LMFDB - Embedded newform 1089.3.b.i.485.2

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			485.2
Root			$-3.25431i$ of defining polynomial
Character	$\chi$	$=$	1089.485
Dual form			1089.3.b.i.485.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25431i q^{2} -6.59053 q^{4} +1.59288i q^{5} -10.8798 q^{7} +8.43039i q^{8} +O(q^{10})$	$q-3.25431i q^{2} -6.59053 q^{4} +1.59288i q^{5} -10.8798 q^{7} +8.43039i q^{8} +5.18373 q^{10} -9.23818 q^{13} +35.4062i q^{14} +1.07299 q^{16} +4.81648i q^{17} -25.0886 q^{19} -10.4979i q^{20} -36.6892i q^{23} +22.4627 q^{25} +30.0639i q^{26} +71.7035 q^{28} +24.4515i q^{29} +58.1821 q^{31} +30.2297i q^{32} +15.6743 q^{34} -17.3302i q^{35} +12.7978 q^{37} +81.6462i q^{38} -13.4286 q^{40} -12.4167i q^{41} +19.6984 q^{43} -119.398 q^{46} +64.9463i q^{47} +69.3696 q^{49} -73.1007i q^{50} +60.8845 q^{52} +49.5059i q^{53} -91.7208i q^{56} +79.5728 q^{58} -24.0917i q^{59} +44.2401 q^{61} -189.343i q^{62} +102.669 q^{64} -14.7153i q^{65} +33.0898 q^{67} -31.7432i q^{68} -56.3978 q^{70} -74.9390i q^{71} +41.3762 q^{73} -41.6480i q^{74} +165.347 q^{76} +63.9703 q^{79} +1.70914i q^{80} -40.4076 q^{82} +42.1858i q^{83} -7.67208 q^{85} -64.1046i q^{86} +111.111i q^{89} +100.509 q^{91} +241.801i q^{92} +211.355 q^{94} -39.9632i q^{95} -108.103 q^{97} -225.750i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 3.25431i	− 1.62715i	−0.581457	−	0.813577i	$-0.697517\pi$
$2$	− 3.25431i	− 1.62715i	0.581457	−	0.813577i	$-0.302483\pi$
$3$	0	0
$3$	0	0
$4$	−6.59053	−1.64763
$5$	1.59288i	0.318576i	0.987232	+	0.159288i	$0.0509198\pi$
$5$	1.59288i	0.318576i	−0.987232	+	0.159288i	$0.949080\pi$
$6$	0	0
$7$	−10.8798	−1.55425	−0.777127	−	0.629344i	$-0.783324\pi$
$7$	−10.8798	−1.55425	−0.777127	+	0.629344i	$0.783324\pi$
$8$	8.43039i	1.05380i
$9$	0	0
$10$	5.18373	0.518373
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−9.23818	−0.710629	−0.355315	−	0.934747i	$-0.615626\pi$
$13$	−9.23818	−0.710629	−0.355315	+	0.934747i	$0.615626\pi$
$14$	35.4062i	2.52901i
$15$	0	0
$16$	1.07299	0.0670616
$17$	4.81648i	0.283322i	0.989915	+	0.141661i	$0.0452444\pi$
$17$	4.81648i	0.283322i	−0.989915	+	0.141661i	$0.954756\pi$
$18$	0	0
$19$	−25.0886	−1.32045	−0.660227	−	0.751066i	$-0.729540\pi$
$19$	−25.0886	−1.32045	−0.660227	+	0.751066i	$0.729540\pi$
$20$	− 10.4979i	− 0.524896i
$21$	0	0
$22$	0	0
$23$	− 36.6892i	− 1.59518i	−0.603199	−	0.797590i	$-0.706108\pi$
$23$	− 36.6892i	− 1.59518i	0.603199	−	0.797590i	$-0.293892\pi$
$24$	0	0
$25$	22.4627	0.898509
$26$	30.0639i	1.15630i
$27$	0	0
$28$	71.7035	2.56084
$29$	24.4515i	0.843156i	0.906792	+	0.421578i	$0.138524\pi$
$29$	24.4515i	0.843156i	−0.906792	+	0.421578i	$0.861476\pi$
$30$	0	0
$31$	58.1821	1.87684	0.938421	−	0.345493i	$-0.112288\pi$
$31$	58.1821	1.87684	0.938421	+	0.345493i	$0.112288\pi$
$32$	30.2297i	0.944680i
$33$	0	0
$34$	15.6743	0.461009
$35$	− 17.3302i	− 0.495148i
$36$	0	0
$37$	12.7978	0.345887	0.172943	−	0.984932i	$-0.444672\pi$
$37$	12.7978	0.345887	0.172943	+	0.984932i	$0.444672\pi$
$38$	81.6462i	2.14858i
$39$	0	0
$40$	−13.4286	−0.335715
$41$	− 12.4167i	− 0.302845i	−0.988469	−	0.151423i	$-0.951615\pi$
$41$	− 12.4167i	− 0.302845i	0.988469	−	0.151423i	$-0.0483854\pi$
$42$	0	0
$43$	19.6984	0.458102	0.229051	−	0.973414i	$-0.426438\pi$
$43$	19.6984	0.458102	0.229051	+	0.973414i	$0.426438\pi$
$44$	0	0
$45$	0	0
$46$	−119.398	−2.59561
$47$	64.9463i	1.38184i	0.722933	+	0.690918i	$0.242793\pi$
$47$	64.9463i	1.38184i	−0.722933	+	0.690918i	$0.757207\pi$
$48$	0	0
$49$	69.3696	1.41571
$50$	− 73.1007i	− 1.46201i
$51$	0	0
$52$	60.8845	1.17086
$53$	49.5059i	0.934073i	0.884238	+	0.467037i	$0.154678\pi$
$53$	49.5059i	0.934073i	−0.884238	+	0.467037i	$0.845322\pi$
$54$	0	0
$55$	0	0
$56$	− 91.7208i	− 1.63787i
$57$	0	0
$58$	79.5728	1.37195
$59$	− 24.0917i	− 0.408333i	−0.978936	−	0.204167i	$-0.934552\pi$
$59$	− 24.0917i	− 0.408333i	0.978936	−	0.204167i	$-0.0654484\pi$
$60$	0	0
$61$	44.2401	0.725248	0.362624	−	0.931936i	$-0.381881\pi$
$61$	44.2401	0.725248	0.362624	+	0.931936i	$0.381881\pi$
$62$	− 189.343i	− 3.05391i
$63$	0	0
$64$	102.669	1.60420
$65$	− 14.7153i	− 0.226389i
$66$	0	0
$67$	33.0898	0.493878	0.246939	−	0.969031i	$-0.420575\pi$
$67$	33.0898	0.493878	0.246939	+	0.969031i	$0.420575\pi$
$68$	− 31.7432i	− 0.466811i
$69$	0	0
$70$	−56.3978	−0.805683
$71$	− 74.9390i	− 1.05548i	−0.849406	−	0.527739i	$-0.823040\pi$
$71$	− 74.9390i	− 1.05548i	0.849406	−	0.527739i	$-0.176960\pi$
$72$	0	0
$73$	41.3762	0.566797	0.283399	−	0.959002i	$-0.408538\pi$
$73$	41.3762	0.566797	0.283399	+	0.959002i	$0.408538\pi$
$74$	− 41.6480i	− 0.562811i
$75$	0	0
$76$	165.347	2.17562
$77$	0	0
$78$	0	0
$79$	63.9703	0.809751	0.404875	−	0.914372i	$-0.367315\pi$
$79$	63.9703	0.809751	0.404875	+	0.914372i	$0.367315\pi$
$80$	1.70914i	0.0213642i
$81$	0	0
$82$	−40.4076	−0.492776
$83$	42.1858i	0.508262i	0.967170	+	0.254131i	$0.0817895\pi$
$83$	42.1858i	0.508262i	−0.967170	+	0.254131i	$0.918211\pi$
$84$	0	0
$85$	−7.67208	−0.0902597
$86$	− 64.1046i	− 0.745402i
$87$	0	0
$88$	0	0
$89$	111.111i	1.24844i	0.781249	+	0.624220i	$0.214583\pi$
$89$	111.111i	1.24844i	−0.781249	+	0.624220i	$0.785417\pi$
$90$	0	0
$91$	100.509	1.10450
$92$	241.801i	2.62827i
$93$	0	0
$94$	211.355	2.24846
$95$	− 39.9632i	− 0.420665i
$96$	0	0
$97$	−108.103	−1.11447	−0.557233	−	0.830356i	$-0.688137\pi$
$97$	−108.103	−1.11447	−0.557233	+	0.830356i	$0.688137\pi$
$98$	− 225.750i	− 2.30357i
$99$	0	0
$100$	−148.041	−1.48041
$101$	116.418i	1.15265i	0.817221	+	0.576324i	$0.195514\pi$
$101$	116.418i	1.15265i	−0.817221	+	0.576324i	$0.804486\pi$
$102$	0	0
$103$	−109.280	−1.06097	−0.530484	−	0.847695i	$-0.677990\pi$
$103$	−109.280	−1.06097	−0.530484	+	0.847695i	$0.677990\pi$
$104$	− 77.8815i	− 0.748861i
$105$	0	0
$106$	161.107	1.51988
$107$	− 43.1743i	− 0.403498i	−0.979437	−	0.201749i	$-0.935337\pi$
$107$	− 43.1743i	− 0.403498i	0.979437	−	0.201749i	$-0.0646626\pi$
$108$	0	0
$109$	75.3063	0.690884	0.345442	−	0.938440i	$-0.387729\pi$
$109$	75.3063	0.690884	0.345442	+	0.938440i	$0.387729\pi$
$110$	0	0
$111$	0	0
$112$	−11.6738	−0.104231
$113$	− 3.52202i	− 0.0311683i	−0.999879	−	0.0155842i	$-0.995039\pi$
$113$	− 3.52202i	− 0.0311683i	0.999879	−	0.0155842i	$-0.00496080\pi$
$114$	0	0
$115$	58.4414	0.508186
$116$	− 161.149i	− 1.38921i
$117$	0	0
$118$	−78.4017	−0.664421
$119$	− 52.4023i	− 0.440355i
$120$	0	0
$121$	0	0
$122$	− 143.971i	− 1.18009i
$123$	0	0
$124$	−383.451	−3.09235
$125$	75.6024i	0.604820i
$126$	0	0
$127$	−237.588	−1.87077	−0.935385	−	0.353630i	$-0.884947\pi$
$127$	−237.588	−1.87077	−0.935385	+	0.353630i	$0.884947\pi$
$128$	− 213.197i	− 1.66561i
$129$	0	0
$130$	−47.8882	−0.368371
$131$	− 64.9832i	− 0.496055i	−0.968753	−	0.248027i	$-0.920218\pi$
$131$	− 64.9832i	− 0.496055i	0.968753	−	0.248027i	$-0.0797823\pi$
$132$	0	0
$133$	272.959	2.05232
$134$	− 107.685i	− 0.803616i
$135$	0	0
$136$	−40.6048	−0.298565
$137$	125.513i	0.916152i	0.888913	+	0.458076i	$0.151461\pi$
$137$	125.513i	0.916152i	−0.888913	+	0.458076i	$0.848539\pi$
$138$	0	0
$139$	83.9520	0.603971	0.301986	−	0.953312i	$-0.402351\pi$
$139$	83.9520	0.603971	0.301986	+	0.953312i	$0.402351\pi$
$140$	114.215i	0.815822i
$141$	0	0
$142$	−243.875	−1.71743
$143$	0	0
$144$	0	0
$145$	−38.9483	−0.268609
$146$	− 134.651i	− 0.922267i
$147$	0	0
$148$	−84.3444	−0.569894
$149$	− 208.842i	− 1.40162i	−0.713346	−	0.700812i	$-0.752821\pi$
$149$	− 208.842i	− 1.40162i	0.713346	−	0.700812i	$-0.247179\pi$
$150$	0	0
$151$	62.1096	0.411322	0.205661	−	0.978623i	$-0.434066\pi$
$151$	62.1096	0.411322	0.205661	+	0.978623i	$0.434066\pi$
$152$	− 211.507i	− 1.39149i
$153$	0	0
$154$	0	0
$155$	92.6771i	0.597917i
$156$	0	0
$157$	−107.364	−0.683847	−0.341923	−	0.939728i	$-0.611078\pi$
$157$	−107.364	−0.683847	−0.341923	+	0.939728i	$0.611078\pi$
$158$	− 208.179i	− 1.31759i
$159$	0	0
$160$	−48.1524	−0.300952
$161$	399.170i	2.47932i
$162$	0	0
$163$	203.397	1.24783	0.623916	−	0.781491i	$-0.285541\pi$
$163$	203.397	1.24783	0.623916	+	0.781491i	$0.285541\pi$
$164$	81.8324i	0.498978i
$165$	0	0
$166$	137.286	0.827021
$167$	316.071i	1.89264i	0.323230	+	0.946320i	$0.395231\pi$
$167$	316.071i	1.89264i	−0.323230	+	0.946320i	$0.604769\pi$
$168$	0	0
$169$	−83.6560	−0.495006
$170$	24.9673i	0.146867i
$171$	0	0
$172$	−129.823	−0.754783
$173$	− 114.001i	− 0.658963i	−0.944162	−	0.329482i	$-0.893126\pi$
$173$	− 114.001i	− 0.658963i	0.944162	−	0.329482i	$-0.106874\pi$
$174$	0	0
$175$	−244.390	−1.39651
$176$	0	0
$177$	0	0
$178$	361.590	2.03140
$179$	145.784i	0.814436i	0.913331	+	0.407218i	$0.133501\pi$
$179$	145.784i	0.814436i	−0.913331	+	0.407218i	$0.866499\pi$
$180$	0	0
$181$	−347.607	−1.92048	−0.960241	−	0.279171i	$-0.909940\pi$
$181$	−347.607	−1.92048	−0.960241	+	0.279171i	$0.909940\pi$
$182$	− 327.089i	− 1.79719i
$183$	0	0
$184$	309.304	1.68100
$185$	20.3854i	0.110191i
$186$	0	0
$187$	0	0
$188$	− 428.030i	− 2.27676i
$189$	0	0
$190$	−130.053	−0.684487
$191$	74.8885i	0.392086i	0.980595	+	0.196043i	$0.0628093\pi$
$191$	74.8885i	0.392086i	−0.980595	+	0.196043i	$0.937191\pi$
$192$	0	0
$193$	−31.8200	−0.164871	−0.0824353	−	0.996596i	$-0.526270\pi$
$193$	−31.8200	−0.164871	−0.0824353	+	0.996596i	$0.526270\pi$
$194$	351.801i	1.81341i
$195$	0	0
$196$	−457.183	−2.33256
$197$	165.104i	0.838089i	0.907966	+	0.419045i	$0.137635\pi$
$197$	165.104i	0.838089i	−0.907966	+	0.419045i	$0.862365\pi$
$198$	0	0
$199$	195.093	0.980365	0.490182	−	0.871620i	$-0.336930\pi$
$199$	195.093	0.980365	0.490182	+	0.871620i	$0.336930\pi$
$200$	189.370i	0.946848i
$201$	0	0
$202$	378.859	1.87554
$203$	− 266.027i	− 1.31048i
$204$	0	0
$205$	19.7782	0.0964792
$206$	355.630i	1.72636i
$207$	0	0
$208$	−9.91243	−0.0476559
$209$	0	0
$210$	0	0
$211$	182.295	0.863959	0.431980	−	0.901883i	$-0.357815\pi$
$211$	182.295	0.863959	0.431980	+	0.901883i	$0.357815\pi$
$212$	− 326.270i	− 1.53901i
$213$	0	0
$214$	−140.503	−0.656554
$215$	31.3771i	0.145940i
$216$	0	0
$217$	−633.009	−2.91709
$218$	− 245.070i	− 1.12417i
$219$	0	0
$220$	0	0
$221$	− 44.4955i	− 0.201337i
$222$	0	0
$223$	−114.587	−0.513843	−0.256921	−	0.966432i	$-0.582708\pi$
$223$	−114.587	−0.513843	−0.256921	+	0.966432i	$0.582708\pi$
$224$	− 328.893i	− 1.46827i
$225$	0	0
$226$	−11.4618	−0.0507157
$227$	− 233.643i	− 1.02926i	−0.857411	−	0.514632i	$-0.827929\pi$
$227$	− 233.643i	− 1.02926i	0.857411	−	0.514632i	$-0.172071\pi$
$228$	0	0
$229$	139.279	0.608206	0.304103	−	0.952639i	$-0.401643\pi$
$229$	139.279	0.608206	0.304103	+	0.952639i	$0.401643\pi$
$230$	− 190.187i	− 0.826898i
$231$	0	0
$232$	−206.136	−0.888517
$233$	118.544i	0.508772i	0.967103	+	0.254386i	$0.0818734\pi$
$233$	118.544i	0.508772i	−0.967103	+	0.254386i	$0.918127\pi$
$234$	0	0
$235$	−103.452	−0.440220
$236$	158.777i	0.672783i
$237$	0	0
$238$	−170.533	−0.716526
$239$	− 288.314i	− 1.20634i	−0.797614	−	0.603168i	$-0.793905\pi$
$239$	− 288.314i	− 1.20634i	0.797614	−	0.603168i	$-0.206095\pi$
$240$	0	0
$241$	−115.517	−0.479324	−0.239662	−	0.970856i	$-0.577037\pi$
$241$	−115.517	−0.479324	−0.239662	+	0.970856i	$0.577037\pi$
$242$	0	0
$243$	0	0
$244$	−291.566	−1.19494
$245$	110.497i	0.451010i
$246$	0	0
$247$	231.773	0.938354
$248$	490.498i	1.97782i
$249$	0	0
$250$	246.034	0.984135
$251$	34.4392i	0.137208i	0.997644	+	0.0686040i	$0.0218545\pi$
$251$	34.4392i	0.137208i	−0.997644	+	0.0686040i	$0.978145\pi$
$252$	0	0
$253$	0	0
$254$	773.185i	3.04403i
$255$	0	0
$256$	−283.135	−1.10600
$257$	208.411i	0.810937i	0.914109	+	0.405469i	$0.132892\pi$
$257$	208.411i	0.810937i	−0.914109	+	0.405469i	$0.867108\pi$
$258$	0	0
$259$	−139.237	−0.537596
$260$	96.9817i	0.373007i
$261$	0	0
$262$	−211.475	−0.807158
$263$	230.901i	0.877952i	0.898499	+	0.438976i	$0.144659\pi$
$263$	230.901i	0.877952i	−0.898499	+	0.438976i	$0.855341\pi$
$264$	0	0
$265$	−78.8569	−0.297573
$266$	− 888.293i	− 3.33945i
$267$	0	0
$268$	−218.080	−0.813730
$269$	11.0843i	0.0412055i	0.999788	+	0.0206027i	$0.00655852\pi$
$269$	11.0843i	0.0412055i	−0.999788	+	0.0206027i	$0.993441\pi$
$270$	0	0
$271$	100.794	0.371935	0.185968	−	0.982556i	$-0.440458\pi$
$271$	100.794	0.371935	0.185968	+	0.982556i	$0.440458\pi$
$272$	5.16801i	0.0190000i
$273$	0	0
$274$	408.458	1.49072
$275$	0	0
$276$	0	0
$277$	−53.4162	−0.192838	−0.0964191	−	0.995341i	$-0.530739\pi$
$277$	−53.4162	−0.192838	−0.0964191	+	0.995341i	$0.530739\pi$
$278$	− 273.206i	− 0.982755i
$279$	0	0
$280$	146.100	0.521787
$281$	369.254i	1.31407i	0.753859	+	0.657036i	$0.228190\pi$
$281$	369.254i	1.31407i	−0.753859	+	0.657036i	$0.771810\pi$
$282$	0	0
$283$	241.063	0.851814	0.425907	−	0.904767i	$-0.359955\pi$
$283$	241.063	0.851814	0.425907	+	0.904767i	$0.359955\pi$
$284$	493.888i	1.73904i
$285$	0	0
$286$	0	0
$287$	135.090i	0.470699i
$288$	0	0
$289$	265.802	0.919728
$290$	126.750i	0.437069i
$291$	0	0
$292$	−272.691	−0.933874
$293$	− 55.1299i	− 0.188157i	−0.995565	−	0.0940783i	$-0.970010\pi$
$293$	− 55.1299i	− 0.188157i	0.995565	−	0.0940783i	$-0.0299904\pi$
$294$	0	0
$295$	38.3751	0.130085
$296$	107.891i	0.364495i
$297$	0	0
$298$	−679.637	−2.28066
$299$	338.941i	1.13358i
$300$	0	0
$301$	−214.314	−0.712006
$302$	− 202.124i	− 0.669285i
$303$	0	0
$304$	−26.9197	−0.0885518
$305$	70.4692i	0.231047i
$306$	0	0
$307$	−12.5104	−0.0407504	−0.0203752	−	0.999792i	$-0.506486\pi$
$307$	−12.5104	−0.0407504	−0.0203752	+	0.999792i	$0.506486\pi$
$308$	0	0
$309$	0	0
$310$	301.600	0.972904
$311$	215.665i	0.693458i	0.937965	+	0.346729i	$0.112708\pi$
$311$	215.665i	0.693458i	−0.937965	+	0.346729i	$0.887292\pi$
$312$	0	0
$313$	435.646	1.39184	0.695920	−	0.718120i	$-0.254997\pi$
$313$	435.646	1.39184	0.695920	+	0.718120i	$0.254997\pi$
$314$	349.396i	1.11272i
$315$	0	0
$316$	−421.598	−1.33417
$317$	313.816i	0.989956i	0.868905	+	0.494978i	$0.164824\pi$
$317$	313.816i	0.989956i	−0.868905	+	0.494978i	$0.835176\pi$
$318$	0	0
$319$	0	0
$320$	163.539i	0.511060i
$321$	0	0
$322$	1299.02	4.03423
$323$	− 120.839i	− 0.374114i
$324$	0	0
$325$	−207.515	−0.638507
$326$	− 661.916i	− 2.03042i
$327$	0	0
$328$	104.677	0.319138
$329$	− 706.601i	− 2.14772i
$330$	0	0
$331$	69.5045	0.209983	0.104992	−	0.994473i	$-0.466518\pi$
$331$	69.5045	0.209983	0.104992	+	0.994473i	$0.466518\pi$
$332$	− 278.027i	− 0.837430i
$333$	0	0
$334$	1028.59	3.07962
$335$	52.7081i	0.157338i
$336$	0	0
$337$	−108.794	−0.322832	−0.161416	−	0.986886i	$-0.551606\pi$
$337$	−108.794	−0.322832	−0.161416	+	0.986886i	$0.551606\pi$
$338$	272.243i	0.805452i
$339$	0	0
$340$	50.5631	0.148715
$341$	0	0
$342$	0	0
$343$	−221.617	−0.646113
$344$	166.065i	0.482747i
$345$	0	0
$346$	−370.993	−1.07224
$347$	382.854i	1.10332i	0.834068	+	0.551662i	$0.186006\pi$
$347$	382.854i	1.10332i	−0.834068	+	0.551662i	$0.813994\pi$
$348$	0	0
$349$	441.623	1.26540	0.632698	−	0.774399i	$-0.281948\pi$
$349$	441.623	1.26540	0.632698	+	0.774399i	$0.281948\pi$
$350$	795.319i	2.27234i
$351$	0	0
$352$	0	0
$353$	237.587i	0.673051i	0.941674	+	0.336525i	$0.109252\pi$
$353$	237.587i	0.673051i	−0.941674	+	0.336525i	$0.890748\pi$
$354$	0	0
$355$	119.369	0.336250
$356$	− 732.281i	− 2.05697i
$357$	0	0
$358$	474.427	1.32521
$359$	385.744i	1.07450i	0.843424	+	0.537248i	$0.180536\pi$
$359$	385.744i	1.07450i	−0.843424	+	0.537248i	$0.819464\pi$
$360$	0	0
$361$	268.440	0.743600
$362$	1131.22i	3.12492i
$363$	0	0
$364$	−662.410	−1.81981
$365$	65.9073i	0.180568i
$366$	0	0
$367$	154.272	0.420361	0.210180	−	0.977663i	$-0.432595\pi$
$367$	154.272	0.420361	0.210180	+	0.977663i	$0.432595\pi$
$368$	− 39.3669i	− 0.106975i
$369$	0	0
$370$	66.3403	0.179298
$371$	− 538.613i	− 1.45179i
$372$	0	0
$373$	440.608	1.18125	0.590627	−	0.806944i	$-0.298880\pi$
$373$	440.608	1.18125	0.590627	+	0.806944i	$0.298880\pi$
$374$	0	0
$375$	0	0
$376$	−547.523	−1.45618
$377$	− 225.888i	− 0.599171i
$378$	0	0
$379$	556.553	1.46848	0.734239	−	0.678891i	$-0.237539\pi$
$379$	556.553	1.46848	0.734239	+	0.678891i	$0.237539\pi$
$380$	263.379i	0.693102i
$381$	0	0
$382$	243.710	0.637985
$383$	− 125.668i	− 0.328115i	−0.986451	−	0.164057i	$-0.947542\pi$
$383$	− 125.668i	− 0.328115i	0.986451	−	0.164057i	$-0.0524582\pi$
$384$	0	0
$385$	0	0
$386$	103.552i	0.268270i
$387$	0	0
$388$	712.457	1.83623
$389$	− 67.1506i	− 0.172624i	−0.996268	−	0.0863119i	$-0.972492\pi$
$389$	− 67.1506i	− 0.172624i	0.996268	−	0.0863119i	$-0.0275081\pi$
$390$	0	0
$391$	176.713	0.451951
$392$	584.813i	1.49187i
$393$	0	0
$394$	537.298	1.36370
$395$	101.897i	0.257967i
$396$	0	0
$397$	342.588	0.862942	0.431471	−	0.902127i	$-0.357995\pi$
$397$	342.588	0.862942	0.431471	+	0.902127i	$0.357995\pi$
$398$	− 634.892i	− 1.59521i
$399$	0	0
$400$	24.1022	0.0602555
$401$	− 370.533i	− 0.924022i	−0.886874	−	0.462011i	$-0.847128\pi$
$401$	− 370.533i	− 0.924022i	0.886874	−	0.462011i	$-0.152872\pi$
$402$	0	0
$403$	−537.497	−1.33374
$404$	− 767.254i	− 1.89914i
$405$	0	0
$406$	−865.735	−2.13235
$407$	0	0
$408$	0	0
$409$	−320.164	−0.782797	−0.391398	−	0.920221i	$-0.628009\pi$
$409$	−320.164	−0.782797	−0.391398	+	0.920221i	$0.628009\pi$
$410$	− 64.3645i	− 0.156987i
$411$	0	0
$412$	720.211	1.74808
$413$	262.112i	0.634653i
$414$	0	0
$415$	−67.1969	−0.161920
$416$	− 279.268i	− 0.671317i
$417$	0	0
$418$	0	0
$419$	− 574.973i	− 1.37225i	−0.727483	−	0.686126i	$-0.759310\pi$
$419$	− 574.973i	− 1.37225i	0.727483	−	0.686126i	$-0.240690\pi$
$420$	0	0
$421$	141.737	0.336668	0.168334	−	0.985730i	$-0.446161\pi$
$421$	141.737	0.336668	0.168334	+	0.985730i	$0.446161\pi$
$422$	− 593.246i	− 1.40580i
$423$	0	0
$424$	−417.354	−0.984326
$425$	108.191i	0.254568i
$426$	0	0
$427$	−481.323	−1.12722
$428$	284.542i	0.664817i
$429$	0	0
$430$	102.111	0.237467
$431$	− 391.659i	− 0.908722i	−0.890818	−	0.454361i	$-0.849868\pi$
$431$	− 391.659i	− 0.908722i	0.890818	−	0.454361i	$-0.150132\pi$
$432$	0	0
$433$	−343.995	−0.794447	−0.397223	−	0.917722i	$-0.630026\pi$
$433$	−343.995	−0.794447	−0.397223	+	0.917722i	$0.630026\pi$
$434$	2060.01i	4.74656i
$435$	0	0
$436$	−496.309	−1.13832
$437$	920.481i	2.10636i
$438$	0	0
$439$	36.1143	0.0822650	0.0411325	−	0.999154i	$-0.486903\pi$
$439$	36.1143	0.0822650	0.0411325	+	0.999154i	$0.486903\pi$
$440$	0	0
$441$	0	0
$442$	−144.802	−0.327607
$443$	382.066i	0.862452i	0.902244	+	0.431226i	$0.141919\pi$
$443$	382.066i	0.862452i	−0.902244	+	0.431226i	$0.858081\pi$
$444$	0	0
$445$	−176.987	−0.397723
$446$	372.901i	0.836101i
$447$	0	0
$448$	−1117.02	−2.49334
$449$	− 30.7104i	− 0.0683974i	−0.999415	−	0.0341987i	$-0.989112\pi$
$449$	− 30.7104i	− 0.0683974i	0.999415	−	0.0341987i	$-0.0108879\pi$
$450$	0	0
$451$	0	0
$452$	23.2120i	0.0513540i
$453$	0	0
$454$	−760.346	−1.67477
$455$	160.099i	0.351867i
$456$	0	0
$457$	412.699	0.903061	0.451531	−	0.892256i	$-0.350878\pi$
$457$	412.699	0.903061	0.451531	+	0.892256i	$0.350878\pi$
$458$	− 453.258i	− 0.989645i
$459$	0	0
$460$	−385.160	−0.837305
$461$	765.072i	1.65959i	0.558066	+	0.829797i	$0.311544\pi$
$461$	765.072i	1.65959i	−0.558066	+	0.829797i	$0.688456\pi$
$462$	0	0
$463$	113.587	0.245328	0.122664	−	0.992448i	$-0.460856\pi$
$463$	113.587	0.245328	0.122664	+	0.992448i	$0.460856\pi$
$464$	26.2361i	0.0565434i
$465$	0	0
$466$	385.779	0.827852
$467$	− 756.817i	− 1.62059i	−0.586019	−	0.810297i	$-0.699306\pi$
$467$	− 756.817i	− 1.62059i	0.586019	−	0.810297i	$-0.300694\pi$
$468$	0	0
$469$	−360.010	−0.767612
$470$	336.664i	0.716305i
$471$	0	0
$472$	203.102	0.430301
$473$	0	0
$474$	0	0
$475$	−563.559	−1.18644
$476$	345.359i	0.725544i
$477$	0	0
$478$	−938.264	−1.96290
$479$	− 364.563i	− 0.761091i	−0.924762	−	0.380546i	$-0.875736\pi$
$479$	− 364.563i	− 0.761091i	0.924762	−	0.380546i	$-0.124264\pi$
$480$	0	0
$481$	−118.228	−0.245797
$482$	375.929i	0.779935i
$483$	0	0
$484$	0	0
$485$	− 172.195i	− 0.355042i
$486$	0	0
$487$	−240.050	−0.492915	−0.246458	−	0.969154i	$-0.579267\pi$
$487$	−240.050	−0.492915	−0.246458	+	0.969154i	$0.579267\pi$
$488$	372.962i	0.764266i
$489$	0	0
$490$	359.593	0.733863
$491$	837.796i	1.70630i	0.521662	+	0.853152i	$0.325312\pi$
$491$	837.796i	1.70630i	−0.521662	+	0.853152i	$0.674688\pi$
$492$	0	0
$493$	−117.770	−0.238885
$494$	− 754.262i	− 1.52685i
$495$	0	0
$496$	62.4286	0.125864
$497$	815.320i	1.64048i
$498$	0	0
$499$	644.663	1.29191	0.645955	−	0.763376i	$-0.276460\pi$
$499$	644.663	1.29191	0.645955	+	0.763376i	$0.276460\pi$
$500$	− 498.260i	− 0.996521i
$501$	0	0
$502$	112.076	0.223259
$503$	− 10.5076i	− 0.0208898i	−0.999945	−	0.0104449i	$-0.996675\pi$
$503$	− 10.5076i	− 0.0208898i	0.999945	−	0.0104449i	$-0.00332478\pi$
$504$	0	0
$505$	−185.439	−0.367206
$506$	0	0
$507$	0	0
$508$	1565.83	3.08234
$509$	915.705i	1.79903i	0.436893	+	0.899514i	$0.356079\pi$
$509$	915.705i	1.79903i	−0.436893	+	0.899514i	$0.643921\pi$
$510$	0	0
$511$	−450.164	−0.880947
$512$	68.6188i	0.134021i
$513$	0	0
$514$	678.233	1.31952
$515$	− 174.069i	− 0.337999i
$516$	0	0
$517$	0	0
$518$	453.121i	0.874752i
$519$	0	0
$520$	124.056	0.238569
$521$	235.989i	0.452953i	0.974017	+	0.226477i	$0.0727207\pi$
$521$	235.989i	0.452953i	−0.974017	+	0.226477i	$0.927279\pi$
$522$	0	0
$523$	−151.331	−0.289351	−0.144676	−	0.989479i	$-0.546214\pi$
$523$	−151.331	−0.289351	−0.144676	+	0.989479i	$0.546214\pi$
$524$	428.274i	0.817316i
$525$	0	0
$526$	751.425	1.42856
$527$	280.233i	0.531752i
$528$	0	0
$529$	−817.095	−1.54460
$530$	256.625i	0.484198i
$531$	0	0
$532$	−1798.94	−3.38147
$533$	114.707i	0.215211i
$534$	0	0
$535$	68.7715	0.128545
$536$	278.960i	0.520448i
$537$	0	0
$538$	36.0716	0.0670477
$539$	0	0
$540$	0	0
$541$	300.434	0.555331	0.277665	−	0.960678i	$-0.410439\pi$
$541$	300.434	0.555331	0.277665	+	0.960678i	$0.410439\pi$
$542$	− 328.016i	− 0.605196i
$543$	0	0
$544$	−145.601	−0.267649
$545$	119.954i	0.220099i
$546$	0	0
$547$	−324.960	−0.594077	−0.297038	−	0.954866i	$-0.595999\pi$
$547$	−324.960	−0.594077	−0.297038	+	0.954866i	$0.595999\pi$
$548$	− 827.196i	− 1.50948i
$549$	0	0
$550$	0	0
$551$	− 613.455i	− 1.11335i
$552$	0	0
$553$	−695.983	−1.25856
$554$	173.833i	0.313778i
$555$	0	0
$556$	−553.289	−0.995123
$557$	45.2931i	0.0813162i	0.999173	+	0.0406581i	$0.0129454\pi$
$557$	45.2931i	0.0813162i	−0.999173	+	0.0406581i	$0.987055\pi$
$558$	0	0
$559$	−181.977	−0.325540
$560$	− 18.5950i	− 0.0332054i
$561$	0	0
$562$	1201.67	2.13820
$563$	− 728.842i	− 1.29457i	−0.762249	−	0.647284i	$-0.775905\pi$
$563$	− 728.842i	− 1.29457i	0.762249	−	0.647284i	$-0.224095\pi$
$564$	0	0
$565$	5.61016	0.00992948
$566$	− 784.495i	− 1.38603i
$567$	0	0
$568$	631.765	1.11226
$569$	146.146i	0.256847i	0.991719	+	0.128423i	$0.0409916\pi$
$569$	146.146i	0.256847i	−0.991719	+	0.128423i	$0.959008\pi$
$570$	0	0
$571$	398.941	0.698672	0.349336	−	0.936998i	$-0.386407\pi$
$571$	398.941	0.698672	0.349336	+	0.936998i	$0.386407\pi$
$572$	0	0
$573$	0	0
$574$	439.626	0.765899
$575$	− 824.139i	− 1.43328i
$576$	0	0
$577$	252.489	0.437589	0.218795	−	0.975771i	$-0.429788\pi$
$577$	252.489	0.437589	0.218795	+	0.975771i	$0.429788\pi$
$578$	− 865.000i	− 1.49654i
$579$	0	0
$580$	256.690	0.442569
$581$	− 458.972i	− 0.789969i
$582$	0	0
$583$	0	0
$584$	348.818i	0.597290i
$585$	0	0
$586$	−179.410	−0.306160
$587$	985.570i	1.67900i	0.543363	+	0.839498i	$0.317151\pi$
$587$	985.570i	1.67900i	−0.543363	+	0.839498i	$0.682849\pi$
$588$	0	0
$589$	−1459.71	−2.47829
$590$	− 124.884i	− 0.211669i
$591$	0	0
$592$	13.7319	0.0231957
$593$	− 795.418i	− 1.34135i	−0.741753	−	0.670673i	$-0.766005\pi$
$593$	− 795.418i	− 1.34135i	0.741753	−	0.670673i	$-0.233995\pi$
$594$	0	0
$595$	83.4705	0.140287
$596$	1376.38i	2.30936i
$597$	0	0
$598$	1103.02	1.84451
$599$	− 114.903i	− 0.191825i	−0.995390	−	0.0959126i	$-0.969423\pi$
$599$	− 114.903i	− 0.191825i	0.995390	−	0.0959126i	$-0.0305769\pi$
$600$	0	0
$601$	−294.608	−0.490196	−0.245098	−	0.969498i	$-0.578820\pi$
$601$	−294.608	−0.490196	−0.245098	+	0.969498i	$0.578820\pi$
$602$	697.444i	1.15854i
$603$	0	0
$604$	−409.336	−0.677708
$605$	0	0
$606$	0	0
$607$	−435.121	−0.716838	−0.358419	−	0.933561i	$-0.616684\pi$
$607$	−435.121	−0.716838	−0.358419	+	0.933561i	$0.616684\pi$
$608$	− 758.423i	− 1.24741i
$609$	0	0
$610$	229.329	0.375949
$611$	− 599.985i	− 0.981973i
$612$	0	0
$613$	628.161	1.02473	0.512366	−	0.858767i	$-0.328769\pi$
$613$	628.161	1.02473	0.512366	+	0.858767i	$0.328769\pi$
$614$	40.7126i	0.0663072i
$615$	0	0
$616$	0	0
$617$	− 215.080i	− 0.348590i	−0.984694	−	0.174295i	$-0.944235\pi$
$617$	− 215.080i	− 0.348590i	0.984694	−	0.174295i	$-0.0557646\pi$
$618$	0	0
$619$	−956.887	−1.54586	−0.772929	−	0.634492i	$-0.781209\pi$
$619$	−956.887	−1.54586	−0.772929	+	0.634492i	$0.781209\pi$
$620$	− 610.792i	− 0.985148i
$621$	0	0
$622$	701.842	1.12836
$623$	− 1208.86i	− 1.94039i
$624$	0	0
$625$	441.143	0.705828
$626$	− 1417.73i	− 2.26474i
$627$	0	0
$628$	707.585	1.12673
$629$	61.6404i	0.0979975i
$630$	0	0
$631$	−637.996	−1.01109	−0.505544	−	0.862801i	$-0.668708\pi$
$631$	−637.996	−1.01109	−0.505544	+	0.862801i	$0.668708\pi$
$632$	539.295i	0.853315i
$633$	0	0
$634$	1021.25	1.61081
$635$	− 378.449i	− 0.595983i
$636$	0	0
$637$	−640.849	−1.00604
$638$	0	0
$639$	0	0
$640$	339.598	0.530622
$641$	− 958.209i	− 1.49487i	−0.664337	−	0.747433i	$-0.731286\pi$
$641$	− 958.209i	− 1.49487i	0.664337	−	0.747433i	$-0.268714\pi$
$642$	0	0
$643$	−294.864	−0.458575	−0.229288	−	0.973359i	$-0.573640\pi$
$643$	−294.864	−0.458575	−0.229288	+	0.973359i	$0.573640\pi$
$644$	− 2630.74i	− 4.08500i
$645$	0	0
$646$	−393.247	−0.608742
$647$	1005.11i	1.55350i	0.629809	+	0.776750i	$0.283133\pi$
$647$	1005.11i	1.55350i	−0.629809	+	0.776750i	$0.716867\pi$
$648$	0	0
$649$	0	0
$650$	675.317i	1.03895i
$651$	0	0
$652$	−1340.49	−2.05597
$653$	− 1035.63i	− 1.58595i	−0.609253	−	0.792976i	$-0.708531\pi$
$653$	− 1035.63i	− 1.58595i	0.609253	−	0.792976i	$-0.291469\pi$
$654$	0	0
$655$	103.510	0.158031
$656$	− 13.3229i	− 0.0203093i
$657$	0	0
$658$	−2299.50	−3.49468
$659$	− 572.524i	− 0.868777i	−0.900726	−	0.434388i	$-0.856965\pi$
$659$	− 572.524i	− 0.868777i	0.900726	−	0.434388i	$-0.143035\pi$
$660$	0	0
$661$	−717.443	−1.08539	−0.542695	−	0.839930i	$-0.682596\pi$
$661$	−717.443	−1.08539	−0.542695	+	0.839930i	$0.682596\pi$
$662$	− 226.189i	− 0.341675i
$663$	0	0
$664$	−355.643	−0.535606
$665$	434.791i	0.653821i
$666$	0	0
$667$	897.106	1.34499
$668$	− 2083.08i	− 3.11838i
$669$	0	0
$670$	171.529	0.256013
$671$	0	0
$672$	0	0
$673$	−872.035	−1.29574	−0.647872	−	0.761749i	$-0.724341\pi$
$673$	−872.035	−1.29574	−0.647872	+	0.761749i	$0.724341\pi$
$674$	354.050i	0.525297i
$675$	0	0
$676$	551.338	0.815588
$677$	1276.05i	1.88487i	0.334394	+	0.942433i	$0.391468\pi$
$677$	1276.05i	1.88487i	−0.334394	+	0.942433i	$0.608532\pi$
$678$	0	0
$679$	1176.14	1.73216
$680$	− 64.6786i	− 0.0951156i
$681$	0	0
$682$	0	0
$683$	− 411.727i	− 0.602822i	−0.953494	−	0.301411i	$-0.902542\pi$
$683$	− 411.727i	− 0.602822i	0.953494	−	0.301411i	$-0.0974575\pi$
$684$	0	0
$685$	−199.927	−0.291864
$686$	721.210i	1.05133i
$687$	0	0
$688$	21.1361	0.0307210
$689$	− 457.344i	− 0.663780i
$690$	0	0
$691$	−832.518	−1.20480	−0.602401	−	0.798194i	$-0.705789\pi$
$691$	−832.518	−1.20480	−0.602401	+	0.798194i	$0.705789\pi$
$692$	751.325i	1.08573i
$693$	0	0
$694$	1245.92	1.79528
$695$	133.726i	0.192411i
$696$	0	0
$697$	59.8046	0.0858029
$698$	− 1437.18i	− 2.05899i
$699$	0	0
$700$	1610.66	2.30094
$701$	− 90.8137i	− 0.129549i	−0.997900	−	0.0647744i	$-0.979367\pi$
$701$	− 90.8137i	− 0.129549i	0.997900	−	0.0647744i	$-0.0206328\pi$
$702$	0	0
$703$	−321.080	−0.456728
$704$	0	0
$705$	0	0
$706$	773.182	1.09516
$707$	− 1266.60i	− 1.79151i
$708$	0	0
$709$	−684.473	−0.965406	−0.482703	−	0.875784i	$-0.660345\pi$
$709$	−684.473	−0.965406	−0.482703	+	0.875784i	$0.660345\pi$
$710$	− 388.463i	− 0.547131i
$711$	0	0
$712$	−936.711	−1.31560
$713$	− 2134.65i	− 2.99390i
$714$	0	0
$715$	0	0
$716$	− 960.795i	− 1.34189i
$717$	0	0
$718$	1255.33	1.74837
$719$	1246.85i	1.73415i	0.498182	+	0.867073i	$0.334001\pi$
$719$	1246.85i	1.73415i	−0.498182	+	0.867073i	$0.665999\pi$
$720$	0	0
$721$	1188.94	1.64901
$722$	− 873.586i	− 1.20995i
$723$	0	0
$724$	2290.92	3.16425
$725$	549.248i	0.757583i
$726$	0	0
$727$	−837.811	−1.15242	−0.576211	−	0.817301i	$-0.695469\pi$
$727$	−837.811	−1.15242	−0.576211	+	0.817301i	$0.695469\pi$
$728$	847.334i	1.16392i
$729$	0	0
$730$	214.483	0.293812
$731$	94.8768i	0.129790i
$732$	0	0
$733$	−62.9218	−0.0858415	−0.0429208	−	0.999078i	$-0.513666\pi$
$733$	−62.9218	−0.0858415	−0.0429208	+	0.999078i	$0.513666\pi$
$734$	− 502.050i	− 0.683992i
$735$	0	0
$736$	1109.10	1.50694
$737$	0	0
$738$	0	0
$739$	1180.19	1.59701	0.798504	−	0.601989i	$-0.205625\pi$
$739$	1180.19	1.59701	0.798504	+	0.601989i	$0.205625\pi$
$740$	− 134.350i	− 0.181555i
$741$	0	0
$742$	−1752.81	−2.36228
$743$	496.072i	0.667661i	0.942633	+	0.333830i	$0.108341\pi$
$743$	496.072i	0.667661i	−0.942633	+	0.333830i	$0.891659\pi$
$744$	0	0
$745$	332.660	0.446524
$746$	− 1433.88i	− 1.92208i
$747$	0	0
$748$	0	0
$749$	469.727i	0.627139i
$750$	0	0
$751$	408.480	0.543914	0.271957	−	0.962309i	$-0.412329\pi$
$751$	408.480	0.543914	0.271957	+	0.962309i	$0.412329\pi$
$752$	69.6864i	0.0926681i
$753$	0	0
$754$	−735.108	−0.974944
$755$	98.9332i	0.131037i
$756$	0	0
$757$	−757.992	−1.00131	−0.500655	−	0.865647i	$-0.666907\pi$
$757$	−757.992	−1.00131	−0.500655	+	0.865647i	$0.666907\pi$
$758$	− 1811.20i	− 2.38944i
$759$	0	0
$760$	336.905	0.443297
$761$	1353.55i	1.77865i	0.457278	+	0.889324i	$0.348825\pi$
$761$	1353.55i	1.77865i	−0.457278	+	0.889324i	$0.651175\pi$
$762$	0	0
$763$	−819.316	−1.07381
$764$	− 493.555i	− 0.646014i
$765$	0	0
$766$	−408.962	−0.533893
$767$	222.563i	0.290173i
$768$	0	0
$769$	550.877	0.716355	0.358178	−	0.933654i	$-0.383398\pi$
$769$	550.877	0.716355	0.358178	+	0.933654i	$0.383398\pi$
$770$	0	0
$771$	0	0
$772$	209.711	0.271646
$773$	885.139i	1.14507i	0.819880	+	0.572535i	$0.194040\pi$
$773$	885.139i	1.14507i	−0.819880	+	0.572535i	$0.805960\pi$
$774$	0	0
$775$	1306.93	1.68636
$776$	− 911.352i	− 1.17442i
$777$	0	0
$778$	−218.529	−0.280886
$779$	311.517i	0.399893i
$780$	0	0
$781$	0	0
$782$	− 575.078i	− 0.735394i
$783$	0	0
$784$	74.4326	0.0949395
$785$	− 171.018i	− 0.217857i
$786$	0	0
$787$	1481.78	1.88283	0.941413	−	0.337255i	$-0.109498\pi$
$787$	1481.78	1.88283	0.941413	+	0.337255i	$0.109498\pi$
$788$	− 1088.12i	− 1.38086i
$789$	0	0
$790$	331.605	0.419753
$791$	38.3188i	0.0484435i
$792$	0	0
$793$	−408.698	−0.515382
$794$	− 1114.89i	− 1.40414i
$795$	0	0
$796$	−1285.76	−1.61528
$797$	− 69.4562i	− 0.0871471i	−0.999050	−	0.0435735i	$-0.986126\pi$
$797$	− 69.4562i	− 0.0871471i	0.999050	−	0.0435735i	$-0.0138743\pi$
$798$	0	0
$799$	−312.812	−0.391505
$800$	679.043i	0.848803i
$801$	0	0
$802$	−1205.83	−1.50353
$803$	0	0
$804$	0	0
$805$	−635.830	−0.789851
$806$	1749.18i	2.17020i
$807$	0	0
$808$	−981.446	−1.21466
$809$	− 881.324i	− 1.08940i	−0.838631	−	0.544700i	$-0.816644\pi$
$809$	− 881.324i	− 1.08940i	0.838631	−	0.544700i	$-0.183356\pi$
$810$	0	0
$811$	1048.72	1.29312	0.646560	−	0.762863i	$-0.276207\pi$
$811$	1048.72	1.29312	0.646560	+	0.762863i	$0.276207\pi$
$812$	1753.26i	2.15919i
$813$	0	0
$814$	0	0
$815$	323.986i	0.397529i
$816$	0	0
$817$	−494.205	−0.604902
$818$	1041.91i	1.27373i
$819$	0	0
$820$	−130.349	−0.158962
$821$	− 983.682i	− 1.19815i	−0.800692	−	0.599076i	$-0.795535\pi$
$821$	− 983.682i	− 1.19815i	0.800692	−	0.599076i	$-0.204465\pi$
$822$	0	0
$823$	−977.362	−1.18756	−0.593780	−	0.804627i	$-0.702365\pi$
$823$	−977.362	−1.18756	−0.593780	+	0.804627i	$0.702365\pi$
$824$	− 921.270i	− 1.11805i
$825$	0	0
$826$	852.993	1.03268
$827$	− 23.7343i	− 0.0286993i	−0.999897	−	0.0143496i	$-0.995432\pi$
$827$	− 23.7343i	− 0.0286993i	0.999897	−	0.0143496i	$-0.00456779\pi$
$828$	0	0
$829$	908.389	1.09576	0.547882	−	0.836555i	$-0.315434\pi$
$829$	908.389	1.09576	0.547882	+	0.836555i	$0.315434\pi$
$830$	218.679i	0.263469i
$831$	0	0
$832$	−948.474	−1.13999
$833$	334.117i	0.401101i
$834$	0	0
$835$	−503.463	−0.602950
$836$	0	0
$837$	0	0
$838$	−1871.14	−2.23287
$839$	780.010i	0.929690i	0.885392	+	0.464845i	$0.153890\pi$
$839$	780.010i	0.929690i	−0.885392	+	0.464845i	$0.846110\pi$
$840$	0	0
$841$	243.123	0.289088
$842$	− 461.256i	− 0.547810i
$843$	0	0
$844$	−1201.42	−1.42349
$845$	− 133.254i	− 0.157697i
$846$	0	0
$847$	0	0
$848$	53.1191i	0.0626404i
$849$	0	0
$850$	352.088	0.414221
$851$	− 469.541i	− 0.551752i
$852$	0	0
$853$	−642.554	−0.753287	−0.376644	−	0.926358i	$-0.622922\pi$
$853$	−642.554	−0.753287	−0.376644	+	0.926358i	$0.622922\pi$
$854$	1566.37i	1.83416i
$855$	0	0
$856$	363.977	0.425206
$857$	442.968i	0.516883i	0.966027	+	0.258441i	$0.0832088\pi$
$857$	442.968i	0.516883i	−0.966027	+	0.258441i	$0.916791\pi$
$858$	0	0
$859$	−95.7522	−0.111469	−0.0557347	−	0.998446i	$-0.517750\pi$
$859$	−95.7522	−0.111469	−0.0557347	+	0.998446i	$0.517750\pi$
$860$	− 206.792i	− 0.240456i
$861$	0	0
$862$	−1274.58	−1.47863
$863$	576.575i	0.668105i	0.942554	+	0.334053i	$0.108416\pi$
$863$	576.575i	0.668105i	−0.942554	+	0.334053i	$0.891584\pi$
$864$	0	0
$865$	181.589	0.209930
$866$	1119.47i	1.29269i
$867$	0	0
$868$	4171.86	4.80630
$869$	0	0
$870$	0	0
$871$	−305.690	−0.350964
$872$	634.862i	0.728053i
$873$	0	0
$874$	2995.53	3.42738
$875$	− 822.538i	− 0.940043i
$876$	0	0
$877$	−909.291	−1.03682	−0.518410	−	0.855132i	$-0.673476\pi$
$877$	−909.291	−1.03682	−0.518410	+	0.855132i	$0.673476\pi$
$878$	− 117.527i	− 0.133858i
$879$	0	0
$880$	0	0
$881$	− 864.536i	− 0.981312i	−0.871353	−	0.490656i	$-0.836757\pi$
$881$	− 864.536i	− 0.981312i	0.871353	−	0.490656i	$-0.163243\pi$
$882$	0	0
$883$	−1.93310	−0.00218924	−0.00109462	−	0.999999i	$-0.500348\pi$
$883$	−1.93310	−0.00218924	−0.00109462	+	0.999999i	$0.500348\pi$
$884$	293.249i	0.331730i
$885$	0	0
$886$	1243.36	1.40334
$887$	743.312i	0.838007i	0.907985	+	0.419004i	$0.137621\pi$
$887$	743.312i	0.838007i	−0.907985	+	0.419004i	$0.862379\pi$
$888$	0	0
$889$	2584.90	2.90765
$890$	575.970i	0.647157i
$891$	0	0
$892$	755.189	0.846624
$893$	− 1629.41i	− 1.82465i
$894$	0	0
$895$	−232.217	−0.259460
$896$	2319.54i	2.58877i
$897$	0	0
$898$	−99.9413	−0.111293
$899$	1422.64i	1.58247i
$900$	0	0
$901$	−238.444	−0.264644
$902$	0	0
$903$	0	0
$904$	29.6920	0.0328452
$905$	− 553.697i	− 0.611820i
$906$	0	0
$907$	−715.123	−0.788449	−0.394225	−	0.919014i	$-0.628987\pi$
$907$	−715.123	−0.788449	−0.394225	+	0.919014i	$0.628987\pi$
$908$	1539.83i	1.69585i
$909$	0	0
$910$	521.013	0.572542
$911$	382.652i	0.420035i	0.977698	+	0.210018i	$0.0673521\pi$
$911$	382.652i	0.420035i	−0.977698	+	0.210018i	$0.932648\pi$
$912$	0	0
$913$	0	0
$914$	− 1343.05i	− 1.46942i
$915$	0	0
$916$	−917.924	−1.00210
$917$	707.003i	0.770995i
$918$	0	0
$919$	383.352	0.417140	0.208570	−	0.978007i	$-0.433119\pi$
$919$	383.352	0.417140	0.208570	+	0.978007i	$0.433119\pi$
$920$	492.684i	0.535526i
$921$	0	0
$922$	2489.78	2.70042
$923$	692.300i	0.750054i
$924$	0	0
$925$	287.474	0.310782
$926$	− 369.646i	− 0.399186i
$927$	0	0
$928$	−739.163	−0.796512
$929$	− 216.559i	− 0.233110i	−0.993184	−	0.116555i	$-0.962815\pi$
$929$	− 216.559i	− 0.233110i	0.993184	−	0.116555i	$-0.0371851\pi$
$930$	0	0
$931$	−1740.39	−1.86938
$932$	− 781.268i	− 0.838270i
$933$	0	0
$934$	−2462.92	−2.63696
$935$	0	0
$936$	0	0
$937$	−931.108	−0.993712	−0.496856	−	0.867833i	$-0.665512\pi$
$937$	−931.108	−0.993712	−0.496856	+	0.867833i	$0.665512\pi$
$938$	1171.58i	1.24902i
$939$	0	0
$940$	681.801	0.725320
$941$	1364.86i	1.45043i	0.688522	+	0.725215i	$0.258260\pi$
$941$	1364.86i	1.45043i	−0.688522	+	0.725215i	$0.741740\pi$
$942$	0	0
$943$	−455.557	−0.483093
$944$	− 25.8500i	− 0.0273835i
$945$	0	0
$946$	0	0
$947$	− 564.949i	− 0.596567i	−0.954477	−	0.298283i	$-0.903586\pi$
$947$	− 564.949i	− 0.596567i	0.954477	−	0.298283i	$-0.0964141\pi$
$948$	0	0
$949$	−382.241	−0.402783
$950$	1834.00i	1.93052i
$951$	0	0
$952$	441.772	0.464046
$953$	199.814i	0.209668i	0.994490	+	0.104834i	$0.0334311\pi$
$953$	199.814i	0.209668i	−0.994490	+	0.104834i	$0.966569\pi$
$954$	0	0
$955$	−119.288	−0.124909
$956$	1900.15i	1.98760i
$957$	0	0
$958$	−1186.40	−1.23841
$959$	− 1365.55i	− 1.42393i
$960$	0	0
$961$	2424.16	2.52254
$962$	384.752i	0.399950i
$963$	0	0
$964$	761.319	0.789751
$965$	− 50.6855i	− 0.0525238i
$966$	0	0
$967$	267.827	0.276967	0.138483	−	0.990365i	$-0.455777\pi$
$967$	267.827	0.276967	0.138483	+	0.990365i	$0.455777\pi$
$968$	0	0
$969$	0	0
$970$	−560.377	−0.577708
$971$	1748.88i	1.80111i	0.434745	+	0.900554i	$0.356839\pi$
$971$	1748.88i	1.80111i	−0.434745	+	0.900554i	$0.643161\pi$
$972$	0	0
$973$	−913.380	−0.938725
$974$	781.196i	0.802050i
$975$	0	0
$976$	47.4690	0.0486363
$977$	− 917.940i	− 0.939549i	−0.882786	−	0.469775i	$-0.844335\pi$
$977$	− 917.940i	− 0.939549i	0.882786	−	0.469775i	$-0.155665\pi$
$978$	0	0
$979$	0	0
$980$	− 728.237i	− 0.743099i
$981$	0	0
$982$	2726.45	2.77642
$983$	− 652.287i	− 0.663568i	−0.943355	−	0.331784i	$-0.892350\pi$
$983$	− 652.287i	− 0.663568i	0.943355	−	0.331784i	$-0.107650\pi$
$984$	0	0
$985$	−262.990	−0.266995
$986$	383.261i	0.388703i
$987$	0	0
$988$	−1527.51	−1.54606
$989$	− 722.717i	− 0.730755i
$990$	0	0
$991$	573.705	0.578915	0.289458	−	0.957191i	$-0.406525\pi$
$991$	573.705	0.578915	0.289458	+	0.957191i	$0.406525\pi$
$992$	1758.83i	1.77302i
$993$	0	0
$994$	2653.30	2.66932
$995$	310.759i	0.312321i
$996$	0	0
$997$	333.026	0.334028	0.167014	−	0.985955i	$-0.446587\pi$
$997$	333.026	0.334028	0.167014	+	0.985955i	$0.446587\pi$
$998$	− 2097.93i	− 2.10214i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.b.i.485.2		16
3.2	odd	2	inner	1089.3.b.i.485.15		16
11.5	even	5		99.3.l.a.80.8	yes	32
11.9	even	5		99.3.l.a.26.1	✓	32
11.10	odd	2		1089.3.b.j.485.15		16
33.5	odd	10		99.3.l.a.80.1	yes	32
33.20	odd	10		99.3.l.a.26.8	yes	32
33.32	even	2		1089.3.b.j.485.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.l.a.26.1	✓	32	11.9	even	5
99.3.l.a.26.8	yes	32	33.20	odd	10
99.3.l.a.80.1	yes	32	33.5	odd	10
99.3.l.a.80.8	yes	32	11.5	even	5
1089.3.b.i.485.2		16	1.1	even	1	trivial
1089.3.b.i.485.15		16	3.2	odd	2	inner
1089.3.b.j.485.2		16	33.32	even	2
1089.3.b.j.485.15		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)

\(f(q)\)	\(=\)	\(q-3.25431i q^{2} -6.59053 q^{4} +1.59288i q^{5} -10.8798 q^{7} +8.43039i q^{8} +O(q^{10})\)	\(q-3.25431i q^{2} -6.59053 q^{4} +1.59288i q^{5} -10.8798 q^{7} +8.43039i q^{8} +5.18373 q^{10} -9.23818 q^{13} +35.4062i q^{14} +1.07299 q^{16} +4.81648i q^{17} -25.0886 q^{19} -10.4979i q^{20} -36.6892i q^{23} +22.4627 q^{25} +30.0639i q^{26} +71.7035 q^{28} +24.4515i q^{29} +58.1821 q^{31} +30.2297i q^{32} +15.6743 q^{34} -17.3302i q^{35} +12.7978 q^{37} +81.6462i q^{38} -13.4286 q^{40} -12.4167i q^{41} +19.6984 q^{43} -119.398 q^{46} +64.9463i q^{47} +69.3696 q^{49} -73.1007i q^{50} +60.8845 q^{52} +49.5059i q^{53} -91.7208i q^{56} +79.5728 q^{58} -24.0917i q^{59} +44.2401 q^{61} -189.343i q^{62} +102.669 q^{64} -14.7153i q^{65} +33.0898 q^{67} -31.7432i q^{68} -56.3978 q^{70} -74.9390i q^{71} +41.3762 q^{73} -41.6480i q^{74} +165.347 q^{76} +63.9703 q^{79} +1.70914i q^{80} -40.4076 q^{82} +42.1858i q^{83} -7.67208 q^{85} -64.1046i q^{86} +111.111i q^{89} +100.509 q^{91} +241.801i q^{92} +211.355 q^{94} -39.9632i q^{95} -108.103 q^{97} -225.750i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{7}+O(q^{10}) \) 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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