LMFDB - Embedded newform 950.4.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [950,4,Mod(1,950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("950.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,4,-1,8,0,-2,-57,16,35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.0518145055$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +6.15207 q^{3} +4.00000 q^{4} +12.3041 q^{6} -21.8479 q^{7} +8.00000 q^{8} +10.8479 q^{9} -8.30413 q^{11} +24.6083 q^{12} -53.0645 q^{13} -43.6959 q^{14} +16.0000 q^{16} -74.2810 q^{17} +21.6959 q^{18} -19.0000 q^{19} -134.410 q^{21} -16.6083 q^{22} +163.977 q^{23} +49.2165 q^{24} -106.129 q^{26} -99.3686 q^{27} -87.3917 q^{28} -232.410 q^{29} +98.4331 q^{31} +32.0000 q^{32} -51.0876 q^{33} -148.562 q^{34} +43.3917 q^{36} -296.433 q^{37} -38.0000 q^{38} -326.456 q^{39} -434.912 q^{41} -268.820 q^{42} +171.299 q^{43} -33.2165 q^{44} +327.954 q^{46} +366.083 q^{47} +98.4331 q^{48} +134.332 q^{49} -456.982 q^{51} -212.258 q^{52} -138.631 q^{53} -198.737 q^{54} -174.783 q^{56} -116.889 q^{57} -464.820 q^{58} +572.797 q^{59} -632.691 q^{61} +196.866 q^{62} -237.005 q^{63} +64.0000 q^{64} -102.175 q^{66} +183.461 q^{67} -297.124 q^{68} +1008.80 q^{69} +56.6545 q^{71} +86.7835 q^{72} -68.1521 q^{73} -592.866 q^{74} -76.0000 q^{76} +181.428 q^{77} -652.912 q^{78} -332.820 q^{79} -904.217 q^{81} -869.825 q^{82} -1152.91 q^{83} -537.640 q^{84} +342.598 q^{86} -1429.80 q^{87} -66.4331 q^{88} -368.479 q^{89} +1159.35 q^{91} +655.908 q^{92} +605.567 q^{93} +732.165 q^{94} +196.866 q^{96} +426.443 q^{97} +268.664 q^{98} -90.0827 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - q^{3} + 8 q^{4} - 2 q^{6} - 57 q^{7} + 16 q^{8} + 35 q^{9} + 10 q^{11} - 4 q^{12} - 13 q^{13} - 114 q^{14} + 32 q^{16} + 51 q^{17} + 70 q^{18} - 38 q^{19} + 117 q^{21} + 20 q^{22} + 155 q^{23}+ \cdots + 352 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - q^3 + 8 * q^4 - 2 * q^6 - 57 * q^7 + 16 * q^8 + 35 * q^9 + 10 * q^11 - 4 * q^12 - 13 * q^13 - 114 * q^14 + 32 * q^16 + 51 * q^17 + 70 * q^18 - 38 * q^19 + 117 * q^21 + 20 * q^22 + 155 * q^23 - 8 * q^24 - 26 * q^26 - 79 * q^27 - 228 * q^28 - 79 * q^29 - 16 * q^31 + 64 * q^32 - 182 * q^33 + 102 * q^34 + 140 * q^36 - 380 * q^37 - 76 * q^38 - 613 * q^39 - 790 * q^41 + 234 * q^42 - 296 * q^43 + 40 * q^44 + 310 * q^46 + 200 * q^47 - 16 * q^48 + 1027 * q^49 - 1353 * q^51 - 52 * q^52 - 397 * q^53 - 158 * q^54 - 456 * q^56 + 19 * q^57 - 158 * q^58 + 201 * q^59 - 680 * q^61 - 32 * q^62 - 1086 * q^63 + 128 * q^64 - 364 * q^66 + 939 * q^67 + 204 * q^68 + 1073 * q^69 + 406 * q^71 + 280 * q^72 - 123 * q^73 - 760 * q^74 - 152 * q^76 - 462 * q^77 - 1226 * q^78 + 106 * q^79 - 1702 * q^81 - 1580 * q^82 - 2226 * q^83 + 468 * q^84 - 592 * q^86 - 2527 * q^87 + 80 * q^88 - 870 * q^89 - 249 * q^91 + 620 * q^92 + 1424 * q^93 + 400 * q^94 - 32 * q^96 + 1864 * q^97 + 2054 * q^98 + 352 * q^99

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	6.15207	1.18397	0.591983	−	0.805950i	$-0.298345\pi$
$3$	6.15207	1.18397	0.591983	+	0.805950i	$0.298345\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	12.3041	0.837190
$7$	−21.8479	−1.17968	−0.589839	−	0.807521i	$-0.700809\pi$
$7$	−21.8479	−1.17968	−0.589839	+	0.807521i	$0.700809\pi$
$8$	8.00000	0.353553
$9$	10.8479	0.401775
$10$	0	0
$11$	−8.30413	−0.227617	−0.113809	−	0.993503i	$-0.536305\pi$
$11$	−8.30413	−0.227617	−0.113809	+	0.993503i	$0.536305\pi$
$12$	24.6083	0.591983
$13$	−53.0645	−1.13211	−0.566055	−	0.824367i	$-0.691531\pi$
$13$	−53.0645	−1.13211	−0.566055	+	0.824367i	$0.691531\pi$
$14$	−43.6959	−0.834158
$15$	0	0
$16$	16.0000	0.250000
$17$	−74.2810	−1.05975	−0.529876	−	0.848075i	$-0.677762\pi$
$17$	−74.2810	−1.05975	−0.529876	+	0.848075i	$0.677762\pi$
$18$	21.6959	0.284098
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−134.410	−1.39670
$22$	−16.6083	−0.160950
$23$	163.977	1.48659	0.743294	−	0.668964i	$-0.233262\pi$
$23$	163.977	1.48659	0.743294	+	0.668964i	$0.233262\pi$
$24$	49.2165	0.418595
$25$	0	0
$26$	−106.129	−0.800523
$27$	−99.3686	−0.708278
$28$	−87.3917	−0.589839
$29$	−232.410	−1.48819	−0.744094	−	0.668075i	$-0.767118\pi$
$29$	−232.410	−1.48819	−0.744094	+	0.668075i	$0.767118\pi$
$30$	0	0
$31$	98.4331	0.570294	0.285147	−	0.958484i	$-0.407958\pi$
$31$	98.4331	0.570294	0.285147	+	0.958484i	$0.407958\pi$
$32$	32.0000	0.176777
$33$	−51.0876	−0.269491
$34$	−148.562	−0.749358
$35$	0	0
$36$	43.3917	0.200888
$37$	−296.433	−1.31712	−0.658558	−	0.752530i	$-0.728833\pi$
$37$	−296.433	−1.31712	−0.658558	+	0.752530i	$0.728833\pi$
$38$	−38.0000	−0.162221
$39$	−326.456	−1.34038
$40$	0	0
$41$	−434.912	−1.65663	−0.828316	−	0.560261i	$-0.810701\pi$
$41$	−434.912	−1.65663	−0.828316	+	0.560261i	$0.810701\pi$
$42$	−268.820	−0.987615
$43$	171.299	0.607509	0.303755	−	0.952750i	$-0.401760\pi$
$43$	171.299	0.607509	0.303755	+	0.952750i	$0.401760\pi$
$44$	−33.2165	−0.113809
$45$	0	0
$46$	327.954	1.05118
$47$	366.083	1.13614	0.568071	−	0.822980i	$-0.307690\pi$
$47$	366.083	1.13614	0.568071	+	0.822980i	$0.307690\pi$
$48$	98.4331	0.295991
$49$	134.332	0.391639
$50$	0	0
$51$	−456.982	−1.25471
$52$	−212.258	−0.566055
$53$	−138.631	−0.359292	−0.179646	−	0.983731i	$-0.557495\pi$
$53$	−138.631	−0.359292	−0.179646	+	0.983731i	$0.557495\pi$
$54$	−198.737	−0.500828
$55$	0	0
$56$	−174.783	−0.417079
$57$	−116.889	−0.271620
$58$	−464.820	−1.05231
$59$	572.797	1.26393	0.631964	−	0.774997i	$-0.282249\pi$
$59$	572.797	1.26393	0.631964	+	0.774997i	$0.282249\pi$
$60$	0	0
$61$	−632.691	−1.32800	−0.663998	−	0.747734i	$-0.731142\pi$
$61$	−632.691	−1.32800	−0.663998	+	0.747734i	$0.731142\pi$
$62$	196.866	0.403258
$63$	−237.005	−0.473965
$64$	64.0000	0.125000
$65$	0	0
$66$	−102.175	−0.190559
$67$	183.461	0.334527	0.167264	−	0.985912i	$-0.446507\pi$
$67$	183.461	0.334527	0.167264	+	0.985912i	$0.446507\pi$
$68$	−297.124	−0.529876
$69$	1008.80	1.76007
$70$	0	0
$71$	56.6545	0.0946994	0.0473497	−	0.998878i	$-0.484922\pi$
$71$	56.6545	0.0946994	0.0473497	+	0.998878i	$0.484922\pi$
$72$	86.7835	0.142049
$73$	−68.1521	−0.109268	−0.0546342	−	0.998506i	$-0.517399\pi$
$73$	−68.1521	−0.109268	−0.0546342	+	0.998506i	$0.517399\pi$
$74$	−592.866	−0.931342
$75$	0	0
$76$	−76.0000	−0.114708
$77$	181.428	0.268515
$78$	−652.912	−0.947792
$79$	−332.820	−0.473989	−0.236995	−	0.971511i	$-0.576162\pi$
$79$	−332.820	−0.473989	−0.236995	+	0.971511i	$0.576162\pi$
$80$	0	0
$81$	−904.217	−1.24035
$82$	−869.825	−1.17142
$83$	−1152.91	−1.52468	−0.762341	−	0.647176i	$-0.775950\pi$
$83$	−1152.91	−1.52468	−0.762341	+	0.647176i	$0.775950\pi$
$84$	−537.640	−0.698349
$85$	0	0
$86$	342.598	0.429574
$87$	−1429.80	−1.76196
$88$	−66.4331	−0.0804749
$89$	−368.479	−0.438862	−0.219431	−	0.975628i	$-0.570420\pi$
$89$	−368.479	−0.438862	−0.219431	+	0.975628i	$0.570420\pi$
$90$	0	0
$91$	1159.35	1.33553
$92$	655.908	0.743294
$93$	605.567	0.675208
$94$	732.165	0.803373
$95$	0	0
$96$	196.866	0.209298
$97$	426.443	0.446378	0.223189	−	0.974775i	$-0.428353\pi$
$97$	426.443	0.446378	0.223189	+	0.974775i	$0.428353\pi$
$98$	268.664	0.276931
$99$	−90.0827	−0.0914510
$100$	0	0
$101$	−403.124	−0.397152	−0.198576	−	0.980086i	$-0.563632\pi$
$101$	−403.124	−0.397152	−0.198576	+	0.980086i	$0.563632\pi$
$102$	−913.964	−0.887214
$103$	1135.68	1.08642	0.543211	−	0.839596i	$-0.317208\pi$
$103$	1135.68	1.08642	0.543211	+	0.839596i	$0.317208\pi$
$104$	−424.516	−0.400262
$105$	0	0
$106$	−277.263	−0.254058
$107$	380.096	0.343414	0.171707	−	0.985148i	$-0.445072\pi$
$107$	380.096	0.343414	0.171707	+	0.985148i	$0.445072\pi$
$108$	−397.474	−0.354139
$109$	1180.74	1.03756	0.518782	−	0.854907i	$-0.326386\pi$
$109$	1180.74	1.03756	0.518782	+	0.854907i	$0.326386\pi$
$110$	0	0
$111$	−1823.68	−1.55942
$112$	−349.567	−0.294919
$113$	1132.51	0.942807	0.471404	−	0.881918i	$-0.343748\pi$
$113$	1132.51	0.942807	0.471404	+	0.881918i	$0.343748\pi$
$114$	−233.779	−0.192065
$115$	0	0
$116$	−929.640	−0.744094
$117$	−575.640	−0.454854
$118$	1145.59	0.893733
$119$	1622.89	1.25017
$120$	0	0
$121$	−1262.04	−0.948190
$122$	−1265.38	−0.939035
$123$	−2675.61	−1.96140
$124$	393.732	0.285147
$125$	0	0
$126$	−474.010	−0.335144
$127$	40.0462	0.0279806	0.0139903	−	0.999902i	$-0.495547\pi$
$127$	40.0462	0.0279806	0.0139903	+	0.999902i	$0.495547\pi$
$128$	128.000	0.0883883
$129$	1053.84	0.719270
$130$	0	0
$131$	−177.898	−0.118649	−0.0593244	−	0.998239i	$-0.518895\pi$
$131$	−177.898	−0.118649	−0.0593244	+	0.998239i	$0.518895\pi$
$132$	−204.350	−0.134746
$133$	415.111	0.270637
$134$	366.922	0.236547
$135$	0	0
$136$	−594.248	−0.374679
$137$	24.7603	0.0154410	0.00772050	−	0.999970i	$-0.497542\pi$
$137$	24.7603	0.0154410	0.00772050	+	0.999970i	$0.497542\pi$
$138$	2017.59	1.24456
$139$	2867.21	1.74959	0.874796	−	0.484491i	$-0.160995\pi$
$139$	2867.21	1.74959	0.874796	+	0.484491i	$0.160995\pi$
$140$	0	0
$141$	2252.17	1.34515
$142$	113.309	0.0669626
$143$	440.655	0.257688
$144$	173.567	0.100444
$145$	0	0
$146$	−136.304	−0.0772645
$147$	826.421	0.463687
$148$	−1185.73	−0.658558
$149$	1949.35	1.07179	0.535895	−	0.844285i	$-0.319974\pi$
$149$	1949.35	1.07179	0.535895	+	0.844285i	$0.319974\pi$
$150$	0	0
$151$	1120.99	0.604136	0.302068	−	0.953286i	$-0.402323\pi$
$151$	1120.99	0.604136	0.302068	+	0.953286i	$0.402323\pi$
$152$	−152.000	−0.0811107
$153$	−805.795	−0.425782
$154$	362.856	0.189869
$155$	0	0
$156$	−1305.82	−0.670190
$157$	2360.23	1.19979	0.599894	−	0.800079i	$-0.295209\pi$
$157$	2360.23	1.19979	0.599894	+	0.800079i	$0.295209\pi$
$158$	−665.640	−0.335161
$159$	−852.870	−0.425390
$160$	0	0
$161$	−3582.56	−1.75370
$162$	−1808.43	−0.877061
$163$	−861.825	−0.414131	−0.207065	−	0.978327i	$-0.566391\pi$
$163$	−861.825	−0.414131	−0.207065	+	0.978327i	$0.566391\pi$
$164$	−1739.65	−0.828316
$165$	0	0
$166$	−2305.82	−1.07811
$167$	−1686.51	−0.781472	−0.390736	−	0.920503i	$-0.627779\pi$
$167$	−1686.51	−0.781472	−0.390736	+	0.920503i	$0.627779\pi$
$168$	−1075.28	−0.493807
$169$	618.838	0.281674
$170$	0	0
$171$	−206.111	−0.0921736
$172$	685.197	0.303755
$173$	3191.44	1.40255	0.701273	−	0.712893i	$-0.252616\pi$
$173$	3191.44	1.40255	0.701273	+	0.712893i	$0.252616\pi$
$174$	−2859.60	−1.24590
$175$	0	0
$176$	−132.866	−0.0569043
$177$	3523.88	1.49645
$178$	−736.959	−0.310322
$179$	−1229.49	−0.513389	−0.256695	−	0.966493i	$-0.582633\pi$
$179$	−1229.49	−0.513389	−0.256695	+	0.966493i	$0.582633\pi$
$180$	0	0
$181$	−3108.95	−1.27672	−0.638360	−	0.769738i	$-0.720387\pi$
$181$	−3108.95	−1.27672	−0.638360	+	0.769738i	$0.720387\pi$
$182$	2318.70	0.944359
$183$	−3892.36	−1.57230
$184$	1311.82	0.525589
$185$	0	0
$186$	1211.13	0.477444
$187$	616.840	0.241218
$188$	1464.33	0.568071
$189$	2171.00	0.835539
$190$	0	0
$191$	−1415.48	−0.536233	−0.268117	−	0.963386i	$-0.586401\pi$
$191$	−1415.48	−0.536233	−0.268117	+	0.963386i	$0.586401\pi$
$192$	393.732	0.147996
$193$	−1443.40	−0.538333	−0.269167	−	0.963094i	$-0.586748\pi$
$193$	−1443.40	−0.538333	−0.269167	+	0.963094i	$0.586748\pi$
$194$	852.886	0.315637
$195$	0	0
$196$	537.329	0.195819
$197$	−5271.92	−1.90664	−0.953322	−	0.301954i	$-0.902361\pi$
$197$	−5271.92	−1.90664	−0.953322	+	0.301954i	$0.902361\pi$
$198$	−180.165	−0.0646656
$199$	−2510.19	−0.894183	−0.447091	−	0.894488i	$-0.647540\pi$
$199$	−2510.19	−0.894183	−0.447091	+	0.894488i	$0.647540\pi$
$200$	0	0
$201$	1128.67	0.396069
$202$	−806.248	−0.280829
$203$	5077.68	1.75558
$204$	−1827.93	−0.627355
$205$	0	0
$206$	2271.35	0.768217
$207$	1778.81	0.597275
$208$	−849.032	−0.283028
$209$	157.779	0.0522190
$210$	0	0
$211$	1854.44	0.605046	0.302523	−	0.953142i	$-0.402171\pi$
$211$	1854.44	0.605046	0.302523	+	0.953142i	$0.402171\pi$
$212$	−554.526	−0.179646
$213$	348.542	0.112121
$214$	760.192	0.242830
$215$	0	0
$216$	−794.949	−0.250414
$217$	−2150.56	−0.672763
$218$	2361.48	0.733668
$219$	−419.276	−0.129370
$220$	0	0
$221$	3941.68	1.19976
$222$	−3647.35	−1.10268
$223$	−1880.34	−0.564649	−0.282325	−	0.959319i	$-0.591105\pi$
$223$	−1880.34	−0.564649	−0.282325	+	0.959319i	$0.591105\pi$
$224$	−699.134	−0.208539
$225$	0	0
$226$	2265.01	0.666665
$227$	−1799.23	−0.526075	−0.263038	−	0.964786i	$-0.584724\pi$
$227$	−1799.23	−0.526075	−0.263038	+	0.964786i	$0.584724\pi$
$228$	−467.557	−0.135810
$229$	4835.34	1.39532	0.697660	−	0.716429i	$-0.254224\pi$
$229$	4835.34	1.39532	0.697660	+	0.716429i	$0.254224\pi$
$230$	0	0
$231$	1116.16	0.317913
$232$	−1859.28	−0.526154
$233$	−865.299	−0.243295	−0.121647	−	0.992573i	$-0.538818\pi$
$233$	−865.299	−0.243295	−0.121647	+	0.992573i	$0.538818\pi$
$234$	−1151.28	−0.321630
$235$	0	0
$236$	2291.19	0.631964
$237$	−2047.53	−0.561187
$238$	3245.77	0.884001
$239$	−4764.27	−1.28943	−0.644717	−	0.764421i	$-0.723025\pi$
$239$	−4764.27	−1.28943	−0.644717	+	0.764421i	$0.723025\pi$
$240$	0	0
$241$	615.336	0.164470	0.0822350	−	0.996613i	$-0.473794\pi$
$241$	615.336	0.164470	0.0822350	+	0.996613i	$0.473794\pi$
$242$	−2524.08	−0.670472
$243$	−2879.85	−0.760257
$244$	−2530.76	−0.663998
$245$	0	0
$246$	−5351.22	−1.38692
$247$	1008.22	0.259724
$248$	787.465	0.201629
$249$	−7092.79	−1.80517
$250$	0	0
$251$	−1658.08	−0.416959	−0.208480	−	0.978027i	$-0.566852\pi$
$251$	−1658.08	−0.416959	−0.208480	+	0.978027i	$0.566852\pi$
$252$	−948.020	−0.236983
$253$	−1361.69	−0.338373
$254$	80.0925	0.0197852
$255$	0	0
$256$	256.000	0.0625000
$257$	−3446.12	−0.836432	−0.418216	−	0.908348i	$-0.637344\pi$
$257$	−3446.12	−0.836432	−0.418216	+	0.908348i	$0.637344\pi$
$258$	2107.69	0.508601
$259$	6476.45	1.55377
$260$	0	0
$261$	−2521.17	−0.597917
$262$	−355.795	−0.0838974
$263$	−5755.80	−1.34950	−0.674748	−	0.738048i	$-0.735748\pi$
$263$	−5755.80	−1.34950	−0.674748	+	0.738048i	$0.735748\pi$
$264$	−408.701	−0.0952795
$265$	0	0
$266$	830.221	0.191369
$267$	−2266.91	−0.519598
$268$	733.844	0.167264
$269$	2257.28	0.511631	0.255816	−	0.966726i	$-0.417656\pi$
$269$	2257.28	0.511631	0.255816	+	0.966726i	$0.417656\pi$
$270$	0	0
$271$	7012.13	1.57180	0.785898	−	0.618357i	$-0.212201\pi$
$271$	7012.13	1.57180	0.785898	+	0.618357i	$0.212201\pi$
$272$	−1188.50	−0.264938
$273$	7132.39	1.58122
$274$	49.5207	0.0109184
$275$	0	0
$276$	4035.19	0.880035
$277$	372.810	0.0808664	0.0404332	−	0.999182i	$-0.487126\pi$
$277$	372.810	0.0808664	0.0404332	+	0.999182i	$0.487126\pi$
$278$	5734.41	1.23715
$279$	1067.80	0.229130
$280$	0	0
$281$	−1888.96	−0.401017	−0.200508	−	0.979692i	$-0.564259\pi$
$281$	−1888.96	−0.401017	−0.200508	+	0.979692i	$0.564259\pi$
$282$	4504.33	0.951167
$283$	3884.43	0.815920	0.407960	−	0.913000i	$-0.366240\pi$
$283$	3884.43	0.815920	0.407960	+	0.913000i	$0.366240\pi$
$284$	226.618	0.0473497
$285$	0	0
$286$	881.309	0.182213
$287$	9501.94	1.95429
$288$	347.134	0.0710245
$289$	604.668	0.123075
$290$	0	0
$291$	2623.51	0.528497
$292$	−272.608	−0.0546342
$293$	1273.01	0.253823	0.126911	−	0.991914i	$-0.459494\pi$
$293$	1273.01	0.253823	0.126911	+	0.991914i	$0.459494\pi$
$294$	1652.84	0.327876
$295$	0	0
$296$	−2371.46	−0.465671
$297$	825.170	0.161216
$298$	3898.69	0.757869
$299$	−8701.35	−1.68298
$300$	0	0
$301$	−3742.53	−0.716665
$302$	2241.97	0.427188
$303$	−2480.05	−0.470214
$304$	−304.000	−0.0573539
$305$	0	0
$306$	−1611.59	−0.301074
$307$	−819.153	−0.152285	−0.0761426	−	0.997097i	$-0.524260\pi$
$307$	−819.153	−0.152285	−0.0761426	+	0.997097i	$0.524260\pi$
$308$	725.713	0.134258
$309$	6986.76	1.28629
$310$	0	0
$311$	2104.67	0.383745	0.191872	−	0.981420i	$-0.438544\pi$
$311$	2104.67	0.383745	0.191872	+	0.981420i	$0.438544\pi$
$312$	−2611.65	−0.473896
$313$	−2395.47	−0.432587	−0.216293	−	0.976328i	$-0.569397\pi$
$313$	−2395.47	−0.432587	−0.216293	+	0.976328i	$0.569397\pi$
$314$	4720.46	0.848378
$315$	0	0
$316$	−1331.28	−0.236995
$317$	−2158.23	−0.382393	−0.191196	−	0.981552i	$-0.561237\pi$
$317$	−2158.23	−0.382393	−0.191196	+	0.981552i	$0.561237\pi$
$318$	−1705.74	−0.300796
$319$	1929.96	0.338737
$320$	0	0
$321$	2338.38	0.406590
$322$	−7165.11	−1.24005
$323$	1411.34	0.243124
$324$	−3616.87	−0.620176
$325$	0	0
$326$	−1723.65	−0.292835
$327$	7264.00	1.22844
$328$	−3479.30	−0.585708
$329$	−7998.15	−1.34028
$330$	0	0
$331$	517.782	0.0859815	0.0429908	−	0.999075i	$-0.486311\pi$
$331$	517.782	0.0859815	0.0429908	+	0.999075i	$0.486311\pi$
$332$	−4611.65	−0.762341
$333$	−3215.69	−0.529185
$334$	−3373.01	−0.552584
$335$	0	0
$336$	−2150.56	−0.349174
$337$	−6503.63	−1.05126	−0.525631	−	0.850713i	$-0.676171\pi$
$337$	−6503.63	−1.05126	−0.525631	+	0.850713i	$0.676171\pi$
$338$	1237.68	0.199174
$339$	6967.25	1.11625
$340$	0	0
$341$	−817.402	−0.129809
$342$	−412.221	−0.0651766
$343$	4558.96	0.717670
$344$	1370.39	0.214787
$345$	0	0
$346$	6382.87	0.991749
$347$	−6058.33	−0.937257	−0.468628	−	0.883395i	$-0.655252\pi$
$347$	−6058.33	−0.937257	−0.468628	+	0.883395i	$0.655252\pi$
$348$	−5719.21	−0.880982
$349$	−10955.1	−1.68027	−0.840135	−	0.542377i	$-0.817524\pi$
$349$	−10955.1	−1.68027	−0.840135	+	0.542377i	$0.817524\pi$
$350$	0	0
$351$	5272.94	0.801849
$352$	−265.732	−0.0402374
$353$	1806.43	0.272369	0.136185	−	0.990683i	$-0.456516\pi$
$353$	1806.43	0.272369	0.136185	+	0.990683i	$0.456516\pi$
$354$	7047.77	1.05815
$355$	0	0
$356$	−1473.92	−0.219431
$357$	9984.11	1.48015
$358$	−2458.99	−0.363021
$359$	−7964.50	−1.17089	−0.585446	−	0.810711i	$-0.699081\pi$
$359$	−7964.50	−1.17089	−0.585446	+	0.810711i	$0.699081\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−6217.90	−0.902777
$363$	−7764.16	−1.12263
$364$	4637.40	0.667763
$365$	0	0
$366$	−7784.71	−1.11179
$367$	7311.58	1.03995	0.519974	−	0.854182i	$-0.325941\pi$
$367$	7311.58	1.03995	0.519974	+	0.854182i	$0.325941\pi$
$368$	2623.63	0.371647
$369$	−4717.90	−0.665594
$370$	0	0
$371$	3028.81	0.423849
$372$	2422.27	0.337604
$373$	−5518.38	−0.766035	−0.383017	−	0.923741i	$-0.625115\pi$
$373$	−5518.38	−0.766035	−0.383017	+	0.923741i	$0.625115\pi$
$374$	1233.68	0.170567
$375$	0	0
$376$	2928.66	0.401687
$377$	12332.7	1.68479
$378$	4342.00	0.590815
$379$	−1139.97	−0.154502	−0.0772512	−	0.997012i	$-0.524614\pi$
$379$	−1139.97	−0.154502	−0.0772512	+	0.997012i	$0.524614\pi$
$380$	0	0
$381$	246.367	0.0331280
$382$	−2830.96	−0.379174
$383$	10409.5	1.38877	0.694385	−	0.719604i	$-0.255677\pi$
$383$	10409.5	1.38877	0.694385	+	0.719604i	$0.255677\pi$
$384$	787.465	0.104649
$385$	0	0
$386$	−2886.80	−0.380659
$387$	1858.24	0.244082
$388$	1705.77	0.223189
$389$	10471.2	1.36481	0.682404	−	0.730975i	$-0.260934\pi$
$389$	10471.2	1.36481	0.682404	+	0.730975i	$0.260934\pi$
$390$	0	0
$391$	−12180.4	−1.57542
$392$	1074.66	0.138465
$393$	−1094.44	−0.140476
$394$	−10543.8	−1.34820
$395$	0	0
$396$	−360.331	−0.0457255
$397$	9588.68	1.21220	0.606098	−	0.795390i	$-0.292734\pi$
$397$	9588.68	1.21220	0.606098	+	0.795390i	$0.292734\pi$
$398$	−5020.37	−0.632283
$399$	2553.79	0.320424
$400$	0	0
$401$	8549.30	1.06467	0.532334	−	0.846535i	$-0.321315\pi$
$401$	8549.30	1.06467	0.532334	+	0.846535i	$0.321315\pi$
$402$	2257.33	0.280063
$403$	−5223.30	−0.645635
$404$	−1612.50	−0.198576
$405$	0	0
$406$	10155.4	1.24138
$407$	2461.62	0.299798
$408$	−3655.85	−0.443607
$409$	266.960	0.0322746	0.0161373	−	0.999870i	$-0.494863\pi$
$409$	266.960	0.0322746	0.0161373	+	0.999870i	$0.494863\pi$
$410$	0	0
$411$	152.327	0.0182816
$412$	4542.71	0.543211
$413$	−12514.4	−1.49103
$414$	3557.62	0.422337
$415$	0	0
$416$	−1698.06	−0.200131
$417$	17639.2	2.07146
$418$	315.557	0.0369244
$419$	8643.83	1.00783	0.503913	−	0.863755i	$-0.331893\pi$
$419$	8643.83	1.00783	0.503913	+	0.863755i	$0.331893\pi$
$420$	0	0
$421$	−16801.3	−1.94500	−0.972499	−	0.232905i	$-0.925177\pi$
$421$	−16801.3	−1.94500	−0.972499	+	0.232905i	$0.925177\pi$
$422$	3708.87	0.427832
$423$	3971.24	0.456474
$424$	−1109.05	−0.127029
$425$	0	0
$426$	697.085	0.0792814
$427$	13823.0	1.56661
$428$	1520.38	0.171707
$429$	2710.94	0.305094
$430$	0	0
$431$	12053.5	1.34710	0.673548	−	0.739144i	$-0.264769\pi$
$431$	12053.5	1.34710	0.673548	+	0.739144i	$0.264769\pi$
$432$	−1589.90	−0.177069
$433$	−9034.61	−1.00271	−0.501357	−	0.865240i	$-0.667166\pi$
$433$	−9034.61	−1.00271	−0.501357	+	0.865240i	$0.667166\pi$
$434$	−4301.12	−0.475715
$435$	0	0
$436$	4722.96	0.518782
$437$	−3115.56	−0.341047
$438$	−838.552	−0.0914785
$439$	3008.87	0.327120	0.163560	−	0.986533i	$-0.447702\pi$
$439$	3008.87	0.327120	0.163560	+	0.986533i	$0.447702\pi$
$440$	0	0
$441$	1457.23	0.157351
$442$	7883.37	0.848356
$443$	229.594	0.0246237	0.0123119	−	0.999924i	$-0.496081\pi$
$443$	229.594	0.0246237	0.0123119	+	0.999924i	$0.496081\pi$
$444$	−7294.71	−0.779710
$445$	0	0
$446$	−3760.68	−0.399267
$447$	11992.5	1.26896
$448$	−1398.27	−0.147460
$449$	−7559.44	−0.794548	−0.397274	−	0.917700i	$-0.630044\pi$
$449$	−7559.44	−0.794548	−0.397274	+	0.917700i	$0.630044\pi$
$450$	0	0
$451$	3611.57	0.377078
$452$	4530.02	0.471404
$453$	6896.38	0.715276
$454$	−3598.46	−0.371991
$455$	0	0
$456$	−935.114	−0.0960323
$457$	−11556.4	−1.18290	−0.591449	−	0.806343i	$-0.701444\pi$
$457$	−11556.4	−1.18290	−0.591449	+	0.806343i	$0.701444\pi$
$458$	9670.69	0.986641
$459$	7381.20	0.750599
$460$	0	0
$461$	9191.58	0.928622	0.464311	−	0.885672i	$-0.346302\pi$
$461$	9191.58	0.928622	0.464311	+	0.885672i	$0.346302\pi$
$462$	2232.32	0.224798
$463$	−1356.03	−0.136113	−0.0680564	−	0.997681i	$-0.521680\pi$
$463$	−1356.03	−0.136113	−0.0680564	+	0.997681i	$0.521680\pi$
$464$	−3718.56	−0.372047
$465$	0	0
$466$	−1730.60	−0.172035
$467$	14808.9	1.46740	0.733700	−	0.679473i	$-0.237792\pi$
$467$	14808.9	1.46740	0.733700	+	0.679473i	$0.237792\pi$
$468$	−2302.56	−0.227427
$469$	−4008.25	−0.394635
$470$	0	0
$471$	14520.3	1.42051
$472$	4582.37	0.446866
$473$	−1422.49	−0.138280
$474$	−4095.06	−0.396819
$475$	0	0
$476$	6491.55	0.625083
$477$	−1503.86	−0.144355
$478$	−9528.54	−0.911768
$479$	9834.66	0.938115	0.469058	−	0.883168i	$-0.344594\pi$
$479$	9834.66	0.938115	0.469058	+	0.883168i	$0.344594\pi$
$480$	0	0
$481$	15730.1	1.49112
$482$	1230.67	0.116298
$483$	−22040.1	−2.07632
$484$	−5048.17	−0.474095
$485$	0	0
$486$	−5759.70	−0.537583
$487$	−3687.82	−0.343144	−0.171572	−	0.985172i	$-0.554885\pi$
$487$	−3687.82	−0.343144	−0.171572	+	0.985172i	$0.554885\pi$
$488$	−5061.53	−0.469518
$489$	−5302.00	−0.490317
$490$	0	0
$491$	−11197.4	−1.02919	−0.514593	−	0.857435i	$-0.672057\pi$
$491$	−11197.4	−1.02919	−0.514593	+	0.857435i	$0.672057\pi$
$492$	−10702.4	−0.980698
$493$	17263.6	1.57711
$494$	2016.45	0.183653
$495$	0	0
$496$	1574.93	0.142573
$497$	−1237.78	−0.111715
$498$	−14185.6	−1.27645
$499$	−12101.6	−1.08566	−0.542829	−	0.839843i	$-0.682647\pi$
$499$	−12101.6	−1.08566	−0.542829	+	0.839843i	$0.682647\pi$
$500$	0	0
$501$	−10375.5	−0.925236
$502$	−3316.15	−0.294835
$503$	7266.58	0.644136	0.322068	−	0.946716i	$-0.395622\pi$
$503$	7266.58	0.644136	0.322068	+	0.946716i	$0.395622\pi$
$504$	−1896.04	−0.167572
$505$	0	0
$506$	−2723.37	−0.239266
$507$	3807.13	0.333493
$508$	160.185	0.0139903
$509$	2564.99	0.223362	0.111681	−	0.993744i	$-0.464376\pi$
$509$	2564.99	0.223362	0.111681	+	0.993744i	$0.464376\pi$
$510$	0	0
$511$	1488.98	0.128902
$512$	512.000	0.0441942
$513$	1888.00	0.162490
$514$	−6892.24	−0.591447
$515$	0	0
$516$	4215.38	0.359635
$517$	−3040.00	−0.258606
$518$	12952.9	1.09868
$519$	19633.9	1.66057
$520$	0	0
$521$	11254.6	0.946401	0.473201	−	0.880955i	$-0.343099\pi$
$521$	11254.6	0.946401	0.473201	+	0.880955i	$0.343099\pi$
$522$	−5042.34	−0.422791
$523$	18357.1	1.53480	0.767401	−	0.641168i	$-0.221550\pi$
$523$	18357.1	1.53480	0.767401	+	0.641168i	$0.221550\pi$
$524$	−711.591	−0.0593244
$525$	0	0
$526$	−11511.6	−0.954238
$527$	−7311.71	−0.604370
$528$	−817.402	−0.0673728
$529$	14721.4	1.20995
$530$	0	0
$531$	6213.66	0.507815
$532$	1660.44	0.135318
$533$	23078.4	1.87549
$534$	−4533.82	−0.367411
$535$	0	0
$536$	1467.69	0.118273
$537$	−7563.93	−0.607836
$538$	4514.56	0.361778
$539$	−1115.51	−0.0891438
$540$	0	0
$541$	11381.8	0.904510	0.452255	−	0.891889i	$-0.350620\pi$
$541$	11381.8	0.904510	0.452255	+	0.891889i	$0.350620\pi$
$542$	14024.3	1.11143
$543$	−19126.5	−1.51159
$544$	−2376.99	−0.187340
$545$	0	0
$546$	14264.8	1.11809
$547$	8996.84	0.703248	0.351624	−	0.936141i	$-0.385629\pi$
$547$	8996.84	0.703248	0.351624	+	0.936141i	$0.385629\pi$
$548$	99.0413	0.00772050
$549$	−6863.39	−0.533556
$550$	0	0
$551$	4415.79	0.341414
$552$	8070.37	0.622279
$553$	7271.43	0.559155
$554$	745.620	0.0571812
$555$	0	0
$556$	11468.8	0.874796
$557$	−10759.8	−0.818507	−0.409253	−	0.912421i	$-0.634211\pi$
$557$	−10759.8	−0.818507	−0.409253	+	0.912421i	$0.634211\pi$
$558$	2135.59	0.162019
$559$	−9089.90	−0.687767
$560$	0	0
$561$	3794.84	0.285594
$562$	−3777.91	−0.283562
$563$	−25381.8	−1.90003	−0.950015	−	0.312205i	$-0.898932\pi$
$563$	−25381.8	−1.90003	−0.950015	+	0.312205i	$0.898932\pi$
$564$	9008.66	0.672576
$565$	0	0
$566$	7768.86	0.576943
$567$	19755.3	1.46322
$568$	453.236	0.0334813
$569$	−5546.00	−0.408612	−0.204306	−	0.978907i	$-0.565494\pi$
$569$	−5546.00	−0.408612	−0.204306	+	0.978907i	$0.565494\pi$
$570$	0	0
$571$	7714.96	0.565431	0.282715	−	0.959204i	$-0.408765\pi$
$571$	7714.96	0.565431	0.282715	+	0.959204i	$0.408765\pi$
$572$	1762.62	0.128844
$573$	−8708.13	−0.634882
$574$	19003.9	1.38189
$575$	0	0
$576$	694.268	0.0502219
$577$	−21335.1	−1.53933	−0.769665	−	0.638448i	$-0.779577\pi$
$577$	−21335.1	−1.53933	−0.769665	+	0.638448i	$0.779577\pi$
$578$	1209.34	0.0870273
$579$	−8879.90	−0.637368
$580$	0	0
$581$	25188.8	1.79863
$582$	5247.01	0.373704
$583$	1151.21	0.0817811
$584$	−545.217	−0.0386322
$585$	0	0
$586$	2546.02	0.179480
$587$	−12370.9	−0.869846	−0.434923	−	0.900468i	$-0.643224\pi$
$587$	−12370.9	−0.869846	−0.434923	+	0.900468i	$0.643224\pi$
$588$	3305.68	0.231844
$589$	−1870.23	−0.130834
$590$	0	0
$591$	−32433.2	−2.25740
$592$	−4742.93	−0.329279
$593$	−12982.5	−0.899034	−0.449517	−	0.893272i	$-0.648404\pi$
$593$	−12982.5	−0.899034	−0.449517	+	0.893272i	$0.648404\pi$
$594$	1650.34	0.113997
$595$	0	0
$596$	7797.38	0.535895
$597$	−15442.8	−1.05868
$598$	−17402.7	−1.19005
$599$	13163.7	0.897921	0.448960	−	0.893552i	$-0.351794\pi$
$599$	13163.7	0.897921	0.448960	+	0.893552i	$0.351794\pi$
$600$	0	0
$601$	−29325.4	−1.99036	−0.995180	−	0.0980640i	$-0.968735\pi$
$601$	−29325.4	−1.99036	−0.995180	+	0.0980640i	$0.968735\pi$
$602$	−7485.07	−0.506758
$603$	1990.17	0.134405
$604$	4483.94	0.302068
$605$	0	0
$606$	−4960.09	−0.332492
$607$	2170.11	0.145110	0.0725552	−	0.997364i	$-0.476885\pi$
$607$	2170.11	0.145110	0.0725552	+	0.997364i	$0.476885\pi$
$608$	−608.000	−0.0405554
$609$	31238.2	2.07855
$610$	0	0
$611$	−19426.0	−1.28624
$612$	−3223.18	−0.212891
$613$	4917.96	0.324036	0.162018	−	0.986788i	$-0.448200\pi$
$613$	4917.96	0.324036	0.162018	+	0.986788i	$0.448200\pi$
$614$	−1638.31	−0.107682
$615$	0	0
$616$	1451.43	0.0949344
$617$	−24763.5	−1.61579	−0.807894	−	0.589328i	$-0.799393\pi$
$617$	−24763.5	−1.61579	−0.807894	+	0.589328i	$0.799393\pi$
$618$	13973.5	0.909542
$619$	−27552.0	−1.78902	−0.894512	−	0.447043i	$-0.852477\pi$
$619$	−27552.0	−1.78902	−0.894512	+	0.447043i	$0.852477\pi$
$620$	0	0
$621$	−16294.2	−1.05292
$622$	4209.33	0.271348
$623$	8050.51	0.517716
$624$	−5223.30	−0.335095
$625$	0	0
$626$	−4790.93	−0.305885
$627$	970.664	0.0618255
$628$	9440.91	0.599894
$629$	22019.3	1.39582
$630$	0	0
$631$	377.839	0.0238376	0.0119188	−	0.999929i	$-0.496206\pi$
$631$	377.839	0.0238376	0.0119188	+	0.999929i	$0.496206\pi$
$632$	−2662.56	−0.167581
$633$	11408.6	0.716354
$634$	−4316.47	−0.270393
$635$	0	0
$636$	−3411.48	−0.212695
$637$	−7128.27	−0.443379
$638$	3859.93	0.239523
$639$	614.584	0.0380479
$640$	0	0
$641$	−14047.3	−0.865576	−0.432788	−	0.901496i	$-0.642470\pi$
$641$	−14047.3	−0.865576	−0.432788	+	0.901496i	$0.642470\pi$
$642$	4676.75	0.287503
$643$	−5768.94	−0.353818	−0.176909	−	0.984227i	$-0.556610\pi$
$643$	−5768.94	−0.353818	−0.176909	+	0.984227i	$0.556610\pi$
$644$	−14330.2	−0.876848
$645$	0	0
$646$	2822.68	0.171915
$647$	−4568.59	−0.277604	−0.138802	−	0.990320i	$-0.544325\pi$
$647$	−4568.59	−0.277604	−0.138802	+	0.990320i	$0.544325\pi$
$648$	−7233.73	−0.438531
$649$	−4756.58	−0.287692
$650$	0	0
$651$	−13230.4	−0.796528
$652$	−3447.30	−0.207065
$653$	16532.7	0.990774	0.495387	−	0.868672i	$-0.335026\pi$
$653$	16532.7	0.990774	0.495387	+	0.868672i	$0.335026\pi$
$654$	14528.0	0.868638
$655$	0	0
$656$	−6958.60	−0.414158
$657$	−739.309	−0.0439014
$658$	−15996.3	−0.947721
$659$	−18630.0	−1.10125	−0.550624	−	0.834753i	$-0.685610\pi$
$659$	−18630.0	−1.10125	−0.550624	+	0.834753i	$0.685610\pi$
$660$	0	0
$661$	28851.3	1.69771	0.848854	−	0.528627i	$-0.177293\pi$
$661$	28851.3	1.69771	0.848854	+	0.528627i	$0.177293\pi$
$662$	1035.56	0.0607981
$663$	24249.5	1.42047
$664$	−9223.30	−0.539056
$665$	0	0
$666$	−6431.37	−0.374190
$667$	−38109.9	−2.21232
$668$	−6746.02	−0.390736
$669$	−11568.0	−0.668525
$670$	0	0
$671$	5253.95	0.302275
$672$	−4301.12	−0.246904
$673$	3873.41	0.221856	0.110928	−	0.993828i	$-0.464618\pi$
$673$	3873.41	0.221856	0.110928	+	0.993828i	$0.464618\pi$
$674$	−13007.3	−0.743354
$675$	0	0
$676$	2475.35	0.140837
$677$	−5025.24	−0.285282	−0.142641	−	0.989775i	$-0.545559\pi$
$677$	−5025.24	−0.285282	−0.142641	+	0.989775i	$0.545559\pi$
$678$	13934.5	0.789309
$679$	−9316.90	−0.526583
$680$	0	0
$681$	−11069.0	−0.622855
$682$	−1634.80	−0.0917886
$683$	14157.0	0.793122	0.396561	−	0.918008i	$-0.370204\pi$
$683$	14157.0	0.793122	0.396561	+	0.918008i	$0.370204\pi$
$684$	−824.443	−0.0460868
$685$	0	0
$686$	9117.92	0.507469
$687$	29747.4	1.65201
$688$	2740.79	0.151877
$689$	7356.40	0.406758
$690$	0	0
$691$	−2437.65	−0.134200	−0.0671002	−	0.997746i	$-0.521375\pi$
$691$	−2437.65	−0.134200	−0.0671002	+	0.997746i	$0.521375\pi$
$692$	12765.7	0.701273
$693$	1968.12	0.107883
$694$	−12116.7	−0.662741
$695$	0	0
$696$	−11438.4	−0.622948
$697$	32305.7	1.75562
$698$	−21910.2	−1.18813
$699$	−5323.38	−0.288052
$700$	0	0
$701$	9496.96	0.511691	0.255845	−	0.966718i	$-0.417646\pi$
$701$	9496.96	0.511691	0.255845	+	0.966718i	$0.417646\pi$
$702$	10545.9	0.566993
$703$	5632.23	0.302167
$704$	−531.465	−0.0284522
$705$	0	0
$706$	3612.85	0.192594
$707$	8807.43	0.468511
$708$	14095.5	0.748224
$709$	6350.33	0.336377	0.168189	−	0.985755i	$-0.446208\pi$
$709$	6350.33	0.336377	0.168189	+	0.985755i	$0.446208\pi$
$710$	0	0
$711$	−3610.41	−0.190437
$712$	−2947.83	−0.155161
$713$	16140.7	0.847792
$714$	19968.2	1.04663
$715$	0	0
$716$	−4917.98	−0.256695
$717$	−29310.1	−1.52665
$718$	−15929.0	−0.827946
$719$	−19722.6	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$719$	−19722.6	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$720$	0	0
$721$	−24812.2	−1.28163
$722$	722.000	0.0372161
$723$	3785.59	0.194727
$724$	−12435.8	−0.638360
$725$	0	0
$726$	−15528.3	−0.793816
$727$	−28325.8	−1.44504	−0.722520	−	0.691350i	$-0.757016\pi$
$727$	−28325.8	−1.44504	−0.722520	+	0.691350i	$0.757016\pi$
$728$	9274.79	0.472179
$729$	6696.82	0.340234
$730$	0	0
$731$	−12724.3	−0.643809
$732$	−15569.4	−0.786151
$733$	−33981.8	−1.71234	−0.856170	−	0.516694i	$-0.827163\pi$
$733$	−33981.8	−1.71234	−0.856170	+	0.516694i	$0.827163\pi$
$734$	14623.2	0.735355
$735$	0	0
$736$	5247.26	0.262794
$737$	−1523.49	−0.0761443
$738$	−9435.80	−0.470646
$739$	9147.10	0.455320	0.227660	−	0.973741i	$-0.426893\pi$
$739$	9147.10	0.455320	0.227660	+	0.973741i	$0.426893\pi$
$740$	0	0
$741$	6202.67	0.307504
$742$	6057.62	0.299706
$743$	34159.2	1.68665	0.843324	−	0.537405i	$-0.180596\pi$
$743$	34159.2	1.68665	0.843324	+	0.537405i	$0.180596\pi$
$744$	4844.54	0.238722
$745$	0	0
$746$	−11036.8	−0.541668
$747$	−12506.7	−0.612579
$748$	2467.36	0.120609
$749$	−8304.31	−0.405117
$750$	0	0
$751$	28361.8	1.37808	0.689038	−	0.724725i	$-0.258033\pi$
$751$	28361.8	1.37808	0.689038	+	0.724725i	$0.258033\pi$
$752$	5857.32	0.284035
$753$	−10200.6	−0.493666
$754$	24665.4	1.19133
$755$	0	0
$756$	8683.99	0.417770
$757$	−11464.9	−0.550462	−0.275231	−	0.961378i	$-0.588754\pi$
$757$	−11464.9	−0.550462	−0.275231	+	0.961378i	$0.588754\pi$
$758$	−2279.94	−0.109250
$759$	−8377.18	−0.400623
$760$	0	0
$761$	14289.3	0.680666	0.340333	−	0.940305i	$-0.389460\pi$
$761$	14289.3	0.680666	0.340333	+	0.940305i	$0.389460\pi$
$762$	492.734	0.0234250
$763$	−25796.7	−1.22399
$764$	−5661.92	−0.268117
$765$	0	0
$766$	20818.9	0.982008
$767$	−30395.2	−1.43091
$768$	1574.93	0.0739979
$769$	100.811	0.00472734	0.00236367	−	0.999997i	$-0.499248\pi$
$769$	100.811	0.00472734	0.00236367	+	0.999997i	$0.499248\pi$
$770$	0	0
$771$	−21200.8	−0.990307
$772$	−5773.61	−0.269167
$773$	−52.1073	−0.00242454	−0.00121227	−	0.999999i	$-0.500386\pi$
$773$	−52.1073	−0.00242454	−0.00121227	+	0.999999i	$0.500386\pi$
$774$	3716.49	0.172592
$775$	0	0
$776$	3411.54	0.157819
$777$	39843.6	1.83961
$778$	20942.4	0.965065
$779$	8263.34	0.380057
$780$	0	0
$781$	−470.467	−0.0215552
$782$	−24360.7	−1.11399
$783$	23094.3	1.05405
$784$	2149.31	0.0979097
$785$	0	0
$786$	−2188.88	−0.0993316
$787$	−15261.8	−0.691266	−0.345633	−	0.938370i	$-0.612336\pi$
$787$	−15261.8	−0.691266	−0.345633	+	0.938370i	$0.612336\pi$
$788$	−21087.7	−0.953322
$789$	−35410.0	−1.59776
$790$	0	0
$791$	−24742.9	−1.11221
$792$	−720.662	−0.0323328
$793$	33573.4	1.50344
$794$	19177.4	0.857152
$795$	0	0
$796$	−10040.7	−0.447091
$797$	−24432.3	−1.08587	−0.542934	−	0.839775i	$-0.682687\pi$
$797$	−24432.3	−1.08587	−0.542934	+	0.839775i	$0.682687\pi$
$798$	5107.58	0.226574
$799$	−27193.0	−1.20403
$800$	0	0
$801$	−3997.24	−0.176324
$802$	17098.6	0.752834
$803$	565.944	0.0248714
$804$	4514.66	0.198035
$805$	0	0
$806$	−10446.6	−0.456533
$807$	13886.9	0.605754
$808$	−3224.99	−0.140414
$809$	3635.86	0.158010	0.0790050	−	0.996874i	$-0.474826\pi$
$809$	3635.86	0.158010	0.0790050	+	0.996874i	$0.474826\pi$
$810$	0	0
$811$	1087.93	0.0471051	0.0235526	−	0.999723i	$-0.492502\pi$
$811$	1087.93	0.0471051	0.0235526	+	0.999723i	$0.492502\pi$
$812$	20310.7	0.877791
$813$	43139.1	1.86095
$814$	4923.24	0.211990
$815$	0	0
$816$	−7311.71	−0.313678
$817$	−3254.69	−0.139372
$818$	533.920	0.0228216
$819$	12576.5	0.536581
$820$	0	0
$821$	10953.7	0.465636	0.232818	−	0.972520i	$-0.425205\pi$
$821$	10953.7	0.465636	0.232818	+	0.972520i	$0.425205\pi$
$822$	304.655	0.0129271
$823$	−13502.5	−0.571893	−0.285947	−	0.958246i	$-0.592308\pi$
$823$	−13502.5	−0.571893	−0.285947	+	0.958246i	$0.592308\pi$
$824$	9085.41	0.384108
$825$	0	0
$826$	−25028.9	−1.05432
$827$	−26812.4	−1.12740	−0.563699	−	0.825980i	$-0.690622\pi$
$827$	−26812.4	−1.12740	−0.563699	+	0.825980i	$0.690622\pi$
$828$	7115.24	0.298637
$829$	−4497.94	−0.188444	−0.0942218	−	0.995551i	$-0.530036\pi$
$829$	−4497.94	−0.188444	−0.0942218	+	0.995551i	$0.530036\pi$
$830$	0	0
$831$	2293.55	0.0957430
$832$	−3396.13	−0.141514
$833$	−9978.33	−0.415040
$834$	35278.5	1.46474
$835$	0	0
$836$	631.114	0.0261095
$837$	−9781.16	−0.403926
$838$	17287.7	0.712640
$839$	30324.9	1.24783	0.623917	−	0.781491i	$-0.285540\pi$
$839$	30324.9	1.24783	0.623917	+	0.781491i	$0.285540\pi$
$840$	0	0
$841$	29625.4	1.21470
$842$	−33602.6	−1.37532
$843$	−11621.0	−0.474790
$844$	7417.75	0.302523
$845$	0	0
$846$	7942.48	0.322776
$847$	27573.0	1.11856
$848$	−2218.10	−0.0898230
$849$	23897.3	0.966022
$850$	0	0
$851$	−48608.2	−1.95801
$852$	1394.17	0.0560604
$853$	−17230.8	−0.691642	−0.345821	−	0.938300i	$-0.612400\pi$
$853$	−17230.8	−0.691642	−0.345821	+	0.938300i	$0.612400\pi$
$854$	27646.0	1.10776
$855$	0	0
$856$	3040.77	0.121415
$857$	6724.12	0.268018	0.134009	−	0.990980i	$-0.457215\pi$
$857$	6724.12	0.268018	0.134009	+	0.990980i	$0.457215\pi$
$858$	5421.87	0.215734
$859$	−36183.5	−1.43721	−0.718606	−	0.695417i	$-0.755219\pi$
$859$	−36183.5	−1.43721	−0.718606	+	0.695417i	$0.755219\pi$
$860$	0	0
$861$	58456.6	2.31381
$862$	24107.1	0.952541
$863$	−14350.8	−0.566056	−0.283028	−	0.959112i	$-0.591339\pi$
$863$	−14350.8	−0.566056	−0.283028	+	0.959112i	$0.591339\pi$
$864$	−3179.80	−0.125207
$865$	0	0
$866$	−18069.2	−0.709027
$867$	3719.96	0.145717
$868$	−8602.24	−0.336381
$869$	2763.78	0.107888
$870$	0	0
$871$	−9735.27	−0.378722
$872$	9445.93	0.366834
$873$	4626.02	0.179344
$874$	−6231.12	−0.241157
$875$	0	0
$876$	−1677.10	−0.0646851
$877$	42978.6	1.65483	0.827414	−	0.561592i	$-0.189811\pi$
$877$	42978.6	1.65483	0.827414	+	0.561592i	$0.189811\pi$
$878$	6017.74	0.231309
$879$	7831.65	0.300518
$880$	0	0
$881$	−7298.75	−0.279116	−0.139558	−	0.990214i	$-0.544568\pi$
$881$	−7298.75	−0.279116	−0.139558	+	0.990214i	$0.544568\pi$
$882$	2914.45	0.111264
$883$	−46722.1	−1.78066	−0.890331	−	0.455314i	$-0.849527\pi$
$883$	−46722.1	−1.78066	−0.890331	+	0.455314i	$0.849527\pi$
$884$	15766.7	0.599878
$885$	0	0
$886$	459.187	0.0174116
$887$	−35268.7	−1.33507	−0.667535	−	0.744578i	$-0.732651\pi$
$887$	−35268.7	−1.33507	−0.667535	+	0.744578i	$0.732651\pi$
$888$	−14589.4	−0.551338
$889$	−874.928	−0.0330080
$890$	0	0
$891$	7508.74	0.282326
$892$	−7521.35	−0.282325
$893$	−6955.57	−0.260649
$894$	23985.0	0.897292
$895$	0	0
$896$	−2796.54	−0.104270
$897$	−53531.3	−1.99259
$898$	−15118.9	−0.561830
$899$	−22876.8	−0.848704
$900$	0	0
$901$	10297.7	0.380761
$902$	7223.14	0.266635
$903$	−23024.3	−0.848507
$904$	9060.05	0.333333
$905$	0	0
$906$	13792.8	0.505777
$907$	46292.7	1.69474	0.847368	−	0.531007i	$-0.178186\pi$
$907$	46292.7	1.69474	0.847368	+	0.531007i	$0.178186\pi$
$908$	−7196.92	−0.263038
$909$	−4373.06	−0.159566
$910$	0	0
$911$	−42085.5	−1.53058	−0.765288	−	0.643688i	$-0.777403\pi$
$911$	−42085.5	−1.53058	−0.765288	+	0.643688i	$0.777403\pi$
$912$	−1870.23	−0.0679051
$913$	9573.94	0.347044
$914$	−23112.7	−0.836435
$915$	0	0
$916$	19341.4	0.697660
$917$	3886.70	0.139967
$918$	14762.4	0.530754
$919$	21331.8	0.765691	0.382845	−	0.923812i	$-0.374944\pi$
$919$	21331.8	0.765691	0.382845	+	0.923812i	$0.374944\pi$
$920$	0	0
$921$	−5039.49	−0.180300
$922$	18383.2	0.656635
$923$	−3006.34	−0.107210
$924$	4464.63	0.158956
$925$	0	0
$926$	−2712.07	−0.0962463
$927$	12319.7	0.436498
$928$	−7437.12	−0.263077
$929$	−34331.5	−1.21247	−0.606233	−	0.795287i	$-0.707320\pi$
$929$	−34331.5	−1.21247	−0.606233	+	0.795287i	$0.707320\pi$
$930$	0	0
$931$	−2552.31	−0.0898481
$932$	−3461.20	−0.121647
$933$	12948.0	0.454341
$934$	29617.9	1.03761
$935$	0	0
$936$	−4605.12	−0.160815
$937$	−13625.1	−0.475041	−0.237521	−	0.971383i	$-0.576335\pi$
$937$	−13625.1	−0.475041	−0.237521	+	0.971383i	$0.576335\pi$
$938$	−8016.49	−0.279049
$939$	−14737.1	−0.512168
$940$	0	0
$941$	7086.48	0.245497	0.122748	−	0.992438i	$-0.460829\pi$
$941$	7086.48	0.245497	0.122748	+	0.992438i	$0.460829\pi$
$942$	29040.6	1.00445
$943$	−71315.6	−2.46273
$944$	9164.75	0.315982
$945$	0	0
$946$	−2844.98	−0.0977784
$947$	−38735.9	−1.32920	−0.664598	−	0.747202i	$-0.731397\pi$
$947$	−38735.9	−1.32920	−0.664598	+	0.747202i	$0.731397\pi$
$948$	−8190.12	−0.280594
$949$	3616.45	0.123704
$950$	0	0
$951$	−13277.6	−0.452740
$952$	12983.1	0.442000
$953$	−30964.3	−1.05250	−0.526249	−	0.850330i	$-0.676402\pi$
$953$	−30964.3	−1.05250	−0.526249	+	0.850330i	$0.676402\pi$
$954$	−3007.73	−0.102074
$955$	0	0
$956$	−19057.1	−0.644717
$957$	11873.3	0.401053
$958$	19669.3	0.663347
$959$	−540.962	−0.0182154
$960$	0	0
$961$	−20101.9	−0.674765
$962$	31460.1	1.05438
$963$	4123.26	0.137975
$964$	2461.34	0.0822350
$965$	0	0
$966$	−44080.2	−1.46818
$967$	−10968.0	−0.364743	−0.182371	−	0.983230i	$-0.558377\pi$
$967$	−10968.0	−0.364743	−0.182371	+	0.983230i	$0.558377\pi$
$968$	−10096.3	−0.335236
$969$	8682.65	0.287850
$970$	0	0
$971$	−30081.2	−0.994183	−0.497091	−	0.867698i	$-0.665599\pi$
$971$	−30081.2	−0.994183	−0.497091	+	0.867698i	$0.665599\pi$
$972$	−11519.4	−0.380128
$973$	−62642.5	−2.06395
$974$	−7375.64	−0.242640
$975$	0	0
$976$	−10123.1	−0.331999
$977$	−26628.1	−0.871962	−0.435981	−	0.899956i	$-0.643599\pi$
$977$	−26628.1	−0.871962	−0.435981	+	0.899956i	$0.643599\pi$
$978$	−10604.0	−0.346706
$979$	3059.90	0.0998926
$980$	0	0
$981$	12808.6	0.416867
$982$	−22394.7	−0.727744
$983$	−25495.3	−0.827236	−0.413618	−	0.910450i	$-0.635735\pi$
$983$	−25495.3	−0.827236	−0.413618	+	0.910450i	$0.635735\pi$
$984$	−21404.9	−0.693458
$985$	0	0
$986$	34527.3	1.11519
$987$	−49205.2	−1.58685
$988$	4032.90	0.129862
$989$	28089.1	0.903116
$990$	0	0
$991$	−54385.4	−1.74330	−0.871649	−	0.490130i	$-0.836949\pi$
$991$	−54385.4	−1.74330	−0.871649	+	0.490130i	$0.836949\pi$
$992$	3149.86	0.100815
$993$	3185.43	0.101799
$994$	−2475.57	−0.0789942
$995$	0	0
$996$	−28371.2	−0.902586
$997$	36846.5	1.17045	0.585227	−	0.810870i	$-0.301006\pi$
$997$	36846.5	1.17045	0.585227	+	0.810870i	$0.301006\pi$
$998$	−24203.3	−0.767677
$999$	29456.1	0.932884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.4.a.h.1.2		2
5.2	odd	4		950.4.b.g.799.3		4
5.3	odd	4		950.4.b.g.799.2		4
5.4	even	2		38.4.a.b.1.1	✓	2
15.14	odd	2		342.4.a.k.1.1		2
20.19	odd	2		304.4.a.d.1.2		2
35.34	odd	2		1862.4.a.b.1.2		2
40.19	odd	2		1216.4.a.l.1.1		2
40.29	even	2		1216.4.a.j.1.2		2
95.94	odd	2		722.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.b.1.1	✓	2	5.4	even	2
304.4.a.d.1.2		2	20.19	odd	2
342.4.a.k.1.1		2	15.14	odd	2
722.4.a.i.1.2		2	95.94	odd	2
950.4.a.h.1.2		2	1.1	even	1	trivial
950.4.b.g.799.2		4	5.3	odd	4
950.4.b.g.799.3		4	5.2	odd	4
1216.4.a.j.1.2		2	40.29	even	2
1216.4.a.l.1.1		2	40.19	odd	2
1862.4.a.b.1.2		2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	950.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +6.15207 q^{3} +4.00000 q^{4} +12.3041 q^{6} -21.8479 q^{7} +8.00000 q^{8} +10.8479 q^{9} -8.30413 q^{11} +24.6083 q^{12} -53.0645 q^{13} -43.6959 q^{14} +16.0000 q^{16} -74.2810 q^{17} +21.6959 q^{18} -19.0000 q^{19} -134.410 q^{21} -16.6083 q^{22} +163.977 q^{23} +49.2165 q^{24} -106.129 q^{26} -99.3686 q^{27} -87.3917 q^{28} -232.410 q^{29} +98.4331 q^{31} +32.0000 q^{32} -51.0876 q^{33} -148.562 q^{34} +43.3917 q^{36} -296.433 q^{37} -38.0000 q^{38} -326.456 q^{39} -434.912 q^{41} -268.820 q^{42} +171.299 q^{43} -33.2165 q^{44} +327.954 q^{46} +366.083 q^{47} +98.4331 q^{48} +134.332 q^{49} -456.982 q^{51} -212.258 q^{52} -138.631 q^{53} -198.737 q^{54} -174.783 q^{56} -116.889 q^{57} -464.820 q^{58} +572.797 q^{59} -632.691 q^{61} +196.866 q^{62} -237.005 q^{63} +64.0000 q^{64} -102.175 q^{66} +183.461 q^{67} -297.124 q^{68} +1008.80 q^{69} +56.6545 q^{71} +86.7835 q^{72} -68.1521 q^{73} -592.866 q^{74} -76.0000 q^{76} +181.428 q^{77} -652.912 q^{78} -332.820 q^{79} -904.217 q^{81} -869.825 q^{82} -1152.91 q^{83} -537.640 q^{84} +342.598 q^{86} -1429.80 q^{87} -66.4331 q^{88} -368.479 q^{89} +1159.35 q^{91} +655.908 q^{92} +605.567 q^{93} +732.165 q^{94} +196.866 q^{96} +426.443 q^{97} +268.664 q^{98} -90.0827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - q^{3} + 8 q^{4} - 2 q^{6} - 57 q^{7} + 16 q^{8} + 35 q^{9} + 10 q^{11} - 4 q^{12} - 13 q^{13} - 114 q^{14} + 32 q^{16} + 51 q^{17} + 70 q^{18} - 38 q^{19} + 117 q^{21} + 20 q^{22} + 155 q^{23}+ \cdots + 352 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - q^3 + 8 * q^4 - 2 * q^6 - 57 * q^7 + 16 * q^8 + 35 * q^9 + 10 * q^11 - 4 * q^12 - 13 * q^13 - 114 * q^14 + 32 * q^16 + 51 * q^17 + 70 * q^18 - 38 * q^19 + 117 * q^21 + 20 * q^22 + 155 * q^23 - 8 * q^24 - 26 * q^26 - 79 * q^27 - 228 * q^28 - 79 * q^29 - 16 * q^31 + 64 * q^32 - 182 * q^33 + 102 * q^34 + 140 * q^36 - 380 * q^37 - 76 * q^38 - 613 * q^39 - 790 * q^41 + 234 * q^42 - 296 * q^43 + 40 * q^44 + 310 * q^46 + 200 * q^47 - 16 * q^48 + 1027 * q^49 - 1353 * q^51 - 52 * q^52 - 397 * q^53 - 158 * q^54 - 456 * q^56 + 19 * q^57 - 158 * q^58 + 201 * q^59 - 680 * q^61 - 32 * q^62 - 1086 * q^63 + 128 * q^64 - 364 * q^66 + 939 * q^67 + 204 * q^68 + 1073 * q^69 + 406 * q^71 + 280 * q^72 - 123 * q^73 - 760 * q^74 - 152 * q^76 - 462 * q^77 - 1226 * q^78 + 106 * q^79 - 1702 * q^81 - 1580 * q^82 - 2226 * q^83 + 468 * q^84 - 592 * q^86 - 2527 * q^87 + 80 * q^88 - 870 * q^89 - 249 * q^91 + 620 * q^92 + 1424 * q^93 + 400 * q^94 - 32 * q^96 + 1864 * q^97 + 2054 * q^98 + 352 * q^99

Introduction

L-functions

Modular forms

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