Properties

Label 4018.2.a.bc.1.1
Level $4018$
Weight $2$
Character 4018.1
Self dual yes
Analytic conductor $32.084$
Analytic rank $0$
Dimension $2$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, 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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.82843 q^{5} -0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.82843 q^{5} -0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} -1.82843 q^{10} -6.24264 q^{11} -0.414214 q^{12} +0.585786 q^{13} +0.757359 q^{15} +1.00000 q^{16} +3.82843 q^{17} -2.82843 q^{18} +2.82843 q^{19} -1.82843 q^{20} -6.24264 q^{22} +5.65685 q^{23} -0.414214 q^{24} -1.65685 q^{25} +0.585786 q^{26} +2.41421 q^{27} -5.00000 q^{29} +0.757359 q^{30} -5.58579 q^{31} +1.00000 q^{32} +2.58579 q^{33} +3.82843 q^{34} -2.82843 q^{36} +6.58579 q^{37} +2.82843 q^{38} -0.242641 q^{39} -1.82843 q^{40} -1.00000 q^{41} +1.24264 q^{43} -6.24264 q^{44} +5.17157 q^{45} +5.65685 q^{46} +10.2426 q^{47} -0.414214 q^{48} -1.65685 q^{50} -1.58579 q^{51} +0.585786 q^{52} -7.48528 q^{53} +2.41421 q^{54} +11.4142 q^{55} -1.17157 q^{57} -5.00000 q^{58} +10.4853 q^{59} +0.757359 q^{60} +11.4853 q^{61} -5.58579 q^{62} +1.00000 q^{64} -1.07107 q^{65} +2.58579 q^{66} -7.17157 q^{67} +3.82843 q^{68} -2.34315 q^{69} +8.89949 q^{71} -2.82843 q^{72} +6.58579 q^{74} +0.686292 q^{75} +2.82843 q^{76} -0.242641 q^{78} -9.24264 q^{79} -1.82843 q^{80} +7.48528 q^{81} -1.00000 q^{82} +14.2426 q^{83} -7.00000 q^{85} +1.24264 q^{86} +2.07107 q^{87} -6.24264 q^{88} +3.82843 q^{89} +5.17157 q^{90} +5.65685 q^{92} +2.31371 q^{93} +10.2426 q^{94} -5.17157 q^{95} -0.414214 q^{96} +0.171573 q^{97} +17.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 10 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{20} - 4 q^{22} + 2 q^{24} + 8 q^{25} + 4 q^{26} + 2 q^{27} - 10 q^{29} + 10 q^{30} - 14 q^{31} + 2 q^{32} + 8 q^{33} + 2 q^{34} + 16 q^{37} + 8 q^{39} + 2 q^{40} - 2 q^{41} - 6 q^{43} - 4 q^{44} + 16 q^{45} + 12 q^{47} + 2 q^{48} + 8 q^{50} - 6 q^{51} + 4 q^{52} + 2 q^{53} + 2 q^{54} + 20 q^{55} - 8 q^{57} - 10 q^{58} + 4 q^{59} + 10 q^{60} + 6 q^{61} - 14 q^{62} + 2 q^{64} + 12 q^{65} + 8 q^{66} - 20 q^{67} + 2 q^{68} - 16 q^{69} - 2 q^{71} + 16 q^{74} + 24 q^{75} + 8 q^{78} - 10 q^{79} + 2 q^{80} - 2 q^{81} - 2 q^{82} + 20 q^{83} - 14 q^{85} - 6 q^{86} - 10 q^{87} - 4 q^{88} + 2 q^{89} + 16 q^{90} - 18 q^{93} + 12 q^{94} - 16 q^{95} + 2 q^{96} + 6 q^{97} + 24 q^{99}+O(q^{100}) \) 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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.82843 −0.817697 −0.408849 0.912602i \(-0.634070\pi\)
−0.408849 + 0.912602i \(0.634070\pi\)
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.82843 −0.942809
\(10\) −1.82843 −0.578199
\(11\) −6.24264 −1.88223 −0.941113 0.338091i \(-0.890219\pi\)
−0.941113 + 0.338091i \(0.890219\pi\)
\(12\) −0.414214 −0.119573
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) 0.757359 0.195549
\(16\) 1.00000 0.250000
\(17\) 3.82843 0.928530 0.464265 0.885696i \(-0.346319\pi\)
0.464265 + 0.885696i \(0.346319\pi\)
\(18\) −2.82843 −0.666667
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) −1.82843 −0.408849
\(21\) 0 0
\(22\) −6.24264 −1.33094
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) −0.414214 −0.0845510
\(25\) −1.65685 −0.331371
\(26\) 0.585786 0.114882
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0.757359 0.138274
\(31\) −5.58579 −1.00324 −0.501618 0.865089i \(-0.667262\pi\)
−0.501618 + 0.865089i \(0.667262\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.58579 0.450128
\(34\) 3.82843 0.656570
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) 6.58579 1.08270 0.541348 0.840798i \(-0.317914\pi\)
0.541348 + 0.840798i \(0.317914\pi\)
\(38\) 2.82843 0.458831
\(39\) −0.242641 −0.0388536
\(40\) −1.82843 −0.289100
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 1.24264 0.189501 0.0947505 0.995501i \(-0.469795\pi\)
0.0947505 + 0.995501i \(0.469795\pi\)
\(44\) −6.24264 −0.941113
\(45\) 5.17157 0.770933
\(46\) 5.65685 0.834058
\(47\) 10.2426 1.49404 0.747021 0.664800i \(-0.231483\pi\)
0.747021 + 0.664800i \(0.231483\pi\)
\(48\) −0.414214 −0.0597866
\(49\) 0 0
\(50\) −1.65685 −0.234315
\(51\) −1.58579 −0.222055
\(52\) 0.585786 0.0812340
\(53\) −7.48528 −1.02818 −0.514091 0.857736i \(-0.671871\pi\)
−0.514091 + 0.857736i \(0.671871\pi\)
\(54\) 2.41421 0.328533
\(55\) 11.4142 1.53909
\(56\) 0 0
\(57\) −1.17157 −0.155179
\(58\) −5.00000 −0.656532
\(59\) 10.4853 1.36507 0.682534 0.730854i \(-0.260878\pi\)
0.682534 + 0.730854i \(0.260878\pi\)
\(60\) 0.757359 0.0977747
\(61\) 11.4853 1.47054 0.735270 0.677775i \(-0.237055\pi\)
0.735270 + 0.677775i \(0.237055\pi\)
\(62\) −5.58579 −0.709396
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.07107 −0.132850
\(66\) 2.58579 0.318288
\(67\) −7.17157 −0.876147 −0.438074 0.898939i \(-0.644339\pi\)
−0.438074 + 0.898939i \(0.644339\pi\)
\(68\) 3.82843 0.464265
\(69\) −2.34315 −0.282082
\(70\) 0 0
\(71\) 8.89949 1.05618 0.528088 0.849190i \(-0.322909\pi\)
0.528088 + 0.849190i \(0.322909\pi\)
\(72\) −2.82843 −0.333333
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 6.58579 0.765582
\(75\) 0.686292 0.0792461
\(76\) 2.82843 0.324443
\(77\) 0 0
\(78\) −0.242641 −0.0274736
\(79\) −9.24264 −1.03988 −0.519939 0.854203i \(-0.674045\pi\)
−0.519939 + 0.854203i \(0.674045\pi\)
\(80\) −1.82843 −0.204424
\(81\) 7.48528 0.831698
\(82\) −1.00000 −0.110432
\(83\) 14.2426 1.56333 0.781666 0.623697i \(-0.214370\pi\)
0.781666 + 0.623697i \(0.214370\pi\)
\(84\) 0 0
\(85\) −7.00000 −0.759257
\(86\) 1.24264 0.133997
\(87\) 2.07107 0.222042
\(88\) −6.24264 −0.665468
\(89\) 3.82843 0.405812 0.202906 0.979198i \(-0.434961\pi\)
0.202906 + 0.979198i \(0.434961\pi\)
\(90\) 5.17157 0.545132
\(91\) 0 0
\(92\) 5.65685 0.589768
\(93\) 2.31371 0.239920
\(94\) 10.2426 1.05645
\(95\) −5.17157 −0.530592
\(96\) −0.414214 −0.0422755
\(97\) 0.171573 0.0174206 0.00871029 0.999962i \(-0.497227\pi\)
0.00871029 + 0.999962i \(0.497227\pi\)
\(98\) 0 0
\(99\) 17.6569 1.77458
\(100\) −1.65685 −0.165685
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −1.58579 −0.157016
\(103\) −16.4142 −1.61734 −0.808670 0.588262i \(-0.799812\pi\)
−0.808670 + 0.588262i \(0.799812\pi\)
\(104\) 0.585786 0.0574411
\(105\) 0 0
\(106\) −7.48528 −0.727035
\(107\) 8.89949 0.860347 0.430173 0.902746i \(-0.358452\pi\)
0.430173 + 0.902746i \(0.358452\pi\)
\(108\) 2.41421 0.232308
\(109\) 10.8284 1.03718 0.518588 0.855024i \(-0.326458\pi\)
0.518588 + 0.855024i \(0.326458\pi\)
\(110\) 11.4142 1.08830
\(111\) −2.72792 −0.258923
\(112\) 0 0
\(113\) 11.0000 1.03479 0.517396 0.855746i \(-0.326901\pi\)
0.517396 + 0.855746i \(0.326901\pi\)
\(114\) −1.17157 −0.109728
\(115\) −10.3431 −0.964503
\(116\) −5.00000 −0.464238
\(117\) −1.65685 −0.153176
\(118\) 10.4853 0.965248
\(119\) 0 0
\(120\) 0.757359 0.0691371
\(121\) 27.9706 2.54278
\(122\) 11.4853 1.03983
\(123\) 0.414214 0.0373484
\(124\) −5.58579 −0.501618
\(125\) 12.1716 1.08866
\(126\) 0 0
\(127\) −9.07107 −0.804927 −0.402464 0.915436i \(-0.631846\pi\)
−0.402464 + 0.915436i \(0.631846\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.514719 −0.0453184
\(130\) −1.07107 −0.0939389
\(131\) −4.72792 −0.413080 −0.206540 0.978438i \(-0.566220\pi\)
−0.206540 + 0.978438i \(0.566220\pi\)
\(132\) 2.58579 0.225064
\(133\) 0 0
\(134\) −7.17157 −0.619530
\(135\) −4.41421 −0.379915
\(136\) 3.82843 0.328285
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) −2.34315 −0.199462
\(139\) 5.51472 0.467752 0.233876 0.972266i \(-0.424859\pi\)
0.233876 + 0.972266i \(0.424859\pi\)
\(140\) 0 0
\(141\) −4.24264 −0.357295
\(142\) 8.89949 0.746829
\(143\) −3.65685 −0.305802
\(144\) −2.82843 −0.235702
\(145\) 9.14214 0.759213
\(146\) 0 0
\(147\) 0 0
\(148\) 6.58579 0.541348
\(149\) 5.48528 0.449372 0.224686 0.974431i \(-0.427864\pi\)
0.224686 + 0.974431i \(0.427864\pi\)
\(150\) 0.686292 0.0560355
\(151\) 2.75736 0.224391 0.112195 0.993686i \(-0.464212\pi\)
0.112195 + 0.993686i \(0.464212\pi\)
\(152\) 2.82843 0.229416
\(153\) −10.8284 −0.875426
\(154\) 0 0
\(155\) 10.2132 0.820344
\(156\) −0.242641 −0.0194268
\(157\) −7.31371 −0.583697 −0.291849 0.956464i \(-0.594270\pi\)
−0.291849 + 0.956464i \(0.594270\pi\)
\(158\) −9.24264 −0.735305
\(159\) 3.10051 0.245886
\(160\) −1.82843 −0.144550
\(161\) 0 0
\(162\) 7.48528 0.588099
\(163\) 13.3137 1.04281 0.521405 0.853309i \(-0.325408\pi\)
0.521405 + 0.853309i \(0.325408\pi\)
\(164\) −1.00000 −0.0780869
\(165\) −4.72792 −0.368068
\(166\) 14.2426 1.10544
\(167\) 9.65685 0.747270 0.373635 0.927576i \(-0.378111\pi\)
0.373635 + 0.927576i \(0.378111\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) −7.00000 −0.536875
\(171\) −8.00000 −0.611775
\(172\) 1.24264 0.0947505
\(173\) −4.31371 −0.327965 −0.163983 0.986463i \(-0.552434\pi\)
−0.163983 + 0.986463i \(0.552434\pi\)
\(174\) 2.07107 0.157007
\(175\) 0 0
\(176\) −6.24264 −0.470557
\(177\) −4.34315 −0.326451
\(178\) 3.82843 0.286953
\(179\) −21.2132 −1.58555 −0.792775 0.609515i \(-0.791364\pi\)
−0.792775 + 0.609515i \(0.791364\pi\)
\(180\) 5.17157 0.385466
\(181\) 21.8995 1.62778 0.813888 0.581021i \(-0.197347\pi\)
0.813888 + 0.581021i \(0.197347\pi\)
\(182\) 0 0
\(183\) −4.75736 −0.351674
\(184\) 5.65685 0.417029
\(185\) −12.0416 −0.885318
\(186\) 2.31371 0.169649
\(187\) −23.8995 −1.74770
\(188\) 10.2426 0.747021
\(189\) 0 0
\(190\) −5.17157 −0.375185
\(191\) 18.0711 1.30758 0.653788 0.756678i \(-0.273179\pi\)
0.653788 + 0.756678i \(0.273179\pi\)
\(192\) −0.414214 −0.0298933
\(193\) −4.58579 −0.330092 −0.165046 0.986286i \(-0.552777\pi\)
−0.165046 + 0.986286i \(0.552777\pi\)
\(194\) 0.171573 0.0123182
\(195\) 0.443651 0.0317705
\(196\) 0 0
\(197\) −0.686292 −0.0488962 −0.0244481 0.999701i \(-0.507783\pi\)
−0.0244481 + 0.999701i \(0.507783\pi\)
\(198\) 17.6569 1.25482
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.65685 −0.117157
\(201\) 2.97056 0.209527
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −1.58579 −0.111027
\(205\) 1.82843 0.127703
\(206\) −16.4142 −1.14363
\(207\) −16.0000 −1.11208
\(208\) 0.585786 0.0406170
\(209\) −17.6569 −1.22135
\(210\) 0 0
\(211\) 6.34315 0.436680 0.218340 0.975873i \(-0.429936\pi\)
0.218340 + 0.975873i \(0.429936\pi\)
\(212\) −7.48528 −0.514091
\(213\) −3.68629 −0.252581
\(214\) 8.89949 0.608357
\(215\) −2.27208 −0.154954
\(216\) 2.41421 0.164266
\(217\) 0 0
\(218\) 10.8284 0.733394
\(219\) 0 0
\(220\) 11.4142 0.769546
\(221\) 2.24264 0.150856
\(222\) −2.72792 −0.183086
\(223\) −22.5563 −1.51048 −0.755242 0.655446i \(-0.772481\pi\)
−0.755242 + 0.655446i \(0.772481\pi\)
\(224\) 0 0
\(225\) 4.68629 0.312419
\(226\) 11.0000 0.731709
\(227\) −23.7279 −1.57488 −0.787439 0.616393i \(-0.788593\pi\)
−0.787439 + 0.616393i \(0.788593\pi\)
\(228\) −1.17157 −0.0775893
\(229\) 13.0711 0.863760 0.431880 0.901931i \(-0.357850\pi\)
0.431880 + 0.901931i \(0.357850\pi\)
\(230\) −10.3431 −0.682007
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) −6.10051 −0.399657 −0.199829 0.979831i \(-0.564039\pi\)
−0.199829 + 0.979831i \(0.564039\pi\)
\(234\) −1.65685 −0.108312
\(235\) −18.7279 −1.22167
\(236\) 10.4853 0.682534
\(237\) 3.82843 0.248683
\(238\) 0 0
\(239\) 11.5147 0.744825 0.372413 0.928067i \(-0.378531\pi\)
0.372413 + 0.928067i \(0.378531\pi\)
\(240\) 0.757359 0.0488873
\(241\) −19.5563 −1.25974 −0.629868 0.776703i \(-0.716891\pi\)
−0.629868 + 0.776703i \(0.716891\pi\)
\(242\) 27.9706 1.79802
\(243\) −10.3431 −0.663513
\(244\) 11.4853 0.735270
\(245\) 0 0
\(246\) 0.414214 0.0264093
\(247\) 1.65685 0.105423
\(248\) −5.58579 −0.354698
\(249\) −5.89949 −0.373865
\(250\) 12.1716 0.769798
\(251\) 4.24264 0.267793 0.133897 0.990995i \(-0.457251\pi\)
0.133897 + 0.990995i \(0.457251\pi\)
\(252\) 0 0
\(253\) −35.3137 −2.22015
\(254\) −9.07107 −0.569169
\(255\) 2.89949 0.181573
\(256\) 1.00000 0.0625000
\(257\) −17.1421 −1.06930 −0.534649 0.845075i \(-0.679556\pi\)
−0.534649 + 0.845075i \(0.679556\pi\)
\(258\) −0.514719 −0.0320450
\(259\) 0 0
\(260\) −1.07107 −0.0664248
\(261\) 14.1421 0.875376
\(262\) −4.72792 −0.292092
\(263\) 19.3137 1.19093 0.595467 0.803380i \(-0.296967\pi\)
0.595467 + 0.803380i \(0.296967\pi\)
\(264\) 2.58579 0.159144
\(265\) 13.6863 0.840742
\(266\) 0 0
\(267\) −1.58579 −0.0970486
\(268\) −7.17157 −0.438074
\(269\) 28.6274 1.74544 0.872722 0.488217i \(-0.162353\pi\)
0.872722 + 0.488217i \(0.162353\pi\)
\(270\) −4.41421 −0.268640
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 3.82843 0.232132
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) 10.3431 0.623715
\(276\) −2.34315 −0.141041
\(277\) −14.9706 −0.899494 −0.449747 0.893156i \(-0.648486\pi\)
−0.449747 + 0.893156i \(0.648486\pi\)
\(278\) 5.51472 0.330751
\(279\) 15.7990 0.945861
\(280\) 0 0
\(281\) −22.9706 −1.37031 −0.685154 0.728398i \(-0.740265\pi\)
−0.685154 + 0.728398i \(0.740265\pi\)
\(282\) −4.24264 −0.252646
\(283\) 26.4853 1.57439 0.787193 0.616706i \(-0.211533\pi\)
0.787193 + 0.616706i \(0.211533\pi\)
\(284\) 8.89949 0.528088
\(285\) 2.14214 0.126889
\(286\) −3.65685 −0.216234
\(287\) 0 0
\(288\) −2.82843 −0.166667
\(289\) −2.34315 −0.137832
\(290\) 9.14214 0.536845
\(291\) −0.0710678 −0.00416607
\(292\) 0 0
\(293\) 11.6569 0.681001 0.340500 0.940244i \(-0.389404\pi\)
0.340500 + 0.940244i \(0.389404\pi\)
\(294\) 0 0
\(295\) −19.1716 −1.11621
\(296\) 6.58579 0.382791
\(297\) −15.0711 −0.874512
\(298\) 5.48528 0.317754
\(299\) 3.31371 0.191637
\(300\) 0.686292 0.0396231
\(301\) 0 0
\(302\) 2.75736 0.158668
\(303\) −4.14214 −0.237959
\(304\) 2.82843 0.162221
\(305\) −21.0000 −1.20246
\(306\) −10.8284 −0.619020
\(307\) −24.9706 −1.42515 −0.712573 0.701598i \(-0.752470\pi\)
−0.712573 + 0.701598i \(0.752470\pi\)
\(308\) 0 0
\(309\) 6.79899 0.386781
\(310\) 10.2132 0.580071
\(311\) 20.1421 1.14216 0.571078 0.820896i \(-0.306525\pi\)
0.571078 + 0.820896i \(0.306525\pi\)
\(312\) −0.242641 −0.0137368
\(313\) −31.3137 −1.76996 −0.884978 0.465633i \(-0.845827\pi\)
−0.884978 + 0.465633i \(0.845827\pi\)
\(314\) −7.31371 −0.412736
\(315\) 0 0
\(316\) −9.24264 −0.519939
\(317\) 14.6274 0.821558 0.410779 0.911735i \(-0.365257\pi\)
0.410779 + 0.911735i \(0.365257\pi\)
\(318\) 3.10051 0.173868
\(319\) 31.2132 1.74760
\(320\) −1.82843 −0.102212
\(321\) −3.68629 −0.205749
\(322\) 0 0
\(323\) 10.8284 0.602510
\(324\) 7.48528 0.415849
\(325\) −0.970563 −0.0538371
\(326\) 13.3137 0.737378
\(327\) −4.48528 −0.248037
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) −4.72792 −0.260264
\(331\) 20.0416 1.10159 0.550794 0.834641i \(-0.314325\pi\)
0.550794 + 0.834641i \(0.314325\pi\)
\(332\) 14.2426 0.781666
\(333\) −18.6274 −1.02078
\(334\) 9.65685 0.528400
\(335\) 13.1127 0.716423
\(336\) 0 0
\(337\) 23.3431 1.27158 0.635791 0.771861i \(-0.280674\pi\)
0.635791 + 0.771861i \(0.280674\pi\)
\(338\) −12.6569 −0.688442
\(339\) −4.55635 −0.247467
\(340\) −7.00000 −0.379628
\(341\) 34.8701 1.88832
\(342\) −8.00000 −0.432590
\(343\) 0 0
\(344\) 1.24264 0.0669987
\(345\) 4.28427 0.230657
\(346\) −4.31371 −0.231906
\(347\) 28.2843 1.51838 0.759190 0.650870i \(-0.225596\pi\)
0.759190 + 0.650870i \(0.225596\pi\)
\(348\) 2.07107 0.111021
\(349\) 7.51472 0.402254 0.201127 0.979565i \(-0.435540\pi\)
0.201127 + 0.979565i \(0.435540\pi\)
\(350\) 0 0
\(351\) 1.41421 0.0754851
\(352\) −6.24264 −0.332734
\(353\) 28.0000 1.49029 0.745145 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) −4.34315 −0.230836
\(355\) −16.2721 −0.863632
\(356\) 3.82843 0.202906
\(357\) 0 0
\(358\) −21.2132 −1.12115
\(359\) −18.8701 −0.995924 −0.497962 0.867199i \(-0.665918\pi\)
−0.497962 + 0.867199i \(0.665918\pi\)
\(360\) 5.17157 0.272566
\(361\) −11.0000 −0.578947
\(362\) 21.8995 1.15101
\(363\) −11.5858 −0.608096
\(364\) 0 0
\(365\) 0 0
\(366\) −4.75736 −0.248671
\(367\) −30.0711 −1.56970 −0.784848 0.619688i \(-0.787259\pi\)
−0.784848 + 0.619688i \(0.787259\pi\)
\(368\) 5.65685 0.294884
\(369\) 2.82843 0.147242
\(370\) −12.0416 −0.626015
\(371\) 0 0
\(372\) 2.31371 0.119960
\(373\) 10.5858 0.548111 0.274056 0.961714i \(-0.411635\pi\)
0.274056 + 0.961714i \(0.411635\pi\)
\(374\) −23.8995 −1.23581
\(375\) −5.04163 −0.260349
\(376\) 10.2426 0.528224
\(377\) −2.92893 −0.150848
\(378\) 0 0
\(379\) 6.27208 0.322175 0.161088 0.986940i \(-0.448500\pi\)
0.161088 + 0.986940i \(0.448500\pi\)
\(380\) −5.17157 −0.265296
\(381\) 3.75736 0.192495
\(382\) 18.0711 0.924596
\(383\) 14.7279 0.752562 0.376281 0.926506i \(-0.377203\pi\)
0.376281 + 0.926506i \(0.377203\pi\)
\(384\) −0.414214 −0.0211377
\(385\) 0 0
\(386\) −4.58579 −0.233410
\(387\) −3.51472 −0.178663
\(388\) 0.171573 0.00871029
\(389\) 5.02944 0.255003 0.127501 0.991838i \(-0.459304\pi\)
0.127501 + 0.991838i \(0.459304\pi\)
\(390\) 0.443651 0.0224651
\(391\) 21.6569 1.09523
\(392\) 0 0
\(393\) 1.95837 0.0987867
\(394\) −0.686292 −0.0345749
\(395\) 16.8995 0.850306
\(396\) 17.6569 0.887290
\(397\) 9.27208 0.465352 0.232676 0.972554i \(-0.425252\pi\)
0.232676 + 0.972554i \(0.425252\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) −1.65685 −0.0828427
\(401\) 27.6274 1.37965 0.689824 0.723977i \(-0.257688\pi\)
0.689824 + 0.723977i \(0.257688\pi\)
\(402\) 2.97056 0.148158
\(403\) −3.27208 −0.162994
\(404\) 10.0000 0.497519
\(405\) −13.6863 −0.680077
\(406\) 0 0
\(407\) −41.1127 −2.03788
\(408\) −1.58579 −0.0785081
\(409\) 18.1005 0.895012 0.447506 0.894281i \(-0.352312\pi\)
0.447506 + 0.894281i \(0.352312\pi\)
\(410\) 1.82843 0.0902996
\(411\) 1.65685 0.0817266
\(412\) −16.4142 −0.808670
\(413\) 0 0
\(414\) −16.0000 −0.786357
\(415\) −26.0416 −1.27833
\(416\) 0.585786 0.0287205
\(417\) −2.28427 −0.111861
\(418\) −17.6569 −0.863625
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −17.9706 −0.875832 −0.437916 0.899016i \(-0.644283\pi\)
−0.437916 + 0.899016i \(0.644283\pi\)
\(422\) 6.34315 0.308780
\(423\) −28.9706 −1.40860
\(424\) −7.48528 −0.363517
\(425\) −6.34315 −0.307688
\(426\) −3.68629 −0.178601
\(427\) 0 0
\(428\) 8.89949 0.430173
\(429\) 1.51472 0.0731313
\(430\) −2.27208 −0.109569
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 2.41421 0.116154
\(433\) 14.6274 0.702949 0.351474 0.936197i \(-0.385680\pi\)
0.351474 + 0.936197i \(0.385680\pi\)
\(434\) 0 0
\(435\) −3.78680 −0.181563
\(436\) 10.8284 0.518588
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 12.1421 0.579513 0.289756 0.957100i \(-0.406426\pi\)
0.289756 + 0.957100i \(0.406426\pi\)
\(440\) 11.4142 0.544151
\(441\) 0 0
\(442\) 2.24264 0.106672
\(443\) −5.10051 −0.242332 −0.121166 0.992632i \(-0.538663\pi\)
−0.121166 + 0.992632i \(0.538663\pi\)
\(444\) −2.72792 −0.129461
\(445\) −7.00000 −0.331832
\(446\) −22.5563 −1.06807
\(447\) −2.27208 −0.107466
\(448\) 0 0
\(449\) 14.1716 0.668798 0.334399 0.942432i \(-0.391467\pi\)
0.334399 + 0.942432i \(0.391467\pi\)
\(450\) 4.68629 0.220914
\(451\) 6.24264 0.293954
\(452\) 11.0000 0.517396
\(453\) −1.14214 −0.0536622
\(454\) −23.7279 −1.11361
\(455\) 0 0
\(456\) −1.17157 −0.0548639
\(457\) −10.5858 −0.495182 −0.247591 0.968865i \(-0.579639\pi\)
−0.247591 + 0.968865i \(0.579639\pi\)
\(458\) 13.0711 0.610771
\(459\) 9.24264 0.431410
\(460\) −10.3431 −0.482252
\(461\) 40.4558 1.88422 0.942108 0.335309i \(-0.108841\pi\)
0.942108 + 0.335309i \(0.108841\pi\)
\(462\) 0 0
\(463\) 12.6863 0.589582 0.294791 0.955562i \(-0.404750\pi\)
0.294791 + 0.955562i \(0.404750\pi\)
\(464\) −5.00000 −0.232119
\(465\) −4.23045 −0.196182
\(466\) −6.10051 −0.282600
\(467\) 15.1716 0.702057 0.351028 0.936365i \(-0.385832\pi\)
0.351028 + 0.936365i \(0.385832\pi\)
\(468\) −1.65685 −0.0765881
\(469\) 0 0
\(470\) −18.7279 −0.863855
\(471\) 3.02944 0.139589
\(472\) 10.4853 0.482624
\(473\) −7.75736 −0.356684
\(474\) 3.82843 0.175845
\(475\) −4.68629 −0.215022
\(476\) 0 0
\(477\) 21.1716 0.969380
\(478\) 11.5147 0.526671
\(479\) 2.24264 0.102469 0.0512344 0.998687i \(-0.483684\pi\)
0.0512344 + 0.998687i \(0.483684\pi\)
\(480\) 0.757359 0.0345686
\(481\) 3.85786 0.175903
\(482\) −19.5563 −0.890767
\(483\) 0 0
\(484\) 27.9706 1.27139
\(485\) −0.313708 −0.0142448
\(486\) −10.3431 −0.469175
\(487\) −23.5563 −1.06744 −0.533720 0.845661i \(-0.679206\pi\)
−0.533720 + 0.845661i \(0.679206\pi\)
\(488\) 11.4853 0.519914
\(489\) −5.51472 −0.249384
\(490\) 0 0
\(491\) −10.7574 −0.485473 −0.242736 0.970092i \(-0.578045\pi\)
−0.242736 + 0.970092i \(0.578045\pi\)
\(492\) 0.414214 0.0186742
\(493\) −19.1421 −0.862118
\(494\) 1.65685 0.0745454
\(495\) −32.2843 −1.45107
\(496\) −5.58579 −0.250809
\(497\) 0 0
\(498\) −5.89949 −0.264363
\(499\) −32.1421 −1.43888 −0.719440 0.694555i \(-0.755601\pi\)
−0.719440 + 0.694555i \(0.755601\pi\)
\(500\) 12.1716 0.544329
\(501\) −4.00000 −0.178707
\(502\) 4.24264 0.189358
\(503\) 0.686292 0.0306002 0.0153001 0.999883i \(-0.495130\pi\)
0.0153001 + 0.999883i \(0.495130\pi\)
\(504\) 0 0
\(505\) −18.2843 −0.813639
\(506\) −35.3137 −1.56989
\(507\) 5.24264 0.232834
\(508\) −9.07107 −0.402464
\(509\) −18.3431 −0.813046 −0.406523 0.913641i \(-0.633259\pi\)
−0.406523 + 0.913641i \(0.633259\pi\)
\(510\) 2.89949 0.128392
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.82843 0.301482
\(514\) −17.1421 −0.756107
\(515\) 30.0122 1.32250
\(516\) −0.514719 −0.0226592
\(517\) −63.9411 −2.81213
\(518\) 0 0
\(519\) 1.78680 0.0784317
\(520\) −1.07107 −0.0469694
\(521\) −4.48528 −0.196504 −0.0982519 0.995162i \(-0.531325\pi\)
−0.0982519 + 0.995162i \(0.531325\pi\)
\(522\) 14.1421 0.618984
\(523\) 1.65685 0.0724492 0.0362246 0.999344i \(-0.488467\pi\)
0.0362246 + 0.999344i \(0.488467\pi\)
\(524\) −4.72792 −0.206540
\(525\) 0 0
\(526\) 19.3137 0.842118
\(527\) −21.3848 −0.931535
\(528\) 2.58579 0.112532
\(529\) 9.00000 0.391304
\(530\) 13.6863 0.594495
\(531\) −29.6569 −1.28700
\(532\) 0 0
\(533\) −0.585786 −0.0253732
\(534\) −1.58579 −0.0686237
\(535\) −16.2721 −0.703503
\(536\) −7.17157 −0.309765
\(537\) 8.78680 0.379178
\(538\) 28.6274 1.23422
\(539\) 0 0
\(540\) −4.41421 −0.189958
\(541\) 30.9706 1.33153 0.665764 0.746162i \(-0.268106\pi\)
0.665764 + 0.746162i \(0.268106\pi\)
\(542\) −6.00000 −0.257722
\(543\) −9.07107 −0.389277
\(544\) 3.82843 0.164142
\(545\) −19.7990 −0.848096
\(546\) 0 0
\(547\) 36.1421 1.54533 0.772663 0.634816i \(-0.218924\pi\)
0.772663 + 0.634816i \(0.218924\pi\)
\(548\) −4.00000 −0.170872
\(549\) −32.4853 −1.38644
\(550\) 10.3431 0.441033
\(551\) −14.1421 −0.602475
\(552\) −2.34315 −0.0997309
\(553\) 0 0
\(554\) −14.9706 −0.636038
\(555\) 4.98781 0.211721
\(556\) 5.51472 0.233876
\(557\) −36.6569 −1.55320 −0.776600 0.629994i \(-0.783058\pi\)
−0.776600 + 0.629994i \(0.783058\pi\)
\(558\) 15.7990 0.668825
\(559\) 0.727922 0.0307878
\(560\) 0 0
\(561\) 9.89949 0.417957
\(562\) −22.9706 −0.968955
\(563\) −4.62742 −0.195022 −0.0975112 0.995234i \(-0.531088\pi\)
−0.0975112 + 0.995234i \(0.531088\pi\)
\(564\) −4.24264 −0.178647
\(565\) −20.1127 −0.846148
\(566\) 26.4853 1.11326
\(567\) 0 0
\(568\) 8.89949 0.373415
\(569\) −38.1127 −1.59777 −0.798884 0.601486i \(-0.794576\pi\)
−0.798884 + 0.601486i \(0.794576\pi\)
\(570\) 2.14214 0.0897242
\(571\) −36.9706 −1.54717 −0.773585 0.633693i \(-0.781538\pi\)
−0.773585 + 0.633693i \(0.781538\pi\)
\(572\) −3.65685 −0.152901
\(573\) −7.48528 −0.312702
\(574\) 0 0
\(575\) −9.37258 −0.390864
\(576\) −2.82843 −0.117851
\(577\) 32.2843 1.34401 0.672006 0.740546i \(-0.265433\pi\)
0.672006 + 0.740546i \(0.265433\pi\)
\(578\) −2.34315 −0.0974620
\(579\) 1.89949 0.0789403
\(580\) 9.14214 0.379607
\(581\) 0 0
\(582\) −0.0710678 −0.00294586
\(583\) 46.7279 1.93527
\(584\) 0 0
\(585\) 3.02944 0.125252
\(586\) 11.6569 0.481540
\(587\) 0.757359 0.0312596 0.0156298 0.999878i \(-0.495025\pi\)
0.0156298 + 0.999878i \(0.495025\pi\)
\(588\) 0 0
\(589\) −15.7990 −0.650986
\(590\) −19.1716 −0.789281
\(591\) 0.284271 0.0116934
\(592\) 6.58579 0.270674
\(593\) −46.6569 −1.91597 −0.957984 0.286823i \(-0.907401\pi\)
−0.957984 + 0.286823i \(0.907401\pi\)
\(594\) −15.0711 −0.618373
\(595\) 0 0
\(596\) 5.48528 0.224686
\(597\) 1.65685 0.0678105
\(598\) 3.31371 0.135508
\(599\) 35.7574 1.46101 0.730503 0.682909i \(-0.239286\pi\)
0.730503 + 0.682909i \(0.239286\pi\)
\(600\) 0.686292 0.0280177
\(601\) 3.68629 0.150367 0.0751835 0.997170i \(-0.476046\pi\)
0.0751835 + 0.997170i \(0.476046\pi\)
\(602\) 0 0
\(603\) 20.2843 0.826039
\(604\) 2.75736 0.112195
\(605\) −51.1421 −2.07922
\(606\) −4.14214 −0.168263
\(607\) 11.0416 0.448166 0.224083 0.974570i \(-0.428061\pi\)
0.224083 + 0.974570i \(0.428061\pi\)
\(608\) 2.82843 0.114708
\(609\) 0 0
\(610\) −21.0000 −0.850265
\(611\) 6.00000 0.242734
\(612\) −10.8284 −0.437713
\(613\) −22.2426 −0.898372 −0.449186 0.893438i \(-0.648286\pi\)
−0.449186 + 0.893438i \(0.648286\pi\)
\(614\) −24.9706 −1.00773
\(615\) −0.757359 −0.0305397
\(616\) 0 0
\(617\) 23.7990 0.958111 0.479056 0.877785i \(-0.340979\pi\)
0.479056 + 0.877785i \(0.340979\pi\)
\(618\) 6.79899 0.273495
\(619\) −20.9289 −0.841205 −0.420602 0.907245i \(-0.638181\pi\)
−0.420602 + 0.907245i \(0.638181\pi\)
\(620\) 10.2132 0.410172
\(621\) 13.6569 0.548031
\(622\) 20.1421 0.807626
\(623\) 0 0
\(624\) −0.242641 −0.00971340
\(625\) −13.9706 −0.558823
\(626\) −31.3137 −1.25155
\(627\) 7.31371 0.292081
\(628\) −7.31371 −0.291849
\(629\) 25.2132 1.00532
\(630\) 0 0
\(631\) −31.4142 −1.25058 −0.625290 0.780392i \(-0.715019\pi\)
−0.625290 + 0.780392i \(0.715019\pi\)
\(632\) −9.24264 −0.367653
\(633\) −2.62742 −0.104430
\(634\) 14.6274 0.580929
\(635\) 16.5858 0.658187
\(636\) 3.10051 0.122943
\(637\) 0 0
\(638\) 31.2132 1.23574
\(639\) −25.1716 −0.995772
\(640\) −1.82843 −0.0722749
\(641\) 46.6690 1.84332 0.921658 0.388003i \(-0.126835\pi\)
0.921658 + 0.388003i \(0.126835\pi\)
\(642\) −3.68629 −0.145486
\(643\) 5.58579 0.220282 0.110141 0.993916i \(-0.464870\pi\)
0.110141 + 0.993916i \(0.464870\pi\)
\(644\) 0 0
\(645\) 0.941125 0.0370568
\(646\) 10.8284 0.426039
\(647\) −41.3848 −1.62700 −0.813502 0.581563i \(-0.802442\pi\)
−0.813502 + 0.581563i \(0.802442\pi\)
\(648\) 7.48528 0.294050
\(649\) −65.4558 −2.56937
\(650\) −0.970563 −0.0380686
\(651\) 0 0
\(652\) 13.3137 0.521405
\(653\) −17.4853 −0.684252 −0.342126 0.939654i \(-0.611147\pi\)
−0.342126 + 0.939654i \(0.611147\pi\)
\(654\) −4.48528 −0.175388
\(655\) 8.64466 0.337775
\(656\) −1.00000 −0.0390434
\(657\) 0 0
\(658\) 0 0
\(659\) −26.3431 −1.02618 −0.513092 0.858334i \(-0.671500\pi\)
−0.513092 + 0.858334i \(0.671500\pi\)
\(660\) −4.72792 −0.184034
\(661\) −4.20101 −0.163400 −0.0817002 0.996657i \(-0.526035\pi\)
−0.0817002 + 0.996657i \(0.526035\pi\)
\(662\) 20.0416 0.778940
\(663\) −0.928932 −0.0360767
\(664\) 14.2426 0.552722
\(665\) 0 0
\(666\) −18.6274 −0.721798
\(667\) −28.2843 −1.09517
\(668\) 9.65685 0.373635
\(669\) 9.34315 0.361227
\(670\) 13.1127 0.506588
\(671\) −71.6985 −2.76789
\(672\) 0 0
\(673\) −16.9289 −0.652562 −0.326281 0.945273i \(-0.605796\pi\)
−0.326281 + 0.945273i \(0.605796\pi\)
\(674\) 23.3431 0.899144
\(675\) −4.00000 −0.153960
\(676\) −12.6569 −0.486802
\(677\) 14.1421 0.543526 0.271763 0.962364i \(-0.412393\pi\)
0.271763 + 0.962364i \(0.412393\pi\)
\(678\) −4.55635 −0.174986
\(679\) 0 0
\(680\) −7.00000 −0.268438
\(681\) 9.82843 0.376626
\(682\) 34.8701 1.33524
\(683\) 45.6569 1.74701 0.873505 0.486814i \(-0.161841\pi\)
0.873505 + 0.486814i \(0.161841\pi\)
\(684\) −8.00000 −0.305888
\(685\) 7.31371 0.279442
\(686\) 0 0
\(687\) −5.41421 −0.206565
\(688\) 1.24264 0.0473752
\(689\) −4.38478 −0.167047
\(690\) 4.28427 0.163099
\(691\) −20.6985 −0.787407 −0.393704 0.919237i \(-0.628806\pi\)
−0.393704 + 0.919237i \(0.628806\pi\)
\(692\) −4.31371 −0.163983
\(693\) 0 0
\(694\) 28.2843 1.07366
\(695\) −10.0833 −0.382480
\(696\) 2.07107 0.0785036
\(697\) −3.82843 −0.145012
\(698\) 7.51472 0.284436
\(699\) 2.52691 0.0955766
\(700\) 0 0
\(701\) −22.3431 −0.843889 −0.421944 0.906622i \(-0.638652\pi\)
−0.421944 + 0.906622i \(0.638652\pi\)
\(702\) 1.41421 0.0533761
\(703\) 18.6274 0.702546
\(704\) −6.24264 −0.235278
\(705\) 7.75736 0.292159
\(706\) 28.0000 1.05379
\(707\) 0 0
\(708\) −4.34315 −0.163225
\(709\) 23.6274 0.887346 0.443673 0.896189i \(-0.353675\pi\)
0.443673 + 0.896189i \(0.353675\pi\)
\(710\) −16.2721 −0.610680
\(711\) 26.1421 0.980407
\(712\) 3.82843 0.143476
\(713\) −31.5980 −1.18335
\(714\) 0 0
\(715\) 6.68629 0.250053
\(716\) −21.2132 −0.792775
\(717\) −4.76955 −0.178122
\(718\) −18.8701 −0.704224
\(719\) 33.8995 1.26424 0.632119 0.774871i \(-0.282185\pi\)
0.632119 + 0.774871i \(0.282185\pi\)
\(720\) 5.17157 0.192733
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 8.10051 0.301261
\(724\) 21.8995 0.813888
\(725\) 8.28427 0.307670
\(726\) −11.5858 −0.429989
\(727\) 2.58579 0.0959015 0.0479508 0.998850i \(-0.484731\pi\)
0.0479508 + 0.998850i \(0.484731\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 4.75736 0.175957
\(732\) −4.75736 −0.175837
\(733\) 42.6569 1.57557 0.787783 0.615952i \(-0.211229\pi\)
0.787783 + 0.615952i \(0.211229\pi\)
\(734\) −30.0711 −1.10994
\(735\) 0 0
\(736\) 5.65685 0.208514
\(737\) 44.7696 1.64911
\(738\) 2.82843 0.104116
\(739\) 8.89949 0.327373 0.163687 0.986512i \(-0.447661\pi\)
0.163687 + 0.986512i \(0.447661\pi\)
\(740\) −12.0416 −0.442659
\(741\) −0.686292 −0.0252115
\(742\) 0 0
\(743\) 0.970563 0.0356065 0.0178032 0.999842i \(-0.494333\pi\)
0.0178032 + 0.999842i \(0.494333\pi\)
\(744\) 2.31371 0.0848247
\(745\) −10.0294 −0.367450
\(746\) 10.5858 0.387573
\(747\) −40.2843 −1.47392
\(748\) −23.8995 −0.873852
\(749\) 0 0
\(750\) −5.04163 −0.184094
\(751\) −36.3431 −1.32618 −0.663090 0.748540i \(-0.730755\pi\)
−0.663090 + 0.748540i \(0.730755\pi\)
\(752\) 10.2426 0.373511
\(753\) −1.75736 −0.0640417
\(754\) −2.92893 −0.106665
\(755\) −5.04163 −0.183484
\(756\) 0 0
\(757\) 34.9411 1.26996 0.634978 0.772530i \(-0.281009\pi\)
0.634978 + 0.772530i \(0.281009\pi\)
\(758\) 6.27208 0.227812
\(759\) 14.6274 0.530942
\(760\) −5.17157 −0.187593
\(761\) 15.2721 0.553612 0.276806 0.960926i \(-0.410724\pi\)
0.276806 + 0.960926i \(0.410724\pi\)
\(762\) 3.75736 0.136115
\(763\) 0 0
\(764\) 18.0711 0.653788
\(765\) 19.7990 0.715834
\(766\) 14.7279 0.532141
\(767\) 6.14214 0.221780
\(768\) −0.414214 −0.0149466
\(769\) 42.9706 1.54956 0.774779 0.632232i \(-0.217861\pi\)
0.774779 + 0.632232i \(0.217861\pi\)
\(770\) 0 0
\(771\) 7.10051 0.255718
\(772\) −4.58579 −0.165046
\(773\) −25.2132 −0.906856 −0.453428 0.891293i \(-0.649799\pi\)
−0.453428 + 0.891293i \(0.649799\pi\)
\(774\) −3.51472 −0.126334
\(775\) 9.25483 0.332443
\(776\) 0.171573 0.00615911
\(777\) 0 0
\(778\) 5.02944 0.180314
\(779\) −2.82843 −0.101339
\(780\) 0.443651 0.0158852
\(781\) −55.5563 −1.98796
\(782\) 21.6569 0.774448
\(783\) −12.0711 −0.431385
\(784\) 0 0
\(785\) 13.3726 0.477288
\(786\) 1.95837 0.0698527
\(787\) 24.9706 0.890104 0.445052 0.895505i \(-0.353185\pi\)
0.445052 + 0.895505i \(0.353185\pi\)
\(788\) −0.686292 −0.0244481
\(789\) −8.00000 −0.284808
\(790\) 16.8995 0.601257
\(791\) 0 0
\(792\) 17.6569 0.627409
\(793\) 6.72792 0.238916
\(794\) 9.27208 0.329054
\(795\) −5.66905 −0.201060
\(796\) −4.00000 −0.141776
\(797\) 21.7696 0.771117 0.385559 0.922683i \(-0.374009\pi\)
0.385559 + 0.922683i \(0.374009\pi\)
\(798\) 0 0
\(799\) 39.2132 1.38726
\(800\) −1.65685 −0.0585786
\(801\) −10.8284 −0.382604
\(802\) 27.6274 0.975558
\(803\) 0 0
\(804\) 2.97056 0.104764
\(805\) 0 0
\(806\) −3.27208 −0.115254
\(807\) −11.8579 −0.417417
\(808\) 10.0000 0.351799
\(809\) 21.6985 0.762878 0.381439 0.924394i \(-0.375429\pi\)
0.381439 + 0.924394i \(0.375429\pi\)
\(810\) −13.6863 −0.480887
\(811\) −4.72792 −0.166020 −0.0830099 0.996549i \(-0.526453\pi\)
−0.0830099 + 0.996549i \(0.526453\pi\)
\(812\) 0 0
\(813\) 2.48528 0.0871626
\(814\) −41.1127 −1.44100
\(815\) −24.3431 −0.852703
\(816\) −1.58579 −0.0555136
\(817\) 3.51472 0.122964
\(818\) 18.1005 0.632869
\(819\) 0 0
\(820\) 1.82843 0.0638514
\(821\) −35.2132 −1.22895 −0.614475 0.788937i \(-0.710632\pi\)
−0.614475 + 0.788937i \(0.710632\pi\)
\(822\) 1.65685 0.0577894
\(823\) −1.87006 −0.0651861 −0.0325931 0.999469i \(-0.510377\pi\)
−0.0325931 + 0.999469i \(0.510377\pi\)
\(824\) −16.4142 −0.571816
\(825\) −4.28427 −0.149159
\(826\) 0 0
\(827\) 37.1127 1.29053 0.645267 0.763957i \(-0.276746\pi\)
0.645267 + 0.763957i \(0.276746\pi\)
\(828\) −16.0000 −0.556038
\(829\) 38.3137 1.33069 0.665345 0.746536i \(-0.268284\pi\)
0.665345 + 0.746536i \(0.268284\pi\)
\(830\) −26.0416 −0.903918
\(831\) 6.20101 0.215111
\(832\) 0.585786 0.0203085
\(833\) 0 0
\(834\) −2.28427 −0.0790978
\(835\) −17.6569 −0.611041
\(836\) −17.6569 −0.610675
\(837\) −13.4853 −0.466120
\(838\) −24.0416 −0.830504
\(839\) −41.1127 −1.41937 −0.709684 0.704520i \(-0.751162\pi\)
−0.709684 + 0.704520i \(0.751162\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −17.9706 −0.619306
\(843\) 9.51472 0.327704
\(844\) 6.34315 0.218340
\(845\) 23.1421 0.796114
\(846\) −28.9706 −0.996028
\(847\) 0 0
\(848\) −7.48528 −0.257046
\(849\) −10.9706 −0.376509
\(850\) −6.34315 −0.217568
\(851\) 37.2548 1.27708
\(852\) −3.68629 −0.126290
\(853\) −0.171573 −0.00587454 −0.00293727 0.999996i \(-0.500935\pi\)
−0.00293727 + 0.999996i \(0.500935\pi\)
\(854\) 0 0
\(855\) 14.6274 0.500247
\(856\) 8.89949 0.304178
\(857\) 57.5980 1.96751 0.983755 0.179518i \(-0.0574537\pi\)
0.983755 + 0.179518i \(0.0574537\pi\)
\(858\) 1.51472 0.0517116
\(859\) 6.38478 0.217846 0.108923 0.994050i \(-0.465260\pi\)
0.108923 + 0.994050i \(0.465260\pi\)
\(860\) −2.27208 −0.0774772
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −25.4142 −0.865110 −0.432555 0.901608i \(-0.642388\pi\)
−0.432555 + 0.901608i \(0.642388\pi\)
\(864\) 2.41421 0.0821332
\(865\) 7.88730 0.268176
\(866\) 14.6274 0.497060
\(867\) 0.970563 0.0329620
\(868\) 0 0
\(869\) 57.6985 1.95729
\(870\) −3.78680 −0.128384
\(871\) −4.20101 −0.142346
\(872\) 10.8284 0.366697
\(873\) −0.485281 −0.0164243
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 0 0
\(877\) 34.6274 1.16928 0.584642 0.811291i \(-0.301235\pi\)
0.584642 + 0.811291i \(0.301235\pi\)
\(878\) 12.1421 0.409777
\(879\) −4.82843 −0.162859
\(880\) 11.4142 0.384773
\(881\) 4.10051 0.138150 0.0690748 0.997611i \(-0.477995\pi\)
0.0690748 + 0.997611i \(0.477995\pi\)
\(882\) 0 0
\(883\) 4.82843 0.162490 0.0812448 0.996694i \(-0.474110\pi\)
0.0812448 + 0.996694i \(0.474110\pi\)
\(884\) 2.24264 0.0754282
\(885\) 7.94113 0.266938
\(886\) −5.10051 −0.171355
\(887\) −28.9706 −0.972736 −0.486368 0.873754i \(-0.661679\pi\)
−0.486368 + 0.873754i \(0.661679\pi\)
\(888\) −2.72792 −0.0915431
\(889\) 0 0
\(890\) −7.00000 −0.234641
\(891\) −46.7279 −1.56544
\(892\) −22.5563 −0.755242
\(893\) 28.9706 0.969463
\(894\) −2.27208 −0.0759897
\(895\) 38.7868 1.29650
\(896\) 0 0
\(897\) −1.37258 −0.0458292
\(898\) 14.1716 0.472912
\(899\) 27.9289 0.931482
\(900\) 4.68629 0.156210
\(901\) −28.6569 −0.954698
\(902\) 6.24264 0.207857
\(903\) 0 0
\(904\) 11.0000 0.365855
\(905\) −40.0416 −1.33103
\(906\) −1.14214 −0.0379449
\(907\) −18.0711 −0.600040 −0.300020 0.953933i \(-0.596993\pi\)
−0.300020 + 0.953933i \(0.596993\pi\)
\(908\) −23.7279 −0.787439
\(909\) −28.2843 −0.938130
\(910\) 0 0
\(911\) 15.1716 0.502657 0.251328 0.967902i \(-0.419133\pi\)
0.251328 + 0.967902i \(0.419133\pi\)
\(912\) −1.17157 −0.0387947
\(913\) −88.9117 −2.94255
\(914\) −10.5858 −0.350147
\(915\) 8.69848 0.287563
\(916\) 13.0711 0.431880
\(917\) 0 0
\(918\) 9.24264 0.305053
\(919\) −21.2426 −0.700730 −0.350365 0.936613i \(-0.613942\pi\)
−0.350365 + 0.936613i \(0.613942\pi\)
\(920\) −10.3431 −0.341003
\(921\) 10.3431 0.340818
\(922\) 40.4558 1.33234
\(923\) 5.21320 0.171595
\(924\) 0 0
\(925\) −10.9117 −0.358774
\(926\) 12.6863 0.416897
\(927\) 46.4264 1.52484
\(928\) −5.00000 −0.164133
\(929\) −5.85786 −0.192190 −0.0960951 0.995372i \(-0.530635\pi\)
−0.0960951 + 0.995372i \(0.530635\pi\)
\(930\) −4.23045 −0.138722
\(931\) 0 0
\(932\) −6.10051 −0.199829
\(933\) −8.34315 −0.273142
\(934\) 15.1716 0.496429
\(935\) 43.6985 1.42909
\(936\) −1.65685 −0.0541560
\(937\) −0.0294373 −0.000961673 0 −0.000480837 1.00000i \(-0.500153\pi\)
−0.000480837 1.00000i \(0.500153\pi\)
\(938\) 0 0
\(939\) 12.9706 0.423278
\(940\) −18.7279 −0.610837
\(941\) 56.9706 1.85719 0.928594 0.371098i \(-0.121019\pi\)
0.928594 + 0.371098i \(0.121019\pi\)
\(942\) 3.02944 0.0987044
\(943\) −5.65685 −0.184213
\(944\) 10.4853 0.341267
\(945\) 0 0
\(946\) −7.75736 −0.252214
\(947\) −45.4558 −1.47712 −0.738558 0.674190i \(-0.764493\pi\)
−0.738558 + 0.674190i \(0.764493\pi\)
\(948\) 3.82843 0.124342
\(949\) 0 0
\(950\) −4.68629 −0.152043
\(951\) −6.05887 −0.196472
\(952\) 0 0
\(953\) 22.8284 0.739485 0.369743 0.929134i \(-0.379446\pi\)
0.369743 + 0.929134i \(0.379446\pi\)
\(954\) 21.1716 0.685455
\(955\) −33.0416 −1.06920
\(956\) 11.5147 0.372413
\(957\) −12.9289 −0.417933
\(958\) 2.24264 0.0724564
\(959\) 0 0
\(960\) 0.757359 0.0244437
\(961\) 0.201010 0.00648420
\(962\) 3.85786 0.124383
\(963\) −25.1716 −0.811143
\(964\) −19.5563 −0.629868
\(965\) 8.38478 0.269915
\(966\) 0 0
\(967\) 29.5269 0.949521 0.474761 0.880115i \(-0.342535\pi\)
0.474761 + 0.880115i \(0.342535\pi\)
\(968\) 27.9706 0.899008
\(969\) −4.48528 −0.144088
\(970\) −0.313708 −0.0100726
\(971\) −4.12994 −0.132536 −0.0662681 0.997802i \(-0.521109\pi\)
−0.0662681 + 0.997802i \(0.521109\pi\)
\(972\) −10.3431 −0.331757
\(973\) 0 0
\(974\) −23.5563 −0.754794
\(975\) 0.402020 0.0128750
\(976\) 11.4853 0.367635
\(977\) −17.3137 −0.553915 −0.276957 0.960882i \(-0.589326\pi\)
−0.276957 + 0.960882i \(0.589326\pi\)
\(978\) −5.51472 −0.176341
\(979\) −23.8995 −0.763831
\(980\) 0 0
\(981\) −30.6274 −0.977858
\(982\) −10.7574 −0.343281
\(983\) −25.2426 −0.805115 −0.402558 0.915395i \(-0.631879\pi\)
−0.402558 + 0.915395i \(0.631879\pi\)
\(984\) 0.414214 0.0132046
\(985\) 1.25483 0.0399823
\(986\) −19.1421 −0.609610
\(987\) 0 0
\(988\) 1.65685 0.0527116
\(989\) 7.02944 0.223523
\(990\) −32.2843 −1.02606
\(991\) 10.9706 0.348491 0.174246 0.984702i \(-0.444251\pi\)
0.174246 + 0.984702i \(0.444251\pi\)
\(992\) −5.58579 −0.177349
\(993\) −8.30152 −0.263441
\(994\) 0 0
\(995\) 7.31371 0.231860
\(996\) −5.89949 −0.186933
\(997\) −17.9411 −0.568201 −0.284101 0.958794i \(-0.591695\pi\)
−0.284101 + 0.958794i \(0.591695\pi\)
\(998\) −32.1421 −1.01744
\(999\) 15.8995 0.503038
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 4018.2.a.bc.1.1 yes 2
7.6 odd 2 4018.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4018.2.a.y.1.2 2 7.6 odd 2
4018.2.a.bc.1.1 yes 2 1.1 even 1 trivial