Properties

Label 1225.2.a.u.1.2
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.23607 q^{3} +0.618034 q^{4} -2.00000 q^{6} -2.23607 q^{8} -1.47214 q^{9} +4.23607 q^{11} -0.763932 q^{12} +3.23607 q^{13} -4.85410 q^{16} +6.47214 q^{17} -2.38197 q^{18} -4.47214 q^{19} +6.85410 q^{22} +1.76393 q^{23} +2.76393 q^{24} +5.23607 q^{26} +5.52786 q^{27} +5.00000 q^{29} +9.70820 q^{31} -3.38197 q^{32} -5.23607 q^{33} +10.4721 q^{34} -0.909830 q^{36} +3.00000 q^{37} -7.23607 q^{38} -4.00000 q^{39} -9.23607 q^{41} +6.23607 q^{43} +2.61803 q^{44} +2.85410 q^{46} +2.00000 q^{47} +6.00000 q^{48} -8.00000 q^{51} +2.00000 q^{52} -0.472136 q^{53} +8.94427 q^{54} +5.52786 q^{57} +8.09017 q^{58} +1.70820 q^{59} -3.70820 q^{61} +15.7082 q^{62} +4.23607 q^{64} -8.47214 q^{66} +0.236068 q^{67} +4.00000 q^{68} -2.18034 q^{69} -4.70820 q^{71} +3.29180 q^{72} +13.2361 q^{73} +4.85410 q^{74} -2.76393 q^{76} -6.47214 q^{78} +11.1803 q^{79} -2.41641 q^{81} -14.9443 q^{82} -5.70820 q^{83} +10.0902 q^{86} -6.18034 q^{87} -9.47214 q^{88} -12.7639 q^{89} +1.09017 q^{92} -12.0000 q^{93} +3.23607 q^{94} +4.18034 q^{96} -0.763932 q^{97} -6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} - 4q^{6} + 6q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} - 4q^{6} + 6q^{9} + 4q^{11} - 6q^{12} + 2q^{13} - 3q^{16} + 4q^{17} - 7q^{18} + 7q^{22} + 8q^{23} + 10q^{24} + 6q^{26} + 20q^{27} + 10q^{29} + 6q^{31} - 9q^{32} - 6q^{33} + 12q^{34} - 13q^{36} + 6q^{37} - 10q^{38} - 8q^{39} - 14q^{41} + 8q^{43} + 3q^{44} - q^{46} + 4q^{47} + 12q^{48} - 16q^{51} + 4q^{52} + 8q^{53} + 20q^{57} + 5q^{58} - 10q^{59} + 6q^{61} + 18q^{62} + 4q^{64} - 8q^{66} - 4q^{67} + 8q^{68} + 18q^{69} + 4q^{71} + 20q^{72} + 22q^{73} + 3q^{74} - 10q^{76} - 4q^{78} + 22q^{81} - 12q^{82} + 2q^{83} + 9q^{86} + 10q^{87} - 10q^{88} - 30q^{89} - 9q^{92} - 24q^{93} + 2q^{94} - 14q^{96} - 6q^{97} - 8q^{99} + O(q^{100}) \)

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 \(0.306144\pi\)
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 4.23607 1.27722 0.638611 0.769529i \(-0.279509\pi\)
0.638611 + 0.769529i \(0.279509\pi\)
\(12\) −0.763932 −0.220528
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) −2.38197 −0.561435
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 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 \(0.441127\pi\)
\(24\) 2.76393 0.564185
\(25\) 0 0
\(26\) 5.23607 1.02688
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 9.70820 1.74364 0.871822 0.489822i \(-0.162938\pi\)
0.871822 + 0.489822i \(0.162938\pi\)
\(32\) −3.38197 −0.597853
\(33\) −5.23607 −0.911482
\(34\) 10.4721 1.79596
\(35\) 0 0
\(36\) −0.909830 −0.151638
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) −7.23607 −1.17385
\(39\) −4.00000 −0.640513
\(40\) 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 \(-0.342269\pi\)
0.475496 + 0.879718i \(0.342269\pi\)
\(44\) 2.61803 0.394683
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 6.00000 0.866025
\(49\) 0 0
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 2.00000 0.277350
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 8.94427 1.21716
\(55\) 0 0
\(56\) 0 0
\(57\) 5.52786 0.732183
\(58\) 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 \(-0.576293\pi\)
−0.237393 + 0.971414i \(0.576293\pi\)
\(62\) 15.7082 1.99494
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) −8.47214 −1.04285
\(67\) 0.236068 0.0288403 0.0144201 0.999896i \(-0.495410\pi\)
0.0144201 + 0.999896i \(0.495410\pi\)
\(68\) 4.00000 0.485071
\(69\) −2.18034 −0.262482
\(70\) 0 0
\(71\) −4.70820 −0.558761 −0.279381 0.960180i \(-0.590129\pi\)
−0.279381 + 0.960180i \(0.590129\pi\)
\(72\) 3.29180 0.387942
\(73\) 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
\(77\) 0 0
\(78\) −6.47214 −0.732825
\(79\) 11.1803 1.25789 0.628943 0.777451i \(-0.283488\pi\)
0.628943 + 0.777451i \(0.283488\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) −14.9443 −1.65032
\(83\) −5.70820 −0.626557 −0.313278 0.949661i \(-0.601427\pi\)
−0.313278 + 0.949661i \(0.601427\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0902 1.08805
\(87\) −6.18034 −0.662602
\(88\) −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
\(91\) 0 0
\(92\) 1.09017 0.113658
\(93\) −12.0000 −1.24434
\(94\) 3.23607 0.333775
\(95\) 0 0
\(96\) 4.18034 0.426654
\(97\) −0.763932 −0.0775655 −0.0387828 0.999248i \(-0.512348\pi\)
−0.0387828 + 0.999248i \(0.512348\pi\)
\(98\) 0 0
\(99\) −6.23607 −0.626748
\(100\) 0 0
\(101\) −9.23607 −0.919023 −0.459512 0.888172i \(-0.651976\pi\)
−0.459512 + 0.888172i \(0.651976\pi\)
\(102\) −12.9443 −1.28167
\(103\) 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 \(0.373617\pi\)
\(108\) 3.41641 0.328744
\(109\) −18.4164 −1.76397 −0.881986 0.471276i \(-0.843794\pi\)
−0.881986 + 0.471276i \(0.843794\pi\)
\(110\) 0 0
\(111\) −3.70820 −0.351967
\(112\) 0 0
\(113\) 12.4164 1.16804 0.584019 0.811740i \(-0.301479\pi\)
0.584019 + 0.811740i \(0.301479\pi\)
\(114\) 8.94427 0.837708
\(115\) 0 0
\(116\) 3.09017 0.286915
\(117\) −4.76393 −0.440426
\(118\) 2.76393 0.254441
\(119\) 0 0
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) −6.00000 −0.543214
\(123\) 11.4164 1.02938
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) −17.6525 −1.56640 −0.783202 0.621767i \(-0.786415\pi\)
−0.783202 + 0.621767i \(0.786415\pi\)
\(128\) 13.6180 1.20368
\(129\) −7.70820 −0.678670
\(130\) 0 0
\(131\) −0.944272 −0.0825014 −0.0412507 0.999149i \(-0.513134\pi\)
−0.0412507 + 0.999149i \(0.513134\pi\)
\(132\) −3.23607 −0.281664
\(133\) 0 0
\(134\) 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 \(0.404132\pi\)
\(138\) −3.52786 −0.300312
\(139\) 20.6525 1.75172 0.875860 0.482565i \(-0.160295\pi\)
0.875860 + 0.482565i \(0.160295\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) −7.61803 −0.639291
\(143\) 13.7082 1.14634
\(144\) 7.14590 0.595492
\(145\) 0 0
\(146\) 21.4164 1.77243
\(147\) 0 0
\(148\) 1.85410 0.152406
\(149\) −13.9443 −1.14236 −0.571180 0.820825i \(-0.693514\pi\)
−0.571180 + 0.820825i \(0.693514\pi\)
\(150\) 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
\(153\) −9.52786 −0.770282
\(154\) 0 0
\(155\) 0 0
\(156\) −2.47214 −0.197929
\(157\) −5.23607 −0.417884 −0.208942 0.977928i \(-0.567002\pi\)
−0.208942 + 0.977928i \(0.567002\pi\)
\(158\) 18.0902 1.43918
\(159\) 0.583592 0.0462819
\(160\) 0 0
\(161\) 0 0
\(162\) −3.90983 −0.307185
\(163\) −10.4721 −0.820241 −0.410120 0.912031i \(-0.634513\pi\)
−0.410120 + 0.912031i \(0.634513\pi\)
\(164\) −5.70820 −0.445736
\(165\) 0 0
\(166\) −9.23607 −0.716858
\(167\) −0.763932 −0.0591148 −0.0295574 0.999563i \(-0.509410\pi\)
−0.0295574 + 0.999563i \(0.509410\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 6.58359 0.503460
\(172\) 3.85410 0.293873
\(173\) 20.4721 1.55647 0.778234 0.627975i \(-0.216116\pi\)
0.778234 + 0.627975i \(0.216116\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −20.5623 −1.54994
\(177\) −2.11146 −0.158707
\(178\) −20.6525 −1.54797
\(179\) 3.41641 0.255354 0.127677 0.991816i \(-0.459248\pi\)
0.127677 + 0.991816i \(0.459248\pi\)
\(180\) 0 0
\(181\) 14.1803 1.05402 0.527008 0.849860i \(-0.323314\pi\)
0.527008 + 0.849860i \(0.323314\pi\)
\(182\) 0 0
\(183\) 4.58359 0.338829
\(184\) −3.94427 −0.290776
\(185\) 0 0
\(186\) −19.4164 −1.42368
\(187\) 27.4164 2.00489
\(188\) 1.23607 0.0901495
\(189\) 0 0
\(190\) 0 0
\(191\) −2.47214 −0.178877 −0.0894387 0.995992i \(-0.528507\pi\)
−0.0894387 + 0.995992i \(0.528507\pi\)
\(192\) −5.23607 −0.377881
\(193\) −14.4164 −1.03772 −0.518858 0.854861i \(-0.673643\pi\)
−0.518858 + 0.854861i \(0.673643\pi\)
\(194\) −1.23607 −0.0887445
\(195\) 0 0
\(196\) 0 0
\(197\) 7.47214 0.532368 0.266184 0.963922i \(-0.414237\pi\)
0.266184 + 0.963922i \(0.414237\pi\)
\(198\) −10.0902 −0.717077
\(199\) −2.76393 −0.195930 −0.0979650 0.995190i \(-0.531233\pi\)
−0.0979650 + 0.995190i \(0.531233\pi\)
\(200\) 0 0
\(201\) −0.291796 −0.0205817
\(202\) −14.9443 −1.05148
\(203\) 0 0
\(204\) −4.94427 −0.346168
\(205\) 0 0
\(206\) 0.763932 0.0532257
\(207\) −2.59675 −0.180486
\(208\) −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
\(213\) 5.81966 0.398757
\(214\) 12.9443 0.884852
\(215\) 0 0
\(216\) −12.3607 −0.841038
\(217\) 0 0
\(218\) −29.7984 −2.01820
\(219\) −16.3607 −1.10555
\(220\) 0 0
\(221\) 20.9443 1.40886
\(222\) −6.00000 −0.402694
\(223\) 2.18034 0.146006 0.0730032 0.997332i \(-0.476742\pi\)
0.0730032 + 0.997332i \(0.476742\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 20.0902 1.33638
\(227\) 5.41641 0.359500 0.179750 0.983712i \(-0.442471\pi\)
0.179750 + 0.983712i \(0.442471\pi\)
\(228\) 3.41641 0.226257
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 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 \(0.605612\pi\)
\(234\) −7.70820 −0.503901
\(235\) 0 0
\(236\) 1.05573 0.0687220
\(237\) −13.8197 −0.897683
\(238\) 0 0
\(239\) −14.4721 −0.936125 −0.468062 0.883695i \(-0.655048\pi\)
−0.468062 + 0.883695i \(0.655048\pi\)
\(240\) 0 0
\(241\) 12.4721 0.803401 0.401700 0.915771i \(-0.368419\pi\)
0.401700 + 0.915771i \(0.368419\pi\)
\(242\) 11.2361 0.722282
\(243\) −13.5967 −0.872232
\(244\) −2.29180 −0.146717
\(245\) 0 0
\(246\) 18.4721 1.17774
\(247\) −14.4721 −0.920840
\(248\) −21.7082 −1.37847
\(249\) 7.05573 0.447139
\(250\) 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 \(-0.697634\pi\)
−0.581755 + 0.813364i \(0.697634\pi\)
\(258\) −12.4721 −0.776481
\(259\) 0 0
\(260\) 0 0
\(261\) −7.36068 −0.455615
\(262\) −1.52786 −0.0943918
\(263\) 11.7639 0.725395 0.362698 0.931907i \(-0.381856\pi\)
0.362698 + 0.931907i \(0.381856\pi\)
\(264\) 11.7082 0.720590
\(265\) 0 0
\(266\) 0 0
\(267\) 15.7771 0.965542
\(268\) 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 \(-0.601197\pi\)
−0.312591 + 0.949888i \(0.601197\pi\)
\(272\) −31.4164 −1.90490
\(273\) 0 0
\(274\) 11.2361 0.678796
\(275\) 0 0
\(276\) −1.34752 −0.0811114
\(277\) 15.8885 0.954650 0.477325 0.878727i \(-0.341606\pi\)
0.477325 + 0.878727i \(0.341606\pi\)
\(278\) 33.4164 2.00418
\(279\) −14.2918 −0.855627
\(280\) 0 0
\(281\) 29.3607 1.75151 0.875756 0.482755i \(-0.160364\pi\)
0.875756 + 0.482755i \(0.160364\pi\)
\(282\) −4.00000 −0.238197
\(283\) 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
\(287\) 0 0
\(288\) 4.97871 0.293374
\(289\) 24.8885 1.46403
\(290\) 0 0
\(291\) 0.944272 0.0553542
\(292\) 8.18034 0.478718
\(293\) −9.12461 −0.533066 −0.266533 0.963826i \(-0.585878\pi\)
−0.266533 + 0.963826i \(0.585878\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.70820 −0.389906
\(297\) 23.4164 1.35876
\(298\) −22.5623 −1.30700
\(299\) 5.70820 0.330114
\(300\) 0 0
\(301\) 0 0
\(302\) −25.5066 −1.46774
\(303\) 11.4164 0.655855
\(304\) 21.7082 1.24505
\(305\) 0 0
\(306\) −15.4164 −0.881297
\(307\) −31.4164 −1.79303 −0.896515 0.443014i \(-0.853909\pi\)
−0.896515 + 0.443014i \(0.853909\pi\)
\(308\) 0 0
\(309\) −0.583592 −0.0331994
\(310\) 0 0
\(311\) 20.3607 1.15455 0.577274 0.816550i \(-0.304116\pi\)
0.577274 + 0.816550i \(0.304116\pi\)
\(312\) 8.94427 0.506370
\(313\) −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 \(0.682978\pi\)
\(318\) 0.944272 0.0529521
\(319\) 21.1803 1.18587
\(320\) 0 0
\(321\) −9.88854 −0.551925
\(322\) 0 0
\(323\) −28.9443 −1.61050
\(324\) −1.49342 −0.0829679
\(325\) 0 0
\(326\) −16.9443 −0.938456
\(327\) 22.7639 1.25885
\(328\) 20.6525 1.14034
\(329\) 0 0
\(330\) 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
\(333\) −4.41641 −0.242018
\(334\) −1.23607 −0.0676346
\(335\) 0 0
\(336\) 0 0
\(337\) −7.52786 −0.410069 −0.205034 0.978755i \(-0.565731\pi\)
−0.205034 + 0.978755i \(0.565731\pi\)
\(338\) −4.09017 −0.222476
\(339\) −15.3475 −0.833563
\(340\) 0 0
\(341\) 41.1246 2.22702
\(342\) 10.6525 0.576020
\(343\) 0 0
\(344\) −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 \(0.360933\pi\)
\(348\) −3.81966 −0.204755
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) −14.3262 −0.763591
\(353\) −20.1803 −1.07409 −0.537046 0.843553i \(-0.680460\pi\)
−0.537046 + 0.843553i \(0.680460\pi\)
\(354\) −3.41641 −0.181580
\(355\) 0 0
\(356\) −7.88854 −0.418092
\(357\) 0 0
\(358\) 5.52786 0.292157
\(359\) 10.1246 0.534357 0.267178 0.963647i \(-0.413909\pi\)
0.267178 + 0.963647i \(0.413909\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 22.9443 1.20592
\(363\) −8.58359 −0.450522
\(364\) 0 0
\(365\) 0 0
\(366\) 7.41641 0.387662
\(367\) −3.12461 −0.163103 −0.0815517 0.996669i \(-0.525988\pi\)
−0.0815517 + 0.996669i \(0.525988\pi\)
\(368\) −8.56231 −0.446341
\(369\) 13.5967 0.707818
\(370\) 0 0
\(371\) 0 0
\(372\) −7.41641 −0.384523
\(373\) 15.8328 0.819792 0.409896 0.912132i \(-0.365565\pi\)
0.409896 + 0.912132i \(0.365565\pi\)
\(374\) 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
\(381\) 21.8197 1.11786
\(382\) −4.00000 −0.204658
\(383\) 28.7639 1.46977 0.734884 0.678193i \(-0.237237\pi\)
0.734884 + 0.678193i \(0.237237\pi\)
\(384\) −16.8328 −0.858996
\(385\) 0 0
\(386\) −23.3262 −1.18727
\(387\) −9.18034 −0.466663
\(388\) −0.472136 −0.0239691
\(389\) −32.8885 −1.66752 −0.833758 0.552131i \(-0.813815\pi\)
−0.833758 + 0.552131i \(0.813815\pi\)
\(390\) 0 0
\(391\) 11.4164 0.577353
\(392\) 0 0
\(393\) 1.16718 0.0588767
\(394\) 12.0902 0.609094
\(395\) 0 0
\(396\) −3.85410 −0.193676
\(397\) −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 \(-0.407526\pi\)
0.286446 + 0.958097i \(0.407526\pi\)
\(402\) −0.472136 −0.0235480
\(403\) 31.4164 1.56496
\(404\) −5.70820 −0.283994
\(405\) 0 0
\(406\) 0 0
\(407\) 12.7082 0.629922
\(408\) 17.8885 0.885615
\(409\) −15.5279 −0.767803 −0.383902 0.923374i \(-0.625420\pi\)
−0.383902 + 0.923374i \(0.625420\pi\)
\(410\) 0 0
\(411\) −8.58359 −0.423397
\(412\) 0.291796 0.0143758
\(413\) 0 0
\(414\) −4.20163 −0.206499
\(415\) 0 0
\(416\) −10.9443 −0.536587
\(417\) −25.5279 −1.25010
\(418\) −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
\(423\) −2.94427 −0.143155
\(424\) 1.05573 0.0512707
\(425\) 0 0
\(426\) 9.41641 0.456226
\(427\) 0 0
\(428\) 4.94427 0.238990
\(429\) −16.9443 −0.818077
\(430\) 0 0
\(431\) 26.4721 1.27512 0.637559 0.770402i \(-0.279944\pi\)
0.637559 + 0.770402i \(0.279944\pi\)
\(432\) −26.8328 −1.29099
\(433\) −16.3607 −0.786244 −0.393122 0.919486i \(-0.628605\pi\)
−0.393122 + 0.919486i \(0.628605\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −11.3820 −0.545097
\(437\) −7.88854 −0.377360
\(438\) −26.4721 −1.26489
\(439\) 21.7082 1.03608 0.518038 0.855358i \(-0.326663\pi\)
0.518038 + 0.855358i \(0.326663\pi\)
\(440\) 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 \(0.443625\pi\)
\(444\) −2.29180 −0.108764
\(445\) 0 0
\(446\) 3.52786 0.167049
\(447\) 17.2361 0.815238
\(448\) 0 0
\(449\) 29.4721 1.39088 0.695438 0.718586i \(-0.255210\pi\)
0.695438 + 0.718586i \(0.255210\pi\)
\(450\) 0 0
\(451\) −39.1246 −1.84231
\(452\) 7.67376 0.360943
\(453\) 19.4853 0.915499
\(454\) 8.76393 0.411312
\(455\) 0 0
\(456\) −12.3607 −0.578842
\(457\) −21.4721 −1.00442 −0.502212 0.864744i \(-0.667480\pi\)
−0.502212 + 0.864744i \(0.667480\pi\)
\(458\) −7.23607 −0.338119
\(459\) 35.7771 1.66993
\(460\) 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 \(-0.330155\pi\)
0.508623 + 0.860989i \(0.330155\pi\)
\(464\) −24.2705 −1.12673
\(465\) 0 0
\(466\) −16.0902 −0.745363
\(467\) 10.9443 0.506441 0.253220 0.967409i \(-0.418510\pi\)
0.253220 + 0.967409i \(0.418510\pi\)
\(468\) −2.94427 −0.136099
\(469\) 0 0
\(470\) 0 0
\(471\) 6.47214 0.298220
\(472\) −3.81966 −0.175814
\(473\) 26.4164 1.21463
\(474\) −22.3607 −1.02706
\(475\) 0 0
\(476\) 0 0
\(477\) 0.695048 0.0318241
\(478\) −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
\(486\) −22.0000 −0.997940
\(487\) 10.2361 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(488\) 8.29180 0.375352
\(489\) 12.9443 0.585360
\(490\) 0 0
\(491\) −10.2361 −0.461947 −0.230974 0.972960i \(-0.574191\pi\)
−0.230974 + 0.972960i \(0.574191\pi\)
\(492\) 7.05573 0.318097
\(493\) 32.3607 1.45745
\(494\) −23.4164 −1.05355
\(495\) 0 0
\(496\) −47.1246 −2.11596
\(497\) 0 0
\(498\) 11.4164 0.511581
\(499\) 28.9443 1.29572 0.647862 0.761758i \(-0.275663\pi\)
0.647862 + 0.761758i \(0.275663\pi\)
\(500\) 0 0
\(501\) 0.944272 0.0421870
\(502\) 4.00000 0.178529
\(503\) 43.8885 1.95689 0.978447 0.206499i \(-0.0662072\pi\)
0.978447 + 0.206499i \(0.0662072\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12.0902 0.537474
\(507\) 3.12461 0.138769
\(508\) −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
\(511\) 0 0
\(512\) −5.29180 −0.233867
\(513\) −24.7214 −1.09147
\(514\) −30.1803 −1.33120
\(515\) 0 0
\(516\) −4.76393 −0.209720
\(517\) 8.47214 0.372604
\(518\) 0 0
\(519\) −25.3050 −1.11076
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) −11.9098 −0.521279
\(523\) 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
\(528\) 25.4164 1.10611
\(529\) −19.8885 −0.864719
\(530\) 0 0
\(531\) −2.51471 −0.109129
\(532\) 0 0
\(533\) −29.8885 −1.29462
\(534\) 25.5279 1.10470
\(535\) 0 0
\(536\) −0.527864 −0.0228003
\(537\) −4.22291 −0.182232
\(538\) −2.76393 −0.119162
\(539\) 0 0
\(540\) 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
\(543\) −17.5279 −0.752193
\(544\) −21.8885 −0.938464
\(545\) 0 0
\(546\) 0 0
\(547\) −14.2361 −0.608690 −0.304345 0.952562i \(-0.598438\pi\)
−0.304345 + 0.952562i \(0.598438\pi\)
\(548\) 4.29180 0.183336
\(549\) 5.45898 0.232984
\(550\) 0 0
\(551\) −22.3607 −0.952597
\(552\) 4.87539 0.207510
\(553\) 0 0
\(554\) 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 \(0.899936\pi\)
\(558\) −23.1246 −0.978943
\(559\) 20.1803 0.853537
\(560\) 0 0
\(561\) −33.8885 −1.43078
\(562\) 47.5066 2.00394
\(563\) 9.41641 0.396854 0.198427 0.980116i \(-0.436417\pi\)
0.198427 + 0.980116i \(0.436417\pi\)
\(564\) −1.52786 −0.0643347
\(565\) 0 0
\(566\) 15.2361 0.640420
\(567\) 0 0
\(568\) 10.5279 0.441739
\(569\) 13.9443 0.584574 0.292287 0.956331i \(-0.405584\pi\)
0.292287 + 0.956331i \(0.405584\pi\)
\(570\) 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
\(573\) 3.05573 0.127655
\(574\) 0 0
\(575\) 0 0
\(576\) −6.23607 −0.259836
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 40.2705 1.67503
\(579\) 17.8197 0.740560
\(580\) 0 0
\(581\) 0 0
\(582\) 1.52786 0.0633320
\(583\) −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 \(-0.293832\pi\)
0.603351 + 0.797476i \(0.293832\pi\)
\(588\) 0 0
\(589\) −43.4164 −1.78894
\(590\) 0 0
\(591\) −9.23607 −0.379921
\(592\) −14.5623 −0.598507
\(593\) −25.3050 −1.03915 −0.519575 0.854425i \(-0.673910\pi\)
−0.519575 + 0.854425i \(0.673910\pi\)
\(594\) 37.8885 1.55459
\(595\) 0 0
\(596\) −8.61803 −0.353008
\(597\) 3.41641 0.139824
\(598\) 9.23607 0.377691
\(599\) 11.1803 0.456816 0.228408 0.973565i \(-0.426648\pi\)
0.228408 + 0.973565i \(0.426648\pi\)
\(600\) 0 0
\(601\) 19.0557 0.777299 0.388650 0.921386i \(-0.372942\pi\)
0.388650 + 0.921386i \(0.372942\pi\)
\(602\) 0 0
\(603\) −0.347524 −0.0141523
\(604\) −9.74265 −0.396423
\(605\) 0 0
\(606\) 18.4721 0.750379
\(607\) −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
\(612\) −5.88854 −0.238030
\(613\) −17.5836 −0.710195 −0.355097 0.934829i \(-0.615552\pi\)
−0.355097 + 0.934829i \(0.615552\pi\)
\(614\) −50.8328 −2.05145
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9443 0.480858 0.240429 0.970667i \(-0.422712\pi\)
0.240429 + 0.970667i \(0.422712\pi\)
\(618\) −0.944272 −0.0379842
\(619\) −1.70820 −0.0686585 −0.0343293 0.999411i \(-0.510929\pi\)
−0.0343293 + 0.999411i \(0.510929\pi\)
\(620\) 0 0
\(621\) 9.75078 0.391285
\(622\) 32.9443 1.32094
\(623\) 0 0
\(624\) 19.4164 0.777278
\(625\) 0 0
\(626\) −46.0689 −1.84128
\(627\) 23.4164 0.935161
\(628\) −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
\(633\) −14.8328 −0.589551
\(634\) −31.3262 −1.24412
\(635\) 0 0
\(636\) 0.360680 0.0143019
\(637\) 0 0
\(638\) 34.2705 1.35678
\(639\) 6.93112 0.274191
\(640\) 0 0
\(641\) −9.83282 −0.388373 −0.194186 0.980965i \(-0.562207\pi\)
−0.194186 + 0.980965i \(0.562207\pi\)
\(642\) −16.0000 −0.631470
\(643\) −9.52786 −0.375742 −0.187871 0.982194i \(-0.560159\pi\)
−0.187871 + 0.982194i \(0.560159\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −46.8328 −1.84261
\(647\) −15.8885 −0.624643 −0.312322 0.949976i \(-0.601107\pi\)
−0.312322 + 0.949976i \(0.601107\pi\)
\(648\) 5.40325 0.212260
\(649\) 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 \(0.182397\pi\)
\(654\) 36.8328 1.44028
\(655\) 0 0
\(656\) 44.8328 1.75043
\(657\) −19.4853 −0.760194
\(658\) 0 0
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 0 0
\(661\) −46.7214 −1.81725 −0.908625 0.417613i \(-0.862867\pi\)
−0.908625 + 0.417613i \(0.862867\pi\)
\(662\) −18.2705 −0.710104
\(663\) −25.8885 −1.00543
\(664\) 12.7639 0.495337
\(665\) 0 0
\(666\) −7.14590 −0.276898
\(667\) 8.81966 0.341499
\(668\) −0.472136 −0.0182675
\(669\) −2.69505 −0.104197
\(670\) 0 0
\(671\) −15.7082 −0.606408
\(672\) 0 0
\(673\) 28.4721 1.09752 0.548760 0.835980i \(-0.315100\pi\)
0.548760 + 0.835980i \(0.315100\pi\)
\(674\) −12.1803 −0.469169
\(675\) 0 0
\(676\) −1.56231 −0.0600887
\(677\) −30.3607 −1.16686 −0.583428 0.812165i \(-0.698289\pi\)
−0.583428 + 0.812165i \(0.698289\pi\)
\(678\) −24.8328 −0.953699
\(679\) 0 0
\(680\) 0 0
\(681\) −6.69505 −0.256555
\(682\) 66.5410 2.54799
\(683\) −26.1246 −0.999630 −0.499815 0.866132i \(-0.666599\pi\)
−0.499815 + 0.866132i \(0.666599\pi\)
\(684\) 4.06888 0.155578
\(685\) 0 0
\(686\) 0 0
\(687\) 5.52786 0.210901
\(688\) −30.2705 −1.15405
\(689\) −1.52786 −0.0582070
\(690\) 0 0
\(691\) −18.1803 −0.691613 −0.345806 0.938306i \(-0.612395\pi\)
−0.345806 + 0.938306i \(0.612395\pi\)
\(692\) 12.6525 0.480975
\(693\) 0 0
\(694\) 25.5066 0.968216
\(695\) 0 0
\(696\) 13.8197 0.523833
\(697\) −59.7771 −2.26422
\(698\) 7.23607 0.273889
\(699\) 12.2918 0.464918
\(700\) 0 0
\(701\) −46.9443 −1.77306 −0.886530 0.462670i \(-0.846891\pi\)
−0.886530 + 0.462670i \(0.846891\pi\)
\(702\) 28.9443 1.09243
\(703\) −13.4164 −0.506009
\(704\) 17.9443 0.676300
\(705\) 0 0
\(706\) −32.6525 −1.22889
\(707\) 0 0
\(708\) −1.30495 −0.0490431
\(709\) −47.8885 −1.79849 −0.899246 0.437443i \(-0.855884\pi\)
−0.899246 + 0.437443i \(0.855884\pi\)
\(710\) 0 0
\(711\) −16.4590 −0.617260
\(712\) 28.5410 1.06962
\(713\) 17.1246 0.641322
\(714\) 0 0
\(715\) 0 0
\(716\) 2.11146 0.0789088
\(717\) 17.8885 0.668060
\(718\) 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 0
\(722\) 1.61803 0.0602170
\(723\) −15.4164 −0.573342
\(724\) 8.76393 0.325709
\(725\) 0 0
\(726\) −13.8885 −0.515452
\(727\) 20.9443 0.776780 0.388390 0.921495i \(-0.373031\pi\)
0.388390 + 0.921495i \(0.373031\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) 40.3607 1.49279
\(732\) 2.83282 0.104704
\(733\) −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
\(738\) 22.0000 0.809831
\(739\) −5.65248 −0.207930 −0.103965 0.994581i \(-0.533153\pi\)
−0.103965 + 0.994581i \(0.533153\pi\)
\(740\) 0 0
\(741\) 17.8885 0.657152
\(742\) 0 0
\(743\) −1.52786 −0.0560519 −0.0280259 0.999607i \(-0.508922\pi\)
−0.0280259 + 0.999607i \(0.508922\pi\)
\(744\) 26.8328 0.983739
\(745\) 0 0
\(746\) 25.6180 0.937943
\(747\) 8.40325 0.307459
\(748\) 16.9443 0.619544
\(749\) 0 0
\(750\) 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
\(753\) −3.05573 −0.111357
\(754\) 26.1803 0.953432
\(755\) 0 0
\(756\) 0 0
\(757\) 46.4164 1.68703 0.843517 0.537103i \(-0.180481\pi\)
0.843517 + 0.537103i \(0.180481\pi\)
\(758\) 18.0902 0.657065
\(759\) −9.23607 −0.335248
\(760\) 0 0
\(761\) 43.7771 1.58692 0.793459 0.608624i \(-0.208278\pi\)
0.793459 + 0.608624i \(0.208278\pi\)
\(762\) 35.3050 1.27896
\(763\) 0 0
\(764\) −1.52786 −0.0552762
\(765\) 0 0
\(766\) 46.5410 1.68160
\(767\) 5.52786 0.199600
\(768\) −16.7639 −0.604916
\(769\) −33.0132 −1.19048 −0.595242 0.803546i \(-0.702944\pi\)
−0.595242 + 0.803546i \(0.702944\pi\)
\(770\) 0 0
\(771\) 23.0557 0.830332
\(772\) −8.90983 −0.320672
\(773\) −27.8197 −1.00060 −0.500302 0.865851i \(-0.666778\pi\)
−0.500302 + 0.865851i \(0.666778\pi\)
\(774\) −14.8541 −0.533920
\(775\) 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
\(783\) 27.6393 0.987749
\(784\) 0 0
\(785\) 0 0
\(786\) 1.88854 0.0673621
\(787\) −45.2361 −1.61249 −0.806246 0.591581i \(-0.798504\pi\)
−0.806246 + 0.591581i \(0.798504\pi\)
\(788\) 4.61803 0.164511
\(789\) −14.5410 −0.517674
\(790\) 0 0
\(791\) 0 0
\(792\) 13.9443 0.495488
\(793\) −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 \(-0.451421\pi\)
0.152023 + 0.988377i \(0.451421\pi\)
\(798\) 0 0
\(799\) 12.9443 0.457935
\(800\) 0 0
\(801\) 18.7902 0.663921
\(802\) 18.5623 0.655458
\(803\) 56.0689 1.97863
\(804\) −0.180340 −0.00636010
\(805\) 0 0
\(806\) 50.8328 1.79051
\(807\) 2.11146 0.0743268
\(808\) 20.6525 0.726552
\(809\) 20.5279 0.721721 0.360861 0.932620i \(-0.382483\pi\)
0.360861 + 0.932620i \(0.382483\pi\)
\(810\) 0 0
\(811\) −46.7214 −1.64061 −0.820304 0.571927i \(-0.806196\pi\)
−0.820304 + 0.571927i \(0.806196\pi\)
\(812\) 0 0
\(813\) 12.7214 0.446158
\(814\) 20.5623 0.720708
\(815\) 0 0
\(816\) 38.8328 1.35942
\(817\) −27.8885 −0.975697
\(818\) −25.1246 −0.878461
\(819\) 0 0
\(820\) 0 0
\(821\) −24.8328 −0.866671 −0.433336 0.901233i \(-0.642664\pi\)
−0.433336 + 0.901233i \(0.642664\pi\)
\(822\) −13.8885 −0.484419
\(823\) −0.347524 −0.0121139 −0.00605697 0.999982i \(-0.501928\pi\)
−0.00605697 + 0.999982i \(0.501928\pi\)
\(824\) −1.05573 −0.0367780
\(825\) 0 0
\(826\) 0 0
\(827\) −25.5410 −0.888148 −0.444074 0.895990i \(-0.646467\pi\)
−0.444074 + 0.895990i \(0.646467\pi\)
\(828\) −1.60488 −0.0557734
\(829\) 52.3607 1.81856 0.909281 0.416183i \(-0.136632\pi\)
0.909281 + 0.416183i \(0.136632\pi\)
\(830\) 0 0
\(831\) −19.6393 −0.681280
\(832\) 13.7082 0.475246
\(833\) 0 0
\(834\) −41.3050 −1.43027
\(835\) 0 0
\(836\) −11.7082 −0.404937
\(837\) 53.6656 1.85496
\(838\) 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
\(843\) −36.2918 −1.24996
\(844\) 7.41641 0.255283
\(845\) 0 0
\(846\) −4.76393 −0.163787
\(847\) 0 0
\(848\) 2.29180 0.0787006
\(849\) −11.6393 −0.399460
\(850\) 0 0
\(851\) 5.29180 0.181400
\(852\) 3.59675 0.123223
\(853\) −0.583592 −0.0199818 −0.00999091 0.999950i \(-0.503180\pi\)
−0.00999091 + 0.999950i \(0.503180\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −17.8885 −0.611418
\(857\) 38.1803 1.30422 0.652108 0.758126i \(-0.273885\pi\)
0.652108 + 0.758126i \(0.273885\pi\)
\(858\) −27.4164 −0.935981
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 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 \(0.179544\pi\)
\(864\) −18.6950 −0.636018
\(865\) 0 0
\(866\) −26.4721 −0.899560
\(867\) −30.7639 −1.04480
\(868\) 0 0
\(869\) 47.3607 1.60660
\(870\) 0 0
\(871\) 0.763932 0.0258848
\(872\) 41.1803 1.39454
\(873\) 1.12461 0.0380623
\(874\) −12.7639 −0.431746
\(875\) 0 0
\(876\) −10.1115 −0.341634
\(877\) −14.3607 −0.484926 −0.242463 0.970161i \(-0.577955\pi\)
−0.242463 + 0.970161i \(0.577955\pi\)
\(878\) 35.1246 1.18540
\(879\) 11.2786 0.380419
\(880\) 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 \(-0.824220\pi\)
−0.851358 + 0.524585i \(0.824220\pi\)
\(884\) 12.9443 0.435363
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 52.6525 1.76790 0.883949 0.467584i \(-0.154876\pi\)
0.883949 + 0.467584i \(0.154876\pi\)
\(888\) 8.29180 0.278254
\(889\) 0 0
\(890\) 0 0
\(891\) −10.2361 −0.342921
\(892\) 1.34752 0.0451184
\(893\) −8.94427 −0.299309
\(894\) 27.8885 0.932732
\(895\) 0 0
\(896\) 0 0
\(897\) −7.05573 −0.235584
\(898\) 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
\(906\) 31.5279 1.04744
\(907\) −18.8328 −0.625333 −0.312667 0.949863i \(-0.601222\pi\)
−0.312667 + 0.949863i \(0.601222\pi\)
\(908\) 3.34752 0.111091
\(909\) 13.5967 0.450976
\(910\) 0 0
\(911\) 23.1803 0.767999 0.383999 0.923333i \(-0.374546\pi\)
0.383999 + 0.923333i \(0.374546\pi\)
\(912\) −26.8328 −0.888523
\(913\) −24.1803 −0.800252
\(914\) −34.7426 −1.14918
\(915\) 0 0
\(916\) −2.76393 −0.0913229
\(917\) 0 0
\(918\) 57.8885 1.91061
\(919\) −32.2361 −1.06337 −0.531685 0.846942i \(-0.678441\pi\)
−0.531685 + 0.846942i \(0.678441\pi\)
\(920\) 0 0
\(921\) 38.8328 1.27958
\(922\) −13.2361 −0.435907
\(923\) −15.2361 −0.501501
\(924\) 0 0
\(925\) 0 0
\(926\) 35.4164 1.16386
\(927\) −0.695048 −0.0228284
\(928\) −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
\(931\) 0 0
\(932\) −6.14590 −0.201316
\(933\) −25.1672 −0.823937
\(934\) 17.7082 0.579430
\(935\) 0 0
\(936\) 10.6525 0.348187
\(937\) −30.7639 −1.00501 −0.502507 0.864573i \(-0.667589\pi\)
−0.502507 + 0.864573i \(0.667589\pi\)
\(938\) 0 0
\(939\) 35.1935 1.14850
\(940\) 0 0
\(941\) 0.763932 0.0249035 0.0124517 0.999922i \(-0.496036\pi\)
0.0124517 + 0.999922i \(0.496036\pi\)
\(942\) 10.4721 0.341201
\(943\) −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 \(0.598988\pi\)
\(948\) −8.54102 −0.277399
\(949\) 42.8328 1.39041
\(950\) 0 0
\(951\) 23.9311 0.776020
\(952\) 0 0
\(953\) −5.47214 −0.177260 −0.0886299 0.996065i \(-0.528249\pi\)
−0.0886299 + 0.996065i \(0.528249\pi\)
\(954\) 1.12461 0.0364107
\(955\) 0 0
\(956\) −8.94427 −0.289278
\(957\) −26.1803 −0.846290
\(958\) 6.18034 0.199678
\(959\) 0 0
\(960\) 0 0
\(961\) 63.2492 2.04030
\(962\) 15.7082 0.506453
\(963\) −11.7771 −0.379511
\(964\) 7.70820 0.248265
\(965\) 0 0
\(966\) 0 0
\(967\) −49.8885 −1.60431 −0.802154 0.597118i \(-0.796313\pi\)
−0.802154 + 0.597118i \(0.796313\pi\)
\(968\) −15.5279 −0.499084
\(969\) 35.7771 1.14933
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) −8.40325 −0.269534
\(973\) 0 0
\(974\) 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 \(0.512875\pi\)
\(978\) 20.9443 0.669724
\(979\) −54.0689 −1.72805
\(980\) 0 0
\(981\) 27.1115 0.865602
\(982\) −16.5623 −0.528524
\(983\) −32.5410 −1.03790 −0.518949 0.854805i \(-0.673676\pi\)
−0.518949 + 0.854805i \(0.673676\pi\)
\(984\) −25.5279 −0.813799
\(985\) 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
\(993\) 13.9574 0.442926
\(994\) 0 0
\(995\) 0 0
\(996\) 4.36068 0.138173
\(997\) 18.5836 0.588548 0.294274 0.955721i \(-0.404922\pi\)
0.294274 + 0.955721i \(0.404922\pi\)
\(998\) 46.8328 1.48247
\(999\) 16.5836 0.524682
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 1225.2.a.u.1.2 2
5.2 odd 4 1225.2.b.k.99.4 4
5.3 odd 4 1225.2.b.k.99.1 4
5.4 even 2 1225.2.a.n.1.1 2
7.6 odd 2 175.2.a.e.1.2 yes 2
21.20 even 2 1575.2.a.n.1.1 2
28.27 even 2 2800.2.a.bp.1.1 2
35.13 even 4 175.2.b.c.99.1 4
35.27 even 4 175.2.b.c.99.4 4
35.34 odd 2 175.2.a.d.1.1 2
105.62 odd 4 1575.2.d.k.1324.1 4
105.83 odd 4 1575.2.d.k.1324.4 4
105.104 even 2 1575.2.a.s.1.2 2
140.27 odd 4 2800.2.g.s.449.3 4
140.83 odd 4 2800.2.g.s.449.2 4
140.139 even 2 2800.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 35.34 odd 2
175.2.a.e.1.2 yes 2 7.6 odd 2
175.2.b.c.99.1 4 35.13 even 4
175.2.b.c.99.4 4 35.27 even 4
1225.2.a.n.1.1 2 5.4 even 2
1225.2.a.u.1.2 2 1.1 even 1 trivial
1225.2.b.k.99.1 4 5.3 odd 4
1225.2.b.k.99.4 4 5.2 odd 4
1575.2.a.n.1.1 2 21.20 even 2
1575.2.a.s.1.2 2 105.104 even 2
1575.2.d.k.1324.1 4 105.62 odd 4
1575.2.d.k.1324.4 4 105.83 odd 4
2800.2.a.bh.1.2 2 140.139 even 2
2800.2.a.bp.1.1 2 28.27 even 2
2800.2.g.s.449.2 4 140.83 odd 4
2800.2.g.s.449.3 4 140.27 odd 4