Properties

Label 4014.2.d.a.4013.5
Level $4014$
Weight $2$
Character 4014.4013
Analytic conductor $32.052$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4014,2,Mod(4013,4014)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4014.4013");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4013.5
Character \(\chi\) \(=\) 4014.4013
Dual form 4014.2.d.a.4013.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.31709 q^{5} -2.47546 q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -2.31709 q^{5} -2.47546 q^{7} +1.00000i q^{8} +2.31709i q^{10} -4.47607 q^{11} -3.18579i q^{13} +2.47546i q^{14} +1.00000 q^{16} -6.92873i q^{17} -3.20422 q^{19} +2.31709 q^{20} +4.47607i q^{22} -4.79334 q^{23} +0.368920 q^{25} -3.18579 q^{26} +2.47546 q^{28} -4.97704i q^{29} -3.06904 q^{31} -1.00000i q^{32} -6.92873 q^{34} +5.73588 q^{35} +1.43114 q^{37} +3.20422i q^{38} -2.31709i q^{40} +5.12875i q^{41} +4.70388 q^{43} +4.47607 q^{44} +4.79334i q^{46} -5.15156i q^{47} -0.872083 q^{49} -0.368920i q^{50} +3.18579i q^{52} -1.42456i q^{53} +10.3715 q^{55} -2.47546i q^{56} -4.97704 q^{58} -7.24113 q^{59} -3.09020i q^{61} +3.06904i q^{62} -1.00000 q^{64} +7.38178i q^{65} +0.492578i q^{67} +6.92873i q^{68} -5.73588i q^{70} -11.1458 q^{71} +12.2783 q^{73} -1.43114i q^{74} +3.20422 q^{76} +11.0803 q^{77} +7.55115i q^{79} -2.31709 q^{80} +5.12875 q^{82} +5.70648i q^{83} +16.0545i q^{85} -4.70388i q^{86} -4.47607i q^{88} -10.7939i q^{89} +7.88632i q^{91} +4.79334 q^{92} -5.15156 q^{94} +7.42447 q^{95} +7.87729i q^{97} +0.872083i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times\).

\(n\) \(893\) \(2233\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.31709 −1.03624 −0.518118 0.855309i \(-0.673367\pi\)
−0.518118 + 0.855309i \(0.673367\pi\)
\(6\) 0 0
\(7\) −2.47546 −0.935637 −0.467819 0.883825i \(-0.654960\pi\)
−0.467819 + 0.883825i \(0.654960\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.31709i 0.732729i
\(11\) −4.47607 −1.34959 −0.674793 0.738007i \(-0.735767\pi\)
−0.674793 + 0.738007i \(0.735767\pi\)
\(12\) 0 0
\(13\) 3.18579i 0.883580i −0.897118 0.441790i \(-0.854344\pi\)
0.897118 0.441790i \(-0.145656\pi\)
\(14\) 2.47546i 0.661595i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.92873i 1.68046i −0.542227 0.840232i \(-0.682419\pi\)
0.542227 0.840232i \(-0.317581\pi\)
\(18\) 0 0
\(19\) −3.20422 −0.735098 −0.367549 0.930004i \(-0.619803\pi\)
−0.367549 + 0.930004i \(0.619803\pi\)
\(20\) 2.31709 0.518118
\(21\) 0 0
\(22\) 4.47607i 0.954301i
\(23\) −4.79334 −0.999480 −0.499740 0.866175i \(-0.666571\pi\)
−0.499740 + 0.866175i \(0.666571\pi\)
\(24\) 0 0
\(25\) 0.368920 0.0737839
\(26\) −3.18579 −0.624786
\(27\) 0 0
\(28\) 2.47546 0.467819
\(29\) 4.97704i 0.924213i −0.886824 0.462107i \(-0.847094\pi\)
0.886824 0.462107i \(-0.152906\pi\)
\(30\) 0 0
\(31\) −3.06904 −0.551215 −0.275608 0.961270i \(-0.588879\pi\)
−0.275608 + 0.961270i \(0.588879\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.92873 −1.18827
\(35\) 5.73588 0.969540
\(36\) 0 0
\(37\) 1.43114 0.235278 0.117639 0.993056i \(-0.462467\pi\)
0.117639 + 0.993056i \(0.462467\pi\)
\(38\) 3.20422i 0.519793i
\(39\) 0 0
\(40\) 2.31709i 0.366365i
\(41\) 5.12875i 0.800976i 0.916302 + 0.400488i \(0.131159\pi\)
−0.916302 + 0.400488i \(0.868841\pi\)
\(42\) 0 0
\(43\) 4.70388 0.717335 0.358668 0.933465i \(-0.383231\pi\)
0.358668 + 0.933465i \(0.383231\pi\)
\(44\) 4.47607 0.674793
\(45\) 0 0
\(46\) 4.79334i 0.706739i
\(47\) 5.15156i 0.751432i −0.926735 0.375716i \(-0.877397\pi\)
0.926735 0.375716i \(-0.122603\pi\)
\(48\) 0 0
\(49\) −0.872083 −0.124583
\(50\) 0.368920i 0.0521731i
\(51\) 0 0
\(52\) 3.18579i 0.441790i
\(53\) 1.42456i 0.195678i −0.995202 0.0978391i \(-0.968807\pi\)
0.995202 0.0978391i \(-0.0311930\pi\)
\(54\) 0 0
\(55\) 10.3715 1.39849
\(56\) 2.47546i 0.330798i
\(57\) 0 0
\(58\) −4.97704 −0.653518
\(59\) −7.24113 −0.942715 −0.471357 0.881942i \(-0.656236\pi\)
−0.471357 + 0.881942i \(0.656236\pi\)
\(60\) 0 0
\(61\) 3.09020i 0.395659i −0.980236 0.197830i \(-0.936611\pi\)
0.980236 0.197830i \(-0.0633893\pi\)
\(62\) 3.06904i 0.389768i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 7.38178i 0.915597i
\(66\) 0 0
\(67\) 0.492578i 0.0601780i 0.999547 + 0.0300890i \(0.00957907\pi\)
−0.999547 + 0.0300890i \(0.990421\pi\)
\(68\) 6.92873i 0.840232i
\(69\) 0 0
\(70\) 5.73588i 0.685569i
\(71\) −11.1458 −1.32276 −0.661382 0.750049i \(-0.730030\pi\)
−0.661382 + 0.750049i \(0.730030\pi\)
\(72\) 0 0
\(73\) 12.2783 1.43707 0.718536 0.695490i \(-0.244813\pi\)
0.718536 + 0.695490i \(0.244813\pi\)
\(74\) 1.43114i 0.166367i
\(75\) 0 0
\(76\) 3.20422 0.367549
\(77\) 11.0803 1.26272
\(78\) 0 0
\(79\) 7.55115i 0.849570i 0.905294 + 0.424785i \(0.139650\pi\)
−0.905294 + 0.424785i \(0.860350\pi\)
\(80\) −2.31709 −0.259059
\(81\) 0 0
\(82\) 5.12875 0.566376
\(83\) 5.70648i 0.626368i 0.949693 + 0.313184i \(0.101396\pi\)
−0.949693 + 0.313184i \(0.898604\pi\)
\(84\) 0 0
\(85\) 16.0545i 1.74136i
\(86\) 4.70388i 0.507233i
\(87\) 0 0
\(88\) 4.47607i 0.477150i
\(89\) 10.7939i 1.14415i −0.820200 0.572077i \(-0.806138\pi\)
0.820200 0.572077i \(-0.193862\pi\)
\(90\) 0 0
\(91\) 7.88632i 0.826710i
\(92\) 4.79334 0.499740
\(93\) 0 0
\(94\) −5.15156 −0.531342
\(95\) 7.42447 0.761734
\(96\) 0 0
\(97\) 7.87729i 0.799817i 0.916555 + 0.399909i \(0.130958\pi\)
−0.916555 + 0.399909i \(0.869042\pi\)
\(98\) 0.872083i 0.0880937i
\(99\) 0 0
\(100\) −0.368920 −0.0368920
\(101\) 1.15442i 0.114869i 0.998349 + 0.0574343i \(0.0182920\pi\)
−0.998349 + 0.0574343i \(0.981708\pi\)
\(102\) 0 0
\(103\) 11.8044i 1.16312i 0.813504 + 0.581559i \(0.197557\pi\)
−0.813504 + 0.581559i \(0.802443\pi\)
\(104\) 3.18579 0.312393
\(105\) 0 0
\(106\) −1.42456 −0.138365
\(107\) −15.3578 −1.48470 −0.742348 0.670014i \(-0.766288\pi\)
−0.742348 + 0.670014i \(0.766288\pi\)
\(108\) 0 0
\(109\) 9.03262 0.865168 0.432584 0.901594i \(-0.357602\pi\)
0.432584 + 0.901594i \(0.357602\pi\)
\(110\) 10.3715i 0.988881i
\(111\) 0 0
\(112\) −2.47546 −0.233909
\(113\) −5.90600 −0.555590 −0.277795 0.960640i \(-0.589604\pi\)
−0.277795 + 0.960640i \(0.589604\pi\)
\(114\) 0 0
\(115\) 11.1066 1.03570
\(116\) 4.97704i 0.462107i
\(117\) 0 0
\(118\) 7.24113i 0.666600i
\(119\) 17.1518i 1.57230i
\(120\) 0 0
\(121\) 9.03519 0.821381
\(122\) −3.09020 −0.279774
\(123\) 0 0
\(124\) 3.06904 0.275608
\(125\) 10.7306 0.959778
\(126\) 0 0
\(127\) 8.62356 0.765217 0.382609 0.923911i \(-0.375026\pi\)
0.382609 + 0.923911i \(0.375026\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 7.38178 0.647425
\(131\) 5.61511i 0.490594i 0.969448 + 0.245297i \(0.0788855\pi\)
−0.969448 + 0.245297i \(0.921114\pi\)
\(132\) 0 0
\(133\) 7.93192 0.687785
\(134\) 0.492578 0.0425523
\(135\) 0 0
\(136\) 6.92873 0.594134
\(137\) 17.9442 1.53308 0.766539 0.642197i \(-0.221977\pi\)
0.766539 + 0.642197i \(0.221977\pi\)
\(138\) 0 0
\(139\) −3.95347 −0.335329 −0.167665 0.985844i \(-0.553623\pi\)
−0.167665 + 0.985844i \(0.553623\pi\)
\(140\) −5.73588 −0.484770
\(141\) 0 0
\(142\) 11.1458i 0.935335i
\(143\) 14.2598i 1.19247i
\(144\) 0 0
\(145\) 11.5323i 0.957703i
\(146\) 12.2783i 1.01616i
\(147\) 0 0
\(148\) −1.43114 −0.117639
\(149\) 13.7892 1.12965 0.564827 0.825209i \(-0.308943\pi\)
0.564827 + 0.825209i \(0.308943\pi\)
\(150\) 0 0
\(151\) 10.4468i 0.850148i −0.905159 0.425074i \(-0.860248\pi\)
0.905159 0.425074i \(-0.139752\pi\)
\(152\) 3.20422i 0.259896i
\(153\) 0 0
\(154\) 11.0803i 0.892879i
\(155\) 7.11124 0.571189
\(156\) 0 0
\(157\) 12.1420i 0.969034i 0.874782 + 0.484517i \(0.161005\pi\)
−0.874782 + 0.484517i \(0.838995\pi\)
\(158\) 7.55115 0.600737
\(159\) 0 0
\(160\) 2.31709i 0.183182i
\(161\) 11.8657 0.935151
\(162\) 0 0
\(163\) 14.7257i 1.15341i 0.816954 + 0.576703i \(0.195661\pi\)
−0.816954 + 0.576703i \(0.804339\pi\)
\(164\) 5.12875i 0.400488i
\(165\) 0 0
\(166\) 5.70648 0.442909
\(167\) 7.54710 0.584012 0.292006 0.956416i \(-0.405677\pi\)
0.292006 + 0.956416i \(0.405677\pi\)
\(168\) 0 0
\(169\) 2.85072 0.219286
\(170\) 16.0545 1.23133
\(171\) 0 0
\(172\) −4.70388 −0.358668
\(173\) −9.83355 −0.747631 −0.373815 0.927503i \(-0.621951\pi\)
−0.373815 + 0.927503i \(0.621951\pi\)
\(174\) 0 0
\(175\) −0.913247 −0.0690350
\(176\) −4.47607 −0.337396
\(177\) 0 0
\(178\) −10.7939 −0.809039
\(179\) 14.8865i 1.11267i −0.830958 0.556335i \(-0.812207\pi\)
0.830958 0.556335i \(-0.187793\pi\)
\(180\) 0 0
\(181\) −12.6535 −0.940525 −0.470262 0.882527i \(-0.655841\pi\)
−0.470262 + 0.882527i \(0.655841\pi\)
\(182\) 7.88632 0.584573
\(183\) 0 0
\(184\) 4.79334i 0.353370i
\(185\) −3.31608 −0.243803
\(186\) 0 0
\(187\) 31.0135i 2.26793i
\(188\) 5.15156i 0.375716i
\(189\) 0 0
\(190\) 7.42447i 0.538628i
\(191\) −23.5626 −1.70493 −0.852466 0.522783i \(-0.824894\pi\)
−0.852466 + 0.522783i \(0.824894\pi\)
\(192\) 0 0
\(193\) 7.04534i 0.507135i −0.967318 0.253567i \(-0.918396\pi\)
0.967318 0.253567i \(-0.0816039\pi\)
\(194\) 7.87729 0.565556
\(195\) 0 0
\(196\) 0.872083 0.0622916
\(197\) 15.3685i 1.09496i −0.836819 0.547480i \(-0.815587\pi\)
0.836819 0.547480i \(-0.184413\pi\)
\(198\) 0 0
\(199\) −4.89975 −0.347334 −0.173667 0.984804i \(-0.555562\pi\)
−0.173667 + 0.984804i \(0.555562\pi\)
\(200\) 0.368920i 0.0260866i
\(201\) 0 0
\(202\) 1.15442 0.0812244
\(203\) 12.3205i 0.864728i
\(204\) 0 0
\(205\) 11.8838i 0.830000i
\(206\) 11.8044 0.822449
\(207\) 0 0
\(208\) 3.18579i 0.220895i
\(209\) 14.3423 0.992077
\(210\) 0 0
\(211\) 22.7020 1.56287 0.781434 0.623988i \(-0.214489\pi\)
0.781434 + 0.623988i \(0.214489\pi\)
\(212\) 1.42456i 0.0978391i
\(213\) 0 0
\(214\) 15.3578i 1.04984i
\(215\) −10.8993 −0.743328
\(216\) 0 0
\(217\) 7.59728 0.515737
\(218\) 9.03262i 0.611766i
\(219\) 0 0
\(220\) −10.3715 −0.699244
\(221\) −22.0735 −1.48483
\(222\) 0 0
\(223\) 8.80524 12.0610i 0.589642 0.807664i
\(224\) 2.47546i 0.165399i
\(225\) 0 0
\(226\) 5.90600i 0.392861i
\(227\) 10.0503i 0.667062i −0.942739 0.333531i \(-0.891760\pi\)
0.942739 0.333531i \(-0.108240\pi\)
\(228\) 0 0
\(229\) 1.88715i 0.124706i 0.998054 + 0.0623532i \(0.0198605\pi\)
−0.998054 + 0.0623532i \(0.980139\pi\)
\(230\) 11.1066i 0.732348i
\(231\) 0 0
\(232\) 4.97704 0.326759
\(233\) −5.88159 −0.385315 −0.192658 0.981266i \(-0.561711\pi\)
−0.192658 + 0.981266i \(0.561711\pi\)
\(234\) 0 0
\(235\) 11.9366i 0.778660i
\(236\) 7.24113 0.471357
\(237\) 0 0
\(238\) 17.1518 1.11179
\(239\) 8.02633i 0.519180i −0.965719 0.259590i \(-0.916413\pi\)
0.965719 0.259590i \(-0.0835875\pi\)
\(240\) 0 0
\(241\) −3.93627 −0.253557 −0.126779 0.991931i \(-0.540464\pi\)
−0.126779 + 0.991931i \(0.540464\pi\)
\(242\) 9.03519i 0.580804i
\(243\) 0 0
\(244\) 3.09020i 0.197830i
\(245\) 2.02070 0.129098
\(246\) 0 0
\(247\) 10.2080i 0.649518i
\(248\) 3.06904i 0.194884i
\(249\) 0 0
\(250\) 10.7306i 0.678665i
\(251\) 13.6711i 0.862909i −0.902135 0.431455i \(-0.858001\pi\)
0.902135 0.431455i \(-0.141999\pi\)
\(252\) 0 0
\(253\) 21.4553 1.34888
\(254\) 8.62356i 0.541090i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.0907i 1.56511i 0.622580 + 0.782556i \(0.286084\pi\)
−0.622580 + 0.782556i \(0.713916\pi\)
\(258\) 0 0
\(259\) −3.54273 −0.220135
\(260\) 7.38178i 0.457799i
\(261\) 0 0
\(262\) 5.61511 0.346902
\(263\) 21.4145 1.32047 0.660236 0.751058i \(-0.270456\pi\)
0.660236 + 0.751058i \(0.270456\pi\)
\(264\) 0 0
\(265\) 3.30083i 0.202769i
\(266\) 7.93192i 0.486337i
\(267\) 0 0
\(268\) 0.492578i 0.0300890i
\(269\) 11.5205 0.702415 0.351207 0.936298i \(-0.385771\pi\)
0.351207 + 0.936298i \(0.385771\pi\)
\(270\) 0 0
\(271\) 7.80613i 0.474188i 0.971487 + 0.237094i \(0.0761950\pi\)
−0.971487 + 0.237094i \(0.923805\pi\)
\(272\) 6.92873i 0.420116i
\(273\) 0 0
\(274\) 17.9442i 1.08405i
\(275\) −1.65131 −0.0995777
\(276\) 0 0
\(277\) 5.43334i 0.326458i −0.986588 0.163229i \(-0.947809\pi\)
0.986588 0.163229i \(-0.0521909\pi\)
\(278\) 3.95347i 0.237113i
\(279\) 0 0
\(280\) 5.73588i 0.342784i
\(281\) 6.21107i 0.370521i −0.982689 0.185261i \(-0.940687\pi\)
0.982689 0.185261i \(-0.0593130\pi\)
\(282\) 0 0
\(283\) −8.88487 −0.528150 −0.264075 0.964502i \(-0.585067\pi\)
−0.264075 + 0.964502i \(0.585067\pi\)
\(284\) 11.1458 0.661382
\(285\) 0 0
\(286\) 14.2598 0.843202
\(287\) 12.6960i 0.749423i
\(288\) 0 0
\(289\) −31.0073 −1.82396
\(290\) 11.5323 0.677198
\(291\) 0 0
\(292\) −12.2783 −0.718536
\(293\) −18.8347 −1.10033 −0.550167 0.835055i \(-0.685436\pi\)
−0.550167 + 0.835055i \(0.685436\pi\)
\(294\) 0 0
\(295\) 16.7784 0.976875
\(296\) 1.43114i 0.0831833i
\(297\) 0 0
\(298\) 13.7892i 0.798786i
\(299\) 15.2706i 0.883121i
\(300\) 0 0
\(301\) −11.6443 −0.671166
\(302\) −10.4468 −0.601145
\(303\) 0 0
\(304\) −3.20422 −0.183774
\(305\) 7.16028i 0.409996i
\(306\) 0 0
\(307\) 23.3097i 1.33035i −0.746686 0.665177i \(-0.768356\pi\)
0.746686 0.665177i \(-0.231644\pi\)
\(308\) −11.0803 −0.631361
\(309\) 0 0
\(310\) 7.11124i 0.403891i
\(311\) 4.23929 0.240388 0.120194 0.992750i \(-0.461648\pi\)
0.120194 + 0.992750i \(0.461648\pi\)
\(312\) 0 0
\(313\) 9.91668i 0.560524i −0.959924 0.280262i \(-0.909579\pi\)
0.959924 0.280262i \(-0.0904213\pi\)
\(314\) 12.1420 0.685211
\(315\) 0 0
\(316\) 7.55115i 0.424785i
\(317\) 10.4875i 0.589037i −0.955646 0.294518i \(-0.904841\pi\)
0.955646 0.294518i \(-0.0951592\pi\)
\(318\) 0 0
\(319\) 22.2776i 1.24731i
\(320\) 2.31709 0.129529
\(321\) 0 0
\(322\) 11.8657i 0.661251i
\(323\) 22.2012i 1.23531i
\(324\) 0 0
\(325\) 1.17530i 0.0651940i
\(326\) 14.7257 0.815581
\(327\) 0 0
\(328\) −5.12875 −0.283188
\(329\) 12.7525i 0.703067i
\(330\) 0 0
\(331\) 6.02817i 0.331338i −0.986181 0.165669i \(-0.947022\pi\)
0.986181 0.165669i \(-0.0529783\pi\)
\(332\) 5.70648i 0.313184i
\(333\) 0 0
\(334\) 7.54710i 0.412959i
\(335\) 1.14135i 0.0623586i
\(336\) 0 0
\(337\) 18.3363i 0.998840i 0.866360 + 0.499420i \(0.166453\pi\)
−0.866360 + 0.499420i \(0.833547\pi\)
\(338\) 2.85072i 0.155059i
\(339\) 0 0
\(340\) 16.0545i 0.870678i
\(341\) 13.7372 0.743912
\(342\) 0 0
\(343\) 19.4870 1.05220
\(344\) 4.70388i 0.253616i
\(345\) 0 0
\(346\) 9.83355i 0.528655i
\(347\) 22.6952i 1.21834i 0.793039 + 0.609171i \(0.208498\pi\)
−0.793039 + 0.609171i \(0.791502\pi\)
\(348\) 0 0
\(349\) −11.2887 −0.604271 −0.302136 0.953265i \(-0.597700\pi\)
−0.302136 + 0.953265i \(0.597700\pi\)
\(350\) 0.913247i 0.0488151i
\(351\) 0 0
\(352\) 4.47607i 0.238575i
\(353\) 21.2848i 1.13288i 0.824104 + 0.566438i \(0.191679\pi\)
−0.824104 + 0.566438i \(0.808321\pi\)
\(354\) 0 0
\(355\) 25.8259 1.37069
\(356\) 10.7939i 0.572077i
\(357\) 0 0
\(358\) −14.8865 −0.786777
\(359\) 32.9135i 1.73711i −0.495596 0.868553i \(-0.665050\pi\)
0.495596 0.868553i \(-0.334950\pi\)
\(360\) 0 0
\(361\) −8.73299 −0.459631
\(362\) 12.6535i 0.665052i
\(363\) 0 0
\(364\) 7.88632i 0.413355i
\(365\) −28.4501 −1.48914
\(366\) 0 0
\(367\) −17.4416 −0.910444 −0.455222 0.890378i \(-0.650440\pi\)
−0.455222 + 0.890378i \(0.650440\pi\)
\(368\) −4.79334 −0.249870
\(369\) 0 0
\(370\) 3.31608i 0.172395i
\(371\) 3.52644i 0.183084i
\(372\) 0 0
\(373\) 0.833131i 0.0431379i −0.999767 0.0215689i \(-0.993134\pi\)
0.999767 0.0215689i \(-0.00686614\pi\)
\(374\) 31.0135 1.60367
\(375\) 0 0
\(376\) 5.15156 0.265671
\(377\) −15.8558 −0.816617
\(378\) 0 0
\(379\) 26.3549 1.35376 0.676880 0.736093i \(-0.263331\pi\)
0.676880 + 0.736093i \(0.263331\pi\)
\(380\) −7.42447 −0.380867
\(381\) 0 0
\(382\) 23.5626i 1.20557i
\(383\) −22.1406 −1.13133 −0.565666 0.824634i \(-0.691381\pi\)
−0.565666 + 0.824634i \(0.691381\pi\)
\(384\) 0 0
\(385\) −25.6742 −1.30848
\(386\) −7.04534 −0.358598
\(387\) 0 0
\(388\) 7.87729i 0.399909i
\(389\) 7.32613i 0.371449i −0.982602 0.185725i \(-0.940537\pi\)
0.982602 0.185725i \(-0.0594633\pi\)
\(390\) 0 0
\(391\) 33.2118i 1.67959i
\(392\) 0.872083i 0.0440468i
\(393\) 0 0
\(394\) −15.3685 −0.774254
\(395\) 17.4967i 0.880355i
\(396\) 0 0
\(397\) 15.7523i 0.790586i −0.918555 0.395293i \(-0.870643\pi\)
0.918555 0.395293i \(-0.129357\pi\)
\(398\) 4.89975i 0.245602i
\(399\) 0 0
\(400\) 0.368920 0.0184460
\(401\) 12.2999i 0.614227i 0.951673 + 0.307113i \(0.0993631\pi\)
−0.951673 + 0.307113i \(0.900637\pi\)
\(402\) 0 0
\(403\) 9.77732i 0.487043i
\(404\) 1.15442i 0.0574343i
\(405\) 0 0
\(406\) 12.3205 0.611455
\(407\) −6.40588 −0.317527
\(408\) 0 0
\(409\) 14.6362i 0.723715i 0.932233 + 0.361857i \(0.117857\pi\)
−0.932233 + 0.361857i \(0.882143\pi\)
\(410\) −11.8838 −0.586899
\(411\) 0 0
\(412\) 11.8044i 0.581559i
\(413\) 17.9252 0.882039
\(414\) 0 0
\(415\) 13.2224i 0.649064i
\(416\) −3.18579 −0.156196
\(417\) 0 0
\(418\) 14.3423i 0.701505i
\(419\) 25.9262i 1.26658i 0.773916 + 0.633289i \(0.218295\pi\)
−0.773916 + 0.633289i \(0.781705\pi\)
\(420\) 0 0
\(421\) 22.1612i 1.08007i 0.841642 + 0.540036i \(0.181589\pi\)
−0.841642 + 0.540036i \(0.818411\pi\)
\(422\) 22.7020i 1.10511i
\(423\) 0 0
\(424\) 1.42456 0.0691827
\(425\) 2.55614i 0.123991i
\(426\) 0 0
\(427\) 7.64967i 0.370194i
\(428\) 15.3578 0.742348
\(429\) 0 0
\(430\) 10.8993i 0.525613i
\(431\) 32.3460 1.55805 0.779026 0.626992i \(-0.215714\pi\)
0.779026 + 0.626992i \(0.215714\pi\)
\(432\) 0 0
\(433\) −34.4500 −1.65556 −0.827780 0.561053i \(-0.810397\pi\)
−0.827780 + 0.561053i \(0.810397\pi\)
\(434\) 7.59728i 0.364681i
\(435\) 0 0
\(436\) −9.03262 −0.432584
\(437\) 15.3589 0.734716
\(438\) 0 0
\(439\) 16.1050i 0.768650i −0.923198 0.384325i \(-0.874434\pi\)
0.923198 0.384325i \(-0.125566\pi\)
\(440\) 10.3715i 0.494440i
\(441\) 0 0
\(442\) 22.0735i 1.04993i
\(443\) 29.9702i 1.42393i −0.702215 0.711965i \(-0.747806\pi\)
0.702215 0.711965i \(-0.252194\pi\)
\(444\) 0 0
\(445\) 25.0105i 1.18561i
\(446\) −12.0610 8.80524i −0.571105 0.416940i
\(447\) 0 0
\(448\) 2.47546 0.116955
\(449\) 31.4914 1.48617 0.743085 0.669197i \(-0.233362\pi\)
0.743085 + 0.669197i \(0.233362\pi\)
\(450\) 0 0
\(451\) 22.9566i 1.08099i
\(452\) 5.90600 0.277795
\(453\) 0 0
\(454\) −10.0503 −0.471684
\(455\) 18.2733i 0.856667i
\(456\) 0 0
\(457\) 12.6715i 0.592750i −0.955072 0.296375i \(-0.904222\pi\)
0.955072 0.296375i \(-0.0957778\pi\)
\(458\) 1.88715 0.0881807
\(459\) 0 0
\(460\) −11.1066 −0.517848
\(461\) 10.0712i 0.469065i 0.972108 + 0.234532i \(0.0753559\pi\)
−0.972108 + 0.234532i \(0.924644\pi\)
\(462\) 0 0
\(463\) −12.2551 −0.569541 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(464\) 4.97704i 0.231053i
\(465\) 0 0
\(466\) 5.88159i 0.272459i
\(467\) −35.5860 −1.64673 −0.823363 0.567515i \(-0.807905\pi\)
−0.823363 + 0.567515i \(0.807905\pi\)
\(468\) 0 0
\(469\) 1.21936i 0.0563048i
\(470\) 11.9366 0.550596
\(471\) 0 0
\(472\) 7.24113i 0.333300i
\(473\) −21.0549 −0.968105
\(474\) 0 0
\(475\) −1.18210 −0.0542384
\(476\) 17.1518i 0.786152i
\(477\) 0 0
\(478\) −8.02633 −0.367116
\(479\) 20.2002i 0.922972i −0.887147 0.461486i \(-0.847316\pi\)
0.887147 0.461486i \(-0.152684\pi\)
\(480\) 0 0
\(481\) 4.55931i 0.207887i
\(482\) 3.93627i 0.179292i
\(483\) 0 0
\(484\) −9.03519 −0.410690
\(485\) 18.2524i 0.828799i
\(486\) 0 0
\(487\) 14.3808 0.651655 0.325827 0.945429i \(-0.394357\pi\)
0.325827 + 0.945429i \(0.394357\pi\)
\(488\) 3.09020 0.139887
\(489\) 0 0
\(490\) 2.02070i 0.0912858i
\(491\) −14.5010 −0.654423 −0.327211 0.944951i \(-0.606109\pi\)
−0.327211 + 0.944951i \(0.606109\pi\)
\(492\) 0 0
\(493\) −34.4846 −1.55311
\(494\) 10.2080 0.459279
\(495\) 0 0
\(496\) −3.06904 −0.137804
\(497\) 27.5910 1.23763
\(498\) 0 0
\(499\) 0.662177 0.0296431 0.0148216 0.999890i \(-0.495282\pi\)
0.0148216 + 0.999890i \(0.495282\pi\)
\(500\) −10.7306 −0.479889
\(501\) 0 0
\(502\) −13.6711 −0.610169
\(503\) 0.806990 0.0359819 0.0179910 0.999838i \(-0.494273\pi\)
0.0179910 + 0.999838i \(0.494273\pi\)
\(504\) 0 0
\(505\) 2.67489i 0.119031i
\(506\) 21.4553i 0.953805i
\(507\) 0 0
\(508\) −8.62356 −0.382609
\(509\) 38.7822i 1.71899i −0.511144 0.859495i \(-0.670778\pi\)
0.511144 0.859495i \(-0.329222\pi\)
\(510\) 0 0
\(511\) −30.3946 −1.34458
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 25.0907 1.10670
\(515\) 27.3518i 1.20526i
\(516\) 0 0
\(517\) 23.0587i 1.01412i
\(518\) 3.54273i 0.155659i
\(519\) 0 0
\(520\) −7.38178 −0.323713
\(521\) −12.2647 −0.537326 −0.268663 0.963234i \(-0.586582\pi\)
−0.268663 + 0.963234i \(0.586582\pi\)
\(522\) 0 0
\(523\) 3.78942i 0.165700i −0.996562 0.0828498i \(-0.973598\pi\)
0.996562 0.0828498i \(-0.0264022\pi\)
\(524\) 5.61511i 0.245297i
\(525\) 0 0
\(526\) 21.4145i 0.933715i
\(527\) 21.2645i 0.926297i
\(528\) 0 0
\(529\) −0.0239065 −0.00103941
\(530\) 3.30083 0.143379
\(531\) 0 0
\(532\) −7.93192 −0.343892
\(533\) 16.3391 0.707727
\(534\) 0 0
\(535\) 35.5855 1.53849
\(536\) −0.492578 −0.0212761
\(537\) 0 0
\(538\) 11.5205i 0.496682i
\(539\) 3.90350 0.168136
\(540\) 0 0
\(541\) 8.00406i 0.344121i −0.985086 0.172061i \(-0.944957\pi\)
0.985086 0.172061i \(-0.0550425\pi\)
\(542\) 7.80613 0.335302
\(543\) 0 0
\(544\) −6.92873 −0.297067
\(545\) −20.9294 −0.896517
\(546\) 0 0
\(547\) 32.4378 1.38694 0.693469 0.720486i \(-0.256081\pi\)
0.693469 + 0.720486i \(0.256081\pi\)
\(548\) −17.9442 −0.766539
\(549\) 0 0
\(550\) 1.65131i 0.0704121i
\(551\) 15.9475i 0.679387i
\(552\) 0 0
\(553\) 18.6926i 0.794889i
\(554\) −5.43334 −0.230841
\(555\) 0 0
\(556\) 3.95347 0.167665
\(557\) 36.7815 1.55848 0.779241 0.626725i \(-0.215605\pi\)
0.779241 + 0.626725i \(0.215605\pi\)
\(558\) 0 0
\(559\) 14.9856i 0.633823i
\(560\) 5.73588 0.242385
\(561\) 0 0
\(562\) −6.21107 −0.261998
\(563\) −8.32447 −0.350835 −0.175417 0.984494i \(-0.556127\pi\)
−0.175417 + 0.984494i \(0.556127\pi\)
\(564\) 0 0
\(565\) 13.6848 0.575722
\(566\) 8.88487i 0.373459i
\(567\) 0 0
\(568\) 11.1458i 0.467668i
\(569\) −16.6911 −0.699726 −0.349863 0.936801i \(-0.613772\pi\)
−0.349863 + 0.936801i \(0.613772\pi\)
\(570\) 0 0
\(571\) 12.1998i 0.510546i 0.966869 + 0.255273i \(0.0821654\pi\)
−0.966869 + 0.255273i \(0.917835\pi\)
\(572\) 14.2598i 0.596234i
\(573\) 0 0
\(574\) −12.6960 −0.529922
\(575\) −1.76836 −0.0737456
\(576\) 0 0
\(577\) −40.9535 −1.70492 −0.852459 0.522795i \(-0.824889\pi\)
−0.852459 + 0.522795i \(0.824889\pi\)
\(578\) 31.0073i 1.28973i
\(579\) 0 0
\(580\) 11.5323i 0.478851i
\(581\) 14.1262i 0.586053i
\(582\) 0 0
\(583\) 6.37642i 0.264084i
\(584\) 12.2783i 0.508082i
\(585\) 0 0
\(586\) 18.8347i 0.778054i
\(587\) −30.3885 −1.25427 −0.627133 0.778912i \(-0.715772\pi\)
−0.627133 + 0.778912i \(0.715772\pi\)
\(588\) 0 0
\(589\) 9.83386 0.405197
\(590\) 16.7784i 0.690755i
\(591\) 0 0
\(592\) 1.43114 0.0588195
\(593\) 28.9431 1.18855 0.594275 0.804262i \(-0.297439\pi\)
0.594275 + 0.804262i \(0.297439\pi\)
\(594\) 0 0
\(595\) 39.7424i 1.62928i
\(596\) −13.7892 −0.564827
\(597\) 0 0
\(598\) 15.2706 0.624461
\(599\) 12.8836i 0.526410i 0.964740 + 0.263205i \(0.0847796\pi\)
−0.964740 + 0.263205i \(0.915220\pi\)
\(600\) 0 0
\(601\) 32.0957i 1.30921i −0.755971 0.654605i \(-0.772835\pi\)
0.755971 0.654605i \(-0.227165\pi\)
\(602\) 11.6443i 0.474586i
\(603\) 0 0
\(604\) 10.4468i 0.425074i
\(605\) −20.9354 −0.851144
\(606\) 0 0
\(607\) 0.570165i 0.0231423i −0.999933 0.0115711i \(-0.996317\pi\)
0.999933 0.0115711i \(-0.00368329\pi\)
\(608\) 3.20422i 0.129948i
\(609\) 0 0
\(610\) 7.16028 0.289911
\(611\) −16.4118 −0.663950
\(612\) 0 0
\(613\) 32.5731i 1.31561i −0.753186 0.657807i \(-0.771484\pi\)
0.753186 0.657807i \(-0.228516\pi\)
\(614\) −23.3097 −0.940702
\(615\) 0 0
\(616\) 11.0803i 0.446440i
\(617\) 17.8838i 0.719973i 0.932958 + 0.359986i \(0.117219\pi\)
−0.932958 + 0.359986i \(0.882781\pi\)
\(618\) 0 0
\(619\) 30.4967i 1.22576i −0.790174 0.612882i \(-0.790010\pi\)
0.790174 0.612882i \(-0.209990\pi\)
\(620\) −7.11124 −0.285594
\(621\) 0 0
\(622\) 4.23929i 0.169980i
\(623\) 26.7200i 1.07051i
\(624\) 0 0
\(625\) −26.7085 −1.06834
\(626\) −9.91668 −0.396350
\(627\) 0 0
\(628\) 12.1420i 0.484517i
\(629\) 9.91598i 0.395376i
\(630\) 0 0
\(631\) 27.2019i 1.08289i 0.840736 + 0.541445i \(0.182123\pi\)
−0.840736 + 0.541445i \(0.817877\pi\)
\(632\) −7.55115 −0.300368
\(633\) 0 0
\(634\) −10.4875 −0.416512
\(635\) −19.9816 −0.792945
\(636\) 0 0
\(637\) 2.77828i 0.110079i
\(638\) 22.2776 0.881978
\(639\) 0 0
\(640\) 2.31709i 0.0915911i
\(641\) 23.9546 0.946148 0.473074 0.881023i \(-0.343144\pi\)
0.473074 + 0.881023i \(0.343144\pi\)
\(642\) 0 0
\(643\) 19.1127 0.753730 0.376865 0.926268i \(-0.377002\pi\)
0.376865 + 0.926268i \(0.377002\pi\)
\(644\) −11.8657 −0.467575
\(645\) 0 0
\(646\) 22.2012 0.873493
\(647\) 42.5846i 1.67417i 0.547070 + 0.837087i \(0.315743\pi\)
−0.547070 + 0.837087i \(0.684257\pi\)
\(648\) 0 0
\(649\) 32.4118 1.27227
\(650\) −1.17530 −0.0460991
\(651\) 0 0
\(652\) 14.7257i 0.576703i
\(653\) −24.0328 −0.940475 −0.470237 0.882540i \(-0.655832\pi\)
−0.470237 + 0.882540i \(0.655832\pi\)
\(654\) 0 0
\(655\) 13.0107i 0.508371i
\(656\) 5.12875i 0.200244i
\(657\) 0 0
\(658\) 12.7525 0.497144
\(659\) 0.449124i 0.0174954i −0.999962 0.00874770i \(-0.997215\pi\)
0.999962 0.00874770i \(-0.00278452\pi\)
\(660\) 0 0
\(661\) 15.3311i 0.596310i −0.954517 0.298155i \(-0.903629\pi\)
0.954517 0.298155i \(-0.0963713\pi\)
\(662\) −6.02817 −0.234291
\(663\) 0 0
\(664\) −5.70648 −0.221454
\(665\) −18.3790 −0.712707
\(666\) 0 0
\(667\) 23.8566i 0.923733i
\(668\) −7.54710 −0.292006
\(669\) 0 0
\(670\) −1.14135 −0.0440942
\(671\) 13.8319i 0.533976i
\(672\) 0 0
\(673\) −8.03508 −0.309730 −0.154865 0.987936i \(-0.549494\pi\)
−0.154865 + 0.987936i \(0.549494\pi\)
\(674\) 18.3363 0.706286
\(675\) 0 0
\(676\) −2.85072 −0.109643
\(677\) 6.14474i 0.236162i 0.993004 + 0.118081i \(0.0376742\pi\)
−0.993004 + 0.118081i \(0.962326\pi\)
\(678\) 0 0
\(679\) 19.4999i 0.748339i
\(680\) −16.0545 −0.615663
\(681\) 0 0
\(682\) 13.7372i 0.526025i
\(683\) 37.4660i 1.43360i 0.697280 + 0.716799i \(0.254393\pi\)
−0.697280 + 0.716799i \(0.745607\pi\)
\(684\) 0 0
\(685\) −41.5784 −1.58863
\(686\) 19.4870i 0.744019i
\(687\) 0 0
\(688\) 4.70388 0.179334
\(689\) −4.53835 −0.172897
\(690\) 0 0
\(691\) 31.8091i 1.21008i 0.796196 + 0.605038i \(0.206842\pi\)
−0.796196 + 0.605038i \(0.793158\pi\)
\(692\) 9.83355 0.373815
\(693\) 0 0
\(694\) 22.6952 0.861497
\(695\) 9.16056 0.347480
\(696\) 0 0
\(697\) 35.5357 1.34601
\(698\) 11.2887i 0.427284i
\(699\) 0 0
\(700\) 0.913247 0.0345175
\(701\) 8.09104i 0.305594i −0.988258 0.152797i \(-0.951172\pi\)
0.988258 0.152797i \(-0.0488281\pi\)
\(702\) 0 0
\(703\) −4.58568 −0.172952
\(704\) 4.47607 0.168698
\(705\) 0 0
\(706\) 21.2848 0.801064
\(707\) 2.85771i 0.107475i
\(708\) 0 0
\(709\) 27.8290i 1.04514i 0.852596 + 0.522571i \(0.175027\pi\)
−0.852596 + 0.522571i \(0.824973\pi\)
\(710\) 25.8259i 0.969228i
\(711\) 0 0
\(712\) 10.7939 0.404519
\(713\) 14.7109 0.550929
\(714\) 0 0
\(715\) 33.0414i 1.23568i
\(716\) 14.8865i 0.556335i
\(717\) 0 0
\(718\) −32.9135 −1.22832
\(719\) 33.2905i 1.24153i 0.783998 + 0.620763i \(0.213177\pi\)
−0.783998 + 0.620763i \(0.786823\pi\)
\(720\) 0 0
\(721\) 29.2213i 1.08826i
\(722\) 8.73299i 0.325008i
\(723\) 0 0
\(724\) 12.6535 0.470262
\(725\) 1.83613i 0.0681921i
\(726\) 0 0
\(727\) −25.1927 −0.934347 −0.467174 0.884166i \(-0.654728\pi\)
−0.467174 + 0.884166i \(0.654728\pi\)
\(728\) −7.88632 −0.292286
\(729\) 0 0
\(730\) 28.4501i 1.05298i
\(731\) 32.5919i 1.20546i
\(732\) 0 0
\(733\) 40.2307 1.48596 0.742978 0.669316i \(-0.233413\pi\)
0.742978 + 0.669316i \(0.233413\pi\)
\(734\) 17.4416i 0.643781i
\(735\) 0 0
\(736\) 4.79334i 0.176685i
\(737\) 2.20481i 0.0812154i
\(738\) 0 0
\(739\) 7.91269i 0.291073i 0.989353 + 0.145536i \(0.0464908\pi\)
−0.989353 + 0.145536i \(0.953509\pi\)
\(740\) 3.31608 0.121902
\(741\) 0 0
\(742\) 3.52644 0.129460
\(743\) 53.3888i 1.95865i 0.202304 + 0.979323i \(0.435157\pi\)
−0.202304 + 0.979323i \(0.564843\pi\)
\(744\) 0 0
\(745\) −31.9508 −1.17059
\(746\) −0.833131 −0.0305031
\(747\) 0 0
\(748\) 31.0135i 1.13397i
\(749\) 38.0177 1.38914
\(750\) 0 0
\(751\) −20.5434 −0.749638 −0.374819 0.927098i \(-0.622295\pi\)
−0.374819 + 0.927098i \(0.622295\pi\)
\(752\) 5.15156i 0.187858i
\(753\) 0 0
\(754\) 15.8558i 0.577435i
\(755\) 24.2062i 0.880953i
\(756\) 0 0
\(757\) 20.1514i 0.732416i −0.930533 0.366208i \(-0.880656\pi\)
0.930533 0.366208i \(-0.119344\pi\)
\(758\) 26.3549i 0.957253i
\(759\) 0 0
\(760\) 7.42447i 0.269314i
\(761\) −38.5714 −1.39821 −0.699107 0.715018i \(-0.746419\pi\)
−0.699107 + 0.715018i \(0.746419\pi\)
\(762\) 0 0
\(763\) −22.3599 −0.809483
\(764\) 23.5626 0.852466
\(765\) 0 0
\(766\) 22.1406i 0.799973i
\(767\) 23.0688i 0.832964i
\(768\) 0 0
\(769\) 33.2421 1.19874 0.599371 0.800471i \(-0.295417\pi\)
0.599371 + 0.800471i \(0.295417\pi\)
\(770\) 25.6742i 0.925233i
\(771\) 0 0
\(772\) 7.04534i 0.253567i
\(773\) 38.5412 1.38623 0.693115 0.720827i \(-0.256238\pi\)
0.693115 + 0.720827i \(0.256238\pi\)
\(774\) 0 0
\(775\) −1.13223 −0.0406708
\(776\) −7.87729 −0.282778
\(777\) 0 0
\(778\) −7.32613 −0.262654
\(779\) 16.4336i 0.588796i
\(780\) 0 0
\(781\) 49.8894 1.78518
\(782\) 33.2118 1.18765
\(783\) 0 0
\(784\) −0.872083 −0.0311458
\(785\) 28.1341i 1.00415i
\(786\) 0 0
\(787\) 19.6474i 0.700355i −0.936683 0.350178i \(-0.886121\pi\)
0.936683 0.350178i \(-0.113879\pi\)
\(788\) 15.3685i 0.547480i
\(789\) 0 0
\(790\) −17.4967 −0.622505
\(791\) 14.6201 0.519831
\(792\) 0 0
\(793\) −9.84474 −0.349597
\(794\) −15.7523 −0.559029
\(795\) 0 0
\(796\) 4.89975 0.173667
\(797\) 15.2579i 0.540461i 0.962796 + 0.270231i \(0.0870999\pi\)
−0.962796 + 0.270231i \(0.912900\pi\)
\(798\) 0 0
\(799\) −35.6937 −1.26275
\(800\) 0.368920i 0.0130433i
\(801\) 0 0
\(802\) 12.2999 0.434324
\(803\) −54.9587 −1.93945
\(804\) 0 0
\(805\) −27.4940 −0.969036
\(806\) 9.77732 0.344391
\(807\) 0 0
\(808\) −1.15442 −0.0406122
\(809\) −32.5186 −1.14329 −0.571646 0.820500i \(-0.693695\pi\)
−0.571646 + 0.820500i \(0.693695\pi\)
\(810\) 0 0
\(811\) 17.3227i 0.608283i −0.952627 0.304142i \(-0.901630\pi\)
0.952627 0.304142i \(-0.0983696\pi\)
\(812\) 12.3205i 0.432364i
\(813\) 0 0
\(814\) 6.40588i 0.224526i
\(815\) 34.1208i 1.19520i
\(816\) 0 0
\(817\) −15.0723 −0.527312
\(818\) 14.6362 0.511744
\(819\) 0 0
\(820\) 11.8838i 0.415000i
\(821\) 27.0772i 0.945000i 0.881331 + 0.472500i \(0.156648\pi\)
−0.881331 + 0.472500i \(0.843352\pi\)
\(822\) 0 0
\(823\) 2.90080i 0.101116i −0.998721 0.0505578i \(-0.983900\pi\)
0.998721 0.0505578i \(-0.0160999\pi\)
\(824\) −11.8044 −0.411224
\(825\) 0 0
\(826\) 17.9252i 0.623696i
\(827\) 26.4616 0.920158 0.460079 0.887878i \(-0.347821\pi\)
0.460079 + 0.887878i \(0.347821\pi\)
\(828\) 0 0
\(829\) 11.9299i 0.414343i −0.978305 0.207172i \(-0.933574\pi\)
0.978305 0.207172i \(-0.0664258\pi\)
\(830\) −13.2224 −0.458958
\(831\) 0 0
\(832\) 3.18579i 0.110448i
\(833\) 6.04243i 0.209358i
\(834\) 0 0
\(835\) −17.4873 −0.605174
\(836\) −14.3423 −0.496039
\(837\) 0 0
\(838\) 25.9262 0.895605
\(839\) 10.2588 0.354175 0.177087 0.984195i \(-0.443332\pi\)
0.177087 + 0.984195i \(0.443332\pi\)
\(840\) 0 0
\(841\) 4.22905 0.145829
\(842\) 22.1612 0.763726
\(843\) 0 0
\(844\) −22.7020 −0.781434
\(845\) −6.60537 −0.227232
\(846\) 0 0
\(847\) −22.3663 −0.768514
\(848\) 1.42456i 0.0489195i
\(849\) 0 0
\(850\) −2.55614 −0.0876750
\(851\) −6.85993 −0.235155
\(852\) 0 0
\(853\) 4.12570i 0.141261i 0.997503 + 0.0706306i \(0.0225012\pi\)
−0.997503 + 0.0706306i \(0.977499\pi\)
\(854\) 7.64967 0.261766
\(855\) 0 0
\(856\) 15.3578i 0.524919i
\(857\) 53.3902i 1.82377i 0.410442 + 0.911886i \(0.365374\pi\)
−0.410442 + 0.911886i \(0.634626\pi\)
\(858\) 0 0
\(859\) 1.84229i 0.0628582i −0.999506 0.0314291i \(-0.989994\pi\)
0.999506 0.0314291i \(-0.0100058\pi\)
\(860\) 10.8993 0.371664
\(861\) 0 0
\(862\) 32.3460i 1.10171i
\(863\) −22.2277 −0.756640 −0.378320 0.925675i \(-0.623498\pi\)
−0.378320 + 0.925675i \(0.623498\pi\)
\(864\) 0 0
\(865\) 22.7852 0.774722
\(866\) 34.4500i 1.17066i
\(867\) 0 0
\(868\) −7.59728 −0.257869
\(869\) 33.7994i 1.14657i
\(870\) 0 0
\(871\) 1.56925 0.0531721
\(872\) 9.03262i 0.305883i
\(873\) 0 0
\(874\) 15.3589i 0.519522i
\(875\) −26.5633 −0.898004
\(876\) 0 0
\(877\) 55.6910i 1.88055i −0.340412 0.940277i \(-0.610566\pi\)
0.340412 0.940277i \(-0.389434\pi\)
\(878\) −16.1050 −0.543518
\(879\) 0 0
\(880\) 10.3715 0.349622
\(881\) 35.2348i 1.18709i 0.804801 + 0.593545i \(0.202272\pi\)
−0.804801 + 0.593545i \(0.797728\pi\)
\(882\) 0 0
\(883\) 51.3102i 1.72673i −0.504583 0.863363i \(-0.668354\pi\)
0.504583 0.863363i \(-0.331646\pi\)
\(884\) 22.0735 0.742413
\(885\) 0 0
\(886\) −29.9702 −1.00687
\(887\) 43.5073i 1.46083i −0.683003 0.730416i \(-0.739326\pi\)
0.683003 0.730416i \(-0.260674\pi\)
\(888\) 0 0
\(889\) −21.3473 −0.715966
\(890\) 25.0105 0.838355
\(891\) 0 0
\(892\) −8.80524 + 12.0610i −0.294821 + 0.403832i
\(893\) 16.5067i 0.552376i
\(894\) 0 0
\(895\) 34.4934i 1.15299i
\(896\) 2.47546i 0.0826994i
\(897\) 0 0
\(898\) 31.4914i 1.05088i
\(899\) 15.2747i 0.509440i
\(900\) 0 0
\(901\) −9.87038 −0.328830
\(902\) −22.9566 −0.764372
\(903\) 0 0
\(904\) 5.90600i 0.196431i
\(905\) 29.3193 0.974605
\(906\) 0 0
\(907\) 7.03310 0.233530 0.116765 0.993160i \(-0.462748\pi\)
0.116765 + 0.993160i \(0.462748\pi\)
\(908\) 10.0503i 0.333531i
\(909\) 0 0
\(910\) −18.2733 −0.605755
\(911\) 3.96055i 0.131219i 0.997845 + 0.0656095i \(0.0208992\pi\)
−0.997845 + 0.0656095i \(0.979101\pi\)
\(912\) 0 0
\(913\) 25.5426i 0.845336i
\(914\) −12.6715 −0.419137
\(915\) 0 0
\(916\) 1.88715i 0.0623532i
\(917\) 13.9000i 0.459018i
\(918\) 0 0
\(919\) 8.35383i 0.275567i −0.990462 0.137784i \(-0.956002\pi\)
0.990462 0.137784i \(-0.0439979\pi\)
\(920\) 11.1066i 0.366174i
\(921\) 0 0
\(922\) 10.0712 0.331679
\(923\) 35.5082i 1.16877i
\(924\) 0 0
\(925\) 0.527975 0.0173597
\(926\) 12.2551i 0.402726i
\(927\) 0 0
\(928\) −4.97704 −0.163379
\(929\) 31.7634i 1.04212i 0.853519 + 0.521061i \(0.174464\pi\)
−0.853519 + 0.521061i \(0.825536\pi\)
\(930\) 0 0
\(931\) 2.79434 0.0915809
\(932\) 5.88159 0.192658
\(933\) 0 0
\(934\) 35.5860i 1.16441i
\(935\) 71.8611i 2.35011i
\(936\) 0 0
\(937\) 4.53245i 0.148069i 0.997256 + 0.0740343i \(0.0235874\pi\)
−0.997256 + 0.0740343i \(0.976413\pi\)
\(938\) −1.21936 −0.0398135
\(939\) 0 0
\(940\) 11.9366i 0.389330i
\(941\) 30.0482i 0.979542i −0.871851 0.489771i \(-0.837080\pi\)
0.871851 0.489771i \(-0.162920\pi\)
\(942\) 0 0
\(943\) 24.5838i 0.800560i
\(944\) −7.24113 −0.235679
\(945\) 0 0
\(946\) 21.0549i 0.684554i
\(947\) 26.9938i 0.877180i 0.898687 + 0.438590i \(0.144522\pi\)
−0.898687 + 0.438590i \(0.855478\pi\)
\(948\) 0 0
\(949\) 39.1163i 1.26977i
\(950\) 1.18210i 0.0383523i
\(951\) 0 0
\(952\) −17.1518 −0.555894
\(953\) −25.5880 −0.828875 −0.414438 0.910078i \(-0.636022\pi\)
−0.414438 + 0.910078i \(0.636022\pi\)
\(954\) 0 0
\(955\) 54.5968 1.76671
\(956\) 8.02633i 0.259590i
\(957\) 0 0
\(958\) −20.2002 −0.652640
\(959\) −44.4203 −1.43441
\(960\) 0 0
\(961\) −21.5810 −0.696162
\(962\) −4.55931 −0.146998
\(963\) 0 0
\(964\) 3.93627 0.126779
\(965\) 16.3247i 0.525511i
\(966\) 0 0
\(967\) 7.46027i 0.239906i −0.992780 0.119953i \(-0.961726\pi\)
0.992780 0.119953i \(-0.0382744\pi\)
\(968\) 9.03519i 0.290402i
\(969\) 0 0
\(970\) −18.2524 −0.586050
\(971\) 32.0804 1.02951 0.514755 0.857337i \(-0.327883\pi\)
0.514755 + 0.857337i \(0.327883\pi\)
\(972\) 0 0
\(973\) 9.78667 0.313746
\(974\) 14.3808i 0.460789i
\(975\) 0 0
\(976\) 3.09020i 0.0989149i
\(977\) −47.8549 −1.53101 −0.765507 0.643428i \(-0.777512\pi\)
−0.765507 + 0.643428i \(0.777512\pi\)
\(978\) 0 0
\(979\) 48.3143i 1.54413i
\(980\) −2.02070 −0.0645488
\(981\) 0 0
\(982\) 14.5010i 0.462747i
\(983\) 9.92674 0.316614 0.158307 0.987390i \(-0.449396\pi\)
0.158307 + 0.987390i \(0.449396\pi\)
\(984\) 0 0
\(985\) 35.6102i 1.13464i
\(986\) 34.4846i 1.09821i
\(987\) 0 0
\(988\) 10.2080i 0.324759i
\(989\) −22.5473 −0.716962
\(990\) 0 0
\(991\) 44.0103i 1.39803i 0.715106 + 0.699016i \(0.246378\pi\)
−0.715106 + 0.699016i \(0.753622\pi\)
\(992\) 3.06904i 0.0974420i
\(993\) 0 0
\(994\) 27.5910i 0.875134i
\(995\) 11.3532 0.359920
\(996\) 0 0
\(997\) 31.3977 0.994375 0.497188 0.867643i \(-0.334366\pi\)
0.497188 + 0.867643i \(0.334366\pi\)
\(998\) 0.662177i 0.0209609i
\(999\) 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.d.a.4013.5 72
3.2 odd 2 inner 4014.2.d.a.4013.32 yes 72
223.222 odd 2 inner 4014.2.d.a.4013.31 yes 72
669.668 even 2 inner 4014.2.d.a.4013.6 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.d.a.4013.5 72 1.1 even 1 trivial
4014.2.d.a.4013.6 yes 72 669.668 even 2 inner
4014.2.d.a.4013.31 yes 72 223.222 odd 2 inner
4014.2.d.a.4013.32 yes 72 3.2 odd 2 inner