Properties

Label 525.4.d.l.274.1
Level 525
Weight 4
Character 525.274
Analytic conductor 30.976
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.l.274.4

$q$-expansion

\(f(q)\) \(=\) \(q-3.82843i q^{2} +3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} -7.00000i q^{7} -5.14214i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-3.82843i q^{2} +3.00000i q^{3} -6.65685 q^{4} +11.4853 q^{6} -7.00000i q^{7} -5.14214i q^{8} -9.00000 q^{9} +48.5685 q^{11} -19.9706i q^{12} +43.6569i q^{13} -26.7990 q^{14} -72.9411 q^{16} -67.6569i q^{17} +34.4558i q^{18} +93.2548 q^{19} +21.0000 q^{21} -185.941i q^{22} +104.167i q^{23} +15.4264 q^{24} +167.137 q^{26} -27.0000i q^{27} +46.5980i q^{28} +58.7351 q^{29} -9.08831 q^{31} +238.113i q^{32} +145.706i q^{33} -259.019 q^{34} +59.9117 q^{36} -252.676i q^{37} -357.019i q^{38} -130.971 q^{39} +276.274 q^{41} -80.3970i q^{42} +92.6375i q^{43} -323.314 q^{44} +398.794 q^{46} -582.794i q^{47} -218.823i q^{48} -49.0000 q^{49} +202.971 q^{51} -290.617i q^{52} -623.019i q^{53} -103.368 q^{54} -35.9949 q^{56} +279.765i q^{57} -224.863i q^{58} +524.999 q^{59} -352.794 q^{61} +34.7939i q^{62} +63.0000i q^{63} +328.068 q^{64} +557.823 q^{66} -736.520i q^{67} +450.382i q^{68} -312.500 q^{69} -492.264 q^{71} +46.2792i q^{72} -1164.75i q^{73} -967.352 q^{74} -620.784 q^{76} -339.980i q^{77} +501.411i q^{78} +872.195 q^{79} +81.0000 q^{81} -1057.70i q^{82} +529.588i q^{83} -139.794 q^{84} +354.656 q^{86} +176.205i q^{87} -249.746i q^{88} +385.216 q^{89} +305.598 q^{91} -693.421i q^{92} -27.2649i q^{93} -2231.18 q^{94} -714.338 q^{96} -463.892i q^{97} +187.593i q^{98} -437.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 12q^{6} - 36q^{9} - 32q^{11} - 28q^{14} - 156q^{16} + 192q^{19} + 84q^{21} - 108q^{24} + 216q^{26} - 376q^{29} - 240q^{31} - 312q^{34} + 36q^{36} - 456q^{39} + 200q^{41} - 1248q^{44} + 1120q^{46} - 196q^{49} + 744q^{51} - 108q^{54} + 252q^{56} - 208q^{59} - 936q^{61} - 68q^{64} + 1824q^{66} - 96q^{69} - 272q^{71} - 2376q^{74} - 1216q^{76} + 864q^{79} + 324q^{81} - 84q^{84} - 912q^{86} + 2808q^{89} + 1064q^{91} - 3200q^{94} - 2484q^{96} + 288q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.82843i − 1.35355i −0.736188 0.676777i \(-0.763376\pi\)
0.736188 0.676777i \(-0.236624\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −6.65685 −0.832107
\(5\) 0 0
\(6\) 11.4853 0.781474
\(7\) − 7.00000i − 0.377964i
\(8\) − 5.14214i − 0.227252i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 48.5685 1.33127 0.665635 0.746278i \(-0.268161\pi\)
0.665635 + 0.746278i \(0.268161\pi\)
\(12\) − 19.9706i − 0.480417i
\(13\) 43.6569i 0.931403i 0.884942 + 0.465701i \(0.154198\pi\)
−0.884942 + 0.465701i \(0.845802\pi\)
\(14\) −26.7990 −0.511595
\(15\) 0 0
\(16\) −72.9411 −1.13971
\(17\) − 67.6569i − 0.965247i −0.875828 0.482623i \(-0.839684\pi\)
0.875828 0.482623i \(-0.160316\pi\)
\(18\) 34.4558i 0.451184i
\(19\) 93.2548 1.12601 0.563003 0.826455i \(-0.309646\pi\)
0.563003 + 0.826455i \(0.309646\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) − 185.941i − 1.80194i
\(23\) 104.167i 0.944357i 0.881503 + 0.472179i \(0.156532\pi\)
−0.881503 + 0.472179i \(0.843468\pi\)
\(24\) 15.4264 0.131204
\(25\) 0 0
\(26\) 167.137 1.26070
\(27\) − 27.0000i − 0.192450i
\(28\) 46.5980i 0.314507i
\(29\) 58.7351 0.376098 0.188049 0.982160i \(-0.439784\pi\)
0.188049 + 0.982160i \(0.439784\pi\)
\(30\) 0 0
\(31\) −9.08831 −0.0526551 −0.0263276 0.999653i \(-0.508381\pi\)
−0.0263276 + 0.999653i \(0.508381\pi\)
\(32\) 238.113i 1.31540i
\(33\) 145.706i 0.768609i
\(34\) −259.019 −1.30651
\(35\) 0 0
\(36\) 59.9117 0.277369
\(37\) − 252.676i − 1.12269i −0.827580 0.561347i \(-0.810283\pi\)
0.827580 0.561347i \(-0.189717\pi\)
\(38\) − 357.019i − 1.52411i
\(39\) −130.971 −0.537745
\(40\) 0 0
\(41\) 276.274 1.05236 0.526180 0.850373i \(-0.323624\pi\)
0.526180 + 0.850373i \(0.323624\pi\)
\(42\) − 80.3970i − 0.295370i
\(43\) 92.6375i 0.328537i 0.986416 + 0.164268i \(0.0525263\pi\)
−0.986416 + 0.164268i \(0.947474\pi\)
\(44\) −323.314 −1.10776
\(45\) 0 0
\(46\) 398.794 1.27824
\(47\) − 582.794i − 1.80871i −0.426784 0.904354i \(-0.640354\pi\)
0.426784 0.904354i \(-0.359646\pi\)
\(48\) − 218.823i − 0.658009i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 202.971 0.557286
\(52\) − 290.617i − 0.775026i
\(53\) − 623.019i − 1.61468i −0.590083 0.807342i \(-0.700905\pi\)
0.590083 0.807342i \(-0.299095\pi\)
\(54\) −103.368 −0.260491
\(55\) 0 0
\(56\) −35.9949 −0.0858933
\(57\) 279.765i 0.650100i
\(58\) − 224.863i − 0.509068i
\(59\) 524.999 1.15846 0.579229 0.815165i \(-0.303354\pi\)
0.579229 + 0.815165i \(0.303354\pi\)
\(60\) 0 0
\(61\) −352.794 −0.740502 −0.370251 0.928932i \(-0.620728\pi\)
−0.370251 + 0.928932i \(0.620728\pi\)
\(62\) 34.7939i 0.0712715i
\(63\) 63.0000i 0.125988i
\(64\) 328.068 0.640758
\(65\) 0 0
\(66\) 557.823 1.04035
\(67\) − 736.520i − 1.34299i −0.741010 0.671494i \(-0.765653\pi\)
0.741010 0.671494i \(-0.234347\pi\)
\(68\) 450.382i 0.803188i
\(69\) −312.500 −0.545225
\(70\) 0 0
\(71\) −492.264 −0.822831 −0.411415 0.911448i \(-0.634965\pi\)
−0.411415 + 0.911448i \(0.634965\pi\)
\(72\) 46.2792i 0.0757508i
\(73\) − 1164.75i − 1.86745i −0.357987 0.933727i \(-0.616537\pi\)
0.357987 0.933727i \(-0.383463\pi\)
\(74\) −967.352 −1.51963
\(75\) 0 0
\(76\) −620.784 −0.936958
\(77\) − 339.980i − 0.503173i
\(78\) 501.411i 0.727867i
\(79\) 872.195 1.24215 0.621074 0.783752i \(-0.286697\pi\)
0.621074 + 0.783752i \(0.286697\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 1057.70i − 1.42443i
\(83\) 529.588i 0.700359i 0.936683 + 0.350180i \(0.113879\pi\)
−0.936683 + 0.350180i \(0.886121\pi\)
\(84\) −139.794 −0.181581
\(85\) 0 0
\(86\) 354.656 0.444692
\(87\) 176.205i 0.217140i
\(88\) − 249.746i − 0.302534i
\(89\) 385.216 0.458796 0.229398 0.973333i \(-0.426324\pi\)
0.229398 + 0.973333i \(0.426324\pi\)
\(90\) 0 0
\(91\) 305.598 0.352037
\(92\) − 693.421i − 0.785806i
\(93\) − 27.2649i − 0.0304005i
\(94\) −2231.18 −2.44818
\(95\) 0 0
\(96\) −714.338 −0.759446
\(97\) − 463.892i − 0.485579i −0.970079 0.242789i \(-0.921938\pi\)
0.970079 0.242789i \(-0.0780624\pi\)
\(98\) 187.593i 0.193365i
\(99\) −437.117 −0.443757
\(100\) 0 0
\(101\) −432.725 −0.426314 −0.213157 0.977018i \(-0.568375\pi\)
−0.213157 + 0.977018i \(0.568375\pi\)
\(102\) − 777.058i − 0.754316i
\(103\) − 512.626i − 0.490393i −0.969473 0.245197i \(-0.921147\pi\)
0.969473 0.245197i \(-0.0788525\pi\)
\(104\) 224.489 0.211663
\(105\) 0 0
\(106\) −2385.18 −2.18556
\(107\) 1963.09i 1.77363i 0.462122 + 0.886817i \(0.347088\pi\)
−0.462122 + 0.886817i \(0.652912\pi\)
\(108\) 179.735i 0.160139i
\(109\) −545.176 −0.479068 −0.239534 0.970888i \(-0.576995\pi\)
−0.239534 + 0.970888i \(0.576995\pi\)
\(110\) 0 0
\(111\) 758.029 0.648188
\(112\) 510.588i 0.430768i
\(113\) 231.823i 0.192992i 0.995333 + 0.0964961i \(0.0307635\pi\)
−0.995333 + 0.0964961i \(0.969236\pi\)
\(114\) 1071.06 0.879945
\(115\) 0 0
\(116\) −390.991 −0.312953
\(117\) − 392.912i − 0.310468i
\(118\) − 2009.92i − 1.56804i
\(119\) −473.598 −0.364829
\(120\) 0 0
\(121\) 1027.90 0.772279
\(122\) 1350.65i 1.00231i
\(123\) 828.823i 0.607581i
\(124\) 60.4996 0.0438147
\(125\) 0 0
\(126\) 241.191 0.170532
\(127\) 2372.90i 1.65796i 0.559280 + 0.828979i \(0.311078\pi\)
−0.559280 + 0.828979i \(0.688922\pi\)
\(128\) 648.917i 0.448099i
\(129\) −277.913 −0.189681
\(130\) 0 0
\(131\) 1200.04 0.800364 0.400182 0.916436i \(-0.368947\pi\)
0.400182 + 0.916436i \(0.368947\pi\)
\(132\) − 969.941i − 0.639565i
\(133\) − 652.784i − 0.425591i
\(134\) −2819.71 −1.81781
\(135\) 0 0
\(136\) −347.901 −0.219355
\(137\) 2781.25i 1.73444i 0.497924 + 0.867221i \(0.334096\pi\)
−0.497924 + 0.867221i \(0.665904\pi\)
\(138\) 1196.38i 0.737991i
\(139\) 1245.60 0.760078 0.380039 0.924971i \(-0.375911\pi\)
0.380039 + 0.924971i \(0.375911\pi\)
\(140\) 0 0
\(141\) 1748.38 1.04426
\(142\) 1884.60i 1.11375i
\(143\) 2120.35i 1.23995i
\(144\) 656.470 0.379902
\(145\) 0 0
\(146\) −4459.17 −2.52770
\(147\) − 147.000i − 0.0824786i
\(148\) 1682.03i 0.934202i
\(149\) −19.4046 −0.0106690 −0.00533452 0.999986i \(-0.501698\pi\)
−0.00533452 + 0.999986i \(0.501698\pi\)
\(150\) 0 0
\(151\) −2349.80 −1.26638 −0.633192 0.773995i \(-0.718256\pi\)
−0.633192 + 0.773995i \(0.718256\pi\)
\(152\) − 479.529i − 0.255888i
\(153\) 608.912i 0.321749i
\(154\) −1301.59 −0.681071
\(155\) 0 0
\(156\) 871.852 0.447462
\(157\) − 3898.46i − 1.98172i −0.134880 0.990862i \(-0.543065\pi\)
0.134880 0.990862i \(-0.456935\pi\)
\(158\) − 3339.14i − 1.68131i
\(159\) 1869.06 0.932239
\(160\) 0 0
\(161\) 729.166 0.356934
\(162\) − 310.103i − 0.150395i
\(163\) − 1527.54i − 0.734024i −0.930216 0.367012i \(-0.880381\pi\)
0.930216 0.367012i \(-0.119619\pi\)
\(164\) −1839.12 −0.875676
\(165\) 0 0
\(166\) 2027.49 0.947974
\(167\) − 998.518i − 0.462681i −0.972873 0.231340i \(-0.925689\pi\)
0.972873 0.231340i \(-0.0743111\pi\)
\(168\) − 107.985i − 0.0495905i
\(169\) 291.079 0.132489
\(170\) 0 0
\(171\) −839.294 −0.375336
\(172\) − 616.674i − 0.273378i
\(173\) − 685.253i − 0.301149i −0.988599 0.150575i \(-0.951888\pi\)
0.988599 0.150575i \(-0.0481124\pi\)
\(174\) 674.589 0.293911
\(175\) 0 0
\(176\) −3542.64 −1.51725
\(177\) 1575.00i 0.668836i
\(178\) − 1474.77i − 0.621005i
\(179\) −1025.58 −0.428245 −0.214122 0.976807i \(-0.568689\pi\)
−0.214122 + 0.976807i \(0.568689\pi\)
\(180\) 0 0
\(181\) 2899.40 1.19067 0.595333 0.803479i \(-0.297020\pi\)
0.595333 + 0.803479i \(0.297020\pi\)
\(182\) − 1169.96i − 0.476501i
\(183\) − 1058.38i − 0.427529i
\(184\) 535.638 0.214608
\(185\) 0 0
\(186\) −104.382 −0.0411486
\(187\) − 3285.99i − 1.28500i
\(188\) 3879.57i 1.50504i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 1074.18 0.406939 0.203469 0.979081i \(-0.434778\pi\)
0.203469 + 0.979081i \(0.434778\pi\)
\(192\) 984.204i 0.369942i
\(193\) − 898.999i − 0.335292i −0.985847 0.167646i \(-0.946383\pi\)
0.985847 0.167646i \(-0.0536166\pi\)
\(194\) −1775.98 −0.657257
\(195\) 0 0
\(196\) 326.186 0.118872
\(197\) 3063.63i 1.10799i 0.832519 + 0.553996i \(0.186898\pi\)
−0.832519 + 0.553996i \(0.813102\pi\)
\(198\) 1673.47i 0.600648i
\(199\) 949.522 0.338240 0.169120 0.985595i \(-0.445907\pi\)
0.169120 + 0.985595i \(0.445907\pi\)
\(200\) 0 0
\(201\) 2209.56 0.775375
\(202\) 1656.66i 0.577039i
\(203\) − 411.145i − 0.142151i
\(204\) −1351.15 −0.463721
\(205\) 0 0
\(206\) −1962.55 −0.663774
\(207\) − 937.499i − 0.314786i
\(208\) − 3184.38i − 1.06152i
\(209\) 4529.25 1.49902
\(210\) 0 0
\(211\) 2306.64 0.752587 0.376294 0.926500i \(-0.377198\pi\)
0.376294 + 0.926500i \(0.377198\pi\)
\(212\) 4147.35i 1.34359i
\(213\) − 1476.79i − 0.475062i
\(214\) 7515.53 2.40071
\(215\) 0 0
\(216\) −138.838 −0.0437348
\(217\) 63.6182i 0.0199018i
\(218\) 2087.17i 0.648444i
\(219\) 3494.26 1.07817
\(220\) 0 0
\(221\) 2953.69 0.899033
\(222\) − 2902.06i − 0.877357i
\(223\) 3227.61i 0.969222i 0.874730 + 0.484611i \(0.161039\pi\)
−0.874730 + 0.484611i \(0.838961\pi\)
\(224\) 1666.79 0.497174
\(225\) 0 0
\(226\) 887.519 0.261225
\(227\) − 637.820i − 0.186492i −0.995643 0.0932458i \(-0.970276\pi\)
0.995643 0.0932458i \(-0.0297242\pi\)
\(228\) − 1862.35i − 0.540953i
\(229\) −544.774 −0.157204 −0.0786019 0.996906i \(-0.525046\pi\)
−0.0786019 + 0.996906i \(0.525046\pi\)
\(230\) 0 0
\(231\) 1019.94 0.290507
\(232\) − 302.024i − 0.0854691i
\(233\) 5748.54i 1.61631i 0.588972 + 0.808154i \(0.299533\pi\)
−0.588972 + 0.808154i \(0.700467\pi\)
\(234\) −1504.23 −0.420234
\(235\) 0 0
\(236\) −3494.84 −0.963961
\(237\) 2616.59i 0.717154i
\(238\) 1813.14i 0.493816i
\(239\) 2678.10 0.724820 0.362410 0.932019i \(-0.381954\pi\)
0.362410 + 0.932019i \(0.381954\pi\)
\(240\) 0 0
\(241\) −2202.16 −0.588604 −0.294302 0.955713i \(-0.595087\pi\)
−0.294302 + 0.955713i \(0.595087\pi\)
\(242\) − 3935.25i − 1.04532i
\(243\) 243.000i 0.0641500i
\(244\) 2348.50 0.616177
\(245\) 0 0
\(246\) 3173.09 0.822393
\(247\) 4071.21i 1.04877i
\(248\) 46.7333i 0.0119660i
\(249\) −1588.76 −0.404353
\(250\) 0 0
\(251\) −5716.90 −1.43764 −0.718820 0.695196i \(-0.755317\pi\)
−0.718820 + 0.695196i \(0.755317\pi\)
\(252\) − 419.382i − 0.104836i
\(253\) 5059.22i 1.25719i
\(254\) 9084.47 2.24413
\(255\) 0 0
\(256\) 5108.88 1.24728
\(257\) 4724.29i 1.14666i 0.819323 + 0.573332i \(0.194350\pi\)
−0.819323 + 0.573332i \(0.805650\pi\)
\(258\) 1063.97i 0.256743i
\(259\) −1768.73 −0.424339
\(260\) 0 0
\(261\) −528.616 −0.125366
\(262\) − 4594.25i − 1.08334i
\(263\) − 5975.36i − 1.40097i −0.713665 0.700487i \(-0.752966\pi\)
0.713665 0.700487i \(-0.247034\pi\)
\(264\) 749.238 0.174668
\(265\) 0 0
\(266\) −2499.14 −0.576059
\(267\) 1155.65i 0.264886i
\(268\) 4902.90i 1.11751i
\(269\) −4486.11 −1.01681 −0.508407 0.861117i \(-0.669766\pi\)
−0.508407 + 0.861117i \(0.669766\pi\)
\(270\) 0 0
\(271\) 3827.68 0.857989 0.428994 0.903307i \(-0.358868\pi\)
0.428994 + 0.903307i \(0.358868\pi\)
\(272\) 4934.97i 1.10010i
\(273\) 916.794i 0.203249i
\(274\) 10647.8 2.34766
\(275\) 0 0
\(276\) 2080.26 0.453685
\(277\) − 3420.54i − 0.741950i −0.928643 0.370975i \(-0.879024\pi\)
0.928643 0.370975i \(-0.120976\pi\)
\(278\) − 4768.71i − 1.02881i
\(279\) 81.7948 0.0175517
\(280\) 0 0
\(281\) 5235.92 1.11156 0.555781 0.831329i \(-0.312419\pi\)
0.555781 + 0.831329i \(0.312419\pi\)
\(282\) − 6693.55i − 1.41346i
\(283\) 6985.88i 1.46738i 0.679486 + 0.733688i \(0.262203\pi\)
−0.679486 + 0.733688i \(0.737797\pi\)
\(284\) 3276.93 0.684683
\(285\) 0 0
\(286\) 8117.60 1.67834
\(287\) − 1933.92i − 0.397755i
\(288\) − 2143.01i − 0.438466i
\(289\) 335.550 0.0682984
\(290\) 0 0
\(291\) 1391.68 0.280349
\(292\) 7753.59i 1.55392i
\(293\) − 7399.70i − 1.47541i −0.675123 0.737705i \(-0.735910\pi\)
0.675123 0.737705i \(-0.264090\pi\)
\(294\) −562.779 −0.111639
\(295\) 0 0
\(296\) −1299.30 −0.255135
\(297\) − 1311.35i − 0.256203i
\(298\) 74.2892i 0.0144411i
\(299\) −4547.58 −0.879577
\(300\) 0 0
\(301\) 648.463 0.124175
\(302\) 8996.04i 1.71412i
\(303\) − 1298.17i − 0.246133i
\(304\) −6802.11 −1.28332
\(305\) 0 0
\(306\) 2331.17 0.435504
\(307\) 2668.64i 0.496116i 0.968745 + 0.248058i \(0.0797923\pi\)
−0.968745 + 0.248058i \(0.920208\pi\)
\(308\) 2263.20i 0.418693i
\(309\) 1537.88 0.283129
\(310\) 0 0
\(311\) 6189.25 1.12849 0.564244 0.825608i \(-0.309168\pi\)
0.564244 + 0.825608i \(0.309168\pi\)
\(312\) 673.468i 0.122204i
\(313\) 2921.59i 0.527598i 0.964578 + 0.263799i \(0.0849755\pi\)
−0.964578 + 0.263799i \(0.915024\pi\)
\(314\) −14925.0 −2.68237
\(315\) 0 0
\(316\) −5806.08 −1.03360
\(317\) 9825.56i 1.74088i 0.492276 + 0.870439i \(0.336165\pi\)
−0.492276 + 0.870439i \(0.663835\pi\)
\(318\) − 7155.55i − 1.26183i
\(319\) 2852.68 0.500687
\(320\) 0 0
\(321\) −5889.26 −1.02401
\(322\) − 2791.56i − 0.483129i
\(323\) − 6309.33i − 1.08687i
\(324\) −539.205 −0.0924563
\(325\) 0 0
\(326\) −5848.07 −0.993541
\(327\) − 1635.53i − 0.276590i
\(328\) − 1420.64i − 0.239151i
\(329\) −4079.56 −0.683627
\(330\) 0 0
\(331\) −9258.17 −1.53739 −0.768693 0.639618i \(-0.779093\pi\)
−0.768693 + 0.639618i \(0.779093\pi\)
\(332\) − 3525.39i − 0.582774i
\(333\) 2274.09i 0.374232i
\(334\) −3822.75 −0.626263
\(335\) 0 0
\(336\) −1531.76 −0.248704
\(337\) − 3693.98i − 0.597103i −0.954394 0.298552i \(-0.903497\pi\)
0.954394 0.298552i \(-0.0965035\pi\)
\(338\) − 1114.38i − 0.179331i
\(339\) −695.470 −0.111424
\(340\) 0 0
\(341\) −441.406 −0.0700982
\(342\) 3213.17i 0.508037i
\(343\) 343.000i 0.0539949i
\(344\) 476.355 0.0746608
\(345\) 0 0
\(346\) −2623.44 −0.407622
\(347\) 3832.83i 0.592960i 0.955039 + 0.296480i \(0.0958128\pi\)
−0.955039 + 0.296480i \(0.904187\pi\)
\(348\) − 1172.97i − 0.180684i
\(349\) −8325.22 −1.27690 −0.638451 0.769662i \(-0.720425\pi\)
−0.638451 + 0.769662i \(0.720425\pi\)
\(350\) 0 0
\(351\) 1178.74 0.179248
\(352\) 11564.8i 1.75115i
\(353\) 8991.52i 1.35572i 0.735189 + 0.677862i \(0.237093\pi\)
−0.735189 + 0.677862i \(0.762907\pi\)
\(354\) 6029.76 0.905306
\(355\) 0 0
\(356\) −2564.33 −0.381767
\(357\) − 1420.79i − 0.210634i
\(358\) 3926.38i 0.579652i
\(359\) 12893.8 1.89557 0.947783 0.318917i \(-0.103319\pi\)
0.947783 + 0.318917i \(0.103319\pi\)
\(360\) 0 0
\(361\) 1837.46 0.267891
\(362\) − 11100.1i − 1.61163i
\(363\) 3083.71i 0.445875i
\(364\) −2034.32 −0.292932
\(365\) 0 0
\(366\) −4051.94 −0.578684
\(367\) − 7480.17i − 1.06393i −0.846767 0.531964i \(-0.821454\pi\)
0.846767 0.531964i \(-0.178546\pi\)
\(368\) − 7598.02i − 1.07629i
\(369\) −2486.47 −0.350787
\(370\) 0 0
\(371\) −4361.14 −0.610293
\(372\) 181.499i 0.0252964i
\(373\) 3523.32i 0.489090i 0.969638 + 0.244545i \(0.0786386\pi\)
−0.969638 + 0.244545i \(0.921361\pi\)
\(374\) −12580.2 −1.73932
\(375\) 0 0
\(376\) −2996.81 −0.411033
\(377\) 2564.19i 0.350298i
\(378\) 723.573i 0.0984565i
\(379\) −13515.4 −1.83177 −0.915886 0.401438i \(-0.868510\pi\)
−0.915886 + 0.401438i \(0.868510\pi\)
\(380\) 0 0
\(381\) −7118.69 −0.957222
\(382\) − 4112.44i − 0.550813i
\(383\) 657.182i 0.0876774i 0.999039 + 0.0438387i \(0.0139588\pi\)
−0.999039 + 0.0438387i \(0.986041\pi\)
\(384\) −1946.75 −0.258710
\(385\) 0 0
\(386\) −3441.75 −0.453836
\(387\) − 833.738i − 0.109512i
\(388\) 3088.06i 0.404053i
\(389\) 9741.87 1.26975 0.634875 0.772615i \(-0.281052\pi\)
0.634875 + 0.772615i \(0.281052\pi\)
\(390\) 0 0
\(391\) 7047.58 0.911538
\(392\) 251.965i 0.0324646i
\(393\) 3600.11i 0.462091i
\(394\) 11728.9 1.49973
\(395\) 0 0
\(396\) 2909.82 0.369253
\(397\) 4407.42i 0.557184i 0.960410 + 0.278592i \(0.0898678\pi\)
−0.960410 + 0.278592i \(0.910132\pi\)
\(398\) − 3635.18i − 0.457827i
\(399\) 1958.35 0.245715
\(400\) 0 0
\(401\) −11569.5 −1.44078 −0.720391 0.693568i \(-0.756037\pi\)
−0.720391 + 0.693568i \(0.756037\pi\)
\(402\) − 8459.14i − 1.04951i
\(403\) − 396.767i − 0.0490431i
\(404\) 2880.59 0.354739
\(405\) 0 0
\(406\) −1574.04 −0.192410
\(407\) − 12272.1i − 1.49461i
\(408\) − 1043.70i − 0.126645i
\(409\) 3083.03 0.372729 0.186364 0.982481i \(-0.440330\pi\)
0.186364 + 0.982481i \(0.440330\pi\)
\(410\) 0 0
\(411\) −8343.76 −1.00138
\(412\) 3412.47i 0.408060i
\(413\) − 3674.99i − 0.437856i
\(414\) −3589.15 −0.426079
\(415\) 0 0
\(416\) −10395.3 −1.22517
\(417\) 3736.81i 0.438831i
\(418\) − 17339.9i − 2.02900i
\(419\) 5415.21 0.631385 0.315692 0.948862i \(-0.397763\pi\)
0.315692 + 0.948862i \(0.397763\pi\)
\(420\) 0 0
\(421\) 4188.34 0.484863 0.242432 0.970168i \(-0.422055\pi\)
0.242432 + 0.970168i \(0.422055\pi\)
\(422\) − 8830.82i − 1.01867i
\(423\) 5245.15i 0.602902i
\(424\) −3203.65 −0.366941
\(425\) 0 0
\(426\) −5653.79 −0.643021
\(427\) 2469.56i 0.279884i
\(428\) − 13068.0i − 1.47585i
\(429\) −6361.05 −0.715884
\(430\) 0 0
\(431\) −9108.41 −1.01795 −0.508975 0.860781i \(-0.669976\pi\)
−0.508975 + 0.860781i \(0.669976\pi\)
\(432\) 1969.41i 0.219336i
\(433\) 16847.0i 1.86979i 0.354930 + 0.934893i \(0.384505\pi\)
−0.354930 + 0.934893i \(0.615495\pi\)
\(434\) 243.558 0.0269381
\(435\) 0 0
\(436\) 3629.16 0.398635
\(437\) 9714.03i 1.06335i
\(438\) − 13377.5i − 1.45937i
\(439\) −8434.14 −0.916946 −0.458473 0.888708i \(-0.651603\pi\)
−0.458473 + 0.888708i \(0.651603\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 11308.0i − 1.21689i
\(443\) 4298.49i 0.461010i 0.973071 + 0.230505i \(0.0740378\pi\)
−0.973071 + 0.230505i \(0.925962\pi\)
\(444\) −5046.09 −0.539362
\(445\) 0 0
\(446\) 12356.7 1.31189
\(447\) − 58.2139i − 0.00615978i
\(448\) − 2296.48i − 0.242184i
\(449\) −10545.0 −1.10835 −0.554173 0.832402i \(-0.686965\pi\)
−0.554173 + 0.832402i \(0.686965\pi\)
\(450\) 0 0
\(451\) 13418.2 1.40098
\(452\) − 1543.21i − 0.160590i
\(453\) − 7049.40i − 0.731147i
\(454\) −2441.85 −0.252426
\(455\) 0 0
\(456\) 1438.59 0.147737
\(457\) 11952.4i 1.22344i 0.791075 + 0.611719i \(0.209522\pi\)
−0.791075 + 0.611719i \(0.790478\pi\)
\(458\) 2085.63i 0.212784i
\(459\) −1826.74 −0.185762
\(460\) 0 0
\(461\) −17200.9 −1.73780 −0.868900 0.494988i \(-0.835173\pi\)
−0.868900 + 0.494988i \(0.835173\pi\)
\(462\) − 3904.76i − 0.393217i
\(463\) 10368.7i 1.04076i 0.853934 + 0.520381i \(0.174210\pi\)
−0.853934 + 0.520381i \(0.825790\pi\)
\(464\) −4284.20 −0.428640
\(465\) 0 0
\(466\) 22007.9 2.18776
\(467\) − 16879.5i − 1.67257i −0.548296 0.836284i \(-0.684723\pi\)
0.548296 0.836284i \(-0.315277\pi\)
\(468\) 2615.56i 0.258342i
\(469\) −5155.64 −0.507602
\(470\) 0 0
\(471\) 11695.4 1.14415
\(472\) − 2699.62i − 0.263263i
\(473\) 4499.27i 0.437371i
\(474\) 10017.4 0.970706
\(475\) 0 0
\(476\) 3152.67 0.303577
\(477\) 5607.17i 0.538228i
\(478\) − 10252.9i − 0.981082i
\(479\) −7329.12 −0.699115 −0.349558 0.936915i \(-0.613668\pi\)
−0.349558 + 0.936915i \(0.613668\pi\)
\(480\) 0 0
\(481\) 11031.0 1.04568
\(482\) 8430.80i 0.796706i
\(483\) 2187.50i 0.206076i
\(484\) −6842.60 −0.642619
\(485\) 0 0
\(486\) 930.308 0.0868305
\(487\) − 17209.5i − 1.60131i −0.599125 0.800655i \(-0.704485\pi\)
0.599125 0.800655i \(-0.295515\pi\)
\(488\) 1814.11i 0.168281i
\(489\) 4582.61 0.423789
\(490\) 0 0
\(491\) 11392.5 1.04712 0.523560 0.851989i \(-0.324604\pi\)
0.523560 + 0.851989i \(0.324604\pi\)
\(492\) − 5517.35i − 0.505572i
\(493\) − 3973.83i − 0.363027i
\(494\) 15586.3 1.41956
\(495\) 0 0
\(496\) 662.912 0.0600113
\(497\) 3445.85i 0.311001i
\(498\) 6082.47i 0.547313i
\(499\) −19079.4 −1.71164 −0.855822 0.517271i \(-0.826948\pi\)
−0.855822 + 0.517271i \(0.826948\pi\)
\(500\) 0 0
\(501\) 2995.55 0.267129
\(502\) 21886.7i 1.94592i
\(503\) 13499.4i 1.19663i 0.801259 + 0.598317i \(0.204164\pi\)
−0.801259 + 0.598317i \(0.795836\pi\)
\(504\) 323.955 0.0286311
\(505\) 0 0
\(506\) 19368.8 1.70168
\(507\) 873.237i 0.0764928i
\(508\) − 15796.0i − 1.37960i
\(509\) 4328.85 0.376960 0.188480 0.982077i \(-0.439644\pi\)
0.188480 + 0.982077i \(0.439644\pi\)
\(510\) 0 0
\(511\) −8153.27 −0.705831
\(512\) − 14367.6i − 1.24017i
\(513\) − 2517.88i − 0.216700i
\(514\) 18086.6 1.55207
\(515\) 0 0
\(516\) 1850.02 0.157835
\(517\) − 28305.5i − 2.40788i
\(518\) 6771.47i 0.574365i
\(519\) 2055.76 0.173869
\(520\) 0 0
\(521\) 19395.7 1.63098 0.815490 0.578771i \(-0.196468\pi\)
0.815490 + 0.578771i \(0.196468\pi\)
\(522\) 2023.77i 0.169689i
\(523\) − 20413.8i − 1.70675i −0.521294 0.853377i \(-0.674550\pi\)
0.521294 0.853377i \(-0.325450\pi\)
\(524\) −7988.47 −0.665989
\(525\) 0 0
\(526\) −22876.2 −1.89629
\(527\) 614.887i 0.0508252i
\(528\) − 10627.9i − 0.875987i
\(529\) 1316.34 0.108189
\(530\) 0 0
\(531\) −4724.99 −0.386153
\(532\) 4345.49i 0.354137i
\(533\) 12061.3i 0.980171i
\(534\) 4424.32 0.358537
\(535\) 0 0
\(536\) −3787.28 −0.305197
\(537\) − 3076.75i − 0.247247i
\(538\) 17174.8i 1.37631i
\(539\) −2379.86 −0.190181
\(540\) 0 0
\(541\) −4919.18 −0.390928 −0.195464 0.980711i \(-0.562621\pi\)
−0.195464 + 0.980711i \(0.562621\pi\)
\(542\) − 14654.0i − 1.16133i
\(543\) 8698.19i 0.687431i
\(544\) 16110.0 1.26969
\(545\) 0 0
\(546\) 3509.88 0.275108
\(547\) 15334.2i 1.19862i 0.800518 + 0.599308i \(0.204558\pi\)
−0.800518 + 0.599308i \(0.795442\pi\)
\(548\) − 18514.4i − 1.44324i
\(549\) 3175.15 0.246834
\(550\) 0 0
\(551\) 5477.33 0.423488
\(552\) 1606.92i 0.123904i
\(553\) − 6105.37i − 0.469487i
\(554\) −13095.3 −1.00427
\(555\) 0 0
\(556\) −8291.81 −0.632466
\(557\) − 8613.78i − 0.655256i −0.944807 0.327628i \(-0.893751\pi\)
0.944807 0.327628i \(-0.106249\pi\)
\(558\) − 313.145i − 0.0237572i
\(559\) −4044.26 −0.306000
\(560\) 0 0
\(561\) 9857.98 0.741897
\(562\) − 20045.3i − 1.50456i
\(563\) 2320.81i 0.173731i 0.996220 + 0.0868654i \(0.0276850\pi\)
−0.996220 + 0.0868654i \(0.972315\pi\)
\(564\) −11638.7 −0.868934
\(565\) 0 0
\(566\) 26744.9 1.98617
\(567\) − 567.000i − 0.0419961i
\(568\) 2531.29i 0.186990i
\(569\) 1736.04 0.127906 0.0639529 0.997953i \(-0.479629\pi\)
0.0639529 + 0.997953i \(0.479629\pi\)
\(570\) 0 0
\(571\) 23897.8 1.75148 0.875738 0.482786i \(-0.160375\pi\)
0.875738 + 0.482786i \(0.160375\pi\)
\(572\) − 14114.9i − 1.03177i
\(573\) 3222.55i 0.234946i
\(574\) −7403.87 −0.538382
\(575\) 0 0
\(576\) −2952.61 −0.213586
\(577\) 8029.26i 0.579311i 0.957131 + 0.289655i \(0.0935407\pi\)
−0.957131 + 0.289655i \(0.906459\pi\)
\(578\) − 1284.63i − 0.0924455i
\(579\) 2697.00 0.193581
\(580\) 0 0
\(581\) 3707.12 0.264711
\(582\) − 5327.93i − 0.379467i
\(583\) − 30259.1i − 2.14958i
\(584\) −5989.32 −0.424383
\(585\) 0 0
\(586\) −28329.2 −1.99705
\(587\) − 8015.14i − 0.563578i −0.959476 0.281789i \(-0.909072\pi\)
0.959476 0.281789i \(-0.0909278\pi\)
\(588\) 978.558i 0.0686310i
\(589\) −847.529 −0.0592900
\(590\) 0 0
\(591\) −9190.88 −0.639700
\(592\) 18430.5i 1.27954i
\(593\) − 12820.3i − 0.887801i −0.896076 0.443901i \(-0.853594\pi\)
0.896076 0.443901i \(-0.146406\pi\)
\(594\) −5020.41 −0.346784
\(595\) 0 0
\(596\) 129.174 0.00887779
\(597\) 2848.57i 0.195283i
\(598\) 17410.1i 1.19055i
\(599\) 15330.5 1.04572 0.522860 0.852418i \(-0.324865\pi\)
0.522860 + 0.852418i \(0.324865\pi\)
\(600\) 0 0
\(601\) 107.658 0.00730694 0.00365347 0.999993i \(-0.498837\pi\)
0.00365347 + 0.999993i \(0.498837\pi\)
\(602\) − 2482.59i − 0.168078i
\(603\) 6628.68i 0.447663i
\(604\) 15642.3 1.05377
\(605\) 0 0
\(606\) −4969.97 −0.333154
\(607\) − 8213.06i − 0.549189i −0.961560 0.274595i \(-0.911456\pi\)
0.961560 0.274595i \(-0.0885436\pi\)
\(608\) 22205.2i 1.48115i
\(609\) 1233.44 0.0820712
\(610\) 0 0
\(611\) 25443.0 1.68463
\(612\) − 4053.44i − 0.267729i
\(613\) − 1242.60i − 0.0818732i −0.999162 0.0409366i \(-0.986966\pi\)
0.999162 0.0409366i \(-0.0130342\pi\)
\(614\) 10216.7 0.671519
\(615\) 0 0
\(616\) −1748.22 −0.114347
\(617\) − 13170.6i − 0.859367i −0.902980 0.429683i \(-0.858625\pi\)
0.902980 0.429683i \(-0.141375\pi\)
\(618\) − 5887.65i − 0.383230i
\(619\) −19774.1 −1.28399 −0.641993 0.766711i \(-0.721892\pi\)
−0.641993 + 0.766711i \(0.721892\pi\)
\(620\) 0 0
\(621\) 2812.50 0.181742
\(622\) − 23695.1i − 1.52747i
\(623\) − 2696.51i − 0.173409i
\(624\) 9553.14 0.612871
\(625\) 0 0
\(626\) 11185.1 0.714132
\(627\) 13587.8i 0.865459i
\(628\) 25951.5i 1.64901i
\(629\) −17095.3 −1.08368
\(630\) 0 0
\(631\) −14308.8 −0.902735 −0.451367 0.892338i \(-0.649064\pi\)
−0.451367 + 0.892338i \(0.649064\pi\)
\(632\) − 4484.95i − 0.282281i
\(633\) 6919.93i 0.434507i
\(634\) 37616.4 2.35637
\(635\) 0 0
\(636\) −12442.0 −0.775722
\(637\) − 2139.19i − 0.133058i
\(638\) − 10921.3i − 0.677707i
\(639\) 4430.38 0.274277
\(640\) 0 0
\(641\) 11537.5 0.710925 0.355463 0.934691i \(-0.384323\pi\)
0.355463 + 0.934691i \(0.384323\pi\)
\(642\) 22546.6i 1.38605i
\(643\) − 19603.0i − 1.20228i −0.799144 0.601139i \(-0.794714\pi\)
0.799144 0.601139i \(-0.205286\pi\)
\(644\) −4853.95 −0.297007
\(645\) 0 0
\(646\) −24154.8 −1.47114
\(647\) 21650.0i 1.31553i 0.753223 + 0.657765i \(0.228498\pi\)
−0.753223 + 0.657765i \(0.771502\pi\)
\(648\) − 416.513i − 0.0252503i
\(649\) 25498.4 1.54222
\(650\) 0 0
\(651\) −190.855 −0.0114903
\(652\) 10168.6i 0.610787i
\(653\) 2927.33i 0.175429i 0.996146 + 0.0877145i \(0.0279563\pi\)
−0.996146 + 0.0877145i \(0.972044\pi\)
\(654\) −6261.50 −0.374379
\(655\) 0 0
\(656\) −20151.7 −1.19938
\(657\) 10482.8i 0.622484i
\(658\) 15618.3i 0.925326i
\(659\) 4778.76 0.282480 0.141240 0.989975i \(-0.454891\pi\)
0.141240 + 0.989975i \(0.454891\pi\)
\(660\) 0 0
\(661\) −31510.3 −1.85417 −0.927086 0.374849i \(-0.877695\pi\)
−0.927086 + 0.374849i \(0.877695\pi\)
\(662\) 35444.2i 2.08093i
\(663\) 8861.06i 0.519057i
\(664\) 2723.21 0.159158
\(665\) 0 0
\(666\) 8706.17 0.506542
\(667\) 6118.23i 0.355170i
\(668\) 6646.99i 0.385000i
\(669\) −9682.82 −0.559581
\(670\) 0 0
\(671\) −17134.7 −0.985808
\(672\) 5000.37i 0.287044i
\(673\) 8992.15i 0.515040i 0.966273 + 0.257520i \(0.0829054\pi\)
−0.966273 + 0.257520i \(0.917095\pi\)
\(674\) −14142.1 −0.808211
\(675\) 0 0
\(676\) −1937.67 −0.110245
\(677\) 19340.8i 1.09797i 0.835832 + 0.548985i \(0.184986\pi\)
−0.835832 + 0.548985i \(0.815014\pi\)
\(678\) 2662.56i 0.150818i
\(679\) −3247.25 −0.183531
\(680\) 0 0
\(681\) 1913.46 0.107671
\(682\) 1689.89i 0.0948816i
\(683\) 4255.14i 0.238387i 0.992871 + 0.119194i \(0.0380309\pi\)
−0.992871 + 0.119194i \(0.961969\pi\)
\(684\) 5587.05 0.312319
\(685\) 0 0
\(686\) 1313.15 0.0730850
\(687\) − 1634.32i − 0.0907616i
\(688\) − 6757.08i − 0.374435i
\(689\) 27199.1 1.50392
\(690\) 0 0
\(691\) −17505.5 −0.963733 −0.481867 0.876245i \(-0.660041\pi\)
−0.481867 + 0.876245i \(0.660041\pi\)
\(692\) 4561.63i 0.250588i
\(693\) 3059.82i 0.167724i
\(694\) 14673.7 0.802603
\(695\) 0 0
\(696\) 906.071 0.0493456
\(697\) − 18691.8i − 1.01579i
\(698\) 31872.5i 1.72836i
\(699\) −17245.6 −0.933175
\(700\) 0 0
\(701\) −3240.77 −0.174611 −0.0873054 0.996182i \(-0.527826\pi\)
−0.0873054 + 0.996182i \(0.527826\pi\)
\(702\) − 4512.70i − 0.242622i
\(703\) − 23563.3i − 1.26416i
\(704\) 15933.8 0.853022
\(705\) 0 0
\(706\) 34423.4 1.83504
\(707\) 3029.07i 0.161132i
\(708\) − 10484.5i − 0.556543i
\(709\) −19949.3 −1.05672 −0.528358 0.849022i \(-0.677192\pi\)
−0.528358 + 0.849022i \(0.677192\pi\)
\(710\) 0 0
\(711\) −7849.76 −0.414049
\(712\) − 1980.83i − 0.104262i
\(713\) − 946.698i − 0.0497253i
\(714\) −5439.41 −0.285105
\(715\) 0 0
\(716\) 6827.17 0.356345
\(717\) 8034.30i 0.418475i
\(718\) − 49362.9i − 2.56575i
\(719\) 11259.4 0.584011 0.292006 0.956417i \(-0.405677\pi\)
0.292006 + 0.956417i \(0.405677\pi\)
\(720\) 0 0
\(721\) −3588.38 −0.185351
\(722\) − 7034.60i − 0.362605i
\(723\) − 6606.47i − 0.339830i
\(724\) −19300.9 −0.990761
\(725\) 0 0
\(726\) 11805.8 0.603516
\(727\) − 12228.5i − 0.623840i −0.950108 0.311920i \(-0.899028\pi\)
0.950108 0.311920i \(-0.100972\pi\)
\(728\) − 1571.43i − 0.0800013i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 6267.56 0.317119
\(732\) 7045.49i 0.355750i
\(733\) 26635.1i 1.34214i 0.741392 + 0.671072i \(0.234166\pi\)
−0.741392 + 0.671072i \(0.765834\pi\)
\(734\) −28637.3 −1.44008
\(735\) 0 0
\(736\) −24803.4 −1.24221
\(737\) − 35771.7i − 1.78788i
\(738\) 9519.26i 0.474809i
\(739\) 6074.00 0.302349 0.151174 0.988507i \(-0.451694\pi\)
0.151174 + 0.988507i \(0.451694\pi\)
\(740\) 0 0
\(741\) −12213.6 −0.605505
\(742\) 16696.3i 0.826065i
\(743\) 4016.87i 0.198337i 0.995071 + 0.0991686i \(0.0316183\pi\)
−0.995071 + 0.0991686i \(0.968382\pi\)
\(744\) −140.200 −0.00690858
\(745\) 0 0
\(746\) 13488.8 0.662010
\(747\) − 4766.29i − 0.233453i
\(748\) 21874.4i 1.06926i
\(749\) 13741.6 0.670370
\(750\) 0 0
\(751\) −23913.2 −1.16192 −0.580962 0.813931i \(-0.697324\pi\)
−0.580962 + 0.813931i \(0.697324\pi\)
\(752\) 42509.6i 2.06139i
\(753\) − 17150.7i − 0.830022i
\(754\) 9816.81 0.474147
\(755\) 0 0
\(756\) 1258.15 0.0605269
\(757\) 31044.9i 1.49055i 0.666758 + 0.745275i \(0.267682\pi\)
−0.666758 + 0.745275i \(0.732318\pi\)
\(758\) 51742.9i 2.47940i
\(759\) −15177.6 −0.725842
\(760\) 0 0
\(761\) 14011.8 0.667446 0.333723 0.942671i \(-0.391695\pi\)
0.333723 + 0.942671i \(0.391695\pi\)
\(762\) 27253.4i 1.29565i
\(763\) 3816.23i 0.181071i
\(764\) −7150.69 −0.338616
\(765\) 0 0
\(766\) 2515.97 0.118676
\(767\) 22919.8i 1.07899i
\(768\) 15326.6i 0.720120i
\(769\) −3342.49 −0.156740 −0.0783701 0.996924i \(-0.524972\pi\)
−0.0783701 + 0.996924i \(0.524972\pi\)
\(770\) 0 0
\(771\) −14172.9 −0.662027
\(772\) 5984.51i 0.278999i
\(773\) 21074.6i 0.980594i 0.871555 + 0.490297i \(0.163112\pi\)
−0.871555 + 0.490297i \(0.836888\pi\)
\(774\) −3191.90 −0.148231
\(775\) 0 0
\(776\) −2385.40 −0.110349
\(777\) − 5306.20i − 0.244992i
\(778\) − 37296.0i − 1.71867i
\(779\) 25763.9 1.18496
\(780\) 0 0
\(781\) −23908.5 −1.09541
\(782\) − 26981.1i − 1.23382i
\(783\) − 1585.85i − 0.0723800i
\(784\) 3574.12 0.162815
\(785\) 0 0
\(786\) 13782.8 0.625464
\(787\) 21394.8i 0.969048i 0.874778 + 0.484524i \(0.161007\pi\)
−0.874778 + 0.484524i \(0.838993\pi\)
\(788\) − 20394.1i − 0.921968i
\(789\) 17926.1 0.808853
\(790\) 0 0
\(791\) 1622.76 0.0729442
\(792\) 2247.71i 0.100845i
\(793\) − 15401.9i − 0.689706i
\(794\) 16873.5 0.754179
\(795\) 0 0
\(796\) −6320.83 −0.281452
\(797\) 20645.0i 0.917547i 0.888553 + 0.458773i \(0.151711\pi\)
−0.888553 + 0.458773i \(0.848289\pi\)
\(798\) − 7497.41i − 0.332588i
\(799\) −39430.0 −1.74585
\(800\) 0 0
\(801\) −3466.95 −0.152932
\(802\) 44293.0i 1.95017i
\(803\) − 56570.4i − 2.48608i
\(804\) −14708.7 −0.645194
\(805\) 0 0
\(806\) −1518.99 −0.0663825
\(807\) − 13458.3i − 0.587058i
\(808\) 2225.13i 0.0968810i
\(809\) 15939.0 0.692688 0.346344 0.938108i \(-0.387423\pi\)
0.346344 + 0.938108i \(0.387423\pi\)
\(810\) 0 0
\(811\) 22829.2 0.988460 0.494230 0.869331i \(-0.335450\pi\)
0.494230 + 0.869331i \(0.335450\pi\)
\(812\) 2736.94i 0.118285i
\(813\) 11483.0i 0.495360i
\(814\) −46982.9 −2.02303
\(815\) 0 0
\(816\) −14804.9 −0.635141
\(817\) 8638.90i 0.369935i
\(818\) − 11803.2i − 0.504508i
\(819\) −2750.38 −0.117346
\(820\) 0 0
\(821\) −5700.22 −0.242313 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(822\) 31943.5i 1.35542i
\(823\) 32438.0i 1.37390i 0.726707 + 0.686948i \(0.241050\pi\)
−0.726707 + 0.686948i \(0.758950\pi\)
\(824\) −2635.99 −0.111443
\(825\) 0 0
\(826\) −14069.4 −0.592662
\(827\) − 12762.6i − 0.536638i −0.963330 0.268319i \(-0.913532\pi\)
0.963330 0.268319i \(-0.0864681\pi\)
\(828\) 6240.79i 0.261935i
\(829\) 30766.7 1.28899 0.644494 0.764609i \(-0.277068\pi\)
0.644494 + 0.764609i \(0.277068\pi\)
\(830\) 0 0
\(831\) 10261.6 0.428365
\(832\) 14322.4i 0.596804i
\(833\) 3315.19i 0.137892i
\(834\) 14306.1 0.593981
\(835\) 0 0
\(836\) −30150.6 −1.24734
\(837\) 245.384i 0.0101335i
\(838\) − 20731.7i − 0.854613i
\(839\) 9779.71 0.402423 0.201212 0.979548i \(-0.435512\pi\)
0.201212 + 0.979548i \(0.435512\pi\)
\(840\) 0 0
\(841\) −20939.2 −0.858551
\(842\) − 16034.8i − 0.656288i
\(843\) 15707.8i 0.641760i
\(844\) −15355.0 −0.626233
\(845\) 0 0
\(846\) 20080.7 0.816061
\(847\) − 7195.32i − 0.291894i
\(848\) 45443.7i 1.84026i
\(849\) −20957.6 −0.847190
\(850\) 0 0
\(851\) 26320.4 1.06023
\(852\) 9830.79i 0.395302i
\(853\) − 24201.5i − 0.971445i −0.874113 0.485723i \(-0.838556\pi\)
0.874113 0.485723i \(-0.161444\pi\)
\(854\) 9454.52 0.378837
\(855\) 0 0
\(856\) 10094.5 0.403062
\(857\) 21036.7i 0.838507i 0.907869 + 0.419254i \(0.137708\pi\)
−0.907869 + 0.419254i \(0.862292\pi\)
\(858\) 24352.8i 0.968988i
\(859\) 6179.19 0.245438 0.122719 0.992441i \(-0.460839\pi\)
0.122719 + 0.992441i \(0.460839\pi\)
\(860\) 0 0
\(861\) 5801.76 0.229644
\(862\) 34870.9i 1.37785i
\(863\) − 50256.2i − 1.98232i −0.132671 0.991160i \(-0.542356\pi\)
0.132671 0.991160i \(-0.457644\pi\)
\(864\) 6429.04 0.253149
\(865\) 0 0
\(866\) 64497.7 2.53085
\(867\) 1006.65i 0.0394321i
\(868\) − 423.497i − 0.0165604i
\(869\) 42361.2 1.65363
\(870\) 0 0
\(871\) 32154.1 1.25086
\(872\) 2803.37i 0.108869i
\(873\) 4175.03i 0.161860i
\(874\) 37189.5 1.43930
\(875\) 0 0
\(876\) −23260.8 −0.897156
\(877\) − 9175.95i − 0.353306i −0.984273 0.176653i \(-0.943473\pi\)
0.984273 0.176653i \(-0.0565271\pi\)
\(878\) 32289.5i 1.24114i
\(879\) 22199.1 0.851828
\(880\) 0 0
\(881\) −26172.7 −1.00089 −0.500444 0.865769i \(-0.666830\pi\)
−0.500444 + 0.865769i \(0.666830\pi\)
\(882\) − 1688.34i − 0.0644549i
\(883\) − 18615.5i − 0.709471i −0.934967 0.354736i \(-0.884571\pi\)
0.934967 0.354736i \(-0.115429\pi\)
\(884\) −19662.3 −0.748092
\(885\) 0 0
\(886\) 16456.4 0.624001
\(887\) − 12837.8i − 0.485964i −0.970031 0.242982i \(-0.921874\pi\)
0.970031 0.242982i \(-0.0781256\pi\)
\(888\) − 3897.89i − 0.147302i
\(889\) 16610.3 0.626649
\(890\) 0 0
\(891\) 3934.05 0.147919
\(892\) − 21485.7i − 0.806496i
\(893\) − 54348.4i − 2.03662i
\(894\) −222.868 −0.00833759
\(895\) 0 0
\(896\) 4542.42 0.169366
\(897\) − 13642.7i − 0.507824i
\(898\) 40370.6i 1.50020i
\(899\) −533.803 −0.0198035
\(900\) 0 0
\(901\) −42151.5 −1.55857
\(902\) − 51370.7i − 1.89630i
\(903\) 1945.39i 0.0716926i
\(904\) 1192.07 0.0438579
\(905\) 0 0
\(906\) −26988.1 −0.989647
\(907\) − 26766.1i − 0.979885i −0.871755 0.489942i \(-0.837018\pi\)
0.871755 0.489942i \(-0.162982\pi\)
\(908\) 4245.87i 0.155181i
\(909\) 3894.52 0.142105
\(910\) 0 0
\(911\) 5022.67 0.182666 0.0913328 0.995820i \(-0.470887\pi\)
0.0913328 + 0.995820i \(0.470887\pi\)
\(912\) − 20406.3i − 0.740923i
\(913\) 25721.3i 0.932367i
\(914\) 45759.0 1.65599
\(915\) 0 0
\(916\) 3626.48 0.130810
\(917\) − 8400.26i − 0.302509i
\(918\) 6993.52i 0.251439i
\(919\) −9541.79 −0.342497 −0.171248 0.985228i \(-0.554780\pi\)
−0.171248 + 0.985228i \(0.554780\pi\)
\(920\) 0 0
\(921\) −8005.93 −0.286432
\(922\) 65852.4i 2.35220i
\(923\) − 21490.7i − 0.766387i
\(924\) −6789.59 −0.241733
\(925\) 0 0
\(926\) 39695.7 1.40873
\(927\) 4613.63i 0.163464i
\(928\) 13985.6i 0.494718i
\(929\) 25479.6 0.899846 0.449923 0.893067i \(-0.351451\pi\)
0.449923 + 0.893067i \(0.351451\pi\)
\(930\) 0 0
\(931\) −4569.49 −0.160858
\(932\) − 38267.2i − 1.34494i
\(933\) 18567.7i 0.651533i
\(934\) −64621.9 −2.26391
\(935\) 0 0
\(936\) −2020.41 −0.0705545
\(937\) − 33608.3i − 1.17176i −0.810399 0.585878i \(-0.800750\pi\)
0.810399 0.585878i \(-0.199250\pi\)
\(938\) 19738.0i 0.687066i
\(939\) −8764.77 −0.304609
\(940\) 0 0
\(941\) −19173.6 −0.664232 −0.332116 0.943239i \(-0.607762\pi\)
−0.332116 + 0.943239i \(0.607762\pi\)
\(942\) − 44774.9i − 1.54867i
\(943\) 28778.5i 0.993804i
\(944\) −38294.0 −1.32030
\(945\) 0 0
\(946\) 17225.1 0.592005
\(947\) − 979.315i − 0.0336045i −0.999859 0.0168023i \(-0.994651\pi\)
0.999859 0.0168023i \(-0.00534857\pi\)
\(948\) − 17418.2i − 0.596749i
\(949\) 50849.5 1.73935
\(950\) 0 0
\(951\) −29476.7 −1.00510
\(952\) 2435.31i 0.0829083i
\(953\) − 3048.61i − 0.103624i −0.998657 0.0518122i \(-0.983500\pi\)
0.998657 0.0518122i \(-0.0164997\pi\)
\(954\) 21466.7 0.728521
\(955\) 0 0
\(956\) −17827.7 −0.603127
\(957\) 8558.03i 0.289072i
\(958\) 28059.0i 0.946290i
\(959\) 19468.8 0.655557
\(960\) 0 0
\(961\) −29708.4 −0.997227
\(962\) − 42231.6i − 1.41538i
\(963\) − 17667.8i − 0.591211i
\(964\) 14659.4 0.489781
\(965\) 0 0
\(966\) 8374.67 0.278934
\(967\) − 14467.9i − 0.481133i −0.970633 0.240567i \(-0.922667\pi\)
0.970633 0.240567i \(-0.0773332\pi\)
\(968\) − 5285.62i − 0.175502i
\(969\) 18928.0 0.627507
\(970\) 0 0
\(971\) 12952.8 0.428090 0.214045 0.976824i \(-0.431336\pi\)
0.214045 + 0.976824i \(0.431336\pi\)
\(972\) − 1617.62i − 0.0533797i
\(973\) − 8719.23i − 0.287282i
\(974\) −65885.4 −2.16746
\(975\) 0 0
\(976\) 25733.2 0.843954
\(977\) 47244.2i 1.54706i 0.633760 + 0.773529i \(0.281511\pi\)
−0.633760 + 0.773529i \(0.718489\pi\)
\(978\) − 17544.2i − 0.573621i
\(979\) 18709.4 0.610781
\(980\) 0 0
\(981\) 4906.58 0.159689
\(982\) − 43615.3i − 1.41733i
\(983\) 1536.84i 0.0498651i 0.999689 + 0.0249326i \(0.00793711\pi\)
−0.999689 + 0.0249326i \(0.992063\pi\)
\(984\) 4261.92 0.138074
\(985\) 0 0
\(986\) −15213.5 −0.491376
\(987\) − 12238.7i − 0.394692i
\(988\) − 27101.5i − 0.872685i
\(989\) −9649.73 −0.310256
\(990\) 0 0
\(991\) 3785.22 0.121334 0.0606668 0.998158i \(-0.480677\pi\)
0.0606668 + 0.998158i \(0.480677\pi\)
\(992\) − 2164.04i − 0.0692625i
\(993\) − 27774.5i − 0.887610i
\(994\) 13192.2 0.420956
\(995\) 0 0
\(996\) 10576.2 0.336465
\(997\) − 25894.9i − 0.822566i −0.911508 0.411283i \(-0.865081\pi\)
0.911508 0.411283i \(-0.134919\pi\)
\(998\) 73044.0i 2.31680i
\(999\) −6822.26 −0.216063
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 525.4.d.l.274.1 4
5.2 odd 4 525.4.a.l.1.2 2
5.3 odd 4 105.4.a.e.1.1 2
5.4 even 2 inner 525.4.d.l.274.4 4
15.2 even 4 1575.4.a.q.1.1 2
15.8 even 4 315.4.a.k.1.2 2
20.3 even 4 1680.4.a.bo.1.1 2
35.13 even 4 735.4.a.o.1.1 2
105.83 odd 4 2205.4.a.bb.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.e.1.1 2 5.3 odd 4
315.4.a.k.1.2 2 15.8 even 4
525.4.a.l.1.2 2 5.2 odd 4
525.4.d.l.274.1 4 1.1 even 1 trivial
525.4.d.l.274.4 4 5.4 even 2 inner
735.4.a.o.1.1 2 35.13 even 4
1575.4.a.q.1.1 2 15.2 even 4
1680.4.a.bo.1.1 2 20.3 even 4
2205.4.a.bb.1.2 2 105.83 odd 4