Properties

Label 1155.2.a.k.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.22272143346\)
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 \(\chi\) \(=\) 1155.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +5.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -3.00000 q^{40} -2.00000 q^{41} -1.00000 q^{42} -12.0000 q^{43} +1.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} -3.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +1.00000 q^{60} -14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} -2.00000 q^{65} +1.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} +1.00000 q^{70} -16.0000 q^{71} -3.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -1.00000 q^{77} +2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -6.00000 q^{85} -12.0000 q^{86} +2.00000 q^{87} +3.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +4.00000 q^{92} +4.00000 q^{93} +4.00000 q^{95} -5.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −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
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000 0.883883
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −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
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.00000 0.149071
\(46\) −4.00000 −0.589768
\(47\) 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
\(51\) 6.00000 0.840168
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) −3.00000 −0.400892
\(57\) −4.00000 −0.529813
\(58\) −2.00000 −0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) −2.00000 −0.248069
\(66\) 1.00000 0.123091
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) 4.00000 0.481543
\(70\) 1.00000 0.119523
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) −3.00000 −0.353553
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 2.00000 0.226455
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) −12.0000 −1.29399
\(87\) 2.00000 0.214423
\(88\) 3.00000 0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 1.00000 0.105409
\(91\) −2.00000 −0.209657
\(92\) 4.00000 0.417029
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −5.00000 −0.510310
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 6.00000 0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 6.00000 0.588348
\(105\) −1.00000 −0.0975900
\(106\) −2.00000 −0.194257
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 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 \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) −4.00000 −0.373002
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 3.00000 0.273861
\(121\) 1.00000 0.0909091
\(122\) −14.0000 −1.26750
\(123\) 2.00000 0.180334
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −3.00000 −0.265165
\(129\) 12.0000 1.05654
\(130\) −2.00000 −0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 4.00000 0.346844
\(134\) −8.00000 −0.691095
\(135\) −1.00000 −0.0860663
\(136\) 18.0000 1.54349
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 4.00000 0.340503
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 2.00000 0.167248
\(144\) −1.00000 −0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −12.0000 −0.973329
\(153\) −6.00000 −0.485071
\(154\) −1.00000 −0.0805823
\(155\) −4.00000 −0.321288
\(156\) −2.00000 −0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 16.0000 1.27289
\(159\) 2.00000 0.158610
\(160\) 5.00000 0.395285
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 2.00000 0.156174
\(165\) 1.00000 0.0778499
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 4.00000 0.305888
\(172\) 12.0000 0.914991
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −2.00000 −0.148250
\(183\) 14.0000 1.03491
\(184\) 12.0000 0.884652
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) 6.00000 0.438763
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 4.00000 0.290191
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −7.00000 −0.505181
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) −1.00000 −0.0714286
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 28.0000 1.98487 0.992434 0.122782i \(-0.0391815\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) −3.00000 −0.212132
\(201\) 8.00000 0.564276
\(202\) 10.0000 0.703598
\(203\) −2.00000 −0.140372
\(204\) −6.00000 −0.420084
\(205\) −2.00000 −0.139686
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) −1.00000 −0.0690066
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000 0.137361
\(213\) 16.0000 1.09630
\(214\) 4.00000 0.273434
\(215\) −12.0000 −0.818393
\(216\) 3.00000 0.204124
\(217\) −4.00000 −0.271538
\(218\) −10.0000 −0.677285
\(219\) −10.0000 −0.675737
\(220\) 1.00000 0.0674200
\(221\) 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
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 4.00000 0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −4.00000 −0.263752
\(231\) 1.00000 0.0657952
\(232\) 6.00000 0.393919
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) −6.00000 −0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 1.00000 0.0645497
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 14.0000 0.896258
\(245\) 1.00000 0.0638877
\(246\) 2.00000 0.127515
\(247\) −8.00000 −0.509028
\(248\) 12.0000 0.762001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 4.00000 0.251478
\(254\) 8.00000 0.501965
\(255\) 6.00000 0.375735
\(256\) −17.0000 −1.06250
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 12.0000 0.747087
\(259\) −2.00000 −0.124274
\(260\) 2.00000 0.124035
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −3.00000 −0.184637
\(265\) −2.00000 −0.122859
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) 8.00000 0.488678
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 2.00000 0.121046
\(274\) 2.00000 0.120824
\(275\) −1.00000 −0.0603023
\(276\) −4.00000 −0.240772
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 20.0000 1.19952
\(279\) −4.00000 −0.239474
\(280\) −3.00000 −0.179284
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 16.0000 0.949425
\(285\) −4.00000 −0.236940
\(286\) 2.00000 0.118262
\(287\) −2.00000 −0.118056
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) −2.00000 −0.117242
\(292\) −10.0000 −0.585206
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 1.00000 0.0580259
\(298\) 6.00000 0.347571
\(299\) 8.00000 0.462652
\(300\) 1.00000 0.0577350
\(301\) −12.0000 −0.691669
\(302\) −16.0000 −0.920697
\(303\) −10.0000 −0.574485
\(304\) −4.00000 −0.229416
\(305\) −14.0000 −0.801638
\(306\) −6.00000 −0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 1.00000 0.0569803
\(309\) −8.00000 −0.455104
\(310\) −4.00000 −0.227185
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) −6.00000 −0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 1.00000 0.0563436
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 2.00000 0.112154
\(319\) 2.00000 0.111979
\(320\) 7.00000 0.391312
\(321\) −4.00000 −0.223258
\(322\) −4.00000 −0.222911
\(323\) −24.0000 −1.33540
\(324\) −1.00000 −0.0555556
\(325\) −2.00000 −0.110940
\(326\) −8.00000 −0.443079
\(327\) 10.0000 0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 4.00000 0.219529
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 1.00000 0.0545545
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −9.00000 −0.489535
\(339\) 6.00000 0.325875
\(340\) 6.00000 0.325396
\(341\) 4.00000 0.216612
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) 36.0000 1.94099
\(345\) 4.00000 0.215353
\(346\) −10.0000 −0.537603
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) −2.00000 −0.107211
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) −5.00000 −0.266501
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 6.00000 0.317999
\(357\) 6.00000 0.317554
\(358\) 20.0000 1.05703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −3.00000 −0.158114
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) −1.00000 −0.0524864
\(364\) 2.00000 0.104828
\(365\) 10.0000 0.523424
\(366\) 14.0000 0.731792
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 4.00000 0.208514
\(369\) −2.00000 −0.104116
\(370\) −2.00000 −0.103975
\(371\) −2.00000 −0.103835
\(372\) −4.00000 −0.207390
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 6.00000 0.310253
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −4.00000 −0.205196
\(381\) −8.00000 −0.409852
\(382\) −8.00000 −0.409316
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 3.00000 0.153093
\(385\) −1.00000 −0.0509647
\(386\) 14.0000 0.712581
\(387\) −12.0000 −0.609994
\(388\) −2.00000 −0.101535
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 2.00000 0.101274
\(391\) 24.0000 1.21373
\(392\) −3.00000 −0.151523
\(393\) −20.0000 −1.00887
\(394\) −22.0000 −1.10834
\(395\) 16.0000 0.805047
\(396\) 1.00000 0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 28.0000 1.40351
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 8.00000 0.399004
\(403\) 8.00000 0.398508
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) −2.00000 −0.0992583
\(407\) 2.00000 0.0991363
\(408\) −18.0000 −0.891133
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −2.00000 −0.0986527
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) −4.00000 −0.196352
\(416\) −10.0000 −0.490290
\(417\) −20.0000 −0.979404
\(418\) −4.00000 −0.195646
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 1.00000 0.0487950
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) 16.0000 0.775203
\(427\) −14.0000 −0.677507
\(428\) −4.00000 −0.193347
\(429\) −2.00000 −0.0965609
\(430\) −12.0000 −0.578691
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −4.00000 −0.192006
\(435\) 2.00000 0.0958927
\(436\) 10.0000 0.478913
\(437\) −16.0000 −0.765384
\(438\) −10.0000 −0.477818
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 3.00000 0.143019
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) −8.00000 −0.380091 −0.190046 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −6.00000 −0.284427
\(446\) 24.0000 1.13643
\(447\) −6.00000 −0.283790
\(448\) 7.00000 0.330719
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) 2.00000 0.0941763
\(452\) 6.00000 0.282216
\(453\) 16.0000 0.751746
\(454\) 20.0000 0.938647
\(455\) −2.00000 −0.0937614
\(456\) 12.0000 0.561951
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 4.00000 0.186501
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 1.00000 0.0465242
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 2.00000 0.0928477
\(465\) 4.00000 0.185496
\(466\) −26.0000 −1.20443
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 2.00000 0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) −16.0000 −0.734904
\(475\) 4.00000 0.183533
\(476\) 6.00000 0.275010
\(477\) −2.00000 −0.0915737
\(478\) −24.0000 −1.09773
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) −5.00000 −0.228218
\(481\) 4.00000 0.182384
\(482\) −10.0000 −0.455488
\(483\) 4.00000 0.182006
\(484\) −1.00000 −0.0454545
\(485\) 2.00000 0.0908153
\(486\) −1.00000 −0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 42.0000 1.90125
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) −1.00000 −0.0449467
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −3.00000 −0.133631
\(505\) 10.0000 0.444994
\(506\) 4.00000 0.177822
\(507\) 9.00000 0.399704
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 6.00000 0.265684
\(511\) 10.0000 0.442374
\(512\) −11.0000 −0.486136
\(513\) −4.00000 −0.176604
\(514\) −14.0000 −0.617514
\(515\) 8.00000 0.352522
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) −2.00000 −0.0878750
\(519\) 10.0000 0.438951
\(520\) 6.00000 0.263117
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −20.0000 −0.873704
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) −1.00000 −0.0435194
\(529\) −7.00000 −0.304348
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 4.00000 0.173259
\(534\) 6.00000 0.259645
\(535\) 4.00000 0.172935
\(536\) 24.0000 1.03664
\(537\) −20.0000 −0.863064
\(538\) −26.0000 −1.12094
\(539\) −1.00000 −0.0430730
\(540\) 1.00000 0.0430331
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −8.00000 −0.343629
\(543\) −14.0000 −0.600798
\(544\) −30.0000 −1.28624
\(545\) −10.0000 −0.428353
\(546\) 2.00000 0.0855921
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −14.0000 −0.597505
\(550\) −1.00000 −0.0426401
\(551\) −8.00000 −0.340811
\(552\) −12.0000 −0.510754
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) 2.00000 0.0848953
\(556\) −20.0000 −0.848189
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −4.00000 −0.169334
\(559\) 24.0000 1.01509
\(560\) −1.00000 −0.0422577
\(561\) −6.00000 −0.253320
\(562\) −6.00000 −0.253095
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 48.0000 2.01404
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) −4.00000 −0.167542
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 8.00000 0.334205
\(574\) −2.00000 −0.0834784
\(575\) −4.00000 −0.166812
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 19.0000 0.790296
\(579\) −14.0000 −0.581820
\(580\) 2.00000 0.0830455
\(581\) −4.00000 −0.165948
\(582\) −2.00000 −0.0829027
\(583\) 2.00000 0.0828315
\(584\) −30.0000 −1.24141
\(585\) −2.00000 −0.0826898
\(586\) −18.0000 −0.743573
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 1.00000 0.0412393
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 2.00000 0.0821995
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 1.00000 0.0410305
\(595\) −6.00000 −0.245976
\(596\) −6.00000 −0.245770
\(597\) −28.0000 −1.14596
\(598\) 8.00000 0.327144
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 3.00000 0.122474
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) −12.0000 −0.489083
\(603\) −8.00000 −0.325785
\(604\) 16.0000 0.651031
\(605\) 1.00000 0.0406558
\(606\) −10.0000 −0.406222
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 20.0000 0.811107
\(609\) 2.00000 0.0810441
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 12.0000 0.484281
\(615\) 2.00000 0.0806478
\(616\) 3.00000 0.120873
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) −8.00000 −0.321807
\(619\) −48.0000 −1.92928 −0.964641 0.263566i \(-0.915101\pi\)
−0.964641 + 0.263566i \(0.915101\pi\)
\(620\) 4.00000 0.160644
\(621\) 4.00000 0.160514
\(622\) 20.0000 0.801927
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 4.00000 0.159745
\(628\) 2.00000 0.0798087
\(629\) 12.0000 0.478471
\(630\) 1.00000 0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −48.0000 −1.90934
\(633\) −20.0000 −0.794929
\(634\) −18.0000 −0.714871
\(635\) 8.00000 0.317470
\(636\) −2.00000 −0.0793052
\(637\) −2.00000 −0.0792429
\(638\) 2.00000 0.0791808
\(639\) −16.0000 −0.632950
\(640\) −3.00000 −0.118585
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 4.00000 0.157622
\(645\) 12.0000 0.472500
\(646\) −24.0000 −0.944267
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 4.00000 0.156772
\(652\) 8.00000 0.313304
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 10.0000 0.391031
\(655\) 20.0000 0.781465
\(656\) 2.00000 0.0780869
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −12.0000 −0.466393
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) −2.00000 −0.0774984
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) −8.00000 −0.309067
\(671\) 14.0000 0.540464
\(672\) −5.00000 −0.192879
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) 9.00000 0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 6.00000 0.230429
\(679\) 2.00000 0.0767530
\(680\) 18.0000 0.690268
\(681\) −20.0000 −0.766402
\(682\) 4.00000 0.153168
\(683\) −16.0000 −0.612223 −0.306111 0.951996i \(-0.599028\pi\)
−0.306111 + 0.951996i \(0.599028\pi\)
\(684\) −4.00000 −0.152944
\(685\) 2.00000 0.0764161
\(686\) 1.00000 0.0381802
\(687\) −14.0000 −0.534133
\(688\) 12.0000 0.457496
\(689\) 4.00000 0.152388
\(690\) 4.00000 0.152277
\(691\) 16.0000 0.608669 0.304334 0.952565i \(-0.401566\pi\)
0.304334 + 0.952565i \(0.401566\pi\)
\(692\) 10.0000 0.380143
\(693\) −1.00000 −0.0379869
\(694\) −4.00000 −0.151838
\(695\) 20.0000 0.758643
\(696\) −6.00000 −0.227429
\(697\) 12.0000 0.454532
\(698\) 18.0000 0.681310
\(699\) 26.0000 0.983410
\(700\) −1.00000 −0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 2.00000 0.0754851
\(703\) −8.00000 −0.301726
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −16.0000 −0.600469
\(711\) 16.0000 0.600047
\(712\) 18.0000 0.674579
\(713\) 16.0000 0.599205
\(714\) 6.00000 0.224544
\(715\) 2.00000 0.0747958
\(716\) −20.0000 −0.747435
\(717\) 24.0000 0.896296
\(718\) 24.0000 0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) −3.00000 −0.111648
\(723\) 10.0000 0.371904
\(724\) −14.0000 −0.520306
\(725\) −2.00000 −0.0742781
\(726\) −1.00000 −0.0371135
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 72.0000 2.66302
\(732\) −14.0000 −0.517455
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 16.0000 0.590571
\(735\) −1.00000 −0.0368856
\(736\) −20.0000 −0.737210
\(737\) 8.00000 0.294684
\(738\) −2.00000 −0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 2.00000 0.0735215
\(741\) 8.00000 0.293887
\(742\) −2.00000 −0.0734223
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −12.0000 −0.439941
\(745\) 6.00000 0.219823
\(746\) −6.00000 −0.219676
\(747\) −4.00000 −0.146352
\(748\) −6.00000 −0.219382
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) −16.0000 −0.582300
\(756\) 1.00000 0.0363696
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −20.0000 −0.726433
\(759\) −4.00000 −0.145191
\(760\) −12.0000 −0.435286
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −8.00000 −0.289809
\(763\) −10.0000 −0.362024
\(764\) 8.00000 0.289430
\(765\) −6.00000 −0.216930
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 14.0000 0.504198
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −12.0000 −0.431331
\(775\) −4.00000 −0.143684
\(776\) −6.00000 −0.215387
\(777\) 2.00000 0.0717496
\(778\) −26.0000 −0.932145
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 16.0000 0.572525
\(782\) 24.0000 0.858238
\(783\) 2.00000 0.0714742
\(784\) −1.00000 −0.0357143
\(785\) −2.00000 −0.0713831
\(786\) −20.0000 −0.713376
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 16.0000 0.569254
\(791\) −6.00000 −0.213335
\(792\) 3.00000 0.106600
\(793\) 28.0000 0.994309
\(794\) −18.0000 −0.638796
\(795\) 2.00000 0.0709327
\(796\) −28.0000 −0.992434
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) 18.0000 0.635602
\(803\) −10.0000 −0.352892
\(804\) −8.00000 −0.282138
\(805\) −4.00000 −0.140981
\(806\) 8.00000 0.281788
\(807\) 26.0000 0.915243
\(808\) −30.0000 −1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 1.00000 0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 2.00000 0.0701862
\(813\) 8.00000 0.280572
\(814\) 2.00000 0.0701000
\(815\) −8.00000 −0.280228
\(816\) −6.00000 −0.210042
\(817\) −48.0000 −1.67931
\(818\) 14.0000 0.489499
\(819\) −2.00000 −0.0698857
\(820\) 2.00000 0.0698430
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −24.0000 −0.836080
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 4.00000 0.139010
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −4.00000 −0.138842
\(831\) −2.00000 −0.0693792
\(832\) −14.0000 −0.485363
\(833\) −6.00000 −0.207888
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) 4.00000 0.138260
\(838\) 16.0000 0.552711
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 3.00000 0.103510
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) 6.00000 0.206651
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) 4.00000 0.137280
\(850\) −6.00000 −0.205798
\(851\) 8.00000 0.274236
\(852\) −16.0000 −0.548151
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −14.0000 −0.479070
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) 10.0000 0.341593 0.170797 0.985306i \(-0.445366\pi\)
0.170797 + 0.985306i \(0.445366\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 12.0000 0.409197
\(861\) 2.00000 0.0681598
\(862\) 16.0000 0.544962
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) −5.00000 −0.170103
\(865\) −10.0000 −0.340010
\(866\) −14.0000 −0.475739
\(867\) −19.0000 −0.645274
\(868\) 4.00000 0.135769
\(869\) −16.0000 −0.542763
\(870\) 2.00000 0.0678064
\(871\) 16.0000 0.542139
\(872\) 30.0000 1.01593
\(873\) 2.00000 0.0676897
\(874\) −16.0000 −0.541208
\(875\) 1.00000 0.0338062
\(876\) 10.0000 0.337869
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) −32.0000 −1.07995
\(879\) 18.0000 0.607125
\(880\) 1.00000 0.0337100
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) 1.00000 0.0336718
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −6.00000 −0.201347
\(889\) 8.00000 0.268311
\(890\) −6.00000 −0.201120
\(891\) −1.00000 −0.0335013
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 20.0000 0.668526
\(896\) −3.00000 −0.100223
\(897\) −8.00000 −0.267112
\(898\) 18.0000 0.600668
\(899\) 8.00000 0.266815
\(900\) −1.00000 −0.0333333
\(901\) 12.0000 0.399778
\(902\) 2.00000 0.0665927
\(903\) 12.0000 0.399335
\(904\) 18.0000 0.598671
\(905\) 14.0000 0.465376
\(906\) 16.0000 0.531564
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) −20.0000 −0.663723
\(909\) 10.0000 0.331679
\(910\) −2.00000 −0.0662994
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 4.00000 0.132453
\(913\) 4.00000 0.132381
\(914\) 30.0000 0.992312
\(915\) 14.0000 0.462826
\(916\) −14.0000 −0.462573
\(917\) 20.0000 0.660458
\(918\) 6.00000 0.198030
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 12.0000 0.395628
\(921\) −12.0000 −0.395413
\(922\) −6.00000 −0.197599
\(923\) 32.0000 1.05329
\(924\) −1.00000 −0.0328976
\(925\) −2.00000 −0.0657596
\(926\) 4.00000 0.131448
\(927\) 8.00000 0.262754
\(928\) −10.0000 −0.328266
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 4.00000 0.131165
\(931\) 4.00000 0.131095
\(932\) 26.0000 0.851658
\(933\) −20.0000 −0.654771
\(934\) −28.0000 −0.916188
\(935\) 6.00000 0.196221
\(936\) 6.00000 0.196116
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) −8.00000 −0.261209
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 2.00000 0.0651635
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 12.0000 0.390154
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 16.0000 0.519656
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) 18.0000 0.583690
\(952\) 18.0000 0.583383
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −8.00000 −0.258874
\(956\) 24.0000 0.776215
\(957\) −2.00000 −0.0646508
\(958\) −32.0000 −1.03387
\(959\) 2.00000 0.0645834
\(960\) −7.00000 −0.225924
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) 4.00000 0.128898
\(964\) 10.0000 0.322078
\(965\) 14.0000 0.450676
\(966\) 4.00000 0.128698
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 24.0000 0.770991
\(970\) 2.00000 0.0642161
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) 4.00000 0.128168
\(975\) 2.00000 0.0640513
\(976\) 14.0000 0.448129
\(977\) 58.0000 1.85558 0.927792 0.373097i \(-0.121704\pi\)
0.927792 + 0.373097i \(0.121704\pi\)
\(978\) 8.00000 0.255812
\(979\) 6.00000 0.191761
\(980\) −1.00000 −0.0319438
\(981\) −10.0000 −0.319275
\(982\) −36.0000 −1.14881
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) −6.00000 −0.191273
\(985\) −22.0000 −0.700978
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 48.0000 1.52631
\(990\) −1.00000 −0.0317821
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −20.0000 −0.635001
\(993\) 12.0000 0.380808
\(994\) −16.0000 −0.507489
\(995\) 28.0000 0.887660
\(996\) −4.00000 −0.126745
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 1155.2.a.k.1.1 1
3.2 odd 2 3465.2.a.b.1.1 1
5.4 even 2 5775.2.a.g.1.1 1
7.6 odd 2 8085.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.k.1.1 1 1.1 even 1 trivial
3465.2.a.b.1.1 1 3.2 odd 2
5775.2.a.g.1.1 1 5.4 even 2
8085.2.a.u.1.1 1 7.6 odd 2