Properties

Label 294.4.a.l.1.1
Level $294$
Weight $4$
Character 294.1
Self dual yes
Analytic conductor $17.347$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3465615417\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.41421 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -7.41421 q^{5} -6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} -14.8284 q^{10} +10.4853 q^{11} -12.0000 q^{12} -2.78680 q^{13} +22.2426 q^{15} +16.0000 q^{16} -50.4437 q^{17} +18.0000 q^{18} -125.054 q^{19} -29.6569 q^{20} +20.9706 q^{22} -182.250 q^{23} -24.0000 q^{24} -70.0294 q^{25} -5.57359 q^{26} -27.0000 q^{27} +156.132 q^{29} +44.4853 q^{30} -139.632 q^{31} +32.0000 q^{32} -31.4558 q^{33} -100.887 q^{34} +36.0000 q^{36} -394.558 q^{37} -250.108 q^{38} +8.36039 q^{39} -59.3137 q^{40} -197.605 q^{41} +343.294 q^{43} +41.9411 q^{44} -66.7279 q^{45} -364.500 q^{46} +610.004 q^{47} -48.0000 q^{48} -140.059 q^{50} +151.331 q^{51} -11.1472 q^{52} -137.529 q^{53} -54.0000 q^{54} -77.7401 q^{55} +375.161 q^{57} +312.264 q^{58} -589.436 q^{59} +88.9706 q^{60} -247.217 q^{61} -279.265 q^{62} +64.0000 q^{64} +20.6619 q^{65} -62.9117 q^{66} -395.647 q^{67} -201.775 q^{68} +546.749 q^{69} +285.661 q^{71} +72.0000 q^{72} +997.457 q^{73} -789.117 q^{74} +210.088 q^{75} -500.215 q^{76} +16.7208 q^{78} -848.264 q^{79} -118.627 q^{80} +81.0000 q^{81} -395.210 q^{82} +210.863 q^{83} +374.000 q^{85} +686.587 q^{86} -468.396 q^{87} +83.8823 q^{88} +553.487 q^{89} -133.456 q^{90} -728.999 q^{92} +418.897 q^{93} +1220.01 q^{94} +927.176 q^{95} -96.0000 q^{96} +903.910 q^{97} +94.3675 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} + 16q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 4q^{2} - 6q^{3} + 8q^{4} - 12q^{5} - 12q^{6} + 16q^{8} + 18q^{9} - 24q^{10} + 4q^{11} - 24q^{12} - 48q^{13} + 36q^{15} + 32q^{16} - 132q^{17} + 36q^{18} - 120q^{19} - 48q^{20} + 8q^{22} - 76q^{23} - 48q^{24} - 174q^{25} - 96q^{26} - 54q^{27} - 112q^{29} + 72q^{30} - 432q^{31} + 64q^{32} - 12q^{33} - 264q^{34} + 72q^{36} - 280q^{37} - 240q^{38} + 144q^{39} - 96q^{40} - 36q^{41} - 128q^{43} + 16q^{44} - 108q^{45} - 152q^{46} + 264q^{47} - 96q^{48} - 348q^{50} + 396q^{51} - 192q^{52} + 268q^{53} - 108q^{54} - 48q^{55} + 360q^{57} - 224q^{58} - 336q^{59} + 144q^{60} + 504q^{61} - 864q^{62} + 128q^{64} + 228q^{65} - 24q^{66} - 384q^{67} - 528q^{68} + 228q^{69} - 396q^{71} + 144q^{72} + 312q^{73} - 560q^{74} + 522q^{75} - 480q^{76} + 288q^{78} - 848q^{79} - 192q^{80} + 162q^{81} - 72q^{82} + 648q^{83} + 748q^{85} - 256q^{86} + 336q^{87} + 32q^{88} + 612q^{89} - 216q^{90} - 304q^{92} + 1296q^{93} + 528q^{94} + 904q^{95} - 192q^{96} + 2184q^{97} + 36q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −7.41421 −0.663147 −0.331574 0.943429i \(-0.607580\pi\)
−0.331574 + 0.943429i \(0.607580\pi\)
\(6\) −6.00000 −0.408248
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −14.8284 −0.468916
\(11\) 10.4853 0.287403 0.143701 0.989621i \(-0.454100\pi\)
0.143701 + 0.989621i \(0.454100\pi\)
\(12\) −12.0000 −0.288675
\(13\) −2.78680 −0.0594553 −0.0297276 0.999558i \(-0.509464\pi\)
−0.0297276 + 0.999558i \(0.509464\pi\)
\(14\) 0 0
\(15\) 22.2426 0.382868
\(16\) 16.0000 0.250000
\(17\) −50.4437 −0.719670 −0.359835 0.933016i \(-0.617167\pi\)
−0.359835 + 0.933016i \(0.617167\pi\)
\(18\) 18.0000 0.235702
\(19\) −125.054 −1.50996 −0.754982 0.655746i \(-0.772354\pi\)
−0.754982 + 0.655746i \(0.772354\pi\)
\(20\) −29.6569 −0.331574
\(21\) 0 0
\(22\) 20.9706 0.203225
\(23\) −182.250 −1.65225 −0.826124 0.563488i \(-0.809459\pi\)
−0.826124 + 0.563488i \(0.809459\pi\)
\(24\) −24.0000 −0.204124
\(25\) −70.0294 −0.560235
\(26\) −5.57359 −0.0420412
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 156.132 0.999758 0.499879 0.866095i \(-0.333378\pi\)
0.499879 + 0.866095i \(0.333378\pi\)
\(30\) 44.4853 0.270729
\(31\) −139.632 −0.808991 −0.404496 0.914540i \(-0.632553\pi\)
−0.404496 + 0.914540i \(0.632553\pi\)
\(32\) 32.0000 0.176777
\(33\) −31.4558 −0.165932
\(34\) −100.887 −0.508883
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) −394.558 −1.75311 −0.876554 0.481303i \(-0.840164\pi\)
−0.876554 + 0.481303i \(0.840164\pi\)
\(38\) −250.108 −1.06771
\(39\) 8.36039 0.0343265
\(40\) −59.3137 −0.234458
\(41\) −197.605 −0.752701 −0.376350 0.926477i \(-0.622821\pi\)
−0.376350 + 0.926477i \(0.622821\pi\)
\(42\) 0 0
\(43\) 343.294 1.21748 0.608741 0.793369i \(-0.291675\pi\)
0.608741 + 0.793369i \(0.291675\pi\)
\(44\) 41.9411 0.143701
\(45\) −66.7279 −0.221049
\(46\) −364.500 −1.16832
\(47\) 610.004 1.89315 0.946577 0.322477i \(-0.104516\pi\)
0.946577 + 0.322477i \(0.104516\pi\)
\(48\) −48.0000 −0.144338
\(49\) 0 0
\(50\) −140.059 −0.396146
\(51\) 151.331 0.415501
\(52\) −11.1472 −0.0297276
\(53\) −137.529 −0.356435 −0.178218 0.983991i \(-0.557033\pi\)
−0.178218 + 0.983991i \(0.557033\pi\)
\(54\) −54.0000 −0.136083
\(55\) −77.7401 −0.190590
\(56\) 0 0
\(57\) 375.161 0.871778
\(58\) 312.264 0.706936
\(59\) −589.436 −1.30064 −0.650322 0.759659i \(-0.725366\pi\)
−0.650322 + 0.759659i \(0.725366\pi\)
\(60\) 88.9706 0.191434
\(61\) −247.217 −0.518901 −0.259450 0.965756i \(-0.583541\pi\)
−0.259450 + 0.965756i \(0.583541\pi\)
\(62\) −279.265 −0.572043
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 20.6619 0.0394276
\(66\) −62.9117 −0.117332
\(67\) −395.647 −0.721432 −0.360716 0.932676i \(-0.617468\pi\)
−0.360716 + 0.932676i \(0.617468\pi\)
\(68\) −201.775 −0.359835
\(69\) 546.749 0.953926
\(70\) 0 0
\(71\) 285.661 0.477489 0.238745 0.971082i \(-0.423264\pi\)
0.238745 + 0.971082i \(0.423264\pi\)
\(72\) 72.0000 0.117851
\(73\) 997.457 1.59923 0.799613 0.600515i \(-0.205038\pi\)
0.799613 + 0.600515i \(0.205038\pi\)
\(74\) −789.117 −1.23963
\(75\) 210.088 0.323452
\(76\) −500.215 −0.754982
\(77\) 0 0
\(78\) 16.7208 0.0242725
\(79\) −848.264 −1.20807 −0.604033 0.796960i \(-0.706440\pi\)
−0.604033 + 0.796960i \(0.706440\pi\)
\(80\) −118.627 −0.165787
\(81\) 81.0000 0.111111
\(82\) −395.210 −0.532240
\(83\) 210.863 0.278858 0.139429 0.990232i \(-0.455473\pi\)
0.139429 + 0.990232i \(0.455473\pi\)
\(84\) 0 0
\(85\) 374.000 0.477247
\(86\) 686.587 0.860890
\(87\) −468.396 −0.577211
\(88\) 83.8823 0.101612
\(89\) 553.487 0.659208 0.329604 0.944119i \(-0.393085\pi\)
0.329604 + 0.944119i \(0.393085\pi\)
\(90\) −133.456 −0.156305
\(91\) 0 0
\(92\) −728.999 −0.826124
\(93\) 418.897 0.467071
\(94\) 1220.01 1.33866
\(95\) 927.176 1.00133
\(96\) −96.0000 −0.102062
\(97\) 903.910 0.946166 0.473083 0.881018i \(-0.343141\pi\)
0.473083 + 0.881018i \(0.343141\pi\)
\(98\) 0 0
\(99\) 94.3675 0.0958009
\(100\) −280.118 −0.280118
\(101\) 313.611 0.308965 0.154482 0.987996i \(-0.450629\pi\)
0.154482 + 0.987996i \(0.450629\pi\)
\(102\) 302.662 0.293804
\(103\) −230.975 −0.220957 −0.110479 0.993878i \(-0.535238\pi\)
−0.110479 + 0.993878i \(0.535238\pi\)
\(104\) −22.2944 −0.0210206
\(105\) 0 0
\(106\) −275.058 −0.252038
\(107\) 125.486 0.113376 0.0566879 0.998392i \(-0.481946\pi\)
0.0566879 + 0.998392i \(0.481946\pi\)
\(108\) −108.000 −0.0962250
\(109\) 745.527 0.655124 0.327562 0.944830i \(-0.393773\pi\)
0.327562 + 0.944830i \(0.393773\pi\)
\(110\) −155.480 −0.134768
\(111\) 1183.68 1.01216
\(112\) 0 0
\(113\) −1043.76 −0.868929 −0.434464 0.900689i \(-0.643062\pi\)
−0.434464 + 0.900689i \(0.643062\pi\)
\(114\) 750.323 0.616440
\(115\) 1351.24 1.09568
\(116\) 624.528 0.499879
\(117\) −25.0812 −0.0198184
\(118\) −1178.87 −0.919694
\(119\) 0 0
\(120\) 177.941 0.135364
\(121\) −1221.06 −0.917400
\(122\) −494.435 −0.366918
\(123\) 592.815 0.434572
\(124\) −558.530 −0.404496
\(125\) 1445.99 1.03467
\(126\) 0 0
\(127\) −2080.17 −1.45343 −0.726715 0.686939i \(-0.758954\pi\)
−0.726715 + 0.686939i \(0.758954\pi\)
\(128\) 128.000 0.0883883
\(129\) −1029.88 −0.702914
\(130\) 41.3238 0.0278795
\(131\) 1269.53 0.846711 0.423355 0.905964i \(-0.360852\pi\)
0.423355 + 0.905964i \(0.360852\pi\)
\(132\) −125.823 −0.0829661
\(133\) 0 0
\(134\) −791.294 −0.510129
\(135\) 200.184 0.127623
\(136\) −403.549 −0.254442
\(137\) 3073.19 1.91650 0.958249 0.285935i \(-0.0923043\pi\)
0.958249 + 0.285935i \(0.0923043\pi\)
\(138\) 1093.50 0.674527
\(139\) −1013.60 −0.618504 −0.309252 0.950980i \(-0.600079\pi\)
−0.309252 + 0.950980i \(0.600079\pi\)
\(140\) 0 0
\(141\) −1830.01 −1.09301
\(142\) 571.322 0.337636
\(143\) −29.2203 −0.0170876
\(144\) 144.000 0.0833333
\(145\) −1157.60 −0.662987
\(146\) 1994.91 1.13082
\(147\) 0 0
\(148\) −1578.23 −0.876554
\(149\) 1231.47 0.677087 0.338544 0.940951i \(-0.390066\pi\)
0.338544 + 0.940951i \(0.390066\pi\)
\(150\) 420.177 0.228715
\(151\) −2244.74 −1.20976 −0.604881 0.796316i \(-0.706779\pi\)
−0.604881 + 0.796316i \(0.706779\pi\)
\(152\) −1000.43 −0.533853
\(153\) −453.993 −0.239890
\(154\) 0 0
\(155\) 1035.26 0.536481
\(156\) 33.4416 0.0171633
\(157\) 3787.16 1.92515 0.962573 0.271021i \(-0.0873614\pi\)
0.962573 + 0.271021i \(0.0873614\pi\)
\(158\) −1696.53 −0.854231
\(159\) 412.587 0.205788
\(160\) −237.255 −0.117229
\(161\) 0 0
\(162\) 162.000 0.0785674
\(163\) −2108.56 −1.01322 −0.506611 0.862175i \(-0.669102\pi\)
−0.506611 + 0.862175i \(0.669102\pi\)
\(164\) −790.420 −0.376350
\(165\) 233.220 0.110037
\(166\) 421.726 0.197182
\(167\) −1502.41 −0.696170 −0.348085 0.937463i \(-0.613168\pi\)
−0.348085 + 0.937463i \(0.613168\pi\)
\(168\) 0 0
\(169\) −2189.23 −0.996465
\(170\) 748.000 0.337465
\(171\) −1125.48 −0.503321
\(172\) 1373.17 0.608741
\(173\) −471.256 −0.207104 −0.103552 0.994624i \(-0.533021\pi\)
−0.103552 + 0.994624i \(0.533021\pi\)
\(174\) −936.792 −0.408150
\(175\) 0 0
\(176\) 167.765 0.0718507
\(177\) 1768.31 0.750927
\(178\) 1106.97 0.466131
\(179\) 1332.49 0.556395 0.278198 0.960524i \(-0.410263\pi\)
0.278198 + 0.960524i \(0.410263\pi\)
\(180\) −266.912 −0.110525
\(181\) 997.727 0.409726 0.204863 0.978791i \(-0.434325\pi\)
0.204863 + 0.978791i \(0.434325\pi\)
\(182\) 0 0
\(183\) 741.652 0.299587
\(184\) −1458.00 −0.584158
\(185\) 2925.34 1.16257
\(186\) 837.795 0.330269
\(187\) −528.916 −0.206835
\(188\) 2440.02 0.946577
\(189\) 0 0
\(190\) 1854.35 0.708046
\(191\) 1226.72 0.464725 0.232362 0.972629i \(-0.425354\pi\)
0.232362 + 0.972629i \(0.425354\pi\)
\(192\) −192.000 −0.0721688
\(193\) −3479.29 −1.29764 −0.648821 0.760941i \(-0.724738\pi\)
−0.648821 + 0.760941i \(0.724738\pi\)
\(194\) 1807.82 0.669040
\(195\) −61.9857 −0.0227635
\(196\) 0 0
\(197\) 3193.47 1.15495 0.577476 0.816408i \(-0.304038\pi\)
0.577476 + 0.816408i \(0.304038\pi\)
\(198\) 188.735 0.0677415
\(199\) −1065.15 −0.379429 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(200\) −560.235 −0.198073
\(201\) 1186.94 0.416519
\(202\) 627.222 0.218471
\(203\) 0 0
\(204\) 605.324 0.207751
\(205\) 1465.09 0.499152
\(206\) −461.949 −0.156241
\(207\) −1640.25 −0.550749
\(208\) −44.5887 −0.0148638
\(209\) −1311.22 −0.433968
\(210\) 0 0
\(211\) 2057.50 0.671298 0.335649 0.941987i \(-0.391044\pi\)
0.335649 + 0.941987i \(0.391044\pi\)
\(212\) −550.116 −0.178218
\(213\) −856.983 −0.275678
\(214\) 250.972 0.0801688
\(215\) −2545.25 −0.807371
\(216\) −216.000 −0.0680414
\(217\) 0 0
\(218\) 1491.05 0.463243
\(219\) −2992.37 −0.923314
\(220\) −310.960 −0.0952952
\(221\) 140.576 0.0427881
\(222\) 2367.35 0.715703
\(223\) −2028.27 −0.609071 −0.304536 0.952501i \(-0.598501\pi\)
−0.304536 + 0.952501i \(0.598501\pi\)
\(224\) 0 0
\(225\) −630.265 −0.186745
\(226\) −2087.53 −0.614425
\(227\) −2423.96 −0.708739 −0.354369 0.935106i \(-0.615304\pi\)
−0.354369 + 0.935106i \(0.615304\pi\)
\(228\) 1500.65 0.435889
\(229\) 1967.17 0.567660 0.283830 0.958875i \(-0.408395\pi\)
0.283830 + 0.958875i \(0.408395\pi\)
\(230\) 2702.48 0.774766
\(231\) 0 0
\(232\) 1249.06 0.353468
\(233\) 4478.33 1.25916 0.629582 0.776934i \(-0.283226\pi\)
0.629582 + 0.776934i \(0.283226\pi\)
\(234\) −50.1623 −0.0140137
\(235\) −4522.70 −1.25544
\(236\) −2357.74 −0.650322
\(237\) 2544.79 0.697477
\(238\) 0 0
\(239\) 6116.92 1.65553 0.827763 0.561078i \(-0.189613\pi\)
0.827763 + 0.561078i \(0.189613\pi\)
\(240\) 355.882 0.0957171
\(241\) −6228.38 −1.66475 −0.832376 0.554211i \(-0.813020\pi\)
−0.832376 + 0.554211i \(0.813020\pi\)
\(242\) −2442.12 −0.648699
\(243\) −243.000 −0.0641500
\(244\) −988.870 −0.259450
\(245\) 0 0
\(246\) 1185.63 0.307289
\(247\) 348.500 0.0897753
\(248\) −1117.06 −0.286022
\(249\) −632.589 −0.160999
\(250\) 2891.98 0.731619
\(251\) −5904.42 −1.48479 −0.742397 0.669960i \(-0.766311\pi\)
−0.742397 + 0.669960i \(0.766311\pi\)
\(252\) 0 0
\(253\) −1910.94 −0.474861
\(254\) −4160.35 −1.02773
\(255\) −1122.00 −0.275539
\(256\) 256.000 0.0625000
\(257\) 408.223 0.0990827 0.0495414 0.998772i \(-0.484224\pi\)
0.0495414 + 0.998772i \(0.484224\pi\)
\(258\) −2059.76 −0.497035
\(259\) 0 0
\(260\) 82.6476 0.0197138
\(261\) 1405.19 0.333253
\(262\) 2539.05 0.598715
\(263\) −4626.01 −1.08461 −0.542304 0.840182i \(-0.682448\pi\)
−0.542304 + 0.840182i \(0.682448\pi\)
\(264\) −251.647 −0.0586659
\(265\) 1019.67 0.236369
\(266\) 0 0
\(267\) −1660.46 −0.380594
\(268\) −1582.59 −0.360716
\(269\) 871.293 0.197486 0.0987429 0.995113i \(-0.468518\pi\)
0.0987429 + 0.995113i \(0.468518\pi\)
\(270\) 400.368 0.0902429
\(271\) −6472.35 −1.45080 −0.725401 0.688327i \(-0.758345\pi\)
−0.725401 + 0.688327i \(0.758345\pi\)
\(272\) −807.098 −0.179917
\(273\) 0 0
\(274\) 6146.38 1.35517
\(275\) −734.278 −0.161013
\(276\) 2187.00 0.476963
\(277\) 4711.88 1.02206 0.511028 0.859564i \(-0.329265\pi\)
0.511028 + 0.859564i \(0.329265\pi\)
\(278\) −2027.19 −0.437349
\(279\) −1256.69 −0.269664
\(280\) 0 0
\(281\) −7165.66 −1.52124 −0.760618 0.649200i \(-0.775104\pi\)
−0.760618 + 0.649200i \(0.775104\pi\)
\(282\) −3660.03 −0.772877
\(283\) −6347.00 −1.33318 −0.666590 0.745424i \(-0.732247\pi\)
−0.666590 + 0.745424i \(0.732247\pi\)
\(284\) 1142.64 0.238745
\(285\) −2781.53 −0.578117
\(286\) −58.4407 −0.0120828
\(287\) 0 0
\(288\) 288.000 0.0589256
\(289\) −2368.44 −0.482076
\(290\) −2315.19 −0.468803
\(291\) −2711.73 −0.546269
\(292\) 3989.83 0.799613
\(293\) −9233.78 −1.84110 −0.920552 0.390621i \(-0.872260\pi\)
−0.920552 + 0.390621i \(0.872260\pi\)
\(294\) 0 0
\(295\) 4370.20 0.862519
\(296\) −3156.47 −0.619817
\(297\) −283.103 −0.0553107
\(298\) 2462.94 0.478773
\(299\) 507.893 0.0982348
\(300\) 840.353 0.161726
\(301\) 0 0
\(302\) −4489.47 −0.855430
\(303\) −940.833 −0.178381
\(304\) −2000.86 −0.377491
\(305\) 1832.92 0.344108
\(306\) −907.986 −0.169628
\(307\) −6786.53 −1.26165 −0.630827 0.775923i \(-0.717284\pi\)
−0.630827 + 0.775923i \(0.717284\pi\)
\(308\) 0 0
\(309\) 692.924 0.127570
\(310\) 2070.53 0.379349
\(311\) −5136.77 −0.936590 −0.468295 0.883572i \(-0.655132\pi\)
−0.468295 + 0.883572i \(0.655132\pi\)
\(312\) 66.8831 0.0121363
\(313\) 3763.56 0.679645 0.339822 0.940490i \(-0.389633\pi\)
0.339822 + 0.940490i \(0.389633\pi\)
\(314\) 7574.31 1.36128
\(315\) 0 0
\(316\) −3393.06 −0.604033
\(317\) −1034.95 −0.183370 −0.0916851 0.995788i \(-0.529225\pi\)
−0.0916851 + 0.995788i \(0.529225\pi\)
\(318\) 825.174 0.145514
\(319\) 1637.09 0.287333
\(320\) −474.510 −0.0828934
\(321\) −376.458 −0.0654575
\(322\) 0 0
\(323\) 6308.17 1.08668
\(324\) 324.000 0.0555556
\(325\) 195.158 0.0333089
\(326\) −4217.12 −0.716456
\(327\) −2236.58 −0.378236
\(328\) −1580.84 −0.266120
\(329\) 0 0
\(330\) 466.441 0.0778082
\(331\) 8800.06 1.46131 0.730657 0.682744i \(-0.239214\pi\)
0.730657 + 0.682744i \(0.239214\pi\)
\(332\) 843.452 0.139429
\(333\) −3551.03 −0.584369
\(334\) −3004.83 −0.492266
\(335\) 2933.41 0.478416
\(336\) 0 0
\(337\) −5859.78 −0.947189 −0.473595 0.880743i \(-0.657044\pi\)
−0.473595 + 0.880743i \(0.657044\pi\)
\(338\) −4378.47 −0.704607
\(339\) 3131.29 0.501676
\(340\) 1496.00 0.238624
\(341\) −1464.09 −0.232506
\(342\) −2250.97 −0.355902
\(343\) 0 0
\(344\) 2746.35 0.430445
\(345\) −4053.72 −0.632593
\(346\) −942.512 −0.146444
\(347\) 7938.54 1.22814 0.614068 0.789253i \(-0.289532\pi\)
0.614068 + 0.789253i \(0.289532\pi\)
\(348\) −1873.58 −0.288605
\(349\) 9927.75 1.52269 0.761347 0.648344i \(-0.224538\pi\)
0.761347 + 0.648344i \(0.224538\pi\)
\(350\) 0 0
\(351\) 75.2435 0.0114422
\(352\) 335.529 0.0508061
\(353\) 10103.0 1.52332 0.761658 0.647979i \(-0.224386\pi\)
0.761658 + 0.647979i \(0.224386\pi\)
\(354\) 3536.61 0.530986
\(355\) −2117.95 −0.316646
\(356\) 2213.95 0.329604
\(357\) 0 0
\(358\) 2664.97 0.393431
\(359\) −4825.27 −0.709382 −0.354691 0.934984i \(-0.615414\pi\)
−0.354691 + 0.934984i \(0.615414\pi\)
\(360\) −533.823 −0.0781527
\(361\) 8779.46 1.27999
\(362\) 1995.45 0.289720
\(363\) 3663.18 0.529661
\(364\) 0 0
\(365\) −7395.36 −1.06052
\(366\) 1483.30 0.211840
\(367\) −6935.30 −0.986430 −0.493215 0.869907i \(-0.664178\pi\)
−0.493215 + 0.869907i \(0.664178\pi\)
\(368\) −2916.00 −0.413062
\(369\) −1778.45 −0.250900
\(370\) 5850.68 0.822061
\(371\) 0 0
\(372\) 1675.59 0.233536
\(373\) 14093.1 1.95634 0.978168 0.207816i \(-0.0666355\pi\)
0.978168 + 0.207816i \(0.0666355\pi\)
\(374\) −1057.83 −0.146254
\(375\) −4337.97 −0.597365
\(376\) 4880.03 0.669331
\(377\) −435.108 −0.0594409
\(378\) 0 0
\(379\) 5354.17 0.725661 0.362830 0.931855i \(-0.381810\pi\)
0.362830 + 0.931855i \(0.381810\pi\)
\(380\) 3708.70 0.500664
\(381\) 6240.52 0.839138
\(382\) 2453.44 0.328610
\(383\) 8970.47 1.19679 0.598394 0.801202i \(-0.295806\pi\)
0.598394 + 0.801202i \(0.295806\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −6958.58 −0.917571
\(387\) 3089.64 0.405828
\(388\) 3615.64 0.473083
\(389\) −3702.58 −0.482592 −0.241296 0.970452i \(-0.577572\pi\)
−0.241296 + 0.970452i \(0.577572\pi\)
\(390\) −123.971 −0.0160962
\(391\) 9193.34 1.18907
\(392\) 0 0
\(393\) −3808.58 −0.488849
\(394\) 6386.94 0.816674
\(395\) 6289.21 0.801125
\(396\) 377.470 0.0479005
\(397\) 2083.24 0.263362 0.131681 0.991292i \(-0.457963\pi\)
0.131681 + 0.991292i \(0.457963\pi\)
\(398\) −2130.30 −0.268297
\(399\) 0 0
\(400\) −1120.47 −0.140059
\(401\) −10634.0 −1.32428 −0.662141 0.749379i \(-0.730352\pi\)
−0.662141 + 0.749379i \(0.730352\pi\)
\(402\) 2373.88 0.294523
\(403\) 389.127 0.0480988
\(404\) 1254.44 0.154482
\(405\) −600.551 −0.0736830
\(406\) 0 0
\(407\) −4137.06 −0.503848
\(408\) 1210.65 0.146902
\(409\) 6516.37 0.787808 0.393904 0.919152i \(-0.371124\pi\)
0.393904 + 0.919152i \(0.371124\pi\)
\(410\) 2930.17 0.352954
\(411\) −9219.56 −1.10649
\(412\) −923.899 −0.110479
\(413\) 0 0
\(414\) −3280.50 −0.389439
\(415\) −1563.38 −0.184924
\(416\) −89.1775 −0.0105103
\(417\) 3040.79 0.357094
\(418\) −2622.45 −0.306862
\(419\) −6079.92 −0.708887 −0.354443 0.935077i \(-0.615330\pi\)
−0.354443 + 0.935077i \(0.615330\pi\)
\(420\) 0 0
\(421\) −5631.58 −0.651939 −0.325969 0.945380i \(-0.605691\pi\)
−0.325969 + 0.945380i \(0.605691\pi\)
\(422\) 4114.99 0.474679
\(423\) 5490.04 0.631051
\(424\) −1100.23 −0.126019
\(425\) 3532.54 0.403184
\(426\) −1713.97 −0.194934
\(427\) 0 0
\(428\) 501.945 0.0566879
\(429\) 87.6610 0.00986554
\(430\) −5090.50 −0.570897
\(431\) −3736.90 −0.417633 −0.208817 0.977955i \(-0.566961\pi\)
−0.208817 + 0.977955i \(0.566961\pi\)
\(432\) −432.000 −0.0481125
\(433\) 5757.46 0.638998 0.319499 0.947587i \(-0.396485\pi\)
0.319499 + 0.947587i \(0.396485\pi\)
\(434\) 0 0
\(435\) 3472.79 0.382776
\(436\) 2982.11 0.327562
\(437\) 22791.0 2.49484
\(438\) −5984.74 −0.652881
\(439\) 9812.76 1.06683 0.533414 0.845854i \(-0.320909\pi\)
0.533414 + 0.845854i \(0.320909\pi\)
\(440\) −621.921 −0.0673839
\(441\) 0 0
\(442\) 281.152 0.0302558
\(443\) 5830.30 0.625296 0.312648 0.949869i \(-0.398784\pi\)
0.312648 + 0.949869i \(0.398784\pi\)
\(444\) 4734.70 0.506079
\(445\) −4103.67 −0.437152
\(446\) −4056.54 −0.430678
\(447\) −3694.41 −0.390916
\(448\) 0 0
\(449\) −8674.94 −0.911794 −0.455897 0.890033i \(-0.650681\pi\)
−0.455897 + 0.890033i \(0.650681\pi\)
\(450\) −1260.53 −0.132049
\(451\) −2071.95 −0.216328
\(452\) −4175.05 −0.434464
\(453\) 6734.21 0.698456
\(454\) −4847.91 −0.501154
\(455\) 0 0
\(456\) 3001.29 0.308220
\(457\) 9106.82 0.932165 0.466082 0.884741i \(-0.345665\pi\)
0.466082 + 0.884741i \(0.345665\pi\)
\(458\) 3934.34 0.401396
\(459\) 1361.98 0.138500
\(460\) 5404.96 0.547842
\(461\) 8729.69 0.881957 0.440979 0.897518i \(-0.354631\pi\)
0.440979 + 0.897518i \(0.354631\pi\)
\(462\) 0 0
\(463\) −1795.62 −0.180237 −0.0901184 0.995931i \(-0.528725\pi\)
−0.0901184 + 0.995931i \(0.528725\pi\)
\(464\) 2498.11 0.249940
\(465\) −3105.79 −0.309737
\(466\) 8956.67 0.890364
\(467\) −130.559 −0.0129370 −0.00646849 0.999979i \(-0.502059\pi\)
−0.00646849 + 0.999979i \(0.502059\pi\)
\(468\) −100.325 −0.00990921
\(469\) 0 0
\(470\) −9045.40 −0.887730
\(471\) −11361.5 −1.11148
\(472\) −4715.49 −0.459847
\(473\) 3599.53 0.349908
\(474\) 5089.58 0.493191
\(475\) 8757.45 0.845935
\(476\) 0 0
\(477\) −1237.76 −0.118812
\(478\) 12233.8 1.17063
\(479\) −11845.8 −1.12996 −0.564978 0.825106i \(-0.691115\pi\)
−0.564978 + 0.825106i \(0.691115\pi\)
\(480\) 711.765 0.0676822
\(481\) 1099.55 0.104232
\(482\) −12456.8 −1.17716
\(483\) 0 0
\(484\) −4884.24 −0.458700
\(485\) −6701.78 −0.627448
\(486\) −486.000 −0.0453609
\(487\) −4807.21 −0.447301 −0.223650 0.974669i \(-0.571797\pi\)
−0.223650 + 0.974669i \(0.571797\pi\)
\(488\) −1977.74 −0.183459
\(489\) 6325.68 0.584984
\(490\) 0 0
\(491\) 6068.04 0.557733 0.278866 0.960330i \(-0.410041\pi\)
0.278866 + 0.960330i \(0.410041\pi\)
\(492\) 2371.26 0.217286
\(493\) −7875.87 −0.719496
\(494\) 696.999 0.0634807
\(495\) −699.661 −0.0635302
\(496\) −2234.12 −0.202248
\(497\) 0 0
\(498\) −1265.18 −0.113843
\(499\) −15506.4 −1.39111 −0.695554 0.718474i \(-0.744841\pi\)
−0.695554 + 0.718474i \(0.744841\pi\)
\(500\) 5783.96 0.517333
\(501\) 4507.24 0.401934
\(502\) −11808.8 −1.04991
\(503\) 1496.79 0.132681 0.0663405 0.997797i \(-0.478868\pi\)
0.0663405 + 0.997797i \(0.478868\pi\)
\(504\) 0 0
\(505\) −2325.18 −0.204889
\(506\) −3821.88 −0.335777
\(507\) 6567.70 0.575309
\(508\) −8320.70 −0.726715
\(509\) 5053.74 0.440084 0.220042 0.975490i \(-0.429380\pi\)
0.220042 + 0.975490i \(0.429380\pi\)
\(510\) −2244.00 −0.194835
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 3376.45 0.290593
\(514\) 816.446 0.0700621
\(515\) 1712.50 0.146527
\(516\) −4119.52 −0.351457
\(517\) 6396.07 0.544098
\(518\) 0 0
\(519\) 1413.77 0.119571
\(520\) 165.295 0.0139398
\(521\) −9736.73 −0.818760 −0.409380 0.912364i \(-0.634255\pi\)
−0.409380 + 0.912364i \(0.634255\pi\)
\(522\) 2810.38 0.235645
\(523\) 11796.7 0.986295 0.493148 0.869946i \(-0.335846\pi\)
0.493148 + 0.869946i \(0.335846\pi\)
\(524\) 5078.11 0.423355
\(525\) 0 0
\(526\) −9252.01 −0.766933
\(527\) 7043.57 0.582206
\(528\) −503.294 −0.0414830
\(529\) 21048.0 1.72992
\(530\) 2039.34 0.167138
\(531\) −5304.92 −0.433548
\(532\) 0 0
\(533\) 550.685 0.0447520
\(534\) −3320.92 −0.269121
\(535\) −930.381 −0.0751848
\(536\) −3165.17 −0.255065
\(537\) −3997.46 −0.321235
\(538\) 1742.59 0.139644
\(539\) 0 0
\(540\) 800.735 0.0638114
\(541\) −4310.82 −0.342581 −0.171291 0.985221i \(-0.554794\pi\)
−0.171291 + 0.985221i \(0.554794\pi\)
\(542\) −12944.7 −1.02587
\(543\) −2993.18 −0.236556
\(544\) −1614.20 −0.127221
\(545\) −5527.50 −0.434444
\(546\) 0 0
\(547\) 17015.9 1.33007 0.665034 0.746813i \(-0.268417\pi\)
0.665034 + 0.746813i \(0.268417\pi\)
\(548\) 12292.8 0.958249
\(549\) −2224.96 −0.172967
\(550\) −1468.56 −0.113854
\(551\) −19524.9 −1.50960
\(552\) 4373.99 0.337264
\(553\) 0 0
\(554\) 9423.76 0.722703
\(555\) −8776.02 −0.671210
\(556\) −4054.38 −0.309252
\(557\) 1741.18 0.132452 0.0662262 0.997805i \(-0.478904\pi\)
0.0662262 + 0.997805i \(0.478904\pi\)
\(558\) −2513.38 −0.190681
\(559\) −956.689 −0.0723858
\(560\) 0 0
\(561\) 1586.75 0.119416
\(562\) −14331.3 −1.07568
\(563\) −11346.8 −0.849401 −0.424700 0.905334i \(-0.639621\pi\)
−0.424700 + 0.905334i \(0.639621\pi\)
\(564\) −7320.05 −0.546507
\(565\) 7738.68 0.576228
\(566\) −12694.0 −0.942701
\(567\) 0 0
\(568\) 2285.29 0.168818
\(569\) −18417.4 −1.35694 −0.678470 0.734628i \(-0.737357\pi\)
−0.678470 + 0.734628i \(0.737357\pi\)
\(570\) −5563.05 −0.408791
\(571\) 9998.49 0.732791 0.366396 0.930459i \(-0.380592\pi\)
0.366396 + 0.930459i \(0.380592\pi\)
\(572\) −116.881 −0.00854380
\(573\) −3680.16 −0.268309
\(574\) 0 0
\(575\) 12762.8 0.925648
\(576\) 576.000 0.0416667
\(577\) 1401.71 0.101133 0.0505667 0.998721i \(-0.483897\pi\)
0.0505667 + 0.998721i \(0.483897\pi\)
\(578\) −4736.88 −0.340879
\(579\) 10437.9 0.749194
\(580\) −4630.38 −0.331494
\(581\) 0 0
\(582\) −5423.46 −0.386271
\(583\) −1442.03 −0.102440
\(584\) 7979.66 0.565412
\(585\) 185.957 0.0131425
\(586\) −18467.6 −1.30186
\(587\) 10851.3 0.763001 0.381500 0.924369i \(-0.375407\pi\)
0.381500 + 0.924369i \(0.375407\pi\)
\(588\) 0 0
\(589\) 17461.6 1.22155
\(590\) 8740.40 0.609893
\(591\) −9580.42 −0.666812
\(592\) −6312.94 −0.438277
\(593\) −20462.0 −1.41699 −0.708493 0.705717i \(-0.750625\pi\)
−0.708493 + 0.705717i \(0.750625\pi\)
\(594\) −566.205 −0.0391106
\(595\) 0 0
\(596\) 4925.88 0.338544
\(597\) 3195.45 0.219064
\(598\) 1015.79 0.0694625
\(599\) 9990.42 0.681465 0.340733 0.940160i \(-0.389325\pi\)
0.340733 + 0.940160i \(0.389325\pi\)
\(600\) 1680.71 0.114358
\(601\) −17435.9 −1.18341 −0.591703 0.806156i \(-0.701544\pi\)
−0.591703 + 0.806156i \(0.701544\pi\)
\(602\) 0 0
\(603\) −3560.82 −0.240477
\(604\) −8978.94 −0.604881
\(605\) 9053.19 0.608371
\(606\) −1881.67 −0.126134
\(607\) −16700.3 −1.11671 −0.558356 0.829601i \(-0.688568\pi\)
−0.558356 + 0.829601i \(0.688568\pi\)
\(608\) −4001.72 −0.266926
\(609\) 0 0
\(610\) 3665.85 0.243321
\(611\) −1699.96 −0.112558
\(612\) −1815.97 −0.119945
\(613\) −26702.9 −1.75941 −0.879707 0.475515i \(-0.842262\pi\)
−0.879707 + 0.475515i \(0.842262\pi\)
\(614\) −13573.1 −0.892124
\(615\) −4395.26 −0.288185
\(616\) 0 0
\(617\) 27790.4 1.81329 0.906645 0.421894i \(-0.138635\pi\)
0.906645 + 0.421894i \(0.138635\pi\)
\(618\) 1385.85 0.0902055
\(619\) −1736.08 −0.112728 −0.0563642 0.998410i \(-0.517951\pi\)
−0.0563642 + 0.998410i \(0.517951\pi\)
\(620\) 4141.06 0.268240
\(621\) 4920.74 0.317975
\(622\) −10273.5 −0.662269
\(623\) 0 0
\(624\) 133.766 0.00858163
\(625\) −1967.20 −0.125901
\(626\) 7527.12 0.480582
\(627\) 3933.67 0.250552
\(628\) 15148.6 0.962573
\(629\) 19903.0 1.26166
\(630\) 0 0
\(631\) −8990.27 −0.567190 −0.283595 0.958944i \(-0.591527\pi\)
−0.283595 + 0.958944i \(0.591527\pi\)
\(632\) −6786.11 −0.427116
\(633\) −6172.49 −0.387574
\(634\) −2069.89 −0.129662
\(635\) 15422.9 0.963839
\(636\) 1650.35 0.102894
\(637\) 0 0
\(638\) 3274.18 0.203175
\(639\) 2570.95 0.159163
\(640\) −949.019 −0.0586145
\(641\) −13769.6 −0.848467 −0.424233 0.905553i \(-0.639456\pi\)
−0.424233 + 0.905553i \(0.639456\pi\)
\(642\) −752.917 −0.0462855
\(643\) −26969.9 −1.65411 −0.827053 0.562124i \(-0.809985\pi\)
−0.827053 + 0.562124i \(0.809985\pi\)
\(644\) 0 0
\(645\) 7635.75 0.466136
\(646\) 12616.3 0.768395
\(647\) −24401.7 −1.48274 −0.741368 0.671098i \(-0.765823\pi\)
−0.741368 + 0.671098i \(0.765823\pi\)
\(648\) 648.000 0.0392837
\(649\) −6180.40 −0.373809
\(650\) 390.316 0.0235530
\(651\) 0 0
\(652\) −8434.24 −0.506611
\(653\) 15969.6 0.957026 0.478513 0.878080i \(-0.341176\pi\)
0.478513 + 0.878080i \(0.341176\pi\)
\(654\) −4473.16 −0.267453
\(655\) −9412.55 −0.561494
\(656\) −3161.68 −0.188175
\(657\) 8977.11 0.533075
\(658\) 0 0
\(659\) −11596.2 −0.685467 −0.342733 0.939433i \(-0.611353\pi\)
−0.342733 + 0.939433i \(0.611353\pi\)
\(660\) 932.881 0.0550187
\(661\) −12602.2 −0.741558 −0.370779 0.928721i \(-0.620909\pi\)
−0.370779 + 0.928721i \(0.620909\pi\)
\(662\) 17600.1 1.03331
\(663\) −421.729 −0.0247037
\(664\) 1686.90 0.0985912
\(665\) 0 0
\(666\) −7102.05 −0.413212
\(667\) −28455.0 −1.65185
\(668\) −6009.66 −0.348085
\(669\) 6084.80 0.351647
\(670\) 5866.82 0.338291
\(671\) −2592.14 −0.149134
\(672\) 0 0
\(673\) 2126.29 0.121787 0.0608934 0.998144i \(-0.480605\pi\)
0.0608934 + 0.998144i \(0.480605\pi\)
\(674\) −11719.6 −0.669764
\(675\) 1890.79 0.107817
\(676\) −8756.94 −0.498233
\(677\) −2619.38 −0.148702 −0.0743508 0.997232i \(-0.523688\pi\)
−0.0743508 + 0.997232i \(0.523688\pi\)
\(678\) 6262.58 0.354739
\(679\) 0 0
\(680\) 2992.00 0.168732
\(681\) 7271.87 0.409190
\(682\) −2928.17 −0.164407
\(683\) −29929.8 −1.67677 −0.838383 0.545082i \(-0.816499\pi\)
−0.838383 + 0.545082i \(0.816499\pi\)
\(684\) −4501.94 −0.251661
\(685\) −22785.3 −1.27092
\(686\) 0 0
\(687\) −5901.51 −0.327739
\(688\) 5492.70 0.304371
\(689\) 383.265 0.0211919
\(690\) −8107.43 −0.447311
\(691\) −6760.90 −0.372209 −0.186105 0.982530i \(-0.559586\pi\)
−0.186105 + 0.982530i \(0.559586\pi\)
\(692\) −1885.02 −0.103552
\(693\) 0 0
\(694\) 15877.1 0.868423
\(695\) 7515.02 0.410160
\(696\) −3747.17 −0.204075
\(697\) 9967.92 0.541696
\(698\) 19855.5 1.07671
\(699\) −13435.0 −0.726979
\(700\) 0 0
\(701\) 467.205 0.0251727 0.0125864 0.999921i \(-0.495994\pi\)
0.0125864 + 0.999921i \(0.495994\pi\)
\(702\) 150.487 0.00809084
\(703\) 49341.0 2.64713
\(704\) 671.058 0.0359254
\(705\) 13568.1 0.724829
\(706\) 20206.1 1.07715
\(707\) 0 0
\(708\) 7073.23 0.375464
\(709\) 8824.64 0.467442 0.233721 0.972304i \(-0.424910\pi\)
0.233721 + 0.972304i \(0.424910\pi\)
\(710\) −4235.90 −0.223902
\(711\) −7634.38 −0.402688
\(712\) 4427.90 0.233065
\(713\) 25448.0 1.33665
\(714\) 0 0
\(715\) 216.646 0.0113316
\(716\) 5329.95 0.278198
\(717\) −18350.8 −0.955818
\(718\) −9650.54 −0.501609
\(719\) 21098.9 1.09438 0.547188 0.837010i \(-0.315698\pi\)
0.547188 + 0.837010i \(0.315698\pi\)
\(720\) −1067.65 −0.0552623
\(721\) 0 0
\(722\) 17558.9 0.905090
\(723\) 18685.1 0.961145
\(724\) 3990.91 0.204863
\(725\) −10933.8 −0.560100
\(726\) 7326.35 0.374527
\(727\) 4616.48 0.235510 0.117755 0.993043i \(-0.462430\pi\)
0.117755 + 0.993043i \(0.462430\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −14790.7 −0.749903
\(731\) −17317.0 −0.876185
\(732\) 2966.61 0.149794
\(733\) 10688.1 0.538571 0.269285 0.963060i \(-0.413212\pi\)
0.269285 + 0.963060i \(0.413212\pi\)
\(734\) −13870.6 −0.697511
\(735\) 0 0
\(736\) −5831.99 −0.292079
\(737\) −4148.47 −0.207342
\(738\) −3556.89 −0.177413
\(739\) −15367.6 −0.764959 −0.382480 0.923964i \(-0.624930\pi\)
−0.382480 + 0.923964i \(0.624930\pi\)
\(740\) 11701.4 0.581285
\(741\) −1045.50 −0.0518318
\(742\) 0 0
\(743\) 6502.58 0.321072 0.160536 0.987030i \(-0.448678\pi\)
0.160536 + 0.987030i \(0.448678\pi\)
\(744\) 3351.18 0.165135
\(745\) −9130.38 −0.449009
\(746\) 28186.2 1.38334
\(747\) 1897.77 0.0929527
\(748\) −2115.66 −0.103418
\(749\) 0 0
\(750\) −8675.94 −0.422401
\(751\) −19874.1 −0.965670 −0.482835 0.875711i \(-0.660393\pi\)
−0.482835 + 0.875711i \(0.660393\pi\)
\(752\) 9760.07 0.473289
\(753\) 17713.2 0.857247
\(754\) −870.216 −0.0420311
\(755\) 16642.9 0.802250
\(756\) 0 0
\(757\) −15157.8 −0.727765 −0.363883 0.931445i \(-0.618549\pi\)
−0.363883 + 0.931445i \(0.618549\pi\)
\(758\) 10708.3 0.513120
\(759\) 5732.82 0.274161
\(760\) 7417.41 0.354023
\(761\) −35219.8 −1.67768 −0.838842 0.544375i \(-0.816767\pi\)
−0.838842 + 0.544375i \(0.816767\pi\)
\(762\) 12481.0 0.593361
\(763\) 0 0
\(764\) 4906.88 0.232362
\(765\) 3366.00 0.159082
\(766\) 17940.9 0.846257
\(767\) 1642.64 0.0773301
\(768\) −768.000 −0.0360844
\(769\) 2264.35 0.106183 0.0530915 0.998590i \(-0.483093\pi\)
0.0530915 + 0.998590i \(0.483093\pi\)
\(770\) 0 0
\(771\) −1224.67 −0.0572054
\(772\) −13917.2 −0.648821
\(773\) 9532.79 0.443558 0.221779 0.975097i \(-0.428814\pi\)
0.221779 + 0.975097i \(0.428814\pi\)
\(774\) 6179.28 0.286963
\(775\) 9778.38 0.453226
\(776\) 7231.28 0.334520
\(777\) 0 0
\(778\) −7405.16 −0.341244
\(779\) 24711.3 1.13655
\(780\) −247.943 −0.0113818
\(781\) 2995.24 0.137232
\(782\) 18386.7 0.840801
\(783\) −4215.56 −0.192404
\(784\) 0 0
\(785\) −28078.8 −1.27666
\(786\) −7617.16 −0.345668
\(787\) 33213.0 1.50434 0.752170 0.658969i \(-0.229007\pi\)
0.752170 + 0.658969i \(0.229007\pi\)
\(788\) 12773.9 0.577476
\(789\) 13878.0 0.626199
\(790\) 12578.4 0.566481
\(791\) 0 0
\(792\) 754.940 0.0338708
\(793\) 688.945 0.0308514
\(794\) 4166.47 0.186225
\(795\) −3059.01 −0.136468
\(796\) −4260.60 −0.189715
\(797\) −42065.6 −1.86956 −0.934781 0.355223i \(-0.884405\pi\)
−0.934781 + 0.355223i \(0.884405\pi\)
\(798\) 0 0
\(799\) −30770.8 −1.36245
\(800\) −2240.94 −0.0990366
\(801\) 4981.39 0.219736
\(802\) −21268.0 −0.936409
\(803\) 10458.6 0.459622
\(804\) 4747.76 0.208259
\(805\) 0 0
\(806\) 778.255 0.0340110
\(807\) −2613.88 −0.114018
\(808\) 2508.89 0.109236
\(809\) −777.560 −0.0337918 −0.0168959 0.999857i \(-0.505378\pi\)
−0.0168959 + 0.999857i \(0.505378\pi\)
\(810\) −1201.10 −0.0521018
\(811\) −16559.4 −0.716991 −0.358496 0.933531i \(-0.616710\pi\)
−0.358496 + 0.933531i \(0.616710\pi\)
\(812\) 0 0
\(813\) 19417.0 0.837620
\(814\) −8274.11 −0.356275
\(815\) 15633.3 0.671916
\(816\) 2421.30 0.103875
\(817\) −42930.2 −1.83836
\(818\) 13032.7 0.557065
\(819\) 0 0
\(820\) 5860.35 0.249576
\(821\) 231.224 0.00982921 0.00491460 0.999988i \(-0.498436\pi\)
0.00491460 + 0.999988i \(0.498436\pi\)
\(822\) −18439.1 −0.782407
\(823\) −5222.86 −0.221212 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(824\) −1847.80 −0.0781203
\(825\) 2202.84 0.0929611
\(826\) 0 0
\(827\) −46225.5 −1.94367 −0.971836 0.235658i \(-0.924275\pi\)
−0.971836 + 0.235658i \(0.924275\pi\)
\(828\) −6560.99 −0.275375
\(829\) 39594.2 1.65882 0.829410 0.558640i \(-0.188677\pi\)
0.829410 + 0.558640i \(0.188677\pi\)
\(830\) −3126.77 −0.130761
\(831\) −14135.6 −0.590084
\(832\) −178.355 −0.00743191
\(833\) 0 0
\(834\) 6081.58 0.252503
\(835\) 11139.2 0.461663
\(836\) −5244.90 −0.216984
\(837\) 3770.08 0.155690
\(838\) −12159.8 −0.501259
\(839\) 45737.6 1.88205 0.941023 0.338344i \(-0.109867\pi\)
0.941023 + 0.338344i \(0.109867\pi\)
\(840\) 0 0
\(841\) −11.7878 −0.000483326 0
\(842\) −11263.2 −0.460990
\(843\) 21497.0 0.878286
\(844\) 8229.98 0.335649
\(845\) 16231.4 0.660803
\(846\) 10980.1 0.446221
\(847\) 0 0
\(848\) −2200.46 −0.0891088
\(849\) 19041.0 0.769712
\(850\) 7065.08 0.285094
\(851\) 71908.2 2.89657
\(852\) −3427.93 −0.137839
\(853\) 8795.55 0.353053 0.176526 0.984296i \(-0.443514\pi\)
0.176526 + 0.984296i \(0.443514\pi\)
\(854\) 0 0
\(855\) 8344.58 0.333776
\(856\) 1003.89 0.0400844
\(857\) −30254.9 −1.20594 −0.602969 0.797764i \(-0.706016\pi\)
−0.602969 + 0.797764i \(0.706016\pi\)
\(858\) 175.322 0.00697599
\(859\) −44993.5 −1.78715 −0.893573 0.448918i \(-0.851809\pi\)
−0.893573 + 0.448918i \(0.851809\pi\)
\(860\) −10181.0 −0.403685
\(861\) 0 0
\(862\) −7473.79 −0.295311
\(863\) 44667.9 1.76189 0.880946 0.473217i \(-0.156907\pi\)
0.880946 + 0.473217i \(0.156907\pi\)
\(864\) −864.000 −0.0340207
\(865\) 3493.99 0.137340
\(866\) 11514.9 0.451840
\(867\) 7105.31 0.278327
\(868\) 0 0
\(869\) −8894.29 −0.347201
\(870\) 6945.58 0.270663
\(871\) 1102.59 0.0428929
\(872\) 5964.22 0.231621
\(873\) 8135.19 0.315389
\(874\) 45582.1 1.76411
\(875\) 0 0
\(876\) −11969.5 −0.461657
\(877\) 5986.84 0.230515 0.115257 0.993336i \(-0.463231\pi\)
0.115257 + 0.993336i \(0.463231\pi\)
\(878\) 19625.5 0.754361
\(879\) 27701.3 1.06296
\(880\) −1243.84 −0.0476476
\(881\) −37911.8 −1.44981 −0.724904 0.688850i \(-0.758116\pi\)
−0.724904 + 0.688850i \(0.758116\pi\)
\(882\) 0 0
\(883\) −16293.6 −0.620978 −0.310489 0.950577i \(-0.600493\pi\)
−0.310489 + 0.950577i \(0.600493\pi\)
\(884\) 562.305 0.0213941
\(885\) −13110.6 −0.497975
\(886\) 11660.6 0.442151
\(887\) −8853.41 −0.335139 −0.167570 0.985860i \(-0.553592\pi\)
−0.167570 + 0.985860i \(0.553592\pi\)
\(888\) 9469.40 0.357852
\(889\) 0 0
\(890\) −8207.35 −0.309113
\(891\) 849.308 0.0319336
\(892\) −8113.07 −0.304536
\(893\) −76283.4 −2.85859
\(894\) −7388.82 −0.276420
\(895\) −9879.34 −0.368972
\(896\) 0 0
\(897\) −1523.68 −0.0567159
\(898\) −17349.9 −0.644736
\(899\) −21801.1 −0.808796
\(900\) −2521.06 −0.0933726
\(901\) 6937.47 0.256516
\(902\) −4143.89 −0.152967
\(903\) 0 0
\(904\) −8350.10 −0.307213
\(905\) −7397.36 −0.271709
\(906\) 13468.4 0.493883
\(907\) 16662.5 0.609998 0.304999 0.952353i \(-0.401344\pi\)
0.304999 + 0.952353i \(0.401344\pi\)
\(908\) −9695.83 −0.354369
\(909\) 2822.50 0.102988
\(910\) 0 0
\(911\) −29071.2 −1.05727 −0.528635 0.848849i \(-0.677296\pi\)
−0.528635 + 0.848849i \(0.677296\pi\)
\(912\) 6002.58 0.217945
\(913\) 2210.96 0.0801446
\(914\) 18213.6 0.659140
\(915\) −5498.77 −0.198671
\(916\) 7868.67 0.283830
\(917\) 0 0
\(918\) 2723.96 0.0979346
\(919\) 7973.39 0.286200 0.143100 0.989708i \(-0.454293\pi\)
0.143100 + 0.989708i \(0.454293\pi\)
\(920\) 10809.9 0.387383
\(921\) 20359.6 0.728416
\(922\) 17459.4 0.623638
\(923\) −796.079 −0.0283892
\(924\) 0 0
\(925\) 27630.7 0.982154
\(926\) −3591.24 −0.127447
\(927\) −2078.77 −0.0736525
\(928\) 4996.23 0.176734
\(929\) −30141.5 −1.06449 −0.532244 0.846591i \(-0.678651\pi\)
−0.532244 + 0.846591i \(0.678651\pi\)
\(930\) −6211.59 −0.219017
\(931\) 0 0
\(932\) 17913.3 0.629582
\(933\) 15410.3 0.540740
\(934\) −261.119 −0.00914783
\(935\) 3921.50 0.137162
\(936\) −200.649 −0.00700687
\(937\) −1126.37 −0.0392709 −0.0196354 0.999807i \(-0.506251\pi\)
−0.0196354 + 0.999807i \(0.506251\pi\)
\(938\) 0 0
\(939\) −11290.7 −0.392393
\(940\) −18090.8 −0.627720
\(941\) 10333.3 0.357976 0.178988 0.983851i \(-0.442718\pi\)
0.178988 + 0.983851i \(0.442718\pi\)
\(942\) −22722.9 −0.785938
\(943\) 36013.5 1.24365
\(944\) −9430.97 −0.325161
\(945\) 0 0
\(946\) 7199.06 0.247422
\(947\) 24810.3 0.851347 0.425673 0.904877i \(-0.360037\pi\)
0.425673 + 0.904877i \(0.360037\pi\)
\(948\) 10179.2 0.348738
\(949\) −2779.71 −0.0950824
\(950\) 17514.9 0.598167
\(951\) 3104.84 0.105869
\(952\) 0 0
\(953\) −15048.6 −0.511513 −0.255757 0.966741i \(-0.582325\pi\)
−0.255757 + 0.966741i \(0.582325\pi\)
\(954\) −2475.52 −0.0840126
\(955\) −9095.17 −0.308181
\(956\) 24467.7 0.827763
\(957\) −4911.26 −0.165892
\(958\) −23691.6 −0.798999
\(959\) 0 0
\(960\) 1423.53 0.0478585
\(961\) −10293.8 −0.345533
\(962\) 2199.11 0.0737028
\(963\) 1129.38 0.0377919
\(964\) −24913.5 −0.832376
\(965\) 25796.2 0.860528
\(966\) 0 0
\(967\) −15619.9 −0.519442 −0.259721 0.965684i \(-0.583631\pi\)
−0.259721 + 0.965684i \(0.583631\pi\)
\(968\) −9768.47 −0.324350
\(969\) −18924.5 −0.627392
\(970\) −13403.6 −0.443672
\(971\) 24833.0 0.820730 0.410365 0.911921i \(-0.365401\pi\)
0.410365 + 0.911921i \(0.365401\pi\)
\(972\) −972.000 −0.0320750
\(973\) 0 0
\(974\) −9614.42 −0.316289
\(975\) −585.473 −0.0192309
\(976\) −3955.48 −0.129725
\(977\) 38656.8 1.26586 0.632928 0.774211i \(-0.281853\pi\)
0.632928 + 0.774211i \(0.281853\pi\)
\(978\) 12651.4 0.413646
\(979\) 5803.47 0.189458
\(980\) 0 0
\(981\) 6709.75 0.218375
\(982\) 12136.1 0.394377
\(983\) −26664.1 −0.865162 −0.432581 0.901595i \(-0.642397\pi\)
−0.432581 + 0.901595i \(0.642397\pi\)
\(984\) 4742.52 0.153644
\(985\) −23677.1 −0.765903
\(986\) −15751.7 −0.508760
\(987\) 0 0
\(988\) 1394.00 0.0448876
\(989\) −62565.2 −2.01158
\(990\) −1399.32 −0.0449226
\(991\) 27228.2 0.872786 0.436393 0.899756i \(-0.356256\pi\)
0.436393 + 0.899756i \(0.356256\pi\)
\(992\) −4468.24 −0.143011
\(993\) −26400.2 −0.843690
\(994\) 0 0
\(995\) 7897.24 0.251617
\(996\) −2530.35 −0.0804994
\(997\) −5162.91 −0.164003 −0.0820016 0.996632i \(-0.526131\pi\)
−0.0820016 + 0.996632i \(0.526131\pi\)
\(998\) −31012.8 −0.983662
\(999\) 10653.1 0.337386
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 294.4.a.l.1.1 2
3.2 odd 2 882.4.a.bb.1.2 2
4.3 odd 2 2352.4.a.bw.1.1 2
7.2 even 3 294.4.e.m.67.2 4
7.3 odd 6 294.4.e.k.79.1 4
7.4 even 3 294.4.e.m.79.2 4
7.5 odd 6 294.4.e.k.67.1 4
7.6 odd 2 294.4.a.o.1.2 yes 2
21.2 odd 6 882.4.g.be.361.1 4
21.5 even 6 882.4.g.bk.361.2 4
21.11 odd 6 882.4.g.be.667.1 4
21.17 even 6 882.4.g.bk.667.2 4
21.20 even 2 882.4.a.t.1.1 2
28.27 even 2 2352.4.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.l.1.1 2 1.1 even 1 trivial
294.4.a.o.1.2 yes 2 7.6 odd 2
294.4.e.k.67.1 4 7.5 odd 6
294.4.e.k.79.1 4 7.3 odd 6
294.4.e.m.67.2 4 7.2 even 3
294.4.e.m.79.2 4 7.4 even 3
882.4.a.t.1.1 2 21.20 even 2
882.4.a.bb.1.2 2 3.2 odd 2
882.4.g.be.361.1 4 21.2 odd 6
882.4.g.be.667.1 4 21.11 odd 6
882.4.g.bk.361.2 4 21.5 even 6
882.4.g.bk.667.2 4 21.17 even 6
2352.4.a.bu.1.2 2 28.27 even 2
2352.4.a.bw.1.1 2 4.3 odd 2