Properties

Label 245.4.a.i.1.2
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.31662 q^{2} -5.00000 q^{3} +10.6332 q^{4} -5.00000 q^{5} -21.5831 q^{6} +11.3668 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+4.31662 q^{2} -5.00000 q^{3} +10.6332 q^{4} -5.00000 q^{5} -21.5831 q^{6} +11.3668 q^{8} -2.00000 q^{9} -21.5831 q^{10} +19.7335 q^{11} -53.1662 q^{12} -71.3325 q^{13} +25.0000 q^{15} -36.0000 q^{16} +31.3325 q^{17} -8.63325 q^{18} -136.332 q^{19} -53.1662 q^{20} +85.1821 q^{22} -100.865 q^{23} -56.8338 q^{24} +25.0000 q^{25} -307.916 q^{26} +145.000 q^{27} -288.198 q^{29} +107.916 q^{30} +208.997 q^{31} -246.332 q^{32} -98.6675 q^{33} +135.251 q^{34} -21.2665 q^{36} +309.931 q^{37} -588.496 q^{38} +356.662 q^{39} -56.8338 q^{40} +181.662 q^{41} -18.2005 q^{43} +209.831 q^{44} +10.0000 q^{45} -435.398 q^{46} +147.665 q^{47} +180.000 q^{48} +107.916 q^{50} -156.662 q^{51} -758.496 q^{52} -127.995 q^{53} +625.911 q^{54} -98.6675 q^{55} +681.662 q^{57} -1244.04 q^{58} -322.665 q^{59} +265.831 q^{60} -341.003 q^{61} +902.164 q^{62} -775.325 q^{64} +356.662 q^{65} -425.911 q^{66} -84.3960 q^{67} +333.166 q^{68} +504.327 q^{69} -315.736 q^{71} -22.7335 q^{72} +1093.32 q^{73} +1337.86 q^{74} -125.000 q^{75} -1449.66 q^{76} +1539.58 q^{78} +1233.19 q^{79} +180.000 q^{80} -671.000 q^{81} +784.169 q^{82} +643.325 q^{83} -156.662 q^{85} -78.5647 q^{86} +1440.99 q^{87} +224.306 q^{88} -1140.32 q^{89} +43.1662 q^{90} -1072.53 q^{92} -1044.99 q^{93} +637.414 q^{94} +681.662 q^{95} +1231.66 q^{96} -1411.99 q^{97} -39.4670 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9} - 10 q^{10} + 66 q^{11} - 40 q^{12} - 10 q^{13} + 50 q^{15} - 72 q^{16} - 70 q^{17} - 4 q^{18} - 140 q^{19} - 40 q^{20} - 22 q^{22} - 16 q^{23} - 180 q^{24} + 50 q^{25} - 450 q^{26} + 290 q^{27} - 258 q^{29} + 50 q^{30} + 20 q^{31} - 360 q^{32} - 330 q^{33} + 370 q^{34} - 16 q^{36} + 328 q^{37} - 580 q^{38} + 50 q^{39} - 180 q^{40} - 300 q^{41} - 116 q^{43} + 88 q^{44} + 20 q^{45} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 50 q^{50} + 350 q^{51} - 920 q^{52} + 540 q^{53} + 290 q^{54} - 330 q^{55} + 700 q^{57} - 1314 q^{58} - 380 q^{59} + 200 q^{60} - 1080 q^{61} + 1340 q^{62} - 224 q^{64} + 50 q^{65} + 110 q^{66} + 468 q^{67} + 600 q^{68} + 80 q^{69} - 1056 q^{71} - 72 q^{72} + 860 q^{73} + 1296 q^{74} - 250 q^{75} - 1440 q^{76} + 2250 q^{78} + 158 q^{79} + 360 q^{80} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 350 q^{85} + 148 q^{86} + 1290 q^{87} + 1364 q^{88} + 240 q^{89} + 20 q^{90} - 1296 q^{92} - 100 q^{93} + 910 q^{94} + 700 q^{95} + 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.31662 1.52616 0.763079 0.646306i \(-0.223687\pi\)
0.763079 + 0.646306i \(0.223687\pi\)
\(3\) −5.00000 −0.962250 −0.481125 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) 10.6332 1.32916
\(5\) −5.00000 −0.447214
\(6\) −21.5831 −1.46855
\(7\) 0 0
\(8\) 11.3668 0.502344
\(9\) −2.00000 −0.0740741
\(10\) −21.5831 −0.682518
\(11\) 19.7335 0.540898 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(12\) −53.1662 −1.27898
\(13\) −71.3325 −1.52185 −0.760926 0.648839i \(-0.775255\pi\)
−0.760926 + 0.648839i \(0.775255\pi\)
\(14\) 0 0
\(15\) 25.0000 0.430331
\(16\) −36.0000 −0.562500
\(17\) 31.3325 0.447014 0.223507 0.974702i \(-0.428249\pi\)
0.223507 + 0.974702i \(0.428249\pi\)
\(18\) −8.63325 −0.113049
\(19\) −136.332 −1.64615 −0.823074 0.567934i \(-0.807743\pi\)
−0.823074 + 0.567934i \(0.807743\pi\)
\(20\) −53.1662 −0.594417
\(21\) 0 0
\(22\) 85.1821 0.825495
\(23\) −100.865 −0.914431 −0.457215 0.889356i \(-0.651153\pi\)
−0.457215 + 0.889356i \(0.651153\pi\)
\(24\) −56.8338 −0.483381
\(25\) 25.0000 0.200000
\(26\) −307.916 −2.32259
\(27\) 145.000 1.03353
\(28\) 0 0
\(29\) −288.198 −1.84541 −0.922707 0.385501i \(-0.874029\pi\)
−0.922707 + 0.385501i \(0.874029\pi\)
\(30\) 107.916 0.656754
\(31\) 208.997 1.21087 0.605436 0.795894i \(-0.292999\pi\)
0.605436 + 0.795894i \(0.292999\pi\)
\(32\) −246.332 −1.36081
\(33\) −98.6675 −0.520479
\(34\) 135.251 0.682214
\(35\) 0 0
\(36\) −21.2665 −0.0984560
\(37\) 309.931 1.37709 0.688546 0.725192i \(-0.258249\pi\)
0.688546 + 0.725192i \(0.258249\pi\)
\(38\) −588.496 −2.51228
\(39\) 356.662 1.46440
\(40\) −56.8338 −0.224655
\(41\) 181.662 0.691973 0.345987 0.938239i \(-0.387544\pi\)
0.345987 + 0.938239i \(0.387544\pi\)
\(42\) 0 0
\(43\) −18.2005 −0.0645477 −0.0322738 0.999479i \(-0.510275\pi\)
−0.0322738 + 0.999479i \(0.510275\pi\)
\(44\) 209.831 0.718937
\(45\) 10.0000 0.0331269
\(46\) −435.398 −1.39557
\(47\) 147.665 0.458280 0.229140 0.973393i \(-0.426409\pi\)
0.229140 + 0.973393i \(0.426409\pi\)
\(48\) 180.000 0.541266
\(49\) 0 0
\(50\) 107.916 0.305231
\(51\) −156.662 −0.430140
\(52\) −758.496 −2.02278
\(53\) −127.995 −0.331726 −0.165863 0.986149i \(-0.553041\pi\)
−0.165863 + 0.986149i \(0.553041\pi\)
\(54\) 625.911 1.57733
\(55\) −98.6675 −0.241897
\(56\) 0 0
\(57\) 681.662 1.58401
\(58\) −1244.04 −2.81639
\(59\) −322.665 −0.711990 −0.355995 0.934488i \(-0.615858\pi\)
−0.355995 + 0.934488i \(0.615858\pi\)
\(60\) 265.831 0.571978
\(61\) −341.003 −0.715752 −0.357876 0.933769i \(-0.616499\pi\)
−0.357876 + 0.933769i \(0.616499\pi\)
\(62\) 902.164 1.84798
\(63\) 0 0
\(64\) −775.325 −1.51431
\(65\) 356.662 0.680593
\(66\) −425.911 −0.794333
\(67\) −84.3960 −0.153890 −0.0769449 0.997035i \(-0.524517\pi\)
−0.0769449 + 0.997035i \(0.524517\pi\)
\(68\) 333.166 0.594152
\(69\) 504.327 0.879911
\(70\) 0 0
\(71\) −315.736 −0.527760 −0.263880 0.964555i \(-0.585002\pi\)
−0.263880 + 0.964555i \(0.585002\pi\)
\(72\) −22.7335 −0.0372107
\(73\) 1093.32 1.75293 0.876466 0.481464i \(-0.159895\pi\)
0.876466 + 0.481464i \(0.159895\pi\)
\(74\) 1337.86 2.10166
\(75\) −125.000 −0.192450
\(76\) −1449.66 −2.18799
\(77\) 0 0
\(78\) 1539.58 2.23491
\(79\) 1233.19 1.75626 0.878128 0.478426i \(-0.158793\pi\)
0.878128 + 0.478426i \(0.158793\pi\)
\(80\) 180.000 0.251558
\(81\) −671.000 −0.920439
\(82\) 784.169 1.05606
\(83\) 643.325 0.850772 0.425386 0.905012i \(-0.360138\pi\)
0.425386 + 0.905012i \(0.360138\pi\)
\(84\) 0 0
\(85\) −156.662 −0.199911
\(86\) −78.5647 −0.0985099
\(87\) 1440.99 1.77575
\(88\) 224.306 0.271717
\(89\) −1140.32 −1.35813 −0.679064 0.734079i \(-0.737614\pi\)
−0.679064 + 0.734079i \(0.737614\pi\)
\(90\) 43.1662 0.0505569
\(91\) 0 0
\(92\) −1072.53 −1.21542
\(93\) −1044.99 −1.16516
\(94\) 637.414 0.699407
\(95\) 681.662 0.736180
\(96\) 1231.66 1.30944
\(97\) −1411.99 −1.47800 −0.739001 0.673705i \(-0.764702\pi\)
−0.739001 + 0.673705i \(0.764702\pi\)
\(98\) 0 0
\(99\) −39.4670 −0.0400665
\(100\) 265.831 0.265831
\(101\) 545.673 0.537589 0.268794 0.963198i \(-0.413375\pi\)
0.268794 + 0.963198i \(0.413375\pi\)
\(102\) −676.253 −0.656461
\(103\) −780.990 −0.747119 −0.373559 0.927606i \(-0.621863\pi\)
−0.373559 + 0.927606i \(0.621863\pi\)
\(104\) −810.819 −0.764493
\(105\) 0 0
\(106\) −552.506 −0.506266
\(107\) −620.660 −0.560761 −0.280381 0.959889i \(-0.590461\pi\)
−0.280381 + 0.959889i \(0.590461\pi\)
\(108\) 1541.82 1.37372
\(109\) 4.18794 0.00368011 0.00184005 0.999998i \(-0.499414\pi\)
0.00184005 + 0.999998i \(0.499414\pi\)
\(110\) −425.911 −0.369173
\(111\) −1549.66 −1.32511
\(112\) 0 0
\(113\) 1413.53 1.17676 0.588379 0.808585i \(-0.299766\pi\)
0.588379 + 0.808585i \(0.299766\pi\)
\(114\) 2942.48 2.41744
\(115\) 504.327 0.408946
\(116\) −3064.48 −2.45284
\(117\) 142.665 0.112730
\(118\) −1392.82 −1.08661
\(119\) 0 0
\(120\) 284.169 0.216175
\(121\) −941.589 −0.707430
\(122\) −1471.98 −1.09235
\(123\) −908.312 −0.665852
\(124\) 2222.32 1.60944
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −2046.26 −1.42974 −0.714868 0.699259i \(-0.753513\pi\)
−0.714868 + 0.699259i \(0.753513\pi\)
\(128\) −1376.13 −0.950262
\(129\) 91.0025 0.0621110
\(130\) 1539.58 1.03869
\(131\) −1140.32 −0.760534 −0.380267 0.924877i \(-0.624168\pi\)
−0.380267 + 0.924877i \(0.624168\pi\)
\(132\) −1049.16 −0.691798
\(133\) 0 0
\(134\) −364.306 −0.234860
\(135\) −725.000 −0.462208
\(136\) 356.149 0.224555
\(137\) −490.343 −0.305787 −0.152893 0.988243i \(-0.548859\pi\)
−0.152893 + 0.988243i \(0.548859\pi\)
\(138\) 2176.99 1.34288
\(139\) −2800.00 −1.70858 −0.854291 0.519795i \(-0.826008\pi\)
−0.854291 + 0.519795i \(0.826008\pi\)
\(140\) 0 0
\(141\) −738.325 −0.440980
\(142\) −1362.91 −0.805445
\(143\) −1407.64 −0.823166
\(144\) 72.0000 0.0416667
\(145\) 1440.99 0.825294
\(146\) 4719.47 2.67525
\(147\) 0 0
\(148\) 3295.58 1.83037
\(149\) 1166.12 0.641154 0.320577 0.947222i \(-0.396123\pi\)
0.320577 + 0.947222i \(0.396123\pi\)
\(150\) −539.578 −0.293709
\(151\) 959.581 0.517150 0.258575 0.965991i \(-0.416747\pi\)
0.258575 + 0.965991i \(0.416747\pi\)
\(152\) −1549.66 −0.826933
\(153\) −62.6650 −0.0331122
\(154\) 0 0
\(155\) −1044.99 −0.541519
\(156\) 3792.48 1.94642
\(157\) 1020.66 0.518838 0.259419 0.965765i \(-0.416469\pi\)
0.259419 + 0.965765i \(0.416469\pi\)
\(158\) 5323.20 2.68032
\(159\) 639.975 0.319203
\(160\) 1231.66 0.608572
\(161\) 0 0
\(162\) −2896.46 −1.40473
\(163\) −1566.02 −0.752514 −0.376257 0.926515i \(-0.622789\pi\)
−0.376257 + 0.926515i \(0.622789\pi\)
\(164\) 1931.66 0.919741
\(165\) 493.338 0.232765
\(166\) 2776.99 1.29841
\(167\) 1130.30 0.523746 0.261873 0.965102i \(-0.415660\pi\)
0.261873 + 0.965102i \(0.415660\pi\)
\(168\) 0 0
\(169\) 2891.32 1.31603
\(170\) −676.253 −0.305096
\(171\) 272.665 0.121937
\(172\) −193.530 −0.0857940
\(173\) −2543.34 −1.11772 −0.558862 0.829260i \(-0.688762\pi\)
−0.558862 + 0.829260i \(0.688762\pi\)
\(174\) 6220.21 2.71008
\(175\) 0 0
\(176\) −710.406 −0.304255
\(177\) 1613.32 0.685113
\(178\) −4922.32 −2.07272
\(179\) −1210.65 −0.505521 −0.252760 0.967529i \(-0.581338\pi\)
−0.252760 + 0.967529i \(0.581338\pi\)
\(180\) 106.332 0.0440309
\(181\) 3031.32 1.24484 0.622421 0.782683i \(-0.286149\pi\)
0.622421 + 0.782683i \(0.286149\pi\)
\(182\) 0 0
\(183\) 1705.01 0.688733
\(184\) −1146.51 −0.459359
\(185\) −1549.66 −0.615854
\(186\) −4510.82 −1.77822
\(187\) 618.300 0.241789
\(188\) 1570.16 0.609125
\(189\) 0 0
\(190\) 2942.48 1.12353
\(191\) −2168.64 −0.821555 −0.410778 0.911736i \(-0.634743\pi\)
−0.410778 + 0.911736i \(0.634743\pi\)
\(192\) 3876.62 1.45714
\(193\) 1490.48 0.555892 0.277946 0.960597i \(-0.410346\pi\)
0.277946 + 0.960597i \(0.410346\pi\)
\(194\) −6095.04 −2.25566
\(195\) −1783.31 −0.654901
\(196\) 0 0
\(197\) 3380.57 1.22262 0.611308 0.791393i \(-0.290644\pi\)
0.611308 + 0.791393i \(0.290644\pi\)
\(198\) −170.364 −0.0611478
\(199\) −4595.33 −1.63696 −0.818478 0.574538i \(-0.805182\pi\)
−0.818478 + 0.574538i \(0.805182\pi\)
\(200\) 284.169 0.100469
\(201\) 421.980 0.148080
\(202\) 2355.46 0.820445
\(203\) 0 0
\(204\) −1665.83 −0.571723
\(205\) −908.312 −0.309460
\(206\) −3371.24 −1.14022
\(207\) 201.731 0.0677356
\(208\) 2567.97 0.856042
\(209\) −2690.32 −0.890398
\(210\) 0 0
\(211\) −4988.66 −1.62765 −0.813824 0.581112i \(-0.802618\pi\)
−0.813824 + 0.581112i \(0.802618\pi\)
\(212\) −1361.00 −0.440915
\(213\) 1578.68 0.507837
\(214\) −2679.16 −0.855810
\(215\) 91.0025 0.0288666
\(216\) 1648.18 0.519187
\(217\) 0 0
\(218\) 18.0778 0.00561643
\(219\) −5466.62 −1.68676
\(220\) −1049.16 −0.321519
\(221\) −2235.03 −0.680290
\(222\) −6689.29 −2.02232
\(223\) −3792.97 −1.13900 −0.569498 0.821993i \(-0.692862\pi\)
−0.569498 + 0.821993i \(0.692862\pi\)
\(224\) 0 0
\(225\) −50.0000 −0.0148148
\(226\) 6101.68 1.79592
\(227\) −3910.96 −1.14352 −0.571762 0.820420i \(-0.693740\pi\)
−0.571762 + 0.820420i \(0.693740\pi\)
\(228\) 7248.29 2.10539
\(229\) −354.327 −0.102247 −0.0511236 0.998692i \(-0.516280\pi\)
−0.0511236 + 0.998692i \(0.516280\pi\)
\(230\) 2176.99 0.624116
\(231\) 0 0
\(232\) −3275.87 −0.927033
\(233\) 6492.48 1.82548 0.912739 0.408543i \(-0.133963\pi\)
0.912739 + 0.408543i \(0.133963\pi\)
\(234\) 615.831 0.172043
\(235\) −738.325 −0.204949
\(236\) −3430.98 −0.946346
\(237\) −6165.93 −1.68996
\(238\) 0 0
\(239\) −342.688 −0.0927474 −0.0463737 0.998924i \(-0.514766\pi\)
−0.0463737 + 0.998924i \(0.514766\pi\)
\(240\) −900.000 −0.242061
\(241\) −2313.67 −0.618408 −0.309204 0.950996i \(-0.600063\pi\)
−0.309204 + 0.950996i \(0.600063\pi\)
\(242\) −4064.49 −1.07965
\(243\) −560.000 −0.147835
\(244\) −3625.96 −0.951347
\(245\) 0 0
\(246\) −3920.84 −1.01619
\(247\) 9724.94 2.50519
\(248\) 2375.62 0.608275
\(249\) −3216.62 −0.818656
\(250\) −539.578 −0.136504
\(251\) −3989.29 −1.00319 −0.501597 0.865101i \(-0.667254\pi\)
−0.501597 + 0.865101i \(0.667254\pi\)
\(252\) 0 0
\(253\) −1990.43 −0.494614
\(254\) −8832.95 −2.18200
\(255\) 783.312 0.192364
\(256\) 262.376 0.0640566
\(257\) −2291.32 −0.556142 −0.278071 0.960560i \(-0.589695\pi\)
−0.278071 + 0.960560i \(0.589695\pi\)
\(258\) 392.824 0.0947912
\(259\) 0 0
\(260\) 3792.48 0.904614
\(261\) 576.396 0.136697
\(262\) −4922.32 −1.16070
\(263\) 6360.47 1.49127 0.745634 0.666356i \(-0.232147\pi\)
0.745634 + 0.666356i \(0.232147\pi\)
\(264\) −1121.53 −0.261460
\(265\) 639.975 0.148352
\(266\) 0 0
\(267\) 5701.59 1.30686
\(268\) −897.404 −0.204543
\(269\) 991.345 0.224697 0.112348 0.993669i \(-0.464163\pi\)
0.112348 + 0.993669i \(0.464163\pi\)
\(270\) −3129.55 −0.705402
\(271\) −730.977 −0.163851 −0.0819257 0.996638i \(-0.526107\pi\)
−0.0819257 + 0.996638i \(0.526107\pi\)
\(272\) −1127.97 −0.251446
\(273\) 0 0
\(274\) −2116.62 −0.466679
\(275\) 493.338 0.108180
\(276\) 5362.64 1.16954
\(277\) −3538.63 −0.767566 −0.383783 0.923423i \(-0.625379\pi\)
−0.383783 + 0.923423i \(0.625379\pi\)
\(278\) −12086.5 −2.60756
\(279\) −417.995 −0.0896943
\(280\) 0 0
\(281\) −4663.20 −0.989975 −0.494988 0.868900i \(-0.664827\pi\)
−0.494988 + 0.868900i \(0.664827\pi\)
\(282\) −3187.07 −0.673005
\(283\) 2104.95 0.442142 0.221071 0.975258i \(-0.429045\pi\)
0.221071 + 0.975258i \(0.429045\pi\)
\(284\) −3357.30 −0.701476
\(285\) −3408.31 −0.708389
\(286\) −6076.25 −1.25628
\(287\) 0 0
\(288\) 492.665 0.100801
\(289\) −3931.27 −0.800178
\(290\) 6220.21 1.25953
\(291\) 7059.96 1.42221
\(292\) 11625.6 2.32992
\(293\) 6594.66 1.31489 0.657447 0.753501i \(-0.271636\pi\)
0.657447 + 0.753501i \(0.271636\pi\)
\(294\) 0 0
\(295\) 1613.32 0.318412
\(296\) 3522.91 0.691774
\(297\) 2861.36 0.559033
\(298\) 5033.69 0.978503
\(299\) 7194.99 1.39163
\(300\) −1329.16 −0.255796
\(301\) 0 0
\(302\) 4142.15 0.789252
\(303\) −2728.36 −0.517295
\(304\) 4907.97 0.925958
\(305\) 1705.01 0.320094
\(306\) −270.501 −0.0505344
\(307\) 2672.97 0.496920 0.248460 0.968642i \(-0.420076\pi\)
0.248460 + 0.968642i \(0.420076\pi\)
\(308\) 0 0
\(309\) 3904.95 0.718915
\(310\) −4510.82 −0.826443
\(311\) 855.698 0.156020 0.0780099 0.996953i \(-0.475143\pi\)
0.0780099 + 0.996953i \(0.475143\pi\)
\(312\) 4054.09 0.735634
\(313\) 3349.99 0.604960 0.302480 0.953156i \(-0.402185\pi\)
0.302480 + 0.953156i \(0.402185\pi\)
\(314\) 4405.81 0.791828
\(315\) 0 0
\(316\) 13112.8 2.33434
\(317\) 7633.05 1.35241 0.676206 0.736712i \(-0.263623\pi\)
0.676206 + 0.736712i \(0.263623\pi\)
\(318\) 2762.53 0.487154
\(319\) −5687.16 −0.998180
\(320\) 3876.62 0.677218
\(321\) 3103.30 0.539593
\(322\) 0 0
\(323\) −4271.64 −0.735852
\(324\) −7134.91 −1.22341
\(325\) −1783.31 −0.304370
\(326\) −6759.90 −1.14845
\(327\) −20.9397 −0.00354119
\(328\) 2064.91 0.347609
\(329\) 0 0
\(330\) 2129.55 0.355236
\(331\) −3321.70 −0.551593 −0.275796 0.961216i \(-0.588942\pi\)
−0.275796 + 0.961216i \(0.588942\pi\)
\(332\) 6840.63 1.13081
\(333\) −619.863 −0.102007
\(334\) 4879.10 0.799319
\(335\) 421.980 0.0688216
\(336\) 0 0
\(337\) −2233.98 −0.361107 −0.180553 0.983565i \(-0.557789\pi\)
−0.180553 + 0.983565i \(0.557789\pi\)
\(338\) 12480.8 2.00847
\(339\) −7067.65 −1.13234
\(340\) −1665.83 −0.265713
\(341\) 4124.25 0.654958
\(342\) 1176.99 0.186095
\(343\) 0 0
\(344\) −206.881 −0.0324252
\(345\) −2521.64 −0.393508
\(346\) −10978.6 −1.70582
\(347\) 2528.61 0.391190 0.195595 0.980685i \(-0.437336\pi\)
0.195595 + 0.980685i \(0.437336\pi\)
\(348\) 15322.4 2.36025
\(349\) −1291.00 −0.198011 −0.0990054 0.995087i \(-0.531566\pi\)
−0.0990054 + 0.995087i \(0.531566\pi\)
\(350\) 0 0
\(351\) −10343.2 −1.57288
\(352\) −4861.00 −0.736058
\(353\) 7768.64 1.17134 0.585670 0.810550i \(-0.300831\pi\)
0.585670 + 0.810550i \(0.300831\pi\)
\(354\) 6964.12 1.04559
\(355\) 1578.68 0.236022
\(356\) −12125.3 −1.80516
\(357\) 0 0
\(358\) −5225.92 −0.771504
\(359\) −2284.14 −0.335800 −0.167900 0.985804i \(-0.553699\pi\)
−0.167900 + 0.985804i \(0.553699\pi\)
\(360\) 113.668 0.0166411
\(361\) 11727.5 1.70980
\(362\) 13085.1 1.89982
\(363\) 4707.94 0.680725
\(364\) 0 0
\(365\) −5466.62 −0.783935
\(366\) 7359.90 1.05112
\(367\) −10707.0 −1.52289 −0.761446 0.648229i \(-0.775510\pi\)
−0.761446 + 0.648229i \(0.775510\pi\)
\(368\) 3631.16 0.514367
\(369\) −363.325 −0.0512573
\(370\) −6689.29 −0.939891
\(371\) 0 0
\(372\) −11111.6 −1.54868
\(373\) −830.429 −0.115276 −0.0576381 0.998338i \(-0.518357\pi\)
−0.0576381 + 0.998338i \(0.518357\pi\)
\(374\) 2668.97 0.369008
\(375\) 625.000 0.0860663
\(376\) 1678.47 0.230214
\(377\) 20557.9 2.80845
\(378\) 0 0
\(379\) 5253.17 0.711972 0.355986 0.934491i \(-0.384145\pi\)
0.355986 + 0.934491i \(0.384145\pi\)
\(380\) 7248.29 0.978498
\(381\) 10231.3 1.37576
\(382\) −9361.19 −1.25382
\(383\) 11243.9 1.50010 0.750048 0.661383i \(-0.230030\pi\)
0.750048 + 0.661383i \(0.230030\pi\)
\(384\) 6880.63 0.914390
\(385\) 0 0
\(386\) 6433.84 0.848378
\(387\) 36.4010 0.00478131
\(388\) −15014.1 −1.96449
\(389\) −8506.85 −1.10878 −0.554389 0.832258i \(-0.687048\pi\)
−0.554389 + 0.832258i \(0.687048\pi\)
\(390\) −7697.89 −0.999482
\(391\) −3160.37 −0.408764
\(392\) 0 0
\(393\) 5701.59 0.731824
\(394\) 14592.6 1.86590
\(395\) −6165.93 −0.785421
\(396\) −419.662 −0.0532546
\(397\) −3123.24 −0.394838 −0.197419 0.980319i \(-0.563256\pi\)
−0.197419 + 0.980319i \(0.563256\pi\)
\(398\) −19836.3 −2.49825
\(399\) 0 0
\(400\) −900.000 −0.112500
\(401\) −11255.3 −1.40166 −0.700828 0.713330i \(-0.747186\pi\)
−0.700828 + 0.713330i \(0.747186\pi\)
\(402\) 1821.53 0.225994
\(403\) −14908.3 −1.84277
\(404\) 5802.27 0.714539
\(405\) 3355.00 0.411633
\(406\) 0 0
\(407\) 6116.03 0.744866
\(408\) −1780.74 −0.216078
\(409\) 7919.92 0.957494 0.478747 0.877953i \(-0.341091\pi\)
0.478747 + 0.877953i \(0.341091\pi\)
\(410\) −3920.84 −0.472285
\(411\) 2451.71 0.294243
\(412\) −8304.46 −0.993037
\(413\) 0 0
\(414\) 870.797 0.103375
\(415\) −3216.62 −0.380477
\(416\) 17571.5 2.07095
\(417\) 14000.0 1.64408
\(418\) −11613.1 −1.35889
\(419\) 5257.28 0.612972 0.306486 0.951875i \(-0.400847\pi\)
0.306486 + 0.951875i \(0.400847\pi\)
\(420\) 0 0
\(421\) 1457.36 0.168711 0.0843556 0.996436i \(-0.473117\pi\)
0.0843556 + 0.996436i \(0.473117\pi\)
\(422\) −21534.2 −2.48405
\(423\) −295.330 −0.0339467
\(424\) −1454.89 −0.166640
\(425\) 783.312 0.0894029
\(426\) 6814.57 0.775040
\(427\) 0 0
\(428\) −6599.63 −0.745339
\(429\) 7038.20 0.792092
\(430\) 392.824 0.0440550
\(431\) −15291.2 −1.70893 −0.854467 0.519506i \(-0.826116\pi\)
−0.854467 + 0.519506i \(0.826116\pi\)
\(432\) −5220.00 −0.581360
\(433\) −187.260 −0.0207832 −0.0103916 0.999946i \(-0.503308\pi\)
−0.0103916 + 0.999946i \(0.503308\pi\)
\(434\) 0 0
\(435\) −7204.95 −0.794140
\(436\) 44.5314 0.00489144
\(437\) 13751.2 1.50529
\(438\) −23597.4 −2.57426
\(439\) −3587.92 −0.390073 −0.195037 0.980796i \(-0.562483\pi\)
−0.195037 + 0.980796i \(0.562483\pi\)
\(440\) −1121.53 −0.121515
\(441\) 0 0
\(442\) −9647.76 −1.03823
\(443\) 4915.65 0.527200 0.263600 0.964632i \(-0.415090\pi\)
0.263600 + 0.964632i \(0.415090\pi\)
\(444\) −16477.9 −1.76128
\(445\) 5701.59 0.607373
\(446\) −16372.8 −1.73829
\(447\) −5830.58 −0.616951
\(448\) 0 0
\(449\) 7091.12 0.745324 0.372662 0.927967i \(-0.378445\pi\)
0.372662 + 0.927967i \(0.378445\pi\)
\(450\) −215.831 −0.0226097
\(451\) 3584.84 0.374287
\(452\) 15030.4 1.56410
\(453\) −4797.91 −0.497628
\(454\) −16882.2 −1.74520
\(455\) 0 0
\(456\) 7748.29 0.795717
\(457\) 5051.81 0.517098 0.258549 0.965998i \(-0.416756\pi\)
0.258549 + 0.965998i \(0.416756\pi\)
\(458\) −1529.50 −0.156045
\(459\) 4543.21 0.462002
\(460\) 5362.64 0.543553
\(461\) 16681.3 1.68531 0.842653 0.538456i \(-0.180992\pi\)
0.842653 + 0.538456i \(0.180992\pi\)
\(462\) 0 0
\(463\) 15569.6 1.56280 0.781402 0.624027i \(-0.214505\pi\)
0.781402 + 0.624027i \(0.214505\pi\)
\(464\) 10375.1 1.03805
\(465\) 5224.94 0.521077
\(466\) 28025.6 2.78597
\(467\) −3328.35 −0.329802 −0.164901 0.986310i \(-0.552731\pi\)
−0.164901 + 0.986310i \(0.552731\pi\)
\(468\) 1516.99 0.149835
\(469\) 0 0
\(470\) −3187.07 −0.312784
\(471\) −5103.30 −0.499252
\(472\) −3667.65 −0.357664
\(473\) −359.160 −0.0349137
\(474\) −26616.0 −2.57914
\(475\) −3408.31 −0.329230
\(476\) 0 0
\(477\) 255.990 0.0245723
\(478\) −1479.25 −0.141547
\(479\) 14607.5 1.39339 0.696695 0.717367i \(-0.254653\pi\)
0.696695 + 0.717367i \(0.254653\pi\)
\(480\) −6158.31 −0.585598
\(481\) −22108.2 −2.09573
\(482\) −9987.23 −0.943789
\(483\) 0 0
\(484\) −10012.2 −0.940285
\(485\) 7059.96 0.660982
\(486\) −2417.31 −0.225620
\(487\) −1879.49 −0.174882 −0.0874412 0.996170i \(-0.527869\pi\)
−0.0874412 + 0.996170i \(0.527869\pi\)
\(488\) −3876.09 −0.359554
\(489\) 7830.08 0.724107
\(490\) 0 0
\(491\) 3221.13 0.296064 0.148032 0.988983i \(-0.452706\pi\)
0.148032 + 0.988983i \(0.452706\pi\)
\(492\) −9658.31 −0.885021
\(493\) −9029.96 −0.824927
\(494\) 41978.9 3.82332
\(495\) 197.335 0.0179183
\(496\) −7523.91 −0.681116
\(497\) 0 0
\(498\) −13885.0 −1.24940
\(499\) 9713.81 0.871443 0.435721 0.900082i \(-0.356493\pi\)
0.435721 + 0.900082i \(0.356493\pi\)
\(500\) −1329.16 −0.118883
\(501\) −5651.52 −0.503975
\(502\) −17220.3 −1.53103
\(503\) −2078.32 −0.184230 −0.0921152 0.995748i \(-0.529363\pi\)
−0.0921152 + 0.995748i \(0.529363\pi\)
\(504\) 0 0
\(505\) −2728.36 −0.240417
\(506\) −8591.94 −0.754858
\(507\) −14456.6 −1.26635
\(508\) −21758.4 −1.90034
\(509\) 18974.5 1.65232 0.826158 0.563439i \(-0.190522\pi\)
0.826158 + 0.563439i \(0.190522\pi\)
\(510\) 3381.27 0.293578
\(511\) 0 0
\(512\) 12141.6 1.04802
\(513\) −19768.2 −1.70134
\(514\) −9890.77 −0.848761
\(515\) 3904.95 0.334122
\(516\) 967.652 0.0825553
\(517\) 2913.95 0.247883
\(518\) 0 0
\(519\) 12716.7 1.07553
\(520\) 4054.09 0.341892
\(521\) −17523.6 −1.47355 −0.736777 0.676136i \(-0.763653\pi\)
−0.736777 + 0.676136i \(0.763653\pi\)
\(522\) 2488.09 0.208622
\(523\) −15218.6 −1.27239 −0.636197 0.771527i \(-0.719493\pi\)
−0.636197 + 0.771527i \(0.719493\pi\)
\(524\) −12125.3 −1.01087
\(525\) 0 0
\(526\) 27455.8 2.27591
\(527\) 6548.41 0.541278
\(528\) 3552.03 0.292769
\(529\) −1993.15 −0.163816
\(530\) 2762.53 0.226409
\(531\) 645.330 0.0527400
\(532\) 0 0
\(533\) −12958.4 −1.05308
\(534\) 24611.6 1.99447
\(535\) 3103.30 0.250780
\(536\) −959.308 −0.0773056
\(537\) 6053.25 0.486438
\(538\) 4279.26 0.342922
\(539\) 0 0
\(540\) −7709.11 −0.614346
\(541\) −15559.3 −1.23650 −0.618249 0.785983i \(-0.712158\pi\)
−0.618249 + 0.785983i \(0.712158\pi\)
\(542\) −3155.36 −0.250063
\(543\) −15156.6 −1.19785
\(544\) −7718.21 −0.608301
\(545\) −20.9397 −0.00164579
\(546\) 0 0
\(547\) −8690.70 −0.679319 −0.339660 0.940548i \(-0.610312\pi\)
−0.339660 + 0.940548i \(0.610312\pi\)
\(548\) −5213.93 −0.406438
\(549\) 682.005 0.0530187
\(550\) 2129.55 0.165099
\(551\) 39290.8 3.03783
\(552\) 5732.56 0.442018
\(553\) 0 0
\(554\) −15274.9 −1.17143
\(555\) 7748.29 0.592606
\(556\) −29773.1 −2.27097
\(557\) −7376.26 −0.561117 −0.280559 0.959837i \(-0.590520\pi\)
−0.280559 + 0.959837i \(0.590520\pi\)
\(558\) −1804.33 −0.136888
\(559\) 1298.29 0.0982320
\(560\) 0 0
\(561\) −3091.50 −0.232662
\(562\) −20129.3 −1.51086
\(563\) −12875.4 −0.963825 −0.481913 0.876219i \(-0.660058\pi\)
−0.481913 + 0.876219i \(0.660058\pi\)
\(564\) −7850.79 −0.586131
\(565\) −7067.65 −0.526263
\(566\) 9086.28 0.674779
\(567\) 0 0
\(568\) −3588.89 −0.265117
\(569\) −12064.6 −0.888882 −0.444441 0.895808i \(-0.646598\pi\)
−0.444441 + 0.895808i \(0.646598\pi\)
\(570\) −14712.4 −1.08111
\(571\) 23745.6 1.74032 0.870158 0.492772i \(-0.164016\pi\)
0.870158 + 0.492772i \(0.164016\pi\)
\(572\) −14967.8 −1.09412
\(573\) 10843.2 0.790542
\(574\) 0 0
\(575\) −2521.64 −0.182886
\(576\) 1550.65 0.112171
\(577\) 9846.08 0.710394 0.355197 0.934791i \(-0.384414\pi\)
0.355197 + 0.934791i \(0.384414\pi\)
\(578\) −16969.8 −1.22120
\(579\) −7452.40 −0.534907
\(580\) 15322.4 1.09695
\(581\) 0 0
\(582\) 30475.2 2.17051
\(583\) −2525.79 −0.179430
\(584\) 12427.6 0.880575
\(585\) −713.325 −0.0504143
\(586\) 28466.7 2.00674
\(587\) −10074.7 −0.708392 −0.354196 0.935171i \(-0.615245\pi\)
−0.354196 + 0.935171i \(0.615245\pi\)
\(588\) 0 0
\(589\) −28493.1 −1.99328
\(590\) 6964.12 0.485946
\(591\) −16902.8 −1.17646
\(592\) −11157.5 −0.774615
\(593\) 7387.25 0.511565 0.255782 0.966734i \(-0.417667\pi\)
0.255782 + 0.966734i \(0.417667\pi\)
\(594\) 12351.4 0.853172
\(595\) 0 0
\(596\) 12399.6 0.852194
\(597\) 22976.6 1.57516
\(598\) 31058.1 2.12384
\(599\) 1252.73 0.0854510 0.0427255 0.999087i \(-0.486396\pi\)
0.0427255 + 0.999087i \(0.486396\pi\)
\(600\) −1420.84 −0.0966762
\(601\) 1800.81 0.122224 0.0611120 0.998131i \(-0.480535\pi\)
0.0611120 + 0.998131i \(0.480535\pi\)
\(602\) 0 0
\(603\) 168.792 0.0113992
\(604\) 10203.5 0.687373
\(605\) 4707.94 0.316372
\(606\) −11777.3 −0.789473
\(607\) −2497.06 −0.166973 −0.0834863 0.996509i \(-0.526605\pi\)
−0.0834863 + 0.996509i \(0.526605\pi\)
\(608\) 33583.1 2.24009
\(609\) 0 0
\(610\) 7359.90 0.488514
\(611\) −10533.3 −0.697434
\(612\) −666.332 −0.0440113
\(613\) −19750.8 −1.30135 −0.650674 0.759357i \(-0.725513\pi\)
−0.650674 + 0.759357i \(0.725513\pi\)
\(614\) 11538.2 0.758378
\(615\) 4541.56 0.297778
\(616\) 0 0
\(617\) 16797.4 1.09601 0.548004 0.836476i \(-0.315388\pi\)
0.548004 + 0.836476i \(0.315388\pi\)
\(618\) 16856.2 1.09718
\(619\) −26547.4 −1.72379 −0.861897 0.507084i \(-0.830723\pi\)
−0.861897 + 0.507084i \(0.830723\pi\)
\(620\) −11111.6 −0.719763
\(621\) −14625.5 −0.945090
\(622\) 3693.73 0.238111
\(623\) 0 0
\(624\) −12839.8 −0.823727
\(625\) 625.000 0.0400000
\(626\) 14460.6 0.923264
\(627\) 13451.6 0.856786
\(628\) 10852.9 0.689616
\(629\) 9710.93 0.615580
\(630\) 0 0
\(631\) 5394.86 0.340358 0.170179 0.985413i \(-0.445565\pi\)
0.170179 + 0.985413i \(0.445565\pi\)
\(632\) 14017.3 0.882245
\(633\) 24943.3 1.56620
\(634\) 32949.0 2.06399
\(635\) 10231.3 0.639398
\(636\) 6805.01 0.424271
\(637\) 0 0
\(638\) −24549.3 −1.52338
\(639\) 631.472 0.0390933
\(640\) 6880.63 0.424970
\(641\) 2452.41 0.151114 0.0755572 0.997141i \(-0.475926\pi\)
0.0755572 + 0.997141i \(0.475926\pi\)
\(642\) 13395.8 0.823504
\(643\) −7074.97 −0.433919 −0.216959 0.976181i \(-0.569614\pi\)
−0.216959 + 0.976181i \(0.569614\pi\)
\(644\) 0 0
\(645\) −455.013 −0.0277769
\(646\) −18439.1 −1.12303
\(647\) −3341.37 −0.203034 −0.101517 0.994834i \(-0.532370\pi\)
−0.101517 + 0.994834i \(0.532370\pi\)
\(648\) −7627.09 −0.462377
\(649\) −6367.31 −0.385114
\(650\) −7697.89 −0.464517
\(651\) 0 0
\(652\) −16651.8 −1.00021
\(653\) −23061.6 −1.38204 −0.691019 0.722837i \(-0.742838\pi\)
−0.691019 + 0.722837i \(0.742838\pi\)
\(654\) −90.3888 −0.00540441
\(655\) 5701.59 0.340121
\(656\) −6539.85 −0.389235
\(657\) −2186.65 −0.129847
\(658\) 0 0
\(659\) 1742.64 0.103010 0.0515049 0.998673i \(-0.483598\pi\)
0.0515049 + 0.998673i \(0.483598\pi\)
\(660\) 5245.78 0.309381
\(661\) 12576.5 0.740046 0.370023 0.929023i \(-0.379350\pi\)
0.370023 + 0.929023i \(0.379350\pi\)
\(662\) −14338.5 −0.841817
\(663\) 11175.1 0.654609
\(664\) 7312.51 0.427380
\(665\) 0 0
\(666\) −2675.72 −0.155679
\(667\) 29069.2 1.68750
\(668\) 12018.8 0.696141
\(669\) 18964.8 1.09600
\(670\) 1821.53 0.105033
\(671\) −6729.17 −0.387149
\(672\) 0 0
\(673\) −10680.8 −0.611760 −0.305880 0.952070i \(-0.598951\pi\)
−0.305880 + 0.952070i \(0.598951\pi\)
\(674\) −9643.27 −0.551105
\(675\) 3625.00 0.206706
\(676\) 30744.2 1.74921
\(677\) 29559.1 1.67806 0.839032 0.544082i \(-0.183122\pi\)
0.839032 + 0.544082i \(0.183122\pi\)
\(678\) −30508.4 −1.72812
\(679\) 0 0
\(680\) −1780.74 −0.100424
\(681\) 19554.8 1.10036
\(682\) 17802.8 0.999569
\(683\) 10250.4 0.574263 0.287132 0.957891i \(-0.407298\pi\)
0.287132 + 0.957891i \(0.407298\pi\)
\(684\) 2899.31 0.162073
\(685\) 2451.71 0.136752
\(686\) 0 0
\(687\) 1771.64 0.0983875
\(688\) 655.218 0.0363081
\(689\) 9130.20 0.504837
\(690\) −10885.0 −0.600556
\(691\) 8874.04 0.488544 0.244272 0.969707i \(-0.421451\pi\)
0.244272 + 0.969707i \(0.421451\pi\)
\(692\) −27043.9 −1.48563
\(693\) 0 0
\(694\) 10915.1 0.597018
\(695\) 14000.0 0.764101
\(696\) 16379.4 0.892038
\(697\) 5691.94 0.309322
\(698\) −5572.77 −0.302196
\(699\) −32462.4 −1.75657
\(700\) 0 0
\(701\) −22086.2 −1.18999 −0.594996 0.803729i \(-0.702846\pi\)
−0.594996 + 0.803729i \(0.702846\pi\)
\(702\) −44647.8 −2.40046
\(703\) −42253.7 −2.26690
\(704\) −15299.9 −0.819085
\(705\) 3691.62 0.197212
\(706\) 33534.3 1.78765
\(707\) 0 0
\(708\) 17154.9 0.910622
\(709\) −27878.9 −1.47675 −0.738373 0.674392i \(-0.764406\pi\)
−0.738373 + 0.674392i \(0.764406\pi\)
\(710\) 6814.57 0.360206
\(711\) −2466.37 −0.130093
\(712\) −12961.7 −0.682248
\(713\) −21080.6 −1.10726
\(714\) 0 0
\(715\) 7038.20 0.368131
\(716\) −12873.1 −0.671916
\(717\) 1713.44 0.0892462
\(718\) −9859.76 −0.512483
\(719\) −25863.3 −1.34150 −0.670750 0.741684i \(-0.734027\pi\)
−0.670750 + 0.741684i \(0.734027\pi\)
\(720\) −360.000 −0.0186339
\(721\) 0 0
\(722\) 50623.4 2.60943
\(723\) 11568.3 0.595064
\(724\) 32232.8 1.65459
\(725\) −7204.95 −0.369083
\(726\) 20322.4 1.03889
\(727\) −29157.0 −1.48744 −0.743722 0.668489i \(-0.766941\pi\)
−0.743722 + 0.668489i \(0.766941\pi\)
\(728\) 0 0
\(729\) 20917.0 1.06269
\(730\) −23597.4 −1.19641
\(731\) −570.267 −0.0288538
\(732\) 18129.8 0.915434
\(733\) −11006.1 −0.554595 −0.277297 0.960784i \(-0.589439\pi\)
−0.277297 + 0.960784i \(0.589439\pi\)
\(734\) −46218.1 −2.32417
\(735\) 0 0
\(736\) 24846.4 1.24436
\(737\) −1665.43 −0.0832386
\(738\) −1568.34 −0.0782267
\(739\) −37214.4 −1.85244 −0.926221 0.376982i \(-0.876962\pi\)
−0.926221 + 0.376982i \(0.876962\pi\)
\(740\) −16477.9 −0.818567
\(741\) −48624.7 −2.41062
\(742\) 0 0
\(743\) 11214.5 0.553730 0.276865 0.960909i \(-0.410704\pi\)
0.276865 + 0.960909i \(0.410704\pi\)
\(744\) −11878.1 −0.585313
\(745\) −5830.58 −0.286733
\(746\) −3584.65 −0.175930
\(747\) −1286.65 −0.0630202
\(748\) 6574.54 0.321375
\(749\) 0 0
\(750\) 2697.89 0.131351
\(751\) 6965.26 0.338437 0.169218 0.985579i \(-0.445876\pi\)
0.169218 + 0.985579i \(0.445876\pi\)
\(752\) −5315.94 −0.257782
\(753\) 19946.4 0.965324
\(754\) 88740.7 4.28613
\(755\) −4797.91 −0.231276
\(756\) 0 0
\(757\) −19352.8 −0.929180 −0.464590 0.885526i \(-0.653798\pi\)
−0.464590 + 0.885526i \(0.653798\pi\)
\(758\) 22676.0 1.08658
\(759\) 9952.15 0.475942
\(760\) 7748.29 0.369816
\(761\) −32383.6 −1.54258 −0.771291 0.636483i \(-0.780388\pi\)
−0.771291 + 0.636483i \(0.780388\pi\)
\(762\) 44164.8 2.09963
\(763\) 0 0
\(764\) −23059.7 −1.09198
\(765\) 313.325 0.0148082
\(766\) 48535.7 2.28938
\(767\) 23016.5 1.08354
\(768\) −1311.88 −0.0616385
\(769\) 25353.9 1.18893 0.594463 0.804123i \(-0.297365\pi\)
0.594463 + 0.804123i \(0.297365\pi\)
\(770\) 0 0
\(771\) 11456.6 0.535148
\(772\) 15848.6 0.738867
\(773\) −26117.0 −1.21522 −0.607610 0.794236i \(-0.707872\pi\)
−0.607610 + 0.794236i \(0.707872\pi\)
\(774\) 157.129 0.00729703
\(775\) 5224.94 0.242175
\(776\) −16049.8 −0.742465
\(777\) 0 0
\(778\) −36720.9 −1.69217
\(779\) −24766.5 −1.13909
\(780\) −18962.4 −0.870465
\(781\) −6230.58 −0.285464
\(782\) −13642.1 −0.623838
\(783\) −41788.7 −1.90729
\(784\) 0 0
\(785\) −5103.30 −0.232031
\(786\) 24611.6 1.11688
\(787\) 2273.38 0.102970 0.0514849 0.998674i \(-0.483605\pi\)
0.0514849 + 0.998674i \(0.483605\pi\)
\(788\) 35946.4 1.62505
\(789\) −31802.4 −1.43497
\(790\) −26616.0 −1.19868
\(791\) 0 0
\(792\) −448.612 −0.0201272
\(793\) 24324.6 1.08927
\(794\) −13481.8 −0.602585
\(795\) −3199.87 −0.142752
\(796\) −48863.3 −2.17577
\(797\) −2937.42 −0.130551 −0.0652753 0.997867i \(-0.520793\pi\)
−0.0652753 + 0.997867i \(0.520793\pi\)
\(798\) 0 0
\(799\) 4626.71 0.204858
\(800\) −6158.31 −0.272162
\(801\) 2280.63 0.100602
\(802\) −48585.0 −2.13915
\(803\) 21575.1 0.948157
\(804\) 4487.02 0.196822
\(805\) 0 0
\(806\) −64353.6 −2.81236
\(807\) −4956.73 −0.216214
\(808\) 6202.52 0.270054
\(809\) −4317.51 −0.187634 −0.0938169 0.995589i \(-0.529907\pi\)
−0.0938169 + 0.995589i \(0.529907\pi\)
\(810\) 14482.3 0.628216
\(811\) −1286.12 −0.0556863 −0.0278432 0.999612i \(-0.508864\pi\)
−0.0278432 + 0.999612i \(0.508864\pi\)
\(812\) 0 0
\(813\) 3654.89 0.157666
\(814\) 26400.6 1.13678
\(815\) 7830.08 0.336534
\(816\) 5639.85 0.241954
\(817\) 2481.32 0.106255
\(818\) 34187.3 1.46129
\(819\) 0 0
\(820\) −9658.31 −0.411321
\(821\) 26350.2 1.12013 0.560066 0.828448i \(-0.310776\pi\)
0.560066 + 0.828448i \(0.310776\pi\)
\(822\) 10583.1 0.449062
\(823\) 9820.05 0.415924 0.207962 0.978137i \(-0.433317\pi\)
0.207962 + 0.978137i \(0.433317\pi\)
\(824\) −8877.32 −0.375311
\(825\) −2466.69 −0.104096
\(826\) 0 0
\(827\) −30370.7 −1.27702 −0.638509 0.769615i \(-0.720448\pi\)
−0.638509 + 0.769615i \(0.720448\pi\)
\(828\) 2145.06 0.0900312
\(829\) 30817.7 1.29112 0.645562 0.763708i \(-0.276623\pi\)
0.645562 + 0.763708i \(0.276623\pi\)
\(830\) −13885.0 −0.580668
\(831\) 17693.1 0.738590
\(832\) 55305.9 2.30455
\(833\) 0 0
\(834\) 60432.7 2.50913
\(835\) −5651.52 −0.234226
\(836\) −28606.8 −1.18348
\(837\) 30304.6 1.25147
\(838\) 22693.7 0.935491
\(839\) 24746.0 1.01827 0.509134 0.860688i \(-0.329966\pi\)
0.509134 + 0.860688i \(0.329966\pi\)
\(840\) 0 0
\(841\) 58669.1 2.40556
\(842\) 6290.88 0.257480
\(843\) 23316.0 0.952604
\(844\) −53045.7 −2.16340
\(845\) −14456.6 −0.588548
\(846\) −1274.83 −0.0518079
\(847\) 0 0
\(848\) 4607.82 0.186596
\(849\) −10524.7 −0.425452
\(850\) 3381.27 0.136443
\(851\) −31261.4 −1.25926
\(852\) 16786.5 0.674995
\(853\) 11812.1 0.474135 0.237067 0.971493i \(-0.423814\pi\)
0.237067 + 0.971493i \(0.423814\pi\)
\(854\) 0 0
\(855\) −1363.32 −0.0545318
\(856\) −7054.89 −0.281695
\(857\) 23440.6 0.934322 0.467161 0.884172i \(-0.345277\pi\)
0.467161 + 0.884172i \(0.345277\pi\)
\(858\) 30381.3 1.20886
\(859\) −8945.58 −0.355319 −0.177660 0.984092i \(-0.556853\pi\)
−0.177660 + 0.984092i \(0.556853\pi\)
\(860\) 967.652 0.0383682
\(861\) 0 0
\(862\) −66006.3 −2.60810
\(863\) 19313.5 0.761806 0.380903 0.924615i \(-0.375613\pi\)
0.380903 + 0.924615i \(0.375613\pi\)
\(864\) −35718.2 −1.40643
\(865\) 12716.7 0.499862
\(866\) −808.330 −0.0317184
\(867\) 19656.4 0.769972
\(868\) 0 0
\(869\) 24335.1 0.949955
\(870\) −31101.1 −1.21198
\(871\) 6020.18 0.234197
\(872\) 47.6033 0.00184868
\(873\) 2823.98 0.109482
\(874\) 59359.0 2.29731
\(875\) 0 0
\(876\) −58128.0 −2.24197
\(877\) −12154.8 −0.468001 −0.234001 0.972236i \(-0.575182\pi\)
−0.234001 + 0.972236i \(0.575182\pi\)
\(878\) −15487.7 −0.595313
\(879\) −32973.3 −1.26526
\(880\) 3552.03 0.136067
\(881\) 29390.4 1.12394 0.561968 0.827159i \(-0.310044\pi\)
0.561968 + 0.827159i \(0.310044\pi\)
\(882\) 0 0
\(883\) 4180.02 0.159308 0.0796540 0.996823i \(-0.474618\pi\)
0.0796540 + 0.996823i \(0.474618\pi\)
\(884\) −23765.6 −0.904211
\(885\) −8066.62 −0.306392
\(886\) 21219.0 0.804590
\(887\) −21825.8 −0.826198 −0.413099 0.910686i \(-0.635554\pi\)
−0.413099 + 0.910686i \(0.635554\pi\)
\(888\) −17614.6 −0.665660
\(889\) 0 0
\(890\) 24611.6 0.926947
\(891\) −13241.2 −0.497863
\(892\) −40331.6 −1.51390
\(893\) −20131.5 −0.754397
\(894\) −25168.4 −0.941565
\(895\) 6053.25 0.226076
\(896\) 0 0
\(897\) −35974.9 −1.33909
\(898\) 30609.7 1.13748
\(899\) −60232.7 −2.23456
\(900\) −531.662 −0.0196912
\(901\) −4010.40 −0.148286
\(902\) 15474.4 0.571221
\(903\) 0 0
\(904\) 16067.2 0.591138
\(905\) −15156.6 −0.556710
\(906\) −20710.8 −0.759458
\(907\) −8356.11 −0.305910 −0.152955 0.988233i \(-0.548879\pi\)
−0.152955 + 0.988233i \(0.548879\pi\)
\(908\) −41586.3 −1.51992
\(909\) −1091.35 −0.0398214
\(910\) 0 0
\(911\) −4419.80 −0.160740 −0.0803701 0.996765i \(-0.525610\pi\)
−0.0803701 + 0.996765i \(0.525610\pi\)
\(912\) −24539.8 −0.891004
\(913\) 12695.1 0.460181
\(914\) 21806.8 0.789173
\(915\) −8525.06 −0.308011
\(916\) −3767.65 −0.135903
\(917\) 0 0
\(918\) 19611.3 0.705088
\(919\) −39257.6 −1.40913 −0.704563 0.709641i \(-0.748857\pi\)
−0.704563 + 0.709641i \(0.748857\pi\)
\(920\) 5732.56 0.205432
\(921\) −13364.8 −0.478162
\(922\) 72007.0 2.57204
\(923\) 22522.2 0.803173
\(924\) 0 0
\(925\) 7748.29 0.275419
\(926\) 67207.9 2.38509
\(927\) 1561.98 0.0553421
\(928\) 70992.5 2.51125
\(929\) 13399.9 0.473235 0.236618 0.971603i \(-0.423961\pi\)
0.236618 + 0.971603i \(0.423961\pi\)
\(930\) 22554.1 0.795245
\(931\) 0 0
\(932\) 69036.1 2.42635
\(933\) −4278.49 −0.150130
\(934\) −14367.2 −0.503330
\(935\) −3091.50 −0.108131
\(936\) 1621.64 0.0566291
\(937\) −27539.8 −0.960176 −0.480088 0.877220i \(-0.659395\pi\)
−0.480088 + 0.877220i \(0.659395\pi\)
\(938\) 0 0
\(939\) −16749.9 −0.582123
\(940\) −7850.79 −0.272409
\(941\) −14363.8 −0.497605 −0.248802 0.968554i \(-0.580037\pi\)
−0.248802 + 0.968554i \(0.580037\pi\)
\(942\) −22029.0 −0.761937
\(943\) −18323.5 −0.632762
\(944\) 11615.9 0.400494
\(945\) 0 0
\(946\) −1550.36 −0.0532838
\(947\) −6372.12 −0.218655 −0.109327 0.994006i \(-0.534870\pi\)
−0.109327 + 0.994006i \(0.534870\pi\)
\(948\) −65563.8 −2.24622
\(949\) −77989.6 −2.66770
\(950\) −14712.4 −0.502456
\(951\) −38165.3 −1.30136
\(952\) 0 0
\(953\) 958.776 0.0325895 0.0162948 0.999867i \(-0.494813\pi\)
0.0162948 + 0.999867i \(0.494813\pi\)
\(954\) 1105.01 0.0375012
\(955\) 10843.2 0.367411
\(956\) −3643.88 −0.123276
\(957\) 28435.8 0.960500
\(958\) 63055.1 2.12653
\(959\) 0 0
\(960\) −19383.1 −0.651654
\(961\) 13888.9 0.466213
\(962\) −95432.7 −3.19842
\(963\) 1241.32 0.0415379
\(964\) −24601.8 −0.821961
\(965\) −7452.40 −0.248602
\(966\) 0 0
\(967\) −40104.5 −1.33369 −0.666843 0.745198i \(-0.732355\pi\)
−0.666843 + 0.745198i \(0.732355\pi\)
\(968\) −10702.8 −0.355373
\(969\) 21358.2 0.708074
\(970\) 30475.2 1.00876
\(971\) −12397.4 −0.409732 −0.204866 0.978790i \(-0.565676\pi\)
−0.204866 + 0.978790i \(0.565676\pi\)
\(972\) −5954.62 −0.196496
\(973\) 0 0
\(974\) −8113.05 −0.266898
\(975\) 8916.56 0.292881
\(976\) 12276.1 0.402611
\(977\) 44982.9 1.47301 0.736506 0.676432i \(-0.236475\pi\)
0.736506 + 0.676432i \(0.236475\pi\)
\(978\) 33799.5 1.10510
\(979\) −22502.5 −0.734608
\(980\) 0 0
\(981\) −8.37588 −0.000272601 0
\(982\) 13904.4 0.451841
\(983\) 7895.76 0.256191 0.128095 0.991762i \(-0.459114\pi\)
0.128095 + 0.991762i \(0.459114\pi\)
\(984\) −10324.6 −0.334487
\(985\) −16902.8 −0.546771
\(986\) −38979.0 −1.25897
\(987\) 0 0
\(988\) 103408. 3.32979
\(989\) 1835.80 0.0590244
\(990\) 851.821 0.0273461
\(991\) 54534.9 1.74809 0.874046 0.485844i \(-0.161488\pi\)
0.874046 + 0.485844i \(0.161488\pi\)
\(992\) −51482.9 −1.64776
\(993\) 16608.5 0.530770
\(994\) 0 0
\(995\) 22976.6 0.732069
\(996\) −34203.2 −1.08812
\(997\) 6028.06 0.191485 0.0957425 0.995406i \(-0.469477\pi\)
0.0957425 + 0.995406i \(0.469477\pi\)
\(998\) 41930.9 1.32996
\(999\) 44940.1 1.42326
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 245.4.a.i.1.2 2
3.2 odd 2 2205.4.a.x.1.1 2
5.4 even 2 1225.4.a.q.1.1 2
7.2 even 3 245.4.e.k.116.1 4
7.3 odd 6 245.4.e.j.226.1 4
7.4 even 3 245.4.e.k.226.1 4
7.5 odd 6 245.4.e.j.116.1 4
7.6 odd 2 245.4.a.j.1.2 yes 2
21.20 even 2 2205.4.a.w.1.1 2
35.34 odd 2 1225.4.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.4.a.i.1.2 2 1.1 even 1 trivial
245.4.a.j.1.2 yes 2 7.6 odd 2
245.4.e.j.116.1 4 7.5 odd 6
245.4.e.j.226.1 4 7.3 odd 6
245.4.e.k.116.1 4 7.2 even 3
245.4.e.k.226.1 4 7.4 even 3
1225.4.a.p.1.1 2 35.34 odd 2
1225.4.a.q.1.1 2 5.4 even 2
2205.4.a.w.1.1 2 21.20 even 2
2205.4.a.x.1.1 2 3.2 odd 2