Properties

Label 1575.2.a.s.1.2
Level $1575$
Weight $2$
Character 1575.1
Self dual yes
Analytic conductor $12.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{7} -2.23607 q^{8} -4.23607 q^{11} +3.23607 q^{13} -1.61803 q^{14} -4.85410 q^{16} -6.47214 q^{17} +4.47214 q^{19} -6.85410 q^{22} +1.76393 q^{23} +5.23607 q^{26} -0.618034 q^{28} -5.00000 q^{29} -9.70820 q^{31} -3.38197 q^{32} -10.4721 q^{34} -3.00000 q^{37} +7.23607 q^{38} -9.23607 q^{41} -6.23607 q^{43} -2.61803 q^{44} +2.85410 q^{46} -2.00000 q^{47} +1.00000 q^{49} +2.00000 q^{52} -0.472136 q^{53} +2.23607 q^{56} -8.09017 q^{58} +1.70820 q^{59} +3.70820 q^{61} -15.7082 q^{62} +4.23607 q^{64} -0.236068 q^{67} -4.00000 q^{68} +4.70820 q^{71} +13.2361 q^{73} -4.85410 q^{74} +2.76393 q^{76} +4.23607 q^{77} +11.1803 q^{79} -14.9443 q^{82} +5.70820 q^{83} -10.0902 q^{86} +9.47214 q^{88} -12.7639 q^{89} -3.23607 q^{91} +1.09017 q^{92} -3.23607 q^{94} -0.763932 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{7} - 4q^{11} + 2q^{13} - q^{14} - 3q^{16} - 4q^{17} - 7q^{22} + 8q^{23} + 6q^{26} + q^{28} - 10q^{29} - 6q^{31} - 9q^{32} - 12q^{34} - 6q^{37} + 10q^{38} - 14q^{41} - 8q^{43} - 3q^{44} - q^{46} - 4q^{47} + 2q^{49} + 4q^{52} + 8q^{53} - 5q^{58} - 10q^{59} - 6q^{61} - 18q^{62} + 4q^{64} + 4q^{67} - 8q^{68} - 4q^{71} + 22q^{73} - 3q^{74} + 10q^{76} + 4q^{77} - 12q^{82} - 2q^{83} - 9q^{86} + 10q^{88} - 30q^{89} - 2q^{91} - 9q^{92} - 2q^{94} - 6q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) −4.23607 −1.27722 −0.638611 0.769529i \(-0.720491\pi\)
−0.638611 + 0.769529i \(0.720491\pi\)
\(12\) 0 0
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) −1.61803 −0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −6.47214 −1.56972 −0.784862 0.619671i \(-0.787266\pi\)
−0.784862 + 0.619671i \(0.787266\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.85410 −1.46130
\(23\) 1.76393 0.367805 0.183903 0.982944i \(-0.441127\pi\)
0.183903 + 0.982944i \(0.441127\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.23607 1.02688
\(27\) 0 0
\(28\) −0.618034 −0.116797
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) −9.70820 −1.74364 −0.871822 0.489822i \(-0.837062\pi\)
−0.871822 + 0.489822i \(0.837062\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −10.4721 −1.79596
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) 7.23607 1.17385
\(39\) 0 0
\(40\) 0 0
\(41\) −9.23607 −1.44243 −0.721216 0.692711i \(-0.756416\pi\)
−0.721216 + 0.692711i \(0.756416\pi\)
\(42\) 0 0
\(43\) −6.23607 −0.950991 −0.475496 0.879718i \(-0.657731\pi\)
−0.475496 + 0.879718i \(0.657731\pi\)
\(44\) −2.61803 −0.394683
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) −8.09017 −1.06229
\(59\) 1.70820 0.222389 0.111195 0.993799i \(-0.464532\pi\)
0.111195 + 0.993799i \(0.464532\pi\)
\(60\) 0 0
\(61\) 3.70820 0.474787 0.237393 0.971414i \(-0.423707\pi\)
0.237393 + 0.971414i \(0.423707\pi\)
\(62\) −15.7082 −1.99494
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) −0.236068 −0.0288403 −0.0144201 0.999896i \(-0.504590\pi\)
−0.0144201 + 0.999896i \(0.504590\pi\)
\(68\) −4.00000 −0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) 4.70820 0.558761 0.279381 0.960180i \(-0.409871\pi\)
0.279381 + 0.960180i \(0.409871\pi\)
\(72\) 0 0
\(73\) 13.2361 1.54916 0.774582 0.632473i \(-0.217960\pi\)
0.774582 + 0.632473i \(0.217960\pi\)
\(74\) −4.85410 −0.564278
\(75\) 0 0
\(76\) 2.76393 0.317045
\(77\) 4.23607 0.482745
\(78\) 0 0
\(79\) 11.1803 1.25789 0.628943 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.9443 −1.65032
\(83\) 5.70820 0.626557 0.313278 0.949661i \(-0.398573\pi\)
0.313278 + 0.949661i \(0.398573\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0902 −1.08805
\(87\) 0 0
\(88\) 9.47214 1.00973
\(89\) −12.7639 −1.35297 −0.676487 0.736455i \(-0.736499\pi\)
−0.676487 + 0.736455i \(0.736499\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) 1.09017 0.113658
\(93\) 0 0
\(94\) −3.23607 −0.333775
\(95\) 0 0
\(96\) 0 0
\(97\) −0.763932 −0.0775655 −0.0387828 0.999248i \(-0.512348\pi\)
−0.0387828 + 0.999248i \(0.512348\pi\)
\(98\) 1.61803 0.163446
\(99\) 0 0
\(100\) 0 0
\(101\) −9.23607 −0.919023 −0.459512 0.888172i \(-0.651976\pi\)
−0.459512 + 0.888172i \(0.651976\pi\)
\(102\) 0 0
\(103\) 0.472136 0.0465209 0.0232605 0.999729i \(-0.492595\pi\)
0.0232605 + 0.999729i \(0.492595\pi\)
\(104\) −7.23607 −0.709555
\(105\) 0 0
\(106\) −0.763932 −0.0741996
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −18.4164 −1.76397 −0.881986 0.471276i \(-0.843794\pi\)
−0.881986 + 0.471276i \(0.843794\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.85410 0.458670
\(113\) 12.4164 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.09017 −0.286915
\(117\) 0 0
\(118\) 2.76393 0.254441
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) 17.6525 1.56640 0.783202 0.621767i \(-0.213585\pi\)
0.783202 + 0.621767i \(0.213585\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) 0 0
\(131\) −0.944272 −0.0825014 −0.0412507 0.999149i \(-0.513134\pi\)
−0.0412507 + 0.999149i \(0.513134\pi\)
\(132\) 0 0
\(133\) −4.47214 −0.387783
\(134\) −0.381966 −0.0329968
\(135\) 0 0
\(136\) 14.4721 1.24098
\(137\) 6.94427 0.593289 0.296645 0.954988i \(-0.404132\pi\)
0.296645 + 0.954988i \(0.404132\pi\)
\(138\) 0 0
\(139\) −20.6525 −1.75172 −0.875860 0.482565i \(-0.839705\pi\)
−0.875860 + 0.482565i \(0.839705\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.61803 0.639291
\(143\) −13.7082 −1.14634
\(144\) 0 0
\(145\) 0 0
\(146\) 21.4164 1.77243
\(147\) 0 0
\(148\) −1.85410 −0.152406
\(149\) 13.9443 1.14236 0.571180 0.820825i \(-0.306486\pi\)
0.571180 + 0.820825i \(0.306486\pi\)
\(150\) 0 0
\(151\) −15.7639 −1.28285 −0.641425 0.767185i \(-0.721657\pi\)
−0.641425 + 0.767185i \(0.721657\pi\)
\(152\) −10.0000 −0.811107
\(153\) 0 0
\(154\) 6.85410 0.552319
\(155\) 0 0
\(156\) 0 0
\(157\) −5.23607 −0.417884 −0.208942 0.977928i \(-0.567002\pi\)
−0.208942 + 0.977928i \(0.567002\pi\)
\(158\) 18.0902 1.43918
\(159\) 0 0
\(160\) 0 0
\(161\) −1.76393 −0.139017
\(162\) 0 0
\(163\) 10.4721 0.820241 0.410120 0.912031i \(-0.365487\pi\)
0.410120 + 0.912031i \(0.365487\pi\)
\(164\) −5.70820 −0.445736
\(165\) 0 0
\(166\) 9.23607 0.716858
\(167\) 0.763932 0.0591148 0.0295574 0.999563i \(-0.490590\pi\)
0.0295574 + 0.999563i \(0.490590\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 0 0
\(172\) −3.85410 −0.293873
\(173\) −20.4721 −1.55647 −0.778234 0.627975i \(-0.783884\pi\)
−0.778234 + 0.627975i \(0.783884\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 20.5623 1.54994
\(177\) 0 0
\(178\) −20.6525 −1.54797
\(179\) −3.41641 −0.255354 −0.127677 0.991816i \(-0.540752\pi\)
−0.127677 + 0.991816i \(0.540752\pi\)
\(180\) 0 0
\(181\) −14.1803 −1.05402 −0.527008 0.849860i \(-0.676686\pi\)
−0.527008 + 0.849860i \(0.676686\pi\)
\(182\) −5.23607 −0.388123
\(183\) 0 0
\(184\) −3.94427 −0.290776
\(185\) 0 0
\(186\) 0 0
\(187\) 27.4164 2.00489
\(188\) −1.23607 −0.0901495
\(189\) 0 0
\(190\) 0 0
\(191\) 2.47214 0.178877 0.0894387 0.995992i \(-0.471493\pi\)
0.0894387 + 0.995992i \(0.471493\pi\)
\(192\) 0 0
\(193\) 14.4164 1.03772 0.518858 0.854861i \(-0.326357\pi\)
0.518858 + 0.854861i \(0.326357\pi\)
\(194\) −1.23607 −0.0887445
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) 7.47214 0.532368 0.266184 0.963922i \(-0.414237\pi\)
0.266184 + 0.963922i \(0.414237\pi\)
\(198\) 0 0
\(199\) 2.76393 0.195930 0.0979650 0.995190i \(-0.468767\pi\)
0.0979650 + 0.995190i \(0.468767\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −14.9443 −1.05148
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 0 0
\(206\) 0.763932 0.0532257
\(207\) 0 0
\(208\) −15.7082 −1.08917
\(209\) −18.9443 −1.31040
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −0.291796 −0.0200406
\(213\) 0 0
\(214\) 12.9443 0.884852
\(215\) 0 0
\(216\) 0 0
\(217\) 9.70820 0.659036
\(218\) −29.7984 −2.01820
\(219\) 0 0
\(220\) 0 0
\(221\) −20.9443 −1.40886
\(222\) 0 0
\(223\) 2.18034 0.146006 0.0730032 0.997332i \(-0.476742\pi\)
0.0730032 + 0.997332i \(0.476742\pi\)
\(224\) 3.38197 0.225967
\(225\) 0 0
\(226\) 20.0902 1.33638
\(227\) −5.41641 −0.359500 −0.179750 0.983712i \(-0.557529\pi\)
−0.179750 + 0.983712i \(0.557529\pi\)
\(228\) 0 0
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 11.1803 0.734025
\(233\) −9.94427 −0.651471 −0.325735 0.945461i \(-0.605612\pi\)
−0.325735 + 0.945461i \(0.605612\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.05573 0.0687220
\(237\) 0 0
\(238\) 10.4721 0.678808
\(239\) 14.4721 0.936125 0.468062 0.883695i \(-0.344952\pi\)
0.468062 + 0.883695i \(0.344952\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 11.2361 0.722282
\(243\) 0 0
\(244\) 2.29180 0.146717
\(245\) 0 0
\(246\) 0 0
\(247\) 14.4721 0.920840
\(248\) 21.7082 1.37847
\(249\) 0 0
\(250\) 0 0
\(251\) 2.47214 0.156040 0.0780199 0.996952i \(-0.475140\pi\)
0.0780199 + 0.996952i \(0.475140\pi\)
\(252\) 0 0
\(253\) −7.47214 −0.469769
\(254\) 28.5623 1.79216
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) 18.6525 1.16351 0.581755 0.813364i \(-0.302366\pi\)
0.581755 + 0.813364i \(0.302366\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) −1.52786 −0.0943918
\(263\) 11.7639 0.725395 0.362698 0.931907i \(-0.381856\pi\)
0.362698 + 0.931907i \(0.381856\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −7.23607 −0.443672
\(267\) 0 0
\(268\) −0.145898 −0.00891214
\(269\) −1.70820 −0.104151 −0.0520755 0.998643i \(-0.516584\pi\)
−0.0520755 + 0.998643i \(0.516584\pi\)
\(270\) 0 0
\(271\) 10.2918 0.625182 0.312591 0.949888i \(-0.398803\pi\)
0.312591 + 0.949888i \(0.398803\pi\)
\(272\) 31.4164 1.90490
\(273\) 0 0
\(274\) 11.2361 0.678796
\(275\) 0 0
\(276\) 0 0
\(277\) −15.8885 −0.954650 −0.477325 0.878727i \(-0.658394\pi\)
−0.477325 + 0.878727i \(0.658394\pi\)
\(278\) −33.4164 −2.00418
\(279\) 0 0
\(280\) 0 0
\(281\) −29.3607 −1.75151 −0.875756 0.482755i \(-0.839636\pi\)
−0.875756 + 0.482755i \(0.839636\pi\)
\(282\) 0 0
\(283\) 9.41641 0.559747 0.279874 0.960037i \(-0.409707\pi\)
0.279874 + 0.960037i \(0.409707\pi\)
\(284\) 2.90983 0.172667
\(285\) 0 0
\(286\) −22.1803 −1.31155
\(287\) 9.23607 0.545188
\(288\) 0 0
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 0 0
\(292\) 8.18034 0.478718
\(293\) 9.12461 0.533066 0.266533 0.963826i \(-0.414122\pi\)
0.266533 + 0.963826i \(0.414122\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.70820 0.389906
\(297\) 0 0
\(298\) 22.5623 1.30700
\(299\) 5.70820 0.330114
\(300\) 0 0
\(301\) 6.23607 0.359441
\(302\) −25.5066 −1.46774
\(303\) 0 0
\(304\) −21.7082 −1.24505
\(305\) 0 0
\(306\) 0 0
\(307\) −31.4164 −1.79303 −0.896515 0.443014i \(-0.853909\pi\)
−0.896515 + 0.443014i \(0.853909\pi\)
\(308\) 2.61803 0.149176
\(309\) 0 0
\(310\) 0 0
\(311\) 20.3607 1.15455 0.577274 0.816550i \(-0.304116\pi\)
0.577274 + 0.816550i \(0.304116\pi\)
\(312\) 0 0
\(313\) −28.4721 −1.60934 −0.804670 0.593722i \(-0.797658\pi\)
−0.804670 + 0.593722i \(0.797658\pi\)
\(314\) −8.47214 −0.478110
\(315\) 0 0
\(316\) 6.90983 0.388708
\(317\) −19.3607 −1.08740 −0.543702 0.839278i \(-0.682978\pi\)
−0.543702 + 0.839278i \(0.682978\pi\)
\(318\) 0 0
\(319\) 21.1803 1.18587
\(320\) 0 0
\(321\) 0 0
\(322\) −2.85410 −0.159053
\(323\) −28.9443 −1.61050
\(324\) 0 0
\(325\) 0 0
\(326\) 16.9443 0.938456
\(327\) 0 0
\(328\) 20.6525 1.14034
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −11.2918 −0.620653 −0.310327 0.950630i \(-0.600438\pi\)
−0.310327 + 0.950630i \(0.600438\pi\)
\(332\) 3.52786 0.193617
\(333\) 0 0
\(334\) 1.23607 0.0676346
\(335\) 0 0
\(336\) 0 0
\(337\) 7.52786 0.410069 0.205034 0.978755i \(-0.434269\pi\)
0.205034 + 0.978755i \(0.434269\pi\)
\(338\) −4.09017 −0.222476
\(339\) 0 0
\(340\) 0 0
\(341\) 41.1246 2.22702
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 13.9443 0.751825
\(345\) 0 0
\(346\) −33.1246 −1.78079
\(347\) 15.7639 0.846252 0.423126 0.906071i \(-0.360933\pi\)
0.423126 + 0.906071i \(0.360933\pi\)
\(348\) 0 0
\(349\) −4.47214 −0.239388 −0.119694 0.992811i \(-0.538191\pi\)
−0.119694 + 0.992811i \(0.538191\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 14.3262 0.763591
\(353\) 20.1803 1.07409 0.537046 0.843553i \(-0.319540\pi\)
0.537046 + 0.843553i \(0.319540\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.88854 −0.418092
\(357\) 0 0
\(358\) −5.52786 −0.292157
\(359\) −10.1246 −0.534357 −0.267178 0.963647i \(-0.586091\pi\)
−0.267178 + 0.963647i \(0.586091\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −22.9443 −1.20592
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) −3.12461 −0.163103 −0.0815517 0.996669i \(-0.525988\pi\)
−0.0815517 + 0.996669i \(0.525988\pi\)
\(368\) −8.56231 −0.446341
\(369\) 0 0
\(370\) 0 0
\(371\) 0.472136 0.0245121
\(372\) 0 0
\(373\) −15.8328 −0.819792 −0.409896 0.912132i \(-0.634435\pi\)
−0.409896 + 0.912132i \(0.634435\pi\)
\(374\) 44.3607 2.29384
\(375\) 0 0
\(376\) 4.47214 0.230633
\(377\) −16.1803 −0.833330
\(378\) 0 0
\(379\) 11.1803 0.574295 0.287148 0.957886i \(-0.407293\pi\)
0.287148 + 0.957886i \(0.407293\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.00000 0.204658
\(383\) −28.7639 −1.46977 −0.734884 0.678193i \(-0.762763\pi\)
−0.734884 + 0.678193i \(0.762763\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.3262 1.18727
\(387\) 0 0
\(388\) −0.472136 −0.0239691
\(389\) 32.8885 1.66752 0.833758 0.552131i \(-0.186185\pi\)
0.833758 + 0.552131i \(0.186185\pi\)
\(390\) 0 0
\(391\) −11.4164 −0.577353
\(392\) −2.23607 −0.112938
\(393\) 0 0
\(394\) 12.0902 0.609094
\(395\) 0 0
\(396\) 0 0
\(397\) −26.9443 −1.35229 −0.676147 0.736767i \(-0.736352\pi\)
−0.676147 + 0.736767i \(0.736352\pi\)
\(398\) 4.47214 0.224168
\(399\) 0 0
\(400\) 0 0
\(401\) −11.4721 −0.572891 −0.286446 0.958097i \(-0.592474\pi\)
−0.286446 + 0.958097i \(0.592474\pi\)
\(402\) 0 0
\(403\) −31.4164 −1.56496
\(404\) −5.70820 −0.283994
\(405\) 0 0
\(406\) 8.09017 0.401508
\(407\) 12.7082 0.629922
\(408\) 0 0
\(409\) 15.5279 0.767803 0.383902 0.923374i \(-0.374580\pi\)
0.383902 + 0.923374i \(0.374580\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.291796 0.0143758
\(413\) −1.70820 −0.0840552
\(414\) 0 0
\(415\) 0 0
\(416\) −10.9443 −0.536587
\(417\) 0 0
\(418\) −30.6525 −1.49926
\(419\) 3.81966 0.186603 0.0933013 0.995638i \(-0.470258\pi\)
0.0933013 + 0.995638i \(0.470258\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 19.4164 0.945176
\(423\) 0 0
\(424\) 1.05573 0.0512707
\(425\) 0 0
\(426\) 0 0
\(427\) −3.70820 −0.179453
\(428\) 4.94427 0.238990
\(429\) 0 0
\(430\) 0 0
\(431\) −26.4721 −1.27512 −0.637559 0.770402i \(-0.720056\pi\)
−0.637559 + 0.770402i \(0.720056\pi\)
\(432\) 0 0
\(433\) −16.3607 −0.786244 −0.393122 0.919486i \(-0.628605\pi\)
−0.393122 + 0.919486i \(0.628605\pi\)
\(434\) 15.7082 0.754018
\(435\) 0 0
\(436\) −11.3820 −0.545097
\(437\) 7.88854 0.377360
\(438\) 0 0
\(439\) −21.7082 −1.03608 −0.518038 0.855358i \(-0.673337\pi\)
−0.518038 + 0.855358i \(0.673337\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −33.8885 −1.61191
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.52786 0.167049
\(447\) 0 0
\(448\) −4.23607 −0.200135
\(449\) −29.4721 −1.39088 −0.695438 0.718586i \(-0.744790\pi\)
−0.695438 + 0.718586i \(0.744790\pi\)
\(450\) 0 0
\(451\) 39.1246 1.84231
\(452\) 7.67376 0.360943
\(453\) 0 0
\(454\) −8.76393 −0.411312
\(455\) 0 0
\(456\) 0 0
\(457\) 21.4721 1.00442 0.502212 0.864744i \(-0.332520\pi\)
0.502212 + 0.864744i \(0.332520\pi\)
\(458\) 7.23607 0.338119
\(459\) 0 0
\(460\) 0 0
\(461\) −8.18034 −0.380996 −0.190498 0.981688i \(-0.561010\pi\)
−0.190498 + 0.981688i \(0.561010\pi\)
\(462\) 0 0
\(463\) −21.8885 −1.01725 −0.508623 0.860989i \(-0.669845\pi\)
−0.508623 + 0.860989i \(0.669845\pi\)
\(464\) 24.2705 1.12673
\(465\) 0 0
\(466\) −16.0902 −0.745363
\(467\) −10.9443 −0.506441 −0.253220 0.967409i \(-0.581490\pi\)
−0.253220 + 0.967409i \(0.581490\pi\)
\(468\) 0 0
\(469\) 0.236068 0.0109006
\(470\) 0 0
\(471\) 0 0
\(472\) −3.81966 −0.175814
\(473\) 26.4164 1.21463
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 23.4164 1.07104
\(479\) 3.81966 0.174525 0.0872624 0.996185i \(-0.472188\pi\)
0.0872624 + 0.996185i \(0.472188\pi\)
\(480\) 0 0
\(481\) −9.70820 −0.442656
\(482\) −20.1803 −0.919189
\(483\) 0 0
\(484\) 4.29180 0.195082
\(485\) 0 0
\(486\) 0 0
\(487\) −10.2361 −0.463841 −0.231920 0.972735i \(-0.574501\pi\)
−0.231920 + 0.972735i \(0.574501\pi\)
\(488\) −8.29180 −0.375352
\(489\) 0 0
\(490\) 0 0
\(491\) 10.2361 0.461947 0.230974 0.972960i \(-0.425809\pi\)
0.230974 + 0.972960i \(0.425809\pi\)
\(492\) 0 0
\(493\) 32.3607 1.45745
\(494\) 23.4164 1.05355
\(495\) 0 0
\(496\) 47.1246 2.11596
\(497\) −4.70820 −0.211192
\(498\) 0 0
\(499\) 28.9443 1.29572 0.647862 0.761758i \(-0.275663\pi\)
0.647862 + 0.761758i \(0.275663\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −43.8885 −1.95689 −0.978447 0.206499i \(-0.933793\pi\)
−0.978447 + 0.206499i \(0.933793\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0902 −0.537474
\(507\) 0 0
\(508\) 10.9098 0.484045
\(509\) −9.34752 −0.414322 −0.207161 0.978307i \(-0.566422\pi\)
−0.207161 + 0.978307i \(0.566422\pi\)
\(510\) 0 0
\(511\) −13.2361 −0.585529
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) 30.1803 1.33120
\(515\) 0 0
\(516\) 0 0
\(517\) 8.47214 0.372604
\(518\) 4.85410 0.213277
\(519\) 0 0
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 28.3607 1.24013 0.620063 0.784552i \(-0.287107\pi\)
0.620063 + 0.784552i \(0.287107\pi\)
\(524\) −0.583592 −0.0254943
\(525\) 0 0
\(526\) 19.0344 0.829941
\(527\) 62.8328 2.73704
\(528\) 0 0
\(529\) −19.8885 −0.864719
\(530\) 0 0
\(531\) 0 0
\(532\) −2.76393 −0.119832
\(533\) −29.8885 −1.29462
\(534\) 0 0
\(535\) 0 0
\(536\) 0.527864 0.0228003
\(537\) 0 0
\(538\) −2.76393 −0.119162
\(539\) −4.23607 −0.182460
\(540\) 0 0
\(541\) −1.94427 −0.0835908 −0.0417954 0.999126i \(-0.513308\pi\)
−0.0417954 + 0.999126i \(0.513308\pi\)
\(542\) 16.6525 0.715285
\(543\) 0 0
\(544\) 21.8885 0.938464
\(545\) 0 0
\(546\) 0 0
\(547\) 14.2361 0.608690 0.304345 0.952562i \(-0.401562\pi\)
0.304345 + 0.952562i \(0.401562\pi\)
\(548\) 4.29180 0.183336
\(549\) 0 0
\(550\) 0 0
\(551\) −22.3607 −0.952597
\(552\) 0 0
\(553\) −11.1803 −0.475436
\(554\) −25.7082 −1.09224
\(555\) 0 0
\(556\) −12.7639 −0.541311
\(557\) −44.8885 −1.90199 −0.950994 0.309208i \(-0.899936\pi\)
−0.950994 + 0.309208i \(0.899936\pi\)
\(558\) 0 0
\(559\) −20.1803 −0.853537
\(560\) 0 0
\(561\) 0 0
\(562\) −47.5066 −2.00394
\(563\) −9.41641 −0.396854 −0.198427 0.980116i \(-0.563583\pi\)
−0.198427 + 0.980116i \(0.563583\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 15.2361 0.640420
\(567\) 0 0
\(568\) −10.5279 −0.441739
\(569\) −13.9443 −0.584574 −0.292287 0.956331i \(-0.594416\pi\)
−0.292287 + 0.956331i \(0.594416\pi\)
\(570\) 0 0
\(571\) −12.5967 −0.527157 −0.263579 0.964638i \(-0.584903\pi\)
−0.263579 + 0.964638i \(0.584903\pi\)
\(572\) −8.47214 −0.354238
\(573\) 0 0
\(574\) 14.9443 0.623762
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 40.2705 1.67503
\(579\) 0 0
\(580\) 0 0
\(581\) −5.70820 −0.236816
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) −29.5967 −1.22472
\(585\) 0 0
\(586\) 14.7639 0.609892
\(587\) −29.2361 −1.20670 −0.603351 0.797476i \(-0.706168\pi\)
−0.603351 + 0.797476i \(0.706168\pi\)
\(588\) 0 0
\(589\) −43.4164 −1.78894
\(590\) 0 0
\(591\) 0 0
\(592\) 14.5623 0.598507
\(593\) 25.3050 1.03915 0.519575 0.854425i \(-0.326090\pi\)
0.519575 + 0.854425i \(0.326090\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.61803 0.353008
\(597\) 0 0
\(598\) 9.23607 0.377691
\(599\) −11.1803 −0.456816 −0.228408 0.973565i \(-0.573352\pi\)
−0.228408 + 0.973565i \(0.573352\pi\)
\(600\) 0 0
\(601\) −19.0557 −0.777299 −0.388650 0.921386i \(-0.627058\pi\)
−0.388650 + 0.921386i \(0.627058\pi\)
\(602\) 10.0902 0.411245
\(603\) 0 0
\(604\) −9.74265 −0.396423
\(605\) 0 0
\(606\) 0 0
\(607\) −33.1246 −1.34449 −0.672243 0.740330i \(-0.734669\pi\)
−0.672243 + 0.740330i \(0.734669\pi\)
\(608\) −15.1246 −0.613384
\(609\) 0 0
\(610\) 0 0
\(611\) −6.47214 −0.261835
\(612\) 0 0
\(613\) 17.5836 0.710195 0.355097 0.934829i \(-0.384448\pi\)
0.355097 + 0.934829i \(0.384448\pi\)
\(614\) −50.8328 −2.05145
\(615\) 0 0
\(616\) −9.47214 −0.381643
\(617\) 11.9443 0.480858 0.240429 0.970667i \(-0.422712\pi\)
0.240429 + 0.970667i \(0.422712\pi\)
\(618\) 0 0
\(619\) 1.70820 0.0686585 0.0343293 0.999411i \(-0.489071\pi\)
0.0343293 + 0.999411i \(0.489071\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 32.9443 1.32094
\(623\) 12.7639 0.511376
\(624\) 0 0
\(625\) 0 0
\(626\) −46.0689 −1.84128
\(627\) 0 0
\(628\) −3.23607 −0.129133
\(629\) 19.4164 0.774183
\(630\) 0 0
\(631\) −3.65248 −0.145403 −0.0727014 0.997354i \(-0.523162\pi\)
−0.0727014 + 0.997354i \(0.523162\pi\)
\(632\) −25.0000 −0.994447
\(633\) 0 0
\(634\) −31.3262 −1.24412
\(635\) 0 0
\(636\) 0 0
\(637\) 3.23607 0.128218
\(638\) 34.2705 1.35678
\(639\) 0 0
\(640\) 0 0
\(641\) 9.83282 0.388373 0.194186 0.980965i \(-0.437793\pi\)
0.194186 + 0.980965i \(0.437793\pi\)
\(642\) 0 0
\(643\) −9.52786 −0.375742 −0.187871 0.982194i \(-0.560159\pi\)
−0.187871 + 0.982194i \(0.560159\pi\)
\(644\) −1.09017 −0.0429587
\(645\) 0 0
\(646\) −46.8328 −1.84261
\(647\) 15.8885 0.624643 0.312322 0.949976i \(-0.398893\pi\)
0.312322 + 0.949976i \(0.398893\pi\)
\(648\) 0 0
\(649\) −7.23607 −0.284041
\(650\) 0 0
\(651\) 0 0
\(652\) 6.47214 0.253468
\(653\) 42.9443 1.68054 0.840270 0.542169i \(-0.182397\pi\)
0.840270 + 0.542169i \(0.182397\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 44.8328 1.75043
\(657\) 0 0
\(658\) 3.23607 0.126155
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) 46.7214 1.81725 0.908625 0.417613i \(-0.137133\pi\)
0.908625 + 0.417613i \(0.137133\pi\)
\(662\) −18.2705 −0.710104
\(663\) 0 0
\(664\) −12.7639 −0.495337
\(665\) 0 0
\(666\) 0 0
\(667\) −8.81966 −0.341499
\(668\) 0.472136 0.0182675
\(669\) 0 0
\(670\) 0 0
\(671\) −15.7082 −0.606408
\(672\) 0 0
\(673\) −28.4721 −1.09752 −0.548760 0.835980i \(-0.684900\pi\)
−0.548760 + 0.835980i \(0.684900\pi\)
\(674\) 12.1803 0.469169
\(675\) 0 0
\(676\) −1.56231 −0.0600887
\(677\) 30.3607 1.16686 0.583428 0.812165i \(-0.301711\pi\)
0.583428 + 0.812165i \(0.301711\pi\)
\(678\) 0 0
\(679\) 0.763932 0.0293170
\(680\) 0 0
\(681\) 0 0
\(682\) 66.5410 2.54799
\(683\) −26.1246 −0.999630 −0.499815 0.866132i \(-0.666599\pi\)
−0.499815 + 0.866132i \(0.666599\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.61803 −0.0617768
\(687\) 0 0
\(688\) 30.2705 1.15405
\(689\) −1.52786 −0.0582070
\(690\) 0 0
\(691\) 18.1803 0.691613 0.345806 0.938306i \(-0.387605\pi\)
0.345806 + 0.938306i \(0.387605\pi\)
\(692\) −12.6525 −0.480975
\(693\) 0 0
\(694\) 25.5066 0.968216
\(695\) 0 0
\(696\) 0 0
\(697\) 59.7771 2.26422
\(698\) −7.23607 −0.273889
\(699\) 0 0
\(700\) 0 0
\(701\) 46.9443 1.77306 0.886530 0.462670i \(-0.153109\pi\)
0.886530 + 0.462670i \(0.153109\pi\)
\(702\) 0 0
\(703\) −13.4164 −0.506009
\(704\) −17.9443 −0.676300
\(705\) 0 0
\(706\) 32.6525 1.22889
\(707\) 9.23607 0.347358
\(708\) 0 0
\(709\) −47.8885 −1.79849 −0.899246 0.437443i \(-0.855884\pi\)
−0.899246 + 0.437443i \(0.855884\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 28.5410 1.06962
\(713\) −17.1246 −0.641322
\(714\) 0 0
\(715\) 0 0
\(716\) −2.11146 −0.0789088
\(717\) 0 0
\(718\) −16.3820 −0.611370
\(719\) 6.18034 0.230488 0.115244 0.993337i \(-0.463235\pi\)
0.115244 + 0.993337i \(0.463235\pi\)
\(720\) 0 0
\(721\) −0.472136 −0.0175833
\(722\) 1.61803 0.0602170
\(723\) 0 0
\(724\) −8.76393 −0.325709
\(725\) 0 0
\(726\) 0 0
\(727\) 20.9443 0.776780 0.388390 0.921495i \(-0.373031\pi\)
0.388390 + 0.921495i \(0.373031\pi\)
\(728\) 7.23607 0.268187
\(729\) 0 0
\(730\) 0 0
\(731\) 40.3607 1.49279
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) −5.05573 −0.186610
\(735\) 0 0
\(736\) −5.96556 −0.219893
\(737\) 1.00000 0.0368355
\(738\) 0 0
\(739\) −5.65248 −0.207930 −0.103965 0.994581i \(-0.533153\pi\)
−0.103965 + 0.994581i \(0.533153\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.763932 0.0280448
\(743\) −1.52786 −0.0560519 −0.0280259 0.999607i \(-0.508922\pi\)
−0.0280259 + 0.999607i \(0.508922\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −25.6180 −0.937943
\(747\) 0 0
\(748\) 16.9443 0.619544
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 20.9443 0.764267 0.382134 0.924107i \(-0.375189\pi\)
0.382134 + 0.924107i \(0.375189\pi\)
\(752\) 9.70820 0.354022
\(753\) 0 0
\(754\) −26.1803 −0.953432
\(755\) 0 0
\(756\) 0 0
\(757\) −46.4164 −1.68703 −0.843517 0.537103i \(-0.819519\pi\)
−0.843517 + 0.537103i \(0.819519\pi\)
\(758\) 18.0902 0.657065
\(759\) 0 0
\(760\) 0 0
\(761\) 43.7771 1.58692 0.793459 0.608624i \(-0.208278\pi\)
0.793459 + 0.608624i \(0.208278\pi\)
\(762\) 0 0
\(763\) 18.4164 0.666719
\(764\) 1.52786 0.0552762
\(765\) 0 0
\(766\) −46.5410 −1.68160
\(767\) 5.52786 0.199600
\(768\) 0 0
\(769\) 33.0132 1.19048 0.595242 0.803546i \(-0.297056\pi\)
0.595242 + 0.803546i \(0.297056\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.90983 0.320672
\(773\) 27.8197 1.00060 0.500302 0.865851i \(-0.333222\pi\)
0.500302 + 0.865851i \(0.333222\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.70820 0.0613209
\(777\) 0 0
\(778\) 53.2148 1.90784
\(779\) −41.3050 −1.47990
\(780\) 0 0
\(781\) −19.9443 −0.713662
\(782\) −18.4721 −0.660562
\(783\) 0 0
\(784\) −4.85410 −0.173361
\(785\) 0 0
\(786\) 0 0
\(787\) −45.2361 −1.61249 −0.806246 0.591581i \(-0.798504\pi\)
−0.806246 + 0.591581i \(0.798504\pi\)
\(788\) 4.61803 0.164511
\(789\) 0 0
\(790\) 0 0
\(791\) −12.4164 −0.441477
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −43.5967 −1.54719
\(795\) 0 0
\(796\) 1.70820 0.0605457
\(797\) −8.58359 −0.304046 −0.152023 0.988377i \(-0.548579\pi\)
−0.152023 + 0.988377i \(0.548579\pi\)
\(798\) 0 0
\(799\) 12.9443 0.457935
\(800\) 0 0
\(801\) 0 0
\(802\) −18.5623 −0.655458
\(803\) −56.0689 −1.97863
\(804\) 0 0
\(805\) 0 0
\(806\) −50.8328 −1.79051
\(807\) 0 0
\(808\) 20.6525 0.726552
\(809\) −20.5279 −0.721721 −0.360861 0.932620i \(-0.617517\pi\)
−0.360861 + 0.932620i \(0.617517\pi\)
\(810\) 0 0
\(811\) 46.7214 1.64061 0.820304 0.571927i \(-0.193804\pi\)
0.820304 + 0.571927i \(0.193804\pi\)
\(812\) 3.09017 0.108444
\(813\) 0 0
\(814\) 20.5623 0.720708
\(815\) 0 0
\(816\) 0 0
\(817\) −27.8885 −0.975697
\(818\) 25.1246 0.878461
\(819\) 0 0
\(820\) 0 0
\(821\) 24.8328 0.866671 0.433336 0.901233i \(-0.357336\pi\)
0.433336 + 0.901233i \(0.357336\pi\)
\(822\) 0 0
\(823\) 0.347524 0.0121139 0.00605697 0.999982i \(-0.498072\pi\)
0.00605697 + 0.999982i \(0.498072\pi\)
\(824\) −1.05573 −0.0367780
\(825\) 0 0
\(826\) −2.76393 −0.0961695
\(827\) −25.5410 −0.888148 −0.444074 0.895990i \(-0.646467\pi\)
−0.444074 + 0.895990i \(0.646467\pi\)
\(828\) 0 0
\(829\) −52.3607 −1.81856 −0.909281 0.416183i \(-0.863368\pi\)
−0.909281 + 0.416183i \(0.863368\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 13.7082 0.475246
\(833\) −6.47214 −0.224246
\(834\) 0 0
\(835\) 0 0
\(836\) −11.7082 −0.404937
\(837\) 0 0
\(838\) 6.18034 0.213496
\(839\) −0.652476 −0.0225260 −0.0112630 0.999937i \(-0.503585\pi\)
−0.0112630 + 0.999937i \(0.503585\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −21.0344 −0.724895
\(843\) 0 0
\(844\) 7.41641 0.255283
\(845\) 0 0
\(846\) 0 0
\(847\) −6.94427 −0.238608
\(848\) 2.29180 0.0787006
\(849\) 0 0
\(850\) 0 0
\(851\) −5.29180 −0.181400
\(852\) 0 0
\(853\) −0.583592 −0.0199818 −0.00999091 0.999950i \(-0.503180\pi\)
−0.00999091 + 0.999950i \(0.503180\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) −17.8885 −0.611418
\(857\) −38.1803 −1.30422 −0.652108 0.758126i \(-0.726115\pi\)
−0.652108 + 0.758126i \(0.726115\pi\)
\(858\) 0 0
\(859\) −22.3607 −0.762937 −0.381468 0.924382i \(-0.624581\pi\)
−0.381468 + 0.924382i \(0.624581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −42.8328 −1.45889
\(863\) 49.6525 1.69019 0.845095 0.534616i \(-0.179544\pi\)
0.845095 + 0.534616i \(0.179544\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −26.4721 −0.899560
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) −47.3607 −1.60660
\(870\) 0 0
\(871\) −0.763932 −0.0258848
\(872\) 41.1803 1.39454
\(873\) 0 0
\(874\) 12.7639 0.431746
\(875\) 0 0
\(876\) 0 0
\(877\) 14.3607 0.484926 0.242463 0.970161i \(-0.422045\pi\)
0.242463 + 0.970161i \(0.422045\pi\)
\(878\) −35.1246 −1.18540
\(879\) 0 0
\(880\) 0 0
\(881\) −28.1803 −0.949420 −0.474710 0.880142i \(-0.657447\pi\)
−0.474710 + 0.880142i \(0.657447\pi\)
\(882\) 0 0
\(883\) 50.5967 1.70272 0.851358 0.524585i \(-0.175780\pi\)
0.851358 + 0.524585i \(0.175780\pi\)
\(884\) −12.9443 −0.435363
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) −52.6525 −1.76790 −0.883949 0.467584i \(-0.845124\pi\)
−0.883949 + 0.467584i \(0.845124\pi\)
\(888\) 0 0
\(889\) −17.6525 −0.592045
\(890\) 0 0
\(891\) 0 0
\(892\) 1.34752 0.0451184
\(893\) −8.94427 −0.299309
\(894\) 0 0
\(895\) 0 0
\(896\) −13.6180 −0.454947
\(897\) 0 0
\(898\) −47.6869 −1.59133
\(899\) 48.5410 1.61893
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) 63.3050 2.10782
\(903\) 0 0
\(904\) −27.7639 −0.923415
\(905\) 0 0
\(906\) 0 0
\(907\) 18.8328 0.625333 0.312667 0.949863i \(-0.398778\pi\)
0.312667 + 0.949863i \(0.398778\pi\)
\(908\) −3.34752 −0.111091
\(909\) 0 0
\(910\) 0 0
\(911\) −23.1803 −0.767999 −0.383999 0.923333i \(-0.625454\pi\)
−0.383999 + 0.923333i \(0.625454\pi\)
\(912\) 0 0
\(913\) −24.1803 −0.800252
\(914\) 34.7426 1.14918
\(915\) 0 0
\(916\) 2.76393 0.0913229
\(917\) 0.944272 0.0311826
\(918\) 0 0
\(919\) −32.2361 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −13.2361 −0.435907
\(923\) 15.2361 0.501501
\(924\) 0 0
\(925\) 0 0
\(926\) −35.4164 −1.16386
\(927\) 0 0
\(928\) 16.9098 0.555092
\(929\) −51.7082 −1.69649 −0.848246 0.529603i \(-0.822341\pi\)
−0.848246 + 0.529603i \(0.822341\pi\)
\(930\) 0 0
\(931\) 4.47214 0.146568
\(932\) −6.14590 −0.201316
\(933\) 0 0
\(934\) −17.7082 −0.579430
\(935\) 0 0
\(936\) 0 0
\(937\) −30.7639 −1.00501 −0.502507 0.864573i \(-0.667589\pi\)
−0.502507 + 0.864573i \(0.667589\pi\)
\(938\) 0.381966 0.0124716
\(939\) 0 0
\(940\) 0 0
\(941\) 0.763932 0.0249035 0.0124517 0.999922i \(-0.496036\pi\)
0.0124517 + 0.999922i \(0.496036\pi\)
\(942\) 0 0
\(943\) −16.2918 −0.530534
\(944\) −8.29180 −0.269875
\(945\) 0 0
\(946\) 42.7426 1.38968
\(947\) −18.8328 −0.611984 −0.305992 0.952034i \(-0.598988\pi\)
−0.305992 + 0.952034i \(0.598988\pi\)
\(948\) 0 0
\(949\) 42.8328 1.39041
\(950\) 0 0
\(951\) 0 0
\(952\) −14.4721 −0.469045
\(953\) −5.47214 −0.177260 −0.0886299 0.996065i \(-0.528249\pi\)
−0.0886299 + 0.996065i \(0.528249\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.94427 0.289278
\(957\) 0 0
\(958\) 6.18034 0.199678
\(959\) −6.94427 −0.224242
\(960\) 0 0
\(961\) 63.2492 2.04030
\(962\) −15.7082 −0.506453
\(963\) 0 0
\(964\) −7.70820 −0.248265
\(965\) 0 0
\(966\) 0 0
\(967\) 49.8885 1.60431 0.802154 0.597118i \(-0.203687\pi\)
0.802154 + 0.597118i \(0.203687\pi\)
\(968\) −15.5279 −0.499084
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 20.6525 0.662088
\(974\) −16.5623 −0.530691
\(975\) 0 0
\(976\) −18.0000 −0.576166
\(977\) −2.52786 −0.0808735 −0.0404368 0.999182i \(-0.512875\pi\)
−0.0404368 + 0.999182i \(0.512875\pi\)
\(978\) 0 0
\(979\) 54.0689 1.72805
\(980\) 0 0
\(981\) 0 0
\(982\) 16.5623 0.528524
\(983\) 32.5410 1.03790 0.518949 0.854805i \(-0.326324\pi\)
0.518949 + 0.854805i \(0.326324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 52.3607 1.66750
\(987\) 0 0
\(988\) 8.94427 0.284555
\(989\) −11.0000 −0.349780
\(990\) 0 0
\(991\) −9.18034 −0.291623 −0.145812 0.989312i \(-0.546579\pi\)
−0.145812 + 0.989312i \(0.546579\pi\)
\(992\) 32.8328 1.04244
\(993\) 0 0
\(994\) −7.61803 −0.241629
\(995\) 0 0
\(996\) 0 0
\(997\) 18.5836 0.588548 0.294274 0.955721i \(-0.404922\pi\)
0.294274 + 0.955721i \(0.404922\pi\)
\(998\) 46.8328 1.48247
\(999\) 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 1575.2.a.s.1.2 2
3.2 odd 2 175.2.a.d.1.1 2
5.2 odd 4 1575.2.d.k.1324.4 4
5.3 odd 4 1575.2.d.k.1324.1 4
5.4 even 2 1575.2.a.n.1.1 2
12.11 even 2 2800.2.a.bh.1.2 2
15.2 even 4 175.2.b.c.99.1 4
15.8 even 4 175.2.b.c.99.4 4
15.14 odd 2 175.2.a.e.1.2 yes 2
21.20 even 2 1225.2.a.n.1.1 2
60.23 odd 4 2800.2.g.s.449.3 4
60.47 odd 4 2800.2.g.s.449.2 4
60.59 even 2 2800.2.a.bp.1.1 2
105.62 odd 4 1225.2.b.k.99.1 4
105.83 odd 4 1225.2.b.k.99.4 4
105.104 even 2 1225.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 3.2 odd 2
175.2.a.e.1.2 yes 2 15.14 odd 2
175.2.b.c.99.1 4 15.2 even 4
175.2.b.c.99.4 4 15.8 even 4
1225.2.a.n.1.1 2 21.20 even 2
1225.2.a.u.1.2 2 105.104 even 2
1225.2.b.k.99.1 4 105.62 odd 4
1225.2.b.k.99.4 4 105.83 odd 4
1575.2.a.n.1.1 2 5.4 even 2
1575.2.a.s.1.2 2 1.1 even 1 trivial
1575.2.d.k.1324.1 4 5.3 odd 4
1575.2.d.k.1324.4 4 5.2 odd 4
2800.2.a.bh.1.2 2 12.11 even 2
2800.2.a.bp.1.1 2 60.59 even 2
2800.2.g.s.449.2 4 60.47 odd 4
2800.2.g.s.449.3 4 60.23 odd 4