Properties

Label 45.6.a.b.1.1
Level $45$
Weight $6$
Character 45.1
Self dual yes
Analytic conductor $7.217$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.21727189158\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 45.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -28.0000 q^{4} -25.0000 q^{5} +192.000 q^{7} +120.000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} -28.0000 q^{4} -25.0000 q^{5} +192.000 q^{7} +120.000 q^{8} +50.0000 q^{10} +148.000 q^{11} +286.000 q^{13} -384.000 q^{14} +656.000 q^{16} +1678.00 q^{17} +1060.00 q^{19} +700.000 q^{20} -296.000 q^{22} -2976.00 q^{23} +625.000 q^{25} -572.000 q^{26} -5376.00 q^{28} +3410.00 q^{29} -2448.00 q^{31} -5152.00 q^{32} -3356.00 q^{34} -4800.00 q^{35} +182.000 q^{37} -2120.00 q^{38} -3000.00 q^{40} +9398.00 q^{41} -1244.00 q^{43} -4144.00 q^{44} +5952.00 q^{46} +12088.0 q^{47} +20057.0 q^{49} -1250.00 q^{50} -8008.00 q^{52} -23846.0 q^{53} -3700.00 q^{55} +23040.0 q^{56} -6820.00 q^{58} +20020.0 q^{59} +32302.0 q^{61} +4896.00 q^{62} -10688.0 q^{64} -7150.00 q^{65} +60972.0 q^{67} -46984.0 q^{68} +9600.00 q^{70} +32648.0 q^{71} -38774.0 q^{73} -364.000 q^{74} -29680.0 q^{76} +28416.0 q^{77} -33360.0 q^{79} -16400.0 q^{80} -18796.0 q^{82} -16716.0 q^{83} -41950.0 q^{85} +2488.00 q^{86} +17760.0 q^{88} -101370. q^{89} +54912.0 q^{91} +83328.0 q^{92} -24176.0 q^{94} -26500.0 q^{95} -119038. q^{97} -40114.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −28.0000 −0.875000
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 192.000 1.48100 0.740502 0.672054i \(-0.234588\pi\)
0.740502 + 0.672054i \(0.234588\pi\)
\(8\) 120.000 0.662913
\(9\) 0 0
\(10\) 50.0000 0.158114
\(11\) 148.000 0.368791 0.184395 0.982852i \(-0.440967\pi\)
0.184395 + 0.982852i \(0.440967\pi\)
\(12\) 0 0
\(13\) 286.000 0.469362 0.234681 0.972072i \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) −384.000 −0.523614
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) 1678.00 1.40822 0.704109 0.710092i \(-0.251347\pi\)
0.704109 + 0.710092i \(0.251347\pi\)
\(18\) 0 0
\(19\) 1060.00 0.673631 0.336815 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(20\) 700.000 0.391312
\(21\) 0 0
\(22\) −296.000 −0.130387
\(23\) −2976.00 −1.17304 −0.586521 0.809934i \(-0.699503\pi\)
−0.586521 + 0.809934i \(0.699503\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −572.000 −0.165944
\(27\) 0 0
\(28\) −5376.00 −1.29588
\(29\) 3410.00 0.752938 0.376469 0.926429i \(-0.377138\pi\)
0.376469 + 0.926429i \(0.377138\pi\)
\(30\) 0 0
\(31\) −2448.00 −0.457517 −0.228758 0.973483i \(-0.573467\pi\)
−0.228758 + 0.973483i \(0.573467\pi\)
\(32\) −5152.00 −0.889408
\(33\) 0 0
\(34\) −3356.00 −0.497880
\(35\) −4800.00 −0.662325
\(36\) 0 0
\(37\) 182.000 0.0218558 0.0109279 0.999940i \(-0.496521\pi\)
0.0109279 + 0.999940i \(0.496521\pi\)
\(38\) −2120.00 −0.238164
\(39\) 0 0
\(40\) −3000.00 −0.296464
\(41\) 9398.00 0.873124 0.436562 0.899674i \(-0.356196\pi\)
0.436562 + 0.899674i \(0.356196\pi\)
\(42\) 0 0
\(43\) −1244.00 −0.102600 −0.0513002 0.998683i \(-0.516337\pi\)
−0.0513002 + 0.998683i \(0.516337\pi\)
\(44\) −4144.00 −0.322692
\(45\) 0 0
\(46\) 5952.00 0.414733
\(47\) 12088.0 0.798196 0.399098 0.916908i \(-0.369323\pi\)
0.399098 + 0.916908i \(0.369323\pi\)
\(48\) 0 0
\(49\) 20057.0 1.19337
\(50\) −1250.00 −0.0707107
\(51\) 0 0
\(52\) −8008.00 −0.410691
\(53\) −23846.0 −1.16607 −0.583037 0.812446i \(-0.698136\pi\)
−0.583037 + 0.812446i \(0.698136\pi\)
\(54\) 0 0
\(55\) −3700.00 −0.164928
\(56\) 23040.0 0.981776
\(57\) 0 0
\(58\) −6820.00 −0.266204
\(59\) 20020.0 0.748745 0.374373 0.927278i \(-0.377858\pi\)
0.374373 + 0.927278i \(0.377858\pi\)
\(60\) 0 0
\(61\) 32302.0 1.11149 0.555744 0.831353i \(-0.312433\pi\)
0.555744 + 0.831353i \(0.312433\pi\)
\(62\) 4896.00 0.161757
\(63\) 0 0
\(64\) −10688.0 −0.326172
\(65\) −7150.00 −0.209905
\(66\) 0 0
\(67\) 60972.0 1.65937 0.829685 0.558231i \(-0.188520\pi\)
0.829685 + 0.558231i \(0.188520\pi\)
\(68\) −46984.0 −1.23219
\(69\) 0 0
\(70\) 9600.00 0.234167
\(71\) 32648.0 0.768618 0.384309 0.923204i \(-0.374440\pi\)
0.384309 + 0.923204i \(0.374440\pi\)
\(72\) 0 0
\(73\) −38774.0 −0.851596 −0.425798 0.904818i \(-0.640007\pi\)
−0.425798 + 0.904818i \(0.640007\pi\)
\(74\) −364.000 −0.00772720
\(75\) 0 0
\(76\) −29680.0 −0.589427
\(77\) 28416.0 0.546180
\(78\) 0 0
\(79\) −33360.0 −0.601393 −0.300696 0.953720i \(-0.597219\pi\)
−0.300696 + 0.953720i \(0.597219\pi\)
\(80\) −16400.0 −0.286496
\(81\) 0 0
\(82\) −18796.0 −0.308696
\(83\) −16716.0 −0.266340 −0.133170 0.991093i \(-0.542516\pi\)
−0.133170 + 0.991093i \(0.542516\pi\)
\(84\) 0 0
\(85\) −41950.0 −0.629774
\(86\) 2488.00 0.0362747
\(87\) 0 0
\(88\) 17760.0 0.244476
\(89\) −101370. −1.35655 −0.678273 0.734810i \(-0.737271\pi\)
−0.678273 + 0.734810i \(0.737271\pi\)
\(90\) 0 0
\(91\) 54912.0 0.695126
\(92\) 83328.0 1.02641
\(93\) 0 0
\(94\) −24176.0 −0.282205
\(95\) −26500.0 −0.301257
\(96\) 0 0
\(97\) −119038. −1.28457 −0.642283 0.766468i \(-0.722013\pi\)
−0.642283 + 0.766468i \(0.722013\pi\)
\(98\) −40114.0 −0.421921
\(99\) 0 0
\(100\) −17500.0 −0.175000
\(101\) 89898.0 0.876893 0.438446 0.898757i \(-0.355529\pi\)
0.438446 + 0.898757i \(0.355529\pi\)
\(102\) 0 0
\(103\) −19504.0 −0.181147 −0.0905734 0.995890i \(-0.528870\pi\)
−0.0905734 + 0.995890i \(0.528870\pi\)
\(104\) 34320.0 0.311146
\(105\) 0 0
\(106\) 47692.0 0.412269
\(107\) −158292. −1.33659 −0.668297 0.743895i \(-0.732976\pi\)
−0.668297 + 0.743895i \(0.732976\pi\)
\(108\) 0 0
\(109\) 36830.0 0.296917 0.148459 0.988919i \(-0.452569\pi\)
0.148459 + 0.988919i \(0.452569\pi\)
\(110\) 7400.00 0.0583109
\(111\) 0 0
\(112\) 125952. 0.948768
\(113\) −11186.0 −0.0824098 −0.0412049 0.999151i \(-0.513120\pi\)
−0.0412049 + 0.999151i \(0.513120\pi\)
\(114\) 0 0
\(115\) 74400.0 0.524600
\(116\) −95480.0 −0.658821
\(117\) 0 0
\(118\) −40040.0 −0.264721
\(119\) 322176. 2.08557
\(120\) 0 0
\(121\) −139147. −0.863993
\(122\) −64604.0 −0.392970
\(123\) 0 0
\(124\) 68544.0 0.400327
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 70552.0 0.388150 0.194075 0.980987i \(-0.437829\pi\)
0.194075 + 0.980987i \(0.437829\pi\)
\(128\) 186240. 1.00473
\(129\) 0 0
\(130\) 14300.0 0.0742126
\(131\) −76452.0 −0.389234 −0.194617 0.980879i \(-0.562346\pi\)
−0.194617 + 0.980879i \(0.562346\pi\)
\(132\) 0 0
\(133\) 203520. 0.997650
\(134\) −121944. −0.586676
\(135\) 0 0
\(136\) 201360. 0.933525
\(137\) 144918. 0.659661 0.329831 0.944040i \(-0.393008\pi\)
0.329831 + 0.944040i \(0.393008\pi\)
\(138\) 0 0
\(139\) 112220. 0.492644 0.246322 0.969188i \(-0.420778\pi\)
0.246322 + 0.969188i \(0.420778\pi\)
\(140\) 134400. 0.579534
\(141\) 0 0
\(142\) −65296.0 −0.271748
\(143\) 42328.0 0.173096
\(144\) 0 0
\(145\) −85250.0 −0.336724
\(146\) 77548.0 0.301085
\(147\) 0 0
\(148\) −5096.00 −0.0191238
\(149\) −403750. −1.48986 −0.744932 0.667140i \(-0.767518\pi\)
−0.744932 + 0.667140i \(0.767518\pi\)
\(150\) 0 0
\(151\) −446648. −1.59413 −0.797064 0.603895i \(-0.793615\pi\)
−0.797064 + 0.603895i \(0.793615\pi\)
\(152\) 127200. 0.446558
\(153\) 0 0
\(154\) −56832.0 −0.193104
\(155\) 61200.0 0.204608
\(156\) 0 0
\(157\) −262258. −0.849141 −0.424570 0.905395i \(-0.639575\pi\)
−0.424570 + 0.905395i \(0.639575\pi\)
\(158\) 66720.0 0.212625
\(159\) 0 0
\(160\) 128800. 0.397755
\(161\) −571392. −1.73728
\(162\) 0 0
\(163\) −154564. −0.455658 −0.227829 0.973701i \(-0.573163\pi\)
−0.227829 + 0.973701i \(0.573163\pi\)
\(164\) −263144. −0.763983
\(165\) 0 0
\(166\) 33432.0 0.0941656
\(167\) −396672. −1.10063 −0.550314 0.834958i \(-0.685492\pi\)
−0.550314 + 0.834958i \(0.685492\pi\)
\(168\) 0 0
\(169\) −289497. −0.779700
\(170\) 83900.0 0.222659
\(171\) 0 0
\(172\) 34832.0 0.0897754
\(173\) 573474. 1.45680 0.728398 0.685155i \(-0.240265\pi\)
0.728398 + 0.685155i \(0.240265\pi\)
\(174\) 0 0
\(175\) 120000. 0.296201
\(176\) 97088.0 0.236257
\(177\) 0 0
\(178\) 202740. 0.479611
\(179\) 594460. 1.38672 0.693362 0.720589i \(-0.256129\pi\)
0.693362 + 0.720589i \(0.256129\pi\)
\(180\) 0 0
\(181\) −107098. −0.242988 −0.121494 0.992592i \(-0.538769\pi\)
−0.121494 + 0.992592i \(0.538769\pi\)
\(182\) −109824. −0.245764
\(183\) 0 0
\(184\) −357120. −0.777624
\(185\) −4550.00 −0.00977422
\(186\) 0 0
\(187\) 248344. 0.519337
\(188\) −338464. −0.698422
\(189\) 0 0
\(190\) 53000.0 0.106510
\(191\) −469552. −0.931323 −0.465661 0.884963i \(-0.654184\pi\)
−0.465661 + 0.884963i \(0.654184\pi\)
\(192\) 0 0
\(193\) 52706.0 0.101851 0.0509257 0.998702i \(-0.483783\pi\)
0.0509257 + 0.998702i \(0.483783\pi\)
\(194\) 238076. 0.454163
\(195\) 0 0
\(196\) −561596. −1.04420
\(197\) −455862. −0.836889 −0.418444 0.908242i \(-0.637425\pi\)
−0.418444 + 0.908242i \(0.637425\pi\)
\(198\) 0 0
\(199\) 865000. 1.54840 0.774200 0.632940i \(-0.218152\pi\)
0.774200 + 0.632940i \(0.218152\pi\)
\(200\) 75000.0 0.132583
\(201\) 0 0
\(202\) −179796. −0.310028
\(203\) 654720. 1.11510
\(204\) 0 0
\(205\) −234950. −0.390473
\(206\) 39008.0 0.0640451
\(207\) 0 0
\(208\) 187616. 0.300685
\(209\) 156880. 0.248429
\(210\) 0 0
\(211\) 1.10565e6 1.70967 0.854835 0.518900i \(-0.173658\pi\)
0.854835 + 0.518900i \(0.173658\pi\)
\(212\) 667688. 1.02031
\(213\) 0 0
\(214\) 316584. 0.472557
\(215\) 31100.0 0.0458843
\(216\) 0 0
\(217\) −470016. −0.677584
\(218\) −73660.0 −0.104976
\(219\) 0 0
\(220\) 103600. 0.144312
\(221\) 479908. 0.660963
\(222\) 0 0
\(223\) 1.12158e6 1.51031 0.755156 0.655545i \(-0.227561\pi\)
0.755156 + 0.655545i \(0.227561\pi\)
\(224\) −989184. −1.31722
\(225\) 0 0
\(226\) 22372.0 0.0291363
\(227\) 23348.0 0.0300736 0.0150368 0.999887i \(-0.495213\pi\)
0.0150368 + 0.999887i \(0.495213\pi\)
\(228\) 0 0
\(229\) −596010. −0.751043 −0.375522 0.926814i \(-0.622536\pi\)
−0.375522 + 0.926814i \(0.622536\pi\)
\(230\) −148800. −0.185474
\(231\) 0 0
\(232\) 409200. 0.499132
\(233\) 485334. 0.585667 0.292834 0.956163i \(-0.405402\pi\)
0.292834 + 0.956163i \(0.405402\pi\)
\(234\) 0 0
\(235\) −302200. −0.356964
\(236\) −560560. −0.655152
\(237\) 0 0
\(238\) −644352. −0.737362
\(239\) 48880.0 0.0553524 0.0276762 0.999617i \(-0.491189\pi\)
0.0276762 + 0.999617i \(0.491189\pi\)
\(240\) 0 0
\(241\) −110798. −0.122882 −0.0614411 0.998111i \(-0.519570\pi\)
−0.0614411 + 0.998111i \(0.519570\pi\)
\(242\) 278294. 0.305468
\(243\) 0 0
\(244\) −904456. −0.972552
\(245\) −501425. −0.533692
\(246\) 0 0
\(247\) 303160. 0.316176
\(248\) −293760. −0.303294
\(249\) 0 0
\(250\) 31250.0 0.0316228
\(251\) 1.64375e6 1.64684 0.823419 0.567434i \(-0.192064\pi\)
0.823419 + 0.567434i \(0.192064\pi\)
\(252\) 0 0
\(253\) −440448. −0.432607
\(254\) −141104. −0.137232
\(255\) 0 0
\(256\) −30464.0 −0.0290527
\(257\) −1.30624e6 −1.23365 −0.616823 0.787102i \(-0.711581\pi\)
−0.616823 + 0.787102i \(0.711581\pi\)
\(258\) 0 0
\(259\) 34944.0 0.0323685
\(260\) 200200. 0.183667
\(261\) 0 0
\(262\) 152904. 0.137615
\(263\) −2.12834e6 −1.89736 −0.948682 0.316231i \(-0.897583\pi\)
−0.948682 + 0.316231i \(0.897583\pi\)
\(264\) 0 0
\(265\) 596150. 0.521484
\(266\) −407040. −0.352722
\(267\) 0 0
\(268\) −1.70722e6 −1.45195
\(269\) 1.44109e6 1.21426 0.607128 0.794604i \(-0.292321\pi\)
0.607128 + 0.794604i \(0.292321\pi\)
\(270\) 0 0
\(271\) −93248.0 −0.0771288 −0.0385644 0.999256i \(-0.512278\pi\)
−0.0385644 + 0.999256i \(0.512278\pi\)
\(272\) 1.10077e6 0.902139
\(273\) 0 0
\(274\) −289836. −0.233225
\(275\) 92500.0 0.0737581
\(276\) 0 0
\(277\) −110298. −0.0863711 −0.0431855 0.999067i \(-0.513751\pi\)
−0.0431855 + 0.999067i \(0.513751\pi\)
\(278\) −224440. −0.174176
\(279\) 0 0
\(280\) −576000. −0.439064
\(281\) 192198. 0.145205 0.0726027 0.997361i \(-0.476869\pi\)
0.0726027 + 0.997361i \(0.476869\pi\)
\(282\) 0 0
\(283\) −331884. −0.246332 −0.123166 0.992386i \(-0.539305\pi\)
−0.123166 + 0.992386i \(0.539305\pi\)
\(284\) −914144. −0.672541
\(285\) 0 0
\(286\) −84656.0 −0.0611988
\(287\) 1.80442e6 1.29310
\(288\) 0 0
\(289\) 1.39583e6 0.983076
\(290\) 170500. 0.119050
\(291\) 0 0
\(292\) 1.08567e6 0.745146
\(293\) −2.19481e6 −1.49358 −0.746788 0.665063i \(-0.768405\pi\)
−0.746788 + 0.665063i \(0.768405\pi\)
\(294\) 0 0
\(295\) −500500. −0.334849
\(296\) 21840.0 0.0144885
\(297\) 0 0
\(298\) 807500. 0.526747
\(299\) −851136. −0.550581
\(300\) 0 0
\(301\) −238848. −0.151952
\(302\) 893296. 0.563609
\(303\) 0 0
\(304\) 695360. 0.431545
\(305\) −807550. −0.497073
\(306\) 0 0
\(307\) −2.37751e6 −1.43971 −0.719857 0.694123i \(-0.755793\pi\)
−0.719857 + 0.694123i \(0.755793\pi\)
\(308\) −795648. −0.477908
\(309\) 0 0
\(310\) −122400. −0.0723398
\(311\) 2.37305e6 1.39125 0.695626 0.718405i \(-0.255127\pi\)
0.695626 + 0.718405i \(0.255127\pi\)
\(312\) 0 0
\(313\) −1.42941e6 −0.824702 −0.412351 0.911025i \(-0.635292\pi\)
−0.412351 + 0.911025i \(0.635292\pi\)
\(314\) 524516. 0.300217
\(315\) 0 0
\(316\) 934080. 0.526219
\(317\) −2.12462e6 −1.18750 −0.593750 0.804650i \(-0.702353\pi\)
−0.593750 + 0.804650i \(0.702353\pi\)
\(318\) 0 0
\(319\) 504680. 0.277677
\(320\) 267200. 0.145868
\(321\) 0 0
\(322\) 1.14278e6 0.614221
\(323\) 1.77868e6 0.948618
\(324\) 0 0
\(325\) 178750. 0.0938723
\(326\) 309128. 0.161100
\(327\) 0 0
\(328\) 1.12776e6 0.578805
\(329\) 2.32090e6 1.18213
\(330\) 0 0
\(331\) 3.09985e6 1.55515 0.777573 0.628793i \(-0.216451\pi\)
0.777573 + 0.628793i \(0.216451\pi\)
\(332\) 468048. 0.233048
\(333\) 0 0
\(334\) 793344. 0.389131
\(335\) −1.52430e6 −0.742093
\(336\) 0 0
\(337\) 2.40008e6 1.15120 0.575601 0.817731i \(-0.304768\pi\)
0.575601 + 0.817731i \(0.304768\pi\)
\(338\) 578994. 0.275665
\(339\) 0 0
\(340\) 1.17460e6 0.551052
\(341\) −362304. −0.168728
\(342\) 0 0
\(343\) 624000. 0.286384
\(344\) −149280. −0.0680151
\(345\) 0 0
\(346\) −1.14695e6 −0.515055
\(347\) −1.77741e6 −0.792436 −0.396218 0.918156i \(-0.629678\pi\)
−0.396218 + 0.918156i \(0.629678\pi\)
\(348\) 0 0
\(349\) −2.14805e6 −0.944019 −0.472010 0.881593i \(-0.656471\pi\)
−0.472010 + 0.881593i \(0.656471\pi\)
\(350\) −240000. −0.104723
\(351\) 0 0
\(352\) −762496. −0.328005
\(353\) 661854. 0.282700 0.141350 0.989960i \(-0.454856\pi\)
0.141350 + 0.989960i \(0.454856\pi\)
\(354\) 0 0
\(355\) −816200. −0.343737
\(356\) 2.83836e6 1.18698
\(357\) 0 0
\(358\) −1.18892e6 −0.490281
\(359\) 259320. 0.106194 0.0530970 0.998589i \(-0.483091\pi\)
0.0530970 + 0.998589i \(0.483091\pi\)
\(360\) 0 0
\(361\) −1.35250e6 −0.546222
\(362\) 214196. 0.0859093
\(363\) 0 0
\(364\) −1.53754e6 −0.608236
\(365\) 969350. 0.380845
\(366\) 0 0
\(367\) −1.49993e6 −0.581307 −0.290653 0.956828i \(-0.593873\pi\)
−0.290653 + 0.956828i \(0.593873\pi\)
\(368\) −1.95226e6 −0.751480
\(369\) 0 0
\(370\) 9100.00 0.00345571
\(371\) −4.57843e6 −1.72696
\(372\) 0 0
\(373\) −2.23807e6 −0.832918 −0.416459 0.909154i \(-0.636729\pi\)
−0.416459 + 0.909154i \(0.636729\pi\)
\(374\) −496688. −0.183614
\(375\) 0 0
\(376\) 1.45056e6 0.529135
\(377\) 975260. 0.353400
\(378\) 0 0
\(379\) 3.15934e6 1.12979 0.564896 0.825162i \(-0.308916\pi\)
0.564896 + 0.825162i \(0.308916\pi\)
\(380\) 742000. 0.263600
\(381\) 0 0
\(382\) 939104. 0.329272
\(383\) −342216. −0.119207 −0.0596037 0.998222i \(-0.518984\pi\)
−0.0596037 + 0.998222i \(0.518984\pi\)
\(384\) 0 0
\(385\) −710400. −0.244259
\(386\) −105412. −0.0360099
\(387\) 0 0
\(388\) 3.33306e6 1.12399
\(389\) −88470.0 −0.0296430 −0.0148215 0.999890i \(-0.504718\pi\)
−0.0148215 + 0.999890i \(0.504718\pi\)
\(390\) 0 0
\(391\) −4.99373e6 −1.65190
\(392\) 2.40684e6 0.791101
\(393\) 0 0
\(394\) 911724. 0.295885
\(395\) 834000. 0.268951
\(396\) 0 0
\(397\) −5.45674e6 −1.73763 −0.868814 0.495138i \(-0.835117\pi\)
−0.868814 + 0.495138i \(0.835117\pi\)
\(398\) −1.73000e6 −0.547442
\(399\) 0 0
\(400\) 410000. 0.128125
\(401\) −4.04680e6 −1.25676 −0.628378 0.777908i \(-0.716281\pi\)
−0.628378 + 0.777908i \(0.716281\pi\)
\(402\) 0 0
\(403\) −700128. −0.214741
\(404\) −2.51714e6 −0.767281
\(405\) 0 0
\(406\) −1.30944e6 −0.394249
\(407\) 26936.0 0.00806022
\(408\) 0 0
\(409\) −2.71207e6 −0.801664 −0.400832 0.916151i \(-0.631279\pi\)
−0.400832 + 0.916151i \(0.631279\pi\)
\(410\) 469900. 0.138053
\(411\) 0 0
\(412\) 546112. 0.158503
\(413\) 3.84384e6 1.10889
\(414\) 0 0
\(415\) 417900. 0.119111
\(416\) −1.47347e6 −0.417454
\(417\) 0 0
\(418\) −313760. −0.0878328
\(419\) −3.71746e6 −1.03445 −0.517227 0.855848i \(-0.673036\pi\)
−0.517227 + 0.855848i \(0.673036\pi\)
\(420\) 0 0
\(421\) 3.55250e6 0.976853 0.488426 0.872605i \(-0.337571\pi\)
0.488426 + 0.872605i \(0.337571\pi\)
\(422\) −2.21130e6 −0.604460
\(423\) 0 0
\(424\) −2.86152e6 −0.773005
\(425\) 1.04875e6 0.281643
\(426\) 0 0
\(427\) 6.20198e6 1.64612
\(428\) 4.43218e6 1.16952
\(429\) 0 0
\(430\) −62200.0 −0.0162226
\(431\) 4.06205e6 1.05330 0.526650 0.850082i \(-0.323448\pi\)
0.526650 + 0.850082i \(0.323448\pi\)
\(432\) 0 0
\(433\) 7.26287e6 1.86161 0.930804 0.365518i \(-0.119108\pi\)
0.930804 + 0.365518i \(0.119108\pi\)
\(434\) 940032. 0.239562
\(435\) 0 0
\(436\) −1.03124e6 −0.259803
\(437\) −3.15456e6 −0.790197
\(438\) 0 0
\(439\) −5.41028e6 −1.33986 −0.669928 0.742426i \(-0.733675\pi\)
−0.669928 + 0.742426i \(0.733675\pi\)
\(440\) −444000. −0.109333
\(441\) 0 0
\(442\) −959816. −0.233686
\(443\) 6.51524e6 1.57733 0.788663 0.614826i \(-0.210774\pi\)
0.788663 + 0.614826i \(0.210774\pi\)
\(444\) 0 0
\(445\) 2.53425e6 0.606666
\(446\) −2.24315e6 −0.533976
\(447\) 0 0
\(448\) −2.05210e6 −0.483062
\(449\) 509950. 0.119375 0.0596873 0.998217i \(-0.480990\pi\)
0.0596873 + 0.998217i \(0.480990\pi\)
\(450\) 0 0
\(451\) 1.39090e6 0.322000
\(452\) 313208. 0.0721085
\(453\) 0 0
\(454\) −46696.0 −0.0106326
\(455\) −1.37280e6 −0.310870
\(456\) 0 0
\(457\) 1.22084e6 0.273444 0.136722 0.990609i \(-0.456343\pi\)
0.136722 + 0.990609i \(0.456343\pi\)
\(458\) 1.19202e6 0.265534
\(459\) 0 0
\(460\) −2.08320e6 −0.459025
\(461\) 4.07210e6 0.892413 0.446207 0.894930i \(-0.352775\pi\)
0.446207 + 0.894930i \(0.352775\pi\)
\(462\) 0 0
\(463\) 2.02294e6 0.438561 0.219280 0.975662i \(-0.429629\pi\)
0.219280 + 0.975662i \(0.429629\pi\)
\(464\) 2.23696e6 0.482351
\(465\) 0 0
\(466\) −970668. −0.207065
\(467\) −3.25097e6 −0.689797 −0.344898 0.938640i \(-0.612087\pi\)
−0.344898 + 0.938640i \(0.612087\pi\)
\(468\) 0 0
\(469\) 1.17066e7 2.45753
\(470\) 604400. 0.126206
\(471\) 0 0
\(472\) 2.40240e6 0.496353
\(473\) −184112. −0.0378381
\(474\) 0 0
\(475\) 662500. 0.134726
\(476\) −9.02093e6 −1.82488
\(477\) 0 0
\(478\) −97760.0 −0.0195700
\(479\) 3.27936e6 0.653056 0.326528 0.945188i \(-0.394121\pi\)
0.326528 + 0.945188i \(0.394121\pi\)
\(480\) 0 0
\(481\) 52052.0 0.0102583
\(482\) 221596. 0.0434455
\(483\) 0 0
\(484\) 3.89612e6 0.755994
\(485\) 2.97595e6 0.574475
\(486\) 0 0
\(487\) −8.53197e6 −1.63015 −0.815074 0.579357i \(-0.803304\pi\)
−0.815074 + 0.579357i \(0.803304\pi\)
\(488\) 3.87624e6 0.736819
\(489\) 0 0
\(490\) 1.00285e6 0.188689
\(491\) −1.51265e6 −0.283162 −0.141581 0.989927i \(-0.545219\pi\)
−0.141581 + 0.989927i \(0.545219\pi\)
\(492\) 0 0
\(493\) 5.72198e6 1.06030
\(494\) −606320. −0.111785
\(495\) 0 0
\(496\) −1.60589e6 −0.293097
\(497\) 6.26842e6 1.13833
\(498\) 0 0
\(499\) −6.49190e6 −1.16713 −0.583567 0.812065i \(-0.698343\pi\)
−0.583567 + 0.812065i \(0.698343\pi\)
\(500\) 437500. 0.0782624
\(501\) 0 0
\(502\) −3.28750e6 −0.582245
\(503\) −8.61770e6 −1.51870 −0.759349 0.650684i \(-0.774482\pi\)
−0.759349 + 0.650684i \(0.774482\pi\)
\(504\) 0 0
\(505\) −2.24745e6 −0.392158
\(506\) 880896. 0.152950
\(507\) 0 0
\(508\) −1.97546e6 −0.339632
\(509\) −2.67323e6 −0.457343 −0.228671 0.973504i \(-0.573438\pi\)
−0.228671 + 0.973504i \(0.573438\pi\)
\(510\) 0 0
\(511\) −7.44461e6 −1.26122
\(512\) −5.89875e6 −0.994455
\(513\) 0 0
\(514\) 2.61248e6 0.436160
\(515\) 487600. 0.0810113
\(516\) 0 0
\(517\) 1.78902e6 0.294367
\(518\) −69888.0 −0.0114440
\(519\) 0 0
\(520\) −858000. −0.139149
\(521\) −6.18500e6 −0.998264 −0.499132 0.866526i \(-0.666348\pi\)
−0.499132 + 0.866526i \(0.666348\pi\)
\(522\) 0 0
\(523\) −6.89452e6 −1.10217 −0.551087 0.834448i \(-0.685787\pi\)
−0.551087 + 0.834448i \(0.685787\pi\)
\(524\) 2.14066e6 0.340580
\(525\) 0 0
\(526\) 4.25667e6 0.670820
\(527\) −4.10774e6 −0.644283
\(528\) 0 0
\(529\) 2.42023e6 0.376026
\(530\) −1.19230e6 −0.184372
\(531\) 0 0
\(532\) −5.69856e6 −0.872943
\(533\) 2.68783e6 0.409811
\(534\) 0 0
\(535\) 3.95730e6 0.597743
\(536\) 7.31664e6 1.10002
\(537\) 0 0
\(538\) −2.88218e6 −0.429304
\(539\) 2.96844e6 0.440104
\(540\) 0 0
\(541\) 155502. 0.0228425 0.0114212 0.999935i \(-0.496364\pi\)
0.0114212 + 0.999935i \(0.496364\pi\)
\(542\) 186496. 0.0272691
\(543\) 0 0
\(544\) −8.64506e6 −1.25248
\(545\) −920750. −0.132785
\(546\) 0 0
\(547\) 1.26544e7 1.80831 0.904157 0.427201i \(-0.140500\pi\)
0.904157 + 0.427201i \(0.140500\pi\)
\(548\) −4.05770e6 −0.577204
\(549\) 0 0
\(550\) −185000. −0.0260774
\(551\) 3.61460e6 0.507202
\(552\) 0 0
\(553\) −6.40512e6 −0.890665
\(554\) 220596. 0.0305368
\(555\) 0 0
\(556\) −3.14216e6 −0.431064
\(557\) 7.07786e6 0.966638 0.483319 0.875444i \(-0.339431\pi\)
0.483319 + 0.875444i \(0.339431\pi\)
\(558\) 0 0
\(559\) −355784. −0.0481567
\(560\) −3.14880e6 −0.424302
\(561\) 0 0
\(562\) −384396. −0.0513379
\(563\) −846636. −0.112571 −0.0562854 0.998415i \(-0.517926\pi\)
−0.0562854 + 0.998415i \(0.517926\pi\)
\(564\) 0 0
\(565\) 279650. 0.0368548
\(566\) 663768. 0.0870914
\(567\) 0 0
\(568\) 3.91776e6 0.509527
\(569\) −4.96041e6 −0.642299 −0.321149 0.947029i \(-0.604069\pi\)
−0.321149 + 0.947029i \(0.604069\pi\)
\(570\) 0 0
\(571\) 8.96505e6 1.15070 0.575351 0.817907i \(-0.304866\pi\)
0.575351 + 0.817907i \(0.304866\pi\)
\(572\) −1.18518e6 −0.151459
\(573\) 0 0
\(574\) −3.60883e6 −0.457180
\(575\) −1.86000e6 −0.234608
\(576\) 0 0
\(577\) −2.86080e6 −0.357724 −0.178862 0.983874i \(-0.557242\pi\)
−0.178862 + 0.983874i \(0.557242\pi\)
\(578\) −2.79165e6 −0.347570
\(579\) 0 0
\(580\) 2.38700e6 0.294634
\(581\) −3.20947e6 −0.394451
\(582\) 0 0
\(583\) −3.52921e6 −0.430037
\(584\) −4.65288e6 −0.564534
\(585\) 0 0
\(586\) 4.38961e6 0.528059
\(587\) 6.74027e6 0.807387 0.403694 0.914894i \(-0.367726\pi\)
0.403694 + 0.914894i \(0.367726\pi\)
\(588\) 0 0
\(589\) −2.59488e6 −0.308197
\(590\) 1.00100e6 0.118387
\(591\) 0 0
\(592\) 119392. 0.0140014
\(593\) 1.78609e6 0.208578 0.104289 0.994547i \(-0.466743\pi\)
0.104289 + 0.994547i \(0.466743\pi\)
\(594\) 0 0
\(595\) −8.05440e6 −0.932697
\(596\) 1.13050e7 1.30363
\(597\) 0 0
\(598\) 1.70227e6 0.194660
\(599\) −4.94620e6 −0.563254 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(600\) 0 0
\(601\) −4.58100e6 −0.517337 −0.258669 0.965966i \(-0.583284\pi\)
−0.258669 + 0.965966i \(0.583284\pi\)
\(602\) 477696. 0.0537230
\(603\) 0 0
\(604\) 1.25061e7 1.39486
\(605\) 3.47868e6 0.386390
\(606\) 0 0
\(607\) 7.07999e6 0.779940 0.389970 0.920828i \(-0.372485\pi\)
0.389970 + 0.920828i \(0.372485\pi\)
\(608\) −5.46112e6 −0.599132
\(609\) 0 0
\(610\) 1.61510e6 0.175742
\(611\) 3.45717e6 0.374643
\(612\) 0 0
\(613\) 5.09609e6 0.547754 0.273877 0.961765i \(-0.411694\pi\)
0.273877 + 0.961765i \(0.411694\pi\)
\(614\) 4.75502e6 0.509016
\(615\) 0 0
\(616\) 3.40992e6 0.362070
\(617\) 1.30003e7 1.37480 0.687400 0.726279i \(-0.258752\pi\)
0.687400 + 0.726279i \(0.258752\pi\)
\(618\) 0 0
\(619\) 4.84406e6 0.508139 0.254070 0.967186i \(-0.418231\pi\)
0.254070 + 0.967186i \(0.418231\pi\)
\(620\) −1.71360e6 −0.179032
\(621\) 0 0
\(622\) −4.74610e6 −0.491882
\(623\) −1.94630e7 −2.00905
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 2.85883e6 0.291576
\(627\) 0 0
\(628\) 7.34322e6 0.742998
\(629\) 305396. 0.0307777
\(630\) 0 0
\(631\) 6.22775e6 0.622670 0.311335 0.950300i \(-0.399224\pi\)
0.311335 + 0.950300i \(0.399224\pi\)
\(632\) −4.00320e6 −0.398671
\(633\) 0 0
\(634\) 4.24924e6 0.419845
\(635\) −1.76380e6 −0.173586
\(636\) 0 0
\(637\) 5.73630e6 0.560123
\(638\) −1.00936e6 −0.0981735
\(639\) 0 0
\(640\) −4.65600e6 −0.449328
\(641\) −1.53280e6 −0.147347 −0.0736734 0.997282i \(-0.523472\pi\)
−0.0736734 + 0.997282i \(0.523472\pi\)
\(642\) 0 0
\(643\) −1.74382e7 −1.66332 −0.831659 0.555287i \(-0.812609\pi\)
−0.831659 + 0.555287i \(0.812609\pi\)
\(644\) 1.59990e7 1.52012
\(645\) 0 0
\(646\) −3.55736e6 −0.335387
\(647\) 4.25469e6 0.399583 0.199792 0.979838i \(-0.435974\pi\)
0.199792 + 0.979838i \(0.435974\pi\)
\(648\) 0 0
\(649\) 2.96296e6 0.276130
\(650\) −357500. −0.0331889
\(651\) 0 0
\(652\) 4.32779e6 0.398701
\(653\) −3.01085e6 −0.276316 −0.138158 0.990410i \(-0.544118\pi\)
−0.138158 + 0.990410i \(0.544118\pi\)
\(654\) 0 0
\(655\) 1.91130e6 0.174071
\(656\) 6.16509e6 0.559345
\(657\) 0 0
\(658\) −4.64179e6 −0.417947
\(659\) 8.11462e6 0.727871 0.363936 0.931424i \(-0.381433\pi\)
0.363936 + 0.931424i \(0.381433\pi\)
\(660\) 0 0
\(661\) 2.47370e6 0.220213 0.110107 0.993920i \(-0.464881\pi\)
0.110107 + 0.993920i \(0.464881\pi\)
\(662\) −6.19970e6 −0.549827
\(663\) 0 0
\(664\) −2.00592e6 −0.176560
\(665\) −5.08800e6 −0.446162
\(666\) 0 0
\(667\) −1.01482e7 −0.883228
\(668\) 1.11068e7 0.963049
\(669\) 0 0
\(670\) 3.04860e6 0.262370
\(671\) 4.78070e6 0.409907
\(672\) 0 0
\(673\) 5.77063e6 0.491117 0.245559 0.969382i \(-0.421029\pi\)
0.245559 + 0.969382i \(0.421029\pi\)
\(674\) −4.80016e6 −0.407011
\(675\) 0 0
\(676\) 8.10592e6 0.682237
\(677\) −1.67197e7 −1.40203 −0.701014 0.713147i \(-0.747269\pi\)
−0.701014 + 0.713147i \(0.747269\pi\)
\(678\) 0 0
\(679\) −2.28553e7 −1.90245
\(680\) −5.03400e6 −0.417485
\(681\) 0 0
\(682\) 724608. 0.0596544
\(683\) −7.14532e6 −0.586097 −0.293049 0.956098i \(-0.594670\pi\)
−0.293049 + 0.956098i \(0.594670\pi\)
\(684\) 0 0
\(685\) −3.62295e6 −0.295009
\(686\) −1.24800e6 −0.101252
\(687\) 0 0
\(688\) −816064. −0.0657284
\(689\) −6.81996e6 −0.547310
\(690\) 0 0
\(691\) −8.78395e6 −0.699833 −0.349917 0.936781i \(-0.613790\pi\)
−0.349917 + 0.936781i \(0.613790\pi\)
\(692\) −1.60573e7 −1.27470
\(693\) 0 0
\(694\) 3.55482e6 0.280169
\(695\) −2.80550e6 −0.220317
\(696\) 0 0
\(697\) 1.57698e7 1.22955
\(698\) 4.29610e6 0.333761
\(699\) 0 0
\(700\) −3.36000e6 −0.259176
\(701\) 1.60141e7 1.23086 0.615428 0.788193i \(-0.288983\pi\)
0.615428 + 0.788193i \(0.288983\pi\)
\(702\) 0 0
\(703\) 192920. 0.0147228
\(704\) −1.58182e6 −0.120289
\(705\) 0 0
\(706\) −1.32371e6 −0.0999495
\(707\) 1.72604e7 1.29868
\(708\) 0 0
\(709\) −1.91354e7 −1.42962 −0.714811 0.699318i \(-0.753487\pi\)
−0.714811 + 0.699318i \(0.753487\pi\)
\(710\) 1.63240e6 0.121529
\(711\) 0 0
\(712\) −1.21644e7 −0.899271
\(713\) 7.28525e6 0.536686
\(714\) 0 0
\(715\) −1.05820e6 −0.0774110
\(716\) −1.66449e7 −1.21338
\(717\) 0 0
\(718\) −518640. −0.0375452
\(719\) −1.02934e7 −0.742566 −0.371283 0.928520i \(-0.621082\pi\)
−0.371283 + 0.928520i \(0.621082\pi\)
\(720\) 0 0
\(721\) −3.74477e6 −0.268279
\(722\) 2.70500e6 0.193119
\(723\) 0 0
\(724\) 2.99874e6 0.212615
\(725\) 2.13125e6 0.150588
\(726\) 0 0
\(727\) −1.93264e7 −1.35618 −0.678088 0.734981i \(-0.737191\pi\)
−0.678088 + 0.734981i \(0.737191\pi\)
\(728\) 6.58944e6 0.460808
\(729\) 0 0
\(730\) −1.93870e6 −0.134649
\(731\) −2.08743e6 −0.144484
\(732\) 0 0
\(733\) 5.26197e6 0.361733 0.180866 0.983508i \(-0.442110\pi\)
0.180866 + 0.983508i \(0.442110\pi\)
\(734\) 2.99986e6 0.205523
\(735\) 0 0
\(736\) 1.53324e7 1.04331
\(737\) 9.02386e6 0.611961
\(738\) 0 0
\(739\) 2.82944e7 1.90585 0.952927 0.303199i \(-0.0980548\pi\)
0.952927 + 0.303199i \(0.0980548\pi\)
\(740\) 127400. 0.00855244
\(741\) 0 0
\(742\) 9.15686e6 0.610572
\(743\) −2.09863e7 −1.39464 −0.697321 0.716759i \(-0.745625\pi\)
−0.697321 + 0.716759i \(0.745625\pi\)
\(744\) 0 0
\(745\) 1.00938e7 0.666288
\(746\) 4.47615e6 0.294481
\(747\) 0 0
\(748\) −6.95363e6 −0.454420
\(749\) −3.03921e7 −1.97950
\(750\) 0 0
\(751\) −1.89668e7 −1.22714 −0.613572 0.789639i \(-0.710268\pi\)
−0.613572 + 0.789639i \(0.710268\pi\)
\(752\) 7.92973e6 0.511345
\(753\) 0 0
\(754\) −1.95052e6 −0.124946
\(755\) 1.11662e7 0.712915
\(756\) 0 0
\(757\) −1.08257e7 −0.686617 −0.343309 0.939223i \(-0.611548\pi\)
−0.343309 + 0.939223i \(0.611548\pi\)
\(758\) −6.31868e6 −0.399442
\(759\) 0 0
\(760\) −3.18000e6 −0.199707
\(761\) −1.90534e7 −1.19264 −0.596322 0.802745i \(-0.703372\pi\)
−0.596322 + 0.802745i \(0.703372\pi\)
\(762\) 0 0
\(763\) 7.07136e6 0.439736
\(764\) 1.31475e7 0.814908
\(765\) 0 0
\(766\) 684432. 0.0421462
\(767\) 5.72572e6 0.351432
\(768\) 0 0
\(769\) −1.57826e7 −0.962415 −0.481208 0.876607i \(-0.659802\pi\)
−0.481208 + 0.876607i \(0.659802\pi\)
\(770\) 1.42080e6 0.0863587
\(771\) 0 0
\(772\) −1.47577e6 −0.0891199
\(773\) 2.44049e7 1.46902 0.734510 0.678598i \(-0.237412\pi\)
0.734510 + 0.678598i \(0.237412\pi\)
\(774\) 0 0
\(775\) −1.53000e6 −0.0915034
\(776\) −1.42846e7 −0.851555
\(777\) 0 0
\(778\) 176940. 0.0104804
\(779\) 9.96188e6 0.588163
\(780\) 0 0
\(781\) 4.83190e6 0.283459
\(782\) 9.98746e6 0.584034
\(783\) 0 0
\(784\) 1.31574e7 0.764504
\(785\) 6.55645e6 0.379747
\(786\) 0 0
\(787\) 3.37607e7 1.94301 0.971505 0.237019i \(-0.0761704\pi\)
0.971505 + 0.237019i \(0.0761704\pi\)
\(788\) 1.27641e7 0.732278
\(789\) 0 0
\(790\) −1.66800e6 −0.0950886
\(791\) −2.14771e6 −0.122049
\(792\) 0 0
\(793\) 9.23837e6 0.521690
\(794\) 1.09135e7 0.614344
\(795\) 0 0
\(796\) −2.42200e7 −1.35485
\(797\) −2.19885e7 −1.22617 −0.613083 0.790019i \(-0.710071\pi\)
−0.613083 + 0.790019i \(0.710071\pi\)
\(798\) 0 0
\(799\) 2.02837e7 1.12403
\(800\) −3.22000e6 −0.177882
\(801\) 0 0
\(802\) 8.09360e6 0.444330
\(803\) −5.73855e6 −0.314061
\(804\) 0 0
\(805\) 1.42848e7 0.776935
\(806\) 1.40026e6 0.0759224
\(807\) 0 0
\(808\) 1.07878e7 0.581303
\(809\) 2.93597e7 1.57717 0.788587 0.614923i \(-0.210813\pi\)
0.788587 + 0.614923i \(0.210813\pi\)
\(810\) 0 0
\(811\) 3.17703e7 1.69617 0.848083 0.529863i \(-0.177757\pi\)
0.848083 + 0.529863i \(0.177757\pi\)
\(812\) −1.83322e7 −0.975716
\(813\) 0 0
\(814\) −53872.0 −0.00284972
\(815\) 3.86410e6 0.203777
\(816\) 0 0
\(817\) −1.31864e6 −0.0691148
\(818\) 5.42414e6 0.283431
\(819\) 0 0
\(820\) 6.57860e6 0.341664
\(821\) 2.71430e6 0.140540 0.0702699 0.997528i \(-0.477614\pi\)
0.0702699 + 0.997528i \(0.477614\pi\)
\(822\) 0 0
\(823\) −1.25866e7 −0.647753 −0.323877 0.946099i \(-0.604986\pi\)
−0.323877 + 0.946099i \(0.604986\pi\)
\(824\) −2.34048e6 −0.120084
\(825\) 0 0
\(826\) −7.68768e6 −0.392053
\(827\) 8.72355e6 0.443537 0.221768 0.975099i \(-0.428817\pi\)
0.221768 + 0.975099i \(0.428817\pi\)
\(828\) 0 0
\(829\) −1.06178e7 −0.536597 −0.268299 0.963336i \(-0.586461\pi\)
−0.268299 + 0.963336i \(0.586461\pi\)
\(830\) −835800. −0.0421121
\(831\) 0 0
\(832\) −3.05677e6 −0.153093
\(833\) 3.36556e7 1.68053
\(834\) 0 0
\(835\) 9.91680e6 0.492216
\(836\) −4.39264e6 −0.217375
\(837\) 0 0
\(838\) 7.43492e6 0.365735
\(839\) −1.67765e7 −0.822805 −0.411403 0.911454i \(-0.634961\pi\)
−0.411403 + 0.911454i \(0.634961\pi\)
\(840\) 0 0
\(841\) −8.88305e6 −0.433084
\(842\) −7.10500e6 −0.345370
\(843\) 0 0
\(844\) −3.09583e7 −1.49596
\(845\) 7.23742e6 0.348692
\(846\) 0 0
\(847\) −2.67162e7 −1.27958
\(848\) −1.56430e7 −0.747016
\(849\) 0 0
\(850\) −2.09750e6 −0.0995760
\(851\) −541632. −0.0256378
\(852\) 0 0
\(853\) −2.20186e7 −1.03613 −0.518067 0.855340i \(-0.673348\pi\)
−0.518067 + 0.855340i \(0.673348\pi\)
\(854\) −1.24040e7 −0.581991
\(855\) 0 0
\(856\) −1.89950e7 −0.886045
\(857\) −3.16676e7 −1.47287 −0.736434 0.676510i \(-0.763492\pi\)
−0.736434 + 0.676510i \(0.763492\pi\)
\(858\) 0 0
\(859\) 1.58064e7 0.730886 0.365443 0.930834i \(-0.380918\pi\)
0.365443 + 0.930834i \(0.380918\pi\)
\(860\) −870800. −0.0401488
\(861\) 0 0
\(862\) −8.12410e6 −0.372398
\(863\) 1.44287e7 0.659476 0.329738 0.944072i \(-0.393040\pi\)
0.329738 + 0.944072i \(0.393040\pi\)
\(864\) 0 0
\(865\) −1.43368e7 −0.651499
\(866\) −1.45257e7 −0.658178
\(867\) 0 0
\(868\) 1.31604e7 0.592886
\(869\) −4.93728e6 −0.221788
\(870\) 0 0
\(871\) 1.74380e7 0.778845
\(872\) 4.41960e6 0.196830
\(873\) 0 0
\(874\) 6.30912e6 0.279377
\(875\) −3.00000e6 −0.132465
\(876\) 0 0
\(877\) 247902. 0.0108838 0.00544191 0.999985i \(-0.498268\pi\)
0.00544191 + 0.999985i \(0.498268\pi\)
\(878\) 1.08206e7 0.473711
\(879\) 0 0
\(880\) −2.42720e6 −0.105657
\(881\) −4.10268e7 −1.78085 −0.890426 0.455128i \(-0.849594\pi\)
−0.890426 + 0.455128i \(0.849594\pi\)
\(882\) 0 0
\(883\) 4.18015e7 1.80422 0.902112 0.431503i \(-0.142016\pi\)
0.902112 + 0.431503i \(0.142016\pi\)
\(884\) −1.34374e7 −0.578343
\(885\) 0 0
\(886\) −1.30305e7 −0.557669
\(887\) 2.10476e7 0.898241 0.449120 0.893471i \(-0.351737\pi\)
0.449120 + 0.893471i \(0.351737\pi\)
\(888\) 0 0
\(889\) 1.35460e7 0.574852
\(890\) −5.06850e6 −0.214489
\(891\) 0 0
\(892\) −3.14041e7 −1.32152
\(893\) 1.28133e7 0.537690
\(894\) 0 0
\(895\) −1.48615e7 −0.620162
\(896\) 3.57581e7 1.48800
\(897\) 0 0
\(898\) −1.01990e6 −0.0422053
\(899\) −8.34768e6 −0.344482
\(900\) 0 0
\(901\) −4.00136e7 −1.64208
\(902\) −2.78181e6 −0.113844
\(903\) 0 0
\(904\) −1.34232e6 −0.0546305
\(905\) 2.67745e6 0.108668
\(906\) 0 0
\(907\) 7.48309e6 0.302039 0.151019 0.988531i \(-0.451744\pi\)
0.151019 + 0.988531i \(0.451744\pi\)
\(908\) −653744. −0.0263144
\(909\) 0 0
\(910\) 2.74560e6 0.109909
\(911\) 6.63165e6 0.264744 0.132372 0.991200i \(-0.457741\pi\)
0.132372 + 0.991200i \(0.457741\pi\)
\(912\) 0 0
\(913\) −2.47397e6 −0.0982239
\(914\) −2.44168e6 −0.0966772
\(915\) 0 0
\(916\) 1.66883e7 0.657163
\(917\) −1.46788e7 −0.576457
\(918\) 0 0
\(919\) −1.68976e7 −0.659990 −0.329995 0.943983i \(-0.607047\pi\)
−0.329995 + 0.943983i \(0.607047\pi\)
\(920\) 8.92800e6 0.347764
\(921\) 0 0
\(922\) −8.14420e6 −0.315516
\(923\) 9.33733e6 0.360760
\(924\) 0 0
\(925\) 113750. 0.00437116
\(926\) −4.04587e6 −0.155055
\(927\) 0 0
\(928\) −1.75683e7 −0.669669
\(929\) 1.28653e7 0.489081 0.244541 0.969639i \(-0.421363\pi\)
0.244541 + 0.969639i \(0.421363\pi\)
\(930\) 0 0
\(931\) 2.12604e7 0.803892
\(932\) −1.35894e7 −0.512459
\(933\) 0 0
\(934\) 6.50194e6 0.243880
\(935\) −6.20860e6 −0.232255
\(936\) 0 0
\(937\) 1.06887e7 0.397718 0.198859 0.980028i \(-0.436276\pi\)
0.198859 + 0.980028i \(0.436276\pi\)
\(938\) −2.34132e7 −0.868870
\(939\) 0 0
\(940\) 8.46160e6 0.312344
\(941\) −2.82455e7 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(942\) 0 0
\(943\) −2.79684e7 −1.02421
\(944\) 1.31331e7 0.479665
\(945\) 0 0
\(946\) 368224. 0.0133778
\(947\) 1.70892e7 0.619222 0.309611 0.950863i \(-0.399801\pi\)
0.309611 + 0.950863i \(0.399801\pi\)
\(948\) 0 0
\(949\) −1.10894e7 −0.399706
\(950\) −1.32500e6 −0.0476329
\(951\) 0 0
\(952\) 3.86611e7 1.38255
\(953\) −2.22259e7 −0.792735 −0.396367 0.918092i \(-0.629729\pi\)
−0.396367 + 0.918092i \(0.629729\pi\)
\(954\) 0 0
\(955\) 1.17388e7 0.416500
\(956\) −1.36864e6 −0.0484333
\(957\) 0 0
\(958\) −6.55872e6 −0.230890
\(959\) 2.78243e7 0.976961
\(960\) 0 0
\(961\) −2.26364e7 −0.790678
\(962\) −104104. −0.00362685
\(963\) 0 0
\(964\) 3.10234e6 0.107522
\(965\) −1.31765e6 −0.0455493
\(966\) 0 0
\(967\) 2.41551e7 0.830696 0.415348 0.909663i \(-0.363660\pi\)
0.415348 + 0.909663i \(0.363660\pi\)
\(968\) −1.66976e7 −0.572752
\(969\) 0 0
\(970\) −5.95190e6 −0.203108
\(971\) 5.48313e7 1.86630 0.933149 0.359491i \(-0.117050\pi\)
0.933149 + 0.359491i \(0.117050\pi\)
\(972\) 0 0
\(973\) 2.15462e7 0.729608
\(974\) 1.70639e7 0.576344
\(975\) 0 0
\(976\) 2.11901e7 0.712047
\(977\) 1.56612e7 0.524915 0.262457 0.964944i \(-0.415467\pi\)
0.262457 + 0.964944i \(0.415467\pi\)
\(978\) 0 0
\(979\) −1.50028e7 −0.500281
\(980\) 1.40399e7 0.466981
\(981\) 0 0
\(982\) 3.02530e6 0.100113
\(983\) 1.63420e7 0.539412 0.269706 0.962943i \(-0.413073\pi\)
0.269706 + 0.962943i \(0.413073\pi\)
\(984\) 0 0
\(985\) 1.13966e7 0.374268
\(986\) −1.14440e7 −0.374873
\(987\) 0 0
\(988\) −8.48848e6 −0.276654
\(989\) 3.70214e6 0.120355
\(990\) 0 0
\(991\) 1.37576e7 0.444997 0.222498 0.974933i \(-0.428579\pi\)
0.222498 + 0.974933i \(0.428579\pi\)
\(992\) 1.26121e7 0.406919
\(993\) 0 0
\(994\) −1.25368e7 −0.402459
\(995\) −2.16250e7 −0.692466
\(996\) 0 0
\(997\) −1.29097e7 −0.411320 −0.205660 0.978624i \(-0.565934\pi\)
−0.205660 + 0.978624i \(0.565934\pi\)
\(998\) 1.29838e7 0.412644
\(999\) 0 0
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 45.6.a.b.1.1 1
3.2 odd 2 5.6.a.a.1.1 1
4.3 odd 2 720.6.a.a.1.1 1
5.2 odd 4 225.6.b.e.199.1 2
5.3 odd 4 225.6.b.e.199.2 2
5.4 even 2 225.6.a.f.1.1 1
12.11 even 2 80.6.a.e.1.1 1
15.2 even 4 25.6.b.a.24.2 2
15.8 even 4 25.6.b.a.24.1 2
15.14 odd 2 25.6.a.a.1.1 1
21.20 even 2 245.6.a.b.1.1 1
24.5 odd 2 320.6.a.j.1.1 1
24.11 even 2 320.6.a.g.1.1 1
33.32 even 2 605.6.a.a.1.1 1
39.38 odd 2 845.6.a.b.1.1 1
60.23 odd 4 400.6.c.j.49.2 2
60.47 odd 4 400.6.c.j.49.1 2
60.59 even 2 400.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 3.2 odd 2
25.6.a.a.1.1 1 15.14 odd 2
25.6.b.a.24.1 2 15.8 even 4
25.6.b.a.24.2 2 15.2 even 4
45.6.a.b.1.1 1 1.1 even 1 trivial
80.6.a.e.1.1 1 12.11 even 2
225.6.a.f.1.1 1 5.4 even 2
225.6.b.e.199.1 2 5.2 odd 4
225.6.b.e.199.2 2 5.3 odd 4
245.6.a.b.1.1 1 21.20 even 2
320.6.a.g.1.1 1 24.11 even 2
320.6.a.j.1.1 1 24.5 odd 2
400.6.a.g.1.1 1 60.59 even 2
400.6.c.j.49.1 2 60.47 odd 4
400.6.c.j.49.2 2 60.23 odd 4
605.6.a.a.1.1 1 33.32 even 2
720.6.a.a.1.1 1 4.3 odd 2
845.6.a.b.1.1 1 39.38 odd 2