Properties

Label 8880.2.a.z.1.1
Level $8880$
Weight $2$
Character 8880.1
Self dual yes
Analytic conductor $70.907$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.9071569949\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8880.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} -2.00000 q^{13} +1.00000 q^{15} -7.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -5.00000 q^{29} +7.00000 q^{31} +5.00000 q^{33} -3.00000 q^{35} +1.00000 q^{37} -2.00000 q^{39} +1.00000 q^{41} -9.00000 q^{43} +1.00000 q^{45} +2.00000 q^{49} -7.00000 q^{51} +3.00000 q^{53} +5.00000 q^{55} +2.00000 q^{57} +14.0000 q^{59} -7.00000 q^{61} -3.00000 q^{63} -2.00000 q^{65} +4.00000 q^{67} -4.00000 q^{69} -8.00000 q^{71} -12.0000 q^{73} +1.00000 q^{75} -15.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +10.0000 q^{83} -7.00000 q^{85} -5.00000 q^{87} -14.0000 q^{89} +6.00000 q^{91} +7.00000 q^{93} +2.00000 q^{95} +1.00000 q^{97} +5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −15.0000 −1.70941
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) −7.00000 −0.759257
\(86\) 0 0
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 7.00000 0.725866
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 21.0000 1.92507
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 1.00000 0.0901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) 7.00000 0.562254
\(156\) 0 0
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 15.0000 1.17489 0.587445 0.809264i \(-0.300134\pi\)
0.587445 + 0.809264i \(0.300134\pi\)
\(164\) 0 0
\(165\) 5.00000 0.389249
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 19.0000 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) 0 0
\(175\) −3.00000 −0.226779
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −35.0000 −2.55945
\(188\) 0 0
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 7.00000 0.506502 0.253251 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 15.0000 1.05279
\(204\) 0 0
\(205\) 1.00000 0.0698430
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −9.00000 −0.613795
\(216\) 0 0
\(217\) −21.0000 −1.42557
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 14.0000 0.941742
\(222\) 0 0
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 7.00000 0.464606 0.232303 0.972643i \(-0.425374\pi\)
0.232303 + 0.972643i \(0.425374\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 0 0
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −17.0000 −1.09964 −0.549819 0.835284i \(-0.685303\pi\)
−0.549819 + 0.835284i \(0.685303\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 0 0
\(255\) −7.00000 −0.438357
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) 0 0
\(265\) 3.00000 0.184289
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 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 \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) 0 0
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 14.0000 0.815112
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 27.0000 1.55625
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) −25.0000 −1.39973
\(320\) 0 0
\(321\) −14.0000 −0.781404
\(322\) 0 0
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) −7.00000 −0.387101
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 1.00000 0.0547997
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −5.00000 −0.271563
\(340\) 0 0
\(341\) 35.0000 1.89536
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 17.0000 0.904819 0.452409 0.891810i \(-0.350565\pi\)
0.452409 + 0.891810i \(0.350565\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 21.0000 1.11144
\(358\) 0 0
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 14.0000 0.734809
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) 0 0
\(369\) 1.00000 0.0520579
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) −15.0000 −0.764471
\(386\) 0 0
\(387\) −9.00000 −0.457496
\(388\) 0 0
\(389\) −31.0000 −1.57176 −0.785881 0.618378i \(-0.787790\pi\)
−0.785881 + 0.618378i \(0.787790\pi\)
\(390\) 0 0
\(391\) 28.0000 1.41602
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) 0 0
\(413\) −42.0000 −2.06668
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 0 0
\(417\) 3.00000 0.146911
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.00000 −0.339550
\(426\) 0 0
\(427\) 21.0000 1.01626
\(428\) 0 0
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) −5.00000 −0.239732
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 33.0000 1.57500 0.787502 0.616312i \(-0.211374\pi\)
0.787502 + 0.616312i \(0.211374\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −42.0000 −1.99548 −0.997740 0.0671913i \(-0.978596\pi\)
−0.997740 + 0.0671913i \(0.978596\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 5.00000 0.235441
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −39.0000 −1.82434 −0.912172 0.409809i \(-0.865595\pi\)
−0.912172 + 0.409809i \(0.865595\pi\)
\(458\) 0 0
\(459\) −7.00000 −0.326732
\(460\) 0 0
\(461\) −15.0000 −0.698620 −0.349310 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) 0 0
\(465\) 7.00000 0.324617
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) 0 0
\(473\) −45.0000 −2.06910
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) 30.0000 1.35943 0.679715 0.733476i \(-0.262104\pi\)
0.679715 + 0.733476i \(0.262104\pi\)
\(488\) 0 0
\(489\) 15.0000 0.678323
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 35.0000 1.57632
\(494\) 0 0
\(495\) 5.00000 0.224733
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 36.0000 1.59255
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 19.0000 0.834007
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −3.00000 −0.130931
\(526\) 0 0
\(527\) −49.0000 −2.13447
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −14.0000 −0.605273
\(536\) 0 0
\(537\) 24.0000 1.03568
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 7.00000 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(548\) 0 0
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) 1.00000 0.0424476
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) −35.0000 −1.47770
\(562\) 0 0
\(563\) 3.00000 0.126435 0.0632175 0.998000i \(-0.479864\pi\)
0.0632175 + 0.998000i \(0.479864\pi\)
\(564\) 0 0
\(565\) −5.00000 −0.210352
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −27.0000 −1.12991 −0.564957 0.825120i \(-0.691107\pi\)
−0.564957 + 0.825120i \(0.691107\pi\)
\(572\) 0 0
\(573\) 7.00000 0.292429
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −30.0000 −1.24461
\(582\) 0 0
\(583\) 15.0000 0.621237
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) 21.0000 0.866763 0.433381 0.901211i \(-0.357320\pi\)
0.433381 + 0.901211i \(0.357320\pi\)
\(588\) 0 0
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 46.0000 1.88899 0.944497 0.328521i \(-0.106550\pi\)
0.944497 + 0.328521i \(0.106550\pi\)
\(594\) 0 0
\(595\) 21.0000 0.860916
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 15.0000 0.607831
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 43.0000 1.73675 0.868377 0.495905i \(-0.165164\pi\)
0.868377 + 0.495905i \(0.165164\pi\)
\(614\) 0 0
\(615\) 1.00000 0.0403239
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −5.00000 −0.200967 −0.100483 0.994939i \(-0.532039\pi\)
−0.100483 + 0.994939i \(0.532039\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 42.0000 1.68269
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 10.0000 0.399362
\(628\) 0 0
\(629\) −7.00000 −0.279108
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 0 0
\(643\) 43.0000 1.69575 0.847877 0.530193i \(-0.177880\pi\)
0.847877 + 0.530193i \(0.177880\pi\)
\(644\) 0 0
\(645\) −9.00000 −0.354375
\(646\) 0 0
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) 0 0
\(649\) 70.0000 2.74774
\(650\) 0 0
\(651\) −21.0000 −0.823055
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 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
\(664\) 0 0
\(665\) −6.00000 −0.232670
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) 0 0
\(669\) −9.00000 −0.347960
\(670\) 0 0
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) 0 0
\(683\) 47.0000 1.79841 0.899203 0.437533i \(-0.144148\pi\)
0.899203 + 0.437533i \(0.144148\pi\)
\(684\) 0 0
\(685\) −22.0000 −0.840577
\(686\) 0 0
\(687\) −28.0000 −1.06827
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 41.0000 1.55971 0.779857 0.625958i \(-0.215292\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 0 0
\(693\) −15.0000 −0.569803
\(694\) 0 0
\(695\) 3.00000 0.113796
\(696\) 0 0
\(697\) −7.00000 −0.265144
\(698\) 0 0
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −27.0000 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) −10.0000 −0.373979
\(716\) 0 0
\(717\) −17.0000 −0.634877
\(718\) 0 0
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 63.0000 2.33014
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 20.0000 0.736709
\(738\) 0 0
\(739\) −33.0000 −1.21392 −0.606962 0.794731i \(-0.707612\pi\)
−0.606962 + 0.794731i \(0.707612\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 13.0000 0.476924 0.238462 0.971152i \(-0.423357\pi\)
0.238462 + 0.971152i \(0.423357\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 10.0000 0.365881
\(748\) 0 0
\(749\) 42.0000 1.53465
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) −47.0000 −1.70375 −0.851874 0.523746i \(-0.824534\pi\)
−0.851874 + 0.523746i \(0.824534\pi\)
\(762\) 0 0
\(763\) 21.0000 0.760251
\(764\) 0 0
\(765\) −7.00000 −0.253086
\(766\) 0 0
\(767\) −28.0000 −1.01102
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 33.0000 1.18693 0.593464 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) 0 0
\(777\) −3.00000 −0.107624
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) −11.0000 −0.392607
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 0 0
\(795\) 3.00000 0.106399
\(796\) 0 0
\(797\) 32.0000 1.13350 0.566749 0.823890i \(-0.308201\pi\)
0.566749 + 0.823890i \(0.308201\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −60.0000 −2.11735
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 16.0000 0.563227
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) 15.0000 0.525427
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) 0 0
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) −21.0000 −0.730242 −0.365121 0.930960i \(-0.618972\pi\)
−0.365121 + 0.930960i \(0.618972\pi\)
\(828\) 0 0
\(829\) 29.0000 1.00721 0.503606 0.863934i \(-0.332006\pi\)
0.503606 + 0.863934i \(0.332006\pi\)
\(830\) 0 0
\(831\) 16.0000 0.555034
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) −20.0000 −0.692129
\(836\) 0 0
\(837\) 7.00000 0.241955
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 10.0000 0.344418
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) 39.0000 1.33221 0.666107 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(858\) 0 0
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 0 0
\(861\) −3.00000 −0.102240
\(862\) 0 0
\(863\) 51.0000 1.73606 0.868030 0.496512i \(-0.165386\pi\)
0.868030 + 0.496512i \(0.165386\pi\)
\(864\) 0 0
\(865\) 19.0000 0.646019
\(866\) 0 0
\(867\) 32.0000 1.08678
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 1.00000 0.0338449
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −3.00000 −0.101303 −0.0506514 0.998716i \(-0.516130\pi\)
−0.0506514 + 0.998716i \(0.516130\pi\)
\(878\) 0 0
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 49.0000 1.64898 0.824491 0.565876i \(-0.191462\pi\)
0.824491 + 0.565876i \(0.191462\pi\)
\(884\) 0 0
\(885\) 14.0000 0.470605
\(886\) 0 0
\(887\) −15.0000 −0.503651 −0.251825 0.967773i \(-0.581031\pi\)
−0.251825 + 0.967773i \(0.581031\pi\)
\(888\) 0 0
\(889\) 48.0000 1.60987
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) −35.0000 −1.16732
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) 0 0
\(903\) 27.0000 0.898504
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 36.0000 1.19536 0.597680 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 50.0000 1.65476
\(914\) 0 0
\(915\) −7.00000 −0.231413
\(916\) 0 0
\(917\) −18.0000 −0.594412
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) 0 0
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −41.0000 −1.34517 −0.672583 0.740022i \(-0.734815\pi\)
−0.672583 + 0.740022i \(0.734815\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) −21.0000 −0.687509
\(934\) 0 0
\(935\) −35.0000 −1.14462
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) −4.00000 −0.130258
\(944\) 0 0
\(945\) −3.00000 −0.0975900
\(946\) 0 0
\(947\) −49.0000 −1.59229 −0.796143 0.605108i \(-0.793130\pi\)
−0.796143 + 0.605108i \(0.793130\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) −21.0000 −0.680972
\(952\) 0 0
\(953\) −4.00000 −0.129573 −0.0647864 0.997899i \(-0.520637\pi\)
−0.0647864 + 0.997899i \(0.520637\pi\)
\(954\) 0 0
\(955\) 7.00000 0.226515
\(956\) 0 0
\(957\) −25.0000 −0.808135
\(958\) 0 0
\(959\) 66.0000 2.13125
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −14.0000 −0.451144
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) −14.0000 −0.449745
\(970\) 0 0
\(971\) −53.0000 −1.70085 −0.850425 0.526096i \(-0.823655\pi\)
−0.850425 + 0.526096i \(0.823655\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) 5.00000 0.159964 0.0799821 0.996796i \(-0.474514\pi\)
0.0799821 + 0.996796i \(0.474514\pi\)
\(978\) 0 0
\(979\) −70.0000 −2.23721
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 0 0
\(999\) 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 8880.2.a.z.1.1 1
4.3 odd 2 1110.2.a.d.1.1 1
12.11 even 2 3330.2.a.q.1.1 1
20.19 odd 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 4.3 odd 2
3330.2.a.q.1.1 1 12.11 even 2
5550.2.a.bf.1.1 1 20.19 odd 2
8880.2.a.z.1.1 1 1.1 even 1 trivial