Properties

Label 4225.2.a.h.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 $1$

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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 \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −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 \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 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 \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −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 \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) −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 \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 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 \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 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 \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 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 \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 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 \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 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 \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 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 \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 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 \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 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 \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 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 \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 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 \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 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 \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −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 \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 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 \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) −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 \(-0.617396\pi\)
−0.360505 + 0.932757i \(0.617396\pi\)
\(278\) −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 \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 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 \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 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 \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 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 \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 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 \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 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 \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 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 \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) −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 \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 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 \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 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 \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 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 \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 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 \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 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 \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 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 \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) 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 \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −24.0000 −1.11178
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 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 \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 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 \(-0.129590\pi\)
0.918266 + 0.395964i \(0.129590\pi\)
\(524\) 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 \(-0.625734\pi\)
−0.384812 + 0.922995i \(0.625734\pi\)
\(548\) 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 \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) −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 \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) −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 \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 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 \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) 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 \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) −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 \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) −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 \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 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 \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) −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 \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 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 \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) −32.0000 −1.23259
\(675\) 0 0
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 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 \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −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 \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) 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 \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) −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 \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 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 \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 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 \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) −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 \(-0.568609\pi\)
−0.213877 + 0.976861i \(0.568609\pi\)
\(788\) 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 \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 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 \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) −18.0000 −0.627060
\(825\) 0 0
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 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 \(-0.124514\pi\)
0.924462 + 0.381273i \(0.124514\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 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 \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) 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 \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −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 \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) −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 \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 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 \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) 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 \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 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 \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 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 \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −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 \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −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 \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) −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 \(-0.100971\pi\)
0.950110 + 0.311916i \(0.100971\pi\)
\(998\) −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.h.1.1 1
5.2 odd 4 845.2.b.b.339.1 2
5.3 odd 4 845.2.b.b.339.2 2
5.4 even 2 4225.2.a.k.1.1 1
13.5 odd 4 325.2.c.e.51.2 2
13.8 odd 4 325.2.c.e.51.1 2
13.12 even 2 4225.2.a.m.1.1 1
65.2 even 12 845.2.l.b.654.2 4
65.3 odd 12 845.2.n.b.529.1 4
65.7 even 12 845.2.l.a.699.1 4
65.8 even 4 65.2.d.a.64.2 yes 2
65.12 odd 4 845.2.b.a.339.2 2
65.17 odd 12 845.2.n.a.484.2 4
65.18 even 4 65.2.d.b.64.2 yes 2
65.22 odd 12 845.2.n.b.484.1 4
65.23 odd 12 845.2.n.a.529.2 4
65.28 even 12 845.2.l.a.654.1 4
65.32 even 12 845.2.l.b.699.1 4
65.33 even 12 845.2.l.b.699.2 4
65.34 odd 4 325.2.c.b.51.2 2
65.37 even 12 845.2.l.a.654.2 4
65.38 odd 4 845.2.b.a.339.1 2
65.42 odd 12 845.2.n.b.529.2 4
65.43 odd 12 845.2.n.a.484.1 4
65.44 odd 4 325.2.c.b.51.1 2
65.47 even 4 65.2.d.b.64.1 yes 2
65.48 odd 12 845.2.n.b.484.2 4
65.57 even 4 65.2.d.a.64.1 2
65.58 even 12 845.2.l.a.699.2 4
65.62 odd 12 845.2.n.a.529.1 4
65.63 even 12 845.2.l.b.654.1 4
65.64 even 2 4225.2.a.e.1.1 1
195.8 odd 4 585.2.h.c.64.1 2
195.47 odd 4 585.2.h.b.64.1 2
195.83 odd 4 585.2.h.b.64.2 2
195.122 odd 4 585.2.h.c.64.2 2
260.47 odd 4 1040.2.f.b.129.2 2
260.83 odd 4 1040.2.f.b.129.1 2
260.187 odd 4 1040.2.f.a.129.2 2
260.203 odd 4 1040.2.f.a.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.d.a.64.1 2 65.57 even 4
65.2.d.a.64.2 yes 2 65.8 even 4
65.2.d.b.64.1 yes 2 65.47 even 4
65.2.d.b.64.2 yes 2 65.18 even 4
325.2.c.b.51.1 2 65.44 odd 4
325.2.c.b.51.2 2 65.34 odd 4
325.2.c.e.51.1 2 13.8 odd 4
325.2.c.e.51.2 2 13.5 odd 4
585.2.h.b.64.1 2 195.47 odd 4
585.2.h.b.64.2 2 195.83 odd 4
585.2.h.c.64.1 2 195.8 odd 4
585.2.h.c.64.2 2 195.122 odd 4
845.2.b.a.339.1 2 65.38 odd 4
845.2.b.a.339.2 2 65.12 odd 4
845.2.b.b.339.1 2 5.2 odd 4
845.2.b.b.339.2 2 5.3 odd 4
845.2.l.a.654.1 4 65.28 even 12
845.2.l.a.654.2 4 65.37 even 12
845.2.l.a.699.1 4 65.7 even 12
845.2.l.a.699.2 4 65.58 even 12
845.2.l.b.654.1 4 65.63 even 12
845.2.l.b.654.2 4 65.2 even 12
845.2.l.b.699.1 4 65.32 even 12
845.2.l.b.699.2 4 65.33 even 12
845.2.n.a.484.1 4 65.43 odd 12
845.2.n.a.484.2 4 65.17 odd 12
845.2.n.a.529.1 4 65.62 odd 12
845.2.n.a.529.2 4 65.23 odd 12
845.2.n.b.484.1 4 65.22 odd 12
845.2.n.b.484.2 4 65.48 odd 12
845.2.n.b.529.1 4 65.3 odd 12
845.2.n.b.529.2 4 65.42 odd 12
1040.2.f.a.129.1 2 260.203 odd 4
1040.2.f.a.129.2 2 260.187 odd 4
1040.2.f.b.129.1 2 260.83 odd 4
1040.2.f.b.129.2 2 260.47 odd 4
4225.2.a.e.1.1 1 65.64 even 2
4225.2.a.h.1.1 1 1.1 even 1 trivial
4225.2.a.k.1.1 1 5.4 even 2
4225.2.a.m.1.1 1 13.12 even 2