Properties

Label 507.4.b.k.337.8
Level $507$
Weight $4$
Character 507.337
Analytic conductor $29.914$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \( x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + 6703200 x^{4} + 137781 x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 13^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.8
Root \(0.107680i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.4.b.k.337.11

$q$-expansion

\(f(q)\) \(=\) \(q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-0.447278i q^{2} +3.00000 q^{3} +7.79994 q^{4} -1.93073i q^{5} -1.34183i q^{6} -8.14537i q^{7} -7.06697i q^{8} +9.00000 q^{9} -0.863573 q^{10} +8.40842i q^{11} +23.3998 q^{12} -3.64325 q^{14} -5.79219i q^{15} +59.2386 q^{16} +52.1271 q^{17} -4.02550i q^{18} -48.8304i q^{19} -15.0596i q^{20} -24.4361i q^{21} +3.76090 q^{22} -88.9229 q^{23} -21.2009i q^{24} +121.272 q^{25} +27.0000 q^{27} -63.5334i q^{28} +191.979 q^{29} -2.59072 q^{30} -115.257i q^{31} -83.0319i q^{32} +25.2252i q^{33} -23.3153i q^{34} -15.7265 q^{35} +70.1995 q^{36} -136.716i q^{37} -21.8408 q^{38} -13.6444 q^{40} -436.077i q^{41} -10.9297 q^{42} -202.048 q^{43} +65.5852i q^{44} -17.3766i q^{45} +39.7733i q^{46} +618.160i q^{47} +177.716 q^{48} +276.653 q^{49} -54.2425i q^{50} +156.381 q^{51} -453.170 q^{53} -12.0765i q^{54} +16.2344 q^{55} -57.5631 q^{56} -146.491i q^{57} -85.8679i q^{58} +500.044i q^{59} -45.1787i q^{60} +480.502 q^{61} -51.5522 q^{62} -73.3083i q^{63} +436.771 q^{64} +11.2827 q^{66} -886.769i q^{67} +406.589 q^{68} -266.769 q^{69} +7.03412i q^{70} -123.732i q^{71} -63.6027i q^{72} +673.168i q^{73} -61.1501 q^{74} +363.817 q^{75} -380.875i q^{76} +68.4897 q^{77} -681.298 q^{79} -114.374i q^{80} +81.0000 q^{81} -195.048 q^{82} +939.418i q^{83} -190.600i q^{84} -100.643i q^{85} +90.3716i q^{86} +575.936 q^{87} +59.4220 q^{88} -754.979i q^{89} -7.77215 q^{90} -693.594 q^{92} -345.772i q^{93} +276.489 q^{94} -94.2783 q^{95} -249.096i q^{96} -1051.10i q^{97} -123.741i q^{98} +75.6757i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.447278i − 0.158137i −0.996869 0.0790684i \(-0.974805\pi\)
0.996869 0.0790684i \(-0.0251945\pi\)
\(3\) 3.00000 0.577350
\(4\) 7.79994 0.974993
\(5\) − 1.93073i − 0.172690i −0.996265 0.0863448i \(-0.972481\pi\)
0.996265 0.0863448i \(-0.0275187\pi\)
\(6\) − 1.34183i − 0.0913003i
\(7\) − 8.14537i − 0.439809i −0.975521 0.219904i \(-0.929425\pi\)
0.975521 0.219904i \(-0.0705745\pi\)
\(8\) − 7.06697i − 0.312319i
\(9\) 9.00000 0.333333
\(10\) −0.863573 −0.0273086
\(11\) 8.40842i 0.230476i 0.993338 + 0.115238i \(0.0367630\pi\)
−0.993338 + 0.115238i \(0.963237\pi\)
\(12\) 23.3998 0.562912
\(13\) 0 0
\(14\) −3.64325 −0.0695499
\(15\) − 5.79219i − 0.0997024i
\(16\) 59.2386 0.925604
\(17\) 52.1271 0.743688 0.371844 0.928295i \(-0.378726\pi\)
0.371844 + 0.928295i \(0.378726\pi\)
\(18\) − 4.02550i − 0.0527122i
\(19\) − 48.8304i − 0.589604i −0.955558 0.294802i \(-0.904746\pi\)
0.955558 0.294802i \(-0.0952537\pi\)
\(20\) − 15.0596i − 0.168371i
\(21\) − 24.4361i − 0.253924i
\(22\) 3.76090 0.0364467
\(23\) −88.9229 −0.806162 −0.403081 0.915164i \(-0.632061\pi\)
−0.403081 + 0.915164i \(0.632061\pi\)
\(24\) − 21.2009i − 0.180317i
\(25\) 121.272 0.970178
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) − 63.5334i − 0.428810i
\(29\) 191.979 1.22930 0.614648 0.788802i \(-0.289298\pi\)
0.614648 + 0.788802i \(0.289298\pi\)
\(30\) −2.59072 −0.0157666
\(31\) − 115.257i − 0.667769i −0.942614 0.333885i \(-0.891640\pi\)
0.942614 0.333885i \(-0.108360\pi\)
\(32\) − 83.0319i − 0.458691i
\(33\) 25.2252i 0.133065i
\(34\) − 23.3153i − 0.117604i
\(35\) −15.7265 −0.0759504
\(36\) 70.1995 0.324998
\(37\) − 136.716i − 0.607459i −0.952758 0.303729i \(-0.901768\pi\)
0.952758 0.303729i \(-0.0982319\pi\)
\(38\) −21.8408 −0.0932380
\(39\) 0 0
\(40\) −13.6444 −0.0539342
\(41\) − 436.077i − 1.66107i −0.556967 0.830535i \(-0.688035\pi\)
0.556967 0.830535i \(-0.311965\pi\)
\(42\) −10.9297 −0.0401547
\(43\) −202.048 −0.716559 −0.358279 0.933614i \(-0.616636\pi\)
−0.358279 + 0.933614i \(0.616636\pi\)
\(44\) 65.5852i 0.224712i
\(45\) − 17.3766i − 0.0575632i
\(46\) 39.7733i 0.127484i
\(47\) 618.160i 1.91847i 0.282617 + 0.959233i \(0.408798\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(48\) 177.716 0.534398
\(49\) 276.653 0.806568
\(50\) − 54.2425i − 0.153421i
\(51\) 156.381 0.429368
\(52\) 0 0
\(53\) −453.170 −1.17448 −0.587242 0.809411i \(-0.699786\pi\)
−0.587242 + 0.809411i \(0.699786\pi\)
\(54\) − 12.0765i − 0.0304334i
\(55\) 16.2344 0.0398008
\(56\) −57.5631 −0.137361
\(57\) − 146.491i − 0.340408i
\(58\) − 85.8679i − 0.194397i
\(59\) 500.044i 1.10339i 0.834045 + 0.551697i \(0.186019\pi\)
−0.834045 + 0.551697i \(0.813981\pi\)
\(60\) − 45.1787i − 0.0972091i
\(61\) 480.502 1.00856 0.504279 0.863541i \(-0.331758\pi\)
0.504279 + 0.863541i \(0.331758\pi\)
\(62\) −51.5522 −0.105599
\(63\) − 73.3083i − 0.146603i
\(64\) 436.771 0.853068
\(65\) 0 0
\(66\) 11.2827 0.0210425
\(67\) − 886.769i − 1.61696i −0.588526 0.808478i \(-0.700292\pi\)
0.588526 0.808478i \(-0.299708\pi\)
\(68\) 406.589 0.725090
\(69\) −266.769 −0.465438
\(70\) 7.03412i 0.0120105i
\(71\) − 123.732i − 0.206821i −0.994639 0.103410i \(-0.967025\pi\)
0.994639 0.103410i \(-0.0329755\pi\)
\(72\) − 63.6027i − 0.104106i
\(73\) 673.168i 1.07929i 0.841892 + 0.539646i \(0.181442\pi\)
−0.841892 + 0.539646i \(0.818558\pi\)
\(74\) −61.1501 −0.0960616
\(75\) 363.817 0.560133
\(76\) − 380.875i − 0.574859i
\(77\) 68.4897 0.101365
\(78\) 0 0
\(79\) −681.298 −0.970279 −0.485139 0.874437i \(-0.661231\pi\)
−0.485139 + 0.874437i \(0.661231\pi\)
\(80\) − 114.374i − 0.159842i
\(81\) 81.0000 0.111111
\(82\) −195.048 −0.262676
\(83\) 939.418i 1.24234i 0.783675 + 0.621172i \(0.213343\pi\)
−0.783675 + 0.621172i \(0.786657\pi\)
\(84\) − 190.600i − 0.247574i
\(85\) − 100.643i − 0.128427i
\(86\) 90.3716i 0.113314i
\(87\) 575.936 0.709734
\(88\) 59.4220 0.0719819
\(89\) − 754.979i − 0.899187i −0.893233 0.449593i \(-0.851569\pi\)
0.893233 0.449593i \(-0.148431\pi\)
\(90\) −7.77215 −0.00910286
\(91\) 0 0
\(92\) −693.594 −0.786002
\(93\) − 345.772i − 0.385537i
\(94\) 276.489 0.303380
\(95\) −94.2783 −0.101818
\(96\) − 249.096i − 0.264825i
\(97\) − 1051.10i − 1.10024i −0.835087 0.550118i \(-0.814583\pi\)
0.835087 0.550118i \(-0.185417\pi\)
\(98\) − 123.741i − 0.127548i
\(99\) 75.6757i 0.0768252i
\(100\) 945.917 0.945917
\(101\) −599.873 −0.590986 −0.295493 0.955345i \(-0.595484\pi\)
−0.295493 + 0.955345i \(0.595484\pi\)
\(102\) − 69.9460i − 0.0678989i
\(103\) 293.312 0.280591 0.140295 0.990110i \(-0.455195\pi\)
0.140295 + 0.990110i \(0.455195\pi\)
\(104\) 0 0
\(105\) −47.1795 −0.0438500
\(106\) 202.693i 0.185729i
\(107\) 1533.69 1.38568 0.692839 0.721092i \(-0.256360\pi\)
0.692839 + 0.721092i \(0.256360\pi\)
\(108\) 210.598 0.187637
\(109\) − 590.718i − 0.519088i −0.965731 0.259544i \(-0.916428\pi\)
0.965731 0.259544i \(-0.0835722\pi\)
\(110\) − 7.26128i − 0.00629396i
\(111\) − 410.148i − 0.350717i
\(112\) − 482.521i − 0.407089i
\(113\) −653.785 −0.544274 −0.272137 0.962259i \(-0.587730\pi\)
−0.272137 + 0.962259i \(0.587730\pi\)
\(114\) −65.5224 −0.0538310
\(115\) 171.686i 0.139216i
\(116\) 1497.42 1.19855
\(117\) 0 0
\(118\) 223.659 0.174487
\(119\) − 424.595i − 0.327080i
\(120\) −40.9332 −0.0311389
\(121\) 1260.30 0.946881
\(122\) − 214.918i − 0.159490i
\(123\) − 1308.23i − 0.959019i
\(124\) − 899.002i − 0.651070i
\(125\) − 475.485i − 0.340229i
\(126\) −32.7892 −0.0231833
\(127\) 141.203 0.0986595 0.0493297 0.998783i \(-0.484291\pi\)
0.0493297 + 0.998783i \(0.484291\pi\)
\(128\) − 859.613i − 0.593592i
\(129\) −606.144 −0.413705
\(130\) 0 0
\(131\) −731.910 −0.488147 −0.244074 0.969757i \(-0.578484\pi\)
−0.244074 + 0.969757i \(0.578484\pi\)
\(132\) 196.755i 0.129738i
\(133\) −397.742 −0.259313
\(134\) −396.633 −0.255700
\(135\) − 52.1297i − 0.0332341i
\(136\) − 368.381i − 0.232268i
\(137\) 1762.27i 1.09898i 0.835499 + 0.549492i \(0.185179\pi\)
−0.835499 + 0.549492i \(0.814821\pi\)
\(138\) 119.320i 0.0736028i
\(139\) −664.776 −0.405652 −0.202826 0.979215i \(-0.565013\pi\)
−0.202826 + 0.979215i \(0.565013\pi\)
\(140\) −122.666 −0.0740511
\(141\) 1854.48i 1.10763i
\(142\) −55.3425 −0.0327059
\(143\) 0 0
\(144\) 533.148 0.308535
\(145\) − 370.659i − 0.212286i
\(146\) 301.094 0.170676
\(147\) 829.959 0.465672
\(148\) − 1066.38i − 0.592268i
\(149\) 3300.71i 1.81480i 0.420270 + 0.907399i \(0.361936\pi\)
−0.420270 + 0.907399i \(0.638064\pi\)
\(150\) − 162.727i − 0.0885776i
\(151\) 1464.15i 0.789079i 0.918879 + 0.394540i \(0.129096\pi\)
−0.918879 + 0.394540i \(0.870904\pi\)
\(152\) −345.083 −0.184144
\(153\) 469.144 0.247896
\(154\) − 30.6339i − 0.0160296i
\(155\) −222.531 −0.115317
\(156\) 0 0
\(157\) 1535.32 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(158\) 304.730i 0.153437i
\(159\) −1359.51 −0.678089
\(160\) −160.312 −0.0792111
\(161\) 724.310i 0.354557i
\(162\) − 36.2295i − 0.0175707i
\(163\) 793.554i 0.381325i 0.981656 + 0.190663i \(0.0610636\pi\)
−0.981656 + 0.190663i \(0.938936\pi\)
\(164\) − 3401.38i − 1.61953i
\(165\) 48.7031 0.0229790
\(166\) 420.181 0.196460
\(167\) − 1336.87i − 0.619460i −0.950825 0.309730i \(-0.899761\pi\)
0.950825 0.309730i \(-0.100239\pi\)
\(168\) −172.689 −0.0793052
\(169\) 0 0
\(170\) −45.0156 −0.0203090
\(171\) − 439.474i − 0.196535i
\(172\) −1575.96 −0.698640
\(173\) −568.460 −0.249822 −0.124911 0.992168i \(-0.539865\pi\)
−0.124911 + 0.992168i \(0.539865\pi\)
\(174\) − 257.604i − 0.112235i
\(175\) − 987.808i − 0.426693i
\(176\) 498.103i 0.213329i
\(177\) 1500.13i 0.637044i
\(178\) −337.686 −0.142194
\(179\) −1546.00 −0.645552 −0.322776 0.946475i \(-0.604616\pi\)
−0.322776 + 0.946475i \(0.604616\pi\)
\(180\) − 135.536i − 0.0561237i
\(181\) −3408.96 −1.39992 −0.699960 0.714182i \(-0.746799\pi\)
−0.699960 + 0.714182i \(0.746799\pi\)
\(182\) 0 0
\(183\) 1441.51 0.582291
\(184\) 628.416i 0.251780i
\(185\) −263.962 −0.104902
\(186\) −154.656 −0.0609675
\(187\) 438.307i 0.171402i
\(188\) 4821.61i 1.87049i
\(189\) − 219.925i − 0.0846412i
\(190\) 42.1686i 0.0161012i
\(191\) −3464.71 −1.31255 −0.656277 0.754520i \(-0.727870\pi\)
−0.656277 + 0.754520i \(0.727870\pi\)
\(192\) 1310.31 0.492519
\(193\) 4652.24i 1.73510i 0.497346 + 0.867552i \(0.334308\pi\)
−0.497346 + 0.867552i \(0.665692\pi\)
\(194\) −470.134 −0.173988
\(195\) 0 0
\(196\) 2157.88 0.786398
\(197\) 2870.98i 1.03832i 0.854677 + 0.519160i \(0.173755\pi\)
−0.854677 + 0.519160i \(0.826245\pi\)
\(198\) 33.8481 0.0121489
\(199\) −25.0330 −0.00891730 −0.00445865 0.999990i \(-0.501419\pi\)
−0.00445865 + 0.999990i \(0.501419\pi\)
\(200\) − 857.028i − 0.303005i
\(201\) − 2660.31i − 0.933550i
\(202\) 268.310i 0.0934565i
\(203\) − 1563.74i − 0.540655i
\(204\) 1219.77 0.418631
\(205\) −841.947 −0.286849
\(206\) − 131.192i − 0.0443717i
\(207\) −800.307 −0.268721
\(208\) 0 0
\(209\) 410.587 0.135889
\(210\) 21.1024i 0.00693429i
\(211\) −3605.91 −1.17650 −0.588249 0.808679i \(-0.700183\pi\)
−0.588249 + 0.808679i \(0.700183\pi\)
\(212\) −3534.70 −1.14511
\(213\) − 371.195i − 0.119408i
\(214\) − 685.987i − 0.219127i
\(215\) 390.100i 0.123742i
\(216\) − 190.808i − 0.0601058i
\(217\) −938.815 −0.293691
\(218\) −264.215 −0.0820868
\(219\) 2019.50i 0.623130i
\(220\) 126.627 0.0388054
\(221\) 0 0
\(222\) −183.450 −0.0554612
\(223\) 4304.37i 1.29256i 0.763098 + 0.646282i \(0.223677\pi\)
−0.763098 + 0.646282i \(0.776323\pi\)
\(224\) −676.326 −0.201736
\(225\) 1091.45 0.323393
\(226\) 292.424i 0.0860697i
\(227\) 5475.19i 1.60089i 0.599409 + 0.800443i \(0.295402\pi\)
−0.599409 + 0.800443i \(0.704598\pi\)
\(228\) − 1142.62i − 0.331895i
\(229\) 378.108i 0.109110i 0.998511 + 0.0545548i \(0.0173740\pi\)
−0.998511 + 0.0545548i \(0.982626\pi\)
\(230\) 76.7914 0.0220151
\(231\) 205.469 0.0585232
\(232\) − 1356.71i − 0.383932i
\(233\) −2547.96 −0.716405 −0.358202 0.933644i \(-0.616610\pi\)
−0.358202 + 0.933644i \(0.616610\pi\)
\(234\) 0 0
\(235\) 1193.50 0.331299
\(236\) 3900.32i 1.07580i
\(237\) −2043.89 −0.560191
\(238\) −189.912 −0.0517234
\(239\) 6313.91i 1.70884i 0.519583 + 0.854420i \(0.326087\pi\)
−0.519583 + 0.854420i \(0.673913\pi\)
\(240\) − 343.121i − 0.0922849i
\(241\) − 6763.73i − 1.80784i −0.427699 0.903921i \(-0.640676\pi\)
0.427699 0.903921i \(-0.359324\pi\)
\(242\) − 563.704i − 0.149737i
\(243\) 243.000 0.0641500
\(244\) 3747.89 0.983336
\(245\) − 534.142i − 0.139286i
\(246\) −585.144 −0.151656
\(247\) 0 0
\(248\) −814.521 −0.208557
\(249\) 2818.25i 0.717267i
\(250\) −212.674 −0.0538027
\(251\) 3416.00 0.859028 0.429514 0.903060i \(-0.358685\pi\)
0.429514 + 0.903060i \(0.358685\pi\)
\(252\) − 571.801i − 0.142937i
\(253\) − 747.701i − 0.185801i
\(254\) − 63.1571i − 0.0156017i
\(255\) − 301.930i − 0.0741474i
\(256\) 3109.68 0.759199
\(257\) −7002.36 −1.69959 −0.849796 0.527112i \(-0.823275\pi\)
−0.849796 + 0.527112i \(0.823275\pi\)
\(258\) 271.115i 0.0654220i
\(259\) −1113.60 −0.267166
\(260\) 0 0
\(261\) 1727.81 0.409765
\(262\) 327.367i 0.0771940i
\(263\) 3369.76 0.790071 0.395035 0.918666i \(-0.370732\pi\)
0.395035 + 0.918666i \(0.370732\pi\)
\(264\) 178.266 0.0415588
\(265\) 874.948i 0.202821i
\(266\) 177.901i 0.0410069i
\(267\) − 2264.94i − 0.519146i
\(268\) − 6916.75i − 1.57652i
\(269\) −7801.70 −1.76832 −0.884160 0.467185i \(-0.845268\pi\)
−0.884160 + 0.467185i \(0.845268\pi\)
\(270\) −23.3165 −0.00525554
\(271\) 4410.05i 0.988528i 0.869312 + 0.494264i \(0.164562\pi\)
−0.869312 + 0.494264i \(0.835438\pi\)
\(272\) 3087.94 0.688360
\(273\) 0 0
\(274\) 788.225 0.173790
\(275\) 1019.71i 0.223603i
\(276\) −2080.78 −0.453798
\(277\) 4636.44 1.00569 0.502846 0.864376i \(-0.332286\pi\)
0.502846 + 0.864376i \(0.332286\pi\)
\(278\) 297.340i 0.0641485i
\(279\) − 1037.32i − 0.222590i
\(280\) 111.139i 0.0237207i
\(281\) 1090.50i 0.231508i 0.993278 + 0.115754i \(0.0369285\pi\)
−0.993278 + 0.115754i \(0.963072\pi\)
\(282\) 829.468 0.175156
\(283\) −1140.52 −0.239565 −0.119782 0.992800i \(-0.538220\pi\)
−0.119782 + 0.992800i \(0.538220\pi\)
\(284\) − 965.101i − 0.201649i
\(285\) −282.835 −0.0587849
\(286\) 0 0
\(287\) −3552.01 −0.730553
\(288\) − 747.287i − 0.152897i
\(289\) −2195.76 −0.446929
\(290\) −165.788 −0.0335703
\(291\) − 3153.30i − 0.635222i
\(292\) 5250.67i 1.05230i
\(293\) − 335.079i − 0.0668106i −0.999442 0.0334053i \(-0.989365\pi\)
0.999442 0.0334053i \(-0.0106352\pi\)
\(294\) − 371.223i − 0.0736399i
\(295\) 965.449 0.190545
\(296\) −966.168 −0.189721
\(297\) 227.027i 0.0443551i
\(298\) 1476.34 0.286986
\(299\) 0 0
\(300\) 2837.75 0.546125
\(301\) 1645.76i 0.315149i
\(302\) 654.883 0.124782
\(303\) −1799.62 −0.341206
\(304\) − 2892.65i − 0.545739i
\(305\) − 927.719i − 0.174167i
\(306\) − 209.838i − 0.0392014i
\(307\) 2540.04i 0.472207i 0.971728 + 0.236104i \(0.0758705\pi\)
−0.971728 + 0.236104i \(0.924130\pi\)
\(308\) 534.215 0.0988303
\(309\) 879.935 0.161999
\(310\) 99.5332i 0.0182358i
\(311\) −2376.36 −0.433283 −0.216642 0.976251i \(-0.569510\pi\)
−0.216642 + 0.976251i \(0.569510\pi\)
\(312\) 0 0
\(313\) 1315.78 0.237612 0.118806 0.992917i \(-0.462093\pi\)
0.118806 + 0.992917i \(0.462093\pi\)
\(314\) − 686.713i − 0.123419i
\(315\) −141.538 −0.0253168
\(316\) −5314.09 −0.946015
\(317\) 6043.13i 1.07071i 0.844626 + 0.535357i \(0.179823\pi\)
−0.844626 + 0.535357i \(0.820177\pi\)
\(318\) 608.079i 0.107231i
\(319\) 1614.24i 0.283323i
\(320\) − 843.286i − 0.147316i
\(321\) 4601.08 0.800022
\(322\) 323.968 0.0560685
\(323\) − 2545.39i − 0.438481i
\(324\) 631.795 0.108333
\(325\) 0 0
\(326\) 354.940 0.0603015
\(327\) − 1772.16i − 0.299695i
\(328\) −3081.75 −0.518784
\(329\) 5035.14 0.843758
\(330\) − 21.7838i − 0.00363382i
\(331\) − 8168.18i − 1.35639i −0.734884 0.678193i \(-0.762763\pi\)
0.734884 0.678193i \(-0.237237\pi\)
\(332\) 7327.40i 1.21128i
\(333\) − 1230.44i − 0.202486i
\(334\) −597.951 −0.0979594
\(335\) −1712.11 −0.279232
\(336\) − 1447.56i − 0.235033i
\(337\) −3076.32 −0.497263 −0.248631 0.968598i \(-0.579981\pi\)
−0.248631 + 0.968598i \(0.579981\pi\)
\(338\) 0 0
\(339\) −1961.36 −0.314237
\(340\) − 785.012i − 0.125215i
\(341\) 969.133 0.153905
\(342\) −196.567 −0.0310793
\(343\) − 5047.30i − 0.794544i
\(344\) 1427.87i 0.223795i
\(345\) 515.058i 0.0803762i
\(346\) 254.260i 0.0395061i
\(347\) −8754.07 −1.35430 −0.677152 0.735844i \(-0.736786\pi\)
−0.677152 + 0.735844i \(0.736786\pi\)
\(348\) 4492.27 0.691985
\(349\) 1064.97i 0.163343i 0.996659 + 0.0816715i \(0.0260258\pi\)
−0.996659 + 0.0816715i \(0.973974\pi\)
\(350\) −441.825 −0.0674758
\(351\) 0 0
\(352\) 698.167 0.105717
\(353\) − 5047.62i − 0.761070i −0.924767 0.380535i \(-0.875740\pi\)
0.924767 0.380535i \(-0.124260\pi\)
\(354\) 670.977 0.100740
\(355\) −238.893 −0.0357158
\(356\) − 5888.79i − 0.876700i
\(357\) − 1273.78i − 0.188840i
\(358\) 691.494i 0.102085i
\(359\) 8152.39i 1.19851i 0.800557 + 0.599257i \(0.204537\pi\)
−0.800557 + 0.599257i \(0.795463\pi\)
\(360\) −122.800 −0.0179781
\(361\) 4474.59 0.652367
\(362\) 1524.75i 0.221379i
\(363\) 3780.90 0.546682
\(364\) 0 0
\(365\) 1299.71 0.186383
\(366\) − 644.754i − 0.0920816i
\(367\) 11636.3 1.65507 0.827537 0.561412i \(-0.189742\pi\)
0.827537 + 0.561412i \(0.189742\pi\)
\(368\) −5267.67 −0.746186
\(369\) − 3924.70i − 0.553690i
\(370\) 118.064i 0.0165888i
\(371\) 3691.24i 0.516548i
\(372\) − 2697.00i − 0.375896i
\(373\) 4720.59 0.655289 0.327644 0.944801i \(-0.393745\pi\)
0.327644 + 0.944801i \(0.393745\pi\)
\(374\) 196.045 0.0271049
\(375\) − 1426.45i − 0.196431i
\(376\) 4368.52 0.599173
\(377\) 0 0
\(378\) −98.3677 −0.0133849
\(379\) − 12933.7i − 1.75293i −0.481467 0.876464i \(-0.659896\pi\)
0.481467 0.876464i \(-0.340104\pi\)
\(380\) −735.365 −0.0992722
\(381\) 423.609 0.0569611
\(382\) 1549.69i 0.207563i
\(383\) − 554.945i − 0.0740376i −0.999315 0.0370188i \(-0.988214\pi\)
0.999315 0.0370188i \(-0.0117861\pi\)
\(384\) − 2578.84i − 0.342711i
\(385\) − 132.235i − 0.0175047i
\(386\) 2080.84 0.274384
\(387\) −1818.43 −0.238853
\(388\) − 8198.51i − 1.07272i
\(389\) −5091.35 −0.663604 −0.331802 0.943349i \(-0.607657\pi\)
−0.331802 + 0.943349i \(0.607657\pi\)
\(390\) 0 0
\(391\) −4635.30 −0.599532
\(392\) − 1955.10i − 0.251907i
\(393\) −2195.73 −0.281832
\(394\) 1284.13 0.164196
\(395\) 1315.40i 0.167557i
\(396\) 590.266i 0.0749040i
\(397\) 10254.6i 1.29638i 0.761480 + 0.648188i \(0.224473\pi\)
−0.761480 + 0.648188i \(0.775527\pi\)
\(398\) 11.1967i 0.00141015i
\(399\) −1193.23 −0.149714
\(400\) 7184.00 0.898001
\(401\) 505.816i 0.0629906i 0.999504 + 0.0314953i \(0.0100269\pi\)
−0.999504 + 0.0314953i \(0.989973\pi\)
\(402\) −1189.90 −0.147629
\(403\) 0 0
\(404\) −4678.97 −0.576207
\(405\) − 156.389i − 0.0191877i
\(406\) −699.426 −0.0854974
\(407\) 1149.57 0.140005
\(408\) − 1105.14i − 0.134100i
\(409\) − 2219.21i − 0.268295i −0.990961 0.134148i \(-0.957170\pi\)
0.990961 0.134148i \(-0.0428296\pi\)
\(410\) 376.585i 0.0453614i
\(411\) 5286.81i 0.634499i
\(412\) 2287.82 0.273574
\(413\) 4073.04 0.485282
\(414\) 357.960i 0.0424946i
\(415\) 1813.76 0.214540
\(416\) 0 0
\(417\) −1994.33 −0.234203
\(418\) − 183.646i − 0.0214891i
\(419\) 10249.3 1.19501 0.597506 0.801864i \(-0.296158\pi\)
0.597506 + 0.801864i \(0.296158\pi\)
\(420\) −367.997 −0.0427534
\(421\) − 97.9194i − 0.0113356i −0.999984 0.00566781i \(-0.998196\pi\)
0.999984 0.00566781i \(-0.00180413\pi\)
\(422\) 1612.85i 0.186048i
\(423\) 5563.44i 0.639489i
\(424\) 3202.54i 0.366814i
\(425\) 6321.58 0.721510
\(426\) −166.028 −0.0188828
\(427\) − 3913.87i − 0.443572i
\(428\) 11962.7 1.35103
\(429\) 0 0
\(430\) 174.483 0.0195682
\(431\) − 11727.2i − 1.31062i −0.755360 0.655310i \(-0.772538\pi\)
0.755360 0.655310i \(-0.227462\pi\)
\(432\) 1599.44 0.178133
\(433\) −6945.13 −0.770812 −0.385406 0.922747i \(-0.625939\pi\)
−0.385406 + 0.922747i \(0.625939\pi\)
\(434\) 419.911i 0.0464433i
\(435\) − 1111.98i − 0.122564i
\(436\) − 4607.57i − 0.506107i
\(437\) 4342.15i 0.475316i
\(438\) 903.281 0.0985398
\(439\) −11446.3 −1.24442 −0.622210 0.782851i \(-0.713765\pi\)
−0.622210 + 0.782851i \(0.713765\pi\)
\(440\) − 114.728i − 0.0124305i
\(441\) 2489.88 0.268856
\(442\) 0 0
\(443\) −10513.6 −1.12758 −0.563789 0.825919i \(-0.690657\pi\)
−0.563789 + 0.825919i \(0.690657\pi\)
\(444\) − 3199.13i − 0.341946i
\(445\) −1457.66 −0.155280
\(446\) 1925.25 0.204402
\(447\) 9902.14i 1.04777i
\(448\) − 3557.66i − 0.375187i
\(449\) − 9862.37i − 1.03660i −0.855199 0.518300i \(-0.826565\pi\)
0.855199 0.518300i \(-0.173435\pi\)
\(450\) − 488.182i − 0.0511403i
\(451\) 3666.72 0.382836
\(452\) −5099.49 −0.530663
\(453\) 4392.45i 0.455575i
\(454\) 2448.93 0.253159
\(455\) 0 0
\(456\) −1035.25 −0.106316
\(457\) 5990.48i 0.613179i 0.951842 + 0.306590i \(0.0991879\pi\)
−0.951842 + 0.306590i \(0.900812\pi\)
\(458\) 169.120 0.0172542
\(459\) 1407.43 0.143123
\(460\) 1339.14i 0.135734i
\(461\) − 16579.5i − 1.67502i −0.546423 0.837510i \(-0.684011\pi\)
0.546423 0.837510i \(-0.315989\pi\)
\(462\) − 91.9018i − 0.00925467i
\(463\) 3606.65i 0.362020i 0.983481 + 0.181010i \(0.0579367\pi\)
−0.983481 + 0.181010i \(0.942063\pi\)
\(464\) 11372.6 1.13784
\(465\) −667.593 −0.0665782
\(466\) 1139.65i 0.113290i
\(467\) 6789.61 0.672775 0.336387 0.941724i \(-0.390795\pi\)
0.336387 + 0.941724i \(0.390795\pi\)
\(468\) 0 0
\(469\) −7223.06 −0.711152
\(470\) − 533.826i − 0.0523906i
\(471\) 4605.95 0.450596
\(472\) 3533.80 0.344611
\(473\) − 1698.90i − 0.165149i
\(474\) 914.189i 0.0885867i
\(475\) − 5921.78i − 0.572021i
\(476\) − 3311.82i − 0.318901i
\(477\) −4078.53 −0.391495
\(478\) 2824.07 0.270230
\(479\) − 12688.3i − 1.21032i −0.796103 0.605161i \(-0.793109\pi\)
0.796103 0.605161i \(-0.206891\pi\)
\(480\) −480.936 −0.0457326
\(481\) 0 0
\(482\) −3025.27 −0.285886
\(483\) 2172.93i 0.204703i
\(484\) 9830.26 0.923202
\(485\) −2029.39 −0.189999
\(486\) − 108.689i − 0.0101445i
\(487\) 1217.28i 0.113265i 0.998395 + 0.0566325i \(0.0180363\pi\)
−0.998395 + 0.0566325i \(0.981964\pi\)
\(488\) − 3395.69i − 0.314991i
\(489\) 2380.66i 0.220158i
\(490\) −238.910 −0.0220262
\(491\) 10496.3 0.964747 0.482374 0.875966i \(-0.339775\pi\)
0.482374 + 0.875966i \(0.339775\pi\)
\(492\) − 10204.1i − 0.935037i
\(493\) 10007.3 0.914211
\(494\) 0 0
\(495\) 146.109 0.0132669
\(496\) − 6827.69i − 0.618090i
\(497\) −1007.84 −0.0909615
\(498\) 1260.54 0.113426
\(499\) 10516.5i 0.943450i 0.881746 + 0.471725i \(0.156368\pi\)
−0.881746 + 0.471725i \(0.843632\pi\)
\(500\) − 3708.75i − 0.331721i
\(501\) − 4010.60i − 0.357645i
\(502\) − 1527.90i − 0.135844i
\(503\) 13400.9 1.18790 0.593952 0.804500i \(-0.297567\pi\)
0.593952 + 0.804500i \(0.297567\pi\)
\(504\) −518.068 −0.0457869
\(505\) 1158.19i 0.102057i
\(506\) −334.430 −0.0293819
\(507\) 0 0
\(508\) 1101.38 0.0961923
\(509\) − 4683.90i − 0.407878i −0.978984 0.203939i \(-0.934626\pi\)
0.978984 0.203939i \(-0.0653745\pi\)
\(510\) −135.047 −0.0117254
\(511\) 5483.20 0.474682
\(512\) − 8267.80i − 0.713649i
\(513\) − 1318.42i − 0.113469i
\(514\) 3132.00i 0.268768i
\(515\) − 566.305i − 0.0484551i
\(516\) −4727.89 −0.403360
\(517\) −5197.75 −0.442160
\(518\) 498.090i 0.0422487i
\(519\) −1705.38 −0.144235
\(520\) 0 0
\(521\) 15034.6 1.26426 0.632128 0.774864i \(-0.282182\pi\)
0.632128 + 0.774864i \(0.282182\pi\)
\(522\) − 772.811i − 0.0647989i
\(523\) 8009.33 0.669643 0.334822 0.942282i \(-0.391324\pi\)
0.334822 + 0.942282i \(0.391324\pi\)
\(524\) −5708.86 −0.475940
\(525\) − 2963.42i − 0.246351i
\(526\) − 1507.22i − 0.124939i
\(527\) − 6008.04i − 0.496612i
\(528\) 1494.31i 0.123166i
\(529\) −4259.71 −0.350103
\(530\) 391.345 0.0320735
\(531\) 4500.40i 0.367798i
\(532\) −3102.36 −0.252828
\(533\) 0 0
\(534\) −1013.06 −0.0820960
\(535\) − 2961.14i − 0.239292i
\(536\) −6266.77 −0.505006
\(537\) −4638.01 −0.372710
\(538\) 3489.53i 0.279636i
\(539\) 2326.21i 0.185894i
\(540\) − 406.608i − 0.0324030i
\(541\) 20833.2i 1.65561i 0.561013 + 0.827807i \(0.310412\pi\)
−0.561013 + 0.827807i \(0.689588\pi\)
\(542\) 1972.52 0.156323
\(543\) −10226.9 −0.808245
\(544\) − 4328.22i − 0.341123i
\(545\) −1140.52 −0.0896411
\(546\) 0 0
\(547\) −14247.2 −1.11365 −0.556825 0.830630i \(-0.687981\pi\)
−0.556825 + 0.830630i \(0.687981\pi\)
\(548\) 13745.6i 1.07150i
\(549\) 4324.52 0.336186
\(550\) 456.093 0.0353598
\(551\) − 9374.41i − 0.724797i
\(552\) 1885.25i 0.145365i
\(553\) 5549.43i 0.426737i
\(554\) − 2073.78i − 0.159037i
\(555\) −791.885 −0.0605651
\(556\) −5185.22 −0.395508
\(557\) 2007.94i 0.152745i 0.997079 + 0.0763725i \(0.0243338\pi\)
−0.997079 + 0.0763725i \(0.975666\pi\)
\(558\) −463.969 −0.0351996
\(559\) 0 0
\(560\) −931.616 −0.0703000
\(561\) 1314.92i 0.0989589i
\(562\) 487.758 0.0366100
\(563\) −10562.1 −0.790659 −0.395330 0.918539i \(-0.629370\pi\)
−0.395330 + 0.918539i \(0.629370\pi\)
\(564\) 14464.8i 1.07993i
\(565\) 1262.28i 0.0939905i
\(566\) 510.129i 0.0378840i
\(567\) − 659.775i − 0.0488676i
\(568\) −874.409 −0.0645940
\(569\) 23207.2 1.70983 0.854916 0.518766i \(-0.173608\pi\)
0.854916 + 0.518766i \(0.173608\pi\)
\(570\) 126.506i 0.00929605i
\(571\) −7987.78 −0.585426 −0.292713 0.956200i \(-0.594558\pi\)
−0.292713 + 0.956200i \(0.594558\pi\)
\(572\) 0 0
\(573\) −10394.1 −0.757803
\(574\) 1588.74i 0.115527i
\(575\) −10783.9 −0.782120
\(576\) 3930.94 0.284356
\(577\) − 3224.41i − 0.232641i −0.993212 0.116321i \(-0.962890\pi\)
0.993212 0.116321i \(-0.0371100\pi\)
\(578\) 982.116i 0.0706759i
\(579\) 13956.7i 1.00176i
\(580\) − 2891.12i − 0.206978i
\(581\) 7651.90 0.546393
\(582\) −1410.40 −0.100452
\(583\) − 3810.44i − 0.270690i
\(584\) 4757.26 0.337084
\(585\) 0 0
\(586\) −149.873 −0.0105652
\(587\) 661.034i 0.0464800i 0.999730 + 0.0232400i \(0.00739819\pi\)
−0.999730 + 0.0232400i \(0.992602\pi\)
\(588\) 6473.63 0.454027
\(589\) −5628.07 −0.393719
\(590\) − 431.824i − 0.0301321i
\(591\) 8612.94i 0.599474i
\(592\) − 8098.87i − 0.562266i
\(593\) − 10172.5i − 0.704443i −0.935917 0.352221i \(-0.885426\pi\)
0.935917 0.352221i \(-0.114574\pi\)
\(594\) 101.544 0.00701417
\(595\) −819.777 −0.0564834
\(596\) 25745.4i 1.76942i
\(597\) −75.0990 −0.00514840
\(598\) 0 0
\(599\) 23462.2 1.60040 0.800200 0.599733i \(-0.204726\pi\)
0.800200 + 0.599733i \(0.204726\pi\)
\(600\) − 2571.08i − 0.174940i
\(601\) −10482.6 −0.711474 −0.355737 0.934586i \(-0.615770\pi\)
−0.355737 + 0.934586i \(0.615770\pi\)
\(602\) 736.111 0.0498366
\(603\) − 7980.92i − 0.538986i
\(604\) 11420.3i 0.769347i
\(605\) − 2433.29i − 0.163516i
\(606\) 804.930i 0.0539572i
\(607\) −24949.8 −1.66834 −0.834169 0.551508i \(-0.814053\pi\)
−0.834169 + 0.551508i \(0.814053\pi\)
\(608\) −4054.48 −0.270446
\(609\) − 4691.21i − 0.312147i
\(610\) −414.949 −0.0275423
\(611\) 0 0
\(612\) 3659.30 0.241697
\(613\) − 8238.86i − 0.542846i −0.962460 0.271423i \(-0.912506\pi\)
0.962460 0.271423i \(-0.0874942\pi\)
\(614\) 1136.10 0.0746733
\(615\) −2525.84 −0.165613
\(616\) − 484.014i − 0.0316583i
\(617\) − 5248.85i − 0.342481i −0.985229 0.171241i \(-0.945222\pi\)
0.985229 0.171241i \(-0.0547775\pi\)
\(618\) − 393.576i − 0.0256180i
\(619\) − 17820.9i − 1.15716i −0.815626 0.578580i \(-0.803607\pi\)
0.815626 0.578580i \(-0.196393\pi\)
\(620\) −1735.73 −0.112433
\(621\) −2400.92 −0.155146
\(622\) 1062.89i 0.0685180i
\(623\) −6149.58 −0.395470
\(624\) 0 0
\(625\) 14241.0 0.911424
\(626\) − 588.522i − 0.0375752i
\(627\) 1231.76 0.0784557
\(628\) 11975.4 0.760938
\(629\) − 7126.62i − 0.451760i
\(630\) 63.3071i 0.00400351i
\(631\) 25445.0i 1.60531i 0.596446 + 0.802653i \(0.296579\pi\)
−0.596446 + 0.802653i \(0.703421\pi\)
\(632\) 4814.71i 0.303036i
\(633\) −10817.7 −0.679252
\(634\) 2702.96 0.169319
\(635\) − 272.625i − 0.0170375i
\(636\) −10604.1 −0.661132
\(637\) 0 0
\(638\) 722.013 0.0448037
\(639\) − 1113.59i − 0.0689402i
\(640\) −1659.68 −0.102507
\(641\) 18701.0 1.15233 0.576167 0.817332i \(-0.304548\pi\)
0.576167 + 0.817332i \(0.304548\pi\)
\(642\) − 2057.96i − 0.126513i
\(643\) 28465.9i 1.74586i 0.487849 + 0.872928i \(0.337782\pi\)
−0.487849 + 0.872928i \(0.662218\pi\)
\(644\) 5649.58i 0.345690i
\(645\) 1170.30i 0.0714426i
\(646\) −1138.50 −0.0693399
\(647\) 10924.6 0.663818 0.331909 0.943311i \(-0.392307\pi\)
0.331909 + 0.943311i \(0.392307\pi\)
\(648\) − 572.425i − 0.0347021i
\(649\) −4204.58 −0.254305
\(650\) 0 0
\(651\) −2816.44 −0.169562
\(652\) 6189.68i 0.371789i
\(653\) −27110.9 −1.62470 −0.812351 0.583169i \(-0.801813\pi\)
−0.812351 + 0.583169i \(0.801813\pi\)
\(654\) −792.646 −0.0473929
\(655\) 1413.12i 0.0842980i
\(656\) − 25832.6i − 1.53749i
\(657\) 6058.51i 0.359764i
\(658\) − 2252.11i − 0.133429i
\(659\) 1727.08 0.102090 0.0510452 0.998696i \(-0.483745\pi\)
0.0510452 + 0.998696i \(0.483745\pi\)
\(660\) 379.881 0.0224043
\(661\) 20651.1i 1.21518i 0.794249 + 0.607592i \(0.207864\pi\)
−0.794249 + 0.607592i \(0.792136\pi\)
\(662\) −3653.45 −0.214494
\(663\) 0 0
\(664\) 6638.84 0.388007
\(665\) 767.932i 0.0447806i
\(666\) −550.351 −0.0320205
\(667\) −17071.3 −0.991010
\(668\) − 10427.5i − 0.603969i
\(669\) 12913.1i 0.746263i
\(670\) 765.790i 0.0441568i
\(671\) 4040.26i 0.232448i
\(672\) −2028.98 −0.116472
\(673\) −10858.0 −0.621907 −0.310954 0.950425i \(-0.600648\pi\)
−0.310954 + 0.950425i \(0.600648\pi\)
\(674\) 1375.97i 0.0786355i
\(675\) 3274.35 0.186711
\(676\) 0 0
\(677\) 29132.9 1.65387 0.826933 0.562301i \(-0.190084\pi\)
0.826933 + 0.562301i \(0.190084\pi\)
\(678\) 877.272i 0.0496924i
\(679\) −8561.59 −0.483894
\(680\) −711.244 −0.0401102
\(681\) 16425.6i 0.924272i
\(682\) − 433.472i − 0.0243380i
\(683\) − 9443.65i − 0.529065i −0.964377 0.264532i \(-0.914782\pi\)
0.964377 0.264532i \(-0.0852176\pi\)
\(684\) − 3427.87i − 0.191620i
\(685\) 3402.46 0.189783
\(686\) −2257.55 −0.125647
\(687\) 1134.33i 0.0629945i
\(688\) −11969.0 −0.663249
\(689\) 0 0
\(690\) 230.374 0.0127104
\(691\) 23700.1i 1.30477i 0.757889 + 0.652384i \(0.226231\pi\)
−0.757889 + 0.652384i \(0.773769\pi\)
\(692\) −4433.96 −0.243575
\(693\) 616.407 0.0337884
\(694\) 3915.51i 0.214165i
\(695\) 1283.50i 0.0700519i
\(696\) − 4070.12i − 0.221663i
\(697\) − 22731.5i − 1.23532i
\(698\) 476.340 0.0258305
\(699\) −7643.88 −0.413616
\(700\) − 7704.84i − 0.416022i
\(701\) −21643.1 −1.16612 −0.583059 0.812430i \(-0.698144\pi\)
−0.583059 + 0.812430i \(0.698144\pi\)
\(702\) 0 0
\(703\) −6675.91 −0.358160
\(704\) 3672.55i 0.196611i
\(705\) 3580.50 0.191276
\(706\) −2257.69 −0.120353
\(707\) 4886.18i 0.259921i
\(708\) 11700.9i 0.621114i
\(709\) − 28965.3i − 1.53429i −0.641472 0.767146i \(-0.721676\pi\)
0.641472 0.767146i \(-0.278324\pi\)
\(710\) 106.851i 0.00564798i
\(711\) −6131.68 −0.323426
\(712\) −5335.41 −0.280833
\(713\) 10249.0i 0.538330i
\(714\) −569.736 −0.0298625
\(715\) 0 0
\(716\) −12058.7 −0.629408
\(717\) 18941.7i 0.986599i
\(718\) 3646.39 0.189529
\(719\) 21256.5 1.10255 0.551276 0.834323i \(-0.314141\pi\)
0.551276 + 0.834323i \(0.314141\pi\)
\(720\) − 1029.36i − 0.0532807i
\(721\) − 2389.13i − 0.123406i
\(722\) − 2001.39i − 0.103163i
\(723\) − 20291.2i − 1.04376i
\(724\) −26589.7 −1.36491
\(725\) 23281.7 1.19264
\(726\) − 1691.11i − 0.0864505i
\(727\) 11868.9 0.605494 0.302747 0.953071i \(-0.402096\pi\)
0.302747 + 0.953071i \(0.402096\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) − 581.330i − 0.0294739i
\(731\) −10532.2 −0.532896
\(732\) 11243.7 0.567729
\(733\) − 27296.4i − 1.37546i −0.725965 0.687732i \(-0.758607\pi\)
0.725965 0.687732i \(-0.241393\pi\)
\(734\) − 5204.68i − 0.261728i
\(735\) − 1602.43i − 0.0804168i
\(736\) 7383.44i 0.369779i
\(737\) 7456.32 0.372669
\(738\) −1755.43 −0.0875587
\(739\) − 26424.6i − 1.31535i −0.753302 0.657674i \(-0.771540\pi\)
0.753302 0.657674i \(-0.228460\pi\)
\(740\) −2058.89 −0.102279
\(741\) 0 0
\(742\) 1651.01 0.0816853
\(743\) 25974.8i 1.28253i 0.767318 + 0.641267i \(0.221591\pi\)
−0.767318 + 0.641267i \(0.778409\pi\)
\(744\) −2443.56 −0.120410
\(745\) 6372.78 0.313397
\(746\) − 2111.42i − 0.103625i
\(747\) 8454.76i 0.414114i
\(748\) 3418.77i 0.167116i
\(749\) − 12492.5i − 0.609433i
\(750\) −638.022 −0.0310630
\(751\) −20234.8 −0.983195 −0.491598 0.870823i \(-0.663587\pi\)
−0.491598 + 0.870823i \(0.663587\pi\)
\(752\) 36619.0i 1.77574i
\(753\) 10248.0 0.495960
\(754\) 0 0
\(755\) 2826.88 0.136266
\(756\) − 1715.40i − 0.0825246i
\(757\) −3363.48 −0.161490 −0.0807449 0.996735i \(-0.525730\pi\)
−0.0807449 + 0.996735i \(0.525730\pi\)
\(758\) −5784.97 −0.277202
\(759\) − 2243.10i − 0.107272i
\(760\) 666.262i 0.0317998i
\(761\) − 21311.3i − 1.01516i −0.861605 0.507579i \(-0.830541\pi\)
0.861605 0.507579i \(-0.169459\pi\)
\(762\) − 189.471i − 0.00900764i
\(763\) −4811.62 −0.228299
\(764\) −27024.5 −1.27973
\(765\) − 905.790i − 0.0428090i
\(766\) −248.215 −0.0117081
\(767\) 0 0
\(768\) 9329.04 0.438324
\(769\) 34989.6i 1.64078i 0.571807 + 0.820388i \(0.306242\pi\)
−0.571807 + 0.820388i \(0.693758\pi\)
\(770\) −59.1458 −0.00276814
\(771\) −21007.1 −0.981260
\(772\) 36287.2i 1.69171i
\(773\) − 20034.1i − 0.932181i −0.884737 0.466091i \(-0.845662\pi\)
0.884737 0.466091i \(-0.154338\pi\)
\(774\) 813.345i 0.0377714i
\(775\) − 13977.5i − 0.647855i
\(776\) −7428.09 −0.343625
\(777\) −3340.81 −0.154248
\(778\) 2277.25i 0.104940i
\(779\) −21293.9 −0.979373
\(780\) 0 0
\(781\) 1040.39 0.0476671
\(782\) 2073.27i 0.0948081i
\(783\) 5183.43 0.236578
\(784\) 16388.5 0.746563
\(785\) − 2964.28i − 0.134777i
\(786\) 982.102i 0.0445680i
\(787\) 5591.98i 0.253282i 0.991949 + 0.126641i \(0.0404196\pi\)
−0.991949 + 0.126641i \(0.959580\pi\)
\(788\) 22393.5i 1.01235i
\(789\) 10109.3 0.456147
\(790\) 588.351 0.0264969
\(791\) 5325.32i 0.239376i
\(792\) 534.798 0.0239940
\(793\) 0 0
\(794\) 4586.64 0.205005
\(795\) 2624.84i 0.117099i
\(796\) −195.256 −0.00869430
\(797\) 12738.6 0.566152 0.283076 0.959098i \(-0.408645\pi\)
0.283076 + 0.959098i \(0.408645\pi\)
\(798\) 533.704i 0.0236753i
\(799\) 32222.9i 1.42674i
\(800\) − 10069.5i − 0.445012i
\(801\) − 6794.81i − 0.299729i
\(802\) 226.240 0.00996113
\(803\) −5660.28 −0.248751
\(804\) − 20750.2i − 0.910205i
\(805\) 1398.45 0.0612283
\(806\) 0 0
\(807\) −23405.1 −1.02094
\(808\) 4239.28i 0.184576i
\(809\) −8806.48 −0.382719 −0.191359 0.981520i \(-0.561290\pi\)
−0.191359 + 0.981520i \(0.561290\pi\)
\(810\) −69.9494 −0.00303429
\(811\) 10565.8i 0.457478i 0.973488 + 0.228739i \(0.0734603\pi\)
−0.973488 + 0.228739i \(0.926540\pi\)
\(812\) − 12197.1i − 0.527134i
\(813\) 13230.1i 0.570727i
\(814\) − 514.176i − 0.0221399i
\(815\) 1532.14 0.0658509
\(816\) 9263.82 0.397425
\(817\) 9866.09i 0.422486i
\(818\) −992.604 −0.0424273
\(819\) 0 0
\(820\) −6567.14 −0.279676
\(821\) − 422.966i − 0.0179800i −0.999960 0.00899002i \(-0.997138\pi\)
0.999960 0.00899002i \(-0.00286165\pi\)
\(822\) 2364.67 0.100338
\(823\) −11148.8 −0.472202 −0.236101 0.971728i \(-0.575870\pi\)
−0.236101 + 0.971728i \(0.575870\pi\)
\(824\) − 2072.83i − 0.0876339i
\(825\) 3059.12i 0.129097i
\(826\) − 1821.78i − 0.0767409i
\(827\) − 813.158i − 0.0341914i −0.999854 0.0170957i \(-0.994558\pi\)
0.999854 0.0170957i \(-0.00544200\pi\)
\(828\) −6242.34 −0.262001
\(829\) 28320.8 1.18652 0.593258 0.805012i \(-0.297841\pi\)
0.593258 + 0.805012i \(0.297841\pi\)
\(830\) − 811.255i − 0.0339266i
\(831\) 13909.3 0.580637
\(832\) 0 0
\(833\) 14421.1 0.599835
\(834\) 892.020i 0.0370361i
\(835\) −2581.13 −0.106974
\(836\) 3202.55 0.132491
\(837\) − 3111.95i − 0.128512i
\(838\) − 4584.28i − 0.188975i
\(839\) − 22139.3i − 0.911006i −0.890234 0.455503i \(-0.849459\pi\)
0.890234 0.455503i \(-0.150541\pi\)
\(840\) 333.416i 0.0136952i
\(841\) 12466.8 0.511166
\(842\) −43.7972 −0.00179258
\(843\) 3271.50i 0.133661i
\(844\) −28125.9 −1.14708
\(845\) 0 0
\(846\) 2488.41 0.101127
\(847\) − 10265.6i − 0.416446i
\(848\) −26845.2 −1.08711
\(849\) −3421.56 −0.138313
\(850\) − 2827.50i − 0.114097i
\(851\) 12157.2i 0.489710i
\(852\) − 2895.30i − 0.116422i
\(853\) − 14080.5i − 0.565191i −0.959239 0.282595i \(-0.908805\pi\)
0.959239 0.282595i \(-0.0911953\pi\)
\(854\) −1750.59 −0.0701451
\(855\) −848.505 −0.0339395
\(856\) − 10838.6i − 0.432774i
\(857\) −10907.7 −0.434772 −0.217386 0.976086i \(-0.569753\pi\)
−0.217386 + 0.976086i \(0.569753\pi\)
\(858\) 0 0
\(859\) 7739.24 0.307403 0.153702 0.988117i \(-0.450881\pi\)
0.153702 + 0.988117i \(0.450881\pi\)
\(860\) 3042.76i 0.120648i
\(861\) −10656.0 −0.421785
\(862\) −5245.30 −0.207257
\(863\) 29072.5i 1.14674i 0.819295 + 0.573372i \(0.194365\pi\)
−0.819295 + 0.573372i \(0.805635\pi\)
\(864\) − 2241.86i − 0.0882751i
\(865\) 1097.54i 0.0431417i
\(866\) 3106.40i 0.121894i
\(867\) −6587.28 −0.258034
\(868\) −7322.70 −0.286346
\(869\) − 5728.64i − 0.223626i
\(870\) −497.363 −0.0193818
\(871\) 0 0
\(872\) −4174.59 −0.162121
\(873\) − 9459.89i − 0.366746i
\(874\) 1942.15 0.0751649
\(875\) −3873.00 −0.149636
\(876\) 15752.0i 0.607547i
\(877\) − 2391.96i − 0.0920988i −0.998939 0.0460494i \(-0.985337\pi\)
0.998939 0.0460494i \(-0.0146632\pi\)
\(878\) 5119.66i 0.196788i
\(879\) − 1005.24i − 0.0385731i
\(880\) 961.702 0.0368397
\(881\) 11975.8 0.457974 0.228987 0.973430i \(-0.426459\pi\)
0.228987 + 0.973430i \(0.426459\pi\)
\(882\) − 1113.67i − 0.0425160i
\(883\) 44712.4 1.70407 0.852035 0.523485i \(-0.175368\pi\)
0.852035 + 0.523485i \(0.175368\pi\)
\(884\) 0 0
\(885\) 2896.35 0.110011
\(886\) 4702.51i 0.178311i
\(887\) 17023.3 0.644403 0.322202 0.946671i \(-0.395577\pi\)
0.322202 + 0.946671i \(0.395577\pi\)
\(888\) −2898.51 −0.109535
\(889\) − 1150.15i − 0.0433913i
\(890\) 651.979i 0.0245555i
\(891\) 681.082i 0.0256084i
\(892\) 33573.8i 1.26024i
\(893\) 30185.0 1.13113
\(894\) 4429.01 0.165692
\(895\) 2984.91i 0.111480i
\(896\) −7001.87 −0.261067
\(897\) 0 0
\(898\) −4411.22 −0.163925
\(899\) − 22127.0i − 0.820886i
\(900\) 8513.25 0.315306
\(901\) −23622.5 −0.873449
\(902\) − 1640.04i − 0.0605405i
\(903\) 4937.27i 0.181951i
\(904\) 4620.28i 0.169987i
\(905\) 6581.77i 0.241752i
\(906\) 1964.65 0.0720432
\(907\) −2303.59 −0.0843324 −0.0421662 0.999111i \(-0.513426\pi\)
−0.0421662 + 0.999111i \(0.513426\pi\)
\(908\) 42706.2i 1.56085i
\(909\) −5398.85 −0.196995
\(910\) 0 0
\(911\) −12897.0 −0.469042 −0.234521 0.972111i \(-0.575352\pi\)
−0.234521 + 0.972111i \(0.575352\pi\)
\(912\) − 8677.95i − 0.315083i
\(913\) −7899.01 −0.286330
\(914\) 2679.41 0.0969662
\(915\) − 2783.16i − 0.100556i
\(916\) 2949.22i 0.106381i
\(917\) 5961.68i 0.214691i
\(918\) − 629.514i − 0.0226330i
\(919\) 16785.2 0.602496 0.301248 0.953546i \(-0.402597\pi\)
0.301248 + 0.953546i \(0.402597\pi\)
\(920\) 1213.30 0.0434797
\(921\) 7620.11i 0.272629i
\(922\) −7415.64 −0.264882
\(923\) 0 0
\(924\) 1602.65 0.0570597
\(925\) − 16579.9i − 0.589344i
\(926\) 1613.18 0.0572487
\(927\) 2639.81 0.0935303
\(928\) − 15940.4i − 0.563866i
\(929\) − 16348.0i − 0.577352i −0.957427 0.288676i \(-0.906785\pi\)
0.957427 0.288676i \(-0.0932150\pi\)
\(930\) 298.600i 0.0105285i
\(931\) − 13509.1i − 0.475556i
\(932\) −19873.9 −0.698489
\(933\) −7129.08 −0.250156
\(934\) − 3036.85i − 0.106390i
\(935\) 846.251 0.0295993
\(936\) 0 0
\(937\) −26247.5 −0.915121 −0.457560 0.889179i \(-0.651277\pi\)
−0.457560 + 0.889179i \(0.651277\pi\)
\(938\) 3230.72i 0.112459i
\(939\) 3947.35 0.137185
\(940\) 9309.22 0.323014
\(941\) − 43838.4i − 1.51869i −0.650686 0.759347i \(-0.725519\pi\)
0.650686 0.759347i \(-0.274481\pi\)
\(942\) − 2060.14i − 0.0712558i
\(943\) 38777.3i 1.33909i
\(944\) 29621.9i 1.02130i
\(945\) −424.615 −0.0146167
\(946\) −759.882 −0.0261162
\(947\) − 45077.2i − 1.54679i −0.633924 0.773395i \(-0.718557\pi\)
0.633924 0.773395i \(-0.281443\pi\)
\(948\) −15942.3 −0.546182
\(949\) 0 0
\(950\) −2648.68 −0.0904575
\(951\) 18129.4i 0.618177i
\(952\) −3000.60 −0.102153
\(953\) −26479.1 −0.900046 −0.450023 0.893017i \(-0.648584\pi\)
−0.450023 + 0.893017i \(0.648584\pi\)
\(954\) 1824.24i 0.0619097i
\(955\) 6689.41i 0.226664i
\(956\) 49248.1i 1.66611i
\(957\) 4842.71i 0.163576i
\(958\) −5675.21 −0.191396
\(959\) 14354.3 0.483343
\(960\) − 2529.86i − 0.0850529i
\(961\) 16506.7 0.554084
\(962\) 0 0
\(963\) 13803.2 0.461893
\(964\) − 52756.7i − 1.76263i
\(965\) 8982.20 0.299635
\(966\) 971.905 0.0323711
\(967\) 602.475i 0.0200355i 0.999950 + 0.0100177i \(0.00318880\pi\)
−0.999950 + 0.0100177i \(0.996811\pi\)
\(968\) − 8906.49i − 0.295729i
\(969\) − 7636.17i − 0.253157i
\(970\) 907.701i 0.0300459i
\(971\) −11330.2 −0.374464 −0.187232 0.982316i \(-0.559952\pi\)
−0.187232 + 0.982316i \(0.559952\pi\)
\(972\) 1895.39 0.0625458
\(973\) 5414.85i 0.178409i
\(974\) 544.461 0.0179114
\(975\) 0 0
\(976\) 28464.3 0.933524
\(977\) − 55633.6i − 1.82178i −0.412653 0.910888i \(-0.635398\pi\)
0.412653 0.910888i \(-0.364602\pi\)
\(978\) 1064.82 0.0348151
\(979\) 6348.18 0.207241
\(980\) − 4166.27i − 0.135803i
\(981\) − 5316.47i − 0.173029i
\(982\) − 4694.76i − 0.152562i
\(983\) − 6884.90i − 0.223392i −0.993742 0.111696i \(-0.964372\pi\)
0.993742 0.111696i \(-0.0356282\pi\)
\(984\) −9245.24 −0.299520
\(985\) 5543.08 0.179307
\(986\) − 4476.05i − 0.144570i
\(987\) 15105.4 0.487144
\(988\) 0 0
\(989\) 17966.7 0.577662
\(990\) − 65.3515i − 0.00209799i
\(991\) 52646.1 1.68755 0.843773 0.536700i \(-0.180329\pi\)
0.843773 + 0.536700i \(0.180329\pi\)
\(992\) −9570.05 −0.306300
\(993\) − 24504.5i − 0.783109i
\(994\) 450.786i 0.0143844i
\(995\) 48.3319i 0.00153992i
\(996\) 21982.2i 0.699330i
\(997\) 26100.7 0.829106 0.414553 0.910025i \(-0.363938\pi\)
0.414553 + 0.910025i \(0.363938\pi\)
\(998\) 4703.78 0.149194
\(999\) − 3691.33i − 0.116906i
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 507.4.b.k.337.8 18
13.5 odd 4 507.4.a.o.1.5 9
13.8 odd 4 507.4.a.p.1.5 yes 9
13.12 even 2 inner 507.4.b.k.337.11 18
39.5 even 4 1521.4.a.bi.1.5 9
39.8 even 4 1521.4.a.bf.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.4.a.o.1.5 9 13.5 odd 4
507.4.a.p.1.5 yes 9 13.8 odd 4
507.4.b.k.337.8 18 1.1 even 1 trivial
507.4.b.k.337.11 18 13.12 even 2 inner
1521.4.a.bf.1.5 9 39.8 even 4
1521.4.a.bi.1.5 9 39.5 even 4