Properties

Label 115.3.d.a.91.4
Level $115$
Weight $3$
Character 115.91
Analytic conductor $3.134$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(91,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.13352304014\)
Analytic rank: \(0\)
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: \( x^{6} - 2x^{5} - 34x^{4} + 50x^{3} + 690x^{2} - 600x + 4725 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.4
Root \(0.396209 - 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 115.91
Dual form 115.3.d.a.91.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.396209 q^{3} -3.00000 q^{4} +2.23607i q^{5} -0.396209 q^{6} -8.16531i q^{7} +7.00000 q^{8} -8.84302 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.396209 q^{3} -3.00000 q^{4} +2.23607i q^{5} -0.396209 q^{6} -8.16531i q^{7} +7.00000 q^{8} -8.84302 q^{9} -2.23607i q^{10} -18.9885i q^{11} -1.18863 q^{12} -18.6354 q^{13} +8.16531i q^{14} +0.885949i q^{15} +5.00000 q^{16} +5.57206i q^{17} +8.84302 q^{18} -5.25110i q^{19} -6.70820i q^{20} -3.23517i q^{21} +18.9885i q^{22} +(-18.4468 - 13.7374i) q^{23} +2.77346 q^{24} -5.00000 q^{25} +18.6354 q^{26} -7.06956 q^{27} +24.4959i q^{28} +11.6227 q^{29} -0.885949i q^{30} +49.5291 q^{31} -33.0000 q^{32} -7.52339i q^{33} -5.57206i q^{34} +18.2582 q^{35} +26.5291 q^{36} +29.7470i q^{37} +5.25110i q^{38} -7.38352 q^{39} +15.6525i q^{40} -10.4595 q^{41} +3.23517i q^{42} +57.2218i q^{43} +56.9654i q^{44} -19.7736i q^{45} +(18.4468 + 13.7374i) q^{46} -15.2709 q^{47} +1.98104 q^{48} -17.6723 q^{49} +5.00000 q^{50} +2.20770i q^{51} +55.9063 q^{52} -40.2492i q^{53} +7.06956 q^{54} +42.4595 q^{55} -57.1572i q^{56} -2.08053i q^{57} -11.6227 q^{58} -27.7872 q^{59} -2.65785i q^{60} -110.631i q^{61} -49.5291 q^{62} +72.2060i q^{63} +13.0000 q^{64} -41.6701i q^{65} +7.52339i q^{66} +35.0627i q^{67} -16.7162i q^{68} +(-7.30879 - 5.44286i) q^{69} -18.2582 q^{70} -70.4595 q^{71} -61.9011 q^{72} -0.377252 q^{73} -29.7470i q^{74} -1.98104 q^{75} +15.7533i q^{76} -155.047 q^{77} +7.38352 q^{78} -102.657i q^{79} +11.1803i q^{80} +76.7861 q^{81} +10.4595 q^{82} -116.845i q^{83} +9.70550i q^{84} -12.4595 q^{85} -57.2218i q^{86} +4.60503 q^{87} -132.919i q^{88} -49.6338i q^{89} +19.7736i q^{90} +152.164i q^{91} +(55.3404 + 41.2121i) q^{92} +19.6238 q^{93} +15.2709 q^{94} +11.7418 q^{95} -13.0749 q^{96} +69.6107i q^{97} +17.6723 q^{98} +167.915i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 2 q^{3} - 18 q^{4} - 2 q^{6} + 42 q^{8} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 2 q^{3} - 18 q^{4} - 2 q^{6} + 42 q^{8} + 48 q^{9} - 6 q^{12} - 10 q^{13} + 30 q^{16} - 48 q^{18} - 10 q^{23} + 14 q^{24} - 30 q^{25} + 10 q^{26} + 56 q^{27} + 72 q^{29} - 6 q^{31} - 198 q^{32} + 10 q^{35} - 144 q^{36} - 148 q^{39} + 142 q^{41} + 10 q^{46} + 112 q^{47} + 10 q^{48} - 304 q^{49} + 30 q^{50} + 30 q^{52} - 56 q^{54} + 50 q^{55} - 72 q^{58} + 236 q^{59} + 6 q^{62} + 78 q^{64} + 156 q^{69} - 10 q^{70} - 218 q^{71} + 336 q^{72} - 10 q^{75} + 184 q^{77} + 148 q^{78} + 354 q^{81} - 142 q^{82} + 130 q^{85} - 584 q^{87} + 30 q^{92} - 176 q^{93} - 112 q^{94} + 170 q^{95} - 66 q^{96} + 304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.500000 −0.250000 0.968246i \(-0.580431\pi\)
−0.250000 + 0.968246i \(0.580431\pi\)
\(3\) 0.396209 0.132070 0.0660348 0.997817i \(-0.478965\pi\)
0.0660348 + 0.997817i \(0.478965\pi\)
\(4\) −3.00000 −0.750000
\(5\) 2.23607i 0.447214i
\(6\) −0.396209 −0.0660348
\(7\) 8.16531i 1.16647i −0.812303 0.583236i \(-0.801786\pi\)
0.812303 0.583236i \(-0.198214\pi\)
\(8\) 7.00000 0.875000
\(9\) −8.84302 −0.982558
\(10\) 2.23607i 0.223607i
\(11\) 18.9885i 1.72622i −0.505012 0.863112i \(-0.668512\pi\)
0.505012 0.863112i \(-0.331488\pi\)
\(12\) −1.18863 −0.0990522
\(13\) −18.6354 −1.43350 −0.716748 0.697333i \(-0.754370\pi\)
−0.716748 + 0.697333i \(0.754370\pi\)
\(14\) 8.16531i 0.583236i
\(15\) 0.885949i 0.0590633i
\(16\) 5.00000 0.312500
\(17\) 5.57206i 0.327768i 0.986480 + 0.163884i \(0.0524023\pi\)
−0.986480 + 0.163884i \(0.947598\pi\)
\(18\) 8.84302 0.491279
\(19\) 5.25110i 0.276374i −0.990406 0.138187i \(-0.955873\pi\)
0.990406 0.138187i \(-0.0441274\pi\)
\(20\) 6.70820i 0.335410i
\(21\) 3.23517i 0.154056i
\(22\) 18.9885i 0.863112i
\(23\) −18.4468 13.7374i −0.802035 0.597277i
\(24\) 2.77346 0.115561
\(25\) −5.00000 −0.200000
\(26\) 18.6354 0.716748
\(27\) −7.06956 −0.261835
\(28\) 24.4959i 0.874854i
\(29\) 11.6227 0.400784 0.200392 0.979716i \(-0.435778\pi\)
0.200392 + 0.979716i \(0.435778\pi\)
\(30\) 0.885949i 0.0295316i
\(31\) 49.5291 1.59771 0.798856 0.601523i \(-0.205439\pi\)
0.798856 + 0.601523i \(0.205439\pi\)
\(32\) −33.0000 −1.03125
\(33\) 7.52339i 0.227982i
\(34\) 5.57206i 0.163884i
\(35\) 18.2582 0.521662
\(36\) 26.5291 0.736918
\(37\) 29.7470i 0.803974i 0.915645 + 0.401987i \(0.131680\pi\)
−0.915645 + 0.401987i \(0.868320\pi\)
\(38\) 5.25110i 0.138187i
\(39\) −7.38352 −0.189321
\(40\) 15.6525i 0.391312i
\(41\) −10.4595 −0.255110 −0.127555 0.991832i \(-0.540713\pi\)
−0.127555 + 0.991832i \(0.540713\pi\)
\(42\) 3.23517i 0.0770278i
\(43\) 57.2218i 1.33074i 0.746514 + 0.665369i \(0.231726\pi\)
−0.746514 + 0.665369i \(0.768274\pi\)
\(44\) 56.9654i 1.29467i
\(45\) 19.7736i 0.439413i
\(46\) 18.4468 + 13.7374i 0.401018 + 0.298638i
\(47\) −15.2709 −0.324912 −0.162456 0.986716i \(-0.551942\pi\)
−0.162456 + 0.986716i \(0.551942\pi\)
\(48\) 1.98104 0.0412717
\(49\) −17.6723 −0.360658
\(50\) 5.00000 0.100000
\(51\) 2.20770i 0.0432882i
\(52\) 55.9063 1.07512
\(53\) 40.2492i 0.759419i −0.925106 0.379710i \(-0.876024\pi\)
0.925106 0.379710i \(-0.123976\pi\)
\(54\) 7.06956 0.130918
\(55\) 42.4595 0.771991
\(56\) 57.1572i 1.02066i
\(57\) 2.08053i 0.0365005i
\(58\) −11.6227 −0.200392
\(59\) −27.7872 −0.470970 −0.235485 0.971878i \(-0.575668\pi\)
−0.235485 + 0.971878i \(0.575668\pi\)
\(60\) 2.65785i 0.0442975i
\(61\) 110.631i 1.81362i −0.421536 0.906812i \(-0.638509\pi\)
0.421536 0.906812i \(-0.361491\pi\)
\(62\) −49.5291 −0.798856
\(63\) 72.2060i 1.14613i
\(64\) 13.0000 0.203125
\(65\) 41.6701i 0.641078i
\(66\) 7.52339i 0.113991i
\(67\) 35.0627i 0.523324i 0.965160 + 0.261662i \(0.0842706\pi\)
−0.965160 + 0.261662i \(0.915729\pi\)
\(68\) 16.7162i 0.245826i
\(69\) −7.30879 5.44286i −0.105924 0.0788821i
\(70\) −18.2582 −0.260831
\(71\) −70.4595 −0.992387 −0.496194 0.868212i \(-0.665269\pi\)
−0.496194 + 0.868212i \(0.665269\pi\)
\(72\) −61.9011 −0.859738
\(73\) −0.377252 −0.00516783 −0.00258392 0.999997i \(-0.500822\pi\)
−0.00258392 + 0.999997i \(0.500822\pi\)
\(74\) 29.7470i 0.401987i
\(75\) −1.98104 −0.0264139
\(76\) 15.7533i 0.207280i
\(77\) −155.047 −2.01359
\(78\) 7.38352 0.0946605
\(79\) 102.657i 1.29946i −0.760164 0.649731i \(-0.774882\pi\)
0.760164 0.649731i \(-0.225118\pi\)
\(80\) 11.1803i 0.139754i
\(81\) 76.7861 0.947977
\(82\) 10.4595 0.127555
\(83\) 116.845i 1.40777i −0.710313 0.703886i \(-0.751447\pi\)
0.710313 0.703886i \(-0.248553\pi\)
\(84\) 9.70550i 0.115542i
\(85\) −12.4595 −0.146582
\(86\) 57.2218i 0.665369i
\(87\) 4.60503 0.0529314
\(88\) 132.919i 1.51045i
\(89\) 49.6338i 0.557683i −0.960337 0.278841i \(-0.910050\pi\)
0.960337 0.278841i \(-0.0899504\pi\)
\(90\) 19.7736i 0.219707i
\(91\) 152.164i 1.67213i
\(92\) 55.3404 + 41.2121i 0.601526 + 0.447958i
\(93\) 19.6238 0.211009
\(94\) 15.2709 0.162456
\(95\) 11.7418 0.123598
\(96\) −13.0749 −0.136197
\(97\) 69.6107i 0.717636i 0.933408 + 0.358818i \(0.116820\pi\)
−0.933408 + 0.358818i \(0.883180\pi\)
\(98\) 17.6723 0.180329
\(99\) 167.915i 1.69611i
\(100\) 15.0000 0.150000
\(101\) 146.919 1.45464 0.727322 0.686297i \(-0.240765\pi\)
0.727322 + 0.686297i \(0.240765\pi\)
\(102\) 2.20770i 0.0216441i
\(103\) 120.016i 1.16520i 0.812759 + 0.582600i \(0.197964\pi\)
−0.812759 + 0.582600i \(0.802036\pi\)
\(104\) −130.448 −1.25431
\(105\) 7.23405 0.0688957
\(106\) 40.2492i 0.379710i
\(107\) 191.386i 1.78865i −0.447415 0.894326i \(-0.647655\pi\)
0.447415 0.894326i \(-0.352345\pi\)
\(108\) 21.2087 0.196377
\(109\) 134.741i 1.23616i 0.786115 + 0.618080i \(0.212089\pi\)
−0.786115 + 0.618080i \(0.787911\pi\)
\(110\) −42.4595 −0.385995
\(111\) 11.7860i 0.106180i
\(112\) 40.8265i 0.364523i
\(113\) 86.9688i 0.769635i −0.922993 0.384818i \(-0.874264\pi\)
0.922993 0.384818i \(-0.125736\pi\)
\(114\) 2.08053i 0.0182503i
\(115\) 30.7177 41.2483i 0.267110 0.358681i
\(116\) −34.8682 −0.300588
\(117\) 164.794 1.40849
\(118\) 27.7872 0.235485
\(119\) 45.4976 0.382332
\(120\) 6.20165i 0.0516804i
\(121\) −239.562 −1.97985
\(122\) 110.631i 0.906812i
\(123\) −4.14414 −0.0336922
\(124\) −148.587 −1.19828
\(125\) 11.1803i 0.0894427i
\(126\) 72.2060i 0.573063i
\(127\) 71.0835 0.559713 0.279856 0.960042i \(-0.409713\pi\)
0.279856 + 0.960042i \(0.409713\pi\)
\(128\) 119.000 0.929688
\(129\) 22.6718i 0.175750i
\(130\) 41.6701i 0.320539i
\(131\) 72.1391 0.550680 0.275340 0.961347i \(-0.411210\pi\)
0.275340 + 0.961347i \(0.411210\pi\)
\(132\) 22.5702i 0.170986i
\(133\) −42.8768 −0.322382
\(134\) 35.0627i 0.261662i
\(135\) 15.8080i 0.117096i
\(136\) 39.0044i 0.286797i
\(137\) 56.1317i 0.409721i 0.978791 + 0.204860i \(0.0656740\pi\)
−0.978791 + 0.204860i \(0.934326\pi\)
\(138\) 7.30879 + 5.44286i 0.0529622 + 0.0394410i
\(139\) 266.091 1.91432 0.957161 0.289555i \(-0.0935074\pi\)
0.957161 + 0.289555i \(0.0935074\pi\)
\(140\) −54.7746 −0.391247
\(141\) −6.05045 −0.0429110
\(142\) 70.4595 0.496194
\(143\) 353.858i 2.47453i
\(144\) −44.2151 −0.307049
\(145\) 25.9893i 0.179236i
\(146\) 0.377252 0.00258392
\(147\) −7.00190 −0.0476320
\(148\) 89.2411i 0.602980i
\(149\) 222.352i 1.49230i −0.665780 0.746148i \(-0.731901\pi\)
0.665780 0.746148i \(-0.268099\pi\)
\(150\) 1.98104 0.0132070
\(151\) −194.788 −1.28999 −0.644995 0.764187i \(-0.723141\pi\)
−0.644995 + 0.764187i \(0.723141\pi\)
\(152\) 36.7577i 0.241827i
\(153\) 49.2738i 0.322051i
\(154\) 155.047 1.00680
\(155\) 110.750i 0.714518i
\(156\) 22.1506 0.141991
\(157\) 106.214i 0.676520i −0.941053 0.338260i \(-0.890162\pi\)
0.941053 0.338260i \(-0.109838\pi\)
\(158\) 102.657i 0.649731i
\(159\) 15.9471i 0.100296i
\(160\) 73.7902i 0.461189i
\(161\) −112.170 + 150.624i −0.696707 + 0.935552i
\(162\) −76.7861 −0.473989
\(163\) 39.9063 0.244824 0.122412 0.992479i \(-0.460937\pi\)
0.122412 + 0.992479i \(0.460937\pi\)
\(164\) 31.3785 0.191332
\(165\) 16.8228 0.101956
\(166\) 116.845i 0.703886i
\(167\) 79.7872 0.477768 0.238884 0.971048i \(-0.423218\pi\)
0.238884 + 0.971048i \(0.423218\pi\)
\(168\) 22.6462i 0.134799i
\(169\) 178.279 1.05491
\(170\) 12.4595 0.0732912
\(171\) 46.4356i 0.271553i
\(172\) 171.665i 0.998054i
\(173\) −298.323 −1.72441 −0.862205 0.506560i \(-0.830917\pi\)
−0.862205 + 0.506560i \(0.830917\pi\)
\(174\) −4.60503 −0.0264657
\(175\) 40.8265i 0.233295i
\(176\) 94.9423i 0.539445i
\(177\) −11.0095 −0.0622008
\(178\) 49.6338i 0.278841i
\(179\) 48.1391 0.268934 0.134467 0.990918i \(-0.457068\pi\)
0.134467 + 0.990918i \(0.457068\pi\)
\(180\) 59.3208i 0.329560i
\(181\) 34.5224i 0.190731i 0.995442 + 0.0953657i \(0.0304020\pi\)
−0.995442 + 0.0953657i \(0.969598\pi\)
\(182\) 152.164i 0.836066i
\(183\) 43.8330i 0.239524i
\(184\) −129.128 96.1616i −0.701781 0.522617i
\(185\) −66.5164 −0.359548
\(186\) −19.6238 −0.105505
\(187\) 105.805 0.565801
\(188\) 45.8126 0.243684
\(189\) 57.7251i 0.305424i
\(190\) −11.7418 −0.0617990
\(191\) 172.912i 0.905299i 0.891689 + 0.452650i \(0.149521\pi\)
−0.891689 + 0.452650i \(0.850479\pi\)
\(192\) 5.15071 0.0268266
\(193\) −114.255 −0.591997 −0.295998 0.955188i \(-0.595652\pi\)
−0.295998 + 0.955188i \(0.595652\pi\)
\(194\) 69.6107i 0.358818i
\(195\) 16.5101i 0.0846669i
\(196\) 53.0168 0.270494
\(197\) 139.191 0.706554 0.353277 0.935519i \(-0.385067\pi\)
0.353277 + 0.935519i \(0.385067\pi\)
\(198\) 167.915i 0.848057i
\(199\) 245.347i 1.23290i −0.787395 0.616449i \(-0.788571\pi\)
0.787395 0.616449i \(-0.211429\pi\)
\(200\) −35.0000 −0.175000
\(201\) 13.8922i 0.0691152i
\(202\) −146.919 −0.727322
\(203\) 94.9033i 0.467504i
\(204\) 6.62309i 0.0324661i
\(205\) 23.3881i 0.114089i
\(206\) 120.016i 0.582600i
\(207\) 163.125 + 121.480i 0.788046 + 0.586859i
\(208\) −93.1772 −0.447967
\(209\) −99.7103 −0.477083
\(210\) −7.23405 −0.0344479
\(211\) −184.076 −0.872399 −0.436199 0.899850i \(-0.643676\pi\)
−0.436199 + 0.899850i \(0.643676\pi\)
\(212\) 120.748i 0.569564i
\(213\) −27.9167 −0.131064
\(214\) 191.386i 0.894326i
\(215\) −127.952 −0.595124
\(216\) −49.4869 −0.229106
\(217\) 404.420i 1.86369i
\(218\) 134.741i 0.618080i
\(219\) −0.149470 −0.000682513
\(220\) −127.378 −0.578993
\(221\) 103.838i 0.469854i
\(222\) 11.7860i 0.0530902i
\(223\) −31.1064 −0.139490 −0.0697452 0.997565i \(-0.522219\pi\)
−0.0697452 + 0.997565i \(0.522219\pi\)
\(224\) 269.455i 1.20292i
\(225\) 44.2151 0.196512
\(226\) 86.9688i 0.384818i
\(227\) 192.411i 0.847627i 0.905750 + 0.423813i \(0.139309\pi\)
−0.905750 + 0.423813i \(0.860691\pi\)
\(228\) 6.24159i 0.0273754i
\(229\) 365.836i 1.59754i −0.601638 0.798769i \(-0.705485\pi\)
0.601638 0.798769i \(-0.294515\pi\)
\(230\) −30.7177 + 41.2483i −0.133555 + 0.179341i
\(231\) −61.4308 −0.265934
\(232\) 81.3592 0.350686
\(233\) 235.271 1.00975 0.504873 0.863194i \(-0.331539\pi\)
0.504873 + 0.863194i \(0.331539\pi\)
\(234\) −164.794 −0.704246
\(235\) 34.1467i 0.145305i
\(236\) 83.3617 0.353228
\(237\) 40.6738i 0.171619i
\(238\) −45.4976 −0.191166
\(239\) 178.278 0.745934 0.372967 0.927845i \(-0.378340\pi\)
0.372967 + 0.927845i \(0.378340\pi\)
\(240\) 4.42975i 0.0184573i
\(241\) 32.9156i 0.136579i −0.997666 0.0682896i \(-0.978246\pi\)
0.997666 0.0682896i \(-0.0217542\pi\)
\(242\) 239.562 0.989925
\(243\) 94.0494 0.387034
\(244\) 331.893i 1.36022i
\(245\) 39.5164i 0.161291i
\(246\) 4.14414 0.0168461
\(247\) 97.8565i 0.396180i
\(248\) 346.703 1.39800
\(249\) 46.2950i 0.185924i
\(250\) 11.1803i 0.0447214i
\(251\) 193.339i 0.770276i −0.922859 0.385138i \(-0.874154\pi\)
0.922859 0.385138i \(-0.125846\pi\)
\(252\) 216.618i 0.859595i
\(253\) −260.851 + 350.277i −1.03103 + 1.38449i
\(254\) −71.0835 −0.279856
\(255\) −4.93656 −0.0193591
\(256\) −171.000 −0.667969
\(257\) −88.0147 −0.342470 −0.171235 0.985230i \(-0.554776\pi\)
−0.171235 + 0.985230i \(0.554776\pi\)
\(258\) 22.6718i 0.0878750i
\(259\) 242.894 0.937813
\(260\) 125.010i 0.480809i
\(261\) −102.780 −0.393794
\(262\) −72.1391 −0.275340
\(263\) 98.7568i 0.375501i −0.982217 0.187751i \(-0.939880\pi\)
0.982217 0.187751i \(-0.0601197\pi\)
\(264\) 52.6638i 0.199484i
\(265\) 90.0000 0.339623
\(266\) 42.8768 0.161191
\(267\) 19.6653i 0.0736529i
\(268\) 105.188i 0.392493i
\(269\) −337.903 −1.25615 −0.628073 0.778154i \(-0.716156\pi\)
−0.628073 + 0.778154i \(0.716156\pi\)
\(270\) 15.8080i 0.0585482i
\(271\) 150.086 0.553821 0.276911 0.960896i \(-0.410689\pi\)
0.276911 + 0.960896i \(0.410689\pi\)
\(272\) 27.8603i 0.102428i
\(273\) 60.2887i 0.220838i
\(274\) 56.1317i 0.204860i
\(275\) 94.9423i 0.345245i
\(276\) 21.9264 + 16.3286i 0.0794433 + 0.0591616i
\(277\) −246.856 −0.891177 −0.445588 0.895238i \(-0.647006\pi\)
−0.445588 + 0.895238i \(0.647006\pi\)
\(278\) −266.091 −0.957161
\(279\) −437.986 −1.56984
\(280\) 127.807 0.456455
\(281\) 495.802i 1.76442i 0.470856 + 0.882210i \(0.343945\pi\)
−0.470856 + 0.882210i \(0.656055\pi\)
\(282\) 6.05045 0.0214555
\(283\) 191.002i 0.674920i 0.941340 + 0.337460i \(0.109568\pi\)
−0.941340 + 0.337460i \(0.890432\pi\)
\(284\) 211.378 0.744290
\(285\) 4.65221 0.0163235
\(286\) 353.858i 1.23727i
\(287\) 85.4050i 0.297578i
\(288\) 291.820 1.01326
\(289\) 257.952 0.892568
\(290\) 25.9893i 0.0896181i
\(291\) 27.5804i 0.0947779i
\(292\) 1.13176 0.00387588
\(293\) 302.569i 1.03266i −0.856391 0.516329i \(-0.827298\pi\)
0.856391 0.516329i \(-0.172702\pi\)
\(294\) 7.00190 0.0238160
\(295\) 62.1342i 0.210624i
\(296\) 208.229i 0.703477i
\(297\) 134.240i 0.451987i
\(298\) 222.352i 0.746148i
\(299\) 343.764 + 256.002i 1.14971 + 0.856193i
\(300\) 5.94313 0.0198104
\(301\) 467.233 1.55227
\(302\) 194.788 0.644995
\(303\) 58.2106 0.192114
\(304\) 26.2555i 0.0863668i
\(305\) 247.378 0.811077
\(306\) 49.2738i 0.161026i
\(307\) 137.421 0.447626 0.223813 0.974632i \(-0.428150\pi\)
0.223813 + 0.974632i \(0.428150\pi\)
\(308\) 465.140 1.51019
\(309\) 47.5512i 0.153887i
\(310\) 110.750i 0.357259i
\(311\) 106.051 0.340999 0.170500 0.985358i \(-0.445462\pi\)
0.170500 + 0.985358i \(0.445462\pi\)
\(312\) −51.6846 −0.165656
\(313\) 376.278i 1.20217i 0.799187 + 0.601083i \(0.205264\pi\)
−0.799187 + 0.601083i \(0.794736\pi\)
\(314\) 106.214i 0.338260i
\(315\) −161.457 −0.512563
\(316\) 307.972i 0.974596i
\(317\) −519.005 −1.63724 −0.818619 0.574337i \(-0.805260\pi\)
−0.818619 + 0.574337i \(0.805260\pi\)
\(318\) 15.9471i 0.0501481i
\(319\) 220.698i 0.691844i
\(320\) 29.0689i 0.0908403i
\(321\) 75.8287i 0.236227i
\(322\) 112.170 150.624i 0.348353 0.467776i
\(323\) 29.2594 0.0905865
\(324\) −230.358 −0.710983
\(325\) 93.1772 0.286699
\(326\) −39.9063 −0.122412
\(327\) 53.3857i 0.163259i
\(328\) −73.2165 −0.223221
\(329\) 124.691i 0.379001i
\(330\) −16.8228 −0.0509782
\(331\) −468.696 −1.41600 −0.707999 0.706213i \(-0.750402\pi\)
−0.707999 + 0.706213i \(0.750402\pi\)
\(332\) 350.535i 1.05583i
\(333\) 263.053i 0.789950i
\(334\) −79.7872 −0.238884
\(335\) −78.4026 −0.234038
\(336\) 16.1758i 0.0481423i
\(337\) 212.787i 0.631416i 0.948856 + 0.315708i \(0.102242\pi\)
−0.948856 + 0.315708i \(0.897758\pi\)
\(338\) −178.279 −0.527454
\(339\) 34.4578i 0.101645i
\(340\) 37.3785 0.109937
\(341\) 940.481i 2.75801i
\(342\) 46.4356i 0.135777i
\(343\) 255.801i 0.745775i
\(344\) 400.552i 1.16440i
\(345\) 12.1706 16.3429i 0.0352771 0.0473708i
\(346\) 298.323 0.862205
\(347\) −300.161 −0.865017 −0.432508 0.901630i \(-0.642371\pi\)
−0.432508 + 0.901630i \(0.642371\pi\)
\(348\) −13.8151 −0.0396986
\(349\) −54.4534 −0.156027 −0.0780134 0.996952i \(-0.524858\pi\)
−0.0780134 + 0.996952i \(0.524858\pi\)
\(350\) 40.8265i 0.116647i
\(351\) 131.744 0.375340
\(352\) 626.619i 1.78017i
\(353\) −51.6964 −0.146449 −0.0732244 0.997315i \(-0.523329\pi\)
−0.0732244 + 0.997315i \(0.523329\pi\)
\(354\) 11.0095 0.0311004
\(355\) 157.552i 0.443809i
\(356\) 148.901i 0.418262i
\(357\) 18.0265 0.0504945
\(358\) −48.1391 −0.134467
\(359\) 277.495i 0.772968i −0.922296 0.386484i \(-0.873690\pi\)
0.922296 0.386484i \(-0.126310\pi\)
\(360\) 138.415i 0.384486i
\(361\) 333.426 0.923618
\(362\) 34.5224i 0.0953657i
\(363\) −94.9165 −0.261478
\(364\) 456.492i 1.25410i
\(365\) 0.843561i 0.00231113i
\(366\) 43.8330i 0.119762i
\(367\) 138.584i 0.377612i 0.982014 + 0.188806i \(0.0604617\pi\)
−0.982014 + 0.188806i \(0.939538\pi\)
\(368\) −92.2341 68.6868i −0.250636 0.186649i
\(369\) 92.4935 0.250660
\(370\) 66.5164 0.179774
\(371\) −328.647 −0.885842
\(372\) −58.8715 −0.158257
\(373\) 62.0247i 0.166286i −0.996538 0.0831431i \(-0.973504\pi\)
0.996538 0.0831431i \(-0.0264958\pi\)
\(374\) −105.805 −0.282901
\(375\) 4.42975i 0.0118127i
\(376\) −106.896 −0.284298
\(377\) −216.595 −0.574522
\(378\) 57.7251i 0.152712i
\(379\) 411.108i 1.08472i 0.840147 + 0.542358i \(0.182469\pi\)
−0.840147 + 0.542358i \(0.817531\pi\)
\(380\) −35.2254 −0.0926985
\(381\) 28.1639 0.0739210
\(382\) 172.912i 0.452650i
\(383\) 576.379i 1.50490i −0.658647 0.752452i \(-0.728871\pi\)
0.658647 0.752452i \(-0.271129\pi\)
\(384\) 47.1488 0.122783
\(385\) 346.695i 0.900506i
\(386\) 114.255 0.295998
\(387\) 506.013i 1.30753i
\(388\) 208.832i 0.538227i
\(389\) 233.138i 0.599327i −0.954045 0.299664i \(-0.903126\pi\)
0.954045 0.299664i \(-0.0968744\pi\)
\(390\) 16.5101i 0.0423335i
\(391\) 76.5454 102.787i 0.195768 0.262882i
\(392\) −123.706 −0.315576
\(393\) 28.5821 0.0727281
\(394\) −139.191 −0.353277
\(395\) 229.549 0.581137
\(396\) 503.746i 1.27209i
\(397\) −596.355 −1.50215 −0.751077 0.660215i \(-0.770465\pi\)
−0.751077 + 0.660215i \(0.770465\pi\)
\(398\) 245.347i 0.616449i
\(399\) −16.9882 −0.0425769
\(400\) −25.0000 −0.0625000
\(401\) 114.366i 0.285201i 0.989780 + 0.142600i \(0.0455464\pi\)
−0.989780 + 0.142600i \(0.954454\pi\)
\(402\) 13.8922i 0.0345576i
\(403\) −922.996 −2.29031
\(404\) −440.757 −1.09098
\(405\) 171.699i 0.423948i
\(406\) 94.9033i 0.233752i
\(407\) 564.850 1.38784
\(408\) 15.4539i 0.0378772i
\(409\) 149.858 0.366401 0.183201 0.983076i \(-0.441354\pi\)
0.183201 + 0.983076i \(0.441354\pi\)
\(410\) 23.3881i 0.0570443i
\(411\) 22.2399i 0.0541116i
\(412\) 360.047i 0.873900i
\(413\) 226.891i 0.549374i
\(414\) −163.125 121.480i −0.394023 0.293429i
\(415\) 261.273 0.629574
\(416\) 614.969 1.47829
\(417\) 105.427 0.252824
\(418\) 99.7103 0.238541
\(419\) 195.750i 0.467184i 0.972335 + 0.233592i \(0.0750480\pi\)
−0.972335 + 0.233592i \(0.924952\pi\)
\(420\) −21.7022 −0.0516718
\(421\) 348.840i 0.828599i −0.910141 0.414299i \(-0.864027\pi\)
0.910141 0.414299i \(-0.135973\pi\)
\(422\) 184.076 0.436199
\(423\) 135.041 0.319245
\(424\) 281.745i 0.664492i
\(425\) 27.8603i 0.0655536i
\(426\) 27.9167 0.0655321
\(427\) −903.336 −2.11554
\(428\) 574.157i 1.34149i
\(429\) 140.202i 0.326811i
\(430\) 127.952 0.297562
\(431\) 293.133i 0.680123i 0.940403 + 0.340061i \(0.110448\pi\)
−0.940403 + 0.340061i \(0.889552\pi\)
\(432\) −35.3478 −0.0818236
\(433\) 345.030i 0.796835i 0.917204 + 0.398418i \(0.130440\pi\)
−0.917204 + 0.398418i \(0.869560\pi\)
\(434\) 404.420i 0.931843i
\(435\) 10.2972i 0.0236716i
\(436\) 404.224i 0.927120i
\(437\) −72.1363 + 96.8660i −0.165072 + 0.221661i
\(438\) 0.149470 0.000341257
\(439\) −570.096 −1.29862 −0.649312 0.760522i \(-0.724943\pi\)
−0.649312 + 0.760522i \(0.724943\pi\)
\(440\) 297.216 0.675492
\(441\) 156.276 0.354368
\(442\) 103.838i 0.234927i
\(443\) 146.539 0.330788 0.165394 0.986228i \(-0.447110\pi\)
0.165394 + 0.986228i \(0.447110\pi\)
\(444\) 35.3581i 0.0796353i
\(445\) 110.984 0.249403
\(446\) 31.1064 0.0697452
\(447\) 88.0978i 0.197087i
\(448\) 106.149i 0.236940i
\(449\) 258.527 0.575783 0.287892 0.957663i \(-0.407046\pi\)
0.287892 + 0.957663i \(0.407046\pi\)
\(450\) −44.2151 −0.0982558
\(451\) 198.610i 0.440377i
\(452\) 260.906i 0.577226i
\(453\) −77.1769 −0.170368
\(454\) 192.411i 0.423813i
\(455\) −340.249 −0.747800
\(456\) 14.5637i 0.0319380i
\(457\) 712.852i 1.55985i 0.625872 + 0.779926i \(0.284743\pi\)
−0.625872 + 0.779926i \(0.715257\pi\)
\(458\) 365.836i 0.798769i
\(459\) 39.3920i 0.0858213i
\(460\) −92.1530 + 123.745i −0.200333 + 0.269011i
\(461\) 295.332 0.640634 0.320317 0.947310i \(-0.396211\pi\)
0.320317 + 0.947310i \(0.396211\pi\)
\(462\) 61.4308 0.132967
\(463\) 728.570 1.57358 0.786792 0.617218i \(-0.211740\pi\)
0.786792 + 0.617218i \(0.211740\pi\)
\(464\) 58.1137 0.125245
\(465\) 43.8802i 0.0943661i
\(466\) −235.271 −0.504873
\(467\) 754.057i 1.61468i −0.590085 0.807341i \(-0.700906\pi\)
0.590085 0.807341i \(-0.299094\pi\)
\(468\) −494.381 −1.05637
\(469\) 286.298 0.610443
\(470\) 34.1467i 0.0726526i
\(471\) 42.0827i 0.0893477i
\(472\) −194.511 −0.412099
\(473\) 1086.55 2.29715
\(474\) 40.6738i 0.0858097i
\(475\) 26.2555i 0.0552747i
\(476\) −136.493 −0.286749
\(477\) 355.925i 0.746173i
\(478\) −178.278 −0.372967
\(479\) 129.310i 0.269958i 0.990848 + 0.134979i \(0.0430967\pi\)
−0.990848 + 0.134979i \(0.956903\pi\)
\(480\) 29.2363i 0.0609090i
\(481\) 554.349i 1.15249i
\(482\) 32.9156i 0.0682896i
\(483\) −44.4427 + 59.6785i −0.0920138 + 0.123558i
\(484\) 718.685 1.48489
\(485\) −155.654 −0.320937
\(486\) −94.0494 −0.193517
\(487\) 532.507 1.09344 0.546721 0.837315i \(-0.315876\pi\)
0.546721 + 0.837315i \(0.315876\pi\)
\(488\) 774.417i 1.58692i
\(489\) 15.8112 0.0323338
\(490\) 39.5164i 0.0806457i
\(491\) 274.430 0.558922 0.279461 0.960157i \(-0.409844\pi\)
0.279461 + 0.960157i \(0.409844\pi\)
\(492\) 12.4324 0.0252692
\(493\) 64.7626i 0.131364i
\(494\) 97.8565i 0.198090i
\(495\) −375.470 −0.758526
\(496\) 247.645 0.499285
\(497\) 575.324i 1.15759i
\(498\) 46.2950i 0.0929618i
\(499\) 8.15135 0.0163354 0.00816769 0.999967i \(-0.497400\pi\)
0.00816769 + 0.999967i \(0.497400\pi\)
\(500\) 33.5410i 0.0670820i
\(501\) 31.6124 0.0630986
\(502\) 193.339i 0.385138i
\(503\) 202.082i 0.401753i 0.979617 + 0.200877i \(0.0643790\pi\)
−0.979617 + 0.200877i \(0.935621\pi\)
\(504\) 505.442i 1.00286i
\(505\) 328.521i 0.650536i
\(506\) 260.851 350.277i 0.515517 0.692246i
\(507\) 70.6359 0.139321
\(508\) −213.250 −0.419784
\(509\) −640.462 −1.25828 −0.629138 0.777294i \(-0.716592\pi\)
−0.629138 + 0.777294i \(0.716592\pi\)
\(510\) 4.93656 0.00967953
\(511\) 3.08038i 0.00602814i
\(512\) −305.000 −0.595703
\(513\) 37.1230i 0.0723644i
\(514\) 88.0147 0.171235
\(515\) −268.363 −0.521093
\(516\) 68.0153i 0.131813i
\(517\) 289.970i 0.560871i
\(518\) −242.894 −0.468907
\(519\) −118.198 −0.227742
\(520\) 291.691i 0.560944i
\(521\) 471.404i 0.904806i 0.891814 + 0.452403i \(0.149433\pi\)
−0.891814 + 0.452403i \(0.850567\pi\)
\(522\) 102.780 0.196897
\(523\) 141.188i 0.269958i 0.990848 + 0.134979i \(0.0430967\pi\)
−0.990848 + 0.134979i \(0.956903\pi\)
\(524\) −216.417 −0.413010
\(525\) 16.1758i 0.0308111i
\(526\) 98.7568i 0.187751i
\(527\) 275.979i 0.523679i
\(528\) 37.6170i 0.0712443i
\(529\) 151.570 + 506.821i 0.286521 + 0.958074i
\(530\) −90.0000 −0.169811
\(531\) 245.723 0.462755
\(532\) 128.631 0.241787
\(533\) 194.917 0.365699
\(534\) 19.6653i 0.0368265i
\(535\) 427.952 0.799910
\(536\) 245.439i 0.457909i
\(537\) 19.0731 0.0355179
\(538\) 337.903 0.628073
\(539\) 335.569i 0.622577i
\(540\) 47.4240i 0.0878223i
\(541\) 645.381 1.19294 0.596471 0.802635i \(-0.296569\pi\)
0.596471 + 0.802635i \(0.296569\pi\)
\(542\) −150.086 −0.276911
\(543\) 13.6781i 0.0251898i
\(544\) 183.878i 0.338011i
\(545\) −301.291 −0.552827
\(546\) 60.2887i 0.110419i
\(547\) 704.099 1.28720 0.643600 0.765362i \(-0.277440\pi\)
0.643600 + 0.765362i \(0.277440\pi\)
\(548\) 168.395i 0.307290i
\(549\) 978.312i 1.78199i
\(550\) 94.9423i 0.172622i
\(551\) 61.0322i 0.110766i
\(552\) −51.1615 38.1000i −0.0926839 0.0690218i
\(553\) −838.230 −1.51579
\(554\) 246.856 0.445588
\(555\) −26.3544 −0.0474853
\(556\) −798.273 −1.43574
\(557\) 377.571i 0.677865i −0.940811 0.338933i \(-0.889934\pi\)
0.940811 0.338933i \(-0.110066\pi\)
\(558\) 437.986 0.784922
\(559\) 1066.35i 1.90761i
\(560\) 91.2909 0.163020
\(561\) 41.9208 0.0747251
\(562\) 495.802i 0.882210i
\(563\) 281.177i 0.499426i −0.968320 0.249713i \(-0.919664\pi\)
0.968320 0.249713i \(-0.0803362\pi\)
\(564\) 18.1514 0.0321833
\(565\) 194.468 0.344191
\(566\) 191.002i 0.337460i
\(567\) 626.983i 1.10579i
\(568\) −493.216 −0.868339
\(569\) 717.960i 1.26179i −0.775867 0.630897i \(-0.782687\pi\)
0.775867 0.630897i \(-0.217313\pi\)
\(570\) −4.65221 −0.00816177
\(571\) 828.411i 1.45081i 0.688324 + 0.725404i \(0.258347\pi\)
−0.688324 + 0.725404i \(0.741653\pi\)
\(572\) 1061.57i 1.85590i
\(573\) 68.5093i 0.119562i
\(574\) 85.4050i 0.148789i
\(575\) 92.2341 + 68.6868i 0.160407 + 0.119455i
\(576\) −114.959 −0.199582
\(577\) 387.299 0.671228 0.335614 0.942000i \(-0.391056\pi\)
0.335614 + 0.942000i \(0.391056\pi\)
\(578\) −257.952 −0.446284
\(579\) −45.2690 −0.0781847
\(580\) 77.9678i 0.134427i
\(581\) −954.075 −1.64213
\(582\) 27.5804i 0.0473889i
\(583\) −764.271 −1.31093
\(584\) −2.64076 −0.00452185
\(585\) 368.489i 0.629897i
\(586\) 302.569i 0.516329i
\(587\) 996.553 1.69770 0.848852 0.528630i \(-0.177294\pi\)
0.848852 + 0.528630i \(0.177294\pi\)
\(588\) 21.0057 0.0357240
\(589\) 260.082i 0.441565i
\(590\) 62.1342i 0.105312i
\(591\) 55.1487 0.0933142
\(592\) 148.735i 0.251242i
\(593\) 524.228 0.884028 0.442014 0.897008i \(-0.354264\pi\)
0.442014 + 0.897008i \(0.354264\pi\)
\(594\) 134.240i 0.225993i
\(595\) 101.736i 0.170984i
\(596\) 667.056i 1.11922i
\(597\) 97.2085i 0.162828i
\(598\) −343.764 256.002i −0.574857 0.428097i
\(599\) −667.991 −1.11518 −0.557588 0.830118i \(-0.688273\pi\)
−0.557588 + 0.830118i \(0.688273\pi\)
\(600\) −13.8673 −0.0231122
\(601\) −18.8858 −0.0314240 −0.0157120 0.999877i \(-0.505001\pi\)
−0.0157120 + 0.999877i \(0.505001\pi\)
\(602\) −467.233 −0.776135
\(603\) 310.060i 0.514196i
\(604\) 584.365 0.967492
\(605\) 535.676i 0.885416i
\(606\) −58.2106 −0.0960571
\(607\) −142.493 −0.234750 −0.117375 0.993088i \(-0.537448\pi\)
−0.117375 + 0.993088i \(0.537448\pi\)
\(608\) 173.286i 0.285010i
\(609\) 37.6015i 0.0617430i
\(610\) −247.378 −0.405539
\(611\) 284.579 0.465760
\(612\) 147.821i 0.241538i
\(613\) 787.223i 1.28421i 0.766615 + 0.642106i \(0.221939\pi\)
−0.766615 + 0.642106i \(0.778061\pi\)
\(614\) −137.421 −0.223813
\(615\) 9.26659i 0.0150676i
\(616\) −1085.33 −1.76189
\(617\) 47.1461i 0.0764119i −0.999270 0.0382059i \(-0.987836\pi\)
0.999270 0.0382059i \(-0.0121643\pi\)
\(618\) 47.5512i 0.0769437i
\(619\) 738.262i 1.19267i −0.802736 0.596335i \(-0.796623\pi\)
0.802736 0.596335i \(-0.203377\pi\)
\(620\) 332.251i 0.535889i
\(621\) 130.411 + 97.1171i 0.210001 + 0.156388i
\(622\) −106.051 −0.170500
\(623\) −405.275 −0.650522
\(624\) −36.9176 −0.0591628
\(625\) 25.0000 0.0400000
\(626\) 376.278i 0.601083i
\(627\) −39.5061 −0.0630081
\(628\) 318.641i 0.507390i
\(629\) −165.752 −0.263517
\(630\) 161.457 0.256282
\(631\) 692.184i 1.09696i 0.836163 + 0.548482i \(0.184794\pi\)
−0.836163 + 0.548482i \(0.815206\pi\)
\(632\) 718.602i 1.13703i
\(633\) −72.9326 −0.115217
\(634\) 519.005 0.818619
\(635\) 158.948i 0.250311i
\(636\) 47.8413i 0.0752221i
\(637\) 329.330 0.517002
\(638\) 220.698i 0.345922i
\(639\) 623.075 0.975078
\(640\) 266.092i 0.415769i
\(641\) 582.771i 0.909159i −0.890706 0.454580i \(-0.849790\pi\)
0.890706 0.454580i \(-0.150210\pi\)
\(642\) 75.8287i 0.118113i
\(643\) 600.348i 0.933668i −0.884345 0.466834i \(-0.845395\pi\)
0.884345 0.466834i \(-0.154605\pi\)
\(644\) 336.509 451.872i 0.522530 0.701664i
\(645\) −50.6956 −0.0785978
\(646\) −29.2594 −0.0452932
\(647\) −595.814 −0.920886 −0.460443 0.887689i \(-0.652310\pi\)
−0.460443 + 0.887689i \(0.652310\pi\)
\(648\) 537.503 0.829480
\(649\) 527.637i 0.813000i
\(650\) −93.1772 −0.143350
\(651\) 160.235i 0.246136i
\(652\) −119.719 −0.183618
\(653\) 148.378 0.227225 0.113612 0.993525i \(-0.463758\pi\)
0.113612 + 0.993525i \(0.463758\pi\)
\(654\) 53.3857i 0.0816295i
\(655\) 161.308i 0.246272i
\(656\) −52.2975 −0.0797218
\(657\) 3.33605 0.00507769
\(658\) 124.691i 0.189501i
\(659\) 94.1732i 0.142903i 0.997444 + 0.0714516i \(0.0227632\pi\)
−0.997444 + 0.0714516i \(0.977237\pi\)
\(660\) −50.4685 −0.0764674
\(661\) 705.908i 1.06794i −0.845504 0.533970i \(-0.820700\pi\)
0.845504 0.533970i \(-0.179300\pi\)
\(662\) 468.696 0.707999
\(663\) 41.1414i 0.0620534i
\(664\) 817.915i 1.23180i
\(665\) 95.8755i 0.144174i
\(666\) 263.053i 0.394975i
\(667\) −214.403 159.666i −0.321443 0.239379i
\(668\) −239.362 −0.358326
\(669\) −12.3246 −0.0184224
\(670\) 78.4026 0.117019
\(671\) −2100.71 −3.13072
\(672\) 106.760i 0.158870i
\(673\) 617.516 0.917558 0.458779 0.888550i \(-0.348287\pi\)
0.458779 + 0.888550i \(0.348287\pi\)
\(674\) 212.787i 0.315708i
\(675\) 35.3478 0.0523671
\(676\) −534.838 −0.791181
\(677\) 923.117i 1.36354i 0.731566 + 0.681771i \(0.238790\pi\)
−0.731566 + 0.681771i \(0.761210\pi\)
\(678\) 34.4578i 0.0508227i
\(679\) 568.393 0.837103
\(680\) −87.2165 −0.128260
\(681\) 76.2350i 0.111946i
\(682\) 940.481i 1.37900i
\(683\) 440.495 0.644941 0.322471 0.946580i \(-0.395487\pi\)
0.322471 + 0.946580i \(0.395487\pi\)
\(684\) 139.307i 0.203665i
\(685\) −125.514 −0.183233
\(686\) 255.801i 0.372887i
\(687\) 144.947i 0.210986i
\(688\) 286.109i 0.415856i
\(689\) 750.062i 1.08862i
\(690\) −12.1706 + 16.3429i −0.0176386 + 0.0236854i
\(691\) −934.136 −1.35186 −0.675930 0.736965i \(-0.736258\pi\)
−0.675930 + 0.736965i \(0.736258\pi\)
\(692\) 894.969 1.29331
\(693\) 1371.08 1.97847
\(694\) 300.161 0.432508
\(695\) 594.997i 0.856111i
\(696\) 32.2352 0.0463150
\(697\) 58.2809i 0.0836168i
\(698\) 54.4534 0.0780134
\(699\) 93.2164 0.133357
\(700\) 122.480i 0.174971i
\(701\) 462.313i 0.659505i −0.944067 0.329753i \(-0.893035\pi\)
0.944067 0.329753i \(-0.106965\pi\)
\(702\) −131.744 −0.187670
\(703\) 156.205 0.222197
\(704\) 246.850i 0.350639i
\(705\) 13.5292i 0.0191904i
\(706\) 51.6964 0.0732244
\(707\) 1199.64i 1.69680i
\(708\) 33.0286 0.0466506
\(709\) 959.421i 1.35320i 0.736349 + 0.676601i \(0.236548\pi\)
−0.736349 + 0.676601i \(0.763452\pi\)
\(710\) 157.552i 0.221905i
\(711\) 907.802i 1.27680i
\(712\) 347.436i 0.487972i
\(713\) −913.653 680.399i −1.28142 0.954276i
\(714\) −18.0265 −0.0252472
\(715\) −791.251 −1.10665
\(716\) −144.417 −0.201700
\(717\) 70.6354 0.0985152
\(718\) 277.495i 0.386484i
\(719\) 195.959 0.272543 0.136272 0.990672i \(-0.456488\pi\)
0.136272 + 0.990672i \(0.456488\pi\)
\(720\) 98.8680i 0.137317i
\(721\) 979.964 1.35917
\(722\) −333.426 −0.461809
\(723\) 13.0414i 0.0180379i
\(724\) 103.567i 0.143049i
\(725\) −58.1137 −0.0801569
\(726\) 94.9165 0.130739
\(727\) 959.462i 1.31975i −0.751373 0.659877i \(-0.770608\pi\)
0.751373 0.659877i \(-0.229392\pi\)
\(728\) 1065.15i 1.46312i
\(729\) −653.812 −0.896862
\(730\) 0.843561i 0.00115556i
\(731\) −318.843 −0.436174
\(732\) 131.499i 0.179643i
\(733\) 1454.43i 1.98421i −0.125412 0.992105i \(-0.540025\pi\)
0.125412 0.992105i \(-0.459975\pi\)
\(734\) 138.584i 0.188806i
\(735\) 15.6567i 0.0213017i
\(736\) 608.745 + 453.333i 0.827099 + 0.615942i
\(737\) 665.787 0.903375
\(738\) −92.4935 −0.125330
\(739\) −430.714 −0.582834 −0.291417 0.956596i \(-0.594127\pi\)
−0.291417 + 0.956596i \(0.594127\pi\)
\(740\) 199.549 0.269661
\(741\) 38.7716i 0.0523234i
\(742\) 328.647 0.442921
\(743\) 331.646i 0.446361i −0.974777 0.223180i \(-0.928356\pi\)
0.974777 0.223180i \(-0.0716439\pi\)
\(744\) 137.367 0.184633
\(745\) 497.194 0.667375
\(746\) 62.0247i 0.0831431i
\(747\) 1033.26i 1.38322i
\(748\) −317.414 −0.424351
\(749\) −1562.72 −2.08641
\(750\) 4.42975i 0.00590633i
\(751\) 55.4622i 0.0738511i −0.999318 0.0369256i \(-0.988244\pi\)
0.999318 0.0369256i \(-0.0117564\pi\)
\(752\) −76.3544 −0.101535
\(753\) 76.6027i 0.101730i
\(754\) 216.595 0.287261
\(755\) 435.560i 0.576901i
\(756\) 173.175i 0.229068i
\(757\) 253.679i 0.335111i −0.985863 0.167556i \(-0.946413\pi\)
0.985863 0.167556i \(-0.0535874\pi\)
\(758\) 411.108i 0.542358i
\(759\) −103.352 + 138.783i −0.136168 + 0.182849i
\(760\) 82.1927 0.108148
\(761\) 474.238 0.623178 0.311589 0.950217i \(-0.399139\pi\)
0.311589 + 0.950217i \(0.399139\pi\)
\(762\) −28.1639 −0.0369605
\(763\) 1100.21 1.44195
\(764\) 518.736i 0.678974i
\(765\) 110.180 0.144026
\(766\) 576.379i 0.752452i
\(767\) 517.827 0.675133
\(768\) −67.7517 −0.0882183
\(769\) 859.116i 1.11719i 0.829442 + 0.558593i \(0.188659\pi\)
−0.829442 + 0.558593i \(0.811341\pi\)
\(770\) 346.695i 0.450253i
\(771\) −34.8722 −0.0452298
\(772\) 342.766 0.443997
\(773\) 1350.61i 1.74724i −0.486612 0.873618i \(-0.661768\pi\)
0.486612 0.873618i \(-0.338232\pi\)
\(774\) 506.013i 0.653764i
\(775\) −247.645 −0.319542
\(776\) 487.275i 0.627932i
\(777\) 96.2366 0.123857
\(778\) 233.138i 0.299664i
\(779\) 54.9239i 0.0705056i
\(780\) 49.5302i 0.0635002i
\(781\) 1337.92i 1.71308i
\(782\) −76.5454 + 102.787i −0.0978841 + 0.131441i
\(783\) −82.1677 −0.104940
\(784\) −88.3613 −0.112706
\(785\) 237.501 0.302549
\(786\) −28.5821 −0.0363640
\(787\) 1010.62i 1.28414i −0.766644 0.642072i \(-0.778075\pi\)
0.766644 0.642072i \(-0.221925\pi\)
\(788\) −417.573 −0.529915
\(789\) 39.1283i 0.0495923i
\(790\) −229.549 −0.290568
\(791\) −710.127 −0.897758
\(792\) 1175.41i 1.48410i
\(793\) 2061.66i 2.59982i
\(794\) 596.355 0.751077
\(795\) 35.6588 0.0448538
\(796\) 736.040i 0.924674i
\(797\) 1079.08i 1.35393i −0.736017 0.676963i \(-0.763296\pi\)
0.736017 0.676963i \(-0.236704\pi\)
\(798\) 16.9882 0.0212884
\(799\) 85.0902i 0.106496i
\(800\) 165.000 0.206250
\(801\) 438.912i 0.547955i
\(802\) 114.366i 0.142600i
\(803\) 7.16343i 0.00892084i
\(804\) 41.6765i 0.0518364i
\(805\) −336.805 250.819i −0.418392 0.311577i
\(806\) 922.996 1.14516
\(807\) −133.880 −0.165899
\(808\) 1028.43 1.27281
\(809\) 577.335 0.713641 0.356820 0.934173i \(-0.383861\pi\)
0.356820 + 0.934173i \(0.383861\pi\)
\(810\) 171.699i 0.211974i
\(811\) −655.395 −0.808132 −0.404066 0.914730i \(-0.632403\pi\)
−0.404066 + 0.914730i \(0.632403\pi\)
\(812\) 284.710i 0.350628i
\(813\) 59.4652 0.0731429
\(814\) −564.850 −0.693919
\(815\) 89.2332i 0.109489i
\(816\) 11.0385i 0.0135276i
\(817\) 300.477 0.367781
\(818\) −149.858 −0.183201
\(819\) 1345.59i 1.64297i
\(820\) 70.1644i 0.0855664i
\(821\) 1117.23 1.36082 0.680410 0.732832i \(-0.261802\pi\)
0.680410 + 0.732832i \(0.261802\pi\)
\(822\) 22.2399i 0.0270558i
\(823\) 331.810 0.403171 0.201586 0.979471i \(-0.435391\pi\)
0.201586 + 0.979471i \(0.435391\pi\)
\(824\) 840.109i 1.01955i
\(825\) 37.6170i 0.0455963i
\(826\) 226.891i 0.274687i
\(827\) 158.794i 0.192013i 0.995381 + 0.0960063i \(0.0306069\pi\)
−0.995381 + 0.0960063i \(0.969393\pi\)
\(828\) −489.376 364.439i −0.591034 0.440144i
\(829\) 941.980 1.13629 0.568143 0.822930i \(-0.307662\pi\)
0.568143 + 0.822930i \(0.307662\pi\)
\(830\) −261.273 −0.314787
\(831\) −97.8065 −0.117697
\(832\) −242.261 −0.291179
\(833\) 98.4708i 0.118212i
\(834\) −105.427 −0.126412
\(835\) 178.410i 0.213664i
\(836\) 299.131 0.357812
\(837\) −350.149 −0.418338
\(838\) 195.750i 0.233592i
\(839\) 817.388i 0.974241i 0.873335 + 0.487120i \(0.161953\pi\)
−0.873335 + 0.487120i \(0.838047\pi\)
\(840\) 50.6384 0.0602838
\(841\) −705.912 −0.839372
\(842\) 348.840i 0.414299i
\(843\) 196.441i 0.233026i
\(844\) 552.228 0.654299
\(845\) 398.645i 0.471769i
\(846\) −135.041 −0.159622
\(847\) 1956.10i 2.30944i
\(848\) 201.246i 0.237319i
\(849\) 75.6768i 0.0891364i
\(850\) 27.8603i 0.0327768i
\(851\) 408.646 548.738i 0.480195 0.644815i
\(852\) 83.7500 0.0982981
\(853\) −251.633 −0.294998 −0.147499 0.989062i \(-0.547122\pi\)
−0.147499 + 0.989062i \(0.547122\pi\)
\(854\) 903.336 1.05777
\(855\) −103.833 −0.121442
\(856\) 1339.70i 1.56507i
\(857\) −264.475 −0.308605 −0.154303 0.988024i \(-0.549313\pi\)
−0.154303 + 0.988024i \(0.549313\pi\)
\(858\) 140.202i 0.163405i
\(859\) 368.069 0.428485 0.214243 0.976780i \(-0.431272\pi\)
0.214243 + 0.976780i \(0.431272\pi\)
\(860\) 383.855 0.446343
\(861\) 33.8382i 0.0393011i
\(862\) 293.133i 0.340061i
\(863\) −1521.94 −1.76355 −0.881775 0.471669i \(-0.843652\pi\)
−0.881775 + 0.471669i \(0.843652\pi\)
\(864\) 233.295 0.270018
\(865\) 667.070i 0.771179i
\(866\) 345.030i 0.398418i
\(867\) 102.203 0.117881
\(868\) 1213.26i 1.39777i
\(869\) −1949.31 −2.24316
\(870\) 10.2972i 0.0118358i
\(871\) 653.409i 0.750183i
\(872\) 943.190i 1.08164i
\(873\) 615.569i 0.705119i
\(874\) 72.1363 96.8660i 0.0825358 0.110831i
\(875\) −91.2909 −0.104332
\(876\) 0.448411 0.000511885
\(877\) −1696.01 −1.93388 −0.966938 0.255011i \(-0.917921\pi\)
−0.966938 + 0.255011i \(0.917921\pi\)
\(878\) 570.096 0.649312
\(879\) 119.880i 0.136383i
\(880\) 212.297 0.241247
\(881\) 258.150i 0.293020i −0.989209 0.146510i \(-0.953196\pi\)
0.989209 0.146510i \(-0.0468040\pi\)
\(882\) −156.276 −0.177184
\(883\) 1286.20 1.45663 0.728315 0.685243i \(-0.240304\pi\)
0.728315 + 0.685243i \(0.240304\pi\)
\(884\) 311.513i 0.352390i
\(885\) 24.6181i 0.0278171i
\(886\) −146.539 −0.165394
\(887\) 828.723 0.934299 0.467150 0.884178i \(-0.345281\pi\)
0.467150 + 0.884178i \(0.345281\pi\)
\(888\) 82.5022i 0.0929079i
\(889\) 580.419i 0.652889i
\(890\) −110.984 −0.124702
\(891\) 1458.05i 1.63642i
\(892\) 93.3191 0.104618
\(893\) 80.1889i 0.0897972i
\(894\) 88.0978i 0.0985434i
\(895\) 107.642i 0.120271i
\(896\) 971.672i 1.08445i
\(897\) 136.202 + 101.430i 0.151842 + 0.113077i
\(898\) −258.527 −0.287892
\(899\) 575.664 0.640338
\(900\) −132.645 −0.147384
\(901\) 224.271 0.248913
\(902\) 198.610i 0.220188i
\(903\) 185.122 0.205008
\(904\) 608.781i 0.673431i
\(905\) −77.1944 −0.0852977
\(906\) 77.1769 0.0851842
\(907\) 973.777i 1.07362i −0.843702 0.536812i \(-0.819629\pi\)
0.843702 0.536812i \(-0.180371\pi\)
\(908\) 577.234i 0.635720i
\(909\) −1299.21 −1.42927
\(910\) 340.249 0.373900
\(911\) 564.279i 0.619406i −0.950833 0.309703i \(-0.899770\pi\)
0.950833 0.309703i \(-0.100230\pi\)
\(912\) 10.4027i 0.0114064i
\(913\) −2218.71 −2.43013
\(914\) 712.852i 0.779926i
\(915\) 98.0135 0.107119
\(916\) 1097.51i 1.19815i
\(917\) 589.038i 0.642353i
\(918\) 39.3920i 0.0429107i
\(919\) 119.220i 0.129728i 0.997894 + 0.0648638i \(0.0206613\pi\)
−0.997894 + 0.0648638i \(0.979339\pi\)
\(920\) 215.024 288.738i 0.233721 0.313846i
\(921\) 54.4474 0.0591177
\(922\) −295.332 −0.320317
\(923\) 1313.04 1.42258
\(924\) 184.292 0.199451
\(925\) 148.735i 0.160795i
\(926\) −728.570 −0.786792
\(927\) 1061.30i 1.14488i
\(928\) −383.551 −0.413309
\(929\) −1117.77 −1.20319 −0.601596 0.798800i \(-0.705468\pi\)
−0.601596 + 0.798800i \(0.705468\pi\)
\(930\) 43.8802i 0.0471831i
\(931\) 92.7988i 0.0996765i
\(932\) −705.813 −0.757310
\(933\) 42.0182 0.0450356
\(934\) 754.057i 0.807341i
\(935\) 236.587i 0.253034i
\(936\) 1153.55 1.23243
\(937\) 226.429i 0.241653i −0.992674 0.120827i \(-0.961445\pi\)
0.992674 0.120827i \(-0.0385545\pi\)
\(938\) −286.298 −0.305222
\(939\) 149.085i 0.158769i
\(940\) 102.440i 0.108979i
\(941\) 523.829i 0.556672i −0.960484 0.278336i \(-0.910217\pi\)
0.960484 0.278336i \(-0.0897829\pi\)
\(942\) 42.0827i 0.0446738i
\(943\) 192.944 + 143.686i 0.204607 + 0.152371i
\(944\) −138.936 −0.147178
\(945\) −129.077 −0.136590
\(946\) −1086.55 −1.14858
\(947\) −978.726 −1.03350 −0.516751 0.856136i \(-0.672859\pi\)
−0.516751 + 0.856136i \(0.672859\pi\)
\(948\) 122.021i 0.128714i
\(949\) 7.03025 0.00740806
\(950\) 26.2555i 0.0276374i
\(951\) −205.634 −0.216229
\(952\) 318.483 0.334541
\(953\) 1859.65i 1.95136i 0.219192 + 0.975682i \(0.429658\pi\)
−0.219192 + 0.975682i \(0.570342\pi\)
\(954\) 355.925i 0.373087i
\(955\) −386.643 −0.404862
\(956\) −534.835 −0.559451
\(957\) 87.4425i 0.0913715i
\(958\) 129.310i 0.134979i
\(959\) 458.333 0.477928
\(960\) 11.5173i 0.0119972i
\(961\) 1492.13 1.55268
\(962\) 554.349i 0.576246i
\(963\) 1692.43i 1.75745i
\(964\) 98.7467i 0.102434i
\(965\) 255.483i 0.264749i
\(966\) 44.4427 59.6785i 0.0460069 0.0617790i
\(967\) 598.954 0.619394 0.309697 0.950835i \(-0.399772\pi\)
0.309697 + 0.950835i \(0.399772\pi\)
\(968\) −1676.93 −1.73237
\(969\) 11.5928 0.0119637
\(970\) 155.654 0.160468
\(971\) 678.046i 0.698296i 0.937068 + 0.349148i \(0.113529\pi\)
−0.937068 + 0.349148i \(0.886471\pi\)
\(972\) −282.148 −0.290276
\(973\) 2172.71i 2.23300i
\(974\) −532.507 −0.546721
\(975\) 36.9176 0.0378642
\(976\) 553.155i 0.566757i
\(977\) 302.051i 0.309161i 0.987980 + 0.154581i \(0.0494026\pi\)
−0.987980 + 0.154581i \(0.950597\pi\)
\(978\) −15.8112 −0.0161669
\(979\) −942.469 −0.962685
\(980\) 118.549i 0.120968i
\(981\) 1191.52i 1.21460i
\(982\) −274.430 −0.279461
\(983\) 19.2099i 0.0195421i −0.999952 0.00977105i \(-0.996890\pi\)
0.999952 0.00977105i \(-0.00311027\pi\)
\(984\) −29.0090 −0.0294807
\(985\) 311.241i 0.315980i
\(986\) 64.7626i 0.0656822i
\(987\) 49.4038i 0.0500545i
\(988\) 293.570i 0.297135i
\(989\) 786.076 1055.56i 0.794819 1.06730i
\(990\) 375.470 0.379263
\(991\) 1291.94 1.30367 0.651837 0.758359i \(-0.273999\pi\)
0.651837 + 0.758359i \(0.273999\pi\)
\(992\) −1634.46 −1.64764
\(993\) −185.701 −0.187010
\(994\) 575.324i 0.578796i
\(995\) 548.612 0.551369
\(996\) 138.885i 0.139443i
\(997\) −1343.41 −1.34745 −0.673727 0.738981i \(-0.735307\pi\)
−0.673727 + 0.738981i \(0.735307\pi\)
\(998\) −8.15135 −0.00816769
\(999\) 210.298i 0.210509i
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 115.3.d.a.91.4 yes 6
3.2 odd 2 1035.3.g.a.91.1 6
4.3 odd 2 1840.3.k.a.321.4 6
5.2 odd 4 575.3.c.c.574.4 12
5.3 odd 4 575.3.c.c.574.9 12
5.4 even 2 575.3.d.e.551.4 6
23.22 odd 2 inner 115.3.d.a.91.3 6
69.68 even 2 1035.3.g.a.91.6 6
92.91 even 2 1840.3.k.a.321.3 6
115.22 even 4 575.3.c.c.574.3 12
115.68 even 4 575.3.c.c.574.10 12
115.114 odd 2 575.3.d.e.551.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.3.d.a.91.3 6 23.22 odd 2 inner
115.3.d.a.91.4 yes 6 1.1 even 1 trivial
575.3.c.c.574.3 12 115.22 even 4
575.3.c.c.574.4 12 5.2 odd 4
575.3.c.c.574.9 12 5.3 odd 4
575.3.c.c.574.10 12 115.68 even 4
575.3.d.e.551.3 6 115.114 odd 2
575.3.d.e.551.4 6 5.4 even 2
1035.3.g.a.91.1 6 3.2 odd 2
1035.3.g.a.91.6 6 69.68 even 2
1840.3.k.a.321.3 6 92.91 even 2
1840.3.k.a.321.4 6 4.3 odd 2