LMFDB - Embedded newform 2151.2.a.e.1.4

Newspace parameters

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1767625.1
Defining polynomial:	$x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.39213$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

$f(q)$	$=$	$q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} +O(q^{10})$	\(q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} -1.50593 q^{10} +5.08858 q^{11} +6.43675 q^{13} -1.58137 q^{14} -0.201602 q^{16} +2.24001 q^{17} -3.96067 q^{19} +1.99270 q^{20} +4.57824 q^{22} -7.71459 q^{23} -2.19839 q^{25} +5.79120 q^{26} +2.09253 q^{28} -0.101311 q^{29} -4.63283 q^{31} +5.55970 q^{32} +2.01536 q^{34} +2.94196 q^{35} -7.78041 q^{37} -3.56345 q^{38} +4.80472 q^{40} -1.42586 q^{41} -3.65358 q^{43} -6.05808 q^{44} -6.94088 q^{46} +7.73754 q^{47} -3.91066 q^{49} -1.97791 q^{50} -7.66311 q^{52} -1.61074 q^{53} -8.51728 q^{55} +5.04542 q^{56} -0.0911499 q^{58} +2.46261 q^{59} -4.77088 q^{61} -4.16819 q^{62} +5.40531 q^{64} -10.7738 q^{65} -3.42357 q^{67} -2.66679 q^{68} +2.64691 q^{70} +13.1223 q^{71} -14.4582 q^{73} -7.00010 q^{74} +4.71528 q^{76} -8.94396 q^{77} -13.0644 q^{79} +0.337442 q^{80} -1.28286 q^{82} +0.423954 q^{83} -3.74933 q^{85} -3.28716 q^{86} -14.6070 q^{88} -17.1426 q^{89} -11.3136 q^{91} +9.18440 q^{92} +6.96153 q^{94} +6.62938 q^{95} +6.06424 q^{97} -3.51845 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) $	$ 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{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 $

$2$	0.899709	0.636190	0.318095	−	0.948059i	$-0.396957\pi$
$2$	0.899709	0.636190	0.318095	+	0.948059i	$0.396957\pi$
$3$	0	0
$3$	0	0
$4$	−1.19052	−0.595262
$5$	−1.67380	−0.748547	−0.374273	−	0.927318i	$-0.622108\pi$
$5$	−1.67380	−0.748547	−0.374273	+	0.927318i	$0.622108\pi$
$6$	0	0
$7$	−1.75765	−0.664330	−0.332165	−	0.943221i	$-0.607779\pi$
$7$	−1.75765	−0.664330	−0.332165	+	0.943221i	$0.607779\pi$
$8$	−2.87054	−1.01489
$9$	0	0
$10$	−1.50593	−0.476218
$11$	5.08858	1.53427	0.767133	−	0.641488i	$-0.221683\pi$
$11$	5.08858	1.53427	0.767133	+	0.641488i	$0.221683\pi$
$12$	0	0
$13$	6.43675	1.78523	0.892617	−	0.450816i	$-0.148867\pi$
$13$	6.43675	1.78523	0.892617	+	0.450816i	$0.148867\pi$
$14$	−1.58137	−0.422640
$15$	0	0
$16$	−0.201602	−0.0504005
$17$	2.24001	0.543283	0.271641	−	0.962399i	$-0.412434\pi$
$17$	2.24001	0.543283	0.271641	+	0.962399i	$0.412434\pi$
$18$	0	0
$19$	−3.96067	−0.908641	−0.454320	−	0.890838i	$-0.650118\pi$
$19$	−3.96067	−0.908641	−0.454320	+	0.890838i	$0.650118\pi$
$20$	1.99270	0.445582
$21$	0	0
$22$	4.57824	0.976085
$23$	−7.71459	−1.60860	−0.804301	−	0.594222i	$-0.797460\pi$
$23$	−7.71459	−1.60860	−0.804301	+	0.594222i	$0.797460\pi$
$24$	0	0
$25$	−2.19839	−0.439678
$26$	5.79120	1.13575
$27$	0	0
$28$	2.09253	0.395451
$29$	−0.101311	−0.0188129	−0.00940644	−	0.999956i	$-0.502994\pi$
$29$	−0.101311	−0.0188129	−0.00940644	+	0.999956i	$0.502994\pi$
$30$	0	0
$31$	−4.63283	−0.832080	−0.416040	−	0.909346i	$-0.636582\pi$
$31$	−4.63283	−0.832080	−0.416040	+	0.909346i	$0.636582\pi$
$32$	5.55970	0.982826
$33$	0	0
$34$	2.01536	0.345631
$35$	2.94196	0.497282
$36$	0	0
$37$	−7.78041	−1.27909	−0.639546	−	0.768753i	$-0.720878\pi$
$37$	−7.78041	−1.27909	−0.639546	+	0.768753i	$0.720878\pi$
$38$	−3.56345	−0.578068
$39$	0	0
$40$	4.80472	0.759692
$41$	−1.42586	−0.222682	−0.111341	−	0.993782i	$-0.535515\pi$
$41$	−1.42586	−0.222682	−0.111341	+	0.993782i	$0.535515\pi$
$42$	0	0
$43$	−3.65358	−0.557166	−0.278583	−	0.960412i	$-0.589865\pi$
$43$	−3.65358	−0.557166	−0.278583	+	0.960412i	$0.589865\pi$
$44$	−6.05808	−0.913291
$45$	0	0
$46$	−6.94088	−1.02338
$47$	7.73754	1.12864	0.564318	−	0.825558i	$-0.309139\pi$
$47$	7.73754	1.12864	0.564318	+	0.825558i	$0.309139\pi$
$48$	0	0
$49$	−3.91066	−0.558665
$50$	−1.97791	−0.279719
$51$	0	0
$52$	−7.66311	−1.06268
$53$	−1.61074	−0.221252	−0.110626	−	0.993862i	$-0.535285\pi$
$53$	−1.61074	−0.221252	−0.110626	+	0.993862i	$0.535285\pi$
$54$	0	0
$55$	−8.51728	−1.14847
$56$	5.04542	0.674222
$57$	0	0
$58$	−0.0911499	−0.0119686
$59$	2.46261	0.320604	0.160302	−	0.987068i	$-0.448753\pi$
$59$	2.46261	0.320604	0.160302	+	0.987068i	$0.448753\pi$
$60$	0	0
$61$	−4.77088	−0.610849	−0.305424	−	0.952216i	$-0.598798\pi$
$61$	−4.77088	−0.610849	−0.305424	+	0.952216i	$0.598798\pi$
$62$	−4.16819	−0.529361
$63$	0	0
$64$	5.40531	0.675664
$65$	−10.7738	−1.33633
$66$	0	0
$67$	−3.42357	−0.418255	−0.209128	−	0.977888i	$-0.567062\pi$
$67$	−3.42357	−0.418255	−0.209128	+	0.977888i	$0.567062\pi$
$68$	−2.66679	−0.323396
$69$	0	0
$70$	2.64691	0.316366
$71$	13.1223	1.55733	0.778665	−	0.627440i	$-0.215897\pi$
$71$	13.1223	1.55733	0.778665	+	0.627440i	$0.215897\pi$
$72$	0	0
$73$	−14.4582	−1.69220	−0.846100	−	0.533025i	$-0.821055\pi$
$73$	−14.4582	−1.69220	−0.846100	+	0.533025i	$0.821055\pi$
$74$	−7.00010	−0.813745
$75$	0	0
$76$	4.71528	0.540880
$77$	−8.94396	−1.01926
$78$	0	0
$79$	−13.0644	−1.46986	−0.734930	−	0.678143i	$-0.762785\pi$
$79$	−13.0644	−1.46986	−0.734930	+	0.678143i	$0.762785\pi$
$80$	0.337442	0.0377271
$81$	0	0
$82$	−1.28286	−0.141668
$83$	0.423954	0.0465350	0.0232675	−	0.999729i	$-0.492593\pi$
$83$	0.423954	0.0465350	0.0232675	+	0.999729i	$0.492593\pi$
$84$	0	0
$85$	−3.74933	−0.406672
$86$	−3.28716	−0.354463
$87$	0	0
$88$	−14.6070	−1.55711
$89$	−17.1426	−1.81711	−0.908556	−	0.417763i	$-0.862814\pi$
$89$	−17.1426	−1.81711	−0.908556	+	0.417763i	$0.862814\pi$
$90$	0	0
$91$	−11.3136	−1.18598
$92$	9.18440	0.957540
$93$	0	0
$94$	6.96153	0.718027
$95$	6.62938	0.680160
$96$	0	0
$97$	6.06424	0.615730	0.307865	−	0.951430i	$-0.400386\pi$
$97$	6.06424	0.615730	0.307865	+	0.951430i	$0.400386\pi$
$98$	−3.51845	−0.355417
$99$	0	0
$100$	2.61724	0.261724
$101$	−10.1166	−1.00664	−0.503320	−	0.864100i	$-0.667888\pi$
$101$	−10.1166	−1.00664	−0.503320	+	0.864100i	$0.667888\pi$
$102$	0	0
$103$	2.53914	0.250189	0.125094	−	0.992145i	$-0.460077\pi$
$103$	2.53914	0.250189	0.125094	+	0.992145i	$0.460077\pi$
$104$	−18.4770	−1.81182
$105$	0	0
$106$	−1.44919	−0.140758
$107$	−9.82497	−0.949816	−0.474908	−	0.880035i	$-0.657519\pi$
$107$	−9.82497	−0.949816	−0.474908	+	0.880035i	$0.657519\pi$
$108$	0	0
$109$	−15.1110	−1.44738	−0.723688	−	0.690127i	$-0.757555\pi$
$109$	−15.1110	−1.44738	−0.723688	+	0.690127i	$0.757555\pi$
$110$	−7.66307	−0.730645
$111$	0	0
$112$	0.354346	0.0334826
$113$	−2.58916	−0.243568	−0.121784	−	0.992557i	$-0.538861\pi$
$113$	−2.58916	−0.243568	−0.121784	+	0.992557i	$0.538861\pi$
$114$	0	0
$115$	12.9127	1.20411
$116$	0.120613	0.0111986
$117$	0	0
$118$	2.21563	0.203965
$119$	−3.93716	−0.360919
$120$	0	0
$121$	14.8937	1.35397
$122$	−4.29240	−0.388616
$123$	0	0
$124$	5.51549	0.495306
$125$	12.0487	1.07767
$126$	0	0
$127$	7.24277	0.642692	0.321346	−	0.946962i	$-0.395865\pi$
$127$	7.24277	0.642692	0.321346	+	0.946962i	$0.395865\pi$
$128$	−6.25619	−0.552975
$129$	0	0
$130$	−9.69332	−0.850160
$131$	−2.14842	−0.187709	−0.0938543	−	0.995586i	$-0.529919\pi$
$131$	−2.14842	−0.187709	−0.0938543	+	0.995586i	$0.529919\pi$
$132$	0	0
$133$	6.96149	0.603637
$134$	−3.08021	−0.266090
$135$	0	0
$136$	−6.43005	−0.551372
$137$	−6.53872	−0.558640	−0.279320	−	0.960198i	$-0.590109\pi$
$137$	−6.53872	−0.558640	−0.279320	+	0.960198i	$0.590109\pi$
$138$	0	0
$139$	9.78192	0.829691	0.414846	−	0.909892i	$-0.363836\pi$
$139$	9.78192	0.829691	0.414846	+	0.909892i	$0.363836\pi$
$140$	−3.50248	−0.296013
$141$	0	0
$142$	11.8062	0.990758
$143$	32.7540	2.73902
$144$	0	0
$145$	0.169574	0.0140823
$146$	−13.0081	−1.07656
$147$	0	0
$148$	9.26277	0.761395
$149$	12.0860	0.990120	0.495060	−	0.868859i	$-0.335146\pi$
$149$	12.0860	0.990120	0.495060	+	0.868859i	$0.335146\pi$
$150$	0	0
$151$	12.6923	1.03288	0.516442	−	0.856322i	$-0.327256\pi$
$151$	12.6923	1.03288	0.516442	+	0.856322i	$0.327256\pi$
$152$	11.3693	0.922170
$153$	0	0
$154$	−8.04696	−0.648442
$155$	7.75443	0.622851
$156$	0	0
$157$	−20.3326	−1.62272	−0.811359	−	0.584548i	$-0.801272\pi$
$157$	−20.3326	−1.62272	−0.811359	+	0.584548i	$0.801272\pi$
$158$	−11.7542	−0.935110
$159$	0	0
$160$	−9.30583	−0.735691
$161$	13.5596	1.06864
$162$	0	0
$163$	−14.3750	−1.12593	−0.562967	−	0.826480i	$-0.690340\pi$
$163$	−14.3750	−1.12593	−0.562967	+	0.826480i	$0.690340\pi$
$164$	1.69752	0.132554
$165$	0	0
$166$	0.381435	0.0296051
$167$	18.7294	1.44933	0.724663	−	0.689103i	$-0.241995\pi$
$167$	18.7294	1.44933	0.724663	+	0.689103i	$0.241995\pi$
$168$	0	0
$169$	28.4318	2.18706
$170$	−3.37331	−0.258721
$171$	0	0
$172$	4.34968	0.331660
$173$	11.7428	0.892786	0.446393	−	0.894837i	$-0.352708\pi$
$173$	11.7428	0.892786	0.446393	+	0.894837i	$0.352708\pi$
$174$	0	0
$175$	3.86401	0.292091
$176$	−1.02587	−0.0773278
$177$	0	0
$178$	−15.4233	−1.15603
$179$	−24.5193	−1.83266	−0.916328	−	0.400429i	$-0.868861\pi$
$179$	−24.5193	−1.83266	−0.916328	+	0.400429i	$0.868861\pi$
$180$	0	0
$181$	2.13689	0.158833	0.0794167	−	0.996842i	$-0.474694\pi$
$181$	2.13689	0.158833	0.0794167	+	0.996842i	$0.474694\pi$
$182$	−10.1789	−0.754512
$183$	0	0
$184$	22.1450	1.63255
$185$	13.0229	0.957460
$186$	0	0
$187$	11.3985	0.833540
$188$	−9.21173	−0.671834
$189$	0	0
$190$	5.96451	0.432711
$191$	−11.6936	−0.846118	−0.423059	−	0.906102i	$-0.639044\pi$
$191$	−11.6936	−0.846118	−0.423059	+	0.906102i	$0.639044\pi$
$192$	0	0
$193$	2.15548	0.155155	0.0775774	−	0.996986i	$-0.475282\pi$
$193$	2.15548	0.155155	0.0775774	+	0.996986i	$0.475282\pi$
$194$	5.45605	0.391721
$195$	0	0
$196$	4.65573	0.332552
$197$	−6.46675	−0.460737	−0.230368	−	0.973104i	$-0.573993\pi$
$197$	−6.46675	−0.460737	−0.230368	+	0.973104i	$0.573993\pi$
$198$	0	0
$199$	21.8975	1.55228	0.776138	−	0.630563i	$-0.217176\pi$
$199$	21.8975	1.55228	0.776138	+	0.630563i	$0.217176\pi$
$200$	6.31057	0.446225
$201$	0	0
$202$	−9.10200	−0.640415
$203$	0.178069	0.0124980
$204$	0	0
$205$	2.38661	0.166688
$206$	2.28449	0.159168
$207$	0	0
$208$	−1.29766	−0.0899767
$209$	−20.1542	−1.39410
$210$	0	0
$211$	11.2465	0.774243	0.387122	−	0.922029i	$-0.373469\pi$
$211$	11.2465	0.774243	0.387122	+	0.922029i	$0.373469\pi$
$212$	1.91762	0.131703
$213$	0	0
$214$	−8.83961	−0.604263
$215$	6.11537	0.417065
$216$	0	0
$217$	8.14290	0.552776
$218$	−13.5955	−0.920806
$219$	0	0
$220$	10.1400	0.683641
$221$	14.4184	0.969887
$222$	0	0
$223$	−19.8849	−1.33159	−0.665797	−	0.746133i	$-0.731908\pi$
$223$	−19.8849	−1.33159	−0.665797	+	0.746133i	$0.731908\pi$
$224$	−9.77202	−0.652921
$225$	0	0
$226$	−2.32949	−0.154955
$227$	6.20415	0.411784	0.205892	−	0.978575i	$-0.433990\pi$
$227$	6.20415	0.411784	0.205892	+	0.978575i	$0.433990\pi$
$228$	0	0
$229$	−6.29386	−0.415910	−0.207955	−	0.978138i	$-0.566681\pi$
$229$	−6.29386	−0.415910	−0.207955	+	0.978138i	$0.566681\pi$
$230$	11.6177	0.766045
$231$	0	0
$232$	0.290816	0.0190930
$233$	1.37434	0.0900362	0.0450181	−	0.998986i	$-0.485665\pi$
$233$	1.37434	0.0900362	0.0450181	+	0.998986i	$0.485665\pi$
$234$	0	0
$235$	−12.9511	−0.844836
$236$	−2.93179	−0.190843
$237$	0	0
$238$	−3.54230	−0.229613
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−16.6512	−1.07260	−0.536299	−	0.844028i	$-0.680178\pi$
$241$	−16.6512	−1.07260	−0.536299	+	0.844028i	$0.680178\pi$
$242$	13.4000	0.861383
$243$	0	0
$244$	5.67985	0.363615
$245$	6.54566	0.418187
$246$	0	0
$247$	−25.4939	−1.62214
$248$	13.2987	0.844470
$249$	0	0
$250$	10.8403	0.685600
$251$	−8.70369	−0.549372	−0.274686	−	0.961534i	$-0.588574\pi$
$251$	−8.70369	−0.549372	−0.274686	+	0.961534i	$0.588574\pi$
$252$	0	0
$253$	−39.2563	−2.46802
$254$	6.51638	0.408874
$255$	0	0
$256$	−16.4394	−1.02746
$257$	17.8087	1.11087	0.555437	−	0.831559i	$-0.312551\pi$
$257$	17.8087	1.11087	0.555437	+	0.831559i	$0.312551\pi$
$258$	0	0
$259$	13.6753	0.849739
$260$	12.8265	0.795467
$261$	0	0
$262$	−1.93295	−0.119418
$263$	−21.2660	−1.31132	−0.655658	−	0.755058i	$-0.727609\pi$
$263$	−21.2660	−1.31132	−0.655658	+	0.755058i	$0.727609\pi$
$264$	0	0
$265$	2.69605	0.165617
$266$	6.26331	0.384028
$267$	0	0
$268$	4.07584	0.248971
$269$	17.1887	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$269$	17.1887	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$270$	0	0
$271$	19.1030	1.16043	0.580213	−	0.814465i	$-0.302969\pi$
$271$	19.1030	1.16043	0.580213	+	0.814465i	$0.302969\pi$
$272$	−0.451591	−0.0273817
$273$	0	0
$274$	−5.88294	−0.355401
$275$	−11.1867	−0.674583
$276$	0	0
$277$	19.3315	1.16152	0.580760	−	0.814075i	$-0.302755\pi$
$277$	19.3315	1.16152	0.580760	+	0.814075i	$0.302755\pi$
$278$	8.80087	0.527841
$279$	0	0
$280$	−8.44502	−0.504687
$281$	4.05588	0.241954	0.120977	−	0.992655i	$-0.461397\pi$
$281$	4.05588	0.241954	0.120977	+	0.992655i	$0.461397\pi$
$282$	0	0
$283$	−10.8007	−0.642035	−0.321018	−	0.947073i	$-0.604025\pi$
$283$	−10.8007	−0.642035	−0.321018	+	0.947073i	$0.604025\pi$
$284$	−15.6224	−0.927020
$285$	0	0
$286$	29.4690	1.74254
$287$	2.50617	0.147934
$288$	0	0
$289$	−11.9823	−0.704844
$290$	0.152567	0.00895903
$291$	0	0
$292$	17.2128	1.00730
$293$	−5.01609	−0.293043	−0.146522	−	0.989207i	$-0.546808\pi$
$293$	−5.01609	−0.293043	−0.146522	+	0.989207i	$0.546808\pi$
$294$	0	0
$295$	−4.12191	−0.239987
$296$	22.3340	1.29814
$297$	0	0
$298$	10.8738	0.629904
$299$	−49.6569	−2.87173
$300$	0	0
$301$	6.42173	0.370142
$302$	11.4194	0.657110
$303$	0	0
$304$	0.798480	0.0457959
$305$	7.98550	0.457249
$306$	0	0
$307$	−9.70591	−0.553945	−0.276973	−	0.960878i	$-0.589331\pi$
$307$	−9.70591	−0.553945	−0.276973	+	0.960878i	$0.589331\pi$
$308$	10.6480	0.606727
$309$	0	0
$310$	6.97673	0.396251
$311$	10.3411	0.586389	0.293195	−	0.956053i	$-0.405282\pi$
$311$	10.3411	0.586389	0.293195	+	0.956053i	$0.405282\pi$
$312$	0	0
$313$	−23.8987	−1.35084	−0.675418	−	0.737435i	$-0.736037\pi$
$313$	−23.8987	−1.35084	−0.675418	+	0.737435i	$0.736037\pi$
$314$	−18.2934	−1.03236
$315$	0	0
$316$	15.5535	0.874952
$317$	1.48619	0.0834729	0.0417365	−	0.999129i	$-0.486711\pi$
$317$	1.48619	0.0834729	0.0417365	+	0.999129i	$0.486711\pi$
$318$	0	0
$319$	−0.515527	−0.0288640
$320$	−9.04742	−0.505766
$321$	0	0
$322$	12.1997	0.679860
$323$	−8.87196	−0.493649
$324$	0	0
$325$	−14.1505	−0.784928
$326$	−12.9333	−0.716307
$327$	0	0
$328$	4.09299	0.225998
$329$	−13.5999	−0.749787
$330$	0	0
$331$	−17.6038	−0.967593	−0.483797	−	0.875180i	$-0.660743\pi$
$331$	−17.6038	−0.967593	−0.483797	+	0.875180i	$0.660743\pi$
$332$	−0.504728	−0.0277006
$333$	0	0
$334$	16.8510	0.922047
$335$	5.73037	0.313083
$336$	0	0
$337$	18.9294	1.03115	0.515576	−	0.856844i	$-0.327578\pi$
$337$	18.9294	1.03115	0.515576	+	0.856844i	$0.327578\pi$
$338$	25.5803	1.39139
$339$	0	0
$340$	4.46368	0.242077
$341$	−23.5745	−1.27663
$342$	0	0
$343$	19.1771	1.03547
$344$	10.4878	0.565462
$345$	0	0
$346$	10.5651	0.567982
$347$	1.63077	0.0875444	0.0437722	−	0.999042i	$-0.486062\pi$
$347$	1.63077	0.0875444	0.0437722	+	0.999042i	$0.486062\pi$
$348$	0	0
$349$	8.00213	0.428344	0.214172	−	0.976796i	$-0.431295\pi$
$349$	8.00213	0.428344	0.214172	+	0.976796i	$0.431295\pi$
$350$	3.47648	0.185826
$351$	0	0
$352$	28.2910	1.50792
$353$	−9.60124	−0.511022	−0.255511	−	0.966806i	$-0.582244\pi$
$353$	−9.60124	−0.511022	−0.255511	+	0.966806i	$0.582244\pi$
$354$	0	0
$355$	−21.9641	−1.16573
$356$	20.4087	1.08166
$357$	0	0
$358$	−22.0602	−1.16592
$359$	31.4338	1.65901	0.829506	−	0.558498i	$-0.188622\pi$
$359$	31.4338	1.65901	0.829506	+	0.558498i	$0.188622\pi$
$360$	0	0
$361$	−3.31307	−0.174372
$362$	1.92257	0.101048
$363$	0	0
$364$	13.4691	0.705972
$365$	24.2001	1.26669
$366$	0	0
$367$	−9.76744	−0.509856	−0.254928	−	0.966960i	$-0.582052\pi$
$367$	−9.76744	−0.509856	−0.254928	+	0.966960i	$0.582052\pi$
$368$	1.55528	0.0810744
$369$	0	0
$370$	11.7168	0.609126
$371$	2.83111	0.146984
$372$	0	0
$373$	18.4876	0.957252	0.478626	−	0.878019i	$-0.341135\pi$
$373$	18.4876	0.957252	0.478626	+	0.878019i	$0.341135\pi$
$374$	10.2553	0.530290
$375$	0	0
$376$	−22.2109	−1.14544
$377$	−0.652111	−0.0335854
$378$	0	0
$379$	−1.80626	−0.0927813	−0.0463907	−	0.998923i	$-0.514772\pi$
$379$	−1.80626	−0.0927813	−0.0463907	+	0.998923i	$0.514772\pi$
$380$	−7.89244	−0.404874
$381$	0	0
$382$	−10.5208	−0.538292
$383$	25.2177	1.28856	0.644281	−	0.764789i	$-0.277157\pi$
$383$	25.2177	1.28856	0.644281	+	0.764789i	$0.277157\pi$
$384$	0	0
$385$	14.9704	0.762963
$386$	1.93930	0.0987079
$387$	0	0
$388$	−7.21962	−0.366521
$389$	−29.1698	−1.47897	−0.739485	−	0.673173i	$-0.764931\pi$
$389$	−29.1698	−1.47897	−0.739485	+	0.673173i	$0.764931\pi$
$390$	0	0
$391$	−17.2808	−0.873926
$392$	11.2257	0.566984
$393$	0	0
$394$	−5.81819	−0.293116
$395$	21.8672	1.10026
$396$	0	0
$397$	37.3463	1.87435	0.937177	−	0.348853i	$-0.113429\pi$
$397$	37.3463	1.87435	0.937177	+	0.348853i	$0.113429\pi$
$398$	19.7014	0.987542
$399$	0	0
$400$	0.443200	0.0221600
$401$	−5.50916	−0.275114	−0.137557	−	0.990494i	$-0.543925\pi$
$401$	−5.50916	−0.275114	−0.137557	+	0.990494i	$0.543925\pi$
$402$	0	0
$403$	−29.8204	−1.48546
$404$	12.0441	0.599215
$405$	0	0
$406$	0.160210	0.00795108
$407$	−39.5913	−1.96247
$408$	0	0
$409$	−28.2723	−1.39797	−0.698987	−	0.715134i	$-0.746366\pi$
$409$	−28.2723	−1.39797	−0.698987	+	0.715134i	$0.746366\pi$
$410$	2.14725	0.106045
$411$	0	0
$412$	−3.02291	−0.148928
$413$	−4.32840	−0.212987
$414$	0	0
$415$	−0.709615	−0.0348337
$416$	35.7864	1.75457
$417$	0	0
$418$	−18.1329	−0.886910
$419$	−12.0357	−0.587983	−0.293992	−	0.955808i	$-0.594984\pi$
$419$	−12.0357	−0.587983	−0.293992	+	0.955808i	$0.594984\pi$
$420$	0	0
$421$	−29.9983	−1.46203	−0.731015	−	0.682362i	$-0.760953\pi$
$421$	−29.9983	−1.46203	−0.731015	+	0.682362i	$0.760953\pi$
$422$	10.1186	0.492566
$423$	0	0
$424$	4.62368	0.224546
$425$	−4.92442	−0.238869
$426$	0	0
$427$	8.38555	0.405805
$428$	11.6969	0.565390
$429$	0	0
$430$	5.50205	0.265332
$431$	17.8098	0.857870	0.428935	−	0.903335i	$-0.358889\pi$
$431$	17.8098	0.857870	0.428935	+	0.903335i	$0.358889\pi$
$432$	0	0
$433$	20.2488	0.973094	0.486547	−	0.873654i	$-0.338256\pi$
$433$	20.2488	0.973094	0.486547	+	0.873654i	$0.338256\pi$
$434$	7.32623	0.351670
$435$	0	0
$436$	17.9901	0.861568
$437$	30.5550	1.46164
$438$	0	0
$439$	−34.7608	−1.65904	−0.829521	−	0.558475i	$-0.811387\pi$
$439$	−34.7608	−1.65904	−0.829521	+	0.558475i	$0.811387\pi$
$440$	24.4492	1.16557
$441$	0	0
$442$	12.9724	0.617032
$443$	25.7595	1.22387	0.611935	−	0.790908i	$-0.290392\pi$
$443$	25.7595	1.22387	0.611935	+	0.790908i	$0.290392\pi$
$444$	0	0
$445$	28.6933	1.36019
$446$	−17.8907	−0.847147
$447$	0	0
$448$	−9.50066	−0.448864
$449$	30.2122	1.42580	0.712901	−	0.701265i	$-0.247381\pi$
$449$	30.2122	1.42580	0.712901	+	0.701265i	$0.247381\pi$
$450$	0	0
$451$	−7.25561	−0.341653
$452$	3.08246	0.144987
$453$	0	0
$454$	5.58193	0.261973
$455$	18.9367	0.887765
$456$	0	0
$457$	1.33578	0.0624850	0.0312425	−	0.999512i	$-0.490054\pi$
$457$	1.33578	0.0624850	0.0312425	+	0.999512i	$0.490054\pi$
$458$	−5.66264	−0.264598
$459$	0	0
$460$	−15.3729	−0.716764
$461$	−16.6243	−0.774272	−0.387136	−	0.922023i	$-0.626536\pi$
$461$	−16.6243	−0.774272	−0.387136	+	0.922023i	$0.626536\pi$
$462$	0	0
$463$	29.4246	1.36748	0.683739	−	0.729726i	$-0.260353\pi$
$463$	29.4246	1.36748	0.683739	+	0.729726i	$0.260353\pi$
$464$	0.0204244	0.000948179 0
$465$	0	0
$466$	1.23651	0.0572801
$467$	2.53660	0.117380	0.0586899	−	0.998276i	$-0.481308\pi$
$467$	2.53660	0.117380	0.0586899	+	0.998276i	$0.481308\pi$
$468$	0	0
$469$	6.01744	0.277859
$470$	−11.6522	−0.537476
$471$	0	0
$472$	−7.06901	−0.325378
$473$	−18.5916	−0.854841
$474$	0	0
$475$	8.70710	0.399509
$476$	4.68729	0.214842
$477$	0	0
$478$	0.899709	0.0411517
$479$	−38.6620	−1.76651	−0.883255	−	0.468893i	$-0.844653\pi$
$479$	−38.6620	−1.76651	−0.883255	+	0.468893i	$0.844653\pi$
$480$	0	0
$481$	−50.0806	−2.28348
$482$	−14.9812	−0.682376
$483$	0	0
$484$	−17.7313	−0.805968
$485$	−10.1503	−0.460903
$486$	0	0
$487$	9.29522	0.421207	0.210603	−	0.977572i	$-0.432457\pi$
$487$	9.29522	0.421207	0.210603	+	0.977572i	$0.432457\pi$
$488$	13.6950	0.619944
$489$	0	0
$490$	5.88919	0.266046
$491$	−11.3475	−0.512105	−0.256052	−	0.966663i	$-0.582422\pi$
$491$	−11.3475	−0.512105	−0.256052	+	0.966663i	$0.582422\pi$
$492$	0	0
$493$	−0.226937	−0.0102207
$494$	−22.9371	−1.03199
$495$	0	0
$496$	0.933987	0.0419372
$497$	−23.0644	−1.03458
$498$	0	0
$499$	24.8745	1.11353	0.556767	−	0.830669i	$-0.312042\pi$
$499$	24.8745	1.11353	0.556767	+	0.830669i	$0.312042\pi$
$500$	−14.3442	−0.641494
$501$	0	0
$502$	−7.83078	−0.349505
$503$	−13.2786	−0.592064	−0.296032	−	0.955178i	$-0.595663\pi$
$503$	−13.2786	−0.592064	−0.296032	+	0.955178i	$0.595663\pi$
$504$	0	0
$505$	16.9332	0.753517
$506$	−35.3192	−1.57013
$507$	0	0
$508$	−8.62270	−0.382570
$509$	−33.4053	−1.48067	−0.740333	−	0.672241i	$-0.765332\pi$
$509$	−33.4053	−1.48067	−0.740333	+	0.672241i	$0.765332\pi$
$510$	0	0
$511$	25.4124	1.12418
$512$	−2.27826	−0.100686
$513$	0	0
$514$	16.0226	0.706727
$515$	−4.25001	−0.187278
$516$	0	0
$517$	39.3731	1.73163
$518$	12.3037	0.540596
$519$	0	0
$520$	30.9268	1.35623
$521$	−33.4430	−1.46516	−0.732582	−	0.680678i	$-0.761685\pi$
$521$	−33.4430	−1.46516	−0.732582	+	0.680678i	$0.761685\pi$
$522$	0	0
$523$	−25.7941	−1.12790	−0.563948	−	0.825810i	$-0.690718\pi$
$523$	−25.7941	−1.12790	−0.563948	+	0.825810i	$0.690718\pi$
$524$	2.55775	0.111736
$525$	0	0
$526$	−19.1332	−0.834246
$527$	−10.3776	−0.452055
$528$	0	0
$529$	36.5148	1.58760
$530$	2.42566	0.105364
$531$	0	0
$532$	−8.28782	−0.359323
$533$	−9.17791	−0.397540
$534$	0	0
$535$	16.4451	0.710982
$536$	9.82749	0.424483
$537$	0	0
$538$	15.4648	0.666734
$539$	−19.8997	−0.857141
$540$	0	0
$541$	4.49430	0.193225	0.0966126	−	0.995322i	$-0.469199\pi$
$541$	4.49430	0.193225	0.0966126	+	0.995322i	$0.469199\pi$
$542$	17.1871	0.738251
$543$	0	0
$544$	12.4538	0.533952
$545$	25.2929	1.08343
$546$	0	0
$547$	−24.5160	−1.04823	−0.524114	−	0.851648i	$-0.675603\pi$
$547$	−24.5160	−1.04823	−0.524114	+	0.851648i	$0.675603\pi$
$548$	7.78450	0.332537
$549$	0	0
$550$	−10.0648	−0.429163
$551$	0.401258	0.0170942
$552$	0	0
$553$	22.9627	0.976472
$554$	17.3928	0.738947
$555$	0	0
$556$	−11.6456	−0.493884
$557$	−25.5502	−1.08260	−0.541299	−	0.840830i	$-0.682067\pi$
$557$	−25.5502	−1.08260	−0.541299	+	0.840830i	$0.682067\pi$
$558$	0	0
$559$	−23.5172	−0.994672
$560$	−0.593105	−0.0250633
$561$	0	0
$562$	3.64911	0.153929
$563$	13.4776	0.568011	0.284006	−	0.958823i	$-0.408337\pi$
$563$	13.4776	0.568011	0.284006	+	0.958823i	$0.408337\pi$
$564$	0	0
$565$	4.33374	0.182322
$566$	−9.71749	−0.408456
$567$	0	0
$568$	−37.6681	−1.58052
$569$	6.39428	0.268062	0.134031	−	0.990977i	$-0.457208\pi$
$569$	6.39428	0.268062	0.134031	+	0.990977i	$0.457208\pi$
$570$	0	0
$571$	−17.9506	−0.751210	−0.375605	−	0.926780i	$-0.622565\pi$
$571$	−17.9506	−0.751210	−0.375605	+	0.926780i	$0.622565\pi$
$572$	−38.9944	−1.63044
$573$	0	0
$574$	2.25482	0.0941144
$575$	16.9597	0.707267
$576$	0	0
$577$	−37.1291	−1.54571	−0.772853	−	0.634585i	$-0.781171\pi$
$577$	−37.1291	−1.54571	−0.772853	+	0.634585i	$0.781171\pi$
$578$	−10.7806	−0.448415
$579$	0	0
$580$	−0.201882	−0.00838268
$581$	−0.745165	−0.0309146
$582$	0	0
$583$	−8.19636	−0.339459
$584$	41.5027	1.71740
$585$	0	0
$586$	−4.51302	−0.186431
$587$	36.6232	1.51160	0.755800	−	0.654803i	$-0.227248\pi$
$587$	36.6232	1.51160	0.755800	+	0.654803i	$0.227248\pi$
$588$	0	0
$589$	18.3491	0.756062
$590$	−3.70852	−0.152677
$591$	0	0
$592$	1.56855	0.0644669
$593$	−38.1568	−1.56691	−0.783456	−	0.621447i	$-0.786545\pi$
$593$	−38.1568	−1.56691	−0.783456	+	0.621447i	$0.786545\pi$
$594$	0	0
$595$	6.59003	0.270165
$596$	−14.3886	−0.589381
$597$	0	0
$598$	−44.6767	−1.82697
$599$	−48.4161	−1.97823	−0.989115	−	0.147148i	$-0.952991\pi$
$599$	−48.4161	−1.97823	−0.989115	+	0.147148i	$0.952991\pi$
$600$	0	0
$601$	22.2469	0.907472	0.453736	−	0.891136i	$-0.350091\pi$
$601$	22.2469	0.907472	0.453736	+	0.891136i	$0.350091\pi$
$602$	5.77768	0.235481
$603$	0	0
$604$	−15.1105	−0.614837
$605$	−24.9291	−1.01351
$606$	0	0
$607$	40.3888	1.63933	0.819665	−	0.572842i	$-0.194159\pi$
$607$	40.3888	1.63933	0.819665	+	0.572842i	$0.194159\pi$
$608$	−22.0202	−0.893035
$609$	0	0
$610$	7.18463	0.290897
$611$	49.8046	2.01488
$612$	0	0
$613$	−16.3156	−0.658979	−0.329490	−	0.944159i	$-0.606877\pi$
$613$	−16.3156	−0.658979	−0.329490	+	0.944159i	$0.606877\pi$
$614$	−8.73249	−0.352415
$615$	0	0
$616$	25.6740	1.03444
$617$	−5.46752	−0.220114	−0.110057	−	0.993925i	$-0.535103\pi$
$617$	−5.46752	−0.220114	−0.110057	+	0.993925i	$0.535103\pi$
$618$	0	0
$619$	43.6271	1.75352	0.876761	−	0.480926i	$-0.159700\pi$
$619$	43.6271	1.75352	0.876761	+	0.480926i	$0.159700\pi$
$620$	−9.23184	−0.370760
$621$	0	0
$622$	9.30397	0.373055
$623$	30.1307	1.20716
$624$	0	0
$625$	−9.17513	−0.367005
$626$	−21.5019	−0.859389
$627$	0	0
$628$	24.2065	0.965943
$629$	−17.4282	−0.694908
$630$	0	0
$631$	25.7573	1.02538	0.512692	−	0.858573i	$-0.328648\pi$
$631$	25.7573	1.02538	0.512692	+	0.858573i	$0.328648\pi$
$632$	37.5019	1.49175
$633$	0	0
$634$	1.33714	0.0531046
$635$	−12.1230	−0.481085
$636$	0	0
$637$	−25.1719	−0.997349
$638$	−0.463824	−0.0183630
$639$	0	0
$640$	10.4716	0.413927
$641$	−2.85693	−0.112842	−0.0564210	−	0.998407i	$-0.517969\pi$
$641$	−2.85693	−0.112842	−0.0564210	+	0.998407i	$0.517969\pi$
$642$	0	0
$643$	10.7053	0.422176	0.211088	−	0.977467i	$-0.432299\pi$
$643$	10.7053	0.422176	0.211088	+	0.977467i	$0.432299\pi$
$644$	−16.1430	−0.636123
$645$	0	0
$646$	−7.98217	−0.314054
$647$	−10.9862	−0.431913	−0.215956	−	0.976403i	$-0.569287\pi$
$647$	−10.9862	−0.431913	−0.215956	+	0.976403i	$0.569287\pi$
$648$	0	0
$649$	12.5312	0.491892
$650$	−12.7313	−0.499363
$651$	0	0
$652$	17.1137	0.670226
$653$	41.1768	1.61137	0.805687	−	0.592342i	$-0.201797\pi$
$653$	41.1768	1.61137	0.805687	+	0.592342i	$0.201797\pi$
$654$	0	0
$655$	3.59603	0.140509
$656$	0.287456	0.0112233
$657$	0	0
$658$	−12.2359	−0.477007
$659$	−1.39962	−0.0545216	−0.0272608	−	0.999628i	$-0.508678\pi$
$659$	−1.39962	−0.0545216	−0.0272608	+	0.999628i	$0.508678\pi$
$660$	0	0
$661$	26.9112	1.04673	0.523363	−	0.852110i	$-0.324677\pi$
$661$	26.9112	1.04673	0.523363	+	0.852110i	$0.324677\pi$
$662$	−15.8383	−0.615573
$663$	0	0
$664$	−1.21698	−0.0472280
$665$	−11.6521	−0.451851
$666$	0	0
$667$	0.781569	0.0302625
$668$	−22.2978	−0.862729
$669$	0	0
$670$	5.15566	0.199181
$671$	−24.2770	−0.937204
$672$	0	0
$673$	35.4943	1.36821	0.684103	−	0.729386i	$-0.260194\pi$
$673$	35.4943	1.36821	0.684103	+	0.729386i	$0.260194\pi$
$674$	17.0310	0.656009
$675$	0	0
$676$	−33.8487	−1.30187
$677$	−7.66192	−0.294471	−0.147236	−	0.989101i	$-0.547038\pi$
$677$	−7.66192	−0.294471	−0.147236	+	0.989101i	$0.547038\pi$
$678$	0	0
$679$	−10.6588	−0.409048
$680$	10.7626	0.412728
$681$	0	0
$682$	−21.2102	−0.812180
$683$	11.5533	0.442074	0.221037	−	0.975265i	$-0.429056\pi$
$683$	11.5533	0.442074	0.221037	+	0.975265i	$0.429056\pi$
$684$	0	0
$685$	10.9445	0.418168
$686$	17.2538	0.658755
$687$	0	0
$688$	0.736569	0.0280814
$689$	−10.3679	−0.394986
$690$	0	0
$691$	9.48943	0.360995	0.180498	−	0.983575i	$-0.442229\pi$
$691$	9.48943	0.360995	0.180498	+	0.983575i	$0.442229\pi$
$692$	−13.9801	−0.531442
$693$	0	0
$694$	1.46722	0.0556949
$695$	−16.3730	−0.621063
$696$	0	0
$697$	−3.19395	−0.120979
$698$	7.19959	0.272508
$699$	0	0
$700$	−4.60019	−0.173871
$701$	33.6461	1.27079	0.635397	−	0.772186i	$-0.280836\pi$
$701$	33.6461	1.27079	0.635397	+	0.772186i	$0.280836\pi$
$702$	0	0
$703$	30.8157	1.16223
$704$	27.5054	1.03665
$705$	0	0
$706$	−8.63831	−0.325107
$707$	17.7815	0.668742
$708$	0	0
$709$	−27.7519	−1.04224	−0.521122	−	0.853482i	$-0.674486\pi$
$709$	−27.7519	−1.04224	−0.521122	+	0.853482i	$0.674486\pi$
$710$	−19.7613	−0.741629
$711$	0	0
$712$	49.2085	1.84417
$713$	35.7403	1.33849
$714$	0	0
$715$	−54.8236	−2.05029
$716$	29.1908	1.09091
$717$	0	0
$718$	28.2812	1.05545
$719$	28.9767	1.08065	0.540324	−	0.841457i	$-0.318302\pi$
$719$	28.9767	1.08065	0.540324	+	0.841457i	$0.318302\pi$
$720$	0	0
$721$	−4.46292	−0.166208
$722$	−2.98080	−0.110934
$723$	0	0
$724$	−2.54402	−0.0945476
$725$	0.222720	0.00827161
$726$	0	0
$727$	−17.6229	−0.653599	−0.326799	−	0.945094i	$-0.605970\pi$
$727$	−17.6229	−0.653599	−0.326799	+	0.945094i	$0.605970\pi$
$728$	32.4761	1.20364
$729$	0	0
$730$	21.7730	0.805855
$731$	−8.18407	−0.302699
$732$	0	0
$733$	10.2978	0.380359	0.190180	−	0.981749i	$-0.439093\pi$
$733$	10.2978	0.380359	0.190180	+	0.981749i	$0.439093\pi$
$734$	−8.78785	−0.324366
$735$	0	0
$736$	−42.8908	−1.58098
$737$	−17.4211	−0.641714
$738$	0	0
$739$	13.2229	0.486411	0.243206	−	0.969975i	$-0.421801\pi$
$739$	13.2229	0.486411	0.243206	+	0.969975i	$0.421801\pi$
$740$	−15.5040	−0.569940
$741$	0	0
$742$	2.54718	0.0935098
$743$	−7.80394	−0.286299	−0.143149	−	0.989701i	$-0.545723\pi$
$743$	−7.80394	−0.286299	−0.143149	+	0.989701i	$0.545723\pi$
$744$	0	0
$745$	−20.2295	−0.741151
$746$	16.6335	0.608994
$747$	0	0
$748$	−13.5702	−0.496175
$749$	17.2689	0.630991
$750$	0	0
$751$	50.3886	1.83871	0.919353	−	0.393435i	$-0.128713\pi$
$751$	50.3886	1.83871	0.919353	+	0.393435i	$0.128713\pi$
$752$	−1.55990	−0.0568838
$753$	0	0
$754$	−0.586710	−0.0213667
$755$	−21.2444	−0.773162
$756$	0	0
$757$	−46.6796	−1.69660	−0.848299	−	0.529517i	$-0.822373\pi$
$757$	−46.6796	−1.69660	−0.848299	+	0.529517i	$0.822373\pi$
$758$	−1.62511	−0.0590265
$759$	0	0
$760$	−19.0299	−0.690287
$761$	−24.2370	−0.878591	−0.439296	−	0.898343i	$-0.644772\pi$
$761$	−24.2370	−0.878591	−0.439296	+	0.898343i	$0.644772\pi$
$762$	0	0
$763$	26.5600	0.961536
$764$	13.9215	0.503662
$765$	0	0
$766$	22.6886	0.819771
$767$	15.8512	0.572353
$768$	0	0
$769$	23.9656	0.864222	0.432111	−	0.901820i	$-0.357769\pi$
$769$	23.9656	0.864222	0.432111	+	0.901820i	$0.357769\pi$
$770$	13.4690	0.485389
$771$	0	0
$772$	−2.56615	−0.0923578
$773$	−1.40235	−0.0504391	−0.0252195	−	0.999682i	$-0.508028\pi$
$773$	−1.40235	−0.0504391	−0.0252195	+	0.999682i	$0.508028\pi$
$774$	0	0
$775$	10.1848	0.365847
$776$	−17.4076	−0.624898
$777$	0	0
$778$	−26.2444	−0.940906
$779$	5.64737	0.202338
$780$	0	0
$781$	66.7739	2.38936
$782$	−15.5477	−0.555983
$783$	0	0
$784$	0.788396	0.0281570
$785$	34.0327	1.21468
$786$	0	0
$787$	−22.5141	−0.802540	−0.401270	−	0.915960i	$-0.631431\pi$
$787$	−22.5141	−0.802540	−0.401270	+	0.915960i	$0.631431\pi$
$788$	7.69882	0.274259
$789$	0	0
$790$	19.6741	0.699974
$791$	4.55085	0.161809
$792$	0	0
$793$	−30.7090	−1.09051
$794$	33.6007	1.19245
$795$	0	0
$796$	−26.0696	−0.924011
$797$	35.7937	1.26788	0.633939	−	0.773383i	$-0.281437\pi$
$797$	35.7937	1.26788	0.633939	+	0.773383i	$0.281437\pi$
$798$	0	0
$799$	17.3322	0.613168
$800$	−12.2224	−0.432127
$801$	0	0
$802$	−4.95664	−0.175025
$803$	−73.5715	−2.59628
$804$	0	0
$805$	−22.6960	−0.799929
$806$	−26.8296	−0.945033
$807$	0	0
$808$	29.0402	1.02163
$809$	22.3480	0.785712	0.392856	−	0.919600i	$-0.371487\pi$
$809$	22.3480	0.785712	0.392856	+	0.919600i	$0.371487\pi$
$810$	0	0
$811$	12.5070	0.439182	0.219591	−	0.975592i	$-0.429528\pi$
$811$	12.5070	0.439182	0.219591	+	0.975592i	$0.429528\pi$
$812$	−0.211995	−0.00743957
$813$	0	0
$814$	−35.6206	−1.24850
$815$	24.0608	0.842813
$816$	0	0
$817$	14.4706	0.506264
$818$	−25.4368	−0.889377
$819$	0	0
$820$	−2.84131	−0.0992230
$821$	38.7711	1.35312	0.676560	−	0.736387i	$-0.263470\pi$
$821$	38.7711	1.35312	0.676560	+	0.736387i	$0.263470\pi$
$822$	0	0
$823$	51.3284	1.78920	0.894598	−	0.446872i	$-0.147462\pi$
$823$	51.3284	1.78920	0.894598	+	0.446872i	$0.147462\pi$
$824$	−7.28871	−0.253914
$825$	0	0
$826$	−3.89430	−0.135500
$827$	15.0423	0.523072	0.261536	−	0.965194i	$-0.415771\pi$
$827$	15.0423	0.523072	0.261536	+	0.965194i	$0.415771\pi$
$828$	0	0
$829$	−24.9052	−0.864992	−0.432496	−	0.901636i	$-0.642367\pi$
$829$	−24.9052	−0.864992	−0.432496	+	0.901636i	$0.642367\pi$
$830$	−0.638447	−0.0221608
$831$	0	0
$832$	34.7927	1.20622
$833$	−8.75992	−0.303513
$834$	0	0
$835$	−31.3493	−1.08489
$836$	23.9941	0.829853
$837$	0	0
$838$	−10.8286	−0.374069
$839$	−14.5488	−0.502279	−0.251139	−	0.967951i	$-0.580805\pi$
$839$	−14.5488	−0.502279	−0.251139	+	0.967951i	$0.580805\pi$
$840$	0	0
$841$	−28.9897	−0.999646
$842$	−26.9898	−0.930129
$843$	0	0
$844$	−13.3893	−0.460878
$845$	−47.5892	−1.63712
$846$	0	0
$847$	−26.1779	−0.899484
$848$	0.324727	0.0111512
$849$	0	0
$850$	−4.43054	−0.151966
$851$	60.0226	2.05755
$852$	0	0
$853$	−33.7550	−1.15575	−0.577875	−	0.816125i	$-0.696118\pi$
$853$	−33.7550	−1.15575	−0.577875	+	0.816125i	$0.696118\pi$
$854$	7.54455	0.258169
$855$	0	0
$856$	28.2030	0.963959
$857$	−47.0497	−1.60719	−0.803593	−	0.595179i	$-0.797081\pi$
$857$	−47.0497	−1.60719	−0.803593	+	0.595179i	$0.797081\pi$
$858$	0	0
$859$	−47.7888	−1.63053	−0.815267	−	0.579085i	$-0.803410\pi$
$859$	−47.7888	−1.63053	−0.815267	+	0.579085i	$0.803410\pi$
$860$	−7.28050	−0.248263
$861$	0	0
$862$	16.0237	0.545768
$863$	−30.8008	−1.04847	−0.524236	−	0.851573i	$-0.675649\pi$
$863$	−30.8008	−1.04847	−0.524236	+	0.851573i	$0.675649\pi$
$864$	0	0
$865$	−19.6551	−0.668292
$866$	18.2180	0.619072
$867$	0	0
$868$	−9.69432	−0.329047
$869$	−66.4793	−2.25516
$870$	0	0
$871$	−22.0366	−0.746683
$872$	43.3769	1.46893
$873$	0	0
$874$	27.4906	0.929882
$875$	−21.1774	−0.715926
$876$	0	0
$877$	−43.0660	−1.45424	−0.727118	−	0.686512i	$-0.759141\pi$
$877$	−43.0660	−1.45424	−0.727118	+	0.686512i	$0.759141\pi$
$878$	−31.2746	−1.05547
$879$	0	0
$880$	1.71710	0.0578834
$881$	−30.2543	−1.01929	−0.509647	−	0.860384i	$-0.670224\pi$
$881$	−30.2543	−1.01929	−0.509647	+	0.860384i	$0.670224\pi$
$882$	0	0
$883$	33.4773	1.12660	0.563300	−	0.826252i	$-0.309531\pi$
$883$	33.4773	1.12660	0.563300	+	0.826252i	$0.309531\pi$
$884$	−17.1655	−0.577337
$885$	0	0
$886$	23.1760	0.778613
$887$	34.7429	1.16655	0.583277	−	0.812274i	$-0.301770\pi$
$887$	34.7429	1.16655	0.583277	+	0.812274i	$0.301770\pi$
$888$	0	0
$889$	−12.7303	−0.426960
$890$	25.8156	0.865341
$891$	0	0
$892$	23.6735	0.792648
$893$	−30.6458	−1.02552
$894$	0	0
$895$	41.0404	1.37183
$896$	10.9962	0.367358
$897$	0	0
$898$	27.1822	0.907081
$899$	0.469354	0.0156538
$900$	0	0
$901$	−3.60807	−0.120202
$902$	−6.52794	−0.217356
$903$	0	0
$904$	7.43230	0.247194
$905$	−3.57672	−0.118894
$906$	0	0
$907$	−18.9277	−0.628484	−0.314242	−	0.949343i	$-0.601750\pi$
$907$	−18.9277	−0.628484	−0.314242	+	0.949343i	$0.601750\pi$
$908$	−7.38620	−0.245120
$909$	0	0
$910$	17.0375	0.564787
$911$	53.6024	1.77593	0.887964	−	0.459914i	$-0.152120\pi$
$911$	53.6024	1.77593	0.887964	+	0.459914i	$0.152120\pi$
$912$	0	0
$913$	2.15733	0.0713971
$914$	1.20181	0.0397524
$915$	0	0
$916$	7.49300	0.247576
$917$	3.77618	0.124701
$918$	0	0
$919$	−38.1249	−1.25762	−0.628812	−	0.777558i	$-0.716458\pi$
$919$	−38.1249	−1.25762	−0.628812	+	0.777558i	$0.716458\pi$
$920$	−37.0664	−1.22204
$921$	0	0
$922$	−14.9571	−0.492584
$923$	84.4650	2.78020
$924$	0	0
$925$	17.1044	0.562388
$926$	26.4736	0.869976
$927$	0	0
$928$	−0.563256	−0.0184898
$929$	−23.1631	−0.759957	−0.379978	−	0.924995i	$-0.624069\pi$
$929$	−23.1631	−0.759957	−0.379978	+	0.924995i	$0.624069\pi$
$930$	0	0
$931$	15.4888	0.507626
$932$	−1.63619	−0.0535952
$933$	0	0
$934$	2.28220	0.0746758
$935$	−19.0788	−0.623944
$936$	0	0
$937$	3.99000	0.130348	0.0651738	−	0.997874i	$-0.479240\pi$
$937$	3.99000	0.130348	0.0651738	+	0.997874i	$0.479240\pi$
$938$	5.41394	0.176771
$939$	0	0
$940$	15.4186	0.502899
$941$	−15.0860	−0.491790	−0.245895	−	0.969296i	$-0.579082\pi$
$941$	−15.0860	−0.491790	−0.245895	+	0.969296i	$0.579082\pi$
$942$	0	0
$943$	10.9999	0.358207
$944$	−0.496466	−0.0161586
$945$	0	0
$946$	−16.7270	−0.543841
$947$	55.4188	1.80087	0.900435	−	0.434991i	$-0.143248\pi$
$947$	55.4188	1.80087	0.900435	+	0.434991i	$0.143248\pi$
$948$	0	0
$949$	−93.0636	−3.02097
$950$	7.83385	0.254164
$951$	0	0
$952$	11.3018	0.366293
$953$	46.8160	1.51652	0.758259	−	0.651953i	$-0.226050\pi$
$953$	46.8160	1.51652	0.758259	+	0.651953i	$0.226050\pi$
$954$	0	0
$955$	19.5727	0.633359
$956$	−1.19052	−0.0385043
$957$	0	0
$958$	−34.7845	−1.12384
$959$	11.4928	0.371122
$960$	0	0
$961$	−9.53693	−0.307643
$962$	−45.0579	−1.45273
$963$	0	0
$964$	19.8237	0.638477
$965$	−3.60784	−0.116141
$966$	0	0
$967$	−32.6031	−1.04844	−0.524222	−	0.851582i	$-0.675644\pi$
$967$	−32.6031	−1.04844	−0.524222	+	0.851582i	$0.675644\pi$
$968$	−42.7530	−1.37413
$969$	0	0
$970$	−9.13233	−0.293222
$971$	22.8194	0.732310	0.366155	−	0.930554i	$-0.380674\pi$
$971$	22.8194	0.732310	0.366155	+	0.930554i	$0.380674\pi$
$972$	0	0
$973$	−17.1932	−0.551189
$974$	8.36299	0.267967
$975$	0	0
$976$	0.961819	0.0307871
$977$	13.1827	0.421751	0.210875	−	0.977513i	$-0.432369\pi$
$977$	13.1827	0.421751	0.210875	+	0.977513i	$0.432369\pi$
$978$	0	0
$979$	−87.2315	−2.78793
$980$	−7.79277	−0.248931
$981$	0	0
$982$	−10.2094	−0.325796
$983$	26.5649	0.847290	0.423645	−	0.905828i	$-0.360750\pi$
$983$	26.5649	0.847290	0.423645	+	0.905828i	$0.360750\pi$
$984$	0	0
$985$	10.8240	0.344883
$986$	−0.204177	−0.00650232
$987$	0	0
$988$	30.3511	0.965597
$989$	28.1859	0.896259
$990$	0	0
$991$	29.8250	0.947423	0.473711	−	0.880680i	$-0.342914\pi$
$991$	29.8250	0.947423	0.473711	+	0.880680i	$0.342914\pi$
$992$	−25.7571	−0.817790
$993$	0	0
$994$	−20.7513	−0.658191
$995$	−36.6521	−1.16195
$996$	0	0
$997$	21.3442	0.675976	0.337988	−	0.941150i	$-0.390254\pi$
$997$	21.3442	0.675976	0.337988	+	0.941150i	$0.390254\pi$
$998$	22.3798	0.708419
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.e.1.4		6
3.2	odd	2		717.2.a.d.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.d.1.3	✓	6	3.2	odd	2
2151.2.a.e.1.4		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} +O(q^{10})\)	\(q+0.899709 q^{2} -1.19052 q^{4} -1.67380 q^{5} -1.75765 q^{7} -2.87054 q^{8} -1.50593 q^{10} +5.08858 q^{11} +6.43675 q^{13} -1.58137 q^{14} -0.201602 q^{16} +2.24001 q^{17} -3.96067 q^{19} +1.99270 q^{20} +4.57824 q^{22} -7.71459 q^{23} -2.19839 q^{25} +5.79120 q^{26} +2.09253 q^{28} -0.101311 q^{29} -4.63283 q^{31} +5.55970 q^{32} +2.01536 q^{34} +2.94196 q^{35} -7.78041 q^{37} -3.56345 q^{38} +4.80472 q^{40} -1.42586 q^{41} -3.65358 q^{43} -6.05808 q^{44} -6.94088 q^{46} +7.73754 q^{47} -3.91066 q^{49} -1.97791 q^{50} -7.66311 q^{52} -1.61074 q^{53} -8.51728 q^{55} +5.04542 q^{56} -0.0911499 q^{58} +2.46261 q^{59} -4.77088 q^{61} -4.16819 q^{62} +5.40531 q^{64} -10.7738 q^{65} -3.42357 q^{67} -2.66679 q^{68} +2.64691 q^{70} +13.1223 q^{71} -14.4582 q^{73} -7.00010 q^{74} +4.71528 q^{76} -8.94396 q^{77} -13.0644 q^{79} +0.337442 q^{80} -1.28286 q^{82} +0.423954 q^{83} -3.74933 q^{85} -3.28716 q^{86} -14.6070 q^{88} -17.1426 q^{89} -11.3136 q^{91} +9.18440 q^{92} +6.96153 q^{94} +6.62938 q^{95} +6.06424 q^{97} -3.51845 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \)	\( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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