Properties

Label 475.3.d.c.474.20
Level $475$
Weight $3$
Character 475.474
Analytic conductor $12.943$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,3,Mod(474,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.474");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.9428125571\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 474.20
Character \(\chi\) \(=\) 475.474
Dual form 475.3.d.c.474.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.36559 q^{2} +3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} -5.68687 q^{8} +3.64250 q^{9} +O(q^{10})\) \(q+2.36559 q^{2} +3.55563 q^{3} +1.59600 q^{4} +8.41115 q^{6} -11.0785i q^{7} -5.68687 q^{8} +3.64250 q^{9} +16.7704 q^{11} +5.67479 q^{12} +14.3035 q^{13} -26.2070i q^{14} -19.8368 q^{16} +20.2160i q^{17} +8.61666 q^{18} +(16.4480 - 9.51127i) q^{19} -39.3909i q^{21} +39.6717 q^{22} -13.5882i q^{23} -20.2204 q^{24} +33.8362 q^{26} -19.0493 q^{27} -17.6812i q^{28} -30.3972i q^{29} +16.1256i q^{31} -24.1782 q^{32} +59.6292 q^{33} +47.8227i q^{34} +5.81344 q^{36} -51.2678 q^{37} +(38.9091 - 22.4997i) q^{38} +50.8580 q^{39} +74.6652i q^{41} -93.1825i q^{42} +6.23197i q^{43} +26.7655 q^{44} -32.1441i q^{46} +44.0252i q^{47} -70.5323 q^{48} -73.7321 q^{49} +71.8806i q^{51} +22.8284 q^{52} -30.8058 q^{53} -45.0627 q^{54} +63.0017i q^{56} +(58.4829 - 33.8185i) q^{57} -71.9073i q^{58} +73.0098i q^{59} -17.7828 q^{61} +38.1466i q^{62} -40.3533i q^{63} +22.1516 q^{64} +141.058 q^{66} +20.5719 q^{67} +32.2647i q^{68} -48.3147i q^{69} +7.48928i q^{71} -20.7144 q^{72} -29.2302i q^{73} -121.278 q^{74} +(26.2510 - 15.1800i) q^{76} -185.790i q^{77} +120.309 q^{78} +8.78894i q^{79} -100.515 q^{81} +176.627i q^{82} -63.7886i q^{83} -62.8679i q^{84} +14.7423i q^{86} -108.081i q^{87} -95.3708 q^{88} +162.908i q^{89} -158.461i q^{91} -21.6868i q^{92} +57.3367i q^{93} +104.145i q^{94} -85.9686 q^{96} -75.5974 q^{97} -174.420 q^{98} +61.0861 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 24 q^{4} - 56 q^{6} + 96 q^{9} + 64 q^{11} - 88 q^{16} - 16 q^{19} - 200 q^{24} + 216 q^{26} - 160 q^{36} - 152 q^{39} + 512 q^{44} - 144 q^{49} + 152 q^{54} - 592 q^{61} - 376 q^{64} + 304 q^{66} - 272 q^{74} + 496 q^{76} - 744 q^{81} - 88 q^{96} + 624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.36559 1.18279 0.591397 0.806381i \(-0.298577\pi\)
0.591397 + 0.806381i \(0.298577\pi\)
\(3\) 3.55563 1.18521 0.592605 0.805493i \(-0.298100\pi\)
0.592605 + 0.805493i \(0.298100\pi\)
\(4\) 1.59600 0.399000
\(5\) 0 0
\(6\) 8.41115 1.40186
\(7\) 11.0785i 1.58264i −0.611405 0.791318i \(-0.709395\pi\)
0.611405 0.791318i \(-0.290605\pi\)
\(8\) −5.68687 −0.710859
\(9\) 3.64250 0.404723
\(10\) 0 0
\(11\) 16.7704 1.52458 0.762289 0.647237i \(-0.224076\pi\)
0.762289 + 0.647237i \(0.224076\pi\)
\(12\) 5.67479 0.472899
\(13\) 14.3035 1.10027 0.550136 0.835075i \(-0.314576\pi\)
0.550136 + 0.835075i \(0.314576\pi\)
\(14\) 26.2070i 1.87193i
\(15\) 0 0
\(16\) −19.8368 −1.23980
\(17\) 20.2160i 1.18918i 0.804031 + 0.594588i \(0.202685\pi\)
−0.804031 + 0.594588i \(0.797315\pi\)
\(18\) 8.61666 0.478703
\(19\) 16.4480 9.51127i 0.865683 0.500593i
\(20\) 0 0
\(21\) 39.3909i 1.87576i
\(22\) 39.6717 1.80326
\(23\) 13.5882i 0.590792i −0.955375 0.295396i \(-0.904548\pi\)
0.955375 0.295396i \(-0.0954516\pi\)
\(24\) −20.2204 −0.842517
\(25\) 0 0
\(26\) 33.8362 1.30139
\(27\) −19.0493 −0.705529
\(28\) 17.6812i 0.631472i
\(29\) 30.3972i 1.04818i −0.851663 0.524090i \(-0.824406\pi\)
0.851663 0.524090i \(-0.175594\pi\)
\(30\) 0 0
\(31\) 16.1256i 0.520181i 0.965584 + 0.260091i \(0.0837525\pi\)
−0.965584 + 0.260091i \(0.916248\pi\)
\(32\) −24.1782 −0.755567
\(33\) 59.6292 1.80694
\(34\) 47.8227i 1.40655i
\(35\) 0 0
\(36\) 5.81344 0.161484
\(37\) −51.2678 −1.38562 −0.692808 0.721122i \(-0.743627\pi\)
−0.692808 + 0.721122i \(0.743627\pi\)
\(38\) 38.9091 22.4997i 1.02392 0.592098i
\(39\) 50.8580 1.30405
\(40\) 0 0
\(41\) 74.6652i 1.82110i 0.413395 + 0.910552i \(0.364343\pi\)
−0.413395 + 0.910552i \(0.635657\pi\)
\(42\) 93.1825i 2.21863i
\(43\) 6.23197i 0.144929i 0.997371 + 0.0724647i \(0.0230865\pi\)
−0.997371 + 0.0724647i \(0.976914\pi\)
\(44\) 26.7655 0.608307
\(45\) 0 0
\(46\) 32.1441i 0.698785i
\(47\) 44.0252i 0.936706i 0.883541 + 0.468353i \(0.155152\pi\)
−0.883541 + 0.468353i \(0.844848\pi\)
\(48\) −70.5323 −1.46942
\(49\) −73.7321 −1.50474
\(50\) 0 0
\(51\) 71.8806i 1.40942i
\(52\) 22.8284 0.439008
\(53\) −30.8058 −0.581241 −0.290620 0.956838i \(-0.593862\pi\)
−0.290620 + 0.956838i \(0.593862\pi\)
\(54\) −45.0627 −0.834495
\(55\) 0 0
\(56\) 63.0017i 1.12503i
\(57\) 58.4829 33.8185i 1.02602 0.593308i
\(58\) 71.9073i 1.23978i
\(59\) 73.0098i 1.23745i 0.785606 + 0.618727i \(0.212351\pi\)
−0.785606 + 0.618727i \(0.787649\pi\)
\(60\) 0 0
\(61\) −17.7828 −0.291522 −0.145761 0.989320i \(-0.546563\pi\)
−0.145761 + 0.989320i \(0.546563\pi\)
\(62\) 38.1466i 0.615267i
\(63\) 40.3533i 0.640528i
\(64\) 22.1516 0.346119
\(65\) 0 0
\(66\) 141.058 2.13724
\(67\) 20.5719 0.307044 0.153522 0.988145i \(-0.450938\pi\)
0.153522 + 0.988145i \(0.450938\pi\)
\(68\) 32.2647i 0.474481i
\(69\) 48.3147i 0.700212i
\(70\) 0 0
\(71\) 7.48928i 0.105483i 0.998608 + 0.0527414i \(0.0167959\pi\)
−0.998608 + 0.0527414i \(0.983204\pi\)
\(72\) −20.7144 −0.287700
\(73\) 29.2302i 0.400414i −0.979754 0.200207i \(-0.935838\pi\)
0.979754 0.200207i \(-0.0641615\pi\)
\(74\) −121.278 −1.63890
\(75\) 0 0
\(76\) 26.2510 15.1800i 0.345408 0.199737i
\(77\) 185.790i 2.41285i
\(78\) 120.309 1.54242
\(79\) 8.78894i 0.111252i 0.998452 + 0.0556262i \(0.0177155\pi\)
−0.998452 + 0.0556262i \(0.982284\pi\)
\(80\) 0 0
\(81\) −100.515 −1.24092
\(82\) 176.627i 2.15399i
\(83\) 63.7886i 0.768537i −0.923221 0.384269i \(-0.874454\pi\)
0.923221 0.384269i \(-0.125546\pi\)
\(84\) 62.8679i 0.748427i
\(85\) 0 0
\(86\) 14.7423i 0.171422i
\(87\) 108.081i 1.24231i
\(88\) −95.3708 −1.08376
\(89\) 162.908i 1.83043i 0.402965 + 0.915215i \(0.367980\pi\)
−0.402965 + 0.915215i \(0.632020\pi\)
\(90\) 0 0
\(91\) 158.461i 1.74133i
\(92\) 21.6868i 0.235726i
\(93\) 57.3367i 0.616524i
\(94\) 104.145i 1.10793i
\(95\) 0 0
\(96\) −85.9686 −0.895506
\(97\) −75.5974 −0.779355 −0.389677 0.920951i \(-0.627413\pi\)
−0.389677 + 0.920951i \(0.627413\pi\)
\(98\) −174.420 −1.77979
\(99\) 61.0861 0.617031
\(100\) 0 0
\(101\) 15.7692 0.156131 0.0780656 0.996948i \(-0.475126\pi\)
0.0780656 + 0.996948i \(0.475126\pi\)
\(102\) 170.040i 1.66706i
\(103\) −7.15141 −0.0694312 −0.0347156 0.999397i \(-0.511053\pi\)
−0.0347156 + 0.999397i \(0.511053\pi\)
\(104\) −81.3423 −0.782137
\(105\) 0 0
\(106\) −72.8737 −0.687488
\(107\) 85.9872 0.803619 0.401809 0.915723i \(-0.368381\pi\)
0.401809 + 0.915723i \(0.368381\pi\)
\(108\) −30.4027 −0.281506
\(109\) 95.0893i 0.872379i −0.899855 0.436190i \(-0.856328\pi\)
0.899855 0.436190i \(-0.143672\pi\)
\(110\) 0 0
\(111\) −182.289 −1.64225
\(112\) 219.761i 1.96215i
\(113\) −19.0355 −0.168456 −0.0842281 0.996447i \(-0.526842\pi\)
−0.0842281 + 0.996447i \(0.526842\pi\)
\(114\) 138.346 80.0007i 1.21356 0.701761i
\(115\) 0 0
\(116\) 48.5140i 0.418224i
\(117\) 52.1006 0.445304
\(118\) 172.711i 1.46365i
\(119\) 223.962 1.88203
\(120\) 0 0
\(121\) 160.245 1.32434
\(122\) −42.0668 −0.344810
\(123\) 265.482i 2.15839i
\(124\) 25.7365i 0.207552i
\(125\) 0 0
\(126\) 95.4592i 0.757613i
\(127\) 93.4107 0.735518 0.367759 0.929921i \(-0.380125\pi\)
0.367759 + 0.929921i \(0.380125\pi\)
\(128\) 149.114 1.16495
\(129\) 22.1586i 0.171772i
\(130\) 0 0
\(131\) 126.496 0.965619 0.482810 0.875725i \(-0.339616\pi\)
0.482810 + 0.875725i \(0.339616\pi\)
\(132\) 95.1682 0.720971
\(133\) −105.370 182.218i −0.792257 1.37006i
\(134\) 48.6647 0.363170
\(135\) 0 0
\(136\) 114.966i 0.845336i
\(137\) 93.8813i 0.685265i −0.939470 0.342632i \(-0.888681\pi\)
0.939470 0.342632i \(-0.111319\pi\)
\(138\) 114.293i 0.828207i
\(139\) −41.0986 −0.295673 −0.147837 0.989012i \(-0.547231\pi\)
−0.147837 + 0.989012i \(0.547231\pi\)
\(140\) 0 0
\(141\) 156.537i 1.11019i
\(142\) 17.7165i 0.124764i
\(143\) 239.875 1.67745
\(144\) −72.2555 −0.501775
\(145\) 0 0
\(146\) 69.1467i 0.473607i
\(147\) −262.164 −1.78343
\(148\) −81.8235 −0.552861
\(149\) −92.2911 −0.619404 −0.309702 0.950834i \(-0.600229\pi\)
−0.309702 + 0.950834i \(0.600229\pi\)
\(150\) 0 0
\(151\) 7.26853i 0.0481359i 0.999710 + 0.0240680i \(0.00766181\pi\)
−0.999710 + 0.0240680i \(0.992338\pi\)
\(152\) −93.5375 + 54.0893i −0.615378 + 0.355851i
\(153\) 73.6368i 0.481286i
\(154\) 439.501i 2.85390i
\(155\) 0 0
\(156\) 81.1695 0.520317
\(157\) 229.214i 1.45996i −0.683468 0.729981i \(-0.739529\pi\)
0.683468 0.729981i \(-0.260471\pi\)
\(158\) 20.7910i 0.131589i
\(159\) −109.534 −0.688892
\(160\) 0 0
\(161\) −150.536 −0.935009
\(162\) −237.776 −1.46775
\(163\) 34.0426i 0.208850i 0.994533 + 0.104425i \(0.0333002\pi\)
−0.994533 + 0.104425i \(0.966700\pi\)
\(164\) 119.166i 0.726621i
\(165\) 0 0
\(166\) 150.897i 0.909021i
\(167\) 162.274 0.971702 0.485851 0.874042i \(-0.338510\pi\)
0.485851 + 0.874042i \(0.338510\pi\)
\(168\) 224.011i 1.33340i
\(169\) 35.5908 0.210597
\(170\) 0 0
\(171\) 59.9118 34.6448i 0.350361 0.202601i
\(172\) 9.94623i 0.0578269i
\(173\) −29.7956 −0.172229 −0.0861145 0.996285i \(-0.527445\pi\)
−0.0861145 + 0.996285i \(0.527445\pi\)
\(174\) 255.676i 1.46940i
\(175\) 0 0
\(176\) −332.670 −1.89017
\(177\) 259.596i 1.46664i
\(178\) 385.374i 2.16502i
\(179\) 59.2752i 0.331146i 0.986197 + 0.165573i \(0.0529474\pi\)
−0.986197 + 0.165573i \(0.947053\pi\)
\(180\) 0 0
\(181\) 205.700i 1.13647i 0.822868 + 0.568233i \(0.192373\pi\)
−0.822868 + 0.568233i \(0.807627\pi\)
\(182\) 374.853i 2.05963i
\(183\) −63.2291 −0.345514
\(184\) 77.2744i 0.419970i
\(185\) 0 0
\(186\) 135.635i 0.729221i
\(187\) 339.029i 1.81299i
\(188\) 70.2642i 0.373746i
\(189\) 211.037i 1.11660i
\(190\) 0 0
\(191\) −253.682 −1.32818 −0.664089 0.747654i \(-0.731180\pi\)
−0.664089 + 0.747654i \(0.731180\pi\)
\(192\) 78.7629 0.410223
\(193\) 289.646 1.50076 0.750378 0.661009i \(-0.229872\pi\)
0.750378 + 0.661009i \(0.229872\pi\)
\(194\) −178.832 −0.921816
\(195\) 0 0
\(196\) −117.676 −0.600390
\(197\) 178.190i 0.904515i −0.891887 0.452258i \(-0.850619\pi\)
0.891887 0.452258i \(-0.149381\pi\)
\(198\) 144.504 0.729820
\(199\) 154.602 0.776893 0.388446 0.921471i \(-0.373012\pi\)
0.388446 + 0.921471i \(0.373012\pi\)
\(200\) 0 0
\(201\) 73.1462 0.363912
\(202\) 37.3035 0.184671
\(203\) −336.754 −1.65889
\(204\) 114.721i 0.562360i
\(205\) 0 0
\(206\) −16.9173 −0.0821228
\(207\) 49.4951i 0.239107i
\(208\) −283.736 −1.36412
\(209\) 275.838 159.507i 1.31980 0.763193i
\(210\) 0 0
\(211\) 273.846i 1.29785i −0.760854 0.648923i \(-0.775220\pi\)
0.760854 0.648923i \(-0.224780\pi\)
\(212\) −49.1660 −0.231915
\(213\) 26.6291i 0.125019i
\(214\) 203.410 0.950515
\(215\) 0 0
\(216\) 108.331 0.501531
\(217\) 178.647 0.823258
\(218\) 224.942i 1.03184i
\(219\) 103.932i 0.474575i
\(220\) 0 0
\(221\) 289.160i 1.30842i
\(222\) −431.221 −1.94244
\(223\) −229.796 −1.03048 −0.515238 0.857047i \(-0.672296\pi\)
−0.515238 + 0.857047i \(0.672296\pi\)
\(224\) 267.857i 1.19579i
\(225\) 0 0
\(226\) −45.0302 −0.199249
\(227\) −366.166 −1.61306 −0.806532 0.591190i \(-0.798658\pi\)
−0.806532 + 0.591190i \(0.798658\pi\)
\(228\) 93.3387 53.9744i 0.409380 0.236730i
\(229\) −334.098 −1.45894 −0.729471 0.684012i \(-0.760234\pi\)
−0.729471 + 0.684012i \(0.760234\pi\)
\(230\) 0 0
\(231\) 660.599i 2.85974i
\(232\) 172.865i 0.745108i
\(233\) 83.9900i 0.360472i 0.983623 + 0.180236i \(0.0576861\pi\)
−0.983623 + 0.180236i \(0.942314\pi\)
\(234\) 123.249 0.526703
\(235\) 0 0
\(236\) 116.524i 0.493745i
\(237\) 31.2502i 0.131857i
\(238\) 529.801 2.22606
\(239\) 110.020 0.460335 0.230168 0.973151i \(-0.426073\pi\)
0.230168 + 0.973151i \(0.426073\pi\)
\(240\) 0 0
\(241\) 321.641i 1.33461i 0.744785 + 0.667305i \(0.232552\pi\)
−0.744785 + 0.667305i \(0.767448\pi\)
\(242\) 379.073 1.56642
\(243\) −185.950 −0.765225
\(244\) −28.3814 −0.116317
\(245\) 0 0
\(246\) 628.021i 2.55293i
\(247\) 235.264 136.045i 0.952486 0.550788i
\(248\) 91.7043i 0.369775i
\(249\) 226.809i 0.910878i
\(250\) 0 0
\(251\) 143.410 0.571353 0.285676 0.958326i \(-0.407782\pi\)
0.285676 + 0.958326i \(0.407782\pi\)
\(252\) 64.4039i 0.255571i
\(253\) 227.879i 0.900708i
\(254\) 220.971 0.869965
\(255\) 0 0
\(256\) 264.136 1.03178
\(257\) −171.202 −0.666156 −0.333078 0.942899i \(-0.608087\pi\)
−0.333078 + 0.942899i \(0.608087\pi\)
\(258\) 52.4180i 0.203171i
\(259\) 567.968i 2.19293i
\(260\) 0 0
\(261\) 110.722i 0.424222i
\(262\) 299.238 1.14213
\(263\) 184.993i 0.703395i −0.936114 0.351697i \(-0.885605\pi\)
0.936114 0.351697i \(-0.114395\pi\)
\(264\) −339.103 −1.28448
\(265\) 0 0
\(266\) −249.262 431.053i −0.937076 1.62050i
\(267\) 579.242i 2.16944i
\(268\) 32.8328 0.122511
\(269\) 453.194i 1.68473i 0.538904 + 0.842367i \(0.318839\pi\)
−0.538904 + 0.842367i \(0.681161\pi\)
\(270\) 0 0
\(271\) 336.795 1.24279 0.621393 0.783499i \(-0.286567\pi\)
0.621393 + 0.783499i \(0.286567\pi\)
\(272\) 401.020i 1.47434i
\(273\) 563.428i 2.06384i
\(274\) 222.084i 0.810527i
\(275\) 0 0
\(276\) 77.1102i 0.279385i
\(277\) 270.449i 0.976351i −0.872746 0.488175i \(-0.837663\pi\)
0.872746 0.488175i \(-0.162337\pi\)
\(278\) −97.2223 −0.349720
\(279\) 58.7376i 0.210529i
\(280\) 0 0
\(281\) 151.335i 0.538558i −0.963062 0.269279i \(-0.913215\pi\)
0.963062 0.269279i \(-0.0867853\pi\)
\(282\) 370.303i 1.31313i
\(283\) 437.891i 1.54732i −0.633603 0.773658i \(-0.718425\pi\)
0.633603 0.773658i \(-0.281575\pi\)
\(284\) 11.9529i 0.0420876i
\(285\) 0 0
\(286\) 567.446 1.98408
\(287\) 827.175 2.88214
\(288\) −88.0690 −0.305795
\(289\) −119.686 −0.414139
\(290\) 0 0
\(291\) −268.796 −0.923699
\(292\) 46.6515i 0.159765i
\(293\) 265.247 0.905279 0.452640 0.891694i \(-0.350482\pi\)
0.452640 + 0.891694i \(0.350482\pi\)
\(294\) −620.172 −2.10943
\(295\) 0 0
\(296\) 291.553 0.984977
\(297\) −319.463 −1.07563
\(298\) −218.323 −0.732627
\(299\) 194.359i 0.650031i
\(300\) 0 0
\(301\) 69.0406 0.229371
\(302\) 17.1943i 0.0569349i
\(303\) 56.0696 0.185048
\(304\) −326.275 + 188.673i −1.07327 + 0.620635i
\(305\) 0 0
\(306\) 174.194i 0.569262i
\(307\) 405.065 1.31943 0.659714 0.751517i \(-0.270677\pi\)
0.659714 + 0.751517i \(0.270677\pi\)
\(308\) 296.520i 0.962728i
\(309\) −25.4278 −0.0822906
\(310\) 0 0
\(311\) −313.906 −1.00934 −0.504672 0.863311i \(-0.668387\pi\)
−0.504672 + 0.863311i \(0.668387\pi\)
\(312\) −289.223 −0.926997
\(313\) 513.442i 1.64039i 0.572085 + 0.820194i \(0.306135\pi\)
−0.572085 + 0.820194i \(0.693865\pi\)
\(314\) 542.225i 1.72683i
\(315\) 0 0
\(316\) 14.0271i 0.0443897i
\(317\) −25.8051 −0.0814042 −0.0407021 0.999171i \(-0.512959\pi\)
−0.0407021 + 0.999171i \(0.512959\pi\)
\(318\) −259.112 −0.814817
\(319\) 509.772i 1.59803i
\(320\) 0 0
\(321\) 305.739 0.952457
\(322\) −356.107 −1.10592
\(323\) 192.280 + 332.512i 0.595293 + 1.02945i
\(324\) −160.422 −0.495128
\(325\) 0 0
\(326\) 80.5307i 0.247027i
\(327\) 338.102i 1.03395i
\(328\) 424.611i 1.29455i
\(329\) 487.731 1.48247
\(330\) 0 0
\(331\) 171.635i 0.518534i −0.965806 0.259267i \(-0.916519\pi\)
0.965806 0.259267i \(-0.0834810\pi\)
\(332\) 101.807i 0.306647i
\(333\) −186.743 −0.560790
\(334\) 383.874 1.14932
\(335\) 0 0
\(336\) 781.388i 2.32556i
\(337\) −206.114 −0.611613 −0.305807 0.952094i \(-0.598926\pi\)
−0.305807 + 0.952094i \(0.598926\pi\)
\(338\) 84.1932 0.249092
\(339\) −67.6833 −0.199656
\(340\) 0 0
\(341\) 270.432i 0.793057i
\(342\) 141.727 81.9553i 0.414405 0.239635i
\(343\) 273.994i 0.798815i
\(344\) 35.4404i 0.103024i
\(345\) 0 0
\(346\) −70.4841 −0.203711
\(347\) 7.91893i 0.0228211i 0.999935 + 0.0114106i \(0.00363217\pi\)
−0.999935 + 0.0114106i \(0.996368\pi\)
\(348\) 172.498i 0.495683i
\(349\) −409.448 −1.17320 −0.586602 0.809876i \(-0.699535\pi\)
−0.586602 + 0.809876i \(0.699535\pi\)
\(350\) 0 0
\(351\) −272.472 −0.776273
\(352\) −405.476 −1.15192
\(353\) 637.919i 1.80714i −0.428445 0.903568i \(-0.640939\pi\)
0.428445 0.903568i \(-0.359061\pi\)
\(354\) 614.097i 1.73474i
\(355\) 0 0
\(356\) 260.002i 0.730342i
\(357\) 796.325 2.23060
\(358\) 140.221i 0.391678i
\(359\) 237.272 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(360\) 0 0
\(361\) 180.072 312.882i 0.498813 0.866710i
\(362\) 486.602i 1.34420i
\(363\) 569.771 1.56962
\(364\) 252.904i 0.694790i
\(365\) 0 0
\(366\) −149.574 −0.408672
\(367\) 542.889i 1.47926i −0.673013 0.739631i \(-0.735000\pi\)
0.673013 0.739631i \(-0.265000\pi\)
\(368\) 269.546i 0.732463i
\(369\) 271.968i 0.737042i
\(370\) 0 0
\(371\) 341.280i 0.919892i
\(372\) 91.5095i 0.245993i
\(373\) −444.746 −1.19235 −0.596175 0.802855i \(-0.703313\pi\)
−0.596175 + 0.802855i \(0.703313\pi\)
\(374\) 802.003i 2.14439i
\(375\) 0 0
\(376\) 250.366i 0.665866i
\(377\) 434.787i 1.15328i
\(378\) 499.225i 1.32070i
\(379\) 150.221i 0.396362i 0.980165 + 0.198181i \(0.0635035\pi\)
−0.980165 + 0.198181i \(0.936497\pi\)
\(380\) 0 0
\(381\) 332.134 0.871743
\(382\) −600.106 −1.57096
\(383\) −85.8011 −0.224024 −0.112012 0.993707i \(-0.535729\pi\)
−0.112012 + 0.993707i \(0.535729\pi\)
\(384\) 530.195 1.38072
\(385\) 0 0
\(386\) 685.182 1.77508
\(387\) 22.7000i 0.0586562i
\(388\) −120.654 −0.310963
\(389\) 249.669 0.641822 0.320911 0.947109i \(-0.396011\pi\)
0.320911 + 0.947109i \(0.396011\pi\)
\(390\) 0 0
\(391\) 274.699 0.702555
\(392\) 419.305 1.06966
\(393\) 449.773 1.14446
\(394\) 421.523i 1.06985i
\(395\) 0 0
\(396\) 97.4934 0.246195
\(397\) 340.476i 0.857622i 0.903394 + 0.428811i \(0.141067\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(398\) 365.724 0.918904
\(399\) −374.657 647.900i −0.938990 1.62381i
\(400\) 0 0
\(401\) 491.002i 1.22444i −0.790686 0.612221i \(-0.790276\pi\)
0.790686 0.612221i \(-0.209724\pi\)
\(402\) 173.034 0.430432
\(403\) 230.653i 0.572341i
\(404\) 25.1677 0.0622964
\(405\) 0 0
\(406\) −796.621 −1.96212
\(407\) −859.779 −2.11248
\(408\) 408.775i 1.00190i
\(409\) 115.833i 0.283210i 0.989923 + 0.141605i \(0.0452263\pi\)
−0.989923 + 0.141605i \(0.954774\pi\)
\(410\) 0 0
\(411\) 333.807i 0.812183i
\(412\) −11.4137 −0.0277031
\(413\) 808.836 1.95844
\(414\) 117.085i 0.282814i
\(415\) 0 0
\(416\) −345.833 −0.831329
\(417\) −146.131 −0.350435
\(418\) 652.519 377.328i 1.56105 0.902700i
\(419\) −81.4969 −0.194503 −0.0972516 0.995260i \(-0.531005\pi\)
−0.0972516 + 0.995260i \(0.531005\pi\)
\(420\) 0 0
\(421\) 221.652i 0.526490i −0.964729 0.263245i \(-0.915207\pi\)
0.964729 0.263245i \(-0.0847927\pi\)
\(422\) 647.806i 1.53508i
\(423\) 160.362i 0.379106i
\(424\) 175.188 0.413180
\(425\) 0 0
\(426\) 62.9934i 0.147872i
\(427\) 197.006i 0.461372i
\(428\) 137.236 0.320644
\(429\) 852.907 1.98813
\(430\) 0 0
\(431\) 211.834i 0.491494i 0.969334 + 0.245747i \(0.0790333\pi\)
−0.969334 + 0.245747i \(0.920967\pi\)
\(432\) 377.876 0.874714
\(433\) −389.534 −0.899616 −0.449808 0.893125i \(-0.648508\pi\)
−0.449808 + 0.893125i \(0.648508\pi\)
\(434\) 422.605 0.973744
\(435\) 0 0
\(436\) 151.763i 0.348079i
\(437\) −129.241 223.499i −0.295746 0.511438i
\(438\) 245.860i 0.561324i
\(439\) 465.691i 1.06080i −0.847748 0.530400i \(-0.822042\pi\)
0.847748 0.530400i \(-0.177958\pi\)
\(440\) 0 0
\(441\) −268.569 −0.609001
\(442\) 684.033i 1.54759i
\(443\) 382.479i 0.863383i −0.902021 0.431692i \(-0.857917\pi\)
0.902021 0.431692i \(-0.142083\pi\)
\(444\) −290.934 −0.655257
\(445\) 0 0
\(446\) −543.603 −1.21884
\(447\) −328.153 −0.734123
\(448\) 245.406i 0.547780i
\(449\) 666.393i 1.48417i −0.670305 0.742086i \(-0.733837\pi\)
0.670305 0.742086i \(-0.266163\pi\)
\(450\) 0 0
\(451\) 1252.16i 2.77641i
\(452\) −30.3807 −0.0672140
\(453\) 25.8442i 0.0570512i
\(454\) −866.196 −1.90792
\(455\) 0 0
\(456\) −332.585 + 192.322i −0.729352 + 0.421758i
\(457\) 627.200i 1.37243i 0.727399 + 0.686214i \(0.240729\pi\)
−0.727399 + 0.686214i \(0.759271\pi\)
\(458\) −790.337 −1.72563
\(459\) 385.100i 0.838998i
\(460\) 0 0
\(461\) 798.338 1.73175 0.865876 0.500258i \(-0.166762\pi\)
0.865876 + 0.500258i \(0.166762\pi\)
\(462\) 1562.70i 3.38248i
\(463\) 791.786i 1.71012i 0.518528 + 0.855061i \(0.326480\pi\)
−0.518528 + 0.855061i \(0.673520\pi\)
\(464\) 602.983i 1.29953i
\(465\) 0 0
\(466\) 198.686i 0.426364i
\(467\) 418.723i 0.896622i 0.893878 + 0.448311i \(0.147974\pi\)
−0.893878 + 0.448311i \(0.852026\pi\)
\(468\) 83.1526 0.177677
\(469\) 227.905i 0.485939i
\(470\) 0 0
\(471\) 815.000i 1.73036i
\(472\) 415.197i 0.879655i
\(473\) 104.512i 0.220956i
\(474\) 73.9251i 0.155960i
\(475\) 0 0
\(476\) 357.443 0.750931
\(477\) −112.210 −0.235241
\(478\) 260.262 0.544481
\(479\) −63.6160 −0.132810 −0.0664050 0.997793i \(-0.521153\pi\)
−0.0664050 + 0.997793i \(0.521153\pi\)
\(480\) 0 0
\(481\) −733.310 −1.52455
\(482\) 760.869i 1.57857i
\(483\) −535.252 −1.10818
\(484\) 255.751 0.528411
\(485\) 0 0
\(486\) −439.880 −0.905102
\(487\) 556.997 1.14373 0.571866 0.820347i \(-0.306220\pi\)
0.571866 + 0.820347i \(0.306220\pi\)
\(488\) 101.129 0.207231
\(489\) 121.043i 0.247531i
\(490\) 0 0
\(491\) 856.809 1.74503 0.872514 0.488589i \(-0.162488\pi\)
0.872514 + 0.488589i \(0.162488\pi\)
\(492\) 423.709i 0.861198i
\(493\) 614.510 1.24647
\(494\) 556.537 321.825i 1.12659 0.651469i
\(495\) 0 0
\(496\) 319.881i 0.644920i
\(497\) 82.9696 0.166941
\(498\) 536.536i 1.07738i
\(499\) −761.632 −1.52632 −0.763158 0.646211i \(-0.776352\pi\)
−0.763158 + 0.646211i \(0.776352\pi\)
\(500\) 0 0
\(501\) 576.987 1.15167
\(502\) 339.248 0.675792
\(503\) 34.7192i 0.0690242i −0.999404 0.0345121i \(-0.989012\pi\)
0.999404 0.0345121i \(-0.0109877\pi\)
\(504\) 229.484i 0.455325i
\(505\) 0 0
\(506\) 539.068i 1.06535i
\(507\) 126.548 0.249601
\(508\) 149.084 0.293472
\(509\) 9.31181i 0.0182943i 0.999958 + 0.00914716i \(0.00291167\pi\)
−0.999958 + 0.00914716i \(0.997088\pi\)
\(510\) 0 0
\(511\) −323.826 −0.633710
\(512\) 28.3801 0.0554300
\(513\) −313.322 + 181.183i −0.610764 + 0.353183i
\(514\) −404.993 −0.787925
\(515\) 0 0
\(516\) 35.3651i 0.0685370i
\(517\) 738.318i 1.42808i
\(518\) 1343.58i 2.59378i
\(519\) −105.942 −0.204128
\(520\) 0 0
\(521\) 88.0858i 0.169071i −0.996420 0.0845353i \(-0.973059\pi\)
0.996420 0.0845353i \(-0.0269406\pi\)
\(522\) 261.922i 0.501767i
\(523\) −377.616 −0.722019 −0.361009 0.932562i \(-0.617568\pi\)
−0.361009 + 0.932562i \(0.617568\pi\)
\(524\) 201.888 0.385282
\(525\) 0 0
\(526\) 437.617i 0.831971i
\(527\) −325.995 −0.618587
\(528\) −1182.85 −2.24025
\(529\) 344.360 0.650965
\(530\) 0 0
\(531\) 265.938i 0.500826i
\(532\) −168.171 290.820i −0.316111 0.546654i
\(533\) 1067.98i 2.00371i
\(534\) 1370.25i 2.56600i
\(535\) 0 0
\(536\) −116.990 −0.218265
\(537\) 210.761i 0.392478i
\(538\) 1072.07i 1.99269i
\(539\) −1236.51 −2.29409
\(540\) 0 0
\(541\) −225.756 −0.417293 −0.208647 0.977991i \(-0.566906\pi\)
−0.208647 + 0.977991i \(0.566906\pi\)
\(542\) 796.718 1.46996
\(543\) 731.394i 1.34695i
\(544\) 488.785i 0.898502i
\(545\) 0 0
\(546\) 1332.84i 2.44110i
\(547\) −1050.58 −1.92062 −0.960309 0.278938i \(-0.910018\pi\)
−0.960309 + 0.278938i \(0.910018\pi\)
\(548\) 149.835i 0.273421i
\(549\) −64.7739 −0.117985
\(550\) 0 0
\(551\) −289.116 499.973i −0.524712 0.907391i
\(552\) 274.759i 0.497752i
\(553\) 97.3678 0.176072
\(554\) 639.771i 1.15482i
\(555\) 0 0
\(556\) −65.5934 −0.117974
\(557\) 580.999i 1.04309i −0.853225 0.521543i \(-0.825356\pi\)
0.853225 0.521543i \(-0.174644\pi\)
\(558\) 138.949i 0.249012i
\(559\) 89.1391i 0.159462i
\(560\) 0 0
\(561\) 1205.46i 2.14877i
\(562\) 357.995i 0.637002i
\(563\) −796.409 −1.41458 −0.707291 0.706923i \(-0.750083\pi\)
−0.707291 + 0.706923i \(0.750083\pi\)
\(564\) 249.834i 0.442967i
\(565\) 0 0
\(566\) 1035.87i 1.83016i
\(567\) 1113.55i 1.96393i
\(568\) 42.5905i 0.0749833i
\(569\) 171.428i 0.301280i 0.988589 + 0.150640i \(0.0481334\pi\)
−0.988589 + 0.150640i \(0.951867\pi\)
\(570\) 0 0
\(571\) −673.960 −1.18031 −0.590157 0.807288i \(-0.700934\pi\)
−0.590157 + 0.807288i \(0.700934\pi\)
\(572\) 382.841 0.669302
\(573\) −901.999 −1.57417
\(574\) 1956.76 3.40898
\(575\) 0 0
\(576\) 80.6873 0.140082
\(577\) 795.916i 1.37940i 0.724094 + 0.689702i \(0.242258\pi\)
−0.724094 + 0.689702i \(0.757742\pi\)
\(578\) −283.128 −0.489840
\(579\) 1029.87 1.77871
\(580\) 0 0
\(581\) −706.679 −1.21632
\(582\) −635.861 −1.09254
\(583\) −516.623 −0.886146
\(584\) 166.229i 0.284638i
\(585\) 0 0
\(586\) 627.464 1.07076
\(587\) 19.6929i 0.0335484i 0.999859 + 0.0167742i \(0.00533965\pi\)
−0.999859 + 0.0167742i \(0.994660\pi\)
\(588\) −418.414 −0.711589
\(589\) 153.375 + 265.234i 0.260399 + 0.450312i
\(590\) 0 0
\(591\) 633.576i 1.07204i
\(592\) 1016.99 1.71789
\(593\) 636.058i 1.07261i −0.844024 0.536306i \(-0.819819\pi\)
0.844024 0.536306i \(-0.180181\pi\)
\(594\) −755.718 −1.27225
\(595\) 0 0
\(596\) −147.297 −0.247142
\(597\) 549.706 0.920781
\(598\) 459.774i 0.768853i
\(599\) 936.525i 1.56348i 0.623604 + 0.781741i \(0.285668\pi\)
−0.623604 + 0.781741i \(0.714332\pi\)
\(600\) 0 0
\(601\) 252.090i 0.419451i 0.977760 + 0.209725i \(0.0672570\pi\)
−0.977760 + 0.209725i \(0.932743\pi\)
\(602\) 163.321 0.271298
\(603\) 74.9334 0.124268
\(604\) 11.6006i 0.0192062i
\(605\) 0 0
\(606\) 132.638 0.218874
\(607\) −498.526 −0.821295 −0.410648 0.911794i \(-0.634697\pi\)
−0.410648 + 0.911794i \(0.634697\pi\)
\(608\) −397.682 + 229.965i −0.654082 + 0.378232i
\(609\) −1197.37 −1.96613
\(610\) 0 0
\(611\) 629.715i 1.03063i
\(612\) 117.524i 0.192033i
\(613\) 638.709i 1.04194i −0.853575 0.520969i \(-0.825571\pi\)
0.853575 0.520969i \(-0.174429\pi\)
\(614\) 958.215 1.56061
\(615\) 0 0
\(616\) 1056.56i 1.71520i
\(617\) 702.709i 1.13891i −0.822021 0.569457i \(-0.807154\pi\)
0.822021 0.569457i \(-0.192846\pi\)
\(618\) −60.1516 −0.0973327
\(619\) −195.693 −0.316144 −0.158072 0.987428i \(-0.550528\pi\)
−0.158072 + 0.987428i \(0.550528\pi\)
\(620\) 0 0
\(621\) 258.846i 0.416821i
\(622\) −742.572 −1.19385
\(623\) 1804.77 2.89691
\(624\) −1008.86 −1.61676
\(625\) 0 0
\(626\) 1214.59i 1.94024i
\(627\) 980.779 567.149i 1.56424 0.904544i
\(628\) 365.826i 0.582525i
\(629\) 1036.43i 1.64774i
\(630\) 0 0
\(631\) −1063.16 −1.68489 −0.842444 0.538784i \(-0.818884\pi\)
−0.842444 + 0.538784i \(0.818884\pi\)
\(632\) 49.9815i 0.0790847i
\(633\) 973.694i 1.53822i
\(634\) −61.0443 −0.0962843
\(635\) 0 0
\(636\) −174.816 −0.274868
\(637\) −1054.63 −1.65562
\(638\) 1205.91i 1.89014i
\(639\) 27.2797i 0.0426913i
\(640\) 0 0
\(641\) 744.630i 1.16167i −0.814021 0.580835i \(-0.802726\pi\)
0.814021 0.580835i \(-0.197274\pi\)
\(642\) 723.251 1.12656
\(643\) 195.660i 0.304293i 0.988358 + 0.152146i \(0.0486185\pi\)
−0.988358 + 0.152146i \(0.951382\pi\)
\(644\) −240.256 −0.373069
\(645\) 0 0
\(646\) 454.854 + 786.586i 0.704109 + 1.21763i
\(647\) 128.045i 0.197906i 0.995092 + 0.0989529i \(0.0315493\pi\)
−0.995092 + 0.0989529i \(0.968451\pi\)
\(648\) 571.614 0.882120
\(649\) 1224.40i 1.88660i
\(650\) 0 0
\(651\) 635.202 0.975733
\(652\) 54.3320i 0.0833313i
\(653\) 717.897i 1.09938i −0.835368 0.549692i \(-0.814745\pi\)
0.835368 0.549692i \(-0.185255\pi\)
\(654\) 799.811i 1.22295i
\(655\) 0 0
\(656\) 1481.12i 2.25780i
\(657\) 106.471i 0.162057i
\(658\) 1153.77 1.75345
\(659\) 544.617i 0.826430i 0.910633 + 0.413215i \(0.135594\pi\)
−0.910633 + 0.413215i \(0.864406\pi\)
\(660\) 0 0
\(661\) 435.420i 0.658729i 0.944203 + 0.329365i \(0.106835\pi\)
−0.944203 + 0.329365i \(0.893165\pi\)
\(662\) 406.017i 0.613319i
\(663\) 1028.15i 1.55075i
\(664\) 362.757i 0.546321i
\(665\) 0 0
\(666\) −441.757 −0.663299
\(667\) −413.044 −0.619256
\(668\) 258.990 0.387709
\(669\) −817.070 −1.22133
\(670\) 0 0
\(671\) −298.224 −0.444447
\(672\) 952.399i 1.41726i
\(673\) 397.296 0.590336 0.295168 0.955445i \(-0.404624\pi\)
0.295168 + 0.955445i \(0.404624\pi\)
\(674\) −487.580 −0.723412
\(675\) 0 0
\(676\) 56.8030 0.0840281
\(677\) 456.859 0.674829 0.337415 0.941356i \(-0.390448\pi\)
0.337415 + 0.941356i \(0.390448\pi\)
\(678\) −160.111 −0.236152
\(679\) 837.502i 1.23343i
\(680\) 0 0
\(681\) −1301.95 −1.91182
\(682\) 639.731i 0.938022i
\(683\) −880.553 −1.28924 −0.644621 0.764502i \(-0.722985\pi\)
−0.644621 + 0.764502i \(0.722985\pi\)
\(684\) 95.6192 55.2932i 0.139794 0.0808379i
\(685\) 0 0
\(686\) 648.155i 0.944833i
\(687\) −1187.93 −1.72915
\(688\) 123.622i 0.179683i
\(689\) −440.631 −0.639522
\(690\) 0 0
\(691\) −34.3920 −0.0497713 −0.0248857 0.999690i \(-0.507922\pi\)
−0.0248857 + 0.999690i \(0.507922\pi\)
\(692\) −47.5538 −0.0687194
\(693\) 676.739i 0.976535i
\(694\) 18.7329i 0.0269927i
\(695\) 0 0
\(696\) 614.644i 0.883109i
\(697\) −1509.43 −2.16561
\(698\) −968.585 −1.38766
\(699\) 298.637i 0.427235i
\(700\) 0 0
\(701\) 419.321 0.598175 0.299088 0.954226i \(-0.403318\pi\)
0.299088 + 0.954226i \(0.403318\pi\)
\(702\) −644.556 −0.918171
\(703\) −843.251 + 487.622i −1.19950 + 0.693630i
\(704\) 371.490 0.527685
\(705\) 0 0
\(706\) 1509.05i 2.13747i
\(707\) 174.699i 0.247099i
\(708\) 414.315i 0.585191i
\(709\) −373.670 −0.527038 −0.263519 0.964654i \(-0.584883\pi\)
−0.263519 + 0.964654i \(0.584883\pi\)
\(710\) 0 0
\(711\) 32.0137i 0.0450263i
\(712\) 926.438i 1.30118i
\(713\) 219.118 0.307319
\(714\) 1883.78 2.63834
\(715\) 0 0
\(716\) 94.6033i 0.132127i
\(717\) 391.191 0.545594
\(718\) 561.289 0.781739
\(719\) 328.089 0.456312 0.228156 0.973625i \(-0.426730\pi\)
0.228156 + 0.973625i \(0.426730\pi\)
\(720\) 0 0
\(721\) 79.2266i 0.109884i
\(722\) 425.975 740.150i 0.589993 1.02514i
\(723\) 1143.64i 1.58179i
\(724\) 328.298i 0.453450i
\(725\) 0 0
\(726\) 1347.84 1.85653
\(727\) 1131.05i 1.55578i −0.628402 0.777889i \(-0.716291\pi\)
0.628402 0.777889i \(-0.283709\pi\)
\(728\) 901.146i 1.23784i
\(729\) 243.465 0.333971
\(730\) 0 0
\(731\) −125.985 −0.172347
\(732\) −100.914 −0.137860
\(733\) 450.895i 0.615137i −0.951526 0.307568i \(-0.900485\pi\)
0.951526 0.307568i \(-0.0995153\pi\)
\(734\) 1284.25i 1.74966i
\(735\) 0 0
\(736\) 328.538i 0.446383i
\(737\) 344.999 0.468112
\(738\) 643.365i 0.871768i
\(739\) 195.478 0.264517 0.132258 0.991215i \(-0.457777\pi\)
0.132258 + 0.991215i \(0.457777\pi\)
\(740\) 0 0
\(741\) 836.512 483.724i 1.12890 0.652800i
\(742\) 807.328i 1.08804i
\(743\) −194.450 −0.261710 −0.130855 0.991402i \(-0.541772\pi\)
−0.130855 + 0.991402i \(0.541772\pi\)
\(744\) 326.067i 0.438262i
\(745\) 0 0
\(746\) −1052.09 −1.41030
\(747\) 232.350i 0.311044i
\(748\) 541.091i 0.723383i
\(749\) 952.605i 1.27184i
\(750\) 0 0
\(751\) 943.646i 1.25652i 0.778004 + 0.628259i \(0.216232\pi\)
−0.778004 + 0.628259i \(0.783768\pi\)
\(752\) 873.318i 1.16133i
\(753\) 509.911 0.677173
\(754\) 1028.53i 1.36409i
\(755\) 0 0
\(756\) 336.814i 0.445522i
\(757\) 891.082i 1.17712i −0.808453 0.588561i \(-0.799695\pi\)
0.808453 0.588561i \(-0.200305\pi\)
\(758\) 355.362i 0.468815i
\(759\) 810.254i 1.06753i
\(760\) 0 0
\(761\) 759.322 0.997795 0.498898 0.866661i \(-0.333738\pi\)
0.498898 + 0.866661i \(0.333738\pi\)
\(762\) 785.692 1.03109
\(763\) −1053.44 −1.38066
\(764\) −404.876 −0.529943
\(765\) 0 0
\(766\) −202.970 −0.264974
\(767\) 1044.30i 1.36154i
\(768\) 939.170 1.22288
\(769\) −225.596 −0.293363 −0.146681 0.989184i \(-0.546859\pi\)
−0.146681 + 0.989184i \(0.546859\pi\)
\(770\) 0 0
\(771\) −608.731 −0.789535
\(772\) 462.275 0.598802
\(773\) 1189.39 1.53867 0.769337 0.638843i \(-0.220587\pi\)
0.769337 + 0.638843i \(0.220587\pi\)
\(774\) 53.6987i 0.0693782i
\(775\) 0 0
\(776\) 429.913 0.554011
\(777\) 2019.48i 2.59908i
\(778\) 590.613 0.759143
\(779\) 710.161 + 1228.09i 0.911632 + 1.57650i
\(780\) 0 0
\(781\) 125.598i 0.160817i
\(782\) 649.825 0.830978
\(783\) 579.045i 0.739521i
\(784\) 1462.61 1.86557
\(785\) 0 0
\(786\) 1063.98 1.35366
\(787\) 254.533 0.323422 0.161711 0.986838i \(-0.448299\pi\)
0.161711 + 0.986838i \(0.448299\pi\)
\(788\) 284.391i 0.360902i
\(789\) 657.766i 0.833670i
\(790\) 0 0
\(791\) 210.884i 0.266605i
\(792\) −347.388 −0.438622
\(793\) −254.357 −0.320753
\(794\) 805.425i 1.01439i
\(795\) 0 0
\(796\) 246.744 0.309980
\(797\) 719.295 0.902503 0.451251 0.892397i \(-0.350978\pi\)
0.451251 + 0.892397i \(0.350978\pi\)
\(798\) −886.284 1532.66i −1.11063 1.92063i
\(799\) −890.013 −1.11391
\(800\) 0 0
\(801\) 593.394i 0.740816i
\(802\) 1161.51i 1.44826i
\(803\) 490.202i 0.610463i
\(804\) 116.741 0.145201
\(805\) 0 0
\(806\) 545.630i 0.676961i
\(807\) 1611.39i 1.99676i
\(808\) −89.6776 −0.110987
\(809\) −67.3018 −0.0831914 −0.0415957 0.999135i \(-0.513244\pi\)
−0.0415957 + 0.999135i \(0.513244\pi\)
\(810\) 0 0
\(811\) 401.142i 0.494626i −0.968936 0.247313i \(-0.920452\pi\)
0.968936 0.247313i \(-0.0795477\pi\)
\(812\) −537.460 −0.661896
\(813\) 1197.52 1.47296
\(814\) −2033.88 −2.49863
\(815\) 0 0
\(816\) 1425.88i 1.74740i
\(817\) 59.2739 + 102.503i 0.0725507 + 0.125463i
\(818\) 274.013i 0.334979i
\(819\) 577.194i 0.704755i
\(820\) 0 0
\(821\) −99.8212 −0.121585 −0.0607925 0.998150i \(-0.519363\pi\)
−0.0607925 + 0.998150i \(0.519363\pi\)
\(822\) 789.650i 0.960644i
\(823\) 93.5359i 0.113652i −0.998384 0.0568262i \(-0.981902\pi\)
0.998384 0.0568262i \(-0.0180981\pi\)
\(824\) 40.6692 0.0493558
\(825\) 0 0
\(826\) 1913.37 2.31643
\(827\) −398.725 −0.482135 −0.241067 0.970508i \(-0.577497\pi\)
−0.241067 + 0.970508i \(0.577497\pi\)
\(828\) 78.9942i 0.0954036i
\(829\) 805.348i 0.971470i 0.874106 + 0.485735i \(0.161448\pi\)
−0.874106 + 0.485735i \(0.838552\pi\)
\(830\) 0 0
\(831\) 961.617i 1.15718i
\(832\) 316.846 0.380825
\(833\) 1490.57i 1.78940i
\(834\) −345.686 −0.414492
\(835\) 0 0
\(836\) 440.238 254.574i 0.526601 0.304514i
\(837\) 307.181i 0.367003i
\(838\) −192.788 −0.230057
\(839\) 1547.08i 1.84396i −0.387235 0.921981i \(-0.626570\pi\)
0.387235 0.921981i \(-0.373430\pi\)
\(840\) 0 0
\(841\) −82.9912 −0.0986815
\(842\) 524.338i 0.622729i
\(843\) 538.090i 0.638304i
\(844\) 437.058i 0.517841i
\(845\) 0 0
\(846\) 379.350i 0.448404i
\(847\) 1775.26i 2.09594i
\(848\) 611.087 0.720622
\(849\) 1556.98i 1.83390i
\(850\) 0 0
\(851\) 696.638i 0.818611i
\(852\) 42.5001i 0.0498827i
\(853\) 252.303i 0.295784i −0.989004 0.147892i \(-0.952751\pi\)
0.989004 0.147892i \(-0.0472487\pi\)
\(854\) 466.035i 0.545708i
\(855\) 0 0
\(856\) −488.998 −0.571259
\(857\) 1397.35 1.63051 0.815255 0.579102i \(-0.196597\pi\)
0.815255 + 0.579102i \(0.196597\pi\)
\(858\) 2017.63 2.35155
\(859\) 1472.96 1.71473 0.857366 0.514706i \(-0.172099\pi\)
0.857366 + 0.514706i \(0.172099\pi\)
\(860\) 0 0
\(861\) 2941.13 3.41595
\(862\) 501.112i 0.581336i
\(863\) −63.9533 −0.0741058 −0.0370529 0.999313i \(-0.511797\pi\)
−0.0370529 + 0.999313i \(0.511797\pi\)
\(864\) 460.576 0.533075
\(865\) 0 0
\(866\) −921.476 −1.06406
\(867\) −425.559 −0.490841
\(868\) 285.121 0.328480
\(869\) 147.394i 0.169613i
\(870\) 0 0
\(871\) 294.251 0.337832
\(872\) 540.761i 0.620138i
\(873\) −275.364 −0.315422
\(874\) −305.731 528.705i −0.349807 0.604926i
\(875\) 0 0
\(876\) 165.875i 0.189356i
\(877\) 399.492 0.455522 0.227761 0.973717i \(-0.426860\pi\)
0.227761 + 0.973717i \(0.426860\pi\)
\(878\) 1101.63i 1.25471i
\(879\) 943.119 1.07295
\(880\) 0 0
\(881\) 514.180 0.583632 0.291816 0.956474i \(-0.405740\pi\)
0.291816 + 0.956474i \(0.405740\pi\)
\(882\) −635.324 −0.720322
\(883\) 323.869i 0.366783i 0.983040 + 0.183392i \(0.0587076\pi\)
−0.983040 + 0.183392i \(0.941292\pi\)
\(884\) 461.499i 0.522058i
\(885\) 0 0
\(886\) 904.787i 1.02120i
\(887\) −277.999 −0.313415 −0.156707 0.987645i \(-0.550088\pi\)
−0.156707 + 0.987645i \(0.550088\pi\)
\(888\) 1036.66 1.16740
\(889\) 1034.85i 1.16406i
\(890\) 0 0
\(891\) −1685.67 −1.89188
\(892\) −366.755 −0.411160
\(893\) 418.735 + 724.125i 0.468909 + 0.810890i
\(894\) −776.275 −0.868316
\(895\) 0 0
\(896\) 1651.95i 1.84370i
\(897\) 691.070i 0.770424i
\(898\) 1576.41i 1.75547i
\(899\) 490.174 0.545244
\(900\) 0 0
\(901\) 622.769i 0.691197i
\(902\) 2962.10i 3.28392i
\(903\) 245.483 0.271852
\(904\) 108.253 0.119749
\(905\) 0 0
\(906\) 61.1367i 0.0674798i
\(907\) 401.356 0.442509 0.221255 0.975216i \(-0.428985\pi\)
0.221255 + 0.975216i \(0.428985\pi\)
\(908\) −584.400 −0.643613
\(909\) 57.4395 0.0631898
\(910\) 0 0
\(911\) 378.750i 0.415752i 0.978155 + 0.207876i \(0.0666551\pi\)
−0.978155 + 0.207876i \(0.933345\pi\)
\(912\) −1160.11 + 670.851i −1.27205 + 0.735583i
\(913\) 1069.76i 1.17170i
\(914\) 1483.70i 1.62330i
\(915\) 0 0
\(916\) −533.220 −0.582118
\(917\) 1401.38i 1.52822i
\(918\) 910.987i 0.992361i
\(919\) 102.234 0.111244 0.0556222 0.998452i \(-0.482286\pi\)
0.0556222 + 0.998452i \(0.482286\pi\)
\(920\) 0 0
\(921\) 1440.26 1.56380
\(922\) 1888.54 2.04831
\(923\) 107.123i 0.116060i
\(924\) 1054.32i 1.14103i
\(925\) 0 0
\(926\) 1873.04i 2.02272i
\(927\) −26.0490 −0.0281004
\(928\) 734.949i 0.791971i
\(929\) −1794.45 −1.93160 −0.965798 0.259295i \(-0.916510\pi\)
−0.965798 + 0.259295i \(0.916510\pi\)
\(930\) 0 0
\(931\) −1212.74 + 701.286i −1.30262 + 0.753261i
\(932\) 134.048i 0.143828i
\(933\) −1116.13 −1.19629
\(934\) 990.525i 1.06052i
\(935\) 0 0
\(936\) −296.289 −0.316549
\(937\) 823.194i 0.878542i 0.898355 + 0.439271i \(0.144763\pi\)
−0.898355 + 0.439271i \(0.855237\pi\)
\(938\) 539.130i 0.574765i
\(939\) 1825.61i 1.94421i
\(940\) 0 0
\(941\) 270.097i 0.287032i 0.989648 + 0.143516i \(0.0458409\pi\)
−0.989648 + 0.143516i \(0.954159\pi\)
\(942\) 1927.95i 2.04666i
\(943\) 1014.57 1.07589
\(944\) 1448.28i 1.53420i
\(945\) 0 0
\(946\) 247.233i 0.261346i
\(947\) 1412.14i 1.49118i 0.666407 + 0.745588i \(0.267831\pi\)
−0.666407 + 0.745588i \(0.732169\pi\)
\(948\) 49.8753i 0.0526111i
\(949\) 418.096i 0.440564i
\(950\) 0 0
\(951\) −91.7535 −0.0964811
\(952\) −1273.64 −1.33786
\(953\) 432.598 0.453933 0.226966 0.973903i \(-0.427119\pi\)
0.226966 + 0.973903i \(0.427119\pi\)
\(954\) −265.443 −0.278242
\(955\) 0 0
\(956\) 175.592 0.183674
\(957\) 1812.56i 1.89400i
\(958\) −150.489 −0.157087
\(959\) −1040.06 −1.08452
\(960\) 0 0
\(961\) 700.964 0.729411
\(962\) −1734.71 −1.80323
\(963\) 313.209 0.325243
\(964\) 513.339i 0.532509i
\(965\) 0 0
\(966\) −1266.18 −1.31075
\(967\) 10.0222i 0.0103642i −0.999987 0.00518210i \(-0.998350\pi\)
0.999987 0.00518210i \(-0.00164952\pi\)
\(968\) −911.291 −0.941416
\(969\) 683.675 + 1182.29i 0.705547 + 1.22011i
\(970\) 0 0
\(971\) 339.417i 0.349554i 0.984608 + 0.174777i \(0.0559205\pi\)
−0.984608 + 0.174777i \(0.944080\pi\)
\(972\) −296.776 −0.305325
\(973\) 455.309i 0.467943i
\(974\) 1317.63 1.35280
\(975\) 0 0
\(976\) 352.754 0.361428
\(977\) 673.762 0.689623 0.344812 0.938672i \(-0.387943\pi\)
0.344812 + 0.938672i \(0.387943\pi\)
\(978\) 286.337i 0.292778i
\(979\) 2732.03i 2.79063i
\(980\) 0 0
\(981\) 346.363i 0.353071i
\(982\) 2026.86 2.06401
\(983\) −428.230 −0.435636 −0.217818 0.975989i \(-0.569894\pi\)
−0.217818 + 0.975989i \(0.569894\pi\)
\(984\) 1509.76i 1.53431i
\(985\) 0 0
\(986\) 1453.68 1.47432
\(987\) 1734.19 1.75703
\(988\) 375.481 217.127i 0.380042 0.219765i
\(989\) 84.6813 0.0856232
\(990\) 0 0
\(991\) 1310.89i 1.32280i 0.750033 + 0.661400i \(0.230037\pi\)
−0.750033 + 0.661400i \(0.769963\pi\)
\(992\) 389.888i 0.393032i
\(993\) 610.270i 0.614572i
\(994\) 196.272 0.197457
\(995\) 0 0
\(996\) 361.987i 0.363441i
\(997\) 690.240i 0.692316i 0.938176 + 0.346158i \(0.112514\pi\)
−0.938176 + 0.346158i \(0.887486\pi\)
\(998\) −1801.71 −1.80532
\(999\) 976.615 0.977592
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 475.3.d.c.474.20 24
5.2 odd 4 95.3.c.a.56.10 yes 12
5.3 odd 4 475.3.c.g.151.3 12
5.4 even 2 inner 475.3.d.c.474.5 24
15.2 even 4 855.3.e.a.721.3 12
19.18 odd 2 inner 475.3.d.c.474.6 24
20.7 even 4 1520.3.h.a.721.9 12
95.18 even 4 475.3.c.g.151.10 12
95.37 even 4 95.3.c.a.56.3 12
95.94 odd 2 inner 475.3.d.c.474.19 24
285.227 odd 4 855.3.e.a.721.10 12
380.227 odd 4 1520.3.h.a.721.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.3.c.a.56.3 12 95.37 even 4
95.3.c.a.56.10 yes 12 5.2 odd 4
475.3.c.g.151.3 12 5.3 odd 4
475.3.c.g.151.10 12 95.18 even 4
475.3.d.c.474.5 24 5.4 even 2 inner
475.3.d.c.474.6 24 19.18 odd 2 inner
475.3.d.c.474.19 24 95.94 odd 2 inner
475.3.d.c.474.20 24 1.1 even 1 trivial
855.3.e.a.721.3 12 15.2 even 4
855.3.e.a.721.10 12 285.227 odd 4
1520.3.h.a.721.4 12 380.227 odd 4
1520.3.h.a.721.9 12 20.7 even 4