Properties

Label 1110.2.a.d.1.1
Level $1110$
Weight $2$
Character 1110.1
Self dual yes
Analytic conductor $8.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.86339462436\)
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 \(\chi\) \(=\) 1110.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -5.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} -3.00000 q^{21} +5.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +3.00000 q^{28} -5.00000 q^{29} +1.00000 q^{30} -7.00000 q^{31} -1.00000 q^{32} +5.00000 q^{33} +7.00000 q^{34} +3.00000 q^{35} +1.00000 q^{36} +1.00000 q^{37} +2.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +1.00000 q^{41} +3.00000 q^{42} +9.00000 q^{43} -5.00000 q^{44} +1.00000 q^{45} -4.00000 q^{46} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +7.00000 q^{51} -2.00000 q^{52} +3.00000 q^{53} +1.00000 q^{54} -5.00000 q^{55} -3.00000 q^{56} +2.00000 q^{57} +5.00000 q^{58} -14.0000 q^{59} -1.00000 q^{60} -7.00000 q^{61} +7.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -5.00000 q^{66} -4.00000 q^{67} -7.00000 q^{68} -4.00000 q^{69} -3.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -12.0000 q^{73} -1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -15.0000 q^{77} -2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.00000 q^{82} -10.0000 q^{83} -3.00000 q^{84} -7.00000 q^{85} -9.00000 q^{86} +5.00000 q^{87} +5.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} -6.00000 q^{91} +4.00000 q^{92} +7.00000 q^{93} -2.00000 q^{95} +1.00000 q^{96} +1.00000 q^{97} -2.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −3.00000 −0.801784
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.00000 −0.654654
\(22\) 5.00000 1.06600
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 3.00000 0.566947
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 1.00000 0.182574
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.00000 0.870388
\(34\) 7.00000 1.20049
\(35\) 3.00000 0.507093
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 3.00000 0.462910
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) −5.00000 −0.753778
\(45\) 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
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 7.00000 0.980196
\(52\) −2.00000 −0.277350
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000 0.136083
\(55\) −5.00000 −0.674200
\(56\) −3.00000 −0.400892
\(57\) 2.00000 0.264906
\(58\) 5.00000 0.656532
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) −1.00000 −0.129099
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 7.00000 0.889001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −5.00000 −0.615457
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −7.00000 −0.848875
\(69\) −4.00000 −0.481543
\(70\) −3.00000 −0.358569
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) −15.0000 −1.70941
\(78\) −2.00000 −0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.00000 −0.110432
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) −3.00000 −0.327327
\(85\) −7.00000 −0.759257
\(86\) −9.00000 −0.970495
\(87\) 5.00000 0.536056
\(88\) 5.00000 0.533002
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.00000 −0.105409
\(91\) −6.00000 −0.628971
\(92\) 4.00000 0.417029
\(93\) 7.00000 0.725866
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 1.00000 0.102062
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) −2.00000 −0.202031
\(99\) −5.00000 −0.502519
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −7.00000 −0.693103
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.00000 0.196116
\(105\) −3.00000 −0.292770
\(106\) −3.00000 −0.291386
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 5.00000 0.476731
\(111\) −1.00000 −0.0949158
\(112\) 3.00000 0.283473
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) −2.00000 −0.187317
\(115\) 4.00000 0.373002
\(116\) −5.00000 −0.464238
\(117\) −2.00000 −0.184900
\(118\) 14.0000 1.28880
\(119\) −21.0000 −1.92507
\(120\) 1.00000 0.0912871
\(121\) 14.0000 1.27273
\(122\) 7.00000 0.633750
\(123\) −1.00000 −0.0901670
\(124\) −7.00000 −0.628619
\(125\) 1.00000 0.0894427
\(126\) −3.00000 −0.267261
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.00000 −0.792406
\(130\) 2.00000 0.175412
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 5.00000 0.435194
\(133\) −6.00000 −0.520266
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 7.00000 0.600245
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 4.00000 0.340503
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 3.00000 0.253546
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 10.0000 0.836242
\(144\) 1.00000 0.0833333
\(145\) −5.00000 −0.415227
\(146\) 12.0000 0.993127
\(147\) −2.00000 −0.164957
\(148\) 1.00000 0.0821995
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 2.00000 0.162221
\(153\) −7.00000 −0.565916
\(154\) 15.0000 1.20873
\(155\) −7.00000 −0.562254
\(156\) 2.00000 0.160128
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) −4.00000 −0.318223
\(159\) −3.00000 −0.237915
\(160\) −1.00000 −0.0790569
\(161\) 12.0000 0.945732
\(162\) −1.00000 −0.0785674
\(163\) −15.0000 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(164\) 1.00000 0.0780869
\(165\) 5.00000 0.389249
\(166\) 10.0000 0.776151
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 3.00000 0.231455
\(169\) −9.00000 −0.692308
\(170\) 7.00000 0.536875
\(171\) −2.00000 −0.152944
\(172\) 9.00000 0.686244
\(173\) 19.0000 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) −5.00000 −0.379049
\(175\) 3.00000 0.226779
\(176\) −5.00000 −0.376889
\(177\) 14.0000 1.05230
\(178\) 14.0000 1.04934
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 6.00000 0.444750
\(183\) 7.00000 0.517455
\(184\) −4.00000 −0.294884
\(185\) 1.00000 0.0735215
\(186\) −7.00000 −0.513265
\(187\) 35.0000 2.55945
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 2.00000 0.145095
\(191\) −7.00000 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −1.00000 −0.0717958
\(195\) 2.00000 0.143223
\(196\) 2.00000 0.142857
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 5.00000 0.355335
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −15.0000 −1.05279
\(204\) 7.00000 0.490098
\(205\) 1.00000 0.0698430
\(206\) −8.00000 −0.557386
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) 10.0000 0.691714
\(210\) 3.00000 0.207020
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 3.00000 0.206041
\(213\) −8.00000 −0.548151
\(214\) −14.0000 −0.957020
\(215\) 9.00000 0.613795
\(216\) 1.00000 0.0680414
\(217\) −21.0000 −1.42557
\(218\) 7.00000 0.474100
\(219\) 12.0000 0.810885
\(220\) −5.00000 −0.337100
\(221\) 14.0000 0.941742
\(222\) 1.00000 0.0671156
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) −3.00000 −0.200446
\(225\) 1.00000 0.0666667
\(226\) 5.00000 0.332595
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) 2.00000 0.132453
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) −4.00000 −0.263752
\(231\) 15.0000 0.986928
\(232\) 5.00000 0.328266
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) −4.00000 −0.259828
\(238\) 21.0000 1.36123
\(239\) 17.0000 1.09964 0.549819 0.835284i \(-0.314697\pi\)
0.549819 + 0.835284i \(0.314697\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −14.0000 −0.899954
\(243\) −1.00000 −0.0641500
\(244\) −7.00000 −0.448129
\(245\) 2.00000 0.127775
\(246\) 1.00000 0.0637577
\(247\) 4.00000 0.254514
\(248\) 7.00000 0.444500
\(249\) 10.0000 0.633724
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 3.00000 0.188982
\(253\) −20.0000 −1.25739
\(254\) −16.0000 −1.00393
\(255\) 7.00000 0.438357
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 9.00000 0.560316
\(259\) 3.00000 0.186411
\(260\) −2.00000 −0.124035
\(261\) −5.00000 −0.309492
\(262\) 6.00000 0.370681
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) −5.00000 −0.307729
\(265\) 3.00000 0.184289
\(266\) 6.00000 0.367884
\(267\) 14.0000 0.856786
\(268\) −4.00000 −0.244339
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −7.00000 −0.424437
\(273\) 6.00000 0.363137
\(274\) 22.0000 1.32907
\(275\) −5.00000 −0.301511
\(276\) −4.00000 −0.240772
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 3.00000 0.179928
\(279\) −7.00000 −0.419079
\(280\) −3.00000 −0.179284
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 8.00000 0.474713
\(285\) 2.00000 0.118470
\(286\) −10.0000 −0.591312
\(287\) 3.00000 0.177084
\(288\) −1.00000 −0.0589256
\(289\) 32.0000 1.88235
\(290\) 5.00000 0.293610
\(291\) −1.00000 −0.0586210
\(292\) −12.0000 −0.702247
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 2.00000 0.116642
\(295\) −14.0000 −0.815112
\(296\) −1.00000 −0.0581238
\(297\) 5.00000 0.290129
\(298\) 6.00000 0.347571
\(299\) −8.00000 −0.462652
\(300\) −1.00000 −0.0577350
\(301\) 27.0000 1.55625
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) −7.00000 −0.400819
\(306\) 7.00000 0.400163
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) −15.0000 −0.854704
\(309\) −8.00000 −0.455104
\(310\) 7.00000 0.397573
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) −2.00000 −0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 11.0000 0.620766
\(315\) 3.00000 0.169031
\(316\) 4.00000 0.225018
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 3.00000 0.168232
\(319\) 25.0000 1.39973
\(320\) 1.00000 0.0559017
\(321\) −14.0000 −0.781404
\(322\) −12.0000 −0.668734
\(323\) 14.0000 0.778981
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 15.0000 0.830773
\(327\) 7.00000 0.387101
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) −5.00000 −0.275241
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) −10.0000 −0.548821
\(333\) 1.00000 0.0547997
\(334\) −20.0000 −1.09435
\(335\) −4.00000 −0.218543
\(336\) −3.00000 −0.163663
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 5.00000 0.271563
\(340\) −7.00000 −0.379628
\(341\) 35.0000 1.89536
\(342\) 2.00000 0.108148
\(343\) −15.0000 −0.809924
\(344\) −9.00000 −0.485247
\(345\) −4.00000 −0.215353
\(346\) −19.0000 −1.02145
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 5.00000 0.268028
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) −3.00000 −0.160357
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 17.0000 0.904819 0.452409 0.891810i \(-0.350565\pi\)
0.452409 + 0.891810i \(0.350565\pi\)
\(354\) −14.0000 −0.744092
\(355\) 8.00000 0.424596
\(356\) −14.0000 −0.741999
\(357\) 21.0000 1.11144
\(358\) 24.0000 1.26844
\(359\) 34.0000 1.79445 0.897226 0.441572i \(-0.145579\pi\)
0.897226 + 0.441572i \(0.145579\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) −14.0000 −0.735824
\(363\) −14.0000 −0.734809
\(364\) −6.00000 −0.314485
\(365\) −12.0000 −0.628109
\(366\) −7.00000 −0.365896
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) 4.00000 0.208514
\(369\) 1.00000 0.0520579
\(370\) −1.00000 −0.0519875
\(371\) 9.00000 0.467257
\(372\) 7.00000 0.362933
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −35.0000 −1.80981
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 3.00000 0.154303
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −2.00000 −0.102598
\(381\) −16.0000 −0.819705
\(382\) 7.00000 0.358151
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) −15.0000 −0.764471
\(386\) −14.0000 −0.712581
\(387\) 9.00000 0.457496
\(388\) 1.00000 0.0507673
\(389\) −31.0000 −1.57176 −0.785881 0.618378i \(-0.787790\pi\)
−0.785881 + 0.618378i \(0.787790\pi\)
\(390\) −2.00000 −0.101274
\(391\) −28.0000 −1.41602
\(392\) −2.00000 −0.101015
\(393\) 6.00000 0.302660
\(394\) −10.0000 −0.503793
\(395\) 4.00000 0.201262
\(396\) −5.00000 −0.251259
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) −4.00000 −0.200502
\(399\) 6.00000 0.300376
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) −4.00000 −0.199502
\(403\) 14.0000 0.697390
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 15.0000 0.744438
\(407\) −5.00000 −0.247841
\(408\) −7.00000 −0.346552
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) −1.00000 −0.0493865
\(411\) 22.0000 1.08518
\(412\) 8.00000 0.394132
\(413\) −42.0000 −2.06668
\(414\) −4.00000 −0.196589
\(415\) −10.0000 −0.490881
\(416\) 2.00000 0.0980581
\(417\) 3.00000 0.146911
\(418\) −10.0000 −0.489116
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) −3.00000 −0.146385
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 13.0000 0.632830
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) −7.00000 −0.339550
\(426\) 8.00000 0.387601
\(427\) −21.0000 −1.01626
\(428\) 14.0000 0.676716
\(429\) −10.0000 −0.482805
\(430\) −9.00000 −0.434019
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 21.0000 1.00803
\(435\) 5.00000 0.239732
\(436\) −7.00000 −0.335239
\(437\) −8.00000 −0.382692
\(438\) −12.0000 −0.573382
\(439\) −33.0000 −1.57500 −0.787502 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(440\) 5.00000 0.238366
\(441\) 2.00000 0.0952381
\(442\) −14.0000 −0.665912
\(443\) 42.0000 1.99548 0.997740 0.0671913i \(-0.0214038\pi\)
0.997740 + 0.0671913i \(0.0214038\pi\)
\(444\) −1.00000 −0.0474579
\(445\) −14.0000 −0.663664
\(446\) −9.00000 −0.426162
\(447\) 6.00000 0.283790
\(448\) 3.00000 0.141737
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −5.00000 −0.235441
\(452\) −5.00000 −0.235180
\(453\) −10.0000 −0.469841
\(454\) 7.00000 0.328526
\(455\) −6.00000 −0.281284
\(456\) −2.00000 −0.0936586
\(457\) −39.0000 −1.82434 −0.912172 0.409809i \(-0.865595\pi\)
−0.912172 + 0.409809i \(0.865595\pi\)
\(458\) 28.0000 1.30835
\(459\) 7.00000 0.326732
\(460\) 4.00000 0.186501
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) −15.0000 −0.697863
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) −5.00000 −0.232119
\(465\) 7.00000 0.324617
\(466\) 20.0000 0.926482
\(467\) 3.00000 0.138823 0.0694117 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 11.0000 0.506853
\(472\) 14.0000 0.644402
\(473\) −45.0000 −2.06910
\(474\) 4.00000 0.183726
\(475\) −2.00000 −0.0917663
\(476\) −21.0000 −0.962533
\(477\) 3.00000 0.137361
\(478\) −17.0000 −0.777562
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) −2.00000 −0.0911922
\(482\) 2.00000 0.0910975
\(483\) −12.0000 −0.546019
\(484\) 14.0000 0.636364
\(485\) 1.00000 0.0454077
\(486\) 1.00000 0.0453609
\(487\) −30.0000 −1.35943 −0.679715 0.733476i \(-0.737896\pi\)
−0.679715 + 0.733476i \(0.737896\pi\)
\(488\) 7.00000 0.316875
\(489\) 15.0000 0.678323
\(490\) −2.00000 −0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −1.00000 −0.0450835
\(493\) 35.0000 1.57632
\(494\) −4.00000 −0.179969
\(495\) −5.00000 −0.224733
\(496\) −7.00000 −0.314309
\(497\) 24.0000 1.07655
\(498\) −10.0000 −0.448111
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 1.00000 0.0447214
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 20.0000 0.889108
\(507\) 9.00000 0.399704
\(508\) 16.0000 0.709885
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) −7.00000 −0.309965
\(511\) −36.0000 −1.59255
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) −14.0000 −0.617514
\(515\) 8.00000 0.352522
\(516\) −9.00000 −0.396203
\(517\) 0 0
\(518\) −3.00000 −0.131812
\(519\) −19.0000 −0.834007
\(520\) 2.00000 0.0877058
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 5.00000 0.218844
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −6.00000 −0.262111
\(525\) −3.00000 −0.130931
\(526\) −11.0000 −0.479623
\(527\) 49.0000 2.13447
\(528\) 5.00000 0.217597
\(529\) −7.00000 −0.304348
\(530\) −3.00000 −0.130312
\(531\) −14.0000 −0.607548
\(532\) −6.00000 −0.260133
\(533\) −2.00000 −0.0866296
\(534\) −14.0000 −0.605839
\(535\) 14.0000 0.605273
\(536\) 4.00000 0.172774
\(537\) 24.0000 1.03568
\(538\) −16.0000 −0.689809
\(539\) −10.0000 −0.430730
\(540\) −1.00000 −0.0430331
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) −20.0000 −0.859074
\(543\) −14.0000 −0.600798
\(544\) 7.00000 0.300123
\(545\) −7.00000 −0.299847
\(546\) −6.00000 −0.256776
\(547\) −7.00000 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(548\) −22.0000 −0.939793
\(549\) −7.00000 −0.298753
\(550\) 5.00000 0.213201
\(551\) 10.0000 0.426014
\(552\) 4.00000 0.170251
\(553\) 12.0000 0.510292
\(554\) −16.0000 −0.679775
\(555\) −1.00000 −0.0424476
\(556\) −3.00000 −0.127228
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 7.00000 0.296334
\(559\) −18.0000 −0.761319
\(560\) 3.00000 0.126773
\(561\) −35.0000 −1.47770
\(562\) −10.0000 −0.421825
\(563\) −3.00000 −0.126435 −0.0632175 0.998000i \(-0.520136\pi\)
−0.0632175 + 0.998000i \(0.520136\pi\)
\(564\) 0 0
\(565\) −5.00000 −0.210352
\(566\) 12.0000 0.504398
\(567\) 3.00000 0.125988
\(568\) −8.00000 −0.335673
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 27.0000 1.12991 0.564957 0.825120i \(-0.308893\pi\)
0.564957 + 0.825120i \(0.308893\pi\)
\(572\) 10.0000 0.418121
\(573\) 7.00000 0.292429
\(574\) −3.00000 −0.125218
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −32.0000 −1.33102
\(579\) −14.0000 −0.581820
\(580\) −5.00000 −0.207614
\(581\) −30.0000 −1.24461
\(582\) 1.00000 0.0414513
\(583\) −15.0000 −0.621237
\(584\) 12.0000 0.496564
\(585\) −2.00000 −0.0826898
\(586\) −9.00000 −0.371787
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 14.0000 0.576860
\(590\) 14.0000 0.576371
\(591\) −10.0000 −0.411345
\(592\) 1.00000 0.0410997
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) −5.00000 −0.205152
\(595\) −21.0000 −0.860916
\(596\) −6.00000 −0.245770
\(597\) −4.00000 −0.163709
\(598\) 8.00000 0.327144
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000 0.0408248
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) −27.0000 −1.10044
\(603\) −4.00000 −0.162893
\(604\) 10.0000 0.406894
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 2.00000 0.0811107
\(609\) 15.0000 0.607831
\(610\) 7.00000 0.283422
\(611\) 0 0
\(612\) −7.00000 −0.282958
\(613\) 43.0000 1.73675 0.868377 0.495905i \(-0.165164\pi\)
0.868377 + 0.495905i \(0.165164\pi\)
\(614\) −6.00000 −0.242140
\(615\) −1.00000 −0.0403239
\(616\) 15.0000 0.604367
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 8.00000 0.321807
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) −7.00000 −0.281127
\(621\) −4.00000 −0.160514
\(622\) −21.0000 −0.842023
\(623\) −42.0000 −1.68269
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) −10.0000 −0.399362
\(628\) −11.0000 −0.438948
\(629\) −7.00000 −0.279108
\(630\) −3.00000 −0.119523
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −4.00000 −0.159111
\(633\) 13.0000 0.516704
\(634\) 21.0000 0.834017
\(635\) 16.0000 0.634941
\(636\) −3.00000 −0.118958
\(637\) −4.00000 −0.158486
\(638\) −25.0000 −0.989759
\(639\) 8.00000 0.316475
\(640\) −1.00000 −0.0395285
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 14.0000 0.552536
\(643\) −43.0000 −1.69575 −0.847877 0.530193i \(-0.822120\pi\)
−0.847877 + 0.530193i \(0.822120\pi\)
\(644\) 12.0000 0.472866
\(645\) −9.00000 −0.354375
\(646\) −14.0000 −0.550823
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 70.0000 2.74774
\(650\) 2.00000 0.0784465
\(651\) 21.0000 0.823055
\(652\) −15.0000 −0.587445
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) −7.00000 −0.273722
\(655\) −6.00000 −0.234439
\(656\) 1.00000 0.0390434
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 5.00000 0.194625
\(661\) 31.0000 1.20576 0.602880 0.797832i \(-0.294020\pi\)
0.602880 + 0.797832i \(0.294020\pi\)
\(662\) 0 0
\(663\) −14.0000 −0.543715
\(664\) 10.0000 0.388075
\(665\) −6.00000 −0.232670
\(666\) −1.00000 −0.0387492
\(667\) −20.0000 −0.774403
\(668\) 20.0000 0.773823
\(669\) −9.00000 −0.347960
\(670\) 4.00000 0.154533
\(671\) 35.0000 1.35116
\(672\) 3.00000 0.115728
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 22.0000 0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −5.00000 −0.192024
\(679\) 3.00000 0.115129
\(680\) 7.00000 0.268438
\(681\) 7.00000 0.268241
\(682\) −35.0000 −1.34022
\(683\) −47.0000 −1.79841 −0.899203 0.437533i \(-0.855852\pi\)
−0.899203 + 0.437533i \(0.855852\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −22.0000 −0.840577
\(686\) 15.0000 0.572703
\(687\) 28.0000 1.06827
\(688\) 9.00000 0.343122
\(689\) −6.00000 −0.228582
\(690\) 4.00000 0.152277
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 19.0000 0.722272
\(693\) −15.0000 −0.569803
\(694\) 12.0000 0.455514
\(695\) −3.00000 −0.113796
\(696\) −5.00000 −0.189525
\(697\) −7.00000 −0.265144
\(698\) 16.0000 0.605609
\(699\) 20.0000 0.756469
\(700\) 3.00000 0.113389
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −2.00000 −0.0754314
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −17.0000 −0.639803
\(707\) 0 0
\(708\) 14.0000 0.526152
\(709\) −27.0000 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) −8.00000 −0.300235
\(711\) 4.00000 0.150012
\(712\) 14.0000 0.524672
\(713\) −28.0000 −1.04861
\(714\) −21.0000 −0.785905
\(715\) 10.0000 0.373979
\(716\) −24.0000 −0.896922
\(717\) −17.0000 −0.634877
\(718\) −34.0000 −1.26887
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 1.00000 0.0372678
\(721\) 24.0000 0.893807
\(722\) 15.0000 0.558242
\(723\) 2.00000 0.0743808
\(724\) 14.0000 0.520306
\(725\) −5.00000 −0.185695
\(726\) 14.0000 0.519589
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −63.0000 −2.33014
\(732\) 7.00000 0.258727
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 13.0000 0.479839
\(735\) −2.00000 −0.0737711
\(736\) −4.00000 −0.147442
\(737\) 20.0000 0.736709
\(738\) −1.00000 −0.0368105
\(739\) 33.0000 1.21392 0.606962 0.794731i \(-0.292388\pi\)
0.606962 + 0.794731i \(0.292388\pi\)
\(740\) 1.00000 0.0367607
\(741\) −4.00000 −0.146944
\(742\) −9.00000 −0.330400
\(743\) −13.0000 −0.476924 −0.238462 0.971152i \(-0.576643\pi\)
−0.238462 + 0.971152i \(0.576643\pi\)
\(744\) −7.00000 −0.256632
\(745\) −6.00000 −0.219823
\(746\) −22.0000 −0.805477
\(747\) −10.0000 −0.365881
\(748\) 35.0000 1.27973
\(749\) 42.0000 1.53465
\(750\) 1.00000 0.0365148
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −10.0000 −0.364179
\(755\) 10.0000 0.363937
\(756\) −3.00000 −0.109109
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −16.0000 −0.581146
\(759\) 20.0000 0.725954
\(760\) 2.00000 0.0725476
\(761\) −47.0000 −1.70375 −0.851874 0.523746i \(-0.824534\pi\)
−0.851874 + 0.523746i \(0.824534\pi\)
\(762\) 16.0000 0.579619
\(763\) −21.0000 −0.760251
\(764\) −7.00000 −0.253251
\(765\) −7.00000 −0.253086
\(766\) −4.00000 −0.144526
\(767\) 28.0000 1.01102
\(768\) −1.00000 −0.0360844
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 15.0000 0.540562
\(771\) −14.0000 −0.504198
\(772\) 14.0000 0.503871
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) −9.00000 −0.323498
\(775\) −7.00000 −0.251447
\(776\) −1.00000 −0.0358979
\(777\) −3.00000 −0.107624
\(778\) 31.0000 1.11140
\(779\) −2.00000 −0.0716574
\(780\) 2.00000 0.0716115
\(781\) −40.0000 −1.43131
\(782\) 28.0000 1.00128
\(783\) 5.00000 0.178685
\(784\) 2.00000 0.0714286
\(785\) −11.0000 −0.392607
\(786\) −6.00000 −0.214013
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 10.0000 0.356235
\(789\) −11.0000 −0.391610
\(790\) −4.00000 −0.142314
\(791\) −15.0000 −0.533339
\(792\) 5.00000 0.177667
\(793\) 14.0000 0.497155
\(794\) 34.0000 1.20661
\(795\) −3.00000 −0.106399
\(796\) 4.00000 0.141776
\(797\) 32.0000 1.13350 0.566749 0.823890i \(-0.308201\pi\)
0.566749 + 0.823890i \(0.308201\pi\)
\(798\) −6.00000 −0.212398
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) −22.0000 −0.776847
\(803\) 60.0000 2.11735
\(804\) 4.00000 0.141069
\(805\) 12.0000 0.422944
\(806\) −14.0000 −0.493129
\(807\) −16.0000 −0.563227
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −15.0000 −0.526397
\(813\) −20.0000 −0.701431
\(814\) 5.00000 0.175250
\(815\) −15.0000 −0.525427
\(816\) 7.00000 0.245049
\(817\) −18.0000 −0.629740
\(818\) −34.0000 −1.18878
\(819\) −6.00000 −0.209657
\(820\) 1.00000 0.0349215
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −22.0000 −0.767338
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −8.00000 −0.278693
\(825\) 5.00000 0.174078
\(826\) 42.0000 1.46137
\(827\) 21.0000 0.730242 0.365121 0.930960i \(-0.381028\pi\)
0.365121 + 0.930960i \(0.381028\pi\)
\(828\) 4.00000 0.139010
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 10.0000 0.347105
\(831\) −16.0000 −0.555034
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) −3.00000 −0.103882
\(835\) 20.0000 0.692129
\(836\) 10.0000 0.345857
\(837\) 7.00000 0.241955
\(838\) −20.0000 −0.690889
\(839\) 18.0000 0.621429 0.310715 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 3.00000 0.103510
\(841\) −4.00000 −0.137931
\(842\) −10.0000 −0.344623
\(843\) −10.0000 −0.344418
\(844\) −13.0000 −0.447478
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 42.0000 1.44314
\(848\) 3.00000 0.103020
\(849\) 12.0000 0.411839
\(850\) 7.00000 0.240098
\(851\) 4.00000 0.137118
\(852\) −8.00000 −0.274075
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 21.0000 0.718605
\(855\) −2.00000 −0.0683986
\(856\) −14.0000 −0.478510
\(857\) 39.0000 1.33221 0.666107 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(858\) 10.0000 0.341394
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 9.00000 0.306897
\(861\) −3.00000 −0.102240
\(862\) −21.0000 −0.715263
\(863\) −51.0000 −1.73606 −0.868030 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) 1.00000 0.0340207
\(865\) 19.0000 0.646019
\(866\) 16.0000 0.543702
\(867\) −32.0000 −1.08678
\(868\) −21.0000 −0.712786
\(869\) −20.0000 −0.678454
\(870\) −5.00000 −0.169516
\(871\) 8.00000 0.271070
\(872\) 7.00000 0.237050
\(873\) 1.00000 0.0338449
\(874\) 8.00000 0.270604
\(875\) 3.00000 0.101419
\(876\) 12.0000 0.405442
\(877\) −3.00000 −0.101303 −0.0506514 0.998716i \(-0.516130\pi\)
−0.0506514 + 0.998716i \(0.516130\pi\)
\(878\) 33.0000 1.11370
\(879\) −9.00000 −0.303562
\(880\) −5.00000 −0.168550
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −49.0000 −1.64898 −0.824491 0.565876i \(-0.808538\pi\)
−0.824491 + 0.565876i \(0.808538\pi\)
\(884\) 14.0000 0.470871
\(885\) 14.0000 0.470605
\(886\) −42.0000 −1.41102
\(887\) 15.0000 0.503651 0.251825 0.967773i \(-0.418969\pi\)
0.251825 + 0.967773i \(0.418969\pi\)
\(888\) 1.00000 0.0335578
\(889\) 48.0000 1.60987
\(890\) 14.0000 0.469281
\(891\) −5.00000 −0.167506
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) −24.0000 −0.802232
\(896\) −3.00000 −0.100223
\(897\) 8.00000 0.267112
\(898\) 12.0000 0.400445
\(899\) 35.0000 1.16732
\(900\) 1.00000 0.0333333
\(901\) −21.0000 −0.699611
\(902\) 5.00000 0.166482
\(903\) −27.0000 −0.898504
\(904\) 5.00000 0.166298
\(905\) 14.0000 0.465376
\(906\) 10.0000 0.332228
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −7.00000 −0.232303
\(909\) 0 0
\(910\) 6.00000 0.198898
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 2.00000 0.0662266
\(913\) 50.0000 1.65476
\(914\) 39.0000 1.29001
\(915\) 7.00000 0.231413
\(916\) −28.0000 −0.925146
\(917\) −18.0000 −0.594412
\(918\) −7.00000 −0.231034
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −4.00000 −0.131876
\(921\) −6.00000 −0.197707
\(922\) 15.0000 0.493999
\(923\) −16.0000 −0.526646
\(924\) 15.0000 0.493464
\(925\) 1.00000 0.0328798
\(926\) 6.00000 0.197172
\(927\) 8.00000 0.262754
\(928\) 5.00000 0.164133
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) −7.00000 −0.229539
\(931\) −4.00000 −0.131095
\(932\) −20.0000 −0.655122
\(933\) −21.0000 −0.687509
\(934\) −3.00000 −0.0981630
\(935\) 35.0000 1.14462
\(936\) 2.00000 0.0653720
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 12.0000 0.391814
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −11.0000 −0.358399
\(943\) 4.00000 0.130258
\(944\) −14.0000 −0.455661
\(945\) −3.00000 −0.0975900
\(946\) 45.0000 1.46308
\(947\) 49.0000 1.59229 0.796143 0.605108i \(-0.206870\pi\)
0.796143 + 0.605108i \(0.206870\pi\)
\(948\) −4.00000 −0.129914
\(949\) 24.0000 0.779073
\(950\) 2.00000 0.0648886
\(951\) 21.0000 0.680972
\(952\) 21.0000 0.680614
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) −3.00000 −0.0971286
\(955\) −7.00000 −0.226515
\(956\) 17.0000 0.549819
\(957\) −25.0000 −0.808135
\(958\) 24.0000 0.775405
\(959\) −66.0000 −2.13125
\(960\) −1.00000 −0.0322749
\(961\) 18.0000 0.580645
\(962\) 2.00000 0.0644826
\(963\) 14.0000 0.451144
\(964\) −2.00000 −0.0644157
\(965\) 14.0000 0.450676
\(966\) 12.0000 0.386094
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −14.0000 −0.449977
\(969\) −14.0000 −0.449745
\(970\) −1.00000 −0.0321081
\(971\) 53.0000 1.70085 0.850425 0.526096i \(-0.176345\pi\)
0.850425 + 0.526096i \(0.176345\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −9.00000 −0.288527
\(974\) 30.0000 0.961262
\(975\) 2.00000 0.0640513
\(976\) −7.00000 −0.224065
\(977\) 5.00000 0.159964 0.0799821 0.996796i \(-0.474514\pi\)
0.0799821 + 0.996796i \(0.474514\pi\)
\(978\) −15.0000 −0.479647
\(979\) 70.0000 2.23721
\(980\) 2.00000 0.0638877
\(981\) −7.00000 −0.223493
\(982\) −28.0000 −0.893516
\(983\) 45.0000 1.43528 0.717639 0.696416i \(-0.245223\pi\)
0.717639 + 0.696416i \(0.245223\pi\)
\(984\) 1.00000 0.0318788
\(985\) 10.0000 0.318626
\(986\) −35.0000 −1.11463
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 36.0000 1.14473
\(990\) 5.00000 0.158910
\(991\) 19.0000 0.603555 0.301777 0.953378i \(-0.402420\pi\)
0.301777 + 0.953378i \(0.402420\pi\)
\(992\) 7.00000 0.222250
\(993\) 0 0
\(994\) −24.0000 −0.761234
\(995\) 4.00000 0.126809
\(996\) 10.0000 0.316862
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) −4.00000 −0.126618
\(999\) −1.00000 −0.0316386
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 1110.2.a.d.1.1 1
3.2 odd 2 3330.2.a.q.1.1 1
4.3 odd 2 8880.2.a.z.1.1 1
5.4 even 2 5550.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.d.1.1 1 1.1 even 1 trivial
3330.2.a.q.1.1 1 3.2 odd 2
5550.2.a.bf.1.1 1 5.4 even 2
8880.2.a.z.1.1 1 4.3 odd 2