LMFDB - Embedded newform 1519.4.a.h.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1519,4,Mod(1,1519)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1519, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1519.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1519.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23,5,-6,91,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.6239012987$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	1519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.77516 q^{2} +3.21029 q^{3} +14.8022 q^{4} -4.53317 q^{5} +15.3297 q^{6} +32.4816 q^{8} -16.6940 q^{9} -21.6466 q^{10} -42.1235 q^{11} +47.5194 q^{12} +36.9198 q^{13} -14.5528 q^{15} +36.6875 q^{16} -89.6498 q^{17} -79.7168 q^{18} -107.955 q^{19} -67.1009 q^{20} -201.146 q^{22} +102.456 q^{23} +104.275 q^{24} -104.450 q^{25} +176.298 q^{26} -140.271 q^{27} +169.760 q^{29} -69.4920 q^{30} -31.0000 q^{31} -84.6640 q^{32} -135.229 q^{33} -428.093 q^{34} -247.108 q^{36} +303.679 q^{37} -515.502 q^{38} +118.523 q^{39} -147.245 q^{40} -464.571 q^{41} +235.343 q^{43} -623.520 q^{44} +75.6769 q^{45} +489.242 q^{46} -530.532 q^{47} +117.778 q^{48} -498.768 q^{50} -287.802 q^{51} +546.495 q^{52} -354.534 q^{53} -669.815 q^{54} +190.953 q^{55} -346.566 q^{57} +810.632 q^{58} -531.716 q^{59} -215.413 q^{60} +98.2079 q^{61} -148.030 q^{62} -697.785 q^{64} -167.364 q^{65} -645.739 q^{66} +620.188 q^{67} -1327.01 q^{68} +328.912 q^{69} -397.571 q^{71} -542.250 q^{72} +865.656 q^{73} +1450.12 q^{74} -335.316 q^{75} -1597.97 q^{76} +565.969 q^{78} +630.131 q^{79} -166.311 q^{80} +0.429822 q^{81} -2218.40 q^{82} +904.487 q^{83} +406.398 q^{85} +1123.80 q^{86} +544.979 q^{87} -1368.24 q^{88} +209.978 q^{89} +361.370 q^{90} +1516.57 q^{92} -99.5190 q^{93} -2533.38 q^{94} +489.377 q^{95} -271.796 q^{96} +310.503 q^{97} +703.211 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) $ 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * 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 $

$2$	4.77516	1.68828	0.844138	−	0.536126i	$-0.180113\pi$
$2$	4.77516	1.68828	0.844138	+	0.536126i	$0.180113\pi$
$3$	3.21029	0.617821	0.308910	−	0.951091i	$-0.400036\pi$
$3$	3.21029	0.617821	0.308910	+	0.951091i	$0.400036\pi$
$4$	14.8022	1.85028
$5$	−4.53317	−0.405459	−0.202730	−	0.979235i	$-0.564981\pi$
$5$	−4.53317	−0.405459	−0.202730	+	0.979235i	$0.564981\pi$
$6$	15.3297	1.04305
$7$	0	0
$7$	0	0
$8$	32.4816	1.43550
$9$	−16.6940	−0.618298
$10$	−21.6466	−0.684527
$11$	−42.1235	−1.15461	−0.577305	−	0.816529i	$-0.695895\pi$
$11$	−42.1235	−1.15461	−0.577305	+	0.816529i	$0.695895\pi$
$12$	47.5194	1.14314
$13$	36.9198	0.787671	0.393835	−	0.919181i	$-0.371148\pi$
$13$	36.9198	0.787671	0.393835	+	0.919181i	$0.371148\pi$
$14$	0	0
$15$	−14.5528	−0.250501
$16$	36.6875	0.573243
$17$	−89.6498	−1.27902	−0.639508	−	0.768784i	$-0.720862\pi$
$17$	−89.6498	−1.27902	−0.639508	+	0.768784i	$0.720862\pi$
$18$	−79.7168	−1.04386
$19$	−107.955	−1.30350	−0.651750	−	0.758434i	$-0.725965\pi$
$19$	−107.955	−1.30350	−0.651750	+	0.758434i	$0.725965\pi$
$20$	−67.1009	−0.750211
$21$	0	0
$22$	−201.146	−1.94930
$23$	102.456	0.928846	0.464423	−	0.885613i	$-0.346262\pi$
$23$	102.456	0.928846	0.464423	+	0.885613i	$0.346262\pi$
$24$	104.275	0.886881
$25$	−104.450	−0.835603
$26$	176.298	1.32981
$27$	−140.271	−0.999818
$28$	0	0
$29$	169.760	1.08702	0.543511	−	0.839402i	$-0.317095\pi$
$29$	169.760	1.08702	0.543511	+	0.839402i	$0.317095\pi$
$30$	−69.4920	−0.422915
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−84.6640	−0.467707
$33$	−135.229	−0.713341
$34$	−428.093	−2.15933
$35$	0	0
$36$	−247.108	−1.14402
$37$	303.679	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$37$	303.679	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$38$	−515.502	−2.20067
$39$	118.523	0.486639
$40$	−147.245	−0.582036
$41$	−464.571	−1.76961	−0.884803	−	0.465966i	$-0.845707\pi$
$41$	−464.571	−1.76961	−0.884803	+	0.465966i	$0.845707\pi$
$42$	0	0
$43$	235.343	0.834637	0.417319	−	0.908760i	$-0.362970\pi$
$43$	235.343	0.834637	0.417319	+	0.908760i	$0.362970\pi$
$44$	−623.520	−2.13634
$45$	75.6769	0.250694
$46$	489.242	1.56815
$47$	−530.532	−1.64651	−0.823256	−	0.567670i	$-0.807845\pi$
$47$	−530.532	−1.64651	−0.823256	+	0.567670i	$0.807845\pi$
$48$	117.778	0.354161
$49$	0	0
$50$	−498.768	−1.41073
$51$	−287.802	−0.790203
$52$	546.495	1.45741
$53$	−354.534	−0.918850	−0.459425	−	0.888217i	$-0.651944\pi$
$53$	−354.534	−0.918850	−0.459425	+	0.888217i	$0.651944\pi$
$54$	−669.815	−1.68797
$55$	190.953	0.468147
$56$	0	0
$57$	−346.566	−0.805330
$58$	810.632	1.83519
$59$	−531.716	−1.17328	−0.586640	−	0.809848i	$-0.699550\pi$
$59$	−531.716	−1.17328	−0.586640	+	0.809848i	$0.699550\pi$
$60$	−215.413	−0.463496
$61$	98.2079	0.206135	0.103067	−	0.994674i	$-0.467134\pi$
$61$	98.2079	0.206135	0.103067	+	0.994674i	$0.467134\pi$
$62$	−148.030	−0.303223
$63$	0	0
$64$	−697.785	−1.36286
$65$	−167.364	−0.319368
$66$	−645.739	−1.20432
$67$	620.188	1.13087	0.565433	−	0.824794i	$-0.308709\pi$
$67$	620.188	1.13087	0.565433	+	0.824794i	$0.308709\pi$
$68$	−1327.01	−2.36653
$69$	328.912	0.573860
$70$	0	0
$71$	−397.571	−0.664550	−0.332275	−	0.943183i	$-0.607816\pi$
$71$	−397.571	−0.664550	−0.332275	+	0.943183i	$0.607816\pi$
$72$	−542.250	−0.887566
$73$	865.656	1.38791	0.693955	−	0.720019i	$-0.255867\pi$
$73$	865.656	1.38791	0.693955	+	0.720019i	$0.255867\pi$
$74$	1450.12	2.27801
$75$	−335.316	−0.516253
$76$	−1597.97	−2.41183
$77$	0	0
$78$	565.969	0.821581
$79$	630.131	0.897409	0.448704	−	0.893680i	$-0.351886\pi$
$79$	630.131	0.897409	0.448704	+	0.893680i	$0.351886\pi$
$80$	−166.311	−0.232427
$81$	0.429822	0.000589605 0
$82$	−2218.40	−2.98758
$83$	904.487	1.19615	0.598075	−	0.801440i	$-0.295933\pi$
$83$	904.487	1.19615	0.598075	+	0.801440i	$0.295933\pi$
$84$	0	0
$85$	406.398	0.518589
$86$	1123.80	1.40910
$87$	544.979	0.671585
$88$	−1368.24	−1.65744
$89$	209.978	0.250086	0.125043	−	0.992151i	$-0.460093\pi$
$89$	209.978	0.250086	0.125043	+	0.992151i	$0.460093\pi$
$90$	361.370	0.423241
$91$	0	0
$92$	1516.57	1.71862
$93$	−99.5190	−0.110964
$94$	−2533.38	−2.77977
$95$	489.377	0.528516
$96$	−271.796	−0.288959
$97$	310.503	0.325019	0.162509	−	0.986707i	$-0.448041\pi$
$97$	310.503	0.325019	0.162509	+	0.986707i	$0.448041\pi$
$98$	0	0
$99$	703.211	0.713892
$100$	−1546.10	−1.54610
$101$	−1209.38	−1.19146	−0.595732	−	0.803183i	$-0.703138\pi$
$101$	−1209.38	−1.19146	−0.595732	+	0.803183i	$0.703138\pi$
$102$	−1374.30	−1.33408
$103$	−318.757	−0.304932	−0.152466	−	0.988309i	$-0.548722\pi$
$103$	−318.757	−0.304932	−0.152466	+	0.988309i	$0.548722\pi$
$104$	1199.22	1.13070
$105$	0	0
$106$	−1692.96	−1.55127
$107$	−1186.78	−1.07225	−0.536124	−	0.844139i	$-0.680112\pi$
$107$	−1186.78	−1.07225	−0.536124	+	0.844139i	$0.680112\pi$
$108$	−2076.31	−1.84994
$109$	−401.940	−0.353200	−0.176600	−	0.984283i	$-0.556510\pi$
$109$	−401.940	−0.353200	−0.176600	+	0.984283i	$0.556510\pi$
$110$	911.831	0.790361
$111$	974.899	0.833633
$112$	0	0
$113$	−1275.58	−1.06191	−0.530957	−	0.847399i	$-0.678167\pi$
$113$	−1275.58	−1.06191	−0.530957	+	0.847399i	$0.678167\pi$
$114$	−1654.91	−1.35962
$115$	−464.449	−0.376609
$116$	2512.82	2.01129
$117$	−616.341	−0.487015
$118$	−2539.03	−1.98082
$119$	0	0
$120$	−472.699	−0.359594
$121$	443.386	0.333122
$122$	468.959	0.348013
$123$	−1491.41	−1.09330
$124$	−458.868	−0.332319
$125$	1040.14	0.744262
$126$	0	0
$127$	−186.457	−0.130278	−0.0651391	−	0.997876i	$-0.520749\pi$
$127$	−186.457	−0.130278	−0.0651391	+	0.997876i	$0.520749\pi$
$128$	−2654.73	−1.83318
$129$	755.518	0.515656
$130$	−799.190	−0.539182
$131$	2737.22	1.82559	0.912793	−	0.408423i	$-0.133921\pi$
$131$	2737.22	1.82559	0.912793	+	0.408423i	$0.133921\pi$
$132$	−2001.68	−1.31988
$133$	0	0
$134$	2961.50	1.90921
$135$	635.870	0.405385
$136$	−2911.97	−1.83603
$137$	512.517	0.319615	0.159808	−	0.987148i	$-0.448913\pi$
$137$	512.517	0.319615	0.159808	+	0.987148i	$0.448913\pi$
$138$	1570.61	0.968835
$139$	−235.108	−0.143465	−0.0717323	−	0.997424i	$-0.522853\pi$
$139$	−235.108	−0.143465	−0.0717323	+	0.997424i	$0.522853\pi$
$140$	0	0
$141$	−1703.16	−1.01725
$142$	−1898.47	−1.12194
$143$	−1555.19	−0.909452
$144$	−612.463	−0.354435
$145$	−769.551	−0.440743
$146$	4133.65	2.34317
$147$	0	0
$148$	4495.12	2.49660
$149$	2962.89	1.62906	0.814529	−	0.580122i	$-0.196995\pi$
$149$	2962.89	1.62906	0.814529	+	0.580122i	$0.196995\pi$
$150$	−1601.19	−0.871577
$151$	−322.954	−0.174051	−0.0870253	−	0.996206i	$-0.527736\pi$
$151$	−322.954	−0.174051	−0.0870253	+	0.996206i	$0.527736\pi$
$152$	−3506.55	−1.87117
$153$	1496.62	0.790813
$154$	0	0
$155$	140.528	0.0728226
$156$	1754.41	0.900416
$157$	−2545.93	−1.29419	−0.647093	−	0.762411i	$-0.724016\pi$
$157$	−2545.93	−1.29419	−0.647093	+	0.762411i	$0.724016\pi$
$158$	3008.98	1.51507
$159$	−1138.16	−0.567684
$160$	383.796	0.189636
$161$	0	0
$162$	2.05247	0.000995415 0
$163$	1522.29	0.731502	0.365751	−	0.930713i	$-0.380812\pi$
$163$	1522.29	0.731502	0.365751	+	0.930713i	$0.380812\pi$
$164$	−6876.68	−3.27426
$165$	613.014	0.289231
$166$	4319.08	2.01943
$167$	1259.72	0.583712	0.291856	−	0.956462i	$-0.405727\pi$
$167$	1259.72	0.583712	0.291856	+	0.956462i	$0.405727\pi$
$168$	0	0
$169$	−833.927	−0.379575
$170$	1940.62	0.875521
$171$	1802.20	0.805951
$172$	3483.59	1.54431
$173$	137.299	0.0603389	0.0301694	−	0.999545i	$-0.490395\pi$
$173$	137.299	0.0603389	0.0301694	+	0.999545i	$0.490395\pi$
$174$	2602.36	1.13382
$175$	0	0
$176$	−1545.41	−0.661871
$177$	−1706.96	−0.724876
$178$	1002.68	0.422213
$179$	−3589.96	−1.49903	−0.749515	−	0.661988i	$-0.769713\pi$
$179$	−3589.96	−1.49903	−0.749515	+	0.661988i	$0.769713\pi$
$180$	1120.18	0.463854
$181$	−2124.54	−0.872465	−0.436233	−	0.899834i	$-0.643687\pi$
$181$	−2124.54	−0.872465	−0.436233	+	0.899834i	$0.643687\pi$
$182$	0	0
$183$	315.276	0.127354
$184$	3327.92	1.33336
$185$	−1376.63	−0.547091
$186$	−475.220	−0.187338
$187$	3776.36	1.47676
$188$	−7853.04	−3.04650
$189$	0	0
$190$	2336.86	0.892281
$191$	−483.797	−0.183279	−0.0916395	−	0.995792i	$-0.529211\pi$
$191$	−483.797	−0.183279	−0.0916395	+	0.995792i	$0.529211\pi$
$192$	−2240.09	−0.842004
$193$	5247.65	1.95717	0.978586	−	0.205836i	$-0.0659915\pi$
$193$	5247.65	1.95717	0.978586	+	0.205836i	$0.0659915\pi$
$194$	1482.70	0.548721
$195$	−537.287	−0.197312
$196$	0	0
$197$	−2639.11	−0.954460	−0.477230	−	0.878779i	$-0.658359\pi$
$197$	−2639.11	−0.954460	−0.477230	+	0.878779i	$0.658359\pi$
$198$	3357.95	1.20525
$199$	−4472.22	−1.59310	−0.796551	−	0.604572i	$-0.793344\pi$
$199$	−4472.22	−1.59310	−0.796551	+	0.604572i	$0.793344\pi$
$200$	−3392.72	−1.19951
$201$	1990.98	0.698672
$202$	−5774.99	−2.01152
$203$	0	0
$204$	−4260.10	−1.46209
$205$	2105.98	0.717503
$206$	−1522.12	−0.514810
$207$	−1710.40	−0.574304
$208$	1354.50	0.451526
$209$	4547.43	1.50503
$210$	0	0
$211$	3388.07	1.10542	0.552711	−	0.833373i	$-0.313593\pi$
$211$	3388.07	1.10542	0.552711	+	0.833373i	$0.313593\pi$
$212$	−5247.89	−1.70013
$213$	−1276.32	−0.410573
$214$	−5667.08	−1.81025
$215$	−1066.85	−0.338411
$216$	−4556.22	−1.43524
$217$	0	0
$218$	−1919.33	−0.596300
$219$	2779.01	0.857479
$220$	2826.52	0.866200
$221$	−3309.86	−1.00744
$222$	4655.30	1.40740
$223$	3850.41	1.15625	0.578123	−	0.815950i	$-0.303785\pi$
$223$	3850.41	1.15625	0.578123	+	0.815950i	$0.303785\pi$
$224$	0	0
$225$	1743.70	0.516651
$226$	−6091.09	−1.79280
$227$	−5893.87	−1.72330	−0.861652	−	0.507499i	$-0.830570\pi$
$227$	−5893.87	−1.72330	−0.861652	+	0.507499i	$0.830570\pi$
$228$	−5129.94	−1.49008
$229$	−4219.17	−1.21751	−0.608757	−	0.793356i	$-0.708332\pi$
$229$	−4219.17	−1.21751	−0.608757	+	0.793356i	$0.708332\pi$
$230$	−2217.82	−0.635820
$231$	0	0
$232$	5514.08	1.56042
$233$	3478.49	0.978042	0.489021	−	0.872272i	$-0.337354\pi$
$233$	3478.49	0.978042	0.489021	+	0.872272i	$0.337354\pi$
$234$	−2943.13	−0.822215
$235$	2404.99	0.667593
$236$	−7870.56	−2.17089
$237$	2022.90	0.554438
$238$	0	0
$239$	1188.46	0.321654	0.160827	−	0.986983i	$-0.448584\pi$
$239$	1188.46	0.321654	0.160827	+	0.986983i	$0.448584\pi$
$240$	−533.906	−0.143598
$241$	3976.01	1.06273	0.531363	−	0.847144i	$-0.321680\pi$
$241$	3976.01	1.06273	0.531363	+	0.847144i	$0.321680\pi$
$242$	2117.24	0.562402
$243$	3788.68	1.00018
$244$	1453.69	0.381406
$245$	0	0
$246$	−7121.72	−1.84579
$247$	−3985.67	−1.02673
$248$	−1006.93	−0.257823
$249$	2903.67	0.739006
$250$	4966.83	1.25652
$251$	−2475.92	−0.622623	−0.311312	−	0.950308i	$-0.600768\pi$
$251$	−2475.92	−0.622623	−0.311312	+	0.950308i	$0.600768\pi$
$252$	0	0
$253$	−4315.78	−1.07245
$254$	−890.361	−0.219946
$255$	1304.66	0.320395
$256$	−7094.47	−1.73205
$257$	−6536.64	−1.58655	−0.793277	−	0.608861i	$-0.791627\pi$
$257$	−6536.64	−1.58655	−0.793277	+	0.608861i	$0.791627\pi$
$258$	3607.72	0.870570
$259$	0	0
$260$	−2477.35	−0.590919
$261$	−2833.98	−0.672103
$262$	13070.7	3.08209
$263$	1241.60	0.291104	0.145552	−	0.989351i	$-0.453504\pi$
$263$	1241.60	0.291104	0.145552	+	0.989351i	$0.453504\pi$
$264$	−4392.44	−1.02400
$265$	1607.17	0.372556
$266$	0	0
$267$	674.090	0.154508
$268$	9180.15	2.09241
$269$	378.504	0.0857911	0.0428956	−	0.999080i	$-0.486342\pi$
$269$	378.504	0.0857911	0.0428956	+	0.999080i	$0.486342\pi$
$270$	3036.39	0.684402
$271$	5061.39	1.13453	0.567265	−	0.823535i	$-0.308001\pi$
$271$	5061.39	1.13453	0.567265	+	0.823535i	$0.308001\pi$
$272$	−3289.03	−0.733187
$273$	0	0
$274$	2447.35	0.539598
$275$	4399.81	0.964795
$276$	4868.62	1.06180
$277$	−67.5856	−0.0146600	−0.00733001	−	0.999973i	$-0.502333\pi$
$277$	−67.5856	−0.0146600	−0.00733001	+	0.999973i	$0.502333\pi$
$278$	−1122.68	−0.242208
$279$	517.515	0.111050
$280$	0	0
$281$	−2432.05	−0.516312	−0.258156	−	0.966103i	$-0.583115\pi$
$281$	−2432.05	−0.516312	−0.258156	+	0.966103i	$0.583115\pi$
$282$	−8132.88	−1.71740
$283$	2803.95	0.588966	0.294483	−	0.955657i	$-0.404852\pi$
$283$	2803.95	0.588966	0.294483	+	0.955657i	$0.404852\pi$
$284$	−5884.93	−1.22960
$285$	1571.04	0.326528
$286$	−7426.29	−1.53541
$287$	0	0
$288$	1413.38	0.289182
$289$	3124.09	0.635883
$290$	−3674.73	−0.744096
$291$	996.805	0.200803
$292$	12813.6	2.56801
$293$	3534.24	0.704684	0.352342	−	0.935871i	$-0.385385\pi$
$293$	3534.24	0.704684	0.352342	+	0.935871i	$0.385385\pi$
$294$	0	0
$295$	2410.36	0.475717
$296$	9864.00	1.93694
$297$	5908.68	1.15440
$298$	14148.3	2.75030
$299$	3782.64	0.731625
$300$	−4963.41	−0.955210
$301$	0	0
$302$	−1542.16	−0.293845
$303$	−3882.46	−0.736111
$304$	−3960.59	−0.747222
$305$	−445.193	−0.0835793
$306$	7146.60	1.33511
$307$	−8447.96	−1.57052	−0.785261	−	0.619164i	$-0.787471\pi$
$307$	−8447.96	−1.57052	−0.785261	+	0.619164i	$0.787471\pi$
$308$	0	0
$309$	−1023.30	−0.188394
$310$	671.046	0.122945
$311$	1143.95	0.208578	0.104289	−	0.994547i	$-0.466743\pi$
$311$	1143.95	0.208578	0.104289	+	0.994547i	$0.466743\pi$
$312$	3849.83	0.698570
$313$	−1828.74	−0.330244	−0.165122	−	0.986273i	$-0.552802\pi$
$313$	−1828.74	−0.330244	−0.165122	+	0.986273i	$0.552802\pi$
$314$	−12157.2	−2.18494
$315$	0	0
$316$	9327.33	1.66045
$317$	2927.26	0.518648	0.259324	−	0.965790i	$-0.416500\pi$
$317$	2927.26	0.518648	0.259324	+	0.965790i	$0.416500\pi$
$318$	−5434.89	−0.958408
$319$	−7150.88	−1.25509
$320$	3163.18	0.552584
$321$	−3809.92	−0.662457
$322$	0	0
$323$	9678.12	1.66720
$324$	6.36231	0.00109093
$325$	−3856.29	−0.658180
$326$	7269.18	1.23498
$327$	−1290.34	−0.218214
$328$	−15090.0	−2.54027
$329$	0	0
$330$	2927.24	0.488301
$331$	−4091.87	−0.679486	−0.339743	−	0.940518i	$-0.610340\pi$
$331$	−4091.87	−0.679486	−0.339743	+	0.940518i	$0.610340\pi$
$332$	13388.4	2.21321
$333$	−5069.63	−0.834277
$334$	6015.36	0.985467
$335$	−2811.42	−0.458520
$336$	0	0
$337$	4590.80	0.742068	0.371034	−	0.928619i	$-0.379003\pi$
$337$	4590.80	0.742068	0.371034	+	0.928619i	$0.379003\pi$
$338$	−3982.14	−0.640827
$339$	−4094.97	−0.656072
$340$	6015.58	0.959532
$341$	1305.83	0.207374
$342$	8605.80	1.36067
$343$	0	0
$344$	7644.31	1.19812
$345$	−1491.02	−0.232677
$346$	655.624	0.101869
$347$	−1097.66	−0.169814	−0.0849068	−	0.996389i	$-0.527059\pi$
$347$	−1097.66	−0.169814	−0.0849068	+	0.996389i	$0.527059\pi$
$348$	8066.89	1.24262
$349$	−5451.71	−0.836170	−0.418085	−	0.908408i	$-0.637299\pi$
$349$	−5451.71	−0.836170	−0.418085	+	0.908408i	$0.637299\pi$
$350$	0	0
$351$	−5178.76	−0.787527
$352$	3566.34	0.540019
$353$	−4456.73	−0.671976	−0.335988	−	0.941866i	$-0.609070\pi$
$353$	−4456.73	−0.671976	−0.335988	+	0.941866i	$0.609070\pi$
$354$	−8151.02	−1.22379
$355$	1802.26	0.269448
$356$	3108.14	0.462727
$357$	0	0
$358$	−17142.7	−2.53078
$359$	11421.2	1.67908	0.839541	−	0.543296i	$-0.182824\pi$
$359$	11421.2	1.67908	0.839541	+	0.543296i	$0.182824\pi$
$360$	2458.11	0.359872
$361$	4795.22	0.699114
$362$	−10145.1	−1.47296
$363$	1423.40	0.205810
$364$	0	0
$365$	−3924.17	−0.562741
$366$	1505.49	0.215009
$367$	−2783.90	−0.395963	−0.197982	−	0.980206i	$-0.563439\pi$
$367$	−2783.90	−0.395963	−0.197982	+	0.980206i	$0.563439\pi$
$368$	3758.84	0.532454
$369$	7755.57	1.09414
$370$	−6573.64	−0.923641
$371$	0	0
$372$	−1473.10	−0.205314
$373$	−11756.6	−1.63199	−0.815994	−	0.578060i	$-0.803810\pi$
$373$	−11756.6	−1.63199	−0.815994	+	0.578060i	$0.803810\pi$
$374$	18032.7	2.49318
$375$	3339.14	0.459820
$376$	−17232.5	−2.36357
$377$	6267.51	0.856215
$378$	0	0
$379$	−1926.33	−0.261079	−0.130540	−	0.991443i	$-0.541671\pi$
$379$	−1926.33	−0.261079	−0.130540	+	0.991443i	$0.541671\pi$
$380$	7243.86	0.977900
$381$	−598.580	−0.0804886
$382$	−2310.21	−0.309426
$383$	8852.34	1.18103	0.590514	−	0.807028i	$-0.298925\pi$
$383$	8852.34	1.18103	0.590514	+	0.807028i	$0.298925\pi$
$384$	−8522.44	−1.13258
$385$	0	0
$386$	25058.4	3.30425
$387$	−3928.82	−0.516054
$388$	4596.13	0.601374
$389$	−3795.37	−0.494686	−0.247343	−	0.968928i	$-0.579558\pi$
$389$	−3795.37	−0.494686	−0.247343	+	0.968928i	$0.579558\pi$
$390$	−2565.63	−0.333118
$391$	−9185.13	−1.18801
$392$	0	0
$393$	8787.26	1.12788
$394$	−12602.2	−1.61139
$395$	−2856.49	−0.363863
$396$	10409.1	1.32090
$397$	−702.612	−0.0888239	−0.0444120	−	0.999013i	$-0.514141\pi$
$397$	−702.612	−0.0888239	−0.0444120	+	0.999013i	$0.514141\pi$
$398$	−21355.6	−2.68959
$399$	0	0
$400$	−3832.03	−0.479003
$401$	−11900.4	−1.48199	−0.740995	−	0.671511i	$-0.765646\pi$
$401$	−11900.4	−1.48199	−0.740995	+	0.671511i	$0.765646\pi$
$402$	9507.28	1.17955
$403$	−1144.51	−0.141470
$404$	−17901.5	−2.20454
$405$	−1.94846	−0.000239061 0
$406$	0	0
$407$	−12792.0	−1.55793
$408$	−9348.28	−1.13434
$409$	−3364.72	−0.406784	−0.203392	−	0.979097i	$-0.565197\pi$
$409$	−3364.72	−0.406784	−0.203392	+	0.979097i	$0.565197\pi$
$410$	10056.4	1.21134
$411$	1645.33	0.197465
$412$	−4718.30	−0.564209
$413$	0	0
$414$	−8167.43	−0.969583
$415$	−4100.20	−0.484990
$416$	−3125.78	−0.368399
$417$	−754.764	−0.0886354
$418$	21714.7	2.54091
$419$	7695.93	0.897305	0.448652	−	0.893706i	$-0.351904\pi$
$419$	7695.93	0.897305	0.448652	+	0.893706i	$0.351904\pi$
$420$	0	0
$421$	−6842.42	−0.792111	−0.396056	−	0.918226i	$-0.629621\pi$
$421$	−6842.42	−0.792111	−0.396056	+	0.918226i	$0.629621\pi$
$422$	16178.6	1.86626
$423$	8856.72	1.01803
$424$	−11515.9	−1.31901
$425$	9363.96	1.06875
$426$	−6094.64	−0.693160
$427$	0	0
$428$	−17567.0	−1.98395
$429$	−4992.61	−0.561878
$430$	−5094.38	−0.571332
$431$	−3846.91	−0.429928	−0.214964	−	0.976622i	$-0.568963\pi$
$431$	−3846.91	−0.429928	−0.214964	+	0.976622i	$0.568963\pi$
$432$	−5146.18	−0.573138
$433$	6271.87	0.696090	0.348045	−	0.937478i	$-0.386846\pi$
$433$	6271.87	0.696090	0.348045	+	0.937478i	$0.386846\pi$
$434$	0	0
$435$	−2470.48	−0.272300
$436$	−5949.59	−0.653518
$437$	−11060.6	−1.21075
$438$	13270.2	1.44766
$439$	−522.595	−0.0568157	−0.0284078	−	0.999596i	$-0.509044\pi$
$439$	−522.595	−0.0568157	−0.0284078	+	0.999596i	$0.509044\pi$
$440$	6202.46	0.672024
$441$	0	0
$442$	−15805.1	−1.70084
$443$	−9818.56	−1.05303	−0.526517	−	0.850165i	$-0.676502\pi$
$443$	−9818.56	−1.05303	−0.526517	+	0.850165i	$0.676502\pi$
$444$	14430.6	1.54245
$445$	−951.866	−0.101399
$446$	18386.4	1.95206
$447$	9511.75	1.00647
$448$	0	0
$449$	3995.19	0.419922	0.209961	−	0.977710i	$-0.432666\pi$
$449$	3995.19	0.419922	0.209961	+	0.977710i	$0.432666\pi$
$450$	8326.45	0.872250
$451$	19569.3	2.04320
$452$	−18881.3	−1.96483
$453$	−1036.78	−0.107532
$454$	−28144.2	−2.90941
$455$	0	0
$456$	−11257.0	−1.15605
$457$	14028.1	1.43590	0.717949	−	0.696095i	$-0.245081\pi$
$457$	14028.1	1.43590	0.717949	+	0.696095i	$0.245081\pi$
$458$	−20147.3	−2.05550
$459$	12575.2	1.27878
$460$	−6874.86	−0.696831
$461$	19561.3	1.97627	0.988134	−	0.153592i	$-0.0490841\pi$
$461$	19561.3	1.97627	0.988134	+	0.153592i	$0.0490841\pi$
$462$	0	0
$463$	14173.4	1.42266	0.711332	−	0.702856i	$-0.248092\pi$
$463$	14173.4	1.42266	0.711332	+	0.702856i	$0.248092\pi$
$464$	6228.08	0.623127
$465$	451.137	0.0449913
$466$	16610.4	1.65120
$467$	−11183.3	−1.10814	−0.554070	−	0.832470i	$-0.686926\pi$
$467$	−11183.3	−1.10814	−0.554070	+	0.832470i	$0.686926\pi$
$468$	−9123.20	−0.901111
$469$	0	0
$470$	11484.2	1.12708
$471$	−8173.17	−0.799575
$472$	−17271.0	−1.68424
$473$	−9913.44	−0.963680
$474$	9659.70	0.936044
$475$	11275.9	1.08921
$476$	0	0
$477$	5918.61	0.568123
$478$	5675.12	0.543041
$479$	1100.76	0.105000	0.0525000	−	0.998621i	$-0.483281\pi$
$479$	1100.76	0.105000	0.0525000	+	0.998621i	$0.483281\pi$
$480$	1232.10	0.117161
$481$	11211.8	1.06281
$482$	18986.1	1.79417
$483$	0	0
$484$	6563.08	0.616368
$485$	−1407.56	−0.131782
$486$	18091.6	1.68858
$487$	−20214.1	−1.88088	−0.940442	−	0.339954i	$-0.889588\pi$
$487$	−20214.1	−1.88088	−0.940442	+	0.339954i	$0.889588\pi$
$488$	3189.95	0.295906
$489$	4886.99	0.451937
$490$	0	0
$491$	−14121.1	−1.29792	−0.648959	−	0.760824i	$-0.724795\pi$
$491$	−14121.1	−1.29792	−0.648959	+	0.760824i	$0.724795\pi$
$492$	−22076.1	−2.02290
$493$	−15219.0	−1.39032
$494$	−19032.2	−1.73340
$495$	−3187.77	−0.289454
$496$	−1137.31	−0.102957
$497$	0	0
$498$	13865.5	1.24765
$499$	731.102	0.0655884	0.0327942	−	0.999462i	$-0.489559\pi$
$499$	731.102	0.0655884	0.0327942	+	0.999462i	$0.489559\pi$
$500$	15396.3	1.37709
$501$	4044.06	0.360629
$502$	−11822.9	−1.05116
$503$	7911.95	0.701345	0.350672	−	0.936498i	$-0.385953\pi$
$503$	7911.95	0.701345	0.350672	+	0.936498i	$0.385953\pi$
$504$	0	0
$505$	5482.33	0.483090
$506$	−20608.6	−1.81060
$507$	−2677.15	−0.234509
$508$	−2759.97	−0.241051
$509$	−19327.7	−1.68307	−0.841535	−	0.540202i	$-0.818348\pi$
$509$	−19327.7	−1.68307	−0.841535	+	0.540202i	$0.818348\pi$
$510$	6229.95	0.540915
$511$	0	0
$512$	−12639.5	−1.09100
$513$	15142.9	1.30326
$514$	−31213.5	−2.67854
$515$	1444.98	0.123638
$516$	11183.3	0.954106
$517$	22347.8	1.90108
$518$	0	0
$519$	440.769	0.0372786
$520$	−5436.25	−0.458453
$521$	1133.21	0.0952917	0.0476459	−	0.998864i	$-0.484828\pi$
$521$	1133.21	0.0952917	0.0476459	+	0.998864i	$0.484828\pi$
$522$	−13532.7	−1.13470
$523$	18739.8	1.56679	0.783396	−	0.621523i	$-0.213486\pi$
$523$	18739.8	1.56679	0.783396	+	0.621523i	$0.213486\pi$
$524$	40516.8	3.37783
$525$	0	0
$526$	5928.85	0.491465
$527$	2779.14	0.229718
$528$	−4961.20	−0.408918
$529$	−1669.85	−0.137244
$530$	7674.48	0.628977
$531$	8876.48	0.725436
$532$	0	0
$533$	−17151.9	−1.39387
$534$	3218.89	0.260852
$535$	5379.89	0.434753
$536$	20144.7	1.62336
$537$	−11524.8	−0.926132
$538$	1807.42	0.144839
$539$	0	0
$540$	9412.28	0.750074
$541$	21335.2	1.69551	0.847757	−	0.530384i	$-0.177952\pi$
$541$	21335.2	1.69551	0.847757	+	0.530384i	$0.177952\pi$
$542$	24169.0	1.91540
$543$	−6820.41	−0.539027
$544$	7590.11	0.598205
$545$	1822.06	0.143208
$546$	0	0
$547$	−16423.9	−1.28379	−0.641896	−	0.766792i	$-0.721852\pi$
$547$	−16423.9	−1.28379	−0.641896	+	0.766792i	$0.721852\pi$
$548$	7586.38	0.591376
$549$	−1639.49	−0.127453
$550$	21009.8	1.62884
$551$	−18326.4	−1.41693
$552$	10683.6	0.823776
$553$	0	0
$554$	−322.733	−0.0247502
$555$	−4419.38	−0.338004
$556$	−3480.11	−0.265449
$557$	15181.7	1.15489	0.577443	−	0.816431i	$-0.304051\pi$
$557$	15181.7	1.15489	0.577443	+	0.816431i	$0.304051\pi$
$558$	2471.22	0.187482
$559$	8688.81	0.657419
$560$	0	0
$561$	12123.2	0.912375
$562$	−11613.4	−0.871677
$563$	−24173.9	−1.80960	−0.904802	−	0.425832i	$-0.859981\pi$
$563$	−24173.9	−1.80960	−0.904802	+	0.425832i	$0.859981\pi$
$564$	−25210.5	−1.88219
$565$	5782.41	0.430562
$566$	13389.3	0.994338
$567$	0	0
$568$	−12913.8	−0.953961
$569$	4211.30	0.310276	0.155138	−	0.987893i	$-0.450418\pi$
$569$	4211.30	0.310276	0.155138	+	0.987893i	$0.450418\pi$
$570$	7501.99	0.551270
$571$	−23848.5	−1.74786	−0.873931	−	0.486050i	$-0.838437\pi$
$571$	−23848.5	−1.74786	−0.873931	+	0.486050i	$0.838437\pi$
$572$	−23020.2	−1.68274
$573$	−1553.13	−0.113234
$574$	0	0
$575$	−10701.5	−0.776147
$576$	11648.8	0.842654
$577$	−10608.0	−0.765366	−0.382683	−	0.923880i	$-0.625000\pi$
$577$	−10608.0	−0.765366	−0.382683	+	0.923880i	$0.625000\pi$
$578$	14918.1	1.07355
$579$	16846.5	1.20918
$580$	−11391.0	−0.815496
$581$	0	0
$582$	4759.91	0.339011
$583$	14934.2	1.06091
$584$	28117.9	1.99234
$585$	2793.98	0.197465
$586$	16876.6	1.18970
$587$	21712.2	1.52668	0.763338	−	0.645999i	$-0.223559\pi$
$587$	21712.2	1.52668	0.763338	+	0.645999i	$0.223559\pi$
$588$	0	0
$589$	3346.60	0.234116
$590$	11509.9	0.803141
$591$	−8472.30	−0.589685
$592$	11141.2	0.773484
$593$	−4769.12	−0.330260	−0.165130	−	0.986272i	$-0.552804\pi$
$593$	−4769.12	−0.330260	−0.165130	+	0.986272i	$0.552804\pi$
$594$	28214.9	1.94894
$595$	0	0
$596$	43857.4	3.01421
$597$	−14357.1	−0.984251
$598$	18062.7	1.23518
$599$	3351.27	0.228596	0.114298	−	0.993446i	$-0.463538\pi$
$599$	3351.27	0.228596	0.114298	+	0.993446i	$0.463538\pi$
$600$	−10891.6	−0.741080
$601$	28127.1	1.90903	0.954515	−	0.298164i	$-0.0963743\pi$
$601$	28127.1	1.90903	0.954515	+	0.298164i	$0.0963743\pi$
$602$	0	0
$603$	−10353.4	−0.699212
$604$	−4780.43	−0.322041
$605$	−2009.94	−0.135067
$606$	−18539.4	−1.24276
$607$	−16808.2	−1.12392	−0.561962	−	0.827163i	$-0.689953\pi$
$607$	−16808.2	−1.12392	−0.561962	+	0.827163i	$0.689953\pi$
$608$	9139.88	0.609656
$609$	0	0
$610$	−2125.87	−0.141105
$611$	−19587.2	−1.29691
$612$	22153.2	1.46322
$613$	−15033.8	−0.990556	−0.495278	−	0.868734i	$-0.664934\pi$
$613$	−15033.8	−0.990556	−0.495278	+	0.868734i	$0.664934\pi$
$614$	−40340.4	−2.65148
$615$	6760.81	0.443288
$616$	0	0
$617$	−1222.93	−0.0797944	−0.0398972	−	0.999204i	$-0.512703\pi$
$617$	−1222.93	−0.0797944	−0.0398972	+	0.999204i	$0.512703\pi$
$618$	−4886.44	−0.318060
$619$	−15787.3	−1.02511	−0.512557	−	0.858653i	$-0.671302\pi$
$619$	−15787.3	−1.02511	−0.512557	+	0.858653i	$0.671302\pi$
$620$	2080.13	0.134742
$621$	−14371.5	−0.928677
$622$	5462.57	0.352137
$623$	0	0
$624$	4348.33	0.278962
$625$	8341.17	0.533835
$626$	−8732.52	−0.557543
$627$	14598.6	0.929841
$628$	−37685.4	−2.39460
$629$	−27224.8	−1.72579
$630$	0	0
$631$	−6302.31	−0.397608	−0.198804	−	0.980039i	$-0.563706\pi$
$631$	−6302.31	−0.397608	−0.198804	+	0.980039i	$0.563706\pi$
$632$	20467.7	1.28823
$633$	10876.7	0.682953
$634$	13978.2	0.875622
$635$	845.239	0.0528225
$636$	−16847.2	−1.05037
$637$	0	0
$638$	−34146.6	−2.11893
$639$	6637.07	0.410890
$640$	12034.3	0.743279
$641$	5784.13	0.356411	0.178205	−	0.983993i	$-0.442971\pi$
$641$	5784.13	0.356411	0.178205	+	0.983993i	$0.442971\pi$
$642$	−18193.0	−1.11841
$643$	7889.01	0.483845	0.241922	−	0.970296i	$-0.422222\pi$
$643$	7889.01	0.483845	0.241922	+	0.970296i	$0.422222\pi$
$644$	0	0
$645$	−3424.89	−0.209078
$646$	46214.6	2.81469
$647$	−9604.80	−0.583623	−0.291811	−	0.956476i	$-0.594258\pi$
$647$	−9604.80	−0.583623	−0.291811	+	0.956476i	$0.594258\pi$
$648$	13.9613	0.000846377 0
$649$	22397.7	1.35468
$650$	−18414.4	−1.11119
$651$	0	0
$652$	22533.2	1.35348
$653$	−23654.4	−1.41756	−0.708782	−	0.705428i	$-0.750755\pi$
$653$	−23654.4	−1.41756	−0.708782	+	0.705428i	$0.750755\pi$
$654$	−6161.60	−0.368406
$655$	−12408.3	−0.740200
$656$	−17044.0	−1.01441
$657$	−14451.3	−0.858141
$658$	0	0
$659$	21660.3	1.28038	0.640188	−	0.768219i	$-0.278857\pi$
$659$	21660.3	1.28038	0.640188	+	0.768219i	$0.278857\pi$
$660$	9073.96	0.535157
$661$	−16974.1	−0.998813	−0.499407	−	0.866368i	$-0.666449\pi$
$661$	−16974.1	−0.998813	−0.499407	+	0.866368i	$0.666449\pi$
$662$	−19539.4	−1.14716
$663$	−10625.6	−0.622419
$664$	29379.2	1.71707
$665$	0	0
$666$	−24208.3	−1.40849
$667$	17392.9	1.00968
$668$	18646.6	1.08003
$669$	12360.9	0.714353
$670$	−13425.0	−0.774108
$671$	−4136.85	−0.238005
$672$	0	0
$673$	20032.1	1.14737	0.573685	−	0.819076i	$-0.305513\pi$
$673$	20032.1	1.14737	0.573685	+	0.819076i	$0.305513\pi$
$674$	21921.8	1.25282
$675$	14651.3	0.835451
$676$	−12343.9	−0.702318
$677$	−23865.4	−1.35483	−0.677415	−	0.735601i	$-0.736900\pi$
$677$	−23865.4	−1.35483	−0.677415	+	0.735601i	$0.736900\pi$
$678$	−19554.2	−1.10763
$679$	0	0
$680$	13200.5	0.744434
$681$	−18921.0	−1.06469
$682$	6235.54	0.350104
$683$	20669.9	1.15800	0.578999	−	0.815328i	$-0.303444\pi$
$683$	20669.9	1.15800	0.578999	+	0.815328i	$0.303444\pi$
$684$	26676.5	1.49123
$685$	−2323.33	−0.129591
$686$	0	0
$687$	−13544.8	−0.752206
$688$	8634.14	0.478450
$689$	−13089.3	−0.723751
$690$	−7119.84	−0.392823
$691$	19569.7	1.07738	0.538689	−	0.842505i	$-0.318920\pi$
$691$	19569.7	1.07738	0.538689	+	0.842505i	$0.318920\pi$
$692$	2032.32	0.111644
$693$	0	0
$694$	−5241.49	−0.286692
$695$	1065.78	0.0581691
$696$	17701.8	0.964059
$697$	41648.7	2.26335
$698$	−26032.8	−1.41169
$699$	11167.0	0.604254
$700$	0	0
$701$	20506.9	1.10490	0.552450	−	0.833546i	$-0.313693\pi$
$701$	20506.9	1.10490	0.552450	+	0.833546i	$0.313693\pi$
$702$	−24729.5	−1.32956
$703$	−32783.6	−1.75883
$704$	29393.1	1.57357
$705$	7720.72	0.412453
$706$	−21281.6	−1.13448
$707$	0	0
$708$	−25266.8	−1.34122
$709$	−22997.0	−1.21815	−0.609077	−	0.793111i	$-0.708460\pi$
$709$	−22997.0	−1.21815	−0.609077	+	0.793111i	$0.708460\pi$
$710$	8606.08	0.454902
$711$	−10519.4	−0.554866
$712$	6820.43	0.358998
$713$	−3176.12	−0.166826
$714$	0	0
$715$	7049.95	0.368745
$716$	−53139.3	−2.77362
$717$	3815.32	0.198725
$718$	54538.3	2.83475
$719$	16742.8	0.868427	0.434214	−	0.900810i	$-0.357026\pi$
$719$	16742.8	0.868427	0.434214	+	0.900810i	$0.357026\pi$
$720$	2776.40	0.143709
$721$	0	0
$722$	22898.0	1.18030
$723$	12764.1	0.656574
$724$	−31447.9	−1.61430
$725$	−17731.5	−0.908319
$726$	6796.95	0.347464
$727$	−12842.0	−0.655136	−0.327568	−	0.944828i	$-0.606229\pi$
$727$	−12842.0	−0.655136	−0.327568	+	0.944828i	$0.606229\pi$
$728$	0	0
$729$	12151.2	0.617343
$730$	−18738.5	−0.950061
$731$	−21098.4	−1.06751
$732$	4666.77	0.235641
$733$	−3756.07	−0.189268	−0.0946341	−	0.995512i	$-0.530168\pi$
$733$	−3756.07	−0.189268	−0.0946341	+	0.995512i	$0.530168\pi$
$734$	−13293.6	−0.668495
$735$	0	0
$736$	−8674.30	−0.434428
$737$	−26124.5	−1.30571
$738$	37034.1	1.84722
$739$	201.583	0.0100343	0.00501714	−	0.999987i	$-0.498403\pi$
$739$	201.583	0.0100343	0.00501714	+	0.999987i	$0.498403\pi$
$740$	−20377.2	−1.01227
$741$	−12795.2	−0.634334
$742$	0	0
$743$	3397.05	0.167733	0.0838666	−	0.996477i	$-0.473273\pi$
$743$	3397.05	0.167733	0.0838666	+	0.996477i	$0.473273\pi$
$744$	−3232.54	−0.159288
$745$	−13431.3	−0.660517
$746$	−56139.5	−2.75525
$747$	−15099.5	−0.739576
$748$	55898.4	2.73242
$749$	0	0
$750$	15945.0	0.776304
$751$	−9145.31	−0.444364	−0.222182	−	0.975005i	$-0.571318\pi$
$751$	−9145.31	−0.444364	−0.222182	+	0.975005i	$0.571318\pi$
$752$	−19463.9	−0.943851
$753$	−7948.41	−0.384670
$754$	29928.4	1.44553
$755$	1464.01	0.0705704
$756$	0	0
$757$	−16040.4	−0.770145	−0.385073	−	0.922886i	$-0.625824\pi$
$757$	−16040.4	−0.770145	−0.385073	+	0.922886i	$0.625824\pi$
$758$	−9198.56	−0.440774
$759$	−13854.9	−0.662585
$760$	15895.8	0.758684
$761$	−33464.5	−1.59407	−0.797035	−	0.603934i	$-0.793599\pi$
$761$	−33464.5	−1.59407	−0.797035	+	0.603934i	$0.793599\pi$
$762$	−2858.32	−0.135887
$763$	0	0
$764$	−7161.25	−0.339117
$765$	−6784.42	−0.320642
$766$	42271.4	1.99390
$767$	−19630.8	−0.924158
$768$	−22775.3	−1.07010
$769$	20030.6	0.939302	0.469651	−	0.882852i	$-0.344380\pi$
$769$	20030.6	0.939302	0.469651	+	0.882852i	$0.344380\pi$
$770$	0	0
$771$	−20984.5	−0.980206
$772$	77676.8	3.62131
$773$	23050.4	1.07253	0.536265	−	0.844050i	$-0.319835\pi$
$773$	23050.4	1.07253	0.536265	+	0.844050i	$0.319835\pi$
$774$	−18760.8	−0.871242
$775$	3237.96	0.150079
$776$	10085.6	0.466564
$777$	0	0
$778$	−18123.5	−0.835167
$779$	50152.7	2.30668
$780$	−7953.02	−0.365082
$781$	16747.1	0.767295
$782$	−43860.5	−2.00569
$783$	−23812.3	−1.08682
$784$	0	0
$785$	11541.1	0.524740
$786$	41960.6	1.90418
$787$	−19091.7	−0.864735	−0.432367	−	0.901698i	$-0.642322\pi$
$787$	−19091.7	−0.864735	−0.432367	+	0.901698i	$0.642322\pi$
$788$	−39064.6	−1.76601
$789$	3985.90	0.179850
$790$	−13640.2	−0.614300
$791$	0	0
$792$	22841.4	1.02479
$793$	3625.82	0.162366
$794$	−3355.09	−0.149959
$795$	5159.47	0.230173
$796$	−66198.7	−2.94768
$797$	−20103.3	−0.893471	−0.446736	−	0.894666i	$-0.647414\pi$
$797$	−20103.3	−0.893471	−0.446736	+	0.894666i	$0.647414\pi$
$798$	0	0
$799$	47562.1	2.10592
$800$	8843.18	0.390817
$801$	−3505.38	−0.154627
$802$	−56826.4	−2.50201
$803$	−36464.4	−1.60249
$804$	29470.9	1.29274
$805$	0	0
$806$	−5465.25	−0.238840
$807$	1215.11	0.0530035
$808$	−39282.7	−1.71035
$809$	30318.7	1.31761	0.658806	−	0.752313i	$-0.271062\pi$
$809$	30318.7	1.31761	0.658806	+	0.752313i	$0.271062\pi$
$810$	−9.30420	−0.000403600 0
$811$	5917.70	0.256225	0.128113	−	0.991760i	$-0.459108\pi$
$811$	5917.70	0.256225	0.128113	+	0.991760i	$0.459108\pi$
$812$	0	0
$813$	16248.5	0.700936
$814$	−61084.0	−2.63021
$815$	−6900.79	−0.296594
$816$	−10558.7	−0.452978
$817$	−25406.3	−1.08795
$818$	−16067.1	−0.686764
$819$	0	0
$820$	31173.1	1.32758
$821$	38691.9	1.64477	0.822385	−	0.568931i	$-0.192643\pi$
$821$	38691.9	1.64477	0.822385	+	0.568931i	$0.192643\pi$
$822$	7856.71	0.333375
$823$	−1876.14	−0.0794631	−0.0397316	−	0.999210i	$-0.512650\pi$
$823$	−1876.14	−0.0794631	−0.0397316	+	0.999210i	$0.512650\pi$
$824$	−10353.7	−0.437730
$825$	14124.7	0.596070
$826$	0	0
$827$	−15766.5	−0.662946	−0.331473	−	0.943465i	$-0.607546\pi$
$827$	−15766.5	−0.662946	−0.331473	+	0.943465i	$0.607546\pi$
$828$	−25317.6	−1.06262
$829$	−517.002	−0.0216601	−0.0108301	−	0.999941i	$-0.503447\pi$
$829$	−517.002	−0.0216601	−0.0108301	+	0.999941i	$0.503447\pi$
$830$	−19579.1	−0.818796
$831$	−216.970	−0.00905727
$832$	−25762.1	−1.07349
$833$	0	0
$834$	−3604.12	−0.149641
$835$	−5710.52	−0.236671
$836$	67311.9	2.78473
$837$	4348.39	0.179573
$838$	36749.3	1.51490
$839$	4552.67	0.187337	0.0936684	−	0.995603i	$-0.470141\pi$
$839$	4552.67	0.187337	0.0936684	+	0.995603i	$0.470141\pi$
$840$	0	0
$841$	4429.45	0.181617
$842$	−32673.7	−1.33730
$843$	−7807.57	−0.318988
$844$	50150.8	2.04534
$845$	3780.33	0.153902
$846$	42292.3	1.71872
$847$	0	0
$848$	−13007.0	−0.526724
$849$	9001.49	0.363876
$850$	44714.4	1.80434
$851$	31113.6	1.25330
$852$	−18892.3	−0.759672
$853$	−16827.0	−0.675436	−0.337718	−	0.941247i	$-0.609655\pi$
$853$	−16827.0	−0.675436	−0.337718	+	0.941247i	$0.609655\pi$
$854$	0	0
$855$	−8169.68	−0.326780
$856$	−38548.6	−1.53921
$857$	36750.8	1.46486	0.732429	−	0.680843i	$-0.238387\pi$
$857$	36750.8	1.46486	0.732429	+	0.680843i	$0.238387\pi$
$858$	−23840.6	−0.948605
$859$	7707.99	0.306162	0.153081	−	0.988214i	$-0.451080\pi$
$859$	7707.99	0.306162	0.153081	+	0.988214i	$0.451080\pi$
$860$	−15791.7	−0.626154
$861$	0	0
$862$	−18369.6	−0.725837
$863$	−25903.3	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$863$	−25903.3	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$864$	11875.9	0.467622
$865$	−622.399	−0.0244650
$866$	29949.2	1.17519
$867$	10029.2	0.392861
$868$	0	0
$869$	−26543.3	−1.03616
$870$	−11797.0	−0.459718
$871$	22897.2	0.890750
$872$	−13055.7	−0.507019
$873$	−5183.55	−0.200958
$874$	−52816.0	−2.04408
$875$	0	0
$876$	41135.4	1.58657
$877$	20643.9	0.794861	0.397431	−	0.917632i	$-0.369902\pi$
$877$	20643.9	0.794861	0.397431	+	0.917632i	$0.369902\pi$
$878$	−2495.48	−0.0959205
$879$	11345.9	0.435368
$880$	7005.59	0.268362
$881$	−20652.4	−0.789779	−0.394890	−	0.918729i	$-0.629217\pi$
$881$	−20652.4	−0.789779	−0.394890	+	0.918729i	$0.629217\pi$
$882$	0	0
$883$	20538.5	0.782759	0.391380	−	0.920229i	$-0.371998\pi$
$883$	20538.5	0.782759	0.391380	+	0.920229i	$0.371998\pi$
$884$	−48993.2	−1.86405
$885$	7737.95	0.293908
$886$	−46885.2	−1.77781
$887$	−41376.4	−1.56627	−0.783135	−	0.621851i	$-0.786381\pi$
$887$	−41376.4	−1.56627	−0.783135	+	0.621851i	$0.786381\pi$
$888$	31666.3	1.19668
$889$	0	0
$890$	−4545.32	−0.171190
$891$	−18.1056	−0.000680763 0
$892$	56994.6	2.13937
$893$	57273.4	2.14623
$894$	45420.2	1.69919
$895$	16273.9	0.607795
$896$	0	0
$897$	12143.4	0.452013
$898$	19077.7	0.708944
$899$	−5262.56	−0.195235
$900$	25810.6	0.955947
$901$	31784.0	1.17522
$902$	93446.9	3.44949
$903$	0	0
$904$	−41432.8	−1.52438
$905$	9630.92	0.353749
$906$	−4950.78	−0.181544
$907$	4443.24	0.162663	0.0813315	−	0.996687i	$-0.474083\pi$
$907$	4443.24	0.162663	0.0813315	+	0.996687i	$0.474083\pi$
$908$	−87242.3	−3.18859
$909$	20189.4	0.736679
$910$	0	0
$911$	422.602	0.0153693	0.00768466	−	0.999970i	$-0.497554\pi$
$911$	422.602	0.0153693	0.00768466	+	0.999970i	$0.497554\pi$
$912$	−12714.7	−0.461649
$913$	−38100.1	−1.38108
$914$	66986.4	2.42419
$915$	−1429.20	−0.0516370
$916$	−62453.1	−2.25274
$917$	0	0
$918$	60048.8	2.15894
$919$	−16620.0	−0.596565	−0.298283	−	0.954478i	$-0.596414\pi$
$919$	−16620.0	−0.596565	−0.298283	+	0.954478i	$0.596414\pi$
$920$	−15086.1	−0.540622
$921$	−27120.4	−0.970301
$922$	93408.4	3.33649
$923$	−14678.3	−0.523446
$924$	0	0
$925$	−31719.4	−1.12749
$926$	67680.3	2.40185
$927$	5321.34	0.188539
$928$	−14372.6	−0.508408
$929$	24878.4	0.878617	0.439308	−	0.898336i	$-0.355224\pi$
$929$	24878.4	0.878617	0.439308	+	0.898336i	$0.355224\pi$
$930$	2154.25	0.0759577
$931$	0	0
$932$	51489.4	1.80965
$933$	3672.42	0.128864
$934$	−53402.1	−1.87085
$935$	−17118.9	−0.598767
$936$	−20019.8	−0.699109
$937$	−14356.7	−0.500546	−0.250273	−	0.968175i	$-0.580520\pi$
$937$	−14356.7	−0.500546	−0.250273	+	0.968175i	$0.580520\pi$
$938$	0	0
$939$	−5870.78	−0.204032
$940$	35599.2	1.23523
$941$	43079.9	1.49242	0.746208	−	0.665713i	$-0.231873\pi$
$941$	43079.9	1.49242	0.746208	+	0.665713i	$0.231873\pi$
$942$	−39028.2	−1.34990
$943$	−47597.9	−1.64369
$944$	−19507.3	−0.672574
$945$	0	0
$946$	−47338.3	−1.62696
$947$	−7501.33	−0.257403	−0.128701	−	0.991683i	$-0.541081\pi$
$947$	−7501.33	−0.257403	−0.128701	+	0.991683i	$0.541081\pi$
$948$	29943.4	1.02586
$949$	31959.9	1.09322
$950$	53844.3	1.83888
$951$	9397.37	0.320432
$952$	0	0
$953$	−37150.2	−1.26276	−0.631381	−	0.775473i	$-0.717512\pi$
$953$	−37150.2	−1.26276	−0.631381	+	0.775473i	$0.717512\pi$
$954$	28262.3	0.959148
$955$	2193.13	0.0743122
$956$	17591.9	0.595149
$957$	−22956.4	−0.775418
$958$	5256.31	0.177269
$959$	0	0
$960$	10154.7	0.341398
$961$	961.000	0.0322581
$962$	53538.1	1.79432
$963$	19812.2	0.662969
$964$	58853.6	1.96634
$965$	−23788.5	−0.793554
$966$	0	0
$967$	15771.8	0.524494	0.262247	−	0.965001i	$-0.415536\pi$
$967$	15771.8	0.524494	0.262247	+	0.965001i	$0.415536\pi$
$968$	14401.9	0.478197
$969$	31069.6	1.03003
$970$	−6721.35	−0.222484
$971$	8802.10	0.290909	0.145455	−	0.989365i	$-0.453536\pi$
$971$	8802.10	0.290909	0.145455	+	0.989365i	$0.453536\pi$
$972$	56080.9	1.85061
$973$	0	0
$974$	−96525.9	−3.17545
$975$	−12379.8	−0.406637
$976$	3603.00	0.118165
$977$	−45922.4	−1.50377	−0.751887	−	0.659292i	$-0.770856\pi$
$977$	−45922.4	−1.50377	−0.751887	+	0.659292i	$0.770856\pi$
$978$	23336.2	0.762994
$979$	−8845.00	−0.288751
$980$	0	0
$981$	6710.00	0.218383
$982$	−67430.7	−2.19124
$983$	−11654.9	−0.378163	−0.189082	−	0.981961i	$-0.560551\pi$
$983$	−11654.9	−0.378163	−0.189082	+	0.981961i	$0.560551\pi$
$984$	−48443.4	−1.56943
$985$	11963.5	0.386994
$986$	−72673.0	−2.34724
$987$	0	0
$988$	−58996.7	−1.89973
$989$	24112.2	0.775250
$990$	−15222.1	−0.488678
$991$	24451.5	0.783781	0.391890	−	0.920012i	$-0.371821\pi$
$991$	24451.5	0.783781	0.391890	+	0.920012i	$0.371821\pi$
$992$	2624.58	0.0840026
$993$	−13136.1	−0.419800
$994$	0	0
$995$	20273.3	0.645937
$996$	42980.7	1.36736
$997$	13644.7	0.433431	0.216716	−	0.976235i	$-0.430466\pi$
$997$	13644.7	0.433431	0.216716	+	0.976235i	$0.430466\pi$
$998$	3491.13	0.110731
$999$	−42597.3	−1.34907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.20	✓	23
7.6	odd	2		1519.4.a.i.1.20	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.20	✓	23	1.1	even	1	trivial
1519.4.a.i.1.20	yes	23	7.6	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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q+4.77516 q^{2} +3.21029 q^{3} +14.8022 q^{4} -4.53317 q^{5} +15.3297 q^{6} +32.4816 q^{8} -16.6940 q^{9} -21.6466 q^{10} -42.1235 q^{11} +47.5194 q^{12} +36.9198 q^{13} -14.5528 q^{15} +36.6875 q^{16} -89.6498 q^{17} -79.7168 q^{18} -107.955 q^{19} -67.1009 q^{20} -201.146 q^{22} +102.456 q^{23} +104.275 q^{24} -104.450 q^{25} +176.298 q^{26} -140.271 q^{27} +169.760 q^{29} -69.4920 q^{30} -31.0000 q^{31} -84.6640 q^{32} -135.229 q^{33} -428.093 q^{34} -247.108 q^{36} +303.679 q^{37} -515.502 q^{38} +118.523 q^{39} -147.245 q^{40} -464.571 q^{41} +235.343 q^{43} -623.520 q^{44} +75.6769 q^{45} +489.242 q^{46} -530.532 q^{47} +117.778 q^{48} -498.768 q^{50} -287.802 q^{51} +546.495 q^{52} -354.534 q^{53} -669.815 q^{54} +190.953 q^{55} -346.566 q^{57} +810.632 q^{58} -531.716 q^{59} -215.413 q^{60} +98.2079 q^{61} -148.030 q^{62} -697.785 q^{64} -167.364 q^{65} -645.739 q^{66} +620.188 q^{67} -1327.01 q^{68} +328.912 q^{69} -397.571 q^{71} -542.250 q^{72} +865.656 q^{73} +1450.12 q^{74} -335.316 q^{75} -1597.97 q^{76} +565.969 q^{78} +630.131 q^{79} -166.311 q^{80} +0.429822 q^{81} -2218.40 q^{82} +904.487 q^{83} +406.398 q^{85} +1123.80 q^{86} +544.979 q^{87} -1368.24 q^{88} +209.978 q^{89} +361.370 q^{90} +1516.57 q^{92} -99.5190 q^{93} -2533.38 q^{94} +489.377 q^{95} -271.796 q^{96} +310.503 q^{97} +703.211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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