Properties

Label 5.6.a.a.1.1
Level $5$
Weight $6$
Character 5.1
Self dual yes
Analytic conductor $0.802$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.801919099065\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -4.00000 q^{3} -28.0000 q^{4} +25.0000 q^{5} -8.00000 q^{6} +192.000 q^{7} -120.000 q^{8} -227.000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -4.00000 q^{3} -28.0000 q^{4} +25.0000 q^{5} -8.00000 q^{6} +192.000 q^{7} -120.000 q^{8} -227.000 q^{9} +50.0000 q^{10} -148.000 q^{11} +112.000 q^{12} +286.000 q^{13} +384.000 q^{14} -100.000 q^{15} +656.000 q^{16} -1678.00 q^{17} -454.000 q^{18} +1060.00 q^{19} -700.000 q^{20} -768.000 q^{21} -296.000 q^{22} +2976.00 q^{23} +480.000 q^{24} +625.000 q^{25} +572.000 q^{26} +1880.00 q^{27} -5376.00 q^{28} -3410.00 q^{29} -200.000 q^{30} -2448.00 q^{31} +5152.00 q^{32} +592.000 q^{33} -3356.00 q^{34} +4800.00 q^{35} +6356.00 q^{36} +182.000 q^{37} +2120.00 q^{38} -1144.00 q^{39} -3000.00 q^{40} -9398.00 q^{41} -1536.00 q^{42} -1244.00 q^{43} +4144.00 q^{44} -5675.00 q^{45} +5952.00 q^{46} -12088.0 q^{47} -2624.00 q^{48} +20057.0 q^{49} +1250.00 q^{50} +6712.00 q^{51} -8008.00 q^{52} +23846.0 q^{53} +3760.00 q^{54} -3700.00 q^{55} -23040.0 q^{56} -4240.00 q^{57} -6820.00 q^{58} -20020.0 q^{59} +2800.00 q^{60} +32302.0 q^{61} -4896.00 q^{62} -43584.0 q^{63} -10688.0 q^{64} +7150.00 q^{65} +1184.00 q^{66} +60972.0 q^{67} +46984.0 q^{68} -11904.0 q^{69} +9600.00 q^{70} -32648.0 q^{71} +27240.0 q^{72} -38774.0 q^{73} +364.000 q^{74} -2500.00 q^{75} -29680.0 q^{76} -28416.0 q^{77} -2288.00 q^{78} -33360.0 q^{79} +16400.0 q^{80} +47641.0 q^{81} -18796.0 q^{82} +16716.0 q^{83} +21504.0 q^{84} -41950.0 q^{85} -2488.00 q^{86} +13640.0 q^{87} +17760.0 q^{88} +101370. q^{89} -11350.0 q^{90} +54912.0 q^{91} -83328.0 q^{92} +9792.00 q^{93} -24176.0 q^{94} +26500.0 q^{95} -20608.0 q^{96} -119038. q^{97} +40114.0 q^{98} +33596.0 q^{99} +O(q^{100})\)

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 \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) −4.00000 −0.256600 −0.128300 0.991735i \(-0.540952\pi\)
−0.128300 + 0.991735i \(0.540952\pi\)
\(4\) −28.0000 −0.875000
\(5\) 25.0000 0.447214
\(6\) −8.00000 −0.0907218
\(7\) 192.000 1.48100 0.740502 0.672054i \(-0.234588\pi\)
0.740502 + 0.672054i \(0.234588\pi\)
\(8\) −120.000 −0.662913
\(9\) −227.000 −0.934156
\(10\) 50.0000 0.158114
\(11\) −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 \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) 384.000 0.523614
\(15\) −100.000 −0.114755
\(16\) 656.000 0.640625
\(17\) −1678.00 −1.40822 −0.704109 0.710092i \(-0.748653\pi\)
−0.704109 + 0.710092i \(0.748653\pi\)
\(18\) −454.000 −0.330274
\(19\) 1060.00 0.673631 0.336815 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(20\) −700.000 −0.391312
\(21\) −768.000 −0.380026
\(22\) −296.000 −0.130387
\(23\) 2976.00 1.17304 0.586521 0.809934i \(-0.300497\pi\)
0.586521 + 0.809934i \(0.300497\pi\)
\(24\) 480.000 0.170103
\(25\) 625.000 0.200000
\(26\) 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 \(0.622862\pi\)
\(30\) −200.000 −0.0405720
\(31\) −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
\(35\) 4800.00 0.662325
\(36\) 6356.00 0.817387
\(37\) 182.000 0.0218558 0.0109279 0.999940i \(-0.496521\pi\)
0.0109279 + 0.999940i \(0.496521\pi\)
\(38\) 2120.00 0.238164
\(39\) −1144.00 −0.120438
\(40\) −3000.00 −0.296464
\(41\) −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 \(-0.516337\pi\)
−0.0513002 + 0.998683i \(0.516337\pi\)
\(44\) 4144.00 0.322692
\(45\) −5675.00 −0.417767
\(46\) 5952.00 0.414733
\(47\) −12088.0 −0.798196 −0.399098 0.916908i \(-0.630677\pi\)
−0.399098 + 0.916908i \(0.630677\pi\)
\(48\) −2624.00 −0.164384
\(49\) 20057.0 1.19337
\(50\) 1250.00 0.0707107
\(51\) 6712.00 0.361349
\(52\) −8008.00 −0.410691
\(53\) 23846.0 1.16607 0.583037 0.812446i \(-0.301864\pi\)
0.583037 + 0.812446i \(0.301864\pi\)
\(54\) 3760.00 0.175470
\(55\) −3700.00 −0.164928
\(56\) −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 \(0.622142\pi\)
\(60\) 2800.00 0.100411
\(61\) 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
\(65\) 7150.00 0.209905
\(66\) 1184.00 0.0334574
\(67\) 60972.0 1.65937 0.829685 0.558231i \(-0.188520\pi\)
0.829685 + 0.558231i \(0.188520\pi\)
\(68\) 46984.0 1.23219
\(69\) −11904.0 −0.301003
\(70\) 9600.00 0.234167
\(71\) −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 \(-0.640007\pi\)
−0.425798 + 0.904818i \(0.640007\pi\)
\(74\) 364.000 0.00772720
\(75\) −2500.00 −0.0513200
\(76\) −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 \(0.597219\pi\)
\(80\) 16400.0 0.286496
\(81\) 47641.0 0.806805
\(82\) −18796.0 −0.308696
\(83\) 16716.0 0.266340 0.133170 0.991093i \(-0.457484\pi\)
0.133170 + 0.991093i \(0.457484\pi\)
\(84\) 21504.0 0.332522
\(85\) −41950.0 −0.629774
\(86\) −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 \(0.262729\pi\)
\(90\) −11350.0 −0.147703
\(91\) 54912.0 0.695126
\(92\) −83328.0 −1.02641
\(93\) 9792.00 0.117399
\(94\) −24176.0 −0.282205
\(95\) 26500.0 0.301257
\(96\) −20608.0 −0.228222
\(97\) −119038. −1.28457 −0.642283 0.766468i \(-0.722013\pi\)
−0.642283 + 0.766468i \(0.722013\pi\)
\(98\) 40114.0 0.421921
\(99\) 33596.0 0.344508
\(100\) −17500.0 −0.175000
\(101\) −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 \(-0.528870\pi\)
−0.0905734 + 0.995890i \(0.528870\pi\)
\(104\) −34320.0 −0.311146
\(105\) −19200.0 −0.169953
\(106\) 47692.0 0.412269
\(107\) 158292. 1.33659 0.668297 0.743895i \(-0.267024\pi\)
0.668297 + 0.743895i \(0.267024\pi\)
\(108\) −52640.0 −0.434267
\(109\) 36830.0 0.296917 0.148459 0.988919i \(-0.452569\pi\)
0.148459 + 0.988919i \(0.452569\pi\)
\(110\) −7400.00 −0.0583109
\(111\) −728.000 −0.00560821
\(112\) 125952. 0.948768
\(113\) 11186.0 0.0824098 0.0412049 0.999151i \(-0.486880\pi\)
0.0412049 + 0.999151i \(0.486880\pi\)
\(114\) −8480.00 −0.0611130
\(115\) 74400.0 0.524600
\(116\) 95480.0 0.658821
\(117\) −64922.0 −0.438457
\(118\) −40040.0 −0.264721
\(119\) −322176. −2.08557
\(120\) 12000.0 0.0760726
\(121\) −139147. −0.863993
\(122\) 64604.0 0.392970
\(123\) 37592.0 0.224044
\(124\) 68544.0 0.400327
\(125\) 15625.0 0.0894427
\(126\) −87168.0 −0.489137
\(127\) 70552.0 0.388150 0.194075 0.980987i \(-0.437829\pi\)
0.194075 + 0.980987i \(0.437829\pi\)
\(128\) −186240. −1.00473
\(129\) 4976.00 0.0263273
\(130\) 14300.0 0.0742126
\(131\) 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
\(135\) 47000.0 0.221954
\(136\) 201360. 0.933525
\(137\) −144918. −0.659661 −0.329831 0.944040i \(-0.606992\pi\)
−0.329831 + 0.944040i \(0.606992\pi\)
\(138\) −23808.0 −0.106420
\(139\) 112220. 0.492644 0.246322 0.969188i \(-0.420778\pi\)
0.246322 + 0.969188i \(0.420778\pi\)
\(140\) −134400. −0.579534
\(141\) 48352.0 0.204817
\(142\) −65296.0 −0.271748
\(143\) −42328.0 −0.173096
\(144\) −148912. −0.598444
\(145\) −85250.0 −0.336724
\(146\) −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 \(0.232482\pi\)
\(150\) −5000.00 −0.0181444
\(151\) −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
\(155\) −61200.0 −0.204608
\(156\) 32032.0 0.105383
\(157\) −262258. −0.849141 −0.424570 0.905395i \(-0.639575\pi\)
−0.424570 + 0.905395i \(0.639575\pi\)
\(158\) −66720.0 −0.212625
\(159\) −95384.0 −0.299215
\(160\) 128800. 0.397755
\(161\) 571392. 1.73728
\(162\) 95282.0 0.285248
\(163\) −154564. −0.455658 −0.227829 0.973701i \(-0.573163\pi\)
−0.227829 + 0.973701i \(0.573163\pi\)
\(164\) 263144. 0.763983
\(165\) 14800.0 0.0423206
\(166\) 33432.0 0.0941656
\(167\) 396672. 1.10063 0.550314 0.834958i \(-0.314508\pi\)
0.550314 + 0.834958i \(0.314508\pi\)
\(168\) 92160.0 0.251924
\(169\) −289497. −0.779700
\(170\) −83900.0 −0.222659
\(171\) −240620. −0.629276
\(172\) 34832.0 0.0897754
\(173\) −573474. −1.45680 −0.728398 0.685155i \(-0.759735\pi\)
−0.728398 + 0.685155i \(0.759735\pi\)
\(174\) 27280.0 0.0683079
\(175\) 120000. 0.296201
\(176\) −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 \(0.743871\pi\)
\(180\) 158900. 0.365547
\(181\) −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
\(185\) 4550.00 0.00977422
\(186\) 19584.0 0.0415068
\(187\) 248344. 0.519337
\(188\) 338464. 0.698422
\(189\) 360960. 0.735029
\(190\) 53000.0 0.106510
\(191\) 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 \(-0.483783\pi\)
0.0509257 + 0.998702i \(0.483783\pi\)
\(194\) −238076. −0.454163
\(195\) −28600.0 −0.0538616
\(196\) −561596. −1.04420
\(197\) 455862. 0.836889 0.418444 0.908242i \(-0.362575\pi\)
0.418444 + 0.908242i \(0.362575\pi\)
\(198\) 67192.0 0.121802
\(199\) 865000. 1.54840 0.774200 0.632940i \(-0.218152\pi\)
0.774200 + 0.632940i \(0.218152\pi\)
\(200\) −75000.0 −0.132583
\(201\) −243888. −0.425795
\(202\) −179796. −0.310028
\(203\) −654720. −1.11510
\(204\) −187936. −0.316180
\(205\) −234950. −0.390473
\(206\) −39008.0 −0.0640451
\(207\) −675552. −1.09580
\(208\) 187616. 0.300685
\(209\) −156880. −0.248429
\(210\) −38400.0 −0.0600873
\(211\) 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
\(215\) −31100.0 −0.0458843
\(216\) −225600. −0.329007
\(217\) −470016. −0.677584
\(218\) 73660.0 0.104976
\(219\) 155096. 0.218520
\(220\) 103600. 0.144312
\(221\) −479908. −0.660963
\(222\) −1456.00 −0.00198280
\(223\) 1.12158e6 1.51031 0.755156 0.655545i \(-0.227561\pi\)
0.755156 + 0.655545i \(0.227561\pi\)
\(224\) 989184. 1.31722
\(225\) −141875. −0.186831
\(226\) 22372.0 0.0291363
\(227\) −23348.0 −0.0300736 −0.0150368 0.999887i \(-0.504787\pi\)
−0.0150368 + 0.999887i \(0.504787\pi\)
\(228\) 118720. 0.151247
\(229\) −596010. −0.751043 −0.375522 0.926814i \(-0.622536\pi\)
−0.375522 + 0.926814i \(0.622536\pi\)
\(230\) 148800. 0.185474
\(231\) 113664. 0.140150
\(232\) 409200. 0.499132
\(233\) −485334. −0.585667 −0.292834 0.956163i \(-0.594598\pi\)
−0.292834 + 0.956163i \(0.594598\pi\)
\(234\) −129844. −0.155018
\(235\) −302200. −0.356964
\(236\) 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 \(0.508811\pi\)
\(240\) −65600.0 −0.0735150
\(241\) −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
\(245\) 501425. 0.533692
\(246\) 75184.0 0.0792114
\(247\) 303160. 0.316176
\(248\) 293760. 0.303294
\(249\) −66864.0 −0.0683430
\(250\) 31250.0 0.0316228
\(251\) −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
\(255\) 167800. 0.161600
\(256\) −30464.0 −0.0290527
\(257\) 1.30624e6 1.23365 0.616823 0.787102i \(-0.288419\pi\)
0.616823 + 0.787102i \(0.288419\pi\)
\(258\) 9952.00 0.00930810
\(259\) 34944.0 0.0323685
\(260\) −200200. −0.183667
\(261\) 774070. 0.703362
\(262\) 152904. 0.137615
\(263\) 2.12834e6 1.89736 0.948682 0.316231i \(-0.102417\pi\)
0.948682 + 0.316231i \(0.102417\pi\)
\(264\) −71040.0 −0.0627326
\(265\) 596150. 0.521484
\(266\) 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 \(0.707679\pi\)
\(270\) 94000.0 0.0784727
\(271\) −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
\(275\) −92500.0 −0.0737581
\(276\) 333312. 0.263377
\(277\) −110298. −0.0863711 −0.0431855 0.999067i \(-0.513751\pi\)
−0.0431855 + 0.999067i \(0.513751\pi\)
\(278\) 224440. 0.174176
\(279\) 555696. 0.427392
\(280\) −576000. −0.439064
\(281\) −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 \(-0.539305\pi\)
−0.123166 + 0.992386i \(0.539305\pi\)
\(284\) 914144. 0.672541
\(285\) −106000. −0.0773025
\(286\) −84656.0 −0.0611988
\(287\) −1.80442e6 −1.29310
\(288\) −1.16950e6 −0.830846
\(289\) 1.39583e6 0.983076
\(290\) −170500. −0.119050
\(291\) 476152. 0.329620
\(292\) 1.08567e6 0.745146
\(293\) 2.19481e6 1.49358 0.746788 0.665063i \(-0.231595\pi\)
0.746788 + 0.665063i \(0.231595\pi\)
\(294\) −160456. −0.108265
\(295\) −500500. −0.334849
\(296\) −21840.0 −0.0144885
\(297\) −278240. −0.183033
\(298\) 807500. 0.526747
\(299\) 851136. 0.550581
\(300\) 70000.0 0.0449050
\(301\) −238848. −0.151952
\(302\) −893296. −0.563609
\(303\) 359592. 0.225011
\(304\) 695360. 0.431545
\(305\) 807550. 0.497073
\(306\) 761812. 0.465098
\(307\) −2.37751e6 −1.43971 −0.719857 0.694123i \(-0.755793\pi\)
−0.719857 + 0.694123i \(0.755793\pi\)
\(308\) 795648. 0.477908
\(309\) 78016.0 0.0464823
\(310\) −122400. −0.0723398
\(311\) −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 \(-0.635292\pi\)
−0.412351 + 0.911025i \(0.635292\pi\)
\(314\) −524516. −0.300217
\(315\) −1.08960e6 −0.618715
\(316\) 934080. 0.526219
\(317\) 2.12462e6 1.18750 0.593750 0.804650i \(-0.297647\pi\)
0.593750 + 0.804650i \(0.297647\pi\)
\(318\) −190768. −0.105788
\(319\) 504680. 0.277677
\(320\) −267200. −0.145868
\(321\) −633168. −0.342970
\(322\) 1.14278e6 0.614221
\(323\) −1.77868e6 −0.948618
\(324\) −1.33395e6 −0.705954
\(325\) 178750. 0.0938723
\(326\) −309128. −0.161100
\(327\) −147320. −0.0761890
\(328\) 1.12776e6 0.578805
\(329\) −2.32090e6 −1.18213
\(330\) 29600.0 0.0149626
\(331\) 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
\(335\) 1.52430e6 0.742093
\(336\) −503808. −0.243454
\(337\) 2.40008e6 1.15120 0.575601 0.817731i \(-0.304768\pi\)
0.575601 + 0.817731i \(0.304768\pi\)
\(338\) −578994. −0.275665
\(339\) −44744.0 −0.0211464
\(340\) 1.17460e6 0.551052
\(341\) 362304. 0.168728
\(342\) −481240. −0.222483
\(343\) 624000. 0.286384
\(344\) 149280. 0.0680151
\(345\) −297600. −0.134612
\(346\) −1.14695e6 −0.515055
\(347\) 1.77741e6 0.792436 0.396218 0.918156i \(-0.370322\pi\)
0.396218 + 0.918156i \(0.370322\pi\)
\(348\) −381920. −0.169054
\(349\) −2.14805e6 −0.944019 −0.472010 0.881593i \(-0.656471\pi\)
−0.472010 + 0.881593i \(0.656471\pi\)
\(350\) 240000. 0.104723
\(351\) 537680. 0.232946
\(352\) −762496. −0.328005
\(353\) −661854. −0.282700 −0.141350 0.989960i \(-0.545144\pi\)
−0.141350 + 0.989960i \(0.545144\pi\)
\(354\) 160160. 0.0679275
\(355\) −816200. −0.343737
\(356\) −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 \(0.516909\pi\)
\(360\) 681000. 0.276943
\(361\) −1.35250e6 −0.546222
\(362\) −214196. −0.0859093
\(363\) 556588. 0.221701
\(364\) −1.53754e6 −0.608236
\(365\) −969350. −0.380845
\(366\) −258416. −0.100836
\(367\) −1.49993e6 −0.581307 −0.290653 0.956828i \(-0.593873\pi\)
−0.290653 + 0.956828i \(0.593873\pi\)
\(368\) 1.95226e6 0.751480
\(369\) 2.13335e6 0.815634
\(370\) 9100.00 0.00345571
\(371\) 4.57843e6 1.72696
\(372\) −274176. −0.102724
\(373\) −2.23807e6 −0.832918 −0.416459 0.909154i \(-0.636729\pi\)
−0.416459 + 0.909154i \(0.636729\pi\)
\(374\) 496688. 0.183614
\(375\) −62500.0 −0.0229510
\(376\) 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 \(0.308916\pi\)
\(380\) −742000. −0.263600
\(381\) −282208. −0.0995994
\(382\) 939104. 0.329272
\(383\) 342216. 0.119207 0.0596037 0.998222i \(-0.481016\pi\)
0.0596037 + 0.998222i \(0.481016\pi\)
\(384\) 744960. 0.257813
\(385\) −710400. −0.244259
\(386\) 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 \(0.495282\pi\)
\(390\) −57200.0 −0.0190430
\(391\) −4.99373e6 −1.65190
\(392\) −2.40684e6 −0.791101
\(393\) −305808. −0.0998775
\(394\) 911724. 0.295885
\(395\) −834000. −0.268951
\(396\) −940688. −0.301445
\(397\) −5.45674e6 −1.73763 −0.868814 0.495138i \(-0.835117\pi\)
−0.868814 + 0.495138i \(0.835117\pi\)
\(398\) 1.73000e6 0.547442
\(399\) −814080. −0.255997
\(400\) 410000. 0.128125
\(401\) 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
\(405\) 1.19102e6 0.360814
\(406\) −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 \(0.631279\pi\)
\(410\) −469900. −0.138053
\(411\) 579672. 0.169269
\(412\) 546112. 0.158503
\(413\) −3.84384e6 −1.10889
\(414\) −1.35110e6 −0.387425
\(415\) 417900. 0.119111
\(416\) 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 \(0.326964\pi\)
\(420\) 537600. 0.148709
\(421\) 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
\(425\) −1.04875e6 −0.281643
\(426\) 261184. 0.0697305
\(427\) 6.20198e6 1.64612
\(428\) −4.43218e6 −1.16952
\(429\) 169312. 0.0444165
\(430\) −62200.0 −0.0162226
\(431\) −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 \(-0.119108\pi\)
0.930804 + 0.365518i \(0.119108\pi\)
\(434\) −940032. −0.239562
\(435\) 341000. 0.0864035
\(436\) −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 \(0.733675\pi\)
\(440\) 444000. 0.109333
\(441\) −4.55294e6 −1.11480
\(442\) −959816. −0.233686
\(443\) −6.51524e6 −1.57733 −0.788663 0.614826i \(-0.789226\pi\)
−0.788663 + 0.614826i \(0.789226\pi\)
\(444\) 20384.0 0.00490718
\(445\) 2.53425e6 0.606666
\(446\) 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 \(0.519010\pi\)
\(450\) −283750. −0.0660548
\(451\) 1.39090e6 0.322000
\(452\) −313208. −0.0721085
\(453\) 1.78659e6 0.409053
\(454\) −46696.0 −0.0106326
\(455\) 1.37280e6 0.310870
\(456\) 508800. 0.114587
\(457\) 1.22084e6 0.273444 0.136722 0.990609i \(-0.456343\pi\)
0.136722 + 0.990609i \(0.456343\pi\)
\(458\) −1.19202e6 −0.265534
\(459\) −3.15464e6 −0.698905
\(460\) −2.08320e6 −0.459025
\(461\) −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 \(-0.429629\pi\)
0.219280 + 0.975662i \(0.429629\pi\)
\(464\) −2.23696e6 −0.482351
\(465\) 244800. 0.0525024
\(466\) −970668. −0.207065
\(467\) 3.25097e6 0.689797 0.344898 0.938640i \(-0.387913\pi\)
0.344898 + 0.938640i \(0.387913\pi\)
\(468\) 1.81782e6 0.383650
\(469\) 1.17066e7 2.45753
\(470\) −604400. −0.126206
\(471\) 1.04903e6 0.217890
\(472\) 2.40240e6 0.496353
\(473\) 184112. 0.0378381
\(474\) 266880. 0.0545595
\(475\) 662500. 0.134726
\(476\) 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 \(0.605879\pi\)
\(480\) −515200. −0.102064
\(481\) 52052.0 0.0102583
\(482\) −221596. −0.0434455
\(483\) −2.28557e6 −0.445786
\(484\) 3.89612e6 0.755994
\(485\) −2.97595e6 −0.574475
\(486\) −1.29481e6 −0.248665
\(487\) −8.53197e6 −1.63015 −0.815074 0.579357i \(-0.803304\pi\)
−0.815074 + 0.579357i \(0.803304\pi\)
\(488\) −3.87624e6 −0.736819
\(489\) 618256. 0.116922
\(490\) 1.00285e6 0.188689
\(491\) 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
\(495\) 839900. 0.154069
\(496\) −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 \(0.698343\pi\)
\(500\) −437500. −0.0782624
\(501\) −1.58669e6 −0.282421
\(502\) −3.28750e6 −0.582245
\(503\) 8.61770e6 1.51870 0.759349 0.650684i \(-0.225518\pi\)
0.759349 + 0.650684i \(0.225518\pi\)
\(504\) 5.23008e6 0.917132
\(505\) −2.24745e6 −0.392158
\(506\) −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 \(0.426562\pi\)
\(510\) 335600. 0.0571342
\(511\) −7.44461e6 −1.26122
\(512\) 5.89875e6 0.994455
\(513\) 1.99280e6 0.334326
\(514\) 2.61248e6 0.436160
\(515\) −487600. −0.0810113
\(516\) −139328. −0.0230364
\(517\) 1.78902e6 0.294367
\(518\) 69888.0 0.0114440
\(519\) 2.29390e6 0.373814
\(520\) −858000. −0.139149
\(521\) 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 \(-0.685787\pi\)
−0.551087 + 0.834448i \(0.685787\pi\)
\(524\) −2.14066e6 −0.340580
\(525\) −480000. −0.0760051
\(526\) 4.25667e6 0.670820
\(527\) 4.10774e6 0.644283
\(528\) 388352. 0.0606235
\(529\) 2.42023e6 0.376026
\(530\) 1.19230e6 0.184372
\(531\) 4.54454e6 0.699445
\(532\) −5.69856e6 −0.872943
\(533\) −2.68783e6 −0.409811
\(534\) −810960. −0.123068
\(535\) 3.95730e6 0.597743
\(536\) −7.31664e6 −1.10002
\(537\) 2.37784e6 0.355834
\(538\) −2.88218e6 −0.429304
\(539\) −2.96844e6 −0.440104
\(540\) −1.31600e6 −0.194210
\(541\) 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
\(545\) 920750. 0.132785
\(546\) −439296. −0.0630631
\(547\) 1.26544e7 1.80831 0.904157 0.427201i \(-0.140500\pi\)
0.904157 + 0.427201i \(0.140500\pi\)
\(548\) 4.05770e6 0.577204
\(549\) −7.33255e6 −1.03830
\(550\) −185000. −0.0260774
\(551\) −3.61460e6 −0.507202
\(552\) 1.42848e6 0.199538
\(553\) −6.40512e6 −0.890665
\(554\) −220596. −0.0305368
\(555\) −18200.0 −0.00250807
\(556\) −3.14216e6 −0.431064
\(557\) −7.07786e6 −0.966638 −0.483319 0.875444i \(-0.660569\pi\)
−0.483319 + 0.875444i \(0.660569\pi\)
\(558\) 1.11139e6 0.151106
\(559\) −355784. −0.0481567
\(560\) 3.14880e6 0.424302
\(561\) −993376. −0.133262
\(562\) −384396. −0.0513379
\(563\) 846636. 0.112571 0.0562854 0.998415i \(-0.482074\pi\)
0.0562854 + 0.998415i \(0.482074\pi\)
\(564\) −1.35386e6 −0.179215
\(565\) 279650. 0.0368548
\(566\) −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 \(0.395931\pi\)
\(570\) −212000. −0.0273306
\(571\) 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
\(575\) 1.86000e6 0.234608
\(576\) 2.42618e6 0.304696
\(577\) −2.86080e6 −0.357724 −0.178862 0.983874i \(-0.557242\pi\)
−0.178862 + 0.983874i \(0.557242\pi\)
\(578\) 2.79165e6 0.347570
\(579\) −210824. −0.0261351
\(580\) 2.38700e6 0.294634
\(581\) 3.20947e6 0.394451
\(582\) 952304. 0.116538
\(583\) −3.52921e6 −0.430037
\(584\) 4.65288e6 0.564534
\(585\) −1.62305e6 −0.196084
\(586\) 4.38961e6 0.528059
\(587\) −6.74027e6 −0.807387 −0.403694 0.914894i \(-0.632274\pi\)
−0.403694 + 0.914894i \(0.632274\pi\)
\(588\) 2.24638e6 0.267942
\(589\) −2.59488e6 −0.308197
\(590\) −1.00100e6 −0.118387
\(591\) −1.82345e6 −0.214746
\(592\) 119392. 0.0140014
\(593\) −1.78609e6 −0.208578 −0.104289 0.994547i \(-0.533257\pi\)
−0.104289 + 0.994547i \(0.533257\pi\)
\(594\) −556480. −0.0647118
\(595\) −8.05440e6 −0.932697
\(596\) −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 \(0.409126\pi\)
\(600\) 300000. 0.0340207
\(601\) −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
\(605\) −3.47868e6 −0.386390
\(606\) 719184. 0.0795533
\(607\) 7.07999e6 0.779940 0.389970 0.920828i \(-0.372485\pi\)
0.389970 + 0.920828i \(0.372485\pi\)
\(608\) 5.46112e6 0.599132
\(609\) 2.61888e6 0.286136
\(610\) 1.61510e6 0.175742
\(611\) −3.45717e6 −0.374643
\(612\) −1.06654e7 −1.15106
\(613\) 5.09609e6 0.547754 0.273877 0.961765i \(-0.411694\pi\)
0.273877 + 0.961765i \(0.411694\pi\)
\(614\) −4.75502e6 −0.509016
\(615\) 939800. 0.100195
\(616\) 3.40992e6 0.362070
\(617\) −1.30003e7 −1.37480 −0.687400 0.726279i \(-0.741248\pi\)
−0.687400 + 0.726279i \(0.741248\pi\)
\(618\) 156032. 0.0164340
\(619\) 4.84406e6 0.508139 0.254070 0.967186i \(-0.418231\pi\)
0.254070 + 0.967186i \(0.418231\pi\)
\(620\) 1.71360e6 0.179032
\(621\) 5.59488e6 0.582186
\(622\) −4.74610e6 −0.491882
\(623\) 1.94630e7 2.00905
\(624\) −750464. −0.0771558
\(625\) 390625. 0.0400000
\(626\) −2.85883e6 −0.291576
\(627\) 627520. 0.0637468
\(628\) 7.34322e6 0.742998
\(629\) −305396. −0.0307777
\(630\) −2.17920e6 −0.218749
\(631\) 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
\(635\) 1.76380e6 0.173586
\(636\) 2.67075e6 0.261813
\(637\) 5.73630e6 0.560123
\(638\) 1.00936e6 0.0981735
\(639\) 7.41110e6 0.718010
\(640\) −4.65600e6 −0.449328
\(641\) 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 \(-0.812609\pi\)
−0.831659 + 0.555287i \(0.812609\pi\)
\(644\) −1.59990e7 −1.52012
\(645\) 124400. 0.0117739
\(646\) −3.55736e6 −0.335387
\(647\) −4.25469e6 −0.399583 −0.199792 0.979838i \(-0.564026\pi\)
−0.199792 + 0.979838i \(0.564026\pi\)
\(648\) −5.71692e6 −0.534841
\(649\) 2.96296e6 0.276130
\(650\) 357500. 0.0331889
\(651\) 1.88006e6 0.173868
\(652\) 4.32779e6 0.398701
\(653\) 3.01085e6 0.276316 0.138158 0.990410i \(-0.455882\pi\)
0.138158 + 0.990410i \(0.455882\pi\)
\(654\) −294640. −0.0269369
\(655\) 1.91130e6 0.174071
\(656\) −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 \(0.618567\pi\)
\(660\) −414400. −0.0370305
\(661\) 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
\(665\) 5.08800e6 0.446162
\(666\) −82628.0 −0.00721841
\(667\) −1.01482e7 −0.883228
\(668\) −1.11068e7 −0.963049
\(669\) −4.48630e6 −0.387546
\(670\) 3.04860e6 0.262370
\(671\) −4.78070e6 −0.409907
\(672\) −3.95674e6 −0.337998
\(673\) 5.77063e6 0.491117 0.245559 0.969382i \(-0.421029\pi\)
0.245559 + 0.969382i \(0.421029\pi\)
\(674\) 4.80016e6 0.407011
\(675\) 1.17500e6 0.0992610
\(676\) 8.10592e6 0.682237
\(677\) 1.67197e7 1.40203 0.701014 0.713147i \(-0.252731\pi\)
0.701014 + 0.713147i \(0.252731\pi\)
\(678\) −89488.0 −0.00747637
\(679\) −2.28553e7 −1.90245
\(680\) 5.03400e6 0.417485
\(681\) 93392.0 0.00771688
\(682\) 724608. 0.0596544
\(683\) 7.14532e6 0.586097 0.293049 0.956098i \(-0.405330\pi\)
0.293049 + 0.956098i \(0.405330\pi\)
\(684\) 6.73736e6 0.550617
\(685\) −3.62295e6 −0.295009
\(686\) 1.24800e6 0.101252
\(687\) 2.38404e6 0.192718
\(688\) −816064. −0.0657284
\(689\) 6.81996e6 0.547310
\(690\) −595200. −0.0475927
\(691\) −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
\(695\) 2.80550e6 0.220317
\(696\) −1.63680e6 −0.128077
\(697\) 1.57698e7 1.22955
\(698\) −4.29610e6 −0.333761
\(699\) 1.94134e6 0.150282
\(700\) −3.36000e6 −0.259176
\(701\) −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
\(705\) 1.20880e6 0.0915971
\(706\) −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 \(0.753487\pi\)
\(710\) −1.63240e6 −0.121529
\(711\) 7.57272e6 0.561795
\(712\) −1.21644e7 −0.899271
\(713\) −7.28525e6 −0.536686
\(714\) 2.57741e6 0.189207
\(715\) −1.05820e6 −0.0774110
\(716\) 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 \(0.378918\pi\)
\(720\) −3.72280e6 −0.267632
\(721\) −3.74477e6 −0.268279
\(722\) −2.70500e6 −0.193119
\(723\) 443192. 0.0315316
\(724\) 2.99874e6 0.212615
\(725\) −2.13125e6 −0.150588
\(726\) 1.11318e6 0.0783831
\(727\) −1.93264e7 −1.35618 −0.678088 0.734981i \(-0.737191\pi\)
−0.678088 + 0.734981i \(0.737191\pi\)
\(728\) −6.58944e6 −0.460808
\(729\) −8.98715e6 −0.626330
\(730\) −1.93870e6 −0.134649
\(731\) 2.08743e6 0.144484
\(732\) 3.61782e6 0.249557
\(733\) 5.26197e6 0.361733 0.180866 0.983508i \(-0.442110\pi\)
0.180866 + 0.983508i \(0.442110\pi\)
\(734\) −2.99986e6 −0.205523
\(735\) −2.00570e6 −0.136945
\(736\) 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 \(0.0980548\pi\)
\(740\) −127400. −0.00855244
\(741\) −1.21264e6 −0.0811309
\(742\) 9.15686e6 0.610572
\(743\) 2.09863e7 1.39464 0.697321 0.716759i \(-0.254375\pi\)
0.697321 + 0.716759i \(0.254375\pi\)
\(744\) −1.17504e6 −0.0778252
\(745\) 1.00938e7 0.666288
\(746\) −4.47615e6 −0.294481
\(747\) −3.79453e6 −0.248804
\(748\) −6.95363e6 −0.454420
\(749\) 3.03921e7 1.97950
\(750\) −125000. −0.00811441
\(751\) −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
\(755\) −1.11662e7 −0.712915
\(756\) −1.01069e7 −0.643151
\(757\) −1.08257e7 −0.686617 −0.343309 0.939223i \(-0.611548\pi\)
−0.343309 + 0.939223i \(0.611548\pi\)
\(758\) 6.31868e6 0.399442
\(759\) 1.76179e6 0.111007
\(760\) −3.18000e6 −0.199707
\(761\) 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
\(765\) 9.52265e6 0.588307
\(766\) 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 \(0.659802\pi\)
\(770\) −1.42080e6 −0.0863587
\(771\) −5.22497e6 −0.316554
\(772\) −1.47577e6 −0.0891199
\(773\) −2.44049e7 −1.46902 −0.734510 0.678598i \(-0.762588\pi\)
−0.734510 + 0.678598i \(0.762588\pi\)
\(774\) 564776. 0.0338863
\(775\) −1.53000e6 −0.0915034
\(776\) 1.42846e7 0.851555
\(777\) −139776. −0.00830577
\(778\) 176940. 0.0104804
\(779\) −9.96188e6 −0.588163
\(780\) 800800. 0.0471289
\(781\) 4.83190e6 0.283459
\(782\) −9.98746e6 −0.584034
\(783\) −6.41080e6 −0.373687
\(784\) 1.31574e7 0.764504
\(785\) −6.55645e6 −0.379747
\(786\) −611616. −0.0353120
\(787\) 3.37607e7 1.94301 0.971505 0.237019i \(-0.0761704\pi\)
0.971505 + 0.237019i \(0.0761704\pi\)
\(788\) −1.27641e7 −0.732278
\(789\) −8.51334e6 −0.486864
\(790\) −1.66800e6 −0.0950886
\(791\) 2.14771e6 0.122049
\(792\) −4.03152e6 −0.228379
\(793\) 9.23837e6 0.521690
\(794\) −1.09135e7 −0.614344
\(795\) −2.38460e6 −0.133813
\(796\) −2.42200e7 −1.35485
\(797\) 2.19885e7 1.22617 0.613083 0.790019i \(-0.289929\pi\)
0.613083 + 0.790019i \(0.289929\pi\)
\(798\) −1.62816e6 −0.0905086
\(799\) 2.02837e7 1.12403
\(800\) 3.22000e6 0.177882
\(801\) −2.30110e7 −1.26723
\(802\) 8.09360e6 0.444330
\(803\) 5.73855e6 0.314061
\(804\) 6.82886e6 0.372570
\(805\) 1.42848e7 0.776935
\(806\) −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 \(0.789187\pi\)
\(810\) 2.38205e6 0.127567
\(811\) 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
\(815\) −3.86410e6 −0.203777
\(816\) 4.40307e6 0.231489
\(817\) −1.31864e6 −0.0691148
\(818\) −5.42414e6 −0.283431
\(819\) −1.24650e7 −0.649357
\(820\) 6.57860e6 0.341664
\(821\) −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 \(-0.604986\pi\)
−0.323877 + 0.946099i \(0.604986\pi\)
\(824\) 2.34048e6 0.120084
\(825\) 370000. 0.0189263
\(826\) −7.68768e6 −0.392053
\(827\) −8.72355e6 −0.443537 −0.221768 0.975099i \(-0.571183\pi\)
−0.221768 + 0.975099i \(0.571183\pi\)
\(828\) 1.89155e7 0.958829
\(829\) −1.06178e7 −0.536597 −0.268299 0.963336i \(-0.586461\pi\)
−0.268299 + 0.963336i \(0.586461\pi\)
\(830\) 835800. 0.0421121
\(831\) 441192. 0.0221628
\(832\) −3.05677e6 −0.153093
\(833\) −3.36556e7 −1.68053
\(834\) −897760. −0.0446936
\(835\) 9.91680e6 0.492216
\(836\) 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 \(0.365039\pi\)
\(840\) 2.30400e6 0.112664
\(841\) −8.88305e6 −0.433084
\(842\) 7.10500e6 0.345370
\(843\) 768792. 0.0372597
\(844\) −3.09583e7 −1.49596
\(845\) −7.23742e6 −0.348692
\(846\) 5.48795e6 0.263624
\(847\) −2.67162e7 −1.27958
\(848\) 1.56430e7 0.747016
\(849\) 1.32754e6 0.0632087
\(850\) −2.09750e6 −0.0995760
\(851\) 541632. 0.0256378
\(852\) −3.65658e6 −0.172574
\(853\) −2.20186e7 −1.03613 −0.518067 0.855340i \(-0.673348\pi\)
−0.518067 + 0.855340i \(0.673348\pi\)
\(854\) 1.24040e7 0.581991
\(855\) −6.01550e6 −0.281421
\(856\) −1.89950e7 −0.886045
\(857\) 3.16676e7 1.47287 0.736434 0.676510i \(-0.236508\pi\)
0.736434 + 0.676510i \(0.236508\pi\)
\(858\) 338624. 0.0157036
\(859\) 1.58064e7 0.730886 0.365443 0.930834i \(-0.380918\pi\)
0.365443 + 0.930834i \(0.380918\pi\)
\(860\) 870800. 0.0401488
\(861\) 7.21766e6 0.331809
\(862\) −8.12410e6 −0.372398
\(863\) −1.44287e7 −0.659476 −0.329738 0.944072i \(-0.606960\pi\)
−0.329738 + 0.944072i \(0.606960\pi\)
\(864\) 9.68576e6 0.441417
\(865\) −1.43368e7 −0.651499
\(866\) 1.45257e7 0.658178
\(867\) −5.58331e6 −0.252257
\(868\) 1.31604e7 0.592886
\(869\) 4.93728e6 0.221788
\(870\) 682000. 0.0305482
\(871\) 1.74380e7 0.778845
\(872\) −4.41960e6 −0.196830
\(873\) 2.70216e7 1.19999
\(874\) 6.30912e6 0.279377
\(875\) 3.00000e6 0.132465
\(876\) −4.34269e6 −0.191205
\(877\) 247902. 0.0108838 0.00544191 0.999985i \(-0.498268\pi\)
0.00544191 + 0.999985i \(0.498268\pi\)
\(878\) −1.08206e7 −0.473711
\(879\) −8.77922e6 −0.383252
\(880\) −2.42720e6 −0.105657
\(881\) 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 \(-0.142016\pi\)
0.902112 + 0.431503i \(0.142016\pi\)
\(884\) 1.34374e7 0.578343
\(885\) 2.00200e6 0.0859223
\(886\) −1.30305e7 −0.557669
\(887\) −2.10476e7 −0.898241 −0.449120 0.893471i \(-0.648263\pi\)
−0.449120 + 0.893471i \(0.648263\pi\)
\(888\) 87360.0 0.00371775
\(889\) 1.35460e7 0.574852
\(890\) 5.06850e6 0.214489
\(891\) −7.05087e6 −0.297542
\(892\) −3.14041e7 −1.32152
\(893\) −1.28133e7 −0.537690
\(894\) −3.23000e6 −0.135163
\(895\) −1.48615e7 −0.620162
\(896\) −3.57581e7 −1.48800
\(897\) −3.40454e6 −0.141279
\(898\) −1.01990e6 −0.0422053
\(899\) 8.34768e6 0.344482
\(900\) 3.97250e6 0.163477
\(901\) −4.00136e7 −1.64208
\(902\) 2.78181e6 0.113844
\(903\) 955392. 0.0389908
\(904\) −1.34232e6 −0.0546305
\(905\) −2.67745e6 −0.108668
\(906\) 3.57318e6 0.144622
\(907\) 7.48309e6 0.302039 0.151019 0.988531i \(-0.451744\pi\)
0.151019 + 0.988531i \(0.451744\pi\)
\(908\) 653744. 0.0263144
\(909\) 2.04068e7 0.819155
\(910\) 2.74560e6 0.109909
\(911\) −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
\(915\) −3.23020e6 −0.127549
\(916\) 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 \(0.607047\pi\)
\(920\) −8.92800e6 −0.347764
\(921\) 9.51003e6 0.369431
\(922\) −8.14420e6 −0.315516
\(923\) −9.33733e6 −0.360760
\(924\) −3.18259e6 −0.122631
\(925\) 113750. 0.00437116
\(926\) 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 \(0.578637\pi\)
\(930\) 489600. 0.0185624
\(931\) 2.12604e7 0.803892
\(932\) 1.35894e7 0.512459
\(933\) 9.49219e6 0.356995
\(934\) 6.50194e6 0.243880
\(935\) 6.20860e6 0.232255
\(936\) 7.79064e6 0.290659
\(937\) 1.06887e7 0.397718 0.198859 0.980028i \(-0.436276\pi\)
0.198859 + 0.980028i \(0.436276\pi\)
\(938\) 2.34132e7 0.868870
\(939\) 5.71766e6 0.211619
\(940\) 8.46160e6 0.312344
\(941\) 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
\(945\) 9.02400e6 0.328715
\(946\) 368224. 0.0133778
\(947\) −1.70892e7 −0.619222 −0.309611 0.950863i \(-0.600199\pi\)
−0.309611 + 0.950863i \(0.600199\pi\)
\(948\) −3.73632e6 −0.135028
\(949\) −1.10894e7 −0.399706
\(950\) 1.32500e6 0.0476329
\(951\) −8.49849e6 −0.304713
\(952\) 3.86611e7 1.38255
\(953\) 2.22259e7 0.792735 0.396367 0.918092i \(-0.370271\pi\)
0.396367 + 0.918092i \(0.370271\pi\)
\(954\) −1.08261e7 −0.385124
\(955\) 1.17388e7 0.416500
\(956\) 1.36864e6 0.0484333
\(957\) −2.01872e6 −0.0712519
\(958\) −6.55872e6 −0.230890
\(959\) −2.78243e7 −0.976961
\(960\) 1.06880e6 0.0374299
\(961\) −2.26364e7 −0.790678
\(962\) 104104. 0.00362685
\(963\) −3.59323e7 −1.24859
\(964\) 3.10234e6 0.107522
\(965\) 1.31765e6 0.0455493
\(966\) −4.57114e6 −0.157609
\(967\) 2.41551e7 0.830696 0.415348 0.909663i \(-0.363660\pi\)
0.415348 + 0.909663i \(0.363660\pi\)
\(968\) 1.66976e7 0.572752
\(969\) 7.11472e6 0.243416
\(970\) −5.95190e6 −0.203108
\(971\) −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
\(975\) −715000. −0.0240877
\(976\) 2.11901e7 0.712047
\(977\) −1.56612e7 −0.524915 −0.262457 0.964944i \(-0.584533\pi\)
−0.262457 + 0.964944i \(0.584533\pi\)
\(978\) 1.23651e6 0.0413382
\(979\) −1.50028e7 −0.500281
\(980\) −1.40399e7 −0.466981
\(981\) −8.36041e6 −0.277367
\(982\) 3.02530e6 0.100113
\(983\) −1.63420e7 −0.539412 −0.269706 0.962943i \(-0.586927\pi\)
−0.269706 + 0.962943i \(0.586927\pi\)
\(984\) −4.51104e6 −0.148521
\(985\) 1.13966e7 0.374268
\(986\) 1.14440e7 0.374873
\(987\) 9.28358e6 0.303335
\(988\) −8.48848e6 −0.276654
\(989\) −3.70214e6 −0.120355
\(990\) 1.67980e6 0.0544715
\(991\) 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
\(995\) 2.16250e7 0.692466
\(996\) 1.87219e6 0.0598001
\(997\) −1.29097e7 −0.411320 −0.205660 0.978624i \(-0.565934\pi\)
−0.205660 + 0.978624i \(0.565934\pi\)
\(998\) −1.29838e7 −0.412644
\(999\) 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 5.6.a.a.1.1 1
3.2 odd 2 45.6.a.b.1.1 1
4.3 odd 2 80.6.a.e.1.1 1
5.2 odd 4 25.6.b.a.24.2 2
5.3 odd 4 25.6.b.a.24.1 2
5.4 even 2 25.6.a.a.1.1 1
7.6 odd 2 245.6.a.b.1.1 1
8.3 odd 2 320.6.a.g.1.1 1
8.5 even 2 320.6.a.j.1.1 1
11.10 odd 2 605.6.a.a.1.1 1
12.11 even 2 720.6.a.a.1.1 1
13.12 even 2 845.6.a.b.1.1 1
15.2 even 4 225.6.b.e.199.1 2
15.8 even 4 225.6.b.e.199.2 2
15.14 odd 2 225.6.a.f.1.1 1
20.3 even 4 400.6.c.j.49.2 2
20.7 even 4 400.6.c.j.49.1 2
20.19 odd 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 1.1 even 1 trivial
25.6.a.a.1.1 1 5.4 even 2
25.6.b.a.24.1 2 5.3 odd 4
25.6.b.a.24.2 2 5.2 odd 4
45.6.a.b.1.1 1 3.2 odd 2
80.6.a.e.1.1 1 4.3 odd 2
225.6.a.f.1.1 1 15.14 odd 2
225.6.b.e.199.1 2 15.2 even 4
225.6.b.e.199.2 2 15.8 even 4
245.6.a.b.1.1 1 7.6 odd 2
320.6.a.g.1.1 1 8.3 odd 2
320.6.a.j.1.1 1 8.5 even 2
400.6.a.g.1.1 1 20.19 odd 2
400.6.c.j.49.1 2 20.7 even 4
400.6.c.j.49.2 2 20.3 even 4
605.6.a.a.1.1 1 11.10 odd 2
720.6.a.a.1.1 1 12.11 even 2
845.6.a.b.1.1 1 13.12 even 2