Properties

Label 175.2.b.c.99.4
Level $175$
Weight $2$
Character 175.99
Analytic conductor $1.397$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.2.b.c.99.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803i q^{2} -1.23607i q^{3} -0.618034 q^{4} +2.00000 q^{6} +1.00000i q^{7} +2.23607i q^{8} +1.47214 q^{9} +O(q^{10})\) \(q+1.61803i q^{2} -1.23607i q^{3} -0.618034 q^{4} +2.00000 q^{6} +1.00000i q^{7} +2.23607i q^{8} +1.47214 q^{9} +4.23607 q^{11} +0.763932i q^{12} +3.23607i q^{13} -1.61803 q^{14} -4.85410 q^{16} -6.47214i q^{17} +2.38197i q^{18} -4.47214 q^{19} +1.23607 q^{21} +6.85410i q^{22} -1.76393i q^{23} +2.76393 q^{24} -5.23607 q^{26} -5.52786i q^{27} -0.618034i q^{28} -5.00000 q^{29} -9.70820 q^{31} -3.38197i q^{32} -5.23607i q^{33} +10.4721 q^{34} -0.909830 q^{36} +3.00000i q^{37} -7.23607i q^{38} +4.00000 q^{39} +9.23607 q^{41} +2.00000i q^{42} -6.23607i q^{43} -2.61803 q^{44} +2.85410 q^{46} -2.00000i q^{47} +6.00000i q^{48} -1.00000 q^{49} -8.00000 q^{51} -2.00000i q^{52} +0.472136i q^{53} +8.94427 q^{54} -2.23607 q^{56} +5.52786i q^{57} -8.09017i q^{58} +1.70820 q^{59} +3.70820 q^{61} -15.7082i q^{62} +1.47214i q^{63} -4.23607 q^{64} +8.47214 q^{66} +0.236068i q^{67} +4.00000i q^{68} -2.18034 q^{69} -4.70820 q^{71} +3.29180i q^{72} +13.2361i q^{73} -4.85410 q^{74} +2.76393 q^{76} +4.23607i q^{77} +6.47214i q^{78} -11.1803 q^{79} -2.41641 q^{81} +14.9443i q^{82} -5.70820i q^{83} -0.763932 q^{84} +10.0902 q^{86} +6.18034i q^{87} +9.47214i q^{88} -12.7639 q^{89} -3.23607 q^{91} +1.09017i q^{92} +12.0000i q^{93} +3.23607 q^{94} -4.18034 q^{96} +0.763932i q^{97} -1.61803i q^{98} +6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 8q^{6} - 12q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 8q^{6} - 12q^{9} + 8q^{11} - 2q^{14} - 6q^{16} - 4q^{21} + 20q^{24} - 12q^{26} - 20q^{29} - 12q^{31} + 24q^{34} - 26q^{36} + 16q^{39} + 28q^{41} - 6q^{44} - 2q^{46} - 4q^{49} - 32q^{51} - 20q^{59} - 12q^{61} - 8q^{64} + 16q^{66} + 36q^{69} + 8q^{71} - 6q^{74} + 20q^{76} + 44q^{81} - 12q^{84} + 18q^{86} - 60q^{89} - 4q^{91} + 4q^{94} + 28q^{96} + 16q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.193856\pi\)
−0.820211 + 0.572061i \(0.806144\pi\)
\(3\) − 1.23607i − 0.713644i −0.934172 0.356822i \(-0.883860\pi\)
0.934172 0.356822i \(-0.116140\pi\)
\(4\) −0.618034 −0.309017
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000i 0.377964i
\(8\) 2.23607i 0.790569i
\(9\) 1.47214 0.490712
\(10\) 0 0
\(11\) 4.23607 1.27722 0.638611 0.769529i \(-0.279509\pi\)
0.638611 + 0.769529i \(0.279509\pi\)
\(12\) 0.763932i 0.220528i
\(13\) 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 \(-0.712734\pi\)
0.619671 0.784862i \(-0.287266\pi\)
\(18\) 2.38197i 0.561435i
\(19\) −4.47214 −1.02598 −0.512989 0.858395i \(-0.671462\pi\)
−0.512989 + 0.858395i \(0.671462\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) 6.85410i 1.46130i
\(23\) − 1.76393i − 0.367805i −0.982944 0.183903i \(-0.941127\pi\)
0.982944 0.183903i \(-0.0588731\pi\)
\(24\) 2.76393 0.564185
\(25\) 0 0
\(26\) −5.23607 −1.02688
\(27\) − 5.52786i − 1.06384i
\(28\) − 0.618034i − 0.116797i
\(29\) −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 \(0.837062\pi\)
\(32\) − 3.38197i − 0.597853i
\(33\) − 5.23607i − 0.911482i
\(34\) 10.4721 1.79596
\(35\) 0 0
\(36\) −0.909830 −0.151638
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) − 7.23607i − 1.17385i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 9.23607 1.44243 0.721216 0.692711i \(-0.243584\pi\)
0.721216 + 0.692711i \(0.243584\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 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 \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) 6.00000i 0.866025i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) − 2.00000i − 0.277350i
\(53\) 0.472136i 0.0648529i 0.999474 + 0.0324264i \(0.0103235\pi\)
−0.999474 + 0.0324264i \(0.989677\pi\)
\(54\) 8.94427 1.21716
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 5.52786i 0.732183i
\(58\) − 8.09017i − 1.06229i
\(59\) 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 \(0.423707\pi\)
\(62\) − 15.7082i − 1.99494i
\(63\) 1.47214i 0.185472i
\(64\) −4.23607 −0.529508
\(65\) 0 0
\(66\) 8.47214 1.04285
\(67\) 0.236068i 0.0288403i 0.999896 + 0.0144201i \(0.00459023\pi\)
−0.999896 + 0.0144201i \(0.995410\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −2.18034 −0.262482
\(70\) 0 0
\(71\) −4.70820 −0.558761 −0.279381 0.960180i \(-0.590129\pi\)
−0.279381 + 0.960180i \(0.590129\pi\)
\(72\) 3.29180i 0.387942i
\(73\) 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
\(78\) 6.47214i 0.732825i
\(79\) −11.1803 −1.25789 −0.628943 0.777451i \(-0.716512\pi\)
−0.628943 + 0.777451i \(0.716512\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 14.9443i 1.65032i
\(83\) − 5.70820i − 0.626557i −0.949661 0.313278i \(-0.898573\pi\)
0.949661 0.313278i \(-0.101427\pi\)
\(84\) −0.763932 −0.0833518
\(85\) 0 0
\(86\) 10.0902 1.08805
\(87\) 6.18034i 0.662602i
\(88\) 9.47214i 1.00973i
\(89\) −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
\(92\) 1.09017i 0.113658i
\(93\) 12.0000i 1.24434i
\(94\) 3.23607 0.333775
\(95\) 0 0
\(96\) −4.18034 −0.426654
\(97\) 0.763932i 0.0775655i 0.999248 + 0.0387828i \(0.0123480\pi\)
−0.999248 + 0.0387828i \(0.987652\pi\)
\(98\) − 1.61803i − 0.163446i
\(99\) 6.23607 0.626748
\(100\) 0 0
\(101\) 9.23607 0.919023 0.459512 0.888172i \(-0.348024\pi\)
0.459512 + 0.888172i \(0.348024\pi\)
\(102\) − 12.9443i − 1.28167i
\(103\) 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 \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 3.41641i 0.328744i
\(109\) 18.4164 1.76397 0.881986 0.471276i \(-0.156206\pi\)
0.881986 + 0.471276i \(0.156206\pi\)
\(110\) 0 0
\(111\) 3.70820 0.351967
\(112\) − 4.85410i − 0.458670i
\(113\) − 12.4164i − 1.16804i −0.811740 0.584019i \(-0.801479\pi\)
0.811740 0.584019i \(-0.198521\pi\)
\(114\) −8.94427 −0.837708
\(115\) 0 0
\(116\) 3.09017 0.286915
\(117\) 4.76393i 0.440426i
\(118\) 2.76393i 0.254441i
\(119\) 6.47214 0.593300
\(120\) 0 0
\(121\) 6.94427 0.631297
\(122\) 6.00000i 0.543214i
\(123\) − 11.4164i − 1.02938i
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) −2.38197 −0.212202
\(127\) − 17.6525i − 1.56640i −0.621767 0.783202i \(-0.713585\pi\)
0.621767 0.783202i \(-0.286415\pi\)
\(128\) − 13.6180i − 1.20368i
\(129\) −7.70820 −0.678670
\(130\) 0 0
\(131\) 0.944272 0.0825014 0.0412507 0.999149i \(-0.486866\pi\)
0.0412507 + 0.999149i \(0.486866\pi\)
\(132\) 3.23607i 0.281664i
\(133\) − 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 \(0.0958677\pi\)
−0.954988 + 0.296645i \(0.904132\pi\)
\(138\) − 3.52786i − 0.300312i
\(139\) 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
\(142\) − 7.61803i − 0.639291i
\(143\) 13.7082i 1.14634i
\(144\) −7.14590 −0.595492
\(145\) 0 0
\(146\) −21.4164 −1.77243
\(147\) 1.23607i 0.101949i
\(148\) − 1.85410i − 0.152406i
\(149\) 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 \(0.721657\pi\)
\(152\) − 10.0000i − 0.811107i
\(153\) − 9.52786i − 0.770282i
\(154\) −6.85410 −0.552319
\(155\) 0 0
\(156\) −2.47214 −0.197929
\(157\) 5.23607i 0.417884i 0.977928 + 0.208942i \(0.0670019\pi\)
−0.977928 + 0.208942i \(0.932998\pi\)
\(158\) − 18.0902i − 1.43918i
\(159\) 0.583592 0.0462819
\(160\) 0 0
\(161\) 1.76393 0.139017
\(162\) − 3.90983i − 0.307185i
\(163\) 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 \(0.00940979\pi\)
−0.999563 + 0.0295574i \(0.990590\pi\)
\(168\) 2.76393i 0.213242i
\(169\) 2.52786 0.194451
\(170\) 0 0
\(171\) −6.58359 −0.503460
\(172\) 3.85410i 0.293873i
\(173\) 20.4721i 1.55647i 0.627975 + 0.778234i \(0.283884\pi\)
−0.627975 + 0.778234i \(0.716116\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −20.5623 −1.54994
\(177\) − 2.11146i − 0.158707i
\(178\) − 20.6525i − 1.54797i
\(179\) −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 \(0.676686\pi\)
\(182\) − 5.23607i − 0.388123i
\(183\) − 4.58359i − 0.338829i
\(184\) 3.94427 0.290776
\(185\) 0 0
\(186\) −19.4164 −1.42368
\(187\) − 27.4164i − 2.00489i
\(188\) 1.23607i 0.0901495i
\(189\) 5.52786 0.402093
\(190\) 0 0
\(191\) −2.47214 −0.178877 −0.0894387 0.995992i \(-0.528507\pi\)
−0.0894387 + 0.995992i \(0.528507\pi\)
\(192\) 5.23607i 0.377881i
\(193\) 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 \(0.0857628\pi\)
−0.963922 + 0.266184i \(0.914237\pi\)
\(198\) 10.0902i 0.717077i
\(199\) −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
\(202\) 14.9443i 1.05148i
\(203\) − 5.00000i − 0.350931i
\(204\) 4.94427 0.346168
\(205\) 0 0
\(206\) −0.763932 −0.0532257
\(207\) − 2.59675i − 0.180486i
\(208\) − 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
\(213\) 5.81966i 0.398757i
\(214\) −12.9443 −0.884852
\(215\) 0 0
\(216\) 12.3607 0.841038
\(217\) − 9.70820i − 0.659036i
\(218\) 29.7984i 2.01820i
\(219\) 16.3607 1.10555
\(220\) 0 0
\(221\) 20.9443 1.40886
\(222\) 6.00000i 0.402694i
\(223\) 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 \(-0.942471\pi\)
0.983712 0.179750i \(-0.0575288\pi\)
\(228\) − 3.41641i − 0.226257i
\(229\) −4.47214 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(230\) 0 0
\(231\) 5.23607 0.344508
\(232\) − 11.1803i − 0.734025i
\(233\) 9.94427i 0.651471i 0.945461 + 0.325735i \(0.105612\pi\)
−0.945461 + 0.325735i \(0.894388\pi\)
\(234\) −7.70820 −0.503901
\(235\) 0 0
\(236\) −1.05573 −0.0687220
\(237\) 13.8197i 0.897683i
\(238\) 10.4721i 0.678808i
\(239\) 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 \(0.631581\pi\)
\(242\) 11.2361i 0.722282i
\(243\) − 13.5967i − 0.872232i
\(244\) −2.29180 −0.146717
\(245\) 0 0
\(246\) 18.4721 1.17774
\(247\) − 14.4721i − 0.920840i
\(248\) − 21.7082i − 1.37847i
\(249\) −7.05573 −0.447139
\(250\) 0 0
\(251\) −2.47214 −0.156040 −0.0780199 0.996952i \(-0.524860\pi\)
−0.0780199 + 0.996952i \(0.524860\pi\)
\(252\) − 0.909830i − 0.0573139i
\(253\) − 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 \(0.197634\pi\)
−0.813364 + 0.581755i \(0.802366\pi\)
\(258\) − 12.4721i − 0.776481i
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) −7.36068 −0.455615
\(262\) 1.52786i 0.0943918i
\(263\) − 11.7639i − 0.725395i −0.931907 0.362698i \(-0.881856\pi\)
0.931907 0.362698i \(-0.118144\pi\)
\(264\) 11.7082 0.720590
\(265\) 0 0
\(266\) 7.23607 0.443672
\(267\) 15.7771i 0.965542i
\(268\) − 0.145898i − 0.00891214i
\(269\) −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 \(0.398803\pi\)
\(272\) 31.4164i 1.90490i
\(273\) 4.00000i 0.242091i
\(274\) −11.2361 −0.678796
\(275\) 0 0
\(276\) 1.34752 0.0811114
\(277\) 15.8885i 0.954650i 0.878727 + 0.477325i \(0.158394\pi\)
−0.878727 + 0.477325i \(0.841606\pi\)
\(278\) 33.4164i 2.00418i
\(279\) −14.2918 −0.855627
\(280\) 0 0
\(281\) 29.3607 1.75151 0.875756 0.482755i \(-0.160364\pi\)
0.875756 + 0.482755i \(0.160364\pi\)
\(282\) − 4.00000i − 0.238197i
\(283\) 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
\(288\) − 4.97871i − 0.293374i
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) 0.944272 0.0553542
\(292\) − 8.18034i − 0.478718i
\(293\) − 9.12461i − 0.533066i −0.963826 0.266533i \(-0.914122\pi\)
0.963826 0.266533i \(-0.0858781\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −6.70820 −0.389906
\(297\) − 23.4164i − 1.35876i
\(298\) 22.5623i 1.30700i
\(299\) 5.70820 0.330114
\(300\) 0 0
\(301\) 6.23607 0.359441
\(302\) − 25.5066i − 1.46774i
\(303\) − 11.4164i − 0.655855i
\(304\) 21.7082 1.24505
\(305\) 0 0
\(306\) 15.4164 0.881297
\(307\) 31.4164i 1.79303i 0.443014 + 0.896515i \(0.353909\pi\)
−0.443014 + 0.896515i \(0.646091\pi\)
\(308\) − 2.61803i − 0.149176i
\(309\) 0.583592 0.0331994
\(310\) 0 0
\(311\) −20.3607 −1.15455 −0.577274 0.816550i \(-0.695884\pi\)
−0.577274 + 0.816550i \(0.695884\pi\)
\(312\) 8.94427i 0.506370i
\(313\) − 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 \(-0.817022\pi\)
0.839278 0.543702i \(-0.182978\pi\)
\(318\) 0.944272i 0.0529521i
\(319\) −21.1803 −1.18587
\(320\) 0 0
\(321\) 9.88854 0.551925
\(322\) 2.85410i 0.159053i
\(323\) 28.9443i 1.61050i
\(324\) 1.49342 0.0829679
\(325\) 0 0
\(326\) −16.9443 −0.938456
\(327\) − 22.7639i − 1.25885i
\(328\) 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
\(333\) 4.41641i 0.242018i
\(334\) −1.23607 −0.0676346
\(335\) 0 0
\(336\) −6.00000 −0.327327
\(337\) − 7.52786i − 0.410069i −0.978755 0.205034i \(-0.934269\pi\)
0.978755 0.205034i \(-0.0657306\pi\)
\(338\) 4.09017i 0.222476i
\(339\) −15.3475 −0.833563
\(340\) 0 0
\(341\) −41.1246 −2.22702
\(342\) − 10.6525i − 0.576020i
\(343\) − 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 \(0.139067\pi\)
−0.906071 + 0.423126i \(0.860933\pi\)
\(348\) − 3.81966i − 0.204755i
\(349\) 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
\(352\) − 14.3262i − 0.763591i
\(353\) − 20.1803i − 1.07409i −0.843553 0.537046i \(-0.819540\pi\)
0.843553 0.537046i \(-0.180460\pi\)
\(354\) 3.41641 0.181580
\(355\) 0 0
\(356\) 7.88854 0.418092
\(357\) − 8.00000i − 0.423405i
\(358\) − 5.52786i − 0.292157i
\(359\) −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
\(362\) − 22.9443i − 1.20592i
\(363\) − 8.58359i − 0.450522i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 7.41641 0.387662
\(367\) 3.12461i 0.163103i 0.996669 + 0.0815517i \(0.0259876\pi\)
−0.996669 + 0.0815517i \(0.974012\pi\)
\(368\) 8.56231i 0.446341i
\(369\) 13.5967 0.707818
\(370\) 0 0
\(371\) −0.472136 −0.0245121
\(372\) − 7.41641i − 0.384523i
\(373\) − 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
\(378\) 8.94427i 0.460044i
\(379\) −11.1803 −0.574295 −0.287148 0.957886i \(-0.592707\pi\)
−0.287148 + 0.957886i \(0.592707\pi\)
\(380\) 0 0
\(381\) −21.8197 −1.11786
\(382\) − 4.00000i − 0.204658i
\(383\) 28.7639i 1.46977i 0.678193 + 0.734884i \(0.262763\pi\)
−0.678193 + 0.734884i \(0.737237\pi\)
\(384\) −16.8328 −0.858996
\(385\) 0 0
\(386\) −23.3262 −1.18727
\(387\) − 9.18034i − 0.466663i
\(388\) − 0.472136i − 0.0239691i
\(389\) 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
\(392\) − 2.23607i − 0.112938i
\(393\) − 1.16718i − 0.0588767i
\(394\) −12.0902 −0.609094
\(395\) 0 0
\(396\) −3.85410 −0.193676
\(397\) 26.9443i 1.35229i 0.736767 + 0.676147i \(0.236352\pi\)
−0.736767 + 0.676147i \(0.763648\pi\)
\(398\) − 4.47214i − 0.224168i
\(399\) −5.52786 −0.276739
\(400\) 0 0
\(401\) 11.4721 0.572891 0.286446 0.958097i \(-0.407526\pi\)
0.286446 + 0.958097i \(0.407526\pi\)
\(402\) 0.472136i 0.0235480i
\(403\) − 31.4164i − 1.56496i
\(404\) −5.70820 −0.283994
\(405\) 0 0
\(406\) 8.09017 0.401508
\(407\) 12.7082i 0.629922i
\(408\) − 17.8885i − 0.885615i
\(409\) −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
\(412\) − 0.291796i − 0.0143758i
\(413\) 1.70820i 0.0840552i
\(414\) 4.20163 0.206499
\(415\) 0 0
\(416\) 10.9443 0.536587
\(417\) − 25.5279i − 1.25010i
\(418\) − 30.6525i − 1.49926i
\(419\) 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 \(0.602605\pi\)
\(422\) 19.4164i 0.945176i
\(423\) − 2.94427i − 0.143155i
\(424\) −1.05573 −0.0512707
\(425\) 0 0
\(426\) −9.41641 −0.456226
\(427\) 3.70820i 0.179453i
\(428\) − 4.94427i − 0.238990i
\(429\) 16.9443 0.818077
\(430\) 0 0
\(431\) 26.4721 1.27512 0.637559 0.770402i \(-0.279944\pi\)
0.637559 + 0.770402i \(0.279944\pi\)
\(432\) 26.8328i 1.29099i
\(433\) − 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
\(438\) 26.4721i 1.26489i
\(439\) 21.7082 1.03608 0.518038 0.855358i \(-0.326663\pi\)
0.518038 + 0.855358i \(0.326663\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 33.8885i 1.61191i
\(443\) − 7.41641i − 0.352364i −0.984358 0.176182i \(-0.943625\pi\)
0.984358 0.176182i \(-0.0563748\pi\)
\(444\) −2.29180 −0.108764
\(445\) 0 0
\(446\) −3.52786 −0.167049
\(447\) − 17.2361i − 0.815238i
\(448\) − 4.23607i − 0.200135i
\(449\) −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
\(452\) 7.67376i 0.360943i
\(453\) 19.4853i 0.915499i
\(454\) 8.76393 0.411312
\(455\) 0 0
\(456\) −12.3607 −0.578842
\(457\) − 21.4721i − 1.00442i −0.864744 0.502212i \(-0.832520\pi\)
0.864744 0.502212i \(-0.167480\pi\)
\(458\) − 7.23607i − 0.338119i
\(459\) −35.7771 −1.66993
\(460\) 0 0
\(461\) 8.18034 0.380996 0.190498 0.981688i \(-0.438990\pi\)
0.190498 + 0.981688i \(0.438990\pi\)
\(462\) 8.47214i 0.394159i
\(463\) − 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 \(-0.918510\pi\)
0.967409 0.253220i \(-0.0814897\pi\)
\(468\) − 2.94427i − 0.136099i
\(469\) −0.236068 −0.0109006
\(470\) 0 0
\(471\) 6.47214 0.298220
\(472\) 3.81966i 0.175814i
\(473\) − 26.4164i − 1.21463i
\(474\) −22.3607 −1.02706
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) 0.695048i 0.0318241i
\(478\) 23.4164i 1.07104i
\(479\) 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
\(482\) − 20.1803i − 0.919189i
\(483\) − 2.18034i − 0.0992089i
\(484\) −4.29180 −0.195082
\(485\) 0 0
\(486\) 22.0000 0.997940
\(487\) 10.2361i 0.463841i 0.972735 + 0.231920i \(0.0745008\pi\)
−0.972735 + 0.231920i \(0.925499\pi\)
\(488\) 8.29180i 0.375352i
\(489\) 12.9443 0.585360
\(490\) 0 0
\(491\) −10.2361 −0.461947 −0.230974 0.972960i \(-0.574191\pi\)
−0.230974 + 0.972960i \(0.574191\pi\)
\(492\) 7.05573i 0.318097i
\(493\) 32.3607i 1.45745i
\(494\) 23.4164 1.05355
\(495\) 0 0
\(496\) 47.1246 2.11596
\(497\) − 4.70820i − 0.211192i
\(498\) − 11.4164i − 0.511581i
\(499\) −28.9443 −1.29572 −0.647862 0.761758i \(-0.724337\pi\)
−0.647862 + 0.761758i \(0.724337\pi\)
\(500\) 0 0
\(501\) 0.944272 0.0421870
\(502\) − 4.00000i − 0.178529i
\(503\) 43.8885i 1.95689i 0.206499 + 0.978447i \(0.433793\pi\)
−0.206499 + 0.978447i \(0.566207\pi\)
\(504\) −3.29180 −0.146628
\(505\) 0 0
\(506\) 12.0902 0.537474
\(507\) − 3.12461i − 0.138769i
\(508\) 10.9098i 0.484045i
\(509\) −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
\(512\) − 5.29180i − 0.233867i
\(513\) 24.7214i 1.09147i
\(514\) −30.1803 −1.33120
\(515\) 0 0
\(516\) 4.76393 0.209720
\(517\) − 8.47214i − 0.372604i
\(518\) − 4.85410i − 0.213277i
\(519\) 25.3050 1.11076
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) − 11.9098i − 0.521279i
\(523\) 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
\(528\) 25.4164i 1.10611i
\(529\) 19.8885 0.864719
\(530\) 0 0
\(531\) 2.51471 0.109129
\(532\) 2.76393i 0.119832i
\(533\) 29.8885i 1.29462i
\(534\) −25.5279 −1.10470
\(535\) 0 0
\(536\) −0.527864 −0.0228003
\(537\) 4.22291i 0.182232i
\(538\) − 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
\(543\) 17.5279i 0.752193i
\(544\) −21.8885 −0.938464
\(545\) 0 0
\(546\) −6.47214 −0.276982
\(547\) − 14.2361i − 0.608690i −0.952562 0.304345i \(-0.901562\pi\)
0.952562 0.304345i \(-0.0984376\pi\)
\(548\) − 4.29180i − 0.183336i
\(549\) 5.45898 0.232984
\(550\) 0 0
\(551\) 22.3607 0.952597
\(552\) − 4.87539i − 0.207510i
\(553\) − 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 \(-0.600064\pi\)
0.309208 0.950994i \(-0.399936\pi\)
\(558\) − 23.1246i − 0.978943i
\(559\) 20.1803 0.853537
\(560\) 0 0
\(561\) −33.8885 −1.43078
\(562\) 47.5066i 2.00394i
\(563\) 9.41641i 0.396854i 0.980116 + 0.198427i \(0.0635833\pi\)
−0.980116 + 0.198427i \(0.936417\pi\)
\(564\) 1.52786 0.0643347
\(565\) 0 0
\(566\) −15.2361 −0.640420
\(567\) − 2.41641i − 0.101480i
\(568\) − 10.5279i − 0.441739i
\(569\) −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 \(0.584903\pi\)
\(572\) − 8.47214i − 0.354238i
\(573\) 3.05573i 0.127655i
\(574\) −14.9443 −0.623762
\(575\) 0 0
\(576\) −6.23607 −0.259836
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 40.2705i − 1.67503i
\(579\) 17.8197 0.740560
\(580\) 0 0
\(581\) 5.70820 0.236816
\(582\) 1.52786i 0.0633320i
\(583\) 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 \(-0.793832\pi\)
0.797476 0.603351i \(-0.206168\pi\)
\(588\) − 0.763932i − 0.0315040i
\(589\) 43.4164 1.78894
\(590\) 0 0
\(591\) 9.23607 0.379921
\(592\) − 14.5623i − 0.598507i
\(593\) − 25.3050i − 1.03915i −0.854425 0.519575i \(-0.826090\pi\)
0.854425 0.519575i \(-0.173910\pi\)
\(594\) 37.8885 1.55459
\(595\) 0 0
\(596\) −8.61803 −0.353008
\(597\) 3.41641i 0.139824i
\(598\) 9.23607i 0.377691i
\(599\) −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 \(0.627058\pi\)
\(602\) 10.0902i 0.411245i
\(603\) 0.347524i 0.0141523i
\(604\) 9.74265 0.396423
\(605\) 0 0
\(606\) 18.4721 0.750379
\(607\) 33.1246i 1.34449i 0.740330 + 0.672243i \(0.234669\pi\)
−0.740330 + 0.672243i \(0.765331\pi\)
\(608\) 15.1246i 0.613384i
\(609\) −6.18034 −0.250440
\(610\) 0 0
\(611\) 6.47214 0.261835
\(612\) 5.88854i 0.238030i
\(613\) 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 \(0.0772882\pi\)
−0.970667 + 0.240429i \(0.922712\pi\)
\(618\) 0.944272i 0.0379842i
\(619\) −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
\(622\) − 32.9443i − 1.32094i
\(623\) − 12.7639i − 0.511376i
\(624\) −19.4164 −0.777278
\(625\) 0 0
\(626\) 46.0689 1.84128
\(627\) 23.4164i 0.935161i
\(628\) − 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
\(633\) − 14.8328i − 0.589551i
\(634\) 31.3262 1.24412
\(635\) 0 0
\(636\) −0.360680 −0.0143019
\(637\) − 3.23607i − 0.128218i
\(638\) − 34.2705i − 1.35678i
\(639\) −6.93112 −0.274191
\(640\) 0 0
\(641\) −9.83282 −0.388373 −0.194186 0.980965i \(-0.562207\pi\)
−0.194186 + 0.980965i \(0.562207\pi\)
\(642\) 16.0000i 0.631470i
\(643\) − 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 \(0.101107\pi\)
−0.949976 + 0.312322i \(0.898893\pi\)
\(648\) − 5.40325i − 0.212260i
\(649\) 7.23607 0.284041
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) − 6.47214i − 0.253468i
\(653\) − 42.9443i − 1.68054i −0.542169 0.840270i \(-0.682397\pi\)
0.542169 0.840270i \(-0.317603\pi\)
\(654\) 36.8328 1.44028
\(655\) 0 0
\(656\) −44.8328 −1.75043
\(657\) 19.4853i 0.760194i
\(658\) 3.23607i 0.126155i
\(659\) 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 \(0.137133\pi\)
\(662\) − 18.2705i − 0.710104i
\(663\) − 25.8885i − 1.00543i
\(664\) 12.7639 0.495337
\(665\) 0 0
\(666\) −7.14590 −0.276898
\(667\) 8.81966i 0.341499i
\(668\) − 0.472136i − 0.0182675i
\(669\) 2.69505 0.104197
\(670\) 0 0
\(671\) 15.7082 0.606408
\(672\) − 4.18034i − 0.161260i
\(673\) − 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 \(0.198289\pi\)
−0.812165 + 0.583428i \(0.801711\pi\)
\(678\) − 24.8328i − 0.953699i
\(679\) −0.763932 −0.0293170
\(680\) 0 0
\(681\) −6.69505 −0.256555
\(682\) − 66.5410i − 2.54799i
\(683\) 26.1246i 0.999630i 0.866132 + 0.499815i \(0.166599\pi\)
−0.866132 + 0.499815i \(0.833401\pi\)
\(684\) 4.06888 0.155578
\(685\) 0 0
\(686\) 1.61803 0.0617768
\(687\) 5.52786i 0.210901i
\(688\) 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
\(693\) 6.23607i 0.236889i
\(694\) −25.5066 −0.968216
\(695\) 0 0
\(696\) −13.8197 −0.523833
\(697\) − 59.7771i − 2.26422i
\(698\) 7.23607i 0.273889i
\(699\) 12.2918 0.464918
\(700\) 0 0
\(701\) −46.9443 −1.77306 −0.886530 0.462670i \(-0.846891\pi\)
−0.886530 + 0.462670i \(0.846891\pi\)
\(702\) 28.9443i 1.09243i
\(703\) − 13.4164i − 0.506009i
\(704\) −17.9443 −0.676300
\(705\) 0 0
\(706\) 32.6525 1.22889
\(707\) 9.23607i 0.347358i
\(708\) 1.30495i 0.0490431i
\(709\) 47.8885 1.79849 0.899246 0.437443i \(-0.144116\pi\)
0.899246 + 0.437443i \(0.144116\pi\)
\(710\) 0 0
\(711\) −16.4590 −0.617260
\(712\) − 28.5410i − 1.06962i
\(713\) 17.1246i 0.641322i
\(714\) 12.9443 0.484427
\(715\) 0 0
\(716\) 2.11146 0.0789088
\(717\) − 17.8885i − 0.668060i
\(718\) − 16.3820i − 0.611370i
\(719\) 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
\(722\) 1.61803i 0.0602170i
\(723\) 15.4164i 0.573342i
\(724\) 8.76393 0.325709
\(725\) 0 0
\(726\) 13.8885 0.515452
\(727\) − 20.9443i − 0.776780i −0.921495 0.388390i \(-0.873031\pi\)
0.921495 0.388390i \(-0.126969\pi\)
\(728\) − 7.23607i − 0.268187i
\(729\) −24.0557 −0.890953
\(730\) 0 0
\(731\) −40.3607 −1.49279
\(732\) 2.83282i 0.104704i
\(733\) − 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
\(738\) 22.0000i 0.809831i
\(739\) 5.65248 0.207930 0.103965 0.994581i \(-0.466847\pi\)
0.103965 + 0.994581i \(0.466847\pi\)
\(740\) 0 0
\(741\) −17.8885 −0.657152
\(742\) − 0.763932i − 0.0280448i
\(743\) 1.52786i 0.0560519i 0.999607 + 0.0280259i \(0.00892210\pi\)
−0.999607 + 0.0280259i \(0.991078\pi\)
\(744\) −26.8328 −0.983739
\(745\) 0 0
\(746\) 25.6180 0.937943
\(747\) − 8.40325i − 0.307459i
\(748\) 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
\(753\) 3.05573i 0.111357i
\(754\) 26.1803 0.953432
\(755\) 0 0
\(756\) −3.41641 −0.124254
\(757\) 46.4164i 1.68703i 0.537103 + 0.843517i \(0.319519\pi\)
−0.537103 + 0.843517i \(0.680481\pi\)
\(758\) − 18.0902i − 0.657065i
\(759\) −9.23607 −0.335248
\(760\) 0 0
\(761\) −43.7771 −1.58692 −0.793459 0.608624i \(-0.791722\pi\)
−0.793459 + 0.608624i \(0.791722\pi\)
\(762\) − 35.3050i − 1.27896i
\(763\) 18.4164i 0.666719i
\(764\) 1.52786 0.0552762
\(765\) 0 0
\(766\) −46.5410 −1.68160
\(767\) 5.52786i 0.199600i
\(768\) − 16.7639i − 0.604916i
\(769\) −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
\(772\) − 8.90983i − 0.320672i
\(773\) − 27.8197i − 1.00060i −0.865851 0.500302i \(-0.833222\pi\)
0.865851 0.500302i \(-0.166778\pi\)
\(774\) 14.8541 0.533920
\(775\) 0 0
\(776\) −1.70820 −0.0613209
\(777\) 3.70820i 0.133031i
\(778\) 53.2148i 1.90784i
\(779\) −41.3050 −1.47990
\(780\) 0 0
\(781\) −19.9443 −0.713662
\(782\) − 18.4721i − 0.660562i
\(783\) 27.6393i 0.987749i
\(784\) 4.85410 0.173361
\(785\) 0 0
\(786\) 1.88854 0.0673621
\(787\) 45.2361i 1.61249i 0.591581 + 0.806246i \(0.298504\pi\)
−0.591581 + 0.806246i \(0.701496\pi\)
\(788\) − 4.61803i − 0.164511i
\(789\) −14.5410 −0.517674
\(790\) 0 0
\(791\) 12.4164 0.441477
\(792\) 13.9443i 0.495488i
\(793\) 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 \(-0.951421\pi\)
0.988377 0.152023i \(-0.0485789\pi\)
\(798\) − 8.94427i − 0.316624i
\(799\) −12.9443 −0.457935
\(800\) 0 0
\(801\) −18.7902 −0.663921
\(802\) 18.5623i 0.655458i
\(803\) 56.0689i 1.97863i
\(804\) −0.180340 −0.00636010
\(805\) 0 0
\(806\) 50.8328 1.79051
\(807\) 2.11146i 0.0743268i
\(808\) 20.6525i 0.726552i
\(809\) −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 \(0.193804\pi\)
\(812\) 3.09017i 0.108444i
\(813\) − 12.7214i − 0.446158i
\(814\) −20.5623 −0.720708
\(815\) 0 0
\(816\) 38.8328 1.35942
\(817\) 27.8885i 0.975697i
\(818\) − 25.1246i − 0.878461i
\(819\) −4.76393 −0.166465
\(820\) 0 0
\(821\) −24.8328 −0.866671 −0.433336 0.901233i \(-0.642664\pi\)
−0.433336 + 0.901233i \(0.642664\pi\)
\(822\) 13.8885i 0.484419i
\(823\) 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 \(-0.853533\pi\)
0.895990 0.444074i \(-0.146467\pi\)
\(828\) 1.60488i 0.0557734i
\(829\) 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
\(832\) − 13.7082i − 0.475246i
\(833\) 6.47214i 0.224246i
\(834\) 41.3050 1.43027
\(835\) 0 0
\(836\) 11.7082 0.404937
\(837\) 53.6656i 1.85496i
\(838\) 6.18034i 0.213496i
\(839\) −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
\(842\) − 21.0344i − 0.724895i
\(843\) − 36.2918i − 1.24996i
\(844\) −7.41641 −0.255283
\(845\) 0 0
\(846\) 4.76393 0.163787
\(847\) 6.94427i 0.238608i
\(848\) − 2.29180i − 0.0787006i
\(849\) 11.6393 0.399460
\(850\) 0 0
\(851\) 5.29180 0.181400
\(852\) − 3.59675i − 0.123223i
\(853\) − 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 \(-0.773885\pi\)
0.758126 0.652108i \(-0.226115\pi\)
\(858\) 27.4164i 0.935981i
\(859\) 22.3607 0.762937 0.381468 0.924382i \(-0.375419\pi\)
0.381468 + 0.924382i \(0.375419\pi\)
\(860\) 0 0
\(861\) 11.4164 0.389070
\(862\) 42.8328i 1.45889i
\(863\) − 49.6525i − 1.69019i −0.534616 0.845095i \(-0.679544\pi\)
0.534616 0.845095i \(-0.320456\pi\)
\(864\) −18.6950 −0.636018
\(865\) 0 0
\(866\) 26.4721 0.899560
\(867\) 30.7639i 1.04480i
\(868\) 6.00000i 0.203653i
\(869\) −47.3607 −1.60660
\(870\) 0 0
\(871\) −0.763932 −0.0258848
\(872\) 41.1803i 1.39454i
\(873\) 1.12461i 0.0380623i
\(874\) −12.7639 −0.431746
\(875\) 0 0
\(876\) −10.1115 −0.341634
\(877\) − 14.3607i − 0.484926i −0.970161 0.242463i \(-0.922045\pi\)
0.970161 0.242463i \(-0.0779553\pi\)
\(878\) 35.1246i 1.18540i
\(879\) −11.2786 −0.380419
\(880\) 0 0
\(881\) 28.1803 0.949420 0.474710 0.880142i \(-0.342553\pi\)
0.474710 + 0.880142i \(0.342553\pi\)
\(882\) − 2.38197i − 0.0802050i
\(883\) 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 \(-0.654876\pi\)
0.467584 0.883949i \(-0.345124\pi\)
\(888\) 8.29180i 0.278254i
\(889\) 17.6525 0.592045
\(890\) 0 0
\(891\) −10.2361 −0.342921
\(892\) − 1.34752i − 0.0451184i
\(893\) 8.94427i 0.299309i
\(894\) 27.8885 0.932732
\(895\) 0 0
\(896\) 13.6180 0.454947
\(897\) − 7.05573i − 0.235584i
\(898\) − 47.6869i − 1.59133i
\(899\) 48.5410 1.61893
\(900\) 0 0
\(901\) 3.05573 0.101801
\(902\) 63.3050i 2.10782i
\(903\) − 7.70820i − 0.256513i
\(904\) 27.7639 0.923415
\(905\) 0 0
\(906\) −31.5279 −1.04744
\(907\) − 18.8328i − 0.625333i −0.949863 0.312667i \(-0.898778\pi\)
0.949863 0.312667i \(-0.101222\pi\)
\(908\) 3.34752i 0.111091i
\(909\) 13.5967 0.450976
\(910\) 0 0
\(911\) 23.1803 0.767999 0.383999 0.923333i \(-0.374546\pi\)
0.383999 + 0.923333i \(0.374546\pi\)
\(912\) − 26.8328i − 0.888523i
\(913\) − 24.1803i − 0.800252i
\(914\) 34.7426 1.14918
\(915\) 0 0
\(916\) 2.76393 0.0913229
\(917\) 0.944272i 0.0311826i
\(918\) − 57.8885i − 1.91061i
\(919\) 32.2361 1.06337 0.531685 0.846942i \(-0.321559\pi\)
0.531685 + 0.846942i \(0.321559\pi\)
\(920\) 0 0
\(921\) 38.8328 1.27958
\(922\) 13.2361i 0.435907i
\(923\) − 15.2361i − 0.501501i
\(924\) −3.23607 −0.106459
\(925\) 0 0
\(926\) 35.4164 1.16386
\(927\) 0.695048i 0.0228284i
\(928\) 16.9098i 0.555092i
\(929\) −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
\(932\) − 6.14590i − 0.201316i
\(933\) 25.1672i 0.823937i
\(934\) 17.7082 0.579430
\(935\) 0 0
\(936\) −10.6525 −0.348187
\(937\) 30.7639i 1.00501i 0.864573 + 0.502507i \(0.167589\pi\)
−0.864573 + 0.502507i \(0.832411\pi\)
\(938\) − 0.381966i − 0.0124716i
\(939\) −35.1935 −1.14850
\(940\) 0 0
\(941\) −0.763932 −0.0249035 −0.0124517 0.999922i \(-0.503964\pi\)
−0.0124517 + 0.999922i \(0.503964\pi\)
\(942\) 10.4721i 0.341201i
\(943\) − 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 \(-0.901012\pi\)
0.952034 0.305992i \(-0.0989881\pi\)
\(948\) − 8.54102i − 0.277399i
\(949\) −42.8328 −1.39041
\(950\) 0 0
\(951\) −23.9311 −0.776020
\(952\) 14.4721i 0.469045i
\(953\) 5.47214i 0.177260i 0.996065 + 0.0886299i \(0.0282489\pi\)
−0.996065 + 0.0886299i \(0.971751\pi\)
\(954\) −1.12461 −0.0364107
\(955\) 0 0
\(956\) −8.94427 −0.289278
\(957\) 26.1803i 0.846290i
\(958\) 6.18034i 0.199678i
\(959\) −6.94427 −0.224242
\(960\) 0 0
\(961\) 63.2492 2.04030
\(962\) − 15.7082i − 0.506453i
\(963\) 11.7771i 0.379511i
\(964\) 7.70820 0.248265
\(965\) 0 0
\(966\) 3.52786 0.113507
\(967\) − 49.8885i − 1.60431i −0.597118 0.802154i \(-0.703687\pi\)
0.597118 0.802154i \(-0.296313\pi\)
\(968\) 15.5279i 0.499084i
\(969\) 35.7771 1.14933
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 8.40325i 0.269534i
\(973\) 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 \(-0.987125\pi\)
0.999182 0.0404368i \(-0.0128749\pi\)
\(978\) 20.9443i 0.669724i
\(979\) −54.0689 −1.72805
\(980\) 0 0
\(981\) 27.1115 0.865602
\(982\) − 16.5623i − 0.528524i
\(983\) − 32.5410i − 1.03790i −0.854805 0.518949i \(-0.826324\pi\)
0.854805 0.518949i \(-0.173676\pi\)
\(984\) 25.5279 0.813799
\(985\) 0 0
\(986\) −52.3607 −1.66750
\(987\) − 2.47214i − 0.0786890i
\(988\) 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
\(993\) 13.9574i 0.442926i
\(994\) 7.61803 0.241629
\(995\) 0 0
\(996\) 4.36068 0.138173
\(997\) − 18.5836i − 0.588548i −0.955721 0.294274i \(-0.904922\pi\)
0.955721 0.294274i \(-0.0950779\pi\)
\(998\) − 46.8328i − 1.48247i
\(999\) 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 175.2.b.c.99.4 4
3.2 odd 2 1575.2.d.k.1324.1 4
4.3 odd 2 2800.2.g.s.449.3 4
5.2 odd 4 175.2.a.d.1.1 2
5.3 odd 4 175.2.a.e.1.2 yes 2
5.4 even 2 inner 175.2.b.c.99.1 4
7.6 odd 2 1225.2.b.k.99.4 4
15.2 even 4 1575.2.a.s.1.2 2
15.8 even 4 1575.2.a.n.1.1 2
15.14 odd 2 1575.2.d.k.1324.4 4
20.3 even 4 2800.2.a.bp.1.1 2
20.7 even 4 2800.2.a.bh.1.2 2
20.19 odd 2 2800.2.g.s.449.2 4
35.13 even 4 1225.2.a.u.1.2 2
35.27 even 4 1225.2.a.n.1.1 2
35.34 odd 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 5.2 odd 4
175.2.a.e.1.2 yes 2 5.3 odd 4
175.2.b.c.99.1 4 5.4 even 2 inner
175.2.b.c.99.4 4 1.1 even 1 trivial
1225.2.a.n.1.1 2 35.27 even 4
1225.2.a.u.1.2 2 35.13 even 4
1225.2.b.k.99.1 4 35.34 odd 2
1225.2.b.k.99.4 4 7.6 odd 2
1575.2.a.n.1.1 2 15.8 even 4
1575.2.a.s.1.2 2 15.2 even 4
1575.2.d.k.1324.1 4 3.2 odd 2
1575.2.d.k.1324.4 4 15.14 odd 2
2800.2.a.bh.1.2 2 20.7 even 4
2800.2.a.bp.1.1 2 20.3 even 4
2800.2.g.s.449.2 4 20.19 odd 2
2800.2.g.s.449.3 4 4.3 odd 2