - 32) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 $$2 q + q^{2} + q^{4} - 16 q^{5} - 2 q^{7} + 9 q^{8} + 58 q^{10} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} + 54 q^{19} + 58 q^{20} - 224 q^{23} + 142 q^{25} - 94 q^{26} + 32 q^{28} + 222 q^{29} - 40 q^{31} - 247 q^{32} + 482 q^{34} + 148 q^{35} - 48 q^{37} + 324 q^{38} - 534 q^{40} - 494 q^{41} + 66 q^{43} - 112 q^{46} + 64 q^{47} - 618 q^{49} - 985 q^{50} - 94 q^{52} + 84 q^{53} - 240 q^{56} + 870 q^{58} - 392 q^{59} + 1104 q^{61} + 1696 q^{62} + 713 q^{64} - 1136 q^{65} + 928 q^{67} + 482 q^{68} - 256 q^{70} - 456 q^{71} + 592 q^{73} + 702 q^{74} + 324 q^{76} + 230 q^{79} + 490 q^{80} - 214 q^{82} + 348 q^{83} + 2188 q^{85} + 1452 q^{86} - 972 q^{89} - 340 q^{91} - 112 q^{92} + 824 q^{94} + 756 q^{95} - 1184 q^{97} - 375 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 - 16 * q^5 - 2 * q^7 + 9 * q^8 + 58 * q^10 + 76 * q^13 + 32 * q^14 - 119 * q^16 - 26 * q^17 + 54 * q^19 + 58 * q^20 - 224 * q^23 + 142 * q^25 - 94 * q^26 + 32 * q^28 + 222 * q^29 - 40 * q^31 - 247 * q^32 + 482 * q^34 + 148 * q^35 - 48 * q^37 + 324 * q^38 - 534 * q^40 - 494 * q^41 + 66 * q^43 - 112 * q^46 + 64 * q^47 - 618 * q^49 - 985 * q^50 - 94 * q^52 + 84 * q^53 - 240 * q^56 + 870 * q^58 - 392 * q^59 + 1104 * q^61 + 1696 * q^62 + 713 * q^64 - 1136 * q^65 + 928 * q^67 + 482 * q^68 - 256 * q^70 - 456 * q^71 + 592 * q^73 + 702 * q^74 + 324 * q^76 + 230 * q^79 + 490 * q^80 - 214 * q^82 + 348 * q^83 + 2188 * q^85 + 1452 * q^86 - 972 * q^89 - 340 * q^91 - 112 * q^92 + 824 * q^94 + 756 * q^95 - 1184 * q^97 - 375 * 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
 −2.37228 3.37228
−2.37228 0 −2.37228 −19.4891 0 −6.74456 24.6060 0 46.2337
1.2 3.37228 0 3.37228 3.48913 0 4.74456 −15.6060 0 11.7663
 $$n$$: 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.t 2
3.b odd 2 1 363.4.a.j 2
11.b odd 2 1 99.4.a.e 2
33.d even 2 1 33.4.a.d 2
44.c even 2 1 1584.4.a.x 2
55.d odd 2 1 2475.4.a.o 2
132.d odd 2 1 528.4.a.o 2
165.d even 2 1 825.4.a.k 2
165.l odd 4 2 825.4.c.i 4
231.h odd 2 1 1617.4.a.j 2
264.m even 2 1 2112.4.a.ba 2
264.p odd 2 1 2112.4.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 33.d even 2 1
99.4.a.e 2 11.b odd 2 1
363.4.a.j 2 3.b odd 2 1
528.4.a.o 2 132.d odd 2 1
825.4.a.k 2 165.d even 2 1
825.4.c.i 4 165.l odd 4 2
1089.4.a.t 2 1.a even 1 1 trivial
1584.4.a.x 2 44.c even 2 1
1617.4.a.j 2 231.h odd 2 1
2112.4.a.ba 2 264.m even 2 1
2112.4.a.bh 2 264.p odd 2 1
2475.4.a.o 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - T_{2} - 8$$ T2^2 - T2 - 8 $$T_{5}^{2} + 16T_{5} - 68$$ T5^2 + 16*T5 - 68 $$T_{7}^{2} + 2T_{7} - 32$$ T7^2 + 2*T7 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 8$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 16T - 68$$
$7$ $$T^{2} + 2T - 32$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 76T + 916$$
$17$ $$T^{2} + 26T - 7256$$
$19$ $$T^{2} - 54T - 1944$$
$23$ $$(T + 112)^{2}$$
$29$ $$T^{2} - 222T - 5136$$
$31$ $$T^{2} + 40T - 88832$$
$37$ $$T^{2} + 48T - 15396$$
$41$ $$T^{2} + 494T + 60976$$
$43$ $$T^{2} - 66T - 59928$$
$47$ $$T^{2} - 64T - 17984$$
$53$ $$T^{2} - 84T - 133404$$
$59$ $$(T + 196)^{2}$$
$61$ $$T^{2} - 1104 T + 282396$$
$67$ $$T^{2} - 928T + 24688$$
$71$ $$T^{2} + 456T - 227328$$
$73$ $$T^{2} - 592T - 436292$$
$79$ $$T^{2} - 230T - 31952$$
$83$ $$T^{2} - 348T - 835776$$
$89$ $$T^{2} + 972T + 235668$$
$97$ $$T^{2} + 1184 T - 1104836$$
show more
show less