LMFDB - Embedded newform 1089.3.b.i.485.3

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			485.3
Root			$-2.99491i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.i.485.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +O(q^{10})$	$q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +4.82698 q^{10} +7.56350 q^{13} +9.97250i q^{14} -11.1821 q^{16} -28.3526i q^{17} -26.0667 q^{19} -8.00948i q^{20} +19.5656i q^{23} +22.4023 q^{25} -22.6520i q^{26} +16.5475 q^{28} +2.15288i q^{29} -9.22840 q^{31} +45.1036i q^{32} -84.9135 q^{34} -5.36675i q^{35} -67.5342 q^{37} +78.0674i q^{38} -4.67977 q^{40} +29.0597i q^{41} +0.719802 q^{43} +58.5973 q^{46} +24.5249i q^{47} -37.9123 q^{49} -67.0930i q^{50} -37.5868 q^{52} +5.79176i q^{53} -9.66837i q^{56} +6.44768 q^{58} +73.9896i q^{59} +72.1664 q^{61} +27.6383i q^{62} +90.3531 q^{64} +12.1903i q^{65} -79.8310 q^{67} +140.898i q^{68} -16.0730 q^{70} +107.780i q^{71} -90.3799 q^{73} +202.259i q^{74} +129.538 q^{76} +114.802 q^{79} -18.0224i q^{80} +87.0314 q^{82} +125.752i q^{83} +45.6966 q^{85} -2.15574i q^{86} -83.3041i q^{89} -25.1850 q^{91} -97.2313i q^{92} +73.4499 q^{94} -42.0123i q^{95} -78.0275 q^{97} +113.544i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.99491i	− 1.49746i	−0.662877	−	0.748728i	$-0.730665\pi$
$2$	− 2.99491i	− 1.49746i	0.662877	−	0.748728i	$-0.269335\pi$
$3$	0	0
$3$	0	0
$4$	−4.96950	−1.24238
$5$	1.61173i	0.322345i	0.986926	+	0.161173i	$0.0515276\pi$
$5$	1.61173i	0.322345i	−0.986926	+	0.161173i	$0.948472\pi$
$6$	0	0
$7$	−3.32981	−0.475688	−0.237844	−	0.971303i	$-0.576441\pi$
$7$	−3.32981	−0.475688	−0.237844	+	0.971303i	$0.576441\pi$
$8$	2.90358i	0.362947i
$9$	0	0
$10$	4.82698	0.482698
$11$	0	0
$11$	0	0
$12$	0	0
$13$	7.56350	0.581807	0.290904	−	0.956752i	$-0.406044\pi$
$13$	7.56350	0.581807	0.290904	+	0.956752i	$0.406044\pi$
$14$	9.97250i	0.712322i
$15$	0	0
$16$	−11.1821	−0.698878
$17$	− 28.3526i	− 1.66780i	−0.551917	−	0.833899i	$-0.686104\pi$
$17$	− 28.3526i	− 1.66780i	0.551917	−	0.833899i	$-0.313896\pi$
$18$	0	0
$19$	−26.0667	−1.37193	−0.685965	−	0.727635i	$-0.740620\pi$
$19$	−26.0667	−1.37193	−0.685965	+	0.727635i	$0.740620\pi$
$20$	− 8.00948i	− 0.400474i
$21$	0	0
$22$	0	0
$23$	19.5656i	0.850679i	0.905034	+	0.425339i	$0.139845\pi$
$23$	19.5656i	0.850679i	−0.905034	+	0.425339i	$0.860155\pi$
$24$	0	0
$25$	22.4023	0.896093
$26$	− 22.6520i	− 0.871231i
$27$	0	0
$28$	16.5475	0.590983
$29$	2.15288i	0.0742372i	0.999311	+	0.0371186i	$0.0118179\pi$
$29$	2.15288i	0.0742372i	−0.999311	+	0.0371186i	$0.988182\pi$
$30$	0	0
$31$	−9.22840	−0.297690	−0.148845	−	0.988861i	$-0.547556\pi$
$31$	−9.22840	−0.297690	−0.148845	+	0.988861i	$0.547556\pi$
$32$	45.1036i	1.40949i
$33$	0	0
$34$	−84.9135	−2.49745
$35$	− 5.36675i	− 0.153336i
$36$	0	0
$37$	−67.5342	−1.82525	−0.912624	−	0.408800i	$-0.865948\pi$
$37$	−67.5342	−1.82525	−0.912624	+	0.408800i	$0.865948\pi$
$38$	78.0674i	2.05440i
$39$	0	0
$40$	−4.67977	−0.116994
$41$	29.0597i	0.708774i	0.935099	+	0.354387i	$0.115310\pi$
$41$	29.0597i	0.708774i	−0.935099	+	0.354387i	$0.884690\pi$
$42$	0	0
$43$	0.719802	0.0167396	0.00836979	−	0.999965i	$-0.497336\pi$
$43$	0.719802	0.0167396	0.00836979	+	0.999965i	$0.497336\pi$
$44$	0	0
$45$	0	0
$46$	58.5973	1.27385
$47$	24.5249i	0.521806i	0.965365	+	0.260903i	$0.0840203\pi$
$47$	24.5249i	0.521806i	−0.965365	+	0.260903i	$0.915980\pi$
$48$	0	0
$49$	−37.9123	−0.773721
$50$	− 67.0930i	− 1.34186i
$51$	0	0
$52$	−37.5868	−0.722823
$53$	5.79176i	0.109278i	0.998506	+	0.0546392i	$0.0174009\pi$
$53$	5.79176i	0.109278i	−0.998506	+	0.0546392i	$0.982599\pi$
$54$	0	0
$55$	0	0
$56$	− 9.66837i	− 0.172650i
$57$	0	0
$58$	6.44768	0.111167
$59$	73.9896i	1.25406i	0.778994	+	0.627031i	$0.215730\pi$
$59$	73.9896i	1.25406i	−0.778994	+	0.627031i	$0.784270\pi$
$60$	0	0
$61$	72.1664	1.18306	0.591528	−	0.806284i	$-0.298525\pi$
$61$	72.1664	1.18306	0.591528	+	0.806284i	$0.298525\pi$
$62$	27.6383i	0.445778i
$63$	0	0
$64$	90.3531	1.41177
$65$	12.1903i	0.187543i
$66$	0	0
$67$	−79.8310	−1.19151	−0.595754	−	0.803167i	$-0.703146\pi$
$67$	−79.8310	−1.19151	−0.595754	+	0.803167i	$0.703146\pi$
$68$	140.898i	2.07203i
$69$	0	0
$70$	−16.0730	−0.229614
$71$	107.780i	1.51803i	0.651072	+	0.759016i	$0.274319\pi$
$71$	107.780i	1.51803i	−0.651072	+	0.759016i	$0.725681\pi$
$72$	0	0
$73$	−90.3799	−1.23808	−0.619041	−	0.785359i	$-0.712478\pi$
$73$	−90.3799	−1.23808	−0.619041	+	0.785359i	$0.712478\pi$
$74$	202.259i	2.73323i
$75$	0	0
$76$	129.538	1.70445
$77$	0	0
$78$	0	0
$79$	114.802	1.45319	0.726595	−	0.687066i	$-0.241102\pi$
$79$	114.802	1.45319	0.726595	+	0.687066i	$0.241102\pi$
$80$	− 18.0224i	− 0.225280i
$81$	0	0
$82$	87.0314	1.06136
$83$	125.752i	1.51509i	0.652784	+	0.757544i	$0.273601\pi$
$83$	125.752i	1.51509i	−0.652784	+	0.757544i	$0.726399\pi$
$84$	0	0
$85$	45.6966	0.537607
$86$	− 2.15574i	− 0.0250668i
$87$	0	0
$88$	0	0
$89$	− 83.3041i	− 0.936001i	−0.883728	−	0.468000i	$-0.844975\pi$
$89$	− 83.3041i	− 0.936001i	0.883728	−	0.468000i	$-0.155025\pi$
$90$	0	0
$91$	−25.1850	−0.276759
$92$	− 97.2313i	− 1.05686i
$93$	0	0
$94$	73.4499	0.781382
$95$	− 42.0123i	− 0.442235i
$96$	0	0
$97$	−78.0275	−0.804407	−0.402203	−	0.915550i	$-0.631756\pi$
$97$	−78.0275	−0.804407	−0.402203	+	0.915550i	$0.631756\pi$
$98$	113.544i	1.15861i
$99$	0	0
$100$	−111.328	−1.11328
$101$	38.9858i	0.385998i	0.981199	+	0.192999i	$0.0618215\pi$
$101$	38.9858i	0.385998i	−0.981199	+	0.192999i	$0.938179\pi$
$102$	0	0
$103$	30.1474	0.292694	0.146347	−	0.989233i	$-0.453248\pi$
$103$	30.1474	0.292694	0.146347	+	0.989233i	$0.453248\pi$
$104$	21.9612i	0.211165i
$105$	0	0
$106$	17.3458	0.163640
$107$	− 49.7544i	− 0.464994i	−0.972597	−	0.232497i	$-0.925310\pi$
$107$	− 49.7544i	− 0.464994i	0.972597	−	0.232497i	$-0.0746897\pi$
$108$	0	0
$109$	−149.823	−1.37453	−0.687263	−	0.726409i	$-0.741188\pi$
$109$	−149.823	−1.37453	−0.687263	+	0.726409i	$0.741188\pi$
$110$	0	0
$111$	0	0
$112$	37.2342	0.332448
$113$	− 127.469i	− 1.12805i	−0.825759	−	0.564024i	$-0.809253\pi$
$113$	− 127.469i	− 1.12805i	0.825759	−	0.564024i	$-0.190747\pi$
$114$	0	0
$115$	−31.5344	−0.274212
$116$	− 10.6987i	− 0.0922305i
$117$	0	0
$118$	221.593	1.87790
$119$	94.4088i	0.793351i
$120$	0	0
$121$	0	0
$122$	− 216.132i	− 1.77157i
$123$	0	0
$124$	45.8606	0.369843
$125$	76.3996i	0.611197i
$126$	0	0
$127$	−237.975	−1.87382	−0.936910	−	0.349572i	$-0.886327\pi$
$127$	−237.975	−1.87382	−0.936910	+	0.349572i	$0.886327\pi$
$128$	− 90.1853i	− 0.704573i
$129$	0	0
$130$	36.5089	0.280837
$131$	148.535i	1.13386i	0.823767	+	0.566929i	$0.191869\pi$
$131$	148.535i	1.13386i	−0.823767	+	0.566929i	$0.808131\pi$
$132$	0	0
$133$	86.7972	0.652610
$134$	239.087i	1.78423i
$135$	0	0
$136$	82.3239	0.605322
$137$	20.1586i	0.147143i	0.997290	+	0.0735716i	$0.0234397\pi$
$137$	20.1586i	0.147143i	−0.997290	+	0.0735716i	$0.976560\pi$
$138$	0	0
$139$	148.878	1.07106	0.535532	−	0.844515i	$-0.320111\pi$
$139$	148.878	1.07106	0.535532	+	0.844515i	$0.320111\pi$
$140$	26.6701i	0.190501i
$141$	0	0
$142$	322.792	2.27319
$143$	0	0
$144$	0	0
$145$	−3.46985	−0.0239300
$146$	270.680i	1.85397i
$147$	0	0
$148$	335.611	2.26764
$149$	− 44.8984i	− 0.301332i	−0.988585	−	0.150666i	$-0.951858\pi$
$149$	− 44.8984i	− 0.301332i	0.988585	−	0.150666i	$-0.0481418\pi$
$150$	0	0
$151$	46.5427	0.308230	0.154115	−	0.988053i	$-0.450747\pi$
$151$	46.5427	0.308230	0.154115	+	0.988053i	$0.450747\pi$
$152$	− 75.6866i	− 0.497938i
$153$	0	0
$154$	0	0
$155$	− 14.8737i	− 0.0959591i
$156$	0	0
$157$	18.1734	0.115754	0.0578769	−	0.998324i	$-0.481567\pi$
$157$	18.1734	0.115754	0.0578769	+	0.998324i	$0.481567\pi$
$158$	− 343.822i	− 2.17609i
$159$	0	0
$160$	−72.6947	−0.454342
$161$	− 65.1498i	− 0.404657i
$162$	0	0
$163$	−139.162	−0.853753	−0.426876	−	0.904310i	$-0.640386\pi$
$163$	−139.162	−0.853753	−0.426876	+	0.904310i	$0.640386\pi$
$164$	− 144.412i	− 0.880564i
$165$	0	0
$166$	376.617	2.26878
$167$	− 145.214i	− 0.869544i	−0.900541	−	0.434772i	$-0.856829\pi$
$167$	− 145.214i	− 0.869544i	0.900541	−	0.434772i	$-0.143171\pi$
$168$	0	0
$169$	−111.794	−0.661500
$170$	− 136.857i	− 0.805043i
$171$	0	0
$172$	−3.57706	−0.0207968
$173$	− 290.416i	− 1.67871i	−0.543586	−	0.839353i	$-0.682934\pi$
$173$	− 290.416i	− 1.67871i	0.543586	−	0.839353i	$-0.317066\pi$
$174$	0	0
$175$	−74.5956	−0.426261
$176$	0	0
$177$	0	0
$178$	−249.488	−1.40162
$179$	146.411i	0.817940i	0.912548	+	0.408970i	$0.134112\pi$
$179$	146.411i	0.817940i	−0.912548	+	0.408970i	$0.865888\pi$
$180$	0	0
$181$	223.461	1.23459	0.617295	−	0.786732i	$-0.288228\pi$
$181$	223.461	1.23459	0.617295	+	0.786732i	$0.288228\pi$
$182$	75.4270i	0.414434i
$183$	0	0
$184$	−56.8102	−0.308751
$185$	− 108.847i	− 0.588361i
$186$	0	0
$187$	0	0
$188$	− 121.876i	− 0.648279i
$189$	0	0
$190$	−125.823	−0.662228
$191$	− 23.2178i	− 0.121559i	−0.998151	−	0.0607795i	$-0.980641\pi$
$191$	− 23.2178i	− 0.121559i	0.998151	−	0.0607795i	$-0.0193586\pi$
$192$	0	0
$193$	−125.437	−0.649934	−0.324967	−	0.945725i	$-0.605353\pi$
$193$	−125.437	−0.649934	−0.324967	+	0.945725i	$0.605353\pi$
$194$	233.685i	1.20456i
$195$	0	0
$196$	188.405	0.961252
$197$	− 12.3847i	− 0.0628665i	−0.999506	−	0.0314333i	$-0.989993\pi$
$197$	− 12.3847i	− 0.0628665i	0.999506	−	0.0314333i	$-0.0100072\pi$
$198$	0	0
$199$	112.611	0.565885	0.282943	−	0.959137i	$-0.408689\pi$
$199$	112.611	0.565885	0.282943	+	0.959137i	$0.408689\pi$
$200$	65.0469i	0.325235i
$201$	0	0
$202$	116.759	0.578016
$203$	− 7.16868i	− 0.0353137i
$204$	0	0
$205$	−46.8364	−0.228470
$206$	− 90.2890i	− 0.438296i
$207$	0	0
$208$	−84.5754	−0.406613
$209$	0	0
$210$	0	0
$211$	153.910	0.729433	0.364716	−	0.931119i	$-0.381166\pi$
$211$	153.910	0.729433	0.364716	+	0.931119i	$0.381166\pi$
$212$	− 28.7822i	− 0.135765i
$213$	0	0
$214$	−149.010	−0.696309
$215$	1.16012i	0.00539593i
$216$	0	0
$217$	30.7289	0.141608
$218$	448.708i	2.05829i
$219$	0	0
$220$	0	0
$221$	− 214.445i	− 0.970337i
$222$	0	0
$223$	282.384	1.26630	0.633148	−	0.774031i	$-0.281763\pi$
$223$	282.384	1.26630	0.633148	+	0.774031i	$0.281763\pi$
$224$	− 150.187i	− 0.670476i
$225$	0	0
$226$	−381.760	−1.68920
$227$	− 202.865i	− 0.893677i	−0.894615	−	0.446839i	$-0.852550\pi$
$227$	− 202.865i	− 0.893677i	0.894615	−	0.446839i	$-0.147450\pi$
$228$	0	0
$229$	−419.120	−1.83022	−0.915108	−	0.403208i	$-0.867895\pi$
$229$	−419.120	−1.83022	−0.915108	+	0.403208i	$0.867895\pi$
$230$	94.4428i	0.410621i
$231$	0	0
$232$	−6.25105	−0.0269442
$233$	− 111.121i	− 0.476916i	−0.971153	−	0.238458i	$-0.923358\pi$
$233$	− 111.121i	− 0.476916i	0.971153	−	0.238458i	$-0.0766420\pi$
$234$	0	0
$235$	−39.5274	−0.168202
$236$	− 367.692i	− 1.55802i
$237$	0	0
$238$	282.746	1.18801
$239$	− 314.056i	− 1.31404i	−0.753872	−	0.657021i	$-0.771816\pi$
$239$	− 314.056i	− 1.31404i	0.753872	−	0.657021i	$-0.228184\pi$
$240$	0	0
$241$	−104.293	−0.432753	−0.216376	−	0.976310i	$-0.569424\pi$
$241$	−104.293	−0.432753	−0.216376	+	0.976310i	$0.569424\pi$
$242$	0	0
$243$	0	0
$244$	−358.631	−1.46980
$245$	− 61.1043i	− 0.249405i
$246$	0	0
$247$	−197.155	−0.798199
$248$	− 26.7954i	− 0.108046i
$249$	0	0
$250$	228.810	0.915241
$251$	− 333.537i	− 1.32883i	−0.747363	−	0.664416i	$-0.768680\pi$
$251$	− 333.537i	− 1.32883i	0.747363	−	0.664416i	$-0.231320\pi$
$252$	0	0
$253$	0	0
$254$	712.714i	2.80596i
$255$	0	0
$256$	91.3152	0.356700
$257$	306.989i	1.19451i	0.802052	+	0.597254i	$0.203742\pi$
$257$	306.989i	1.19451i	−0.802052	+	0.597254i	$0.796258\pi$
$258$	0	0
$259$	224.876	0.868248
$260$	− 60.5797i	− 0.232999i
$261$	0	0
$262$	444.851	1.69790
$263$	− 249.060i	− 0.946997i	−0.880794	−	0.473499i	$-0.842991\pi$
$263$	− 249.060i	− 0.946997i	0.880794	−	0.473499i	$-0.157009\pi$
$264$	0	0
$265$	−9.33473	−0.0352254
$266$	− 259.950i	− 0.977255i
$267$	0	0
$268$	396.720	1.48030
$269$	196.927i	0.732071i	0.930601	+	0.366035i	$0.119285\pi$
$269$	196.927i	0.732071i	−0.930601	+	0.366035i	$0.880715\pi$
$270$	0	0
$271$	−463.464	−1.71020	−0.855099	−	0.518465i	$-0.826504\pi$
$271$	−463.464	−1.71020	−0.855099	+	0.518465i	$0.826504\pi$
$272$	317.040i	1.16559i
$273$	0	0
$274$	60.3733	0.220340
$275$	0	0
$276$	0	0
$277$	−23.4550	−0.0846751	−0.0423376	−	0.999103i	$-0.513480\pi$
$277$	−23.4550	−0.0846751	−0.0423376	+	0.999103i	$0.513480\pi$
$278$	− 445.876i	− 1.60387i
$279$	0	0
$280$	15.5828	0.0556528
$281$	263.447i	0.937534i	0.883322	+	0.468767i	$0.155302\pi$
$281$	263.447i	0.937534i	−0.883322	+	0.468767i	$0.844698\pi$
$282$	0	0
$283$	−35.8237	−0.126585	−0.0632927	−	0.997995i	$-0.520160\pi$
$283$	−35.8237	−0.126585	−0.0632927	+	0.997995i	$0.520160\pi$
$284$	− 535.614i	− 1.88597i
$285$	0	0
$286$	0	0
$287$	− 96.7636i	− 0.337155i
$288$	0	0
$289$	−514.868	−1.78155
$290$	10.3919i	0.0358342i
$291$	0	0
$292$	449.143	1.53816
$293$	114.195i	0.389745i	0.980829	+	0.194873i	$0.0624294\pi$
$293$	114.195i	0.389745i	−0.980829	+	0.194873i	$0.937571\pi$
$294$	0	0
$295$	−119.251	−0.404241
$296$	− 196.091i	− 0.662469i
$297$	0	0
$298$	−134.467	−0.451231
$299$	147.984i	0.494931i
$300$	0	0
$301$	−2.39681	−0.00796281
$302$	− 139.391i	− 0.461560i
$303$	0	0
$304$	291.479	0.958812
$305$	116.313i	0.381353i
$306$	0	0
$307$	−224.438	−0.731070	−0.365535	−	0.930798i	$-0.619114\pi$
$307$	−224.438	−0.731070	−0.365535	+	0.930798i	$0.619114\pi$
$308$	0	0
$309$	0	0
$310$	−44.5453	−0.143695
$311$	− 171.175i	− 0.550401i	−0.961387	−	0.275201i	$-0.911256\pi$
$311$	− 171.175i	− 0.550401i	0.961387	−	0.275201i	$-0.0887442\pi$
$312$	0	0
$313$	−274.110	−0.875749	−0.437875	−	0.899036i	$-0.644269\pi$
$313$	−274.110	−0.875749	−0.437875	+	0.899036i	$0.644269\pi$
$314$	− 54.4276i	− 0.173336i
$315$	0	0
$316$	−570.509	−1.80541
$317$	618.549i	1.95126i	0.219427	+	0.975629i	$0.429581\pi$
$317$	618.549i	1.95126i	−0.219427	+	0.975629i	$0.570419\pi$
$318$	0	0
$319$	0	0
$320$	145.625i	0.455077i
$321$	0	0
$322$	−195.118	−0.605957
$323$	739.057i	2.28810i
$324$	0	0
$325$	169.440	0.521354
$326$	416.777i	1.27846i
$327$	0	0
$328$	−84.3772	−0.257248
$329$	− 81.6633i	− 0.248217i
$330$	0	0
$331$	−160.328	−0.484376	−0.242188	−	0.970229i	$-0.577865\pi$
$331$	−160.328	−0.484376	−0.242188	+	0.970229i	$0.577865\pi$
$332$	− 624.926i	− 1.88231i
$333$	0	0
$334$	−434.903	−1.30210
$335$	− 128.666i	− 0.384077i
$336$	0	0
$337$	−110.004	−0.326422	−0.163211	−	0.986591i	$-0.552185\pi$
$337$	−110.004	−0.326422	−0.163211	+	0.986591i	$0.552185\pi$
$338$	334.812i	0.990568i
$339$	0	0
$340$	−227.089	−0.667910
$341$	0	0
$342$	0	0
$343$	289.402	0.843738
$344$	2.09000i	0.00607558i
$345$	0	0
$346$	−869.771	−2.51379
$347$	− 221.736i	− 0.639008i	−0.947585	−	0.319504i	$-0.896484\pi$
$347$	− 221.736i	− 0.639008i	0.947585	−	0.319504i	$-0.103516\pi$
$348$	0	0
$349$	−466.352	−1.33625	−0.668126	−	0.744049i	$-0.732903\pi$
$349$	−466.352	−1.33625	−0.668126	+	0.744049i	$0.732903\pi$
$350$	223.407i	0.638307i
$351$	0	0
$352$	0	0
$353$	494.404i	1.40058i	0.713860	+	0.700289i	$0.246945\pi$
$353$	494.404i	1.40058i	−0.713860	+	0.700289i	$0.753055\pi$
$354$	0	0
$355$	−173.712	−0.489330
$356$	413.980i	1.16286i
$357$	0	0
$358$	438.489	1.22483
$359$	− 28.1698i	− 0.0784673i	−0.999230	−	0.0392336i	$-0.987508\pi$
$359$	− 28.1698i	− 0.0784673i	0.999230	−	0.0392336i	$-0.0124917\pi$
$360$	0	0
$361$	318.471	0.882191
$362$	− 669.246i	− 1.84875i
$363$	0	0
$364$	125.157	0.343838
$365$	− 145.668i	− 0.399090i
$366$	0	0
$367$	29.7976	0.0811925	0.0405962	−	0.999176i	$-0.487074\pi$
$367$	29.7976	0.0811925	0.0405962	+	0.999176i	$0.487074\pi$
$368$	− 218.784i	− 0.594521i
$369$	0	0
$370$	−325.986	−0.881044
$371$	− 19.2855i	− 0.0519824i
$372$	0	0
$373$	−313.887	−0.841520	−0.420760	−	0.907172i	$-0.638237\pi$
$373$	−313.887	−0.841520	−0.420760	+	0.907172i	$0.638237\pi$
$374$	0	0
$375$	0	0
$376$	−71.2099	−0.189388
$377$	16.2833i	0.0431917i
$378$	0	0
$379$	−31.4155	−0.0828904	−0.0414452	−	0.999141i	$-0.513196\pi$
$379$	−31.4155	−0.0828904	−0.0414452	+	0.999141i	$0.513196\pi$
$380$	208.780i	0.549422i
$381$	0	0
$382$	−69.5352	−0.182029
$383$	− 474.897i	− 1.23994i	−0.784625	−	0.619970i	$-0.787145\pi$
$383$	− 474.897i	− 1.23994i	0.784625	−	0.619970i	$-0.212855\pi$
$384$	0	0
$385$	0	0
$386$	375.674i	0.973249i
$387$	0	0
$388$	387.758	0.999376
$389$	42.3470i	0.108861i	0.998518	+	0.0544306i	$0.0173344\pi$
$389$	42.3470i	0.108861i	−0.998518	+	0.0544306i	$0.982666\pi$
$390$	0	0
$391$	554.735	1.41876
$392$	− 110.081i	− 0.280820i
$393$	0	0
$394$	−37.0911	−0.0941399
$395$	185.029i	0.468429i
$396$	0	0
$397$	−560.270	−1.41126	−0.705630	−	0.708580i	$-0.749336\pi$
$397$	−560.270	−1.41126	−0.705630	+	0.708580i	$0.749336\pi$
$398$	− 337.261i	− 0.847389i
$399$	0	0
$400$	−250.504	−0.626260
$401$	565.289i	1.40970i	0.709357	+	0.704850i	$0.248986\pi$
$401$	565.289i	1.40970i	−0.709357	+	0.704850i	$0.751014\pi$
$402$	0	0
$403$	−69.7990	−0.173198
$404$	− 193.740i	− 0.479555i
$405$	0	0
$406$	−21.4696	−0.0528807
$407$	0	0
$408$	0	0
$409$	174.254	0.426048	0.213024	−	0.977047i	$-0.431669\pi$
$409$	174.254	0.426048	0.213024	+	0.977047i	$0.431669\pi$
$410$	140.271i	0.342124i
$411$	0	0
$412$	−149.818	−0.363635
$413$	− 246.372i	− 0.596542i
$414$	0	0
$415$	−202.678	−0.488382
$416$	341.141i	0.820050i
$417$	0	0
$418$	0	0
$419$	279.267i	0.666508i	0.942837	+	0.333254i	$0.108147\pi$
$419$	279.267i	0.666508i	−0.942837	+	0.333254i	$0.891853\pi$
$420$	0	0
$421$	387.008	0.919259	0.459630	−	0.888111i	$-0.347982\pi$
$421$	387.008	0.919259	0.459630	+	0.888111i	$0.347982\pi$
$422$	− 460.948i	− 1.09229i
$423$	0	0
$424$	−16.8168	−0.0396623
$425$	− 635.164i	− 1.49450i
$426$	0	0
$427$	−240.301	−0.562765
$428$	247.255i	0.577698i
$429$	0	0
$430$	3.47447	0.00808017
$431$	− 319.975i	− 0.742402i	−0.928552	−	0.371201i	$-0.878946\pi$
$431$	− 319.975i	− 0.742402i	0.928552	−	0.371201i	$-0.121054\pi$
$432$	0	0
$433$	526.145	1.21512	0.607558	−	0.794276i	$-0.292149\pi$
$433$	526.145	1.21512	0.607558	+	0.794276i	$0.292149\pi$
$434$	− 92.0303i	− 0.212051i
$435$	0	0
$436$	744.547	1.70768
$437$	− 510.010i	− 1.16707i
$438$	0	0
$439$	22.7817	0.0518946	0.0259473	−	0.999663i	$-0.491740\pi$
$439$	22.7817	0.0518946	0.0259473	+	0.999663i	$0.491740\pi$
$440$	0	0
$441$	0	0
$442$	−642.243	−1.45304
$443$	60.0022i	0.135445i	0.997704	+	0.0677226i	$0.0215733\pi$
$443$	60.0022i	0.135445i	−0.997704	+	0.0677226i	$0.978427\pi$
$444$	0	0
$445$	134.263	0.301716
$446$	− 845.715i	− 1.89622i
$447$	0	0
$448$	−300.859	−0.671560
$449$	− 761.224i	− 1.69538i	−0.530495	−	0.847688i	$-0.677994\pi$
$449$	− 761.224i	− 1.69538i	0.530495	−	0.847688i	$-0.322006\pi$
$450$	0	0
$451$	0	0
$452$	633.459i	1.40146i
$453$	0	0
$454$	−607.562	−1.33824
$455$	− 40.5914i	− 0.0892119i
$456$	0	0
$457$	−788.938	−1.72634	−0.863171	−	0.504912i	$-0.831525\pi$
$457$	−788.938	−1.72634	−0.863171	+	0.504912i	$0.831525\pi$
$458$	1255.23i	2.74067i
$459$	0	0
$460$	156.710	0.340675
$461$	− 726.842i	− 1.57666i	−0.615250	−	0.788332i	$-0.710945\pi$
$461$	− 726.842i	− 1.57666i	0.615250	−	0.788332i	$-0.289055\pi$
$462$	0	0
$463$	301.548	0.651291	0.325646	−	0.945492i	$-0.394418\pi$
$463$	301.548	0.651291	0.325646	+	0.945492i	$0.394418\pi$
$464$	− 24.0736i	− 0.0518827i
$465$	0	0
$466$	−332.799	−0.714161
$467$	− 292.485i	− 0.626306i	−0.949703	−	0.313153i	$-0.898615\pi$
$467$	− 292.485i	− 0.626306i	0.949703	−	0.313153i	$-0.101385\pi$
$468$	0	0
$469$	265.822	0.566785
$470$	118.381i	0.251875i
$471$	0	0
$472$	−214.835	−0.455158
$473$	0	0
$474$	0	0
$475$	−583.954	−1.22938
$476$	− 469.165i	− 0.985640i
$477$	0	0
$478$	−940.571	−1.96772
$479$	496.880i	1.03733i	0.854978	+	0.518664i	$0.173570\pi$
$479$	496.880i	1.03733i	−0.854978	+	0.518664i	$0.826430\pi$
$480$	0	0
$481$	−510.795	−1.06194
$482$	312.350i	0.648029i
$483$	0	0
$484$	0	0
$485$	− 125.759i	− 0.259297i
$486$	0	0
$487$	691.718	1.42037	0.710183	−	0.704017i	$-0.248612\pi$
$487$	691.718	1.42037	0.710183	+	0.704017i	$0.248612\pi$
$488$	209.541i	0.429387i
$489$	0	0
$490$	−183.002	−0.373474
$491$	296.243i	0.603346i	0.953412	+	0.301673i	$0.0975450\pi$
$491$	296.243i	0.603346i	−0.953412	+	0.301673i	$0.902455\pi$
$492$	0	0
$493$	61.0396	0.123813
$494$	590.462i	1.19527i
$495$	0	0
$496$	103.192	0.208049
$497$	− 358.888i	− 0.722109i
$498$	0	0
$499$	−134.004	−0.268544	−0.134272	−	0.990944i	$-0.542870\pi$
$499$	−134.004	−0.268544	−0.134272	+	0.990944i	$0.542870\pi$
$500$	− 379.668i	− 0.759336i
$501$	0	0
$502$	−998.914	−1.98987
$503$	− 190.003i	− 0.377740i	−0.982002	−	0.188870i	$-0.939518\pi$
$503$	− 190.003i	− 0.377740i	0.982002	−	0.188870i	$-0.0604824\pi$
$504$	0	0
$505$	−62.8345	−0.124425
$506$	0	0
$507$	0	0
$508$	1182.62	2.32799
$509$	− 119.100i	− 0.233989i	−0.993133	−	0.116995i	$-0.962674\pi$
$509$	− 119.100i	− 0.233989i	0.993133	−	0.116995i	$-0.0373260\pi$
$510$	0	0
$511$	300.948	0.588940
$512$	− 634.222i	− 1.23872i
$513$	0	0
$514$	919.404	1.78872
$515$	48.5895i	0.0943485i
$516$	0	0
$517$	0	0
$518$	− 673.485i	− 1.30016i
$519$	0	0
$520$	−35.3955	−0.0680682
$521$	− 637.297i	− 1.22322i	−0.791160	−	0.611609i	$-0.790522\pi$
$521$	− 637.297i	− 1.22322i	0.791160	−	0.611609i	$-0.209478\pi$
$522$	0	0
$523$	129.698	0.247988	0.123994	−	0.992283i	$-0.460430\pi$
$523$	129.698	0.247988	0.123994	+	0.992283i	$0.460430\pi$
$524$	− 738.147i	− 1.40868i
$525$	0	0
$526$	−745.914	−1.41809
$527$	261.649i	0.496487i
$528$	0	0
$529$	146.187	0.276346
$530$	27.9567i	0.0527485i
$531$	0	0
$532$	−431.339	−0.810787
$533$	219.793i	0.412370i
$534$	0	0
$535$	80.1905	0.149889
$536$	− 231.795i	− 0.432454i
$537$	0	0
$538$	589.779	1.09624
$539$	0	0
$540$	0	0
$541$	−493.991	−0.913108	−0.456554	−	0.889696i	$-0.650916\pi$
$541$	−493.991	−0.913108	−0.456554	+	0.889696i	$0.650916\pi$
$542$	1388.03i	2.56095i
$543$	0	0
$544$	1278.80	2.35074
$545$	− 241.474i	− 0.443072i
$546$	0	0
$547$	−106.128	−0.194018	−0.0970089	−	0.995284i	$-0.530928\pi$
$547$	−106.128	−0.194018	−0.0970089	+	0.995284i	$0.530928\pi$
$548$	− 100.178i	− 0.182807i
$549$	0	0
$550$	0	0
$551$	− 56.1183i	− 0.101848i
$552$	0	0
$553$	−382.269	−0.691264
$554$	70.2457i	0.126797i
$555$	0	0
$556$	−739.849	−1.33066
$557$	− 827.637i	− 1.48588i	−0.669356	−	0.742942i	$-0.733430\pi$
$557$	− 827.637i	− 1.48588i	0.669356	−	0.742942i	$-0.266570\pi$
$558$	0	0
$559$	5.44422	0.00973921
$560$	60.0113i	0.107163i
$561$	0	0
$562$	789.001	1.40392
$563$	470.917i	0.836442i	0.908345	+	0.418221i	$0.137346\pi$
$563$	470.917i	0.836442i	−0.908345	+	0.418221i	$0.862654\pi$
$564$	0	0
$565$	205.446	0.363621
$566$	107.289i	0.189556i
$567$	0	0
$568$	−312.948	−0.550965
$569$	200.074i	0.351624i	0.984424	+	0.175812i	$0.0562551\pi$
$569$	200.074i	0.351624i	−0.984424	+	0.175812i	$0.943745\pi$
$570$	0	0
$571$	658.692	1.15358	0.576788	−	0.816894i	$-0.304306\pi$
$571$	658.692	1.15358	0.576788	+	0.816894i	$0.304306\pi$
$572$	0	0
$573$	0	0
$574$	−289.798	−0.504875
$575$	438.315i	0.762287i
$576$	0	0
$577$	−100.977	−0.175004	−0.0875020	−	0.996164i	$-0.527888\pi$
$577$	−100.977	−0.175004	−0.0875020	+	0.996164i	$0.527888\pi$
$578$	1541.98i	2.66779i
$579$	0	0
$580$	17.2434	0.0297301
$581$	− 418.732i	− 0.720709i
$582$	0	0
$583$	0	0
$584$	− 262.425i	− 0.449358i
$585$	0	0
$586$	342.005	0.583627
$587$	− 349.390i	− 0.595213i	−0.954689	−	0.297606i	$-0.903812\pi$
$587$	− 349.390i	− 0.595213i	0.954689	−	0.297606i	$-0.0961883\pi$
$588$	0	0
$589$	240.554	0.408410
$590$	357.147i	0.605333i
$591$	0	0
$592$	755.171	1.27563
$593$	− 670.238i	− 1.13025i	−0.825005	−	0.565125i	$-0.808828\pi$
$593$	− 670.238i	− 1.13025i	0.825005	−	0.565125i	$-0.191172\pi$
$594$	0	0
$595$	−152.161	−0.255733
$596$	223.123i	0.374367i
$597$	0	0
$598$	443.200	0.741138
$599$	− 771.328i	− 1.28769i	−0.765155	−	0.643847i	$-0.777338\pi$
$599$	− 771.328i	− 1.28769i	0.765155	−	0.643847i	$-0.222662\pi$
$600$	0	0
$601$	−185.722	−0.309021	−0.154511	−	0.987991i	$-0.549380\pi$
$601$	−185.722	−0.309021	−0.154511	+	0.987991i	$0.549380\pi$
$602$	7.17823i	0.0119240i
$603$	0	0
$604$	−231.294	−0.382937
$605$	0	0
$606$	0	0
$607$	−488.185	−0.804259	−0.402129	−	0.915583i	$-0.631730\pi$
$607$	−488.185	−0.804259	−0.402129	+	0.915583i	$0.631730\pi$
$608$	− 1175.70i	− 1.93372i
$609$	0	0
$610$	348.346	0.571059
$611$	185.494i	0.303591i
$612$	0	0
$613$	510.461	0.832726	0.416363	−	0.909199i	$-0.363305\pi$
$613$	510.461	0.832726	0.416363	+	0.909199i	$0.363305\pi$
$614$	672.174i	1.09475i
$615$	0	0
$616$	0	0
$617$	− 1142.14i	− 1.85111i	−0.378608	−	0.925557i	$-0.623597\pi$
$617$	− 1142.14i	− 1.85111i	0.378608	−	0.925557i	$-0.376403\pi$
$618$	0	0
$619$	−908.213	−1.46723	−0.733613	−	0.679567i	$-0.762168\pi$
$619$	−908.213	−1.46723	−0.733613	+	0.679567i	$0.762168\pi$
$620$	73.9147i	0.119217i
$621$	0	0
$622$	−512.653	−0.824202
$623$	277.387i	0.445244i
$624$	0	0
$625$	436.923	0.699077
$626$	820.934i	1.31140i
$627$	0	0
$628$	−90.3126	−0.143810
$629$	1914.77i	3.04415i
$630$	0	0
$631$	944.402	1.49668	0.748338	−	0.663318i	$-0.230852\pi$
$631$	944.402	1.49668	0.748338	+	0.663318i	$0.230852\pi$
$632$	333.336i	0.527431i
$633$	0	0
$634$	1852.50	2.92192
$635$	− 383.551i	− 0.604017i
$636$	0	0
$637$	−286.750	−0.450157
$638$	0	0
$639$	0	0
$640$	145.354	0.227116
$641$	− 359.081i	− 0.560188i	−0.959973	−	0.280094i	$-0.909634\pi$
$641$	− 359.081i	− 0.560188i	0.959973	−	0.280094i	$-0.0903657\pi$
$642$	0	0
$643$	928.146	1.44346	0.721731	−	0.692174i	$-0.243347\pi$
$643$	928.146	1.44346	0.721731	+	0.692174i	$0.243347\pi$
$644$	323.762i	0.502737i
$645$	0	0
$646$	2213.41	3.42633
$647$	436.289i	0.674326i	0.941446	+	0.337163i	$0.109467\pi$
$647$	436.289i	0.674326i	−0.941446	+	0.337163i	$0.890533\pi$
$648$	0	0
$649$	0	0
$650$	− 507.458i	− 0.780705i
$651$	0	0
$652$	691.565	1.06068
$653$	− 243.417i	− 0.372767i	−0.982477	−	0.186383i	$-0.940323\pi$
$653$	− 243.417i	− 0.372767i	0.982477	−	0.186383i	$-0.0596767\pi$
$654$	0	0
$655$	−239.399	−0.365494
$656$	− 324.948i	− 0.495347i
$657$	0	0
$658$	−244.574	−0.371694
$659$	− 348.053i	− 0.528153i	−0.964502	−	0.264076i	$-0.914933\pi$
$659$	− 348.053i	− 0.528153i	0.964502	−	0.264076i	$-0.0850671\pi$
$660$	0	0
$661$	601.901	0.910592	0.455296	−	0.890340i	$-0.349533\pi$
$661$	601.901	0.910592	0.455296	+	0.890340i	$0.349533\pi$
$662$	480.170i	0.725332i
$663$	0	0
$664$	−365.131	−0.549897
$665$	139.893i	0.210366i
$666$	0	0
$667$	−42.1224	−0.0631520
$668$	721.640i	1.08030i
$669$	0	0
$670$	−385.343	−0.575138
$671$	0	0
$672$	0	0
$673$	738.885	1.09790	0.548949	−	0.835856i	$-0.315028\pi$
$673$	738.885	1.09790	0.548949	+	0.835856i	$0.315028\pi$
$674$	329.453i	0.488803i
$675$	0	0
$676$	555.558	0.821832
$677$	42.1168i	0.0622109i	0.999516	+	0.0311055i	$0.00990277\pi$
$677$	42.1168i	0.0622109i	−0.999516	+	0.0311055i	$0.990097\pi$
$678$	0	0
$679$	259.817	0.382647
$680$	132.684i	0.195123i
$681$	0	0
$682$	0	0
$683$	− 543.110i	− 0.795183i	−0.917562	−	0.397592i	$-0.869846\pi$
$683$	− 543.110i	− 0.795183i	0.917562	−	0.397592i	$-0.130154\pi$
$684$	0	0
$685$	−32.4902	−0.0474309
$686$	− 866.734i	− 1.26346i
$687$	0	0
$688$	−8.04886	−0.0116989
$689$	43.8059i	0.0635790i
$690$	0	0
$691$	−428.173	−0.619643	−0.309821	−	0.950795i	$-0.600269\pi$
$691$	−428.173	−0.619643	−0.309821	+	0.950795i	$0.600269\pi$
$692$	1443.22i	2.08558i
$693$	0	0
$694$	−664.080	−0.956887
$695$	239.951i	0.345253i
$696$	0	0
$697$	823.918	1.18209
$698$	1396.68i	2.00098i
$699$	0	0
$700$	370.703	0.529576
$701$	2.78056i	0.00396656i	0.999998	+	0.00198328i	$0.000631299\pi$
$701$	2.78056i	0.00396656i	−0.999998	+	0.00198328i	$0.999369\pi$
$702$	0	0
$703$	1760.39	2.50411
$704$	0	0
$705$	0	0
$706$	1480.70	2.09730
$707$	− 129.816i	− 0.183615i
$708$	0	0
$709$	1077.51	1.51976	0.759878	−	0.650065i	$-0.225258\pi$
$709$	1077.51	1.51976	0.759878	+	0.650065i	$0.225258\pi$
$710$	520.253i	0.732751i
$711$	0	0
$712$	241.880	0.339719
$713$	− 180.559i	− 0.253239i
$714$	0	0
$715$	0	0
$716$	− 727.591i	− 1.01619i
$717$	0	0
$718$	−84.3660	−0.117501
$719$	1073.42i	1.49293i	0.665424	+	0.746465i	$0.268251\pi$
$719$	1073.42i	1.49293i	−0.665424	+	0.746465i	$0.731749\pi$
$720$	0	0
$721$	−100.385	−0.139231
$722$	− 953.792i	− 1.32104i
$723$	0	0
$724$	−1110.49	−1.53383
$725$	48.2295i	0.0665234i
$726$	0	0
$727$	−212.371	−0.292119	−0.146060	−	0.989276i	$-0.546659\pi$
$727$	−212.371	−0.292119	−0.146060	+	0.989276i	$0.546659\pi$
$728$	− 73.1267i	− 0.100449i
$729$	0	0
$730$	−436.262	−0.597620
$731$	− 20.4082i	− 0.0279182i
$732$	0	0
$733$	−942.223	−1.28543	−0.642717	−	0.766104i	$-0.722193\pi$
$733$	−942.223	−1.28543	−0.642717	+	0.766104i	$0.722193\pi$
$734$	− 89.2414i	− 0.121582i
$735$	0	0
$736$	−882.479	−1.19902
$737$	0	0
$738$	0	0
$739$	−720.862	−0.975456	−0.487728	−	0.872996i	$-0.662174\pi$
$739$	−720.862	−0.975456	−0.487728	+	0.872996i	$0.662174\pi$
$740$	540.914i	0.730965i
$741$	0	0
$742$	−57.7583	−0.0778414
$743$	836.564i	1.12593i	0.826482	+	0.562963i	$0.190339\pi$
$743$	836.564i	1.12593i	−0.826482	+	0.562963i	$0.809661\pi$
$744$	0	0
$745$	72.3640	0.0971329
$746$	940.064i	1.26014i
$747$	0	0
$748$	0	0
$749$	165.673i	0.221192i
$750$	0	0
$751$	−35.2668	−0.0469598	−0.0234799	−	0.999724i	$-0.507475\pi$
$751$	−35.2668	−0.0469598	−0.0234799	+	0.999724i	$0.507475\pi$
$752$	− 274.238i	− 0.364679i
$753$	0	0
$754$	48.7670	0.0646777
$755$	75.0141i	0.0993564i
$756$	0	0
$757$	454.700	0.600660	0.300330	−	0.953835i	$-0.402903\pi$
$757$	454.700	0.600660	0.300330	+	0.953835i	$0.402903\pi$
$758$	94.0866i	0.124125i
$759$	0	0
$760$	121.986	0.160508
$761$	− 104.689i	− 0.137568i	−0.997632	−	0.0687840i	$-0.978088\pi$
$761$	− 104.689i	− 0.137568i	0.997632	−	0.0687840i	$-0.0219119\pi$
$762$	0	0
$763$	498.884	0.653845
$764$	115.381i	0.151022i
$765$	0	0
$766$	−1422.28	−1.85676
$767$	559.620i	0.729622i
$768$	0	0
$769$	−93.7806	−0.121951	−0.0609757	−	0.998139i	$-0.519421\pi$
$769$	−93.7806	−0.121951	−0.0609757	+	0.998139i	$0.519421\pi$
$770$	0	0
$771$	0	0
$772$	623.361	0.807463
$773$	− 358.726i	− 0.464070i	−0.972707	−	0.232035i	$-0.925462\pi$
$773$	− 358.726i	− 0.464070i	0.972707	−	0.232035i	$-0.0745384\pi$
$774$	0	0
$775$	−206.738	−0.266758
$776$	− 226.559i	− 0.291957i
$777$	0	0
$778$	126.826	0.163015
$779$	− 757.491i	− 0.972388i
$780$	0	0
$781$	0	0
$782$	− 1661.38i	− 2.12453i
$783$	0	0
$784$	423.938	0.540737
$785$	29.2905i	0.0373127i
$786$	0	0
$787$	276.916	0.351863	0.175931	−	0.984402i	$-0.443706\pi$
$787$	276.916	0.351863	0.175931	+	0.984402i	$0.443706\pi$
$788$	61.5458i	0.0781038i
$789$	0	0
$790$	554.147	0.701452
$791$	424.449i	0.536598i
$792$	0	0
$793$	545.830	0.688311
$794$	1677.96i	2.11330i
$795$	0	0
$796$	−559.622	−0.703042
$797$	− 528.612i	− 0.663252i	−0.943411	−	0.331626i	$-0.892403\pi$
$797$	− 528.612i	− 0.663252i	0.943411	−	0.331626i	$-0.107597\pi$
$798$	0	0
$799$	695.343	0.870267
$800$	1010.43i	1.26303i
$801$	0	0
$802$	1692.99	2.11096
$803$	0	0
$804$	0	0
$805$	105.004	0.130439
$806$	209.042i	0.259357i
$807$	0	0
$808$	−113.198	−0.140097
$809$	− 1257.86i	− 1.55483i	−0.628985	−	0.777417i	$-0.716529\pi$
$809$	− 1257.86i	− 1.55483i	0.628985	−	0.777417i	$-0.283471\pi$
$810$	0	0
$811$	−1041.19	−1.28383	−0.641916	−	0.766775i	$-0.721860\pi$
$811$	−1041.19	−1.28383	−0.641916	+	0.766775i	$0.721860\pi$
$812$	35.6248i	0.0438729i
$813$	0	0
$814$	0	0
$815$	− 224.291i	− 0.275203i
$816$	0	0
$817$	−18.7628	−0.0229655
$818$	− 521.875i	− 0.637989i
$819$	0	0
$820$	232.754	0.283846
$821$	435.664i	0.530650i	0.964159	+	0.265325i	$0.0854793\pi$
$821$	435.664i	0.530650i	−0.964159	+	0.265325i	$0.914521\pi$
$822$	0	0
$823$	587.220	0.713512	0.356756	−	0.934198i	$-0.383883\pi$
$823$	587.220	0.713512	0.356756	+	0.934198i	$0.383883\pi$
$824$	87.5354i	0.106232i
$825$	0	0
$826$	−737.862	−0.893296
$827$	− 274.955i	− 0.332473i	−0.986086	−	0.166236i	$-0.946839\pi$
$827$	− 274.955i	− 0.332473i	0.986086	−	0.166236i	$-0.0531615\pi$
$828$	0	0
$829$	−916.540	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$829$	−916.540	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$830$	607.004i	0.731330i
$831$	0	0
$832$	683.385	0.821376
$833$	1074.91i	1.29041i
$834$	0	0
$835$	234.045	0.280293
$836$	0	0
$837$	0	0
$838$	836.380	0.998067
$839$	1058.83i	1.26202i	0.775777	+	0.631008i	$0.217358\pi$
$839$	1058.83i	1.26202i	−0.775777	+	0.631008i	$0.782642\pi$
$840$	0	0
$841$	836.365	0.994489
$842$	− 1159.06i	− 1.37655i
$843$	0	0
$844$	−764.858	−0.906229
$845$	− 180.181i	− 0.213232i
$846$	0	0
$847$	0	0
$848$	− 64.7637i	− 0.0763723i
$849$	0	0
$850$	−1902.26	−2.23795
$851$	− 1321.35i	− 1.55270i
$852$	0	0
$853$	−262.534	−0.307777	−0.153888	−	0.988088i	$-0.549180\pi$
$853$	−262.534	−0.307777	−0.153888	+	0.988088i	$0.549180\pi$
$854$	719.680i	0.842716i
$855$	0	0
$856$	144.466	0.168768
$857$	− 139.937i	− 0.163287i	−0.996662	−	0.0816437i	$-0.973983\pi$
$857$	− 139.937i	− 0.163287i	0.996662	−	0.0816437i	$-0.0260169\pi$
$858$	0	0
$859$	−466.124	−0.542635	−0.271318	−	0.962490i	$-0.587459\pi$
$859$	−466.124	−0.542635	−0.271318	+	0.962490i	$0.587459\pi$
$860$	− 5.76524i	− 0.00670377i
$861$	0	0
$862$	−958.299	−1.11172
$863$	223.415i	0.258882i	0.991587	+	0.129441i	$0.0413183\pi$
$863$	223.415i	0.258882i	−0.991587	+	0.129441i	$0.958682\pi$
$864$	0	0
$865$	468.072	0.541123
$866$	− 1575.76i	− 1.81958i
$867$	0	0
$868$	−152.707	−0.175930
$869$	0	0
$870$	0	0
$871$	−603.801	−0.693228
$872$	− 435.023i	− 0.498880i
$873$	0	0
$874$	−1527.44	−1.74764
$875$	− 254.397i	− 0.290739i
$876$	0	0
$877$	446.074	0.508637	0.254318	−	0.967121i	$-0.418149\pi$
$877$	446.074	0.508637	0.254318	+	0.967121i	$0.418149\pi$
$878$	− 68.2292i	− 0.0777099i
$879$	0	0
$880$	0	0
$881$	1631.61i	1.85200i	0.377522	+	0.926001i	$0.376776\pi$
$881$	1631.61i	1.85200i	−0.377522	+	0.926001i	$0.623224\pi$
$882$	0	0
$883$	1491.22	1.68881	0.844404	−	0.535707i	$-0.179955\pi$
$883$	1491.22	1.68881	0.844404	+	0.535707i	$0.179955\pi$
$884$	1065.68i	1.20552i
$885$	0	0
$886$	179.701	0.202823
$887$	− 623.153i	− 0.702540i	−0.936274	−	0.351270i	$-0.885750\pi$
$887$	− 623.153i	− 0.702540i	0.936274	−	0.351270i	$-0.114250\pi$
$888$	0	0
$889$	792.413	0.891353
$890$	− 402.107i	− 0.451806i
$891$	0	0
$892$	−1403.31	−1.57321
$893$	− 639.282i	− 0.715881i
$894$	0	0
$895$	−235.975	−0.263659
$896$	300.300i	0.335157i
$897$	0	0
$898$	−2279.80	−2.53875
$899$	− 19.8676i	− 0.0220997i
$900$	0	0
$901$	164.211	0.182254
$902$	0	0
$903$	0	0
$904$	370.117	0.409422
$905$	360.158i	0.397965i
$906$	0	0
$907$	−966.557	−1.06566	−0.532832	−	0.846221i	$-0.678872\pi$
$907$	−966.557	−1.06566	−0.532832	+	0.846221i	$0.678872\pi$
$908$	1008.14i	1.11028i
$909$	0	0
$910$	−121.568	−0.133591
$911$	− 617.042i	− 0.677324i	−0.940908	−	0.338662i	$-0.890026\pi$
$911$	− 617.042i	− 0.677324i	0.940908	−	0.338662i	$-0.109974\pi$
$912$	0	0
$913$	0	0
$914$	2362.80i	2.58512i
$915$	0	0
$916$	2082.82	2.27382
$917$	− 494.595i	− 0.539362i
$918$	0	0
$919$	1104.16	1.20148	0.600742	−	0.799443i	$-0.294872\pi$
$919$	1104.16	1.20148	0.600742	+	0.799443i	$0.294872\pi$
$920$	− 91.5626i	− 0.0995246i
$921$	0	0
$922$	−2176.83	−2.36099
$923$	815.195i	0.883202i
$924$	0	0
$925$	−1512.92	−1.63559
$926$	− 903.109i	− 0.975280i
$927$	0	0
$928$	−97.1025	−0.104636
$929$	1078.61i	1.16104i	0.814245	+	0.580521i	$0.197151\pi$
$929$	1078.61i	1.16104i	−0.814245	+	0.580521i	$0.802849\pi$
$930$	0	0
$931$	988.248	1.06149
$932$	552.218i	0.592509i
$933$	0	0
$934$	−875.967	−0.937866
$935$	0	0
$936$	0	0
$937$	−587.691	−0.627205	−0.313602	−	0.949554i	$-0.601536\pi$
$937$	−587.691	−0.627205	−0.313602	+	0.949554i	$0.601536\pi$
$938$	− 796.115i	− 0.848736i
$939$	0	0
$940$	196.432	0.208970
$941$	− 1069.27i	− 1.13631i	−0.822921	−	0.568155i	$-0.807657\pi$
$941$	− 1069.27i	− 1.13631i	0.822921	−	0.568155i	$-0.192343\pi$
$942$	0	0
$943$	−568.572	−0.602939
$944$	− 827.356i	− 0.876437i
$945$	0	0
$946$	0	0
$947$	435.924i	0.460321i	0.973153	+	0.230160i	$0.0739251\pi$
$947$	435.924i	0.460321i	−0.973153	+	0.230160i	$0.926075\pi$
$948$	0	0
$949$	−683.588	−0.720325
$950$	1748.89i	1.84094i
$951$	0	0
$952$	−274.123	−0.287945
$953$	− 670.977i	− 0.704068i	−0.935987	−	0.352034i	$-0.885490\pi$
$953$	− 670.977i	− 0.704068i	0.935987	−	0.352034i	$-0.114510\pi$
$954$	0	0
$955$	37.4207	0.0391840
$956$	1560.70i	1.63253i
$957$	0	0
$958$	1488.11	1.55335
$959$	− 67.1244i	− 0.0699942i
$960$	0	0
$961$	−875.837	−0.911380
$962$	1529.79i	1.59021i
$963$	0	0
$964$	518.287	0.537642
$965$	− 202.171i	− 0.209503i
$966$	0	0
$967$	−940.682	−0.972784	−0.486392	−	0.873741i	$-0.661687\pi$
$967$	−940.682	−0.972784	−0.486392	+	0.873741i	$0.661687\pi$
$968$	0	0
$969$	0	0
$970$	−376.637	−0.388286
$971$	1077.62i	1.10980i	0.831916	+	0.554901i	$0.187244\pi$
$971$	1077.62i	1.10980i	−0.831916	+	0.554901i	$0.812756\pi$
$972$	0	0
$973$	−495.736	−0.509492
$974$	− 2071.64i	− 2.12694i
$975$	0	0
$976$	−806.969	−0.826812
$977$	1006.05i	1.02973i	0.857270	+	0.514866i	$0.172158\pi$
$977$	1006.05i	1.02973i	−0.857270	+	0.514866i	$0.827842\pi$
$978$	0	0
$979$	0	0
$980$	303.658i	0.309855i
$981$	0	0
$982$	887.221	0.903484
$983$	− 1173.82i	− 1.19412i	−0.802197	−	0.597060i	$-0.796335\pi$
$983$	− 1173.82i	− 1.19412i	0.802197	−	0.597060i	$-0.203665\pi$
$984$	0	0
$985$	19.9608	0.0202647
$986$	− 182.808i	− 0.185404i
$987$	0	0
$988$	979.763	0.991663
$989$	14.0834i	0.0142400i
$990$	0	0
$991$	−989.386	−0.998372	−0.499186	−	0.866495i	$-0.666368\pi$
$991$	−989.386	−0.998372	−0.499186	+	0.866495i	$0.666368\pi$
$992$	− 416.234i	− 0.419591i
$993$	0	0
$994$	−1074.84	−1.08133
$995$	181.499i	0.182411i
$996$	0	0
$997$	−1465.59	−1.47000	−0.735000	−	0.678067i	$-0.762818\pi$
$997$	−1465.59	−1.47000	−0.735000	+	0.678067i	$0.762818\pi$
$998$	401.329i	0.402133i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.i.485.3		16
3.2	odd	2	inner	1089.3.b.i.485.14		16
11.3	even	5		99.3.l.a.53.2	✓	32
11.4	even	5		99.3.l.a.71.7	yes	32
11.10	odd	2		1089.3.b.j.485.14		16
33.14	odd	10		99.3.l.a.53.7	yes	32
33.26	odd	10		99.3.l.a.71.2	yes	32
33.32	even	2		1089.3.b.j.485.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.53.2	✓	32	11.3	even	5
99.3.l.a.53.7	yes	32	33.14	odd	10
99.3.l.a.71.2	yes	32	33.26	odd	10
99.3.l.a.71.7	yes	32	11.4	even	5
1089.3.b.i.485.3		16	1.1	even	1	trivial
1089.3.b.i.485.14		16	3.2	odd	2	inner
1089.3.b.j.485.3		16	33.32	even	2
1089.3.b.j.485.14		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)

\(f(q)\)	\(=\)	\(q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +O(q^{10})\)	\(q-2.99491i q^{2} -4.96950 q^{4} +1.61173i q^{5} -3.32981 q^{7} +2.90358i q^{8} +4.82698 q^{10} +7.56350 q^{13} +9.97250i q^{14} -11.1821 q^{16} -28.3526i q^{17} -26.0667 q^{19} -8.00948i q^{20} +19.5656i q^{23} +22.4023 q^{25} -22.6520i q^{26} +16.5475 q^{28} +2.15288i q^{29} -9.22840 q^{31} +45.1036i q^{32} -84.9135 q^{34} -5.36675i q^{35} -67.5342 q^{37} +78.0674i q^{38} -4.67977 q^{40} +29.0597i q^{41} +0.719802 q^{43} +58.5973 q^{46} +24.5249i q^{47} -37.9123 q^{49} -67.0930i q^{50} -37.5868 q^{52} +5.79176i q^{53} -9.66837i q^{56} +6.44768 q^{58} +73.9896i q^{59} +72.1664 q^{61} +27.6383i q^{62} +90.3531 q^{64} +12.1903i q^{65} -79.8310 q^{67} +140.898i q^{68} -16.0730 q^{70} +107.780i q^{71} -90.3799 q^{73} +202.259i q^{74} +129.538 q^{76} +114.802 q^{79} -18.0224i q^{80} +87.0314 q^{82} +125.752i q^{83} +45.6966 q^{85} -2.15574i q^{86} -83.3041i q^{89} -25.1850 q^{91} -97.2313i q^{92} +73.4499 q^{94} -42.0123i q^{95} -78.0275 q^{97} +113.544i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more