Properties

Label 825.6.a.v.1.10
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(4.62206\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.62206 q^{2} -9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} +168.040 q^{7} -184.293 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.62206 q^{2} -9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} +168.040 q^{7} -184.293 q^{8} +81.0000 q^{9} +121.000 q^{11} +169.926 q^{12} -155.005 q^{13} +608.652 q^{14} -63.3377 q^{16} -426.797 q^{17} +293.387 q^{18} -1674.14 q^{19} -1512.36 q^{21} +438.269 q^{22} -11.2327 q^{23} +1658.64 q^{24} -561.437 q^{26} -729.000 q^{27} -3172.72 q^{28} -1107.81 q^{29} +7186.50 q^{31} +5667.96 q^{32} -1089.00 q^{33} -1545.88 q^{34} -1529.34 q^{36} -4576.30 q^{37} -6063.84 q^{38} +1395.05 q^{39} -14041.6 q^{41} -5477.87 q^{42} +20306.0 q^{43} -2284.56 q^{44} -40.6856 q^{46} +10551.8 q^{47} +570.039 q^{48} +11430.6 q^{49} +3841.17 q^{51} +2926.60 q^{52} +27069.4 q^{53} -2640.48 q^{54} -30968.6 q^{56} +15067.3 q^{57} -4012.57 q^{58} +22773.1 q^{59} -916.341 q^{61} +26029.9 q^{62} +13611.3 q^{63} +22556.5 q^{64} -3944.42 q^{66} -25328.4 q^{67} +8058.22 q^{68} +101.095 q^{69} -16788.6 q^{71} -14927.7 q^{72} -55976.9 q^{73} -16575.6 q^{74} +31608.9 q^{76} +20332.9 q^{77} +5052.94 q^{78} -47517.8 q^{79} +6561.00 q^{81} -50859.4 q^{82} -24415.3 q^{83} +28554.5 q^{84} +73549.4 q^{86} +9970.33 q^{87} -22299.4 q^{88} -22081.6 q^{89} -26047.1 q^{91} +212.082 q^{92} -64678.5 q^{93} +38219.1 q^{94} -51011.6 q^{96} -15706.3 q^{97} +41402.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.62206 0.640296 0.320148 0.947368i \(-0.396267\pi\)
0.320148 + 0.947368i \(0.396267\pi\)
\(3\) −9.00000 −0.577350
\(4\) −18.8807 −0.590021
\(5\) 0 0
\(6\) −32.5985 −0.369675
\(7\) 168.040 1.29619 0.648095 0.761560i \(-0.275566\pi\)
0.648095 + 0.761560i \(0.275566\pi\)
\(8\) −184.293 −1.01808
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 169.926 0.340649
\(13\) −155.005 −0.254383 −0.127191 0.991878i \(-0.540596\pi\)
−0.127191 + 0.991878i \(0.540596\pi\)
\(14\) 608.652 0.829944
\(15\) 0 0
\(16\) −63.3377 −0.0618532
\(17\) −426.797 −0.358178 −0.179089 0.983833i \(-0.557315\pi\)
−0.179089 + 0.983833i \(0.557315\pi\)
\(18\) 293.387 0.213432
\(19\) −1674.14 −1.06392 −0.531959 0.846770i \(-0.678544\pi\)
−0.531959 + 0.846770i \(0.678544\pi\)
\(20\) 0 0
\(21\) −1512.36 −0.748355
\(22\) 438.269 0.193056
\(23\) −11.2327 −0.00442757 −0.00221379 0.999998i \(-0.500705\pi\)
−0.00221379 + 0.999998i \(0.500705\pi\)
\(24\) 1658.64 0.587791
\(25\) 0 0
\(26\) −561.437 −0.162880
\(27\) −729.000 −0.192450
\(28\) −3172.72 −0.764780
\(29\) −1107.81 −0.244609 −0.122304 0.992493i \(-0.539028\pi\)
−0.122304 + 0.992493i \(0.539028\pi\)
\(30\) 0 0
\(31\) 7186.50 1.34312 0.671558 0.740952i \(-0.265625\pi\)
0.671558 + 0.740952i \(0.265625\pi\)
\(32\) 5667.96 0.978480
\(33\) −1089.00 −0.174078
\(34\) −1545.88 −0.229340
\(35\) 0 0
\(36\) −1529.34 −0.196674
\(37\) −4576.30 −0.549554 −0.274777 0.961508i \(-0.588604\pi\)
−0.274777 + 0.961508i \(0.588604\pi\)
\(38\) −6063.84 −0.681222
\(39\) 1395.05 0.146868
\(40\) 0 0
\(41\) −14041.6 −1.30454 −0.652268 0.757988i \(-0.726182\pi\)
−0.652268 + 0.757988i \(0.726182\pi\)
\(42\) −5477.87 −0.479169
\(43\) 20306.0 1.67476 0.837380 0.546621i \(-0.184086\pi\)
0.837380 + 0.546621i \(0.184086\pi\)
\(44\) −2284.56 −0.177898
\(45\) 0 0
\(46\) −40.6856 −0.00283495
\(47\) 10551.8 0.696756 0.348378 0.937354i \(-0.386733\pi\)
0.348378 + 0.937354i \(0.386733\pi\)
\(48\) 570.039 0.0357110
\(49\) 11430.6 0.680107
\(50\) 0 0
\(51\) 3841.17 0.206794
\(52\) 2926.60 0.150091
\(53\) 27069.4 1.32370 0.661849 0.749637i \(-0.269772\pi\)
0.661849 + 0.749637i \(0.269772\pi\)
\(54\) −2640.48 −0.123225
\(55\) 0 0
\(56\) −30968.6 −1.31963
\(57\) 15067.3 0.614253
\(58\) −4012.57 −0.156622
\(59\) 22773.1 0.851709 0.425854 0.904792i \(-0.359974\pi\)
0.425854 + 0.904792i \(0.359974\pi\)
\(60\) 0 0
\(61\) −916.341 −0.0315306 −0.0157653 0.999876i \(-0.505018\pi\)
−0.0157653 + 0.999876i \(0.505018\pi\)
\(62\) 26029.9 0.859991
\(63\) 13611.3 0.432063
\(64\) 22556.5 0.688369
\(65\) 0 0
\(66\) −3944.42 −0.111461
\(67\) −25328.4 −0.689321 −0.344660 0.938727i \(-0.612006\pi\)
−0.344660 + 0.938727i \(0.612006\pi\)
\(68\) 8058.22 0.211333
\(69\) 101.095 0.00255626
\(70\) 0 0
\(71\) −16788.6 −0.395248 −0.197624 0.980278i \(-0.563323\pi\)
−0.197624 + 0.980278i \(0.563323\pi\)
\(72\) −14927.7 −0.339361
\(73\) −55976.9 −1.22942 −0.614712 0.788752i \(-0.710728\pi\)
−0.614712 + 0.788752i \(0.710728\pi\)
\(74\) −16575.6 −0.351877
\(75\) 0 0
\(76\) 31608.9 0.627735
\(77\) 20332.9 0.390816
\(78\) 5052.94 0.0940389
\(79\) −47517.8 −0.856621 −0.428310 0.903632i \(-0.640891\pi\)
−0.428310 + 0.903632i \(0.640891\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −50859.4 −0.835289
\(83\) −24415.3 −0.389016 −0.194508 0.980901i \(-0.562311\pi\)
−0.194508 + 0.980901i \(0.562311\pi\)
\(84\) 28554.5 0.441546
\(85\) 0 0
\(86\) 73549.4 1.07234
\(87\) 9970.33 0.141225
\(88\) −22299.4 −0.306964
\(89\) −22081.6 −0.295499 −0.147749 0.989025i \(-0.547203\pi\)
−0.147749 + 0.989025i \(0.547203\pi\)
\(90\) 0 0
\(91\) −26047.1 −0.329728
\(92\) 212.082 0.00261236
\(93\) −64678.5 −0.775448
\(94\) 38219.1 0.446130
\(95\) 0 0
\(96\) −51011.6 −0.564925
\(97\) −15706.3 −0.169490 −0.0847449 0.996403i \(-0.527008\pi\)
−0.0847449 + 0.996403i \(0.527008\pi\)
\(98\) 41402.1 0.435469
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 10765.2 0.105007 0.0525033 0.998621i \(-0.483280\pi\)
0.0525033 + 0.998621i \(0.483280\pi\)
\(102\) 13913.0 0.132409
\(103\) −28684.5 −0.266412 −0.133206 0.991088i \(-0.542527\pi\)
−0.133206 + 0.991088i \(0.542527\pi\)
\(104\) 28566.3 0.258983
\(105\) 0 0
\(106\) 98046.9 0.847558
\(107\) 121918. 1.02946 0.514728 0.857354i \(-0.327893\pi\)
0.514728 + 0.857354i \(0.327893\pi\)
\(108\) 13764.0 0.113550
\(109\) −169782. −1.36876 −0.684378 0.729128i \(-0.739926\pi\)
−0.684378 + 0.729128i \(0.739926\pi\)
\(110\) 0 0
\(111\) 41186.7 0.317285
\(112\) −10643.3 −0.0801735
\(113\) −220268. −1.62276 −0.811380 0.584519i \(-0.801283\pi\)
−0.811380 + 0.584519i \(0.801283\pi\)
\(114\) 54574.6 0.393304
\(115\) 0 0
\(116\) 20916.3 0.144324
\(117\) −12555.4 −0.0847942
\(118\) 82485.3 0.545346
\(119\) −71719.1 −0.464267
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −3319.04 −0.0201889
\(123\) 126374. 0.753174
\(124\) −135686. −0.792467
\(125\) 0 0
\(126\) 49300.8 0.276648
\(127\) −230988. −1.27081 −0.635404 0.772180i \(-0.719166\pi\)
−0.635404 + 0.772180i \(0.719166\pi\)
\(128\) −99673.7 −0.537720
\(129\) −182754. −0.966923
\(130\) 0 0
\(131\) 175894. 0.895513 0.447757 0.894155i \(-0.352223\pi\)
0.447757 + 0.894155i \(0.352223\pi\)
\(132\) 20561.1 0.102710
\(133\) −281323. −1.37904
\(134\) −91741.1 −0.441369
\(135\) 0 0
\(136\) 78655.7 0.364655
\(137\) −8437.54 −0.0384073 −0.0192037 0.999816i \(-0.506113\pi\)
−0.0192037 + 0.999816i \(0.506113\pi\)
\(138\) 366.170 0.00163676
\(139\) −75054.6 −0.329489 −0.164744 0.986336i \(-0.552680\pi\)
−0.164744 + 0.986336i \(0.552680\pi\)
\(140\) 0 0
\(141\) −94965.9 −0.402272
\(142\) −60809.5 −0.253076
\(143\) −18755.6 −0.0766992
\(144\) −5130.35 −0.0206177
\(145\) 0 0
\(146\) −202752. −0.787194
\(147\) −102875. −0.392660
\(148\) 86403.7 0.324248
\(149\) −119523. −0.441049 −0.220524 0.975381i \(-0.570777\pi\)
−0.220524 + 0.975381i \(0.570777\pi\)
\(150\) 0 0
\(151\) 372713. 1.33025 0.665124 0.746733i \(-0.268379\pi\)
0.665124 + 0.746733i \(0.268379\pi\)
\(152\) 308532. 1.08316
\(153\) −34570.6 −0.119393
\(154\) 73646.9 0.250238
\(155\) 0 0
\(156\) −26339.4 −0.0866552
\(157\) −407799. −1.32038 −0.660188 0.751101i \(-0.729523\pi\)
−0.660188 + 0.751101i \(0.729523\pi\)
\(158\) −172112. −0.548491
\(159\) −243624. −0.764237
\(160\) 0 0
\(161\) −1887.55 −0.00573897
\(162\) 23764.3 0.0711440
\(163\) −360328. −1.06225 −0.531127 0.847292i \(-0.678231\pi\)
−0.531127 + 0.847292i \(0.678231\pi\)
\(164\) 265115. 0.769704
\(165\) 0 0
\(166\) −88433.8 −0.249085
\(167\) −69666.3 −0.193300 −0.0966500 0.995318i \(-0.530813\pi\)
−0.0966500 + 0.995318i \(0.530813\pi\)
\(168\) 278718. 0.761888
\(169\) −347266. −0.935289
\(170\) 0 0
\(171\) −135605. −0.354639
\(172\) −383391. −0.988144
\(173\) −627131. −1.59310 −0.796550 0.604572i \(-0.793344\pi\)
−0.796550 + 0.604572i \(0.793344\pi\)
\(174\) 36113.1 0.0904257
\(175\) 0 0
\(176\) −7663.86 −0.0186494
\(177\) −204957. −0.491734
\(178\) −79980.8 −0.189206
\(179\) −662582. −1.54564 −0.772818 0.634628i \(-0.781154\pi\)
−0.772818 + 0.634628i \(0.781154\pi\)
\(180\) 0 0
\(181\) −525680. −1.19268 −0.596341 0.802731i \(-0.703379\pi\)
−0.596341 + 0.802731i \(0.703379\pi\)
\(182\) −94344.1 −0.211123
\(183\) 8247.07 0.0182042
\(184\) 2070.11 0.00450764
\(185\) 0 0
\(186\) −234269. −0.496516
\(187\) −51642.5 −0.107995
\(188\) −199225. −0.411101
\(189\) −122501. −0.249452
\(190\) 0 0
\(191\) −824338. −1.63502 −0.817508 0.575918i \(-0.804645\pi\)
−0.817508 + 0.575918i \(0.804645\pi\)
\(192\) −203008. −0.397430
\(193\) 783372. 1.51382 0.756911 0.653518i \(-0.226707\pi\)
0.756911 + 0.653518i \(0.226707\pi\)
\(194\) −56889.0 −0.108524
\(195\) 0 0
\(196\) −215817. −0.401278
\(197\) 272361. 0.500010 0.250005 0.968245i \(-0.419568\pi\)
0.250005 + 0.968245i \(0.419568\pi\)
\(198\) 35499.8 0.0643521
\(199\) 982522. 1.75877 0.879386 0.476109i \(-0.157953\pi\)
0.879386 + 0.476109i \(0.157953\pi\)
\(200\) 0 0
\(201\) 227956. 0.397979
\(202\) 38992.0 0.0672353
\(203\) −186157. −0.317059
\(204\) −72524.0 −0.122013
\(205\) 0 0
\(206\) −103897. −0.170583
\(207\) −909.851 −0.00147586
\(208\) 9817.66 0.0157344
\(209\) −202571. −0.320783
\(210\) 0 0
\(211\) −430400. −0.665527 −0.332764 0.943010i \(-0.607981\pi\)
−0.332764 + 0.943010i \(0.607981\pi\)
\(212\) −511089. −0.781010
\(213\) 151098. 0.228197
\(214\) 441593. 0.659156
\(215\) 0 0
\(216\) 134349. 0.195930
\(217\) 1.20762e6 1.74093
\(218\) −614961. −0.876408
\(219\) 503792. 0.709808
\(220\) 0 0
\(221\) 66155.7 0.0911143
\(222\) 149181. 0.203156
\(223\) −532033. −0.716434 −0.358217 0.933638i \(-0.616615\pi\)
−0.358217 + 0.933638i \(0.616615\pi\)
\(224\) 952446. 1.26829
\(225\) 0 0
\(226\) −797822. −1.03905
\(227\) −792684. −1.02102 −0.510512 0.859871i \(-0.670544\pi\)
−0.510512 + 0.859871i \(0.670544\pi\)
\(228\) −284481. −0.362423
\(229\) 286558. 0.361097 0.180549 0.983566i \(-0.442213\pi\)
0.180549 + 0.983566i \(0.442213\pi\)
\(230\) 0 0
\(231\) −182996. −0.225638
\(232\) 204162. 0.249032
\(233\) −1.41434e6 −1.70673 −0.853366 0.521312i \(-0.825443\pi\)
−0.853366 + 0.521312i \(0.825443\pi\)
\(234\) −45476.4 −0.0542934
\(235\) 0 0
\(236\) −429971. −0.502527
\(237\) 427660. 0.494570
\(238\) −259771. −0.297268
\(239\) 1.00694e6 1.14028 0.570138 0.821549i \(-0.306890\pi\)
0.570138 + 0.821549i \(0.306890\pi\)
\(240\) 0 0
\(241\) 1.08716e6 1.20573 0.602866 0.797843i \(-0.294025\pi\)
0.602866 + 0.797843i \(0.294025\pi\)
\(242\) 53030.6 0.0582087
\(243\) −59049.0 −0.0641500
\(244\) 17301.1 0.0186037
\(245\) 0 0
\(246\) 457735. 0.482254
\(247\) 259500. 0.270642
\(248\) −1.32442e6 −1.36740
\(249\) 219738. 0.224598
\(250\) 0 0
\(251\) 312788. 0.313376 0.156688 0.987648i \(-0.449918\pi\)
0.156688 + 0.987648i \(0.449918\pi\)
\(252\) −256990. −0.254927
\(253\) −1359.16 −0.00133496
\(254\) −836652. −0.813693
\(255\) 0 0
\(256\) −1.08283e6 −1.03267
\(257\) −1.50645e6 −1.42273 −0.711366 0.702822i \(-0.751923\pi\)
−0.711366 + 0.702822i \(0.751923\pi\)
\(258\) −661945. −0.619117
\(259\) −769003. −0.712326
\(260\) 0 0
\(261\) −89732.9 −0.0815362
\(262\) 637097. 0.573393
\(263\) −189662. −0.169079 −0.0845397 0.996420i \(-0.526942\pi\)
−0.0845397 + 0.996420i \(0.526942\pi\)
\(264\) 200695. 0.177226
\(265\) 0 0
\(266\) −1.01897e6 −0.882993
\(267\) 198734. 0.170606
\(268\) 478218. 0.406714
\(269\) 1.60463e6 1.35206 0.676029 0.736875i \(-0.263699\pi\)
0.676029 + 0.736875i \(0.263699\pi\)
\(270\) 0 0
\(271\) 1.71899e6 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(272\) 27032.3 0.0221545
\(273\) 234424. 0.190369
\(274\) −30561.3 −0.0245921
\(275\) 0 0
\(276\) −1908.73 −0.00150825
\(277\) −756133. −0.592105 −0.296053 0.955172i \(-0.595670\pi\)
−0.296053 + 0.955172i \(0.595670\pi\)
\(278\) −271852. −0.210970
\(279\) 582107. 0.447705
\(280\) 0 0
\(281\) 772332. 0.583497 0.291748 0.956495i \(-0.405763\pi\)
0.291748 + 0.956495i \(0.405763\pi\)
\(282\) −343972. −0.257573
\(283\) −1.93103e6 −1.43325 −0.716625 0.697458i \(-0.754314\pi\)
−0.716625 + 0.697458i \(0.754314\pi\)
\(284\) 316981. 0.233205
\(285\) 0 0
\(286\) −67933.9 −0.0491102
\(287\) −2.35955e6 −1.69093
\(288\) 459105. 0.326160
\(289\) −1.23770e6 −0.871708
\(290\) 0 0
\(291\) 141356. 0.0978550
\(292\) 1.05688e6 0.725386
\(293\) 1.70779e6 1.16216 0.581080 0.813847i \(-0.302630\pi\)
0.581080 + 0.813847i \(0.302630\pi\)
\(294\) −372619. −0.251418
\(295\) 0 0
\(296\) 843379. 0.559492
\(297\) −88209.0 −0.0580259
\(298\) −432920. −0.282402
\(299\) 1741.13 0.00112630
\(300\) 0 0
\(301\) 3.41222e6 2.17081
\(302\) 1.34999e6 0.851752
\(303\) −96886.4 −0.0606256
\(304\) 106036. 0.0658068
\(305\) 0 0
\(306\) −125217. −0.0764467
\(307\) 256601. 0.155386 0.0776931 0.996977i \(-0.475245\pi\)
0.0776931 + 0.996977i \(0.475245\pi\)
\(308\) −383899. −0.230590
\(309\) 258160. 0.153813
\(310\) 0 0
\(311\) 2.08034e6 1.21964 0.609822 0.792539i \(-0.291241\pi\)
0.609822 + 0.792539i \(0.291241\pi\)
\(312\) −257097. −0.149524
\(313\) −1.05439e6 −0.608331 −0.304166 0.952619i \(-0.598378\pi\)
−0.304166 + 0.952619i \(0.598378\pi\)
\(314\) −1.47707e6 −0.845431
\(315\) 0 0
\(316\) 897169. 0.505425
\(317\) −2.35341e6 −1.31538 −0.657688 0.753291i \(-0.728465\pi\)
−0.657688 + 0.753291i \(0.728465\pi\)
\(318\) −882422. −0.489338
\(319\) −134045. −0.0737523
\(320\) 0 0
\(321\) −1.09726e6 −0.594356
\(322\) −6836.82 −0.00367464
\(323\) 714519. 0.381072
\(324\) −123876. −0.0655579
\(325\) 0 0
\(326\) −1.30513e6 −0.680157
\(327\) 1.52804e6 0.790251
\(328\) 2.58776e6 1.32813
\(329\) 1.77312e6 0.903127
\(330\) 0 0
\(331\) −376896. −0.189083 −0.0945414 0.995521i \(-0.530138\pi\)
−0.0945414 + 0.995521i \(0.530138\pi\)
\(332\) 460978. 0.229528
\(333\) −370680. −0.183185
\(334\) −252336. −0.123769
\(335\) 0 0
\(336\) 95789.6 0.0462882
\(337\) 3.13928e6 1.50576 0.752879 0.658159i \(-0.228665\pi\)
0.752879 + 0.658159i \(0.228665\pi\)
\(338\) −1.25782e6 −0.598862
\(339\) 1.98241e6 0.936901
\(340\) 0 0
\(341\) 869567. 0.404965
\(342\) −491171. −0.227074
\(343\) −903460. −0.414642
\(344\) −3.74225e6 −1.70505
\(345\) 0 0
\(346\) −2.27151e6 −1.02006
\(347\) −1.74141e6 −0.776384 −0.388192 0.921579i \(-0.626900\pi\)
−0.388192 + 0.921579i \(0.626900\pi\)
\(348\) −188247. −0.0833257
\(349\) −4.40916e6 −1.93772 −0.968862 0.247602i \(-0.920357\pi\)
−0.968862 + 0.247602i \(0.920357\pi\)
\(350\) 0 0
\(351\) 112999. 0.0489560
\(352\) 685823. 0.295023
\(353\) −1.97624e6 −0.844116 −0.422058 0.906569i \(-0.638692\pi\)
−0.422058 + 0.906569i \(0.638692\pi\)
\(354\) −742368. −0.314855
\(355\) 0 0
\(356\) 416916. 0.174350
\(357\) 645472. 0.268045
\(358\) −2.39991e6 −0.989664
\(359\) 340087. 0.139269 0.0696343 0.997573i \(-0.477817\pi\)
0.0696343 + 0.997573i \(0.477817\pi\)
\(360\) 0 0
\(361\) 326652. 0.131922
\(362\) −1.90404e6 −0.763669
\(363\) −131769. −0.0524864
\(364\) 491787. 0.194547
\(365\) 0 0
\(366\) 29871.4 0.0116561
\(367\) −3.60337e6 −1.39651 −0.698254 0.715850i \(-0.746039\pi\)
−0.698254 + 0.715850i \(0.746039\pi\)
\(368\) 711.455 0.000273859 0
\(369\) −1.13737e6 −0.434845
\(370\) 0 0
\(371\) 4.54875e6 1.71576
\(372\) 1.22118e6 0.457531
\(373\) −1.94994e6 −0.725687 −0.362844 0.931850i \(-0.618194\pi\)
−0.362844 + 0.931850i \(0.618194\pi\)
\(374\) −187052. −0.0691486
\(375\) 0 0
\(376\) −1.94462e6 −0.709356
\(377\) 171717. 0.0622242
\(378\) −443707. −0.159723
\(379\) 2.15699e6 0.771349 0.385674 0.922635i \(-0.373969\pi\)
0.385674 + 0.922635i \(0.373969\pi\)
\(380\) 0 0
\(381\) 2.07889e6 0.733701
\(382\) −2.98580e6 −1.04689
\(383\) 1.66805e6 0.581049 0.290525 0.956868i \(-0.406170\pi\)
0.290525 + 0.956868i \(0.406170\pi\)
\(384\) 897064. 0.310453
\(385\) 0 0
\(386\) 2.83742e6 0.969294
\(387\) 1.64478e6 0.558253
\(388\) 296545. 0.100003
\(389\) 1.58917e6 0.532473 0.266236 0.963908i \(-0.414220\pi\)
0.266236 + 0.963908i \(0.414220\pi\)
\(390\) 0 0
\(391\) 4794.09 0.00158586
\(392\) −2.10657e6 −0.692406
\(393\) −1.58304e6 −0.517025
\(394\) 986506. 0.320154
\(395\) 0 0
\(396\) −185050. −0.0592994
\(397\) 4.07780e6 1.29852 0.649261 0.760566i \(-0.275078\pi\)
0.649261 + 0.760566i \(0.275078\pi\)
\(398\) 3.55875e6 1.12613
\(399\) 2.53191e6 0.796189
\(400\) 0 0
\(401\) −4.89367e6 −1.51976 −0.759878 0.650066i \(-0.774741\pi\)
−0.759878 + 0.650066i \(0.774741\pi\)
\(402\) 825670. 0.254824
\(403\) −1.11394e6 −0.341665
\(404\) −203254. −0.0619562
\(405\) 0 0
\(406\) −674273. −0.203012
\(407\) −553732. −0.165697
\(408\) −707901. −0.210534
\(409\) 508943. 0.150439 0.0752196 0.997167i \(-0.476034\pi\)
0.0752196 + 0.997167i \(0.476034\pi\)
\(410\) 0 0
\(411\) 75937.8 0.0221745
\(412\) 541583. 0.157189
\(413\) 3.82679e6 1.10398
\(414\) −3295.53 −0.000944985 0
\(415\) 0 0
\(416\) −878562. −0.248908
\(417\) 675492. 0.190230
\(418\) −733725. −0.205396
\(419\) 6.78930e6 1.88925 0.944626 0.328148i \(-0.106425\pi\)
0.944626 + 0.328148i \(0.106425\pi\)
\(420\) 0 0
\(421\) −3.93204e6 −1.08122 −0.540608 0.841275i \(-0.681806\pi\)
−0.540608 + 0.841275i \(0.681806\pi\)
\(422\) −1.55893e6 −0.426134
\(423\) 854693. 0.232252
\(424\) −4.98869e6 −1.34764
\(425\) 0 0
\(426\) 547285. 0.146113
\(427\) −153982. −0.0408697
\(428\) −2.30189e6 −0.607401
\(429\) 168800. 0.0442823
\(430\) 0 0
\(431\) −3.24173e6 −0.840589 −0.420295 0.907388i \(-0.638073\pi\)
−0.420295 + 0.907388i \(0.638073\pi\)
\(432\) 46173.2 0.0119037
\(433\) −5.96730e6 −1.52953 −0.764765 0.644309i \(-0.777145\pi\)
−0.764765 + 0.644309i \(0.777145\pi\)
\(434\) 4.37408e6 1.11471
\(435\) 0 0
\(436\) 3.20560e6 0.807595
\(437\) 18805.2 0.00471057
\(438\) 1.82476e6 0.454487
\(439\) 2.85969e6 0.708203 0.354101 0.935207i \(-0.384787\pi\)
0.354101 + 0.935207i \(0.384787\pi\)
\(440\) 0 0
\(441\) 925875. 0.226702
\(442\) 239620. 0.0583401
\(443\) 2.52822e6 0.612077 0.306038 0.952019i \(-0.400996\pi\)
0.306038 + 0.952019i \(0.400996\pi\)
\(444\) −777633. −0.187205
\(445\) 0 0
\(446\) −1.92705e6 −0.458730
\(447\) 1.07571e6 0.254640
\(448\) 3.79040e6 0.892257
\(449\) 5.72753e6 1.34076 0.670380 0.742018i \(-0.266131\pi\)
0.670380 + 0.742018i \(0.266131\pi\)
\(450\) 0 0
\(451\) −1.69903e6 −0.393333
\(452\) 4.15880e6 0.957464
\(453\) −3.35442e6 −0.768019
\(454\) −2.87115e6 −0.653757
\(455\) 0 0
\(456\) −2.77679e6 −0.625362
\(457\) −4.94045e6 −1.10656 −0.553281 0.832994i \(-0.686625\pi\)
−0.553281 + 0.832994i \(0.686625\pi\)
\(458\) 1.03793e6 0.231209
\(459\) 311135. 0.0689314
\(460\) 0 0
\(461\) 6.18580e6 1.35564 0.677818 0.735229i \(-0.262926\pi\)
0.677818 + 0.735229i \(0.262926\pi\)
\(462\) −662822. −0.144475
\(463\) −1.23230e6 −0.267155 −0.133577 0.991038i \(-0.542646\pi\)
−0.133577 + 0.991038i \(0.542646\pi\)
\(464\) 70166.4 0.0151298
\(465\) 0 0
\(466\) −5.12284e6 −1.09281
\(467\) −7.74380e6 −1.64309 −0.821546 0.570142i \(-0.806888\pi\)
−0.821546 + 0.570142i \(0.806888\pi\)
\(468\) 237055. 0.0500304
\(469\) −4.25620e6 −0.893490
\(470\) 0 0
\(471\) 3.67019e6 0.762319
\(472\) −4.19691e6 −0.867111
\(473\) 2.45702e6 0.504959
\(474\) 1.54901e6 0.316671
\(475\) 0 0
\(476\) 1.35411e6 0.273927
\(477\) 2.19262e6 0.441233
\(478\) 3.64721e6 0.730114
\(479\) −3.36135e6 −0.669384 −0.334692 0.942328i \(-0.608632\pi\)
−0.334692 + 0.942328i \(0.608632\pi\)
\(480\) 0 0
\(481\) 709349. 0.139797
\(482\) 3.93776e6 0.772025
\(483\) 16988.0 0.00331340
\(484\) −276432. −0.0536383
\(485\) 0 0
\(486\) −213879. −0.0410750
\(487\) 397760. 0.0759973 0.0379987 0.999278i \(-0.487902\pi\)
0.0379987 + 0.999278i \(0.487902\pi\)
\(488\) 168875. 0.0321008
\(489\) 3.24295e6 0.613293
\(490\) 0 0
\(491\) −585823. −0.109664 −0.0548318 0.998496i \(-0.517462\pi\)
−0.0548318 + 0.998496i \(0.517462\pi\)
\(492\) −2.38603e6 −0.444389
\(493\) 472812. 0.0876135
\(494\) 939926. 0.173291
\(495\) 0 0
\(496\) −455177. −0.0830760
\(497\) −2.82117e6 −0.512316
\(498\) 795904. 0.143809
\(499\) 7.82230e6 1.40632 0.703158 0.711034i \(-0.251773\pi\)
0.703158 + 0.711034i \(0.251773\pi\)
\(500\) 0 0
\(501\) 626997. 0.111602
\(502\) 1.13294e6 0.200653
\(503\) 4.04320e6 0.712534 0.356267 0.934384i \(-0.384049\pi\)
0.356267 + 0.934384i \(0.384049\pi\)
\(504\) −2.50846e6 −0.439876
\(505\) 0 0
\(506\) −4922.96 −0.000854771 0
\(507\) 3.12540e6 0.539990
\(508\) 4.36121e6 0.749804
\(509\) −1.07585e7 −1.84060 −0.920299 0.391216i \(-0.872054\pi\)
−0.920299 + 0.391216i \(0.872054\pi\)
\(510\) 0 0
\(511\) −9.40637e6 −1.59357
\(512\) −732521. −0.123494
\(513\) 1.22045e6 0.204751
\(514\) −5.45646e6 −0.910969
\(515\) 0 0
\(516\) 3.45052e6 0.570505
\(517\) 1.27676e6 0.210080
\(518\) −2.78537e6 −0.456099
\(519\) 5.64418e6 0.919777
\(520\) 0 0
\(521\) −3.48830e6 −0.563014 −0.281507 0.959559i \(-0.590834\pi\)
−0.281507 + 0.959559i \(0.590834\pi\)
\(522\) −325018. −0.0522073
\(523\) 5.47692e6 0.875552 0.437776 0.899084i \(-0.355766\pi\)
0.437776 + 0.899084i \(0.355766\pi\)
\(524\) −3.32099e6 −0.528372
\(525\) 0 0
\(526\) −686967. −0.108261
\(527\) −3.06718e6 −0.481075
\(528\) 68974.7 0.0107673
\(529\) −6.43622e6 −0.999980
\(530\) 0 0
\(531\) 1.84462e6 0.283903
\(532\) 5.31158e6 0.813663
\(533\) 2.17651e6 0.331851
\(534\) 719828. 0.109238
\(535\) 0 0
\(536\) 4.66785e6 0.701786
\(537\) 5.96324e6 0.892374
\(538\) 5.81208e6 0.865717
\(539\) 1.38310e6 0.205060
\(540\) 0 0
\(541\) −7.78523e6 −1.14361 −0.571806 0.820389i \(-0.693757\pi\)
−0.571806 + 0.820389i \(0.693757\pi\)
\(542\) 6.22630e6 0.910399
\(543\) 4.73112e6 0.688596
\(544\) −2.41907e6 −0.350470
\(545\) 0 0
\(546\) 849097. 0.121892
\(547\) 3.92034e6 0.560215 0.280108 0.959969i \(-0.409630\pi\)
0.280108 + 0.959969i \(0.409630\pi\)
\(548\) 159307. 0.0226612
\(549\) −74223.6 −0.0105102
\(550\) 0 0
\(551\) 1.85464e6 0.260244
\(552\) −18631.0 −0.00260249
\(553\) −7.98491e6 −1.11034
\(554\) −2.73876e6 −0.379122
\(555\) 0 0
\(556\) 1.41708e6 0.194405
\(557\) 5.56520e6 0.760051 0.380026 0.924976i \(-0.375915\pi\)
0.380026 + 0.924976i \(0.375915\pi\)
\(558\) 2.10843e6 0.286664
\(559\) −3.14753e6 −0.426030
\(560\) 0 0
\(561\) 464782. 0.0623508
\(562\) 2.79743e6 0.373610
\(563\) −6.49216e6 −0.863213 −0.431607 0.902062i \(-0.642053\pi\)
−0.431607 + 0.902062i \(0.642053\pi\)
\(564\) 1.79302e6 0.237349
\(565\) 0 0
\(566\) −6.99429e6 −0.917704
\(567\) 1.10251e6 0.144021
\(568\) 3.09403e6 0.402396
\(569\) −1.02437e7 −1.32641 −0.663204 0.748439i \(-0.730804\pi\)
−0.663204 + 0.748439i \(0.730804\pi\)
\(570\) 0 0
\(571\) −1.42962e7 −1.83498 −0.917488 0.397764i \(-0.869786\pi\)
−0.917488 + 0.397764i \(0.869786\pi\)
\(572\) 354119. 0.0452542
\(573\) 7.41904e6 0.943976
\(574\) −8.54643e6 −1.08269
\(575\) 0 0
\(576\) 1.82708e6 0.229456
\(577\) −5.37436e6 −0.672027 −0.336014 0.941857i \(-0.609079\pi\)
−0.336014 + 0.941857i \(0.609079\pi\)
\(578\) −4.48303e6 −0.558151
\(579\) −7.05035e6 −0.874006
\(580\) 0 0
\(581\) −4.10276e6 −0.504238
\(582\) 512001. 0.0626561
\(583\) 3.27540e6 0.399110
\(584\) 1.03161e7 1.25166
\(585\) 0 0
\(586\) 6.18572e6 0.744126
\(587\) 1.17715e7 1.41006 0.705030 0.709178i \(-0.250934\pi\)
0.705030 + 0.709178i \(0.250934\pi\)
\(588\) 1.94235e6 0.231678
\(589\) −1.20312e7 −1.42897
\(590\) 0 0
\(591\) −2.45125e6 −0.288681
\(592\) 289852. 0.0339917
\(593\) −3.16466e6 −0.369564 −0.184782 0.982780i \(-0.559158\pi\)
−0.184782 + 0.982780i \(0.559158\pi\)
\(594\) −319498. −0.0371537
\(595\) 0 0
\(596\) 2.25668e6 0.260228
\(597\) −8.84270e6 −1.01543
\(598\) 6306.47 0.000721163 0
\(599\) 4.34303e6 0.494567 0.247284 0.968943i \(-0.420462\pi\)
0.247284 + 0.968943i \(0.420462\pi\)
\(600\) 0 0
\(601\) 8.05946e6 0.910165 0.455082 0.890449i \(-0.349610\pi\)
0.455082 + 0.890449i \(0.349610\pi\)
\(602\) 1.23593e7 1.38996
\(603\) −2.05160e6 −0.229774
\(604\) −7.03708e6 −0.784874
\(605\) 0 0
\(606\) −350928. −0.0388183
\(607\) −284560. −0.0313475 −0.0156737 0.999877i \(-0.504989\pi\)
−0.0156737 + 0.999877i \(0.504989\pi\)
\(608\) −9.48897e6 −1.04102
\(609\) 1.67542e6 0.183054
\(610\) 0 0
\(611\) −1.63558e6 −0.177243
\(612\) 652716. 0.0704443
\(613\) 1.43183e7 1.53901 0.769503 0.638644i \(-0.220504\pi\)
0.769503 + 0.638644i \(0.220504\pi\)
\(614\) 929424. 0.0994931
\(615\) 0 0
\(616\) −3.74720e6 −0.397883
\(617\) −1.35489e7 −1.43282 −0.716411 0.697679i \(-0.754216\pi\)
−0.716411 + 0.697679i \(0.754216\pi\)
\(618\) 935072. 0.0984859
\(619\) −1.25027e7 −1.31152 −0.655762 0.754968i \(-0.727652\pi\)
−0.655762 + 0.754968i \(0.727652\pi\)
\(620\) 0 0
\(621\) 8188.65 0.000852086 0
\(622\) 7.53511e6 0.780932
\(623\) −3.71060e6 −0.383022
\(624\) −88358.9 −0.00908425
\(625\) 0 0
\(626\) −3.81906e6 −0.389512
\(627\) 1.82314e6 0.185204
\(628\) 7.69953e6 0.779050
\(629\) 1.95315e6 0.196838
\(630\) 0 0
\(631\) 272233. 0.0272187 0.0136093 0.999907i \(-0.495668\pi\)
0.0136093 + 0.999907i \(0.495668\pi\)
\(632\) 8.75719e6 0.872112
\(633\) 3.87360e6 0.384242
\(634\) −8.52419e6 −0.842229
\(635\) 0 0
\(636\) 4.59980e6 0.450916
\(637\) −1.77179e6 −0.173007
\(638\) −485521. −0.0472233
\(639\) −1.35988e6 −0.131749
\(640\) 0 0
\(641\) 1.57242e7 1.51156 0.755779 0.654827i \(-0.227258\pi\)
0.755779 + 0.654827i \(0.227258\pi\)
\(642\) −3.97434e6 −0.380564
\(643\) 5.47692e6 0.522407 0.261203 0.965284i \(-0.415881\pi\)
0.261203 + 0.965284i \(0.415881\pi\)
\(644\) 35638.2 0.00338611
\(645\) 0 0
\(646\) 2.58803e6 0.243999
\(647\) 3.91671e6 0.367841 0.183921 0.982941i \(-0.441121\pi\)
0.183921 + 0.982941i \(0.441121\pi\)
\(648\) −1.20915e6 −0.113120
\(649\) 2.75554e6 0.256800
\(650\) 0 0
\(651\) −1.08686e7 −1.00513
\(652\) 6.80323e6 0.626753
\(653\) 1.51457e7 1.38997 0.694985 0.719024i \(-0.255411\pi\)
0.694985 + 0.719024i \(0.255411\pi\)
\(654\) 5.53465e6 0.505995
\(655\) 0 0
\(656\) 889361. 0.0806898
\(657\) −4.53413e6 −0.409808
\(658\) 6.42235e6 0.578269
\(659\) 1.56274e6 0.140176 0.0700880 0.997541i \(-0.477672\pi\)
0.0700880 + 0.997541i \(0.477672\pi\)
\(660\) 0 0
\(661\) 1.33712e7 1.19033 0.595163 0.803605i \(-0.297088\pi\)
0.595163 + 0.803605i \(0.297088\pi\)
\(662\) −1.36514e6 −0.121069
\(663\) −595401. −0.0526049
\(664\) 4.49957e6 0.396051
\(665\) 0 0
\(666\) −1.34263e6 −0.117292
\(667\) 12443.8 0.00108302
\(668\) 1.31535e6 0.114051
\(669\) 4.78829e6 0.413633
\(670\) 0 0
\(671\) −110877. −0.00950684
\(672\) −8.57201e6 −0.732250
\(673\) 1.30303e7 1.10896 0.554481 0.832196i \(-0.312917\pi\)
0.554481 + 0.832196i \(0.312917\pi\)
\(674\) 1.13707e7 0.964130
\(675\) 0 0
\(676\) 6.55663e6 0.551841
\(677\) −1.20799e7 −1.01296 −0.506478 0.862253i \(-0.669053\pi\)
−0.506478 + 0.862253i \(0.669053\pi\)
\(678\) 7.18040e6 0.599894
\(679\) −2.63929e6 −0.219691
\(680\) 0 0
\(681\) 7.13416e6 0.589488
\(682\) 3.14962e6 0.259297
\(683\) 1.84029e7 1.50950 0.754752 0.656011i \(-0.227757\pi\)
0.754752 + 0.656011i \(0.227757\pi\)
\(684\) 2.56032e6 0.209245
\(685\) 0 0
\(686\) −3.27238e6 −0.265494
\(687\) −2.57903e6 −0.208480
\(688\) −1.28613e6 −0.103589
\(689\) −4.19589e6 −0.336726
\(690\) 0 0
\(691\) 5.23001e6 0.416684 0.208342 0.978056i \(-0.433193\pi\)
0.208342 + 0.978056i \(0.433193\pi\)
\(692\) 1.18407e7 0.939963
\(693\) 1.64696e6 0.130272
\(694\) −6.30748e6 −0.497115
\(695\) 0 0
\(696\) −1.83746e6 −0.143779
\(697\) 5.99290e6 0.467257
\(698\) −1.59702e7 −1.24072
\(699\) 1.27291e7 0.985382
\(700\) 0 0
\(701\) 4.83055e6 0.371280 0.185640 0.982618i \(-0.440564\pi\)
0.185640 + 0.982618i \(0.440564\pi\)
\(702\) 409288. 0.0313463
\(703\) 7.66137e6 0.584680
\(704\) 2.72934e6 0.207551
\(705\) 0 0
\(706\) −7.15804e6 −0.540484
\(707\) 1.80898e6 0.136109
\(708\) 3.86974e6 0.290134
\(709\) −6.13980e6 −0.458711 −0.229355 0.973343i \(-0.573662\pi\)
−0.229355 + 0.973343i \(0.573662\pi\)
\(710\) 0 0
\(711\) −3.84894e6 −0.285540
\(712\) 4.06948e6 0.300842
\(713\) −80724.0 −0.00594674
\(714\) 2.33794e6 0.171628
\(715\) 0 0
\(716\) 1.25100e7 0.911959
\(717\) −9.06249e6 −0.658339
\(718\) 1.23181e6 0.0891731
\(719\) 1.31130e7 0.945974 0.472987 0.881069i \(-0.343176\pi\)
0.472987 + 0.881069i \(0.343176\pi\)
\(720\) 0 0
\(721\) −4.82015e6 −0.345321
\(722\) 1.18315e6 0.0844690
\(723\) −9.78444e6 −0.696130
\(724\) 9.92520e6 0.703708
\(725\) 0 0
\(726\) −477275. −0.0336068
\(727\) −2.62296e7 −1.84059 −0.920293 0.391229i \(-0.872050\pi\)
−0.920293 + 0.391229i \(0.872050\pi\)
\(728\) 4.80029e6 0.335691
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −8.66653e6 −0.599863
\(732\) −155710. −0.0107409
\(733\) 261993. 0.0180106 0.00900532 0.999959i \(-0.497133\pi\)
0.00900532 + 0.999959i \(0.497133\pi\)
\(734\) −1.30516e7 −0.894178
\(735\) 0 0
\(736\) −63666.6 −0.00433229
\(737\) −3.06474e6 −0.207838
\(738\) −4.11961e6 −0.278430
\(739\) 2.14527e7 1.44501 0.722506 0.691365i \(-0.242990\pi\)
0.722506 + 0.691365i \(0.242990\pi\)
\(740\) 0 0
\(741\) −2.33550e6 −0.156255
\(742\) 1.64758e7 1.09860
\(743\) −1.11848e7 −0.743283 −0.371642 0.928376i \(-0.621205\pi\)
−0.371642 + 0.928376i \(0.621205\pi\)
\(744\) 1.19198e7 0.789471
\(745\) 0 0
\(746\) −7.06280e6 −0.464654
\(747\) −1.97764e6 −0.129672
\(748\) 975045. 0.0637192
\(749\) 2.04871e7 1.33437
\(750\) 0 0
\(751\) 5.63342e6 0.364479 0.182239 0.983254i \(-0.441665\pi\)
0.182239 + 0.983254i \(0.441665\pi\)
\(752\) −668325. −0.0430966
\(753\) −2.81509e6 −0.180928
\(754\) 621968. 0.0398419
\(755\) 0 0
\(756\) 2.31291e6 0.147182
\(757\) 7.69867e6 0.488288 0.244144 0.969739i \(-0.421493\pi\)
0.244144 + 0.969739i \(0.421493\pi\)
\(758\) 7.81276e6 0.493891
\(759\) 12232.4 0.000770741 0
\(760\) 0 0
\(761\) 2.39899e7 1.50164 0.750822 0.660505i \(-0.229658\pi\)
0.750822 + 0.660505i \(0.229658\pi\)
\(762\) 7.52987e6 0.469786
\(763\) −2.85303e7 −1.77417
\(764\) 1.55641e7 0.964694
\(765\) 0 0
\(766\) 6.04179e6 0.372043
\(767\) −3.52994e6 −0.216660
\(768\) 9.74549e6 0.596212
\(769\) −2.03436e7 −1.24054 −0.620272 0.784387i \(-0.712978\pi\)
−0.620272 + 0.784387i \(0.712978\pi\)
\(770\) 0 0
\(771\) 1.35581e7 0.821415
\(772\) −1.47906e7 −0.893188
\(773\) 1.86605e7 1.12324 0.561622 0.827394i \(-0.310178\pi\)
0.561622 + 0.827394i \(0.310178\pi\)
\(774\) 5.95750e6 0.357447
\(775\) 0 0
\(776\) 2.89455e6 0.172555
\(777\) 6.92102e6 0.411261
\(778\) 5.75608e6 0.340940
\(779\) 2.35076e7 1.38792
\(780\) 0 0
\(781\) −2.03143e6 −0.119172
\(782\) 17364.5 0.00101542
\(783\) 807596. 0.0470750
\(784\) −723985. −0.0420668
\(785\) 0 0
\(786\) −5.73388e6 −0.331049
\(787\) 2.98644e7 1.71877 0.859383 0.511333i \(-0.170848\pi\)
0.859383 + 0.511333i \(0.170848\pi\)
\(788\) −5.14235e6 −0.295017
\(789\) 1.70696e6 0.0976180
\(790\) 0 0
\(791\) −3.70138e7 −2.10341
\(792\) −1.80625e6 −0.102321
\(793\) 142037. 0.00802084
\(794\) 1.47700e7 0.831438
\(795\) 0 0
\(796\) −1.85507e7 −1.03771
\(797\) 2.60350e7 1.45182 0.725909 0.687791i \(-0.241419\pi\)
0.725909 + 0.687791i \(0.241419\pi\)
\(798\) 9.17073e6 0.509796
\(799\) −4.50346e6 −0.249563
\(800\) 0 0
\(801\) −1.78861e6 −0.0984995
\(802\) −1.77252e7 −0.973093
\(803\) −6.77320e6 −0.370685
\(804\) −4.30396e6 −0.234816
\(805\) 0 0
\(806\) −4.03477e6 −0.218767
\(807\) −1.44417e7 −0.780611
\(808\) −1.98394e6 −0.106906
\(809\) −1.97670e7 −1.06186 −0.530932 0.847414i \(-0.678158\pi\)
−0.530932 + 0.847414i \(0.678158\pi\)
\(810\) 0 0
\(811\) 7.26108e6 0.387658 0.193829 0.981035i \(-0.437909\pi\)
0.193829 + 0.981035i \(0.437909\pi\)
\(812\) 3.51478e6 0.187072
\(813\) −1.54710e7 −0.820901
\(814\) −2.00565e6 −0.106095
\(815\) 0 0
\(816\) −243291. −0.0127909
\(817\) −3.39951e7 −1.78181
\(818\) 1.84342e6 0.0963256
\(819\) −2.10981e6 −0.109909
\(820\) 0 0
\(821\) 3.10030e7 1.60526 0.802631 0.596476i \(-0.203433\pi\)
0.802631 + 0.596476i \(0.203433\pi\)
\(822\) 275051. 0.0141982
\(823\) −2.56767e7 −1.32142 −0.660709 0.750643i \(-0.729744\pi\)
−0.660709 + 0.750643i \(0.729744\pi\)
\(824\) 5.28635e6 0.271230
\(825\) 0 0
\(826\) 1.38609e7 0.706871
\(827\) −1.60345e7 −0.815254 −0.407627 0.913149i \(-0.633644\pi\)
−0.407627 + 0.913149i \(0.633644\pi\)
\(828\) 17178.6 0.000870787 0
\(829\) −1.90352e7 −0.961992 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(830\) 0 0
\(831\) 6.80520e6 0.341852
\(832\) −3.49637e6 −0.175109
\(833\) −4.87853e6 −0.243599
\(834\) 2.44667e6 0.121804
\(835\) 0 0
\(836\) 3.82468e6 0.189269
\(837\) −5.23896e6 −0.258483
\(838\) 2.45912e7 1.20968
\(839\) 2.86604e7 1.40565 0.702826 0.711362i \(-0.251921\pi\)
0.702826 + 0.711362i \(0.251921\pi\)
\(840\) 0 0
\(841\) −1.92839e7 −0.940167
\(842\) −1.42421e7 −0.692297
\(843\) −6.95099e6 −0.336882
\(844\) 8.12625e6 0.392676
\(845\) 0 0
\(846\) 3.09575e6 0.148710
\(847\) 2.46028e6 0.117835
\(848\) −1.71451e6 −0.0818749
\(849\) 1.73792e7 0.827488
\(850\) 0 0
\(851\) 51404.3 0.00243319
\(852\) −2.85283e6 −0.134641
\(853\) −5.22750e6 −0.245992 −0.122996 0.992407i \(-0.539250\pi\)
−0.122996 + 0.992407i \(0.539250\pi\)
\(854\) −557733. −0.0261687
\(855\) 0 0
\(856\) −2.24686e7 −1.04807
\(857\) −9.88397e6 −0.459705 −0.229853 0.973225i \(-0.573824\pi\)
−0.229853 + 0.973225i \(0.573824\pi\)
\(858\) 611405. 0.0283538
\(859\) −1.37127e7 −0.634076 −0.317038 0.948413i \(-0.602688\pi\)
−0.317038 + 0.948413i \(0.602688\pi\)
\(860\) 0 0
\(861\) 2.12360e7 0.976257
\(862\) −1.17417e7 −0.538226
\(863\) 3.39850e7 1.55332 0.776659 0.629921i \(-0.216913\pi\)
0.776659 + 0.629921i \(0.216913\pi\)
\(864\) −4.13194e6 −0.188308
\(865\) 0 0
\(866\) −2.16139e7 −0.979352
\(867\) 1.11393e7 0.503281
\(868\) −2.28007e7 −1.02719
\(869\) −5.74965e6 −0.258281
\(870\) 0 0
\(871\) 3.92603e6 0.175351
\(872\) 3.12896e7 1.39351
\(873\) −1.27221e6 −0.0564966
\(874\) 68113.4 0.00301616
\(875\) 0 0
\(876\) −9.51193e6 −0.418802
\(877\) 1.66649e7 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(878\) 1.03580e7 0.453459
\(879\) −1.53701e7 −0.670973
\(880\) 0 0
\(881\) −1.85938e7 −0.807103 −0.403551 0.914957i \(-0.632224\pi\)
−0.403551 + 0.914957i \(0.632224\pi\)
\(882\) 3.35357e6 0.145156
\(883\) −1.41760e7 −0.611859 −0.305929 0.952054i \(-0.598967\pi\)
−0.305929 + 0.952054i \(0.598967\pi\)
\(884\) −1.24907e6 −0.0537594
\(885\) 0 0
\(886\) 9.15737e6 0.391910
\(887\) −1.10264e7 −0.470570 −0.235285 0.971926i \(-0.575602\pi\)
−0.235285 + 0.971926i \(0.575602\pi\)
\(888\) −7.59041e6 −0.323023
\(889\) −3.88153e7 −1.64721
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 1.00451e7 0.422711
\(893\) −1.76652e7 −0.741291
\(894\) 3.89628e6 0.163045
\(895\) 0 0
\(896\) −1.67492e7 −0.696986
\(897\) −15670.2 −0.000650268 0
\(898\) 2.07454e7 0.858483
\(899\) −7.96131e6 −0.328538
\(900\) 0 0
\(901\) −1.15531e7 −0.474120
\(902\) −6.15399e6 −0.251849
\(903\) −3.07100e7 −1.25332
\(904\) 4.05937e7 1.65211
\(905\) 0 0
\(906\) −1.21499e7 −0.491759
\(907\) −1.14062e7 −0.460388 −0.230194 0.973145i \(-0.573936\pi\)
−0.230194 + 0.973145i \(0.573936\pi\)
\(908\) 1.49664e7 0.602426
\(909\) 871978. 0.0350022
\(910\) 0 0
\(911\) 2.17910e7 0.869922 0.434961 0.900449i \(-0.356762\pi\)
0.434961 + 0.900449i \(0.356762\pi\)
\(912\) −954326. −0.0379935
\(913\) −2.95425e6 −0.117293
\(914\) −1.78946e7 −0.708527
\(915\) 0 0
\(916\) −5.41042e6 −0.213055
\(917\) 2.95572e7 1.16075
\(918\) 1.12695e6 0.0441365
\(919\) 3.12982e7 1.22245 0.611225 0.791457i \(-0.290677\pi\)
0.611225 + 0.791457i \(0.290677\pi\)
\(920\) 0 0
\(921\) −2.30941e6 −0.0897123
\(922\) 2.24053e7 0.868008
\(923\) 2.60232e6 0.100544
\(924\) 3.45509e6 0.133131
\(925\) 0 0
\(926\) −4.46345e6 −0.171058
\(927\) −2.32344e6 −0.0888040
\(928\) −6.27904e6 −0.239345
\(929\) −3.00058e7 −1.14069 −0.570343 0.821407i \(-0.693190\pi\)
−0.570343 + 0.821407i \(0.693190\pi\)
\(930\) 0 0
\(931\) −1.91364e7 −0.723578
\(932\) 2.67038e7 1.00701
\(933\) −1.87230e7 −0.704161
\(934\) −2.80485e7 −1.05206
\(935\) 0 0
\(936\) 2.31387e6 0.0863276
\(937\) −2.12772e7 −0.791711 −0.395855 0.918313i \(-0.629552\pi\)
−0.395855 + 0.918313i \(0.629552\pi\)
\(938\) −1.54162e7 −0.572098
\(939\) 9.48950e6 0.351220
\(940\) 0 0
\(941\) 4.12864e7 1.51996 0.759982 0.649944i \(-0.225208\pi\)
0.759982 + 0.649944i \(0.225208\pi\)
\(942\) 1.32937e7 0.488110
\(943\) 157725. 0.00577593
\(944\) −1.44239e6 −0.0526809
\(945\) 0 0
\(946\) 8.89948e6 0.323323
\(947\) −1.15296e7 −0.417771 −0.208886 0.977940i \(-0.566984\pi\)
−0.208886 + 0.977940i \(0.566984\pi\)
\(948\) −8.07452e6 −0.291807
\(949\) 8.67670e6 0.312744
\(950\) 0 0
\(951\) 2.11807e7 0.759432
\(952\) 1.32173e7 0.472663
\(953\) 9.67798e6 0.345185 0.172593 0.984993i \(-0.444786\pi\)
0.172593 + 0.984993i \(0.444786\pi\)
\(954\) 7.94180e6 0.282519
\(955\) 0 0
\(956\) −1.90118e7 −0.672788
\(957\) 1.20641e6 0.0425809
\(958\) −1.21750e7 −0.428604
\(959\) −1.41785e6 −0.0497832
\(960\) 0 0
\(961\) 2.30167e7 0.803959
\(962\) 2.56931e6 0.0895113
\(963\) 9.87534e6 0.343152
\(964\) −2.05263e7 −0.711408
\(965\) 0 0
\(966\) 61531.4 0.00212155
\(967\) −3.88946e7 −1.33759 −0.668795 0.743447i \(-0.733189\pi\)
−0.668795 + 0.743447i \(0.733189\pi\)
\(968\) −2.69823e6 −0.0925531
\(969\) −6.43067e6 −0.220012
\(970\) 0 0
\(971\) 2.79477e7 0.951259 0.475629 0.879646i \(-0.342220\pi\)
0.475629 + 0.879646i \(0.342220\pi\)
\(972\) 1.11489e6 0.0378499
\(973\) −1.26122e7 −0.427080
\(974\) 1.44071e6 0.0486608
\(975\) 0 0
\(976\) 58038.9 0.00195027
\(977\) −7.01873e6 −0.235246 −0.117623 0.993058i \(-0.537527\pi\)
−0.117623 + 0.993058i \(0.537527\pi\)
\(978\) 1.17462e7 0.392689
\(979\) −2.67187e6 −0.0890962
\(980\) 0 0
\(981\) −1.37524e7 −0.456252
\(982\) −2.12188e6 −0.0702171
\(983\) 4.12903e7 1.36290 0.681450 0.731864i \(-0.261350\pi\)
0.681450 + 0.731864i \(0.261350\pi\)
\(984\) −2.32899e7 −0.766795
\(985\) 0 0
\(986\) 1.71255e6 0.0560985
\(987\) −1.59581e7 −0.521421
\(988\) −4.89955e6 −0.159685
\(989\) −228091. −0.00741512
\(990\) 0 0
\(991\) 2.63627e6 0.0852717 0.0426359 0.999091i \(-0.486424\pi\)
0.0426359 + 0.999091i \(0.486424\pi\)
\(992\) 4.07328e7 1.31421
\(993\) 3.39207e6 0.109167
\(994\) −1.02184e7 −0.328034
\(995\) 0 0
\(996\) −4.14880e6 −0.132518
\(997\) −4.09967e7 −1.30620 −0.653102 0.757270i \(-0.726533\pi\)
−0.653102 + 0.757270i \(0.726533\pi\)
\(998\) 2.83328e7 0.900458
\(999\) 3.33612e6 0.105762
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.v.1.10 13
5.2 odd 4 165.6.c.b.34.17 yes 26
5.3 odd 4 165.6.c.b.34.10 26
5.4 even 2 825.6.a.y.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.10 26 5.3 odd 4
165.6.c.b.34.17 yes 26 5.2 odd 4
825.6.a.v.1.10 13 1.1 even 1 trivial
825.6.a.y.1.4 13 5.4 even 2