LMFDB - Embedded newform 1089.3.b.i.485.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(485,1089)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.b (of order $2$, degree $1$, 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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 921x^{12} + 8986x^{10} + 46812x^{8} + 125072x^{6} + 152129x^{4} + 65614x^{2} + 5041 $ x^16 + 48x^14 + 921x^12 + 8986x^10 + 46812x^8 + 125072x^6 + 152129x^4 + 65614*x^2 + 5041
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.1
Root			$-3.62039i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.i.485.16

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62039i q^{2} -9.10725 q^{4} +0.165198i q^{5} +10.2989 q^{7} +18.4903i q^{8} +O(q^{10})$	$q-3.62039i q^{2} -9.10725 q^{4} +0.165198i q^{5} +10.2989 q^{7} +18.4903i q^{8} +0.598083 q^{10} -20.3554 q^{13} -37.2859i q^{14} +30.5130 q^{16} +23.0147i q^{17} -4.13897 q^{19} -1.50450i q^{20} +13.5095i q^{23} +24.9727 q^{25} +73.6945i q^{26} -93.7943 q^{28} +4.01646i q^{29} -20.5033 q^{31} -36.5080i q^{32} +83.3223 q^{34} +1.70135i q^{35} +43.5939 q^{37} +14.9847i q^{38} -3.05456 q^{40} +52.0734i q^{41} +62.5698 q^{43} +48.9097 q^{46} +38.7052i q^{47} +57.0665 q^{49} -90.4110i q^{50} +185.382 q^{52} +35.3780i q^{53} +190.428i q^{56} +14.5412 q^{58} +19.6883i q^{59} -6.42431 q^{61} +74.2301i q^{62} -10.1214 q^{64} -3.36267i q^{65} -3.98708 q^{67} -209.601i q^{68} +6.15957 q^{70} +54.7712i q^{71} -22.8087 q^{73} -157.827i q^{74} +37.6946 q^{76} -57.1563 q^{79} +5.04069i q^{80} +188.526 q^{82} -56.8060i q^{83} -3.80199 q^{85} -226.527i q^{86} +28.1602i q^{89} -209.637 q^{91} -123.034i q^{92} +140.128 q^{94} -0.683750i q^{95} +82.9323 q^{97} -206.603i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 - 8 * q^7	$ 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 - 8 * q^7 + 24 * q^10 + 4 * q^13 + 28 * q^16 - 20 * q^19 - 44 * q^25 + 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 + 224 * q^40 + 272 * q^43 + 208 * q^46 + 348 * q^49 + 520 * q^52 - 44 * q^58 + 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 - 4 * q^73 + 1052 * q^76 + 216 * q^79 + 348 * q^82 + 416 * q^85 - 168 * q^91 + 1140 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$244$	$848$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 3.62039i	− 1.81020i	−0.425202	−	0.905098i	$-0.639797\pi$
$2$	− 3.62039i	− 1.81020i	0.425202	−	0.905098i	$-0.360203\pi$
$3$	0	0
$3$	0	0
$4$	−9.10725	−2.27681
$5$	0.165198i	0.0330396i	0.999864	+	0.0165198i	$0.00525866\pi$
$5$	0.165198i	0.0330396i	−0.999864	+	0.0165198i	$0.994741\pi$
$6$	0	0
$7$	10.2989	1.47127	0.735633	−	0.677381i	$-0.236885\pi$
$7$	10.2989	1.47127	0.735633	+	0.677381i	$0.236885\pi$
$8$	18.4903i	2.31128i
$9$	0	0
$10$	0.598083	0.0598083
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−20.3554	−1.56580	−0.782899	−	0.622148i	$-0.786260\pi$
$13$	−20.3554	−1.56580	−0.782899	+	0.622148i	$0.786260\pi$
$14$	− 37.2859i	− 2.66328i
$15$	0	0
$16$	30.5130	1.90706
$17$	23.0147i	1.35381i	0.736072	+	0.676903i	$0.236679\pi$
$17$	23.0147i	1.35381i	−0.736072	+	0.676903i	$0.763321\pi$
$18$	0	0
$19$	−4.13897	−0.217840	−0.108920	−	0.994051i	$-0.534739\pi$
$19$	−4.13897	−0.217840	−0.108920	+	0.994051i	$0.534739\pi$
$20$	− 1.50450i	− 0.0752251i
$21$	0	0
$22$	0	0
$23$	13.5095i	0.587369i	0.955902	+	0.293685i	$0.0948816\pi$
$23$	13.5095i	0.587369i	−0.955902	+	0.293685i	$0.905118\pi$
$24$	0	0
$25$	24.9727	0.998908
$26$	73.6945i	2.83440i
$27$	0	0
$28$	−93.7943	−3.34979
$29$	4.01646i	0.138499i	0.997599	+	0.0692494i	$0.0220604\pi$
$29$	4.01646i	0.138499i	−0.997599	+	0.0692494i	$0.977940\pi$
$30$	0	0
$31$	−20.5033	−0.661397	−0.330699	−	0.943736i	$-0.607284\pi$
$31$	−20.5033	−0.661397	−0.330699	+	0.943736i	$0.607284\pi$
$32$	− 36.5080i	− 1.14088i
$33$	0	0
$34$	83.3223	2.45066
$35$	1.70135i	0.0486101i
$36$	0	0
$37$	43.5939	1.17821	0.589107	−	0.808055i	$-0.299480\pi$
$37$	43.5939	1.17821	0.589107	+	0.808055i	$0.299480\pi$
$38$	14.9847i	0.394334i
$39$	0	0
$40$	−3.05456	−0.0763639
$41$	52.0734i	1.27008i	0.772478	+	0.635042i	$0.219017\pi$
$41$	52.0734i	1.27008i	−0.772478	+	0.635042i	$0.780983\pi$
$42$	0	0
$43$	62.5698	1.45511	0.727556	−	0.686048i	$-0.240656\pi$
$43$	62.5698	1.45511	0.727556	+	0.686048i	$0.240656\pi$
$44$	0	0
$45$	0	0
$46$	48.9097	1.06325
$47$	38.7052i	0.823515i	0.911293	+	0.411758i	$0.135085\pi$
$47$	38.7052i	0.823515i	−0.911293	+	0.411758i	$0.864915\pi$
$48$	0	0
$49$	57.0665	1.16462
$50$	− 90.4110i	− 1.80822i
$51$	0	0
$52$	185.382	3.56503
$53$	35.3780i	0.667510i	0.942660	+	0.333755i	$0.108316\pi$
$53$	35.3780i	0.667510i	−0.942660	+	0.333755i	$0.891684\pi$
$54$	0	0
$55$	0	0
$56$	190.428i	3.40051i
$57$	0	0
$58$	14.5412	0.250710
$59$	19.6883i	0.333699i	0.985982	+	0.166850i	$0.0533594\pi$
$59$	19.6883i	0.333699i	−0.985982	+	0.166850i	$0.946641\pi$
$60$	0	0
$61$	−6.42431	−0.105317	−0.0526583	−	0.998613i	$-0.516769\pi$
$61$	−6.42431	−0.105317	−0.0526583	+	0.998613i	$0.516769\pi$
$62$	74.2301i	1.19726i
$63$	0	0
$64$	−10.1214	−0.158147
$65$	− 3.36267i	− 0.0517334i
$66$	0	0
$67$	−3.98708	−0.0595087	−0.0297543	−	0.999557i	$-0.509472\pi$
$67$	−3.98708	−0.0595087	−0.0297543	+	0.999557i	$0.509472\pi$
$68$	− 209.601i	− 3.08236i
$69$	0	0
$70$	6.15957	0.0879938
$71$	54.7712i	0.771426i	0.922619	+	0.385713i	$0.126044\pi$
$71$	54.7712i	0.771426i	−0.922619	+	0.385713i	$0.873956\pi$
$72$	0	0
$73$	−22.8087	−0.312448	−0.156224	−	0.987722i	$-0.549932\pi$
$73$	−22.8087	−0.312448	−0.156224	+	0.987722i	$0.549932\pi$
$74$	− 157.827i	− 2.13280i
$75$	0	0
$76$	37.6946	0.495981
$77$	0	0
$78$	0	0
$79$	−57.1563	−0.723497	−0.361748	−	0.932276i	$-0.617820\pi$
$79$	−57.1563	−0.723497	−0.361748	+	0.932276i	$0.617820\pi$
$80$	5.04069i	0.0630086i
$81$	0	0
$82$	188.526	2.29910
$83$	− 56.8060i	− 0.684410i	−0.939625	−	0.342205i	$-0.888826\pi$
$83$	− 56.8060i	− 0.684410i	0.939625	−	0.342205i	$-0.111174\pi$
$84$	0	0
$85$	−3.80199	−0.0447293
$86$	− 226.527i	− 2.63404i
$87$	0	0
$88$	0	0
$89$	28.1602i	0.316407i	0.987407	+	0.158203i	$0.0505701\pi$
$89$	28.1602i	0.316407i	−0.987407	+	0.158203i	$0.949430\pi$
$90$	0	0
$91$	−209.637	−2.30370
$92$	− 123.034i	− 1.33733i
$93$	0	0
$94$	140.128	1.49073
$95$	− 0.683750i	− 0.00719737i
$96$	0	0
$97$	82.9323	0.854972	0.427486	−	0.904022i	$-0.359399\pi$
$97$	82.9323	0.854972	0.427486	+	0.904022i	$0.359399\pi$
$98$	− 206.603i	− 2.10819i
$99$	0	0
$100$	−227.433	−2.27433
$101$	− 153.046i	− 1.51531i	−0.652658	−	0.757653i	$-0.726346\pi$
$101$	− 153.046i	− 1.51531i	0.652658	−	0.757653i	$-0.273654\pi$
$102$	0	0
$103$	105.119	1.02057	0.510287	−	0.860004i	$-0.329539\pi$
$103$	105.119	1.02057	0.510287	+	0.860004i	$0.329539\pi$
$104$	− 376.376i	− 3.61900i
$105$	0	0
$106$	128.082	1.20832
$107$	− 112.763i	− 1.05386i	−0.849910	−	0.526929i	$-0.823344\pi$
$107$	− 112.763i	− 1.05386i	0.849910	−	0.526929i	$-0.176656\pi$
$108$	0	0
$109$	−96.1530	−0.882137	−0.441069	−	0.897473i	$-0.645400\pi$
$109$	−96.1530	−0.882137	−0.441069	+	0.897473i	$0.645400\pi$
$110$	0	0
$111$	0	0
$112$	314.249	2.80579
$113$	− 104.101i	− 0.921249i	−0.887595	−	0.460625i	$-0.847625\pi$
$113$	− 104.101i	− 0.921249i	0.887595	−	0.460625i	$-0.152375\pi$
$114$	0	0
$115$	−2.23175	−0.0194065
$116$	− 36.5789i	− 0.315336i
$117$	0	0
$118$	71.2792	0.604061
$119$	237.025i	1.99181i
$120$	0	0
$121$	0	0
$122$	23.2585i	0.190644i
$123$	0	0
$124$	186.729	1.50588
$125$	8.25540i	0.0660432i
$126$	0	0
$127$	−8.05817	−0.0634502	−0.0317251	−	0.999497i	$-0.510100\pi$
$127$	−8.05817	−0.0634502	−0.0317251	+	0.999497i	$0.510100\pi$
$128$	− 109.388i	− 0.854597i
$129$	0	0
$130$	−12.1742	−0.0936477
$131$	240.588i	1.83655i	0.395947	+	0.918273i	$0.370416\pi$
$131$	240.588i	1.83655i	−0.395947	+	0.918273i	$0.629584\pi$
$132$	0	0
$133$	−42.6266	−0.320501
$134$	14.4348i	0.107722i
$135$	0	0
$136$	−425.548	−3.12903
$137$	− 18.8680i	− 0.137723i	−0.997626	−	0.0688614i	$-0.978063\pi$
$137$	− 18.8680i	− 0.137723i	0.997626	−	0.0688614i	$-0.0219366\pi$
$138$	0	0
$139$	−33.4773	−0.240844	−0.120422	−	0.992723i	$-0.538425\pi$
$139$	−33.4773	−0.240844	−0.120422	+	0.992723i	$0.538425\pi$
$140$	− 15.4946i	− 0.110676i
$141$	0	0
$142$	198.293	1.39643
$143$	0	0
$144$	0	0
$145$	−0.663513	−0.00457595
$146$	82.5765i	0.565593i
$147$	0	0
$148$	−397.020	−2.68257
$149$	103.421i	0.694098i	0.937847	+	0.347049i	$0.112816\pi$
$149$	103.421i	0.694098i	−0.937847	+	0.347049i	$0.887184\pi$
$150$	0	0
$151$	−286.382	−1.89657	−0.948283	−	0.317425i	$-0.897182\pi$
$151$	−286.382	−1.89657	−0.948283	+	0.317425i	$0.897182\pi$
$152$	− 76.5305i	− 0.503490i
$153$	0	0
$154$	0	0
$155$	− 3.38711i	− 0.0218523i
$156$	0	0
$157$	261.304	1.66436	0.832179	−	0.554508i	$-0.187093\pi$
$157$	261.304	1.66436	0.832179	+	0.554508i	$0.187093\pi$
$158$	206.928i	1.30967i
$159$	0	0
$160$	6.03106	0.0376941
$161$	139.132i	0.864176i
$162$	0	0
$163$	192.496	1.18096	0.590480	−	0.807052i	$-0.298938\pi$
$163$	192.496	1.18096	0.590480	+	0.807052i	$0.298938\pi$
$164$	− 474.246i	− 2.89174i
$165$	0	0
$166$	−205.660	−1.23892
$167$	211.194i	1.26463i	0.774710	+	0.632316i	$0.217896\pi$
$167$	211.194i	1.26463i	−0.774710	+	0.632316i	$0.782104\pi$
$168$	0	0
$169$	245.342	1.45173
$170$	13.7647i	0.0809688i
$171$	0	0
$172$	−569.839	−3.31302
$173$	103.773i	0.599844i	0.953964	+	0.299922i	$0.0969607\pi$
$173$	103.773i	0.599844i	−0.953964	+	0.299922i	$0.903039\pi$
$174$	0	0
$175$	257.190	1.46966
$176$	0	0
$177$	0	0
$178$	101.951	0.572758
$179$	11.0578i	0.0617755i	0.999523	+	0.0308877i	$0.00983344\pi$
$179$	11.0578i	0.0617755i	−0.999523	+	0.0308877i	$0.990167\pi$
$180$	0	0
$181$	124.377	0.687164	0.343582	−	0.939123i	$-0.388360\pi$
$181$	124.377	0.687164	0.343582	+	0.939123i	$0.388360\pi$
$182$	758.969i	4.17016i
$183$	0	0
$184$	−249.794	−1.35758
$185$	7.20163i	0.0389278i
$186$	0	0
$187$	0	0
$188$	− 352.498i	− 1.87499i
$189$	0	0
$190$	−2.47544	−0.0130286
$191$	177.560i	0.929634i	0.885407	+	0.464817i	$0.153880\pi$
$191$	177.560i	0.929634i	−0.885407	+	0.464817i	$0.846120\pi$
$192$	0	0
$193$	243.331	1.26078	0.630391	−	0.776278i	$-0.282895\pi$
$193$	243.331	1.26078	0.630391	+	0.776278i	$0.282895\pi$
$194$	− 300.248i	− 1.54767i
$195$	0	0
$196$	−519.718	−2.65162
$197$	215.588i	1.09435i	0.837017	+	0.547177i	$0.184298\pi$
$197$	215.588i	1.09435i	−0.837017	+	0.547177i	$0.815702\pi$
$198$	0	0
$199$	−44.0830	−0.221523	−0.110761	−	0.993847i	$-0.535329\pi$
$199$	−44.0830	−0.221523	−0.110761	+	0.993847i	$0.535329\pi$
$200$	461.752i	2.30876i
$201$	0	0
$202$	−554.086	−2.74300
$203$	41.3650i	0.203768i
$204$	0	0
$205$	−8.60244	−0.0419631
$206$	− 380.573i	− 1.84744i
$207$	0	0
$208$	−621.103	−2.98607
$209$	0	0
$210$	0	0
$211$	−264.305	−1.25263	−0.626315	−	0.779570i	$-0.715438\pi$
$211$	−264.305	−1.25263	−0.626315	+	0.779570i	$0.715438\pi$
$212$	− 322.197i	− 1.51980i
$213$	0	0
$214$	−408.245	−1.90769
$215$	10.3364i	0.0480764i
$216$	0	0
$217$	−211.161	−0.973091
$218$	348.112i	1.59684i
$219$	0	0
$220$	0	0
$221$	− 468.473i	− 2.11979i
$222$	0	0
$223$	14.5149	0.0650890	0.0325445	−	0.999470i	$-0.489639\pi$
$223$	14.5149	0.0650890	0.0325445	+	0.999470i	$0.489639\pi$
$224$	− 375.991i	− 1.67853i
$225$	0	0
$226$	−376.887	−1.66764
$227$	122.517i	0.539722i	0.962899	+	0.269861i	$0.0869778\pi$
$227$	122.517i	0.539722i	−0.962899	+	0.269861i	$0.913022\pi$
$228$	0	0
$229$	3.89370	0.0170030	0.00850152	−	0.999964i	$-0.497294\pi$
$229$	3.89370	0.0170030	0.00850152	+	0.999964i	$0.497294\pi$
$230$	8.07980i	0.0351295i
$231$	0	0
$232$	−74.2654	−0.320110
$233$	− 365.767i	− 1.56982i	−0.619612	−	0.784908i	$-0.712710\pi$
$233$	− 365.767i	− 1.56982i	0.619612	−	0.784908i	$-0.287290\pi$
$234$	0	0
$235$	−6.39404	−0.0272087
$236$	− 179.306i	− 0.759770i
$237$	0	0
$238$	858.125	3.60557
$239$	147.298i	0.616308i	0.951336	+	0.308154i	$0.0997112\pi$
$239$	147.298i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$240$	0	0
$241$	278.601	1.15602	0.578010	−	0.816029i	$-0.303829\pi$
$241$	278.601	1.15602	0.578010	+	0.816029i	$0.303829\pi$
$242$	0	0
$243$	0	0
$244$	58.5078	0.239786
$245$	9.42728i	0.0384787i
$246$	0	0
$247$	84.2502	0.341094
$248$	− 379.112i	− 1.52868i
$249$	0	0
$250$	29.8878	0.119551
$251$	− 48.3215i	− 0.192516i	−0.995356	−	0.0962580i	$-0.969313\pi$
$251$	− 48.3215i	− 0.192516i	0.995356	−	0.0962580i	$-0.0306874\pi$
$252$	0	0
$253$	0	0
$254$	29.1737i	0.114857i
$255$	0	0
$256$	−436.515	−1.70514
$257$	− 199.225i	− 0.775195i	−0.921829	−	0.387597i	$-0.873305\pi$
$257$	− 199.225i	− 0.775195i	0.921829	−	0.387597i	$-0.126695\pi$
$258$	0	0
$259$	448.967	1.73346
$260$	30.6247i	0.117787i
$261$	0	0
$262$	871.022	3.32451
$263$	− 69.3106i	− 0.263538i	−0.991280	−	0.131769i	$-0.957934\pi$
$263$	− 69.3106i	− 0.263538i	0.991280	−	0.131769i	$-0.0420658\pi$
$264$	0	0
$265$	−5.84439	−0.0220543
$266$	154.325i	0.580170i
$267$	0	0
$268$	36.3113	0.135490
$269$	− 184.094i	− 0.684363i	−0.939634	−	0.342181i	$-0.888834\pi$
$269$	− 184.094i	− 0.684363i	0.939634	−	0.342181i	$-0.111166\pi$
$270$	0	0
$271$	−87.7148	−0.323671	−0.161835	−	0.986818i	$-0.551741\pi$
$271$	−87.7148	−0.323671	−0.161835	+	0.986818i	$0.551741\pi$
$272$	702.248i	2.58179i
$273$	0	0
$274$	−68.3096	−0.249305
$275$	0	0
$276$	0	0
$277$	−249.233	−0.899757	−0.449878	−	0.893090i	$-0.648533\pi$
$277$	−249.233	−0.899757	−0.449878	+	0.893090i	$0.648533\pi$
$278$	121.201i	0.435975i
$279$	0	0
$280$	−31.4584	−0.112352
$281$	16.8466i	0.0599524i	0.999551	+	0.0299762i	$0.00954314\pi$
$281$	16.8466i	0.0599524i	−0.999551	+	0.0299762i	$0.990457\pi$
$282$	0	0
$283$	47.6126	0.168242	0.0841212	−	0.996456i	$-0.473192\pi$
$283$	47.6126	0.168242	0.0841212	+	0.996456i	$0.473192\pi$
$284$	− 498.815i	− 1.75639i
$285$	0	0
$286$	0	0
$287$	536.297i	1.86863i
$288$	0	0
$289$	−240.677	−0.832793
$290$	2.40218i	0.00828337i
$291$	0	0
$292$	207.725	0.711386
$293$	425.104i	1.45087i	0.688293	+	0.725433i	$0.258360\pi$
$293$	425.104i	1.45087i	−0.688293	+	0.725433i	$0.741640\pi$
$294$	0	0
$295$	−3.25246	−0.0110253
$296$	806.062i	2.72318i
$297$	0	0
$298$	374.423	1.25645
$299$	− 274.991i	− 0.919702i
$300$	0	0
$301$	644.398	2.14086
$302$	1036.81i	3.43316i
$303$	0	0
$304$	−126.292	−0.415435
$305$	− 1.06129i	− 0.00347962i
$306$	0	0
$307$	−463.469	−1.50967	−0.754836	−	0.655914i	$-0.772284\pi$
$307$	−463.469	−1.50967	−0.754836	+	0.655914i	$0.772284\pi$
$308$	0	0
$309$	0	0
$310$	−12.2627	−0.0395570
$311$	143.144i	0.460269i	0.973159	+	0.230135i	$0.0739166\pi$
$311$	143.144i	0.460269i	−0.973159	+	0.230135i	$0.926083\pi$
$312$	0	0
$313$	237.744	0.759565	0.379783	−	0.925076i	$-0.375999\pi$
$313$	237.744	0.759565	0.379783	+	0.925076i	$0.375999\pi$
$314$	− 946.024i	− 3.01281i
$315$	0	0
$316$	520.536	1.64727
$317$	313.909i	0.990249i	0.868822	+	0.495125i	$0.164878\pi$
$317$	313.909i	0.990249i	−0.868822	+	0.495125i	$0.835122\pi$
$318$	0	0
$319$	0	0
$320$	− 1.67204i	− 0.00522514i
$321$	0	0
$322$	503.714	1.56433
$323$	− 95.2571i	− 0.294914i
$324$	0	0
$325$	−508.329	−1.56409
$326$	− 696.913i	− 2.13777i
$327$	0	0
$328$	−962.851	−2.93552
$329$	398.620i	1.21161i
$330$	0	0
$331$	−52.4920	−0.158586	−0.0792931	−	0.996851i	$-0.525266\pi$
$331$	−52.4920	−0.158586	−0.0792931	+	0.996851i	$0.525266\pi$
$332$	517.347i	1.55827i
$333$	0	0
$334$	764.604	2.28923
$335$	− 0.658659i	− 0.00196615i
$336$	0	0
$337$	−493.841	−1.46540	−0.732702	−	0.680550i	$-0.761741\pi$
$337$	−493.841	−1.46540	−0.732702	+	0.680550i	$0.761741\pi$
$338$	− 888.233i	− 2.62791i
$339$	0	0
$340$	34.6257	0.101840
$341$	0	0
$342$	0	0
$343$	83.0752	0.242202
$344$	1156.93i	3.36317i
$345$	0	0
$346$	375.699	1.08584
$347$	− 451.711i	− 1.30176i	−0.759181	−	0.650880i	$-0.774400\pi$
$347$	− 451.711i	− 1.30176i	0.759181	−	0.650880i	$-0.225600\pi$
$348$	0	0
$349$	177.153	0.507603	0.253802	−	0.967256i	$-0.418319\pi$
$349$	177.153	0.507603	0.253802	+	0.967256i	$0.418319\pi$
$350$	− 931.130i	− 2.66037i
$351$	0	0
$352$	0	0
$353$	590.460i	1.67269i	0.548203	+	0.836346i	$0.315312\pi$
$353$	590.460i	1.67269i	−0.548203	+	0.836346i	$0.684688\pi$
$354$	0	0
$355$	−9.04811	−0.0254876
$356$	− 256.462i	− 0.720399i
$357$	0	0
$358$	40.0336	0.111826
$359$	− 592.141i	− 1.64942i	−0.565558	−	0.824709i	$-0.691339\pi$
$359$	− 592.141i	− 1.64942i	0.565558	−	0.824709i	$-0.308661\pi$
$360$	0	0
$361$	−343.869	−0.952546
$362$	− 450.292i	− 1.24390i
$363$	0	0
$364$	1909.22	5.24510
$365$	− 3.76796i	− 0.0103232i
$366$	0	0
$367$	226.762	0.617880	0.308940	−	0.951082i	$-0.400026\pi$
$367$	226.762	0.617880	0.308940	+	0.951082i	$0.400026\pi$
$368$	412.215i	1.12015i
$369$	0	0
$370$	26.0727	0.0704669
$371$	364.353i	0.982084i
$372$	0	0
$373$	−76.5478	−0.205222	−0.102611	−	0.994722i	$-0.532720\pi$
$373$	−76.5478	−0.205222	−0.102611	+	0.994722i	$0.532720\pi$
$374$	0	0
$375$	0	0
$376$	−715.669	−1.90338
$377$	− 81.7566i	− 0.216861i
$378$	0	0
$379$	351.579	0.927649	0.463825	−	0.885927i	$-0.346477\pi$
$379$	351.579	0.927649	0.463825	+	0.885927i	$0.346477\pi$
$380$	6.22708i	0.0163871i
$381$	0	0
$382$	642.837	1.68282
$383$	572.874i	1.49575i	0.663837	+	0.747877i	$0.268927\pi$
$383$	572.874i	1.49575i	−0.663837	+	0.747877i	$0.731073\pi$
$384$	0	0
$385$	0	0
$386$	− 880.953i	− 2.28226i
$387$	0	0
$388$	−755.285	−1.94661
$389$	− 196.178i	− 0.504313i	−0.967687	−	0.252156i	$-0.918860\pi$
$389$	− 196.178i	− 0.504313i	0.967687	−	0.252156i	$-0.0811397\pi$
$390$	0	0
$391$	−310.917	−0.795185
$392$	1055.17i	2.69177i
$393$	0	0
$394$	780.512	1.98100
$395$	− 9.44211i	− 0.0239041i
$396$	0	0
$397$	−105.240	−0.265089	−0.132544	−	0.991177i	$-0.542315\pi$
$397$	−105.240	−0.265089	−0.132544	+	0.991177i	$0.542315\pi$
$398$	159.598i	0.401000i
$399$	0	0
$400$	761.992	1.90498
$401$	490.021i	1.22200i	0.791631	+	0.610999i	$0.209232\pi$
$401$	490.021i	1.22200i	−0.791631	+	0.610999i	$0.790768\pi$
$402$	0	0
$403$	417.353	1.03562
$404$	1393.83i	3.45007i
$405$	0	0
$406$	149.757	0.368861
$407$	0	0
$408$	0	0
$409$	533.858	1.30528	0.652639	−	0.757669i	$-0.273662\pi$
$409$	533.858	1.30528	0.652639	+	0.757669i	$0.273662\pi$
$410$	31.1442i	0.0759615i
$411$	0	0
$412$	−957.346	−2.32366
$413$	202.767i	0.490960i
$414$	0	0
$415$	9.38426	0.0226127
$416$	743.135i	1.78638i
$417$	0	0
$418$	0	0
$419$	− 344.904i	− 0.823160i	−0.911374	−	0.411580i	$-0.864977\pi$
$419$	− 344.904i	− 0.823160i	0.911374	−	0.411580i	$-0.135023\pi$
$420$	0	0
$421$	−403.467	−0.958355	−0.479177	−	0.877718i	$-0.659065\pi$
$421$	−403.467	−0.958355	−0.479177	+	0.877718i	$0.659065\pi$
$422$	956.888i	2.26751i
$423$	0	0
$424$	−654.149	−1.54280
$425$	574.740i	1.35233i
$426$	0	0
$427$	−66.1631	−0.154949
$428$	1026.96i	2.39943i
$429$	0	0
$430$	37.4219	0.0870277
$431$	132.691i	0.307868i	0.988081	+	0.153934i	$0.0491944\pi$
$431$	132.691i	0.307868i	−0.988081	+	0.153934i	$0.950806\pi$
$432$	0	0
$433$	−823.887	−1.90274	−0.951370	−	0.308049i	$-0.900324\pi$
$433$	−823.887	−1.90274	−0.951370	+	0.308049i	$0.900324\pi$
$434$	764.485i	1.76149i
$435$	0	0
$436$	875.689	2.00846
$437$	− 55.9154i	− 0.127953i
$438$	0	0
$439$	109.320	0.249021	0.124511	−	0.992218i	$-0.460264\pi$
$439$	109.320	0.249021	0.124511	+	0.992218i	$0.460264\pi$
$440$	0	0
$441$	0	0
$442$	−1696.06	−3.83723
$443$	− 375.738i	− 0.848167i	−0.905623	−	0.424083i	$-0.860596\pi$
$443$	− 375.738i	− 0.848167i	0.905623	−	0.424083i	$-0.139404\pi$
$444$	0	0
$445$	−4.65201	−0.0104540
$446$	− 52.5495i	− 0.117824i
$447$	0	0
$448$	−104.239	−0.232677
$449$	377.057i	0.839770i	0.907577	+	0.419885i	$0.137930\pi$
$449$	377.057i	0.839770i	−0.907577	+	0.419885i	$0.862070\pi$
$450$	0	0
$451$	0	0
$452$	948.075i	2.09751i
$453$	0	0
$454$	443.560	0.977004
$455$	− 34.6317i	− 0.0761136i
$456$	0	0
$457$	−397.734	−0.870316	−0.435158	−	0.900354i	$-0.643307\pi$
$457$	−397.734	−0.870316	−0.435158	+	0.900354i	$0.643307\pi$
$458$	− 14.0967i	− 0.0307789i
$459$	0	0
$460$	20.3251	0.0441849
$461$	− 452.581i	− 0.981737i	−0.871234	−	0.490869i	$-0.836680\pi$
$461$	− 452.581i	− 0.981737i	0.871234	−	0.490869i	$-0.163320\pi$
$462$	0	0
$463$	−125.818	−0.271745	−0.135873	−	0.990726i	$-0.543384\pi$
$463$	−125.818	−0.271745	−0.135873	+	0.990726i	$0.543384\pi$
$464$	122.554i	0.264126i
$465$	0	0
$466$	−1324.22	−2.84168
$467$	− 277.000i	− 0.593147i	−0.955010	−	0.296574i	$-0.904156\pi$
$467$	− 277.000i	− 0.593147i	0.955010	−	0.296574i	$-0.0958440\pi$
$468$	0	0
$469$	−41.0624	−0.0875530
$470$	23.1489i	0.0492530i
$471$	0	0
$472$	−364.041	−0.771273
$473$	0	0
$474$	0	0
$475$	−103.361	−0.217603
$476$	− 2158.65i	− 4.53497i
$477$	0	0
$478$	533.275	1.11564
$479$	500.992i	1.04591i	0.852360	+	0.522956i	$0.175171\pi$
$479$	500.992i	1.04591i	−0.852360	+	0.522956i	$0.824829\pi$
$480$	0	0
$481$	−887.370	−1.84484
$482$	− 1008.65i	− 2.09263i
$483$	0	0
$484$	0	0
$485$	13.7003i	0.0282480i
$486$	0	0
$487$	690.163	1.41717	0.708586	−	0.705624i	$-0.249333\pi$
$487$	690.163	1.41717	0.708586	+	0.705624i	$0.249333\pi$
$488$	− 118.787i	− 0.243416i
$489$	0	0
$490$	34.1305	0.0696540
$491$	355.830i	0.724705i	0.932041	+	0.362353i	$0.118026\pi$
$491$	355.830i	0.724705i	−0.932041	+	0.362353i	$0.881974\pi$
$492$	0	0
$493$	−92.4378	−0.187501
$494$	− 305.019i	− 0.617447i
$495$	0	0
$496$	−625.618	−1.26133
$497$	564.081i	1.13497i
$498$	0	0
$499$	145.012	0.290604	0.145302	−	0.989387i	$-0.453585\pi$
$499$	145.012	0.290604	0.145302	+	0.989387i	$0.453585\pi$
$500$	− 75.1840i	− 0.150368i
$501$	0	0
$502$	−174.943	−0.348492
$503$	868.917i	1.72747i	0.503947	+	0.863734i	$0.331881\pi$
$503$	868.917i	1.72747i	−0.503947	+	0.863734i	$0.668119\pi$
$504$	0	0
$505$	25.2829	0.0500651
$506$	0	0
$507$	0	0
$508$	73.3878	0.144464
$509$	504.109i	0.990391i	0.868782	+	0.495195i	$0.164904\pi$
$509$	504.109i	0.990391i	−0.868782	+	0.495195i	$0.835096\pi$
$510$	0	0
$511$	−234.904	−0.459694
$512$	1142.80i	2.23204i
$513$	0	0
$514$	−721.273	−1.40325
$515$	17.3655i	0.0337194i
$516$	0	0
$517$	0	0
$518$	− 1625.44i	− 3.13791i
$519$	0	0
$520$	62.1767	0.119571
$521$	258.912i	0.496951i	0.968638	+	0.248476i	$0.0799296\pi$
$521$	258.912i	0.496951i	−0.968638	+	0.248476i	$0.920070\pi$
$522$	0	0
$523$	597.180	1.14184	0.570918	−	0.821007i	$-0.306587\pi$
$523$	597.180	1.14184	0.570918	+	0.821007i	$0.306587\pi$
$524$	− 2191.09i	− 4.18147i
$525$	0	0
$526$	−250.932	−0.477056
$527$	− 471.878i	− 0.895404i
$528$	0	0
$529$	346.493	0.654997
$530$	21.1590i	0.0399226i
$531$	0	0
$532$	388.211	0.729720
$533$	− 1059.97i	− 1.98870i
$534$	0	0
$535$	18.6282	0.0348191
$536$	− 73.7221i	− 0.137541i
$537$	0	0
$538$	−666.491	−1.23883
$539$	0	0
$540$	0	0
$541$	241.706	0.446776	0.223388	−	0.974730i	$-0.428288\pi$
$541$	241.706	0.446776	0.223388	+	0.974730i	$0.428288\pi$
$542$	317.562i	0.585908i
$543$	0	0
$544$	840.222	1.54453
$545$	− 15.8843i	− 0.0291455i
$546$	0	0
$547$	−535.306	−0.978622	−0.489311	−	0.872109i	$-0.662752\pi$
$547$	−535.306	−0.978622	−0.489311	+	0.872109i	$0.662752\pi$
$548$	171.836i	0.313569i
$549$	0	0
$550$	0	0
$551$	− 16.6240i	− 0.0301706i
$552$	0	0
$553$	−588.644	−1.06446
$554$	902.320i	1.62874i
$555$	0	0
$556$	304.886	0.548356
$557$	− 38.6536i	− 0.0693961i	−0.999398	−	0.0346980i	$-0.988953\pi$
$557$	− 38.6536i	− 0.0693961i	0.999398	−	0.0346980i	$-0.0110469\pi$
$558$	0	0
$559$	−1273.63	−2.27841
$560$	51.9134i	0.0927024i
$561$	0	0
$562$	60.9914	0.108526
$563$	− 356.051i	− 0.632418i	−0.948690	−	0.316209i	$-0.897590\pi$
$563$	− 356.051i	− 0.632418i	0.948690	−	0.316209i	$-0.102410\pi$
$564$	0	0
$565$	17.1973	0.0304378
$566$	− 172.376i	− 0.304552i
$567$	0	0
$568$	−1012.73	−1.78298
$569$	98.7149i	0.173488i	0.996231	+	0.0867442i	$0.0276463\pi$
$569$	98.7149i	0.173488i	−0.996231	+	0.0867442i	$0.972354\pi$
$570$	0	0
$571$	−37.7221	−0.0660633	−0.0330316	−	0.999454i	$-0.510516\pi$
$571$	−37.7221	−0.0660633	−0.0330316	+	0.999454i	$0.510516\pi$
$572$	0	0
$573$	0	0
$574$	1941.61	3.38259
$575$	337.369i	0.586728i
$576$	0	0
$577$	−978.346	−1.69557	−0.847787	−	0.530337i	$-0.822065\pi$
$577$	−978.346	−1.69557	−0.847787	+	0.530337i	$0.822065\pi$
$578$	871.346i	1.50752i
$579$	0	0
$580$	6.04277	0.0104186
$581$	− 585.037i	− 1.00695i
$582$	0	0
$583$	0	0
$584$	− 421.739i	− 0.722156i
$585$	0	0
$586$	1539.04	2.62635
$587$	− 1115.33i	− 1.90006i	−0.312162	−	0.950029i	$-0.601053\pi$
$587$	− 1115.33i	− 1.90006i	0.312162	−	0.950029i	$-0.398947\pi$
$588$	0	0
$589$	84.8625	0.144079
$590$	11.7752i	0.0199580i
$591$	0	0
$592$	1330.18	2.24693
$593$	1006.79i	1.69779i	0.528563	+	0.848894i	$0.322731\pi$
$593$	1006.79i	1.69779i	−0.528563	+	0.848894i	$0.677269\pi$
$594$	0	0
$595$	−39.1562	−0.0658087
$596$	− 941.877i	− 1.58033i
$597$	0	0
$598$	−995.576	−1.66484
$599$	− 590.141i	− 0.985210i	−0.870253	−	0.492605i	$-0.836045\pi$
$599$	− 590.141i	− 0.985210i	0.870253	−	0.492605i	$-0.163955\pi$
$600$	0	0
$601$	−750.837	−1.24931	−0.624656	−	0.780900i	$-0.714761\pi$
$601$	−750.837	−1.24931	−0.624656	+	0.780900i	$0.714761\pi$
$602$	− 2332.97i	− 3.87537i
$603$	0	0
$604$	2608.15	4.31813
$605$	0	0
$606$	0	0
$607$	176.162	0.290218	0.145109	−	0.989416i	$-0.453647\pi$
$607$	176.162	0.290218	0.145109	+	0.989416i	$0.453647\pi$
$608$	151.105i	0.248529i
$609$	0	0
$610$	−3.84227	−0.00629880
$611$	− 787.860i	− 1.28946i
$612$	0	0
$613$	908.432	1.48194	0.740972	−	0.671535i	$-0.234365\pi$
$613$	908.432	1.48194	0.740972	+	0.671535i	$0.234365\pi$
$614$	1677.94i	2.73280i
$615$	0	0
$616$	0	0
$617$	836.440i	1.35566i	0.735220	+	0.677828i	$0.237079\pi$
$617$	836.440i	1.35566i	−0.735220	+	0.677828i	$0.762921\pi$
$618$	0	0
$619$	960.477	1.55166	0.775830	−	0.630942i	$-0.217332\pi$
$619$	960.477	1.55166	0.775830	+	0.630942i	$0.217332\pi$
$620$	30.8473i	0.0497537i
$621$	0	0
$622$	518.237	0.833178
$623$	290.018i	0.465518i
$624$	0	0
$625$	622.954	0.996726
$626$	− 860.727i	− 1.37496i
$627$	0	0
$628$	−2379.76	−3.78943
$629$	1003.30i	1.59507i
$630$	0	0
$631$	230.308	0.364988	0.182494	−	0.983207i	$-0.441583\pi$
$631$	230.308	0.364988	0.182494	+	0.983207i	$0.441583\pi$
$632$	− 1056.83i	− 1.67220i
$633$	0	0
$634$	1136.47	1.79255
$635$	− 1.33120i	− 0.00209637i
$636$	0	0
$637$	−1161.61	−1.82356
$638$	0	0
$639$	0	0
$640$	18.0708	0.0282356
$641$	478.911i	0.747131i	0.927604	+	0.373565i	$0.121865\pi$
$641$	478.911i	0.747131i	−0.927604	+	0.373565i	$0.878135\pi$
$642$	0	0
$643$	−716.383	−1.11413	−0.557063	−	0.830470i	$-0.688072\pi$
$643$	−716.383	−1.11413	−0.557063	+	0.830470i	$0.688072\pi$
$644$	− 1267.11i	− 1.96757i
$645$	0	0
$646$	−344.868	−0.533852
$647$	− 977.171i	− 1.51031i	−0.655546	−	0.755155i	$-0.727562\pi$
$647$	− 977.171i	− 1.51031i	0.655546	−	0.755155i	$-0.272438\pi$
$648$	0	0
$649$	0	0
$650$	1840.35i	2.83131i
$651$	0	0
$652$	−1753.11	−2.68882
$653$	− 892.901i	− 1.36738i	−0.729771	−	0.683692i	$-0.760373\pi$
$653$	− 892.901i	− 1.36738i	0.729771	−	0.683692i	$-0.239627\pi$
$654$	0	0
$655$	−39.7446	−0.0606788
$656$	1588.92i	2.42213i
$657$	0	0
$658$	1443.16	2.19325
$659$	− 350.792i	− 0.532309i	−0.963930	−	0.266154i	$-0.914247\pi$
$659$	− 350.792i	− 0.532309i	0.963930	−	0.266154i	$-0.0857531\pi$
$660$	0	0
$661$	−285.033	−0.431215	−0.215607	−	0.976480i	$-0.569173\pi$
$661$	−285.033	−0.431215	−0.215607	+	0.976480i	$0.569173\pi$
$662$	190.042i	0.287072i
$663$	0	0
$664$	1050.36	1.58186
$665$	− 7.04184i	− 0.0105892i
$666$	0	0
$667$	−54.2604	−0.0813499
$668$	− 1923.39i	− 2.87933i
$669$	0	0
$670$	−2.38460	−0.00355911
$671$	0	0
$672$	0	0
$673$	101.236	0.150426	0.0752128	−	0.997168i	$-0.476036\pi$
$673$	101.236	0.150426	0.0752128	+	0.997168i	$0.476036\pi$
$674$	1787.90i	2.65267i
$675$	0	0
$676$	−2234.39	−3.30531
$677$	399.059i	0.589453i	0.955582	+	0.294726i	$0.0952285\pi$
$677$	399.059i	0.589453i	−0.955582	+	0.294726i	$0.904772\pi$
$678$	0	0
$679$	854.108	1.25789
$680$	− 70.2998i	− 0.103382i
$681$	0	0
$682$	0	0
$683$	− 781.537i	− 1.14427i	−0.820159	−	0.572135i	$-0.806115\pi$
$683$	− 781.537i	− 1.14427i	0.820159	−	0.572135i	$-0.193885\pi$
$684$	0	0
$685$	3.11696	0.00455031
$686$	− 300.765i	− 0.438433i
$687$	0	0
$688$	1909.19	2.77499
$689$	− 720.133i	− 1.04519i
$690$	0	0
$691$	−551.762	−0.798497	−0.399249	−	0.916843i	$-0.630729\pi$
$691$	−551.762	−0.798497	−0.399249	+	0.916843i	$0.630729\pi$
$692$	− 945.087i	− 1.36573i
$693$	0	0
$694$	−1635.37	−2.35644
$695$	− 5.53039i	− 0.00795740i
$696$	0	0
$697$	−1198.46	−1.71945
$698$	− 641.365i	− 0.918861i
$699$	0	0
$700$	−2342.30	−3.34614
$701$	− 327.564i	− 0.467281i	−0.972323	−	0.233640i	$-0.924936\pi$
$701$	− 327.564i	− 0.467281i	0.972323	−	0.233640i	$-0.0750639\pi$
$702$	0	0
$703$	−180.434	−0.256662
$704$	0	0
$705$	0	0
$706$	2137.70	3.02790
$707$	− 1576.20i	− 2.22942i
$708$	0	0
$709$	1039.76	1.46652	0.733260	−	0.679949i	$-0.237998\pi$
$709$	1039.76	1.46652	0.733260	+	0.679949i	$0.237998\pi$
$710$	32.7577i	0.0461376i
$711$	0	0
$712$	−520.689	−0.731305
$713$	− 276.990i	− 0.388485i
$714$	0	0
$715$	0	0
$716$	− 100.706i	− 0.140651i
$717$	0	0
$718$	−2143.78	−2.98577
$719$	− 909.087i	− 1.26438i	−0.774815	−	0.632188i	$-0.782157\pi$
$719$	− 909.087i	− 1.26438i	0.774815	−	0.632188i	$-0.217843\pi$
$720$	0	0
$721$	1082.61	1.50153
$722$	1244.94i	1.72429i
$723$	0	0
$724$	−1132.73	−1.56454
$725$	100.302i	0.138348i
$726$	0	0
$727$	−709.728	−0.976242	−0.488121	−	0.872776i	$-0.662318\pi$
$727$	−709.728	−0.976242	−0.488121	+	0.872776i	$0.662318\pi$
$728$	− 3876.24i	− 5.32451i
$729$	0	0
$730$	−13.6415	−0.0186870
$731$	1440.03i	1.96994i
$732$	0	0
$733$	495.869	0.676493	0.338246	−	0.941058i	$-0.390166\pi$
$733$	495.869	0.676493	0.338246	+	0.941058i	$0.390166\pi$
$734$	− 820.967i	− 1.11848i
$735$	0	0
$736$	493.205	0.670115
$737$	0	0
$738$	0	0
$739$	1099.53	1.48787	0.743934	−	0.668253i	$-0.232958\pi$
$739$	1099.53	1.48787	0.743934	+	0.668253i	$0.232958\pi$
$740$	− 65.5871i	− 0.0886312i
$741$	0	0
$742$	1319.10	1.77777
$743$	− 723.678i	− 0.973994i	−0.873404	−	0.486997i	$-0.838092\pi$
$743$	− 723.678i	− 0.973994i	0.873404	−	0.486997i	$-0.161908\pi$
$744$	0	0
$745$	−17.0849	−0.0229328
$746$	277.133i	0.371492i
$747$	0	0
$748$	0	0
$749$	− 1161.33i	− 1.55050i
$750$	0	0
$751$	−543.282	−0.723412	−0.361706	−	0.932292i	$-0.617806\pi$
$751$	−543.282	−0.723412	−0.361706	+	0.932292i	$0.617806\pi$
$752$	1181.01i	1.57049i
$753$	0	0
$754$	−295.991	−0.392561
$755$	− 47.3097i	− 0.0626619i
$756$	0	0
$757$	−347.247	−0.458715	−0.229357	−	0.973342i	$-0.573662\pi$
$757$	−347.247	−0.458715	−0.229357	+	0.973342i	$0.573662\pi$
$758$	− 1272.85i	− 1.67923i
$759$	0	0
$760$	12.6427	0.0166351
$761$	− 802.222i	− 1.05417i	−0.849813	−	0.527084i	$-0.823285\pi$
$761$	− 802.222i	− 1.05417i	0.849813	−	0.527084i	$-0.176715\pi$
$762$	0	0
$763$	−990.266	−1.29786
$764$	− 1617.08i	− 2.11660i
$765$	0	0
$766$	2074.03	2.70761
$767$	− 400.762i	− 0.522506i
$768$	0	0
$769$	−1441.35	−1.87432	−0.937158	−	0.348905i	$-0.886554\pi$
$769$	−1441.35	−1.87432	−0.937158	+	0.348905i	$0.886554\pi$
$770$	0	0
$771$	0	0
$772$	−2216.07	−2.87056
$773$	1333.16i	1.72466i	0.506348	+	0.862329i	$0.330995\pi$
$773$	1333.16i	1.72466i	−0.506348	+	0.862329i	$0.669005\pi$
$774$	0	0
$775$	−512.023	−0.660675
$776$	1533.44i	1.97608i
$777$	0	0
$778$	−710.240	−0.912905
$779$	− 215.530i	− 0.276675i
$780$	0	0
$781$	0	0
$782$	1125.64i	1.43944i
$783$	0	0
$784$	1741.27	2.22100
$785$	43.1670i	0.0549898i
$786$	0	0
$787$	−442.103	−0.561757	−0.280878	−	0.959743i	$-0.590626\pi$
$787$	−442.103	−0.561757	−0.280878	+	0.959743i	$0.590626\pi$
$788$	− 1963.41i	− 2.49164i
$789$	0	0
$790$	−34.1842	−0.0432711
$791$	− 1072.12i	− 1.35540i
$792$	0	0
$793$	130.769	0.164905
$794$	381.011i	0.479863i
$795$	0	0
$796$	401.475	0.504365
$797$	− 1298.24i	− 1.62891i	−0.580225	−	0.814456i	$-0.697035\pi$
$797$	− 1298.24i	− 1.62891i	0.580225	−	0.814456i	$-0.302965\pi$
$798$	0	0
$799$	−890.790	−1.11488
$800$	− 911.704i	− 1.13963i
$801$	0	0
$802$	1774.07	2.21206
$803$	0	0
$804$	0	0
$805$	−22.9844	−0.0285521
$806$	− 1510.98i	− 1.87467i
$807$	0	0
$808$	2829.86	3.50230
$809$	− 913.645i	− 1.12935i	−0.825313	−	0.564675i	$-0.809001\pi$
$809$	− 913.645i	− 1.12935i	0.825313	−	0.564675i	$-0.190999\pi$
$810$	0	0
$811$	343.061	0.423010	0.211505	−	0.977377i	$-0.432163\pi$
$811$	343.061	0.423010	0.211505	+	0.977377i	$0.432163\pi$
$812$	− 376.721i	− 0.463942i
$813$	0	0
$814$	0	0
$815$	31.8001i	0.0390185i
$816$	0	0
$817$	−258.974	−0.316982
$818$	− 1932.78i	− 2.36281i
$819$	0	0
$820$	78.3445	0.0955421
$821$	16.3641i	0.0199319i	0.999950	+	0.00996596i	$0.00317232\pi$
$821$	16.3641i	0.0199319i	−0.999950	+	0.00996596i	$0.996828\pi$
$822$	0	0
$823$	−63.5873	−0.0772628	−0.0386314	−	0.999254i	$-0.512300\pi$
$823$	−63.5873	−0.0772628	−0.0386314	+	0.999254i	$0.512300\pi$
$824$	1943.68i	2.35883i
$825$	0	0
$826$	734.095	0.888734
$827$	1174.88i	1.42066i	0.703871	+	0.710328i	$0.251453\pi$
$827$	1174.88i	1.42066i	−0.703871	+	0.710328i	$0.748547\pi$
$828$	0	0
$829$	−277.361	−0.334573	−0.167287	−	0.985908i	$-0.553501\pi$
$829$	−277.361	−0.334573	−0.167287	+	0.985908i	$0.553501\pi$
$830$	− 33.9747i	− 0.0409334i
$831$	0	0
$832$	206.026	0.247627
$833$	1313.37i	1.57667i
$834$	0	0
$835$	−34.8888	−0.0417830
$836$	0	0
$837$	0	0
$838$	−1248.69	−1.49008
$839$	− 932.487i	− 1.11143i	−0.831374	−	0.555713i	$-0.812445\pi$
$839$	− 932.487i	− 1.11143i	0.831374	−	0.555713i	$-0.187555\pi$
$840$	0	0
$841$	824.868	0.980818
$842$	1460.71i	1.73481i
$843$	0	0
$844$	2407.09	2.85200
$845$	40.5300i	0.0479645i
$846$	0	0
$847$	0	0
$848$	1079.49i	1.27298i
$849$	0	0
$850$	2080.78	2.44798
$851$	588.932i	0.692047i
$852$	0	0
$853$	−1539.03	−1.80425	−0.902127	−	0.431471i	$-0.857995\pi$
$853$	−1539.03	−1.80425	−0.902127	+	0.431471i	$0.857995\pi$
$854$	239.536i	0.280488i
$855$	0	0
$856$	2085.01	2.43576
$857$	443.512i	0.517517i	0.965942	+	0.258758i	$0.0833133\pi$
$857$	443.512i	0.517517i	−0.965942	+	0.258758i	$0.916687\pi$
$858$	0	0
$859$	1399.85	1.62962	0.814812	−	0.579725i	$-0.196840\pi$
$859$	1399.85	1.62962	0.814812	+	0.579725i	$0.196840\pi$
$860$	− 94.1364i	− 0.109461i
$861$	0	0
$862$	480.395	0.557302
$863$	1447.63i	1.67744i	0.544566	+	0.838718i	$0.316694\pi$
$863$	1447.63i	1.67744i	−0.544566	+	0.838718i	$0.683306\pi$
$864$	0	0
$865$	−17.1431	−0.0198186
$866$	2982.79i	3.44433i
$867$	0	0
$868$	1923.09	2.21555
$869$	0	0
$870$	0	0
$871$	81.1586	0.0931786
$872$	− 1777.89i	− 2.03887i
$873$	0	0
$874$	−202.436	−0.231620
$875$	85.0212i	0.0971671i
$876$	0	0
$877$	−193.706	−0.220874	−0.110437	−	0.993883i	$-0.535225\pi$
$877$	−193.706	−0.220874	−0.110437	+	0.993883i	$0.535225\pi$
$878$	− 395.782i	− 0.450777i
$879$	0	0
$880$	0	0
$881$	1111.18i	1.26127i	0.776079	+	0.630636i	$0.217206\pi$
$881$	1111.18i	1.26127i	−0.776079	+	0.630636i	$0.782794\pi$
$882$	0	0
$883$	710.379	0.804507	0.402253	−	0.915528i	$-0.368227\pi$
$883$	710.379	0.804507	0.402253	+	0.915528i	$0.368227\pi$
$884$	4266.50i	4.82636i
$885$	0	0
$886$	−1360.32	−1.53535
$887$	1134.08i	1.27856i	0.768975	+	0.639279i	$0.220767\pi$
$887$	1134.08i	1.27856i	−0.768975	+	0.639279i	$0.779233\pi$
$888$	0	0
$889$	−82.9899	−0.0933520
$890$	16.8421i	0.0189237i
$891$	0	0
$892$	−132.190	−0.148196
$893$	− 160.200i	− 0.179395i
$894$	0	0
$895$	−1.82673	−0.00204104
$896$	− 1126.58i	− 1.25734i
$897$	0	0
$898$	1365.09	1.52015
$899$	− 82.3508i	− 0.0916027i
$900$	0	0
$901$	−814.215	−0.903680
$902$	0	0
$903$	0	0
$904$	1924.86	2.12927
$905$	20.5468i	0.0227036i
$906$	0	0
$907$	−987.303	−1.08854	−0.544268	−	0.838911i	$-0.683193\pi$
$907$	−987.303	−1.08854	−0.544268	+	0.838911i	$0.683193\pi$
$908$	− 1115.79i	− 1.22885i
$909$	0	0
$910$	−125.380	−0.137781
$911$	1029.67i	1.13026i	0.825002	+	0.565130i	$0.191174\pi$
$911$	1029.67i	1.13026i	−0.825002	+	0.565130i	$0.808826\pi$
$912$	0	0
$913$	0	0
$914$	1439.95i	1.57544i
$915$	0	0
$916$	−35.4609	−0.0387127
$917$	2477.78i	2.70205i
$918$	0	0
$919$	−555.392	−0.604343	−0.302172	−	0.953254i	$-0.597712\pi$
$919$	−555.392	−0.604343	−0.302172	+	0.953254i	$0.597712\pi$
$920$	− 41.2655i	− 0.0448538i
$921$	0	0
$922$	−1638.52	−1.77714
$923$	− 1114.89i	− 1.20790i
$924$	0	0
$925$	1088.66	1.17693
$926$	455.511i	0.491913i
$927$	0	0
$928$	146.633	0.158010
$929$	− 624.021i	− 0.671713i	−0.941913	−	0.335856i	$-0.890974\pi$
$929$	− 624.021i	− 0.671713i	0.941913	−	0.335856i	$-0.109026\pi$
$930$	0	0
$931$	−236.196	−0.253701
$932$	3331.13i	3.57418i
$933$	0	0
$934$	−1002.85	−1.07371
$935$	0	0
$936$	0	0
$937$	270.217	0.288386	0.144193	−	0.989550i	$-0.453941\pi$
$937$	270.217	0.288386	0.144193	+	0.989550i	$0.453941\pi$
$938$	148.662i	0.158488i
$939$	0	0
$940$	58.2321	0.0619490
$941$	1656.43i	1.76029i	0.474703	+	0.880146i	$0.342555\pi$
$941$	1656.43i	1.76029i	−0.474703	+	0.880146i	$0.657445\pi$
$942$	0	0
$943$	−703.486	−0.746008
$944$	600.747i	0.636385i
$945$	0	0
$946$	0	0
$947$	− 771.207i	− 0.814368i	−0.913346	−	0.407184i	$-0.866511\pi$
$947$	− 771.207i	− 0.814368i	0.913346	−	0.407184i	$-0.133489\pi$
$948$	0	0
$949$	464.280	0.489231
$950$	374.208i	0.393903i
$951$	0	0
$952$	−4382.66	−4.60363
$953$	− 297.525i	− 0.312199i	−0.987741	−	0.156099i	$-0.950108\pi$
$953$	− 297.525i	− 0.312199i	0.987741	−	0.156099i	$-0.0498920\pi$
$954$	0	0
$955$	−29.3326	−0.0307148
$956$	− 1341.48i	− 1.40322i
$957$	0	0
$958$	1813.79	1.89331
$959$	− 194.319i	− 0.202627i
$960$	0	0
$961$	−540.614	−0.562553
$962$	3212.63i	3.33953i
$963$	0	0
$964$	−2537.29	−2.63204
$965$	40.1978i	0.0416558i
$966$	0	0
$967$	1078.20	1.11499	0.557497	−	0.830179i	$-0.311762\pi$
$967$	1078.20	1.11499	0.557497	+	0.830179i	$0.311762\pi$
$968$	0	0
$969$	0	0
$970$	49.6004	0.0511344
$971$	687.460i	0.707992i	0.935247	+	0.353996i	$0.115177\pi$
$971$	687.460i	0.707992i	−0.935247	+	0.353996i	$0.884823\pi$
$972$	0	0
$973$	−344.778	−0.354345
$974$	− 2498.66i	− 2.56536i
$975$	0	0
$976$	−196.025	−0.200845
$977$	− 1597.66i	− 1.63527i	−0.575738	−	0.817634i	$-0.695285\pi$
$977$	− 1597.66i	− 1.63527i	0.575738	−	0.817634i	$-0.304715\pi$
$978$	0	0
$979$	0	0
$980$	− 85.8566i	− 0.0876087i
$981$	0	0
$982$	1288.25	1.31186
$983$	− 1254.84i	− 1.27654i	−0.769811	−	0.638272i	$-0.779650\pi$
$983$	− 1254.84i	− 1.27654i	0.769811	−	0.638272i	$-0.220350\pi$
$984$	0	0
$985$	−35.6147	−0.0361571
$986$	334.661i	0.339413i
$987$	0	0
$988$	−767.288	−0.776607
$989$	845.287i	0.854689i
$990$	0	0
$991$	973.882	0.982727	0.491363	−	0.870955i	$-0.336499\pi$
$991$	973.882	0.982727	0.491363	+	0.870955i	$0.336499\pi$
$992$	748.536i	0.754572i
$993$	0	0
$994$	2042.20	2.05452
$995$	− 7.28243i	− 0.00731903i
$996$	0	0
$997$	1601.03	1.60585	0.802923	−	0.596083i	$-0.203277\pi$
$997$	1601.03	1.60585	0.802923	+	0.596083i	$0.203277\pi$
$998$	− 524.999i	− 0.526051i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.i.485.1		16
3.2	odd	2	inner	1089.3.b.i.485.16		16
11.3	even	5		99.3.l.a.53.1	✓	32
11.4	even	5		99.3.l.a.71.8	yes	32
11.10	odd	2		1089.3.b.j.485.16		16
33.14	odd	10		99.3.l.a.53.8	yes	32
33.26	odd	10		99.3.l.a.71.1	yes	32
33.32	even	2		1089.3.b.j.485.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.53.1	✓	32	11.3	even	5
99.3.l.a.53.8	yes	32	33.14	odd	10
99.3.l.a.71.1	yes	32	33.26	odd	10
99.3.l.a.71.8	yes	32	11.4	even	5
1089.3.b.i.485.1		16	1.1	even	1	trivial
1089.3.b.i.485.16		16	3.2	odd	2	inner
1089.3.b.j.485.1		16	33.32	even	2
1089.3.b.j.485.16		16	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.62039i q^{2} -9.10725 q^{4} +0.165198i q^{5} +10.2989 q^{7} +18.4903i q^{8} +O(q^{10})\)	\(q-3.62039i q^{2} -9.10725 q^{4} +0.165198i q^{5} +10.2989 q^{7} +18.4903i q^{8} +0.598083 q^{10} -20.3554 q^{13} -37.2859i q^{14} +30.5130 q^{16} +23.0147i q^{17} -4.13897 q^{19} -1.50450i q^{20} +13.5095i q^{23} +24.9727 q^{25} +73.6945i q^{26} -93.7943 q^{28} +4.01646i q^{29} -20.5033 q^{31} -36.5080i q^{32} +83.3223 q^{34} +1.70135i q^{35} +43.5939 q^{37} +14.9847i q^{38} -3.05456 q^{40} +52.0734i q^{41} +62.5698 q^{43} +48.9097 q^{46} +38.7052i q^{47} +57.0665 q^{49} -90.4110i q^{50} +185.382 q^{52} +35.3780i q^{53} +190.428i q^{56} +14.5412 q^{58} +19.6883i q^{59} -6.42431 q^{61} +74.2301i q^{62} -10.1214 q^{64} -3.36267i q^{65} -3.98708 q^{67} -209.601i q^{68} +6.15957 q^{70} +54.7712i q^{71} -22.8087 q^{73} -157.827i q^{74} +37.6946 q^{76} -57.1563 q^{79} +5.04069i q^{80} +188.526 q^{82} -56.8060i q^{83} -3.80199 q^{85} -226.527i q^{86} +28.1602i q^{89} -209.637 q^{91} -123.034i q^{92} +140.128 q^{94} -0.683750i q^{95} +82.9323 q^{97} -206.603i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 - 8 * q^7	\( 16 q - 32 q^{4} - 8 q^{7} + 24 q^{10} + 4 q^{13} + 28 q^{16} - 20 q^{19} - 44 q^{25} + 16 q^{28} + 28 q^{31} + 148 q^{34} - 148 q^{37} + 224 q^{40} + 272 q^{43} + 208 q^{46} + 348 q^{49} + 520 q^{52} - 44 q^{58} + 224 q^{61} + 436 q^{64} + 24 q^{67} - 664 q^{70} - 4 q^{73} + 1052 q^{76} + 216 q^{79} + 348 q^{82} + 416 q^{85} - 168 q^{91} + 1140 q^{94} - 44 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 - 8 * q^7 + 24 * q^10 + 4 * q^13 + 28 * q^16 - 20 * q^19 - 44 * q^25 + 16 * q^28 + 28 * q^31 + 148 * q^34 - 148 * q^37 + 224 * q^40 + 272 * q^43 + 208 * q^46 + 348 * q^49 + 520 * q^52 - 44 * q^58 + 224 * q^61 + 436 * q^64 + 24 * q^67 - 664 * q^70 - 4 * q^73 + 1052 * q^76 + 216 * q^79 + 348 * q^82 + 416 * q^85 - 168 * q^91 + 1140 * q^94 - 44 * q^97

Introduction

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