Properties

Label 7350.2.a.bk.1.1
Level 7350
Weight 2
Character 7350.1
Self dual yes
Analytic conductor 58.690
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.6900454856\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7350.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.00000 q^{19} -2.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} +6.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +4.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{57} -6.00000 q^{58} +8.00000 q^{59} +10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} +8.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +4.00000 q^{74} +6.00000 q^{76} -6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} -8.00000 q^{83} +4.00000 q^{86} +6.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} +8.00000 q^{92} +2.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −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
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −6.00000 −0.973329
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) −6.00000 −0.787839
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 4.00000 0.485071
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −2.00000 −0.213201
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 2.00000 0.207390
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −4.00000 −0.396059
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 6.00000 0.554700
\(118\) −8.00000 −0.736460
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −10.0000 −0.905357
\(123\) −2.00000 −0.180334
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −8.00000 −0.681005
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 6.00000 0.503509
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −6.00000 −0.486664
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 12.0000 0.954669
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −4.00000 −0.304997
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 8.00000 0.601317
\(178\) −10.0000 −0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −2.00000 −0.142134
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 8.00000 0.556038
\(208\) 6.00000 0.416025
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −6.00000 −0.412082
\(213\) −6.00000 −0.411113
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 4.00000 0.268462
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 6.00000 0.397360
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) −12.0000 −0.779484
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 36.0000 2.29063
\(248\) −2.00000 −0.127000
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −8.00000 −0.488678
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) −14.0000 −0.839664
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −8.00000 −0.476393
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −14.0000 −0.819288
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 2.00000 0.116052
\(298\) −10.0000 −0.579284
\(299\) 48.0000 2.77591
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −6.00000 −0.339683
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 6.00000 0.336463
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −14.0000 −0.774202
\(328\) 2.00000 0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −8.00000 −0.439057
\(333\) −4.00000 −0.219199
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −23.0000 −1.25104
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −8.00000 −0.430083
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 6.00000 0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) −2.00000 −0.106600
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −10.0000 −0.528516
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 8.00000 0.417029
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 18.0000 0.920960
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 6.00000 0.300753
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 8.00000 0.399004
\(403\) 12.0000 0.597763
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −4.00000 −0.198030
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 14.0000 0.685583
\(418\) −12.0000 −0.586939
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 48.0000 2.29615
\(438\) 14.0000 0.668946
\(439\) 18.0000 0.859093 0.429547 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 14.0000 0.654177
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −8.00000 −0.368230
\(473\) −8.00000 −0.367840
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −26.0000 −1.18921
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 24.0000 1.08091
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 8.00000 0.358489
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 23.0000 1.02147
\(508\) −4.00000 −0.177471
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) 0 0
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 0 0
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −6.00000 −0.262613
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 10.0000 0.431532
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 10.0000 0.429537
\(543\) −2.00000 −0.0858282
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 6.00000 0.256307
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −2.00000 −0.0846668
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −30.0000 −1.26547
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 6.00000 0.251754
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 12.0000 0.501745
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 1.00000 0.0415945
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −12.0000 −0.496989
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) −4.00000 −0.164399
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −6.00000 −0.245564
\(598\) −48.0000 −1.96287
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 4.00000 0.161690
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −8.00000 −0.321807
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 12.0000 0.479234
\(628\) −22.0000 −0.877896
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 12.0000 0.473602
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 8.00000 0.310929
\(663\) 24.0000 0.932083
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 48.0000 1.85857
\(668\) −12.0000 −0.464294
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 40.0000 1.53732 0.768662 0.639655i \(-0.220923\pi\)
0.768662 + 0.639655i \(0.220923\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −4.00000 −0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −4.00000 −0.152499
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 8.00000 0.304114
\(693\) 0 0
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −8.00000 −0.303022
\(698\) −26.0000 −0.984115
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −6.00000 −0.226455
\(703\) −24.0000 −0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −10.0000 −0.374766
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 26.0000 0.970988
\(718\) −6.00000 −0.223918
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 26.0000 0.966950
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 10.0000 0.369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −16.0000 −0.589368
\(738\) 2.00000 0.0736210
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 36.0000 1.32249
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) −8.00000 −0.292705
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) −20.0000 −0.726433
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) −32.0000 −1.14432
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 60.0000 2.13066
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 14.0000 0.494357
\(803\) −28.0000 −0.988099
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −18.0000 −0.633630
\(808\) 10.0000 0.351799
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) −24.0000 −0.839654
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) −6.00000 −0.209274
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 8.00000 0.278019
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 2.00000 0.0691301
\(838\) 20.0000 0.690889
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −34.0000 −1.17172
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) −6.00000 −0.205557
\(853\) 38.0000 1.30110 0.650548 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) −12.0000 −0.409673
\(859\) 10.0000 0.341196 0.170598 0.985341i \(-0.445430\pi\)
0.170598 + 0.985341i \(0.445430\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.00000 −0.0681203
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 26.0000 0.883516
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 14.0000 0.474100
\(873\) −10.0000 −0.338449
\(874\) −48.0000 −1.62362
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −18.0000 −0.607471
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −24.0000 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −8.00000 −0.267860
\(893\) 48.0000 1.60626
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 6.00000 0.200223
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −8.00000 −0.265489
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 6.00000 0.198680
\(913\) −16.0000 −0.529523
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) 60.0000 1.97922 0.989609 0.143787i \(-0.0459280\pi\)
0.989609 + 0.143787i \(0.0459280\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −18.0000 −0.592798
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 8.00000 0.262754
\(928\) −6.00000 −0.196960
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −24.0000 −0.785725
\(934\) 24.0000 0.785304
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 22.0000 0.716799
\(943\) −16.0000 −0.521032
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −12.0000 −0.389742
\(949\) −84.0000 −2.72676
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 12.0000 0.387905
\(958\) 8.00000 0.258468
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 24.0000 0.773791
\(963\) −12.0000 −0.386695
\(964\) 26.0000 0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 7.00000 0.224989
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −4.00000 −0.127906
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 30.0000 0.957338
\(983\) −16.0000 −0.510321 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 36.0000 1.14531
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 12.0000 0.379853
\(999\) −4.00000 −0.126554
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 7350.2.a.bk.1.1 1
5.2 odd 4 1470.2.g.b.589.1 2
5.3 odd 4 1470.2.g.b.589.2 2
5.4 even 2 7350.2.a.bz.1.1 1
7.6 odd 2 1050.2.a.d.1.1 1
21.20 even 2 3150.2.a.bk.1.1 1
28.27 even 2 8400.2.a.bp.1.1 1
35.2 odd 12 1470.2.n.f.949.1 4
35.3 even 12 1470.2.n.b.79.1 4
35.12 even 12 1470.2.n.b.949.1 4
35.13 even 4 210.2.g.b.169.2 yes 2
35.17 even 12 1470.2.n.b.79.2 4
35.18 odd 12 1470.2.n.f.79.1 4
35.23 odd 12 1470.2.n.f.949.2 4
35.27 even 4 210.2.g.b.169.1 2
35.32 odd 12 1470.2.n.f.79.2 4
35.33 even 12 1470.2.n.b.949.2 4
35.34 odd 2 1050.2.a.p.1.1 1
105.62 odd 4 630.2.g.c.379.2 2
105.83 odd 4 630.2.g.c.379.1 2
105.104 even 2 3150.2.a.d.1.1 1
140.27 odd 4 1680.2.t.e.1009.1 2
140.83 odd 4 1680.2.t.e.1009.2 2
140.139 even 2 8400.2.a.w.1.1 1
420.83 even 4 5040.2.t.h.1009.2 2
420.167 even 4 5040.2.t.h.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.b.169.1 2 35.27 even 4
210.2.g.b.169.2 yes 2 35.13 even 4
630.2.g.c.379.1 2 105.83 odd 4
630.2.g.c.379.2 2 105.62 odd 4
1050.2.a.d.1.1 1 7.6 odd 2
1050.2.a.p.1.1 1 35.34 odd 2
1470.2.g.b.589.1 2 5.2 odd 4
1470.2.g.b.589.2 2 5.3 odd 4
1470.2.n.b.79.1 4 35.3 even 12
1470.2.n.b.79.2 4 35.17 even 12
1470.2.n.b.949.1 4 35.12 even 12
1470.2.n.b.949.2 4 35.33 even 12
1470.2.n.f.79.1 4 35.18 odd 12
1470.2.n.f.79.2 4 35.32 odd 12
1470.2.n.f.949.1 4 35.2 odd 12
1470.2.n.f.949.2 4 35.23 odd 12
1680.2.t.e.1009.1 2 140.27 odd 4
1680.2.t.e.1009.2 2 140.83 odd 4
3150.2.a.d.1.1 1 105.104 even 2
3150.2.a.bk.1.1 1 21.20 even 2
5040.2.t.h.1009.1 2 420.167 even 4
5040.2.t.h.1009.2 2 420.83 even 4
7350.2.a.bk.1.1 1 1.1 even 1 trivial
7350.2.a.bz.1.1 1 5.4 even 2
8400.2.a.w.1.1 1 140.139 even 2
8400.2.a.bp.1.1 1 28.27 even 2