LMFDB - Embedded newform 1089.4.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2530799963$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 121)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46410 q^{2} -1.92820 q^{4} +1.53590 q^{5} -28.2487 q^{7} +24.4641 q^{8} +O(q^{10})$	$q-2.46410 q^{2} -1.92820 q^{4} +1.53590 q^{5} -28.2487 q^{7} +24.4641 q^{8} -3.78461 q^{10} +68.4641 q^{13} +69.6077 q^{14} -44.8564 q^{16} -55.3538 q^{17} -55.1769 q^{19} -2.96152 q^{20} +178.315 q^{23} -122.641 q^{25} -168.703 q^{26} +54.4693 q^{28} -113.172 q^{29} +70.8128 q^{31} -85.1821 q^{32} +136.397 q^{34} -43.3872 q^{35} -210.664 q^{37} +135.962 q^{38} +37.5744 q^{40} -191.928 q^{41} -208.210 q^{43} -439.387 q^{46} -512.515 q^{47} +454.990 q^{49} +302.200 q^{50} -132.013 q^{52} +375.449 q^{53} -691.079 q^{56} +278.867 q^{58} +506.508 q^{59} -468.697 q^{61} -174.490 q^{62} +568.749 q^{64} +105.154 q^{65} -289.895 q^{67} +106.733 q^{68} +106.910 q^{70} +394.010 q^{71} +289.538 q^{73} +519.098 q^{74} +106.392 q^{76} -169.587 q^{79} -68.8949 q^{80} +472.931 q^{82} +303.331 q^{83} -85.0179 q^{85} +513.051 q^{86} +1146.68 q^{89} -1934.02 q^{91} -343.828 q^{92} +1262.89 q^{94} -84.7461 q^{95} +641.600 q^{97} -1121.14 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 10 q^{4} + 10 q^{5} - 8 q^{7} + 42 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 - 8 * q^7 + 42 * q^8	$ 2 q + 2 q^{2} + 10 q^{4} + 10 q^{5} - 8 q^{7} + 42 q^{8} + 34 q^{10} + 130 q^{13} + 160 q^{14} - 62 q^{16} + 14 q^{17} - 48 q^{19} + 98 q^{20} + 128 q^{23} - 176 q^{25} + 106 q^{26} + 296 q^{28} + 30 q^{29} - 184 q^{31} - 302 q^{32} + 446 q^{34} + 128 q^{35} + 126 q^{37} + 168 q^{38} + 186 q^{40} - 370 q^{41} - 264 q^{43} - 664 q^{46} - 256 q^{47} + 522 q^{49} + 64 q^{50} + 602 q^{52} + 162 q^{53} - 336 q^{56} + 918 q^{58} + 1304 q^{59} - 300 q^{61} - 1312 q^{62} - 262 q^{64} + 626 q^{65} - 656 q^{67} + 934 q^{68} + 872 q^{70} + 1176 q^{71} - 668 q^{73} + 2022 q^{74} + 192 q^{76} + 416 q^{79} - 214 q^{80} - 322 q^{82} + 960 q^{83} + 502 q^{85} + 264 q^{86} + 1074 q^{89} - 688 q^{91} - 944 q^{92} + 2408 q^{94} - 24 q^{95} - 338 q^{97} - 822 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 - 8 * q^7 + 42 * q^8 + 34 * q^10 + 130 * q^13 + 160 * q^14 - 62 * q^16 + 14 * q^17 - 48 * q^19 + 98 * q^20 + 128 * q^23 - 176 * q^25 + 106 * q^26 + 296 * q^28 + 30 * q^29 - 184 * q^31 - 302 * q^32 + 446 * q^34 + 128 * q^35 + 126 * q^37 + 168 * q^38 + 186 * q^40 - 370 * q^41 - 264 * q^43 - 664 * q^46 - 256 * q^47 + 522 * q^49 + 64 * q^50 + 602 * q^52 + 162 * q^53 - 336 * q^56 + 918 * q^58 + 1304 * q^59 - 300 * q^61 - 1312 * q^62 - 262 * q^64 + 626 * q^65 - 656 * q^67 + 934 * q^68 + 872 * q^70 + 1176 * q^71 - 668 * q^73 + 2022 * q^74 + 192 * q^76 + 416 * q^79 - 214 * q^80 - 322 * q^82 + 960 * q^83 + 502 * q^85 + 264 * q^86 + 1074 * q^89 - 688 * q^91 - 944 * q^92 + 2408 * q^94 - 24 * q^95 - 338 * q^97 - 822 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.46410	−0.871191	−0.435596	−	0.900142i	$-0.643462\pi$
$2$	−2.46410	−0.871191	−0.435596	+	0.900142i	$0.643462\pi$
$3$	0	0
$3$	0	0
$4$	−1.92820	−0.241025
$5$	1.53590	0.137375	0.0686875	−	0.997638i	$-0.478119\pi$
$5$	1.53590	0.137375	0.0686875	+	0.997638i	$0.478119\pi$
$6$	0	0
$7$	−28.2487	−1.52529	−0.762644	−	0.646819i	$-0.776099\pi$
$7$	−28.2487	−1.52529	−0.762644	+	0.646819i	$0.776099\pi$
$8$	24.4641	1.08117
$9$	0	0
$10$	−3.78461	−0.119680
$11$	0	0
$11$	0	0
$12$	0	0
$13$	68.4641	1.46066	0.730328	−	0.683097i	$-0.239367\pi$
$13$	68.4641	1.46066	0.730328	+	0.683097i	$0.239367\pi$
$14$	69.6077	1.32882
$15$	0	0
$16$	−44.8564	−0.700881
$17$	−55.3538	−0.789722	−0.394861	−	0.918741i	$-0.629207\pi$
$17$	−55.3538	−0.789722	−0.394861	+	0.918741i	$0.629207\pi$
$18$	0	0
$19$	−55.1769	−0.666234	−0.333117	−	0.942885i	$-0.608100\pi$
$19$	−55.1769	−0.666234	−0.333117	+	0.942885i	$0.608100\pi$
$20$	−2.96152	−0.0331108
$21$	0	0
$22$	0	0
$23$	178.315	1.61658	0.808290	−	0.588785i	$-0.200394\pi$
$23$	178.315	1.61658	0.808290	+	0.588785i	$0.200394\pi$
$24$	0	0
$25$	−122.641	−0.981128
$26$	−168.703	−1.27251
$27$	0	0
$28$	54.4693	0.367633
$29$	−113.172	−0.724671	−0.362336	−	0.932048i	$-0.618021\pi$
$29$	−113.172	−0.724671	−0.362336	+	0.932048i	$0.618021\pi$
$30$	0	0
$31$	70.8128	0.410269	0.205135	−	0.978734i	$-0.434237\pi$
$31$	70.8128	0.410269	0.205135	+	0.978734i	$0.434237\pi$
$32$	−85.1821	−0.470569
$33$	0	0
$34$	136.397	0.687999
$35$	−43.3872	−0.209536
$36$	0	0
$37$	−210.664	−0.936026	−0.468013	−	0.883722i	$-0.655030\pi$
$37$	−210.664	−0.936026	−0.468013	+	0.883722i	$0.655030\pi$
$38$	135.962	0.580418
$39$	0	0
$40$	37.5744	0.148526
$41$	−191.928	−0.731077	−0.365538	−	0.930796i	$-0.619115\pi$
$41$	−191.928	−0.731077	−0.365538	+	0.930796i	$0.619115\pi$
$42$	0	0
$43$	−208.210	−0.738413	−0.369207	−	0.929347i	$-0.620371\pi$
$43$	−208.210	−0.738413	−0.369207	+	0.929347i	$0.620371\pi$
$44$	0	0
$45$	0	0
$46$	−439.387	−1.40835
$47$	−512.515	−1.59060	−0.795298	−	0.606218i	$-0.792686\pi$
$47$	−512.515	−1.59060	−0.795298	+	0.606218i	$0.792686\pi$
$48$	0	0
$49$	454.990	1.32650
$50$	302.200	0.854750
$51$	0	0
$52$	−132.013	−0.352055
$53$	375.449	0.973054	0.486527	−	0.873666i	$-0.338264\pi$
$53$	375.449	0.973054	0.486527	+	0.873666i	$0.338264\pi$
$54$	0	0
$55$	0	0
$56$	−691.079	−1.64910
$57$	0	0
$58$	278.867	0.631327
$59$	506.508	1.11766	0.558828	−	0.829284i	$-0.311251\pi$
$59$	506.508	1.11766	0.558828	+	0.829284i	$0.311251\pi$
$60$	0	0
$61$	−468.697	−0.983779	−0.491890	−	0.870658i	$-0.663694\pi$
$61$	−468.697	−0.983779	−0.491890	+	0.870658i	$0.663694\pi$
$62$	−174.490	−0.357423
$63$	0	0
$64$	568.749	1.11084
$65$	105.154	0.200657
$66$	0	0
$67$	−289.895	−0.528601	−0.264301	−	0.964440i	$-0.585141\pi$
$67$	−289.895	−0.528601	−0.264301	+	0.964440i	$0.585141\pi$
$68$	106.733	0.190343
$69$	0	0
$70$	106.910	0.182546
$71$	394.010	0.658597	0.329299	−	0.944226i	$-0.393188\pi$
$71$	394.010	0.658597	0.329299	+	0.944226i	$0.393188\pi$
$72$	0	0
$73$	289.538	0.464218	0.232109	−	0.972690i	$-0.425437\pi$
$73$	289.538	0.464218	0.232109	+	0.972690i	$0.425437\pi$
$74$	519.098	0.815458
$75$	0	0
$76$	106.392	0.160579
$77$	0	0
$78$	0	0
$79$	−169.587	−0.241519	−0.120760	−	0.992682i	$-0.538533\pi$
$79$	−169.587	−0.241519	−0.120760	+	0.992682i	$0.538533\pi$
$80$	−68.8949	−0.0962835
$81$	0	0
$82$	472.931	0.636908
$83$	303.331	0.401143	0.200572	−	0.979679i	$-0.435720\pi$
$83$	303.331	0.401143	0.200572	+	0.979679i	$0.435720\pi$
$84$	0	0
$85$	−85.0179	−0.108488
$86$	513.051	0.643299
$87$	0	0
$88$	0	0
$89$	1146.68	1.36571	0.682854	−	0.730555i	$-0.260738\pi$
$89$	1146.68	1.36571	0.682854	+	0.730555i	$0.260738\pi$
$90$	0	0
$91$	−1934.02	−2.22792
$92$	−343.828	−0.389637
$93$	0	0
$94$	1262.89	1.38571
$95$	−84.7461	−0.0915239
$96$	0	0
$97$	641.600	0.671594	0.335797	−	0.941934i	$-0.390994\pi$
$97$	641.600	0.671594	0.335797	+	0.941934i	$0.390994\pi$
$98$	−1121.14	−1.15564
$99$	0	0
$100$	236.477	0.236477
$101$	−1107.57	−1.09116	−0.545580	−	0.838058i	$-0.683691\pi$
$101$	−1107.57	−1.09116	−0.545580	+	0.838058i	$0.683691\pi$
$102$	0	0
$103$	−298.297	−0.285360	−0.142680	−	0.989769i	$-0.545572\pi$
$103$	−298.297	−0.285360	−0.142680	+	0.989769i	$0.545572\pi$
$104$	1674.91	1.57922
$105$	0	0
$106$	−925.144	−0.847716
$107$	598.126	0.540402	0.270201	−	0.962804i	$-0.412910\pi$
$107$	598.126	0.540402	0.270201	+	0.962804i	$0.412910\pi$
$108$	0	0
$109$	−1530.16	−1.34461	−0.672305	−	0.740275i	$-0.734695\pi$
$109$	−1530.16	−1.34461	−0.672305	+	0.740275i	$0.734695\pi$
$110$	0	0
$111$	0	0
$112$	1267.14	1.06905
$113$	151.010	0.125716	0.0628578	−	0.998022i	$-0.479979\pi$
$113$	151.010	0.125716	0.0628578	+	0.998022i	$0.479979\pi$
$114$	0	0
$115$	273.874	0.222077
$116$	218.218	0.174664
$117$	0	0
$118$	−1248.09	−0.973692
$119$	1563.67	1.20455
$120$	0	0
$121$	0	0
$122$	1154.92	0.857060
$123$	0	0
$124$	−136.541	−0.0988853
$125$	−380.351	−0.272157
$126$	0	0
$127$	−695.749	−0.486124	−0.243062	−	0.970011i	$-0.578152\pi$
$127$	−695.749	−0.486124	−0.243062	+	0.970011i	$0.578152\pi$
$128$	−719.998	−0.497183
$129$	0	0
$130$	−259.110	−0.174811
$131$	−1665.68	−1.11093	−0.555463	−	0.831541i	$-0.687459\pi$
$131$	−1665.68	−1.11093	−0.555463	+	0.831541i	$0.687459\pi$
$132$	0	0
$133$	1558.68	1.01620
$134$	714.330	0.460513
$135$	0	0
$136$	−1354.18	−0.853824
$137$	1605.48	1.00121	0.500603	−	0.865677i	$-0.333112\pi$
$137$	1605.48	1.00121	0.500603	+	0.865677i	$0.333112\pi$
$138$	0	0
$139$	−1069.30	−0.652495	−0.326248	−	0.945284i	$-0.605784\pi$
$139$	−1069.30	−0.652495	−0.326248	+	0.945284i	$0.605784\pi$
$140$	83.6592	0.0505035
$141$	0	0
$142$	−970.881	−0.573765
$143$	0	0
$144$	0	0
$145$	−173.820	−0.0995517
$146$	−713.452	−0.404423
$147$	0	0
$148$	406.203	0.225606
$149$	355.172	0.195281	0.0976403	−	0.995222i	$-0.468871\pi$
$149$	355.172	0.195281	0.0976403	+	0.995222i	$0.468871\pi$
$150$	0	0
$151$	1879.55	1.01295	0.506476	−	0.862254i	$-0.330948\pi$
$151$	1879.55	1.01295	0.506476	+	0.862254i	$0.330948\pi$
$152$	−1349.85	−0.720313
$153$	0	0
$154$	0	0
$155$	108.761	0.0563607
$156$	0	0
$157$	2499.99	1.27083	0.635417	−	0.772169i	$-0.280828\pi$
$157$	2499.99	1.27083	0.635417	+	0.772169i	$0.280828\pi$
$158$	417.880	0.210410
$159$	0	0
$160$	−130.831	−0.0646444
$161$	−5037.18	−2.46575
$162$	0	0
$163$	1863.02	0.895235	0.447617	−	0.894225i	$-0.352273\pi$
$163$	1863.02	0.895235	0.447617	+	0.894225i	$0.352273\pi$
$164$	370.077	0.176208
$165$	0	0
$166$	−747.438	−0.349473
$167$	2647.27	1.22666	0.613328	−	0.789828i	$-0.289830\pi$
$167$	2647.27	1.22666	0.613328	+	0.789828i	$0.289830\pi$
$168$	0	0
$169$	2490.33	1.13352
$170$	209.493	0.0945138
$171$	0	0
$172$	401.472	0.177976
$173$	2109.38	0.927015	0.463507	−	0.886093i	$-0.346591\pi$
$173$	2109.38	0.927015	0.463507	+	0.886093i	$0.346591\pi$
$174$	0	0
$175$	3464.45	1.49650
$176$	0	0
$177$	0	0
$178$	−2825.54	−1.18979
$179$	−1391.25	−0.580931	−0.290465	−	0.956886i	$-0.593810\pi$
$179$	−1391.25	−0.580931	−0.290465	+	0.956886i	$0.593810\pi$
$180$	0	0
$181$	3701.40	1.52002	0.760008	−	0.649913i	$-0.225195\pi$
$181$	3701.40	1.52002	0.760008	+	0.649913i	$0.225195\pi$
$182$	4765.63	1.94094
$183$	0	0
$184$	4362.32	1.74780
$185$	−323.559	−0.128586
$186$	0	0
$187$	0	0
$188$	988.234	0.383374
$189$	0	0
$190$	208.823	0.0797348
$191$	3533.03	1.33844	0.669218	−	0.743066i	$-0.266629\pi$
$191$	3533.03	1.33844	0.669218	+	0.743066i	$0.266629\pi$
$192$	0	0
$193$	2605.66	0.971811	0.485906	−	0.874011i	$-0.338490\pi$
$193$	2605.66	0.971811	0.485906	+	0.874011i	$0.338490\pi$
$194$	−1580.97	−0.585087
$195$	0	0
$196$	−877.313	−0.319720
$197$	719.202	0.260107	0.130053	−	0.991507i	$-0.458485\pi$
$197$	719.202	0.260107	0.130053	+	0.991507i	$0.458485\pi$
$198$	0	0
$199$	1035.15	0.368744	0.184372	−	0.982857i	$-0.440975\pi$
$199$	1035.15	0.368744	0.184372	+	0.982857i	$0.440975\pi$
$200$	−3000.30	−1.06077
$201$	0	0
$202$	2729.16	0.950610
$203$	3196.96	1.10533
$204$	0	0
$205$	−294.782	−0.100432
$206$	735.035	0.248604
$207$	0	0
$208$	−3071.05	−1.02375
$209$	0	0
$210$	0	0
$211$	356.297	0.116249	0.0581244	−	0.998309i	$-0.481488\pi$
$211$	356.297	0.116249	0.0581244	+	0.998309i	$0.481488\pi$
$212$	−723.941	−0.234531
$213$	0	0
$214$	−1473.84	−0.470793
$215$	−319.790	−0.101439
$216$	0	0
$217$	−2000.37	−0.625779
$218$	3770.46	1.17141
$219$	0	0
$220$	0	0
$221$	−3789.75	−1.15351
$222$	0	0
$223$	−292.544	−0.0878485	−0.0439242	−	0.999035i	$-0.513986\pi$
$223$	−292.544	−0.0878485	−0.0439242	+	0.999035i	$0.513986\pi$
$224$	2406.28	0.717753
$225$	0	0
$226$	−372.105	−0.109522
$227$	5604.04	1.63856	0.819280	−	0.573394i	$-0.194374\pi$
$227$	5604.04	1.63856	0.819280	+	0.573394i	$0.194374\pi$
$228$	0	0
$229$	−5654.38	−1.63167	−0.815833	−	0.578287i	$-0.803721\pi$
$229$	−5654.38	−1.63167	−0.815833	+	0.578287i	$0.803721\pi$
$230$	−674.854	−0.193472
$231$	0	0
$232$	−2768.65	−0.783493
$233$	−2553.08	−0.717845	−0.358923	−	0.933367i	$-0.616856\pi$
$233$	−2553.08	−0.717845	−0.358923	+	0.933367i	$0.616856\pi$
$234$	0	0
$235$	−787.171	−0.218508
$236$	−976.650	−0.269383
$237$	0	0
$238$	−3853.05	−1.04940
$239$	5297.27	1.43369	0.716845	−	0.697233i	$-0.245586\pi$
$239$	5297.27	1.43369	0.716845	+	0.697233i	$0.245586\pi$
$240$	0	0
$241$	4145.14	1.10793	0.553966	−	0.832539i	$-0.313114\pi$
$241$	4145.14	1.10793	0.553966	+	0.832539i	$0.313114\pi$
$242$	0	0
$243$	0	0
$244$	903.744	0.237116
$245$	698.818	0.182228
$246$	0	0
$247$	−3777.64	−0.973139
$248$	1732.37	0.443571
$249$	0	0
$250$	937.225	0.237101
$251$	1788.13	0.449665	0.224832	−	0.974397i	$-0.427817\pi$
$251$	1788.13	0.449665	0.224832	+	0.974397i	$0.427817\pi$
$252$	0	0
$253$	0	0
$254$	1714.40	0.423507
$255$	0	0
$256$	−2775.84	−0.677696
$257$	−5167.01	−1.25412	−0.627061	−	0.778970i	$-0.715742\pi$
$257$	−5167.01	−1.25412	−0.627061	+	0.778970i	$0.715742\pi$
$258$	0	0
$259$	5950.99	1.42771
$260$	−202.758	−0.0483636
$261$	0	0
$262$	4104.41	0.967829
$263$	57.6791	0.0135234	0.00676169	−	0.999977i	$-0.497848\pi$
$263$	57.6791	0.0135234	0.00676169	+	0.999977i	$0.497848\pi$
$264$	0	0
$265$	576.651	0.133673
$266$	−3840.74	−0.885304
$267$	0	0
$268$	558.976	0.127406
$269$	3028.06	0.686335	0.343167	−	0.939274i	$-0.388500\pi$
$269$	3028.06	0.686335	0.343167	+	0.939274i	$0.388500\pi$
$270$	0	0
$271$	−1487.84	−0.333504	−0.166752	−	0.985999i	$-0.553328\pi$
$271$	−1487.84	−0.333504	−0.166752	+	0.985999i	$0.553328\pi$
$272$	2482.97	0.553501
$273$	0	0
$274$	−3956.06	−0.872241
$275$	0	0
$276$	0	0
$277$	7460.46	1.61825	0.809126	−	0.587635i	$-0.199941\pi$
$277$	7460.46	1.61825	0.809126	+	0.587635i	$0.199941\pi$
$278$	2634.86	0.568448
$279$	0	0
$280$	−1061.43	−0.226544
$281$	−900.155	−0.191099	−0.0955493	−	0.995425i	$-0.530461\pi$
$281$	−900.155	−0.191099	−0.0955493	+	0.995425i	$0.530461\pi$
$282$	0	0
$283$	6486.92	1.36257	0.681285	−	0.732018i	$-0.261422\pi$
$283$	6486.92	1.36257	0.681285	+	0.732018i	$0.261422\pi$
$284$	−759.732	−0.158739
$285$	0	0
$286$	0	0
$287$	5421.72	1.11510
$288$	0	0
$289$	−1848.95	−0.376339
$290$	428.311	0.0867286
$291$	0	0
$292$	−558.289	−0.111888
$293$	−6129.38	−1.22212	−0.611062	−	0.791583i	$-0.709257\pi$
$293$	−6129.38	−1.22212	−0.611062	+	0.791583i	$0.709257\pi$
$294$	0	0
$295$	777.944	0.153538
$296$	−5153.71	−1.01200
$297$	0	0
$298$	−875.179	−0.170127
$299$	12208.2	2.36127
$300$	0	0
$301$	5881.67	1.12629
$302$	−4631.41	−0.882476
$303$	0	0
$304$	2475.04	0.466951
$305$	−719.872	−0.135147
$306$	0	0
$307$	5377.67	0.999740	0.499870	−	0.866101i	$-0.333381\pi$
$307$	5377.67	0.999740	0.499870	+	0.866101i	$0.333381\pi$
$308$	0	0
$309$	0	0
$310$	−267.999	−0.0491010
$311$	6066.41	1.10609	0.553046	−	0.833151i	$-0.313465\pi$
$311$	6066.41	1.10609	0.553046	+	0.833151i	$0.313465\pi$
$312$	0	0
$313$	3241.18	0.585311	0.292655	−	0.956218i	$-0.405461\pi$
$313$	3241.18	0.585311	0.292655	+	0.956218i	$0.405461\pi$
$314$	−6160.23	−1.10714
$315$	0	0
$316$	326.998	0.0582123
$317$	−6519.32	−1.15508	−0.577542	−	0.816361i	$-0.695988\pi$
$317$	−6519.32	−1.15508	−0.577542	+	0.816361i	$0.695988\pi$
$318$	0	0
$319$	0	0
$320$	873.540	0.152601
$321$	0	0
$322$	12412.1	2.14814
$323$	3054.25	0.526140
$324$	0	0
$325$	−8396.51	−1.43309
$326$	−4590.68	−0.779921
$327$	0	0
$328$	−4695.35	−0.790419
$329$	14477.9	2.42612
$330$	0	0
$331$	5879.55	0.976343	0.488171	−	0.872748i	$-0.337664\pi$
$331$	5879.55	0.976343	0.488171	+	0.872748i	$0.337664\pi$
$332$	−584.883	−0.0966857
$333$	0	0
$334$	−6523.13	−1.06865
$335$	−445.249	−0.0726166
$336$	0	0
$337$	1342.66	0.217031	0.108516	−	0.994095i	$-0.465390\pi$
$337$	1342.66	0.217031	0.108516	+	0.994095i	$0.465390\pi$
$338$	−6136.43	−0.987509
$339$	0	0
$340$	163.932	0.0261484
$341$	0	0
$342$	0	0
$343$	−3163.56	−0.498007
$344$	−5093.68	−0.798351
$345$	0	0
$346$	−5197.74	−0.807607
$347$	5650.03	0.874091	0.437046	−	0.899439i	$-0.356025\pi$
$347$	5650.03	0.874091	0.437046	+	0.899439i	$0.356025\pi$
$348$	0	0
$349$	−1249.55	−0.191653	−0.0958266	−	0.995398i	$-0.530549\pi$
$349$	−1249.55	−0.191653	−0.0958266	+	0.995398i	$0.530549\pi$
$350$	−8536.76	−1.30374
$351$	0	0
$352$	0	0
$353$	−5984.25	−0.902293	−0.451147	−	0.892450i	$-0.648985\pi$
$353$	−5984.25	−0.902293	−0.451147	+	0.892450i	$0.648985\pi$
$354$	0	0
$355$	605.160	0.0904748
$356$	−2211.04	−0.329170
$357$	0	0
$358$	3428.17	0.506102
$359$	2176.01	0.319904	0.159952	−	0.987125i	$-0.448866\pi$
$359$	2176.01	0.319904	0.159952	+	0.987125i	$0.448866\pi$
$360$	0	0
$361$	−3814.51	−0.556132
$362$	−9120.63	−1.32423
$363$	0	0
$364$	3729.19	0.536985
$365$	444.701	0.0637719
$366$	0	0
$367$	−8464.39	−1.20392	−0.601958	−	0.798528i	$-0.705613\pi$
$367$	−8464.39	−1.20392	−0.601958	+	0.798528i	$0.705613\pi$
$368$	−7998.59	−1.13303
$369$	0	0
$370$	797.281	0.112023
$371$	−10605.9	−1.48419
$372$	0	0
$373$	−4248.93	−0.589816	−0.294908	−	0.955526i	$-0.595289\pi$
$373$	−4248.93	−0.589816	−0.294908	+	0.955526i	$0.595289\pi$
$374$	0	0
$375$	0	0
$376$	−12538.2	−1.71971
$377$	−7748.20	−1.05850
$378$	0	0
$379$	4852.93	0.657727	0.328863	−	0.944378i	$-0.393334\pi$
$379$	4852.93	0.657727	0.328863	+	0.944378i	$0.393334\pi$
$380$	163.408	0.0220596
$381$	0	0
$382$	−8705.75	−1.16603
$383$	8181.81	1.09157	0.545785	−	0.837926i	$-0.316232\pi$
$383$	8181.81	1.09157	0.545785	+	0.837926i	$0.316232\pi$
$384$	0	0
$385$	0	0
$386$	−6420.61	−0.846634
$387$	0	0
$388$	−1237.13	−0.161871
$389$	−1254.82	−0.163552	−0.0817761	−	0.996651i	$-0.526059\pi$
$389$	−1254.82	−0.163552	−0.0817761	+	0.996651i	$0.526059\pi$
$390$	0	0
$391$	−9870.44	−1.27665
$392$	11130.9	1.43417
$393$	0	0
$394$	−1772.19	−0.226603
$395$	−260.469	−0.0331787
$396$	0	0
$397$	−11519.3	−1.45627	−0.728133	−	0.685436i	$-0.759612\pi$
$397$	−11519.3	−1.45627	−0.728133	+	0.685436i	$0.759612\pi$
$398$	−2550.72	−0.321247
$399$	0	0
$400$	5501.24	0.687654
$401$	1500.66	0.186882	0.0934409	−	0.995625i	$-0.470213\pi$
$401$	1500.66	0.186882	0.0934409	+	0.995625i	$0.470213\pi$
$402$	0	0
$403$	4848.13	0.599262
$404$	2135.62	0.262998
$405$	0	0
$406$	−7877.63	−0.962956
$407$	0	0
$408$	0	0
$409$	91.3936	0.0110492	0.00552460	−	0.999985i	$-0.498241\pi$
$409$	91.3936	0.0110492	0.00552460	+	0.999985i	$0.498241\pi$
$410$	726.373	0.0874952
$411$	0	0
$412$	575.178	0.0687791
$413$	−14308.2	−1.70475
$414$	0	0
$415$	465.885	0.0551070
$416$	−5831.91	−0.687339
$417$	0	0
$418$	0	0
$419$	1880.83	0.219295	0.109648	−	0.993971i	$-0.465028\pi$
$419$	1880.83	0.219295	0.109648	+	0.993971i	$0.465028\pi$
$420$	0	0
$421$	−7279.83	−0.842748	−0.421374	−	0.906887i	$-0.638452\pi$
$421$	−7279.83	−0.842748	−0.421374	+	0.906887i	$0.638452\pi$
$422$	−877.951	−0.101275
$423$	0	0
$424$	9185.01	1.05204
$425$	6788.65	0.774819
$426$	0	0
$427$	13240.1	1.50055
$428$	−1153.31	−0.130251
$429$	0	0
$430$	787.994	0.0883732
$431$	−6870.61	−0.767855	−0.383928	−	0.923363i	$-0.625429\pi$
$431$	−6870.61	−0.767855	−0.383928	+	0.923363i	$0.625429\pi$
$432$	0	0
$433$	1121.32	0.124451	0.0622255	−	0.998062i	$-0.480180\pi$
$433$	1121.32	0.124451	0.0622255	+	0.998062i	$0.480180\pi$
$434$	4929.11	0.545173
$435$	0	0
$436$	2950.45	0.324085
$437$	−9838.89	−1.07702
$438$	0	0
$439$	7114.94	0.773525	0.386763	−	0.922179i	$-0.373593\pi$
$439$	7114.94	0.773525	0.386763	+	0.922179i	$0.373593\pi$
$440$	0	0
$441$	0	0
$442$	9338.33	1.00493
$443$	14057.4	1.50764	0.753822	−	0.657079i	$-0.228208\pi$
$443$	14057.4	1.50764	0.753822	+	0.657079i	$0.228208\pi$
$444$	0	0
$445$	1761.19	0.187614
$446$	720.859	0.0765328
$447$	0	0
$448$	−16066.4	−1.69435
$449$	15323.9	1.61064	0.805320	−	0.592840i	$-0.201993\pi$
$449$	15323.9	1.61064	0.805320	+	0.592840i	$0.201993\pi$
$450$	0	0
$451$	0	0
$452$	−291.179	−0.0303006
$453$	0	0
$454$	−13808.9	−1.42750
$455$	−2970.46	−0.306060
$456$	0	0
$457$	−7212.20	−0.738233	−0.369117	−	0.929383i	$-0.620340\pi$
$457$	−7212.20	−0.738233	−0.369117	+	0.929383i	$0.620340\pi$
$458$	13933.0	1.42149
$459$	0	0
$460$	−528.085	−0.0535263
$461$	10159.6	1.02642	0.513212	−	0.858262i	$-0.328456\pi$
$461$	10159.6	1.02642	0.513212	+	0.858262i	$0.328456\pi$
$462$	0	0
$463$	−10292.0	−1.03306	−0.516532	−	0.856268i	$-0.672777\pi$
$463$	−10292.0	−1.03306	−0.516532	+	0.856268i	$0.672777\pi$
$464$	5076.48	0.507909
$465$	0	0
$466$	6291.05	0.625380
$467$	−18023.0	−1.78588	−0.892938	−	0.450180i	$-0.851360\pi$
$467$	−18023.0	−1.78588	−0.892938	+	0.450180i	$0.851360\pi$
$468$	0	0
$469$	8189.16	0.806269
$470$	1939.67	0.190362
$471$	0	0
$472$	12391.3	1.20838
$473$	0	0
$474$	0	0
$475$	6766.95	0.653661
$476$	−3015.08	−0.290328
$477$	0	0
$478$	−13053.0	−1.24902
$479$	−17083.8	−1.62960	−0.814801	−	0.579741i	$-0.803154\pi$
$479$	−17083.8	−1.62960	−0.814801	+	0.579741i	$0.803154\pi$
$480$	0	0
$481$	−14422.9	−1.36721
$482$	−10214.0	−0.965221
$483$	0	0
$484$	0	0
$485$	985.432	0.0922601
$486$	0	0
$487$	461.804	0.0429699	0.0214850	−	0.999769i	$-0.493161\pi$
$487$	461.804	0.0429699	0.0214850	+	0.999769i	$0.493161\pi$
$488$	−11466.3	−1.06363
$489$	0	0
$490$	−1721.96	−0.158755
$491$	12542.7	1.15284	0.576422	−	0.817152i	$-0.304448\pi$
$491$	12542.7	1.15284	0.576422	+	0.817152i	$0.304448\pi$
$492$	0	0
$493$	6264.49	0.572289
$494$	9308.48	0.847790
$495$	0	0
$496$	−3176.41	−0.287550
$497$	−11130.3	−1.00455
$498$	0	0
$499$	−18277.0	−1.63966	−0.819829	−	0.572608i	$-0.805932\pi$
$499$	−18277.0	−1.63966	−0.819829	+	0.572608i	$0.805932\pi$
$500$	733.395	0.0655968
$501$	0	0
$502$	−4406.14	−0.391744
$503$	2655.18	0.235365	0.117683	−	0.993051i	$-0.462453\pi$
$503$	2655.18	0.235365	0.117683	+	0.993051i	$0.462453\pi$
$504$	0	0
$505$	−1701.11	−0.149898
$506$	0	0
$507$	0	0
$508$	1341.54	0.117168
$509$	4887.16	0.425579	0.212789	−	0.977098i	$-0.431745\pi$
$509$	4887.16	0.425579	0.212789	+	0.977098i	$0.431745\pi$
$510$	0	0
$511$	−8179.08	−0.708065
$512$	12599.9	1.08759
$513$	0	0
$514$	12732.0	1.09258
$515$	−458.155	−0.0392014
$516$	0	0
$517$	0	0
$518$	−14663.8	−1.24381
$519$	0	0
$520$	2572.50	0.216945
$521$	21941.1	1.84502	0.922512	−	0.385969i	$-0.126133\pi$
$521$	21941.1	1.84502	0.922512	+	0.385969i	$0.126133\pi$
$522$	0	0
$523$	−6102.28	−0.510199	−0.255099	−	0.966915i	$-0.582108\pi$
$523$	−6102.28	−0.510199	−0.255099	+	0.966915i	$0.582108\pi$
$524$	3211.77	0.267761
$525$	0	0
$526$	−142.127	−0.0117814
$527$	−3919.76	−0.323999
$528$	0	0
$529$	19629.4	1.61333
$530$	−1420.93	−0.116455
$531$	0	0
$532$	−3005.45	−0.244930
$533$	−13140.2	−1.06785
$534$	0	0
$535$	918.660	0.0742377
$536$	−7092.02	−0.571508
$537$	0	0
$538$	−7461.44	−0.597929
$539$	0	0
$540$	0	0
$541$	−2099.35	−0.166836	−0.0834179	−	0.996515i	$-0.526584\pi$
$541$	−2099.35	−0.166836	−0.0834179	+	0.996515i	$0.526584\pi$
$542$	3666.18	0.290546
$543$	0	0
$544$	4715.15	0.371619
$545$	−2350.16	−0.184716
$546$	0	0
$547$	−9029.06	−0.705767	−0.352884	−	0.935667i	$-0.614799\pi$
$547$	−9029.06	−0.705767	−0.352884	+	0.935667i	$0.614799\pi$
$548$	−3095.68	−0.241316
$549$	0	0
$550$	0	0
$551$	6244.47	0.482801
$552$	0	0
$553$	4790.62	0.368386
$554$	−18383.3	−1.40981
$555$	0	0
$556$	2061.83	0.157268
$557$	−7894.54	−0.600543	−0.300271	−	0.953854i	$-0.597077\pi$
$557$	−7894.54	−0.600543	−0.300271	+	0.953854i	$0.597077\pi$
$558$	0	0
$559$	−14254.9	−1.07857
$560$	1946.19	0.146860
$561$	0	0
$562$	2218.07	0.166484
$563$	22377.6	1.67514	0.837569	−	0.546332i	$-0.183976\pi$
$563$	22377.6	1.67514	0.837569	+	0.546332i	$0.183976\pi$
$564$	0	0
$565$	231.936	0.0172702
$566$	−15984.4	−1.18706
$567$	0	0
$568$	9639.11	0.712056
$569$	−16920.5	−1.24665	−0.623325	−	0.781963i	$-0.714219\pi$
$569$	−16920.5	−1.24665	−0.623325	+	0.781963i	$0.714219\pi$
$570$	0	0
$571$	−16320.0	−1.19609	−0.598047	−	0.801461i	$-0.704056\pi$
$571$	−16320.0	−1.19609	−0.598047	+	0.801461i	$0.704056\pi$
$572$	0	0
$573$	0	0
$574$	−13359.7	−0.971467
$575$	−21868.8	−1.58607
$576$	0	0
$577$	830.262	0.0599034	0.0299517	−	0.999551i	$-0.490465\pi$
$577$	830.262	0.0599034	0.0299517	+	0.999551i	$0.490465\pi$
$578$	4556.01	0.327863
$579$	0	0
$580$	335.161	0.0239945
$581$	−8568.70	−0.611858
$582$	0	0
$583$	0	0
$584$	7083.29	0.501899
$585$	0	0
$586$	15103.4	1.06470
$587$	−21919.4	−1.54124	−0.770621	−	0.637294i	$-0.780054\pi$
$587$	−21919.4	−1.54124	−0.770621	+	0.637294i	$0.780054\pi$
$588$	0	0
$589$	−3907.23	−0.273336
$590$	−1916.93	−0.133761
$591$	0	0
$592$	9449.63	0.656043
$593$	−8236.51	−0.570376	−0.285188	−	0.958472i	$-0.592056\pi$
$593$	−8236.51	−0.570376	−0.285188	+	0.958472i	$0.592056\pi$
$594$	0	0
$595$	2401.64	0.165475
$596$	−684.843	−0.0470676
$597$	0	0
$598$	−30082.2	−2.05711
$599$	10922.0	0.745009	0.372505	−	0.928030i	$-0.378499\pi$
$599$	10922.0	0.745009	0.372505	+	0.928030i	$0.378499\pi$
$600$	0	0
$601$	1386.44	0.0940997	0.0470498	−	0.998893i	$-0.485018\pi$
$601$	1386.44	0.0940997	0.0470498	+	0.998893i	$0.485018\pi$
$602$	−14493.0	−0.981216
$603$	0	0
$604$	−3624.16	−0.244147
$605$	0	0
$606$	0	0
$607$	−1417.78	−0.0948035	−0.0474017	−	0.998876i	$-0.515094\pi$
$607$	−1417.78	−0.0948035	−0.0474017	+	0.998876i	$0.515094\pi$
$608$	4700.08	0.313509
$609$	0	0
$610$	1773.84	0.117739
$611$	−35088.9	−2.32331
$612$	0	0
$613$	15424.9	1.01632	0.508162	−	0.861261i	$-0.330325\pi$
$613$	15424.9	1.01632	0.508162	+	0.861261i	$0.330325\pi$
$614$	−13251.1	−0.870965
$615$	0	0
$616$	0	0
$617$	−15169.5	−0.989793	−0.494897	−	0.868952i	$-0.664794\pi$
$617$	−15169.5	−0.989793	−0.494897	+	0.868952i	$0.664794\pi$
$618$	0	0
$619$	−2081.56	−0.135162	−0.0675809	−	0.997714i	$-0.521528\pi$
$619$	−2081.56	−0.135162	−0.0675809	+	0.997714i	$0.521528\pi$
$620$	−209.714	−0.0135844
$621$	0	0
$622$	−14948.3	−0.963618
$623$	−32392.3	−2.08310
$624$	0	0
$625$	14745.9	0.943741
$626$	−7986.59	−0.509918
$627$	0	0
$628$	−4820.49	−0.306303
$629$	11661.1	0.739200
$630$	0	0
$631$	25249.7	1.59298	0.796492	−	0.604649i	$-0.206687\pi$
$631$	25249.7	1.59298	0.796492	+	0.604649i	$0.206687\pi$
$632$	−4148.80	−0.261124
$633$	0	0
$634$	16064.3	1.00630
$635$	−1068.60	−0.0667812
$636$	0	0
$637$	31150.5	1.93756
$638$	0	0
$639$	0	0
$640$	−1105.84	−0.0683005
$641$	2626.57	0.161846	0.0809231	−	0.996720i	$-0.474213\pi$
$641$	2626.57	0.161846	0.0809231	+	0.996720i	$0.474213\pi$
$642$	0	0
$643$	9229.61	0.566066	0.283033	−	0.959110i	$-0.408659\pi$
$643$	9229.61	0.566066	0.283033	+	0.959110i	$0.408659\pi$
$644$	9712.70	0.594308
$645$	0	0
$646$	−7525.99	−0.458369
$647$	−316.901	−0.0192561	−0.00962803	−	0.999954i	$-0.503065\pi$
$647$	−316.901	−0.0192561	−0.00962803	+	0.999954i	$0.503065\pi$
$648$	0	0
$649$	0	0
$650$	20689.8	1.24850
$651$	0	0
$652$	−3592.29	−0.215774
$653$	5022.66	0.300998	0.150499	−	0.988610i	$-0.451912\pi$
$653$	5022.66	0.300998	0.150499	+	0.988610i	$0.451912\pi$
$654$	0	0
$655$	−2558.32	−0.152613
$656$	8609.21	0.512398
$657$	0	0
$658$	−35675.0	−2.11361
$659$	24927.5	1.47350	0.736752	−	0.676163i	$-0.236358\pi$
$659$	24927.5	1.47350	0.736752	+	0.676163i	$0.236358\pi$
$660$	0	0
$661$	−16440.5	−0.967418	−0.483709	−	0.875229i	$-0.660711\pi$
$661$	−16440.5	−0.967418	−0.483709	+	0.875229i	$0.660711\pi$
$662$	−14487.8	−0.850581
$663$	0	0
$664$	7420.72	0.433704
$665$	2393.97	0.139600
$666$	0	0
$667$	−20180.3	−1.17149
$668$	−5104.47	−0.295655
$669$	0	0
$670$	1097.14	0.0632630
$671$	0	0
$672$	0	0
$673$	−11777.4	−0.674572	−0.337286	−	0.941402i	$-0.609509\pi$
$673$	−11777.4	−0.674572	−0.337286	+	0.941402i	$0.609509\pi$
$674$	−3308.46	−0.189076
$675$	0	0
$676$	−4801.87	−0.273206
$677$	2818.49	0.160005	0.0800025	−	0.996795i	$-0.474507\pi$
$677$	2818.49	0.160005	0.0800025	+	0.996795i	$0.474507\pi$
$678$	0	0
$679$	−18124.4	−1.02437
$680$	−2079.89	−0.117294
$681$	0	0
$682$	0	0
$683$	−15803.2	−0.885346	−0.442673	−	0.896683i	$-0.645970\pi$
$683$	−15803.2	−0.885346	−0.442673	+	0.896683i	$0.645970\pi$
$684$	0	0
$685$	2465.85	0.137540
$686$	7795.34	0.433860
$687$	0	0
$688$	9339.56	0.517540
$689$	25704.8	1.42130
$690$	0	0
$691$	3300.72	0.181716	0.0908578	−	0.995864i	$-0.471039\pi$
$691$	3300.72	0.181716	0.0908578	+	0.995864i	$0.471039\pi$
$692$	−4067.32	−0.223434
$693$	0	0
$694$	−13922.3	−0.761501
$695$	−1642.34	−0.0896365
$696$	0	0
$697$	10624.0	0.577348
$698$	3079.02	0.166967
$699$	0	0
$700$	−6680.16	−0.360695
$701$	29773.2	1.60416	0.802082	−	0.597214i	$-0.203726\pi$
$701$	29773.2	1.60416	0.802082	+	0.597214i	$0.203726\pi$
$702$	0	0
$703$	11623.8	0.623612
$704$	0	0
$705$	0	0
$706$	14745.8	0.786070
$707$	31287.4	1.66433
$708$	0	0
$709$	−24002.5	−1.27141	−0.635707	−	0.771931i	$-0.719291\pi$
$709$	−24002.5	−1.27141	−0.635707	+	0.771931i	$0.719291\pi$
$710$	−1491.18	−0.0788209
$711$	0	0
$712$	28052.5	1.47656
$713$	12627.0	0.663233
$714$	0	0
$715$	0	0
$716$	2682.60	0.140019
$717$	0	0
$718$	−5361.91	−0.278697
$719$	20668.7	1.07206	0.536032	−	0.844198i	$-0.319923\pi$
$719$	20668.7	1.07206	0.536032	+	0.844198i	$0.319923\pi$
$720$	0	0
$721$	8426.52	0.435257
$722$	9399.34	0.484497
$723$	0	0
$724$	−7137.06	−0.366363
$725$	13879.5	0.710995
$726$	0	0
$727$	21928.9	1.11870	0.559351	−	0.828931i	$-0.311050\pi$
$727$	21928.9	1.11870	0.559351	+	0.828931i	$0.311050\pi$
$728$	−47314.1	−2.40876
$729$	0	0
$730$	−1095.79	−0.0555575
$731$	11525.2	0.583141
$732$	0	0
$733$	−25124.0	−1.26600	−0.633000	−	0.774152i	$-0.718177\pi$
$733$	−25124.0	−1.26600	−0.633000	+	0.774152i	$0.718177\pi$
$734$	20857.1	1.04884
$735$	0	0
$736$	−15189.3	−0.760712
$737$	0	0
$738$	0	0
$739$	37038.1	1.84367	0.921833	−	0.387588i	$-0.126692\pi$
$739$	37038.1	1.84367	0.921833	+	0.387588i	$0.126692\pi$
$740$	623.887	0.0309926
$741$	0	0
$742$	26134.1	1.29301
$743$	24798.0	1.22443	0.612213	−	0.790693i	$-0.290279\pi$
$743$	24798.0	1.22443	0.612213	+	0.790693i	$0.290279\pi$
$744$	0	0
$745$	545.508	0.0268267
$746$	10469.8	0.513843
$747$	0	0
$748$	0	0
$749$	−16896.3	−0.824268
$750$	0	0
$751$	13967.2	0.678654	0.339327	−	0.940668i	$-0.389801\pi$
$751$	13967.2	0.678654	0.339327	+	0.940668i	$0.389801\pi$
$752$	22989.6	1.11482
$753$	0	0
$754$	19092.4	0.922152
$755$	2886.80	0.139154
$756$	0	0
$757$	−8515.45	−0.408850	−0.204425	−	0.978882i	$-0.565532\pi$
$757$	−8515.45	−0.408850	−0.204425	+	0.978882i	$0.565532\pi$
$758$	−11958.1	−0.573006
$759$	0	0
$760$	−2073.24	−0.0989530
$761$	31658.0	1.50802	0.754009	−	0.656865i	$-0.228118\pi$
$761$	31658.0	1.50802	0.754009	+	0.656865i	$0.228118\pi$
$762$	0	0
$763$	43224.9	2.05091
$764$	−6812.40	−0.322597
$765$	0	0
$766$	−20160.8	−0.950966
$767$	34677.6	1.63251
$768$	0	0
$769$	−606.519	−0.0284416	−0.0142208	−	0.999899i	$-0.504527\pi$
$769$	−606.519	−0.0284416	−0.0142208	+	0.999899i	$0.504527\pi$
$770$	0	0
$771$	0	0
$772$	−5024.24	−0.234231
$773$	4699.76	0.218679	0.109339	−	0.994004i	$-0.465126\pi$
$773$	4699.76	0.218679	0.109339	+	0.994004i	$0.465126\pi$
$774$	0	0
$775$	−8684.55	−0.402527
$776$	15696.2	0.726107
$777$	0	0
$778$	3092.00	0.142485
$779$	10590.0	0.487068
$780$	0	0
$781$	0	0
$782$	24321.8	1.11221
$783$	0	0
$784$	−20409.2	−0.929720
$785$	3839.73	0.174581
$786$	0	0
$787$	24078.1	1.09059	0.545293	−	0.838245i	$-0.316418\pi$
$787$	24078.1	1.09059	0.545293	+	0.838245i	$0.316418\pi$
$788$	−1386.77	−0.0626923
$789$	0	0
$790$	641.821	0.0289050
$791$	−4265.85	−0.191752
$792$	0	0
$793$	−32088.9	−1.43696
$794$	28384.8	1.26869
$795$	0	0
$796$	−1995.99	−0.0888767
$797$	18977.7	0.843443	0.421722	−	0.906725i	$-0.361426\pi$
$797$	18977.7	0.843443	0.421722	+	0.906725i	$0.361426\pi$
$798$	0	0
$799$	28369.7	1.25613
$800$	10446.8	0.461688
$801$	0	0
$802$	−3697.79	−0.162810
$803$	0	0
$804$	0	0
$805$	−7736.59	−0.338732
$806$	−11946.3	−0.522072
$807$	0	0
$808$	−27095.7	−1.17973
$809$	−9120.70	−0.396374	−0.198187	−	0.980164i	$-0.563505\pi$
$809$	−9120.70	−0.396374	−0.198187	+	0.980164i	$0.563505\pi$
$810$	0	0
$811$	39874.2	1.72648	0.863239	−	0.504796i	$-0.168432\pi$
$811$	39874.2	1.72648	0.863239	+	0.504796i	$0.168432\pi$
$812$	−6164.38	−0.266413
$813$	0	0
$814$	0	0
$815$	2861.41	0.122983
$816$	0	0
$817$	11488.4	0.491956
$818$	−225.203	−0.00962597
$819$	0	0
$820$	568.400	0.0242066
$821$	−4913.94	−0.208889	−0.104444	−	0.994531i	$-0.533306\pi$
$821$	−4913.94	−0.208889	−0.104444	+	0.994531i	$0.533306\pi$
$822$	0	0
$823$	2777.58	0.117643	0.0588215	−	0.998269i	$-0.481266\pi$
$823$	2777.58	0.117643	0.0588215	+	0.998269i	$0.481266\pi$
$824$	−7297.58	−0.308523
$825$	0	0
$826$	35256.8	1.48516
$827$	−20306.5	−0.853842	−0.426921	−	0.904289i	$-0.640402\pi$
$827$	−20306.5	−0.853842	−0.426921	+	0.904289i	$0.640402\pi$
$828$	0	0
$829$	−35572.7	−1.49034	−0.745170	−	0.666875i	$-0.767632\pi$
$829$	−35572.7	−1.49034	−0.745170	+	0.666875i	$0.767632\pi$
$830$	−1147.99	−0.0480088
$831$	0	0
$832$	38938.9	1.62255
$833$	−25185.4	−1.04757
$834$	0	0
$835$	4065.93	0.168512
$836$	0	0
$837$	0	0
$838$	−4634.56	−0.191048
$839$	−7096.72	−0.292021	−0.146011	−	0.989283i	$-0.546643\pi$
$839$	−7096.72	−0.292021	−0.146011	+	0.989283i	$0.546643\pi$
$840$	0	0
$841$	−11581.2	−0.474851
$842$	17938.2	0.734195
$843$	0	0
$844$	−687.013	−0.0280189
$845$	3824.90	0.155717
$846$	0	0
$847$	0	0
$848$	−16841.3	−0.681995
$849$	0	0
$850$	−16727.9	−0.675015
$851$	−37564.6	−1.51316
$852$	0	0
$853$	24157.6	0.969682	0.484841	−	0.874602i	$-0.338877\pi$
$853$	24157.6	0.969682	0.484841	+	0.874602i	$0.338877\pi$
$854$	−32624.9	−1.30726
$855$	0	0
$856$	14632.6	0.584267
$857$	−28806.8	−1.14822	−0.574108	−	0.818779i	$-0.694651\pi$
$857$	−28806.8	−1.14822	−0.574108	+	0.818779i	$0.694651\pi$
$858$	0	0
$859$	11244.4	0.446628	0.223314	−	0.974747i	$-0.428312\pi$
$859$	11244.4	0.446628	0.223314	+	0.974747i	$0.428312\pi$
$860$	616.620	0.0244495
$861$	0	0
$862$	16929.9	0.668949
$863$	1291.92	0.0509589	0.0254794	−	0.999675i	$-0.491889\pi$
$863$	1291.92	0.0509589	0.0254794	+	0.999675i	$0.491889\pi$
$864$	0	0
$865$	3239.80	0.127349
$866$	−2763.05	−0.108421
$867$	0	0
$868$	3857.12	0.150829
$869$	0	0
$870$	0	0
$871$	−19847.4	−0.772105
$872$	−37433.9	−1.45375
$873$	0	0
$874$	24244.0	0.938291
$875$	10744.4	0.415118
$876$	0	0
$877$	26823.1	1.03278	0.516391	−	0.856353i	$-0.327275\pi$
$877$	26823.1	1.03278	0.516391	+	0.856353i	$0.327275\pi$
$878$	−17531.9	−0.673889
$879$	0	0
$880$	0	0
$881$	28515.7	1.09049	0.545243	−	0.838278i	$-0.316437\pi$
$881$	28515.7	1.09049	0.545243	+	0.838278i	$0.316437\pi$
$882$	0	0
$883$	41686.4	1.58874	0.794371	−	0.607433i	$-0.207801\pi$
$883$	41686.4	1.58874	0.794371	+	0.607433i	$0.207801\pi$
$884$	7307.41	0.278026
$885$	0	0
$886$	−34638.8	−1.31345
$887$	−49179.3	−1.86164	−0.930822	−	0.365473i	$-0.880907\pi$
$887$	−49179.3	−1.86164	−0.930822	+	0.365473i	$0.880907\pi$
$888$	0	0
$889$	19654.0	0.741478
$890$	−4339.74	−0.163448
$891$	0	0
$892$	564.085	0.0211737
$893$	28279.0	1.05971
$894$	0	0
$895$	−2136.81	−0.0798053
$896$	20339.0	0.758346
$897$	0	0
$898$	−37759.5	−1.40318
$899$	−8014.01	−0.297310
$900$	0	0
$901$	−20782.5	−0.768442
$902$	0	0
$903$	0	0
$904$	3694.33	0.135920
$905$	5684.98	0.208812
$906$	0	0
$907$	52977.8	1.93947	0.969734	−	0.244163i	$-0.0785133\pi$
$907$	52977.8	1.93947	0.969734	+	0.244163i	$0.0785133\pi$
$908$	−10805.7	−0.394934
$909$	0	0
$910$	7319.52	0.266637
$911$	31469.8	1.14450	0.572251	−	0.820078i	$-0.306070\pi$
$911$	31469.8	1.14450	0.572251	+	0.820078i	$0.306070\pi$
$912$	0	0
$913$	0	0
$914$	17771.6	0.643143
$915$	0	0
$916$	10902.8	0.393273
$917$	47053.4	1.69448
$918$	0	0
$919$	−42860.9	−1.53847	−0.769234	−	0.638967i	$-0.779362\pi$
$919$	−42860.9	−1.53847	−0.769234	+	0.638967i	$0.779362\pi$
$920$	6700.09	0.240104
$921$	0	0
$922$	−25034.4	−0.894211
$923$	26975.6	0.961984
$924$	0	0
$925$	25836.1	0.918361
$926$	25360.5	0.899996
$927$	0	0
$928$	9640.20	0.341008
$929$	20968.3	0.740525	0.370262	−	0.928927i	$-0.379268\pi$
$929$	20968.3	0.740525	0.370262	+	0.928927i	$0.379268\pi$
$930$	0	0
$931$	−25104.9	−0.883760
$932$	4922.86	0.173019
$933$	0	0
$934$	44410.4	1.55584
$935$	0	0
$936$	0	0
$937$	17126.8	0.597127	0.298563	−	0.954390i	$-0.403493\pi$
$937$	17126.8	0.597127	0.298563	+	0.954390i	$0.403493\pi$
$938$	−20178.9	−0.702415
$939$	0	0
$940$	1517.83	0.0526660
$941$	44021.2	1.52503	0.762513	−	0.646973i	$-0.223966\pi$
$941$	44021.2	1.52503	0.762513	+	0.646973i	$0.223966\pi$
$942$	0	0
$943$	−34223.7	−1.18184
$944$	−22720.1	−0.783344
$945$	0	0
$946$	0	0
$947$	−8692.03	−0.298261	−0.149130	−	0.988818i	$-0.547647\pi$
$947$	−8692.03	−0.298261	−0.149130	+	0.988818i	$0.547647\pi$
$948$	0	0
$949$	19823.0	0.678062
$950$	−16674.5	−0.569464
$951$	0	0
$952$	38253.9	1.30233
$953$	−57906.0	−1.96827	−0.984133	−	0.177431i	$-0.943221\pi$
$953$	−57906.0	−1.96827	−0.984133	+	0.177431i	$0.943221\pi$
$954$	0	0
$955$	5426.38	0.183867
$956$	−10214.2	−0.345556
$957$	0	0
$958$	42096.2	1.41969
$959$	−45352.6	−1.52713
$960$	0	0
$961$	−24776.6	−0.831679
$962$	35539.5	1.19110
$963$	0	0
$964$	−7992.67	−0.267040
$965$	4002.03	0.133503
$966$	0	0
$967$	−56564.3	−1.88106	−0.940530	−	0.339711i	$-0.889671\pi$
$967$	−56564.3	−1.88106	−0.940530	+	0.339711i	$0.889671\pi$
$968$	0	0
$969$	0	0
$970$	−2428.20	−0.0803762
$971$	30894.7	1.02107	0.510534	−	0.859857i	$-0.329448\pi$
$971$	30894.7	1.02107	0.510534	+	0.859857i	$0.329448\pi$
$972$	0	0
$973$	30206.3	0.995242
$974$	−1137.93	−0.0374350
$975$	0	0
$976$	21024.1	0.689513
$977$	29998.9	0.982342	0.491171	−	0.871063i	$-0.336569\pi$
$977$	29998.9	0.982342	0.491171	+	0.871063i	$0.336569\pi$
$978$	0	0
$979$	0	0
$980$	−1347.46	−0.0439216
$981$	0	0
$982$	−30906.6	−1.00435
$983$	24485.9	0.794485	0.397242	−	0.917714i	$-0.369967\pi$
$983$	24485.9	0.794485	0.397242	+	0.917714i	$0.369967\pi$
$984$	0	0
$985$	1104.62	0.0357321
$986$	−15436.3	−0.498573
$987$	0	0
$988$	7284.05	0.234551
$989$	−37127.1	−1.19370
$990$	0	0
$991$	−52661.1	−1.68803	−0.844014	−	0.536321i	$-0.819814\pi$
$991$	−52661.1	−1.68803	−0.844014	+	0.536321i	$0.819814\pi$
$992$	−6031.98	−0.193060
$993$	0	0
$994$	27426.1	0.875156
$995$	1589.89	0.0506562
$996$	0	0
$997$	−54748.6	−1.73912	−0.869561	−	0.493826i	$-0.835598\pi$
$997$	−54748.6	−1.73912	−0.869561	+	0.493826i	$0.835598\pi$
$998$	45036.3	1.42846
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.x.1.1		2
3.2	odd	2		121.4.a.b.1.2	✓	2
11.10	odd	2		1089.4.a.k.1.2		2
12.11	even	2		1936.4.a.z.1.1		2
33.2	even	10		121.4.c.d.81.2		8
33.5	odd	10		121.4.c.g.3.1		8
33.8	even	10		121.4.c.d.9.1		8
33.14	odd	10		121.4.c.g.9.2		8
33.17	even	10		121.4.c.d.3.2		8
33.20	odd	10		121.4.c.g.81.1		8
33.26	odd	10		121.4.c.g.27.2		8
33.29	even	10		121.4.c.d.27.1		8
33.32	even	2		121.4.a.e.1.1	yes	2
132.131	odd	2		1936.4.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.b.1.2	✓	2	3.2	odd	2
121.4.a.e.1.1	yes	2	33.32	even	2
121.4.c.d.3.2		8	33.17	even	10
121.4.c.d.9.1		8	33.8	even	10
121.4.c.d.27.1		8	33.29	even	10
121.4.c.d.81.2		8	33.2	even	10
121.4.c.g.3.1		8	33.5	odd	10
121.4.c.g.9.2		8	33.14	odd	10
121.4.c.g.27.2		8	33.26	odd	10
121.4.c.g.81.1		8	33.20	odd	10
1089.4.a.k.1.2		2	11.10	odd	2
1089.4.a.x.1.1		2	1.1	even	1	trivial
1936.4.a.y.1.1		2	132.131	odd	2
1936.4.a.z.1.1		2	12.11	even	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46410 q^{2} -1.92820 q^{4} +1.53590 q^{5} -28.2487 q^{7} +24.4641 q^{8} +O(q^{10})\)	\(q-2.46410 q^{2} -1.92820 q^{4} +1.53590 q^{5} -28.2487 q^{7} +24.4641 q^{8} -3.78461 q^{10} +68.4641 q^{13} +69.6077 q^{14} -44.8564 q^{16} -55.3538 q^{17} -55.1769 q^{19} -2.96152 q^{20} +178.315 q^{23} -122.641 q^{25} -168.703 q^{26} +54.4693 q^{28} -113.172 q^{29} +70.8128 q^{31} -85.1821 q^{32} +136.397 q^{34} -43.3872 q^{35} -210.664 q^{37} +135.962 q^{38} +37.5744 q^{40} -191.928 q^{41} -208.210 q^{43} -439.387 q^{46} -512.515 q^{47} +454.990 q^{49} +302.200 q^{50} -132.013 q^{52} +375.449 q^{53} -691.079 q^{56} +278.867 q^{58} +506.508 q^{59} -468.697 q^{61} -174.490 q^{62} +568.749 q^{64} +105.154 q^{65} -289.895 q^{67} +106.733 q^{68} +106.910 q^{70} +394.010 q^{71} +289.538 q^{73} +519.098 q^{74} +106.392 q^{76} -169.587 q^{79} -68.8949 q^{80} +472.931 q^{82} +303.331 q^{83} -85.0179 q^{85} +513.051 q^{86} +1146.68 q^{89} -1934.02 q^{91} -343.828 q^{92} +1262.89 q^{94} -84.7461 q^{95} +641.600 q^{97} -1121.14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 10 q^{4} + 10 q^{5} - 8 q^{7} + 42 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 - 8 * q^7 + 42 * q^8	\( 2 q + 2 q^{2} + 10 q^{4} + 10 q^{5} - 8 q^{7} + 42 q^{8} + 34 q^{10} + 130 q^{13} + 160 q^{14} - 62 q^{16} + 14 q^{17} - 48 q^{19} + 98 q^{20} + 128 q^{23} - 176 q^{25} + 106 q^{26} + 296 q^{28} + 30 q^{29} - 184 q^{31} - 302 q^{32} + 446 q^{34} + 128 q^{35} + 126 q^{37} + 168 q^{38} + 186 q^{40} - 370 q^{41} - 264 q^{43} - 664 q^{46} - 256 q^{47} + 522 q^{49} + 64 q^{50} + 602 q^{52} + 162 q^{53} - 336 q^{56} + 918 q^{58} + 1304 q^{59} - 300 q^{61} - 1312 q^{62} - 262 q^{64} + 626 q^{65} - 656 q^{67} + 934 q^{68} + 872 q^{70} + 1176 q^{71} - 668 q^{73} + 2022 q^{74} + 192 q^{76} + 416 q^{79} - 214 q^{80} - 322 q^{82} + 960 q^{83} + 502 q^{85} + 264 q^{86} + 1074 q^{89} - 688 q^{91} - 944 q^{92} + 2408 q^{94} - 24 q^{95} - 338 q^{97} - 822 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 10 * q^4 + 10 * q^5 - 8 * q^7 + 42 * q^8 + 34 * q^10 + 130 * q^13 + 160 * q^14 - 62 * q^16 + 14 * q^17 - 48 * q^19 + 98 * q^20 + 128 * q^23 - 176 * q^25 + 106 * q^26 + 296 * q^28 + 30 * q^29 - 184 * q^31 - 302 * q^32 + 446 * q^34 + 128 * q^35 + 126 * q^37 + 168 * q^38 + 186 * q^40 - 370 * q^41 - 264 * q^43 - 664 * q^46 - 256 * q^47 + 522 * q^49 + 64 * q^50 + 602 * q^52 + 162 * q^53 - 336 * q^56 + 918 * q^58 + 1304 * q^59 - 300 * q^61 - 1312 * q^62 - 262 * q^64 + 626 * q^65 - 656 * q^67 + 934 * q^68 + 872 * q^70 + 1176 * q^71 - 668 * q^73 + 2022 * q^74 + 192 * q^76 + 416 * q^79 - 214 * q^80 - 322 * q^82 + 960 * q^83 + 502 * q^85 + 264 * q^86 + 1074 * q^89 - 688 * q^91 - 944 * q^92 + 2408 * q^94 - 24 * q^95 - 338 * q^97 - 822 * q^98

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