LMFDB - Embedded newform 1587.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1587,4,Mod(1,1587)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1587, 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("1587.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1587 = 3 \cdot 23^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1587.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-4,6,2,26,-12,10,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	yes
Analytic conductor:	$93.6360311791$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1587.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.23607 q^{2} +3.00000 q^{3} +9.94427 q^{4} +10.7639 q^{5} -12.7082 q^{6} -10.6525 q^{7} -8.23607 q^{8} +9.00000 q^{9} -45.5967 q^{10} +52.3607 q^{11} +29.8328 q^{12} -79.0820 q^{13} +45.1246 q^{14} +32.2918 q^{15} -44.6656 q^{16} +77.2361 q^{17} -38.1246 q^{18} -50.7902 q^{19} +107.039 q^{20} -31.9574 q^{21} -221.803 q^{22} -24.7082 q^{24} -9.13777 q^{25} +334.997 q^{26} +27.0000 q^{27} -105.931 q^{28} +12.7477 q^{29} -136.790 q^{30} -12.4133 q^{31} +255.095 q^{32} +157.082 q^{33} -327.177 q^{34} -114.663 q^{35} +89.4984 q^{36} +73.1084 q^{37} +215.151 q^{38} -237.246 q^{39} -88.6525 q^{40} -38.8916 q^{41} +135.374 q^{42} -171.787 q^{43} +520.689 q^{44} +96.8754 q^{45} +614.545 q^{47} -133.997 q^{48} -229.525 q^{49} +38.7082 q^{50} +231.708 q^{51} -786.413 q^{52} -269.597 q^{53} -114.374 q^{54} +563.607 q^{55} +87.7345 q^{56} -152.371 q^{57} -54.0000 q^{58} -534.768 q^{59} +321.118 q^{60} +838.604 q^{61} +52.5836 q^{62} -95.8723 q^{63} -723.276 q^{64} -851.234 q^{65} -665.410 q^{66} +448.180 q^{67} +768.056 q^{68} +485.718 q^{70} +628.604 q^{71} -74.1246 q^{72} +925.266 q^{73} -309.692 q^{74} -27.4133 q^{75} -505.072 q^{76} -557.771 q^{77} +1004.99 q^{78} +963.479 q^{79} -480.778 q^{80} +81.0000 q^{81} +164.748 q^{82} -133.358 q^{83} -317.793 q^{84} +831.364 q^{85} +727.702 q^{86} +38.2430 q^{87} -431.246 q^{88} +778.581 q^{89} -410.371 q^{90} +842.420 q^{91} -37.2399 q^{93} -2603.25 q^{94} -546.703 q^{95} +765.286 q^{96} -1603.57 q^{97} +972.282 q^{98} +471.246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} + 26 q^{5} - 12 q^{6} + 10 q^{7} - 12 q^{8} + 18 q^{9} - 42 q^{10} + 60 q^{11} + 6 q^{12} - 24 q^{13} + 50 q^{14} + 78 q^{15} + 18 q^{16} + 150 q^{17} - 36 q^{18} + 46 q^{19}+ \cdots + 540 q^{99}+O(q^{100})$ 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 + 26 * q^5 - 12 * q^6 + 10 * q^7 - 12 * q^8 + 18 * q^9 - 42 * q^10 + 60 * q^11 + 6 * q^12 - 24 * q^13 + 50 * q^14 + 78 * q^15 + 18 * q^16 + 150 * q^17 - 36 * q^18 + 46 * q^19 - 14 * q^20 + 30 * q^21 - 220 * q^22 - 36 * q^24 + 98 * q^25 + 348 * q^26 + 54 * q^27 - 270 * q^28 - 216 * q^29 - 126 * q^30 + 324 * q^31 + 300 * q^32 + 180 * q^33 - 310 * q^34 + 200 * q^35 + 18 * q^36 - 140 * q^37 + 238 * q^38 - 72 * q^39 - 146 * q^40 - 364 * q^41 + 150 * q^42 + 126 * q^43 + 460 * q^44 + 234 * q^45 + 120 * q^47 + 54 * q^48 - 146 * q^49 + 64 * q^50 + 450 * q^51 - 1224 * q^52 - 490 * q^53 - 108 * q^54 + 680 * q^55 + 10 * q^56 + 138 * q^57 - 108 * q^58 - 32 * q^59 - 42 * q^60 + 908 * q^61 + 132 * q^62 + 90 * q^63 - 1214 * q^64 - 12 * q^65 - 660 * q^66 + 874 * q^67 + 190 * q^68 + 560 * q^70 + 488 * q^71 - 108 * q^72 + 652 * q^73 - 360 * q^74 + 294 * q^75 - 1274 * q^76 - 400 * q^77 + 1044 * q^78 + 1198 * q^79 + 474 * q^80 + 162 * q^81 + 88 * q^82 + 100 * q^83 - 810 * q^84 + 1940 * q^85 + 798 * q^86 - 648 * q^87 - 460 * q^88 - 102 * q^89 - 378 * q^90 + 1980 * q^91 + 972 * q^93 - 2720 * q^94 + 928 * q^95 + 900 * q^96 - 1624 * q^97 + 992 * q^98 + 540 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−4.23607	−1.49768	−0.748838	−	0.662753i	$-0.769388\pi$
$2$	−4.23607	−1.49768	−0.748838	+	0.662753i	$0.769388\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	9.94427	1.24303
$5$	10.7639	0.962755	0.481378	−	0.876513i	$-0.340137\pi$
$5$	10.7639	0.962755	0.481378	+	0.876513i	$0.340137\pi$
$6$	−12.7082	−0.864684
$7$	−10.6525	−0.575180	−0.287590	−	0.957754i	$-0.592854\pi$
$7$	−10.6525	−0.575180	−0.287590	+	0.957754i	$0.592854\pi$
$8$	−8.23607	−0.363986
$9$	9.00000	0.333333
$10$	−45.5967	−1.44190
$11$	52.3607	1.43521	0.717606	−	0.696449i	$-0.245238\pi$
$11$	52.3607	1.43521	0.717606	+	0.696449i	$0.245238\pi$
$12$	29.8328	0.717666
$13$	−79.0820	−1.68719	−0.843593	−	0.536984i	$-0.819564\pi$
$13$	−79.0820	−1.68719	−0.843593	+	0.536984i	$0.819564\pi$
$14$	45.1246	0.861433
$15$	32.2918	0.555847
$16$	−44.6656	−0.697900
$17$	77.2361	1.10191	0.550956	−	0.834534i	$-0.314263\pi$
$17$	77.2361	1.10191	0.550956	+	0.834534i	$0.314263\pi$
$18$	−38.1246	−0.499225
$19$	−50.7902	−0.613267	−0.306634	−	0.951828i	$-0.599203\pi$
$19$	−50.7902	−0.613267	−0.306634	+	0.951828i	$0.599203\pi$
$20$	107.039	1.19674
$21$	−31.9574	−0.332080
$22$	−221.803	−2.14948
$23$	0	0
$23$	0	0
$24$	−24.7082	−0.210148
$25$	−9.13777	−0.0731021
$26$	334.997	2.52686
$27$	27.0000	0.192450
$28$	−105.931	−0.714968
$29$	12.7477	0.0816270	0.0408135	−	0.999167i	$-0.487005\pi$
$29$	12.7477	0.0816270	0.0408135	+	0.999167i	$0.487005\pi$
$30$	−136.790	−0.832479
$31$	−12.4133	−0.0719192	−0.0359596	−	0.999353i	$-0.511449\pi$
$31$	−12.4133	−0.0719192	−0.0359596	+	0.999353i	$0.511449\pi$
$32$	255.095	1.40922
$33$	157.082	0.828620
$34$	−327.177	−1.65031
$35$	−114.663	−0.553757
$36$	89.4984	0.414345
$37$	73.1084	0.324836	0.162418	−	0.986722i	$-0.448071\pi$
$37$	73.1084	0.324836	0.162418	+	0.986722i	$0.448071\pi$
$38$	215.151	0.918476
$39$	−237.246	−0.974097
$40$	−88.6525	−0.350430
$41$	−38.8916	−0.148143	−0.0740714	−	0.997253i	$-0.523599\pi$
$41$	−38.8916	−0.148143	−0.0740714	+	0.997253i	$0.523599\pi$
$42$	135.374	0.497348
$43$	−171.787	−0.609239	−0.304620	−	0.952474i	$-0.598529\pi$
$43$	−171.787	−0.609239	−0.304620	+	0.952474i	$0.598529\pi$
$44$	520.689	1.78402
$45$	96.8754	0.320918
$46$	0	0
$47$	614.545	1.90725	0.953623	−	0.301003i	$-0.0973214\pi$
$47$	614.545	1.90725	0.953623	+	0.301003i	$0.0973214\pi$
$48$	−133.997	−0.402933
$49$	−229.525	−0.669168
$50$	38.7082	0.109483
$51$	231.708	0.636189
$52$	−786.413	−2.09723
$53$	−269.597	−0.698716	−0.349358	−	0.936989i	$-0.613600\pi$
$53$	−269.597	−0.698716	−0.349358	+	0.936989i	$0.613600\pi$
$54$	−114.374	−0.288228
$55$	563.607	1.38176
$56$	87.7345	0.209357
$57$	−152.371	−0.354070
$58$	−54.0000	−0.122251
$59$	−534.768	−1.18001	−0.590007	−	0.807398i	$-0.700875\pi$
$59$	−534.768	−1.18001	−0.590007	+	0.807398i	$0.700875\pi$
$60$	321.118	0.690937
$61$	838.604	1.76020	0.880100	−	0.474788i	$-0.157475\pi$
$61$	838.604	1.76020	0.880100	+	0.474788i	$0.157475\pi$
$62$	52.5836	0.107712
$63$	−95.8723	−0.191727
$64$	−723.276	−1.41265
$65$	−851.234	−1.62435
$66$	−665.410	−1.24101
$67$	448.180	0.817223	0.408612	−	0.912708i	$-0.366013\pi$
$67$	448.180	0.817223	0.408612	+	0.912708i	$0.366013\pi$
$68$	768.056	1.36971
$69$	0	0
$70$	485.718	0.829349
$71$	628.604	1.05073	0.525363	−	0.850878i	$-0.323930\pi$
$71$	628.604	1.05073	0.525363	+	0.850878i	$0.323930\pi$
$72$	−74.1246	−0.121329
$73$	925.266	1.48348	0.741741	−	0.670686i	$-0.234000\pi$
$73$	925.266	1.48348	0.741741	+	0.670686i	$0.234000\pi$
$74$	−309.692	−0.486499
$75$	−27.4133	−0.0422055
$76$	−505.072	−0.762312
$77$	−557.771	−0.825505
$78$	1004.99	1.45888
$79$	963.479	1.37215	0.686075	−	0.727531i	$-0.259332\pi$
$79$	963.479	1.37215	0.686075	+	0.727531i	$0.259332\pi$
$80$	−480.778	−0.671907
$81$	81.0000	0.111111
$82$	164.748	0.221870
$83$	−133.358	−0.176360	−0.0881801	−	0.996105i	$-0.528105\pi$
$83$	−133.358	−0.176360	−0.0881801	+	0.996105i	$0.528105\pi$
$84$	−317.793	−0.412787
$85$	831.364	1.06087
$86$	727.702	0.912443
$87$	38.2430	0.0471274
$88$	−431.246	−0.522398
$89$	778.581	0.927297	0.463649	−	0.886019i	$-0.346540\pi$
$89$	778.581	0.927297	0.463649	+	0.886019i	$0.346540\pi$
$90$	−410.371	−0.480632
$91$	842.420	0.970435
$92$	0	0
$93$	−37.2399	−0.0415226
$94$	−2603.25	−2.85644
$95$	−546.703	−0.590426
$96$	765.286	0.813611
$97$	−1603.57	−1.67853	−0.839266	−	0.543721i	$-0.817015\pi$
$97$	−1603.57	−1.67853	−0.839266	+	0.543721i	$0.817015\pi$
$98$	972.282	1.00220
$99$	471.246	0.478404
$100$	−90.8684	−0.0908684
$101$	1229.04	1.21084	0.605418	−	0.795908i	$-0.293006\pi$
$101$	1229.04	1.21084	0.605418	+	0.795908i	$0.293006\pi$
$102$	−981.532	−0.952805
$103$	−215.066	−0.205738	−0.102869	−	0.994695i	$-0.532802\pi$
$103$	−215.066	−0.205738	−0.102869	+	0.994695i	$0.532802\pi$
$104$	651.325	0.614112
$105$	−343.988	−0.319712
$106$	1142.03	1.04645
$107$	1586.58	1.43346	0.716730	−	0.697351i	$-0.245638\pi$
$107$	1586.58	1.43346	0.716730	+	0.697351i	$0.245638\pi$
$108$	268.495	0.239222
$109$	−822.255	−0.722549	−0.361274	−	0.932460i	$-0.617658\pi$
$109$	−822.255	−0.722549	−0.361274	+	0.932460i	$0.617658\pi$
$110$	−2387.48	−2.06943
$111$	219.325	0.187544
$112$	475.800	0.401418
$113$	−19.7005	−0.0164006	−0.00820031	−	0.999966i	$-0.502610\pi$
$113$	−19.7005	−0.0164006	−0.00820031	+	0.999966i	$0.502610\pi$
$114$	645.453	0.530282
$115$	0	0
$116$	126.766	0.101465
$117$	−711.738	−0.562395
$118$	2265.31	1.76728
$119$	−822.755	−0.633797
$120$	−265.957	−0.202321
$121$	1410.64	1.05984
$122$	−3552.38	−2.63621
$123$	−116.675	−0.0855303
$124$	−123.441	−0.0893980
$125$	−1443.85	−1.03313
$126$	406.122	0.287144
$127$	1143.23	0.798784	0.399392	−	0.916780i	$-0.369221\pi$
$127$	1143.23	0.798784	0.399392	+	0.916780i	$0.369221\pi$
$128$	1023.08	0.706473
$129$	−515.361	−0.351745
$130$	3605.88	2.43275
$131$	1154.64	0.770085	0.385043	−	0.922899i	$-0.374187\pi$
$131$	1154.64	0.770085	0.385043	+	0.922899i	$0.374187\pi$
$132$	1562.07	1.03000
$133$	541.042	0.352739
$134$	−1898.52	−1.22394
$135$	290.626	0.185282
$136$	−636.122	−0.401081
$137$	2153.10	1.34271	0.671356	−	0.741135i	$-0.265712\pi$
$137$	2153.10	1.34271	0.671356	+	0.741135i	$0.265712\pi$
$138$	0	0
$139$	−2392.16	−1.45971	−0.729857	−	0.683600i	$-0.760413\pi$
$139$	−2392.16	−1.45971	−0.729857	+	0.683600i	$0.760413\pi$
$140$	−1140.24	−0.688339
$141$	1843.63	1.10115
$142$	−2662.81	−1.57365
$143$	−4140.79	−2.42147
$144$	−401.991	−0.232633
$145$	137.215	0.0785868
$146$	−3919.49	−2.22178
$147$	−688.574	−0.386345
$148$	727.009	0.403782
$149$	2259.80	1.24248	0.621242	−	0.783619i	$-0.286628\pi$
$149$	2259.80	1.24248	0.621242	+	0.783619i	$0.286628\pi$
$150$	116.125	0.0632102
$151$	−2410.84	−1.29928	−0.649639	−	0.760243i	$-0.725080\pi$
$151$	−2410.84	−1.29928	−0.649639	+	0.760243i	$0.725080\pi$
$152$	418.312	0.223221
$153$	695.125	0.367304
$154$	2362.76	1.23634
$155$	−133.616	−0.0692406
$156$	−2359.24	−1.21084
$157$	614.960	0.312606	0.156303	−	0.987709i	$-0.450042\pi$
$157$	614.960	0.312606	0.156303	+	0.987709i	$0.450042\pi$
$158$	−4081.36	−2.05504
$159$	−808.790	−0.403404
$160$	2745.83	1.35673
$161$	0	0
$162$	−343.122	−0.166408
$163$	−50.8127	−0.0244169	−0.0122085	−	0.999925i	$-0.503886\pi$
$163$	−50.8127	−0.0244169	−0.0122085	+	0.999925i	$0.503886\pi$
$164$	−386.749	−0.184147
$165$	1690.82	0.797759
$166$	564.912	0.264130
$167$	−2000.84	−0.927125	−0.463562	−	0.886064i	$-0.653429\pi$
$167$	−2000.84	−0.927125	−0.463562	+	0.886064i	$0.653429\pi$
$168$	263.204	0.120873
$169$	4056.97	1.84659
$170$	−3521.71	−1.58884
$171$	−457.112	−0.204422
$172$	−1708.30	−0.757305
$173$	−3796.78	−1.66858	−0.834288	−	0.551330i	$-0.814121\pi$
$173$	−3796.78	−1.66858	−0.834288	+	0.551330i	$0.814121\pi$
$174$	−162.000	−0.0705815
$175$	97.3398	0.0420469
$176$	−2338.72	−1.00164
$177$	−1604.30	−0.681281
$178$	−3298.12	−1.38879
$179$	−3084.16	−1.28783	−0.643913	−	0.765099i	$-0.722690\pi$
$179$	−3084.16	−1.28783	−0.643913	+	0.765099i	$0.722690\pi$
$180$	963.355	0.398913
$181$	−4733.12	−1.94370	−0.971851	−	0.235597i	$-0.924295\pi$
$181$	−4733.12	−1.94370	−0.971851	+	0.235597i	$0.924295\pi$
$182$	−3568.55	−1.45340
$183$	2515.81	1.01625
$184$	0	0
$185$	786.933	0.312738
$186$	157.751	0.0621874
$187$	4044.13	1.58148
$188$	6111.20	2.37077
$189$	−287.617	−0.110693
$190$	2315.87	0.884268
$191$	1293.13	0.489885	0.244942	−	0.969538i	$-0.421231\pi$
$191$	1293.13	0.489885	0.244942	+	0.969538i	$0.421231\pi$
$192$	−2169.83	−0.815592
$193$	−1171.75	−0.437019	−0.218510	−	0.975835i	$-0.570119\pi$
$193$	−1171.75	−0.437019	−0.218510	+	0.975835i	$0.570119\pi$
$194$	6792.82	2.51390
$195$	−2553.70	−0.937817
$196$	−2282.46	−0.831799
$197$	4559.19	1.64888	0.824438	−	0.565952i	$-0.191491\pi$
$197$	4559.19	1.64888	0.824438	+	0.565952i	$0.191491\pi$
$198$	−1996.23	−0.716495
$199$	473.721	0.168750	0.0843748	−	0.996434i	$-0.473111\pi$
$199$	473.721	0.168750	0.0843748	+	0.996434i	$0.473111\pi$
$200$	75.2593	0.0266082
$201$	1344.54	0.471824
$202$	−5206.31	−1.81344
$203$	−135.794	−0.0469502
$204$	2304.17	0.790805
$205$	−418.627	−0.142625
$206$	911.033	0.308130
$207$	0	0
$208$	3532.25	1.17749
$209$	−2659.41	−0.880169
$210$	1457.15	0.478825
$211$	−1423.56	−0.464465	−0.232232	−	0.972660i	$-0.574603\pi$
$211$	−1423.56	−0.464465	−0.232232	+	0.972660i	$0.574603\pi$
$212$	−2680.94	−0.868528
$213$	1885.81	0.606637
$214$	−6720.85	−2.14686
$215$	−1849.11	−0.586548
$216$	−222.374	−0.0700492
$217$	132.232	0.0413665
$218$	3483.13	1.08214
$219$	2775.80	0.856489
$220$	5604.66	1.71757
$221$	−6107.99	−1.85913
$222$	−929.076	−0.280881
$223$	6088.63	1.82836	0.914181	−	0.405305i	$-0.132835\pi$
$223$	6088.63	1.82836	0.914181	+	0.405305i	$0.132835\pi$
$224$	−2717.40	−0.810552
$225$	−82.2399	−0.0243674
$226$	83.4528	0.0245628
$227$	4463.66	1.30513	0.652563	−	0.757734i	$-0.273694\pi$
$227$	4463.66	1.30513	0.652563	+	0.757734i	$0.273694\pi$
$228$	−1515.22	−0.440121
$229$	−1298.87	−0.374812	−0.187406	−	0.982283i	$-0.560008\pi$
$229$	−1298.87	−0.374812	−0.187406	+	0.982283i	$0.560008\pi$
$230$	0	0
$231$	−1673.31	−0.476606
$232$	−104.991	−0.0297111
$233$	−1257.41	−0.353544	−0.176772	−	0.984252i	$-0.556566\pi$
$233$	−1257.41	−0.353544	−0.176772	+	0.984252i	$0.556566\pi$
$234$	3014.97	0.842286
$235$	6614.92	1.83621
$236$	−5317.88	−1.46680
$237$	2890.44	0.792211
$238$	3485.25	0.949223
$239$	3473.96	0.940217	0.470108	−	0.882609i	$-0.344215\pi$
$239$	3473.96	0.940217	0.470108	+	0.882609i	$0.344215\pi$
$240$	−1442.33	−0.387926
$241$	2140.92	0.572235	0.286118	−	0.958195i	$-0.407635\pi$
$241$	2140.92	0.572235	0.286118	+	0.958195i	$0.407635\pi$
$242$	−5975.57	−1.58729
$243$	243.000	0.0641500
$244$	8339.30	2.18799
$245$	−2470.59	−0.644245
$246$	494.243	0.128097
$247$	4016.60	1.03470
$248$	102.237	0.0261776
$249$	−400.073	−0.101822
$250$	6116.25	1.54730
$251$	5613.17	1.41155	0.705777	−	0.708434i	$-0.250598\pi$
$251$	5613.17	1.41155	0.705777	+	0.708434i	$0.250598\pi$
$252$	−953.380	−0.238323
$253$	0	0
$254$	−4842.82	−1.19632
$255$	2494.09	0.612494
$256$	1452.36	0.354579
$257$	1017.98	0.247082	0.123541	−	0.992339i	$-0.460575\pi$
$257$	1017.98	0.247082	0.123541	+	0.992339i	$0.460575\pi$
$258$	2183.11	0.526799
$259$	−778.785	−0.186839
$260$	−8464.90	−2.01912
$261$	114.729	0.0272090
$262$	−4891.12	−1.15334
$263$	995.531	0.233411	0.116705	−	0.993167i	$-0.462767\pi$
$263$	995.531	0.233411	0.116705	+	0.993167i	$0.462767\pi$
$264$	−1293.74	−0.301606
$265$	−2901.92	−0.672693
$266$	−2291.89	−0.528289
$267$	2335.74	0.535375
$268$	4456.83	1.01584
$269$	−3568.95	−0.808932	−0.404466	−	0.914553i	$-0.632543\pi$
$269$	−3568.95	−0.808932	−0.404466	+	0.914553i	$0.632543\pi$
$270$	−1231.11	−0.277493
$271$	−6294.26	−1.41088	−0.705441	−	0.708768i	$-0.749251\pi$
$271$	−6294.26	−1.41088	−0.705441	+	0.708768i	$0.749251\pi$
$272$	−3449.80	−0.769025
$273$	2527.26	0.560281
$274$	−9120.67	−2.01095
$275$	−478.460	−0.104917
$276$	0	0
$277$	−2263.65	−0.491009	−0.245505	−	0.969395i	$-0.578954\pi$
$277$	−2263.65	−0.491009	−0.245505	+	0.969395i	$0.578954\pi$
$278$	10133.3	2.18618
$279$	−111.720	−0.0239731
$280$	944.368	0.201560
$281$	505.900	0.107400	0.0537001	−	0.998557i	$-0.482898\pi$
$281$	505.900	0.107400	0.0537001	+	0.998557i	$0.482898\pi$
$282$	−7809.76	−1.64916
$283$	7018.20	1.47417	0.737083	−	0.675802i	$-0.236203\pi$
$283$	7018.20	1.47417	0.737083	+	0.675802i	$0.236203\pi$
$284$	6251.01	1.30609
$285$	−1640.11	−0.340883
$286$	17540.7	3.62658
$287$	414.292	0.0852087
$288$	2295.86	0.469738
$289$	1052.41	0.214209
$290$	−581.252	−0.117698
$291$	−4810.70	−0.969101
$292$	9201.10	1.84402
$293$	9488.82	1.89196	0.945978	−	0.324232i	$-0.105106\pi$
$293$	9488.82	1.89196	0.945978	+	0.324232i	$0.105106\pi$
$294$	2916.85	0.578619
$295$	−5756.20	−1.13606
$296$	−602.125	−0.118236
$297$	1413.74	0.276207
$298$	−9572.67	−1.86084
$299$	0	0
$300$	−272.605	−0.0524629
$301$	1829.96	0.350422
$302$	10212.5	1.94590
$303$	3687.13	0.699076
$304$	2268.58	0.428000
$305$	9026.67	1.69464
$306$	−2944.60	−0.550102
$307$	4660.58	0.866427	0.433214	−	0.901291i	$-0.357380\pi$
$307$	4660.58	0.866427	0.433214	+	0.901291i	$0.357380\pi$
$308$	−5546.63	−1.02613
$309$	−645.197	−0.118783
$310$	566.006	0.103700
$311$	4378.51	0.798336	0.399168	−	0.916878i	$-0.369299\pi$
$311$	4378.51	0.798336	0.399168	+	0.916878i	$0.369299\pi$
$312$	1953.98	0.354558
$313$	4444.17	0.802553	0.401277	−	0.915957i	$-0.368567\pi$
$313$	4444.17	0.802553	0.401277	+	0.915957i	$0.368567\pi$
$314$	−2605.01	−0.468182
$315$	−1031.96	−0.184586
$316$	9581.10	1.70563
$317$	7881.61	1.39645	0.698226	−	0.715878i	$-0.253973\pi$
$317$	7881.61	1.39645	0.698226	+	0.715878i	$0.253973\pi$
$318$	3426.09	0.604169
$319$	667.477	0.117152
$320$	−7785.29	−1.36003
$321$	4759.73	0.827609
$322$	0	0
$323$	−3922.84	−0.675767
$324$	805.486	0.138115
$325$	722.633	0.123337
$326$	215.246	0.0365686
$327$	−2466.77	−0.417164
$328$	320.314	0.0539219
$329$	−6546.42	−1.09701
$330$	−7162.43	−1.19478
$331$	−127.060	−0.0210993	−0.0105497	−	0.999944i	$-0.503358\pi$
$331$	−127.060	−0.0210993	−0.0105497	+	0.999944i	$0.503358\pi$
$332$	−1326.14	−0.219222
$333$	657.975	0.108279
$334$	8475.70	1.38853
$335$	4824.18	0.786786
$336$	1427.40	0.231759
$337$	1739.07	0.281107	0.140554	−	0.990073i	$-0.455112\pi$
$337$	1739.07	0.281107	0.140554	+	0.990073i	$0.455112\pi$
$338$	−17185.6	−2.76560
$339$	−59.1016	−0.00946890
$340$	8267.31	1.31870
$341$	−649.969	−0.103219
$342$	1936.36	0.306159
$343$	6098.81	0.960072
$344$	1414.85	0.221755
$345$	0	0
$346$	16083.4	2.49899
$347$	7551.30	1.16823	0.584114	−	0.811672i	$-0.301442\pi$
$347$	7551.30	1.16823	0.584114	+	0.811672i	$0.301442\pi$
$348$	380.299	0.0585809
$349$	−6237.28	−0.956659	−0.478329	−	0.878180i	$-0.658758\pi$
$349$	−6237.28	−0.956659	−0.478329	+	0.878180i	$0.658758\pi$
$350$	−412.338	−0.0629726
$351$	−2135.22	−0.324699
$352$	13357.0	2.02252
$353$	1788.57	0.269678	0.134839	−	0.990868i	$-0.456948\pi$
$353$	1788.57	0.269678	0.134839	+	0.990868i	$0.456948\pi$
$354$	6795.94	1.02034
$355$	6766.25	1.01159
$356$	7742.42	1.15266
$357$	−2468.27	−0.365923
$358$	13064.7	1.92875
$359$	−2234.70	−0.328531	−0.164266	−	0.986416i	$-0.552525\pi$
$359$	−2234.70	−0.328531	−0.164266	+	0.986416i	$0.552525\pi$
$360$	−797.872	−0.116810
$361$	−4279.35	−0.623903
$362$	20049.8	2.91104
$363$	4231.92	0.611896
$364$	8377.25	1.20628
$365$	9959.50	1.42823
$366$	−10657.1	−1.52202
$367$	−10551.7	−1.50080	−0.750399	−	0.660985i	$-0.770139\pi$
$367$	−10551.7	−1.50080	−0.750399	+	0.660985i	$0.770139\pi$
$368$	0	0
$369$	−350.025	−0.0493809
$370$	−3333.50	−0.468380
$371$	2871.87	0.401887
$372$	−370.324	−0.0516140
$373$	996.068	0.138269	0.0691347	−	0.997607i	$-0.477976\pi$
$373$	996.068	0.138269	0.0691347	+	0.997607i	$0.477976\pi$
$374$	−17131.2	−2.36854
$375$	−4331.55	−0.596481
$376$	−5061.43	−0.694211
$377$	−1008.11	−0.137720
$378$	1218.36	0.165783
$379$	11117.0	1.50670	0.753351	−	0.657619i	$-0.228436\pi$
$379$	11117.0	1.50670	0.753351	+	0.657619i	$0.228436\pi$
$380$	−5436.56	−0.733920
$381$	3429.70	0.461178
$382$	−5477.81	−0.733688
$383$	−5315.09	−0.709108	−0.354554	−	0.935035i	$-0.615367\pi$
$383$	−5315.09	−0.709108	−0.354554	+	0.935035i	$0.615367\pi$
$384$	3069.25	0.407883
$385$	−6003.81	−0.794759
$386$	4963.63	0.654513
$387$	−1546.08	−0.203080
$388$	−15946.3	−2.08647
$389$	8438.06	1.09981	0.549906	−	0.835227i	$-0.314664\pi$
$389$	8438.06	1.09981	0.549906	+	0.835227i	$0.314664\pi$
$390$	10817.7	1.40455
$391$	0	0
$392$	1890.38	0.243568
$393$	3463.91	0.444609
$394$	−19313.0	−2.46948
$395$	10370.8	1.32104
$396$	4686.20	0.594673
$397$	11582.0	1.46419	0.732094	−	0.681204i	$-0.238543\pi$
$397$	11582.0	1.46419	0.732094	+	0.681204i	$0.238543\pi$
$398$	−2006.71	−0.252732
$399$	1623.13	0.203654
$400$	408.144	0.0510180
$401$	−4267.85	−0.531487	−0.265744	−	0.964044i	$-0.585617\pi$
$401$	−4267.85	−0.531487	−0.265744	+	0.964044i	$0.585617\pi$
$402$	−5695.57	−0.706639
$403$	981.669	0.121341
$404$	12221.9	1.50511
$405$	871.878	0.106973
$406$	575.234	0.0703162
$407$	3828.00	0.466209
$408$	−1908.36	−0.231564
$409$	6902.51	0.834491	0.417246	−	0.908794i	$-0.362995\pi$
$409$	6902.51	0.834491	0.417246	+	0.908794i	$0.362995\pi$
$410$	1773.33	0.213606
$411$	6459.29	0.775215
$412$	−2138.67	−0.255740
$413$	5696.60	0.678720
$414$	0	0
$415$	−1435.45	−0.169792
$416$	−20173.4	−2.37761
$417$	−7176.47	−0.842766
$418$	11265.4	1.31821
$419$	2241.24	0.261316	0.130658	−	0.991427i	$-0.458291\pi$
$419$	2241.24	0.261316	0.130658	+	0.991427i	$0.458291\pi$
$420$	−3420.71	−0.397413
$421$	−9117.52	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$421$	−9117.52	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$422$	6030.30	0.695618
$423$	5530.90	0.635749
$424$	2220.42	0.254323
$425$	−705.765	−0.0805521
$426$	−7988.42	−0.908546
$427$	−8933.21	−1.01243
$428$	15777.4	1.78184
$429$	−12422.4	−1.39804
$430$	7832.93	0.878460
$431$	1366.55	0.152725	0.0763623	−	0.997080i	$-0.475669\pi$
$431$	1366.55	0.152725	0.0763623	+	0.997080i	$0.475669\pi$
$432$	−1205.97	−0.134311
$433$	−3511.21	−0.389695	−0.194848	−	0.980834i	$-0.562421\pi$
$433$	−3511.21	−0.389695	−0.194848	+	0.980834i	$0.562421\pi$
$434$	−560.145	−0.0619536
$435$	411.645	0.0453721
$436$	−8176.73	−0.898152
$437$	0	0
$438$	−11758.5	−1.28274
$439$	7032.17	0.764526	0.382263	−	0.924054i	$-0.375145\pi$
$439$	7032.17	0.764526	0.382263	+	0.924054i	$0.375145\pi$
$440$	−4641.90	−0.502941
$441$	−2065.72	−0.223056
$442$	25873.8	2.78437
$443$	8748.68	0.938289	0.469145	−	0.883121i	$-0.344562\pi$
$443$	8748.68	0.938289	0.469145	+	0.883121i	$0.344562\pi$
$444$	2181.03	0.233124
$445$	8380.60	0.892760
$446$	−25791.9	−2.73830
$447$	6779.40	0.717348
$448$	7704.68	0.812526
$449$	−805.208	−0.0846327	−0.0423164	−	0.999104i	$-0.513474\pi$
$449$	−805.208	−0.0846327	−0.0423164	+	0.999104i	$0.513474\pi$
$450$	348.374	0.0364944
$451$	−2036.39	−0.212616
$452$	−195.907	−0.0203865
$453$	−7232.51	−0.750139
$454$	−18908.4	−1.95466
$455$	9067.75	0.934291
$456$	1254.94	0.128877
$457$	10496.9	1.07445	0.537226	−	0.843438i	$-0.319472\pi$
$457$	10496.9	1.07445	0.537226	+	0.843438i	$0.319472\pi$
$458$	5502.12	0.561348
$459$	2085.37	0.212063
$460$	0	0
$461$	−3325.68	−0.335992	−0.167996	−	0.985788i	$-0.553730\pi$
$461$	−3325.68	−0.335992	−0.167996	+	0.985788i	$0.553730\pi$
$462$	7088.27	0.713801
$463$	−840.876	−0.0844035	−0.0422018	−	0.999109i	$-0.513437\pi$
$463$	−840.876	−0.0844035	−0.0422018	+	0.999109i	$0.513437\pi$
$464$	−569.383	−0.0569675
$465$	−400.848	−0.0399761
$466$	5326.48	0.529495
$467$	−8450.82	−0.837382	−0.418691	−	0.908129i	$-0.637511\pi$
$467$	−8450.82	−0.837382	−0.418691	+	0.908129i	$0.637511\pi$
$468$	−7077.72	−0.699076
$469$	−4774.23	−0.470050
$470$	−28021.2	−2.75005
$471$	1844.88	0.180483
$472$	4404.38	0.429509
$473$	−8994.89	−0.874388
$474$	−12244.1	−1.18648
$475$	464.109	0.0448312
$476$	−8181.70	−0.787831
$477$	−2426.37	−0.232905
$478$	−14715.9	−1.40814
$479$	5686.86	0.542462	0.271231	−	0.962514i	$-0.412569\pi$
$479$	5686.86	0.542462	0.271231	+	0.962514i	$0.412569\pi$
$480$	8237.48	0.783308
$481$	−5781.56	−0.548059
$482$	−9069.07	−0.857023
$483$	0	0
$484$	14027.8	1.31741
$485$	−17260.7	−1.61602
$486$	−1029.36	−0.0960760
$487$	−13768.2	−1.28110	−0.640551	−	0.767916i	$-0.721294\pi$
$487$	−13768.2	−1.28110	−0.640551	+	0.767916i	$0.721294\pi$
$488$	−6906.80	−0.640689
$489$	−152.438	−0.0140971
$490$	10465.6	0.964871
$491$	8480.14	0.779436	0.389718	−	0.920934i	$-0.372572\pi$
$491$	8480.14	0.779436	0.389718	+	0.920934i	$0.372572\pi$
$492$	−1160.25	−0.106317
$493$	984.580	0.0899457
$494$	−17014.6	−1.54964
$495$	5072.46	0.460586
$496$	554.448	0.0501924
$497$	−6696.19	−0.604356
$498$	1694.74	0.152496
$499$	12472.5	1.11893	0.559466	−	0.828853i	$-0.311006\pi$
$499$	12472.5	1.11893	0.559466	+	0.828853i	$0.311006\pi$
$500$	−14358.0	−1.28422
$501$	−6002.53	−0.535276
$502$	−23777.8	−2.11405
$503$	4676.36	0.414530	0.207265	−	0.978285i	$-0.433544\pi$
$503$	4676.36	0.414530	0.207265	+	0.978285i	$0.433544\pi$
$504$	789.611	0.0697858
$505$	13229.3	1.16574
$506$	0	0
$507$	12170.9	1.06613
$508$	11368.6	0.992916
$509$	5925.58	0.516005	0.258003	−	0.966144i	$-0.416936\pi$
$509$	5925.58	0.516005	0.258003	+	0.966144i	$0.416936\pi$
$510$	−10565.1	−0.917318
$511$	−9856.38	−0.853269
$512$	−14336.9	−1.23752
$513$	−1371.34	−0.118023
$514$	−4312.25	−0.370049
$515$	−2314.95	−0.198076
$516$	−5124.89	−0.437230
$517$	32178.0	2.73730
$518$	3298.99	0.279825
$519$	−11390.3	−0.963352
$520$	7010.82	0.591240
$521$	−2626.25	−0.220841	−0.110421	−	0.993885i	$-0.535220\pi$
$521$	−2626.25	−0.220841	−0.110421	+	0.993885i	$0.535220\pi$
$522$	−486.000	−0.0407503
$523$	−372.504	−0.0311443	−0.0155721	−	0.999879i	$-0.504957\pi$
$523$	−372.504	−0.0311443	−0.0155721	+	0.999879i	$0.504957\pi$
$524$	11482.0	0.957242
$525$	292.020	0.0242758
$526$	−4217.14	−0.349574
$527$	−958.755	−0.0792486
$528$	−7016.17	−0.578295
$529$	0	0
$530$	12292.7	1.00748
$531$	−4812.91	−0.393338
$532$	5380.27	0.438466
$533$	3075.63	0.249944
$534$	−9894.37	−0.801819
$535$	17077.8	1.38007
$536$	−3691.24	−0.297458
$537$	−9252.47	−0.743526
$538$	15118.3	1.21152
$539$	−12018.1	−0.960399
$540$	2890.07	0.230312
$541$	12899.2	1.02510	0.512550	−	0.858657i	$-0.328701\pi$
$541$	12899.2	1.02510	0.512550	+	0.858657i	$0.328701\pi$
$542$	26662.9	2.11305
$543$	−14199.4	−1.12220
$544$	19702.5	1.55283
$545$	−8850.70	−0.695637
$546$	−10705.6	−0.839119
$547$	−22144.7	−1.73097	−0.865485	−	0.500936i	$-0.832989\pi$
$547$	−22144.7	−1.73097	−0.865485	+	0.500936i	$0.832989\pi$
$548$	21411.0	1.66904
$549$	7547.43	0.586733
$550$	2026.79	0.157132
$551$	−647.457	−0.0500592
$552$	0	0
$553$	−10263.4	−0.789233
$554$	9588.98	0.735373
$555$	2360.80	0.180559
$556$	−23788.3	−1.81447
$557$	10236.7	0.778716	0.389358	−	0.921086i	$-0.372697\pi$
$557$	10236.7	0.778716	0.389358	+	0.921086i	$0.372697\pi$
$558$	473.252	0.0359039
$559$	13585.3	1.02790
$560$	5121.47	0.386467
$561$	12132.4	0.913066
$562$	−2143.03	−0.160851
$563$	−5750.83	−0.430495	−0.215247	−	0.976560i	$-0.569056\pi$
$563$	−5750.83	−0.430495	−0.215247	+	0.976560i	$0.569056\pi$
$564$	18333.6	1.36877
$565$	−212.055	−0.0157898
$566$	−29729.6	−2.20782
$567$	−862.851	−0.0639088
$568$	−5177.22	−0.382450
$569$	8285.10	0.610421	0.305210	−	0.952285i	$-0.401273\pi$
$569$	8285.10	0.610421	0.305210	+	0.952285i	$0.401273\pi$
$570$	6947.61	0.510532
$571$	19799.8	1.45113	0.725565	−	0.688154i	$-0.241579\pi$
$571$	19799.8	1.45113	0.725565	+	0.688154i	$0.241579\pi$
$572$	−41177.1	−3.00997
$573$	3879.40	0.282835
$574$	−1754.97	−0.127615
$575$	0	0
$576$	−6509.48	−0.470883
$577$	−250.868	−0.0181001	−0.00905006	−	0.999959i	$-0.502881\pi$
$577$	−250.868	−0.0181001	−0.00905006	+	0.999959i	$0.502881\pi$
$578$	−4458.08	−0.320816
$579$	−3515.26	−0.252313
$580$	1364.50	0.0976861
$581$	1420.59	0.101439
$582$	20378.5	1.45140
$583$	−14116.3	−1.00281
$584$	−7620.56	−0.539967
$585$	−7661.10	−0.541449
$586$	−40195.3	−2.83354
$587$	−9767.11	−0.686766	−0.343383	−	0.939195i	$-0.611573\pi$
$587$	−9767.11	−0.686766	−0.343383	+	0.939195i	$0.611573\pi$
$588$	−6847.37	−0.480239
$589$	630.475	0.0441057
$590$	24383.7	1.70146
$591$	13677.6	0.951979
$592$	−3265.43	−0.226703
$593$	−18853.8	−1.30562	−0.652810	−	0.757521i	$-0.726410\pi$
$593$	−18853.8	−1.30562	−0.652810	+	0.757521i	$0.726410\pi$
$594$	−5988.69	−0.413668
$595$	−8856.08	−0.610192
$596$	22472.1	1.54445
$597$	1421.16	0.0974276
$598$	0	0
$599$	−11356.0	−0.774615	−0.387307	−	0.921951i	$-0.626595\pi$
$599$	−11356.0	−0.774615	−0.387307	+	0.921951i	$0.626595\pi$
$600$	225.778	0.0153622
$601$	20359.1	1.38180	0.690902	−	0.722948i	$-0.257213\pi$
$601$	20359.1	1.38180	0.690902	+	0.722948i	$0.257213\pi$
$602$	−7751.83	−0.524819
$603$	4033.62	0.272408
$604$	−23974.0	−1.61505
$605$	15184.0	1.02036
$606$	−15618.9	−1.04699
$607$	5975.14	0.399545	0.199772	−	0.979842i	$-0.435980\pi$
$607$	5975.14	0.399545	0.199772	+	0.979842i	$0.435980\pi$
$608$	−12956.3	−0.864226
$609$	−407.383	−0.0271067
$610$	−38237.6	−2.53802
$611$	−48599.5	−3.21788
$612$	6912.51	0.456571
$613$	−16997.4	−1.11993	−0.559967	−	0.828515i	$-0.689186\pi$
$613$	−16997.4	−1.11993	−0.559967	+	0.828515i	$0.689186\pi$
$614$	−19742.5	−1.29763
$615$	−1255.88	−0.0823447
$616$	4593.84	0.300472
$617$	13096.0	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$617$	13096.0	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$618$	2733.10	0.177899
$619$	−702.058	−0.0455866	−0.0227933	−	0.999740i	$-0.507256\pi$
$619$	−702.058	−0.0455866	−0.0227933	+	0.999740i	$0.507256\pi$
$620$	−1328.71	−0.0860684
$621$	0	0
$622$	−18547.7	−1.19565
$623$	−8293.82	−0.533362
$624$	10596.7	0.679823
$625$	−14399.3	−0.921554
$626$	−18825.8	−1.20196
$627$	−7978.23	−0.508166
$628$	6115.33	0.388580
$629$	5646.60	0.357941
$630$	4371.46	0.276450
$631$	−23871.5	−1.50604	−0.753019	−	0.657999i	$-0.771403\pi$
$631$	−23871.5	−1.50604	−0.753019	+	0.657999i	$0.771403\pi$
$632$	−7935.28	−0.499444
$633$	−4270.69	−0.268159
$634$	−33387.0	−2.09143
$635$	12305.7	0.769034
$636$	−8042.83	−0.501445
$637$	18151.3	1.12901
$638$	−2827.48	−0.175456
$639$	5657.43	0.350242
$640$	11012.4	0.680161
$641$	−24172.6	−1.48949	−0.744743	−	0.667351i	$-0.767428\pi$
$641$	−24172.6	−1.48949	−0.744743	+	0.667351i	$0.767428\pi$
$642$	−20162.5	−1.23949
$643$	−13648.6	−0.837090	−0.418545	−	0.908196i	$-0.637460\pi$
$643$	−13648.6	−0.837090	−0.418545	+	0.908196i	$0.637460\pi$
$644$	0	0
$645$	−5547.32	−0.338644
$646$	16617.4	1.01208
$647$	−7009.18	−0.425903	−0.212952	−	0.977063i	$-0.568308\pi$
$647$	−7009.18	−0.425903	−0.212952	+	0.977063i	$0.568308\pi$
$648$	−667.122	−0.0404429
$649$	−28000.8	−1.69357
$650$	−3061.12	−0.184719
$651$	396.697	0.0238829
$652$	−505.295	−0.0303511
$653$	4533.57	0.271688	0.135844	−	0.990730i	$-0.456625\pi$
$653$	4533.57	0.271688	0.135844	+	0.990730i	$0.456625\pi$
$654$	10449.4	0.624776
$655$	12428.4	0.741404
$656$	1737.12	0.103389
$657$	8327.40	0.494494
$658$	27731.1	1.64296
$659$	7112.62	0.420438	0.210219	−	0.977654i	$-0.432582\pi$
$659$	7112.62	0.420438	0.210219	+	0.977654i	$0.432582\pi$
$660$	16814.0	0.991641
$661$	−16088.6	−0.946709	−0.473354	−	0.880872i	$-0.656957\pi$
$661$	−16088.6	−0.946709	−0.473354	+	0.880872i	$0.656957\pi$
$662$	538.237	0.0316000
$663$	−18324.0	−1.07337
$664$	1098.34	0.0641927
$665$	5823.74	0.339601
$666$	−2787.23	−0.162166
$667$	0	0
$668$	−19896.9	−1.15245
$669$	18265.9	1.05561
$670$	−20435.6	−1.17835
$671$	43909.9	2.52626
$672$	−8152.19	−0.467972
$673$	22629.7	1.29615	0.648076	−	0.761576i	$-0.275574\pi$
$673$	22629.7	1.29615	0.648076	+	0.761576i	$0.275574\pi$
$674$	−7366.82	−0.421008
$675$	−246.720	−0.0140685
$676$	40343.6	2.29538
$677$	15444.0	0.876749	0.438375	−	0.898792i	$-0.355554\pi$
$677$	15444.0	0.876749	0.438375	+	0.898792i	$0.355554\pi$
$678$	250.358	0.0141813
$679$	17082.0	0.965458
$680$	−6847.17	−0.386143
$681$	13391.0	0.753515
$682$	2753.31	0.154589
$683$	834.894	0.0467736	0.0233868	−	0.999726i	$-0.492555\pi$
$683$	834.894	0.0467736	0.0233868	+	0.999726i	$0.492555\pi$
$684$	−4545.65	−0.254104
$685$	23175.8	1.29270
$686$	−25835.0	−1.43788
$687$	−3896.62	−0.216398
$688$	7672.98	0.425188
$689$	21320.3	1.17886
$690$	0	0
$691$	4982.99	0.274330	0.137165	−	0.990548i	$-0.456201\pi$
$691$	4982.99	0.274330	0.137165	+	0.990548i	$0.456201\pi$
$692$	−37756.2	−2.07410
$693$	−5019.94	−0.275168
$694$	−31987.8	−1.74963
$695$	−25749.0	−1.40535
$696$	−314.972	−0.0171537
$697$	−3003.84	−0.163240
$698$	26421.5	1.43277
$699$	−3772.23	−0.204119
$700$	967.974	0.0522657
$701$	−21916.9	−1.18087	−0.590436	−	0.807084i	$-0.701044\pi$
$701$	−21916.9	−1.18087	−0.590436	+	0.807084i	$0.701044\pi$
$702$	9044.92	0.486294
$703$	−3713.19	−0.199211
$704$	−37871.2	−2.02745
$705$	19844.8	1.06014
$706$	−7576.52	−0.403890
$707$	−13092.4	−0.696448
$708$	−15953.6	−0.846856
$709$	30858.3	1.63456	0.817282	−	0.576237i	$-0.195480\pi$
$709$	30858.3	1.63456	0.817282	+	0.576237i	$0.195480\pi$
$710$	−28662.3	−1.51504
$711$	8671.31	0.457383
$712$	−6412.45	−0.337523
$713$	0	0
$714$	10455.7	0.548034
$715$	−44571.2	−2.33128
$716$	−30669.7	−1.60081
$717$	10421.9	0.542834
$718$	9466.32	0.492033
$719$	19500.3	1.01146	0.505728	−	0.862693i	$-0.331224\pi$
$719$	19500.3	1.01146	0.505728	+	0.862693i	$0.331224\pi$
$720$	−4327.00	−0.223969
$721$	2290.98	0.118337
$722$	18127.6	0.934405
$723$	6422.75	0.330380
$724$	−47067.4	−2.41609
$725$	−116.485	−0.00596711
$726$	−17926.7	−0.916422
$727$	−26922.8	−1.37347	−0.686735	−	0.726908i	$-0.740957\pi$
$727$	−26922.8	−1.37347	−0.686735	+	0.726908i	$0.740957\pi$
$728$	−6938.22	−0.353225
$729$	729.000	0.0370370
$730$	−42189.1	−2.13903
$731$	−13268.2	−0.671328
$732$	25017.9	1.26324
$733$	22120.5	1.11465	0.557326	−	0.830294i	$-0.311827\pi$
$733$	22120.5	1.11465	0.557326	+	0.830294i	$0.311827\pi$
$734$	44697.6	2.24771
$735$	−7411.77	−0.371955
$736$	0	0
$737$	23467.0	1.17289
$738$	1482.73	0.0739567
$739$	−27225.8	−1.35523	−0.677617	−	0.735415i	$-0.736987\pi$
$739$	−27225.8	−1.35523	−0.677617	+	0.735415i	$0.736987\pi$
$740$	7825.48	0.388744
$741$	12049.8	0.597382
$742$	−12165.4	−0.601897
$743$	191.429	0.00945201	0.00472600	−	0.999989i	$-0.498496\pi$
$743$	191.429	0.00945201	0.00472600	+	0.999989i	$0.498496\pi$
$744$	306.710	0.0151136
$745$	24324.3	1.19621
$746$	−4219.41	−0.207083
$747$	−1200.22	−0.0587867
$748$	40216.0	1.96583
$749$	−16901.0	−0.824497
$750$	18348.7	0.893335
$751$	876.709	0.0425986	0.0212993	−	0.999773i	$-0.493220\pi$
$751$	876.709	0.0425986	0.0212993	+	0.999773i	$0.493220\pi$
$752$	−27449.0	−1.33107
$753$	16839.5	0.814961
$754$	4270.43	0.206260
$755$	−25950.1	−1.25089
$756$	−2860.14	−0.137596
$757$	26681.0	1.28103	0.640513	−	0.767948i	$-0.278722\pi$
$757$	26681.0	1.28103	0.640513	+	0.767948i	$0.278722\pi$
$758$	−47092.2	−2.25655
$759$	0	0
$760$	4502.68	0.214907
$761$	−6947.99	−0.330965	−0.165483	−	0.986213i	$-0.552918\pi$
$761$	−6947.99	−0.330965	−0.165483	+	0.986213i	$0.552918\pi$
$762$	−14528.4	−0.690696
$763$	8759.06	0.415595
$764$	12859.3	0.608943
$765$	7482.27	0.353624
$766$	22515.1	1.06201
$767$	42290.5	1.99090
$768$	4357.07	0.204716
$769$	−7189.91	−0.337159	−0.168579	−	0.985688i	$-0.553918\pi$
$769$	−7189.91	−0.337159	−0.168579	+	0.985688i	$0.553918\pi$
$770$	25432.5	1.19029
$771$	3053.95	0.142653
$772$	−11652.2	−0.543230
$773$	27591.1	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$773$	27591.1	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$774$	6549.32	0.304148
$775$	113.430	0.00525745
$776$	13207.1	0.610963
$777$	−2336.35	−0.107872
$778$	−35744.2	−1.64716
$779$	1975.32	0.0908512
$780$	−25394.7	−1.16574
$781$	32914.1	1.50801
$782$	0	0
$783$	344.187	0.0157091
$784$	10251.9	0.467013
$785$	6619.38	0.300963
$786$	−14673.4	−0.665880
$787$	−10322.4	−0.467541	−0.233770	−	0.972292i	$-0.575106\pi$
$787$	−10322.4	−0.467541	−0.233770	+	0.972292i	$0.575106\pi$
$788$	45337.8	2.04961
$789$	2986.59	0.134760
$790$	−43931.5	−1.97850
$791$	209.859	0.00943330
$792$	−3881.22	−0.174133
$793$	−66318.5	−2.96978
$794$	−49062.0	−2.19288
$795$	−8705.76	−0.388379
$796$	4710.81	0.209761
$797$	−40205.9	−1.78691	−0.893455	−	0.449153i	$-0.851726\pi$
$797$	−40205.9	−1.78691	−0.893455	+	0.449153i	$0.851726\pi$
$798$	−6875.67	−0.305008
$799$	47465.0	2.10162
$800$	−2331.00	−0.103017
$801$	7007.23	0.309099
$802$	18078.9	0.795995
$803$	48447.6	2.12911
$804$	13370.5	0.586493
$805$	0	0
$806$	−4158.42	−0.181730
$807$	−10706.8	−0.467037
$808$	−10122.5	−0.440727
$809$	−42375.9	−1.84160	−0.920802	−	0.390031i	$-0.872464\pi$
$809$	−42375.9	−1.84160	−0.920802	+	0.390031i	$0.872464\pi$
$810$	−3693.34	−0.160211
$811$	7082.68	0.306667	0.153333	−	0.988175i	$-0.450999\pi$
$811$	7082.68	0.306667	0.153333	+	0.988175i	$0.450999\pi$
$812$	−1350.37	−0.0583607
$813$	−18882.8	−0.814573
$814$	−16215.7	−0.698230
$815$	−546.945	−0.0235075
$816$	−10349.4	−0.443997
$817$	8725.11	0.373627
$818$	−29239.5	−1.24980
$819$	7581.78	0.323478
$820$	−4162.94	−0.177288
$821$	18154.3	0.771728	0.385864	−	0.922556i	$-0.373903\pi$
$821$	18154.3	0.771728	0.385864	+	0.922556i	$0.373903\pi$
$822$	−27362.0	−1.16102
$823$	10437.1	0.442058	0.221029	−	0.975267i	$-0.429059\pi$
$823$	10437.1	0.442058	0.221029	+	0.975267i	$0.429059\pi$
$824$	1771.30	0.0748860
$825$	−1435.38	−0.0605739
$826$	−24131.2	−1.01650
$827$	1486.57	0.0625068	0.0312534	−	0.999511i	$-0.490050\pi$
$827$	1486.57	0.0625068	0.0312534	+	0.999511i	$0.490050\pi$
$828$	0	0
$829$	−9987.68	−0.418440	−0.209220	−	0.977869i	$-0.567092\pi$
$829$	−9987.68	−0.418440	−0.209220	+	0.977869i	$0.567092\pi$
$830$	6080.67	0.254293
$831$	−6790.95	−0.283484
$832$	57198.1	2.38340
$833$	−17727.6	−0.737364
$834$	30400.0	1.26219
$835$	−21536.9	−0.892594
$836$	−26445.9	−1.09408
$837$	−335.159	−0.0138409
$838$	−9494.03	−0.391367
$839$	2835.68	0.116685	0.0583424	−	0.998297i	$-0.481418\pi$
$839$	2835.68	0.116685	0.0583424	+	0.998297i	$0.481418\pi$
$840$	2833.11	0.116371
$841$	−24226.5	−0.993337
$842$	38622.4	1.58078
$843$	1517.70	0.0620076
$844$	−14156.3	−0.577345
$845$	43668.9	1.77782
$846$	−23429.3	−0.952146
$847$	−15026.8	−0.609596
$848$	12041.7	0.487634
$849$	21054.6	0.851110
$850$	2989.67	0.120641
$851$	0	0
$852$	18753.0	0.754070
$853$	17544.2	0.704223	0.352111	−	0.935958i	$-0.385464\pi$
$853$	17544.2	0.704223	0.352111	+	0.935958i	$0.385464\pi$
$854$	37841.7	1.51629
$855$	−4920.32	−0.196809
$856$	−13067.2	−0.521760
$857$	12827.7	0.511301	0.255651	−	0.966769i	$-0.417710\pi$
$857$	12827.7	0.511301	0.255651	+	0.966769i	$0.417710\pi$
$858$	52622.0	2.09381
$859$	−35061.5	−1.39265	−0.696324	−	0.717728i	$-0.745182\pi$
$859$	−35061.5	−1.39265	−0.696324	+	0.717728i	$0.745182\pi$
$860$	−18388.0	−0.729100
$861$	1242.88	0.0491953
$862$	−5788.79	−0.228732
$863$	−4222.11	−0.166538	−0.0832691	−	0.996527i	$-0.526536\pi$
$863$	−4222.11	−0.166538	−0.0832691	+	0.996527i	$0.526536\pi$
$864$	6887.57	0.271204
$865$	−40868.2	−1.60643
$866$	14873.7	0.583637
$867$	3157.23	0.123674
$868$	1314.95	0.0514199
$869$	50448.4	1.96933
$870$	−1743.76	−0.0679528
$871$	−35443.0	−1.37881
$872$	6772.15	0.262998
$873$	−14432.1	−0.559511
$874$	0	0
$875$	15380.6	0.594238
$876$	27603.3	1.06464
$877$	17535.0	0.675159	0.337579	−	0.941297i	$-0.390392\pi$
$877$	17535.0	0.675159	0.337579	+	0.941297i	$0.390392\pi$
$878$	−29788.7	−1.14501
$879$	28466.5	1.09232
$880$	−25173.9	−0.964330
$881$	−9116.63	−0.348634	−0.174317	−	0.984690i	$-0.555772\pi$
$881$	−9116.63	−0.348634	−0.174317	+	0.984690i	$0.555772\pi$
$882$	8750.54	0.334066
$883$	−41050.4	−1.56450	−0.782251	−	0.622963i	$-0.785929\pi$
$883$	−41050.4	−1.56450	−0.782251	+	0.622963i	$0.785929\pi$
$884$	−60739.5	−2.31096
$885$	−17268.6	−0.655907
$886$	−37060.0	−1.40525
$887$	22519.5	0.852461	0.426230	−	0.904615i	$-0.359841\pi$
$887$	22519.5	0.852461	0.426230	+	0.904615i	$0.359841\pi$
$888$	−1806.38	−0.0682635
$889$	−12178.3	−0.459444
$890$	−35500.8	−1.33707
$891$	4241.22	0.159468
$892$	60547.0	2.27272
$893$	−31212.9	−1.16965
$894$	−28718.0	−1.07436
$895$	−33197.7	−1.23986
$896$	−10898.4	−0.406349
$897$	0	0
$898$	3410.91	0.126752
$899$	−158.241	−0.00587055
$900$	−817.816	−0.0302895
$901$	−20822.6	−0.769924
$902$	8626.30	0.318431
$903$	5489.88	0.202316
$904$	162.255	0.00596960
$905$	−50947.0	−1.87131
$906$	30637.4	1.12346
$907$	32037.9	1.17288	0.586439	−	0.809993i	$-0.300529\pi$
$907$	32037.9	1.17288	0.586439	+	0.809993i	$0.300529\pi$
$908$	44387.9	1.62232
$909$	11061.4	0.403612
$910$	−38411.6	−1.39927
$911$	12881.7	0.468485	0.234242	−	0.972178i	$-0.424739\pi$
$911$	12881.7	0.468485	0.234242	+	0.972178i	$0.424739\pi$
$912$	6805.73	0.247106
$913$	−6982.69	−0.253114
$914$	−44465.6	−1.60918
$915$	27080.0	0.978402
$916$	−12916.4	−0.465905
$917$	−12299.8	−0.442937
$918$	−8833.79	−0.317602
$919$	7311.92	0.262457	0.131229	−	0.991352i	$-0.458108\pi$
$919$	7311.92	0.262457	0.131229	+	0.991352i	$0.458108\pi$
$920$	0	0
$921$	13981.7	0.500232
$922$	14087.8	0.503207
$923$	−49711.3	−1.77277
$924$	−16639.9	−0.592437
$925$	−668.047	−0.0237462
$926$	3562.01	0.126409
$927$	−1935.59	−0.0685795
$928$	3251.87	0.115030
$929$	35059.5	1.23817	0.619087	−	0.785322i	$-0.287503\pi$
$929$	35059.5	1.23817	0.619087	+	0.785322i	$0.287503\pi$
$930$	1698.02	0.0598712
$931$	11657.6	0.410379
$932$	−12504.0	−0.439467
$933$	13135.5	0.460919
$934$	35798.2	1.25413
$935$	43530.8	1.52258
$936$	5861.93	0.204704
$937$	−25471.7	−0.888072	−0.444036	−	0.896009i	$-0.646454\pi$
$937$	−25471.7	−0.888072	−0.444036	+	0.896009i	$0.646454\pi$
$938$	20224.0	0.703983
$939$	13332.5	0.463354
$940$	65780.6	2.28247
$941$	−22426.5	−0.776923	−0.388461	−	0.921465i	$-0.626993\pi$
$941$	−22426.5	−0.776923	−0.388461	+	0.921465i	$0.626993\pi$
$942$	−7815.03	−0.270305
$943$	0	0
$944$	23885.7	0.823532
$945$	−3095.89	−0.106571
$946$	38103.0	1.30955
$947$	−36090.9	−1.23843	−0.619216	−	0.785220i	$-0.712550\pi$
$947$	−36090.9	−1.23843	−0.619216	+	0.785220i	$0.712550\pi$
$948$	28743.3	0.984746
$949$	−73171.9	−2.50291
$950$	−1966.00	−0.0671426
$951$	23644.8	0.806242
$952$	6776.27	0.230693
$953$	16379.8	0.556762	0.278381	−	0.960471i	$-0.410202\pi$
$953$	16379.8	0.556762	0.278381	+	0.960471i	$0.410202\pi$
$954$	10278.3	0.348817
$955$	13919.2	0.471639
$956$	34546.0	1.16872
$957$	2002.43	0.0676378
$958$	−24089.9	−0.812432
$959$	−22935.8	−0.772301
$960$	−23355.9	−0.785216
$961$	−29636.9	−0.994828
$962$	24491.1	0.820815
$963$	14279.2	0.477820
$964$	21289.9	0.711308
$965$	−12612.7	−0.420742
$966$	0	0
$967$	−22686.2	−0.754434	−0.377217	−	0.926125i	$-0.623119\pi$
$967$	−22686.2	−0.754434	−0.377217	+	0.926125i	$0.623119\pi$
$968$	−11618.1	−0.385765
$969$	−11768.5	−0.390154
$970$	73117.5	2.42027
$971$	−53773.7	−1.77722	−0.888610	−	0.458663i	$-0.848328\pi$
$971$	−53773.7	−1.77722	−0.888610	+	0.458663i	$0.848328\pi$
$972$	2416.46	0.0797407
$973$	25482.4	0.839597
$974$	58323.0	1.91867
$975$	2167.90	0.0712086
$976$	−37456.8	−1.22844
$977$	20398.6	0.667972	0.333986	−	0.942578i	$-0.391606\pi$
$977$	20398.6	0.667972	0.333986	+	0.942578i	$0.391606\pi$
$978$	645.738	0.0211129
$979$	40767.0	1.33087
$980$	−24568.2	−0.800819
$981$	−7400.30	−0.240850
$982$	−35922.4	−1.16734
$983$	12118.6	0.393208	0.196604	−	0.980483i	$-0.437009\pi$
$983$	12118.6	0.393208	0.196604	+	0.980483i	$0.437009\pi$
$984$	960.943	0.0311318
$985$	49074.8	1.58746
$986$	−4170.75	−0.134710
$987$	−19639.3	−0.633359
$988$	39942.1	1.28616
$989$	0	0
$990$	−21487.3	−0.689809
$991$	−26885.7	−0.861809	−0.430904	−	0.902398i	$-0.641805\pi$
$991$	−26885.7	−0.861809	−0.430904	+	0.902398i	$0.641805\pi$
$992$	−3166.57	−0.101350
$993$	−381.181	−0.0121817
$994$	28365.5	0.905130
$995$	5099.10	0.162465
$996$	−3978.43	−0.126568
$997$	21022.7	0.667797	0.333899	−	0.942609i	$-0.391636\pi$
$997$	21022.7	0.667797	0.333899	+	0.942609i	$0.391636\pi$
$998$	−52834.5	−1.67580
$999$	1973.93	0.0625148

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1587.4.a.b.1.1		2
23.22	odd	2		69.4.a.a.1.1	✓	2
69.68	even	2		207.4.a.c.1.2		2
92.91	even	2		1104.4.a.h.1.2		2
115.114	odd	2		1725.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.a.1.1	✓	2	23.22	odd	2
207.4.a.c.1.2		2	69.68	even	2
1104.4.a.h.1.2		2	92.91	even	2
1587.4.a.b.1.1		2	1.1	even	1	trivial
1725.4.a.n.1.2		2	115.114	odd	2

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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1587.4.a.b.1.1

Level	$1587$
Weight	$4$
Character	1587.1
Self dual	yes
Analytic conductor	$93.636$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$