Properties

Label 825.6.a.m.1.5
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-3.33276\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.33276 q^{2} +9.00000 q^{3} -26.5582 q^{4} +20.9949 q^{6} -146.260 q^{7} -136.602 q^{8} +81.0000 q^{9} +121.000 q^{11} -239.024 q^{12} +158.241 q^{13} -341.189 q^{14} +531.202 q^{16} +1926.04 q^{17} +188.954 q^{18} -32.9364 q^{19} -1316.34 q^{21} +282.264 q^{22} -2077.06 q^{23} -1229.42 q^{24} +369.139 q^{26} +729.000 q^{27} +3884.40 q^{28} +1641.87 q^{29} -3563.71 q^{31} +5610.45 q^{32} +1089.00 q^{33} +4492.99 q^{34} -2151.22 q^{36} +5396.83 q^{37} -76.8329 q^{38} +1424.17 q^{39} +2803.48 q^{41} -3070.71 q^{42} +1140.23 q^{43} -3213.54 q^{44} -4845.29 q^{46} +15797.7 q^{47} +4780.82 q^{48} +4584.95 q^{49} +17334.4 q^{51} -4202.61 q^{52} -24043.7 q^{53} +1700.58 q^{54} +19979.4 q^{56} -296.428 q^{57} +3830.09 q^{58} +5002.23 q^{59} -4798.86 q^{61} -8313.28 q^{62} -11847.0 q^{63} -3910.63 q^{64} +2540.38 q^{66} -34126.4 q^{67} -51152.2 q^{68} -18693.6 q^{69} +10059.7 q^{71} -11064.8 q^{72} -71743.6 q^{73} +12589.5 q^{74} +874.733 q^{76} -17697.4 q^{77} +3322.25 q^{78} -14305.9 q^{79} +6561.00 q^{81} +6539.84 q^{82} -66464.5 q^{83} +34959.6 q^{84} +2659.88 q^{86} +14776.8 q^{87} -16528.9 q^{88} -11104.6 q^{89} -23144.4 q^{91} +55163.1 q^{92} -32073.4 q^{93} +36852.3 q^{94} +50494.0 q^{96} -56522.6 q^{97} +10695.6 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.33276 0.412378 0.206189 0.978512i \(-0.433894\pi\)
0.206189 + 0.978512i \(0.433894\pi\)
\(3\) 9.00000 0.577350
\(4\) −26.5582 −0.829944
\(5\) 0 0
\(6\) 20.9949 0.238087
\(7\) −146.260 −1.12818 −0.564092 0.825712i \(-0.690774\pi\)
−0.564092 + 0.825712i \(0.690774\pi\)
\(8\) −136.602 −0.754629
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −239.024 −0.479169
\(13\) 158.241 0.259694 0.129847 0.991534i \(-0.458551\pi\)
0.129847 + 0.991534i \(0.458551\pi\)
\(14\) −341.189 −0.465238
\(15\) 0 0
\(16\) 531.202 0.518752
\(17\) 1926.04 1.61638 0.808189 0.588923i \(-0.200448\pi\)
0.808189 + 0.588923i \(0.200448\pi\)
\(18\) 188.954 0.137459
\(19\) −32.9364 −0.0209311 −0.0104656 0.999945i \(-0.503331\pi\)
−0.0104656 + 0.999945i \(0.503331\pi\)
\(20\) 0 0
\(21\) −1316.34 −0.651358
\(22\) 282.264 0.124337
\(23\) −2077.06 −0.818710 −0.409355 0.912375i \(-0.634246\pi\)
−0.409355 + 0.912375i \(0.634246\pi\)
\(24\) −1229.42 −0.435685
\(25\) 0 0
\(26\) 369.139 0.107092
\(27\) 729.000 0.192450
\(28\) 3884.40 0.936330
\(29\) 1641.87 0.362530 0.181265 0.983434i \(-0.441981\pi\)
0.181265 + 0.983434i \(0.441981\pi\)
\(30\) 0 0
\(31\) −3563.71 −0.666036 −0.333018 0.942920i \(-0.608067\pi\)
−0.333018 + 0.942920i \(0.608067\pi\)
\(32\) 5610.45 0.968551
\(33\) 1089.00 0.174078
\(34\) 4492.99 0.666559
\(35\) 0 0
\(36\) −2151.22 −0.276648
\(37\) 5396.83 0.648089 0.324045 0.946042i \(-0.394957\pi\)
0.324045 + 0.946042i \(0.394957\pi\)
\(38\) −76.8329 −0.00863154
\(39\) 1424.17 0.149934
\(40\) 0 0
\(41\) 2803.48 0.260458 0.130229 0.991484i \(-0.458429\pi\)
0.130229 + 0.991484i \(0.458429\pi\)
\(42\) −3070.71 −0.268606
\(43\) 1140.23 0.0940417 0.0470208 0.998894i \(-0.485027\pi\)
0.0470208 + 0.998894i \(0.485027\pi\)
\(44\) −3213.54 −0.250238
\(45\) 0 0
\(46\) −4845.29 −0.337618
\(47\) 15797.7 1.04316 0.521578 0.853204i \(-0.325344\pi\)
0.521578 + 0.853204i \(0.325344\pi\)
\(48\) 4780.82 0.299502
\(49\) 4584.95 0.272800
\(50\) 0 0
\(51\) 17334.4 0.933216
\(52\) −4202.61 −0.215531
\(53\) −24043.7 −1.17574 −0.587871 0.808955i \(-0.700034\pi\)
−0.587871 + 0.808955i \(0.700034\pi\)
\(54\) 1700.58 0.0793622
\(55\) 0 0
\(56\) 19979.4 0.851360
\(57\) −296.428 −0.0120846
\(58\) 3830.09 0.149499
\(59\) 5002.23 0.187083 0.0935414 0.995615i \(-0.470181\pi\)
0.0935414 + 0.995615i \(0.470181\pi\)
\(60\) 0 0
\(61\) −4798.86 −0.165125 −0.0825626 0.996586i \(-0.526310\pi\)
−0.0825626 + 0.996586i \(0.526310\pi\)
\(62\) −8313.28 −0.274659
\(63\) −11847.0 −0.376061
\(64\) −3910.63 −0.119343
\(65\) 0 0
\(66\) 2540.38 0.0717858
\(67\) −34126.4 −0.928759 −0.464379 0.885636i \(-0.653723\pi\)
−0.464379 + 0.885636i \(0.653723\pi\)
\(68\) −51152.2 −1.34150
\(69\) −18693.6 −0.472683
\(70\) 0 0
\(71\) 10059.7 0.236832 0.118416 0.992964i \(-0.462218\pi\)
0.118416 + 0.992964i \(0.462218\pi\)
\(72\) −11064.8 −0.251543
\(73\) −71743.6 −1.57571 −0.787855 0.615861i \(-0.788808\pi\)
−0.787855 + 0.615861i \(0.788808\pi\)
\(74\) 12589.5 0.267258
\(75\) 0 0
\(76\) 874.733 0.0173717
\(77\) −17697.4 −0.340160
\(78\) 3322.25 0.0618296
\(79\) −14305.9 −0.257899 −0.128949 0.991651i \(-0.541160\pi\)
−0.128949 + 0.991651i \(0.541160\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 6539.84 0.107407
\(83\) −66464.5 −1.05900 −0.529498 0.848311i \(-0.677620\pi\)
−0.529498 + 0.848311i \(0.677620\pi\)
\(84\) 34959.6 0.540591
\(85\) 0 0
\(86\) 2659.88 0.0387807
\(87\) 14776.8 0.209307
\(88\) −16528.9 −0.227529
\(89\) −11104.6 −0.148603 −0.0743017 0.997236i \(-0.523673\pi\)
−0.0743017 + 0.997236i \(0.523673\pi\)
\(90\) 0 0
\(91\) −23144.4 −0.292982
\(92\) 55163.1 0.679484
\(93\) −32073.4 −0.384536
\(94\) 36852.3 0.430174
\(95\) 0 0
\(96\) 50494.0 0.559193
\(97\) −56522.6 −0.609948 −0.304974 0.952361i \(-0.598648\pi\)
−0.304974 + 0.952361i \(0.598648\pi\)
\(98\) 10695.6 0.112497
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −21709.1 −0.211758 −0.105879 0.994379i \(-0.533766\pi\)
−0.105879 + 0.994379i \(0.533766\pi\)
\(102\) 40436.9 0.384838
\(103\) −15441.9 −0.143419 −0.0717096 0.997426i \(-0.522845\pi\)
−0.0717096 + 0.997426i \(0.522845\pi\)
\(104\) −21616.1 −0.195972
\(105\) 0 0
\(106\) −56088.3 −0.484850
\(107\) −33358.3 −0.281673 −0.140836 0.990033i \(-0.544979\pi\)
−0.140836 + 0.990033i \(0.544979\pi\)
\(108\) −19360.9 −0.159723
\(109\) −94499.4 −0.761838 −0.380919 0.924608i \(-0.624392\pi\)
−0.380919 + 0.924608i \(0.624392\pi\)
\(110\) 0 0
\(111\) 48571.5 0.374174
\(112\) −77693.6 −0.585248
\(113\) 150985. 1.11234 0.556170 0.831069i \(-0.312270\pi\)
0.556170 + 0.831069i \(0.312270\pi\)
\(114\) −691.496 −0.00498342
\(115\) 0 0
\(116\) −43605.1 −0.300879
\(117\) 12817.5 0.0865646
\(118\) 11669.0 0.0771488
\(119\) −281702. −1.82357
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −11194.6 −0.0680940
\(123\) 25231.3 0.150375
\(124\) 94645.7 0.552773
\(125\) 0 0
\(126\) −27636.3 −0.155079
\(127\) −348511. −1.91737 −0.958687 0.284462i \(-0.908185\pi\)
−0.958687 + 0.284462i \(0.908185\pi\)
\(128\) −188657. −1.01777
\(129\) 10262.0 0.0542950
\(130\) 0 0
\(131\) 59482.3 0.302837 0.151419 0.988470i \(-0.451616\pi\)
0.151419 + 0.988470i \(0.451616\pi\)
\(132\) −28921.9 −0.144475
\(133\) 4817.28 0.0236142
\(134\) −79608.7 −0.383000
\(135\) 0 0
\(136\) −263102. −1.21977
\(137\) 148254. 0.674848 0.337424 0.941353i \(-0.390444\pi\)
0.337424 + 0.941353i \(0.390444\pi\)
\(138\) −43607.7 −0.194924
\(139\) −196689. −0.863462 −0.431731 0.902002i \(-0.642097\pi\)
−0.431731 + 0.902002i \(0.642097\pi\)
\(140\) 0 0
\(141\) 142179. 0.602266
\(142\) 23467.0 0.0976645
\(143\) 19147.2 0.0783006
\(144\) 43027.4 0.172917
\(145\) 0 0
\(146\) −167361. −0.649788
\(147\) 41264.5 0.157501
\(148\) −143330. −0.537878
\(149\) −305815. −1.12848 −0.564239 0.825611i \(-0.690830\pi\)
−0.564239 + 0.825611i \(0.690830\pi\)
\(150\) 0 0
\(151\) −220420. −0.786701 −0.393350 0.919389i \(-0.628684\pi\)
−0.393350 + 0.919389i \(0.628684\pi\)
\(152\) 4499.20 0.0157952
\(153\) 156009. 0.538793
\(154\) −41283.9 −0.140275
\(155\) 0 0
\(156\) −37823.5 −0.124437
\(157\) 585267. 1.89498 0.947491 0.319782i \(-0.103610\pi\)
0.947491 + 0.319782i \(0.103610\pi\)
\(158\) −33372.4 −0.106352
\(159\) −216393. −0.678815
\(160\) 0 0
\(161\) 303791. 0.923656
\(162\) 15305.3 0.0458198
\(163\) −109151. −0.321780 −0.160890 0.986972i \(-0.551436\pi\)
−0.160890 + 0.986972i \(0.551436\pi\)
\(164\) −74455.3 −0.216165
\(165\) 0 0
\(166\) −155046. −0.436707
\(167\) 243334. 0.675168 0.337584 0.941295i \(-0.390390\pi\)
0.337584 + 0.941295i \(0.390390\pi\)
\(168\) 179815. 0.491533
\(169\) −346253. −0.932559
\(170\) 0 0
\(171\) −2667.85 −0.00697704
\(172\) −30282.4 −0.0780494
\(173\) 398905. 1.01334 0.506669 0.862141i \(-0.330877\pi\)
0.506669 + 0.862141i \(0.330877\pi\)
\(174\) 34470.8 0.0863134
\(175\) 0 0
\(176\) 64275.5 0.156410
\(177\) 45020.1 0.108012
\(178\) −25904.5 −0.0612808
\(179\) 198924. 0.464039 0.232019 0.972711i \(-0.425467\pi\)
0.232019 + 0.972711i \(0.425467\pi\)
\(180\) 0 0
\(181\) −474113. −1.07569 −0.537843 0.843045i \(-0.680761\pi\)
−0.537843 + 0.843045i \(0.680761\pi\)
\(182\) −53990.3 −0.120819
\(183\) −43189.7 −0.0953350
\(184\) 283732. 0.617822
\(185\) 0 0
\(186\) −74819.5 −0.158574
\(187\) 233051. 0.487356
\(188\) −419559. −0.865761
\(189\) −106623. −0.217119
\(190\) 0 0
\(191\) 201548. 0.399757 0.199878 0.979821i \(-0.435945\pi\)
0.199878 + 0.979821i \(0.435945\pi\)
\(192\) −35195.7 −0.0689028
\(193\) −205191. −0.396519 −0.198260 0.980150i \(-0.563529\pi\)
−0.198260 + 0.980150i \(0.563529\pi\)
\(194\) −131854. −0.251529
\(195\) 0 0
\(196\) −121768. −0.226409
\(197\) 99895.6 0.183392 0.0916961 0.995787i \(-0.470771\pi\)
0.0916961 + 0.995787i \(0.470771\pi\)
\(198\) 22863.4 0.0414455
\(199\) −371365. −0.664765 −0.332382 0.943145i \(-0.607852\pi\)
−0.332382 + 0.943145i \(0.607852\pi\)
\(200\) 0 0
\(201\) −307137. −0.536219
\(202\) −50642.3 −0.0873242
\(203\) −240139. −0.409000
\(204\) −460370. −0.774518
\(205\) 0 0
\(206\) −36022.3 −0.0591429
\(207\) −168242. −0.272903
\(208\) 84058.1 0.134717
\(209\) −3985.31 −0.00631097
\(210\) 0 0
\(211\) −285877. −0.442052 −0.221026 0.975268i \(-0.570940\pi\)
−0.221026 + 0.975268i \(0.570940\pi\)
\(212\) 638558. 0.975800
\(213\) 90537.7 0.136735
\(214\) −77817.0 −0.116156
\(215\) 0 0
\(216\) −99583.2 −0.145228
\(217\) 521227. 0.751412
\(218\) −220445. −0.314165
\(219\) −645693. −0.909737
\(220\) 0 0
\(221\) 304779. 0.419763
\(222\) 113306. 0.154301
\(223\) −1.12395e6 −1.51351 −0.756756 0.653698i \(-0.773217\pi\)
−0.756756 + 0.653698i \(0.773217\pi\)
\(224\) −820583. −1.09270
\(225\) 0 0
\(226\) 352212. 0.458705
\(227\) −998160. −1.28569 −0.642844 0.765997i \(-0.722246\pi\)
−0.642844 + 0.765997i \(0.722246\pi\)
\(228\) 7872.60 0.0100295
\(229\) −964804. −1.21577 −0.607884 0.794026i \(-0.707981\pi\)
−0.607884 + 0.794026i \(0.707981\pi\)
\(230\) 0 0
\(231\) −159277. −0.196392
\(232\) −224283. −0.273575
\(233\) −550785. −0.664649 −0.332324 0.943165i \(-0.607833\pi\)
−0.332324 + 0.943165i \(0.607833\pi\)
\(234\) 29900.3 0.0356973
\(235\) 0 0
\(236\) −132850. −0.155268
\(237\) −128754. −0.148898
\(238\) −657144. −0.752001
\(239\) −413388. −0.468127 −0.234063 0.972221i \(-0.575202\pi\)
−0.234063 + 0.972221i \(0.575202\pi\)
\(240\) 0 0
\(241\) 213029. 0.236263 0.118132 0.992998i \(-0.462310\pi\)
0.118132 + 0.992998i \(0.462310\pi\)
\(242\) 34154.0 0.0374889
\(243\) 59049.0 0.0641500
\(244\) 127449. 0.137045
\(245\) 0 0
\(246\) 58858.6 0.0620115
\(247\) −5211.90 −0.00543568
\(248\) 486811. 0.502610
\(249\) −598181. −0.611412
\(250\) 0 0
\(251\) 1.62034e6 1.62339 0.811694 0.584082i \(-0.198546\pi\)
0.811694 + 0.584082i \(0.198546\pi\)
\(252\) 314637. 0.312110
\(253\) −251325. −0.246850
\(254\) −812993. −0.790683
\(255\) 0 0
\(256\) −314951. −0.300361
\(257\) 1.03963e6 0.981849 0.490924 0.871202i \(-0.336659\pi\)
0.490924 + 0.871202i \(0.336659\pi\)
\(258\) 23938.9 0.0223901
\(259\) −789340. −0.731164
\(260\) 0 0
\(261\) 132991. 0.120843
\(262\) 138758. 0.124883
\(263\) 207828. 0.185274 0.0926372 0.995700i \(-0.470470\pi\)
0.0926372 + 0.995700i \(0.470470\pi\)
\(264\) −148760. −0.131364
\(265\) 0 0
\(266\) 11237.6 0.00973796
\(267\) −99941.6 −0.0857963
\(268\) 906335. 0.770818
\(269\) 302188. 0.254622 0.127311 0.991863i \(-0.459365\pi\)
0.127311 + 0.991863i \(0.459365\pi\)
\(270\) 0 0
\(271\) 1.17189e6 0.969310 0.484655 0.874705i \(-0.338945\pi\)
0.484655 + 0.874705i \(0.338945\pi\)
\(272\) 1.02312e6 0.838499
\(273\) −208299. −0.169153
\(274\) 345842. 0.278292
\(275\) 0 0
\(276\) 496468. 0.392300
\(277\) 2.15093e6 1.68433 0.842164 0.539221i \(-0.181281\pi\)
0.842164 + 0.539221i \(0.181281\pi\)
\(278\) −458829. −0.356073
\(279\) −288660. −0.222012
\(280\) 0 0
\(281\) −2.31245e6 −1.74705 −0.873527 0.486775i \(-0.838173\pi\)
−0.873527 + 0.486775i \(0.838173\pi\)
\(282\) 331670. 0.248361
\(283\) −655859. −0.486793 −0.243397 0.969927i \(-0.578262\pi\)
−0.243397 + 0.969927i \(0.578262\pi\)
\(284\) −267169. −0.196558
\(285\) 0 0
\(286\) 44665.9 0.0322894
\(287\) −410036. −0.293844
\(288\) 454446. 0.322850
\(289\) 2.28977e6 1.61268
\(290\) 0 0
\(291\) −508703. −0.352154
\(292\) 1.90538e6 1.30775
\(293\) 602815. 0.410218 0.205109 0.978739i \(-0.434245\pi\)
0.205109 + 0.978739i \(0.434245\pi\)
\(294\) 96260.3 0.0649500
\(295\) 0 0
\(296\) −737221. −0.489067
\(297\) 88209.0 0.0580259
\(298\) −713394. −0.465360
\(299\) −328677. −0.212614
\(300\) 0 0
\(301\) −166770. −0.106096
\(302\) −514189. −0.324418
\(303\) −195382. −0.122258
\(304\) −17495.9 −0.0108581
\(305\) 0 0
\(306\) 363932. 0.222186
\(307\) 62538.0 0.0378703 0.0189351 0.999821i \(-0.493972\pi\)
0.0189351 + 0.999821i \(0.493972\pi\)
\(308\) 470013. 0.282314
\(309\) −138977. −0.0828031
\(310\) 0 0
\(311\) −1.16927e6 −0.685510 −0.342755 0.939425i \(-0.611360\pi\)
−0.342755 + 0.939425i \(0.611360\pi\)
\(312\) −194545. −0.113145
\(313\) −2.51021e6 −1.44827 −0.724135 0.689659i \(-0.757761\pi\)
−0.724135 + 0.689659i \(0.757761\pi\)
\(314\) 1.36529e6 0.781449
\(315\) 0 0
\(316\) 379940. 0.214041
\(317\) 1.58370e6 0.885167 0.442584 0.896727i \(-0.354062\pi\)
0.442584 + 0.896727i \(0.354062\pi\)
\(318\) −504794. −0.279928
\(319\) 198666. 0.109307
\(320\) 0 0
\(321\) −300225. −0.162624
\(322\) 708672. 0.380895
\(323\) −63436.9 −0.0338326
\(324\) −174248. −0.0922160
\(325\) 0 0
\(326\) −254623. −0.132695
\(327\) −850495. −0.439848
\(328\) −382961. −0.196549
\(329\) −2.31057e6 −1.17687
\(330\) 0 0
\(331\) −2.73382e6 −1.37151 −0.685757 0.727831i \(-0.740529\pi\)
−0.685757 + 0.727831i \(0.740529\pi\)
\(332\) 1.76518e6 0.878908
\(333\) 437144. 0.216030
\(334\) 567641. 0.278424
\(335\) 0 0
\(336\) −699242. −0.337893
\(337\) 3.08115e6 1.47787 0.738937 0.673774i \(-0.235328\pi\)
0.738937 + 0.673774i \(0.235328\pi\)
\(338\) −807725. −0.384567
\(339\) 1.35886e6 0.642210
\(340\) 0 0
\(341\) −431209. −0.200817
\(342\) −6223.46 −0.00287718
\(343\) 1.78760e6 0.820416
\(344\) −155758. −0.0709666
\(345\) 0 0
\(346\) 930551. 0.417878
\(347\) 1.31963e6 0.588338 0.294169 0.955753i \(-0.404957\pi\)
0.294169 + 0.955753i \(0.404957\pi\)
\(348\) −392446. −0.173713
\(349\) −4.16706e6 −1.83133 −0.915664 0.401944i \(-0.868335\pi\)
−0.915664 + 0.401944i \(0.868335\pi\)
\(350\) 0 0
\(351\) 115358. 0.0499781
\(352\) 678864. 0.292029
\(353\) −3.56475e6 −1.52262 −0.761310 0.648388i \(-0.775444\pi\)
−0.761310 + 0.648388i \(0.775444\pi\)
\(354\) 105021. 0.0445419
\(355\) 0 0
\(356\) 294919. 0.123333
\(357\) −2.53532e6 −1.05284
\(358\) 464042. 0.191359
\(359\) −402049. −0.164643 −0.0823215 0.996606i \(-0.526233\pi\)
−0.0823215 + 0.996606i \(0.526233\pi\)
\(360\) 0 0
\(361\) −2.47501e6 −0.999562
\(362\) −1.10599e6 −0.443589
\(363\) 131769. 0.0524864
\(364\) 614673. 0.243159
\(365\) 0 0
\(366\) −100751. −0.0393141
\(367\) −1.69899e6 −0.658455 −0.329227 0.944251i \(-0.606788\pi\)
−0.329227 + 0.944251i \(0.606788\pi\)
\(368\) −1.10334e6 −0.424708
\(369\) 227081. 0.0868192
\(370\) 0 0
\(371\) 3.51663e6 1.32645
\(372\) 851812. 0.319144
\(373\) 1.78079e6 0.662738 0.331369 0.943501i \(-0.392490\pi\)
0.331369 + 0.943501i \(0.392490\pi\)
\(374\) 543652. 0.200975
\(375\) 0 0
\(376\) −2.15800e6 −0.787195
\(377\) 259811. 0.0941466
\(378\) −248727. −0.0895352
\(379\) −2.12968e6 −0.761580 −0.380790 0.924662i \(-0.624348\pi\)
−0.380790 + 0.924662i \(0.624348\pi\)
\(380\) 0 0
\(381\) −3.13660e6 −1.10700
\(382\) 470165. 0.164851
\(383\) −3.06780e6 −1.06864 −0.534318 0.845284i \(-0.679432\pi\)
−0.534318 + 0.845284i \(0.679432\pi\)
\(384\) −1.69791e6 −0.587607
\(385\) 0 0
\(386\) −478661. −0.163516
\(387\) 92358.4 0.0313472
\(388\) 1.50114e6 0.506223
\(389\) 784173. 0.262747 0.131374 0.991333i \(-0.458061\pi\)
0.131374 + 0.991333i \(0.458061\pi\)
\(390\) 0 0
\(391\) −4.00051e6 −1.32335
\(392\) −626315. −0.205863
\(393\) 535340. 0.174843
\(394\) 233033. 0.0756269
\(395\) 0 0
\(396\) −260297. −0.0834125
\(397\) −4.54374e6 −1.44690 −0.723448 0.690379i \(-0.757444\pi\)
−0.723448 + 0.690379i \(0.757444\pi\)
\(398\) −866306. −0.274134
\(399\) 43355.5 0.0136336
\(400\) 0 0
\(401\) 3.45110e6 1.07176 0.535879 0.844295i \(-0.319980\pi\)
0.535879 + 0.844295i \(0.319980\pi\)
\(402\) −716478. −0.221125
\(403\) −563926. −0.172965
\(404\) 576556. 0.175747
\(405\) 0 0
\(406\) −560188. −0.168663
\(407\) 653017. 0.195406
\(408\) −2.36791e6 −0.704232
\(409\) 3.80281e6 1.12408 0.562039 0.827111i \(-0.310017\pi\)
0.562039 + 0.827111i \(0.310017\pi\)
\(410\) 0 0
\(411\) 1.33429e6 0.389624
\(412\) 410109. 0.119030
\(413\) −731626. −0.211064
\(414\) −392469. −0.112539
\(415\) 0 0
\(416\) 887804. 0.251527
\(417\) −1.77020e6 −0.498520
\(418\) −9296.78 −0.00260251
\(419\) −2.60701e6 −0.725450 −0.362725 0.931896i \(-0.618153\pi\)
−0.362725 + 0.931896i \(0.618153\pi\)
\(420\) 0 0
\(421\) −5.05826e6 −1.39090 −0.695449 0.718575i \(-0.744795\pi\)
−0.695449 + 0.718575i \(0.744795\pi\)
\(422\) −666883. −0.182292
\(423\) 1.27961e6 0.347718
\(424\) 3.28443e6 0.887248
\(425\) 0 0
\(426\) 211203. 0.0563866
\(427\) 701880. 0.186292
\(428\) 885937. 0.233773
\(429\) 172325. 0.0452069
\(430\) 0 0
\(431\) −3.62455e6 −0.939856 −0.469928 0.882705i \(-0.655720\pi\)
−0.469928 + 0.882705i \(0.655720\pi\)
\(432\) 387246. 0.0998339
\(433\) 6.53089e6 1.67399 0.836995 0.547211i \(-0.184310\pi\)
0.836995 + 0.547211i \(0.184310\pi\)
\(434\) 1.21590e6 0.309866
\(435\) 0 0
\(436\) 2.50974e6 0.632283
\(437\) 68411.1 0.0171365
\(438\) −1.50625e6 −0.375155
\(439\) −4.25980e6 −1.05494 −0.527471 0.849573i \(-0.676860\pi\)
−0.527471 + 0.849573i \(0.676860\pi\)
\(440\) 0 0
\(441\) 371381. 0.0909333
\(442\) 710977. 0.173101
\(443\) −1.87861e6 −0.454807 −0.227404 0.973801i \(-0.573024\pi\)
−0.227404 + 0.973801i \(0.573024\pi\)
\(444\) −1.28997e6 −0.310544
\(445\) 0 0
\(446\) −2.62191e6 −0.624139
\(447\) −2.75234e6 −0.651527
\(448\) 571969. 0.134641
\(449\) −296649. −0.0694428 −0.0347214 0.999397i \(-0.511054\pi\)
−0.0347214 + 0.999397i \(0.511054\pi\)
\(450\) 0 0
\(451\) 339220. 0.0785309
\(452\) −4.00989e6 −0.923180
\(453\) −1.98378e6 −0.454202
\(454\) −2.32847e6 −0.530189
\(455\) 0 0
\(456\) 40492.8 0.00911938
\(457\) 2.30010e6 0.515177 0.257588 0.966255i \(-0.417072\pi\)
0.257588 + 0.966255i \(0.417072\pi\)
\(458\) −2.25066e6 −0.501356
\(459\) 1.40408e6 0.311072
\(460\) 0 0
\(461\) 7.72547e6 1.69306 0.846530 0.532340i \(-0.178687\pi\)
0.846530 + 0.532340i \(0.178687\pi\)
\(462\) −371555. −0.0809876
\(463\) 6.54474e6 1.41886 0.709430 0.704776i \(-0.248952\pi\)
0.709430 + 0.704776i \(0.248952\pi\)
\(464\) 872164. 0.188063
\(465\) 0 0
\(466\) −1.28485e6 −0.274087
\(467\) −6.94728e6 −1.47409 −0.737043 0.675846i \(-0.763778\pi\)
−0.737043 + 0.675846i \(0.763778\pi\)
\(468\) −340411. −0.0718438
\(469\) 4.99132e6 1.04781
\(470\) 0 0
\(471\) 5.26741e6 1.09407
\(472\) −683317. −0.141178
\(473\) 137968. 0.0283546
\(474\) −300351. −0.0614022
\(475\) 0 0
\(476\) 7.48151e6 1.51346
\(477\) −1.94754e6 −0.391914
\(478\) −964336. −0.193045
\(479\) −3.98195e6 −0.792971 −0.396485 0.918041i \(-0.629770\pi\)
−0.396485 + 0.918041i \(0.629770\pi\)
\(480\) 0 0
\(481\) 854002. 0.168305
\(482\) 496946. 0.0974298
\(483\) 2.73412e6 0.533273
\(484\) −388839. −0.0754495
\(485\) 0 0
\(486\) 137747. 0.0264541
\(487\) −2.10449e6 −0.402090 −0.201045 0.979582i \(-0.564434\pi\)
−0.201045 + 0.979582i \(0.564434\pi\)
\(488\) 655535. 0.124608
\(489\) −982358. −0.185779
\(490\) 0 0
\(491\) −4.60384e6 −0.861820 −0.430910 0.902395i \(-0.641807\pi\)
−0.430910 + 0.902395i \(0.641807\pi\)
\(492\) −670098. −0.124803
\(493\) 3.16230e6 0.585985
\(494\) −12158.1 −0.00224156
\(495\) 0 0
\(496\) −1.89305e6 −0.345508
\(497\) −1.47134e6 −0.267191
\(498\) −1.39541e6 −0.252133
\(499\) −2.17027e6 −0.390177 −0.195089 0.980786i \(-0.562499\pi\)
−0.195089 + 0.980786i \(0.562499\pi\)
\(500\) 0 0
\(501\) 2.19001e6 0.389808
\(502\) 3.77987e6 0.669450
\(503\) 1.16557e6 0.205409 0.102705 0.994712i \(-0.467250\pi\)
0.102705 + 0.994712i \(0.467250\pi\)
\(504\) 1.61834e6 0.283787
\(505\) 0 0
\(506\) −586281. −0.101796
\(507\) −3.11627e6 −0.538413
\(508\) 9.25583e6 1.59131
\(509\) 498433. 0.0852732 0.0426366 0.999091i \(-0.486424\pi\)
0.0426366 + 0.999091i \(0.486424\pi\)
\(510\) 0 0
\(511\) 1.04932e7 1.77769
\(512\) 5.30231e6 0.893903
\(513\) −24010.7 −0.00402820
\(514\) 2.42520e6 0.404893
\(515\) 0 0
\(516\) −272542. −0.0450618
\(517\) 1.91152e6 0.314523
\(518\) −1.84134e6 −0.301516
\(519\) 3.59015e6 0.585051
\(520\) 0 0
\(521\) 1.11019e6 0.179186 0.0895929 0.995978i \(-0.471443\pi\)
0.0895929 + 0.995978i \(0.471443\pi\)
\(522\) 310237. 0.0498331
\(523\) 5.52154e6 0.882686 0.441343 0.897338i \(-0.354502\pi\)
0.441343 + 0.897338i \(0.354502\pi\)
\(524\) −1.57974e6 −0.251338
\(525\) 0 0
\(526\) 484814. 0.0764031
\(527\) −6.86384e6 −1.07657
\(528\) 578479. 0.0903031
\(529\) −2.12215e6 −0.329714
\(530\) 0 0
\(531\) 405181. 0.0623609
\(532\) −127938. −0.0195984
\(533\) 443626. 0.0676392
\(534\) −233140. −0.0353805
\(535\) 0 0
\(536\) 4.66174e6 0.700868
\(537\) 1.79032e6 0.267913
\(538\) 704933. 0.105001
\(539\) 554779. 0.0822523
\(540\) 0 0
\(541\) −1.17913e7 −1.73208 −0.866039 0.499977i \(-0.833342\pi\)
−0.866039 + 0.499977i \(0.833342\pi\)
\(542\) 2.73373e6 0.399722
\(543\) −4.26702e6 −0.621047
\(544\) 1.08059e7 1.56554
\(545\) 0 0
\(546\) −485912. −0.0697552
\(547\) 1.11424e7 1.59224 0.796121 0.605137i \(-0.206882\pi\)
0.796121 + 0.605137i \(0.206882\pi\)
\(548\) −3.93737e6 −0.560086
\(549\) −388707. −0.0550417
\(550\) 0 0
\(551\) −54077.3 −0.00758815
\(552\) 2.55359e6 0.356700
\(553\) 2.09239e6 0.290957
\(554\) 5.01761e6 0.694580
\(555\) 0 0
\(556\) 5.22371e6 0.716626
\(557\) −8.68222e6 −1.18575 −0.592875 0.805295i \(-0.702007\pi\)
−0.592875 + 0.805295i \(0.702007\pi\)
\(558\) −673376. −0.0915529
\(559\) 180431. 0.0244220
\(560\) 0 0
\(561\) 2.09746e6 0.281375
\(562\) −5.39439e6 −0.720447
\(563\) 2.59561e6 0.345118 0.172559 0.984999i \(-0.444796\pi\)
0.172559 + 0.984999i \(0.444796\pi\)
\(564\) −3.77603e6 −0.499847
\(565\) 0 0
\(566\) −1.52996e6 −0.200743
\(567\) −959611. −0.125354
\(568\) −1.37419e6 −0.178721
\(569\) 1.43008e7 1.85174 0.925869 0.377846i \(-0.123335\pi\)
0.925869 + 0.377846i \(0.123335\pi\)
\(570\) 0 0
\(571\) 3.13885e6 0.402884 0.201442 0.979500i \(-0.435437\pi\)
0.201442 + 0.979500i \(0.435437\pi\)
\(572\) −508515. −0.0649851
\(573\) 1.81394e6 0.230800
\(574\) −956516. −0.121175
\(575\) 0 0
\(576\) −316761. −0.0397810
\(577\) 7.88699e6 0.986215 0.493108 0.869968i \(-0.335861\pi\)
0.493108 + 0.869968i \(0.335861\pi\)
\(578\) 5.34149e6 0.665033
\(579\) −1.84671e6 −0.228930
\(580\) 0 0
\(581\) 9.72109e6 1.19474
\(582\) −1.18668e6 −0.145220
\(583\) −2.90929e6 −0.354499
\(584\) 9.80035e6 1.18908
\(585\) 0 0
\(586\) 1.40622e6 0.169165
\(587\) 1.91771e6 0.229714 0.114857 0.993382i \(-0.463359\pi\)
0.114857 + 0.993382i \(0.463359\pi\)
\(588\) −1.09591e6 −0.130717
\(589\) 117376. 0.0139409
\(590\) 0 0
\(591\) 899061. 0.105882
\(592\) 2.86681e6 0.336198
\(593\) 1.20632e7 1.40872 0.704362 0.709841i \(-0.251233\pi\)
0.704362 + 0.709841i \(0.251233\pi\)
\(594\) 205771. 0.0239286
\(595\) 0 0
\(596\) 8.12190e6 0.936574
\(597\) −3.34228e6 −0.383802
\(598\) −766726. −0.0876773
\(599\) −5.17336e6 −0.589123 −0.294562 0.955633i \(-0.595174\pi\)
−0.294562 + 0.955633i \(0.595174\pi\)
\(600\) 0 0
\(601\) 9.40838e6 1.06250 0.531250 0.847215i \(-0.321723\pi\)
0.531250 + 0.847215i \(0.321723\pi\)
\(602\) −389034. −0.0437518
\(603\) −2.76423e6 −0.309586
\(604\) 5.85398e6 0.652918
\(605\) 0 0
\(606\) −455781. −0.0504167
\(607\) −368372. −0.0405802 −0.0202901 0.999794i \(-0.506459\pi\)
−0.0202901 + 0.999794i \(0.506459\pi\)
\(608\) −184788. −0.0202729
\(609\) −2.16126e6 −0.236136
\(610\) 0 0
\(611\) 2.49985e6 0.270901
\(612\) −4.14333e6 −0.447168
\(613\) 6.57960e6 0.707210 0.353605 0.935395i \(-0.384956\pi\)
0.353605 + 0.935395i \(0.384956\pi\)
\(614\) 145886. 0.0156169
\(615\) 0 0
\(616\) 2.41751e6 0.256695
\(617\) −140601. −0.0148688 −0.00743441 0.999972i \(-0.502366\pi\)
−0.00743441 + 0.999972i \(0.502366\pi\)
\(618\) −324200. −0.0341462
\(619\) 1.69769e7 1.78087 0.890433 0.455115i \(-0.150402\pi\)
0.890433 + 0.455115i \(0.150402\pi\)
\(620\) 0 0
\(621\) −1.51418e6 −0.157561
\(622\) −2.72763e6 −0.282689
\(623\) 1.62416e6 0.167652
\(624\) 756523. 0.0777787
\(625\) 0 0
\(626\) −5.85572e6 −0.597234
\(627\) −35867.8 −0.00364364
\(628\) −1.55437e7 −1.57273
\(629\) 1.03945e7 1.04756
\(630\) 0 0
\(631\) −1.16982e7 −1.16962 −0.584811 0.811170i \(-0.698831\pi\)
−0.584811 + 0.811170i \(0.698831\pi\)
\(632\) 1.95423e6 0.194618
\(633\) −2.57289e6 −0.255219
\(634\) 3.69440e6 0.365023
\(635\) 0 0
\(636\) 5.74702e6 0.563378
\(637\) 725528. 0.0708444
\(638\) 463441. 0.0450757
\(639\) 814839. 0.0789441
\(640\) 0 0
\(641\) 1.21774e6 0.117060 0.0585301 0.998286i \(-0.481359\pi\)
0.0585301 + 0.998286i \(0.481359\pi\)
\(642\) −700353. −0.0670625
\(643\) 1.22181e7 1.16541 0.582703 0.812685i \(-0.301995\pi\)
0.582703 + 0.812685i \(0.301995\pi\)
\(644\) −8.06815e6 −0.766583
\(645\) 0 0
\(646\) −147983. −0.0139518
\(647\) −1.38354e7 −1.29936 −0.649682 0.760206i \(-0.725098\pi\)
−0.649682 + 0.760206i \(0.725098\pi\)
\(648\) −896248. −0.0838476
\(649\) 605270. 0.0564076
\(650\) 0 0
\(651\) 4.69105e6 0.433828
\(652\) 2.89885e6 0.267059
\(653\) −7.07482e6 −0.649280 −0.324640 0.945838i \(-0.605243\pi\)
−0.324640 + 0.945838i \(0.605243\pi\)
\(654\) −1.98400e6 −0.181383
\(655\) 0 0
\(656\) 1.48921e6 0.135113
\(657\) −5.81124e6 −0.525237
\(658\) −5.39001e6 −0.485316
\(659\) −1.42464e7 −1.27788 −0.638942 0.769255i \(-0.720627\pi\)
−0.638942 + 0.769255i \(0.720627\pi\)
\(660\) 0 0
\(661\) −1.44156e7 −1.28330 −0.641651 0.766997i \(-0.721750\pi\)
−0.641651 + 0.766997i \(0.721750\pi\)
\(662\) −6.37735e6 −0.565582
\(663\) 2.74301e6 0.242350
\(664\) 9.07921e6 0.799149
\(665\) 0 0
\(666\) 1.01975e6 0.0890859
\(667\) −3.41026e6 −0.296807
\(668\) −6.46252e6 −0.560352
\(669\) −1.01156e7 −0.873826
\(670\) 0 0
\(671\) −580662. −0.0497871
\(672\) −7.38525e6 −0.630873
\(673\) 4.41234e6 0.375519 0.187759 0.982215i \(-0.439877\pi\)
0.187759 + 0.982215i \(0.439877\pi\)
\(674\) 7.18758e6 0.609443
\(675\) 0 0
\(676\) 9.19586e6 0.773972
\(677\) −2.15482e7 −1.80692 −0.903461 0.428670i \(-0.858982\pi\)
−0.903461 + 0.428670i \(0.858982\pi\)
\(678\) 3.16991e6 0.264833
\(679\) 8.26699e6 0.688134
\(680\) 0 0
\(681\) −8.98344e6 −0.742292
\(682\) −1.00591e6 −0.0828127
\(683\) 1.93575e7 1.58781 0.793903 0.608045i \(-0.208046\pi\)
0.793903 + 0.608045i \(0.208046\pi\)
\(684\) 70853.4 0.00579056
\(685\) 0 0
\(686\) 4.17004e6 0.338321
\(687\) −8.68324e6 −0.701923
\(688\) 605691. 0.0487843
\(689\) −3.80471e6 −0.305333
\(690\) 0 0
\(691\) 7.29763e6 0.581416 0.290708 0.956812i \(-0.406109\pi\)
0.290708 + 0.956812i \(0.406109\pi\)
\(692\) −1.05942e7 −0.841014
\(693\) −1.43349e6 −0.113387
\(694\) 3.07837e6 0.242618
\(695\) 0 0
\(696\) −2.01855e6 −0.157949
\(697\) 5.39960e6 0.420998
\(698\) −9.72076e6 −0.755200
\(699\) −4.95707e6 −0.383735
\(700\) 0 0
\(701\) 1.83574e7 1.41096 0.705481 0.708729i \(-0.250731\pi\)
0.705481 + 0.708729i \(0.250731\pi\)
\(702\) 269103. 0.0206099
\(703\) −177753. −0.0135652
\(704\) −473187. −0.0359833
\(705\) 0 0
\(706\) −8.31570e6 −0.627895
\(707\) 3.17518e6 0.238902
\(708\) −1.19565e6 −0.0896442
\(709\) 749166. 0.0559709 0.0279855 0.999608i \(-0.491091\pi\)
0.0279855 + 0.999608i \(0.491091\pi\)
\(710\) 0 0
\(711\) −1.15878e6 −0.0859662
\(712\) 1.51692e6 0.112140
\(713\) 7.40205e6 0.545291
\(714\) −5.91430e6 −0.434168
\(715\) 0 0
\(716\) −5.28306e6 −0.385126
\(717\) −3.72049e6 −0.270273
\(718\) −937885. −0.0678951
\(719\) 1.65136e7 1.19129 0.595647 0.803246i \(-0.296896\pi\)
0.595647 + 0.803246i \(0.296896\pi\)
\(720\) 0 0
\(721\) 2.25853e6 0.161803
\(722\) −5.77362e6 −0.412197
\(723\) 1.91726e6 0.136407
\(724\) 1.25916e7 0.892759
\(725\) 0 0
\(726\) 307386. 0.0216442
\(727\) 1.79988e7 1.26301 0.631505 0.775372i \(-0.282438\pi\)
0.631505 + 0.775372i \(0.282438\pi\)
\(728\) 3.16157e6 0.221093
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.19612e6 0.152007
\(732\) 1.14704e6 0.0791228
\(733\) 127875. 0.00879075 0.00439537 0.999990i \(-0.498601\pi\)
0.00439537 + 0.999990i \(0.498601\pi\)
\(734\) −3.96334e6 −0.271532
\(735\) 0 0
\(736\) −1.16533e7 −0.792962
\(737\) −4.12929e6 −0.280031
\(738\) 529727. 0.0358023
\(739\) −7.32976e6 −0.493718 −0.246859 0.969051i \(-0.579398\pi\)
−0.246859 + 0.969051i \(0.579398\pi\)
\(740\) 0 0
\(741\) −46907.1 −0.00313829
\(742\) 8.20346e6 0.547000
\(743\) 2.18733e7 1.45359 0.726797 0.686852i \(-0.241008\pi\)
0.726797 + 0.686852i \(0.241008\pi\)
\(744\) 4.38130e6 0.290182
\(745\) 0 0
\(746\) 4.15417e6 0.273298
\(747\) −5.38363e6 −0.352999
\(748\) −6.18941e6 −0.404479
\(749\) 4.87898e6 0.317779
\(750\) 0 0
\(751\) −2.28695e7 −1.47964 −0.739820 0.672805i \(-0.765089\pi\)
−0.739820 + 0.672805i \(0.765089\pi\)
\(752\) 8.39177e6 0.541139
\(753\) 1.45831e7 0.937264
\(754\) 606078. 0.0388240
\(755\) 0 0
\(756\) 2.83173e6 0.180197
\(757\) 1.68271e7 1.06726 0.533629 0.845719i \(-0.320828\pi\)
0.533629 + 0.845719i \(0.320828\pi\)
\(758\) −4.96803e6 −0.314059
\(759\) −2.26192e6 −0.142519
\(760\) 0 0
\(761\) −1.72838e7 −1.08188 −0.540938 0.841062i \(-0.681931\pi\)
−0.540938 + 0.841062i \(0.681931\pi\)
\(762\) −7.31694e6 −0.456501
\(763\) 1.38215e7 0.859494
\(764\) −5.35277e6 −0.331776
\(765\) 0 0
\(766\) −7.15644e6 −0.440682
\(767\) 791560. 0.0485842
\(768\) −2.83456e6 −0.173413
\(769\) 1.11705e7 0.681173 0.340587 0.940213i \(-0.389374\pi\)
0.340587 + 0.940213i \(0.389374\pi\)
\(770\) 0 0
\(771\) 9.35664e6 0.566871
\(772\) 5.44950e6 0.329089
\(773\) −1.77002e7 −1.06544 −0.532719 0.846292i \(-0.678830\pi\)
−0.532719 + 0.846292i \(0.678830\pi\)
\(774\) 215450. 0.0129269
\(775\) 0 0
\(776\) 7.72112e6 0.460284
\(777\) −7.10406e6 −0.422138
\(778\) 1.82929e6 0.108351
\(779\) −92336.5 −0.00545167
\(780\) 0 0
\(781\) 1.21723e6 0.0714076
\(782\) −9.33223e6 −0.545718
\(783\) 1.19692e6 0.0697688
\(784\) 2.43553e6 0.141515
\(785\) 0 0
\(786\) 1.24882e6 0.0721015
\(787\) −3.55985e6 −0.204878 −0.102439 0.994739i \(-0.532665\pi\)
−0.102439 + 0.994739i \(0.532665\pi\)
\(788\) −2.65305e6 −0.152205
\(789\) 1.87045e6 0.106968
\(790\) 0 0
\(791\) −2.20830e7 −1.25492
\(792\) −1.33884e6 −0.0758430
\(793\) −759377. −0.0428820
\(794\) −1.05995e7 −0.596668
\(795\) 0 0
\(796\) 9.86279e6 0.551718
\(797\) 2.09403e7 1.16771 0.583857 0.811856i \(-0.301543\pi\)
0.583857 + 0.811856i \(0.301543\pi\)
\(798\) 101138. 0.00562222
\(799\) 3.04270e7 1.68613
\(800\) 0 0
\(801\) −899475. −0.0495345
\(802\) 8.05060e6 0.441970
\(803\) −8.68098e6 −0.475094
\(804\) 8.15702e6 0.445032
\(805\) 0 0
\(806\) −1.31550e6 −0.0713271
\(807\) 2.71969e6 0.147006
\(808\) 2.96552e6 0.159799
\(809\) −2.22981e7 −1.19784 −0.598918 0.800810i \(-0.704402\pi\)
−0.598918 + 0.800810i \(0.704402\pi\)
\(810\) 0 0
\(811\) −7.95262e6 −0.424578 −0.212289 0.977207i \(-0.568092\pi\)
−0.212289 + 0.977207i \(0.568092\pi\)
\(812\) 6.37768e6 0.339447
\(813\) 1.05470e7 0.559631
\(814\) 1.52333e6 0.0805812
\(815\) 0 0
\(816\) 9.20805e6 0.484108
\(817\) −37555.0 −0.00196840
\(818\) 8.87105e6 0.463545
\(819\) −1.87469e6 −0.0976608
\(820\) 0 0
\(821\) −1.23384e7 −0.638852 −0.319426 0.947611i \(-0.603490\pi\)
−0.319426 + 0.947611i \(0.603490\pi\)
\(822\) 3.11258e6 0.160672
\(823\) −1.87024e6 −0.0962493 −0.0481247 0.998841i \(-0.515324\pi\)
−0.0481247 + 0.998841i \(0.515324\pi\)
\(824\) 2.10940e6 0.108228
\(825\) 0 0
\(826\) −1.70671e6 −0.0870381
\(827\) −2.63791e7 −1.34121 −0.670604 0.741816i \(-0.733965\pi\)
−0.670604 + 0.741816i \(0.733965\pi\)
\(828\) 4.46821e6 0.226495
\(829\) 2.85869e7 1.44471 0.722355 0.691522i \(-0.243060\pi\)
0.722355 + 0.691522i \(0.243060\pi\)
\(830\) 0 0
\(831\) 1.93584e7 0.972447
\(832\) −618824. −0.0309926
\(833\) 8.83079e6 0.440948
\(834\) −4.12946e6 −0.205579
\(835\) 0 0
\(836\) 105843. 0.00523776
\(837\) −2.59794e6 −0.128179
\(838\) −6.08153e6 −0.299159
\(839\) −1.62633e7 −0.797634 −0.398817 0.917031i \(-0.630579\pi\)
−0.398817 + 0.917031i \(0.630579\pi\)
\(840\) 0 0
\(841\) −1.78154e7 −0.868572
\(842\) −1.17997e7 −0.573576
\(843\) −2.08120e7 −1.00866
\(844\) 7.59238e6 0.366878
\(845\) 0 0
\(846\) 2.98503e6 0.143391
\(847\) −2.14139e6 −0.102562
\(848\) −1.27721e7 −0.609918
\(849\) −5.90273e6 −0.281050
\(850\) 0 0
\(851\) −1.12096e7 −0.530597
\(852\) −2.40452e6 −0.113483
\(853\) 1.29208e6 0.0608021 0.0304010 0.999538i \(-0.490322\pi\)
0.0304010 + 0.999538i \(0.490322\pi\)
\(854\) 1.63732e6 0.0768225
\(855\) 0 0
\(856\) 4.55683e6 0.212558
\(857\) −8.96034e6 −0.416747 −0.208374 0.978049i \(-0.566817\pi\)
−0.208374 + 0.978049i \(0.566817\pi\)
\(858\) 401993. 0.0186423
\(859\) −4.85453e6 −0.224473 −0.112236 0.993682i \(-0.535801\pi\)
−0.112236 + 0.993682i \(0.535801\pi\)
\(860\) 0 0
\(861\) −3.69032e6 −0.169651
\(862\) −8.45522e6 −0.387576
\(863\) 6.39862e6 0.292455 0.146228 0.989251i \(-0.453287\pi\)
0.146228 + 0.989251i \(0.453287\pi\)
\(864\) 4.09001e6 0.186398
\(865\) 0 0
\(866\) 1.52350e7 0.690316
\(867\) 2.06079e7 0.931080
\(868\) −1.38429e7 −0.623630
\(869\) −1.73102e6 −0.0777593
\(870\) 0 0
\(871\) −5.40020e6 −0.241193
\(872\) 1.29088e7 0.574905
\(873\) −4.57833e6 −0.203316
\(874\) 159587. 0.00706673
\(875\) 0 0
\(876\) 1.71485e7 0.755031
\(877\) 1.53595e7 0.674340 0.337170 0.941444i \(-0.390530\pi\)
0.337170 + 0.941444i \(0.390530\pi\)
\(878\) −9.93711e6 −0.435035
\(879\) 5.42533e6 0.236840
\(880\) 0 0
\(881\) −2.14534e7 −0.931228 −0.465614 0.884988i \(-0.654167\pi\)
−0.465614 + 0.884988i \(0.654167\pi\)
\(882\) 866343. 0.0374989
\(883\) −3.30136e6 −0.142492 −0.0712460 0.997459i \(-0.522698\pi\)
−0.0712460 + 0.997459i \(0.522698\pi\)
\(884\) −8.09439e6 −0.348380
\(885\) 0 0
\(886\) −4.38235e6 −0.187553
\(887\) −1.34325e7 −0.573256 −0.286628 0.958042i \(-0.592534\pi\)
−0.286628 + 0.958042i \(0.592534\pi\)
\(888\) −6.63498e6 −0.282363
\(889\) 5.09732e7 2.16315
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 2.98502e7 1.25613
\(893\) −520320. −0.0218344
\(894\) −6.42054e6 −0.268675
\(895\) 0 0
\(896\) 2.75929e7 1.14823
\(897\) −2.95809e6 −0.122753
\(898\) −692011. −0.0286367
\(899\) −5.85114e6 −0.241458
\(900\) 0 0
\(901\) −4.63091e7 −1.90044
\(902\) 791321. 0.0323844
\(903\) −1.50093e6 −0.0612548
\(904\) −2.06249e7 −0.839404
\(905\) 0 0
\(906\) −4.62770e6 −0.187303
\(907\) 2.40573e6 0.0971023 0.0485512 0.998821i \(-0.484540\pi\)
0.0485512 + 0.998821i \(0.484540\pi\)
\(908\) 2.65093e7 1.06705
\(909\) −1.75844e6 −0.0705859
\(910\) 0 0
\(911\) −4.38090e7 −1.74891 −0.874455 0.485107i \(-0.838781\pi\)
−0.874455 + 0.485107i \(0.838781\pi\)
\(912\) −157463. −0.00626891
\(913\) −8.04221e6 −0.319300
\(914\) 5.36559e6 0.212448
\(915\) 0 0
\(916\) 2.56235e7 1.00902
\(917\) −8.69987e6 −0.341656
\(918\) 3.27539e6 0.128279
\(919\) 4.10650e6 0.160392 0.0801962 0.996779i \(-0.474445\pi\)
0.0801962 + 0.996779i \(0.474445\pi\)
\(920\) 0 0
\(921\) 562842. 0.0218644
\(922\) 1.80217e7 0.698181
\(923\) 1.59187e6 0.0615039
\(924\) 4.23011e6 0.162994
\(925\) 0 0
\(926\) 1.52673e7 0.585107
\(927\) −1.25079e6 −0.0478064
\(928\) 9.21161e6 0.351128
\(929\) −7.91355e6 −0.300838 −0.150419 0.988622i \(-0.548062\pi\)
−0.150419 + 0.988622i \(0.548062\pi\)
\(930\) 0 0
\(931\) −151012. −0.00571001
\(932\) 1.46279e7 0.551622
\(933\) −1.05234e7 −0.395780
\(934\) −1.62064e7 −0.607880
\(935\) 0 0
\(936\) −1.75091e6 −0.0653241
\(937\) 5.82636e6 0.216795 0.108397 0.994108i \(-0.465428\pi\)
0.108397 + 0.994108i \(0.465428\pi\)
\(938\) 1.16436e7 0.432094
\(939\) −2.25919e7 −0.836159
\(940\) 0 0
\(941\) −4.94213e7 −1.81945 −0.909725 0.415210i \(-0.863708\pi\)
−0.909725 + 0.415210i \(0.863708\pi\)
\(942\) 1.22876e7 0.451170
\(943\) −5.82300e6 −0.213239
\(944\) 2.65720e6 0.0970496
\(945\) 0 0
\(946\) 321845. 0.0116928
\(947\) 1.11808e7 0.405135 0.202567 0.979268i \(-0.435071\pi\)
0.202567 + 0.979268i \(0.435071\pi\)
\(948\) 3.41946e6 0.123577
\(949\) −1.13528e7 −0.409202
\(950\) 0 0
\(951\) 1.42533e7 0.511051
\(952\) 3.84812e7 1.37612
\(953\) −1.89338e7 −0.675315 −0.337658 0.941269i \(-0.609635\pi\)
−0.337658 + 0.941269i \(0.609635\pi\)
\(954\) −4.54315e6 −0.161617
\(955\) 0 0
\(956\) 1.09789e7 0.388519
\(957\) 1.78799e6 0.0631083
\(958\) −9.28894e6 −0.327004
\(959\) −2.16837e7 −0.761353
\(960\) 0 0
\(961\) −1.59291e7 −0.556396
\(962\) 1.99218e6 0.0694052
\(963\) −2.70202e6 −0.0938909
\(964\) −5.65767e6 −0.196085
\(965\) 0 0
\(966\) 6.37805e6 0.219910
\(967\) 2.87893e7 0.990069 0.495034 0.868873i \(-0.335155\pi\)
0.495034 + 0.868873i \(0.335155\pi\)
\(968\) −2.00000e6 −0.0686026
\(969\) −570932. −0.0195333
\(970\) 0 0
\(971\) 5.23023e7 1.78022 0.890108 0.455750i \(-0.150629\pi\)
0.890108 + 0.455750i \(0.150629\pi\)
\(972\) −1.56824e6 −0.0532410
\(973\) 2.87677e7 0.974145
\(974\) −4.90926e6 −0.165813
\(975\) 0 0
\(976\) −2.54916e6 −0.0856590
\(977\) 5.27070e7 1.76657 0.883286 0.468834i \(-0.155326\pi\)
0.883286 + 0.468834i \(0.155326\pi\)
\(978\) −2.29161e6 −0.0766114
\(979\) −1.34366e6 −0.0448056
\(980\) 0 0
\(981\) −7.65445e6 −0.253946
\(982\) −1.07397e7 −0.355396
\(983\) −5.51463e7 −1.82026 −0.910128 0.414328i \(-0.864017\pi\)
−0.910128 + 0.414328i \(0.864017\pi\)
\(984\) −3.44665e6 −0.113478
\(985\) 0 0
\(986\) 7.37690e6 0.241647
\(987\) −2.07951e7 −0.679467
\(988\) 138419. 0.00451131
\(989\) −2.36833e6 −0.0769929
\(990\) 0 0
\(991\) −4.11719e7 −1.33173 −0.665866 0.746071i \(-0.731938\pi\)
−0.665866 + 0.746071i \(0.731938\pi\)
\(992\) −1.99940e7 −0.645090
\(993\) −2.46044e7 −0.791844
\(994\) −3.43228e6 −0.110184
\(995\) 0 0
\(996\) 1.58866e7 0.507438
\(997\) −4.61912e7 −1.47171 −0.735853 0.677141i \(-0.763219\pi\)
−0.735853 + 0.677141i \(0.763219\pi\)
\(998\) −5.06272e6 −0.160900
\(999\) 3.93429e6 0.124725
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 825.6.a.m.1.5 7
5.4 even 2 825.6.a.o.1.3 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.5 7 1.1 even 1 trivial
825.6.a.o.1.3 yes 7 5.4 even 2