Properties

Label 25.6.a.a.1.1
Level $25$
Weight $6$
Character 25.1
Self dual yes
Analytic conductor $4.010$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.00959549532\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{3} -28.0000 q^{4} -8.00000 q^{6} -192.000 q^{7} +120.000 q^{8} -227.000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{3} -28.0000 q^{4} -8.00000 q^{6} -192.000 q^{7} +120.000 q^{8} -227.000 q^{9} -148.000 q^{11} -112.000 q^{12} -286.000 q^{13} +384.000 q^{14} +656.000 q^{16} +1678.00 q^{17} +454.000 q^{18} +1060.00 q^{19} -768.000 q^{21} +296.000 q^{22} -2976.00 q^{23} +480.000 q^{24} +572.000 q^{26} -1880.00 q^{27} +5376.00 q^{28} -3410.00 q^{29} -2448.00 q^{31} -5152.00 q^{32} -592.000 q^{33} -3356.00 q^{34} +6356.00 q^{36} -182.000 q^{37} -2120.00 q^{38} -1144.00 q^{39} -9398.00 q^{41} +1536.00 q^{42} +1244.00 q^{43} +4144.00 q^{44} +5952.00 q^{46} +12088.0 q^{47} +2624.00 q^{48} +20057.0 q^{49} +6712.00 q^{51} +8008.00 q^{52} -23846.0 q^{53} +3760.00 q^{54} -23040.0 q^{56} +4240.00 q^{57} +6820.00 q^{58} -20020.0 q^{59} +32302.0 q^{61} +4896.00 q^{62} +43584.0 q^{63} -10688.0 q^{64} +1184.00 q^{66} -60972.0 q^{67} -46984.0 q^{68} -11904.0 q^{69} -32648.0 q^{71} -27240.0 q^{72} +38774.0 q^{73} +364.000 q^{74} -29680.0 q^{76} +28416.0 q^{77} +2288.00 q^{78} -33360.0 q^{79} +47641.0 q^{81} +18796.0 q^{82} -16716.0 q^{83} +21504.0 q^{84} -2488.00 q^{86} -13640.0 q^{87} -17760.0 q^{88} +101370. q^{89} +54912.0 q^{91} +83328.0 q^{92} -9792.00 q^{93} -24176.0 q^{94} -20608.0 q^{96} +119038. q^{97} -40114.0 q^{98} +33596.0 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 4.00000 0.256600 0.128300 0.991735i \(-0.459048\pi\)
0.128300 + 0.991735i \(0.459048\pi\)
\(4\) −28.0000 −0.875000
\(5\) 0 0
\(6\) −8.00000 −0.0907218
\(7\) −192.000 −1.48100 −0.740502 0.672054i \(-0.765412\pi\)
−0.740502 + 0.672054i \(0.765412\pi\)
\(8\) 120.000 0.662913
\(9\) −227.000 −0.934156
\(10\) 0 0
\(11\) −148.000 −0.368791 −0.184395 0.982852i \(-0.559033\pi\)
−0.184395 + 0.982852i \(0.559033\pi\)
\(12\) −112.000 −0.224525
\(13\) −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 \(-0.251347\pi\)
0.704109 + 0.710092i \(0.251347\pi\)
\(18\) 454.000 0.330274
\(19\) 1060.00 0.673631 0.336815 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(20\) 0 0
\(21\) −768.000 −0.380026
\(22\) 296.000 0.130387
\(23\) −2976.00 −1.17304 −0.586521 0.809934i \(-0.699503\pi\)
−0.586521 + 0.809934i \(0.699503\pi\)
\(24\) 480.000 0.170103
\(25\) 0 0
\(26\) 572.000 0.165944
\(27\) −1880.00 −0.496305
\(28\) 5376.00 1.29588
\(29\) −3410.00 −0.752938 −0.376469 0.926429i \(-0.622862\pi\)
−0.376469 + 0.926429i \(0.622862\pi\)
\(30\) 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
\(33\) −592.000 −0.0946317
\(34\) −3356.00 −0.497880
\(35\) 0 0
\(36\) 6356.00 0.817387
\(37\) −182.000 −0.0218558 −0.0109279 0.999940i \(-0.503479\pi\)
−0.0109279 + 0.999940i \(0.503479\pi\)
\(38\) −2120.00 −0.238164
\(39\) −1144.00 −0.120438
\(40\) 0 0
\(41\) −9398.00 −0.873124 −0.436562 0.899674i \(-0.643804\pi\)
−0.436562 + 0.899674i \(0.643804\pi\)
\(42\) 1536.00 0.134359
\(43\) 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 \(-0.369323\pi\)
0.399098 + 0.916908i \(0.369323\pi\)
\(48\) 2624.00 0.164384
\(49\) 20057.0 1.19337
\(50\) 0 0
\(51\) 6712.00 0.361349
\(52\) 8008.00 0.410691
\(53\) −23846.0 −1.16607 −0.583037 0.812446i \(-0.698136\pi\)
−0.583037 + 0.812446i \(0.698136\pi\)
\(54\) 3760.00 0.175470
\(55\) 0 0
\(56\) −23040.0 −0.981776
\(57\) 4240.00 0.172854
\(58\) 6820.00 0.266204
\(59\) −20020.0 −0.748745 −0.374373 0.927278i \(-0.622142\pi\)
−0.374373 + 0.927278i \(0.622142\pi\)
\(60\) 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
\(63\) 43584.0 1.38349
\(64\) −10688.0 −0.326172
\(65\) 0 0
\(66\) 1184.00 0.0334574
\(67\) −60972.0 −1.65937 −0.829685 0.558231i \(-0.811480\pi\)
−0.829685 + 0.558231i \(0.811480\pi\)
\(68\) −46984.0 −1.23219
\(69\) −11904.0 −0.301003
\(70\) 0 0
\(71\) −32648.0 −0.768618 −0.384309 0.923204i \(-0.625560\pi\)
−0.384309 + 0.923204i \(0.625560\pi\)
\(72\) −27240.0 −0.619264
\(73\) 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
\(78\) 2288.00 0.0425814
\(79\) −33360.0 −0.601393 −0.300696 0.953720i \(-0.597219\pi\)
−0.300696 + 0.953720i \(0.597219\pi\)
\(80\) 0 0
\(81\) 47641.0 0.806805
\(82\) 18796.0 0.308696
\(83\) −16716.0 −0.266340 −0.133170 0.991093i \(-0.542516\pi\)
−0.133170 + 0.991093i \(0.542516\pi\)
\(84\) 21504.0 0.332522
\(85\) 0 0
\(86\) −2488.00 −0.0362747
\(87\) −13640.0 −0.193204
\(88\) −17760.0 −0.244476
\(89\) 101370. 1.35655 0.678273 0.734810i \(-0.262729\pi\)
0.678273 + 0.734810i \(0.262729\pi\)
\(90\) 0 0
\(91\) 54912.0 0.695126
\(92\) 83328.0 1.02641
\(93\) −9792.00 −0.117399
\(94\) −24176.0 −0.282205
\(95\) 0 0
\(96\) −20608.0 −0.228222
\(97\) 119038. 1.28457 0.642283 0.766468i \(-0.277987\pi\)
0.642283 + 0.766468i \(0.277987\pi\)
\(98\) −40114.0 −0.421921
\(99\) 33596.0 0.344508
\(100\) 0 0
\(101\) −89898.0 −0.876893 −0.438446 0.898757i \(-0.644471\pi\)
−0.438446 + 0.898757i \(0.644471\pi\)
\(102\) −13424.0 −0.127756
\(103\) 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 \(-0.732976\pi\)
−0.668297 + 0.743895i \(0.732976\pi\)
\(108\) 52640.0 0.434267
\(109\) 36830.0 0.296917 0.148459 0.988919i \(-0.452569\pi\)
0.148459 + 0.988919i \(0.452569\pi\)
\(110\) 0 0
\(111\) −728.000 −0.00560821
\(112\) −125952. −0.948768
\(113\) −11186.0 −0.0824098 −0.0412049 0.999151i \(-0.513120\pi\)
−0.0412049 + 0.999151i \(0.513120\pi\)
\(114\) −8480.00 −0.0611130
\(115\) 0 0
\(116\) 95480.0 0.658821
\(117\) 64922.0 0.438457
\(118\) 40040.0 0.264721
\(119\) −322176. −2.08557
\(120\) 0 0
\(121\) −139147. −0.863993
\(122\) −64604.0 −0.392970
\(123\) −37592.0 −0.224044
\(124\) 68544.0 0.400327
\(125\) 0 0
\(126\) −87168.0 −0.489137
\(127\) −70552.0 −0.388150 −0.194075 0.980987i \(-0.562171\pi\)
−0.194075 + 0.980987i \(0.562171\pi\)
\(128\) 186240. 1.00473
\(129\) 4976.00 0.0263273
\(130\) 0 0
\(131\) 76452.0 0.389234 0.194617 0.980879i \(-0.437654\pi\)
0.194617 + 0.980879i \(0.437654\pi\)
\(132\) 16576.0 0.0828028
\(133\) −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 \(0.393008\pi\)
\(138\) 23808.0 0.106420
\(139\) 112220. 0.492644 0.246322 0.969188i \(-0.420778\pi\)
0.246322 + 0.969188i \(0.420778\pi\)
\(140\) 0 0
\(141\) 48352.0 0.204817
\(142\) 65296.0 0.271748
\(143\) 42328.0 0.173096
\(144\) −148912. −0.598444
\(145\) 0 0
\(146\) −77548.0 −0.301085
\(147\) 80228.0 0.306219
\(148\) 5096.00 0.0191238
\(149\) 403750. 1.48986 0.744932 0.667140i \(-0.232482\pi\)
0.744932 + 0.667140i \(0.232482\pi\)
\(150\) 0 0
\(151\) −446648. −1.59413 −0.797064 0.603895i \(-0.793615\pi\)
−0.797064 + 0.603895i \(0.793615\pi\)
\(152\) 127200. 0.446558
\(153\) −380906. −1.31550
\(154\) −56832.0 −0.193104
\(155\) 0 0
\(156\) 32032.0 0.105383
\(157\) 262258. 0.849141 0.424570 0.905395i \(-0.360425\pi\)
0.424570 + 0.905395i \(0.360425\pi\)
\(158\) 66720.0 0.212625
\(159\) −95384.0 −0.299215
\(160\) 0 0
\(161\) 571392. 1.73728
\(162\) −95282.0 −0.285248
\(163\) 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 \(-0.685492\pi\)
−0.550314 + 0.834958i \(0.685492\pi\)
\(168\) −92160.0 −0.251924
\(169\) −289497. −0.779700
\(170\) 0 0
\(171\) −240620. −0.629276
\(172\) −34832.0 −0.0897754
\(173\) 573474. 1.45680 0.728398 0.685155i \(-0.240265\pi\)
0.728398 + 0.685155i \(0.240265\pi\)
\(174\) 27280.0 0.0683079
\(175\) 0 0
\(176\) −97088.0 −0.236257
\(177\) −80080.0 −0.192128
\(178\) −202740. −0.479611
\(179\) −594460. −1.38672 −0.693362 0.720589i \(-0.743871\pi\)
−0.693362 + 0.720589i \(0.743871\pi\)
\(180\) 0 0
\(181\) −107098. −0.242988 −0.121494 0.992592i \(-0.538769\pi\)
−0.121494 + 0.992592i \(0.538769\pi\)
\(182\) −109824. −0.245764
\(183\) 129208. 0.285208
\(184\) −357120. −0.777624
\(185\) 0 0
\(186\) 19584.0 0.0415068
\(187\) −248344. −0.519337
\(188\) −338464. −0.698422
\(189\) 360960. 0.735029
\(190\) 0 0
\(191\) 469552. 0.931323 0.465661 0.884963i \(-0.345816\pi\)
0.465661 + 0.884963i \(0.345816\pi\)
\(192\) −42752.0 −0.0836957
\(193\) −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 \(-0.637425\pi\)
−0.418444 + 0.908242i \(0.637425\pi\)
\(198\) −67192.0 −0.121802
\(199\) 865000. 1.54840 0.774200 0.632940i \(-0.218152\pi\)
0.774200 + 0.632940i \(0.218152\pi\)
\(200\) 0 0
\(201\) −243888. −0.425795
\(202\) 179796. 0.310028
\(203\) 654720. 1.11510
\(204\) −187936. −0.316180
\(205\) 0 0
\(206\) −39008.0 −0.0640451
\(207\) 675552. 1.09580
\(208\) −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
\(213\) −130592. −0.197228
\(214\) 316584. 0.472557
\(215\) 0 0
\(216\) −225600. −0.329007
\(217\) 470016. 0.677584
\(218\) −73660.0 −0.104976
\(219\) 155096. 0.218520
\(220\) 0 0
\(221\) −479908. −0.660963
\(222\) 1456.00 0.00198280
\(223\) −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 \(-0.495213\pi\)
0.0150368 + 0.999887i \(0.495213\pi\)
\(228\) −118720. −0.151247
\(229\) −596010. −0.751043 −0.375522 0.926814i \(-0.622536\pi\)
−0.375522 + 0.926814i \(0.622536\pi\)
\(230\) 0 0
\(231\) 113664. 0.140150
\(232\) −409200. −0.499132
\(233\) 485334. 0.585667 0.292834 0.956163i \(-0.405402\pi\)
0.292834 + 0.956163i \(0.405402\pi\)
\(234\) −129844. −0.155018
\(235\) 0 0
\(236\) 560560. 0.655152
\(237\) −133440. −0.154317
\(238\) 644352. 0.737362
\(239\) −48880.0 −0.0553524 −0.0276762 0.999617i \(-0.508811\pi\)
−0.0276762 + 0.999617i \(0.508811\pi\)
\(240\) 0 0
\(241\) −110798. −0.122882 −0.0614411 0.998111i \(-0.519570\pi\)
−0.0614411 + 0.998111i \(0.519570\pi\)
\(242\) 278294. 0.305468
\(243\) 647404. 0.703331
\(244\) −904456. −0.972552
\(245\) 0 0
\(246\) 75184.0 0.0792114
\(247\) −303160. −0.316176
\(248\) −293760. −0.303294
\(249\) −66864.0 −0.0683430
\(250\) 0 0
\(251\) −1.64375e6 −1.64684 −0.823419 0.567434i \(-0.807936\pi\)
−0.823419 + 0.567434i \(0.807936\pi\)
\(252\) −1.22035e6 −1.21055
\(253\) 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 \(0.711581\pi\)
\(258\) −9952.00 −0.00930810
\(259\) 34944.0 0.0323685
\(260\) 0 0
\(261\) 774070. 0.703362
\(262\) −152904. −0.137615
\(263\) −2.12834e6 −1.89736 −0.948682 0.316231i \(-0.897583\pi\)
−0.948682 + 0.316231i \(0.897583\pi\)
\(264\) −71040.0 −0.0627326
\(265\) 0 0
\(266\) 407040. 0.352722
\(267\) 405480. 0.348090
\(268\) 1.70722e6 1.45195
\(269\) −1.44109e6 −1.21426 −0.607128 0.794604i \(-0.707679\pi\)
−0.607128 + 0.794604i \(0.707679\pi\)
\(270\) 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
\(273\) 219648. 0.178370
\(274\) −289836. −0.233225
\(275\) 0 0
\(276\) 333312. 0.263377
\(277\) 110298. 0.0863711 0.0431855 0.999067i \(-0.486249\pi\)
0.0431855 + 0.999067i \(0.486249\pi\)
\(278\) −224440. −0.174176
\(279\) 555696. 0.427392
\(280\) 0 0
\(281\) −192198. −0.145205 −0.0726027 0.997361i \(-0.523131\pi\)
−0.0726027 + 0.997361i \(0.523131\pi\)
\(282\) −96704.0 −0.0724139
\(283\) 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
\(288\) 1.16950e6 0.830846
\(289\) 1.39583e6 0.983076
\(290\) 0 0
\(291\) 476152. 0.329620
\(292\) −1.08567e6 −0.745146
\(293\) −2.19481e6 −1.49358 −0.746788 0.665063i \(-0.768405\pi\)
−0.746788 + 0.665063i \(0.768405\pi\)
\(294\) −160456. −0.108265
\(295\) 0 0
\(296\) −21840.0 −0.0144885
\(297\) 278240. 0.183033
\(298\) −807500. −0.526747
\(299\) 851136. 0.550581
\(300\) 0 0
\(301\) −238848. −0.151952
\(302\) 893296. 0.563609
\(303\) −359592. −0.225011
\(304\) 695360. 0.431545
\(305\) 0 0
\(306\) 761812. 0.465098
\(307\) 2.37751e6 1.43971 0.719857 0.694123i \(-0.244207\pi\)
0.719857 + 0.694123i \(0.244207\pi\)
\(308\) −795648. −0.477908
\(309\) 78016.0 0.0464823
\(310\) 0 0
\(311\) −2.37305e6 −1.39125 −0.695626 0.718405i \(-0.744873\pi\)
−0.695626 + 0.718405i \(0.744873\pi\)
\(312\) −137280. −0.0798400
\(313\) 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 \(-0.702353\pi\)
−0.593750 + 0.804650i \(0.702353\pi\)
\(318\) 190768. 0.105788
\(319\) 504680. 0.277677
\(320\) 0 0
\(321\) −633168. −0.342970
\(322\) −1.14278e6 −0.614221
\(323\) 1.77868e6 0.948618
\(324\) −1.33395e6 −0.705954
\(325\) 0 0
\(326\) −309128. −0.161100
\(327\) 147320. 0.0761890
\(328\) −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
\(333\) 41314.0 0.0204168
\(334\) 793344. 0.389131
\(335\) 0 0
\(336\) −503808. −0.243454
\(337\) −2.40008e6 −1.15120 −0.575601 0.817731i \(-0.695232\pi\)
−0.575601 + 0.817731i \(0.695232\pi\)
\(338\) 578994. 0.275665
\(339\) −44744.0 −0.0211464
\(340\) 0 0
\(341\) 362304. 0.168728
\(342\) 481240. 0.222483
\(343\) −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 \(0.629678\pi\)
\(348\) 381920. 0.169054
\(349\) −2.14805e6 −0.944019 −0.472010 0.881593i \(-0.656471\pi\)
−0.472010 + 0.881593i \(0.656471\pi\)
\(350\) 0 0
\(351\) 537680. 0.232946
\(352\) 762496. 0.328005
\(353\) 661854. 0.282700 0.141350 0.989960i \(-0.454856\pi\)
0.141350 + 0.989960i \(0.454856\pi\)
\(354\) 160160. 0.0679275
\(355\) 0 0
\(356\) −2.83836e6 −1.18698
\(357\) −1.28870e6 −0.535159
\(358\) 1.18892e6 0.490281
\(359\) −259320. −0.106194 −0.0530970 0.998589i \(-0.516909\pi\)
−0.0530970 + 0.998589i \(0.516909\pi\)
\(360\) 0 0
\(361\) −1.35250e6 −0.546222
\(362\) 214196. 0.0859093
\(363\) −556588. −0.221701
\(364\) −1.53754e6 −0.608236
\(365\) 0 0
\(366\) −258416. −0.100836
\(367\) 1.49993e6 0.581307 0.290653 0.956828i \(-0.406127\pi\)
0.290653 + 0.956828i \(0.406127\pi\)
\(368\) −1.95226e6 −0.751480
\(369\) 2.13335e6 0.815634
\(370\) 0 0
\(371\) 4.57843e6 1.72696
\(372\) 274176. 0.102724
\(373\) 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
\(378\) −721920. −0.259872
\(379\) 3.15934e6 1.12979 0.564896 0.825162i \(-0.308916\pi\)
0.564896 + 0.825162i \(0.308916\pi\)
\(380\) 0 0
\(381\) −282208. −0.0995994
\(382\) −939104. −0.329272
\(383\) −342216. −0.119207 −0.0596037 0.998222i \(-0.518984\pi\)
−0.0596037 + 0.998222i \(0.518984\pi\)
\(384\) 744960. 0.257813
\(385\) 0 0
\(386\) 105412. 0.0360099
\(387\) −282388. −0.0958449
\(388\) −3.33306e6 −1.12399
\(389\) 88470.0 0.0296430 0.0148215 0.999890i \(-0.495282\pi\)
0.0148215 + 0.999890i \(0.495282\pi\)
\(390\) 0 0
\(391\) −4.99373e6 −1.65190
\(392\) 2.40684e6 0.791101
\(393\) 305808. 0.0998775
\(394\) 911724. 0.295885
\(395\) 0 0
\(396\) −940688. −0.301445
\(397\) 5.45674e6 1.73763 0.868814 0.495138i \(-0.164883\pi\)
0.868814 + 0.495138i \(0.164883\pi\)
\(398\) −1.73000e6 −0.547442
\(399\) −814080. −0.255997
\(400\) 0 0
\(401\) 4.04680e6 1.25676 0.628378 0.777908i \(-0.283719\pi\)
0.628378 + 0.777908i \(0.283719\pi\)
\(402\) 487776. 0.150541
\(403\) 700128. 0.214741
\(404\) 2.51714e6 0.767281
\(405\) 0 0
\(406\) −1.30944e6 −0.394249
\(407\) 26936.0 0.00806022
\(408\) 805440. 0.239543
\(409\) −2.71207e6 −0.801664 −0.400832 0.916151i \(-0.631279\pi\)
−0.400832 + 0.916151i \(0.631279\pi\)
\(410\) 0 0
\(411\) 579672. 0.169269
\(412\) −546112. −0.158503
\(413\) 3.84384e6 1.10889
\(414\) −1.35110e6 −0.387425
\(415\) 0 0
\(416\) 1.47347e6 0.417454
\(417\) 448880. 0.126413
\(418\) 313760. 0.0878328
\(419\) 3.71746e6 1.03445 0.517227 0.855848i \(-0.326964\pi\)
0.517227 + 0.855848i \(0.326964\pi\)
\(420\) 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
\(423\) −2.74398e6 −0.745640
\(424\) −2.86152e6 −0.773005
\(425\) 0 0
\(426\) 261184. 0.0697305
\(427\) −6.20198e6 −1.64612
\(428\) 4.43218e6 1.16952
\(429\) 169312. 0.0444165
\(430\) 0 0
\(431\) −4.06205e6 −1.05330 −0.526650 0.850082i \(-0.676552\pi\)
−0.526650 + 0.850082i \(0.676552\pi\)
\(432\) −1.23328e6 −0.317945
\(433\) −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
\(438\) −310192. −0.0772583
\(439\) −5.41028e6 −1.33986 −0.669928 0.742426i \(-0.733675\pi\)
−0.669928 + 0.742426i \(0.733675\pi\)
\(440\) 0 0
\(441\) −4.55294e6 −1.11480
\(442\) 959816. 0.233686
\(443\) 6.51524e6 1.57733 0.788663 0.614826i \(-0.210774\pi\)
0.788663 + 0.614826i \(0.210774\pi\)
\(444\) 20384.0 0.00490718
\(445\) 0 0
\(446\) 2.24315e6 0.533976
\(447\) 1.61500e6 0.382299
\(448\) 2.05210e6 0.483062
\(449\) −509950. −0.119375 −0.0596873 0.998217i \(-0.519010\pi\)
−0.0596873 + 0.998217i \(0.519010\pi\)
\(450\) 0 0
\(451\) 1.39090e6 0.322000
\(452\) 313208. 0.0721085
\(453\) −1.78659e6 −0.409053
\(454\) −46696.0 −0.0106326
\(455\) 0 0
\(456\) 508800. 0.114587
\(457\) −1.22084e6 −0.273444 −0.136722 0.990609i \(-0.543657\pi\)
−0.136722 + 0.990609i \(0.543657\pi\)
\(458\) 1.19202e6 0.265534
\(459\) −3.15464e6 −0.698905
\(460\) 0 0
\(461\) −4.07210e6 −0.892413 −0.446207 0.894930i \(-0.647225\pi\)
−0.446207 + 0.894930i \(0.647225\pi\)
\(462\) −227328. −0.0495505
\(463\) −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 \(-0.612087\pi\)
−0.344898 + 0.938640i \(0.612087\pi\)
\(468\) −1.81782e6 −0.383650
\(469\) 1.17066e7 2.45753
\(470\) 0 0
\(471\) 1.04903e6 0.217890
\(472\) −2.40240e6 −0.496353
\(473\) −184112. −0.0378381
\(474\) 266880. 0.0545595
\(475\) 0 0
\(476\) 9.02093e6 1.82488
\(477\) 5.41304e6 1.08929
\(478\) 97760.0 0.0195700
\(479\) −3.27936e6 −0.653056 −0.326528 0.945188i \(-0.605879\pi\)
−0.326528 + 0.945188i \(0.605879\pi\)
\(480\) 0 0
\(481\) 52052.0 0.0102583
\(482\) 221596. 0.0434455
\(483\) 2.28557e6 0.445786
\(484\) 3.89612e6 0.755994
\(485\) 0 0
\(486\) −1.29481e6 −0.248665
\(487\) 8.53197e6 1.63015 0.815074 0.579357i \(-0.196696\pi\)
0.815074 + 0.579357i \(0.196696\pi\)
\(488\) 3.87624e6 0.736819
\(489\) 618256. 0.116922
\(490\) 0 0
\(491\) 1.51265e6 0.283162 0.141581 0.989927i \(-0.454781\pi\)
0.141581 + 0.989927i \(0.454781\pi\)
\(492\) 1.05258e6 0.196038
\(493\) −5.72198e6 −1.06030
\(494\) 606320. 0.111785
\(495\) 0 0
\(496\) −1.60589e6 −0.293097
\(497\) 6.26842e6 1.13833
\(498\) 133728. 0.0241629
\(499\) −6.49190e6 −1.16713 −0.583567 0.812065i \(-0.698343\pi\)
−0.583567 + 0.812065i \(0.698343\pi\)
\(500\) 0 0
\(501\) −1.58669e6 −0.282421
\(502\) 3.28750e6 0.582245
\(503\) −8.61770e6 −1.51870 −0.759349 0.650684i \(-0.774482\pi\)
−0.759349 + 0.650684i \(0.774482\pi\)
\(504\) 5.23008e6 0.917132
\(505\) 0 0
\(506\) −880896. −0.152950
\(507\) −1.15799e6 −0.200071
\(508\) 1.97546e6 0.339632
\(509\) 2.67323e6 0.457343 0.228671 0.973504i \(-0.426562\pi\)
0.228671 + 0.973504i \(0.426562\pi\)
\(510\) 0 0
\(511\) −7.44461e6 −1.26122
\(512\) −5.89875e6 −0.994455
\(513\) −1.99280e6 −0.334326
\(514\) 2.61248e6 0.436160
\(515\) 0 0
\(516\) −139328. −0.0230364
\(517\) −1.78902e6 −0.294367
\(518\) −69888.0 −0.0114440
\(519\) 2.29390e6 0.373814
\(520\) 0 0
\(521\) 6.18500e6 0.998264 0.499132 0.866526i \(-0.333652\pi\)
0.499132 + 0.866526i \(0.333652\pi\)
\(522\) −1.54814e6 −0.248676
\(523\) 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
\(528\) −388352. −0.0606235
\(529\) 2.42023e6 0.376026
\(530\) 0 0
\(531\) 4.54454e6 0.699445
\(532\) 5.69856e6 0.872943
\(533\) 2.68783e6 0.409811
\(534\) −810960. −0.123068
\(535\) 0 0
\(536\) −7.31664e6 −1.10002
\(537\) −2.37784e6 −0.355834
\(538\) 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
\(543\) −428392. −0.0623508
\(544\) −8.64506e6 −1.25248
\(545\) 0 0
\(546\) −439296. −0.0630631
\(547\) −1.26544e7 −1.80831 −0.904157 0.427201i \(-0.859500\pi\)
−0.904157 + 0.427201i \(0.859500\pi\)
\(548\) −4.05770e6 −0.577204
\(549\) −7.33255e6 −1.03830
\(550\) 0 0
\(551\) −3.61460e6 −0.507202
\(552\) −1.42848e6 −0.199538
\(553\) 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 \(0.339431\pi\)
\(558\) −1.11139e6 −0.151106
\(559\) −355784. −0.0481567
\(560\) 0 0
\(561\) −993376. −0.133262
\(562\) 384396. 0.0513379
\(563\) −846636. −0.112571 −0.0562854 0.998415i \(-0.517926\pi\)
−0.0562854 + 0.998415i \(0.517926\pi\)
\(564\) −1.35386e6 −0.179215
\(565\) 0 0
\(566\) −663768. −0.0870914
\(567\) −9.14707e6 −1.19488
\(568\) −3.91776e6 −0.509527
\(569\) 4.96041e6 0.642299 0.321149 0.947029i \(-0.395931\pi\)
0.321149 + 0.947029i \(0.395931\pi\)
\(570\) 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
\(573\) 1.87821e6 0.238978
\(574\) −3.60883e6 −0.457180
\(575\) 0 0
\(576\) 2.42618e6 0.304696
\(577\) 2.86080e6 0.357724 0.178862 0.983874i \(-0.442758\pi\)
0.178862 + 0.983874i \(0.442758\pi\)
\(578\) −2.79165e6 −0.347570
\(579\) −210824. −0.0261351
\(580\) 0 0
\(581\) 3.20947e6 0.394451
\(582\) −952304. −0.116538
\(583\) 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 \(0.367726\pi\)
\(588\) −2.24638e6 −0.267942
\(589\) −2.59488e6 −0.308197
\(590\) 0 0
\(591\) −1.82345e6 −0.214746
\(592\) −119392. −0.0140014
\(593\) 1.78609e6 0.208578 0.104289 0.994547i \(-0.466743\pi\)
0.104289 + 0.994547i \(0.466743\pi\)
\(594\) −556480. −0.0647118
\(595\) 0 0
\(596\) −1.13050e7 −1.30363
\(597\) 3.46000e6 0.397320
\(598\) −1.70227e6 −0.194660
\(599\) 4.94620e6 0.563254 0.281627 0.959524i \(-0.409126\pi\)
0.281627 + 0.959524i \(0.409126\pi\)
\(600\) 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
\(603\) 1.38406e7 1.55011
\(604\) 1.25061e7 1.39486
\(605\) 0 0
\(606\) 719184. 0.0795533
\(607\) −7.07999e6 −0.779940 −0.389970 0.920828i \(-0.627515\pi\)
−0.389970 + 0.920828i \(0.627515\pi\)
\(608\) −5.46112e6 −0.599132
\(609\) 2.61888e6 0.286136
\(610\) 0 0
\(611\) −3.45717e6 −0.374643
\(612\) 1.06654e7 1.15106
\(613\) −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 \(-0.258752\pi\)
0.687400 + 0.726279i \(0.258752\pi\)
\(618\) −156032. −0.0164340
\(619\) 4.84406e6 0.508139 0.254070 0.967186i \(-0.418231\pi\)
0.254070 + 0.967186i \(0.418231\pi\)
\(620\) 0 0
\(621\) 5.59488e6 0.582186
\(622\) 4.74610e6 0.491882
\(623\) −1.94630e7 −2.00905
\(624\) −750464. −0.0771558
\(625\) 0 0
\(626\) −2.85883e6 −0.291576
\(627\) −627520. −0.0637468
\(628\) −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
\(633\) 4.42261e6 0.438702
\(634\) 4.24924e6 0.419845
\(635\) 0 0
\(636\) 2.67075e6 0.261813
\(637\) −5.73630e6 −0.560123
\(638\) −1.00936e6 −0.0981735
\(639\) 7.41110e6 0.718010
\(640\) 0 0
\(641\) 1.53280e6 0.147347 0.0736734 0.997282i \(-0.476528\pi\)
0.0736734 + 0.997282i \(0.476528\pi\)
\(642\) 1.26634e6 0.121258
\(643\) 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 \(-0.435974\pi\)
0.199792 + 0.979838i \(0.435974\pi\)
\(648\) 5.71692e6 0.534841
\(649\) 2.96296e6 0.276130
\(650\) 0 0
\(651\) 1.88006e6 0.173868
\(652\) −4.32779e6 −0.398701
\(653\) −3.01085e6 −0.276316 −0.138158 0.990410i \(-0.544118\pi\)
−0.138158 + 0.990410i \(0.544118\pi\)
\(654\) −294640. −0.0269369
\(655\) 0 0
\(656\) −6.16509e6 −0.559345
\(657\) −8.80170e6 −0.795524
\(658\) 4.64179e6 0.417947
\(659\) −8.11462e6 −0.727871 −0.363936 0.931424i \(-0.618567\pi\)
−0.363936 + 0.931424i \(0.618567\pi\)
\(660\) 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
\(663\) −1.91963e6 −0.169603
\(664\) −2.00592e6 −0.176560
\(665\) 0 0
\(666\) −82628.0 −0.00721841
\(667\) 1.01482e7 0.883228
\(668\) 1.11068e7 0.963049
\(669\) −4.48630e6 −0.387546
\(670\) 0 0
\(671\) −4.78070e6 −0.409907
\(672\) 3.95674e6 0.337998
\(673\) −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 \(-0.747269\pi\)
−0.701014 + 0.713147i \(0.747269\pi\)
\(678\) 89488.0 0.00747637
\(679\) −2.28553e7 −1.90245
\(680\) 0 0
\(681\) 93392.0 0.00771688
\(682\) −724608. −0.0596544
\(683\) −7.14532e6 −0.586097 −0.293049 0.956098i \(-0.594670\pi\)
−0.293049 + 0.956098i \(0.594670\pi\)
\(684\) 6.73736e6 0.550617
\(685\) 0 0
\(686\) 1.24800e6 0.101252
\(687\) −2.38404e6 −0.192718
\(688\) 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
\(693\) −6.45043e6 −0.510218
\(694\) 3.55482e6 0.280169
\(695\) 0 0
\(696\) −1.63680e6 −0.128077
\(697\) −1.57698e7 −1.22955
\(698\) 4.29610e6 0.333761
\(699\) 1.94134e6 0.150282
\(700\) 0 0
\(701\) −1.60141e7 −1.23086 −0.615428 0.788193i \(-0.711017\pi\)
−0.615428 + 0.788193i \(0.711017\pi\)
\(702\) −1.07536e6 −0.0823590
\(703\) −192920. −0.0147228
\(704\) 1.58182e6 0.120289
\(705\) 0 0
\(706\) −1.32371e6 −0.0999495
\(707\) 1.72604e7 1.29868
\(708\) 2.24224e6 0.168112
\(709\) −1.91354e7 −1.42962 −0.714811 0.699318i \(-0.753487\pi\)
−0.714811 + 0.699318i \(0.753487\pi\)
\(710\) 0 0
\(711\) 7.57272e6 0.561795
\(712\) 1.21644e7 0.899271
\(713\) 7.28525e6 0.536686
\(714\) 2.57741e6 0.189207
\(715\) 0 0
\(716\) 1.66449e7 1.21338
\(717\) −195520. −0.0142034
\(718\) 518640. 0.0375452
\(719\) 1.02934e7 0.742566 0.371283 0.928520i \(-0.378918\pi\)
0.371283 + 0.928520i \(0.378918\pi\)
\(720\) 0 0
\(721\) −3.74477e6 −0.268279
\(722\) 2.70500e6 0.193119
\(723\) −443192. −0.0315316
\(724\) 2.99874e6 0.212615
\(725\) 0 0
\(726\) 1.11318e6 0.0783831
\(727\) 1.93264e7 1.35618 0.678088 0.734981i \(-0.262809\pi\)
0.678088 + 0.734981i \(0.262809\pi\)
\(728\) 6.58944e6 0.460808
\(729\) −8.98715e6 −0.626330
\(730\) 0 0
\(731\) 2.08743e6 0.144484
\(732\) −3.61782e6 −0.249557
\(733\) −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
\(738\) −4.26669e6 −0.288370
\(739\) 2.82944e7 1.90585 0.952927 0.303199i \(-0.0980548\pi\)
0.952927 + 0.303199i \(0.0980548\pi\)
\(740\) 0 0
\(741\) −1.21264e6 −0.0811309
\(742\) −9.15686e6 −0.610572
\(743\) −2.09863e7 −1.39464 −0.697321 0.716759i \(-0.745625\pi\)
−0.697321 + 0.716759i \(0.745625\pi\)
\(744\) −1.17504e6 −0.0778252
\(745\) 0 0
\(746\) −4.47615e6 −0.294481
\(747\) 3.79453e6 0.248804
\(748\) 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
\(753\) −6.57499e6 −0.422579
\(754\) −1.95052e6 −0.124946
\(755\) 0 0
\(756\) −1.01069e7 −0.643151
\(757\) 1.08257e7 0.686617 0.343309 0.939223i \(-0.388452\pi\)
0.343309 + 0.939223i \(0.388452\pi\)
\(758\) −6.31868e6 −0.399442
\(759\) 1.76179e6 0.111007
\(760\) 0 0
\(761\) 1.90534e7 1.19264 0.596322 0.802745i \(-0.296628\pi\)
0.596322 + 0.802745i \(0.296628\pi\)
\(762\) 564416. 0.0352137
\(763\) −7.07136e6 −0.439736
\(764\) −1.31475e7 −0.814908
\(765\) 0 0
\(766\) 684432. 0.0421462
\(767\) 5.72572e6 0.351432
\(768\) −121856. −0.00745494
\(769\) −1.57826e7 −0.962415 −0.481208 0.876607i \(-0.659802\pi\)
−0.481208 + 0.876607i \(0.659802\pi\)
\(770\) 0 0
\(771\) −5.22497e6 −0.316554
\(772\) 1.47577e6 0.0891199
\(773\) 2.44049e7 1.46902 0.734510 0.678598i \(-0.237412\pi\)
0.734510 + 0.678598i \(0.237412\pi\)
\(774\) 564776. 0.0338863
\(775\) 0 0
\(776\) 1.42846e7 0.851555
\(777\) 139776. 0.00830577
\(778\) −176940. −0.0104804
\(779\) −9.96188e6 −0.588163
\(780\) 0 0
\(781\) 4.83190e6 0.283459
\(782\) 9.98746e6 0.584034
\(783\) 6.41080e6 0.373687
\(784\) 1.31574e7 0.764504
\(785\) 0 0
\(786\) −611616. −0.0353120
\(787\) −3.37607e7 −1.94301 −0.971505 0.237019i \(-0.923830\pi\)
−0.971505 + 0.237019i \(0.923830\pi\)
\(788\) 1.27641e7 0.732278
\(789\) −8.51334e6 −0.486864
\(790\) 0 0
\(791\) 2.14771e6 0.122049
\(792\) 4.03152e6 0.228379
\(793\) −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 \(0.710071\pi\)
\(798\) 1.62816e6 0.0905086
\(799\) 2.02837e7 1.12403
\(800\) 0 0
\(801\) −2.30110e7 −1.26723
\(802\) −8.09360e6 −0.444330
\(803\) −5.73855e6 −0.314061
\(804\) 6.82886e6 0.372570
\(805\) 0 0
\(806\) −1.40026e6 −0.0759224
\(807\) −5.76436e6 −0.311578
\(808\) −1.07878e7 −0.581303
\(809\) −2.93597e7 −1.57717 −0.788587 0.614923i \(-0.789187\pi\)
−0.788587 + 0.614923i \(0.789187\pi\)
\(810\) 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
\(813\) −372992. −0.0197912
\(814\) −53872.0 −0.00284972
\(815\) 0 0
\(816\) 4.40307e6 0.231489
\(817\) 1.31864e6 0.0691148
\(818\) 5.42414e6 0.283431
\(819\) −1.24650e7 −0.649357
\(820\) 0 0
\(821\) −2.71430e6 −0.140540 −0.0702699 0.997528i \(-0.522386\pi\)
−0.0702699 + 0.997528i \(0.522386\pi\)
\(822\) −1.15934e6 −0.0598457
\(823\) 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 \(-0.428817\pi\)
0.221768 + 0.975099i \(0.428817\pi\)
\(828\) −1.89155e7 −0.958829
\(829\) −1.06178e7 −0.536597 −0.268299 0.963336i \(-0.586461\pi\)
−0.268299 + 0.963336i \(0.586461\pi\)
\(830\) 0 0
\(831\) 441192. 0.0221628
\(832\) 3.05677e6 0.153093
\(833\) 3.36556e7 1.68053
\(834\) −897760. −0.0446936
\(835\) 0 0
\(836\) 4.39264e6 0.217375
\(837\) 4.60224e6 0.227068
\(838\) −7.43492e6 −0.365735
\(839\) 1.67765e7 0.822805 0.411403 0.911454i \(-0.365039\pi\)
0.411403 + 0.911454i \(0.365039\pi\)
\(840\) 0 0
\(841\) −8.88305e6 −0.433084
\(842\) −7.10500e6 −0.345370
\(843\) −768792. −0.0372597
\(844\) −3.09583e7 −1.49596
\(845\) 0 0
\(846\) 5.48795e6 0.263624
\(847\) 2.67162e7 1.27958
\(848\) −1.56430e7 −0.747016
\(849\) 1.32754e6 0.0632087
\(850\) 0 0
\(851\) 541632. 0.0256378
\(852\) 3.65658e6 0.172574
\(853\) 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 \(-0.763492\pi\)
−0.736434 + 0.676510i \(0.763492\pi\)
\(858\) −338624. −0.0157036
\(859\) 1.58064e7 0.730886 0.365443 0.930834i \(-0.380918\pi\)
0.365443 + 0.930834i \(0.380918\pi\)
\(860\) 0 0
\(861\) 7.21766e6 0.331809
\(862\) 8.12410e6 0.372398
\(863\) 1.44287e7 0.659476 0.329738 0.944072i \(-0.393040\pi\)
0.329738 + 0.944072i \(0.393040\pi\)
\(864\) 9.68576e6 0.441417
\(865\) 0 0
\(866\) 1.45257e7 0.658178
\(867\) 5.58331e6 0.252257
\(868\) −1.31604e7 −0.592886
\(869\) 4.93728e6 0.221788
\(870\) 0 0
\(871\) 1.74380e7 0.778845
\(872\) 4.41960e6 0.196830
\(873\) −2.70216e7 −1.19999
\(874\) 6.30912e6 0.279377
\(875\) 0 0
\(876\) −4.34269e6 −0.191205
\(877\) −247902. −0.0108838 −0.00544191 0.999985i \(-0.501732\pi\)
−0.00544191 + 0.999985i \(0.501732\pi\)
\(878\) 1.08206e7 0.473711
\(879\) −8.77922e6 −0.383252
\(880\) 0 0
\(881\) 4.10268e7 1.78085 0.890426 0.455128i \(-0.150406\pi\)
0.890426 + 0.455128i \(0.150406\pi\)
\(882\) 9.10588e6 0.394140
\(883\) −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 \(-0.351737\pi\)
0.449120 + 0.893471i \(0.351737\pi\)
\(888\) −87360.0 −0.00371775
\(889\) 1.35460e7 0.574852
\(890\) 0 0
\(891\) −7.05087e6 −0.297542
\(892\) 3.14041e7 1.32152
\(893\) 1.28133e7 0.537690
\(894\) −3.23000e6 −0.135163
\(895\) 0 0
\(896\) −3.57581e7 −1.48800
\(897\) 3.40454e6 0.141279
\(898\) 1.01990e6 0.0422053
\(899\) 8.34768e6 0.344482
\(900\) 0 0
\(901\) −4.00136e7 −1.64208
\(902\) −2.78181e6 −0.113844
\(903\) −955392. −0.0389908
\(904\) −1.34232e6 −0.0546305
\(905\) 0 0
\(906\) 3.57318e6 0.144622
\(907\) −7.48309e6 −0.302039 −0.151019 0.988531i \(-0.548256\pi\)
−0.151019 + 0.988531i \(0.548256\pi\)
\(908\) −653744. −0.0263144
\(909\) 2.04068e7 0.819155
\(910\) 0 0
\(911\) −6.63165e6 −0.264744 −0.132372 0.991200i \(-0.542259\pi\)
−0.132372 + 0.991200i \(0.542259\pi\)
\(912\) 2.78144e6 0.110734
\(913\) 2.47397e6 0.0982239
\(914\) 2.44168e6 0.0966772
\(915\) 0 0
\(916\) 1.66883e7 0.657163
\(917\) −1.46788e7 −0.576457
\(918\) 6.30928e6 0.247100
\(919\) −1.68976e7 −0.659990 −0.329995 0.943983i \(-0.607047\pi\)
−0.329995 + 0.943983i \(0.607047\pi\)
\(920\) 0 0
\(921\) 9.51003e6 0.369431
\(922\) 8.14420e6 0.315516
\(923\) 9.33733e6 0.360760
\(924\) −3.18259e6 −0.122631
\(925\) 0 0
\(926\) 4.04587e6 0.155055
\(927\) −4.42741e6 −0.169219
\(928\) 1.75683e7 0.669669
\(929\) −1.28653e7 −0.489081 −0.244541 0.969639i \(-0.578637\pi\)
−0.244541 + 0.969639i \(0.578637\pi\)
\(930\) 0 0
\(931\) 2.12604e7 0.803892
\(932\) −1.35894e7 −0.512459
\(933\) −9.49219e6 −0.356995
\(934\) 6.50194e6 0.243880
\(935\) 0 0
\(936\) 7.79064e6 0.290659
\(937\) −1.06887e7 −0.397718 −0.198859 0.980028i \(-0.563724\pi\)
−0.198859 + 0.980028i \(0.563724\pi\)
\(938\) −2.34132e7 −0.868870
\(939\) 5.71766e6 0.211619
\(940\) 0 0
\(941\) 2.82455e7 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(942\) −2.09806e6 −0.0770356
\(943\) 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 \(0.399801\pi\)
\(948\) 3.73632e6 0.135028
\(949\) −1.10894e7 −0.399706
\(950\) 0 0
\(951\) −8.49849e6 −0.304713
\(952\) −3.86611e7 −1.38255
\(953\) −2.22259e7 −0.792735 −0.396367 0.918092i \(-0.629729\pi\)
−0.396367 + 0.918092i \(0.629729\pi\)
\(954\) −1.08261e7 −0.385124
\(955\) 0 0
\(956\) 1.36864e6 0.0484333
\(957\) 2.01872e6 0.0712519
\(958\) 6.55872e6 0.230890
\(959\) −2.78243e7 −0.976961
\(960\) 0 0
\(961\) −2.26364e7 −0.790678
\(962\) −104104. −0.00362685
\(963\) 3.59323e7 1.24859
\(964\) 3.10234e6 0.107522
\(965\) 0 0
\(966\) −4.57114e6 −0.157609
\(967\) −2.41551e7 −0.830696 −0.415348 0.909663i \(-0.636340\pi\)
−0.415348 + 0.909663i \(0.636340\pi\)
\(968\) −1.66976e7 −0.572752
\(969\) 7.11472e6 0.243416
\(970\) 0 0
\(971\) −5.48313e7 −1.86630 −0.933149 0.359491i \(-0.882950\pi\)
−0.933149 + 0.359491i \(0.882950\pi\)
\(972\) −1.81273e7 −0.615415
\(973\) −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 \(0.415467\pi\)
\(978\) −1.23651e6 −0.0413382
\(979\) −1.50028e7 −0.500281
\(980\) 0 0
\(981\) −8.36041e6 −0.277367
\(982\) −3.02530e6 −0.100113
\(983\) 1.63420e7 0.539412 0.269706 0.962943i \(-0.413073\pi\)
0.269706 + 0.962943i \(0.413073\pi\)
\(984\) −4.51104e6 −0.148521
\(985\) 0 0
\(986\) 1.14440e7 0.374873
\(987\) −9.28358e6 −0.303335
\(988\) 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
\(993\) 1.23994e7 0.399050
\(994\) −1.25368e7 −0.402459
\(995\) 0 0
\(996\) 1.87219e6 0.0598001
\(997\) 1.29097e7 0.411320 0.205660 0.978624i \(-0.434066\pi\)
0.205660 + 0.978624i \(0.434066\pi\)
\(998\) 1.29838e7 0.412644
\(999\) 342160. 0.0108471
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 25.6.a.a.1.1 1
3.2 odd 2 225.6.a.f.1.1 1
4.3 odd 2 400.6.a.g.1.1 1
5.2 odd 4 25.6.b.a.24.1 2
5.3 odd 4 25.6.b.a.24.2 2
5.4 even 2 5.6.a.a.1.1 1
15.2 even 4 225.6.b.e.199.2 2
15.8 even 4 225.6.b.e.199.1 2
15.14 odd 2 45.6.a.b.1.1 1
20.3 even 4 400.6.c.j.49.1 2
20.7 even 4 400.6.c.j.49.2 2
20.19 odd 2 80.6.a.e.1.1 1
35.34 odd 2 245.6.a.b.1.1 1
40.19 odd 2 320.6.a.g.1.1 1
40.29 even 2 320.6.a.j.1.1 1
55.54 odd 2 605.6.a.a.1.1 1
60.59 even 2 720.6.a.a.1.1 1
65.64 even 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 5.4 even 2
25.6.a.a.1.1 1 1.1 even 1 trivial
25.6.b.a.24.1 2 5.2 odd 4
25.6.b.a.24.2 2 5.3 odd 4
45.6.a.b.1.1 1 15.14 odd 2
80.6.a.e.1.1 1 20.19 odd 2
225.6.a.f.1.1 1 3.2 odd 2
225.6.b.e.199.1 2 15.8 even 4
225.6.b.e.199.2 2 15.2 even 4
245.6.a.b.1.1 1 35.34 odd 2
320.6.a.g.1.1 1 40.19 odd 2
320.6.a.j.1.1 1 40.29 even 2
400.6.a.g.1.1 1 4.3 odd 2
400.6.c.j.49.1 2 20.3 even 4
400.6.c.j.49.2 2 20.7 even 4
605.6.a.a.1.1 1 55.54 odd 2
720.6.a.a.1.1 1 60.59 even 2
845.6.a.b.1.1 1 65.64 even 2