Properties

Label 825.4.c.l.199.5
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2230106176.1
Defining polynomial: \(x^{6} + 41 x^{4} + 452 x^{2} + 676\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.5
Root \(4.59056i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.l.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+3.59056i q^{2} +3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} -16.1465i q^{7} +11.1590i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+3.59056i q^{2} +3.00000i q^{3} -4.89212 q^{4} -10.7717 q^{6} -16.1465i q^{7} +11.1590i q^{8} -9.00000 q^{9} +11.0000 q^{11} -14.6764i q^{12} +54.1214i q^{13} +57.9749 q^{14} -79.2041 q^{16} -107.010i q^{17} -32.3150i q^{18} -48.7496 q^{19} +48.4394 q^{21} +39.4962i q^{22} -11.9498i q^{23} -33.4771 q^{24} -194.326 q^{26} -27.0000i q^{27} +78.9905i q^{28} -239.733 q^{29} -82.0851 q^{31} -195.115i q^{32} +33.0000i q^{33} +384.224 q^{34} +44.0291 q^{36} -21.7573i q^{37} -175.038i q^{38} -162.364 q^{39} -124.835 q^{41} +173.925i q^{42} -224.459i q^{43} -53.8133 q^{44} +42.9064 q^{46} -186.832i q^{47} -237.612i q^{48} +82.2913 q^{49} +321.029 q^{51} -264.768i q^{52} -233.997i q^{53} +96.9451 q^{54} +180.179 q^{56} -146.249i q^{57} -860.774i q^{58} -232.936 q^{59} +163.849 q^{61} -294.731i q^{62} +145.318i q^{63} +66.9386 q^{64} -118.488 q^{66} -876.918i q^{67} +523.503i q^{68} +35.8493 q^{69} -733.141 q^{71} -100.431i q^{72} -1161.97i q^{73} +78.1208 q^{74} +238.489 q^{76} -177.611i q^{77} -582.978i q^{78} +588.831 q^{79} +81.0000 q^{81} -448.226i q^{82} +1161.06i q^{83} -236.971 q^{84} +805.933 q^{86} -719.198i q^{87} +122.749i q^{88} +1042.16 q^{89} +873.869 q^{91} +58.4597i q^{92} -246.255i q^{93} +670.831 q^{94} +585.345 q^{96} +1546.63i q^{97} +295.472i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + O(q^{10}) \) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + 66q^{11} + 112q^{14} + 100q^{16} - 292q^{19} + 24q^{21} - 288q^{24} - 1016q^{26} - 136q^{29} - 136q^{31} + 352q^{34} + 396q^{36} - 392q^{41} - 484q^{44} + 2320q^{46} + 314q^{49} + 1308q^{51} - 216q^{54} + 2736q^{56} + 2088q^{59} + 1284q^{61} - 2332q^{64} + 264q^{66} - 1200q^{69} - 1088q^{71} - 3072q^{74} + 3992q^{76} + 3172q^{79} + 486q^{81} - 2040q^{84} + 7136q^{86} + 4244q^{89} - 16q^{91} + 4304q^{94} + 4128q^{96} - 594q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.59056i 1.26945i 0.772736 + 0.634727i \(0.218888\pi\)
−0.772736 + 0.634727i \(0.781112\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −4.89212 −0.611515
\(5\) 0 0
\(6\) −10.7717 −0.732920
\(7\) − 16.1465i − 0.871828i −0.899988 0.435914i \(-0.856425\pi\)
0.899988 0.435914i \(-0.143575\pi\)
\(8\) 11.1590i 0.493164i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 14.6764i − 0.353058i
\(13\) 54.1214i 1.15466i 0.816511 + 0.577329i \(0.195905\pi\)
−0.816511 + 0.577329i \(0.804095\pi\)
\(14\) 57.9749 1.10675
\(15\) 0 0
\(16\) −79.2041 −1.23756
\(17\) − 107.010i − 1.52668i −0.645995 0.763342i \(-0.723557\pi\)
0.645995 0.763342i \(-0.276443\pi\)
\(18\) − 32.3150i − 0.423152i
\(19\) −48.7496 −0.588628 −0.294314 0.955709i \(-0.595091\pi\)
−0.294314 + 0.955709i \(0.595091\pi\)
\(20\) 0 0
\(21\) 48.4394 0.503350
\(22\) 39.4962i 0.382755i
\(23\) − 11.9498i − 0.108335i −0.998532 0.0541674i \(-0.982750\pi\)
0.998532 0.0541674i \(-0.0172505\pi\)
\(24\) −33.4771 −0.284729
\(25\) 0 0
\(26\) −194.326 −1.46579
\(27\) − 27.0000i − 0.192450i
\(28\) 78.9905i 0.533136i
\(29\) −239.733 −1.53508 −0.767538 0.641003i \(-0.778519\pi\)
−0.767538 + 0.641003i \(0.778519\pi\)
\(30\) 0 0
\(31\) −82.0851 −0.475578 −0.237789 0.971317i \(-0.576423\pi\)
−0.237789 + 0.971317i \(0.576423\pi\)
\(32\) − 195.115i − 1.07787i
\(33\) 33.0000i 0.174078i
\(34\) 384.224 1.93806
\(35\) 0 0
\(36\) 44.0291 0.203838
\(37\) − 21.7573i − 0.0966723i −0.998831 0.0483361i \(-0.984608\pi\)
0.998831 0.0483361i \(-0.0153919\pi\)
\(38\) − 175.038i − 0.747236i
\(39\) −162.364 −0.666643
\(40\) 0 0
\(41\) −124.835 −0.475510 −0.237755 0.971325i \(-0.576412\pi\)
−0.237755 + 0.971325i \(0.576412\pi\)
\(42\) 173.925i 0.638980i
\(43\) − 224.459i − 0.796039i −0.917377 0.398019i \(-0.869698\pi\)
0.917377 0.398019i \(-0.130302\pi\)
\(44\) −53.8133 −0.184379
\(45\) 0 0
\(46\) 42.9064 0.137526
\(47\) − 186.832i − 0.579835i −0.957052 0.289917i \(-0.906372\pi\)
0.957052 0.289917i \(-0.0936278\pi\)
\(48\) − 237.612i − 0.714508i
\(49\) 82.2913 0.239916
\(50\) 0 0
\(51\) 321.029 0.881431
\(52\) − 264.768i − 0.706091i
\(53\) − 233.997i − 0.606453i −0.952919 0.303226i \(-0.901936\pi\)
0.952919 0.303226i \(-0.0980638\pi\)
\(54\) 96.9451 0.244307
\(55\) 0 0
\(56\) 180.179 0.429954
\(57\) − 146.249i − 0.339844i
\(58\) − 860.774i − 1.94871i
\(59\) −232.936 −0.513996 −0.256998 0.966412i \(-0.582733\pi\)
−0.256998 + 0.966412i \(0.582733\pi\)
\(60\) 0 0
\(61\) 163.849 0.343913 0.171957 0.985105i \(-0.444991\pi\)
0.171957 + 0.985105i \(0.444991\pi\)
\(62\) − 294.731i − 0.603725i
\(63\) 145.318i 0.290609i
\(64\) 66.9386 0.130739
\(65\) 0 0
\(66\) −118.488 −0.220984
\(67\) − 876.918i − 1.59899i −0.600670 0.799497i \(-0.705100\pi\)
0.600670 0.799497i \(-0.294900\pi\)
\(68\) 523.503i 0.933590i
\(69\) 35.8493 0.0625471
\(70\) 0 0
\(71\) −733.141 −1.22546 −0.612731 0.790291i \(-0.709929\pi\)
−0.612731 + 0.790291i \(0.709929\pi\)
\(72\) − 100.431i − 0.164388i
\(73\) − 1161.97i − 1.86299i −0.363750 0.931496i \(-0.618504\pi\)
0.363750 0.931496i \(-0.381496\pi\)
\(74\) 78.1208 0.122721
\(75\) 0 0
\(76\) 238.489 0.359954
\(77\) − 177.611i − 0.262866i
\(78\) − 582.978i − 0.846272i
\(79\) 588.831 0.838591 0.419296 0.907850i \(-0.362277\pi\)
0.419296 + 0.907850i \(0.362277\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 448.226i − 0.603638i
\(83\) 1161.06i 1.53546i 0.640772 + 0.767731i \(0.278614\pi\)
−0.640772 + 0.767731i \(0.721386\pi\)
\(84\) −236.971 −0.307806
\(85\) 0 0
\(86\) 805.933 1.01053
\(87\) − 719.198i − 0.886277i
\(88\) 122.749i 0.148695i
\(89\) 1042.16 1.24122 0.620610 0.784120i \(-0.286885\pi\)
0.620610 + 0.784120i \(0.286885\pi\)
\(90\) 0 0
\(91\) 873.869 1.00666
\(92\) 58.4597i 0.0662483i
\(93\) − 246.255i − 0.274575i
\(94\) 670.831 0.736074
\(95\) 0 0
\(96\) 585.345 0.622307
\(97\) 1546.63i 1.61893i 0.587169 + 0.809464i \(0.300242\pi\)
−0.587169 + 0.809464i \(0.699758\pi\)
\(98\) 295.472i 0.304563i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −662.282 −0.652470 −0.326235 0.945289i \(-0.605780\pi\)
−0.326235 + 0.945289i \(0.605780\pi\)
\(102\) 1152.67i 1.11894i
\(103\) − 399.592i − 0.382262i −0.981565 0.191131i \(-0.938784\pi\)
0.981565 0.191131i \(-0.0612156\pi\)
\(104\) −603.942 −0.569436
\(105\) 0 0
\(106\) 840.181 0.769864
\(107\) − 1591.22i − 1.43765i −0.695189 0.718827i \(-0.744679\pi\)
0.695189 0.718827i \(-0.255321\pi\)
\(108\) 132.087i 0.117686i
\(109\) −755.128 −0.663561 −0.331780 0.943357i \(-0.607649\pi\)
−0.331780 + 0.943357i \(0.607649\pi\)
\(110\) 0 0
\(111\) 65.2718 0.0558138
\(112\) 1278.87i 1.07894i
\(113\) 1145.65i 0.953753i 0.878970 + 0.476876i \(0.158231\pi\)
−0.878970 + 0.476876i \(0.841769\pi\)
\(114\) 525.115 0.431417
\(115\) 0 0
\(116\) 1172.80 0.938722
\(117\) − 487.092i − 0.384886i
\(118\) − 836.372i − 0.652494i
\(119\) −1727.83 −1.33101
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 588.309i 0.436582i
\(123\) − 374.504i − 0.274536i
\(124\) 401.570 0.290823
\(125\) 0 0
\(126\) −521.774 −0.368915
\(127\) 1461.29i 1.02101i 0.859875 + 0.510505i \(0.170542\pi\)
−0.859875 + 0.510505i \(0.829458\pi\)
\(128\) − 1320.57i − 0.911900i
\(129\) 673.377 0.459593
\(130\) 0 0
\(131\) −1524.94 −1.01706 −0.508528 0.861045i \(-0.669810\pi\)
−0.508528 + 0.861045i \(0.669810\pi\)
\(132\) − 161.440i − 0.106451i
\(133\) 787.134i 0.513182i
\(134\) 3148.63 2.02985
\(135\) 0 0
\(136\) 1194.12 0.752906
\(137\) − 2125.68i − 1.32561i −0.748792 0.662805i \(-0.769366\pi\)
0.748792 0.662805i \(-0.230634\pi\)
\(138\) 128.719i 0.0794007i
\(139\) 1774.28 1.08268 0.541339 0.840805i \(-0.317918\pi\)
0.541339 + 0.840805i \(0.317918\pi\)
\(140\) 0 0
\(141\) 560.496 0.334768
\(142\) − 2632.39i − 1.55567i
\(143\) 595.335i 0.348143i
\(144\) 712.837 0.412521
\(145\) 0 0
\(146\) 4172.13 2.36498
\(147\) 246.874i 0.138516i
\(148\) 106.439i 0.0591165i
\(149\) −1575.78 −0.866393 −0.433197 0.901299i \(-0.642614\pi\)
−0.433197 + 0.901299i \(0.642614\pi\)
\(150\) 0 0
\(151\) 420.978 0.226879 0.113439 0.993545i \(-0.463813\pi\)
0.113439 + 0.993545i \(0.463813\pi\)
\(152\) − 543.998i − 0.290290i
\(153\) 963.086i 0.508895i
\(154\) 637.724 0.333696
\(155\) 0 0
\(156\) 794.304 0.407662
\(157\) − 2224.30i − 1.13069i −0.824854 0.565345i \(-0.808743\pi\)
0.824854 0.565345i \(-0.191257\pi\)
\(158\) 2114.23i 1.06455i
\(159\) 701.992 0.350136
\(160\) 0 0
\(161\) −192.947 −0.0944492
\(162\) 290.835i 0.141051i
\(163\) 3093.37i 1.48645i 0.669040 + 0.743226i \(0.266705\pi\)
−0.669040 + 0.743226i \(0.733295\pi\)
\(164\) 610.706 0.290781
\(165\) 0 0
\(166\) −4168.87 −1.94920
\(167\) − 2416.43i − 1.11970i −0.828595 0.559848i \(-0.810860\pi\)
0.828595 0.559848i \(-0.189140\pi\)
\(168\) 540.537i 0.248234i
\(169\) −732.122 −0.333237
\(170\) 0 0
\(171\) 438.746 0.196209
\(172\) 1098.08i 0.486789i
\(173\) 3758.02i 1.65154i 0.564005 + 0.825771i \(0.309260\pi\)
−0.564005 + 0.825771i \(0.690740\pi\)
\(174\) 2582.32 1.12509
\(175\) 0 0
\(176\) −871.245 −0.373140
\(177\) − 698.809i − 0.296756i
\(178\) 3741.93i 1.57567i
\(179\) −2533.99 −1.05810 −0.529049 0.848591i \(-0.677451\pi\)
−0.529049 + 0.848591i \(0.677451\pi\)
\(180\) 0 0
\(181\) −13.8995 −0.00570798 −0.00285399 0.999996i \(-0.500908\pi\)
−0.00285399 + 0.999996i \(0.500908\pi\)
\(182\) 3137.68i 1.27791i
\(183\) 491.547i 0.198558i
\(184\) 133.348 0.0534268
\(185\) 0 0
\(186\) 884.194 0.348561
\(187\) − 1177.10i − 0.460312i
\(188\) 914.004i 0.354577i
\(189\) −435.955 −0.167783
\(190\) 0 0
\(191\) 3495.39 1.32417 0.662087 0.749427i \(-0.269671\pi\)
0.662087 + 0.749427i \(0.269671\pi\)
\(192\) 200.816i 0.0754824i
\(193\) − 3469.33i − 1.29393i −0.762522 0.646963i \(-0.776039\pi\)
0.762522 0.646963i \(-0.223961\pi\)
\(194\) −5553.25 −2.05516
\(195\) 0 0
\(196\) −402.579 −0.146712
\(197\) 3638.39i 1.31586i 0.753079 + 0.657930i \(0.228568\pi\)
−0.753079 + 0.657930i \(0.771432\pi\)
\(198\) − 355.465i − 0.127585i
\(199\) −51.6049 −0.0183828 −0.00919140 0.999958i \(-0.502926\pi\)
−0.00919140 + 0.999958i \(0.502926\pi\)
\(200\) 0 0
\(201\) 2630.75 0.923179
\(202\) − 2377.96i − 0.828281i
\(203\) 3870.84i 1.33832i
\(204\) −1570.51 −0.539008
\(205\) 0 0
\(206\) 1434.76 0.485265
\(207\) 107.548i 0.0361116i
\(208\) − 4286.63i − 1.42896i
\(209\) −536.246 −0.177478
\(210\) 0 0
\(211\) 2084.57 0.680131 0.340065 0.940402i \(-0.389551\pi\)
0.340065 + 0.940402i \(0.389551\pi\)
\(212\) 1144.74i 0.370855i
\(213\) − 2199.42i − 0.707521i
\(214\) 5713.37 1.82504
\(215\) 0 0
\(216\) 301.294 0.0949095
\(217\) 1325.38i 0.414622i
\(218\) − 2711.33i − 0.842361i
\(219\) 3485.91 1.07560
\(220\) 0 0
\(221\) 5791.50 1.76280
\(222\) 234.362i 0.0708531i
\(223\) − 1887.36i − 0.566757i −0.959008 0.283378i \(-0.908545\pi\)
0.959008 0.283378i \(-0.0914552\pi\)
\(224\) −3150.42 −0.939715
\(225\) 0 0
\(226\) −4113.54 −1.21075
\(227\) − 1150.73i − 0.336462i −0.985748 0.168231i \(-0.946194\pi\)
0.985748 0.168231i \(-0.0538055\pi\)
\(228\) 715.466i 0.207820i
\(229\) −4106.79 −1.18508 −0.592542 0.805540i \(-0.701875\pi\)
−0.592542 + 0.805540i \(0.701875\pi\)
\(230\) 0 0
\(231\) 532.834 0.151766
\(232\) − 2675.18i − 0.757045i
\(233\) 5733.58i 1.61210i 0.591848 + 0.806050i \(0.298399\pi\)
−0.591848 + 0.806050i \(0.701601\pi\)
\(234\) 1748.93 0.488596
\(235\) 0 0
\(236\) 1139.55 0.314316
\(237\) 1766.49i 0.484161i
\(238\) − 6203.86i − 1.68965i
\(239\) −6036.18 −1.63367 −0.816837 0.576868i \(-0.804275\pi\)
−0.816837 + 0.576868i \(0.804275\pi\)
\(240\) 0 0
\(241\) 3720.90 0.994540 0.497270 0.867596i \(-0.334336\pi\)
0.497270 + 0.867596i \(0.334336\pi\)
\(242\) 434.458i 0.115405i
\(243\) 243.000i 0.0641500i
\(244\) −801.568 −0.210308
\(245\) 0 0
\(246\) 1344.68 0.348511
\(247\) − 2638.39i − 0.679664i
\(248\) − 915.990i − 0.234538i
\(249\) −3483.19 −0.886500
\(250\) 0 0
\(251\) −3809.88 −0.958077 −0.479038 0.877794i \(-0.659015\pi\)
−0.479038 + 0.877794i \(0.659015\pi\)
\(252\) − 710.914i − 0.177712i
\(253\) − 131.447i − 0.0326641i
\(254\) −5246.84 −1.29613
\(255\) 0 0
\(256\) 5277.10 1.28835
\(257\) 1225.95i 0.297559i 0.988870 + 0.148780i \(0.0475345\pi\)
−0.988870 + 0.148780i \(0.952465\pi\)
\(258\) 2417.80i 0.583433i
\(259\) −351.303 −0.0842816
\(260\) 0 0
\(261\) 2157.59 0.511692
\(262\) − 5475.38i − 1.29111i
\(263\) − 7397.68i − 1.73445i −0.497916 0.867225i \(-0.665901\pi\)
0.497916 0.867225i \(-0.334099\pi\)
\(264\) −368.248 −0.0858489
\(265\) 0 0
\(266\) −2826.25 −0.651461
\(267\) 3126.47i 0.716618i
\(268\) 4289.99i 0.977808i
\(269\) 3214.48 0.728589 0.364295 0.931284i \(-0.381310\pi\)
0.364295 + 0.931284i \(0.381310\pi\)
\(270\) 0 0
\(271\) −7377.37 −1.65367 −0.826833 0.562448i \(-0.809860\pi\)
−0.826833 + 0.562448i \(0.809860\pi\)
\(272\) 8475.60i 1.88937i
\(273\) 2621.61i 0.581197i
\(274\) 7632.36 1.68280
\(275\) 0 0
\(276\) −175.379 −0.0382485
\(277\) − 810.606i − 0.175829i −0.996128 0.0879144i \(-0.971980\pi\)
0.996128 0.0879144i \(-0.0280202\pi\)
\(278\) 6370.65i 1.37441i
\(279\) 738.766 0.158526
\(280\) 0 0
\(281\) 1114.72 0.236651 0.118325 0.992975i \(-0.462247\pi\)
0.118325 + 0.992975i \(0.462247\pi\)
\(282\) 2012.49i 0.424972i
\(283\) − 2265.40i − 0.475844i −0.971284 0.237922i \(-0.923534\pi\)
0.971284 0.237922i \(-0.0764663\pi\)
\(284\) 3586.61 0.749388
\(285\) 0 0
\(286\) −2137.59 −0.441951
\(287\) 2015.64i 0.414563i
\(288\) 1756.03i 0.359289i
\(289\) −6538.04 −1.33076
\(290\) 0 0
\(291\) −4639.88 −0.934689
\(292\) 5684.50i 1.13925i
\(293\) − 3802.06i − 0.758084i −0.925380 0.379042i \(-0.876254\pi\)
0.925380 0.379042i \(-0.123746\pi\)
\(294\) −886.416 −0.175840
\(295\) 0 0
\(296\) 242.790 0.0476753
\(297\) − 297.000i − 0.0580259i
\(298\) − 5657.92i − 1.09985i
\(299\) 646.738 0.125090
\(300\) 0 0
\(301\) −3624.22 −0.694009
\(302\) 1511.55i 0.288013i
\(303\) − 1986.85i − 0.376704i
\(304\) 3861.17 0.728465
\(305\) 0 0
\(306\) −3458.02 −0.646019
\(307\) − 1356.35i − 0.252153i −0.992021 0.126077i \(-0.959761\pi\)
0.992021 0.126077i \(-0.0402385\pi\)
\(308\) 868.895i 0.160746i
\(309\) 1198.78 0.220699
\(310\) 0 0
\(311\) −8078.07 −1.47288 −0.736440 0.676503i \(-0.763495\pi\)
−0.736440 + 0.676503i \(0.763495\pi\)
\(312\) − 1811.83i − 0.328764i
\(313\) − 5761.54i − 1.04045i −0.854029 0.520226i \(-0.825848\pi\)
0.854029 0.520226i \(-0.174152\pi\)
\(314\) 7986.48 1.43536
\(315\) 0 0
\(316\) −2880.63 −0.512811
\(317\) 5107.00i 0.904851i 0.891802 + 0.452426i \(0.149441\pi\)
−0.891802 + 0.452426i \(0.850559\pi\)
\(318\) 2520.54i 0.444481i
\(319\) −2637.06 −0.462843
\(320\) 0 0
\(321\) 4773.66 0.830030
\(322\) − 692.786i − 0.119899i
\(323\) 5216.67i 0.898648i
\(324\) −396.262 −0.0679461
\(325\) 0 0
\(326\) −11106.9 −1.88698
\(327\) − 2265.38i − 0.383107i
\(328\) − 1393.03i − 0.234504i
\(329\) −3016.68 −0.505516
\(330\) 0 0
\(331\) −2780.94 −0.461796 −0.230898 0.972978i \(-0.574166\pi\)
−0.230898 + 0.972978i \(0.574166\pi\)
\(332\) − 5680.06i − 0.938958i
\(333\) 195.816i 0.0322241i
\(334\) 8676.35 1.42140
\(335\) 0 0
\(336\) −3836.60 −0.622928
\(337\) 4939.28i 0.798397i 0.916865 + 0.399198i \(0.130712\pi\)
−0.916865 + 0.399198i \(0.869288\pi\)
\(338\) − 2628.73i − 0.423029i
\(339\) −3436.96 −0.550649
\(340\) 0 0
\(341\) −902.936 −0.143392
\(342\) 1575.34i 0.249079i
\(343\) − 6866.96i − 1.08099i
\(344\) 2504.74 0.392578
\(345\) 0 0
\(346\) −13493.4 −2.09656
\(347\) − 2711.58i − 0.419496i −0.977755 0.209748i \(-0.932736\pi\)
0.977755 0.209748i \(-0.0672644\pi\)
\(348\) 3518.40i 0.541971i
\(349\) −5496.03 −0.842967 −0.421484 0.906836i \(-0.638490\pi\)
−0.421484 + 0.906836i \(0.638490\pi\)
\(350\) 0 0
\(351\) 1461.28 0.222214
\(352\) − 2146.26i − 0.324989i
\(353\) − 6372.23i − 0.960792i −0.877052 0.480396i \(-0.840493\pi\)
0.877052 0.480396i \(-0.159507\pi\)
\(354\) 2509.12 0.376718
\(355\) 0 0
\(356\) −5098.36 −0.759024
\(357\) − 5183.48i − 0.768456i
\(358\) − 9098.46i − 1.34321i
\(359\) −630.622 −0.0927101 −0.0463551 0.998925i \(-0.514761\pi\)
−0.0463551 + 0.998925i \(0.514761\pi\)
\(360\) 0 0
\(361\) −4482.48 −0.653518
\(362\) − 49.9071i − 0.00724602i
\(363\) 363.000i 0.0524864i
\(364\) −4275.07 −0.615590
\(365\) 0 0
\(366\) −1764.93 −0.252061
\(367\) − 5374.49i − 0.764431i −0.924073 0.382216i \(-0.875161\pi\)
0.924073 0.382216i \(-0.124839\pi\)
\(368\) 946.471i 0.134071i
\(369\) 1123.51 0.158503
\(370\) 0 0
\(371\) −3778.23 −0.528722
\(372\) 1204.71i 0.167907i
\(373\) − 7520.12i − 1.04391i −0.852974 0.521953i \(-0.825204\pi\)
0.852974 0.521953i \(-0.174796\pi\)
\(374\) 4226.46 0.584346
\(375\) 0 0
\(376\) 2084.86 0.285954
\(377\) − 12974.7i − 1.77249i
\(378\) − 1565.32i − 0.212993i
\(379\) 12509.1 1.69538 0.847690 0.530492i \(-0.177993\pi\)
0.847690 + 0.530492i \(0.177993\pi\)
\(380\) 0 0
\(381\) −4383.86 −0.589481
\(382\) 12550.4i 1.68098i
\(383\) 11149.9i 1.48755i 0.668430 + 0.743775i \(0.266967\pi\)
−0.668430 + 0.743775i \(0.733033\pi\)
\(384\) 3961.72 0.526486
\(385\) 0 0
\(386\) 12456.8 1.64258
\(387\) 2020.13i 0.265346i
\(388\) − 7566.28i − 0.989999i
\(389\) −3194.22 −0.416333 −0.208166 0.978093i \(-0.566750\pi\)
−0.208166 + 0.978093i \(0.566750\pi\)
\(390\) 0 0
\(391\) −1278.74 −0.165393
\(392\) 918.292i 0.118318i
\(393\) − 4574.81i − 0.587198i
\(394\) −13063.8 −1.67042
\(395\) 0 0
\(396\) 484.320 0.0614596
\(397\) − 584.410i − 0.0738809i −0.999317 0.0369404i \(-0.988239\pi\)
0.999317 0.0369404i \(-0.0117612\pi\)
\(398\) − 185.291i − 0.0233361i
\(399\) −2361.40 −0.296286
\(400\) 0 0
\(401\) −6951.24 −0.865657 −0.432829 0.901476i \(-0.642484\pi\)
−0.432829 + 0.901476i \(0.642484\pi\)
\(402\) 9445.88i 1.17193i
\(403\) − 4442.56i − 0.549130i
\(404\) 3239.96 0.398995
\(405\) 0 0
\(406\) −13898.5 −1.69894
\(407\) − 239.330i − 0.0291478i
\(408\) 3582.37i 0.434690i
\(409\) −11754.1 −1.42104 −0.710519 0.703678i \(-0.751540\pi\)
−0.710519 + 0.703678i \(0.751540\pi\)
\(410\) 0 0
\(411\) 6377.03 0.765342
\(412\) 1954.85i 0.233759i
\(413\) 3761.10i 0.448116i
\(414\) −386.157 −0.0458420
\(415\) 0 0
\(416\) 10559.9 1.24457
\(417\) 5322.83i 0.625084i
\(418\) − 1925.42i − 0.225300i
\(419\) −11829.8 −1.37929 −0.689646 0.724146i \(-0.742234\pi\)
−0.689646 + 0.724146i \(0.742234\pi\)
\(420\) 0 0
\(421\) −2000.07 −0.231538 −0.115769 0.993276i \(-0.536933\pi\)
−0.115769 + 0.993276i \(0.536933\pi\)
\(422\) 7484.76i 0.863395i
\(423\) 1681.49i 0.193278i
\(424\) 2611.18 0.299081
\(425\) 0 0
\(426\) 7897.16 0.898166
\(427\) − 2645.58i − 0.299833i
\(428\) 7784.43i 0.879147i
\(429\) −1786.00 −0.201000
\(430\) 0 0
\(431\) −8064.11 −0.901240 −0.450620 0.892716i \(-0.648797\pi\)
−0.450620 + 0.892716i \(0.648797\pi\)
\(432\) 2138.51i 0.238169i
\(433\) 10710.3i 1.18869i 0.804210 + 0.594345i \(0.202589\pi\)
−0.804210 + 0.594345i \(0.797411\pi\)
\(434\) −4758.87 −0.526344
\(435\) 0 0
\(436\) 3694.18 0.405777
\(437\) 582.546i 0.0637688i
\(438\) 12516.4i 1.36542i
\(439\) −4658.08 −0.506419 −0.253210 0.967411i \(-0.581486\pi\)
−0.253210 + 0.967411i \(0.581486\pi\)
\(440\) 0 0
\(441\) −740.622 −0.0799721
\(442\) 20794.7i 2.23779i
\(443\) − 2094.73i − 0.224658i −0.993671 0.112329i \(-0.964169\pi\)
0.993671 0.112329i \(-0.0358310\pi\)
\(444\) −319.318 −0.0341310
\(445\) 0 0
\(446\) 6776.67 0.719472
\(447\) − 4727.33i − 0.500212i
\(448\) − 1080.82i − 0.113982i
\(449\) 1402.21 0.147382 0.0736909 0.997281i \(-0.476522\pi\)
0.0736909 + 0.997281i \(0.476522\pi\)
\(450\) 0 0
\(451\) −1373.18 −0.143372
\(452\) − 5604.67i − 0.583234i
\(453\) 1262.93i 0.130989i
\(454\) 4131.78 0.427124
\(455\) 0 0
\(456\) 1632.00 0.167599
\(457\) − 4914.19i − 0.503011i −0.967856 0.251506i \(-0.919074\pi\)
0.967856 0.251506i \(-0.0809257\pi\)
\(458\) − 14745.7i − 1.50441i
\(459\) −2889.26 −0.293810
\(460\) 0 0
\(461\) −2214.08 −0.223688 −0.111844 0.993726i \(-0.535676\pi\)
−0.111844 + 0.993726i \(0.535676\pi\)
\(462\) 1913.17i 0.192660i
\(463\) 5567.02i 0.558793i 0.960176 + 0.279396i \(0.0901344\pi\)
−0.960176 + 0.279396i \(0.909866\pi\)
\(464\) 18987.8 1.89976
\(465\) 0 0
\(466\) −20586.8 −2.04649
\(467\) 497.054i 0.0492525i 0.999697 + 0.0246263i \(0.00783957\pi\)
−0.999697 + 0.0246263i \(0.992160\pi\)
\(468\) 2382.91i 0.235364i
\(469\) −14159.1 −1.39405
\(470\) 0 0
\(471\) 6672.90 0.652804
\(472\) − 2599.35i − 0.253484i
\(473\) − 2469.05i − 0.240015i
\(474\) −6342.70 −0.614620
\(475\) 0 0
\(476\) 8452.73 0.813929
\(477\) 2105.97i 0.202151i
\(478\) − 21673.3i − 2.07388i
\(479\) 9349.28 0.891815 0.445908 0.895079i \(-0.352881\pi\)
0.445908 + 0.895079i \(0.352881\pi\)
\(480\) 0 0
\(481\) 1177.53 0.111624
\(482\) 13360.1i 1.26252i
\(483\) − 578.840i − 0.0545303i
\(484\) −591.946 −0.0555923
\(485\) 0 0
\(486\) −872.506 −0.0814355
\(487\) 197.750i 0.0184003i 0.999958 + 0.00920013i \(0.00292853\pi\)
−0.999958 + 0.00920013i \(0.997071\pi\)
\(488\) 1828.40i 0.169606i
\(489\) −9280.12 −0.858204
\(490\) 0 0
\(491\) −9997.05 −0.918861 −0.459430 0.888214i \(-0.651946\pi\)
−0.459430 + 0.888214i \(0.651946\pi\)
\(492\) 1832.12i 0.167883i
\(493\) 25653.7i 2.34358i
\(494\) 9473.31 0.862803
\(495\) 0 0
\(496\) 6501.48 0.588558
\(497\) 11837.6i 1.06839i
\(498\) − 12506.6i − 1.12537i
\(499\) 8714.73 0.781813 0.390907 0.920430i \(-0.372162\pi\)
0.390907 + 0.920430i \(0.372162\pi\)
\(500\) 0 0
\(501\) 7249.30 0.646457
\(502\) − 13679.6i − 1.21623i
\(503\) − 5978.53i − 0.529959i −0.964254 0.264979i \(-0.914635\pi\)
0.964254 0.264979i \(-0.0853652\pi\)
\(504\) −1621.61 −0.143318
\(505\) 0 0
\(506\) 471.970 0.0414657
\(507\) − 2196.36i − 0.192394i
\(508\) − 7148.79i − 0.624363i
\(509\) 8205.79 0.714569 0.357284 0.933996i \(-0.383703\pi\)
0.357284 + 0.933996i \(0.383703\pi\)
\(510\) 0 0
\(511\) −18761.7 −1.62421
\(512\) 8383.17i 0.723608i
\(513\) 1316.24i 0.113281i
\(514\) −4401.85 −0.377738
\(515\) 0 0
\(516\) −3294.24 −0.281048
\(517\) − 2055.15i − 0.174827i
\(518\) − 1261.38i − 0.106992i
\(519\) −11274.1 −0.953519
\(520\) 0 0
\(521\) −5266.06 −0.442822 −0.221411 0.975181i \(-0.571066\pi\)
−0.221411 + 0.975181i \(0.571066\pi\)
\(522\) 7746.97i 0.649570i
\(523\) 22398.6i 1.87270i 0.351068 + 0.936350i \(0.385819\pi\)
−0.351068 + 0.936350i \(0.614181\pi\)
\(524\) 7460.18 0.621945
\(525\) 0 0
\(526\) 26561.8 2.20181
\(527\) 8783.89i 0.726057i
\(528\) − 2613.74i − 0.215432i
\(529\) 12024.2 0.988264
\(530\) 0 0
\(531\) 2096.43 0.171332
\(532\) − 3850.75i − 0.313818i
\(533\) − 6756.22i − 0.549052i
\(534\) −11225.8 −0.909714
\(535\) 0 0
\(536\) 9785.56 0.788566
\(537\) − 7601.98i − 0.610893i
\(538\) 11541.8i 0.924911i
\(539\) 905.205 0.0723375
\(540\) 0 0
\(541\) −13030.2 −1.03551 −0.517757 0.855528i \(-0.673233\pi\)
−0.517757 + 0.855528i \(0.673233\pi\)
\(542\) − 26488.9i − 2.09925i
\(543\) − 41.6986i − 0.00329550i
\(544\) −20879.1 −1.64556
\(545\) 0 0
\(546\) −9413.04 −0.737804
\(547\) 10448.9i 0.816747i 0.912815 + 0.408374i \(0.133904\pi\)
−0.912815 + 0.408374i \(0.866096\pi\)
\(548\) 10399.1i 0.810631i
\(549\) −1474.64 −0.114638
\(550\) 0 0
\(551\) 11686.9 0.903588
\(552\) 400.044i 0.0308460i
\(553\) − 9507.55i − 0.731107i
\(554\) 2910.53 0.223207
\(555\) 0 0
\(556\) −8679.97 −0.662073
\(557\) − 11448.7i − 0.870911i −0.900210 0.435456i \(-0.856587\pi\)
0.900210 0.435456i \(-0.143413\pi\)
\(558\) 2652.58i 0.201242i
\(559\) 12148.0 0.919153
\(560\) 0 0
\(561\) 3531.31 0.265762
\(562\) 4002.49i 0.300418i
\(563\) 26035.8i 1.94898i 0.224422 + 0.974492i \(0.427951\pi\)
−0.224422 + 0.974492i \(0.572049\pi\)
\(564\) −2742.01 −0.204715
\(565\) 0 0
\(566\) 8134.05 0.604063
\(567\) − 1307.86i − 0.0968697i
\(568\) − 8181.15i − 0.604354i
\(569\) −25075.0 −1.84745 −0.923725 0.383057i \(-0.874871\pi\)
−0.923725 + 0.383057i \(0.874871\pi\)
\(570\) 0 0
\(571\) 19056.6 1.39666 0.698330 0.715776i \(-0.253927\pi\)
0.698330 + 0.715776i \(0.253927\pi\)
\(572\) − 2912.45i − 0.212894i
\(573\) 10486.2i 0.764512i
\(574\) −7237.28 −0.526268
\(575\) 0 0
\(576\) −602.447 −0.0435798
\(577\) − 6482.55i − 0.467716i −0.972271 0.233858i \(-0.924865\pi\)
0.972271 0.233858i \(-0.0751351\pi\)
\(578\) − 23475.2i − 1.68934i
\(579\) 10408.0 0.747048
\(580\) 0 0
\(581\) 18747.1 1.33866
\(582\) − 16659.8i − 1.18654i
\(583\) − 2573.97i − 0.182852i
\(584\) 12966.5 0.918762
\(585\) 0 0
\(586\) 13651.5 0.962353
\(587\) 24650.3i 1.73327i 0.498946 + 0.866633i \(0.333720\pi\)
−0.498946 + 0.866633i \(0.666280\pi\)
\(588\) − 1207.74i − 0.0847045i
\(589\) 4001.61 0.279938
\(590\) 0 0
\(591\) −10915.2 −0.759712
\(592\) 1723.27i 0.119638i
\(593\) − 23163.2i − 1.60405i −0.597292 0.802024i \(-0.703757\pi\)
0.597292 0.802024i \(-0.296243\pi\)
\(594\) 1066.40 0.0736612
\(595\) 0 0
\(596\) 7708.88 0.529812
\(597\) − 154.815i − 0.0106133i
\(598\) 2322.15i 0.158796i
\(599\) −5355.44 −0.365304 −0.182652 0.983178i \(-0.558468\pi\)
−0.182652 + 0.983178i \(0.558468\pi\)
\(600\) 0 0
\(601\) −20417.6 −1.38578 −0.692889 0.721045i \(-0.743662\pi\)
−0.692889 + 0.721045i \(0.743662\pi\)
\(602\) − 13013.0i − 0.881012i
\(603\) 7892.26i 0.532998i
\(604\) −2059.48 −0.138740
\(605\) 0 0
\(606\) 7133.89 0.478208
\(607\) − 6749.81i − 0.451345i −0.974203 0.225673i \(-0.927542\pi\)
0.974203 0.225673i \(-0.0724580\pi\)
\(608\) 9511.77i 0.634463i
\(609\) −11612.5 −0.772681
\(610\) 0 0
\(611\) 10111.6 0.669511
\(612\) − 4711.53i − 0.311197i
\(613\) 30321.0i 1.99780i 0.0468613 + 0.998901i \(0.485078\pi\)
−0.0468613 + 0.998901i \(0.514922\pi\)
\(614\) 4870.06 0.320097
\(615\) 0 0
\(616\) 1981.97 0.129636
\(617\) 15236.8i 0.994182i 0.867699 + 0.497091i \(0.165598\pi\)
−0.867699 + 0.497091i \(0.834402\pi\)
\(618\) 4304.28i 0.280168i
\(619\) 20875.4 1.35550 0.677749 0.735293i \(-0.262956\pi\)
0.677749 + 0.735293i \(0.262956\pi\)
\(620\) 0 0
\(621\) −322.644 −0.0208490
\(622\) − 29004.8i − 1.86975i
\(623\) − 16827.2i − 1.08213i
\(624\) 12859.9 0.825013
\(625\) 0 0
\(626\) 20687.2 1.32081
\(627\) − 1608.74i − 0.102467i
\(628\) 10881.5i 0.691434i
\(629\) −2328.24 −0.147588
\(630\) 0 0
\(631\) 27966.1 1.76437 0.882183 0.470907i \(-0.156073\pi\)
0.882183 + 0.470907i \(0.156073\pi\)
\(632\) 6570.79i 0.413563i
\(633\) 6253.70i 0.392674i
\(634\) −18337.0 −1.14867
\(635\) 0 0
\(636\) −3434.23 −0.214113
\(637\) 4453.72i 0.277022i
\(638\) − 9468.52i − 0.587558i
\(639\) 6598.27 0.408487
\(640\) 0 0
\(641\) −17992.0 −1.10865 −0.554323 0.832301i \(-0.687023\pi\)
−0.554323 + 0.832301i \(0.687023\pi\)
\(642\) 17140.1i 1.05369i
\(643\) 9448.64i 0.579499i 0.957102 + 0.289750i \(0.0935721\pi\)
−0.957102 + 0.289750i \(0.906428\pi\)
\(644\) 943.918 0.0577571
\(645\) 0 0
\(646\) −18730.8 −1.14079
\(647\) − 7429.22i − 0.451426i −0.974194 0.225713i \(-0.927529\pi\)
0.974194 0.225713i \(-0.0724712\pi\)
\(648\) 903.882i 0.0547960i
\(649\) −2562.30 −0.154976
\(650\) 0 0
\(651\) −3976.15 −0.239382
\(652\) − 15133.2i − 0.908988i
\(653\) 4488.63i 0.268995i 0.990914 + 0.134497i \(0.0429420\pi\)
−0.990914 + 0.134497i \(0.957058\pi\)
\(654\) 8134.00 0.486337
\(655\) 0 0
\(656\) 9887.42 0.588474
\(657\) 10457.7i 0.620998i
\(658\) − 10831.6i − 0.641730i
\(659\) 25326.7 1.49710 0.748550 0.663078i \(-0.230750\pi\)
0.748550 + 0.663078i \(0.230750\pi\)
\(660\) 0 0
\(661\) 15192.3 0.893969 0.446984 0.894542i \(-0.352498\pi\)
0.446984 + 0.894542i \(0.352498\pi\)
\(662\) − 9985.15i − 0.586229i
\(663\) 17374.5i 1.01775i
\(664\) −12956.4 −0.757235
\(665\) 0 0
\(666\) −703.087 −0.0409070
\(667\) 2864.75i 0.166302i
\(668\) 11821.5i 0.684711i
\(669\) 5662.07 0.327217
\(670\) 0 0
\(671\) 1802.34 0.103694
\(672\) − 9451.25i − 0.542545i
\(673\) − 11718.2i − 0.671180i −0.942008 0.335590i \(-0.891064\pi\)
0.942008 0.335590i \(-0.108936\pi\)
\(674\) −17734.8 −1.01353
\(675\) 0 0
\(676\) 3581.63 0.203779
\(677\) − 15649.1i − 0.888397i −0.895928 0.444198i \(-0.853489\pi\)
0.895928 0.444198i \(-0.146511\pi\)
\(678\) − 12340.6i − 0.699024i
\(679\) 24972.6 1.41143
\(680\) 0 0
\(681\) 3452.20 0.194257
\(682\) − 3242.05i − 0.182030i
\(683\) 18162.8i 1.01754i 0.860903 + 0.508770i \(0.169899\pi\)
−0.860903 + 0.508770i \(0.830101\pi\)
\(684\) −2146.40 −0.119985
\(685\) 0 0
\(686\) 24656.2 1.37227
\(687\) − 12320.4i − 0.684208i
\(688\) 17778.1i 0.985149i
\(689\) 12664.2 0.700246
\(690\) 0 0
\(691\) 29606.7 1.62995 0.814973 0.579500i \(-0.196752\pi\)
0.814973 + 0.579500i \(0.196752\pi\)
\(692\) − 18384.7i − 1.00994i
\(693\) 1598.50i 0.0876220i
\(694\) 9736.08 0.532531
\(695\) 0 0
\(696\) 8025.55 0.437080
\(697\) 13358.5i 0.725953i
\(698\) − 19733.8i − 1.07011i
\(699\) −17200.7 −0.930746
\(700\) 0 0
\(701\) −26164.7 −1.40974 −0.704870 0.709336i \(-0.748995\pi\)
−0.704870 + 0.709336i \(0.748995\pi\)
\(702\) 5246.80i 0.282091i
\(703\) 1060.66i 0.0569040i
\(704\) 736.324 0.0394194
\(705\) 0 0
\(706\) 22879.9 1.21968
\(707\) 10693.5i 0.568842i
\(708\) 3418.66i 0.181470i
\(709\) 14508.9 0.768535 0.384268 0.923222i \(-0.374454\pi\)
0.384268 + 0.923222i \(0.374454\pi\)
\(710\) 0 0
\(711\) −5299.48 −0.279530
\(712\) 11629.5i 0.612125i
\(713\) 980.898i 0.0515216i
\(714\) 18611.6 0.975520
\(715\) 0 0
\(716\) 12396.6 0.647043
\(717\) − 18108.5i − 0.943202i
\(718\) − 2264.28i − 0.117691i
\(719\) 2545.80 0.132048 0.0660239 0.997818i \(-0.478969\pi\)
0.0660239 + 0.997818i \(0.478969\pi\)
\(720\) 0 0
\(721\) −6452.01 −0.333267
\(722\) − 16094.6i − 0.829611i
\(723\) 11162.7i 0.574198i
\(724\) 67.9982 0.00349051
\(725\) 0 0
\(726\) −1303.37 −0.0666291
\(727\) − 18984.8i − 0.968511i −0.874927 0.484255i \(-0.839091\pi\)
0.874927 0.484255i \(-0.160909\pi\)
\(728\) 9751.54i 0.496451i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −24019.2 −1.21530
\(732\) − 2404.70i − 0.121421i
\(733\) − 36740.4i − 1.85134i −0.378326 0.925672i \(-0.623500\pi\)
0.378326 0.925672i \(-0.376500\pi\)
\(734\) 19297.4 0.970411
\(735\) 0 0
\(736\) −2331.58 −0.116770
\(737\) − 9646.10i − 0.482115i
\(738\) 4034.04i 0.201213i
\(739\) −1755.17 −0.0873682 −0.0436841 0.999045i \(-0.513910\pi\)
−0.0436841 + 0.999045i \(0.513910\pi\)
\(740\) 0 0
\(741\) 7915.18 0.392404
\(742\) − 13566.0i − 0.671189i
\(743\) − 17972.6i − 0.887419i −0.896171 0.443709i \(-0.853662\pi\)
0.896171 0.443709i \(-0.146338\pi\)
\(744\) 2747.97 0.135411
\(745\) 0 0
\(746\) 27001.4 1.32519
\(747\) − 10449.6i − 0.511821i
\(748\) 5758.54i 0.281488i
\(749\) −25692.6 −1.25339
\(750\) 0 0
\(751\) 22704.9 1.10321 0.551606 0.834105i \(-0.314015\pi\)
0.551606 + 0.834105i \(0.314015\pi\)
\(752\) 14797.9i 0.717583i
\(753\) − 11429.6i − 0.553146i
\(754\) 46586.3 2.25010
\(755\) 0 0
\(756\) 2132.74 0.102602
\(757\) 39860.3i 1.91380i 0.290414 + 0.956901i \(0.406207\pi\)
−0.290414 + 0.956901i \(0.593793\pi\)
\(758\) 44914.6i 2.15221i
\(759\) 394.342 0.0188587
\(760\) 0 0
\(761\) 19631.4 0.935135 0.467568 0.883957i \(-0.345130\pi\)
0.467568 + 0.883957i \(0.345130\pi\)
\(762\) − 15740.5i − 0.748319i
\(763\) 12192.7i 0.578511i
\(764\) −17099.8 −0.809752
\(765\) 0 0
\(766\) −40034.3 −1.88838
\(767\) − 12606.8i − 0.593490i
\(768\) 15831.3i 0.743832i
\(769\) −35029.9 −1.64267 −0.821333 0.570449i \(-0.806769\pi\)
−0.821333 + 0.570449i \(0.806769\pi\)
\(770\) 0 0
\(771\) −3677.86 −0.171796
\(772\) 16972.4i 0.791255i
\(773\) 26736.9i 1.24406i 0.782992 + 0.622032i \(0.213693\pi\)
−0.782992 + 0.622032i \(0.786307\pi\)
\(774\) −7253.40 −0.336845
\(775\) 0 0
\(776\) −17258.8 −0.798398
\(777\) − 1053.91i − 0.0486600i
\(778\) − 11469.0i − 0.528515i
\(779\) 6085.64 0.279898
\(780\) 0 0
\(781\) −8064.55 −0.369491
\(782\) − 4591.39i − 0.209959i
\(783\) 6472.78i 0.295426i
\(784\) −6517.81 −0.296912
\(785\) 0 0
\(786\) 16426.1 0.745421
\(787\) 4799.45i 0.217385i 0.994075 + 0.108693i \(0.0346664\pi\)
−0.994075 + 0.108693i \(0.965334\pi\)
\(788\) − 17799.4i − 0.804668i
\(789\) 22193.0 1.00139
\(790\) 0 0
\(791\) 18498.3 0.831508
\(792\) − 1104.74i − 0.0495649i
\(793\) 8867.72i 0.397102i
\(794\) 2098.36 0.0937884
\(795\) 0 0
\(796\) 252.457 0.0112413
\(797\) 38438.9i 1.70838i 0.519963 + 0.854189i \(0.325946\pi\)
−0.519963 + 0.854189i \(0.674054\pi\)
\(798\) − 8478.76i − 0.376121i
\(799\) −19992.8 −0.885224
\(800\) 0 0
\(801\) −9379.42 −0.413740
\(802\) − 24958.9i − 1.09891i
\(803\) − 12781.7i − 0.561713i
\(804\) −12870.0 −0.564538
\(805\) 0 0
\(806\) 15951.3 0.697096
\(807\) 9643.45i 0.420651i
\(808\) − 7390.42i − 0.321775i
\(809\) −9960.91 −0.432889 −0.216444 0.976295i \(-0.569446\pi\)
−0.216444 + 0.976295i \(0.569446\pi\)
\(810\) 0 0
\(811\) −34199.9 −1.48079 −0.740396 0.672171i \(-0.765362\pi\)
−0.740396 + 0.672171i \(0.765362\pi\)
\(812\) − 18936.6i − 0.818404i
\(813\) − 22132.1i − 0.954744i
\(814\) 859.329 0.0370018
\(815\) 0 0
\(816\) −25426.8 −1.09083
\(817\) 10942.3i 0.468570i
\(818\) − 42203.9i − 1.80394i
\(819\) −7864.82 −0.335555
\(820\) 0 0
\(821\) −31796.4 −1.35165 −0.675823 0.737064i \(-0.736212\pi\)
−0.675823 + 0.737064i \(0.736212\pi\)
\(822\) 22897.1i 0.971567i
\(823\) 10783.6i 0.456733i 0.973575 + 0.228366i \(0.0733384\pi\)
−0.973575 + 0.228366i \(0.926662\pi\)
\(824\) 4459.07 0.188518
\(825\) 0 0
\(826\) −13504.5 −0.568862
\(827\) 44197.5i 1.85840i 0.369576 + 0.929201i \(0.379503\pi\)
−0.369576 + 0.929201i \(0.620497\pi\)
\(828\) − 526.137i − 0.0220828i
\(829\) −22487.7 −0.942137 −0.471069 0.882097i \(-0.656132\pi\)
−0.471069 + 0.882097i \(0.656132\pi\)
\(830\) 0 0
\(831\) 2431.82 0.101515
\(832\) 3622.81i 0.150959i
\(833\) − 8805.96i − 0.366276i
\(834\) −19111.9 −0.793516
\(835\) 0 0
\(836\) 2623.38 0.108530
\(837\) 2216.30i 0.0915250i
\(838\) − 42475.6i − 1.75095i
\(839\) −22491.0 −0.925477 −0.462738 0.886495i \(-0.653133\pi\)
−0.462738 + 0.886495i \(0.653133\pi\)
\(840\) 0 0
\(841\) 33082.7 1.35646
\(842\) − 7181.37i − 0.293927i
\(843\) 3344.17i 0.136630i
\(844\) −10198.0 −0.415910
\(845\) 0 0
\(846\) −6037.48 −0.245358
\(847\) − 1953.72i − 0.0792571i
\(848\) 18533.5i 0.750524i
\(849\) 6796.19 0.274729
\(850\) 0 0
\(851\) −259.994 −0.0104730
\(852\) 10759.8i 0.432660i
\(853\) − 22327.9i − 0.896241i −0.893973 0.448120i \(-0.852094\pi\)
0.893973 0.448120i \(-0.147906\pi\)
\(854\) 9499.12 0.380624
\(855\) 0 0
\(856\) 17756.5 0.708999
\(857\) − 14505.9i − 0.578193i −0.957300 0.289096i \(-0.906645\pi\)
0.957300 0.289096i \(-0.0933548\pi\)
\(858\) − 6412.76i − 0.255161i
\(859\) 8411.45 0.334104 0.167052 0.985948i \(-0.446575\pi\)
0.167052 + 0.985948i \(0.446575\pi\)
\(860\) 0 0
\(861\) −6046.92 −0.239348
\(862\) − 28954.7i − 1.14408i
\(863\) − 32499.6i − 1.28192i −0.767573 0.640961i \(-0.778536\pi\)
0.767573 0.640961i \(-0.221464\pi\)
\(864\) −5268.10 −0.207436
\(865\) 0 0
\(866\) −38455.9 −1.50899
\(867\) − 19614.1i − 0.768316i
\(868\) − 6483.94i − 0.253548i
\(869\) 6477.15 0.252845
\(870\) 0 0
\(871\) 47460.0 1.84629
\(872\) − 8426.50i − 0.327245i
\(873\) − 13919.6i − 0.539643i
\(874\) −2091.67 −0.0809516
\(875\) 0 0
\(876\) −17053.5 −0.657745
\(877\) − 32183.1i − 1.23916i −0.784933 0.619581i \(-0.787303\pi\)
0.784933 0.619581i \(-0.212697\pi\)
\(878\) − 16725.1i − 0.642876i
\(879\) 11406.2 0.437680
\(880\) 0 0
\(881\) −6246.34 −0.238870 −0.119435 0.992842i \(-0.538108\pi\)
−0.119435 + 0.992842i \(0.538108\pi\)
\(882\) − 2659.25i − 0.101521i
\(883\) 11801.1i 0.449762i 0.974386 + 0.224881i \(0.0721993\pi\)
−0.974386 + 0.224881i \(0.927801\pi\)
\(884\) −28332.7 −1.07798
\(885\) 0 0
\(886\) 7521.24 0.285193
\(887\) 32375.1i 1.22553i 0.790264 + 0.612767i \(0.209944\pi\)
−0.790264 + 0.612767i \(0.790056\pi\)
\(888\) 728.371i 0.0275254i
\(889\) 23594.6 0.890145
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 9233.17i 0.346580i
\(893\) 9107.98i 0.341307i
\(894\) 16973.8 0.634997
\(895\) 0 0
\(896\) −21322.6 −0.795020
\(897\) 1940.21i 0.0722205i
\(898\) 5034.72i 0.187095i
\(899\) 19678.5 0.730049
\(900\) 0 0
\(901\) −25039.9 −0.925861
\(902\) − 4930.49i − 0.182004i
\(903\) − 10872.7i − 0.400686i
\(904\) −12784.4 −0.470357
\(905\) 0 0
\(906\) −4534.64 −0.166284
\(907\) − 19592.8i − 0.717277i −0.933477 0.358638i \(-0.883241\pi\)
0.933477 0.358638i \(-0.116759\pi\)
\(908\) 5629.53i 0.205752i
\(909\) 5960.54 0.217490
\(910\) 0 0
\(911\) 18673.8 0.679133 0.339567 0.940582i \(-0.389720\pi\)
0.339567 + 0.940582i \(0.389720\pi\)
\(912\) 11583.5i 0.420579i
\(913\) 12771.7i 0.462959i
\(914\) 17644.7 0.638550
\(915\) 0 0
\(916\) 20090.9 0.724696
\(917\) 24622.4i 0.886698i
\(918\) − 10374.1i − 0.372979i
\(919\) 4572.90 0.164142 0.0820708 0.996627i \(-0.473847\pi\)
0.0820708 + 0.996627i \(0.473847\pi\)
\(920\) 0 0
\(921\) 4069.05 0.145581
\(922\) − 7949.78i − 0.283961i
\(923\) − 39678.6i − 1.41499i
\(924\) −2606.69 −0.0928070
\(925\) 0 0
\(926\) −19988.7 −0.709362
\(927\) 3596.33i 0.127421i
\(928\) 46775.4i 1.65461i
\(929\) 44222.1 1.56176 0.780882 0.624679i \(-0.214770\pi\)
0.780882 + 0.624679i \(0.214770\pi\)
\(930\) 0 0
\(931\) −4011.67 −0.141221
\(932\) − 28049.3i − 0.985823i
\(933\) − 24234.2i − 0.850367i
\(934\) −1784.70 −0.0625238
\(935\) 0 0
\(936\) 5435.48 0.189812
\(937\) 9218.62i 0.321408i 0.987003 + 0.160704i \(0.0513764\pi\)
−0.987003 + 0.160704i \(0.948624\pi\)
\(938\) − 50839.2i − 1.76968i
\(939\) 17284.6 0.600705
\(940\) 0 0
\(941\) −26484.9 −0.917516 −0.458758 0.888561i \(-0.651706\pi\)
−0.458758 + 0.888561i \(0.651706\pi\)
\(942\) 23959.4i 0.828705i
\(943\) 1491.75i 0.0515142i
\(944\) 18449.5 0.636103
\(945\) 0 0
\(946\) 8865.26 0.304688
\(947\) − 44972.4i − 1.54320i −0.636110 0.771599i \(-0.719457\pi\)
0.636110 0.771599i \(-0.280543\pi\)
\(948\) − 8641.90i − 0.296072i
\(949\) 62887.5 2.15112
\(950\) 0 0
\(951\) −15321.0 −0.522416
\(952\) − 19280.9i − 0.656404i
\(953\) 2052.50i 0.0697659i 0.999391 + 0.0348829i \(0.0111058\pi\)
−0.999391 + 0.0348829i \(0.988894\pi\)
\(954\) −7561.63 −0.256621
\(955\) 0 0
\(956\) 29529.7 0.999016
\(957\) − 7911.18i − 0.267223i
\(958\) 33569.2i 1.13212i
\(959\) −34322.2 −1.15570
\(960\) 0 0
\(961\) −23053.0 −0.773826
\(962\) 4228.00i 0.141701i
\(963\) 14321.0i 0.479218i
\(964\) −18203.1 −0.608176
\(965\) 0 0
\(966\) 2078.36 0.0692237
\(967\) 14950.9i 0.497195i 0.968607 + 0.248598i \(0.0799697\pi\)
−0.968607 + 0.248598i \(0.920030\pi\)
\(968\) 1350.24i 0.0448331i
\(969\) −15650.0 −0.518835
\(970\) 0 0
\(971\) 26751.9 0.884151 0.442076 0.896978i \(-0.354242\pi\)
0.442076 + 0.896978i \(0.354242\pi\)
\(972\) − 1188.78i − 0.0392287i
\(973\) − 28648.3i − 0.943908i
\(974\) −710.034 −0.0233583
\(975\) 0 0
\(976\) −12977.5 −0.425615
\(977\) − 56893.7i − 1.86304i −0.363690 0.931520i \(-0.618483\pi\)
0.363690 0.931520i \(-0.381517\pi\)
\(978\) − 33320.8i − 1.08945i
\(979\) 11463.7 0.374242
\(980\) 0 0
\(981\) 6796.15 0.221187
\(982\) − 35895.0i − 1.16645i
\(983\) 34633.1i 1.12373i 0.827230 + 0.561863i \(0.189915\pi\)
−0.827230 + 0.561863i \(0.810085\pi\)
\(984\) 4179.10 0.135391
\(985\) 0 0
\(986\) −92111.0 −2.97506
\(987\) − 9050.03i − 0.291860i
\(988\) 12907.3i 0.415625i
\(989\) −2682.23 −0.0862386
\(990\) 0 0
\(991\) 1961.79 0.0628843 0.0314422 0.999506i \(-0.489990\pi\)
0.0314422 + 0.999506i \(0.489990\pi\)
\(992\) 16016.0i 0.512610i
\(993\) − 8342.83i − 0.266618i
\(994\) −42503.8 −1.35628
\(995\) 0 0
\(996\) 17040.2 0.542108
\(997\) 13096.5i 0.416017i 0.978127 + 0.208009i \(0.0666982\pi\)
−0.978127 + 0.208009i \(0.933302\pi\)
\(998\) 31290.8i 0.992476i
\(999\) −587.447 −0.0186046
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 825.4.c.l.199.5 6
5.2 odd 4 825.4.a.s.1.1 3
5.3 odd 4 165.4.a.d.1.3 3
5.4 even 2 inner 825.4.c.l.199.2 6
15.2 even 4 2475.4.a.s.1.3 3
15.8 even 4 495.4.a.l.1.1 3
55.43 even 4 1815.4.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.3 3 5.3 odd 4
495.4.a.l.1.1 3 15.8 even 4
825.4.a.s.1.1 3 5.2 odd 4
825.4.c.l.199.2 6 5.4 even 2 inner
825.4.c.l.199.5 6 1.1 even 1 trivial
1815.4.a.s.1.1 3 55.43 even 4
2475.4.a.s.1.3 3 15.2 even 4