Properties

Label 2415.2.a.d.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -2.00000 q^{31} -5.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} +4.00000 q^{37} -4.00000 q^{39} +3.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -1.00000 q^{54} -2.00000 q^{55} +3.00000 q^{56} +6.00000 q^{58} +2.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -4.00000 q^{65} +2.00000 q^{66} -12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} +3.00000 q^{72} +8.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} -2.00000 q^{77} +4.00000 q^{78} +10.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} -1.00000 q^{84} -2.00000 q^{85} +8.00000 q^{86} -6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -1.00000 q^{92} -2.00000 q^{93} +8.00000 q^{94} -5.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 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
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −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
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.00000 −0.348155
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 3.00000 0.474342
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −1.00000 −0.154303
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 4.00000 0.554700
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) −1.00000 −0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 2.00000 0.246183
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 4.00000 0.452911
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.00000 −0.216930
\(86\) 8.00000 0.862662
\(87\) −6.00000 −0.643268
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 −0.419314
\(92\) −1.00000 −0.104257
\(93\) −2.00000 −0.207390
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) −1.00000 −0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −12.0000 −1.17670
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 2.00000 0.190693
\(111\) 4.00000 0.379663
\(112\) −1.00000 −0.0944911
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 6.00000 0.557086
\(117\) −4.00000 −0.369800
\(118\) −2.00000 −0.184115
\(119\) −2.00000 −0.183340
\(120\) 3.00000 0.273861
\(121\) −7.00000 −0.636364
\(122\) 14.0000 1.26750
\(123\) −2.00000 −0.180334
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) −8.00000 −0.704361
\(130\) 4.00000 0.350823
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 1.00000 0.0860663
\(136\) −6.00000 −0.514496
\(137\) −20.0000 −1.70872 −0.854358 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) −1.00000 −0.0833333
\(145\) −6.00000 −0.498273
\(146\) −8.00000 −0.662085
\(147\) 1.00000 0.0824786
\(148\) −4.00000 −0.328798
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 2.00000 0.161165
\(155\) −2.00000 −0.160644
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) −5.00000 −0.395285
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) −2.00000 −0.155700
\(166\) −12.0000 −0.931381
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 3.00000 0.231455
\(169\) 3.00000 0.230769
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) 2.00000 0.150756
\(177\) 2.00000 0.150329
\(178\) −6.00000 −0.449719
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 4.00000 0.296500
\(183\) −14.0000 −1.03491
\(184\) 3.00000 0.221163
\(185\) 4.00000 0.294086
\(186\) 2.00000 0.146647
\(187\) 4.00000 0.292509
\(188\) 8.00000 0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 7.00000 0.505181
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) −4.00000 −0.286446
\(196\) −1.00000 −0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 3.00000 0.212132
\(201\) −12.0000 −0.846415
\(202\) −18.0000 −1.26648
\(203\) −6.00000 −0.421117
\(204\) 2.00000 0.140028
\(205\) −2.00000 −0.139686
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −8.00000 −0.545595
\(216\) 3.00000 0.204124
\(217\) −2.00000 −0.135769
\(218\) −2.00000 −0.135457
\(219\) 8.00000 0.540590
\(220\) 2.00000 0.134840
\(221\) 8.00000 0.538138
\(222\) −4.00000 −0.268462
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −5.00000 −0.334077
\(225\) 1.00000 0.0666667
\(226\) 8.00000 0.532152
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −2.00000 −0.131590
\(232\) −18.0000 −1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) −8.00000 −0.521862
\(236\) −2.00000 −0.130189
\(237\) 10.0000 0.649570
\(238\) 2.00000 0.129641
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 1.00000 0.0638877
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) 16.0000 1.00393
\(255\) −2.00000 −0.125245
\(256\) −17.0000 −1.06250
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 8.00000 0.498058
\(259\) 4.00000 0.248548
\(260\) 4.00000 0.248069
\(261\) −6.00000 −0.371391
\(262\) 6.00000 0.370681
\(263\) 20.0000 1.23325 0.616626 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 2.00000 0.121268
\(273\) −4.00000 −0.242091
\(274\) 20.0000 1.20824
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 10.0000 0.599760
\(279\) −2.00000 −0.119737
\(280\) 3.00000 0.179284
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −2.00000 −0.118056
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 6.00000 0.352332
\(291\) −14.0000 −0.820695
\(292\) −8.00000 −0.468165
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 2.00000 0.116445
\(296\) 12.0000 0.697486
\(297\) −2.00000 −0.116052
\(298\) −2.00000 −0.115857
\(299\) −4.00000 −0.231326
\(300\) −1.00000 −0.0577350
\(301\) −8.00000 −0.461112
\(302\) 20.0000 1.15087
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 2.00000 0.114332
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 2.00000 0.113592
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) −12.0000 −0.679366
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −10.0000 −0.564333
\(315\) 1.00000 0.0563436
\(316\) −10.0000 −0.562544
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 7.00000 0.391312
\(321\) 4.00000 0.223258
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) −4.00000 −0.221880
\(326\) 20.0000 1.10770
\(327\) 2.00000 0.110600
\(328\) −6.00000 −0.331295
\(329\) −8.00000 −0.441054
\(330\) 2.00000 0.110096
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −12.0000 −0.658586
\(333\) 4.00000 0.219199
\(334\) 16.0000 0.875481
\(335\) −12.0000 −0.655630
\(336\) −1.00000 −0.0545545
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −3.00000 −0.163178
\(339\) −8.00000 −0.434500
\(340\) 2.00000 0.108465
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −24.0000 −1.29399
\(345\) 1.00000 0.0538382
\(346\) 4.00000 0.215041
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 6.00000 0.321634
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −4.00000 −0.213504
\(352\) 10.0000 0.533002
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) −2.00000 −0.106299
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −2.00000 −0.105851
\(358\) −4.00000 −0.211407
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 3.00000 0.158114
\(361\) −19.0000 −1.00000
\(362\) 26.0000 1.36653
\(363\) −7.00000 −0.367405
\(364\) 4.00000 0.209657
\(365\) 8.00000 0.418739
\(366\) 14.0000 0.731792
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) 28.0000 1.44979 0.724893 0.688862i \(-0.241889\pi\)
0.724893 + 0.688862i \(0.241889\pi\)
\(374\) −4.00000 −0.206835
\(375\) 1.00000 0.0516398
\(376\) −24.0000 −1.23771
\(377\) 24.0000 1.23606
\(378\) −1.00000 −0.0514344
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 18.0000 0.920960
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 3.00000 0.153093
\(385\) −2.00000 −0.101929
\(386\) −22.0000 −1.11977
\(387\) −8.00000 −0.406663
\(388\) 14.0000 0.710742
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 4.00000 0.202548
\(391\) −2.00000 −0.101144
\(392\) 3.00000 0.151523
\(393\) −6.00000 −0.302660
\(394\) −18.0000 −0.906827
\(395\) 10.0000 0.503155
\(396\) 2.00000 0.100504
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 12.0000 0.598506
\(403\) 8.00000 0.398508
\(404\) −18.0000 −0.895533
\(405\) 1.00000 0.0496904
\(406\) 6.00000 0.297775
\(407\) −8.00000 −0.396545
\(408\) −6.00000 −0.297044
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 2.00000 0.0987730
\(411\) −20.0000 −0.986527
\(412\) −8.00000 −0.394132
\(413\) 2.00000 0.0984136
\(414\) −1.00000 −0.0491473
\(415\) 12.0000 0.589057
\(416\) 20.0000 0.980581
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −8.00000 −0.389434
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −14.0000 −0.677507
\(428\) −4.00000 −0.193347
\(429\) 8.00000 0.386244
\(430\) 8.00000 0.385794
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 2.00000 0.0960031
\(435\) −6.00000 −0.287678
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −8.00000 −0.382255
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) −6.00000 −0.286039
\(441\) 1.00000 0.0476190
\(442\) −8.00000 −0.380521
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −4.00000 −0.189832
\(445\) 6.00000 0.284427
\(446\) −16.0000 −0.757622
\(447\) 2.00000 0.0945968
\(448\) 7.00000 0.330719
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 4.00000 0.188353
\(452\) 8.00000 0.376288
\(453\) −20.0000 −0.939682
\(454\) −4.00000 −0.187729
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −20.0000 −0.935561 −0.467780 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) −1.00000 −0.0466252
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 2.00000 0.0930484
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 6.00000 0.278543
\(465\) −2.00000 −0.0927478
\(466\) −6.00000 −0.277945
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 4.00000 0.184900
\(469\) −12.0000 −0.554109
\(470\) 8.00000 0.369012
\(471\) 10.0000 0.460776
\(472\) 6.00000 0.276172
\(473\) 16.0000 0.735681
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) −5.00000 −0.228218
\(481\) −16.0000 −0.729537
\(482\) 6.00000 0.273293
\(483\) 1.00000 0.0455016
\(484\) 7.00000 0.318182
\(485\) −14.0000 −0.635707
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −42.0000 −1.90125
\(489\) −20.0000 −0.904431
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −16.0000 −0.714827
\(502\) −16.0000 −0.714115
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 3.00000 0.133631
\(505\) 18.0000 0.800989
\(506\) 2.00000 0.0889108
\(507\) 3.00000 0.133235
\(508\) 16.0000 0.709885
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 2.00000 0.0885615
\(511\) 8.00000 0.353899
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 8.00000 0.352180
\(517\) 16.0000 0.703679
\(518\) −4.00000 −0.175750
\(519\) −4.00000 −0.175581
\(520\) −12.0000 −0.526235
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 6.00000 0.262613
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 6.00000 0.262111
\(525\) 1.00000 0.0436436
\(526\) −20.0000 −0.872041
\(527\) 4.00000 0.174243
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.00000 0.0867926
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) −6.00000 −0.259645
\(535\) 4.00000 0.172935
\(536\) −36.0000 −1.55496
\(537\) 4.00000 0.172613
\(538\) −10.0000 −0.431131
\(539\) −2.00000 −0.0861461
\(540\) −1.00000 −0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 10.0000 0.429537
\(543\) −26.0000 −1.11577
\(544\) 10.0000 0.428746
\(545\) 2.00000 0.0856706
\(546\) 4.00000 0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 20.0000 0.854358
\(549\) −14.0000 −0.597505
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 3.00000 0.127688
\(553\) 10.0000 0.425243
\(554\) −10.0000 −0.424859
\(555\) 4.00000 0.169791
\(556\) 10.0000 0.424094
\(557\) 40.0000 1.69485 0.847427 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 2.00000 0.0846668
\(559\) 32.0000 1.35346
\(560\) −1.00000 −0.0422577
\(561\) 4.00000 0.168880
\(562\) 18.0000 0.759284
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 8.00000 0.336861
\(565\) −8.00000 −0.336563
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) −8.00000 −0.334497
\(573\) −18.0000 −0.751961
\(574\) 2.00000 0.0834784
\(575\) 1.00000 0.0417029
\(576\) 7.00000 0.291667
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 13.0000 0.540729
\(579\) 22.0000 0.914289
\(580\) 6.00000 0.249136
\(581\) 12.0000 0.497844
\(582\) 14.0000 0.580319
\(583\) 0 0
\(584\) 24.0000 0.993127
\(585\) −4.00000 −0.165380
\(586\) 10.0000 0.413096
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 0 0
\(590\) −2.00000 −0.0823387
\(591\) 18.0000 0.740421
\(592\) −4.00000 −0.164399
\(593\) 44.0000 1.80686 0.903432 0.428732i \(-0.141040\pi\)
0.903432 + 0.428732i \(0.141040\pi\)
\(594\) 2.00000 0.0820610
\(595\) −2.00000 −0.0819920
\(596\) −2.00000 −0.0819232
\(597\) −20.0000 −0.818546
\(598\) 4.00000 0.163572
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 3.00000 0.122474
\(601\) 6.00000 0.244745 0.122373 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 8.00000 0.326056
\(603\) −12.0000 −0.488678
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) −18.0000 −0.731200
\(607\) 36.0000 1.46119 0.730597 0.682808i \(-0.239242\pi\)
0.730597 + 0.682808i \(0.239242\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 14.0000 0.566843
\(611\) 32.0000 1.29458
\(612\) 2.00000 0.0808452
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 20.0000 0.807134
\(615\) −2.00000 −0.0806478
\(616\) −6.00000 −0.241747
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) −8.00000 −0.321807
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 2.00000 0.0803219
\(621\) 1.00000 0.0401286
\(622\) 6.00000 0.240578
\(623\) 6.00000 0.240385
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −8.00000 −0.318981
\(630\) −1.00000 −0.0398410
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 30.0000 1.19334
\(633\) 8.00000 0.317971
\(634\) −10.0000 −0.397151
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −4.00000 −0.157867
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 3.00000 0.117851
\(649\) −4.00000 −0.157014
\(650\) 4.00000 0.156893
\(651\) −2.00000 −0.0783862
\(652\) 20.0000 0.783260
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −6.00000 −0.234439
\(656\) 2.00000 0.0780869
\(657\) 8.00000 0.312110
\(658\) 8.00000 0.311872
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 2.00000 0.0778499
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 4.00000 0.155464
\(663\) 8.00000 0.310694
\(664\) 36.0000 1.39707
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −6.00000 −0.232321
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) 12.0000 0.463600
\(671\) 28.0000 1.08093
\(672\) −5.00000 −0.192879
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −3.00000 −0.115385
\(677\) −14.0000 −0.538064 −0.269032 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 8.00000 0.307238
\(679\) −14.0000 −0.537271
\(680\) −6.00000 −0.230089
\(681\) 4.00000 0.153280
\(682\) −4.00000 −0.153168
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.764161
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) 14.0000 0.532585 0.266293 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 4.00000 0.152057
\(693\) −2.00000 −0.0759737
\(694\) −36.0000 −1.36654
\(695\) −10.0000 −0.379322
\(696\) −18.0000 −0.682288
\(697\) 4.00000 0.151511
\(698\) 26.0000 0.984115
\(699\) 6.00000 0.226941
\(700\) −1.00000 −0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 4.00000 0.150970
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) −8.00000 −0.301297
\(706\) 4.00000 0.150542
\(707\) 18.0000 0.676960
\(708\) −2.00000 −0.0751646
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 18.0000 0.674579
\(713\) −2.00000 −0.0749006
\(714\) 2.00000 0.0748481
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) 8.00000 0.298765
\(718\) −10.0000 −0.373197
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 19.0000 0.707107
\(723\) −6.00000 −0.223142
\(724\) 26.0000 0.966282
\(725\) −6.00000 −0.222834
\(726\) 7.00000 0.259794
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −12.0000 −0.444750
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) 16.0000 0.591781
\(732\) 14.0000 0.517455
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) −32.0000 −1.18114
\(735\) 1.00000 0.0368856
\(736\) −5.00000 −0.184302
\(737\) 24.0000 0.884051
\(738\) 2.00000 0.0736210
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −6.00000 −0.219971
\(745\) 2.00000 0.0732743
\(746\) −28.0000 −1.02515
\(747\) 12.0000 0.439057
\(748\) −4.00000 −0.146254
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 8.00000 0.291730
\(753\) 16.0000 0.583072
\(754\) −24.0000 −0.874028
\(755\) −20.0000 −0.727875
\(756\) −1.00000 −0.0363696
\(757\) 24.0000 0.872295 0.436147 0.899875i \(-0.356343\pi\)
0.436147 + 0.899875i \(0.356343\pi\)
\(758\) −26.0000 −0.944363
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 16.0000 0.579619
\(763\) 2.00000 0.0724049
\(764\) 18.0000 0.651217
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) −8.00000 −0.288863
\(768\) −17.0000 −0.613435
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 2.00000 0.0720750
\(771\) 0 0
\(772\) −22.0000 −0.791797
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 8.00000 0.287554
\(775\) −2.00000 −0.0718421
\(776\) −42.0000 −1.50771
\(777\) 4.00000 0.143499
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 2.00000 0.0715199
\(783\) −6.00000 −0.214423
\(784\) −1.00000 −0.0357143
\(785\) 10.0000 0.356915
\(786\) 6.00000 0.214013
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) −18.0000 −0.641223
\(789\) 20.0000 0.712019
\(790\) −10.0000 −0.355784
\(791\) −8.00000 −0.284447
\(792\) −6.00000 −0.213201
\(793\) 56.0000 1.98862
\(794\) 0 0
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −5.00000 −0.176777
\(801\) 6.00000 0.212000
\(802\) 14.0000 0.494357
\(803\) −16.0000 −0.564628
\(804\) 12.0000 0.423207
\(805\) 1.00000 0.0352454
\(806\) −8.00000 −0.281788
\(807\) 10.0000 0.352017
\(808\) 54.0000 1.89971
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 6.00000 0.210559
\(813\) −10.0000 −0.350715
\(814\) 8.00000 0.280400
\(815\) −20.0000 −0.700569
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) −34.0000 −1.18878
\(819\) −4.00000 −0.139771
\(820\) 2.00000 0.0698430
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 20.0000 0.697580
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 24.0000 0.836080
\(825\) −2.00000 −0.0696311
\(826\) −2.00000 −0.0695889
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) −12.0000 −0.416526
\(831\) 10.0000 0.346896
\(832\) −28.0000 −0.970725
\(833\) −2.00000 −0.0692959
\(834\) 10.0000 0.346272
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 12.0000 0.414533
\(839\) −4.00000 −0.138095 −0.0690477 0.997613i \(-0.521996\pi\)
−0.0690477 + 0.997613i \(0.521996\pi\)
\(840\) 3.00000 0.103510
\(841\) 7.00000 0.241379
\(842\) 10.0000 0.344623
\(843\) −18.0000 −0.619953
\(844\) −8.00000 −0.275371
\(845\) 3.00000 0.103203
\(846\) 8.00000 0.275046
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 2.00000 0.0685994
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) −8.00000 −0.273115
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 8.00000 0.272798
\(861\) −2.00000 −0.0681598
\(862\) −10.0000 −0.340601
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −5.00000 −0.170103
\(865\) −4.00000 −0.136004
\(866\) −2.00000 −0.0679628
\(867\) −13.0000 −0.441503
\(868\) 2.00000 0.0678844
\(869\) −20.0000 −0.678454
\(870\) 6.00000 0.203419
\(871\) 48.0000 1.62642
\(872\) 6.00000 0.203186
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −8.00000 −0.270295
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 14.0000 0.472477
\(879\) −10.0000 −0.337292
\(880\) 2.00000 0.0674200
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −8.00000 −0.269069
\(885\) 2.00000 0.0672293
\(886\) −4.00000 −0.134383
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 12.0000 0.402694
\(889\) −16.0000 −0.536623
\(890\) −6.00000 −0.201120
\(891\) −2.00000 −0.0670025
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −2.00000 −0.0668900
\(895\) 4.00000 0.133705
\(896\) 3.00000 0.100223
\(897\) −4.00000 −0.133556
\(898\) 10.0000 0.333704
\(899\) 12.0000 0.400222
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) −4.00000 −0.133185
\(903\) −8.00000 −0.266223
\(904\) −24.0000 −0.798228
\(905\) −26.0000 −0.864269
\(906\) 20.0000 0.664455
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −4.00000 −0.132745
\(909\) 18.0000 0.597022
\(910\) 4.00000 0.132599
\(911\) −38.0000 −1.25900 −0.629498 0.777002i \(-0.716739\pi\)
−0.629498 + 0.777002i \(0.716739\pi\)
\(912\) 0 0
\(913\) −24.0000 −0.794284
\(914\) 20.0000 0.661541
\(915\) −14.0000 −0.462826
\(916\) 14.0000 0.462573
\(917\) −6.00000 −0.198137
\(918\) 2.00000 0.0660098
\(919\) 38.0000 1.25350 0.626752 0.779219i \(-0.284384\pi\)
0.626752 + 0.779219i \(0.284384\pi\)
\(920\) 3.00000 0.0989071
\(921\) −20.0000 −0.659022
\(922\) −18.0000 −0.592798
\(923\) 0 0
\(924\) 2.00000 0.0657952
\(925\) 4.00000 0.131519
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) 30.0000 0.984798
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 2.00000 0.0655826
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −6.00000 −0.196431
\(934\) −20.0000 −0.654420
\(935\) 4.00000 0.130814
\(936\) −12.0000 −0.392232
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 12.0000 0.391814
\(939\) −26.0000 −0.848478
\(940\) 8.00000 0.260931
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) −10.0000 −0.325818
\(943\) −2.00000 −0.0651290
\(944\) −2.00000 −0.0650945
\(945\) 1.00000 0.0325300
\(946\) −16.0000 −0.520205
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −10.0000 −0.324785
\(949\) −32.0000 −1.03876
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) −6.00000 −0.194461
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) −8.00000 −0.258738
\(957\) 12.0000 0.387905
\(958\) −28.0000 −0.904639
\(959\) −20.0000 −0.645834
\(960\) 7.00000 0.225924
\(961\) −27.0000 −0.870968
\(962\) 16.0000 0.515861
\(963\) 4.00000 0.128898
\(964\) 6.00000 0.193247
\(965\) 22.0000 0.708205
\(966\) −1.00000 −0.0321745
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) 14.0000 0.449513
\(971\) 56.0000 1.79713 0.898563 0.438845i \(-0.144612\pi\)
0.898563 + 0.438845i \(0.144612\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −10.0000 −0.320585
\(974\) 8.00000 0.256337
\(975\) −4.00000 −0.128103
\(976\) 14.0000 0.448129
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 20.0000 0.639529
\(979\) −12.0000 −0.383522
\(980\) −1.00000 −0.0319438
\(981\) 2.00000 0.0638551
\(982\) 12.0000 0.382935
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) −6.00000 −0.191273
\(985\) 18.0000 0.573528
\(986\) −12.0000 −0.382158
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 2.00000 0.0635642
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 10.0000 0.317500
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) −12.0000 −0.380235
\(997\) 12.0000 0.380044 0.190022 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) 24.0000 0.759707
\(999\) 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 2415.2.a.d.1.1 1
3.2 odd 2 7245.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2415.2.a.d.1.1 1 1.1 even 1 trivial
7245.2.a.m.1.1 1 3.2 odd 2