Properties

Label 3150.2.g.u.2899.1
Level 3150
Weight 2
Character 3150.2899
Analytic conductor 25.153
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(1.00000i\)
Character \(\chi\) = 3150.2899
Dual form 3150.2.g.u.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} -3.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +6.00000 q^{19} -4.00000i q^{22} -9.00000i q^{23} -3.00000 q^{26} -1.00000i q^{28} -3.00000 q^{29} -7.00000 q^{31} -1.00000i q^{32} +7.00000 q^{34} +10.0000i q^{37} -6.00000i q^{38} -1.00000 q^{41} +13.0000i q^{43} -4.00000 q^{44} -9.00000 q^{46} -2.00000i q^{47} -1.00000 q^{49} +3.00000i q^{52} +1.00000i q^{53} -1.00000 q^{56} +3.00000i q^{58} +11.0000 q^{59} +13.0000 q^{61} +7.00000i q^{62} -1.00000 q^{64} -7.00000i q^{68} +8.00000 q^{71} +8.00000i q^{73} +10.0000 q^{74} -6.00000 q^{76} +4.00000i q^{77} -4.00000 q^{79} +1.00000i q^{82} -7.00000i q^{83} +13.0000 q^{86} +4.00000i q^{88} +14.0000 q^{89} +3.00000 q^{91} +9.00000i q^{92} -2.00000 q^{94} -8.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 8q^{11} + 2q^{14} + 2q^{16} + 12q^{19} - 6q^{26} - 6q^{29} - 14q^{31} + 14q^{34} - 2q^{41} - 8q^{44} - 18q^{46} - 2q^{49} - 2q^{56} + 22q^{59} + 26q^{61} - 2q^{64} + 16q^{71} + 20q^{74} - 12q^{76} - 8q^{79} + 26q^{86} + 28q^{89} + 6q^{91} - 4q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) − 3.00000i − 0.832050i −0.909353 0.416025i \(-0.863423\pi\)
0.909353 0.416025i \(-0.136577\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000i 1.69775i 0.528594 + 0.848875i \(0.322719\pi\)
−0.528594 + 0.848875i \(0.677281\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 9.00000i − 1.87663i −0.345782 0.938315i \(-0.612386\pi\)
0.345782 0.938315i \(-0.387614\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −0.156174 −0.0780869 0.996947i \(-0.524881\pi\)
−0.0780869 + 0.996947i \(0.524881\pi\)
\(42\) 0 0
\(43\) 13.0000i 1.98248i 0.132068 + 0.991241i \(0.457838\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 3.00000i 0.416025i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000i 0.393919i
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 7.00000i 0.889001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 7.00000i − 0.848875i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.00000i 0.110432i
\(83\) − 7.00000i − 0.768350i −0.923260 0.384175i \(-0.874486\pi\)
0.923260 0.384175i \(-0.125514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 13.0000 1.40183
\(87\) 0 0
\(88\) 4.00000i 0.426401i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 9.00000i 0.938315i
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) − 11.0000i − 1.08386i −0.840423 0.541931i \(-0.817693\pi\)
0.840423 0.541931i \(-0.182307\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) − 16.0000i − 1.50515i −0.658505 0.752577i \(-0.728811\pi\)
0.658505 0.752577i \(-0.271189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) − 11.0000i − 1.01263i
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 13.0000i − 1.17696i
\(123\) 0 0
\(124\) 7.00000 0.628619
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 8.00000i − 0.671345i
\(143\) − 12.0000i − 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) − 10.0000i − 0.821995i
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 6.00000i 0.486664i
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 4.00000i 0.318223i
\(159\) 0 0
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) 1.00000 0.0780869
\(165\) 0 0
\(166\) −7.00000 −0.543305
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 4.00000 0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) − 13.0000i − 0.991241i
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) − 14.0000i − 1.04934i
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) − 3.00000i − 0.222375i
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) 0 0
\(187\) 28.0000i 2.04756i
\(188\) 2.00000i 0.145865i
\(189\) 0 0
\(190\) 0 0
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 5.00000i − 0.356235i −0.984009 0.178118i \(-0.942999\pi\)
0.984009 0.178118i \(-0.0570008\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000i 0.422159i
\(203\) − 3.00000i − 0.210559i
\(204\) 0 0
\(205\) 0 0
\(206\) −11.0000 −0.766406
\(207\) 0 0
\(208\) − 3.00000i − 0.208013i
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) − 1.00000i − 0.0686803i
\(213\) 0 0
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) − 7.00000i − 0.475191i
\(218\) − 6.00000i − 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) 21.0000 1.41261
\(222\) 0 0
\(223\) 13.0000i 0.870544i 0.900299 + 0.435272i \(0.143348\pi\)
−0.900299 + 0.435272i \(0.856652\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −16.0000 −1.06430
\(227\) − 11.0000i − 0.730096i −0.930989 0.365048i \(-0.881053\pi\)
0.930989 0.365048i \(-0.118947\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −11.0000 −0.716039
\(237\) 0 0
\(238\) 7.00000i 0.453743i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 0 0
\(244\) −13.0000 −0.832240
\(245\) 0 0
\(246\) 0 0
\(247\) − 18.0000i − 1.14531i
\(248\) − 7.00000i − 0.444500i
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) − 36.0000i − 2.26330i
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 17.0000i − 1.06043i −0.847863 0.530215i \(-0.822111\pi\)
0.847863 0.530215i \(-0.177889\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) 0 0
\(262\) − 20.0000i − 1.23560i
\(263\) − 13.0000i − 0.801614i −0.916162 0.400807i \(-0.868730\pi\)
0.916162 0.400807i \(-0.131270\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 7.00000i 0.424437i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) − 12.0000i − 0.721010i −0.932757 0.360505i \(-0.882604\pi\)
0.932757 0.360505i \(-0.117396\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 0 0
\(280\) 0 0
\(281\) 32.0000 1.90896 0.954480 0.298275i \(-0.0964112\pi\)
0.954480 + 0.298275i \(0.0964112\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) − 1.00000i − 0.0590281i
\(288\) 0 0
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) 0 0
\(292\) − 8.00000i − 0.468165i
\(293\) 28.0000i 1.63578i 0.575376 + 0.817889i \(0.304856\pi\)
−0.575376 + 0.817889i \(0.695144\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) − 9.00000i − 0.521356i
\(299\) −27.0000 −1.56145
\(300\) 0 0
\(301\) −13.0000 −0.749308
\(302\) − 2.00000i − 0.115087i
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 0 0
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) − 4.00000i − 0.227921i
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) − 16.0000i − 0.904373i −0.891923 0.452187i \(-0.850644\pi\)
0.891923 0.452187i \(-0.149356\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 11.0000i 0.617822i 0.951091 + 0.308911i \(0.0999645\pi\)
−0.951091 + 0.308911i \(0.900036\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) − 9.00000i − 0.501550i
\(323\) 42.0000i 2.33694i
\(324\) 0 0
\(325\) 0 0
\(326\) 11.0000 0.609234
\(327\) 0 0
\(328\) − 1.00000i − 0.0552158i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −5.00000 −0.274825 −0.137412 0.990514i \(-0.543879\pi\)
−0.137412 + 0.990514i \(0.543879\pi\)
\(332\) 7.00000i 0.384175i
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 5.00000i 0.272367i 0.990684 + 0.136184i \(0.0434837\pi\)
−0.990684 + 0.136184i \(0.956516\pi\)
\(338\) − 4.00000i − 0.217571i
\(339\) 0 0
\(340\) 0 0
\(341\) −28.0000 −1.51629
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −13.0000 −0.700913
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) 0 0
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 4.00000i − 0.213201i
\(353\) − 30.0000i − 1.59674i −0.602168 0.798369i \(-0.705696\pi\)
0.602168 0.798369i \(-0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) − 6.00000i − 0.317110i
\(359\) −11.0000 −0.580558 −0.290279 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000i 0.105118i
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) 0 0
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) − 9.00000i − 0.469157i
\(369\) 0 0
\(370\) 0 0
\(371\) −1.00000 −0.0519174
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 28.0000 1.44785
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) 9.00000i 0.463524i
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 21.0000i 1.07445i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) 8.00000i 0.406138i
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 63.0000 3.18605
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) −5.00000 −0.251896
\(395\) 0 0
\(396\) 0 0
\(397\) − 19.0000i − 0.953583i −0.879017 0.476791i \(-0.841800\pi\)
0.879017 0.476791i \(-0.158200\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) 4.00000 0.199750 0.0998752 0.995000i \(-0.468156\pi\)
0.0998752 + 0.995000i \(0.468156\pi\)
\(402\) 0 0
\(403\) 21.0000i 1.04608i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 11.0000i 0.541931i
\(413\) 11.0000i 0.541275i
\(414\) 0 0
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 0 0
\(418\) − 24.0000i − 1.17388i
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) − 13.0000i − 0.632830i
\(423\) 0 0
\(424\) −1.00000 −0.0485643
\(425\) 0 0
\(426\) 0 0
\(427\) 13.0000i 0.629114i
\(428\) 6.00000i 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) −7.00000 −0.336011
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) − 54.0000i − 2.58317i
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 21.0000i − 0.998868i
\(443\) − 2.00000i − 0.0950229i −0.998871 0.0475114i \(-0.984871\pi\)
0.998871 0.0475114i \(-0.0151291\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.0000 0.615568
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 16.0000i 0.752577i
\(453\) 0 0
\(454\) −11.0000 −0.516256
\(455\) 0 0
\(456\) 0 0
\(457\) 17.0000i 0.795226i 0.917553 + 0.397613i \(0.130161\pi\)
−0.917553 + 0.397613i \(0.869839\pi\)
\(458\) 22.0000i 1.02799i
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 5.00000i − 0.231372i −0.993286 0.115686i \(-0.963093\pi\)
0.993286 0.115686i \(-0.0369067\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 11.0000i 0.506316i
\(473\) 52.0000i 2.39096i
\(474\) 0 0
\(475\) 0 0
\(476\) 7.00000 0.320844
\(477\) 0 0
\(478\) − 8.00000i − 0.365911i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 30.0000 1.36788
\(482\) − 22.0000i − 1.00207i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) 13.0000i 0.588482i
\(489\) 0 0
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) − 21.0000i − 0.945792i
\(494\) −18.0000 −0.809858
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) 21.0000 0.940089 0.470045 0.882643i \(-0.344238\pi\)
0.470045 + 0.882643i \(0.344238\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 21.0000i − 0.937276i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) − 14.0000i − 0.621150i
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −17.0000 −0.749838
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.00000i − 0.351840i
\(518\) 10.0000i 0.439375i
\(519\) 0 0
\(520\) 0 0
\(521\) −43.0000 −1.88386 −0.941932 0.335803i \(-0.890992\pi\)
−0.941932 + 0.335803i \(0.890992\pi\)
\(522\) 0 0
\(523\) − 14.0000i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) −13.0000 −0.566827
\(527\) − 49.0000i − 2.13447i
\(528\) 0 0
\(529\) −58.0000 −2.52174
\(530\) 0 0
\(531\) 0 0
\(532\) − 6.00000i − 0.260133i
\(533\) 3.00000i 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 6.00000i 0.258678i
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) 0 0
\(547\) 9.00000i 0.384812i 0.981315 + 0.192406i \(0.0616291\pi\)
−0.981315 + 0.192406i \(0.938371\pi\)
\(548\) − 12.0000i − 0.512615i
\(549\) 0 0
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) − 4.00000i − 0.170097i
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 39.0000 1.64952
\(560\) 0 0
\(561\) 0 0
\(562\) − 32.0000i − 1.34984i
\(563\) − 19.0000i − 0.800755i −0.916350 0.400377i \(-0.868879\pi\)
0.916350 0.400377i \(-0.131121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 8.00000i 0.335673i
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −39.0000 −1.63210 −0.816050 0.577982i \(-0.803840\pi\)
−0.816050 + 0.577982i \(0.803840\pi\)
\(572\) 12.0000i 0.501745i
\(573\) 0 0
\(574\) −1.00000 −0.0417392
\(575\) 0 0
\(576\) 0 0
\(577\) − 46.0000i − 1.91501i −0.288425 0.957503i \(-0.593132\pi\)
0.288425 0.957503i \(-0.406868\pi\)
\(578\) 32.0000i 1.33102i
\(579\) 0 0
\(580\) 0 0
\(581\) 7.00000 0.290409
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) −8.00000 −0.331042
\(585\) 0 0
\(586\) 28.0000 1.15667
\(587\) 7.00000i 0.288921i 0.989511 + 0.144460i \(0.0461446\pi\)
−0.989511 + 0.144460i \(0.953855\pi\)
\(588\) 0 0
\(589\) −42.0000 −1.73058
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000i 0.410997i
\(593\) − 18.0000i − 0.739171i −0.929197 0.369586i \(-0.879500\pi\)
0.929197 0.369586i \(-0.120500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.00000 −0.368654
\(597\) 0 0
\(598\) 27.0000i 1.10411i
\(599\) 21.0000 0.858037 0.429018 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 13.0000i 0.529840i
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 6.00000i − 0.243332i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 42.0000i 1.69636i 0.529705 + 0.848182i \(0.322303\pi\)
−0.529705 + 0.848182i \(0.677697\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) − 26.0000i − 1.04672i −0.852111 0.523360i \(-0.824678\pi\)
0.852111 0.523360i \(-0.175322\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.0000i 1.20289i
\(623\) 14.0000i 0.560898i
\(624\) 0 0
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −70.0000 −2.79108
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) 0 0
\(634\) 11.0000 0.436866
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000i 0.118864i
\(638\) 12.0000i 0.475085i
\(639\) 0 0
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) −9.00000 −0.354650
\(645\) 0 0
\(646\) 42.0000 1.65247
\(647\) − 28.0000i − 1.10079i −0.834903 0.550397i \(-0.814476\pi\)
0.834903 0.550397i \(-0.185524\pi\)
\(648\) 0 0
\(649\) 44.0000 1.72715
\(650\) 0 0
\(651\) 0 0
\(652\) − 11.0000i − 0.430793i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.00000 −0.0390434
\(657\) 0 0
\(658\) − 2.00000i − 0.0779681i
\(659\) −16.0000 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 5.00000i 0.194331i
\(663\) 0 0
\(664\) 7.00000 0.271653
\(665\) 0 0
\(666\) 0 0
\(667\) 27.0000i 1.04544i
\(668\) − 12.0000i − 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) 52.0000 2.00744
\(672\) 0 0
\(673\) 31.0000i 1.19496i 0.801883 + 0.597481i \(0.203832\pi\)
−0.801883 + 0.597481i \(0.796168\pi\)
\(674\) 5.00000 0.192593
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 28.0000i 1.07218i
\(683\) 38.0000i 1.45403i 0.686622 + 0.727015i \(0.259093\pi\)
−0.686622 + 0.727015i \(0.740907\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 13.0000i 0.495620i
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 10.0000i 0.380143i
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) 0 0
\(697\) − 7.00000i − 0.265144i
\(698\) 19.0000i 0.719161i
\(699\) 0 0
\(700\) 0 0
\(701\) 7.00000 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(702\) 0 0
\(703\) 60.0000i 2.26294i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000i 0.524672i
\(713\) 63.0000i 2.35937i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.00000 −0.224231
\(717\) 0 0
\(718\) 11.0000i 0.410516i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 11.0000 0.409661
\(722\) − 17.0000i − 0.632674i
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000i 0.111264i 0.998451 + 0.0556319i \(0.0177173\pi\)
−0.998451 + 0.0556319i \(0.982283\pi\)
\(728\) 3.00000i 0.111187i
\(729\) 0 0
\(730\) 0 0
\(731\) −91.0000 −3.36576
\(732\) 0 0
\(733\) − 41.0000i − 1.51437i −0.653201 0.757185i \(-0.726574\pi\)
0.653201 0.757185i \(-0.273426\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 0 0
\(738\) 0 0
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.00000i 0.0367112i
\(743\) 45.0000i 1.65089i 0.564483 + 0.825445i \(0.309076\pi\)
−0.564483 + 0.825445i \(0.690924\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) − 28.0000i − 1.02378i
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) − 2.00000i − 0.0729325i
\(753\) 0 0
\(754\) 9.00000 0.327761
\(755\) 0 0
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) − 15.0000i − 0.544825i
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 6.00000i 0.217215i
\(764\) 21.0000 0.759753
\(765\) 0 0
\(766\) 0 0
\(767\) − 33.0000i − 1.19156i
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 14.0000i − 0.503871i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) 26.0000i 0.932145i
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) − 63.0000i − 2.25288i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 5.00000i 0.178118i
\(789\) 0 0
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) − 39.0000i − 1.38493i
\(794\) −19.0000 −0.674285
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 14.0000 0.495284
\(800\) 0 0
\(801\) 0 0
\(802\) − 4.00000i − 0.141245i
\(803\) 32.0000i 1.12926i
\(804\) 0 0
\(805\) 0 0
\(806\) 21.0000 0.739693
\(807\) 0 0
\(808\) − 6.00000i − 0.211079i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 3.00000i 0.105279i
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) 78.0000i 2.72887i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 11.0000 0.383203
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) 34.0000i 1.18230i 0.806563 + 0.591148i \(0.201325\pi\)
−0.806563 + 0.591148i \(0.798675\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.00000i 0.104006i
\(833\) − 7.00000i − 0.242536i
\(834\) 0 0
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) − 9.00000i − 0.310900i
\(839\) −32.0000 −1.10476 −0.552381 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) − 8.00000i − 0.275698i
\(843\) 0 0
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 5.00000i 0.171802i
\(848\) 1.00000i 0.0343401i
\(849\) 0 0
\(850\) 0 0
\(851\) 90.0000 3.08516
\(852\) 0 0
\(853\) − 13.0000i − 0.445112i −0.974920 0.222556i \(-0.928560\pi\)
0.974920 0.222556i \(-0.0714399\pi\)
\(854\) 13.0000 0.444851
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 21.0000i 0.715263i
\(863\) 28.0000i 0.953131i 0.879139 + 0.476566i \(0.158119\pi\)
−0.879139 + 0.476566i \(0.841881\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 7.00000i 0.237595i
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 6.00000i 0.203186i
\(873\) 0 0
\(874\) −54.0000 −1.82658
\(875\) 0 0
\(876\) 0 0
\(877\) − 8.00000i − 0.270141i −0.990836 0.135070i \(-0.956874\pi\)
0.990836 0.135070i \(-0.0431261\pi\)
\(878\) − 35.0000i − 1.18119i
\(879\) 0 0
\(880\) 0 0
\(881\) −13.0000 −0.437981 −0.218991 0.975727i \(-0.570276\pi\)
−0.218991 + 0.975727i \(0.570276\pi\)
\(882\) 0 0
\(883\) − 21.0000i − 0.706706i −0.935490 0.353353i \(-0.885041\pi\)
0.935490 0.353353i \(-0.114959\pi\)
\(884\) −21.0000 −0.706306
\(885\) 0 0
\(886\) −2.00000 −0.0671913
\(887\) − 38.0000i − 1.27592i −0.770072 0.637958i \(-0.779780\pi\)
0.770072 0.637958i \(-0.220220\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) − 13.0000i − 0.435272i
\(893\) − 12.0000i − 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 30.0000i 1.00111i
\(899\) 21.0000 0.700389
\(900\) 0 0
\(901\) −7.00000 −0.233204
\(902\) 4.00000i 0.133185i
\(903\) 0 0
\(904\) 16.0000 0.532152
\(905\) 0 0
\(906\) 0 0
\(907\) − 47.0000i − 1.56061i −0.625400 0.780305i \(-0.715064\pi\)
0.625400 0.780305i \(-0.284936\pi\)
\(908\) 11.0000i 0.365048i
\(909\) 0 0
\(910\) 0 0
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 0 0
\(913\) − 28.0000i − 0.926665i
\(914\) 17.0000 0.562310
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 20.0000i 0.660458i
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.0000i 0.461065i
\(923\) − 24.0000i − 0.789970i
\(924\) 0 0
\(925\) 0 0
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 3.00000i 0.0984798i
\(929\) −31.0000 −1.01708 −0.508539 0.861039i \(-0.669814\pi\)
−0.508539 + 0.861039i \(0.669814\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 18.0000i 0.589610i
\(933\) 0 0
\(934\) −5.00000 −0.163605
\(935\) 0 0
\(936\) 0 0
\(937\) − 38.0000i − 1.24141i −0.784046 0.620703i \(-0.786847\pi\)
0.784046 0.620703i \(-0.213153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 0 0
\(943\) 9.00000i 0.293080i
\(944\) 11.0000 0.358020
\(945\) 0 0
\(946\) 52.0000 1.69067
\(947\) − 2.00000i − 0.0649913i −0.999472 0.0324956i \(-0.989654\pi\)
0.999472 0.0324956i \(-0.0103455\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) − 7.00000i − 0.226871i
\(953\) 12.0000i 0.388718i 0.980930 + 0.194359i \(0.0622627\pi\)
−0.980930 + 0.194359i \(0.937737\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) − 30.0000i − 0.969256i
\(959\) −12.0000 −0.387500
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) − 30.0000i − 0.967239i
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) 26.0000i 0.836104i 0.908423 + 0.418052i \(0.137287\pi\)
−0.908423 + 0.418052i \(0.862713\pi\)
\(968\) 5.00000i 0.160706i
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 0 0
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) 0 0
\(982\) − 6.00000i − 0.191468i
\(983\) − 4.00000i − 0.127580i −0.997963 0.0637901i \(-0.979681\pi\)
0.997963 0.0637901i \(-0.0203188\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −21.0000 −0.668776
\(987\) 0 0
\(988\) 18.0000i 0.572656i
\(989\) 117.000 3.72038
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 7.00000i 0.222250i
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.0000i − 0.443384i −0.975117 0.221692i \(-0.928842\pi\)
0.975117 0.221692i \(-0.0711580\pi\)
\(998\) − 21.0000i − 0.664743i
\(999\) 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 3150.2.g.u.2899.1 2
3.2 odd 2 3150.2.g.d.2899.2 2
5.2 odd 4 3150.2.a.bf.1.1 yes 1
5.3 odd 4 3150.2.a.u.1.1 yes 1
5.4 even 2 inner 3150.2.g.u.2899.2 2
15.2 even 4 3150.2.a.b.1.1 1
15.8 even 4 3150.2.a.bi.1.1 yes 1
15.14 odd 2 3150.2.g.d.2899.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.a.b.1.1 1 15.2 even 4
3150.2.a.u.1.1 yes 1 5.3 odd 4
3150.2.a.bf.1.1 yes 1 5.2 odd 4
3150.2.a.bi.1.1 yes 1 15.8 even 4
3150.2.g.d.2899.1 2 15.14 odd 2
3150.2.g.d.2899.2 2 3.2 odd 2
3150.2.g.u.2899.1 2 1.1 even 1 trivial
3150.2.g.u.2899.2 2 5.4 even 2 inner