Properties

Label 8034.2.a.bc.1.14
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 1.14
Root \(-3.53778\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.53778 q^{5} -1.00000 q^{6} -4.09818 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.53778 q^{5} -1.00000 q^{6} -4.09818 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.53778 q^{10} -3.53153 q^{11} -1.00000 q^{12} -1.00000 q^{13} -4.09818 q^{14} -3.53778 q^{15} +1.00000 q^{16} -2.18686 q^{17} +1.00000 q^{18} -0.266722 q^{19} +3.53778 q^{20} +4.09818 q^{21} -3.53153 q^{22} -9.08925 q^{23} -1.00000 q^{24} +7.51589 q^{25} -1.00000 q^{26} -1.00000 q^{27} -4.09818 q^{28} +6.77872 q^{29} -3.53778 q^{30} +6.69972 q^{31} +1.00000 q^{32} +3.53153 q^{33} -2.18686 q^{34} -14.4985 q^{35} +1.00000 q^{36} +10.3454 q^{37} -0.266722 q^{38} +1.00000 q^{39} +3.53778 q^{40} -5.22997 q^{41} +4.09818 q^{42} +6.71778 q^{43} -3.53153 q^{44} +3.53778 q^{45} -9.08925 q^{46} -8.08805 q^{47} -1.00000 q^{48} +9.79508 q^{49} +7.51589 q^{50} +2.18686 q^{51} -1.00000 q^{52} +5.93846 q^{53} -1.00000 q^{54} -12.4938 q^{55} -4.09818 q^{56} +0.266722 q^{57} +6.77872 q^{58} +1.22445 q^{59} -3.53778 q^{60} -4.83351 q^{61} +6.69972 q^{62} -4.09818 q^{63} +1.00000 q^{64} -3.53778 q^{65} +3.53153 q^{66} +12.7005 q^{67} -2.18686 q^{68} +9.08925 q^{69} -14.4985 q^{70} -8.30262 q^{71} +1.00000 q^{72} +11.4874 q^{73} +10.3454 q^{74} -7.51589 q^{75} -0.266722 q^{76} +14.4729 q^{77} +1.00000 q^{78} +16.9009 q^{79} +3.53778 q^{80} +1.00000 q^{81} -5.22997 q^{82} +8.26872 q^{83} +4.09818 q^{84} -7.73663 q^{85} +6.71778 q^{86} -6.77872 q^{87} -3.53153 q^{88} +11.1766 q^{89} +3.53778 q^{90} +4.09818 q^{91} -9.08925 q^{92} -6.69972 q^{93} -8.08805 q^{94} -0.943606 q^{95} -1.00000 q^{96} +8.23809 q^{97} +9.79508 q^{98} -3.53153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 3.53778 1.58214 0.791072 0.611723i \(-0.209523\pi\)
0.791072 + 0.611723i \(0.209523\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.09818 −1.54897 −0.774483 0.632594i \(-0.781990\pi\)
−0.774483 + 0.632594i \(0.781990\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.53778 1.11874
\(11\) −3.53153 −1.06480 −0.532398 0.846494i \(-0.678709\pi\)
−0.532398 + 0.846494i \(0.678709\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −4.09818 −1.09528
\(15\) −3.53778 −0.913451
\(16\) 1.00000 0.250000
\(17\) −2.18686 −0.530391 −0.265196 0.964195i \(-0.585437\pi\)
−0.265196 + 0.964195i \(0.585437\pi\)
\(18\) 1.00000 0.235702
\(19\) −0.266722 −0.0611903 −0.0305952 0.999532i \(-0.509740\pi\)
−0.0305952 + 0.999532i \(0.509740\pi\)
\(20\) 3.53778 0.791072
\(21\) 4.09818 0.894296
\(22\) −3.53153 −0.752925
\(23\) −9.08925 −1.89524 −0.947619 0.319402i \(-0.896518\pi\)
−0.947619 + 0.319402i \(0.896518\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.51589 1.50318
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −4.09818 −0.774483
\(29\) 6.77872 1.25878 0.629389 0.777091i \(-0.283305\pi\)
0.629389 + 0.777091i \(0.283305\pi\)
\(30\) −3.53778 −0.645907
\(31\) 6.69972 1.20331 0.601653 0.798758i \(-0.294509\pi\)
0.601653 + 0.798758i \(0.294509\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.53153 0.614761
\(34\) −2.18686 −0.375043
\(35\) −14.4985 −2.45069
\(36\) 1.00000 0.166667
\(37\) 10.3454 1.70078 0.850390 0.526153i \(-0.176366\pi\)
0.850390 + 0.526153i \(0.176366\pi\)
\(38\) −0.266722 −0.0432681
\(39\) 1.00000 0.160128
\(40\) 3.53778 0.559372
\(41\) −5.22997 −0.816784 −0.408392 0.912807i \(-0.633910\pi\)
−0.408392 + 0.912807i \(0.633910\pi\)
\(42\) 4.09818 0.632363
\(43\) 6.71778 1.02445 0.512226 0.858851i \(-0.328821\pi\)
0.512226 + 0.858851i \(0.328821\pi\)
\(44\) −3.53153 −0.532398
\(45\) 3.53778 0.527381
\(46\) −9.08925 −1.34014
\(47\) −8.08805 −1.17976 −0.589882 0.807490i \(-0.700826\pi\)
−0.589882 + 0.807490i \(0.700826\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.79508 1.39930
\(50\) 7.51589 1.06291
\(51\) 2.18686 0.306222
\(52\) −1.00000 −0.138675
\(53\) 5.93846 0.815710 0.407855 0.913047i \(-0.366277\pi\)
0.407855 + 0.913047i \(0.366277\pi\)
\(54\) −1.00000 −0.136083
\(55\) −12.4938 −1.68466
\(56\) −4.09818 −0.547642
\(57\) 0.266722 0.0353283
\(58\) 6.77872 0.890090
\(59\) 1.22445 0.159410 0.0797048 0.996819i \(-0.474602\pi\)
0.0797048 + 0.996819i \(0.474602\pi\)
\(60\) −3.53778 −0.456726
\(61\) −4.83351 −0.618867 −0.309434 0.950921i \(-0.600139\pi\)
−0.309434 + 0.950921i \(0.600139\pi\)
\(62\) 6.69972 0.850866
\(63\) −4.09818 −0.516322
\(64\) 1.00000 0.125000
\(65\) −3.53778 −0.438808
\(66\) 3.53153 0.434701
\(67\) 12.7005 1.55161 0.775806 0.630971i \(-0.217343\pi\)
0.775806 + 0.630971i \(0.217343\pi\)
\(68\) −2.18686 −0.265196
\(69\) 9.08925 1.09422
\(70\) −14.4985 −1.73290
\(71\) −8.30262 −0.985340 −0.492670 0.870216i \(-0.663979\pi\)
−0.492670 + 0.870216i \(0.663979\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.4874 1.34449 0.672247 0.740327i \(-0.265329\pi\)
0.672247 + 0.740327i \(0.265329\pi\)
\(74\) 10.3454 1.20263
\(75\) −7.51589 −0.867860
\(76\) −0.266722 −0.0305952
\(77\) 14.4729 1.64933
\(78\) 1.00000 0.113228
\(79\) 16.9009 1.90150 0.950750 0.309960i \(-0.100316\pi\)
0.950750 + 0.309960i \(0.100316\pi\)
\(80\) 3.53778 0.395536
\(81\) 1.00000 0.111111
\(82\) −5.22997 −0.577554
\(83\) 8.26872 0.907610 0.453805 0.891101i \(-0.350066\pi\)
0.453805 + 0.891101i \(0.350066\pi\)
\(84\) 4.09818 0.447148
\(85\) −7.73663 −0.839155
\(86\) 6.71778 0.724396
\(87\) −6.77872 −0.726755
\(88\) −3.53153 −0.376462
\(89\) 11.1766 1.18472 0.592360 0.805673i \(-0.298196\pi\)
0.592360 + 0.805673i \(0.298196\pi\)
\(90\) 3.53778 0.372915
\(91\) 4.09818 0.429606
\(92\) −9.08925 −0.947619
\(93\) −6.69972 −0.694729
\(94\) −8.08805 −0.834219
\(95\) −0.943606 −0.0968119
\(96\) −1.00000 −0.102062
\(97\) 8.23809 0.836451 0.418226 0.908343i \(-0.362652\pi\)
0.418226 + 0.908343i \(0.362652\pi\)
\(98\) 9.79508 0.989452
\(99\) −3.53153 −0.354932
\(100\) 7.51589 0.751589
\(101\) −16.6091 −1.65267 −0.826334 0.563180i \(-0.809578\pi\)
−0.826334 + 0.563180i \(0.809578\pi\)
\(102\) 2.18686 0.216531
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 14.4985 1.41490
\(106\) 5.93846 0.576794
\(107\) 18.4079 1.77956 0.889778 0.456394i \(-0.150859\pi\)
0.889778 + 0.456394i \(0.150859\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 12.5219 1.19938 0.599689 0.800233i \(-0.295291\pi\)
0.599689 + 0.800233i \(0.295291\pi\)
\(110\) −12.4938 −1.19124
\(111\) −10.3454 −0.981945
\(112\) −4.09818 −0.387242
\(113\) 12.1628 1.14418 0.572089 0.820191i \(-0.306133\pi\)
0.572089 + 0.820191i \(0.306133\pi\)
\(114\) 0.266722 0.0249808
\(115\) −32.1558 −2.99854
\(116\) 6.77872 0.629389
\(117\) −1.00000 −0.0924500
\(118\) 1.22445 0.112720
\(119\) 8.96214 0.821558
\(120\) −3.53778 −0.322954
\(121\) 1.47171 0.133792
\(122\) −4.83351 −0.437605
\(123\) 5.22997 0.471571
\(124\) 6.69972 0.601653
\(125\) 8.90067 0.796100
\(126\) −4.09818 −0.365095
\(127\) −8.55557 −0.759184 −0.379592 0.925154i \(-0.623936\pi\)
−0.379592 + 0.925154i \(0.623936\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.71778 −0.591467
\(130\) −3.53778 −0.310284
\(131\) 6.96127 0.608209 0.304105 0.952639i \(-0.401643\pi\)
0.304105 + 0.952639i \(0.401643\pi\)
\(132\) 3.53153 0.307380
\(133\) 1.09308 0.0947818
\(134\) 12.7005 1.09716
\(135\) −3.53778 −0.304484
\(136\) −2.18686 −0.187522
\(137\) 12.7438 1.08878 0.544389 0.838833i \(-0.316762\pi\)
0.544389 + 0.838833i \(0.316762\pi\)
\(138\) 9.08925 0.773728
\(139\) 6.43994 0.546229 0.273114 0.961982i \(-0.411946\pi\)
0.273114 + 0.961982i \(0.411946\pi\)
\(140\) −14.4985 −1.22534
\(141\) 8.08805 0.681137
\(142\) −8.30262 −0.696740
\(143\) 3.53153 0.295321
\(144\) 1.00000 0.0833333
\(145\) 23.9816 1.99157
\(146\) 11.4874 0.950701
\(147\) −9.79508 −0.807885
\(148\) 10.3454 0.850390
\(149\) 1.99904 0.163767 0.0818837 0.996642i \(-0.473906\pi\)
0.0818837 + 0.996642i \(0.473906\pi\)
\(150\) −7.51589 −0.613670
\(151\) 7.74370 0.630173 0.315087 0.949063i \(-0.397966\pi\)
0.315087 + 0.949063i \(0.397966\pi\)
\(152\) −0.266722 −0.0216340
\(153\) −2.18686 −0.176797
\(154\) 14.4729 1.16626
\(155\) 23.7022 1.90380
\(156\) 1.00000 0.0800641
\(157\) −6.61959 −0.528300 −0.264150 0.964482i \(-0.585091\pi\)
−0.264150 + 0.964482i \(0.585091\pi\)
\(158\) 16.9009 1.34456
\(159\) −5.93846 −0.470950
\(160\) 3.53778 0.279686
\(161\) 37.2494 2.93566
\(162\) 1.00000 0.0785674
\(163\) −9.98151 −0.781812 −0.390906 0.920431i \(-0.627838\pi\)
−0.390906 + 0.920431i \(0.627838\pi\)
\(164\) −5.22997 −0.408392
\(165\) 12.4938 0.972640
\(166\) 8.26872 0.641777
\(167\) −23.0834 −1.78625 −0.893125 0.449809i \(-0.851492\pi\)
−0.893125 + 0.449809i \(0.851492\pi\)
\(168\) 4.09818 0.316181
\(169\) 1.00000 0.0769231
\(170\) −7.73663 −0.593372
\(171\) −0.266722 −0.0203968
\(172\) 6.71778 0.512226
\(173\) 4.63859 0.352666 0.176333 0.984331i \(-0.443576\pi\)
0.176333 + 0.984331i \(0.443576\pi\)
\(174\) −6.77872 −0.513894
\(175\) −30.8015 −2.32837
\(176\) −3.53153 −0.266199
\(177\) −1.22445 −0.0920352
\(178\) 11.1766 0.837724
\(179\) −25.0381 −1.87143 −0.935717 0.352751i \(-0.885246\pi\)
−0.935717 + 0.352751i \(0.885246\pi\)
\(180\) 3.53778 0.263691
\(181\) −14.6223 −1.08687 −0.543435 0.839451i \(-0.682877\pi\)
−0.543435 + 0.839451i \(0.682877\pi\)
\(182\) 4.09818 0.303777
\(183\) 4.83351 0.357303
\(184\) −9.08925 −0.670068
\(185\) 36.5999 2.69088
\(186\) −6.69972 −0.491248
\(187\) 7.72296 0.564759
\(188\) −8.08805 −0.589882
\(189\) 4.09818 0.298099
\(190\) −0.943606 −0.0684563
\(191\) 24.7035 1.78748 0.893741 0.448584i \(-0.148071\pi\)
0.893741 + 0.448584i \(0.148071\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −13.9811 −1.00638 −0.503189 0.864176i \(-0.667840\pi\)
−0.503189 + 0.864176i \(0.667840\pi\)
\(194\) 8.23809 0.591460
\(195\) 3.53778 0.253346
\(196\) 9.79508 0.699649
\(197\) −24.1579 −1.72118 −0.860590 0.509299i \(-0.829905\pi\)
−0.860590 + 0.509299i \(0.829905\pi\)
\(198\) −3.53153 −0.250975
\(199\) 13.8851 0.984288 0.492144 0.870514i \(-0.336213\pi\)
0.492144 + 0.870514i \(0.336213\pi\)
\(200\) 7.51589 0.531454
\(201\) −12.7005 −0.895824
\(202\) −16.6091 −1.16861
\(203\) −27.7804 −1.94980
\(204\) 2.18686 0.153111
\(205\) −18.5025 −1.29227
\(206\) −1.00000 −0.0696733
\(207\) −9.08925 −0.631746
\(208\) −1.00000 −0.0693375
\(209\) 0.941939 0.0651553
\(210\) 14.4985 1.00049
\(211\) 13.7740 0.948244 0.474122 0.880459i \(-0.342766\pi\)
0.474122 + 0.880459i \(0.342766\pi\)
\(212\) 5.93846 0.407855
\(213\) 8.30262 0.568886
\(214\) 18.4079 1.25834
\(215\) 23.7660 1.62083
\(216\) −1.00000 −0.0680414
\(217\) −27.4567 −1.86388
\(218\) 12.5219 0.848088
\(219\) −11.4874 −0.776244
\(220\) −12.4938 −0.842331
\(221\) 2.18686 0.147104
\(222\) −10.3454 −0.694340
\(223\) 12.8558 0.860891 0.430445 0.902617i \(-0.358357\pi\)
0.430445 + 0.902617i \(0.358357\pi\)
\(224\) −4.09818 −0.273821
\(225\) 7.51589 0.501059
\(226\) 12.1628 0.809057
\(227\) −12.2505 −0.813097 −0.406548 0.913629i \(-0.633268\pi\)
−0.406548 + 0.913629i \(0.633268\pi\)
\(228\) 0.266722 0.0176641
\(229\) −6.81988 −0.450670 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(230\) −32.1558 −2.12029
\(231\) −14.4729 −0.952244
\(232\) 6.77872 0.445045
\(233\) 18.1244 1.18737 0.593685 0.804697i \(-0.297672\pi\)
0.593685 + 0.804697i \(0.297672\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −28.6137 −1.86655
\(236\) 1.22445 0.0797048
\(237\) −16.9009 −1.09783
\(238\) 8.96214 0.580930
\(239\) 1.67370 0.108263 0.0541313 0.998534i \(-0.482761\pi\)
0.0541313 + 0.998534i \(0.482761\pi\)
\(240\) −3.53778 −0.228363
\(241\) −1.25519 −0.0808540 −0.0404270 0.999182i \(-0.512872\pi\)
−0.0404270 + 0.999182i \(0.512872\pi\)
\(242\) 1.47171 0.0946053
\(243\) −1.00000 −0.0641500
\(244\) −4.83351 −0.309434
\(245\) 34.6528 2.21389
\(246\) 5.22997 0.333451
\(247\) 0.266722 0.0169711
\(248\) 6.69972 0.425433
\(249\) −8.26872 −0.524009
\(250\) 8.90067 0.562928
\(251\) 3.37351 0.212934 0.106467 0.994316i \(-0.466046\pi\)
0.106467 + 0.994316i \(0.466046\pi\)
\(252\) −4.09818 −0.258161
\(253\) 32.0990 2.01804
\(254\) −8.55557 −0.536824
\(255\) 7.73663 0.484487
\(256\) 1.00000 0.0625000
\(257\) 10.8138 0.674549 0.337274 0.941406i \(-0.390495\pi\)
0.337274 + 0.941406i \(0.390495\pi\)
\(258\) −6.71778 −0.418230
\(259\) −42.3975 −2.63445
\(260\) −3.53778 −0.219404
\(261\) 6.77872 0.419592
\(262\) 6.96127 0.430069
\(263\) −10.1360 −0.625012 −0.312506 0.949916i \(-0.601168\pi\)
−0.312506 + 0.949916i \(0.601168\pi\)
\(264\) 3.53153 0.217351
\(265\) 21.0090 1.29057
\(266\) 1.09308 0.0670208
\(267\) −11.1766 −0.683999
\(268\) 12.7005 0.775806
\(269\) −5.13298 −0.312963 −0.156482 0.987681i \(-0.550015\pi\)
−0.156482 + 0.987681i \(0.550015\pi\)
\(270\) −3.53778 −0.215302
\(271\) −13.1844 −0.800897 −0.400449 0.916319i \(-0.631146\pi\)
−0.400449 + 0.916319i \(0.631146\pi\)
\(272\) −2.18686 −0.132598
\(273\) −4.09818 −0.248033
\(274\) 12.7438 0.769882
\(275\) −26.5426 −1.60058
\(276\) 9.08925 0.547108
\(277\) 10.7211 0.644168 0.322084 0.946711i \(-0.395617\pi\)
0.322084 + 0.946711i \(0.395617\pi\)
\(278\) 6.43994 0.386242
\(279\) 6.69972 0.401102
\(280\) −14.4985 −0.866449
\(281\) 18.2550 1.08900 0.544501 0.838760i \(-0.316719\pi\)
0.544501 + 0.838760i \(0.316719\pi\)
\(282\) 8.08805 0.481636
\(283\) 30.7561 1.82826 0.914131 0.405418i \(-0.132874\pi\)
0.914131 + 0.405418i \(0.132874\pi\)
\(284\) −8.30262 −0.492670
\(285\) 0.943606 0.0558944
\(286\) 3.53153 0.208824
\(287\) 21.4334 1.26517
\(288\) 1.00000 0.0589256
\(289\) −12.2176 −0.718685
\(290\) 23.9816 1.40825
\(291\) −8.23809 −0.482925
\(292\) 11.4874 0.672247
\(293\) −24.9096 −1.45524 −0.727619 0.685982i \(-0.759373\pi\)
−0.727619 + 0.685982i \(0.759373\pi\)
\(294\) −9.79508 −0.571261
\(295\) 4.33183 0.252209
\(296\) 10.3454 0.601316
\(297\) 3.53153 0.204920
\(298\) 1.99904 0.115801
\(299\) 9.08925 0.525645
\(300\) −7.51589 −0.433930
\(301\) −27.5307 −1.58684
\(302\) 7.74370 0.445600
\(303\) 16.6091 0.954169
\(304\) −0.266722 −0.0152976
\(305\) −17.0999 −0.979136
\(306\) −2.18686 −0.125014
\(307\) 21.8157 1.24509 0.622544 0.782585i \(-0.286099\pi\)
0.622544 + 0.782585i \(0.286099\pi\)
\(308\) 14.4729 0.824667
\(309\) 1.00000 0.0568880
\(310\) 23.7022 1.34619
\(311\) −31.2060 −1.76953 −0.884765 0.466037i \(-0.845681\pi\)
−0.884765 + 0.466037i \(0.845681\pi\)
\(312\) 1.00000 0.0566139
\(313\) −19.1775 −1.08398 −0.541988 0.840386i \(-0.682328\pi\)
−0.541988 + 0.840386i \(0.682328\pi\)
\(314\) −6.61959 −0.373565
\(315\) −14.4985 −0.816896
\(316\) 16.9009 0.950750
\(317\) −31.2536 −1.75538 −0.877690 0.479230i \(-0.840916\pi\)
−0.877690 + 0.479230i \(0.840916\pi\)
\(318\) −5.93846 −0.333012
\(319\) −23.9393 −1.34034
\(320\) 3.53778 0.197768
\(321\) −18.4079 −1.02743
\(322\) 37.2494 2.07583
\(323\) 0.583285 0.0324548
\(324\) 1.00000 0.0555556
\(325\) −7.51589 −0.416907
\(326\) −9.98151 −0.552824
\(327\) −12.5219 −0.692461
\(328\) −5.22997 −0.288777
\(329\) 33.1463 1.82741
\(330\) 12.4938 0.687760
\(331\) −14.0854 −0.774204 −0.387102 0.922037i \(-0.626524\pi\)
−0.387102 + 0.922037i \(0.626524\pi\)
\(332\) 8.26872 0.453805
\(333\) 10.3454 0.566926
\(334\) −23.0834 −1.26307
\(335\) 44.9316 2.45487
\(336\) 4.09818 0.223574
\(337\) 13.8956 0.756942 0.378471 0.925613i \(-0.376450\pi\)
0.378471 + 0.925613i \(0.376450\pi\)
\(338\) 1.00000 0.0543928
\(339\) −12.1628 −0.660592
\(340\) −7.73663 −0.419578
\(341\) −23.6603 −1.28128
\(342\) −0.266722 −0.0144227
\(343\) −11.4547 −0.618498
\(344\) 6.71778 0.362198
\(345\) 32.1558 1.73121
\(346\) 4.63859 0.249372
\(347\) 0.0369366 0.00198286 0.000991429 1.00000i \(-0.499684\pi\)
0.000991429 1.00000i \(0.499684\pi\)
\(348\) −6.77872 −0.363378
\(349\) 15.1285 0.809810 0.404905 0.914359i \(-0.367305\pi\)
0.404905 + 0.914359i \(0.367305\pi\)
\(350\) −30.8015 −1.64641
\(351\) 1.00000 0.0533761
\(352\) −3.53153 −0.188231
\(353\) −3.79280 −0.201871 −0.100935 0.994893i \(-0.532184\pi\)
−0.100935 + 0.994893i \(0.532184\pi\)
\(354\) −1.22445 −0.0650787
\(355\) −29.3728 −1.55895
\(356\) 11.1766 0.592360
\(357\) −8.96214 −0.474327
\(358\) −25.0381 −1.32330
\(359\) 9.47254 0.499942 0.249971 0.968253i \(-0.419579\pi\)
0.249971 + 0.968253i \(0.419579\pi\)
\(360\) 3.53778 0.186457
\(361\) −18.9289 −0.996256
\(362\) −14.6223 −0.768534
\(363\) −1.47171 −0.0772449
\(364\) 4.09818 0.214803
\(365\) 40.6398 2.12718
\(366\) 4.83351 0.252651
\(367\) 7.47886 0.390393 0.195197 0.980764i \(-0.437465\pi\)
0.195197 + 0.980764i \(0.437465\pi\)
\(368\) −9.08925 −0.473810
\(369\) −5.22997 −0.272261
\(370\) 36.5999 1.90274
\(371\) −24.3369 −1.26351
\(372\) −6.69972 −0.347364
\(373\) −20.7493 −1.07436 −0.537179 0.843468i \(-0.680510\pi\)
−0.537179 + 0.843468i \(0.680510\pi\)
\(374\) 7.72296 0.399345
\(375\) −8.90067 −0.459629
\(376\) −8.08805 −0.417109
\(377\) −6.77872 −0.349122
\(378\) 4.09818 0.210788
\(379\) −7.42769 −0.381535 −0.190767 0.981635i \(-0.561098\pi\)
−0.190767 + 0.981635i \(0.561098\pi\)
\(380\) −0.943606 −0.0484059
\(381\) 8.55557 0.438315
\(382\) 24.7035 1.26394
\(383\) −11.2539 −0.575049 −0.287524 0.957773i \(-0.592832\pi\)
−0.287524 + 0.957773i \(0.592832\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 51.2018 2.60948
\(386\) −13.9811 −0.711617
\(387\) 6.71778 0.341484
\(388\) 8.23809 0.418226
\(389\) 29.4974 1.49558 0.747788 0.663938i \(-0.231116\pi\)
0.747788 + 0.663938i \(0.231116\pi\)
\(390\) 3.53778 0.179142
\(391\) 19.8769 1.00522
\(392\) 9.79508 0.494726
\(393\) −6.96127 −0.351150
\(394\) −24.1579 −1.21706
\(395\) 59.7917 3.00844
\(396\) −3.53153 −0.177466
\(397\) −7.73124 −0.388020 −0.194010 0.981000i \(-0.562149\pi\)
−0.194010 + 0.981000i \(0.562149\pi\)
\(398\) 13.8851 0.695996
\(399\) −1.09308 −0.0547223
\(400\) 7.51589 0.375795
\(401\) −23.0488 −1.15100 −0.575501 0.817801i \(-0.695193\pi\)
−0.575501 + 0.817801i \(0.695193\pi\)
\(402\) −12.7005 −0.633443
\(403\) −6.69972 −0.333737
\(404\) −16.6091 −0.826334
\(405\) 3.53778 0.175794
\(406\) −27.7804 −1.37872
\(407\) −36.5352 −1.81098
\(408\) 2.18686 0.108266
\(409\) −12.7459 −0.630246 −0.315123 0.949051i \(-0.602046\pi\)
−0.315123 + 0.949051i \(0.602046\pi\)
\(410\) −18.5025 −0.913773
\(411\) −12.7438 −0.628606
\(412\) −1.00000 −0.0492665
\(413\) −5.01801 −0.246920
\(414\) −9.08925 −0.446712
\(415\) 29.2529 1.43597
\(416\) −1.00000 −0.0490290
\(417\) −6.43994 −0.315365
\(418\) 0.941939 0.0460717
\(419\) −8.38630 −0.409697 −0.204849 0.978794i \(-0.565670\pi\)
−0.204849 + 0.978794i \(0.565670\pi\)
\(420\) 14.4985 0.707452
\(421\) 24.5437 1.19619 0.598093 0.801426i \(-0.295925\pi\)
0.598093 + 0.801426i \(0.295925\pi\)
\(422\) 13.7740 0.670509
\(423\) −8.08805 −0.393254
\(424\) 5.93846 0.288397
\(425\) −16.4362 −0.797273
\(426\) 8.30262 0.402263
\(427\) 19.8086 0.958604
\(428\) 18.4079 0.889778
\(429\) −3.53153 −0.170504
\(430\) 23.7660 1.14610
\(431\) 23.6338 1.13840 0.569200 0.822199i \(-0.307253\pi\)
0.569200 + 0.822199i \(0.307253\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.3781 −1.65211 −0.826053 0.563593i \(-0.809419\pi\)
−0.826053 + 0.563593i \(0.809419\pi\)
\(434\) −27.4567 −1.31796
\(435\) −23.9816 −1.14983
\(436\) 12.5219 0.599689
\(437\) 2.42431 0.115970
\(438\) −11.4874 −0.548887
\(439\) 41.8211 1.99601 0.998007 0.0631002i \(-0.0200988\pi\)
0.998007 + 0.0631002i \(0.0200988\pi\)
\(440\) −12.4938 −0.595618
\(441\) 9.79508 0.466432
\(442\) 2.18686 0.104018
\(443\) −6.44418 −0.306172 −0.153086 0.988213i \(-0.548921\pi\)
−0.153086 + 0.988213i \(0.548921\pi\)
\(444\) −10.3454 −0.490973
\(445\) 39.5405 1.87440
\(446\) 12.8558 0.608742
\(447\) −1.99904 −0.0945512
\(448\) −4.09818 −0.193621
\(449\) 23.5623 1.11197 0.555986 0.831191i \(-0.312341\pi\)
0.555986 + 0.831191i \(0.312341\pi\)
\(450\) 7.51589 0.354303
\(451\) 18.4698 0.869709
\(452\) 12.1628 0.572089
\(453\) −7.74370 −0.363831
\(454\) −12.2505 −0.574946
\(455\) 14.4985 0.679698
\(456\) 0.266722 0.0124904
\(457\) −19.6464 −0.919021 −0.459511 0.888172i \(-0.651975\pi\)
−0.459511 + 0.888172i \(0.651975\pi\)
\(458\) −6.81988 −0.318672
\(459\) 2.18686 0.102074
\(460\) −32.1558 −1.49927
\(461\) −10.8812 −0.506790 −0.253395 0.967363i \(-0.581547\pi\)
−0.253395 + 0.967363i \(0.581547\pi\)
\(462\) −14.4729 −0.673338
\(463\) −5.81015 −0.270021 −0.135010 0.990844i \(-0.543107\pi\)
−0.135010 + 0.990844i \(0.543107\pi\)
\(464\) 6.77872 0.314694
\(465\) −23.7022 −1.09916
\(466\) 18.1244 0.839597
\(467\) −27.3350 −1.26491 −0.632456 0.774596i \(-0.717953\pi\)
−0.632456 + 0.774596i \(0.717953\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −52.0489 −2.40340
\(470\) −28.6137 −1.31985
\(471\) 6.61959 0.305014
\(472\) 1.22445 0.0563598
\(473\) −23.7240 −1.09083
\(474\) −16.9009 −0.776284
\(475\) −2.00466 −0.0919800
\(476\) 8.96214 0.410779
\(477\) 5.93846 0.271903
\(478\) 1.67370 0.0765532
\(479\) 35.2723 1.61163 0.805816 0.592166i \(-0.201727\pi\)
0.805816 + 0.592166i \(0.201727\pi\)
\(480\) −3.53778 −0.161477
\(481\) −10.3454 −0.471711
\(482\) −1.25519 −0.0571724
\(483\) −37.2494 −1.69490
\(484\) 1.47171 0.0668960
\(485\) 29.1446 1.32339
\(486\) −1.00000 −0.0453609
\(487\) 10.8224 0.490410 0.245205 0.969471i \(-0.421145\pi\)
0.245205 + 0.969471i \(0.421145\pi\)
\(488\) −4.83351 −0.218803
\(489\) 9.98151 0.451379
\(490\) 34.6528 1.56546
\(491\) 23.2696 1.05014 0.525070 0.851059i \(-0.324039\pi\)
0.525070 + 0.851059i \(0.324039\pi\)
\(492\) 5.22997 0.235785
\(493\) −14.8241 −0.667645
\(494\) 0.266722 0.0120004
\(495\) −12.4938 −0.561554
\(496\) 6.69972 0.300826
\(497\) 34.0256 1.52626
\(498\) −8.26872 −0.370530
\(499\) 26.9387 1.20594 0.602971 0.797763i \(-0.293983\pi\)
0.602971 + 0.797763i \(0.293983\pi\)
\(500\) 8.90067 0.398050
\(501\) 23.0834 1.03129
\(502\) 3.37351 0.150567
\(503\) 30.5374 1.36160 0.680798 0.732472i \(-0.261633\pi\)
0.680798 + 0.732472i \(0.261633\pi\)
\(504\) −4.09818 −0.182547
\(505\) −58.7594 −2.61476
\(506\) 32.0990 1.42697
\(507\) −1.00000 −0.0444116
\(508\) −8.55557 −0.379592
\(509\) 18.1183 0.803079 0.401540 0.915842i \(-0.368475\pi\)
0.401540 + 0.915842i \(0.368475\pi\)
\(510\) 7.73663 0.342584
\(511\) −47.0773 −2.08258
\(512\) 1.00000 0.0441942
\(513\) 0.266722 0.0117761
\(514\) 10.8138 0.476978
\(515\) −3.53778 −0.155893
\(516\) −6.71778 −0.295734
\(517\) 28.5632 1.25621
\(518\) −42.3975 −1.86284
\(519\) −4.63859 −0.203612
\(520\) −3.53778 −0.155142
\(521\) 5.68447 0.249041 0.124521 0.992217i \(-0.460261\pi\)
0.124521 + 0.992217i \(0.460261\pi\)
\(522\) 6.77872 0.296697
\(523\) 1.40621 0.0614891 0.0307446 0.999527i \(-0.490212\pi\)
0.0307446 + 0.999527i \(0.490212\pi\)
\(524\) 6.96127 0.304105
\(525\) 30.8015 1.34429
\(526\) −10.1360 −0.441950
\(527\) −14.6514 −0.638223
\(528\) 3.53153 0.153690
\(529\) 59.6144 2.59193
\(530\) 21.0090 0.912571
\(531\) 1.22445 0.0531365
\(532\) 1.09308 0.0473909
\(533\) 5.22997 0.226535
\(534\) −11.1766 −0.483660
\(535\) 65.1230 2.81551
\(536\) 12.7005 0.548578
\(537\) 25.0381 1.08047
\(538\) −5.13298 −0.221298
\(539\) −34.5916 −1.48997
\(540\) −3.53778 −0.152242
\(541\) −13.8989 −0.597561 −0.298781 0.954322i \(-0.596580\pi\)
−0.298781 + 0.954322i \(0.596580\pi\)
\(542\) −13.1844 −0.566320
\(543\) 14.6223 0.627505
\(544\) −2.18686 −0.0937608
\(545\) 44.2997 1.89759
\(546\) −4.09818 −0.175386
\(547\) −5.45224 −0.233121 −0.116560 0.993184i \(-0.537187\pi\)
−0.116560 + 0.993184i \(0.537187\pi\)
\(548\) 12.7438 0.544389
\(549\) −4.83351 −0.206289
\(550\) −26.5426 −1.13178
\(551\) −1.80804 −0.0770250
\(552\) 9.08925 0.386864
\(553\) −69.2629 −2.94536
\(554\) 10.7211 0.455495
\(555\) −36.5999 −1.55358
\(556\) 6.43994 0.273114
\(557\) 14.3364 0.607453 0.303727 0.952759i \(-0.401769\pi\)
0.303727 + 0.952759i \(0.401769\pi\)
\(558\) 6.69972 0.283622
\(559\) −6.71778 −0.284132
\(560\) −14.4985 −0.612672
\(561\) −7.72296 −0.326064
\(562\) 18.2550 0.770040
\(563\) −20.0390 −0.844543 −0.422271 0.906470i \(-0.638767\pi\)
−0.422271 + 0.906470i \(0.638767\pi\)
\(564\) 8.08805 0.340568
\(565\) 43.0293 1.81026
\(566\) 30.7561 1.29278
\(567\) −4.09818 −0.172107
\(568\) −8.30262 −0.348370
\(569\) 38.1201 1.59808 0.799039 0.601279i \(-0.205342\pi\)
0.799039 + 0.601279i \(0.205342\pi\)
\(570\) 0.943606 0.0395233
\(571\) 24.6481 1.03149 0.515746 0.856742i \(-0.327515\pi\)
0.515746 + 0.856742i \(0.327515\pi\)
\(572\) 3.53153 0.147661
\(573\) −24.7035 −1.03200
\(574\) 21.4334 0.894611
\(575\) −68.3138 −2.84888
\(576\) 1.00000 0.0416667
\(577\) 3.64245 0.151637 0.0758185 0.997122i \(-0.475843\pi\)
0.0758185 + 0.997122i \(0.475843\pi\)
\(578\) −12.2176 −0.508187
\(579\) 13.9811 0.581033
\(580\) 23.9816 0.995783
\(581\) −33.8867 −1.40586
\(582\) −8.23809 −0.341480
\(583\) −20.9718 −0.868565
\(584\) 11.4874 0.475350
\(585\) −3.53778 −0.146269
\(586\) −24.9096 −1.02901
\(587\) 15.7341 0.649417 0.324708 0.945814i \(-0.394734\pi\)
0.324708 + 0.945814i \(0.394734\pi\)
\(588\) −9.79508 −0.403942
\(589\) −1.78697 −0.0736307
\(590\) 4.33183 0.178339
\(591\) 24.1579 0.993723
\(592\) 10.3454 0.425195
\(593\) −12.5972 −0.517307 −0.258653 0.965970i \(-0.583279\pi\)
−0.258653 + 0.965970i \(0.583279\pi\)
\(594\) 3.53153 0.144900
\(595\) 31.7061 1.29982
\(596\) 1.99904 0.0818837
\(597\) −13.8851 −0.568279
\(598\) 9.08925 0.371687
\(599\) 20.8720 0.852809 0.426404 0.904533i \(-0.359780\pi\)
0.426404 + 0.904533i \(0.359780\pi\)
\(600\) −7.51589 −0.306835
\(601\) 8.13507 0.331836 0.165918 0.986140i \(-0.446941\pi\)
0.165918 + 0.986140i \(0.446941\pi\)
\(602\) −27.5307 −1.12207
\(603\) 12.7005 0.517204
\(604\) 7.74370 0.315087
\(605\) 5.20660 0.211678
\(606\) 16.6091 0.674699
\(607\) −12.7719 −0.518395 −0.259197 0.965824i \(-0.583458\pi\)
−0.259197 + 0.965824i \(0.583458\pi\)
\(608\) −0.266722 −0.0108170
\(609\) 27.7804 1.12572
\(610\) −17.0999 −0.692354
\(611\) 8.08805 0.327207
\(612\) −2.18686 −0.0883986
\(613\) 23.3889 0.944668 0.472334 0.881420i \(-0.343412\pi\)
0.472334 + 0.881420i \(0.343412\pi\)
\(614\) 21.8157 0.880410
\(615\) 18.5025 0.746093
\(616\) 14.4729 0.583128
\(617\) 48.5879 1.95607 0.978037 0.208431i \(-0.0668357\pi\)
0.978037 + 0.208431i \(0.0668357\pi\)
\(618\) 1.00000 0.0402259
\(619\) −21.5137 −0.864710 −0.432355 0.901703i \(-0.642317\pi\)
−0.432355 + 0.901703i \(0.642317\pi\)
\(620\) 23.7022 0.951901
\(621\) 9.08925 0.364739
\(622\) −31.2060 −1.25125
\(623\) −45.8038 −1.83509
\(624\) 1.00000 0.0400320
\(625\) −6.09083 −0.243633
\(626\) −19.1775 −0.766487
\(627\) −0.941939 −0.0376174
\(628\) −6.61959 −0.264150
\(629\) −22.6240 −0.902079
\(630\) −14.4985 −0.577633
\(631\) −31.0257 −1.23511 −0.617556 0.786527i \(-0.711877\pi\)
−0.617556 + 0.786527i \(0.711877\pi\)
\(632\) 16.9009 0.672282
\(633\) −13.7740 −0.547469
\(634\) −31.2536 −1.24124
\(635\) −30.2677 −1.20114
\(636\) −5.93846 −0.235475
\(637\) −9.79508 −0.388095
\(638\) −23.9393 −0.947765
\(639\) −8.30262 −0.328447
\(640\) 3.53778 0.139843
\(641\) −16.1406 −0.637513 −0.318757 0.947837i \(-0.603265\pi\)
−0.318757 + 0.947837i \(0.603265\pi\)
\(642\) −18.4079 −0.726500
\(643\) −10.6907 −0.421601 −0.210801 0.977529i \(-0.567607\pi\)
−0.210801 + 0.977529i \(0.567607\pi\)
\(644\) 37.2494 1.46783
\(645\) −23.7660 −0.935786
\(646\) 0.583285 0.0229490
\(647\) −17.9354 −0.705114 −0.352557 0.935790i \(-0.614688\pi\)
−0.352557 + 0.935790i \(0.614688\pi\)
\(648\) 1.00000 0.0392837
\(649\) −4.32418 −0.169739
\(650\) −7.51589 −0.294798
\(651\) 27.4567 1.07611
\(652\) −9.98151 −0.390906
\(653\) −26.4130 −1.03362 −0.516810 0.856100i \(-0.672881\pi\)
−0.516810 + 0.856100i \(0.672881\pi\)
\(654\) −12.5219 −0.489644
\(655\) 24.6275 0.962274
\(656\) −5.22997 −0.204196
\(657\) 11.4874 0.448165
\(658\) 33.1463 1.29218
\(659\) −23.5594 −0.917743 −0.458872 0.888503i \(-0.651746\pi\)
−0.458872 + 0.888503i \(0.651746\pi\)
\(660\) 12.4938 0.486320
\(661\) 33.1998 1.29132 0.645662 0.763623i \(-0.276581\pi\)
0.645662 + 0.763623i \(0.276581\pi\)
\(662\) −14.0854 −0.547445
\(663\) −2.18686 −0.0849306
\(664\) 8.26872 0.320889
\(665\) 3.86707 0.149958
\(666\) 10.3454 0.400878
\(667\) −61.6135 −2.38568
\(668\) −23.0834 −0.893125
\(669\) −12.8558 −0.497035
\(670\) 44.9316 1.73586
\(671\) 17.0697 0.658968
\(672\) 4.09818 0.158091
\(673\) −30.4630 −1.17426 −0.587132 0.809491i \(-0.699743\pi\)
−0.587132 + 0.809491i \(0.699743\pi\)
\(674\) 13.8956 0.535239
\(675\) −7.51589 −0.289287
\(676\) 1.00000 0.0384615
\(677\) −14.1077 −0.542202 −0.271101 0.962551i \(-0.587388\pi\)
−0.271101 + 0.962551i \(0.587388\pi\)
\(678\) −12.1628 −0.467109
\(679\) −33.7612 −1.29564
\(680\) −7.73663 −0.296686
\(681\) 12.2505 0.469442
\(682\) −23.6603 −0.905999
\(683\) −2.86939 −0.109794 −0.0548971 0.998492i \(-0.517483\pi\)
−0.0548971 + 0.998492i \(0.517483\pi\)
\(684\) −0.266722 −0.0101984
\(685\) 45.0848 1.72260
\(686\) −11.4547 −0.437344
\(687\) 6.81988 0.260195
\(688\) 6.71778 0.256113
\(689\) −5.93846 −0.226237
\(690\) 32.1558 1.22415
\(691\) 6.46830 0.246066 0.123033 0.992403i \(-0.460738\pi\)
0.123033 + 0.992403i \(0.460738\pi\)
\(692\) 4.63859 0.176333
\(693\) 14.4729 0.549778
\(694\) 0.0369366 0.00140209
\(695\) 22.7831 0.864213
\(696\) −6.77872 −0.256947
\(697\) 11.4372 0.433215
\(698\) 15.1285 0.572622
\(699\) −18.1244 −0.685528
\(700\) −30.8015 −1.16419
\(701\) 25.1756 0.950869 0.475434 0.879751i \(-0.342291\pi\)
0.475434 + 0.879751i \(0.342291\pi\)
\(702\) 1.00000 0.0377426
\(703\) −2.75936 −0.104071
\(704\) −3.53153 −0.133100
\(705\) 28.6137 1.07766
\(706\) −3.79280 −0.142744
\(707\) 68.0672 2.55993
\(708\) −1.22445 −0.0460176
\(709\) 20.2006 0.758649 0.379324 0.925264i \(-0.376156\pi\)
0.379324 + 0.925264i \(0.376156\pi\)
\(710\) −29.3728 −1.10234
\(711\) 16.9009 0.633833
\(712\) 11.1766 0.418862
\(713\) −60.8954 −2.28055
\(714\) −8.96214 −0.335400
\(715\) 12.4938 0.467241
\(716\) −25.0381 −0.935717
\(717\) −1.67370 −0.0625055
\(718\) 9.47254 0.353512
\(719\) 30.4655 1.13617 0.568085 0.822970i \(-0.307684\pi\)
0.568085 + 0.822970i \(0.307684\pi\)
\(720\) 3.53778 0.131845
\(721\) 4.09818 0.152624
\(722\) −18.9289 −0.704459
\(723\) 1.25519 0.0466811
\(724\) −14.6223 −0.543435
\(725\) 50.9481 1.89217
\(726\) −1.47171 −0.0546204
\(727\) −9.45194 −0.350553 −0.175276 0.984519i \(-0.556082\pi\)
−0.175276 + 0.984519i \(0.556082\pi\)
\(728\) 4.09818 0.151889
\(729\) 1.00000 0.0370370
\(730\) 40.6398 1.50415
\(731\) −14.6908 −0.543360
\(732\) 4.83351 0.178652
\(733\) 39.4273 1.45628 0.728139 0.685429i \(-0.240385\pi\)
0.728139 + 0.685429i \(0.240385\pi\)
\(734\) 7.47886 0.276050
\(735\) −34.6528 −1.27819
\(736\) −9.08925 −0.335034
\(737\) −44.8522 −1.65215
\(738\) −5.22997 −0.192518
\(739\) 35.8088 1.31725 0.658624 0.752472i \(-0.271139\pi\)
0.658624 + 0.752472i \(0.271139\pi\)
\(740\) 36.5999 1.34544
\(741\) −0.266722 −0.00979829
\(742\) −24.3369 −0.893435
\(743\) −9.60364 −0.352323 −0.176162 0.984361i \(-0.556368\pi\)
−0.176162 + 0.984361i \(0.556368\pi\)
\(744\) −6.69972 −0.245624
\(745\) 7.07215 0.259104
\(746\) −20.7493 −0.759685
\(747\) 8.26872 0.302537
\(748\) 7.72296 0.282379
\(749\) −75.4387 −2.75647
\(750\) −8.90067 −0.325007
\(751\) −9.79075 −0.357269 −0.178635 0.983915i \(-0.557168\pi\)
−0.178635 + 0.983915i \(0.557168\pi\)
\(752\) −8.08805 −0.294941
\(753\) −3.37351 −0.122937
\(754\) −6.77872 −0.246867
\(755\) 27.3955 0.997025
\(756\) 4.09818 0.149049
\(757\) −43.7464 −1.58999 −0.794996 0.606615i \(-0.792527\pi\)
−0.794996 + 0.606615i \(0.792527\pi\)
\(758\) −7.42769 −0.269786
\(759\) −32.0990 −1.16512
\(760\) −0.943606 −0.0342282
\(761\) 40.5064 1.46836 0.734178 0.678957i \(-0.237568\pi\)
0.734178 + 0.678957i \(0.237568\pi\)
\(762\) 8.55557 0.309935
\(763\) −51.3169 −1.85780
\(764\) 24.7035 0.893741
\(765\) −7.73663 −0.279718
\(766\) −11.2539 −0.406621
\(767\) −1.22445 −0.0442123
\(768\) −1.00000 −0.0360844
\(769\) 13.3773 0.482396 0.241198 0.970476i \(-0.422460\pi\)
0.241198 + 0.970476i \(0.422460\pi\)
\(770\) 51.2018 1.84518
\(771\) −10.8138 −0.389451
\(772\) −13.9811 −0.503189
\(773\) 9.24662 0.332578 0.166289 0.986077i \(-0.446822\pi\)
0.166289 + 0.986077i \(0.446822\pi\)
\(774\) 6.71778 0.241465
\(775\) 50.3544 1.80878
\(776\) 8.23809 0.295730
\(777\) 42.3975 1.52100
\(778\) 29.4974 1.05753
\(779\) 1.39495 0.0499793
\(780\) 3.53778 0.126673
\(781\) 29.3210 1.04919
\(782\) 19.8769 0.710797
\(783\) −6.77872 −0.242252
\(784\) 9.79508 0.349824
\(785\) −23.4186 −0.835847
\(786\) −6.96127 −0.248300
\(787\) −14.5093 −0.517199 −0.258600 0.965985i \(-0.583261\pi\)
−0.258600 + 0.965985i \(0.583261\pi\)
\(788\) −24.1579 −0.860590
\(789\) 10.1360 0.360851
\(790\) 59.7917 2.12729
\(791\) −49.8453 −1.77229
\(792\) −3.53153 −0.125487
\(793\) 4.83351 0.171643
\(794\) −7.73124 −0.274371
\(795\) −21.0090 −0.745111
\(796\) 13.8851 0.492144
\(797\) −17.5455 −0.621493 −0.310746 0.950493i \(-0.600579\pi\)
−0.310746 + 0.950493i \(0.600579\pi\)
\(798\) −1.09308 −0.0386945
\(799\) 17.6874 0.625736
\(800\) 7.51589 0.265727
\(801\) 11.1766 0.394907
\(802\) −23.0488 −0.813882
\(803\) −40.5680 −1.43161
\(804\) −12.7005 −0.447912
\(805\) 131.780 4.64464
\(806\) −6.69972 −0.235988
\(807\) 5.13298 0.180689
\(808\) −16.6091 −0.584307
\(809\) −17.3211 −0.608978 −0.304489 0.952516i \(-0.598486\pi\)
−0.304489 + 0.952516i \(0.598486\pi\)
\(810\) 3.53778 0.124305
\(811\) 55.5789 1.95164 0.975820 0.218574i \(-0.0701406\pi\)
0.975820 + 0.218574i \(0.0701406\pi\)
\(812\) −27.7804 −0.974902
\(813\) 13.1844 0.462398
\(814\) −36.5352 −1.28056
\(815\) −35.3124 −1.23694
\(816\) 2.18686 0.0765554
\(817\) −1.79178 −0.0626865
\(818\) −12.7459 −0.445651
\(819\) 4.09818 0.143202
\(820\) −18.5025 −0.646135
\(821\) −37.1176 −1.29541 −0.647706 0.761890i \(-0.724272\pi\)
−0.647706 + 0.761890i \(0.724272\pi\)
\(822\) −12.7438 −0.444492
\(823\) 11.3125 0.394328 0.197164 0.980370i \(-0.436827\pi\)
0.197164 + 0.980370i \(0.436827\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 26.5426 0.924095
\(826\) −5.01801 −0.174599
\(827\) 15.5065 0.539212 0.269606 0.962971i \(-0.413106\pi\)
0.269606 + 0.962971i \(0.413106\pi\)
\(828\) −9.08925 −0.315873
\(829\) 50.2473 1.74516 0.872580 0.488472i \(-0.162445\pi\)
0.872580 + 0.488472i \(0.162445\pi\)
\(830\) 29.2529 1.01538
\(831\) −10.7211 −0.371910
\(832\) −1.00000 −0.0346688
\(833\) −21.4205 −0.742175
\(834\) −6.43994 −0.222997
\(835\) −81.6641 −2.82610
\(836\) 0.941939 0.0325776
\(837\) −6.69972 −0.231576
\(838\) −8.38630 −0.289700
\(839\) 2.22509 0.0768187 0.0384094 0.999262i \(-0.487771\pi\)
0.0384094 + 0.999262i \(0.487771\pi\)
\(840\) 14.4985 0.500244
\(841\) 16.9511 0.584520
\(842\) 24.5437 0.845832
\(843\) −18.2550 −0.628735
\(844\) 13.7740 0.474122
\(845\) 3.53778 0.121703
\(846\) −8.08805 −0.278073
\(847\) −6.03134 −0.207239
\(848\) 5.93846 0.203927
\(849\) −30.7561 −1.05555
\(850\) −16.4362 −0.563757
\(851\) −94.0322 −3.22338
\(852\) 8.30262 0.284443
\(853\) 34.4691 1.18020 0.590100 0.807330i \(-0.299088\pi\)
0.590100 + 0.807330i \(0.299088\pi\)
\(854\) 19.8086 0.677836
\(855\) −0.943606 −0.0322706
\(856\) 18.4079 0.629168
\(857\) −11.2493 −0.384270 −0.192135 0.981369i \(-0.561541\pi\)
−0.192135 + 0.981369i \(0.561541\pi\)
\(858\) −3.53153 −0.120564
\(859\) 9.47184 0.323175 0.161588 0.986858i \(-0.448339\pi\)
0.161588 + 0.986858i \(0.448339\pi\)
\(860\) 23.7660 0.810414
\(861\) −21.4334 −0.730447
\(862\) 23.6338 0.804971
\(863\) 34.9733 1.19051 0.595253 0.803539i \(-0.297052\pi\)
0.595253 + 0.803539i \(0.297052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 16.4103 0.557968
\(866\) −34.3781 −1.16822
\(867\) 12.2176 0.414933
\(868\) −27.4567 −0.931940
\(869\) −59.6860 −2.02471
\(870\) −23.9816 −0.813053
\(871\) −12.7005 −0.430340
\(872\) 12.5219 0.424044
\(873\) 8.23809 0.278817
\(874\) 2.42431 0.0820034
\(875\) −36.4766 −1.23313
\(876\) −11.4874 −0.388122
\(877\) −54.9994 −1.85720 −0.928599 0.371085i \(-0.878986\pi\)
−0.928599 + 0.371085i \(0.878986\pi\)
\(878\) 41.8211 1.41140
\(879\) 24.9096 0.840181
\(880\) −12.4938 −0.421165
\(881\) 27.7577 0.935180 0.467590 0.883946i \(-0.345122\pi\)
0.467590 + 0.883946i \(0.345122\pi\)
\(882\) 9.79508 0.329817
\(883\) −2.25682 −0.0759481 −0.0379741 0.999279i \(-0.512090\pi\)
−0.0379741 + 0.999279i \(0.512090\pi\)
\(884\) 2.18686 0.0735520
\(885\) −4.33183 −0.145613
\(886\) −6.44418 −0.216496
\(887\) −8.79693 −0.295372 −0.147686 0.989034i \(-0.547183\pi\)
−0.147686 + 0.989034i \(0.547183\pi\)
\(888\) −10.3454 −0.347170
\(889\) 35.0623 1.17595
\(890\) 39.5405 1.32540
\(891\) −3.53153 −0.118311
\(892\) 12.8558 0.430445
\(893\) 2.15726 0.0721901
\(894\) −1.99904 −0.0668578
\(895\) −88.5793 −2.96088
\(896\) −4.09818 −0.136911
\(897\) −9.08925 −0.303481
\(898\) 23.5623 0.786283
\(899\) 45.4156 1.51469
\(900\) 7.51589 0.250530
\(901\) −12.9866 −0.432645
\(902\) 18.4698 0.614977
\(903\) 27.5307 0.916163
\(904\) 12.1628 0.404528
\(905\) −51.7307 −1.71959
\(906\) −7.74370 −0.257267
\(907\) 5.55253 0.184369 0.0921844 0.995742i \(-0.470615\pi\)
0.0921844 + 0.995742i \(0.470615\pi\)
\(908\) −12.2505 −0.406548
\(909\) −16.6091 −0.550890
\(910\) 14.4985 0.480619
\(911\) −26.5524 −0.879720 −0.439860 0.898066i \(-0.644972\pi\)
−0.439860 + 0.898066i \(0.644972\pi\)
\(912\) 0.266722 0.00883206
\(913\) −29.2012 −0.966420
\(914\) −19.6464 −0.649846
\(915\) 17.0999 0.565305
\(916\) −6.81988 −0.225335
\(917\) −28.5285 −0.942095
\(918\) 2.18686 0.0721771
\(919\) −21.4100 −0.706252 −0.353126 0.935576i \(-0.614881\pi\)
−0.353126 + 0.935576i \(0.614881\pi\)
\(920\) −32.1558 −1.06014
\(921\) −21.8157 −0.718852
\(922\) −10.8812 −0.358355
\(923\) 8.30262 0.273284
\(924\) −14.4729 −0.476122
\(925\) 77.7552 2.55657
\(926\) −5.81015 −0.190933
\(927\) −1.00000 −0.0328443
\(928\) 6.77872 0.222522
\(929\) 49.0695 1.60992 0.804960 0.593330i \(-0.202187\pi\)
0.804960 + 0.593330i \(0.202187\pi\)
\(930\) −23.7022 −0.777224
\(931\) −2.61257 −0.0856234
\(932\) 18.1244 0.593685
\(933\) 31.2060 1.02164
\(934\) −27.3350 −0.894428
\(935\) 27.3221 0.893530
\(936\) −1.00000 −0.0326860
\(937\) 6.02397 0.196795 0.0983973 0.995147i \(-0.468628\pi\)
0.0983973 + 0.995147i \(0.468628\pi\)
\(938\) −52.0489 −1.69946
\(939\) 19.1775 0.625834
\(940\) −28.6137 −0.933277
\(941\) 9.03028 0.294379 0.147189 0.989108i \(-0.452977\pi\)
0.147189 + 0.989108i \(0.452977\pi\)
\(942\) 6.61959 0.215678
\(943\) 47.5365 1.54800
\(944\) 1.22445 0.0398524
\(945\) 14.4985 0.471635
\(946\) −23.7240 −0.771335
\(947\) −30.3063 −0.984821 −0.492411 0.870363i \(-0.663884\pi\)
−0.492411 + 0.870363i \(0.663884\pi\)
\(948\) −16.9009 −0.548916
\(949\) −11.4874 −0.372896
\(950\) −2.00466 −0.0650397
\(951\) 31.2536 1.01347
\(952\) 8.96214 0.290465
\(953\) −32.8060 −1.06269 −0.531345 0.847156i \(-0.678313\pi\)
−0.531345 + 0.847156i \(0.678313\pi\)
\(954\) 5.93846 0.192265
\(955\) 87.3955 2.82805
\(956\) 1.67370 0.0541313
\(957\) 23.9393 0.773847
\(958\) 35.2723 1.13960
\(959\) −52.2265 −1.68648
\(960\) −3.53778 −0.114181
\(961\) 13.8863 0.447945
\(962\) −10.3454 −0.333550
\(963\) 18.4079 0.593185
\(964\) −1.25519 −0.0404270
\(965\) −49.4619 −1.59223
\(966\) −37.2494 −1.19848
\(967\) 29.5884 0.951500 0.475750 0.879581i \(-0.342177\pi\)
0.475750 + 0.879581i \(0.342177\pi\)
\(968\) 1.47171 0.0473026
\(969\) −0.583285 −0.0187378
\(970\) 29.1446 0.935775
\(971\) 36.7491 1.17933 0.589667 0.807647i \(-0.299259\pi\)
0.589667 + 0.807647i \(0.299259\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −26.3921 −0.846090
\(974\) 10.8224 0.346772
\(975\) 7.51589 0.240701
\(976\) −4.83351 −0.154717
\(977\) −32.9434 −1.05395 −0.526977 0.849879i \(-0.676675\pi\)
−0.526977 + 0.849879i \(0.676675\pi\)
\(978\) 9.98151 0.319173
\(979\) −39.4706 −1.26149
\(980\) 34.6528 1.10694
\(981\) 12.5219 0.399793
\(982\) 23.2696 0.742562
\(983\) −39.7184 −1.26682 −0.633410 0.773817i \(-0.718345\pi\)
−0.633410 + 0.773817i \(0.718345\pi\)
\(984\) 5.22997 0.166725
\(985\) −85.4654 −2.72315
\(986\) −14.8241 −0.472096
\(987\) −33.1463 −1.05506
\(988\) 0.266722 0.00848557
\(989\) −61.0595 −1.94158
\(990\) −12.4938 −0.397078
\(991\) 60.6842 1.92770 0.963849 0.266450i \(-0.0858506\pi\)
0.963849 + 0.266450i \(0.0858506\pi\)
\(992\) 6.69972 0.212716
\(993\) 14.0854 0.446987
\(994\) 34.0256 1.07923
\(995\) 49.1224 1.55728
\(996\) −8.26872 −0.262004
\(997\) 1.38146 0.0437513 0.0218757 0.999761i \(-0.493036\pi\)
0.0218757 + 0.999761i \(0.493036\pi\)
\(998\) 26.9387 0.852730
\(999\) −10.3454 −0.327315
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.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.14 15 1.1 even 1 trivial