Properties

Label 2415.2.a.a.1.1
Level $2415$
Weight $2$
Character 2415.1
Self dual yes
Analytic conductor $19.284$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2838720881\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 2415.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -1.00000 q^{30} -5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +2.00000 q^{37} +6.00000 q^{39} -3.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} -3.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} +10.0000 q^{61} -1.00000 q^{63} +7.00000 q^{64} +6.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} +4.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{77} -6.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -1.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +12.0000 q^{88} -10.0000 q^{89} +1.00000 q^{90} +6.00000 q^{91} +1.00000 q^{92} +5.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) −1.00000 −0.208514
\(24\) −3.00000 −0.612372
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 2.00000 0.280056
\(52\) 6.00000 0.832050
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.00000 −0.539360
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 6.00000 0.744208
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) −6.00000 −0.679366
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 12.0000 1.27920
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 1.00000 0.105409
\(91\) 6.00000 0.628971
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.00000 0.402015
\(100\) −1.00000 −0.100000
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −2.00000 −0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −18.0000 −1.76505
\(105\) −1.00000 −0.0975900
\(106\) −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
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 4.00000 0.381385
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −6.00000 −0.557086
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) 2.00000 0.183340
\(120\) 3.00000 0.273861
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) −6.00000 −0.526235
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) −6.00000 −0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −24.0000 −2.00698
\(144\) −1.00000 −0.0833333
\(145\) −6.00000 −0.498273
\(146\) 2.00000 0.165521
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 5.00000 0.395285
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 3.00000 0.231455
\(169\) 23.0000 1.76923
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) −6.00000 −0.444750
\(183\) −10.0000 −0.739221
\(184\) −3.00000 −0.221163
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 2.00000 0.143592
\(195\) −6.00000 −0.429669
\(196\) −1.00000 −0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −4.00000 −0.284268
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 3.00000 0.212132
\(201\) −4.00000 −0.282138
\(202\) 14.0000 0.985037
\(203\) −6.00000 −0.421117
\(204\) −2.00000 −0.140028
\(205\) −6.00000 −0.419058
\(206\) 8.00000 0.557386
\(207\) −1.00000 −0.0695048
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) −4.00000 −0.274075
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) 2.00000 0.134231
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 5.00000 0.334077
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 4.00000 0.263181
\(232\) 18.0000 1.18176
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 4.00000 0.258738 0.129369 0.991596i \(-0.458705\pi\)
0.129369 + 0.991596i \(0.458705\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) −4.00000 −0.250982
\(255\) −2.00000 −0.125245
\(256\) −17.0000 −1.06250
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 4.00000 0.249029
\(259\) −2.00000 −0.124274
\(260\) −6.00000 −0.372104
\(261\) 6.00000 0.371391
\(262\) −4.00000 −0.247121
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −12.0000 −0.738549
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −4.00000 −0.244339
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 2.00000 0.121268
\(273\) −6.00000 −0.363137
\(274\) 6.00000 0.362473
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) −6.00000 −0.354169
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) 2.00000 0.117242
\(292\) 2.00000 0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) 14.0000 0.810998
\(299\) 6.00000 0.346989
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 4.00000 0.227921
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 18.0000 1.01905
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −2.00000 −0.112867
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 6.00000 0.336463
\(319\) 24.0000 1.34374
\(320\) −7.00000 −0.391312
\(321\) 12.0000 0.669775
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) −6.00000 −0.332820
\(326\) 16.0000 0.886158
\(327\) 2.00000 0.110600
\(328\) 18.0000 0.993884
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) −8.00000 −0.437741
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −23.0000 −1.25104
\(339\) 6.00000 0.325875
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 12.0000 0.646997
\(345\) −1.00000 −0.0538382
\(346\) 18.0000 0.967686
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 6.00000 0.321634
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 1.00000 0.0534522
\(351\) 6.00000 0.320256
\(352\) −20.0000 −1.06600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) −4.00000 −0.212298
\(356\) 10.0000 0.529999
\(357\) −2.00000 −0.105851
\(358\) 16.0000 0.845626
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −3.00000 −0.158114
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) −5.00000 −0.262432
\(364\) −6.00000 −0.314485
\(365\) 2.00000 0.104685
\(366\) 10.0000 0.522708
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 2.00000 0.103975
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) −1.00000 −0.0514344
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) −3.00000 −0.153093
\(385\) 4.00000 0.203859
\(386\) 6.00000 0.305392
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 6.00000 0.303822
\(391\) 2.00000 0.101144
\(392\) 3.00000 0.151523
\(393\) −4.00000 −0.201773
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 4.00000 0.199502
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) −1.00000 −0.0496904
\(406\) 6.00000 0.297775
\(407\) 8.00000 0.396545
\(408\) 6.00000 0.297044
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 6.00000 0.296319
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 1.00000 0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −2.00000 −0.0970143
\(426\) 4.00000 0.193801
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 24.0000 1.15873
\(430\) 4.00000 0.192897
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −12.0000 −0.572078
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 2.00000 0.0949158
\(445\) 10.0000 0.474045
\(446\) −24.0000 −1.13643
\(447\) 14.0000 0.662177
\(448\) −7.00000 −0.330719
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −10.0000 −0.467269
\(459\) 2.00000 0.0933520
\(460\) −1.00000 −0.0466252
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) −4.00000 −0.186097
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 6.00000 0.277350
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) −36.0000 −1.65703
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) 6.00000 0.274721
\(478\) −4.00000 −0.182956
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −5.00000 −0.228218
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) −1.00000 −0.0455016
\(484\) −5.00000 −0.227273
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 30.0000 1.35804
\(489\) 16.0000 0.723545
\(490\) 1.00000 0.0451754
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) 6.00000 0.270501
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) −28.0000 −1.24970
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −3.00000 −0.133631
\(505\) 14.0000 0.622992
\(506\) 4.00000 0.177822
\(507\) −23.0000 −1.02147
\(508\) −4.00000 −0.177471
\(509\) 42.0000 1.86162 0.930809 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 2.00000 0.0885615
\(511\) 2.00000 0.0884748
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) 8.00000 0.352522
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 18.0000 0.790112
\(520\) 18.0000 0.789352
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −6.00000 −0.262613
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −4.00000 −0.174741
\(525\) 1.00000 0.0436436
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) −10.0000 −0.432742
\(535\) 12.0000 0.518805
\(536\) 12.0000 0.518321
\(537\) 16.0000 0.690451
\(538\) −2.00000 −0.0862261
\(539\) 4.00000 0.172292
\(540\) −1.00000 −0.0430331
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 24.0000 1.03089
\(543\) −10.0000 −0.429141
\(544\) 10.0000 0.428746
\(545\) 2.00000 0.0856706
\(546\) 6.00000 0.256776
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 6.00000 0.256307
\(549\) 10.0000 0.426790
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 2.00000 0.0848953
\(556\) 20.0000 0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) −1.00000 −0.0422577
\(561\) 8.00000 0.337760
\(562\) −6.00000 −0.253095
\(563\) −16.0000 −0.674320 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 4.00000 0.168133
\(567\) −1.00000 −0.0419961
\(568\) 12.0000 0.503509
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 24.0000 1.00349
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) −1.00000 −0.0417029
\(576\) 7.00000 0.291667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 13.0000 0.540729
\(579\) 6.00000 0.249351
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) 6.00000 0.248069
\(586\) 6.00000 0.247858
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) −6.00000 −0.246807
\(592\) −2.00000 −0.0821995
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 4.00000 0.164122
\(595\) −2.00000 −0.0819920
\(596\) 14.0000 0.573462
\(597\) 4.00000 0.163709
\(598\) −6.00000 −0.245358
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) −3.00000 −0.122474
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) −14.0000 −0.568711
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 42.0000 1.69636 0.848182 0.529705i \(-0.177697\pi\)
0.848182 + 0.529705i \(0.177697\pi\)
\(614\) −20.0000 −0.807134
\(615\) 6.00000 0.241943
\(616\) −12.0000 −0.483494
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 8.00000 0.320771
\(623\) 10.0000 0.400642
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −4.00000 −0.159490
\(630\) −1.00000 −0.0398410
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 2.00000 0.0794301
\(635\) −4.00000 −0.158735
\(636\) 6.00000 0.237915
\(637\) −6.00000 −0.237729
\(638\) −24.0000 −0.950169
\(639\) 4.00000 0.158238
\(640\) −3.00000 −0.118585
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) −12.0000 −0.473602
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 3.00000 0.117851
\(649\) −48.0000 −1.88416
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −4.00000 −0.156293
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −4.00000 −0.155700
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 12.0000 0.466393
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −6.00000 −0.232321
\(668\) −8.00000 −0.309529
\(669\) −24.0000 −0.927894
\(670\) 4.00000 0.154533
\(671\) 40.0000 1.54418
\(672\) −5.00000 −0.192879
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 10.0000 0.385186
\(675\) −1.00000 −0.0384900
\(676\) −23.0000 −0.884615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) 2.00000 0.0767530
\(680\) 6.00000 0.230089
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 1.00000 0.0381802
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) −36.0000 −1.37149
\(690\) 1.00000 0.0380693
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 18.0000 0.684257
\(693\) −4.00000 −0.151947
\(694\) −12.0000 −0.455514
\(695\) 20.0000 0.758643
\(696\) −18.0000 −0.682288
\(697\) −12.0000 −0.454532
\(698\) 2.00000 0.0757011
\(699\) 6.00000 0.226941
\(700\) 1.00000 0.0377964
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) −6.00000 −0.226455
\(703\) 0 0
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 14.0000 0.526524
\(708\) −12.0000 −0.450988
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 4.00000 0.150117
\(711\) 0 0
\(712\) −30.0000 −1.12430
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 24.0000 0.897549
\(716\) 16.0000 0.597948
\(717\) −4.00000 −0.149383
\(718\) 16.0000 0.597115
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) 19.0000 0.707107
\(723\) −14.0000 −0.520666
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 5.00000 0.185567
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 18.0000 0.667124
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) −8.00000 −0.295891
\(732\) 10.0000 0.369611
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) −16.0000 −0.590571
\(735\) 1.00000 0.0368856
\(736\) 5.00000 0.184302
\(737\) 16.0000 0.589368
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) −4.00000 −0.145287
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 4.00000 0.144905
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) −28.0000 −1.01168
\(767\) 72.0000 2.59977
\(768\) 17.0000 0.613435
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) −4.00000 −0.144150
\(771\) 22.0000 0.792311
\(772\) 6.00000 0.215945
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 2.00000 0.0717496
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 6.00000 0.214834
\(781\) 16.0000 0.572525
\(782\) −2.00000 −0.0715199
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) −2.00000 −0.0713831
\(786\) 4.00000 0.142675
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −6.00000 −0.213741
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 12.0000 0.426401
\(793\) −60.0000 −2.13066
\(794\) 22.0000 0.780751
\(795\) 6.00000 0.212798
\(796\) 4.00000 0.141776
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) −10.0000 −0.353333
\(802\) 18.0000 0.635602
\(803\) −8.00000 −0.282314
\(804\) 4.00000 0.141069
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) −2.00000 −0.0704033
\(808\) −42.0000 −1.47755
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 6.00000 0.210559
\(813\) 24.0000 0.841717
\(814\) −8.00000 −0.280400
\(815\) 16.0000 0.560456
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 6.00000 0.209785
\(819\) 6.00000 0.209657
\(820\) 6.00000 0.209529
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) −6.00000 −0.209274
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) −24.0000 −0.836080
\(825\) −4.00000 −0.139262
\(826\) −12.0000 −0.417533
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 1.00000 0.0347524
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) −42.0000 −1.45609
\(833\) −2.00000 −0.0692959
\(834\) −20.0000 −0.692543
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 36.0000 1.24360
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) −3.00000 −0.103510
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −6.00000 −0.206651
\(844\) −4.00000 −0.137686
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −5.00000 −0.171802
\(848\) −6.00000 −0.206041
\(849\) 4.00000 0.137280
\(850\) 2.00000 0.0685994
\(851\) −2.00000 −0.0685591
\(852\) 4.00000 0.137038
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −24.0000 −0.819346
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 4.00000 0.136399
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 5.00000 0.170103
\(865\) 18.0000 0.612018
\(866\) 2.00000 0.0679628
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) −6.00000 −0.203419
\(871\) −24.0000 −0.813209
\(872\) −6.00000 −0.203186
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) −24.0000 −0.809961
\(879\) 6.00000 0.202375
\(880\) 4.00000 0.134840
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) −12.0000 −0.403604
\(885\) −12.0000 −0.403376
\(886\) 36.0000 1.20944
\(887\) 56.0000 1.88030 0.940148 0.340766i \(-0.110687\pi\)
0.940148 + 0.340766i \(0.110687\pi\)
\(888\) −6.00000 −0.201347
\(889\) −4.00000 −0.134156
\(890\) −10.0000 −0.335201
\(891\) 4.00000 0.134005
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 16.0000 0.534821
\(896\) −3.00000 −0.100223
\(897\) −6.00000 −0.200334
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) −24.0000 −0.799113
\(903\) 4.00000 0.133112
\(904\) −18.0000 −0.598671
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 24.0000 0.796468
\(909\) −14.0000 −0.464351
\(910\) 6.00000 0.198898
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 18.0000 0.595387
\(915\) 10.0000 0.330590
\(916\) −10.0000 −0.330409
\(917\) −4.00000 −0.132092
\(918\) −2.00000 −0.0660098
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 3.00000 0.0989071
\(921\) −20.0000 −0.659022
\(922\) −26.0000 −0.856264
\(923\) −24.0000 −0.789970
\(924\) −4.00000 −0.131590
\(925\) 2.00000 0.0657596
\(926\) 4.00000 0.131448
\(927\) −8.00000 −0.262754
\(928\) −30.0000 −0.984798
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 8.00000 0.261908
\(934\) −24.0000 −0.785304
\(935\) 8.00000 0.261628
\(936\) −18.0000 −0.588348
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 4.00000 0.130605
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 2.00000 0.0651635
\(943\) −6.00000 −0.195387
\(944\) 12.0000 0.390567
\(945\) −1.00000 −0.0325300
\(946\) −16.0000 −0.520205
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 6.00000 0.194461
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −4.00000 −0.129369
\(957\) −24.0000 −0.775810
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 7.00000 0.225924
\(961\) −31.0000 −1.00000
\(962\) 12.0000 0.386896
\(963\) −12.0000 −0.386695
\(964\) −14.0000 −0.450910
\(965\) 6.00000 0.193147
\(966\) 1.00000 0.0321745
\(967\) 20.0000 0.643157 0.321578 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) 20.0000 0.640841
\(975\) 6.00000 0.192154
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −16.0000 −0.511624
\(979\) −40.0000 −1.27841
\(980\) 1.00000 0.0319438
\(981\) −2.00000 −0.0638551
\(982\) −40.0000 −1.27645
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −18.0000 −0.573819
\(985\) −6.00000 −0.191176
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 4.00000 0.127128
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 4.00000 0.126872
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −28.0000 −0.886325
\(999\) −2.00000 −0.0632772
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 2415.2.a.a.1.1 1
3.2 odd 2 7245.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.a.1.1 1 1.1 even 1 trivial
7245.2.a.n.1.1 1 3.2 odd 2