Properties

Label 6762.2.a.cn.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 14\)
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 \(-1.87996\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.879961 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.879961 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.879961 q^{10} -4.07288 q^{11} -1.00000 q^{12} +2.22129 q^{13} +0.879961 q^{15} +1.00000 q^{16} +7.75992 q^{17} +1.00000 q^{18} +0.585786 q^{19} -0.879961 q^{20} -4.07288 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.22567 q^{25} +2.22129 q^{26} -1.00000 q^{27} +4.60713 q^{29} +0.879961 q^{30} +0.585786 q^{31} +1.00000 q^{32} +4.07288 q^{33} +7.75992 q^{34} +1.00000 q^{36} +3.53863 q^{37} +0.585786 q^{38} -2.22129 q^{39} -0.879961 q^{40} -0.124413 q^{41} -5.85158 q^{43} -4.07288 q^{44} -0.879961 q^{45} -1.00000 q^{46} +0.0515337 q^{47} -1.00000 q^{48} -4.22567 q^{50} -7.75992 q^{51} +2.22129 q^{52} -7.34390 q^{53} -1.00000 q^{54} +3.58397 q^{55} -0.585786 q^{57} +4.60713 q^{58} -4.90131 q^{59} +0.879961 q^{60} +2.14320 q^{61} +0.585786 q^{62} +1.00000 q^{64} -1.95465 q^{65} +4.07288 q^{66} -8.58835 q^{67} +7.75992 q^{68} +1.00000 q^{69} +4.07288 q^{71} +1.00000 q^{72} +4.97162 q^{73} +3.53863 q^{74} +4.22567 q^{75} +0.585786 q^{76} -2.22129 q^{78} -1.92712 q^{79} -0.879961 q^{80} +1.00000 q^{81} -0.124413 q^{82} +6.99819 q^{83} -6.82843 q^{85} -5.85158 q^{86} -4.60713 q^{87} -4.07288 q^{88} +10.4587 q^{89} -0.879961 q^{90} -1.00000 q^{92} -0.585786 q^{93} +0.0515337 q^{94} -0.515469 q^{95} -1.00000 q^{96} +17.5084 q^{97} -4.07288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 6q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + 6q^{10} - 4q^{12} + 10q^{13} - 6q^{15} + 4q^{16} + 12q^{17} + 4q^{18} + 8q^{19} + 6q^{20} - 4q^{23} - 4q^{24} + 6q^{25} + 10q^{26} - 4q^{27} + 6q^{29} - 6q^{30} + 8q^{31} + 4q^{32} + 12q^{34} + 4q^{36} - 6q^{37} + 8q^{38} - 10q^{39} + 6q^{40} + 14q^{41} - 6q^{43} + 6q^{45} - 4q^{46} + 2q^{47} - 4q^{48} + 6q^{50} - 12q^{51} + 10q^{52} - 4q^{53} - 4q^{54} + 8q^{55} - 8q^{57} + 6q^{58} + 8q^{59} - 6q^{60} + 12q^{61} + 8q^{62} + 4q^{64} + 6q^{65} - 4q^{67} + 12q^{68} + 4q^{69} + 4q^{72} + 12q^{73} - 6q^{74} - 6q^{75} + 8q^{76} - 10q^{78} - 24q^{79} + 6q^{80} + 4q^{81} + 14q^{82} + 16q^{83} - 16q^{85} - 6q^{86} - 6q^{87} + 12q^{89} + 6q^{90} - 4q^{92} - 8q^{93} + 2q^{94} + 12q^{95} - 4q^{96} + 30q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.879961 −0.393530 −0.196765 0.980451i \(-0.563044\pi\)
−0.196765 + 0.980451i \(0.563044\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.879961 −0.278268
\(11\) −4.07288 −1.22802 −0.614010 0.789298i \(-0.710444\pi\)
−0.614010 + 0.789298i \(0.710444\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.22129 0.616076 0.308038 0.951374i \(-0.400328\pi\)
0.308038 + 0.951374i \(0.400328\pi\)
\(14\) 0 0
\(15\) 0.879961 0.227205
\(16\) 1.00000 0.250000
\(17\) 7.75992 1.88206 0.941029 0.338327i \(-0.109861\pi\)
0.941029 + 0.338327i \(0.109861\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.585786 0.134389 0.0671943 0.997740i \(-0.478595\pi\)
0.0671943 + 0.997740i \(0.478595\pi\)
\(20\) −0.879961 −0.196765
\(21\) 0 0
\(22\) −4.07288 −0.868341
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.22567 −0.845134
\(26\) 2.22129 0.435632
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.60713 0.855523 0.427762 0.903892i \(-0.359302\pi\)
0.427762 + 0.903892i \(0.359302\pi\)
\(30\) 0.879961 0.160658
\(31\) 0.585786 0.105210 0.0526052 0.998615i \(-0.483248\pi\)
0.0526052 + 0.998615i \(0.483248\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.07288 0.708997
\(34\) 7.75992 1.33082
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 3.53863 0.581747 0.290873 0.956762i \(-0.406054\pi\)
0.290873 + 0.956762i \(0.406054\pi\)
\(38\) 0.585786 0.0950271
\(39\) −2.22129 −0.355692
\(40\) −0.879961 −0.139134
\(41\) −0.124413 −0.0194301 −0.00971505 0.999953i \(-0.503092\pi\)
−0.00971505 + 0.999953i \(0.503092\pi\)
\(42\) 0 0
\(43\) −5.85158 −0.892358 −0.446179 0.894944i \(-0.647216\pi\)
−0.446179 + 0.894944i \(0.647216\pi\)
\(44\) −4.07288 −0.614010
\(45\) −0.879961 −0.131177
\(46\) −1.00000 −0.147442
\(47\) 0.0515337 0.00751696 0.00375848 0.999993i \(-0.498804\pi\)
0.00375848 + 0.999993i \(0.498804\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.22567 −0.597600
\(51\) −7.75992 −1.08661
\(52\) 2.22129 0.308038
\(53\) −7.34390 −1.00876 −0.504381 0.863481i \(-0.668279\pi\)
−0.504381 + 0.863481i \(0.668279\pi\)
\(54\) −1.00000 −0.136083
\(55\) 3.58397 0.483263
\(56\) 0 0
\(57\) −0.585786 −0.0775893
\(58\) 4.60713 0.604946
\(59\) −4.90131 −0.638096 −0.319048 0.947739i \(-0.603363\pi\)
−0.319048 + 0.947739i \(0.603363\pi\)
\(60\) 0.879961 0.113602
\(61\) 2.14320 0.274408 0.137204 0.990543i \(-0.456188\pi\)
0.137204 + 0.990543i \(0.456188\pi\)
\(62\) 0.585786 0.0743950
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.95465 −0.242445
\(66\) 4.07288 0.501337
\(67\) −8.58835 −1.04923 −0.524617 0.851338i \(-0.675791\pi\)
−0.524617 + 0.851338i \(0.675791\pi\)
\(68\) 7.75992 0.941029
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 4.07288 0.483362 0.241681 0.970356i \(-0.422301\pi\)
0.241681 + 0.970356i \(0.422301\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.97162 0.581885 0.290942 0.956741i \(-0.406031\pi\)
0.290942 + 0.956741i \(0.406031\pi\)
\(74\) 3.53863 0.411357
\(75\) 4.22567 0.487938
\(76\) 0.585786 0.0671943
\(77\) 0 0
\(78\) −2.22129 −0.251512
\(79\) −1.92712 −0.216818 −0.108409 0.994106i \(-0.534576\pi\)
−0.108409 + 0.994106i \(0.534576\pi\)
\(80\) −0.879961 −0.0983826
\(81\) 1.00000 0.111111
\(82\) −0.124413 −0.0137392
\(83\) 6.99819 0.768151 0.384075 0.923302i \(-0.374520\pi\)
0.384075 + 0.923302i \(0.374520\pi\)
\(84\) 0 0
\(85\) −6.82843 −0.740647
\(86\) −5.85158 −0.630993
\(87\) −4.60713 −0.493936
\(88\) −4.07288 −0.434170
\(89\) 10.4587 1.10862 0.554311 0.832310i \(-0.312982\pi\)
0.554311 + 0.832310i \(0.312982\pi\)
\(90\) −0.879961 −0.0927560
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −0.585786 −0.0607432
\(94\) 0.0515337 0.00531530
\(95\) −0.515469 −0.0528860
\(96\) −1.00000 −0.102062
\(97\) 17.5084 1.77771 0.888856 0.458186i \(-0.151501\pi\)
0.888856 + 0.458186i \(0.151501\pi\)
\(98\) 0 0
\(99\) −4.07288 −0.409340
\(100\) −4.22567 −0.422567
\(101\) −4.41240 −0.439050 −0.219525 0.975607i \(-0.570451\pi\)
−0.219525 + 0.975607i \(0.570451\pi\)
\(102\) −7.75992 −0.768347
\(103\) 3.77871 0.372327 0.186163 0.982519i \(-0.440395\pi\)
0.186163 + 0.982519i \(0.440395\pi\)
\(104\) 2.22129 0.217816
\(105\) 0 0
\(106\) −7.34390 −0.713302
\(107\) −3.61416 −0.349394 −0.174697 0.984622i \(-0.555895\pi\)
−0.174697 + 0.984622i \(0.555895\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.42381 0.615289 0.307645 0.951501i \(-0.400459\pi\)
0.307645 + 0.951501i \(0.400459\pi\)
\(110\) 3.58397 0.341719
\(111\) −3.53863 −0.335872
\(112\) 0 0
\(113\) 13.6115 1.28046 0.640231 0.768182i \(-0.278838\pi\)
0.640231 + 0.768182i \(0.278838\pi\)
\(114\) −0.585786 −0.0548639
\(115\) 0.879961 0.0820568
\(116\) 4.60713 0.427762
\(117\) 2.22129 0.205359
\(118\) −4.90131 −0.451202
\(119\) 0 0
\(120\) 0.879961 0.0803291
\(121\) 5.58835 0.508032
\(122\) 2.14320 0.194036
\(123\) 0.124413 0.0112180
\(124\) 0.585786 0.0526052
\(125\) 8.11823 0.726116
\(126\) 0 0
\(127\) 5.92446 0.525711 0.262856 0.964835i \(-0.415336\pi\)
0.262856 + 0.964835i \(0.415336\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.85158 0.515203
\(130\) −1.95465 −0.171434
\(131\) 6.76430 0.590999 0.295500 0.955343i \(-0.404514\pi\)
0.295500 + 0.955343i \(0.404514\pi\)
\(132\) 4.07288 0.354499
\(133\) 0 0
\(134\) −8.58835 −0.741920
\(135\) 0.879961 0.0757350
\(136\) 7.75992 0.665408
\(137\) 5.56881 0.475776 0.237888 0.971293i \(-0.423545\pi\)
0.237888 + 0.971293i \(0.423545\pi\)
\(138\) 1.00000 0.0851257
\(139\) 12.8258 1.08787 0.543934 0.839128i \(-0.316934\pi\)
0.543934 + 0.839128i \(0.316934\pi\)
\(140\) 0 0
\(141\) −0.0515337 −0.00433992
\(142\) 4.07288 0.341788
\(143\) −9.04707 −0.756554
\(144\) 1.00000 0.0833333
\(145\) −4.05410 −0.336674
\(146\) 4.97162 0.411455
\(147\) 0 0
\(148\) 3.53863 0.290873
\(149\) 13.9322 1.14137 0.570687 0.821168i \(-0.306677\pi\)
0.570687 + 0.821168i \(0.306677\pi\)
\(150\) 4.22567 0.345024
\(151\) 1.15279 0.0938127 0.0469063 0.998899i \(-0.485064\pi\)
0.0469063 + 0.998899i \(0.485064\pi\)
\(152\) 0.585786 0.0475136
\(153\) 7.75992 0.627352
\(154\) 0 0
\(155\) −0.515469 −0.0414035
\(156\) −2.22129 −0.177846
\(157\) 15.6970 1.25276 0.626378 0.779520i \(-0.284537\pi\)
0.626378 + 0.779520i \(0.284537\pi\)
\(158\) −1.92712 −0.153313
\(159\) 7.34390 0.582409
\(160\) −0.879961 −0.0695670
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.48528 −0.664619 −0.332309 0.943170i \(-0.607828\pi\)
−0.332309 + 0.943170i \(0.607828\pi\)
\(164\) −0.124413 −0.00971505
\(165\) −3.58397 −0.279012
\(166\) 6.99819 0.543165
\(167\) 19.0409 1.47343 0.736714 0.676205i \(-0.236377\pi\)
0.736714 + 0.676205i \(0.236377\pi\)
\(168\) 0 0
\(169\) −8.06585 −0.620450
\(170\) −6.82843 −0.523716
\(171\) 0.585786 0.0447962
\(172\) −5.85158 −0.446179
\(173\) −12.0087 −0.913008 −0.456504 0.889721i \(-0.650899\pi\)
−0.456504 + 0.889721i \(0.650899\pi\)
\(174\) −4.60713 −0.349266
\(175\) 0 0
\(176\) −4.07288 −0.307005
\(177\) 4.90131 0.368405
\(178\) 10.4587 0.783914
\(179\) 10.6461 0.795725 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(180\) −0.879961 −0.0655884
\(181\) −21.5385 −1.60095 −0.800473 0.599368i \(-0.795418\pi\)
−0.800473 + 0.599368i \(0.795418\pi\)
\(182\) 0 0
\(183\) −2.14320 −0.158430
\(184\) −1.00000 −0.0737210
\(185\) −3.11385 −0.228935
\(186\) −0.585786 −0.0429519
\(187\) −31.6052 −2.31120
\(188\) 0.0515337 0.00375848
\(189\) 0 0
\(190\) −0.515469 −0.0373961
\(191\) 25.2194 1.82481 0.912405 0.409287i \(-0.134223\pi\)
0.912405 + 0.409287i \(0.134223\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.50844 −0.540469 −0.270235 0.962794i \(-0.587101\pi\)
−0.270235 + 0.962794i \(0.587101\pi\)
\(194\) 17.5084 1.25703
\(195\) 1.95465 0.139976
\(196\) 0 0
\(197\) 6.66388 0.474782 0.237391 0.971414i \(-0.423708\pi\)
0.237391 + 0.971414i \(0.423708\pi\)
\(198\) −4.07288 −0.289447
\(199\) −4.80964 −0.340947 −0.170473 0.985362i \(-0.554530\pi\)
−0.170473 + 0.985362i \(0.554530\pi\)
\(200\) −4.22567 −0.298800
\(201\) 8.58835 0.605775
\(202\) −4.41240 −0.310456
\(203\) 0 0
\(204\) −7.75992 −0.543303
\(205\) 0.109479 0.00764634
\(206\) 3.77871 0.263275
\(207\) −1.00000 −0.0695048
\(208\) 2.22129 0.154019
\(209\) −2.38584 −0.165032
\(210\) 0 0
\(211\) 4.79824 0.330324 0.165162 0.986266i \(-0.447185\pi\)
0.165162 + 0.986266i \(0.447185\pi\)
\(212\) −7.34390 −0.504381
\(213\) −4.07288 −0.279069
\(214\) −3.61416 −0.247059
\(215\) 5.14917 0.351170
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.42381 0.435075
\(219\) −4.97162 −0.335951
\(220\) 3.58397 0.241632
\(221\) 17.2371 1.15949
\(222\) −3.53863 −0.237497
\(223\) −14.5983 −0.977574 −0.488787 0.872403i \(-0.662560\pi\)
−0.488787 + 0.872403i \(0.662560\pi\)
\(224\) 0 0
\(225\) −4.22567 −0.281711
\(226\) 13.6115 0.905424
\(227\) −9.39543 −0.623597 −0.311798 0.950148i \(-0.600931\pi\)
−0.311798 + 0.950148i \(0.600931\pi\)
\(228\) −0.585786 −0.0387947
\(229\) 7.14757 0.472325 0.236162 0.971714i \(-0.424110\pi\)
0.236162 + 0.971714i \(0.424110\pi\)
\(230\) 0.879961 0.0580229
\(231\) 0 0
\(232\) 4.60713 0.302473
\(233\) 15.3439 1.00521 0.502606 0.864516i \(-0.332375\pi\)
0.502606 + 0.864516i \(0.332375\pi\)
\(234\) 2.22129 0.145211
\(235\) −0.0453476 −0.00295815
\(236\) −4.90131 −0.319048
\(237\) 1.92712 0.125180
\(238\) 0 0
\(239\) 17.7687 1.14936 0.574680 0.818378i \(-0.305127\pi\)
0.574680 + 0.818378i \(0.305127\pi\)
\(240\) 0.879961 0.0568012
\(241\) 5.07629 0.326992 0.163496 0.986544i \(-0.447723\pi\)
0.163496 + 0.986544i \(0.447723\pi\)
\(242\) 5.58835 0.359233
\(243\) −1.00000 −0.0641500
\(244\) 2.14320 0.137204
\(245\) 0 0
\(246\) 0.124413 0.00793231
\(247\) 1.30120 0.0827936
\(248\) 0.585786 0.0371975
\(249\) −6.99819 −0.443492
\(250\) 8.11823 0.513442
\(251\) 13.7473 0.867723 0.433862 0.900979i \(-0.357151\pi\)
0.433862 + 0.900979i \(0.357151\pi\)
\(252\) 0 0
\(253\) 4.07288 0.256060
\(254\) 5.92446 0.371734
\(255\) 6.82843 0.427613
\(256\) 1.00000 0.0625000
\(257\) 11.0069 0.686594 0.343297 0.939227i \(-0.388456\pi\)
0.343297 + 0.939227i \(0.388456\pi\)
\(258\) 5.85158 0.364304
\(259\) 0 0
\(260\) −1.95465 −0.121222
\(261\) 4.60713 0.285174
\(262\) 6.76430 0.417900
\(263\) −26.7153 −1.64734 −0.823669 0.567071i \(-0.808076\pi\)
−0.823669 + 0.567071i \(0.808076\pi\)
\(264\) 4.07288 0.250668
\(265\) 6.46234 0.396978
\(266\) 0 0
\(267\) −10.4587 −0.640063
\(268\) −8.58835 −0.524617
\(269\) 17.1693 1.04683 0.523416 0.852077i \(-0.324657\pi\)
0.523416 + 0.852077i \(0.324657\pi\)
\(270\) 0.879961 0.0535527
\(271\) −17.5260 −1.06463 −0.532315 0.846546i \(-0.678678\pi\)
−0.532315 + 0.846546i \(0.678678\pi\)
\(272\) 7.75992 0.470514
\(273\) 0 0
\(274\) 5.56881 0.336424
\(275\) 17.2106 1.03784
\(276\) 1.00000 0.0601929
\(277\) −5.00075 −0.300466 −0.150233 0.988651i \(-0.548002\pi\)
−0.150233 + 0.988651i \(0.548002\pi\)
\(278\) 12.8258 0.769239
\(279\) 0.585786 0.0350701
\(280\) 0 0
\(281\) 4.73676 0.282572 0.141286 0.989969i \(-0.454876\pi\)
0.141286 + 0.989969i \(0.454876\pi\)
\(282\) −0.0515337 −0.00306879
\(283\) −5.12101 −0.304412 −0.152206 0.988349i \(-0.548638\pi\)
−0.152206 + 0.988349i \(0.548638\pi\)
\(284\) 4.07288 0.241681
\(285\) 0.515469 0.0305338
\(286\) −9.04707 −0.534964
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 43.2164 2.54214
\(290\) −4.05410 −0.238065
\(291\) −17.5084 −1.02636
\(292\) 4.97162 0.290942
\(293\) 9.45315 0.552259 0.276129 0.961120i \(-0.410948\pi\)
0.276129 + 0.961120i \(0.410948\pi\)
\(294\) 0 0
\(295\) 4.31296 0.251110
\(296\) 3.53863 0.205679
\(297\) 4.07288 0.236332
\(298\) 13.9322 0.807073
\(299\) −2.22129 −0.128461
\(300\) 4.22567 0.243969
\(301\) 0 0
\(302\) 1.15279 0.0663356
\(303\) 4.41240 0.253486
\(304\) 0.585786 0.0335972
\(305\) −1.88593 −0.107988
\(306\) 7.75992 0.443605
\(307\) 29.8036 1.70098 0.850490 0.525991i \(-0.176305\pi\)
0.850490 + 0.525991i \(0.176305\pi\)
\(308\) 0 0
\(309\) −3.77871 −0.214963
\(310\) −0.515469 −0.0292767
\(311\) −18.5983 −1.05461 −0.527306 0.849675i \(-0.676798\pi\)
−0.527306 + 0.849675i \(0.676798\pi\)
\(312\) −2.22129 −0.125756
\(313\) 17.1803 0.971089 0.485545 0.874212i \(-0.338621\pi\)
0.485545 + 0.874212i \(0.338621\pi\)
\(314\) 15.6970 0.885832
\(315\) 0 0
\(316\) −1.92712 −0.108409
\(317\) 2.45775 0.138041 0.0690205 0.997615i \(-0.478013\pi\)
0.0690205 + 0.997615i \(0.478013\pi\)
\(318\) 7.34390 0.411825
\(319\) −18.7643 −1.05060
\(320\) −0.879961 −0.0491913
\(321\) 3.61416 0.201723
\(322\) 0 0
\(323\) 4.54566 0.252927
\(324\) 1.00000 0.0555556
\(325\) −9.38646 −0.520667
\(326\) −8.48528 −0.469956
\(327\) −6.42381 −0.355237
\(328\) −0.124413 −0.00686958
\(329\) 0 0
\(330\) −3.58397 −0.197291
\(331\) 19.4595 1.06959 0.534795 0.844982i \(-0.320389\pi\)
0.534795 + 0.844982i \(0.320389\pi\)
\(332\) 6.99819 0.384075
\(333\) 3.53863 0.193916
\(334\) 19.0409 1.04187
\(335\) 7.55741 0.412905
\(336\) 0 0
\(337\) 26.0213 1.41747 0.708734 0.705476i \(-0.249267\pi\)
0.708734 + 0.705476i \(0.249267\pi\)
\(338\) −8.06585 −0.438724
\(339\) −13.6115 −0.739276
\(340\) −6.82843 −0.370323
\(341\) −2.38584 −0.129200
\(342\) 0.585786 0.0316757
\(343\) 0 0
\(344\) −5.85158 −0.315496
\(345\) −0.879961 −0.0473755
\(346\) −12.0087 −0.645594
\(347\) −31.4229 −1.68687 −0.843434 0.537233i \(-0.819470\pi\)
−0.843434 + 0.537233i \(0.819470\pi\)
\(348\) −4.60713 −0.246968
\(349\) −13.3688 −0.715614 −0.357807 0.933796i \(-0.616475\pi\)
−0.357807 + 0.933796i \(0.616475\pi\)
\(350\) 0 0
\(351\) −2.22129 −0.118564
\(352\) −4.07288 −0.217085
\(353\) −15.8005 −0.840973 −0.420487 0.907299i \(-0.638141\pi\)
−0.420487 + 0.907299i \(0.638141\pi\)
\(354\) 4.90131 0.260502
\(355\) −3.58397 −0.190218
\(356\) 10.4587 0.554311
\(357\) 0 0
\(358\) 10.6461 0.562662
\(359\) 26.9642 1.42311 0.711557 0.702629i \(-0.247990\pi\)
0.711557 + 0.702629i \(0.247990\pi\)
\(360\) −0.879961 −0.0463780
\(361\) −18.6569 −0.981940
\(362\) −21.5385 −1.13204
\(363\) −5.58835 −0.293312
\(364\) 0 0
\(365\) −4.37483 −0.228989
\(366\) −2.14320 −0.112027
\(367\) 5.67201 0.296077 0.148038 0.988982i \(-0.452704\pi\)
0.148038 + 0.988982i \(0.452704\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −0.124413 −0.00647670
\(370\) −3.11385 −0.161882
\(371\) 0 0
\(372\) −0.585786 −0.0303716
\(373\) 7.49704 0.388182 0.194091 0.980984i \(-0.437824\pi\)
0.194091 + 0.980984i \(0.437824\pi\)
\(374\) −31.6052 −1.63427
\(375\) −8.11823 −0.419223
\(376\) 0.0515337 0.00265765
\(377\) 10.2338 0.527067
\(378\) 0 0
\(379\) −2.24273 −0.115202 −0.0576008 0.998340i \(-0.518345\pi\)
−0.0576008 + 0.998340i \(0.518345\pi\)
\(380\) −0.515469 −0.0264430
\(381\) −5.92446 −0.303520
\(382\) 25.2194 1.29034
\(383\) 24.0693 1.22988 0.614941 0.788573i \(-0.289180\pi\)
0.614941 + 0.788573i \(0.289180\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −7.50844 −0.382169
\(387\) −5.85158 −0.297453
\(388\) 17.5084 0.888856
\(389\) −11.9021 −0.603458 −0.301729 0.953394i \(-0.597564\pi\)
−0.301729 + 0.953394i \(0.597564\pi\)
\(390\) 1.95465 0.0989777
\(391\) −7.75992 −0.392436
\(392\) 0 0
\(393\) −6.76430 −0.341214
\(394\) 6.66388 0.335722
\(395\) 1.69579 0.0853245
\(396\) −4.07288 −0.204670
\(397\) −12.7305 −0.638925 −0.319462 0.947599i \(-0.603502\pi\)
−0.319462 + 0.947599i \(0.603502\pi\)
\(398\) −4.80964 −0.241086
\(399\) 0 0
\(400\) −4.22567 −0.211283
\(401\) −24.8032 −1.23861 −0.619307 0.785149i \(-0.712586\pi\)
−0.619307 + 0.785149i \(0.712586\pi\)
\(402\) 8.58835 0.428348
\(403\) 1.30120 0.0648176
\(404\) −4.41240 −0.219525
\(405\) −0.879961 −0.0437256
\(406\) 0 0
\(407\) −14.4124 −0.714396
\(408\) −7.75992 −0.384173
\(409\) −13.1174 −0.648613 −0.324306 0.945952i \(-0.605131\pi\)
−0.324306 + 0.945952i \(0.605131\pi\)
\(410\) 0.109479 0.00540678
\(411\) −5.56881 −0.274689
\(412\) 3.77871 0.186163
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −6.15813 −0.302291
\(416\) 2.22129 0.108908
\(417\) −12.8258 −0.628081
\(418\) −2.38584 −0.116695
\(419\) 31.2417 1.52626 0.763129 0.646247i \(-0.223662\pi\)
0.763129 + 0.646247i \(0.223662\pi\)
\(420\) 0 0
\(421\) −38.1129 −1.85751 −0.928755 0.370694i \(-0.879120\pi\)
−0.928755 + 0.370694i \(0.879120\pi\)
\(422\) 4.79824 0.233575
\(423\) 0.0515337 0.00250565
\(424\) −7.34390 −0.356651
\(425\) −32.7909 −1.59059
\(426\) −4.07288 −0.197332
\(427\) 0 0
\(428\) −3.61416 −0.174697
\(429\) 9.04707 0.436796
\(430\) 5.14917 0.248315
\(431\) 29.4031 1.41630 0.708149 0.706063i \(-0.249530\pi\)
0.708149 + 0.706063i \(0.249530\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.8309 −0.712727 −0.356364 0.934347i \(-0.615984\pi\)
−0.356364 + 0.934347i \(0.615984\pi\)
\(434\) 0 0
\(435\) 4.05410 0.194379
\(436\) 6.42381 0.307645
\(437\) −0.585786 −0.0280220
\(438\) −4.97162 −0.237553
\(439\) 0.299392 0.0142892 0.00714461 0.999974i \(-0.497726\pi\)
0.00714461 + 0.999974i \(0.497726\pi\)
\(440\) 3.58397 0.170859
\(441\) 0 0
\(442\) 17.2371 0.819884
\(443\) 10.5202 0.499829 0.249915 0.968268i \(-0.419597\pi\)
0.249915 + 0.968268i \(0.419597\pi\)
\(444\) −3.53863 −0.167936
\(445\) −9.20326 −0.436276
\(446\) −14.5983 −0.691249
\(447\) −13.9322 −0.658973
\(448\) 0 0
\(449\) −40.9505 −1.93257 −0.966287 0.257469i \(-0.917112\pi\)
−0.966287 + 0.257469i \(0.917112\pi\)
\(450\) −4.22567 −0.199200
\(451\) 0.506721 0.0238605
\(452\) 13.6115 0.640231
\(453\) −1.15279 −0.0541628
\(454\) −9.39543 −0.440949
\(455\) 0 0
\(456\) −0.585786 −0.0274320
\(457\) 15.1539 0.708869 0.354435 0.935081i \(-0.384673\pi\)
0.354435 + 0.935081i \(0.384673\pi\)
\(458\) 7.14757 0.333984
\(459\) −7.75992 −0.362202
\(460\) 0.879961 0.0410284
\(461\) 22.0089 1.02506 0.512528 0.858671i \(-0.328709\pi\)
0.512528 + 0.858671i \(0.328709\pi\)
\(462\) 0 0
\(463\) −1.83546 −0.0853009 −0.0426505 0.999090i \(-0.513580\pi\)
−0.0426505 + 0.999090i \(0.513580\pi\)
\(464\) 4.60713 0.213881
\(465\) 0.515469 0.0239043
\(466\) 15.3439 0.710792
\(467\) −22.4586 −1.03926 −0.519631 0.854391i \(-0.673930\pi\)
−0.519631 + 0.854391i \(0.673930\pi\)
\(468\) 2.22129 0.102679
\(469\) 0 0
\(470\) −0.0453476 −0.00209173
\(471\) −15.6970 −0.723279
\(472\) −4.90131 −0.225601
\(473\) 23.8328 1.09583
\(474\) 1.92712 0.0885156
\(475\) −2.47534 −0.113576
\(476\) 0 0
\(477\) −7.34390 −0.336254
\(478\) 17.7687 0.812720
\(479\) −19.6760 −0.899021 −0.449511 0.893275i \(-0.648402\pi\)
−0.449511 + 0.893275i \(0.648402\pi\)
\(480\) 0.879961 0.0401645
\(481\) 7.86033 0.358400
\(482\) 5.07629 0.231218
\(483\) 0 0
\(484\) 5.58835 0.254016
\(485\) −15.4067 −0.699584
\(486\) −1.00000 −0.0453609
\(487\) −6.01154 −0.272409 −0.136204 0.990681i \(-0.543490\pi\)
−0.136204 + 0.990681i \(0.543490\pi\)
\(488\) 2.14320 0.0970180
\(489\) 8.48528 0.383718
\(490\) 0 0
\(491\) 26.2401 1.18420 0.592099 0.805865i \(-0.298299\pi\)
0.592099 + 0.805865i \(0.298299\pi\)
\(492\) 0.124413 0.00560899
\(493\) 35.7510 1.61014
\(494\) 1.30120 0.0585440
\(495\) 3.58397 0.161088
\(496\) 0.585786 0.0263026
\(497\) 0 0
\(498\) −6.99819 −0.313596
\(499\) −6.49934 −0.290950 −0.145475 0.989362i \(-0.546471\pi\)
−0.145475 + 0.989362i \(0.546471\pi\)
\(500\) 8.11823 0.363058
\(501\) −19.0409 −0.850684
\(502\) 13.7473 0.613573
\(503\) −34.4071 −1.53414 −0.767068 0.641566i \(-0.778285\pi\)
−0.767068 + 0.641566i \(0.778285\pi\)
\(504\) 0 0
\(505\) 3.88274 0.172780
\(506\) 4.07288 0.181062
\(507\) 8.06585 0.358217
\(508\) 5.92446 0.262856
\(509\) −7.76368 −0.344119 −0.172059 0.985087i \(-0.555042\pi\)
−0.172059 + 0.985087i \(0.555042\pi\)
\(510\) 6.82843 0.302368
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −0.585786 −0.0258631
\(514\) 11.0069 0.485495
\(515\) −3.32511 −0.146522
\(516\) 5.85158 0.257602
\(517\) −0.209891 −0.00923098
\(518\) 0 0
\(519\) 12.0087 0.527126
\(520\) −1.95465 −0.0857172
\(521\) 31.7347 1.39032 0.695162 0.718853i \(-0.255333\pi\)
0.695162 + 0.718853i \(0.255333\pi\)
\(522\) 4.60713 0.201649
\(523\) 0.0771298 0.00337265 0.00168632 0.999999i \(-0.499463\pi\)
0.00168632 + 0.999999i \(0.499463\pi\)
\(524\) 6.76430 0.295500
\(525\) 0 0
\(526\) −26.7153 −1.16484
\(527\) 4.54566 0.198012
\(528\) 4.07288 0.177249
\(529\) 1.00000 0.0434783
\(530\) 6.46234 0.280706
\(531\) −4.90131 −0.212699
\(532\) 0 0
\(533\) −0.276359 −0.0119704
\(534\) −10.4587 −0.452593
\(535\) 3.18032 0.137497
\(536\) −8.58835 −0.370960
\(537\) −10.6461 −0.459412
\(538\) 17.1693 0.740222
\(539\) 0 0
\(540\) 0.879961 0.0378675
\(541\) 10.9550 0.470992 0.235496 0.971875i \(-0.424328\pi\)
0.235496 + 0.971875i \(0.424328\pi\)
\(542\) −17.5260 −0.752807
\(543\) 21.5385 0.924307
\(544\) 7.75992 0.332704
\(545\) −5.65270 −0.242135
\(546\) 0 0
\(547\) −34.9490 −1.49431 −0.747155 0.664649i \(-0.768581\pi\)
−0.747155 + 0.664649i \(0.768581\pi\)
\(548\) 5.56881 0.237888
\(549\) 2.14320 0.0914694
\(550\) 17.2106 0.733864
\(551\) 2.69880 0.114973
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −5.00075 −0.212462
\(555\) 3.11385 0.132176
\(556\) 12.8258 0.543934
\(557\) 9.65685 0.409174 0.204587 0.978848i \(-0.434415\pi\)
0.204587 + 0.978848i \(0.434415\pi\)
\(558\) 0.585786 0.0247983
\(559\) −12.9981 −0.549761
\(560\) 0 0
\(561\) 31.6052 1.33437
\(562\) 4.73676 0.199808
\(563\) −23.7010 −0.998878 −0.499439 0.866349i \(-0.666461\pi\)
−0.499439 + 0.866349i \(0.666461\pi\)
\(564\) −0.0515337 −0.00216996
\(565\) −11.9776 −0.503901
\(566\) −5.12101 −0.215252
\(567\) 0 0
\(568\) 4.07288 0.170894
\(569\) 20.8825 0.875441 0.437720 0.899111i \(-0.355786\pi\)
0.437720 + 0.899111i \(0.355786\pi\)
\(570\) 0.515469 0.0215906
\(571\) 0.686292 0.0287204 0.0143602 0.999897i \(-0.495429\pi\)
0.0143602 + 0.999897i \(0.495429\pi\)
\(572\) −9.04707 −0.378277
\(573\) −25.2194 −1.05356
\(574\) 0 0
\(575\) 4.22567 0.176223
\(576\) 1.00000 0.0416667
\(577\) 29.9707 1.24770 0.623848 0.781546i \(-0.285568\pi\)
0.623848 + 0.781546i \(0.285568\pi\)
\(578\) 43.2164 1.79756
\(579\) 7.50844 0.312040
\(580\) −4.05410 −0.168337
\(581\) 0 0
\(582\) −17.5084 −0.725748
\(583\) 29.9108 1.23878
\(584\) 4.97162 0.205727
\(585\) −1.95465 −0.0808149
\(586\) 9.45315 0.390506
\(587\) 24.2577 1.00122 0.500611 0.865672i \(-0.333109\pi\)
0.500611 + 0.865672i \(0.333109\pi\)
\(588\) 0 0
\(589\) 0.343146 0.0141391
\(590\) 4.31296 0.177562
\(591\) −6.66388 −0.274116
\(592\) 3.53863 0.145437
\(593\) 20.7179 0.850782 0.425391 0.905010i \(-0.360137\pi\)
0.425391 + 0.905010i \(0.360137\pi\)
\(594\) 4.07288 0.167112
\(595\) 0 0
\(596\) 13.9322 0.570687
\(597\) 4.80964 0.196846
\(598\) −2.22129 −0.0908355
\(599\) 29.9322 1.22300 0.611499 0.791245i \(-0.290567\pi\)
0.611499 + 0.791245i \(0.290567\pi\)
\(600\) 4.22567 0.172512
\(601\) 23.2985 0.950364 0.475182 0.879888i \(-0.342382\pi\)
0.475182 + 0.879888i \(0.342382\pi\)
\(602\) 0 0
\(603\) −8.58835 −0.349745
\(604\) 1.15279 0.0469063
\(605\) −4.91753 −0.199926
\(606\) 4.41240 0.179242
\(607\) −21.6809 −0.879999 −0.439999 0.897998i \(-0.645021\pi\)
−0.439999 + 0.897998i \(0.645021\pi\)
\(608\) 0.585786 0.0237568
\(609\) 0 0
\(610\) −1.88593 −0.0763590
\(611\) 0.114472 0.00463102
\(612\) 7.75992 0.313676
\(613\) −30.9746 −1.25105 −0.625526 0.780203i \(-0.715116\pi\)
−0.625526 + 0.780203i \(0.715116\pi\)
\(614\) 29.8036 1.20277
\(615\) −0.109479 −0.00441462
\(616\) 0 0
\(617\) −12.6274 −0.508361 −0.254180 0.967157i \(-0.581806\pi\)
−0.254180 + 0.967157i \(0.581806\pi\)
\(618\) −3.77871 −0.152002
\(619\) 35.4672 1.42555 0.712773 0.701395i \(-0.247439\pi\)
0.712773 + 0.701395i \(0.247439\pi\)
\(620\) −0.515469 −0.0207017
\(621\) 1.00000 0.0401286
\(622\) −18.5983 −0.745724
\(623\) 0 0
\(624\) −2.22129 −0.0889230
\(625\) 13.9846 0.559385
\(626\) 17.1803 0.686664
\(627\) 2.38584 0.0952812
\(628\) 15.6970 0.626378
\(629\) 27.4595 1.09488
\(630\) 0 0
\(631\) −16.3025 −0.648993 −0.324497 0.945887i \(-0.605195\pi\)
−0.324497 + 0.945887i \(0.605195\pi\)
\(632\) −1.92712 −0.0766567
\(633\) −4.79824 −0.190713
\(634\) 2.45775 0.0976097
\(635\) −5.21330 −0.206883
\(636\) 7.34390 0.291204
\(637\) 0 0
\(638\) −18.7643 −0.742886
\(639\) 4.07288 0.161121
\(640\) −0.879961 −0.0347835
\(641\) −13.8228 −0.545966 −0.272983 0.962019i \(-0.588010\pi\)
−0.272983 + 0.962019i \(0.588010\pi\)
\(642\) 3.61416 0.142640
\(643\) −26.9502 −1.06281 −0.531406 0.847117i \(-0.678336\pi\)
−0.531406 + 0.847117i \(0.678336\pi\)
\(644\) 0 0
\(645\) −5.14917 −0.202748
\(646\) 4.54566 0.178846
\(647\) 25.4495 1.00052 0.500262 0.865874i \(-0.333237\pi\)
0.500262 + 0.865874i \(0.333237\pi\)
\(648\) 1.00000 0.0392837
\(649\) 19.9624 0.783594
\(650\) −9.38646 −0.368167
\(651\) 0 0
\(652\) −8.48528 −0.332309
\(653\) −29.2277 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(654\) −6.42381 −0.251191
\(655\) −5.95232 −0.232576
\(656\) −0.124413 −0.00485753
\(657\) 4.97162 0.193962
\(658\) 0 0
\(659\) 17.6958 0.689330 0.344665 0.938726i \(-0.387992\pi\)
0.344665 + 0.938726i \(0.387992\pi\)
\(660\) −3.58397 −0.139506
\(661\) −4.52334 −0.175938 −0.0879688 0.996123i \(-0.528038\pi\)
−0.0879688 + 0.996123i \(0.528038\pi\)
\(662\) 19.4595 0.756314
\(663\) −17.2371 −0.669432
\(664\) 6.99819 0.271582
\(665\) 0 0
\(666\) 3.53863 0.137119
\(667\) −4.60713 −0.178389
\(668\) 19.0409 0.736714
\(669\) 14.5983 0.564403
\(670\) 7.55741 0.291968
\(671\) −8.72898 −0.336979
\(672\) 0 0
\(673\) −40.9679 −1.57920 −0.789598 0.613624i \(-0.789711\pi\)
−0.789598 + 0.613624i \(0.789711\pi\)
\(674\) 26.0213 1.00230
\(675\) 4.22567 0.162646
\(676\) −8.06585 −0.310225
\(677\) 8.96787 0.344663 0.172332 0.985039i \(-0.444870\pi\)
0.172332 + 0.985039i \(0.444870\pi\)
\(678\) −13.6115 −0.522747
\(679\) 0 0
\(680\) −6.82843 −0.261858
\(681\) 9.39543 0.360034
\(682\) −2.38584 −0.0913584
\(683\) 41.3579 1.58252 0.791258 0.611482i \(-0.209426\pi\)
0.791258 + 0.611482i \(0.209426\pi\)
\(684\) 0.585786 0.0223981
\(685\) −4.90034 −0.187232
\(686\) 0 0
\(687\) −7.14757 −0.272697
\(688\) −5.85158 −0.223090
\(689\) −16.3130 −0.621474
\(690\) −0.879961 −0.0334995
\(691\) −29.6028 −1.12614 −0.563071 0.826409i \(-0.690380\pi\)
−0.563071 + 0.826409i \(0.690380\pi\)
\(692\) −12.0087 −0.456504
\(693\) 0 0
\(694\) −31.4229 −1.19280
\(695\) −11.2862 −0.428109
\(696\) −4.60713 −0.174633
\(697\) −0.965438 −0.0365686
\(698\) −13.3688 −0.506016
\(699\) −15.3439 −0.580359
\(700\) 0 0
\(701\) 31.4306 1.18712 0.593560 0.804790i \(-0.297722\pi\)
0.593560 + 0.804790i \(0.297722\pi\)
\(702\) −2.22129 −0.0838374
\(703\) 2.07288 0.0781801
\(704\) −4.07288 −0.153502
\(705\) 0.0453476 0.00170789
\(706\) −15.8005 −0.594658
\(707\) 0 0
\(708\) 4.90131 0.184202
\(709\) 46.0250 1.72851 0.864253 0.503058i \(-0.167792\pi\)
0.864253 + 0.503058i \(0.167792\pi\)
\(710\) −3.58397 −0.134504
\(711\) −1.92712 −0.0722727
\(712\) 10.4587 0.391957
\(713\) −0.585786 −0.0219379
\(714\) 0 0
\(715\) 7.96106 0.297727
\(716\) 10.6461 0.397862
\(717\) −17.7687 −0.663583
\(718\) 26.9642 1.00629
\(719\) −18.3195 −0.683204 −0.341602 0.939845i \(-0.610969\pi\)
−0.341602 + 0.939845i \(0.610969\pi\)
\(720\) −0.879961 −0.0327942
\(721\) 0 0
\(722\) −18.6569 −0.694336
\(723\) −5.07629 −0.188789
\(724\) −21.5385 −0.800473
\(725\) −19.4682 −0.723031
\(726\) −5.58835 −0.207403
\(727\) −3.36365 −0.124751 −0.0623754 0.998053i \(-0.519868\pi\)
−0.0623754 + 0.998053i \(0.519868\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.37483 −0.161920
\(731\) −45.4078 −1.67947
\(732\) −2.14320 −0.0792148
\(733\) 18.5909 0.686671 0.343335 0.939213i \(-0.388443\pi\)
0.343335 + 0.939213i \(0.388443\pi\)
\(734\) 5.67201 0.209358
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 34.9793 1.28848
\(738\) −0.124413 −0.00457972
\(739\) 13.9802 0.514272 0.257136 0.966375i \(-0.417221\pi\)
0.257136 + 0.966375i \(0.417221\pi\)
\(740\) −3.11385 −0.114468
\(741\) −1.30120 −0.0478009
\(742\) 0 0
\(743\) 29.7015 1.08964 0.544821 0.838552i \(-0.316598\pi\)
0.544821 + 0.838552i \(0.316598\pi\)
\(744\) −0.585786 −0.0214760
\(745\) −12.2598 −0.449165
\(746\) 7.49704 0.274486
\(747\) 6.99819 0.256050
\(748\) −31.6052 −1.15560
\(749\) 0 0
\(750\) −8.11823 −0.296436
\(751\) −24.8409 −0.906458 −0.453229 0.891394i \(-0.649728\pi\)
−0.453229 + 0.891394i \(0.649728\pi\)
\(752\) 0.0515337 0.00187924
\(753\) −13.7473 −0.500980
\(754\) 10.2338 0.372693
\(755\) −1.01441 −0.0369181
\(756\) 0 0
\(757\) −22.9866 −0.835461 −0.417730 0.908571i \(-0.637174\pi\)
−0.417730 + 0.908571i \(0.637174\pi\)
\(758\) −2.24273 −0.0814598
\(759\) −4.07288 −0.147836
\(760\) −0.515469 −0.0186980
\(761\) −41.1614 −1.49210 −0.746051 0.665889i \(-0.768052\pi\)
−0.746051 + 0.665889i \(0.768052\pi\)
\(762\) −5.92446 −0.214621
\(763\) 0 0
\(764\) 25.2194 0.912405
\(765\) −6.82843 −0.246882
\(766\) 24.0693 0.869658
\(767\) −10.8872 −0.393116
\(768\) −1.00000 −0.0360844
\(769\) 24.8876 0.897472 0.448736 0.893664i \(-0.351874\pi\)
0.448736 + 0.893664i \(0.351874\pi\)
\(770\) 0 0
\(771\) −11.0069 −0.396405
\(772\) −7.50844 −0.270235
\(773\) −43.6051 −1.56837 −0.784184 0.620529i \(-0.786918\pi\)
−0.784184 + 0.620529i \(0.786918\pi\)
\(774\) −5.85158 −0.210331
\(775\) −2.47534 −0.0889168
\(776\) 17.5084 0.628516
\(777\) 0 0
\(778\) −11.9021 −0.426709
\(779\) −0.0728797 −0.00261119
\(780\) 1.95465 0.0699878
\(781\) −16.5883 −0.593578
\(782\) −7.75992 −0.277494
\(783\) −4.60713 −0.164645
\(784\) 0 0
\(785\) −13.8127 −0.492998
\(786\) −6.76430 −0.241274
\(787\) 34.6048 1.23353 0.616764 0.787148i \(-0.288443\pi\)
0.616764 + 0.787148i \(0.288443\pi\)
\(788\) 6.66388 0.237391
\(789\) 26.7153 0.951091
\(790\) 1.69579 0.0603335
\(791\) 0 0
\(792\) −4.07288 −0.144723
\(793\) 4.76067 0.169056
\(794\) −12.7305 −0.451788
\(795\) −6.46234 −0.229196
\(796\) −4.80964 −0.170473
\(797\) 38.5951 1.36711 0.683554 0.729900i \(-0.260433\pi\)
0.683554 + 0.729900i \(0.260433\pi\)
\(798\) 0 0
\(799\) 0.399898 0.0141474
\(800\) −4.22567 −0.149400
\(801\) 10.4587 0.369541
\(802\) −24.8032 −0.875833
\(803\) −20.2488 −0.714566
\(804\) 8.58835 0.302888
\(805\) 0 0
\(806\) 1.30120 0.0458330
\(807\) −17.1693 −0.604389
\(808\) −4.41240 −0.155228
\(809\) −1.95505 −0.0687360 −0.0343680 0.999409i \(-0.510942\pi\)
−0.0343680 + 0.999409i \(0.510942\pi\)
\(810\) −0.879961 −0.0309187
\(811\) −5.50694 −0.193375 −0.0966874 0.995315i \(-0.530825\pi\)
−0.0966874 + 0.995315i \(0.530825\pi\)
\(812\) 0 0
\(813\) 17.5260 0.614665
\(814\) −14.4124 −0.505154
\(815\) 7.46672 0.261548
\(816\) −7.75992 −0.271652
\(817\) −3.42778 −0.119923
\(818\) −13.1174 −0.458639
\(819\) 0 0
\(820\) 0.109479 0.00382317
\(821\) −51.6392 −1.80222 −0.901110 0.433591i \(-0.857246\pi\)
−0.901110 + 0.433591i \(0.857246\pi\)
\(822\) −5.56881 −0.194235
\(823\) −29.3892 −1.02444 −0.512222 0.858853i \(-0.671178\pi\)
−0.512222 + 0.858853i \(0.671178\pi\)
\(824\) 3.77871 0.131637
\(825\) −17.2106 −0.599198
\(826\) 0 0
\(827\) −19.1112 −0.664561 −0.332281 0.943181i \(-0.607818\pi\)
−0.332281 + 0.943181i \(0.607818\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −5.30858 −0.184375 −0.0921874 0.995742i \(-0.529386\pi\)
−0.0921874 + 0.995742i \(0.529386\pi\)
\(830\) −6.15813 −0.213752
\(831\) 5.00075 0.173474
\(832\) 2.22129 0.0770095
\(833\) 0 0
\(834\) −12.8258 −0.444120
\(835\) −16.7552 −0.579839
\(836\) −2.38584 −0.0825159
\(837\) −0.585786 −0.0202477
\(838\) 31.2417 1.07923
\(839\) −52.6220 −1.81671 −0.908356 0.418197i \(-0.862662\pi\)
−0.908356 + 0.418197i \(0.862662\pi\)
\(840\) 0 0
\(841\) −7.77433 −0.268080
\(842\) −38.1129 −1.31346
\(843\) −4.73676 −0.163143
\(844\) 4.79824 0.165162
\(845\) 7.09763 0.244166
\(846\) 0.0515337 0.00177177
\(847\) 0 0
\(848\) −7.34390 −0.252190
\(849\) 5.12101 0.175753
\(850\) −32.7909 −1.12472
\(851\) −3.53863 −0.121303
\(852\) −4.07288 −0.139535
\(853\) −25.6808 −0.879293 −0.439646 0.898171i \(-0.644896\pi\)
−0.439646 + 0.898171i \(0.644896\pi\)
\(854\) 0 0
\(855\) −0.515469 −0.0176287
\(856\) −3.61416 −0.123530
\(857\) −51.9661 −1.77513 −0.887564 0.460684i \(-0.847604\pi\)
−0.887564 + 0.460684i \(0.847604\pi\)
\(858\) 9.04707 0.308862
\(859\) −14.5806 −0.497483 −0.248741 0.968570i \(-0.580017\pi\)
−0.248741 + 0.968570i \(0.580017\pi\)
\(860\) 5.14917 0.175585
\(861\) 0 0
\(862\) 29.4031 1.00147
\(863\) −28.1046 −0.956691 −0.478345 0.878172i \(-0.658763\pi\)
−0.478345 + 0.878172i \(0.658763\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 10.5672 0.359297
\(866\) −14.8309 −0.503974
\(867\) −43.2164 −1.46771
\(868\) 0 0
\(869\) 7.84893 0.266257
\(870\) 4.05410 0.137447
\(871\) −19.0773 −0.646408
\(872\) 6.42381 0.217538
\(873\) 17.5084 0.592571
\(874\) −0.585786 −0.0198145
\(875\) 0 0
\(876\) −4.97162 −0.167976
\(877\) −0.645101 −0.0217835 −0.0108917 0.999941i \(-0.503467\pi\)
−0.0108917 + 0.999941i \(0.503467\pi\)
\(878\) 0.299392 0.0101040
\(879\) −9.45315 −0.318847
\(880\) 3.58397 0.120816
\(881\) 11.0581 0.372556 0.186278 0.982497i \(-0.440358\pi\)
0.186278 + 0.982497i \(0.440358\pi\)
\(882\) 0 0
\(883\) 9.56047 0.321735 0.160868 0.986976i \(-0.448571\pi\)
0.160868 + 0.986976i \(0.448571\pi\)
\(884\) 17.2371 0.579746
\(885\) −4.31296 −0.144978
\(886\) 10.5202 0.353433
\(887\) 54.7189 1.83728 0.918640 0.395096i \(-0.129289\pi\)
0.918640 + 0.395096i \(0.129289\pi\)
\(888\) −3.53863 −0.118749
\(889\) 0 0
\(890\) −9.20326 −0.308494
\(891\) −4.07288 −0.136447
\(892\) −14.5983 −0.488787
\(893\) 0.0301877 0.00101019
\(894\) −13.9322 −0.465964
\(895\) −9.36812 −0.313142
\(896\) 0 0
\(897\) 2.22129 0.0741669
\(898\) −40.9505 −1.36654
\(899\) 2.69880 0.0900099
\(900\) −4.22567 −0.140856
\(901\) −56.9881 −1.89855
\(902\) 0.506721 0.0168720
\(903\) 0 0
\(904\) 13.6115 0.452712
\(905\) 18.9531 0.630021
\(906\) −1.15279 −0.0382989
\(907\) −2.58738 −0.0859126 −0.0429563 0.999077i \(-0.513678\pi\)
−0.0429563 + 0.999077i \(0.513678\pi\)
\(908\) −9.39543 −0.311798
\(909\) −4.41240 −0.146350
\(910\) 0 0
\(911\) 42.6050 1.41157 0.705783 0.708428i \(-0.250595\pi\)
0.705783 + 0.708428i \(0.250595\pi\)
\(912\) −0.585786 −0.0193973
\(913\) −28.5028 −0.943304
\(914\) 15.1539 0.501246
\(915\) 1.88593 0.0623469
\(916\) 7.14757 0.236162
\(917\) 0 0
\(918\) −7.75992 −0.256116
\(919\) −5.53823 −0.182689 −0.0913446 0.995819i \(-0.529116\pi\)
−0.0913446 + 0.995819i \(0.529116\pi\)
\(920\) 0.879961 0.0290114
\(921\) −29.8036 −0.982061
\(922\) 22.0089 0.724824
\(923\) 9.04707 0.297788
\(924\) 0 0
\(925\) −14.9531 −0.491654
\(926\) −1.83546 −0.0603169
\(927\) 3.77871 0.124109
\(928\) 4.60713 0.151237
\(929\) −13.9855 −0.458848 −0.229424 0.973327i \(-0.573684\pi\)
−0.229424 + 0.973327i \(0.573684\pi\)
\(930\) 0.515469 0.0169029
\(931\) 0 0
\(932\) 15.3439 0.502606
\(933\) 18.5983 0.608881
\(934\) −22.4586 −0.734869
\(935\) 27.8114 0.909529
\(936\) 2.22129 0.0726053
\(937\) −24.5733 −0.802775 −0.401388 0.915908i \(-0.631472\pi\)
−0.401388 + 0.915908i \(0.631472\pi\)
\(938\) 0 0
\(939\) −17.1803 −0.560659
\(940\) −0.0453476 −0.00147908
\(941\) 3.20030 0.104327 0.0521634 0.998639i \(-0.483388\pi\)
0.0521634 + 0.998639i \(0.483388\pi\)
\(942\) −15.6970 −0.511435
\(943\) 0.124413 0.00405146
\(944\) −4.90131 −0.159524
\(945\) 0 0
\(946\) 23.8328 0.774871
\(947\) −2.21392 −0.0719426 −0.0359713 0.999353i \(-0.511452\pi\)
−0.0359713 + 0.999353i \(0.511452\pi\)
\(948\) 1.92712 0.0625900
\(949\) 11.0434 0.358485
\(950\) −2.47534 −0.0803106
\(951\) −2.45775 −0.0796980
\(952\) 0 0
\(953\) 37.1750 1.20422 0.602108 0.798414i \(-0.294328\pi\)
0.602108 + 0.798414i \(0.294328\pi\)
\(954\) −7.34390 −0.237767
\(955\) −22.1921 −0.718119
\(956\) 17.7687 0.574680
\(957\) 18.7643 0.606564
\(958\) −19.6760 −0.635704
\(959\) 0 0
\(960\) 0.879961 0.0284006
\(961\) −30.6569 −0.988931
\(962\) 7.86033 0.253427
\(963\) −3.61416 −0.116465
\(964\) 5.07629 0.163496
\(965\) 6.60713 0.212691
\(966\) 0 0
\(967\) 39.5449 1.27168 0.635838 0.771822i \(-0.280654\pi\)
0.635838 + 0.771822i \(0.280654\pi\)
\(968\) 5.58835 0.179616
\(969\) −4.54566 −0.146028
\(970\) −15.4067 −0.494681
\(971\) 32.6674 1.04835 0.524174 0.851611i \(-0.324374\pi\)
0.524174 + 0.851611i \(0.324374\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −6.01154 −0.192622
\(975\) 9.38646 0.300607
\(976\) 2.14320 0.0686021
\(977\) −42.7155 −1.36659 −0.683294 0.730143i \(-0.739453\pi\)
−0.683294 + 0.730143i \(0.739453\pi\)
\(978\) 8.48528 0.271329
\(979\) −42.5971 −1.36141
\(980\) 0 0
\(981\) 6.42381 0.205096
\(982\) 26.2401 0.837355
\(983\) −13.7812 −0.439551 −0.219776 0.975550i \(-0.570533\pi\)
−0.219776 + 0.975550i \(0.570533\pi\)
\(984\) 0.124413 0.00396615
\(985\) −5.86396 −0.186841
\(986\) 35.7510 1.13854
\(987\) 0 0
\(988\) 1.30120 0.0413968
\(989\) 5.85158 0.186070
\(990\) 3.58397 0.113906
\(991\) 51.1765 1.62568 0.812838 0.582490i \(-0.197922\pi\)
0.812838 + 0.582490i \(0.197922\pi\)
\(992\) 0.585786 0.0185987
\(993\) −19.4595 −0.617528
\(994\) 0 0
\(995\) 4.23230 0.134173
\(996\) −6.99819 −0.221746
\(997\) 41.3363 1.30913 0.654567 0.756004i \(-0.272851\pi\)
0.654567 + 0.756004i \(0.272851\pi\)
\(998\) −6.49934 −0.205733
\(999\) −3.53863 −0.111957
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 6762.2.a.cn.1.1 4
7.6 odd 2 6762.2.a.co.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.1 4 1.1 even 1 trivial
6762.2.a.co.1.4 yes 4 7.6 odd 2