LMFDB - Embedded newform 1089.3.c.m.604.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (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} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.8
Root			$-0.797732 - 1.94863i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.313054i q^{2} +3.90200 q^{4} +7.41540 q^{5} -10.0271i q^{7} -2.47375i q^{8} +O(q^{10})$	$q-0.313054i q^{2} +3.90200 q^{4} +7.41540 q^{5} -10.0271i q^{7} -2.47375i q^{8} -2.32142i q^{10} -3.40527i q^{13} -3.13902 q^{14} +14.8336 q^{16} +15.5568i q^{17} -30.6419i q^{19} +28.9349 q^{20} -7.67868 q^{23} +29.9882 q^{25} -1.06603 q^{26} -39.1257i q^{28} -3.37604i q^{29} -4.04250 q^{31} -14.5387i q^{32} +4.87012 q^{34} -74.3549i q^{35} -2.37601 q^{37} -9.59256 q^{38} -18.3438i q^{40} +7.03443i q^{41} +3.99630i q^{43} +2.40384i q^{46} -49.4390 q^{47} -51.5425 q^{49} -9.38791i q^{50} -13.2873i q^{52} +59.7343 q^{53} -24.8045 q^{56} -1.05688 q^{58} +10.9980 q^{59} +74.3100i q^{61} +1.26552i q^{62} +54.7829 q^{64} -25.2514i q^{65} -3.22579 q^{67} +60.7027i q^{68} -23.2771 q^{70} -116.884 q^{71} +18.7225i q^{73} +0.743820i q^{74} -119.565i q^{76} +3.51539i q^{79} +109.997 q^{80} +2.20215 q^{82} +147.099i q^{83} +115.360i q^{85} +1.25105 q^{86} +65.8879 q^{89} -34.1449 q^{91} -29.9622 q^{92} +15.4771i q^{94} -227.222i q^{95} +64.8371 q^{97} +16.1356i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * 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 $

$2$	− 0.313054i	− 0.156527i	−0.996933	−	0.0782634i	$-0.975062\pi$
$2$	− 0.313054i	− 0.156527i	0.996933	−	0.0782634i	$-0.0249375\pi$
$3$	0	0
$3$	0	0
$4$	3.90200	0.975499
$5$	7.41540	1.48308	0.741540	−	0.670908i	$-0.234096\pi$
$5$	7.41540	1.48308	0.741540	+	0.670908i	$0.234096\pi$
$6$	0	0
$7$	− 10.0271i	− 1.43244i	−0.697874	−	0.716221i	$-0.745870\pi$
$7$	− 10.0271i	− 1.43244i	0.697874	−	0.716221i	$-0.254130\pi$
$8$	− 2.47375i	− 0.309219i
$9$	0	0
$10$	− 2.32142i	− 0.232142i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 3.40527i	− 0.261944i	−0.991386	−	0.130972i	$-0.958190\pi$
$13$	− 3.40527i	− 0.261944i	0.991386	−	0.130972i	$-0.0418097\pi$
$14$	−3.13902	−0.224215
$15$	0	0
$16$	14.8336	0.927098
$17$	15.5568i	0.915107i	0.889182	+	0.457554i	$0.151274\pi$
$17$	15.5568i	0.915107i	−0.889182	+	0.457554i	$0.848726\pi$
$18$	0	0
$19$	− 30.6419i	− 1.61273i	−0.591416	−	0.806366i	$-0.701431\pi$
$19$	− 30.6419i	− 1.61273i	0.591416	−	0.806366i	$-0.298569\pi$
$20$	28.9349	1.44674
$21$	0	0
$22$	0	0
$23$	−7.67868	−0.333856	−0.166928	−	0.985969i	$-0.553385\pi$
$23$	−7.67868	−0.333856	−0.166928	+	0.985969i	$0.553385\pi$
$24$	0	0
$25$	29.9882	1.19953
$26$	−1.06603	−0.0410012
$27$	0	0
$28$	− 39.1257i	− 1.39735i
$29$	− 3.37604i	− 0.116415i	−0.998305	−	0.0582076i	$-0.981461\pi$
$29$	− 3.37604i	− 0.116415i	0.998305	−	0.0582076i	$-0.0185385\pi$
$30$	0	0
$31$	−4.04250	−0.130403	−0.0652016	−	0.997872i	$-0.520769\pi$
$31$	−4.04250	−0.130403	−0.0652016	+	0.997872i	$0.520769\pi$
$32$	− 14.5387i	− 0.454334i
$33$	0	0
$34$	4.87012	0.143239
$35$	− 74.3549i	− 2.12443i
$36$	0	0
$37$	−2.37601	−0.0642166	−0.0321083	−	0.999484i	$-0.510222\pi$
$37$	−2.37601	−0.0642166	−0.0321083	+	0.999484i	$0.510222\pi$
$38$	−9.59256	−0.252436
$39$	0	0
$40$	− 18.3438i	− 0.458596i
$41$	7.03443i	0.171571i	0.996314	+	0.0857857i	$0.0273400\pi$
$41$	7.03443i	0.171571i	−0.996314	+	0.0857857i	$0.972660\pi$
$42$	0	0
$43$	3.99630i	0.0929371i	0.998920	+	0.0464686i	$0.0147967\pi$
$43$	3.99630i	0.0929371i	−0.998920	+	0.0464686i	$0.985203\pi$
$44$	0	0
$45$	0	0
$46$	2.40384i	0.0522574i
$47$	−49.4390	−1.05189	−0.525947	−	0.850517i	$-0.676289\pi$
$47$	−49.4390	−1.05189	−0.525947	+	0.850517i	$0.676289\pi$
$48$	0	0
$49$	−51.5425	−1.05189
$50$	− 9.38791i	− 0.187758i
$51$	0	0
$52$	− 13.2873i	− 0.255526i
$53$	59.7343	1.12706	0.563531	−	0.826095i	$-0.309443\pi$
$53$	59.7343	1.12706	0.563531	+	0.826095i	$0.309443\pi$
$54$	0	0
$55$	0	0
$56$	−24.8045	−0.442937
$57$	0	0
$58$	−1.05688	−0.0182221
$59$	10.9980	0.186407	0.0932036	−	0.995647i	$-0.470289\pi$
$59$	10.9980	0.186407	0.0932036	+	0.995647i	$0.470289\pi$
$60$	0	0
$61$	74.3100i	1.21820i	0.793095	+	0.609098i	$0.208468\pi$
$61$	74.3100i	1.21820i	−0.793095	+	0.609098i	$0.791532\pi$
$62$	1.26552i	0.0204116i
$63$	0	0
$64$	54.7829	0.855983
$65$	− 25.2514i	− 0.388484i
$66$	0	0
$67$	−3.22579	−0.0481461	−0.0240730	−	0.999710i	$-0.507663\pi$
$67$	−3.22579	−0.0481461	−0.0240730	+	0.999710i	$0.507663\pi$
$68$	60.7027i	0.892686i
$69$	0	0
$70$	−23.2771	−0.332530
$71$	−116.884	−1.64626	−0.823128	−	0.567856i	$-0.807773\pi$
$71$	−116.884	−1.64626	−0.823128	+	0.567856i	$0.807773\pi$
$72$	0	0
$73$	18.7225i	0.256473i	0.991744	+	0.128236i	$0.0409316\pi$
$73$	18.7225i	0.256473i	−0.991744	+	0.128236i	$0.959068\pi$
$74$	0.743820i	0.0100516i
$75$	0	0
$76$	− 119.565i	− 1.57322i
$77$	0	0
$78$	0	0
$79$	3.51539i	0.0444986i	0.999752	+	0.0222493i	$0.00708275\pi$
$79$	3.51539i	0.0444986i	−0.999752	+	0.0222493i	$0.992917\pi$
$80$	109.997	1.37496
$81$	0	0
$82$	2.20215	0.0268555
$83$	147.099i	1.77228i	0.463417	+	0.886140i	$0.346623\pi$
$83$	147.099i	1.77228i	−0.463417	+	0.886140i	$0.653377\pi$
$84$	0	0
$85$	115.360i	1.35718i
$86$	1.25105	0.0145471
$87$	0	0
$88$	0	0
$89$	65.8879	0.740313	0.370156	−	0.928969i	$-0.379304\pi$
$89$	65.8879	0.740313	0.370156	+	0.928969i	$0.379304\pi$
$90$	0	0
$91$	−34.1449	−0.375219
$92$	−29.9622	−0.325676
$93$	0	0
$94$	15.4771i	0.164650i
$95$	− 227.222i	− 2.39181i
$96$	0	0
$97$	64.8371	0.668424	0.334212	−	0.942498i	$-0.391530\pi$
$97$	64.8371	0.668424	0.334212	+	0.942498i	$0.391530\pi$
$98$	16.1356i	0.164649i
$99$	0	0
$100$	117.014	1.17014
$101$	− 111.937i	− 1.10829i	−0.832421	−	0.554144i	$-0.813046\pi$
$101$	− 111.937i	− 1.10829i	0.832421	−	0.554144i	$-0.186954\pi$
$102$	0	0
$103$	−2.81892	−0.0273682	−0.0136841	−	0.999906i	$-0.504356\pi$
$103$	−2.81892	−0.0273682	−0.0136841	+	0.999906i	$0.504356\pi$
$104$	−8.42377	−0.0809978
$105$	0	0
$106$	− 18.7000i	− 0.176415i
$107$	− 36.5234i	− 0.341340i	−0.985328	−	0.170670i	$-0.945407\pi$
$107$	− 36.5234i	− 0.341340i	0.985328	−	0.170670i	$-0.0545932\pi$
$108$	0	0
$109$	185.532i	1.70213i	0.525061	+	0.851064i	$0.324042\pi$
$109$	185.532i	1.70213i	−0.525061	+	0.851064i	$0.675958\pi$
$110$	0	0
$111$	0	0
$112$	− 148.738i	− 1.32801i
$113$	184.439	1.63220	0.816101	−	0.577909i	$-0.196131\pi$
$113$	184.439	1.63220	0.816101	+	0.577909i	$0.196131\pi$
$114$	0	0
$115$	−56.9405	−0.495135
$116$	− 13.1733i	− 0.113563i
$117$	0	0
$118$	− 3.44297i	− 0.0291777i
$119$	155.990	1.31084
$120$	0	0
$121$	0	0
$122$	23.2630	0.190680
$123$	0	0
$124$	−15.7738	−0.127208
$125$	36.9896	0.295917
$126$	0	0
$127$	193.342i	1.52238i	0.648531	+	0.761188i	$0.275384\pi$
$127$	193.342i	1.52238i	−0.648531	+	0.761188i	$0.724616\pi$
$128$	− 75.3048i	− 0.588319i
$129$	0	0
$130$	−7.90505	−0.0608081
$131$	− 168.561i	− 1.28673i	−0.765562	−	0.643363i	$-0.777539\pi$
$131$	− 168.561i	− 1.28673i	0.765562	−	0.643363i	$-0.222461\pi$
$132$	0	0
$133$	−307.249	−2.31015
$134$	1.00984i	0.00753615i
$135$	0	0
$136$	38.4837	0.282968
$137$	0.0145115	0.000105923 0	5.29617e−5	−	1.00000i	$-0.499983\pi$
$137$	0.0145115	0.000105923 0	5.29617e−5		1.00000i	$0.499983\pi$
$138$	0	0
$139$	− 118.395i	− 0.851760i	−0.904780	−	0.425880i	$-0.859965\pi$
$139$	− 118.395i	− 0.851760i	0.904780	−	0.425880i	$-0.140035\pi$
$140$	− 290.133i	− 2.07238i
$141$	0	0
$142$	36.5910i	0.257683i
$143$	0	0
$144$	0	0
$145$	− 25.0347i	− 0.172653i
$146$	5.86115	0.0401449
$147$	0	0
$148$	−9.27120	−0.0626433
$149$	− 182.415i	− 1.22426i	−0.790756	−	0.612131i	$-0.790312\pi$
$149$	− 182.415i	− 1.22426i	0.790756	−	0.612131i	$-0.209688\pi$
$150$	0	0
$151$	− 170.924i	− 1.13195i	−0.824423	−	0.565974i	$-0.808500\pi$
$151$	− 170.924i	− 1.13195i	0.824423	−	0.565974i	$-0.191500\pi$
$152$	−75.8004	−0.498687
$153$	0	0
$154$	0	0
$155$	−29.9767	−0.193398
$156$	0	0
$157$	−27.6279	−0.175974	−0.0879869	−	0.996122i	$-0.528043\pi$
$157$	−27.6279	−0.175974	−0.0879869	+	0.996122i	$0.528043\pi$
$158$	1.10050	0.00696522
$159$	0	0
$160$	− 107.810i	− 0.673814i
$161$	76.9948i	0.478229i
$162$	0	0
$163$	206.764	1.26849	0.634245	−	0.773132i	$-0.281311\pi$
$163$	206.764	1.26849	0.634245	+	0.773132i	$0.281311\pi$
$164$	27.4483i	0.167368i
$165$	0	0
$166$	46.0499	0.277409
$167$	− 63.8163i	− 0.382134i	−0.981577	−	0.191067i	$-0.938805\pi$
$167$	− 63.8163i	− 0.382134i	0.981577	−	0.191067i	$-0.0611947\pi$
$168$	0	0
$169$	157.404	0.931386
$170$	36.1139	0.212435
$171$	0	0
$172$	15.5935i	0.0906601i
$173$	226.810i	1.31104i	0.755177	+	0.655521i	$0.227551\pi$
$173$	226.810i	1.31104i	−0.755177	+	0.655521i	$0.772449\pi$
$174$	0	0
$175$	− 300.694i	− 1.71825i
$176$	0	0
$177$	0	0
$178$	− 20.6264i	− 0.115879i
$179$	−283.704	−1.58494	−0.792469	−	0.609912i	$-0.791205\pi$
$179$	−283.704	−1.58494	−0.792469	+	0.609912i	$0.791205\pi$
$180$	0	0
$181$	134.784	0.744660	0.372330	−	0.928100i	$-0.378559\pi$
$181$	134.784	0.744660	0.372330	+	0.928100i	$0.378559\pi$
$182$	10.6892i	0.0587318i
$183$	0	0
$184$	18.9951i	0.103234i
$185$	−17.6191	−0.0952384
$186$	0	0
$187$	0	0
$188$	−192.911	−1.02612
$189$	0	0
$190$	−71.1327	−0.374383
$191$	99.6465	0.521710	0.260855	−	0.965378i	$-0.415996\pi$
$191$	99.6465	0.521710	0.260855	+	0.965378i	$0.415996\pi$
$192$	0	0
$193$	− 210.795i	− 1.09220i	−0.837719	−	0.546102i	$-0.816111\pi$
$193$	− 210.795i	− 1.09220i	0.837719	−	0.546102i	$-0.183889\pi$
$194$	− 20.2975i	− 0.104626i
$195$	0	0
$196$	−201.119	−1.02612
$197$	− 51.1334i	− 0.259561i	−0.991543	−	0.129780i	$-0.958573\pi$
$197$	− 51.1334i	− 0.259561i	0.991543	−	0.129780i	$-0.0414272\pi$
$198$	0	0
$199$	77.3567	0.388727	0.194364	−	0.980930i	$-0.437736\pi$
$199$	77.3567	0.388727	0.194364	+	0.980930i	$0.437736\pi$
$200$	− 74.1833i	− 0.370916i
$201$	0	0
$202$	−35.0423	−0.173477
$203$	−33.8518	−0.166758
$204$	0	0
$205$	52.1631i	0.254454i
$206$	0.882473i	0.00428385i
$207$	0	0
$208$	− 50.5123i	− 0.242848i
$209$	0	0
$210$	0	0
$211$	4.80058i	0.0227515i	0.999935	+	0.0113758i	$0.00362110\pi$
$211$	4.80058i	0.0227515i	−0.999935	+	0.0113758i	$0.996379\pi$
$212$	233.083	1.09945
$213$	0	0
$214$	−11.4338	−0.0534288
$215$	29.6341i	0.137833i
$216$	0	0
$217$	40.5345i	0.186795i
$218$	58.0815	0.266429
$219$	0	0
$220$	0	0
$221$	52.9751	0.239706
$222$	0	0
$223$	250.735	1.12437	0.562185	−	0.827011i	$-0.309961\pi$
$223$	250.735	1.12437	0.562185	+	0.827011i	$0.309961\pi$
$224$	−145.781	−0.650807
$225$	0	0
$226$	− 57.7393i	− 0.255483i
$227$	341.954i	1.50640i	0.657789	+	0.753202i	$0.271492\pi$
$227$	341.954i	1.50640i	−0.657789	+	0.753202i	$0.728508\pi$
$228$	0	0
$229$	−231.327	−1.01016	−0.505081	−	0.863072i	$-0.668538\pi$
$229$	−231.327	−1.01016	−0.505081	+	0.863072i	$0.668538\pi$
$230$	17.8254i	0.0775019i
$231$	0	0
$232$	−8.35147	−0.0359977
$233$	85.3998i	0.366523i	0.983064	+	0.183261i	$0.0586655\pi$
$233$	85.3998i	0.366523i	−0.983064	+	0.183261i	$0.941335\pi$
$234$	0	0
$235$	−366.610	−1.56004
$236$	42.9143	0.181840
$237$	0	0
$238$	− 48.8331i	− 0.205181i
$239$	11.8302i	0.0494989i	0.999694	+	0.0247495i	$0.00787880\pi$
$239$	11.8302i	0.0494989i	−0.999694	+	0.0247495i	$0.992121\pi$
$240$	0	0
$241$	236.527i	0.981438i	0.871318	+	0.490719i	$0.163266\pi$
$241$	236.527i	0.981438i	−0.871318	+	0.490719i	$0.836734\pi$
$242$	0	0
$243$	0	0
$244$	289.957i	1.18835i
$245$	−382.209	−1.56003
$246$	0	0
$247$	−104.344	−0.422445
$248$	10.0001i	0.0403231i
$249$	0	0
$250$	− 11.5797i	− 0.0463189i
$251$	250.671	0.998690	0.499345	−	0.866403i	$-0.333574\pi$
$251$	250.671	0.998690	0.499345	+	0.866403i	$0.333574\pi$
$252$	0	0
$253$	0	0
$254$	60.5263	0.238293
$255$	0	0
$256$	195.557	0.763895
$257$	−260.932	−1.01530	−0.507650	−	0.861563i	$-0.669486\pi$
$257$	−260.932	−1.01530	−0.507650	+	0.861563i	$0.669486\pi$
$258$	0	0
$259$	23.8245i	0.0919865i
$260$	− 98.5310i	− 0.378965i
$261$	0	0
$262$	−52.7686	−0.201407
$263$	243.396i	0.925459i	0.886499	+	0.462730i	$0.153130\pi$
$263$	243.396i	0.925459i	−0.886499	+	0.462730i	$0.846870\pi$
$264$	0	0
$265$	442.954	1.67152
$266$	96.1855i	0.361600i
$267$	0	0
$268$	−12.5870	−0.0469665
$269$	260.616	0.968831	0.484416	−	0.874838i	$-0.339032\pi$
$269$	260.616	0.968831	0.484416	+	0.874838i	$0.339032\pi$
$270$	0	0
$271$	438.744i	1.61898i	0.587131	+	0.809492i	$0.300257\pi$
$271$	438.744i	1.61898i	−0.587131	+	0.809492i	$0.699743\pi$
$272$	230.763i	0.848394i
$273$	0	0
$274$	− 0.00454288i	0 1.65798e-5i
$275$	0	0
$276$	0	0
$277$	202.271i	0.730220i	0.930964	+	0.365110i	$0.118969\pi$
$277$	202.271i	0.730220i	−0.930964	+	0.365110i	$0.881031\pi$
$278$	−37.0639	−0.133323
$279$	0	0
$280$	−183.935	−0.656912
$281$	431.706i	1.53632i	0.640258	+	0.768160i	$0.278827\pi$
$281$	431.706i	1.53632i	−0.640258	+	0.768160i	$0.721173\pi$
$282$	0	0
$283$	157.515i	0.556592i	0.960495	+	0.278296i	$0.0897696\pi$
$283$	157.515i	0.556592i	−0.960495	+	0.278296i	$0.910230\pi$
$284$	−456.082	−1.60592
$285$	0	0
$286$	0	0
$287$	70.5348	0.245766
$288$	0	0
$289$	46.9853	0.162579
$290$	−7.83720	−0.0270248
$291$	0	0
$292$	73.0552i	0.250189i
$293$	446.793i	1.52489i	0.647053	+	0.762445i	$0.276001\pi$
$293$	446.793i	1.52489i	−0.647053	+	0.762445i	$0.723999\pi$
$294$	0	0
$295$	81.5548	0.276457
$296$	5.87766i	0.0198570i
$297$	0	0
$298$	−57.1057	−0.191630
$299$	26.1480i	0.0874514i
$300$	0	0
$301$	40.0712	0.133127
$302$	−53.5084	−0.177180
$303$	0	0
$304$	− 454.529i	− 1.49516i
$305$	551.038i	1.80668i
$306$	0	0
$307$	− 374.322i	− 1.21929i	−0.792674	−	0.609645i	$-0.791312\pi$
$307$	− 374.322i	− 1.21929i	0.792674	−	0.609645i	$-0.208688\pi$
$308$	0	0
$309$	0	0
$310$	9.38433i	0.0302720i
$311$	−92.0038	−0.295832	−0.147916	−	0.989000i	$-0.547257\pi$
$311$	−92.0038	−0.295832	−0.147916	+	0.989000i	$0.547257\pi$
$312$	0	0
$313$	484.951	1.54936	0.774682	−	0.632352i	$-0.217910\pi$
$313$	484.951	1.54936	0.774682	+	0.632352i	$0.217910\pi$
$314$	8.64900i	0.0275446i
$315$	0	0
$316$	13.7170i	0.0434083i
$317$	−418.385	−1.31983	−0.659913	−	0.751342i	$-0.729407\pi$
$317$	−418.385	−1.31983	−0.659913	+	0.751342i	$0.729407\pi$
$318$	0	0
$319$	0	0
$320$	406.237	1.26949
$321$	0	0
$322$	24.1035	0.0748556
$323$	476.691	1.47582
$324$	0	0
$325$	− 102.118i	− 0.314209i
$326$	− 64.7282i	− 0.198553i
$327$	0	0
$328$	17.4014	0.0530531
$329$	495.730i	1.50678i
$330$	0	0
$331$	−232.490	−0.702387	−0.351194	−	0.936303i	$-0.614224\pi$
$331$	−232.490	−0.702387	−0.351194	+	0.936303i	$0.614224\pi$
$332$	573.981i	1.72886i
$333$	0	0
$334$	−19.9779	−0.0598141
$335$	−23.9205	−0.0714045
$336$	0	0
$337$	151.000i	0.448071i	0.974581	+	0.224035i	$0.0719231\pi$
$337$	151.000i	0.448071i	−0.974581	+	0.224035i	$0.928077\pi$
$338$	− 49.2759i	− 0.145787i
$339$	0	0
$340$	450.135i	1.32393i
$341$	0	0
$342$	0	0
$343$	25.4940i	0.0743267i
$344$	9.88583	0.0287379
$345$	0	0
$346$	71.0038	0.205213
$347$	− 262.765i	− 0.757248i	−0.925551	−	0.378624i	$-0.876397\pi$
$347$	− 262.765i	− 0.757248i	0.925551	−	0.378624i	$-0.123603\pi$
$348$	0	0
$349$	− 439.696i	− 1.25987i	−0.776646	−	0.629937i	$-0.783081\pi$
$349$	− 439.696i	− 1.25987i	0.776646	−	0.629937i	$-0.216919\pi$
$350$	−94.1335	−0.268953
$351$	0	0
$352$	0	0
$353$	−479.476	−1.35829	−0.679145	−	0.734005i	$-0.737649\pi$
$353$	−479.476	−1.35829	−0.679145	+	0.734005i	$0.737649\pi$
$354$	0	0
$355$	−866.743	−2.44153
$356$	257.094	0.722175
$357$	0	0
$358$	88.8145i	0.248085i
$359$	− 209.270i	− 0.582924i	−0.956582	−	0.291462i	$-0.905858\pi$
$359$	− 209.270i	− 0.582924i	0.956582	−	0.291462i	$-0.0941417\pi$
$360$	0	0
$361$	−577.928	−1.60091
$362$	− 42.1945i	− 0.116559i
$363$	0	0
$364$	−133.233	−0.366026
$365$	138.835i	0.380370i
$366$	0	0
$367$	394.113	1.07388	0.536939	−	0.843621i	$-0.319581\pi$
$367$	394.113	1.07388	0.536939	+	0.843621i	$0.319581\pi$
$368$	−113.902	−0.309517
$369$	0	0
$370$	5.51572i	0.0149074i
$371$	− 598.961i	− 1.61445i
$372$	0	0
$373$	497.797i	1.33458i	0.744800	+	0.667288i	$0.232545\pi$
$373$	497.797i	1.33458i	−0.744800	+	0.667288i	$0.767455\pi$
$374$	0	0
$375$	0	0
$376$	122.300i	0.325265i
$377$	−11.4963	−0.0304942
$378$	0	0
$379$	−313.379	−0.826858	−0.413429	−	0.910536i	$-0.635669\pi$
$379$	−313.379	−0.826858	−0.413429	+	0.910536i	$0.635669\pi$
$380$	− 886.621i	− 2.33321i
$381$	0	0
$382$	− 31.1947i	− 0.0816615i
$383$	−269.503	−0.703663	−0.351831	−	0.936063i	$-0.614441\pi$
$383$	−269.503	−0.703663	−0.351831	+	0.936063i	$0.614441\pi$
$384$	0	0
$385$	0	0
$386$	−65.9902	−0.170959
$387$	0	0
$388$	252.994	0.652047
$389$	−231.705	−0.595644	−0.297822	−	0.954621i	$-0.596260\pi$
$389$	−231.705	−0.595644	−0.297822	+	0.954621i	$0.596260\pi$
$390$	0	0
$391$	− 119.456i	− 0.305514i
$392$	127.503i	0.325263i
$393$	0	0
$394$	−16.0075	−0.0406282
$395$	26.0680i	0.0659950i
$396$	0	0
$397$	−256.488	−0.646065	−0.323033	−	0.946388i	$-0.604702\pi$
$397$	−256.488	−0.646065	−0.323033	+	0.946388i	$0.604702\pi$
$398$	− 24.2168i	− 0.0608462i
$399$	0	0
$400$	444.832	1.11208
$401$	307.932	0.767910	0.383955	−	0.923352i	$-0.374562\pi$
$401$	307.932	0.767910	0.383955	+	0.923352i	$0.374562\pi$
$402$	0	0
$403$	13.7658i	0.0341583i
$404$	− 436.779i	− 1.08114i
$405$	0	0
$406$	10.5974i	0.0261021i
$407$	0	0
$408$	0	0
$409$	− 432.083i	− 1.05644i	−0.849108	−	0.528219i	$-0.822860\pi$
$409$	− 432.083i	− 1.05644i	0.849108	−	0.528219i	$-0.177140\pi$
$410$	16.3299	0.0398289
$411$	0	0
$412$	−10.9994	−0.0266976
$413$	− 110.278i	− 0.267017i
$414$	0	0
$415$	1090.80i	2.62843i
$416$	−49.5081	−0.119010
$417$	0	0
$418$	0	0
$419$	−485.955	−1.15980	−0.579898	−	0.814689i	$-0.696908\pi$
$419$	−485.955	−1.15980	−0.579898	+	0.814689i	$0.696908\pi$
$420$	0	0
$421$	−472.666	−1.12272	−0.561361	−	0.827571i	$-0.689722\pi$
$421$	−472.666	−1.12272	−0.561361	+	0.827571i	$0.689722\pi$
$422$	1.50284	0.00356123
$423$	0	0
$424$	− 147.768i	− 0.348508i
$425$	466.521i	1.09770i
$426$	0	0
$427$	745.113	1.74499
$428$	− 142.514i	− 0.332977i
$429$	0	0
$430$	9.27708	0.0215746
$431$	− 254.324i	− 0.590078i	−0.955485	−	0.295039i	$-0.904667\pi$
$431$	− 254.324i	− 0.590078i	0.955485	−	0.295039i	$-0.0953327\pi$
$432$	0	0
$433$	−60.6356	−0.140036	−0.0700180	−	0.997546i	$-0.522306\pi$
$433$	−60.6356	−0.140036	−0.0700180	+	0.997546i	$0.522306\pi$
$434$	12.6895	0.0292384
$435$	0	0
$436$	723.946i	1.66043i
$437$	235.290i	0.538420i
$438$	0	0
$439$	190.549i	0.434053i	0.976166	+	0.217026i	$0.0696358\pi$
$439$	190.549i	0.434053i	−0.976166	+	0.217026i	$0.930364\pi$
$440$	0	0
$441$	0	0
$442$	− 16.5841i	− 0.0375205i
$443$	290.568	0.655910	0.327955	−	0.944693i	$-0.393641\pi$
$443$	290.568	0.655910	0.327955	+	0.944693i	$0.393641\pi$
$444$	0	0
$445$	488.585	1.09794
$446$	− 78.4934i	− 0.175994i
$447$	0	0
$448$	− 549.313i	− 1.22615i
$449$	−459.491	−1.02337	−0.511683	−	0.859174i	$-0.670978\pi$
$449$	−459.491	−1.02337	−0.511683	+	0.859174i	$0.670978\pi$
$450$	0	0
$451$	0	0
$452$	719.680	1.59221
$453$	0	0
$454$	107.050	0.235793
$455$	−253.198	−0.556480
$456$	0	0
$457$	12.7670i	0.0279365i	0.999902	+	0.0139683i	$0.00444638\pi$
$457$	12.7670i	0.0279365i	−0.999902	+	0.0139683i	$0.995554\pi$
$458$	72.4178i	0.158118i
$459$	0	0
$460$	−222.182	−0.483004
$461$	109.347i	0.237195i	0.992942	+	0.118597i	$0.0378397\pi$
$461$	109.347i	0.237195i	−0.992942	+	0.118597i	$0.962160\pi$
$462$	0	0
$463$	419.108	0.905202	0.452601	−	0.891713i	$-0.350496\pi$
$463$	419.108	0.905202	0.452601	+	0.891713i	$0.350496\pi$
$464$	− 50.0787i	− 0.107928i
$465$	0	0
$466$	26.7347	0.0573706
$467$	458.783	0.982405	0.491202	−	0.871045i	$-0.336558\pi$
$467$	458.783	0.982405	0.491202	+	0.871045i	$0.336558\pi$
$468$	0	0
$469$	32.3453i	0.0689664i
$470$	114.769i	0.244189i
$471$	0	0
$472$	− 27.2063i	− 0.0576406i
$473$	0	0
$474$	0	0
$475$	− 918.896i	− 1.93452i
$476$	608.671	1.27872
$477$	0	0
$478$	3.70350	0.00774790
$479$	− 248.786i	− 0.519386i	−0.965691	−	0.259693i	$-0.916379\pi$
$479$	− 248.786i	− 0.519386i	0.965691	−	0.259693i	$-0.0836214\pi$
$480$	0	0
$481$	8.09096i	0.0168211i
$482$	74.0455	0.153621
$483$	0	0
$484$	0	0
$485$	480.793	0.991326
$486$	0	0
$487$	138.842	0.285097	0.142549	−	0.989788i	$-0.454470\pi$
$487$	138.842	0.285097	0.142549	+	0.989788i	$0.454470\pi$
$488$	183.824	0.376689
$489$	0	0
$490$	119.652i	0.244187i
$491$	− 221.833i	− 0.451798i	−0.974151	−	0.225899i	$-0.927468\pi$
$491$	− 221.833i	− 0.451798i	0.974151	−	0.225899i	$-0.0725320\pi$
$492$	0	0
$493$	52.5204	0.106532
$494$	32.6652i	0.0661240i
$495$	0	0
$496$	−59.9647	−0.120897
$497$	1172.01i	2.35817i
$498$	0	0
$499$	−751.672	−1.50636	−0.753179	−	0.657816i	$-0.771480\pi$
$499$	−751.672	−1.50636	−0.753179	+	0.657816i	$0.771480\pi$
$500$	144.333	0.288666
$501$	0	0
$502$	− 78.4735i	− 0.156322i
$503$	− 13.4846i	− 0.0268084i	−0.999910	−	0.0134042i	$-0.995733\pi$
$503$	− 13.4846i	− 0.0268084i	0.999910	−	0.0134042i	$-0.00426681\pi$
$504$	0	0
$505$	− 830.059i	− 1.64368i
$506$	0	0
$507$	0	0
$508$	754.419i	1.48508i
$509$	−731.429	−1.43699	−0.718496	−	0.695531i	$-0.755169\pi$
$509$	−731.429	−1.43699	−0.718496	+	0.695531i	$0.755169\pi$
$510$	0	0
$511$	187.732	0.367382
$512$	− 362.439i	− 0.707889i
$513$	0	0
$514$	81.6857i	0.158922i
$515$	−20.9034	−0.0405892
$516$	0	0
$517$	0	0
$518$	7.45835	0.0143984
$519$	0	0
$520$	−62.4657	−0.120126
$521$	−703.605	−1.35049	−0.675244	−	0.737594i	$-0.735962\pi$
$521$	−703.605	−1.35049	−0.675244	+	0.737594i	$0.735962\pi$
$522$	0	0
$523$	824.898i	1.57724i	0.614879	+	0.788621i	$0.289205\pi$
$523$	824.898i	1.57724i	−0.614879	+	0.788621i	$0.710795\pi$
$524$	− 657.725i	− 1.25520i
$525$	0	0
$526$	76.1959	0.144859
$527$	− 62.8884i	− 0.119333i
$528$	0	0
$529$	−470.038	−0.888540
$530$	− 138.668i	− 0.261638i
$531$	0	0
$532$	−1198.89	−2.25355
$533$	23.9541	0.0449420
$534$	0	0
$535$	− 270.836i	− 0.506235i
$536$	7.97979i	0.0148877i
$537$	0	0
$538$	− 81.5866i	− 0.151648i
$539$	0	0
$540$	0	0
$541$	986.658i	1.82377i	0.410449	+	0.911884i	$0.365372\pi$
$541$	986.658i	1.82377i	−0.410449	+	0.911884i	$0.634628\pi$
$542$	137.351	0.253414
$543$	0	0
$544$	226.176	0.415764
$545$	1375.80i	2.52439i
$546$	0	0
$547$	101.477i	0.185516i	0.995689	+	0.0927579i	$0.0295683\pi$
$547$	101.477i	0.185516i	−0.995689	+	0.0927579i	$0.970432\pi$
$548$	0.0566238	0.000103328 0
$549$	0	0
$550$	0	0
$551$	−103.448	−0.187747
$552$	0	0
$553$	35.2491	0.0637416
$554$	63.3216	0.114299
$555$	0	0
$556$	− 461.976i	− 0.830891i
$557$	875.686i	1.57215i	0.618133	+	0.786074i	$0.287889\pi$
$557$	875.686i	1.57215i	−0.618133	+	0.786074i	$0.712111\pi$
$558$	0	0
$559$	13.6085	0.0243443
$560$	− 1102.95i	− 1.96955i
$561$	0	0
$562$	135.147	0.240475
$563$	− 755.179i	− 1.34135i	−0.741752	−	0.670674i	$-0.766005\pi$
$563$	− 755.179i	− 1.34135i	0.741752	−	0.670674i	$-0.233995\pi$
$564$	0	0
$565$	1367.69	2.42069
$566$	49.3108	0.0871215
$567$	0	0
$568$	289.142i	0.509053i
$569$	− 240.032i	− 0.421849i	−0.977502	−	0.210925i	$-0.932353\pi$
$569$	− 240.032i	− 0.421849i	0.977502	−	0.210925i	$-0.0676475\pi$
$570$	0	0
$571$	343.953i	0.602369i	0.953566	+	0.301184i	$0.0973820\pi$
$571$	343.953i	0.602369i	−0.953566	+	0.301184i	$0.902618\pi$
$572$	0	0
$573$	0	0
$574$	− 22.0812i	− 0.0384690i
$575$	−230.270	−0.400469
$576$	0	0
$577$	−922.811	−1.59933	−0.799663	−	0.600449i	$-0.794988\pi$
$577$	−922.811	−1.59933	−0.799663	+	0.600449i	$0.794988\pi$
$578$	− 14.7089i	− 0.0254480i
$579$	0	0
$580$	− 97.6853i	− 0.168423i
$581$	1474.98	2.53869
$582$	0	0
$583$	0	0
$584$	46.3148	0.0793061
$585$	0	0
$586$	139.870	0.238686
$587$	2.57138	0.00438054	0.00219027	−	0.999998i	$-0.499303\pi$
$587$	2.57138	0.00438054	0.00219027	+	0.999998i	$0.499303\pi$
$588$	0	0
$589$	123.870i	0.210305i
$590$	− 25.5310i	− 0.0432729i
$591$	0	0
$592$	−35.2448	−0.0595351
$593$	115.926i	0.195491i	0.995211	+	0.0977456i	$0.0311631\pi$
$593$	115.926i	0.195491i	−0.995211	+	0.0977456i	$0.968837\pi$
$594$	0	0
$595$	1156.73	1.94408
$596$	− 711.783i	− 1.19427i
$597$	0	0
$598$	8.18571	0.0136885
$599$	−226.525	−0.378172	−0.189086	−	0.981961i	$-0.560552\pi$
$599$	−226.525	−0.378172	−0.189086	+	0.981961i	$0.560552\pi$
$600$	0	0
$601$	− 568.740i	− 0.946323i	−0.880976	−	0.473161i	$-0.843113\pi$
$601$	− 568.740i	− 0.946323i	0.880976	−	0.473161i	$-0.156887\pi$
$602$	− 12.5444i	− 0.0208379i
$603$	0	0
$604$	− 666.946i	− 1.10422i
$605$	0	0
$606$	0	0
$607$	310.107i	0.510885i	0.966824	+	0.255442i	$0.0822211\pi$
$607$	310.107i	0.510885i	−0.966824	+	0.255442i	$0.917779\pi$
$608$	−445.494	−0.732720
$609$	0	0
$610$	172.505	0.282794
$611$	168.353i	0.275537i
$612$	0	0
$613$	224.226i	0.365784i	0.983133	+	0.182892i	$0.0585459\pi$
$613$	224.226i	0.365784i	−0.983133	+	0.182892i	$0.941454\pi$
$614$	−117.183	−0.190852
$615$	0	0
$616$	0	0
$617$	−693.508	−1.12400	−0.562000	−	0.827137i	$-0.689968\pi$
$617$	−693.508	−1.12400	−0.562000	+	0.827137i	$0.689968\pi$
$618$	0	0
$619$	373.028	0.602630	0.301315	−	0.953525i	$-0.402574\pi$
$619$	373.028	0.602630	0.301315	+	0.953525i	$0.402574\pi$
$620$	−116.969	−0.188660
$621$	0	0
$622$	28.8021i	0.0463056i
$623$	− 660.663i	− 1.06045i
$624$	0	0
$625$	−475.413	−0.760660
$626$	− 151.816i	− 0.242517i
$627$	0	0
$628$	−107.804	−0.171662
$629$	− 36.9632i	− 0.0587651i
$630$	0	0
$631$	−694.005	−1.09985	−0.549925	−	0.835214i	$-0.685344\pi$
$631$	−694.005	−1.09985	−0.549925	+	0.835214i	$0.685344\pi$
$632$	8.69618	0.0137598
$633$	0	0
$634$	130.977i	0.206588i
$635$	1433.71i	2.25781i
$636$	0	0
$637$	175.516i	0.275535i
$638$	0	0
$639$	0	0
$640$	− 558.415i	− 0.872524i
$641$	880.771	1.37406	0.687029	−	0.726630i	$-0.258914\pi$
$641$	880.771	1.37406	0.687029	+	0.726630i	$0.258914\pi$
$642$	0	0
$643$	−1032.22	−1.60532	−0.802661	−	0.596436i	$-0.796583\pi$
$643$	−1032.22	−1.60532	−0.802661	+	0.596436i	$0.796583\pi$
$644$	300.434i	0.466512i
$645$	0	0
$646$	− 149.230i	− 0.231006i
$647$	354.884	0.548508	0.274254	−	0.961657i	$-0.411569\pi$
$647$	354.884	0.548508	0.274254	+	0.961657i	$0.411569\pi$
$648$	0	0
$649$	0	0
$650$	−31.9684	−0.0491821
$651$	0	0
$652$	806.792	1.23741
$653$	720.236	1.10296	0.551482	−	0.834187i	$-0.314062\pi$
$653$	720.236	1.10296	0.551482	+	0.834187i	$0.314062\pi$
$654$	0	0
$655$	− 1249.95i	− 1.90832i
$656$	104.346i	0.159064i
$657$	0	0
$658$	155.190	0.235851
$659$	745.955i	1.13195i	0.824423	+	0.565975i	$0.191500\pi$
$659$	745.955i	1.13195i	−0.824423	+	0.565975i	$0.808500\pi$
$660$	0	0
$661$	−655.463	−0.991623	−0.495811	−	0.868430i	$-0.665129\pi$
$661$	−655.463	−0.991623	−0.495811	+	0.868430i	$0.665129\pi$
$662$	72.7819i	0.109942i
$663$	0	0
$664$	363.887	0.548022
$665$	−2278.38	−3.42613
$666$	0	0
$667$	25.9235i	0.0388659i
$668$	− 249.011i	− 0.372771i
$669$	0	0
$670$	7.48840i	0.0111767i
$671$	0	0
$672$	0	0
$673$	− 1136.90i	− 1.68931i	−0.535314	−	0.844653i	$-0.679807\pi$
$673$	− 1136.90i	− 1.68931i	0.535314	−	0.844653i	$-0.320193\pi$
$674$	47.2710	0.0701351
$675$	0	0
$676$	614.191	0.908566
$677$	480.106i	0.709167i	0.935024	+	0.354583i	$0.115377\pi$
$677$	480.106i	0.709167i	−0.935024	+	0.354583i	$0.884623\pi$
$678$	0	0
$679$	− 650.127i	− 0.957478i
$680$	285.372	0.419664
$681$	0	0
$682$	0	0
$683$	110.003	0.161059	0.0805294	−	0.996752i	$-0.474339\pi$
$683$	110.003	0.161059	0.0805294	+	0.996752i	$0.474339\pi$
$684$	0	0
$685$	0.107609	0.000157093 0
$686$	7.98100	0.0116341
$687$	0	0
$688$	59.2794i	0.0861619i
$689$	− 203.411i	− 0.295227i
$690$	0	0
$691$	−539.540	−0.780811	−0.390405	−	0.920643i	$-0.627665\pi$
$691$	−539.540	−0.780811	−0.390405	+	0.920643i	$0.627665\pi$
$692$	885.013i	1.27892i
$693$	0	0
$694$	−82.2595	−0.118530
$695$	− 877.944i	− 1.26323i
$696$	0	0
$697$	−109.433	−0.157006
$698$	−137.648	−0.197204
$699$	0	0
$700$	− 1173.31i	− 1.67616i
$701$	− 416.973i	− 0.594825i	−0.954749	−	0.297413i	$-0.903876\pi$
$701$	− 416.973i	− 0.594825i	0.954749	−	0.297413i	$-0.0961237\pi$
$702$	0	0
$703$	72.8057i	0.103564i
$704$	0	0
$705$	0	0
$706$	150.102i	0.212609i
$707$	−1122.40	−1.58756
$708$	0	0
$709$	−90.2767	−0.127330	−0.0636648	−	0.997971i	$-0.520279\pi$
$709$	−90.2767	−0.127330	−0.0636648	+	0.997971i	$0.520279\pi$
$710$	271.337i	0.382165i
$711$	0	0
$712$	− 162.990i	− 0.228919i
$713$	31.0410	0.0435358
$714$	0	0
$715$	0	0
$716$	−1107.01	−1.54611
$717$	0	0
$718$	−65.5126	−0.0912432
$719$	1334.01	1.85536	0.927682	−	0.373370i	$-0.121798\pi$
$719$	1334.01	1.85536	0.927682	+	0.373370i	$0.121798\pi$
$720$	0	0
$721$	28.2656i	0.0392033i
$722$	180.922i	0.250585i
$723$	0	0
$724$	525.925	0.726416
$725$	− 101.241i	− 0.139643i
$726$	0	0
$727$	−68.6829	−0.0944744	−0.0472372	−	0.998884i	$-0.515042\pi$
$727$	−68.6829	−0.0944744	−0.0472372	+	0.998884i	$0.515042\pi$
$728$	84.4659i	0.116025i
$729$	0	0
$730$	43.4628	0.0595381
$731$	−62.1697	−0.0850474
$732$	0	0
$733$	− 661.840i	− 0.902919i	−0.892292	−	0.451459i	$-0.850904\pi$
$733$	− 661.840i	− 0.902919i	0.892292	−	0.451459i	$-0.149096\pi$
$734$	− 123.378i	− 0.168091i
$735$	0	0
$736$	111.638i	0.151682i
$737$	0	0
$738$	0	0
$739$	180.429i	0.244153i	0.992521	+	0.122076i	$0.0389553\pi$
$739$	180.429i	0.244153i	−0.992521	+	0.122076i	$0.961045\pi$
$740$	−68.7497	−0.0929050
$741$	0	0
$742$	−187.507	−0.252705
$743$	431.776i	0.581126i	0.956856	+	0.290563i	$0.0938425\pi$
$743$	431.776i	0.581126i	−0.956856	+	0.290563i	$0.906157\pi$
$744$	0	0
$745$	− 1352.68i	− 1.81568i
$746$	155.837	0.208897
$747$	0	0
$748$	0	0
$749$	−366.223	−0.488949
$750$	0	0
$751$	662.380	0.881998	0.440999	−	0.897508i	$-0.354624\pi$
$751$	662.380	0.881998	0.440999	+	0.897508i	$0.354624\pi$
$752$	−733.357	−0.975209
$753$	0	0
$754$	3.59896i	0.00477316i
$755$	− 1267.47i	− 1.67877i
$756$	0	0
$757$	1252.41	1.65444	0.827218	−	0.561881i	$-0.189922\pi$
$757$	1252.41	1.65444	0.827218	+	0.561881i	$0.189922\pi$
$758$	98.1044i	0.129425i
$759$	0	0
$760$	−562.091	−0.739593
$761$	− 622.848i	− 0.818461i	−0.912431	−	0.409230i	$-0.865797\pi$
$761$	− 622.848i	− 0.818461i	0.912431	−	0.409230i	$-0.134203\pi$
$762$	0	0
$763$	1860.35	2.43820
$764$	388.821	0.508927
$765$	0	0
$766$	84.3688i	0.110142i
$767$	− 37.4512i	− 0.0488282i
$768$	0	0
$769$	− 39.8312i	− 0.0517960i	−0.999665	−	0.0258980i	$-0.991755\pi$
$769$	− 39.8312i	− 0.0517960i	0.999665	−	0.0258980i	$-0.00824452\pi$
$770$	0	0
$771$	0	0
$772$	− 822.522i	− 1.06544i
$773$	565.206	0.731185	0.365593	−	0.930775i	$-0.380866\pi$
$773$	565.206	0.731185	0.365593	+	0.930775i	$0.380866\pi$
$774$	0	0
$775$	−121.227	−0.156422
$776$	− 160.391i	− 0.206689i
$777$	0	0
$778$	72.5362i	0.0932342i
$779$	215.548	0.276699
$780$	0	0
$781$	0	0
$782$	−37.3961	−0.0478211
$783$	0	0
$784$	−764.560	−0.975204
$785$	−204.872	−0.260983
$786$	0	0
$787$	1016.00i	1.29098i	0.763770	+	0.645488i	$0.223346\pi$
$787$	1016.00i	1.29098i	−0.763770	+	0.645488i	$0.776654\pi$
$788$	− 199.523i	− 0.253201i
$789$	0	0
$790$	8.16068	0.0103300
$791$	− 1849.39i	− 2.33803i
$792$	0	0
$793$	253.045	0.319099
$794$	80.2945i	0.101127i
$795$	0	0
$796$	301.846	0.379203
$797$	353.467	0.443497	0.221748	−	0.975104i	$-0.428824\pi$
$797$	353.467	0.443497	0.221748	+	0.975104i	$0.428824\pi$
$798$	0	0
$799$	− 769.114i	− 0.962596i
$800$	− 435.989i	− 0.544987i
$801$	0	0
$802$	− 96.3992i	− 0.120199i
$803$	0	0
$804$	0	0
$805$	570.948i	0.709252i
$806$	4.30943	0.00534668
$807$	0	0
$808$	−276.904	−0.342703
$809$	− 458.253i	− 0.566444i	−0.959054	−	0.283222i	$-0.908597\pi$
$809$	− 458.253i	− 0.566444i	0.959054	−	0.283222i	$-0.0914033\pi$
$810$	0	0
$811$	1486.30i	1.83267i	0.400407	+	0.916337i	$0.368869\pi$
$811$	1486.30i	1.83267i	−0.400407	+	0.916337i	$0.631131\pi$
$812$	−132.090	−0.162672
$813$	0	0
$814$	0	0
$815$	1533.24	1.88127
$816$	0	0
$817$	122.454	0.149883
$818$	−135.265	−0.165361
$819$	0	0
$820$	203.540i	0.248220i
$821$	− 1338.24i	− 1.63001i	−0.579452	−	0.815007i	$-0.696733\pi$
$821$	− 1338.24i	− 1.63001i	0.579452	−	0.815007i	$-0.303267\pi$
$822$	0	0
$823$	−1035.08	−1.25769	−0.628845	−	0.777531i	$-0.716472\pi$
$823$	−1035.08	−1.25769	−0.628845	+	0.777531i	$0.716472\pi$
$824$	6.97330i	0.00846274i
$825$	0	0
$826$	−34.5230	−0.0417954
$827$	− 903.181i	− 1.09212i	−0.837747	−	0.546058i	$-0.816128\pi$
$827$	− 903.181i	− 1.09212i	0.837747	−	0.546058i	$-0.183872\pi$
$828$	0	0
$829$	−507.692	−0.612416	−0.306208	−	0.951965i	$-0.599060\pi$
$829$	−507.692	−0.612416	−0.306208	+	0.951965i	$0.599060\pi$
$830$	341.479	0.411420
$831$	0	0
$832$	− 186.550i	− 0.224219i
$833$	− 801.838i	− 0.962590i
$834$	0	0
$835$	− 473.224i	− 0.566735i
$836$	0	0
$837$	0	0
$838$	152.130i	0.181539i
$839$	1193.32	1.42231	0.711155	−	0.703036i	$-0.248173\pi$
$839$	1193.32	1.42231	0.711155	+	0.703036i	$0.248173\pi$
$840$	0	0
$841$	829.602	0.986448
$842$	147.970i	0.175736i
$843$	0	0
$844$	18.7318i	0.0221941i
$845$	1167.22	1.38132
$846$	0	0
$847$	0	0
$848$	886.073	1.04490
$849$	0	0
$850$	146.046	0.171819
$851$	18.2447	0.0214391
$852$	0	0
$853$	1175.50i	1.37807i	0.724727	+	0.689037i	$0.241966\pi$
$853$	1175.50i	1.37807i	−0.724727	+	0.689037i	$0.758034\pi$
$854$	− 233.260i	− 0.273138i
$855$	0	0
$856$	−90.3496	−0.105549
$857$	− 921.178i	− 1.07489i	−0.843300	−	0.537444i	$-0.819390\pi$
$857$	− 921.178i	− 1.07489i	0.843300	−	0.537444i	$-0.180610\pi$
$858$	0	0
$859$	−674.852	−0.785625	−0.392813	−	0.919619i	$-0.628498\pi$
$859$	−674.852	−0.785625	−0.392813	+	0.919619i	$0.628498\pi$
$860$	115.632i	0.134456i
$861$	0	0
$862$	−79.6169	−0.0923630
$863$	659.764	0.764500	0.382250	−	0.924059i	$-0.375149\pi$
$863$	659.764	0.764500	0.382250	+	0.924059i	$0.375149\pi$
$864$	0	0
$865$	1681.89i	1.94438i
$866$	18.9822i	0.0219194i
$867$	0	0
$868$	158.165i	0.182218i
$869$	0	0
$870$	0	0
$871$	10.9847i	0.0126116i
$872$	458.960	0.526330
$873$	0	0
$874$	73.6582	0.0842772
$875$	− 370.898i	− 0.423883i
$876$	0	0
$877$	139.314i	0.158853i	0.996841	+	0.0794266i	$0.0253089\pi$
$877$	139.314i	0.158853i	−0.996841	+	0.0794266i	$0.974691\pi$
$878$	59.6521	0.0679409
$879$	0	0
$880$	0	0
$881$	28.6079	0.0324721	0.0162360	−	0.999868i	$-0.494832\pi$
$881$	28.6079	0.0324721	0.0162360	+	0.999868i	$0.494832\pi$
$882$	0	0
$883$	−757.316	−0.857663	−0.428831	−	0.903385i	$-0.641075\pi$
$883$	−757.316	−0.857663	−0.428831	+	0.903385i	$0.641075\pi$
$884$	206.709	0.233833
$885$	0	0
$886$	− 90.9634i	− 0.102667i
$887$	1002.64i	1.13037i	0.824965	+	0.565183i	$0.191195\pi$
$887$	1002.64i	1.13037i	−0.824965	+	0.565183i	$0.808805\pi$
$888$	0	0
$889$	1938.66	2.18072
$890$	− 152.953i	− 0.171858i
$891$	0	0
$892$	978.366	1.09682
$893$	1514.91i	1.69642i
$894$	0	0
$895$	−2103.78	−2.35059
$896$	−755.088	−0.842732
$897$	0	0
$898$	143.845i	0.160184i
$899$	13.6476i	0.0151809i
$900$	0	0
$901$	929.275i	1.03138i
$902$	0	0
$903$	0	0
$904$	− 456.255i	− 0.504707i
$905$	999.474	1.10439
$906$	0	0
$907$	1298.75	1.43192	0.715960	−	0.698141i	$-0.245989\pi$
$907$	1298.75	1.43192	0.715960	+	0.698141i	$0.245989\pi$
$908$	1334.30i	1.46950i
$909$	0	0
$910$	79.2646i	0.0871040i
$911$	775.053	0.850771	0.425386	−	0.905012i	$-0.360138\pi$
$911$	775.053	0.850771	0.425386	+	0.905012i	$0.360138\pi$
$912$	0	0
$913$	0	0
$914$	3.99675	0.00437281
$915$	0	0
$916$	−902.639	−0.985413
$917$	−1690.18	−1.84316
$918$	0	0
$919$	− 108.607i	− 0.118179i	−0.998253	−	0.0590897i	$-0.981180\pi$
$919$	− 108.607i	− 0.118179i	0.998253	−	0.0590897i	$-0.0188198\pi$
$920$	140.857i	0.153105i
$921$	0	0
$922$	34.2314	0.0371273
$923$	398.022i	0.431226i
$924$	0	0
$925$	−71.2524	−0.0770296
$926$	− 131.203i	− 0.141688i
$927$	0	0
$928$	−49.0832	−0.0528914
$929$	−303.150	−0.326319	−0.163159	−	0.986600i	$-0.552168\pi$
$929$	−303.150	−0.326319	−0.163159	+	0.986600i	$0.552168\pi$
$930$	0	0
$931$	1579.36i	1.69641i
$932$	333.230i	0.357543i
$933$	0	0
$934$	− 143.624i	− 0.153773i
$935$	0	0
$936$	0	0
$937$	− 355.654i	− 0.379567i	−0.981826	−	0.189783i	$-0.939221\pi$
$937$	− 355.654i	− 0.379567i	0.981826	−	0.189783i	$-0.0607786\pi$
$938$	10.1258	0.0107951
$939$	0	0
$940$	−1430.51	−1.52182
$941$	− 1250.48i	− 1.32889i	−0.747338	−	0.664444i	$-0.768668\pi$
$941$	− 1250.48i	− 1.32889i	0.747338	−	0.664444i	$-0.231332\pi$
$942$	0	0
$943$	− 54.0151i	− 0.0572801i
$944$	163.140	0.172818
$945$	0	0
$946$	0	0
$947$	373.371	0.394267	0.197134	−	0.980377i	$-0.436837\pi$
$947$	373.371	0.394267	0.197134	+	0.980377i	$0.436837\pi$
$948$	0	0
$949$	63.7552	0.0671814
$950$	−287.664	−0.302804
$951$	0	0
$952$	− 385.879i	− 0.405335i
$953$	− 659.076i	− 0.691580i	−0.938312	−	0.345790i	$-0.887611\pi$
$953$	− 659.076i	− 0.691580i	0.938312	−	0.345790i	$-0.112389\pi$
$954$	0	0
$955$	738.919	0.773737
$956$	46.1616i	0.0482862i
$957$	0	0
$958$	−77.8833	−0.0812979
$959$	− 0.145508i	0 0.000151729i
$960$	0	0
$961$	−944.658	−0.982995
$962$	2.53290	0.00263296
$963$	0	0
$964$	922.926i	0.957392i
$965$	− 1563.13i	− 1.61983i
$966$	0	0
$967$	− 1521.95i	− 1.57389i	−0.617021	−	0.786947i	$-0.711661\pi$
$967$	− 1521.95i	− 1.57389i	0.617021	−	0.786947i	$-0.288339\pi$
$968$	0	0
$969$	0	0
$970$	− 150.514i	− 0.155169i
$971$	−997.740	−1.02754	−0.513769	−	0.857929i	$-0.671751\pi$
$971$	−997.740	−1.02754	−0.513769	+	0.857929i	$0.671751\pi$
$972$	0	0
$973$	−1187.15	−1.22010
$974$	− 43.4651i	− 0.0446254i
$975$	0	0
$976$	1102.28i	1.12939i
$977$	−1730.28	−1.77101	−0.885505	−	0.464630i	$-0.846187\pi$
$977$	−1730.28	−1.77101	−0.885505	+	0.464630i	$0.846187\pi$
$978$	0	0
$979$	0	0
$980$	−1491.38	−1.52181
$981$	0	0
$982$	−69.4456	−0.0707185
$983$	−187.430	−0.190671	−0.0953355	−	0.995445i	$-0.530392\pi$
$983$	−187.430	−0.190671	−0.0953355	+	0.995445i	$0.530392\pi$
$984$	0	0
$985$	− 379.175i	− 0.384949i
$986$	− 16.4417i	− 0.0166752i
$987$	0	0
$988$	−407.150	−0.412095
$989$	− 30.6863i	− 0.0310276i
$990$	0	0
$991$	1229.91	1.24108	0.620540	−	0.784175i	$-0.286913\pi$
$991$	1229.91	1.24108	0.620540	+	0.784175i	$0.286913\pi$
$992$	58.7726i	0.0592466i
$993$	0	0
$994$	366.901	0.369116
$995$	573.631	0.576514
$996$	0	0
$997$	− 1873.95i	− 1.87959i	−0.341737	−	0.939796i	$-0.611015\pi$
$997$	− 1873.95i	− 1.87959i	0.341737	−	0.939796i	$-0.388985\pi$
$998$	235.314i	0.235785i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.8		16
3.2	odd	2		363.3.c.e.241.9		16
11.2	odd	10		99.3.k.c.73.3		16
11.5	even	5		99.3.k.c.19.3		16
11.10	odd	2	inner	1089.3.c.m.604.9		16
33.2	even	10		33.3.g.a.7.2	✓	16
33.5	odd	10		33.3.g.a.19.2	yes	16
33.8	even	10		363.3.g.g.112.2		16
33.14	odd	10		363.3.g.a.112.3		16
33.17	even	10		363.3.g.f.118.3		16
33.20	odd	10		363.3.g.f.40.3		16
33.26	odd	10		363.3.g.g.94.2		16
33.29	even	10		363.3.g.a.94.3		16
33.32	even	2		363.3.c.e.241.8		16
132.35	odd	10		528.3.bf.b.337.2		16
132.71	even	10		528.3.bf.b.481.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.2	✓	16	33.2	even	10
33.3.g.a.19.2	yes	16	33.5	odd	10
99.3.k.c.19.3		16	11.5	even	5
99.3.k.c.73.3		16	11.2	odd	10
363.3.c.e.241.8		16	33.32	even	2
363.3.c.e.241.9		16	3.2	odd	2
363.3.g.a.94.3		16	33.29	even	10
363.3.g.a.112.3		16	33.14	odd	10
363.3.g.f.40.3		16	33.20	odd	10
363.3.g.f.118.3		16	33.17	even	10
363.3.g.g.94.2		16	33.26	odd	10
363.3.g.g.112.2		16	33.8	even	10
528.3.bf.b.337.2		16	132.35	odd	10
528.3.bf.b.481.2		16	132.71	even	10
1089.3.c.m.604.8		16	1.1	even	1	trivial
1089.3.c.m.604.9		16	11.10	odd	2	inner

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.313054i q^{2} +3.90200 q^{4} +7.41540 q^{5} -10.0271i q^{7} -2.47375i q^{8} +O(q^{10})\)	\(q-0.313054i q^{2} +3.90200 q^{4} +7.41540 q^{5} -10.0271i q^{7} -2.47375i q^{8} -2.32142i q^{10} -3.40527i q^{13} -3.13902 q^{14} +14.8336 q^{16} +15.5568i q^{17} -30.6419i q^{19} +28.9349 q^{20} -7.67868 q^{23} +29.9882 q^{25} -1.06603 q^{26} -39.1257i q^{28} -3.37604i q^{29} -4.04250 q^{31} -14.5387i q^{32} +4.87012 q^{34} -74.3549i q^{35} -2.37601 q^{37} -9.59256 q^{38} -18.3438i q^{40} +7.03443i q^{41} +3.99630i q^{43} +2.40384i q^{46} -49.4390 q^{47} -51.5425 q^{49} -9.38791i q^{50} -13.2873i q^{52} +59.7343 q^{53} -24.8045 q^{56} -1.05688 q^{58} +10.9980 q^{59} +74.3100i q^{61} +1.26552i q^{62} +54.7829 q^{64} -25.2514i q^{65} -3.22579 q^{67} +60.7027i q^{68} -23.2771 q^{70} -116.884 q^{71} +18.7225i q^{73} +0.743820i q^{74} -119.565i q^{76} +3.51539i q^{79} +109.997 q^{80} +2.20215 q^{82} +147.099i q^{83} +115.360i q^{85} +1.25105 q^{86} +65.8879 q^{89} -34.1449 q^{91} -29.9622 q^{92} +15.4771i q^{94} -227.222i q^{95} +64.8371 q^{97} +16.1356i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

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