Properties

Label 2475.2.a.bb.1.3
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} +O(q^{10})\) \(q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 5q^{4} - 9q^{8} + O(q^{10}) \) \( 3q - q^{2} + 5q^{4} - 9q^{8} - 3q^{11} + 2q^{13} + 12q^{14} + 13q^{16} - 2q^{17} + 8q^{19} + q^{22} - 10q^{26} + 8q^{28} + 10q^{29} + 8q^{31} - 29q^{32} - 30q^{34} + 6q^{37} + 14q^{41} - 4q^{43} - 5q^{44} + 24q^{46} - 8q^{47} + 11q^{49} + 30q^{52} - 6q^{53} + 4q^{56} + 18q^{58} - 12q^{59} - 6q^{61} + 16q^{62} + 13q^{64} + 4q^{67} + 42q^{68} - 8q^{71} + 14q^{73} - 2q^{74} + 48q^{76} + 12q^{79} - 26q^{82} + 8q^{86} + 9q^{88} + 10q^{89} + 8q^{91} + 16q^{92} - 16q^{94} - 22q^{97} + 39q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.90321 1.34577 0.672887 0.739745i \(-0.265054\pi\)
0.672887 + 0.739745i \(0.265054\pi\)
\(3\) 0 0
\(4\) 1.62222 0.811108
\(5\) 0 0
\(6\) 0 0
\(7\) 4.42864 1.67387 0.836934 0.547304i \(-0.184346\pi\)
0.836934 + 0.547304i \(0.184346\pi\)
\(8\) −0.719004 −0.254206
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.622216 0.172572 0.0862858 0.996270i \(-0.472500\pi\)
0.0862858 + 0.996270i \(0.472500\pi\)
\(14\) 8.42864 2.25265
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) −5.18421 −1.25736 −0.628678 0.777666i \(-0.716403\pi\)
−0.628678 + 0.777666i \(0.716403\pi\)
\(18\) 0 0
\(19\) 7.05086 1.61758 0.808789 0.588100i \(-0.200124\pi\)
0.808789 + 0.588100i \(0.200124\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.90321 −0.405766
\(23\) 8.85728 1.84687 0.923435 0.383754i \(-0.125369\pi\)
0.923435 + 0.383754i \(0.125369\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.18421 0.232242
\(27\) 0 0
\(28\) 7.18421 1.35769
\(29\) 7.80642 1.44962 0.724808 0.688951i \(-0.241928\pi\)
0.724808 + 0.688951i \(0.241928\pi\)
\(30\) 0 0
\(31\) 2.75557 0.494915 0.247457 0.968899i \(-0.420405\pi\)
0.247457 + 0.968899i \(0.420405\pi\)
\(32\) −7.34122 −1.29776
\(33\) 0 0
\(34\) −9.86665 −1.69212
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 13.4193 2.17689
\(39\) 0 0
\(40\) 0 0
\(41\) 0.193576 0.0302315 0.0151158 0.999886i \(-0.495188\pi\)
0.0151158 + 0.999886i \(0.495188\pi\)
\(42\) 0 0
\(43\) −5.67307 −0.865135 −0.432568 0.901602i \(-0.642392\pi\)
−0.432568 + 0.901602i \(0.642392\pi\)
\(44\) −1.62222 −0.244558
\(45\) 0 0
\(46\) 16.8573 2.48547
\(47\) −2.75557 −0.401941 −0.200971 0.979597i \(-0.564410\pi\)
−0.200971 + 0.979597i \(0.564410\pi\)
\(48\) 0 0
\(49\) 12.6128 1.80184
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00937 0.139974
\(53\) −10.8573 −1.49136 −0.745681 0.666303i \(-0.767876\pi\)
−0.745681 + 0.666303i \(0.767876\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.18421 −0.425508
\(57\) 0 0
\(58\) 14.8573 1.95086
\(59\) 4.85728 0.632364 0.316182 0.948699i \(-0.397599\pi\)
0.316182 + 0.948699i \(0.397599\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 5.24443 0.666043
\(63\) 0 0
\(64\) −4.74620 −0.593275
\(65\) 0 0
\(66\) 0 0
\(67\) 1.24443 0.152031 0.0760157 0.997107i \(-0.475780\pi\)
0.0760157 + 0.997107i \(0.475780\pi\)
\(68\) −8.40990 −1.01985
\(69\) 0 0
\(70\) 0 0
\(71\) −2.75557 −0.327026 −0.163513 0.986541i \(-0.552283\pi\)
−0.163513 + 0.986541i \(0.552283\pi\)
\(72\) 0 0
\(73\) −4.23506 −0.495677 −0.247838 0.968801i \(-0.579720\pi\)
−0.247838 + 0.968801i \(0.579720\pi\)
\(74\) 3.80642 0.442488
\(75\) 0 0
\(76\) 11.4380 1.31203
\(77\) −4.42864 −0.504690
\(78\) 0 0
\(79\) 8.56199 0.963299 0.481650 0.876364i \(-0.340038\pi\)
0.481650 + 0.876364i \(0.340038\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.368416 0.0406848
\(83\) 0.133353 0.0146374 0.00731870 0.999973i \(-0.497670\pi\)
0.00731870 + 0.999973i \(0.497670\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.7971 −1.16428
\(87\) 0 0
\(88\) 0.719004 0.0766461
\(89\) −5.61285 −0.594961 −0.297480 0.954728i \(-0.596146\pi\)
−0.297480 + 0.954728i \(0.596146\pi\)
\(90\) 0 0
\(91\) 2.75557 0.288862
\(92\) 14.3684 1.49801
\(93\) 0 0
\(94\) −5.24443 −0.540922
\(95\) 0 0
\(96\) 0 0
\(97\) −7.24443 −0.735561 −0.367780 0.929913i \(-0.619882\pi\)
−0.367780 + 0.929913i \(0.619882\pi\)
\(98\) 24.0049 2.42486
\(99\) 0 0
\(100\) 0 0
\(101\) −4.66370 −0.464056 −0.232028 0.972709i \(-0.574536\pi\)
−0.232028 + 0.972709i \(0.574536\pi\)
\(102\) 0 0
\(103\) 11.6128 1.14425 0.572124 0.820167i \(-0.306120\pi\)
0.572124 + 0.820167i \(0.306120\pi\)
\(104\) −0.447375 −0.0438688
\(105\) 0 0
\(106\) −20.6637 −2.00704
\(107\) 2.62222 0.253499 0.126750 0.991935i \(-0.459546\pi\)
0.126750 + 0.991935i \(0.459546\pi\)
\(108\) 0 0
\(109\) −19.7146 −1.88831 −0.944156 0.329499i \(-0.893120\pi\)
−0.944156 + 0.329499i \(0.893120\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −20.4286 −1.93032
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.6637 1.17580
\(117\) 0 0
\(118\) 9.24443 0.851019
\(119\) −22.9590 −2.10465
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 13.0509 1.18157
\(123\) 0 0
\(124\) 4.47013 0.401429
\(125\) 0 0
\(126\) 0 0
\(127\) 15.1842 1.34738 0.673690 0.739014i \(-0.264708\pi\)
0.673690 + 0.739014i \(0.264708\pi\)
\(128\) 5.64941 0.499342
\(129\) 0 0
\(130\) 0 0
\(131\) −1.24443 −0.108726 −0.0543632 0.998521i \(-0.517313\pi\)
−0.0543632 + 0.998521i \(0.517313\pi\)
\(132\) 0 0
\(133\) 31.2257 2.70761
\(134\) 2.36842 0.204600
\(135\) 0 0
\(136\) 3.72746 0.319627
\(137\) −0.488863 −0.0417663 −0.0208832 0.999782i \(-0.506648\pi\)
−0.0208832 + 0.999782i \(0.506648\pi\)
\(138\) 0 0
\(139\) 17.8064 1.51032 0.755161 0.655540i \(-0.227559\pi\)
0.755161 + 0.655540i \(0.227559\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.24443 −0.440103
\(143\) −0.622216 −0.0520323
\(144\) 0 0
\(145\) 0 0
\(146\) −8.06022 −0.667069
\(147\) 0 0
\(148\) 3.24443 0.266691
\(149\) 1.43801 0.117806 0.0589031 0.998264i \(-0.481240\pi\)
0.0589031 + 0.998264i \(0.481240\pi\)
\(150\) 0 0
\(151\) −12.1748 −0.990774 −0.495387 0.868672i \(-0.664974\pi\)
−0.495387 + 0.868672i \(0.664974\pi\)
\(152\) −5.06959 −0.411198
\(153\) 0 0
\(154\) −8.42864 −0.679199
\(155\) 0 0
\(156\) 0 0
\(157\) −18.4701 −1.47408 −0.737038 0.675851i \(-0.763776\pi\)
−0.737038 + 0.675851i \(0.763776\pi\)
\(158\) 16.2953 1.29638
\(159\) 0 0
\(160\) 0 0
\(161\) 39.2257 3.09142
\(162\) 0 0
\(163\) 10.1017 0.791227 0.395614 0.918417i \(-0.370532\pi\)
0.395614 + 0.918417i \(0.370532\pi\)
\(164\) 0.314022 0.0245210
\(165\) 0 0
\(166\) 0.253799 0.0196986
\(167\) −16.3368 −1.26418 −0.632089 0.774896i \(-0.717802\pi\)
−0.632089 + 0.774896i \(0.717802\pi\)
\(168\) 0 0
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 0 0
\(172\) −9.20294 −0.701718
\(173\) −9.18421 −0.698262 −0.349131 0.937074i \(-0.613523\pi\)
−0.349131 + 0.937074i \(0.613523\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.61285 0.347706
\(177\) 0 0
\(178\) −10.6824 −0.800683
\(179\) 25.3274 1.89306 0.946530 0.322617i \(-0.104563\pi\)
0.946530 + 0.322617i \(0.104563\pi\)
\(180\) 0 0
\(181\) −13.6128 −1.01184 −0.505918 0.862582i \(-0.668846\pi\)
−0.505918 + 0.862582i \(0.668846\pi\)
\(182\) 5.24443 0.388743
\(183\) 0 0
\(184\) −6.36842 −0.469486
\(185\) 0 0
\(186\) 0 0
\(187\) 5.18421 0.379107
\(188\) −4.47013 −0.326017
\(189\) 0 0
\(190\) 0 0
\(191\) −6.10171 −0.441504 −0.220752 0.975330i \(-0.570851\pi\)
−0.220752 + 0.975330i \(0.570851\pi\)
\(192\) 0 0
\(193\) −18.3368 −1.31991 −0.659955 0.751305i \(-0.729425\pi\)
−0.659955 + 0.751305i \(0.729425\pi\)
\(194\) −13.7877 −0.989898
\(195\) 0 0
\(196\) 20.4608 1.46148
\(197\) −6.69535 −0.477024 −0.238512 0.971140i \(-0.576660\pi\)
−0.238512 + 0.971140i \(0.576660\pi\)
\(198\) 0 0
\(199\) 14.1017 0.999644 0.499822 0.866128i \(-0.333399\pi\)
0.499822 + 0.866128i \(0.333399\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.87601 −0.624514
\(203\) 34.5718 2.42647
\(204\) 0 0
\(205\) 0 0
\(206\) 22.1017 1.53990
\(207\) 0 0
\(208\) −2.87019 −0.199012
\(209\) −7.05086 −0.487718
\(210\) 0 0
\(211\) −10.6637 −0.734120 −0.367060 0.930197i \(-0.619636\pi\)
−0.367060 + 0.930197i \(0.619636\pi\)
\(212\) −17.6128 −1.20966
\(213\) 0 0
\(214\) 4.99063 0.341153
\(215\) 0 0
\(216\) 0 0
\(217\) 12.2034 0.828422
\(218\) −37.5210 −2.54124
\(219\) 0 0
\(220\) 0 0
\(221\) −3.22570 −0.216984
\(222\) 0 0
\(223\) −8.85728 −0.593127 −0.296564 0.955013i \(-0.595841\pi\)
−0.296564 + 0.955013i \(0.595841\pi\)
\(224\) −32.5116 −2.17227
\(225\) 0 0
\(226\) −11.4193 −0.759599
\(227\) 13.3778 0.887915 0.443957 0.896048i \(-0.353574\pi\)
0.443957 + 0.896048i \(0.353574\pi\)
\(228\) 0 0
\(229\) 11.5111 0.760677 0.380339 0.924847i \(-0.375807\pi\)
0.380339 + 0.924847i \(0.375807\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −5.61285 −0.368502
\(233\) −4.32693 −0.283467 −0.141733 0.989905i \(-0.545268\pi\)
−0.141733 + 0.989905i \(0.545268\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.87955 0.512915
\(237\) 0 0
\(238\) −43.6958 −2.83238
\(239\) 3.34614 0.216444 0.108222 0.994127i \(-0.465484\pi\)
0.108222 + 0.994127i \(0.465484\pi\)
\(240\) 0 0
\(241\) −1.34614 −0.0867126 −0.0433563 0.999060i \(-0.513805\pi\)
−0.0433563 + 0.999060i \(0.513805\pi\)
\(242\) 1.90321 0.122343
\(243\) 0 0
\(244\) 11.1240 0.712140
\(245\) 0 0
\(246\) 0 0
\(247\) 4.38715 0.279148
\(248\) −1.98126 −0.125810
\(249\) 0 0
\(250\) 0 0
\(251\) −22.7556 −1.43632 −0.718159 0.695879i \(-0.755015\pi\)
−0.718159 + 0.695879i \(0.755015\pi\)
\(252\) 0 0
\(253\) −8.85728 −0.556852
\(254\) 28.8988 1.81327
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) −6.85728 −0.427745 −0.213873 0.976862i \(-0.568608\pi\)
−0.213873 + 0.976862i \(0.568608\pi\)
\(258\) 0 0
\(259\) 8.85728 0.550365
\(260\) 0 0
\(261\) 0 0
\(262\) −2.36842 −0.146321
\(263\) −29.5812 −1.82406 −0.912028 0.410129i \(-0.865484\pi\)
−0.912028 + 0.410129i \(0.865484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 59.4291 3.64383
\(267\) 0 0
\(268\) 2.01874 0.123314
\(269\) −8.48886 −0.517575 −0.258788 0.965934i \(-0.583323\pi\)
−0.258788 + 0.965934i \(0.583323\pi\)
\(270\) 0 0
\(271\) 14.6637 0.890757 0.445378 0.895343i \(-0.353069\pi\)
0.445378 + 0.895343i \(0.353069\pi\)
\(272\) 23.9140 1.45000
\(273\) 0 0
\(274\) −0.930409 −0.0562081
\(275\) 0 0
\(276\) 0 0
\(277\) −14.6035 −0.877438 −0.438719 0.898624i \(-0.644568\pi\)
−0.438719 + 0.898624i \(0.644568\pi\)
\(278\) 33.8894 2.03255
\(279\) 0 0
\(280\) 0 0
\(281\) 0.193576 0.0115478 0.00577389 0.999983i \(-0.498162\pi\)
0.00577389 + 0.999983i \(0.498162\pi\)
\(282\) 0 0
\(283\) −27.1842 −1.61593 −0.807967 0.589228i \(-0.799432\pi\)
−0.807967 + 0.589228i \(0.799432\pi\)
\(284\) −4.47013 −0.265253
\(285\) 0 0
\(286\) −1.18421 −0.0700237
\(287\) 0.857279 0.0506036
\(288\) 0 0
\(289\) 9.87601 0.580942
\(290\) 0 0
\(291\) 0 0
\(292\) −6.87019 −0.402047
\(293\) −2.81579 −0.164500 −0.0822502 0.996612i \(-0.526211\pi\)
−0.0822502 + 0.996612i \(0.526211\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.43801 −0.0835825
\(297\) 0 0
\(298\) 2.73683 0.158540
\(299\) 5.51114 0.318717
\(300\) 0 0
\(301\) −25.1240 −1.44812
\(302\) −23.1713 −1.33336
\(303\) 0 0
\(304\) −32.5245 −1.86541
\(305\) 0 0
\(306\) 0 0
\(307\) 24.4286 1.39422 0.697108 0.716966i \(-0.254470\pi\)
0.697108 + 0.716966i \(0.254470\pi\)
\(308\) −7.18421 −0.409358
\(309\) 0 0
\(310\) 0 0
\(311\) −19.8796 −1.12727 −0.563633 0.826025i \(-0.690597\pi\)
−0.563633 + 0.826025i \(0.690597\pi\)
\(312\) 0 0
\(313\) 15.7146 0.888239 0.444120 0.895967i \(-0.353517\pi\)
0.444120 + 0.895967i \(0.353517\pi\)
\(314\) −35.1526 −1.98377
\(315\) 0 0
\(316\) 13.8894 0.781340
\(317\) 16.4889 0.926107 0.463053 0.886330i \(-0.346754\pi\)
0.463053 + 0.886330i \(0.346754\pi\)
\(318\) 0 0
\(319\) −7.80642 −0.437076
\(320\) 0 0
\(321\) 0 0
\(322\) 74.6548 4.16035
\(323\) −36.5531 −2.03387
\(324\) 0 0
\(325\) 0 0
\(326\) 19.2257 1.06481
\(327\) 0 0
\(328\) −0.139182 −0.00768504
\(329\) −12.2034 −0.672796
\(330\) 0 0
\(331\) 15.3461 0.843500 0.421750 0.906712i \(-0.361416\pi\)
0.421750 + 0.906712i \(0.361416\pi\)
\(332\) 0.216327 0.0118725
\(333\) 0 0
\(334\) −31.0923 −1.70130
\(335\) 0 0
\(336\) 0 0
\(337\) −28.2351 −1.53806 −0.769031 0.639212i \(-0.779261\pi\)
−0.769031 + 0.639212i \(0.779261\pi\)
\(338\) −24.0049 −1.30570
\(339\) 0 0
\(340\) 0 0
\(341\) −2.75557 −0.149222
\(342\) 0 0
\(343\) 24.8573 1.34217
\(344\) 4.07896 0.219923
\(345\) 0 0
\(346\) −17.4795 −0.939703
\(347\) 2.62222 0.140768 0.0703840 0.997520i \(-0.477578\pi\)
0.0703840 + 0.997520i \(0.477578\pi\)
\(348\) 0 0
\(349\) 5.14272 0.275284 0.137642 0.990482i \(-0.456048\pi\)
0.137642 + 0.990482i \(0.456048\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.34122 0.391288
\(353\) −9.34614 −0.497445 −0.248722 0.968575i \(-0.580011\pi\)
−0.248722 + 0.968575i \(0.580011\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.10525 −0.482577
\(357\) 0 0
\(358\) 48.2034 2.54763
\(359\) −10.7556 −0.567657 −0.283829 0.958875i \(-0.591605\pi\)
−0.283829 + 0.958875i \(0.591605\pi\)
\(360\) 0 0
\(361\) 30.7146 1.61656
\(362\) −25.9081 −1.36170
\(363\) 0 0
\(364\) 4.47013 0.234298
\(365\) 0 0
\(366\) 0 0
\(367\) −33.7975 −1.76422 −0.882108 0.471046i \(-0.843876\pi\)
−0.882108 + 0.471046i \(0.843876\pi\)
\(368\) −40.8573 −2.12983
\(369\) 0 0
\(370\) 0 0
\(371\) −48.0830 −2.49634
\(372\) 0 0
\(373\) −33.9496 −1.75784 −0.878922 0.476965i \(-0.841737\pi\)
−0.878922 + 0.476965i \(0.841737\pi\)
\(374\) 9.86665 0.510192
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) 4.85728 0.250163
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −11.6128 −0.594165
\(383\) −14.6351 −0.747820 −0.373910 0.927465i \(-0.621983\pi\)
−0.373910 + 0.927465i \(0.621983\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −34.8988 −1.77630
\(387\) 0 0
\(388\) −11.7520 −0.596619
\(389\) 5.61285 0.284583 0.142291 0.989825i \(-0.454553\pi\)
0.142291 + 0.989825i \(0.454553\pi\)
\(390\) 0 0
\(391\) −45.9180 −2.32217
\(392\) −9.06868 −0.458038
\(393\) 0 0
\(394\) −12.7427 −0.641966
\(395\) 0 0
\(396\) 0 0
\(397\) 12.7556 0.640184 0.320092 0.947387i \(-0.396286\pi\)
0.320092 + 0.947387i \(0.396286\pi\)
\(398\) 26.8385 1.34529
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) 1.71456 0.0854082
\(404\) −7.56553 −0.376399
\(405\) 0 0
\(406\) 65.7975 3.26548
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −7.12399 −0.352258 −0.176129 0.984367i \(-0.556358\pi\)
−0.176129 + 0.984367i \(0.556358\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 18.8385 0.928108
\(413\) 21.5111 1.05849
\(414\) 0 0
\(415\) 0 0
\(416\) −4.56782 −0.223956
\(417\) 0 0
\(418\) −13.4193 −0.656358
\(419\) −15.6128 −0.762738 −0.381369 0.924423i \(-0.624547\pi\)
−0.381369 + 0.924423i \(0.624547\pi\)
\(420\) 0 0
\(421\) 7.89829 0.384939 0.192470 0.981303i \(-0.438350\pi\)
0.192470 + 0.981303i \(0.438350\pi\)
\(422\) −20.2953 −0.987959
\(423\) 0 0
\(424\) 7.80642 0.379113
\(425\) 0 0
\(426\) 0 0
\(427\) 30.3684 1.46963
\(428\) 4.25380 0.205615
\(429\) 0 0
\(430\) 0 0
\(431\) −34.3051 −1.65242 −0.826210 0.563362i \(-0.809508\pi\)
−0.826210 + 0.563362i \(0.809508\pi\)
\(432\) 0 0
\(433\) −14.4701 −0.695390 −0.347695 0.937608i \(-0.613036\pi\)
−0.347695 + 0.937608i \(0.613036\pi\)
\(434\) 23.2257 1.11487
\(435\) 0 0
\(436\) −31.9813 −1.53162
\(437\) 62.4514 2.98746
\(438\) 0 0
\(439\) −19.3176 −0.921977 −0.460988 0.887406i \(-0.652505\pi\)
−0.460988 + 0.887406i \(0.652505\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.13918 −0.292011
\(443\) −13.1240 −0.623539 −0.311770 0.950158i \(-0.600922\pi\)
−0.311770 + 0.950158i \(0.600922\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.8573 −0.798215
\(447\) 0 0
\(448\) −21.0192 −0.993064
\(449\) 32.3051 1.52457 0.762287 0.647240i \(-0.224077\pi\)
0.762287 + 0.647240i \(0.224077\pi\)
\(450\) 0 0
\(451\) −0.193576 −0.00911514
\(452\) −9.73329 −0.457816
\(453\) 0 0
\(454\) 25.4608 1.19493
\(455\) 0 0
\(456\) 0 0
\(457\) 23.4608 1.09745 0.548724 0.836004i \(-0.315114\pi\)
0.548724 + 0.836004i \(0.315114\pi\)
\(458\) 21.9081 1.02370
\(459\) 0 0
\(460\) 0 0
\(461\) 28.8671 1.34448 0.672238 0.740335i \(-0.265333\pi\)
0.672238 + 0.740335i \(0.265333\pi\)
\(462\) 0 0
\(463\) 19.3461 0.899091 0.449546 0.893257i \(-0.351586\pi\)
0.449546 + 0.893257i \(0.351586\pi\)
\(464\) −36.0098 −1.67172
\(465\) 0 0
\(466\) −8.23506 −0.381482
\(467\) 3.14272 0.145428 0.0727139 0.997353i \(-0.476834\pi\)
0.0727139 + 0.997353i \(0.476834\pi\)
\(468\) 0 0
\(469\) 5.51114 0.254481
\(470\) 0 0
\(471\) 0 0
\(472\) −3.49240 −0.160751
\(473\) 5.67307 0.260848
\(474\) 0 0
\(475\) 0 0
\(476\) −37.2444 −1.70710
\(477\) 0 0
\(478\) 6.36842 0.291285
\(479\) 24.8573 1.13576 0.567879 0.823112i \(-0.307764\pi\)
0.567879 + 0.823112i \(0.307764\pi\)
\(480\) 0 0
\(481\) 1.24443 0.0567412
\(482\) −2.56199 −0.116696
\(483\) 0 0
\(484\) 1.62222 0.0737371
\(485\) 0 0
\(486\) 0 0
\(487\) 11.5299 0.522468 0.261234 0.965275i \(-0.415871\pi\)
0.261234 + 0.965275i \(0.415871\pi\)
\(488\) −4.93041 −0.223189
\(489\) 0 0
\(490\) 0 0
\(491\) −16.3872 −0.739542 −0.369771 0.929123i \(-0.620564\pi\)
−0.369771 + 0.929123i \(0.620564\pi\)
\(492\) 0 0
\(493\) −40.4701 −1.82268
\(494\) 8.34968 0.375670
\(495\) 0 0
\(496\) −12.7110 −0.570742
\(497\) −12.2034 −0.547398
\(498\) 0 0
\(499\) −25.3274 −1.13381 −0.566905 0.823783i \(-0.691859\pi\)
−0.566905 + 0.823783i \(0.691859\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −43.3087 −1.93296
\(503\) 19.0923 0.851285 0.425643 0.904891i \(-0.360048\pi\)
0.425643 + 0.904891i \(0.360048\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.8573 −0.749397
\(507\) 0 0
\(508\) 24.6321 1.09287
\(509\) 32.4514 1.43838 0.719191 0.694812i \(-0.244512\pi\)
0.719191 + 0.694812i \(0.244512\pi\)
\(510\) 0 0
\(511\) −18.7556 −0.829698
\(512\) 27.2306 1.20343
\(513\) 0 0
\(514\) −13.0509 −0.575649
\(515\) 0 0
\(516\) 0 0
\(517\) 2.75557 0.121190
\(518\) 16.8573 0.740666
\(519\) 0 0
\(520\) 0 0
\(521\) 29.2257 1.28040 0.640200 0.768208i \(-0.278851\pi\)
0.640200 + 0.768208i \(0.278851\pi\)
\(522\) 0 0
\(523\) −6.71408 −0.293586 −0.146793 0.989167i \(-0.546895\pi\)
−0.146793 + 0.989167i \(0.546895\pi\)
\(524\) −2.01874 −0.0881889
\(525\) 0 0
\(526\) −56.2993 −2.45477
\(527\) −14.2854 −0.622284
\(528\) 0 0
\(529\) 55.4514 2.41093
\(530\) 0 0
\(531\) 0 0
\(532\) 50.6548 2.19616
\(533\) 0.120446 0.00521710
\(534\) 0 0
\(535\) 0 0
\(536\) −0.894751 −0.0386473
\(537\) 0 0
\(538\) −16.1561 −0.696539
\(539\) −12.6128 −0.543274
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 27.9081 1.19876
\(543\) 0 0
\(544\) 38.0584 1.63174
\(545\) 0 0
\(546\) 0 0
\(547\) −41.3689 −1.76881 −0.884403 0.466724i \(-0.845434\pi\)
−0.884403 + 0.466724i \(0.845434\pi\)
\(548\) −0.793040 −0.0338770
\(549\) 0 0
\(550\) 0 0
\(551\) 55.0420 2.34487
\(552\) 0 0
\(553\) 37.9180 1.61244
\(554\) −27.7935 −1.18083
\(555\) 0 0
\(556\) 28.8859 1.22503
\(557\) −20.7971 −0.881200 −0.440600 0.897704i \(-0.645234\pi\)
−0.440600 + 0.897704i \(0.645234\pi\)
\(558\) 0 0
\(559\) −3.52987 −0.149298
\(560\) 0 0
\(561\) 0 0
\(562\) 0.368416 0.0155407
\(563\) 37.7275 1.59002 0.795012 0.606594i \(-0.207465\pi\)
0.795012 + 0.606594i \(0.207465\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −51.7373 −2.17468
\(567\) 0 0
\(568\) 1.98126 0.0831320
\(569\) 7.33630 0.307554 0.153777 0.988106i \(-0.450856\pi\)
0.153777 + 0.988106i \(0.450856\pi\)
\(570\) 0 0
\(571\) 36.6450 1.53354 0.766772 0.641919i \(-0.221862\pi\)
0.766772 + 0.641919i \(0.221862\pi\)
\(572\) −1.00937 −0.0422038
\(573\) 0 0
\(574\) 1.63158 0.0681010
\(575\) 0 0
\(576\) 0 0
\(577\) −4.22216 −0.175771 −0.0878853 0.996131i \(-0.528011\pi\)
−0.0878853 + 0.996131i \(0.528011\pi\)
\(578\) 18.7961 0.781817
\(579\) 0 0
\(580\) 0 0
\(581\) 0.590573 0.0245011
\(582\) 0 0
\(583\) 10.8573 0.449663
\(584\) 3.04503 0.126004
\(585\) 0 0
\(586\) −5.35905 −0.221380
\(587\) 34.3684 1.41854 0.709268 0.704939i \(-0.249026\pi\)
0.709268 + 0.704939i \(0.249026\pi\)
\(588\) 0 0
\(589\) 19.4291 0.800563
\(590\) 0 0
\(591\) 0 0
\(592\) −9.22570 −0.379174
\(593\) 27.9398 1.14735 0.573675 0.819083i \(-0.305517\pi\)
0.573675 + 0.819083i \(0.305517\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.33276 0.0955535
\(597\) 0 0
\(598\) 10.4889 0.428921
\(599\) 31.2257 1.27585 0.637924 0.770100i \(-0.279794\pi\)
0.637924 + 0.770100i \(0.279794\pi\)
\(600\) 0 0
\(601\) −8.75557 −0.357147 −0.178574 0.983927i \(-0.557148\pi\)
−0.178574 + 0.983927i \(0.557148\pi\)
\(602\) −47.8163 −1.94885
\(603\) 0 0
\(604\) −19.7502 −0.803625
\(605\) 0 0
\(606\) 0 0
\(607\) 15.1842 0.616308 0.308154 0.951336i \(-0.400289\pi\)
0.308154 + 0.951336i \(0.400289\pi\)
\(608\) −51.7619 −2.09922
\(609\) 0 0
\(610\) 0 0
\(611\) −1.71456 −0.0693636
\(612\) 0 0
\(613\) −42.7239 −1.72560 −0.862802 0.505543i \(-0.831292\pi\)
−0.862802 + 0.505543i \(0.831292\pi\)
\(614\) 46.4929 1.87630
\(615\) 0 0
\(616\) 3.18421 0.128295
\(617\) −3.51114 −0.141353 −0.0706765 0.997499i \(-0.522516\pi\)
−0.0706765 + 0.997499i \(0.522516\pi\)
\(618\) 0 0
\(619\) 17.5941 0.707167 0.353584 0.935403i \(-0.384963\pi\)
0.353584 + 0.935403i \(0.384963\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −37.8350 −1.51705
\(623\) −24.8573 −0.995886
\(624\) 0 0
\(625\) 0 0
\(626\) 29.9081 1.19537
\(627\) 0 0
\(628\) −29.9625 −1.19564
\(629\) −10.3684 −0.413416
\(630\) 0 0
\(631\) 15.8163 0.629636 0.314818 0.949152i \(-0.398057\pi\)
0.314818 + 0.949152i \(0.398057\pi\)
\(632\) −6.15610 −0.244877
\(633\) 0 0
\(634\) 31.3818 1.24633
\(635\) 0 0
\(636\) 0 0
\(637\) 7.84791 0.310946
\(638\) −14.8573 −0.588205
\(639\) 0 0
\(640\) 0 0
\(641\) −25.8163 −1.01968 −0.509841 0.860269i \(-0.670296\pi\)
−0.509841 + 0.860269i \(0.670296\pi\)
\(642\) 0 0
\(643\) −18.1017 −0.713862 −0.356931 0.934131i \(-0.616177\pi\)
−0.356931 + 0.934131i \(0.616177\pi\)
\(644\) 63.6325 2.50747
\(645\) 0 0
\(646\) −69.5683 −2.73713
\(647\) −47.0420 −1.84941 −0.924705 0.380684i \(-0.875689\pi\)
−0.924705 + 0.380684i \(0.875689\pi\)
\(648\) 0 0
\(649\) −4.85728 −0.190665
\(650\) 0 0
\(651\) 0 0
\(652\) 16.3872 0.641770
\(653\) 30.0830 1.17724 0.588619 0.808411i \(-0.299672\pi\)
0.588619 + 0.808411i \(0.299672\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.892937 −0.0348633
\(657\) 0 0
\(658\) −23.2257 −0.905432
\(659\) 10.2854 0.400664 0.200332 0.979728i \(-0.435798\pi\)
0.200332 + 0.979728i \(0.435798\pi\)
\(660\) 0 0
\(661\) −27.7146 −1.07797 −0.538986 0.842315i \(-0.681192\pi\)
−0.538986 + 0.842315i \(0.681192\pi\)
\(662\) 29.2070 1.13516
\(663\) 0 0
\(664\) −0.0958814 −0.00372092
\(665\) 0 0
\(666\) 0 0
\(667\) 69.1437 2.67725
\(668\) −26.5018 −1.02538
\(669\) 0 0
\(670\) 0 0
\(671\) −6.85728 −0.264722
\(672\) 0 0
\(673\) 9.86665 0.380331 0.190166 0.981752i \(-0.439097\pi\)
0.190166 + 0.981752i \(0.439097\pi\)
\(674\) −53.7373 −2.06988
\(675\) 0 0
\(676\) −20.4608 −0.786952
\(677\) −5.65433 −0.217314 −0.108657 0.994079i \(-0.534655\pi\)
−0.108657 + 0.994079i \(0.534655\pi\)
\(678\) 0 0
\(679\) −32.0830 −1.23123
\(680\) 0 0
\(681\) 0 0
\(682\) −5.24443 −0.200820
\(683\) 34.1847 1.30804 0.654020 0.756477i \(-0.273081\pi\)
0.654020 + 0.756477i \(0.273081\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 47.3087 1.80625
\(687\) 0 0
\(688\) 26.1690 0.997684
\(689\) −6.75557 −0.257367
\(690\) 0 0
\(691\) −19.2257 −0.731380 −0.365690 0.930737i \(-0.619167\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(692\) −14.8988 −0.566366
\(693\) 0 0
\(694\) 4.99063 0.189442
\(695\) 0 0
\(696\) 0 0
\(697\) −1.00354 −0.0380118
\(698\) 9.78769 0.370469
\(699\) 0 0
\(700\) 0 0
\(701\) 29.9081 1.12961 0.564807 0.825223i \(-0.308950\pi\)
0.564807 + 0.825223i \(0.308950\pi\)
\(702\) 0 0
\(703\) 14.1017 0.531856
\(704\) 4.74620 0.178879
\(705\) 0 0
\(706\) −17.7877 −0.669448
\(707\) −20.6539 −0.776768
\(708\) 0 0
\(709\) −15.3274 −0.575633 −0.287816 0.957686i \(-0.592929\pi\)
−0.287816 + 0.957686i \(0.592929\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.03566 0.151243
\(713\) 24.4068 0.914043
\(714\) 0 0
\(715\) 0 0
\(716\) 41.0865 1.53548
\(717\) 0 0
\(718\) −20.4701 −0.763938
\(719\) −23.8163 −0.888197 −0.444098 0.895978i \(-0.646476\pi\)
−0.444098 + 0.895978i \(0.646476\pi\)
\(720\) 0 0
\(721\) 51.4291 1.91532
\(722\) 58.4563 2.17552
\(723\) 0 0
\(724\) −22.0830 −0.820707
\(725\) 0 0
\(726\) 0 0
\(727\) 32.9403 1.22169 0.610843 0.791752i \(-0.290831\pi\)
0.610843 + 0.791752i \(0.290831\pi\)
\(728\) −1.98126 −0.0734305
\(729\) 0 0
\(730\) 0 0
\(731\) 29.4104 1.08778
\(732\) 0 0
\(733\) 29.8666 1.10315 0.551575 0.834125i \(-0.314027\pi\)
0.551575 + 0.834125i \(0.314027\pi\)
\(734\) −64.3239 −2.37424
\(735\) 0 0
\(736\) −65.0232 −2.39679
\(737\) −1.24443 −0.0458392
\(738\) 0 0
\(739\) −5.06959 −0.186488 −0.0932440 0.995643i \(-0.529724\pi\)
−0.0932440 + 0.995643i \(0.529724\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −91.5121 −3.35951
\(743\) 22.4385 0.823188 0.411594 0.911367i \(-0.364972\pi\)
0.411594 + 0.911367i \(0.364972\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −64.6133 −2.36566
\(747\) 0 0
\(748\) 8.40990 0.307497
\(749\) 11.6128 0.424324
\(750\) 0 0
\(751\) −6.63512 −0.242119 −0.121060 0.992645i \(-0.538629\pi\)
−0.121060 + 0.992645i \(0.538629\pi\)
\(752\) 12.7110 0.463523
\(753\) 0 0
\(754\) 9.24443 0.336662
\(755\) 0 0
\(756\) 0 0
\(757\) −8.75557 −0.318227 −0.159113 0.987260i \(-0.550864\pi\)
−0.159113 + 0.987260i \(0.550864\pi\)
\(758\) −38.0642 −1.38256
\(759\) 0 0
\(760\) 0 0
\(761\) −3.15257 −0.114280 −0.0571402 0.998366i \(-0.518198\pi\)
−0.0571402 + 0.998366i \(0.518198\pi\)
\(762\) 0 0
\(763\) −87.3087 −3.16079
\(764\) −9.89829 −0.358108
\(765\) 0 0
\(766\) −27.8537 −1.00640
\(767\) 3.02227 0.109128
\(768\) 0 0
\(769\) −28.9590 −1.04429 −0.522144 0.852857i \(-0.674868\pi\)
−0.522144 + 0.852857i \(0.674868\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −29.7462 −1.07059
\(773\) 29.1427 1.04819 0.524095 0.851660i \(-0.324404\pi\)
0.524095 + 0.851660i \(0.324404\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.20877 0.186984
\(777\) 0 0
\(778\) 10.6824 0.382984
\(779\) 1.36488 0.0489018
\(780\) 0 0
\(781\) 2.75557 0.0986020
\(782\) −87.3916 −3.12512
\(783\) 0 0
\(784\) −58.1811 −2.07790
\(785\) 0 0
\(786\) 0 0
\(787\) 11.2672 0.401632 0.200816 0.979629i \(-0.435641\pi\)
0.200816 + 0.979629i \(0.435641\pi\)
\(788\) −10.8613 −0.386918
\(789\) 0 0
\(790\) 0 0
\(791\) −26.5718 −0.944786
\(792\) 0 0
\(793\) 4.26671 0.151515
\(794\) 24.2766 0.861543
\(795\) 0 0
\(796\) 22.8760 0.810819
\(797\) 41.9625 1.48639 0.743195 0.669075i \(-0.233310\pi\)
0.743195 + 0.669075i \(0.233310\pi\)
\(798\) 0 0
\(799\) 14.2854 0.505383
\(800\) 0 0
\(801\) 0 0
\(802\) −3.80642 −0.134409
\(803\) 4.23506 0.149452
\(804\) 0 0
\(805\) 0 0
\(806\) 3.26317 0.114940
\(807\) 0 0
\(808\) 3.35322 0.117966
\(809\) 27.8064 0.977622 0.488811 0.872390i \(-0.337431\pi\)
0.488811 + 0.872390i \(0.337431\pi\)
\(810\) 0 0
\(811\) 6.78415 0.238224 0.119112 0.992881i \(-0.461995\pi\)
0.119112 + 0.992881i \(0.461995\pi\)
\(812\) 56.0830 1.96813
\(813\) 0 0
\(814\) −3.80642 −0.133415
\(815\) 0 0
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) −13.5585 −0.474060
\(819\) 0 0
\(820\) 0 0
\(821\) −3.62269 −0.126433 −0.0632164 0.998000i \(-0.520136\pi\)
−0.0632164 + 0.998000i \(0.520136\pi\)
\(822\) 0 0
\(823\) −42.0642 −1.46627 −0.733134 0.680085i \(-0.761943\pi\)
−0.733134 + 0.680085i \(0.761943\pi\)
\(824\) −8.34968 −0.290875
\(825\) 0 0
\(826\) 40.9403 1.42449
\(827\) 30.8256 1.07191 0.535956 0.844246i \(-0.319951\pi\)
0.535956 + 0.844246i \(0.319951\pi\)
\(828\) 0 0
\(829\) 7.12399 0.247426 0.123713 0.992318i \(-0.460520\pi\)
0.123713 + 0.992318i \(0.460520\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.95316 −0.102382
\(833\) −65.3876 −2.26555
\(834\) 0 0
\(835\) 0 0
\(836\) −11.4380 −0.395592
\(837\) 0 0
\(838\) −29.7146 −1.02647
\(839\) −3.34614 −0.115522 −0.0577608 0.998330i \(-0.518396\pi\)
−0.0577608 + 0.998330i \(0.518396\pi\)
\(840\) 0 0
\(841\) 31.9403 1.10139
\(842\) 15.0321 0.518041
\(843\) 0 0
\(844\) −17.2988 −0.595450
\(845\) 0 0
\(846\) 0 0
\(847\) 4.42864 0.152170
\(848\) 50.0830 1.71986
\(849\) 0 0
\(850\) 0 0
\(851\) 17.7146 0.607247
\(852\) 0 0
\(853\) 26.4197 0.904595 0.452297 0.891867i \(-0.350605\pi\)
0.452297 + 0.891867i \(0.350605\pi\)
\(854\) 57.7975 1.97779
\(855\) 0 0
\(856\) −1.88538 −0.0644411
\(857\) 38.7783 1.32464 0.662321 0.749220i \(-0.269571\pi\)
0.662321 + 0.749220i \(0.269571\pi\)
\(858\) 0 0
\(859\) −27.3087 −0.931760 −0.465880 0.884848i \(-0.654262\pi\)
−0.465880 + 0.884848i \(0.654262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −65.2899 −2.22378
\(863\) −49.5308 −1.68605 −0.843024 0.537875i \(-0.819227\pi\)
−0.843024 + 0.537875i \(0.819227\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −27.5397 −0.935838
\(867\) 0 0
\(868\) 19.7966 0.671940
\(869\) −8.56199 −0.290446
\(870\) 0 0
\(871\) 0.774305 0.0262363
\(872\) 14.1748 0.480021
\(873\) 0 0
\(874\) 118.858 4.02044
\(875\) 0 0
\(876\) 0 0
\(877\) 4.50177 0.152014 0.0760070 0.997107i \(-0.475783\pi\)
0.0760070 + 0.997107i \(0.475783\pi\)
\(878\) −36.7654 −1.24077
\(879\) 0 0
\(880\) 0 0
\(881\) 15.1240 0.509540 0.254770 0.967002i \(-0.418000\pi\)
0.254770 + 0.967002i \(0.418000\pi\)
\(882\) 0 0
\(883\) 30.2480 1.01793 0.508963 0.860789i \(-0.330029\pi\)
0.508963 + 0.860789i \(0.330029\pi\)
\(884\) −5.23277 −0.175997
\(885\) 0 0
\(886\) −24.9777 −0.839143
\(887\) 57.1941 1.92039 0.960194 0.279333i \(-0.0901135\pi\)
0.960194 + 0.279333i \(0.0901135\pi\)
\(888\) 0 0
\(889\) 67.2454 2.25534
\(890\) 0 0
\(891\) 0 0
\(892\) −14.3684 −0.481090
\(893\) −19.4291 −0.650171
\(894\) 0 0
\(895\) 0 0
\(896\) 25.0192 0.835833
\(897\) 0 0
\(898\) 61.4835 2.05173
\(899\) 21.5111 0.717437
\(900\) 0 0
\(901\) 56.2864 1.87517
\(902\) −0.368416 −0.0122669
\(903\) 0 0
\(904\) 4.31402 0.143482
\(905\) 0 0
\(906\) 0 0
\(907\) 53.2641 1.76861 0.884303 0.466913i \(-0.154634\pi\)
0.884303 + 0.466913i \(0.154634\pi\)
\(908\) 21.7017 0.720195
\(909\) 0 0
\(910\) 0 0
\(911\) 0.590573 0.0195665 0.00978327 0.999952i \(-0.496886\pi\)
0.00978327 + 0.999952i \(0.496886\pi\)
\(912\) 0 0
\(913\) −0.133353 −0.00441334
\(914\) 44.6508 1.47692
\(915\) 0 0
\(916\) 18.6735 0.616991
\(917\) −5.51114 −0.181994
\(918\) 0 0
\(919\) 55.8707 1.84300 0.921502 0.388375i \(-0.126963\pi\)
0.921502 + 0.388375i \(0.126963\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 54.9403 1.80936
\(923\) −1.71456 −0.0564354
\(924\) 0 0
\(925\) 0 0
\(926\) 36.8198 1.20997
\(927\) 0 0
\(928\) −57.3087 −1.88125
\(929\) −15.3274 −0.502876 −0.251438 0.967873i \(-0.580903\pi\)
−0.251438 + 0.967873i \(0.580903\pi\)
\(930\) 0 0
\(931\) 88.9314 2.91461
\(932\) −7.01921 −0.229922
\(933\) 0 0
\(934\) 5.98126 0.195713
\(935\) 0 0
\(936\) 0 0
\(937\) 27.8479 0.909752 0.454876 0.890555i \(-0.349684\pi\)
0.454876 + 0.890555i \(0.349684\pi\)
\(938\) 10.4889 0.342474
\(939\) 0 0
\(940\) 0 0
\(941\) −10.4157 −0.339543 −0.169772 0.985483i \(-0.554303\pi\)
−0.169772 + 0.985483i \(0.554303\pi\)
\(942\) 0 0
\(943\) 1.71456 0.0558337
\(944\) −22.4059 −0.729250
\(945\) 0 0
\(946\) 10.7971 0.351043
\(947\) 8.47013 0.275242 0.137621 0.990485i \(-0.456054\pi\)
0.137621 + 0.990485i \(0.456054\pi\)
\(948\) 0 0
\(949\) −2.63512 −0.0855397
\(950\) 0 0
\(951\) 0 0
\(952\) 16.5076 0.535014
\(953\) −8.71408 −0.282277 −0.141138 0.989990i \(-0.545076\pi\)
−0.141138 + 0.989990i \(0.545076\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 5.42816 0.175559
\(957\) 0 0
\(958\) 47.3087 1.52847
\(959\) −2.16500 −0.0699114
\(960\) 0 0
\(961\) −23.4068 −0.755059
\(962\) 2.36842 0.0763608
\(963\) 0 0
\(964\) −2.18373 −0.0703333
\(965\) 0 0
\(966\) 0 0
\(967\) −44.2449 −1.42282 −0.711410 0.702777i \(-0.751943\pi\)
−0.711410 + 0.702777i \(0.751943\pi\)
\(968\) −0.719004 −0.0231097
\(969\) 0 0
\(970\) 0 0
\(971\) 57.1437 1.83383 0.916914 0.399085i \(-0.130672\pi\)
0.916914 + 0.399085i \(0.130672\pi\)
\(972\) 0 0
\(973\) 78.8582 2.52808
\(974\) 21.9438 0.703124
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) −16.2480 −0.519819 −0.259909 0.965633i \(-0.583693\pi\)
−0.259909 + 0.965633i \(0.583693\pi\)
\(978\) 0 0
\(979\) 5.61285 0.179387
\(980\) 0 0
\(981\) 0 0
\(982\) −31.1882 −0.995256
\(983\) 1.12399 0.0358496 0.0179248 0.999839i \(-0.494294\pi\)
0.0179248 + 0.999839i \(0.494294\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −77.0232 −2.45292
\(987\) 0 0
\(988\) 7.11691 0.226419
\(989\) −50.2480 −1.59779
\(990\) 0 0
\(991\) −53.6513 −1.70429 −0.852144 0.523307i \(-0.824698\pi\)
−0.852144 + 0.523307i \(0.824698\pi\)
\(992\) −20.2292 −0.642279
\(993\) 0 0
\(994\) −23.2257 −0.736674
\(995\) 0 0
\(996\) 0 0
\(997\) 35.7275 1.13150 0.565750 0.824577i \(-0.308587\pi\)
0.565750 + 0.824577i \(0.308587\pi\)
\(998\) −48.2034 −1.52585
\(999\) 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 2475.2.a.bb.1.3 3
3.2 odd 2 825.2.a.k.1.1 3
5.2 odd 4 2475.2.c.r.199.5 6
5.3 odd 4 2475.2.c.r.199.2 6
5.4 even 2 495.2.a.e.1.1 3
15.2 even 4 825.2.c.g.199.2 6
15.8 even 4 825.2.c.g.199.5 6
15.14 odd 2 165.2.a.c.1.3 3
20.19 odd 2 7920.2.a.cj.1.3 3
33.32 even 2 9075.2.a.cf.1.3 3
55.54 odd 2 5445.2.a.z.1.3 3
60.59 even 2 2640.2.a.be.1.3 3
105.104 even 2 8085.2.a.bk.1.3 3
165.164 even 2 1815.2.a.m.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.3 3 15.14 odd 2
495.2.a.e.1.1 3 5.4 even 2
825.2.a.k.1.1 3 3.2 odd 2
825.2.c.g.199.2 6 15.2 even 4
825.2.c.g.199.5 6 15.8 even 4
1815.2.a.m.1.1 3 165.164 even 2
2475.2.a.bb.1.3 3 1.1 even 1 trivial
2475.2.c.r.199.2 6 5.3 odd 4
2475.2.c.r.199.5 6 5.2 odd 4
2640.2.a.be.1.3 3 60.59 even 2
5445.2.a.z.1.3 3 55.54 odd 2
7920.2.a.cj.1.3 3 20.19 odd 2
8085.2.a.bk.1.3 3 105.104 even 2
9075.2.a.cf.1.3 3 33.32 even 2