Properties

Label 225.6.a.f.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.0863594579\)
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 \(\chi\) \(=\) 225.1

$q$-expansion

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

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 \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 0 0
\(4\) −28.0000 −0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) −192.000 −1.48100 −0.740502 0.672054i \(-0.765412\pi\)
−0.740502 + 0.672054i \(0.765412\pi\)
\(8\) −120.000 −0.662913
\(9\) 0 0
\(10\) 0 0
\(11\) 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 \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) −384.000 −0.523614
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) −1678.00 −1.40822 −0.704109 0.710092i \(-0.748653\pi\)
−0.704109 + 0.710092i \(0.748653\pi\)
\(18\) 0 0
\(19\) 1060.00 0.673631 0.336815 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 296.000 0.130387
\(23\) 2976.00 1.17304 0.586521 0.809934i \(-0.300497\pi\)
0.586521 + 0.809934i \(0.300497\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −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
\(35\) 0 0
\(36\) 0 0
\(37\) −182.000 −0.0218558 −0.0109279 0.999940i \(-0.503479\pi\)
−0.0109279 + 0.999940i \(0.503479\pi\)
\(38\) 2120.00 0.238164
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.483663\pi\)
0.0513002 + 0.998683i \(0.483663\pi\)
\(44\) −4144.00 −0.322692
\(45\) 0 0
\(46\) 5952.00 0.414733
\(47\) −12088.0 −0.798196 −0.399098 0.916908i \(-0.630677\pi\)
−0.399098 + 0.916908i \(0.630677\pi\)
\(48\) 0 0
\(49\) 20057.0 1.19337
\(50\) 0 0
\(51\) 0 0
\(52\) 8008.00 0.410691
\(53\) 23846.0 1.16607 0.583037 0.812446i \(-0.301864\pi\)
0.583037 + 0.812446i \(0.301864\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) −60972.0 −1.65937 −0.829685 0.558231i \(-0.811480\pi\)
−0.829685 + 0.558231i \(0.811480\pi\)
\(68\) 46984.0 1.23219
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(-0.359993\pi\)
0.425798 + 0.904818i \(0.359993\pi\)
\(74\) −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 \(0.597219\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 18796.0 0.308696
\(83\) 16716.0 0.266340 0.133170 0.991093i \(-0.457484\pi\)
0.133170 + 0.991093i \(0.457484\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 119038. 1.28457 0.642283 0.766468i \(-0.277987\pi\)
0.642283 + 0.766468i \(0.277987\pi\)
\(98\) 40114.0 0.421921
\(99\) 0 0
\(100\) 0 0
\(101\) 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 \(-0.471130\pi\)
0.0905734 + 0.995890i \(0.471130\pi\)
\(104\) 34320.0 0.311146
\(105\) 0 0
\(106\) 47692.0 0.412269
\(107\) 158292. 1.33659 0.668297 0.743895i \(-0.267024\pi\)
0.668297 + 0.743895i \(0.267024\pi\)
\(108\) 0 0
\(109\) 36830.0 0.296917 0.148459 0.988919i \(-0.452569\pi\)
0.148459 + 0.988919i \(0.452569\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −125952. −0.948768
\(113\) 11186.0 0.0824098 0.0412049 0.999151i \(-0.486880\pi\)
0.0412049 + 0.999151i \(0.486880\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) −70552.0 −0.388150 −0.194075 0.980987i \(-0.562171\pi\)
−0.194075 + 0.980987i \(0.562171\pi\)
\(128\) −186240. −1.00473
\(129\) 0 0
\(130\) 0 0
\(131\) −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 \(-0.606992\pi\)
−0.329831 + 0.944040i \(0.606992\pi\)
\(138\) 0 0
\(139\) 112220. 0.492644 0.246322 0.969188i \(-0.420778\pi\)
0.246322 + 0.969188i \(0.420778\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 65296.0 0.271748
\(143\) −42328.0 −0.173096
\(144\) 0 0
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 262258. 0.849141 0.424570 0.905395i \(-0.360425\pi\)
0.424570 + 0.905395i \(0.360425\pi\)
\(158\) −66720.0 −0.212625
\(159\) 0 0
\(160\) 0 0
\(161\) −571392. −1.73728
\(162\) 0 0
\(163\) 154564. 0.455658 0.227829 0.973701i \(-0.426837\pi\)
0.227829 + 0.973701i \(0.426837\pi\)
\(164\) −263144. −0.763983
\(165\) 0 0
\(166\) 33432.0 0.0941656
\(167\) 396672. 1.10063 0.550314 0.834958i \(-0.314508\pi\)
0.550314 + 0.834958i \(0.314508\pi\)
\(168\) 0 0
\(169\) −289497. −0.779700
\(170\) 0 0
\(171\) 0 0
\(172\) −34832.0 −0.0897754
\(173\) −573474. −1.45680 −0.728398 0.685155i \(-0.759735\pi\)
−0.728398 + 0.685155i \(0.759735\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 0 0
\(187\) −248344. −0.519337
\(188\) 338464. 0.698422
\(189\) 0 0
\(190\) 0 0
\(191\) −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 \(-0.516217\pi\)
−0.0509257 + 0.998702i \(0.516217\pi\)
\(194\) 238076. 0.454163
\(195\) 0 0
\(196\) −561596. −1.04420
\(197\) 455862. 0.836889 0.418444 0.908242i \(-0.362575\pi\)
0.418444 + 0.908242i \(0.362575\pi\)
\(198\) 0 0
\(199\) 865000. 1.54840 0.774200 0.632940i \(-0.218152\pi\)
0.774200 + 0.632940i \(0.218152\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 179796. 0.310028
\(203\) −654720. −1.11510
\(204\) 0 0
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 0 0
\(217\) 470016. 0.677584
\(218\) 73660.0 0.104976
\(219\) 0 0
\(220\) 0 0
\(221\) 479908. 0.660963
\(222\) 0 0
\(223\) −1.12158e6 −1.51031 −0.755156 0.655545i \(-0.772439\pi\)
−0.755156 + 0.655545i \(0.772439\pi\)
\(224\) −989184. −1.31722
\(225\) 0 0
\(226\) 22372.0 0.0291363
\(227\) −23348.0 −0.0300736 −0.0150368 0.999887i \(-0.504787\pi\)
−0.0150368 + 0.999887i \(0.504787\pi\)
\(228\) 0 0
\(229\) −596010. −0.751043 −0.375522 0.926814i \(-0.622536\pi\)
−0.375522 + 0.926814i \(0.622536\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −409200. −0.499132
\(233\) −485334. −0.585667 −0.292834 0.956163i \(-0.594598\pi\)
−0.292834 + 0.956163i \(0.594598\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −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
\(245\) 0 0
\(246\) 0 0
\(247\) −303160. −0.316176
\(248\) 293760. 0.303294
\(249\) 0 0
\(250\) 0 0
\(251\) 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 \(-0.288419\pi\)
0.616823 + 0.787102i \(0.288419\pi\)
\(258\) 0 0
\(259\) 34944.0 0.0323685
\(260\) 0 0
\(261\) 0 0
\(262\) −152904. −0.137615
\(263\) 2.12834e6 1.89736 0.948682 0.316231i \(-0.102417\pi\)
0.948682 + 0.316231i \(0.102417\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −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
\(275\) 0 0
\(276\) 0 0
\(277\) 110298. 0.0863711 0.0431855 0.999067i \(-0.486249\pi\)
0.0431855 + 0.999067i \(0.486249\pi\)
\(278\) 224440. 0.174176
\(279\) 0 0
\(280\) 0 0
\(281\) 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 \(-0.460695\pi\)
0.123166 + 0.992386i \(0.460695\pi\)
\(284\) −914144. −0.672541
\(285\) 0 0
\(286\) −84656.0 −0.0611988
\(287\) −1.80442e6 −1.29310
\(288\) 0 0
\(289\) 1.39583e6 0.983076
\(290\) 0 0
\(291\) 0 0
\(292\) −1.08567e6 −0.745146
\(293\) 2.19481e6 1.49358 0.746788 0.665063i \(-0.231595\pi\)
0.746788 + 0.665063i \(0.231595\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 2.37751e6 1.43971 0.719857 0.694123i \(-0.244207\pi\)
0.719857 + 0.694123i \(0.244207\pi\)
\(308\) 795648. 0.477908
\(309\) 0 0
\(310\) 0 0
\(311\) 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 \(-0.364708\pi\)
0.412351 + 0.911025i \(0.364708\pi\)
\(314\) 524516. 0.300217
\(315\) 0 0
\(316\) 934080. 0.526219
\(317\) 2.12462e6 1.18750 0.593750 0.804650i \(-0.297647\pi\)
0.593750 + 0.804650i \(0.297647\pi\)
\(318\) 0 0
\(319\) 504680. 0.277677
\(320\) 0 0
\(321\) 0 0
\(322\) −1.14278e6 −0.614221
\(323\) −1.77868e6 −0.948618
\(324\) 0 0
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) −2.40008e6 −1.15120 −0.575601 0.817731i \(-0.695232\pi\)
−0.575601 + 0.817731i \(0.695232\pi\)
\(338\) −578994. −0.275665
\(339\) 0 0
\(340\) 0 0
\(341\) −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 \(-0.370322\pi\)
0.396218 + 0.918156i \(0.370322\pi\)
\(348\) 0 0
\(349\) −2.14805e6 −0.944019 −0.472010 0.881593i \(-0.656471\pi\)
−0.472010 + 0.881593i \(0.656471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 762496. 0.328005
\(353\) −661854. −0.282700 −0.141350 0.989960i \(-0.545144\pi\)
−0.141350 + 0.989960i \(0.545144\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 1.49993e6 0.581307 0.290653 0.956828i \(-0.406127\pi\)
0.290653 + 0.956828i \(0.406127\pi\)
\(368\) 1.95226e6 0.751480
\(369\) 0 0
\(370\) 0 0
\(371\) −4.57843e6 −1.72696
\(372\) 0 0
\(373\) 2.23807e6 0.832918 0.416459 0.909154i \(-0.363271\pi\)
0.416459 + 0.909154i \(0.363271\pi\)
\(374\) −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 \(0.308916\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −939104. −0.329272
\(383\) 342216. 0.119207 0.0596037 0.998222i \(-0.481016\pi\)
0.0596037 + 0.998222i \(0.481016\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −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
\(395\) 0 0
\(396\) 0 0
\(397\) 5.45674e6 1.73763 0.868814 0.495138i \(-0.164883\pi\)
0.868814 + 0.495138i \(0.164883\pi\)
\(398\) 1.73000e6 0.547442
\(399\) 0 0
\(400\) 0 0
\(401\) −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 \(0.631279\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −546112. −0.158503
\(413\) −3.84384e6 −1.10889
\(414\) 0 0
\(415\) 0 0
\(416\) −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
\(425\) 0 0
\(426\) 0 0
\(427\) −6.20198e6 −1.64612
\(428\) −4.43218e6 −1.16952
\(429\) 0 0
\(430\) 0 0
\(431\) 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 \(-0.880892\pi\)
−0.930804 + 0.365518i \(0.880892\pi\)
\(434\) 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 \(0.733675\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 959816. 0.233686
\(443\) −6.51524e6 −1.57733 −0.788663 0.614826i \(-0.789226\pi\)
−0.788663 + 0.614826i \(0.789226\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −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
\(455\) 0 0
\(456\) 0 0
\(457\) −1.22084e6 −0.273444 −0.136722 0.990609i \(-0.543657\pi\)
−0.136722 + 0.990609i \(0.543657\pi\)
\(458\) −1.19202e6 −0.265534
\(459\) 0 0
\(460\) 0 0
\(461\) 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 \(-0.570371\pi\)
−0.219280 + 0.975662i \(0.570371\pi\)
\(464\) 2.23696e6 0.482351
\(465\) 0 0
\(466\) −970668. −0.207065
\(467\) 3.25097e6 0.689797 0.344898 0.938640i \(-0.387913\pi\)
0.344898 + 0.938640i \(0.387913\pi\)
\(468\) 0 0
\(469\) 1.17066e7 2.45753
\(470\) 0 0
\(471\) 0 0
\(472\) −2.40240e6 −0.496353
\(473\) 184112. 0.0378381
\(474\) 0 0
\(475\) 0 0
\(476\) −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
\(485\) 0 0
\(486\) 0 0
\(487\) 8.53197e6 1.63015 0.815074 0.579357i \(-0.196696\pi\)
0.815074 + 0.579357i \(0.196696\pi\)
\(488\) −3.87624e6 −0.736819
\(489\) 0 0
\(490\) 0 0
\(491\) −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 \(0.698343\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.28750e6 0.582245
\(503\) 8.61770e6 1.51870 0.759349 0.650684i \(-0.225518\pi\)
0.759349 + 0.650684i \(0.225518\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 0 0
\(517\) −1.78902e6 −0.294367
\(518\) 69888.0 0.0114440
\(519\) 0 0
\(520\) 0 0
\(521\) −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 \(-0.314213\pi\)
0.551087 + 0.834448i \(0.314213\pi\)
\(524\) 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
\(530\) 0 0
\(531\) 0 0
\(532\) 5.69856e6 0.872943
\(533\) −2.68783e6 −0.409811
\(534\) 0 0
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) −1.26544e7 −1.80831 −0.904157 0.427201i \(-0.859500\pi\)
−0.904157 + 0.427201i \(0.859500\pi\)
\(548\) 4.05770e6 0.577204
\(549\) 0 0
\(550\) 0 0
\(551\) 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 \(-0.660569\pi\)
−0.483319 + 0.875444i \(0.660569\pi\)
\(558\) 0 0
\(559\) −355784. −0.0481567
\(560\) 0 0
\(561\) 0 0
\(562\) 384396. 0.0513379
\(563\) 846636. 0.112571 0.0562854 0.998415i \(-0.482074\pi\)
0.0562854 + 0.998415i \(0.482074\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) 2.86080e6 0.357724 0.178862 0.983874i \(-0.442758\pi\)
0.178862 + 0.983874i \(0.442758\pi\)
\(578\) 2.79165e6 0.347570
\(579\) 0 0
\(580\) 0 0
\(581\) −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 \(-0.632274\pi\)
−0.403694 + 0.914894i \(0.632274\pi\)
\(588\) 0 0
\(589\) −2.59488e6 −0.308197
\(590\) 0 0
\(591\) 0 0
\(592\) −119392. −0.0140014
\(593\) −1.78609e6 −0.208578 −0.104289 0.994547i \(-0.533257\pi\)
−0.104289 + 0.994547i \(0.533257\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) −7.07999e6 −0.779940 −0.389970 0.920828i \(-0.627515\pi\)
−0.389970 + 0.920828i \(0.627515\pi\)
\(608\) 5.46112e6 0.599132
\(609\) 0 0
\(610\) 0 0
\(611\) 3.45717e6 0.374643
\(612\) 0 0
\(613\) −5.09609e6 −0.547754 −0.273877 0.961765i \(-0.588306\pi\)
−0.273877 + 0.961765i \(0.588306\pi\)
\(614\) 4.75502e6 0.509016
\(615\) 0 0
\(616\) 3.40992e6 0.362070
\(617\) −1.30003e7 −1.37480 −0.687400 0.726279i \(-0.741248\pi\)
−0.687400 + 0.726279i \(0.741248\pi\)
\(618\) 0 0
\(619\) 4.84406e6 0.508139 0.254070 0.967186i \(-0.418231\pi\)
0.254070 + 0.967186i \(0.418231\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 4.74610e6 0.491882
\(623\) 1.94630e7 2.00905
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 0 0
\(637\) −5.73630e6 −0.560123
\(638\) 1.00936e6 0.0981735
\(639\) 0 0
\(640\) 0 0
\(641\) −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 \(-0.187391\pi\)
0.831659 + 0.555287i \(0.187391\pi\)
\(644\) 1.59990e7 1.52012
\(645\) 0 0
\(646\) −3.55736e6 −0.335387
\(647\) −4.25469e6 −0.399583 −0.199792 0.979838i \(-0.564026\pi\)
−0.199792 + 0.979838i \(0.564026\pi\)
\(648\) 0 0
\(649\) 2.96296e6 0.276130
\(650\) 0 0
\(651\) 0 0
\(652\) −4.32779e6 −0.398701
\(653\) 3.01085e6 0.276316 0.138158 0.990410i \(-0.455882\pi\)
0.138158 + 0.990410i \(0.455882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 0 0
\(667\) 1.01482e7 0.883228
\(668\) −1.11068e7 −0.963049
\(669\) 0 0
\(670\) 0 0
\(671\) 4.78070e6 0.409907
\(672\) 0 0
\(673\) −5.77063e6 −0.491117 −0.245559 0.969382i \(-0.578971\pi\)
−0.245559 + 0.969382i \(0.578971\pi\)
\(674\) −4.80016e6 −0.407011
\(675\) 0 0
\(676\) 8.10592e6 0.682237
\(677\) 1.67197e7 1.40203 0.701014 0.713147i \(-0.252731\pi\)
0.701014 + 0.713147i \(0.252731\pi\)
\(678\) 0 0
\(679\) −2.28553e7 −1.90245
\(680\) 0 0
\(681\) 0 0
\(682\) −724608. −0.0596544
\(683\) 7.14532e6 0.586097 0.293049 0.956098i \(-0.405330\pi\)
0.293049 + 0.956098i \(0.405330\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −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
\(695\) 0 0
\(696\) 0 0
\(697\) −1.57698e7 −1.22955
\(698\) −4.29610e6 −0.333761
\(699\) 0 0
\(700\) 0 0
\(701\) 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 \(0.753487\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.21644e7 0.899271
\(713\) −7.28525e6 −0.536686
\(714\) 0 0
\(715\) 0 0
\(716\) −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
\(725\) 0 0
\(726\) 0 0
\(727\) 1.93264e7 1.35618 0.678088 0.734981i \(-0.262809\pi\)
0.678088 + 0.734981i \(0.262809\pi\)
\(728\) −6.58944e6 −0.460808
\(729\) 0 0
\(730\) 0 0
\(731\) −2.08743e6 −0.144484
\(732\) 0 0
\(733\) −5.26197e6 −0.361733 −0.180866 0.983508i \(-0.557890\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(734\) 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 \(0.0980548\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −9.15686e6 −0.610572
\(743\) 2.09863e7 1.39464 0.697321 0.716759i \(-0.254375\pi\)
0.697321 + 0.716759i \(0.254375\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 1.08257e7 0.686617 0.343309 0.939223i \(-0.388452\pi\)
0.343309 + 0.939223i \(0.388452\pi\)
\(758\) 6.31868e6 0.399442
\(759\) 0 0
\(760\) 0 0
\(761\) −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 \(0.659802\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.47577e6 0.0891199
\(773\) −2.44049e7 −1.46902 −0.734510 0.678598i \(-0.762588\pi\)
−0.734510 + 0.678598i \(0.762588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −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
\(785\) 0 0
\(786\) 0 0
\(787\) −3.37607e7 −1.94301 −0.971505 0.237019i \(-0.923830\pi\)
−0.971505 + 0.237019i \(0.923830\pi\)
\(788\) −1.27641e7 −0.732278
\(789\) 0 0
\(790\) 0 0
\(791\) −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 \(-0.289929\pi\)
0.613083 + 0.790019i \(0.289929\pi\)
\(798\) 0 0
\(799\) 2.02837e7 1.12403
\(800\) 0 0
\(801\) 0 0
\(802\) −8.09360e6 −0.444330
\(803\) 5.73855e6 0.314061
\(804\) 0 0
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 0 0
\(817\) 1.31864e6 0.0691148
\(818\) −5.42414e6 −0.283431
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.395014\pi\)
0.323877 + 0.946099i \(0.395014\pi\)
\(824\) −2.34048e6 −0.120084
\(825\) 0 0
\(826\) −7.68768e6 −0.392053
\(827\) −8.72355e6 −0.443537 −0.221768 0.975099i \(-0.571183\pi\)
−0.221768 + 0.975099i \(0.571183\pi\)
\(828\) 0 0
\(829\) −1.06178e7 −0.536597 −0.268299 0.963336i \(-0.586461\pi\)
−0.268299 + 0.963336i \(0.586461\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.05677e6 0.153093
\(833\) −3.36556e7 −1.68053
\(834\) 0 0
\(835\) 0 0
\(836\) −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
\(845\) 0 0
\(846\) 0 0
\(847\) 2.67162e7 1.27958
\(848\) 1.56430e7 0.747016
\(849\) 0 0
\(850\) 0 0
\(851\) −541632. −0.0256378
\(852\) 0 0
\(853\) 2.20186e7 1.03613 0.518067 0.855340i \(-0.326652\pi\)
0.518067 + 0.855340i \(0.326652\pi\)
\(854\) −1.24040e7 −0.581991
\(855\) 0 0
\(856\) −1.89950e7 −0.886045
\(857\) 3.16676e7 1.47287 0.736434 0.676510i \(-0.236508\pi\)
0.736434 + 0.676510i \(0.236508\pi\)
\(858\) 0 0
\(859\) 1.58064e7 0.730886 0.365443 0.930834i \(-0.380918\pi\)
0.365443 + 0.930834i \(0.380918\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.12410e6 0.372398
\(863\) −1.44287e7 −0.659476 −0.329738 0.944072i \(-0.606960\pi\)
−0.329738 + 0.944072i \(0.606960\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −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
\(875\) 0 0
\(876\) 0 0
\(877\) −247902. −0.0108838 −0.00544191 0.999985i \(-0.501732\pi\)
−0.00544191 + 0.999985i \(0.501732\pi\)
\(878\) −1.08206e7 −0.473711
\(879\) 0 0
\(880\) 0 0
\(881\) −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 \(-0.857984\pi\)
−0.902112 + 0.431503i \(0.857984\pi\)
\(884\) −1.34374e7 −0.578343
\(885\) 0 0
\(886\) −1.30305e7 −0.557669
\(887\) −2.10476e7 −0.898241 −0.449120 0.893471i \(-0.648263\pi\)
−0.449120 + 0.893471i \(0.648263\pi\)
\(888\) 0 0
\(889\) 1.35460e7 0.574852
\(890\) 0 0
\(891\) 0 0
\(892\) 3.14041e7 1.32152
\(893\) −1.28133e7 −0.537690
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) −7.48309e6 −0.302039 −0.151019 0.988531i \(-0.548256\pi\)
−0.151019 + 0.988531i \(0.548256\pi\)
\(908\) 653744. 0.0263144
\(909\) 0 0
\(910\) 0 0
\(911\) 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 \(0.607047\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 8.14420e6 0.315516
\(923\) −9.33733e6 −0.360760
\(924\) 0 0
\(925\) 0 0
\(926\) −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
\(935\) 0 0
\(936\) 0 0
\(937\) −1.06887e7 −0.397718 −0.198859 0.980028i \(-0.563724\pi\)
−0.198859 + 0.980028i \(0.563724\pi\)
\(938\) 2.34132e7 0.868870
\(939\) 0 0
\(940\) 0 0
\(941\) −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 \(-0.600199\pi\)
−0.309611 + 0.950863i \(0.600199\pi\)
\(948\) 0 0
\(949\) −1.10894e7 −0.399706
\(950\) 0 0
\(951\) 0 0
\(952\) −3.86611e7 −1.38255
\(953\) 2.22259e7 0.792735 0.396367 0.918092i \(-0.370271\pi\)
0.396367 + 0.918092i \(0.370271\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −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
\(965\) 0 0
\(966\) 0 0
\(967\) −2.41551e7 −0.830696 −0.415348 0.909663i \(-0.636340\pi\)
−0.415348 + 0.909663i \(0.636340\pi\)
\(968\) 1.66976e7 0.572752
\(969\) 0 0
\(970\) 0 0
\(971\) 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 \(-0.584533\pi\)
−0.262457 + 0.964944i \(0.584533\pi\)
\(978\) 0 0
\(979\) −1.50028e7 −0.500281
\(980\) 0 0
\(981\) 0 0
\(982\) −3.02530e6 −0.100113
\(983\) −1.63420e7 −0.539412 −0.269706 0.962943i \(-0.586927\pi\)
−0.269706 + 0.962943i \(0.586927\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) 0 0
\(997\) 1.29097e7 0.411320 0.205660 0.978624i \(-0.434066\pi\)
0.205660 + 0.978624i \(0.434066\pi\)
\(998\) −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 225.6.a.f.1.1 1
3.2 odd 2 25.6.a.a.1.1 1
5.2 odd 4 225.6.b.e.199.2 2
5.3 odd 4 225.6.b.e.199.1 2
5.4 even 2 45.6.a.b.1.1 1
12.11 even 2 400.6.a.g.1.1 1
15.2 even 4 25.6.b.a.24.1 2
15.8 even 4 25.6.b.a.24.2 2
15.14 odd 2 5.6.a.a.1.1 1
20.19 odd 2 720.6.a.a.1.1 1
60.23 odd 4 400.6.c.j.49.1 2
60.47 odd 4 400.6.c.j.49.2 2
60.59 even 2 80.6.a.e.1.1 1
105.104 even 2 245.6.a.b.1.1 1
120.29 odd 2 320.6.a.j.1.1 1
120.59 even 2 320.6.a.g.1.1 1
165.164 even 2 605.6.a.a.1.1 1
195.194 odd 2 845.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 1 15.14 odd 2
25.6.a.a.1.1 1 3.2 odd 2
25.6.b.a.24.1 2 15.2 even 4
25.6.b.a.24.2 2 15.8 even 4
45.6.a.b.1.1 1 5.4 even 2
80.6.a.e.1.1 1 60.59 even 2
225.6.a.f.1.1 1 1.1 even 1 trivial
225.6.b.e.199.1 2 5.3 odd 4
225.6.b.e.199.2 2 5.2 odd 4
245.6.a.b.1.1 1 105.104 even 2
320.6.a.g.1.1 1 120.59 even 2
320.6.a.j.1.1 1 120.29 odd 2
400.6.a.g.1.1 1 12.11 even 2
400.6.c.j.49.1 2 60.23 odd 4
400.6.c.j.49.2 2 60.47 odd 4
605.6.a.a.1.1 1 165.164 even 2
720.6.a.a.1.1 1 20.19 odd 2
845.6.a.b.1.1 1 195.194 odd 2