LMFDB - Embedded newform 1143.3.b.a.890.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.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:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.20
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.65

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +O(q^{10})$	\(q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +9.67106 q^{10} -7.16818i q^{11} +20.5325 q^{13} +19.4011i q^{14} -19.9630 q^{16} +13.1195i q^{17} -14.0782 q^{19} -7.25426i q^{20} -17.2747 q^{22} -26.8517i q^{23} +8.89554 q^{25} -49.4815i q^{26} +14.5527 q^{28} +1.48580i q^{29} -47.6593 q^{31} +26.9758i q^{32} +31.6169 q^{34} -32.3071i q^{35} -50.2868 q^{37} +33.9272i q^{38} +21.2021 q^{40} -54.4030i q^{41} +38.8264 q^{43} +12.9577i q^{44} -64.7103 q^{46} +59.0925i q^{47} +15.8110 q^{49} -21.4375i q^{50} -37.1160 q^{52} +50.8215i q^{53} +28.7662 q^{55} +42.5334i q^{56} +3.58064 q^{58} -36.7929i q^{59} -111.007 q^{61} +114.855i q^{62} -14.8426 q^{64} +82.3976i q^{65} -37.8794 q^{67} -23.7158i q^{68} -77.8571 q^{70} -60.7250i q^{71} -96.3723 q^{73} +121.187i q^{74} +25.4488 q^{76} +57.7076i q^{77} -148.728 q^{79} -80.1123i q^{80} -131.106 q^{82} -100.833i q^{83} -52.6491 q^{85} -93.5681i q^{86} -37.8717 q^{88} +123.265i q^{89} -165.297 q^{91} +48.5391i q^{92} +142.408 q^{94} -56.4963i q^{95} -69.7434 q^{97} -38.1031i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 2.40991i	− 1.20496i	−0.798136	−	0.602478i	$-0.794180\pi$
$2$	− 2.40991i	− 1.20496i	0.798136	−	0.602478i	$-0.205820\pi$
$3$	0	0
$3$	0	0
$4$	−1.80767	−0.451919
$5$	4.01304i	0.802607i	0.915945	+	0.401304i	$0.131443\pi$
$5$	4.01304i	0.802607i	−0.915945	+	0.401304i	$0.868557\pi$
$6$	0	0
$7$	−8.05053	−1.15008	−0.575038	−	0.818127i	$-0.695013\pi$
$7$	−8.05053	−1.15008	−0.575038	+	0.818127i	$0.695013\pi$
$8$	− 5.28331i	− 0.660414i
$9$	0	0
$10$	9.67106	0.967106
$11$	− 7.16818i	− 0.651653i	−0.945430	−	0.325826i	$-0.894358\pi$
$11$	− 7.16818i	− 0.651653i	0.945430	−	0.325826i	$-0.105642\pi$
$12$	0	0
$13$	20.5325	1.57942	0.789711	−	0.613480i	$-0.210231\pi$
$13$	20.5325	1.57942	0.789711	+	0.613480i	$0.210231\pi$
$14$	19.4011i	1.38579i
$15$	0	0
$16$	−19.9630	−1.24769
$17$	13.1195i	0.771737i	0.922554	+	0.385869i	$0.126098\pi$
$17$	13.1195i	0.771737i	−0.922554	+	0.385869i	$0.873902\pi$
$18$	0	0
$19$	−14.0782	−0.740957	−0.370478	−	0.928841i	$-0.620806\pi$
$19$	−14.0782	−0.740957	−0.370478	+	0.928841i	$0.620806\pi$
$20$	− 7.25426i	− 0.362713i
$21$	0	0
$22$	−17.2747	−0.785213
$23$	− 26.8517i	− 1.16747i	−0.811946	−	0.583733i	$-0.801592\pi$
$23$	− 26.8517i	− 1.16747i	0.811946	−	0.583733i	$-0.198408\pi$
$24$	0	0
$25$	8.89554	0.355822
$26$	− 49.4815i	− 1.90313i
$27$	0	0
$28$	14.5527	0.519740
$29$	1.48580i	0.0512344i	0.999672	+	0.0256172i	$0.00815510\pi$
$29$	1.48580i	0.0512344i	−0.999672	+	0.0256172i	$0.991845\pi$
$30$	0	0
$31$	−47.6593	−1.53740	−0.768698	−	0.639612i	$-0.779095\pi$
$31$	−47.6593	−1.53740	−0.768698	+	0.639612i	$0.779095\pi$
$32$	26.9758i	0.842995i
$33$	0	0
$34$	31.6169	0.929909
$35$	− 32.3071i	− 0.923059i
$36$	0	0
$37$	−50.2868	−1.35910	−0.679551	−	0.733628i	$-0.737826\pi$
$37$	−50.2868	−1.35910	−0.679551	+	0.733628i	$0.737826\pi$
$38$	33.9272i	0.892820i
$39$	0	0
$40$	21.2021	0.530053
$41$	− 54.4030i	− 1.32690i	−0.748220	−	0.663451i	$-0.769091\pi$
$41$	− 54.4030i	− 1.32690i	0.748220	−	0.663451i	$-0.230909\pi$
$42$	0	0
$43$	38.8264	0.902939	0.451469	−	0.892287i	$-0.350900\pi$
$43$	38.8264	0.902939	0.451469	+	0.892287i	$0.350900\pi$
$44$	12.9577i	0.294494i
$45$	0	0
$46$	−64.7103	−1.40674
$47$	59.0925i	1.25729i	0.777694	+	0.628643i	$0.216389\pi$
$47$	59.0925i	1.25729i	−0.777694	+	0.628643i	$0.783611\pi$
$48$	0	0
$49$	15.8110	0.322673
$50$	− 21.4375i	− 0.428749i
$51$	0	0
$52$	−37.1160	−0.713770
$53$	50.8215i	0.958896i	0.877570	+	0.479448i	$0.159163\pi$
$53$	50.8215i	0.958896i	−0.877570	+	0.479448i	$0.840837\pi$
$54$	0	0
$55$	28.7662	0.523021
$56$	42.5334i	0.759526i
$57$	0	0
$58$	3.58064	0.0617352
$59$	− 36.7929i	− 0.623609i	−0.950146	−	0.311805i	$-0.899067\pi$
$59$	− 36.7929i	− 0.623609i	0.950146	−	0.311805i	$-0.100933\pi$
$60$	0	0
$61$	−111.007	−1.81978	−0.909889	−	0.414851i	$-0.863834\pi$
$61$	−111.007	−1.81978	−0.909889	+	0.414851i	$0.863834\pi$
$62$	114.855i	1.85249i
$63$	0	0
$64$	−14.8426	−0.231916
$65$	82.3976i	1.26765i
$66$	0	0
$67$	−37.8794	−0.565364	−0.282682	−	0.959214i	$-0.591224\pi$
$67$	−37.8794	−0.565364	−0.282682	+	0.959214i	$0.591224\pi$
$68$	− 23.7158i	− 0.348762i
$69$	0	0
$70$	−77.8571	−1.11224
$71$	− 60.7250i	− 0.855282i	−0.903949	−	0.427641i	$-0.859345\pi$
$71$	− 60.7250i	− 0.855282i	0.903949	−	0.427641i	$-0.140655\pi$
$72$	0	0
$73$	−96.3723	−1.32017	−0.660085	−	0.751191i	$-0.729480\pi$
$73$	−96.3723	−1.32017	−0.660085	+	0.751191i	$0.729480\pi$
$74$	121.187i	1.63766i
$75$	0	0
$76$	25.4488	0.334852
$77$	57.7076i	0.749450i
$78$	0	0
$79$	−148.728	−1.88263	−0.941315	−	0.337530i	$-0.890409\pi$
$79$	−148.728	−1.88263	−0.941315	+	0.337530i	$0.890409\pi$
$80$	− 80.1123i	− 1.00140i
$81$	0	0
$82$	−131.106	−1.59886
$83$	− 100.833i	− 1.21486i	−0.794374	−	0.607429i	$-0.792201\pi$
$83$	− 100.833i	− 1.21486i	0.794374	−	0.607429i	$-0.207799\pi$
$84$	0	0
$85$	−52.6491	−0.619402
$86$	− 93.5681i	− 1.08800i
$87$	0	0
$88$	−37.8717	−0.430361
$89$	123.265i	1.38500i	0.721417	+	0.692501i	$0.243491\pi$
$89$	123.265i	1.38500i	−0.721417	+	0.692501i	$0.756509\pi$
$90$	0	0
$91$	−165.297	−1.81645
$92$	48.5391i	0.527599i
$93$	0	0
$94$	142.408	1.51498
$95$	− 56.4963i	− 0.594697i
$96$	0	0
$97$	−69.7434	−0.719005	−0.359502	−	0.933144i	$-0.617054\pi$
$97$	−69.7434	−0.719005	−0.359502	+	0.933144i	$0.617054\pi$
$98$	− 38.1031i	− 0.388807i
$99$	0	0
$100$	−16.0802	−0.160802
$101$	69.5740i	0.688851i	0.938814	+	0.344426i	$0.111926\pi$
$101$	69.5740i	0.688851i	−0.938814	+	0.344426i	$0.888074\pi$
$102$	0	0
$103$	47.2476	0.458714	0.229357	−	0.973342i	$-0.426338\pi$
$103$	47.2476	0.458714	0.229357	+	0.973342i	$0.426338\pi$
$104$	− 108.479i	− 1.04307i
$105$	0	0
$106$	122.475	1.15543
$107$	45.4403i	0.424676i	0.977196	+	0.212338i	$0.0681078\pi$
$107$	45.4403i	0.424676i	−0.977196	+	0.212338i	$0.931892\pi$
$108$	0	0
$109$	−36.6880	−0.336587	−0.168294	−	0.985737i	$-0.553826\pi$
$109$	−36.6880	−0.336587	−0.168294	+	0.985737i	$0.553826\pi$
$110$	− 69.3239i	− 0.630217i
$111$	0	0
$112$	160.713	1.43494
$113$	− 85.2429i	− 0.754362i	−0.926140	−	0.377181i	$-0.876893\pi$
$113$	− 85.2429i	− 0.754362i	0.926140	−	0.377181i	$-0.123107\pi$
$114$	0	0
$115$	107.757	0.937016
$116$	− 2.68584i	− 0.0231538i
$117$	0	0
$118$	−88.6678	−0.751422
$119$	− 105.619i	− 0.887556i
$120$	0	0
$121$	69.6172	0.575349
$122$	267.516i	2.19275i
$123$	0	0
$124$	86.1524	0.694777
$125$	136.024i	1.08819i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	143.673i	1.12244i
$129$	0	0
$130$	198.571	1.52747
$131$	− 159.575i	− 1.21813i	−0.793120	−	0.609065i	$-0.791545\pi$
$131$	− 159.575i	− 1.21813i	0.793120	−	0.609065i	$-0.208455\pi$
$132$	0	0
$133$	113.337	0.852156
$134$	91.2860i	0.681239i
$135$	0	0
$136$	69.3146	0.509666
$137$	134.857i	0.984358i	0.870494	+	0.492179i	$0.163799\pi$
$137$	134.857i	0.984358i	−0.870494	+	0.492179i	$0.836201\pi$
$138$	0	0
$139$	160.927	1.15775	0.578874	−	0.815417i	$-0.303492\pi$
$139$	160.927	1.15775	0.578874	+	0.815417i	$0.303492\pi$
$140$	58.4006i	0.417147i
$141$	0	0
$142$	−146.342	−1.03058
$143$	− 147.180i	− 1.02923i
$144$	0	0
$145$	−5.96256	−0.0411211
$146$	232.249i	1.59075i
$147$	0	0
$148$	90.9021	0.614204
$149$	33.7999i	0.226845i	0.993547	+	0.113423i	$0.0361814\pi$
$149$	33.7999i	0.226845i	−0.993547	+	0.113423i	$0.963819\pi$
$150$	0	0
$151$	−227.193	−1.50459	−0.752294	−	0.658827i	$-0.771053\pi$
$151$	−227.193	−1.50459	−0.752294	+	0.658827i	$0.771053\pi$
$152$	74.3794i	0.489338i
$153$	0	0
$154$	139.070	0.903054
$155$	− 191.258i	− 1.23392i
$156$	0	0
$157$	−276.321	−1.76001	−0.880004	−	0.474966i	$-0.842460\pi$
$157$	−276.321	−1.76001	−0.880004	+	0.474966i	$0.842460\pi$
$158$	358.421i	2.26849i
$159$	0	0
$160$	−108.255	−0.676594
$161$	216.170i	1.34267i
$162$	0	0
$163$	96.7045	0.593279	0.296640	−	0.954990i	$-0.404134\pi$
$163$	96.7045	0.593279	0.296640	+	0.954990i	$0.404134\pi$
$164$	98.3429i	0.599652i
$165$	0	0
$166$	−242.999	−1.46385
$167$	− 285.957i	− 1.71232i	−0.516714	−	0.856158i	$-0.672845\pi$
$167$	− 285.957i	− 1.71232i	0.516714	−	0.856158i	$-0.327155\pi$
$168$	0	0
$169$	252.583	1.49457
$170$	126.880i	0.746352i
$171$	0	0
$172$	−70.1854	−0.408055
$173$	− 51.8652i	− 0.299799i	−0.988701	−	0.149900i	$-0.952105\pi$
$173$	− 51.8652i	− 0.299799i	0.988701	−	0.149900i	$-0.0478950\pi$
$174$	0	0
$175$	−71.6138	−0.409222
$176$	143.098i	0.813059i
$177$	0	0
$178$	297.058	1.66887
$179$	135.861i	0.758998i	0.925192	+	0.379499i	$0.123904\pi$
$179$	135.861i	0.758998i	−0.925192	+	0.379499i	$0.876096\pi$
$180$	0	0
$181$	206.452	1.14062	0.570309	−	0.821430i	$-0.306824\pi$
$181$	206.452	1.14062	0.570309	+	0.821430i	$0.306824\pi$
$182$	398.352i	2.18875i
$183$	0	0
$184$	−141.866	−0.771011
$185$	− 201.803i	− 1.09083i
$186$	0	0
$187$	94.0431	0.502904
$188$	− 106.820i	− 0.568191i
$189$	0	0
$190$	−136.151	−0.716584
$191$	1.82849i	0.00957327i	0.999989	+	0.00478663i	$0.00152364\pi$
$191$	1.82849i	0.00957327i	−0.999989	+	0.00478663i	$0.998476\pi$
$192$	0	0
$193$	−155.090	−0.803576	−0.401788	−	0.915733i	$-0.631611\pi$
$193$	−155.090	−0.803576	−0.401788	+	0.915733i	$0.631611\pi$
$194$	168.076i	0.866369i
$195$	0	0
$196$	−28.5811	−0.145822
$197$	− 210.777i	− 1.06993i	−0.844874	−	0.534966i	$-0.820324\pi$
$197$	− 210.777i	− 1.06993i	0.844874	−	0.534966i	$-0.179676\pi$
$198$	0	0
$199$	303.056	1.52290	0.761448	−	0.648226i	$-0.224489\pi$
$199$	303.056	1.52290	0.761448	+	0.648226i	$0.224489\pi$
$200$	− 46.9979i	− 0.234990i
$201$	0	0
$202$	167.667	0.830035
$203$	− 11.9615i	− 0.0589234i
$204$	0	0
$205$	218.321	1.06498
$206$	− 113.862i	− 0.552730i
$207$	0	0
$208$	−409.890	−1.97063
$209$	100.915i	0.482847i
$210$	0	0
$211$	79.4793	0.376679	0.188340	−	0.982104i	$-0.439689\pi$
$211$	79.4793	0.376679	0.188340	+	0.982104i	$0.439689\pi$
$212$	− 91.8687i	− 0.433343i
$213$	0	0
$214$	109.507	0.511716
$215$	155.812i	0.724705i
$216$	0	0
$217$	383.682	1.76812
$218$	88.4148i	0.405573i
$219$	0	0
$220$	−51.9998	−0.236363
$221$	269.376i	1.21890i
$222$	0	0
$223$	37.1959	0.166798	0.0833988	−	0.996516i	$-0.473422\pi$
$223$	37.1959	0.166798	0.0833988	+	0.996516i	$0.473422\pi$
$224$	− 217.170i	− 0.969508i
$225$	0	0
$226$	−205.428	−0.908973
$227$	− 67.1448i	− 0.295792i	−0.989003	−	0.147896i	$-0.952750\pi$
$227$	− 67.1448i	− 0.295792i	0.989003	−	0.147896i	$-0.0472501\pi$
$228$	0	0
$229$	20.3389	0.0888162	0.0444081	−	0.999013i	$-0.485860\pi$
$229$	20.3389	0.0888162	0.0444081	+	0.999013i	$0.485860\pi$
$230$	− 259.685i	− 1.12906i
$231$	0	0
$232$	7.84993	0.0338359
$233$	− 82.5152i	− 0.354142i	−0.984198	−	0.177071i	$-0.943338\pi$
$233$	− 82.5152i	− 0.354142i	0.984198	−	0.177071i	$-0.0566623\pi$
$234$	0	0
$235$	−237.140	−1.00911
$236$	66.5097i	0.281821i
$237$	0	0
$238$	−254.533	−1.06947
$239$	− 142.959i	− 0.598154i	−0.954229	−	0.299077i	$-0.903321\pi$
$239$	− 142.959i	− 0.598154i	0.954229	−	0.299077i	$-0.0966788\pi$
$240$	0	0
$241$	133.180	0.552616	0.276308	−	0.961069i	$-0.410889\pi$
$241$	133.180	0.552616	0.276308	+	0.961069i	$0.410889\pi$
$242$	− 167.771i	− 0.693270i
$243$	0	0
$244$	200.664	0.822392
$245$	63.4501i	0.258980i
$246$	0	0
$247$	−289.060	−1.17028
$248$	251.799i	1.01532i
$249$	0	0
$250$	327.806	1.31122
$251$	− 112.331i	− 0.447534i	−0.974643	−	0.223767i	$-0.928165\pi$
$251$	− 112.331i	− 0.447534i	0.974643	−	0.223767i	$-0.0718354\pi$
$252$	0	0
$253$	−192.478	−0.760782
$254$	27.1583i	0.106923i
$255$	0	0
$256$	286.868	1.12058
$257$	− 433.008i	− 1.68485i	−0.538810	−	0.842427i	$-0.681126\pi$
$257$	− 433.008i	− 1.68485i	0.538810	−	0.842427i	$-0.318874\pi$
$258$	0	0
$259$	404.835	1.56307
$260$	− 148.948i	− 0.572877i
$261$	0	0
$262$	−384.562	−1.46779
$263$	172.140i	0.654526i	0.944933	+	0.327263i	$0.106126\pi$
$263$	172.140i	0.654526i	−0.944933	+	0.327263i	$0.893874\pi$
$264$	0	0
$265$	−203.948	−0.769617
$266$	− 273.132i	− 1.02681i
$267$	0	0
$268$	68.4736	0.255498
$269$	− 103.072i	− 0.383167i	−0.981476	−	0.191584i	$-0.938638\pi$
$269$	− 103.072i	− 0.383167i	0.981476	−	0.191584i	$-0.0613623\pi$
$270$	0	0
$271$	197.651	0.729340	0.364670	−	0.931137i	$-0.381182\pi$
$271$	197.651	0.729340	0.364670	+	0.931137i	$0.381182\pi$
$272$	− 261.905i	− 0.962887i
$273$	0	0
$274$	324.994	1.18611
$275$	− 63.7648i	− 0.231872i
$276$	0	0
$277$	101.105	0.365000	0.182500	−	0.983206i	$-0.441581\pi$
$277$	101.105	0.365000	0.182500	+	0.983206i	$0.441581\pi$
$278$	− 387.820i	− 1.39503i
$279$	0	0
$280$	−170.688	−0.609601
$281$	380.094i	1.35265i	0.736605	+	0.676323i	$0.236428\pi$
$281$	380.094i	1.35265i	−0.736605	+	0.676323i	$0.763572\pi$
$282$	0	0
$283$	332.711	1.17566	0.587829	−	0.808986i	$-0.299983\pi$
$283$	332.711	1.17566	0.587829	+	0.808986i	$0.299983\pi$
$284$	109.771i	0.386518i
$285$	0	0
$286$	−354.692	−1.24018
$287$	437.973i	1.52604i
$288$	0	0
$289$	116.878	0.404422
$290$	14.3692i	0.0495491i
$291$	0	0
$292$	174.210	0.596609
$293$	− 551.613i	− 1.88264i	−0.337517	−	0.941319i	$-0.609587\pi$
$293$	− 551.613i	− 1.88264i	0.337517	−	0.941319i	$-0.390413\pi$
$294$	0	0
$295$	147.651	0.500513
$296$	265.681i	0.897570i
$297$	0	0
$298$	81.4548	0.273338
$299$	− 551.332i	− 1.84392i
$300$	0	0
$301$	−312.573	−1.03845
$302$	547.515i	1.81296i
$303$	0	0
$304$	281.043	0.924483
$305$	− 445.473i	− 1.46057i
$306$	0	0
$307$	−497.277	−1.61980	−0.809898	−	0.586571i	$-0.800478\pi$
$307$	−497.277	−1.61980	−0.809898	+	0.586571i	$0.800478\pi$
$308$	− 104.317i	− 0.338690i
$309$	0	0
$310$	−460.916	−1.48682
$311$	30.6422i	0.0985280i	0.998786	+	0.0492640i	$0.0156876\pi$
$311$	30.6422i	0.0985280i	−0.998786	+	0.0492640i	$0.984312\pi$
$312$	0	0
$313$	−327.233	−1.04547	−0.522737	−	0.852494i	$-0.675089\pi$
$313$	−327.233	−1.04547	−0.522737	+	0.852494i	$0.675089\pi$
$314$	665.910i	2.12073i
$315$	0	0
$316$	268.851	0.850795
$317$	− 457.919i	− 1.44454i	−0.691611	−	0.722270i	$-0.743099\pi$
$317$	− 457.919i	− 1.44454i	0.691611	−	0.722270i	$-0.256901\pi$
$318$	0	0
$319$	10.6505	0.0333870
$320$	− 59.5641i	− 0.186138i
$321$	0	0
$322$	520.952	1.61786
$323$	− 184.699i	− 0.571824i
$324$	0	0
$325$	182.648	0.561992
$326$	− 233.049i	− 0.714875i
$327$	0	0
$328$	−287.428	−0.876305
$329$	− 475.726i	− 1.44597i
$330$	0	0
$331$	391.290	1.18214	0.591072	−	0.806619i	$-0.298705\pi$
$331$	391.290	1.18214	0.591072	+	0.806619i	$0.298705\pi$
$332$	182.274i	0.549017i
$333$	0	0
$334$	−689.130	−2.06326
$335$	− 152.011i	− 0.453765i
$336$	0	0
$337$	−418.213	−1.24099	−0.620494	−	0.784212i	$-0.713068\pi$
$337$	−418.213	−1.24099	−0.620494	+	0.784212i	$0.713068\pi$
$338$	− 608.702i	− 1.80089i
$339$	0	0
$340$	95.1725	0.279919
$341$	341.630i	1.00185i
$342$	0	0
$343$	267.189	0.778977
$344$	− 205.132i	− 0.596313i
$345$	0	0
$346$	−124.991	−0.361245
$347$	436.901i	1.25908i	0.776967	+	0.629541i	$0.216757\pi$
$347$	436.901i	1.25908i	−0.776967	+	0.629541i	$0.783243\pi$
$348$	0	0
$349$	−297.279	−0.851802	−0.425901	−	0.904770i	$-0.640043\pi$
$349$	−297.279	−0.851802	−0.425901	+	0.904770i	$0.640043\pi$
$350$	172.583i	0.493094i
$351$	0	0
$352$	193.368	0.549340
$353$	380.600i	1.07819i	0.842246	+	0.539094i	$0.181233\pi$
$353$	380.600i	1.07819i	−0.842246	+	0.539094i	$0.818767\pi$
$354$	0	0
$355$	243.692	0.686455
$356$	− 222.823i	− 0.625908i
$357$	0	0
$358$	327.412	0.914559
$359$	− 72.2876i	− 0.201358i	−0.994919	−	0.100679i	$-0.967898\pi$
$359$	− 72.2876i	− 0.201358i	0.994919	−	0.100679i	$-0.0321015\pi$
$360$	0	0
$361$	−162.805	−0.450983
$362$	− 497.531i	− 1.37439i
$363$	0	0
$364$	298.804	0.820889
$365$	− 386.746i	− 1.05958i
$366$	0	0
$367$	−567.460	−1.54621	−0.773106	−	0.634277i	$-0.781298\pi$
$367$	−567.460	−1.54621	−0.773106	+	0.634277i	$0.781298\pi$
$368$	536.041i	1.45663i
$369$	0	0
$370$	−486.327	−1.31440
$371$	− 409.140i	− 1.10280i
$372$	0	0
$373$	463.835	1.24353	0.621763	−	0.783206i	$-0.286417\pi$
$373$	463.835	1.24353	0.621763	+	0.783206i	$0.286417\pi$
$374$	− 226.636i	− 0.605978i
$375$	0	0
$376$	312.204	0.830330
$377$	30.5071i	0.0809207i
$378$	0	0
$379$	−427.077	−1.12685	−0.563427	−	0.826166i	$-0.690517\pi$
$379$	−427.077	−1.12685	−0.563427	+	0.826166i	$0.690517\pi$
$380$	102.127i	0.268755i
$381$	0	0
$382$	4.40651	0.0115354
$383$	− 410.113i	− 1.07079i	−0.844602	−	0.535395i	$-0.820163\pi$
$383$	− 410.113i	− 1.07079i	0.844602	−	0.535395i	$-0.179837\pi$
$384$	0	0
$385$	−231.583	−0.601514
$386$	373.753i	0.968273i
$387$	0	0
$388$	126.073	0.324931
$389$	176.669i	0.454162i	0.973876	+	0.227081i	$0.0729182\pi$
$389$	176.669i	0.454162i	−0.973876	+	0.227081i	$0.927082\pi$
$390$	0	0
$391$	352.282	0.900977
$392$	− 83.5344i	− 0.213098i
$393$	0	0
$394$	−507.953	−1.28922
$395$	− 596.850i	− 1.51101i
$396$	0	0
$397$	599.290	1.50955	0.754773	−	0.655986i	$-0.227747\pi$
$397$	599.290	1.50955	0.754773	+	0.655986i	$0.227747\pi$
$398$	− 730.339i	− 1.83502i
$399$	0	0
$400$	−177.582	−0.443955
$401$	677.386i	1.68924i	0.535366	+	0.844620i	$0.320174\pi$
$401$	677.386i	1.68924i	−0.535366	+	0.844620i	$0.679826\pi$
$402$	0	0
$403$	−978.563	−2.42820
$404$	− 125.767i	− 0.311305i
$405$	0	0
$406$	−28.8261	−0.0710001
$407$	360.465i	0.885663i
$408$	0	0
$409$	−187.238	−0.457796	−0.228898	−	0.973450i	$-0.573512\pi$
$409$	−187.238	−0.457796	−0.228898	+	0.973450i	$0.573512\pi$
$410$	− 526.135i	− 1.28326i
$411$	0	0
$412$	−85.4082	−0.207301
$413$	296.203i	0.717198i
$414$	0	0
$415$	404.647	0.975054
$416$	553.881i	1.33144i
$417$	0	0
$418$	243.196	0.581809
$419$	− 111.510i	− 0.266135i	−0.991107	−	0.133067i	$-0.957517\pi$
$419$	− 111.510i	− 0.266135i	0.991107	−	0.133067i	$-0.0424827\pi$
$420$	0	0
$421$	3.66404	0.00870319	0.00435159	−	0.999991i	$-0.498615\pi$
$421$	3.66404	0.00870319	0.00435159	+	0.999991i	$0.498615\pi$
$422$	− 191.538i	− 0.453882i
$423$	0	0
$424$	268.506	0.633268
$425$	116.705i	0.274601i
$426$	0	0
$427$	893.661	2.09288
$428$	− 82.1413i	− 0.191919i
$429$	0	0
$430$	375.492	0.873237
$431$	256.037i	0.594054i	0.954869	+	0.297027i	$0.0959951\pi$
$431$	256.037i	0.594054i	−0.954869	+	0.297027i	$0.904005\pi$
$432$	0	0
$433$	719.675	1.66207	0.831034	−	0.556222i	$-0.187750\pi$
$433$	719.675	1.66207	0.831034	+	0.556222i	$0.187750\pi$
$434$	− 924.640i	− 2.13051i
$435$	0	0
$436$	66.3199	0.152110
$437$	378.023i	0.865042i
$438$	0	0
$439$	−341.817	−0.778625	−0.389313	−	0.921106i	$-0.627287\pi$
$439$	−341.817	−0.778625	−0.389313	+	0.921106i	$0.627287\pi$
$440$	− 151.981i	− 0.345410i
$441$	0	0
$442$	649.173	1.46872
$443$	534.762i	1.20714i	0.797311	+	0.603569i	$0.206255\pi$
$443$	534.762i	1.20714i	−0.797311	+	0.603569i	$0.793745\pi$
$444$	0	0
$445$	−494.668	−1.11161
$446$	− 89.6388i	− 0.200984i
$447$	0	0
$448$	119.491	0.266721
$449$	747.959i	1.66583i	0.553398	+	0.832917i	$0.313331\pi$
$449$	747.959i	1.66583i	−0.553398	+	0.832917i	$0.686669\pi$
$450$	0	0
$451$	−389.970	−0.864679
$452$	154.091i	0.340910i
$453$	0	0
$454$	−161.813	−0.356416
$455$	− 663.344i	− 1.45790i
$456$	0	0
$457$	588.629	1.28803	0.644015	−	0.765013i	$-0.277268\pi$
$457$	588.629	1.28803	0.644015	+	0.765013i	$0.277268\pi$
$458$	− 49.0150i	− 0.107020i
$459$	0	0
$460$	−194.789	−0.423455
$461$	− 521.154i	− 1.13049i	−0.824924	−	0.565243i	$-0.808782\pi$
$461$	− 521.154i	− 1.13049i	0.824924	−	0.565243i	$-0.191218\pi$
$462$	0	0
$463$	−752.738	−1.62578	−0.812892	−	0.582414i	$-0.802108\pi$
$463$	−752.738	−1.62578	−0.812892	+	0.582414i	$0.802108\pi$
$464$	− 29.6610i	− 0.0639246i
$465$	0	0
$466$	−198.854	−0.426726
$467$	− 867.632i	− 1.85789i	−0.370224	−	0.928943i	$-0.620719\pi$
$467$	− 867.632i	− 1.85789i	0.370224	−	0.928943i	$-0.379281\pi$
$468$	0	0
$469$	304.949	0.650211
$470$	571.487i	1.21593i
$471$	0	0
$472$	−194.389	−0.411840
$473$	− 278.314i	− 0.588402i
$474$	0	0
$475$	−125.233	−0.263649
$476$	190.925i	0.401103i
$477$	0	0
$478$	−344.518	−0.720750
$479$	− 672.423i	− 1.40381i	−0.712272	−	0.701903i	$-0.752334\pi$
$479$	− 672.423i	− 1.40381i	0.712272	−	0.701903i	$-0.247666\pi$
$480$	0	0
$481$	−1032.51	−2.14660
$482$	− 320.953i	− 0.665877i
$483$	0	0
$484$	−125.845	−0.260011
$485$	− 279.883i	− 0.577078i
$486$	0	0
$487$	−570.328	−1.17111	−0.585553	−	0.810634i	$-0.699123\pi$
$487$	−570.328	−1.17111	−0.585553	+	0.810634i	$0.699123\pi$
$488$	586.482i	1.20181i
$489$	0	0
$490$	152.909	0.312059
$491$	84.6117i	0.172325i	0.996281	+	0.0861626i	$0.0274605\pi$
$491$	84.6117i	0.172325i	−0.996281	+	0.0861626i	$0.972540\pi$
$492$	0	0
$493$	−19.4930	−0.0395395
$494$	696.609i	1.41014i
$495$	0	0
$496$	951.422	1.91819
$497$	488.868i	0.983639i
$498$	0	0
$499$	515.637	1.03334	0.516670	−	0.856185i	$-0.327171\pi$
$499$	515.637	1.03334	0.516670	+	0.856185i	$0.327171\pi$
$500$	− 245.887i	− 0.491774i
$501$	0	0
$502$	−270.708	−0.539258
$503$	804.967i	1.60033i	0.599779	+	0.800166i	$0.295255\pi$
$503$	804.967i	1.60033i	−0.599779	+	0.800166i	$0.704745\pi$
$504$	0	0
$505$	−279.203	−0.552877
$506$	463.855i	0.916709i
$507$	0	0
$508$	20.3715	0.0401013
$509$	− 213.386i	− 0.419225i	−0.977785	−	0.209612i	$-0.932780\pi$
$509$	− 213.386i	− 0.419225i	0.977785	−	0.209612i	$-0.0672202\pi$
$510$	0	0
$511$	775.848	1.51829
$512$	− 116.636i	− 0.227804i
$513$	0	0
$514$	−1043.51	−2.03018
$515$	189.606i	0.368167i
$516$	0	0
$517$	423.585	0.819314
$518$	− 975.617i	− 1.88343i
$519$	0	0
$520$	435.332	0.837177
$521$	− 643.878i	− 1.23585i	−0.786237	−	0.617925i	$-0.787973\pi$
$521$	− 643.878i	− 1.23585i	0.786237	−	0.617925i	$-0.212027\pi$
$522$	0	0
$523$	−94.4710	−0.180633	−0.0903164	−	0.995913i	$-0.528788\pi$
$523$	−94.4710	−0.180633	−0.0903164	+	0.995913i	$0.528788\pi$
$524$	288.460i	0.550496i
$525$	0	0
$526$	414.843	0.788675
$527$	− 625.267i	− 1.18647i
$528$	0	0
$529$	−192.015	−0.362977
$530$	491.498i	0.927354i
$531$	0	0
$532$	−204.876	−0.385105
$533$	− 1117.03i	− 2.09574i
$534$	0	0
$535$	−182.354	−0.340848
$536$	200.129i	0.373374i
$537$	0	0
$538$	−248.394	−0.461699
$539$	− 113.336i	− 0.210271i
$540$	0	0
$541$	52.3339	0.0967355	0.0483678	−	0.998830i	$-0.484598\pi$
$541$	52.3339	0.0967355	0.0483678	+	0.998830i	$0.484598\pi$
$542$	− 476.322i	− 0.878823i
$543$	0	0
$544$	−353.910	−0.650571
$545$	− 147.230i	− 0.270147i
$546$	0	0
$547$	−726.667	−1.32846	−0.664230	−	0.747528i	$-0.731240\pi$
$547$	−726.667	−1.32846	−0.664230	+	0.747528i	$0.731240\pi$
$548$	− 243.778i	− 0.444850i
$549$	0	0
$550$	−153.668	−0.279396
$551$	− 20.9173i	− 0.0379625i
$552$	0	0
$553$	1197.34	2.16517
$554$	− 243.654i	− 0.439808i
$555$	0	0
$556$	−290.903	−0.523207
$557$	− 363.285i	− 0.652217i	−0.945332	−	0.326108i	$-0.894263\pi$
$557$	− 363.285i	− 0.652217i	0.945332	−	0.326108i	$-0.105737\pi$
$558$	0	0
$559$	797.201	1.42612
$560$	644.946i	1.15169i
$561$	0	0
$562$	915.992	1.62988
$563$	− 719.685i	− 1.27830i	−0.769080	−	0.639152i	$-0.779285\pi$
$563$	− 719.685i	− 1.27830i	0.769080	−	0.639152i	$-0.220715\pi$
$564$	0	0
$565$	342.083	0.605456
$566$	− 801.804i	− 1.41661i
$567$	0	0
$568$	−320.829	−0.564840
$569$	− 432.080i	− 0.759368i	−0.925116	−	0.379684i	$-0.876033\pi$
$569$	− 432.080i	− 0.759368i	0.925116	−	0.379684i	$-0.123967\pi$
$570$	0	0
$571$	521.210	0.912802	0.456401	−	0.889774i	$-0.349138\pi$
$571$	521.210	0.912802	0.456401	+	0.889774i	$0.349138\pi$
$572$	266.054i	0.465130i
$573$	0	0
$574$	1055.48	1.83881
$575$	− 238.861i	− 0.415410i
$576$	0	0
$577$	−1027.18	−1.78021	−0.890103	−	0.455760i	$-0.849368\pi$
$577$	−1027.18	−1.78021	−0.890103	+	0.455760i	$0.849368\pi$
$578$	− 281.666i	− 0.487311i
$579$	0	0
$580$	10.7784	0.0185834
$581$	811.761i	1.39718i
$582$	0	0
$583$	364.297	0.624867
$584$	509.165i	0.871858i
$585$	0	0
$586$	−1329.34	−2.26850
$587$	106.115i	0.180775i	0.995907	+	0.0903876i	$0.0288106\pi$
$587$	106.115i	0.180775i	−0.995907	+	0.0903876i	$0.971189\pi$
$588$	0	0
$589$	670.956	1.13914
$590$	− 355.827i	− 0.603096i
$591$	0	0
$592$	1003.88	1.69574
$593$	− 705.404i	− 1.18955i	−0.803892	−	0.594776i	$-0.797241\pi$
$593$	− 705.404i	− 1.18955i	0.803892	−	0.594776i	$-0.202759\pi$
$594$	0	0
$595$	423.853	0.712359
$596$	− 61.0992i	− 0.102516i
$597$	0	0
$598$	−1328.66	−2.22184
$599$	129.503i	0.216199i	0.994140	+	0.108100i	$0.0344766\pi$
$599$	129.503i	0.216199i	−0.994140	+	0.108100i	$0.965523\pi$
$600$	0	0
$601$	759.823	1.26427	0.632133	−	0.774860i	$-0.282180\pi$
$601$	759.823	1.26427	0.632133	+	0.774860i	$0.282180\pi$
$602$	753.272i	1.25128i
$603$	0	0
$604$	410.691	0.679951
$605$	279.376i	0.461779i
$606$	0	0
$607$	862.669	1.42120	0.710601	−	0.703595i	$-0.248423\pi$
$607$	862.669	1.42120	0.710601	+	0.703595i	$0.248423\pi$
$608$	− 379.771i	− 0.624623i
$609$	0	0
$610$	−1073.55	−1.75992
$611$	1213.31i	1.98579i
$612$	0	0
$613$	443.227	0.723046	0.361523	−	0.932363i	$-0.382257\pi$
$613$	443.227	0.723046	0.361523	+	0.932363i	$0.382257\pi$
$614$	1198.39i	1.95178i
$615$	0	0
$616$	304.887	0.494947
$617$	681.056i	1.10382i	0.833904	+	0.551909i	$0.186101\pi$
$617$	681.056i	1.10382i	−0.833904	+	0.551909i	$0.813899\pi$
$618$	0	0
$619$	−401.105	−0.647988	−0.323994	−	0.946059i	$-0.605026\pi$
$619$	−401.105	−0.647988	−0.323994	+	0.946059i	$0.605026\pi$
$620$	345.733i	0.557633i
$621$	0	0
$622$	73.8450	0.118722
$623$	− 992.350i	− 1.59286i
$624$	0	0
$625$	−323.481	−0.517569
$626$	788.603i	1.25975i
$627$	0	0
$628$	499.499	0.795380
$629$	− 659.739i	− 1.04887i
$630$	0	0
$631$	−697.284	−1.10505	−0.552523	−	0.833498i	$-0.686335\pi$
$631$	−697.284	−1.10505	−0.552523	+	0.833498i	$0.686335\pi$
$632$	785.775i	1.24331i
$633$	0	0
$634$	−1103.55	−1.74061
$635$	− 45.2246i	− 0.0712199i
$636$	0	0
$637$	324.639	0.509637
$638$	− 25.6667i	− 0.0402299i
$639$	0	0
$640$	−576.564	−0.900882
$641$	1045.27i	1.63068i	0.578981	+	0.815341i	$0.303451\pi$
$641$	1045.27i	1.63068i	−0.578981	+	0.815341i	$0.696549\pi$
$642$	0	0
$643$	109.777	0.170726	0.0853630	−	0.996350i	$-0.472795\pi$
$643$	109.777	0.170726	0.0853630	+	0.996350i	$0.472795\pi$
$644$	− 390.766i	− 0.606779i
$645$	0	0
$646$	−445.109	−0.689023
$647$	8.76887i	0.0135531i	0.999977	+	0.00677656i	$0.00215706\pi$
$647$	8.76887i	0.0135531i	−0.999977	+	0.00677656i	$0.997843\pi$
$648$	0	0
$649$	−263.738	−0.406377
$650$	− 440.164i	− 0.677176i
$651$	0	0
$652$	−174.810	−0.268114
$653$	− 239.411i	− 0.366632i	−0.983054	−	0.183316i	$-0.941317\pi$
$653$	− 239.411i	− 0.366632i	0.983054	−	0.183316i	$-0.0586832\pi$
$654$	0	0
$655$	640.381	0.977680
$656$	1086.05i	1.65556i
$657$	0	0
$658$	−1146.46	−1.74234
$659$	32.6693i	0.0495741i	0.999693	+	0.0247870i	$0.00789077\pi$
$659$	32.6693i	0.0495741i	−0.999693	+	0.0247870i	$0.992109\pi$
$660$	0	0
$661$	248.759	0.376337	0.188169	−	0.982137i	$-0.439745\pi$
$661$	248.759	0.376337	0.188169	+	0.982137i	$0.439745\pi$
$662$	− 942.973i	− 1.42443i
$663$	0	0
$664$	−532.733	−0.802309
$665$	454.825i	0.683947i
$666$	0	0
$667$	39.8962	0.0598144
$668$	516.917i	0.773827i
$669$	0	0
$670$	−366.334	−0.546767
$671$	795.715i	1.18586i
$672$	0	0
$673$	38.2884	0.0568921	0.0284461	−	0.999595i	$-0.490944\pi$
$673$	38.2884	0.0568921	0.0284461	+	0.999595i	$0.490944\pi$
$674$	1007.86i	1.49533i
$675$	0	0
$676$	−456.587	−0.675424
$677$	824.364i	1.21767i	0.793296	+	0.608836i	$0.208363\pi$
$677$	824.364i	1.21767i	−0.793296	+	0.608836i	$0.791637\pi$
$678$	0	0
$679$	561.472	0.826909
$680$	278.162i	0.409062i
$681$	0	0
$682$	823.298	1.20718
$683$	1040.88i	1.52398i	0.647590	+	0.761989i	$0.275777\pi$
$683$	1040.88i	1.52398i	−0.647590	+	0.761989i	$0.724223\pi$
$684$	0	0
$685$	−541.186	−0.790053
$686$	− 643.902i	− 0.938633i
$687$	0	0
$688$	−775.091	−1.12659
$689$	1043.49i	1.51450i
$690$	0	0
$691$	133.361	0.192998	0.0964988	−	0.995333i	$-0.469236\pi$
$691$	133.361	0.192998	0.0964988	+	0.995333i	$0.469236\pi$
$692$	93.7555i	0.135485i
$693$	0	0
$694$	1052.89	1.51714
$695$	645.805i	0.929216i
$696$	0	0
$697$	713.742	1.02402
$698$	716.416i	1.02638i
$699$	0	0
$700$	129.454	0.184935
$701$	617.809i	0.881325i	0.897673	+	0.440663i	$0.145256\pi$
$701$	617.809i	0.881325i	−0.897673	+	0.440663i	$0.854744\pi$
$702$	0	0
$703$	707.947	1.00704
$704$	106.395i	0.151129i
$705$	0	0
$706$	917.213	1.29917
$707$	− 560.107i	− 0.792231i
$708$	0	0
$709$	582.110	0.821030	0.410515	−	0.911854i	$-0.365349\pi$
$709$	582.110	0.821030	0.410515	+	0.911854i	$0.365349\pi$
$710$	− 587.275i	− 0.827148i
$711$	0	0
$712$	651.249	0.914675
$713$	1279.73i	1.79486i
$714$	0	0
$715$	590.640	0.826071
$716$	− 245.592i	− 0.343005i
$717$	0	0
$718$	−174.207	−0.242628
$719$	− 76.0348i	− 0.105751i	−0.998601	−	0.0528754i	$-0.983161\pi$
$719$	− 76.0348i	− 0.105751i	0.998601	−	0.0528754i	$-0.0168386\pi$
$720$	0	0
$721$	−380.368	−0.527556
$722$	392.345i	0.543414i
$723$	0	0
$724$	−373.197	−0.515466
$725$	13.2170i	0.0182303i
$726$	0	0
$727$	335.337	0.461261	0.230631	−	0.973041i	$-0.425921\pi$
$727$	335.337	0.461261	0.230631	+	0.973041i	$0.425921\pi$
$728$	873.317i	1.19961i
$729$	0	0
$730$	−932.023	−1.27674
$731$	509.384i	0.696831i
$732$	0	0
$733$	474.692	0.647602	0.323801	−	0.946125i	$-0.395039\pi$
$733$	474.692	0.647602	0.323801	+	0.946125i	$0.395039\pi$
$734$	1367.53i	1.86312i
$735$	0	0
$736$	724.348	0.984168
$737$	271.526i	0.368421i
$738$	0	0
$739$	−181.712	−0.245889	−0.122944	−	0.992414i	$-0.539234\pi$
$739$	−181.712	−0.245889	−0.122944	+	0.992414i	$0.539234\pi$
$740$	364.794i	0.492964i
$741$	0	0
$742$	−985.991	−1.32883
$743$	917.281i	1.23456i	0.786742	+	0.617282i	$0.211766\pi$
$743$	917.281i	1.23456i	−0.786742	+	0.617282i	$0.788234\pi$
$744$	0	0
$745$	−135.640	−0.182068
$746$	− 1117.80i	− 1.49839i
$747$	0	0
$748$	−169.999	−0.227272
$749$	− 365.819i	− 0.488409i
$750$	0	0
$751$	−551.998	−0.735017	−0.367508	−	0.930020i	$-0.619789\pi$
$751$	−551.998	−0.735017	−0.367508	+	0.930020i	$0.619789\pi$
$752$	− 1179.66i	− 1.56870i
$753$	0	0
$754$	73.5194	0.0975059
$755$	− 911.733i	− 1.20759i
$756$	0	0
$757$	137.805	0.182041	0.0910203	−	0.995849i	$-0.470987\pi$
$757$	137.805	0.182041	0.0910203	+	0.995849i	$0.470987\pi$
$758$	1029.22i	1.35781i
$759$	0	0
$760$	−298.487	−0.392746
$761$	− 762.095i	− 1.00144i	−0.865610	−	0.500719i	$-0.833069\pi$
$761$	− 762.095i	− 1.00144i	0.865610	−	0.500719i	$-0.166931\pi$
$762$	0	0
$763$	295.358	0.387100
$764$	− 3.30532i	− 0.00432634i
$765$	0	0
$766$	−988.336	−1.29026
$767$	− 755.450i	− 0.984942i
$768$	0	0
$769$	245.712	0.319522	0.159761	−	0.987156i	$-0.448928\pi$
$769$	245.712	0.319522	0.159761	+	0.987156i	$0.448928\pi$
$770$	558.094i	0.724797i
$771$	0	0
$772$	280.352	0.363151
$773$	− 293.377i	− 0.379531i	−0.981829	−	0.189765i	$-0.939227\pi$
$773$	− 293.377i	− 0.379531i	0.981829	−	0.189765i	$-0.0607727\pi$
$774$	0	0
$775$	−423.955	−0.547039
$776$	368.476i	0.474841i
$777$	0	0
$778$	425.757	0.547245
$779$	765.895i	0.983177i
$780$	0	0
$781$	−435.288	−0.557347
$782$	− 848.968i	− 1.08564i
$783$	0	0
$784$	−315.635	−0.402596
$785$	− 1108.89i	− 1.41260i
$786$	0	0
$787$	−488.046	−0.620135	−0.310068	−	0.950715i	$-0.600352\pi$
$787$	−488.046	−0.620135	−0.310068	+	0.950715i	$0.600352\pi$
$788$	381.015i	0.483522i
$789$	0	0
$790$	−1438.36	−1.82070
$791$	686.250i	0.867573i
$792$	0	0
$793$	−2279.24	−2.87420
$794$	− 1444.24i	− 1.81894i
$795$	0	0
$796$	−547.827	−0.688225
$797$	− 430.259i	− 0.539849i	−0.962882	−	0.269924i	$-0.913001\pi$
$797$	− 430.259i	− 0.539849i	0.962882	−	0.269924i	$-0.0869987\pi$
$798$	0	0
$799$	−775.266	−0.970295
$800$	239.965i	0.299956i
$801$	0	0
$802$	1632.44	2.03546
$803$	690.814i	0.860292i
$804$	0	0
$805$	−867.500	−1.07764
$806$	2358.25i	2.92587i
$807$	0	0
$808$	367.581	0.454927
$809$	144.409i	0.178503i	0.996009	+	0.0892516i	$0.0284475\pi$
$809$	144.409i	0.178503i	−0.996009	+	0.0892516i	$0.971552\pi$
$810$	0	0
$811$	1008.89	1.24401	0.622005	−	0.783014i	$-0.286319\pi$
$811$	1008.89	1.24401	0.622005	+	0.783014i	$0.286319\pi$
$812$	21.6224i	0.0266286i
$813$	0	0
$814$	868.688	1.06718
$815$	388.079i	0.476170i
$816$	0	0
$817$	−546.605	−0.669039
$818$	451.228i	0.551624i
$819$	0	0
$820$	−394.653	−0.481285
$821$	719.054i	0.875827i	0.899017	+	0.437914i	$0.144282\pi$
$821$	719.054i	0.875827i	−0.899017	+	0.437914i	$0.855718\pi$
$822$	0	0
$823$	695.297	0.844832	0.422416	−	0.906402i	$-0.361182\pi$
$823$	695.297	0.844832	0.422416	+	0.906402i	$0.361182\pi$
$824$	− 249.624i	− 0.302941i
$825$	0	0
$826$	713.822	0.864191
$827$	− 1201.83i	− 1.45324i	−0.687041	−	0.726618i	$-0.741091\pi$
$827$	− 1201.83i	− 1.45324i	0.687041	−	0.726618i	$-0.258909\pi$
$828$	0	0
$829$	−129.209	−0.155861	−0.0779307	−	0.996959i	$-0.524831\pi$
$829$	−129.209	−0.155861	−0.0779307	+	0.996959i	$0.524831\pi$
$830$	− 975.164i	− 1.17490i
$831$	0	0
$832$	−304.756	−0.366294
$833$	207.433i	0.249019i
$834$	0	0
$835$	1147.55	1.37432
$836$	− 182.421i	− 0.218207i
$837$	0	0
$838$	−268.730	−0.320681
$839$	− 598.182i	− 0.712970i	−0.934301	−	0.356485i	$-0.883975\pi$
$839$	− 598.182i	− 0.712970i	0.934301	−	0.356485i	$-0.116025\pi$
$840$	0	0
$841$	838.792	0.997375
$842$	− 8.83002i	− 0.0104870i
$843$	0	0
$844$	−143.673	−0.170228
$845$	1013.62i	1.19955i
$846$	0	0
$847$	−560.455	−0.661695
$848$	− 1014.55i	− 1.19640i
$849$	0	0
$850$	281.250	0.330882
$851$	1350.29i	1.58671i
$852$	0	0
$853$	−50.4253	−0.0591153	−0.0295576	−	0.999563i	$-0.509410\pi$
$853$	−50.4253	−0.0591153	−0.0295576	+	0.999563i	$0.509410\pi$
$854$	− 2153.64i	− 2.52183i
$855$	0	0
$856$	240.075	0.280462
$857$	− 1015.39i	− 1.18482i	−0.805637	−	0.592409i	$-0.798177\pi$
$857$	− 1015.39i	− 1.18482i	0.805637	−	0.592409i	$-0.201823\pi$
$858$	0	0
$859$	554.440	0.645448	0.322724	−	0.946493i	$-0.395401\pi$
$859$	554.440	0.645448	0.322724	+	0.946493i	$0.395401\pi$
$860$	− 281.657i	− 0.327508i
$861$	0	0
$862$	617.027	0.715808
$863$	249.538i	0.289151i	0.989494	+	0.144576i	$0.0461817\pi$
$863$	249.538i	0.289151i	−0.989494	+	0.144576i	$0.953818\pi$
$864$	0	0
$865$	208.137	0.240621
$866$	− 1734.35i	− 2.00272i
$867$	0	0
$868$	−693.572	−0.799046
$869$	1066.11i	1.22682i
$870$	0	0
$871$	−777.758	−0.892948
$872$	193.834i	0.222287i
$873$	0	0
$874$	911.003	1.04234
$875$	− 1095.07i	− 1.25150i
$876$	0	0
$877$	462.780	0.527685	0.263843	−	0.964566i	$-0.415010\pi$
$877$	462.780	0.527685	0.263843	+	0.964566i	$0.415010\pi$
$878$	823.748i	0.938209i
$879$	0	0
$880$	−574.259	−0.652567
$881$	− 650.354i	− 0.738200i	−0.929390	−	0.369100i	$-0.879666\pi$
$881$	− 650.354i	− 0.738200i	0.929390	−	0.369100i	$-0.120334\pi$
$882$	0	0
$883$	94.7771	0.107335	0.0536677	−	0.998559i	$-0.482909\pi$
$883$	94.7771	0.107335	0.0536677	+	0.998559i	$0.482909\pi$
$884$	− 486.945i	− 0.550842i
$885$	0	0
$886$	1288.73	1.45455
$887$	− 330.723i	− 0.372855i	−0.982469	−	0.186428i	$-0.940309\pi$
$887$	− 330.723i	− 0.372855i	0.982469	−	0.186428i	$-0.0596910\pi$
$888$	0	0
$889$	90.7248	0.102053
$890$	1192.11i	1.33944i
$891$	0	0
$892$	−67.2380	−0.0753790
$893$	− 831.915i	− 0.931595i
$894$	0	0
$895$	−545.214	−0.609177
$896$	− 1156.64i	− 1.29090i
$897$	0	0
$898$	1802.52	2.00726
$899$	− 70.8120i	− 0.0787676i
$900$	0	0
$901$	−666.754	−0.740015
$902$	939.794i	1.04190i
$903$	0	0
$904$	−450.365	−0.498191
$905$	828.498i	0.915468i
$906$	0	0
$907$	975.660	1.07570	0.537850	−	0.843040i	$-0.319237\pi$
$907$	975.660	1.07570	0.537850	+	0.843040i	$0.319237\pi$
$908$	121.376i	0.133674i
$909$	0	0
$910$	−1598.60	−1.75670
$911$	861.914i	0.946119i	0.881030	+	0.473059i	$0.156850\pi$
$911$	861.914i	0.946119i	−0.881030	+	0.473059i	$0.843150\pi$
$912$	0	0
$913$	−722.791	−0.791666
$914$	− 1418.54i	− 1.55202i
$915$	0	0
$916$	−36.7661	−0.0401377
$917$	1284.66i	1.40094i
$918$	0	0
$919$	−788.050	−0.857508	−0.428754	−	0.903421i	$-0.641047\pi$
$919$	−788.050	−0.857508	−0.428754	+	0.903421i	$0.641047\pi$
$920$	− 569.313i	− 0.618819i
$921$	0	0
$922$	−1255.94	−1.36219
$923$	− 1246.83i	− 1.35085i
$924$	0	0
$925$	−447.328	−0.483598
$926$	1814.03i	1.95900i
$927$	0	0
$928$	−40.0807	−0.0431904
$929$	301.848i	0.324917i	0.986715	+	0.162459i	$0.0519424\pi$
$929$	301.848i	0.324917i	−0.986715	+	0.162459i	$0.948058\pi$
$930$	0	0
$931$	−222.590	−0.239087
$932$	149.161i	0.160044i
$933$	0	0
$934$	−2090.92	−2.23867
$935$	377.398i	0.403635i
$936$	0	0
$937$	1469.40	1.56819	0.784096	−	0.620640i	$-0.213127\pi$
$937$	1469.40	1.56819	0.784096	+	0.620640i	$0.213127\pi$
$938$	− 734.900i	− 0.783476i
$939$	0	0
$940$	428.672	0.456034
$941$	887.430i	0.943071i	0.881847	+	0.471536i	$0.156300\pi$
$941$	887.430i	0.943071i	−0.881847	+	0.471536i	$0.843700\pi$
$942$	0	0
$943$	−1460.81	−1.54911
$944$	734.498i	0.778070i
$945$	0	0
$946$	−670.713	−0.708999
$947$	707.982i	0.747605i	0.927508	+	0.373802i	$0.121946\pi$
$947$	707.982i	0.747605i	−0.927508	+	0.373802i	$0.878054\pi$
$948$	0	0
$949$	−1978.76	−2.08510
$950$	301.801i	0.317685i
$951$	0	0
$952$	−558.019	−0.586154
$953$	512.218i	0.537480i	0.963213	+	0.268740i	$0.0866072\pi$
$953$	512.218i	0.537480i	−0.963213	+	0.268740i	$0.913393\pi$
$954$	0	0
$955$	−7.33781	−0.00768357
$956$	258.423i	0.270317i
$957$	0	0
$958$	−1620.48	−1.69152
$959$	− 1085.67i	− 1.13209i
$960$	0	0
$961$	1310.41	1.36358
$962$	2488.26i	2.58655i
$963$	0	0
$964$	−240.747	−0.249737
$965$	− 622.382i	− 0.644956i
$966$	0	0
$967$	0.643507	0.000665467 0	0.000332734	−	1.00000i	$-0.499894\pi$
$967$	0.643507	0.000665467 0	0.000332734		1.00000i	$0.499894\pi$
$968$	− 367.809i	− 0.379968i
$969$	0	0
$970$	−674.493	−0.695354
$971$	− 706.216i	− 0.727308i	−0.931534	−	0.363654i	$-0.881529\pi$
$971$	− 706.216i	− 0.727308i	0.931534	−	0.363654i	$-0.118471\pi$
$972$	0	0
$973$	−1295.55	−1.33150
$974$	1374.44i	1.41113i
$975$	0	0
$976$	2216.02	2.27052
$977$	− 860.514i	− 0.880772i	−0.897808	−	0.440386i	$-0.854842\pi$
$977$	− 860.514i	− 0.880772i	0.897808	−	0.440386i	$-0.145158\pi$
$978$	0	0
$979$	883.587	0.902541
$980$	− 114.697i	− 0.117038i
$981$	0	0
$982$	203.907	0.207644
$983$	1605.30i	1.63306i	0.577304	+	0.816529i	$0.304105\pi$
$983$	1605.30i	1.63306i	−0.577304	+	0.816529i	$0.695895\pi$
$984$	0	0
$985$	845.854	0.858735
$986$	46.9763i	0.0476433i
$987$	0	0
$988$	522.526	0.528873
$989$	− 1042.55i	− 1.05415i
$990$	0	0
$991$	−542.923	−0.547854	−0.273927	−	0.961750i	$-0.588323\pi$
$991$	−542.923	−0.547854	−0.273927	+	0.961750i	$0.588323\pi$
$992$	− 1285.65i	− 1.29602i
$993$	0	0
$994$	1178.13	1.18524
$995$	1216.18i	1.22229i
$996$	0	0
$997$	−953.401	−0.956269	−0.478135	−	0.878287i	$-0.658687\pi$
$997$	−953.401	−0.956269	−0.478135	+	0.878287i	$0.658687\pi$
$998$	− 1242.64i	− 1.24513i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.20	✓	84
3.2	odd	2	inner	1143.3.b.a.890.65	yes	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.20	✓	84	1.1	even	1	trivial
1143.3.b.a.890.65	yes	84	3.2	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +O(q^{10})\)	\(q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +9.67106 q^{10} -7.16818i q^{11} +20.5325 q^{13} +19.4011i q^{14} -19.9630 q^{16} +13.1195i q^{17} -14.0782 q^{19} -7.25426i q^{20} -17.2747 q^{22} -26.8517i q^{23} +8.89554 q^{25} -49.4815i q^{26} +14.5527 q^{28} +1.48580i q^{29} -47.6593 q^{31} +26.9758i q^{32} +31.6169 q^{34} -32.3071i q^{35} -50.2868 q^{37} +33.9272i q^{38} +21.2021 q^{40} -54.4030i q^{41} +38.8264 q^{43} +12.9577i q^{44} -64.7103 q^{46} +59.0925i q^{47} +15.8110 q^{49} -21.4375i q^{50} -37.1160 q^{52} +50.8215i q^{53} +28.7662 q^{55} +42.5334i q^{56} +3.58064 q^{58} -36.7929i q^{59} -111.007 q^{61} +114.855i q^{62} -14.8426 q^{64} +82.3976i q^{65} -37.8794 q^{67} -23.7158i q^{68} -77.8571 q^{70} -60.7250i q^{71} -96.3723 q^{73} +121.187i q^{74} +25.4488 q^{76} +57.7076i q^{77} -148.728 q^{79} -80.1123i q^{80} -131.106 q^{82} -100.833i q^{83} -52.6491 q^{85} -93.5681i q^{86} -37.8717 q^{88} +123.265i q^{89} -165.297 q^{91} +48.5391i q^{92} +142.408 q^{94} -56.4963i q^{95} -69.7434 q^{97} -38.1031i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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