LMFDB - Embedded Newform 4005.2.a.l.1.4

Newspace parameters

Level:	$ N $	=	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	4005.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.09529$
Character	$\chi$	=	4005.1

$q$-expansion

$f(q)$	$=$	$q+2.39026 q^{2} +3.71333 q^{4} +1.00000 q^{5} +1.95452 q^{7} +4.09529 q^{8} +O(q^{10})$	\(q+2.39026 q^{2} +3.71333 q^{4} +1.00000 q^{5} +1.95452 q^{7} +4.09529 q^{8} +2.39026 q^{10} +4.29496 q^{11} +1.05377 q^{13} +4.67180 q^{14} +2.36215 q^{16} -3.65443 q^{17} -3.07792 q^{19} +3.71333 q^{20} +10.2661 q^{22} +1.46385 q^{23} +1.00000 q^{25} +2.51878 q^{26} +7.25777 q^{28} +8.44007 q^{29} +4.24040 q^{31} -2.54445 q^{32} -8.73503 q^{34} +1.95452 q^{35} -4.61163 q^{37} -7.35702 q^{38} +4.09529 q^{40} -0.785638 q^{41} -3.60895 q^{43} +15.9486 q^{44} +3.49897 q^{46} +10.8220 q^{47} -3.17985 q^{49} +2.39026 q^{50} +3.91300 q^{52} +9.07035 q^{53} +4.29496 q^{55} +8.00433 q^{56} +20.1739 q^{58} +5.36459 q^{59} -5.59950 q^{61} +10.1356 q^{62} -10.8062 q^{64} +1.05377 q^{65} -7.92129 q^{67} -13.5701 q^{68} +4.67180 q^{70} -10.0621 q^{71} +2.47330 q^{73} -11.0230 q^{74} -11.4293 q^{76} +8.39459 q^{77} -11.6532 q^{79} +2.36215 q^{80} -1.87788 q^{82} -12.5329 q^{83} -3.65443 q^{85} -8.62632 q^{86} +17.5891 q^{88} +1.00000 q^{89} +2.05962 q^{91} +5.43574 q^{92} +25.8674 q^{94} -3.07792 q^{95} +5.96756 q^{97} -7.60066 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} + O(q^{10}) $	$ 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} - q^{10} + 14q^{11} - 5q^{13} + 5q^{14} - 3q^{16} + 3q^{17} - q^{19} + 3q^{20} + 2q^{22} + 3q^{23} + 4q^{25} + 9q^{26} + 5q^{28} + 10q^{29} - 11q^{31} + 2q^{32} - 8q^{34} + 2q^{35} + 3q^{37} - 2q^{38} + 9q^{40} + 3q^{41} + 9q^{43} + 9q^{44} - 4q^{46} + 24q^{47} - q^{50} + 8q^{52} - 3q^{53} + 14q^{55} + 13q^{56} + 19q^{58} + 22q^{59} - 3q^{61} + 24q^{62} - 11q^{64} - 5q^{65} - 9q^{67} - 31q^{68} + 5q^{70} - 16q^{71} + 3q^{73} - 31q^{74} - 24q^{76} + 4q^{77} - 27q^{79} - 3q^{80} - 15q^{82} - 6q^{83} + 3q^{85} - 15q^{86} + 30q^{88} + 4q^{89} - 29q^{91} + 17q^{92} + 13q^{94} - q^{95} + 41q^{97} - 22q^{98} + O(q^{100}) $

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 $

$2$	2.39026	1.69017	0.845083	−	0.534634i	$-0.179551\pi$
$2$	2.39026	1.69017	0.845083	+	0.534634i	$0.179551\pi$
$3$	0	0
$3$	0	0
$4$	3.71333	1.85666
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.95452	0.738739	0.369370	−	0.929283i	$-0.379574\pi$
$7$	1.95452	0.738739	0.369370	+	0.929283i	$0.379574\pi$
$8$	4.09529	1.44791
$9$	0	0
$10$	2.39026	0.755866
$11$	4.29496	1.29498	0.647490	−	0.762074i	$-0.275819\pi$
$11$	4.29496	1.29498	0.647490	+	0.762074i	$0.275819\pi$
$12$	0	0
$13$	1.05377	0.292263	0.146132	−	0.989265i	$-0.453318\pi$
$13$	1.05377	0.292263	0.146132	+	0.989265i	$0.453318\pi$
$14$	4.67180	1.24859
$15$	0	0
$16$	2.36215	0.590537
$17$	−3.65443	−0.886330	−0.443165	−	0.896440i	$-0.646144\pi$
$17$	−3.65443	−0.886330	−0.443165	+	0.896440i	$0.646144\pi$
$18$	0	0
$19$	−3.07792	−0.706124	−0.353062	−	0.935600i	$-0.614859\pi$
$19$	−3.07792	−0.706124	−0.353062	+	0.935600i	$0.614859\pi$
$20$	3.71333	0.830325
$21$	0	0
$22$	10.2661	2.18873
$23$	1.46385	0.305233	0.152616	−	0.988286i	$-0.451230\pi$
$23$	1.46385	0.305233	0.152616	+	0.988286i	$0.451230\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.51878	0.493974
$27$	0	0
$28$	7.25777	1.37159
$29$	8.44007	1.56728	0.783641	−	0.621214i	$-0.213360\pi$
$29$	8.44007	1.56728	0.783641	+	0.621214i	$0.213360\pi$
$30$	0	0
$31$	4.24040	0.761599	0.380799	−	0.924658i	$-0.375649\pi$
$31$	4.24040	0.761599	0.380799	+	0.924658i	$0.375649\pi$
$32$	−2.54445	−0.449799
$33$	0	0
$34$	−8.73503	−1.49805
$35$	1.95452	0.330374
$36$	0	0
$37$	−4.61163	−0.758148	−0.379074	−	0.925366i	$-0.623757\pi$
$37$	−4.61163	−0.758148	−0.379074	+	0.925366i	$0.623757\pi$
$38$	−7.35702	−1.19347
$39$	0	0
$40$	4.09529	0.647523
$41$	−0.785638	−0.122696	−0.0613480	−	0.998116i	$-0.519540\pi$
$41$	−0.785638	−0.122696	−0.0613480	+	0.998116i	$0.519540\pi$
$42$	0	0
$43$	−3.60895	−0.550360	−0.275180	−	0.961393i	$-0.588737\pi$
$43$	−3.60895	−0.550360	−0.275180	+	0.961393i	$0.588737\pi$
$44$	15.9486	2.40434
$45$	0	0
$46$	3.49897	0.515894
$47$	10.8220	1.57856	0.789278	−	0.614036i	$-0.210455\pi$
$47$	10.8220	1.57856	0.789278	+	0.614036i	$0.210455\pi$
$48$	0	0
$49$	−3.17985	−0.454265
$50$	2.39026	0.338033
$51$	0	0
$52$	3.91300	0.542635
$53$	9.07035	1.24591	0.622954	−	0.782258i	$-0.285932\pi$
$53$	9.07035	1.24591	0.622954	+	0.782258i	$0.285932\pi$
$54$	0	0
$55$	4.29496	0.579133
$56$	8.00433	1.06962
$57$	0	0
$58$	20.1739	2.64897
$59$	5.36459	0.698411	0.349205	−	0.937046i	$-0.386452\pi$
$59$	5.36459	0.698411	0.349205	+	0.937046i	$0.386452\pi$
$60$	0	0
$61$	−5.59950	−0.716942	−0.358471	−	0.933541i	$-0.616702\pi$
$61$	−5.59950	−0.716942	−0.358471	+	0.933541i	$0.616702\pi$
$62$	10.1356	1.28723
$63$	0	0
$64$	−10.8062	−1.35077
$65$	1.05377	0.130704
$66$	0	0
$67$	−7.92129	−0.967739	−0.483870	−	0.875140i	$-0.660769\pi$
$67$	−7.92129	−0.967739	−0.483870	+	0.875140i	$0.660769\pi$
$68$	−13.5701	−1.64562
$69$	0	0
$70$	4.67180	0.558387
$71$	−10.0621	−1.19415	−0.597074	−	0.802187i	$-0.703670\pi$
$71$	−10.0621	−1.19415	−0.597074	+	0.802187i	$0.703670\pi$
$72$	0	0
$73$	2.47330	0.289478	0.144739	−	0.989470i	$-0.453766\pi$
$73$	2.47330	0.289478	0.144739	+	0.989470i	$0.453766\pi$
$74$	−11.0230	−1.28140
$75$	0	0
$76$	−11.4293	−1.31103
$77$	8.39459	0.956652
$78$	0	0
$79$	−11.6532	−1.31108	−0.655541	−	0.755159i	$-0.727559\pi$
$79$	−11.6532	−1.31108	−0.655541	+	0.755159i	$0.727559\pi$
$80$	2.36215	0.264096
$81$	0	0
$82$	−1.87788	−0.207377
$83$	−12.5329	−1.37567	−0.687833	−	0.725869i	$-0.741438\pi$
$83$	−12.5329	−1.37567	−0.687833	+	0.725869i	$0.741438\pi$
$84$	0	0
$85$	−3.65443	−0.396379
$86$	−8.62632	−0.930201
$87$	0	0
$88$	17.5891	1.87501
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	2.05962	0.215906
$92$	5.43574	0.566715
$93$	0	0
$94$	25.8674	2.66802
$95$	−3.07792	−0.315788
$96$	0	0
$97$	5.96756	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$97$	5.96756	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$98$	−7.60066	−0.767783
$99$	0	0
$100$	3.71333	0.371333
$101$	−6.15863	−0.612807	−0.306403	−	0.951902i	$-0.599126\pi$
$101$	−6.15863	−0.612807	−0.306403	+	0.951902i	$0.599126\pi$
$102$	0	0
$103$	15.0166	1.47963	0.739814	−	0.672812i	$-0.234914\pi$
$103$	15.0166	1.47963	0.739814	+	0.672812i	$0.234914\pi$
$104$	4.31550	0.423170
$105$	0	0
$106$	21.6805	2.10579
$107$	−2.68955	−0.260009	−0.130004	−	0.991513i	$-0.541499\pi$
$107$	−2.68955	−0.260009	−0.130004	+	0.991513i	$0.541499\pi$
$108$	0	0
$109$	−0.671805	−0.0643472	−0.0321736	−	0.999482i	$-0.510243\pi$
$109$	−0.671805	−0.0643472	−0.0321736	+	0.999482i	$0.510243\pi$
$110$	10.2661	0.978831
$111$	0	0
$112$	4.61687	0.436253
$113$	8.72825	0.821085	0.410543	−	0.911841i	$-0.365339\pi$
$113$	8.72825	0.821085	0.410543	+	0.911841i	$0.365339\pi$
$114$	0	0
$115$	1.46385	0.136504
$116$	31.3408	2.90992
$117$	0	0
$118$	12.8228	1.18043
$119$	−7.14266	−0.654767
$120$	0	0
$121$	7.44671	0.676973
$122$	−13.3842	−1.21175
$123$	0	0
$124$	15.7460	1.41403
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.55006	−0.403752	−0.201876	−	0.979411i	$-0.564704\pi$
$127$	−4.55006	−0.403752	−0.201876	+	0.979411i	$0.564704\pi$
$128$	−20.7406	−1.83323
$129$	0	0
$130$	2.51878	0.220912
$131$	9.12736	0.797461	0.398731	−	0.917068i	$-0.369451\pi$
$131$	9.12736	0.797461	0.398731	+	0.917068i	$0.369451\pi$
$132$	0	0
$133$	−6.01586	−0.521641
$134$	−18.9339	−1.63564
$135$	0	0
$136$	−14.9660	−1.28332
$137$	−16.8521	−1.43978	−0.719888	−	0.694090i	$-0.755807\pi$
$137$	−16.8521	−1.43978	−0.719888	+	0.694090i	$0.755807\pi$
$138$	0	0
$139$	3.20205	0.271594	0.135797	−	0.990737i	$-0.456641\pi$
$139$	3.20205	0.271594	0.135797	+	0.990737i	$0.456641\pi$
$140$	7.25777	0.613394
$141$	0	0
$142$	−24.0509	−2.01831
$143$	4.52591	0.378475
$144$	0	0
$145$	8.44007	0.700910
$146$	5.91183	0.489267
$147$	0	0
$148$	−17.1245	−1.40763
$149$	−14.6448	−1.19975	−0.599874	−	0.800094i	$-0.704783\pi$
$149$	−14.6448	−1.19975	−0.599874	+	0.800094i	$0.704783\pi$
$150$	0	0
$151$	−17.8411	−1.45189	−0.725943	−	0.687755i	$-0.758596\pi$
$151$	−17.8411	−1.45189	−0.725943	+	0.687755i	$0.758596\pi$
$152$	−12.6050	−1.02240
$153$	0	0
$154$	20.0652	1.61690
$155$	4.24040	0.340597
$156$	0	0
$157$	15.4767	1.23518	0.617588	−	0.786502i	$-0.288110\pi$
$157$	15.4767	1.23518	0.617588	+	0.786502i	$0.288110\pi$
$158$	−27.8540	−2.21595
$159$	0	0
$160$	−2.54445	−0.201156
$161$	2.86111	0.225487
$162$	0	0
$163$	−2.39177	−0.187338	−0.0936689	−	0.995603i	$-0.529860\pi$
$163$	−2.39177	−0.187338	−0.0936689	+	0.995603i	$0.529860\pi$
$164$	−2.91733	−0.227805
$165$	0	0
$166$	−29.9569	−2.32511
$167$	−0.644864	−0.0499011	−0.0249505	−	0.999689i	$-0.507943\pi$
$167$	−0.644864	−0.0499011	−0.0249505	+	0.999689i	$0.507943\pi$
$168$	0	0
$169$	−11.8896	−0.914582
$170$	−8.73503	−0.669947
$171$	0	0
$172$	−13.4012	−1.02183
$173$	19.2152	1.46091	0.730453	−	0.682963i	$-0.239309\pi$
$173$	19.2152	1.46091	0.730453	+	0.682963i	$0.239309\pi$
$174$	0	0
$175$	1.95452	0.147748
$176$	10.1453	0.764734
$177$	0	0
$178$	2.39026	0.179157
$179$	14.9898	1.12039	0.560193	−	0.828362i	$-0.310727\pi$
$179$	14.9898	1.12039	0.560193	+	0.828362i	$0.310727\pi$
$180$	0	0
$181$	23.1716	1.72233	0.861167	−	0.508322i	$-0.169734\pi$
$181$	23.1716	1.72233	0.861167	+	0.508322i	$0.169734\pi$
$182$	4.92301	0.364918
$183$	0	0
$184$	5.99488	0.441948
$185$	−4.61163	−0.339054
$186$	0	0
$187$	−15.6957	−1.14778
$188$	40.1858	2.93085
$189$	0	0
$190$	−7.35702	−0.533735
$191$	−0.390024	−0.0282211	−0.0141106	−	0.999900i	$-0.504492\pi$
$191$	−0.390024	−0.0282211	−0.0141106	+	0.999900i	$0.504492\pi$
$192$	0	0
$193$	9.91428	0.713645	0.356823	−	0.934172i	$-0.383860\pi$
$193$	9.91428	0.713645	0.356823	+	0.934172i	$0.383860\pi$
$194$	14.2640	1.02410
$195$	0	0
$196$	−11.8078	−0.843417
$197$	1.01097	0.0720286	0.0360143	−	0.999351i	$-0.488534\pi$
$197$	1.01097	0.0720286	0.0360143	+	0.999351i	$0.488534\pi$
$198$	0	0
$199$	−13.3504	−0.946384	−0.473192	−	0.880959i	$-0.656898\pi$
$199$	−13.3504	−0.946384	−0.473192	+	0.880959i	$0.656898\pi$
$200$	4.09529	0.289581
$201$	0	0
$202$	−14.7207	−1.03575
$203$	16.4963	1.15781
$204$	0	0
$205$	−0.785638	−0.0548713
$206$	35.8935	2.50082
$207$	0	0
$208$	2.48916	0.172592
$209$	−13.2196	−0.914416
$210$	0	0
$211$	−25.4818	−1.75424	−0.877118	−	0.480274i	$-0.840537\pi$
$211$	−25.4818	−1.75424	−0.877118	+	0.480274i	$0.840537\pi$
$212$	33.6812	2.31323
$213$	0	0
$214$	−6.42872	−0.439459
$215$	−3.60895	−0.246129
$216$	0	0
$217$	8.28795	0.562623
$218$	−1.60579	−0.108758
$219$	0	0
$220$	15.9486	1.07525
$221$	−3.85094	−0.259042
$222$	0	0
$223$	20.3456	1.36244	0.681222	−	0.732076i	$-0.261449\pi$
$223$	20.3456	1.36244	0.681222	+	0.732076i	$0.261449\pi$
$224$	−4.97317	−0.332284
$225$	0	0
$226$	20.8628	1.38777
$227$	−16.0605	−1.06597	−0.532986	−	0.846124i	$-0.678930\pi$
$227$	−16.0605	−1.06597	−0.532986	+	0.846124i	$0.678930\pi$
$228$	0	0
$229$	15.2661	1.00881	0.504405	−	0.863467i	$-0.331712\pi$
$229$	15.2661	1.00881	0.504405	+	0.863467i	$0.331712\pi$
$230$	3.49897	0.230715
$231$	0	0
$232$	34.5646	2.26928
$233$	−18.5117	−1.21274	−0.606371	−	0.795182i	$-0.707375\pi$
$233$	−18.5117	−1.21274	−0.606371	+	0.795182i	$0.707375\pi$
$234$	0	0
$235$	10.8220	0.705952
$236$	19.9205	1.29671
$237$	0	0
$238$	−17.0728	−1.10667
$239$	27.6511	1.78860	0.894300	−	0.447468i	$-0.147674\pi$
$239$	27.6511	1.78860	0.894300	+	0.447468i	$0.147674\pi$
$240$	0	0
$241$	27.2269	1.75384	0.876920	−	0.480636i	$-0.159594\pi$
$241$	27.2269	1.75384	0.876920	+	0.480636i	$0.159594\pi$
$242$	17.7995	1.14420
$243$	0	0
$244$	−20.7928	−1.33112
$245$	−3.17985	−0.203153
$246$	0	0
$247$	−3.24342	−0.206374
$248$	17.3657	1.10272
$249$	0	0
$250$	2.39026	0.151173
$251$	−12.9645	−0.818310	−0.409155	−	0.912465i	$-0.634176\pi$
$251$	−12.9645	−0.818310	−0.409155	+	0.912465i	$0.634176\pi$
$252$	0	0
$253$	6.28716	0.395270
$254$	−10.8758	−0.682409
$255$	0	0
$256$	−27.9631	−1.74769
$257$	−18.9552	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$257$	−18.9552	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$258$	0	0
$259$	−9.01353	−0.560073
$260$	3.91300	0.242674
$261$	0	0
$262$	21.8167	1.34784
$263$	16.4109	1.01194	0.505971	−	0.862551i	$-0.331134\pi$
$263$	16.4109	1.01194	0.505971	+	0.862551i	$0.331134\pi$
$264$	0	0
$265$	9.07035	0.557187
$266$	−14.3795	−0.881661
$267$	0	0
$268$	−29.4143	−1.79677
$269$	−28.9069	−1.76248	−0.881242	−	0.472664i	$-0.843292\pi$
$269$	−28.9069	−1.76248	−0.881242	+	0.472664i	$0.843292\pi$
$270$	0	0
$271$	6.39573	0.388513	0.194256	−	0.980951i	$-0.437771\pi$
$271$	6.39573	0.388513	0.194256	+	0.980951i	$0.437771\pi$
$272$	−8.63231	−0.523411
$273$	0	0
$274$	−40.2810	−2.43346
$275$	4.29496	0.258996
$276$	0	0
$277$	−2.90811	−0.174731	−0.0873656	−	0.996176i	$-0.527845\pi$
$277$	−2.90811	−0.174731	−0.0873656	+	0.996176i	$0.527845\pi$
$278$	7.65371	0.459039
$279$	0	0
$280$	8.00433	0.478350
$281$	6.27691	0.374449	0.187225	−	0.982317i	$-0.440051\pi$
$281$	6.27691	0.374449	0.187225	+	0.982317i	$0.440051\pi$
$282$	0	0
$283$	11.8814	0.706277	0.353139	−	0.935571i	$-0.385114\pi$
$283$	11.8814	0.706277	0.353139	+	0.935571i	$0.385114\pi$
$284$	−37.3637	−2.21713
$285$	0	0
$286$	10.8181	0.639686
$287$	−1.53554	−0.0906403
$288$	0	0
$289$	−3.64512	−0.214419
$290$	20.1739	1.18465
$291$	0	0
$292$	9.18419	0.537464
$293$	15.4163	0.900630	0.450315	−	0.892870i	$-0.351312\pi$
$293$	15.4163	0.900630	0.450315	+	0.892870i	$0.351312\pi$
$294$	0	0
$295$	5.36459	0.312339
$296$	−18.8860	−1.09773
$297$	0	0
$298$	−35.0048	−2.02777
$299$	1.54256	0.0892084
$300$	0	0
$301$	−7.05377	−0.406573
$302$	−42.6447	−2.45393
$303$	0	0
$304$	−7.27051	−0.416992
$305$	−5.59950	−0.320626
$306$	0	0
$307$	−13.1695	−0.751623	−0.375811	−	0.926696i	$-0.622636\pi$
$307$	−13.1695	−0.751623	−0.375811	+	0.926696i	$0.622636\pi$
$308$	31.1719	1.77618
$309$	0	0
$310$	10.1356	0.575666
$311$	−13.6116	−0.771841	−0.385920	−	0.922532i	$-0.626116\pi$
$311$	−13.6116	−0.771841	−0.385920	+	0.922532i	$0.626116\pi$
$312$	0	0
$313$	−20.0023	−1.13059	−0.565297	−	0.824887i	$-0.691239\pi$
$313$	−20.0023	−1.13059	−0.565297	+	0.824887i	$0.691239\pi$
$314$	36.9933	2.08765
$315$	0	0
$316$	−43.2720	−2.43424
$317$	−17.9215	−1.00657	−0.503286	−	0.864120i	$-0.667876\pi$
$317$	−17.9215	−1.00657	−0.503286	+	0.864120i	$0.667876\pi$
$318$	0	0
$319$	36.2498	2.02960
$320$	−10.8062	−0.604084
$321$	0	0
$322$	6.83880	0.381111
$323$	11.2481	0.625859
$324$	0	0
$325$	1.05377	0.0584527
$326$	−5.71694	−0.316632
$327$	0	0
$328$	−3.21742	−0.177652
$329$	21.1519	1.16614
$330$	0	0
$331$	−20.2646	−1.11384	−0.556920	−	0.830566i	$-0.688017\pi$
$331$	−20.2646	−1.11384	−0.556920	+	0.830566i	$0.688017\pi$
$332$	−46.5388	−2.55415
$333$	0	0
$334$	−1.54139	−0.0843411
$335$	−7.92129	−0.432786
$336$	0	0
$337$	−16.1392	−0.879158	−0.439579	−	0.898204i	$-0.644872\pi$
$337$	−16.1392	−0.879158	−0.439579	+	0.898204i	$0.644872\pi$
$338$	−28.4191	−1.54580
$339$	0	0
$340$	−13.5701	−0.735942
$341$	18.2124	0.986255
$342$	0	0
$343$	−19.8967	−1.07432
$344$	−14.7797	−0.796869
$345$	0	0
$346$	45.9293	2.46918
$347$	−14.3396	−0.769789	−0.384895	−	0.922961i	$-0.625762\pi$
$347$	−14.3396	−0.769789	−0.384895	+	0.922961i	$0.625762\pi$
$348$	0	0
$349$	−31.8954	−1.70732	−0.853660	−	0.520830i	$-0.825622\pi$
$349$	−31.8954	−1.70732	−0.853660	+	0.520830i	$0.825622\pi$
$350$	4.67180	0.249718
$351$	0	0
$352$	−10.9283	−0.582480
$353$	−2.46050	−0.130959	−0.0654796	−	0.997854i	$-0.520858\pi$
$353$	−2.46050	−0.130959	−0.0654796	+	0.997854i	$0.520858\pi$
$354$	0	0
$355$	−10.0621	−0.534039
$356$	3.71333	0.196806
$357$	0	0
$358$	35.8294	1.89364
$359$	−4.79276	−0.252952	−0.126476	−	0.991970i	$-0.540367\pi$
$359$	−4.79276	−0.252952	−0.126476	+	0.991970i	$0.540367\pi$
$360$	0	0
$361$	−9.52640	−0.501389
$362$	55.3862	2.91103
$363$	0	0
$364$	7.64803	0.400866
$365$	2.47330	0.129459
$366$	0	0
$367$	−23.3647	−1.21963	−0.609813	−	0.792545i	$-0.708755\pi$
$367$	−23.3647	−1.21963	−0.609813	+	0.792545i	$0.708755\pi$
$368$	3.45782	0.180251
$369$	0	0
$370$	−11.0230	−0.573058
$371$	17.7282	0.920402
$372$	0	0
$373$	23.7500	1.22973	0.614863	−	0.788634i	$-0.289211\pi$
$373$	23.7500	1.22973	0.614863	+	0.788634i	$0.289211\pi$
$374$	−37.5166	−1.93994
$375$	0	0
$376$	44.3194	2.28560
$377$	8.89390	0.458059
$378$	0	0
$379$	−27.7001	−1.42286	−0.711429	−	0.702758i	$-0.751952\pi$
$379$	−27.7001	−1.42286	−0.711429	+	0.702758i	$0.751952\pi$
$380$	−11.4293	−0.586312
$381$	0	0
$382$	−0.932257	−0.0476984
$383$	−6.03526	−0.308388	−0.154194	−	0.988041i	$-0.549278\pi$
$383$	−6.03526	−0.308388	−0.154194	+	0.988041i	$0.549278\pi$
$384$	0	0
$385$	8.39459	0.427828
$386$	23.6977	1.20618
$387$	0	0
$388$	22.1595	1.12498
$389$	−8.53031	−0.432504	−0.216252	−	0.976338i	$-0.569383\pi$
$389$	−8.53031	−0.432504	−0.216252	+	0.976338i	$0.569383\pi$
$390$	0	0
$391$	−5.34952	−0.270537
$392$	−13.0224	−0.657732
$393$	0	0
$394$	2.41648	0.121740
$395$	−11.6532	−0.586334
$396$	0	0
$397$	−14.6163	−0.733572	−0.366786	−	0.930305i	$-0.619542\pi$
$397$	−14.6163	−0.733572	−0.366786	+	0.930305i	$0.619542\pi$
$398$	−31.9109	−1.59955
$399$	0	0
$400$	2.36215	0.118107
$401$	−17.2987	−0.863858	−0.431929	−	0.901908i	$-0.642167\pi$
$401$	−17.2987	−0.863858	−0.431929	+	0.901908i	$0.642167\pi$
$402$	0	0
$403$	4.46841	0.222587
$404$	−22.8690	−1.13778
$405$	0	0
$406$	39.4304	1.95690
$407$	−19.8068	−0.981786
$408$	0	0
$409$	−2.78264	−0.137592	−0.0687962	−	0.997631i	$-0.521916\pi$
$409$	−2.78264	−0.137592	−0.0687962	+	0.997631i	$0.521916\pi$
$410$	−1.87788	−0.0927417
$411$	0	0
$412$	55.7615	2.74717
$413$	10.4852	0.515943
$414$	0	0
$415$	−12.5329	−0.615217
$416$	−2.68126	−0.131460
$417$	0	0
$418$	−31.5981	−1.54552
$419$	4.20691	0.205521	0.102761	−	0.994706i	$-0.467232\pi$
$419$	4.20691	0.205521	0.102761	+	0.994706i	$0.467232\pi$
$420$	0	0
$421$	19.0766	0.929737	0.464869	−	0.885380i	$-0.346102\pi$
$421$	19.0766	0.929737	0.464869	+	0.885380i	$0.346102\pi$
$422$	−60.9079	−2.96495
$423$	0	0
$424$	37.1458	1.80396
$425$	−3.65443	−0.177266
$426$	0	0
$427$	−10.9443	−0.529633
$428$	−9.98720	−0.482749
$429$	0	0
$430$	−8.62632	−0.415998
$431$	4.11651	0.198285	0.0991427	−	0.995073i	$-0.468390\pi$
$431$	4.11651	0.198285	0.0991427	+	0.995073i	$0.468390\pi$
$432$	0	0
$433$	−34.4248	−1.65435	−0.827176	−	0.561943i	$-0.810054\pi$
$433$	−34.4248	−1.65435	−0.827176	+	0.561943i	$0.810054\pi$
$434$	19.8103	0.950926
$435$	0	0
$436$	−2.49463	−0.119471
$437$	−4.50560	−0.215532
$438$	0	0
$439$	15.1883	0.724896	0.362448	−	0.932004i	$-0.381941\pi$
$439$	15.1883	0.724896	0.362448	+	0.932004i	$0.381941\pi$
$440$	17.5891	0.838529
$441$	0	0
$442$	−9.20472	−0.437824
$443$	−2.14752	−0.102032	−0.0510159	−	0.998698i	$-0.516246\pi$
$443$	−2.14752	−0.102032	−0.0510159	+	0.998698i	$0.516246\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	48.6313	2.30276
$447$	0	0
$448$	−21.1209	−0.997868
$449$	−32.5132	−1.53439	−0.767197	−	0.641411i	$-0.778349\pi$
$449$	−32.5132	−1.53439	−0.767197	+	0.641411i	$0.778349\pi$
$450$	0	0
$451$	−3.37429	−0.158889
$452$	32.4109	1.52448
$453$	0	0
$454$	−38.3887	−1.80167
$455$	2.05962	0.0965563
$456$	0	0
$457$	−15.9654	−0.746831	−0.373415	−	0.927664i	$-0.621813\pi$
$457$	−15.9654	−0.746831	−0.373415	+	0.927664i	$0.621813\pi$
$458$	36.4898	1.70506
$459$	0	0
$460$	5.43574	0.253443
$461$	0.406983	0.0189551	0.00947754	−	0.999955i	$-0.496983\pi$
$461$	0.406983	0.0189551	0.00947754	+	0.999955i	$0.496983\pi$
$462$	0	0
$463$	28.6219	1.33017	0.665087	−	0.746766i	$-0.268395\pi$
$463$	28.6219	1.33017	0.665087	+	0.746766i	$0.268395\pi$
$464$	19.9367	0.925538
$465$	0	0
$466$	−44.2477	−2.04974
$467$	38.5772	1.78514	0.892571	−	0.450907i	$-0.148900\pi$
$467$	38.5772	1.78514	0.892571	+	0.450907i	$0.148900\pi$
$468$	0	0
$469$	−15.4823	−0.714907
$470$	25.8674	1.19318
$471$	0	0
$472$	21.9696	1.01123
$473$	−15.5003	−0.712705
$474$	0	0
$475$	−3.07792	−0.141225
$476$	−26.5230	−1.21568
$477$	0	0
$478$	66.0932	3.02303
$479$	14.1350	0.645843	0.322922	−	0.946426i	$-0.395335\pi$
$479$	14.1350	0.645843	0.322922	+	0.946426i	$0.395335\pi$
$480$	0	0
$481$	−4.85960	−0.221579
$482$	65.0793	2.96428
$483$	0	0
$484$	27.6521	1.25691
$485$	5.96756	0.270973
$486$	0	0
$487$	−12.5869	−0.570369	−0.285184	−	0.958473i	$-0.592055\pi$
$487$	−12.5869	−0.570369	−0.285184	+	0.958473i	$0.592055\pi$
$488$	−22.9316	−1.03806
$489$	0	0
$490$	−7.60066	−0.343363
$491$	13.4541	0.607174	0.303587	−	0.952804i	$-0.401816\pi$
$491$	13.4541	0.607174	0.303587	+	0.952804i	$0.401816\pi$
$492$	0	0
$493$	−30.8437	−1.38913
$494$	−7.75262	−0.348807
$495$	0	0
$496$	10.0165	0.449752
$497$	−19.6665	−0.882163
$498$	0	0
$499$	17.5343	0.784945	0.392472	−	0.919764i	$-0.371620\pi$
$499$	17.5343	0.784945	0.392472	+	0.919764i	$0.371620\pi$
$500$	3.71333	0.166065
$501$	0	0
$502$	−30.9884	−1.38308
$503$	32.4398	1.44642	0.723209	−	0.690629i	$-0.242666\pi$
$503$	32.4398	1.44642	0.723209	+	0.690629i	$0.242666\pi$
$504$	0	0
$505$	−6.15863	−0.274056
$506$	15.0279	0.668073
$507$	0	0
$508$	−16.8959	−0.749632
$509$	−37.9507	−1.68214	−0.841068	−	0.540929i	$-0.818073\pi$
$509$	−37.9507	−1.68214	−0.841068	+	0.540929i	$0.818073\pi$
$510$	0	0
$511$	4.83412	0.213849
$512$	−25.3577	−1.12066
$513$	0	0
$514$	−45.3079	−1.99845
$515$	15.0166	0.661710
$516$	0	0
$517$	46.4802	2.04420
$518$	−21.5446	−0.946617
$519$	0	0
$520$	4.31550	0.189247
$521$	12.0211	0.526656	0.263328	−	0.964706i	$-0.415180\pi$
$521$	12.0211	0.526656	0.263328	+	0.964706i	$0.415180\pi$
$522$	0	0
$523$	−23.7443	−1.03826	−0.519132	−	0.854694i	$-0.673745\pi$
$523$	−23.7443	−1.03826	−0.519132	+	0.854694i	$0.673745\pi$
$524$	33.8929	1.48062
$525$	0	0
$526$	39.2264	1.71035
$527$	−15.4963	−0.675028
$528$	0	0
$529$	−20.8572	−0.906833
$530$	21.6805	0.941740
$531$	0	0
$532$	−22.3389	−0.968513
$533$	−0.827882	−0.0358596
$534$	0	0
$535$	−2.68955	−0.116280
$536$	−32.4400	−1.40119
$537$	0	0
$538$	−69.0949	−2.97889
$539$	−13.6573	−0.588263
$540$	0	0
$541$	3.22247	0.138545	0.0692725	−	0.997598i	$-0.477932\pi$
$541$	3.22247	0.138545	0.0692725	+	0.997598i	$0.477932\pi$
$542$	15.2874	0.656651
$543$	0	0
$544$	9.29851	0.398670
$545$	−0.671805	−0.0287770
$546$	0	0
$547$	−0.363571	−0.0155452	−0.00777259	−	0.999970i	$-0.502474\pi$
$547$	−0.363571	−0.0155452	−0.00777259	+	0.999970i	$0.502474\pi$
$548$	−62.5775	−2.67318
$549$	0	0
$550$	10.2661	0.437746
$551$	−25.9779	−1.10669
$552$	0	0
$553$	−22.7763	−0.968548
$554$	−6.95112	−0.295325
$555$	0	0
$556$	11.8902	0.504259
$557$	10.6808	0.452561	0.226280	−	0.974062i	$-0.427343\pi$
$557$	10.6808	0.452561	0.226280	+	0.974062i	$0.427343\pi$
$558$	0	0
$559$	−3.80301	−0.160850
$560$	4.61687	0.195098
$561$	0	0
$562$	15.0034	0.632882
$563$	16.3376	0.688549	0.344274	−	0.938869i	$-0.388125\pi$
$563$	16.3376	0.688549	0.344274	+	0.938869i	$0.388125\pi$
$564$	0	0
$565$	8.72825	0.367200
$566$	28.3996	1.19373
$567$	0	0
$568$	−41.2071	−1.72901
$569$	40.6357	1.70354	0.851768	−	0.523919i	$-0.175530\pi$
$569$	40.6357	1.70354	0.851768	+	0.523919i	$0.175530\pi$
$570$	0	0
$571$	−38.8677	−1.62656	−0.813281	−	0.581871i	$-0.802321\pi$
$571$	−38.8677	−1.62656	−0.813281	+	0.581871i	$0.802321\pi$
$572$	16.8062	0.702702
$573$	0	0
$574$	−3.67035	−0.153197
$575$	1.46385	0.0610466
$576$	0	0
$577$	−29.1209	−1.21232	−0.606158	−	0.795344i	$-0.707290\pi$
$577$	−29.1209	−1.21232	−0.606158	+	0.795344i	$0.707290\pi$
$578$	−8.71277	−0.362403
$579$	0	0
$580$	31.3408	1.30135
$581$	−24.4958	−1.01626
$582$	0	0
$583$	38.9568	1.61343
$584$	10.1289	0.419137
$585$	0	0
$586$	36.8489	1.52221
$587$	11.3820	0.469787	0.234894	−	0.972021i	$-0.424526\pi$
$587$	11.3820	0.469787	0.234894	+	0.972021i	$0.424526\pi$
$588$	0	0
$589$	−13.0516	−0.537783
$590$	12.8228	0.527905
$591$	0	0
$592$	−10.8934	−0.447714
$593$	5.10274	0.209544	0.104772	−	0.994496i	$-0.466589\pi$
$593$	5.10274	0.209544	0.104772	+	0.994496i	$0.466589\pi$
$594$	0	0
$595$	−7.14266	−0.292821
$596$	−54.3809	−2.22753
$597$	0	0
$598$	3.68711	0.150777
$599$	−22.6299	−0.924631	−0.462316	−	0.886715i	$-0.652981\pi$
$599$	−22.6299	−0.924631	−0.462316	+	0.886715i	$0.652981\pi$
$600$	0	0
$601$	27.2940	1.11335	0.556673	−	0.830732i	$-0.312078\pi$
$601$	27.2940	1.11335	0.556673	+	0.830732i	$0.312078\pi$
$602$	−16.8603	−0.687176
$603$	0	0
$604$	−66.2497	−2.69566
$605$	7.44671	0.302752
$606$	0	0
$607$	34.1969	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$607$	34.1969	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$608$	7.83161	0.317614
$609$	0	0
$610$	−13.3842	−0.541912
$611$	11.4039	0.461354
$612$	0	0
$613$	24.2886	0.981009	0.490504	−	0.871439i	$-0.336813\pi$
$613$	24.2886	0.981009	0.490504	+	0.871439i	$0.336813\pi$
$614$	−31.4785	−1.27037
$615$	0	0
$616$	34.3783	1.38514
$617$	−31.9292	−1.28542	−0.642711	−	0.766109i	$-0.722190\pi$
$617$	−31.9292	−1.28542	−0.642711	+	0.766109i	$0.722190\pi$
$618$	0	0
$619$	−29.1111	−1.17007	−0.585037	−	0.811007i	$-0.698920\pi$
$619$	−29.1111	−1.17007	−0.585037	+	0.811007i	$0.698920\pi$
$620$	15.7460	0.632375
$621$	0	0
$622$	−32.5351	−1.30454
$623$	1.95452	0.0783062
$624$	0	0
$625$	1.00000	0.0400000
$626$	−47.8106	−1.91089
$627$	0	0
$628$	57.4701	2.29331
$629$	16.8529	0.671969
$630$	0	0
$631$	26.7634	1.06543	0.532717	−	0.846293i	$-0.321171\pi$
$631$	26.7634	1.06543	0.532717	+	0.846293i	$0.321171\pi$
$632$	−47.7231	−1.89832
$633$	0	0
$634$	−42.8370	−1.70128
$635$	−4.55006	−0.180564
$636$	0	0
$637$	−3.35084	−0.132765
$638$	86.6463	3.43036
$639$	0	0
$640$	−20.7406	−0.819846
$641$	16.4357	0.649173	0.324586	−	0.945856i	$-0.394775\pi$
$641$	16.4357	0.649173	0.324586	+	0.945856i	$0.394775\pi$
$642$	0	0
$643$	−28.2642	−1.11463	−0.557316	−	0.830300i	$-0.688169\pi$
$643$	−28.2642	−1.11463	−0.557316	+	0.830300i	$0.688169\pi$
$644$	10.6243	0.418654
$645$	0	0
$646$	26.8858	1.05781
$647$	−16.9725	−0.667258	−0.333629	−	0.942704i	$-0.608273\pi$
$647$	−16.9725	−0.667258	−0.333629	+	0.942704i	$0.608273\pi$
$648$	0	0
$649$	23.0407	0.904428
$650$	2.51878	0.0987948
$651$	0	0
$652$	−8.88142	−0.347823
$653$	39.2631	1.53648	0.768242	−	0.640160i	$-0.221132\pi$
$653$	39.2631	1.53648	0.768242	+	0.640160i	$0.221132\pi$
$654$	0	0
$655$	9.12736	0.356635
$656$	−1.85579	−0.0724566
$657$	0	0
$658$	50.5584	1.97097
$659$	−3.77282	−0.146968	−0.0734841	−	0.997296i	$-0.523412\pi$
$659$	−3.77282	−0.146968	−0.0734841	+	0.997296i	$0.523412\pi$
$660$	0	0
$661$	1.17732	0.0457923	0.0228962	−	0.999738i	$-0.492711\pi$
$661$	1.17732	0.0457923	0.0228962	+	0.999738i	$0.492711\pi$
$662$	−48.4375	−1.88258
$663$	0	0
$664$	−51.3260	−1.99183
$665$	−6.01586	−0.233285
$666$	0	0
$667$	12.3550	0.478386
$668$	−2.39459	−0.0926495
$669$	0	0
$670$	−18.9339	−0.731481
$671$	−24.0496	−0.928425
$672$	0	0
$673$	45.6233	1.75865	0.879324	−	0.476224i	$-0.157995\pi$
$673$	45.6233	1.75865	0.879324	+	0.476224i	$0.157995\pi$
$674$	−38.5768	−1.48592
$675$	0	0
$676$	−44.1499	−1.69807
$677$	43.1019	1.65654	0.828270	−	0.560329i	$-0.189325\pi$
$677$	43.1019	1.65654	0.828270	+	0.560329i	$0.189325\pi$
$678$	0	0
$679$	11.6637	0.447612
$680$	−14.9660	−0.573919
$681$	0	0
$682$	43.5322	1.66694
$683$	−24.1509	−0.924108	−0.462054	−	0.886852i	$-0.652887\pi$
$683$	−24.1509	−0.924108	−0.462054	+	0.886852i	$0.652887\pi$
$684$	0	0
$685$	−16.8521	−0.643887
$686$	−47.5583	−1.81578
$687$	0	0
$688$	−8.52488	−0.325008
$689$	9.55807	0.364134
$690$	0	0
$691$	−31.4456	−1.19625	−0.598124	−	0.801404i	$-0.704087\pi$
$691$	−31.4456	−1.19625	−0.598124	+	0.801404i	$0.704087\pi$
$692$	71.3524	2.71241
$693$	0	0
$694$	−34.2753	−1.30107
$695$	3.20205	0.121461
$696$	0	0
$697$	2.87106	0.108749
$698$	−76.2381	−2.88566
$699$	0	0
$700$	7.25777	0.274318
$701$	45.2299	1.70831	0.854155	−	0.520019i	$-0.174075\pi$
$701$	45.2299	1.70831	0.854155	+	0.520019i	$0.174075\pi$
$702$	0	0
$703$	14.1942	0.535346
$704$	−46.4121	−1.74922
$705$	0	0
$706$	−5.88122	−0.221343
$707$	−12.0372	−0.452705
$708$	0	0
$709$	11.5007	0.431919	0.215960	−	0.976402i	$-0.430712\pi$
$709$	11.5007	0.431919	0.215960	+	0.976402i	$0.430712\pi$
$710$	−24.0509	−0.902615
$711$	0	0
$712$	4.09529	0.153478
$713$	6.20729	0.232465
$714$	0	0
$715$	4.52591	0.169259
$716$	55.6619	2.08018
$717$	0	0
$718$	−11.4559	−0.427532
$719$	51.2805	1.91244	0.956220	−	0.292648i	$-0.0945365\pi$
$719$	51.2805	1.91244	0.956220	+	0.292648i	$0.0945365\pi$
$720$	0	0
$721$	29.3502	1.09306
$722$	−22.7705	−0.847431
$723$	0	0
$724$	86.0439	3.19780
$725$	8.44007	0.313456
$726$	0	0
$727$	36.9416	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$727$	36.9416	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$728$	8.43473	0.312612
$729$	0	0
$730$	5.91183	0.218807
$731$	13.1887	0.487801
$732$	0	0
$733$	14.0425	0.518673	0.259337	−	0.965787i	$-0.416496\pi$
$733$	14.0425	0.518673	0.259337	+	0.965787i	$0.416496\pi$
$734$	−55.8476	−2.06137
$735$	0	0
$736$	−3.72467	−0.137293
$737$	−34.0216	−1.25320
$738$	0	0
$739$	−18.4116	−0.677282	−0.338641	−	0.940916i	$-0.609967\pi$
$739$	−18.4116	−0.677282	−0.338641	+	0.940916i	$0.609967\pi$
$740$	−17.1245	−0.629509
$741$	0	0
$742$	42.3749	1.55563
$743$	20.0671	0.736191	0.368096	−	0.929788i	$-0.380010\pi$
$743$	20.0671	0.736191	0.368096	+	0.929788i	$0.380010\pi$
$744$	0	0
$745$	−14.6448	−0.536544
$746$	56.7685	2.07844
$747$	0	0
$748$	−58.2831	−2.13104
$749$	−5.25679	−0.192079
$750$	0	0
$751$	−17.5280	−0.639607	−0.319803	−	0.947484i	$-0.603617\pi$
$751$	−17.5280	−0.639607	−0.319803	+	0.947484i	$0.603617\pi$
$752$	25.5633	0.932196
$753$	0	0
$754$	21.2587	0.774196
$755$	−17.8411	−0.649303
$756$	0	0
$757$	21.0316	0.764406	0.382203	−	0.924078i	$-0.375166\pi$
$757$	21.0316	0.764406	0.382203	+	0.924078i	$0.375166\pi$
$758$	−66.2103	−2.40487
$759$	0	0
$760$	−12.6050	−0.457231
$761$	37.2035	1.34863	0.674313	−	0.738445i	$-0.264440\pi$
$761$	37.2035	1.34863	0.674313	+	0.738445i	$0.264440\pi$
$762$	0	0
$763$	−1.31306	−0.0475358
$764$	−1.44829	−0.0523972
$765$	0	0
$766$	−14.4258	−0.521227
$767$	5.65305	0.204120
$768$	0	0
$769$	−15.2246	−0.549013	−0.274507	−	0.961585i	$-0.588515\pi$
$769$	−15.2246	−0.549013	−0.274507	+	0.961585i	$0.588515\pi$
$770$	20.0652	0.723101
$771$	0	0
$772$	36.8150	1.32500
$773$	−26.7520	−0.962202	−0.481101	−	0.876665i	$-0.659763\pi$
$773$	−26.7520	−0.962202	−0.481101	+	0.876665i	$0.659763\pi$
$774$	0	0
$775$	4.24040	0.152320
$776$	24.4389	0.877306
$777$	0	0
$778$	−20.3896	−0.731004
$779$	2.41813	0.0866386
$780$	0	0
$781$	−43.2162	−1.54640
$782$	−12.7867	−0.457253
$783$	0	0
$784$	−7.51128	−0.268260
$785$	15.4767	0.552387
$786$	0	0
$787$	28.4808	1.01523	0.507615	−	0.861584i	$-0.330527\pi$
$787$	28.4808	1.01523	0.507615	+	0.861584i	$0.330527\pi$
$788$	3.75406	0.133733
$789$	0	0
$790$	−27.8540	−0.991002
$791$	17.0595	0.606568
$792$	0	0
$793$	−5.90058	−0.209536
$794$	−34.9367	−1.23986
$795$	0	0
$796$	−49.5744	−1.75712
$797$	−44.0413	−1.56002	−0.780012	−	0.625764i	$-0.784787\pi$
$797$	−44.0413	−1.56002	−0.780012	+	0.625764i	$0.784787\pi$
$798$	0	0
$799$	−39.5484	−1.39912
$800$	−2.54445	−0.0899597
$801$	0	0
$802$	−41.3484	−1.46006
$803$	10.6227	0.374869
$804$	0	0
$805$	2.86111	0.100841
$806$	10.6807	0.376210
$807$	0	0
$808$	−25.2214	−0.887286
$809$	30.2316	1.06289	0.531444	−	0.847094i	$-0.321650\pi$
$809$	30.2316	1.06289	0.531444	+	0.847094i	$0.321650\pi$
$810$	0	0
$811$	−34.5210	−1.21220	−0.606099	−	0.795389i	$-0.707266\pi$
$811$	−34.5210	−1.21220	−0.606099	+	0.795389i	$0.707266\pi$
$812$	61.2561	2.14967
$813$	0	0
$814$	−47.3433	−1.65938
$815$	−2.39177	−0.0837800
$816$	0	0
$817$	11.1081	0.388622
$818$	−6.65121	−0.232554
$819$	0	0
$820$	−2.91733	−0.101878
$821$	−35.2307	−1.22956	−0.614780	−	0.788698i	$-0.710755\pi$
$821$	−35.2307	−1.22956	−0.614780	+	0.788698i	$0.710755\pi$
$822$	0	0
$823$	−40.5379	−1.41306	−0.706532	−	0.707681i	$-0.749741\pi$
$823$	−40.5379	−1.41306	−0.706532	+	0.707681i	$0.749741\pi$
$824$	61.4973	2.14236
$825$	0	0
$826$	25.0623	0.872030
$827$	30.0036	1.04333	0.521664	−	0.853151i	$-0.325312\pi$
$827$	30.0036	1.04333	0.521664	+	0.853151i	$0.325312\pi$
$828$	0	0
$829$	11.3949	0.395761	0.197881	−	0.980226i	$-0.436594\pi$
$829$	11.3949	0.395761	0.197881	+	0.980226i	$0.436594\pi$
$830$	−29.9569	−1.03982
$831$	0	0
$832$	−11.3872	−0.394781
$833$	11.6206	0.402628
$834$	0	0
$835$	−0.644864	−0.0223164
$836$	−49.0886	−1.69776
$837$	0	0
$838$	10.0556	0.347365
$839$	−38.3055	−1.32245	−0.661226	−	0.750187i	$-0.729963\pi$
$839$	−38.3055	−1.32245	−0.661226	+	0.750187i	$0.729963\pi$
$840$	0	0
$841$	42.2348	1.45637
$842$	45.5980	1.57141
$843$	0	0
$844$	−94.6221	−3.25703
$845$	−11.8896	−0.409014
$846$	0	0
$847$	14.5547	0.500107
$848$	21.4255	0.735755
$849$	0	0
$850$	−8.73503	−0.299609
$851$	−6.75071	−0.231412
$852$	0	0
$853$	−5.70914	−0.195477	−0.0977386	−	0.995212i	$-0.531161\pi$
$853$	−5.70914	−0.195477	−0.0977386	+	0.995212i	$0.531161\pi$
$854$	−26.1597	−0.895168
$855$	0	0
$856$	−11.0145	−0.376468
$857$	18.5138	0.632419	0.316210	−	0.948689i	$-0.397590\pi$
$857$	18.5138	0.632419	0.316210	+	0.948689i	$0.397590\pi$
$858$	0	0
$859$	36.1682	1.23404	0.617021	−	0.786947i	$-0.288339\pi$
$859$	36.1682	1.23404	0.617021	+	0.786947i	$0.288339\pi$
$860$	−13.4012	−0.456978
$861$	0	0
$862$	9.83952	0.335135
$863$	48.1408	1.63873	0.819366	−	0.573271i	$-0.194326\pi$
$863$	48.1408	1.63873	0.819366	+	0.573271i	$0.194326\pi$
$864$	0	0
$865$	19.2152	0.653337
$866$	−82.2842	−2.79613
$867$	0	0
$868$	30.7759	1.04460
$869$	−50.0499	−1.69783
$870$	0	0
$871$	−8.34722	−0.282835
$872$	−2.75124	−0.0931687
$873$	0	0
$874$	−10.7695	−0.364285
$875$	1.95452	0.0660748
$876$	0	0
$877$	16.7949	0.567122	0.283561	−	0.958954i	$-0.408484\pi$
$877$	16.7949	0.567122	0.283561	+	0.958954i	$0.408484\pi$
$878$	36.3038	1.22519
$879$	0	0
$880$	10.1453	0.341999
$881$	−10.4309	−0.351425	−0.175713	−	0.984442i	$-0.556223\pi$
$881$	−10.4309	−0.351425	−0.175713	+	0.984442i	$0.556223\pi$
$882$	0	0
$883$	29.5575	0.994690	0.497345	−	0.867553i	$-0.334308\pi$
$883$	29.5575	0.994690	0.497345	+	0.867553i	$0.334308\pi$
$884$	−14.2998	−0.480954
$885$	0	0
$886$	−5.13313	−0.172451
$887$	−30.1297	−1.01166	−0.505828	−	0.862634i	$-0.668813\pi$
$887$	−30.1297	−1.01166	−0.505828	+	0.862634i	$0.668813\pi$
$888$	0	0
$889$	−8.89318	−0.298268
$890$	2.39026	0.0801216
$891$	0	0
$892$	75.5500	2.52960
$893$	−33.3094	−1.11466
$894$	0	0
$895$	14.9898	0.501052
$896$	−40.5380	−1.35428
$897$	0	0
$898$	−77.7150	−2.59338
$899$	35.7893	1.19364
$900$	0	0
$901$	−33.1470	−1.10429
$902$	−8.06541	−0.268549
$903$	0	0
$904$	35.7448	1.18885
$905$	23.1716	0.770251
$906$	0	0
$907$	34.4041	1.14237	0.571185	−	0.820821i	$-0.306484\pi$
$907$	34.4041	1.14237	0.571185	+	0.820821i	$0.306484\pi$
$908$	−59.6378	−1.97915
$909$	0	0
$910$	4.92301	0.163196
$911$	48.0626	1.59239	0.796193	−	0.605043i	$-0.206844\pi$
$911$	48.0626	1.59239	0.796193	+	0.605043i	$0.206844\pi$
$912$	0	0
$913$	−53.8284	−1.78146
$914$	−38.1615	−1.26227
$915$	0	0
$916$	56.6879	1.87302
$917$	17.8396	0.589116
$918$	0	0
$919$	9.64985	0.318319	0.159160	−	0.987253i	$-0.449122\pi$
$919$	9.64985	0.318319	0.159160	+	0.987253i	$0.449122\pi$
$920$	5.99488	0.197645
$921$	0	0
$922$	0.972793	0.0320372
$923$	−10.6031	−0.349006
$924$	0	0
$925$	−4.61163	−0.151630
$926$	68.4137	2.24822
$927$	0	0
$928$	−21.4753	−0.704961
$929$	4.87453	0.159928	0.0799641	−	0.996798i	$-0.474519\pi$
$929$	4.87453	0.159928	0.0799641	+	0.996798i	$0.474519\pi$
$930$	0	0
$931$	9.78734	0.320767
$932$	−68.7400	−2.25165
$933$	0	0
$934$	92.2095	3.01719
$935$	−15.6957	−0.513303
$936$	0	0
$937$	6.79035	0.221831	0.110916	−	0.993830i	$-0.464622\pi$
$937$	6.79035	0.221831	0.110916	+	0.993830i	$0.464622\pi$
$938$	−37.0067	−1.20831
$939$	0	0
$940$	40.1858	1.31072
$941$	26.9378	0.878147	0.439074	−	0.898451i	$-0.355307\pi$
$941$	26.9378	0.878147	0.439074	+	0.898451i	$0.355307\pi$
$942$	0	0
$943$	−1.15005	−0.0374508
$944$	12.6720	0.412437
$945$	0	0
$946$	−37.0497	−1.20459
$947$	−0.0394219	−0.00128104	−0.000640519	−	1.00000i	$-0.500204\pi$
$947$	−0.0394219	−0.00128104	−0.000640519		1.00000i	$0.500204\pi$
$948$	0	0
$949$	2.60629	0.0846039
$950$	−7.35702	−0.238693
$951$	0	0
$952$	−29.2513	−0.948040
$953$	3.27013	0.105930	0.0529650	−	0.998596i	$-0.483133\pi$
$953$	3.27013	0.105930	0.0529650	+	0.998596i	$0.483133\pi$
$954$	0	0
$955$	−0.390024	−0.0126209
$956$	102.678	3.32083
$957$	0	0
$958$	33.7862	1.09158
$959$	−32.9379	−1.06362
$960$	0	0
$961$	−13.0190	−0.419967
$962$	−11.6157	−0.374505
$963$	0	0
$964$	101.102	3.25629
$965$	9.91428	0.319152
$966$	0	0
$967$	47.5329	1.52855	0.764277	−	0.644888i	$-0.223096\pi$
$967$	47.5329	1.52855	0.764277	+	0.644888i	$0.223096\pi$
$968$	30.4965	0.980193
$969$	0	0
$970$	14.2640	0.457989
$971$	25.1811	0.808101	0.404050	−	0.914737i	$-0.367602\pi$
$971$	25.1811	0.808101	0.404050	+	0.914737i	$0.367602\pi$
$972$	0	0
$973$	6.25846	0.200637
$974$	−30.0860	−0.964019
$975$	0	0
$976$	−13.2268	−0.423381
$977$	33.1228	1.05969	0.529846	−	0.848094i	$-0.322250\pi$
$977$	33.1228	1.05969	0.529846	+	0.848094i	$0.322250\pi$
$978$	0	0
$979$	4.29496	0.137268
$980$	−11.8078	−0.377187
$981$	0	0
$982$	32.1587	1.02623
$983$	22.3428	0.712625	0.356313	−	0.934367i	$-0.384034\pi$
$983$	22.3428	0.712625	0.356313	+	0.934367i	$0.384034\pi$
$984$	0	0
$985$	1.01097	0.0322122
$986$	−73.7243	−2.34786
$987$	0	0
$988$	−12.0439	−0.383167
$989$	−5.28295	−0.167988
$990$	0	0
$991$	37.5767	1.19366	0.596832	−	0.802366i	$-0.296426\pi$
$991$	37.5767	1.19366	0.596832	+	0.802366i	$0.296426\pi$
$992$	−10.7895	−0.342566
$993$	0	0
$994$	−47.0080	−1.49100
$995$	−13.3504	−0.423236
$996$	0	0
$997$	21.4843	0.680416	0.340208	−	0.940350i	$-0.389502\pi$
$997$	21.4843	0.680416	0.340208	+	0.940350i	$0.389502\pi$
$998$	41.9116	1.32669
$999$	0	0

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39026 q^{2} +3.71333 q^{4} +1.00000 q^{5} +1.95452 q^{7} +4.09529 q^{8} +O(q^{10})\)	\(q+2.39026 q^{2} +3.71333 q^{4} +1.00000 q^{5} +1.95452 q^{7} +4.09529 q^{8} +2.39026 q^{10} +4.29496 q^{11} +1.05377 q^{13} +4.67180 q^{14} +2.36215 q^{16} -3.65443 q^{17} -3.07792 q^{19} +3.71333 q^{20} +10.2661 q^{22} +1.46385 q^{23} +1.00000 q^{25} +2.51878 q^{26} +7.25777 q^{28} +8.44007 q^{29} +4.24040 q^{31} -2.54445 q^{32} -8.73503 q^{34} +1.95452 q^{35} -4.61163 q^{37} -7.35702 q^{38} +4.09529 q^{40} -0.785638 q^{41} -3.60895 q^{43} +15.9486 q^{44} +3.49897 q^{46} +10.8220 q^{47} -3.17985 q^{49} +2.39026 q^{50} +3.91300 q^{52} +9.07035 q^{53} +4.29496 q^{55} +8.00433 q^{56} +20.1739 q^{58} +5.36459 q^{59} -5.59950 q^{61} +10.1356 q^{62} -10.8062 q^{64} +1.05377 q^{65} -7.92129 q^{67} -13.5701 q^{68} +4.67180 q^{70} -10.0621 q^{71} +2.47330 q^{73} -11.0230 q^{74} -11.4293 q^{76} +8.39459 q^{77} -11.6532 q^{79} +2.36215 q^{80} -1.87788 q^{82} -12.5329 q^{83} -3.65443 q^{85} -8.62632 q^{86} +17.5891 q^{88} +1.00000 q^{89} +2.05962 q^{91} +5.43574 q^{92} +25.8674 q^{94} -3.07792 q^{95} +5.96756 q^{97} -7.60066 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} + O(q^{10}) \)	\( 4q - q^{2} + 3q^{4} + 4q^{5} + 2q^{7} + 9q^{8} - q^{10} + 14q^{11} - 5q^{13} + 5q^{14} - 3q^{16} + 3q^{17} - q^{19} + 3q^{20} + 2q^{22} + 3q^{23} + 4q^{25} + 9q^{26} + 5q^{28} + 10q^{29} - 11q^{31} + 2q^{32} - 8q^{34} + 2q^{35} + 3q^{37} - 2q^{38} + 9q^{40} + 3q^{41} + 9q^{43} + 9q^{44} - 4q^{46} + 24q^{47} - q^{50} + 8q^{52} - 3q^{53} + 14q^{55} + 13q^{56} + 19q^{58} + 22q^{59} - 3q^{61} + 24q^{62} - 11q^{64} - 5q^{65} - 9q^{67} - 31q^{68} + 5q^{70} - 16q^{71} + 3q^{73} - 31q^{74} - 24q^{76} + 4q^{77} - 27q^{79} - 3q^{80} - 15q^{82} - 6q^{83} + 3q^{85} - 15q^{86} + 30q^{88} + 4q^{89} - 29q^{91} + 17q^{92} + 13q^{94} - q^{95} + 41q^{97} - 22q^{98} + O(q^{100}) \)

Introduction and more

L-functions

Modular Forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$-expansion

Coefficient data

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4005.2.a.l.1.4

Level	4005
Weight	2
Character	4005.1
Self dual	Yes
Analytic conductor	31.980
Analytic rank	0
Dimension	4
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(31.9800860095\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	4.4.725.1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$