Properties

Label 825.6.a.s.1.7
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $9$
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(6.71211\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} -121.000 q^{11} -117.472 q^{12} -783.433 q^{13} -1189.20 q^{14} -1271.31 q^{16} -1335.12 q^{17} +543.681 q^{18} -915.496 q^{19} +1594.55 q^{21} -812.166 q^{22} -1684.59 q^{23} +1144.60 q^{24} -5258.49 q^{26} -729.000 q^{27} -2312.53 q^{28} +1720.16 q^{29} +7741.92 q^{31} -4463.49 q^{32} +1089.00 q^{33} -8961.50 q^{34} +1057.25 q^{36} -9013.63 q^{37} -6144.91 q^{38} +7050.90 q^{39} +13835.8 q^{41} +10702.8 q^{42} +11885.9 q^{43} -1579.35 q^{44} -11307.2 q^{46} -20567.8 q^{47} +11441.8 q^{48} +14583.0 q^{49} +12016.1 q^{51} -10225.7 q^{52} -3483.63 q^{53} -4893.13 q^{54} +22532.4 q^{56} +8239.46 q^{57} +11545.9 q^{58} -22223.3 q^{59} -40834.7 q^{61} +51964.6 q^{62} -14350.9 q^{63} +10722.5 q^{64} +7309.49 q^{66} +2692.02 q^{67} -17426.6 q^{68} +15161.3 q^{69} +13211.7 q^{71} -10301.4 q^{72} -37232.0 q^{73} -60500.5 q^{74} -11949.5 q^{76} +21437.8 q^{77} +47326.4 q^{78} -46277.2 q^{79} +6561.00 q^{81} +92867.6 q^{82} +401.818 q^{83} +20812.8 q^{84} +79779.7 q^{86} -15481.5 q^{87} +15388.5 q^{88} -33472.8 q^{89} +138803. q^{91} -21988.1 q^{92} -69677.3 q^{93} -138053. q^{94} +40171.4 q^{96} +168527. q^{97} +97882.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.71211 1.18655 0.593273 0.805002i \(-0.297836\pi\)
0.593273 + 0.805002i \(0.297836\pi\)
\(3\) −9.00000 −0.577350
\(4\) 13.0525 0.407889
\(5\) 0 0
\(6\) −60.4090 −0.685052
\(7\) −177.172 −1.36663 −0.683314 0.730124i \(-0.739462\pi\)
−0.683314 + 0.730124i \(0.739462\pi\)
\(8\) −127.178 −0.702566
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −117.472 −0.235495
\(13\) −783.433 −1.28571 −0.642856 0.765987i \(-0.722251\pi\)
−0.642856 + 0.765987i \(0.722251\pi\)
\(14\) −1189.20 −1.62157
\(15\) 0 0
\(16\) −1271.31 −1.24152
\(17\) −1335.12 −1.12047 −0.560233 0.828335i \(-0.689289\pi\)
−0.560233 + 0.828335i \(0.689289\pi\)
\(18\) 543.681 0.395515
\(19\) −915.496 −0.581798 −0.290899 0.956754i \(-0.593954\pi\)
−0.290899 + 0.956754i \(0.593954\pi\)
\(20\) 0 0
\(21\) 1594.55 0.789023
\(22\) −812.166 −0.357757
\(23\) −1684.59 −0.664011 −0.332006 0.943277i \(-0.607725\pi\)
−0.332006 + 0.943277i \(0.607725\pi\)
\(24\) 1144.60 0.405627
\(25\) 0 0
\(26\) −5258.49 −1.52555
\(27\) −729.000 −0.192450
\(28\) −2312.53 −0.557433
\(29\) 1720.16 0.379817 0.189909 0.981802i \(-0.439181\pi\)
0.189909 + 0.981802i \(0.439181\pi\)
\(30\) 0 0
\(31\) 7741.92 1.44692 0.723460 0.690366i \(-0.242551\pi\)
0.723460 + 0.690366i \(0.242551\pi\)
\(32\) −4463.49 −0.770548
\(33\) 1089.00 0.174078
\(34\) −8961.50 −1.32948
\(35\) 0 0
\(36\) 1057.25 0.135963
\(37\) −9013.63 −1.08242 −0.541209 0.840888i \(-0.682033\pi\)
−0.541209 + 0.840888i \(0.682033\pi\)
\(38\) −6144.91 −0.690330
\(39\) 7050.90 0.742306
\(40\) 0 0
\(41\) 13835.8 1.28542 0.642710 0.766109i \(-0.277810\pi\)
0.642710 + 0.766109i \(0.277810\pi\)
\(42\) 10702.8 0.936212
\(43\) 11885.9 0.980307 0.490153 0.871636i \(-0.336941\pi\)
0.490153 + 0.871636i \(0.336941\pi\)
\(44\) −1579.35 −0.122983
\(45\) 0 0
\(46\) −11307.2 −0.787880
\(47\) −20567.8 −1.35814 −0.679068 0.734075i \(-0.737616\pi\)
−0.679068 + 0.734075i \(0.737616\pi\)
\(48\) 11441.8 0.716789
\(49\) 14583.0 0.867673
\(50\) 0 0
\(51\) 12016.1 0.646902
\(52\) −10225.7 −0.524428
\(53\) −3483.63 −0.170350 −0.0851750 0.996366i \(-0.527145\pi\)
−0.0851750 + 0.996366i \(0.527145\pi\)
\(54\) −4893.13 −0.228351
\(55\) 0 0
\(56\) 22532.4 0.960147
\(57\) 8239.46 0.335901
\(58\) 11545.9 0.450670
\(59\) −22223.3 −0.831147 −0.415574 0.909560i \(-0.636419\pi\)
−0.415574 + 0.909560i \(0.636419\pi\)
\(60\) 0 0
\(61\) −40834.7 −1.40509 −0.702546 0.711638i \(-0.747954\pi\)
−0.702546 + 0.711638i \(0.747954\pi\)
\(62\) 51964.6 1.71684
\(63\) −14350.9 −0.455543
\(64\) 10722.5 0.327225
\(65\) 0 0
\(66\) 7309.49 0.206551
\(67\) 2692.02 0.0732640 0.0366320 0.999329i \(-0.488337\pi\)
0.0366320 + 0.999329i \(0.488337\pi\)
\(68\) −17426.6 −0.457026
\(69\) 15161.3 0.383367
\(70\) 0 0
\(71\) 13211.7 0.311039 0.155519 0.987833i \(-0.450295\pi\)
0.155519 + 0.987833i \(0.450295\pi\)
\(72\) −10301.4 −0.234189
\(73\) −37232.0 −0.817728 −0.408864 0.912595i \(-0.634075\pi\)
−0.408864 + 0.912595i \(0.634075\pi\)
\(74\) −60500.5 −1.28434
\(75\) 0 0
\(76\) −11949.5 −0.237309
\(77\) 21437.8 0.412054
\(78\) 47326.4 0.880779
\(79\) −46277.2 −0.834257 −0.417128 0.908848i \(-0.636963\pi\)
−0.417128 + 0.908848i \(0.636963\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 92867.6 1.52521
\(83\) 401.818 0.00640227 0.00320113 0.999995i \(-0.498981\pi\)
0.00320113 + 0.999995i \(0.498981\pi\)
\(84\) 20812.8 0.321834
\(85\) 0 0
\(86\) 79779.7 1.16318
\(87\) −15481.5 −0.219288
\(88\) 15388.5 0.211832
\(89\) −33472.8 −0.447937 −0.223969 0.974596i \(-0.571901\pi\)
−0.223969 + 0.974596i \(0.571901\pi\)
\(90\) 0 0
\(91\) 138803. 1.75709
\(92\) −21988.1 −0.270843
\(93\) −69677.3 −0.835380
\(94\) −138053. −1.61149
\(95\) 0 0
\(96\) 40171.4 0.444876
\(97\) 168527. 1.81861 0.909305 0.416130i \(-0.136614\pi\)
0.909305 + 0.416130i \(0.136614\pi\)
\(98\) 97882.6 1.02953
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 176215. 1.71885 0.859425 0.511261i \(-0.170821\pi\)
0.859425 + 0.511261i \(0.170821\pi\)
\(102\) 80653.5 0.767578
\(103\) −66485.7 −0.617497 −0.308749 0.951144i \(-0.599910\pi\)
−0.308749 + 0.951144i \(0.599910\pi\)
\(104\) 99635.5 0.903297
\(105\) 0 0
\(106\) −23382.5 −0.202128
\(107\) 87124.1 0.735663 0.367831 0.929892i \(-0.380100\pi\)
0.367831 + 0.929892i \(0.380100\pi\)
\(108\) −9515.24 −0.0784983
\(109\) 17894.5 0.144263 0.0721314 0.997395i \(-0.477020\pi\)
0.0721314 + 0.997395i \(0.477020\pi\)
\(110\) 0 0
\(111\) 81122.7 0.624935
\(112\) 225241. 1.69669
\(113\) −32973.4 −0.242922 −0.121461 0.992596i \(-0.538758\pi\)
−0.121461 + 0.992596i \(0.538758\pi\)
\(114\) 55304.2 0.398562
\(115\) 0 0
\(116\) 22452.4 0.154923
\(117\) −63458.1 −0.428570
\(118\) −149165. −0.986194
\(119\) 236547. 1.53126
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −274087. −1.66721
\(123\) −124522. −0.742138
\(124\) 101051. 0.590183
\(125\) 0 0
\(126\) −96325.2 −0.540522
\(127\) 297307. 1.63567 0.817836 0.575451i \(-0.195174\pi\)
0.817836 + 0.575451i \(0.195174\pi\)
\(128\) 214803. 1.15882
\(129\) −106973. −0.565980
\(130\) 0 0
\(131\) 44531.9 0.226722 0.113361 0.993554i \(-0.463838\pi\)
0.113361 + 0.993554i \(0.463838\pi\)
\(132\) 14214.1 0.0710044
\(133\) 162200. 0.795102
\(134\) 18069.1 0.0869310
\(135\) 0 0
\(136\) 169798. 0.787202
\(137\) 193093. 0.878953 0.439476 0.898254i \(-0.355164\pi\)
0.439476 + 0.898254i \(0.355164\pi\)
\(138\) 101765. 0.454882
\(139\) −45294.3 −0.198841 −0.0994206 0.995046i \(-0.531699\pi\)
−0.0994206 + 0.995046i \(0.531699\pi\)
\(140\) 0 0
\(141\) 185110. 0.784120
\(142\) 88678.7 0.369061
\(143\) 94795.4 0.387657
\(144\) −102976. −0.413839
\(145\) 0 0
\(146\) −249905. −0.970271
\(147\) −131247. −0.500951
\(148\) −117650. −0.441507
\(149\) −461266. −1.70210 −0.851051 0.525083i \(-0.824034\pi\)
−0.851051 + 0.525083i \(0.824034\pi\)
\(150\) 0 0
\(151\) −356514. −1.27243 −0.636216 0.771511i \(-0.719501\pi\)
−0.636216 + 0.771511i \(0.719501\pi\)
\(152\) 116431. 0.408752
\(153\) −108145. −0.373489
\(154\) 143893. 0.488921
\(155\) 0 0
\(156\) 92031.5 0.302779
\(157\) −92075.8 −0.298124 −0.149062 0.988828i \(-0.547625\pi\)
−0.149062 + 0.988828i \(0.547625\pi\)
\(158\) −310618. −0.989883
\(159\) 31352.7 0.0983517
\(160\) 0 0
\(161\) 298463. 0.907457
\(162\) 44038.2 0.131838
\(163\) −497333. −1.46615 −0.733074 0.680149i \(-0.761915\pi\)
−0.733074 + 0.680149i \(0.761915\pi\)
\(164\) 180591. 0.524309
\(165\) 0 0
\(166\) 2697.05 0.00759658
\(167\) −292934. −0.812792 −0.406396 0.913697i \(-0.633215\pi\)
−0.406396 + 0.913697i \(0.633215\pi\)
\(168\) −202792. −0.554341
\(169\) 242474. 0.653054
\(170\) 0 0
\(171\) −74155.1 −0.193933
\(172\) 155141. 0.399857
\(173\) 446012. 1.13300 0.566502 0.824060i \(-0.308296\pi\)
0.566502 + 0.824060i \(0.308296\pi\)
\(174\) −103913. −0.260195
\(175\) 0 0
\(176\) 153829. 0.374331
\(177\) 200009. 0.479863
\(178\) −224673. −0.531498
\(179\) −414071. −0.965922 −0.482961 0.875642i \(-0.660439\pi\)
−0.482961 + 0.875642i \(0.660439\pi\)
\(180\) 0 0
\(181\) −560334. −1.27131 −0.635654 0.771974i \(-0.719269\pi\)
−0.635654 + 0.771974i \(0.719269\pi\)
\(182\) 931658. 2.08487
\(183\) 367512. 0.811230
\(184\) 214243. 0.466512
\(185\) 0 0
\(186\) −467682. −0.991216
\(187\) 161550. 0.337833
\(188\) −268460. −0.553969
\(189\) 129159. 0.263008
\(190\) 0 0
\(191\) 720559. 1.42918 0.714589 0.699545i \(-0.246614\pi\)
0.714589 + 0.699545i \(0.246614\pi\)
\(192\) −96502.7 −0.188924
\(193\) −432118. −0.835044 −0.417522 0.908667i \(-0.637101\pi\)
−0.417522 + 0.908667i \(0.637101\pi\)
\(194\) 1.13117e6 2.15786
\(195\) 0 0
\(196\) 190344. 0.353915
\(197\) 786560. 1.44400 0.721998 0.691895i \(-0.243224\pi\)
0.721998 + 0.691895i \(0.243224\pi\)
\(198\) −65785.4 −0.119252
\(199\) 284214. 0.508761 0.254380 0.967104i \(-0.418129\pi\)
0.254380 + 0.967104i \(0.418129\pi\)
\(200\) 0 0
\(201\) −24228.1 −0.0422990
\(202\) 1.18277e6 2.03949
\(203\) −304765. −0.519069
\(204\) 156840. 0.263864
\(205\) 0 0
\(206\) −446259. −0.732688
\(207\) −136452. −0.221337
\(208\) 995988. 1.59623
\(209\) 110775. 0.175419
\(210\) 0 0
\(211\) 1.03608e6 1.60209 0.801045 0.598604i \(-0.204278\pi\)
0.801045 + 0.598604i \(0.204278\pi\)
\(212\) −45469.9 −0.0694840
\(213\) −118906. −0.179578
\(214\) 584787. 0.872897
\(215\) 0 0
\(216\) 92712.8 0.135209
\(217\) −1.37165e6 −1.97740
\(218\) 120110. 0.171174
\(219\) 335088. 0.472116
\(220\) 0 0
\(221\) 1.04598e6 1.44060
\(222\) 544504. 0.741513
\(223\) 1.13339e6 1.52622 0.763110 0.646269i \(-0.223672\pi\)
0.763110 + 0.646269i \(0.223672\pi\)
\(224\) 790807. 1.05305
\(225\) 0 0
\(226\) −221321. −0.288238
\(227\) −681222. −0.877453 −0.438727 0.898621i \(-0.644570\pi\)
−0.438727 + 0.898621i \(0.644570\pi\)
\(228\) 107545. 0.137011
\(229\) −620686. −0.782137 −0.391069 0.920361i \(-0.627895\pi\)
−0.391069 + 0.920361i \(0.627895\pi\)
\(230\) 0 0
\(231\) −192941. −0.237899
\(232\) −218767. −0.266847
\(233\) −31388.4 −0.0378774 −0.0189387 0.999821i \(-0.506029\pi\)
−0.0189387 + 0.999821i \(0.506029\pi\)
\(234\) −425938. −0.508518
\(235\) 0 0
\(236\) −290068. −0.339016
\(237\) 416495. 0.481658
\(238\) 1.58773e6 1.81691
\(239\) −195859. −0.221794 −0.110897 0.993832i \(-0.535372\pi\)
−0.110897 + 0.993832i \(0.535372\pi\)
\(240\) 0 0
\(241\) −1.33370e6 −1.47916 −0.739578 0.673071i \(-0.764975\pi\)
−0.739578 + 0.673071i \(0.764975\pi\)
\(242\) 98272.0 0.107868
\(243\) −59049.0 −0.0641500
\(244\) −532993. −0.573122
\(245\) 0 0
\(246\) −835808. −0.880580
\(247\) 717229. 0.748024
\(248\) −984602. −1.01656
\(249\) −3616.36 −0.00369635
\(250\) 0 0
\(251\) −98131.5 −0.0983160 −0.0491580 0.998791i \(-0.515654\pi\)
−0.0491580 + 0.998791i \(0.515654\pi\)
\(252\) −187315. −0.185811
\(253\) 203836. 0.200207
\(254\) 1.99556e6 1.94080
\(255\) 0 0
\(256\) 1.09866e6 1.04776
\(257\) −717413. −0.677543 −0.338771 0.940869i \(-0.610011\pi\)
−0.338771 + 0.940869i \(0.610011\pi\)
\(258\) −718017. −0.671561
\(259\) 1.59696e6 1.47926
\(260\) 0 0
\(261\) 139333. 0.126606
\(262\) 298903. 0.269015
\(263\) −1.90624e6 −1.69937 −0.849685 0.527290i \(-0.823208\pi\)
−0.849685 + 0.527290i \(0.823208\pi\)
\(264\) −138497. −0.122301
\(265\) 0 0
\(266\) 1.08871e6 0.943424
\(267\) 301255. 0.258617
\(268\) 35137.4 0.0298836
\(269\) 1.80790e6 1.52333 0.761665 0.647971i \(-0.224382\pi\)
0.761665 + 0.647971i \(0.224382\pi\)
\(270\) 0 0
\(271\) 1.27169e6 1.05186 0.525929 0.850529i \(-0.323718\pi\)
0.525929 + 0.850529i \(0.323718\pi\)
\(272\) 1.69736e6 1.39108
\(273\) −1.24922e6 −1.01446
\(274\) 1.29606e6 1.04292
\(275\) 0 0
\(276\) 197893. 0.156371
\(277\) 1.80095e6 1.41027 0.705134 0.709074i \(-0.250887\pi\)
0.705134 + 0.709074i \(0.250887\pi\)
\(278\) −304020. −0.235934
\(279\) 627096. 0.482307
\(280\) 0 0
\(281\) 1.54030e6 1.16370 0.581849 0.813297i \(-0.302330\pi\)
0.581849 + 0.813297i \(0.302330\pi\)
\(282\) 1.24248e6 0.930394
\(283\) −1.87910e6 −1.39471 −0.697354 0.716727i \(-0.745640\pi\)
−0.697354 + 0.716727i \(0.745640\pi\)
\(284\) 172446. 0.126869
\(285\) 0 0
\(286\) 636277. 0.459972
\(287\) −2.45132e6 −1.75669
\(288\) −361543. −0.256849
\(289\) 362696. 0.255445
\(290\) 0 0
\(291\) −1.51674e6 −1.04998
\(292\) −485969. −0.333543
\(293\) −307140. −0.209010 −0.104505 0.994524i \(-0.533326\pi\)
−0.104505 + 0.994524i \(0.533326\pi\)
\(294\) −880944. −0.594401
\(295\) 0 0
\(296\) 1.14634e6 0.760471
\(297\) 88209.0 0.0580259
\(298\) −3.09607e6 −2.01962
\(299\) 1.31977e6 0.853727
\(300\) 0 0
\(301\) −2.10586e6 −1.33971
\(302\) −2.39296e6 −1.50980
\(303\) −1.58593e6 −0.992379
\(304\) 1.16388e6 0.722311
\(305\) 0 0
\(306\) −725881. −0.443161
\(307\) −1.32237e6 −0.800771 −0.400386 0.916347i \(-0.631124\pi\)
−0.400386 + 0.916347i \(0.631124\pi\)
\(308\) 279816. 0.168072
\(309\) 598371. 0.356512
\(310\) 0 0
\(311\) 2.22049e6 1.30181 0.650904 0.759160i \(-0.274390\pi\)
0.650904 + 0.759160i \(0.274390\pi\)
\(312\) −896719. −0.521519
\(313\) −978241. −0.564397 −0.282199 0.959356i \(-0.591064\pi\)
−0.282199 + 0.959356i \(0.591064\pi\)
\(314\) −618023. −0.353737
\(315\) 0 0
\(316\) −604032. −0.340284
\(317\) −2.64915e6 −1.48067 −0.740335 0.672239i \(-0.765333\pi\)
−0.740335 + 0.672239i \(0.765333\pi\)
\(318\) 210443. 0.116699
\(319\) −208140. −0.114519
\(320\) 0 0
\(321\) −784117. −0.424735
\(322\) 2.00332e6 1.07674
\(323\) 1.22230e6 0.651885
\(324\) 85637.2 0.0453210
\(325\) 0 0
\(326\) −3.33815e6 −1.73965
\(327\) −161051. −0.0832901
\(328\) −1.75961e6 −0.903093
\(329\) 3.64404e6 1.85607
\(330\) 0 0
\(331\) −555272. −0.278571 −0.139286 0.990252i \(-0.544481\pi\)
−0.139286 + 0.990252i \(0.544481\pi\)
\(332\) 5244.71 0.00261142
\(333\) −730104. −0.360806
\(334\) −1.96621e6 −0.964414
\(335\) 0 0
\(336\) −2.02717e6 −0.979585
\(337\) −1.96065e6 −0.940427 −0.470213 0.882553i \(-0.655823\pi\)
−0.470213 + 0.882553i \(0.655823\pi\)
\(338\) 1.62751e6 0.774878
\(339\) 296761. 0.140251
\(340\) 0 0
\(341\) −936772. −0.436263
\(342\) −497738. −0.230110
\(343\) 394034. 0.180841
\(344\) −1.51163e6 −0.688730
\(345\) 0 0
\(346\) 2.99368e6 1.34436
\(347\) 899300. 0.400942 0.200471 0.979700i \(-0.435753\pi\)
0.200471 + 0.979700i \(0.435753\pi\)
\(348\) −202071. −0.0894451
\(349\) 852454. 0.374634 0.187317 0.982299i \(-0.440021\pi\)
0.187317 + 0.982299i \(0.440021\pi\)
\(350\) 0 0
\(351\) 571123. 0.247435
\(352\) 540083. 0.232329
\(353\) −58325.4 −0.0249127 −0.0124564 0.999922i \(-0.503965\pi\)
−0.0124564 + 0.999922i \(0.503965\pi\)
\(354\) 1.34249e6 0.569379
\(355\) 0 0
\(356\) −436903. −0.182709
\(357\) −2.12892e6 −0.884074
\(358\) −2.77929e6 −1.14611
\(359\) 4.35963e6 1.78531 0.892654 0.450743i \(-0.148841\pi\)
0.892654 + 0.450743i \(0.148841\pi\)
\(360\) 0 0
\(361\) −1.63797e6 −0.661511
\(362\) −3.76103e6 −1.50846
\(363\) −131769. −0.0524864
\(364\) 1.81171e6 0.716698
\(365\) 0 0
\(366\) 2.46678e6 0.962562
\(367\) −1.51064e6 −0.585459 −0.292730 0.956195i \(-0.594564\pi\)
−0.292730 + 0.956195i \(0.594564\pi\)
\(368\) 2.14164e6 0.824381
\(369\) 1.12070e6 0.428474
\(370\) 0 0
\(371\) 617202. 0.232805
\(372\) −909460. −0.340742
\(373\) 1.33021e6 0.495050 0.247525 0.968881i \(-0.420383\pi\)
0.247525 + 0.968881i \(0.420383\pi\)
\(374\) 1.08434e6 0.400855
\(375\) 0 0
\(376\) 2.61577e6 0.954180
\(377\) −1.34763e6 −0.488335
\(378\) 866927. 0.312071
\(379\) −4.17882e6 −1.49436 −0.747181 0.664621i \(-0.768593\pi\)
−0.747181 + 0.664621i \(0.768593\pi\)
\(380\) 0 0
\(381\) −2.67577e6 −0.944356
\(382\) 4.83647e6 1.69578
\(383\) −1.31585e6 −0.458362 −0.229181 0.973384i \(-0.573605\pi\)
−0.229181 + 0.973384i \(0.573605\pi\)
\(384\) −1.93322e6 −0.669043
\(385\) 0 0
\(386\) −2.90043e6 −0.990818
\(387\) 962760. 0.326769
\(388\) 2.19969e6 0.741792
\(389\) −2.78879e6 −0.934420 −0.467210 0.884146i \(-0.654741\pi\)
−0.467210 + 0.884146i \(0.654741\pi\)
\(390\) 0 0
\(391\) 2.24914e6 0.744003
\(392\) −1.85464e6 −0.609598
\(393\) −400787. −0.130898
\(394\) 5.27948e6 1.71337
\(395\) 0 0
\(396\) −127927. −0.0409944
\(397\) −265601. −0.0845772 −0.0422886 0.999105i \(-0.513465\pi\)
−0.0422886 + 0.999105i \(0.513465\pi\)
\(398\) 1.90768e6 0.603667
\(399\) −1.45980e6 −0.459052
\(400\) 0 0
\(401\) −69100.4 −0.0214595 −0.0107298 0.999942i \(-0.503415\pi\)
−0.0107298 + 0.999942i \(0.503415\pi\)
\(402\) −162622. −0.0501896
\(403\) −6.06528e6 −1.86032
\(404\) 2.30003e6 0.701101
\(405\) 0 0
\(406\) −2.04562e6 −0.615899
\(407\) 1.09065e6 0.326362
\(408\) −1.52818e6 −0.454491
\(409\) −5.05776e6 −1.49503 −0.747515 0.664244i \(-0.768753\pi\)
−0.747515 + 0.664244i \(0.768753\pi\)
\(410\) 0 0
\(411\) −1.73784e6 −0.507464
\(412\) −867801. −0.251870
\(413\) 3.93735e6 1.13587
\(414\) −915882. −0.262627
\(415\) 0 0
\(416\) 3.49685e6 0.990703
\(417\) 407648. 0.114801
\(418\) 743534. 0.208142
\(419\) −4.26079e6 −1.18565 −0.592824 0.805332i \(-0.701987\pi\)
−0.592824 + 0.805332i \(0.701987\pi\)
\(420\) 0 0
\(421\) −6.24929e6 −1.71840 −0.859202 0.511636i \(-0.829040\pi\)
−0.859202 + 0.511636i \(0.829040\pi\)
\(422\) 6.95428e6 1.90095
\(423\) −1.66599e6 −0.452712
\(424\) 443041. 0.119682
\(425\) 0 0
\(426\) −798108. −0.213078
\(427\) 7.23477e6 1.92024
\(428\) 1.13718e6 0.300069
\(429\) −853159. −0.223814
\(430\) 0 0
\(431\) 1.70549e6 0.442238 0.221119 0.975247i \(-0.429029\pi\)
0.221119 + 0.975247i \(0.429029\pi\)
\(432\) 926786. 0.238930
\(433\) −605782. −0.155273 −0.0776367 0.996982i \(-0.524737\pi\)
−0.0776367 + 0.996982i \(0.524737\pi\)
\(434\) −9.20669e6 −2.34628
\(435\) 0 0
\(436\) 233568. 0.0588432
\(437\) 1.54224e6 0.386321
\(438\) 2.24915e6 0.560186
\(439\) −2.77526e6 −0.687293 −0.343647 0.939099i \(-0.611662\pi\)
−0.343647 + 0.939099i \(0.611662\pi\)
\(440\) 0 0
\(441\) 1.18122e6 0.289224
\(442\) 7.02073e6 1.70933
\(443\) −394928. −0.0956111 −0.0478055 0.998857i \(-0.515223\pi\)
−0.0478055 + 0.998857i \(0.515223\pi\)
\(444\) 1.05885e6 0.254904
\(445\) 0 0
\(446\) 7.60744e6 1.81093
\(447\) 4.15139e6 0.982709
\(448\) −1.89973e6 −0.447196
\(449\) 7.55318e6 1.76813 0.884064 0.467366i \(-0.154797\pi\)
0.884064 + 0.467366i \(0.154797\pi\)
\(450\) 0 0
\(451\) −1.67413e6 −0.387569
\(452\) −430384. −0.0990854
\(453\) 3.20863e6 0.734639
\(454\) −4.57244e6 −1.04114
\(455\) 0 0
\(456\) −1.04788e6 −0.235993
\(457\) −2.81470e6 −0.630436 −0.315218 0.949019i \(-0.602078\pi\)
−0.315218 + 0.949019i \(0.602078\pi\)
\(458\) −4.16611e6 −0.928041
\(459\) 973305. 0.215634
\(460\) 0 0
\(461\) 1.15222e6 0.252512 0.126256 0.991998i \(-0.459704\pi\)
0.126256 + 0.991998i \(0.459704\pi\)
\(462\) −1.29504e6 −0.282278
\(463\) 2.64702e6 0.573857 0.286929 0.957952i \(-0.407366\pi\)
0.286929 + 0.957952i \(0.407366\pi\)
\(464\) −2.18686e6 −0.471549
\(465\) 0 0
\(466\) −210683. −0.0449432
\(467\) −8.05020e6 −1.70810 −0.854052 0.520188i \(-0.825862\pi\)
−0.854052 + 0.520188i \(0.825862\pi\)
\(468\) −828284. −0.174809
\(469\) −476950. −0.100125
\(470\) 0 0
\(471\) 828682. 0.172122
\(472\) 2.82631e6 0.583936
\(473\) −1.43820e6 −0.295574
\(474\) 2.79556e6 0.571509
\(475\) 0 0
\(476\) 3.08751e6 0.624585
\(477\) −282174. −0.0567834
\(478\) −1.31463e6 −0.263168
\(479\) 2.33516e6 0.465026 0.232513 0.972593i \(-0.425305\pi\)
0.232513 + 0.972593i \(0.425305\pi\)
\(480\) 0 0
\(481\) 7.06157e6 1.39168
\(482\) −8.95191e6 −1.75509
\(483\) −2.68617e6 −0.523921
\(484\) 191101. 0.0370808
\(485\) 0 0
\(486\) −396344. −0.0761169
\(487\) 8.09784e6 1.54720 0.773600 0.633674i \(-0.218454\pi\)
0.773600 + 0.633674i \(0.218454\pi\)
\(488\) 5.19328e6 0.987170
\(489\) 4.47599e6 0.846481
\(490\) 0 0
\(491\) 5.53902e6 1.03688 0.518441 0.855114i \(-0.326513\pi\)
0.518441 + 0.855114i \(0.326513\pi\)
\(492\) −1.62532e6 −0.302710
\(493\) −2.29663e6 −0.425573
\(494\) 4.81413e6 0.887565
\(495\) 0 0
\(496\) −9.84240e6 −1.79637
\(497\) −2.34075e6 −0.425074
\(498\) −24273.4 −0.00438589
\(499\) −1.00373e7 −1.80454 −0.902271 0.431169i \(-0.858101\pi\)
−0.902271 + 0.431169i \(0.858101\pi\)
\(500\) 0 0
\(501\) 2.63641e6 0.469266
\(502\) −658670. −0.116656
\(503\) 2.38538e6 0.420376 0.210188 0.977661i \(-0.432592\pi\)
0.210188 + 0.977661i \(0.432592\pi\)
\(504\) 1.82513e6 0.320049
\(505\) 0 0
\(506\) 1.36817e6 0.237555
\(507\) −2.18227e6 −0.377041
\(508\) 3.88059e6 0.667173
\(509\) −8.86436e6 −1.51654 −0.758268 0.651943i \(-0.773954\pi\)
−0.758268 + 0.651943i \(0.773954\pi\)
\(510\) 0 0
\(511\) 6.59647e6 1.11753
\(512\) 500636. 0.0844009
\(513\) 667396. 0.111967
\(514\) −4.81536e6 −0.803935
\(515\) 0 0
\(516\) −1.39626e6 −0.230857
\(517\) 2.48870e6 0.409493
\(518\) 1.07190e7 1.75521
\(519\) −4.01411e6 −0.654140
\(520\) 0 0
\(521\) 5.39366e6 0.870542 0.435271 0.900300i \(-0.356653\pi\)
0.435271 + 0.900300i \(0.356653\pi\)
\(522\) 935220. 0.150223
\(523\) 850386. 0.135945 0.0679723 0.997687i \(-0.478347\pi\)
0.0679723 + 0.997687i \(0.478347\pi\)
\(524\) 581250. 0.0924773
\(525\) 0 0
\(526\) −1.27949e7 −2.01638
\(527\) −1.03364e7 −1.62123
\(528\) −1.38446e6 −0.216120
\(529\) −3.59849e6 −0.559089
\(530\) 0 0
\(531\) −1.80008e6 −0.277049
\(532\) 2.11711e6 0.324313
\(533\) −1.08394e7 −1.65268
\(534\) 2.02206e6 0.306860
\(535\) 0 0
\(536\) −342365. −0.0514728
\(537\) 3.72664e6 0.557675
\(538\) 1.21348e7 1.80750
\(539\) −1.76454e6 −0.261613
\(540\) 0 0
\(541\) 7.13219e6 1.04768 0.523841 0.851816i \(-0.324499\pi\)
0.523841 + 0.851816i \(0.324499\pi\)
\(542\) 8.53570e6 1.24808
\(543\) 5.04301e6 0.733990
\(544\) 5.95931e6 0.863374
\(545\) 0 0
\(546\) −8.38492e6 −1.20370
\(547\) −1.39768e6 −0.199729 −0.0998644 0.995001i \(-0.531841\pi\)
−0.0998644 + 0.995001i \(0.531841\pi\)
\(548\) 2.52034e6 0.358515
\(549\) −3.30761e6 −0.468364
\(550\) 0 0
\(551\) −1.57480e6 −0.220977
\(552\) −1.92819e6 −0.269341
\(553\) 8.19904e6 1.14012
\(554\) 1.20882e7 1.67335
\(555\) 0 0
\(556\) −591201. −0.0811052
\(557\) 1.91000e6 0.260852 0.130426 0.991458i \(-0.458365\pi\)
0.130426 + 0.991458i \(0.458365\pi\)
\(558\) 4.20914e6 0.572279
\(559\) −9.31183e6 −1.26039
\(560\) 0 0
\(561\) −1.45395e6 −0.195048
\(562\) 1.03387e7 1.38078
\(563\) 1.33043e7 1.76898 0.884488 0.466563i \(-0.154508\pi\)
0.884488 + 0.466563i \(0.154508\pi\)
\(564\) 2.41614e6 0.319834
\(565\) 0 0
\(566\) −1.26127e7 −1.65488
\(567\) −1.16243e6 −0.151848
\(568\) −1.68024e6 −0.218525
\(569\) 692234. 0.0896339 0.0448170 0.998995i \(-0.485730\pi\)
0.0448170 + 0.998995i \(0.485730\pi\)
\(570\) 0 0
\(571\) 7.25834e6 0.931638 0.465819 0.884880i \(-0.345760\pi\)
0.465819 + 0.884880i \(0.345760\pi\)
\(572\) 1.23731e6 0.158121
\(573\) −6.48503e6 −0.825136
\(574\) −1.64536e7 −2.08439
\(575\) 0 0
\(576\) 868524. 0.109075
\(577\) −1.41479e7 −1.76910 −0.884550 0.466446i \(-0.845534\pi\)
−0.884550 + 0.466446i \(0.845534\pi\)
\(578\) 2.43446e6 0.303098
\(579\) 3.88907e6 0.482113
\(580\) 0 0
\(581\) −71190.9 −0.00874952
\(582\) −1.01805e7 −1.24584
\(583\) 421519. 0.0513625
\(584\) 4.73509e6 0.574508
\(585\) 0 0
\(586\) −2.06156e6 −0.248000
\(587\) 5.01437e6 0.600649 0.300325 0.953837i \(-0.402905\pi\)
0.300325 + 0.953837i \(0.402905\pi\)
\(588\) −1.71309e6 −0.204333
\(589\) −7.08769e6 −0.841815
\(590\) 0 0
\(591\) −7.07904e6 −0.833692
\(592\) 1.14591e7 1.34384
\(593\) −6.71458e6 −0.784119 −0.392060 0.919940i \(-0.628237\pi\)
−0.392060 + 0.919940i \(0.628237\pi\)
\(594\) 592069. 0.0688503
\(595\) 0 0
\(596\) −6.02065e6 −0.694269
\(597\) −2.55793e6 −0.293733
\(598\) 8.85842e6 1.01299
\(599\) −1.12021e7 −1.27565 −0.637825 0.770182i \(-0.720166\pi\)
−0.637825 + 0.770182i \(0.720166\pi\)
\(600\) 0 0
\(601\) 1.05901e7 1.19596 0.597978 0.801512i \(-0.295971\pi\)
0.597978 + 0.801512i \(0.295971\pi\)
\(602\) −1.41347e7 −1.58963
\(603\) 218053. 0.0244213
\(604\) −4.65339e6 −0.519011
\(605\) 0 0
\(606\) −1.06449e7 −1.17750
\(607\) 1.39277e7 1.53429 0.767147 0.641472i \(-0.221676\pi\)
0.767147 + 0.641472i \(0.221676\pi\)
\(608\) 4.08631e6 0.448303
\(609\) 2.74289e6 0.299685
\(610\) 0 0
\(611\) 1.61135e7 1.74617
\(612\) −1.41156e6 −0.152342
\(613\) 1.91560e6 0.205898 0.102949 0.994687i \(-0.467172\pi\)
0.102949 + 0.994687i \(0.467172\pi\)
\(614\) −8.87592e6 −0.950151
\(615\) 0 0
\(616\) −2.72642e6 −0.289495
\(617\) −524280. −0.0554435 −0.0277217 0.999616i \(-0.508825\pi\)
−0.0277217 + 0.999616i \(0.508825\pi\)
\(618\) 4.01633e6 0.423018
\(619\) 9.20908e6 0.966028 0.483014 0.875613i \(-0.339542\pi\)
0.483014 + 0.875613i \(0.339542\pi\)
\(620\) 0 0
\(621\) 1.22807e6 0.127789
\(622\) 1.49042e7 1.54465
\(623\) 5.93045e6 0.612164
\(624\) −8.96389e6 −0.921584
\(625\) 0 0
\(626\) −6.56606e6 −0.669683
\(627\) −996975. −0.101278
\(628\) −1.20182e6 −0.121601
\(629\) 1.20343e7 1.21281
\(630\) 0 0
\(631\) 1.98666e6 0.198633 0.0993164 0.995056i \(-0.468334\pi\)
0.0993164 + 0.995056i \(0.468334\pi\)
\(632\) 5.88545e6 0.586121
\(633\) −9.32471e6 −0.924967
\(634\) −1.77814e7 −1.75688
\(635\) 0 0
\(636\) 409229. 0.0401166
\(637\) −1.14248e7 −1.11558
\(638\) −1.39706e6 −0.135882
\(639\) 1.07015e6 0.103680
\(640\) 0 0
\(641\) −1.31651e7 −1.26555 −0.632776 0.774335i \(-0.718085\pi\)
−0.632776 + 0.774335i \(0.718085\pi\)
\(642\) −5.26308e6 −0.503967
\(643\) 7.46182e6 0.711733 0.355867 0.934537i \(-0.384186\pi\)
0.355867 + 0.934537i \(0.384186\pi\)
\(644\) 3.89568e6 0.370142
\(645\) 0 0
\(646\) 8.20421e6 0.773491
\(647\) 1.12027e7 1.05212 0.526058 0.850449i \(-0.323669\pi\)
0.526058 + 0.850449i \(0.323669\pi\)
\(648\) −834415. −0.0780629
\(649\) 2.68902e6 0.250600
\(650\) 0 0
\(651\) 1.23449e7 1.14165
\(652\) −6.49141e6 −0.598026
\(653\) −5.01727e6 −0.460452 −0.230226 0.973137i \(-0.573947\pi\)
−0.230226 + 0.973137i \(0.573947\pi\)
\(654\) −1.08099e6 −0.0988275
\(655\) 0 0
\(656\) −1.75896e7 −1.59587
\(657\) −3.01579e6 −0.272576
\(658\) 2.44592e7 2.20231
\(659\) 1.59087e6 0.142699 0.0713495 0.997451i \(-0.477269\pi\)
0.0713495 + 0.997451i \(0.477269\pi\)
\(660\) 0 0
\(661\) 1.38029e7 1.22876 0.614378 0.789012i \(-0.289407\pi\)
0.614378 + 0.789012i \(0.289407\pi\)
\(662\) −3.72705e6 −0.330537
\(663\) −9.41381e6 −0.831729
\(664\) −51102.4 −0.00449802
\(665\) 0 0
\(666\) −4.90054e6 −0.428113
\(667\) −2.89778e6 −0.252203
\(668\) −3.82351e6 −0.331529
\(669\) −1.02005e7 −0.881163
\(670\) 0 0
\(671\) 4.94100e6 0.423651
\(672\) −7.11726e6 −0.607980
\(673\) −1.14669e7 −0.975911 −0.487955 0.872869i \(-0.662257\pi\)
−0.487955 + 0.872869i \(0.662257\pi\)
\(674\) −1.31601e7 −1.11586
\(675\) 0 0
\(676\) 3.16488e6 0.266374
\(677\) −1.82188e7 −1.52774 −0.763869 0.645371i \(-0.776703\pi\)
−0.763869 + 0.645371i \(0.776703\pi\)
\(678\) 1.99189e6 0.166414
\(679\) −2.98583e7 −2.48536
\(680\) 0 0
\(681\) 6.13100e6 0.506598
\(682\) −6.28772e6 −0.517645
\(683\) 1.75425e7 1.43893 0.719464 0.694530i \(-0.244388\pi\)
0.719464 + 0.694530i \(0.244388\pi\)
\(684\) −967907. −0.0791031
\(685\) 0 0
\(686\) 2.64480e6 0.214577
\(687\) 5.58617e6 0.451567
\(688\) −1.51107e7 −1.21707
\(689\) 2.72919e6 0.219021
\(690\) 0 0
\(691\) 7.62835e6 0.607765 0.303882 0.952710i \(-0.401717\pi\)
0.303882 + 0.952710i \(0.401717\pi\)
\(692\) 5.82155e6 0.462140
\(693\) 1.73646e6 0.137351
\(694\) 6.03621e6 0.475735
\(695\) 0 0
\(696\) 1.96890e6 0.154064
\(697\) −1.84725e7 −1.44027
\(698\) 5.72177e6 0.444520
\(699\) 282496. 0.0218685
\(700\) 0 0
\(701\) −1.99597e7 −1.53412 −0.767059 0.641577i \(-0.778281\pi\)
−0.767059 + 0.641577i \(0.778281\pi\)
\(702\) 3.83344e6 0.293593
\(703\) 8.25194e6 0.629749
\(704\) −1.29743e6 −0.0986622
\(705\) 0 0
\(706\) −391487. −0.0295601
\(707\) −3.12203e7 −2.34903
\(708\) 2.61061e6 0.195731
\(709\) 8.79537e6 0.657110 0.328555 0.944485i \(-0.393438\pi\)
0.328555 + 0.944485i \(0.393438\pi\)
\(710\) 0 0
\(711\) −3.74846e6 −0.278086
\(712\) 4.25701e6 0.314706
\(713\) −1.30420e7 −0.960771
\(714\) −1.42895e7 −1.04899
\(715\) 0 0
\(716\) −5.40464e6 −0.393989
\(717\) 1.76273e6 0.128053
\(718\) 2.92623e7 2.11835
\(719\) −1.04268e7 −0.752190 −0.376095 0.926581i \(-0.622733\pi\)
−0.376095 + 0.926581i \(0.622733\pi\)
\(720\) 0 0
\(721\) 1.17794e7 0.843889
\(722\) −1.09942e7 −0.784913
\(723\) 1.20033e7 0.853991
\(724\) −7.31374e6 −0.518553
\(725\) 0 0
\(726\) −884448. −0.0622775
\(727\) 3.28724e6 0.230672 0.115336 0.993327i \(-0.463205\pi\)
0.115336 + 0.993327i \(0.463205\pi\)
\(728\) −1.76526e7 −1.23447
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.58692e7 −1.09840
\(732\) 4.79694e6 0.330892
\(733\) −5.45350e6 −0.374900 −0.187450 0.982274i \(-0.560022\pi\)
−0.187450 + 0.982274i \(0.560022\pi\)
\(734\) −1.01396e7 −0.694674
\(735\) 0 0
\(736\) 7.51917e6 0.511653
\(737\) −325734. −0.0220899
\(738\) 7.52227e6 0.508403
\(739\) 2.58376e7 1.74037 0.870183 0.492729i \(-0.164001\pi\)
0.870183 + 0.492729i \(0.164001\pi\)
\(740\) 0 0
\(741\) −6.45507e6 −0.431872
\(742\) 4.14273e6 0.276234
\(743\) 7.94543e6 0.528014 0.264007 0.964521i \(-0.414956\pi\)
0.264007 + 0.964521i \(0.414956\pi\)
\(744\) 8.86142e6 0.586909
\(745\) 0 0
\(746\) 8.92854e6 0.587399
\(747\) 32547.2 0.00213409
\(748\) 2.10862e6 0.137799
\(749\) −1.54360e7 −1.00538
\(750\) 0 0
\(751\) 1.16319e7 0.752574 0.376287 0.926503i \(-0.377201\pi\)
0.376287 + 0.926503i \(0.377201\pi\)
\(752\) 2.61481e7 1.68615
\(753\) 883184. 0.0567628
\(754\) −9.04546e6 −0.579432
\(755\) 0 0
\(756\) 1.68584e6 0.107278
\(757\) 1.46923e7 0.931856 0.465928 0.884823i \(-0.345721\pi\)
0.465928 + 0.884823i \(0.345721\pi\)
\(758\) −2.80487e7 −1.77313
\(759\) −1.83452e6 −0.115590
\(760\) 0 0
\(761\) −1.52759e6 −0.0956194 −0.0478097 0.998856i \(-0.515224\pi\)
−0.0478097 + 0.998856i \(0.515224\pi\)
\(762\) −1.79600e7 −1.12052
\(763\) −3.17041e6 −0.197154
\(764\) 9.40507e6 0.582946
\(765\) 0 0
\(766\) −8.83212e6 −0.543867
\(767\) 1.74104e7 1.06862
\(768\) −9.88792e6 −0.604926
\(769\) 1.53066e7 0.933389 0.466694 0.884419i \(-0.345445\pi\)
0.466694 + 0.884419i \(0.345445\pi\)
\(770\) 0 0
\(771\) 6.45672e6 0.391180
\(772\) −5.64021e6 −0.340606
\(773\) −2.13061e7 −1.28249 −0.641245 0.767336i \(-0.721582\pi\)
−0.641245 + 0.767336i \(0.721582\pi\)
\(774\) 6.46215e6 0.387726
\(775\) 0 0
\(776\) −2.14329e7 −1.27769
\(777\) −1.43727e7 −0.854054
\(778\) −1.87187e7 −1.10873
\(779\) −1.26666e7 −0.747855
\(780\) 0 0
\(781\) −1.59862e6 −0.0937817
\(782\) 1.50965e7 0.882793
\(783\) −1.25400e6 −0.0730959
\(784\) −1.85395e7 −1.07723
\(785\) 0 0
\(786\) −2.69013e6 −0.155316
\(787\) 1.60779e7 0.925319 0.462659 0.886536i \(-0.346895\pi\)
0.462659 + 0.886536i \(0.346895\pi\)
\(788\) 1.02665e7 0.588991
\(789\) 1.71562e7 0.981132
\(790\) 0 0
\(791\) 5.84197e6 0.331985
\(792\) 1.24647e6 0.0706105
\(793\) 3.19913e7 1.80654
\(794\) −1.78274e6 −0.100355
\(795\) 0 0
\(796\) 3.70970e6 0.207518
\(797\) −3.34929e7 −1.86770 −0.933849 0.357668i \(-0.883572\pi\)
−0.933849 + 0.357668i \(0.883572\pi\)
\(798\) −9.79836e6 −0.544686
\(799\) 2.74605e7 1.52175
\(800\) 0 0
\(801\) −2.71130e6 −0.149312
\(802\) −463810. −0.0254627
\(803\) 4.50507e6 0.246554
\(804\) −316237. −0.0172533
\(805\) 0 0
\(806\) −4.07108e7 −2.20736
\(807\) −1.62711e7 −0.879495
\(808\) −2.24106e7 −1.20761
\(809\) −3.65008e7 −1.96079 −0.980396 0.197037i \(-0.936868\pi\)
−0.980396 + 0.197037i \(0.936868\pi\)
\(810\) 0 0
\(811\) −1.57753e7 −0.842222 −0.421111 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(812\) −3.97793e6 −0.211723
\(813\) −1.14452e7 −0.607290
\(814\) 7.32056e6 0.387243
\(815\) 0 0
\(816\) −1.52762e7 −0.803139
\(817\) −1.08815e7 −0.570340
\(818\) −3.39483e7 −1.77392
\(819\) 1.12430e7 0.585697
\(820\) 0 0
\(821\) −1.08891e7 −0.563811 −0.281906 0.959442i \(-0.590966\pi\)
−0.281906 + 0.959442i \(0.590966\pi\)
\(822\) −1.16646e7 −0.602129
\(823\) −4.77752e6 −0.245869 −0.122934 0.992415i \(-0.539230\pi\)
−0.122934 + 0.992415i \(0.539230\pi\)
\(824\) 8.45552e6 0.433832
\(825\) 0 0
\(826\) 2.64279e7 1.34776
\(827\) 1.24682e7 0.633927 0.316964 0.948438i \(-0.397337\pi\)
0.316964 + 0.948438i \(0.397337\pi\)
\(828\) −1.78104e6 −0.0902811
\(829\) −2.92619e7 −1.47882 −0.739412 0.673253i \(-0.764897\pi\)
−0.739412 + 0.673253i \(0.764897\pi\)
\(830\) 0 0
\(831\) −1.62085e7 −0.814219
\(832\) −8.40038e6 −0.420717
\(833\) −1.94701e7 −0.972199
\(834\) 2.73618e6 0.136217
\(835\) 0 0
\(836\) 1.44589e6 0.0715514
\(837\) −5.64386e6 −0.278460
\(838\) −2.85989e7 −1.40682
\(839\) 1.47124e7 0.721571 0.360786 0.932649i \(-0.382509\pi\)
0.360786 + 0.932649i \(0.382509\pi\)
\(840\) 0 0
\(841\) −1.75522e7 −0.855739
\(842\) −4.19459e7 −2.03896
\(843\) −1.38627e7 −0.671861
\(844\) 1.35234e7 0.653475
\(845\) 0 0
\(846\) −1.11823e7 −0.537163
\(847\) −2.59398e6 −0.124239
\(848\) 4.42878e6 0.211492
\(849\) 1.69119e7 0.805235
\(850\) 0 0
\(851\) 1.51843e7 0.718739
\(852\) −1.55201e6 −0.0732480
\(853\) 4.14062e7 1.94846 0.974232 0.225548i \(-0.0724171\pi\)
0.974232 + 0.225548i \(0.0724171\pi\)
\(854\) 4.85606e7 2.27845
\(855\) 0 0
\(856\) −1.10803e7 −0.516852
\(857\) −1.96593e7 −0.914359 −0.457179 0.889375i \(-0.651140\pi\)
−0.457179 + 0.889375i \(0.651140\pi\)
\(858\) −5.72650e6 −0.265565
\(859\) 3.89269e7 1.79997 0.899987 0.435916i \(-0.143575\pi\)
0.899987 + 0.435916i \(0.143575\pi\)
\(860\) 0 0
\(861\) 2.20619e7 1.01423
\(862\) 1.14475e7 0.524736
\(863\) −2.21492e7 −1.01235 −0.506176 0.862430i \(-0.668941\pi\)
−0.506176 + 0.862430i \(0.668941\pi\)
\(864\) 3.25389e6 0.148292
\(865\) 0 0
\(866\) −4.06608e6 −0.184239
\(867\) −3.26426e6 −0.147482
\(868\) −1.79034e7 −0.806561
\(869\) 5.59955e6 0.251538
\(870\) 0 0
\(871\) −2.10901e6 −0.0941963
\(872\) −2.27579e6 −0.101354
\(873\) 1.36507e7 0.606203
\(874\) 1.03517e7 0.458387
\(875\) 0 0
\(876\) 4.37372e6 0.192571
\(877\) 1.54661e7 0.679017 0.339509 0.940603i \(-0.389739\pi\)
0.339509 + 0.940603i \(0.389739\pi\)
\(878\) −1.86278e7 −0.815505
\(879\) 2.76426e6 0.120672
\(880\) 0 0
\(881\) 2.87591e7 1.24835 0.624174 0.781286i \(-0.285436\pi\)
0.624174 + 0.781286i \(0.285436\pi\)
\(882\) 7.92849e6 0.343178
\(883\) 2.15386e7 0.929642 0.464821 0.885405i \(-0.346119\pi\)
0.464821 + 0.885405i \(0.346119\pi\)
\(884\) 1.36526e7 0.587604
\(885\) 0 0
\(886\) −2.65080e6 −0.113447
\(887\) −2.32877e7 −0.993844 −0.496922 0.867795i \(-0.665537\pi\)
−0.496922 + 0.867795i \(0.665537\pi\)
\(888\) −1.03170e7 −0.439058
\(889\) −5.26746e7 −2.23536
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 1.47935e7 0.622529
\(893\) 1.88297e7 0.790161
\(894\) 2.78646e7 1.16603
\(895\) 0 0
\(896\) −3.80570e7 −1.58367
\(897\) −1.18779e7 −0.492900
\(898\) 5.06978e7 2.09796
\(899\) 1.33174e7 0.549565
\(900\) 0 0
\(901\) 4.65107e6 0.190872
\(902\) −1.12370e7 −0.459868
\(903\) 1.89527e7 0.773485
\(904\) 4.19349e6 0.170669
\(905\) 0 0
\(906\) 2.15367e7 0.871682
\(907\) 6.21299e6 0.250774 0.125387 0.992108i \(-0.459983\pi\)
0.125387 + 0.992108i \(0.459983\pi\)
\(908\) −8.89162e6 −0.357904
\(909\) 1.42734e7 0.572950
\(910\) 0 0
\(911\) 2.40141e7 0.958674 0.479337 0.877631i \(-0.340877\pi\)
0.479337 + 0.877631i \(0.340877\pi\)
\(912\) −1.04749e7 −0.417027
\(913\) −48620.0 −0.00193036
\(914\) −1.88926e7 −0.748041
\(915\) 0 0
\(916\) −8.10147e6 −0.319025
\(917\) −7.88981e6 −0.309844
\(918\) 6.53293e6 0.255859
\(919\) 1.20620e7 0.471117 0.235559 0.971860i \(-0.424308\pi\)
0.235559 + 0.971860i \(0.424308\pi\)
\(920\) 0 0
\(921\) 1.19014e7 0.462325
\(922\) 7.73380e6 0.299616
\(923\) −1.03505e7 −0.399906
\(924\) −2.51835e6 −0.0970366
\(925\) 0 0
\(926\) 1.77671e7 0.680908
\(927\) −5.38534e6 −0.205832
\(928\) −7.67794e6 −0.292668
\(929\) 5.56585e6 0.211589 0.105794 0.994388i \(-0.466262\pi\)
0.105794 + 0.994388i \(0.466262\pi\)
\(930\) 0 0
\(931\) −1.33507e7 −0.504811
\(932\) −409696. −0.0154498
\(933\) −1.99844e7 −0.751600
\(934\) −5.40338e7 −2.02674
\(935\) 0 0
\(936\) 8.07047e6 0.301099
\(937\) −1.39429e7 −0.518806 −0.259403 0.965769i \(-0.583526\pi\)
−0.259403 + 0.965769i \(0.583526\pi\)
\(938\) −3.20134e6 −0.118802
\(939\) 8.80417e6 0.325855
\(940\) 0 0
\(941\) −1.76975e7 −0.651536 −0.325768 0.945450i \(-0.605623\pi\)
−0.325768 + 0.945450i \(0.605623\pi\)
\(942\) 5.56221e6 0.204230
\(943\) −2.33077e7 −0.853534
\(944\) 2.82527e7 1.03188
\(945\) 0 0
\(946\) −9.65334e6 −0.350711
\(947\) 1.08090e7 0.391659 0.195830 0.980638i \(-0.437260\pi\)
0.195830 + 0.980638i \(0.437260\pi\)
\(948\) 5.43629e6 0.196463
\(949\) 2.91688e7 1.05136
\(950\) 0 0
\(951\) 2.38423e7 0.854865
\(952\) −3.00835e7 −1.07581
\(953\) 2.79930e7 0.998428 0.499214 0.866479i \(-0.333622\pi\)
0.499214 + 0.866479i \(0.333622\pi\)
\(954\) −1.89398e6 −0.0673760
\(955\) 0 0
\(956\) −2.55644e6 −0.0904673
\(957\) 1.87326e6 0.0661177
\(958\) 1.56738e7 0.551775
\(959\) −3.42108e7 −1.20120
\(960\) 0 0
\(961\) 3.13082e7 1.09358
\(962\) 4.73981e7 1.65129
\(963\) 7.05705e6 0.245221
\(964\) −1.74080e7 −0.603332
\(965\) 0 0
\(966\) −1.80299e7 −0.621655
\(967\) 3.30769e7 1.13752 0.568760 0.822504i \(-0.307423\pi\)
0.568760 + 0.822504i \(0.307423\pi\)
\(968\) −1.86201e6 −0.0638696
\(969\) −1.10007e7 −0.376366
\(970\) 0 0
\(971\) −3.56721e7 −1.21417 −0.607086 0.794636i \(-0.707662\pi\)
−0.607086 + 0.794636i \(0.707662\pi\)
\(972\) −770735. −0.0261661
\(973\) 8.02488e6 0.271742
\(974\) 5.43536e7 1.83582
\(975\) 0 0
\(976\) 5.19137e7 1.74444
\(977\) 4.13863e7 1.38714 0.693571 0.720389i \(-0.256037\pi\)
0.693571 + 0.720389i \(0.256037\pi\)
\(978\) 3.00434e7 1.00439
\(979\) 4.05021e6 0.135058
\(980\) 0 0
\(981\) 1.44946e6 0.0480876
\(982\) 3.71785e7 1.23031
\(983\) 2.54427e7 0.839805 0.419903 0.907569i \(-0.362064\pi\)
0.419903 + 0.907569i \(0.362064\pi\)
\(984\) 1.58365e7 0.521401
\(985\) 0 0
\(986\) −1.54152e7 −0.504961
\(987\) −3.27964e7 −1.07160
\(988\) 9.36161e6 0.305111
\(989\) −2.00230e7 −0.650935
\(990\) 0 0
\(991\) 2.30502e7 0.745573 0.372786 0.927917i \(-0.378402\pi\)
0.372786 + 0.927917i \(0.378402\pi\)
\(992\) −3.45560e7 −1.11492
\(993\) 4.99745e6 0.160833
\(994\) −1.57114e7 −0.504370
\(995\) 0 0
\(996\) −47202.4 −0.00150770
\(997\) 1.12684e7 0.359026 0.179513 0.983756i \(-0.442548\pi\)
0.179513 + 0.983756i \(0.442548\pi\)
\(998\) −6.73717e7 −2.14117
\(999\) 6.57093e6 0.208312
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.s.1.7 yes 9
5.4 even 2 825.6.a.r.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.3 9 5.4 even 2
825.6.a.s.1.7 yes 9 1.1 even 1 trivial