Properties

Label 2475.2.a.x.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{7} +4.41421 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} +3.82843 q^{4} +2.00000 q^{7} +4.41421 q^{8} -1.00000 q^{11} +1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843 q^{17} -2.41421 q^{22} -2.82843 q^{23} +2.82843 q^{26} +7.65685 q^{28} +3.65685 q^{29} -1.58579 q^{32} +16.4853 q^{34} +7.65685 q^{37} -6.00000 q^{41} +6.00000 q^{43} -3.82843 q^{44} -6.82843 q^{46} +2.82843 q^{47} -3.00000 q^{49} +4.48528 q^{52} +11.6569 q^{53} +8.82843 q^{56} +8.82843 q^{58} -1.65685 q^{59} -9.31371 q^{61} -9.82843 q^{64} -12.4853 q^{67} +26.1421 q^{68} -11.3137 q^{71} +1.17157 q^{73} +18.4853 q^{74} -2.00000 q^{77} +4.00000 q^{79} -14.4853 q^{82} -6.00000 q^{83} +14.4853 q^{86} -4.41421 q^{88} +13.3137 q^{89} +2.34315 q^{91} -10.8284 q^{92} +6.82843 q^{94} -3.65685 q^{97} -7.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 4.41421 1.56066
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.17157 0.324936 0.162468 0.986714i \(-0.448055\pi\)
0.162468 + 0.986714i \(0.448055\pi\)
\(14\) 4.82843 1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.41421 −0.514712
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.82843 0.554700
\(27\) 0 0
\(28\) 7.65685 1.44701
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.58579 −0.280330
\(33\) 0 0
\(34\) 16.4853 2.82720
\(35\) 0 0
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −3.82843 −0.577157
\(45\) 0 0
\(46\) −6.82843 −1.00680
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 4.48528 0.621997
\(53\) 11.6569 1.60119 0.800596 0.599204i \(-0.204516\pi\)
0.800596 + 0.599204i \(0.204516\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.82843 1.17975
\(57\) 0 0
\(58\) 8.82843 1.15923
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4853 −1.52532 −0.762660 0.646800i \(-0.776107\pi\)
−0.762660 + 0.646800i \(0.776107\pi\)
\(68\) 26.1421 3.17020
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) 1.17157 0.137122 0.0685611 0.997647i \(-0.478159\pi\)
0.0685611 + 0.997647i \(0.478159\pi\)
\(74\) 18.4853 2.14887
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.4853 −1.59963
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.4853 1.56199
\(87\) 0 0
\(88\) −4.41421 −0.470557
\(89\) 13.3137 1.41125 0.705625 0.708585i \(-0.250666\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(90\) 0 0
\(91\) 2.34315 0.245628
\(92\) −10.8284 −1.12894
\(93\) 0 0
\(94\) 6.82843 0.704298
\(95\) 0 0
\(96\) 0 0
\(97\) −3.65685 −0.371297 −0.185649 0.982616i \(-0.559439\pi\)
−0.185649 + 0.982616i \(0.559439\pi\)
\(98\) −7.24264 −0.731617
\(99\) 0 0
\(100\) 0 0
\(101\) −9.31371 −0.926749 −0.463374 0.886163i \(-0.653361\pi\)
−0.463374 + 0.886163i \(0.653361\pi\)
\(102\) 0 0
\(103\) −6.82843 −0.672825 −0.336412 0.941715i \(-0.609214\pi\)
−0.336412 + 0.941715i \(0.609214\pi\)
\(104\) 5.17157 0.507114
\(105\) 0 0
\(106\) 28.1421 2.73341
\(107\) 7.65685 0.740216 0.370108 0.928989i \(-0.379321\pi\)
0.370108 + 0.928989i \(0.379321\pi\)
\(108\) 0 0
\(109\) −7.65685 −0.733394 −0.366697 0.930341i \(-0.619511\pi\)
−0.366697 + 0.930341i \(0.619511\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.00000 0.566947
\(113\) 19.6569 1.84916 0.924581 0.380986i \(-0.124416\pi\)
0.924581 + 0.380986i \(0.124416\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −22.4853 −2.03572
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.34315 −0.385392 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(128\) −20.5563 −1.81694
\(129\) 0 0
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −30.1421 −2.60388
\(135\) 0 0
\(136\) 30.1421 2.58467
\(137\) −10.9706 −0.937278 −0.468639 0.883390i \(-0.655256\pi\)
−0.468639 + 0.883390i \(0.655256\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −27.3137 −2.29212
\(143\) −1.17157 −0.0979718
\(144\) 0 0
\(145\) 0 0
\(146\) 2.82843 0.234082
\(147\) 0 0
\(148\) 29.3137 2.40957
\(149\) −0.343146 −0.0281116 −0.0140558 0.999901i \(-0.504474\pi\)
−0.0140558 + 0.999901i \(0.504474\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −4.82843 −0.389086
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 9.65685 0.768258
\(159\) 0 0
\(160\) 0 0
\(161\) −5.65685 −0.445823
\(162\) 0 0
\(163\) −16.4853 −1.29123 −0.645613 0.763664i \(-0.723398\pi\)
−0.645613 + 0.763664i \(0.723398\pi\)
\(164\) −22.9706 −1.79370
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) −22.9706 −1.77752 −0.888758 0.458377i \(-0.848431\pi\)
−0.888758 + 0.458377i \(0.848431\pi\)
\(168\) 0 0
\(169\) −11.6274 −0.894417
\(170\) 0 0
\(171\) 0 0
\(172\) 22.9706 1.75149
\(173\) −22.1421 −1.68344 −0.841718 0.539918i \(-0.818455\pi\)
−0.841718 + 0.539918i \(0.818455\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 32.1421 2.40915
\(179\) −9.65685 −0.721787 −0.360894 0.932607i \(-0.617528\pi\)
−0.360894 + 0.932607i \(0.617528\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 5.65685 0.419314
\(183\) 0 0
\(184\) −12.4853 −0.920427
\(185\) 0 0
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) 10.8284 0.789744
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) 1.17157 0.0843317 0.0421658 0.999111i \(-0.486574\pi\)
0.0421658 + 0.999111i \(0.486574\pi\)
\(194\) −8.82843 −0.633844
\(195\) 0 0
\(196\) −11.4853 −0.820377
\(197\) −10.8284 −0.771493 −0.385747 0.922605i \(-0.626056\pi\)
−0.385747 + 0.922605i \(0.626056\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −22.4853 −1.58206
\(203\) 7.31371 0.513322
\(204\) 0 0
\(205\) 0 0
\(206\) −16.4853 −1.14858
\(207\) 0 0
\(208\) 3.51472 0.243702
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 44.6274 3.06502
\(213\) 0 0
\(214\) 18.4853 1.26363
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −18.4853 −1.25198
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 10.8284 0.725125 0.362563 0.931959i \(-0.381902\pi\)
0.362563 + 0.931959i \(0.381902\pi\)
\(224\) −3.17157 −0.211910
\(225\) 0 0
\(226\) 47.4558 3.15672
\(227\) 25.3137 1.68013 0.840065 0.542486i \(-0.182517\pi\)
0.840065 + 0.542486i \(0.182517\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 16.1421 1.05978
\(233\) −6.14214 −0.402385 −0.201192 0.979552i \(-0.564482\pi\)
−0.201192 + 0.979552i \(0.564482\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.34315 −0.412904
\(237\) 0 0
\(238\) 32.9706 2.13716
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 2.41421 0.155192
\(243\) 0 0
\(244\) −35.6569 −2.28270
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.82843 0.177822
\(254\) −10.4853 −0.657905
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) −9.31371 −0.580973 −0.290487 0.956879i \(-0.593817\pi\)
−0.290487 + 0.956879i \(0.593817\pi\)
\(258\) 0 0
\(259\) 15.3137 0.951548
\(260\) 0 0
\(261\) 0 0
\(262\) 27.3137 1.68745
\(263\) −10.9706 −0.676474 −0.338237 0.941061i \(-0.609831\pi\)
−0.338237 + 0.941061i \(0.609831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −47.7990 −2.91979
\(269\) −17.3137 −1.05564 −0.527818 0.849358i \(-0.676990\pi\)
−0.527818 + 0.849358i \(0.676990\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) 20.4853 1.24210
\(273\) 0 0
\(274\) −26.4853 −1.60003
\(275\) 0 0
\(276\) 0 0
\(277\) −6.82843 −0.410280 −0.205140 0.978733i \(-0.565765\pi\)
−0.205140 + 0.978733i \(0.565765\pi\)
\(278\) −9.65685 −0.579180
\(279\) 0 0
\(280\) 0 0
\(281\) −17.3137 −1.03285 −0.516425 0.856333i \(-0.672737\pi\)
−0.516425 + 0.856333i \(0.672737\pi\)
\(282\) 0 0
\(283\) −32.6274 −1.93950 −0.969749 0.244103i \(-0.921507\pi\)
−0.969749 + 0.244103i \(0.921507\pi\)
\(284\) −43.3137 −2.57020
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 0 0
\(292\) 4.48528 0.262481
\(293\) −9.17157 −0.535809 −0.267905 0.963445i \(-0.586331\pi\)
−0.267905 + 0.963445i \(0.586331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 33.7990 1.96453
\(297\) 0 0
\(298\) −0.828427 −0.0479895
\(299\) −3.31371 −0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −28.9706 −1.66707
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.3431 0.932753 0.466376 0.884586i \(-0.345559\pi\)
0.466376 + 0.884586i \(0.345559\pi\)
\(308\) −7.65685 −0.436290
\(309\) 0 0
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) 1.31371 0.0742552 0.0371276 0.999311i \(-0.488179\pi\)
0.0371276 + 0.999311i \(0.488179\pi\)
\(314\) 33.7990 1.90739
\(315\) 0 0
\(316\) 15.3137 0.861463
\(317\) −1.31371 −0.0737852 −0.0368926 0.999319i \(-0.511746\pi\)
−0.0368926 + 0.999319i \(0.511746\pi\)
\(318\) 0 0
\(319\) −3.65685 −0.204745
\(320\) 0 0
\(321\) 0 0
\(322\) −13.6569 −0.761067
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −39.7990 −2.20426
\(327\) 0 0
\(328\) −26.4853 −1.46241
\(329\) 5.65685 0.311872
\(330\) 0 0
\(331\) −7.31371 −0.401998 −0.200999 0.979591i \(-0.564419\pi\)
−0.200999 + 0.979591i \(0.564419\pi\)
\(332\) −22.9706 −1.26067
\(333\) 0 0
\(334\) −55.4558 −3.03441
\(335\) 0 0
\(336\) 0 0
\(337\) 20.4853 1.11590 0.557952 0.829873i \(-0.311587\pi\)
0.557952 + 0.829873i \(0.311587\pi\)
\(338\) −28.0711 −1.52686
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 26.4853 1.42799
\(345\) 0 0
\(346\) −53.4558 −2.87380
\(347\) 10.9706 0.588931 0.294465 0.955662i \(-0.404858\pi\)
0.294465 + 0.955662i \(0.404858\pi\)
\(348\) 0 0
\(349\) 26.9706 1.44370 0.721851 0.692049i \(-0.243292\pi\)
0.721851 + 0.692049i \(0.243292\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.58579 0.0845227
\(353\) 21.3137 1.13441 0.567207 0.823575i \(-0.308024\pi\)
0.567207 + 0.823575i \(0.308024\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 50.9706 2.70143
\(357\) 0 0
\(358\) −23.3137 −1.23217
\(359\) −0.686292 −0.0362211 −0.0181105 0.999836i \(-0.505765\pi\)
−0.0181105 + 0.999836i \(0.505765\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 51.4558 2.70446
\(363\) 0 0
\(364\) 8.97056 0.470185
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528 0.442928 0.221464 0.975169i \(-0.428916\pi\)
0.221464 + 0.975169i \(0.428916\pi\)
\(368\) −8.48528 −0.442326
\(369\) 0 0
\(370\) 0 0
\(371\) 23.3137 1.21039
\(372\) 0 0
\(373\) −35.7990 −1.85360 −0.926801 0.375554i \(-0.877453\pi\)
−0.926801 + 0.375554i \(0.877453\pi\)
\(374\) −16.4853 −0.852434
\(375\) 0 0
\(376\) 12.4853 0.643879
\(377\) 4.28427 0.220651
\(378\) 0 0
\(379\) 33.6569 1.72884 0.864418 0.502773i \(-0.167687\pi\)
0.864418 + 0.502773i \(0.167687\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −5.85786 −0.299323 −0.149661 0.988737i \(-0.547818\pi\)
−0.149661 + 0.988737i \(0.547818\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.82843 0.143963
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) −20.6274 −1.04585 −0.522926 0.852378i \(-0.675160\pi\)
−0.522926 + 0.852378i \(0.675160\pi\)
\(390\) 0 0
\(391\) −19.3137 −0.976736
\(392\) −13.2426 −0.668854
\(393\) 0 0
\(394\) −26.1421 −1.31702
\(395\) 0 0
\(396\) 0 0
\(397\) 9.31371 0.467442 0.233721 0.972304i \(-0.424910\pi\)
0.233721 + 0.972304i \(0.424910\pi\)
\(398\) 24.9706 1.25166
\(399\) 0 0
\(400\) 0 0
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −35.6569 −1.77399
\(405\) 0 0
\(406\) 17.6569 0.876295
\(407\) −7.65685 −0.379536
\(408\) 0 0
\(409\) 1.02944 0.0509024 0.0254512 0.999676i \(-0.491898\pi\)
0.0254512 + 0.999676i \(0.491898\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −26.1421 −1.28793
\(413\) −3.31371 −0.163057
\(414\) 0 0
\(415\) 0 0
\(416\) −1.85786 −0.0910893
\(417\) 0 0
\(418\) 0 0
\(419\) 25.6569 1.25342 0.626710 0.779253i \(-0.284401\pi\)
0.626710 + 0.779253i \(0.284401\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −38.6274 −1.88035
\(423\) 0 0
\(424\) 51.4558 2.49892
\(425\) 0 0
\(426\) 0 0
\(427\) −18.6274 −0.901444
\(428\) 29.3137 1.41693
\(429\) 0 0
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 0 0
\(433\) 7.65685 0.367965 0.183982 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −29.3137 −1.40387
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 19.3137 0.918659
\(443\) −26.8284 −1.27466 −0.637329 0.770592i \(-0.719961\pi\)
−0.637329 + 0.770592i \(0.719961\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 26.1421 1.23787
\(447\) 0 0
\(448\) −19.6569 −0.928699
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 75.2548 3.53969
\(453\) 0 0
\(454\) 61.1127 2.86816
\(455\) 0 0
\(456\) 0 0
\(457\) −0.485281 −0.0227005 −0.0113503 0.999936i \(-0.503613\pi\)
−0.0113503 + 0.999936i \(0.503613\pi\)
\(458\) 3.17157 0.148198
\(459\) 0 0
\(460\) 0 0
\(461\) −12.6274 −0.588117 −0.294059 0.955787i \(-0.595006\pi\)
−0.294059 + 0.955787i \(0.595006\pi\)
\(462\) 0 0
\(463\) 6.14214 0.285449 0.142725 0.989762i \(-0.454414\pi\)
0.142725 + 0.989762i \(0.454414\pi\)
\(464\) 10.9706 0.509296
\(465\) 0 0
\(466\) −14.8284 −0.686914
\(467\) −14.8284 −0.686178 −0.343089 0.939303i \(-0.611473\pi\)
−0.343089 + 0.939303i \(0.611473\pi\)
\(468\) 0 0
\(469\) −24.9706 −1.15303
\(470\) 0 0
\(471\) 0 0
\(472\) −7.31371 −0.336641
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 52.2843 2.39645
\(477\) 0 0
\(478\) 56.2843 2.57438
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) 14.4853 0.659786
\(483\) 0 0
\(484\) 3.82843 0.174019
\(485\) 0 0
\(486\) 0 0
\(487\) 24.4853 1.10953 0.554767 0.832006i \(-0.312807\pi\)
0.554767 + 0.832006i \(0.312807\pi\)
\(488\) −41.1127 −1.86108
\(489\) 0 0
\(490\) 0 0
\(491\) 0.686292 0.0309719 0.0154860 0.999880i \(-0.495070\pi\)
0.0154860 + 0.999880i \(0.495070\pi\)
\(492\) 0 0
\(493\) 24.9706 1.12462
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −22.6274 −1.01498
\(498\) 0 0
\(499\) 9.65685 0.432300 0.216150 0.976360i \(-0.430650\pi\)
0.216150 + 0.976360i \(0.430650\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −28.9706 −1.29302
\(503\) 16.6274 0.741380 0.370690 0.928757i \(-0.379121\pi\)
0.370690 + 0.928757i \(0.379121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.82843 0.303561
\(507\) 0 0
\(508\) −16.6274 −0.737722
\(509\) 13.3137 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(510\) 0 0
\(511\) 2.34315 0.103655
\(512\) −31.2426 −1.38074
\(513\) 0 0
\(514\) −22.4853 −0.991783
\(515\) 0 0
\(516\) 0 0
\(517\) −2.82843 −0.124394
\(518\) 36.9706 1.62439
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0 0
\(523\) 41.5980 1.81895 0.909476 0.415756i \(-0.136483\pi\)
0.909476 + 0.415756i \(0.136483\pi\)
\(524\) 43.3137 1.89217
\(525\) 0 0
\(526\) −26.4853 −1.15481
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.02944 −0.304479
\(534\) 0 0
\(535\) 0 0
\(536\) −55.1127 −2.38051
\(537\) 0 0
\(538\) −41.7990 −1.80208
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 17.6569 0.758427
\(543\) 0 0
\(544\) −10.8284 −0.464265
\(545\) 0 0
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) −42.0000 −1.79415
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −16.4853 −0.700392
\(555\) 0 0
\(556\) −15.3137 −0.649446
\(557\) 9.85786 0.417691 0.208846 0.977949i \(-0.433029\pi\)
0.208846 + 0.977949i \(0.433029\pi\)
\(558\) 0 0
\(559\) 7.02944 0.297314
\(560\) 0 0
\(561\) 0 0
\(562\) −41.7990 −1.76318
\(563\) 0.343146 0.0144619 0.00723093 0.999974i \(-0.497698\pi\)
0.00723093 + 0.999974i \(0.497698\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −78.7696 −3.31093
\(567\) 0 0
\(568\) −49.9411 −2.09548
\(569\) −31.6569 −1.32712 −0.663562 0.748121i \(-0.730956\pi\)
−0.663562 + 0.748121i \(0.730956\pi\)
\(570\) 0 0
\(571\) −21.9411 −0.918208 −0.459104 0.888383i \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(572\) −4.48528 −0.187539
\(573\) 0 0
\(574\) −28.9706 −1.20921
\(575\) 0 0
\(576\) 0 0
\(577\) 26.9706 1.12280 0.561400 0.827545i \(-0.310263\pi\)
0.561400 + 0.827545i \(0.310263\pi\)
\(578\) 71.5269 2.97513
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −11.6569 −0.482778
\(584\) 5.17157 0.214001
\(585\) 0 0
\(586\) −22.1421 −0.914683
\(587\) −2.14214 −0.0884154 −0.0442077 0.999022i \(-0.514076\pi\)
−0.0442077 + 0.999022i \(0.514076\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 22.9706 0.944084
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.31371 −0.0538116
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) 28.9706 1.18075
\(603\) 0 0
\(604\) −45.9411 −1.86932
\(605\) 0 0
\(606\) 0 0
\(607\) −38.2843 −1.55391 −0.776955 0.629556i \(-0.783237\pi\)
−0.776955 + 0.629556i \(0.783237\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.31371 0.134058
\(612\) 0 0
\(613\) 25.4558 1.02815 0.514076 0.857745i \(-0.328135\pi\)
0.514076 + 0.857745i \(0.328135\pi\)
\(614\) 39.4558 1.59231
\(615\) 0 0
\(616\) −8.82843 −0.355707
\(617\) 0.343146 0.0138145 0.00690726 0.999976i \(-0.497801\pi\)
0.00690726 + 0.999976i \(0.497801\pi\)
\(618\) 0 0
\(619\) −14.3431 −0.576500 −0.288250 0.957555i \(-0.593073\pi\)
−0.288250 + 0.957555i \(0.593073\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −11.3137 −0.453638
\(623\) 26.6274 1.06680
\(624\) 0 0
\(625\) 0 0
\(626\) 3.17157 0.126762
\(627\) 0 0
\(628\) 53.5980 2.13879
\(629\) 52.2843 2.08471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 17.6569 0.702352
\(633\) 0 0
\(634\) −3.17157 −0.125959
\(635\) 0 0
\(636\) 0 0
\(637\) −3.51472 −0.139258
\(638\) −8.82843 −0.349521
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 1.45584 0.0574129 0.0287064 0.999588i \(-0.490861\pi\)
0.0287064 + 0.999588i \(0.490861\pi\)
\(644\) −21.6569 −0.853400
\(645\) 0 0
\(646\) 0 0
\(647\) 27.1127 1.06591 0.532955 0.846144i \(-0.321081\pi\)
0.532955 + 0.846144i \(0.321081\pi\)
\(648\) 0 0
\(649\) 1.65685 0.0650372
\(650\) 0 0
\(651\) 0 0
\(652\) −63.1127 −2.47168
\(653\) 11.6569 0.456168 0.228084 0.973641i \(-0.426754\pi\)
0.228084 + 0.973641i \(0.426754\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 0 0
\(658\) 13.6569 0.532400
\(659\) −45.9411 −1.78961 −0.894806 0.446455i \(-0.852686\pi\)
−0.894806 + 0.446455i \(0.852686\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) −17.6569 −0.686253
\(663\) 0 0
\(664\) −26.4853 −1.02783
\(665\) 0 0
\(666\) 0 0
\(667\) −10.3431 −0.400488
\(668\) −87.9411 −3.40254
\(669\) 0 0
\(670\) 0 0
\(671\) 9.31371 0.359552
\(672\) 0 0
\(673\) 12.4853 0.481272 0.240636 0.970615i \(-0.422644\pi\)
0.240636 + 0.970615i \(0.422644\pi\)
\(674\) 49.4558 1.90497
\(675\) 0 0
\(676\) −44.5147 −1.71210
\(677\) 22.8284 0.877368 0.438684 0.898641i \(-0.355445\pi\)
0.438684 + 0.898641i \(0.355445\pi\)
\(678\) 0 0
\(679\) −7.31371 −0.280674
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.79899 −0.298420 −0.149210 0.988806i \(-0.547673\pi\)
−0.149210 + 0.988806i \(0.547673\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −48.2843 −1.84350
\(687\) 0 0
\(688\) 18.0000 0.686244
\(689\) 13.6569 0.520285
\(690\) 0 0
\(691\) −39.3137 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(692\) −84.7696 −3.22245
\(693\) 0 0
\(694\) 26.4853 1.00537
\(695\) 0 0
\(696\) 0 0
\(697\) −40.9706 −1.55187
\(698\) 65.1127 2.46455
\(699\) 0 0
\(700\) 0 0
\(701\) 12.6274 0.476931 0.238465 0.971151i \(-0.423356\pi\)
0.238465 + 0.971151i \(0.423356\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 9.82843 0.370423
\(705\) 0 0
\(706\) 51.4558 1.93657
\(707\) −18.6274 −0.700556
\(708\) 0 0
\(709\) 24.6274 0.924902 0.462451 0.886645i \(-0.346970\pi\)
0.462451 + 0.886645i \(0.346970\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 58.7696 2.20248
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −36.9706 −1.38165
\(717\) 0 0
\(718\) −1.65685 −0.0618333
\(719\) −18.3431 −0.684084 −0.342042 0.939685i \(-0.611118\pi\)
−0.342042 + 0.939685i \(0.611118\pi\)
\(720\) 0 0
\(721\) −13.6569 −0.508608
\(722\) −45.8701 −1.70711
\(723\) 0 0
\(724\) 81.5980 3.03257
\(725\) 0 0
\(726\) 0 0
\(727\) 19.5147 0.723761 0.361880 0.932225i \(-0.382135\pi\)
0.361880 + 0.932225i \(0.382135\pi\)
\(728\) 10.3431 0.383342
\(729\) 0 0
\(730\) 0 0
\(731\) 40.9706 1.51535
\(732\) 0 0
\(733\) 17.4558 0.644746 0.322373 0.946613i \(-0.395519\pi\)
0.322373 + 0.946613i \(0.395519\pi\)
\(734\) 20.4853 0.756126
\(735\) 0 0
\(736\) 4.48528 0.165330
\(737\) 12.4853 0.459901
\(738\) 0 0
\(739\) 29.9411 1.10140 0.550701 0.834703i \(-0.314360\pi\)
0.550701 + 0.834703i \(0.314360\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 56.2843 2.06626
\(743\) −49.5980 −1.81957 −0.909787 0.415076i \(-0.863755\pi\)
−0.909787 + 0.415076i \(0.863755\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −86.4264 −3.16430
\(747\) 0 0
\(748\) −26.1421 −0.955851
\(749\) 15.3137 0.559551
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.48528 0.309426
\(753\) 0 0
\(754\) 10.3431 0.376675
\(755\) 0 0
\(756\) 0 0
\(757\) −13.3137 −0.483895 −0.241947 0.970289i \(-0.577786\pi\)
−0.241947 + 0.970289i \(0.577786\pi\)
\(758\) 81.2548 2.95131
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) −15.3137 −0.554393
\(764\) −12.6863 −0.458974
\(765\) 0 0
\(766\) −14.1421 −0.510976
\(767\) −1.94113 −0.0700900
\(768\) 0 0
\(769\) −18.9706 −0.684096 −0.342048 0.939682i \(-0.611121\pi\)
−0.342048 + 0.939682i \(0.611121\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.48528 0.161429
\(773\) 26.2843 0.945380 0.472690 0.881229i \(-0.343283\pi\)
0.472690 + 0.881229i \(0.343283\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.1421 −0.579469
\(777\) 0 0
\(778\) −49.7990 −1.78538
\(779\) 0 0
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) −46.6274 −1.66739
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 14.9706 0.533643 0.266821 0.963746i \(-0.414027\pi\)
0.266821 + 0.963746i \(0.414027\pi\)
\(788\) −41.4558 −1.47680
\(789\) 0 0
\(790\) 0 0
\(791\) 39.3137 1.39783
\(792\) 0 0
\(793\) −10.9117 −0.387485
\(794\) 22.4853 0.797973
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) 32.6274 1.15572 0.577861 0.816135i \(-0.303887\pi\)
0.577861 + 0.816135i \(0.303887\pi\)
\(798\) 0 0
\(799\) 19.3137 0.683270
\(800\) 0 0
\(801\) 0 0
\(802\) 12.8284 0.452988
\(803\) −1.17157 −0.0413439
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −41.1127 −1.44634
\(809\) 10.9706 0.385704 0.192852 0.981228i \(-0.438226\pi\)
0.192852 + 0.981228i \(0.438226\pi\)
\(810\) 0 0
\(811\) 53.9411 1.89413 0.947065 0.321043i \(-0.104033\pi\)
0.947065 + 0.321043i \(0.104033\pi\)
\(812\) 28.0000 0.982607
\(813\) 0 0
\(814\) −18.4853 −0.647909
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 2.48528 0.0868958
\(819\) 0 0
\(820\) 0 0
\(821\) 41.3137 1.44186 0.720929 0.693009i \(-0.243715\pi\)
0.720929 + 0.693009i \(0.243715\pi\)
\(822\) 0 0
\(823\) −19.5147 −0.680240 −0.340120 0.940382i \(-0.610468\pi\)
−0.340120 + 0.940382i \(0.610468\pi\)
\(824\) −30.1421 −1.05005
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 22.2843 0.774900 0.387450 0.921891i \(-0.373356\pi\)
0.387450 + 0.921891i \(0.373356\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −11.5147 −0.399201
\(833\) −20.4853 −0.709773
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 61.9411 2.13972
\(839\) −26.3431 −0.909466 −0.454733 0.890628i \(-0.650265\pi\)
−0.454733 + 0.890628i \(0.650265\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) −14.4853 −0.499196
\(843\) 0 0
\(844\) −61.2548 −2.10848
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 34.9706 1.20089
\(849\) 0 0
\(850\) 0 0
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) 15.5147 0.531214 0.265607 0.964081i \(-0.414428\pi\)
0.265607 + 0.964081i \(0.414428\pi\)
\(854\) −44.9706 −1.53886
\(855\) 0 0
\(856\) 33.7990 1.15523
\(857\) 24.7696 0.846112 0.423056 0.906104i \(-0.360957\pi\)
0.423056 + 0.906104i \(0.360957\pi\)
\(858\) 0 0
\(859\) 24.2843 0.828569 0.414284 0.910148i \(-0.364032\pi\)
0.414284 + 0.910148i \(0.364032\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 27.3137 0.930309
\(863\) −9.17157 −0.312204 −0.156102 0.987741i \(-0.549893\pi\)
−0.156102 + 0.987741i \(0.549893\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 18.4853 0.628155
\(867\) 0 0
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −14.6274 −0.495631
\(872\) −33.7990 −1.14458
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 49.4558 1.67001 0.835003 0.550246i \(-0.185466\pi\)
0.835003 + 0.550246i \(0.185466\pi\)
\(878\) −38.6274 −1.30361
\(879\) 0 0
\(880\) 0 0
\(881\) 7.37258 0.248389 0.124194 0.992258i \(-0.460365\pi\)
0.124194 + 0.992258i \(0.460365\pi\)
\(882\) 0 0
\(883\) −37.1716 −1.25092 −0.625462 0.780255i \(-0.715089\pi\)
−0.625462 + 0.780255i \(0.715089\pi\)
\(884\) 30.6274 1.03011
\(885\) 0 0
\(886\) −64.7696 −2.17598
\(887\) 38.2843 1.28546 0.642730 0.766093i \(-0.277802\pi\)
0.642730 + 0.766093i \(0.277802\pi\)
\(888\) 0 0
\(889\) −8.68629 −0.291329
\(890\) 0 0
\(891\) 0 0
\(892\) 41.4558 1.38804
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −41.1127 −1.37348
\(897\) 0 0
\(898\) −69.1127 −2.30632
\(899\) 0 0
\(900\) 0 0
\(901\) 79.5980 2.65179
\(902\) 14.4853 0.482307
\(903\) 0 0
\(904\) 86.7696 2.88591
\(905\) 0 0
\(906\) 0 0
\(907\) 27.5147 0.913611 0.456806 0.889567i \(-0.348993\pi\)
0.456806 + 0.889567i \(0.348993\pi\)
\(908\) 96.9117 3.21613
\(909\) 0 0
\(910\) 0 0
\(911\) 9.94113 0.329364 0.164682 0.986347i \(-0.447340\pi\)
0.164682 + 0.986347i \(0.447340\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) −1.17157 −0.0387522
\(915\) 0 0
\(916\) 5.02944 0.166177
\(917\) 22.6274 0.747223
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −30.4853 −1.00398
\(923\) −13.2548 −0.436288
\(924\) 0 0
\(925\) 0 0
\(926\) 14.8284 0.487292
\(927\) 0 0
\(928\) −5.79899 −0.190361
\(929\) −5.31371 −0.174337 −0.0871686 0.996194i \(-0.527782\pi\)
−0.0871686 + 0.996194i \(0.527782\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.5147 −0.770250
\(933\) 0 0
\(934\) −35.7990 −1.17138
\(935\) 0 0
\(936\) 0 0
\(937\) 1.45584 0.0475604 0.0237802 0.999717i \(-0.492430\pi\)
0.0237802 + 0.999717i \(0.492430\pi\)
\(938\) −60.2843 −1.96835
\(939\) 0 0
\(940\) 0 0
\(941\) 6.68629 0.217967 0.108983 0.994044i \(-0.465240\pi\)
0.108983 + 0.994044i \(0.465240\pi\)
\(942\) 0 0
\(943\) 16.9706 0.552638
\(944\) −4.97056 −0.161778
\(945\) 0 0
\(946\) −14.4853 −0.470957
\(947\) 41.1716 1.33790 0.668948 0.743309i \(-0.266745\pi\)
0.668948 + 0.743309i \(0.266745\pi\)
\(948\) 0 0
\(949\) 1.37258 0.0445559
\(950\) 0 0
\(951\) 0 0
\(952\) 60.2843 1.95382
\(953\) 53.1716 1.72240 0.861198 0.508269i \(-0.169715\pi\)
0.861198 + 0.508269i \(0.169715\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 89.2548 2.88671
\(957\) 0 0
\(958\) 86.9117 2.80799
\(959\) −21.9411 −0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 21.6569 0.698245
\(963\) 0 0
\(964\) 22.9706 0.739832
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9706 0.481421 0.240710 0.970597i \(-0.422620\pi\)
0.240710 + 0.970597i \(0.422620\pi\)
\(968\) 4.41421 0.141878
\(969\) 0 0
\(970\) 0 0
\(971\) 8.68629 0.278756 0.139378 0.990239i \(-0.455490\pi\)
0.139378 + 0.990239i \(0.455490\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 59.1127 1.89409
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) 32.3431 1.03475 0.517374 0.855759i \(-0.326909\pi\)
0.517374 + 0.855759i \(0.326909\pi\)
\(978\) 0 0
\(979\) −13.3137 −0.425508
\(980\) 0 0
\(981\) 0 0
\(982\) 1.65685 0.0528723
\(983\) −21.8579 −0.697158 −0.348579 0.937279i \(-0.613336\pi\)
−0.348579 + 0.937279i \(0.613336\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 60.2843 1.91984
\(987\) 0 0
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 0 0
\(991\) −57.9411 −1.84056 −0.920280 0.391260i \(-0.872039\pi\)
−0.920280 + 0.391260i \(0.872039\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −54.6274 −1.73268
\(995\) 0 0
\(996\) 0 0
\(997\) 41.4558 1.31292 0.656460 0.754361i \(-0.272053\pi\)
0.656460 + 0.754361i \(0.272053\pi\)
\(998\) 23.3137 0.737983
\(999\) 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.x.1.2 2
3.2 odd 2 275.2.a.c.1.1 2
5.2 odd 4 2475.2.c.l.199.4 4
5.3 odd 4 2475.2.c.l.199.1 4
5.4 even 2 495.2.a.b.1.1 2
12.11 even 2 4400.2.a.bn.1.1 2
15.2 even 4 275.2.b.d.199.1 4
15.8 even 4 275.2.b.d.199.4 4
15.14 odd 2 55.2.a.b.1.2 2
20.19 odd 2 7920.2.a.ch.1.2 2
33.32 even 2 3025.2.a.o.1.2 2
55.54 odd 2 5445.2.a.y.1.2 2
60.23 odd 4 4400.2.b.q.4049.2 4
60.47 odd 4 4400.2.b.q.4049.3 4
60.59 even 2 880.2.a.m.1.2 2
105.104 even 2 2695.2.a.f.1.2 2
120.29 odd 2 3520.2.a.bn.1.2 2
120.59 even 2 3520.2.a.bo.1.1 2
165.14 odd 10 605.2.g.f.251.2 8
165.29 even 10 605.2.g.l.511.1 8
165.59 odd 10 605.2.g.f.511.2 8
165.74 even 10 605.2.g.l.251.1 8
165.104 odd 10 605.2.g.f.366.1 8
165.119 odd 10 605.2.g.f.81.1 8
165.134 even 10 605.2.g.l.81.2 8
165.149 even 10 605.2.g.l.366.2 8
165.164 even 2 605.2.a.d.1.1 2
195.194 odd 2 9295.2.a.g.1.1 2
660.659 odd 2 9680.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 15.14 odd 2
275.2.a.c.1.1 2 3.2 odd 2
275.2.b.d.199.1 4 15.2 even 4
275.2.b.d.199.4 4 15.8 even 4
495.2.a.b.1.1 2 5.4 even 2
605.2.a.d.1.1 2 165.164 even 2
605.2.g.f.81.1 8 165.119 odd 10
605.2.g.f.251.2 8 165.14 odd 10
605.2.g.f.366.1 8 165.104 odd 10
605.2.g.f.511.2 8 165.59 odd 10
605.2.g.l.81.2 8 165.134 even 10
605.2.g.l.251.1 8 165.74 even 10
605.2.g.l.366.2 8 165.149 even 10
605.2.g.l.511.1 8 165.29 even 10
880.2.a.m.1.2 2 60.59 even 2
2475.2.a.x.1.2 2 1.1 even 1 trivial
2475.2.c.l.199.1 4 5.3 odd 4
2475.2.c.l.199.4 4 5.2 odd 4
2695.2.a.f.1.2 2 105.104 even 2
3025.2.a.o.1.2 2 33.32 even 2
3520.2.a.bn.1.2 2 120.29 odd 2
3520.2.a.bo.1.1 2 120.59 even 2
4400.2.a.bn.1.1 2 12.11 even 2
4400.2.b.q.4049.2 4 60.23 odd 4
4400.2.b.q.4049.3 4 60.47 odd 4
5445.2.a.y.1.2 2 55.54 odd 2
7920.2.a.ch.1.2 2 20.19 odd 2
9295.2.a.g.1.1 2 195.194 odd 2
9680.2.a.bn.1.2 2 660.659 odd 2