Properties

Label 1840.4.a.n.1.2
Level $1840$
Weight $4$
Character 1840.1
Self dual yes
Analytic conductor $108.564$
Analytic rank $0$
Dimension $5$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.41740\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.84147 q^{3} -5.00000 q^{5} +8.97260 q^{7} +34.4886 q^{9} +O(q^{10})\) \(q-7.84147 q^{3} -5.00000 q^{5} +8.97260 q^{7} +34.4886 q^{9} +28.9011 q^{11} +16.0879 q^{13} +39.2073 q^{15} +25.1771 q^{17} +35.6298 q^{19} -70.3583 q^{21} +23.0000 q^{23} +25.0000 q^{25} -58.7218 q^{27} +138.272 q^{29} -40.1277 q^{31} -226.627 q^{33} -44.8630 q^{35} +379.745 q^{37} -126.153 q^{39} +412.514 q^{41} +402.095 q^{43} -172.443 q^{45} -110.070 q^{47} -262.492 q^{49} -197.426 q^{51} -421.300 q^{53} -144.506 q^{55} -279.390 q^{57} -755.913 q^{59} -307.032 q^{61} +309.453 q^{63} -80.4396 q^{65} -319.974 q^{67} -180.354 q^{69} +554.138 q^{71} -705.131 q^{73} -196.037 q^{75} +259.318 q^{77} -1170.51 q^{79} -470.728 q^{81} +455.978 q^{83} -125.886 q^{85} -1084.26 q^{87} -1495.57 q^{89} +144.351 q^{91} +314.660 q^{93} -178.149 q^{95} +1041.24 q^{97} +996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{3} - 25 q^{5} + 3 q^{7} + 77 q^{9} - 23 q^{11} + 132 q^{13} + 20 q^{15} + 23 q^{17} + 161 q^{19} - 60 q^{21} + 115 q^{23} + 125 q^{25} - 577 q^{27} + 401 q^{29} - 32 q^{31} + 189 q^{33} - 15 q^{35} - 38 q^{37} - 335 q^{39} - 12 q^{41} + 566 q^{43} - 385 q^{45} - 919 q^{47} - 738 q^{49} + 993 q^{51} + 1156 q^{53} + 115 q^{55} + 114 q^{57} - 1324 q^{59} - 1673 q^{61} - 270 q^{63} - 660 q^{65} - 558 q^{67} - 92 q^{69} + 108 q^{71} + 1173 q^{73} - 100 q^{75} + 2608 q^{77} - 656 q^{79} - 319 q^{81} + 82 q^{83} - 115 q^{85} - 2389 q^{87} + 570 q^{89} + 1589 q^{91} + 911 q^{93} - 805 q^{95} + 633 q^{97} - 2021 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −7.84147 −1.50909 −0.754546 0.656248i \(-0.772143\pi\)
−0.754546 + 0.656248i \(0.772143\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 8.97260 0.484475 0.242237 0.970217i \(-0.422119\pi\)
0.242237 + 0.970217i \(0.422119\pi\)
\(8\) 0 0
\(9\) 34.4886 1.27736
\(10\) 0 0
\(11\) 28.9011 0.792183 0.396092 0.918211i \(-0.370366\pi\)
0.396092 + 0.918211i \(0.370366\pi\)
\(12\) 0 0
\(13\) 16.0879 0.343230 0.171615 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(14\) 0 0
\(15\) 39.2073 0.674886
\(16\) 0 0
\(17\) 25.1771 0.359197 0.179599 0.983740i \(-0.442520\pi\)
0.179599 + 0.983740i \(0.442520\pi\)
\(18\) 0 0
\(19\) 35.6298 0.430212 0.215106 0.976591i \(-0.430990\pi\)
0.215106 + 0.976591i \(0.430990\pi\)
\(20\) 0 0
\(21\) −70.3583 −0.731117
\(22\) 0 0
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −58.7218 −0.418556
\(28\) 0 0
\(29\) 138.272 0.885395 0.442698 0.896671i \(-0.354022\pi\)
0.442698 + 0.896671i \(0.354022\pi\)
\(30\) 0 0
\(31\) −40.1277 −0.232489 −0.116244 0.993221i \(-0.537086\pi\)
−0.116244 + 0.993221i \(0.537086\pi\)
\(32\) 0 0
\(33\) −226.627 −1.19548
\(34\) 0 0
\(35\) −44.8630 −0.216664
\(36\) 0 0
\(37\) 379.745 1.68729 0.843645 0.536902i \(-0.180405\pi\)
0.843645 + 0.536902i \(0.180405\pi\)
\(38\) 0 0
\(39\) −126.153 −0.517965
\(40\) 0 0
\(41\) 412.514 1.57132 0.785658 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) 0 0
\(43\) 402.095 1.42602 0.713011 0.701153i \(-0.247331\pi\)
0.713011 + 0.701153i \(0.247331\pi\)
\(44\) 0 0
\(45\) −172.443 −0.571251
\(46\) 0 0
\(47\) −110.070 −0.341603 −0.170801 0.985305i \(-0.554636\pi\)
−0.170801 + 0.985305i \(0.554636\pi\)
\(48\) 0 0
\(49\) −262.492 −0.765284
\(50\) 0 0
\(51\) −197.426 −0.542061
\(52\) 0 0
\(53\) −421.300 −1.09189 −0.545943 0.837822i \(-0.683829\pi\)
−0.545943 + 0.837822i \(0.683829\pi\)
\(54\) 0 0
\(55\) −144.506 −0.354275
\(56\) 0 0
\(57\) −279.390 −0.649229
\(58\) 0 0
\(59\) −755.913 −1.66799 −0.833996 0.551770i \(-0.813952\pi\)
−0.833996 + 0.551770i \(0.813952\pi\)
\(60\) 0 0
\(61\) −307.032 −0.644450 −0.322225 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) 0 0
\(63\) 309.453 0.618847
\(64\) 0 0
\(65\) −80.4396 −0.153497
\(66\) 0 0
\(67\) −319.974 −0.583448 −0.291724 0.956502i \(-0.594229\pi\)
−0.291724 + 0.956502i \(0.594229\pi\)
\(68\) 0 0
\(69\) −180.354 −0.314667
\(70\) 0 0
\(71\) 554.138 0.926254 0.463127 0.886292i \(-0.346727\pi\)
0.463127 + 0.886292i \(0.346727\pi\)
\(72\) 0 0
\(73\) −705.131 −1.13054 −0.565269 0.824907i \(-0.691228\pi\)
−0.565269 + 0.824907i \(0.691228\pi\)
\(74\) 0 0
\(75\) −196.037 −0.301818
\(76\) 0 0
\(77\) 259.318 0.383793
\(78\) 0 0
\(79\) −1170.51 −1.66699 −0.833495 0.552526i \(-0.813664\pi\)
−0.833495 + 0.552526i \(0.813664\pi\)
\(80\) 0 0
\(81\) −470.728 −0.645717
\(82\) 0 0
\(83\) 455.978 0.603014 0.301507 0.953464i \(-0.402510\pi\)
0.301507 + 0.953464i \(0.402510\pi\)
\(84\) 0 0
\(85\) −125.886 −0.160638
\(86\) 0 0
\(87\) −1084.26 −1.33614
\(88\) 0 0
\(89\) −1495.57 −1.78124 −0.890621 0.454746i \(-0.849730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(90\) 0 0
\(91\) 144.351 0.166286
\(92\) 0 0
\(93\) 314.660 0.350847
\(94\) 0 0
\(95\) −178.149 −0.192397
\(96\) 0 0
\(97\) 1041.24 1.08992 0.544958 0.838463i \(-0.316545\pi\)
0.544958 + 0.838463i \(0.316545\pi\)
\(98\) 0 0
\(99\) 996.760 1.01190
\(100\) 0 0
\(101\) 1450.22 1.42873 0.714367 0.699771i \(-0.246715\pi\)
0.714367 + 0.699771i \(0.246715\pi\)
\(102\) 0 0
\(103\) −220.283 −0.210729 −0.105365 0.994434i \(-0.533601\pi\)
−0.105365 + 0.994434i \(0.533601\pi\)
\(104\) 0 0
\(105\) 351.792 0.326965
\(106\) 0 0
\(107\) −59.2679 −0.0535481 −0.0267741 0.999642i \(-0.508523\pi\)
−0.0267741 + 0.999642i \(0.508523\pi\)
\(108\) 0 0
\(109\) 1966.29 1.72786 0.863930 0.503612i \(-0.167996\pi\)
0.863930 + 0.503612i \(0.167996\pi\)
\(110\) 0 0
\(111\) −2977.76 −2.54627
\(112\) 0 0
\(113\) 1819.68 1.51488 0.757439 0.652905i \(-0.226450\pi\)
0.757439 + 0.652905i \(0.226450\pi\)
\(114\) 0 0
\(115\) −115.000 −0.0932505
\(116\) 0 0
\(117\) 554.851 0.438427
\(118\) 0 0
\(119\) 225.904 0.174022
\(120\) 0 0
\(121\) −495.725 −0.372445
\(122\) 0 0
\(123\) −3234.72 −2.37126
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −834.517 −0.583082 −0.291541 0.956558i \(-0.594168\pi\)
−0.291541 + 0.956558i \(0.594168\pi\)
\(128\) 0 0
\(129\) −3153.02 −2.15200
\(130\) 0 0
\(131\) −510.293 −0.340340 −0.170170 0.985415i \(-0.554432\pi\)
−0.170170 + 0.985415i \(0.554432\pi\)
\(132\) 0 0
\(133\) 319.692 0.208427
\(134\) 0 0
\(135\) 293.609 0.187184
\(136\) 0 0
\(137\) 11.2339 0.00700567 0.00350284 0.999994i \(-0.498885\pi\)
0.00350284 + 0.999994i \(0.498885\pi\)
\(138\) 0 0
\(139\) 1656.81 1.01100 0.505501 0.862826i \(-0.331308\pi\)
0.505501 + 0.862826i \(0.331308\pi\)
\(140\) 0 0
\(141\) 863.109 0.515510
\(142\) 0 0
\(143\) 464.959 0.271901
\(144\) 0 0
\(145\) −691.360 −0.395961
\(146\) 0 0
\(147\) 2058.33 1.15488
\(148\) 0 0
\(149\) −510.206 −0.280522 −0.140261 0.990115i \(-0.544794\pi\)
−0.140261 + 0.990115i \(0.544794\pi\)
\(150\) 0 0
\(151\) 2337.38 1.25969 0.629846 0.776720i \(-0.283118\pi\)
0.629846 + 0.776720i \(0.283118\pi\)
\(152\) 0 0
\(153\) 868.325 0.458823
\(154\) 0 0
\(155\) 200.639 0.103972
\(156\) 0 0
\(157\) 146.592 0.0745178 0.0372589 0.999306i \(-0.488137\pi\)
0.0372589 + 0.999306i \(0.488137\pi\)
\(158\) 0 0
\(159\) 3303.61 1.64776
\(160\) 0 0
\(161\) 206.370 0.101020
\(162\) 0 0
\(163\) 3278.19 1.57526 0.787630 0.616149i \(-0.211308\pi\)
0.787630 + 0.616149i \(0.211308\pi\)
\(164\) 0 0
\(165\) 1133.14 0.534634
\(166\) 0 0
\(167\) 1555.42 0.720732 0.360366 0.932811i \(-0.382652\pi\)
0.360366 + 0.932811i \(0.382652\pi\)
\(168\) 0 0
\(169\) −1938.18 −0.882193
\(170\) 0 0
\(171\) 1228.82 0.549534
\(172\) 0 0
\(173\) −472.392 −0.207603 −0.103801 0.994598i \(-0.533101\pi\)
−0.103801 + 0.994598i \(0.533101\pi\)
\(174\) 0 0
\(175\) 224.315 0.0968950
\(176\) 0 0
\(177\) 5927.47 2.51715
\(178\) 0 0
\(179\) −2429.45 −1.01444 −0.507222 0.861815i \(-0.669328\pi\)
−0.507222 + 0.861815i \(0.669328\pi\)
\(180\) 0 0
\(181\) −982.359 −0.403415 −0.201708 0.979446i \(-0.564649\pi\)
−0.201708 + 0.979446i \(0.564649\pi\)
\(182\) 0 0
\(183\) 2407.58 0.972533
\(184\) 0 0
\(185\) −1898.73 −0.754579
\(186\) 0 0
\(187\) 727.648 0.284550
\(188\) 0 0
\(189\) −526.887 −0.202780
\(190\) 0 0
\(191\) −1361.14 −0.515649 −0.257824 0.966192i \(-0.583006\pi\)
−0.257824 + 0.966192i \(0.583006\pi\)
\(192\) 0 0
\(193\) −1456.29 −0.543142 −0.271571 0.962418i \(-0.587543\pi\)
−0.271571 + 0.962418i \(0.587543\pi\)
\(194\) 0 0
\(195\) 630.765 0.231641
\(196\) 0 0
\(197\) 1071.99 0.387695 0.193847 0.981032i \(-0.437903\pi\)
0.193847 + 0.981032i \(0.437903\pi\)
\(198\) 0 0
\(199\) 2879.31 1.02567 0.512836 0.858486i \(-0.328595\pi\)
0.512836 + 0.858486i \(0.328595\pi\)
\(200\) 0 0
\(201\) 2509.07 0.880477
\(202\) 0 0
\(203\) 1240.66 0.428952
\(204\) 0 0
\(205\) −2062.57 −0.702713
\(206\) 0 0
\(207\) 793.238 0.266347
\(208\) 0 0
\(209\) 1029.74 0.340807
\(210\) 0 0
\(211\) 4746.47 1.54863 0.774313 0.632802i \(-0.218095\pi\)
0.774313 + 0.632802i \(0.218095\pi\)
\(212\) 0 0
\(213\) −4345.25 −1.39780
\(214\) 0 0
\(215\) −2010.48 −0.637737
\(216\) 0 0
\(217\) −360.050 −0.112635
\(218\) 0 0
\(219\) 5529.26 1.70609
\(220\) 0 0
\(221\) 405.048 0.123287
\(222\) 0 0
\(223\) −4874.04 −1.46363 −0.731815 0.681503i \(-0.761327\pi\)
−0.731815 + 0.681503i \(0.761327\pi\)
\(224\) 0 0
\(225\) 862.216 0.255471
\(226\) 0 0
\(227\) 2742.09 0.801757 0.400879 0.916131i \(-0.368705\pi\)
0.400879 + 0.916131i \(0.368705\pi\)
\(228\) 0 0
\(229\) 1528.38 0.441041 0.220520 0.975382i \(-0.429224\pi\)
0.220520 + 0.975382i \(0.429224\pi\)
\(230\) 0 0
\(231\) −2033.44 −0.579179
\(232\) 0 0
\(233\) 5552.78 1.56126 0.780632 0.624991i \(-0.214897\pi\)
0.780632 + 0.624991i \(0.214897\pi\)
\(234\) 0 0
\(235\) 550.349 0.152769
\(236\) 0 0
\(237\) 9178.49 2.51564
\(238\) 0 0
\(239\) −2779.63 −0.752299 −0.376149 0.926559i \(-0.622752\pi\)
−0.376149 + 0.926559i \(0.622752\pi\)
\(240\) 0 0
\(241\) −5568.82 −1.48846 −0.744231 0.667922i \(-0.767184\pi\)
−0.744231 + 0.667922i \(0.767184\pi\)
\(242\) 0 0
\(243\) 5276.68 1.39300
\(244\) 0 0
\(245\) 1312.46 0.342245
\(246\) 0 0
\(247\) 573.209 0.147662
\(248\) 0 0
\(249\) −3575.54 −0.910003
\(250\) 0 0
\(251\) 387.065 0.0973360 0.0486680 0.998815i \(-0.484502\pi\)
0.0486680 + 0.998815i \(0.484502\pi\)
\(252\) 0 0
\(253\) 664.726 0.165182
\(254\) 0 0
\(255\) 987.129 0.242417
\(256\) 0 0
\(257\) 1476.07 0.358267 0.179133 0.983825i \(-0.442671\pi\)
0.179133 + 0.983825i \(0.442671\pi\)
\(258\) 0 0
\(259\) 3407.30 0.817449
\(260\) 0 0
\(261\) 4768.81 1.13097
\(262\) 0 0
\(263\) −203.984 −0.0478259 −0.0239130 0.999714i \(-0.507612\pi\)
−0.0239130 + 0.999714i \(0.507612\pi\)
\(264\) 0 0
\(265\) 2106.50 0.488306
\(266\) 0 0
\(267\) 11727.5 2.68806
\(268\) 0 0
\(269\) −6973.71 −1.58065 −0.790324 0.612689i \(-0.790088\pi\)
−0.790324 + 0.612689i \(0.790088\pi\)
\(270\) 0 0
\(271\) 2164.97 0.485287 0.242643 0.970116i \(-0.421985\pi\)
0.242643 + 0.970116i \(0.421985\pi\)
\(272\) 0 0
\(273\) −1131.92 −0.250941
\(274\) 0 0
\(275\) 722.528 0.158437
\(276\) 0 0
\(277\) 1194.00 0.258992 0.129496 0.991580i \(-0.458664\pi\)
0.129496 + 0.991580i \(0.458664\pi\)
\(278\) 0 0
\(279\) −1383.95 −0.296971
\(280\) 0 0
\(281\) 6485.98 1.37694 0.688472 0.725263i \(-0.258282\pi\)
0.688472 + 0.725263i \(0.258282\pi\)
\(282\) 0 0
\(283\) 3214.41 0.675182 0.337591 0.941293i \(-0.390388\pi\)
0.337591 + 0.941293i \(0.390388\pi\)
\(284\) 0 0
\(285\) 1396.95 0.290344
\(286\) 0 0
\(287\) 3701.33 0.761263
\(288\) 0 0
\(289\) −4279.11 −0.870977
\(290\) 0 0
\(291\) −8164.85 −1.64478
\(292\) 0 0
\(293\) 5585.47 1.11367 0.556837 0.830622i \(-0.312015\pi\)
0.556837 + 0.830622i \(0.312015\pi\)
\(294\) 0 0
\(295\) 3779.57 0.745949
\(296\) 0 0
\(297\) −1697.13 −0.331573
\(298\) 0 0
\(299\) 370.022 0.0715684
\(300\) 0 0
\(301\) 3607.84 0.690872
\(302\) 0 0
\(303\) −11371.8 −2.15609
\(304\) 0 0
\(305\) 1535.16 0.288207
\(306\) 0 0
\(307\) −2849.51 −0.529740 −0.264870 0.964284i \(-0.585329\pi\)
−0.264870 + 0.964284i \(0.585329\pi\)
\(308\) 0 0
\(309\) 1727.34 0.318010
\(310\) 0 0
\(311\) 6617.84 1.20664 0.603318 0.797501i \(-0.293845\pi\)
0.603318 + 0.797501i \(0.293845\pi\)
\(312\) 0 0
\(313\) −6271.26 −1.13250 −0.566250 0.824233i \(-0.691606\pi\)
−0.566250 + 0.824233i \(0.691606\pi\)
\(314\) 0 0
\(315\) −1547.26 −0.276757
\(316\) 0 0
\(317\) 5650.73 1.00119 0.500594 0.865682i \(-0.333115\pi\)
0.500594 + 0.865682i \(0.333115\pi\)
\(318\) 0 0
\(319\) 3996.22 0.701395
\(320\) 0 0
\(321\) 464.748 0.0808090
\(322\) 0 0
\(323\) 897.055 0.154531
\(324\) 0 0
\(325\) 402.198 0.0686460
\(326\) 0 0
\(327\) −15418.6 −2.60750
\(328\) 0 0
\(329\) −987.612 −0.165498
\(330\) 0 0
\(331\) 6391.40 1.06134 0.530669 0.847579i \(-0.321941\pi\)
0.530669 + 0.847579i \(0.321941\pi\)
\(332\) 0 0
\(333\) 13096.9 2.15527
\(334\) 0 0
\(335\) 1599.87 0.260926
\(336\) 0 0
\(337\) 4153.87 0.671441 0.335721 0.941962i \(-0.391020\pi\)
0.335721 + 0.941962i \(0.391020\pi\)
\(338\) 0 0
\(339\) −14269.0 −2.28609
\(340\) 0 0
\(341\) −1159.74 −0.184174
\(342\) 0 0
\(343\) −5432.84 −0.855236
\(344\) 0 0
\(345\) 901.769 0.140723
\(346\) 0 0
\(347\) 3071.86 0.475234 0.237617 0.971359i \(-0.423634\pi\)
0.237617 + 0.971359i \(0.423634\pi\)
\(348\) 0 0
\(349\) −8358.91 −1.28207 −0.641035 0.767512i \(-0.721495\pi\)
−0.641035 + 0.767512i \(0.721495\pi\)
\(350\) 0 0
\(351\) −944.712 −0.143661
\(352\) 0 0
\(353\) 8018.12 1.20896 0.604478 0.796622i \(-0.293382\pi\)
0.604478 + 0.796622i \(0.293382\pi\)
\(354\) 0 0
\(355\) −2770.69 −0.414234
\(356\) 0 0
\(357\) −1771.42 −0.262615
\(358\) 0 0
\(359\) −2138.46 −0.314384 −0.157192 0.987568i \(-0.550244\pi\)
−0.157192 + 0.987568i \(0.550244\pi\)
\(360\) 0 0
\(361\) −5589.52 −0.814918
\(362\) 0 0
\(363\) 3887.21 0.562054
\(364\) 0 0
\(365\) 3525.65 0.505592
\(366\) 0 0
\(367\) 4509.96 0.641467 0.320733 0.947170i \(-0.396071\pi\)
0.320733 + 0.947170i \(0.396071\pi\)
\(368\) 0 0
\(369\) 14227.1 2.00713
\(370\) 0 0
\(371\) −3780.15 −0.528991
\(372\) 0 0
\(373\) −3362.09 −0.466709 −0.233355 0.972392i \(-0.574970\pi\)
−0.233355 + 0.972392i \(0.574970\pi\)
\(374\) 0 0
\(375\) 980.184 0.134977
\(376\) 0 0
\(377\) 2224.51 0.303894
\(378\) 0 0
\(379\) −2107.16 −0.285587 −0.142793 0.989753i \(-0.545608\pi\)
−0.142793 + 0.989753i \(0.545608\pi\)
\(380\) 0 0
\(381\) 6543.84 0.879924
\(382\) 0 0
\(383\) −1373.05 −0.183184 −0.0915919 0.995797i \(-0.529196\pi\)
−0.0915919 + 0.995797i \(0.529196\pi\)
\(384\) 0 0
\(385\) −1296.59 −0.171637
\(386\) 0 0
\(387\) 13867.7 1.82154
\(388\) 0 0
\(389\) 7008.05 0.913425 0.456713 0.889614i \(-0.349027\pi\)
0.456713 + 0.889614i \(0.349027\pi\)
\(390\) 0 0
\(391\) 579.074 0.0748978
\(392\) 0 0
\(393\) 4001.44 0.513604
\(394\) 0 0
\(395\) 5852.53 0.745501
\(396\) 0 0
\(397\) 2402.53 0.303727 0.151864 0.988401i \(-0.451473\pi\)
0.151864 + 0.988401i \(0.451473\pi\)
\(398\) 0 0
\(399\) −2506.85 −0.314535
\(400\) 0 0
\(401\) −11902.0 −1.48218 −0.741091 0.671404i \(-0.765691\pi\)
−0.741091 + 0.671404i \(0.765691\pi\)
\(402\) 0 0
\(403\) −645.572 −0.0797971
\(404\) 0 0
\(405\) 2353.64 0.288773
\(406\) 0 0
\(407\) 10975.1 1.33664
\(408\) 0 0
\(409\) 5277.96 0.638089 0.319044 0.947740i \(-0.396638\pi\)
0.319044 + 0.947740i \(0.396638\pi\)
\(410\) 0 0
\(411\) −88.0903 −0.0105722
\(412\) 0 0
\(413\) −6782.51 −0.808100
\(414\) 0 0
\(415\) −2279.89 −0.269676
\(416\) 0 0
\(417\) −12991.9 −1.52569
\(418\) 0 0
\(419\) 11196.4 1.30545 0.652723 0.757597i \(-0.273627\pi\)
0.652723 + 0.757597i \(0.273627\pi\)
\(420\) 0 0
\(421\) 5176.82 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(422\) 0 0
\(423\) −3796.16 −0.436349
\(424\) 0 0
\(425\) 629.428 0.0718394
\(426\) 0 0
\(427\) −2754.88 −0.312220
\(428\) 0 0
\(429\) −3645.96 −0.410324
\(430\) 0 0
\(431\) −9348.93 −1.04483 −0.522415 0.852691i \(-0.674969\pi\)
−0.522415 + 0.852691i \(0.674969\pi\)
\(432\) 0 0
\(433\) 4320.91 0.479560 0.239780 0.970827i \(-0.422925\pi\)
0.239780 + 0.970827i \(0.422925\pi\)
\(434\) 0 0
\(435\) 5421.28 0.597541
\(436\) 0 0
\(437\) 819.484 0.0897054
\(438\) 0 0
\(439\) 7016.03 0.762772 0.381386 0.924416i \(-0.375447\pi\)
0.381386 + 0.924416i \(0.375447\pi\)
\(440\) 0 0
\(441\) −9053.00 −0.977541
\(442\) 0 0
\(443\) −12343.6 −1.32384 −0.661921 0.749574i \(-0.730259\pi\)
−0.661921 + 0.749574i \(0.730259\pi\)
\(444\) 0 0
\(445\) 7477.87 0.796596
\(446\) 0 0
\(447\) 4000.77 0.423333
\(448\) 0 0
\(449\) 18146.9 1.90737 0.953683 0.300815i \(-0.0972587\pi\)
0.953683 + 0.300815i \(0.0972587\pi\)
\(450\) 0 0
\(451\) 11922.1 1.24477
\(452\) 0 0
\(453\) −18328.5 −1.90099
\(454\) 0 0
\(455\) −721.753 −0.0743655
\(456\) 0 0
\(457\) 5233.82 0.535728 0.267864 0.963457i \(-0.413682\pi\)
0.267864 + 0.963457i \(0.413682\pi\)
\(458\) 0 0
\(459\) −1478.45 −0.150344
\(460\) 0 0
\(461\) −2335.01 −0.235905 −0.117952 0.993019i \(-0.537633\pi\)
−0.117952 + 0.993019i \(0.537633\pi\)
\(462\) 0 0
\(463\) −13644.1 −1.36954 −0.684769 0.728760i \(-0.740097\pi\)
−0.684769 + 0.728760i \(0.740097\pi\)
\(464\) 0 0
\(465\) −1573.30 −0.156903
\(466\) 0 0
\(467\) 5246.31 0.519851 0.259925 0.965629i \(-0.416302\pi\)
0.259925 + 0.965629i \(0.416302\pi\)
\(468\) 0 0
\(469\) −2871.00 −0.282666
\(470\) 0 0
\(471\) −1149.50 −0.112454
\(472\) 0 0
\(473\) 11621.0 1.12967
\(474\) 0 0
\(475\) 890.744 0.0860424
\(476\) 0 0
\(477\) −14530.0 −1.39473
\(478\) 0 0
\(479\) −12278.3 −1.17121 −0.585603 0.810598i \(-0.699142\pi\)
−0.585603 + 0.810598i \(0.699142\pi\)
\(480\) 0 0
\(481\) 6109.31 0.579128
\(482\) 0 0
\(483\) −1618.24 −0.152448
\(484\) 0 0
\(485\) −5206.20 −0.487426
\(486\) 0 0
\(487\) 8945.24 0.832336 0.416168 0.909288i \(-0.363373\pi\)
0.416168 + 0.909288i \(0.363373\pi\)
\(488\) 0 0
\(489\) −25705.8 −2.37721
\(490\) 0 0
\(491\) 7792.94 0.716274 0.358137 0.933669i \(-0.383412\pi\)
0.358137 + 0.933669i \(0.383412\pi\)
\(492\) 0 0
\(493\) 3481.29 0.318031
\(494\) 0 0
\(495\) −4983.80 −0.452536
\(496\) 0 0
\(497\) 4972.06 0.448747
\(498\) 0 0
\(499\) 8577.71 0.769521 0.384761 0.923016i \(-0.374284\pi\)
0.384761 + 0.923016i \(0.374284\pi\)
\(500\) 0 0
\(501\) −12196.8 −1.08765
\(502\) 0 0
\(503\) −7784.99 −0.690091 −0.345045 0.938586i \(-0.612136\pi\)
−0.345045 + 0.938586i \(0.612136\pi\)
\(504\) 0 0
\(505\) −7251.09 −0.638949
\(506\) 0 0
\(507\) 15198.2 1.33131
\(508\) 0 0
\(509\) −11390.4 −0.991891 −0.495946 0.868354i \(-0.665178\pi\)
−0.495946 + 0.868354i \(0.665178\pi\)
\(510\) 0 0
\(511\) −6326.85 −0.547717
\(512\) 0 0
\(513\) −2092.24 −0.180068
\(514\) 0 0
\(515\) 1101.42 0.0942411
\(516\) 0 0
\(517\) −3181.14 −0.270612
\(518\) 0 0
\(519\) 3704.24 0.313292
\(520\) 0 0
\(521\) −12824.5 −1.07841 −0.539206 0.842174i \(-0.681276\pi\)
−0.539206 + 0.842174i \(0.681276\pi\)
\(522\) 0 0
\(523\) −13087.4 −1.09421 −0.547107 0.837063i \(-0.684271\pi\)
−0.547107 + 0.837063i \(0.684271\pi\)
\(524\) 0 0
\(525\) −1758.96 −0.146223
\(526\) 0 0
\(527\) −1010.30 −0.0835093
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) 0 0
\(531\) −26070.4 −2.13062
\(532\) 0 0
\(533\) 6636.50 0.539322
\(534\) 0 0
\(535\) 296.340 0.0239474
\(536\) 0 0
\(537\) 19050.4 1.53089
\(538\) 0 0
\(539\) −7586.33 −0.606245
\(540\) 0 0
\(541\) 15464.3 1.22895 0.614473 0.788938i \(-0.289369\pi\)
0.614473 + 0.788938i \(0.289369\pi\)
\(542\) 0 0
\(543\) 7703.14 0.608791
\(544\) 0 0
\(545\) −9831.47 −0.772722
\(546\) 0 0
\(547\) −6981.32 −0.545703 −0.272852 0.962056i \(-0.587967\pi\)
−0.272852 + 0.962056i \(0.587967\pi\)
\(548\) 0 0
\(549\) −10589.1 −0.823192
\(550\) 0 0
\(551\) 4926.60 0.380908
\(552\) 0 0
\(553\) −10502.5 −0.807615
\(554\) 0 0
\(555\) 14888.8 1.13873
\(556\) 0 0
\(557\) 24964.8 1.89909 0.949543 0.313637i \(-0.101547\pi\)
0.949543 + 0.313637i \(0.101547\pi\)
\(558\) 0 0
\(559\) 6468.88 0.489454
\(560\) 0 0
\(561\) −5705.83 −0.429412
\(562\) 0 0
\(563\) 10820.3 0.809986 0.404993 0.914320i \(-0.367274\pi\)
0.404993 + 0.914320i \(0.367274\pi\)
\(564\) 0 0
\(565\) −9098.41 −0.677474
\(566\) 0 0
\(567\) −4223.65 −0.312834
\(568\) 0 0
\(569\) 8438.73 0.621740 0.310870 0.950453i \(-0.399380\pi\)
0.310870 + 0.950453i \(0.399380\pi\)
\(570\) 0 0
\(571\) −23107.6 −1.69356 −0.846781 0.531942i \(-0.821462\pi\)
−0.846781 + 0.531942i \(0.821462\pi\)
\(572\) 0 0
\(573\) 10673.4 0.778161
\(574\) 0 0
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 24848.7 1.79283 0.896416 0.443215i \(-0.146162\pi\)
0.896416 + 0.443215i \(0.146162\pi\)
\(578\) 0 0
\(579\) 11419.5 0.819651
\(580\) 0 0
\(581\) 4091.31 0.292145
\(582\) 0 0
\(583\) −12176.0 −0.864974
\(584\) 0 0
\(585\) −2774.25 −0.196070
\(586\) 0 0
\(587\) −8663.94 −0.609198 −0.304599 0.952481i \(-0.598522\pi\)
−0.304599 + 0.952481i \(0.598522\pi\)
\(588\) 0 0
\(589\) −1429.74 −0.100019
\(590\) 0 0
\(591\) −8405.94 −0.585066
\(592\) 0 0
\(593\) −24678.9 −1.70901 −0.854503 0.519446i \(-0.826138\pi\)
−0.854503 + 0.519446i \(0.826138\pi\)
\(594\) 0 0
\(595\) −1129.52 −0.0778250
\(596\) 0 0
\(597\) −22578.0 −1.54783
\(598\) 0 0
\(599\) −19698.4 −1.34367 −0.671833 0.740702i \(-0.734493\pi\)
−0.671833 + 0.740702i \(0.734493\pi\)
\(600\) 0 0
\(601\) −18449.1 −1.25217 −0.626086 0.779754i \(-0.715344\pi\)
−0.626086 + 0.779754i \(0.715344\pi\)
\(602\) 0 0
\(603\) −11035.5 −0.745272
\(604\) 0 0
\(605\) 2478.62 0.166563
\(606\) 0 0
\(607\) 11321.0 0.757012 0.378506 0.925599i \(-0.376438\pi\)
0.378506 + 0.925599i \(0.376438\pi\)
\(608\) 0 0
\(609\) −9728.59 −0.647327
\(610\) 0 0
\(611\) −1770.80 −0.117248
\(612\) 0 0
\(613\) 712.219 0.0469270 0.0234635 0.999725i \(-0.492531\pi\)
0.0234635 + 0.999725i \(0.492531\pi\)
\(614\) 0 0
\(615\) 16173.6 1.06046
\(616\) 0 0
\(617\) 3234.25 0.211031 0.105515 0.994418i \(-0.466351\pi\)
0.105515 + 0.994418i \(0.466351\pi\)
\(618\) 0 0
\(619\) −26905.1 −1.74703 −0.873513 0.486801i \(-0.838164\pi\)
−0.873513 + 0.486801i \(0.838164\pi\)
\(620\) 0 0
\(621\) −1350.60 −0.0872750
\(622\) 0 0
\(623\) −13419.2 −0.862967
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −8074.68 −0.514309
\(628\) 0 0
\(629\) 9560.90 0.606070
\(630\) 0 0
\(631\) −16199.5 −1.02202 −0.511008 0.859576i \(-0.670728\pi\)
−0.511008 + 0.859576i \(0.670728\pi\)
\(632\) 0 0
\(633\) −37219.3 −2.33702
\(634\) 0 0
\(635\) 4172.59 0.260762
\(636\) 0 0
\(637\) −4222.96 −0.262668
\(638\) 0 0
\(639\) 19111.5 1.18316
\(640\) 0 0
\(641\) 19943.5 1.22889 0.614446 0.788959i \(-0.289380\pi\)
0.614446 + 0.788959i \(0.289380\pi\)
\(642\) 0 0
\(643\) 27691.0 1.69833 0.849164 0.528129i \(-0.177106\pi\)
0.849164 + 0.528129i \(0.177106\pi\)
\(644\) 0 0
\(645\) 15765.1 0.962403
\(646\) 0 0
\(647\) 23560.3 1.43161 0.715805 0.698300i \(-0.246060\pi\)
0.715805 + 0.698300i \(0.246060\pi\)
\(648\) 0 0
\(649\) −21846.7 −1.32136
\(650\) 0 0
\(651\) 2823.32 0.169976
\(652\) 0 0
\(653\) 5571.58 0.333894 0.166947 0.985966i \(-0.446609\pi\)
0.166947 + 0.985966i \(0.446609\pi\)
\(654\) 0 0
\(655\) 2551.46 0.152205
\(656\) 0 0
\(657\) −24319.0 −1.44410
\(658\) 0 0
\(659\) −10177.4 −0.601602 −0.300801 0.953687i \(-0.597254\pi\)
−0.300801 + 0.953687i \(0.597254\pi\)
\(660\) 0 0
\(661\) −27413.8 −1.61312 −0.806561 0.591151i \(-0.798674\pi\)
−0.806561 + 0.591151i \(0.798674\pi\)
\(662\) 0 0
\(663\) −3176.17 −0.186052
\(664\) 0 0
\(665\) −1598.46 −0.0932113
\(666\) 0 0
\(667\) 3180.25 0.184618
\(668\) 0 0
\(669\) 38219.6 2.20875
\(670\) 0 0
\(671\) −8873.57 −0.510522
\(672\) 0 0
\(673\) 12767.5 0.731277 0.365639 0.930757i \(-0.380851\pi\)
0.365639 + 0.930757i \(0.380851\pi\)
\(674\) 0 0
\(675\) −1468.05 −0.0837112
\(676\) 0 0
\(677\) 17036.9 0.967180 0.483590 0.875295i \(-0.339333\pi\)
0.483590 + 0.875295i \(0.339333\pi\)
\(678\) 0 0
\(679\) 9342.63 0.528037
\(680\) 0 0
\(681\) −21502.0 −1.20993
\(682\) 0 0
\(683\) 510.213 0.0285838 0.0142919 0.999898i \(-0.495451\pi\)
0.0142919 + 0.999898i \(0.495451\pi\)
\(684\) 0 0
\(685\) −56.1695 −0.00313303
\(686\) 0 0
\(687\) −11984.8 −0.665570
\(688\) 0 0
\(689\) −6777.84 −0.374768
\(690\) 0 0
\(691\) −22793.1 −1.25483 −0.627416 0.778684i \(-0.715888\pi\)
−0.627416 + 0.778684i \(0.715888\pi\)
\(692\) 0 0
\(693\) 8943.53 0.490240
\(694\) 0 0
\(695\) −8284.07 −0.452134
\(696\) 0 0
\(697\) 10385.9 0.564412
\(698\) 0 0
\(699\) −43541.9 −2.35609
\(700\) 0 0
\(701\) 26324.3 1.41834 0.709168 0.705039i \(-0.249071\pi\)
0.709168 + 0.705039i \(0.249071\pi\)
\(702\) 0 0
\(703\) 13530.2 0.725892
\(704\) 0 0
\(705\) −4315.55 −0.230543
\(706\) 0 0
\(707\) 13012.2 0.692185
\(708\) 0 0
\(709\) −10961.3 −0.580620 −0.290310 0.956933i \(-0.593758\pi\)
−0.290310 + 0.956933i \(0.593758\pi\)
\(710\) 0 0
\(711\) −40369.2 −2.12934
\(712\) 0 0
\(713\) −922.938 −0.0484773
\(714\) 0 0
\(715\) −2324.80 −0.121598
\(716\) 0 0
\(717\) 21796.4 1.13529
\(718\) 0 0
\(719\) 1304.68 0.0676723 0.0338362 0.999427i \(-0.489228\pi\)
0.0338362 + 0.999427i \(0.489228\pi\)
\(720\) 0 0
\(721\) −1976.51 −0.102093
\(722\) 0 0
\(723\) 43667.7 2.24622
\(724\) 0 0
\(725\) 3456.80 0.177079
\(726\) 0 0
\(727\) −1583.59 −0.0807869 −0.0403934 0.999184i \(-0.512861\pi\)
−0.0403934 + 0.999184i \(0.512861\pi\)
\(728\) 0 0
\(729\) −28667.3 −1.45645
\(730\) 0 0
\(731\) 10123.6 0.512223
\(732\) 0 0
\(733\) −34351.2 −1.73095 −0.865477 0.500948i \(-0.832985\pi\)
−0.865477 + 0.500948i \(0.832985\pi\)
\(734\) 0 0
\(735\) −10291.6 −0.516480
\(736\) 0 0
\(737\) −9247.61 −0.462198
\(738\) 0 0
\(739\) −14996.6 −0.746494 −0.373247 0.927732i \(-0.621756\pi\)
−0.373247 + 0.927732i \(0.621756\pi\)
\(740\) 0 0
\(741\) −4494.80 −0.222835
\(742\) 0 0
\(743\) 32781.7 1.61863 0.809317 0.587372i \(-0.199838\pi\)
0.809317 + 0.587372i \(0.199838\pi\)
\(744\) 0 0
\(745\) 2551.03 0.125453
\(746\) 0 0
\(747\) 15726.1 0.770263
\(748\) 0 0
\(749\) −531.787 −0.0259427
\(750\) 0 0
\(751\) 24044.4 1.16830 0.584149 0.811647i \(-0.301429\pi\)
0.584149 + 0.811647i \(0.301429\pi\)
\(752\) 0 0
\(753\) −3035.16 −0.146889
\(754\) 0 0
\(755\) −11686.9 −0.563351
\(756\) 0 0
\(757\) −21680.9 −1.04096 −0.520479 0.853874i \(-0.674247\pi\)
−0.520479 + 0.853874i \(0.674247\pi\)
\(758\) 0 0
\(759\) −5212.43 −0.249274
\(760\) 0 0
\(761\) −10299.1 −0.490594 −0.245297 0.969448i \(-0.578885\pi\)
−0.245297 + 0.969448i \(0.578885\pi\)
\(762\) 0 0
\(763\) 17642.8 0.837105
\(764\) 0 0
\(765\) −4341.62 −0.205192
\(766\) 0 0
\(767\) −12161.1 −0.572505
\(768\) 0 0
\(769\) 28377.9 1.33073 0.665365 0.746518i \(-0.268276\pi\)
0.665365 + 0.746518i \(0.268276\pi\)
\(770\) 0 0
\(771\) −11574.5 −0.540657
\(772\) 0 0
\(773\) 35409.5 1.64759 0.823797 0.566885i \(-0.191852\pi\)
0.823797 + 0.566885i \(0.191852\pi\)
\(774\) 0 0
\(775\) −1003.19 −0.0464978
\(776\) 0 0
\(777\) −26718.2 −1.23361
\(778\) 0 0
\(779\) 14697.8 0.675999
\(780\) 0 0
\(781\) 16015.2 0.733763
\(782\) 0 0
\(783\) −8119.58 −0.370588
\(784\) 0 0
\(785\) −732.959 −0.0333254
\(786\) 0 0
\(787\) −6117.56 −0.277087 −0.138544 0.990356i \(-0.544242\pi\)
−0.138544 + 0.990356i \(0.544242\pi\)
\(788\) 0 0
\(789\) 1599.54 0.0721737
\(790\) 0 0
\(791\) 16327.3 0.733921
\(792\) 0 0
\(793\) −4939.51 −0.221194
\(794\) 0 0
\(795\) −16518.0 −0.736899
\(796\) 0 0
\(797\) 4099.46 0.182196 0.0910980 0.995842i \(-0.470962\pi\)
0.0910980 + 0.995842i \(0.470962\pi\)
\(798\) 0 0
\(799\) −2771.24 −0.122703
\(800\) 0 0
\(801\) −51580.3 −2.27528
\(802\) 0 0
\(803\) −20379.1 −0.895594
\(804\) 0 0
\(805\) −1031.85 −0.0451775
\(806\) 0 0
\(807\) 54684.1 2.38534
\(808\) 0 0
\(809\) 21358.6 0.928216 0.464108 0.885779i \(-0.346375\pi\)
0.464108 + 0.885779i \(0.346375\pi\)
\(810\) 0 0
\(811\) −13967.7 −0.604776 −0.302388 0.953185i \(-0.597784\pi\)
−0.302388 + 0.953185i \(0.597784\pi\)
\(812\) 0 0
\(813\) −16976.6 −0.732342
\(814\) 0 0
\(815\) −16390.9 −0.704478
\(816\) 0 0
\(817\) 14326.6 0.613492
\(818\) 0 0
\(819\) 4978.45 0.212407
\(820\) 0 0
\(821\) −22387.6 −0.951684 −0.475842 0.879531i \(-0.657857\pi\)
−0.475842 + 0.879531i \(0.657857\pi\)
\(822\) 0 0
\(823\) 22615.7 0.957877 0.478939 0.877848i \(-0.341022\pi\)
0.478939 + 0.877848i \(0.341022\pi\)
\(824\) 0 0
\(825\) −5665.68 −0.239095
\(826\) 0 0
\(827\) −10878.0 −0.457394 −0.228697 0.973498i \(-0.573446\pi\)
−0.228697 + 0.973498i \(0.573446\pi\)
\(828\) 0 0
\(829\) 27382.3 1.14720 0.573600 0.819136i \(-0.305547\pi\)
0.573600 + 0.819136i \(0.305547\pi\)
\(830\) 0 0
\(831\) −9362.74 −0.390842
\(832\) 0 0
\(833\) −6608.81 −0.274888
\(834\) 0 0
\(835\) −7777.12 −0.322321
\(836\) 0 0
\(837\) 2356.37 0.0973096
\(838\) 0 0
\(839\) 31799.7 1.30852 0.654260 0.756270i \(-0.272980\pi\)
0.654260 + 0.756270i \(0.272980\pi\)
\(840\) 0 0
\(841\) −5269.87 −0.216076
\(842\) 0 0
\(843\) −50859.6 −2.07793
\(844\) 0 0
\(845\) 9690.89 0.394529
\(846\) 0 0
\(847\) −4447.94 −0.180440
\(848\) 0 0
\(849\) −25205.7 −1.01891
\(850\) 0 0
\(851\) 8734.14 0.351824
\(852\) 0 0
\(853\) 32016.4 1.28514 0.642568 0.766229i \(-0.277869\pi\)
0.642568 + 0.766229i \(0.277869\pi\)
\(854\) 0 0
\(855\) −6144.11 −0.245759
\(856\) 0 0
\(857\) 25280.1 1.00764 0.503822 0.863807i \(-0.331927\pi\)
0.503822 + 0.863807i \(0.331927\pi\)
\(858\) 0 0
\(859\) 22313.5 0.886296 0.443148 0.896448i \(-0.353862\pi\)
0.443148 + 0.896448i \(0.353862\pi\)
\(860\) 0 0
\(861\) −29023.8 −1.14881
\(862\) 0 0
\(863\) 1478.28 0.0583096 0.0291548 0.999575i \(-0.490718\pi\)
0.0291548 + 0.999575i \(0.490718\pi\)
\(864\) 0 0
\(865\) 2361.96 0.0928428
\(866\) 0 0
\(867\) 33554.5 1.31438
\(868\) 0 0
\(869\) −33829.0 −1.32056
\(870\) 0 0
\(871\) −5147.72 −0.200257
\(872\) 0 0
\(873\) 35910.9 1.39221
\(874\) 0 0
\(875\) −1121.57 −0.0433327
\(876\) 0 0
\(877\) 32974.6 1.26964 0.634819 0.772661i \(-0.281075\pi\)
0.634819 + 0.772661i \(0.281075\pi\)
\(878\) 0 0
\(879\) −43798.3 −1.68064
\(880\) 0 0
\(881\) −32000.7 −1.22376 −0.611879 0.790951i \(-0.709586\pi\)
−0.611879 + 0.790951i \(0.709586\pi\)
\(882\) 0 0
\(883\) 44218.6 1.68525 0.842623 0.538503i \(-0.181010\pi\)
0.842623 + 0.538503i \(0.181010\pi\)
\(884\) 0 0
\(885\) −29637.4 −1.12570
\(886\) 0 0
\(887\) −27374.8 −1.03625 −0.518126 0.855304i \(-0.673370\pi\)
−0.518126 + 0.855304i \(0.673370\pi\)
\(888\) 0 0
\(889\) −7487.79 −0.282489
\(890\) 0 0
\(891\) −13604.6 −0.511526
\(892\) 0 0
\(893\) −3921.76 −0.146962
\(894\) 0 0
\(895\) 12147.2 0.453673
\(896\) 0 0
\(897\) −2901.52 −0.108003
\(898\) 0 0
\(899\) −5548.54 −0.205844
\(900\) 0 0
\(901\) −10607.1 −0.392203
\(902\) 0 0
\(903\) −28290.8 −1.04259
\(904\) 0 0
\(905\) 4911.80 0.180413
\(906\) 0 0
\(907\) −7835.16 −0.286838 −0.143419 0.989662i \(-0.545810\pi\)
−0.143419 + 0.989662i \(0.545810\pi\)
\(908\) 0 0
\(909\) 50016.0 1.82500
\(910\) 0 0
\(911\) 36528.6 1.32848 0.664241 0.747519i \(-0.268755\pi\)
0.664241 + 0.747519i \(0.268755\pi\)
\(912\) 0 0
\(913\) 13178.3 0.477698
\(914\) 0 0
\(915\) −12037.9 −0.434930
\(916\) 0 0
\(917\) −4578.65 −0.164886
\(918\) 0 0
\(919\) 15741.5 0.565030 0.282515 0.959263i \(-0.408831\pi\)
0.282515 + 0.959263i \(0.408831\pi\)
\(920\) 0 0
\(921\) 22344.4 0.799427
\(922\) 0 0
\(923\) 8914.93 0.317918
\(924\) 0 0
\(925\) 9493.63 0.337458
\(926\) 0 0
\(927\) −7597.26 −0.269177
\(928\) 0 0
\(929\) 47428.9 1.67502 0.837510 0.546422i \(-0.184011\pi\)
0.837510 + 0.546422i \(0.184011\pi\)
\(930\) 0 0
\(931\) −9352.54 −0.329234
\(932\) 0 0
\(933\) −51893.6 −1.82092
\(934\) 0 0
\(935\) −3638.24 −0.127255
\(936\) 0 0
\(937\) −34978.0 −1.21951 −0.609756 0.792589i \(-0.708733\pi\)
−0.609756 + 0.792589i \(0.708733\pi\)
\(938\) 0 0
\(939\) 49175.9 1.70905
\(940\) 0 0
\(941\) 45144.0 1.56392 0.781962 0.623326i \(-0.214219\pi\)
0.781962 + 0.623326i \(0.214219\pi\)
\(942\) 0 0
\(943\) 9487.83 0.327642
\(944\) 0 0
\(945\) 2634.44 0.0906859
\(946\) 0 0
\(947\) −26123.6 −0.896413 −0.448207 0.893930i \(-0.647937\pi\)
−0.448207 + 0.893930i \(0.647937\pi\)
\(948\) 0 0
\(949\) −11344.1 −0.388035
\(950\) 0 0
\(951\) −44310.0 −1.51088
\(952\) 0 0
\(953\) −22143.5 −0.752673 −0.376336 0.926483i \(-0.622816\pi\)
−0.376336 + 0.926483i \(0.622816\pi\)
\(954\) 0 0
\(955\) 6805.72 0.230605
\(956\) 0 0
\(957\) −31336.2 −1.05847
\(958\) 0 0
\(959\) 100.797 0.00339407
\(960\) 0 0
\(961\) −28180.8 −0.945949
\(962\) 0 0
\(963\) −2044.07 −0.0684000
\(964\) 0 0
\(965\) 7281.47 0.242900
\(966\) 0 0
\(967\) −44869.8 −1.49216 −0.746078 0.665858i \(-0.768066\pi\)
−0.746078 + 0.665858i \(0.768066\pi\)
\(968\) 0 0
\(969\) −7034.23 −0.233201
\(970\) 0 0
\(971\) 25048.3 0.827845 0.413923 0.910312i \(-0.364158\pi\)
0.413923 + 0.910312i \(0.364158\pi\)
\(972\) 0 0
\(973\) 14865.9 0.489805
\(974\) 0 0
\(975\) −3153.82 −0.103593
\(976\) 0 0
\(977\) 37320.7 1.22210 0.611052 0.791590i \(-0.290747\pi\)
0.611052 + 0.791590i \(0.290747\pi\)
\(978\) 0 0
\(979\) −43223.8 −1.41107
\(980\) 0 0
\(981\) 67814.7 2.20709
\(982\) 0 0
\(983\) 17189.4 0.557737 0.278869 0.960329i \(-0.410041\pi\)
0.278869 + 0.960329i \(0.410041\pi\)
\(984\) 0 0
\(985\) −5359.93 −0.173382
\(986\) 0 0
\(987\) 7744.33 0.249752
\(988\) 0 0
\(989\) 9248.19 0.297346
\(990\) 0 0
\(991\) 57797.1 1.85266 0.926330 0.376712i \(-0.122945\pi\)
0.926330 + 0.376712i \(0.122945\pi\)
\(992\) 0 0
\(993\) −50117.9 −1.60166
\(994\) 0 0
\(995\) −14396.5 −0.458695
\(996\) 0 0
\(997\) 46801.0 1.48666 0.743331 0.668923i \(-0.233245\pi\)
0.743331 + 0.668923i \(0.233245\pi\)
\(998\) 0 0
\(999\) −22299.3 −0.706226
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 1840.4.a.n.1.2 5
4.3 odd 2 115.4.a.e.1.4 5
12.11 even 2 1035.4.a.k.1.2 5
20.3 even 4 575.4.b.i.24.2 10
20.7 even 4 575.4.b.i.24.9 10
20.19 odd 2 575.4.a.j.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.4 5 4.3 odd 2
575.4.a.j.1.2 5 20.19 odd 2
575.4.b.i.24.2 10 20.3 even 4
575.4.b.i.24.9 10 20.7 even 4
1035.4.a.k.1.2 5 12.11 even 2
1840.4.a.n.1.2 5 1.1 even 1 trivial