Properties

Label 2475.2.a.x.1.1
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: \( x^{2} - 2 \) Copy content Toggle raw display
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{7} +1.58579 q^{8} -1.00000 q^{11} +6.82843 q^{13} -0.828427 q^{14} +3.00000 q^{16} +1.17157 q^{17} +0.414214 q^{22} +2.82843 q^{23} -2.82843 q^{26} -3.65685 q^{28} -7.65685 q^{29} -4.41421 q^{32} -0.485281 q^{34} -3.65685 q^{37} -6.00000 q^{41} +6.00000 q^{43} +1.82843 q^{44} -1.17157 q^{46} -2.82843 q^{47} -3.00000 q^{49} -12.4853 q^{52} +0.343146 q^{53} +3.17157 q^{56} +3.17157 q^{58} +9.65685 q^{59} +13.3137 q^{61} -4.17157 q^{64} +4.48528 q^{67} -2.14214 q^{68} +11.3137 q^{71} +6.82843 q^{73} +1.51472 q^{74} -2.00000 q^{77} +4.00000 q^{79} +2.48528 q^{82} -6.00000 q^{83} -2.48528 q^{86} -1.58579 q^{88} -9.31371 q^{89} +13.6569 q^{91} -5.17157 q^{92} +1.17157 q^{94} +7.65685 q^{97} +1.24264 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 6 q^{8} - 2 q^{11} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} - 2 q^{22} + 4 q^{28} - 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{37} - 12 q^{41} + 12 q^{43} - 2 q^{44} - 8 q^{46} - 6 q^{49} - 8 q^{52} + 12 q^{53} + 12 q^{56} + 12 q^{58} + 8 q^{59} + 4 q^{61} - 14 q^{64} - 8 q^{67} + 24 q^{68} + 8 q^{73} + 20 q^{74} - 4 q^{77} + 8 q^{79} - 12 q^{82} - 12 q^{83} + 12 q^{86} - 6 q^{88} + 4 q^{89} + 16 q^{91} - 16 q^{92} + 8 q^{94} + 4 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.82843 1.89386 0.946932 0.321433i \(-0.104164\pi\)
0.946932 + 0.321433i \(0.104164\pi\)
\(14\) −0.828427 −0.221406
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.414214 0.0883106
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.82843 −0.554700
\(27\) 0 0
\(28\) −3.65685 −0.691080
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) −0.485281 −0.0832251
\(35\) 0 0
\(36\) 0 0
\(37\) −3.65685 −0.601183 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(38\) 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 \(0.348747\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −1.17157 −0.172739
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) −12.4853 −1.73140
\(53\) 0.343146 0.0471347 0.0235673 0.999722i \(-0.492498\pi\)
0.0235673 + 0.999722i \(0.492498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.17157 0.423819
\(57\) 0 0
\(58\) 3.17157 0.416448
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) 4.48528 0.547964 0.273982 0.961735i \(-0.411659\pi\)
0.273982 + 0.961735i \(0.411659\pi\)
\(68\) −2.14214 −0.259772
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) 6.82843 0.799207 0.399603 0.916688i \(-0.369148\pi\)
0.399603 + 0.916688i \(0.369148\pi\)
\(74\) 1.51472 0.176082
\(75\) 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
\(82\) 2.48528 0.274453
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.48528 −0.267995
\(87\) 0 0
\(88\) −1.58579 −0.169045
\(89\) −9.31371 −0.987251 −0.493626 0.869675i \(-0.664329\pi\)
−0.493626 + 0.869675i \(0.664329\pi\)
\(90\) 0 0
\(91\) 13.6569 1.43163
\(92\) −5.17157 −0.539174
\(93\) 0 0
\(94\) 1.17157 0.120839
\(95\) 0 0
\(96\) 0 0
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 1.24264 0.125526
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3137 1.32476 0.662382 0.749166i \(-0.269546\pi\)
0.662382 + 0.749166i \(0.269546\pi\)
\(102\) 0 0
\(103\) −1.17157 −0.115439 −0.0577193 0.998333i \(-0.518383\pi\)
−0.0577193 + 0.998333i \(0.518383\pi\)
\(104\) 10.8284 1.06181
\(105\) 0 0
\(106\) −0.142136 −0.0138054
\(107\) −3.65685 −0.353521 −0.176761 0.984254i \(-0.556562\pi\)
−0.176761 + 0.984254i \(0.556562\pi\)
\(108\) 0 0
\(109\) 3.65685 0.350263 0.175132 0.984545i \(-0.443965\pi\)
0.175132 + 0.984545i \(0.443965\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.00000 0.566947
\(113\) 8.34315 0.784857 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.51472 −0.499279
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −15.6569 −1.38932 −0.694661 0.719338i \(-0.744445\pi\)
−0.694661 + 0.719338i \(0.744445\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1.85786 −0.160495
\(135\) 0 0
\(136\) 1.85786 0.159311
\(137\) 22.9706 1.96251 0.981254 0.192720i \(-0.0617309\pi\)
0.981254 + 0.192720i \(0.0617309\pi\)
\(138\) 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
\(142\) −4.68629 −0.393265
\(143\) −6.82843 −0.571022
\(144\) 0 0
\(145\) 0 0
\(146\) −2.82843 −0.234082
\(147\) 0 0
\(148\) 6.68629 0.549610
\(149\) −11.6569 −0.954967 −0.477483 0.878641i \(-0.658451\pi\)
−0.477483 + 0.878641i \(0.658451\pi\)
\(150\) 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
\(154\) 0.828427 0.0667566
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −1.65685 −0.131812
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685 0.445823
\(162\) 0 0
\(163\) 0.485281 0.0380102 0.0190051 0.999819i \(-0.493950\pi\)
0.0190051 + 0.999819i \(0.493950\pi\)
\(164\) 10.9706 0.856657
\(165\) 0 0
\(166\) 2.48528 0.192895
\(167\) 10.9706 0.848928 0.424464 0.905445i \(-0.360463\pi\)
0.424464 + 0.905445i \(0.360463\pi\)
\(168\) 0 0
\(169\) 33.6274 2.58672
\(170\) 0 0
\(171\) 0 0
\(172\) −10.9706 −0.836498
\(173\) 6.14214 0.466978 0.233489 0.972359i \(-0.424986\pi\)
0.233489 + 0.972359i \(0.424986\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 3.85786 0.289159
\(179\) 1.65685 0.123839 0.0619196 0.998081i \(-0.480278\pi\)
0.0619196 + 0.998081i \(0.480278\pi\)
\(180\) 0 0
\(181\) −1.31371 −0.0976472 −0.0488236 0.998807i \(-0.515547\pi\)
−0.0488236 + 0.998807i \(0.515547\pi\)
\(182\) −5.65685 −0.419314
\(183\) 0 0
\(184\) 4.48528 0.330659
\(185\) 0 0
\(186\) 0 0
\(187\) −1.17157 −0.0856739
\(188\) 5.17157 0.377176
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 0 0
\(193\) 6.82843 0.491521 0.245760 0.969331i \(-0.420962\pi\)
0.245760 + 0.969331i \(0.420962\pi\)
\(194\) −3.17157 −0.227706
\(195\) 0 0
\(196\) 5.48528 0.391806
\(197\) −5.17157 −0.368459 −0.184230 0.982883i \(-0.558979\pi\)
−0.184230 + 0.982883i \(0.558979\pi\)
\(198\) 0 0
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −5.51472 −0.388014
\(203\) −15.3137 −1.07481
\(204\) 0 0
\(205\) 0 0
\(206\) 0.485281 0.0338112
\(207\) 0 0
\(208\) 20.4853 1.42040
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −0.627417 −0.0430912
\(213\) 0 0
\(214\) 1.51472 0.103544
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.51472 −0.102590
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 5.17157 0.346314 0.173157 0.984894i \(-0.444603\pi\)
0.173157 + 0.984894i \(0.444603\pi\)
\(224\) −8.82843 −0.589874
\(225\) 0 0
\(226\) −3.45584 −0.229879
\(227\) 2.68629 0.178295 0.0891477 0.996018i \(-0.471586\pi\)
0.0891477 + 0.996018i \(0.471586\pi\)
\(228\) 0 0
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −12.1421 −0.797170
\(233\) 22.1421 1.45058 0.725290 0.688444i \(-0.241706\pi\)
0.725290 + 0.688444i \(0.241706\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −17.6569 −1.14936
\(237\) 0 0
\(238\) −0.970563 −0.0629122
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 0 0
\(244\) −24.3431 −1.55841
\(245\) 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
\(254\) 6.48528 0.406923
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 13.3137 0.830486 0.415243 0.909710i \(-0.363696\pi\)
0.415243 + 0.909710i \(0.363696\pi\)
\(258\) 0 0
\(259\) −7.31371 −0.454452
\(260\) 0 0
\(261\) 0 0
\(262\) 4.68629 0.289520
\(263\) 22.9706 1.41643 0.708213 0.705999i \(-0.249502\pi\)
0.708213 + 0.705999i \(0.249502\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −8.20101 −0.500956
\(269\) 5.31371 0.323983 0.161991 0.986792i \(-0.448208\pi\)
0.161991 + 0.986792i \(0.448208\pi\)
\(270\) 0 0
\(271\) −15.3137 −0.930242 −0.465121 0.885247i \(-0.653989\pi\)
−0.465121 + 0.885247i \(0.653989\pi\)
\(272\) 3.51472 0.213111
\(273\) 0 0
\(274\) −9.51472 −0.574805
\(275\) 0 0
\(276\) 0 0
\(277\) −1.17157 −0.0703930 −0.0351965 0.999380i \(-0.511206\pi\)
−0.0351965 + 0.999380i \(0.511206\pi\)
\(278\) 1.65685 0.0993715
\(279\) 0 0
\(280\) 0 0
\(281\) 5.31371 0.316989 0.158495 0.987360i \(-0.449336\pi\)
0.158495 + 0.987360i \(0.449336\pi\)
\(282\) 0 0
\(283\) 12.6274 0.750622 0.375311 0.926899i \(-0.377536\pi\)
0.375311 + 0.926899i \(0.377536\pi\)
\(284\) −20.6863 −1.22751
\(285\) 0 0
\(286\) 2.82843 0.167248
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) −12.4853 −0.730646
\(293\) −14.8284 −0.866286 −0.433143 0.901325i \(-0.642595\pi\)
−0.433143 + 0.901325i \(0.642595\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.79899 −0.337059
\(297\) 0 0
\(298\) 4.82843 0.279703
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 4.97056 0.286024
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.6569 1.57846 0.789230 0.614098i \(-0.210480\pi\)
0.789230 + 0.614098i \(0.210480\pi\)
\(308\) 3.65685 0.208369
\(309\) 0 0
\(310\) 0 0
\(311\) −27.3137 −1.54882 −0.774409 0.632685i \(-0.781953\pi\)
−0.774409 + 0.632685i \(0.781953\pi\)
\(312\) 0 0
\(313\) −21.3137 −1.20472 −0.602361 0.798224i \(-0.705773\pi\)
−0.602361 + 0.798224i \(0.705773\pi\)
\(314\) −5.79899 −0.327256
\(315\) 0 0
\(316\) −7.31371 −0.411428
\(317\) 21.3137 1.19710 0.598549 0.801087i \(-0.295744\pi\)
0.598549 + 0.801087i \(0.295744\pi\)
\(318\) 0 0
\(319\) 7.65685 0.428702
\(320\) 0 0
\(321\) 0 0
\(322\) −2.34315 −0.130578
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −0.201010 −0.0111329
\(327\) 0 0
\(328\) −9.51472 −0.525362
\(329\) −5.65685 −0.311872
\(330\) 0 0
\(331\) 15.3137 0.841718 0.420859 0.907126i \(-0.361729\pi\)
0.420859 + 0.907126i \(0.361729\pi\)
\(332\) 10.9706 0.602088
\(333\) 0 0
\(334\) −4.54416 −0.248645
\(335\) 0 0
\(336\) 0 0
\(337\) 3.51472 0.191459 0.0957295 0.995407i \(-0.469482\pi\)
0.0957295 + 0.995407i \(0.469482\pi\)
\(338\) −13.9289 −0.757634
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 9.51472 0.512999
\(345\) 0 0
\(346\) −2.54416 −0.136775
\(347\) −22.9706 −1.23312 −0.616562 0.787306i \(-0.711475\pi\)
−0.616562 + 0.787306i \(0.711475\pi\)
\(348\) 0 0
\(349\) −6.97056 −0.373126 −0.186563 0.982443i \(-0.559735\pi\)
−0.186563 + 0.982443i \(0.559735\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.41421 0.235278
\(353\) −1.31371 −0.0699216 −0.0349608 0.999389i \(-0.511131\pi\)
−0.0349608 + 0.999389i \(0.511131\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 17.0294 0.902558
\(357\) 0 0
\(358\) −0.686292 −0.0362716
\(359\) −23.3137 −1.23045 −0.615225 0.788351i \(-0.710935\pi\)
−0.615225 + 0.788351i \(0.710935\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0.544156 0.0286002
\(363\) 0 0
\(364\) −24.9706 −1.30881
\(365\) 0 0
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 8.48528 0.442326
\(369\) 0 0
\(370\) 0 0
\(371\) 0.686292 0.0356305
\(372\) 0 0
\(373\) 3.79899 0.196704 0.0983521 0.995152i \(-0.468643\pi\)
0.0983521 + 0.995152i \(0.468643\pi\)
\(374\) 0.485281 0.0250933
\(375\) 0 0
\(376\) −4.48528 −0.231311
\(377\) −52.2843 −2.69278
\(378\) 0 0
\(379\) 22.3431 1.14769 0.573845 0.818964i \(-0.305451\pi\)
0.573845 + 0.818964i \(0.305451\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −34.1421 −1.74458 −0.872291 0.488987i \(-0.837366\pi\)
−0.872291 + 0.488987i \(0.837366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.82843 −0.143963
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 24.6274 1.24866 0.624330 0.781161i \(-0.285372\pi\)
0.624330 + 0.781161i \(0.285372\pi\)
\(390\) 0 0
\(391\) 3.31371 0.167581
\(392\) −4.75736 −0.240283
\(393\) 0 0
\(394\) 2.14214 0.107919
\(395\) 0 0
\(396\) 0 0
\(397\) −13.3137 −0.668196 −0.334098 0.942538i \(-0.608432\pi\)
−0.334098 + 0.942538i \(0.608432\pi\)
\(398\) −8.97056 −0.449654
\(399\) 0 0
\(400\) 0 0
\(401\) −17.3137 −0.864605 −0.432303 0.901729i \(-0.642299\pi\)
−0.432303 + 0.901729i \(0.642299\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −24.3431 −1.21112
\(405\) 0 0
\(406\) 6.34315 0.314805
\(407\) 3.65685 0.181264
\(408\) 0 0
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.14214 0.105535
\(413\) 19.3137 0.950365
\(414\) 0 0
\(415\) 0 0
\(416\) −30.1421 −1.47784
\(417\) 0 0
\(418\) 0 0
\(419\) 14.3431 0.700709 0.350354 0.936617i \(-0.386061\pi\)
0.350354 + 0.936617i \(0.386061\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 6.62742 0.322618
\(423\) 0 0
\(424\) 0.544156 0.0264265
\(425\) 0 0
\(426\) 0 0
\(427\) 26.6274 1.28859
\(428\) 6.68629 0.323194
\(429\) 0 0
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) −3.65685 −0.175737 −0.0878686 0.996132i \(-0.528006\pi\)
−0.0878686 + 0.996132i \(0.528006\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.68629 −0.320215
\(437\) 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
\(442\) −3.31371 −0.157617
\(443\) −21.1716 −1.00589 −0.502946 0.864318i \(-0.667751\pi\)
−0.502946 + 0.864318i \(0.667751\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2.14214 −0.101433
\(447\) 0 0
\(448\) −8.34315 −0.394177
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) −15.2548 −0.717527
\(453\) 0 0
\(454\) −1.11270 −0.0522215
\(455\) 0 0
\(456\) 0 0
\(457\) 16.4853 0.771149 0.385574 0.922677i \(-0.374003\pi\)
0.385574 + 0.922677i \(0.374003\pi\)
\(458\) 8.82843 0.412525
\(459\) 0 0
\(460\) 0 0
\(461\) 32.6274 1.51961 0.759805 0.650151i \(-0.225294\pi\)
0.759805 + 0.650151i \(0.225294\pi\)
\(462\) 0 0
\(463\) −22.1421 −1.02903 −0.514516 0.857481i \(-0.672028\pi\)
−0.514516 + 0.857481i \(0.672028\pi\)
\(464\) −22.9706 −1.06638
\(465\) 0 0
\(466\) −9.17157 −0.424865
\(467\) −9.17157 −0.424410 −0.212205 0.977225i \(-0.568064\pi\)
−0.212205 + 0.977225i \(0.568064\pi\)
\(468\) 0 0
\(469\) 8.97056 0.414222
\(470\) 0 0
\(471\) 0 0
\(472\) 15.3137 0.704871
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) −4.28427 −0.196369
\(477\) 0 0
\(478\) −0.284271 −0.0130023
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −24.9706 −1.13856
\(482\) −2.48528 −0.113201
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) 0 0
\(487\) 7.51472 0.340524 0.170262 0.985399i \(-0.445539\pi\)
0.170262 + 0.985399i \(0.445539\pi\)
\(488\) 21.1127 0.955727
\(489\) 0 0
\(490\) 0 0
\(491\) 23.3137 1.05213 0.526066 0.850443i \(-0.323666\pi\)
0.526066 + 0.850443i \(0.323666\pi\)
\(492\) 0 0
\(493\) −8.97056 −0.404014
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274 1.01498
\(498\) 0 0
\(499\) −1.65685 −0.0741710 −0.0370855 0.999312i \(-0.511807\pi\)
−0.0370855 + 0.999312i \(0.511807\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.97056 0.221847
\(503\) −28.6274 −1.27643 −0.638217 0.769857i \(-0.720328\pi\)
−0.638217 + 0.769857i \(0.720328\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.17157 0.0520828
\(507\) 0 0
\(508\) 28.6274 1.27014
\(509\) −9.31371 −0.412823 −0.206411 0.978465i \(-0.566179\pi\)
−0.206411 + 0.978465i \(0.566179\pi\)
\(510\) 0 0
\(511\) 13.6569 0.604144
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) −5.51472 −0.243244
\(515\) 0 0
\(516\) 0 0
\(517\) 2.82843 0.124394
\(518\) 3.02944 0.133106
\(519\) 0 0
\(520\) 0 0
\(521\) −2.68629 −0.117689 −0.0588443 0.998267i \(-0.518742\pi\)
−0.0588443 + 0.998267i \(0.518742\pi\)
\(522\) 0 0
\(523\) −37.5980 −1.64404 −0.822022 0.569455i \(-0.807154\pi\)
−0.822022 + 0.569455i \(0.807154\pi\)
\(524\) 20.6863 0.903685
\(525\) 0 0
\(526\) −9.51472 −0.414861
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −40.9706 −1.77463
\(534\) 0 0
\(535\) 0 0
\(536\) 7.11270 0.307222
\(537\) 0 0
\(538\) −2.20101 −0.0948923
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 6.34315 0.272461
\(543\) 0 0
\(544\) −5.17157 −0.221729
\(545\) 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
\(554\) 0.485281 0.0206176
\(555\) 0 0
\(556\) 7.31371 0.310170
\(557\) 38.1421 1.61613 0.808067 0.589090i \(-0.200514\pi\)
0.808067 + 0.589090i \(0.200514\pi\)
\(558\) 0 0
\(559\) 40.9706 1.73287
\(560\) 0 0
\(561\) 0 0
\(562\) −2.20101 −0.0928440
\(563\) 11.6569 0.491278 0.245639 0.969361i \(-0.421002\pi\)
0.245639 + 0.969361i \(0.421002\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −5.23045 −0.219852
\(567\) 0 0
\(568\) 17.9411 0.752793
\(569\) −20.3431 −0.852829 −0.426415 0.904528i \(-0.640224\pi\)
−0.426415 + 0.904528i \(0.640224\pi\)
\(570\) 0 0
\(571\) 45.9411 1.92258 0.961288 0.275545i \(-0.0888584\pi\)
0.961288 + 0.275545i \(0.0888584\pi\)
\(572\) 12.4853 0.522036
\(573\) 0 0
\(574\) 4.97056 0.207467
\(575\) 0 0
\(576\) 0 0
\(577\) −6.97056 −0.290188 −0.145094 0.989418i \(-0.546349\pi\)
−0.145094 + 0.989418i \(0.546349\pi\)
\(578\) 6.47309 0.269245
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −0.343146 −0.0142116
\(584\) 10.8284 0.448084
\(585\) 0 0
\(586\) 6.14214 0.253729
\(587\) 26.1421 1.07900 0.539501 0.841985i \(-0.318613\pi\)
0.539501 + 0.841985i \(0.318613\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −10.9706 −0.450887
\(593\) 20.4853 0.841230 0.420615 0.907239i \(-0.361814\pi\)
0.420615 + 0.907239i \(0.361814\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 21.3137 0.873044
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) −43.9411 −1.79240 −0.896198 0.443654i \(-0.853682\pi\)
−0.896198 + 0.443654i \(0.853682\pi\)
\(602\) −4.97056 −0.202585
\(603\) 0 0
\(604\) 21.9411 0.892772
\(605\) 0 0
\(606\) 0 0
\(607\) 18.2843 0.742136 0.371068 0.928606i \(-0.378992\pi\)
0.371068 + 0.928606i \(0.378992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.3137 −0.781349
\(612\) 0 0
\(613\) −25.4558 −1.02815 −0.514076 0.857745i \(-0.671865\pi\)
−0.514076 + 0.857745i \(0.671865\pi\)
\(614\) −11.4558 −0.462320
\(615\) 0 0
\(616\) −3.17157 −0.127786
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) 0 0
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 11.3137 0.453638
\(623\) −18.6274 −0.746292
\(624\) 0 0
\(625\) 0 0
\(626\) 8.82843 0.352855
\(627\) 0 0
\(628\) −25.5980 −1.02147
\(629\) −4.28427 −0.170825
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 6.34315 0.252317
\(633\) 0 0
\(634\) −8.82843 −0.350622
\(635\) 0 0
\(636\) 0 0
\(637\) −20.4853 −0.811656
\(638\) −3.17157 −0.125564
\(639\) 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
\(643\) −49.4558 −1.95035 −0.975174 0.221440i \(-0.928924\pi\)
−0.975174 + 0.221440i \(0.928924\pi\)
\(644\) −10.3431 −0.407577
\(645\) 0 0
\(646\) 0 0
\(647\) −35.1127 −1.38042 −0.690211 0.723608i \(-0.742482\pi\)
−0.690211 + 0.723608i \(0.742482\pi\)
\(648\) 0 0
\(649\) −9.65685 −0.379065
\(650\) 0 0
\(651\) 0 0
\(652\) −0.887302 −0.0347494
\(653\) 0.343146 0.0134283 0.00671417 0.999977i \(-0.497863\pi\)
0.00671417 + 0.999977i \(0.497863\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 0 0
\(658\) 2.34315 0.0913453
\(659\) 21.9411 0.854705 0.427352 0.904085i \(-0.359446\pi\)
0.427352 + 0.904085i \(0.359446\pi\)
\(660\) 0 0
\(661\) −0.627417 −0.0244037 −0.0122018 0.999926i \(-0.503884\pi\)
−0.0122018 + 0.999926i \(0.503884\pi\)
\(662\) −6.34315 −0.246533
\(663\) 0 0
\(664\) −9.51472 −0.369243
\(665\) 0 0
\(666\) 0 0
\(667\) −21.6569 −0.838557
\(668\) −20.0589 −0.776101
\(669\) 0 0
\(670\) 0 0
\(671\) −13.3137 −0.513970
\(672\) 0 0
\(673\) −4.48528 −0.172895 −0.0864474 0.996256i \(-0.527551\pi\)
−0.0864474 + 0.996256i \(0.527551\pi\)
\(674\) −1.45584 −0.0560770
\(675\) 0 0
\(676\) −61.4853 −2.36482
\(677\) 17.1716 0.659957 0.329979 0.943988i \(-0.392958\pi\)
0.329979 + 0.943988i \(0.392958\pi\)
\(678\) 0 0
\(679\) 15.3137 0.587686
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 31.7990 1.21675 0.608377 0.793648i \(-0.291821\pi\)
0.608377 + 0.793648i \(0.291821\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.28427 0.316295
\(687\) 0 0
\(688\) 18.0000 0.686244
\(689\) 2.34315 0.0892667
\(690\) 0 0
\(691\) −16.6863 −0.634776 −0.317388 0.948296i \(-0.602806\pi\)
−0.317388 + 0.948296i \(0.602806\pi\)
\(692\) −11.2304 −0.426918
\(693\) 0 0
\(694\) 9.51472 0.361174
\(695\) 0 0
\(696\) 0 0
\(697\) −7.02944 −0.266259
\(698\) 2.88730 0.109286
\(699\) 0 0
\(700\) 0 0
\(701\) −32.6274 −1.23232 −0.616160 0.787621i \(-0.711313\pi\)
−0.616160 + 0.787621i \(0.711313\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) 0.544156 0.0204796
\(707\) 26.6274 1.00143
\(708\) 0 0
\(709\) −20.6274 −0.774679 −0.387339 0.921937i \(-0.626606\pi\)
−0.387339 + 0.921937i \(0.626606\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −14.7696 −0.553512
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −3.02944 −0.113215
\(717\) 0 0
\(718\) 9.65685 0.360391
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 0 0
\(721\) −2.34315 −0.0872633
\(722\) 7.87006 0.292893
\(723\) 0 0
\(724\) 2.40202 0.0892704
\(725\) 0 0
\(726\) 0 0
\(727\) 36.4853 1.35316 0.676582 0.736367i \(-0.263460\pi\)
0.676582 + 0.736367i \(0.263460\pi\)
\(728\) 21.6569 0.802656
\(729\) 0 0
\(730\) 0 0
\(731\) 7.02944 0.259993
\(732\) 0 0
\(733\) −33.4558 −1.23572 −0.617860 0.786288i \(-0.712000\pi\)
−0.617860 + 0.786288i \(0.712000\pi\)
\(734\) 3.51472 0.129731
\(735\) 0 0
\(736\) −12.4853 −0.460214
\(737\) −4.48528 −0.165217
\(738\) 0 0
\(739\) −37.9411 −1.39569 −0.697843 0.716250i \(-0.745857\pi\)
−0.697843 + 0.716250i \(0.745857\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.284271 −0.0104359
\(743\) 29.5980 1.08584 0.542922 0.839783i \(-0.317318\pi\)
0.542922 + 0.839783i \(0.317318\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.57359 −0.0576133
\(747\) 0 0
\(748\) 2.14214 0.0783242
\(749\) −7.31371 −0.267237
\(750\) 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
\(754\) 21.6569 0.788696
\(755\) 0 0
\(756\) 0 0
\(757\) 9.31371 0.338512 0.169256 0.985572i \(-0.445863\pi\)
0.169256 + 0.985572i \(0.445863\pi\)
\(758\) −9.25483 −0.336151
\(759\) 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
\(763\) 7.31371 0.264774
\(764\) −35.3137 −1.27761
\(765\) 0 0
\(766\) 14.1421 0.510976
\(767\) 65.9411 2.38100
\(768\) 0 0
\(769\) 14.9706 0.539852 0.269926 0.962881i \(-0.413001\pi\)
0.269926 + 0.962881i \(0.413001\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.4853 −0.449355
\(773\) −30.2843 −1.08925 −0.544625 0.838680i \(-0.683328\pi\)
−0.544625 + 0.838680i \(0.683328\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 12.1421 0.435877
\(777\) 0 0
\(778\) −10.2010 −0.365724
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) −1.37258 −0.0490835
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) −18.9706 −0.676228 −0.338114 0.941105i \(-0.609789\pi\)
−0.338114 + 0.941105i \(0.609789\pi\)
\(788\) 9.45584 0.336850
\(789\) 0 0
\(790\) 0 0
\(791\) 16.6863 0.593296
\(792\) 0 0
\(793\) 90.9117 3.22837
\(794\) 5.51472 0.195710
\(795\) 0 0
\(796\) −39.5980 −1.40351
\(797\) −12.6274 −0.447286 −0.223643 0.974671i \(-0.571795\pi\)
−0.223643 + 0.974671i \(0.571795\pi\)
\(798\) 0 0
\(799\) −3.31371 −0.117231
\(800\) 0 0
\(801\) 0 0
\(802\) 7.17157 0.253237
\(803\) −6.82843 −0.240970
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 21.1127 0.742742
\(809\) −22.9706 −0.807602 −0.403801 0.914847i \(-0.632311\pi\)
−0.403801 + 0.914847i \(0.632311\pi\)
\(810\) 0 0
\(811\) −13.9411 −0.489539 −0.244770 0.969581i \(-0.578712\pi\)
−0.244770 + 0.969581i \(0.578712\pi\)
\(812\) 28.0000 0.982607
\(813\) 0 0
\(814\) −1.51472 −0.0530909
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −14.4853 −0.506466
\(819\) 0 0
\(820\) 0 0
\(821\) 18.6863 0.652156 0.326078 0.945343i \(-0.394273\pi\)
0.326078 + 0.945343i \(0.394273\pi\)
\(822\) 0 0
\(823\) −36.4853 −1.27180 −0.635898 0.771773i \(-0.719370\pi\)
−0.635898 + 0.771773i \(0.719370\pi\)
\(824\) −1.85786 −0.0647218
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −34.2843 −1.19218 −0.596090 0.802917i \(-0.703280\pi\)
−0.596090 + 0.802917i \(0.703280\pi\)
\(828\) 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
\(832\) −28.4853 −0.987549
\(833\) −3.51472 −0.121778
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −5.94113 −0.205233
\(839\) −37.6569 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 2.48528 0.0856485
\(843\) 0 0
\(844\) 29.2548 1.00699
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 1.02944 0.0353510
\(849\) 0 0
\(850\) 0 0
\(851\) −10.3431 −0.354558
\(852\) 0 0
\(853\) 32.4853 1.11227 0.556137 0.831090i \(-0.312283\pi\)
0.556137 + 0.831090i \(0.312283\pi\)
\(854\) −11.0294 −0.377420
\(855\) 0 0
\(856\) −5.79899 −0.198205
\(857\) −48.7696 −1.66594 −0.832968 0.553321i \(-0.813360\pi\)
−0.832968 + 0.553321i \(0.813360\pi\)
\(858\) 0 0
\(859\) −32.2843 −1.10153 −0.550763 0.834662i \(-0.685663\pi\)
−0.550763 + 0.834662i \(0.685663\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 4.68629 0.159616
\(863\) −14.8284 −0.504766 −0.252383 0.967627i \(-0.581214\pi\)
−0.252383 + 0.967627i \(0.581214\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.51472 0.0514722
\(867\) 0 0
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) 30.6274 1.03777
\(872\) 5.79899 0.196379
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.45584 −0.0491604 −0.0245802 0.999698i \(-0.507825\pi\)
−0.0245802 + 0.999698i \(0.507825\pi\)
\(878\) 6.62742 0.223664
\(879\) 0 0
\(880\) 0 0
\(881\) 52.6274 1.77306 0.886531 0.462668i \(-0.153108\pi\)
0.886531 + 0.462668i \(0.153108\pi\)
\(882\) 0 0
\(883\) −42.8284 −1.44129 −0.720646 0.693304i \(-0.756155\pi\)
−0.720646 + 0.693304i \(0.756155\pi\)
\(884\) −14.6274 −0.491973
\(885\) 0 0
\(886\) 8.76955 0.294619
\(887\) −18.2843 −0.613926 −0.306963 0.951721i \(-0.599313\pi\)
−0.306963 + 0.951721i \(0.599313\pi\)
\(888\) 0 0
\(889\) −31.3137 −1.05023
\(890\) 0 0
\(891\) 0 0
\(892\) −9.45584 −0.316605
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 21.1127 0.705326
\(897\) 0 0
\(898\) −6.88730 −0.229832
\(899\) 0 0
\(900\) 0 0
\(901\) 0.402020 0.0133932
\(902\) −2.48528 −0.0827508
\(903\) 0 0
\(904\) 13.2304 0.440038
\(905\) 0 0
\(906\) 0 0
\(907\) 44.4853 1.47711 0.738555 0.674193i \(-0.235509\pi\)
0.738555 + 0.674193i \(0.235509\pi\)
\(908\) −4.91169 −0.163000
\(909\) 0 0
\(910\) 0 0
\(911\) −57.9411 −1.91968 −0.959838 0.280556i \(-0.909481\pi\)
−0.959838 + 0.280556i \(0.909481\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) −6.82843 −0.225864
\(915\) 0 0
\(916\) 38.9706 1.28762
\(917\) −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
\(922\) −13.5147 −0.445084
\(923\) 77.2548 2.54287
\(924\) 0 0
\(925\) 0 0
\(926\) 9.17157 0.301397
\(927\) 0 0
\(928\) 33.7990 1.10951
\(929\) 17.3137 0.568044 0.284022 0.958818i \(-0.408331\pi\)
0.284022 + 0.958818i \(0.408331\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −40.4853 −1.32614
\(933\) 0 0
\(934\) 3.79899 0.124307
\(935\) 0 0
\(936\) 0 0
\(937\) −49.4558 −1.61565 −0.807826 0.589421i \(-0.799356\pi\)
−0.807826 + 0.589421i \(0.799356\pi\)
\(938\) −3.71573 −0.121323
\(939\) 0 0
\(940\) 0 0
\(941\) 29.3137 0.955600 0.477800 0.878469i \(-0.341434\pi\)
0.477800 + 0.878469i \(0.341434\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 28.9706 0.942912
\(945\) 0 0
\(946\) 2.48528 0.0808035
\(947\) 46.8284 1.52172 0.760860 0.648916i \(-0.224778\pi\)
0.760860 + 0.648916i \(0.224778\pi\)
\(948\) 0 0
\(949\) 46.6274 1.51359
\(950\) 0 0
\(951\) 0 0
\(952\) 3.71573 0.120427
\(953\) 58.8284 1.90564 0.952820 0.303536i \(-0.0981674\pi\)
0.952820 + 0.303536i \(0.0981674\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.25483 −0.0405842
\(957\) 0 0
\(958\) −14.9117 −0.481775
\(959\) 45.9411 1.48352
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 10.3431 0.333476
\(963\) 0 0
\(964\) −10.9706 −0.353338
\(965\) 0 0
\(966\) 0 0
\(967\) −18.9706 −0.610052 −0.305026 0.952344i \(-0.598665\pi\)
−0.305026 + 0.952344i \(0.598665\pi\)
\(968\) 1.58579 0.0509691
\(969\) 0 0
\(970\) 0 0
\(971\) 31.3137 1.00490 0.502452 0.864605i \(-0.332431\pi\)
0.502452 + 0.864605i \(0.332431\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) −3.11270 −0.0997373
\(975\) 0 0
\(976\) 39.9411 1.27848
\(977\) 43.6569 1.39671 0.698353 0.715753i \(-0.253916\pi\)
0.698353 + 0.715753i \(0.253916\pi\)
\(978\) 0 0
\(979\) 9.31371 0.297667
\(980\) 0 0
\(981\) 0 0
\(982\) −9.65685 −0.308163
\(983\) −50.1421 −1.59929 −0.799643 0.600476i \(-0.794978\pi\)
−0.799643 + 0.600476i \(0.794978\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 3.71573 0.118333
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) 9.94113 0.315790 0.157895 0.987456i \(-0.449529\pi\)
0.157895 + 0.987456i \(0.449529\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −9.37258 −0.297280
\(995\) 0 0
\(996\) 0 0
\(997\) −9.45584 −0.299470 −0.149735 0.988726i \(-0.547842\pi\)
−0.149735 + 0.988726i \(0.547842\pi\)
\(998\) 0.686292 0.0217242
\(999\) 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.1.1 2
3.2 odd 2 275.2.a.c.1.2 2
5.2 odd 4 2475.2.c.l.199.2 4
5.3 odd 4 2475.2.c.l.199.3 4
5.4 even 2 495.2.a.b.1.2 2
12.11 even 2 4400.2.a.bn.1.2 2
15.2 even 4 275.2.b.d.199.3 4
15.8 even 4 275.2.b.d.199.2 4
15.14 odd 2 55.2.a.b.1.1 2
20.19 odd 2 7920.2.a.ch.1.1 2
33.32 even 2 3025.2.a.o.1.1 2
55.54 odd 2 5445.2.a.y.1.1 2
60.23 odd 4 4400.2.b.q.4049.4 4
60.47 odd 4 4400.2.b.q.4049.1 4
60.59 even 2 880.2.a.m.1.1 2
105.104 even 2 2695.2.a.f.1.1 2
120.29 odd 2 3520.2.a.bn.1.1 2
120.59 even 2 3520.2.a.bo.1.2 2
165.14 odd 10 605.2.g.f.251.1 8
165.29 even 10 605.2.g.l.511.2 8
165.59 odd 10 605.2.g.f.511.1 8
165.74 even 10 605.2.g.l.251.2 8
165.104 odd 10 605.2.g.f.366.2 8
165.119 odd 10 605.2.g.f.81.2 8
165.134 even 10 605.2.g.l.81.1 8
165.149 even 10 605.2.g.l.366.1 8
165.164 even 2 605.2.a.d.1.2 2
195.194 odd 2 9295.2.a.g.1.2 2
660.659 odd 2 9680.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 15.14 odd 2
275.2.a.c.1.2 2 3.2 odd 2
275.2.b.d.199.2 4 15.8 even 4
275.2.b.d.199.3 4 15.2 even 4
495.2.a.b.1.2 2 5.4 even 2
605.2.a.d.1.2 2 165.164 even 2
605.2.g.f.81.2 8 165.119 odd 10
605.2.g.f.251.1 8 165.14 odd 10
605.2.g.f.366.2 8 165.104 odd 10
605.2.g.f.511.1 8 165.59 odd 10
605.2.g.l.81.1 8 165.134 even 10
605.2.g.l.251.2 8 165.74 even 10
605.2.g.l.366.1 8 165.149 even 10
605.2.g.l.511.2 8 165.29 even 10
880.2.a.m.1.1 2 60.59 even 2
2475.2.a.x.1.1 2 1.1 even 1 trivial
2475.2.c.l.199.2 4 5.2 odd 4
2475.2.c.l.199.3 4 5.3 odd 4
2695.2.a.f.1.1 2 105.104 even 2
3025.2.a.o.1.1 2 33.32 even 2
3520.2.a.bn.1.1 2 120.29 odd 2
3520.2.a.bo.1.2 2 120.59 even 2
4400.2.a.bn.1.2 2 12.11 even 2
4400.2.b.q.4049.1 4 60.47 odd 4
4400.2.b.q.4049.4 4 60.23 odd 4
5445.2.a.y.1.1 2 55.54 odd 2
7920.2.a.ch.1.1 2 20.19 odd 2
9295.2.a.g.1.2 2 195.194 odd 2
9680.2.a.bn.1.1 2 660.659 odd 2