Properties

Label 483.6.a.b.1.2
Level $483$
Weight $6$
Character 483.1
Self dual yes
Analytic conductor $77.465$
Analytic rank $1$
Dimension $12$
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] = [483,6,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(77.4653849697\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 268 x^{10} + 83 x^{9} + 25315 x^{8} + 5134 x^{7} - 993368 x^{6} - 511968 x^{5} + 14212480 x^{4} + 10085312 x^{3} - 18833856 x^{2} + \cdots + 102912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.45158\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.45158 q^{2} +9.00000 q^{3} +57.3324 q^{4} +43.5642 q^{5} -85.0642 q^{6} -49.0000 q^{7} -239.431 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.45158 q^{2} +9.00000 q^{3} +57.3324 q^{4} +43.5642 q^{5} -85.0642 q^{6} -49.0000 q^{7} -239.431 q^{8} +81.0000 q^{9} -411.751 q^{10} -280.372 q^{11} +515.992 q^{12} -66.9269 q^{13} +463.127 q^{14} +392.078 q^{15} +428.367 q^{16} +1147.68 q^{17} -765.578 q^{18} +832.663 q^{19} +2497.64 q^{20} -441.000 q^{21} +2649.96 q^{22} +529.000 q^{23} -2154.88 q^{24} -1227.16 q^{25} +632.565 q^{26} +729.000 q^{27} -2809.29 q^{28} -7403.94 q^{29} -3705.76 q^{30} +3551.90 q^{31} +3613.05 q^{32} -2523.35 q^{33} -10847.4 q^{34} -2134.65 q^{35} +4643.92 q^{36} -997.957 q^{37} -7869.98 q^{38} -602.342 q^{39} -10430.6 q^{40} -11509.2 q^{41} +4168.15 q^{42} -9987.91 q^{43} -16074.4 q^{44} +3528.70 q^{45} -4999.89 q^{46} -3589.49 q^{47} +3855.30 q^{48} +2401.00 q^{49} +11598.6 q^{50} +10329.1 q^{51} -3837.08 q^{52} -1360.92 q^{53} -6890.20 q^{54} -12214.2 q^{55} +11732.1 q^{56} +7493.96 q^{57} +69979.0 q^{58} -41158.9 q^{59} +22478.8 q^{60} +17699.0 q^{61} -33571.1 q^{62} -3969.00 q^{63} -47856.8 q^{64} -2915.62 q^{65} +23849.6 q^{66} -24228.6 q^{67} +65799.0 q^{68} +4761.00 q^{69} +20175.8 q^{70} +10714.2 q^{71} -19393.9 q^{72} +52743.5 q^{73} +9432.27 q^{74} -11044.4 q^{75} +47738.5 q^{76} +13738.2 q^{77} +5693.08 q^{78} +94926.3 q^{79} +18661.5 q^{80} +6561.00 q^{81} +108780. q^{82} -38010.8 q^{83} -25283.6 q^{84} +49997.7 q^{85} +94401.5 q^{86} -66635.5 q^{87} +67129.7 q^{88} -48776.9 q^{89} -33351.8 q^{90} +3279.42 q^{91} +30328.8 q^{92} +31967.1 q^{93} +33926.4 q^{94} +36274.3 q^{95} +32517.5 q^{96} +46593.3 q^{97} -22693.2 q^{98} -22710.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 108 q^{3} + 153 q^{4} - 162 q^{5} - 9 q^{6} - 588 q^{7} - 492 q^{8} + 972 q^{9} + 528 q^{10} - 1425 q^{11} + 1377 q^{12} - 70 q^{13} + 49 q^{14} - 1458 q^{15} + 3865 q^{16} - 398 q^{17} - 81 q^{18} - 1293 q^{19} - 8593 q^{20} - 5292 q^{21} + 4961 q^{22} + 6348 q^{23} - 4428 q^{24} + 5830 q^{25} - 5187 q^{26} + 8748 q^{27} - 7497 q^{28} - 5127 q^{29} + 4752 q^{30} + 6498 q^{31} - 28485 q^{32} - 12825 q^{33} - 14527 q^{34} + 7938 q^{35} + 12393 q^{36} - 35545 q^{37} - 32617 q^{38} - 630 q^{39} + 35789 q^{40} - 7806 q^{41} + 441 q^{42} - 66142 q^{43} - 83253 q^{44} - 13122 q^{45} - 529 q^{46} - 16432 q^{47} + 34785 q^{48} + 28812 q^{49} - 177328 q^{50} - 3582 q^{51} - 187010 q^{52} - 67456 q^{53} - 729 q^{54} - 10453 q^{55} + 24108 q^{56} - 11637 q^{57} - 92677 q^{58} - 36346 q^{59} - 77337 q^{60} - 8768 q^{61} - 141813 q^{62} - 47628 q^{63} - 24604 q^{64} + 121875 q^{65} + 44649 q^{66} - 123617 q^{67} + 17217 q^{68} + 57132 q^{69} - 25872 q^{70} - 108667 q^{71} - 39852 q^{72} - 107406 q^{73} - 87825 q^{74} + 52470 q^{75} + 120191 q^{76} + 69825 q^{77} - 46683 q^{78} - 39470 q^{79} - 513682 q^{80} + 78732 q^{81} + 150219 q^{82} - 181838 q^{83} - 67473 q^{84} - 52633 q^{85} + 125713 q^{86} - 46143 q^{87} + 120642 q^{88} - 277361 q^{89} + 42768 q^{90} + 3430 q^{91} + 80937 q^{92} + 58482 q^{93} - 40880 q^{94} - 272491 q^{95} - 256365 q^{96} - 169005 q^{97} - 2401 q^{98} - 115425 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.45158 −1.67082 −0.835410 0.549628i \(-0.814770\pi\)
−0.835410 + 0.549628i \(0.814770\pi\)
\(3\) 9.00000 0.577350
\(4\) 57.3324 1.79164
\(5\) 43.5642 0.779301 0.389650 0.920963i \(-0.372596\pi\)
0.389650 + 0.920963i \(0.372596\pi\)
\(6\) −85.0642 −0.964648
\(7\) −49.0000 −0.377964
\(8\) −239.431 −1.32268
\(9\) 81.0000 0.333333
\(10\) −411.751 −1.30207
\(11\) −280.372 −0.698638 −0.349319 0.937004i \(-0.613587\pi\)
−0.349319 + 0.937004i \(0.613587\pi\)
\(12\) 515.992 1.03440
\(13\) −66.9269 −0.109835 −0.0549177 0.998491i \(-0.517490\pi\)
−0.0549177 + 0.998491i \(0.517490\pi\)
\(14\) 463.127 0.631510
\(15\) 392.078 0.449929
\(16\) 428.367 0.418327
\(17\) 1147.68 0.963157 0.481579 0.876403i \(-0.340064\pi\)
0.481579 + 0.876403i \(0.340064\pi\)
\(18\) −765.578 −0.556940
\(19\) 832.663 0.529158 0.264579 0.964364i \(-0.414767\pi\)
0.264579 + 0.964364i \(0.414767\pi\)
\(20\) 2497.64 1.39622
\(21\) −441.000 −0.218218
\(22\) 2649.96 1.16730
\(23\) 529.000 0.208514
\(24\) −2154.88 −0.763651
\(25\) −1227.16 −0.392690
\(26\) 632.565 0.183515
\(27\) 729.000 0.192450
\(28\) −2809.29 −0.677175
\(29\) −7403.94 −1.63481 −0.817406 0.576062i \(-0.804589\pi\)
−0.817406 + 0.576062i \(0.804589\pi\)
\(30\) −3705.76 −0.751751
\(31\) 3551.90 0.663829 0.331915 0.943309i \(-0.392305\pi\)
0.331915 + 0.943309i \(0.392305\pi\)
\(32\) 3613.05 0.623734
\(33\) −2523.35 −0.403359
\(34\) −10847.4 −1.60926
\(35\) −2134.65 −0.294548
\(36\) 4643.92 0.597212
\(37\) −997.957 −0.119842 −0.0599208 0.998203i \(-0.519085\pi\)
−0.0599208 + 0.998203i \(0.519085\pi\)
\(38\) −7869.98 −0.884127
\(39\) −602.342 −0.0634135
\(40\) −10430.6 −1.03077
\(41\) −11509.2 −1.06926 −0.534631 0.845086i \(-0.679549\pi\)
−0.534631 + 0.845086i \(0.679549\pi\)
\(42\) 4168.15 0.364603
\(43\) −9987.91 −0.823765 −0.411882 0.911237i \(-0.635129\pi\)
−0.411882 + 0.911237i \(0.635129\pi\)
\(44\) −16074.4 −1.25171
\(45\) 3528.70 0.259767
\(46\) −4999.89 −0.348390
\(47\) −3589.49 −0.237022 −0.118511 0.992953i \(-0.537812\pi\)
−0.118511 + 0.992953i \(0.537812\pi\)
\(48\) 3855.30 0.241521
\(49\) 2401.00 0.142857
\(50\) 11598.6 0.656115
\(51\) 10329.1 0.556079
\(52\) −3837.08 −0.196785
\(53\) −1360.92 −0.0665491 −0.0332745 0.999446i \(-0.510594\pi\)
−0.0332745 + 0.999446i \(0.510594\pi\)
\(54\) −6890.20 −0.321549
\(55\) −12214.2 −0.544449
\(56\) 11732.1 0.499927
\(57\) 7493.96 0.305509
\(58\) 69979.0 2.73148
\(59\) −41158.9 −1.53934 −0.769668 0.638445i \(-0.779578\pi\)
−0.769668 + 0.638445i \(0.779578\pi\)
\(60\) 22478.8 0.806110
\(61\) 17699.0 0.609009 0.304505 0.952511i \(-0.401509\pi\)
0.304505 + 0.952511i \(0.401509\pi\)
\(62\) −33571.1 −1.10914
\(63\) −3969.00 −0.125988
\(64\) −47856.8 −1.46047
\(65\) −2915.62 −0.0855948
\(66\) 23849.6 0.673940
\(67\) −24228.6 −0.659387 −0.329694 0.944088i \(-0.606945\pi\)
−0.329694 + 0.944088i \(0.606945\pi\)
\(68\) 65799.0 1.72563
\(69\) 4761.00 0.120386
\(70\) 20175.8 0.492136
\(71\) 10714.2 0.252239 0.126120 0.992015i \(-0.459748\pi\)
0.126120 + 0.992015i \(0.459748\pi\)
\(72\) −19393.9 −0.440894
\(73\) 52743.5 1.15841 0.579205 0.815182i \(-0.303363\pi\)
0.579205 + 0.815182i \(0.303363\pi\)
\(74\) 9432.27 0.200234
\(75\) −11044.4 −0.226720
\(76\) 47738.5 0.948058
\(77\) 13738.2 0.264060
\(78\) 5693.08 0.105952
\(79\) 94926.3 1.71127 0.855635 0.517579i \(-0.173167\pi\)
0.855635 + 0.517579i \(0.173167\pi\)
\(80\) 18661.5 0.326002
\(81\) 6561.00 0.111111
\(82\) 108780. 1.78654
\(83\) −38010.8 −0.605636 −0.302818 0.953048i \(-0.597927\pi\)
−0.302818 + 0.953048i \(0.597927\pi\)
\(84\) −25283.6 −0.390967
\(85\) 49997.7 0.750589
\(86\) 94401.5 1.37636
\(87\) −66635.5 −0.943860
\(88\) 67129.7 0.924077
\(89\) −48776.9 −0.652738 −0.326369 0.945242i \(-0.605825\pi\)
−0.326369 + 0.945242i \(0.605825\pi\)
\(90\) −33351.8 −0.434024
\(91\) 3279.42 0.0415139
\(92\) 30328.8 0.373582
\(93\) 31967.1 0.383262
\(94\) 33926.4 0.396020
\(95\) 36274.3 0.412373
\(96\) 32517.5 0.360113
\(97\) 46593.3 0.502799 0.251399 0.967883i \(-0.419109\pi\)
0.251399 + 0.967883i \(0.419109\pi\)
\(98\) −22693.2 −0.238688
\(99\) −22710.1 −0.232879
\(100\) −70355.9 −0.703559
\(101\) 66377.4 0.647466 0.323733 0.946148i \(-0.395062\pi\)
0.323733 + 0.946148i \(0.395062\pi\)
\(102\) −97626.2 −0.929108
\(103\) −27114.2 −0.251828 −0.125914 0.992041i \(-0.540186\pi\)
−0.125914 + 0.992041i \(0.540186\pi\)
\(104\) 16024.4 0.145277
\(105\) −19211.8 −0.170057
\(106\) 12862.8 0.111191
\(107\) −67083.0 −0.566439 −0.283220 0.959055i \(-0.591403\pi\)
−0.283220 + 0.959055i \(0.591403\pi\)
\(108\) 41795.3 0.344801
\(109\) −62940.7 −0.507417 −0.253709 0.967281i \(-0.581650\pi\)
−0.253709 + 0.967281i \(0.581650\pi\)
\(110\) 115443. 0.909677
\(111\) −8981.61 −0.0691905
\(112\) −20990.0 −0.158113
\(113\) 171135. 1.26079 0.630394 0.776275i \(-0.282893\pi\)
0.630394 + 0.776275i \(0.282893\pi\)
\(114\) −70829.8 −0.510451
\(115\) 23045.5 0.162495
\(116\) −424486. −2.92899
\(117\) −5421.08 −0.0366118
\(118\) 389016. 2.57195
\(119\) −56236.2 −0.364039
\(120\) −93875.7 −0.595114
\(121\) −82442.7 −0.511904
\(122\) −167283. −1.01754
\(123\) −103582. −0.617339
\(124\) 203639. 1.18934
\(125\) −189598. −1.08532
\(126\) 37513.3 0.210503
\(127\) 142616. 0.784617 0.392309 0.919834i \(-0.371677\pi\)
0.392309 + 0.919834i \(0.371677\pi\)
\(128\) 336705. 1.81645
\(129\) −89891.2 −0.475601
\(130\) 27557.2 0.143013
\(131\) −306953. −1.56276 −0.781381 0.624054i \(-0.785485\pi\)
−0.781381 + 0.624054i \(0.785485\pi\)
\(132\) −144669. −0.722673
\(133\) −40800.5 −0.200003
\(134\) 228998. 1.10172
\(135\) 31758.3 0.149976
\(136\) −274790. −1.27395
\(137\) −258887. −1.17844 −0.589221 0.807972i \(-0.700565\pi\)
−0.589221 + 0.807972i \(0.700565\pi\)
\(138\) −44999.0 −0.201143
\(139\) 6303.56 0.0276726 0.0138363 0.999904i \(-0.495596\pi\)
0.0138363 + 0.999904i \(0.495596\pi\)
\(140\) −122384. −0.527723
\(141\) −32305.4 −0.136845
\(142\) −101266. −0.421446
\(143\) 18764.4 0.0767352
\(144\) 34697.7 0.139442
\(145\) −322547. −1.27401
\(146\) −498510. −1.93549
\(147\) 21609.0 0.0824786
\(148\) −57215.2 −0.214713
\(149\) 61201.8 0.225839 0.112919 0.993604i \(-0.463980\pi\)
0.112919 + 0.993604i \(0.463980\pi\)
\(150\) 104387. 0.378808
\(151\) −270243. −0.964523 −0.482262 0.876027i \(-0.660185\pi\)
−0.482262 + 0.876027i \(0.660185\pi\)
\(152\) −199365. −0.699908
\(153\) 92961.8 0.321052
\(154\) −129848. −0.441197
\(155\) 154736. 0.517323
\(156\) −34533.7 −0.113614
\(157\) −97760.9 −0.316531 −0.158265 0.987397i \(-0.550590\pi\)
−0.158265 + 0.987397i \(0.550590\pi\)
\(158\) −897203. −2.85922
\(159\) −12248.3 −0.0384221
\(160\) 157400. 0.486077
\(161\) −25921.0 −0.0788110
\(162\) −62011.8 −0.185647
\(163\) 40884.2 0.120527 0.0602637 0.998182i \(-0.480806\pi\)
0.0602637 + 0.998182i \(0.480806\pi\)
\(164\) −659848. −1.91573
\(165\) −109928. −0.314338
\(166\) 359262. 1.01191
\(167\) −132106. −0.366547 −0.183274 0.983062i \(-0.558669\pi\)
−0.183274 + 0.983062i \(0.558669\pi\)
\(168\) 105589. 0.288633
\(169\) −366814. −0.987936
\(170\) −472557. −1.25410
\(171\) 67445.7 0.176386
\(172\) −572631. −1.47589
\(173\) −686455. −1.74380 −0.871900 0.489683i \(-0.837112\pi\)
−0.871900 + 0.489683i \(0.837112\pi\)
\(174\) 629811. 1.57702
\(175\) 60130.7 0.148423
\(176\) −120102. −0.292259
\(177\) −370430. −0.888736
\(178\) 461019. 1.09061
\(179\) −625254. −1.45856 −0.729279 0.684217i \(-0.760144\pi\)
−0.729279 + 0.684217i \(0.760144\pi\)
\(180\) 202309. 0.465408
\(181\) −26029.4 −0.0590564 −0.0295282 0.999564i \(-0.509400\pi\)
−0.0295282 + 0.999564i \(0.509400\pi\)
\(182\) −30995.7 −0.0693622
\(183\) 159291. 0.351612
\(184\) −126659. −0.275798
\(185\) −43475.2 −0.0933926
\(186\) −302140. −0.640362
\(187\) −321776. −0.672899
\(188\) −205794. −0.424657
\(189\) −35721.0 −0.0727393
\(190\) −342850. −0.689001
\(191\) 131095. 0.260018 0.130009 0.991513i \(-0.458499\pi\)
0.130009 + 0.991513i \(0.458499\pi\)
\(192\) −430711. −0.843205
\(193\) 410139. 0.792571 0.396285 0.918127i \(-0.370299\pi\)
0.396285 + 0.918127i \(0.370299\pi\)
\(194\) −440381. −0.840086
\(195\) −26240.6 −0.0494182
\(196\) 137655. 0.255948
\(197\) −621690. −1.14132 −0.570661 0.821186i \(-0.693313\pi\)
−0.570661 + 0.821186i \(0.693313\pi\)
\(198\) 214646. 0.389100
\(199\) 921494. 1.64953 0.824764 0.565477i \(-0.191308\pi\)
0.824764 + 0.565477i \(0.191308\pi\)
\(200\) 293820. 0.519405
\(201\) −218057. −0.380697
\(202\) −627372. −1.08180
\(203\) 362793. 0.617901
\(204\) 592191. 0.996292
\(205\) −501388. −0.833277
\(206\) 256272. 0.420759
\(207\) 42849.0 0.0695048
\(208\) −28669.2 −0.0459471
\(209\) −233455. −0.369690
\(210\) 181582. 0.284135
\(211\) 863819. 1.33572 0.667862 0.744285i \(-0.267210\pi\)
0.667862 + 0.744285i \(0.267210\pi\)
\(212\) −78024.6 −0.119232
\(213\) 96427.6 0.145630
\(214\) 634041. 0.946418
\(215\) −435115. −0.641961
\(216\) −174545. −0.254550
\(217\) −174043. −0.250904
\(218\) 594889. 0.847803
\(219\) 474692. 0.668808
\(220\) −700268. −0.975456
\(221\) −76810.4 −0.105789
\(222\) 84890.4 0.115605
\(223\) −345030. −0.464616 −0.232308 0.972642i \(-0.574628\pi\)
−0.232308 + 0.972642i \(0.574628\pi\)
\(224\) −177040. −0.235749
\(225\) −99399.8 −0.130897
\(226\) −1.61749e6 −2.10655
\(227\) −542412. −0.698658 −0.349329 0.937000i \(-0.613590\pi\)
−0.349329 + 0.937000i \(0.613590\pi\)
\(228\) 429647. 0.547362
\(229\) 94472.2 0.119046 0.0595231 0.998227i \(-0.481042\pi\)
0.0595231 + 0.998227i \(0.481042\pi\)
\(230\) −217816. −0.271501
\(231\) 123644. 0.152455
\(232\) 1.77273e6 2.16234
\(233\) −727236. −0.877578 −0.438789 0.898590i \(-0.644592\pi\)
−0.438789 + 0.898590i \(0.644592\pi\)
\(234\) 51237.7 0.0611717
\(235\) −156373. −0.184711
\(236\) −2.35974e6 −2.75793
\(237\) 854336. 0.988003
\(238\) 531521. 0.608244
\(239\) 1.26515e6 1.43268 0.716340 0.697752i \(-0.245816\pi\)
0.716340 + 0.697752i \(0.245816\pi\)
\(240\) 167953. 0.188218
\(241\) 655951. 0.727493 0.363746 0.931498i \(-0.381498\pi\)
0.363746 + 0.931498i \(0.381498\pi\)
\(242\) 779214. 0.855300
\(243\) 59049.0 0.0641500
\(244\) 1.01473e6 1.09112
\(245\) 104598. 0.111329
\(246\) 979018. 1.03146
\(247\) −55727.5 −0.0581202
\(248\) −850435. −0.878036
\(249\) −342097. −0.349664
\(250\) 1.79200e6 1.81338
\(251\) 341516. 0.342158 0.171079 0.985257i \(-0.445275\pi\)
0.171079 + 0.985257i \(0.445275\pi\)
\(252\) −227552. −0.225725
\(253\) −148317. −0.145676
\(254\) −1.34794e6 −1.31095
\(255\) 449979. 0.433353
\(256\) −1.65098e6 −1.57449
\(257\) 248822. 0.234993 0.117497 0.993073i \(-0.462513\pi\)
0.117497 + 0.993073i \(0.462513\pi\)
\(258\) 849614. 0.794643
\(259\) 48899.9 0.0452958
\(260\) −167159. −0.153355
\(261\) −599719. −0.544938
\(262\) 2.90119e6 2.61109
\(263\) −1.66537e6 −1.48464 −0.742321 0.670044i \(-0.766275\pi\)
−0.742321 + 0.670044i \(0.766275\pi\)
\(264\) 604167. 0.533516
\(265\) −59287.3 −0.0518617
\(266\) 385629. 0.334169
\(267\) −438992. −0.376858
\(268\) −1.38908e6 −1.18138
\(269\) −1.52750e6 −1.28707 −0.643534 0.765417i \(-0.722533\pi\)
−0.643534 + 0.765417i \(0.722533\pi\)
\(270\) −300166. −0.250584
\(271\) −887513. −0.734094 −0.367047 0.930202i \(-0.619631\pi\)
−0.367047 + 0.930202i \(0.619631\pi\)
\(272\) 491626. 0.402915
\(273\) 29514.7 0.0239680
\(274\) 2.44689e6 1.96896
\(275\) 344060. 0.274349
\(276\) 272960. 0.215688
\(277\) −568262. −0.444989 −0.222495 0.974934i \(-0.571420\pi\)
−0.222495 + 0.974934i \(0.571420\pi\)
\(278\) −59578.7 −0.0462358
\(279\) 287704. 0.221276
\(280\) 511101. 0.389594
\(281\) 85997.1 0.0649708 0.0324854 0.999472i \(-0.489658\pi\)
0.0324854 + 0.999472i \(0.489658\pi\)
\(282\) 305337. 0.228643
\(283\) −1.71331e6 −1.27165 −0.635827 0.771832i \(-0.719341\pi\)
−0.635827 + 0.771832i \(0.719341\pi\)
\(284\) 614269. 0.451921
\(285\) 326469. 0.238084
\(286\) −177353. −0.128211
\(287\) 563949. 0.404143
\(288\) 292657. 0.207911
\(289\) −102695. −0.0723280
\(290\) 3.04858e6 2.12864
\(291\) 419340. 0.290291
\(292\) 3.02391e6 2.07545
\(293\) 775969. 0.528050 0.264025 0.964516i \(-0.414950\pi\)
0.264025 + 0.964516i \(0.414950\pi\)
\(294\) −204239. −0.137807
\(295\) −1.79305e6 −1.19961
\(296\) 238942. 0.158512
\(297\) −204391. −0.134453
\(298\) −578454. −0.377336
\(299\) −35404.3 −0.0229023
\(300\) −633203. −0.406200
\(301\) 489407. 0.311354
\(302\) 2.55423e6 1.61154
\(303\) 597397. 0.373815
\(304\) 356685. 0.221361
\(305\) 771043. 0.474601
\(306\) −878636. −0.536421
\(307\) −125127. −0.0757714 −0.0378857 0.999282i \(-0.512062\pi\)
−0.0378857 + 0.999282i \(0.512062\pi\)
\(308\) 787645. 0.473101
\(309\) −244028. −0.145393
\(310\) −1.46250e6 −0.864353
\(311\) −1.06602e6 −0.624977 −0.312489 0.949921i \(-0.601163\pi\)
−0.312489 + 0.949921i \(0.601163\pi\)
\(312\) 144219. 0.0838759
\(313\) −2.00246e6 −1.15532 −0.577660 0.816278i \(-0.696034\pi\)
−0.577660 + 0.816278i \(0.696034\pi\)
\(314\) 923996. 0.528866
\(315\) −172906. −0.0981827
\(316\) 5.44235e6 3.06598
\(317\) −3.07793e6 −1.72032 −0.860162 0.510021i \(-0.829638\pi\)
−0.860162 + 0.510021i \(0.829638\pi\)
\(318\) 115765. 0.0641964
\(319\) 2.07586e6 1.14214
\(320\) −2.08485e6 −1.13815
\(321\) −603747. −0.327034
\(322\) 244994. 0.131679
\(323\) 955627. 0.509662
\(324\) 376158. 0.199071
\(325\) 82129.8 0.0431313
\(326\) −386420. −0.201380
\(327\) −566466. −0.292957
\(328\) 2.75565e6 1.41429
\(329\) 175885. 0.0895858
\(330\) 1.03899e6 0.525202
\(331\) −2.80698e6 −1.40822 −0.704108 0.710093i \(-0.748653\pi\)
−0.704108 + 0.710093i \(0.748653\pi\)
\(332\) −2.17925e6 −1.08508
\(333\) −80834.5 −0.0399472
\(334\) 1.24861e6 0.612434
\(335\) −1.05550e6 −0.513861
\(336\) −188910. −0.0912864
\(337\) 1.07786e6 0.516995 0.258497 0.966012i \(-0.416773\pi\)
0.258497 + 0.966012i \(0.416773\pi\)
\(338\) 3.46697e6 1.65066
\(339\) 1.54021e6 0.727916
\(340\) 2.86649e6 1.34478
\(341\) −995852. −0.463777
\(342\) −637468. −0.294709
\(343\) −117649. −0.0539949
\(344\) 2.39142e6 1.08958
\(345\) 207409. 0.0938168
\(346\) 6.48809e6 2.91358
\(347\) 1.11455e6 0.496909 0.248454 0.968644i \(-0.420077\pi\)
0.248454 + 0.968644i \(0.420077\pi\)
\(348\) −3.82037e6 −1.69105
\(349\) −1.51049e6 −0.663825 −0.331912 0.943310i \(-0.607694\pi\)
−0.331912 + 0.943310i \(0.607694\pi\)
\(350\) −568330. −0.247988
\(351\) −48789.7 −0.0211378
\(352\) −1.01300e6 −0.435765
\(353\) 2.25980e6 0.965235 0.482618 0.875831i \(-0.339686\pi\)
0.482618 + 0.875831i \(0.339686\pi\)
\(354\) 3.50115e6 1.48492
\(355\) 466755. 0.196570
\(356\) −2.79649e6 −1.16947
\(357\) −506125. −0.210178
\(358\) 5.90963e6 2.43699
\(359\) 3.83389e6 1.57001 0.785007 0.619487i \(-0.212659\pi\)
0.785007 + 0.619487i \(0.212659\pi\)
\(360\) −844881. −0.343589
\(361\) −1.78277e6 −0.719992
\(362\) 246019. 0.0986726
\(363\) −741984. −0.295548
\(364\) 188017. 0.0743778
\(365\) 2.29773e6 0.902750
\(366\) −1.50555e6 −0.587479
\(367\) −3.92836e6 −1.52246 −0.761231 0.648480i \(-0.775405\pi\)
−0.761231 + 0.648480i \(0.775405\pi\)
\(368\) 226606. 0.0872272
\(369\) −932242. −0.356421
\(370\) 410910. 0.156042
\(371\) 66684.9 0.0251532
\(372\) 1.83275e6 0.686666
\(373\) −1.95136e6 −0.726215 −0.363107 0.931747i \(-0.618284\pi\)
−0.363107 + 0.931747i \(0.618284\pi\)
\(374\) 3.04129e6 1.12429
\(375\) −1.70639e6 −0.626612
\(376\) 859436. 0.313505
\(377\) 495523. 0.179560
\(378\) 337620. 0.121534
\(379\) −1.20683e6 −0.431567 −0.215784 0.976441i \(-0.569231\pi\)
−0.215784 + 0.976441i \(0.569231\pi\)
\(380\) 2.07969e6 0.738823
\(381\) 1.28354e6 0.452999
\(382\) −1.23906e6 −0.434443
\(383\) 2.15608e6 0.751049 0.375525 0.926812i \(-0.377463\pi\)
0.375525 + 0.926812i \(0.377463\pi\)
\(384\) 3.03034e6 1.04873
\(385\) 598495. 0.205783
\(386\) −3.87646e6 −1.32424
\(387\) −809020. −0.274588
\(388\) 2.67131e6 0.900833
\(389\) 79987.4 0.0268008 0.0134004 0.999910i \(-0.495734\pi\)
0.0134004 + 0.999910i \(0.495734\pi\)
\(390\) 248015. 0.0825688
\(391\) 607121. 0.200832
\(392\) −574874. −0.188955
\(393\) −2.76257e6 −0.902261
\(394\) 5.87595e6 1.90694
\(395\) 4.13539e6 1.33359
\(396\) −1.30202e6 −0.417236
\(397\) −3.48448e6 −1.10959 −0.554795 0.831987i \(-0.687203\pi\)
−0.554795 + 0.831987i \(0.687203\pi\)
\(398\) −8.70957e6 −2.75606
\(399\) −367204. −0.115472
\(400\) −525673. −0.164273
\(401\) −798809. −0.248074 −0.124037 0.992278i \(-0.539584\pi\)
−0.124037 + 0.992278i \(0.539584\pi\)
\(402\) 2.06098e6 0.636076
\(403\) −237717. −0.0729119
\(404\) 3.80558e6 1.16002
\(405\) 285825. 0.0865890
\(406\) −3.42897e6 −1.03240
\(407\) 279799. 0.0837259
\(408\) −2.47311e6 −0.735516
\(409\) 3.44063e6 1.01702 0.508510 0.861056i \(-0.330196\pi\)
0.508510 + 0.861056i \(0.330196\pi\)
\(410\) 4.73891e6 1.39225
\(411\) −2.32998e6 −0.680374
\(412\) −1.55452e6 −0.451185
\(413\) 2.01678e6 0.581814
\(414\) −404991. −0.116130
\(415\) −1.65591e6 −0.471973
\(416\) −241810. −0.0685081
\(417\) 56732.1 0.0159768
\(418\) 2.20652e6 0.617685
\(419\) −3.88408e6 −1.08082 −0.540409 0.841402i \(-0.681731\pi\)
−0.540409 + 0.841402i \(0.681731\pi\)
\(420\) −1.10146e6 −0.304681
\(421\) 1.03769e6 0.285340 0.142670 0.989770i \(-0.454431\pi\)
0.142670 + 0.989770i \(0.454431\pi\)
\(422\) −8.16446e6 −2.23175
\(423\) −290749. −0.0790072
\(424\) 325846. 0.0880233
\(425\) −1.40838e6 −0.378223
\(426\) −911393. −0.243322
\(427\) −867250. −0.230184
\(428\) −3.84603e6 −1.01485
\(429\) 168880. 0.0443031
\(430\) 4.11253e6 1.07260
\(431\) −1.16941e6 −0.303232 −0.151616 0.988439i \(-0.548448\pi\)
−0.151616 + 0.988439i \(0.548448\pi\)
\(432\) 312279. 0.0805070
\(433\) −787842. −0.201939 −0.100969 0.994890i \(-0.532194\pi\)
−0.100969 + 0.994890i \(0.532194\pi\)
\(434\) 1.64498e6 0.419215
\(435\) −2.90292e6 −0.735550
\(436\) −3.60854e6 −0.909108
\(437\) 440479. 0.110337
\(438\) −4.48659e6 −1.11746
\(439\) 1.47818e6 0.366072 0.183036 0.983106i \(-0.441407\pi\)
0.183036 + 0.983106i \(0.441407\pi\)
\(440\) 2.92445e6 0.720134
\(441\) 194481. 0.0476190
\(442\) 725980. 0.176754
\(443\) 3.46335e6 0.838470 0.419235 0.907878i \(-0.362298\pi\)
0.419235 + 0.907878i \(0.362298\pi\)
\(444\) −514937. −0.123964
\(445\) −2.12493e6 −0.508679
\(446\) 3.26108e6 0.776290
\(447\) 550816. 0.130388
\(448\) 2.34498e6 0.552007
\(449\) 502703. 0.117678 0.0588391 0.998267i \(-0.481260\pi\)
0.0588391 + 0.998267i \(0.481260\pi\)
\(450\) 939485. 0.218705
\(451\) 3.22684e6 0.747027
\(452\) 9.81157e6 2.25888
\(453\) −2.43219e6 −0.556868
\(454\) 5.12665e6 1.16733
\(455\) 142865. 0.0323518
\(456\) −1.79429e6 −0.404092
\(457\) 6.33402e6 1.41870 0.709348 0.704859i \(-0.248990\pi\)
0.709348 + 0.704859i \(0.248990\pi\)
\(458\) −892912. −0.198905
\(459\) 836656. 0.185360
\(460\) 1.32125e6 0.291133
\(461\) −6.42752e6 −1.40861 −0.704306 0.709897i \(-0.748742\pi\)
−0.704306 + 0.709897i \(0.748742\pi\)
\(462\) −1.16863e6 −0.254725
\(463\) 1.01387e6 0.219800 0.109900 0.993943i \(-0.464947\pi\)
0.109900 + 0.993943i \(0.464947\pi\)
\(464\) −3.17160e6 −0.683886
\(465\) 1.39262e6 0.298676
\(466\) 6.87353e6 1.46627
\(467\) −4.23941e6 −0.899524 −0.449762 0.893148i \(-0.648491\pi\)
−0.449762 + 0.893148i \(0.648491\pi\)
\(468\) −310803. −0.0655950
\(469\) 1.18720e6 0.249225
\(470\) 1.47798e6 0.308619
\(471\) −879848. −0.182749
\(472\) 9.85471e6 2.03605
\(473\) 2.80033e6 0.575514
\(474\) −8.07483e6 −1.65077
\(475\) −1.02181e6 −0.207795
\(476\) −3.22415e6 −0.652226
\(477\) −110234. −0.0221830
\(478\) −1.19577e7 −2.39375
\(479\) 4.42492e6 0.881185 0.440592 0.897707i \(-0.354768\pi\)
0.440592 + 0.897707i \(0.354768\pi\)
\(480\) 1.41660e6 0.280636
\(481\) 66790.1 0.0131628
\(482\) −6.19977e6 −1.21551
\(483\) −233289. −0.0455016
\(484\) −4.72664e6 −0.917147
\(485\) 2.02980e6 0.391832
\(486\) −558106. −0.107183
\(487\) 783310. 0.149662 0.0748309 0.997196i \(-0.476158\pi\)
0.0748309 + 0.997196i \(0.476158\pi\)
\(488\) −4.23769e6 −0.805526
\(489\) 367957. 0.0695866
\(490\) −988614. −0.186010
\(491\) 5.03694e6 0.942894 0.471447 0.881895i \(-0.343732\pi\)
0.471447 + 0.881895i \(0.343732\pi\)
\(492\) −5.93863e6 −1.10605
\(493\) −8.49733e6 −1.57458
\(494\) 526713. 0.0971084
\(495\) −989348. −0.181483
\(496\) 1.52152e6 0.277698
\(497\) −524995. −0.0953375
\(498\) 3.23336e6 0.584226
\(499\) −2.06678e6 −0.371573 −0.185786 0.982590i \(-0.559483\pi\)
−0.185786 + 0.982590i \(0.559483\pi\)
\(500\) −1.08701e7 −1.94451
\(501\) −1.18895e6 −0.211626
\(502\) −3.22786e6 −0.571684
\(503\) −1.31778e6 −0.232232 −0.116116 0.993236i \(-0.537044\pi\)
−0.116116 + 0.993236i \(0.537044\pi\)
\(504\) 950302. 0.166642
\(505\) 2.89168e6 0.504571
\(506\) 1.40183e6 0.243399
\(507\) −3.30132e6 −0.570385
\(508\) 8.17649e6 1.40575
\(509\) −1.06692e7 −1.82532 −0.912660 0.408720i \(-0.865975\pi\)
−0.912660 + 0.408720i \(0.865975\pi\)
\(510\) −4.25301e6 −0.724054
\(511\) −2.58443e6 −0.437838
\(512\) 4.82977e6 0.814239
\(513\) 607011. 0.101836
\(514\) −2.35176e6 −0.392631
\(515\) −1.18121e6 −0.196250
\(516\) −5.15368e6 −0.852104
\(517\) 1.00639e6 0.165592
\(518\) −462181. −0.0756812
\(519\) −6.17810e6 −1.00678
\(520\) 698090. 0.113215
\(521\) −9.37667e6 −1.51340 −0.756701 0.653761i \(-0.773190\pi\)
−0.756701 + 0.653761i \(0.773190\pi\)
\(522\) 5.66830e6 0.910492
\(523\) −8.94470e6 −1.42992 −0.714960 0.699165i \(-0.753555\pi\)
−0.714960 + 0.699165i \(0.753555\pi\)
\(524\) −1.75983e7 −2.79990
\(525\) 541176. 0.0856921
\(526\) 1.57404e7 2.48057
\(527\) 4.07643e6 0.639372
\(528\) −1.08092e6 −0.168736
\(529\) 279841. 0.0434783
\(530\) 560359. 0.0866516
\(531\) −3.33387e6 −0.513112
\(532\) −2.33919e6 −0.358332
\(533\) 770272. 0.117443
\(534\) 4.14917e6 0.629662
\(535\) −2.92242e6 −0.441426
\(536\) 5.80107e6 0.872160
\(537\) −5.62728e6 −0.842099
\(538\) 1.44373e7 2.15046
\(539\) −673172. −0.0998055
\(540\) 1.82078e6 0.268703
\(541\) −1.48099e6 −0.217550 −0.108775 0.994066i \(-0.534693\pi\)
−0.108775 + 0.994066i \(0.534693\pi\)
\(542\) 8.38840e6 1.22654
\(543\) −234264. −0.0340963
\(544\) 4.14662e6 0.600754
\(545\) −2.74196e6 −0.395431
\(546\) −278961. −0.0400463
\(547\) −3.38141e6 −0.483203 −0.241602 0.970376i \(-0.577673\pi\)
−0.241602 + 0.970376i \(0.577673\pi\)
\(548\) −1.48426e7 −2.11134
\(549\) 1.43362e6 0.203003
\(550\) −3.25191e6 −0.458387
\(551\) −6.16499e6 −0.865074
\(552\) −1.13993e6 −0.159232
\(553\) −4.65139e6 −0.646800
\(554\) 5.37098e6 0.743496
\(555\) −391277. −0.0539202
\(556\) 361398. 0.0495792
\(557\) −1.17512e7 −1.60488 −0.802442 0.596730i \(-0.796466\pi\)
−0.802442 + 0.596730i \(0.796466\pi\)
\(558\) −2.71926e6 −0.369713
\(559\) 668459. 0.0904785
\(560\) −914412. −0.123217
\(561\) −2.89598e6 −0.388498
\(562\) −812809. −0.108554
\(563\) 3.08008e6 0.409535 0.204767 0.978811i \(-0.434356\pi\)
0.204767 + 0.978811i \(0.434356\pi\)
\(564\) −1.85215e6 −0.245176
\(565\) 7.45536e6 0.982533
\(566\) 1.61935e7 2.12470
\(567\) −321489. −0.0419961
\(568\) −2.56531e6 −0.333633
\(569\) 2.42613e6 0.314148 0.157074 0.987587i \(-0.449794\pi\)
0.157074 + 0.987587i \(0.449794\pi\)
\(570\) −3.08565e6 −0.397795
\(571\) 6.72920e6 0.863721 0.431860 0.901940i \(-0.357857\pi\)
0.431860 + 0.901940i \(0.357857\pi\)
\(572\) 1.07581e6 0.137482
\(573\) 1.17986e6 0.150121
\(574\) −5.33021e6 −0.675250
\(575\) −649166. −0.0818816
\(576\) −3.87640e6 −0.486825
\(577\) −1.07527e7 −1.34456 −0.672280 0.740297i \(-0.734685\pi\)
−0.672280 + 0.740297i \(0.734685\pi\)
\(578\) 970633. 0.120847
\(579\) 3.69125e6 0.457591
\(580\) −1.84924e7 −2.28256
\(581\) 1.86253e6 0.228909
\(582\) −3.96343e6 −0.485024
\(583\) 381563. 0.0464937
\(584\) −1.26284e7 −1.53221
\(585\) −236165. −0.0285316
\(586\) −7.33413e6 −0.882276
\(587\) 3.00125e6 0.359507 0.179754 0.983712i \(-0.442470\pi\)
0.179754 + 0.983712i \(0.442470\pi\)
\(588\) 1.23890e6 0.147772
\(589\) 2.95753e6 0.351270
\(590\) 1.69472e7 2.00432
\(591\) −5.59521e6 −0.658942
\(592\) −427491. −0.0501329
\(593\) −8.19366e6 −0.956844 −0.478422 0.878130i \(-0.658791\pi\)
−0.478422 + 0.878130i \(0.658791\pi\)
\(594\) 1.93182e6 0.224647
\(595\) −2.44988e6 −0.283696
\(596\) 3.50884e6 0.404621
\(597\) 8.29344e6 0.952355
\(598\) 334627. 0.0382655
\(599\) 1.11379e7 1.26835 0.634173 0.773191i \(-0.281341\pi\)
0.634173 + 0.773191i \(0.281341\pi\)
\(600\) 2.64438e6 0.299879
\(601\) 813772. 0.0919002 0.0459501 0.998944i \(-0.485368\pi\)
0.0459501 + 0.998944i \(0.485368\pi\)
\(602\) −4.62567e6 −0.520216
\(603\) −1.96251e6 −0.219796
\(604\) −1.54937e7 −1.72808
\(605\) −3.59155e6 −0.398927
\(606\) −5.64635e6 −0.624577
\(607\) 1.11311e7 1.22621 0.613107 0.790000i \(-0.289919\pi\)
0.613107 + 0.790000i \(0.289919\pi\)
\(608\) 3.00846e6 0.330054
\(609\) 3.26514e6 0.356745
\(610\) −7.28757e6 −0.792973
\(611\) 240233. 0.0260334
\(612\) 5.32972e6 0.575210
\(613\) 497094. 0.0534302 0.0267151 0.999643i \(-0.491495\pi\)
0.0267151 + 0.999643i \(0.491495\pi\)
\(614\) 1.18265e6 0.126600
\(615\) −4.51249e6 −0.481092
\(616\) −3.28936e6 −0.349268
\(617\) 1.33391e7 1.41064 0.705318 0.708891i \(-0.250804\pi\)
0.705318 + 0.708891i \(0.250804\pi\)
\(618\) 2.30645e6 0.242925
\(619\) 1.40742e6 0.147638 0.0738188 0.997272i \(-0.476481\pi\)
0.0738188 + 0.997272i \(0.476481\pi\)
\(620\) 8.87137e6 0.926854
\(621\) 385641. 0.0401286
\(622\) 1.00756e7 1.04422
\(623\) 2.39007e6 0.246712
\(624\) −258023. −0.0265276
\(625\) −4.42484e6 −0.453104
\(626\) 1.89264e7 1.93033
\(627\) −2.10110e6 −0.213441
\(628\) −5.60487e6 −0.567109
\(629\) −1.14533e6 −0.115426
\(630\) 1.63424e6 0.164045
\(631\) 1.52181e7 1.52155 0.760777 0.649014i \(-0.224818\pi\)
0.760777 + 0.649014i \(0.224818\pi\)
\(632\) −2.27283e7 −2.26347
\(633\) 7.77438e6 0.771181
\(634\) 2.90913e7 2.87435
\(635\) 6.21294e6 0.611453
\(636\) −702222. −0.0688385
\(637\) −160691. −0.0156908
\(638\) −1.96201e7 −1.90831
\(639\) 867848. 0.0840798
\(640\) 1.46683e7 1.41556
\(641\) 1.94866e7 1.87323 0.936613 0.350366i \(-0.113943\pi\)
0.936613 + 0.350366i \(0.113943\pi\)
\(642\) 5.70637e6 0.546414
\(643\) −6.13077e6 −0.584774 −0.292387 0.956300i \(-0.594450\pi\)
−0.292387 + 0.956300i \(0.594450\pi\)
\(644\) −1.48611e6 −0.141201
\(645\) −3.91604e6 −0.370636
\(646\) −9.03219e6 −0.851553
\(647\) 267659. 0.0251374 0.0125687 0.999921i \(-0.495999\pi\)
0.0125687 + 0.999921i \(0.495999\pi\)
\(648\) −1.57091e6 −0.146965
\(649\) 1.15398e7 1.07544
\(650\) −776257. −0.0720646
\(651\) −1.56639e6 −0.144859
\(652\) 2.34399e6 0.215942
\(653\) 4.89649e6 0.449367 0.224684 0.974432i \(-0.427865\pi\)
0.224684 + 0.974432i \(0.427865\pi\)
\(654\) 5.35400e6 0.489479
\(655\) −1.33722e7 −1.21786
\(656\) −4.93014e6 −0.447301
\(657\) 4.27223e6 0.386137
\(658\) −1.66239e6 −0.149682
\(659\) 1.29049e7 1.15755 0.578775 0.815487i \(-0.303531\pi\)
0.578775 + 0.815487i \(0.303531\pi\)
\(660\) −6.30241e6 −0.563180
\(661\) −5.68181e6 −0.505805 −0.252902 0.967492i \(-0.581385\pi\)
−0.252902 + 0.967492i \(0.581385\pi\)
\(662\) 2.65304e7 2.35287
\(663\) −691294. −0.0610771
\(664\) 9.10097e6 0.801065
\(665\) −1.77744e6 −0.155862
\(666\) 764014. 0.0667445
\(667\) −3.91669e6 −0.340882
\(668\) −7.57393e6 −0.656719
\(669\) −3.10527e6 −0.268246
\(670\) 9.97613e6 0.858569
\(671\) −4.96229e6 −0.425477
\(672\) −1.59336e6 −0.136110
\(673\) 2.09836e7 1.78584 0.892918 0.450219i \(-0.148654\pi\)
0.892918 + 0.450219i \(0.148654\pi\)
\(674\) −1.01874e7 −0.863805
\(675\) −894598. −0.0755733
\(676\) −2.10303e7 −1.77002
\(677\) 1.55182e7 1.30128 0.650640 0.759386i \(-0.274501\pi\)
0.650640 + 0.759386i \(0.274501\pi\)
\(678\) −1.45574e7 −1.21622
\(679\) −2.28307e6 −0.190040
\(680\) −1.19710e7 −0.992791
\(681\) −4.88171e6 −0.403370
\(682\) 9.41238e6 0.774887
\(683\) 6.40256e6 0.525172 0.262586 0.964909i \(-0.415425\pi\)
0.262586 + 0.964909i \(0.415425\pi\)
\(684\) 3.86682e6 0.316019
\(685\) −1.12782e7 −0.918361
\(686\) 1.11197e6 0.0902158
\(687\) 850250. 0.0687314
\(688\) −4.27849e6 −0.344603
\(689\) 91081.9 0.00730944
\(690\) −1.96035e6 −0.156751
\(691\) 448684. 0.0357475 0.0178738 0.999840i \(-0.494310\pi\)
0.0178738 + 0.999840i \(0.494310\pi\)
\(692\) −3.93561e7 −3.12426
\(693\) 1.11280e6 0.0880202
\(694\) −1.05343e7 −0.830244
\(695\) 274610. 0.0215652
\(696\) 1.59546e7 1.24843
\(697\) −1.32088e7 −1.02987
\(698\) 1.42765e7 1.10913
\(699\) −6.54513e6 −0.506670
\(700\) 3.44744e6 0.265920
\(701\) 2.63513e6 0.202538 0.101269 0.994859i \(-0.467710\pi\)
0.101269 + 0.994859i \(0.467710\pi\)
\(702\) 461140. 0.0353175
\(703\) −830961. −0.0634151
\(704\) 1.34177e7 1.02034
\(705\) −1.40736e6 −0.106643
\(706\) −2.13587e7 −1.61273
\(707\) −3.25249e6 −0.244719
\(708\) −2.12376e7 −1.59229
\(709\) −5.40731e6 −0.403985 −0.201993 0.979387i \(-0.564742\pi\)
−0.201993 + 0.979387i \(0.564742\pi\)
\(710\) −4.41157e6 −0.328433
\(711\) 7.68903e6 0.570424
\(712\) 1.16787e7 0.863365
\(713\) 1.87895e6 0.138418
\(714\) 4.78369e6 0.351170
\(715\) 817457. 0.0597998
\(716\) −3.58473e7 −2.61321
\(717\) 1.13864e7 0.827158
\(718\) −3.62363e7 −2.62321
\(719\) 2.14255e7 1.54564 0.772820 0.634626i \(-0.218846\pi\)
0.772820 + 0.634626i \(0.218846\pi\)
\(720\) 1.51158e6 0.108667
\(721\) 1.32860e6 0.0951821
\(722\) 1.68500e7 1.20298
\(723\) 5.90356e6 0.420018
\(724\) −1.49233e6 −0.105808
\(725\) 9.08580e6 0.641975
\(726\) 7.01293e6 0.493808
\(727\) −8.23407e6 −0.577801 −0.288901 0.957359i \(-0.593290\pi\)
−0.288901 + 0.957359i \(0.593290\pi\)
\(728\) −785194. −0.0549097
\(729\) 531441. 0.0370370
\(730\) −2.17172e7 −1.50833
\(731\) −1.14629e7 −0.793415
\(732\) 9.13253e6 0.629960
\(733\) 640189. 0.0440097 0.0220048 0.999758i \(-0.492995\pi\)
0.0220048 + 0.999758i \(0.492995\pi\)
\(734\) 3.71293e7 2.54376
\(735\) 941380. 0.0642756
\(736\) 1.91131e6 0.130058
\(737\) 6.79300e6 0.460673
\(738\) 8.81116e6 0.595515
\(739\) 1.46203e7 0.984794 0.492397 0.870371i \(-0.336121\pi\)
0.492397 + 0.870371i \(0.336121\pi\)
\(740\) −2.49254e6 −0.167326
\(741\) −501547. −0.0335557
\(742\) −630278. −0.0420264
\(743\) 1.97843e7 1.31477 0.657383 0.753557i \(-0.271664\pi\)
0.657383 + 0.753557i \(0.271664\pi\)
\(744\) −7.65392e6 −0.506934
\(745\) 2.66621e6 0.175996
\(746\) 1.84434e7 1.21337
\(747\) −3.07887e6 −0.201879
\(748\) −1.84482e7 −1.20559
\(749\) 3.28707e6 0.214094
\(750\) 1.61280e7 1.04696
\(751\) −1.13853e7 −0.736619 −0.368310 0.929703i \(-0.620063\pi\)
−0.368310 + 0.929703i \(0.620063\pi\)
\(752\) −1.53762e6 −0.0991525
\(753\) 3.07364e6 0.197545
\(754\) −4.68347e6 −0.300013
\(755\) −1.17729e7 −0.751654
\(756\) −2.04797e6 −0.130322
\(757\) 4.36996e6 0.277165 0.138582 0.990351i \(-0.455745\pi\)
0.138582 + 0.990351i \(0.455745\pi\)
\(758\) 1.14065e7 0.721071
\(759\) −1.33485e6 −0.0841062
\(760\) −8.68520e6 −0.545439
\(761\) −3.29286e6 −0.206116 −0.103058 0.994675i \(-0.532863\pi\)
−0.103058 + 0.994675i \(0.532863\pi\)
\(762\) −1.21315e7 −0.756879
\(763\) 3.08409e6 0.191786
\(764\) 7.51600e6 0.465858
\(765\) 4.04981e6 0.250196
\(766\) −2.03784e7 −1.25487
\(767\) 2.75463e6 0.169073
\(768\) −1.48588e7 −0.909034
\(769\) 1.15568e7 0.704726 0.352363 0.935863i \(-0.385378\pi\)
0.352363 + 0.935863i \(0.385378\pi\)
\(770\) −5.65672e6 −0.343825
\(771\) 2.23940e6 0.135673
\(772\) 2.35143e7 1.42000
\(773\) 2.15820e7 1.29910 0.649550 0.760319i \(-0.274957\pi\)
0.649550 + 0.760319i \(0.274957\pi\)
\(774\) 7.64652e6 0.458787
\(775\) −4.35874e6 −0.260679
\(776\) −1.11559e7 −0.665043
\(777\) 440099. 0.0261516
\(778\) −756008. −0.0447793
\(779\) −9.58325e6 −0.565808
\(780\) −1.50443e6 −0.0885394
\(781\) −3.00395e6 −0.176224
\(782\) −5.73825e6 −0.335554
\(783\) −5.39747e6 −0.314620
\(784\) 1.02851e6 0.0597610
\(785\) −4.25888e6 −0.246673
\(786\) 2.61107e7 1.50752
\(787\) 3.76994e6 0.216969 0.108484 0.994098i \(-0.465400\pi\)
0.108484 + 0.994098i \(0.465400\pi\)
\(788\) −3.56430e7 −2.04483
\(789\) −1.49883e7 −0.857159
\(790\) −3.90860e7 −2.22820
\(791\) −8.38560e6 −0.476533
\(792\) 5.43751e6 0.308026
\(793\) −1.18454e6 −0.0668907
\(794\) 3.29339e7 1.85392
\(795\) −533586. −0.0299424
\(796\) 5.28314e7 2.95536
\(797\) 4.11076e6 0.229232 0.114616 0.993410i \(-0.463436\pi\)
0.114616 + 0.993410i \(0.463436\pi\)
\(798\) 3.47066e6 0.192932
\(799\) −4.11957e6 −0.228289
\(800\) −4.43379e6 −0.244934
\(801\) −3.95093e6 −0.217579
\(802\) 7.55001e6 0.414487
\(803\) −1.47878e7 −0.809309
\(804\) −1.25017e7 −0.682072
\(805\) −1.12923e6 −0.0614175
\(806\) 2.24681e6 0.121823
\(807\) −1.37475e7 −0.743089
\(808\) −1.58928e7 −0.856392
\(809\) 2.33800e7 1.25595 0.627977 0.778232i \(-0.283883\pi\)
0.627977 + 0.778232i \(0.283883\pi\)
\(810\) −2.70150e6 −0.144675
\(811\) 1.89532e7 1.01188 0.505942 0.862567i \(-0.331145\pi\)
0.505942 + 0.862567i \(0.331145\pi\)
\(812\) 2.07998e7 1.10705
\(813\) −7.98762e6 −0.423829
\(814\) −2.64454e6 −0.139891
\(815\) 1.78109e6 0.0939272
\(816\) 4.42464e6 0.232623
\(817\) −8.31656e6 −0.435902
\(818\) −3.25194e7 −1.69926
\(819\) 265633. 0.0138380
\(820\) −2.87458e7 −1.49293
\(821\) −2.37660e7 −1.23055 −0.615273 0.788314i \(-0.710954\pi\)
−0.615273 + 0.788314i \(0.710954\pi\)
\(822\) 2.20220e7 1.13678
\(823\) −2.72873e7 −1.40430 −0.702151 0.712028i \(-0.747777\pi\)
−0.702151 + 0.712028i \(0.747777\pi\)
\(824\) 6.49199e6 0.333089
\(825\) 3.09654e6 0.158395
\(826\) −1.90618e7 −0.972106
\(827\) −1.87246e7 −0.952023 −0.476012 0.879439i \(-0.657918\pi\)
−0.476012 + 0.879439i \(0.657918\pi\)
\(828\) 2.45664e6 0.124527
\(829\) 2.88037e7 1.45567 0.727833 0.685754i \(-0.240527\pi\)
0.727833 + 0.685754i \(0.240527\pi\)
\(830\) 1.56510e7 0.788581
\(831\) −5.11436e6 −0.256915
\(832\) 3.20291e6 0.160412
\(833\) 2.75557e6 0.137594
\(834\) −536208. −0.0266943
\(835\) −5.75508e6 −0.285650
\(836\) −1.33845e7 −0.662350
\(837\) 2.58933e6 0.127754
\(838\) 3.67107e7 1.80585
\(839\) 1.80901e7 0.887228 0.443614 0.896218i \(-0.353696\pi\)
0.443614 + 0.896218i \(0.353696\pi\)
\(840\) 4.59991e6 0.224932
\(841\) 3.43072e7 1.67261
\(842\) −9.80780e6 −0.476751
\(843\) 773974. 0.0375109
\(844\) 4.95248e7 2.39313
\(845\) −1.59800e7 −0.769899
\(846\) 2.74804e6 0.132007
\(847\) 4.03969e6 0.193482
\(848\) −582972. −0.0278393
\(849\) −1.54198e7 −0.734190
\(850\) 1.33114e7 0.631942
\(851\) −527919. −0.0249887
\(852\) 5.52842e6 0.260917
\(853\) 2.27481e6 0.107047 0.0535233 0.998567i \(-0.482955\pi\)
0.0535233 + 0.998567i \(0.482955\pi\)
\(854\) 8.19689e6 0.384596
\(855\) 2.93822e6 0.137458
\(856\) 1.60618e7 0.749219
\(857\) 8.68568e6 0.403972 0.201986 0.979388i \(-0.435260\pi\)
0.201986 + 0.979388i \(0.435260\pi\)
\(858\) −1.59618e6 −0.0740224
\(859\) 2.67321e7 1.23609 0.618045 0.786142i \(-0.287925\pi\)
0.618045 + 0.786142i \(0.287925\pi\)
\(860\) −2.49462e7 −1.15016
\(861\) 5.07554e6 0.233332
\(862\) 1.10528e7 0.506646
\(863\) −1.82060e7 −0.832122 −0.416061 0.909337i \(-0.636590\pi\)
−0.416061 + 0.909337i \(0.636590\pi\)
\(864\) 2.63392e6 0.120038
\(865\) −2.99049e7 −1.35895
\(866\) 7.44636e6 0.337403
\(867\) −924258. −0.0417586
\(868\) −9.97831e6 −0.449529
\(869\) −2.66146e7 −1.19556
\(870\) 2.74372e7 1.22897
\(871\) 1.62154e6 0.0724240
\(872\) 1.50700e7 0.671152
\(873\) 3.77406e6 0.167600
\(874\) −4.16322e6 −0.184353
\(875\) 9.29032e6 0.410214
\(876\) 2.72152e7 1.19826
\(877\) 5.85269e6 0.256955 0.128477 0.991712i \(-0.458991\pi\)
0.128477 + 0.991712i \(0.458991\pi\)
\(878\) −1.39712e7 −0.611641
\(879\) 6.98372e6 0.304870
\(880\) −5.23215e6 −0.227758
\(881\) −2.29356e7 −0.995567 −0.497783 0.867301i \(-0.665853\pi\)
−0.497783 + 0.867301i \(0.665853\pi\)
\(882\) −1.83815e6 −0.0795628
\(883\) −1.27042e6 −0.0548335 −0.0274167 0.999624i \(-0.508728\pi\)
−0.0274167 + 0.999624i \(0.508728\pi\)
\(884\) −4.40372e6 −0.189535
\(885\) −1.61375e7 −0.692592
\(886\) −3.27342e7 −1.40093
\(887\) −3.72691e6 −0.159052 −0.0795261 0.996833i \(-0.525341\pi\)
−0.0795261 + 0.996833i \(0.525341\pi\)
\(888\) 2.15048e6 0.0915171
\(889\) −6.98817e6 −0.296557
\(890\) 2.00839e7 0.849911
\(891\) −1.83952e6 −0.0776265
\(892\) −1.97814e7 −0.832424
\(893\) −2.98883e6 −0.125422
\(894\) −5.20608e6 −0.217855
\(895\) −2.72387e7 −1.13665
\(896\) −1.64985e7 −0.686555
\(897\) −318639. −0.0132226
\(898\) −4.75134e6 −0.196619
\(899\) −2.62981e7 −1.08524
\(900\) −5.69883e6 −0.234520
\(901\) −1.56189e6 −0.0640972
\(902\) −3.04988e7 −1.24815
\(903\) 4.40467e6 0.179760
\(904\) −4.09750e7 −1.66762
\(905\) −1.13395e6 −0.0460227
\(906\) 2.29881e7 0.930426
\(907\) 1.38899e7 0.560636 0.280318 0.959907i \(-0.409560\pi\)
0.280318 + 0.959907i \(0.409560\pi\)
\(908\) −3.10978e7 −1.25174
\(909\) 5.37657e6 0.215822
\(910\) −1.35030e6 −0.0540540
\(911\) −5.94594e6 −0.237369 −0.118685 0.992932i \(-0.537868\pi\)
−0.118685 + 0.992932i \(0.537868\pi\)
\(912\) 3.21016e6 0.127803
\(913\) 1.06572e7 0.423121
\(914\) −5.98665e7 −2.37038
\(915\) 6.93938e6 0.274011
\(916\) 5.41632e6 0.213288
\(917\) 1.50407e7 0.590669
\(918\) −7.90772e6 −0.309703
\(919\) 4.55301e7 1.77832 0.889161 0.457595i \(-0.151289\pi\)
0.889161 + 0.457595i \(0.151289\pi\)
\(920\) −5.51781e6 −0.214930
\(921\) −1.12614e6 −0.0437466
\(922\) 6.07502e7 2.35353
\(923\) −717066. −0.0277048
\(924\) 7.08880e6 0.273145
\(925\) 1.22465e6 0.0470606
\(926\) −9.58263e6 −0.367246
\(927\) −2.19625e6 −0.0839427
\(928\) −2.67508e7 −1.01969
\(929\) −5.10171e7 −1.93944 −0.969720 0.244219i \(-0.921468\pi\)
−0.969720 + 0.244219i \(0.921468\pi\)
\(930\) −1.31625e7 −0.499034
\(931\) 1.99922e6 0.0755939
\(932\) −4.16942e7 −1.57230
\(933\) −9.59418e6 −0.360831
\(934\) 4.00691e7 1.50294
\(935\) −1.40179e7 −0.524390
\(936\) 1.29797e6 0.0484258
\(937\) −1.78745e7 −0.665097 −0.332549 0.943086i \(-0.607909\pi\)
−0.332549 + 0.943086i \(0.607909\pi\)
\(938\) −1.12209e7 −0.416410
\(939\) −1.80221e7 −0.667024
\(940\) −8.96526e6 −0.330935
\(941\) −9.76160e6 −0.359374 −0.179687 0.983724i \(-0.557509\pi\)
−0.179687 + 0.983724i \(0.557509\pi\)
\(942\) 8.31596e6 0.305341
\(943\) −6.08835e6 −0.222957
\(944\) −1.76311e7 −0.643945
\(945\) −1.55616e6 −0.0566858
\(946\) −2.64675e7 −0.961580
\(947\) −1.90931e7 −0.691835 −0.345918 0.938265i \(-0.612432\pi\)
−0.345918 + 0.938265i \(0.612432\pi\)
\(948\) 4.89812e7 1.77014
\(949\) −3.52996e6 −0.127234
\(950\) 9.65770e6 0.347188
\(951\) −2.77014e7 −0.993230
\(952\) 1.34647e7 0.481508
\(953\) −4.18953e7 −1.49428 −0.747142 0.664665i \(-0.768574\pi\)
−0.747142 + 0.664665i \(0.768574\pi\)
\(954\) 1.04189e6 0.0370638
\(955\) 5.71106e6 0.202632
\(956\) 7.25344e7 2.56684
\(957\) 1.86827e7 0.659416
\(958\) −4.18225e7 −1.47230
\(959\) 1.26854e7 0.445409
\(960\) −1.87636e7 −0.657110
\(961\) −1.60132e7 −0.559331
\(962\) −631272. −0.0219927
\(963\) −5.43373e6 −0.188813
\(964\) 3.76072e7 1.30340
\(965\) 1.78674e7 0.617651
\(966\) 2.20495e6 0.0760249
\(967\) −7.15505e6 −0.246063 −0.123032 0.992403i \(-0.539262\pi\)
−0.123032 + 0.992403i \(0.539262\pi\)
\(968\) 1.97394e7 0.677087
\(969\) 8.60065e6 0.294254
\(970\) −1.91848e7 −0.654680
\(971\) 3.97454e7 1.35282 0.676408 0.736527i \(-0.263536\pi\)
0.676408 + 0.736527i \(0.263536\pi\)
\(972\) 3.38542e6 0.114934
\(973\) −308875. −0.0104592
\(974\) −7.40352e6 −0.250058
\(975\) 739168. 0.0249019
\(976\) 7.58166e6 0.254765
\(977\) −3.04507e6 −0.102061 −0.0510306 0.998697i \(-0.516251\pi\)
−0.0510306 + 0.998697i \(0.516251\pi\)
\(978\) −3.47778e6 −0.116267
\(979\) 1.36757e7 0.456028
\(980\) 5.99684e6 0.199461
\(981\) −5.09820e6 −0.169139
\(982\) −4.76070e7 −1.57540
\(983\) −5.24955e7 −1.73276 −0.866380 0.499385i \(-0.833559\pi\)
−0.866380 + 0.499385i \(0.833559\pi\)
\(984\) 2.48009e7 0.816543
\(985\) −2.70834e7 −0.889433
\(986\) 8.03132e7 2.63084
\(987\) 1.58297e6 0.0517224
\(988\) −3.19499e6 −0.104130
\(989\) −5.28360e6 −0.171767
\(990\) 9.35091e6 0.303226
\(991\) −6.26275e6 −0.202573 −0.101286 0.994857i \(-0.532296\pi\)
−0.101286 + 0.994857i \(0.532296\pi\)
\(992\) 1.28332e7 0.414053
\(993\) −2.52628e7 −0.813034
\(994\) 4.96203e6 0.159292
\(995\) 4.01442e7 1.28548
\(996\) −1.96133e7 −0.626471
\(997\) 1.28695e7 0.410036 0.205018 0.978758i \(-0.434275\pi\)
0.205018 + 0.978758i \(0.434275\pi\)
\(998\) 1.95344e7 0.620831
\(999\) −727510. −0.0230635
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 483.6.a.b.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.6.a.b.1.2 12 1.1 even 1 trivial