Properties

Label 930.2.a.a.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} -3.00000 q^{22} +5.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -3.00000 q^{28} +4.00000 q^{29} -1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +4.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +3.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} +1.00000 q^{43} +3.00000 q^{44} -1.00000 q^{45} -5.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -3.00000 q^{55} +3.00000 q^{56} +3.00000 q^{57} -4.00000 q^{58} +6.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -1.00000 q^{62} -3.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} +2.00000 q^{67} -4.00000 q^{68} -5.00000 q^{69} -3.00000 q^{70} +7.00000 q^{71} -1.00000 q^{72} +5.00000 q^{73} -1.00000 q^{75} -3.00000 q^{76} -9.00000 q^{77} -2.00000 q^{78} -1.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +12.0000 q^{83} +3.00000 q^{84} +4.00000 q^{85} -1.00000 q^{86} -4.00000 q^{87} -3.00000 q^{88} +1.00000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +5.00000 q^{92} -1.00000 q^{93} -10.0000 q^{94} +3.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -2.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.00000 0.801784
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −1.00000 −0.235702
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.00000 0.654654
\(22\) −3.00000 −0.639602
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 4.00000 0.685994
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 3.00000 0.486664
\(39\) 2.00000 0.320256
\(40\) 1.00000 0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) −3.00000 −0.462910
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 3.00000 0.452267
\(45\) −1.00000 −0.149071
\(46\) −5.00000 −0.737210
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.00000 −0.404520
\(56\) 3.00000 0.400892
\(57\) 3.00000 0.397360
\(58\) −4.00000 −0.525226
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −1.00000 −0.127000
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 3.00000 0.369274
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −4.00000 −0.485071
\(69\) −5.00000 −0.601929
\(70\) −3.00000 −0.358569
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) −3.00000 −0.344124
\(77\) −9.00000 −1.02565
\(78\) −2.00000 −0.226455
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 3.00000 0.327327
\(85\) 4.00000 0.433861
\(86\) −1.00000 −0.107833
\(87\) −4.00000 −0.428845
\(88\) −3.00000 −0.319801
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 1.00000 0.105409
\(91\) 6.00000 0.628971
\(92\) 5.00000 0.521286
\(93\) −1.00000 −0.103695
\(94\) −10.0000 −1.03142
\(95\) 3.00000 0.307794
\(96\) 1.00000 0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −2.00000 −0.202031
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) −4.00000 −0.396059
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 2.00000 0.196116
\(105\) −3.00000 −0.292770
\(106\) −3.00000 −0.291386
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) −3.00000 −0.283473
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) −3.00000 −0.280976
\(115\) −5.00000 −0.466252
\(116\) 4.00000 0.371391
\(117\) −2.00000 −0.184900
\(118\) −6.00000 −0.552345
\(119\) 12.0000 1.10004
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) 2.00000 0.181071
\(123\) −4.00000 −0.360668
\(124\) 1.00000 0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 3.00000 0.267261
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) −2.00000 −0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −3.00000 −0.261116
\(133\) 9.00000 0.780399
\(134\) −2.00000 −0.172774
\(135\) 1.00000 0.0860663
\(136\) 4.00000 0.342997
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 5.00000 0.425628
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 3.00000 0.253546
\(141\) −10.0000 −0.842152
\(142\) −7.00000 −0.587427
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) −5.00000 −0.413803
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 3.00000 0.243332
\(153\) −4.00000 −0.323381
\(154\) 9.00000 0.725241
\(155\) −1.00000 −0.0803219
\(156\) 2.00000 0.160128
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 1.00000 0.0795557
\(159\) −3.00000 −0.237915
\(160\) 1.00000 0.0790569
\(161\) −15.0000 −1.18217
\(162\) −1.00000 −0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 4.00000 0.312348
\(165\) 3.00000 0.233550
\(166\) −12.0000 −0.931381
\(167\) −19.0000 −1.47026 −0.735132 0.677924i \(-0.762880\pi\)
−0.735132 + 0.677924i \(0.762880\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) −3.00000 −0.229416
\(172\) 1.00000 0.0762493
\(173\) −22.0000 −1.67263 −0.836315 0.548250i \(-0.815294\pi\)
−0.836315 + 0.548250i \(0.815294\pi\)
\(174\) 4.00000 0.303239
\(175\) −3.00000 −0.226779
\(176\) 3.00000 0.226134
\(177\) −6.00000 −0.450988
\(178\) −1.00000 −0.0749532
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) −6.00000 −0.444750
\(183\) 2.00000 0.147844
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) −12.0000 −0.877527
\(188\) 10.0000 0.729325
\(189\) 3.00000 0.218218
\(190\) −3.00000 −0.217643
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) 2.00000 0.142857
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −3.00000 −0.213201
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) 1.00000 0.0703598
\(203\) −12.0000 −0.842235
\(204\) 4.00000 0.280056
\(205\) −4.00000 −0.279372
\(206\) −16.0000 −1.11477
\(207\) 5.00000 0.347524
\(208\) −2.00000 −0.138675
\(209\) −9.00000 −0.622543
\(210\) 3.00000 0.207020
\(211\) −19.0000 −1.30801 −0.654007 0.756489i \(-0.726913\pi\)
−0.654007 + 0.756489i \(0.726913\pi\)
\(212\) 3.00000 0.206041
\(213\) −7.00000 −0.479632
\(214\) −9.00000 −0.615227
\(215\) −1.00000 −0.0681994
\(216\) 1.00000 0.0680414
\(217\) −3.00000 −0.203653
\(218\) −20.0000 −1.35457
\(219\) −5.00000 −0.337869
\(220\) −3.00000 −0.202260
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −6.00000 −0.401790 −0.200895 0.979613i \(-0.564385\pi\)
−0.200895 + 0.979613i \(0.564385\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) −9.00000 −0.598671
\(227\) −27.0000 −1.79205 −0.896026 0.444001i \(-0.853559\pi\)
−0.896026 + 0.444001i \(0.853559\pi\)
\(228\) 3.00000 0.198680
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 5.00000 0.329690
\(231\) 9.00000 0.592157
\(232\) −4.00000 −0.262613
\(233\) 15.0000 0.982683 0.491341 0.870967i \(-0.336507\pi\)
0.491341 + 0.870967i \(0.336507\pi\)
\(234\) 2.00000 0.130744
\(235\) −10.0000 −0.652328
\(236\) 6.00000 0.390567
\(237\) 1.00000 0.0649570
\(238\) −12.0000 −0.777844
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 1.00000 0.0645497
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 2.00000 0.128565
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −2.00000 −0.127775
\(246\) 4.00000 0.255031
\(247\) 6.00000 0.381771
\(248\) −1.00000 −0.0635001
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −3.00000 −0.188982
\(253\) 15.0000 0.943042
\(254\) −8.00000 −0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −23.0000 −1.43470 −0.717350 0.696713i \(-0.754645\pi\)
−0.717350 + 0.696713i \(0.754645\pi\)
\(258\) 1.00000 0.0622573
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 4.00000 0.247594
\(262\) −6.00000 −0.370681
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 3.00000 0.184637
\(265\) −3.00000 −0.184289
\(266\) −9.00000 −0.551825
\(267\) −1.00000 −0.0611990
\(268\) 2.00000 0.122169
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) −4.00000 −0.242536
\(273\) −6.00000 −0.363137
\(274\) 10.0000 0.604122
\(275\) 3.00000 0.180907
\(276\) −5.00000 −0.300965
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 14.0000 0.839664
\(279\) 1.00000 0.0598684
\(280\) −3.00000 −0.179284
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 10.0000 0.595491
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 7.00000 0.415374
\(285\) −3.00000 −0.177705
\(286\) 6.00000 0.354787
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 4.00000 0.234888
\(291\) 10.0000 0.586210
\(292\) 5.00000 0.292603
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) 2.00000 0.116642
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 11.0000 0.637213
\(299\) −10.0000 −0.578315
\(300\) −1.00000 −0.0577350
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) 1.00000 0.0574485
\(304\) −3.00000 −0.172062
\(305\) 2.00000 0.114520
\(306\) 4.00000 0.228665
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) −9.00000 −0.512823
\(309\) −16.0000 −0.910208
\(310\) 1.00000 0.0567962
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −2.00000 −0.113228
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 5.00000 0.282166
\(315\) 3.00000 0.169031
\(316\) −1.00000 −0.0562544
\(317\) −8.00000 −0.449325 −0.224662 0.974437i \(-0.572128\pi\)
−0.224662 + 0.974437i \(0.572128\pi\)
\(318\) 3.00000 0.168232
\(319\) 12.0000 0.671871
\(320\) −1.00000 −0.0559017
\(321\) −9.00000 −0.502331
\(322\) 15.0000 0.835917
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −14.0000 −0.775388
\(327\) −20.0000 −1.10600
\(328\) −4.00000 −0.220863
\(329\) −30.0000 −1.65395
\(330\) −3.00000 −0.165145
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 19.0000 1.03963
\(335\) −2.00000 −0.109272
\(336\) 3.00000 0.163663
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 9.00000 0.489535
\(339\) −9.00000 −0.488813
\(340\) 4.00000 0.216930
\(341\) 3.00000 0.162459
\(342\) 3.00000 0.162221
\(343\) 15.0000 0.809924
\(344\) −1.00000 −0.0539164
\(345\) 5.00000 0.269191
\(346\) 22.0000 1.18273
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) −4.00000 −0.214423
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 3.00000 0.160357
\(351\) 2.00000 0.106752
\(352\) −3.00000 −0.159901
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 6.00000 0.318896
\(355\) −7.00000 −0.371521
\(356\) 1.00000 0.0529999
\(357\) −12.0000 −0.635107
\(358\) −4.00000 −0.211407
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 1.00000 0.0527046
\(361\) −10.0000 −0.526316
\(362\) −5.00000 −0.262794
\(363\) 2.00000 0.104973
\(364\) 6.00000 0.314485
\(365\) −5.00000 −0.261712
\(366\) −2.00000 −0.104542
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 5.00000 0.260643
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) −1.00000 −0.0518476
\(373\) −9.00000 −0.466002 −0.233001 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) 12.0000 0.620505
\(375\) 1.00000 0.0516398
\(376\) −10.0000 −0.515711
\(377\) −8.00000 −0.412021
\(378\) −3.00000 −0.154303
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 3.00000 0.153897
\(381\) −8.00000 −0.409852
\(382\) −16.0000 −0.818631
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 1.00000 0.0510310
\(385\) 9.00000 0.458682
\(386\) 6.00000 0.305392
\(387\) 1.00000 0.0508329
\(388\) −10.0000 −0.507673
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 2.00000 0.101274
\(391\) −20.0000 −1.01144
\(392\) −2.00000 −0.101015
\(393\) −6.00000 −0.302660
\(394\) 6.00000 0.302276
\(395\) 1.00000 0.0503155
\(396\) 3.00000 0.150756
\(397\) 35.0000 1.75660 0.878300 0.478110i \(-0.158678\pi\)
0.878300 + 0.478110i \(0.158678\pi\)
\(398\) −7.00000 −0.350878
\(399\) −9.00000 −0.450564
\(400\) 1.00000 0.0500000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 2.00000 0.0997509
\(403\) −2.00000 −0.0996271
\(404\) −1.00000 −0.0497519
\(405\) −1.00000 −0.0496904
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) 8.00000 0.395575 0.197787 0.980245i \(-0.436624\pi\)
0.197787 + 0.980245i \(0.436624\pi\)
\(410\) 4.00000 0.197546
\(411\) 10.0000 0.493264
\(412\) 16.0000 0.788263
\(413\) −18.0000 −0.885722
\(414\) −5.00000 −0.245737
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) 14.0000 0.685583
\(418\) 9.00000 0.440204
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −3.00000 −0.146385
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 19.0000 0.924906
\(423\) 10.0000 0.486217
\(424\) −3.00000 −0.145693
\(425\) −4.00000 −0.194029
\(426\) 7.00000 0.339151
\(427\) 6.00000 0.290360
\(428\) 9.00000 0.435031
\(429\) 6.00000 0.289683
\(430\) 1.00000 0.0482243
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 3.00000 0.144005
\(435\) 4.00000 0.191785
\(436\) 20.0000 0.957826
\(437\) −15.0000 −0.717547
\(438\) 5.00000 0.238909
\(439\) 18.0000 0.859093 0.429547 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 3.00000 0.143019
\(441\) 2.00000 0.0952381
\(442\) −8.00000 −0.380521
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) 6.00000 0.284108
\(447\) 11.0000 0.520282
\(448\) −3.00000 −0.141737
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 9.00000 0.423324
\(453\) 0 0
\(454\) 27.0000 1.26717
\(455\) −6.00000 −0.281284
\(456\) −3.00000 −0.140488
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −13.0000 −0.607450
\(459\) 4.00000 0.186704
\(460\) −5.00000 −0.233126
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) −9.00000 −0.418718
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 4.00000 0.185695
\(465\) 1.00000 0.0463739
\(466\) −15.0000 −0.694862
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −6.00000 −0.277054
\(470\) 10.0000 0.461266
\(471\) 5.00000 0.230388
\(472\) −6.00000 −0.276172
\(473\) 3.00000 0.137940
\(474\) −1.00000 −0.0459315
\(475\) −3.00000 −0.137649
\(476\) 12.0000 0.550019
\(477\) 3.00000 0.137361
\(478\) −12.0000 −0.548867
\(479\) 3.00000 0.137073 0.0685367 0.997649i \(-0.478167\pi\)
0.0685367 + 0.997649i \(0.478167\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 18.0000 0.819878
\(483\) 15.0000 0.682524
\(484\) −2.00000 −0.0909091
\(485\) 10.0000 0.454077
\(486\) 1.00000 0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 2.00000 0.0905357
\(489\) −14.0000 −0.633102
\(490\) 2.00000 0.0903508
\(491\) −37.0000 −1.66979 −0.834893 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) −4.00000 −0.180334
\(493\) −16.0000 −0.720604
\(494\) −6.00000 −0.269953
\(495\) −3.00000 −0.134840
\(496\) 1.00000 0.0449013
\(497\) −21.0000 −0.941979
\(498\) 12.0000 0.537733
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 19.0000 0.848857
\(502\) 12.0000 0.535586
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 3.00000 0.133631
\(505\) 1.00000 0.0444994
\(506\) −15.0000 −0.666831
\(507\) 9.00000 0.399704
\(508\) 8.00000 0.354943
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 4.00000 0.177123
\(511\) −15.0000 −0.663561
\(512\) −1.00000 −0.0441942
\(513\) 3.00000 0.132453
\(514\) 23.0000 1.01449
\(515\) −16.0000 −0.705044
\(516\) −1.00000 −0.0440225
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) 22.0000 0.965693
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −4.00000 −0.175075
\(523\) −43.0000 −1.88026 −0.940129 0.340818i \(-0.889296\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 6.00000 0.262111
\(525\) 3.00000 0.130931
\(526\) −24.0000 −1.04645
\(527\) −4.00000 −0.174243
\(528\) −3.00000 −0.130558
\(529\) 2.00000 0.0869565
\(530\) 3.00000 0.130312
\(531\) 6.00000 0.260378
\(532\) 9.00000 0.390199
\(533\) −8.00000 −0.346518
\(534\) 1.00000 0.0432742
\(535\) −9.00000 −0.389104
\(536\) −2.00000 −0.0863868
\(537\) −4.00000 −0.172613
\(538\) −2.00000 −0.0862261
\(539\) 6.00000 0.258438
\(540\) 1.00000 0.0430331
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −13.0000 −0.558398
\(543\) −5.00000 −0.214571
\(544\) 4.00000 0.171499
\(545\) −20.0000 −0.856706
\(546\) 6.00000 0.256776
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −10.0000 −0.427179
\(549\) −2.00000 −0.0853579
\(550\) −3.00000 −0.127920
\(551\) −12.0000 −0.511217
\(552\) 5.00000 0.212814
\(553\) 3.00000 0.127573
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) −1.00000 −0.0423714 −0.0211857 0.999776i \(-0.506744\pi\)
−0.0211857 + 0.999776i \(0.506744\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −2.00000 −0.0845910
\(560\) 3.00000 0.126773
\(561\) 12.0000 0.506640
\(562\) −16.0000 −0.674919
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −10.0000 −0.421076
\(565\) −9.00000 −0.378633
\(566\) 24.0000 1.00880
\(567\) −3.00000 −0.125988
\(568\) −7.00000 −0.293713
\(569\) −23.0000 −0.964210 −0.482105 0.876113i \(-0.660128\pi\)
−0.482105 + 0.876113i \(0.660128\pi\)
\(570\) 3.00000 0.125656
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) −6.00000 −0.250873
\(573\) −16.0000 −0.668410
\(574\) 12.0000 0.500870
\(575\) 5.00000 0.208514
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 1.00000 0.0415945
\(579\) 6.00000 0.249351
\(580\) −4.00000 −0.166091
\(581\) −36.0000 −1.49353
\(582\) −10.0000 −0.414513
\(583\) 9.00000 0.372742
\(584\) −5.00000 −0.206901
\(585\) 2.00000 0.0826898
\(586\) −30.0000 −1.23929
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −3.00000 −0.123613
\(590\) 6.00000 0.247016
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 3.00000 0.123091
\(595\) −12.0000 −0.491952
\(596\) −11.0000 −0.450578
\(597\) −7.00000 −0.286491
\(598\) 10.0000 0.408930
\(599\) −27.0000 −1.10319 −0.551595 0.834112i \(-0.685981\pi\)
−0.551595 + 0.834112i \(0.685981\pi\)
\(600\) 1.00000 0.0408248
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 3.00000 0.122271
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) −1.00000 −0.0406222
\(607\) 29.0000 1.17707 0.588537 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 3.00000 0.121666
\(609\) 12.0000 0.486265
\(610\) −2.00000 −0.0809776
\(611\) −20.0000 −0.809113
\(612\) −4.00000 −0.161690
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −16.0000 −0.645707
\(615\) 4.00000 0.161296
\(616\) 9.00000 0.362620
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 16.0000 0.643614
\(619\) −38.0000 −1.52735 −0.763674 0.645601i \(-0.776607\pi\)
−0.763674 + 0.645601i \(0.776607\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −3.00000 −0.120192
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) 9.00000 0.359425
\(628\) −5.00000 −0.199522
\(629\) 0 0
\(630\) −3.00000 −0.119523
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 1.00000 0.0397779
\(633\) 19.0000 0.755182
\(634\) 8.00000 0.317721
\(635\) −8.00000 −0.317470
\(636\) −3.00000 −0.118958
\(637\) −4.00000 −0.158486
\(638\) −12.0000 −0.475085
\(639\) 7.00000 0.276916
\(640\) 1.00000 0.0395285
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 9.00000 0.355202
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) −15.0000 −0.591083
\(645\) 1.00000 0.0393750
\(646\) −12.0000 −0.472134
\(647\) 3.00000 0.117942 0.0589711 0.998260i \(-0.481218\pi\)
0.0589711 + 0.998260i \(0.481218\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 18.0000 0.706562
\(650\) 2.00000 0.0784465
\(651\) 3.00000 0.117579
\(652\) 14.0000 0.548282
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 20.0000 0.782062
\(655\) −6.00000 −0.234439
\(656\) 4.00000 0.156174
\(657\) 5.00000 0.195069
\(658\) 30.0000 1.16952
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 3.00000 0.116775
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) −12.0000 −0.466393
\(663\) −8.00000 −0.310694
\(664\) −12.0000 −0.465690
\(665\) −9.00000 −0.349005
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) −19.0000 −0.735132
\(669\) 6.00000 0.231973
\(670\) 2.00000 0.0772667
\(671\) −6.00000 −0.231627
\(672\) −3.00000 −0.115728
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 27.0000 1.03769 0.518847 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 9.00000 0.345643
\(679\) 30.0000 1.15129
\(680\) −4.00000 −0.153393
\(681\) 27.0000 1.03464
\(682\) −3.00000 −0.114876
\(683\) −19.0000 −0.727015 −0.363507 0.931591i \(-0.618421\pi\)
−0.363507 + 0.931591i \(0.618421\pi\)
\(684\) −3.00000 −0.114708
\(685\) 10.0000 0.382080
\(686\) −15.0000 −0.572703
\(687\) −13.0000 −0.495981
\(688\) 1.00000 0.0381246
\(689\) −6.00000 −0.228582
\(690\) −5.00000 −0.190347
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) −22.0000 −0.836315
\(693\) −9.00000 −0.341882
\(694\) 32.0000 1.21470
\(695\) 14.0000 0.531050
\(696\) 4.00000 0.151620
\(697\) −16.0000 −0.606043
\(698\) 24.0000 0.908413
\(699\) −15.0000 −0.567352
\(700\) −3.00000 −0.113389
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 10.0000 0.376622
\(706\) −18.0000 −0.677439
\(707\) 3.00000 0.112827
\(708\) −6.00000 −0.225494
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 7.00000 0.262705
\(711\) −1.00000 −0.0375029
\(712\) −1.00000 −0.0374766
\(713\) 5.00000 0.187251
\(714\) 12.0000 0.449089
\(715\) 6.00000 0.224387
\(716\) 4.00000 0.149487
\(717\) −12.0000 −0.448148
\(718\) −15.0000 −0.559795
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −48.0000 −1.78761
\(722\) 10.0000 0.372161
\(723\) 18.0000 0.669427
\(724\) 5.00000 0.185824
\(725\) 4.00000 0.148556
\(726\) −2.00000 −0.0742270
\(727\) 45.0000 1.66896 0.834479 0.551040i \(-0.185769\pi\)
0.834479 + 0.551040i \(0.185769\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 5.00000 0.185058
\(731\) −4.00000 −0.147945
\(732\) 2.00000 0.0739221
\(733\) −18.0000 −0.664845 −0.332423 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 2.00000 0.0737711
\(736\) −5.00000 −0.184302
\(737\) 6.00000 0.221013
\(738\) −4.00000 −0.147242
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 9.00000 0.330400
\(743\) −37.0000 −1.35740 −0.678699 0.734416i \(-0.737456\pi\)
−0.678699 + 0.734416i \(0.737456\pi\)
\(744\) 1.00000 0.0366618
\(745\) 11.0000 0.403009
\(746\) 9.00000 0.329513
\(747\) 12.0000 0.439057
\(748\) −12.0000 −0.438763
\(749\) −27.0000 −0.986559
\(750\) −1.00000 −0.0365148
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 10.0000 0.364662
\(753\) 12.0000 0.437304
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) −15.0000 −0.544825
\(759\) −15.0000 −0.544466
\(760\) −3.00000 −0.108821
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 8.00000 0.289809
\(763\) −60.0000 −2.17215
\(764\) 16.0000 0.578860
\(765\) 4.00000 0.144620
\(766\) −12.0000 −0.433578
\(767\) −12.0000 −0.433295
\(768\) −1.00000 −0.0360844
\(769\) 9.00000 0.324548 0.162274 0.986746i \(-0.448117\pi\)
0.162274 + 0.986746i \(0.448117\pi\)
\(770\) −9.00000 −0.324337
\(771\) 23.0000 0.828325
\(772\) −6.00000 −0.215945
\(773\) −31.0000 −1.11499 −0.557496 0.830179i \(-0.688238\pi\)
−0.557496 + 0.830179i \(0.688238\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 1.00000 0.0359211
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) −12.0000 −0.429945
\(780\) −2.00000 −0.0716115
\(781\) 21.0000 0.751439
\(782\) 20.0000 0.715199
\(783\) −4.00000 −0.142948
\(784\) 2.00000 0.0714286
\(785\) 5.00000 0.178458
\(786\) 6.00000 0.214013
\(787\) −35.0000 −1.24762 −0.623808 0.781578i \(-0.714415\pi\)
−0.623808 + 0.781578i \(0.714415\pi\)
\(788\) −6.00000 −0.213741
\(789\) −24.0000 −0.854423
\(790\) −1.00000 −0.0355784
\(791\) −27.0000 −0.960009
\(792\) −3.00000 −0.106600
\(793\) 4.00000 0.142044
\(794\) −35.0000 −1.24210
\(795\) 3.00000 0.106399
\(796\) 7.00000 0.248108
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 9.00000 0.318597
\(799\) −40.0000 −1.41510
\(800\) −1.00000 −0.0353553
\(801\) 1.00000 0.0353333
\(802\) −25.0000 −0.882781
\(803\) 15.0000 0.529339
\(804\) −2.00000 −0.0705346
\(805\) 15.0000 0.528681
\(806\) 2.00000 0.0704470
\(807\) −2.00000 −0.0704033
\(808\) 1.00000 0.0351799
\(809\) 17.0000 0.597688 0.298844 0.954302i \(-0.403399\pi\)
0.298844 + 0.954302i \(0.403399\pi\)
\(810\) 1.00000 0.0351364
\(811\) −27.0000 −0.948098 −0.474049 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) −12.0000 −0.421117
\(813\) −13.0000 −0.455930
\(814\) 0 0
\(815\) −14.0000 −0.490399
\(816\) 4.00000 0.140028
\(817\) −3.00000 −0.104957
\(818\) −8.00000 −0.279713
\(819\) 6.00000 0.209657
\(820\) −4.00000 −0.139686
\(821\) 40.0000 1.39601 0.698005 0.716093i \(-0.254071\pi\)
0.698005 + 0.716093i \(0.254071\pi\)
\(822\) −10.0000 −0.348790
\(823\) −18.0000 −0.627441 −0.313720 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) −16.0000 −0.557386
\(825\) −3.00000 −0.104447
\(826\) 18.0000 0.626300
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 5.00000 0.173762
\(829\) −35.0000 −1.21560 −0.607800 0.794090i \(-0.707948\pi\)
−0.607800 + 0.794090i \(0.707948\pi\)
\(830\) 12.0000 0.416526
\(831\) −12.0000 −0.416275
\(832\) −2.00000 −0.0693375
\(833\) −8.00000 −0.277184
\(834\) −14.0000 −0.484780
\(835\) 19.0000 0.657522
\(836\) −9.00000 −0.311272
\(837\) −1.00000 −0.0345651
\(838\) −12.0000 −0.414533
\(839\) 51.0000 1.76072 0.880358 0.474310i \(-0.157302\pi\)
0.880358 + 0.474310i \(0.157302\pi\)
\(840\) 3.00000 0.103510
\(841\) −13.0000 −0.448276
\(842\) 22.0000 0.758170
\(843\) −16.0000 −0.551069
\(844\) −19.0000 −0.654007
\(845\) 9.00000 0.309609
\(846\) −10.0000 −0.343807
\(847\) 6.00000 0.206162
\(848\) 3.00000 0.103020
\(849\) 24.0000 0.823678
\(850\) 4.00000 0.137199
\(851\) 0 0
\(852\) −7.00000 −0.239816
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) −6.00000 −0.205316
\(855\) 3.00000 0.102598
\(856\) −9.00000 −0.307614
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −6.00000 −0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 12.0000 0.408959
\(862\) −16.0000 −0.544962
\(863\) 43.0000 1.46374 0.731869 0.681446i \(-0.238649\pi\)
0.731869 + 0.681446i \(0.238649\pi\)
\(864\) 1.00000 0.0340207
\(865\) 22.0000 0.748022
\(866\) 1.00000 0.0339814
\(867\) 1.00000 0.0339618
\(868\) −3.00000 −0.101827
\(869\) −3.00000 −0.101768
\(870\) −4.00000 −0.135613
\(871\) −4.00000 −0.135535
\(872\) −20.0000 −0.677285
\(873\) −10.0000 −0.338449
\(874\) 15.0000 0.507383
\(875\) 3.00000 0.101419
\(876\) −5.00000 −0.168934
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) −18.0000 −0.607471
\(879\) −30.0000 −1.01187
\(880\) −3.00000 −0.101130
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 11.0000 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(884\) 8.00000 0.269069
\(885\) 6.00000 0.201688
\(886\) −15.0000 −0.503935
\(887\) −46.0000 −1.54453 −0.772264 0.635301i \(-0.780876\pi\)
−0.772264 + 0.635301i \(0.780876\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 1.00000 0.0335201
\(891\) 3.00000 0.100504
\(892\) −6.00000 −0.200895
\(893\) −30.0000 −1.00391
\(894\) −11.0000 −0.367895
\(895\) −4.00000 −0.133705
\(896\) 3.00000 0.100223
\(897\) 10.0000 0.333890
\(898\) −2.00000 −0.0667409
\(899\) 4.00000 0.133407
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −12.0000 −0.399556
\(903\) 3.00000 0.0998337
\(904\) −9.00000 −0.299336
\(905\) −5.00000 −0.166206
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −27.0000 −0.896026
\(909\) −1.00000 −0.0331679
\(910\) 6.00000 0.198898
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 3.00000 0.0993399
\(913\) 36.0000 1.19143
\(914\) −10.0000 −0.330771
\(915\) −2.00000 −0.0661180
\(916\) 13.0000 0.429532
\(917\) −18.0000 −0.594412
\(918\) −4.00000 −0.132020
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 5.00000 0.164845
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) −14.0000 −0.460816
\(924\) 9.00000 0.296078
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 16.0000 0.525509
\(928\) −4.00000 −0.131306
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) −1.00000 −0.0327913
\(931\) −6.00000 −0.196642
\(932\) 15.0000 0.491341
\(933\) 0 0
\(934\) −24.0000 −0.785304
\(935\) 12.0000 0.392442
\(936\) 2.00000 0.0653720
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) 6.00000 0.195907
\(939\) −30.0000 −0.979013
\(940\) −10.0000 −0.326164
\(941\) 40.0000 1.30396 0.651981 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) −5.00000 −0.162909
\(943\) 20.0000 0.651290
\(944\) 6.00000 0.195283
\(945\) −3.00000 −0.0975900
\(946\) −3.00000 −0.0975384
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 1.00000 0.0324785
\(949\) −10.0000 −0.324614
\(950\) 3.00000 0.0973329
\(951\) 8.00000 0.259418
\(952\) −12.0000 −0.388922
\(953\) −52.0000 −1.68445 −0.842223 0.539130i \(-0.818753\pi\)
−0.842223 + 0.539130i \(0.818753\pi\)
\(954\) −3.00000 −0.0971286
\(955\) −16.0000 −0.517748
\(956\) 12.0000 0.388108
\(957\) −12.0000 −0.387905
\(958\) −3.00000 −0.0969256
\(959\) 30.0000 0.968751
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 9.00000 0.290021
\(964\) −18.0000 −0.579741
\(965\) 6.00000 0.193147
\(966\) −15.0000 −0.482617
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 2.00000 0.0642824
\(969\) −12.0000 −0.385496
\(970\) −10.0000 −0.321081
\(971\) −54.0000 −1.73294 −0.866471 0.499227i \(-0.833617\pi\)
−0.866471 + 0.499227i \(0.833617\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 42.0000 1.34646
\(974\) −32.0000 −1.02535
\(975\) 2.00000 0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 14.0000 0.447671
\(979\) 3.00000 0.0958804
\(980\) −2.00000 −0.0638877
\(981\) 20.0000 0.638551
\(982\) 37.0000 1.18072
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 4.00000 0.127515
\(985\) 6.00000 0.191176
\(986\) 16.0000 0.509544
\(987\) 30.0000 0.954911
\(988\) 6.00000 0.190885
\(989\) 5.00000 0.158991
\(990\) 3.00000 0.0953463
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −12.0000 −0.380808
\(994\) 21.0000 0.666080
\(995\) −7.00000 −0.221915
\(996\) −12.0000 −0.380235
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 930.2.a.a.1.1 1
3.2 odd 2 2790.2.a.y.1.1 1
4.3 odd 2 7440.2.a.v.1.1 1
5.2 odd 4 4650.2.d.y.3349.1 2
5.3 odd 4 4650.2.d.y.3349.2 2
5.4 even 2 4650.2.a.bv.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.a.1.1 1 1.1 even 1 trivial
2790.2.a.y.1.1 1 3.2 odd 2
4650.2.a.bv.1.1 1 5.4 even 2
4650.2.d.y.3349.1 2 5.2 odd 4
4650.2.d.y.3349.2 2 5.3 odd 4
7440.2.a.v.1.1 1 4.3 odd 2