Properties

Label 102.8.a.b.1.1
Level $102$
Weight $8$
Character 102.1
Self dual yes
Analytic conductor $31.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,8,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 102.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.8632725994\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 102.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -235.000 q^{5} +216.000 q^{6} -344.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -235.000 q^{5} +216.000 q^{6} -344.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1880.00 q^{10} -4185.00 q^{11} +1728.00 q^{12} -1833.00 q^{13} -2752.00 q^{14} -6345.00 q^{15} +4096.00 q^{16} +4913.00 q^{17} +5832.00 q^{18} -37829.0 q^{19} -15040.0 q^{20} -9288.00 q^{21} -33480.0 q^{22} -85437.0 q^{23} +13824.0 q^{24} -22900.0 q^{25} -14664.0 q^{26} +19683.0 q^{27} -22016.0 q^{28} +54794.0 q^{29} -50760.0 q^{30} -89586.0 q^{31} +32768.0 q^{32} -112995. q^{33} +39304.0 q^{34} +80840.0 q^{35} +46656.0 q^{36} +30392.0 q^{37} -302632. q^{38} -49491.0 q^{39} -120320. q^{40} +550715. q^{41} -74304.0 q^{42} -434107. q^{43} -267840. q^{44} -171315. q^{45} -683496. q^{46} +259378. q^{47} +110592. q^{48} -705207. q^{49} -183200. q^{50} +132651. q^{51} -117312. q^{52} -923366. q^{53} +157464. q^{54} +983475. q^{55} -176128. q^{56} -1.02138e6 q^{57} +438352. q^{58} -1.32046e6 q^{59} -406080. q^{60} +1.19316e6 q^{61} -716688. q^{62} -250776. q^{63} +262144. q^{64} +430755. q^{65} -903960. q^{66} -369324. q^{67} +314432. q^{68} -2.30680e6 q^{69} +646720. q^{70} -2.74228e6 q^{71} +373248. q^{72} +1.10217e6 q^{73} +243136. q^{74} -618300. q^{75} -2.42106e6 q^{76} +1.43964e6 q^{77} -395928. q^{78} +5.53807e6 q^{79} -962560. q^{80} +531441. q^{81} +4.40572e6 q^{82} +5.35331e6 q^{83} -594432. q^{84} -1.15456e6 q^{85} -3.47286e6 q^{86} +1.47944e6 q^{87} -2.14272e6 q^{88} -4.26979e6 q^{89} -1.37052e6 q^{90} +630552. q^{91} -5.46797e6 q^{92} -2.41882e6 q^{93} +2.07502e6 q^{94} +8.88982e6 q^{95} +884736. q^{96} -1.00334e7 q^{97} -5.64166e6 q^{98} -3.05087e6 q^{99} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −235.000 −0.840762 −0.420381 0.907348i \(-0.638104\pi\)
−0.420381 + 0.907348i \(0.638104\pi\)
\(6\) 216.000 0.408248
\(7\) −344.000 −0.379066 −0.189533 0.981874i \(-0.560697\pi\)
−0.189533 + 0.981874i \(0.560697\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1880.00 −0.594508
\(11\) −4185.00 −0.948028 −0.474014 0.880517i \(-0.657195\pi\)
−0.474014 + 0.880517i \(0.657195\pi\)
\(12\) 1728.00 0.288675
\(13\) −1833.00 −0.231399 −0.115699 0.993284i \(-0.536911\pi\)
−0.115699 + 0.993284i \(0.536911\pi\)
\(14\) −2752.00 −0.268040
\(15\) −6345.00 −0.485414
\(16\) 4096.00 0.250000
\(17\) 4913.00 0.242536
\(18\) 5832.00 0.235702
\(19\) −37829.0 −1.26528 −0.632641 0.774445i \(-0.718029\pi\)
−0.632641 + 0.774445i \(0.718029\pi\)
\(20\) −15040.0 −0.420381
\(21\) −9288.00 −0.218854
\(22\) −33480.0 −0.670357
\(23\) −85437.0 −1.46419 −0.732097 0.681201i \(-0.761458\pi\)
−0.732097 + 0.681201i \(0.761458\pi\)
\(24\) 13824.0 0.204124
\(25\) −22900.0 −0.293120
\(26\) −14664.0 −0.163624
\(27\) 19683.0 0.192450
\(28\) −22016.0 −0.189533
\(29\) 54794.0 0.417196 0.208598 0.978001i \(-0.433110\pi\)
0.208598 + 0.978001i \(0.433110\pi\)
\(30\) −50760.0 −0.343239
\(31\) −89586.0 −0.540100 −0.270050 0.962846i \(-0.587040\pi\)
−0.270050 + 0.962846i \(0.587040\pi\)
\(32\) 32768.0 0.176777
\(33\) −112995. −0.547344
\(34\) 39304.0 0.171499
\(35\) 80840.0 0.318704
\(36\) 46656.0 0.166667
\(37\) 30392.0 0.0986400 0.0493200 0.998783i \(-0.484295\pi\)
0.0493200 + 0.998783i \(0.484295\pi\)
\(38\) −302632. −0.894689
\(39\) −49491.0 −0.133598
\(40\) −120320. −0.297254
\(41\) 550715. 1.24791 0.623955 0.781460i \(-0.285525\pi\)
0.623955 + 0.781460i \(0.285525\pi\)
\(42\) −74304.0 −0.154753
\(43\) −434107. −0.832640 −0.416320 0.909218i \(-0.636680\pi\)
−0.416320 + 0.909218i \(0.636680\pi\)
\(44\) −267840. −0.474014
\(45\) −171315. −0.280254
\(46\) −683496. −1.03534
\(47\) 259378. 0.364410 0.182205 0.983261i \(-0.441677\pi\)
0.182205 + 0.983261i \(0.441677\pi\)
\(48\) 110592. 0.144338
\(49\) −705207. −0.856309
\(50\) −183200. −0.207267
\(51\) 132651. 0.140028
\(52\) −117312. −0.115699
\(53\) −923366. −0.851939 −0.425969 0.904738i \(-0.640067\pi\)
−0.425969 + 0.904738i \(0.640067\pi\)
\(54\) 157464. 0.136083
\(55\) 983475. 0.797065
\(56\) −176128. −0.134020
\(57\) −1.02138e6 −0.730511
\(58\) 438352. 0.295002
\(59\) −1.32046e6 −0.837035 −0.418518 0.908209i \(-0.637450\pi\)
−0.418518 + 0.908209i \(0.637450\pi\)
\(60\) −406080. −0.242707
\(61\) 1.19316e6 0.673043 0.336521 0.941676i \(-0.390750\pi\)
0.336521 + 0.941676i \(0.390750\pi\)
\(62\) −716688. −0.381908
\(63\) −250776. −0.126355
\(64\) 262144. 0.125000
\(65\) 430755. 0.194551
\(66\) −903960. −0.387031
\(67\) −369324. −0.150019 −0.0750094 0.997183i \(-0.523899\pi\)
−0.0750094 + 0.997183i \(0.523899\pi\)
\(68\) 314432. 0.121268
\(69\) −2.30680e6 −0.845353
\(70\) 646720. 0.225358
\(71\) −2.74228e6 −0.909299 −0.454650 0.890670i \(-0.650236\pi\)
−0.454650 + 0.890670i \(0.650236\pi\)
\(72\) 373248. 0.117851
\(73\) 1.10217e6 0.331602 0.165801 0.986159i \(-0.446979\pi\)
0.165801 + 0.986159i \(0.446979\pi\)
\(74\) 243136. 0.0697490
\(75\) −618300. −0.169233
\(76\) −2.42106e6 −0.632641
\(77\) 1.43964e6 0.359365
\(78\) −395928. −0.0944681
\(79\) 5.53807e6 1.26376 0.631879 0.775067i \(-0.282284\pi\)
0.631879 + 0.775067i \(0.282284\pi\)
\(80\) −962560. −0.210190
\(81\) 531441. 0.111111
\(82\) 4.40572e6 0.882406
\(83\) 5.35331e6 1.02766 0.513830 0.857892i \(-0.328226\pi\)
0.513830 + 0.857892i \(0.328226\pi\)
\(84\) −594432. −0.109427
\(85\) −1.15456e6 −0.203915
\(86\) −3.47286e6 −0.588765
\(87\) 1.47944e6 0.240868
\(88\) −2.14272e6 −0.335178
\(89\) −4.26979e6 −0.642009 −0.321005 0.947078i \(-0.604021\pi\)
−0.321005 + 0.947078i \(0.604021\pi\)
\(90\) −1.37052e6 −0.198169
\(91\) 630552. 0.0877154
\(92\) −5.46797e6 −0.732097
\(93\) −2.41882e6 −0.311827
\(94\) 2.07502e6 0.257677
\(95\) 8.88982e6 1.06380
\(96\) 884736. 0.102062
\(97\) −1.00334e7 −1.11621 −0.558105 0.829771i \(-0.688471\pi\)
−0.558105 + 0.829771i \(0.688471\pi\)
\(98\) −5.64166e6 −0.605502
\(99\) −3.05087e6 −0.316009
\(100\) −1.46560e6 −0.146560
\(101\) 927704. 0.0895952 0.0447976 0.998996i \(-0.485736\pi\)
0.0447976 + 0.998996i \(0.485736\pi\)
\(102\) 1.06121e6 0.0990148
\(103\) −1.45382e6 −0.131094 −0.0655468 0.997849i \(-0.520879\pi\)
−0.0655468 + 0.997849i \(0.520879\pi\)
\(104\) −938496. −0.0818118
\(105\) 2.18268e6 0.184004
\(106\) −7.38693e6 −0.602412
\(107\) 2.19897e7 1.73531 0.867654 0.497169i \(-0.165627\pi\)
0.867654 + 0.497169i \(0.165627\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.27832e7 0.945471 0.472736 0.881204i \(-0.343267\pi\)
0.472736 + 0.881204i \(0.343267\pi\)
\(110\) 7.86780e6 0.563610
\(111\) 820584. 0.0569499
\(112\) −1.40902e6 −0.0947666
\(113\) 2.54158e7 1.65703 0.828513 0.559970i \(-0.189187\pi\)
0.828513 + 0.559970i \(0.189187\pi\)
\(114\) −8.17106e6 −0.516549
\(115\) 2.00777e7 1.23104
\(116\) 3.50682e6 0.208598
\(117\) −1.33626e6 −0.0771329
\(118\) −1.05637e7 −0.591873
\(119\) −1.69007e6 −0.0919371
\(120\) −3.24864e6 −0.171620
\(121\) −1.97295e6 −0.101243
\(122\) 9.54525e6 0.475913
\(123\) 1.48693e7 0.720481
\(124\) −5.73350e6 −0.270050
\(125\) 2.37409e7 1.08721
\(126\) −2.00621e6 −0.0893468
\(127\) 1.10221e7 0.477474 0.238737 0.971084i \(-0.423267\pi\)
0.238737 + 0.971084i \(0.423267\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.17209e7 −0.480725
\(130\) 3.44604e6 0.137568
\(131\) 3.83804e7 1.49163 0.745813 0.666155i \(-0.232061\pi\)
0.745813 + 0.666155i \(0.232061\pi\)
\(132\) −7.23168e6 −0.273672
\(133\) 1.30132e7 0.479626
\(134\) −2.95459e6 −0.106079
\(135\) −4.62550e6 −0.161805
\(136\) 2.51546e6 0.0857493
\(137\) −1.53813e7 −0.511058 −0.255529 0.966801i \(-0.582250\pi\)
−0.255529 + 0.966801i \(0.582250\pi\)
\(138\) −1.84544e7 −0.597755
\(139\) −1.23072e7 −0.388694 −0.194347 0.980933i \(-0.562259\pi\)
−0.194347 + 0.980933i \(0.562259\pi\)
\(140\) 5.17376e6 0.159352
\(141\) 7.00321e6 0.210392
\(142\) −2.19382e7 −0.642972
\(143\) 7.67110e6 0.219372
\(144\) 2.98598e6 0.0833333
\(145\) −1.28766e7 −0.350762
\(146\) 8.81733e6 0.234478
\(147\) −1.90406e7 −0.494390
\(148\) 1.94509e6 0.0493200
\(149\) 1.72111e7 0.426242 0.213121 0.977026i \(-0.431637\pi\)
0.213121 + 0.977026i \(0.431637\pi\)
\(150\) −4.94640e6 −0.119666
\(151\) −2.69730e7 −0.637544 −0.318772 0.947831i \(-0.603270\pi\)
−0.318772 + 0.947831i \(0.603270\pi\)
\(152\) −1.93684e7 −0.447345
\(153\) 3.58158e6 0.0808452
\(154\) 1.15171e7 0.254110
\(155\) 2.10527e7 0.454095
\(156\) −3.16742e6 −0.0667990
\(157\) −2.24669e7 −0.463333 −0.231667 0.972795i \(-0.574418\pi\)
−0.231667 + 0.972795i \(0.574418\pi\)
\(158\) 4.43046e7 0.893611
\(159\) −2.49309e7 −0.491867
\(160\) −7.70048e6 −0.148627
\(161\) 2.93903e7 0.555027
\(162\) 4.25153e6 0.0785674
\(163\) −7.32660e7 −1.32509 −0.662546 0.749021i \(-0.730524\pi\)
−0.662546 + 0.749021i \(0.730524\pi\)
\(164\) 3.52458e7 0.623955
\(165\) 2.65538e7 0.460186
\(166\) 4.28265e7 0.726665
\(167\) −2.23422e7 −0.371208 −0.185604 0.982625i \(-0.559424\pi\)
−0.185604 + 0.982625i \(0.559424\pi\)
\(168\) −4.75546e6 −0.0773766
\(169\) −5.93886e7 −0.946455
\(170\) −9.23644e6 −0.144189
\(171\) −2.75773e7 −0.421761
\(172\) −2.77828e7 −0.416320
\(173\) 3.10862e7 0.456464 0.228232 0.973607i \(-0.426705\pi\)
0.228232 + 0.973607i \(0.426705\pi\)
\(174\) 1.18355e7 0.170320
\(175\) 7.87760e6 0.111112
\(176\) −1.71418e7 −0.237007
\(177\) −3.56525e7 −0.483263
\(178\) −3.41583e7 −0.453969
\(179\) −4.13577e7 −0.538978 −0.269489 0.963003i \(-0.586855\pi\)
−0.269489 + 0.963003i \(0.586855\pi\)
\(180\) −1.09642e7 −0.140127
\(181\) −1.23423e7 −0.154711 −0.0773554 0.997004i \(-0.524648\pi\)
−0.0773554 + 0.997004i \(0.524648\pi\)
\(182\) 5.04442e6 0.0620242
\(183\) 3.22152e7 0.388582
\(184\) −4.37437e7 −0.517671
\(185\) −7.14212e6 −0.0829328
\(186\) −1.93506e7 −0.220495
\(187\) −2.05609e7 −0.229931
\(188\) 1.66002e7 0.182205
\(189\) −6.77095e6 −0.0729514
\(190\) 7.11185e7 0.752220
\(191\) 4.70786e7 0.488885 0.244443 0.969664i \(-0.421395\pi\)
0.244443 + 0.969664i \(0.421395\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −6.59166e7 −0.660001 −0.330000 0.943981i \(-0.607049\pi\)
−0.330000 + 0.943981i \(0.607049\pi\)
\(194\) −8.02669e7 −0.789279
\(195\) 1.16304e7 0.112324
\(196\) −4.51332e7 −0.428154
\(197\) −1.44442e8 −1.34605 −0.673025 0.739620i \(-0.735005\pi\)
−0.673025 + 0.739620i \(0.735005\pi\)
\(198\) −2.44069e7 −0.223452
\(199\) −2.13401e8 −1.91960 −0.959800 0.280685i \(-0.909438\pi\)
−0.959800 + 0.280685i \(0.909438\pi\)
\(200\) −1.17248e7 −0.103634
\(201\) −9.97175e6 −0.0866134
\(202\) 7.42163e6 0.0633534
\(203\) −1.88491e7 −0.158145
\(204\) 8.48966e6 0.0700140
\(205\) −1.29418e8 −1.04920
\(206\) −1.16306e7 −0.0926971
\(207\) −6.22836e7 −0.488065
\(208\) −7.50797e6 −0.0578497
\(209\) 1.58314e8 1.19952
\(210\) 1.74614e7 0.130111
\(211\) −1.77492e8 −1.30074 −0.650369 0.759618i \(-0.725386\pi\)
−0.650369 + 0.759618i \(0.725386\pi\)
\(212\) −5.90954e7 −0.425969
\(213\) −7.40415e7 −0.524984
\(214\) 1.75918e8 1.22705
\(215\) 1.02015e8 0.700052
\(216\) 1.00777e7 0.0680414
\(217\) 3.08176e7 0.204734
\(218\) 1.02266e8 0.668549
\(219\) 2.97585e7 0.191450
\(220\) 6.29424e7 0.398533
\(221\) −9.00553e6 −0.0561224
\(222\) 6.56467e6 0.0402696
\(223\) −1.84303e8 −1.11292 −0.556462 0.830873i \(-0.687841\pi\)
−0.556462 + 0.830873i \(0.687841\pi\)
\(224\) −1.12722e7 −0.0670101
\(225\) −1.66941e7 −0.0977067
\(226\) 2.03326e8 1.17169
\(227\) 8.22979e6 0.0466980 0.0233490 0.999727i \(-0.492567\pi\)
0.0233490 + 0.999727i \(0.492567\pi\)
\(228\) −6.53685e7 −0.365255
\(229\) 3.09325e8 1.70212 0.851062 0.525065i \(-0.175959\pi\)
0.851062 + 0.525065i \(0.175959\pi\)
\(230\) 1.60622e8 0.870475
\(231\) 3.88703e7 0.207480
\(232\) 2.80545e7 0.147501
\(233\) 1.25573e8 0.650355 0.325177 0.945653i \(-0.394576\pi\)
0.325177 + 0.945653i \(0.394576\pi\)
\(234\) −1.06901e7 −0.0545412
\(235\) −6.09538e7 −0.306382
\(236\) −8.45096e7 −0.418518
\(237\) 1.49528e8 0.729631
\(238\) −1.35206e7 −0.0650094
\(239\) 3.88888e8 1.84260 0.921301 0.388849i \(-0.127127\pi\)
0.921301 + 0.388849i \(0.127127\pi\)
\(240\) −2.59891e7 −0.121353
\(241\) −2.00123e8 −0.920952 −0.460476 0.887672i \(-0.652321\pi\)
−0.460476 + 0.887672i \(0.652321\pi\)
\(242\) −1.57836e7 −0.0715898
\(243\) 1.43489e7 0.0641500
\(244\) 7.63620e7 0.336521
\(245\) 1.65724e8 0.719951
\(246\) 1.18954e8 0.509457
\(247\) 6.93406e7 0.292784
\(248\) −4.58680e7 −0.190954
\(249\) 1.44539e8 0.593320
\(250\) 1.89927e8 0.768770
\(251\) −1.45992e8 −0.582736 −0.291368 0.956611i \(-0.594110\pi\)
−0.291368 + 0.956611i \(0.594110\pi\)
\(252\) −1.60497e7 −0.0631777
\(253\) 3.57554e8 1.38810
\(254\) 8.81766e7 0.337625
\(255\) −3.11730e7 −0.117730
\(256\) 1.67772e7 0.0625000
\(257\) −4.23238e8 −1.55532 −0.777658 0.628688i \(-0.783592\pi\)
−0.777658 + 0.628688i \(0.783592\pi\)
\(258\) −9.37671e7 −0.339924
\(259\) −1.04548e7 −0.0373911
\(260\) 2.75683e7 0.0972755
\(261\) 3.99448e7 0.139065
\(262\) 3.07043e8 1.05474
\(263\) 4.67569e7 0.158490 0.0792448 0.996855i \(-0.474749\pi\)
0.0792448 + 0.996855i \(0.474749\pi\)
\(264\) −5.78534e7 −0.193515
\(265\) 2.16991e8 0.716277
\(266\) 1.04105e8 0.339147
\(267\) −1.15284e8 −0.370664
\(268\) −2.36367e7 −0.0750094
\(269\) −2.85453e8 −0.894132 −0.447066 0.894501i \(-0.647531\pi\)
−0.447066 + 0.894501i \(0.647531\pi\)
\(270\) −3.70040e7 −0.114413
\(271\) −1.72044e8 −0.525107 −0.262554 0.964917i \(-0.584565\pi\)
−0.262554 + 0.964917i \(0.584565\pi\)
\(272\) 2.01236e7 0.0606339
\(273\) 1.70249e7 0.0506425
\(274\) −1.23050e8 −0.361373
\(275\) 9.58365e7 0.277886
\(276\) −1.47635e8 −0.422676
\(277\) 2.88317e8 0.815062 0.407531 0.913191i \(-0.366390\pi\)
0.407531 + 0.913191i \(0.366390\pi\)
\(278\) −9.84576e7 −0.274848
\(279\) −6.53082e7 −0.180033
\(280\) 4.13901e7 0.112679
\(281\) 5.53413e7 0.148791 0.0743956 0.997229i \(-0.476297\pi\)
0.0743956 + 0.997229i \(0.476297\pi\)
\(282\) 5.60256e7 0.148770
\(283\) 6.75308e7 0.177113 0.0885563 0.996071i \(-0.471775\pi\)
0.0885563 + 0.996071i \(0.471775\pi\)
\(284\) −1.75506e8 −0.454650
\(285\) 2.40025e8 0.614185
\(286\) 6.13688e7 0.155120
\(287\) −1.89446e8 −0.473041
\(288\) 2.38879e7 0.0589256
\(289\) 2.41376e7 0.0588235
\(290\) −1.03013e8 −0.248026
\(291\) −2.70901e8 −0.644444
\(292\) 7.05386e7 0.165801
\(293\) −1.54735e8 −0.359379 −0.179689 0.983723i \(-0.557509\pi\)
−0.179689 + 0.983723i \(0.557509\pi\)
\(294\) −1.52325e8 −0.349587
\(295\) 3.10309e8 0.703747
\(296\) 1.55607e7 0.0348745
\(297\) −8.23734e7 −0.182448
\(298\) 1.37689e8 0.301398
\(299\) 1.56606e8 0.338812
\(300\) −3.95712e7 −0.0846165
\(301\) 1.49333e8 0.315626
\(302\) −2.15784e8 −0.450812
\(303\) 2.50480e7 0.0517278
\(304\) −1.54948e8 −0.316320
\(305\) −2.80392e8 −0.565869
\(306\) 2.86526e7 0.0571662
\(307\) −4.94943e8 −0.976272 −0.488136 0.872767i \(-0.662323\pi\)
−0.488136 + 0.872767i \(0.662323\pi\)
\(308\) 9.21370e7 0.179683
\(309\) −3.92532e7 −0.0756869
\(310\) 1.68422e8 0.321094
\(311\) −6.81862e8 −1.28539 −0.642695 0.766122i \(-0.722184\pi\)
−0.642695 + 0.766122i \(0.722184\pi\)
\(312\) −2.53394e7 −0.0472340
\(313\) −2.57050e7 −0.0473818 −0.0236909 0.999719i \(-0.507542\pi\)
−0.0236909 + 0.999719i \(0.507542\pi\)
\(314\) −1.79735e8 −0.327626
\(315\) 5.89324e7 0.106235
\(316\) 3.54436e8 0.631879
\(317\) 7.39662e8 1.30415 0.652073 0.758157i \(-0.273900\pi\)
0.652073 + 0.758157i \(0.273900\pi\)
\(318\) −1.99447e8 −0.347803
\(319\) −2.29313e8 −0.395513
\(320\) −6.16038e7 −0.105095
\(321\) 5.93722e8 1.00188
\(322\) 2.35123e8 0.392463
\(323\) −1.85854e8 −0.306876
\(324\) 3.40122e7 0.0555556
\(325\) 4.19757e7 0.0678276
\(326\) −5.86128e8 −0.936981
\(327\) 3.45148e8 0.545868
\(328\) 2.81966e8 0.441203
\(329\) −8.92260e7 −0.138136
\(330\) 2.12431e8 0.325401
\(331\) −6.53666e8 −0.990735 −0.495368 0.868683i \(-0.664967\pi\)
−0.495368 + 0.868683i \(0.664967\pi\)
\(332\) 3.42612e8 0.513830
\(333\) 2.21558e7 0.0328800
\(334\) −1.78737e8 −0.262484
\(335\) 8.67911e7 0.126130
\(336\) −3.80436e7 −0.0547135
\(337\) −5.68180e8 −0.808689 −0.404345 0.914607i \(-0.632500\pi\)
−0.404345 + 0.914607i \(0.632500\pi\)
\(338\) −4.75109e8 −0.669245
\(339\) 6.86227e8 0.956684
\(340\) −7.38915e7 −0.101957
\(341\) 3.74917e8 0.512030
\(342\) −2.20619e8 −0.298230
\(343\) 5.25890e8 0.703664
\(344\) −2.22263e8 −0.294383
\(345\) 5.42098e8 0.710740
\(346\) 2.48690e8 0.322769
\(347\) −8.79203e8 −1.12963 −0.564815 0.825218i \(-0.691052\pi\)
−0.564815 + 0.825218i \(0.691052\pi\)
\(348\) 9.46840e7 0.120434
\(349\) −3.28412e8 −0.413551 −0.206776 0.978388i \(-0.566297\pi\)
−0.206776 + 0.978388i \(0.566297\pi\)
\(350\) 6.30208e7 0.0785680
\(351\) −3.60789e7 −0.0445327
\(352\) −1.37134e8 −0.167589
\(353\) −5.34888e8 −0.647219 −0.323610 0.946191i \(-0.604896\pi\)
−0.323610 + 0.946191i \(0.604896\pi\)
\(354\) −2.85220e8 −0.341718
\(355\) 6.44435e8 0.764504
\(356\) −2.73266e8 −0.321005
\(357\) −4.56319e7 −0.0530799
\(358\) −3.30862e8 −0.381115
\(359\) −5.70927e8 −0.651253 −0.325627 0.945498i \(-0.605575\pi\)
−0.325627 + 0.945498i \(0.605575\pi\)
\(360\) −8.77133e7 −0.0990847
\(361\) 5.37162e8 0.600938
\(362\) −9.87383e7 −0.109397
\(363\) −5.32695e7 −0.0584529
\(364\) 4.03553e7 0.0438577
\(365\) −2.59009e8 −0.278798
\(366\) 2.57722e8 0.274769
\(367\) 4.94074e8 0.521748 0.260874 0.965373i \(-0.415989\pi\)
0.260874 + 0.965373i \(0.415989\pi\)
\(368\) −3.49950e8 −0.366048
\(369\) 4.01471e8 0.415970
\(370\) −5.71370e7 −0.0586423
\(371\) 3.17638e8 0.322941
\(372\) −1.54805e8 −0.155913
\(373\) −1.58450e8 −0.158092 −0.0790461 0.996871i \(-0.525187\pi\)
−0.0790461 + 0.996871i \(0.525187\pi\)
\(374\) −1.64487e8 −0.162585
\(375\) 6.41004e8 0.627698
\(376\) 1.32802e8 0.128838
\(377\) −1.00437e8 −0.0965385
\(378\) −5.41676e7 −0.0515844
\(379\) −1.54108e9 −1.45408 −0.727038 0.686597i \(-0.759104\pi\)
−0.727038 + 0.686597i \(0.759104\pi\)
\(380\) 5.68948e8 0.531900
\(381\) 2.97596e8 0.275670
\(382\) 3.76629e8 0.345694
\(383\) −1.33583e9 −1.21495 −0.607473 0.794340i \(-0.707817\pi\)
−0.607473 + 0.794340i \(0.707817\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −3.38315e8 −0.302141
\(386\) −5.27333e8 −0.466691
\(387\) −3.16464e8 −0.277547
\(388\) −6.42136e8 −0.558105
\(389\) 1.68908e9 1.45488 0.727441 0.686170i \(-0.240709\pi\)
0.727441 + 0.686170i \(0.240709\pi\)
\(390\) 9.30431e7 0.0794251
\(391\) −4.19752e8 −0.355119
\(392\) −3.61066e8 −0.302751
\(393\) 1.03627e9 0.861191
\(394\) −1.15553e9 −0.951800
\(395\) −1.30145e9 −1.06252
\(396\) −1.95255e8 −0.158005
\(397\) 8.70907e8 0.698562 0.349281 0.937018i \(-0.386426\pi\)
0.349281 + 0.937018i \(0.386426\pi\)
\(398\) −1.70721e9 −1.35736
\(399\) 3.51356e8 0.276912
\(400\) −9.37984e7 −0.0732800
\(401\) 1.97489e9 1.52946 0.764731 0.644350i \(-0.222872\pi\)
0.764731 + 0.644350i \(0.222872\pi\)
\(402\) −7.97740e7 −0.0612449
\(403\) 1.64211e8 0.124978
\(404\) 5.93731e7 0.0447976
\(405\) −1.24889e8 −0.0934180
\(406\) −1.50793e8 −0.111825
\(407\) −1.27191e8 −0.0935135
\(408\) 6.79173e7 0.0495074
\(409\) −7.48222e8 −0.540753 −0.270376 0.962755i \(-0.587148\pi\)
−0.270376 + 0.962755i \(0.587148\pi\)
\(410\) −1.03534e9 −0.741893
\(411\) −4.15295e8 −0.295060
\(412\) −9.30447e7 −0.0655468
\(413\) 4.54239e8 0.317292
\(414\) −4.98269e8 −0.345114
\(415\) −1.25803e9 −0.864017
\(416\) −6.00637e7 −0.0409059
\(417\) −3.32294e8 −0.224412
\(418\) 1.26651e9 0.848190
\(419\) −7.91638e8 −0.525748 −0.262874 0.964830i \(-0.584670\pi\)
−0.262874 + 0.964830i \(0.584670\pi\)
\(420\) 1.39692e8 0.0920021
\(421\) 1.09245e8 0.0713534 0.0356767 0.999363i \(-0.488641\pi\)
0.0356767 + 0.999363i \(0.488641\pi\)
\(422\) −1.41993e9 −0.919761
\(423\) 1.89087e8 0.121470
\(424\) −4.72763e8 −0.301206
\(425\) −1.12508e8 −0.0710920
\(426\) −5.92332e8 −0.371220
\(427\) −4.10446e8 −0.255128
\(428\) 1.40734e9 0.867654
\(429\) 2.07120e8 0.126655
\(430\) 8.16121e8 0.495011
\(431\) 2.94715e9 1.77310 0.886548 0.462636i \(-0.153096\pi\)
0.886548 + 0.462636i \(0.153096\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.60626e9 0.950841 0.475420 0.879759i \(-0.342296\pi\)
0.475420 + 0.879759i \(0.342296\pi\)
\(434\) 2.46541e8 0.144769
\(435\) −3.47668e8 −0.202513
\(436\) 8.18128e8 0.472736
\(437\) 3.23200e9 1.85262
\(438\) 2.38068e8 0.135376
\(439\) −1.11330e9 −0.628038 −0.314019 0.949417i \(-0.601676\pi\)
−0.314019 + 0.949417i \(0.601676\pi\)
\(440\) 5.03539e8 0.281805
\(441\) −5.14096e8 −0.285436
\(442\) −7.20442e7 −0.0396845
\(443\) 7.84768e8 0.428873 0.214436 0.976738i \(-0.431209\pi\)
0.214436 + 0.976738i \(0.431209\pi\)
\(444\) 5.25174e7 0.0284749
\(445\) 1.00340e9 0.539777
\(446\) −1.47442e9 −0.786956
\(447\) 4.64699e8 0.246091
\(448\) −9.01775e7 −0.0473833
\(449\) −2.36655e9 −1.23382 −0.616911 0.787033i \(-0.711616\pi\)
−0.616911 + 0.787033i \(0.711616\pi\)
\(450\) −1.33553e8 −0.0690890
\(451\) −2.30474e9 −1.18305
\(452\) 1.62661e9 0.828513
\(453\) −7.28272e8 −0.368086
\(454\) 6.58383e7 0.0330205
\(455\) −1.48180e8 −0.0737478
\(456\) −5.22948e8 −0.258275
\(457\) 1.62527e9 0.796558 0.398279 0.917264i \(-0.369608\pi\)
0.398279 + 0.917264i \(0.369608\pi\)
\(458\) 2.47460e9 1.20358
\(459\) 9.67026e7 0.0466760
\(460\) 1.28497e9 0.615519
\(461\) −6.34505e8 −0.301635 −0.150817 0.988562i \(-0.548191\pi\)
−0.150817 + 0.988562i \(0.548191\pi\)
\(462\) 3.10962e8 0.146710
\(463\) −2.54209e9 −1.19030 −0.595152 0.803613i \(-0.702908\pi\)
−0.595152 + 0.803613i \(0.702908\pi\)
\(464\) 2.24436e8 0.104299
\(465\) 5.68423e8 0.262172
\(466\) 1.00458e9 0.459870
\(467\) −1.92009e8 −0.0872392 −0.0436196 0.999048i \(-0.513889\pi\)
−0.0436196 + 0.999048i \(0.513889\pi\)
\(468\) −8.55204e7 −0.0385664
\(469\) 1.27047e8 0.0568671
\(470\) −4.87631e8 −0.216645
\(471\) −6.06605e8 −0.267506
\(472\) −6.76077e8 −0.295937
\(473\) 1.81674e9 0.789366
\(474\) 1.19622e9 0.515927
\(475\) 8.66284e8 0.370879
\(476\) −1.08165e8 −0.0459686
\(477\) −6.73134e8 −0.283980
\(478\) 3.11110e9 1.30292
\(479\) −8.56817e8 −0.356216 −0.178108 0.984011i \(-0.556998\pi\)
−0.178108 + 0.984011i \(0.556998\pi\)
\(480\) −2.07913e8 −0.0858099
\(481\) −5.57085e7 −0.0228252
\(482\) −1.60098e9 −0.651212
\(483\) 7.93539e8 0.320445
\(484\) −1.26269e8 −0.0506217
\(485\) 2.35784e9 0.938466
\(486\) 1.14791e8 0.0453609
\(487\) 4.95013e9 1.94207 0.971036 0.238934i \(-0.0767981\pi\)
0.971036 + 0.238934i \(0.0767981\pi\)
\(488\) 6.10896e8 0.237957
\(489\) −1.97818e9 −0.765042
\(490\) 1.32579e9 0.509083
\(491\) −4.17893e8 −0.159324 −0.0796618 0.996822i \(-0.525384\pi\)
−0.0796618 + 0.996822i \(0.525384\pi\)
\(492\) 9.51636e8 0.360241
\(493\) 2.69203e8 0.101185
\(494\) 5.54724e8 0.207030
\(495\) 7.16953e8 0.265688
\(496\) −3.66944e8 −0.135025
\(497\) 9.43343e8 0.344685
\(498\) 1.15632e9 0.419540
\(499\) −1.60682e9 −0.578917 −0.289459 0.957191i \(-0.593475\pi\)
−0.289459 + 0.957191i \(0.593475\pi\)
\(500\) 1.51942e9 0.543603
\(501\) −6.03239e8 −0.214317
\(502\) −1.16794e9 −0.412057
\(503\) −4.37126e8 −0.153151 −0.0765754 0.997064i \(-0.524399\pi\)
−0.0765754 + 0.997064i \(0.524399\pi\)
\(504\) −1.28397e8 −0.0446734
\(505\) −2.18010e8 −0.0753282
\(506\) 2.86043e9 0.981532
\(507\) −1.60349e9 −0.546436
\(508\) 7.05413e8 0.238737
\(509\) 9.17987e8 0.308549 0.154275 0.988028i \(-0.450696\pi\)
0.154275 + 0.988028i \(0.450696\pi\)
\(510\) −2.49384e8 −0.0832478
\(511\) −3.79145e8 −0.125699
\(512\) 1.34218e8 0.0441942
\(513\) −7.44588e8 −0.243504
\(514\) −3.38590e9 −1.09977
\(515\) 3.41648e8 0.110218
\(516\) −7.50137e8 −0.240362
\(517\) −1.08550e9 −0.345471
\(518\) −8.36388e7 −0.0264395
\(519\) 8.39328e8 0.263540
\(520\) 2.20547e8 0.0687842
\(521\) −1.07807e9 −0.333975 −0.166987 0.985959i \(-0.553404\pi\)
−0.166987 + 0.985959i \(0.553404\pi\)
\(522\) 3.19559e8 0.0983340
\(523\) 2.82939e9 0.864841 0.432421 0.901672i \(-0.357660\pi\)
0.432421 + 0.901672i \(0.357660\pi\)
\(524\) 2.45635e9 0.745813
\(525\) 2.12695e8 0.0641505
\(526\) 3.74056e8 0.112069
\(527\) −4.40136e8 −0.130994
\(528\) −4.62828e8 −0.136836
\(529\) 3.89466e9 1.14386
\(530\) 1.73593e9 0.506485
\(531\) −9.62617e8 −0.279012
\(532\) 8.32843e8 0.239813
\(533\) −1.00946e9 −0.288765
\(534\) −9.22274e8 −0.262099
\(535\) −5.16758e9 −1.45898
\(536\) −1.89094e8 −0.0530397
\(537\) −1.11666e9 −0.311179
\(538\) −2.28362e9 −0.632247
\(539\) 2.95129e9 0.811804
\(540\) −2.96032e8 −0.0809023
\(541\) −4.17298e9 −1.13307 −0.566534 0.824038i \(-0.691716\pi\)
−0.566534 + 0.824038i \(0.691716\pi\)
\(542\) −1.37635e9 −0.371307
\(543\) −3.33242e8 −0.0893223
\(544\) 1.60989e8 0.0428746
\(545\) −3.00406e9 −0.794916
\(546\) 1.36199e8 0.0358097
\(547\) −4.42040e9 −1.15480 −0.577399 0.816462i \(-0.695932\pi\)
−0.577399 + 0.816462i \(0.695932\pi\)
\(548\) −9.84402e8 −0.255529
\(549\) 8.69811e8 0.224348
\(550\) 7.66692e8 0.196495
\(551\) −2.07280e9 −0.527870
\(552\) −1.18108e9 −0.298877
\(553\) −1.90510e9 −0.479048
\(554\) 2.30654e9 0.576336
\(555\) −1.92837e8 −0.0478812
\(556\) −7.87661e8 −0.194347
\(557\) 6.40759e9 1.57109 0.785546 0.618804i \(-0.212382\pi\)
0.785546 + 0.618804i \(0.212382\pi\)
\(558\) −5.22466e8 −0.127303
\(559\) 7.95718e8 0.192672
\(560\) 3.31121e8 0.0796761
\(561\) −5.55144e8 −0.132750
\(562\) 4.42730e8 0.105211
\(563\) −2.40800e9 −0.568693 −0.284346 0.958722i \(-0.591777\pi\)
−0.284346 + 0.958722i \(0.591777\pi\)
\(564\) 4.48205e8 0.105196
\(565\) −5.97271e9 −1.39316
\(566\) 5.40246e8 0.125238
\(567\) −1.82816e8 −0.0421185
\(568\) −1.40405e9 −0.321486
\(569\) −6.98204e9 −1.58887 −0.794437 0.607346i \(-0.792234\pi\)
−0.794437 + 0.607346i \(0.792234\pi\)
\(570\) 1.92020e9 0.434295
\(571\) 2.32773e9 0.523245 0.261623 0.965170i \(-0.415742\pi\)
0.261623 + 0.965170i \(0.415742\pi\)
\(572\) 4.90951e8 0.109686
\(573\) 1.27112e9 0.282258
\(574\) −1.51557e9 −0.334490
\(575\) 1.95651e9 0.429184
\(576\) 1.91103e8 0.0416667
\(577\) 7.31947e9 1.58622 0.793111 0.609077i \(-0.208460\pi\)
0.793111 + 0.609077i \(0.208460\pi\)
\(578\) 1.93101e8 0.0415945
\(579\) −1.77975e9 −0.381052
\(580\) −8.24102e8 −0.175381
\(581\) −1.84154e9 −0.389551
\(582\) −2.16721e9 −0.455691
\(583\) 3.86429e9 0.807662
\(584\) 5.64309e8 0.117239
\(585\) 3.14020e8 0.0648503
\(586\) −1.23788e9 −0.254119
\(587\) −1.98324e9 −0.404707 −0.202354 0.979313i \(-0.564859\pi\)
−0.202354 + 0.979313i \(0.564859\pi\)
\(588\) −1.21860e9 −0.247195
\(589\) 3.38895e9 0.683379
\(590\) 2.48247e9 0.497624
\(591\) −3.89993e9 −0.777142
\(592\) 1.24486e8 0.0246600
\(593\) −2.61002e9 −0.513988 −0.256994 0.966413i \(-0.582732\pi\)
−0.256994 + 0.966413i \(0.582732\pi\)
\(594\) −6.58987e8 −0.129010
\(595\) 3.97167e8 0.0772972
\(596\) 1.10151e9 0.213121
\(597\) −5.76183e9 −1.10828
\(598\) 1.25285e9 0.239577
\(599\) −3.38989e9 −0.644453 −0.322227 0.946663i \(-0.604431\pi\)
−0.322227 + 0.946663i \(0.604431\pi\)
\(600\) −3.16570e8 −0.0598329
\(601\) 2.12243e9 0.398816 0.199408 0.979917i \(-0.436098\pi\)
0.199408 + 0.979917i \(0.436098\pi\)
\(602\) 1.19466e9 0.223181
\(603\) −2.69237e8 −0.0500063
\(604\) −1.72627e9 −0.318772
\(605\) 4.63642e8 0.0851215
\(606\) 2.00384e8 0.0365771
\(607\) 4.83573e9 0.877610 0.438805 0.898582i \(-0.355402\pi\)
0.438805 + 0.898582i \(0.355402\pi\)
\(608\) −1.23958e9 −0.223672
\(609\) −5.08927e8 −0.0913050
\(610\) −2.24313e9 −0.400130
\(611\) −4.75440e8 −0.0843240
\(612\) 2.29221e8 0.0404226
\(613\) −8.79032e9 −1.54132 −0.770661 0.637246i \(-0.780074\pi\)
−0.770661 + 0.637246i \(0.780074\pi\)
\(614\) −3.95955e9 −0.690329
\(615\) −3.49429e9 −0.605753
\(616\) 7.37096e8 0.127055
\(617\) −3.23362e9 −0.554231 −0.277116 0.960837i \(-0.589379\pi\)
−0.277116 + 0.960837i \(0.589379\pi\)
\(618\) −3.14026e8 −0.0535187
\(619\) 4.02310e9 0.681780 0.340890 0.940103i \(-0.389272\pi\)
0.340890 + 0.940103i \(0.389272\pi\)
\(620\) 1.34737e9 0.227048
\(621\) −1.68166e9 −0.281784
\(622\) −5.45490e9 −0.908908
\(623\) 1.46881e9 0.243364
\(624\) −2.02715e8 −0.0333995
\(625\) −3.79004e9 −0.620961
\(626\) −2.05640e8 −0.0335040
\(627\) 4.27449e9 0.692545
\(628\) −1.43788e9 −0.231667
\(629\) 1.49316e8 0.0239237
\(630\) 4.71459e8 0.0751194
\(631\) −1.15616e10 −1.83195 −0.915977 0.401231i \(-0.868582\pi\)
−0.915977 + 0.401231i \(0.868582\pi\)
\(632\) 2.83549e9 0.446806
\(633\) −4.79228e9 −0.750981
\(634\) 5.91730e9 0.922170
\(635\) −2.59019e9 −0.401442
\(636\) −1.59558e9 −0.245934
\(637\) 1.29264e9 0.198149
\(638\) −1.83450e9 −0.279670
\(639\) −1.99912e9 −0.303100
\(640\) −4.92831e8 −0.0743135
\(641\) 1.28560e10 1.92798 0.963990 0.265938i \(-0.0856818\pi\)
0.963990 + 0.265938i \(0.0856818\pi\)
\(642\) 4.74978e9 0.708436
\(643\) −2.59277e9 −0.384614 −0.192307 0.981335i \(-0.561597\pi\)
−0.192307 + 0.981335i \(0.561597\pi\)
\(644\) 1.88098e9 0.277513
\(645\) 2.75441e9 0.404175
\(646\) −1.48683e9 −0.216994
\(647\) −1.18826e9 −0.172483 −0.0862417 0.996274i \(-0.527486\pi\)
−0.0862417 + 0.996274i \(0.527486\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 5.52613e9 0.793533
\(650\) 3.35806e8 0.0479613
\(651\) 8.32075e8 0.118203
\(652\) −4.68902e9 −0.662546
\(653\) 2.53925e9 0.356870 0.178435 0.983952i \(-0.442897\pi\)
0.178435 + 0.983952i \(0.442897\pi\)
\(654\) 2.76118e9 0.385987
\(655\) −9.01940e9 −1.25410
\(656\) 2.25573e9 0.311978
\(657\) 8.03479e8 0.110534
\(658\) −7.13808e8 −0.0976767
\(659\) 8.64850e9 1.17718 0.588588 0.808433i \(-0.299684\pi\)
0.588588 + 0.808433i \(0.299684\pi\)
\(660\) 1.69944e9 0.230093
\(661\) 7.25099e9 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(662\) −5.22932e9 −0.700556
\(663\) −2.43149e8 −0.0324023
\(664\) 2.74090e9 0.363333
\(665\) −3.05810e9 −0.403251
\(666\) 1.77246e8 0.0232497
\(667\) −4.68143e9 −0.610856
\(668\) −1.42990e9 −0.185604
\(669\) −4.97618e9 −0.642547
\(670\) 6.94329e8 0.0891874
\(671\) −4.99336e9 −0.638063
\(672\) −3.04349e8 −0.0386883
\(673\) 5.66492e9 0.716376 0.358188 0.933650i \(-0.383395\pi\)
0.358188 + 0.933650i \(0.383395\pi\)
\(674\) −4.54544e9 −0.571830
\(675\) −4.50741e8 −0.0564110
\(676\) −3.80087e9 −0.473227
\(677\) 9.24182e9 1.14471 0.572357 0.820004i \(-0.306029\pi\)
0.572357 + 0.820004i \(0.306029\pi\)
\(678\) 5.48981e9 0.676478
\(679\) 3.45148e9 0.423117
\(680\) −5.91132e8 −0.0720947
\(681\) 2.22204e8 0.0269611
\(682\) 2.99934e9 0.362060
\(683\) 2.38058e9 0.285897 0.142949 0.989730i \(-0.454342\pi\)
0.142949 + 0.989730i \(0.454342\pi\)
\(684\) −1.76495e9 −0.210880
\(685\) 3.61460e9 0.429678
\(686\) 4.20712e9 0.497566
\(687\) 8.35178e9 0.982722
\(688\) −1.77810e9 −0.208160
\(689\) 1.69253e9 0.197137
\(690\) 4.33678e9 0.502569
\(691\) −4.57058e8 −0.0526985 −0.0263492 0.999653i \(-0.508388\pi\)
−0.0263492 + 0.999653i \(0.508388\pi\)
\(692\) 1.98952e9 0.228232
\(693\) 1.04950e9 0.119788
\(694\) −7.03362e9 −0.798769
\(695\) 2.89219e9 0.326799
\(696\) 7.57472e8 0.0851598
\(697\) 2.70566e9 0.302663
\(698\) −2.62729e9 −0.292425
\(699\) 3.39047e9 0.375483
\(700\) 5.04166e8 0.0555560
\(701\) −3.87966e9 −0.425383 −0.212692 0.977119i \(-0.568223\pi\)
−0.212692 + 0.977119i \(0.568223\pi\)
\(702\) −2.88632e8 −0.0314894
\(703\) −1.14970e9 −0.124807
\(704\) −1.09707e9 −0.118503
\(705\) −1.64575e9 −0.176890
\(706\) −4.27910e9 −0.457653
\(707\) −3.19130e8 −0.0339625
\(708\) −2.28176e9 −0.241631
\(709\) 1.50551e10 1.58643 0.793215 0.608942i \(-0.208406\pi\)
0.793215 + 0.608942i \(0.208406\pi\)
\(710\) 5.15548e9 0.540586
\(711\) 4.03725e9 0.421252
\(712\) −2.18613e9 −0.226985
\(713\) 7.65396e9 0.790811
\(714\) −3.65056e8 −0.0375332
\(715\) −1.80271e9 −0.184440
\(716\) −2.64689e9 −0.269489
\(717\) 1.05000e10 1.06383
\(718\) −4.56741e9 −0.460505
\(719\) 1.19938e10 1.20339 0.601696 0.798725i \(-0.294492\pi\)
0.601696 + 0.798725i \(0.294492\pi\)
\(720\) −7.01706e8 −0.0700635
\(721\) 5.00115e8 0.0496932
\(722\) 4.29729e9 0.424927
\(723\) −5.40332e9 −0.531712
\(724\) −7.89907e8 −0.0773554
\(725\) −1.25478e9 −0.122288
\(726\) −4.26156e8 −0.0413324
\(727\) 1.80966e10 1.74674 0.873368 0.487060i \(-0.161931\pi\)
0.873368 + 0.487060i \(0.161931\pi\)
\(728\) 3.22843e8 0.0310121
\(729\) 3.87420e8 0.0370370
\(730\) −2.07207e9 −0.197140
\(731\) −2.13277e9 −0.201945
\(732\) 2.06177e9 0.194291
\(733\) 4.43663e9 0.416092 0.208046 0.978119i \(-0.433290\pi\)
0.208046 + 0.978119i \(0.433290\pi\)
\(734\) 3.95259e9 0.368932
\(735\) 4.47454e9 0.415664
\(736\) −2.79960e9 −0.258835
\(737\) 1.54562e9 0.142222
\(738\) 3.21177e9 0.294135
\(739\) −3.90799e9 −0.356203 −0.178102 0.984012i \(-0.556996\pi\)
−0.178102 + 0.984012i \(0.556996\pi\)
\(740\) −4.57096e8 −0.0414664
\(741\) 1.87220e9 0.169039
\(742\) 2.54110e9 0.228354
\(743\) 4.26083e9 0.381095 0.190548 0.981678i \(-0.438974\pi\)
0.190548 + 0.981678i \(0.438974\pi\)
\(744\) −1.23844e9 −0.110247
\(745\) −4.04460e9 −0.358368
\(746\) −1.26760e9 −0.111788
\(747\) 3.90257e9 0.342553
\(748\) −1.31590e9 −0.114965
\(749\) −7.56446e9 −0.657797
\(750\) 5.12803e9 0.443850
\(751\) 2.18517e10 1.88255 0.941274 0.337643i \(-0.109630\pi\)
0.941274 + 0.337643i \(0.109630\pi\)
\(752\) 1.06241e9 0.0911026
\(753\) −3.94179e9 −0.336443
\(754\) −8.03499e8 −0.0682631
\(755\) 6.33866e9 0.536023
\(756\) −4.33341e8 −0.0364757
\(757\) −3.81516e9 −0.319652 −0.159826 0.987145i \(-0.551093\pi\)
−0.159826 + 0.987145i \(0.551093\pi\)
\(758\) −1.23286e10 −1.02819
\(759\) 9.65395e9 0.801418
\(760\) 4.55159e9 0.376110
\(761\) −1.45183e10 −1.19418 −0.597090 0.802174i \(-0.703676\pi\)
−0.597090 + 0.802174i \(0.703676\pi\)
\(762\) 2.38077e9 0.194928
\(763\) −4.39744e9 −0.358396
\(764\) 3.01303e9 0.244443
\(765\) −8.41671e8 −0.0679715
\(766\) −1.06867e10 −0.859096
\(767\) 2.42041e9 0.193689
\(768\) 4.52985e8 0.0360844
\(769\) 1.36277e10 1.08064 0.540318 0.841461i \(-0.318304\pi\)
0.540318 + 0.841461i \(0.318304\pi\)
\(770\) −2.70652e9 −0.213646
\(771\) −1.14274e10 −0.897962
\(772\) −4.21867e9 −0.330000
\(773\) −2.29018e10 −1.78337 −0.891685 0.452657i \(-0.850476\pi\)
−0.891685 + 0.452657i \(0.850476\pi\)
\(774\) −2.53171e9 −0.196255
\(775\) 2.05152e9 0.158314
\(776\) −5.13708e9 −0.394640
\(777\) −2.82281e8 −0.0215878
\(778\) 1.35127e10 1.02876
\(779\) −2.08330e10 −1.57896
\(780\) 7.44345e8 0.0561621
\(781\) 1.14764e10 0.862041
\(782\) −3.35802e9 −0.251107
\(783\) 1.07851e9 0.0802894
\(784\) −2.88853e9 −0.214077
\(785\) 5.27971e9 0.389553
\(786\) 8.29017e9 0.608954
\(787\) −7.48603e8 −0.0547444 −0.0273722 0.999625i \(-0.508714\pi\)
−0.0273722 + 0.999625i \(0.508714\pi\)
\(788\) −9.24427e9 −0.673025
\(789\) 1.26244e9 0.0915041
\(790\) −1.04116e10 −0.751314
\(791\) −8.74304e9 −0.628123
\(792\) −1.56204e9 −0.111726
\(793\) −2.18705e9 −0.155741
\(794\) 6.96725e9 0.493958
\(795\) 5.85876e9 0.413543
\(796\) −1.36577e10 −0.959800
\(797\) −1.65797e10 −1.16004 −0.580020 0.814602i \(-0.696955\pi\)
−0.580020 + 0.814602i \(0.696955\pi\)
\(798\) 2.81085e9 0.195806
\(799\) 1.27432e9 0.0883825
\(800\) −7.50387e8 −0.0518168
\(801\) −3.11268e9 −0.214003
\(802\) 1.57992e10 1.08149
\(803\) −4.61256e9 −0.314368
\(804\) −6.38192e8 −0.0433067
\(805\) −6.90673e9 −0.466645
\(806\) 1.31369e9 0.0883731
\(807\) −7.70723e9 −0.516227
\(808\) 4.74984e8 0.0316767
\(809\) 1.50355e9 0.0998387 0.0499194 0.998753i \(-0.484104\pi\)
0.0499194 + 0.998753i \(0.484104\pi\)
\(810\) −9.99109e8 −0.0660565
\(811\) −2.23679e10 −1.47249 −0.736243 0.676717i \(-0.763402\pi\)
−0.736243 + 0.676717i \(0.763402\pi\)
\(812\) −1.20634e9 −0.0790725
\(813\) −4.64520e9 −0.303171
\(814\) −1.01752e9 −0.0661240
\(815\) 1.72175e10 1.11409
\(816\) 5.43338e8 0.0350070
\(817\) 1.64218e10 1.05352
\(818\) −5.98577e9 −0.382370
\(819\) 4.59672e8 0.0292385
\(820\) −8.28275e9 −0.524598
\(821\) −1.53966e10 −0.971008 −0.485504 0.874234i \(-0.661364\pi\)
−0.485504 + 0.874234i \(0.661364\pi\)
\(822\) −3.32236e9 −0.208639
\(823\) 1.18952e10 0.743829 0.371914 0.928267i \(-0.378701\pi\)
0.371914 + 0.928267i \(0.378701\pi\)
\(824\) −7.44357e8 −0.0463486
\(825\) 2.58759e9 0.160438
\(826\) 3.63391e9 0.224359
\(827\) 1.84279e10 1.13294 0.566468 0.824084i \(-0.308309\pi\)
0.566468 + 0.824084i \(0.308309\pi\)
\(828\) −3.98615e9 −0.244032
\(829\) 6.56187e9 0.400025 0.200012 0.979793i \(-0.435902\pi\)
0.200012 + 0.979793i \(0.435902\pi\)
\(830\) −1.00642e10 −0.610952
\(831\) 7.78456e9 0.470577
\(832\) −4.80510e8 −0.0289248
\(833\) −3.46468e9 −0.207685
\(834\) −2.65835e9 −0.158683
\(835\) 5.25041e9 0.312098
\(836\) 1.01321e10 0.599761
\(837\) −1.76332e9 −0.103942
\(838\) −6.33311e9 −0.371760
\(839\) −2.58040e10 −1.50841 −0.754205 0.656639i \(-0.771978\pi\)
−0.754205 + 0.656639i \(0.771978\pi\)
\(840\) 1.11753e9 0.0650553
\(841\) −1.42475e10 −0.825948
\(842\) 8.73961e8 0.0504545
\(843\) 1.49422e9 0.0859047
\(844\) −1.13595e10 −0.650369
\(845\) 1.39563e10 0.795743
\(846\) 1.51269e9 0.0858923
\(847\) 6.78693e8 0.0383779
\(848\) −3.78211e9 −0.212985
\(849\) 1.82333e9 0.102256
\(850\) −9.00062e8 −0.0502697
\(851\) −2.59660e9 −0.144428
\(852\) −4.73865e9 −0.262492
\(853\) 9.66749e7 0.00533325 0.00266663 0.999996i \(-0.499151\pi\)
0.00266663 + 0.999996i \(0.499151\pi\)
\(854\) −3.28357e9 −0.180403
\(855\) 6.48068e9 0.354600
\(856\) 1.12587e10 0.613524
\(857\) −5.64116e9 −0.306151 −0.153075 0.988215i \(-0.548918\pi\)
−0.153075 + 0.988215i \(0.548918\pi\)
\(858\) 1.65696e9 0.0895584
\(859\) −1.62139e10 −0.872792 −0.436396 0.899755i \(-0.643745\pi\)
−0.436396 + 0.899755i \(0.643745\pi\)
\(860\) 6.52897e9 0.350026
\(861\) −5.11504e9 −0.273110
\(862\) 2.35772e10 1.25377
\(863\) −2.17887e10 −1.15397 −0.576983 0.816756i \(-0.695770\pi\)
−0.576983 + 0.816756i \(0.695770\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −7.30526e9 −0.383778
\(866\) 1.28501e10 0.672346
\(867\) 6.51714e8 0.0339618
\(868\) 1.97233e9 0.102367
\(869\) −2.31768e10 −1.19808
\(870\) −2.78134e9 −0.143198
\(871\) 6.76971e8 0.0347141
\(872\) 6.54502e9 0.334275
\(873\) −7.31433e9 −0.372070
\(874\) 2.58560e10 1.31000
\(875\) −8.16686e9 −0.412123
\(876\) 1.90454e9 0.0957252
\(877\) 1.57947e10 0.790700 0.395350 0.918531i \(-0.370623\pi\)
0.395350 + 0.918531i \(0.370623\pi\)
\(878\) −8.90640e9 −0.444090
\(879\) −4.17785e9 −0.207487
\(880\) 4.02831e9 0.199266
\(881\) −3.46306e10 −1.70626 −0.853128 0.521702i \(-0.825297\pi\)
−0.853128 + 0.521702i \(0.825297\pi\)
\(882\) −4.11277e9 −0.201834
\(883\) 7.20168e9 0.352023 0.176012 0.984388i \(-0.443680\pi\)
0.176012 + 0.984388i \(0.443680\pi\)
\(884\) −5.76354e8 −0.0280612
\(885\) 8.37833e9 0.406309
\(886\) 6.27814e9 0.303259
\(887\) −7.76600e8 −0.0373650 −0.0186825 0.999825i \(-0.505947\pi\)
−0.0186825 + 0.999825i \(0.505947\pi\)
\(888\) 4.20139e8 0.0201348
\(889\) −3.79159e9 −0.180995
\(890\) 8.02720e9 0.381680
\(891\) −2.22408e9 −0.105336
\(892\) −1.17954e10 −0.556462
\(893\) −9.81201e9 −0.461082
\(894\) 3.71759e9 0.174012
\(895\) 9.71906e9 0.453152
\(896\) −7.21420e8 −0.0335051
\(897\) 4.22836e9 0.195613
\(898\) −1.89324e10 −0.872444
\(899\) −4.90878e9 −0.225328
\(900\) −1.06842e9 −0.0488533
\(901\) −4.53650e9 −0.206625
\(902\) −1.84379e10 −0.836545
\(903\) 4.03199e9 0.182227
\(904\) 1.30129e10 0.585847
\(905\) 2.90044e9 0.130075
\(906\) −5.82617e9 −0.260276
\(907\) 7.02650e9 0.312690 0.156345 0.987703i \(-0.450029\pi\)
0.156345 + 0.987703i \(0.450029\pi\)
\(908\) 5.26707e8 0.0233490
\(909\) 6.76296e8 0.0298651
\(910\) −1.18544e9 −0.0521475
\(911\) 3.87256e10 1.69701 0.848503 0.529190i \(-0.177504\pi\)
0.848503 + 0.529190i \(0.177504\pi\)
\(912\) −4.18358e9 −0.182628
\(913\) −2.24036e10 −0.974250
\(914\) 1.30021e10 0.563252
\(915\) −7.57057e9 −0.326704
\(916\) 1.97968e10 0.851062
\(917\) −1.32029e10 −0.565425
\(918\) 7.73621e8 0.0330049
\(919\) 2.34371e10 0.996094 0.498047 0.867150i \(-0.334051\pi\)
0.498047 + 0.867150i \(0.334051\pi\)
\(920\) 1.02798e10 0.435238
\(921\) −1.33635e10 −0.563651
\(922\) −5.07604e9 −0.213288
\(923\) 5.02659e9 0.210411
\(924\) 2.48770e9 0.103740
\(925\) −6.95977e8 −0.0289134
\(926\) −2.03367e10 −0.841672
\(927\) −1.05984e9 −0.0436978
\(928\) 1.79549e9 0.0737505
\(929\) 1.70755e10 0.698744 0.349372 0.936984i \(-0.386395\pi\)
0.349372 + 0.936984i \(0.386395\pi\)
\(930\) 4.54739e9 0.185384
\(931\) 2.66773e10 1.08347
\(932\) 8.03667e9 0.325177
\(933\) −1.84103e10 −0.742121
\(934\) −1.53607e9 −0.0616874
\(935\) 4.83181e9 0.193317
\(936\) −6.84164e8 −0.0272706
\(937\) −4.17866e10 −1.65939 −0.829694 0.558218i \(-0.811485\pi\)
−0.829694 + 0.558218i \(0.811485\pi\)
\(938\) 1.01638e9 0.0402111
\(939\) −6.94034e8 −0.0273559
\(940\) −3.90105e9 −0.153191
\(941\) −1.20231e10 −0.470383 −0.235192 0.971949i \(-0.575572\pi\)
−0.235192 + 0.971949i \(0.575572\pi\)
\(942\) −4.85284e9 −0.189155
\(943\) −4.70514e10 −1.82718
\(944\) −5.40861e9 −0.209259
\(945\) 1.59117e9 0.0613347
\(946\) 1.45339e10 0.558166
\(947\) 4.29691e10 1.64411 0.822056 0.569407i \(-0.192827\pi\)
0.822056 + 0.569407i \(0.192827\pi\)
\(948\) 9.56978e9 0.364815
\(949\) −2.02027e9 −0.0767322
\(950\) 6.93027e9 0.262251
\(951\) 1.99709e10 0.752949
\(952\) −8.65317e8 −0.0325047
\(953\) −3.01447e10 −1.12820 −0.564100 0.825706i \(-0.690777\pi\)
−0.564100 + 0.825706i \(0.690777\pi\)
\(954\) −5.38507e9 −0.200804
\(955\) −1.10635e10 −0.411036
\(956\) 2.48888e10 0.921301
\(957\) −6.19145e9 −0.228350
\(958\) −6.85453e9 −0.251883
\(959\) 5.29116e9 0.193725
\(960\) −1.66330e9 −0.0606767
\(961\) −1.94870e10 −0.708292
\(962\) −4.45668e8 −0.0161398
\(963\) 1.60305e10 0.578436
\(964\) −1.28079e10 −0.460476
\(965\) 1.54904e10 0.554903
\(966\) 6.34831e9 0.226589
\(967\) −1.06854e10 −0.380014 −0.190007 0.981783i \(-0.560851\pi\)
−0.190007 + 0.981783i \(0.560851\pi\)
\(968\) −1.01015e9 −0.0357949
\(969\) −5.01805e9 −0.177175
\(970\) 1.88627e10 0.663596
\(971\) 5.10197e10 1.78842 0.894212 0.447643i \(-0.147737\pi\)
0.894212 + 0.447643i \(0.147737\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 4.23368e9 0.147341
\(974\) 3.96010e10 1.37325
\(975\) 1.13334e9 0.0391603
\(976\) 4.88717e9 0.168261
\(977\) −5.11414e10 −1.75445 −0.877226 0.480079i \(-0.840608\pi\)
−0.877226 + 0.480079i \(0.840608\pi\)
\(978\) −1.58255e10 −0.540966
\(979\) 1.78691e10 0.608643
\(980\) 1.06063e10 0.359976
\(981\) 9.31898e9 0.315157
\(982\) −3.34315e9 −0.112659
\(983\) −5.07243e10 −1.70325 −0.851626 0.524149i \(-0.824383\pi\)
−0.851626 + 0.524149i \(0.824383\pi\)
\(984\) 7.61308e9 0.254729
\(985\) 3.39438e10 1.13171
\(986\) 2.15362e9 0.0715485
\(987\) −2.40910e9 −0.0797527
\(988\) 4.43780e9 0.146392
\(989\) 3.70888e10 1.21915
\(990\) 5.73563e9 0.187870
\(991\) −1.71204e10 −0.558800 −0.279400 0.960175i \(-0.590136\pi\)
−0.279400 + 0.960175i \(0.590136\pi\)
\(992\) −2.93555e9 −0.0954771
\(993\) −1.76490e10 −0.572001
\(994\) 7.54674e9 0.243729
\(995\) 5.01492e10 1.61393
\(996\) 9.25053e9 0.296660
\(997\) −6.20246e9 −0.198213 −0.0991063 0.995077i \(-0.531598\pi\)
−0.0991063 + 0.995077i \(0.531598\pi\)
\(998\) −1.28546e10 −0.409356
\(999\) 5.98206e8 0.0189833
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 102.8.a.b.1.1 1
3.2 odd 2 306.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.8.a.b.1.1 1 1.1 even 1 trivial
306.8.a.b.1.1 1 3.2 odd 2