Properties

Label 3330.2.a.q.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +5.00000 q^{11} -2.00000 q^{13} +3.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} -2.00000 q^{19} -1.00000 q^{20} +5.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +3.00000 q^{28} +5.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} +7.00000 q^{34} -3.00000 q^{35} +1.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} -1.00000 q^{41} +9.00000 q^{43} +5.00000 q^{44} -4.00000 q^{46} +2.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -3.00000 q^{53} -5.00000 q^{55} +3.00000 q^{56} +5.00000 q^{58} +14.0000 q^{59} -7.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} +7.00000 q^{68} -3.00000 q^{70} -8.00000 q^{71} -12.0000 q^{73} +1.00000 q^{74} -2.00000 q^{76} +15.0000 q^{77} +4.00000 q^{79} -1.00000 q^{80} -1.00000 q^{82} +10.0000 q^{83} -7.00000 q^{85} +9.00000 q^{86} +5.00000 q^{88} +14.0000 q^{89} -6.00000 q^{91} -4.00000 q^{92} +2.00000 q^{95} +1.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 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 \(0.589456\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) −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
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −1.00000 −0.156174 −0.0780869 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) 5.00000 0.656532
\(59\) 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 \(0.647910\pi\)
\(62\) −7.00000 −0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 7.00000 0.848875
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −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 \(0.747820\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 15.0000 1.70941
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −1.00000 −0.110432
\(83\) 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
\(86\) 9.00000 0.970495
\(87\) 0 0
\(88\) 5.00000 0.533002
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 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 \(0.483833\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −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 \(0.608817\pi\)
\(110\) −5.00000 −0.476731
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) 5.00000 0.470360 0.235180 0.971952i \(-0.424432\pi\)
0.235180 + 0.971952i \(0.424432\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 14.0000 1.28880
\(119\) 21.0000 1.92507
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −7.00000 −0.633750
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 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
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) −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
\(144\) 0 0
\(145\) −5.00000 −0.415227
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) 0 0
\(154\) 15.0000 1.20873
\(155\) 7.00000 0.562254
\(156\) 0 0
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −15.0000 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) −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
\(170\) −7.00000 −0.536875
\(171\) 0 0
\(172\) 9.00000 0.686244
\(173\) −19.0000 −1.44454 −0.722272 0.691609i \(-0.756902\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 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 \(0.325818\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 35.0000 2.55945
\(188\) 0 0
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 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 \(0.331909\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 0 0
\(203\) 15.0000 1.05279
\(204\) 0 0
\(205\) 1.00000 0.0698430
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −3.00000 −0.206041
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) −9.00000 −0.613795
\(216\) 0 0
\(217\) −21.0000 −1.42557
\(218\) −7.00000 −0.474100
\(219\) 0 0
\(220\) −5.00000 −0.337100
\(221\) −14.0000 −0.941742
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) 5.00000 0.332595
\(227\) 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 \(0.876058\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 5.00000 0.328266
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) 21.0000 1.36123
\(239\) −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 \(0.520518\pi\)
\(242\) 14.0000 0.899954
\(243\) 0 0
\(244\) −7.00000 −0.448129
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) −7.00000 −0.444500
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −20.0000 −1.25739
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −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
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −16.0000 −0.975537 −0.487769 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 7.00000 0.424437
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) −3.00000 −0.179928
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 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
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) −5.00000 −0.293610
\(291\) 0 0
\(292\) −12.0000 −0.702247
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 27.0000 1.55625
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 15.0000 0.854704
\(309\) 0 0
\(310\) 7.00000 0.397573
\(311\) −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 \(0.591202\pi\)
\(314\) −11.0000 −0.620766
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 25.0000 1.39973
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −12.0000 −0.668734
\(323\) −14.0000 −0.778981
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −15.0000 −0.830773
\(327\) 0 0
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 10.0000 0.548821
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −7.00000 −0.379628
\(341\) −35.0000 −1.89536
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 9.00000 0.485247
\(345\) 0 0
\(346\) −19.0000 −1.02145
\(347\) 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 \(0.640863\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) 5.00000 0.266501
\(353\) −17.0000 −0.904819 −0.452409 0.891810i \(-0.649435\pi\)
−0.452409 + 0.891810i \(0.649435\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −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
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 35.0000 1.80981
\(375\) 0 0
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 7.00000 0.358151
\(383\) −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
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 1.00000 0.0507673
\(389\) 31.0000 1.57176 0.785881 0.618378i \(-0.212210\pi\)
0.785881 + 0.618378i \(0.212210\pi\)
\(390\) 0 0
\(391\) −28.0000 −1.41602
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 14.0000 0.697390
\(404\) 0 0
\(405\) 0 0
\(406\) 15.0000 0.744438
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 1.00000 0.0493865
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 42.0000 2.06668
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) −10.0000 −0.489116
\(419\) −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 \(0.421644\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) −3.00000 −0.145693
\(425\) 7.00000 0.339550
\(426\) 0 0
\(427\) −21.0000 −1.01626
\(428\) −14.0000 −0.676716
\(429\) 0 0
\(430\) −9.00000 −0.434019
\(431\) −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 \(0.625611\pi\)
\(434\) −21.0000 −1.00803
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −33.0000 −1.57500 −0.787502 0.616312i \(-0.788626\pi\)
−0.787502 + 0.616312i \(0.788626\pi\)
\(440\) −5.00000 −0.238366
\(441\) 0 0
\(442\) −14.0000 −0.665912
\(443\) −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
\(446\) 9.00000 0.426162
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) 5.00000 0.235180
\(453\) 0 0
\(454\) 7.00000 0.328526
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) −39.0000 −1.82434 −0.912172 0.409809i \(-0.865595\pi\)
−0.912172 + 0.409809i \(0.865595\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 15.0000 0.698620 0.349310 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) −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
\(471\) 0 0
\(472\) 14.0000 0.644402
\(473\) 45.0000 2.06910
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 21.0000 0.962533
\(477\) 0 0
\(478\) −17.0000 −0.777562
\(479\) 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
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) −30.0000 −1.35943 −0.679715 0.733476i \(-0.737896\pi\)
−0.679715 + 0.733476i \(0.737896\pi\)
\(488\) −7.00000 −0.316875
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −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
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 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
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.0000 −0.889108
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 3.00000 0.131812
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −11.0000 −0.479623
\(527\) −49.0000 −2.13447
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 3.00000 0.130312
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 14.0000 0.605273
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) −16.0000 −0.689809
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −7.00000 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(548\) 22.0000 0.939793
\(549\) 0 0
\(550\) 5.00000 0.213201
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 16.0000 0.679775
\(555\) 0 0
\(556\) −3.00000 −0.127228
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) 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
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 27.0000 1.12991 0.564957 0.825120i \(-0.308893\pi\)
0.564957 + 0.825120i \(0.308893\pi\)
\(572\) −10.0000 −0.418121
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) −5.00000 −0.207614
\(581\) 30.0000 1.24461
\(582\) 0 0
\(583\) −15.0000 −0.621237
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) 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
\(590\) −14.0000 −0.576371
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) −21.0000 −0.860916
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 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 \(0.545600\pi\)
\(602\) 27.0000 1.10044
\(603\) 0 0
\(604\) 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
\(609\) 0 0
\(610\) 7.00000 0.283422
\(611\) 0 0
\(612\) 0 0
\(613\) 43.0000 1.73675 0.868377 0.495905i \(-0.165164\pi\)
0.868377 + 0.495905i \(0.165164\pi\)
\(614\) 6.00000 0.242140
\(615\) 0 0
\(616\) 15.0000 0.604367
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 5.00000 0.200967 0.100483 0.994939i \(-0.467961\pi\)
0.100483 + 0.994939i \(0.467961\pi\)
\(620\) 7.00000 0.281127
\(621\) 0 0
\(622\) −21.0000 −0.842023
\(623\) 42.0000 1.68269
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −11.0000 −0.438948
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 21.0000 0.834017
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 25.0000 0.989759
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −41.0000 −1.61940 −0.809701 0.586842i \(-0.800371\pi\)
−0.809701 + 0.586842i \(0.800371\pi\)
\(642\) 0 0
\(643\) −43.0000 −1.69575 −0.847877 0.530193i \(-0.822120\pi\)
−0.847877 + 0.530193i \(0.822120\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −14.0000 −0.550823
\(647\) −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
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −15.0000 −0.587445
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) −1.00000 −0.0390434
\(657\) 0 0
\(658\) 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
\(663\) 0 0
\(664\) 10.0000 0.388075
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −20.0000 −0.774403
\(668\) −20.0000 −0.773823
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −35.0000 −1.35116
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) 3.00000 0.115129
\(680\) −7.00000 −0.268438
\(681\) 0 0
\(682\) −35.0000 −1.34022
\(683\) 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
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 9.00000 0.343122
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) −19.0000 −0.722272
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 3.00000 0.113796
\(696\) 0 0
\(697\) −7.00000 −0.265144
\(698\) −16.0000 −0.605609
\(699\) 0 0
\(700\) 3.00000 0.113389
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −17.0000 −0.639803
\(707\) 0 0
\(708\) 0 0
\(709\) −27.0000 −1.01401 −0.507003 0.861944i \(-0.669247\pi\)
−0.507003 + 0.861944i \(0.669247\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 10.0000 0.373979
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) −34.0000 −1.26887
\(719\) −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
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 5.00000 0.185695
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) −6.00000 −0.222375
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) 63.0000 2.33014
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) −13.0000 −0.479839
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) 33.0000 1.21392 0.606962 0.794731i \(-0.292388\pi\)
0.606962 + 0.794731i \(0.292388\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −9.00000 −0.330400
\(743\) 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
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 35.0000 1.27973
\(749\) −42.0000 −1.53465
\(750\) 0 0
\(751\) 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
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 47.0000 1.70375 0.851874 0.523746i \(-0.175466\pi\)
0.851874 + 0.523746i \(0.175466\pi\)
\(762\) 0 0
\(763\) −21.0000 −0.760251
\(764\) 7.00000 0.253251
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) −28.0000 −1.01102
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) −15.0000 −0.540562
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) −33.0000 −1.18693 −0.593464 0.804861i \(-0.702240\pi\)
−0.593464 + 0.804861i \(0.702240\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) 31.0000 1.11140
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) −28.0000 −1.00128
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 11.0000 0.392607
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −32.0000 −1.13350 −0.566749 0.823890i \(-0.691799\pi\)
−0.566749 + 0.823890i \(0.691799\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −22.0000 −0.776847
\(803\) −60.0000 −2.11735
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 14.0000 0.493129
\(807\) 0 0
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 15.0000 0.526397
\(813\) 0 0
\(814\) 5.00000 0.175250
\(815\) 15.0000 0.525427
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) 34.0000 1.18878
\(819\) 0 0
\(820\) 1.00000 0.0349215
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 42.0000 1.46137
\(827\) −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 \(0.332006\pi\)
\(830\) −10.0000 −0.347105
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 14.0000 0.485071
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) −10.0000 −0.345857
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −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
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 42.0000 1.44314
\(848\) −3.00000 −0.103020
\(849\) 0 0
\(850\) 7.00000 0.240098
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −21.0000 −0.718605
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) −39.0000 −1.33221 −0.666107 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) −9.00000 −0.306897
\(861\) 0 0
\(862\) −21.0000 −0.715263
\(863\) 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
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) −21.0000 −0.712786
\(869\) 20.0000 0.678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −7.00000 −0.237050
\(873\) 0 0
\(874\) 8.00000 0.270604
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −3.00000 −0.101303 −0.0506514 0.998716i \(-0.516130\pi\)
−0.0506514 + 0.998716i \(0.516130\pi\)
\(878\) −33.0000 −1.11370
\(879\) 0 0
\(880\) −5.00000 −0.168550
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) −49.0000 −1.64898 −0.824491 0.565876i \(-0.808538\pi\)
−0.824491 + 0.565876i \(0.808538\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −42.0000 −1.41102
\(887\) −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
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −35.0000 −1.16732
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) −5.00000 −0.166482
\(903\) 0 0
\(904\) 5.00000 0.166298
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −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 \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 50.0000 1.65476
\(914\) −39.0000 −1.29001
\(915\) 0 0
\(916\) −28.0000 −0.925146
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 15.0000 0.493999
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −6.00000 −0.197172
\(927\) 0 0
\(928\) 5.00000 0.164133
\(929\) 41.0000 1.34517 0.672583 0.740022i \(-0.265185\pi\)
0.672583 + 0.740022i \(0.265185\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 20.0000 0.655122
\(933\) 0 0
\(934\) −3.00000 −0.0981630
\(935\) −35.0000 −1.14462
\(936\) 0 0
\(937\) 32.0000 1.04539 0.522697 0.852518i \(-0.324926\pi\)
0.522697 + 0.852518i \(0.324926\pi\)
\(938\) −12.0000 −0.391814
\(939\) 0 0
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 14.0000 0.455661
\(945\) 0 0
\(946\) 45.0000 1.46308
\(947\) −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
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 21.0000 0.680614
\(953\) 4.00000 0.129573 0.0647864 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) 0 0
\(955\) −7.00000 −0.226515
\(956\) −17.0000 −0.549819
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 66.0000 2.13125
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −2.00000 −0.0644826
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) 14.0000 0.449977
\(969\) 0 0
\(970\) −1.00000 −0.0321081
\(971\) −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
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −5.00000 −0.159964 −0.0799821 0.996796i \(-0.525486\pi\)
−0.0799821 + 0.996796i \(0.525486\pi\)
\(978\) 0 0
\(979\) 70.0000 2.23721
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) −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
\(986\) 35.0000 1.11463
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −36.0000 −1.14473
\(990\) 0 0
\(991\) 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
\(996\) 0 0
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
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 3330.2.a.q.1.1 1
3.2 odd 2 1110.2.a.d.1.1 1
12.11 even 2 8880.2.a.z.1.1 1
15.14 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 3.2 odd 2
3330.2.a.q.1.1 1 1.1 even 1 trivial
5550.2.a.bf.1.1 1 15.14 odd 2
8880.2.a.z.1.1 1 12.11 even 2