Properties

Label 304.6.a.f.1.1
Level $304$
Weight $6$
Character 304.1
Self dual yes
Analytic conductor $48.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.7566812231\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.4803\) of defining polynomial
Character \(\chi\) \(=\) 304.1

$q$-expansion

\(f(q)\) \(=\) \(q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} +O(q^{10})\) \(q-20.4803 q^{3} +34.4408 q^{5} +18.9210 q^{7} +176.441 q^{9} -349.480 q^{11} +711.599 q^{13} -705.355 q^{15} +221.803 q^{17} -361.000 q^{19} -387.507 q^{21} +662.468 q^{23} -1938.83 q^{25} +1363.15 q^{27} -7219.28 q^{29} -5407.76 q^{31} +7157.44 q^{33} +651.654 q^{35} +1979.40 q^{37} -14573.7 q^{39} -3111.11 q^{41} -318.049 q^{43} +6076.75 q^{45} +27240.5 q^{47} -16449.0 q^{49} -4542.57 q^{51} -1114.63 q^{53} -12036.4 q^{55} +7393.37 q^{57} +37904.9 q^{59} +37469.2 q^{61} +3338.44 q^{63} +24508.0 q^{65} +54955.3 q^{67} -13567.5 q^{69} +7177.04 q^{71} +64746.1 q^{73} +39707.8 q^{75} -6612.52 q^{77} -36104.4 q^{79} -70792.8 q^{81} +51782.7 q^{83} +7639.05 q^{85} +147853. q^{87} +145254. q^{89} +13464.2 q^{91} +110752. q^{93} -12433.1 q^{95} +39512.8 q^{97} -61662.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} - 45q^{5} - 114q^{7} + 239q^{9} + O(q^{10}) \) \( 2q - 3q^{3} - 45q^{5} - 114q^{7} + 239q^{9} - 661q^{11} + 1613q^{13} - 2094q^{15} + 64q^{17} - 722q^{19} - 2711q^{21} + 3185q^{23} + 1247q^{25} - 1791q^{27} - 2481q^{29} + 1180q^{31} + 1712q^{33} + 11211q^{35} + 10488q^{37} + 1183q^{39} + 16630q^{41} - 11303q^{43} + 1107q^{45} + 12155q^{47} - 15588q^{49} - 7301q^{51} + 20585q^{53} + 12711q^{55} + 1083q^{57} + 78581q^{59} + 43621q^{61} - 4977q^{63} - 47100q^{65} - 7805q^{67} + 30527q^{69} + 62488q^{71} + 16218q^{73} + 95397q^{75} + 34795q^{77} - 67122q^{79} - 141130q^{81} + 10714q^{83} + 20175q^{85} + 230679q^{87} + 128188q^{89} - 106351q^{91} + 225908q^{93} + 16245q^{95} + 178558q^{97} - 81151q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −20.4803 −1.31381 −0.656904 0.753974i \(-0.728135\pi\)
−0.656904 + 0.753974i \(0.728135\pi\)
\(4\) 0 0
\(5\) 34.4408 0.616095 0.308048 0.951371i \(-0.400324\pi\)
0.308048 + 0.951371i \(0.400324\pi\)
\(6\) 0 0
\(7\) 18.9210 0.145948 0.0729742 0.997334i \(-0.476751\pi\)
0.0729742 + 0.997334i \(0.476751\pi\)
\(8\) 0 0
\(9\) 176.441 0.726094
\(10\) 0 0
\(11\) −349.480 −0.870845 −0.435423 0.900226i \(-0.643401\pi\)
−0.435423 + 0.900226i \(0.643401\pi\)
\(12\) 0 0
\(13\) 711.599 1.16782 0.583911 0.811818i \(-0.301522\pi\)
0.583911 + 0.811818i \(0.301522\pi\)
\(14\) 0 0
\(15\) −705.355 −0.809431
\(16\) 0 0
\(17\) 221.803 0.186142 0.0930710 0.995659i \(-0.470332\pi\)
0.0930710 + 0.995659i \(0.470332\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) −387.507 −0.191748
\(22\) 0 0
\(23\) 662.468 0.261123 0.130561 0.991440i \(-0.458322\pi\)
0.130561 + 0.991440i \(0.458322\pi\)
\(24\) 0 0
\(25\) −1938.83 −0.620427
\(26\) 0 0
\(27\) 1363.15 0.359861
\(28\) 0 0
\(29\) −7219.28 −1.59404 −0.797019 0.603954i \(-0.793591\pi\)
−0.797019 + 0.603954i \(0.793591\pi\)
\(30\) 0 0
\(31\) −5407.76 −1.01068 −0.505339 0.862921i \(-0.668633\pi\)
−0.505339 + 0.862921i \(0.668633\pi\)
\(32\) 0 0
\(33\) 7157.44 1.14412
\(34\) 0 0
\(35\) 651.654 0.0899181
\(36\) 0 0
\(37\) 1979.40 0.237700 0.118850 0.992912i \(-0.462079\pi\)
0.118850 + 0.992912i \(0.462079\pi\)
\(38\) 0 0
\(39\) −14573.7 −1.53430
\(40\) 0 0
\(41\) −3111.11 −0.289039 −0.144519 0.989502i \(-0.546164\pi\)
−0.144519 + 0.989502i \(0.546164\pi\)
\(42\) 0 0
\(43\) −318.049 −0.0262315 −0.0131157 0.999914i \(-0.504175\pi\)
−0.0131157 + 0.999914i \(0.504175\pi\)
\(44\) 0 0
\(45\) 6076.75 0.447343
\(46\) 0 0
\(47\) 27240.5 1.79875 0.899374 0.437181i \(-0.144023\pi\)
0.899374 + 0.437181i \(0.144023\pi\)
\(48\) 0 0
\(49\) −16449.0 −0.978699
\(50\) 0 0
\(51\) −4542.57 −0.244555
\(52\) 0 0
\(53\) −1114.63 −0.0545057 −0.0272528 0.999629i \(-0.508676\pi\)
−0.0272528 + 0.999629i \(0.508676\pi\)
\(54\) 0 0
\(55\) −12036.4 −0.536523
\(56\) 0 0
\(57\) 7393.37 0.301408
\(58\) 0 0
\(59\) 37904.9 1.41764 0.708820 0.705390i \(-0.249228\pi\)
0.708820 + 0.705390i \(0.249228\pi\)
\(60\) 0 0
\(61\) 37469.2 1.28929 0.644644 0.764483i \(-0.277006\pi\)
0.644644 + 0.764483i \(0.277006\pi\)
\(62\) 0 0
\(63\) 3338.44 0.105972
\(64\) 0 0
\(65\) 24508.0 0.719490
\(66\) 0 0
\(67\) 54955.3 1.49562 0.747812 0.663911i \(-0.231105\pi\)
0.747812 + 0.663911i \(0.231105\pi\)
\(68\) 0 0
\(69\) −13567.5 −0.343066
\(70\) 0 0
\(71\) 7177.04 0.168966 0.0844830 0.996425i \(-0.473076\pi\)
0.0844830 + 0.996425i \(0.473076\pi\)
\(72\) 0 0
\(73\) 64746.1 1.42202 0.711011 0.703181i \(-0.248238\pi\)
0.711011 + 0.703181i \(0.248238\pi\)
\(74\) 0 0
\(75\) 39707.8 0.815122
\(76\) 0 0
\(77\) −6612.52 −0.127098
\(78\) 0 0
\(79\) −36104.4 −0.650866 −0.325433 0.945565i \(-0.605510\pi\)
−0.325433 + 0.945565i \(0.605510\pi\)
\(80\) 0 0
\(81\) −70792.8 −1.19888
\(82\) 0 0
\(83\) 51782.7 0.825067 0.412534 0.910942i \(-0.364644\pi\)
0.412534 + 0.910942i \(0.364644\pi\)
\(84\) 0 0
\(85\) 7639.05 0.114681
\(86\) 0 0
\(87\) 147853. 2.09426
\(88\) 0 0
\(89\) 145254. 1.94380 0.971900 0.235392i \(-0.0756375\pi\)
0.971900 + 0.235392i \(0.0756375\pi\)
\(90\) 0 0
\(91\) 13464.2 0.170442
\(92\) 0 0
\(93\) 110752. 1.32784
\(94\) 0 0
\(95\) −12433.1 −0.141342
\(96\) 0 0
\(97\) 39512.8 0.426391 0.213196 0.977010i \(-0.431613\pi\)
0.213196 + 0.977010i \(0.431613\pi\)
\(98\) 0 0
\(99\) −61662.6 −0.632315
\(100\) 0 0
\(101\) −16722.9 −0.163120 −0.0815602 0.996668i \(-0.525990\pi\)
−0.0815602 + 0.996668i \(0.525990\pi\)
\(102\) 0 0
\(103\) −92664.4 −0.860636 −0.430318 0.902677i \(-0.641599\pi\)
−0.430318 + 0.902677i \(0.641599\pi\)
\(104\) 0 0
\(105\) −13346.0 −0.118135
\(106\) 0 0
\(107\) 47156.3 0.398180 0.199090 0.979981i \(-0.436201\pi\)
0.199090 + 0.979981i \(0.436201\pi\)
\(108\) 0 0
\(109\) 52450.1 0.422844 0.211422 0.977395i \(-0.432191\pi\)
0.211422 + 0.977395i \(0.432191\pi\)
\(110\) 0 0
\(111\) −40538.5 −0.312292
\(112\) 0 0
\(113\) 150596. 1.10947 0.554737 0.832026i \(-0.312819\pi\)
0.554737 + 0.832026i \(0.312819\pi\)
\(114\) 0 0
\(115\) 22815.9 0.160877
\(116\) 0 0
\(117\) 125555. 0.847948
\(118\) 0 0
\(119\) 4196.73 0.0271671
\(120\) 0 0
\(121\) −38914.6 −0.241629
\(122\) 0 0
\(123\) 63716.4 0.379742
\(124\) 0 0
\(125\) −174402. −0.998337
\(126\) 0 0
\(127\) −352240. −1.93789 −0.968945 0.247276i \(-0.920465\pi\)
−0.968945 + 0.247276i \(0.920465\pi\)
\(128\) 0 0
\(129\) 6513.72 0.0344632
\(130\) 0 0
\(131\) −19070.2 −0.0970907 −0.0485453 0.998821i \(-0.515459\pi\)
−0.0485453 + 0.998821i \(0.515459\pi\)
\(132\) 0 0
\(133\) −6830.49 −0.0334829
\(134\) 0 0
\(135\) 46947.9 0.221708
\(136\) 0 0
\(137\) −266677. −1.21390 −0.606952 0.794738i \(-0.707608\pi\)
−0.606952 + 0.794738i \(0.707608\pi\)
\(138\) 0 0
\(139\) 294888. 1.29456 0.647278 0.762254i \(-0.275907\pi\)
0.647278 + 0.762254i \(0.275907\pi\)
\(140\) 0 0
\(141\) −557892. −2.36321
\(142\) 0 0
\(143\) −248690. −1.01699
\(144\) 0 0
\(145\) −248637. −0.982079
\(146\) 0 0
\(147\) 336880. 1.28582
\(148\) 0 0
\(149\) −103990. −0.383729 −0.191864 0.981421i \(-0.561453\pi\)
−0.191864 + 0.981421i \(0.561453\pi\)
\(150\) 0 0
\(151\) 477343. 1.70368 0.851840 0.523802i \(-0.175487\pi\)
0.851840 + 0.523802i \(0.175487\pi\)
\(152\) 0 0
\(153\) 39135.0 0.135156
\(154\) 0 0
\(155\) −186247. −0.622674
\(156\) 0 0
\(157\) 434955. 1.40830 0.704151 0.710051i \(-0.251328\pi\)
0.704151 + 0.710051i \(0.251328\pi\)
\(158\) 0 0
\(159\) 22827.9 0.0716101
\(160\) 0 0
\(161\) 12534.6 0.0381105
\(162\) 0 0
\(163\) 255083. 0.751990 0.375995 0.926622i \(-0.377301\pi\)
0.375995 + 0.926622i \(0.377301\pi\)
\(164\) 0 0
\(165\) 246508. 0.704889
\(166\) 0 0
\(167\) −510501. −1.41646 −0.708232 0.705980i \(-0.750507\pi\)
−0.708232 + 0.705980i \(0.750507\pi\)
\(168\) 0 0
\(169\) 135080. 0.363809
\(170\) 0 0
\(171\) −63695.1 −0.166577
\(172\) 0 0
\(173\) 773453. 1.96480 0.982401 0.186783i \(-0.0598062\pi\)
0.982401 + 0.186783i \(0.0598062\pi\)
\(174\) 0 0
\(175\) −36684.7 −0.0905503
\(176\) 0 0
\(177\) −776303. −1.86251
\(178\) 0 0
\(179\) 477664. 1.11427 0.557135 0.830422i \(-0.311901\pi\)
0.557135 + 0.830422i \(0.311901\pi\)
\(180\) 0 0
\(181\) 729114. 1.65424 0.827121 0.562024i \(-0.189977\pi\)
0.827121 + 0.562024i \(0.189977\pi\)
\(182\) 0 0
\(183\) −767379. −1.69388
\(184\) 0 0
\(185\) 68171.9 0.146446
\(186\) 0 0
\(187\) −77515.6 −0.162101
\(188\) 0 0
\(189\) 25792.2 0.0525211
\(190\) 0 0
\(191\) 285584. 0.566435 0.283218 0.959056i \(-0.408598\pi\)
0.283218 + 0.959056i \(0.408598\pi\)
\(192\) 0 0
\(193\) −53690.2 −0.103753 −0.0518766 0.998654i \(-0.516520\pi\)
−0.0518766 + 0.998654i \(0.516520\pi\)
\(194\) 0 0
\(195\) −501930. −0.945272
\(196\) 0 0
\(197\) 14538.9 0.0266910 0.0133455 0.999911i \(-0.495752\pi\)
0.0133455 + 0.999911i \(0.495752\pi\)
\(198\) 0 0
\(199\) 143700. 0.257232 0.128616 0.991695i \(-0.458947\pi\)
0.128616 + 0.991695i \(0.458947\pi\)
\(200\) 0 0
\(201\) −1.12550e6 −1.96496
\(202\) 0 0
\(203\) −136596. −0.232647
\(204\) 0 0
\(205\) −107149. −0.178075
\(206\) 0 0
\(207\) 116886. 0.189600
\(208\) 0 0
\(209\) 126162. 0.199786
\(210\) 0 0
\(211\) −83653.6 −0.129354 −0.0646768 0.997906i \(-0.520602\pi\)
−0.0646768 + 0.997906i \(0.520602\pi\)
\(212\) 0 0
\(213\) −146988. −0.221989
\(214\) 0 0
\(215\) −10953.8 −0.0161611
\(216\) 0 0
\(217\) −102320. −0.147507
\(218\) 0 0
\(219\) −1.32602e6 −1.86826
\(220\) 0 0
\(221\) 157834. 0.217381
\(222\) 0 0
\(223\) 665414. 0.896045 0.448022 0.894022i \(-0.352128\pi\)
0.448022 + 0.894022i \(0.352128\pi\)
\(224\) 0 0
\(225\) −342089. −0.450488
\(226\) 0 0
\(227\) −1.15382e6 −1.48619 −0.743096 0.669185i \(-0.766643\pi\)
−0.743096 + 0.669185i \(0.766643\pi\)
\(228\) 0 0
\(229\) −433691. −0.546502 −0.273251 0.961943i \(-0.588099\pi\)
−0.273251 + 0.961943i \(0.588099\pi\)
\(230\) 0 0
\(231\) 135426. 0.166983
\(232\) 0 0
\(233\) 387173. 0.467214 0.233607 0.972331i \(-0.424947\pi\)
0.233607 + 0.972331i \(0.424947\pi\)
\(234\) 0 0
\(235\) 938183. 1.10820
\(236\) 0 0
\(237\) 739426. 0.855114
\(238\) 0 0
\(239\) 622463. 0.704886 0.352443 0.935833i \(-0.385351\pi\)
0.352443 + 0.935833i \(0.385351\pi\)
\(240\) 0 0
\(241\) −371454. −0.411967 −0.205984 0.978555i \(-0.566039\pi\)
−0.205984 + 0.978555i \(0.566039\pi\)
\(242\) 0 0
\(243\) 1.11861e6 1.21524
\(244\) 0 0
\(245\) −566516. −0.602972
\(246\) 0 0
\(247\) −256887. −0.267917
\(248\) 0 0
\(249\) −1.06052e6 −1.08398
\(250\) 0 0
\(251\) −376098. −0.376805 −0.188403 0.982092i \(-0.560331\pi\)
−0.188403 + 0.982092i \(0.560331\pi\)
\(252\) 0 0
\(253\) −231519. −0.227398
\(254\) 0 0
\(255\) −156450. −0.150669
\(256\) 0 0
\(257\) 1.64382e6 1.55247 0.776233 0.630446i \(-0.217128\pi\)
0.776233 + 0.630446i \(0.217128\pi\)
\(258\) 0 0
\(259\) 37452.2 0.0346919
\(260\) 0 0
\(261\) −1.27378e6 −1.15742
\(262\) 0 0
\(263\) −775299. −0.691162 −0.345581 0.938389i \(-0.612318\pi\)
−0.345581 + 0.938389i \(0.612318\pi\)
\(264\) 0 0
\(265\) −38388.8 −0.0335807
\(266\) 0 0
\(267\) −2.97483e6 −2.55378
\(268\) 0 0
\(269\) 334851. 0.282144 0.141072 0.989999i \(-0.454945\pi\)
0.141072 + 0.989999i \(0.454945\pi\)
\(270\) 0 0
\(271\) −893377. −0.738944 −0.369472 0.929242i \(-0.620461\pi\)
−0.369472 + 0.929242i \(0.620461\pi\)
\(272\) 0 0
\(273\) −275750. −0.223928
\(274\) 0 0
\(275\) 677584. 0.540296
\(276\) 0 0
\(277\) −1.13462e6 −0.888483 −0.444242 0.895907i \(-0.646527\pi\)
−0.444242 + 0.895907i \(0.646527\pi\)
\(278\) 0 0
\(279\) −954149. −0.733847
\(280\) 0 0
\(281\) −1.98111e6 −1.49672 −0.748362 0.663290i \(-0.769159\pi\)
−0.748362 + 0.663290i \(0.769159\pi\)
\(282\) 0 0
\(283\) −1.32192e6 −0.981161 −0.490580 0.871396i \(-0.663215\pi\)
−0.490580 + 0.871396i \(0.663215\pi\)
\(284\) 0 0
\(285\) 254633. 0.185696
\(286\) 0 0
\(287\) −58865.4 −0.0421847
\(288\) 0 0
\(289\) −1.37066e6 −0.965351
\(290\) 0 0
\(291\) −809232. −0.560197
\(292\) 0 0
\(293\) 1.35808e6 0.924179 0.462089 0.886833i \(-0.347100\pi\)
0.462089 + 0.886833i \(0.347100\pi\)
\(294\) 0 0
\(295\) 1.30547e6 0.873401
\(296\) 0 0
\(297\) −476394. −0.313383
\(298\) 0 0
\(299\) 471411. 0.304945
\(300\) 0 0
\(301\) −6017.81 −0.00382844
\(302\) 0 0
\(303\) 342489. 0.214309
\(304\) 0 0
\(305\) 1.29047e6 0.794324
\(306\) 0 0
\(307\) 1.23842e6 0.749930 0.374965 0.927039i \(-0.377655\pi\)
0.374965 + 0.927039i \(0.377655\pi\)
\(308\) 0 0
\(309\) 1.89779e6 1.13071
\(310\) 0 0
\(311\) 1.78565e6 1.04688 0.523439 0.852063i \(-0.324649\pi\)
0.523439 + 0.852063i \(0.324649\pi\)
\(312\) 0 0
\(313\) −3.01608e6 −1.74013 −0.870066 0.492935i \(-0.835924\pi\)
−0.870066 + 0.492935i \(0.835924\pi\)
\(314\) 0 0
\(315\) 114978. 0.0652889
\(316\) 0 0
\(317\) 515839. 0.288314 0.144157 0.989555i \(-0.453953\pi\)
0.144157 + 0.989555i \(0.453953\pi\)
\(318\) 0 0
\(319\) 2.52300e6 1.38816
\(320\) 0 0
\(321\) −965772. −0.523133
\(322\) 0 0
\(323\) −80070.7 −0.0427039
\(324\) 0 0
\(325\) −1.37967e6 −0.724548
\(326\) 0 0
\(327\) −1.07419e6 −0.555536
\(328\) 0 0
\(329\) 515417. 0.262524
\(330\) 0 0
\(331\) 2.25449e6 1.13104 0.565521 0.824734i \(-0.308675\pi\)
0.565521 + 0.824734i \(0.308675\pi\)
\(332\) 0 0
\(333\) 349246. 0.172592
\(334\) 0 0
\(335\) 1.89270e6 0.921446
\(336\) 0 0
\(337\) 2.40261e6 1.15242 0.576208 0.817303i \(-0.304532\pi\)
0.576208 + 0.817303i \(0.304532\pi\)
\(338\) 0 0
\(339\) −3.08424e6 −1.45764
\(340\) 0 0
\(341\) 1.88991e6 0.880145
\(342\) 0 0
\(343\) −629237. −0.288788
\(344\) 0 0
\(345\) −467275. −0.211361
\(346\) 0 0
\(347\) 684693. 0.305262 0.152631 0.988283i \(-0.451225\pi\)
0.152631 + 0.988283i \(0.451225\pi\)
\(348\) 0 0
\(349\) 2.19857e6 0.966220 0.483110 0.875560i \(-0.339507\pi\)
0.483110 + 0.875560i \(0.339507\pi\)
\(350\) 0 0
\(351\) 970016. 0.420253
\(352\) 0 0
\(353\) 2.03446e6 0.868987 0.434493 0.900675i \(-0.356927\pi\)
0.434493 + 0.900675i \(0.356927\pi\)
\(354\) 0 0
\(355\) 247183. 0.104099
\(356\) 0 0
\(357\) −85950.1 −0.0356924
\(358\) 0 0
\(359\) −2.30592e6 −0.944298 −0.472149 0.881519i \(-0.656522\pi\)
−0.472149 + 0.881519i \(0.656522\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 796980. 0.317454
\(364\) 0 0
\(365\) 2.22990e6 0.876101
\(366\) 0 0
\(367\) −2.64475e6 −1.02499 −0.512495 0.858690i \(-0.671279\pi\)
−0.512495 + 0.858690i \(0.671279\pi\)
\(368\) 0 0
\(369\) −548927. −0.209869
\(370\) 0 0
\(371\) −21090.0 −0.00795502
\(372\) 0 0
\(373\) 173405. 0.0645342 0.0322671 0.999479i \(-0.489727\pi\)
0.0322671 + 0.999479i \(0.489727\pi\)
\(374\) 0 0
\(375\) 3.57180e6 1.31162
\(376\) 0 0
\(377\) −5.13723e6 −1.86155
\(378\) 0 0
\(379\) −2.23408e6 −0.798915 −0.399458 0.916752i \(-0.630802\pi\)
−0.399458 + 0.916752i \(0.630802\pi\)
\(380\) 0 0
\(381\) 7.21396e6 2.54602
\(382\) 0 0
\(383\) 4.44976e6 1.55003 0.775014 0.631944i \(-0.217743\pi\)
0.775014 + 0.631944i \(0.217743\pi\)
\(384\) 0 0
\(385\) −227740. −0.0783047
\(386\) 0 0
\(387\) −56116.8 −0.0190465
\(388\) 0 0
\(389\) 2.43083e6 0.814480 0.407240 0.913321i \(-0.366491\pi\)
0.407240 + 0.913321i \(0.366491\pi\)
\(390\) 0 0
\(391\) 146937. 0.0486059
\(392\) 0 0
\(393\) 390563. 0.127559
\(394\) 0 0
\(395\) −1.24346e6 −0.400996
\(396\) 0 0
\(397\) 2.61205e6 0.831775 0.415887 0.909416i \(-0.363471\pi\)
0.415887 + 0.909416i \(0.363471\pi\)
\(398\) 0 0
\(399\) 139890. 0.0439901
\(400\) 0 0
\(401\) 1.42660e6 0.443038 0.221519 0.975156i \(-0.428899\pi\)
0.221519 + 0.975156i \(0.428899\pi\)
\(402\) 0 0
\(403\) −3.84816e6 −1.18029
\(404\) 0 0
\(405\) −2.43816e6 −0.738625
\(406\) 0 0
\(407\) −691760. −0.207000
\(408\) 0 0
\(409\) 4.73321e6 1.39910 0.699548 0.714586i \(-0.253385\pi\)
0.699548 + 0.714586i \(0.253385\pi\)
\(410\) 0 0
\(411\) 5.46162e6 1.59484
\(412\) 0 0
\(413\) 717200. 0.206902
\(414\) 0 0
\(415\) 1.78344e6 0.508320
\(416\) 0 0
\(417\) −6.03939e6 −1.70080
\(418\) 0 0
\(419\) 357759. 0.0995531 0.0497766 0.998760i \(-0.484149\pi\)
0.0497766 + 0.998760i \(0.484149\pi\)
\(420\) 0 0
\(421\) 652504. 0.179423 0.0897115 0.995968i \(-0.471405\pi\)
0.0897115 + 0.995968i \(0.471405\pi\)
\(422\) 0 0
\(423\) 4.80633e6 1.30606
\(424\) 0 0
\(425\) −430038. −0.115487
\(426\) 0 0
\(427\) 708955. 0.188169
\(428\) 0 0
\(429\) 5.09323e6 1.33613
\(430\) 0 0
\(431\) 6.25148e6 1.62103 0.810513 0.585721i \(-0.199188\pi\)
0.810513 + 0.585721i \(0.199188\pi\)
\(432\) 0 0
\(433\) 4.45832e6 1.14275 0.571375 0.820689i \(-0.306410\pi\)
0.571375 + 0.820689i \(0.306410\pi\)
\(434\) 0 0
\(435\) 5.09216e6 1.29026
\(436\) 0 0
\(437\) −239151. −0.0599057
\(438\) 0 0
\(439\) −5.00652e6 −1.23986 −0.619932 0.784655i \(-0.712840\pi\)
−0.619932 + 0.784655i \(0.712840\pi\)
\(440\) 0 0
\(441\) −2.90227e6 −0.710627
\(442\) 0 0
\(443\) −715908. −0.173320 −0.0866599 0.996238i \(-0.527619\pi\)
−0.0866599 + 0.996238i \(0.527619\pi\)
\(444\) 0 0
\(445\) 5.00264e6 1.19757
\(446\) 0 0
\(447\) 2.12974e6 0.504147
\(448\) 0 0
\(449\) −4.83183e6 −1.13109 −0.565544 0.824718i \(-0.691334\pi\)
−0.565544 + 0.824718i \(0.691334\pi\)
\(450\) 0 0
\(451\) 1.08727e6 0.251708
\(452\) 0 0
\(453\) −9.77610e6 −2.23831
\(454\) 0 0
\(455\) 463716. 0.105008
\(456\) 0 0
\(457\) −6.44410e6 −1.44335 −0.721675 0.692232i \(-0.756628\pi\)
−0.721675 + 0.692232i \(0.756628\pi\)
\(458\) 0 0
\(459\) 302350. 0.0669851
\(460\) 0 0
\(461\) −4.10108e6 −0.898764 −0.449382 0.893340i \(-0.648356\pi\)
−0.449382 + 0.893340i \(0.648356\pi\)
\(462\) 0 0
\(463\) −8.98683e6 −1.94829 −0.974146 0.225919i \(-0.927461\pi\)
−0.974146 + 0.225919i \(0.927461\pi\)
\(464\) 0 0
\(465\) 3.81439e6 0.818075
\(466\) 0 0
\(467\) 8.84409e6 1.87655 0.938276 0.345886i \(-0.112422\pi\)
0.938276 + 0.345886i \(0.112422\pi\)
\(468\) 0 0
\(469\) 1.03981e6 0.218284
\(470\) 0 0
\(471\) −8.90800e6 −1.85024
\(472\) 0 0
\(473\) 111152. 0.0228436
\(474\) 0 0
\(475\) 699919. 0.142336
\(476\) 0 0
\(477\) −196667. −0.0395762
\(478\) 0 0
\(479\) −2.89320e6 −0.576155 −0.288077 0.957607i \(-0.593016\pi\)
−0.288077 + 0.957607i \(0.593016\pi\)
\(480\) 0 0
\(481\) 1.40854e6 0.277591
\(482\) 0 0
\(483\) −256711. −0.0500699
\(484\) 0 0
\(485\) 1.36085e6 0.262697
\(486\) 0 0
\(487\) 3.18166e6 0.607900 0.303950 0.952688i \(-0.401694\pi\)
0.303950 + 0.952688i \(0.401694\pi\)
\(488\) 0 0
\(489\) −5.22416e6 −0.987971
\(490\) 0 0
\(491\) 4.45509e6 0.833975 0.416987 0.908912i \(-0.363086\pi\)
0.416987 + 0.908912i \(0.363086\pi\)
\(492\) 0 0
\(493\) −1.60125e6 −0.296717
\(494\) 0 0
\(495\) −2.12371e6 −0.389566
\(496\) 0 0
\(497\) 135797. 0.0246603
\(498\) 0 0
\(499\) −9.31472e6 −1.67463 −0.837315 0.546721i \(-0.815876\pi\)
−0.837315 + 0.546721i \(0.815876\pi\)
\(500\) 0 0
\(501\) 1.04552e7 1.86096
\(502\) 0 0
\(503\) 2.04835e6 0.360980 0.180490 0.983577i \(-0.442232\pi\)
0.180490 + 0.983577i \(0.442232\pi\)
\(504\) 0 0
\(505\) −575949. −0.100498
\(506\) 0 0
\(507\) −2.76647e6 −0.477976
\(508\) 0 0
\(509\) −166211. −0.0284358 −0.0142179 0.999899i \(-0.504526\pi\)
−0.0142179 + 0.999899i \(0.504526\pi\)
\(510\) 0 0
\(511\) 1.22506e6 0.207542
\(512\) 0 0
\(513\) −492097. −0.0825577
\(514\) 0 0
\(515\) −3.19143e6 −0.530234
\(516\) 0 0
\(517\) −9.52001e6 −1.56643
\(518\) 0 0
\(519\) −1.58405e7 −2.58137
\(520\) 0 0
\(521\) 2.64602e6 0.427069 0.213535 0.976935i \(-0.431502\pi\)
0.213535 + 0.976935i \(0.431502\pi\)
\(522\) 0 0
\(523\) 7.02287e6 1.12269 0.561346 0.827581i \(-0.310284\pi\)
0.561346 + 0.827581i \(0.310284\pi\)
\(524\) 0 0
\(525\) 751312. 0.118966
\(526\) 0 0
\(527\) −1.19945e6 −0.188130
\(528\) 0 0
\(529\) −5.99748e6 −0.931815
\(530\) 0 0
\(531\) 6.68798e6 1.02934
\(532\) 0 0
\(533\) −2.21386e6 −0.337546
\(534\) 0 0
\(535\) 1.62410e6 0.245317
\(536\) 0 0
\(537\) −9.78268e6 −1.46394
\(538\) 0 0
\(539\) 5.74860e6 0.852295
\(540\) 0 0
\(541\) 4.35066e6 0.639090 0.319545 0.947571i \(-0.396470\pi\)
0.319545 + 0.947571i \(0.396470\pi\)
\(542\) 0 0
\(543\) −1.49324e7 −2.17336
\(544\) 0 0
\(545\) 1.80642e6 0.260512
\(546\) 0 0
\(547\) −9.77794e6 −1.39727 −0.698633 0.715480i \(-0.746208\pi\)
−0.698633 + 0.715480i \(0.746208\pi\)
\(548\) 0 0
\(549\) 6.61110e6 0.936144
\(550\) 0 0
\(551\) 2.60616e6 0.365698
\(552\) 0 0
\(553\) −683131. −0.0949929
\(554\) 0 0
\(555\) −1.39618e6 −0.192401
\(556\) 0 0
\(557\) 6.69323e6 0.914108 0.457054 0.889439i \(-0.348905\pi\)
0.457054 + 0.889439i \(0.348905\pi\)
\(558\) 0 0
\(559\) −226323. −0.0306337
\(560\) 0 0
\(561\) 1.58754e6 0.212969
\(562\) 0 0
\(563\) −7.54034e6 −1.00258 −0.501291 0.865279i \(-0.667141\pi\)
−0.501291 + 0.865279i \(0.667141\pi\)
\(564\) 0 0
\(565\) 5.18664e6 0.683542
\(566\) 0 0
\(567\) −1.33947e6 −0.174975
\(568\) 0 0
\(569\) 1.34726e6 0.174450 0.0872252 0.996189i \(-0.472200\pi\)
0.0872252 + 0.996189i \(0.472200\pi\)
\(570\) 0 0
\(571\) −2.05762e6 −0.264104 −0.132052 0.991243i \(-0.542157\pi\)
−0.132052 + 0.991243i \(0.542157\pi\)
\(572\) 0 0
\(573\) −5.84883e6 −0.744188
\(574\) 0 0
\(575\) −1.28441e6 −0.162008
\(576\) 0 0
\(577\) 7.88561e6 0.986043 0.493021 0.870017i \(-0.335892\pi\)
0.493021 + 0.870017i \(0.335892\pi\)
\(578\) 0 0
\(579\) 1.09959e6 0.136312
\(580\) 0 0
\(581\) 979781. 0.120417
\(582\) 0 0
\(583\) 389542. 0.0474660
\(584\) 0 0
\(585\) 4.32421e6 0.522417
\(586\) 0 0
\(587\) −1.90439e6 −0.228119 −0.114059 0.993474i \(-0.536385\pi\)
−0.114059 + 0.993474i \(0.536385\pi\)
\(588\) 0 0
\(589\) 1.95220e6 0.231866
\(590\) 0 0
\(591\) −297760. −0.0350669
\(592\) 0 0
\(593\) −1.35835e7 −1.58626 −0.793132 0.609050i \(-0.791551\pi\)
−0.793132 + 0.609050i \(0.791551\pi\)
\(594\) 0 0
\(595\) 144539. 0.0167375
\(596\) 0 0
\(597\) −2.94301e6 −0.337953
\(598\) 0 0
\(599\) 1.04361e7 1.18842 0.594212 0.804309i \(-0.297464\pi\)
0.594212 + 0.804309i \(0.297464\pi\)
\(600\) 0 0
\(601\) −1.17196e7 −1.32351 −0.661753 0.749722i \(-0.730187\pi\)
−0.661753 + 0.749722i \(0.730187\pi\)
\(602\) 0 0
\(603\) 9.69635e6 1.08596
\(604\) 0 0
\(605\) −1.34025e6 −0.148866
\(606\) 0 0
\(607\) 7.53524e6 0.830091 0.415045 0.909801i \(-0.363766\pi\)
0.415045 + 0.909801i \(0.363766\pi\)
\(608\) 0 0
\(609\) 2.79752e6 0.305654
\(610\) 0 0
\(611\) 1.93843e7 2.10062
\(612\) 0 0
\(613\) −4.37292e6 −0.470025 −0.235012 0.971992i \(-0.575513\pi\)
−0.235012 + 0.971992i \(0.575513\pi\)
\(614\) 0 0
\(615\) 2.19444e6 0.233957
\(616\) 0 0
\(617\) −7.50595e6 −0.793767 −0.396883 0.917869i \(-0.629908\pi\)
−0.396883 + 0.917869i \(0.629908\pi\)
\(618\) 0 0
\(619\) 1.30877e7 1.37289 0.686447 0.727180i \(-0.259169\pi\)
0.686447 + 0.727180i \(0.259169\pi\)
\(620\) 0 0
\(621\) 903043. 0.0939679
\(622\) 0 0
\(623\) 2.74834e6 0.283695
\(624\) 0 0
\(625\) 52309.5 0.00535649
\(626\) 0 0
\(627\) −2.58384e6 −0.262480
\(628\) 0 0
\(629\) 439035. 0.0442459
\(630\) 0 0
\(631\) −8.92096e6 −0.891945 −0.445972 0.895047i \(-0.647142\pi\)
−0.445972 + 0.895047i \(0.647142\pi\)
\(632\) 0 0
\(633\) 1.71325e6 0.169946
\(634\) 0 0
\(635\) −1.21314e7 −1.19392
\(636\) 0 0
\(637\) −1.17051e7 −1.14295
\(638\) 0 0
\(639\) 1.26632e6 0.122685
\(640\) 0 0
\(641\) 1.45438e7 1.39808 0.699041 0.715082i \(-0.253611\pi\)
0.699041 + 0.715082i \(0.253611\pi\)
\(642\) 0 0
\(643\) −1.45757e7 −1.39028 −0.695140 0.718874i \(-0.744658\pi\)
−0.695140 + 0.718874i \(0.744658\pi\)
\(644\) 0 0
\(645\) 224338. 0.0212326
\(646\) 0 0
\(647\) 2.24522e6 0.210862 0.105431 0.994427i \(-0.466378\pi\)
0.105431 + 0.994427i \(0.466378\pi\)
\(648\) 0 0
\(649\) −1.32470e7 −1.23454
\(650\) 0 0
\(651\) 2.09555e6 0.193796
\(652\) 0 0
\(653\) 1.54461e7 1.41754 0.708770 0.705440i \(-0.249251\pi\)
0.708770 + 0.705440i \(0.249251\pi\)
\(654\) 0 0
\(655\) −656793. −0.0598171
\(656\) 0 0
\(657\) 1.14238e7 1.03252
\(658\) 0 0
\(659\) −8.68940e6 −0.779429 −0.389714 0.920936i \(-0.627426\pi\)
−0.389714 + 0.920936i \(0.627426\pi\)
\(660\) 0 0
\(661\) −2.03442e7 −1.81108 −0.905538 0.424266i \(-0.860532\pi\)
−0.905538 + 0.424266i \(0.860532\pi\)
\(662\) 0 0
\(663\) −3.23249e6 −0.285597
\(664\) 0 0
\(665\) −235247. −0.0206286
\(666\) 0 0
\(667\) −4.78254e6 −0.416240
\(668\) 0 0
\(669\) −1.36278e7 −1.17723
\(670\) 0 0
\(671\) −1.30947e7 −1.12277
\(672\) 0 0
\(673\) 1.71139e7 1.45650 0.728251 0.685310i \(-0.240333\pi\)
0.728251 + 0.685310i \(0.240333\pi\)
\(674\) 0 0
\(675\) −2.64292e6 −0.223267
\(676\) 0 0
\(677\) 2.25299e6 0.188924 0.0944621 0.995528i \(-0.469887\pi\)
0.0944621 + 0.995528i \(0.469887\pi\)
\(678\) 0 0
\(679\) 747622. 0.0622311
\(680\) 0 0
\(681\) 2.36306e7 1.95257
\(682\) 0 0
\(683\) −5.33481e6 −0.437590 −0.218795 0.975771i \(-0.570213\pi\)
−0.218795 + 0.975771i \(0.570213\pi\)
\(684\) 0 0
\(685\) −9.18457e6 −0.747881
\(686\) 0 0
\(687\) 8.88211e6 0.718000
\(688\) 0 0
\(689\) −793171. −0.0636530
\(690\) 0 0
\(691\) −8.08495e6 −0.644143 −0.322071 0.946715i \(-0.604379\pi\)
−0.322071 + 0.946715i \(0.604379\pi\)
\(692\) 0 0
\(693\) −1.16672e6 −0.0922854
\(694\) 0 0
\(695\) 1.01562e7 0.797569
\(696\) 0 0
\(697\) −690053. −0.0538022
\(698\) 0 0
\(699\) −7.92941e6 −0.613829
\(700\) 0 0
\(701\) 7.88116e6 0.605752 0.302876 0.953030i \(-0.402053\pi\)
0.302876 + 0.953030i \(0.402053\pi\)
\(702\) 0 0
\(703\) −714562. −0.0545320
\(704\) 0 0
\(705\) −1.92142e7 −1.45596
\(706\) 0 0
\(707\) −316414. −0.0238071
\(708\) 0 0
\(709\) −2.07701e7 −1.55175 −0.775876 0.630885i \(-0.782692\pi\)
−0.775876 + 0.630885i \(0.782692\pi\)
\(710\) 0 0
\(711\) −6.37028e6 −0.472590
\(712\) 0 0
\(713\) −3.58247e6 −0.263911
\(714\) 0 0
\(715\) −8.56506e6 −0.626564
\(716\) 0 0
\(717\) −1.27482e7 −0.926086
\(718\) 0 0
\(719\) −1.03826e7 −0.749004 −0.374502 0.927226i \(-0.622186\pi\)
−0.374502 + 0.927226i \(0.622186\pi\)
\(720\) 0 0
\(721\) −1.75330e6 −0.125608
\(722\) 0 0
\(723\) 7.60748e6 0.541246
\(724\) 0 0
\(725\) 1.39970e7 0.988985
\(726\) 0 0
\(727\) 6.52486e6 0.457863 0.228931 0.973443i \(-0.426477\pi\)
0.228931 + 0.973443i \(0.426477\pi\)
\(728\) 0 0
\(729\) −5.70674e6 −0.397712
\(730\) 0 0
\(731\) −70544.1 −0.00488278
\(732\) 0 0
\(733\) −2.46101e6 −0.169182 −0.0845908 0.996416i \(-0.526958\pi\)
−0.0845908 + 0.996416i \(0.526958\pi\)
\(734\) 0 0
\(735\) 1.16024e7 0.792189
\(736\) 0 0
\(737\) −1.92058e7 −1.30246
\(738\) 0 0
\(739\) 7.33586e6 0.494128 0.247064 0.968999i \(-0.420534\pi\)
0.247064 + 0.968999i \(0.420534\pi\)
\(740\) 0 0
\(741\) 5.26111e6 0.351991
\(742\) 0 0
\(743\) −2.44394e7 −1.62412 −0.812060 0.583574i \(-0.801654\pi\)
−0.812060 + 0.583574i \(0.801654\pi\)
\(744\) 0 0
\(745\) −3.58148e6 −0.236414
\(746\) 0 0
\(747\) 9.13658e6 0.599076
\(748\) 0 0
\(749\) 892244. 0.0581138
\(750\) 0 0
\(751\) 2.11169e6 0.136625 0.0683126 0.997664i \(-0.478238\pi\)
0.0683126 + 0.997664i \(0.478238\pi\)
\(752\) 0 0
\(753\) 7.70259e6 0.495050
\(754\) 0 0
\(755\) 1.64401e7 1.04963
\(756\) 0 0
\(757\) −1.47627e7 −0.936325 −0.468162 0.883642i \(-0.655084\pi\)
−0.468162 + 0.883642i \(0.655084\pi\)
\(758\) 0 0
\(759\) 4.74157e6 0.298757
\(760\) 0 0
\(761\) 2.67074e7 1.67174 0.835871 0.548926i \(-0.184963\pi\)
0.835871 + 0.548926i \(0.184963\pi\)
\(762\) 0 0
\(763\) 992408. 0.0617133
\(764\) 0 0
\(765\) 1.34784e6 0.0832692
\(766\) 0 0
\(767\) 2.69731e7 1.65555
\(768\) 0 0
\(769\) 3.05104e6 0.186051 0.0930256 0.995664i \(-0.470346\pi\)
0.0930256 + 0.995664i \(0.470346\pi\)
\(770\) 0 0
\(771\) −3.36659e7 −2.03964
\(772\) 0 0
\(773\) −2.18723e7 −1.31657 −0.658287 0.752767i \(-0.728719\pi\)
−0.658287 + 0.752767i \(0.728719\pi\)
\(774\) 0 0
\(775\) 1.04847e7 0.627052
\(776\) 0 0
\(777\) −767030. −0.0455785
\(778\) 0 0
\(779\) 1.12311e6 0.0663100
\(780\) 0 0
\(781\) −2.50823e6 −0.147143
\(782\) 0 0
\(783\) −9.84096e6 −0.573632
\(784\) 0 0
\(785\) 1.49802e7 0.867647
\(786\) 0 0
\(787\) 1.26837e7 0.729975 0.364987 0.931013i \(-0.381073\pi\)
0.364987 + 0.931013i \(0.381073\pi\)
\(788\) 0 0
\(789\) 1.58783e7 0.908055
\(790\) 0 0
\(791\) 2.84943e6 0.161926
\(792\) 0 0
\(793\) 2.66630e7 1.50566
\(794\) 0 0
\(795\) 786212. 0.0441186
\(796\) 0 0
\(797\) 7.28989e6 0.406514 0.203257 0.979125i \(-0.434847\pi\)
0.203257 + 0.979125i \(0.434847\pi\)
\(798\) 0 0
\(799\) 6.04201e6 0.334822
\(800\) 0 0
\(801\) 2.56286e7 1.41138
\(802\) 0 0
\(803\) −2.26275e7 −1.23836
\(804\) 0 0
\(805\) 431700. 0.0234797
\(806\) 0 0
\(807\) −6.85783e6 −0.370683
\(808\) 0 0
\(809\) −2.72506e7 −1.46388 −0.731939 0.681370i \(-0.761385\pi\)
−0.731939 + 0.681370i \(0.761385\pi\)
\(810\) 0 0
\(811\) −1.45954e7 −0.779229 −0.389615 0.920978i \(-0.627392\pi\)
−0.389615 + 0.920978i \(0.627392\pi\)
\(812\) 0 0
\(813\) 1.82966e7 0.970831
\(814\) 0 0
\(815\) 8.78524e6 0.463297
\(816\) 0 0
\(817\) 114816. 0.00601791
\(818\) 0 0
\(819\) 2.37563e6 0.123757
\(820\) 0 0
\(821\) 4.16570e6 0.215690 0.107845 0.994168i \(-0.465605\pi\)
0.107845 + 0.994168i \(0.465605\pi\)
\(822\) 0 0
\(823\) 9.74487e6 0.501506 0.250753 0.968051i \(-0.419322\pi\)
0.250753 + 0.968051i \(0.419322\pi\)
\(824\) 0 0
\(825\) −1.38771e7 −0.709845
\(826\) 0 0
\(827\) −1.04970e6 −0.0533707 −0.0266854 0.999644i \(-0.508495\pi\)
−0.0266854 + 0.999644i \(0.508495\pi\)
\(828\) 0 0
\(829\) 1.87001e7 0.945056 0.472528 0.881316i \(-0.343342\pi\)
0.472528 + 0.881316i \(0.343342\pi\)
\(830\) 0 0
\(831\) 2.32372e7 1.16730
\(832\) 0 0
\(833\) −3.64843e6 −0.182177
\(834\) 0 0
\(835\) −1.75820e7 −0.872676
\(836\) 0 0
\(837\) −7.37159e6 −0.363703
\(838\) 0 0
\(839\) −1.96094e7 −0.961742 −0.480871 0.876791i \(-0.659680\pi\)
−0.480871 + 0.876791i \(0.659680\pi\)
\(840\) 0 0
\(841\) 3.16068e7 1.54096
\(842\) 0 0
\(843\) 4.05735e7 1.96641
\(844\) 0 0
\(845\) 4.65225e6 0.224141
\(846\) 0 0
\(847\) −736303. −0.0352653
\(848\) 0 0
\(849\) 2.70733e7 1.28906
\(850\) 0 0
\(851\) 1.31129e6 0.0620688
\(852\) 0 0
\(853\) −5.00288e6 −0.235422 −0.117711 0.993048i \(-0.537556\pi\)
−0.117711 + 0.993048i \(0.537556\pi\)
\(854\) 0 0
\(855\) −2.19371e6 −0.102627
\(856\) 0 0
\(857\) 1.36218e7 0.633553 0.316776 0.948500i \(-0.397400\pi\)
0.316776 + 0.948500i \(0.397400\pi\)
\(858\) 0 0
\(859\) −1.88600e7 −0.872084 −0.436042 0.899926i \(-0.643620\pi\)
−0.436042 + 0.899926i \(0.643620\pi\)
\(860\) 0 0
\(861\) 1.20558e6 0.0554227
\(862\) 0 0
\(863\) 3.62744e7 1.65796 0.828979 0.559280i \(-0.188922\pi\)
0.828979 + 0.559280i \(0.188922\pi\)
\(864\) 0 0
\(865\) 2.66383e7 1.21050
\(866\) 0 0
\(867\) 2.80715e7 1.26829
\(868\) 0 0
\(869\) 1.26178e7 0.566804
\(870\) 0 0
\(871\) 3.91061e7 1.74662
\(872\) 0 0
\(873\) 6.97166e6 0.309600
\(874\) 0 0
\(875\) −3.29987e6 −0.145706
\(876\) 0 0
\(877\) −1.47531e7 −0.647715 −0.323857 0.946106i \(-0.604980\pi\)
−0.323857 + 0.946106i \(0.604980\pi\)
\(878\) 0 0
\(879\) −2.78138e7 −1.21419
\(880\) 0 0
\(881\) 1.41746e7 0.615277 0.307638 0.951503i \(-0.400461\pi\)
0.307638 + 0.951503i \(0.400461\pi\)
\(882\) 0 0
\(883\) −2.05570e7 −0.887276 −0.443638 0.896206i \(-0.646312\pi\)
−0.443638 + 0.896206i \(0.646312\pi\)
\(884\) 0 0
\(885\) −2.67365e7 −1.14748
\(886\) 0 0
\(887\) 9.48962e6 0.404986 0.202493 0.979284i \(-0.435096\pi\)
0.202493 + 0.979284i \(0.435096\pi\)
\(888\) 0 0
\(889\) −6.66473e6 −0.282832
\(890\) 0 0
\(891\) 2.47407e7 1.04404
\(892\) 0 0
\(893\) −9.83381e6 −0.412661
\(894\) 0 0
\(895\) 1.64511e7 0.686496
\(896\) 0 0
\(897\) −9.65462e6 −0.400640
\(898\) 0 0
\(899\) 3.90401e7 1.61106
\(900\) 0 0
\(901\) −247228. −0.0101458
\(902\) 0 0
\(903\) 123246. 0.00502984
\(904\) 0 0
\(905\) 2.51112e7 1.01917
\(906\) 0 0
\(907\) −1.30672e7 −0.527428 −0.263714 0.964601i \(-0.584947\pi\)
−0.263714 + 0.964601i \(0.584947\pi\)
\(908\) 0 0
\(909\) −2.95060e6 −0.118441
\(910\) 0 0
\(911\) 4.60549e7 1.83857 0.919284 0.393594i \(-0.128768\pi\)
0.919284 + 0.393594i \(0.128768\pi\)
\(912\) 0 0
\(913\) −1.80970e7 −0.718506
\(914\) 0 0
\(915\) −2.64291e7 −1.04359
\(916\) 0 0
\(917\) −360828. −0.0141702
\(918\) 0 0
\(919\) 6.67836e6 0.260844 0.130422 0.991459i \(-0.458367\pi\)
0.130422 + 0.991459i \(0.458367\pi\)
\(920\) 0 0
\(921\) −2.53631e7 −0.985264
\(922\) 0 0
\(923\) 5.10717e6 0.197322
\(924\) 0 0
\(925\) −3.83772e6 −0.147475
\(926\) 0 0
\(927\) −1.63498e7 −0.624903
\(928\) 0 0
\(929\) −84737.5 −0.00322134 −0.00161067 0.999999i \(-0.500513\pi\)
−0.00161067 + 0.999999i \(0.500513\pi\)
\(930\) 0 0
\(931\) 5.93809e6 0.224529
\(932\) 0 0
\(933\) −3.65706e7 −1.37540
\(934\) 0 0
\(935\) −2.66970e6 −0.0998695
\(936\) 0 0
\(937\) −3.12820e7 −1.16398 −0.581989 0.813196i \(-0.697725\pi\)
−0.581989 + 0.813196i \(0.697725\pi\)
\(938\) 0 0
\(939\) 6.17701e7 2.28620
\(940\) 0 0
\(941\) 2.88841e7 1.06337 0.531685 0.846942i \(-0.321559\pi\)
0.531685 + 0.846942i \(0.321559\pi\)
\(942\) 0 0
\(943\) −2.06101e6 −0.0754746
\(944\) 0 0
\(945\) 888302. 0.0323580
\(946\) 0 0
\(947\) −1.99925e7 −0.724422 −0.362211 0.932096i \(-0.617978\pi\)
−0.362211 + 0.932096i \(0.617978\pi\)
\(948\) 0 0
\(949\) 4.60732e7 1.66067
\(950\) 0 0
\(951\) −1.05645e7 −0.378790
\(952\) 0 0
\(953\) 5.22187e7 1.86249 0.931244 0.364396i \(-0.118724\pi\)
0.931244 + 0.364396i \(0.118724\pi\)
\(954\) 0 0
\(955\) 9.83573e6 0.348978
\(956\) 0 0
\(957\) −5.16716e7 −1.82378
\(958\) 0 0
\(959\) −5.04580e6 −0.177167
\(960\) 0 0
\(961\) 614716. 0.0214717
\(962\) 0 0
\(963\) 8.32029e6 0.289116
\(964\) 0 0
\(965\) −1.84913e6 −0.0639218
\(966\) 0 0
\(967\) −2.66303e7 −0.915818 −0.457909 0.888999i \(-0.651402\pi\)
−0.457909 + 0.888999i \(0.651402\pi\)
\(968\) 0 0
\(969\) 1.63987e6 0.0561047
\(970\) 0 0
\(971\) −1.04206e7 −0.354686 −0.177343 0.984149i \(-0.556750\pi\)
−0.177343 + 0.984149i \(0.556750\pi\)
\(972\) 0 0
\(973\) 5.57959e6 0.188938
\(974\) 0 0
\(975\) 2.82560e7 0.951918
\(976\) 0 0
\(977\) 1.82795e7 0.612673 0.306336 0.951923i \(-0.400897\pi\)
0.306336 + 0.951923i \(0.400897\pi\)
\(978\) 0 0
\(979\) −5.07633e7 −1.69275
\(980\) 0 0
\(981\) 9.25433e6 0.307024
\(982\) 0 0
\(983\) −2.25159e7 −0.743199 −0.371600 0.928393i \(-0.621191\pi\)
−0.371600 + 0.928393i \(0.621191\pi\)
\(984\) 0 0
\(985\) 500730. 0.0164442
\(986\) 0 0
\(987\) −1.05559e7 −0.344907
\(988\) 0 0
\(989\) −210697. −0.00684964
\(990\) 0 0
\(991\) −3.42068e7 −1.10644 −0.553221 0.833034i \(-0.686602\pi\)
−0.553221 + 0.833034i \(0.686602\pi\)
\(992\) 0 0
\(993\) −4.61726e7 −1.48597
\(994\) 0 0
\(995\) 4.94914e6 0.158479
\(996\) 0 0
\(997\) −1.12682e7 −0.359018 −0.179509 0.983756i \(-0.557451\pi\)
−0.179509 + 0.983756i \(0.557451\pi\)
\(998\) 0 0
\(999\) 2.69821e6 0.0855387
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 304.6.a.f.1.1 2
4.3 odd 2 38.6.a.c.1.2 2
12.11 even 2 342.6.a.i.1.1 2
20.19 odd 2 950.6.a.d.1.1 2
76.75 even 2 722.6.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.6.a.c.1.2 2 4.3 odd 2
304.6.a.f.1.1 2 1.1 even 1 trivial
342.6.a.i.1.1 2 12.11 even 2
722.6.a.c.1.1 2 76.75 even 2
950.6.a.d.1.1 2 20.19 odd 2