Properties

 Label 1089.4.a.v 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: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + (2 \beta - 4) q^{4} + ( - 8 \beta - 1) q^{5} + (4 \beta - 10) q^{7} + ( - 10 \beta - 6) q^{8}+O(q^{10})$$ q + (b + 1) * q^2 + (2*b - 4) * q^4 + (-8*b - 1) * q^5 + (4*b - 10) * q^7 + (-10*b - 6) * q^8 $$q + (\beta + 1) q^{2} + (2 \beta - 4) q^{4} + ( - 8 \beta - 1) q^{5} + (4 \beta - 10) q^{7} + ( - 10 \beta - 6) q^{8} + ( - 9 \beta - 25) q^{10} + (20 \beta - 40) q^{13} + ( - 6 \beta + 2) q^{14} + ( - 32 \beta - 4) q^{16} + (12 \beta - 62) q^{17} + ( - 60 \beta - 36) q^{19} + (30 \beta - 44) q^{20} + (36 \beta + 49) q^{23} + (16 \beta + 68) q^{25} + ( - 20 \beta + 20) q^{26} + ( - 36 \beta + 64) q^{28} + ( - 56 \beta + 72) q^{29} + (28 \beta - 17) q^{31} + (44 \beta - 52) q^{32} + ( - 50 \beta - 26) q^{34} + (76 \beta - 86) q^{35} + ( - 8 \beta + 27) q^{37} + ( - 96 \beta - 216) q^{38} + (58 \beta + 246) q^{40} + ( - 4 \beta + 268) q^{41} + (16 \beta + 30) q^{43} + (85 \beta + 157) q^{46} + (120 \beta + 136) q^{47} + ( - 80 \beta - 195) q^{49} + (84 \beta + 116) q^{50} + ( - 160 \beta + 280) q^{52} + (56 \beta + 246) q^{53} + (76 \beta - 60) q^{56} + (16 \beta - 96) q^{58} + (132 \beta - 317) q^{59} + ( - 184 \beta - 420) q^{61} + (11 \beta + 67) q^{62} + (248 \beta + 112) q^{64} + (300 \beta - 440) q^{65} + ( - 20 \beta + 377) q^{67} + ( - 172 \beta + 320) q^{68} + ( - 10 \beta + 142) q^{70} + ( - 76 \beta + 339) q^{71} + (468 \beta + 200) q^{73} + (19 \beta + 3) q^{74} + (168 \beta - 216) q^{76} + ( - 656 \beta - 158) q^{79} + (64 \beta + 772) q^{80} + (264 \beta + 256) q^{82} + (120 \beta + 234) q^{83} + (484 \beta - 226) q^{85} + (46 \beta + 78) q^{86} + (328 \beta + 921) q^{89} + ( - 360 \beta + 640) q^{91} + ( - 46 \beta + 20) q^{92} + (256 \beta + 496) q^{94} + (348 \beta + 1476) q^{95} + (144 \beta + 1097) q^{97} + ( - 275 \beta - 435) q^{98}+O(q^{100})$$ q + (b + 1) * q^2 + (2*b - 4) * q^4 + (-8*b - 1) * q^5 + (4*b - 10) * q^7 + (-10*b - 6) * q^8 + (-9*b - 25) * q^10 + (20*b - 40) * q^13 + (-6*b + 2) * q^14 + (-32*b - 4) * q^16 + (12*b - 62) * q^17 + (-60*b - 36) * q^19 + (30*b - 44) * q^20 + (36*b + 49) * q^23 + (16*b + 68) * q^25 + (-20*b + 20) * q^26 + (-36*b + 64) * q^28 + (-56*b + 72) * q^29 + (28*b - 17) * q^31 + (44*b - 52) * q^32 + (-50*b - 26) * q^34 + (76*b - 86) * q^35 + (-8*b + 27) * q^37 + (-96*b - 216) * q^38 + (58*b + 246) * q^40 + (-4*b + 268) * q^41 + (16*b + 30) * q^43 + (85*b + 157) * q^46 + (120*b + 136) * q^47 + (-80*b - 195) * q^49 + (84*b + 116) * q^50 + (-160*b + 280) * q^52 + (56*b + 246) * q^53 + (76*b - 60) * q^56 + (16*b - 96) * q^58 + (132*b - 317) * q^59 + (-184*b - 420) * q^61 + (11*b + 67) * q^62 + (248*b + 112) * q^64 + (300*b - 440) * q^65 + (-20*b + 377) * q^67 + (-172*b + 320) * q^68 + (-10*b + 142) * q^70 + (-76*b + 339) * q^71 + (468*b + 200) * q^73 + (19*b + 3) * q^74 + (168*b - 216) * q^76 + (-656*b - 158) * q^79 + (64*b + 772) * q^80 + (264*b + 256) * q^82 + (120*b + 234) * q^83 + (484*b - 226) * q^85 + (46*b + 78) * q^86 + (328*b + 921) * q^89 + (-360*b + 640) * q^91 + (-46*b + 20) * q^92 + (256*b + 496) * q^94 + (348*b + 1476) * q^95 + (144*b + 1097) * q^97 + (-275*b - 435) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 8 q^{4} - 2 q^{5} - 20 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 8 * q^4 - 2 * q^5 - 20 * q^7 - 12 * q^8 $$2 q + 2 q^{2} - 8 q^{4} - 2 q^{5} - 20 q^{7} - 12 q^{8} - 50 q^{10} - 80 q^{13} + 4 q^{14} - 8 q^{16} - 124 q^{17} - 72 q^{19} - 88 q^{20} + 98 q^{23} + 136 q^{25} + 40 q^{26} + 128 q^{28} + 144 q^{29} - 34 q^{31} - 104 q^{32} - 52 q^{34} - 172 q^{35} + 54 q^{37} - 432 q^{38} + 492 q^{40} + 536 q^{41} + 60 q^{43} + 314 q^{46} + 272 q^{47} - 390 q^{49} + 232 q^{50} + 560 q^{52} + 492 q^{53} - 120 q^{56} - 192 q^{58} - 634 q^{59} - 840 q^{61} + 134 q^{62} + 224 q^{64} - 880 q^{65} + 754 q^{67} + 640 q^{68} + 284 q^{70} + 678 q^{71} + 400 q^{73} + 6 q^{74} - 432 q^{76} - 316 q^{79} + 1544 q^{80} + 512 q^{82} + 468 q^{83} - 452 q^{85} + 156 q^{86} + 1842 q^{89} + 1280 q^{91} + 40 q^{92} + 992 q^{94} + 2952 q^{95} + 2194 q^{97} - 870 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 8 * q^4 - 2 * q^5 - 20 * q^7 - 12 * q^8 - 50 * q^10 - 80 * q^13 + 4 * q^14 - 8 * q^16 - 124 * q^17 - 72 * q^19 - 88 * q^20 + 98 * q^23 + 136 * q^25 + 40 * q^26 + 128 * q^28 + 144 * q^29 - 34 * q^31 - 104 * q^32 - 52 * q^34 - 172 * q^35 + 54 * q^37 - 432 * q^38 + 492 * q^40 + 536 * q^41 + 60 * q^43 + 314 * q^46 + 272 * q^47 - 390 * q^49 + 232 * q^50 + 560 * q^52 + 492 * q^53 - 120 * q^56 - 192 * q^58 - 634 * q^59 - 840 * q^61 + 134 * q^62 + 224 * q^64 - 880 * q^65 + 754 * q^67 + 640 * q^68 + 284 * q^70 + 678 * q^71 + 400 * q^73 + 6 * q^74 - 432 * q^76 - 316 * q^79 + 1544 * q^80 + 512 * q^82 + 468 * q^83 - 452 * q^85 + 156 * q^86 + 1842 * q^89 + 1280 * q^91 + 40 * q^92 + 992 * q^94 + 2952 * q^95 + 2194 * q^97 - 870 * 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
 −1.73205 1.73205
−0.732051 0 −7.46410 12.8564 0 −16.9282 11.3205 0 −9.41154
1.2 2.73205 0 −0.535898 −14.8564 0 −3.07180 −23.3205 0 −40.5885
 $$n$$: 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.v 2
3.b odd 2 1 121.4.a.c 2
11.b odd 2 1 99.4.a.c 2
12.b even 2 1 1936.4.a.w 2
33.d even 2 1 11.4.a.a 2
33.f even 10 4 121.4.c.c 8
33.h odd 10 4 121.4.c.f 8
44.c even 2 1 1584.4.a.bc 2
55.d odd 2 1 2475.4.a.q 2
132.d odd 2 1 176.4.a.i 2
165.d even 2 1 275.4.a.b 2
165.l odd 4 2 275.4.b.c 4
231.h odd 2 1 539.4.a.e 2
264.m even 2 1 704.4.a.p 2
264.p odd 2 1 704.4.a.n 2
429.e even 2 1 1859.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 33.d even 2 1
99.4.a.c 2 11.b odd 2 1
121.4.a.c 2 3.b odd 2 1
121.4.c.c 8 33.f even 10 4
121.4.c.f 8 33.h odd 10 4
176.4.a.i 2 132.d odd 2 1
275.4.a.b 2 165.d even 2 1
275.4.b.c 4 165.l odd 4 2
539.4.a.e 2 231.h odd 2 1
704.4.a.n 2 264.p odd 2 1
704.4.a.p 2 264.m even 2 1
1089.4.a.v 2 1.a even 1 1 trivial
1584.4.a.bc 2 44.c even 2 1
1859.4.a.a 2 429.e even 2 1
1936.4.a.w 2 12.b even 2 1
2475.4.a.q 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - 2T_{2} - 2$$ T2^2 - 2*T2 - 2 $$T_{5}^{2} + 2T_{5} - 191$$ T5^2 + 2*T5 - 191 $$T_{7}^{2} + 20T_{7} + 52$$ T7^2 + 20*T7 + 52

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T - 191$$
$7$ $$T^{2} + 20T + 52$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 80T + 400$$
$17$ $$T^{2} + 124T + 3412$$
$19$ $$T^{2} + 72T - 9504$$
$23$ $$T^{2} - 98T - 1487$$
$29$ $$T^{2} - 144T - 4224$$
$31$ $$T^{2} + 34T - 2063$$
$37$ $$T^{2} - 54T + 537$$
$41$ $$T^{2} - 536T + 71776$$
$43$ $$T^{2} - 60T + 132$$
$47$ $$T^{2} - 272T - 24704$$
$53$ $$T^{2} - 492T + 51108$$
$59$ $$T^{2} + 634T + 48217$$
$61$ $$T^{2} + 840T + 74832$$
$67$ $$T^{2} - 754T + 140929$$
$71$ $$T^{2} - 678T + 97593$$
$73$ $$T^{2} - 400T - 617072$$
$79$ $$T^{2} + 316 T - 1266044$$
$83$ $$T^{2} - 468T + 11556$$
$89$ $$T^{2} - 1842 T + 525489$$
$97$ $$T^{2} - 2194 T + 1141201$$