Properties

Label 4225.2.a.k.1.1
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4225.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +2.00000 q^{12} -1.00000 q^{16} +1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{22} +6.00000 q^{23} +6.00000 q^{24} +4.00000 q^{27} -6.00000 q^{29} +6.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} -1.00000 q^{36} +6.00000 q^{37} +6.00000 q^{38} -8.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} +6.00000 q^{46} -8.00000 q^{47} +2.00000 q^{48} -7.00000 q^{49} +12.0000 q^{53} +4.00000 q^{54} -12.0000 q^{57} -6.00000 q^{58} +2.00000 q^{59} +6.00000 q^{61} +6.00000 q^{62} +7.00000 q^{64} +4.00000 q^{66} -12.0000 q^{67} -12.0000 q^{69} -2.00000 q^{71} -3.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -6.00000 q^{76} -11.0000 q^{81} -8.00000 q^{82} -4.00000 q^{83} -6.00000 q^{86} +12.0000 q^{87} +6.00000 q^{88} -8.00000 q^{89} -6.00000 q^{92} -12.0000 q^{93} -8.00000 q^{94} -10.0000 q^{96} +6.00000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 6.00000 1.22474
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 5.00000 0.883883
\(33\) 4.00000 0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 2.00000 0.288675
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) −6.00000 −0.787839
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −3.00000 −0.353553
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −8.00000 −0.883452
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 12.0000 1.28654
\(88\) 6.00000 0.639602
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) −12.0000 −1.24434
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −10.0000 −1.02062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −7.00000 −0.707107
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −4.00000 −0.384900
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 16.0000 1.44267
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −3.00000 −0.265165
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −12.0000 −1.02151
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 16.0000 1.34744
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 14.0000 1.15470
\(148\) −6.00000 −0.493197
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) −18.0000 −1.45999
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.0000 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 6.00000 0.457496
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) −8.00000 −0.599625
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) −18.0000 −1.32698
\(185\) 0 0
\(186\) −12.0000 −0.879883
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −14.0000 −1.01036
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −2.00000 −0.142134
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 24.0000 1.69283
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −12.0000 −0.824163
\(213\) 4.00000 0.274075
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) −12.0000 −0.816497
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) −12.0000 −0.805387
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 12.0000 0.794719
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −7.00000 −0.449977
\(243\) 10.0000 0.641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) −18.0000 −1.14300
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 12.0000 0.747087
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 0 0
\(267\) 16.0000 0.979184
\(268\) 12.0000 0.733017
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 4.00000 0.239904
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 16.0000 0.952786
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) −6.00000 −0.351123
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 14.0000 0.816497
\(295\) 0 0
\(296\) −18.0000 −1.04623
\(297\) −8.00000 −0.464207
\(298\) −20.0000 −1.15857
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 18.0000 1.03578
\(303\) 12.0000 0.689382
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −24.0000 −1.34585
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 24.0000 1.32720
\(328\) 24.0000 1.32518
\(329\) 0 0
\(330\) 0 0
\(331\) −30.0000 −1.64895 −0.824475 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) −16.0000 −0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −12.0000 −0.643268
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −10.0000 −0.533002
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) −12.0000 −0.627250
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −6.00000 −0.312772
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 0 0
\(379\) −18.0000 −0.924598 −0.462299 0.886724i \(-0.652975\pi\)
−0.462299 + 0.886724i \(0.652975\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 6.00000 0.306186
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) −6.00000 −0.304997
\(388\) −6.00000 −0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 21.0000 1.06066
\(393\) 24.0000 1.21064
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −24.0000 −1.20301
\(399\) 0 0
\(400\) 0 0
\(401\) −16.0000 −0.799002 −0.399501 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 24.0000 1.19701
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) −6.00000 −0.295599
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) −12.0000 −0.586939
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 36.0000 1.75453 0.877266 0.480004i \(-0.159365\pi\)
0.877266 + 0.480004i \(0.159365\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) −36.0000 −1.74831
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) −4.00000 −0.192450
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 36.0000 1.72211
\(438\) −12.0000 −0.573382
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 12.0000 0.569495
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 40.0000 1.89194
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 0 0
\(453\) −36.0000 −1.69143
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 36.0000 1.68585
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0 0
\(460\) 0 0
\(461\) −4.00000 −0.186299 −0.0931493 0.995652i \(-0.529693\pi\)
−0.0931493 + 0.995652i \(0.529693\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) −6.00000 −0.276172
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 10.0000 0.457389
\(479\) 22.0000 1.00521 0.502603 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −18.0000 −0.814822
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −16.0000 −0.721336
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) 32.0000 1.42965
\(502\) −12.0000 −0.535586
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −6.00000 −0.262613
\(523\) −42.0000 −1.83653 −0.918266 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 0 0
\(534\) 16.0000 0.692388
\(535\) 0 0
\(536\) 36.0000 1.55496
\(537\) 24.0000 1.03568
\(538\) 18.0000 0.776035
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) −12.0000 −0.515920 −0.257960 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) −6.00000 −0.257722
\(543\) −4.00000 −0.171656
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.0000 0.769624 0.384812 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 36.0000 1.53226
\(553\) 0 0
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 6.00000 0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) −16.0000 −0.673722
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) −17.0000 −0.707107
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) 0 0
\(582\) −12.0000 −0.497416
\(583\) −24.0000 −0.993978
\(584\) −18.0000 −0.744845
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) −14.0000 −0.577350
\(589\) 36.0000 1.48335
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) −6.00000 −0.246598
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) −8.00000 −0.328244
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 48.0000 1.96451
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 30.0000 1.21666
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 30.0000 1.21169 0.605844 0.795583i \(-0.292835\pi\)
0.605844 + 0.795583i \(0.292835\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) −12.0000 −0.482711
\(619\) 18.0000 0.723481 0.361741 0.932279i \(-0.382183\pi\)
0.361741 + 0.932279i \(0.382183\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 24.0000 0.958468
\(628\) −12.0000 −0.478852
\(629\) 0 0
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 24.0000 0.951662
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 12.0000 0.473602
\(643\) −36.0000 −1.41970 −0.709851 0.704352i \(-0.751238\pi\)
−0.709851 + 0.704352i \(0.751238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 33.0000 1.29636
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 24.0000 0.938474
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) −30.0000 −1.16598
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −36.0000 −1.39393
\(668\) 16.0000 0.619059
\(669\) −48.0000 −1.85579
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 0 0
\(677\) 36.0000 1.38359 0.691796 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −12.0000 −0.459504
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) 24.0000 0.915657
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) −36.0000 −1.36458
\(697\) 0 0
\(698\) 12.0000 0.454207
\(699\) 48.0000 1.81553
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 24.0000 0.899438
\(713\) 36.0000 1.34821
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −20.0000 −0.746914
\(718\) −2.00000 −0.0746393
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 14.0000 0.519589
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 12.0000 0.443533
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) 24.0000 0.884051
\(738\) −8.00000 −0.294484
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 36.0000 1.31982
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 8.00000 0.291730
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) −18.0000 −0.653789
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) −4.00000 −0.144905
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 0 0
\(768\) 34.0000 1.22687
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) −24.0000 −0.857690
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) −16.0000 −0.564980
\(803\) −12.0000 −0.423471
\(804\) −24.0000 −0.846415
\(805\) 0 0
\(806\) 0 0
\(807\) −36.0000 −1.26726
\(808\) 18.0000 0.633238
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 24.0000 0.839140
\(819\) 0 0
\(820\) 0 0
\(821\) 20.0000 0.698005 0.349002 0.937122i \(-0.386521\pi\)
0.349002 + 0.937122i \(0.386521\pi\)
\(822\) −4.00000 −0.139516
\(823\) −42.0000 −1.46403 −0.732014 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) −18.0000 −0.627060
\(825\) 0 0
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) −6.00000 −0.208514
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 24.0000 0.829561
\(838\) 12.0000 0.414533
\(839\) 46.0000 1.58810 0.794048 0.607855i \(-0.207970\pi\)
0.794048 + 0.607855i \(0.207970\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 36.0000 1.24064
\(843\) 16.0000 0.551069
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 44.0000 1.51008
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) −4.00000 −0.137038
\(853\) −54.0000 −1.84892 −0.924462 0.381273i \(-0.875486\pi\)
−0.924462 + 0.381273i \(0.875486\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 20.0000 0.680414
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) 34.0000 1.15470
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 36.0000 1.21911
\(873\) 6.00000 0.203069
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 8.00000 0.269987
\(879\) −52.0000 −1.75392
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) −7.00000 −0.235702
\(883\) 2.00000 0.0673054 0.0336527 0.999434i \(-0.489286\pi\)
0.0336527 + 0.999434i \(0.489286\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 36.0000 1.20808
\(889\) 0 0
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) −24.0000 −0.803579
\(893\) −48.0000 −1.60626
\(894\) 40.0000 1.33780
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −16.0000 −0.533927
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 0 0
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −36.0000 −1.19602
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 12.0000 0.397360
\(913\) 8.00000 0.264761
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 12.0000 0.396491
\(917\) 0 0
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) −4.00000 −0.131733
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) 6.00000 0.197066
\(928\) −30.0000 −0.984798
\(929\) −16.0000 −0.524943 −0.262471 0.964940i \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 24.0000 0.786146
\(933\) 48.0000 1.57145
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 0 0
\(937\) −56.0000 −1.82944 −0.914720 0.404088i \(-0.867589\pi\)
−0.914720 + 0.404088i \(0.867589\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) 28.0000 0.912774 0.456387 0.889781i \(-0.349143\pi\)
0.456387 + 0.889781i \(0.349143\pi\)
\(942\) −24.0000 −0.781962
\(943\) −48.0000 −1.56310
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −10.0000 −0.323423
\(957\) −24.0000 −0.775810
\(958\) 22.0000 0.710788
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 21.0000 0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −10.0000 −0.320750
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 24.0000 0.767435
\(979\) 16.0000 0.511362
\(980\) 0 0
\(981\) −12.0000 −0.383131
\(982\) −12.0000 −0.382935
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) −48.0000 −1.53018
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 30.0000 0.952501
\(993\) 60.0000 1.90404
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −60.0000 −1.90022 −0.950110 0.311916i \(-0.899029\pi\)
−0.950110 + 0.311916i \(0.899029\pi\)
\(998\) 6.00000 0.189927
\(999\) 24.0000 0.759326
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 4225.2.a.k.1.1 1
5.2 odd 4 845.2.b.b.339.2 2
5.3 odd 4 845.2.b.b.339.1 2
5.4 even 2 4225.2.a.h.1.1 1
13.5 odd 4 325.2.c.b.51.1 2
13.8 odd 4 325.2.c.b.51.2 2
13.12 even 2 4225.2.a.e.1.1 1
65.2 even 12 845.2.l.a.654.1 4
65.3 odd 12 845.2.n.b.529.2 4
65.7 even 12 845.2.l.b.699.2 4
65.8 even 4 65.2.d.b.64.1 yes 2
65.12 odd 4 845.2.b.a.339.1 2
65.17 odd 12 845.2.n.a.484.1 4
65.18 even 4 65.2.d.a.64.1 2
65.22 odd 12 845.2.n.b.484.2 4
65.23 odd 12 845.2.n.a.529.1 4
65.28 even 12 845.2.l.b.654.2 4
65.32 even 12 845.2.l.a.699.2 4
65.33 even 12 845.2.l.a.699.1 4
65.34 odd 4 325.2.c.e.51.1 2
65.37 even 12 845.2.l.b.654.1 4
65.38 odd 4 845.2.b.a.339.2 2
65.42 odd 12 845.2.n.b.529.1 4
65.43 odd 12 845.2.n.a.484.2 4
65.44 odd 4 325.2.c.e.51.2 2
65.47 even 4 65.2.d.a.64.2 yes 2
65.48 odd 12 845.2.n.b.484.1 4
65.57 even 4 65.2.d.b.64.2 yes 2
65.58 even 12 845.2.l.b.699.1 4
65.62 odd 12 845.2.n.a.529.2 4
65.63 even 12 845.2.l.a.654.2 4
65.64 even 2 4225.2.a.m.1.1 1
195.8 odd 4 585.2.h.b.64.1 2
195.47 odd 4 585.2.h.c.64.1 2
195.83 odd 4 585.2.h.c.64.2 2
195.122 odd 4 585.2.h.b.64.2 2
260.47 odd 4 1040.2.f.a.129.1 2
260.83 odd 4 1040.2.f.a.129.2 2
260.187 odd 4 1040.2.f.b.129.1 2
260.203 odd 4 1040.2.f.b.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.d.a.64.1 2 65.18 even 4
65.2.d.a.64.2 yes 2 65.47 even 4
65.2.d.b.64.1 yes 2 65.8 even 4
65.2.d.b.64.2 yes 2 65.57 even 4
325.2.c.b.51.1 2 13.5 odd 4
325.2.c.b.51.2 2 13.8 odd 4
325.2.c.e.51.1 2 65.34 odd 4
325.2.c.e.51.2 2 65.44 odd 4
585.2.h.b.64.1 2 195.8 odd 4
585.2.h.b.64.2 2 195.122 odd 4
585.2.h.c.64.1 2 195.47 odd 4
585.2.h.c.64.2 2 195.83 odd 4
845.2.b.a.339.1 2 65.12 odd 4
845.2.b.a.339.2 2 65.38 odd 4
845.2.b.b.339.1 2 5.3 odd 4
845.2.b.b.339.2 2 5.2 odd 4
845.2.l.a.654.1 4 65.2 even 12
845.2.l.a.654.2 4 65.63 even 12
845.2.l.a.699.1 4 65.33 even 12
845.2.l.a.699.2 4 65.32 even 12
845.2.l.b.654.1 4 65.37 even 12
845.2.l.b.654.2 4 65.28 even 12
845.2.l.b.699.1 4 65.58 even 12
845.2.l.b.699.2 4 65.7 even 12
845.2.n.a.484.1 4 65.17 odd 12
845.2.n.a.484.2 4 65.43 odd 12
845.2.n.a.529.1 4 65.23 odd 12
845.2.n.a.529.2 4 65.62 odd 12
845.2.n.b.484.1 4 65.48 odd 12
845.2.n.b.484.2 4 65.22 odd 12
845.2.n.b.529.1 4 65.42 odd 12
845.2.n.b.529.2 4 65.3 odd 12
1040.2.f.a.129.1 2 260.47 odd 4
1040.2.f.a.129.2 2 260.83 odd 4
1040.2.f.b.129.1 2 260.187 odd 4
1040.2.f.b.129.2 2 260.203 odd 4
4225.2.a.e.1.1 1 13.12 even 2
4225.2.a.h.1.1 1 5.4 even 2
4225.2.a.k.1.1 1 1.1 even 1 trivial
4225.2.a.m.1.1 1 65.64 even 2