Properties

 Label 1089.4.a.e Level $1089$ Weight $4$ Character orbit 1089.a Self dual yes Analytic conductor $64.253$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 7 q^{4} + 4 q^{5} + 26 q^{7} + 15 q^{8}+O(q^{10})$$ q - q^2 - 7 * q^4 + 4 * q^5 + 26 * q^7 + 15 * q^8 $$q - q^{2} - 7 q^{4} + 4 q^{5} + 26 q^{7} + 15 q^{8} - 4 q^{10} + 32 q^{13} - 26 q^{14} + 41 q^{16} + 74 q^{17} + 60 q^{19} - 28 q^{20} + 182 q^{23} - 109 q^{25} - 32 q^{26} - 182 q^{28} - 90 q^{29} - 8 q^{31} - 161 q^{32} - 74 q^{34} + 104 q^{35} - 66 q^{37} - 60 q^{38} + 60 q^{40} + 422 q^{41} - 408 q^{43} - 182 q^{46} + 506 q^{47} + 333 q^{49} + 109 q^{50} - 224 q^{52} - 348 q^{53} + 390 q^{56} + 90 q^{58} + 200 q^{59} - 132 q^{61} + 8 q^{62} - 167 q^{64} + 128 q^{65} - 1036 q^{67} - 518 q^{68} - 104 q^{70} - 762 q^{71} + 542 q^{73} + 66 q^{74} - 420 q^{76} + 550 q^{79} + 164 q^{80} - 422 q^{82} - 132 q^{83} + 296 q^{85} + 408 q^{86} - 570 q^{89} + 832 q^{91} - 1274 q^{92} - 506 q^{94} + 240 q^{95} + 14 q^{97} - 333 q^{98}+O(q^{100})$$ q - q^2 - 7 * q^4 + 4 * q^5 + 26 * q^7 + 15 * q^8 - 4 * q^10 + 32 * q^13 - 26 * q^14 + 41 * q^16 + 74 * q^17 + 60 * q^19 - 28 * q^20 + 182 * q^23 - 109 * q^25 - 32 * q^26 - 182 * q^28 - 90 * q^29 - 8 * q^31 - 161 * q^32 - 74 * q^34 + 104 * q^35 - 66 * q^37 - 60 * q^38 + 60 * q^40 + 422 * q^41 - 408 * q^43 - 182 * q^46 + 506 * q^47 + 333 * q^49 + 109 * q^50 - 224 * q^52 - 348 * q^53 + 390 * q^56 + 90 * q^58 + 200 * q^59 - 132 * q^61 + 8 * q^62 - 167 * q^64 + 128 * q^65 - 1036 * q^67 - 518 * q^68 - 104 * q^70 - 762 * q^71 + 542 * q^73 + 66 * q^74 - 420 * q^76 + 550 * q^79 + 164 * q^80 - 422 * q^82 - 132 * q^83 + 296 * q^85 + 408 * q^86 - 570 * q^89 + 832 * q^91 - 1274 * q^92 - 506 * q^94 + 240 * q^95 + 14 * q^97 - 333 * 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
 0
−1.00000 0 −7.00000 4.00000 0 26.0000 15.0000 0 −4.00000
 $$n$$: 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.e 1
3.b odd 2 1 363.4.a.d 1
11.b odd 2 1 99.4.a.a 1
33.d even 2 1 33.4.a.b 1
44.c even 2 1 1584.4.a.l 1
55.d odd 2 1 2475.4.a.e 1
132.d odd 2 1 528.4.a.h 1
165.d even 2 1 825.4.a.f 1
165.l odd 4 2 825.4.c.f 2
231.h odd 2 1 1617.4.a.d 1
264.m even 2 1 2112.4.a.u 1
264.p odd 2 1 2112.4.a.h 1

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

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 26$$ T7 - 26

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 4$$
$7$ $$T - 26$$
$11$ $$T$$
$13$ $$T - 32$$
$17$ $$T - 74$$
$19$ $$T - 60$$
$23$ $$T - 182$$
$29$ $$T + 90$$
$31$ $$T + 8$$
$37$ $$T + 66$$
$41$ $$T - 422$$
$43$ $$T + 408$$
$47$ $$T - 506$$
$53$ $$T + 348$$
$59$ $$T - 200$$
$61$ $$T + 132$$
$67$ $$T + 1036$$
$71$ $$T + 762$$
$73$ $$T - 542$$
$79$ $$T - 550$$
$83$ $$T + 132$$
$89$ $$T + 570$$
$97$ $$T - 14$$