Properties

Label 8034.2.a.bc.1.1
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.06882\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.06882 q^{5} -1.00000 q^{6} +4.31915 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.06882 q^{5} -1.00000 q^{6} +4.31915 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.06882 q^{10} -3.45943 q^{11} -1.00000 q^{12} -1.00000 q^{13} +4.31915 q^{14} +4.06882 q^{15} +1.00000 q^{16} -7.56875 q^{17} +1.00000 q^{18} -1.84870 q^{19} -4.06882 q^{20} -4.31915 q^{21} -3.45943 q^{22} -8.50665 q^{23} -1.00000 q^{24} +11.5553 q^{25} -1.00000 q^{26} -1.00000 q^{27} +4.31915 q^{28} +6.61550 q^{29} +4.06882 q^{30} -5.55957 q^{31} +1.00000 q^{32} +3.45943 q^{33} -7.56875 q^{34} -17.5738 q^{35} +1.00000 q^{36} +10.0080 q^{37} -1.84870 q^{38} +1.00000 q^{39} -4.06882 q^{40} +0.695896 q^{41} -4.31915 q^{42} -3.09807 q^{43} -3.45943 q^{44} -4.06882 q^{45} -8.50665 q^{46} -1.17792 q^{47} -1.00000 q^{48} +11.6551 q^{49} +11.5553 q^{50} +7.56875 q^{51} -1.00000 q^{52} +5.76954 q^{53} -1.00000 q^{54} +14.0758 q^{55} +4.31915 q^{56} +1.84870 q^{57} +6.61550 q^{58} +6.88464 q^{59} +4.06882 q^{60} +12.3520 q^{61} -5.55957 q^{62} +4.31915 q^{63} +1.00000 q^{64} +4.06882 q^{65} +3.45943 q^{66} -14.3536 q^{67} -7.56875 q^{68} +8.50665 q^{69} -17.5738 q^{70} +4.40758 q^{71} +1.00000 q^{72} -8.29562 q^{73} +10.0080 q^{74} -11.5553 q^{75} -1.84870 q^{76} -14.9418 q^{77} +1.00000 q^{78} +6.21617 q^{79} -4.06882 q^{80} +1.00000 q^{81} +0.695896 q^{82} -11.6646 q^{83} -4.31915 q^{84} +30.7958 q^{85} -3.09807 q^{86} -6.61550 q^{87} -3.45943 q^{88} -2.88091 q^{89} -4.06882 q^{90} -4.31915 q^{91} -8.50665 q^{92} +5.55957 q^{93} -1.17792 q^{94} +7.52201 q^{95} -1.00000 q^{96} +4.85929 q^{97} +11.6551 q^{98} -3.45943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −4.06882 −1.81963 −0.909815 0.415015i \(-0.863776\pi\)
−0.909815 + 0.415015i \(0.863776\pi\)
\(6\) −1.00000 −0.408248
\(7\) 4.31915 1.63249 0.816243 0.577708i \(-0.196053\pi\)
0.816243 + 0.577708i \(0.196053\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −4.06882 −1.28667
\(11\) −3.45943 −1.04306 −0.521529 0.853234i \(-0.674638\pi\)
−0.521529 + 0.853234i \(0.674638\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.31915 1.15434
\(15\) 4.06882 1.05056
\(16\) 1.00000 0.250000
\(17\) −7.56875 −1.83569 −0.917845 0.396938i \(-0.870073\pi\)
−0.917845 + 0.396938i \(0.870073\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.84870 −0.424120 −0.212060 0.977257i \(-0.568017\pi\)
−0.212060 + 0.977257i \(0.568017\pi\)
\(20\) −4.06882 −0.909815
\(21\) −4.31915 −0.942517
\(22\) −3.45943 −0.737553
\(23\) −8.50665 −1.77376 −0.886879 0.462001i \(-0.847132\pi\)
−0.886879 + 0.462001i \(0.847132\pi\)
\(24\) −1.00000 −0.204124
\(25\) 11.5553 2.31105
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.31915 0.816243
\(29\) 6.61550 1.22847 0.614234 0.789124i \(-0.289465\pi\)
0.614234 + 0.789124i \(0.289465\pi\)
\(30\) 4.06882 0.742861
\(31\) −5.55957 −0.998528 −0.499264 0.866450i \(-0.666396\pi\)
−0.499264 + 0.866450i \(0.666396\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.45943 0.602210
\(34\) −7.56875 −1.29803
\(35\) −17.5738 −2.97052
\(36\) 1.00000 0.166667
\(37\) 10.0080 1.64530 0.822652 0.568545i \(-0.192493\pi\)
0.822652 + 0.568545i \(0.192493\pi\)
\(38\) −1.84870 −0.299898
\(39\) 1.00000 0.160128
\(40\) −4.06882 −0.643336
\(41\) 0.695896 0.108681 0.0543403 0.998522i \(-0.482694\pi\)
0.0543403 + 0.998522i \(0.482694\pi\)
\(42\) −4.31915 −0.666460
\(43\) −3.09807 −0.472452 −0.236226 0.971698i \(-0.575910\pi\)
−0.236226 + 0.971698i \(0.575910\pi\)
\(44\) −3.45943 −0.521529
\(45\) −4.06882 −0.606543
\(46\) −8.50665 −1.25424
\(47\) −1.17792 −0.171818 −0.0859089 0.996303i \(-0.527379\pi\)
−0.0859089 + 0.996303i \(0.527379\pi\)
\(48\) −1.00000 −0.144338
\(49\) 11.6551 1.66501
\(50\) 11.5553 1.63416
\(51\) 7.56875 1.05984
\(52\) −1.00000 −0.138675
\(53\) 5.76954 0.792507 0.396253 0.918141i \(-0.370310\pi\)
0.396253 + 0.918141i \(0.370310\pi\)
\(54\) −1.00000 −0.136083
\(55\) 14.0758 1.89798
\(56\) 4.31915 0.577171
\(57\) 1.84870 0.244866
\(58\) 6.61550 0.868657
\(59\) 6.88464 0.896303 0.448152 0.893958i \(-0.352082\pi\)
0.448152 + 0.893958i \(0.352082\pi\)
\(60\) 4.06882 0.525282
\(61\) 12.3520 1.58151 0.790755 0.612133i \(-0.209688\pi\)
0.790755 + 0.612133i \(0.209688\pi\)
\(62\) −5.55957 −0.706066
\(63\) 4.31915 0.544162
\(64\) 1.00000 0.125000
\(65\) 4.06882 0.504674
\(66\) 3.45943 0.425827
\(67\) −14.3536 −1.75357 −0.876786 0.480881i \(-0.840317\pi\)
−0.876786 + 0.480881i \(0.840317\pi\)
\(68\) −7.56875 −0.917845
\(69\) 8.50665 1.02408
\(70\) −17.5738 −2.10048
\(71\) 4.40758 0.523083 0.261542 0.965192i \(-0.415769\pi\)
0.261542 + 0.965192i \(0.415769\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.29562 −0.970928 −0.485464 0.874257i \(-0.661349\pi\)
−0.485464 + 0.874257i \(0.661349\pi\)
\(74\) 10.0080 1.16341
\(75\) −11.5553 −1.33429
\(76\) −1.84870 −0.212060
\(77\) −14.9418 −1.70278
\(78\) 1.00000 0.113228
\(79\) 6.21617 0.699373 0.349687 0.936867i \(-0.386288\pi\)
0.349687 + 0.936867i \(0.386288\pi\)
\(80\) −4.06882 −0.454907
\(81\) 1.00000 0.111111
\(82\) 0.695896 0.0768488
\(83\) −11.6646 −1.28036 −0.640179 0.768226i \(-0.721140\pi\)
−0.640179 + 0.768226i \(0.721140\pi\)
\(84\) −4.31915 −0.471258
\(85\) 30.7958 3.34028
\(86\) −3.09807 −0.334074
\(87\) −6.61550 −0.709256
\(88\) −3.45943 −0.368777
\(89\) −2.88091 −0.305375 −0.152688 0.988274i \(-0.548793\pi\)
−0.152688 + 0.988274i \(0.548793\pi\)
\(90\) −4.06882 −0.428891
\(91\) −4.31915 −0.452770
\(92\) −8.50665 −0.886879
\(93\) 5.55957 0.576501
\(94\) −1.17792 −0.121494
\(95\) 7.52201 0.771742
\(96\) −1.00000 −0.102062
\(97\) 4.85929 0.493386 0.246693 0.969094i \(-0.420656\pi\)
0.246693 + 0.969094i \(0.420656\pi\)
\(98\) 11.6551 1.17734
\(99\) −3.45943 −0.347686
\(100\) 11.5553 1.15553
\(101\) 9.41575 0.936903 0.468451 0.883489i \(-0.344812\pi\)
0.468451 + 0.883489i \(0.344812\pi\)
\(102\) 7.56875 0.749418
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 17.5738 1.71503
\(106\) 5.76954 0.560387
\(107\) 3.81409 0.368722 0.184361 0.982859i \(-0.440978\pi\)
0.184361 + 0.982859i \(0.440978\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 18.7539 1.79630 0.898149 0.439691i \(-0.144912\pi\)
0.898149 + 0.439691i \(0.144912\pi\)
\(110\) 14.0758 1.34207
\(111\) −10.0080 −0.949917
\(112\) 4.31915 0.408122
\(113\) −10.5876 −0.995998 −0.497999 0.867178i \(-0.665932\pi\)
−0.497999 + 0.867178i \(0.665932\pi\)
\(114\) 1.84870 0.173146
\(115\) 34.6120 3.22758
\(116\) 6.61550 0.614234
\(117\) −1.00000 −0.0924500
\(118\) 6.88464 0.633782
\(119\) −32.6906 −2.99674
\(120\) 4.06882 0.371430
\(121\) 0.967666 0.0879696
\(122\) 12.3520 1.11830
\(123\) −0.695896 −0.0627468
\(124\) −5.55957 −0.499264
\(125\) −26.6721 −2.38563
\(126\) 4.31915 0.384781
\(127\) −21.5905 −1.91585 −0.957923 0.287026i \(-0.907333\pi\)
−0.957923 + 0.287026i \(0.907333\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.09807 0.272770
\(130\) 4.06882 0.356859
\(131\) −2.47190 −0.215971 −0.107985 0.994152i \(-0.534440\pi\)
−0.107985 + 0.994152i \(0.534440\pi\)
\(132\) 3.45943 0.301105
\(133\) −7.98481 −0.692370
\(134\) −14.3536 −1.23996
\(135\) 4.06882 0.350188
\(136\) −7.56875 −0.649015
\(137\) 7.60382 0.649638 0.324819 0.945776i \(-0.394697\pi\)
0.324819 + 0.945776i \(0.394697\pi\)
\(138\) 8.50665 0.724134
\(139\) −0.701284 −0.0594821 −0.0297410 0.999558i \(-0.509468\pi\)
−0.0297410 + 0.999558i \(0.509468\pi\)
\(140\) −17.5738 −1.48526
\(141\) 1.17792 0.0991990
\(142\) 4.40758 0.369876
\(143\) 3.45943 0.289292
\(144\) 1.00000 0.0833333
\(145\) −26.9172 −2.23535
\(146\) −8.29562 −0.686550
\(147\) −11.6551 −0.961296
\(148\) 10.0080 0.822652
\(149\) 18.0995 1.48276 0.741382 0.671083i \(-0.234171\pi\)
0.741382 + 0.671083i \(0.234171\pi\)
\(150\) −11.5553 −0.943483
\(151\) 16.5279 1.34502 0.672512 0.740086i \(-0.265215\pi\)
0.672512 + 0.740086i \(0.265215\pi\)
\(152\) −1.84870 −0.149949
\(153\) −7.56875 −0.611897
\(154\) −14.9418 −1.20405
\(155\) 22.6209 1.81695
\(156\) 1.00000 0.0800641
\(157\) −8.57471 −0.684336 −0.342168 0.939639i \(-0.611161\pi\)
−0.342168 + 0.939639i \(0.611161\pi\)
\(158\) 6.21617 0.494532
\(159\) −5.76954 −0.457554
\(160\) −4.06882 −0.321668
\(161\) −36.7415 −2.89564
\(162\) 1.00000 0.0785674
\(163\) −9.70287 −0.759987 −0.379994 0.924989i \(-0.624074\pi\)
−0.379994 + 0.924989i \(0.624074\pi\)
\(164\) 0.695896 0.0543403
\(165\) −14.0758 −1.09580
\(166\) −11.6646 −0.905350
\(167\) 13.4484 1.04067 0.520335 0.853962i \(-0.325807\pi\)
0.520335 + 0.853962i \(0.325807\pi\)
\(168\) −4.31915 −0.333230
\(169\) 1.00000 0.0769231
\(170\) 30.7958 2.36193
\(171\) −1.84870 −0.141373
\(172\) −3.09807 −0.236226
\(173\) 0.334934 0.0254646 0.0127323 0.999919i \(-0.495947\pi\)
0.0127323 + 0.999919i \(0.495947\pi\)
\(174\) −6.61550 −0.501520
\(175\) 49.9089 3.77276
\(176\) −3.45943 −0.260764
\(177\) −6.88464 −0.517481
\(178\) −2.88091 −0.215933
\(179\) 18.9801 1.41864 0.709321 0.704886i \(-0.249002\pi\)
0.709321 + 0.704886i \(0.249002\pi\)
\(180\) −4.06882 −0.303272
\(181\) −4.69449 −0.348938 −0.174469 0.984663i \(-0.555821\pi\)
−0.174469 + 0.984663i \(0.555821\pi\)
\(182\) −4.31915 −0.320157
\(183\) −12.3520 −0.913085
\(184\) −8.50665 −0.627118
\(185\) −40.7207 −2.99385
\(186\) 5.55957 0.407647
\(187\) 26.1836 1.91473
\(188\) −1.17792 −0.0859089
\(189\) −4.31915 −0.314172
\(190\) 7.52201 0.545704
\(191\) 12.4462 0.900579 0.450289 0.892883i \(-0.351321\pi\)
0.450289 + 0.892883i \(0.351321\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.02744 0.145939 0.0729693 0.997334i \(-0.476752\pi\)
0.0729693 + 0.997334i \(0.476752\pi\)
\(194\) 4.85929 0.348877
\(195\) −4.06882 −0.291374
\(196\) 11.6551 0.832507
\(197\) 16.8314 1.19919 0.599595 0.800303i \(-0.295328\pi\)
0.599595 + 0.800303i \(0.295328\pi\)
\(198\) −3.45943 −0.245851
\(199\) −0.301529 −0.0213748 −0.0106874 0.999943i \(-0.503402\pi\)
−0.0106874 + 0.999943i \(0.503402\pi\)
\(200\) 11.5553 0.817080
\(201\) 14.3536 1.01243
\(202\) 9.41575 0.662490
\(203\) 28.5734 2.00546
\(204\) 7.56875 0.529918
\(205\) −2.83147 −0.197759
\(206\) −1.00000 −0.0696733
\(207\) −8.50665 −0.591253
\(208\) −1.00000 −0.0693375
\(209\) 6.39544 0.442382
\(210\) 17.5738 1.21271
\(211\) −3.92060 −0.269905 −0.134953 0.990852i \(-0.543088\pi\)
−0.134953 + 0.990852i \(0.543088\pi\)
\(212\) 5.76954 0.396253
\(213\) −4.40758 −0.302002
\(214\) 3.81409 0.260726
\(215\) 12.6055 0.859687
\(216\) −1.00000 −0.0680414
\(217\) −24.0126 −1.63008
\(218\) 18.7539 1.27017
\(219\) 8.29562 0.560566
\(220\) 14.0758 0.948989
\(221\) 7.56875 0.509129
\(222\) −10.0080 −0.671693
\(223\) 13.3625 0.894819 0.447410 0.894329i \(-0.352347\pi\)
0.447410 + 0.894329i \(0.352347\pi\)
\(224\) 4.31915 0.288586
\(225\) 11.5553 0.770351
\(226\) −10.5876 −0.704277
\(227\) −1.25297 −0.0831623 −0.0415811 0.999135i \(-0.513240\pi\)
−0.0415811 + 0.999135i \(0.513240\pi\)
\(228\) 1.84870 0.122433
\(229\) 8.24201 0.544647 0.272324 0.962206i \(-0.412208\pi\)
0.272324 + 0.962206i \(0.412208\pi\)
\(230\) 34.6120 2.28225
\(231\) 14.9418 0.983099
\(232\) 6.61550 0.434329
\(233\) 12.8913 0.844538 0.422269 0.906471i \(-0.361234\pi\)
0.422269 + 0.906471i \(0.361234\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 4.79275 0.312645
\(236\) 6.88464 0.448152
\(237\) −6.21617 −0.403783
\(238\) −32.6906 −2.11902
\(239\) 20.1551 1.30373 0.651864 0.758336i \(-0.273987\pi\)
0.651864 + 0.758336i \(0.273987\pi\)
\(240\) 4.06882 0.262641
\(241\) −2.43963 −0.157150 −0.0785751 0.996908i \(-0.525037\pi\)
−0.0785751 + 0.996908i \(0.525037\pi\)
\(242\) 0.967666 0.0622039
\(243\) −1.00000 −0.0641500
\(244\) 12.3520 0.790755
\(245\) −47.4224 −3.02971
\(246\) −0.695896 −0.0443687
\(247\) 1.84870 0.117630
\(248\) −5.55957 −0.353033
\(249\) 11.6646 0.739215
\(250\) −26.6721 −1.68689
\(251\) −7.14154 −0.450770 −0.225385 0.974270i \(-0.572364\pi\)
−0.225385 + 0.974270i \(0.572364\pi\)
\(252\) 4.31915 0.272081
\(253\) 29.4282 1.85013
\(254\) −21.5905 −1.35471
\(255\) −30.7958 −1.92851
\(256\) 1.00000 0.0625000
\(257\) 9.93685 0.619844 0.309922 0.950762i \(-0.399697\pi\)
0.309922 + 0.950762i \(0.399697\pi\)
\(258\) 3.09807 0.192878
\(259\) 43.2261 2.68594
\(260\) 4.06882 0.252337
\(261\) 6.61550 0.409489
\(262\) −2.47190 −0.152715
\(263\) −24.6988 −1.52299 −0.761497 0.648168i \(-0.775535\pi\)
−0.761497 + 0.648168i \(0.775535\pi\)
\(264\) 3.45943 0.212913
\(265\) −23.4752 −1.44207
\(266\) −7.98481 −0.489580
\(267\) 2.88091 0.176309
\(268\) −14.3536 −0.876786
\(269\) 25.6107 1.56151 0.780756 0.624836i \(-0.214834\pi\)
0.780756 + 0.624836i \(0.214834\pi\)
\(270\) 4.06882 0.247620
\(271\) −6.42255 −0.390142 −0.195071 0.980789i \(-0.562494\pi\)
−0.195071 + 0.980789i \(0.562494\pi\)
\(272\) −7.56875 −0.458923
\(273\) 4.31915 0.261407
\(274\) 7.60382 0.459364
\(275\) −39.9746 −2.41056
\(276\) 8.50665 0.512040
\(277\) 18.0070 1.08194 0.540969 0.841043i \(-0.318058\pi\)
0.540969 + 0.841043i \(0.318058\pi\)
\(278\) −0.701284 −0.0420602
\(279\) −5.55957 −0.332843
\(280\) −17.5738 −1.05024
\(281\) −12.7632 −0.761386 −0.380693 0.924701i \(-0.624315\pi\)
−0.380693 + 0.924701i \(0.624315\pi\)
\(282\) 1.17792 0.0701443
\(283\) 10.4087 0.618732 0.309366 0.950943i \(-0.399883\pi\)
0.309366 + 0.950943i \(0.399883\pi\)
\(284\) 4.40758 0.261542
\(285\) −7.52201 −0.445565
\(286\) 3.45943 0.204560
\(287\) 3.00568 0.177420
\(288\) 1.00000 0.0589256
\(289\) 40.2859 2.36976
\(290\) −26.9172 −1.58063
\(291\) −4.85929 −0.284857
\(292\) −8.29562 −0.485464
\(293\) −13.9122 −0.812758 −0.406379 0.913705i \(-0.633209\pi\)
−0.406379 + 0.913705i \(0.633209\pi\)
\(294\) −11.6551 −0.679739
\(295\) −28.0123 −1.63094
\(296\) 10.0080 0.581703
\(297\) 3.45943 0.200737
\(298\) 18.0995 1.04847
\(299\) 8.50665 0.491952
\(300\) −11.5553 −0.667143
\(301\) −13.3811 −0.771271
\(302\) 16.5279 0.951076
\(303\) −9.41575 −0.540921
\(304\) −1.84870 −0.106030
\(305\) −50.2580 −2.87776
\(306\) −7.56875 −0.432676
\(307\) 7.65826 0.437080 0.218540 0.975828i \(-0.429871\pi\)
0.218540 + 0.975828i \(0.429871\pi\)
\(308\) −14.9418 −0.851389
\(309\) 1.00000 0.0568880
\(310\) 22.6209 1.28478
\(311\) 8.22932 0.466642 0.233321 0.972400i \(-0.425041\pi\)
0.233321 + 0.972400i \(0.425041\pi\)
\(312\) 1.00000 0.0566139
\(313\) 0.937649 0.0529991 0.0264995 0.999649i \(-0.491564\pi\)
0.0264995 + 0.999649i \(0.491564\pi\)
\(314\) −8.57471 −0.483899
\(315\) −17.5738 −0.990174
\(316\) 6.21617 0.349687
\(317\) −13.7271 −0.770991 −0.385496 0.922710i \(-0.625969\pi\)
−0.385496 + 0.922710i \(0.625969\pi\)
\(318\) −5.76954 −0.323540
\(319\) −22.8859 −1.28136
\(320\) −4.06882 −0.227454
\(321\) −3.81409 −0.212882
\(322\) −36.7415 −2.04752
\(323\) 13.9923 0.778553
\(324\) 1.00000 0.0555556
\(325\) −11.5553 −0.640970
\(326\) −9.70287 −0.537392
\(327\) −18.7539 −1.03709
\(328\) 0.695896 0.0384244
\(329\) −5.08763 −0.280490
\(330\) −14.0758 −0.774847
\(331\) 33.3238 1.83164 0.915822 0.401585i \(-0.131540\pi\)
0.915822 + 0.401585i \(0.131540\pi\)
\(332\) −11.6646 −0.640179
\(333\) 10.0080 0.548435
\(334\) 13.4484 0.735865
\(335\) 58.4022 3.19085
\(336\) −4.31915 −0.235629
\(337\) −11.5951 −0.631623 −0.315811 0.948822i \(-0.602277\pi\)
−0.315811 + 0.948822i \(0.602277\pi\)
\(338\) 1.00000 0.0543928
\(339\) 10.5876 0.575040
\(340\) 30.7958 1.67014
\(341\) 19.2330 1.04152
\(342\) −1.84870 −0.0999661
\(343\) 20.1061 1.08563
\(344\) −3.09807 −0.167037
\(345\) −34.6120 −1.86345
\(346\) 0.334934 0.0180062
\(347\) −11.6647 −0.626192 −0.313096 0.949721i \(-0.601366\pi\)
−0.313096 + 0.949721i \(0.601366\pi\)
\(348\) −6.61550 −0.354628
\(349\) 9.80093 0.524632 0.262316 0.964982i \(-0.415514\pi\)
0.262316 + 0.964982i \(0.415514\pi\)
\(350\) 49.9089 2.66775
\(351\) 1.00000 0.0533761
\(352\) −3.45943 −0.184388
\(353\) 6.38978 0.340094 0.170047 0.985436i \(-0.445608\pi\)
0.170047 + 0.985436i \(0.445608\pi\)
\(354\) −6.88464 −0.365914
\(355\) −17.9336 −0.951817
\(356\) −2.88091 −0.152688
\(357\) 32.6906 1.73017
\(358\) 18.9801 1.00313
\(359\) −14.0791 −0.743064 −0.371532 0.928420i \(-0.621167\pi\)
−0.371532 + 0.928420i \(0.621167\pi\)
\(360\) −4.06882 −0.214445
\(361\) −15.5823 −0.820122
\(362\) −4.69449 −0.246737
\(363\) −0.967666 −0.0507893
\(364\) −4.31915 −0.226385
\(365\) 33.7533 1.76673
\(366\) −12.3520 −0.645649
\(367\) −25.9658 −1.35540 −0.677701 0.735338i \(-0.737023\pi\)
−0.677701 + 0.735338i \(0.737023\pi\)
\(368\) −8.50665 −0.443440
\(369\) 0.695896 0.0362269
\(370\) −40.7207 −2.11697
\(371\) 24.9195 1.29376
\(372\) 5.55957 0.288250
\(373\) 28.3171 1.46620 0.733101 0.680120i \(-0.238072\pi\)
0.733101 + 0.680120i \(0.238072\pi\)
\(374\) 26.1836 1.35392
\(375\) 26.6721 1.37734
\(376\) −1.17792 −0.0607468
\(377\) −6.61550 −0.340715
\(378\) −4.31915 −0.222153
\(379\) 5.80714 0.298293 0.149146 0.988815i \(-0.452347\pi\)
0.149146 + 0.988815i \(0.452347\pi\)
\(380\) 7.52201 0.385871
\(381\) 21.5905 1.10611
\(382\) 12.4462 0.636805
\(383\) 22.5633 1.15293 0.576467 0.817121i \(-0.304431\pi\)
0.576467 + 0.817121i \(0.304431\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 60.7955 3.09843
\(386\) 2.02744 0.103194
\(387\) −3.09807 −0.157484
\(388\) 4.85929 0.246693
\(389\) 23.6159 1.19737 0.598686 0.800984i \(-0.295690\pi\)
0.598686 + 0.800984i \(0.295690\pi\)
\(390\) −4.06882 −0.206032
\(391\) 64.3847 3.25607
\(392\) 11.6551 0.588671
\(393\) 2.47190 0.124691
\(394\) 16.8314 0.847956
\(395\) −25.2924 −1.27260
\(396\) −3.45943 −0.173843
\(397\) 18.8538 0.946246 0.473123 0.880996i \(-0.343127\pi\)
0.473123 + 0.880996i \(0.343127\pi\)
\(398\) −0.301529 −0.0151143
\(399\) 7.98481 0.399740
\(400\) 11.5553 0.577763
\(401\) 10.3423 0.516472 0.258236 0.966082i \(-0.416859\pi\)
0.258236 + 0.966082i \(0.416859\pi\)
\(402\) 14.3536 0.715893
\(403\) 5.55957 0.276942
\(404\) 9.41575 0.468451
\(405\) −4.06882 −0.202181
\(406\) 28.5734 1.41807
\(407\) −34.6220 −1.71615
\(408\) 7.56875 0.374709
\(409\) −18.2409 −0.901956 −0.450978 0.892535i \(-0.648925\pi\)
−0.450978 + 0.892535i \(0.648925\pi\)
\(410\) −2.83147 −0.139836
\(411\) −7.60382 −0.375069
\(412\) −1.00000 −0.0492665
\(413\) 29.7358 1.46320
\(414\) −8.50665 −0.418079
\(415\) 47.4612 2.32978
\(416\) −1.00000 −0.0490290
\(417\) 0.701284 0.0343420
\(418\) 6.39544 0.312811
\(419\) −22.7734 −1.11255 −0.556277 0.830997i \(-0.687771\pi\)
−0.556277 + 0.830997i \(0.687771\pi\)
\(420\) 17.5738 0.857516
\(421\) 19.3529 0.943204 0.471602 0.881811i \(-0.343676\pi\)
0.471602 + 0.881811i \(0.343676\pi\)
\(422\) −3.92060 −0.190852
\(423\) −1.17792 −0.0572726
\(424\) 5.76954 0.280193
\(425\) −87.4588 −4.24238
\(426\) −4.40758 −0.213548
\(427\) 53.3501 2.58179
\(428\) 3.81409 0.184361
\(429\) −3.45943 −0.167023
\(430\) 12.6055 0.607890
\(431\) −23.1867 −1.11687 −0.558433 0.829550i \(-0.688597\pi\)
−0.558433 + 0.829550i \(0.688597\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 5.73037 0.275384 0.137692 0.990475i \(-0.456032\pi\)
0.137692 + 0.990475i \(0.456032\pi\)
\(434\) −24.0126 −1.15264
\(435\) 26.9172 1.29058
\(436\) 18.7539 0.898149
\(437\) 15.7262 0.752287
\(438\) 8.29562 0.396380
\(439\) −3.84780 −0.183645 −0.0918226 0.995775i \(-0.529269\pi\)
−0.0918226 + 0.995775i \(0.529269\pi\)
\(440\) 14.0758 0.671037
\(441\) 11.6551 0.555004
\(442\) 7.56875 0.360009
\(443\) 31.2358 1.48406 0.742029 0.670367i \(-0.233864\pi\)
0.742029 + 0.670367i \(0.233864\pi\)
\(444\) −10.0080 −0.474959
\(445\) 11.7219 0.555670
\(446\) 13.3625 0.632733
\(447\) −18.0995 −0.856075
\(448\) 4.31915 0.204061
\(449\) 3.02705 0.142855 0.0714277 0.997446i \(-0.477244\pi\)
0.0714277 + 0.997446i \(0.477244\pi\)
\(450\) 11.5553 0.544720
\(451\) −2.40740 −0.113360
\(452\) −10.5876 −0.497999
\(453\) −16.5279 −0.776550
\(454\) −1.25297 −0.0588046
\(455\) 17.5738 0.823874
\(456\) 1.84870 0.0865732
\(457\) −38.0751 −1.78108 −0.890540 0.454905i \(-0.849673\pi\)
−0.890540 + 0.454905i \(0.849673\pi\)
\(458\) 8.24201 0.385124
\(459\) 7.56875 0.353279
\(460\) 34.6120 1.61379
\(461\) −16.0253 −0.746373 −0.373186 0.927756i \(-0.621735\pi\)
−0.373186 + 0.927756i \(0.621735\pi\)
\(462\) 14.9418 0.695156
\(463\) −7.54870 −0.350818 −0.175409 0.984496i \(-0.556125\pi\)
−0.175409 + 0.984496i \(0.556125\pi\)
\(464\) 6.61550 0.307117
\(465\) −22.6209 −1.04902
\(466\) 12.8913 0.597178
\(467\) −32.7890 −1.51729 −0.758646 0.651503i \(-0.774139\pi\)
−0.758646 + 0.651503i \(0.774139\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −61.9954 −2.86268
\(470\) 4.79275 0.221073
\(471\) 8.57471 0.395102
\(472\) 6.88464 0.316891
\(473\) 10.7176 0.492794
\(474\) −6.21617 −0.285518
\(475\) −21.3622 −0.980164
\(476\) −32.6906 −1.49837
\(477\) 5.76954 0.264169
\(478\) 20.1551 0.921874
\(479\) 38.4984 1.75904 0.879519 0.475863i \(-0.157864\pi\)
0.879519 + 0.475863i \(0.157864\pi\)
\(480\) 4.06882 0.185715
\(481\) −10.0080 −0.456325
\(482\) −2.43963 −0.111122
\(483\) 36.7415 1.67180
\(484\) 0.967666 0.0439848
\(485\) −19.7716 −0.897781
\(486\) −1.00000 −0.0453609
\(487\) −6.04136 −0.273760 −0.136880 0.990588i \(-0.543707\pi\)
−0.136880 + 0.990588i \(0.543707\pi\)
\(488\) 12.3520 0.559148
\(489\) 9.70287 0.438779
\(490\) −47.4224 −2.14233
\(491\) 26.8174 1.21025 0.605126 0.796130i \(-0.293123\pi\)
0.605126 + 0.796130i \(0.293123\pi\)
\(492\) −0.695896 −0.0313734
\(493\) −50.0710 −2.25509
\(494\) 1.84870 0.0831768
\(495\) 14.0758 0.632660
\(496\) −5.55957 −0.249632
\(497\) 19.0370 0.853926
\(498\) 11.6646 0.522704
\(499\) 8.59968 0.384975 0.192487 0.981299i \(-0.438345\pi\)
0.192487 + 0.981299i \(0.438345\pi\)
\(500\) −26.6721 −1.19281
\(501\) −13.4484 −0.600832
\(502\) −7.14154 −0.318743
\(503\) 31.4358 1.40165 0.700827 0.713331i \(-0.252814\pi\)
0.700827 + 0.713331i \(0.252814\pi\)
\(504\) 4.31915 0.192390
\(505\) −38.3110 −1.70482
\(506\) 29.4282 1.30824
\(507\) −1.00000 −0.0444116
\(508\) −21.5905 −0.957923
\(509\) 31.8853 1.41329 0.706646 0.707567i \(-0.250207\pi\)
0.706646 + 0.707567i \(0.250207\pi\)
\(510\) −30.7958 −1.36366
\(511\) −35.8300 −1.58503
\(512\) 1.00000 0.0441942
\(513\) 1.84870 0.0816220
\(514\) 9.93685 0.438296
\(515\) 4.06882 0.179293
\(516\) 3.09807 0.136385
\(517\) 4.07494 0.179216
\(518\) 43.2261 1.89925
\(519\) −0.334934 −0.0147020
\(520\) 4.06882 0.178429
\(521\) −10.1483 −0.444606 −0.222303 0.974978i \(-0.571357\pi\)
−0.222303 + 0.974978i \(0.571357\pi\)
\(522\) 6.61550 0.289552
\(523\) 27.9932 1.22406 0.612028 0.790836i \(-0.290354\pi\)
0.612028 + 0.790836i \(0.290354\pi\)
\(524\) −2.47190 −0.107985
\(525\) −49.9089 −2.17820
\(526\) −24.6988 −1.07692
\(527\) 42.0790 1.83299
\(528\) 3.45943 0.150552
\(529\) 49.3630 2.14622
\(530\) −23.4752 −1.01970
\(531\) 6.88464 0.298768
\(532\) −7.98481 −0.346185
\(533\) −0.695896 −0.0301426
\(534\) 2.88091 0.124669
\(535\) −15.5188 −0.670937
\(536\) −14.3536 −0.619981
\(537\) −18.9801 −0.819053
\(538\) 25.6107 1.10416
\(539\) −40.3200 −1.73670
\(540\) 4.06882 0.175094
\(541\) −31.0959 −1.33692 −0.668459 0.743749i \(-0.733046\pi\)
−0.668459 + 0.743749i \(0.733046\pi\)
\(542\) −6.42255 −0.275872
\(543\) 4.69449 0.201460
\(544\) −7.56875 −0.324507
\(545\) −76.3062 −3.26860
\(546\) 4.31915 0.184843
\(547\) −13.9231 −0.595311 −0.297655 0.954673i \(-0.596205\pi\)
−0.297655 + 0.954673i \(0.596205\pi\)
\(548\) 7.60382 0.324819
\(549\) 12.3520 0.527170
\(550\) −39.9746 −1.70452
\(551\) −12.2300 −0.521018
\(552\) 8.50665 0.362067
\(553\) 26.8486 1.14172
\(554\) 18.0070 0.765045
\(555\) 40.7207 1.72850
\(556\) −0.701284 −0.0297410
\(557\) 40.1013 1.69915 0.849573 0.527471i \(-0.176860\pi\)
0.849573 + 0.527471i \(0.176860\pi\)
\(558\) −5.55957 −0.235355
\(559\) 3.09807 0.131034
\(560\) −17.5738 −0.742630
\(561\) −26.1836 −1.10547
\(562\) −12.7632 −0.538382
\(563\) 16.0210 0.675204 0.337602 0.941289i \(-0.390384\pi\)
0.337602 + 0.941289i \(0.390384\pi\)
\(564\) 1.17792 0.0495995
\(565\) 43.0790 1.81235
\(566\) 10.4087 0.437510
\(567\) 4.31915 0.181387
\(568\) 4.40758 0.184938
\(569\) −0.0543899 −0.00228014 −0.00114007 0.999999i \(-0.500363\pi\)
−0.00114007 + 0.999999i \(0.500363\pi\)
\(570\) −7.52201 −0.315062
\(571\) 3.23201 0.135256 0.0676278 0.997711i \(-0.478457\pi\)
0.0676278 + 0.997711i \(0.478457\pi\)
\(572\) 3.45943 0.144646
\(573\) −12.4462 −0.519949
\(574\) 3.00568 0.125455
\(575\) −98.2965 −4.09925
\(576\) 1.00000 0.0416667
\(577\) −17.2957 −0.720029 −0.360015 0.932947i \(-0.617228\pi\)
−0.360015 + 0.932947i \(0.617228\pi\)
\(578\) 40.2859 1.67567
\(579\) −2.02744 −0.0842577
\(580\) −26.9172 −1.11768
\(581\) −50.3813 −2.09017
\(582\) −4.85929 −0.201424
\(583\) −19.9593 −0.826630
\(584\) −8.29562 −0.343275
\(585\) 4.06882 0.168225
\(586\) −13.9122 −0.574707
\(587\) −20.0181 −0.826237 −0.413119 0.910677i \(-0.635561\pi\)
−0.413119 + 0.910677i \(0.635561\pi\)
\(588\) −11.6551 −0.480648
\(589\) 10.2780 0.423496
\(590\) −28.0123 −1.15325
\(591\) −16.8314 −0.692353
\(592\) 10.0080 0.411326
\(593\) 0.139648 0.00573467 0.00286733 0.999996i \(-0.499087\pi\)
0.00286733 + 0.999996i \(0.499087\pi\)
\(594\) 3.45943 0.141942
\(595\) 133.012 5.45296
\(596\) 18.0995 0.741382
\(597\) 0.301529 0.0123408
\(598\) 8.50665 0.347863
\(599\) −38.6375 −1.57869 −0.789343 0.613952i \(-0.789579\pi\)
−0.789343 + 0.613952i \(0.789579\pi\)
\(600\) −11.5553 −0.471741
\(601\) 4.72512 0.192742 0.0963708 0.995346i \(-0.469277\pi\)
0.0963708 + 0.995346i \(0.469277\pi\)
\(602\) −13.3811 −0.545371
\(603\) −14.3536 −0.584524
\(604\) 16.5279 0.672512
\(605\) −3.93725 −0.160072
\(606\) −9.41575 −0.382489
\(607\) 13.3446 0.541639 0.270819 0.962630i \(-0.412705\pi\)
0.270819 + 0.962630i \(0.412705\pi\)
\(608\) −1.84870 −0.0749746
\(609\) −28.5734 −1.15785
\(610\) −50.2580 −2.03489
\(611\) 1.17792 0.0476537
\(612\) −7.56875 −0.305948
\(613\) −20.0161 −0.808442 −0.404221 0.914661i \(-0.632457\pi\)
−0.404221 + 0.914661i \(0.632457\pi\)
\(614\) 7.65826 0.309062
\(615\) 2.83147 0.114176
\(616\) −14.9418 −0.602023
\(617\) −26.6433 −1.07262 −0.536309 0.844021i \(-0.680182\pi\)
−0.536309 + 0.844021i \(0.680182\pi\)
\(618\) 1.00000 0.0402259
\(619\) 35.7643 1.43749 0.718745 0.695274i \(-0.244717\pi\)
0.718745 + 0.695274i \(0.244717\pi\)
\(620\) 22.6209 0.908476
\(621\) 8.50665 0.341360
\(622\) 8.22932 0.329966
\(623\) −12.4431 −0.498521
\(624\) 1.00000 0.0400320
\(625\) 50.7477 2.02991
\(626\) 0.937649 0.0374760
\(627\) −6.39544 −0.255409
\(628\) −8.57471 −0.342168
\(629\) −75.7480 −3.02027
\(630\) −17.5738 −0.700159
\(631\) 1.98763 0.0791262 0.0395631 0.999217i \(-0.487403\pi\)
0.0395631 + 0.999217i \(0.487403\pi\)
\(632\) 6.21617 0.247266
\(633\) 3.92060 0.155830
\(634\) −13.7271 −0.545173
\(635\) 87.8477 3.48613
\(636\) −5.76954 −0.228777
\(637\) −11.6551 −0.461792
\(638\) −22.8859 −0.906060
\(639\) 4.40758 0.174361
\(640\) −4.06882 −0.160834
\(641\) 25.8792 1.02217 0.511084 0.859531i \(-0.329244\pi\)
0.511084 + 0.859531i \(0.329244\pi\)
\(642\) −3.81409 −0.150530
\(643\) −14.3720 −0.566778 −0.283389 0.959005i \(-0.591459\pi\)
−0.283389 + 0.959005i \(0.591459\pi\)
\(644\) −36.7415 −1.44782
\(645\) −12.6055 −0.496340
\(646\) 13.9923 0.550520
\(647\) 42.4147 1.66749 0.833747 0.552147i \(-0.186191\pi\)
0.833747 + 0.552147i \(0.186191\pi\)
\(648\) 1.00000 0.0392837
\(649\) −23.8169 −0.934896
\(650\) −11.5553 −0.453235
\(651\) 24.0126 0.941130
\(652\) −9.70287 −0.379994
\(653\) −9.61493 −0.376261 −0.188131 0.982144i \(-0.560243\pi\)
−0.188131 + 0.982144i \(0.560243\pi\)
\(654\) −18.7539 −0.733336
\(655\) 10.0577 0.392987
\(656\) 0.695896 0.0271702
\(657\) −8.29562 −0.323643
\(658\) −5.08763 −0.198337
\(659\) 0.623573 0.0242910 0.0121455 0.999926i \(-0.496134\pi\)
0.0121455 + 0.999926i \(0.496134\pi\)
\(660\) −14.0758 −0.547899
\(661\) −7.46895 −0.290508 −0.145254 0.989394i \(-0.546400\pi\)
−0.145254 + 0.989394i \(0.546400\pi\)
\(662\) 33.3238 1.29517
\(663\) −7.56875 −0.293946
\(664\) −11.6646 −0.452675
\(665\) 32.4887 1.25986
\(666\) 10.0080 0.387802
\(667\) −56.2757 −2.17900
\(668\) 13.4484 0.520335
\(669\) −13.3625 −0.516624
\(670\) 58.4022 2.25627
\(671\) −42.7309 −1.64961
\(672\) −4.31915 −0.166615
\(673\) 45.3201 1.74696 0.873481 0.486858i \(-0.161857\pi\)
0.873481 + 0.486858i \(0.161857\pi\)
\(674\) −11.5951 −0.446625
\(675\) −11.5553 −0.444762
\(676\) 1.00000 0.0384615
\(677\) 41.0621 1.57814 0.789072 0.614301i \(-0.210562\pi\)
0.789072 + 0.614301i \(0.210562\pi\)
\(678\) 10.5876 0.406614
\(679\) 20.9880 0.805447
\(680\) 30.7958 1.18097
\(681\) 1.25297 0.0480138
\(682\) 19.2330 0.736468
\(683\) −1.15076 −0.0440324 −0.0220162 0.999758i \(-0.507009\pi\)
−0.0220162 + 0.999758i \(0.507009\pi\)
\(684\) −1.84870 −0.0706867
\(685\) −30.9385 −1.18210
\(686\) 20.1061 0.767653
\(687\) −8.24201 −0.314452
\(688\) −3.09807 −0.118113
\(689\) −5.76954 −0.219802
\(690\) −34.6120 −1.31766
\(691\) 28.6259 1.08898 0.544490 0.838767i \(-0.316723\pi\)
0.544490 + 0.838767i \(0.316723\pi\)
\(692\) 0.334934 0.0127323
\(693\) −14.9418 −0.567593
\(694\) −11.6647 −0.442785
\(695\) 2.85339 0.108235
\(696\) −6.61550 −0.250760
\(697\) −5.26706 −0.199504
\(698\) 9.80093 0.370971
\(699\) −12.8913 −0.487594
\(700\) 49.9089 1.88638
\(701\) −40.7619 −1.53955 −0.769777 0.638313i \(-0.779633\pi\)
−0.769777 + 0.638313i \(0.779633\pi\)
\(702\) 1.00000 0.0377426
\(703\) −18.5018 −0.697807
\(704\) −3.45943 −0.130382
\(705\) −4.79275 −0.180505
\(706\) 6.38978 0.240482
\(707\) 40.6681 1.52948
\(708\) −6.88464 −0.258740
\(709\) −32.4818 −1.21988 −0.609940 0.792448i \(-0.708806\pi\)
−0.609940 + 0.792448i \(0.708806\pi\)
\(710\) −17.9336 −0.673037
\(711\) 6.21617 0.233124
\(712\) −2.88091 −0.107966
\(713\) 47.2933 1.77115
\(714\) 32.6906 1.22341
\(715\) −14.0758 −0.526405
\(716\) 18.9801 0.709321
\(717\) −20.1551 −0.752707
\(718\) −14.0791 −0.525426
\(719\) −36.3983 −1.35743 −0.678714 0.734403i \(-0.737462\pi\)
−0.678714 + 0.734403i \(0.737462\pi\)
\(720\) −4.06882 −0.151636
\(721\) −4.31915 −0.160854
\(722\) −15.5823 −0.579914
\(723\) 2.43963 0.0907307
\(724\) −4.69449 −0.174469
\(725\) 76.4438 2.83905
\(726\) −0.967666 −0.0359134
\(727\) −27.7079 −1.02763 −0.513814 0.857902i \(-0.671768\pi\)
−0.513814 + 0.857902i \(0.671768\pi\)
\(728\) −4.31915 −0.160079
\(729\) 1.00000 0.0370370
\(730\) 33.7533 1.24927
\(731\) 23.4485 0.867275
\(732\) −12.3520 −0.456543
\(733\) 31.1625 1.15101 0.575507 0.817797i \(-0.304805\pi\)
0.575507 + 0.817797i \(0.304805\pi\)
\(734\) −25.9658 −0.958414
\(735\) 47.4224 1.74920
\(736\) −8.50665 −0.313559
\(737\) 49.6553 1.82908
\(738\) 0.695896 0.0256163
\(739\) −11.1385 −0.409736 −0.204868 0.978790i \(-0.565676\pi\)
−0.204868 + 0.978790i \(0.565676\pi\)
\(740\) −40.7207 −1.49692
\(741\) −1.84870 −0.0679136
\(742\) 24.9195 0.914824
\(743\) −32.7056 −1.19985 −0.599925 0.800056i \(-0.704803\pi\)
−0.599925 + 0.800056i \(0.704803\pi\)
\(744\) 5.55957 0.203824
\(745\) −73.6433 −2.69808
\(746\) 28.3171 1.03676
\(747\) −11.6646 −0.426786
\(748\) 26.1836 0.957366
\(749\) 16.4736 0.601934
\(750\) 26.6721 0.973929
\(751\) 5.02809 0.183478 0.0917389 0.995783i \(-0.470757\pi\)
0.0917389 + 0.995783i \(0.470757\pi\)
\(752\) −1.17792 −0.0429544
\(753\) 7.14154 0.260252
\(754\) −6.61550 −0.240922
\(755\) −67.2491 −2.44745
\(756\) −4.31915 −0.157086
\(757\) 49.6215 1.80352 0.901762 0.432233i \(-0.142274\pi\)
0.901762 + 0.432233i \(0.142274\pi\)
\(758\) 5.80714 0.210925
\(759\) −29.4282 −1.06817
\(760\) 7.52201 0.272852
\(761\) 53.9627 1.95615 0.978074 0.208258i \(-0.0667793\pi\)
0.978074 + 0.208258i \(0.0667793\pi\)
\(762\) 21.5905 0.782141
\(763\) 81.0010 2.93243
\(764\) 12.4462 0.450289
\(765\) 30.7958 1.11343
\(766\) 22.5633 0.815247
\(767\) −6.88464 −0.248590
\(768\) −1.00000 −0.0360844
\(769\) −26.3748 −0.951098 −0.475549 0.879689i \(-0.657751\pi\)
−0.475549 + 0.879689i \(0.657751\pi\)
\(770\) 60.7955 2.19092
\(771\) −9.93685 −0.357867
\(772\) 2.02744 0.0729693
\(773\) −13.9060 −0.500165 −0.250083 0.968225i \(-0.580458\pi\)
−0.250083 + 0.968225i \(0.580458\pi\)
\(774\) −3.09807 −0.111358
\(775\) −64.2423 −2.30765
\(776\) 4.85929 0.174438
\(777\) −43.2261 −1.55073
\(778\) 23.6159 0.846670
\(779\) −1.28650 −0.0460937
\(780\) −4.06882 −0.145687
\(781\) −15.2477 −0.545606
\(782\) 64.3847 2.30239
\(783\) −6.61550 −0.236419
\(784\) 11.6551 0.416253
\(785\) 34.8889 1.24524
\(786\) 2.47190 0.0881698
\(787\) −29.1344 −1.03853 −0.519265 0.854613i \(-0.673794\pi\)
−0.519265 + 0.854613i \(0.673794\pi\)
\(788\) 16.8314 0.599595
\(789\) 24.6988 0.879301
\(790\) −25.2924 −0.899864
\(791\) −45.7295 −1.62595
\(792\) −3.45943 −0.122926
\(793\) −12.3520 −0.438632
\(794\) 18.8538 0.669097
\(795\) 23.4752 0.832579
\(796\) −0.301529 −0.0106874
\(797\) −0.139563 −0.00494358 −0.00247179 0.999997i \(-0.500787\pi\)
−0.00247179 + 0.999997i \(0.500787\pi\)
\(798\) 7.98481 0.282659
\(799\) 8.91540 0.315404
\(800\) 11.5553 0.408540
\(801\) −2.88091 −0.101792
\(802\) 10.3423 0.365201
\(803\) 28.6981 1.01273
\(804\) 14.3536 0.506213
\(805\) 149.494 5.26899
\(806\) 5.55957 0.195828
\(807\) −25.6107 −0.901540
\(808\) 9.41575 0.331245
\(809\) −32.0202 −1.12577 −0.562884 0.826536i \(-0.690308\pi\)
−0.562884 + 0.826536i \(0.690308\pi\)
\(810\) −4.06882 −0.142964
\(811\) −40.4544 −1.42055 −0.710273 0.703926i \(-0.751428\pi\)
−0.710273 + 0.703926i \(0.751428\pi\)
\(812\) 28.5734 1.00273
\(813\) 6.42255 0.225249
\(814\) −34.6220 −1.21350
\(815\) 39.4792 1.38290
\(816\) 7.56875 0.264959
\(817\) 5.72740 0.200376
\(818\) −18.2409 −0.637779
\(819\) −4.31915 −0.150923
\(820\) −2.83147 −0.0988793
\(821\) −33.3453 −1.16376 −0.581880 0.813274i \(-0.697683\pi\)
−0.581880 + 0.813274i \(0.697683\pi\)
\(822\) −7.60382 −0.265214
\(823\) −49.9130 −1.73986 −0.869928 0.493179i \(-0.835835\pi\)
−0.869928 + 0.493179i \(0.835835\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 39.9746 1.39174
\(826\) 29.7358 1.03464
\(827\) 10.8878 0.378606 0.189303 0.981919i \(-0.439377\pi\)
0.189303 + 0.981919i \(0.439377\pi\)
\(828\) −8.50665 −0.295626
\(829\) 32.1946 1.11817 0.559083 0.829112i \(-0.311153\pi\)
0.559083 + 0.829112i \(0.311153\pi\)
\(830\) 47.4612 1.64740
\(831\) −18.0070 −0.624657
\(832\) −1.00000 −0.0346688
\(833\) −88.2144 −3.05645
\(834\) 0.701284 0.0242835
\(835\) −54.7192 −1.89364
\(836\) 6.39544 0.221191
\(837\) 5.55957 0.192167
\(838\) −22.7734 −0.786695
\(839\) 19.0612 0.658065 0.329033 0.944319i \(-0.393277\pi\)
0.329033 + 0.944319i \(0.393277\pi\)
\(840\) 17.5738 0.606355
\(841\) 14.7648 0.509131
\(842\) 19.3529 0.666946
\(843\) 12.7632 0.439587
\(844\) −3.92060 −0.134953
\(845\) −4.06882 −0.139972
\(846\) −1.17792 −0.0404978
\(847\) 4.17950 0.143609
\(848\) 5.76954 0.198127
\(849\) −10.4087 −0.357225
\(850\) −87.4588 −2.99981
\(851\) −85.1345 −2.91837
\(852\) −4.40758 −0.151001
\(853\) −6.03350 −0.206583 −0.103292 0.994651i \(-0.532937\pi\)
−0.103292 + 0.994651i \(0.532937\pi\)
\(854\) 53.3501 1.82560
\(855\) 7.52201 0.257247
\(856\) 3.81409 0.130363
\(857\) 39.3766 1.34508 0.672540 0.740061i \(-0.265203\pi\)
0.672540 + 0.740061i \(0.265203\pi\)
\(858\) −3.45943 −0.118103
\(859\) −37.2844 −1.27213 −0.636063 0.771637i \(-0.719438\pi\)
−0.636063 + 0.771637i \(0.719438\pi\)
\(860\) 12.6055 0.429843
\(861\) −3.00568 −0.102433
\(862\) −23.1867 −0.789743
\(863\) −55.0793 −1.87492 −0.937461 0.348090i \(-0.886830\pi\)
−0.937461 + 0.348090i \(0.886830\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.36278 −0.0463361
\(866\) 5.73037 0.194726
\(867\) −40.2859 −1.36818
\(868\) −24.0126 −0.815042
\(869\) −21.5044 −0.729487
\(870\) 26.9172 0.912580
\(871\) 14.3536 0.486353
\(872\) 18.7539 0.635087
\(873\) 4.85929 0.164462
\(874\) 15.7262 0.531947
\(875\) −115.201 −3.89451
\(876\) 8.29562 0.280283
\(877\) −17.2810 −0.583539 −0.291769 0.956489i \(-0.594244\pi\)
−0.291769 + 0.956489i \(0.594244\pi\)
\(878\) −3.84780 −0.129857
\(879\) 13.9122 0.469246
\(880\) 14.0758 0.474495
\(881\) 2.51424 0.0847068 0.0423534 0.999103i \(-0.486514\pi\)
0.0423534 + 0.999103i \(0.486514\pi\)
\(882\) 11.6551 0.392447
\(883\) −46.5170 −1.56542 −0.782711 0.622385i \(-0.786164\pi\)
−0.782711 + 0.622385i \(0.786164\pi\)
\(884\) 7.56875 0.254565
\(885\) 28.0123 0.941624
\(886\) 31.2358 1.04939
\(887\) 50.6035 1.69910 0.849550 0.527508i \(-0.176874\pi\)
0.849550 + 0.527508i \(0.176874\pi\)
\(888\) −10.0080 −0.335846
\(889\) −93.2526 −3.12759
\(890\) 11.7219 0.392918
\(891\) −3.45943 −0.115895
\(892\) 13.3625 0.447410
\(893\) 2.17762 0.0728714
\(894\) −18.0995 −0.605336
\(895\) −77.2267 −2.58140
\(896\) 4.31915 0.144293
\(897\) −8.50665 −0.284029
\(898\) 3.02705 0.101014
\(899\) −36.7793 −1.22666
\(900\) 11.5553 0.385175
\(901\) −43.6682 −1.45480
\(902\) −2.40740 −0.0801578
\(903\) 13.3811 0.445293
\(904\) −10.5876 −0.352138
\(905\) 19.1010 0.634939
\(906\) −16.5279 −0.549104
\(907\) −31.9748 −1.06171 −0.530853 0.847464i \(-0.678128\pi\)
−0.530853 + 0.847464i \(0.678128\pi\)
\(908\) −1.25297 −0.0415811
\(909\) 9.41575 0.312301
\(910\) 17.5738 0.582567
\(911\) 39.0255 1.29297 0.646486 0.762926i \(-0.276238\pi\)
0.646486 + 0.762926i \(0.276238\pi\)
\(912\) 1.84870 0.0612165
\(913\) 40.3529 1.33549
\(914\) −38.0751 −1.25941
\(915\) 50.2580 1.66148
\(916\) 8.24201 0.272324
\(917\) −10.6765 −0.352570
\(918\) 7.56875 0.249806
\(919\) 31.8283 1.04992 0.524959 0.851127i \(-0.324081\pi\)
0.524959 + 0.851127i \(0.324081\pi\)
\(920\) 34.6120 1.14112
\(921\) −7.65826 −0.252348
\(922\) −16.0253 −0.527765
\(923\) −4.40758 −0.145077
\(924\) 14.9418 0.491550
\(925\) 115.645 3.80238
\(926\) −7.54870 −0.248066
\(927\) −1.00000 −0.0328443
\(928\) 6.61550 0.217164
\(929\) 28.7737 0.944033 0.472017 0.881590i \(-0.343526\pi\)
0.472017 + 0.881590i \(0.343526\pi\)
\(930\) −22.6209 −0.741767
\(931\) −21.5467 −0.706166
\(932\) 12.8913 0.422269
\(933\) −8.22932 −0.269416
\(934\) −32.7890 −1.07289
\(935\) −106.536 −3.48410
\(936\) −1.00000 −0.0326860
\(937\) −52.1487 −1.70362 −0.851812 0.523847i \(-0.824496\pi\)
−0.851812 + 0.523847i \(0.824496\pi\)
\(938\) −61.9954 −2.02422
\(939\) −0.937649 −0.0305990
\(940\) 4.79275 0.156322
\(941\) −46.7160 −1.52290 −0.761449 0.648225i \(-0.775512\pi\)
−0.761449 + 0.648225i \(0.775512\pi\)
\(942\) 8.57471 0.279379
\(943\) −5.91974 −0.192773
\(944\) 6.88464 0.224076
\(945\) 17.5738 0.571677
\(946\) 10.7176 0.348458
\(947\) 60.8358 1.97690 0.988450 0.151549i \(-0.0484263\pi\)
0.988450 + 0.151549i \(0.0484263\pi\)
\(948\) −6.21617 −0.201892
\(949\) 8.29562 0.269287
\(950\) −21.3622 −0.693080
\(951\) 13.7271 0.445132
\(952\) −32.6906 −1.05951
\(953\) 18.6575 0.604375 0.302187 0.953249i \(-0.402283\pi\)
0.302187 + 0.953249i \(0.402283\pi\)
\(954\) 5.76954 0.186796
\(955\) −50.6415 −1.63872
\(956\) 20.1551 0.651864
\(957\) 22.8859 0.739795
\(958\) 38.4984 1.24383
\(959\) 32.8421 1.06053
\(960\) 4.06882 0.131320
\(961\) −0.0911780 −0.00294123
\(962\) −10.0080 −0.322671
\(963\) 3.81409 0.122907
\(964\) −2.43963 −0.0785751
\(965\) −8.24930 −0.265554
\(966\) 36.7415 1.18214
\(967\) 22.4665 0.722474 0.361237 0.932474i \(-0.382355\pi\)
0.361237 + 0.932474i \(0.382355\pi\)
\(968\) 0.967666 0.0311020
\(969\) −13.9923 −0.449498
\(970\) −19.7716 −0.634827
\(971\) −60.8053 −1.95133 −0.975667 0.219258i \(-0.929636\pi\)
−0.975667 + 0.219258i \(0.929636\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −3.02895 −0.0971037
\(974\) −6.04136 −0.193578
\(975\) 11.5553 0.370064
\(976\) 12.3520 0.395378
\(977\) −18.3313 −0.586471 −0.293235 0.956040i \(-0.594732\pi\)
−0.293235 + 0.956040i \(0.594732\pi\)
\(978\) 9.70287 0.310263
\(979\) 9.96629 0.318524
\(980\) −47.4224 −1.51485
\(981\) 18.7539 0.598766
\(982\) 26.8174 0.855778
\(983\) 18.8713 0.601902 0.300951 0.953640i \(-0.402696\pi\)
0.300951 + 0.953640i \(0.402696\pi\)
\(984\) −0.695896 −0.0221843
\(985\) −68.4840 −2.18208
\(986\) −50.0710 −1.59459
\(987\) 5.08763 0.161941
\(988\) 1.84870 0.0588149
\(989\) 26.3542 0.838015
\(990\) 14.0758 0.447358
\(991\) 43.5025 1.38190 0.690951 0.722902i \(-0.257192\pi\)
0.690951 + 0.722902i \(0.257192\pi\)
\(992\) −5.55957 −0.176517
\(993\) −33.3238 −1.05750
\(994\) 19.0370 0.603817
\(995\) 1.22687 0.0388943
\(996\) 11.6646 0.369608
\(997\) −46.1647 −1.46205 −0.731025 0.682351i \(-0.760958\pi\)
−0.731025 + 0.682351i \(0.760958\pi\)
\(998\) 8.59968 0.272218
\(999\) −10.0080 −0.316639
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 8034.2.a.bc.1.1 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.1 15 1.1 even 1 trivial