Properties

Label 507.4.a.e.1.1
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{5} +9.00000 q^{6} +2.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -9.00000 q^{5} +9.00000 q^{6} +2.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} -27.0000 q^{10} +30.0000 q^{11} +3.00000 q^{12} +6.00000 q^{14} -27.0000 q^{15} -71.0000 q^{16} -111.000 q^{17} +27.0000 q^{18} -46.0000 q^{19} -9.00000 q^{20} +6.00000 q^{21} +90.0000 q^{22} -6.00000 q^{23} -63.0000 q^{24} -44.0000 q^{25} +27.0000 q^{27} +2.00000 q^{28} -105.000 q^{29} -81.0000 q^{30} -100.000 q^{31} -45.0000 q^{32} +90.0000 q^{33} -333.000 q^{34} -18.0000 q^{35} +9.00000 q^{36} +17.0000 q^{37} -138.000 q^{38} +189.000 q^{40} -231.000 q^{41} +18.0000 q^{42} -514.000 q^{43} +30.0000 q^{44} -81.0000 q^{45} -18.0000 q^{46} -162.000 q^{47} -213.000 q^{48} -339.000 q^{49} -132.000 q^{50} -333.000 q^{51} +639.000 q^{53} +81.0000 q^{54} -270.000 q^{55} -42.0000 q^{56} -138.000 q^{57} -315.000 q^{58} +600.000 q^{59} -27.0000 q^{60} +233.000 q^{61} -300.000 q^{62} +18.0000 q^{63} +433.000 q^{64} +270.000 q^{66} +926.000 q^{67} -111.000 q^{68} -18.0000 q^{69} -54.0000 q^{70} -930.000 q^{71} -189.000 q^{72} -253.000 q^{73} +51.0000 q^{74} -132.000 q^{75} -46.0000 q^{76} +60.0000 q^{77} -1324.00 q^{79} +639.000 q^{80} +81.0000 q^{81} -693.000 q^{82} +810.000 q^{83} +6.00000 q^{84} +999.000 q^{85} -1542.00 q^{86} -315.000 q^{87} -630.000 q^{88} +498.000 q^{89} -243.000 q^{90} -6.00000 q^{92} -300.000 q^{93} -486.000 q^{94} +414.000 q^{95} -135.000 q^{96} +1358.00 q^{97} -1017.00 q^{98} +270.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 1.06066 0.530330 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 3.00000 0.577350
\(4\) 1.00000 0.125000
\(5\) −9.00000 −0.804984 −0.402492 0.915423i \(-0.631856\pi\)
−0.402492 + 0.915423i \(0.631856\pi\)
\(6\) 9.00000 0.612372
\(7\) 2.00000 0.107990 0.0539949 0.998541i \(-0.482805\pi\)
0.0539949 + 0.998541i \(0.482805\pi\)
\(8\) −21.0000 −0.928078
\(9\) 9.00000 0.333333
\(10\) −27.0000 −0.853815
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 3.00000 0.0721688
\(13\) 0 0
\(14\) 6.00000 0.114541
\(15\) −27.0000 −0.464758
\(16\) −71.0000 −1.10938
\(17\) −111.000 −1.58361 −0.791807 0.610771i \(-0.790860\pi\)
−0.791807 + 0.610771i \(0.790860\pi\)
\(18\) 27.0000 0.353553
\(19\) −46.0000 −0.555428 −0.277714 0.960664i \(-0.589577\pi\)
−0.277714 + 0.960664i \(0.589577\pi\)
\(20\) −9.00000 −0.100623
\(21\) 6.00000 0.0623480
\(22\) 90.0000 0.872185
\(23\) −6.00000 −0.0543951 −0.0271975 0.999630i \(-0.508658\pi\)
−0.0271975 + 0.999630i \(0.508658\pi\)
\(24\) −63.0000 −0.535826
\(25\) −44.0000 −0.352000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 2.00000 0.0134987
\(29\) −105.000 −0.672345 −0.336173 0.941800i \(-0.609133\pi\)
−0.336173 + 0.941800i \(0.609133\pi\)
\(30\) −81.0000 −0.492950
\(31\) −100.000 −0.579372 −0.289686 0.957122i \(-0.593551\pi\)
−0.289686 + 0.957122i \(0.593551\pi\)
\(32\) −45.0000 −0.248592
\(33\) 90.0000 0.474757
\(34\) −333.000 −1.67968
\(35\) −18.0000 −0.0869302
\(36\) 9.00000 0.0416667
\(37\) 17.0000 0.0755347 0.0377673 0.999287i \(-0.487975\pi\)
0.0377673 + 0.999287i \(0.487975\pi\)
\(38\) −138.000 −0.589120
\(39\) 0 0
\(40\) 189.000 0.747088
\(41\) −231.000 −0.879906 −0.439953 0.898021i \(-0.645005\pi\)
−0.439953 + 0.898021i \(0.645005\pi\)
\(42\) 18.0000 0.0661300
\(43\) −514.000 −1.82289 −0.911445 0.411422i \(-0.865032\pi\)
−0.911445 + 0.411422i \(0.865032\pi\)
\(44\) 30.0000 0.102788
\(45\) −81.0000 −0.268328
\(46\) −18.0000 −0.0576947
\(47\) −162.000 −0.502769 −0.251384 0.967887i \(-0.580886\pi\)
−0.251384 + 0.967887i \(0.580886\pi\)
\(48\) −213.000 −0.640498
\(49\) −339.000 −0.988338
\(50\) −132.000 −0.373352
\(51\) −333.000 −0.914301
\(52\) 0 0
\(53\) 639.000 1.65610 0.828051 0.560653i \(-0.189450\pi\)
0.828051 + 0.560653i \(0.189450\pi\)
\(54\) 81.0000 0.204124
\(55\) −270.000 −0.661942
\(56\) −42.0000 −0.100223
\(57\) −138.000 −0.320676
\(58\) −315.000 −0.713130
\(59\) 600.000 1.32396 0.661978 0.749524i \(-0.269717\pi\)
0.661978 + 0.749524i \(0.269717\pi\)
\(60\) −27.0000 −0.0580948
\(61\) 233.000 0.489059 0.244529 0.969642i \(-0.421367\pi\)
0.244529 + 0.969642i \(0.421367\pi\)
\(62\) −300.000 −0.614517
\(63\) 18.0000 0.0359966
\(64\) 433.000 0.845703
\(65\) 0 0
\(66\) 270.000 0.503556
\(67\) 926.000 1.68849 0.844246 0.535957i \(-0.180049\pi\)
0.844246 + 0.535957i \(0.180049\pi\)
\(68\) −111.000 −0.197952
\(69\) −18.0000 −0.0314050
\(70\) −54.0000 −0.0922033
\(71\) −930.000 −1.55452 −0.777258 0.629182i \(-0.783390\pi\)
−0.777258 + 0.629182i \(0.783390\pi\)
\(72\) −189.000 −0.309359
\(73\) −253.000 −0.405636 −0.202818 0.979216i \(-0.565010\pi\)
−0.202818 + 0.979216i \(0.565010\pi\)
\(74\) 51.0000 0.0801166
\(75\) −132.000 −0.203227
\(76\) −46.0000 −0.0694284
\(77\) 60.0000 0.0888004
\(78\) 0 0
\(79\) −1324.00 −1.88559 −0.942795 0.333373i \(-0.891813\pi\)
−0.942795 + 0.333373i \(0.891813\pi\)
\(80\) 639.000 0.893030
\(81\) 81.0000 0.111111
\(82\) −693.000 −0.933281
\(83\) 810.000 1.07119 0.535597 0.844474i \(-0.320087\pi\)
0.535597 + 0.844474i \(0.320087\pi\)
\(84\) 6.00000 0.00779350
\(85\) 999.000 1.27479
\(86\) −1542.00 −1.93347
\(87\) −315.000 −0.388179
\(88\) −630.000 −0.763162
\(89\) 498.000 0.593122 0.296561 0.955014i \(-0.404160\pi\)
0.296561 + 0.955014i \(0.404160\pi\)
\(90\) −243.000 −0.284605
\(91\) 0 0
\(92\) −6.00000 −0.00679938
\(93\) −300.000 −0.334501
\(94\) −486.000 −0.533267
\(95\) 414.000 0.447111
\(96\) −135.000 −0.143525
\(97\) 1358.00 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1017.00 −1.04829
\(99\) 270.000 0.274101
\(100\) −44.0000 −0.0440000
\(101\) −357.000 −0.351711 −0.175856 0.984416i \(-0.556269\pi\)
−0.175856 + 0.984416i \(0.556269\pi\)
\(102\) −999.000 −0.969762
\(103\) 1118.00 1.06951 0.534756 0.845006i \(-0.320403\pi\)
0.534756 + 0.845006i \(0.320403\pi\)
\(104\) 0 0
\(105\) −54.0000 −0.0501891
\(106\) 1917.00 1.75656
\(107\) 714.000 0.645093 0.322547 0.946554i \(-0.395461\pi\)
0.322547 + 0.946554i \(0.395461\pi\)
\(108\) 27.0000 0.0240563
\(109\) 2006.00 1.76275 0.881376 0.472416i \(-0.156618\pi\)
0.881376 + 0.472416i \(0.156618\pi\)
\(110\) −810.000 −0.702095
\(111\) 51.0000 0.0436100
\(112\) −142.000 −0.119801
\(113\) −1119.00 −0.931563 −0.465782 0.884900i \(-0.654227\pi\)
−0.465782 + 0.884900i \(0.654227\pi\)
\(114\) −414.000 −0.340129
\(115\) 54.0000 0.0437872
\(116\) −105.000 −0.0840431
\(117\) 0 0
\(118\) 1800.00 1.40427
\(119\) −222.000 −0.171014
\(120\) 567.000 0.431332
\(121\) −431.000 −0.323817
\(122\) 699.000 0.518725
\(123\) −693.000 −0.508014
\(124\) −100.000 −0.0724215
\(125\) 1521.00 1.08834
\(126\) 54.0000 0.0381802
\(127\) −604.000 −0.422018 −0.211009 0.977484i \(-0.567675\pi\)
−0.211009 + 0.977484i \(0.567675\pi\)
\(128\) 1659.00 1.14560
\(129\) −1542.00 −1.05245
\(130\) 0 0
\(131\) −1584.00 −1.05645 −0.528224 0.849105i \(-0.677142\pi\)
−0.528224 + 0.849105i \(0.677142\pi\)
\(132\) 90.0000 0.0593447
\(133\) −92.0000 −0.0599805
\(134\) 2778.00 1.79092
\(135\) −243.000 −0.154919
\(136\) 2331.00 1.46972
\(137\) 717.000 0.447135 0.223567 0.974688i \(-0.428230\pi\)
0.223567 + 0.974688i \(0.428230\pi\)
\(138\) −54.0000 −0.0333100
\(139\) −820.000 −0.500370 −0.250185 0.968198i \(-0.580492\pi\)
−0.250185 + 0.968198i \(0.580492\pi\)
\(140\) −18.0000 −0.0108663
\(141\) −486.000 −0.290274
\(142\) −2790.00 −1.64881
\(143\) 0 0
\(144\) −639.000 −0.369792
\(145\) 945.000 0.541227
\(146\) −759.000 −0.430242
\(147\) −1017.00 −0.570617
\(148\) 17.0000 0.00944183
\(149\) −1749.00 −0.961635 −0.480818 0.876821i \(-0.659660\pi\)
−0.480818 + 0.876821i \(0.659660\pi\)
\(150\) −396.000 −0.215555
\(151\) −370.000 −0.199405 −0.0997026 0.995017i \(-0.531789\pi\)
−0.0997026 + 0.995017i \(0.531789\pi\)
\(152\) 966.000 0.515480
\(153\) −999.000 −0.527872
\(154\) 180.000 0.0941871
\(155\) 900.000 0.466385
\(156\) 0 0
\(157\) −2611.00 −1.32726 −0.663632 0.748059i \(-0.730986\pi\)
−0.663632 + 0.748059i \(0.730986\pi\)
\(158\) −3972.00 −1.99997
\(159\) 1917.00 0.956151
\(160\) 405.000 0.200113
\(161\) −12.0000 −0.00587411
\(162\) 243.000 0.117851
\(163\) −1636.00 −0.786144 −0.393072 0.919508i \(-0.628588\pi\)
−0.393072 + 0.919508i \(0.628588\pi\)
\(164\) −231.000 −0.109988
\(165\) −810.000 −0.382172
\(166\) 2430.00 1.13617
\(167\) 264.000 0.122329 0.0611645 0.998128i \(-0.480519\pi\)
0.0611645 + 0.998128i \(0.480519\pi\)
\(168\) −126.000 −0.0578638
\(169\) 0 0
\(170\) 2997.00 1.35211
\(171\) −414.000 −0.185143
\(172\) −514.000 −0.227861
\(173\) 1410.00 0.619655 0.309827 0.950793i \(-0.399729\pi\)
0.309827 + 0.950793i \(0.399729\pi\)
\(174\) −945.000 −0.411726
\(175\) −88.0000 −0.0380124
\(176\) −2130.00 −0.912243
\(177\) 1800.00 0.764386
\(178\) 1494.00 0.629101
\(179\) −474.000 −0.197924 −0.0989621 0.995091i \(-0.531552\pi\)
−0.0989621 + 0.995091i \(0.531552\pi\)
\(180\) −81.0000 −0.0335410
\(181\) 2249.00 0.923574 0.461787 0.886991i \(-0.347208\pi\)
0.461787 + 0.886991i \(0.347208\pi\)
\(182\) 0 0
\(183\) 699.000 0.282358
\(184\) 126.000 0.0504828
\(185\) −153.000 −0.0608042
\(186\) −900.000 −0.354791
\(187\) −3330.00 −1.30221
\(188\) −162.000 −0.0628461
\(189\) 54.0000 0.0207827
\(190\) 1242.00 0.474232
\(191\) 3444.00 1.30471 0.652354 0.757915i \(-0.273782\pi\)
0.652354 + 0.757915i \(0.273782\pi\)
\(192\) 1299.00 0.488267
\(193\) −4273.00 −1.59366 −0.796832 0.604201i \(-0.793493\pi\)
−0.796832 + 0.604201i \(0.793493\pi\)
\(194\) 4074.00 1.50771
\(195\) 0 0
\(196\) −339.000 −0.123542
\(197\) −1986.00 −0.718257 −0.359129 0.933288i \(-0.616926\pi\)
−0.359129 + 0.933288i \(0.616926\pi\)
\(198\) 810.000 0.290728
\(199\) −2386.00 −0.849945 −0.424973 0.905206i \(-0.639716\pi\)
−0.424973 + 0.905206i \(0.639716\pi\)
\(200\) 924.000 0.326683
\(201\) 2778.00 0.974851
\(202\) −1071.00 −0.373046
\(203\) −210.000 −0.0726065
\(204\) −333.000 −0.114288
\(205\) 2079.00 0.708311
\(206\) 3354.00 1.13439
\(207\) −54.0000 −0.0181317
\(208\) 0 0
\(209\) −1380.00 −0.456730
\(210\) −162.000 −0.0532336
\(211\) −1600.00 −0.522031 −0.261016 0.965335i \(-0.584057\pi\)
−0.261016 + 0.965335i \(0.584057\pi\)
\(212\) 639.000 0.207013
\(213\) −2790.00 −0.897501
\(214\) 2142.00 0.684225
\(215\) 4626.00 1.46740
\(216\) −567.000 −0.178609
\(217\) −200.000 −0.0625663
\(218\) 6018.00 1.86968
\(219\) −759.000 −0.234194
\(220\) −270.000 −0.0827427
\(221\) 0 0
\(222\) 153.000 0.0462553
\(223\) −3832.00 −1.15072 −0.575358 0.817902i \(-0.695137\pi\)
−0.575358 + 0.817902i \(0.695137\pi\)
\(224\) −90.0000 −0.0268454
\(225\) −396.000 −0.117333
\(226\) −3357.00 −0.988072
\(227\) −1398.00 −0.408760 −0.204380 0.978892i \(-0.565518\pi\)
−0.204380 + 0.978892i \(0.565518\pi\)
\(228\) −138.000 −0.0400845
\(229\) 4466.00 1.28874 0.644370 0.764714i \(-0.277120\pi\)
0.644370 + 0.764714i \(0.277120\pi\)
\(230\) 162.000 0.0464433
\(231\) 180.000 0.0512690
\(232\) 2205.00 0.623989
\(233\) −1638.00 −0.460553 −0.230277 0.973125i \(-0.573963\pi\)
−0.230277 + 0.973125i \(0.573963\pi\)
\(234\) 0 0
\(235\) 1458.00 0.404721
\(236\) 600.000 0.165494
\(237\) −3972.00 −1.08865
\(238\) −666.000 −0.181388
\(239\) −594.000 −0.160764 −0.0803821 0.996764i \(-0.525614\pi\)
−0.0803821 + 0.996764i \(0.525614\pi\)
\(240\) 1917.00 0.515591
\(241\) 2303.00 0.615557 0.307779 0.951458i \(-0.400414\pi\)
0.307779 + 0.951458i \(0.400414\pi\)
\(242\) −1293.00 −0.343459
\(243\) 243.000 0.0641500
\(244\) 233.000 0.0611324
\(245\) 3051.00 0.795597
\(246\) −2079.00 −0.538830
\(247\) 0 0
\(248\) 2100.00 0.537702
\(249\) 2430.00 0.618454
\(250\) 4563.00 1.15436
\(251\) 6324.00 1.59031 0.795154 0.606407i \(-0.207390\pi\)
0.795154 + 0.606407i \(0.207390\pi\)
\(252\) 18.0000 0.00449958
\(253\) −180.000 −0.0447293
\(254\) −1812.00 −0.447618
\(255\) 2997.00 0.735998
\(256\) 1513.00 0.369385
\(257\) 7833.00 1.90120 0.950601 0.310414i \(-0.100468\pi\)
0.950601 + 0.310414i \(0.100468\pi\)
\(258\) −4626.00 −1.11629
\(259\) 34.0000 0.00815698
\(260\) 0 0
\(261\) −945.000 −0.224115
\(262\) −4752.00 −1.12053
\(263\) −3030.00 −0.710410 −0.355205 0.934788i \(-0.615589\pi\)
−0.355205 + 0.934788i \(0.615589\pi\)
\(264\) −1890.00 −0.440612
\(265\) −5751.00 −1.33314
\(266\) −276.000 −0.0636190
\(267\) 1494.00 0.342439
\(268\) 926.000 0.211061
\(269\) −534.000 −0.121036 −0.0605178 0.998167i \(-0.519275\pi\)
−0.0605178 + 0.998167i \(0.519275\pi\)
\(270\) −729.000 −0.164317
\(271\) −3688.00 −0.826679 −0.413340 0.910577i \(-0.635638\pi\)
−0.413340 + 0.910577i \(0.635638\pi\)
\(272\) 7881.00 1.75682
\(273\) 0 0
\(274\) 2151.00 0.474258
\(275\) −1320.00 −0.289451
\(276\) −18.0000 −0.00392563
\(277\) 1865.00 0.404538 0.202269 0.979330i \(-0.435168\pi\)
0.202269 + 0.979330i \(0.435168\pi\)
\(278\) −2460.00 −0.530723
\(279\) −900.000 −0.193124
\(280\) 378.000 0.0806779
\(281\) 2997.00 0.636249 0.318125 0.948049i \(-0.396947\pi\)
0.318125 + 0.948049i \(0.396947\pi\)
\(282\) −1458.00 −0.307882
\(283\) −4114.00 −0.864141 −0.432071 0.901840i \(-0.642217\pi\)
−0.432071 + 0.901840i \(0.642217\pi\)
\(284\) −930.000 −0.194315
\(285\) 1242.00 0.258139
\(286\) 0 0
\(287\) −462.000 −0.0950209
\(288\) −405.000 −0.0828641
\(289\) 7408.00 1.50784
\(290\) 2835.00 0.574058
\(291\) 4074.00 0.820695
\(292\) −253.000 −0.0507045
\(293\) −4665.00 −0.930144 −0.465072 0.885273i \(-0.653972\pi\)
−0.465072 + 0.885273i \(0.653972\pi\)
\(294\) −3051.00 −0.605231
\(295\) −5400.00 −1.06576
\(296\) −357.000 −0.0701020
\(297\) 810.000 0.158252
\(298\) −5247.00 −1.01997
\(299\) 0 0
\(300\) −132.000 −0.0254034
\(301\) −1028.00 −0.196854
\(302\) −1110.00 −0.211501
\(303\) −1071.00 −0.203061
\(304\) 3266.00 0.616177
\(305\) −2097.00 −0.393685
\(306\) −2997.00 −0.559892
\(307\) 1502.00 0.279230 0.139615 0.990206i \(-0.455413\pi\)
0.139615 + 0.990206i \(0.455413\pi\)
\(308\) 60.0000 0.0111001
\(309\) 3354.00 0.617483
\(310\) 2700.00 0.494676
\(311\) 2106.00 0.383988 0.191994 0.981396i \(-0.438505\pi\)
0.191994 + 0.981396i \(0.438505\pi\)
\(312\) 0 0
\(313\) −3898.00 −0.703923 −0.351962 0.936014i \(-0.614485\pi\)
−0.351962 + 0.936014i \(0.614485\pi\)
\(314\) −7833.00 −1.40778
\(315\) −162.000 −0.0289767
\(316\) −1324.00 −0.235699
\(317\) 9351.00 1.65680 0.828398 0.560140i \(-0.189253\pi\)
0.828398 + 0.560140i \(0.189253\pi\)
\(318\) 5751.00 1.01415
\(319\) −3150.00 −0.552872
\(320\) −3897.00 −0.680778
\(321\) 2142.00 0.372445
\(322\) −36.0000 −0.00623044
\(323\) 5106.00 0.879583
\(324\) 81.0000 0.0138889
\(325\) 0 0
\(326\) −4908.00 −0.833831
\(327\) 6018.00 1.01773
\(328\) 4851.00 0.816621
\(329\) −324.000 −0.0542939
\(330\) −2430.00 −0.405355
\(331\) −9172.00 −1.52308 −0.761539 0.648119i \(-0.775556\pi\)
−0.761539 + 0.648119i \(0.775556\pi\)
\(332\) 810.000 0.133899
\(333\) 153.000 0.0251782
\(334\) 792.000 0.129749
\(335\) −8334.00 −1.35921
\(336\) −426.000 −0.0691673
\(337\) −11089.0 −1.79245 −0.896226 0.443598i \(-0.853702\pi\)
−0.896226 + 0.443598i \(0.853702\pi\)
\(338\) 0 0
\(339\) −3357.00 −0.537838
\(340\) 999.000 0.159348
\(341\) −3000.00 −0.476420
\(342\) −1242.00 −0.196373
\(343\) −1364.00 −0.214720
\(344\) 10794.0 1.69178
\(345\) 162.000 0.0252805
\(346\) 4230.00 0.657243
\(347\) 9762.00 1.51024 0.755118 0.655589i \(-0.227580\pi\)
0.755118 + 0.655589i \(0.227580\pi\)
\(348\) −315.000 −0.0485223
\(349\) −8290.00 −1.27150 −0.635750 0.771895i \(-0.719309\pi\)
−0.635750 + 0.771895i \(0.719309\pi\)
\(350\) −264.000 −0.0403183
\(351\) 0 0
\(352\) −1350.00 −0.204418
\(353\) 12405.0 1.87040 0.935200 0.354119i \(-0.115219\pi\)
0.935200 + 0.354119i \(0.115219\pi\)
\(354\) 5400.00 0.810754
\(355\) 8370.00 1.25136
\(356\) 498.000 0.0741403
\(357\) −666.000 −0.0987352
\(358\) −1422.00 −0.209930
\(359\) −1098.00 −0.161421 −0.0807106 0.996738i \(-0.525719\pi\)
−0.0807106 + 0.996738i \(0.525719\pi\)
\(360\) 1701.00 0.249029
\(361\) −4743.00 −0.691500
\(362\) 6747.00 0.979598
\(363\) −1293.00 −0.186956
\(364\) 0 0
\(365\) 2277.00 0.326530
\(366\) 2097.00 0.299486
\(367\) −5734.00 −0.815565 −0.407783 0.913079i \(-0.633698\pi\)
−0.407783 + 0.913079i \(0.633698\pi\)
\(368\) 426.000 0.0603445
\(369\) −2079.00 −0.293302
\(370\) −459.000 −0.0644926
\(371\) 1278.00 0.178842
\(372\) −300.000 −0.0418126
\(373\) −8971.00 −1.24531 −0.622655 0.782496i \(-0.713946\pi\)
−0.622655 + 0.782496i \(0.713946\pi\)
\(374\) −9990.00 −1.38120
\(375\) 4563.00 0.628353
\(376\) 3402.00 0.466608
\(377\) 0 0
\(378\) 162.000 0.0220433
\(379\) 7244.00 0.981792 0.490896 0.871218i \(-0.336669\pi\)
0.490896 + 0.871218i \(0.336669\pi\)
\(380\) 414.000 0.0558888
\(381\) −1812.00 −0.243652
\(382\) 10332.0 1.38385
\(383\) −6312.00 −0.842110 −0.421055 0.907035i \(-0.638340\pi\)
−0.421055 + 0.907035i \(0.638340\pi\)
\(384\) 4977.00 0.661410
\(385\) −540.000 −0.0714830
\(386\) −12819.0 −1.69034
\(387\) −4626.00 −0.607630
\(388\) 1358.00 0.177686
\(389\) 3627.00 0.472741 0.236370 0.971663i \(-0.424042\pi\)
0.236370 + 0.971663i \(0.424042\pi\)
\(390\) 0 0
\(391\) 666.000 0.0861408
\(392\) 7119.00 0.917255
\(393\) −4752.00 −0.609941
\(394\) −5958.00 −0.761827
\(395\) 11916.0 1.51787
\(396\) 270.000 0.0342627
\(397\) −3898.00 −0.492783 −0.246392 0.969170i \(-0.579245\pi\)
−0.246392 + 0.969170i \(0.579245\pi\)
\(398\) −7158.00 −0.901503
\(399\) −276.000 −0.0346298
\(400\) 3124.00 0.390500
\(401\) −5703.00 −0.710210 −0.355105 0.934826i \(-0.615555\pi\)
−0.355105 + 0.934826i \(0.615555\pi\)
\(402\) 8334.00 1.03399
\(403\) 0 0
\(404\) −357.000 −0.0439639
\(405\) −729.000 −0.0894427
\(406\) −630.000 −0.0770108
\(407\) 510.000 0.0621124
\(408\) 6993.00 0.848542
\(409\) 6311.00 0.762980 0.381490 0.924373i \(-0.375411\pi\)
0.381490 + 0.924373i \(0.375411\pi\)
\(410\) 6237.00 0.751277
\(411\) 2151.00 0.258153
\(412\) 1118.00 0.133689
\(413\) 1200.00 0.142974
\(414\) −162.000 −0.0192316
\(415\) −7290.00 −0.862294
\(416\) 0 0
\(417\) −2460.00 −0.288889
\(418\) −4140.00 −0.484435
\(419\) −2328.00 −0.271433 −0.135716 0.990748i \(-0.543334\pi\)
−0.135716 + 0.990748i \(0.543334\pi\)
\(420\) −54.0000 −0.00627364
\(421\) 2045.00 0.236739 0.118370 0.992970i \(-0.462233\pi\)
0.118370 + 0.992970i \(0.462233\pi\)
\(422\) −4800.00 −0.553697
\(423\) −1458.00 −0.167590
\(424\) −13419.0 −1.53699
\(425\) 4884.00 0.557432
\(426\) −8370.00 −0.951943
\(427\) 466.000 0.0528134
\(428\) 714.000 0.0806367
\(429\) 0 0
\(430\) 13878.0 1.55641
\(431\) 5034.00 0.562597 0.281298 0.959620i \(-0.409235\pi\)
0.281298 + 0.959620i \(0.409235\pi\)
\(432\) −1917.00 −0.213499
\(433\) 4283.00 0.475353 0.237676 0.971344i \(-0.423614\pi\)
0.237676 + 0.971344i \(0.423614\pi\)
\(434\) −600.000 −0.0663616
\(435\) 2835.00 0.312478
\(436\) 2006.00 0.220344
\(437\) 276.000 0.0302125
\(438\) −2277.00 −0.248400
\(439\) −1306.00 −0.141986 −0.0709931 0.997477i \(-0.522617\pi\)
−0.0709931 + 0.997477i \(0.522617\pi\)
\(440\) 5670.00 0.614333
\(441\) −3051.00 −0.329446
\(442\) 0 0
\(443\) −5796.00 −0.621617 −0.310808 0.950473i \(-0.600600\pi\)
−0.310808 + 0.950473i \(0.600600\pi\)
\(444\) 51.0000 0.00545125
\(445\) −4482.00 −0.477454
\(446\) −11496.0 −1.22052
\(447\) −5247.00 −0.555200
\(448\) 866.000 0.0913274
\(449\) 2706.00 0.284419 0.142209 0.989837i \(-0.454579\pi\)
0.142209 + 0.989837i \(0.454579\pi\)
\(450\) −1188.00 −0.124451
\(451\) −6930.00 −0.723550
\(452\) −1119.00 −0.116445
\(453\) −1110.00 −0.115127
\(454\) −4194.00 −0.433555
\(455\) 0 0
\(456\) 2898.00 0.297612
\(457\) −829.000 −0.0848555 −0.0424278 0.999100i \(-0.513509\pi\)
−0.0424278 + 0.999100i \(0.513509\pi\)
\(458\) 13398.0 1.36692
\(459\) −2997.00 −0.304767
\(460\) 54.0000 0.00547340
\(461\) −5493.00 −0.554956 −0.277478 0.960732i \(-0.589498\pi\)
−0.277478 + 0.960732i \(0.589498\pi\)
\(462\) 540.000 0.0543789
\(463\) −15346.0 −1.54037 −0.770183 0.637823i \(-0.779835\pi\)
−0.770183 + 0.637823i \(0.779835\pi\)
\(464\) 7455.00 0.745883
\(465\) 2700.00 0.269268
\(466\) −4914.00 −0.488491
\(467\) −9594.00 −0.950658 −0.475329 0.879808i \(-0.657671\pi\)
−0.475329 + 0.879808i \(0.657671\pi\)
\(468\) 0 0
\(469\) 1852.00 0.182340
\(470\) 4374.00 0.429271
\(471\) −7833.00 −0.766296
\(472\) −12600.0 −1.22873
\(473\) −15420.0 −1.49897
\(474\) −11916.0 −1.15468
\(475\) 2024.00 0.195511
\(476\) −222.000 −0.0213768
\(477\) 5751.00 0.552034
\(478\) −1782.00 −0.170516
\(479\) −12840.0 −1.22479 −0.612395 0.790552i \(-0.709794\pi\)
−0.612395 + 0.790552i \(0.709794\pi\)
\(480\) 1215.00 0.115535
\(481\) 0 0
\(482\) 6909.00 0.652897
\(483\) −36.0000 −0.00339142
\(484\) −431.000 −0.0404771
\(485\) −12222.0 −1.14427
\(486\) 729.000 0.0680414
\(487\) −14086.0 −1.31067 −0.655336 0.755337i \(-0.727473\pi\)
−0.655336 + 0.755337i \(0.727473\pi\)
\(488\) −4893.00 −0.453885
\(489\) −4908.00 −0.453880
\(490\) 9153.00 0.843858
\(491\) 11694.0 1.07483 0.537416 0.843317i \(-0.319400\pi\)
0.537416 + 0.843317i \(0.319400\pi\)
\(492\) −693.000 −0.0635017
\(493\) 11655.0 1.06474
\(494\) 0 0
\(495\) −2430.00 −0.220647
\(496\) 7100.00 0.642741
\(497\) −1860.00 −0.167872
\(498\) 7290.00 0.655969
\(499\) −3688.00 −0.330857 −0.165428 0.986222i \(-0.552901\pi\)
−0.165428 + 0.986222i \(0.552901\pi\)
\(500\) 1521.00 0.136042
\(501\) 792.000 0.0706266
\(502\) 18972.0 1.68678
\(503\) −4746.00 −0.420703 −0.210352 0.977626i \(-0.567461\pi\)
−0.210352 + 0.977626i \(0.567461\pi\)
\(504\) −378.000 −0.0334077
\(505\) 3213.00 0.283122
\(506\) −540.000 −0.0474425
\(507\) 0 0
\(508\) −604.000 −0.0527523
\(509\) −14505.0 −1.26311 −0.631555 0.775331i \(-0.717583\pi\)
−0.631555 + 0.775331i \(0.717583\pi\)
\(510\) 8991.00 0.780643
\(511\) −506.000 −0.0438045
\(512\) −8733.00 −0.753804
\(513\) −1242.00 −0.106892
\(514\) 23499.0 2.01653
\(515\) −10062.0 −0.860941
\(516\) −1542.00 −0.131556
\(517\) −4860.00 −0.413429
\(518\) 102.000 0.00865178
\(519\) 4230.00 0.357758
\(520\) 0 0
\(521\) 5085.00 0.427597 0.213798 0.976878i \(-0.431416\pi\)
0.213798 + 0.976878i \(0.431416\pi\)
\(522\) −2835.00 −0.237710
\(523\) −10882.0 −0.909821 −0.454911 0.890537i \(-0.650329\pi\)
−0.454911 + 0.890537i \(0.650329\pi\)
\(524\) −1584.00 −0.132056
\(525\) −264.000 −0.0219465
\(526\) −9090.00 −0.753503
\(527\) 11100.0 0.917502
\(528\) −6390.00 −0.526684
\(529\) −12131.0 −0.997041
\(530\) −17253.0 −1.41400
\(531\) 5400.00 0.441318
\(532\) −92.0000 −0.00749757
\(533\) 0 0
\(534\) 4482.00 0.363212
\(535\) −6426.00 −0.519290
\(536\) −19446.0 −1.56705
\(537\) −1422.00 −0.114272
\(538\) −1602.00 −0.128378
\(539\) −10170.0 −0.812714
\(540\) −243.000 −0.0193649
\(541\) −4699.00 −0.373430 −0.186715 0.982414i \(-0.559784\pi\)
−0.186715 + 0.982414i \(0.559784\pi\)
\(542\) −11064.0 −0.876826
\(543\) 6747.00 0.533226
\(544\) 4995.00 0.393674
\(545\) −18054.0 −1.41899
\(546\) 0 0
\(547\) 8270.00 0.646434 0.323217 0.946325i \(-0.395236\pi\)
0.323217 + 0.946325i \(0.395236\pi\)
\(548\) 717.000 0.0558918
\(549\) 2097.00 0.163020
\(550\) −3960.00 −0.307009
\(551\) 4830.00 0.373439
\(552\) 378.000 0.0291463
\(553\) −2648.00 −0.203625
\(554\) 5595.00 0.429077
\(555\) −459.000 −0.0351053
\(556\) −820.000 −0.0625463
\(557\) −22785.0 −1.73327 −0.866635 0.498943i \(-0.833722\pi\)
−0.866635 + 0.498943i \(0.833722\pi\)
\(558\) −2700.00 −0.204839
\(559\) 0 0
\(560\) 1278.00 0.0964381
\(561\) −9990.00 −0.751833
\(562\) 8991.00 0.674844
\(563\) −11928.0 −0.892905 −0.446452 0.894807i \(-0.647313\pi\)
−0.446452 + 0.894807i \(0.647313\pi\)
\(564\) −486.000 −0.0362842
\(565\) 10071.0 0.749894
\(566\) −12342.0 −0.916560
\(567\) 162.000 0.0119989
\(568\) 19530.0 1.44271
\(569\) −7962.00 −0.586616 −0.293308 0.956018i \(-0.594756\pi\)
−0.293308 + 0.956018i \(0.594756\pi\)
\(570\) 3726.00 0.273798
\(571\) 20618.0 1.51110 0.755549 0.655093i \(-0.227370\pi\)
0.755549 + 0.655093i \(0.227370\pi\)
\(572\) 0 0
\(573\) 10332.0 0.753273
\(574\) −1386.00 −0.100785
\(575\) 264.000 0.0191471
\(576\) 3897.00 0.281901
\(577\) −3493.00 −0.252020 −0.126010 0.992029i \(-0.540217\pi\)
−0.126010 + 0.992029i \(0.540217\pi\)
\(578\) 22224.0 1.59930
\(579\) −12819.0 −0.920103
\(580\) 945.000 0.0676534
\(581\) 1620.00 0.115678
\(582\) 12222.0 0.870478
\(583\) 19170.0 1.36182
\(584\) 5313.00 0.376461
\(585\) 0 0
\(586\) −13995.0 −0.986567
\(587\) 10416.0 0.732392 0.366196 0.930538i \(-0.380660\pi\)
0.366196 + 0.930538i \(0.380660\pi\)
\(588\) −1017.00 −0.0713272
\(589\) 4600.00 0.321799
\(590\) −16200.0 −1.13041
\(591\) −5958.00 −0.414686
\(592\) −1207.00 −0.0837963
\(593\) 2061.00 0.142724 0.0713618 0.997450i \(-0.477266\pi\)
0.0713618 + 0.997450i \(0.477266\pi\)
\(594\) 2430.00 0.167852
\(595\) 1998.00 0.137664
\(596\) −1749.00 −0.120204
\(597\) −7158.00 −0.490716
\(598\) 0 0
\(599\) 12456.0 0.849647 0.424823 0.905276i \(-0.360336\pi\)
0.424823 + 0.905276i \(0.360336\pi\)
\(600\) 2772.00 0.188611
\(601\) −781.000 −0.0530077 −0.0265039 0.999649i \(-0.508437\pi\)
−0.0265039 + 0.999649i \(0.508437\pi\)
\(602\) −3084.00 −0.208795
\(603\) 8334.00 0.562830
\(604\) −370.000 −0.0249256
\(605\) 3879.00 0.260667
\(606\) −3213.00 −0.215378
\(607\) 19304.0 1.29082 0.645408 0.763838i \(-0.276687\pi\)
0.645408 + 0.763838i \(0.276687\pi\)
\(608\) 2070.00 0.138075
\(609\) −630.000 −0.0419194
\(610\) −6291.00 −0.417566
\(611\) 0 0
\(612\) −999.000 −0.0659840
\(613\) 12041.0 0.793363 0.396681 0.917956i \(-0.370162\pi\)
0.396681 + 0.917956i \(0.370162\pi\)
\(614\) 4506.00 0.296168
\(615\) 6237.00 0.408943
\(616\) −1260.00 −0.0824137
\(617\) 9717.00 0.634022 0.317011 0.948422i \(-0.397321\pi\)
0.317011 + 0.948422i \(0.397321\pi\)
\(618\) 10062.0 0.654940
\(619\) −21040.0 −1.36619 −0.683093 0.730332i \(-0.739366\pi\)
−0.683093 + 0.730332i \(0.739366\pi\)
\(620\) 900.000 0.0582982
\(621\) −162.000 −0.0104683
\(622\) 6318.00 0.407281
\(623\) 996.000 0.0640512
\(624\) 0 0
\(625\) −8189.00 −0.524096
\(626\) −11694.0 −0.746623
\(627\) −4140.00 −0.263693
\(628\) −2611.00 −0.165908
\(629\) −1887.00 −0.119618
\(630\) −486.000 −0.0307344
\(631\) −5068.00 −0.319737 −0.159868 0.987138i \(-0.551107\pi\)
−0.159868 + 0.987138i \(0.551107\pi\)
\(632\) 27804.0 1.74997
\(633\) −4800.00 −0.301395
\(634\) 28053.0 1.75730
\(635\) 5436.00 0.339718
\(636\) 1917.00 0.119519
\(637\) 0 0
\(638\) −9450.00 −0.586409
\(639\) −8370.00 −0.518172
\(640\) −14931.0 −0.922187
\(641\) 10185.0 0.627587 0.313794 0.949491i \(-0.398400\pi\)
0.313794 + 0.949491i \(0.398400\pi\)
\(642\) 6426.00 0.395037
\(643\) 25928.0 1.59020 0.795101 0.606476i \(-0.207418\pi\)
0.795101 + 0.606476i \(0.207418\pi\)
\(644\) −12.0000 −0.000734264 0
\(645\) 13878.0 0.847203
\(646\) 15318.0 0.932939
\(647\) 23160.0 1.40729 0.703643 0.710554i \(-0.251556\pi\)
0.703643 + 0.710554i \(0.251556\pi\)
\(648\) −1701.00 −0.103120
\(649\) 18000.0 1.08869
\(650\) 0 0
\(651\) −600.000 −0.0361227
\(652\) −1636.00 −0.0982680
\(653\) 16626.0 0.996364 0.498182 0.867073i \(-0.334001\pi\)
0.498182 + 0.867073i \(0.334001\pi\)
\(654\) 18054.0 1.07946
\(655\) 14256.0 0.850424
\(656\) 16401.0 0.976146
\(657\) −2277.00 −0.135212
\(658\) −972.000 −0.0575874
\(659\) −14808.0 −0.875323 −0.437661 0.899140i \(-0.644193\pi\)
−0.437661 + 0.899140i \(0.644193\pi\)
\(660\) −810.000 −0.0477715
\(661\) 4853.00 0.285567 0.142784 0.989754i \(-0.454395\pi\)
0.142784 + 0.989754i \(0.454395\pi\)
\(662\) −27516.0 −1.61547
\(663\) 0 0
\(664\) −17010.0 −0.994151
\(665\) 828.000 0.0482834
\(666\) 459.000 0.0267055
\(667\) 630.000 0.0365723
\(668\) 264.000 0.0152911
\(669\) −11496.0 −0.664366
\(670\) −25002.0 −1.44166
\(671\) 6990.00 0.402155
\(672\) −270.000 −0.0154992
\(673\) −16165.0 −0.925877 −0.462938 0.886391i \(-0.653205\pi\)
−0.462938 + 0.886391i \(0.653205\pi\)
\(674\) −33267.0 −1.90118
\(675\) −1188.00 −0.0677424
\(676\) 0 0
\(677\) −25686.0 −1.45819 −0.729094 0.684414i \(-0.760058\pi\)
−0.729094 + 0.684414i \(0.760058\pi\)
\(678\) −10071.0 −0.570464
\(679\) 2716.00 0.153506
\(680\) −20979.0 −1.18310
\(681\) −4194.00 −0.235998
\(682\) −9000.00 −0.505319
\(683\) −19056.0 −1.06758 −0.533790 0.845617i \(-0.679233\pi\)
−0.533790 + 0.845617i \(0.679233\pi\)
\(684\) −414.000 −0.0231428
\(685\) −6453.00 −0.359936
\(686\) −4092.00 −0.227745
\(687\) 13398.0 0.744055
\(688\) 36494.0 2.02227
\(689\) 0 0
\(690\) 486.000 0.0268141
\(691\) −16390.0 −0.902323 −0.451161 0.892442i \(-0.648990\pi\)
−0.451161 + 0.892442i \(0.648990\pi\)
\(692\) 1410.00 0.0774569
\(693\) 540.000 0.0296001
\(694\) 29286.0 1.60185
\(695\) 7380.00 0.402790
\(696\) 6615.00 0.360260
\(697\) 25641.0 1.39343
\(698\) −24870.0 −1.34863
\(699\) −4914.00 −0.265901
\(700\) −88.0000 −0.00475155
\(701\) −27846.0 −1.50033 −0.750163 0.661253i \(-0.770025\pi\)
−0.750163 + 0.661253i \(0.770025\pi\)
\(702\) 0 0
\(703\) −782.000 −0.0419540
\(704\) 12990.0 0.695425
\(705\) 4374.00 0.233666
\(706\) 37215.0 1.98386
\(707\) −714.000 −0.0379812
\(708\) 1800.00 0.0955482
\(709\) −12283.0 −0.650632 −0.325316 0.945605i \(-0.605471\pi\)
−0.325316 + 0.945605i \(0.605471\pi\)
\(710\) 25110.0 1.32727
\(711\) −11916.0 −0.628530
\(712\) −10458.0 −0.550464
\(713\) 600.000 0.0315150
\(714\) −1998.00 −0.104724
\(715\) 0 0
\(716\) −474.000 −0.0247405
\(717\) −1782.00 −0.0928173
\(718\) −3294.00 −0.171213
\(719\) 25512.0 1.32328 0.661639 0.749822i \(-0.269861\pi\)
0.661639 + 0.749822i \(0.269861\pi\)
\(720\) 5751.00 0.297677
\(721\) 2236.00 0.115497
\(722\) −14229.0 −0.733447
\(723\) 6909.00 0.355392
\(724\) 2249.00 0.115447
\(725\) 4620.00 0.236666
\(726\) −3879.00 −0.198296
\(727\) 6110.00 0.311702 0.155851 0.987781i \(-0.450188\pi\)
0.155851 + 0.987781i \(0.450188\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 6831.00 0.346338
\(731\) 57054.0 2.88676
\(732\) 699.000 0.0352948
\(733\) −27127.0 −1.36693 −0.683464 0.729984i \(-0.739527\pi\)
−0.683464 + 0.729984i \(0.739527\pi\)
\(734\) −17202.0 −0.865037
\(735\) 9153.00 0.459338
\(736\) 270.000 0.0135222
\(737\) 27780.0 1.38845
\(738\) −6237.00 −0.311094
\(739\) −880.000 −0.0438042 −0.0219021 0.999760i \(-0.506972\pi\)
−0.0219021 + 0.999760i \(0.506972\pi\)
\(740\) −153.000 −0.00760053
\(741\) 0 0
\(742\) 3834.00 0.189691
\(743\) −21876.0 −1.08015 −0.540076 0.841616i \(-0.681604\pi\)
−0.540076 + 0.841616i \(0.681604\pi\)
\(744\) 6300.00 0.310442
\(745\) 15741.0 0.774102
\(746\) −26913.0 −1.32085
\(747\) 7290.00 0.357064
\(748\) −3330.00 −0.162777
\(749\) 1428.00 0.0696635
\(750\) 13689.0 0.666469
\(751\) 11798.0 0.573256 0.286628 0.958042i \(-0.407466\pi\)
0.286628 + 0.958042i \(0.407466\pi\)
\(752\) 11502.0 0.557759
\(753\) 18972.0 0.918165
\(754\) 0 0
\(755\) 3330.00 0.160518
\(756\) 54.0000 0.00259783
\(757\) −8074.00 −0.387655 −0.193827 0.981036i \(-0.562090\pi\)
−0.193827 + 0.981036i \(0.562090\pi\)
\(758\) 21732.0 1.04135
\(759\) −540.000 −0.0258245
\(760\) −8694.00 −0.414953
\(761\) 19554.0 0.931448 0.465724 0.884930i \(-0.345794\pi\)
0.465724 + 0.884930i \(0.345794\pi\)
\(762\) −5436.00 −0.258432
\(763\) 4012.00 0.190359
\(764\) 3444.00 0.163088
\(765\) 8991.00 0.424928
\(766\) −18936.0 −0.893193
\(767\) 0 0
\(768\) 4539.00 0.213264
\(769\) 14030.0 0.657913 0.328956 0.944345i \(-0.393303\pi\)
0.328956 + 0.944345i \(0.393303\pi\)
\(770\) −1620.00 −0.0758192
\(771\) 23499.0 1.09766
\(772\) −4273.00 −0.199208
\(773\) 36042.0 1.67703 0.838513 0.544882i \(-0.183426\pi\)
0.838513 + 0.544882i \(0.183426\pi\)
\(774\) −13878.0 −0.644489
\(775\) 4400.00 0.203939
\(776\) −28518.0 −1.31925
\(777\) 102.000 0.00470943
\(778\) 10881.0 0.501417
\(779\) 10626.0 0.488724
\(780\) 0 0
\(781\) −27900.0 −1.27828
\(782\) 1998.00 0.0913662
\(783\) −2835.00 −0.129393
\(784\) 24069.0 1.09644
\(785\) 23499.0 1.06843
\(786\) −14256.0 −0.646940
\(787\) 28628.0 1.29667 0.648334 0.761356i \(-0.275466\pi\)
0.648334 + 0.761356i \(0.275466\pi\)
\(788\) −1986.00 −0.0897821
\(789\) −9090.00 −0.410155
\(790\) 35748.0 1.60995
\(791\) −2238.00 −0.100599
\(792\) −5670.00 −0.254387
\(793\) 0 0
\(794\) −11694.0 −0.522676
\(795\) −17253.0 −0.769687
\(796\) −2386.00 −0.106243
\(797\) −37434.0 −1.66371 −0.831857 0.554990i \(-0.812722\pi\)
−0.831857 + 0.554990i \(0.812722\pi\)
\(798\) −828.000 −0.0367304
\(799\) 17982.0 0.796192
\(800\) 1980.00 0.0875045
\(801\) 4482.00 0.197707
\(802\) −17109.0 −0.753292
\(803\) −7590.00 −0.333556
\(804\) 2778.00 0.121856
\(805\) 108.000 0.00472857
\(806\) 0 0
\(807\) −1602.00 −0.0698799
\(808\) 7497.00 0.326415
\(809\) 37569.0 1.63270 0.816351 0.577556i \(-0.195994\pi\)
0.816351 + 0.577556i \(0.195994\pi\)
\(810\) −2187.00 −0.0948683
\(811\) 5516.00 0.238832 0.119416 0.992844i \(-0.461898\pi\)
0.119416 + 0.992844i \(0.461898\pi\)
\(812\) −210.000 −0.00907581
\(813\) −11064.0 −0.477283
\(814\) 1530.00 0.0658802
\(815\) 14724.0 0.632833
\(816\) 23643.0 1.01430
\(817\) 23644.0 1.01248
\(818\) 18933.0 0.809263
\(819\) 0 0
\(820\) 2079.00 0.0885388
\(821\) 8778.00 0.373148 0.186574 0.982441i \(-0.440262\pi\)
0.186574 + 0.982441i \(0.440262\pi\)
\(822\) 6453.00 0.273813
\(823\) −3088.00 −0.130791 −0.0653955 0.997859i \(-0.520831\pi\)
−0.0653955 + 0.997859i \(0.520831\pi\)
\(824\) −23478.0 −0.992591
\(825\) −3960.00 −0.167115
\(826\) 3600.00 0.151647
\(827\) 13176.0 0.554020 0.277010 0.960867i \(-0.410657\pi\)
0.277010 + 0.960867i \(0.410657\pi\)
\(828\) −54.0000 −0.00226646
\(829\) −2359.00 −0.0988317 −0.0494158 0.998778i \(-0.515736\pi\)
−0.0494158 + 0.998778i \(0.515736\pi\)
\(830\) −21870.0 −0.914601
\(831\) 5595.00 0.233560
\(832\) 0 0
\(833\) 37629.0 1.56515
\(834\) −7380.00 −0.306413
\(835\) −2376.00 −0.0984729
\(836\) −1380.00 −0.0570913
\(837\) −2700.00 −0.111500
\(838\) −6984.00 −0.287898
\(839\) −2676.00 −0.110114 −0.0550571 0.998483i \(-0.517534\pi\)
−0.0550571 + 0.998483i \(0.517534\pi\)
\(840\) 1134.00 0.0465794
\(841\) −13364.0 −0.547952
\(842\) 6135.00 0.251100
\(843\) 8991.00 0.367339
\(844\) −1600.00 −0.0652539
\(845\) 0 0
\(846\) −4374.00 −0.177756
\(847\) −862.000 −0.0349689
\(848\) −45369.0 −1.83724
\(849\) −12342.0 −0.498912
\(850\) 14652.0 0.591246
\(851\) −102.000 −0.00410871
\(852\) −2790.00 −0.112188
\(853\) 2477.00 0.0994266 0.0497133 0.998764i \(-0.484169\pi\)
0.0497133 + 0.998764i \(0.484169\pi\)
\(854\) 1398.00 0.0560171
\(855\) 3726.00 0.149037
\(856\) −14994.0 −0.598697
\(857\) −17199.0 −0.685539 −0.342769 0.939420i \(-0.611365\pi\)
−0.342769 + 0.939420i \(0.611365\pi\)
\(858\) 0 0
\(859\) 24338.0 0.966708 0.483354 0.875425i \(-0.339418\pi\)
0.483354 + 0.875425i \(0.339418\pi\)
\(860\) 4626.00 0.183425
\(861\) −1386.00 −0.0548603
\(862\) 15102.0 0.596724
\(863\) 25146.0 0.991865 0.495933 0.868361i \(-0.334826\pi\)
0.495933 + 0.868361i \(0.334826\pi\)
\(864\) −1215.00 −0.0478416
\(865\) −12690.0 −0.498813
\(866\) 12849.0 0.504188
\(867\) 22224.0 0.870550
\(868\) −200.000 −0.00782079
\(869\) −39720.0 −1.55053
\(870\) 8505.00 0.331433
\(871\) 0 0
\(872\) −42126.0 −1.63597
\(873\) 12222.0 0.473828
\(874\) 828.000 0.0320452
\(875\) 3042.00 0.117530
\(876\) −759.000 −0.0292742
\(877\) 18089.0 0.696490 0.348245 0.937403i \(-0.386778\pi\)
0.348245 + 0.937403i \(0.386778\pi\)
\(878\) −3918.00 −0.150599
\(879\) −13995.0 −0.537019
\(880\) 19170.0 0.734342
\(881\) −15099.0 −0.577410 −0.288705 0.957418i \(-0.593225\pi\)
−0.288705 + 0.957418i \(0.593225\pi\)
\(882\) −9153.00 −0.349430
\(883\) 33488.0 1.27629 0.638143 0.769918i \(-0.279703\pi\)
0.638143 + 0.769918i \(0.279703\pi\)
\(884\) 0 0
\(885\) −16200.0 −0.615319
\(886\) −17388.0 −0.659324
\(887\) −39768.0 −1.50539 −0.752694 0.658371i \(-0.771246\pi\)
−0.752694 + 0.658371i \(0.771246\pi\)
\(888\) −1071.00 −0.0404734
\(889\) −1208.00 −0.0455737
\(890\) −13446.0 −0.506417
\(891\) 2430.00 0.0913671
\(892\) −3832.00 −0.143840
\(893\) 7452.00 0.279252
\(894\) −15741.0 −0.588879
\(895\) 4266.00 0.159326
\(896\) 3318.00 0.123713
\(897\) 0 0
\(898\) 8118.00 0.301672
\(899\) 10500.0 0.389538
\(900\) −396.000 −0.0146667
\(901\) −70929.0 −2.62263
\(902\) −20790.0 −0.767440
\(903\) −3084.00 −0.113653
\(904\) 23499.0 0.864563
\(905\) −20241.0 −0.743463
\(906\) −3330.00 −0.122110
\(907\) 32156.0 1.17720 0.588601 0.808424i \(-0.299679\pi\)
0.588601 + 0.808424i \(0.299679\pi\)
\(908\) −1398.00 −0.0510950
\(909\) −3213.00 −0.117237
\(910\) 0 0
\(911\) 11520.0 0.418962 0.209481 0.977813i \(-0.432823\pi\)
0.209481 + 0.977813i \(0.432823\pi\)
\(912\) 9798.00 0.355750
\(913\) 24300.0 0.880846
\(914\) −2487.00 −0.0900029
\(915\) −6291.00 −0.227294
\(916\) 4466.00 0.161093
\(917\) −3168.00 −0.114086
\(918\) −8991.00 −0.323254
\(919\) 4952.00 0.177749 0.0888745 0.996043i \(-0.471673\pi\)
0.0888745 + 0.996043i \(0.471673\pi\)
\(920\) −1134.00 −0.0406379
\(921\) 4506.00 0.161214
\(922\) −16479.0 −0.588619
\(923\) 0 0
\(924\) 180.000 0.00640862
\(925\) −748.000 −0.0265882
\(926\) −46038.0 −1.63380
\(927\) 10062.0 0.356504
\(928\) 4725.00 0.167140
\(929\) 8781.00 0.310113 0.155057 0.987906i \(-0.450444\pi\)
0.155057 + 0.987906i \(0.450444\pi\)
\(930\) 8100.00 0.285602
\(931\) 15594.0 0.548950
\(932\) −1638.00 −0.0575692
\(933\) 6318.00 0.221696
\(934\) −28782.0 −1.00833
\(935\) 29970.0 1.04826
\(936\) 0 0
\(937\) 50039.0 1.74461 0.872307 0.488959i \(-0.162623\pi\)
0.872307 + 0.488959i \(0.162623\pi\)
\(938\) 5556.00 0.193401
\(939\) −11694.0 −0.406410
\(940\) 1458.00 0.0505901
\(941\) −50670.0 −1.75536 −0.877681 0.479246i \(-0.840910\pi\)
−0.877681 + 0.479246i \(0.840910\pi\)
\(942\) −23499.0 −0.812780
\(943\) 1386.00 0.0478625
\(944\) −42600.0 −1.46876
\(945\) −486.000 −0.0167297
\(946\) −46260.0 −1.58990
\(947\) 42384.0 1.45438 0.727188 0.686438i \(-0.240827\pi\)
0.727188 + 0.686438i \(0.240827\pi\)
\(948\) −3972.00 −0.136081
\(949\) 0 0
\(950\) 6072.00 0.207370
\(951\) 28053.0 0.956552
\(952\) 4662.00 0.158715
\(953\) −50538.0 −1.71782 −0.858912 0.512123i \(-0.828859\pi\)
−0.858912 + 0.512123i \(0.828859\pi\)
\(954\) 17253.0 0.585520
\(955\) −30996.0 −1.05027
\(956\) −594.000 −0.0200955
\(957\) −9450.00 −0.319201
\(958\) −38520.0 −1.29909
\(959\) 1434.00 0.0482860
\(960\) −11691.0 −0.393047
\(961\) −19791.0 −0.664328
\(962\) 0 0
\(963\) 6426.00 0.215031
\(964\) 2303.00 0.0769446
\(965\) 38457.0 1.28288
\(966\) −108.000 −0.00359715
\(967\) −6886.00 −0.228996 −0.114498 0.993423i \(-0.536526\pi\)
−0.114498 + 0.993423i \(0.536526\pi\)
\(968\) 9051.00 0.300527
\(969\) 15318.0 0.507828
\(970\) −36666.0 −1.21368
\(971\) 9060.00 0.299433 0.149716 0.988729i \(-0.452164\pi\)
0.149716 + 0.988729i \(0.452164\pi\)
\(972\) 243.000 0.00801875
\(973\) −1640.00 −0.0540349
\(974\) −42258.0 −1.39018
\(975\) 0 0
\(976\) −16543.0 −0.542550
\(977\) −28311.0 −0.927072 −0.463536 0.886078i \(-0.653419\pi\)
−0.463536 + 0.886078i \(0.653419\pi\)
\(978\) −14724.0 −0.481413
\(979\) 14940.0 0.487727
\(980\) 3051.00 0.0994496
\(981\) 18054.0 0.587584
\(982\) 35082.0 1.14003
\(983\) 4284.00 0.139001 0.0695007 0.997582i \(-0.477859\pi\)
0.0695007 + 0.997582i \(0.477859\pi\)
\(984\) 14553.0 0.471476
\(985\) 17874.0 0.578186
\(986\) 34965.0 1.12932
\(987\) −972.000 −0.0313466
\(988\) 0 0
\(989\) 3084.00 0.0991562
\(990\) −7290.00 −0.234032
\(991\) −2458.00 −0.0787901 −0.0393950 0.999224i \(-0.512543\pi\)
−0.0393950 + 0.999224i \(0.512543\pi\)
\(992\) 4500.00 0.144027
\(993\) −27516.0 −0.879349
\(994\) −5580.00 −0.178055
\(995\) 21474.0 0.684193
\(996\) 2430.00 0.0773067
\(997\) 24101.0 0.765583 0.382792 0.923835i \(-0.374963\pi\)
0.382792 + 0.923835i \(0.374963\pi\)
\(998\) −11064.0 −0.350927
\(999\) 459.000 0.0145367
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.a.e.1.1 1
3.2 odd 2 1521.4.a.c.1.1 1
13.3 even 3 39.4.e.a.22.1 yes 2
13.5 odd 4 507.4.b.c.337.1 2
13.8 odd 4 507.4.b.c.337.2 2
13.9 even 3 39.4.e.a.16.1 2
13.12 even 2 507.4.a.a.1.1 1
39.29 odd 6 117.4.g.b.100.1 2
39.35 odd 6 117.4.g.b.55.1 2
39.38 odd 2 1521.4.a.j.1.1 1
52.3 odd 6 624.4.q.b.529.1 2
52.35 odd 6 624.4.q.b.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.a.16.1 2 13.9 even 3
39.4.e.a.22.1 yes 2 13.3 even 3
117.4.g.b.55.1 2 39.35 odd 6
117.4.g.b.100.1 2 39.29 odd 6
507.4.a.a.1.1 1 13.12 even 2
507.4.a.e.1.1 1 1.1 even 1 trivial
507.4.b.c.337.1 2 13.5 odd 4
507.4.b.c.337.2 2 13.8 odd 4
624.4.q.b.289.1 2 52.35 odd 6
624.4.q.b.529.1 2 52.3 odd 6
1521.4.a.c.1.1 1 3.2 odd 2
1521.4.a.j.1.1 1 39.38 odd 2