LMFDB - Embedded newform 1573.4.a.q.1.8

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			1.8
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)

$f(q)$	$=$	\(q-3.98792 q^{2} -2.76589 q^{3} +7.90349 q^{4} -14.0062 q^{5} +11.0301 q^{6} +21.9303 q^{7} +0.384876 q^{8} -19.3499 q^{9} +55.8554 q^{10} -21.8602 q^{12} -13.0000 q^{13} -87.4562 q^{14} +38.7395 q^{15} -64.7628 q^{16} +72.0473 q^{17} +77.1656 q^{18} +87.9826 q^{19} -110.697 q^{20} -60.6568 q^{21} -7.78478 q^{23} -1.06452 q^{24} +71.1724 q^{25} +51.8429 q^{26} +128.199 q^{27} +173.326 q^{28} -59.3694 q^{29} -154.490 q^{30} +69.0151 q^{31} +255.190 q^{32} -287.319 q^{34} -307.159 q^{35} -152.931 q^{36} -146.741 q^{37} -350.868 q^{38} +35.9566 q^{39} -5.39063 q^{40} +465.434 q^{41} +241.894 q^{42} +156.756 q^{43} +271.017 q^{45} +31.0451 q^{46} +89.3598 q^{47} +179.127 q^{48} +137.937 q^{49} -283.830 q^{50} -199.275 q^{51} -102.745 q^{52} +237.991 q^{53} -511.245 q^{54} +8.44044 q^{56} -243.350 q^{57} +236.760 q^{58} +756.132 q^{59} +306.177 q^{60} -665.147 q^{61} -275.227 q^{62} -424.348 q^{63} -499.573 q^{64} +182.080 q^{65} -706.040 q^{67} +569.425 q^{68} +21.5318 q^{69} +1224.92 q^{70} -93.1411 q^{71} -7.44730 q^{72} -790.781 q^{73} +585.191 q^{74} -196.855 q^{75} +695.370 q^{76} -143.392 q^{78} -226.942 q^{79} +907.077 q^{80} +167.863 q^{81} -1856.11 q^{82} +911.939 q^{83} -479.400 q^{84} -1009.11 q^{85} -625.131 q^{86} +164.209 q^{87} -936.077 q^{89} -1080.79 q^{90} -285.094 q^{91} -61.5269 q^{92} -190.888 q^{93} -356.360 q^{94} -1232.30 q^{95} -705.826 q^{96} +1063.80 q^{97} -550.083 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.98792	−1.40994	−0.704971	−	0.709236i	$-0.749040\pi$
$2$	−3.98792	−1.40994	−0.704971	+	0.709236i	$0.749040\pi$
$3$	−2.76589	−0.532296	−0.266148	−	0.963932i	$-0.585751\pi$
$3$	−2.76589	−0.532296	−0.266148	+	0.963932i	$0.585751\pi$
$4$	7.90349	0.987936
$5$	−14.0062	−1.25275	−0.626374	−	0.779522i	$-0.715462\pi$
$5$	−14.0062	−1.25275	−0.626374	+	0.779522i	$0.715462\pi$
$6$	11.0301	0.750506
$7$	21.9303	1.18412	0.592062	−	0.805892i	$-0.298314\pi$
$7$	21.9303	1.18412	0.592062	+	0.805892i	$0.298314\pi$
$8$	0.384876	0.0170093
$9$	−19.3499	−0.716661
$10$	55.8554	1.76630
$11$	0	0
$11$	0	0
$12$	−21.8602	−0.525874
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−87.4562	−1.66955
$15$	38.7395	0.666833
$16$	−64.7628	−1.01192
$17$	72.0473	1.02789	0.513943	−	0.857825i	$-0.328184\pi$
$17$	72.0473	1.02789	0.513943	+	0.857825i	$0.328184\pi$
$18$	77.1656	1.01045
$19$	87.9826	1.06235	0.531174	−	0.847263i	$-0.321751\pi$
$19$	87.9826	1.06235	0.531174	+	0.847263i	$0.321751\pi$
$20$	−110.697	−1.23764
$21$	−60.6568	−0.630304
$22$	0	0
$23$	−7.78478	−0.0705756	−0.0352878	−	0.999377i	$-0.511235\pi$
$23$	−7.78478	−0.0705756	−0.0352878	+	0.999377i	$0.511235\pi$
$24$	−1.06452	−0.00905397
$25$	71.1724	0.569379
$26$	51.8429	0.391048
$27$	128.199	0.913771
$28$	173.326	1.16984
$29$	−59.3694	−0.380159	−0.190080	−	0.981769i	$-0.560875\pi$
$29$	−59.3694	−0.380159	−0.190080	+	0.981769i	$0.560875\pi$
$30$	−154.490	−0.940195
$31$	69.0151	0.399854	0.199927	−	0.979811i	$-0.435929\pi$
$31$	69.0151	0.399854	0.199927	+	0.979811i	$0.435929\pi$
$32$	255.190	1.40974
$33$	0	0
$34$	−287.319	−1.44926
$35$	−307.159	−1.48341
$36$	−152.931	−0.708016
$37$	−146.741	−0.652001	−0.326001	−	0.945370i	$-0.605701\pi$
$37$	−146.741	−0.652001	−0.326001	+	0.945370i	$0.605701\pi$
$38$	−350.868	−1.49785
$39$	35.9566	0.147632
$40$	−5.39063	−0.0213083
$41$	465.434	1.77289	0.886446	−	0.462832i	$-0.153167\pi$
$41$	465.434	1.77289	0.886446	+	0.462832i	$0.153167\pi$
$42$	241.894	0.888692
$43$	156.756	0.555933	0.277966	−	0.960591i	$-0.410340\pi$
$43$	156.756	0.555933	0.277966	+	0.960591i	$0.410340\pi$
$44$	0	0
$45$	271.017	0.897796
$46$	31.0451	0.0995075
$47$	89.3598	0.277329	0.138665	−	0.990339i	$-0.455719\pi$
$47$	89.3598	0.277329	0.138665	+	0.990339i	$0.455719\pi$
$48$	179.127	0.538640
$49$	137.937	0.402150
$50$	−283.830	−0.802791
$51$	−199.275	−0.547139
$52$	−102.745	−0.274004
$53$	237.991	0.616803	0.308402	−	0.951256i	$-0.400206\pi$
$53$	237.991	0.616803	0.308402	+	0.951256i	$0.400206\pi$
$54$	−511.245	−1.28836
$55$	0	0
$56$	8.44044	0.0201411
$57$	−243.350	−0.565483
$58$	236.760	0.536002
$59$	756.132	1.66847	0.834237	−	0.551406i	$-0.185908\pi$
$59$	756.132	1.66847	0.834237	+	0.551406i	$0.185908\pi$
$60$	306.177	0.658788
$61$	−665.147	−1.39612	−0.698061	−	0.716039i	$-0.745953\pi$
$61$	−665.147	−1.39612	−0.698061	+	0.716039i	$0.745953\pi$
$62$	−275.227	−0.563771
$63$	−424.348	−0.848616
$64$	−499.573	−0.975729
$65$	182.080	0.347450
$66$	0	0
$67$	−706.040	−1.28741	−0.643705	−	0.765274i	$-0.722604\pi$
$67$	−706.040	−1.28741	−0.643705	+	0.765274i	$0.722604\pi$
$68$	569.425	1.01548
$69$	21.5318	0.0375671
$70$	1224.92	2.09152
$71$	−93.1411	−0.155688	−0.0778438	−	0.996966i	$-0.524804\pi$
$71$	−93.1411	−0.155688	−0.0778438	+	0.996966i	$0.524804\pi$
$72$	−7.44730	−0.0121899
$73$	−790.781	−1.26786	−0.633931	−	0.773390i	$-0.718560\pi$
$73$	−790.781	−1.26786	−0.633931	+	0.773390i	$0.718560\pi$
$74$	585.191	0.919284
$75$	−196.855	−0.303078
$76$	695.370	1.04953
$77$	0	0
$78$	−143.392	−0.208153
$79$	−226.942	−0.323203	−0.161601	−	0.986856i	$-0.551666\pi$
$79$	−226.942	−0.323203	−0.161601	+	0.986856i	$0.551666\pi$
$80$	907.077	1.26768
$81$	167.863	0.230265
$82$	−1856.11	−2.49968
$83$	911.939	1.20600	0.603002	−	0.797740i	$-0.293971\pi$
$83$	911.939	1.20600	0.603002	+	0.797740i	$0.293971\pi$
$84$	−479.400	−0.622700
$85$	−1009.11	−1.28768
$86$	−625.131	−0.783833
$87$	164.209	0.202357
$88$	0	0
$89$	−936.077	−1.11488	−0.557438	−	0.830219i	$-0.688215\pi$
$89$	−936.077	−1.11488	−0.557438	+	0.830219i	$0.688215\pi$
$90$	−1080.79	−1.26584
$91$	−285.094	−0.328417
$92$	−61.5269	−0.0697242
$93$	−190.888	−0.212841
$94$	−356.360	−0.391018
$95$	−1232.30	−1.33085
$96$	−705.826	−0.750397
$97$	1063.80	1.11353	0.556765	−	0.830670i	$-0.312042\pi$
$97$	1063.80	1.11353	0.556765	+	0.830670i	$0.312042\pi$
$98$	−550.083	−0.567008
$99$	0	0
$100$	562.510	0.562510
$101$	60.8741	0.0599722	0.0299861	−	0.999550i	$-0.490454\pi$
$101$	60.8741	0.0599722	0.0299861	+	0.999550i	$0.490454\pi$
$102$	794.692	0.771434
$103$	−2012.17	−1.92491	−0.962453	−	0.271449i	$-0.912497\pi$
$103$	−2012.17	−1.92491	−0.962453	+	0.271449i	$0.912497\pi$
$104$	−5.00339	−0.00471753
$105$	849.568	0.789613
$106$	−949.088	−0.869656
$107$	−1717.68	−1.55191	−0.775955	−	0.630788i	$-0.782732\pi$
$107$	−1717.68	−1.55191	−0.775955	+	0.630788i	$0.782732\pi$
$108$	1013.22	0.902748
$109$	−188.442	−0.165592	−0.0827959	−	0.996567i	$-0.526385\pi$
$109$	−188.442	−0.165592	−0.0827959	+	0.996567i	$0.526385\pi$
$110$	0	0
$111$	405.869	0.347058
$112$	−1420.27	−1.19824
$113$	−963.767	−0.802332	−0.401166	−	0.916005i	$-0.631395\pi$
$113$	−963.767	−0.802332	−0.401166	+	0.916005i	$0.631395\pi$
$114$	970.461	0.797298
$115$	109.035	0.0884135
$116$	−469.225	−0.375573
$117$	251.548	0.198766
$118$	−3015.39	−2.35245
$119$	1580.02	1.21714
$120$	14.9099	0.0113423
$121$	0	0
$122$	2652.55	1.96845
$123$	−1287.34	−0.943703
$124$	545.460	0.395030
$125$	753.918	0.539460
$126$	1692.26	1.19650
$127$	−401.758	−0.280710	−0.140355	−	0.990101i	$-0.544824\pi$
$127$	−401.758	−0.280710	−0.140355	+	0.990101i	$0.544824\pi$
$128$	−49.2606	−0.0340161
$129$	−433.571	−0.295921
$130$	−726.120	−0.489884
$131$	1887.43	1.25882	0.629411	−	0.777072i	$-0.283296\pi$
$131$	1887.43	1.25882	0.629411	+	0.777072i	$0.283296\pi$
$132$	0	0
$133$	1929.48	1.25795
$134$	2815.63	1.81517
$135$	−1795.57	−1.14473
$136$	27.7293	0.0174836
$137$	1099.53	0.685687	0.342844	−	0.939392i	$-0.388610\pi$
$137$	1099.53	0.685687	0.342844	+	0.939392i	$0.388610\pi$
$138$	−85.8672	−0.0529674
$139$	3249.44	1.98284	0.991418	−	0.130730i	$-0.0417320\pi$
$139$	3249.44	1.98284	0.991418	+	0.130730i	$0.0417320\pi$
$140$	−2427.63	−1.46551
$141$	−247.159	−0.147621
$142$	371.439	0.219510
$143$	0	0
$144$	1253.15	0.725203
$145$	831.536	0.476244
$146$	3153.57	1.78761
$147$	−381.520	−0.214063
$148$	−1159.77	−0.644136
$149$	−1025.14	−0.563642	−0.281821	−	0.959467i	$-0.590939\pi$
$149$	−1025.14	−0.563642	−0.281821	+	0.959467i	$0.590939\pi$
$150$	785.041	0.427322
$151$	−2449.66	−1.32020	−0.660102	−	0.751176i	$-0.729487\pi$
$151$	−2449.66	−1.32020	−0.660102	+	0.751176i	$0.729487\pi$
$152$	33.8624	0.0180698
$153$	−1394.11	−0.736645
$154$	0	0
$155$	−966.636	−0.500917
$156$	284.182	0.145851
$157$	−2794.24	−1.42041	−0.710207	−	0.703993i	$-0.751399\pi$
$157$	−2794.24	−1.42041	−0.710207	+	0.703993i	$0.751399\pi$
$158$	905.027	0.455697
$159$	−658.256	−0.328322
$160$	−3574.22	−1.76605
$161$	−170.722	−0.0835703
$162$	−669.424	−0.324660
$163$	340.459	0.163600	0.0818001	−	0.996649i	$-0.473933\pi$
$163$	340.459	0.163600	0.0818001	+	0.996649i	$0.473933\pi$
$164$	3678.55	1.75150
$165$	0	0
$166$	−3636.74	−1.70040
$167$	−1918.62	−0.889024	−0.444512	−	0.895773i	$-0.646623\pi$
$167$	−1918.62	−0.889024	−0.444512	+	0.895773i	$0.646623\pi$
$168$	−23.3453	−0.0107210
$169$	169.000	0.0769231
$170$	4024.23	1.81556
$171$	−1702.45	−0.761343
$172$	1238.92	0.549226
$173$	14.9325	0.00656242	0.00328121	−	0.999995i	$-0.498956\pi$
$173$	14.9325	0.00656242	0.00328121	+	0.999995i	$0.498956\pi$
$174$	−654.852	−0.285312
$175$	1560.83	0.674215
$176$	0	0
$177$	−2091.38	−0.888122
$178$	3733.00	1.57191
$179$	−3290.52	−1.37399	−0.686997	−	0.726661i	$-0.741071\pi$
$179$	−3290.52	−1.37399	−0.686997	+	0.726661i	$0.741071\pi$
$180$	2141.98	0.886966
$181$	−491.416	−0.201805	−0.100902	−	0.994896i	$-0.532173\pi$
$181$	−491.416	−0.201805	−0.100902	+	0.994896i	$0.532173\pi$
$182$	1136.93	0.463049
$183$	1839.72	0.743149
$184$	−2.99617	−0.00120044
$185$	2055.28	0.816794
$186$	761.246	0.300093
$187$	0	0
$188$	706.254	0.273983
$189$	2811.43	1.08202
$190$	4914.30	1.87643
$191$	−3298.60	−1.24962	−0.624811	−	0.780776i	$-0.714824\pi$
$191$	−3298.60	−1.24962	−0.624811	+	0.780776i	$0.714824\pi$
$192$	1381.76	0.519376
$193$	−2058.66	−0.767800	−0.383900	−	0.923375i	$-0.625419\pi$
$193$	−2058.66	−0.767800	−0.383900	+	0.923375i	$0.625419\pi$
$194$	−4242.34	−1.57001
$195$	−503.613	−0.184946
$196$	1090.19	0.397299
$197$	3612.04	1.30633	0.653165	−	0.757215i	$-0.273441\pi$
$197$	3612.04	1.30633	0.653165	+	0.757215i	$0.273441\pi$
$198$	0	0
$199$	4744.84	1.69021	0.845107	−	0.534598i	$-0.179537\pi$
$199$	4744.84	1.69021	0.845107	+	0.534598i	$0.179537\pi$
$200$	27.3925	0.00968472
$201$	1952.83	0.685283
$202$	−242.761	−0.0845574
$203$	−1301.99	−0.450156
$204$	−1574.97	−0.540538
$205$	−6518.94	−2.22099
$206$	8024.38	2.71400
$207$	150.634	0.0505788
$208$	841.916	0.280656
$209$	0	0
$210$	−3388.01	−1.11331
$211$	2045.95	0.667530	0.333765	−	0.942656i	$-0.391681\pi$
$211$	2045.95	0.667530	0.333765	+	0.942656i	$0.391681\pi$
$212$	1880.96	0.609362
$213$	257.618	0.0828718
$214$	6849.97	2.18810
$215$	−2195.55	−0.696444
$216$	49.3406	0.0155426
$217$	1513.52	0.473477
$218$	751.492	0.233475
$219$	2187.21	0.674878
$220$	0	0
$221$	−936.615	−0.285084
$222$	−1618.57	−0.489331
$223$	−1193.57	−0.358419	−0.179210	−	0.983811i	$-0.557354\pi$
$223$	−1193.57	−0.358419	−0.179210	+	0.983811i	$0.557354\pi$
$224$	5596.38	1.66930
$225$	−1377.17	−0.408052
$226$	3843.42	1.13124
$227$	−163.172	−0.0477096	−0.0238548	−	0.999715i	$-0.507594\pi$
$227$	−163.172	−0.0477096	−0.0238548	+	0.999715i	$0.507594\pi$
$228$	−1923.32	−0.558661
$229$	−1012.59	−0.292200	−0.146100	−	0.989270i	$-0.546672\pi$
$229$	−1012.59	−0.292200	−0.146100	+	0.989270i	$0.546672\pi$
$230$	−434.822	−0.124658
$231$	0	0
$232$	−22.8498	−0.00646623
$233$	5696.86	1.60178	0.800888	−	0.598814i	$-0.204361\pi$
$233$	5696.86	1.60178	0.800888	+	0.598814i	$0.204361\pi$
$234$	−1003.15	−0.280249
$235$	−1251.59	−0.347424
$236$	5976.08	1.64835
$237$	627.697	0.172039
$238$	−6300.99	−1.71610
$239$	1013.13	0.274202	0.137101	−	0.990557i	$-0.456222\pi$
$239$	1013.13	0.274202	0.137101	+	0.990557i	$0.456222\pi$
$240$	−2508.88	−0.674780
$241$	6252.27	1.67114	0.835569	−	0.549386i	$-0.185138\pi$
$241$	6252.27	1.67114	0.835569	+	0.549386i	$0.185138\pi$
$242$	0	0
$243$	−3925.65	−1.03634
$244$	−5256.98	−1.37928
$245$	−1931.97	−0.503793
$246$	5133.80	1.33057
$247$	−1143.77	−0.294642
$248$	26.5623	0.00680123
$249$	−2522.32	−0.641951
$250$	−3006.56	−0.760607
$251$	954.792	0.240103	0.120052	−	0.992768i	$-0.461694\pi$
$251$	954.792	0.240103	0.120052	+	0.992768i	$0.461694\pi$
$252$	−3353.83	−0.838378
$253$	0	0
$254$	1602.18	0.395785
$255$	2791.08	0.685427
$256$	4193.03	1.02369
$257$	8101.41	1.96635	0.983175	−	0.182667i	$-0.0584729\pi$
$257$	8101.41	1.96635	0.983175	+	0.182667i	$0.0584729\pi$
$258$	1729.04	0.417231
$259$	−3218.07	−0.772051
$260$	1439.07	0.343258
$261$	1148.79	0.272445
$262$	−7526.92	−1.77487
$263$	−4836.19	−1.13389	−0.566944	−	0.823756i	$-0.691874\pi$
$263$	−4836.19	−1.13389	−0.566944	+	0.823756i	$0.691874\pi$
$264$	0	0
$265$	−3333.34	−0.772699
$266$	−7694.63	−1.77364
$267$	2589.08	0.593443
$268$	−5580.18	−1.27188
$269$	630.562	0.142922	0.0714611	−	0.997443i	$-0.477234\pi$
$269$	630.562	0.142922	0.0714611	+	0.997443i	$0.477234\pi$
$270$	7160.58	1.61400
$271$	−2035.41	−0.456244	−0.228122	−	0.973633i	$-0.573259\pi$
$271$	−2035.41	−0.456244	−0.228122	+	0.973633i	$0.573259\pi$
$272$	−4665.99	−1.04014
$273$	788.538	0.174815
$274$	−4384.83	−0.966779
$275$	0	0
$276$	170.177	0.0371139
$277$	2761.67	0.599034	0.299517	−	0.954091i	$-0.403174\pi$
$277$	2761.67	0.599034	0.299517	+	0.954091i	$0.403174\pi$
$278$	−12958.5	−2.79568
$279$	−1335.43	−0.286560
$280$	−118.218	−0.0252317
$281$	2663.84	0.565522	0.282761	−	0.959190i	$-0.408750\pi$
$281$	2663.84	0.565522	0.282761	+	0.959190i	$0.408750\pi$
$282$	985.651	0.208137
$283$	3276.34	0.688191	0.344096	−	0.938935i	$-0.388186\pi$
$283$	3276.34	0.688191	0.344096	+	0.938935i	$0.388186\pi$
$284$	−736.140	−0.153809
$285$	3408.40	0.708408
$286$	0	0
$287$	10207.1	2.09932
$288$	−4937.88	−1.01030
$289$	277.819	0.0565477
$290$	−3316.10	−0.671476
$291$	−2942.35	−0.592727
$292$	−6249.93	−1.25257
$293$	9125.53	1.81952	0.909760	−	0.415134i	$-0.136265\pi$
$293$	9125.53	1.81952	0.909760	+	0.415134i	$0.136265\pi$
$294$	1521.47	0.301816
$295$	−10590.5	−2.09018
$296$	−56.4771	−0.0110901
$297$	0	0
$298$	4088.17	0.794703
$299$	101.202	0.0195741
$300$	−1555.84	−0.299422
$301$	3437.71	0.658294
$302$	9769.05	1.86141
$303$	−168.371	−0.0319230
$304$	−5698.00	−1.07501
$305$	9316.16	1.74899
$306$	5559.58	1.03863
$307$	704.496	0.130970	0.0654849	−	0.997854i	$-0.479141\pi$
$307$	704.496	0.130970	0.0654849	+	0.997854i	$0.479141\pi$
$308$	0	0
$309$	5565.45	1.02462
$310$	3854.87	0.706264
$311$	−5288.36	−0.964230	−0.482115	−	0.876108i	$-0.660131\pi$
$311$	−5288.36	−0.964230	−0.482115	+	0.876108i	$0.660131\pi$
$312$	13.8388	0.00251112
$313$	3054.95	0.551681	0.275840	−	0.961203i	$-0.411044\pi$
$313$	3054.95	0.551681	0.275840	+	0.961203i	$0.411044\pi$
$314$	11143.2	2.00270
$315$	5943.48	1.06310
$316$	−1793.64	−0.319304
$317$	2867.34	0.508031	0.254016	−	0.967200i	$-0.418249\pi$
$317$	2867.34	0.508031	0.254016	+	0.967200i	$0.418249\pi$
$318$	2625.07	0.462914
$319$	0	0
$320$	6997.10	1.22234
$321$	4750.92	0.826076
$322$	680.827	0.117829
$323$	6338.91	1.09197
$324$	1326.70	0.227487
$325$	−925.241	−0.157917
$326$	−1357.72	−0.230667
$327$	521.211	0.0881438
$328$	179.134	0.0301556
$329$	1959.69	0.328392
$330$	0	0
$331$	−417.323	−0.0692996	−0.0346498	−	0.999400i	$-0.511032\pi$
$331$	−417.323	−0.0692996	−0.0346498	+	0.999400i	$0.511032\pi$
$332$	7207.50	1.19145
$333$	2839.41	0.467264
$334$	7651.28	1.25347
$335$	9888.90	1.61280
$336$	3928.30	0.637816
$337$	372.741	0.0602507	0.0301254	−	0.999546i	$-0.490409\pi$
$337$	372.741	0.0602507	0.0301254	+	0.999546i	$0.490409\pi$
$338$	−673.958	−0.108457
$339$	2665.67	0.427078
$340$	−7975.46	−1.27215
$341$	0	0
$342$	6789.24	1.07345
$343$	−4497.08	−0.707929
$344$	60.3318	0.00945602
$345$	−301.578	−0.0470621
$346$	−59.5497	−0.00925264
$347$	10140.9	1.56886	0.784428	−	0.620220i	$-0.212957\pi$
$347$	10140.9	1.56886	0.784428	+	0.620220i	$0.212957\pi$
$348$	1297.82	0.199916
$349$	12457.3	1.91067	0.955334	−	0.295529i	$-0.0954960\pi$
$349$	12457.3	1.91067	0.955334	+	0.295529i	$0.0954960\pi$
$350$	−6224.46	−0.950604
$351$	−1666.58	−0.253435
$352$	0	0
$353$	−6489.19	−0.978427	−0.489213	−	0.872164i	$-0.662716\pi$
$353$	−6489.19	−0.978427	−0.489213	+	0.872164i	$0.662716\pi$
$354$	8340.24	1.25220
$355$	1304.55	0.195037
$356$	−7398.27	−1.10143
$357$	−4370.16	−0.647880
$358$	13122.3	1.93725
$359$	−9703.27	−1.42652	−0.713258	−	0.700902i	$-0.752781\pi$
$359$	−9703.27	−1.42652	−0.713258	+	0.700902i	$0.752781\pi$
$360$	104.308	0.0152709
$361$	881.943	0.128582
$362$	1959.73	0.284533
$363$	0	0
$364$	−2253.24	−0.324455
$365$	11075.8	1.58831
$366$	−7336.67	−1.04780
$367$	3648.50	0.518938	0.259469	−	0.965751i	$-0.416452\pi$
$367$	3648.50	0.518938	0.259469	+	0.965751i	$0.416452\pi$
$368$	504.164	0.0714167
$369$	−9006.08	−1.27056
$370$	−8196.27	−1.15163
$371$	5219.21	0.730371
$372$	−1508.68	−0.210273
$373$	−10383.5	−1.44139	−0.720695	−	0.693252i	$-0.756177\pi$
$373$	−10383.5	−1.44139	−0.720695	+	0.693252i	$0.756177\pi$
$374$	0	0
$375$	−2085.25	−0.287152
$376$	34.3924	0.00471717
$377$	771.802	0.105437
$378$	−11211.8	−1.52558
$379$	5982.07	0.810761	0.405380	−	0.914148i	$-0.367139\pi$
$379$	5982.07	0.810761	0.405380	+	0.914148i	$0.367139\pi$
$380$	−9739.46	−1.31480
$381$	1111.22	0.149421
$382$	13154.5	1.76190
$383$	5961.49	0.795346	0.397673	−	0.917527i	$-0.369818\pi$
$383$	5961.49	0.795346	0.397673	+	0.917527i	$0.369818\pi$
$384$	136.249	0.0181066
$385$	0	0
$386$	8209.76	1.08255
$387$	−3033.21	−0.398416
$388$	8407.73	1.10010
$389$	1644.85	0.214389	0.107195	−	0.994238i	$-0.465813\pi$
$389$	1644.85	0.214389	0.107195	+	0.994238i	$0.465813\pi$
$390$	2008.37	0.260763
$391$	−560.873	−0.0725436
$392$	53.0888	0.00684028
$393$	−5220.43	−0.670066
$394$	−14404.5	−1.84185
$395$	3178.59	0.404892
$396$	0	0
$397$	11314.3	1.43035	0.715177	−	0.698943i	$-0.246346\pi$
$397$	11314.3	1.43035	0.715177	+	0.698943i	$0.246346\pi$
$398$	−18922.0	−2.38310
$399$	−5336.74	−0.669602
$400$	−4609.32	−0.576165
$401$	6274.46	0.781375	0.390688	−	0.920523i	$-0.372237\pi$
$401$	6274.46	0.781375	0.390688	+	0.920523i	$0.372237\pi$
$402$	−7787.72	−0.966209
$403$	−897.197	−0.110900
$404$	481.118	0.0592488
$405$	−2351.11	−0.288464
$406$	5192.22	0.634693
$407$	0	0
$408$	−76.6962	−0.00930644
$409$	3817.55	0.461530	0.230765	−	0.973009i	$-0.425877\pi$
$409$	3817.55	0.461530	0.230765	+	0.973009i	$0.425877\pi$
$410$	25997.0	3.13146
$411$	−3041.18	−0.364988
$412$	−15903.2	−1.90168
$413$	16582.2	1.97568
$414$	−600.717	−0.0713132
$415$	−12772.8	−1.51082
$416$	−3317.46	−0.390991
$417$	−8987.60	−1.05546
$418$	0	0
$419$	8909.28	1.03878	0.519388	−	0.854539i	$-0.326160\pi$
$419$	8909.28	1.03878	0.519388	+	0.854539i	$0.326160\pi$
$420$	6714.55	0.780087
$421$	6642.54	0.768973	0.384486	−	0.923131i	$-0.374379\pi$
$421$	6642.54	0.768973	0.384486	+	0.923131i	$0.374379\pi$
$422$	−8159.07	−0.941178
$423$	−1729.10	−0.198751
$424$	91.5970	0.0104914
$425$	5127.78	0.585256
$426$	−1027.36	−0.116844
$427$	−14586.9	−1.65318
$428$	−13575.7	−1.53319
$429$	0	0
$430$	8755.69	0.981946
$431$	−3196.51	−0.357240	−0.178620	−	0.983918i	$-0.557163\pi$
$431$	−3196.51	−0.357240	−0.178620	+	0.983918i	$0.557163\pi$
$432$	−8302.50	−0.924662
$433$	14183.1	1.57413	0.787064	−	0.616871i	$-0.211600\pi$
$433$	14183.1	1.57413	0.787064	+	0.616871i	$0.211600\pi$
$434$	−6035.80	−0.667575
$435$	−2299.94	−0.253502
$436$	−1489.35	−0.163594
$437$	−684.925	−0.0749758
$438$	−8722.43	−0.951538
$439$	12267.1	1.33366	0.666830	−	0.745210i	$-0.267651\pi$
$439$	12267.1	1.33366	0.666830	+	0.745210i	$0.267651\pi$
$440$	0	0
$441$	−2669.07	−0.288205
$442$	3735.15	0.401952
$443$	5691.71	0.610432	0.305216	−	0.952283i	$-0.401271\pi$
$443$	5691.71	0.610432	0.305216	+	0.952283i	$0.401271\pi$
$444$	3207.78	0.342871
$445$	13110.8	1.39666
$446$	4759.87	0.505351
$447$	2835.42	0.300024
$448$	−10955.8	−1.15538
$449$	9363.89	0.984207	0.492104	−	0.870537i	$-0.336228\pi$
$449$	9363.89	0.984207	0.492104	+	0.870537i	$0.336228\pi$
$450$	5492.06	0.575329
$451$	0	0
$452$	−7617.12	−0.792653
$453$	6775.50	0.702738
$454$	650.715	0.0672678
$455$	3993.07	0.411424
$456$	−93.6597	−0.00961846
$457$	−15970.1	−1.63468	−0.817342	−	0.576153i	$-0.804553\pi$
$457$	−15970.1	−1.63468	−0.817342	+	0.576153i	$0.804553\pi$
$458$	4038.12	0.411985
$459$	9236.37	0.939252
$460$	861.756	0.0873469
$461$	−3190.62	−0.322347	−0.161174	−	0.986926i	$-0.551528\pi$
$461$	−3190.62	−0.322347	−0.161174	+	0.986926i	$0.551528\pi$
$462$	0	0
$463$	−7312.92	−0.734040	−0.367020	−	0.930213i	$-0.619622\pi$
$463$	−7312.92	−0.734040	−0.367020	+	0.930213i	$0.619622\pi$
$464$	3844.92	0.384690
$465$	2673.61	0.266636
$466$	−22718.6	−2.25841
$467$	4148.42	0.411062	0.205531	−	0.978651i	$-0.434108\pi$
$467$	4148.42	0.411062	0.205531	+	0.978651i	$0.434108\pi$
$468$	1988.11	0.196368
$469$	−15483.7	−1.52445
$470$	4991.23	0.489847
$471$	7728.57	0.756080
$472$	291.017	0.0283795
$473$	0	0
$474$	−2503.21	−0.242565
$475$	6261.93	0.604878
$476$	12487.7	1.20246
$477$	−4605.09	−0.442039
$478$	−4040.30	−0.386609
$479$	−11439.8	−1.09122	−0.545612	−	0.838038i	$-0.683703\pi$
$479$	−11439.8	−1.09122	−0.545612	+	0.838038i	$0.683703\pi$
$480$	9885.91	0.940058
$481$	1907.63	0.180833
$482$	−24933.5	−2.35621
$483$	472.199	0.0444841
$484$	0	0
$485$	−14899.7	−1.39497
$486$	15655.2	1.46118
$487$	−15956.5	−1.48472	−0.742358	−	0.670003i	$-0.766293\pi$
$487$	−15956.5	−1.48472	−0.742358	+	0.670003i	$0.766293\pi$
$488$	−255.999	−0.0237470
$489$	−941.673	−0.0870836
$490$	7704.55	0.710319
$491$	1092.59	0.100423	0.0502116	−	0.998739i	$-0.484010\pi$
$491$	1092.59	0.100423	0.0502116	+	0.998739i	$0.484010\pi$
$492$	−10174.5	−0.932318
$493$	−4277.40	−0.390760
$494$	4561.28	0.415428
$495$	0	0
$496$	−4469.61	−0.404620
$497$	−2042.61	−0.184353
$498$	10058.8	0.905113
$499$	13033.5	1.16926	0.584630	−	0.811300i	$-0.301240\pi$
$499$	13033.5	1.16926	0.584630	+	0.811300i	$0.301240\pi$
$500$	5958.58	0.532952
$501$	5306.68	0.473223
$502$	−3807.63	−0.338532
$503$	10099.2	0.895227	0.447613	−	0.894227i	$-0.352274\pi$
$503$	10099.2	0.895227	0.447613	+	0.894227i	$0.352274\pi$
$504$	−163.321	−0.0144343
$505$	−852.612	−0.0751301
$506$	0	0
$507$	−467.435	−0.0409458
$508$	−3175.29	−0.277324
$509$	−656.690	−0.0571852	−0.0285926	−	0.999591i	$-0.509103\pi$
$509$	−656.690	−0.0571852	−0.0285926	+	0.999591i	$0.509103\pi$
$510$	−11130.6	−0.966413
$511$	−17342.1	−1.50131
$512$	−16327.4	−1.40933
$513$	11279.2	0.970743
$514$	−32307.7	−2.77244
$515$	28182.8	2.41142
$516$	−3426.72	−0.292351
$517$	0	0
$518$	12833.4	1.08855
$519$	−41.3017	−0.00349315
$520$	70.0782	0.00590987
$521$	−17612.0	−1.48099	−0.740493	−	0.672064i	$-0.765408\pi$
$521$	−17612.0	−1.48099	−0.740493	+	0.672064i	$0.765408\pi$
$522$	−4581.27	−0.384132
$523$	−4314.06	−0.360689	−0.180345	−	0.983603i	$-0.557721\pi$
$523$	−4314.06	−0.360689	−0.180345	+	0.983603i	$0.557721\pi$
$524$	14917.3	1.24364
$525$	−4317.08	−0.358882
$526$	19286.3	1.59872
$527$	4972.36	0.411004
$528$	0	0
$529$	−12106.4	−0.995019
$530$	13293.1	1.08946
$531$	−14631.0	−1.19573
$532$	15249.7	1.24278
$533$	−6050.64	−0.491712
$534$	−10325.1	−0.836721
$535$	24058.1	1.94415
$536$	−271.738	−0.0218979
$537$	9101.21	0.731371
$538$	−2514.63	−0.201512
$539$	0	0
$540$	−14191.3	−1.13092
$541$	−12190.4	−0.968773	−0.484386	−	0.874854i	$-0.660957\pi$
$541$	−12190.4	−0.968773	−0.484386	+	0.874854i	$0.660957\pi$
$542$	8117.04	0.643278
$543$	1359.20	0.107420
$544$	18385.7	1.44905
$545$	2639.35	0.207445
$546$	−3144.62	−0.246479
$547$	25117.8	1.96336	0.981680	−	0.190536i	$-0.0610227\pi$
$547$	25117.8	1.96336	0.981680	+	0.190536i	$0.0610227\pi$
$548$	8690.12	0.677415
$549$	12870.5	1.00055
$550$	0	0
$551$	−5223.47	−0.403861
$552$	8.28709	0.000638989 0
$553$	−4976.91	−0.382712
$554$	−11013.3	−0.844603
$555$	−5684.67	−0.434776
$556$	25681.9	1.95892
$557$	−13898.6	−1.05728	−0.528639	−	0.848847i	$-0.677297\pi$
$557$	−13898.6	−1.05728	−0.528639	+	0.848847i	$0.677297\pi$
$558$	5325.60	0.404033
$559$	−2037.83	−0.154188
$560$	19892.5	1.50109
$561$	0	0
$562$	−10623.2	−0.797353
$563$	−7221.55	−0.540590	−0.270295	−	0.962778i	$-0.587121\pi$
$563$	−7221.55	−0.540590	−0.270295	+	0.962778i	$0.587121\pi$
$564$	−1953.42	−0.145840
$565$	13498.7	1.00512
$566$	−13065.8	−0.970310
$567$	3681.28	0.272662
$568$	−35.8478	−0.00264813
$569$	177.828	0.0131018	0.00655091	−	0.999979i	$-0.497915\pi$
$569$	177.828	0.0131018	0.00655091	+	0.999979i	$0.497915\pi$
$570$	−13592.4	−0.998814
$571$	−9260.20	−0.678682	−0.339341	−	0.940663i	$-0.610204\pi$
$571$	−9260.20	−0.678682	−0.339341	+	0.940663i	$0.610204\pi$
$572$	0	0
$573$	9123.55	0.665169
$574$	−40705.1	−2.95993
$575$	−554.061	−0.0401843
$576$	9666.67	0.699267
$577$	−831.744	−0.0600104	−0.0300052	−	0.999550i	$-0.509552\pi$
$577$	−831.744	−0.0600104	−0.0300052	+	0.999550i	$0.509552\pi$
$578$	−1107.92	−0.0797290
$579$	5694.02	0.408697
$580$	6572.04	0.470498
$581$	19999.1	1.42806
$582$	11733.9	0.835711
$583$	0	0
$584$	−304.353	−0.0215654
$585$	−3523.22	−0.249004
$586$	−36391.9	−2.56542
$587$	−24833.1	−1.74612	−0.873061	−	0.487611i	$-0.837868\pi$
$587$	−24833.1	−1.74612	−0.873061	+	0.487611i	$0.837868\pi$
$588$	−3015.34	−0.211480
$589$	6072.13	0.424784
$590$	42234.0	2.94703
$591$	−9990.50	−0.695354
$592$	9503.35	0.659772
$593$	7993.15	0.553523	0.276762	−	0.960939i	$-0.410739\pi$
$593$	7993.15	0.553523	0.276762	+	0.960939i	$0.410739\pi$
$594$	0	0
$595$	−22130.0	−1.52477
$596$	−8102.18	−0.556843
$597$	−13123.7	−0.899693
$598$	−403.586	−0.0275984
$599$	13500.6	0.920902	0.460451	−	0.887685i	$-0.347688\pi$
$599$	13500.6	0.920902	0.460451	+	0.887685i	$0.347688\pi$
$600$	−75.7647	−0.00515514
$601$	−8880.38	−0.602726	−0.301363	−	0.953510i	$-0.597442\pi$
$601$	−8880.38	−0.602726	−0.301363	+	0.953510i	$0.597442\pi$
$602$	−13709.3	−0.928156
$603$	13661.8	0.922637
$604$	−19360.9	−1.30428
$605$	0	0
$606$	671.450	0.0450095
$607$	−239.150	−0.0159914	−0.00799572	−	0.999968i	$-0.502545\pi$
$607$	−239.150	−0.0159914	−0.00799572	+	0.999968i	$0.502545\pi$
$608$	22452.3	1.49763
$609$	3601.15	0.239616
$610$	−37152.1	−2.46597
$611$	−1161.68	−0.0769172
$612$	−11018.3	−0.727759
$613$	−1699.06	−0.111949	−0.0559743	−	0.998432i	$-0.517826\pi$
$613$	−1699.06	−0.111949	−0.0559743	+	0.998432i	$0.517826\pi$
$614$	−2809.47	−0.184660
$615$	18030.7	1.18222
$616$	0	0
$617$	29406.1	1.91871	0.959355	−	0.282201i	$-0.0910646\pi$
$617$	29406.1	1.91871	0.959355	+	0.282201i	$0.0910646\pi$
$618$	−22194.5	−1.44465
$619$	−15649.0	−1.01613	−0.508067	−	0.861317i	$-0.669640\pi$
$619$	−15649.0	−1.01613	−0.508067	+	0.861317i	$0.669640\pi$
$620$	−7639.80	−0.494874
$621$	−997.998	−0.0644900
$622$	21089.6	1.35951
$623$	−20528.4	−1.32015
$624$	−2328.65	−0.149392
$625$	−19456.0	−1.24519
$626$	−12182.9	−0.777838
$627$	0	0
$628$	−22084.3	−1.40328
$629$	−10572.3	−0.670183
$630$	−23702.1	−1.49891
$631$	−4387.50	−0.276804	−0.138402	−	0.990376i	$-0.544197\pi$
$631$	−4387.50	−0.276804	−0.138402	+	0.990376i	$0.544197\pi$
$632$	−87.3447	−0.00549744
$633$	−5658.86	−0.355323
$634$	−11434.7	−0.716294
$635$	5627.08	0.351659
$636$	−5202.52	−0.324361
$637$	−1793.19	−0.111536
$638$	0	0
$639$	1802.27	0.111575
$640$	689.951	0.0426136
$641$	−7916.94	−0.487832	−0.243916	−	0.969796i	$-0.578432\pi$
$641$	−7916.94	−0.487832	−0.243916	+	0.969796i	$0.578432\pi$
$642$	−18946.3	−1.16472
$643$	−15655.6	−0.960182	−0.480091	−	0.877219i	$-0.659396\pi$
$643$	−15655.6	−0.960182	−0.480091	+	0.877219i	$0.659396\pi$
$644$	−1349.30	−0.0825621
$645$	6072.66	0.370714
$646$	−25279.1	−1.53962
$647$	7568.09	0.459864	0.229932	−	0.973207i	$-0.426150\pi$
$647$	7568.09	0.459864	0.229932	+	0.973207i	$0.426150\pi$
$648$	64.6064	0.00391664
$649$	0	0
$650$	3689.78	0.222654
$651$	−4186.23	−0.252030
$652$	2690.82	0.161626
$653$	7199.90	0.431476	0.215738	−	0.976451i	$-0.430784\pi$
$653$	7199.90	0.431476	0.215738	+	0.976451i	$0.430784\pi$
$654$	−2078.55	−0.124278
$655$	−26435.7	−1.57699
$656$	−30142.8	−1.79402
$657$	15301.5	0.908628
$658$	−7815.07	−0.463014
$659$	−14818.4	−0.875937	−0.437969	−	0.898990i	$-0.644302\pi$
$659$	−14818.4	−0.875937	−0.437969	+	0.898990i	$0.644302\pi$
$660$	0	0
$661$	7810.40	0.459591	0.229795	−	0.973239i	$-0.426194\pi$
$661$	7810.40	0.459591	0.229795	+	0.973239i	$0.426194\pi$
$662$	1664.25	0.0977084
$663$	2590.57	0.151749
$664$	350.983	0.0205133
$665$	−27024.7	−1.57590
$666$	−11323.4	−0.658815
$667$	462.177	0.0268300
$668$	−15163.8	−0.878299
$669$	3301.29	0.190785
$670$	−39436.1	−2.27396
$671$	0	0
$672$	−15479.0	−0.888563
$673$	25053.8	1.43500	0.717498	−	0.696560i	$-0.245287\pi$
$673$	25053.8	1.43500	0.717498	+	0.696560i	$0.245287\pi$
$674$	−1486.46	−0.0849500
$675$	9124.20	0.520282
$676$	1335.69	0.0759951
$677$	28543.2	1.62039	0.810196	−	0.586159i	$-0.199361\pi$
$677$	28543.2	1.62039	0.810196	+	0.586159i	$0.199361\pi$
$678$	−10630.5	−0.602155
$679$	23329.4	1.31856
$680$	−388.381	−0.0219025
$681$	451.315	0.0253956
$682$	0	0
$683$	−328.421	−0.0183993	−0.00919963	−	0.999958i	$-0.502928\pi$
$683$	−328.421	−0.0183993	−0.00919963	+	0.999958i	$0.502928\pi$
$684$	−13455.3	−0.752159
$685$	−15400.2	−0.858994
$686$	17934.0	0.998138
$687$	2800.71	0.155537
$688$	−10152.0	−0.562559
$689$	−3093.88	−0.171070
$690$	1202.67	0.0663548
$691$	12861.1	0.708046	0.354023	−	0.935237i	$-0.384813\pi$
$691$	12861.1	0.708046	0.354023	+	0.935237i	$0.384813\pi$
$692$	118.019	0.00648326
$693$	0	0
$694$	−40441.1	−2.21200
$695$	−45512.2	−2.48399
$696$	63.2001	0.00344195
$697$	33533.3	1.82233
$698$	−49678.6	−2.69393
$699$	−15756.9	−0.852618
$700$	12336.0	0.666082
$701$	−27498.4	−1.48160	−0.740800	−	0.671725i	$-0.765554\pi$
$701$	−27498.4	−1.48160	−0.740800	+	0.671725i	$0.765554\pi$
$702$	6646.19	0.357328
$703$	−12910.6	−0.692652
$704$	0	0
$705$	3461.75	0.184932
$706$	25878.3	1.37952
$707$	1334.99	0.0710146
$708$	−16529.2	−0.877408
$709$	−17149.3	−0.908401	−0.454201	−	0.890899i	$-0.650075\pi$
$709$	−17149.3	−0.908401	−0.454201	+	0.890899i	$0.650075\pi$
$710$	−5202.43	−0.274991
$711$	4391.30	0.231627
$712$	−360.273	−0.0189632
$713$	−537.267	−0.0282199
$714$	17427.8	0.913474
$715$	0	0
$716$	−26006.6	−1.35742
$717$	−2802.22	−0.145956
$718$	38695.8	2.01130
$719$	−22358.8	−1.15973	−0.579863	−	0.814714i	$-0.696894\pi$
$719$	−22358.8	−1.15973	−0.579863	+	0.814714i	$0.696894\pi$
$720$	−17551.8	−0.908497
$721$	−44127.5	−2.27933
$722$	−3517.12	−0.181293
$723$	−17293.1	−0.889539
$724$	−3883.90	−0.199370
$725$	−4225.46	−0.216455
$726$	0	0
$727$	2092.48	0.106748	0.0533740	−	0.998575i	$-0.483002\pi$
$727$	2092.48	0.106748	0.0533740	+	0.998575i	$0.483002\pi$
$728$	−109.726	−0.00558614
$729$	6325.62	0.321375
$730$	−44169.4	−2.23943
$731$	11293.9	0.571435
$732$	14540.2	0.734184
$733$	30294.7	1.52655	0.763274	−	0.646075i	$-0.223591\pi$
$733$	30294.7	1.52655	0.763274	+	0.646075i	$0.223591\pi$
$734$	−14549.9	−0.731672
$735$	5343.63	0.268167
$736$	−1986.59	−0.0994930
$737$	0	0
$738$	35915.5	1.79142
$739$	25035.7	1.24622	0.623109	−	0.782135i	$-0.285869\pi$
$739$	25035.7	1.24622	0.623109	+	0.782135i	$0.285869\pi$
$740$	16243.8	0.806940
$741$	3163.55	0.156837
$742$	−20813.8	−1.02978
$743$	−30841.6	−1.52284	−0.761418	−	0.648261i	$-0.775497\pi$
$743$	−30841.6	−1.52284	−0.761418	+	0.648261i	$0.775497\pi$
$744$	−73.4683	−0.00362027
$745$	14358.3	0.706102
$746$	41408.6	2.03228
$747$	−17645.9	−0.864296
$748$	0	0
$749$	−37669.2	−1.83766
$750$	8315.82	0.404868
$751$	29409.4	1.42898	0.714491	−	0.699645i	$-0.246658\pi$
$751$	29409.4	1.42898	0.714491	+	0.699645i	$0.246658\pi$
$752$	−5787.19	−0.280634
$753$	−2640.85	−0.127806
$754$	−3077.88	−0.148660
$755$	34310.3	1.65388
$756$	22220.1	1.06897
$757$	335.724	0.0161190	0.00805951	−	0.999968i	$-0.497435\pi$
$757$	335.724	0.0161190	0.00805951	+	0.999968i	$0.497435\pi$
$758$	−23856.0	−1.14313
$759$	0	0
$760$	−474.282	−0.0226369
$761$	−17446.5	−0.831058	−0.415529	−	0.909580i	$-0.636404\pi$
$761$	−17446.5	−0.831058	−0.415529	+	0.909580i	$0.636404\pi$
$762$	−4431.44	−0.210675
$763$	−4132.59	−0.196081
$764$	−26070.4	−1.23455
$765$	19526.1	0.922831
$766$	−23773.9	−1.12139
$767$	−9829.71	−0.462752
$768$	−11597.5	−0.544905
$769$	−3547.25	−0.166342	−0.0831712	−	0.996535i	$-0.526505\pi$
$769$	−3547.25	−0.166342	−0.0831712	+	0.996535i	$0.526505\pi$
$770$	0	0
$771$	−22407.6	−1.04668
$772$	−16270.6	−0.758537
$773$	16503.0	0.767879	0.383940	−	0.923358i	$-0.374567\pi$
$773$	16503.0	0.767879	0.383940	+	0.923358i	$0.374567\pi$
$774$	12096.2	0.561743
$775$	4911.97	0.227669
$776$	409.431	0.0189403
$777$	8900.83	0.410959
$778$	−6559.55	−0.302276
$779$	40950.1	1.88343
$780$	−3980.30	−0.182715
$781$	0	0
$782$	2236.71	0.102282
$783$	−7611.07	−0.347379
$784$	−8933.21	−0.406943
$785$	39136.6	1.77942
$786$	20818.6	0.944754
$787$	12301.4	0.557174	0.278587	−	0.960411i	$-0.410134\pi$
$787$	12301.4	0.557174	0.278587	+	0.960411i	$0.410134\pi$
$788$	28547.7	1.29057
$789$	13376.4	0.603563
$790$	−12676.0	−0.570874
$791$	−21135.7	−0.950061
$792$	0	0
$793$	8646.91	0.387214
$794$	−45120.7	−2.01672
$795$	9219.64	0.411304
$796$	37500.8	1.66982
$797$	29385.4	1.30600	0.653002	−	0.757356i	$-0.273509\pi$
$797$	29385.4	1.30600	0.653002	+	0.757356i	$0.273509\pi$
$798$	21282.5	0.944100
$799$	6438.14	0.285062
$800$	18162.4	0.802674
$801$	18112.9	0.798988
$802$	−25022.0	−1.10169
$803$	0	0
$804$	15434.2	0.677016
$805$	2391.16	0.104693
$806$	3577.95	0.156362
$807$	−1744.07	−0.0760768
$808$	23.4290	0.00102008
$809$	17689.0	0.768742	0.384371	−	0.923179i	$-0.374418\pi$
$809$	17689.0	0.768742	0.384371	+	0.923179i	$0.374418\pi$
$810$	9376.05	0.406717
$811$	14454.9	0.625870	0.312935	−	0.949775i	$-0.398688\pi$
$811$	14454.9	0.625870	0.312935	+	0.949775i	$0.398688\pi$
$812$	−10290.2	−0.444725
$813$	5629.71	0.242857
$814$	0	0
$815$	−4768.52	−0.204950
$816$	12905.6	0.553660
$817$	13791.8	0.590594
$818$	−15224.1	−0.650731
$819$	5516.52	0.235364
$820$	−51522.4	−2.19419
$821$	−20081.4	−0.853648	−0.426824	−	0.904335i	$-0.640368\pi$
$821$	−20081.4	−0.853648	−0.426824	+	0.904335i	$0.640368\pi$
$822$	12128.0	0.514612
$823$	−11680.6	−0.494728	−0.247364	−	0.968923i	$-0.579564\pi$
$823$	−11680.6	−0.494728	−0.247364	+	0.968923i	$0.579564\pi$
$824$	−774.437	−0.0327413
$825$	0	0
$826$	−66128.4	−2.78560
$827$	−17617.3	−0.740764	−0.370382	−	0.928880i	$-0.620773\pi$
$827$	−17617.3	−0.740764	−0.370382	+	0.928880i	$0.620773\pi$
$828$	1190.54	0.0499686
$829$	19029.6	0.797256	0.398628	−	0.917113i	$-0.369486\pi$
$829$	19029.6	0.797256	0.398628	+	0.917113i	$0.369486\pi$
$830$	50936.7	2.13017
$831$	−7638.46	−0.318863
$832$	6494.45	0.270618
$833$	9938.03	0.413364
$834$	35841.8	1.48813
$835$	26872.4	1.11372
$836$	0	0
$837$	8847.64	0.365375
$838$	−35529.5	−1.46461
$839$	−26848.2	−1.10477	−0.552386	−	0.833588i	$-0.686283\pi$
$839$	−26848.2	−1.10477	−0.552386	+	0.833588i	$0.686283\pi$
$840$	326.978	0.0134307
$841$	−20864.3	−0.855479
$842$	−26489.9	−1.08421
$843$	−7367.90	−0.301025
$844$	16170.1	0.659477
$845$	−2367.04	−0.0963653
$846$	6895.51	0.280227
$847$	0	0
$848$	−15412.9	−0.624154
$849$	−9061.99	−0.366321
$850$	−20449.2	−0.825177
$851$	1142.35	0.0460154
$852$	2036.08	0.0818721
$853$	−7605.31	−0.305276	−0.152638	−	0.988282i	$-0.548777\pi$
$853$	−7605.31	−0.305276	−0.152638	+	0.988282i	$0.548777\pi$
$854$	58171.2	2.33089
$855$	23844.8	0.953772
$856$	−661.094	−0.0263969
$857$	48579.1	1.93632	0.968162	−	0.250324i	$-0.0805370\pi$
$857$	48579.1	1.93632	0.968162	+	0.250324i	$0.0805370\pi$
$858$	0	0
$859$	23198.7	0.921453	0.460727	−	0.887542i	$-0.347589\pi$
$859$	23198.7	0.921453	0.460727	+	0.887542i	$0.347589\pi$
$860$	−17352.5	−0.688042
$861$	−28231.7	−1.11746
$862$	12747.4	0.503688
$863$	−28873.7	−1.13890	−0.569450	−	0.822026i	$-0.692844\pi$
$863$	−28873.7	−1.13890	−0.569450	+	0.822026i	$0.692844\pi$
$864$	32714.9	1.28818
$865$	−209.147	−0.00822107
$866$	−56561.1	−2.21943
$867$	−768.417	−0.0301001
$868$	11962.1	0.467765
$869$	0	0
$870$	9171.96	0.357424
$871$	9178.52	0.357063
$872$	−72.5269	−0.00281660
$873$	−20584.4	−0.798024
$874$	2731.43	0.105712
$875$	16533.6	0.638788
$876$	17286.6	0.666736
$877$	−45146.2	−1.73829	−0.869144	−	0.494559i	$-0.835329\pi$
$877$	−45146.2	−1.73829	−0.869144	+	0.494559i	$0.835329\pi$
$878$	−48920.2	−1.88038
$879$	−25240.2	−0.968523
$880$	0	0
$881$	46983.1	1.79671	0.898354	−	0.439273i	$-0.144764\pi$
$881$	46983.1	1.79671	0.898354	+	0.439273i	$0.144764\pi$
$882$	10644.0	0.406353
$883$	14096.0	0.537223	0.268611	−	0.963249i	$-0.413435\pi$
$883$	14096.0	0.537223	0.268611	+	0.963249i	$0.413435\pi$
$884$	−7402.53	−0.281645
$885$	29292.2	1.11259
$886$	−22698.1	−0.860674
$887$	29429.2	1.11402	0.557010	−	0.830506i	$-0.311948\pi$
$887$	29429.2	1.11402	0.557010	+	0.830506i	$0.311948\pi$
$888$	156.209	0.00590320
$889$	−8810.66	−0.332396
$890$	−52284.9	−1.96921
$891$	0	0
$892$	−9433.39	−0.354096
$893$	7862.11	0.294620
$894$	−11307.4	−0.423017
$895$	46087.5	1.72127
$896$	−1080.30	−0.0402793
$897$	−279.914	−0.0104192
$898$	−37342.4	−1.38768
$899$	−4097.38	−0.152008
$900$	−10884.5	−0.403129
$901$	17146.6	0.634003
$902$	0	0
$903$	−9508.33	−0.350407
$904$	−370.931	−0.0136471
$905$	6882.85	0.252811
$906$	−27020.1	−0.990820
$907$	4151.90	0.151997	0.0759986	−	0.997108i	$-0.475786\pi$
$907$	4151.90	0.151997	0.0759986	+	0.997108i	$0.475786\pi$
$908$	−1289.63	−0.0471340
$909$	−1177.90	−0.0429798
$910$	−15924.0	−0.580084
$911$	43442.1	1.57991	0.789957	−	0.613163i	$-0.210103\pi$
$911$	43442.1	1.57991	0.789957	+	0.613163i	$0.210103\pi$
$912$	15760.0	0.572223
$913$	0	0
$914$	63687.5	2.30481
$915$	−25767.5	−0.930979
$916$	−8002.99	−0.288675
$917$	41391.9	1.49060
$918$	−36833.9	−1.32429
$919$	45186.9	1.62196	0.810978	−	0.585076i	$-0.198936\pi$
$919$	45186.9	1.62196	0.810978	+	0.585076i	$0.198936\pi$
$920$	41.9649	0.00150385
$921$	−1948.56	−0.0697147
$922$	12723.9	0.454491
$923$	1210.83	0.0431800
$924$	0	0
$925$	−10443.9	−0.371236
$926$	29163.3	1.03495
$927$	38935.3	1.37951
$928$	−15150.4	−0.535924
$929$	−27366.8	−0.966496	−0.483248	−	0.875484i	$-0.660543\pi$
$929$	−27366.8	−0.966496	−0.483248	+	0.875484i	$0.660543\pi$
$930$	−10662.1	−0.375941
$931$	12136.1	0.427223
$932$	45025.1	1.58245
$933$	14627.0	0.513256
$934$	−16543.6	−0.579574
$935$	0	0
$936$	96.8148	0.00338087
$937$	12531.7	0.436917	0.218459	−	0.975846i	$-0.429897\pi$
$937$	12531.7	0.436917	0.218459	+	0.975846i	$0.429897\pi$
$938$	61747.5	2.14939
$939$	−8449.66	−0.293657
$940$	−9891.91	−0.343232
$941$	44374.1	1.53725	0.768626	−	0.639698i	$-0.220941\pi$
$941$	44374.1	1.53725	0.768626	+	0.639698i	$0.220941\pi$
$942$	−30820.9	−1.06603
$943$	−3623.30	−0.125123
$944$	−48969.2	−1.68836
$945$	−39377.3	−1.35550
$946$	0	0
$947$	53305.9	1.82915	0.914576	−	0.404413i	$-0.132524\pi$
$947$	53305.9	1.82915	0.914576	+	0.404413i	$0.132524\pi$
$948$	4961.00	0.169964
$949$	10280.2	0.351642
$950$	−24972.1	−0.852843
$951$	−7930.75	−0.270423
$952$	608.111	0.0207027
$953$	19033.3	0.646957	0.323478	−	0.946236i	$-0.395148\pi$
$953$	19033.3	0.646957	0.323478	+	0.946236i	$0.395148\pi$
$954$	18364.7	0.623249
$955$	46200.6	1.56546
$956$	8007.30	0.270894
$957$	0	0
$958$	45620.8	1.53856
$959$	24113.0	0.811939
$960$	−19353.2	−0.650648
$961$	−25027.9	−0.840117
$962$	−7607.48	−0.254964
$963$	33236.9	1.11219
$964$	49414.7	1.65098
$965$	28833.9	0.961860
$966$	−1883.09	−0.0627200
$967$	−2711.27	−0.0901639	−0.0450819	−	0.998983i	$-0.514355\pi$
$967$	−2711.27	−0.0901639	−0.0450819	+	0.998983i	$0.514355\pi$
$968$	0	0
$969$	−17532.7	−0.581251
$970$	59418.9	1.96683
$971$	2443.70	0.0807641	0.0403821	−	0.999184i	$-0.487142\pi$
$971$	2443.70	0.0807641	0.0403821	+	0.999184i	$0.487142\pi$
$972$	−31026.3	−1.02384
$973$	71261.2	2.34792
$974$	63633.1	2.09336
$975$	2559.11	0.0840587
$976$	43076.8	1.41276
$977$	4846.21	0.158694	0.0793470	−	0.996847i	$-0.474716\pi$
$977$	4846.21	0.158694	0.0793470	+	0.996847i	$0.474716\pi$
$978$	3755.31	0.122783
$979$	0	0
$980$	−15269.3	−0.497715
$981$	3646.33	0.118673
$982$	−4357.15	−0.141591
$983$	−37418.3	−1.21410	−0.607048	−	0.794665i	$-0.707647\pi$
$983$	−37418.3	−1.21410	−0.607048	+	0.794665i	$0.707647\pi$
$984$	−495.466	−0.0160517
$985$	−50590.8	−1.63650
$986$	17057.9	0.550949
$987$	−5420.27	−0.174802
$988$	−9039.81	−0.291088
$989$	−1220.31	−0.0392353
$990$	0	0
$991$	−2405.42	−0.0771048	−0.0385524	−	0.999257i	$-0.512275\pi$
$991$	−2405.42	−0.0771048	−0.0385524	+	0.999257i	$0.512275\pi$
$992$	17611.9	0.563689
$993$	1154.27	0.0368879
$994$	8145.77	0.259928
$995$	−66456.9	−2.11741
$996$	−19935.1	−0.634206
$997$	12892.8	0.409549	0.204775	−	0.978809i	$-0.434354\pi$
$997$	12892.8	0.409549	0.204775	+	0.978809i	$0.434354\pi$
$998$	−51976.6	−1.64859
$999$	−18812.0	−0.595780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1573.4.a.q.1.8		38
11.5	even	5		143.4.h.b.14.16	✓	76
11.9	even	5		143.4.h.b.92.16	yes	76
11.10	odd	2		1573.4.a.r.1.31		38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.h.b.14.16	✓	76	11.5	even	5
143.4.h.b.92.16	yes	76	11.9	even	5
1573.4.a.q.1.8		38	1.1	even	1	trivial
1573.4.a.r.1.31		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)

\(f(q)\)	\(=\)	\(q-3.98792 q^{2} -2.76589 q^{3} +7.90349 q^{4} -14.0062 q^{5} +11.0301 q^{6} +21.9303 q^{7} +0.384876 q^{8} -19.3499 q^{9} +55.8554 q^{10} -21.8602 q^{12} -13.0000 q^{13} -87.4562 q^{14} +38.7395 q^{15} -64.7628 q^{16} +72.0473 q^{17} +77.1656 q^{18} +87.9826 q^{19} -110.697 q^{20} -60.6568 q^{21} -7.78478 q^{23} -1.06452 q^{24} +71.1724 q^{25} +51.8429 q^{26} +128.199 q^{27} +173.326 q^{28} -59.3694 q^{29} -154.490 q^{30} +69.0151 q^{31} +255.190 q^{32} -287.319 q^{34} -307.159 q^{35} -152.931 q^{36} -146.741 q^{37} -350.868 q^{38} +35.9566 q^{39} -5.39063 q^{40} +465.434 q^{41} +241.894 q^{42} +156.756 q^{43} +271.017 q^{45} +31.0451 q^{46} +89.3598 q^{47} +179.127 q^{48} +137.937 q^{49} -283.830 q^{50} -199.275 q^{51} -102.745 q^{52} +237.991 q^{53} -511.245 q^{54} +8.44044 q^{56} -243.350 q^{57} +236.760 q^{58} +756.132 q^{59} +306.177 q^{60} -665.147 q^{61} -275.227 q^{62} -424.348 q^{63} -499.573 q^{64} +182.080 q^{65} -706.040 q^{67} +569.425 q^{68} +21.5318 q^{69} +1224.92 q^{70} -93.1411 q^{71} -7.44730 q^{72} -790.781 q^{73} +585.191 q^{74} -196.855 q^{75} +695.370 q^{76} -143.392 q^{78} -226.942 q^{79} +907.077 q^{80} +167.863 q^{81} -1856.11 q^{82} +911.939 q^{83} -479.400 q^{84} -1009.11 q^{85} -625.131 q^{86} +164.209 q^{87} -936.077 q^{89} -1080.79 q^{90} -285.094 q^{91} -61.5269 q^{92} -190.888 q^{93} -356.360 q^{94} -1232.30 q^{95} -705.826 q^{96} +1063.80 q^{97} -550.083 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 3 q^{2} + 19 q^{3} + 181 q^{4} + 52 q^{5} - 104 q^{6} - 12 q^{7} - 57 q^{8} + 477 q^{9} + 30 q^{10} + 122 q^{12} - 494 q^{13} + 181 q^{14} + 264 q^{15} + 961 q^{16} + 33 q^{17} - 28 q^{18} - 107 q^{19}+ \cdots - 2043 q^{98}+O(q^{100}) \) 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

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