Properties

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

$q$-expansion

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

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 \(-0.781112\pi\)
0.772736 0.634727i \(-0.218888\pi\)
\(3\) − 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 \(0.143575\pi\)
−0.899988 + 0.435914i \(0.856425\pi\)
\(8\) − 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 \(-0.804095\pi\)
0.816511 0.577329i \(-0.195905\pi\)
\(14\) 57.9749 1.10675
\(15\) 0 0
\(16\) −79.2041 −1.23756
\(17\) 107.010i 1.52668i 0.645995 + 0.763342i \(0.276443\pi\)
−0.645995 + 0.763342i \(0.723557\pi\)
\(18\) 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 \(0.0172505\pi\)
−0.998532 + 0.0541674i \(0.982750\pi\)
\(24\) −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 \(0.0153919\pi\)
−0.998831 + 0.0483361i \(0.984608\pi\)
\(38\) 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 \(0.130302\pi\)
−0.917377 + 0.398019i \(0.869698\pi\)
\(44\) −53.8133 −0.184379
\(45\) 0 0
\(46\) 42.9064 0.137526
\(47\) 186.832i 0.579835i 0.957052 + 0.289917i \(0.0936278\pi\)
−0.957052 + 0.289917i \(0.906372\pi\)
\(48\) 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 \(0.0980638\pi\)
−0.952919 + 0.303226i \(0.901936\pi\)
\(54\) 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 \(0.294900\pi\)
−0.600670 + 0.799497i \(0.705100\pi\)
\(68\) − 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 \(0.381496\pi\)
−0.363750 + 0.931496i \(0.618504\pi\)
\(74\) 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 \(-0.721386\pi\)
0.640772 0.767731i \(-0.278614\pi\)
\(84\) −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 \(-0.699758\pi\)
0.587169 0.809464i \(-0.300242\pi\)
\(98\) − 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 \(0.0612156\pi\)
−0.981565 + 0.191131i \(0.938784\pi\)
\(104\) −603.942 −0.569436
\(105\) 0 0
\(106\) 840.181 0.769864
\(107\) 1591.22i 1.43765i 0.695189 + 0.718827i \(0.255321\pi\)
−0.695189 + 0.718827i \(0.744679\pi\)
\(108\) − 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 \(-0.841769\pi\)
0.878970 0.476876i \(-0.158231\pi\)
\(114\) 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 \(-0.829458\pi\)
0.859875 0.510505i \(-0.170542\pi\)
\(128\) 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 \(0.230634\pi\)
−0.748792 + 0.662805i \(0.769366\pi\)
\(138\) − 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 \(0.191257\pi\)
−0.824854 + 0.565345i \(0.808743\pi\)
\(158\) − 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 \(-0.733295\pi\)
0.669040 0.743226i \(-0.266705\pi\)
\(164\) 610.706 0.290781
\(165\) 0 0
\(166\) −4168.87 −1.94920
\(167\) 2416.43i 1.11970i 0.828595 + 0.559848i \(0.189140\pi\)
−0.828595 + 0.559848i \(0.810860\pi\)
\(168\) − 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 \(-0.690740\pi\)
0.564005 0.825771i \(-0.309260\pi\)
\(174\) 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 \(0.223961\pi\)
−0.762522 + 0.646963i \(0.776039\pi\)
\(194\) −5553.25 −2.05516
\(195\) 0 0
\(196\) −402.579 −0.146712
\(197\) − 3638.39i − 1.31586i −0.753079 0.657930i \(-0.771432\pi\)
0.753079 0.657930i \(-0.228568\pi\)
\(198\) 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 \(0.0914552\pi\)
−0.959008 + 0.283378i \(0.908545\pi\)
\(224\) −3150.42 −0.939715
\(225\) 0 0
\(226\) −4113.54 −1.21075
\(227\) 1150.73i 0.336462i 0.985748 + 0.168231i \(0.0538055\pi\)
−0.985748 + 0.168231i \(0.946194\pi\)
\(228\) − 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 \(-0.701601\pi\)
0.591848 0.806050i \(-0.298399\pi\)
\(234\) 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 \(-0.952465\pi\)
0.988870 0.148780i \(-0.0475345\pi\)
\(258\) − 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 \(0.334099\pi\)
−0.497916 + 0.867225i \(0.665901\pi\)
\(264\) −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 \(0.0280202\pi\)
−0.996128 + 0.0879144i \(0.971980\pi\)
\(278\) − 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 \(0.0764663\pi\)
−0.971284 + 0.237922i \(0.923534\pi\)
\(284\) 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 \(0.123746\pi\)
−0.925380 + 0.379042i \(0.876254\pi\)
\(294\) −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 \(0.0402385\pi\)
−0.992021 + 0.126077i \(0.959761\pi\)
\(308\) − 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 \(0.174152\pi\)
−0.854029 + 0.520226i \(0.825848\pi\)
\(314\) 7986.48 1.43536
\(315\) 0 0
\(316\) −2880.63 −0.512811
\(317\) − 5107.00i − 0.904851i −0.891802 0.452426i \(-0.850559\pi\)
0.891802 0.452426i \(-0.149441\pi\)
\(318\) − 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 \(-0.869288\pi\)
0.916865 0.399198i \(-0.130712\pi\)
\(338\) 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 \(0.0672644\pi\)
−0.977755 + 0.209748i \(0.932736\pi\)
\(348\) − 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 \(0.159507\pi\)
−0.877052 + 0.480396i \(0.840493\pi\)
\(354\) 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 \(0.124839\pi\)
−0.924073 + 0.382216i \(0.875161\pi\)
\(368\) − 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 \(0.174796\pi\)
−0.852974 + 0.521953i \(0.825204\pi\)
\(374\) 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 \(-0.733033\pi\)
0.668430 0.743775i \(-0.266967\pi\)
\(384\) 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 \(0.0117612\pi\)
−0.999317 + 0.0369404i \(0.988239\pi\)
\(398\) 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 \(-0.797411\pi\)
0.804210 0.594345i \(-0.202589\pi\)
\(434\) −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 \(0.0358310\pi\)
−0.993671 + 0.112329i \(0.964169\pi\)
\(444\) −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 \(0.0809257\pi\)
−0.967856 + 0.251506i \(0.919074\pi\)
\(458\) 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 \(-0.909866\pi\)
0.960176 0.279396i \(-0.0901344\pi\)
\(464\) 18987.8 1.89976
\(465\) 0 0
\(466\) −20586.8 −2.04649
\(467\) − 497.054i − 0.0492525i −0.999697 0.0246263i \(-0.992160\pi\)
0.999697 0.0246263i \(-0.00783957\pi\)
\(468\) − 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 \(-0.997071\pi\)
0.999958 0.00920013i \(-0.00292853\pi\)
\(488\) − 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 \(0.0853652\pi\)
−0.964254 + 0.264979i \(0.914635\pi\)
\(504\) −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 \(-0.614181\pi\)
0.351068 0.936350i \(-0.385819\pi\)
\(524\) 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 \(-0.866096\pi\)
0.912815 0.408374i \(-0.133904\pi\)
\(548\) − 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 \(0.143413\pi\)
−0.900210 + 0.435456i \(0.856587\pi\)
\(558\) − 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 \(-0.572049\pi\)
0.224422 0.974492i \(-0.427951\pi\)
\(564\) −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 \(0.0751351\pi\)
−0.972271 + 0.233858i \(0.924865\pi\)
\(578\) 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 \(-0.666280\pi\)
0.498946 0.866633i \(-0.333720\pi\)
\(588\) 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 \(0.296243\pi\)
−0.597292 + 0.802024i \(0.703757\pi\)
\(594\) 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 \(0.0724580\pi\)
−0.974203 + 0.225673i \(0.927542\pi\)
\(608\) − 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 \(-0.514922\pi\)
0.0468613 0.998901i \(-0.485078\pi\)
\(614\) 4870.06 0.320097
\(615\) 0 0
\(616\) 1981.97 0.129636
\(617\) − 15236.8i − 0.994182i −0.867699 0.497091i \(-0.834402\pi\)
0.867699 0.497091i \(-0.165598\pi\)
\(618\) − 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 \(-0.906428\pi\)
0.957102 0.289750i \(-0.0935721\pi\)
\(644\) 943.918 0.0577571
\(645\) 0 0
\(646\) −18730.8 −1.14079
\(647\) 7429.22i 0.451426i 0.974194 + 0.225713i \(0.0724712\pi\)
−0.974194 + 0.225713i \(0.927529\pi\)
\(648\) − 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 \(-0.957058\pi\)
0.990914 0.134497i \(-0.0429420\pi\)
\(654\) 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 \(0.108936\pi\)
−0.942008 + 0.335590i \(0.891064\pi\)
\(674\) −17734.8 −1.01353
\(675\) 0 0
\(676\) 3581.63 0.203779
\(677\) 15649.1i 0.888397i 0.895928 + 0.444198i \(0.146511\pi\)
−0.895928 + 0.444198i \(0.853489\pi\)
\(678\) 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 \(-0.830101\pi\)
0.860903 0.508770i \(-0.169899\pi\)
\(684\) −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 \(0.160909\pi\)
−0.874927 + 0.484255i \(0.839091\pi\)
\(728\) − 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 \(0.376500\pi\)
−0.378326 + 0.925672i \(0.623500\pi\)
\(734\) 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 \(0.146338\pi\)
−0.896171 + 0.443709i \(0.853662\pi\)
\(744\) 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 \(-0.593793\pi\)
0.290414 0.956901i \(-0.406207\pi\)
\(758\) − 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 \(-0.786307\pi\)
0.782992 0.622032i \(-0.213693\pi\)
\(774\) −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 \(-0.965334\pi\)
0.994075 0.108693i \(-0.0346664\pi\)
\(788\) 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 \(-0.674054\pi\)
0.519963 0.854189i \(-0.325946\pi\)
\(798\) 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 \(-0.926662\pi\)
0.973575 0.228366i \(-0.0733384\pi\)
\(824\) 4459.07 0.188518
\(825\) 0 0
\(826\) −13504.5 −0.568862
\(827\) − 44197.5i − 1.85840i −0.369576 0.929201i \(-0.620497\pi\)
0.369576 0.929201i \(-0.379503\pi\)
\(828\) 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 \(0.147906\pi\)
−0.893973 + 0.448120i \(0.852094\pi\)
\(854\) 9499.12 0.380624
\(855\) 0 0
\(856\) 17756.5 0.708999
\(857\) 14505.9i 0.578193i 0.957300 + 0.289096i \(0.0933548\pi\)
−0.957300 + 0.289096i \(0.906645\pi\)
\(858\) 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 \(0.221464\pi\)
−0.767573 + 0.640961i \(0.778536\pi\)
\(864\) −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 \(0.212697\pi\)
−0.784933 + 0.619581i \(0.787303\pi\)
\(878\) 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 \(-0.927801\pi\)
0.974386 0.224881i \(-0.0721993\pi\)
\(884\) −28332.7 −1.07798
\(885\) 0 0
\(886\) 7521.24 0.285193
\(887\) − 32375.1i − 1.22553i −0.790264 0.612767i \(-0.790056\pi\)
0.790264 0.612767i \(-0.209944\pi\)
\(888\) − 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 \(0.116759\pi\)
−0.933477 + 0.358638i \(0.883241\pi\)
\(908\) − 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 \(-0.948624\pi\)
0.987003 0.160704i \(-0.0513764\pi\)
\(938\) 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 \(0.280543\pi\)
−0.636110 + 0.771599i \(0.719457\pi\)
\(948\) 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 \(-0.988894\pi\)
0.999391 0.0348829i \(-0.0111058\pi\)
\(954\) −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 \(-0.920030\pi\)
0.968607 0.248598i \(-0.0799697\pi\)
\(968\) − 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 \(0.381517\pi\)
−0.363690 + 0.931520i \(0.618483\pi\)
\(978\) 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 \(-0.810085\pi\)
0.827230 0.561863i \(-0.189915\pi\)
\(984\) 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 \(-0.933302\pi\)
0.978127 0.208009i \(-0.0666982\pi\)
\(998\) − 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.199.2 6
5.2 odd 4 165.4.a.d.1.3 3
5.3 odd 4 825.4.a.s.1.1 3
5.4 even 2 inner 825.4.c.l.199.5 6
15.2 even 4 495.4.a.l.1.1 3
15.8 even 4 2475.4.a.s.1.3 3
55.32 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 5.2 odd 4
495.4.a.l.1.1 3 15.2 even 4
825.4.a.s.1.1 3 5.3 odd 4
825.4.c.l.199.2 6 1.1 even 1 trivial
825.4.c.l.199.5 6 5.4 even 2 inner
1815.4.a.s.1.1 3 55.32 even 4
2475.4.a.s.1.3 3 15.8 even 4