Properties

Label 4014.2.a.x.1.4
Level $4014$
Weight $2$
Character 4014.1
Self dual yes
Analytic conductor $32.052$
Analytic rank $0$
Dimension $8$
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] = [4014,2,Mod(1,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, 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("4014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.34121\) of defining polynomial
Character \(\chi\) \(=\) 4014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.350247 q^{5} +3.21437 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.350247 q^{5} +3.21437 q^{7} -1.00000 q^{8} -0.350247 q^{10} +3.33410 q^{11} -4.79431 q^{13} -3.21437 q^{14} +1.00000 q^{16} -1.65967 q^{17} -6.44882 q^{19} +0.350247 q^{20} -3.33410 q^{22} +7.13194 q^{23} -4.87733 q^{25} +4.79431 q^{26} +3.21437 q^{28} +2.80668 q^{29} +6.66261 q^{31} -1.00000 q^{32} +1.65967 q^{34} +1.12582 q^{35} -0.669487 q^{37} +6.44882 q^{38} -0.350247 q^{40} +0.589936 q^{41} -4.08795 q^{43} +3.33410 q^{44} -7.13194 q^{46} +7.50921 q^{47} +3.33217 q^{49} +4.87733 q^{50} -4.79431 q^{52} +7.08046 q^{53} +1.16776 q^{55} -3.21437 q^{56} -2.80668 q^{58} +5.66381 q^{59} -1.64659 q^{61} -6.66261 q^{62} +1.00000 q^{64} -1.67919 q^{65} +9.99722 q^{67} -1.65967 q^{68} -1.12582 q^{70} +10.6585 q^{71} +6.63197 q^{73} +0.669487 q^{74} -6.44882 q^{76} +10.7170 q^{77} +14.4826 q^{79} +0.350247 q^{80} -0.589936 q^{82} +8.59600 q^{83} -0.581295 q^{85} +4.08795 q^{86} -3.33410 q^{88} +0.107172 q^{89} -15.4107 q^{91} +7.13194 q^{92} -7.50921 q^{94} -2.25868 q^{95} -0.0990721 q^{97} -3.33217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 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 0
\(4\) 1.00000 0.500000
\(5\) 0.350247 0.156635 0.0783176 0.996928i \(-0.475045\pi\)
0.0783176 + 0.996928i \(0.475045\pi\)
\(6\) 0 0
\(7\) 3.21437 1.21492 0.607459 0.794351i \(-0.292189\pi\)
0.607459 + 0.794351i \(0.292189\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.350247 −0.110758
\(11\) 3.33410 1.00527 0.502634 0.864499i \(-0.332364\pi\)
0.502634 + 0.864499i \(0.332364\pi\)
\(12\) 0 0
\(13\) −4.79431 −1.32970 −0.664852 0.746975i \(-0.731505\pi\)
−0.664852 + 0.746975i \(0.731505\pi\)
\(14\) −3.21437 −0.859076
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.65967 −0.402530 −0.201265 0.979537i \(-0.564505\pi\)
−0.201265 + 0.979537i \(0.564505\pi\)
\(18\) 0 0
\(19\) −6.44882 −1.47946 −0.739730 0.672903i \(-0.765047\pi\)
−0.739730 + 0.672903i \(0.765047\pi\)
\(20\) 0.350247 0.0783176
\(21\) 0 0
\(22\) −3.33410 −0.710832
\(23\) 7.13194 1.48711 0.743556 0.668674i \(-0.233138\pi\)
0.743556 + 0.668674i \(0.233138\pi\)
\(24\) 0 0
\(25\) −4.87733 −0.975465
\(26\) 4.79431 0.940242
\(27\) 0 0
\(28\) 3.21437 0.607459
\(29\) 2.80668 0.521187 0.260593 0.965449i \(-0.416082\pi\)
0.260593 + 0.965449i \(0.416082\pi\)
\(30\) 0 0
\(31\) 6.66261 1.19664 0.598320 0.801257i \(-0.295835\pi\)
0.598320 + 0.801257i \(0.295835\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.65967 0.284632
\(35\) 1.12582 0.190299
\(36\) 0 0
\(37\) −0.669487 −0.110063 −0.0550315 0.998485i \(-0.517526\pi\)
−0.0550315 + 0.998485i \(0.517526\pi\)
\(38\) 6.44882 1.04614
\(39\) 0 0
\(40\) −0.350247 −0.0553789
\(41\) 0.589936 0.0921326 0.0460663 0.998938i \(-0.485331\pi\)
0.0460663 + 0.998938i \(0.485331\pi\)
\(42\) 0 0
\(43\) −4.08795 −0.623406 −0.311703 0.950180i \(-0.600899\pi\)
−0.311703 + 0.950180i \(0.600899\pi\)
\(44\) 3.33410 0.502634
\(45\) 0 0
\(46\) −7.13194 −1.05155
\(47\) 7.50921 1.09533 0.547665 0.836697i \(-0.315517\pi\)
0.547665 + 0.836697i \(0.315517\pi\)
\(48\) 0 0
\(49\) 3.33217 0.476024
\(50\) 4.87733 0.689758
\(51\) 0 0
\(52\) −4.79431 −0.664852
\(53\) 7.08046 0.972576 0.486288 0.873799i \(-0.338351\pi\)
0.486288 + 0.873799i \(0.338351\pi\)
\(54\) 0 0
\(55\) 1.16776 0.157460
\(56\) −3.21437 −0.429538
\(57\) 0 0
\(58\) −2.80668 −0.368535
\(59\) 5.66381 0.737365 0.368682 0.929555i \(-0.379809\pi\)
0.368682 + 0.929555i \(0.379809\pi\)
\(60\) 0 0
\(61\) −1.64659 −0.210825 −0.105412 0.994429i \(-0.533616\pi\)
−0.105412 + 0.994429i \(0.533616\pi\)
\(62\) −6.66261 −0.846152
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.67919 −0.208278
\(66\) 0 0
\(67\) 9.99722 1.22135 0.610677 0.791880i \(-0.290897\pi\)
0.610677 + 0.791880i \(0.290897\pi\)
\(68\) −1.65967 −0.201265
\(69\) 0 0
\(70\) −1.12582 −0.134562
\(71\) 10.6585 1.26493 0.632466 0.774588i \(-0.282043\pi\)
0.632466 + 0.774588i \(0.282043\pi\)
\(72\) 0 0
\(73\) 6.63197 0.776213 0.388107 0.921614i \(-0.373129\pi\)
0.388107 + 0.921614i \(0.373129\pi\)
\(74\) 0.669487 0.0778263
\(75\) 0 0
\(76\) −6.44882 −0.739730
\(77\) 10.7170 1.22132
\(78\) 0 0
\(79\) 14.4826 1.62942 0.814712 0.579866i \(-0.196895\pi\)
0.814712 + 0.579866i \(0.196895\pi\)
\(80\) 0.350247 0.0391588
\(81\) 0 0
\(82\) −0.589936 −0.0651476
\(83\) 8.59600 0.943533 0.471767 0.881723i \(-0.343616\pi\)
0.471767 + 0.881723i \(0.343616\pi\)
\(84\) 0 0
\(85\) −0.581295 −0.0630503
\(86\) 4.08795 0.440815
\(87\) 0 0
\(88\) −3.33410 −0.355416
\(89\) 0.107172 0.0113602 0.00568011 0.999984i \(-0.498192\pi\)
0.00568011 + 0.999984i \(0.498192\pi\)
\(90\) 0 0
\(91\) −15.4107 −1.61548
\(92\) 7.13194 0.743556
\(93\) 0 0
\(94\) −7.50921 −0.774516
\(95\) −2.25868 −0.231736
\(96\) 0 0
\(97\) −0.0990721 −0.0100593 −0.00502963 0.999987i \(-0.501601\pi\)
−0.00502963 + 0.999987i \(0.501601\pi\)
\(98\) −3.33217 −0.336600
\(99\) 0 0
\(100\) −4.87733 −0.487733
\(101\) −18.0388 −1.79493 −0.897465 0.441086i \(-0.854594\pi\)
−0.897465 + 0.441086i \(0.854594\pi\)
\(102\) 0 0
\(103\) 1.76745 0.174152 0.0870761 0.996202i \(-0.472248\pi\)
0.0870761 + 0.996202i \(0.472248\pi\)
\(104\) 4.79431 0.470121
\(105\) 0 0
\(106\) −7.08046 −0.687715
\(107\) −19.3478 −1.87043 −0.935213 0.354085i \(-0.884792\pi\)
−0.935213 + 0.354085i \(0.884792\pi\)
\(108\) 0 0
\(109\) −1.73637 −0.166314 −0.0831572 0.996536i \(-0.526500\pi\)
−0.0831572 + 0.996536i \(0.526500\pi\)
\(110\) −1.16776 −0.111341
\(111\) 0 0
\(112\) 3.21437 0.303729
\(113\) −10.6213 −0.999164 −0.499582 0.866267i \(-0.666513\pi\)
−0.499582 + 0.866267i \(0.666513\pi\)
\(114\) 0 0
\(115\) 2.49794 0.232934
\(116\) 2.80668 0.260593
\(117\) 0 0
\(118\) −5.66381 −0.521396
\(119\) −5.33480 −0.489040
\(120\) 0 0
\(121\) 0.116202 0.0105638
\(122\) 1.64659 0.149075
\(123\) 0 0
\(124\) 6.66261 0.598320
\(125\) −3.45950 −0.309427
\(126\) 0 0
\(127\) 6.09477 0.540823 0.270412 0.962745i \(-0.412840\pi\)
0.270412 + 0.962745i \(0.412840\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.67919 0.147275
\(131\) 10.3181 0.901499 0.450749 0.892651i \(-0.351157\pi\)
0.450749 + 0.892651i \(0.351157\pi\)
\(132\) 0 0
\(133\) −20.7289 −1.79742
\(134\) −9.99722 −0.863628
\(135\) 0 0
\(136\) 1.65967 0.142316
\(137\) −2.82222 −0.241119 −0.120559 0.992706i \(-0.538469\pi\)
−0.120559 + 0.992706i \(0.538469\pi\)
\(138\) 0 0
\(139\) −1.98204 −0.168114 −0.0840572 0.996461i \(-0.526788\pi\)
−0.0840572 + 0.996461i \(0.526788\pi\)
\(140\) 1.12582 0.0951494
\(141\) 0 0
\(142\) −10.6585 −0.894441
\(143\) −15.9847 −1.33671
\(144\) 0 0
\(145\) 0.983030 0.0816362
\(146\) −6.63197 −0.548866
\(147\) 0 0
\(148\) −0.669487 −0.0550315
\(149\) 13.7074 1.12296 0.561479 0.827491i \(-0.310233\pi\)
0.561479 + 0.827491i \(0.310233\pi\)
\(150\) 0 0
\(151\) 17.4598 1.42086 0.710428 0.703770i \(-0.248501\pi\)
0.710428 + 0.703770i \(0.248501\pi\)
\(152\) 6.44882 0.523068
\(153\) 0 0
\(154\) −10.7170 −0.863602
\(155\) 2.33356 0.187436
\(156\) 0 0
\(157\) 13.3567 1.06598 0.532989 0.846122i \(-0.321069\pi\)
0.532989 + 0.846122i \(0.321069\pi\)
\(158\) −14.4826 −1.15218
\(159\) 0 0
\(160\) −0.350247 −0.0276895
\(161\) 22.9247 1.80672
\(162\) 0 0
\(163\) 5.76239 0.451345 0.225673 0.974203i \(-0.427542\pi\)
0.225673 + 0.974203i \(0.427542\pi\)
\(164\) 0.589936 0.0460663
\(165\) 0 0
\(166\) −8.59600 −0.667179
\(167\) 19.3205 1.49506 0.747532 0.664226i \(-0.231239\pi\)
0.747532 + 0.664226i \(0.231239\pi\)
\(168\) 0 0
\(169\) 9.98545 0.768111
\(170\) 0.581295 0.0445833
\(171\) 0 0
\(172\) −4.08795 −0.311703
\(173\) 9.74463 0.740870 0.370435 0.928858i \(-0.379209\pi\)
0.370435 + 0.928858i \(0.379209\pi\)
\(174\) 0 0
\(175\) −15.6775 −1.18511
\(176\) 3.33410 0.251317
\(177\) 0 0
\(178\) −0.107172 −0.00803289
\(179\) 1.08363 0.0809945 0.0404973 0.999180i \(-0.487106\pi\)
0.0404973 + 0.999180i \(0.487106\pi\)
\(180\) 0 0
\(181\) −13.7767 −1.02401 −0.512006 0.858982i \(-0.671097\pi\)
−0.512006 + 0.858982i \(0.671097\pi\)
\(182\) 15.4107 1.14232
\(183\) 0 0
\(184\) −7.13194 −0.525773
\(185\) −0.234486 −0.0172397
\(186\) 0 0
\(187\) −5.53351 −0.404650
\(188\) 7.50921 0.547665
\(189\) 0 0
\(190\) 2.25868 0.163862
\(191\) 11.9517 0.864795 0.432397 0.901683i \(-0.357668\pi\)
0.432397 + 0.901683i \(0.357668\pi\)
\(192\) 0 0
\(193\) −24.0497 −1.73113 −0.865567 0.500793i \(-0.833042\pi\)
−0.865567 + 0.500793i \(0.833042\pi\)
\(194\) 0.0990721 0.00711296
\(195\) 0 0
\(196\) 3.33217 0.238012
\(197\) −7.92447 −0.564595 −0.282298 0.959327i \(-0.591097\pi\)
−0.282298 + 0.959327i \(0.591097\pi\)
\(198\) 0 0
\(199\) −0.100817 −0.00714674 −0.00357337 0.999994i \(-0.501137\pi\)
−0.00357337 + 0.999994i \(0.501137\pi\)
\(200\) 4.87733 0.344879
\(201\) 0 0
\(202\) 18.0388 1.26921
\(203\) 9.02169 0.633199
\(204\) 0 0
\(205\) 0.206623 0.0144312
\(206\) −1.76745 −0.123144
\(207\) 0 0
\(208\) −4.79431 −0.332426
\(209\) −21.5010 −1.48725
\(210\) 0 0
\(211\) −15.3981 −1.06005 −0.530024 0.847983i \(-0.677817\pi\)
−0.530024 + 0.847983i \(0.677817\pi\)
\(212\) 7.08046 0.486288
\(213\) 0 0
\(214\) 19.3478 1.32259
\(215\) −1.43179 −0.0976473
\(216\) 0 0
\(217\) 21.4161 1.45382
\(218\) 1.73637 0.117602
\(219\) 0 0
\(220\) 1.16776 0.0787302
\(221\) 7.95699 0.535245
\(222\) 0 0
\(223\) 1.00000 0.0669650
\(224\) −3.21437 −0.214769
\(225\) 0 0
\(226\) 10.6213 0.706516
\(227\) −3.07847 −0.204325 −0.102163 0.994768i \(-0.532576\pi\)
−0.102163 + 0.994768i \(0.532576\pi\)
\(228\) 0 0
\(229\) 18.5684 1.22703 0.613516 0.789682i \(-0.289755\pi\)
0.613516 + 0.789682i \(0.289755\pi\)
\(230\) −2.49794 −0.164709
\(231\) 0 0
\(232\) −2.80668 −0.184267
\(233\) −19.5683 −1.28196 −0.640981 0.767557i \(-0.721472\pi\)
−0.640981 + 0.767557i \(0.721472\pi\)
\(234\) 0 0
\(235\) 2.63008 0.171567
\(236\) 5.66381 0.368682
\(237\) 0 0
\(238\) 5.33480 0.345804
\(239\) 18.5672 1.20101 0.600505 0.799621i \(-0.294966\pi\)
0.600505 + 0.799621i \(0.294966\pi\)
\(240\) 0 0
\(241\) −0.634708 −0.0408852 −0.0204426 0.999791i \(-0.506508\pi\)
−0.0204426 + 0.999791i \(0.506508\pi\)
\(242\) −0.116202 −0.00746972
\(243\) 0 0
\(244\) −1.64659 −0.105412
\(245\) 1.16708 0.0745621
\(246\) 0 0
\(247\) 30.9177 1.96724
\(248\) −6.66261 −0.423076
\(249\) 0 0
\(250\) 3.45950 0.218798
\(251\) 9.07437 0.572769 0.286385 0.958115i \(-0.407546\pi\)
0.286385 + 0.958115i \(0.407546\pi\)
\(252\) 0 0
\(253\) 23.7786 1.49495
\(254\) −6.09477 −0.382420
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.7825 1.17162 0.585810 0.810448i \(-0.300776\pi\)
0.585810 + 0.810448i \(0.300776\pi\)
\(258\) 0 0
\(259\) −2.15198 −0.133717
\(260\) −1.67919 −0.104139
\(261\) 0 0
\(262\) −10.3181 −0.637456
\(263\) −0.476195 −0.0293634 −0.0146817 0.999892i \(-0.504674\pi\)
−0.0146817 + 0.999892i \(0.504674\pi\)
\(264\) 0 0
\(265\) 2.47991 0.152340
\(266\) 20.7289 1.27097
\(267\) 0 0
\(268\) 9.99722 0.610677
\(269\) 16.0647 0.979482 0.489741 0.871868i \(-0.337091\pi\)
0.489741 + 0.871868i \(0.337091\pi\)
\(270\) 0 0
\(271\) 14.6687 0.891057 0.445529 0.895268i \(-0.353016\pi\)
0.445529 + 0.895268i \(0.353016\pi\)
\(272\) −1.65967 −0.100632
\(273\) 0 0
\(274\) 2.82222 0.170497
\(275\) −16.2615 −0.980604
\(276\) 0 0
\(277\) −12.1320 −0.728942 −0.364471 0.931215i \(-0.618750\pi\)
−0.364471 + 0.931215i \(0.618750\pi\)
\(278\) 1.98204 0.118875
\(279\) 0 0
\(280\) −1.12582 −0.0672808
\(281\) 7.26142 0.433180 0.216590 0.976263i \(-0.430507\pi\)
0.216590 + 0.976263i \(0.430507\pi\)
\(282\) 0 0
\(283\) −32.8202 −1.95096 −0.975480 0.220089i \(-0.929365\pi\)
−0.975480 + 0.220089i \(0.929365\pi\)
\(284\) 10.6585 0.632466
\(285\) 0 0
\(286\) 15.9847 0.945196
\(287\) 1.89627 0.111933
\(288\) 0 0
\(289\) −14.2455 −0.837970
\(290\) −0.983030 −0.0577255
\(291\) 0 0
\(292\) 6.63197 0.388107
\(293\) 0.225898 0.0131971 0.00659855 0.999978i \(-0.497900\pi\)
0.00659855 + 0.999978i \(0.497900\pi\)
\(294\) 0 0
\(295\) 1.98373 0.115497
\(296\) 0.669487 0.0389131
\(297\) 0 0
\(298\) −13.7074 −0.794051
\(299\) −34.1927 −1.97742
\(300\) 0 0
\(301\) −13.1402 −0.757387
\(302\) −17.4598 −1.00470
\(303\) 0 0
\(304\) −6.44882 −0.369865
\(305\) −0.576714 −0.0330226
\(306\) 0 0
\(307\) −13.7531 −0.784930 −0.392465 0.919767i \(-0.628378\pi\)
−0.392465 + 0.919767i \(0.628378\pi\)
\(308\) 10.7170 0.610659
\(309\) 0 0
\(310\) −2.33356 −0.132537
\(311\) −6.47772 −0.367318 −0.183659 0.982990i \(-0.558794\pi\)
−0.183659 + 0.982990i \(0.558794\pi\)
\(312\) 0 0
\(313\) −13.0231 −0.736109 −0.368055 0.929804i \(-0.619976\pi\)
−0.368055 + 0.929804i \(0.619976\pi\)
\(314\) −13.3567 −0.753760
\(315\) 0 0
\(316\) 14.4826 0.814712
\(317\) 9.33785 0.524466 0.262233 0.965005i \(-0.415541\pi\)
0.262233 + 0.965005i \(0.415541\pi\)
\(318\) 0 0
\(319\) 9.35773 0.523932
\(320\) 0.350247 0.0195794
\(321\) 0 0
\(322\) −22.9247 −1.27754
\(323\) 10.7029 0.595527
\(324\) 0 0
\(325\) 23.3834 1.29708
\(326\) −5.76239 −0.319149
\(327\) 0 0
\(328\) −0.589936 −0.0325738
\(329\) 24.1374 1.33074
\(330\) 0 0
\(331\) 28.0987 1.54444 0.772221 0.635354i \(-0.219146\pi\)
0.772221 + 0.635354i \(0.219146\pi\)
\(332\) 8.59600 0.471767
\(333\) 0 0
\(334\) −19.3205 −1.05717
\(335\) 3.50149 0.191307
\(336\) 0 0
\(337\) 18.9674 1.03322 0.516609 0.856221i \(-0.327194\pi\)
0.516609 + 0.856221i \(0.327194\pi\)
\(338\) −9.98545 −0.543137
\(339\) 0 0
\(340\) −0.581295 −0.0315252
\(341\) 22.2138 1.20294
\(342\) 0 0
\(343\) −11.7898 −0.636588
\(344\) 4.08795 0.220407
\(345\) 0 0
\(346\) −9.74463 −0.523874
\(347\) 18.3610 0.985669 0.492834 0.870123i \(-0.335961\pi\)
0.492834 + 0.870123i \(0.335961\pi\)
\(348\) 0 0
\(349\) −28.1613 −1.50744 −0.753720 0.657196i \(-0.771743\pi\)
−0.753720 + 0.657196i \(0.771743\pi\)
\(350\) 15.6775 0.837999
\(351\) 0 0
\(352\) −3.33410 −0.177708
\(353\) 5.84323 0.311004 0.155502 0.987836i \(-0.450301\pi\)
0.155502 + 0.987836i \(0.450301\pi\)
\(354\) 0 0
\(355\) 3.73311 0.198133
\(356\) 0.107172 0.00568011
\(357\) 0 0
\(358\) −1.08363 −0.0572718
\(359\) −15.8671 −0.837435 −0.418717 0.908117i \(-0.637520\pi\)
−0.418717 + 0.908117i \(0.637520\pi\)
\(360\) 0 0
\(361\) 22.5873 1.18880
\(362\) 13.7767 0.724086
\(363\) 0 0
\(364\) −15.4107 −0.807740
\(365\) 2.32283 0.121582
\(366\) 0 0
\(367\) 7.90055 0.412405 0.206203 0.978509i \(-0.433889\pi\)
0.206203 + 0.978509i \(0.433889\pi\)
\(368\) 7.13194 0.371778
\(369\) 0 0
\(370\) 0.234486 0.0121903
\(371\) 22.7592 1.18160
\(372\) 0 0
\(373\) −11.3810 −0.589285 −0.294643 0.955608i \(-0.595201\pi\)
−0.294643 + 0.955608i \(0.595201\pi\)
\(374\) 5.53351 0.286131
\(375\) 0 0
\(376\) −7.50921 −0.387258
\(377\) −13.4561 −0.693024
\(378\) 0 0
\(379\) 14.6721 0.753658 0.376829 0.926283i \(-0.377014\pi\)
0.376829 + 0.926283i \(0.377014\pi\)
\(380\) −2.25868 −0.115868
\(381\) 0 0
\(382\) −11.9517 −0.611502
\(383\) −19.0623 −0.974036 −0.487018 0.873392i \(-0.661915\pi\)
−0.487018 + 0.873392i \(0.661915\pi\)
\(384\) 0 0
\(385\) 3.75360 0.191301
\(386\) 24.0497 1.22410
\(387\) 0 0
\(388\) −0.0990721 −0.00502963
\(389\) 6.04386 0.306436 0.153218 0.988192i \(-0.451036\pi\)
0.153218 + 0.988192i \(0.451036\pi\)
\(390\) 0 0
\(391\) −11.8367 −0.598607
\(392\) −3.33217 −0.168300
\(393\) 0 0
\(394\) 7.92447 0.399229
\(395\) 5.07250 0.255225
\(396\) 0 0
\(397\) 28.8374 1.44731 0.723653 0.690164i \(-0.242462\pi\)
0.723653 + 0.690164i \(0.242462\pi\)
\(398\) 0.100817 0.00505351
\(399\) 0 0
\(400\) −4.87733 −0.243866
\(401\) 4.13124 0.206304 0.103152 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(402\) 0 0
\(403\) −31.9426 −1.59118
\(404\) −18.0388 −0.897465
\(405\) 0 0
\(406\) −9.02169 −0.447739
\(407\) −2.23213 −0.110643
\(408\) 0 0
\(409\) 20.1566 0.996681 0.498340 0.866981i \(-0.333943\pi\)
0.498340 + 0.866981i \(0.333943\pi\)
\(410\) −0.206623 −0.0102044
\(411\) 0 0
\(412\) 1.76745 0.0870761
\(413\) 18.2056 0.895837
\(414\) 0 0
\(415\) 3.01072 0.147791
\(416\) 4.79431 0.235061
\(417\) 0 0
\(418\) 21.5010 1.05165
\(419\) −13.7607 −0.672253 −0.336127 0.941817i \(-0.609117\pi\)
−0.336127 + 0.941817i \(0.609117\pi\)
\(420\) 0 0
\(421\) −1.59634 −0.0778007 −0.0389003 0.999243i \(-0.512385\pi\)
−0.0389003 + 0.999243i \(0.512385\pi\)
\(422\) 15.3981 0.749567
\(423\) 0 0
\(424\) −7.08046 −0.343857
\(425\) 8.09477 0.392654
\(426\) 0 0
\(427\) −5.29276 −0.256134
\(428\) −19.3478 −0.935213
\(429\) 0 0
\(430\) 1.43179 0.0690471
\(431\) 36.7445 1.76992 0.884959 0.465669i \(-0.154186\pi\)
0.884959 + 0.465669i \(0.154186\pi\)
\(432\) 0 0
\(433\) 13.4318 0.645490 0.322745 0.946486i \(-0.395394\pi\)
0.322745 + 0.946486i \(0.395394\pi\)
\(434\) −21.4161 −1.02800
\(435\) 0 0
\(436\) −1.73637 −0.0831572
\(437\) −45.9926 −2.20012
\(438\) 0 0
\(439\) 21.9136 1.04588 0.522938 0.852371i \(-0.324836\pi\)
0.522938 + 0.852371i \(0.324836\pi\)
\(440\) −1.16776 −0.0556706
\(441\) 0 0
\(442\) −7.95699 −0.378476
\(443\) −1.80133 −0.0855839 −0.0427919 0.999084i \(-0.513625\pi\)
−0.0427919 + 0.999084i \(0.513625\pi\)
\(444\) 0 0
\(445\) 0.0375367 0.00177941
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) 3.21437 0.151865
\(449\) 19.3399 0.912709 0.456354 0.889798i \(-0.349155\pi\)
0.456354 + 0.889798i \(0.349155\pi\)
\(450\) 0 0
\(451\) 1.96690 0.0926179
\(452\) −10.6213 −0.499582
\(453\) 0 0
\(454\) 3.07847 0.144480
\(455\) −5.39755 −0.253041
\(456\) 0 0
\(457\) 14.7852 0.691624 0.345812 0.938304i \(-0.387603\pi\)
0.345812 + 0.938304i \(0.387603\pi\)
\(458\) −18.5684 −0.867642
\(459\) 0 0
\(460\) 2.49794 0.116467
\(461\) −11.6231 −0.541339 −0.270670 0.962672i \(-0.587245\pi\)
−0.270670 + 0.962672i \(0.587245\pi\)
\(462\) 0 0
\(463\) −18.0117 −0.837074 −0.418537 0.908200i \(-0.637457\pi\)
−0.418537 + 0.908200i \(0.637457\pi\)
\(464\) 2.80668 0.130297
\(465\) 0 0
\(466\) 19.5683 0.906484
\(467\) 28.5900 1.32299 0.661493 0.749951i \(-0.269923\pi\)
0.661493 + 0.749951i \(0.269923\pi\)
\(468\) 0 0
\(469\) 32.1347 1.48384
\(470\) −2.63008 −0.121316
\(471\) 0 0
\(472\) −5.66381 −0.260698
\(473\) −13.6296 −0.626690
\(474\) 0 0
\(475\) 31.4530 1.44316
\(476\) −5.33480 −0.244520
\(477\) 0 0
\(478\) −18.5672 −0.849242
\(479\) −36.8616 −1.68425 −0.842124 0.539284i \(-0.818695\pi\)
−0.842124 + 0.539284i \(0.818695\pi\)
\(480\) 0 0
\(481\) 3.20973 0.146351
\(482\) 0.634708 0.0289102
\(483\) 0 0
\(484\) 0.116202 0.00528189
\(485\) −0.0346997 −0.00157563
\(486\) 0 0
\(487\) 9.16658 0.415378 0.207689 0.978195i \(-0.433406\pi\)
0.207689 + 0.978195i \(0.433406\pi\)
\(488\) 1.64659 0.0745377
\(489\) 0 0
\(490\) −1.16708 −0.0527234
\(491\) −21.2418 −0.958630 −0.479315 0.877643i \(-0.659115\pi\)
−0.479315 + 0.877643i \(0.659115\pi\)
\(492\) 0 0
\(493\) −4.65816 −0.209793
\(494\) −30.9177 −1.39105
\(495\) 0 0
\(496\) 6.66261 0.299160
\(497\) 34.2604 1.53679
\(498\) 0 0
\(499\) −2.75001 −0.123107 −0.0615536 0.998104i \(-0.519606\pi\)
−0.0615536 + 0.998104i \(0.519606\pi\)
\(500\) −3.45950 −0.154714
\(501\) 0 0
\(502\) −9.07437 −0.405009
\(503\) −10.5226 −0.469178 −0.234589 0.972095i \(-0.575374\pi\)
−0.234589 + 0.972095i \(0.575374\pi\)
\(504\) 0 0
\(505\) −6.31804 −0.281149
\(506\) −23.7786 −1.05709
\(507\) 0 0
\(508\) 6.09477 0.270412
\(509\) −21.7076 −0.962172 −0.481086 0.876673i \(-0.659758\pi\)
−0.481086 + 0.876673i \(0.659758\pi\)
\(510\) 0 0
\(511\) 21.3176 0.943035
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.7825 −0.828461
\(515\) 0.619045 0.0272784
\(516\) 0 0
\(517\) 25.0364 1.10110
\(518\) 2.15198 0.0945525
\(519\) 0 0
\(520\) 1.67919 0.0736375
\(521\) −18.2299 −0.798666 −0.399333 0.916806i \(-0.630758\pi\)
−0.399333 + 0.916806i \(0.630758\pi\)
\(522\) 0 0
\(523\) 15.2016 0.664720 0.332360 0.943153i \(-0.392155\pi\)
0.332360 + 0.943153i \(0.392155\pi\)
\(524\) 10.3181 0.450749
\(525\) 0 0
\(526\) 0.476195 0.0207631
\(527\) −11.0577 −0.481683
\(528\) 0 0
\(529\) 27.8645 1.21150
\(530\) −2.47991 −0.107720
\(531\) 0 0
\(532\) −20.7289 −0.898711
\(533\) −2.82834 −0.122509
\(534\) 0 0
\(535\) −6.77652 −0.292975
\(536\) −9.99722 −0.431814
\(537\) 0 0
\(538\) −16.0647 −0.692599
\(539\) 11.1098 0.478532
\(540\) 0 0
\(541\) 26.4860 1.13872 0.569362 0.822087i \(-0.307190\pi\)
0.569362 + 0.822087i \(0.307190\pi\)
\(542\) −14.6687 −0.630073
\(543\) 0 0
\(544\) 1.65967 0.0711579
\(545\) −0.608159 −0.0260507
\(546\) 0 0
\(547\) −28.6800 −1.22627 −0.613134 0.789979i \(-0.710092\pi\)
−0.613134 + 0.789979i \(0.710092\pi\)
\(548\) −2.82222 −0.120559
\(549\) 0 0
\(550\) 16.2615 0.693392
\(551\) −18.0997 −0.771075
\(552\) 0 0
\(553\) 46.5525 1.97962
\(554\) 12.1320 0.515440
\(555\) 0 0
\(556\) −1.98204 −0.0840572
\(557\) −12.9805 −0.550000 −0.275000 0.961444i \(-0.588678\pi\)
−0.275000 + 0.961444i \(0.588678\pi\)
\(558\) 0 0
\(559\) 19.5989 0.828945
\(560\) 1.12582 0.0475747
\(561\) 0 0
\(562\) −7.26142 −0.306304
\(563\) 10.8955 0.459191 0.229596 0.973286i \(-0.426260\pi\)
0.229596 + 0.973286i \(0.426260\pi\)
\(564\) 0 0
\(565\) −3.72006 −0.156504
\(566\) 32.8202 1.37954
\(567\) 0 0
\(568\) −10.6585 −0.447221
\(569\) 0.263419 0.0110431 0.00552156 0.999985i \(-0.498242\pi\)
0.00552156 + 0.999985i \(0.498242\pi\)
\(570\) 0 0
\(571\) 13.6277 0.570303 0.285151 0.958483i \(-0.407956\pi\)
0.285151 + 0.958483i \(0.407956\pi\)
\(572\) −15.9847 −0.668354
\(573\) 0 0
\(574\) −1.89627 −0.0791489
\(575\) −34.7848 −1.45063
\(576\) 0 0
\(577\) 40.6443 1.69204 0.846022 0.533149i \(-0.178991\pi\)
0.846022 + 0.533149i \(0.178991\pi\)
\(578\) 14.2455 0.592534
\(579\) 0 0
\(580\) 0.983030 0.0408181
\(581\) 27.6307 1.14631
\(582\) 0 0
\(583\) 23.6069 0.977699
\(584\) −6.63197 −0.274433
\(585\) 0 0
\(586\) −0.225898 −0.00933176
\(587\) −26.5700 −1.09666 −0.548330 0.836262i \(-0.684736\pi\)
−0.548330 + 0.836262i \(0.684736\pi\)
\(588\) 0 0
\(589\) −42.9660 −1.77038
\(590\) −1.98373 −0.0816689
\(591\) 0 0
\(592\) −0.669487 −0.0275157
\(593\) −0.233160 −0.00957474 −0.00478737 0.999989i \(-0.501524\pi\)
−0.00478737 + 0.999989i \(0.501524\pi\)
\(594\) 0 0
\(595\) −1.86850 −0.0766009
\(596\) 13.7074 0.561479
\(597\) 0 0
\(598\) 34.1927 1.39825
\(599\) −30.6750 −1.25335 −0.626673 0.779282i \(-0.715584\pi\)
−0.626673 + 0.779282i \(0.715584\pi\)
\(600\) 0 0
\(601\) −12.5223 −0.510797 −0.255399 0.966836i \(-0.582207\pi\)
−0.255399 + 0.966836i \(0.582207\pi\)
\(602\) 13.1402 0.535553
\(603\) 0 0
\(604\) 17.4598 0.710428
\(605\) 0.0406993 0.00165466
\(606\) 0 0
\(607\) −25.0716 −1.01762 −0.508812 0.860878i \(-0.669915\pi\)
−0.508812 + 0.860878i \(0.669915\pi\)
\(608\) 6.44882 0.261534
\(609\) 0 0
\(610\) 0.576714 0.0233505
\(611\) −36.0015 −1.45647
\(612\) 0 0
\(613\) −20.0877 −0.811336 −0.405668 0.914021i \(-0.632961\pi\)
−0.405668 + 0.914021i \(0.632961\pi\)
\(614\) 13.7531 0.555029
\(615\) 0 0
\(616\) −10.7170 −0.431801
\(617\) −7.66162 −0.308445 −0.154223 0.988036i \(-0.549287\pi\)
−0.154223 + 0.988036i \(0.549287\pi\)
\(618\) 0 0
\(619\) −17.5002 −0.703390 −0.351695 0.936115i \(-0.614395\pi\)
−0.351695 + 0.936115i \(0.614395\pi\)
\(620\) 2.33356 0.0937179
\(621\) 0 0
\(622\) 6.47772 0.259733
\(623\) 0.344491 0.0138017
\(624\) 0 0
\(625\) 23.1750 0.926998
\(626\) 13.0231 0.520508
\(627\) 0 0
\(628\) 13.3567 0.532989
\(629\) 1.11113 0.0443036
\(630\) 0 0
\(631\) 4.04290 0.160945 0.0804726 0.996757i \(-0.474357\pi\)
0.0804726 + 0.996757i \(0.474357\pi\)
\(632\) −14.4826 −0.576088
\(633\) 0 0
\(634\) −9.33785 −0.370853
\(635\) 2.13468 0.0847120
\(636\) 0 0
\(637\) −15.9755 −0.632971
\(638\) −9.35773 −0.370476
\(639\) 0 0
\(640\) −0.350247 −0.0138447
\(641\) −39.0608 −1.54281 −0.771404 0.636346i \(-0.780445\pi\)
−0.771404 + 0.636346i \(0.780445\pi\)
\(642\) 0 0
\(643\) −49.2168 −1.94092 −0.970461 0.241258i \(-0.922440\pi\)
−0.970461 + 0.241258i \(0.922440\pi\)
\(644\) 22.9247 0.903359
\(645\) 0 0
\(646\) −10.7029 −0.421101
\(647\) 34.3816 1.35168 0.675840 0.737049i \(-0.263781\pi\)
0.675840 + 0.737049i \(0.263781\pi\)
\(648\) 0 0
\(649\) 18.8837 0.741249
\(650\) −23.3834 −0.917174
\(651\) 0 0
\(652\) 5.76239 0.225673
\(653\) 36.9090 1.44436 0.722180 0.691705i \(-0.243140\pi\)
0.722180 + 0.691705i \(0.243140\pi\)
\(654\) 0 0
\(655\) 3.61389 0.141206
\(656\) 0.589936 0.0230331
\(657\) 0 0
\(658\) −24.1374 −0.940973
\(659\) 27.6927 1.07876 0.539378 0.842064i \(-0.318660\pi\)
0.539378 + 0.842064i \(0.318660\pi\)
\(660\) 0 0
\(661\) 0.290938 0.0113162 0.00565808 0.999984i \(-0.498199\pi\)
0.00565808 + 0.999984i \(0.498199\pi\)
\(662\) −28.0987 −1.09209
\(663\) 0 0
\(664\) −8.59600 −0.333589
\(665\) −7.26023 −0.281540
\(666\) 0 0
\(667\) 20.0170 0.775063
\(668\) 19.3205 0.747532
\(669\) 0 0
\(670\) −3.50149 −0.135275
\(671\) −5.48990 −0.211935
\(672\) 0 0
\(673\) −20.3379 −0.783968 −0.391984 0.919972i \(-0.628211\pi\)
−0.391984 + 0.919972i \(0.628211\pi\)
\(674\) −18.9674 −0.730596
\(675\) 0 0
\(676\) 9.98545 0.384056
\(677\) −37.5843 −1.44448 −0.722240 0.691642i \(-0.756888\pi\)
−0.722240 + 0.691642i \(0.756888\pi\)
\(678\) 0 0
\(679\) −0.318454 −0.0122212
\(680\) 0.581295 0.0222917
\(681\) 0 0
\(682\) −22.2138 −0.850610
\(683\) 29.3744 1.12398 0.561991 0.827144i \(-0.310036\pi\)
0.561991 + 0.827144i \(0.310036\pi\)
\(684\) 0 0
\(685\) −0.988475 −0.0377677
\(686\) 11.7898 0.450135
\(687\) 0 0
\(688\) −4.08795 −0.155852
\(689\) −33.9459 −1.29324
\(690\) 0 0
\(691\) −34.7978 −1.32377 −0.661885 0.749605i \(-0.730243\pi\)
−0.661885 + 0.749605i \(0.730243\pi\)
\(692\) 9.74463 0.370435
\(693\) 0 0
\(694\) −18.3610 −0.696973
\(695\) −0.694203 −0.0263326
\(696\) 0 0
\(697\) −0.979101 −0.0370861
\(698\) 28.1613 1.06592
\(699\) 0 0
\(700\) −15.6775 −0.592555
\(701\) 25.3023 0.955655 0.477827 0.878454i \(-0.341424\pi\)
0.477827 + 0.878454i \(0.341424\pi\)
\(702\) 0 0
\(703\) 4.31740 0.162834
\(704\) 3.33410 0.125659
\(705\) 0 0
\(706\) −5.84323 −0.219913
\(707\) −57.9834 −2.18069
\(708\) 0 0
\(709\) 25.8439 0.970588 0.485294 0.874351i \(-0.338713\pi\)
0.485294 + 0.874351i \(0.338713\pi\)
\(710\) −3.73311 −0.140101
\(711\) 0 0
\(712\) −0.107172 −0.00401645
\(713\) 47.5173 1.77954
\(714\) 0 0
\(715\) −5.59860 −0.209376
\(716\) 1.08363 0.0404973
\(717\) 0 0
\(718\) 15.8671 0.592156
\(719\) 16.0692 0.599280 0.299640 0.954052i \(-0.403133\pi\)
0.299640 + 0.954052i \(0.403133\pi\)
\(720\) 0 0
\(721\) 5.68124 0.211581
\(722\) −22.5873 −0.840611
\(723\) 0 0
\(724\) −13.7767 −0.512006
\(725\) −13.6891 −0.508400
\(726\) 0 0
\(727\) 39.4748 1.46404 0.732019 0.681284i \(-0.238578\pi\)
0.732019 + 0.681284i \(0.238578\pi\)
\(728\) 15.4107 0.571158
\(729\) 0 0
\(730\) −2.32283 −0.0859717
\(731\) 6.78465 0.250939
\(732\) 0 0
\(733\) −43.7914 −1.61747 −0.808735 0.588173i \(-0.799848\pi\)
−0.808735 + 0.588173i \(0.799848\pi\)
\(734\) −7.90055 −0.291614
\(735\) 0 0
\(736\) −7.13194 −0.262887
\(737\) 33.3317 1.22779
\(738\) 0 0
\(739\) −6.68664 −0.245972 −0.122986 0.992408i \(-0.539247\pi\)
−0.122986 + 0.992408i \(0.539247\pi\)
\(740\) −0.234486 −0.00861987
\(741\) 0 0
\(742\) −22.7592 −0.835517
\(743\) 24.1544 0.886138 0.443069 0.896488i \(-0.353890\pi\)
0.443069 + 0.896488i \(0.353890\pi\)
\(744\) 0 0
\(745\) 4.80099 0.175895
\(746\) 11.3810 0.416688
\(747\) 0 0
\(748\) −5.53351 −0.202325
\(749\) −62.1911 −2.27241
\(750\) 0 0
\(751\) 3.32458 0.121316 0.0606578 0.998159i \(-0.480680\pi\)
0.0606578 + 0.998159i \(0.480680\pi\)
\(752\) 7.50921 0.273833
\(753\) 0 0
\(754\) 13.4561 0.490042
\(755\) 6.11523 0.222556
\(756\) 0 0
\(757\) 36.9677 1.34361 0.671806 0.740727i \(-0.265519\pi\)
0.671806 + 0.740727i \(0.265519\pi\)
\(758\) −14.6721 −0.532916
\(759\) 0 0
\(760\) 2.25868 0.0819309
\(761\) −37.4329 −1.35694 −0.678471 0.734627i \(-0.737357\pi\)
−0.678471 + 0.734627i \(0.737357\pi\)
\(762\) 0 0
\(763\) −5.58134 −0.202058
\(764\) 11.9517 0.432397
\(765\) 0 0
\(766\) 19.0623 0.688747
\(767\) −27.1541 −0.980477
\(768\) 0 0
\(769\) −31.2615 −1.12732 −0.563660 0.826007i \(-0.690607\pi\)
−0.563660 + 0.826007i \(0.690607\pi\)
\(770\) −3.75360 −0.135270
\(771\) 0 0
\(772\) −24.0497 −0.865567
\(773\) −6.58037 −0.236679 −0.118340 0.992973i \(-0.537757\pi\)
−0.118340 + 0.992973i \(0.537757\pi\)
\(774\) 0 0
\(775\) −32.4957 −1.16728
\(776\) 0.0990721 0.00355648
\(777\) 0 0
\(778\) −6.04386 −0.216683
\(779\) −3.80439 −0.136307
\(780\) 0 0
\(781\) 35.5365 1.27159
\(782\) 11.8367 0.423279
\(783\) 0 0
\(784\) 3.33217 0.119006
\(785\) 4.67813 0.166970
\(786\) 0 0
\(787\) −34.8855 −1.24353 −0.621767 0.783202i \(-0.713585\pi\)
−0.621767 + 0.783202i \(0.713585\pi\)
\(788\) −7.92447 −0.282298
\(789\) 0 0
\(790\) −5.07250 −0.180471
\(791\) −34.1407 −1.21390
\(792\) 0 0
\(793\) 7.89428 0.280334
\(794\) −28.8374 −1.02340
\(795\) 0 0
\(796\) −0.100817 −0.00357337
\(797\) −49.6370 −1.75823 −0.879116 0.476608i \(-0.841866\pi\)
−0.879116 + 0.476608i \(0.841866\pi\)
\(798\) 0 0
\(799\) −12.4628 −0.440903
\(800\) 4.87733 0.172440
\(801\) 0 0
\(802\) −4.13124 −0.145879
\(803\) 22.1116 0.780303
\(804\) 0 0
\(805\) 8.02930 0.282996
\(806\) 31.9426 1.12513
\(807\) 0 0
\(808\) 18.0388 0.634604
\(809\) 19.2533 0.676911 0.338455 0.940982i \(-0.390096\pi\)
0.338455 + 0.940982i \(0.390096\pi\)
\(810\) 0 0
\(811\) 19.4871 0.684286 0.342143 0.939648i \(-0.388847\pi\)
0.342143 + 0.939648i \(0.388847\pi\)
\(812\) 9.02169 0.316599
\(813\) 0 0
\(814\) 2.23213 0.0782363
\(815\) 2.01826 0.0706966
\(816\) 0 0
\(817\) 26.3624 0.922305
\(818\) −20.1566 −0.704760
\(819\) 0 0
\(820\) 0.206623 0.00721560
\(821\) 17.7292 0.618754 0.309377 0.950939i \(-0.399879\pi\)
0.309377 + 0.950939i \(0.399879\pi\)
\(822\) 0 0
\(823\) −36.0400 −1.25628 −0.628138 0.778102i \(-0.716183\pi\)
−0.628138 + 0.778102i \(0.716183\pi\)
\(824\) −1.76745 −0.0615721
\(825\) 0 0
\(826\) −18.2056 −0.633453
\(827\) 35.3287 1.22850 0.614250 0.789111i \(-0.289459\pi\)
0.614250 + 0.789111i \(0.289459\pi\)
\(828\) 0 0
\(829\) 15.5468 0.539964 0.269982 0.962865i \(-0.412982\pi\)
0.269982 + 0.962865i \(0.412982\pi\)
\(830\) −3.01072 −0.104504
\(831\) 0 0
\(832\) −4.79431 −0.166213
\(833\) −5.53031 −0.191614
\(834\) 0 0
\(835\) 6.76694 0.234180
\(836\) −21.5010 −0.743627
\(837\) 0 0
\(838\) 13.7607 0.475355
\(839\) −43.6500 −1.50697 −0.753483 0.657467i \(-0.771628\pi\)
−0.753483 + 0.657467i \(0.771628\pi\)
\(840\) 0 0
\(841\) −21.1226 −0.728364
\(842\) 1.59634 0.0550134
\(843\) 0 0
\(844\) −15.3981 −0.530024
\(845\) 3.49737 0.120313
\(846\) 0 0
\(847\) 0.373515 0.0128341
\(848\) 7.08046 0.243144
\(849\) 0 0
\(850\) −8.09477 −0.277648
\(851\) −4.77474 −0.163676
\(852\) 0 0
\(853\) 22.5327 0.771506 0.385753 0.922602i \(-0.373942\pi\)
0.385753 + 0.922602i \(0.373942\pi\)
\(854\) 5.29276 0.181114
\(855\) 0 0
\(856\) 19.3478 0.661296
\(857\) −24.7850 −0.846641 −0.423321 0.905980i \(-0.639136\pi\)
−0.423321 + 0.905980i \(0.639136\pi\)
\(858\) 0 0
\(859\) −28.8346 −0.983824 −0.491912 0.870645i \(-0.663702\pi\)
−0.491912 + 0.870645i \(0.663702\pi\)
\(860\) −1.43179 −0.0488237
\(861\) 0 0
\(862\) −36.7445 −1.25152
\(863\) 39.4609 1.34326 0.671632 0.740885i \(-0.265593\pi\)
0.671632 + 0.740885i \(0.265593\pi\)
\(864\) 0 0
\(865\) 3.41303 0.116046
\(866\) −13.4318 −0.456430
\(867\) 0 0
\(868\) 21.4161 0.726909
\(869\) 48.2865 1.63801
\(870\) 0 0
\(871\) −47.9298 −1.62404
\(872\) 1.73637 0.0588010
\(873\) 0 0
\(874\) 45.9926 1.55572
\(875\) −11.1201 −0.375929
\(876\) 0 0
\(877\) −37.5489 −1.26793 −0.633967 0.773360i \(-0.718575\pi\)
−0.633967 + 0.773360i \(0.718575\pi\)
\(878\) −21.9136 −0.739547
\(879\) 0 0
\(880\) 1.16776 0.0393651
\(881\) −33.8608 −1.14080 −0.570399 0.821368i \(-0.693211\pi\)
−0.570399 + 0.821368i \(0.693211\pi\)
\(882\) 0 0
\(883\) −34.0649 −1.14637 −0.573187 0.819425i \(-0.694293\pi\)
−0.573187 + 0.819425i \(0.694293\pi\)
\(884\) 7.95699 0.267623
\(885\) 0 0
\(886\) 1.80133 0.0605169
\(887\) −19.0289 −0.638928 −0.319464 0.947598i \(-0.603503\pi\)
−0.319464 + 0.947598i \(0.603503\pi\)
\(888\) 0 0
\(889\) 19.5908 0.657056
\(890\) −0.0375367 −0.00125823
\(891\) 0 0
\(892\) 1.00000 0.0334825
\(893\) −48.4255 −1.62050
\(894\) 0 0
\(895\) 0.379539 0.0126866
\(896\) −3.21437 −0.107385
\(897\) 0 0
\(898\) −19.3399 −0.645382
\(899\) 18.6998 0.623673
\(900\) 0 0
\(901\) −11.7512 −0.391491
\(902\) −1.96690 −0.0654908
\(903\) 0 0
\(904\) 10.6213 0.353258
\(905\) −4.82524 −0.160396
\(906\) 0 0
\(907\) −39.3405 −1.30628 −0.653141 0.757237i \(-0.726549\pi\)
−0.653141 + 0.757237i \(0.726549\pi\)
\(908\) −3.07847 −0.102163
\(909\) 0 0
\(910\) 5.39755 0.178927
\(911\) 45.3243 1.50166 0.750831 0.660495i \(-0.229653\pi\)
0.750831 + 0.660495i \(0.229653\pi\)
\(912\) 0 0
\(913\) 28.6599 0.948504
\(914\) −14.7852 −0.489052
\(915\) 0 0
\(916\) 18.5684 0.613516
\(917\) 33.1663 1.09525
\(918\) 0 0
\(919\) −33.4526 −1.10350 −0.551749 0.834010i \(-0.686039\pi\)
−0.551749 + 0.834010i \(0.686039\pi\)
\(920\) −2.49794 −0.0823546
\(921\) 0 0
\(922\) 11.6231 0.382785
\(923\) −51.1002 −1.68198
\(924\) 0 0
\(925\) 3.26531 0.107363
\(926\) 18.0117 0.591901
\(927\) 0 0
\(928\) −2.80668 −0.0921337
\(929\) 0.357135 0.0117172 0.00585860 0.999983i \(-0.498135\pi\)
0.00585860 + 0.999983i \(0.498135\pi\)
\(930\) 0 0
\(931\) −21.4885 −0.704259
\(932\) −19.5683 −0.640981
\(933\) 0 0
\(934\) −28.5900 −0.935492
\(935\) −1.93809 −0.0633825
\(936\) 0 0
\(937\) 12.9547 0.423212 0.211606 0.977355i \(-0.432131\pi\)
0.211606 + 0.977355i \(0.432131\pi\)
\(938\) −32.1347 −1.04924
\(939\) 0 0
\(940\) 2.63008 0.0857837
\(941\) 21.7300 0.708377 0.354189 0.935174i \(-0.384757\pi\)
0.354189 + 0.935174i \(0.384757\pi\)
\(942\) 0 0
\(943\) 4.20739 0.137011
\(944\) 5.66381 0.184341
\(945\) 0 0
\(946\) 13.6296 0.443137
\(947\) −40.0061 −1.30002 −0.650012 0.759924i \(-0.725236\pi\)
−0.650012 + 0.759924i \(0.725236\pi\)
\(948\) 0 0
\(949\) −31.7958 −1.03213
\(950\) −31.4530 −1.02047
\(951\) 0 0
\(952\) 5.33480 0.172902
\(953\) −40.6002 −1.31517 −0.657585 0.753380i \(-0.728422\pi\)
−0.657585 + 0.753380i \(0.728422\pi\)
\(954\) 0 0
\(955\) 4.18605 0.135457
\(956\) 18.5672 0.600505
\(957\) 0 0
\(958\) 36.8616 1.19094
\(959\) −9.07167 −0.292939
\(960\) 0 0
\(961\) 13.3903 0.431946
\(962\) −3.20973 −0.103486
\(963\) 0 0
\(964\) −0.634708 −0.0204426
\(965\) −8.42333 −0.271157
\(966\) 0 0
\(967\) −27.5187 −0.884940 −0.442470 0.896783i \(-0.645898\pi\)
−0.442470 + 0.896783i \(0.645898\pi\)
\(968\) −0.116202 −0.00373486
\(969\) 0 0
\(970\) 0.0346997 0.00111414
\(971\) −7.20664 −0.231272 −0.115636 0.993292i \(-0.536891\pi\)
−0.115636 + 0.993292i \(0.536891\pi\)
\(972\) 0 0
\(973\) −6.37100 −0.204245
\(974\) −9.16658 −0.293716
\(975\) 0 0
\(976\) −1.64659 −0.0527061
\(977\) 46.0399 1.47295 0.736474 0.676466i \(-0.236489\pi\)
0.736474 + 0.676466i \(0.236489\pi\)
\(978\) 0 0
\(979\) 0.357322 0.0114201
\(980\) 1.16708 0.0372810
\(981\) 0 0
\(982\) 21.2418 0.677854
\(983\) −31.4981 −1.00463 −0.502316 0.864684i \(-0.667519\pi\)
−0.502316 + 0.864684i \(0.667519\pi\)
\(984\) 0 0
\(985\) −2.77552 −0.0884355
\(986\) 4.65816 0.148346
\(987\) 0 0
\(988\) 30.9177 0.983622
\(989\) −29.1550 −0.927074
\(990\) 0 0
\(991\) −49.4685 −1.57142 −0.785709 0.618597i \(-0.787702\pi\)
−0.785709 + 0.618597i \(0.787702\pi\)
\(992\) −6.66261 −0.211538
\(993\) 0 0
\(994\) −34.2604 −1.08667
\(995\) −0.0353109 −0.00111943
\(996\) 0 0
\(997\) 1.19903 0.0379735 0.0189868 0.999820i \(-0.493956\pi\)
0.0189868 + 0.999820i \(0.493956\pi\)
\(998\) 2.75001 0.0870499
\(999\) 0 0
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 4014.2.a.x.1.4 8
3.2 odd 2 4014.2.a.y.1.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.a.x.1.4 8 1.1 even 1 trivial
4014.2.a.y.1.5 yes 8 3.2 odd 2