LMFDB - Embedded newform 1573.4.a.q.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1573,4,Mod(1,1573)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1573, 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("1573.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1573 = 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1573.a (trivial)

Newform invariants

comment:select newform

sage:traces = [38,-3,19,181,52] 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:	$92.8100044390$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	1573.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.15222 q^{2} -8.58190 q^{3} +9.24094 q^{4} +6.15304 q^{5} +35.6340 q^{6} -2.19831 q^{7} -5.15264 q^{8} +46.6491 q^{9} -25.5488 q^{10} -79.3048 q^{12} -13.0000 q^{13} +9.12788 q^{14} -52.8048 q^{15} -52.5326 q^{16} +4.33204 q^{17} -193.697 q^{18} -26.5681 q^{19} +56.8599 q^{20} +18.8657 q^{21} +0.674151 q^{23} +44.2195 q^{24} -87.1401 q^{25} +53.9789 q^{26} -168.627 q^{27} -20.3145 q^{28} +80.7315 q^{29} +219.257 q^{30} +290.951 q^{31} +259.348 q^{32} -17.9876 q^{34} -13.5263 q^{35} +431.081 q^{36} +336.533 q^{37} +110.317 q^{38} +111.565 q^{39} -31.7044 q^{40} -162.354 q^{41} -78.3346 q^{42} +258.374 q^{43} +287.034 q^{45} -2.79922 q^{46} +83.6576 q^{47} +450.830 q^{48} -338.167 q^{49} +361.825 q^{50} -37.1772 q^{51} -120.132 q^{52} -253.507 q^{53} +700.175 q^{54} +11.3271 q^{56} +228.005 q^{57} -335.215 q^{58} -25.0547 q^{59} -487.966 q^{60} +433.646 q^{61} -1208.09 q^{62} -102.549 q^{63} -656.610 q^{64} -79.9895 q^{65} +852.622 q^{67} +40.0321 q^{68} -5.78550 q^{69} +56.1642 q^{70} -451.381 q^{71} -240.366 q^{72} -486.807 q^{73} -1397.36 q^{74} +747.828 q^{75} -245.515 q^{76} -463.241 q^{78} +130.035 q^{79} -323.235 q^{80} +187.612 q^{81} +674.130 q^{82} -553.122 q^{83} +174.337 q^{84} +26.6552 q^{85} -1072.83 q^{86} -692.830 q^{87} +1549.44 q^{89} -1191.83 q^{90} +28.5781 q^{91} +6.22979 q^{92} -2496.92 q^{93} -347.365 q^{94} -163.475 q^{95} -2225.70 q^{96} +270.885 q^{97} +1404.15 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 3 q^{2} + 19 q^{3} + 181 q^{4} + 52 q^{5} - 104 q^{6} - 12 q^{7} - 57 q^{8} + 477 q^{9} + 30 q^{10} + 122 q^{12} - 494 q^{13} + 181 q^{14} + 264 q^{15} + 961 q^{16} + 33 q^{17} - 28 q^{18} - 107 q^{19}+ \cdots - 2043 q^{98}+O(q^{100}) $ 38 * q - 3 * q^2 + 19 * q^3 + 181 * q^4 + 52 * q^5 - 104 * q^6 - 12 * q^7 - 57 * q^8 + 477 * q^9 + 30 * q^10 + 122 * q^12 - 494 * q^13 + 181 * q^14 + 264 * q^15 + 961 * q^16 + 33 * q^17 - 28 * q^18 - 107 * q^19 + 595 * q^20 - 148 * q^21 + 840 * q^23 - 968 * q^24 + 1428 * q^25 + 39 * q^26 + 1108 * q^27 + 353 * q^28 + 162 * q^29 + 48 * q^30 + 62 * q^31 - 1204 * q^32 - 58 * q^34 + 352 * q^35 + 3357 * q^36 + 304 * q^37 + 963 * q^38 - 247 * q^39 + 571 * q^40 - 361 * q^41 + 587 * q^42 - 483 * q^43 + 1136 * q^45 + 1860 * q^46 + 2892 * q^47 + 945 * q^48 + 2270 * q^49 + 782 * q^50 + 682 * q^51 - 2353 * q^52 + 1622 * q^53 - 3273 * q^54 + 3269 * q^56 - 62 * q^57 + 1683 * q^58 + 3417 * q^59 + 4279 * q^60 + 1714 * q^61 - 508 * q^62 - 3730 * q^63 + 4823 * q^64 - 676 * q^65 + 4939 * q^67 + 3699 * q^68 + 1818 * q^69 + 592 * q^70 + 1544 * q^71 + 5570 * q^72 - 617 * q^73 + 1398 * q^74 + 5457 * q^75 - 3316 * q^76 + 1352 * q^78 + 3176 * q^79 + 5091 * q^80 + 5994 * q^81 + 1238 * q^82 - 1659 * q^83 - 5157 * q^84 - 1074 * q^85 + 1144 * q^86 + 906 * q^87 + 3727 * q^89 + 6753 * q^90 + 156 * q^91 + 11004 * q^92 + 6956 * q^93 + 9255 * q^94 - 1768 * q^95 - 4812 * q^96 + 5723 * q^97 - 2043 * q^98

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 $

$2$	−4.15222	−1.46803	−0.734016	−	0.679132i	$-0.762356\pi$
$2$	−4.15222	−1.46803	−0.734016	+	0.679132i	$0.762356\pi$
$3$	−8.58190	−1.65159	−0.825794	−	0.563972i	$-0.809273\pi$
$3$	−8.58190	−1.65159	−0.825794	+	0.563972i	$0.809273\pi$
$4$	9.24094	1.15512
$5$	6.15304	0.550345	0.275172	−	0.961395i	$-0.411265\pi$
$5$	6.15304	0.550345	0.275172	+	0.961395i	$0.411265\pi$
$6$	35.6340	2.42458
$7$	−2.19831	−0.118698	−0.0593489	−	0.998237i	$-0.518902\pi$
$7$	−2.19831	−0.118698	−0.0593489	+	0.998237i	$0.518902\pi$
$8$	−5.15264	−0.227717
$9$	46.6491	1.72774
$10$	−25.5488	−0.807923
$11$	0	0
$11$	0	0
$12$	−79.3048	−1.90778
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	9.12788	0.174252
$15$	−52.8048	−0.908943
$16$	−52.5326	−0.820822
$17$	4.33204	0.0618044	0.0309022	−	0.999522i	$-0.490162\pi$
$17$	4.33204	0.0618044	0.0309022	+	0.999522i	$0.490162\pi$
$18$	−193.697	−2.53638
$19$	−26.5681	−0.320797	−0.160399	−	0.987052i	$-0.551278\pi$
$19$	−26.5681	−0.320797	−0.160399	+	0.987052i	$0.551278\pi$
$20$	56.8599	0.635713
$21$	18.8657	0.196040
$22$	0	0
$23$	0.674151	0.00611175	0.00305587	−	0.999995i	$-0.499027\pi$
$23$	0.674151	0.00611175	0.00305587	+	0.999995i	$0.499027\pi$
$24$	44.2195	0.376094
$25$	−87.1401	−0.697121
$26$	53.9789	0.407159
$27$	−168.627	−1.20193
$28$	−20.3145	−0.137110
$29$	80.7315	0.516947	0.258473	−	0.966018i	$-0.416781\pi$
$29$	80.7315	0.516947	0.258473	+	0.966018i	$0.416781\pi$
$30$	219.257	1.33436
$31$	290.951	1.68569	0.842845	−	0.538156i	$-0.180879\pi$
$31$	290.951	1.68569	0.842845	+	0.538156i	$0.180879\pi$
$32$	259.348	1.43271
$33$	0	0
$34$	−17.9876	−0.0907308
$35$	−13.5263	−0.0653247
$36$	431.081	1.99575
$37$	336.533	1.49529	0.747644	−	0.664100i	$-0.231185\pi$
$37$	336.533	1.49529	0.747644	+	0.664100i	$0.231185\pi$
$38$	110.317	0.470941
$39$	111.565	0.458068
$40$	−31.7044	−0.125323
$41$	−162.354	−0.618426	−0.309213	−	0.950993i	$-0.600066\pi$
$41$	−162.354	−0.618426	−0.309213	+	0.950993i	$0.600066\pi$
$42$	−78.3346	−0.287793
$43$	258.374	0.916317	0.458159	−	0.888870i	$-0.348509\pi$
$43$	258.374	0.916317	0.458159	+	0.888870i	$0.348509\pi$
$44$	0	0
$45$	287.034	0.950855
$46$	−2.79922	−0.00897224
$47$	83.6576	0.259632	0.129816	−	0.991538i	$-0.458561\pi$
$47$	83.6576	0.259632	0.129816	+	0.991538i	$0.458561\pi$
$48$	450.830	1.35566
$49$	−338.167	−0.985911
$50$	361.825	1.02340
$51$	−37.1772	−0.102075
$52$	−120.132	−0.320372
$53$	−253.507	−0.657016	−0.328508	−	0.944501i	$-0.606546\pi$
$53$	−253.507	−0.657016	−0.328508	+	0.944501i	$0.606546\pi$
$54$	700.175	1.76448
$55$	0	0
$56$	11.3271	0.0270295
$57$	228.005	0.529825
$58$	−335.215	−0.758894
$59$	−25.0547	−0.0552855	−0.0276427	−	0.999618i	$-0.508800\pi$
$59$	−25.0547	−0.0552855	−0.0276427	+	0.999618i	$0.508800\pi$
$60$	−487.966	−1.04994
$61$	433.646	0.910208	0.455104	−	0.890438i	$-0.349602\pi$
$61$	433.646	0.910208	0.455104	+	0.890438i	$0.349602\pi$
$62$	−1208.09	−2.47465
$63$	−102.549	−0.205079
$64$	−656.610	−1.28244
$65$	−79.9895	−0.152638
$66$	0	0
$67$	852.622	1.55469	0.777346	−	0.629073i	$-0.216565\pi$
$67$	852.622	1.55469	0.777346	+	0.629073i	$0.216565\pi$
$68$	40.0321	0.0713913
$69$	−5.78550	−0.0100941
$70$	56.1642	0.0958987
$71$	−451.381	−0.754493	−0.377247	−	0.926113i	$-0.623129\pi$
$71$	−451.381	−0.754493	−0.377247	+	0.926113i	$0.623129\pi$
$72$	−240.366	−0.393436
$73$	−486.807	−0.780499	−0.390250	−	0.920709i	$-0.627611\pi$
$73$	−486.807	−0.780499	−0.390250	+	0.920709i	$0.627611\pi$
$74$	−1397.36	−2.19513
$75$	747.828	1.15136
$76$	−245.515	−0.370559
$77$	0	0
$78$	−463.241	−0.672459
$79$	130.035	0.185191	0.0925955	−	0.995704i	$-0.470484\pi$
$79$	130.035	0.185191	0.0925955	+	0.995704i	$0.470484\pi$
$80$	−323.235	−0.451735
$81$	187.612	0.257355
$82$	674.130	0.907869
$83$	−553.122	−0.731482	−0.365741	−	0.930717i	$-0.619184\pi$
$83$	−553.122	−0.731482	−0.365741	+	0.930717i	$0.619184\pi$
$84$	174.337	0.226449
$85$	26.6552	0.0340137
$86$	−1072.83	−1.34518
$87$	−692.830	−0.853783
$88$	0	0
$89$	1549.44	1.84540	0.922700	−	0.385519i	$-0.125978\pi$
$89$	1549.44	1.84540	0.922700	+	0.385519i	$0.125978\pi$
$90$	−1191.83	−1.39588
$91$	28.5781	0.0329208
$92$	6.22979	0.00705978
$93$	−2496.92	−2.78407
$94$	−347.365	−0.381148
$95$	−163.475	−0.176549
$96$	−2225.70	−2.36625
$97$	270.885	0.283548	0.141774	−	0.989899i	$-0.454719\pi$
$97$	270.885	0.283548	0.141774	+	0.989899i	$0.454719\pi$
$98$	1404.15	1.44735
$99$	0	0
$100$	−805.256	−0.805256
$101$	−1082.85	−1.06681	−0.533405	−	0.845860i	$-0.679088\pi$
$101$	−1082.85	−1.06681	−0.533405	+	0.845860i	$0.679088\pi$
$102$	154.368	0.149850
$103$	111.118	0.106299	0.0531495	−	0.998587i	$-0.483074\pi$
$103$	111.118	0.106299	0.0531495	+	0.998587i	$0.483074\pi$
$104$	66.9843	0.0631573
$105$	116.081	0.107889
$106$	1052.62	0.964521
$107$	−408.439	−0.369021	−0.184511	−	0.982831i	$-0.559070\pi$
$107$	−408.439	−0.369021	−0.184511	+	0.982831i	$0.559070\pi$
$108$	−1558.27	−1.38837
$109$	−785.113	−0.689910	−0.344955	−	0.938619i	$-0.612106\pi$
$109$	−785.113	−0.689910	−0.344955	+	0.938619i	$0.612106\pi$
$110$	0	0
$111$	−2888.09	−2.46960
$112$	115.483	0.0974297
$113$	2253.49	1.87602	0.938009	−	0.346610i	$-0.112667\pi$
$113$	2253.49	1.87602	0.938009	+	0.346610i	$0.112667\pi$
$114$	−946.728	−0.777800
$115$	4.14808	0.00336357
$116$	746.034	0.597134
$117$	−606.438	−0.479190
$118$	104.033	0.0811608
$119$	−9.52318	−0.00733604
$120$	272.084	0.206981
$121$	0	0
$122$	−1800.59	−1.33621
$123$	1393.31	1.02139
$124$	2688.66	1.94717
$125$	−1305.31	−0.934001
$126$	425.807	0.301063
$127$	−489.084	−0.341726	−0.170863	−	0.985295i	$-0.554656\pi$
$127$	−489.084	−0.341726	−0.170863	+	0.985295i	$0.554656\pi$
$128$	651.603	0.449954
$129$	−2217.34	−1.51338
$130$	332.134	0.224078
$131$	−2008.91	−1.33984	−0.669920	−	0.742433i	$-0.733672\pi$
$131$	−2008.91	−1.33984	−0.669920	+	0.742433i	$0.733672\pi$
$132$	0	0
$133$	58.4051	0.0380779
$134$	−3540.27	−2.28234
$135$	−1037.57	−0.661477
$136$	−22.3215	−0.0140739
$137$	912.528	0.569069	0.284535	−	0.958666i	$-0.408161\pi$
$137$	912.528	0.569069	0.284535	+	0.958666i	$0.408161\pi$
$138$	24.0227	0.0148184
$139$	−2447.16	−1.49328	−0.746638	−	0.665231i	$-0.768333\pi$
$139$	−2447.16	−1.49328	−0.746638	+	0.665231i	$0.768333\pi$
$140$	−124.996	−0.0754576
$141$	−717.942	−0.428806
$142$	1874.23	1.10762
$143$	0	0
$144$	−2450.60	−1.41817
$145$	496.744	0.284499
$146$	2021.33	1.14580
$147$	2902.12	1.62832
$148$	3109.88	1.72723
$149$	−2682.91	−1.47512	−0.737560	−	0.675282i	$-0.764022\pi$
$149$	−2682.91	−1.47512	−0.737560	+	0.675282i	$0.764022\pi$
$150$	−3105.15	−1.69023
$151$	−3419.00	−1.84261	−0.921305	−	0.388841i	$-0.872876\pi$
$151$	−3419.00	−1.84261	−0.921305	+	0.388841i	$0.872876\pi$
$152$	136.896	0.0730509
$153$	202.086	0.106782
$154$	0	0
$155$	1790.24	0.927711
$156$	1030.96	0.529122
$157$	−3034.47	−1.54253	−0.771265	−	0.636514i	$-0.780376\pi$
$157$	−3034.47	−1.54253	−0.771265	+	0.636514i	$0.780376\pi$
$158$	−539.934	−0.271866
$159$	2175.57	1.08512
$160$	1595.78	0.788484
$161$	−1.48199	−0.000725451 0
$162$	−779.004	−0.377805
$163$	3602.67	1.73118	0.865592	−	0.500750i	$-0.166942\pi$
$163$	3602.67	1.73118	0.865592	+	0.500750i	$0.166942\pi$
$164$	−1500.30	−0.714354
$165$	0	0
$166$	2296.68	1.07384
$167$	3514.90	1.62869	0.814346	−	0.580380i	$-0.197096\pi$
$167$	3514.90	1.62869	0.814346	+	0.580380i	$0.197096\pi$
$168$	−97.2082	−0.0446415
$169$	169.000	0.0769231
$170$	−110.678	−0.0499332
$171$	−1239.38	−0.554256
$172$	2387.62	1.05845
$173$	−739.068	−0.324800	−0.162400	−	0.986725i	$-0.551923\pi$
$173$	−739.068	−0.324800	−0.162400	+	0.986725i	$0.551923\pi$
$174$	2876.78	1.25338
$175$	191.561	0.0827466
$176$	0	0
$177$	215.017	0.0913088
$178$	−6433.63	−2.70911
$179$	1016.07	0.424273	0.212137	−	0.977240i	$-0.431958\pi$
$179$	1016.07	0.424273	0.212137	+	0.977240i	$0.431958\pi$
$180$	2652.46	1.09835
$181$	−662.625	−0.272113	−0.136057	−	0.990701i	$-0.543443\pi$
$181$	−662.625	−0.272113	−0.136057	+	0.990701i	$0.543443\pi$
$182$	−118.662	−0.0483288
$183$	−3721.51	−1.50329
$184$	−3.47366	−0.00139175
$185$	2070.70	0.822924
$186$	10367.7	4.08710
$187$	0	0
$188$	773.075	0.299906
$189$	370.694	0.142667
$190$	678.784	0.259180
$191$	−40.6513	−0.0154001	−0.00770006	−	0.999970i	$-0.502451\pi$
$191$	−40.6513	−0.0154001	−0.00770006	+	0.999970i	$0.502451\pi$
$192$	5634.96	2.11806
$193$	−2375.84	−0.886098	−0.443049	−	0.896497i	$-0.646103\pi$
$193$	−2375.84	−0.886098	−0.443049	+	0.896497i	$0.646103\pi$
$194$	−1124.77	−0.416258
$195$	686.462	0.252095
$196$	−3124.98	−1.13884
$197$	3903.38	1.41170	0.705849	−	0.708362i	$-0.250566\pi$
$197$	3903.38	1.41170	0.705849	+	0.708362i	$0.250566\pi$
$198$	0	0
$199$	3133.72	1.11630	0.558150	−	0.829740i	$-0.311511\pi$
$199$	3133.72	1.11630	0.558150	+	0.829740i	$0.311511\pi$
$200$	449.002	0.158746
$201$	−7317.12	−2.56771
$202$	4496.24	1.56611
$203$	−177.473	−0.0613604
$204$	−343.552	−0.117909
$205$	−998.972	−0.340347
$206$	−461.387	−0.156050
$207$	31.4485	0.0105595
$208$	682.924	0.227655
$209$	0	0
$210$	−481.996	−0.158385
$211$	−445.565	−0.145374	−0.0726871	−	0.997355i	$-0.523157\pi$
$211$	−445.565	−0.145374	−0.0726871	+	0.997355i	$0.523157\pi$
$212$	−2342.64	−0.758931
$213$	3873.70	1.24611
$214$	1695.93	0.541735
$215$	1589.78	0.504290
$216$	868.872	0.273700
$217$	−639.602	−0.200088
$218$	3259.96	1.01281
$219$	4177.73	1.28906
$220$	0	0
$221$	−56.3166	−0.0171415
$222$	11992.0	3.62545
$223$	−4170.38	−1.25233	−0.626164	−	0.779692i	$-0.715376\pi$
$223$	−4170.38	−1.25233	−0.626164	+	0.779692i	$0.715376\pi$
$224$	−570.128	−0.170059
$225$	−4065.01	−1.20445
$226$	−9356.97	−2.75405
$227$	940.040	0.274858	0.137429	−	0.990512i	$-0.456116\pi$
$227$	940.040	0.274858	0.137429	+	0.990512i	$0.456116\pi$
$228$	2106.98	0.612010
$229$	−6082.15	−1.75511	−0.877554	−	0.479479i	$-0.840826\pi$
$229$	−6082.15	−1.75511	−0.877554	+	0.479479i	$0.840826\pi$
$230$	−17.2237	−0.00493782
$231$	0	0
$232$	−415.980	−0.117717
$233$	−449.840	−0.126481	−0.0632404	−	0.997998i	$-0.520143\pi$
$233$	−449.840	−0.126481	−0.0632404	+	0.997998i	$0.520143\pi$
$234$	2518.06	0.703466
$235$	514.749	0.142887
$236$	−231.529	−0.0638612
$237$	−1115.95	−0.305859
$238$	39.5424	0.0107695
$239$	5700.21	1.54274	0.771372	−	0.636384i	$-0.219571\pi$
$239$	5700.21	1.54274	0.771372	+	0.636384i	$0.219571\pi$
$240$	2773.97	0.746080
$241$	2914.08	0.778889	0.389444	−	0.921050i	$-0.372667\pi$
$241$	2914.08	0.778889	0.389444	+	0.921050i	$0.372667\pi$
$242$	0	0
$243$	2942.85	0.776889
$244$	4007.30	1.05140
$245$	−2080.76	−0.542591
$246$	−5785.32	−1.49943
$247$	345.386	0.0889732
$248$	−1499.17	−0.383860
$249$	4746.84	1.20811
$250$	5419.92	1.37114
$251$	4162.22	1.04668	0.523340	−	0.852124i	$-0.324686\pi$
$251$	4162.22	1.04668	0.523340	+	0.852124i	$0.324686\pi$
$252$	−947.651	−0.236891
$253$	0	0
$254$	2030.78	0.501664
$255$	−228.753	−0.0561767
$256$	2547.27	0.621893
$257$	−4211.23	−1.02214	−0.511069	−	0.859540i	$-0.670750\pi$
$257$	−4211.23	−1.02214	−0.511069	+	0.859540i	$0.670750\pi$
$258$	9206.88	2.22169
$259$	−739.804	−0.177487
$260$	−739.178	−0.176315
$261$	3766.05	0.893151
$262$	8341.43	1.96693
$263$	−6038.85	−1.41586	−0.707931	−	0.706282i	$-0.750371\pi$
$263$	−6038.85	−1.41586	−0.707931	+	0.706282i	$0.750371\pi$
$264$	0	0
$265$	−1559.84	−0.361585
$266$	−242.511	−0.0558996
$267$	−13297.2	−3.04784
$268$	7879.03	1.79585
$269$	3558.00	0.806450	0.403225	−	0.915101i	$-0.367889\pi$
$269$	3558.00	0.806450	0.403225	+	0.915101i	$0.367889\pi$
$270$	4308.20	0.971070
$271$	7773.31	1.74242	0.871209	−	0.490913i	$-0.163337\pi$
$271$	7773.31	1.74242	0.871209	+	0.490913i	$0.163337\pi$
$272$	−227.573	−0.0507304
$273$	−245.254	−0.0543716
$274$	−3789.02	−0.835412
$275$	0	0
$276$	−53.4634	−0.0116599
$277$	2639.76	0.572592	0.286296	−	0.958141i	$-0.407576\pi$
$277$	2639.76	0.572592	0.286296	+	0.958141i	$0.407576\pi$
$278$	10161.1	2.19218
$279$	13572.6	2.91244
$280$	69.6962	0.0148755
$281$	−1102.23	−0.233999	−0.116999	−	0.993132i	$-0.537328\pi$
$281$	−1102.23	−0.233999	−0.116999	+	0.993132i	$0.537328\pi$
$282$	2981.05	0.629500
$283$	3958.36	0.831450	0.415725	−	0.909490i	$-0.363528\pi$
$283$	3958.36	0.831450	0.415725	+	0.909490i	$0.363528\pi$
$284$	−4171.18	−0.871528
$285$	1402.93	0.291587
$286$	0	0
$287$	356.905	0.0734057
$288$	12098.3	2.47535
$289$	−4894.23	−0.996180
$290$	−2062.59	−0.417653
$291$	−2324.71	−0.468305
$292$	−4498.55	−0.901568
$293$	−2149.93	−0.428670	−0.214335	−	0.976760i	$-0.568758\pi$
$293$	−2149.93	−0.428670	−0.214335	+	0.976760i	$0.568758\pi$
$294$	−12050.2	−2.39042
$295$	−154.162	−0.0304261
$296$	−1734.03	−0.340502
$297$	0	0
$298$	11140.0	2.16552
$299$	−8.76396	−0.00169509
$300$	6910.63	1.32995
$301$	−567.986	−0.108765
$302$	14196.4	2.70501
$303$	9292.94	1.76193
$304$	1395.69	0.263317
$305$	2668.24	0.500928
$306$	−839.105	−0.156760
$307$	−4334.13	−0.805738	−0.402869	−	0.915258i	$-0.631987\pi$
$307$	−4334.13	−0.805738	−0.402869	+	0.915258i	$0.631987\pi$
$308$	0	0
$309$	−953.605	−0.175562
$310$	−7433.45	−1.36191
$311$	1932.81	0.352411	0.176206	−	0.984353i	$-0.443618\pi$
$311$	1932.81	0.352411	0.176206	+	0.984353i	$0.443618\pi$
$312$	−574.853	−0.104310
$313$	−3991.03	−0.720722	−0.360361	−	0.932813i	$-0.617347\pi$
$313$	−3991.03	−0.720722	−0.360361	+	0.932813i	$0.617347\pi$
$314$	12599.8	2.26448
$315$	−630.990	−0.112864
$316$	1201.65	0.213917
$317$	7134.01	1.26399	0.631996	−	0.774971i	$-0.282236\pi$
$317$	7134.01	1.26399	0.631996	+	0.774971i	$0.282236\pi$
$318$	−9033.46	−1.59299
$319$	0	0
$320$	−4040.14	−0.705784
$321$	3505.18	0.609471
$322$	6.15357	0.00106498
$323$	−115.094	−0.0198267
$324$	1733.71	0.297275
$325$	1132.82	0.193347
$326$	−14959.1	−2.54143
$327$	6737.77	1.13945
$328$	836.553	0.140826
$329$	−183.906	−0.0308178
$330$	0	0
$331$	5833.01	0.968614	0.484307	−	0.874898i	$-0.339072\pi$
$331$	5833.01	0.968614	0.484307	+	0.874898i	$0.339072\pi$
$332$	−5111.36	−0.844948
$333$	15698.9	2.58347
$334$	−14594.7	−2.39097
$335$	5246.22	0.855616
$336$	−991.064	−0.160914
$337$	−6846.13	−1.10662	−0.553312	−	0.832974i	$-0.686636\pi$
$337$	−6846.13	−1.10662	−0.553312	+	0.832974i	$0.686636\pi$
$338$	−701.725	−0.112926
$339$	−19339.2	−3.09841
$340$	246.319	0.0392898
$341$	0	0
$342$	5146.18	0.813665
$343$	1497.42	0.235723
$344$	−1331.31	−0.208661
$345$	−35.5984	−0.00555523
$346$	3068.77	0.476816
$347$	1639.33	0.253613	0.126807	−	0.991927i	$-0.459527\pi$
$347$	1639.33	0.253613	0.126807	+	0.991927i	$0.459527\pi$
$348$	−6402.39	−0.986219
$349$	6097.13	0.935162	0.467581	−	0.883950i	$-0.345126\pi$
$349$	6097.13	0.935162	0.467581	+	0.883950i	$0.345126\pi$
$350$	−795.404	−0.121475
$351$	2192.14	0.333356
$352$	0	0
$353$	1745.44	0.263174	0.131587	−	0.991305i	$-0.457993\pi$
$353$	1745.44	0.263174	0.131587	+	0.991305i	$0.457993\pi$
$354$	−892.798	−0.134044
$355$	−2777.36	−0.415231
$356$	14318.3	2.13165
$357$	81.7270	0.0121161
$358$	−4218.96	−0.622847
$359$	4390.14	0.645412	0.322706	−	0.946499i	$-0.395407\pi$
$359$	4390.14	0.645412	0.322706	+	0.946499i	$0.395407\pi$
$360$	−1478.98	−0.216525
$361$	−6153.13	−0.897089
$362$	2751.36	0.399471
$363$	0	0
$364$	264.088	0.0380274
$365$	−2995.34	−0.429544
$366$	15452.5	2.20688
$367$	8713.43	1.23934	0.619669	−	0.784863i	$-0.287267\pi$
$367$	8713.43	1.23934	0.619669	+	0.784863i	$0.287267\pi$
$368$	−35.4149	−0.00501666
$369$	−7573.67	−1.06848
$370$	−8598.01	−1.20808
$371$	557.288	0.0779863
$372$	−23073.8	−3.21592
$373$	−10129.8	−1.40616	−0.703081	−	0.711109i	$-0.748193\pi$
$373$	−10129.8	−1.40616	−0.703081	+	0.711109i	$0.748193\pi$
$374$	0	0
$375$	11202.0	1.54259
$376$	−431.058	−0.0591226
$377$	−1049.51	−0.143375
$378$	−1539.20	−0.209439
$379$	7551.08	1.02341	0.511705	−	0.859161i	$-0.329014\pi$
$379$	7551.08	1.02341	0.511705	+	0.859161i	$0.329014\pi$
$380$	−1510.66	−0.203935
$381$	4197.27	0.564390
$382$	168.793	0.0226079
$383$	9046.18	1.20689	0.603444	−	0.797405i	$-0.293795\pi$
$383$	9046.18	1.20689	0.603444	+	0.797405i	$0.293795\pi$
$384$	−5592.00	−0.743139
$385$	0	0
$386$	9865.02	1.30082
$387$	12052.9	1.58316
$388$	2503.23	0.327532
$389$	−8470.15	−1.10399	−0.551997	−	0.833846i	$-0.686134\pi$
$389$	−8470.15	−1.10399	−0.551997	+	0.833846i	$0.686134\pi$
$390$	−2850.34	−0.370084
$391$	2.92045	0.000377733 0
$392$	1742.46	0.224508
$393$	17240.3	2.21287
$394$	−16207.7	−2.07242
$395$	800.111	0.101919
$396$	0	0
$397$	−7496.00	−0.947641	−0.473820	−	0.880621i	$-0.657125\pi$
$397$	−7496.00	−0.947641	−0.473820	+	0.880621i	$0.657125\pi$
$398$	−13011.9	−1.63876
$399$	−501.227	−0.0628890
$400$	4577.69	0.572212
$401$	11070.5	1.37864	0.689318	−	0.724459i	$-0.257910\pi$
$401$	11070.5	1.37864	0.689318	+	0.724459i	$0.257910\pi$
$402$	30382.3	3.76948
$403$	−3782.37	−0.467526
$404$	−10006.6	−1.23229
$405$	1154.38	0.141634
$406$	736.907	0.0900790
$407$	0	0
$408$	191.561	0.0232443
$409$	−7152.52	−0.864717	−0.432359	−	0.901702i	$-0.642319\pi$
$409$	−7152.52	−0.864717	−0.432359	+	0.901702i	$0.642319\pi$
$410$	4147.95	0.499641
$411$	−7831.22	−0.939868
$412$	1026.84	0.122788
$413$	55.0780	0.00656226
$414$	−130.581	−0.0155017
$415$	−3403.38	−0.402567
$416$	−3371.52	−0.397362
$417$	21001.3	2.46628
$418$	0	0
$419$	−6604.83	−0.770088	−0.385044	−	0.922898i	$-0.625814\pi$
$419$	−6604.83	−0.770088	−0.385044	+	0.922898i	$0.625814\pi$
$420$	1072.70	0.124625
$421$	−16638.4	−1.92614	−0.963070	−	0.269250i	$-0.913224\pi$
$421$	−16638.4	−1.92614	−0.963070	+	0.269250i	$0.913224\pi$
$422$	1850.08	0.213414
$423$	3902.55	0.448578
$424$	1306.23	0.149614
$425$	−377.495	−0.0430851
$426$	−16084.5	−1.82933
$427$	−953.290	−0.108040
$428$	−3774.35	−0.426262
$429$	0	0
$430$	−6601.14	−0.740314
$431$	−9392.11	−1.04966	−0.524829	−	0.851208i	$-0.675871\pi$
$431$	−9392.11	−1.04966	−0.524829	+	0.851208i	$0.675871\pi$
$432$	8858.39	0.986573
$433$	−2478.75	−0.275107	−0.137554	−	0.990494i	$-0.543924\pi$
$433$	−2478.75	−0.275107	−0.137554	+	0.990494i	$0.543924\pi$
$434$	2655.77	0.293735
$435$	−4263.01	−0.469875
$436$	−7255.18	−0.796927
$437$	−17.9109	−0.00196063
$438$	−17346.9	−1.89239
$439$	−14103.0	−1.53326	−0.766630	−	0.642089i	$-0.778068\pi$
$439$	−14103.0	−1.53326	−0.766630	+	0.642089i	$0.778068\pi$
$440$	0	0
$441$	−15775.2	−1.70340
$442$	233.839	0.0251642
$443$	16203.3	1.73779	0.868896	−	0.494995i	$-0.164830\pi$
$443$	16203.3	1.73779	0.868896	+	0.494995i	$0.164830\pi$
$444$	−26688.7	−2.85268
$445$	9533.78	1.01561
$446$	17316.3	1.83846
$447$	23024.5	2.43629
$448$	1443.43	0.152223
$449$	−10516.6	−1.10537	−0.552685	−	0.833390i	$-0.686397\pi$
$449$	−10516.6	−1.10537	−0.552685	+	0.833390i	$0.686397\pi$
$450$	16878.8	1.76816
$451$	0	0
$452$	20824.3	2.16702
$453$	29341.5	3.04323
$454$	−3903.26	−0.403500
$455$	175.842	0.0181178
$456$	−1174.83	−0.120650
$457$	9539.65	0.976468	0.488234	−	0.872713i	$-0.337641\pi$
$457$	9539.65	0.976468	0.488234	+	0.872713i	$0.337641\pi$
$458$	25254.4	2.57655
$459$	−730.497	−0.0742847
$460$	38.3321	0.00388531
$461$	10538.3	1.06468	0.532339	−	0.846531i	$-0.321313\pi$
$461$	10538.3	1.06468	0.532339	+	0.846531i	$0.321313\pi$
$462$	0	0
$463$	−2071.48	−0.207926	−0.103963	−	0.994581i	$-0.533152\pi$
$463$	−2071.48	−0.207926	−0.103963	+	0.994581i	$0.533152\pi$
$464$	−4241.03	−0.424321
$465$	−15363.6	−1.53220
$466$	1867.84	0.185678
$467$	−13278.8	−1.31578	−0.657892	−	0.753112i	$-0.728552\pi$
$467$	−13278.8	−1.31578	−0.657892	+	0.753112i	$0.728552\pi$
$468$	−5604.06	−0.553520
$469$	−1874.33	−0.184538
$470$	−2137.35	−0.209763
$471$	26041.6	2.54763
$472$	129.098	0.0125894
$473$	0	0
$474$	4633.66	0.449011
$475$	2315.15	0.223635
$476$	−88.0031	−0.00847399
$477$	−11825.9	−1.13516
$478$	−23668.5	−2.26480
$479$	2165.10	0.206526	0.103263	−	0.994654i	$-0.467072\pi$
$479$	2165.10	0.206526	0.103263	+	0.994654i	$0.467072\pi$
$480$	−13694.8	−1.30225
$481$	−4374.93	−0.414718
$482$	−12099.9	−1.14343
$483$	12.7183	0.00119815
$484$	0	0
$485$	1666.77	0.156049
$486$	−12219.4	−1.14050
$487$	8970.67	0.834702	0.417351	−	0.908745i	$-0.362959\pi$
$487$	8970.67	0.834702	0.417351	+	0.908745i	$0.362959\pi$
$488$	−2234.42	−0.207270
$489$	−30917.8	−2.85920
$490$	8639.77	0.796540
$491$	−15379.8	−1.41361	−0.706803	−	0.707410i	$-0.749863\pi$
$491$	−15379.8	−1.41361	−0.706803	+	0.707410i	$0.749863\pi$
$492$	12875.5	1.17982
$493$	349.732	0.0319496
$494$	−1434.12	−0.130615
$495$	0	0
$496$	−15284.4	−1.38365
$497$	992.276	0.0895566
$498$	−19709.9	−1.77354
$499$	6023.52	0.540380	0.270190	−	0.962807i	$-0.412913\pi$
$499$	6023.52	0.540380	0.270190	+	0.962807i	$0.412913\pi$
$500$	−12062.3	−1.07888
$501$	−30164.6	−2.68993
$502$	−17282.5	−1.53656
$503$	17429.9	1.54505	0.772526	−	0.634983i	$-0.218993\pi$
$503$	17429.9	1.54505	0.772526	+	0.634983i	$0.218993\pi$
$504$	528.400	0.0467000
$505$	−6662.84	−0.587114
$506$	0	0
$507$	−1450.34	−0.127045
$508$	−4519.59	−0.394733
$509$	−10440.7	−0.909190	−0.454595	−	0.890698i	$-0.650216\pi$
$509$	−10440.7	−0.909190	−0.454595	+	0.890698i	$0.650216\pi$
$510$	949.832	0.0824691
$511$	1070.15	0.0926435
$512$	−15789.7	−1.36291
$513$	4480.09	0.385577
$514$	17486.0	1.50053
$515$	683.714	0.0585011
$516$	−20490.3	−1.74813
$517$	0	0
$518$	3071.83	0.260557
$519$	6342.61	0.536435
$520$	412.157	0.0347583
$521$	−2331.22	−0.196032	−0.0980158	−	0.995185i	$-0.531250\pi$
$521$	−2331.22	−0.196032	−0.0980158	+	0.995185i	$0.531250\pi$
$522$	−15637.5	−1.31117
$523$	−17700.7	−1.47992	−0.739961	−	0.672650i	$-0.765156\pi$
$523$	−17700.7	−1.47992	−0.739961	+	0.672650i	$0.765156\pi$
$524$	−18564.2	−1.54767
$525$	−1643.96	−0.136663
$526$	25074.7	2.07853
$527$	1260.41	0.104183
$528$	0	0
$529$	−12166.5	−0.999963
$530$	6476.80	0.530819
$531$	−1168.78	−0.0955191
$532$	539.718	0.0439845
$533$	2110.60	0.171520
$534$	55212.8	4.47433
$535$	−2513.14	−0.203089
$536$	−4393.26	−0.354029
$537$	−8719.85	−0.700725
$538$	−14773.6	−1.18389
$539$	0	0
$540$	−9588.08	−0.764084
$541$	13245.5	1.05262	0.526312	−	0.850291i	$-0.323574\pi$
$541$	13245.5	1.05262	0.526312	+	0.850291i	$0.323574\pi$
$542$	−32276.5	−2.55792
$543$	5686.58	0.449419
$544$	1123.51	0.0885477
$545$	−4830.83	−0.379688
$546$	1018.35	0.0798193
$547$	9311.56	0.727850	0.363925	−	0.931428i	$-0.381437\pi$
$547$	9311.56	0.727850	0.363925	+	0.931428i	$0.381437\pi$
$548$	8432.61	0.657342
$549$	20229.2	1.57261
$550$	0	0
$551$	−2144.89	−0.165835
$552$	29.8106	0.00229859
$553$	−285.858	−0.0219817
$554$	−10960.9	−0.840584
$555$	−17770.6	−1.35913
$556$	−22614.0	−1.72491
$557$	8674.13	0.659847	0.329923	−	0.944008i	$-0.392977\pi$
$557$	8674.13	0.659847	0.329923	+	0.944008i	$0.392977\pi$
$558$	−56356.5	−4.27556
$559$	−3358.86	−0.254141
$560$	710.572	0.0536199
$561$	0	0
$562$	4576.71	0.343517
$563$	3724.96	0.278843	0.139421	−	0.990233i	$-0.455476\pi$
$563$	3724.96	0.278843	0.139421	+	0.990233i	$0.455476\pi$
$564$	−6634.45	−0.495321
$565$	13865.8	1.03246
$566$	−16436.0	−1.22059
$567$	−412.429	−0.0305474
$568$	2325.80	0.171811
$569$	12865.5	0.947893	0.473947	−	0.880554i	$-0.342829\pi$
$569$	12865.5	0.947893	0.473947	+	0.880554i	$0.342829\pi$
$570$	−5825.26	−0.428058
$571$	23526.9	1.72429	0.862146	−	0.506661i	$-0.169120\pi$
$571$	23526.9	1.72429	0.862146	+	0.506661i	$0.169120\pi$
$572$	0	0
$573$	348.865	0.0254347
$574$	−1481.95	−0.107762
$575$	−58.7456	−0.00426063
$576$	−30630.2	−2.21573
$577$	8584.19	0.619349	0.309675	−	0.950843i	$-0.399780\pi$
$577$	8584.19	0.619349	0.309675	+	0.950843i	$0.399780\pi$
$578$	20321.9	1.46242
$579$	20389.2	1.46347
$580$	4590.38	0.328629
$581$	1215.93	0.0868253
$582$	9652.70	0.687487
$583$	0	0
$584$	2508.34	0.177733
$585$	−3731.44	−0.263720
$586$	8927.00	0.629302
$587$	−1675.18	−0.117789	−0.0588945	−	0.998264i	$-0.518758\pi$
$587$	−1675.18	−0.117789	−0.0588945	+	0.998264i	$0.518758\pi$
$588$	26818.3	1.88090
$589$	−7730.04	−0.540765
$590$	640.117	0.0446664
$591$	−33498.5	−2.33154
$592$	−17678.9	−1.22737
$593$	27794.4	1.92476	0.962378	−	0.271716i	$-0.0875911\pi$
$593$	27794.4	1.92476	0.962378	+	0.271716i	$0.0875911\pi$
$594$	0	0
$595$	−58.5965	−0.00403735
$596$	−24792.6	−1.70394
$597$	−26893.3	−1.84367
$598$	36.3899	0.00248845
$599$	−5242.04	−0.357569	−0.178785	−	0.983888i	$-0.557217\pi$
$599$	−5242.04	−0.357569	−0.178785	+	0.983888i	$0.557217\pi$
$600$	−3853.29	−0.262183
$601$	19010.7	1.29029	0.645143	−	0.764062i	$-0.276798\pi$
$601$	19010.7	1.29029	0.645143	+	0.764062i	$0.276798\pi$
$602$	2358.41	0.159670
$603$	39774.0	2.68611
$604$	−31594.7	−2.12843
$605$	0	0
$606$	−38586.3	−2.58657
$607$	27139.4	1.81475	0.907375	−	0.420323i	$-0.138083\pi$
$607$	27139.4	1.81475	0.907375	+	0.420323i	$0.138083\pi$
$608$	−6890.40	−0.459609
$609$	1523.06	0.101342
$610$	−11079.1	−0.735379
$611$	−1087.55	−0.0720090
$612$	1867.46	0.123346
$613$	14792.4	0.974648	0.487324	−	0.873221i	$-0.337973\pi$
$613$	14792.4	0.974648	0.487324	+	0.873221i	$0.337973\pi$
$614$	17996.2	1.18285
$615$	8573.08	0.562114
$616$	0	0
$617$	−603.355	−0.0393682	−0.0196841	−	0.999806i	$-0.506266\pi$
$617$	−603.355	−0.0393682	−0.0196841	+	0.999806i	$0.506266\pi$
$618$	3959.58	0.257731
$619$	12639.2	0.820697	0.410348	−	0.911929i	$-0.365407\pi$
$619$	12639.2	0.820697	0.410348	+	0.911929i	$0.365407\pi$
$620$	16543.5	1.07161
$621$	−113.680	−0.00734591
$622$	−8025.47	−0.517351
$623$	−3406.16	−0.219045
$624$	−5860.79	−0.375992
$625$	2860.91	0.183098
$626$	16571.6	1.05804
$627$	0	0
$628$	−28041.4	−1.78180
$629$	1457.87	0.0924154
$630$	2620.01	0.165688
$631$	−2187.41	−0.138002	−0.0690012	−	0.997617i	$-0.521981\pi$
$631$	−2187.41	−0.138002	−0.0690012	+	0.997617i	$0.521981\pi$
$632$	−670.024	−0.0421711
$633$	3823.80	0.240098
$634$	−29622.0	−1.85558
$635$	−3009.35	−0.188067
$636$	20104.3	1.25344
$637$	4396.18	0.273442
$638$	0	0
$639$	−21056.5	−1.30357
$640$	4009.34	0.247630
$641$	29850.8	1.83937	0.919684	−	0.392659i	$-0.128445\pi$
$641$	29850.8	1.83937	0.919684	+	0.392659i	$0.128445\pi$
$642$	−14554.3	−0.894722
$643$	21406.6	1.31290	0.656448	−	0.754371i	$-0.272058\pi$
$643$	21406.6	1.31290	0.656448	+	0.754371i	$0.272058\pi$
$644$	−13.6950	−0.000837980 0
$645$	−13643.4	−0.832880
$646$	477.897	0.0291062
$647$	11212.9	0.681335	0.340667	−	0.940184i	$-0.389347\pi$
$647$	11212.9	0.681335	0.340667	+	0.940184i	$0.389347\pi$
$648$	−966.695	−0.0586039
$649$	0	0
$650$	−4703.72	−0.283839
$651$	5489.00	0.330462
$652$	33292.1	1.99972
$653$	25749.4	1.54311	0.771555	−	0.636162i	$-0.219479\pi$
$653$	25749.4	1.54311	0.771555	+	0.636162i	$0.219479\pi$
$654$	−27976.7	−1.67274
$655$	−12360.9	−0.737374
$656$	8528.89	0.507617
$657$	−22709.1	−1.34850
$658$	763.616	0.0452414
$659$	8761.33	0.517895	0.258948	−	0.965891i	$-0.416624\pi$
$659$	8761.33	0.517895	0.258948	+	0.965891i	$0.416624\pi$
$660$	0	0
$661$	5810.78	0.341926	0.170963	−	0.985277i	$-0.445312\pi$
$661$	5810.78	0.341926	0.170963	+	0.985277i	$0.445312\pi$
$662$	−24219.9	−1.42196
$663$	483.303	0.0283106
$664$	2850.04	0.166571
$665$	359.369	0.0209560
$666$	−65185.5	−3.79262
$667$	54.4252	0.00315945
$668$	32481.0	1.88133
$669$	35789.8	2.06833
$670$	−21783.5	−1.25607
$671$	0	0
$672$	4892.78	0.280868
$673$	−31867.0	−1.82523	−0.912616	−	0.408817i	$-0.865941\pi$
$673$	−31867.0	−1.82523	−0.912616	+	0.408817i	$0.865941\pi$
$674$	28426.6	1.62456
$675$	14694.1	0.837892
$676$	1561.72	0.0888552
$677$	14491.7	0.822689	0.411345	−	0.911480i	$-0.365059\pi$
$677$	14491.7	0.822689	0.411345	+	0.911480i	$0.365059\pi$
$678$	80300.6	4.54856
$679$	−595.490	−0.0336565
$680$	−137.345	−0.00774549
$681$	−8067.34	−0.453952
$682$	0	0
$683$	17072.9	0.956478	0.478239	−	0.878230i	$-0.341275\pi$
$683$	17072.9	0.956478	0.478239	+	0.878230i	$0.341275\pi$
$684$	−11453.0	−0.640230
$685$	5614.82	0.313184
$686$	−6217.61	−0.346049
$687$	52196.4	2.89871
$688$	−13573.0	−0.752133
$689$	3295.59	0.182224
$690$	147.812	0.00815525
$691$	−9887.90	−0.544361	−0.272181	−	0.962246i	$-0.587745\pi$
$691$	−9887.90	−0.544361	−0.272181	+	0.962246i	$0.587745\pi$
$692$	−6829.68	−0.375182
$693$	0	0
$694$	−6806.86	−0.372312
$695$	−15057.5	−0.821816
$696$	3569.90	0.194421
$697$	−703.325	−0.0382214
$698$	−25316.6	−1.37285
$699$	3860.49	0.208894
$700$	1770.20	0.0955821
$701$	−3055.12	−0.164608	−0.0823041	−	0.996607i	$-0.526228\pi$
$701$	−3055.12	−0.164608	−0.0823041	+	0.996607i	$0.526228\pi$
$702$	−9102.27	−0.489377
$703$	−8941.06	−0.479685
$704$	0	0
$705$	−4417.52	−0.235991
$706$	−7247.45	−0.386347
$707$	2380.45	0.126628
$708$	1986.96	0.105472
$709$	−10654.2	−0.564354	−0.282177	−	0.959362i	$-0.591057\pi$
$709$	−10654.2	−0.564354	−0.282177	+	0.959362i	$0.591057\pi$
$710$	11532.2	0.609573
$711$	6066.02	0.319963
$712$	−7983.72	−0.420228
$713$	196.145	0.0103025
$714$	−339.349	−0.0177868
$715$	0	0
$716$	9389.47	0.490085
$717$	−48918.6	−2.54798
$718$	−18228.8	−0.947485
$719$	−2879.46	−0.149355	−0.0746773	−	0.997208i	$-0.523793\pi$
$719$	−2879.46	−0.149355	−0.0746773	+	0.997208i	$0.523793\pi$
$720$	−15078.6	−0.780482
$721$	−244.272	−0.0126174
$722$	25549.2	1.31696
$723$	−25008.3	−1.28640
$724$	−6123.27	−0.314323
$725$	−7034.95	−0.360374
$726$	0	0
$727$	13697.2	0.698761	0.349381	−	0.936981i	$-0.386392\pi$
$727$	13697.2	0.698761	0.349381	+	0.936981i	$0.386392\pi$
$728$	−147.253	−0.00749662
$729$	−30320.8	−1.54046
$730$	12437.3	0.630584
$731$	1119.29	0.0566324
$732$	−34390.2	−1.73647
$733$	15736.7	0.792971	0.396485	−	0.918041i	$-0.370230\pi$
$733$	15736.7	0.792971	0.396485	+	0.918041i	$0.370230\pi$
$734$	−36180.1	−1.81939
$735$	17856.9	0.896137
$736$	174.840	0.00875636
$737$	0	0
$738$	31447.6	1.56856
$739$	8499.92	0.423105	0.211552	−	0.977367i	$-0.432148\pi$
$739$	8499.92	0.423105	0.211552	+	0.977367i	$0.432148\pi$
$740$	19135.2	0.950573
$741$	−2964.07	−0.146947
$742$	−2313.98	−0.114486
$743$	−1692.91	−0.0835891	−0.0417945	−	0.999126i	$-0.513307\pi$
$743$	−1692.91	−0.0835891	−0.0417945	+	0.999126i	$0.513307\pi$
$744$	12865.7	0.633978
$745$	−16508.1	−0.811824
$746$	42061.0	2.06429
$747$	−25802.6	−1.26381
$748$	0	0
$749$	897.876	0.0438020
$750$	−46513.2	−2.26456
$751$	−4258.56	−0.206920	−0.103460	−	0.994634i	$-0.532991\pi$
$751$	−4258.56	−0.206920	−0.103460	+	0.994634i	$0.532991\pi$
$752$	−4394.75	−0.213112
$753$	−35719.8	−1.72869
$754$	4357.79	0.210479
$755$	−21037.2	−1.01407
$756$	3425.56	0.164797
$757$	6981.59	0.335205	0.167603	−	0.985855i	$-0.446397\pi$
$757$	6981.59	0.335205	0.167603	+	0.985855i	$0.446397\pi$
$758$	−31353.7	−1.50240
$759$	0	0
$760$	842.327	0.0402032
$761$	−33895.2	−1.61459	−0.807293	−	0.590150i	$-0.799068\pi$
$761$	−33895.2	−1.61459	−0.807293	+	0.590150i	$0.799068\pi$
$762$	−17428.0	−0.828543
$763$	1725.92	0.0818908
$764$	−375.656	−0.0177889
$765$	1243.44	0.0587670
$766$	−37561.7	−1.77175
$767$	325.711	0.0153334
$768$	−21860.5	−1.02711
$769$	−33303.6	−1.56172	−0.780858	−	0.624709i	$-0.785218\pi$
$769$	−33303.6	−1.56172	−0.780858	+	0.624709i	$0.785218\pi$
$770$	0	0
$771$	36140.4	1.68815
$772$	−21955.0	−1.02355
$773$	−2414.53	−0.112347	−0.0561736	−	0.998421i	$-0.517890\pi$
$773$	−2414.53	−0.112347	−0.0561736	+	0.998421i	$0.517890\pi$
$774$	−50046.3	−2.32413
$775$	−25353.5	−1.17513
$776$	−1395.77	−0.0645687
$777$	6348.93	0.293136
$778$	35169.9	1.62070
$779$	4313.45	0.198389
$780$	6343.56	0.291200
$781$	0	0
$782$	−12.1264	−0.000554524 0
$783$	−13613.5	−0.621335
$784$	17764.8	0.809257
$785$	−18671.2	−0.848923
$786$	−71585.4	−3.24856
$787$	6065.42	0.274725	0.137363	−	0.990521i	$-0.456137\pi$
$787$	6065.42	0.274725	0.137363	+	0.990521i	$0.456137\pi$
$788$	36070.9	1.63068
$789$	51824.9	2.33842
$790$	−3322.24	−0.149620
$791$	−4953.87	−0.222679
$792$	0	0
$793$	−5637.40	−0.252446
$794$	31125.0	1.39117
$795$	13386.4	0.597190
$796$	28958.6	1.28946
$797$	18480.2	0.821331	0.410666	−	0.911786i	$-0.365296\pi$
$797$	18480.2	0.821331	0.410666	+	0.911786i	$0.365296\pi$
$798$	2081.20	0.0923231
$799$	362.408	0.0160464
$800$	−22599.6	−0.998771
$801$	72280.1	3.18838
$802$	−45967.0	−2.02388
$803$	0	0
$804$	−67617.0	−2.96601
$805$	−9.11877	−0.000399248 0
$806$	15705.2	0.686344
$807$	−30534.4	−1.33192
$808$	5579.55	0.242931
$809$	26647.9	1.15809	0.579043	−	0.815297i	$-0.303426\pi$
$809$	26647.9	1.15809	0.579043	+	0.815297i	$0.303426\pi$
$810$	−4793.25	−0.207923
$811$	−17176.2	−0.743698	−0.371849	−	0.928293i	$-0.621276\pi$
$811$	−17176.2	−0.743698	−0.371849	+	0.928293i	$0.621276\pi$
$812$	−1640.02	−0.0708784
$813$	−66709.8	−2.87776
$814$	0	0
$815$	22167.4	0.952748
$816$	1953.01	0.0837857
$817$	−6864.51	−0.293952
$818$	29698.8	1.26943
$819$	1333.14	0.0568787
$820$	−9231.44	−0.393141
$821$	34685.8	1.47448	0.737238	−	0.675633i	$-0.236130\pi$
$821$	34685.8	1.47448	0.737238	+	0.675633i	$0.236130\pi$
$822$	32517.0	1.37976
$823$	7272.88	0.308040	0.154020	−	0.988068i	$-0.450778\pi$
$823$	7272.88	0.308040	0.154020	+	0.988068i	$0.450778\pi$
$824$	−572.552	−0.0242061
$825$	0	0
$826$	−228.696	−0.00963360
$827$	27211.7	1.14419	0.572094	−	0.820188i	$-0.306131\pi$
$827$	27211.7	1.14419	0.572094	+	0.820188i	$0.306131\pi$
$828$	290.614	0.0121975
$829$	30947.2	1.29655	0.648276	−	0.761406i	$-0.275490\pi$
$829$	30947.2	1.29655	0.648276	+	0.761406i	$0.275490\pi$
$830$	14131.6	0.590982
$831$	−22654.2	−0.945687
$832$	8535.92	0.355685
$833$	−1464.96	−0.0609336
$834$	−87201.9	−3.62057
$835$	21627.3	0.896341
$836$	0	0
$837$	−49062.1	−2.02609
$838$	27424.7	1.13051
$839$	8581.68	0.353126	0.176563	−	0.984289i	$-0.443502\pi$
$839$	8581.68	0.353126	0.176563	+	0.984289i	$0.443502\pi$
$840$	−598.126	−0.0245682
$841$	−17871.4	−0.732766
$842$	69086.2	2.82764
$843$	9459.24	0.386469
$844$	−4117.44	−0.167924
$845$	1039.86	0.0423342
$846$	−16204.3	−0.658527
$847$	0	0
$848$	13317.4	0.539293
$849$	−33970.3	−1.37321
$850$	1567.44	0.0632503
$851$	226.874	0.00913883
$852$	35796.7	1.43941
$853$	−13722.2	−0.550808	−0.275404	−	0.961329i	$-0.588812\pi$
$853$	−13722.2	−0.550808	−0.275404	+	0.961329i	$0.588812\pi$
$854$	3958.27	0.158606
$855$	−7625.95	−0.305032
$856$	2104.54	0.0840323
$857$	−14377.6	−0.573080	−0.286540	−	0.958068i	$-0.592505\pi$
$857$	−14377.6	−0.573080	−0.286540	+	0.958068i	$0.592505\pi$
$858$	0	0
$859$	−2556.76	−0.101555	−0.0507773	−	0.998710i	$-0.516170\pi$
$859$	−2556.76	−0.101555	−0.0507773	+	0.998710i	$0.516170\pi$
$860$	14691.1	0.582514
$861$	−3062.93	−0.121236
$862$	38998.1	1.54093
$863$	−9864.89	−0.389113	−0.194557	−	0.980891i	$-0.562327\pi$
$863$	−9864.89	−0.389113	−0.194557	+	0.980891i	$0.562327\pi$
$864$	−43733.0	−1.72202
$865$	−4547.52	−0.178752
$866$	10292.3	0.403866
$867$	42001.8	1.64528
$868$	−5910.52	−0.231125
$869$	0	0
$870$	17701.0	0.689791
$871$	−11084.1	−0.431194
$872$	4045.41	0.157104
$873$	12636.5	0.489899
$874$	74.3702	0.00287827
$875$	2869.47	0.110864
$876$	38606.1	1.48902
$877$	9690.40	0.373115	0.186557	−	0.982444i	$-0.440267\pi$
$877$	9690.40	0.373115	0.186557	+	0.982444i	$0.440267\pi$
$878$	58558.9	2.25088
$879$	18450.5	0.707987
$880$	0	0
$881$	−8946.66	−0.342135	−0.171067	−	0.985259i	$-0.554722\pi$
$881$	−8946.66	−0.342135	−0.171067	+	0.985259i	$0.554722\pi$
$882$	65502.1	2.50065
$883$	7357.38	0.280403	0.140201	−	0.990123i	$-0.455225\pi$
$883$	7357.38	0.280403	0.140201	+	0.990123i	$0.455225\pi$
$884$	−520.418	−0.0198004
$885$	1323.01	0.0502513
$886$	−67279.7	−2.55113
$887$	−7039.29	−0.266467	−0.133233	−	0.991085i	$-0.542536\pi$
$887$	−7039.29	−0.266467	−0.133233	+	0.991085i	$0.542536\pi$
$888$	14881.3	0.562369
$889$	1075.16	0.0405621
$890$	−39586.4	−1.49094
$891$	0	0
$892$	−38538.2	−1.44658
$893$	−2222.63	−0.0832894
$894$	−95602.8	−3.57655
$895$	6251.94	0.233497
$896$	−1432.43	−0.0534085
$897$	75.2115	0.00279960
$898$	43667.4	1.62272
$899$	23488.9	0.871412
$900$	−37564.5	−1.39128
$901$	−1098.20	−0.0406065
$902$	0	0
$903$	4874.41	0.179635
$904$	−11611.4	−0.427201
$905$	−4077.16	−0.149756
$906$	−121832.	−4.46756
$907$	34483.1	1.26240	0.631198	−	0.775622i	$-0.282563\pi$
$907$	34483.1	1.26240	0.631198	+	0.775622i	$0.282563\pi$
$908$	8686.85	0.317493
$909$	−50514.1	−1.84318
$910$	−730.135	−0.0265975
$911$	−10157.5	−0.369412	−0.184706	−	0.982794i	$-0.559133\pi$
$911$	−10157.5	−0.369412	−0.184706	+	0.982794i	$0.559133\pi$
$912$	−11977.7	−0.434892
$913$	0	0
$914$	−39610.7	−1.43349
$915$	−22898.6	−0.827327
$916$	−56204.7	−2.02735
$917$	4416.21	0.159036
$918$	3033.19	0.109052
$919$	27753.8	0.996208	0.498104	−	0.867117i	$-0.334030\pi$
$919$	27753.8	0.996208	0.498104	+	0.867117i	$0.334030\pi$
$920$	−21.3736	−0.000765941 0
$921$	37195.1	1.33075
$922$	−43757.3	−1.56298
$923$	5867.95	0.209259
$924$	0	0
$925$	−29325.5	−1.04240
$926$	8601.25	0.305243
$927$	5183.56	0.183657
$928$	20937.5	0.740634
$929$	17784.9	0.628097	0.314048	−	0.949407i	$-0.398315\pi$
$929$	17784.9	0.628097	0.314048	+	0.949407i	$0.398315\pi$
$930$	63793.2	2.24931
$931$	8984.48	0.316278
$932$	−4156.95	−0.146100
$933$	−16587.2	−0.582038
$934$	55136.6	1.93161
$935$	0	0
$936$	3124.76	0.109120
$937$	37947.5	1.32304	0.661522	−	0.749926i	$-0.269911\pi$
$937$	37947.5	1.32304	0.661522	+	0.749926i	$0.269911\pi$
$938$	7782.63	0.270908
$939$	34250.6	1.19034
$940$	4756.76	0.165051
$941$	8536.63	0.295735	0.147867	−	0.989007i	$-0.452759\pi$
$941$	8536.63	0.295735	0.147867	+	0.989007i	$0.452759\pi$
$942$	−108130.	−3.73999
$943$	−109.451	−0.00377966
$944$	1316.19	0.0453795
$945$	2280.89	0.0785159
$946$	0	0
$947$	39690.2	1.36194	0.680970	−	0.732311i	$-0.261558\pi$
$947$	39690.2	1.36194	0.680970	+	0.732311i	$0.261558\pi$
$948$	−10312.4	−0.353303
$949$	6328.49	0.216472
$950$	−9613.02	−0.328303
$951$	−61223.4	−2.08760
$952$	49.0695	0.00167054
$953$	−28989.3	−0.985367	−0.492683	−	0.870209i	$-0.663984\pi$
$953$	−28989.3	−0.985367	−0.492683	+	0.870209i	$0.663984\pi$
$954$	49103.6	1.66644
$955$	−250.129	−0.00847537
$956$	52675.3	1.78205
$957$	0	0
$958$	−8989.98	−0.303187
$959$	−2006.02	−0.0675472
$960$	34672.1	1.16567
$961$	54861.7	1.84155
$962$	18165.7	0.608820
$963$	−19053.3	−0.637574
$964$	26928.8	0.899707
$965$	−14618.7	−0.487659
$966$	−52.8093	−0.00175892
$967$	7443.57	0.247538	0.123769	−	0.992311i	$-0.460502\pi$
$967$	7443.57	0.247538	0.123769	+	0.992311i	$0.460502\pi$
$968$	0	0
$969$	987.729	0.0327455
$970$	−6920.78	−0.229085
$971$	4715.89	0.155860	0.0779300	−	0.996959i	$-0.475169\pi$
$971$	4715.89	0.155860	0.0779300	+	0.996959i	$0.475169\pi$
$972$	27194.7	0.897398
$973$	5379.62	0.177248
$974$	−37248.2	−1.22537
$975$	−9721.76	−0.319329
$976$	−22780.6	−0.747119
$977$	−39450.6	−1.29185	−0.645925	−	0.763401i	$-0.723528\pi$
$977$	−39450.6	−1.29185	−0.645925	+	0.763401i	$0.723528\pi$
$978$	128377.	4.19740
$979$	0	0
$980$	−19228.2	−0.626756
$981$	−36624.8	−1.19199
$982$	63860.3	2.07522
$983$	13652.1	0.442963	0.221482	−	0.975165i	$-0.428911\pi$
$983$	13652.1	0.442963	0.221482	+	0.975165i	$0.428911\pi$
$984$	−7179.22	−0.232586
$985$	24017.7	0.776920
$986$	−1452.16	−0.0469030
$987$	1578.26	0.0508982
$988$	3191.69	0.102774
$989$	174.183	0.00560030
$990$	0	0
$991$	−37372.6	−1.19796	−0.598981	−	0.800763i	$-0.704428\pi$
$991$	−37372.6	−1.19796	−0.598981	+	0.800763i	$0.704428\pi$
$992$	75457.7	2.41510
$993$	−50058.3	−1.59975
$994$	−4120.15	−0.131472
$995$	19281.9	0.614350
$996$	43865.2	1.39551
$997$	−10266.1	−0.326109	−0.163054	−	0.986617i	$-0.552135\pi$
$997$	−10266.1	−0.326109	−0.163054	+	0.986617i	$0.552135\pi$
$998$	−25011.0	−0.793295
$999$	−56748.4	−1.79724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1573.4.a.q.1.7		38
11.5	even	5		143.4.h.b.14.17	✓	76
11.9	even	5		143.4.h.b.92.17	yes	76
11.10	odd	2		1573.4.a.r.1.32		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.h.b.14.17	✓	76	11.5	even	5
143.4.h.b.92.17	yes	76	11.9	even	5
1573.4.a.q.1.7		38	1.1	even	1	trivial
1573.4.a.r.1.32		38	11.10	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:	\( N \)	\(=\)	\( 1573 = 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1573.a (trivial)

\(f(q)\)	\(=\)	\(q-4.15222 q^{2} -8.58190 q^{3} +9.24094 q^{4} +6.15304 q^{5} +35.6340 q^{6} -2.19831 q^{7} -5.15264 q^{8} +46.6491 q^{9} -25.5488 q^{10} -79.3048 q^{12} -13.0000 q^{13} +9.12788 q^{14} -52.8048 q^{15} -52.5326 q^{16} +4.33204 q^{17} -193.697 q^{18} -26.5681 q^{19} +56.8599 q^{20} +18.8657 q^{21} +0.674151 q^{23} +44.2195 q^{24} -87.1401 q^{25} +53.9789 q^{26} -168.627 q^{27} -20.3145 q^{28} +80.7315 q^{29} +219.257 q^{30} +290.951 q^{31} +259.348 q^{32} -17.9876 q^{34} -13.5263 q^{35} +431.081 q^{36} +336.533 q^{37} +110.317 q^{38} +111.565 q^{39} -31.7044 q^{40} -162.354 q^{41} -78.3346 q^{42} +258.374 q^{43} +287.034 q^{45} -2.79922 q^{46} +83.6576 q^{47} +450.830 q^{48} -338.167 q^{49} +361.825 q^{50} -37.1772 q^{51} -120.132 q^{52} -253.507 q^{53} +700.175 q^{54} +11.3271 q^{56} +228.005 q^{57} -335.215 q^{58} -25.0547 q^{59} -487.966 q^{60} +433.646 q^{61} -1208.09 q^{62} -102.549 q^{63} -656.610 q^{64} -79.9895 q^{65} +852.622 q^{67} +40.0321 q^{68} -5.78550 q^{69} +56.1642 q^{70} -451.381 q^{71} -240.366 q^{72} -486.807 q^{73} -1397.36 q^{74} +747.828 q^{75} -245.515 q^{76} -463.241 q^{78} +130.035 q^{79} -323.235 q^{80} +187.612 q^{81} +674.130 q^{82} -553.122 q^{83} +174.337 q^{84} +26.6552 q^{85} -1072.83 q^{86} -692.830 q^{87} +1549.44 q^{89} -1191.83 q^{90} +28.5781 q^{91} +6.22979 q^{92} -2496.92 q^{93} -347.365 q^{94} -163.475 q^{95} -2225.70 q^{96} +270.885 q^{97} +1404.15 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 3 q^{2} + 19 q^{3} + 181 q^{4} + 52 q^{5} - 104 q^{6} - 12 q^{7} - 57 q^{8} + 477 q^{9} + 30 q^{10} + 122 q^{12} - 494 q^{13} + 181 q^{14} + 264 q^{15} + 961 q^{16} + 33 q^{17} - 28 q^{18} - 107 q^{19}+ \cdots - 2043 q^{98}+O(q^{100}) \) 38 * q - 3 * q^2 + 19 * q^3 + 181 * q^4 + 52 * q^5 - 104 * q^6 - 12 * q^7 - 57 * q^8 + 477 * q^9 + 30 * q^10 + 122 * q^12 - 494 * q^13 + 181 * q^14 + 264 * q^15 + 961 * q^16 + 33 * q^17 - 28 * q^18 - 107 * q^19 + 595 * q^20 - 148 * q^21 + 840 * q^23 - 968 * q^24 + 1428 * q^25 + 39 * q^26 + 1108 * q^27 + 353 * q^28 + 162 * q^29 + 48 * q^30 + 62 * q^31 - 1204 * q^32 - 58 * q^34 + 352 * q^35 + 3357 * q^36 + 304 * q^37 + 963 * q^38 - 247 * q^39 + 571 * q^40 - 361 * q^41 + 587 * q^42 - 483 * q^43 + 1136 * q^45 + 1860 * q^46 + 2892 * q^47 + 945 * q^48 + 2270 * q^49 + 782 * q^50 + 682 * q^51 - 2353 * q^52 + 1622 * q^53 - 3273 * q^54 + 3269 * q^56 - 62 * q^57 + 1683 * q^58 + 3417 * q^59 + 4279 * q^60 + 1714 * q^61 - 508 * q^62 - 3730 * q^63 + 4823 * q^64 - 676 * q^65 + 4939 * q^67 + 3699 * q^68 + 1818 * q^69 + 592 * q^70 + 1544 * q^71 + 5570 * q^72 - 617 * q^73 + 1398 * q^74 + 5457 * q^75 - 3316 * q^76 + 1352 * q^78 + 3176 * q^79 + 5091 * q^80 + 5994 * q^81 + 1238 * q^82 - 1659 * q^83 - 5157 * q^84 - 1074 * q^85 + 1144 * q^86 + 906 * q^87 + 3727 * q^89 + 6753 * q^90 + 156 * q^91 + 11004 * q^92 + 6956 * q^93 + 9255 * q^94 - 1768 * q^95 - 4812 * q^96 + 5723 * q^97 - 2043 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more