Properties

Label 1575.2.d.k.1324.1
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.k.1324.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803i q^{2} -0.618034 q^{4} +1.00000i q^{7} -2.23607i q^{8} +O(q^{10})\) \(q-1.61803i q^{2} -0.618034 q^{4} +1.00000i q^{7} -2.23607i q^{8} -4.23607 q^{11} +3.23607i q^{13} +1.61803 q^{14} -4.85410 q^{16} +6.47214i q^{17} -4.47214 q^{19} +6.85410i q^{22} +1.76393i q^{23} +5.23607 q^{26} -0.618034i q^{28} +5.00000 q^{29} -9.70820 q^{31} +3.38197i q^{32} +10.4721 q^{34} +3.00000i q^{37} +7.23607i q^{38} -9.23607 q^{41} -6.23607i q^{43} +2.61803 q^{44} +2.85410 q^{46} +2.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} -0.472136i q^{53} +2.23607 q^{56} -8.09017i q^{58} -1.70820 q^{59} +3.70820 q^{61} +15.7082i q^{62} -4.23607 q^{64} +0.236068i q^{67} -4.00000i q^{68} +4.70820 q^{71} +13.2361i q^{73} +4.85410 q^{74} +2.76393 q^{76} -4.23607i q^{77} -11.1803 q^{79} +14.9443i q^{82} +5.70820i q^{83} -10.0902 q^{86} +9.47214i q^{88} +12.7639 q^{89} -3.23607 q^{91} -1.09017i q^{92} +3.23607 q^{94} +0.763932i q^{97} +1.61803i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{11} + 2q^{14} - 6q^{16} + 12q^{26} + 20q^{29} - 12q^{31} + 24q^{34} - 28q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 20q^{59} - 12q^{61} - 8q^{64} - 8q^{71} + 6q^{74} + 20q^{76} - 18q^{86} + 60q^{89} - 4q^{91} + 4q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.61803i − 1.14412i −0.820211 0.572061i \(-0.806144\pi\)
0.820211 0.572061i \(-0.193856\pi\)
\(3\) 0 0
\(4\) −0.618034 −0.309017
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) − 2.23607i − 0.790569i
\(9\) 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
\(13\) 3.23607i 0.897524i 0.893651 + 0.448762i \(0.148135\pi\)
−0.893651 + 0.448762i \(0.851865\pi\)
\(14\) 1.61803 0.432438
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 6.47214i 1.56972i 0.619671 + 0.784862i \(0.287266\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(18\) 0 0
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.85410i 1.46130i
\(23\) 1.76393i 0.367805i 0.982944 + 0.183903i \(0.0588731\pi\)
−0.982944 + 0.183903i \(0.941127\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.23607 1.02688
\(27\) 0 0
\(28\) − 0.618034i − 0.116797i
\(29\) 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 \(-0.837062\pi\)
−0.871822 + 0.489822i \(0.837062\pi\)
\(32\) 3.38197i 0.597853i
\(33\) 0 0
\(34\) 10.4721 1.79596
\(35\) 0 0
\(36\) 0 0
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 7.23607i 1.17385i
\(39\) 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
\(43\) − 6.23607i − 0.950991i −0.879718 0.475496i \(-0.842269\pi\)
0.879718 0.475496i \(-0.157731\pi\)
\(44\) 2.61803 0.394683
\(45\) 0 0
\(46\) 2.85410 0.420814
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) − 0.472136i − 0.0648529i −0.999474 0.0324264i \(-0.989677\pi\)
0.999474 0.0324264i \(-0.0103235\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) − 8.09017i − 1.06229i
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) 3.70820 0.474787 0.237393 0.971414i \(-0.423707\pi\)
0.237393 + 0.971414i \(0.423707\pi\)
\(62\) 15.7082i 1.99494i
\(63\) 0 0
\(64\) −4.23607 −0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) 0.236068i 0.0288403i 0.999896 + 0.0144201i \(0.00459023\pi\)
−0.999896 + 0.0144201i \(0.995410\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 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
\(73\) 13.2361i 1.54916i 0.632473 + 0.774582i \(0.282040\pi\)
−0.632473 + 0.774582i \(0.717960\pi\)
\(74\) 4.85410 0.564278
\(75\) 0 0
\(76\) 2.76393 0.317045
\(77\) − 4.23607i − 0.482745i
\(78\) 0 0
\(79\) −11.1803 −1.25789 −0.628943 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 14.9443i 1.65032i
\(83\) 5.70820i 0.626557i 0.949661 + 0.313278i \(0.101427\pi\)
−0.949661 + 0.313278i \(0.898573\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −10.0902 −1.08805
\(87\) 0 0
\(88\) 9.47214i 1.00973i
\(89\) 12.7639 1.35297 0.676487 0.736455i \(-0.263501\pi\)
0.676487 + 0.736455i \(0.263501\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) − 1.09017i − 0.113658i
\(93\) 0 0
\(94\) 3.23607 0.333775
\(95\) 0 0
\(96\) 0 0
\(97\) 0.763932i 0.0775655i 0.999248 + 0.0387828i \(0.0123480\pi\)
−0.999248 + 0.0387828i \(0.987652\pi\)
\(98\) 1.61803i 0.163446i
\(99\) 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
\(103\) 0.472136i 0.0465209i 0.999729 + 0.0232605i \(0.00740471\pi\)
−0.999729 + 0.0232605i \(0.992595\pi\)
\(104\) 7.23607 0.709555
\(105\) 0 0
\(106\) −0.763932 −0.0741996
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 18.4164 1.76397 0.881986 0.471276i \(-0.156206\pi\)
0.881986 + 0.471276i \(0.156206\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 4.85410i − 0.458670i
\(113\) 12.4164i 1.16804i 0.811740 + 0.584019i \(0.198521\pi\)
−0.811740 + 0.584019i \(0.801479\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.09017 −0.286915
\(117\) 0 0
\(118\) 2.76393i 0.254441i
\(119\) −6.47214 −0.593300
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) − 6.00000i − 0.543214i
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) − 17.6525i − 1.56640i −0.621767 0.783202i \(-0.713585\pi\)
0.621767 0.783202i \(-0.286415\pi\)
\(128\) 13.6180i 1.20368i
\(129\) 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
\(133\) − 4.47214i − 0.387783i
\(134\) 0.381966 0.0329968
\(135\) 0 0
\(136\) 14.4721 1.24098
\(137\) − 6.94427i − 0.593289i −0.954988 0.296645i \(-0.904132\pi\)
0.954988 0.296645i \(-0.0958677\pi\)
\(138\) 0 0
\(139\) 20.6525 1.75172 0.875860 0.482565i \(-0.160295\pi\)
0.875860 + 0.482565i \(0.160295\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 7.61803i − 0.639291i
\(143\) − 13.7082i − 1.14634i
\(144\) 0 0
\(145\) 0 0
\(146\) 21.4164 1.77243
\(147\) 0 0
\(148\) − 1.85410i − 0.152406i
\(149\) −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 \(0.721657\pi\)
\(152\) 10.0000i 0.811107i
\(153\) 0 0
\(154\) −6.85410 −0.552319
\(155\) 0 0
\(156\) 0 0
\(157\) 5.23607i 0.417884i 0.977928 + 0.208942i \(0.0670019\pi\)
−0.977928 + 0.208942i \(0.932998\pi\)
\(158\) 18.0902i 1.43918i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.76393 −0.139017
\(162\) 0 0
\(163\) 10.4721i 0.820241i 0.912031 + 0.410120i \(0.134513\pi\)
−0.912031 + 0.410120i \(0.865487\pi\)
\(164\) 5.70820 0.445736
\(165\) 0 0
\(166\) 9.23607 0.716858
\(167\) − 0.763932i − 0.0591148i −0.999563 0.0295574i \(-0.990590\pi\)
0.999563 0.0295574i \(-0.00940979\pi\)
\(168\) 0 0
\(169\) 2.52786 0.194451
\(170\) 0 0
\(171\) 0 0
\(172\) 3.85410i 0.293873i
\(173\) − 20.4721i − 1.55647i −0.627975 0.778234i \(-0.716116\pi\)
0.627975 0.778234i \(-0.283884\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 20.5623 1.54994
\(177\) 0 0
\(178\) − 20.6525i − 1.54797i
\(179\) 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 \(-0.676686\pi\)
−0.527008 + 0.849860i \(0.676686\pi\)
\(182\) 5.23607i 0.388123i
\(183\) 0 0
\(184\) 3.94427 0.290776
\(185\) 0 0
\(186\) 0 0
\(187\) − 27.4164i − 2.00489i
\(188\) − 1.23607i − 0.0901495i
\(189\) 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
\(193\) 14.4164i 1.03772i 0.854861 + 0.518858i \(0.173643\pi\)
−0.854861 + 0.518858i \(0.826357\pi\)
\(194\) 1.23607 0.0887445
\(195\) 0 0
\(196\) 0.618034 0.0441453
\(197\) − 7.47214i − 0.532368i −0.963922 0.266184i \(-0.914237\pi\)
0.963922 0.266184i \(-0.0857628\pi\)
\(198\) 0 0
\(199\) −2.76393 −0.195930 −0.0979650 0.995190i \(-0.531233\pi\)
−0.0979650 + 0.995190i \(0.531233\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.9443i 1.05148i
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0.763932 0.0532257
\(207\) 0 0
\(208\) − 15.7082i − 1.08917i
\(209\) 18.9443 1.31040
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0.291796i 0.0200406i
\(213\) 0 0
\(214\) −12.9443 −0.884852
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.70820i − 0.659036i
\(218\) − 29.7984i − 2.01820i
\(219\) 0 0
\(220\) 0 0
\(221\) −20.9443 −1.40886
\(222\) 0 0
\(223\) 2.18034i 0.146006i 0.997332 + 0.0730032i \(0.0232583\pi\)
−0.997332 + 0.0730032i \(0.976742\pi\)
\(224\) −3.38197 −0.225967
\(225\) 0 0
\(226\) 20.0902 1.33638
\(227\) 5.41641i 0.359500i 0.983712 + 0.179750i \(0.0575288\pi\)
−0.983712 + 0.179750i \(0.942471\pi\)
\(228\) 0 0
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 11.1803i − 0.734025i
\(233\) − 9.94427i − 0.651471i −0.945461 0.325735i \(-0.894388\pi\)
0.945461 0.325735i \(-0.105612\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.05573 0.0687220
\(237\) 0 0
\(238\) 10.4721i 0.678808i
\(239\) −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 \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) − 11.2361i − 0.722282i
\(243\) 0 0
\(244\) −2.29180 −0.146717
\(245\) 0 0
\(246\) 0 0
\(247\) − 14.4721i − 0.920840i
\(248\) 21.7082i 1.37847i
\(249\) 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
\(253\) − 7.47214i − 0.469769i
\(254\) −28.5623 −1.79216
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) − 18.6525i − 1.16351i −0.813364 0.581755i \(-0.802366\pi\)
0.813364 0.581755i \(-0.197634\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 1.52786i 0.0943918i
\(263\) 11.7639i 0.725395i 0.931907 + 0.362698i \(0.118144\pi\)
−0.931907 + 0.362698i \(0.881856\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −7.23607 −0.443672
\(267\) 0 0
\(268\) − 0.145898i − 0.00891214i
\(269\) 1.70820 0.104151 0.0520755 0.998643i \(-0.483416\pi\)
0.0520755 + 0.998643i \(0.483416\pi\)
\(270\) 0 0
\(271\) 10.2918 0.625182 0.312591 0.949888i \(-0.398803\pi\)
0.312591 + 0.949888i \(0.398803\pi\)
\(272\) − 31.4164i − 1.90490i
\(273\) 0 0
\(274\) −11.2361 −0.678796
\(275\) 0 0
\(276\) 0 0
\(277\) 15.8885i 0.954650i 0.878727 + 0.477325i \(0.158394\pi\)
−0.878727 + 0.477325i \(0.841606\pi\)
\(278\) − 33.4164i − 2.00418i
\(279\) 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
\(283\) 9.41641i 0.559747i 0.960037 + 0.279874i \(0.0902926\pi\)
−0.960037 + 0.279874i \(0.909707\pi\)
\(284\) −2.90983 −0.172667
\(285\) 0 0
\(286\) −22.1803 −1.31155
\(287\) − 9.23607i − 0.545188i
\(288\) 0 0
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) 0 0
\(292\) − 8.18034i − 0.478718i
\(293\) 9.12461i 0.533066i 0.963826 + 0.266533i \(0.0858781\pi\)
−0.963826 + 0.266533i \(0.914122\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.70820 0.389906
\(297\) 0 0
\(298\) 22.5623i 1.30700i
\(299\) −5.70820 −0.330114
\(300\) 0 0
\(301\) 6.23607 0.359441
\(302\) 25.5066i 1.46774i
\(303\) 0 0
\(304\) 21.7082 1.24505
\(305\) 0 0
\(306\) 0 0
\(307\) 31.4164i 1.79303i 0.443014 + 0.896515i \(0.353909\pi\)
−0.443014 + 0.896515i \(0.646091\pi\)
\(308\) 2.61803i 0.149176i
\(309\) 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
\(313\) − 28.4721i − 1.60934i −0.593722 0.804670i \(-0.702342\pi\)
0.593722 0.804670i \(-0.297658\pi\)
\(314\) 8.47214 0.478110
\(315\) 0 0
\(316\) 6.90983 0.388708
\(317\) 19.3607i 1.08740i 0.839278 + 0.543702i \(0.182978\pi\)
−0.839278 + 0.543702i \(0.817022\pi\)
\(318\) 0 0
\(319\) −21.1803 −1.18587
\(320\) 0 0
\(321\) 0 0
\(322\) 2.85410i 0.159053i
\(323\) − 28.9443i − 1.61050i
\(324\) 0 0
\(325\) 0 0
\(326\) 16.9443 0.938456
\(327\) 0 0
\(328\) 20.6525i 1.14034i
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) −11.2918 −0.620653 −0.310327 0.950630i \(-0.600438\pi\)
−0.310327 + 0.950630i \(0.600438\pi\)
\(332\) − 3.52786i − 0.193617i
\(333\) 0 0
\(334\) −1.23607 −0.0676346
\(335\) 0 0
\(336\) 0 0
\(337\) − 7.52786i − 0.410069i −0.978755 0.205034i \(-0.934269\pi\)
0.978755 0.205034i \(-0.0657306\pi\)
\(338\) − 4.09017i − 0.222476i
\(339\) 0 0
\(340\) 0 0
\(341\) 41.1246 2.22702
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −13.9443 −0.751825
\(345\) 0 0
\(346\) −33.1246 −1.78079
\(347\) − 15.7639i − 0.846252i −0.906071 0.423126i \(-0.860933\pi\)
0.906071 0.423126i \(-0.139067\pi\)
\(348\) 0 0
\(349\) 4.47214 0.239388 0.119694 0.992811i \(-0.461809\pi\)
0.119694 + 0.992811i \(0.461809\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 14.3262i − 0.763591i
\(353\) 20.1803i 1.07409i 0.843553 + 0.537046i \(0.180460\pi\)
−0.843553 + 0.537046i \(0.819540\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.88854 −0.418092
\(357\) 0 0
\(358\) − 5.52786i − 0.292157i
\(359\) 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
\(362\) 22.9443i 1.20592i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 3.12461i 0.163103i 0.996669 + 0.0815517i \(0.0259876\pi\)
−0.996669 + 0.0815517i \(0.974012\pi\)
\(368\) − 8.56231i − 0.446341i
\(369\) 0 0
\(370\) 0 0
\(371\) 0.472136 0.0245121
\(372\) 0 0
\(373\) − 15.8328i − 0.819792i −0.912132 0.409896i \(-0.865565\pi\)
0.912132 0.409896i \(-0.134435\pi\)
\(374\) −44.3607 −2.29384
\(375\) 0 0
\(376\) 4.47214 0.230633
\(377\) 16.1803i 0.833330i
\(378\) 0 0
\(379\) −11.1803 −0.574295 −0.287148 0.957886i \(-0.592707\pi\)
−0.287148 + 0.957886i \(0.592707\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 4.00000i − 0.204658i
\(383\) − 28.7639i − 1.46977i −0.678193 0.734884i \(-0.737237\pi\)
0.678193 0.734884i \(-0.262763\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.3262 1.18727
\(387\) 0 0
\(388\) − 0.472136i − 0.0239691i
\(389\) −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
\(392\) 2.23607i 0.112938i
\(393\) 0 0
\(394\) −12.0902 −0.609094
\(395\) 0 0
\(396\) 0 0
\(397\) 26.9443i 1.35229i 0.736767 + 0.676147i \(0.236352\pi\)
−0.736767 + 0.676147i \(0.763648\pi\)
\(398\) 4.47214i 0.224168i
\(399\) 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
\(403\) − 31.4164i − 1.56496i
\(404\) 5.70820 0.283994
\(405\) 0 0
\(406\) 8.09017 0.401508
\(407\) − 12.7082i − 0.629922i
\(408\) 0 0
\(409\) −15.5279 −0.767803 −0.383902 0.923374i \(-0.625420\pi\)
−0.383902 + 0.923374i \(0.625420\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 0.291796i − 0.0143758i
\(413\) − 1.70820i − 0.0840552i
\(414\) 0 0
\(415\) 0 0
\(416\) −10.9443 −0.536587
\(417\) 0 0
\(418\) − 30.6525i − 1.49926i
\(419\) −3.81966 −0.186603 −0.0933013 0.995638i \(-0.529742\pi\)
−0.0933013 + 0.995638i \(0.529742\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) − 19.4164i − 0.945176i
\(423\) 0 0
\(424\) −1.05573 −0.0512707
\(425\) 0 0
\(426\) 0 0
\(427\) 3.70820i 0.179453i
\(428\) 4.94427i 0.238990i
\(429\) 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
\(433\) − 16.3607i − 0.786244i −0.919486 0.393122i \(-0.871395\pi\)
0.919486 0.393122i \(-0.128605\pi\)
\(434\) −15.7082 −0.754018
\(435\) 0 0
\(436\) −11.3820 −0.545097
\(437\) − 7.88854i − 0.377360i
\(438\) 0 0
\(439\) 21.7082 1.03608 0.518038 0.855358i \(-0.326663\pi\)
0.518038 + 0.855358i \(0.326663\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 33.8885i 1.61191i
\(443\) 7.41641i 0.352364i 0.984358 + 0.176182i \(0.0563748\pi\)
−0.984358 + 0.176182i \(0.943625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.52786 0.167049
\(447\) 0 0
\(448\) − 4.23607i − 0.200135i
\(449\) 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
\(452\) − 7.67376i − 0.360943i
\(453\) 0 0
\(454\) 8.76393 0.411312
\(455\) 0 0
\(456\) 0 0
\(457\) − 21.4721i − 1.00442i −0.864744 0.502212i \(-0.832520\pi\)
0.864744 0.502212i \(-0.167480\pi\)
\(458\) 7.23607i 0.338119i
\(459\) 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
\(463\) − 21.8885i − 1.01725i −0.860989 0.508623i \(-0.830155\pi\)
0.860989 0.508623i \(-0.169845\pi\)
\(464\) −24.2705 −1.12673
\(465\) 0 0
\(466\) −16.0902 −0.745363
\(467\) 10.9443i 0.506441i 0.967409 + 0.253220i \(0.0814897\pi\)
−0.967409 + 0.253220i \(0.918510\pi\)
\(468\) 0 0
\(469\) −0.236068 −0.0109006
\(470\) 0 0
\(471\) 0 0
\(472\) 3.81966i 0.175814i
\(473\) 26.4164i 1.21463i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 23.4164i 1.07104i
\(479\) −3.81966 −0.174525 −0.0872624 0.996185i \(-0.527812\pi\)
−0.0872624 + 0.996185i \(0.527812\pi\)
\(480\) 0 0
\(481\) −9.70820 −0.442656
\(482\) 20.1803i 0.919189i
\(483\) 0 0
\(484\) −4.29180 −0.195082
\(485\) 0 0
\(486\) 0 0
\(487\) 10.2361i 0.463841i 0.972735 + 0.231920i \(0.0745008\pi\)
−0.972735 + 0.231920i \(0.925499\pi\)
\(488\) − 8.29180i − 0.375352i
\(489\) 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
\(493\) 32.3607i 1.45745i
\(494\) −23.4164 −1.05355
\(495\) 0 0
\(496\) 47.1246 2.11596
\(497\) 4.70820i 0.211192i
\(498\) 0 0
\(499\) −28.9443 −1.29572 −0.647862 0.761758i \(-0.724337\pi\)
−0.647862 + 0.761758i \(0.724337\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.00000i − 0.178529i
\(503\) − 43.8885i − 1.95689i −0.206499 0.978447i \(-0.566207\pi\)
0.206499 0.978447i \(-0.433793\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0902 −0.537474
\(507\) 0 0
\(508\) 10.9098i 0.484045i
\(509\) 9.34752 0.414322 0.207161 0.978307i \(-0.433578\pi\)
0.207161 + 0.978307i \(0.433578\pi\)
\(510\) 0 0
\(511\) −13.2361 −0.585529
\(512\) 5.29180i 0.233867i
\(513\) 0 0
\(514\) −30.1803 −1.33120
\(515\) 0 0
\(516\) 0 0
\(517\) − 8.47214i − 0.372604i
\(518\) 4.85410i 0.213277i
\(519\) 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
\(523\) 28.3607i 1.24013i 0.784552 + 0.620063i \(0.212893\pi\)
−0.784552 + 0.620063i \(0.787107\pi\)
\(524\) 0.583592 0.0254943
\(525\) 0 0
\(526\) 19.0344 0.829941
\(527\) − 62.8328i − 2.73704i
\(528\) 0 0
\(529\) 19.8885 0.864719
\(530\) 0 0
\(531\) 0 0
\(532\) 2.76393i 0.119832i
\(533\) − 29.8885i − 1.29462i
\(534\) 0 0
\(535\) 0 0
\(536\) 0.527864 0.0228003
\(537\) 0 0
\(538\) − 2.76393i − 0.119162i
\(539\) 4.23607 0.182460
\(540\) 0 0
\(541\) −1.94427 −0.0835908 −0.0417954 0.999126i \(-0.513308\pi\)
−0.0417954 + 0.999126i \(0.513308\pi\)
\(542\) − 16.6525i − 0.715285i
\(543\) 0 0
\(544\) −21.8885 −0.938464
\(545\) 0 0
\(546\) 0 0
\(547\) − 14.2361i − 0.608690i −0.952562 0.304345i \(-0.901562\pi\)
0.952562 0.304345i \(-0.0984376\pi\)
\(548\) 4.29180i 0.183336i
\(549\) 0 0
\(550\) 0 0
\(551\) −22.3607 −0.952597
\(552\) 0 0
\(553\) − 11.1803i − 0.475436i
\(554\) 25.7082 1.09224
\(555\) 0 0
\(556\) −12.7639 −0.541311
\(557\) 44.8885i 1.90199i 0.309208 + 0.950994i \(0.399936\pi\)
−0.309208 + 0.950994i \(0.600064\pi\)
\(558\) 0 0
\(559\) 20.1803 0.853537
\(560\) 0 0
\(561\) 0 0
\(562\) 47.5066i 2.00394i
\(563\) − 9.41641i − 0.396854i −0.980116 0.198427i \(-0.936417\pi\)
0.980116 0.198427i \(-0.0635833\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 15.2361 0.640420
\(567\) 0 0
\(568\) − 10.5279i − 0.441739i
\(569\) 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 \(0.584903\pi\)
\(572\) 8.47214i 0.354238i
\(573\) 0 0
\(574\) −14.9443 −0.623762
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 40.2705i 1.67503i
\(579\) 0 0
\(580\) 0 0
\(581\) −5.70820 −0.236816
\(582\) 0 0
\(583\) 2.00000i 0.0828315i
\(584\) 29.5967 1.22472
\(585\) 0 0
\(586\) 14.7639 0.609892
\(587\) 29.2361i 1.20670i 0.797476 + 0.603351i \(0.206168\pi\)
−0.797476 + 0.603351i \(0.793832\pi\)
\(588\) 0 0
\(589\) 43.4164 1.78894
\(590\) 0 0
\(591\) 0 0
\(592\) − 14.5623i − 0.598507i
\(593\) 25.3050i 1.03915i 0.854425 + 0.519575i \(0.173910\pi\)
−0.854425 + 0.519575i \(0.826090\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.61803 0.353008
\(597\) 0 0
\(598\) 9.23607i 0.377691i
\(599\) 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 \(-0.627058\pi\)
−0.388650 + 0.921386i \(0.627058\pi\)
\(602\) − 10.0902i − 0.411245i
\(603\) 0 0
\(604\) 9.74265 0.396423
\(605\) 0 0
\(606\) 0 0
\(607\) 33.1246i 1.34449i 0.740330 + 0.672243i \(0.234669\pi\)
−0.740330 + 0.672243i \(0.765331\pi\)
\(608\) − 15.1246i − 0.613384i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.47214 −0.261835
\(612\) 0 0
\(613\) 17.5836i 0.710195i 0.934829 + 0.355097i \(0.115552\pi\)
−0.934829 + 0.355097i \(0.884448\pi\)
\(614\) 50.8328 2.05145
\(615\) 0 0
\(616\) −9.47214 −0.381643
\(617\) − 11.9443i − 0.480858i −0.970667 0.240429i \(-0.922712\pi\)
0.970667 0.240429i \(-0.0772882\pi\)
\(618\) 0 0
\(619\) −1.70820 −0.0686585 −0.0343293 0.999411i \(-0.510929\pi\)
−0.0343293 + 0.999411i \(0.510929\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 32.9443i − 1.32094i
\(623\) 12.7639i 0.511376i
\(624\) 0 0
\(625\) 0 0
\(626\) −46.0689 −1.84128
\(627\) 0 0
\(628\) − 3.23607i − 0.129133i
\(629\) −19.4164 −0.774183
\(630\) 0 0
\(631\) −3.65248 −0.145403 −0.0727014 0.997354i \(-0.523162\pi\)
−0.0727014 + 0.997354i \(0.523162\pi\)
\(632\) 25.0000i 0.994447i
\(633\) 0 0
\(634\) 31.3262 1.24412
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.23607i − 0.128218i
\(638\) 34.2705i 1.35678i
\(639\) 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
\(643\) − 9.52786i − 0.375742i −0.982194 0.187871i \(-0.939841\pi\)
0.982194 0.187871i \(-0.0601587\pi\)
\(644\) 1.09017 0.0429587
\(645\) 0 0
\(646\) −46.8328 −1.84261
\(647\) − 15.8885i − 0.624643i −0.949976 0.312322i \(-0.898893\pi\)
0.949976 0.312322i \(-0.101107\pi\)
\(648\) 0 0
\(649\) 7.23607 0.284041
\(650\) 0 0
\(651\) 0 0
\(652\) − 6.47214i − 0.253468i
\(653\) 42.9443i 1.68054i 0.542169 + 0.840270i \(0.317603\pi\)
−0.542169 + 0.840270i \(0.682397\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 44.8328 1.75043
\(657\) 0 0
\(658\) 3.23607i 0.126155i
\(659\) −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 \(-0.137133\pi\)
0.908625 + 0.417613i \(0.137133\pi\)
\(662\) 18.2705i 0.710104i
\(663\) 0 0
\(664\) 12.7639 0.495337
\(665\) 0 0
\(666\) 0 0
\(667\) 8.81966i 0.341499i
\(668\) 0.472136i 0.0182675i
\(669\) 0 0
\(670\) 0 0
\(671\) −15.7082 −0.606408
\(672\) 0 0
\(673\) − 28.4721i − 1.09752i −0.835980 0.548760i \(-0.815100\pi\)
0.835980 0.548760i \(-0.184900\pi\)
\(674\) −12.1803 −0.469169
\(675\) 0 0
\(676\) −1.56231 −0.0600887
\(677\) − 30.3607i − 1.16686i −0.812165 0.583428i \(-0.801711\pi\)
0.812165 0.583428i \(-0.198289\pi\)
\(678\) 0 0
\(679\) −0.763932 −0.0293170
\(680\) 0 0
\(681\) 0 0
\(682\) − 66.5410i − 2.54799i
\(683\) − 26.1246i − 0.999630i −0.866132 0.499815i \(-0.833401\pi\)
0.866132 0.499815i \(-0.166599\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.61803 −0.0617768
\(687\) 0 0
\(688\) 30.2705i 1.15405i
\(689\) 1.52786 0.0582070
\(690\) 0 0
\(691\) 18.1803 0.691613 0.345806 0.938306i \(-0.387605\pi\)
0.345806 + 0.938306i \(0.387605\pi\)
\(692\) 12.6525i 0.480975i
\(693\) 0 0
\(694\) −25.5066 −0.968216
\(695\) 0 0
\(696\) 0 0
\(697\) − 59.7771i − 2.26422i
\(698\) − 7.23607i − 0.273889i
\(699\) 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
\(703\) − 13.4164i − 0.506009i
\(704\) 17.9443 0.676300
\(705\) 0 0
\(706\) 32.6525 1.22889
\(707\) − 9.23607i − 0.347358i
\(708\) 0 0
\(709\) 47.8885 1.79849 0.899246 0.437443i \(-0.144116\pi\)
0.899246 + 0.437443i \(0.144116\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 28.5410i − 1.06962i
\(713\) − 17.1246i − 0.641322i
\(714\) 0 0
\(715\) 0 0
\(716\) −2.11146 −0.0789088
\(717\) 0 0
\(718\) − 16.3820i − 0.611370i
\(719\) −6.18034 −0.230488 −0.115244 0.993337i \(-0.536765\pi\)
−0.115244 + 0.993337i \(0.536765\pi\)
\(720\) 0 0
\(721\) −0.472136 −0.0175833
\(722\) − 1.61803i − 0.0602170i
\(723\) 0 0
\(724\) 8.76393 0.325709
\(725\) 0 0
\(726\) 0 0
\(727\) − 20.9443i − 0.776780i −0.921495 0.388390i \(-0.873031\pi\)
0.921495 0.388390i \(-0.126969\pi\)
\(728\) 7.23607i 0.268187i
\(729\) 0 0
\(730\) 0 0
\(731\) 40.3607 1.49279
\(732\) 0 0
\(733\) − 4.00000i − 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 5.05573 0.186610
\(735\) 0 0
\(736\) −5.96556 −0.219893
\(737\) − 1.00000i − 0.0368355i
\(738\) 0 0
\(739\) 5.65248 0.207930 0.103965 0.994581i \(-0.466847\pi\)
0.103965 + 0.994581i \(0.466847\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 0.763932i − 0.0280448i
\(743\) − 1.52786i − 0.0560519i −0.999607 0.0280259i \(-0.991078\pi\)
0.999607 0.0280259i \(-0.00892210\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −25.6180 −0.937943
\(747\) 0 0
\(748\) 16.9443i 0.619544i
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) 20.9443 0.764267 0.382134 0.924107i \(-0.375189\pi\)
0.382134 + 0.924107i \(0.375189\pi\)
\(752\) − 9.70820i − 0.354022i
\(753\) 0 0
\(754\) 26.1803 0.953432
\(755\) 0 0
\(756\) 0 0
\(757\) 46.4164i 1.68703i 0.537103 + 0.843517i \(0.319519\pi\)
−0.537103 + 0.843517i \(0.680481\pi\)
\(758\) 18.0902i 0.657065i
\(759\) 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
\(763\) 18.4164i 0.666719i
\(764\) −1.52786 −0.0552762
\(765\) 0 0
\(766\) −46.5410 −1.68160
\(767\) − 5.52786i − 0.199600i
\(768\) 0 0
\(769\) −33.0132 −1.19048 −0.595242 0.803546i \(-0.702944\pi\)
−0.595242 + 0.803546i \(0.702944\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 8.90983i − 0.320672i
\(773\) 27.8197i 1.00060i 0.865851 + 0.500302i \(0.166778\pi\)
−0.865851 + 0.500302i \(0.833222\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.70820 0.0613209
\(777\) 0 0
\(778\) 53.2148i 1.90784i
\(779\) 41.3050 1.47990
\(780\) 0 0
\(781\) −19.9443 −0.713662
\(782\) 18.4721i 0.660562i
\(783\) 0 0
\(784\) 4.85410 0.173361
\(785\) 0 0
\(786\) 0 0
\(787\) 45.2361i 1.61249i 0.591581 + 0.806246i \(0.298504\pi\)
−0.591581 + 0.806246i \(0.701496\pi\)
\(788\) 4.61803i 0.164511i
\(789\) 0 0
\(790\) 0 0
\(791\) −12.4164 −0.441477
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 43.5967 1.54719
\(795\) 0 0
\(796\) 1.70820 0.0605457
\(797\) 8.58359i 0.304046i 0.988377 + 0.152023i \(0.0485789\pi\)
−0.988377 + 0.152023i \(0.951421\pi\)
\(798\) 0 0
\(799\) −12.9443 −0.457935
\(800\) 0 0
\(801\) 0 0
\(802\) 18.5623i 0.655458i
\(803\) − 56.0689i − 1.97863i
\(804\) 0 0
\(805\) 0 0
\(806\) −50.8328 −1.79051
\(807\) 0 0
\(808\) 20.6525i 0.726552i
\(809\) 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 \(-0.193804\pi\)
0.820304 + 0.571927i \(0.193804\pi\)
\(812\) − 3.09017i − 0.108444i
\(813\) 0 0
\(814\) −20.5623 −0.720708
\(815\) 0 0
\(816\) 0 0
\(817\) 27.8885i 0.975697i
\(818\) 25.1246i 0.878461i
\(819\) 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
\(823\) 0.347524i 0.0121139i 0.999982 + 0.00605697i \(0.00192800\pi\)
−0.999982 + 0.00605697i \(0.998072\pi\)
\(824\) 1.05573 0.0367780
\(825\) 0 0
\(826\) −2.76393 −0.0961695
\(827\) 25.5410i 0.888148i 0.895990 + 0.444074i \(0.146467\pi\)
−0.895990 + 0.444074i \(0.853533\pi\)
\(828\) 0 0
\(829\) 52.3607 1.81856 0.909281 0.416183i \(-0.136632\pi\)
0.909281 + 0.416183i \(0.136632\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 13.7082i − 0.475246i
\(833\) − 6.47214i − 0.224246i
\(834\) 0 0
\(835\) 0 0
\(836\) −11.7082 −0.404937
\(837\) 0 0
\(838\) 6.18034i 0.213496i
\(839\) 0.652476 0.0225260 0.0112630 0.999937i \(-0.496415\pi\)
0.0112630 + 0.999937i \(0.496415\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 21.0344i 0.724895i
\(843\) 0 0
\(844\) −7.41641 −0.255283
\(845\) 0 0
\(846\) 0 0
\(847\) 6.94427i 0.238608i
\(848\) 2.29180i 0.0787006i
\(849\) 0 0
\(850\) 0 0
\(851\) −5.29180 −0.181400
\(852\) 0 0
\(853\) − 0.583592i − 0.0199818i −0.999950 0.00999091i \(-0.996820\pi\)
0.999950 0.00999091i \(-0.00318026\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) −17.8885 −0.611418
\(857\) 38.1803i 1.30422i 0.758126 + 0.652108i \(0.226115\pi\)
−0.758126 + 0.652108i \(0.773885\pi\)
\(858\) 0 0
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 42.8328i 1.45889i
\(863\) 49.6525i 1.69019i 0.534616 + 0.845095i \(0.320456\pi\)
−0.534616 + 0.845095i \(0.679544\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −26.4721 −0.899560
\(867\) 0 0
\(868\) 6.00000i 0.203653i
\(869\) 47.3607 1.60660
\(870\) 0 0
\(871\) −0.763932 −0.0258848
\(872\) − 41.1803i − 1.39454i
\(873\) 0 0
\(874\) −12.7639 −0.431746
\(875\) 0 0
\(876\) 0 0
\(877\) − 14.3607i − 0.484926i −0.970161 0.242463i \(-0.922045\pi\)
0.970161 0.242463i \(-0.0779553\pi\)
\(878\) − 35.1246i − 1.18540i
\(879\) 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
\(883\) 50.5967i 1.70272i 0.524585 + 0.851358i \(0.324220\pi\)
−0.524585 + 0.851358i \(0.675780\pi\)
\(884\) 12.9443 0.435363
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 52.6525i 1.76790i 0.467584 + 0.883949i \(0.345124\pi\)
−0.467584 + 0.883949i \(0.654876\pi\)
\(888\) 0 0
\(889\) 17.6525 0.592045
\(890\) 0 0
\(891\) 0 0
\(892\) − 1.34752i − 0.0451184i
\(893\) − 8.94427i − 0.299309i
\(894\) 0 0
\(895\) 0 0
\(896\) −13.6180 −0.454947
\(897\) 0 0
\(898\) − 47.6869i − 1.59133i
\(899\) −48.5410 −1.61893
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) − 63.3050i − 2.10782i
\(903\) 0 0
\(904\) 27.7639 0.923415
\(905\) 0 0
\(906\) 0 0
\(907\) − 18.8328i − 0.625333i −0.949863 0.312667i \(-0.898778\pi\)
0.949863 0.312667i \(-0.101222\pi\)
\(908\) − 3.34752i − 0.111091i
\(909\) 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
\(913\) − 24.1803i − 0.800252i
\(914\) −34.7426 −1.14918
\(915\) 0 0
\(916\) 2.76393 0.0913229
\(917\) − 0.944272i − 0.0311826i
\(918\) 0 0
\(919\) 32.2361 1.06337 0.531685 0.846942i \(-0.321559\pi\)
0.531685 + 0.846942i \(0.321559\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 13.2361i 0.435907i
\(923\) 15.2361i 0.501501i
\(924\) 0 0
\(925\) 0 0
\(926\) −35.4164 −1.16386
\(927\) 0 0
\(928\) 16.9098i 0.555092i
\(929\) 51.7082 1.69649 0.848246 0.529603i \(-0.177659\pi\)
0.848246 + 0.529603i \(0.177659\pi\)
\(930\) 0 0
\(931\) 4.47214 0.146568
\(932\) 6.14590i 0.201316i
\(933\) 0 0
\(934\) 17.7082 0.579430
\(935\) 0 0
\(936\) 0 0
\(937\) 30.7639i 1.00501i 0.864573 + 0.502507i \(0.167589\pi\)
−0.864573 + 0.502507i \(0.832411\pi\)
\(938\) 0.381966i 0.0124716i
\(939\) 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
\(943\) − 16.2918i − 0.530534i
\(944\) 8.29180 0.269875
\(945\) 0 0
\(946\) 42.7426 1.38968
\(947\) 18.8328i 0.611984i 0.952034 + 0.305992i \(0.0989881\pi\)
−0.952034 + 0.305992i \(0.901012\pi\)
\(948\) 0 0
\(949\) −42.8328 −1.39041
\(950\) 0 0
\(951\) 0 0
\(952\) 14.4721i 0.469045i
\(953\) − 5.47214i − 0.177260i −0.996065 0.0886299i \(-0.971751\pi\)
0.996065 0.0886299i \(-0.0282489\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 8.94427 0.289278
\(957\) 0 0
\(958\) 6.18034i 0.199678i
\(959\) 6.94427 0.224242
\(960\) 0 0
\(961\) 63.2492 2.04030
\(962\) 15.7082i 0.506453i
\(963\) 0 0
\(964\) 7.70820 0.248265
\(965\) 0 0
\(966\) 0 0
\(967\) − 49.8885i − 1.60431i −0.597118 0.802154i \(-0.703687\pi\)
0.597118 0.802154i \(-0.296313\pi\)
\(968\) − 15.5279i − 0.499084i
\(969\) 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
\(973\) 20.6525i 0.662088i
\(974\) 16.5623 0.530691
\(975\) 0 0
\(976\) −18.0000 −0.576166
\(977\) 2.52786i 0.0808735i 0.999182 + 0.0404368i \(0.0128749\pi\)
−0.999182 + 0.0404368i \(0.987125\pi\)
\(978\) 0 0
\(979\) −54.0689 −1.72805
\(980\) 0 0
\(981\) 0 0
\(982\) − 16.5623i − 0.528524i
\(983\) 32.5410i 1.03790i 0.854805 + 0.518949i \(0.173676\pi\)
−0.854805 + 0.518949i \(0.826324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 52.3607 1.66750
\(987\) 0 0
\(988\) 8.94427i 0.284555i
\(989\) 11.0000 0.349780
\(990\) 0 0
\(991\) −9.18034 −0.291623 −0.145812 0.989312i \(-0.546579\pi\)
−0.145812 + 0.989312i \(0.546579\pi\)
\(992\) − 32.8328i − 1.04244i
\(993\) 0 0
\(994\) 7.61803 0.241629
\(995\) 0 0
\(996\) 0 0
\(997\) − 18.5836i − 0.588548i −0.955721 0.294274i \(-0.904922\pi\)
0.955721 0.294274i \(-0.0950779\pi\)
\(998\) 46.8328i 1.48247i
\(999\) 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.d.k.1324.1 4
3.2 odd 2 175.2.b.c.99.4 4
5.2 odd 4 1575.2.a.s.1.2 2
5.3 odd 4 1575.2.a.n.1.1 2
5.4 even 2 inner 1575.2.d.k.1324.4 4
12.11 even 2 2800.2.g.s.449.3 4
15.2 even 4 175.2.a.d.1.1 2
15.8 even 4 175.2.a.e.1.2 yes 2
15.14 odd 2 175.2.b.c.99.1 4
21.20 even 2 1225.2.b.k.99.4 4
60.23 odd 4 2800.2.a.bp.1.1 2
60.47 odd 4 2800.2.a.bh.1.2 2
60.59 even 2 2800.2.g.s.449.2 4
105.62 odd 4 1225.2.a.n.1.1 2
105.83 odd 4 1225.2.a.u.1.2 2
105.104 even 2 1225.2.b.k.99.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.2.a.d.1.1 2 15.2 even 4
175.2.a.e.1.2 yes 2 15.8 even 4
175.2.b.c.99.1 4 15.14 odd 2
175.2.b.c.99.4 4 3.2 odd 2
1225.2.a.n.1.1 2 105.62 odd 4
1225.2.a.u.1.2 2 105.83 odd 4
1225.2.b.k.99.1 4 105.104 even 2
1225.2.b.k.99.4 4 21.20 even 2
1575.2.a.n.1.1 2 5.3 odd 4
1575.2.a.s.1.2 2 5.2 odd 4
1575.2.d.k.1324.1 4 1.1 even 1 trivial
1575.2.d.k.1324.4 4 5.4 even 2 inner
2800.2.a.bh.1.2 2 60.47 odd 4
2800.2.a.bp.1.1 2 60.23 odd 4
2800.2.g.s.449.2 4 60.59 even 2
2800.2.g.s.449.3 4 12.11 even 2