Properties

Label 798.4.a.k.1.3
Level $798$
Weight $4$
Character 798.1
Self dual yes
Analytic conductor $47.084$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,4,Mod(1,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, 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("798.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 798.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: \(47.0835241846\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 310x^{2} - 1444x - 1712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.35201\) of defining polynomial
Character \(\chi\) \(=\) 798.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +4.70401 q^{10} -65.1379 q^{11} -12.0000 q^{12} -31.4764 q^{13} +14.0000 q^{14} +7.05602 q^{15} +16.0000 q^{16} -43.0831 q^{17} -18.0000 q^{18} -19.0000 q^{19} -9.40803 q^{20} +21.0000 q^{21} +130.276 q^{22} +92.8218 q^{23} +24.0000 q^{24} -119.468 q^{25} +62.9528 q^{26} -27.0000 q^{27} -28.0000 q^{28} -95.7140 q^{29} -14.1120 q^{30} +161.087 q^{31} -32.0000 q^{32} +195.414 q^{33} +86.1662 q^{34} +16.4641 q^{35} +36.0000 q^{36} +5.04480 q^{37} +38.0000 q^{38} +94.4291 q^{39} +18.8161 q^{40} -434.908 q^{41} -42.0000 q^{42} +54.7476 q^{43} -260.552 q^{44} -21.1681 q^{45} -185.644 q^{46} -220.621 q^{47} -48.0000 q^{48} +49.0000 q^{49} +238.936 q^{50} +129.249 q^{51} -125.906 q^{52} +175.142 q^{53} +54.0000 q^{54} +153.205 q^{55} +56.0000 q^{56} +57.0000 q^{57} +191.428 q^{58} -383.026 q^{59} +28.2241 q^{60} +431.625 q^{61} -322.174 q^{62} -63.0000 q^{63} +64.0000 q^{64} +74.0327 q^{65} -390.828 q^{66} -989.711 q^{67} -172.332 q^{68} -278.465 q^{69} -32.9281 q^{70} -456.366 q^{71} -72.0000 q^{72} +92.3693 q^{73} -10.0896 q^{74} +358.404 q^{75} -76.0000 q^{76} +455.965 q^{77} -188.858 q^{78} +620.873 q^{79} -37.6321 q^{80} +81.0000 q^{81} +869.816 q^{82} +519.728 q^{83} +84.0000 q^{84} +101.332 q^{85} -109.495 q^{86} +287.142 q^{87} +521.103 q^{88} +1049.89 q^{89} +42.3361 q^{90} +220.335 q^{91} +371.287 q^{92} -483.261 q^{93} +441.241 q^{94} +44.6881 q^{95} +96.0000 q^{96} -166.305 q^{97} -98.0000 q^{98} -586.241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 66 q^{11} - 48 q^{12} + 10 q^{13} + 56 q^{14} + 64 q^{16} - 8 q^{17} - 72 q^{18} - 76 q^{19} + 84 q^{21} + 132 q^{22} - 250 q^{23} + 96 q^{24} + 120 q^{25} - 20 q^{26} - 108 q^{27} - 112 q^{28} - 66 q^{29} - 52 q^{31} - 128 q^{32} + 198 q^{33} + 16 q^{34} + 144 q^{36} + 288 q^{37} + 152 q^{38} - 30 q^{39} - 196 q^{41} - 168 q^{42} - 280 q^{43} - 264 q^{44} + 500 q^{46} + 498 q^{47} - 192 q^{48} + 196 q^{49} - 240 q^{50} + 24 q^{51} + 40 q^{52} - 250 q^{53} + 216 q^{54} - 1432 q^{55} + 224 q^{56} + 228 q^{57} + 132 q^{58} - 872 q^{59} + 600 q^{61} + 104 q^{62} - 252 q^{63} + 256 q^{64} - 908 q^{65} - 396 q^{66} + 1174 q^{67} - 32 q^{68} + 750 q^{69} - 1766 q^{71} - 288 q^{72} + 556 q^{73} - 576 q^{74} - 360 q^{75} - 304 q^{76} + 462 q^{77} + 60 q^{78} + 26 q^{79} + 324 q^{81} + 392 q^{82} + 674 q^{83} + 336 q^{84} + 2052 q^{85} + 560 q^{86} + 198 q^{87} + 528 q^{88} - 324 q^{89} - 70 q^{91} - 1000 q^{92} + 156 q^{93} - 996 q^{94} + 384 q^{96} + 2902 q^{97} - 392 q^{98} - 594 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) −2.35201 −0.210370 −0.105185 0.994453i \(-0.533543\pi\)
−0.105185 + 0.994453i \(0.533543\pi\)
\(6\) 6.00000 0.408248
\(7\) −7.00000 −0.377964
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 4.70401 0.148754
\(11\) −65.1379 −1.78544 −0.892719 0.450613i \(-0.851205\pi\)
−0.892719 + 0.450613i \(0.851205\pi\)
\(12\) −12.0000 −0.288675
\(13\) −31.4764 −0.671537 −0.335768 0.941945i \(-0.608996\pi\)
−0.335768 + 0.941945i \(0.608996\pi\)
\(14\) 14.0000 0.267261
\(15\) 7.05602 0.121457
\(16\) 16.0000 0.250000
\(17\) −43.0831 −0.614658 −0.307329 0.951603i \(-0.599435\pi\)
−0.307329 + 0.951603i \(0.599435\pi\)
\(18\) −18.0000 −0.235702
\(19\) −19.0000 −0.229416
\(20\) −9.40803 −0.105185
\(21\) 21.0000 0.218218
\(22\) 130.276 1.26250
\(23\) 92.8218 0.841508 0.420754 0.907175i \(-0.361766\pi\)
0.420754 + 0.907175i \(0.361766\pi\)
\(24\) 24.0000 0.204124
\(25\) −119.468 −0.955744
\(26\) 62.9528 0.474848
\(27\) −27.0000 −0.192450
\(28\) −28.0000 −0.188982
\(29\) −95.7140 −0.612884 −0.306442 0.951889i \(-0.599139\pi\)
−0.306442 + 0.951889i \(0.599139\pi\)
\(30\) −14.1120 −0.0858832
\(31\) 161.087 0.933294 0.466647 0.884444i \(-0.345462\pi\)
0.466647 + 0.884444i \(0.345462\pi\)
\(32\) −32.0000 −0.176777
\(33\) 195.414 1.03082
\(34\) 86.1662 0.434629
\(35\) 16.4641 0.0795124
\(36\) 36.0000 0.166667
\(37\) 5.04480 0.0224151 0.0112076 0.999937i \(-0.496432\pi\)
0.0112076 + 0.999937i \(0.496432\pi\)
\(38\) 38.0000 0.162221
\(39\) 94.4291 0.387712
\(40\) 18.8161 0.0743770
\(41\) −434.908 −1.65661 −0.828307 0.560274i \(-0.810696\pi\)
−0.828307 + 0.560274i \(0.810696\pi\)
\(42\) −42.0000 −0.154303
\(43\) 54.7476 0.194161 0.0970805 0.995277i \(-0.469050\pi\)
0.0970805 + 0.995277i \(0.469050\pi\)
\(44\) −260.552 −0.892719
\(45\) −21.1681 −0.0701233
\(46\) −185.644 −0.595036
\(47\) −220.621 −0.684698 −0.342349 0.939573i \(-0.611223\pi\)
−0.342349 + 0.939573i \(0.611223\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) 238.936 0.675813
\(51\) 129.249 0.354873
\(52\) −125.906 −0.335768
\(53\) 175.142 0.453917 0.226959 0.973904i \(-0.427122\pi\)
0.226959 + 0.973904i \(0.427122\pi\)
\(54\) 54.0000 0.136083
\(55\) 153.205 0.375603
\(56\) 56.0000 0.133631
\(57\) 57.0000 0.132453
\(58\) 191.428 0.433375
\(59\) −383.026 −0.845181 −0.422591 0.906321i \(-0.638879\pi\)
−0.422591 + 0.906321i \(0.638879\pi\)
\(60\) 28.2241 0.0607286
\(61\) 431.625 0.905965 0.452983 0.891519i \(-0.350360\pi\)
0.452983 + 0.891519i \(0.350360\pi\)
\(62\) −322.174 −0.659938
\(63\) −63.0000 −0.125988
\(64\) 64.0000 0.125000
\(65\) 74.0327 0.141271
\(66\) −390.828 −0.728902
\(67\) −989.711 −1.80466 −0.902332 0.431042i \(-0.858146\pi\)
−0.902332 + 0.431042i \(0.858146\pi\)
\(68\) −172.332 −0.307329
\(69\) −278.465 −0.485845
\(70\) −32.9281 −0.0562237
\(71\) −456.366 −0.762827 −0.381413 0.924405i \(-0.624563\pi\)
−0.381413 + 0.924405i \(0.624563\pi\)
\(72\) −72.0000 −0.117851
\(73\) 92.3693 0.148096 0.0740480 0.997255i \(-0.476408\pi\)
0.0740480 + 0.997255i \(0.476408\pi\)
\(74\) −10.0896 −0.0158499
\(75\) 358.404 0.551799
\(76\) −76.0000 −0.114708
\(77\) 455.965 0.674832
\(78\) −188.858 −0.274154
\(79\) 620.873 0.884224 0.442112 0.896960i \(-0.354229\pi\)
0.442112 + 0.896960i \(0.354229\pi\)
\(80\) −37.6321 −0.0525925
\(81\) 81.0000 0.111111
\(82\) 869.816 1.17140
\(83\) 519.728 0.687320 0.343660 0.939094i \(-0.388333\pi\)
0.343660 + 0.939094i \(0.388333\pi\)
\(84\) 84.0000 0.109109
\(85\) 101.332 0.129306
\(86\) −109.495 −0.137293
\(87\) 287.142 0.353849
\(88\) 521.103 0.631248
\(89\) 1049.89 1.25042 0.625212 0.780455i \(-0.285013\pi\)
0.625212 + 0.780455i \(0.285013\pi\)
\(90\) 42.3361 0.0495847
\(91\) 220.335 0.253817
\(92\) 371.287 0.420754
\(93\) −483.261 −0.538837
\(94\) 441.241 0.484155
\(95\) 44.6881 0.0482622
\(96\) 96.0000 0.102062
\(97\) −166.305 −0.174079 −0.0870396 0.996205i \(-0.527741\pi\)
−0.0870396 + 0.996205i \(0.527741\pi\)
\(98\) −98.0000 −0.101015
\(99\) −586.241 −0.595146
\(100\) −477.872 −0.477872
\(101\) 1504.58 1.48229 0.741147 0.671343i \(-0.234282\pi\)
0.741147 + 0.671343i \(0.234282\pi\)
\(102\) −258.499 −0.250933
\(103\) −1145.68 −1.09599 −0.547994 0.836482i \(-0.684608\pi\)
−0.547994 + 0.836482i \(0.684608\pi\)
\(104\) 251.811 0.237424
\(105\) −49.3922 −0.0459065
\(106\) −350.284 −0.320968
\(107\) −804.550 −0.726905 −0.363452 0.931613i \(-0.618402\pi\)
−0.363452 + 0.931613i \(0.618402\pi\)
\(108\) −108.000 −0.0962250
\(109\) 2146.46 1.88618 0.943089 0.332541i \(-0.107906\pi\)
0.943089 + 0.332541i \(0.107906\pi\)
\(110\) −306.410 −0.265591
\(111\) −15.1344 −0.0129414
\(112\) −112.000 −0.0944911
\(113\) −456.239 −0.379818 −0.189909 0.981802i \(-0.560819\pi\)
−0.189909 + 0.981802i \(0.560819\pi\)
\(114\) −114.000 −0.0936586
\(115\) −218.318 −0.177028
\(116\) −382.856 −0.306442
\(117\) −283.287 −0.223846
\(118\) 766.051 0.597633
\(119\) 301.582 0.232319
\(120\) −56.4482 −0.0429416
\(121\) 2911.95 2.18779
\(122\) −863.250 −0.640614
\(123\) 1304.72 0.956447
\(124\) 644.349 0.466647
\(125\) 574.991 0.411430
\(126\) 126.000 0.0890871
\(127\) 1257.44 0.878582 0.439291 0.898345i \(-0.355230\pi\)
0.439291 + 0.898345i \(0.355230\pi\)
\(128\) −128.000 −0.0883883
\(129\) −164.243 −0.112099
\(130\) −148.065 −0.0998938
\(131\) −2329.46 −1.55363 −0.776817 0.629726i \(-0.783167\pi\)
−0.776817 + 0.629726i \(0.783167\pi\)
\(132\) 781.655 0.515412
\(133\) 133.000 0.0867110
\(134\) 1979.42 1.27609
\(135\) 63.5042 0.0404857
\(136\) 344.665 0.217315
\(137\) 1728.59 1.07798 0.538990 0.842312i \(-0.318806\pi\)
0.538990 + 0.842312i \(0.318806\pi\)
\(138\) 556.931 0.343544
\(139\) −691.205 −0.421779 −0.210889 0.977510i \(-0.567636\pi\)
−0.210889 + 0.977510i \(0.567636\pi\)
\(140\) 65.8562 0.0397562
\(141\) 661.862 0.395311
\(142\) 912.733 0.539400
\(143\) 2050.31 1.19899
\(144\) 144.000 0.0833333
\(145\) 225.120 0.128932
\(146\) −184.739 −0.104720
\(147\) −147.000 −0.0824786
\(148\) 20.1792 0.0112076
\(149\) −2337.76 −1.28535 −0.642674 0.766140i \(-0.722175\pi\)
−0.642674 + 0.766140i \(0.722175\pi\)
\(150\) −716.808 −0.390181
\(151\) 2681.77 1.44529 0.722646 0.691218i \(-0.242926\pi\)
0.722646 + 0.691218i \(0.242926\pi\)
\(152\) 152.000 0.0811107
\(153\) −387.748 −0.204886
\(154\) −911.931 −0.477178
\(155\) −378.878 −0.196337
\(156\) 377.717 0.193856
\(157\) 762.000 0.387352 0.193676 0.981066i \(-0.437959\pi\)
0.193676 + 0.981066i \(0.437959\pi\)
\(158\) −1241.75 −0.625241
\(159\) −525.426 −0.262069
\(160\) 75.2642 0.0371885
\(161\) −649.752 −0.318060
\(162\) −162.000 −0.0785674
\(163\) 3452.06 1.65881 0.829406 0.558647i \(-0.188679\pi\)
0.829406 + 0.558647i \(0.188679\pi\)
\(164\) −1739.63 −0.828307
\(165\) −459.615 −0.216854
\(166\) −1039.46 −0.486009
\(167\) 3589.50 1.66326 0.831628 0.555333i \(-0.187409\pi\)
0.831628 + 0.555333i \(0.187409\pi\)
\(168\) −168.000 −0.0771517
\(169\) −1206.24 −0.549039
\(170\) −202.664 −0.0914329
\(171\) −171.000 −0.0764719
\(172\) 218.990 0.0970805
\(173\) 1795.76 0.789187 0.394593 0.918856i \(-0.370885\pi\)
0.394593 + 0.918856i \(0.370885\pi\)
\(174\) −574.284 −0.250209
\(175\) 836.276 0.361237
\(176\) −1042.21 −0.446360
\(177\) 1149.08 0.487966
\(178\) −2099.77 −0.884183
\(179\) −2476.29 −1.03400 −0.517002 0.855984i \(-0.672952\pi\)
−0.517002 + 0.855984i \(0.672952\pi\)
\(180\) −84.6723 −0.0350617
\(181\) 2402.09 0.986441 0.493220 0.869904i \(-0.335820\pi\)
0.493220 + 0.869904i \(0.335820\pi\)
\(182\) −440.669 −0.179476
\(183\) −1294.87 −0.523059
\(184\) −742.574 −0.297518
\(185\) −11.8654 −0.00471547
\(186\) 966.523 0.381016
\(187\) 2806.34 1.09743
\(188\) −882.482 −0.342349
\(189\) 189.000 0.0727393
\(190\) −89.3763 −0.0341265
\(191\) −1635.57 −0.619610 −0.309805 0.950800i \(-0.600264\pi\)
−0.309805 + 0.950800i \(0.600264\pi\)
\(192\) −192.000 −0.0721688
\(193\) 4432.79 1.65326 0.826630 0.562746i \(-0.190255\pi\)
0.826630 + 0.562746i \(0.190255\pi\)
\(194\) 332.609 0.123093
\(195\) −222.098 −0.0815629
\(196\) 196.000 0.0714286
\(197\) 2601.65 0.940914 0.470457 0.882423i \(-0.344089\pi\)
0.470457 + 0.882423i \(0.344089\pi\)
\(198\) 1172.48 0.420832
\(199\) 1975.71 0.703791 0.351895 0.936039i \(-0.385537\pi\)
0.351895 + 0.936039i \(0.385537\pi\)
\(200\) 955.744 0.337907
\(201\) 2969.13 1.04192
\(202\) −3009.17 −1.04814
\(203\) 669.998 0.231649
\(204\) 516.997 0.177437
\(205\) 1022.91 0.348502
\(206\) 2291.35 0.774980
\(207\) 835.396 0.280503
\(208\) −503.622 −0.167884
\(209\) 1237.62 0.409608
\(210\) 98.7843 0.0324608
\(211\) −3735.55 −1.21880 −0.609398 0.792864i \(-0.708589\pi\)
−0.609398 + 0.792864i \(0.708589\pi\)
\(212\) 700.568 0.226959
\(213\) 1369.10 0.440418
\(214\) 1609.10 0.513999
\(215\) −128.767 −0.0408456
\(216\) 216.000 0.0680414
\(217\) −1127.61 −0.352752
\(218\) −4292.91 −1.33373
\(219\) −277.108 −0.0855033
\(220\) 612.819 0.187801
\(221\) 1356.10 0.412766
\(222\) 30.2688 0.00915094
\(223\) 5621.09 1.68796 0.843982 0.536372i \(-0.180205\pi\)
0.843982 + 0.536372i \(0.180205\pi\)
\(224\) 224.000 0.0668153
\(225\) −1075.21 −0.318581
\(226\) 912.479 0.268572
\(227\) −88.9122 −0.0259970 −0.0129985 0.999916i \(-0.504138\pi\)
−0.0129985 + 0.999916i \(0.504138\pi\)
\(228\) 228.000 0.0662266
\(229\) −3591.62 −1.03642 −0.518211 0.855253i \(-0.673402\pi\)
−0.518211 + 0.855253i \(0.673402\pi\)
\(230\) 436.635 0.125178
\(231\) −1367.90 −0.389615
\(232\) 765.712 0.216687
\(233\) −1783.34 −0.501417 −0.250709 0.968063i \(-0.580664\pi\)
−0.250709 + 0.968063i \(0.580664\pi\)
\(234\) 566.575 0.158283
\(235\) 518.901 0.144040
\(236\) −1532.10 −0.422591
\(237\) −1862.62 −0.510507
\(238\) −603.164 −0.164274
\(239\) −5280.63 −1.42919 −0.714594 0.699540i \(-0.753388\pi\)
−0.714594 + 0.699540i \(0.753388\pi\)
\(240\) 112.896 0.0303643
\(241\) 7111.18 1.90071 0.950355 0.311167i \(-0.100720\pi\)
0.950355 + 0.311167i \(0.100720\pi\)
\(242\) −5823.90 −1.54700
\(243\) −243.000 −0.0641500
\(244\) 1726.50 0.452983
\(245\) −115.248 −0.0300528
\(246\) −2609.45 −0.676310
\(247\) 598.051 0.154061
\(248\) −1288.70 −0.329969
\(249\) −1559.18 −0.396824
\(250\) −1149.98 −0.290925
\(251\) −2522.95 −0.634450 −0.317225 0.948350i \(-0.602751\pi\)
−0.317225 + 0.948350i \(0.602751\pi\)
\(252\) −252.000 −0.0629941
\(253\) −6046.22 −1.50246
\(254\) −2514.88 −0.621251
\(255\) −303.995 −0.0746546
\(256\) 256.000 0.0625000
\(257\) −2126.61 −0.516165 −0.258082 0.966123i \(-0.583091\pi\)
−0.258082 + 0.966123i \(0.583091\pi\)
\(258\) 328.485 0.0792659
\(259\) −35.3136 −0.00847213
\(260\) 296.131 0.0706356
\(261\) −861.426 −0.204295
\(262\) 4658.92 1.09859
\(263\) −2709.59 −0.635287 −0.317644 0.948210i \(-0.602892\pi\)
−0.317644 + 0.948210i \(0.602892\pi\)
\(264\) −1563.31 −0.364451
\(265\) −411.935 −0.0954905
\(266\) −266.000 −0.0613139
\(267\) −3149.66 −0.721932
\(268\) −3958.84 −0.902332
\(269\) −1063.61 −0.241076 −0.120538 0.992709i \(-0.538462\pi\)
−0.120538 + 0.992709i \(0.538462\pi\)
\(270\) −127.008 −0.0286277
\(271\) 2415.84 0.541519 0.270759 0.962647i \(-0.412725\pi\)
0.270759 + 0.962647i \(0.412725\pi\)
\(272\) −689.330 −0.153665
\(273\) −661.004 −0.146541
\(274\) −3457.18 −0.762247
\(275\) 7781.90 1.70642
\(276\) −1113.86 −0.242922
\(277\) −2923.91 −0.634226 −0.317113 0.948388i \(-0.602713\pi\)
−0.317113 + 0.948388i \(0.602713\pi\)
\(278\) 1382.41 0.298243
\(279\) 1449.78 0.311098
\(280\) −131.712 −0.0281119
\(281\) −7064.52 −1.49976 −0.749882 0.661571i \(-0.769890\pi\)
−0.749882 + 0.661571i \(0.769890\pi\)
\(282\) −1323.72 −0.279527
\(283\) −7199.32 −1.51221 −0.756104 0.654451i \(-0.772900\pi\)
−0.756104 + 0.654451i \(0.772900\pi\)
\(284\) −1825.47 −0.381413
\(285\) −134.064 −0.0278642
\(286\) −4100.61 −0.847812
\(287\) 3044.35 0.626141
\(288\) −288.000 −0.0589256
\(289\) −3056.85 −0.622195
\(290\) −450.240 −0.0911690
\(291\) 498.914 0.100505
\(292\) 369.477 0.0740480
\(293\) 3001.16 0.598395 0.299198 0.954191i \(-0.403281\pi\)
0.299198 + 0.954191i \(0.403281\pi\)
\(294\) 294.000 0.0583212
\(295\) 900.879 0.177801
\(296\) −40.3584 −0.00792495
\(297\) 1758.72 0.343608
\(298\) 4675.52 0.908878
\(299\) −2921.69 −0.565103
\(300\) 1433.62 0.275900
\(301\) −383.233 −0.0733860
\(302\) −5363.54 −1.02198
\(303\) −4513.75 −0.855803
\(304\) −304.000 −0.0573539
\(305\) −1015.18 −0.190588
\(306\) 775.496 0.144876
\(307\) 1737.38 0.322988 0.161494 0.986874i \(-0.448369\pi\)
0.161494 + 0.986874i \(0.448369\pi\)
\(308\) 1823.86 0.337416
\(309\) 3437.03 0.632769
\(310\) 757.756 0.138831
\(311\) 3951.77 0.720529 0.360264 0.932850i \(-0.382686\pi\)
0.360264 + 0.932850i \(0.382686\pi\)
\(312\) −755.433 −0.137077
\(313\) 2496.86 0.450898 0.225449 0.974255i \(-0.427615\pi\)
0.225449 + 0.974255i \(0.427615\pi\)
\(314\) −1524.00 −0.273899
\(315\) 148.176 0.0265041
\(316\) 2483.49 0.442112
\(317\) 2515.69 0.445725 0.222863 0.974850i \(-0.428460\pi\)
0.222863 + 0.974850i \(0.428460\pi\)
\(318\) 1050.85 0.185311
\(319\) 6234.61 1.09427
\(320\) −150.528 −0.0262962
\(321\) 2413.65 0.419679
\(322\) 1299.50 0.224902
\(323\) 818.579 0.141012
\(324\) 324.000 0.0555556
\(325\) 3760.42 0.641817
\(326\) −6904.12 −1.17296
\(327\) −6439.37 −1.08898
\(328\) 3479.26 0.585702
\(329\) 1544.34 0.258792
\(330\) 919.229 0.153339
\(331\) 980.008 0.162738 0.0813688 0.996684i \(-0.474071\pi\)
0.0813688 + 0.996684i \(0.474071\pi\)
\(332\) 2078.91 0.343660
\(333\) 45.4032 0.00747171
\(334\) −7179.00 −1.17610
\(335\) 2327.81 0.379647
\(336\) 336.000 0.0545545
\(337\) −8806.36 −1.42348 −0.711740 0.702443i \(-0.752093\pi\)
−0.711740 + 0.702443i \(0.752093\pi\)
\(338\) 2412.48 0.388229
\(339\) 1368.72 0.219288
\(340\) 405.327 0.0646528
\(341\) −10492.9 −1.66634
\(342\) 342.000 0.0540738
\(343\) −343.000 −0.0539949
\(344\) −437.980 −0.0686463
\(345\) 654.953 0.102207
\(346\) −3591.53 −0.558039
\(347\) −4506.05 −0.697110 −0.348555 0.937288i \(-0.613328\pi\)
−0.348555 + 0.937288i \(0.613328\pi\)
\(348\) 1148.57 0.176924
\(349\) −11683.9 −1.79204 −0.896021 0.444011i \(-0.853555\pi\)
−0.896021 + 0.444011i \(0.853555\pi\)
\(350\) −1672.55 −0.255433
\(351\) 849.862 0.129237
\(352\) 2084.41 0.315624
\(353\) 10542.7 1.58960 0.794802 0.606868i \(-0.207574\pi\)
0.794802 + 0.606868i \(0.207574\pi\)
\(354\) −2298.15 −0.345044
\(355\) 1073.38 0.160476
\(356\) 4199.54 0.625212
\(357\) −904.745 −0.134129
\(358\) 4952.58 0.731151
\(359\) −365.565 −0.0537432 −0.0268716 0.999639i \(-0.508555\pi\)
−0.0268716 + 0.999639i \(0.508555\pi\)
\(360\) 169.345 0.0247923
\(361\) 361.000 0.0526316
\(362\) −4804.18 −0.697519
\(363\) −8735.85 −1.26312
\(364\) 881.339 0.126909
\(365\) −217.253 −0.0311550
\(366\) 2589.75 0.369859
\(367\) −3287.08 −0.467532 −0.233766 0.972293i \(-0.575105\pi\)
−0.233766 + 0.972293i \(0.575105\pi\)
\(368\) 1485.15 0.210377
\(369\) −3914.17 −0.552205
\(370\) 23.7308 0.00333434
\(371\) −1225.99 −0.171565
\(372\) −1933.05 −0.269419
\(373\) 8899.17 1.23534 0.617669 0.786438i \(-0.288077\pi\)
0.617669 + 0.786438i \(0.288077\pi\)
\(374\) −5612.69 −0.776003
\(375\) −1724.97 −0.237539
\(376\) 1764.96 0.242077
\(377\) 3012.73 0.411574
\(378\) −378.000 −0.0514344
\(379\) −11650.9 −1.57907 −0.789536 0.613705i \(-0.789679\pi\)
−0.789536 + 0.613705i \(0.789679\pi\)
\(380\) 178.753 0.0241311
\(381\) −3772.32 −0.507249
\(382\) 3271.14 0.438131
\(383\) 3378.50 0.450739 0.225370 0.974273i \(-0.427641\pi\)
0.225370 + 0.974273i \(0.427641\pi\)
\(384\) 384.000 0.0510310
\(385\) −1072.43 −0.141964
\(386\) −8865.58 −1.16903
\(387\) 492.728 0.0647203
\(388\) −665.219 −0.0870396
\(389\) −4069.59 −0.530428 −0.265214 0.964190i \(-0.585443\pi\)
−0.265214 + 0.964190i \(0.585443\pi\)
\(390\) 444.196 0.0576737
\(391\) −3999.05 −0.517240
\(392\) −392.000 −0.0505076
\(393\) 6988.39 0.896991
\(394\) −5203.30 −0.665327
\(395\) −1460.30 −0.186014
\(396\) −2344.97 −0.297573
\(397\) 9273.75 1.17238 0.586191 0.810173i \(-0.300627\pi\)
0.586191 + 0.810173i \(0.300627\pi\)
\(398\) −3951.42 −0.497655
\(399\) −399.000 −0.0500626
\(400\) −1911.49 −0.238936
\(401\) 5544.49 0.690471 0.345235 0.938516i \(-0.387799\pi\)
0.345235 + 0.938516i \(0.387799\pi\)
\(402\) −5938.27 −0.736751
\(403\) −5070.44 −0.626741
\(404\) 6018.34 0.741147
\(405\) −190.513 −0.0233744
\(406\) −1340.00 −0.163800
\(407\) −328.608 −0.0400209
\(408\) −1033.99 −0.125467
\(409\) −4863.21 −0.587947 −0.293973 0.955814i \(-0.594978\pi\)
−0.293973 + 0.955814i \(0.594978\pi\)
\(410\) −2045.81 −0.246428
\(411\) −5185.77 −0.622372
\(412\) −4582.70 −0.547994
\(413\) 2681.18 0.319449
\(414\) −1670.79 −0.198345
\(415\) −1222.40 −0.144591
\(416\) 1007.24 0.118712
\(417\) 2073.61 0.243514
\(418\) −2475.24 −0.289636
\(419\) −2633.77 −0.307084 −0.153542 0.988142i \(-0.549068\pi\)
−0.153542 + 0.988142i \(0.549068\pi\)
\(420\) −197.569 −0.0229532
\(421\) 12763.2 1.47753 0.738767 0.673961i \(-0.235409\pi\)
0.738767 + 0.673961i \(0.235409\pi\)
\(422\) 7471.10 0.861819
\(423\) −1985.59 −0.228233
\(424\) −1401.14 −0.160484
\(425\) 5147.06 0.587456
\(426\) −2738.20 −0.311423
\(427\) −3021.37 −0.342423
\(428\) −3218.20 −0.363452
\(429\) −6150.92 −0.692236
\(430\) 257.533 0.0288822
\(431\) −3351.91 −0.374607 −0.187304 0.982302i \(-0.559975\pi\)
−0.187304 + 0.982302i \(0.559975\pi\)
\(432\) −432.000 −0.0481125
\(433\) −3364.43 −0.373405 −0.186702 0.982417i \(-0.559780\pi\)
−0.186702 + 0.982417i \(0.559780\pi\)
\(434\) 2255.22 0.249433
\(435\) −675.360 −0.0744392
\(436\) 8585.83 0.943089
\(437\) −1763.61 −0.193055
\(438\) 554.216 0.0604600
\(439\) −4991.62 −0.542682 −0.271341 0.962483i \(-0.587467\pi\)
−0.271341 + 0.962483i \(0.587467\pi\)
\(440\) −1225.64 −0.132796
\(441\) 441.000 0.0476190
\(442\) −2712.20 −0.291869
\(443\) −13437.4 −1.44115 −0.720574 0.693378i \(-0.756122\pi\)
−0.720574 + 0.693378i \(0.756122\pi\)
\(444\) −60.5376 −0.00647069
\(445\) −2469.34 −0.263051
\(446\) −11242.2 −1.19357
\(447\) 7013.28 0.742096
\(448\) −448.000 −0.0472456
\(449\) 3090.05 0.324785 0.162392 0.986726i \(-0.448079\pi\)
0.162392 + 0.986726i \(0.448079\pi\)
\(450\) 2150.43 0.225271
\(451\) 28329.0 2.95778
\(452\) −1824.96 −0.189909
\(453\) −8045.30 −0.834440
\(454\) 177.824 0.0183826
\(455\) −518.229 −0.0533955
\(456\) −456.000 −0.0468293
\(457\) 9282.31 0.950127 0.475063 0.879952i \(-0.342425\pi\)
0.475063 + 0.879952i \(0.342425\pi\)
\(458\) 7183.23 0.732861
\(459\) 1163.24 0.118291
\(460\) −873.270 −0.0885140
\(461\) −14936.7 −1.50904 −0.754522 0.656275i \(-0.772131\pi\)
−0.754522 + 0.656275i \(0.772131\pi\)
\(462\) 2735.79 0.275499
\(463\) 7024.39 0.705078 0.352539 0.935797i \(-0.385318\pi\)
0.352539 + 0.935797i \(0.385318\pi\)
\(464\) −1531.42 −0.153221
\(465\) 1136.63 0.113355
\(466\) 3566.67 0.354556
\(467\) 800.205 0.0792914 0.0396457 0.999214i \(-0.487377\pi\)
0.0396457 + 0.999214i \(0.487377\pi\)
\(468\) −1133.15 −0.111923
\(469\) 6927.98 0.682099
\(470\) −1037.80 −0.101852
\(471\) −2286.00 −0.223638
\(472\) 3064.21 0.298817
\(473\) −3566.14 −0.346663
\(474\) 3725.24 0.360983
\(475\) 2269.89 0.219263
\(476\) 1206.33 0.116159
\(477\) 1576.28 0.151306
\(478\) 10561.3 1.01059
\(479\) 3573.45 0.340867 0.170433 0.985369i \(-0.445483\pi\)
0.170433 + 0.985369i \(0.445483\pi\)
\(480\) −225.793 −0.0214708
\(481\) −158.792 −0.0150526
\(482\) −14222.4 −1.34401
\(483\) 1949.26 0.183632
\(484\) 11647.8 1.09389
\(485\) 391.150 0.0366210
\(486\) 486.000 0.0453609
\(487\) −13085.1 −1.21755 −0.608773 0.793345i \(-0.708338\pi\)
−0.608773 + 0.793345i \(0.708338\pi\)
\(488\) −3453.00 −0.320307
\(489\) −10356.2 −0.957715
\(490\) 230.497 0.0212506
\(491\) −9072.15 −0.833850 −0.416925 0.908941i \(-0.636892\pi\)
−0.416925 + 0.908941i \(0.636892\pi\)
\(492\) 5218.89 0.478223
\(493\) 4123.66 0.376714
\(494\) −1196.10 −0.108938
\(495\) 1378.84 0.125201
\(496\) 2577.39 0.233323
\(497\) 3194.56 0.288321
\(498\) 3118.37 0.280597
\(499\) −1920.38 −0.172281 −0.0861403 0.996283i \(-0.527453\pi\)
−0.0861403 + 0.996283i \(0.527453\pi\)
\(500\) 2299.96 0.205715
\(501\) −10768.5 −0.960281
\(502\) 5045.89 0.448624
\(503\) −14852.4 −1.31657 −0.658285 0.752769i \(-0.728718\pi\)
−0.658285 + 0.752769i \(0.728718\pi\)
\(504\) 504.000 0.0445435
\(505\) −3538.79 −0.311830
\(506\) 12092.4 1.06240
\(507\) 3618.71 0.316988
\(508\) 5029.76 0.439291
\(509\) 14779.7 1.28703 0.643514 0.765434i \(-0.277476\pi\)
0.643514 + 0.765434i \(0.277476\pi\)
\(510\) 607.991 0.0527888
\(511\) −646.585 −0.0559751
\(512\) −512.000 −0.0441942
\(513\) 513.000 0.0441511
\(514\) 4253.22 0.364983
\(515\) 2694.64 0.230563
\(516\) −656.971 −0.0560495
\(517\) 14370.8 1.22249
\(518\) 70.6272 0.00599070
\(519\) −5387.29 −0.455637
\(520\) −592.261 −0.0499469
\(521\) 6548.34 0.550648 0.275324 0.961351i \(-0.411215\pi\)
0.275324 + 0.961351i \(0.411215\pi\)
\(522\) 1722.85 0.144458
\(523\) 783.385 0.0654972 0.0327486 0.999464i \(-0.489574\pi\)
0.0327486 + 0.999464i \(0.489574\pi\)
\(524\) −9317.85 −0.776817
\(525\) −2508.83 −0.208561
\(526\) 5419.18 0.449216
\(527\) −6940.14 −0.573657
\(528\) 3126.62 0.257706
\(529\) −3551.12 −0.291865
\(530\) 823.871 0.0675220
\(531\) −3447.23 −0.281727
\(532\) 532.000 0.0433555
\(533\) 13689.3 1.11248
\(534\) 6299.32 0.510483
\(535\) 1892.31 0.152919
\(536\) 7917.69 0.638045
\(537\) 7428.88 0.596983
\(538\) 2127.22 0.170466
\(539\) −3191.76 −0.255063
\(540\) 254.017 0.0202429
\(541\) 11120.2 0.883720 0.441860 0.897084i \(-0.354319\pi\)
0.441860 + 0.897084i \(0.354319\pi\)
\(542\) −4831.67 −0.382912
\(543\) −7206.26 −0.569522
\(544\) 1378.66 0.108657
\(545\) −5048.48 −0.396795
\(546\) 1322.01 0.103620
\(547\) −16006.1 −1.25114 −0.625569 0.780169i \(-0.715133\pi\)
−0.625569 + 0.780169i \(0.715133\pi\)
\(548\) 6914.36 0.538990
\(549\) 3884.62 0.301988
\(550\) −15563.8 −1.20662
\(551\) 1818.57 0.140605
\(552\) 2227.72 0.171772
\(553\) −4346.11 −0.334205
\(554\) 5847.82 0.448466
\(555\) 35.5962 0.00272248
\(556\) −2764.82 −0.210889
\(557\) −13422.4 −1.02105 −0.510524 0.859863i \(-0.670549\pi\)
−0.510524 + 0.859863i \(0.670549\pi\)
\(558\) −2899.57 −0.219979
\(559\) −1723.25 −0.130386
\(560\) 263.425 0.0198781
\(561\) −8419.03 −0.633604
\(562\) 14129.0 1.06049
\(563\) 18317.0 1.37117 0.685586 0.727991i \(-0.259546\pi\)
0.685586 + 0.727991i \(0.259546\pi\)
\(564\) 2647.45 0.197655
\(565\) 1073.08 0.0799022
\(566\) 14398.6 1.06929
\(567\) −567.000 −0.0419961
\(568\) 3650.93 0.269700
\(569\) 10513.3 0.774589 0.387295 0.921956i \(-0.373410\pi\)
0.387295 + 0.921956i \(0.373410\pi\)
\(570\) 268.129 0.0197029
\(571\) −12420.2 −0.910277 −0.455138 0.890421i \(-0.650410\pi\)
−0.455138 + 0.890421i \(0.650410\pi\)
\(572\) 8201.22 0.599494
\(573\) 4906.70 0.357732
\(574\) −6088.71 −0.442749
\(575\) −11089.2 −0.804266
\(576\) 576.000 0.0416667
\(577\) 7149.46 0.515833 0.257917 0.966167i \(-0.416964\pi\)
0.257917 + 0.966167i \(0.416964\pi\)
\(578\) 6113.69 0.439958
\(579\) −13298.4 −0.954510
\(580\) 900.480 0.0644662
\(581\) −3638.10 −0.259782
\(582\) −997.828 −0.0710675
\(583\) −11408.4 −0.810441
\(584\) −738.955 −0.0523599
\(585\) 666.294 0.0470904
\(586\) −6002.32 −0.423129
\(587\) 25413.4 1.78692 0.893461 0.449140i \(-0.148270\pi\)
0.893461 + 0.449140i \(0.148270\pi\)
\(588\) −588.000 −0.0412393
\(589\) −3060.66 −0.214112
\(590\) −1801.76 −0.125724
\(591\) −7804.96 −0.543237
\(592\) 80.7168 0.00560379
\(593\) 11905.5 0.824452 0.412226 0.911082i \(-0.364751\pi\)
0.412226 + 0.911082i \(0.364751\pi\)
\(594\) −3517.45 −0.242967
\(595\) −709.323 −0.0488729
\(596\) −9351.04 −0.642674
\(597\) −5927.13 −0.406334
\(598\) 5843.39 0.399588
\(599\) 22430.9 1.53005 0.765027 0.643998i \(-0.222725\pi\)
0.765027 + 0.643998i \(0.222725\pi\)
\(600\) −2867.23 −0.195091
\(601\) −9645.59 −0.654662 −0.327331 0.944910i \(-0.606149\pi\)
−0.327331 + 0.944910i \(0.606149\pi\)
\(602\) 766.466 0.0518917
\(603\) −8907.40 −0.601555
\(604\) 10727.1 0.722646
\(605\) −6848.92 −0.460245
\(606\) 9027.51 0.605144
\(607\) −2535.39 −0.169536 −0.0847679 0.996401i \(-0.527015\pi\)
−0.0847679 + 0.996401i \(0.527015\pi\)
\(608\) 608.000 0.0405554
\(609\) −2009.99 −0.133742
\(610\) 2030.37 0.134766
\(611\) 6944.34 0.459800
\(612\) −1550.99 −0.102443
\(613\) 17163.0 1.13084 0.565421 0.824802i \(-0.308714\pi\)
0.565421 + 0.824802i \(0.308714\pi\)
\(614\) −3474.75 −0.228387
\(615\) −3068.72 −0.201208
\(616\) −3647.72 −0.238589
\(617\) 2852.64 0.186131 0.0930656 0.995660i \(-0.470333\pi\)
0.0930656 + 0.995660i \(0.470333\pi\)
\(618\) −6874.05 −0.447435
\(619\) −15912.1 −1.03322 −0.516608 0.856222i \(-0.672805\pi\)
−0.516608 + 0.856222i \(0.672805\pi\)
\(620\) −1515.51 −0.0981685
\(621\) −2506.19 −0.161948
\(622\) −7903.54 −0.509491
\(623\) −7349.20 −0.472616
\(624\) 1510.87 0.0969280
\(625\) 13581.1 0.869192
\(626\) −4993.73 −0.318833
\(627\) −3712.86 −0.236487
\(628\) 3048.00 0.193676
\(629\) −217.346 −0.0137777
\(630\) −296.353 −0.0187412
\(631\) −6717.70 −0.423815 −0.211908 0.977290i \(-0.567968\pi\)
−0.211908 + 0.977290i \(0.567968\pi\)
\(632\) −4966.98 −0.312620
\(633\) 11206.7 0.703672
\(634\) −5031.37 −0.315175
\(635\) −2957.51 −0.184827
\(636\) −2101.70 −0.131035
\(637\) −1542.34 −0.0959338
\(638\) −12469.2 −0.773764
\(639\) −4107.30 −0.254276
\(640\) 301.057 0.0185943
\(641\) 23563.0 1.45192 0.725961 0.687736i \(-0.241395\pi\)
0.725961 + 0.687736i \(0.241395\pi\)
\(642\) −4827.30 −0.296758
\(643\) 12722.7 0.780303 0.390152 0.920751i \(-0.372423\pi\)
0.390152 + 0.920751i \(0.372423\pi\)
\(644\) −2599.01 −0.159030
\(645\) 386.300 0.0235822
\(646\) −1637.16 −0.0997107
\(647\) 20322.2 1.23485 0.617424 0.786631i \(-0.288176\pi\)
0.617424 + 0.786631i \(0.288176\pi\)
\(648\) −648.000 −0.0392837
\(649\) 24949.5 1.50902
\(650\) −7520.84 −0.453833
\(651\) 3382.83 0.203661
\(652\) 13808.2 0.829406
\(653\) −19287.3 −1.15585 −0.577924 0.816091i \(-0.696137\pi\)
−0.577924 + 0.816091i \(0.696137\pi\)
\(654\) 12878.7 0.770029
\(655\) 5478.91 0.326838
\(656\) −6958.52 −0.414154
\(657\) 831.324 0.0493654
\(658\) −3088.69 −0.182993
\(659\) −7186.76 −0.424820 −0.212410 0.977181i \(-0.568131\pi\)
−0.212410 + 0.977181i \(0.568131\pi\)
\(660\) −1838.46 −0.108427
\(661\) −23799.5 −1.40044 −0.700220 0.713927i \(-0.746915\pi\)
−0.700220 + 0.713927i \(0.746915\pi\)
\(662\) −1960.02 −0.115073
\(663\) −4068.30 −0.238310
\(664\) −4157.82 −0.243004
\(665\) −312.817 −0.0182414
\(666\) −90.8064 −0.00528330
\(667\) −8884.35 −0.515747
\(668\) 14358.0 0.831628
\(669\) −16863.3 −0.974546
\(670\) −4655.62 −0.268451
\(671\) −28115.1 −1.61755
\(672\) −672.000 −0.0385758
\(673\) −6426.09 −0.368065 −0.184032 0.982920i \(-0.558915\pi\)
−0.184032 + 0.982920i \(0.558915\pi\)
\(674\) 17612.7 1.00655
\(675\) 3225.64 0.183933
\(676\) −4824.95 −0.274519
\(677\) 5295.09 0.300601 0.150301 0.988640i \(-0.451976\pi\)
0.150301 + 0.988640i \(0.451976\pi\)
\(678\) −2737.44 −0.155060
\(679\) 1164.13 0.0657958
\(680\) −810.654 −0.0457164
\(681\) 266.737 0.0150094
\(682\) 20985.8 1.17828
\(683\) 27838.6 1.55961 0.779807 0.626020i \(-0.215317\pi\)
0.779807 + 0.626020i \(0.215317\pi\)
\(684\) −684.000 −0.0382360
\(685\) −4065.65 −0.226775
\(686\) 686.000 0.0381802
\(687\) 10774.9 0.598379
\(688\) 875.961 0.0485403
\(689\) −5512.84 −0.304822
\(690\) −1309.91 −0.0722714
\(691\) −19503.3 −1.07372 −0.536859 0.843672i \(-0.680389\pi\)
−0.536859 + 0.843672i \(0.680389\pi\)
\(692\) 7183.05 0.394593
\(693\) 4103.69 0.224944
\(694\) 9012.10 0.492932
\(695\) 1625.72 0.0887296
\(696\) −2297.14 −0.125104
\(697\) 18737.2 1.01825
\(698\) 23367.7 1.26717
\(699\) 5350.01 0.289494
\(700\) 3345.11 0.180619
\(701\) 6387.15 0.344136 0.172068 0.985085i \(-0.444955\pi\)
0.172068 + 0.985085i \(0.444955\pi\)
\(702\) −1699.72 −0.0913846
\(703\) −95.8512 −0.00514239
\(704\) −4168.83 −0.223180
\(705\) −1556.70 −0.0831615
\(706\) −21085.4 −1.12402
\(707\) −10532.1 −0.560255
\(708\) 4596.31 0.243983
\(709\) 24946.9 1.32144 0.660720 0.750632i \(-0.270251\pi\)
0.660720 + 0.750632i \(0.270251\pi\)
\(710\) −2146.75 −0.113474
\(711\) 5587.86 0.294741
\(712\) −8399.09 −0.442091
\(713\) 14952.4 0.785374
\(714\) 1809.49 0.0948438
\(715\) −4822.33 −0.252231
\(716\) −9905.17 −0.517002
\(717\) 15841.9 0.825142
\(718\) 731.130 0.0380022
\(719\) 32315.2 1.67615 0.838077 0.545552i \(-0.183680\pi\)
0.838077 + 0.545552i \(0.183680\pi\)
\(720\) −338.689 −0.0175308
\(721\) 8019.73 0.414244
\(722\) −722.000 −0.0372161
\(723\) −21333.5 −1.09738
\(724\) 9608.35 0.493220
\(725\) 11434.8 0.585761
\(726\) 17471.7 0.893162
\(727\) 21195.9 1.08131 0.540656 0.841244i \(-0.318176\pi\)
0.540656 + 0.841244i \(0.318176\pi\)
\(728\) −1762.68 −0.0897379
\(729\) 729.000 0.0370370
\(730\) 434.507 0.0220299
\(731\) −2358.70 −0.119343
\(732\) −5179.50 −0.261530
\(733\) −11952.5 −0.602287 −0.301144 0.953579i \(-0.597368\pi\)
−0.301144 + 0.953579i \(0.597368\pi\)
\(734\) 6574.16 0.330595
\(735\) 345.745 0.0173510
\(736\) −2970.30 −0.148759
\(737\) 64467.7 3.22212
\(738\) 7828.34 0.390468
\(739\) −12355.2 −0.615010 −0.307505 0.951546i \(-0.599494\pi\)
−0.307505 + 0.951546i \(0.599494\pi\)
\(740\) −47.4616 −0.00235774
\(741\) −1794.15 −0.0889472
\(742\) 2451.99 0.121314
\(743\) 545.630 0.0269411 0.0134705 0.999909i \(-0.495712\pi\)
0.0134705 + 0.999909i \(0.495712\pi\)
\(744\) 3866.09 0.190508
\(745\) 5498.43 0.270399
\(746\) −17798.3 −0.873516
\(747\) 4677.55 0.229107
\(748\) 11225.4 0.548717
\(749\) 5631.85 0.274744
\(750\) 3449.94 0.167966
\(751\) −24657.5 −1.19809 −0.599044 0.800716i \(-0.704453\pi\)
−0.599044 + 0.800716i \(0.704453\pi\)
\(752\) −3529.93 −0.171175
\(753\) 7568.84 0.366300
\(754\) −6025.46 −0.291027
\(755\) −6307.54 −0.304046
\(756\) 756.000 0.0363696
\(757\) −9802.82 −0.470660 −0.235330 0.971916i \(-0.575617\pi\)
−0.235330 + 0.971916i \(0.575617\pi\)
\(758\) 23301.9 1.11657
\(759\) 18138.7 0.867446
\(760\) −357.505 −0.0170633
\(761\) −38995.6 −1.85754 −0.928770 0.370656i \(-0.879133\pi\)
−0.928770 + 0.370656i \(0.879133\pi\)
\(762\) 7544.65 0.358679
\(763\) −15025.2 −0.712908
\(764\) −6542.27 −0.309805
\(765\) 911.986 0.0431019
\(766\) −6756.99 −0.318721
\(767\) 12056.3 0.567570
\(768\) −768.000 −0.0360844
\(769\) 11656.9 0.546631 0.273315 0.961924i \(-0.411880\pi\)
0.273315 + 0.961924i \(0.411880\pi\)
\(770\) 2144.87 0.100384
\(771\) 6379.83 0.298008
\(772\) 17731.2 0.826630
\(773\) 18534.7 0.862414 0.431207 0.902253i \(-0.358088\pi\)
0.431207 + 0.902253i \(0.358088\pi\)
\(774\) −985.456 −0.0457642
\(775\) −19244.8 −0.891990
\(776\) 1330.44 0.0615463
\(777\) 105.941 0.00489139
\(778\) 8139.19 0.375069
\(779\) 8263.25 0.380053
\(780\) −888.392 −0.0407815
\(781\) 29726.7 1.36198
\(782\) 7998.10 0.365744
\(783\) 2584.28 0.117950
\(784\) 784.000 0.0357143
\(785\) −1792.23 −0.0814872
\(786\) −13976.8 −0.634268
\(787\) 21029.9 0.952521 0.476261 0.879304i \(-0.341992\pi\)
0.476261 + 0.879304i \(0.341992\pi\)
\(788\) 10406.6 0.470457
\(789\) 8128.77 0.366783
\(790\) 2920.60 0.131532
\(791\) 3193.68 0.143558
\(792\) 4689.93 0.210416
\(793\) −13586.0 −0.608389
\(794\) −18547.5 −0.829000
\(795\) 1235.81 0.0551315
\(796\) 7902.84 0.351895
\(797\) −16890.1 −0.750661 −0.375330 0.926891i \(-0.622471\pi\)
−0.375330 + 0.926891i \(0.622471\pi\)
\(798\) 798.000 0.0353996
\(799\) 9505.02 0.420856
\(800\) 3822.98 0.168953
\(801\) 9448.97 0.416808
\(802\) −11089.0 −0.488237
\(803\) −6016.75 −0.264416
\(804\) 11876.5 0.520962
\(805\) 1528.22 0.0669103
\(806\) 10140.9 0.443173
\(807\) 3190.83 0.139185
\(808\) −12036.7 −0.524070
\(809\) −26270.6 −1.14169 −0.570843 0.821059i \(-0.693384\pi\)
−0.570843 + 0.821059i \(0.693384\pi\)
\(810\) 381.025 0.0165282
\(811\) −886.827 −0.0383979 −0.0191990 0.999816i \(-0.506112\pi\)
−0.0191990 + 0.999816i \(0.506112\pi\)
\(812\) 2679.99 0.115824
\(813\) −7247.51 −0.312646
\(814\) 657.216 0.0282990
\(815\) −8119.27 −0.348964
\(816\) 2067.99 0.0887183
\(817\) −1040.20 −0.0445436
\(818\) 9726.42 0.415741
\(819\) 1983.01 0.0846057
\(820\) 4091.63 0.174251
\(821\) −16810.5 −0.714605 −0.357303 0.933989i \(-0.616304\pi\)
−0.357303 + 0.933989i \(0.616304\pi\)
\(822\) 10371.5 0.440084
\(823\) −40374.7 −1.71005 −0.855027 0.518583i \(-0.826460\pi\)
−0.855027 + 0.518583i \(0.826460\pi\)
\(824\) 9165.40 0.387490
\(825\) −23345.7 −0.985204
\(826\) −5362.36 −0.225884
\(827\) −21653.2 −0.910468 −0.455234 0.890372i \(-0.650444\pi\)
−0.455234 + 0.890372i \(0.650444\pi\)
\(828\) 3341.58 0.140251
\(829\) 9284.49 0.388979 0.194490 0.980905i \(-0.437695\pi\)
0.194490 + 0.980905i \(0.437695\pi\)
\(830\) 2444.81 0.102242
\(831\) 8771.73 0.366171
\(832\) −2014.49 −0.0839421
\(833\) −2111.07 −0.0878083
\(834\) −4147.23 −0.172190
\(835\) −8442.53 −0.349899
\(836\) 4950.48 0.204804
\(837\) −4349.35 −0.179612
\(838\) 5267.54 0.217141
\(839\) 30110.7 1.23902 0.619509 0.784989i \(-0.287332\pi\)
0.619509 + 0.784989i \(0.287332\pi\)
\(840\) 395.137 0.0162304
\(841\) −15227.8 −0.624373
\(842\) −25526.4 −1.04477
\(843\) 21193.6 0.865890
\(844\) −14942.2 −0.609398
\(845\) 2837.08 0.115501
\(846\) 3971.17 0.161385
\(847\) −20383.6 −0.826907
\(848\) 2802.27 0.113479
\(849\) 21598.0 0.873074
\(850\) −10294.1 −0.415394
\(851\) 468.268 0.0188625
\(852\) 5476.40 0.220209
\(853\) −28220.7 −1.13278 −0.566389 0.824138i \(-0.691660\pi\)
−0.566389 + 0.824138i \(0.691660\pi\)
\(854\) 6042.75 0.242129
\(855\) 402.193 0.0160874
\(856\) 6436.40 0.257000
\(857\) 37317.9 1.48746 0.743731 0.668479i \(-0.233054\pi\)
0.743731 + 0.668479i \(0.233054\pi\)
\(858\) 12301.8 0.489485
\(859\) −15643.5 −0.621363 −0.310681 0.950514i \(-0.600557\pi\)
−0.310681 + 0.950514i \(0.600557\pi\)
\(860\) −515.067 −0.0204228
\(861\) −9133.06 −0.361503
\(862\) 6703.82 0.264887
\(863\) −17616.3 −0.694860 −0.347430 0.937706i \(-0.612946\pi\)
−0.347430 + 0.937706i \(0.612946\pi\)
\(864\) 864.000 0.0340207
\(865\) −4223.65 −0.166021
\(866\) 6728.87 0.264037
\(867\) 9170.54 0.359225
\(868\) −4510.44 −0.176376
\(869\) −40442.4 −1.57873
\(870\) 1350.72 0.0526364
\(871\) 31152.5 1.21190
\(872\) −17171.7 −0.666864
\(873\) −1496.74 −0.0580264
\(874\) 3527.23 0.136511
\(875\) −4024.93 −0.155506
\(876\) −1108.43 −0.0427517
\(877\) 12573.4 0.484121 0.242060 0.970261i \(-0.422177\pi\)
0.242060 + 0.970261i \(0.422177\pi\)
\(878\) 9983.25 0.383734
\(879\) −9003.49 −0.345484
\(880\) 2451.28 0.0939006
\(881\) −9079.75 −0.347224 −0.173612 0.984814i \(-0.555544\pi\)
−0.173612 + 0.984814i \(0.555544\pi\)
\(882\) −882.000 −0.0336718
\(883\) 30666.4 1.16875 0.584376 0.811483i \(-0.301340\pi\)
0.584376 + 0.811483i \(0.301340\pi\)
\(884\) 5424.40 0.206383
\(885\) −2702.64 −0.102653
\(886\) 26874.7 1.01905
\(887\) 24420.8 0.924431 0.462216 0.886768i \(-0.347055\pi\)
0.462216 + 0.886768i \(0.347055\pi\)
\(888\) 121.075 0.00457547
\(889\) −8802.09 −0.332073
\(890\) 4938.68 0.186005
\(891\) −5276.17 −0.198382
\(892\) 22484.4 0.843982
\(893\) 4191.79 0.157081
\(894\) −14026.6 −0.524741
\(895\) 5824.26 0.217523
\(896\) 896.000 0.0334077
\(897\) 8765.08 0.326263
\(898\) −6180.09 −0.229657
\(899\) −15418.3 −0.572001
\(900\) −4300.85 −0.159291
\(901\) −7545.67 −0.279004
\(902\) −56658.0 −2.09147
\(903\) 1149.70 0.0423694
\(904\) 3649.92 0.134286
\(905\) −5649.73 −0.207517
\(906\) 16090.6 0.590038
\(907\) −23548.1 −0.862075 −0.431037 0.902334i \(-0.641852\pi\)
−0.431037 + 0.902334i \(0.641852\pi\)
\(908\) −355.649 −0.0129985
\(909\) 13541.3 0.494098
\(910\) 1036.46 0.0377563
\(911\) −26956.7 −0.980368 −0.490184 0.871619i \(-0.663070\pi\)
−0.490184 + 0.871619i \(0.663070\pi\)
\(912\) 912.000 0.0331133
\(913\) −33854.0 −1.22717
\(914\) −18564.6 −0.671841
\(915\) 3045.55 0.110036
\(916\) −14366.5 −0.518211
\(917\) 16306.2 0.587218
\(918\) −2326.49 −0.0836444
\(919\) −8314.91 −0.298459 −0.149229 0.988803i \(-0.547679\pi\)
−0.149229 + 0.988803i \(0.547679\pi\)
\(920\) 1746.54 0.0625888
\(921\) −5212.13 −0.186477
\(922\) 29873.3 1.06706
\(923\) 14364.8 0.512266
\(924\) −5471.59 −0.194807
\(925\) −602.693 −0.0214231
\(926\) −14048.8 −0.498566
\(927\) −10311.1 −0.365329
\(928\) 3062.85 0.108344
\(929\) −716.122 −0.0252908 −0.0126454 0.999920i \(-0.504025\pi\)
−0.0126454 + 0.999920i \(0.504025\pi\)
\(930\) −2273.27 −0.0801542
\(931\) −931.000 −0.0327737
\(932\) −7133.35 −0.250709
\(933\) −11855.3 −0.415997
\(934\) −1600.41 −0.0560675
\(935\) −6600.54 −0.230867
\(936\) 2266.30 0.0791414
\(937\) −29545.2 −1.03010 −0.515048 0.857161i \(-0.672226\pi\)
−0.515048 + 0.857161i \(0.672226\pi\)
\(938\) −13856.0 −0.482317
\(939\) −7490.59 −0.260326
\(940\) 2075.61 0.0720200
\(941\) −12021.2 −0.416452 −0.208226 0.978081i \(-0.566769\pi\)
−0.208226 + 0.978081i \(0.566769\pi\)
\(942\) 4572.00 0.158136
\(943\) −40368.9 −1.39405
\(944\) −6128.41 −0.211295
\(945\) −444.529 −0.0153022
\(946\) 7132.28 0.245127
\(947\) 32892.4 1.12868 0.564340 0.825542i \(-0.309131\pi\)
0.564340 + 0.825542i \(0.309131\pi\)
\(948\) −7450.48 −0.255253
\(949\) −2907.45 −0.0994519
\(950\) −4539.79 −0.155042
\(951\) −7547.06 −0.257340
\(952\) −2412.65 −0.0821372
\(953\) −25757.2 −0.875505 −0.437752 0.899096i \(-0.644225\pi\)
−0.437752 + 0.899096i \(0.644225\pi\)
\(954\) −3152.56 −0.106989
\(955\) 3846.87 0.130347
\(956\) −21122.5 −0.714594
\(957\) −18703.8 −0.631775
\(958\) −7146.91 −0.241029
\(959\) −12100.1 −0.407438
\(960\) 451.585 0.0151821
\(961\) −3841.94 −0.128963
\(962\) 317.584 0.0106438
\(963\) −7240.95 −0.242302
\(964\) 28444.7 0.950355
\(965\) −10426.0 −0.347796
\(966\) −3898.51 −0.129847
\(967\) −18681.0 −0.621243 −0.310621 0.950534i \(-0.600537\pi\)
−0.310621 + 0.950534i \(0.600537\pi\)
\(968\) −23295.6 −0.773501
\(969\) −2455.74 −0.0814135
\(970\) −782.300 −0.0258950
\(971\) 20470.4 0.676546 0.338273 0.941048i \(-0.390157\pi\)
0.338273 + 0.941048i \(0.390157\pi\)
\(972\) −972.000 −0.0320750
\(973\) 4838.43 0.159417
\(974\) 26170.3 0.860934
\(975\) −11281.3 −0.370553
\(976\) 6906.00 0.226491
\(977\) −57784.2 −1.89220 −0.946100 0.323874i \(-0.895015\pi\)
−0.946100 + 0.323874i \(0.895015\pi\)
\(978\) 20712.4 0.677207
\(979\) −68387.4 −2.23255
\(980\) −460.993 −0.0150264
\(981\) 19318.1 0.628726
\(982\) 18144.3 0.589621
\(983\) 37476.2 1.21598 0.607989 0.793945i \(-0.291976\pi\)
0.607989 + 0.793945i \(0.291976\pi\)
\(984\) −10437.8 −0.338155
\(985\) −6119.10 −0.197940
\(986\) −8247.32 −0.266377
\(987\) −4633.03 −0.149413
\(988\) 2392.20 0.0770305
\(989\) 5081.77 0.163388
\(990\) −2757.69 −0.0885304
\(991\) −20663.9 −0.662373 −0.331186 0.943565i \(-0.607449\pi\)
−0.331186 + 0.943565i \(0.607449\pi\)
\(992\) −5154.79 −0.164985
\(993\) −2940.02 −0.0939566
\(994\) −6389.13 −0.203874
\(995\) −4646.88 −0.148056
\(996\) −6236.73 −0.198412
\(997\) 41487.0 1.31786 0.658930 0.752205i \(-0.271009\pi\)
0.658930 + 0.752205i \(0.271009\pi\)
\(998\) 3840.76 0.121821
\(999\) −136.210 −0.00431380
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 798.4.a.k.1.3 4
3.2 odd 2 2394.4.a.x.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.4.a.k.1.3 4 1.1 even 1 trivial
2394.4.a.x.1.2 4 3.2 odd 2