Properties

Label 1452.4.a.t.1.6
Level $1452$
Weight $4$
Character 1452.1
Self dual yes
Analytic conductor $85.671$
Analytic rank $1$
Dimension $6$
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] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.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: \(85.6707733283\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.44098\) of defining polynomial
Character \(\chi\) \(=\) 1452.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +21.4270 q^{5} -28.4163 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +21.4270 q^{5} -28.4163 q^{7} +9.00000 q^{9} -53.5450 q^{13} +64.2810 q^{15} +44.9597 q^{17} -118.492 q^{19} -85.2490 q^{21} -148.060 q^{23} +334.117 q^{25} +27.0000 q^{27} +124.664 q^{29} -24.8101 q^{31} -608.877 q^{35} -20.0709 q^{37} -160.635 q^{39} -35.8849 q^{41} -290.976 q^{43} +192.843 q^{45} -216.389 q^{47} +464.488 q^{49} +134.879 q^{51} -656.216 q^{53} -355.475 q^{57} -204.402 q^{59} -8.56858 q^{61} -255.747 q^{63} -1147.31 q^{65} +134.926 q^{67} -444.181 q^{69} -260.697 q^{71} -528.386 q^{73} +1002.35 q^{75} +834.880 q^{79} +81.0000 q^{81} -617.359 q^{83} +963.353 q^{85} +373.991 q^{87} -403.992 q^{89} +1521.55 q^{91} -74.4304 q^{93} -2538.92 q^{95} -1186.66 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 21.4270 1.91649 0.958245 0.285949i \(-0.0923088\pi\)
0.958245 + 0.285949i \(0.0923088\pi\)
\(6\) 0 0
\(7\) −28.4163 −1.53434 −0.767169 0.641445i \(-0.778335\pi\)
−0.767169 + 0.641445i \(0.778335\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −53.5450 −1.14236 −0.571181 0.820824i \(-0.693515\pi\)
−0.571181 + 0.820824i \(0.693515\pi\)
\(14\) 0 0
\(15\) 64.2810 1.10649
\(16\) 0 0
\(17\) 44.9597 0.641432 0.320716 0.947175i \(-0.396077\pi\)
0.320716 + 0.947175i \(0.396077\pi\)
\(18\) 0 0
\(19\) −118.492 −1.43073 −0.715364 0.698752i \(-0.753739\pi\)
−0.715364 + 0.698752i \(0.753739\pi\)
\(20\) 0 0
\(21\) −85.2490 −0.885851
\(22\) 0 0
\(23\) −148.060 −1.34229 −0.671146 0.741325i \(-0.734198\pi\)
−0.671146 + 0.741325i \(0.734198\pi\)
\(24\) 0 0
\(25\) 334.117 2.67293
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 124.664 0.798257 0.399128 0.916895i \(-0.369313\pi\)
0.399128 + 0.916895i \(0.369313\pi\)
\(30\) 0 0
\(31\) −24.8101 −0.143743 −0.0718715 0.997414i \(-0.522897\pi\)
−0.0718715 + 0.997414i \(0.522897\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −608.877 −2.94054
\(36\) 0 0
\(37\) −20.0709 −0.0891795 −0.0445898 0.999005i \(-0.514198\pi\)
−0.0445898 + 0.999005i \(0.514198\pi\)
\(38\) 0 0
\(39\) −160.635 −0.659543
\(40\) 0 0
\(41\) −35.8849 −0.136690 −0.0683449 0.997662i \(-0.521772\pi\)
−0.0683449 + 0.997662i \(0.521772\pi\)
\(42\) 0 0
\(43\) −290.976 −1.03194 −0.515970 0.856607i \(-0.672568\pi\)
−0.515970 + 0.856607i \(0.672568\pi\)
\(44\) 0 0
\(45\) 192.843 0.638830
\(46\) 0 0
\(47\) −216.389 −0.671567 −0.335783 0.941939i \(-0.609001\pi\)
−0.335783 + 0.941939i \(0.609001\pi\)
\(48\) 0 0
\(49\) 464.488 1.35419
\(50\) 0 0
\(51\) 134.879 0.370331
\(52\) 0 0
\(53\) −656.216 −1.70072 −0.850360 0.526201i \(-0.823616\pi\)
−0.850360 + 0.526201i \(0.823616\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −355.475 −0.826032
\(58\) 0 0
\(59\) −204.402 −0.451032 −0.225516 0.974239i \(-0.572407\pi\)
−0.225516 + 0.974239i \(0.572407\pi\)
\(60\) 0 0
\(61\) −8.56858 −0.0179852 −0.00899258 0.999960i \(-0.502862\pi\)
−0.00899258 + 0.999960i \(0.502862\pi\)
\(62\) 0 0
\(63\) −255.747 −0.511446
\(64\) 0 0
\(65\) −1147.31 −2.18933
\(66\) 0 0
\(67\) 134.926 0.246028 0.123014 0.992405i \(-0.460744\pi\)
0.123014 + 0.992405i \(0.460744\pi\)
\(68\) 0 0
\(69\) −444.181 −0.774973
\(70\) 0 0
\(71\) −260.697 −0.435761 −0.217880 0.975975i \(-0.569914\pi\)
−0.217880 + 0.975975i \(0.569914\pi\)
\(72\) 0 0
\(73\) −528.386 −0.847163 −0.423582 0.905858i \(-0.639227\pi\)
−0.423582 + 0.905858i \(0.639227\pi\)
\(74\) 0 0
\(75\) 1002.35 1.54322
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 834.880 1.18900 0.594502 0.804094i \(-0.297349\pi\)
0.594502 + 0.804094i \(0.297349\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −617.359 −0.816433 −0.408217 0.912885i \(-0.633849\pi\)
−0.408217 + 0.912885i \(0.633849\pi\)
\(84\) 0 0
\(85\) 963.353 1.22930
\(86\) 0 0
\(87\) 373.991 0.460874
\(88\) 0 0
\(89\) −403.992 −0.481158 −0.240579 0.970630i \(-0.577337\pi\)
−0.240579 + 0.970630i \(0.577337\pi\)
\(90\) 0 0
\(91\) 1521.55 1.75277
\(92\) 0 0
\(93\) −74.4304 −0.0829900
\(94\) 0 0
\(95\) −2538.92 −2.74198
\(96\) 0 0
\(97\) −1186.66 −1.24213 −0.621065 0.783759i \(-0.713300\pi\)
−0.621065 + 0.783759i \(0.713300\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1040.22 1.02481 0.512404 0.858745i \(-0.328755\pi\)
0.512404 + 0.858745i \(0.328755\pi\)
\(102\) 0 0
\(103\) −1958.13 −1.87321 −0.936604 0.350390i \(-0.886049\pi\)
−0.936604 + 0.350390i \(0.886049\pi\)
\(104\) 0 0
\(105\) −1826.63 −1.69772
\(106\) 0 0
\(107\) 211.062 0.190693 0.0953465 0.995444i \(-0.469604\pi\)
0.0953465 + 0.995444i \(0.469604\pi\)
\(108\) 0 0
\(109\) 1506.70 1.32399 0.661997 0.749506i \(-0.269709\pi\)
0.661997 + 0.749506i \(0.269709\pi\)
\(110\) 0 0
\(111\) −60.2128 −0.0514878
\(112\) 0 0
\(113\) −1443.82 −1.20198 −0.600989 0.799257i \(-0.705227\pi\)
−0.600989 + 0.799257i \(0.705227\pi\)
\(114\) 0 0
\(115\) −3172.49 −2.57249
\(116\) 0 0
\(117\) −481.905 −0.380787
\(118\) 0 0
\(119\) −1277.59 −0.984173
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −107.655 −0.0789179
\(124\) 0 0
\(125\) 4480.74 3.20616
\(126\) 0 0
\(127\) 246.030 0.171902 0.0859512 0.996299i \(-0.472607\pi\)
0.0859512 + 0.996299i \(0.472607\pi\)
\(128\) 0 0
\(129\) −872.928 −0.595791
\(130\) 0 0
\(131\) 451.257 0.300966 0.150483 0.988613i \(-0.451917\pi\)
0.150483 + 0.988613i \(0.451917\pi\)
\(132\) 0 0
\(133\) 3367.10 2.19522
\(134\) 0 0
\(135\) 578.529 0.368829
\(136\) 0 0
\(137\) −179.563 −0.111979 −0.0559896 0.998431i \(-0.517831\pi\)
−0.0559896 + 0.998431i \(0.517831\pi\)
\(138\) 0 0
\(139\) −1037.11 −0.632851 −0.316425 0.948617i \(-0.602483\pi\)
−0.316425 + 0.948617i \(0.602483\pi\)
\(140\) 0 0
\(141\) −649.168 −0.387729
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2671.17 1.52985
\(146\) 0 0
\(147\) 1393.47 0.781844
\(148\) 0 0
\(149\) −1496.14 −0.822610 −0.411305 0.911498i \(-0.634927\pi\)
−0.411305 + 0.911498i \(0.634927\pi\)
\(150\) 0 0
\(151\) 2491.92 1.34298 0.671490 0.741014i \(-0.265655\pi\)
0.671490 + 0.741014i \(0.265655\pi\)
\(152\) 0 0
\(153\) 404.638 0.213811
\(154\) 0 0
\(155\) −531.607 −0.275482
\(156\) 0 0
\(157\) 2772.94 1.40959 0.704793 0.709413i \(-0.251040\pi\)
0.704793 + 0.709413i \(0.251040\pi\)
\(158\) 0 0
\(159\) −1968.65 −0.981911
\(160\) 0 0
\(161\) 4207.33 2.05953
\(162\) 0 0
\(163\) 1225.74 0.589004 0.294502 0.955651i \(-0.404846\pi\)
0.294502 + 0.955651i \(0.404846\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1162.87 0.538834 0.269417 0.963024i \(-0.413169\pi\)
0.269417 + 0.963024i \(0.413169\pi\)
\(168\) 0 0
\(169\) 670.065 0.304991
\(170\) 0 0
\(171\) −1066.42 −0.476910
\(172\) 0 0
\(173\) 1031.86 0.453473 0.226736 0.973956i \(-0.427194\pi\)
0.226736 + 0.973956i \(0.427194\pi\)
\(174\) 0 0
\(175\) −9494.37 −4.10118
\(176\) 0 0
\(177\) −613.206 −0.260403
\(178\) 0 0
\(179\) 2690.33 1.12338 0.561689 0.827348i \(-0.310152\pi\)
0.561689 + 0.827348i \(0.310152\pi\)
\(180\) 0 0
\(181\) −3313.07 −1.36054 −0.680272 0.732959i \(-0.738138\pi\)
−0.680272 + 0.732959i \(0.738138\pi\)
\(182\) 0 0
\(183\) −25.7058 −0.0103837
\(184\) 0 0
\(185\) −430.060 −0.170912
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −767.241 −0.295284
\(190\) 0 0
\(191\) −355.584 −0.134708 −0.0673538 0.997729i \(-0.521456\pi\)
−0.0673538 + 0.997729i \(0.521456\pi\)
\(192\) 0 0
\(193\) 2116.73 0.789458 0.394729 0.918798i \(-0.370838\pi\)
0.394729 + 0.918798i \(0.370838\pi\)
\(194\) 0 0
\(195\) −3441.93 −1.26401
\(196\) 0 0
\(197\) −2583.36 −0.934299 −0.467149 0.884178i \(-0.654719\pi\)
−0.467149 + 0.884178i \(0.654719\pi\)
\(198\) 0 0
\(199\) 2570.40 0.915633 0.457817 0.889047i \(-0.348632\pi\)
0.457817 + 0.889047i \(0.348632\pi\)
\(200\) 0 0
\(201\) 404.779 0.142044
\(202\) 0 0
\(203\) −3542.48 −1.22480
\(204\) 0 0
\(205\) −768.906 −0.261964
\(206\) 0 0
\(207\) −1332.54 −0.447431
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3962.24 −1.29276 −0.646379 0.763017i \(-0.723717\pi\)
−0.646379 + 0.763017i \(0.723717\pi\)
\(212\) 0 0
\(213\) −782.090 −0.251587
\(214\) 0 0
\(215\) −6234.74 −1.97770
\(216\) 0 0
\(217\) 705.013 0.220550
\(218\) 0 0
\(219\) −1585.16 −0.489110
\(220\) 0 0
\(221\) −2407.37 −0.732747
\(222\) 0 0
\(223\) 3667.20 1.10123 0.550614 0.834760i \(-0.314394\pi\)
0.550614 + 0.834760i \(0.314394\pi\)
\(224\) 0 0
\(225\) 3007.05 0.890978
\(226\) 0 0
\(227\) −2406.31 −0.703580 −0.351790 0.936079i \(-0.614427\pi\)
−0.351790 + 0.936079i \(0.614427\pi\)
\(228\) 0 0
\(229\) −483.451 −0.139508 −0.0697540 0.997564i \(-0.522221\pi\)
−0.0697540 + 0.997564i \(0.522221\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5376.45 −1.51169 −0.755843 0.654753i \(-0.772773\pi\)
−0.755843 + 0.654753i \(0.772773\pi\)
\(234\) 0 0
\(235\) −4636.58 −1.28705
\(236\) 0 0
\(237\) 2504.64 0.686472
\(238\) 0 0
\(239\) −521.222 −0.141067 −0.0705336 0.997509i \(-0.522470\pi\)
−0.0705336 + 0.997509i \(0.522470\pi\)
\(240\) 0 0
\(241\) 729.004 0.194852 0.0974259 0.995243i \(-0.468939\pi\)
0.0974259 + 0.995243i \(0.468939\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 9952.60 2.59530
\(246\) 0 0
\(247\) 6344.63 1.63441
\(248\) 0 0
\(249\) −1852.08 −0.471368
\(250\) 0 0
\(251\) 972.904 0.244658 0.122329 0.992490i \(-0.460964\pi\)
0.122329 + 0.992490i \(0.460964\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2890.06 0.709735
\(256\) 0 0
\(257\) 2225.28 0.540114 0.270057 0.962844i \(-0.412957\pi\)
0.270057 + 0.962844i \(0.412957\pi\)
\(258\) 0 0
\(259\) 570.343 0.136832
\(260\) 0 0
\(261\) 1121.97 0.266086
\(262\) 0 0
\(263\) −2283.89 −0.535478 −0.267739 0.963492i \(-0.586276\pi\)
−0.267739 + 0.963492i \(0.586276\pi\)
\(264\) 0 0
\(265\) −14060.7 −3.25941
\(266\) 0 0
\(267\) −1211.98 −0.277797
\(268\) 0 0
\(269\) 32.2355 0.00730644 0.00365322 0.999993i \(-0.498837\pi\)
0.00365322 + 0.999993i \(0.498837\pi\)
\(270\) 0 0
\(271\) −2393.47 −0.536506 −0.268253 0.963348i \(-0.586446\pi\)
−0.268253 + 0.963348i \(0.586446\pi\)
\(272\) 0 0
\(273\) 4564.66 1.01196
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3468.31 0.752313 0.376156 0.926556i \(-0.377246\pi\)
0.376156 + 0.926556i \(0.377246\pi\)
\(278\) 0 0
\(279\) −223.291 −0.0479143
\(280\) 0 0
\(281\) −4818.51 −1.02295 −0.511474 0.859299i \(-0.670900\pi\)
−0.511474 + 0.859299i \(0.670900\pi\)
\(282\) 0 0
\(283\) −6824.91 −1.43357 −0.716783 0.697296i \(-0.754386\pi\)
−0.716783 + 0.697296i \(0.754386\pi\)
\(284\) 0 0
\(285\) −7616.76 −1.58308
\(286\) 0 0
\(287\) 1019.72 0.209728
\(288\) 0 0
\(289\) −2891.62 −0.588565
\(290\) 0 0
\(291\) −3559.97 −0.717144
\(292\) 0 0
\(293\) 4435.82 0.884448 0.442224 0.896905i \(-0.354190\pi\)
0.442224 + 0.896905i \(0.354190\pi\)
\(294\) 0 0
\(295\) −4379.72 −0.864398
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7927.89 1.53338
\(300\) 0 0
\(301\) 8268.47 1.58334
\(302\) 0 0
\(303\) 3120.65 0.591673
\(304\) 0 0
\(305\) −183.599 −0.0344684
\(306\) 0 0
\(307\) −4551.33 −0.846117 −0.423059 0.906102i \(-0.639044\pi\)
−0.423059 + 0.906102i \(0.639044\pi\)
\(308\) 0 0
\(309\) −5874.39 −1.08150
\(310\) 0 0
\(311\) −2037.42 −0.371484 −0.185742 0.982599i \(-0.559469\pi\)
−0.185742 + 0.982599i \(0.559469\pi\)
\(312\) 0 0
\(313\) −9108.84 −1.64493 −0.822463 0.568818i \(-0.807401\pi\)
−0.822463 + 0.568818i \(0.807401\pi\)
\(314\) 0 0
\(315\) −5479.89 −0.980181
\(316\) 0 0
\(317\) −6293.63 −1.11510 −0.557548 0.830145i \(-0.688258\pi\)
−0.557548 + 0.830145i \(0.688258\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 633.187 0.110097
\(322\) 0 0
\(323\) −5327.35 −0.917715
\(324\) 0 0
\(325\) −17890.3 −3.05346
\(326\) 0 0
\(327\) 4520.09 0.764409
\(328\) 0 0
\(329\) 6149.00 1.03041
\(330\) 0 0
\(331\) 942.916 0.156578 0.0782891 0.996931i \(-0.475054\pi\)
0.0782891 + 0.996931i \(0.475054\pi\)
\(332\) 0 0
\(333\) −180.638 −0.0297265
\(334\) 0 0
\(335\) 2891.07 0.471510
\(336\) 0 0
\(337\) −8612.17 −1.39209 −0.696046 0.717997i \(-0.745059\pi\)
−0.696046 + 0.717997i \(0.745059\pi\)
\(338\) 0 0
\(339\) −4331.47 −0.693963
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −3452.26 −0.543453
\(344\) 0 0
\(345\) −9517.47 −1.48523
\(346\) 0 0
\(347\) 8816.60 1.36398 0.681989 0.731363i \(-0.261115\pi\)
0.681989 + 0.731363i \(0.261115\pi\)
\(348\) 0 0
\(349\) −8596.61 −1.31853 −0.659263 0.751912i \(-0.729132\pi\)
−0.659263 + 0.751912i \(0.729132\pi\)
\(350\) 0 0
\(351\) −1445.71 −0.219848
\(352\) 0 0
\(353\) 10942.2 1.64985 0.824923 0.565245i \(-0.191218\pi\)
0.824923 + 0.565245i \(0.191218\pi\)
\(354\) 0 0
\(355\) −5585.95 −0.835131
\(356\) 0 0
\(357\) −3832.77 −0.568213
\(358\) 0 0
\(359\) 429.508 0.0631436 0.0315718 0.999501i \(-0.489949\pi\)
0.0315718 + 0.999501i \(0.489949\pi\)
\(360\) 0 0
\(361\) 7181.27 1.04698
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11321.7 −1.62358
\(366\) 0 0
\(367\) 6903.84 0.981955 0.490978 0.871172i \(-0.336640\pi\)
0.490978 + 0.871172i \(0.336640\pi\)
\(368\) 0 0
\(369\) −322.964 −0.0455632
\(370\) 0 0
\(371\) 18647.3 2.60948
\(372\) 0 0
\(373\) −5121.25 −0.710907 −0.355454 0.934694i \(-0.615674\pi\)
−0.355454 + 0.934694i \(0.615674\pi\)
\(374\) 0 0
\(375\) 13442.2 1.85108
\(376\) 0 0
\(377\) −6675.11 −0.911898
\(378\) 0 0
\(379\) −201.217 −0.0272713 −0.0136356 0.999907i \(-0.504340\pi\)
−0.0136356 + 0.999907i \(0.504340\pi\)
\(380\) 0 0
\(381\) 738.089 0.0992478
\(382\) 0 0
\(383\) −3789.38 −0.505557 −0.252779 0.967524i \(-0.581344\pi\)
−0.252779 + 0.967524i \(0.581344\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2618.78 −0.343980
\(388\) 0 0
\(389\) 1530.36 0.199467 0.0997333 0.995014i \(-0.468201\pi\)
0.0997333 + 0.995014i \(0.468201\pi\)
\(390\) 0 0
\(391\) −6656.76 −0.860989
\(392\) 0 0
\(393\) 1353.77 0.173763
\(394\) 0 0
\(395\) 17889.0 2.27871
\(396\) 0 0
\(397\) 7169.61 0.906378 0.453189 0.891414i \(-0.350286\pi\)
0.453189 + 0.891414i \(0.350286\pi\)
\(398\) 0 0
\(399\) 10101.3 1.26741
\(400\) 0 0
\(401\) −242.325 −0.0301773 −0.0150887 0.999886i \(-0.504803\pi\)
−0.0150887 + 0.999886i \(0.504803\pi\)
\(402\) 0 0
\(403\) 1328.46 0.164206
\(404\) 0 0
\(405\) 1735.59 0.212943
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14246.6 1.72237 0.861186 0.508289i \(-0.169722\pi\)
0.861186 + 0.508289i \(0.169722\pi\)
\(410\) 0 0
\(411\) −538.690 −0.0646512
\(412\) 0 0
\(413\) 5808.36 0.692035
\(414\) 0 0
\(415\) −13228.2 −1.56469
\(416\) 0 0
\(417\) −3111.32 −0.365376
\(418\) 0 0
\(419\) 8847.37 1.03156 0.515778 0.856722i \(-0.327503\pi\)
0.515778 + 0.856722i \(0.327503\pi\)
\(420\) 0 0
\(421\) −558.535 −0.0646587 −0.0323294 0.999477i \(-0.510293\pi\)
−0.0323294 + 0.999477i \(0.510293\pi\)
\(422\) 0 0
\(423\) −1947.50 −0.223856
\(424\) 0 0
\(425\) 15021.8 1.71450
\(426\) 0 0
\(427\) 243.488 0.0275953
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13294.7 1.48581 0.742904 0.669398i \(-0.233448\pi\)
0.742904 + 0.669398i \(0.233448\pi\)
\(432\) 0 0
\(433\) −1672.51 −0.185625 −0.0928125 0.995684i \(-0.529586\pi\)
−0.0928125 + 0.995684i \(0.529586\pi\)
\(434\) 0 0
\(435\) 8013.50 0.883260
\(436\) 0 0
\(437\) 17543.9 1.92046
\(438\) 0 0
\(439\) 14049.6 1.52745 0.763723 0.645544i \(-0.223369\pi\)
0.763723 + 0.645544i \(0.223369\pi\)
\(440\) 0 0
\(441\) 4180.40 0.451398
\(442\) 0 0
\(443\) 5593.23 0.599870 0.299935 0.953960i \(-0.403035\pi\)
0.299935 + 0.953960i \(0.403035\pi\)
\(444\) 0 0
\(445\) −8656.33 −0.922134
\(446\) 0 0
\(447\) −4488.43 −0.474934
\(448\) 0 0
\(449\) −2391.88 −0.251402 −0.125701 0.992068i \(-0.540118\pi\)
−0.125701 + 0.992068i \(0.540118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 7475.77 0.775370
\(454\) 0 0
\(455\) 32602.3 3.35917
\(456\) 0 0
\(457\) 8836.02 0.904445 0.452222 0.891905i \(-0.350631\pi\)
0.452222 + 0.891905i \(0.350631\pi\)
\(458\) 0 0
\(459\) 1213.91 0.123444
\(460\) 0 0
\(461\) −16057.8 −1.62231 −0.811154 0.584832i \(-0.801160\pi\)
−0.811154 + 0.584832i \(0.801160\pi\)
\(462\) 0 0
\(463\) 7291.22 0.731862 0.365931 0.930642i \(-0.380751\pi\)
0.365931 + 0.930642i \(0.380751\pi\)
\(464\) 0 0
\(465\) −1594.82 −0.159050
\(466\) 0 0
\(467\) 6614.83 0.655456 0.327728 0.944772i \(-0.393717\pi\)
0.327728 + 0.944772i \(0.393717\pi\)
\(468\) 0 0
\(469\) −3834.11 −0.377490
\(470\) 0 0
\(471\) 8318.83 0.813825
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −39590.0 −3.82424
\(476\) 0 0
\(477\) −5905.94 −0.566907
\(478\) 0 0
\(479\) −18333.6 −1.74881 −0.874407 0.485193i \(-0.838749\pi\)
−0.874407 + 0.485193i \(0.838749\pi\)
\(480\) 0 0
\(481\) 1074.70 0.101875
\(482\) 0 0
\(483\) 12622.0 1.18907
\(484\) 0 0
\(485\) −25426.5 −2.38053
\(486\) 0 0
\(487\) −7024.19 −0.653586 −0.326793 0.945096i \(-0.605968\pi\)
−0.326793 + 0.945096i \(0.605968\pi\)
\(488\) 0 0
\(489\) 3677.23 0.340061
\(490\) 0 0
\(491\) −7427.99 −0.682730 −0.341365 0.939931i \(-0.610889\pi\)
−0.341365 + 0.939931i \(0.610889\pi\)
\(492\) 0 0
\(493\) 5604.84 0.512027
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7408.05 0.668604
\(498\) 0 0
\(499\) 4877.37 0.437557 0.218779 0.975775i \(-0.429793\pi\)
0.218779 + 0.975775i \(0.429793\pi\)
\(500\) 0 0
\(501\) 3488.60 0.311096
\(502\) 0 0
\(503\) 20878.9 1.85079 0.925393 0.379010i \(-0.123735\pi\)
0.925393 + 0.379010i \(0.123735\pi\)
\(504\) 0 0
\(505\) 22288.8 1.96403
\(506\) 0 0
\(507\) 2010.20 0.176087
\(508\) 0 0
\(509\) 11834.6 1.03057 0.515283 0.857020i \(-0.327687\pi\)
0.515283 + 0.857020i \(0.327687\pi\)
\(510\) 0 0
\(511\) 15014.8 1.29984
\(512\) 0 0
\(513\) −3199.27 −0.275344
\(514\) 0 0
\(515\) −41956.9 −3.58998
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 3095.58 0.261813
\(520\) 0 0
\(521\) −8565.08 −0.720236 −0.360118 0.932907i \(-0.617264\pi\)
−0.360118 + 0.932907i \(0.617264\pi\)
\(522\) 0 0
\(523\) 3992.61 0.333814 0.166907 0.985973i \(-0.446622\pi\)
0.166907 + 0.985973i \(0.446622\pi\)
\(524\) 0 0
\(525\) −28483.1 −2.36782
\(526\) 0 0
\(527\) −1115.46 −0.0922013
\(528\) 0 0
\(529\) 9754.88 0.801749
\(530\) 0 0
\(531\) −1839.62 −0.150344
\(532\) 0 0
\(533\) 1921.46 0.156149
\(534\) 0 0
\(535\) 4522.43 0.365461
\(536\) 0 0
\(537\) 8070.99 0.648583
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8402.62 0.667758 0.333879 0.942616i \(-0.391642\pi\)
0.333879 + 0.942616i \(0.391642\pi\)
\(542\) 0 0
\(543\) −9939.21 −0.785511
\(544\) 0 0
\(545\) 32284.0 2.53742
\(546\) 0 0
\(547\) −11011.7 −0.860741 −0.430371 0.902652i \(-0.641617\pi\)
−0.430371 + 0.902652i \(0.641617\pi\)
\(548\) 0 0
\(549\) −77.1173 −0.00599505
\(550\) 0 0
\(551\) −14771.6 −1.14209
\(552\) 0 0
\(553\) −23724.2 −1.82433
\(554\) 0 0
\(555\) −1290.18 −0.0986759
\(556\) 0 0
\(557\) 12563.7 0.955725 0.477863 0.878435i \(-0.341412\pi\)
0.477863 + 0.878435i \(0.341412\pi\)
\(558\) 0 0
\(559\) 15580.3 1.17885
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17222.8 −1.28926 −0.644630 0.764495i \(-0.722989\pi\)
−0.644630 + 0.764495i \(0.722989\pi\)
\(564\) 0 0
\(565\) −30936.8 −2.30358
\(566\) 0 0
\(567\) −2301.72 −0.170482
\(568\) 0 0
\(569\) −24109.4 −1.77630 −0.888152 0.459549i \(-0.848011\pi\)
−0.888152 + 0.459549i \(0.848011\pi\)
\(570\) 0 0
\(571\) −15580.9 −1.14193 −0.570963 0.820976i \(-0.693430\pi\)
−0.570963 + 0.820976i \(0.693430\pi\)
\(572\) 0 0
\(573\) −1066.75 −0.0777734
\(574\) 0 0
\(575\) −49469.4 −3.58786
\(576\) 0 0
\(577\) 7156.93 0.516372 0.258186 0.966095i \(-0.416875\pi\)
0.258186 + 0.966095i \(0.416875\pi\)
\(578\) 0 0
\(579\) 6350.18 0.455794
\(580\) 0 0
\(581\) 17543.1 1.25268
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −10325.8 −0.729775
\(586\) 0 0
\(587\) −4158.73 −0.292417 −0.146209 0.989254i \(-0.546707\pi\)
−0.146209 + 0.989254i \(0.546707\pi\)
\(588\) 0 0
\(589\) 2939.79 0.205657
\(590\) 0 0
\(591\) −7750.08 −0.539418
\(592\) 0 0
\(593\) 25422.3 1.76049 0.880244 0.474521i \(-0.157379\pi\)
0.880244 + 0.474521i \(0.157379\pi\)
\(594\) 0 0
\(595\) −27375.0 −1.88616
\(596\) 0 0
\(597\) 7711.21 0.528641
\(598\) 0 0
\(599\) 17882.5 1.21980 0.609898 0.792480i \(-0.291211\pi\)
0.609898 + 0.792480i \(0.291211\pi\)
\(600\) 0 0
\(601\) 17528.8 1.18971 0.594854 0.803834i \(-0.297210\pi\)
0.594854 + 0.803834i \(0.297210\pi\)
\(602\) 0 0
\(603\) 1214.34 0.0820093
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12970.2 0.867292 0.433646 0.901083i \(-0.357227\pi\)
0.433646 + 0.901083i \(0.357227\pi\)
\(608\) 0 0
\(609\) −10627.5 −0.707136
\(610\) 0 0
\(611\) 11586.6 0.767172
\(612\) 0 0
\(613\) 1926.35 0.126925 0.0634623 0.997984i \(-0.479786\pi\)
0.0634623 + 0.997984i \(0.479786\pi\)
\(614\) 0 0
\(615\) −2306.72 −0.151245
\(616\) 0 0
\(617\) −29846.1 −1.94742 −0.973711 0.227786i \(-0.926851\pi\)
−0.973711 + 0.227786i \(0.926851\pi\)
\(618\) 0 0
\(619\) 18879.1 1.22587 0.612936 0.790132i \(-0.289988\pi\)
0.612936 + 0.790132i \(0.289988\pi\)
\(620\) 0 0
\(621\) −3997.63 −0.258324
\(622\) 0 0
\(623\) 11480.0 0.738259
\(624\) 0 0
\(625\) 54244.3 3.47164
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −902.384 −0.0572026
\(630\) 0 0
\(631\) −23569.9 −1.48701 −0.743503 0.668732i \(-0.766837\pi\)
−0.743503 + 0.668732i \(0.766837\pi\)
\(632\) 0 0
\(633\) −11886.7 −0.746374
\(634\) 0 0
\(635\) 5271.68 0.329449
\(636\) 0 0
\(637\) −24871.0 −1.54698
\(638\) 0 0
\(639\) −2346.27 −0.145254
\(640\) 0 0
\(641\) 601.652 0.0370730 0.0185365 0.999828i \(-0.494099\pi\)
0.0185365 + 0.999828i \(0.494099\pi\)
\(642\) 0 0
\(643\) 23686.5 1.45273 0.726364 0.687310i \(-0.241209\pi\)
0.726364 + 0.687310i \(0.241209\pi\)
\(644\) 0 0
\(645\) −18704.2 −1.14183
\(646\) 0 0
\(647\) −19089.3 −1.15993 −0.579966 0.814641i \(-0.696934\pi\)
−0.579966 + 0.814641i \(0.696934\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 2115.04 0.127335
\(652\) 0 0
\(653\) 5932.18 0.355504 0.177752 0.984075i \(-0.443117\pi\)
0.177752 + 0.984075i \(0.443117\pi\)
\(654\) 0 0
\(655\) 9669.09 0.576798
\(656\) 0 0
\(657\) −4755.48 −0.282388
\(658\) 0 0
\(659\) 10639.0 0.628886 0.314443 0.949276i \(-0.398182\pi\)
0.314443 + 0.949276i \(0.398182\pi\)
\(660\) 0 0
\(661\) 29164.4 1.71614 0.858068 0.513537i \(-0.171665\pi\)
0.858068 + 0.513537i \(0.171665\pi\)
\(662\) 0 0
\(663\) −7222.11 −0.423052
\(664\) 0 0
\(665\) 72146.9 4.20712
\(666\) 0 0
\(667\) −18457.7 −1.07149
\(668\) 0 0
\(669\) 11001.6 0.635794
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7401.20 −0.423915 −0.211958 0.977279i \(-0.567984\pi\)
−0.211958 + 0.977279i \(0.567984\pi\)
\(674\) 0 0
\(675\) 9021.15 0.514406
\(676\) 0 0
\(677\) −7102.23 −0.403192 −0.201596 0.979469i \(-0.564613\pi\)
−0.201596 + 0.979469i \(0.564613\pi\)
\(678\) 0 0
\(679\) 33720.4 1.90585
\(680\) 0 0
\(681\) −7218.94 −0.406212
\(682\) 0 0
\(683\) 9654.49 0.540877 0.270438 0.962737i \(-0.412831\pi\)
0.270438 + 0.962737i \(0.412831\pi\)
\(684\) 0 0
\(685\) −3847.51 −0.214607
\(686\) 0 0
\(687\) −1450.35 −0.0805449
\(688\) 0 0
\(689\) 35137.1 1.94284
\(690\) 0 0
\(691\) −17074.2 −0.939991 −0.469996 0.882669i \(-0.655745\pi\)
−0.469996 + 0.882669i \(0.655745\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22222.1 −1.21285
\(696\) 0 0
\(697\) −1613.38 −0.0876771
\(698\) 0 0
\(699\) −16129.3 −0.872772
\(700\) 0 0
\(701\) 11623.9 0.626290 0.313145 0.949705i \(-0.398617\pi\)
0.313145 + 0.949705i \(0.398617\pi\)
\(702\) 0 0
\(703\) 2378.24 0.127592
\(704\) 0 0
\(705\) −13909.7 −0.743079
\(706\) 0 0
\(707\) −29559.2 −1.57240
\(708\) 0 0
\(709\) −28176.8 −1.49253 −0.746264 0.665650i \(-0.768154\pi\)
−0.746264 + 0.665650i \(0.768154\pi\)
\(710\) 0 0
\(711\) 7513.92 0.396335
\(712\) 0 0
\(713\) 3673.40 0.192945
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1563.67 −0.0814452
\(718\) 0 0
\(719\) 7309.60 0.379141 0.189570 0.981867i \(-0.439290\pi\)
0.189570 + 0.981867i \(0.439290\pi\)
\(720\) 0 0
\(721\) 55642.9 2.87413
\(722\) 0 0
\(723\) 2187.01 0.112498
\(724\) 0 0
\(725\) 41652.2 2.13369
\(726\) 0 0
\(727\) −13595.7 −0.693586 −0.346793 0.937942i \(-0.612729\pi\)
−0.346793 + 0.937942i \(0.612729\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −13082.2 −0.661919
\(732\) 0 0
\(733\) 5776.01 0.291053 0.145527 0.989354i \(-0.453512\pi\)
0.145527 + 0.989354i \(0.453512\pi\)
\(734\) 0 0
\(735\) 29857.8 1.49840
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5272.27 −0.262441 −0.131220 0.991353i \(-0.541890\pi\)
−0.131220 + 0.991353i \(0.541890\pi\)
\(740\) 0 0
\(741\) 19033.9 0.943627
\(742\) 0 0
\(743\) 292.215 0.0144284 0.00721422 0.999974i \(-0.497704\pi\)
0.00721422 + 0.999974i \(0.497704\pi\)
\(744\) 0 0
\(745\) −32057.9 −1.57652
\(746\) 0 0
\(747\) −5556.23 −0.272144
\(748\) 0 0
\(749\) −5997.62 −0.292588
\(750\) 0 0
\(751\) 18247.4 0.886625 0.443312 0.896367i \(-0.353803\pi\)
0.443312 + 0.896367i \(0.353803\pi\)
\(752\) 0 0
\(753\) 2918.71 0.141253
\(754\) 0 0
\(755\) 53394.5 2.57381
\(756\) 0 0
\(757\) −39592.4 −1.90094 −0.950468 0.310822i \(-0.899396\pi\)
−0.950468 + 0.310822i \(0.899396\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8965.92 0.427088 0.213544 0.976933i \(-0.431499\pi\)
0.213544 + 0.976933i \(0.431499\pi\)
\(762\) 0 0
\(763\) −42814.8 −2.03146
\(764\) 0 0
\(765\) 8670.17 0.409766
\(766\) 0 0
\(767\) 10944.7 0.515242
\(768\) 0 0
\(769\) 31954.2 1.49844 0.749219 0.662322i \(-0.230429\pi\)
0.749219 + 0.662322i \(0.230429\pi\)
\(770\) 0 0
\(771\) 6675.85 0.311835
\(772\) 0 0
\(773\) 17755.8 0.826173 0.413087 0.910692i \(-0.364451\pi\)
0.413087 + 0.910692i \(0.364451\pi\)
\(774\) 0 0
\(775\) −8289.48 −0.384215
\(776\) 0 0
\(777\) 1711.03 0.0789997
\(778\) 0 0
\(779\) 4252.06 0.195566
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3365.92 0.153625
\(784\) 0 0
\(785\) 59415.9 2.70146
\(786\) 0 0
\(787\) 20110.3 0.910872 0.455436 0.890269i \(-0.349483\pi\)
0.455436 + 0.890269i \(0.349483\pi\)
\(788\) 0 0
\(789\) −6851.67 −0.309158
\(790\) 0 0
\(791\) 41028.2 1.84424
\(792\) 0 0
\(793\) 458.805 0.0205456
\(794\) 0 0
\(795\) −42182.2 −1.88182
\(796\) 0 0
\(797\) −42485.5 −1.88822 −0.944111 0.329627i \(-0.893077\pi\)
−0.944111 + 0.329627i \(0.893077\pi\)
\(798\) 0 0
\(799\) −9728.81 −0.430764
\(800\) 0 0
\(801\) −3635.93 −0.160386
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 90150.6 3.94707
\(806\) 0 0
\(807\) 96.7064 0.00421837
\(808\) 0 0
\(809\) 35286.5 1.53351 0.766753 0.641942i \(-0.221871\pi\)
0.766753 + 0.641942i \(0.221871\pi\)
\(810\) 0 0
\(811\) −29927.5 −1.29580 −0.647902 0.761724i \(-0.724353\pi\)
−0.647902 + 0.761724i \(0.724353\pi\)
\(812\) 0 0
\(813\) −7180.42 −0.309752
\(814\) 0 0
\(815\) 26264.0 1.12882
\(816\) 0 0
\(817\) 34478.2 1.47643
\(818\) 0 0
\(819\) 13694.0 0.584257
\(820\) 0 0
\(821\) −16053.7 −0.682433 −0.341217 0.939985i \(-0.610839\pi\)
−0.341217 + 0.939985i \(0.610839\pi\)
\(822\) 0 0
\(823\) 11924.8 0.505070 0.252535 0.967588i \(-0.418736\pi\)
0.252535 + 0.967588i \(0.418736\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24286.1 −1.02117 −0.510587 0.859826i \(-0.670572\pi\)
−0.510587 + 0.859826i \(0.670572\pi\)
\(828\) 0 0
\(829\) 660.347 0.0276656 0.0138328 0.999904i \(-0.495597\pi\)
0.0138328 + 0.999904i \(0.495597\pi\)
\(830\) 0 0
\(831\) 10404.9 0.434348
\(832\) 0 0
\(833\) 20883.3 0.868623
\(834\) 0 0
\(835\) 24916.7 1.03267
\(836\) 0 0
\(837\) −669.873 −0.0276633
\(838\) 0 0
\(839\) 39503.6 1.62553 0.812763 0.582595i \(-0.197963\pi\)
0.812763 + 0.582595i \(0.197963\pi\)
\(840\) 0 0
\(841\) −8847.99 −0.362786
\(842\) 0 0
\(843\) −14455.5 −0.590599
\(844\) 0 0
\(845\) 14357.5 0.584512
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −20474.7 −0.827670
\(850\) 0 0
\(851\) 2971.71 0.119705
\(852\) 0 0
\(853\) 42377.1 1.70101 0.850507 0.525963i \(-0.176295\pi\)
0.850507 + 0.525963i \(0.176295\pi\)
\(854\) 0 0
\(855\) −22850.3 −0.913992
\(856\) 0 0
\(857\) −24969.5 −0.995266 −0.497633 0.867388i \(-0.665797\pi\)
−0.497633 + 0.867388i \(0.665797\pi\)
\(858\) 0 0
\(859\) 38577.1 1.53229 0.766143 0.642670i \(-0.222173\pi\)
0.766143 + 0.642670i \(0.222173\pi\)
\(860\) 0 0
\(861\) 3059.15 0.121087
\(862\) 0 0
\(863\) −28307.3 −1.11656 −0.558280 0.829652i \(-0.688539\pi\)
−0.558280 + 0.829652i \(0.688539\pi\)
\(864\) 0 0
\(865\) 22109.7 0.869076
\(866\) 0 0
\(867\) −8674.87 −0.339808
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −7224.62 −0.281053
\(872\) 0 0
\(873\) −10679.9 −0.414043
\(874\) 0 0
\(875\) −127326. −4.91933
\(876\) 0 0
\(877\) 49117.8 1.89121 0.945605 0.325318i \(-0.105471\pi\)
0.945605 + 0.325318i \(0.105471\pi\)
\(878\) 0 0
\(879\) 13307.4 0.510636
\(880\) 0 0
\(881\) 21619.1 0.826750 0.413375 0.910561i \(-0.364350\pi\)
0.413375 + 0.910561i \(0.364350\pi\)
\(882\) 0 0
\(883\) −26505.8 −1.01018 −0.505092 0.863066i \(-0.668541\pi\)
−0.505092 + 0.863066i \(0.668541\pi\)
\(884\) 0 0
\(885\) −13139.2 −0.499060
\(886\) 0 0
\(887\) −22733.5 −0.860558 −0.430279 0.902696i \(-0.641585\pi\)
−0.430279 + 0.902696i \(0.641585\pi\)
\(888\) 0 0
\(889\) −6991.26 −0.263756
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25640.3 0.960830
\(894\) 0 0
\(895\) 57645.7 2.15294
\(896\) 0 0
\(897\) 23783.7 0.885300
\(898\) 0 0
\(899\) −3092.92 −0.114744
\(900\) 0 0
\(901\) −29503.3 −1.09090
\(902\) 0 0
\(903\) 24805.4 0.914144
\(904\) 0 0
\(905\) −70989.2 −2.60747
\(906\) 0 0
\(907\) 18797.6 0.688164 0.344082 0.938940i \(-0.388190\pi\)
0.344082 + 0.938940i \(0.388190\pi\)
\(908\) 0 0
\(909\) 9361.96 0.341603
\(910\) 0 0
\(911\) 815.780 0.0296685 0.0148342 0.999890i \(-0.495278\pi\)
0.0148342 + 0.999890i \(0.495278\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −550.797 −0.0199003
\(916\) 0 0
\(917\) −12823.1 −0.461783
\(918\) 0 0
\(919\) −13795.2 −0.495171 −0.247585 0.968866i \(-0.579637\pi\)
−0.247585 + 0.968866i \(0.579637\pi\)
\(920\) 0 0
\(921\) −13654.0 −0.488506
\(922\) 0 0
\(923\) 13959.0 0.497796
\(924\) 0 0
\(925\) −6706.03 −0.238371
\(926\) 0 0
\(927\) −17623.2 −0.624403
\(928\) 0 0
\(929\) 6512.02 0.229981 0.114991 0.993367i \(-0.463316\pi\)
0.114991 + 0.993367i \(0.463316\pi\)
\(930\) 0 0
\(931\) −55038.0 −1.93748
\(932\) 0 0
\(933\) −6112.25 −0.214476
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −10649.6 −0.371299 −0.185650 0.982616i \(-0.559439\pi\)
−0.185650 + 0.982616i \(0.559439\pi\)
\(938\) 0 0
\(939\) −27326.5 −0.949699
\(940\) 0 0
\(941\) −38859.7 −1.34622 −0.673109 0.739543i \(-0.735042\pi\)
−0.673109 + 0.739543i \(0.735042\pi\)
\(942\) 0 0
\(943\) 5313.13 0.183478
\(944\) 0 0
\(945\) −16439.7 −0.565908
\(946\) 0 0
\(947\) 12389.6 0.425142 0.212571 0.977146i \(-0.431816\pi\)
0.212571 + 0.977146i \(0.431816\pi\)
\(948\) 0 0
\(949\) 28292.4 0.967767
\(950\) 0 0
\(951\) −18880.9 −0.643801
\(952\) 0 0
\(953\) 58421.7 1.98580 0.992899 0.118965i \(-0.0379575\pi\)
0.992899 + 0.118965i \(0.0379575\pi\)
\(954\) 0 0
\(955\) −7619.09 −0.258166
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5102.54 0.171814
\(960\) 0 0
\(961\) −29175.5 −0.979338
\(962\) 0 0
\(963\) 1899.56 0.0635644
\(964\) 0 0
\(965\) 45355.1 1.51299
\(966\) 0 0
\(967\) −5133.64 −0.170721 −0.0853603 0.996350i \(-0.527204\pi\)
−0.0853603 + 0.996350i \(0.527204\pi\)
\(968\) 0 0
\(969\) −15982.1 −0.529843
\(970\) 0 0
\(971\) 12399.5 0.409804 0.204902 0.978782i \(-0.434312\pi\)
0.204902 + 0.978782i \(0.434312\pi\)
\(972\) 0 0
\(973\) 29470.8 0.971007
\(974\) 0 0
\(975\) −53670.8 −1.76291
\(976\) 0 0
\(977\) −10150.9 −0.332401 −0.166201 0.986092i \(-0.553150\pi\)
−0.166201 + 0.986092i \(0.553150\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 13560.3 0.441331
\(982\) 0 0
\(983\) 9449.37 0.306600 0.153300 0.988180i \(-0.451010\pi\)
0.153300 + 0.988180i \(0.451010\pi\)
\(984\) 0 0
\(985\) −55353.7 −1.79057
\(986\) 0 0
\(987\) 18447.0 0.594908
\(988\) 0 0
\(989\) 43082.0 1.38516
\(990\) 0 0
\(991\) 4182.05 0.134054 0.0670269 0.997751i \(-0.478649\pi\)
0.0670269 + 0.997751i \(0.478649\pi\)
\(992\) 0 0
\(993\) 2828.75 0.0904004
\(994\) 0 0
\(995\) 55076.0 1.75480
\(996\) 0 0
\(997\) −10878.8 −0.345571 −0.172786 0.984959i \(-0.555277\pi\)
−0.172786 + 0.984959i \(0.555277\pi\)
\(998\) 0 0
\(999\) −541.915 −0.0171626
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 1452.4.a.t.1.6 6
11.7 odd 10 132.4.i.c.49.3 12
11.8 odd 10 132.4.i.c.97.3 yes 12
11.10 odd 2 1452.4.a.u.1.6 6
33.8 even 10 396.4.j.c.361.1 12
33.29 even 10 396.4.j.c.181.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
132.4.i.c.49.3 12 11.7 odd 10
132.4.i.c.97.3 yes 12 11.8 odd 10
396.4.j.c.181.1 12 33.29 even 10
396.4.j.c.361.1 12 33.8 even 10
1452.4.a.t.1.6 6 1.1 even 1 trivial
1452.4.a.u.1.6 6 11.10 odd 2