Properties

Label 9702.2.a.ec.1.4
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
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] = [9702,2,Mod(1,9702)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9702.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.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: \(77.4708600410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.32685\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{8} +3.29066 q^{10} +1.00000 q^{11} +6.06791 q^{13} +1.00000 q^{16} +6.11908 q^{17} +0.0511786 q^{19} +3.29066 q^{20} +1.00000 q^{22} +6.75605 q^{23} +5.82843 q^{25} +6.06791 q^{26} -2.82843 q^{29} +5.87644 q^{31} +1.00000 q^{32} +6.11908 q^{34} -8.31055 q^{37} +0.0511786 q^{38} +3.29066 q^{40} -6.11908 q^{41} +2.90080 q^{43} +1.00000 q^{44} +6.75605 q^{46} -1.22275 q^{47} +5.82843 q^{50} +6.06791 q^{52} -3.00316 q^{53} +3.29066 q^{55} -2.82843 q^{58} -10.8284 q^{59} +7.23948 q^{61} +5.87644 q^{62} +1.00000 q^{64} +19.9674 q^{65} -1.68051 q^{67} +6.11908 q^{68} +6.47896 q^{71} -11.6736 q^{73} -8.31055 q^{74} +0.0511786 q^{76} -9.13897 q^{79} +3.29066 q^{80} -6.11908 q^{82} -0.951983 q^{83} +20.1358 q^{85} +2.90080 q^{86} +1.00000 q^{88} -16.5469 q^{89} +6.75605 q^{92} -1.22275 q^{94} +0.168411 q^{95} -14.7216 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{11} + 4 q^{16} + 4 q^{22} + 8 q^{23} + 12 q^{25} + 16 q^{31} + 4 q^{32} + 8 q^{37} + 8 q^{43} + 4 q^{44} + 8 q^{46} - 16 q^{47} + 12 q^{50} - 8 q^{53} - 32 q^{59} + 16 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{65} + 16 q^{67} + 8 q^{74} + 16 q^{79} + 32 q^{85} + 8 q^{86} + 4 q^{88} - 16 q^{89} + 8 q^{92} - 16 q^{94} + 16 q^{95} - 16 q^{97}+O(q^{100}) \) 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
\(5\) 3.29066 1.47163 0.735813 0.677184i \(-0.236800\pi\)
0.735813 + 0.677184i \(0.236800\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.29066 1.04060
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 6.06791 1.68293 0.841467 0.540308i \(-0.181692\pi\)
0.841467 + 0.540308i \(0.181692\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.11908 1.48410 0.742048 0.670347i \(-0.233855\pi\)
0.742048 + 0.670347i \(0.233855\pi\)
\(18\) 0 0
\(19\) 0.0511786 0.0117412 0.00587059 0.999983i \(-0.498131\pi\)
0.00587059 + 0.999983i \(0.498131\pi\)
\(20\) 3.29066 0.735813
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 6.75605 1.40873 0.704367 0.709836i \(-0.251231\pi\)
0.704367 + 0.709836i \(0.251231\pi\)
\(24\) 0 0
\(25\) 5.82843 1.16569
\(26\) 6.06791 1.19001
\(27\) 0 0
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 5.87644 1.05544 0.527720 0.849418i \(-0.323047\pi\)
0.527720 + 0.849418i \(0.323047\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.11908 1.04941
\(35\) 0 0
\(36\) 0 0
\(37\) −8.31055 −1.36625 −0.683123 0.730304i \(-0.739379\pi\)
−0.683123 + 0.730304i \(0.739379\pi\)
\(38\) 0.0511786 0.00830226
\(39\) 0 0
\(40\) 3.29066 0.520299
\(41\) −6.11908 −0.955640 −0.477820 0.878458i \(-0.658573\pi\)
−0.477820 + 0.878458i \(0.658573\pi\)
\(42\) 0 0
\(43\) 2.90080 0.442369 0.221184 0.975232i \(-0.429008\pi\)
0.221184 + 0.975232i \(0.429008\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 6.75605 0.996125
\(47\) −1.22275 −0.178357 −0.0891783 0.996016i \(-0.528424\pi\)
−0.0891783 + 0.996016i \(0.528424\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 5.82843 0.824264
\(51\) 0 0
\(52\) 6.06791 0.841467
\(53\) −3.00316 −0.412516 −0.206258 0.978498i \(-0.566129\pi\)
−0.206258 + 0.978498i \(0.566129\pi\)
\(54\) 0 0
\(55\) 3.29066 0.443712
\(56\) 0 0
\(57\) 0 0
\(58\) −2.82843 −0.371391
\(59\) −10.8284 −1.40974 −0.704871 0.709336i \(-0.748995\pi\)
−0.704871 + 0.709336i \(0.748995\pi\)
\(60\) 0 0
\(61\) 7.23948 0.926920 0.463460 0.886118i \(-0.346608\pi\)
0.463460 + 0.886118i \(0.346608\pi\)
\(62\) 5.87644 0.746309
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 19.9674 2.47665
\(66\) 0 0
\(67\) −1.68051 −0.205307 −0.102654 0.994717i \(-0.532733\pi\)
−0.102654 + 0.994717i \(0.532733\pi\)
\(68\) 6.11908 0.742048
\(69\) 0 0
\(70\) 0 0
\(71\) 6.47896 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(72\) 0 0
\(73\) −11.6736 −1.36629 −0.683145 0.730283i \(-0.739388\pi\)
−0.683145 + 0.730283i \(0.739388\pi\)
\(74\) −8.31055 −0.966081
\(75\) 0 0
\(76\) 0.0511786 0.00587059
\(77\) 0 0
\(78\) 0 0
\(79\) −9.13897 −1.02821 −0.514107 0.857726i \(-0.671877\pi\)
−0.514107 + 0.857726i \(0.671877\pi\)
\(80\) 3.29066 0.367907
\(81\) 0 0
\(82\) −6.11908 −0.675740
\(83\) −0.951983 −0.104494 −0.0522469 0.998634i \(-0.516638\pi\)
−0.0522469 + 0.998634i \(0.516638\pi\)
\(84\) 0 0
\(85\) 20.1358 2.18404
\(86\) 2.90080 0.312802
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −16.5469 −1.75396 −0.876982 0.480523i \(-0.840447\pi\)
−0.876982 + 0.480523i \(0.840447\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.75605 0.704367
\(93\) 0 0
\(94\) −1.22275 −0.126117
\(95\) 0.168411 0.0172786
\(96\) 0 0
\(97\) −14.7216 −1.49475 −0.747376 0.664401i \(-0.768687\pi\)
−0.747376 + 0.664401i \(0.768687\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.82843 0.582843
\(101\) −5.51657 −0.548919 −0.274460 0.961599i \(-0.588499\pi\)
−0.274460 + 0.961599i \(0.588499\pi\)
\(102\) 0 0
\(103\) 11.6781 1.15067 0.575336 0.817917i \(-0.304871\pi\)
0.575336 + 0.817917i \(0.304871\pi\)
\(104\) 6.06791 0.595007
\(105\) 0 0
\(106\) −3.00316 −0.291693
\(107\) −15.9611 −1.54302 −0.771508 0.636220i \(-0.780497\pi\)
−0.771508 + 0.636220i \(0.780497\pi\)
\(108\) 0 0
\(109\) −5.65053 −0.541223 −0.270611 0.962689i \(-0.587226\pi\)
−0.270611 + 0.962689i \(0.587226\pi\)
\(110\) 3.29066 0.313752
\(111\) 0 0
\(112\) 0 0
\(113\) 0.247112 0.0232463 0.0116232 0.999932i \(-0.496300\pi\)
0.0116232 + 0.999932i \(0.496300\pi\)
\(114\) 0 0
\(115\) 22.2318 2.07313
\(116\) −2.82843 −0.262613
\(117\) 0 0
\(118\) −10.8284 −0.996838
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 7.23948 0.655432
\(123\) 0 0
\(124\) 5.87644 0.527720
\(125\) 2.72607 0.243827
\(126\) 0 0
\(127\) 13.3310 1.18294 0.591469 0.806328i \(-0.298548\pi\)
0.591469 + 0.806328i \(0.298548\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 19.9674 1.75126
\(131\) −3.04802 −0.266306 −0.133153 0.991095i \(-0.542510\pi\)
−0.133153 + 0.991095i \(0.542510\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.68051 −0.145174
\(135\) 0 0
\(136\) 6.11908 0.524707
\(137\) −8.47896 −0.724406 −0.362203 0.932099i \(-0.617975\pi\)
−0.362203 + 0.932099i \(0.617975\pi\)
\(138\) 0 0
\(139\) −3.80407 −0.322657 −0.161328 0.986901i \(-0.551578\pi\)
−0.161328 + 0.986901i \(0.551578\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.47896 0.543702
\(143\) 6.06791 0.507424
\(144\) 0 0
\(145\) −9.30739 −0.772936
\(146\) −11.6736 −0.966113
\(147\) 0 0
\(148\) −8.31055 −0.683123
\(149\) −24.1269 −1.97655 −0.988275 0.152684i \(-0.951208\pi\)
−0.988275 + 0.152684i \(0.951208\pi\)
\(150\) 0 0
\(151\) 7.85525 0.639251 0.319625 0.947544i \(-0.396443\pi\)
0.319625 + 0.947544i \(0.396443\pi\)
\(152\) 0.0511786 0.00415113
\(153\) 0 0
\(154\) 0 0
\(155\) 19.3374 1.55321
\(156\) 0 0
\(157\) −18.2549 −1.45690 −0.728450 0.685099i \(-0.759759\pi\)
−0.728450 + 0.685099i \(0.759759\pi\)
\(158\) −9.13897 −0.727058
\(159\) 0 0
\(160\) 3.29066 0.260149
\(161\) 0 0
\(162\) 0 0
\(163\) 2.34315 0.183529 0.0917647 0.995781i \(-0.470749\pi\)
0.0917647 + 0.995781i \(0.470749\pi\)
\(164\) −6.11908 −0.477820
\(165\) 0 0
\(166\) −0.951983 −0.0738882
\(167\) −19.7863 −1.53111 −0.765557 0.643369i \(-0.777536\pi\)
−0.765557 + 0.643369i \(0.777536\pi\)
\(168\) 0 0
\(169\) 23.8195 1.83227
\(170\) 20.1358 1.54435
\(171\) 0 0
\(172\) 2.90080 0.221184
\(173\) −10.5768 −0.804143 −0.402071 0.915608i \(-0.631710\pi\)
−0.402071 + 0.915608i \(0.631710\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −16.5469 −1.24024
\(179\) −18.3016 −1.36793 −0.683963 0.729517i \(-0.739745\pi\)
−0.683963 + 0.729517i \(0.739745\pi\)
\(180\) 0 0
\(181\) 25.0499 1.86194 0.930971 0.365093i \(-0.118963\pi\)
0.930971 + 0.365093i \(0.118963\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.75605 0.498063
\(185\) −27.3472 −2.01060
\(186\) 0 0
\(187\) 6.11908 0.447472
\(188\) −1.22275 −0.0891783
\(189\) 0 0
\(190\) 0.168411 0.0122178
\(191\) 21.8650 1.58210 0.791050 0.611752i \(-0.209535\pi\)
0.791050 + 0.611752i \(0.209535\pi\)
\(192\) 0 0
\(193\) 20.8195 1.49862 0.749310 0.662220i \(-0.230385\pi\)
0.749310 + 0.662220i \(0.230385\pi\)
\(194\) −14.7216 −1.05695
\(195\) 0 0
\(196\) 0 0
\(197\) 20.4790 1.45907 0.729533 0.683946i \(-0.239738\pi\)
0.729533 + 0.683946i \(0.239738\pi\)
\(198\) 0 0
\(199\) 15.5333 1.10113 0.550563 0.834794i \(-0.314413\pi\)
0.550563 + 0.834794i \(0.314413\pi\)
\(200\) 5.82843 0.412132
\(201\) 0 0
\(202\) −5.51657 −0.388145
\(203\) 0 0
\(204\) 0 0
\(205\) −20.1358 −1.40635
\(206\) 11.6781 0.813649
\(207\) 0 0
\(208\) 6.06791 0.420734
\(209\) 0.0511786 0.00354010
\(210\) 0 0
\(211\) −6.40658 −0.441047 −0.220524 0.975382i \(-0.570777\pi\)
−0.220524 + 0.975382i \(0.570777\pi\)
\(212\) −3.00316 −0.206258
\(213\) 0 0
\(214\) −15.9611 −1.09108
\(215\) 9.54555 0.651001
\(216\) 0 0
\(217\) 0 0
\(218\) −5.65053 −0.382702
\(219\) 0 0
\(220\) 3.29066 0.221856
\(221\) 37.1300 2.49764
\(222\) 0 0
\(223\) 28.4912 1.90791 0.953956 0.299945i \(-0.0969684\pi\)
0.953956 + 0.299945i \(0.0969684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.247112 0.0164376
\(227\) 2.90326 0.192696 0.0963481 0.995348i \(-0.469284\pi\)
0.0963481 + 0.995348i \(0.469284\pi\)
\(228\) 0 0
\(229\) 6.11908 0.404360 0.202180 0.979348i \(-0.435197\pi\)
0.202180 + 0.979348i \(0.435197\pi\)
\(230\) 22.2318 1.46592
\(231\) 0 0
\(232\) −2.82843 −0.185695
\(233\) 11.6569 0.763666 0.381833 0.924231i \(-0.375293\pi\)
0.381833 + 0.924231i \(0.375293\pi\)
\(234\) 0 0
\(235\) −4.02366 −0.262474
\(236\) −10.8284 −0.704871
\(237\) 0 0
\(238\) 0 0
\(239\) 21.3074 1.37826 0.689130 0.724638i \(-0.257993\pi\)
0.689130 + 0.724638i \(0.257993\pi\)
\(240\) 0 0
\(241\) −11.8783 −0.765148 −0.382574 0.923925i \(-0.624962\pi\)
−0.382574 + 0.923925i \(0.624962\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 7.23948 0.463460
\(245\) 0 0
\(246\) 0 0
\(247\) 0.310547 0.0197596
\(248\) 5.87644 0.373155
\(249\) 0 0
\(250\) 2.72607 0.172412
\(251\) 7.21135 0.455176 0.227588 0.973757i \(-0.426916\pi\)
0.227588 + 0.973757i \(0.426916\pi\)
\(252\) 0 0
\(253\) 6.75605 0.424749
\(254\) 13.3310 0.836464
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.08840 −0.566919 −0.283459 0.958984i \(-0.591482\pi\)
−0.283459 + 0.958984i \(0.591482\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 19.9674 1.23833
\(261\) 0 0
\(262\) −3.04802 −0.188307
\(263\) −5.28373 −0.325809 −0.162904 0.986642i \(-0.552086\pi\)
−0.162904 + 0.986642i \(0.552086\pi\)
\(264\) 0 0
\(265\) −9.88238 −0.607070
\(266\) 0 0
\(267\) 0 0
\(268\) −1.68051 −0.102654
\(269\) −20.2151 −1.23254 −0.616269 0.787536i \(-0.711356\pi\)
−0.616269 + 0.787536i \(0.711356\pi\)
\(270\) 0 0
\(271\) 11.1779 0.679009 0.339504 0.940604i \(-0.389741\pi\)
0.339504 + 0.940604i \(0.389741\pi\)
\(272\) 6.11908 0.371024
\(273\) 0 0
\(274\) −8.47896 −0.512233
\(275\) 5.82843 0.351467
\(276\) 0 0
\(277\) 25.7376 1.54642 0.773212 0.634148i \(-0.218649\pi\)
0.773212 + 0.634148i \(0.218649\pi\)
\(278\) −3.80407 −0.228153
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0063 0.716237 0.358119 0.933676i \(-0.383418\pi\)
0.358119 + 0.933676i \(0.383418\pi\)
\(282\) 0 0
\(283\) −28.7746 −1.71047 −0.855237 0.518237i \(-0.826589\pi\)
−0.855237 + 0.518237i \(0.826589\pi\)
\(284\) 6.47896 0.384455
\(285\) 0 0
\(286\) 6.06791 0.358803
\(287\) 0 0
\(288\) 0 0
\(289\) 20.4432 1.20254
\(290\) −9.30739 −0.546548
\(291\) 0 0
\(292\) −11.6736 −0.683145
\(293\) −20.2337 −1.18207 −0.591033 0.806648i \(-0.701280\pi\)
−0.591033 + 0.806648i \(0.701280\pi\)
\(294\) 0 0
\(295\) −35.6326 −2.07461
\(296\) −8.31055 −0.483041
\(297\) 0 0
\(298\) −24.1269 −1.39763
\(299\) 40.9951 2.37081
\(300\) 0 0
\(301\) 0 0
\(302\) 7.85525 0.452019
\(303\) 0 0
\(304\) 0.0511786 0.00293529
\(305\) 23.8226 1.36408
\(306\) 0 0
\(307\) −17.5033 −0.998967 −0.499484 0.866323i \(-0.666477\pi\)
−0.499484 + 0.866323i \(0.666477\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 19.3374 1.09829
\(311\) −30.0486 −1.70390 −0.851949 0.523625i \(-0.824579\pi\)
−0.851949 + 0.523625i \(0.824579\pi\)
\(312\) 0 0
\(313\) 24.6256 1.39192 0.695960 0.718081i \(-0.254979\pi\)
0.695960 + 0.718081i \(0.254979\pi\)
\(314\) −18.2549 −1.03018
\(315\) 0 0
\(316\) −9.13897 −0.514107
\(317\) −11.1479 −0.626129 −0.313065 0.949732i \(-0.601356\pi\)
−0.313065 + 0.949732i \(0.601356\pi\)
\(318\) 0 0
\(319\) −2.82843 −0.158362
\(320\) 3.29066 0.183953
\(321\) 0 0
\(322\) 0 0
\(323\) 0.313166 0.0174250
\(324\) 0 0
\(325\) 35.3663 1.96177
\(326\) 2.34315 0.129775
\(327\) 0 0
\(328\) −6.11908 −0.337870
\(329\) 0 0
\(330\) 0 0
\(331\) −4.16841 −0.229117 −0.114558 0.993417i \(-0.536545\pi\)
−0.114558 + 0.993417i \(0.536545\pi\)
\(332\) −0.951983 −0.0522469
\(333\) 0 0
\(334\) −19.7863 −1.08266
\(335\) −5.52998 −0.302135
\(336\) 0 0
\(337\) −15.8163 −0.861570 −0.430785 0.902455i \(-0.641763\pi\)
−0.430785 + 0.902455i \(0.641763\pi\)
\(338\) 23.8195 1.29561
\(339\) 0 0
\(340\) 20.1358 1.09202
\(341\) 5.87644 0.318227
\(342\) 0 0
\(343\) 0 0
\(344\) 2.90080 0.156401
\(345\) 0 0
\(346\) −10.5768 −0.568615
\(347\) 16.1058 0.864606 0.432303 0.901728i \(-0.357701\pi\)
0.432303 + 0.901728i \(0.357701\pi\)
\(348\) 0 0
\(349\) −9.19076 −0.491970 −0.245985 0.969274i \(-0.579111\pi\)
−0.245985 + 0.969274i \(0.579111\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −24.8874 −1.32462 −0.662311 0.749229i \(-0.730424\pi\)
−0.662311 + 0.749229i \(0.730424\pi\)
\(354\) 0 0
\(355\) 21.3200 1.13155
\(356\) −16.5469 −0.876982
\(357\) 0 0
\(358\) −18.3016 −0.967270
\(359\) 6.37945 0.336694 0.168347 0.985728i \(-0.446157\pi\)
0.168347 + 0.985728i \(0.446157\pi\)
\(360\) 0 0
\(361\) −18.9974 −0.999862
\(362\) 25.0499 1.31659
\(363\) 0 0
\(364\) 0 0
\(365\) −38.4138 −2.01067
\(366\) 0 0
\(367\) 6.26831 0.327203 0.163602 0.986526i \(-0.447689\pi\)
0.163602 + 0.986526i \(0.447689\pi\)
\(368\) 6.75605 0.352183
\(369\) 0 0
\(370\) −27.3472 −1.42171
\(371\) 0 0
\(372\) 0 0
\(373\) −10.8348 −0.561002 −0.280501 0.959854i \(-0.590501\pi\)
−0.280501 + 0.959854i \(0.590501\pi\)
\(374\) 6.11908 0.316410
\(375\) 0 0
\(376\) −1.22275 −0.0630586
\(377\) −17.1626 −0.883920
\(378\) 0 0
\(379\) 12.7658 0.655738 0.327869 0.944723i \(-0.393670\pi\)
0.327869 + 0.944723i \(0.393670\pi\)
\(380\) 0.168411 0.00863931
\(381\) 0 0
\(382\) 21.8650 1.11871
\(383\) 32.6633 1.66902 0.834509 0.550994i \(-0.185751\pi\)
0.834509 + 0.550994i \(0.185751\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.8195 1.05968
\(387\) 0 0
\(388\) −14.7216 −0.747376
\(389\) −31.0179 −1.57267 −0.786334 0.617801i \(-0.788024\pi\)
−0.786334 + 0.617801i \(0.788024\pi\)
\(390\) 0 0
\(391\) 41.3408 2.09070
\(392\) 0 0
\(393\) 0 0
\(394\) 20.4790 1.03171
\(395\) −30.0732 −1.51315
\(396\) 0 0
\(397\) −15.0986 −0.757777 −0.378888 0.925442i \(-0.623694\pi\)
−0.378888 + 0.925442i \(0.623694\pi\)
\(398\) 15.5333 0.778614
\(399\) 0 0
\(400\) 5.82843 0.291421
\(401\) 9.16525 0.457691 0.228845 0.973463i \(-0.426505\pi\)
0.228845 + 0.973463i \(0.426505\pi\)
\(402\) 0 0
\(403\) 35.6577 1.77624
\(404\) −5.51657 −0.274460
\(405\) 0 0
\(406\) 0 0
\(407\) −8.31055 −0.411939
\(408\) 0 0
\(409\) −9.32149 −0.460918 −0.230459 0.973082i \(-0.574023\pi\)
−0.230459 + 0.973082i \(0.574023\pi\)
\(410\) −20.1358 −0.994437
\(411\) 0 0
\(412\) 11.6781 0.575336
\(413\) 0 0
\(414\) 0 0
\(415\) −3.13265 −0.153776
\(416\) 6.06791 0.297504
\(417\) 0 0
\(418\) 0.0511786 0.00250323
\(419\) 13.5456 0.661744 0.330872 0.943676i \(-0.392657\pi\)
0.330872 + 0.943676i \(0.392657\pi\)
\(420\) 0 0
\(421\) −9.10899 −0.443945 −0.221973 0.975053i \(-0.571250\pi\)
−0.221973 + 0.975053i \(0.571250\pi\)
\(422\) −6.40658 −0.311867
\(423\) 0 0
\(424\) −3.00316 −0.145846
\(425\) 35.6646 1.72999
\(426\) 0 0
\(427\) 0 0
\(428\) −15.9611 −0.771508
\(429\) 0 0
\(430\) 9.54555 0.460328
\(431\) −4.02366 −0.193813 −0.0969064 0.995294i \(-0.530895\pi\)
−0.0969064 + 0.995294i \(0.530895\pi\)
\(432\) 0 0
\(433\) −8.09789 −0.389160 −0.194580 0.980887i \(-0.562334\pi\)
−0.194580 + 0.980887i \(0.562334\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.65053 −0.270611
\(437\) 0.345765 0.0165402
\(438\) 0 0
\(439\) −6.82843 −0.325903 −0.162952 0.986634i \(-0.552101\pi\)
−0.162952 + 0.986634i \(0.552101\pi\)
\(440\) 3.29066 0.156876
\(441\) 0 0
\(442\) 37.1300 1.76610
\(443\) 1.62055 0.0769947 0.0384974 0.999259i \(-0.487743\pi\)
0.0384974 + 0.999259i \(0.487743\pi\)
\(444\) 0 0
\(445\) −54.4501 −2.58118
\(446\) 28.4912 1.34910
\(447\) 0 0
\(448\) 0 0
\(449\) −30.1756 −1.42407 −0.712037 0.702142i \(-0.752227\pi\)
−0.712037 + 0.702142i \(0.752227\pi\)
\(450\) 0 0
\(451\) −6.11908 −0.288136
\(452\) 0.247112 0.0116232
\(453\) 0 0
\(454\) 2.90326 0.136257
\(455\) 0 0
\(456\) 0 0
\(457\) 11.3200 0.529529 0.264764 0.964313i \(-0.414706\pi\)
0.264764 + 0.964313i \(0.414706\pi\)
\(458\) 6.11908 0.285926
\(459\) 0 0
\(460\) 22.2318 1.03657
\(461\) 7.51025 0.349787 0.174894 0.984587i \(-0.444042\pi\)
0.174894 + 0.984587i \(0.444042\pi\)
\(462\) 0 0
\(463\) −3.10899 −0.144487 −0.0722436 0.997387i \(-0.523016\pi\)
−0.0722436 + 0.997387i \(0.523016\pi\)
\(464\) −2.82843 −0.131306
\(465\) 0 0
\(466\) 11.6569 0.539993
\(467\) −36.3189 −1.68064 −0.840320 0.542091i \(-0.817633\pi\)
−0.840320 + 0.542091i \(0.817633\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.02366 −0.185597
\(471\) 0 0
\(472\) −10.8284 −0.498419
\(473\) 2.90080 0.133379
\(474\) 0 0
\(475\) 0.298291 0.0136865
\(476\) 0 0
\(477\) 0 0
\(478\) 21.3074 0.974577
\(479\) 30.5035 1.39374 0.696870 0.717198i \(-0.254576\pi\)
0.696870 + 0.717198i \(0.254576\pi\)
\(480\) 0 0
\(481\) −50.4276 −2.29930
\(482\) −11.8783 −0.541042
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −48.4437 −2.19972
\(486\) 0 0
\(487\) −12.6001 −0.570963 −0.285482 0.958384i \(-0.592154\pi\)
−0.285482 + 0.958384i \(0.592154\pi\)
\(488\) 7.23948 0.327716
\(489\) 0 0
\(490\) 0 0
\(491\) 31.3137 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(492\) 0 0
\(493\) −17.3074 −0.779485
\(494\) 0.310547 0.0139722
\(495\) 0 0
\(496\) 5.87644 0.263860
\(497\) 0 0
\(498\) 0 0
\(499\) −3.49477 −0.156447 −0.0782236 0.996936i \(-0.524925\pi\)
−0.0782236 + 0.996936i \(0.524925\pi\)
\(500\) 2.72607 0.121914
\(501\) 0 0
\(502\) 7.21135 0.321858
\(503\) −20.2627 −0.903468 −0.451734 0.892153i \(-0.649194\pi\)
−0.451734 + 0.892153i \(0.649194\pi\)
\(504\) 0 0
\(505\) −18.1531 −0.807804
\(506\) 6.75605 0.300343
\(507\) 0 0
\(508\) 13.3310 0.591469
\(509\) 18.4123 0.816111 0.408055 0.912957i \(-0.366207\pi\)
0.408055 + 0.912957i \(0.366207\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.08840 −0.400872
\(515\) 38.4285 1.69336
\(516\) 0 0
\(517\) −1.22275 −0.0537765
\(518\) 0 0
\(519\) 0 0
\(520\) 19.9674 0.875628
\(521\) 24.0053 1.05169 0.525846 0.850580i \(-0.323749\pi\)
0.525846 + 0.850580i \(0.323749\pi\)
\(522\) 0 0
\(523\) 39.3894 1.72238 0.861189 0.508285i \(-0.169720\pi\)
0.861189 + 0.508285i \(0.169720\pi\)
\(524\) −3.04802 −0.133153
\(525\) 0 0
\(526\) −5.28373 −0.230382
\(527\) 35.9585 1.56638
\(528\) 0 0
\(529\) 22.6442 0.984531
\(530\) −9.88238 −0.429263
\(531\) 0 0
\(532\) 0 0
\(533\) −37.1300 −1.60828
\(534\) 0 0
\(535\) −52.5224 −2.27074
\(536\) −1.68051 −0.0725870
\(537\) 0 0
\(538\) −20.2151 −0.871536
\(539\) 0 0
\(540\) 0 0
\(541\) 35.1064 1.50934 0.754670 0.656104i \(-0.227797\pi\)
0.754670 + 0.656104i \(0.227797\pi\)
\(542\) 11.1779 0.480132
\(543\) 0 0
\(544\) 6.11908 0.262354
\(545\) −18.5940 −0.796478
\(546\) 0 0
\(547\) 30.3342 1.29700 0.648498 0.761216i \(-0.275397\pi\)
0.648498 + 0.761216i \(0.275397\pi\)
\(548\) −8.47896 −0.362203
\(549\) 0 0
\(550\) 5.82843 0.248525
\(551\) −0.144755 −0.00616676
\(552\) 0 0
\(553\) 0 0
\(554\) 25.7376 1.09349
\(555\) 0 0
\(556\) −3.80407 −0.161328
\(557\) 36.0643 1.52809 0.764047 0.645161i \(-0.223210\pi\)
0.764047 + 0.645161i \(0.223210\pi\)
\(558\) 0 0
\(559\) 17.6018 0.744477
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0063 0.506456
\(563\) −38.7908 −1.63484 −0.817418 0.576046i \(-0.804595\pi\)
−0.817418 + 0.576046i \(0.804595\pi\)
\(564\) 0 0
\(565\) 0.813161 0.0342099
\(566\) −28.7746 −1.20949
\(567\) 0 0
\(568\) 6.47896 0.271851
\(569\) 12.4101 0.520257 0.260128 0.965574i \(-0.416235\pi\)
0.260128 + 0.965574i \(0.416235\pi\)
\(570\) 0 0
\(571\) −35.7863 −1.49761 −0.748806 0.662789i \(-0.769372\pi\)
−0.748806 + 0.662789i \(0.769372\pi\)
\(572\) 6.06791 0.253712
\(573\) 0 0
\(574\) 0 0
\(575\) 39.3771 1.64214
\(576\) 0 0
\(577\) −24.1313 −1.00460 −0.502300 0.864693i \(-0.667513\pi\)
−0.502300 + 0.864693i \(0.667513\pi\)
\(578\) 20.4432 0.850325
\(579\) 0 0
\(580\) −9.30739 −0.386468
\(581\) 0 0
\(582\) 0 0
\(583\) −3.00316 −0.124378
\(584\) −11.6736 −0.483056
\(585\) 0 0
\(586\) −20.2337 −0.835846
\(587\) 31.0539 1.28173 0.640867 0.767652i \(-0.278575\pi\)
0.640867 + 0.767652i \(0.278575\pi\)
\(588\) 0 0
\(589\) 0.300748 0.0123921
\(590\) −35.6326 −1.46697
\(591\) 0 0
\(592\) −8.31055 −0.341561
\(593\) 23.5378 0.966580 0.483290 0.875460i \(-0.339442\pi\)
0.483290 + 0.875460i \(0.339442\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −24.1269 −0.988275
\(597\) 0 0
\(598\) 40.9951 1.67641
\(599\) 20.0037 0.817329 0.408665 0.912685i \(-0.365995\pi\)
0.408665 + 0.912685i \(0.365995\pi\)
\(600\) 0 0
\(601\) −0.799053 −0.0325940 −0.0162970 0.999867i \(-0.505188\pi\)
−0.0162970 + 0.999867i \(0.505188\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 7.85525 0.319625
\(605\) 3.29066 0.133784
\(606\) 0 0
\(607\) 21.3200 0.865353 0.432677 0.901549i \(-0.357569\pi\)
0.432677 + 0.901549i \(0.357569\pi\)
\(608\) 0.0511786 0.00207557
\(609\) 0 0
\(610\) 23.8226 0.964551
\(611\) −7.41954 −0.300162
\(612\) 0 0
\(613\) −8.37398 −0.338222 −0.169111 0.985597i \(-0.554090\pi\)
−0.169111 + 0.985597i \(0.554090\pi\)
\(614\) −17.5033 −0.706376
\(615\) 0 0
\(616\) 0 0
\(617\) −3.74395 −0.150726 −0.0753628 0.997156i \(-0.524011\pi\)
−0.0753628 + 0.997156i \(0.524011\pi\)
\(618\) 0 0
\(619\) 0.924461 0.0371572 0.0185786 0.999827i \(-0.494086\pi\)
0.0185786 + 0.999827i \(0.494086\pi\)
\(620\) 19.3374 0.776607
\(621\) 0 0
\(622\) −30.0486 −1.20484
\(623\) 0 0
\(624\) 0 0
\(625\) −20.1716 −0.806863
\(626\) 24.6256 0.984236
\(627\) 0 0
\(628\) −18.2549 −0.728450
\(629\) −50.8529 −2.02764
\(630\) 0 0
\(631\) 40.4374 1.60979 0.804894 0.593419i \(-0.202222\pi\)
0.804894 + 0.593419i \(0.202222\pi\)
\(632\) −9.13897 −0.363529
\(633\) 0 0
\(634\) −11.1479 −0.442740
\(635\) 43.8679 1.74084
\(636\) 0 0
\(637\) 0 0
\(638\) −2.82843 −0.111979
\(639\) 0 0
\(640\) 3.29066 0.130075
\(641\) −13.6777 −0.540235 −0.270118 0.962827i \(-0.587063\pi\)
−0.270118 + 0.962827i \(0.587063\pi\)
\(642\) 0 0
\(643\) −40.2627 −1.58781 −0.793903 0.608045i \(-0.791954\pi\)
−0.793903 + 0.608045i \(0.791954\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.313166 0.0123214
\(647\) 6.61985 0.260253 0.130127 0.991497i \(-0.458462\pi\)
0.130127 + 0.991497i \(0.458462\pi\)
\(648\) 0 0
\(649\) −10.8284 −0.425053
\(650\) 35.3663 1.38718
\(651\) 0 0
\(652\) 2.34315 0.0917647
\(653\) 15.3200 0.599519 0.299760 0.954015i \(-0.403094\pi\)
0.299760 + 0.954015i \(0.403094\pi\)
\(654\) 0 0
\(655\) −10.0300 −0.391904
\(656\) −6.11908 −0.238910
\(657\) 0 0
\(658\) 0 0
\(659\) −31.7690 −1.23754 −0.618772 0.785570i \(-0.712370\pi\)
−0.618772 + 0.785570i \(0.712370\pi\)
\(660\) 0 0
\(661\) −29.8296 −1.16024 −0.580118 0.814532i \(-0.696994\pi\)
−0.580118 + 0.814532i \(0.696994\pi\)
\(662\) −4.16841 −0.162010
\(663\) 0 0
\(664\) −0.951983 −0.0369441
\(665\) 0 0
\(666\) 0 0
\(667\) −19.1090 −0.739903
\(668\) −19.7863 −0.765557
\(669\) 0 0
\(670\) −5.52998 −0.213642
\(671\) 7.23948 0.279477
\(672\) 0 0
\(673\) 23.7690 0.916228 0.458114 0.888893i \(-0.348525\pi\)
0.458114 + 0.888893i \(0.348525\pi\)
\(674\) −15.8163 −0.609222
\(675\) 0 0
\(676\) 23.8195 0.916134
\(677\) 22.6368 0.870003 0.435002 0.900430i \(-0.356748\pi\)
0.435002 + 0.900430i \(0.356748\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 20.1358 0.772173
\(681\) 0 0
\(682\) 5.87644 0.225021
\(683\) −6.79583 −0.260035 −0.130018 0.991512i \(-0.541503\pi\)
−0.130018 + 0.991512i \(0.541503\pi\)
\(684\) 0 0
\(685\) −27.9013 −1.06606
\(686\) 0 0
\(687\) 0 0
\(688\) 2.90080 0.110592
\(689\) −18.2229 −0.694237
\(690\) 0 0
\(691\) 16.2292 0.617389 0.308694 0.951161i \(-0.400108\pi\)
0.308694 + 0.951161i \(0.400108\pi\)
\(692\) −10.5768 −0.402071
\(693\) 0 0
\(694\) 16.1058 0.611369
\(695\) −12.5179 −0.474830
\(696\) 0 0
\(697\) −37.4432 −1.41826
\(698\) −9.19076 −0.347875
\(699\) 0 0
\(700\) 0 0
\(701\) 32.3253 1.22091 0.610454 0.792052i \(-0.290987\pi\)
0.610454 + 0.792052i \(0.290987\pi\)
\(702\) 0 0
\(703\) −0.425322 −0.0160413
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −24.8874 −0.936649
\(707\) 0 0
\(708\) 0 0
\(709\) −10.8458 −0.407321 −0.203661 0.979042i \(-0.565284\pi\)
−0.203661 + 0.979042i \(0.565284\pi\)
\(710\) 21.3200 0.800127
\(711\) 0 0
\(712\) −16.5469 −0.620120
\(713\) 39.7015 1.48683
\(714\) 0 0
\(715\) 19.9674 0.746738
\(716\) −18.3016 −0.683963
\(717\) 0 0
\(718\) 6.37945 0.238079
\(719\) −3.69277 −0.137717 −0.0688585 0.997626i \(-0.521936\pi\)
−0.0688585 + 0.997626i \(0.521936\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.9974 −0.707009
\(723\) 0 0
\(724\) 25.0499 0.930971
\(725\) −16.4853 −0.612248
\(726\) 0 0
\(727\) 34.8470 1.29240 0.646202 0.763166i \(-0.276356\pi\)
0.646202 + 0.763166i \(0.276356\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −38.4138 −1.42176
\(731\) 17.7503 0.656517
\(732\) 0 0
\(733\) −39.9168 −1.47436 −0.737181 0.675696i \(-0.763843\pi\)
−0.737181 + 0.675696i \(0.763843\pi\)
\(734\) 6.26831 0.231368
\(735\) 0 0
\(736\) 6.75605 0.249031
\(737\) −1.68051 −0.0619024
\(738\) 0 0
\(739\) 4.77479 0.175644 0.0878218 0.996136i \(-0.472009\pi\)
0.0878218 + 0.996136i \(0.472009\pi\)
\(740\) −27.3472 −1.00530
\(741\) 0 0
\(742\) 0 0
\(743\) 25.1153 0.921392 0.460696 0.887558i \(-0.347600\pi\)
0.460696 + 0.887558i \(0.347600\pi\)
\(744\) 0 0
\(745\) −79.3933 −2.90874
\(746\) −10.8348 −0.396688
\(747\) 0 0
\(748\) 6.11908 0.223736
\(749\) 0 0
\(750\) 0 0
\(751\) 54.7154 1.99659 0.998296 0.0583532i \(-0.0185850\pi\)
0.998296 + 0.0583532i \(0.0185850\pi\)
\(752\) −1.22275 −0.0445892
\(753\) 0 0
\(754\) −17.1626 −0.625026
\(755\) 25.8489 0.940739
\(756\) 0 0
\(757\) 16.9316 0.615391 0.307695 0.951485i \(-0.400442\pi\)
0.307695 + 0.951485i \(0.400442\pi\)
\(758\) 12.7658 0.463676
\(759\) 0 0
\(760\) 0.168411 0.00610892
\(761\) 4.16781 0.151083 0.0755414 0.997143i \(-0.475931\pi\)
0.0755414 + 0.997143i \(0.475931\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 21.8650 0.791050
\(765\) 0 0
\(766\) 32.6633 1.18017
\(767\) −65.7059 −2.37250
\(768\) 0 0
\(769\) 16.8273 0.606807 0.303403 0.952862i \(-0.401877\pi\)
0.303403 + 0.952862i \(0.401877\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.8195 0.749310
\(773\) 1.60929 0.0578822 0.0289411 0.999581i \(-0.490786\pi\)
0.0289411 + 0.999581i \(0.490786\pi\)
\(774\) 0 0
\(775\) 34.2504 1.23031
\(776\) −14.7216 −0.528475
\(777\) 0 0
\(778\) −31.0179 −1.11204
\(779\) −0.313166 −0.0112203
\(780\) 0 0
\(781\) 6.47896 0.231835
\(782\) 41.3408 1.47835
\(783\) 0 0
\(784\) 0 0
\(785\) −60.0706 −2.14401
\(786\) 0 0
\(787\) −32.6123 −1.16250 −0.581252 0.813724i \(-0.697437\pi\)
−0.581252 + 0.813724i \(0.697437\pi\)
\(788\) 20.4790 0.729533
\(789\) 0 0
\(790\) −30.0732 −1.06996
\(791\) 0 0
\(792\) 0 0
\(793\) 43.9285 1.55995
\(794\) −15.0986 −0.535829
\(795\) 0 0
\(796\) 15.5333 0.550563
\(797\) −35.4599 −1.25605 −0.628027 0.778191i \(-0.716137\pi\)
−0.628027 + 0.778191i \(0.716137\pi\)
\(798\) 0 0
\(799\) −7.48212 −0.264698
\(800\) 5.82843 0.206066
\(801\) 0 0
\(802\) 9.16525 0.323636
\(803\) −11.6736 −0.411952
\(804\) 0 0
\(805\) 0 0
\(806\) 35.6577 1.25599
\(807\) 0 0
\(808\) −5.51657 −0.194072
\(809\) 53.3953 1.87728 0.938640 0.344899i \(-0.112087\pi\)
0.938640 + 0.344899i \(0.112087\pi\)
\(810\) 0 0
\(811\) 0.178048 0.00625211 0.00312606 0.999995i \(-0.499005\pi\)
0.00312606 + 0.999995i \(0.499005\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.31055 −0.291285
\(815\) 7.71049 0.270087
\(816\) 0 0
\(817\) 0.148459 0.00519392
\(818\) −9.32149 −0.325918
\(819\) 0 0
\(820\) −20.1358 −0.703173
\(821\) −4.15740 −0.145094 −0.0725472 0.997365i \(-0.523113\pi\)
−0.0725472 + 0.997365i \(0.523113\pi\)
\(822\) 0 0
\(823\) 0.0210370 0.000733305 0 0.000366653 1.00000i \(-0.499883\pi\)
0.000366653 1.00000i \(0.499883\pi\)
\(824\) 11.6781 0.406824
\(825\) 0 0
\(826\) 0 0
\(827\) 43.7774 1.52229 0.761145 0.648582i \(-0.224638\pi\)
0.761145 + 0.648582i \(0.224638\pi\)
\(828\) 0 0
\(829\) 32.0602 1.11350 0.556749 0.830681i \(-0.312049\pi\)
0.556749 + 0.830681i \(0.312049\pi\)
\(830\) −3.13265 −0.108736
\(831\) 0 0
\(832\) 6.06791 0.210367
\(833\) 0 0
\(834\) 0 0
\(835\) −65.1101 −2.25323
\(836\) 0.0511786 0.00177005
\(837\) 0 0
\(838\) 13.5456 0.467923
\(839\) −6.92200 −0.238974 −0.119487 0.992836i \(-0.538125\pi\)
−0.119487 + 0.992836i \(0.538125\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) −9.10899 −0.313917
\(843\) 0 0
\(844\) −6.40658 −0.220524
\(845\) 78.3818 2.69641
\(846\) 0 0
\(847\) 0 0
\(848\) −3.00316 −0.103129
\(849\) 0 0
\(850\) 35.6646 1.22329
\(851\) −56.1465 −1.92468
\(852\) 0 0
\(853\) −27.4727 −0.940648 −0.470324 0.882494i \(-0.655863\pi\)
−0.470324 + 0.882494i \(0.655863\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −15.9611 −0.545538
\(857\) 30.1949 1.03144 0.515720 0.856757i \(-0.327525\pi\)
0.515720 + 0.856757i \(0.327525\pi\)
\(858\) 0 0
\(859\) 42.2869 1.44281 0.721405 0.692513i \(-0.243497\pi\)
0.721405 + 0.692513i \(0.243497\pi\)
\(860\) 9.54555 0.325501
\(861\) 0 0
\(862\) −4.02366 −0.137046
\(863\) −45.0340 −1.53298 −0.766488 0.642259i \(-0.777997\pi\)
−0.766488 + 0.642259i \(0.777997\pi\)
\(864\) 0 0
\(865\) −34.8048 −1.18340
\(866\) −8.09789 −0.275177
\(867\) 0 0
\(868\) 0 0
\(869\) −9.13897 −0.310018
\(870\) 0 0
\(871\) −10.1972 −0.345518
\(872\) −5.65053 −0.191351
\(873\) 0 0
\(874\) 0.345765 0.0116957
\(875\) 0 0
\(876\) 0 0
\(877\) −42.6684 −1.44081 −0.720405 0.693554i \(-0.756044\pi\)
−0.720405 + 0.693554i \(0.756044\pi\)
\(878\) −6.82843 −0.230448
\(879\) 0 0
\(880\) 3.29066 0.110928
\(881\) −11.1844 −0.376813 −0.188407 0.982091i \(-0.560332\pi\)
−0.188407 + 0.982091i \(0.560332\pi\)
\(882\) 0 0
\(883\) −24.2843 −0.817231 −0.408615 0.912707i \(-0.633988\pi\)
−0.408615 + 0.912707i \(0.633988\pi\)
\(884\) 37.1300 1.24882
\(885\) 0 0
\(886\) 1.62055 0.0544435
\(887\) 26.2779 0.882327 0.441164 0.897427i \(-0.354566\pi\)
0.441164 + 0.897427i \(0.354566\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −54.4501 −1.82517
\(891\) 0 0
\(892\) 28.4912 0.953956
\(893\) −0.0625787 −0.00209412
\(894\) 0 0
\(895\) −60.2243 −2.01308
\(896\) 0 0
\(897\) 0 0
\(898\) −30.1756 −1.00697
\(899\) −16.6211 −0.554345
\(900\) 0 0
\(901\) −18.3766 −0.612213
\(902\) −6.11908 −0.203743
\(903\) 0 0
\(904\) 0.247112 0.00821882
\(905\) 82.4305 2.74008
\(906\) 0 0
\(907\) −48.9164 −1.62424 −0.812121 0.583489i \(-0.801687\pi\)
−0.812121 + 0.583489i \(0.801687\pi\)
\(908\) 2.90326 0.0963481
\(909\) 0 0
\(910\) 0 0
\(911\) −8.95444 −0.296674 −0.148337 0.988937i \(-0.547392\pi\)
−0.148337 + 0.988937i \(0.547392\pi\)
\(912\) 0 0
\(913\) −0.951983 −0.0315060
\(914\) 11.3200 0.374433
\(915\) 0 0
\(916\) 6.11908 0.202180
\(917\) 0 0
\(918\) 0 0
\(919\) −11.9222 −0.393276 −0.196638 0.980476i \(-0.563002\pi\)
−0.196638 + 0.980476i \(0.563002\pi\)
\(920\) 22.2318 0.732962
\(921\) 0 0
\(922\) 7.51025 0.247337
\(923\) 39.3137 1.29403
\(924\) 0 0
\(925\) −48.4374 −1.59261
\(926\) −3.10899 −0.102168
\(927\) 0 0
\(928\) −2.82843 −0.0928477
\(929\) 13.9105 0.456389 0.228194 0.973616i \(-0.426718\pi\)
0.228194 + 0.973616i \(0.426718\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 11.6569 0.381833
\(933\) 0 0
\(934\) −36.3189 −1.18839
\(935\) 20.1358 0.658511
\(936\) 0 0
\(937\) 20.0104 0.653711 0.326856 0.945074i \(-0.394011\pi\)
0.326856 + 0.945074i \(0.394011\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.02366 −0.131237
\(941\) 54.9084 1.78996 0.894982 0.446103i \(-0.147188\pi\)
0.894982 + 0.446103i \(0.147188\pi\)
\(942\) 0 0
\(943\) −41.3408 −1.34624
\(944\) −10.8284 −0.352435
\(945\) 0 0
\(946\) 2.90080 0.0943133
\(947\) 7.39850 0.240419 0.120210 0.992749i \(-0.461643\pi\)
0.120210 + 0.992749i \(0.461643\pi\)
\(948\) 0 0
\(949\) −70.8342 −2.29938
\(950\) 0.298291 0.00967782
\(951\) 0 0
\(952\) 0 0
\(953\) −21.1416 −0.684843 −0.342422 0.939546i \(-0.611247\pi\)
−0.342422 + 0.939546i \(0.611247\pi\)
\(954\) 0 0
\(955\) 71.9504 2.32826
\(956\) 21.3074 0.689130
\(957\) 0 0
\(958\) 30.5035 0.985522
\(959\) 0 0
\(960\) 0 0
\(961\) 3.53259 0.113955
\(962\) −50.4276 −1.62585
\(963\) 0 0
\(964\) −11.8783 −0.382574
\(965\) 68.5098 2.20541
\(966\) 0 0
\(967\) 11.7174 0.376807 0.188404 0.982092i \(-0.439669\pi\)
0.188404 + 0.982092i \(0.439669\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −48.4437 −1.55543
\(971\) −35.9487 −1.15365 −0.576824 0.816869i \(-0.695708\pi\)
−0.576824 + 0.816869i \(0.695708\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −12.6001 −0.403732
\(975\) 0 0
\(976\) 7.23948 0.231730
\(977\) 4.33561 0.138709 0.0693543 0.997592i \(-0.477906\pi\)
0.0693543 + 0.997592i \(0.477906\pi\)
\(978\) 0 0
\(979\) −16.5469 −0.528840
\(980\) 0 0
\(981\) 0 0
\(982\) 31.3137 0.999261
\(983\) 16.3444 0.521305 0.260653 0.965433i \(-0.416062\pi\)
0.260653 + 0.965433i \(0.416062\pi\)
\(984\) 0 0
\(985\) 67.3892 2.14720
\(986\) −17.3074 −0.551179
\(987\) 0 0
\(988\) 0.310547 0.00987981
\(989\) 19.5980 0.623180
\(990\) 0 0
\(991\) 11.5657 0.367398 0.183699 0.982983i \(-0.441193\pi\)
0.183699 + 0.982983i \(0.441193\pi\)
\(992\) 5.87644 0.186577
\(993\) 0 0
\(994\) 0 0
\(995\) 51.1148 1.62045
\(996\) 0 0
\(997\) 41.6956 1.32051 0.660257 0.751040i \(-0.270447\pi\)
0.660257 + 0.751040i \(0.270447\pi\)
\(998\) −3.49477 −0.110625
\(999\) 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 9702.2.a.ec.1.4 4
3.2 odd 2 3234.2.a.bk.1.1 yes 4
7.6 odd 2 9702.2.a.eb.1.1 4
21.20 even 2 3234.2.a.bj.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bj.1.4 4 21.20 even 2
3234.2.a.bk.1.1 yes 4 3.2 odd 2
9702.2.a.eb.1.1 4 7.6 odd 2
9702.2.a.ec.1.4 4 1.1 even 1 trivial