Properties

Label 115.4.a.e.1.4
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
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] = [115,4,Mod(1,115)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(6.78521965066\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.41740\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.41740 q^{2} +7.84147 q^{3} +11.5134 q^{4} -5.00000 q^{5} +34.6389 q^{6} -8.97260 q^{7} +15.5200 q^{8} +34.4886 q^{9} +O(q^{10})\) \(q+4.41740 q^{2} +7.84147 q^{3} +11.5134 q^{4} -5.00000 q^{5} +34.6389 q^{6} -8.97260 q^{7} +15.5200 q^{8} +34.4886 q^{9} -22.0870 q^{10} -28.9011 q^{11} +90.2818 q^{12} +16.0879 q^{13} -39.6355 q^{14} -39.2073 q^{15} -23.5490 q^{16} +25.1771 q^{17} +152.350 q^{18} -35.6298 q^{19} -57.5669 q^{20} -70.3583 q^{21} -127.668 q^{22} -23.0000 q^{23} +121.700 q^{24} +25.0000 q^{25} +71.0668 q^{26} +58.7218 q^{27} -103.305 q^{28} +138.272 q^{29} -173.194 q^{30} +40.1277 q^{31} -228.186 q^{32} -226.627 q^{33} +111.217 q^{34} +44.8630 q^{35} +397.081 q^{36} +379.745 q^{37} -157.391 q^{38} +126.153 q^{39} -77.6001 q^{40} +412.514 q^{41} -310.801 q^{42} -402.095 q^{43} -332.750 q^{44} -172.443 q^{45} -101.600 q^{46} +110.070 q^{47} -184.659 q^{48} -262.492 q^{49} +110.435 q^{50} +197.426 q^{51} +185.227 q^{52} -421.300 q^{53} +259.397 q^{54} +144.506 q^{55} -139.255 q^{56} -279.390 q^{57} +610.802 q^{58} +755.913 q^{59} -451.409 q^{60} -307.032 q^{61} +177.260 q^{62} -309.453 q^{63} -819.594 q^{64} -80.4396 q^{65} -1001.10 q^{66} +319.974 q^{67} +289.874 q^{68} -180.354 q^{69} +198.178 q^{70} -554.138 q^{71} +535.264 q^{72} -705.131 q^{73} +1677.48 q^{74} +196.037 q^{75} -410.219 q^{76} +259.318 q^{77} +557.268 q^{78} +1170.51 q^{79} +117.745 q^{80} -470.728 q^{81} +1822.24 q^{82} -455.978 q^{83} -810.063 q^{84} -125.886 q^{85} -1776.21 q^{86} +1084.26 q^{87} -448.546 q^{88} -1495.57 q^{89} -761.749 q^{90} -144.351 q^{91} -264.808 q^{92} +314.660 q^{93} +486.222 q^{94} +178.149 q^{95} -1789.31 q^{96} +1041.24 q^{97} -1159.53 q^{98} -996.760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19} - 110 q^{20} - 60 q^{21} + 193 q^{22} - 115 q^{23} + 105 q^{24} + 125 q^{25} - 257 q^{26} + 577 q^{27} + 17 q^{28} + 401 q^{29} - 95 q^{30} + 32 q^{31} + 670 q^{32} + 189 q^{33} - 663 q^{34} + 15 q^{35} - 659 q^{36} - 38 q^{37} - 875 q^{38} + 335 q^{39} - 690 q^{40} - 12 q^{41} - 798 q^{42} - 566 q^{43} + 47 q^{44} - 385 q^{45} - 138 q^{46} + 919 q^{47} - 773 q^{48} - 738 q^{49} + 150 q^{50} - 993 q^{51} - 305 q^{52} + 1156 q^{53} - 8 q^{54} - 115 q^{55} + 343 q^{56} + 114 q^{57} - 1042 q^{58} + 1324 q^{59} - 235 q^{60} - 1673 q^{61} + 565 q^{62} + 270 q^{63} + 2466 q^{64} - 660 q^{65} - 2781 q^{66} + 558 q^{67} - 2267 q^{68} - 92 q^{69} - 465 q^{70} - 108 q^{71} - 789 q^{72} + 1173 q^{73} + 1458 q^{74} + 100 q^{75} - 3477 q^{76} + 2608 q^{77} + 704 q^{78} + 656 q^{79} - 1410 q^{80} - 319 q^{81} + 3505 q^{82} - 82 q^{83} - 718 q^{84} - 115 q^{85} + 112 q^{86} + 2389 q^{87} + 2397 q^{88} + 570 q^{89} + 75 q^{90} - 1589 q^{91} - 506 q^{92} + 911 q^{93} - 948 q^{94} + 805 q^{95} - 5991 q^{96} + 633 q^{97} - 2555 q^{98} + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.41740 1.56179 0.780893 0.624665i \(-0.214765\pi\)
0.780893 + 0.624665i \(0.214765\pi\)
\(3\) 7.84147 1.50909 0.754546 0.656248i \(-0.227857\pi\)
0.754546 + 0.656248i \(0.227857\pi\)
\(4\) 11.5134 1.43917
\(5\) −5.00000 −0.447214
\(6\) 34.6389 2.35688
\(7\) −8.97260 −0.484475 −0.242237 0.970217i \(-0.577881\pi\)
−0.242237 + 0.970217i \(0.577881\pi\)
\(8\) 15.5200 0.685894
\(9\) 34.4886 1.27736
\(10\) −22.0870 −0.698452
\(11\) −28.9011 −0.792183 −0.396092 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(12\) 90.2818 2.17184
\(13\) 16.0879 0.343230 0.171615 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(14\) −39.6355 −0.756646
\(15\) −39.2073 −0.674886
\(16\) −23.5490 −0.367954
\(17\) 25.1771 0.359197 0.179599 0.983740i \(-0.442520\pi\)
0.179599 + 0.983740i \(0.442520\pi\)
\(18\) 152.350 1.99496
\(19\) −35.6298 −0.430212 −0.215106 0.976591i \(-0.569010\pi\)
−0.215106 + 0.976591i \(0.569010\pi\)
\(20\) −57.5669 −0.643618
\(21\) −70.3583 −0.731117
\(22\) −127.668 −1.23722
\(23\) −23.0000 −0.208514
\(24\) 121.700 1.03508
\(25\) 25.0000 0.200000
\(26\) 71.0668 0.536051
\(27\) 58.7218 0.418556
\(28\) −103.305 −0.697243
\(29\) 138.272 0.885395 0.442698 0.896671i \(-0.354022\pi\)
0.442698 + 0.896671i \(0.354022\pi\)
\(30\) −173.194 −1.05403
\(31\) 40.1277 0.232489 0.116244 0.993221i \(-0.462914\pi\)
0.116244 + 0.993221i \(0.462914\pi\)
\(32\) −228.186 −1.26056
\(33\) −226.627 −1.19548
\(34\) 111.217 0.560989
\(35\) 44.8630 0.216664
\(36\) 397.081 1.83834
\(37\) 379.745 1.68729 0.843645 0.536902i \(-0.180405\pi\)
0.843645 + 0.536902i \(0.180405\pi\)
\(38\) −157.391 −0.671899
\(39\) 126.153 0.517965
\(40\) −77.6001 −0.306741
\(41\) 412.514 1.57132 0.785658 0.618662i \(-0.212325\pi\)
0.785658 + 0.618662i \(0.212325\pi\)
\(42\) −310.801 −1.14185
\(43\) −402.095 −1.42602 −0.713011 0.701153i \(-0.752669\pi\)
−0.713011 + 0.701153i \(0.752669\pi\)
\(44\) −332.750 −1.14009
\(45\) −172.443 −0.571251
\(46\) −101.600 −0.325655
\(47\) 110.070 0.341603 0.170801 0.985305i \(-0.445364\pi\)
0.170801 + 0.985305i \(0.445364\pi\)
\(48\) −184.659 −0.555276
\(49\) −262.492 −0.765284
\(50\) 110.435 0.312357
\(51\) 197.426 0.542061
\(52\) 185.227 0.493967
\(53\) −421.300 −1.09189 −0.545943 0.837822i \(-0.683829\pi\)
−0.545943 + 0.837822i \(0.683829\pi\)
\(54\) 259.397 0.653695
\(55\) 144.506 0.354275
\(56\) −139.255 −0.332298
\(57\) −279.390 −0.649229
\(58\) 610.802 1.38280
\(59\) 755.913 1.66799 0.833996 0.551770i \(-0.186048\pi\)
0.833996 + 0.551770i \(0.186048\pi\)
\(60\) −451.409 −0.971278
\(61\) −307.032 −0.644450 −0.322225 0.946663i \(-0.604431\pi\)
−0.322225 + 0.946663i \(0.604431\pi\)
\(62\) 177.260 0.363098
\(63\) −309.453 −0.618847
\(64\) −819.594 −1.60077
\(65\) −80.4396 −0.153497
\(66\) −1001.10 −1.86708
\(67\) 319.974 0.583448 0.291724 0.956502i \(-0.405771\pi\)
0.291724 + 0.956502i \(0.405771\pi\)
\(68\) 289.874 0.516947
\(69\) −180.354 −0.314667
\(70\) 198.178 0.338382
\(71\) −554.138 −0.926254 −0.463127 0.886292i \(-0.653273\pi\)
−0.463127 + 0.886292i \(0.653273\pi\)
\(72\) 535.264 0.876131
\(73\) −705.131 −1.13054 −0.565269 0.824907i \(-0.691228\pi\)
−0.565269 + 0.824907i \(0.691228\pi\)
\(74\) 1677.48 2.63518
\(75\) 196.037 0.301818
\(76\) −410.219 −0.619149
\(77\) 259.318 0.383793
\(78\) 557.268 0.808950
\(79\) 1170.51 1.66699 0.833495 0.552526i \(-0.186336\pi\)
0.833495 + 0.552526i \(0.186336\pi\)
\(80\) 117.745 0.164554
\(81\) −470.728 −0.645717
\(82\) 1822.24 2.45406
\(83\) −455.978 −0.603014 −0.301507 0.953464i \(-0.597490\pi\)
−0.301507 + 0.953464i \(0.597490\pi\)
\(84\) −810.063 −1.05220
\(85\) −125.886 −0.160638
\(86\) −1776.21 −2.22714
\(87\) 1084.26 1.33614
\(88\) −448.546 −0.543354
\(89\) −1495.57 −1.78124 −0.890621 0.454746i \(-0.849730\pi\)
−0.890621 + 0.454746i \(0.849730\pi\)
\(90\) −761.749 −0.892172
\(91\) −144.351 −0.166286
\(92\) −264.808 −0.300088
\(93\) 314.660 0.350847
\(94\) 486.222 0.533510
\(95\) 178.149 0.192397
\(96\) −1789.31 −1.90230
\(97\) 1041.24 1.08992 0.544958 0.838463i \(-0.316545\pi\)
0.544958 + 0.838463i \(0.316545\pi\)
\(98\) −1159.53 −1.19521
\(99\) −996.760 −1.01190
\(100\) 287.835 0.287835
\(101\) 1450.22 1.42873 0.714367 0.699771i \(-0.246715\pi\)
0.714367 + 0.699771i \(0.246715\pi\)
\(102\) 872.108 0.846584
\(103\) 220.283 0.210729 0.105365 0.994434i \(-0.466399\pi\)
0.105365 + 0.994434i \(0.466399\pi\)
\(104\) 249.685 0.235419
\(105\) 351.792 0.326965
\(106\) −1861.05 −1.70529
\(107\) 59.2679 0.0535481 0.0267741 0.999642i \(-0.491477\pi\)
0.0267741 + 0.999642i \(0.491477\pi\)
\(108\) 676.087 0.602375
\(109\) 1966.29 1.72786 0.863930 0.503612i \(-0.167996\pi\)
0.863930 + 0.503612i \(0.167996\pi\)
\(110\) 638.339 0.553302
\(111\) 2977.76 2.54627
\(112\) 211.296 0.178264
\(113\) 1819.68 1.51488 0.757439 0.652905i \(-0.226450\pi\)
0.757439 + 0.652905i \(0.226450\pi\)
\(114\) −1234.17 −1.01396
\(115\) 115.000 0.0932505
\(116\) 1591.98 1.27424
\(117\) 554.851 0.438427
\(118\) 3339.17 2.60505
\(119\) −225.904 −0.174022
\(120\) −608.498 −0.462900
\(121\) −495.725 −0.372445
\(122\) −1356.28 −1.00649
\(123\) 3234.72 2.37126
\(124\) 462.006 0.334592
\(125\) −125.000 −0.0894427
\(126\) −1366.97 −0.966506
\(127\) 834.517 0.583082 0.291541 0.956558i \(-0.405832\pi\)
0.291541 + 0.956558i \(0.405832\pi\)
\(128\) −1794.98 −1.23950
\(129\) −3153.02 −2.15200
\(130\) −355.334 −0.239729
\(131\) 510.293 0.340340 0.170170 0.985415i \(-0.445568\pi\)
0.170170 + 0.985415i \(0.445568\pi\)
\(132\) −2609.25 −1.72050
\(133\) 319.692 0.208427
\(134\) 1413.45 0.911221
\(135\) −293.609 −0.187184
\(136\) 390.750 0.246371
\(137\) 11.2339 0.00700567 0.00350284 0.999994i \(-0.498885\pi\)
0.00350284 + 0.999994i \(0.498885\pi\)
\(138\) −796.694 −0.491443
\(139\) −1656.81 −1.01100 −0.505501 0.862826i \(-0.668692\pi\)
−0.505501 + 0.862826i \(0.668692\pi\)
\(140\) 516.525 0.311817
\(141\) 863.109 0.515510
\(142\) −2447.85 −1.44661
\(143\) −464.959 −0.271901
\(144\) −812.174 −0.470008
\(145\) −691.360 −0.395961
\(146\) −3114.84 −1.76566
\(147\) −2058.33 −1.15488
\(148\) 4372.15 2.42830
\(149\) −510.206 −0.280522 −0.140261 0.990115i \(-0.544794\pi\)
−0.140261 + 0.990115i \(0.544794\pi\)
\(150\) 865.972 0.471375
\(151\) −2337.38 −1.25969 −0.629846 0.776720i \(-0.716882\pi\)
−0.629846 + 0.776720i \(0.716882\pi\)
\(152\) −552.974 −0.295080
\(153\) 868.325 0.458823
\(154\) 1145.51 0.599402
\(155\) −200.639 −0.103972
\(156\) 1452.45 0.745442
\(157\) 146.592 0.0745178 0.0372589 0.999306i \(-0.488137\pi\)
0.0372589 + 0.999306i \(0.488137\pi\)
\(158\) 5170.59 2.60348
\(159\) −3303.61 −1.64776
\(160\) 1140.93 0.563739
\(161\) 206.370 0.101020
\(162\) −2079.39 −1.00847
\(163\) −3278.19 −1.57526 −0.787630 0.616149i \(-0.788692\pi\)
−0.787630 + 0.616149i \(0.788692\pi\)
\(164\) 4749.44 2.26139
\(165\) 1133.14 0.534634
\(166\) −2014.24 −0.941778
\(167\) −1555.42 −0.720732 −0.360366 0.932811i \(-0.617348\pi\)
−0.360366 + 0.932811i \(0.617348\pi\)
\(168\) −1091.96 −0.501469
\(169\) −1938.18 −0.882193
\(170\) −556.087 −0.250882
\(171\) −1228.82 −0.549534
\(172\) −4629.48 −2.05229
\(173\) −472.392 −0.207603 −0.103801 0.994598i \(-0.533101\pi\)
−0.103801 + 0.994598i \(0.533101\pi\)
\(174\) 4789.58 2.08677
\(175\) −224.315 −0.0968950
\(176\) 680.594 0.291487
\(177\) 5927.47 2.51715
\(178\) −6606.54 −2.78192
\(179\) 2429.45 1.01444 0.507222 0.861815i \(-0.330672\pi\)
0.507222 + 0.861815i \(0.330672\pi\)
\(180\) −1985.40 −0.822129
\(181\) −982.359 −0.403415 −0.201708 0.979446i \(-0.564649\pi\)
−0.201708 + 0.979446i \(0.564649\pi\)
\(182\) −637.653 −0.259703
\(183\) −2407.58 −0.972533
\(184\) −356.960 −0.143019
\(185\) −1898.73 −0.754579
\(186\) 1389.98 0.547947
\(187\) −727.648 −0.284550
\(188\) 1267.28 0.491626
\(189\) −526.887 −0.202780
\(190\) 786.954 0.300482
\(191\) 1361.14 0.515649 0.257824 0.966192i \(-0.416994\pi\)
0.257824 + 0.966192i \(0.416994\pi\)
\(192\) −6426.82 −2.41571
\(193\) −1456.29 −0.543142 −0.271571 0.962418i \(-0.587543\pi\)
−0.271571 + 0.962418i \(0.587543\pi\)
\(194\) 4599.57 1.70222
\(195\) −630.765 −0.231641
\(196\) −3022.18 −1.10138
\(197\) 1071.99 0.387695 0.193847 0.981032i \(-0.437903\pi\)
0.193847 + 0.981032i \(0.437903\pi\)
\(198\) −4403.08 −1.58037
\(199\) −2879.31 −1.02567 −0.512836 0.858486i \(-0.671405\pi\)
−0.512836 + 0.858486i \(0.671405\pi\)
\(200\) 388.000 0.137179
\(201\) 2509.07 0.880477
\(202\) 6406.19 2.23138
\(203\) −1240.66 −0.428952
\(204\) 2273.04 0.780120
\(205\) −2062.57 −0.702713
\(206\) 973.077 0.329114
\(207\) −793.238 −0.266347
\(208\) −378.855 −0.126293
\(209\) 1029.74 0.340807
\(210\) 1554.00 0.510650
\(211\) −4746.47 −1.54863 −0.774313 0.632802i \(-0.781905\pi\)
−0.774313 + 0.632802i \(0.781905\pi\)
\(212\) −4850.59 −1.57141
\(213\) −4345.25 −1.39780
\(214\) 261.810 0.0836306
\(215\) 2010.48 0.637737
\(216\) 911.363 0.287085
\(217\) −360.050 −0.112635
\(218\) 8685.90 2.69855
\(219\) −5529.26 −1.70609
\(220\) 1663.75 0.509863
\(221\) 405.048 0.123287
\(222\) 13153.9 3.97673
\(223\) 4874.04 1.46363 0.731815 0.681503i \(-0.238673\pi\)
0.731815 + 0.681503i \(0.238673\pi\)
\(224\) 2047.42 0.610709
\(225\) 862.216 0.255471
\(226\) 8038.26 2.36592
\(227\) −2742.09 −0.801757 −0.400879 0.916131i \(-0.631295\pi\)
−0.400879 + 0.916131i \(0.631295\pi\)
\(228\) −3216.72 −0.934353
\(229\) 1528.38 0.441041 0.220520 0.975382i \(-0.429224\pi\)
0.220520 + 0.975382i \(0.429224\pi\)
\(230\) 508.001 0.145637
\(231\) 2033.44 0.579179
\(232\) 2145.98 0.607287
\(233\) 5552.78 1.56126 0.780632 0.624991i \(-0.214897\pi\)
0.780632 + 0.624991i \(0.214897\pi\)
\(234\) 2450.99 0.684729
\(235\) −550.349 −0.152769
\(236\) 8703.12 2.40053
\(237\) 9178.49 2.51564
\(238\) −997.909 −0.271785
\(239\) 2779.63 0.752299 0.376149 0.926559i \(-0.377248\pi\)
0.376149 + 0.926559i \(0.377248\pi\)
\(240\) 923.295 0.248327
\(241\) −5568.82 −1.48846 −0.744231 0.667922i \(-0.767184\pi\)
−0.744231 + 0.667922i \(0.767184\pi\)
\(242\) −2189.81 −0.581680
\(243\) −5276.68 −1.39300
\(244\) −3534.98 −0.927475
\(245\) 1312.46 0.342245
\(246\) 14289.0 3.70340
\(247\) −573.209 −0.147662
\(248\) 622.783 0.159463
\(249\) −3575.54 −0.910003
\(250\) −552.174 −0.139690
\(251\) −387.065 −0.0973360 −0.0486680 0.998815i \(-0.515498\pi\)
−0.0486680 + 0.998815i \(0.515498\pi\)
\(252\) −3562.85 −0.890628
\(253\) 664.726 0.165182
\(254\) 3686.39 0.910649
\(255\) −987.129 −0.242417
\(256\) −1372.41 −0.335061
\(257\) 1476.07 0.358267 0.179133 0.983825i \(-0.442671\pi\)
0.179133 + 0.983825i \(0.442671\pi\)
\(258\) −13928.1 −3.36096
\(259\) −3407.30 −0.817449
\(260\) −926.133 −0.220909
\(261\) 4768.81 1.13097
\(262\) 2254.17 0.531537
\(263\) 203.984 0.0478259 0.0239130 0.999714i \(-0.492388\pi\)
0.0239130 + 0.999714i \(0.492388\pi\)
\(264\) −3517.26 −0.819971
\(265\) 2106.50 0.488306
\(266\) 1412.20 0.325518
\(267\) −11727.5 −2.68806
\(268\) 3683.98 0.839683
\(269\) −6973.71 −1.58065 −0.790324 0.612689i \(-0.790088\pi\)
−0.790324 + 0.612689i \(0.790088\pi\)
\(270\) −1296.99 −0.292341
\(271\) −2164.97 −0.485287 −0.242643 0.970116i \(-0.578015\pi\)
−0.242643 + 0.970116i \(0.578015\pi\)
\(272\) −592.898 −0.132168
\(273\) −1131.92 −0.250941
\(274\) 49.6246 0.0109414
\(275\) −722.528 −0.158437
\(276\) −2076.48 −0.452861
\(277\) 1194.00 0.258992 0.129496 0.991580i \(-0.458664\pi\)
0.129496 + 0.991580i \(0.458664\pi\)
\(278\) −7318.81 −1.57897
\(279\) 1383.95 0.296971
\(280\) 696.274 0.148608
\(281\) 6485.98 1.37694 0.688472 0.725263i \(-0.258282\pi\)
0.688472 + 0.725263i \(0.258282\pi\)
\(282\) 3812.69 0.805116
\(283\) −3214.41 −0.675182 −0.337591 0.941293i \(-0.609612\pi\)
−0.337591 + 0.941293i \(0.609612\pi\)
\(284\) −6380.00 −1.33304
\(285\) 1396.95 0.290344
\(286\) −2053.91 −0.424651
\(287\) −3701.33 −0.761263
\(288\) −7869.81 −1.61018
\(289\) −4279.11 −0.870977
\(290\) −3054.01 −0.618406
\(291\) 8164.85 1.64478
\(292\) −8118.44 −1.62704
\(293\) 5585.47 1.11367 0.556837 0.830622i \(-0.312015\pi\)
0.556837 + 0.830622i \(0.312015\pi\)
\(294\) −9092.44 −1.80368
\(295\) −3779.57 −0.745949
\(296\) 5893.65 1.15730
\(297\) −1697.13 −0.331573
\(298\) −2253.78 −0.438115
\(299\) −370.022 −0.0715684
\(300\) 2257.05 0.434369
\(301\) 3607.84 0.690872
\(302\) −10325.1 −1.96737
\(303\) 11371.8 2.15609
\(304\) 839.047 0.158298
\(305\) 1535.16 0.288207
\(306\) 3835.73 0.716583
\(307\) 2849.51 0.529740 0.264870 0.964284i \(-0.414671\pi\)
0.264870 + 0.964284i \(0.414671\pi\)
\(308\) 2985.63 0.552344
\(309\) 1727.34 0.318010
\(310\) −886.300 −0.162382
\(311\) −6617.84 −1.20664 −0.603318 0.797501i \(-0.706155\pi\)
−0.603318 + 0.797501i \(0.706155\pi\)
\(312\) 1957.90 0.355269
\(313\) −6271.26 −1.13250 −0.566250 0.824233i \(-0.691606\pi\)
−0.566250 + 0.824233i \(0.691606\pi\)
\(314\) 647.554 0.116381
\(315\) 1547.26 0.276757
\(316\) 13476.5 2.39909
\(317\) 5650.73 1.00119 0.500594 0.865682i \(-0.333115\pi\)
0.500594 + 0.865682i \(0.333115\pi\)
\(318\) −14593.3 −2.57344
\(319\) −3996.22 −0.701395
\(320\) 4097.97 0.715885
\(321\) 464.748 0.0808090
\(322\) 911.617 0.157772
\(323\) −897.055 −0.154531
\(324\) −5419.67 −0.929299
\(325\) 402.198 0.0686460
\(326\) −14481.0 −2.46022
\(327\) 15418.6 2.60750
\(328\) 6402.23 1.07776
\(329\) −987.612 −0.165498
\(330\) 5005.51 0.834983
\(331\) −6391.40 −1.06134 −0.530669 0.847579i \(-0.678059\pi\)
−0.530669 + 0.847579i \(0.678059\pi\)
\(332\) −5249.86 −0.867841
\(333\) 13096.9 2.15527
\(334\) −6870.92 −1.12563
\(335\) −1599.87 −0.260926
\(336\) 1656.87 0.269017
\(337\) 4153.87 0.671441 0.335721 0.941962i \(-0.391020\pi\)
0.335721 + 0.941962i \(0.391020\pi\)
\(338\) −8561.70 −1.37780
\(339\) 14269.0 2.28609
\(340\) −1449.37 −0.231186
\(341\) −1159.74 −0.184174
\(342\) −5428.19 −0.858254
\(343\) 5432.84 0.855236
\(344\) −6240.52 −0.978100
\(345\) 901.769 0.140723
\(346\) −2086.74 −0.324231
\(347\) −3071.86 −0.475234 −0.237617 0.971359i \(-0.576366\pi\)
−0.237617 + 0.971359i \(0.576366\pi\)
\(348\) 12483.4 1.92294
\(349\) −8358.91 −1.28207 −0.641035 0.767512i \(-0.721495\pi\)
−0.641035 + 0.767512i \(0.721495\pi\)
\(350\) −990.888 −0.151329
\(351\) 944.712 0.143661
\(352\) 6594.82 0.998594
\(353\) 8018.12 1.20896 0.604478 0.796622i \(-0.293382\pi\)
0.604478 + 0.796622i \(0.293382\pi\)
\(354\) 26184.0 3.93125
\(355\) 2770.69 0.414234
\(356\) −17219.1 −2.56352
\(357\) −1771.42 −0.262615
\(358\) 10731.8 1.58434
\(359\) 2138.46 0.314384 0.157192 0.987568i \(-0.449756\pi\)
0.157192 + 0.987568i \(0.449756\pi\)
\(360\) −2676.32 −0.391818
\(361\) −5589.52 −0.814918
\(362\) −4339.47 −0.630048
\(363\) −3887.21 −0.562054
\(364\) −1661.96 −0.239315
\(365\) 3525.65 0.505592
\(366\) −10635.2 −1.51889
\(367\) −4509.96 −0.641467 −0.320733 0.947170i \(-0.603929\pi\)
−0.320733 + 0.947170i \(0.603929\pi\)
\(368\) 541.628 0.0767237
\(369\) 14227.1 2.00713
\(370\) −8387.42 −1.17849
\(371\) 3780.15 0.528991
\(372\) 3622.80 0.504929
\(373\) −3362.09 −0.466709 −0.233355 0.972392i \(-0.574970\pi\)
−0.233355 + 0.972392i \(0.574970\pi\)
\(374\) −3214.31 −0.444406
\(375\) −980.184 −0.134977
\(376\) 1708.29 0.234303
\(377\) 2224.51 0.303894
\(378\) −2327.47 −0.316699
\(379\) 2107.16 0.285587 0.142793 0.989753i \(-0.454392\pi\)
0.142793 + 0.989753i \(0.454392\pi\)
\(380\) 2051.10 0.276892
\(381\) 6543.84 0.879924
\(382\) 6012.71 0.805333
\(383\) 1373.05 0.183184 0.0915919 0.995797i \(-0.470804\pi\)
0.0915919 + 0.995797i \(0.470804\pi\)
\(384\) −14075.3 −1.87052
\(385\) −1296.59 −0.171637
\(386\) −6433.03 −0.848271
\(387\) −13867.7 −1.82154
\(388\) 11988.2 1.56858
\(389\) 7008.05 0.913425 0.456713 0.889614i \(-0.349027\pi\)
0.456713 + 0.889614i \(0.349027\pi\)
\(390\) −2786.34 −0.361774
\(391\) −579.074 −0.0748978
\(392\) −4073.89 −0.524904
\(393\) 4001.44 0.513604
\(394\) 4735.39 0.605496
\(395\) −5852.53 −0.745501
\(396\) −11476.1 −1.45630
\(397\) 2402.53 0.303727 0.151864 0.988401i \(-0.451473\pi\)
0.151864 + 0.988401i \(0.451473\pi\)
\(398\) −12719.0 −1.60188
\(399\) 2506.85 0.314535
\(400\) −588.726 −0.0735908
\(401\) −11902.0 −1.48218 −0.741091 0.671404i \(-0.765691\pi\)
−0.741091 + 0.671404i \(0.765691\pi\)
\(402\) 11083.5 1.37512
\(403\) 645.572 0.0797971
\(404\) 16696.9 2.05620
\(405\) 2353.64 0.288773
\(406\) −5480.48 −0.669930
\(407\) −10975.1 −1.33664
\(408\) 3064.05 0.371797
\(409\) 5277.96 0.638089 0.319044 0.947740i \(-0.396638\pi\)
0.319044 + 0.947740i \(0.396638\pi\)
\(410\) −9111.20 −1.09749
\(411\) 88.0903 0.0105722
\(412\) 2536.20 0.303276
\(413\) −6782.51 −0.808100
\(414\) −3504.05 −0.415977
\(415\) 2279.89 0.269676
\(416\) −3671.03 −0.432662
\(417\) −12991.9 −1.52569
\(418\) 4548.77 0.532267
\(419\) −11196.4 −1.30545 −0.652723 0.757597i \(-0.726373\pi\)
−0.652723 + 0.757597i \(0.726373\pi\)
\(420\) 4050.31 0.470560
\(421\) 5176.82 0.599293 0.299647 0.954050i \(-0.403131\pi\)
0.299647 + 0.954050i \(0.403131\pi\)
\(422\) −20967.0 −2.41862
\(423\) 3796.16 0.436349
\(424\) −6538.58 −0.748918
\(425\) 629.428 0.0718394
\(426\) −19194.7 −2.18307
\(427\) 2754.88 0.312220
\(428\) 682.375 0.0770650
\(429\) −3645.96 −0.410324
\(430\) 8881.07 0.996008
\(431\) 9348.93 1.04483 0.522415 0.852691i \(-0.325031\pi\)
0.522415 + 0.852691i \(0.325031\pi\)
\(432\) −1382.84 −0.154009
\(433\) 4320.91 0.479560 0.239780 0.970827i \(-0.422925\pi\)
0.239780 + 0.970827i \(0.422925\pi\)
\(434\) −1590.48 −0.175912
\(435\) −5421.28 −0.597541
\(436\) 22638.7 2.48669
\(437\) 819.484 0.0897054
\(438\) −24424.9 −2.66454
\(439\) −7016.03 −0.762772 −0.381386 0.924416i \(-0.624553\pi\)
−0.381386 + 0.924416i \(0.624553\pi\)
\(440\) 2242.73 0.242995
\(441\) −9053.00 −0.977541
\(442\) 1789.26 0.192548
\(443\) 12343.6 1.32384 0.661921 0.749574i \(-0.269741\pi\)
0.661921 + 0.749574i \(0.269741\pi\)
\(444\) 34284.1 3.66453
\(445\) 7477.87 0.796596
\(446\) 21530.5 2.28588
\(447\) −4000.77 −0.423333
\(448\) 7353.88 0.775532
\(449\) 18146.9 1.90737 0.953683 0.300815i \(-0.0972587\pi\)
0.953683 + 0.300815i \(0.0972587\pi\)
\(450\) 3808.75 0.398991
\(451\) −11922.1 −1.24477
\(452\) 20950.7 2.18017
\(453\) −18328.5 −1.90099
\(454\) −12112.9 −1.25217
\(455\) 721.753 0.0743655
\(456\) −4336.13 −0.445302
\(457\) 5233.82 0.535728 0.267864 0.963457i \(-0.413682\pi\)
0.267864 + 0.963457i \(0.413682\pi\)
\(458\) 6751.47 0.688811
\(459\) 1478.45 0.150344
\(460\) 1324.04 0.134204
\(461\) −2335.01 −0.235905 −0.117952 0.993019i \(-0.537633\pi\)
−0.117952 + 0.993019i \(0.537633\pi\)
\(462\) 8982.49 0.904553
\(463\) 13644.1 1.36954 0.684769 0.728760i \(-0.259903\pi\)
0.684769 + 0.728760i \(0.259903\pi\)
\(464\) −3256.17 −0.325784
\(465\) −1573.30 −0.156903
\(466\) 24528.8 2.43836
\(467\) −5246.31 −0.519851 −0.259925 0.965629i \(-0.583698\pi\)
−0.259925 + 0.965629i \(0.583698\pi\)
\(468\) 6388.21 0.630972
\(469\) −2871.00 −0.282666
\(470\) −2431.11 −0.238593
\(471\) 1149.50 0.112454
\(472\) 11731.8 1.14407
\(473\) 11621.0 1.12967
\(474\) 40545.0 3.92889
\(475\) −890.744 −0.0860424
\(476\) −2600.92 −0.250448
\(477\) −14530.0 −1.39473
\(478\) 12278.7 1.17493
\(479\) 12278.3 1.17121 0.585603 0.810598i \(-0.300858\pi\)
0.585603 + 0.810598i \(0.300858\pi\)
\(480\) 8946.55 0.850734
\(481\) 6109.31 0.579128
\(482\) −24599.7 −2.32466
\(483\) 1618.24 0.152448
\(484\) −5707.47 −0.536013
\(485\) −5206.20 −0.487426
\(486\) −23309.2 −2.17557
\(487\) −8945.24 −0.832336 −0.416168 0.909288i \(-0.636627\pi\)
−0.416168 + 0.909288i \(0.636627\pi\)
\(488\) −4765.14 −0.442024
\(489\) −25705.8 −2.37721
\(490\) 5797.67 0.534514
\(491\) −7792.94 −0.716274 −0.358137 0.933669i \(-0.616588\pi\)
−0.358137 + 0.933669i \(0.616588\pi\)
\(492\) 37242.6 3.41265
\(493\) 3481.29 0.318031
\(494\) −2532.09 −0.230616
\(495\) 4983.80 0.452536
\(496\) −944.970 −0.0855451
\(497\) 4972.06 0.448747
\(498\) −15794.6 −1.42123
\(499\) −8577.71 −0.769521 −0.384761 0.923016i \(-0.625716\pi\)
−0.384761 + 0.923016i \(0.625716\pi\)
\(500\) −1439.17 −0.128724
\(501\) −12196.8 −1.08765
\(502\) −1709.82 −0.152018
\(503\) 7784.99 0.690091 0.345045 0.938586i \(-0.387864\pi\)
0.345045 + 0.938586i \(0.387864\pi\)
\(504\) −4802.71 −0.424464
\(505\) −7251.09 −0.638949
\(506\) 2936.36 0.257978
\(507\) −15198.2 −1.33131
\(508\) 9608.12 0.839156
\(509\) −11390.4 −0.991891 −0.495946 0.868354i \(-0.665178\pi\)
−0.495946 + 0.868354i \(0.665178\pi\)
\(510\) −4360.54 −0.378604
\(511\) 6326.85 0.547717
\(512\) 8297.40 0.716205
\(513\) −2092.24 −0.180068
\(514\) 6520.37 0.559535
\(515\) −1101.42 −0.0942411
\(516\) −36301.9 −3.09710
\(517\) −3181.14 −0.270612
\(518\) −15051.4 −1.27668
\(519\) −3704.24 −0.313292
\(520\) −1248.42 −0.105283
\(521\) −12824.5 −1.07841 −0.539206 0.842174i \(-0.681276\pi\)
−0.539206 + 0.842174i \(0.681276\pi\)
\(522\) 21065.7 1.76632
\(523\) 13087.4 1.09421 0.547107 0.837063i \(-0.315729\pi\)
0.547107 + 0.837063i \(0.315729\pi\)
\(524\) 5875.20 0.489808
\(525\) −1758.96 −0.146223
\(526\) 901.080 0.0746938
\(527\) 1010.30 0.0835093
\(528\) 5336.86 0.439880
\(529\) 529.000 0.0434783
\(530\) 9305.24 0.762630
\(531\) 26070.4 2.13062
\(532\) 3680.73 0.299962
\(533\) 6636.50 0.539322
\(534\) −51805.0 −4.19817
\(535\) −296.340 −0.0239474
\(536\) 4966.00 0.400184
\(537\) 19050.4 1.53089
\(538\) −30805.6 −2.46863
\(539\) 7586.33 0.606245
\(540\) −3380.43 −0.269390
\(541\) 15464.3 1.22895 0.614473 0.788938i \(-0.289369\pi\)
0.614473 + 0.788938i \(0.289369\pi\)
\(542\) −9563.54 −0.757914
\(543\) −7703.14 −0.608791
\(544\) −5745.06 −0.452789
\(545\) −9831.47 −0.772722
\(546\) −5000.14 −0.391916
\(547\) 6981.32 0.545703 0.272852 0.962056i \(-0.412033\pi\)
0.272852 + 0.962056i \(0.412033\pi\)
\(548\) 129.340 0.0100824
\(549\) −10589.1 −0.823192
\(550\) −3191.69 −0.247444
\(551\) −4926.60 −0.380908
\(552\) −2799.09 −0.215828
\(553\) −10502.5 −0.807615
\(554\) 5274.38 0.404489
\(555\) −14888.8 −1.13873
\(556\) −19075.5 −1.45501
\(557\) 24964.8 1.89909 0.949543 0.313637i \(-0.101547\pi\)
0.949543 + 0.313637i \(0.101547\pi\)
\(558\) 6113.45 0.463805
\(559\) −6468.88 −0.489454
\(560\) −1056.48 −0.0797222
\(561\) −5705.83 −0.429412
\(562\) 28651.2 2.15049
\(563\) −10820.3 −0.809986 −0.404993 0.914320i \(-0.632726\pi\)
−0.404993 + 0.914320i \(0.632726\pi\)
\(564\) 9937.31 0.741908
\(565\) −9098.41 −0.677474
\(566\) −14199.3 −1.05449
\(567\) 4223.65 0.312834
\(568\) −8600.23 −0.635312
\(569\) 8438.73 0.621740 0.310870 0.950453i \(-0.399380\pi\)
0.310870 + 0.950453i \(0.399380\pi\)
\(570\) 6170.87 0.453455
\(571\) 23107.6 1.69356 0.846781 0.531942i \(-0.178538\pi\)
0.846781 + 0.531942i \(0.178538\pi\)
\(572\) −5353.26 −0.391313
\(573\) 10673.4 0.778161
\(574\) −16350.2 −1.18893
\(575\) −575.000 −0.0417029
\(576\) −28266.7 −2.04475
\(577\) 24848.7 1.79283 0.896416 0.443215i \(-0.146162\pi\)
0.896416 + 0.443215i \(0.146162\pi\)
\(578\) −18902.5 −1.36028
\(579\) −11419.5 −0.819651
\(580\) −7959.89 −0.569856
\(581\) 4091.31 0.292145
\(582\) 36067.4 2.56880
\(583\) 12176.0 0.864974
\(584\) −10943.6 −0.775430
\(585\) −2774.25 −0.196070
\(586\) 24673.2 1.73932
\(587\) 8663.94 0.609198 0.304599 0.952481i \(-0.401478\pi\)
0.304599 + 0.952481i \(0.401478\pi\)
\(588\) −23698.3 −1.66208
\(589\) −1429.74 −0.100019
\(590\) −16695.8 −1.16501
\(591\) 8405.94 0.585066
\(592\) −8942.64 −0.620845
\(593\) −24678.9 −1.70901 −0.854503 0.519446i \(-0.826138\pi\)
−0.854503 + 0.519446i \(0.826138\pi\)
\(594\) −7496.88 −0.517846
\(595\) 1129.52 0.0778250
\(596\) −5874.20 −0.403719
\(597\) −22578.0 −1.54783
\(598\) −1634.54 −0.111774
\(599\) 19698.4 1.34367 0.671833 0.740702i \(-0.265507\pi\)
0.671833 + 0.740702i \(0.265507\pi\)
\(600\) 3042.49 0.207015
\(601\) −18449.1 −1.25217 −0.626086 0.779754i \(-0.715344\pi\)
−0.626086 + 0.779754i \(0.715344\pi\)
\(602\) 15937.3 1.07899
\(603\) 11035.5 0.745272
\(604\) −26911.2 −1.81291
\(605\) 2478.62 0.166563
\(606\) 50233.9 3.36735
\(607\) −11321.0 −0.757012 −0.378506 0.925599i \(-0.623562\pi\)
−0.378506 + 0.925599i \(0.623562\pi\)
\(608\) 8130.20 0.542308
\(609\) −9728.59 −0.647327
\(610\) 6781.41 0.450117
\(611\) 1770.80 0.117248
\(612\) 9997.36 0.660326
\(613\) 712.219 0.0469270 0.0234635 0.999725i \(-0.492531\pi\)
0.0234635 + 0.999725i \(0.492531\pi\)
\(614\) 12587.4 0.827341
\(615\) −16173.6 −1.06046
\(616\) 4024.62 0.263241
\(617\) 3234.25 0.211031 0.105515 0.994418i \(-0.466351\pi\)
0.105515 + 0.994418i \(0.466351\pi\)
\(618\) 7630.36 0.496663
\(619\) 26905.1 1.74703 0.873513 0.486801i \(-0.161836\pi\)
0.873513 + 0.486801i \(0.161836\pi\)
\(620\) −2310.03 −0.149634
\(621\) −1350.60 −0.0872750
\(622\) −29233.6 −1.88451
\(623\) 13419.2 0.862967
\(624\) −2970.78 −0.190587
\(625\) 625.000 0.0400000
\(626\) −27702.6 −1.76872
\(627\) 8074.68 0.514309
\(628\) 1687.77 0.107244
\(629\) 9560.90 0.606070
\(630\) 6834.87 0.432235
\(631\) 16199.5 1.02202 0.511008 0.859576i \(-0.329272\pi\)
0.511008 + 0.859576i \(0.329272\pi\)
\(632\) 18166.3 1.14338
\(633\) −37219.3 −2.33702
\(634\) 24961.5 1.56364
\(635\) −4172.59 −0.260762
\(636\) −38035.7 −2.37141
\(637\) −4222.96 −0.262668
\(638\) −17652.9 −1.09543
\(639\) −19111.5 −1.18316
\(640\) 8974.92 0.554320
\(641\) 19943.5 1.22889 0.614446 0.788959i \(-0.289380\pi\)
0.614446 + 0.788959i \(0.289380\pi\)
\(642\) 2052.97 0.126206
\(643\) −27691.0 −1.69833 −0.849164 0.528129i \(-0.822894\pi\)
−0.849164 + 0.528129i \(0.822894\pi\)
\(644\) 2376.01 0.145385
\(645\) 15765.1 0.962403
\(646\) −3962.65 −0.241344
\(647\) −23560.3 −1.43161 −0.715805 0.698300i \(-0.753940\pi\)
−0.715805 + 0.698300i \(0.753940\pi\)
\(648\) −7305.70 −0.442893
\(649\) −21846.7 −1.32136
\(650\) 1776.67 0.107210
\(651\) −2823.32 −0.169976
\(652\) −37743.0 −2.26707
\(653\) 5571.58 0.333894 0.166947 0.985966i \(-0.446609\pi\)
0.166947 + 0.985966i \(0.446609\pi\)
\(654\) 68110.2 4.07235
\(655\) −2551.46 −0.152205
\(656\) −9714.32 −0.578171
\(657\) −24319.0 −1.44410
\(658\) −4362.68 −0.258472
\(659\) 10177.4 0.601602 0.300801 0.953687i \(-0.402746\pi\)
0.300801 + 0.953687i \(0.402746\pi\)
\(660\) 13046.2 0.769430
\(661\) −27413.8 −1.61312 −0.806561 0.591151i \(-0.798674\pi\)
−0.806561 + 0.591151i \(0.798674\pi\)
\(662\) −28233.3 −1.65758
\(663\) 3176.17 0.186052
\(664\) −7076.79 −0.413604
\(665\) −1598.46 −0.0932113
\(666\) 57854.1 3.36607
\(667\) −3180.25 −0.184618
\(668\) −17908.2 −1.03726
\(669\) 38219.6 2.20875
\(670\) −7067.26 −0.407511
\(671\) 8873.57 0.510522
\(672\) 16054.8 0.921616
\(673\) 12767.5 0.731277 0.365639 0.930757i \(-0.380851\pi\)
0.365639 + 0.930757i \(0.380851\pi\)
\(674\) 18349.3 1.04865
\(675\) 1468.05 0.0837112
\(676\) −22315.0 −1.26963
\(677\) 17036.9 0.967180 0.483590 0.875295i \(-0.339333\pi\)
0.483590 + 0.875295i \(0.339333\pi\)
\(678\) 63031.7 3.57038
\(679\) −9342.63 −0.528037
\(680\) −1953.75 −0.110181
\(681\) −21502.0 −1.20993
\(682\) −5123.02 −0.287640
\(683\) −510.213 −0.0285838 −0.0142919 0.999898i \(-0.504549\pi\)
−0.0142919 + 0.999898i \(0.504549\pi\)
\(684\) −14147.9 −0.790875
\(685\) −56.1695 −0.00313303
\(686\) 23999.0 1.33569
\(687\) 11984.8 0.665570
\(688\) 9468.96 0.524710
\(689\) −6777.84 −0.374768
\(690\) 3983.47 0.219780
\(691\) 22793.1 1.25483 0.627416 0.778684i \(-0.284112\pi\)
0.627416 + 0.778684i \(0.284112\pi\)
\(692\) −5438.83 −0.298776
\(693\) 8943.53 0.490240
\(694\) −13569.6 −0.742213
\(695\) 8284.07 0.452134
\(696\) 16827.7 0.916452
\(697\) 10385.9 0.564412
\(698\) −36924.6 −2.00232
\(699\) 43541.9 2.35609
\(700\) −2582.62 −0.139449
\(701\) 26324.3 1.41834 0.709168 0.705039i \(-0.249071\pi\)
0.709168 + 0.705039i \(0.249071\pi\)
\(702\) 4173.17 0.224368
\(703\) −13530.2 −0.725892
\(704\) 23687.2 1.26810
\(705\) −4315.55 −0.230543
\(706\) 35419.2 1.88813
\(707\) −13012.2 −0.692185
\(708\) 68245.2 3.62262
\(709\) −10961.3 −0.580620 −0.290310 0.956933i \(-0.593758\pi\)
−0.290310 + 0.956933i \(0.593758\pi\)
\(710\) 12239.2 0.646944
\(711\) 40369.2 2.12934
\(712\) −23211.3 −1.22174
\(713\) −922.938 −0.0484773
\(714\) −7825.07 −0.410148
\(715\) 2324.80 0.121598
\(716\) 27971.2 1.45996
\(717\) 21796.4 1.13529
\(718\) 9446.44 0.491000
\(719\) −1304.68 −0.0676723 −0.0338362 0.999427i \(-0.510772\pi\)
−0.0338362 + 0.999427i \(0.510772\pi\)
\(720\) 4060.87 0.210194
\(721\) −1976.51 −0.102093
\(722\) −24691.1 −1.27273
\(723\) −43667.7 −2.24622
\(724\) −11310.3 −0.580585
\(725\) 3456.80 0.177079
\(726\) −17171.3 −0.877808
\(727\) 1583.59 0.0807869 0.0403934 0.999184i \(-0.487139\pi\)
0.0403934 + 0.999184i \(0.487139\pi\)
\(728\) −2240.32 −0.114055
\(729\) −28667.3 −1.45645
\(730\) 15574.2 0.789626
\(731\) −10123.6 −0.512223
\(732\) −27719.4 −1.39964
\(733\) −34351.2 −1.73095 −0.865477 0.500948i \(-0.832985\pi\)
−0.865477 + 0.500948i \(0.832985\pi\)
\(734\) −19922.3 −1.00183
\(735\) 10291.6 0.516480
\(736\) 5248.27 0.262845
\(737\) −9247.61 −0.462198
\(738\) 62846.5 3.13471
\(739\) 14996.6 0.746494 0.373247 0.927732i \(-0.378244\pi\)
0.373247 + 0.927732i \(0.378244\pi\)
\(740\) −21860.8 −1.08597
\(741\) −4494.80 −0.222835
\(742\) 16698.4 0.826171
\(743\) −32781.7 −1.61863 −0.809317 0.587372i \(-0.800162\pi\)
−0.809317 + 0.587372i \(0.800162\pi\)
\(744\) 4883.53 0.240644
\(745\) 2551.03 0.125453
\(746\) −14851.7 −0.728899
\(747\) −15726.1 −0.770263
\(748\) −8377.69 −0.409517
\(749\) −531.787 −0.0259427
\(750\) −4329.86 −0.210805
\(751\) −24044.4 −1.16830 −0.584149 0.811647i \(-0.698571\pi\)
−0.584149 + 0.811647i \(0.698571\pi\)
\(752\) −2592.04 −0.125694
\(753\) −3035.16 −0.146889
\(754\) 9826.54 0.474617
\(755\) 11686.9 0.563351
\(756\) −6066.26 −0.291835
\(757\) −21680.9 −1.04096 −0.520479 0.853874i \(-0.674247\pi\)
−0.520479 + 0.853874i \(0.674247\pi\)
\(758\) 9308.15 0.446025
\(759\) 5212.43 0.249274
\(760\) 2764.87 0.131964
\(761\) −10299.1 −0.490594 −0.245297 0.969448i \(-0.578885\pi\)
−0.245297 + 0.969448i \(0.578885\pi\)
\(762\) 28906.7 1.37425
\(763\) −17642.8 −0.837105
\(764\) 15671.4 0.742108
\(765\) −4341.62 −0.205192
\(766\) 6065.29 0.286094
\(767\) 12161.1 0.572505
\(768\) −10761.7 −0.505637
\(769\) 28377.9 1.33073 0.665365 0.746518i \(-0.268276\pi\)
0.665365 + 0.746518i \(0.268276\pi\)
\(770\) −5727.56 −0.268061
\(771\) 11574.5 0.540657
\(772\) −16766.9 −0.781675
\(773\) 35409.5 1.64759 0.823797 0.566885i \(-0.191852\pi\)
0.823797 + 0.566885i \(0.191852\pi\)
\(774\) −61259.2 −2.84485
\(775\) 1003.19 0.0464978
\(776\) 16160.1 0.747568
\(777\) −26718.2 −1.23361
\(778\) 30957.3 1.42657
\(779\) −14697.8 −0.675999
\(780\) −7262.24 −0.333372
\(781\) 16015.2 0.733763
\(782\) −2558.00 −0.116974
\(783\) 8119.58 0.370588
\(784\) 6181.45 0.281589
\(785\) −732.959 −0.0333254
\(786\) 17676.0 0.802138
\(787\) 6117.56 0.277087 0.138544 0.990356i \(-0.455758\pi\)
0.138544 + 0.990356i \(0.455758\pi\)
\(788\) 12342.2 0.557960
\(789\) 1599.54 0.0721737
\(790\) −25853.0 −1.16431
\(791\) −16327.3 −0.733921
\(792\) −15469.7 −0.694057
\(793\) −4939.51 −0.221194
\(794\) 10612.9 0.474357
\(795\) 16518.0 0.736899
\(796\) −33150.6 −1.47612
\(797\) 4099.46 0.182196 0.0910980 0.995842i \(-0.470962\pi\)
0.0910980 + 0.995842i \(0.470962\pi\)
\(798\) 11073.8 0.491236
\(799\) 2771.24 0.122703
\(800\) −5704.64 −0.252112
\(801\) −51580.3 −2.27528
\(802\) −52575.6 −2.31485
\(803\) 20379.1 0.895594
\(804\) 28887.8 1.26716
\(805\) −1031.85 −0.0451775
\(806\) 2851.75 0.124626
\(807\) −54684.1 −2.38534
\(808\) 22507.4 0.979960
\(809\) 21358.6 0.928216 0.464108 0.885779i \(-0.346375\pi\)
0.464108 + 0.885779i \(0.346375\pi\)
\(810\) 10397.0 0.451002
\(811\) 13967.7 0.604776 0.302388 0.953185i \(-0.402216\pi\)
0.302388 + 0.953185i \(0.402216\pi\)
\(812\) −14284.2 −0.617336
\(813\) −16976.6 −0.732342
\(814\) −48481.2 −2.08755
\(815\) 16390.9 0.704478
\(816\) −4649.19 −0.199454
\(817\) 14326.6 0.613492
\(818\) 23314.8 0.996557
\(819\) −4978.45 −0.212407
\(820\) −23747.2 −1.01133
\(821\) −22387.6 −0.951684 −0.475842 0.879531i \(-0.657857\pi\)
−0.475842 + 0.879531i \(0.657857\pi\)
\(822\) 389.130 0.0165115
\(823\) −22615.7 −0.957877 −0.478939 0.877848i \(-0.658978\pi\)
−0.478939 + 0.877848i \(0.658978\pi\)
\(824\) 3418.80 0.144538
\(825\) −5665.68 −0.239095
\(826\) −29961.0 −1.26208
\(827\) 10878.0 0.457394 0.228697 0.973498i \(-0.426554\pi\)
0.228697 + 0.973498i \(0.426554\pi\)
\(828\) −9132.86 −0.383320
\(829\) 27382.3 1.14720 0.573600 0.819136i \(-0.305547\pi\)
0.573600 + 0.819136i \(0.305547\pi\)
\(830\) 10071.2 0.421176
\(831\) 9362.74 0.390842
\(832\) −13185.6 −0.549432
\(833\) −6608.81 −0.274888
\(834\) −57390.2 −2.38281
\(835\) 7777.12 0.322321
\(836\) 11855.8 0.490480
\(837\) 2356.37 0.0973096
\(838\) −49459.1 −2.03883
\(839\) −31799.7 −1.30852 −0.654260 0.756270i \(-0.727020\pi\)
−0.654260 + 0.756270i \(0.727020\pi\)
\(840\) 5459.81 0.224264
\(841\) −5269.87 −0.216076
\(842\) 22868.0 0.935968
\(843\) 50859.6 2.07793
\(844\) −54647.9 −2.22874
\(845\) 9690.89 0.394529
\(846\) 16769.1 0.681483
\(847\) 4447.94 0.180440
\(848\) 9921.21 0.401764
\(849\) −25205.7 −1.01891
\(850\) 2780.43 0.112198
\(851\) −8734.14 −0.351824
\(852\) −50028.6 −2.01168
\(853\) 32016.4 1.28514 0.642568 0.766229i \(-0.277869\pi\)
0.642568 + 0.766229i \(0.277869\pi\)
\(854\) 12169.4 0.487620
\(855\) 6144.11 0.245759
\(856\) 919.839 0.0367283
\(857\) 25280.1 1.00764 0.503822 0.863807i \(-0.331927\pi\)
0.503822 + 0.863807i \(0.331927\pi\)
\(858\) −16105.7 −0.640837
\(859\) −22313.5 −0.886296 −0.443148 0.896448i \(-0.646138\pi\)
−0.443148 + 0.896448i \(0.646138\pi\)
\(860\) 23147.4 0.917813
\(861\) −29023.8 −1.14881
\(862\) 41297.9 1.63180
\(863\) −1478.28 −0.0583096 −0.0291548 0.999575i \(-0.509282\pi\)
−0.0291548 + 0.999575i \(0.509282\pi\)
\(864\) −13399.5 −0.527615
\(865\) 2361.96 0.0928428
\(866\) 19087.2 0.748970
\(867\) −33554.5 −1.31438
\(868\) −4145.39 −0.162101
\(869\) −33829.0 −1.32056
\(870\) −23947.9 −0.933230
\(871\) 5147.72 0.200257
\(872\) 30516.9 1.18513
\(873\) 35910.9 1.39221
\(874\) 3619.99 0.140101
\(875\) 1121.57 0.0433327
\(876\) −63660.5 −2.45535
\(877\) 32974.6 1.26964 0.634819 0.772661i \(-0.281075\pi\)
0.634819 + 0.772661i \(0.281075\pi\)
\(878\) −30992.6 −1.19129
\(879\) 43798.3 1.68064
\(880\) −3402.97 −0.130357
\(881\) −32000.7 −1.22376 −0.611879 0.790951i \(-0.709586\pi\)
−0.611879 + 0.790951i \(0.709586\pi\)
\(882\) −39990.7 −1.52671
\(883\) −44218.6 −1.68525 −0.842623 0.538503i \(-0.818990\pi\)
−0.842623 + 0.538503i \(0.818990\pi\)
\(884\) 4663.47 0.177432
\(885\) −29637.4 −1.12570
\(886\) 54526.6 2.06756
\(887\) 27374.8 1.03625 0.518126 0.855304i \(-0.326630\pi\)
0.518126 + 0.855304i \(0.326630\pi\)
\(888\) 46214.9 1.74647
\(889\) −7487.79 −0.282489
\(890\) 33032.7 1.24411
\(891\) 13604.6 0.511526
\(892\) 56116.7 2.10642
\(893\) −3921.76 −0.146962
\(894\) −17673.0 −0.661155
\(895\) −12147.2 −0.453673
\(896\) 16105.7 0.600505
\(897\) −2901.52 −0.108003
\(898\) 80162.2 2.97889
\(899\) 5548.54 0.205844
\(900\) 9927.02 0.367667
\(901\) −10607.1 −0.392203
\(902\) −52664.8 −1.94406
\(903\) 28290.8 1.04259
\(904\) 28241.5 1.03905
\(905\) 4911.80 0.180413
\(906\) −80964.3 −2.96894
\(907\) 7835.16 0.286838 0.143419 0.989662i \(-0.454190\pi\)
0.143419 + 0.989662i \(0.454190\pi\)
\(908\) −31570.7 −1.15387
\(909\) 50016.0 1.82500
\(910\) 3188.27 0.116143
\(911\) −36528.6 −1.32848 −0.664241 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(912\) 6579.36 0.238886
\(913\) 13178.3 0.477698
\(914\) 23119.8 0.836692
\(915\) 12037.9 0.434930
\(916\) 17596.8 0.634734
\(917\) −4578.65 −0.164886
\(918\) 6530.89 0.234805
\(919\) −15741.5 −0.565030 −0.282515 0.959263i \(-0.591169\pi\)
−0.282515 + 0.959263i \(0.591169\pi\)
\(920\) 1784.80 0.0639600
\(921\) 22344.4 0.799427
\(922\) −10314.7 −0.368433
\(923\) −8914.93 −0.317918
\(924\) 23411.7 0.833538
\(925\) 9493.63 0.337458
\(926\) 60271.5 2.13892
\(927\) 7597.26 0.269177
\(928\) −31551.7 −1.11609
\(929\) 47428.9 1.67502 0.837510 0.546422i \(-0.184011\pi\)
0.837510 + 0.546422i \(0.184011\pi\)
\(930\) −6949.89 −0.245049
\(931\) 9352.54 0.329234
\(932\) 63931.3 2.24693
\(933\) −51893.6 −1.82092
\(934\) −23175.0 −0.811895
\(935\) 3638.24 0.127255
\(936\) 8611.29 0.300714
\(937\) −34978.0 −1.21951 −0.609756 0.792589i \(-0.708733\pi\)
−0.609756 + 0.792589i \(0.708733\pi\)
\(938\) −12682.3 −0.441464
\(939\) −49175.9 −1.70905
\(940\) −6336.38 −0.219862
\(941\) 45144.0 1.56392 0.781962 0.623326i \(-0.214219\pi\)
0.781962 + 0.623326i \(0.214219\pi\)
\(942\) 5077.78 0.175629
\(943\) −9487.83 −0.327642
\(944\) −17801.0 −0.613744
\(945\) 2634.44 0.0906859
\(946\) 51334.6 1.76430
\(947\) 26123.6 0.896413 0.448207 0.893930i \(-0.352063\pi\)
0.448207 + 0.893930i \(0.352063\pi\)
\(948\) 105675. 3.62044
\(949\) −11344.1 −0.388035
\(950\) −3934.77 −0.134380
\(951\) 44310.0 1.51088
\(952\) −3506.04 −0.119361
\(953\) −22143.5 −0.752673 −0.376336 0.926483i \(-0.622816\pi\)
−0.376336 + 0.926483i \(0.622816\pi\)
\(954\) −64185.0 −2.17827
\(955\) −6805.72 −0.230605
\(956\) 32003.0 1.08269
\(957\) −31336.2 −1.05847
\(958\) 54237.9 1.82917
\(959\) −100.797 −0.00339407
\(960\) 32134.1 1.08034
\(961\) −28180.8 −0.945949
\(962\) 26987.3 0.904474
\(963\) 2044.07 0.0684000
\(964\) −64116.0 −2.14215
\(965\) 7281.47 0.242900
\(966\) 7148.42 0.238092
\(967\) 44869.8 1.49216 0.746078 0.665858i \(-0.231934\pi\)
0.746078 + 0.665858i \(0.231934\pi\)
\(968\) −7693.65 −0.255458
\(969\) −7034.23 −0.233201
\(970\) −22997.9 −0.761254
\(971\) −25048.3 −0.827845 −0.413923 0.910312i \(-0.635842\pi\)
−0.413923 + 0.910312i \(0.635842\pi\)
\(972\) −60752.5 −2.00477
\(973\) 14865.9 0.489805
\(974\) −39514.7 −1.29993
\(975\) 3153.82 0.103593
\(976\) 7230.31 0.237128
\(977\) 37320.7 1.22210 0.611052 0.791590i \(-0.290747\pi\)
0.611052 + 0.791590i \(0.290747\pi\)
\(978\) −113553. −3.71269
\(979\) 43223.8 1.41107
\(980\) 15110.9 0.492551
\(981\) 67814.7 2.20709
\(982\) −34424.5 −1.11867
\(983\) −17189.4 −0.557737 −0.278869 0.960329i \(-0.589959\pi\)
−0.278869 + 0.960329i \(0.589959\pi\)
\(984\) 50202.9 1.62643
\(985\) −5359.93 −0.173382
\(986\) 15378.2 0.496697
\(987\) −7744.33 −0.249752
\(988\) −6599.58 −0.212511
\(989\) 9248.19 0.297346
\(990\) 22015.4 0.706764
\(991\) −57797.1 −1.85266 −0.926330 0.376712i \(-0.877055\pi\)
−0.926330 + 0.376712i \(0.877055\pi\)
\(992\) −9156.57 −0.293066
\(993\) −50117.9 −1.60166
\(994\) 21963.5 0.700846
\(995\) 14396.5 0.458695
\(996\) −41166.6 −1.30965
\(997\) 46801.0 1.48666 0.743331 0.668923i \(-0.233245\pi\)
0.743331 + 0.668923i \(0.233245\pi\)
\(998\) −37891.2 −1.20183
\(999\) 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 115.4.a.e.1.4 5
3.2 odd 2 1035.4.a.k.1.2 5
4.3 odd 2 1840.4.a.n.1.2 5
5.2 odd 4 575.4.b.i.24.9 10
5.3 odd 4 575.4.b.i.24.2 10
5.4 even 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 1.1 even 1 trivial
575.4.a.j.1.2 5 5.4 even 2
575.4.b.i.24.2 10 5.3 odd 4
575.4.b.i.24.9 10 5.2 odd 4
1035.4.a.k.1.2 5 3.2 odd 2
1840.4.a.n.1.2 5 4.3 odd 2