Properties

Label 8034.2.a.bc.1.4
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 1.4
Root \(2.43337\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.43337 q^{5} -1.00000 q^{6} -2.53842 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.43337 q^{10} +5.77661 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.53842 q^{14} +2.43337 q^{15} +1.00000 q^{16} -4.04922 q^{17} +1.00000 q^{18} -2.00627 q^{19} -2.43337 q^{20} +2.53842 q^{21} +5.77661 q^{22} +8.05359 q^{23} -1.00000 q^{24} +0.921312 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.53842 q^{28} +0.475455 q^{29} +2.43337 q^{30} -6.32058 q^{31} +1.00000 q^{32} -5.77661 q^{33} -4.04922 q^{34} +6.17693 q^{35} +1.00000 q^{36} -2.13431 q^{37} -2.00627 q^{38} +1.00000 q^{39} -2.43337 q^{40} +1.29937 q^{41} +2.53842 q^{42} -1.58896 q^{43} +5.77661 q^{44} -2.43337 q^{45} +8.05359 q^{46} +8.72962 q^{47} -1.00000 q^{48} -0.556418 q^{49} +0.921312 q^{50} +4.04922 q^{51} -1.00000 q^{52} +0.740185 q^{53} -1.00000 q^{54} -14.0566 q^{55} -2.53842 q^{56} +2.00627 q^{57} +0.475455 q^{58} -2.73337 q^{59} +2.43337 q^{60} -6.04981 q^{61} -6.32058 q^{62} -2.53842 q^{63} +1.00000 q^{64} +2.43337 q^{65} -5.77661 q^{66} -3.11178 q^{67} -4.04922 q^{68} -8.05359 q^{69} +6.17693 q^{70} +8.83959 q^{71} +1.00000 q^{72} -8.59636 q^{73} -2.13431 q^{74} -0.921312 q^{75} -2.00627 q^{76} -14.6635 q^{77} +1.00000 q^{78} -2.59198 q^{79} -2.43337 q^{80} +1.00000 q^{81} +1.29937 q^{82} -3.99096 q^{83} +2.53842 q^{84} +9.85327 q^{85} -1.58896 q^{86} -0.475455 q^{87} +5.77661 q^{88} -9.66240 q^{89} -2.43337 q^{90} +2.53842 q^{91} +8.05359 q^{92} +6.32058 q^{93} +8.72962 q^{94} +4.88201 q^{95} -1.00000 q^{96} +12.2300 q^{97} -0.556418 q^{98} +5.77661 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) −2.43337 −1.08824 −0.544119 0.839008i \(-0.683136\pi\)
−0.544119 + 0.839008i \(0.683136\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.53842 −0.959433 −0.479717 0.877424i \(-0.659260\pi\)
−0.479717 + 0.877424i \(0.659260\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.43337 −0.769501
\(11\) 5.77661 1.74171 0.870856 0.491538i \(-0.163565\pi\)
0.870856 + 0.491538i \(0.163565\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −2.53842 −0.678422
\(15\) 2.43337 0.628295
\(16\) 1.00000 0.250000
\(17\) −4.04922 −0.982081 −0.491040 0.871137i \(-0.663383\pi\)
−0.491040 + 0.871137i \(0.663383\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00627 −0.460270 −0.230135 0.973159i \(-0.573917\pi\)
−0.230135 + 0.973159i \(0.573917\pi\)
\(20\) −2.43337 −0.544119
\(21\) 2.53842 0.553929
\(22\) 5.77661 1.23158
\(23\) 8.05359 1.67929 0.839644 0.543136i \(-0.182763\pi\)
0.839644 + 0.543136i \(0.182763\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0.921312 0.184262
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −2.53842 −0.479717
\(29\) 0.475455 0.0882898 0.0441449 0.999025i \(-0.485944\pi\)
0.0441449 + 0.999025i \(0.485944\pi\)
\(30\) 2.43337 0.444271
\(31\) −6.32058 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.77661 −1.00558
\(34\) −4.04922 −0.694436
\(35\) 6.17693 1.04409
\(36\) 1.00000 0.166667
\(37\) −2.13431 −0.350879 −0.175439 0.984490i \(-0.556135\pi\)
−0.175439 + 0.984490i \(0.556135\pi\)
\(38\) −2.00627 −0.325460
\(39\) 1.00000 0.160128
\(40\) −2.43337 −0.384750
\(41\) 1.29937 0.202928 0.101464 0.994839i \(-0.467647\pi\)
0.101464 + 0.994839i \(0.467647\pi\)
\(42\) 2.53842 0.391687
\(43\) −1.58896 −0.242314 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(44\) 5.77661 0.870856
\(45\) −2.43337 −0.362746
\(46\) 8.05359 1.18744
\(47\) 8.72962 1.27335 0.636673 0.771134i \(-0.280310\pi\)
0.636673 + 0.771134i \(0.280310\pi\)
\(48\) −1.00000 −0.144338
\(49\) −0.556418 −0.0794883
\(50\) 0.921312 0.130293
\(51\) 4.04922 0.567005
\(52\) −1.00000 −0.138675
\(53\) 0.740185 0.101672 0.0508361 0.998707i \(-0.483811\pi\)
0.0508361 + 0.998707i \(0.483811\pi\)
\(54\) −1.00000 −0.136083
\(55\) −14.0566 −1.89540
\(56\) −2.53842 −0.339211
\(57\) 2.00627 0.265737
\(58\) 0.475455 0.0624303
\(59\) −2.73337 −0.355854 −0.177927 0.984044i \(-0.556939\pi\)
−0.177927 + 0.984044i \(0.556939\pi\)
\(60\) 2.43337 0.314147
\(61\) −6.04981 −0.774599 −0.387300 0.921954i \(-0.626592\pi\)
−0.387300 + 0.921954i \(0.626592\pi\)
\(62\) −6.32058 −0.802715
\(63\) −2.53842 −0.319811
\(64\) 1.00000 0.125000
\(65\) 2.43337 0.301823
\(66\) −5.77661 −0.711051
\(67\) −3.11178 −0.380164 −0.190082 0.981768i \(-0.560875\pi\)
−0.190082 + 0.981768i \(0.560875\pi\)
\(68\) −4.04922 −0.491040
\(69\) −8.05359 −0.969538
\(70\) 6.17693 0.738284
\(71\) 8.83959 1.04907 0.524533 0.851390i \(-0.324240\pi\)
0.524533 + 0.851390i \(0.324240\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.59636 −1.00613 −0.503064 0.864249i \(-0.667794\pi\)
−0.503064 + 0.864249i \(0.667794\pi\)
\(74\) −2.13431 −0.248109
\(75\) −0.921312 −0.106384
\(76\) −2.00627 −0.230135
\(77\) −14.6635 −1.67106
\(78\) 1.00000 0.113228
\(79\) −2.59198 −0.291620 −0.145810 0.989313i \(-0.546579\pi\)
−0.145810 + 0.989313i \(0.546579\pi\)
\(80\) −2.43337 −0.272060
\(81\) 1.00000 0.111111
\(82\) 1.29937 0.143492
\(83\) −3.99096 −0.438064 −0.219032 0.975718i \(-0.570290\pi\)
−0.219032 + 0.975718i \(0.570290\pi\)
\(84\) 2.53842 0.276964
\(85\) 9.85327 1.06874
\(86\) −1.58896 −0.171342
\(87\) −0.475455 −0.0509741
\(88\) 5.77661 0.615788
\(89\) −9.66240 −1.02421 −0.512106 0.858922i \(-0.671135\pi\)
−0.512106 + 0.858922i \(0.671135\pi\)
\(90\) −2.43337 −0.256500
\(91\) 2.53842 0.266099
\(92\) 8.05359 0.839644
\(93\) 6.32058 0.655414
\(94\) 8.72962 0.900391
\(95\) 4.88201 0.500884
\(96\) −1.00000 −0.102062
\(97\) 12.2300 1.24176 0.620882 0.783904i \(-0.286774\pi\)
0.620882 + 0.783904i \(0.286774\pi\)
\(98\) −0.556418 −0.0562067
\(99\) 5.77661 0.580571
\(100\) 0.921312 0.0921312
\(101\) 11.1817 1.11262 0.556311 0.830974i \(-0.312216\pi\)
0.556311 + 0.830974i \(0.312216\pi\)
\(102\) 4.04922 0.400933
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −6.17693 −0.602807
\(106\) 0.740185 0.0718931
\(107\) 10.8476 1.04868 0.524338 0.851510i \(-0.324313\pi\)
0.524338 + 0.851510i \(0.324313\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 13.9607 1.33719 0.668596 0.743626i \(-0.266896\pi\)
0.668596 + 0.743626i \(0.266896\pi\)
\(110\) −14.0566 −1.34025
\(111\) 2.13431 0.202580
\(112\) −2.53842 −0.239858
\(113\) −9.77485 −0.919540 −0.459770 0.888038i \(-0.652068\pi\)
−0.459770 + 0.888038i \(0.652068\pi\)
\(114\) 2.00627 0.187905
\(115\) −19.5974 −1.82747
\(116\) 0.475455 0.0441449
\(117\) −1.00000 −0.0924500
\(118\) −2.73337 −0.251627
\(119\) 10.2786 0.942241
\(120\) 2.43337 0.222136
\(121\) 22.3692 2.03356
\(122\) −6.04981 −0.547724
\(123\) −1.29937 −0.117161
\(124\) −6.32058 −0.567605
\(125\) 9.92498 0.887717
\(126\) −2.53842 −0.226141
\(127\) −0.741066 −0.0657589 −0.0328795 0.999459i \(-0.510468\pi\)
−0.0328795 + 0.999459i \(0.510468\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.58896 0.139900
\(130\) 2.43337 0.213421
\(131\) 12.8407 1.12189 0.560947 0.827851i \(-0.310437\pi\)
0.560947 + 0.827851i \(0.310437\pi\)
\(132\) −5.77661 −0.502789
\(133\) 5.09276 0.441598
\(134\) −3.11178 −0.268817
\(135\) 2.43337 0.209432
\(136\) −4.04922 −0.347218
\(137\) 0.740211 0.0632405 0.0316202 0.999500i \(-0.489933\pi\)
0.0316202 + 0.999500i \(0.489933\pi\)
\(138\) −8.05359 −0.685567
\(139\) 7.89186 0.669379 0.334689 0.942329i \(-0.391369\pi\)
0.334689 + 0.942329i \(0.391369\pi\)
\(140\) 6.17693 0.522046
\(141\) −8.72962 −0.735166
\(142\) 8.83959 0.741802
\(143\) −5.77661 −0.483064
\(144\) 1.00000 0.0833333
\(145\) −1.15696 −0.0960803
\(146\) −8.59636 −0.711440
\(147\) 0.556418 0.0458926
\(148\) −2.13431 −0.175439
\(149\) −11.9282 −0.977194 −0.488597 0.872510i \(-0.662491\pi\)
−0.488597 + 0.872510i \(0.662491\pi\)
\(150\) −0.921312 −0.0752248
\(151\) 22.9927 1.87112 0.935559 0.353171i \(-0.114897\pi\)
0.935559 + 0.353171i \(0.114897\pi\)
\(152\) −2.00627 −0.162730
\(153\) −4.04922 −0.327360
\(154\) −14.6635 −1.18162
\(155\) 15.3803 1.23538
\(156\) 1.00000 0.0800641
\(157\) 7.41021 0.591399 0.295700 0.955281i \(-0.404447\pi\)
0.295700 + 0.955281i \(0.404447\pi\)
\(158\) −2.59198 −0.206207
\(159\) −0.740185 −0.0587005
\(160\) −2.43337 −0.192375
\(161\) −20.4434 −1.61117
\(162\) 1.00000 0.0785674
\(163\) 22.3303 1.74904 0.874520 0.484989i \(-0.161176\pi\)
0.874520 + 0.484989i \(0.161176\pi\)
\(164\) 1.29937 0.101464
\(165\) 14.0566 1.09431
\(166\) −3.99096 −0.309758
\(167\) −4.53339 −0.350804 −0.175402 0.984497i \(-0.556123\pi\)
−0.175402 + 0.984497i \(0.556123\pi\)
\(168\) 2.53842 0.195843
\(169\) 1.00000 0.0769231
\(170\) 9.85327 0.755712
\(171\) −2.00627 −0.153423
\(172\) −1.58896 −0.121157
\(173\) −9.08335 −0.690595 −0.345297 0.938493i \(-0.612222\pi\)
−0.345297 + 0.938493i \(0.612222\pi\)
\(174\) −0.475455 −0.0360441
\(175\) −2.33868 −0.176788
\(176\) 5.77661 0.435428
\(177\) 2.73337 0.205453
\(178\) −9.66240 −0.724228
\(179\) −17.1575 −1.28241 −0.641207 0.767368i \(-0.721566\pi\)
−0.641207 + 0.767368i \(0.721566\pi\)
\(180\) −2.43337 −0.181373
\(181\) −10.9658 −0.815084 −0.407542 0.913187i \(-0.633614\pi\)
−0.407542 + 0.913187i \(0.633614\pi\)
\(182\) 2.53842 0.188160
\(183\) 6.04981 0.447215
\(184\) 8.05359 0.593718
\(185\) 5.19358 0.381840
\(186\) 6.32058 0.463448
\(187\) −23.3908 −1.71050
\(188\) 8.72962 0.636673
\(189\) 2.53842 0.184643
\(190\) 4.88201 0.354178
\(191\) 19.7366 1.42809 0.714047 0.700098i \(-0.246860\pi\)
0.714047 + 0.700098i \(0.246860\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.9310 −1.07475 −0.537377 0.843342i \(-0.680585\pi\)
−0.537377 + 0.843342i \(0.680585\pi\)
\(194\) 12.2300 0.878060
\(195\) −2.43337 −0.174258
\(196\) −0.556418 −0.0397442
\(197\) −21.4303 −1.52685 −0.763424 0.645898i \(-0.776483\pi\)
−0.763424 + 0.645898i \(0.776483\pi\)
\(198\) 5.77661 0.410526
\(199\) −4.79816 −0.340133 −0.170066 0.985433i \(-0.554398\pi\)
−0.170066 + 0.985433i \(0.554398\pi\)
\(200\) 0.921312 0.0651466
\(201\) 3.11178 0.219488
\(202\) 11.1817 0.786743
\(203\) −1.20690 −0.0847081
\(204\) 4.04922 0.283502
\(205\) −3.16186 −0.220834
\(206\) −1.00000 −0.0696733
\(207\) 8.05359 0.559763
\(208\) −1.00000 −0.0693375
\(209\) −11.5894 −0.801658
\(210\) −6.17693 −0.426249
\(211\) 15.9089 1.09521 0.547606 0.836736i \(-0.315539\pi\)
0.547606 + 0.836736i \(0.315539\pi\)
\(212\) 0.740185 0.0508361
\(213\) −8.83959 −0.605679
\(214\) 10.8476 0.741526
\(215\) 3.86653 0.263695
\(216\) −1.00000 −0.0680414
\(217\) 16.0443 1.08916
\(218\) 13.9607 0.945537
\(219\) 8.59636 0.580888
\(220\) −14.0566 −0.947699
\(221\) 4.04922 0.272380
\(222\) 2.13431 0.143246
\(223\) 8.61008 0.576573 0.288287 0.957544i \(-0.406914\pi\)
0.288287 + 0.957544i \(0.406914\pi\)
\(224\) −2.53842 −0.169605
\(225\) 0.921312 0.0614208
\(226\) −9.77485 −0.650213
\(227\) 16.4584 1.09238 0.546190 0.837661i \(-0.316078\pi\)
0.546190 + 0.837661i \(0.316078\pi\)
\(228\) 2.00627 0.132869
\(229\) 16.3135 1.07803 0.539013 0.842297i \(-0.318797\pi\)
0.539013 + 0.842297i \(0.318797\pi\)
\(230\) −19.5974 −1.29221
\(231\) 14.6635 0.964785
\(232\) 0.475455 0.0312151
\(233\) 9.69358 0.635047 0.317524 0.948250i \(-0.397149\pi\)
0.317524 + 0.948250i \(0.397149\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −21.2424 −1.38570
\(236\) −2.73337 −0.177927
\(237\) 2.59198 0.168367
\(238\) 10.2786 0.666265
\(239\) 14.6164 0.945454 0.472727 0.881209i \(-0.343270\pi\)
0.472727 + 0.881209i \(0.343270\pi\)
\(240\) 2.43337 0.157074
\(241\) 19.0765 1.22883 0.614413 0.788985i \(-0.289393\pi\)
0.614413 + 0.788985i \(0.289393\pi\)
\(242\) 22.3692 1.43795
\(243\) −1.00000 −0.0641500
\(244\) −6.04981 −0.387300
\(245\) 1.35397 0.0865022
\(246\) −1.29937 −0.0828450
\(247\) 2.00627 0.127656
\(248\) −6.32058 −0.401357
\(249\) 3.99096 0.252917
\(250\) 9.92498 0.627711
\(251\) 12.6704 0.799750 0.399875 0.916570i \(-0.369054\pi\)
0.399875 + 0.916570i \(0.369054\pi\)
\(252\) −2.53842 −0.159906
\(253\) 46.5224 2.92484
\(254\) −0.741066 −0.0464986
\(255\) −9.85327 −0.617036
\(256\) 1.00000 0.0625000
\(257\) 26.2911 1.63999 0.819997 0.572368i \(-0.193975\pi\)
0.819997 + 0.572368i \(0.193975\pi\)
\(258\) 1.58896 0.0989241
\(259\) 5.41778 0.336644
\(260\) 2.43337 0.150911
\(261\) 0.475455 0.0294299
\(262\) 12.8407 0.793299
\(263\) 3.53155 0.217765 0.108882 0.994055i \(-0.465273\pi\)
0.108882 + 0.994055i \(0.465273\pi\)
\(264\) −5.77661 −0.355526
\(265\) −1.80115 −0.110644
\(266\) 5.09276 0.312257
\(267\) 9.66240 0.591329
\(268\) −3.11178 −0.190082
\(269\) 17.9185 1.09251 0.546254 0.837619i \(-0.316053\pi\)
0.546254 + 0.837619i \(0.316053\pi\)
\(270\) 2.43337 0.148090
\(271\) 1.14289 0.0694255 0.0347127 0.999397i \(-0.488948\pi\)
0.0347127 + 0.999397i \(0.488948\pi\)
\(272\) −4.04922 −0.245520
\(273\) −2.53842 −0.153632
\(274\) 0.740211 0.0447178
\(275\) 5.32206 0.320932
\(276\) −8.05359 −0.484769
\(277\) −23.1873 −1.39319 −0.696595 0.717465i \(-0.745302\pi\)
−0.696595 + 0.717465i \(0.745302\pi\)
\(278\) 7.89186 0.473322
\(279\) −6.32058 −0.378403
\(280\) 6.17693 0.369142
\(281\) 30.2126 1.80234 0.901168 0.433471i \(-0.142711\pi\)
0.901168 + 0.433471i \(0.142711\pi\)
\(282\) −8.72962 −0.519841
\(283\) −11.4266 −0.679241 −0.339621 0.940562i \(-0.610299\pi\)
−0.339621 + 0.940562i \(0.610299\pi\)
\(284\) 8.83959 0.524533
\(285\) −4.88201 −0.289185
\(286\) −5.77661 −0.341578
\(287\) −3.29836 −0.194696
\(288\) 1.00000 0.0589256
\(289\) −0.603800 −0.0355176
\(290\) −1.15696 −0.0679390
\(291\) −12.2300 −0.716933
\(292\) −8.59636 −0.503064
\(293\) 11.3285 0.661815 0.330908 0.943663i \(-0.392645\pi\)
0.330908 + 0.943663i \(0.392645\pi\)
\(294\) 0.556418 0.0324510
\(295\) 6.65131 0.387254
\(296\) −2.13431 −0.124054
\(297\) −5.77661 −0.335193
\(298\) −11.9282 −0.690980
\(299\) −8.05359 −0.465751
\(300\) −0.921312 −0.0531920
\(301\) 4.03344 0.232484
\(302\) 22.9927 1.32308
\(303\) −11.1817 −0.642373
\(304\) −2.00627 −0.115068
\(305\) 14.7215 0.842948
\(306\) −4.04922 −0.231479
\(307\) 19.6190 1.11972 0.559859 0.828588i \(-0.310855\pi\)
0.559859 + 0.828588i \(0.310855\pi\)
\(308\) −14.6635 −0.835528
\(309\) 1.00000 0.0568880
\(310\) 15.3803 0.873545
\(311\) −26.0380 −1.47648 −0.738239 0.674539i \(-0.764343\pi\)
−0.738239 + 0.674539i \(0.764343\pi\)
\(312\) 1.00000 0.0566139
\(313\) 1.14456 0.0646941 0.0323471 0.999477i \(-0.489702\pi\)
0.0323471 + 0.999477i \(0.489702\pi\)
\(314\) 7.41021 0.418183
\(315\) 6.17693 0.348031
\(316\) −2.59198 −0.145810
\(317\) −24.8040 −1.39313 −0.696566 0.717493i \(-0.745290\pi\)
−0.696566 + 0.717493i \(0.745290\pi\)
\(318\) −0.740185 −0.0415075
\(319\) 2.74652 0.153775
\(320\) −2.43337 −0.136030
\(321\) −10.8476 −0.605454
\(322\) −20.4434 −1.13927
\(323\) 8.12384 0.452023
\(324\) 1.00000 0.0555556
\(325\) −0.921312 −0.0511052
\(326\) 22.3303 1.23676
\(327\) −13.9607 −0.772028
\(328\) 1.29937 0.0717459
\(329\) −22.1594 −1.22169
\(330\) 14.0566 0.773793
\(331\) 26.1663 1.43823 0.719116 0.694890i \(-0.244547\pi\)
0.719116 + 0.694890i \(0.244547\pi\)
\(332\) −3.99096 −0.219032
\(333\) −2.13431 −0.116960
\(334\) −4.53339 −0.248056
\(335\) 7.57212 0.413709
\(336\) 2.53842 0.138482
\(337\) 18.2750 0.995502 0.497751 0.867320i \(-0.334159\pi\)
0.497751 + 0.867320i \(0.334159\pi\)
\(338\) 1.00000 0.0543928
\(339\) 9.77485 0.530897
\(340\) 9.85327 0.534369
\(341\) −36.5115 −1.97721
\(342\) −2.00627 −0.108487
\(343\) 19.1814 1.03570
\(344\) −1.58896 −0.0856708
\(345\) 19.5974 1.05509
\(346\) −9.08335 −0.488324
\(347\) 28.6629 1.53870 0.769351 0.638826i \(-0.220580\pi\)
0.769351 + 0.638826i \(0.220580\pi\)
\(348\) −0.475455 −0.0254871
\(349\) −6.88183 −0.368376 −0.184188 0.982891i \(-0.558966\pi\)
−0.184188 + 0.982891i \(0.558966\pi\)
\(350\) −2.33868 −0.125008
\(351\) 1.00000 0.0533761
\(352\) 5.77661 0.307894
\(353\) −1.99704 −0.106292 −0.0531459 0.998587i \(-0.516925\pi\)
−0.0531459 + 0.998587i \(0.516925\pi\)
\(354\) 2.73337 0.145277
\(355\) −21.5100 −1.14163
\(356\) −9.66240 −0.512106
\(357\) −10.2786 −0.544003
\(358\) −17.1575 −0.906804
\(359\) 11.0783 0.584689 0.292345 0.956313i \(-0.405565\pi\)
0.292345 + 0.956313i \(0.405565\pi\)
\(360\) −2.43337 −0.128250
\(361\) −14.9749 −0.788151
\(362\) −10.9658 −0.576351
\(363\) −22.3692 −1.17408
\(364\) 2.53842 0.133049
\(365\) 20.9182 1.09491
\(366\) 6.04981 0.316229
\(367\) −5.65883 −0.295388 −0.147694 0.989033i \(-0.547185\pi\)
−0.147694 + 0.989033i \(0.547185\pi\)
\(368\) 8.05359 0.419822
\(369\) 1.29937 0.0676427
\(370\) 5.19358 0.270001
\(371\) −1.87890 −0.0975477
\(372\) 6.32058 0.327707
\(373\) −15.2596 −0.790114 −0.395057 0.918657i \(-0.629275\pi\)
−0.395057 + 0.918657i \(0.629275\pi\)
\(374\) −23.3908 −1.20951
\(375\) −9.92498 −0.512524
\(376\) 8.72962 0.450196
\(377\) −0.475455 −0.0244872
\(378\) 2.53842 0.130562
\(379\) −31.9693 −1.64215 −0.821077 0.570818i \(-0.806626\pi\)
−0.821077 + 0.570818i \(0.806626\pi\)
\(380\) 4.88201 0.250442
\(381\) 0.741066 0.0379659
\(382\) 19.7366 1.00981
\(383\) −23.7745 −1.21482 −0.607411 0.794388i \(-0.707792\pi\)
−0.607411 + 0.794388i \(0.707792\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 35.6817 1.81851
\(386\) −14.9310 −0.759966
\(387\) −1.58896 −0.0807712
\(388\) 12.2300 0.620882
\(389\) 25.7378 1.30496 0.652479 0.757807i \(-0.273729\pi\)
0.652479 + 0.757807i \(0.273729\pi\)
\(390\) −2.43337 −0.123219
\(391\) −32.6108 −1.64920
\(392\) −0.556418 −0.0281034
\(393\) −12.8407 −0.647726
\(394\) −21.4303 −1.07964
\(395\) 6.30725 0.317352
\(396\) 5.77661 0.290285
\(397\) 34.3402 1.72348 0.861742 0.507346i \(-0.169374\pi\)
0.861742 + 0.507346i \(0.169374\pi\)
\(398\) −4.79816 −0.240510
\(399\) −5.09276 −0.254957
\(400\) 0.921312 0.0460656
\(401\) 21.4238 1.06985 0.534927 0.844898i \(-0.320339\pi\)
0.534927 + 0.844898i \(0.320339\pi\)
\(402\) 3.11178 0.155201
\(403\) 6.32058 0.314851
\(404\) 11.1817 0.556311
\(405\) −2.43337 −0.120915
\(406\) −1.20690 −0.0598977
\(407\) −12.3291 −0.611130
\(408\) 4.04922 0.200466
\(409\) 30.7711 1.52153 0.760765 0.649027i \(-0.224824\pi\)
0.760765 + 0.649027i \(0.224824\pi\)
\(410\) −3.16186 −0.156153
\(411\) −0.740211 −0.0365119
\(412\) −1.00000 −0.0492665
\(413\) 6.93844 0.341419
\(414\) 8.05359 0.395812
\(415\) 9.71149 0.476718
\(416\) −1.00000 −0.0490290
\(417\) −7.89186 −0.386466
\(418\) −11.5894 −0.566858
\(419\) −3.50389 −0.171176 −0.0855881 0.996331i \(-0.527277\pi\)
−0.0855881 + 0.996331i \(0.527277\pi\)
\(420\) −6.17693 −0.301403
\(421\) 10.2446 0.499294 0.249647 0.968337i \(-0.419685\pi\)
0.249647 + 0.968337i \(0.419685\pi\)
\(422\) 15.9089 0.774432
\(423\) 8.72962 0.424449
\(424\) 0.740185 0.0359466
\(425\) −3.73060 −0.180961
\(426\) −8.83959 −0.428280
\(427\) 15.3570 0.743176
\(428\) 10.8476 0.524338
\(429\) 5.77661 0.278897
\(430\) 3.86653 0.186460
\(431\) −21.0949 −1.01611 −0.508053 0.861326i \(-0.669635\pi\)
−0.508053 + 0.861326i \(0.669635\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.0426 0.674847 0.337423 0.941353i \(-0.390445\pi\)
0.337423 + 0.941353i \(0.390445\pi\)
\(434\) 16.0443 0.770151
\(435\) 1.15696 0.0554720
\(436\) 13.9607 0.668596
\(437\) −16.1577 −0.772927
\(438\) 8.59636 0.410750
\(439\) 36.2834 1.73171 0.865857 0.500291i \(-0.166774\pi\)
0.865857 + 0.500291i \(0.166774\pi\)
\(440\) −14.0566 −0.670124
\(441\) −0.556418 −0.0264961
\(442\) 4.04922 0.192602
\(443\) −20.4398 −0.971123 −0.485561 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(444\) 2.13431 0.101290
\(445\) 23.5122 1.11459
\(446\) 8.61008 0.407699
\(447\) 11.9282 0.564183
\(448\) −2.53842 −0.119929
\(449\) 8.93778 0.421800 0.210900 0.977508i \(-0.432361\pi\)
0.210900 + 0.977508i \(0.432361\pi\)
\(450\) 0.921312 0.0434311
\(451\) 7.50597 0.353442
\(452\) −9.77485 −0.459770
\(453\) −22.9927 −1.08029
\(454\) 16.4584 0.772429
\(455\) −6.17693 −0.289579
\(456\) 2.00627 0.0939523
\(457\) −6.45931 −0.302154 −0.151077 0.988522i \(-0.548274\pi\)
−0.151077 + 0.988522i \(0.548274\pi\)
\(458\) 16.3135 0.762280
\(459\) 4.04922 0.189002
\(460\) −19.5974 −0.913733
\(461\) 19.3065 0.899193 0.449597 0.893232i \(-0.351568\pi\)
0.449597 + 0.893232i \(0.351568\pi\)
\(462\) 14.6635 0.682206
\(463\) 16.9074 0.785752 0.392876 0.919591i \(-0.371480\pi\)
0.392876 + 0.919591i \(0.371480\pi\)
\(464\) 0.475455 0.0220724
\(465\) −15.3803 −0.713246
\(466\) 9.69358 0.449046
\(467\) 26.3793 1.22069 0.610345 0.792136i \(-0.291031\pi\)
0.610345 + 0.792136i \(0.291031\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 7.89900 0.364742
\(470\) −21.2424 −0.979840
\(471\) −7.41021 −0.341445
\(472\) −2.73337 −0.125814
\(473\) −9.17877 −0.422041
\(474\) 2.59198 0.119053
\(475\) −1.84840 −0.0848105
\(476\) 10.2786 0.471120
\(477\) 0.740185 0.0338907
\(478\) 14.6164 0.668537
\(479\) −20.7552 −0.948328 −0.474164 0.880437i \(-0.657250\pi\)
−0.474164 + 0.880437i \(0.657250\pi\)
\(480\) 2.43337 0.111068
\(481\) 2.13431 0.0973162
\(482\) 19.0765 0.868911
\(483\) 20.4434 0.930207
\(484\) 22.3692 1.01678
\(485\) −29.7601 −1.35134
\(486\) −1.00000 −0.0453609
\(487\) −26.9324 −1.22042 −0.610211 0.792239i \(-0.708915\pi\)
−0.610211 + 0.792239i \(0.708915\pi\)
\(488\) −6.04981 −0.273862
\(489\) −22.3303 −1.00981
\(490\) 1.35397 0.0611663
\(491\) 13.5361 0.610876 0.305438 0.952212i \(-0.401197\pi\)
0.305438 + 0.952212i \(0.401197\pi\)
\(492\) −1.29937 −0.0585803
\(493\) −1.92522 −0.0867077
\(494\) 2.00627 0.0902664
\(495\) −14.0566 −0.631799
\(496\) −6.32058 −0.283802
\(497\) −22.4386 −1.00651
\(498\) 3.99096 0.178839
\(499\) 5.86009 0.262334 0.131167 0.991360i \(-0.458128\pi\)
0.131167 + 0.991360i \(0.458128\pi\)
\(500\) 9.92498 0.443858
\(501\) 4.53339 0.202537
\(502\) 12.6704 0.565508
\(503\) −15.7566 −0.702552 −0.351276 0.936272i \(-0.614252\pi\)
−0.351276 + 0.936272i \(0.614252\pi\)
\(504\) −2.53842 −0.113070
\(505\) −27.2093 −1.21080
\(506\) 46.5224 2.06817
\(507\) −1.00000 −0.0444116
\(508\) −0.741066 −0.0328795
\(509\) −36.1402 −1.60189 −0.800943 0.598741i \(-0.795668\pi\)
−0.800943 + 0.598741i \(0.795668\pi\)
\(510\) −9.85327 −0.436310
\(511\) 21.8212 0.965312
\(512\) 1.00000 0.0441942
\(513\) 2.00627 0.0885791
\(514\) 26.2911 1.15965
\(515\) 2.43337 0.107227
\(516\) 1.58896 0.0699499
\(517\) 50.4276 2.21780
\(518\) 5.41778 0.238044
\(519\) 9.08335 0.398715
\(520\) 2.43337 0.106711
\(521\) −2.58432 −0.113221 −0.0566105 0.998396i \(-0.518029\pi\)
−0.0566105 + 0.998396i \(0.518029\pi\)
\(522\) 0.475455 0.0208101
\(523\) 0.904305 0.0395425 0.0197712 0.999805i \(-0.493706\pi\)
0.0197712 + 0.999805i \(0.493706\pi\)
\(524\) 12.8407 0.560947
\(525\) 2.33868 0.102068
\(526\) 3.53155 0.153983
\(527\) 25.5934 1.11487
\(528\) −5.77661 −0.251395
\(529\) 41.8603 1.82001
\(530\) −1.80115 −0.0782369
\(531\) −2.73337 −0.118618
\(532\) 5.09276 0.220799
\(533\) −1.29937 −0.0562821
\(534\) 9.66240 0.418133
\(535\) −26.3963 −1.14121
\(536\) −3.11178 −0.134408
\(537\) 17.1575 0.740402
\(538\) 17.9185 0.772520
\(539\) −3.21421 −0.138446
\(540\) 2.43337 0.104716
\(541\) −42.6924 −1.83549 −0.917746 0.397168i \(-0.869993\pi\)
−0.917746 + 0.397168i \(0.869993\pi\)
\(542\) 1.14289 0.0490912
\(543\) 10.9658 0.470589
\(544\) −4.04922 −0.173609
\(545\) −33.9716 −1.45518
\(546\) −2.53842 −0.108634
\(547\) −31.1663 −1.33257 −0.666287 0.745695i \(-0.732117\pi\)
−0.666287 + 0.745695i \(0.732117\pi\)
\(548\) 0.740211 0.0316202
\(549\) −6.04981 −0.258200
\(550\) 5.32206 0.226933
\(551\) −0.953892 −0.0406371
\(552\) −8.05359 −0.342783
\(553\) 6.57953 0.279790
\(554\) −23.1873 −0.985134
\(555\) −5.19358 −0.220455
\(556\) 7.89186 0.334689
\(557\) 17.3499 0.735138 0.367569 0.929996i \(-0.380190\pi\)
0.367569 + 0.929996i \(0.380190\pi\)
\(558\) −6.32058 −0.267572
\(559\) 1.58896 0.0672057
\(560\) 6.17693 0.261023
\(561\) 23.3908 0.987559
\(562\) 30.2126 1.27444
\(563\) −2.56647 −0.108164 −0.0540819 0.998537i \(-0.517223\pi\)
−0.0540819 + 0.998537i \(0.517223\pi\)
\(564\) −8.72962 −0.367583
\(565\) 23.7859 1.00068
\(566\) −11.4266 −0.480296
\(567\) −2.53842 −0.106604
\(568\) 8.83959 0.370901
\(569\) −36.2540 −1.51984 −0.759922 0.650014i \(-0.774763\pi\)
−0.759922 + 0.650014i \(0.774763\pi\)
\(570\) −4.88201 −0.204485
\(571\) −2.03912 −0.0853346 −0.0426673 0.999089i \(-0.513586\pi\)
−0.0426673 + 0.999089i \(0.513586\pi\)
\(572\) −5.77661 −0.241532
\(573\) −19.7366 −0.824510
\(574\) −3.29836 −0.137671
\(575\) 7.41987 0.309430
\(576\) 1.00000 0.0416667
\(577\) 27.1431 1.12998 0.564992 0.825097i \(-0.308879\pi\)
0.564992 + 0.825097i \(0.308879\pi\)
\(578\) −0.603800 −0.0251148
\(579\) 14.9310 0.620509
\(580\) −1.15696 −0.0480401
\(581\) 10.1307 0.420293
\(582\) −12.2300 −0.506948
\(583\) 4.27576 0.177084
\(584\) −8.59636 −0.355720
\(585\) 2.43337 0.100608
\(586\) 11.3285 0.467974
\(587\) 18.2302 0.752442 0.376221 0.926530i \(-0.377223\pi\)
0.376221 + 0.926530i \(0.377223\pi\)
\(588\) 0.556418 0.0229463
\(589\) 12.6808 0.522503
\(590\) 6.65131 0.273830
\(591\) 21.4303 0.881526
\(592\) −2.13431 −0.0877196
\(593\) 4.48531 0.184190 0.0920948 0.995750i \(-0.470644\pi\)
0.0920948 + 0.995750i \(0.470644\pi\)
\(594\) −5.77661 −0.237017
\(595\) −25.0118 −1.02538
\(596\) −11.9282 −0.488597
\(597\) 4.79816 0.196376
\(598\) −8.05359 −0.329336
\(599\) −6.04937 −0.247170 −0.123585 0.992334i \(-0.539439\pi\)
−0.123585 + 0.992334i \(0.539439\pi\)
\(600\) −0.921312 −0.0376124
\(601\) −13.5231 −0.551619 −0.275810 0.961212i \(-0.588946\pi\)
−0.275810 + 0.961212i \(0.588946\pi\)
\(602\) 4.03344 0.164391
\(603\) −3.11178 −0.126721
\(604\) 22.9927 0.935559
\(605\) −54.4326 −2.21300
\(606\) −11.1817 −0.454226
\(607\) 7.52273 0.305338 0.152669 0.988277i \(-0.451213\pi\)
0.152669 + 0.988277i \(0.451213\pi\)
\(608\) −2.00627 −0.0813651
\(609\) 1.20690 0.0489062
\(610\) 14.7215 0.596054
\(611\) −8.72962 −0.353163
\(612\) −4.04922 −0.163680
\(613\) −25.2747 −1.02084 −0.510418 0.859926i \(-0.670509\pi\)
−0.510418 + 0.859926i \(0.670509\pi\)
\(614\) 19.6190 0.791760
\(615\) 3.16186 0.127499
\(616\) −14.6635 −0.590808
\(617\) 11.9571 0.481373 0.240687 0.970603i \(-0.422627\pi\)
0.240687 + 0.970603i \(0.422627\pi\)
\(618\) 1.00000 0.0402259
\(619\) 20.8781 0.839161 0.419581 0.907718i \(-0.362177\pi\)
0.419581 + 0.907718i \(0.362177\pi\)
\(620\) 15.3803 0.617689
\(621\) −8.05359 −0.323179
\(622\) −26.0380 −1.04403
\(623\) 24.5272 0.982663
\(624\) 1.00000 0.0400320
\(625\) −28.7577 −1.15031
\(626\) 1.14456 0.0457456
\(627\) 11.5894 0.462838
\(628\) 7.41021 0.295700
\(629\) 8.64230 0.344591
\(630\) 6.17693 0.246095
\(631\) 20.3175 0.808826 0.404413 0.914576i \(-0.367476\pi\)
0.404413 + 0.914576i \(0.367476\pi\)
\(632\) −2.59198 −0.103103
\(633\) −15.9089 −0.632321
\(634\) −24.8040 −0.985093
\(635\) 1.80329 0.0715614
\(636\) −0.740185 −0.0293502
\(637\) 0.556418 0.0220461
\(638\) 2.74652 0.108736
\(639\) 8.83959 0.349689
\(640\) −2.43337 −0.0961876
\(641\) 35.7464 1.41190 0.705948 0.708263i \(-0.250521\pi\)
0.705948 + 0.708263i \(0.250521\pi\)
\(642\) −10.8476 −0.428121
\(643\) −31.5474 −1.24411 −0.622054 0.782974i \(-0.713702\pi\)
−0.622054 + 0.782974i \(0.713702\pi\)
\(644\) −20.4434 −0.805583
\(645\) −3.86653 −0.152244
\(646\) 8.12384 0.319628
\(647\) −44.6583 −1.75570 −0.877849 0.478937i \(-0.841022\pi\)
−0.877849 + 0.478937i \(0.841022\pi\)
\(648\) 1.00000 0.0392837
\(649\) −15.7896 −0.619796
\(650\) −0.921312 −0.0361368
\(651\) −16.0443 −0.628826
\(652\) 22.3303 0.874520
\(653\) −16.2905 −0.637495 −0.318748 0.947840i \(-0.603262\pi\)
−0.318748 + 0.947840i \(0.603262\pi\)
\(654\) −13.9607 −0.545906
\(655\) −31.2462 −1.22089
\(656\) 1.29937 0.0507320
\(657\) −8.59636 −0.335376
\(658\) −22.1594 −0.863865
\(659\) 36.9170 1.43808 0.719040 0.694968i \(-0.244582\pi\)
0.719040 + 0.694968i \(0.244582\pi\)
\(660\) 14.0566 0.547154
\(661\) 16.0315 0.623552 0.311776 0.950156i \(-0.399076\pi\)
0.311776 + 0.950156i \(0.399076\pi\)
\(662\) 26.1663 1.01698
\(663\) −4.04922 −0.157259
\(664\) −3.99096 −0.154879
\(665\) −12.3926 −0.480564
\(666\) −2.13431 −0.0827029
\(667\) 3.82912 0.148264
\(668\) −4.53339 −0.175402
\(669\) −8.61008 −0.332885
\(670\) 7.57212 0.292537
\(671\) −34.9474 −1.34913
\(672\) 2.53842 0.0979217
\(673\) 15.0154 0.578800 0.289400 0.957208i \(-0.406544\pi\)
0.289400 + 0.957208i \(0.406544\pi\)
\(674\) 18.2750 0.703926
\(675\) −0.921312 −0.0354613
\(676\) 1.00000 0.0384615
\(677\) 29.3815 1.12922 0.564611 0.825357i \(-0.309026\pi\)
0.564611 + 0.825357i \(0.309026\pi\)
\(678\) 9.77485 0.375401
\(679\) −31.0448 −1.19139
\(680\) 9.85327 0.377856
\(681\) −16.4584 −0.630686
\(682\) −36.5115 −1.39810
\(683\) −8.35468 −0.319683 −0.159842 0.987143i \(-0.551098\pi\)
−0.159842 + 0.987143i \(0.551098\pi\)
\(684\) −2.00627 −0.0767117
\(685\) −1.80121 −0.0688207
\(686\) 19.1814 0.732348
\(687\) −16.3135 −0.622399
\(688\) −1.58896 −0.0605784
\(689\) −0.740185 −0.0281988
\(690\) 19.5974 0.746060
\(691\) −20.9229 −0.795943 −0.397971 0.917398i \(-0.630286\pi\)
−0.397971 + 0.917398i \(0.630286\pi\)
\(692\) −9.08335 −0.345297
\(693\) −14.6635 −0.557019
\(694\) 28.6629 1.08803
\(695\) −19.2038 −0.728443
\(696\) −0.475455 −0.0180221
\(697\) −5.26145 −0.199292
\(698\) −6.88183 −0.260481
\(699\) −9.69358 −0.366645
\(700\) −2.33868 −0.0883938
\(701\) −15.3988 −0.581605 −0.290803 0.956783i \(-0.593922\pi\)
−0.290803 + 0.956783i \(0.593922\pi\)
\(702\) 1.00000 0.0377426
\(703\) 4.28201 0.161499
\(704\) 5.77661 0.217714
\(705\) 21.2424 0.800036
\(706\) −1.99704 −0.0751597
\(707\) −28.3839 −1.06749
\(708\) 2.73337 0.102726
\(709\) 31.4589 1.18146 0.590731 0.806868i \(-0.298839\pi\)
0.590731 + 0.806868i \(0.298839\pi\)
\(710\) −21.5100 −0.807258
\(711\) −2.59198 −0.0972067
\(712\) −9.66240 −0.362114
\(713\) −50.9033 −1.90635
\(714\) −10.2786 −0.384668
\(715\) 14.0566 0.525689
\(716\) −17.1575 −0.641207
\(717\) −14.6164 −0.545858
\(718\) 11.0783 0.413438
\(719\) −35.5732 −1.32666 −0.663328 0.748329i \(-0.730856\pi\)
−0.663328 + 0.748329i \(0.730856\pi\)
\(720\) −2.43337 −0.0906865
\(721\) 2.53842 0.0945357
\(722\) −14.9749 −0.557307
\(723\) −19.0765 −0.709463
\(724\) −10.9658 −0.407542
\(725\) 0.438043 0.0162685
\(726\) −22.3692 −0.830198
\(727\) −19.6665 −0.729390 −0.364695 0.931127i \(-0.618827\pi\)
−0.364695 + 0.931127i \(0.618827\pi\)
\(728\) 2.53842 0.0940801
\(729\) 1.00000 0.0370370
\(730\) 20.9182 0.774216
\(731\) 6.43404 0.237971
\(732\) 6.04981 0.223607
\(733\) 5.59025 0.206481 0.103240 0.994656i \(-0.467079\pi\)
0.103240 + 0.994656i \(0.467079\pi\)
\(734\) −5.65883 −0.208871
\(735\) −1.35397 −0.0499421
\(736\) 8.05359 0.296859
\(737\) −17.9755 −0.662137
\(738\) 1.29937 0.0478306
\(739\) −10.6477 −0.391683 −0.195841 0.980636i \(-0.562744\pi\)
−0.195841 + 0.980636i \(0.562744\pi\)
\(740\) 5.19358 0.190920
\(741\) −2.00627 −0.0737022
\(742\) −1.87890 −0.0689766
\(743\) −30.2932 −1.11135 −0.555674 0.831400i \(-0.687540\pi\)
−0.555674 + 0.831400i \(0.687540\pi\)
\(744\) 6.32058 0.231724
\(745\) 29.0257 1.06342
\(746\) −15.2596 −0.558695
\(747\) −3.99096 −0.146021
\(748\) −23.3908 −0.855251
\(749\) −27.5358 −1.00614
\(750\) −9.92498 −0.362409
\(751\) 12.2962 0.448695 0.224348 0.974509i \(-0.427975\pi\)
0.224348 + 0.974509i \(0.427975\pi\)
\(752\) 8.72962 0.318336
\(753\) −12.6704 −0.461736
\(754\) −0.475455 −0.0173150
\(755\) −55.9498 −2.03622
\(756\) 2.53842 0.0923215
\(757\) −16.9986 −0.617824 −0.308912 0.951091i \(-0.599965\pi\)
−0.308912 + 0.951091i \(0.599965\pi\)
\(758\) −31.9693 −1.16118
\(759\) −46.5224 −1.68866
\(760\) 4.88201 0.177089
\(761\) −43.5302 −1.57797 −0.788984 0.614414i \(-0.789392\pi\)
−0.788984 + 0.614414i \(0.789392\pi\)
\(762\) 0.741066 0.0268460
\(763\) −35.4381 −1.28295
\(764\) 19.7366 0.714047
\(765\) 9.85327 0.356246
\(766\) −23.7745 −0.859009
\(767\) 2.73337 0.0986963
\(768\) −1.00000 −0.0360844
\(769\) 22.4858 0.810858 0.405429 0.914126i \(-0.367122\pi\)
0.405429 + 0.914126i \(0.367122\pi\)
\(770\) 35.6817 1.28588
\(771\) −26.2911 −0.946851
\(772\) −14.9310 −0.537377
\(773\) 12.7123 0.457230 0.228615 0.973517i \(-0.426580\pi\)
0.228615 + 0.973517i \(0.426580\pi\)
\(774\) −1.58896 −0.0571139
\(775\) −5.82323 −0.209177
\(776\) 12.2300 0.439030
\(777\) −5.41778 −0.194362
\(778\) 25.7378 0.922745
\(779\) −2.60690 −0.0934017
\(780\) −2.43337 −0.0871288
\(781\) 51.0629 1.82717
\(782\) −32.6108 −1.16616
\(783\) −0.475455 −0.0169914
\(784\) −0.556418 −0.0198721
\(785\) −18.0318 −0.643583
\(786\) −12.8407 −0.458012
\(787\) 34.7227 1.23773 0.618866 0.785497i \(-0.287592\pi\)
0.618866 + 0.785497i \(0.287592\pi\)
\(788\) −21.4303 −0.763424
\(789\) −3.53155 −0.125727
\(790\) 6.30725 0.224402
\(791\) 24.8127 0.882237
\(792\) 5.77661 0.205263
\(793\) 6.04981 0.214835
\(794\) 34.3402 1.21869
\(795\) 1.80115 0.0638801
\(796\) −4.79816 −0.170066
\(797\) −46.3092 −1.64036 −0.820179 0.572108i \(-0.806126\pi\)
−0.820179 + 0.572108i \(0.806126\pi\)
\(798\) −5.09276 −0.180282
\(799\) −35.3482 −1.25053
\(800\) 0.921312 0.0325733
\(801\) −9.66240 −0.341404
\(802\) 21.4238 0.756501
\(803\) −49.6578 −1.75239
\(804\) 3.11178 0.109744
\(805\) 49.7464 1.75333
\(806\) 6.32058 0.222633
\(807\) −17.9185 −0.630760
\(808\) 11.1817 0.393372
\(809\) 31.4300 1.10502 0.552510 0.833506i \(-0.313670\pi\)
0.552510 + 0.833506i \(0.313670\pi\)
\(810\) −2.43337 −0.0855001
\(811\) −22.5263 −0.791007 −0.395503 0.918465i \(-0.629430\pi\)
−0.395503 + 0.918465i \(0.629430\pi\)
\(812\) −1.20690 −0.0423541
\(813\) −1.14289 −0.0400828
\(814\) −12.3291 −0.432134
\(815\) −54.3379 −1.90337
\(816\) 4.04922 0.141751
\(817\) 3.18788 0.111530
\(818\) 30.7711 1.07588
\(819\) 2.53842 0.0886996
\(820\) −3.16186 −0.110417
\(821\) 17.9755 0.627350 0.313675 0.949530i \(-0.398440\pi\)
0.313675 + 0.949530i \(0.398440\pi\)
\(822\) −0.740211 −0.0258178
\(823\) 23.0890 0.804833 0.402417 0.915457i \(-0.368170\pi\)
0.402417 + 0.915457i \(0.368170\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −5.32206 −0.185290
\(826\) 6.93844 0.241419
\(827\) 4.81901 0.167573 0.0837866 0.996484i \(-0.473299\pi\)
0.0837866 + 0.996484i \(0.473299\pi\)
\(828\) 8.05359 0.279881
\(829\) 14.9496 0.519222 0.259611 0.965713i \(-0.416406\pi\)
0.259611 + 0.965713i \(0.416406\pi\)
\(830\) 9.71149 0.337091
\(831\) 23.1873 0.804358
\(832\) −1.00000 −0.0346688
\(833\) 2.25306 0.0780639
\(834\) −7.89186 −0.273273
\(835\) 11.0314 0.381759
\(836\) −11.5894 −0.400829
\(837\) 6.32058 0.218471
\(838\) −3.50389 −0.121040
\(839\) −14.0849 −0.486264 −0.243132 0.969993i \(-0.578175\pi\)
−0.243132 + 0.969993i \(0.578175\pi\)
\(840\) −6.17693 −0.213124
\(841\) −28.7739 −0.992205
\(842\) 10.2446 0.353054
\(843\) −30.2126 −1.04058
\(844\) 15.9089 0.547606
\(845\) −2.43337 −0.0837106
\(846\) 8.72962 0.300130
\(847\) −56.7824 −1.95107
\(848\) 0.740185 0.0254181
\(849\) 11.4266 0.392160
\(850\) −3.73060 −0.127958
\(851\) −17.1889 −0.589227
\(852\) −8.83959 −0.302839
\(853\) 14.4201 0.493736 0.246868 0.969049i \(-0.420599\pi\)
0.246868 + 0.969049i \(0.420599\pi\)
\(854\) 15.3570 0.525505
\(855\) 4.88201 0.166961
\(856\) 10.8476 0.370763
\(857\) 39.1070 1.33587 0.667935 0.744219i \(-0.267178\pi\)
0.667935 + 0.744219i \(0.267178\pi\)
\(858\) 5.77661 0.197210
\(859\) 21.0370 0.717774 0.358887 0.933381i \(-0.383156\pi\)
0.358887 + 0.933381i \(0.383156\pi\)
\(860\) 3.86653 0.131847
\(861\) 3.29836 0.112408
\(862\) −21.0949 −0.718495
\(863\) 5.08200 0.172993 0.0864966 0.996252i \(-0.472433\pi\)
0.0864966 + 0.996252i \(0.472433\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 22.1032 0.751532
\(866\) 14.0426 0.477189
\(867\) 0.603800 0.0205061
\(868\) 16.0443 0.544579
\(869\) −14.9728 −0.507918
\(870\) 1.15696 0.0392246
\(871\) 3.11178 0.105439
\(872\) 13.9607 0.472769
\(873\) 12.2300 0.413922
\(874\) −16.1577 −0.546542
\(875\) −25.1938 −0.851705
\(876\) 8.59636 0.290444
\(877\) −2.14792 −0.0725301 −0.0362650 0.999342i \(-0.511546\pi\)
−0.0362650 + 0.999342i \(0.511546\pi\)
\(878\) 36.2834 1.22451
\(879\) −11.3285 −0.382099
\(880\) −14.0566 −0.473849
\(881\) 32.0163 1.07866 0.539329 0.842095i \(-0.318678\pi\)
0.539329 + 0.842095i \(0.318678\pi\)
\(882\) −0.556418 −0.0187356
\(883\) −12.9803 −0.436820 −0.218410 0.975857i \(-0.570087\pi\)
−0.218410 + 0.975857i \(0.570087\pi\)
\(884\) 4.04922 0.136190
\(885\) −6.65131 −0.223581
\(886\) −20.4398 −0.686687
\(887\) 57.7282 1.93832 0.969162 0.246426i \(-0.0792564\pi\)
0.969162 + 0.246426i \(0.0792564\pi\)
\(888\) 2.13431 0.0716228
\(889\) 1.88114 0.0630913
\(890\) 23.5122 0.788132
\(891\) 5.77661 0.193524
\(892\) 8.61008 0.288287
\(893\) −17.5140 −0.586083
\(894\) 11.9282 0.398938
\(895\) 41.7507 1.39557
\(896\) −2.53842 −0.0848027
\(897\) 8.05359 0.268901
\(898\) 8.93778 0.298257
\(899\) −3.00515 −0.100227
\(900\) 0.921312 0.0307104
\(901\) −2.99717 −0.0998503
\(902\) 7.50597 0.249921
\(903\) −4.03344 −0.134224
\(904\) −9.77485 −0.325107
\(905\) 26.6840 0.887005
\(906\) −22.9927 −0.763881
\(907\) −20.2845 −0.673537 −0.336769 0.941587i \(-0.609334\pi\)
−0.336769 + 0.941587i \(0.609334\pi\)
\(908\) 16.4584 0.546190
\(909\) 11.1817 0.370874
\(910\) −6.17693 −0.204763
\(911\) 40.8680 1.35402 0.677008 0.735976i \(-0.263276\pi\)
0.677008 + 0.735976i \(0.263276\pi\)
\(912\) 2.00627 0.0664343
\(913\) −23.0542 −0.762982
\(914\) −6.45931 −0.213655
\(915\) −14.7215 −0.486676
\(916\) 16.3135 0.539013
\(917\) −32.5950 −1.07638
\(918\) 4.04922 0.133644
\(919\) −1.12941 −0.0372557 −0.0186279 0.999826i \(-0.505930\pi\)
−0.0186279 + 0.999826i \(0.505930\pi\)
\(920\) −19.5974 −0.646107
\(921\) −19.6190 −0.646469
\(922\) 19.3065 0.635826
\(923\) −8.83959 −0.290959
\(924\) 14.6635 0.482392
\(925\) −1.96637 −0.0646538
\(926\) 16.9074 0.555611
\(927\) −1.00000 −0.0328443
\(928\) 0.475455 0.0156076
\(929\) 10.3220 0.338652 0.169326 0.985560i \(-0.445841\pi\)
0.169326 + 0.985560i \(0.445841\pi\)
\(930\) −15.3803 −0.504341
\(931\) 1.11633 0.0365861
\(932\) 9.69358 0.317524
\(933\) 26.0380 0.852445
\(934\) 26.3793 0.863158
\(935\) 56.9185 1.86143
\(936\) −1.00000 −0.0326860
\(937\) −21.1152 −0.689804 −0.344902 0.938639i \(-0.612088\pi\)
−0.344902 + 0.938639i \(0.612088\pi\)
\(938\) 7.89900 0.257912
\(939\) −1.14456 −0.0373512
\(940\) −21.2424 −0.692852
\(941\) −33.5643 −1.09416 −0.547082 0.837079i \(-0.684262\pi\)
−0.547082 + 0.837079i \(0.684262\pi\)
\(942\) −7.41021 −0.241438
\(943\) 10.4646 0.340775
\(944\) −2.73337 −0.0889636
\(945\) −6.17693 −0.200936
\(946\) −9.17877 −0.298428
\(947\) 6.52226 0.211945 0.105973 0.994369i \(-0.466204\pi\)
0.105973 + 0.994369i \(0.466204\pi\)
\(948\) 2.59198 0.0841835
\(949\) 8.59636 0.279050
\(950\) −1.84840 −0.0599701
\(951\) 24.8040 0.804325
\(952\) 10.2786 0.333132
\(953\) −40.4482 −1.31025 −0.655124 0.755522i \(-0.727383\pi\)
−0.655124 + 0.755522i \(0.727383\pi\)
\(954\) 0.740185 0.0239644
\(955\) −48.0267 −1.55411
\(956\) 14.6164 0.472727
\(957\) −2.74652 −0.0887822
\(958\) −20.7552 −0.670569
\(959\) −1.87897 −0.0606750
\(960\) 2.43337 0.0785368
\(961\) 8.94975 0.288702
\(962\) 2.13431 0.0688130
\(963\) 10.8476 0.349559
\(964\) 19.0765 0.614413
\(965\) 36.3326 1.16959
\(966\) 20.4434 0.657755
\(967\) −18.6062 −0.598334 −0.299167 0.954201i \(-0.596709\pi\)
−0.299167 + 0.954201i \(0.596709\pi\)
\(968\) 22.3692 0.718973
\(969\) −8.12384 −0.260975
\(970\) −29.7601 −0.955539
\(971\) 6.51579 0.209102 0.104551 0.994520i \(-0.466660\pi\)
0.104551 + 0.994520i \(0.466660\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −20.0329 −0.642224
\(974\) −26.9324 −0.862968
\(975\) 0.921312 0.0295056
\(976\) −6.04981 −0.193650
\(977\) −41.9897 −1.34337 −0.671684 0.740838i \(-0.734429\pi\)
−0.671684 + 0.740838i \(0.734429\pi\)
\(978\) −22.3303 −0.714043
\(979\) −55.8159 −1.78388
\(980\) 1.35397 0.0432511
\(981\) 13.9607 0.445731
\(982\) 13.5361 0.431954
\(983\) −58.5253 −1.86667 −0.933334 0.359010i \(-0.883114\pi\)
−0.933334 + 0.359010i \(0.883114\pi\)
\(984\) −1.29937 −0.0414225
\(985\) 52.1480 1.66157
\(986\) −1.92522 −0.0613116
\(987\) 22.1594 0.705343
\(988\) 2.00627 0.0638280
\(989\) −12.7968 −0.406914
\(990\) −14.0566 −0.446750
\(991\) −36.4884 −1.15909 −0.579546 0.814939i \(-0.696770\pi\)
−0.579546 + 0.814939i \(0.696770\pi\)
\(992\) −6.32058 −0.200679
\(993\) −26.1663 −0.830363
\(994\) −22.4386 −0.711710
\(995\) 11.6757 0.370145
\(996\) 3.99096 0.126458
\(997\) −13.1037 −0.414999 −0.207499 0.978235i \(-0.566532\pi\)
−0.207499 + 0.978235i \(0.566532\pi\)
\(998\) 5.86009 0.185498
\(999\) 2.13431 0.0675266
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.1.4 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.4 15 1.1 even 1 trivial