Properties

Label 13.4.a.a.1.1
Level $13$
Weight $4$
Character 13.1
Self dual yes
Analytic conductor $0.767$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.767024830075\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 13.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{2} -7.00000 q^{3} +17.0000 q^{4} -7.00000 q^{5} +35.0000 q^{6} -13.0000 q^{7} -45.0000 q^{8} +22.0000 q^{9} +O(q^{10})\) \(q-5.00000 q^{2} -7.00000 q^{3} +17.0000 q^{4} -7.00000 q^{5} +35.0000 q^{6} -13.0000 q^{7} -45.0000 q^{8} +22.0000 q^{9} +35.0000 q^{10} -26.0000 q^{11} -119.000 q^{12} +13.0000 q^{13} +65.0000 q^{14} +49.0000 q^{15} +89.0000 q^{16} +77.0000 q^{17} -110.000 q^{18} -126.000 q^{19} -119.000 q^{20} +91.0000 q^{21} +130.000 q^{22} -96.0000 q^{23} +315.000 q^{24} -76.0000 q^{25} -65.0000 q^{26} +35.0000 q^{27} -221.000 q^{28} -82.0000 q^{29} -245.000 q^{30} +196.000 q^{31} -85.0000 q^{32} +182.000 q^{33} -385.000 q^{34} +91.0000 q^{35} +374.000 q^{36} -131.000 q^{37} +630.000 q^{38} -91.0000 q^{39} +315.000 q^{40} +336.000 q^{41} -455.000 q^{42} -201.000 q^{43} -442.000 q^{44} -154.000 q^{45} +480.000 q^{46} -105.000 q^{47} -623.000 q^{48} -174.000 q^{49} +380.000 q^{50} -539.000 q^{51} +221.000 q^{52} -432.000 q^{53} -175.000 q^{54} +182.000 q^{55} +585.000 q^{56} +882.000 q^{57} +410.000 q^{58} -294.000 q^{59} +833.000 q^{60} -56.0000 q^{61} -980.000 q^{62} -286.000 q^{63} -287.000 q^{64} -91.0000 q^{65} -910.000 q^{66} +478.000 q^{67} +1309.00 q^{68} +672.000 q^{69} -455.000 q^{70} +9.00000 q^{71} -990.000 q^{72} +98.0000 q^{73} +655.000 q^{74} +532.000 q^{75} -2142.00 q^{76} +338.000 q^{77} +455.000 q^{78} +1304.00 q^{79} -623.000 q^{80} -839.000 q^{81} -1680.00 q^{82} -308.000 q^{83} +1547.00 q^{84} -539.000 q^{85} +1005.00 q^{86} +574.000 q^{87} +1170.00 q^{88} -1190.00 q^{89} +770.000 q^{90} -169.000 q^{91} -1632.00 q^{92} -1372.00 q^{93} +525.000 q^{94} +882.000 q^{95} +595.000 q^{96} +70.0000 q^{97} +870.000 q^{98} -572.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) −7.00000 −1.34715 −0.673575 0.739119i \(-0.735242\pi\)
−0.673575 + 0.739119i \(0.735242\pi\)
\(4\) 17.0000 2.12500
\(5\) −7.00000 −0.626099 −0.313050 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) 35.0000 2.38145
\(7\) −13.0000 −0.701934 −0.350967 0.936388i \(-0.614147\pi\)
−0.350967 + 0.936388i \(0.614147\pi\)
\(8\) −45.0000 −1.98874
\(9\) 22.0000 0.814815
\(10\) 35.0000 1.10680
\(11\) −26.0000 −0.712663 −0.356332 0.934360i \(-0.615973\pi\)
−0.356332 + 0.934360i \(0.615973\pi\)
\(12\) −119.000 −2.86270
\(13\) 13.0000 0.277350
\(14\) 65.0000 1.24086
\(15\) 49.0000 0.843450
\(16\) 89.0000 1.39062
\(17\) 77.0000 1.09854 0.549272 0.835644i \(-0.314905\pi\)
0.549272 + 0.835644i \(0.314905\pi\)
\(18\) −110.000 −1.44040
\(19\) −126.000 −1.52139 −0.760694 0.649110i \(-0.775141\pi\)
−0.760694 + 0.649110i \(0.775141\pi\)
\(20\) −119.000 −1.33046
\(21\) 91.0000 0.945611
\(22\) 130.000 1.25982
\(23\) −96.0000 −0.870321 −0.435161 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 315.000 2.67913
\(25\) −76.0000 −0.608000
\(26\) −65.0000 −0.490290
\(27\) 35.0000 0.249472
\(28\) −221.000 −1.49161
\(29\) −82.0000 −0.525070 −0.262535 0.964923i \(-0.584558\pi\)
−0.262535 + 0.964923i \(0.584558\pi\)
\(30\) −245.000 −1.49102
\(31\) 196.000 1.13557 0.567785 0.823177i \(-0.307801\pi\)
0.567785 + 0.823177i \(0.307801\pi\)
\(32\) −85.0000 −0.469563
\(33\) 182.000 0.960065
\(34\) −385.000 −1.94197
\(35\) 91.0000 0.439480
\(36\) 374.000 1.73148
\(37\) −131.000 −0.582061 −0.291031 0.956714i \(-0.593998\pi\)
−0.291031 + 0.956714i \(0.593998\pi\)
\(38\) 630.000 2.68946
\(39\) −91.0000 −0.373632
\(40\) 315.000 1.24515
\(41\) 336.000 1.27986 0.639932 0.768432i \(-0.278963\pi\)
0.639932 + 0.768432i \(0.278963\pi\)
\(42\) −455.000 −1.67162
\(43\) −201.000 −0.712842 −0.356421 0.934325i \(-0.616003\pi\)
−0.356421 + 0.934325i \(0.616003\pi\)
\(44\) −442.000 −1.51441
\(45\) −154.000 −0.510155
\(46\) 480.000 1.53852
\(47\) −105.000 −0.325869 −0.162934 0.986637i \(-0.552096\pi\)
−0.162934 + 0.986637i \(0.552096\pi\)
\(48\) −623.000 −1.87338
\(49\) −174.000 −0.507289
\(50\) 380.000 1.07480
\(51\) −539.000 −1.47990
\(52\) 221.000 0.589369
\(53\) −432.000 −1.11962 −0.559809 0.828622i \(-0.689126\pi\)
−0.559809 + 0.828622i \(0.689126\pi\)
\(54\) −175.000 −0.441009
\(55\) 182.000 0.446198
\(56\) 585.000 1.39596
\(57\) 882.000 2.04954
\(58\) 410.000 0.928201
\(59\) −294.000 −0.648738 −0.324369 0.945931i \(-0.605152\pi\)
−0.324369 + 0.945931i \(0.605152\pi\)
\(60\) 833.000 1.79233
\(61\) −56.0000 −0.117542 −0.0587710 0.998271i \(-0.518718\pi\)
−0.0587710 + 0.998271i \(0.518718\pi\)
\(62\) −980.000 −2.00742
\(63\) −286.000 −0.571946
\(64\) −287.000 −0.560547
\(65\) −91.0000 −0.173649
\(66\) −910.000 −1.69717
\(67\) 478.000 0.871597 0.435798 0.900044i \(-0.356466\pi\)
0.435798 + 0.900044i \(0.356466\pi\)
\(68\) 1309.00 2.33441
\(69\) 672.000 1.17245
\(70\) −455.000 −0.776899
\(71\) 9.00000 0.0150437 0.00752186 0.999972i \(-0.497606\pi\)
0.00752186 + 0.999972i \(0.497606\pi\)
\(72\) −990.000 −1.62045
\(73\) 98.0000 0.157124 0.0785619 0.996909i \(-0.474967\pi\)
0.0785619 + 0.996909i \(0.474967\pi\)
\(74\) 655.000 1.02895
\(75\) 532.000 0.819068
\(76\) −2142.00 −3.23295
\(77\) 338.000 0.500243
\(78\) 455.000 0.660495
\(79\) 1304.00 1.85711 0.928554 0.371198i \(-0.121053\pi\)
0.928554 + 0.371198i \(0.121053\pi\)
\(80\) −623.000 −0.870669
\(81\) −839.000 −1.15089
\(82\) −1680.00 −2.26250
\(83\) −308.000 −0.407318 −0.203659 0.979042i \(-0.565283\pi\)
−0.203659 + 0.979042i \(0.565283\pi\)
\(84\) 1547.00 2.00942
\(85\) −539.000 −0.687797
\(86\) 1005.00 1.26014
\(87\) 574.000 0.707348
\(88\) 1170.00 1.41730
\(89\) −1190.00 −1.41730 −0.708650 0.705560i \(-0.750696\pi\)
−0.708650 + 0.705560i \(0.750696\pi\)
\(90\) 770.000 0.901835
\(91\) −169.000 −0.194681
\(92\) −1632.00 −1.84943
\(93\) −1372.00 −1.52978
\(94\) 525.000 0.576060
\(95\) 882.000 0.952540
\(96\) 595.000 0.632572
\(97\) 70.0000 0.0732724 0.0366362 0.999329i \(-0.488336\pi\)
0.0366362 + 0.999329i \(0.488336\pi\)
\(98\) 870.000 0.896768
\(99\) −572.000 −0.580689
\(100\) −1292.00 −1.29200
\(101\) 420.000 0.413778 0.206889 0.978364i \(-0.433666\pi\)
0.206889 + 0.978364i \(0.433666\pi\)
\(102\) 2695.00 2.61613
\(103\) 588.000 0.562499 0.281249 0.959635i \(-0.409251\pi\)
0.281249 + 0.959635i \(0.409251\pi\)
\(104\) −585.000 −0.551577
\(105\) −637.000 −0.592046
\(106\) 2160.00 1.97922
\(107\) −684.000 −0.617989 −0.308994 0.951064i \(-0.599992\pi\)
−0.308994 + 0.951064i \(0.599992\pi\)
\(108\) 595.000 0.530129
\(109\) 373.000 0.327770 0.163885 0.986479i \(-0.447597\pi\)
0.163885 + 0.986479i \(0.447597\pi\)
\(110\) −910.000 −0.788774
\(111\) 917.000 0.784124
\(112\) −1157.00 −0.976127
\(113\) −1734.00 −1.44355 −0.721774 0.692128i \(-0.756673\pi\)
−0.721774 + 0.692128i \(0.756673\pi\)
\(114\) −4410.00 −3.62311
\(115\) 672.000 0.544907
\(116\) −1394.00 −1.11577
\(117\) 286.000 0.225989
\(118\) 1470.00 1.14682
\(119\) −1001.00 −0.771105
\(120\) −2205.00 −1.67740
\(121\) −655.000 −0.492111
\(122\) 280.000 0.207787
\(123\) −2352.00 −1.72417
\(124\) 3332.00 2.41308
\(125\) 1407.00 1.00677
\(126\) 1430.00 1.01107
\(127\) 1892.00 1.32195 0.660976 0.750407i \(-0.270143\pi\)
0.660976 + 0.750407i \(0.270143\pi\)
\(128\) 2115.00 1.46048
\(129\) 1407.00 0.960306
\(130\) 455.000 0.306970
\(131\) 1435.00 0.957073 0.478536 0.878068i \(-0.341167\pi\)
0.478536 + 0.878068i \(0.341167\pi\)
\(132\) 3094.00 2.04014
\(133\) 1638.00 1.06791
\(134\) −2390.00 −1.54078
\(135\) −245.000 −0.156194
\(136\) −3465.00 −2.18472
\(137\) −1776.00 −1.10755 −0.553773 0.832667i \(-0.686813\pi\)
−0.553773 + 0.832667i \(0.686813\pi\)
\(138\) −3360.00 −2.07262
\(139\) −1869.00 −1.14048 −0.570239 0.821479i \(-0.693150\pi\)
−0.570239 + 0.821479i \(0.693150\pi\)
\(140\) 1547.00 0.933895
\(141\) 735.000 0.438994
\(142\) −45.0000 −0.0265938
\(143\) −338.000 −0.197657
\(144\) 1958.00 1.13310
\(145\) 574.000 0.328746
\(146\) −490.000 −0.277758
\(147\) 1218.00 0.683394
\(148\) −2227.00 −1.23688
\(149\) 2466.00 1.35586 0.677928 0.735128i \(-0.262878\pi\)
0.677928 + 0.735128i \(0.262878\pi\)
\(150\) −2660.00 −1.44792
\(151\) −3323.00 −1.79087 −0.895437 0.445189i \(-0.853137\pi\)
−0.895437 + 0.445189i \(0.853137\pi\)
\(152\) 5670.00 3.02564
\(153\) 1694.00 0.895110
\(154\) −1690.00 −0.884312
\(155\) −1372.00 −0.710979
\(156\) −1547.00 −0.793969
\(157\) −2730.00 −1.38776 −0.693878 0.720092i \(-0.744099\pi\)
−0.693878 + 0.720092i \(0.744099\pi\)
\(158\) −6520.00 −3.28293
\(159\) 3024.00 1.50829
\(160\) 595.000 0.293993
\(161\) 1248.00 0.610908
\(162\) 4195.00 2.03451
\(163\) −544.000 −0.261407 −0.130704 0.991421i \(-0.541724\pi\)
−0.130704 + 0.991421i \(0.541724\pi\)
\(164\) 5712.00 2.71971
\(165\) −1274.00 −0.601096
\(166\) 1540.00 0.720043
\(167\) 1624.00 0.752508 0.376254 0.926516i \(-0.377212\pi\)
0.376254 + 0.926516i \(0.377212\pi\)
\(168\) −4095.00 −1.88057
\(169\) 169.000 0.0769231
\(170\) 2695.00 1.21587
\(171\) −2772.00 −1.23965
\(172\) −3417.00 −1.51479
\(173\) −336.000 −0.147662 −0.0738312 0.997271i \(-0.523523\pi\)
−0.0738312 + 0.997271i \(0.523523\pi\)
\(174\) −2870.00 −1.25043
\(175\) 988.000 0.426776
\(176\) −2314.00 −0.991047
\(177\) 2058.00 0.873948
\(178\) 5950.00 2.50546
\(179\) −3029.00 −1.26479 −0.632397 0.774645i \(-0.717929\pi\)
−0.632397 + 0.774645i \(0.717929\pi\)
\(180\) −2618.00 −1.08408
\(181\) −28.0000 −0.0114985 −0.00574924 0.999983i \(-0.501830\pi\)
−0.00574924 + 0.999983i \(0.501830\pi\)
\(182\) 845.000 0.344151
\(183\) 392.000 0.158347
\(184\) 4320.00 1.73084
\(185\) 917.000 0.364428
\(186\) 6860.00 2.70430
\(187\) −2002.00 −0.782892
\(188\) −1785.00 −0.692471
\(189\) −455.000 −0.175113
\(190\) −4410.00 −1.68387
\(191\) 422.000 0.159868 0.0799342 0.996800i \(-0.474529\pi\)
0.0799342 + 0.996800i \(0.474529\pi\)
\(192\) 2009.00 0.755141
\(193\) 492.000 0.183497 0.0917485 0.995782i \(-0.470754\pi\)
0.0917485 + 0.995782i \(0.470754\pi\)
\(194\) −350.000 −0.129529
\(195\) 637.000 0.233931
\(196\) −2958.00 −1.07799
\(197\) 2991.00 1.08173 0.540863 0.841111i \(-0.318098\pi\)
0.540863 + 0.841111i \(0.318098\pi\)
\(198\) 2860.00 1.02652
\(199\) −70.0000 −0.0249355 −0.0124678 0.999922i \(-0.503969\pi\)
−0.0124678 + 0.999922i \(0.503969\pi\)
\(200\) 3420.00 1.20915
\(201\) −3346.00 −1.17417
\(202\) −2100.00 −0.731463
\(203\) 1066.00 0.368564
\(204\) −9163.00 −3.14480
\(205\) −2352.00 −0.801321
\(206\) −2940.00 −0.994367
\(207\) −2112.00 −0.709150
\(208\) 1157.00 0.385690
\(209\) 3276.00 1.08424
\(210\) 3185.00 1.04660
\(211\) 2851.00 0.930194 0.465097 0.885260i \(-0.346019\pi\)
0.465097 + 0.885260i \(0.346019\pi\)
\(212\) −7344.00 −2.37919
\(213\) −63.0000 −0.0202661
\(214\) 3420.00 1.09246
\(215\) 1407.00 0.446310
\(216\) −1575.00 −0.496135
\(217\) −2548.00 −0.797095
\(218\) −1865.00 −0.579421
\(219\) −686.000 −0.211669
\(220\) 3094.00 0.948170
\(221\) 1001.00 0.304681
\(222\) −4585.00 −1.38615
\(223\) 217.000 0.0651632 0.0325816 0.999469i \(-0.489627\pi\)
0.0325816 + 0.999469i \(0.489627\pi\)
\(224\) 1105.00 0.329602
\(225\) −1672.00 −0.495407
\(226\) 8670.00 2.55186
\(227\) −2576.00 −0.753194 −0.376597 0.926377i \(-0.622906\pi\)
−0.376597 + 0.926377i \(0.622906\pi\)
\(228\) 14994.0 4.35527
\(229\) 455.000 0.131298 0.0656490 0.997843i \(-0.479088\pi\)
0.0656490 + 0.997843i \(0.479088\pi\)
\(230\) −3360.00 −0.963269
\(231\) −2366.00 −0.673902
\(232\) 3690.00 1.04423
\(233\) 3061.00 0.860656 0.430328 0.902673i \(-0.358398\pi\)
0.430328 + 0.902673i \(0.358398\pi\)
\(234\) −1430.00 −0.399496
\(235\) 735.000 0.204026
\(236\) −4998.00 −1.37857
\(237\) −9128.00 −2.50180
\(238\) 5005.00 1.36313
\(239\) −3477.00 −0.941039 −0.470520 0.882389i \(-0.655934\pi\)
−0.470520 + 0.882389i \(0.655934\pi\)
\(240\) 4361.00 1.17292
\(241\) −1610.00 −0.430329 −0.215164 0.976578i \(-0.569029\pi\)
−0.215164 + 0.976578i \(0.569029\pi\)
\(242\) 3275.00 0.869938
\(243\) 4928.00 1.30095
\(244\) −952.000 −0.249777
\(245\) 1218.00 0.317613
\(246\) 11760.0 3.04793
\(247\) −1638.00 −0.421957
\(248\) −8820.00 −2.25835
\(249\) 2156.00 0.548719
\(250\) −7035.00 −1.77973
\(251\) 1008.00 0.253484 0.126742 0.991936i \(-0.459548\pi\)
0.126742 + 0.991936i \(0.459548\pi\)
\(252\) −4862.00 −1.21539
\(253\) 2496.00 0.620246
\(254\) −9460.00 −2.33690
\(255\) 3773.00 0.926566
\(256\) −8279.00 −2.02124
\(257\) 6041.00 1.46625 0.733127 0.680092i \(-0.238060\pi\)
0.733127 + 0.680092i \(0.238060\pi\)
\(258\) −7035.00 −1.69760
\(259\) 1703.00 0.408569
\(260\) −1547.00 −0.369003
\(261\) −1804.00 −0.427834
\(262\) −7175.00 −1.69188
\(263\) −3708.00 −0.869373 −0.434686 0.900582i \(-0.643141\pi\)
−0.434686 + 0.900582i \(0.643141\pi\)
\(264\) −8190.00 −1.90932
\(265\) 3024.00 0.700992
\(266\) −8190.00 −1.88782
\(267\) 8330.00 1.90932
\(268\) 8126.00 1.85214
\(269\) 8344.00 1.89124 0.945618 0.325278i \(-0.105458\pi\)
0.945618 + 0.325278i \(0.105458\pi\)
\(270\) 1225.00 0.276115
\(271\) −1617.00 −0.362457 −0.181228 0.983441i \(-0.558007\pi\)
−0.181228 + 0.983441i \(0.558007\pi\)
\(272\) 6853.00 1.52766
\(273\) 1183.00 0.262265
\(274\) 8880.00 1.95788
\(275\) 1976.00 0.433299
\(276\) 11424.0 2.49146
\(277\) −3820.00 −0.828598 −0.414299 0.910141i \(-0.635973\pi\)
−0.414299 + 0.910141i \(0.635973\pi\)
\(278\) 9345.00 2.01610
\(279\) 4312.00 0.925278
\(280\) −4095.00 −0.874011
\(281\) −6214.00 −1.31920 −0.659602 0.751615i \(-0.729275\pi\)
−0.659602 + 0.751615i \(0.729275\pi\)
\(282\) −3675.00 −0.776039
\(283\) −5292.00 −1.11158 −0.555789 0.831323i \(-0.687584\pi\)
−0.555789 + 0.831323i \(0.687584\pi\)
\(284\) 153.000 0.0319679
\(285\) −6174.00 −1.28321
\(286\) 1690.00 0.349412
\(287\) −4368.00 −0.898379
\(288\) −1870.00 −0.382607
\(289\) 1016.00 0.206798
\(290\) −2870.00 −0.581146
\(291\) −490.000 −0.0987090
\(292\) 1666.00 0.333888
\(293\) −903.000 −0.180047 −0.0900236 0.995940i \(-0.528694\pi\)
−0.0900236 + 0.995940i \(0.528694\pi\)
\(294\) −6090.00 −1.20808
\(295\) 2058.00 0.406174
\(296\) 5895.00 1.15757
\(297\) −910.000 −0.177790
\(298\) −12330.0 −2.39684
\(299\) −1248.00 −0.241384
\(300\) 9044.00 1.74052
\(301\) 2613.00 0.500368
\(302\) 16615.0 3.16585
\(303\) −2940.00 −0.557421
\(304\) −11214.0 −2.11568
\(305\) 392.000 0.0735930
\(306\) −8470.00 −1.58235
\(307\) 2114.00 0.393004 0.196502 0.980503i \(-0.437042\pi\)
0.196502 + 0.980503i \(0.437042\pi\)
\(308\) 5746.00 1.06302
\(309\) −4116.00 −0.757770
\(310\) 6860.00 1.25684
\(311\) 3402.00 0.620288 0.310144 0.950690i \(-0.399623\pi\)
0.310144 + 0.950690i \(0.399623\pi\)
\(312\) 4095.00 0.743057
\(313\) −10689.0 −1.93028 −0.965141 0.261732i \(-0.915706\pi\)
−0.965141 + 0.261732i \(0.915706\pi\)
\(314\) 13650.0 2.45323
\(315\) 2002.00 0.358095
\(316\) 22168.0 3.94635
\(317\) −7054.00 −1.24982 −0.624909 0.780698i \(-0.714864\pi\)
−0.624909 + 0.780698i \(0.714864\pi\)
\(318\) −15120.0 −2.66631
\(319\) 2132.00 0.374198
\(320\) 2009.00 0.350958
\(321\) 4788.00 0.832524
\(322\) −6240.00 −1.07994
\(323\) −9702.00 −1.67131
\(324\) −14263.0 −2.44564
\(325\) −988.000 −0.168629
\(326\) 2720.00 0.462107
\(327\) −2611.00 −0.441555
\(328\) −15120.0 −2.54531
\(329\) 1365.00 0.228738
\(330\) 6370.00 1.06260
\(331\) 9704.00 1.61142 0.805710 0.592310i \(-0.201784\pi\)
0.805710 + 0.592310i \(0.201784\pi\)
\(332\) −5236.00 −0.865551
\(333\) −2882.00 −0.474272
\(334\) −8120.00 −1.33026
\(335\) −3346.00 −0.545706
\(336\) 8099.00 1.31499
\(337\) −10449.0 −1.68900 −0.844500 0.535555i \(-0.820103\pi\)
−0.844500 + 0.535555i \(0.820103\pi\)
\(338\) −845.000 −0.135982
\(339\) 12138.0 1.94468
\(340\) −9163.00 −1.46157
\(341\) −5096.00 −0.809278
\(342\) 13860.0 2.19141
\(343\) 6721.00 1.05802
\(344\) 9045.00 1.41766
\(345\) −4704.00 −0.734072
\(346\) 1680.00 0.261033
\(347\) −621.000 −0.0960721 −0.0480361 0.998846i \(-0.515296\pi\)
−0.0480361 + 0.998846i \(0.515296\pi\)
\(348\) 9758.00 1.50311
\(349\) 12481.0 1.91431 0.957153 0.289584i \(-0.0935168\pi\)
0.957153 + 0.289584i \(0.0935168\pi\)
\(350\) −4940.00 −0.754440
\(351\) 455.000 0.0691912
\(352\) 2210.00 0.334640
\(353\) −1400.00 −0.211089 −0.105545 0.994415i \(-0.533659\pi\)
−0.105545 + 0.994415i \(0.533659\pi\)
\(354\) −10290.0 −1.54494
\(355\) −63.0000 −0.00941885
\(356\) −20230.0 −3.01176
\(357\) 7007.00 1.03879
\(358\) 15145.0 2.23586
\(359\) −4968.00 −0.730365 −0.365182 0.930936i \(-0.618993\pi\)
−0.365182 + 0.930936i \(0.618993\pi\)
\(360\) 6930.00 1.01456
\(361\) 9017.00 1.31462
\(362\) 140.000 0.0203266
\(363\) 4585.00 0.662948
\(364\) −2873.00 −0.413698
\(365\) −686.000 −0.0983750
\(366\) −1960.00 −0.279920
\(367\) 8722.00 1.24056 0.620279 0.784381i \(-0.287019\pi\)
0.620279 + 0.784381i \(0.287019\pi\)
\(368\) −8544.00 −1.21029
\(369\) 7392.00 1.04285
\(370\) −4585.00 −0.644224
\(371\) 5616.00 0.785898
\(372\) −23324.0 −3.25079
\(373\) 10012.0 1.38982 0.694908 0.719098i \(-0.255445\pi\)
0.694908 + 0.719098i \(0.255445\pi\)
\(374\) 10010.0 1.38397
\(375\) −9849.00 −1.35627
\(376\) 4725.00 0.648067
\(377\) −1066.00 −0.145628
\(378\) 2275.00 0.309559
\(379\) −3372.00 −0.457013 −0.228507 0.973542i \(-0.573384\pi\)
−0.228507 + 0.973542i \(0.573384\pi\)
\(380\) 14994.0 2.02415
\(381\) −13244.0 −1.78087
\(382\) −2110.00 −0.282610
\(383\) −847.000 −0.113002 −0.0565009 0.998403i \(-0.517994\pi\)
−0.0565009 + 0.998403i \(0.517994\pi\)
\(384\) −14805.0 −1.96749
\(385\) −2366.00 −0.313201
\(386\) −2460.00 −0.324380
\(387\) −4422.00 −0.580834
\(388\) 1190.00 0.155704
\(389\) 11314.0 1.47466 0.737330 0.675533i \(-0.236086\pi\)
0.737330 + 0.675533i \(0.236086\pi\)
\(390\) −3185.00 −0.413535
\(391\) −7392.00 −0.956086
\(392\) 7830.00 1.00886
\(393\) −10045.0 −1.28932
\(394\) −14955.0 −1.91224
\(395\) −9128.00 −1.16273
\(396\) −9724.00 −1.23396
\(397\) 1862.00 0.235393 0.117697 0.993050i \(-0.462449\pi\)
0.117697 + 0.993050i \(0.462449\pi\)
\(398\) 350.000 0.0440802
\(399\) −11466.0 −1.43864
\(400\) −6764.00 −0.845500
\(401\) 6820.00 0.849313 0.424657 0.905355i \(-0.360395\pi\)
0.424657 + 0.905355i \(0.360395\pi\)
\(402\) 16730.0 2.07566
\(403\) 2548.00 0.314950
\(404\) 7140.00 0.879278
\(405\) 5873.00 0.720572
\(406\) −5330.00 −0.651536
\(407\) 3406.00 0.414814
\(408\) 24255.0 2.94314
\(409\) −12992.0 −1.57069 −0.785346 0.619057i \(-0.787515\pi\)
−0.785346 + 0.619057i \(0.787515\pi\)
\(410\) 11760.0 1.41655
\(411\) 12432.0 1.49203
\(412\) 9996.00 1.19531
\(413\) 3822.00 0.455371
\(414\) 10560.0 1.25361
\(415\) 2156.00 0.255021
\(416\) −1105.00 −0.130233
\(417\) 13083.0 1.53640
\(418\) −16380.0 −1.91668
\(419\) −7343.00 −0.856155 −0.428078 0.903742i \(-0.640809\pi\)
−0.428078 + 0.903742i \(0.640809\pi\)
\(420\) −10829.0 −1.25810
\(421\) −5059.00 −0.585655 −0.292827 0.956165i \(-0.594596\pi\)
−0.292827 + 0.956165i \(0.594596\pi\)
\(422\) −14255.0 −1.64437
\(423\) −2310.00 −0.265523
\(424\) 19440.0 2.22663
\(425\) −5852.00 −0.667915
\(426\) 315.000 0.0358258
\(427\) 728.000 0.0825068
\(428\) −11628.0 −1.31323
\(429\) 2366.00 0.266274
\(430\) −7035.00 −0.788972
\(431\) 3243.00 0.362436 0.181218 0.983443i \(-0.441996\pi\)
0.181218 + 0.983443i \(0.441996\pi\)
\(432\) 3115.00 0.346922
\(433\) 11599.0 1.28733 0.643663 0.765309i \(-0.277414\pi\)
0.643663 + 0.765309i \(0.277414\pi\)
\(434\) 12740.0 1.40908
\(435\) −4018.00 −0.442870
\(436\) 6341.00 0.696511
\(437\) 12096.0 1.32410
\(438\) 3430.00 0.374182
\(439\) −17374.0 −1.88887 −0.944437 0.328692i \(-0.893392\pi\)
−0.944437 + 0.328692i \(0.893392\pi\)
\(440\) −8190.00 −0.887370
\(441\) −3828.00 −0.413346
\(442\) −5005.00 −0.538605
\(443\) 989.000 0.106070 0.0530348 0.998593i \(-0.483111\pi\)
0.0530348 + 0.998593i \(0.483111\pi\)
\(444\) 15589.0 1.66626
\(445\) 8330.00 0.887370
\(446\) −1085.00 −0.115193
\(447\) −17262.0 −1.82654
\(448\) 3731.00 0.393467
\(449\) −14474.0 −1.52131 −0.760657 0.649154i \(-0.775123\pi\)
−0.760657 + 0.649154i \(0.775123\pi\)
\(450\) 8360.00 0.875765
\(451\) −8736.00 −0.912111
\(452\) −29478.0 −3.06754
\(453\) 23261.0 2.41258
\(454\) 12880.0 1.33147
\(455\) 1183.00 0.121890
\(456\) −39690.0 −4.07600
\(457\) −1594.00 −0.163160 −0.0815801 0.996667i \(-0.525997\pi\)
−0.0815801 + 0.996667i \(0.525997\pi\)
\(458\) −2275.00 −0.232104
\(459\) 2695.00 0.274056
\(460\) 11424.0 1.15793
\(461\) −5915.00 −0.597590 −0.298795 0.954317i \(-0.596585\pi\)
−0.298795 + 0.954317i \(0.596585\pi\)
\(462\) 11830.0 1.19130
\(463\) −11072.0 −1.11136 −0.555680 0.831396i \(-0.687542\pi\)
−0.555680 + 0.831396i \(0.687542\pi\)
\(464\) −7298.00 −0.730175
\(465\) 9604.00 0.957795
\(466\) −15305.0 −1.52144
\(467\) 1260.00 0.124852 0.0624260 0.998050i \(-0.480116\pi\)
0.0624260 + 0.998050i \(0.480116\pi\)
\(468\) 4862.00 0.480227
\(469\) −6214.00 −0.611804
\(470\) −3675.00 −0.360670
\(471\) 19110.0 1.86952
\(472\) 13230.0 1.29017
\(473\) 5226.00 0.508016
\(474\) 45640.0 4.42260
\(475\) 9576.00 0.925004
\(476\) −17017.0 −1.63860
\(477\) −9504.00 −0.912281
\(478\) 17385.0 1.66354
\(479\) −12033.0 −1.14781 −0.573906 0.818921i \(-0.694572\pi\)
−0.573906 + 0.818921i \(0.694572\pi\)
\(480\) −4165.00 −0.396053
\(481\) −1703.00 −0.161435
\(482\) 8050.00 0.760721
\(483\) −8736.00 −0.822985
\(484\) −11135.0 −1.04574
\(485\) −490.000 −0.0458758
\(486\) −24640.0 −2.29978
\(487\) −2280.00 −0.212149 −0.106075 0.994358i \(-0.533828\pi\)
−0.106075 + 0.994358i \(0.533828\pi\)
\(488\) 2520.00 0.233760
\(489\) 3808.00 0.352155
\(490\) −6090.00 −0.561466
\(491\) 16767.0 1.54111 0.770554 0.637375i \(-0.219980\pi\)
0.770554 + 0.637375i \(0.219980\pi\)
\(492\) −39984.0 −3.66386
\(493\) −6314.00 −0.576812
\(494\) 8190.00 0.745922
\(495\) 4004.00 0.363569
\(496\) 17444.0 1.57915
\(497\) −117.000 −0.0105597
\(498\) −10780.0 −0.970007
\(499\) 12840.0 1.15190 0.575949 0.817485i \(-0.304633\pi\)
0.575949 + 0.817485i \(0.304633\pi\)
\(500\) 23919.0 2.13938
\(501\) −11368.0 −1.01374
\(502\) −5040.00 −0.448100
\(503\) −2198.00 −0.194839 −0.0974195 0.995243i \(-0.531059\pi\)
−0.0974195 + 0.995243i \(0.531059\pi\)
\(504\) 12870.0 1.13745
\(505\) −2940.00 −0.259066
\(506\) −12480.0 −1.09645
\(507\) −1183.00 −0.103627
\(508\) 32164.0 2.80915
\(509\) −17066.0 −1.48612 −0.743062 0.669223i \(-0.766627\pi\)
−0.743062 + 0.669223i \(0.766627\pi\)
\(510\) −18865.0 −1.63795
\(511\) −1274.00 −0.110290
\(512\) 24475.0 2.11260
\(513\) −4410.00 −0.379544
\(514\) −30205.0 −2.59200
\(515\) −4116.00 −0.352180
\(516\) 23919.0 2.04065
\(517\) 2730.00 0.232235
\(518\) −8515.00 −0.722254
\(519\) 2352.00 0.198924
\(520\) 4095.00 0.345342
\(521\) 2583.00 0.217204 0.108602 0.994085i \(-0.465363\pi\)
0.108602 + 0.994085i \(0.465363\pi\)
\(522\) 9020.00 0.756312
\(523\) 18620.0 1.55678 0.778390 0.627781i \(-0.216037\pi\)
0.778390 + 0.627781i \(0.216037\pi\)
\(524\) 24395.0 2.03378
\(525\) −6916.00 −0.574931
\(526\) 18540.0 1.53685
\(527\) 15092.0 1.24747
\(528\) 16198.0 1.33509
\(529\) −2951.00 −0.242541
\(530\) −15120.0 −1.23919
\(531\) −6468.00 −0.528601
\(532\) 27846.0 2.26932
\(533\) 4368.00 0.354970
\(534\) −41650.0 −3.37523
\(535\) 4788.00 0.386922
\(536\) −21510.0 −1.73338
\(537\) 21203.0 1.70387
\(538\) −41720.0 −3.34327
\(539\) 4524.00 0.361526
\(540\) −4165.00 −0.331913
\(541\) −16833.0 −1.33772 −0.668861 0.743388i \(-0.733218\pi\)
−0.668861 + 0.743388i \(0.733218\pi\)
\(542\) 8085.00 0.640739
\(543\) 196.000 0.0154902
\(544\) −6545.00 −0.515836
\(545\) −2611.00 −0.205216
\(546\) −5915.00 −0.463624
\(547\) −8615.00 −0.673402 −0.336701 0.941612i \(-0.609311\pi\)
−0.336701 + 0.941612i \(0.609311\pi\)
\(548\) −30192.0 −2.35354
\(549\) −1232.00 −0.0957750
\(550\) −9880.00 −0.765972
\(551\) 10332.0 0.798835
\(552\) −30240.0 −2.33170
\(553\) −16952.0 −1.30357
\(554\) 19100.0 1.46477
\(555\) −6419.00 −0.490939
\(556\) −31773.0 −2.42352
\(557\) 8535.00 0.649263 0.324632 0.945841i \(-0.394760\pi\)
0.324632 + 0.945841i \(0.394760\pi\)
\(558\) −21560.0 −1.63568
\(559\) −2613.00 −0.197707
\(560\) 8099.00 0.611152
\(561\) 14014.0 1.05467
\(562\) 31070.0 2.33204
\(563\) −4641.00 −0.347415 −0.173708 0.984797i \(-0.555575\pi\)
−0.173708 + 0.984797i \(0.555575\pi\)
\(564\) 12495.0 0.932862
\(565\) 12138.0 0.903804
\(566\) 26460.0 1.96501
\(567\) 10907.0 0.807850
\(568\) −405.000 −0.0299180
\(569\) −4793.00 −0.353134 −0.176567 0.984289i \(-0.556499\pi\)
−0.176567 + 0.984289i \(0.556499\pi\)
\(570\) 30870.0 2.26842
\(571\) −5563.00 −0.407713 −0.203857 0.979001i \(-0.565348\pi\)
−0.203857 + 0.979001i \(0.565348\pi\)
\(572\) −5746.00 −0.420022
\(573\) −2954.00 −0.215367
\(574\) 21840.0 1.58813
\(575\) 7296.00 0.529155
\(576\) −6314.00 −0.456742
\(577\) 24038.0 1.73434 0.867171 0.498011i \(-0.165936\pi\)
0.867171 + 0.498011i \(0.165936\pi\)
\(578\) −5080.00 −0.365571
\(579\) −3444.00 −0.247198
\(580\) 9758.00 0.698584
\(581\) 4004.00 0.285910
\(582\) 2450.00 0.174494
\(583\) 11232.0 0.797911
\(584\) −4410.00 −0.312478
\(585\) −2002.00 −0.141491
\(586\) 4515.00 0.318281
\(587\) −21224.0 −1.49235 −0.746174 0.665751i \(-0.768111\pi\)
−0.746174 + 0.665751i \(0.768111\pi\)
\(588\) 20706.0 1.45221
\(589\) −24696.0 −1.72764
\(590\) −10290.0 −0.718021
\(591\) −20937.0 −1.45725
\(592\) −11659.0 −0.809429
\(593\) 4354.00 0.301513 0.150757 0.988571i \(-0.451829\pi\)
0.150757 + 0.988571i \(0.451829\pi\)
\(594\) 4550.00 0.314291
\(595\) 7007.00 0.482788
\(596\) 41922.0 2.88119
\(597\) 490.000 0.0335919
\(598\) 6240.00 0.426710
\(599\) 7310.00 0.498629 0.249314 0.968423i \(-0.419795\pi\)
0.249314 + 0.968423i \(0.419795\pi\)
\(600\) −23940.0 −1.62891
\(601\) −7595.00 −0.515485 −0.257743 0.966214i \(-0.582979\pi\)
−0.257743 + 0.966214i \(0.582979\pi\)
\(602\) −13065.0 −0.884534
\(603\) 10516.0 0.710190
\(604\) −56491.0 −3.80561
\(605\) 4585.00 0.308110
\(606\) 14700.0 0.985391
\(607\) −826.000 −0.0552328 −0.0276164 0.999619i \(-0.508792\pi\)
−0.0276164 + 0.999619i \(0.508792\pi\)
\(608\) 10710.0 0.714388
\(609\) −7462.00 −0.496511
\(610\) −1960.00 −0.130095
\(611\) −1365.00 −0.0903797
\(612\) 28798.0 1.90211
\(613\) 14590.0 0.961312 0.480656 0.876909i \(-0.340398\pi\)
0.480656 + 0.876909i \(0.340398\pi\)
\(614\) −10570.0 −0.694740
\(615\) 16464.0 1.07950
\(616\) −15210.0 −0.994851
\(617\) 4888.00 0.318936 0.159468 0.987203i \(-0.449022\pi\)
0.159468 + 0.987203i \(0.449022\pi\)
\(618\) 20580.0 1.33956
\(619\) −11004.0 −0.714520 −0.357260 0.934005i \(-0.616289\pi\)
−0.357260 + 0.934005i \(0.616289\pi\)
\(620\) −23324.0 −1.51083
\(621\) −3360.00 −0.217121
\(622\) −17010.0 −1.09653
\(623\) 15470.0 0.994851
\(624\) −8099.00 −0.519582
\(625\) −349.000 −0.0223360
\(626\) 53445.0 3.41229
\(627\) −22932.0 −1.46063
\(628\) −46410.0 −2.94898
\(629\) −10087.0 −0.639420
\(630\) −10010.0 −0.633028
\(631\) −4975.00 −0.313869 −0.156935 0.987609i \(-0.550161\pi\)
−0.156935 + 0.987609i \(0.550161\pi\)
\(632\) −58680.0 −3.69330
\(633\) −19957.0 −1.25311
\(634\) 35270.0 2.20939
\(635\) −13244.0 −0.827673
\(636\) 51408.0 3.20513
\(637\) −2262.00 −0.140697
\(638\) −10660.0 −0.661494
\(639\) 198.000 0.0122578
\(640\) −14805.0 −0.914405
\(641\) 3950.00 0.243394 0.121697 0.992567i \(-0.461166\pi\)
0.121697 + 0.992567i \(0.461166\pi\)
\(642\) −23940.0 −1.47171
\(643\) −3682.00 −0.225823 −0.112911 0.993605i \(-0.536018\pi\)
−0.112911 + 0.993605i \(0.536018\pi\)
\(644\) 21216.0 1.29818
\(645\) −9849.00 −0.601247
\(646\) 48510.0 2.95449
\(647\) 10402.0 0.632063 0.316032 0.948749i \(-0.397649\pi\)
0.316032 + 0.948749i \(0.397649\pi\)
\(648\) 37755.0 2.28882
\(649\) 7644.00 0.462332
\(650\) 4940.00 0.298097
\(651\) 17836.0 1.07381
\(652\) −9248.00 −0.555490
\(653\) −31680.0 −1.89852 −0.949260 0.314491i \(-0.898166\pi\)
−0.949260 + 0.314491i \(0.898166\pi\)
\(654\) 13055.0 0.780567
\(655\) −10045.0 −0.599222
\(656\) 29904.0 1.77981
\(657\) 2156.00 0.128027
\(658\) −6825.00 −0.404356
\(659\) 21940.0 1.29691 0.648453 0.761255i \(-0.275416\pi\)
0.648453 + 0.761255i \(0.275416\pi\)
\(660\) −21658.0 −1.27733
\(661\) −31374.0 −1.84615 −0.923077 0.384616i \(-0.874334\pi\)
−0.923077 + 0.384616i \(0.874334\pi\)
\(662\) −48520.0 −2.84862
\(663\) −7007.00 −0.410451
\(664\) 13860.0 0.810049
\(665\) −11466.0 −0.668620
\(666\) 14410.0 0.838403
\(667\) 7872.00 0.456979
\(668\) 27608.0 1.59908
\(669\) −1519.00 −0.0877847
\(670\) 16730.0 0.964681
\(671\) 1456.00 0.0837679
\(672\) −7735.00 −0.444024
\(673\) 18013.0 1.03172 0.515862 0.856672i \(-0.327472\pi\)
0.515862 + 0.856672i \(0.327472\pi\)
\(674\) 52245.0 2.98576
\(675\) −2660.00 −0.151679
\(676\) 2873.00 0.163462
\(677\) −10640.0 −0.604030 −0.302015 0.953303i \(-0.597659\pi\)
−0.302015 + 0.953303i \(0.597659\pi\)
\(678\) −60690.0 −3.43774
\(679\) −910.000 −0.0514324
\(680\) 24255.0 1.36785
\(681\) 18032.0 1.01467
\(682\) 25480.0 1.43062
\(683\) −9336.00 −0.523034 −0.261517 0.965199i \(-0.584223\pi\)
−0.261517 + 0.965199i \(0.584223\pi\)
\(684\) −47124.0 −2.63426
\(685\) 12432.0 0.693434
\(686\) −33605.0 −1.87033
\(687\) −3185.00 −0.176878
\(688\) −17889.0 −0.991296
\(689\) −5616.00 −0.310526
\(690\) 23520.0 1.29767
\(691\) 4200.00 0.231224 0.115612 0.993294i \(-0.463117\pi\)
0.115612 + 0.993294i \(0.463117\pi\)
\(692\) −5712.00 −0.313783
\(693\) 7436.00 0.407605
\(694\) 3105.00 0.169833
\(695\) 13083.0 0.714052
\(696\) −25830.0 −1.40673
\(697\) 25872.0 1.40599
\(698\) −62405.0 −3.38405
\(699\) −21427.0 −1.15943
\(700\) 16796.0 0.906899
\(701\) 9872.00 0.531898 0.265949 0.963987i \(-0.414315\pi\)
0.265949 + 0.963987i \(0.414315\pi\)
\(702\) −2275.00 −0.122314
\(703\) 16506.0 0.885541
\(704\) 7462.00 0.399481
\(705\) −5145.00 −0.274854
\(706\) 7000.00 0.373156
\(707\) −5460.00 −0.290445
\(708\) 34986.0 1.85714
\(709\) 28450.0 1.50700 0.753499 0.657449i \(-0.228364\pi\)
0.753499 + 0.657449i \(0.228364\pi\)
\(710\) 315.000 0.0166503
\(711\) 28688.0 1.51320
\(712\) 53550.0 2.81864
\(713\) −18816.0 −0.988310
\(714\) −35035.0 −1.83635
\(715\) 2366.00 0.123753
\(716\) −51493.0 −2.68769
\(717\) 24339.0 1.26772
\(718\) 24840.0 1.29111
\(719\) 32718.0 1.69705 0.848523 0.529159i \(-0.177493\pi\)
0.848523 + 0.529159i \(0.177493\pi\)
\(720\) −13706.0 −0.709434
\(721\) −7644.00 −0.394837
\(722\) −45085.0 −2.32395
\(723\) 11270.0 0.579718
\(724\) −476.000 −0.0244343
\(725\) 6232.00 0.319242
\(726\) −22925.0 −1.17194
\(727\) −22834.0 −1.16488 −0.582439 0.812874i \(-0.697901\pi\)
−0.582439 + 0.812874i \(0.697901\pi\)
\(728\) 7605.00 0.387170
\(729\) −11843.0 −0.601687
\(730\) 3430.00 0.173904
\(731\) −15477.0 −0.783088
\(732\) 6664.00 0.336487
\(733\) 7875.00 0.396821 0.198410 0.980119i \(-0.436422\pi\)
0.198410 + 0.980119i \(0.436422\pi\)
\(734\) −43610.0 −2.19302
\(735\) −8526.00 −0.427872
\(736\) 8160.00 0.408671
\(737\) −12428.0 −0.621155
\(738\) −36960.0 −1.84352
\(739\) −2140.00 −0.106524 −0.0532620 0.998581i \(-0.516962\pi\)
−0.0532620 + 0.998581i \(0.516962\pi\)
\(740\) 15589.0 0.774410
\(741\) 11466.0 0.568440
\(742\) −28080.0 −1.38928
\(743\) 31971.0 1.57860 0.789302 0.614006i \(-0.210443\pi\)
0.789302 + 0.614006i \(0.210443\pi\)
\(744\) 61740.0 3.04234
\(745\) −17262.0 −0.848900
\(746\) −50060.0 −2.45687
\(747\) −6776.00 −0.331889
\(748\) −34034.0 −1.66364
\(749\) 8892.00 0.433787
\(750\) 49245.0 2.39756
\(751\) −7432.00 −0.361115 −0.180558 0.983564i \(-0.557790\pi\)
−0.180558 + 0.983564i \(0.557790\pi\)
\(752\) −9345.00 −0.453161
\(753\) −7056.00 −0.341481
\(754\) 5330.00 0.257437
\(755\) 23261.0 1.12126
\(756\) −7735.00 −0.372115
\(757\) 20176.0 0.968704 0.484352 0.874873i \(-0.339055\pi\)
0.484352 + 0.874873i \(0.339055\pi\)
\(758\) 16860.0 0.807893
\(759\) −17472.0 −0.835564
\(760\) −39690.0 −1.89435
\(761\) −9478.00 −0.451481 −0.225741 0.974187i \(-0.572480\pi\)
−0.225741 + 0.974187i \(0.572480\pi\)
\(762\) 66220.0 3.14816
\(763\) −4849.00 −0.230073
\(764\) 7174.00 0.339720
\(765\) −11858.0 −0.560427
\(766\) 4235.00 0.199761
\(767\) −3822.00 −0.179928
\(768\) 57953.0 2.72292
\(769\) −12096.0 −0.567221 −0.283610 0.958940i \(-0.591532\pi\)
−0.283610 + 0.958940i \(0.591532\pi\)
\(770\) 11830.0 0.553667
\(771\) −42287.0 −1.97526
\(772\) 8364.00 0.389931
\(773\) 17941.0 0.834790 0.417395 0.908725i \(-0.362943\pi\)
0.417395 + 0.908725i \(0.362943\pi\)
\(774\) 22110.0 1.02678
\(775\) −14896.0 −0.690426
\(776\) −3150.00 −0.145720
\(777\) −11921.0 −0.550403
\(778\) −56570.0 −2.60685
\(779\) −42336.0 −1.94717
\(780\) 10829.0 0.497103
\(781\) −234.000 −0.0107211
\(782\) 36960.0 1.69014
\(783\) −2870.00 −0.130990
\(784\) −15486.0 −0.705448
\(785\) 19110.0 0.868873
\(786\) 50225.0 2.27922
\(787\) 6664.00 0.301837 0.150919 0.988546i \(-0.451777\pi\)
0.150919 + 0.988546i \(0.451777\pi\)
\(788\) 50847.0 2.29867
\(789\) 25956.0 1.17118
\(790\) 45640.0 2.05544
\(791\) 22542.0 1.01328
\(792\) 25740.0 1.15484
\(793\) −728.000 −0.0326003
\(794\) −9310.00 −0.416120
\(795\) −21168.0 −0.944342
\(796\) −1190.00 −0.0529880
\(797\) −1442.00 −0.0640882 −0.0320441 0.999486i \(-0.510202\pi\)
−0.0320441 + 0.999486i \(0.510202\pi\)
\(798\) 57330.0 2.54318
\(799\) −8085.00 −0.357981
\(800\) 6460.00 0.285494
\(801\) −26180.0 −1.15484
\(802\) −34100.0 −1.50139
\(803\) −2548.00 −0.111976
\(804\) −56882.0 −2.49512
\(805\) −8736.00 −0.382489
\(806\) −12740.0 −0.556759
\(807\) −58408.0 −2.54778
\(808\) −18900.0 −0.822896
\(809\) 30207.0 1.31276 0.656379 0.754431i \(-0.272087\pi\)
0.656379 + 0.754431i \(0.272087\pi\)
\(810\) −29365.0 −1.27380
\(811\) 21140.0 0.915322 0.457661 0.889127i \(-0.348687\pi\)
0.457661 + 0.889127i \(0.348687\pi\)
\(812\) 18122.0 0.783199
\(813\) 11319.0 0.488284
\(814\) −17030.0 −0.733294
\(815\) 3808.00 0.163667
\(816\) −47971.0 −2.05799
\(817\) 25326.0 1.08451
\(818\) 64960.0 2.77662
\(819\) −3718.00 −0.158629
\(820\) −39984.0 −1.70281
\(821\) 569.000 0.0241879 0.0120939 0.999927i \(-0.496150\pi\)
0.0120939 + 0.999927i \(0.496150\pi\)
\(822\) −62160.0 −2.63757
\(823\) −8538.00 −0.361623 −0.180812 0.983518i \(-0.557872\pi\)
−0.180812 + 0.983518i \(0.557872\pi\)
\(824\) −26460.0 −1.11866
\(825\) −13832.0 −0.583719
\(826\) −19110.0 −0.804990
\(827\) −32702.0 −1.37504 −0.687521 0.726164i \(-0.741301\pi\)
−0.687521 + 0.726164i \(0.741301\pi\)
\(828\) −35904.0 −1.50694
\(829\) −21154.0 −0.886259 −0.443130 0.896458i \(-0.646132\pi\)
−0.443130 + 0.896458i \(0.646132\pi\)
\(830\) −10780.0 −0.450818
\(831\) 26740.0 1.11625
\(832\) −3731.00 −0.155468
\(833\) −13398.0 −0.557279
\(834\) −65415.0 −2.71599
\(835\) −11368.0 −0.471145
\(836\) 55692.0 2.30400
\(837\) 6860.00 0.283293
\(838\) 36715.0 1.51348
\(839\) −2184.00 −0.0898690 −0.0449345 0.998990i \(-0.514308\pi\)
−0.0449345 + 0.998990i \(0.514308\pi\)
\(840\) 28665.0 1.17742
\(841\) −17665.0 −0.724302
\(842\) 25295.0 1.03530
\(843\) 43498.0 1.77717
\(844\) 48467.0 1.97666
\(845\) −1183.00 −0.0481615
\(846\) 11550.0 0.469382
\(847\) 8515.00 0.345430
\(848\) −38448.0 −1.55697
\(849\) 37044.0 1.49746
\(850\) 29260.0 1.18072
\(851\) 12576.0 0.506580
\(852\) −1071.00 −0.0430656
\(853\) 36687.0 1.47261 0.736307 0.676648i \(-0.236568\pi\)
0.736307 + 0.676648i \(0.236568\pi\)
\(854\) −3640.00 −0.145853
\(855\) 19404.0 0.776144
\(856\) 30780.0 1.22902
\(857\) 36806.0 1.46706 0.733529 0.679658i \(-0.237872\pi\)
0.733529 + 0.679658i \(0.237872\pi\)
\(858\) −11830.0 −0.470710
\(859\) 4900.00 0.194628 0.0973142 0.995254i \(-0.468975\pi\)
0.0973142 + 0.995254i \(0.468975\pi\)
\(860\) 23919.0 0.948408
\(861\) 30576.0 1.21025
\(862\) −16215.0 −0.640702
\(863\) −13697.0 −0.540268 −0.270134 0.962823i \(-0.587068\pi\)
−0.270134 + 0.962823i \(0.587068\pi\)
\(864\) −2975.00 −0.117143
\(865\) 2352.00 0.0924513
\(866\) −57995.0 −2.27569
\(867\) −7112.00 −0.278588
\(868\) −43316.0 −1.69383
\(869\) −33904.0 −1.32349
\(870\) 20090.0 0.782891
\(871\) 6214.00 0.241737
\(872\) −16785.0 −0.651848
\(873\) 1540.00 0.0597034
\(874\) −60480.0 −2.34069
\(875\) −18291.0 −0.706684
\(876\) −11662.0 −0.449797
\(877\) 6239.00 0.240224 0.120112 0.992760i \(-0.461675\pi\)
0.120112 + 0.992760i \(0.461675\pi\)
\(878\) 86870.0 3.33909
\(879\) 6321.00 0.242551
\(880\) 16198.0 0.620494
\(881\) 133.000 0.00508613 0.00254307 0.999997i \(-0.499191\pi\)
0.00254307 + 0.999997i \(0.499191\pi\)
\(882\) 19140.0 0.730700
\(883\) −26003.0 −0.991020 −0.495510 0.868602i \(-0.665019\pi\)
−0.495510 + 0.868602i \(0.665019\pi\)
\(884\) 17017.0 0.647448
\(885\) −14406.0 −0.547178
\(886\) −4945.00 −0.187506
\(887\) −31248.0 −1.18287 −0.591435 0.806353i \(-0.701438\pi\)
−0.591435 + 0.806353i \(0.701438\pi\)
\(888\) −41265.0 −1.55942
\(889\) −24596.0 −0.927923
\(890\) −41650.0 −1.56866
\(891\) 21814.0 0.820198
\(892\) 3689.00 0.138472
\(893\) 13230.0 0.495773
\(894\) 86310.0 3.22890
\(895\) 21203.0 0.791886
\(896\) −27495.0 −1.02516
\(897\) 8736.00 0.325180
\(898\) 72370.0 2.68933
\(899\) −16072.0 −0.596253
\(900\) −28424.0 −1.05274
\(901\) −33264.0 −1.22995
\(902\) 43680.0 1.61240
\(903\) −18291.0 −0.674071
\(904\) 78030.0 2.87084
\(905\) 196.000 0.00719918
\(906\) −116305. −4.26487
\(907\) −38253.0 −1.40041 −0.700204 0.713943i \(-0.746908\pi\)
−0.700204 + 0.713943i \(0.746908\pi\)
\(908\) −43792.0 −1.60054
\(909\) 9240.00 0.337152
\(910\) −5915.00 −0.215473
\(911\) 36374.0 1.32286 0.661429 0.750007i \(-0.269950\pi\)
0.661429 + 0.750007i \(0.269950\pi\)
\(912\) 78498.0 2.85014
\(913\) 8008.00 0.290281
\(914\) 7970.00 0.288429
\(915\) −2744.00 −0.0991408
\(916\) 7735.00 0.279008
\(917\) −18655.0 −0.671802
\(918\) −13475.0 −0.484468
\(919\) −27648.0 −0.992408 −0.496204 0.868206i \(-0.665273\pi\)
−0.496204 + 0.868206i \(0.665273\pi\)
\(920\) −30240.0 −1.08368
\(921\) −14798.0 −0.529436
\(922\) 29575.0 1.05640
\(923\) 117.000 0.00417237
\(924\) −40222.0 −1.43204
\(925\) 9956.00 0.353893
\(926\) 55360.0 1.96462
\(927\) 12936.0 0.458332
\(928\) 6970.00 0.246553
\(929\) 756.000 0.0266992 0.0133496 0.999911i \(-0.495751\pi\)
0.0133496 + 0.999911i \(0.495751\pi\)
\(930\) −48020.0 −1.69316
\(931\) 21924.0 0.771783
\(932\) 52037.0 1.82889
\(933\) −23814.0 −0.835622
\(934\) −6300.00 −0.220709
\(935\) 14014.0 0.490168
\(936\) −12870.0 −0.449433
\(937\) 20846.0 0.726797 0.363399 0.931634i \(-0.381616\pi\)
0.363399 + 0.931634i \(0.381616\pi\)
\(938\) 31070.0 1.08153
\(939\) 74823.0 2.60038
\(940\) 12495.0 0.433555
\(941\) −41321.0 −1.43148 −0.715742 0.698365i \(-0.753911\pi\)
−0.715742 + 0.698365i \(0.753911\pi\)
\(942\) −95550.0 −3.30487
\(943\) −32256.0 −1.11389
\(944\) −26166.0 −0.902151
\(945\) 3185.00 0.109638
\(946\) −26130.0 −0.898055
\(947\) 54966.0 1.88612 0.943060 0.332624i \(-0.107934\pi\)
0.943060 + 0.332624i \(0.107934\pi\)
\(948\) −155176. −5.31633
\(949\) 1274.00 0.0435783
\(950\) −47880.0 −1.63519
\(951\) 49378.0 1.68369
\(952\) 45045.0 1.53353
\(953\) −44553.0 −1.51439 −0.757195 0.653189i \(-0.773431\pi\)
−0.757195 + 0.653189i \(0.773431\pi\)
\(954\) 47520.0 1.61270
\(955\) −2954.00 −0.100093
\(956\) −59109.0 −1.99971
\(957\) −14924.0 −0.504101
\(958\) 60165.0 2.02906
\(959\) 23088.0 0.777425
\(960\) −14063.0 −0.472793
\(961\) 8625.00 0.289517
\(962\) 8515.00 0.285379
\(963\) −15048.0 −0.503546
\(964\) −27370.0 −0.914448
\(965\) −3444.00 −0.114887
\(966\) 43680.0 1.45485
\(967\) −27907.0 −0.928054 −0.464027 0.885821i \(-0.653596\pi\)
−0.464027 + 0.885821i \(0.653596\pi\)
\(968\) 29475.0 0.978680
\(969\) 67914.0 2.25151
\(970\) 2450.00 0.0810977
\(971\) −16443.0 −0.543441 −0.271720 0.962376i \(-0.587593\pi\)
−0.271720 + 0.962376i \(0.587593\pi\)
\(972\) 83776.0 2.76452
\(973\) 24297.0 0.800541
\(974\) 11400.0 0.375030
\(975\) 6916.00 0.227168
\(976\) −4984.00 −0.163457
\(977\) −45414.0 −1.48713 −0.743563 0.668666i \(-0.766866\pi\)
−0.743563 + 0.668666i \(0.766866\pi\)
\(978\) −19040.0 −0.622528
\(979\) 30940.0 1.01006
\(980\) 20706.0 0.674927
\(981\) 8206.00 0.267072
\(982\) −83835.0 −2.72432
\(983\) −8981.00 −0.291403 −0.145702 0.989329i \(-0.546544\pi\)
−0.145702 + 0.989329i \(0.546544\pi\)
\(984\) 105840. 3.42892
\(985\) −20937.0 −0.677267
\(986\) 31570.0 1.01967
\(987\) −9555.00 −0.308145
\(988\) −27846.0 −0.896659
\(989\) 19296.0 0.620402
\(990\) −20020.0 −0.642704
\(991\) −17414.0 −0.558198 −0.279099 0.960262i \(-0.590036\pi\)
−0.279099 + 0.960262i \(0.590036\pi\)
\(992\) −16660.0 −0.533221
\(993\) −67928.0 −2.17083
\(994\) 585.000 0.0186671
\(995\) 490.000 0.0156121
\(996\) 36652.0 1.16603
\(997\) −23702.0 −0.752909 −0.376454 0.926435i \(-0.622857\pi\)
−0.376454 + 0.926435i \(0.622857\pi\)
\(998\) −64200.0 −2.03629
\(999\) −4585.00 −0.145208
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 13.4.a.a.1.1 1
3.2 odd 2 117.4.a.b.1.1 1
4.3 odd 2 208.4.a.g.1.1 1
5.2 odd 4 325.4.b.b.274.1 2
5.3 odd 4 325.4.b.b.274.2 2
5.4 even 2 325.4.a.d.1.1 1
7.6 odd 2 637.4.a.a.1.1 1
8.3 odd 2 832.4.a.a.1.1 1
8.5 even 2 832.4.a.r.1.1 1
11.10 odd 2 1573.4.a.a.1.1 1
12.11 even 2 1872.4.a.k.1.1 1
13.2 odd 12 169.4.e.e.147.1 4
13.3 even 3 169.4.c.e.22.1 2
13.4 even 6 169.4.c.a.146.1 2
13.5 odd 4 169.4.b.a.168.2 2
13.6 odd 12 169.4.e.e.23.2 4
13.7 odd 12 169.4.e.e.23.1 4
13.8 odd 4 169.4.b.a.168.1 2
13.9 even 3 169.4.c.e.146.1 2
13.10 even 6 169.4.c.a.22.1 2
13.11 odd 12 169.4.e.e.147.2 4
13.12 even 2 169.4.a.e.1.1 1
39.38 odd 2 1521.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.a.a.1.1 1 1.1 even 1 trivial
117.4.a.b.1.1 1 3.2 odd 2
169.4.a.e.1.1 1 13.12 even 2
169.4.b.a.168.1 2 13.8 odd 4
169.4.b.a.168.2 2 13.5 odd 4
169.4.c.a.22.1 2 13.10 even 6
169.4.c.a.146.1 2 13.4 even 6
169.4.c.e.22.1 2 13.3 even 3
169.4.c.e.146.1 2 13.9 even 3
169.4.e.e.23.1 4 13.7 odd 12
169.4.e.e.23.2 4 13.6 odd 12
169.4.e.e.147.1 4 13.2 odd 12
169.4.e.e.147.2 4 13.11 odd 12
208.4.a.g.1.1 1 4.3 odd 2
325.4.a.d.1.1 1 5.4 even 2
325.4.b.b.274.1 2 5.2 odd 4
325.4.b.b.274.2 2 5.3 odd 4
637.4.a.a.1.1 1 7.6 odd 2
832.4.a.a.1.1 1 8.3 odd 2
832.4.a.r.1.1 1 8.5 even 2
1521.4.a.a.1.1 1 39.38 odd 2
1573.4.a.a.1.1 1 11.10 odd 2
1872.4.a.k.1.1 1 12.11 even 2