Properties

Label 378.2.a.a.1.1
Level $378$
Weight $2$
Character 378.1
Self dual yes
Analytic conductor $3.018$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 378.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{10} +4.00000 q^{11} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +2.00000 q^{19} -4.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} -3.00000 q^{26} -1.00000 q^{28} -1.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} +4.00000 q^{35} +2.00000 q^{37} -2.00000 q^{38} +4.00000 q^{40} -6.00000 q^{41} +11.0000 q^{43} +4.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} +3.00000 q^{52} +9.00000 q^{53} -16.0000 q^{55} +1.00000 q^{56} +1.00000 q^{58} +5.00000 q^{59} -6.00000 q^{61} +9.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +7.00000 q^{67} +7.00000 q^{68} -4.00000 q^{70} +7.00000 q^{71} -14.0000 q^{73} -2.00000 q^{74} +2.00000 q^{76} -4.00000 q^{77} -6.00000 q^{79} -4.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} -28.0000 q^{85} -11.0000 q^{86} -4.00000 q^{88} +3.00000 q^{89} -3.00000 q^{91} +1.00000 q^{92} -6.00000 q^{94} -8.00000 q^{95} -8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −4.00000 −0.894427
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) −3.00000 −0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −7.00000 −1.20049
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 4.00000 0.632456
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 11.0000 1.67748 0.838742 0.544529i \(-0.183292\pi\)
0.838742 + 0.544529i \(0.183292\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) 3.00000 0.416025
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 9.00000 1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) −4.00000 −0.478091
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −28.0000 −3.03703
\(86\) −11.0000 −1.18616
\(87\) 0 0
\(88\) −4.00000 −0.426401
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 11.0000 1.10000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) 17.0000 1.67506 0.837530 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 16.0000 1.52554
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) −5.00000 −0.460287
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −9.00000 −0.808224
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) −7.00000 −0.604708
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −7.00000 −0.587427
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 36.0000 2.89159
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) 4.00000 0.316228
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 28.0000 2.14750
\(171\) 0 0
\(172\) 11.0000 0.838742
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) −11.0000 −0.831522
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −3.00000 −0.224860
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −9.00000 −0.668965 −0.334482 0.942402i \(-0.608561\pi\)
−0.334482 + 0.942402i \(0.608561\pi\)
\(182\) 3.00000 0.222375
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 28.0000 2.04756
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 5.00000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −11.0000 −0.777817
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 1.00000 0.0701862
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) −17.0000 −1.18445
\(207\) 0 0
\(208\) 3.00000 0.208013
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −44.0000 −3.00078
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) −16.0000 −1.07872
\(221\) 21.0000 1.41261
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) 9.00000 0.597351 0.298675 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 5.00000 0.325472
\(237\) 0 0
\(238\) 7.00000 0.453743
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) −4.00000 −0.255551
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 9.00000 0.571501
\(249\) 0 0
\(250\) 24.0000 1.51789
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −12.0000 −0.744208
\(261\) 0 0
\(262\) 1.00000 0.0617802
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 7.00000 0.427593
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) 44.0000 2.65330
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −10.0000 −0.599760
\(279\) 0 0
\(280\) −4.00000 −0.239046
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −14.0000 −0.819288
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −20.0000 −1.16445
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −3.00000 −0.173785
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) −11.0000 −0.634029
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) −36.0000 −2.04466
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −17.0000 −0.959366
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −4.00000 −0.223607
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 14.0000 0.778981
\(324\) 0 0
\(325\) 33.0000 1.83051
\(326\) 1.00000 0.0553849
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −31.0000 −1.70391 −0.851957 0.523612i \(-0.824584\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) 27.0000 1.47078 0.735392 0.677642i \(-0.236998\pi\)
0.735392 + 0.677642i \(0.236998\pi\)
\(338\) 4.00000 0.217571
\(339\) 0 0
\(340\) −28.0000 −1.51851
\(341\) −36.0000 −1.94951
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −11.0000 −0.593080
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 11.0000 0.587975
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) −28.0000 −1.48609
\(356\) 3.00000 0.159000
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 11.0000 0.580558 0.290279 0.956942i \(-0.406252\pi\)
0.290279 + 0.956942i \(0.406252\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 9.00000 0.473029
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 56.0000 2.93117
\(366\) 0 0
\(367\) −25.0000 −1.30499 −0.652495 0.757793i \(-0.726278\pi\)
−0.652495 + 0.757793i \(0.726278\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) −28.0000 −1.44785
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) −5.00000 −0.254493
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 7.00000 0.354005
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 26.0000 1.30986
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 11.0000 0.550000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −27.0000 −1.34497
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) −24.0000 −1.18528
\(411\) 0 0
\(412\) 17.0000 0.837530
\(413\) −5.00000 −0.246034
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) −3.00000 −0.147087
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) 33.0000 1.61216 0.806078 0.591810i \(-0.201586\pi\)
0.806078 + 0.591810i \(0.201586\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 1.00000 0.0486792
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 77.0000 3.73505
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 44.0000 2.12187
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) −9.00000 −0.432014
\(435\) 0 0
\(436\) 16.0000 0.766261
\(437\) 2.00000 0.0956730
\(438\) 0 0
\(439\) 9.00000 0.429547 0.214773 0.976664i \(-0.431099\pi\)
0.214773 + 0.976664i \(0.431099\pi\)
\(440\) 16.0000 0.762770
\(441\) 0 0
\(442\) −21.0000 −0.998868
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 8.00000 0.376288
\(453\) 0 0
\(454\) −9.00000 −0.422391
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −7.00000 −0.323230
\(470\) 24.0000 1.10704
\(471\) 0 0
\(472\) −5.00000 −0.230144
\(473\) 44.0000 2.02312
\(474\) 0 0
\(475\) 22.0000 1.00943
\(476\) −7.00000 −0.320844
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 32.0000 1.45305
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 4.00000 0.180702
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) −7.00000 −0.315264
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −9.00000 −0.404112
\(497\) −7.00000 −0.313993
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −24.0000 −1.07331
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 30.0000 1.32324
\(515\) −68.0000 −2.99644
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 2.00000 0.0878750
\(519\) 0 0
\(520\) 12.0000 0.526235
\(521\) 15.0000 0.657162 0.328581 0.944476i \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) −1.00000 −0.0436852
\(525\) 0 0
\(526\) 9.00000 0.392419
\(527\) −63.0000 −2.74432
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 36.0000 1.56374
\(531\) 0 0
\(532\) −2.00000 −0.0867110
\(533\) −18.0000 −0.779667
\(534\) 0 0
\(535\) 48.0000 2.07522
\(536\) −7.00000 −0.302354
\(537\) 0 0
\(538\) 4.00000 0.172452
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −36.0000 −1.54776 −0.773880 0.633332i \(-0.781687\pi\)
−0.773880 + 0.633332i \(0.781687\pi\)
\(542\) −3.00000 −0.128861
\(543\) 0 0
\(544\) −7.00000 −0.300123
\(545\) −64.0000 −2.74146
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −22.0000 −0.939793
\(549\) 0 0
\(550\) −44.0000 −1.87617
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) 6.00000 0.255146
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) −1.00000 −0.0423714 −0.0211857 0.999776i \(-0.506744\pi\)
−0.0211857 + 0.999776i \(0.506744\pi\)
\(558\) 0 0
\(559\) 33.0000 1.39575
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 0 0
\(565\) −32.0000 −1.34625
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −7.00000 −0.293713
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −39.0000 −1.63210 −0.816050 0.577982i \(-0.803840\pi\)
−0.816050 + 0.577982i \(0.803840\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 11.0000 0.458732
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 25.0000 1.03186 0.515930 0.856631i \(-0.327446\pi\)
0.515930 + 0.856631i \(0.327446\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 20.0000 0.823387
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 28.0000 1.14789
\(596\) 3.00000 0.122885
\(597\) 0 0
\(598\) −3.00000 −0.122679
\(599\) −21.0000 −0.858037 −0.429018 0.903296i \(-0.641140\pi\)
−0.429018 + 0.903296i \(0.641140\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 11.0000 0.448327
\(603\) 0 0
\(604\) 10.0000 0.406894
\(605\) −20.0000 −0.813116
\(606\) 0 0
\(607\) 39.0000 1.58296 0.791481 0.611194i \(-0.209311\pi\)
0.791481 + 0.611194i \(0.209311\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) −24.0000 −0.971732
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 36.0000 1.44579
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) −3.00000 −0.120192
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 17.0000 0.678374
\(629\) 14.0000 0.558217
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 6.00000 0.238667
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) 4.00000 0.158114
\(641\) 28.0000 1.10593 0.552967 0.833203i \(-0.313496\pi\)
0.552967 + 0.833203i \(0.313496\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) −33.0000 −1.29437
\(651\) 0 0
\(652\) −1.00000 −0.0391630
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 6.00000 0.233904
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 31.0000 1.20485
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 28.0000 1.08173
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) −27.0000 −1.04000
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 28.0000 1.07375
\(681\) 0 0
\(682\) 36.0000 1.37851
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 88.0000 3.36231
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 11.0000 0.419371
\(689\) 27.0000 1.02862
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) −42.0000 −1.59086
\(698\) −25.0000 −0.946264
\(699\) 0 0
\(700\) −11.0000 −0.415761
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 28.0000 1.05082
\(711\) 0 0
\(712\) −3.00000 −0.112430
\(713\) −9.00000 −0.337053
\(714\) 0 0
\(715\) −48.0000 −1.79510
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −11.0000 −0.410516
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −17.0000 −0.633113
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) −9.00000 −0.334482
\(725\) −11.0000 −0.408530
\(726\) 0 0
\(727\) −29.0000 −1.07555 −0.537775 0.843088i \(-0.680735\pi\)
−0.537775 + 0.843088i \(0.680735\pi\)
\(728\) 3.00000 0.111187
\(729\) 0 0
\(730\) −56.0000 −2.07265
\(731\) 77.0000 2.84795
\(732\) 0 0
\(733\) −15.0000 −0.554038 −0.277019 0.960864i \(-0.589346\pi\)
−0.277019 + 0.960864i \(0.589346\pi\)
\(734\) 25.0000 0.922767
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 28.0000 1.03139
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 9.00000 0.330400
\(743\) −51.0000 −1.87101 −0.935504 0.353315i \(-0.885054\pi\)
−0.935504 + 0.353315i \(0.885054\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 38.0000 1.39128
\(747\) 0 0
\(748\) 28.0000 1.02378
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 6.00000 0.218943 0.109472 0.993990i \(-0.465084\pi\)
0.109472 + 0.993990i \(0.465084\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) −40.0000 −1.45575
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 13.0000 0.471250 0.235625 0.971844i \(-0.424286\pi\)
0.235625 + 0.971844i \(0.424286\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 15.0000 0.541619
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) −16.0000 −0.576600
\(771\) 0 0
\(772\) 5.00000 0.179954
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 0 0
\(775\) −99.0000 −3.55618
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 28.0000 1.00192
\(782\) −7.00000 −0.250319
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −68.0000 −2.42702
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −26.0000 −0.926212
\(789\) 0 0
\(790\) −24.0000 −0.853882
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 42.0000 1.48585
\(800\) −11.0000 −0.388909
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) −56.0000 −1.97620
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 27.0000 0.951034
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 32.0000 1.12506 0.562530 0.826777i \(-0.309828\pi\)
0.562530 + 0.826777i \(0.309828\pi\)
\(810\) 0 0
\(811\) −26.0000 −0.912983 −0.456492 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) 1.00000 0.0350931
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 22.0000 0.769683
\(818\) −2.00000 −0.0699284
\(819\) 0 0
\(820\) 24.0000 0.838116
\(821\) 13.0000 0.453703 0.226852 0.973929i \(-0.427157\pi\)
0.226852 + 0.973929i \(0.427157\pi\)
\(822\) 0 0
\(823\) −38.0000 −1.32460 −0.662298 0.749240i \(-0.730419\pi\)
−0.662298 + 0.749240i \(0.730419\pi\)
\(824\) −17.0000 −0.592223
\(825\) 0 0
\(826\) 5.00000 0.173972
\(827\) −42.0000 −1.46048 −0.730242 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) 3.00000 0.104006
\(833\) 7.00000 0.242536
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) −33.0000 −1.13997
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −1.00000 −0.0344214
\(845\) 16.0000 0.550417
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) 9.00000 0.309061
\(849\) 0 0
\(850\) −77.0000 −2.64108
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 53.0000 1.81469 0.907343 0.420392i \(-0.138107\pi\)
0.907343 + 0.420392i \(0.138107\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 0 0
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) −44.0000 −1.50039
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −1.00000 −0.0340404 −0.0170202 0.999855i \(-0.505418\pi\)
−0.0170202 + 0.999855i \(0.505418\pi\)
\(864\) 0 0
\(865\) −32.0000 −1.08803
\(866\) −8.00000 −0.271851
\(867\) 0 0
\(868\) 9.00000 0.305480
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 21.0000 0.711558
\(872\) −16.0000 −0.541828
\(873\) 0 0
\(874\) −2.00000 −0.0676510
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −9.00000 −0.303735
\(879\) 0 0
\(880\) −16.0000 −0.539360
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) 21.0000 0.706306
\(885\) 0 0
\(886\) 26.0000 0.873487
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.0000 0.402241
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 4.00000 0.133482
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 63.0000 2.09883
\(902\) 24.0000 0.799113
\(903\) 0 0
\(904\) −8.00000 −0.266076
\(905\) 36.0000 1.19668
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 9.00000 0.298675
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 29.0000 0.959235
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 1.00000 0.0330229
\(918\) 0 0
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) −14.0000 −0.461065
\(923\) 21.0000 0.691223
\(924\) 0 0
\(925\) 22.0000 0.723356
\(926\) −22.0000 −0.722965
\(927\) 0 0
\(928\) 1.00000 0.0328266
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 4.00000 0.131024
\(933\) 0 0
\(934\) −8.00000 −0.261768
\(935\) −112.000 −3.66279
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) 7.00000 0.228558
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) −44.0000 −1.43056
\(947\) 2.00000 0.0649913 0.0324956 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) 0 0
\(949\) −42.0000 −1.36338
\(950\) −22.0000 −0.713774
\(951\) 0 0
\(952\) 7.00000 0.226871
\(953\) −8.00000 −0.259145 −0.129573 0.991570i \(-0.541361\pi\)
−0.129573 + 0.991570i \(0.541361\pi\)
\(954\) 0 0
\(955\) 48.0000 1.55324
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 22.0000 0.710417
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) −6.00000 −0.193448
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −32.0000 −1.02746
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) −10.0000 −0.320585
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) −4.00000 −0.127775
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 104.000 3.31372
\(986\) 7.00000 0.222925
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 9.00000 0.285750
\(993\) 0 0
\(994\) 7.00000 0.222027
\(995\) 28.0000 0.887660
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) −28.0000 −0.886325
\(999\) 0 0
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 378.2.a.a.1.1 1
3.2 odd 2 378.2.a.h.1.1 yes 1
4.3 odd 2 3024.2.a.a.1.1 1
5.4 even 2 9450.2.a.dv.1.1 1
7.6 odd 2 2646.2.a.o.1.1 1
9.2 odd 6 1134.2.f.a.757.1 2
9.4 even 3 1134.2.f.p.379.1 2
9.5 odd 6 1134.2.f.a.379.1 2
9.7 even 3 1134.2.f.p.757.1 2
12.11 even 2 3024.2.a.bd.1.1 1
15.14 odd 2 9450.2.a.bc.1.1 1
21.20 even 2 2646.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.a.1.1 1 1.1 even 1 trivial
378.2.a.h.1.1 yes 1 3.2 odd 2
1134.2.f.a.379.1 2 9.5 odd 6
1134.2.f.a.757.1 2 9.2 odd 6
1134.2.f.p.379.1 2 9.4 even 3
1134.2.f.p.757.1 2 9.7 even 3
2646.2.a.o.1.1 1 7.6 odd 2
2646.2.a.p.1.1 1 21.20 even 2
3024.2.a.a.1.1 1 4.3 odd 2
3024.2.a.bd.1.1 1 12.11 even 2
9450.2.a.bc.1.1 1 15.14 odd 2
9450.2.a.dv.1.1 1 5.4 even 2