Properties

Label 6014.2.a.e.1.20
Level $6014$
Weight $2$
Character 6014.1
Self dual yes
Analytic conductor $48.022$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6014,2,Mod(1,6014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6014 = 2 \cdot 31 \cdot 97 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6014.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.0220317756\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 6014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.22590 q^{3} +1.00000 q^{4} +1.86093 q^{5} +2.22590 q^{6} -2.90456 q^{7} +1.00000 q^{8} +1.95462 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.22590 q^{3} +1.00000 q^{4} +1.86093 q^{5} +2.22590 q^{6} -2.90456 q^{7} +1.00000 q^{8} +1.95462 q^{9} +1.86093 q^{10} -5.12448 q^{11} +2.22590 q^{12} -6.02174 q^{13} -2.90456 q^{14} +4.14223 q^{15} +1.00000 q^{16} +4.14179 q^{17} +1.95462 q^{18} -4.74903 q^{19} +1.86093 q^{20} -6.46525 q^{21} -5.12448 q^{22} -5.93550 q^{23} +2.22590 q^{24} -1.53695 q^{25} -6.02174 q^{26} -2.32691 q^{27} -2.90456 q^{28} +2.07949 q^{29} +4.14223 q^{30} +1.00000 q^{31} +1.00000 q^{32} -11.4066 q^{33} +4.14179 q^{34} -5.40517 q^{35} +1.95462 q^{36} -7.19584 q^{37} -4.74903 q^{38} -13.4038 q^{39} +1.86093 q^{40} +4.62169 q^{41} -6.46525 q^{42} +6.37322 q^{43} -5.12448 q^{44} +3.63741 q^{45} -5.93550 q^{46} +10.0122 q^{47} +2.22590 q^{48} +1.43646 q^{49} -1.53695 q^{50} +9.21920 q^{51} -6.02174 q^{52} -12.1450 q^{53} -2.32691 q^{54} -9.53628 q^{55} -2.90456 q^{56} -10.5709 q^{57} +2.07949 q^{58} +2.76103 q^{59} +4.14223 q^{60} -12.0007 q^{61} +1.00000 q^{62} -5.67731 q^{63} +1.00000 q^{64} -11.2060 q^{65} -11.4066 q^{66} +10.0168 q^{67} +4.14179 q^{68} -13.2118 q^{69} -5.40517 q^{70} -5.90655 q^{71} +1.95462 q^{72} +8.57239 q^{73} -7.19584 q^{74} -3.42109 q^{75} -4.74903 q^{76} +14.8843 q^{77} -13.4038 q^{78} +2.29281 q^{79} +1.86093 q^{80} -11.0433 q^{81} +4.62169 q^{82} +3.20119 q^{83} -6.46525 q^{84} +7.70757 q^{85} +6.37322 q^{86} +4.62874 q^{87} -5.12448 q^{88} +12.4242 q^{89} +3.63741 q^{90} +17.4905 q^{91} -5.93550 q^{92} +2.22590 q^{93} +10.0122 q^{94} -8.83760 q^{95} +2.22590 q^{96} -1.00000 q^{97} +1.43646 q^{98} -10.0164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 21 q^{2} - 10 q^{3} + 21 q^{4} - 10 q^{5} - 10 q^{6} - 11 q^{7} + 21 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 21 q^{2} - 10 q^{3} + 21 q^{4} - 10 q^{5} - 10 q^{6} - 11 q^{7} + 21 q^{8} + 7 q^{9} - 10 q^{10} - 12 q^{11} - 10 q^{12} - 14 q^{13} - 11 q^{14} + q^{15} + 21 q^{16} + 7 q^{18} - 27 q^{19} - 10 q^{20} - 6 q^{21} - 12 q^{22} - 4 q^{23} - 10 q^{24} + q^{25} - 14 q^{26} - 19 q^{27} - 11 q^{28} - 13 q^{29} + q^{30} + 21 q^{31} + 21 q^{32} - 20 q^{33} - 20 q^{35} + 7 q^{36} - 13 q^{37} - 27 q^{38} - 4 q^{39} - 10 q^{40} - 19 q^{41} - 6 q^{42} - 19 q^{43} - 12 q^{44} + 13 q^{45} - 4 q^{46} - 18 q^{47} - 10 q^{48} - 20 q^{49} + q^{50} - 33 q^{51} - 14 q^{52} - 15 q^{53} - 19 q^{54} - 26 q^{55} - 11 q^{56} + 9 q^{57} - 13 q^{58} - 32 q^{59} + q^{60} - 36 q^{61} + 21 q^{62} - 27 q^{63} + 21 q^{64} - 20 q^{65} - 20 q^{66} - 43 q^{67} - 11 q^{69} - 20 q^{70} - 31 q^{71} + 7 q^{72} - 11 q^{73} - 13 q^{74} - 22 q^{75} - 27 q^{76} + 12 q^{77} - 4 q^{78} - 31 q^{79} - 10 q^{80} - 39 q^{81} - 19 q^{82} - 26 q^{83} - 6 q^{84} - 12 q^{85} - 19 q^{86} + 19 q^{87} - 12 q^{88} - 14 q^{89} + 13 q^{90} - 30 q^{91} - 4 q^{92} - 10 q^{93} - 18 q^{94} - 40 q^{95} - 10 q^{96} - 21 q^{97} - 20 q^{98} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.22590 1.28512 0.642561 0.766234i \(-0.277872\pi\)
0.642561 + 0.766234i \(0.277872\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.86093 0.832232 0.416116 0.909312i \(-0.363391\pi\)
0.416116 + 0.909312i \(0.363391\pi\)
\(6\) 2.22590 0.908719
\(7\) −2.90456 −1.09782 −0.548910 0.835882i \(-0.684957\pi\)
−0.548910 + 0.835882i \(0.684957\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.95462 0.651540
\(10\) 1.86093 0.588477
\(11\) −5.12448 −1.54509 −0.772544 0.634961i \(-0.781016\pi\)
−0.772544 + 0.634961i \(0.781016\pi\)
\(12\) 2.22590 0.642561
\(13\) −6.02174 −1.67013 −0.835066 0.550150i \(-0.814570\pi\)
−0.835066 + 0.550150i \(0.814570\pi\)
\(14\) −2.90456 −0.776276
\(15\) 4.14223 1.06952
\(16\) 1.00000 0.250000
\(17\) 4.14179 1.00453 0.502266 0.864713i \(-0.332500\pi\)
0.502266 + 0.864713i \(0.332500\pi\)
\(18\) 1.95462 0.460709
\(19\) −4.74903 −1.08950 −0.544751 0.838598i \(-0.683376\pi\)
−0.544751 + 0.838598i \(0.683376\pi\)
\(20\) 1.86093 0.416116
\(21\) −6.46525 −1.41083
\(22\) −5.12448 −1.09254
\(23\) −5.93550 −1.23764 −0.618819 0.785534i \(-0.712389\pi\)
−0.618819 + 0.785534i \(0.712389\pi\)
\(24\) 2.22590 0.454360
\(25\) −1.53695 −0.307390
\(26\) −6.02174 −1.18096
\(27\) −2.32691 −0.447813
\(28\) −2.90456 −0.548910
\(29\) 2.07949 0.386152 0.193076 0.981184i \(-0.438154\pi\)
0.193076 + 0.981184i \(0.438154\pi\)
\(30\) 4.14223 0.756265
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) −11.4066 −1.98563
\(34\) 4.14179 0.710311
\(35\) −5.40517 −0.913641
\(36\) 1.95462 0.325770
\(37\) −7.19584 −1.18299 −0.591495 0.806309i \(-0.701462\pi\)
−0.591495 + 0.806309i \(0.701462\pi\)
\(38\) −4.74903 −0.770394
\(39\) −13.4038 −2.14632
\(40\) 1.86093 0.294238
\(41\) 4.62169 0.721786 0.360893 0.932607i \(-0.382472\pi\)
0.360893 + 0.932607i \(0.382472\pi\)
\(42\) −6.46525 −0.997610
\(43\) 6.37322 0.971907 0.485954 0.873985i \(-0.338472\pi\)
0.485954 + 0.873985i \(0.338472\pi\)
\(44\) −5.12448 −0.772544
\(45\) 3.63741 0.542233
\(46\) −5.93550 −0.875142
\(47\) 10.0122 1.46043 0.730217 0.683216i \(-0.239419\pi\)
0.730217 + 0.683216i \(0.239419\pi\)
\(48\) 2.22590 0.321281
\(49\) 1.43646 0.205208
\(50\) −1.53695 −0.217358
\(51\) 9.21920 1.29095
\(52\) −6.02174 −0.835066
\(53\) −12.1450 −1.66824 −0.834119 0.551584i \(-0.814023\pi\)
−0.834119 + 0.551584i \(0.814023\pi\)
\(54\) −2.32691 −0.316652
\(55\) −9.53628 −1.28587
\(56\) −2.90456 −0.388138
\(57\) −10.5709 −1.40014
\(58\) 2.07949 0.273051
\(59\) 2.76103 0.359455 0.179728 0.983716i \(-0.442478\pi\)
0.179728 + 0.983716i \(0.442478\pi\)
\(60\) 4.14223 0.534760
\(61\) −12.0007 −1.53653 −0.768264 0.640133i \(-0.778879\pi\)
−0.768264 + 0.640133i \(0.778879\pi\)
\(62\) 1.00000 0.127000
\(63\) −5.67731 −0.715274
\(64\) 1.00000 0.125000
\(65\) −11.2060 −1.38994
\(66\) −11.4066 −1.40405
\(67\) 10.0168 1.22375 0.611874 0.790956i \(-0.290416\pi\)
0.611874 + 0.790956i \(0.290416\pi\)
\(68\) 4.14179 0.502266
\(69\) −13.2118 −1.59052
\(70\) −5.40517 −0.646042
\(71\) −5.90655 −0.700979 −0.350489 0.936567i \(-0.613985\pi\)
−0.350489 + 0.936567i \(0.613985\pi\)
\(72\) 1.95462 0.230354
\(73\) 8.57239 1.00332 0.501661 0.865064i \(-0.332722\pi\)
0.501661 + 0.865064i \(0.332722\pi\)
\(74\) −7.19584 −0.836500
\(75\) −3.42109 −0.395034
\(76\) −4.74903 −0.544751
\(77\) 14.8843 1.69623
\(78\) −13.4038 −1.51768
\(79\) 2.29281 0.257961 0.128981 0.991647i \(-0.458829\pi\)
0.128981 + 0.991647i \(0.458829\pi\)
\(80\) 1.86093 0.208058
\(81\) −11.0433 −1.22704
\(82\) 4.62169 0.510380
\(83\) 3.20119 0.351376 0.175688 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(84\) −6.46525 −0.705417
\(85\) 7.70757 0.836003
\(86\) 6.37322 0.687242
\(87\) 4.62874 0.496253
\(88\) −5.12448 −0.546271
\(89\) 12.4242 1.31696 0.658481 0.752598i \(-0.271199\pi\)
0.658481 + 0.752598i \(0.271199\pi\)
\(90\) 3.63741 0.383416
\(91\) 17.4905 1.83350
\(92\) −5.93550 −0.618819
\(93\) 2.22590 0.230815
\(94\) 10.0122 1.03268
\(95\) −8.83760 −0.906718
\(96\) 2.22590 0.227180
\(97\) −1.00000 −0.101535
\(98\) 1.43646 0.145104
\(99\) −10.0164 −1.00669
\(100\) −1.53695 −0.153695
\(101\) −8.55427 −0.851181 −0.425591 0.904916i \(-0.639934\pi\)
−0.425591 + 0.904916i \(0.639934\pi\)
\(102\) 9.21920 0.912837
\(103\) −8.87731 −0.874708 −0.437354 0.899290i \(-0.644084\pi\)
−0.437354 + 0.899290i \(0.644084\pi\)
\(104\) −6.02174 −0.590481
\(105\) −12.0314 −1.17414
\(106\) −12.1450 −1.17962
\(107\) −15.8211 −1.52948 −0.764742 0.644336i \(-0.777134\pi\)
−0.764742 + 0.644336i \(0.777134\pi\)
\(108\) −2.32691 −0.223907
\(109\) 8.57088 0.820941 0.410471 0.911874i \(-0.365364\pi\)
0.410471 + 0.911874i \(0.365364\pi\)
\(110\) −9.53628 −0.909249
\(111\) −16.0172 −1.52029
\(112\) −2.90456 −0.274455
\(113\) −7.21645 −0.678867 −0.339433 0.940630i \(-0.610235\pi\)
−0.339433 + 0.940630i \(0.610235\pi\)
\(114\) −10.5709 −0.990051
\(115\) −11.0455 −1.03000
\(116\) 2.07949 0.193076
\(117\) −11.7702 −1.08816
\(118\) 2.76103 0.254173
\(119\) −12.0301 −1.10279
\(120\) 4.14223 0.378132
\(121\) 15.2603 1.38730
\(122\) −12.0007 −1.08649
\(123\) 10.2874 0.927584
\(124\) 1.00000 0.0898027
\(125\) −12.1648 −1.08805
\(126\) −5.67731 −0.505775
\(127\) 8.95218 0.794378 0.397189 0.917737i \(-0.369986\pi\)
0.397189 + 0.917737i \(0.369986\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.1861 1.24902
\(130\) −11.2060 −0.982834
\(131\) 2.78253 0.243111 0.121555 0.992585i \(-0.461212\pi\)
0.121555 + 0.992585i \(0.461212\pi\)
\(132\) −11.4066 −0.992814
\(133\) 13.7938 1.19608
\(134\) 10.0168 0.865320
\(135\) −4.33020 −0.372684
\(136\) 4.14179 0.355155
\(137\) 16.2287 1.38651 0.693255 0.720692i \(-0.256176\pi\)
0.693255 + 0.720692i \(0.256176\pi\)
\(138\) −13.2118 −1.12467
\(139\) 20.8267 1.76650 0.883249 0.468905i \(-0.155351\pi\)
0.883249 + 0.468905i \(0.155351\pi\)
\(140\) −5.40517 −0.456820
\(141\) 22.2862 1.87684
\(142\) −5.90655 −0.495667
\(143\) 30.8583 2.58050
\(144\) 1.95462 0.162885
\(145\) 3.86978 0.321368
\(146\) 8.57239 0.709456
\(147\) 3.19741 0.263718
\(148\) −7.19584 −0.591495
\(149\) −8.72274 −0.714595 −0.357297 0.933991i \(-0.616302\pi\)
−0.357297 + 0.933991i \(0.616302\pi\)
\(150\) −3.42109 −0.279331
\(151\) −15.3256 −1.24718 −0.623591 0.781751i \(-0.714327\pi\)
−0.623591 + 0.781751i \(0.714327\pi\)
\(152\) −4.74903 −0.385197
\(153\) 8.09563 0.654493
\(154\) 14.8843 1.19941
\(155\) 1.86093 0.149473
\(156\) −13.4038 −1.07316
\(157\) 9.80769 0.782739 0.391370 0.920234i \(-0.372001\pi\)
0.391370 + 0.920234i \(0.372001\pi\)
\(158\) 2.29281 0.182406
\(159\) −27.0334 −2.14389
\(160\) 1.86093 0.147119
\(161\) 17.2400 1.35870
\(162\) −11.0433 −0.867645
\(163\) −5.37431 −0.420949 −0.210474 0.977599i \(-0.567501\pi\)
−0.210474 + 0.977599i \(0.567501\pi\)
\(164\) 4.62169 0.360893
\(165\) −21.2268 −1.65250
\(166\) 3.20119 0.248460
\(167\) −19.6741 −1.52243 −0.761213 0.648502i \(-0.775396\pi\)
−0.761213 + 0.648502i \(0.775396\pi\)
\(168\) −6.46525 −0.498805
\(169\) 23.2614 1.78934
\(170\) 7.70757 0.591143
\(171\) −9.28256 −0.709855
\(172\) 6.37322 0.485954
\(173\) −9.66667 −0.734943 −0.367472 0.930035i \(-0.619776\pi\)
−0.367472 + 0.930035i \(0.619776\pi\)
\(174\) 4.62874 0.350904
\(175\) 4.46416 0.337459
\(176\) −5.12448 −0.386272
\(177\) 6.14577 0.461944
\(178\) 12.4242 0.931232
\(179\) −5.71265 −0.426984 −0.213492 0.976945i \(-0.568484\pi\)
−0.213492 + 0.976945i \(0.568484\pi\)
\(180\) 3.63741 0.271116
\(181\) 13.4152 0.997142 0.498571 0.866849i \(-0.333858\pi\)
0.498571 + 0.866849i \(0.333858\pi\)
\(182\) 17.4905 1.29648
\(183\) −26.7123 −1.97463
\(184\) −5.93550 −0.437571
\(185\) −13.3909 −0.984522
\(186\) 2.22590 0.163211
\(187\) −21.2245 −1.55209
\(188\) 10.0122 0.730217
\(189\) 6.75863 0.491618
\(190\) −8.83760 −0.641147
\(191\) −8.50762 −0.615590 −0.307795 0.951453i \(-0.599591\pi\)
−0.307795 + 0.951453i \(0.599591\pi\)
\(192\) 2.22590 0.160640
\(193\) −12.6740 −0.912295 −0.456147 0.889904i \(-0.650771\pi\)
−0.456147 + 0.889904i \(0.650771\pi\)
\(194\) −1.00000 −0.0717958
\(195\) −24.9435 −1.78624
\(196\) 1.43646 0.102604
\(197\) 19.0356 1.35623 0.678115 0.734956i \(-0.262797\pi\)
0.678115 + 0.734956i \(0.262797\pi\)
\(198\) −10.0164 −0.711836
\(199\) 8.99317 0.637509 0.318754 0.947837i \(-0.396735\pi\)
0.318754 + 0.947837i \(0.396735\pi\)
\(200\) −1.53695 −0.108679
\(201\) 22.2964 1.57267
\(202\) −8.55427 −0.601876
\(203\) −6.04001 −0.423925
\(204\) 9.21920 0.645473
\(205\) 8.60062 0.600693
\(206\) −8.87731 −0.618512
\(207\) −11.6017 −0.806371
\(208\) −6.02174 −0.417533
\(209\) 24.3363 1.68338
\(210\) −12.0314 −0.830243
\(211\) −22.1304 −1.52352 −0.761760 0.647860i \(-0.775664\pi\)
−0.761760 + 0.647860i \(0.775664\pi\)
\(212\) −12.1450 −0.834119
\(213\) −13.1474 −0.900844
\(214\) −15.8211 −1.08151
\(215\) 11.8601 0.808852
\(216\) −2.32691 −0.158326
\(217\) −2.90456 −0.197174
\(218\) 8.57088 0.580493
\(219\) 19.0813 1.28939
\(220\) −9.53628 −0.642936
\(221\) −24.9408 −1.67770
\(222\) −16.0172 −1.07500
\(223\) −16.6847 −1.11729 −0.558644 0.829407i \(-0.688678\pi\)
−0.558644 + 0.829407i \(0.688678\pi\)
\(224\) −2.90456 −0.194069
\(225\) −3.00416 −0.200277
\(226\) −7.21645 −0.480031
\(227\) −0.220021 −0.0146033 −0.00730165 0.999973i \(-0.502324\pi\)
−0.00730165 + 0.999973i \(0.502324\pi\)
\(228\) −10.5709 −0.700072
\(229\) −21.3554 −1.41121 −0.705603 0.708607i \(-0.749324\pi\)
−0.705603 + 0.708607i \(0.749324\pi\)
\(230\) −11.0455 −0.728321
\(231\) 33.1310 2.17986
\(232\) 2.07949 0.136525
\(233\) −20.7877 −1.36185 −0.680925 0.732353i \(-0.738422\pi\)
−0.680925 + 0.732353i \(0.738422\pi\)
\(234\) −11.7702 −0.769444
\(235\) 18.6320 1.21542
\(236\) 2.76103 0.179728
\(237\) 5.10356 0.331512
\(238\) −12.0301 −0.779793
\(239\) −9.73993 −0.630024 −0.315012 0.949088i \(-0.602008\pi\)
−0.315012 + 0.949088i \(0.602008\pi\)
\(240\) 4.14223 0.267380
\(241\) −6.86444 −0.442177 −0.221089 0.975254i \(-0.570961\pi\)
−0.221089 + 0.975254i \(0.570961\pi\)
\(242\) 15.2603 0.980968
\(243\) −17.6006 −1.12908
\(244\) −12.0007 −0.768264
\(245\) 2.67314 0.170781
\(246\) 10.2874 0.655901
\(247\) 28.5974 1.81961
\(248\) 1.00000 0.0635001
\(249\) 7.12551 0.451561
\(250\) −12.1648 −0.769369
\(251\) 20.8765 1.31771 0.658857 0.752268i \(-0.271040\pi\)
0.658857 + 0.752268i \(0.271040\pi\)
\(252\) −5.67731 −0.357637
\(253\) 30.4164 1.91226
\(254\) 8.95218 0.561710
\(255\) 17.1563 1.07437
\(256\) 1.00000 0.0625000
\(257\) 9.98085 0.622589 0.311294 0.950314i \(-0.399238\pi\)
0.311294 + 0.950314i \(0.399238\pi\)
\(258\) 14.1861 0.883190
\(259\) 20.9007 1.29871
\(260\) −11.2060 −0.694968
\(261\) 4.06462 0.251594
\(262\) 2.78253 0.171905
\(263\) −1.17374 −0.0723761 −0.0361881 0.999345i \(-0.511522\pi\)
−0.0361881 + 0.999345i \(0.511522\pi\)
\(264\) −11.4066 −0.702026
\(265\) −22.6009 −1.38836
\(266\) 13.7938 0.845754
\(267\) 27.6550 1.69246
\(268\) 10.0168 0.611874
\(269\) 23.5745 1.43737 0.718683 0.695338i \(-0.244745\pi\)
0.718683 + 0.695338i \(0.244745\pi\)
\(270\) −4.33020 −0.263528
\(271\) 17.3609 1.05460 0.527299 0.849680i \(-0.323205\pi\)
0.527299 + 0.849680i \(0.323205\pi\)
\(272\) 4.14179 0.251133
\(273\) 38.9321 2.35628
\(274\) 16.2287 0.980411
\(275\) 7.87607 0.474945
\(276\) −13.2118 −0.795258
\(277\) 26.5456 1.59497 0.797486 0.603337i \(-0.206163\pi\)
0.797486 + 0.603337i \(0.206163\pi\)
\(278\) 20.8267 1.24910
\(279\) 1.95462 0.117020
\(280\) −5.40517 −0.323021
\(281\) −22.9362 −1.36826 −0.684131 0.729359i \(-0.739818\pi\)
−0.684131 + 0.729359i \(0.739818\pi\)
\(282\) 22.2862 1.32712
\(283\) −10.7949 −0.641689 −0.320845 0.947132i \(-0.603967\pi\)
−0.320845 + 0.947132i \(0.603967\pi\)
\(284\) −5.90655 −0.350489
\(285\) −19.6716 −1.16524
\(286\) 30.8583 1.82469
\(287\) −13.4240 −0.792391
\(288\) 1.95462 0.115177
\(289\) 0.154408 0.00908281
\(290\) 3.86978 0.227242
\(291\) −2.22590 −0.130484
\(292\) 8.57239 0.501661
\(293\) 25.4982 1.48962 0.744810 0.667276i \(-0.232540\pi\)
0.744810 + 0.667276i \(0.232540\pi\)
\(294\) 3.19741 0.186477
\(295\) 5.13807 0.299150
\(296\) −7.19584 −0.418250
\(297\) 11.9242 0.691911
\(298\) −8.72274 −0.505295
\(299\) 35.7421 2.06702
\(300\) −3.42109 −0.197517
\(301\) −18.5114 −1.06698
\(302\) −15.3256 −0.881891
\(303\) −19.0409 −1.09387
\(304\) −4.74903 −0.272376
\(305\) −22.3324 −1.27875
\(306\) 8.09563 0.462796
\(307\) −28.3402 −1.61746 −0.808730 0.588180i \(-0.799845\pi\)
−0.808730 + 0.588180i \(0.799845\pi\)
\(308\) 14.8843 0.848114
\(309\) −19.7600 −1.12411
\(310\) 1.86093 0.105694
\(311\) 2.90990 0.165005 0.0825027 0.996591i \(-0.473709\pi\)
0.0825027 + 0.996591i \(0.473709\pi\)
\(312\) −13.4038 −0.758840
\(313\) 26.4882 1.49720 0.748601 0.663021i \(-0.230726\pi\)
0.748601 + 0.663021i \(0.230726\pi\)
\(314\) 9.80769 0.553480
\(315\) −10.5651 −0.595274
\(316\) 2.29281 0.128981
\(317\) −16.6807 −0.936883 −0.468442 0.883495i \(-0.655184\pi\)
−0.468442 + 0.883495i \(0.655184\pi\)
\(318\) −27.0334 −1.51596
\(319\) −10.6563 −0.596639
\(320\) 1.86093 0.104029
\(321\) −35.2162 −1.96558
\(322\) 17.2400 0.960748
\(323\) −19.6695 −1.09444
\(324\) −11.0433 −0.613518
\(325\) 9.25512 0.513382
\(326\) −5.37431 −0.297656
\(327\) 19.0779 1.05501
\(328\) 4.62169 0.255190
\(329\) −29.0811 −1.60329
\(330\) −21.2268 −1.16850
\(331\) −30.8112 −1.69354 −0.846769 0.531961i \(-0.821455\pi\)
−0.846769 + 0.531961i \(0.821455\pi\)
\(332\) 3.20119 0.175688
\(333\) −14.0652 −0.770766
\(334\) −19.6741 −1.07652
\(335\) 18.6405 1.01844
\(336\) −6.46525 −0.352708
\(337\) 27.9078 1.52024 0.760119 0.649784i \(-0.225141\pi\)
0.760119 + 0.649784i \(0.225141\pi\)
\(338\) 23.2614 1.26525
\(339\) −16.0631 −0.872427
\(340\) 7.70757 0.418001
\(341\) −5.12448 −0.277506
\(342\) −9.28256 −0.501943
\(343\) 16.1596 0.872538
\(344\) 6.37322 0.343621
\(345\) −24.5862 −1.32368
\(346\) −9.66667 −0.519683
\(347\) −24.0042 −1.28861 −0.644306 0.764768i \(-0.722854\pi\)
−0.644306 + 0.764768i \(0.722854\pi\)
\(348\) 4.62874 0.248126
\(349\) 33.8405 1.81144 0.905721 0.423874i \(-0.139330\pi\)
0.905721 + 0.423874i \(0.139330\pi\)
\(350\) 4.46416 0.238619
\(351\) 14.0120 0.747907
\(352\) −5.12448 −0.273136
\(353\) 27.0985 1.44231 0.721154 0.692774i \(-0.243612\pi\)
0.721154 + 0.692774i \(0.243612\pi\)
\(354\) 6.14577 0.326644
\(355\) −10.9917 −0.583377
\(356\) 12.4242 0.658481
\(357\) −26.7777 −1.41723
\(358\) −5.71265 −0.301923
\(359\) 21.9690 1.15948 0.579738 0.814803i \(-0.303155\pi\)
0.579738 + 0.814803i \(0.303155\pi\)
\(360\) 3.63741 0.191708
\(361\) 3.55328 0.187015
\(362\) 13.4152 0.705086
\(363\) 33.9678 1.78285
\(364\) 17.4905 0.916752
\(365\) 15.9526 0.834997
\(366\) −26.7123 −1.39627
\(367\) 14.5511 0.759562 0.379781 0.925076i \(-0.375999\pi\)
0.379781 + 0.925076i \(0.375999\pi\)
\(368\) −5.93550 −0.309410
\(369\) 9.03365 0.470273
\(370\) −13.3909 −0.696162
\(371\) 35.2757 1.83142
\(372\) 2.22590 0.115407
\(373\) 30.6032 1.58458 0.792288 0.610148i \(-0.208890\pi\)
0.792288 + 0.610148i \(0.208890\pi\)
\(374\) −21.2245 −1.09749
\(375\) −27.0776 −1.39828
\(376\) 10.0122 0.516341
\(377\) −12.5222 −0.644925
\(378\) 6.75863 0.347627
\(379\) −35.0954 −1.80273 −0.901366 0.433059i \(-0.857434\pi\)
−0.901366 + 0.433059i \(0.857434\pi\)
\(380\) −8.83760 −0.453359
\(381\) 19.9266 1.02087
\(382\) −8.50762 −0.435288
\(383\) −32.8082 −1.67642 −0.838210 0.545347i \(-0.816398\pi\)
−0.838210 + 0.545347i \(0.816398\pi\)
\(384\) 2.22590 0.113590
\(385\) 27.6987 1.41166
\(386\) −12.6740 −0.645090
\(387\) 12.4572 0.633237
\(388\) −1.00000 −0.0507673
\(389\) −16.3780 −0.830399 −0.415200 0.909730i \(-0.636288\pi\)
−0.415200 + 0.909730i \(0.636288\pi\)
\(390\) −24.9435 −1.26306
\(391\) −24.5836 −1.24325
\(392\) 1.43646 0.0725521
\(393\) 6.19363 0.312427
\(394\) 19.0356 0.958999
\(395\) 4.26675 0.214684
\(396\) −10.0164 −0.503344
\(397\) −3.52150 −0.176739 −0.0883695 0.996088i \(-0.528166\pi\)
−0.0883695 + 0.996088i \(0.528166\pi\)
\(398\) 8.99317 0.450787
\(399\) 30.7037 1.53711
\(400\) −1.53695 −0.0768475
\(401\) 24.1396 1.20547 0.602736 0.797940i \(-0.294077\pi\)
0.602736 + 0.797940i \(0.294077\pi\)
\(402\) 22.2964 1.11204
\(403\) −6.02174 −0.299964
\(404\) −8.55427 −0.425591
\(405\) −20.5508 −1.02118
\(406\) −6.04001 −0.299760
\(407\) 36.8749 1.82782
\(408\) 9.21920 0.456418
\(409\) −7.46759 −0.369249 −0.184624 0.982809i \(-0.559107\pi\)
−0.184624 + 0.982809i \(0.559107\pi\)
\(410\) 8.60062 0.424754
\(411\) 36.1234 1.78184
\(412\) −8.87731 −0.437354
\(413\) −8.01957 −0.394617
\(414\) −11.6017 −0.570191
\(415\) 5.95717 0.292426
\(416\) −6.02174 −0.295240
\(417\) 46.3581 2.27017
\(418\) 24.3363 1.19033
\(419\) −6.60184 −0.322521 −0.161261 0.986912i \(-0.551556\pi\)
−0.161261 + 0.986912i \(0.551556\pi\)
\(420\) −12.0314 −0.587070
\(421\) −2.85657 −0.139221 −0.0696104 0.997574i \(-0.522176\pi\)
−0.0696104 + 0.997574i \(0.522176\pi\)
\(422\) −22.1304 −1.07729
\(423\) 19.5701 0.951531
\(424\) −12.1450 −0.589811
\(425\) −6.36572 −0.308783
\(426\) −13.1474 −0.636993
\(427\) 34.8567 1.68683
\(428\) −15.8211 −0.764742
\(429\) 68.6874 3.31626
\(430\) 11.8601 0.571945
\(431\) 20.1215 0.969216 0.484608 0.874731i \(-0.338962\pi\)
0.484608 + 0.874731i \(0.338962\pi\)
\(432\) −2.32691 −0.111953
\(433\) 0.802879 0.0385839 0.0192920 0.999814i \(-0.493859\pi\)
0.0192920 + 0.999814i \(0.493859\pi\)
\(434\) −2.90456 −0.139423
\(435\) 8.61374 0.412997
\(436\) 8.57088 0.410471
\(437\) 28.1879 1.34841
\(438\) 19.0813 0.911738
\(439\) 1.79535 0.0856874 0.0428437 0.999082i \(-0.486358\pi\)
0.0428437 + 0.999082i \(0.486358\pi\)
\(440\) −9.53628 −0.454624
\(441\) 2.80773 0.133702
\(442\) −24.9408 −1.18631
\(443\) −38.9530 −1.85071 −0.925357 0.379096i \(-0.876235\pi\)
−0.925357 + 0.379096i \(0.876235\pi\)
\(444\) −16.0172 −0.760143
\(445\) 23.1205 1.09602
\(446\) −16.6847 −0.790042
\(447\) −19.4159 −0.918342
\(448\) −2.90456 −0.137227
\(449\) −18.4435 −0.870404 −0.435202 0.900333i \(-0.643323\pi\)
−0.435202 + 0.900333i \(0.643323\pi\)
\(450\) −3.00416 −0.141617
\(451\) −23.6837 −1.11522
\(452\) −7.21645 −0.339433
\(453\) −34.1133 −1.60278
\(454\) −0.220021 −0.0103261
\(455\) 32.5486 1.52590
\(456\) −10.5709 −0.495026
\(457\) 27.3308 1.27848 0.639240 0.769007i \(-0.279249\pi\)
0.639240 + 0.769007i \(0.279249\pi\)
\(458\) −21.3554 −0.997874
\(459\) −9.63755 −0.449842
\(460\) −11.0455 −0.515001
\(461\) −18.5691 −0.864851 −0.432426 0.901670i \(-0.642342\pi\)
−0.432426 + 0.901670i \(0.642342\pi\)
\(462\) 33.1310 1.54140
\(463\) 11.0375 0.512956 0.256478 0.966550i \(-0.417438\pi\)
0.256478 + 0.966550i \(0.417438\pi\)
\(464\) 2.07949 0.0965380
\(465\) 4.14223 0.192091
\(466\) −20.7877 −0.962973
\(467\) −38.3447 −1.77438 −0.887191 0.461402i \(-0.847347\pi\)
−0.887191 + 0.461402i \(0.847347\pi\)
\(468\) −11.7702 −0.544079
\(469\) −29.0944 −1.34345
\(470\) 18.6320 0.859431
\(471\) 21.8309 1.00592
\(472\) 2.76103 0.127087
\(473\) −32.6594 −1.50168
\(474\) 5.10356 0.234414
\(475\) 7.29902 0.334902
\(476\) −12.0301 −0.551397
\(477\) −23.7388 −1.08692
\(478\) −9.73993 −0.445494
\(479\) −6.87617 −0.314180 −0.157090 0.987584i \(-0.550211\pi\)
−0.157090 + 0.987584i \(0.550211\pi\)
\(480\) 4.14223 0.189066
\(481\) 43.3315 1.97575
\(482\) −6.86444 −0.312667
\(483\) 38.3745 1.74610
\(484\) 15.2603 0.693649
\(485\) −1.86093 −0.0845003
\(486\) −17.6006 −0.798379
\(487\) 2.83930 0.128661 0.0643306 0.997929i \(-0.479509\pi\)
0.0643306 + 0.997929i \(0.479509\pi\)
\(488\) −12.0007 −0.543245
\(489\) −11.9627 −0.540971
\(490\) 2.67314 0.120760
\(491\) 24.7191 1.11556 0.557779 0.829989i \(-0.311654\pi\)
0.557779 + 0.829989i \(0.311654\pi\)
\(492\) 10.2874 0.463792
\(493\) 8.61282 0.387902
\(494\) 28.5974 1.28666
\(495\) −18.6398 −0.837798
\(496\) 1.00000 0.0449013
\(497\) 17.1559 0.769548
\(498\) 7.12551 0.319302
\(499\) −33.9999 −1.52204 −0.761022 0.648726i \(-0.775302\pi\)
−0.761022 + 0.648726i \(0.775302\pi\)
\(500\) −12.1648 −0.544026
\(501\) −43.7925 −1.95650
\(502\) 20.8765 0.931764
\(503\) −25.1300 −1.12049 −0.560245 0.828327i \(-0.689293\pi\)
−0.560245 + 0.828327i \(0.689293\pi\)
\(504\) −5.67731 −0.252888
\(505\) −15.9189 −0.708380
\(506\) 30.4164 1.35217
\(507\) 51.7775 2.29952
\(508\) 8.95218 0.397189
\(509\) 2.64582 0.117274 0.0586370 0.998279i \(-0.481325\pi\)
0.0586370 + 0.998279i \(0.481325\pi\)
\(510\) 17.1563 0.759692
\(511\) −24.8990 −1.10147
\(512\) 1.00000 0.0441942
\(513\) 11.0505 0.487893
\(514\) 9.98085 0.440237
\(515\) −16.5200 −0.727960
\(516\) 14.1861 0.624510
\(517\) −51.3074 −2.25650
\(518\) 20.9007 0.918326
\(519\) −21.5170 −0.944492
\(520\) −11.2060 −0.491417
\(521\) 8.99781 0.394201 0.197101 0.980383i \(-0.436847\pi\)
0.197101 + 0.980383i \(0.436847\pi\)
\(522\) 4.06462 0.177904
\(523\) 3.03895 0.132884 0.0664420 0.997790i \(-0.478835\pi\)
0.0664420 + 0.997790i \(0.478835\pi\)
\(524\) 2.78253 0.121555
\(525\) 9.93677 0.433676
\(526\) −1.17374 −0.0511776
\(527\) 4.14179 0.180419
\(528\) −11.4066 −0.496407
\(529\) 12.2302 0.531748
\(530\) −22.6009 −0.981719
\(531\) 5.39676 0.234200
\(532\) 13.7938 0.598039
\(533\) −27.8306 −1.20548
\(534\) 27.6550 1.19675
\(535\) −29.4419 −1.27289
\(536\) 10.0168 0.432660
\(537\) −12.7158 −0.548726
\(538\) 23.5745 1.01637
\(539\) −7.36110 −0.317065
\(540\) −4.33020 −0.186342
\(541\) −34.4093 −1.47937 −0.739686 0.672953i \(-0.765026\pi\)
−0.739686 + 0.672953i \(0.765026\pi\)
\(542\) 17.3609 0.745713
\(543\) 29.8608 1.28145
\(544\) 4.14179 0.177578
\(545\) 15.9498 0.683214
\(546\) 38.9321 1.66614
\(547\) −33.5442 −1.43425 −0.717124 0.696946i \(-0.754542\pi\)
−0.717124 + 0.696946i \(0.754542\pi\)
\(548\) 16.2287 0.693255
\(549\) −23.4568 −1.00111
\(550\) 7.87607 0.335837
\(551\) −9.87557 −0.420713
\(552\) −13.2118 −0.562333
\(553\) −6.65960 −0.283195
\(554\) 26.5456 1.12782
\(555\) −29.8069 −1.26523
\(556\) 20.8267 0.883249
\(557\) −1.24871 −0.0529096 −0.0264548 0.999650i \(-0.508422\pi\)
−0.0264548 + 0.999650i \(0.508422\pi\)
\(558\) 1.95462 0.0827457
\(559\) −38.3779 −1.62321
\(560\) −5.40517 −0.228410
\(561\) −47.2436 −1.99463
\(562\) −22.9362 −0.967507
\(563\) 25.2000 1.06205 0.531027 0.847355i \(-0.321806\pi\)
0.531027 + 0.847355i \(0.321806\pi\)
\(564\) 22.2862 0.938418
\(565\) −13.4293 −0.564974
\(566\) −10.7949 −0.453743
\(567\) 32.0760 1.34706
\(568\) −5.90655 −0.247833
\(569\) −27.6058 −1.15729 −0.578647 0.815578i \(-0.696419\pi\)
−0.578647 + 0.815578i \(0.696419\pi\)
\(570\) −19.6716 −0.823952
\(571\) 6.72831 0.281571 0.140785 0.990040i \(-0.455037\pi\)
0.140785 + 0.990040i \(0.455037\pi\)
\(572\) 30.8583 1.29025
\(573\) −18.9371 −0.791108
\(574\) −13.4240 −0.560305
\(575\) 9.12257 0.380438
\(576\) 1.95462 0.0814426
\(577\) −13.7163 −0.571016 −0.285508 0.958376i \(-0.592162\pi\)
−0.285508 + 0.958376i \(0.592162\pi\)
\(578\) 0.154408 0.00642251
\(579\) −28.2110 −1.17241
\(580\) 3.86978 0.160684
\(581\) −9.29803 −0.385747
\(582\) −2.22590 −0.0922664
\(583\) 62.2366 2.57758
\(584\) 8.57239 0.354728
\(585\) −21.9035 −0.905600
\(586\) 25.4982 1.05332
\(587\) −36.2542 −1.49637 −0.748186 0.663489i \(-0.769075\pi\)
−0.748186 + 0.663489i \(0.769075\pi\)
\(588\) 3.19741 0.131859
\(589\) −4.74903 −0.195680
\(590\) 5.13807 0.211531
\(591\) 42.3713 1.74292
\(592\) −7.19584 −0.295747
\(593\) 23.2040 0.952876 0.476438 0.879208i \(-0.341928\pi\)
0.476438 + 0.879208i \(0.341928\pi\)
\(594\) 11.9242 0.489255
\(595\) −22.3871 −0.917781
\(596\) −8.72274 −0.357297
\(597\) 20.0179 0.819277
\(598\) 35.7421 1.46160
\(599\) −24.4955 −1.00086 −0.500429 0.865777i \(-0.666824\pi\)
−0.500429 + 0.865777i \(0.666824\pi\)
\(600\) −3.42109 −0.139666
\(601\) 47.4663 1.93619 0.968095 0.250584i \(-0.0806227\pi\)
0.968095 + 0.250584i \(0.0806227\pi\)
\(602\) −18.5114 −0.754468
\(603\) 19.5791 0.797321
\(604\) −15.3256 −0.623591
\(605\) 28.3983 1.15455
\(606\) −19.0409 −0.773485
\(607\) −29.6031 −1.20155 −0.600775 0.799418i \(-0.705141\pi\)
−0.600775 + 0.799418i \(0.705141\pi\)
\(608\) −4.74903 −0.192599
\(609\) −13.4444 −0.544796
\(610\) −22.3324 −0.904212
\(611\) −60.2911 −2.43912
\(612\) 8.09563 0.327246
\(613\) −13.2539 −0.535322 −0.267661 0.963513i \(-0.586251\pi\)
−0.267661 + 0.963513i \(0.586251\pi\)
\(614\) −28.3402 −1.14372
\(615\) 19.1441 0.771965
\(616\) 14.8843 0.599707
\(617\) 23.0324 0.927251 0.463625 0.886031i \(-0.346548\pi\)
0.463625 + 0.886031i \(0.346548\pi\)
\(618\) −19.7600 −0.794863
\(619\) 40.8776 1.64301 0.821505 0.570201i \(-0.193135\pi\)
0.821505 + 0.570201i \(0.193135\pi\)
\(620\) 1.86093 0.0747366
\(621\) 13.8114 0.554231
\(622\) 2.90990 0.116676
\(623\) −36.0868 −1.44579
\(624\) −13.4038 −0.536581
\(625\) −14.9530 −0.598121
\(626\) 26.4882 1.05868
\(627\) 54.1701 2.16335
\(628\) 9.80769 0.391370
\(629\) −29.8037 −1.18835
\(630\) −10.5651 −0.420922
\(631\) −12.8023 −0.509652 −0.254826 0.966987i \(-0.582018\pi\)
−0.254826 + 0.966987i \(0.582018\pi\)
\(632\) 2.29281 0.0912031
\(633\) −49.2600 −1.95791
\(634\) −16.6807 −0.662476
\(635\) 16.6594 0.661106
\(636\) −27.0334 −1.07195
\(637\) −8.64998 −0.342725
\(638\) −10.6563 −0.421887
\(639\) −11.5451 −0.456716
\(640\) 1.86093 0.0735596
\(641\) 24.7532 0.977694 0.488847 0.872369i \(-0.337418\pi\)
0.488847 + 0.872369i \(0.337418\pi\)
\(642\) −35.2162 −1.38987
\(643\) 9.09483 0.358665 0.179332 0.983789i \(-0.442606\pi\)
0.179332 + 0.983789i \(0.442606\pi\)
\(644\) 17.2400 0.679352
\(645\) 26.3994 1.03947
\(646\) −19.6695 −0.773885
\(647\) −37.5760 −1.47726 −0.738632 0.674109i \(-0.764528\pi\)
−0.738632 + 0.674109i \(0.764528\pi\)
\(648\) −11.0433 −0.433823
\(649\) −14.1488 −0.555390
\(650\) 9.25512 0.363016
\(651\) −6.46525 −0.253393
\(652\) −5.37431 −0.210474
\(653\) −20.2981 −0.794328 −0.397164 0.917748i \(-0.630006\pi\)
−0.397164 + 0.917748i \(0.630006\pi\)
\(654\) 19.0779 0.746005
\(655\) 5.17809 0.202325
\(656\) 4.62169 0.180447
\(657\) 16.7558 0.653705
\(658\) −29.0811 −1.13370
\(659\) 13.6465 0.531593 0.265796 0.964029i \(-0.414365\pi\)
0.265796 + 0.964029i \(0.414365\pi\)
\(660\) −21.2268 −0.826252
\(661\) −31.3995 −1.22130 −0.610650 0.791900i \(-0.709092\pi\)
−0.610650 + 0.791900i \(0.709092\pi\)
\(662\) −30.8112 −1.19751
\(663\) −55.5157 −2.15605
\(664\) 3.20119 0.124230
\(665\) 25.6693 0.995413
\(666\) −14.0652 −0.545014
\(667\) −12.3428 −0.477916
\(668\) −19.6741 −0.761213
\(669\) −37.1384 −1.43585
\(670\) 18.6405 0.720147
\(671\) 61.4972 2.37407
\(672\) −6.46525 −0.249402
\(673\) 26.8564 1.03524 0.517620 0.855611i \(-0.326818\pi\)
0.517620 + 0.855611i \(0.326818\pi\)
\(674\) 27.9078 1.07497
\(675\) 3.57634 0.137653
\(676\) 23.2614 0.894670
\(677\) 22.0246 0.846475 0.423238 0.906019i \(-0.360894\pi\)
0.423238 + 0.906019i \(0.360894\pi\)
\(678\) −16.0631 −0.616899
\(679\) 2.90456 0.111467
\(680\) 7.70757 0.295572
\(681\) −0.489744 −0.0187670
\(682\) −5.12448 −0.196226
\(683\) −1.60535 −0.0614270 −0.0307135 0.999528i \(-0.509778\pi\)
−0.0307135 + 0.999528i \(0.509778\pi\)
\(684\) −9.28256 −0.354927
\(685\) 30.2004 1.15390
\(686\) 16.1596 0.616978
\(687\) −47.5350 −1.81357
\(688\) 6.37322 0.242977
\(689\) 73.1338 2.78618
\(690\) −24.5862 −0.935982
\(691\) −49.2775 −1.87460 −0.937302 0.348519i \(-0.886685\pi\)
−0.937302 + 0.348519i \(0.886685\pi\)
\(692\) −9.66667 −0.367472
\(693\) 29.0933 1.10516
\(694\) −24.0042 −0.911187
\(695\) 38.7570 1.47014
\(696\) 4.62874 0.175452
\(697\) 19.1420 0.725057
\(698\) 33.8405 1.28088
\(699\) −46.2714 −1.75014
\(700\) 4.46416 0.168729
\(701\) 1.41448 0.0534243 0.0267122 0.999643i \(-0.491496\pi\)
0.0267122 + 0.999643i \(0.491496\pi\)
\(702\) 14.0120 0.528850
\(703\) 34.1733 1.28887
\(704\) −5.12448 −0.193136
\(705\) 41.4730 1.56196
\(706\) 27.0985 1.01987
\(707\) 24.8464 0.934444
\(708\) 6.14577 0.230972
\(709\) −35.2843 −1.32513 −0.662565 0.749004i \(-0.730532\pi\)
−0.662565 + 0.749004i \(0.730532\pi\)
\(710\) −10.9917 −0.412510
\(711\) 4.48158 0.168072
\(712\) 12.4242 0.465616
\(713\) −5.93550 −0.222286
\(714\) −26.7777 −1.00213
\(715\) 57.4250 2.14757
\(716\) −5.71265 −0.213492
\(717\) −21.6801 −0.809658
\(718\) 21.9690 0.819874
\(719\) 16.1506 0.602317 0.301159 0.953574i \(-0.402627\pi\)
0.301159 + 0.953574i \(0.402627\pi\)
\(720\) 3.63741 0.135558
\(721\) 25.7847 0.960271
\(722\) 3.55328 0.132239
\(723\) −15.2795 −0.568252
\(724\) 13.4152 0.498571
\(725\) −3.19608 −0.118699
\(726\) 33.9678 1.26066
\(727\) −12.0890 −0.448355 −0.224178 0.974548i \(-0.571970\pi\)
−0.224178 + 0.974548i \(0.571970\pi\)
\(728\) 17.4905 0.648241
\(729\) −6.04714 −0.223968
\(730\) 15.9526 0.590432
\(731\) 26.3965 0.976311
\(732\) −26.7123 −0.987314
\(733\) −26.2618 −0.970002 −0.485001 0.874514i \(-0.661181\pi\)
−0.485001 + 0.874514i \(0.661181\pi\)
\(734\) 14.5511 0.537092
\(735\) 5.95015 0.219474
\(736\) −5.93550 −0.218786
\(737\) −51.3309 −1.89080
\(738\) 9.03365 0.332533
\(739\) −45.7405 −1.68259 −0.841295 0.540576i \(-0.818206\pi\)
−0.841295 + 0.540576i \(0.818206\pi\)
\(740\) −13.3909 −0.492261
\(741\) 63.6550 2.33842
\(742\) 35.2757 1.29501
\(743\) −27.6677 −1.01503 −0.507514 0.861643i \(-0.669436\pi\)
−0.507514 + 0.861643i \(0.669436\pi\)
\(744\) 2.22590 0.0816054
\(745\) −16.2324 −0.594709
\(746\) 30.6032 1.12046
\(747\) 6.25711 0.228936
\(748\) −21.2245 −0.776045
\(749\) 45.9533 1.67910
\(750\) −27.0776 −0.988733
\(751\) 18.9725 0.692317 0.346159 0.938176i \(-0.387486\pi\)
0.346159 + 0.938176i \(0.387486\pi\)
\(752\) 10.0122 0.365108
\(753\) 46.4690 1.69342
\(754\) −12.5222 −0.456031
\(755\) −28.5199 −1.03795
\(756\) 6.75863 0.245809
\(757\) −11.2387 −0.408479 −0.204239 0.978921i \(-0.565472\pi\)
−0.204239 + 0.978921i \(0.565472\pi\)
\(758\) −35.0954 −1.27472
\(759\) 67.7037 2.45749
\(760\) −8.83760 −0.320573
\(761\) −28.5372 −1.03447 −0.517236 0.855843i \(-0.673039\pi\)
−0.517236 + 0.855843i \(0.673039\pi\)
\(762\) 19.9266 0.721866
\(763\) −24.8946 −0.901246
\(764\) −8.50762 −0.307795
\(765\) 15.0654 0.544690
\(766\) −32.8082 −1.18541
\(767\) −16.6262 −0.600337
\(768\) 2.22590 0.0803202
\(769\) 40.3108 1.45364 0.726822 0.686826i \(-0.240996\pi\)
0.726822 + 0.686826i \(0.240996\pi\)
\(770\) 27.6987 0.998191
\(771\) 22.2164 0.800103
\(772\) −12.6740 −0.456147
\(773\) 19.9414 0.717241 0.358621 0.933483i \(-0.383247\pi\)
0.358621 + 0.933483i \(0.383247\pi\)
\(774\) 12.4572 0.447766
\(775\) −1.53695 −0.0552089
\(776\) −1.00000 −0.0358979
\(777\) 46.5229 1.66900
\(778\) −16.3780 −0.587181
\(779\) −21.9485 −0.786388
\(780\) −24.9435 −0.893120
\(781\) 30.2680 1.08307
\(782\) −24.5836 −0.879108
\(783\) −4.83878 −0.172924
\(784\) 1.43646 0.0513021
\(785\) 18.2514 0.651420
\(786\) 6.19363 0.220919
\(787\) 24.0739 0.858141 0.429070 0.903271i \(-0.358841\pi\)
0.429070 + 0.903271i \(0.358841\pi\)
\(788\) 19.0356 0.678115
\(789\) −2.61263 −0.0930122
\(790\) 4.26675 0.151804
\(791\) 20.9606 0.745273
\(792\) −10.0164 −0.355918
\(793\) 72.2650 2.56620
\(794\) −3.52150 −0.124973
\(795\) −50.3073 −1.78421
\(796\) 8.99317 0.318754
\(797\) −49.5991 −1.75689 −0.878445 0.477843i \(-0.841419\pi\)
−0.878445 + 0.477843i \(0.841419\pi\)
\(798\) 30.7037 1.08690
\(799\) 41.4685 1.46705
\(800\) −1.53695 −0.0543394
\(801\) 24.2846 0.858054
\(802\) 24.1396 0.852398
\(803\) −43.9290 −1.55022
\(804\) 22.2964 0.786333
\(805\) 32.0824 1.13076
\(806\) −6.02174 −0.212107
\(807\) 52.4745 1.84719
\(808\) −8.55427 −0.300938
\(809\) −10.8271 −0.380662 −0.190331 0.981720i \(-0.560956\pi\)
−0.190331 + 0.981720i \(0.560956\pi\)
\(810\) −20.5508 −0.722082
\(811\) 3.88132 0.136292 0.0681458 0.997675i \(-0.478292\pi\)
0.0681458 + 0.997675i \(0.478292\pi\)
\(812\) −6.04001 −0.211963
\(813\) 38.6435 1.35529
\(814\) 36.8749 1.29247
\(815\) −10.0012 −0.350327
\(816\) 9.21920 0.322736
\(817\) −30.2666 −1.05889
\(818\) −7.46759 −0.261098
\(819\) 34.1873 1.19460
\(820\) 8.60062 0.300347
\(821\) 9.25937 0.323154 0.161577 0.986860i \(-0.448342\pi\)
0.161577 + 0.986860i \(0.448342\pi\)
\(822\) 36.1234 1.25995
\(823\) −47.6119 −1.65965 −0.829823 0.558027i \(-0.811559\pi\)
−0.829823 + 0.558027i \(0.811559\pi\)
\(824\) −8.87731 −0.309256
\(825\) 17.5313 0.610362
\(826\) −8.01957 −0.279036
\(827\) −1.27464 −0.0443237 −0.0221618 0.999754i \(-0.507055\pi\)
−0.0221618 + 0.999754i \(0.507055\pi\)
\(828\) −11.6017 −0.403186
\(829\) 44.9630 1.56163 0.780815 0.624762i \(-0.214804\pi\)
0.780815 + 0.624762i \(0.214804\pi\)
\(830\) 5.95717 0.206777
\(831\) 59.0879 2.04973
\(832\) −6.02174 −0.208766
\(833\) 5.94951 0.206138
\(834\) 46.3581 1.60525
\(835\) −36.6120 −1.26701
\(836\) 24.3363 0.841689
\(837\) −2.32691 −0.0804296
\(838\) −6.60184 −0.228057
\(839\) −15.1908 −0.524445 −0.262222 0.965007i \(-0.584455\pi\)
−0.262222 + 0.965007i \(0.584455\pi\)
\(840\) −12.0314 −0.415121
\(841\) −24.6757 −0.850887
\(842\) −2.85657 −0.0984440
\(843\) −51.0537 −1.75838
\(844\) −22.1304 −0.761760
\(845\) 43.2878 1.48915
\(846\) 19.5701 0.672834
\(847\) −44.3244 −1.52300
\(848\) −12.1450 −0.417060
\(849\) −24.0283 −0.824649
\(850\) −6.36572 −0.218342
\(851\) 42.7110 1.46411
\(852\) −13.1474 −0.450422
\(853\) 23.5345 0.805805 0.402903 0.915243i \(-0.368001\pi\)
0.402903 + 0.915243i \(0.368001\pi\)
\(854\) 34.8567 1.19277
\(855\) −17.2742 −0.590764
\(856\) −15.8211 −0.540754
\(857\) 12.8332 0.438374 0.219187 0.975683i \(-0.429660\pi\)
0.219187 + 0.975683i \(0.429660\pi\)
\(858\) 68.6874 2.34495
\(859\) −30.1974 −1.03032 −0.515161 0.857094i \(-0.672268\pi\)
−0.515161 + 0.857094i \(0.672268\pi\)
\(860\) 11.8601 0.404426
\(861\) −29.8804 −1.01832
\(862\) 20.1215 0.685339
\(863\) 7.29981 0.248488 0.124244 0.992252i \(-0.460349\pi\)
0.124244 + 0.992252i \(0.460349\pi\)
\(864\) −2.32691 −0.0791629
\(865\) −17.9890 −0.611643
\(866\) 0.802879 0.0272829
\(867\) 0.343696 0.0116725
\(868\) −2.90456 −0.0985871
\(869\) −11.7495 −0.398573
\(870\) 8.61374 0.292033
\(871\) −60.3186 −2.04382
\(872\) 8.57088 0.290247
\(873\) −1.95462 −0.0661539
\(874\) 28.1879 0.953469
\(875\) 35.3333 1.19448
\(876\) 19.0813 0.644696
\(877\) 20.9776 0.708364 0.354182 0.935177i \(-0.384759\pi\)
0.354182 + 0.935177i \(0.384759\pi\)
\(878\) 1.79535 0.0605901
\(879\) 56.7564 1.91435
\(880\) −9.53628 −0.321468
\(881\) −41.9668 −1.41390 −0.706949 0.707264i \(-0.749929\pi\)
−0.706949 + 0.707264i \(0.749929\pi\)
\(882\) 2.80773 0.0945413
\(883\) −6.63476 −0.223278 −0.111639 0.993749i \(-0.535610\pi\)
−0.111639 + 0.993749i \(0.535610\pi\)
\(884\) −24.9408 −0.838850
\(885\) 11.4368 0.384445
\(886\) −38.9530 −1.30865
\(887\) −49.0448 −1.64676 −0.823381 0.567489i \(-0.807915\pi\)
−0.823381 + 0.567489i \(0.807915\pi\)
\(888\) −16.0172 −0.537502
\(889\) −26.0021 −0.872083
\(890\) 23.1205 0.775001
\(891\) 56.5912 1.89588
\(892\) −16.6847 −0.558644
\(893\) −47.5484 −1.59115
\(894\) −19.4159 −0.649366
\(895\) −10.6308 −0.355349
\(896\) −2.90456 −0.0970345
\(897\) 79.5582 2.65637
\(898\) −18.4435 −0.615469
\(899\) 2.07949 0.0693550
\(900\) −3.00416 −0.100139
\(901\) −50.3018 −1.67580
\(902\) −23.6837 −0.788582
\(903\) −41.2045 −1.37120
\(904\) −7.21645 −0.240016
\(905\) 24.9647 0.829854
\(906\) −34.1133 −1.13334
\(907\) 53.7675 1.78532 0.892661 0.450729i \(-0.148836\pi\)
0.892661 + 0.450729i \(0.148836\pi\)
\(908\) −0.220021 −0.00730165
\(909\) −16.7204 −0.554579
\(910\) 32.5486 1.07897
\(911\) 40.1498 1.33022 0.665112 0.746744i \(-0.268384\pi\)
0.665112 + 0.746744i \(0.268384\pi\)
\(912\) −10.5709 −0.350036
\(913\) −16.4044 −0.542907
\(914\) 27.3308 0.904022
\(915\) −49.7096 −1.64335
\(916\) −21.3554 −0.705603
\(917\) −8.08202 −0.266892
\(918\) −9.63755 −0.318087
\(919\) 24.3967 0.804772 0.402386 0.915470i \(-0.368181\pi\)
0.402386 + 0.915470i \(0.368181\pi\)
\(920\) −11.0455 −0.364161
\(921\) −63.0824 −2.07863
\(922\) −18.5691 −0.611542
\(923\) 35.5677 1.17073
\(924\) 33.1310 1.08993
\(925\) 11.0597 0.363639
\(926\) 11.0375 0.362715
\(927\) −17.3518 −0.569907
\(928\) 2.07949 0.0682627
\(929\) −24.3967 −0.800431 −0.400216 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(930\) 4.14223 0.135829
\(931\) −6.82178 −0.223575
\(932\) −20.7877 −0.680925
\(933\) 6.47714 0.212052
\(934\) −38.3447 −1.25468
\(935\) −39.4973 −1.29170
\(936\) −11.7702 −0.384722
\(937\) −24.2097 −0.790896 −0.395448 0.918488i \(-0.629411\pi\)
−0.395448 + 0.918488i \(0.629411\pi\)
\(938\) −29.0944 −0.949965
\(939\) 58.9600 1.92409
\(940\) 18.6320 0.607710
\(941\) −41.7715 −1.36171 −0.680856 0.732418i \(-0.738392\pi\)
−0.680856 + 0.732418i \(0.738392\pi\)
\(942\) 21.8309 0.711290
\(943\) −27.4320 −0.893310
\(944\) 2.76103 0.0898638
\(945\) 12.5773 0.409140
\(946\) −32.6594 −1.06185
\(947\) 4.29317 0.139509 0.0697547 0.997564i \(-0.477778\pi\)
0.0697547 + 0.997564i \(0.477778\pi\)
\(948\) 5.10356 0.165756
\(949\) −51.6207 −1.67568
\(950\) 7.29902 0.236812
\(951\) −37.1296 −1.20401
\(952\) −12.0301 −0.389897
\(953\) 25.8948 0.838816 0.419408 0.907798i \(-0.362238\pi\)
0.419408 + 0.907798i \(0.362238\pi\)
\(954\) −23.7388 −0.768572
\(955\) −15.8321 −0.512313
\(956\) −9.73993 −0.315012
\(957\) −23.7199 −0.766754
\(958\) −6.87617 −0.222159
\(959\) −47.1372 −1.52214
\(960\) 4.14223 0.133690
\(961\) 1.00000 0.0322581
\(962\) 43.3315 1.39706
\(963\) −30.9243 −0.996521
\(964\) −6.86444 −0.221089
\(965\) −23.5854 −0.759241
\(966\) 38.3745 1.23468
\(967\) −27.6241 −0.888332 −0.444166 0.895945i \(-0.646500\pi\)
−0.444166 + 0.895945i \(0.646500\pi\)
\(968\) 15.2603 0.490484
\(969\) −43.7822 −1.40649
\(970\) −1.86093 −0.0597508
\(971\) 38.5280 1.23642 0.618212 0.786012i \(-0.287858\pi\)
0.618212 + 0.786012i \(0.287858\pi\)
\(972\) −17.6006 −0.564539
\(973\) −60.4923 −1.93930
\(974\) 2.83930 0.0909772
\(975\) 20.6010 0.659759
\(976\) −12.0007 −0.384132
\(977\) 9.86730 0.315683 0.157841 0.987464i \(-0.449547\pi\)
0.157841 + 0.987464i \(0.449547\pi\)
\(978\) −11.9627 −0.382524
\(979\) −63.6675 −2.03482
\(980\) 2.67314 0.0853905
\(981\) 16.7528 0.534877
\(982\) 24.7191 0.788819
\(983\) 57.9841 1.84941 0.924703 0.380689i \(-0.124313\pi\)
0.924703 + 0.380689i \(0.124313\pi\)
\(984\) 10.2874 0.327950
\(985\) 35.4238 1.12870
\(986\) 8.61282 0.274288
\(987\) −64.7315 −2.06043
\(988\) 28.5974 0.909806
\(989\) −37.8283 −1.20287
\(990\) −18.6398 −0.592412
\(991\) −8.64927 −0.274753 −0.137377 0.990519i \(-0.543867\pi\)
−0.137377 + 0.990519i \(0.543867\pi\)
\(992\) 1.00000 0.0317500
\(993\) −68.5826 −2.17640
\(994\) 17.1559 0.544153
\(995\) 16.7356 0.530555
\(996\) 7.12551 0.225781
\(997\) −8.74945 −0.277098 −0.138549 0.990356i \(-0.544244\pi\)
−0.138549 + 0.990356i \(0.544244\pi\)
\(998\) −33.9999 −1.07625
\(999\) 16.7441 0.529758
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 6014.2.a.e.1.20 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6014.2.a.e.1.20 21 1.1 even 1 trivial