Properties

Label 7500.2.a.m.1.12
Level $7500$
Weight $2$
Character 7500.1
Self dual yes
Analytic conductor $59.888$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7500,2,Mod(1,7500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7500.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.8878015160\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 11 x^{10} + 94 x^{9} + 27 x^{8} - 460 x^{7} + 55 x^{6} + 812 x^{5} - 127 x^{4} + \cdots - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{3} \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(4.54905\) of defining polynomial
Character \(\chi\) \(=\) 7500.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +4.13266 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +4.13266 q^{7} +1.00000 q^{9} -3.76632 q^{11} +0.698665 q^{13} +5.29251 q^{17} +5.73494 q^{19} -4.13266 q^{21} -5.46649 q^{23} -1.00000 q^{27} -7.02438 q^{29} -10.0920 q^{31} +3.76632 q^{33} -5.78488 q^{37} -0.698665 q^{39} -6.59552 q^{41} -4.79668 q^{43} -9.67379 q^{47} +10.0789 q^{49} -5.29251 q^{51} +3.39638 q^{53} -5.73494 q^{57} +0.745297 q^{59} +11.8079 q^{61} +4.13266 q^{63} +4.79759 q^{67} +5.46649 q^{69} -3.14256 q^{71} -10.7473 q^{73} -15.5649 q^{77} -9.12492 q^{79} +1.00000 q^{81} -8.44098 q^{83} +7.02438 q^{87} -3.38783 q^{89} +2.88735 q^{91} +10.0920 q^{93} -10.4351 q^{97} -3.76632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} - 8 q^{7} + 12 q^{9} + 2 q^{11} - 8 q^{17} + 10 q^{19} + 8 q^{21} - 18 q^{23} - 12 q^{27} + 8 q^{29} - 2 q^{31} - 2 q^{33} - 4 q^{37} + 10 q^{41} - 28 q^{43} - 22 q^{47} + 28 q^{49} + 8 q^{51} - 16 q^{53} - 10 q^{57} - 2 q^{59} + 34 q^{61} - 8 q^{63} - 32 q^{67} + 18 q^{69} - 24 q^{73} - 18 q^{77} + 6 q^{79} + 12 q^{81} - 28 q^{83} - 8 q^{87} + 10 q^{89} + 20 q^{91} + 2 q^{93} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.13266 1.56200 0.780999 0.624532i \(-0.214710\pi\)
0.780999 + 0.624532i \(0.214710\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.76632 −1.13559 −0.567794 0.823171i \(-0.692203\pi\)
−0.567794 + 0.823171i \(0.692203\pi\)
\(12\) 0 0
\(13\) 0.698665 0.193775 0.0968875 0.995295i \(-0.469111\pi\)
0.0968875 + 0.995295i \(0.469111\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.29251 1.28362 0.641811 0.766863i \(-0.278184\pi\)
0.641811 + 0.766863i \(0.278184\pi\)
\(18\) 0 0
\(19\) 5.73494 1.31568 0.657842 0.753156i \(-0.271469\pi\)
0.657842 + 0.753156i \(0.271469\pi\)
\(20\) 0 0
\(21\) −4.13266 −0.901820
\(22\) 0 0
\(23\) −5.46649 −1.13984 −0.569921 0.821699i \(-0.693026\pi\)
−0.569921 + 0.821699i \(0.693026\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.02438 −1.30439 −0.652197 0.758049i \(-0.726153\pi\)
−0.652197 + 0.758049i \(0.726153\pi\)
\(30\) 0 0
\(31\) −10.0920 −1.81258 −0.906288 0.422660i \(-0.861096\pi\)
−0.906288 + 0.422660i \(0.861096\pi\)
\(32\) 0 0
\(33\) 3.76632 0.655632
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.78488 −0.951029 −0.475515 0.879708i \(-0.657738\pi\)
−0.475515 + 0.879708i \(0.657738\pi\)
\(38\) 0 0
\(39\) −0.698665 −0.111876
\(40\) 0 0
\(41\) −6.59552 −1.03005 −0.515024 0.857176i \(-0.672217\pi\)
−0.515024 + 0.857176i \(0.672217\pi\)
\(42\) 0 0
\(43\) −4.79668 −0.731488 −0.365744 0.930716i \(-0.619185\pi\)
−0.365744 + 0.930716i \(0.619185\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.67379 −1.41107 −0.705534 0.708676i \(-0.749293\pi\)
−0.705534 + 0.708676i \(0.749293\pi\)
\(48\) 0 0
\(49\) 10.0789 1.43984
\(50\) 0 0
\(51\) −5.29251 −0.741099
\(52\) 0 0
\(53\) 3.39638 0.466528 0.233264 0.972413i \(-0.425059\pi\)
0.233264 + 0.972413i \(0.425059\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.73494 −0.759611
\(58\) 0 0
\(59\) 0.745297 0.0970294 0.0485147 0.998822i \(-0.484551\pi\)
0.0485147 + 0.998822i \(0.484551\pi\)
\(60\) 0 0
\(61\) 11.8079 1.51185 0.755926 0.654657i \(-0.227187\pi\)
0.755926 + 0.654657i \(0.227187\pi\)
\(62\) 0 0
\(63\) 4.13266 0.520666
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.79759 0.586119 0.293059 0.956094i \(-0.405327\pi\)
0.293059 + 0.956094i \(0.405327\pi\)
\(68\) 0 0
\(69\) 5.46649 0.658088
\(70\) 0 0
\(71\) −3.14256 −0.372953 −0.186477 0.982459i \(-0.559707\pi\)
−0.186477 + 0.982459i \(0.559707\pi\)
\(72\) 0 0
\(73\) −10.7473 −1.25788 −0.628939 0.777455i \(-0.716510\pi\)
−0.628939 + 0.777455i \(0.716510\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.5649 −1.77379
\(78\) 0 0
\(79\) −9.12492 −1.02663 −0.513317 0.858199i \(-0.671583\pi\)
−0.513317 + 0.858199i \(0.671583\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.44098 −0.926518 −0.463259 0.886223i \(-0.653320\pi\)
−0.463259 + 0.886223i \(0.653320\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.02438 0.753093
\(88\) 0 0
\(89\) −3.38783 −0.359109 −0.179555 0.983748i \(-0.557466\pi\)
−0.179555 + 0.983748i \(0.557466\pi\)
\(90\) 0 0
\(91\) 2.88735 0.302676
\(92\) 0 0
\(93\) 10.0920 1.04649
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.4351 −1.05952 −0.529760 0.848147i \(-0.677718\pi\)
−0.529760 + 0.848147i \(0.677718\pi\)
\(98\) 0 0
\(99\) −3.76632 −0.378529
\(100\) 0 0
\(101\) 7.10799 0.707272 0.353636 0.935383i \(-0.384945\pi\)
0.353636 + 0.935383i \(0.384945\pi\)
\(102\) 0 0
\(103\) 12.2091 1.20299 0.601497 0.798875i \(-0.294571\pi\)
0.601497 + 0.798875i \(0.294571\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.04303 −0.390855 −0.195427 0.980718i \(-0.562609\pi\)
−0.195427 + 0.980718i \(0.562609\pi\)
\(108\) 0 0
\(109\) −0.773815 −0.0741180 −0.0370590 0.999313i \(-0.511799\pi\)
−0.0370590 + 0.999313i \(0.511799\pi\)
\(110\) 0 0
\(111\) 5.78488 0.549077
\(112\) 0 0
\(113\) −1.13943 −0.107189 −0.0535944 0.998563i \(-0.517068\pi\)
−0.0535944 + 0.998563i \(0.517068\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.698665 0.0645916
\(118\) 0 0
\(119\) 21.8721 2.00501
\(120\) 0 0
\(121\) 3.18514 0.289559
\(122\) 0 0
\(123\) 6.59552 0.594698
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.537868 0.0477281 0.0238640 0.999715i \(-0.492403\pi\)
0.0238640 + 0.999715i \(0.492403\pi\)
\(128\) 0 0
\(129\) 4.79668 0.422325
\(130\) 0 0
\(131\) −6.72242 −0.587341 −0.293670 0.955907i \(-0.594877\pi\)
−0.293670 + 0.955907i \(0.594877\pi\)
\(132\) 0 0
\(133\) 23.7005 2.05510
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0.183379 0.0156671 0.00783355 0.999969i \(-0.497506\pi\)
0.00783355 + 0.999969i \(0.497506\pi\)
\(138\) 0 0
\(139\) 8.22006 0.697217 0.348608 0.937268i \(-0.386654\pi\)
0.348608 + 0.937268i \(0.386654\pi\)
\(140\) 0 0
\(141\) 9.67379 0.814680
\(142\) 0 0
\(143\) −2.63140 −0.220048
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.0789 −0.831292
\(148\) 0 0
\(149\) 5.53790 0.453682 0.226841 0.973932i \(-0.427160\pi\)
0.226841 + 0.973932i \(0.427160\pi\)
\(150\) 0 0
\(151\) 12.8736 1.04764 0.523821 0.851828i \(-0.324506\pi\)
0.523821 + 0.851828i \(0.324506\pi\)
\(152\) 0 0
\(153\) 5.29251 0.427874
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.6395 −1.16836 −0.584179 0.811625i \(-0.698583\pi\)
−0.584179 + 0.811625i \(0.698583\pi\)
\(158\) 0 0
\(159\) −3.39638 −0.269350
\(160\) 0 0
\(161\) −22.5912 −1.78043
\(162\) 0 0
\(163\) −4.44424 −0.348099 −0.174050 0.984737i \(-0.555685\pi\)
−0.174050 + 0.984737i \(0.555685\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.7655 −1.83903 −0.919514 0.393057i \(-0.871417\pi\)
−0.919514 + 0.393057i \(0.871417\pi\)
\(168\) 0 0
\(169\) −12.5119 −0.962451
\(170\) 0 0
\(171\) 5.73494 0.438562
\(172\) 0 0
\(173\) −15.0868 −1.14703 −0.573513 0.819196i \(-0.694420\pi\)
−0.573513 + 0.819196i \(0.694420\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.745297 −0.0560200
\(178\) 0 0
\(179\) 16.7163 1.24944 0.624718 0.780850i \(-0.285214\pi\)
0.624718 + 0.780850i \(0.285214\pi\)
\(180\) 0 0
\(181\) 10.3923 0.772454 0.386227 0.922404i \(-0.373778\pi\)
0.386227 + 0.922404i \(0.373778\pi\)
\(182\) 0 0
\(183\) −11.8079 −0.872869
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −19.9333 −1.45766
\(188\) 0 0
\(189\) −4.13266 −0.300607
\(190\) 0 0
\(191\) −19.7557 −1.42947 −0.714737 0.699393i \(-0.753454\pi\)
−0.714737 + 0.699393i \(0.753454\pi\)
\(192\) 0 0
\(193\) −2.23549 −0.160914 −0.0804571 0.996758i \(-0.525638\pi\)
−0.0804571 + 0.996758i \(0.525638\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.7568 1.62135 0.810677 0.585494i \(-0.199099\pi\)
0.810677 + 0.585494i \(0.199099\pi\)
\(198\) 0 0
\(199\) 22.8171 1.61746 0.808731 0.588179i \(-0.200155\pi\)
0.808731 + 0.588179i \(0.200155\pi\)
\(200\) 0 0
\(201\) −4.79759 −0.338396
\(202\) 0 0
\(203\) −29.0294 −2.03746
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.46649 −0.379947
\(208\) 0 0
\(209\) −21.5996 −1.49408
\(210\) 0 0
\(211\) 20.0966 1.38351 0.691753 0.722134i \(-0.256839\pi\)
0.691753 + 0.722134i \(0.256839\pi\)
\(212\) 0 0
\(213\) 3.14256 0.215325
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −41.7068 −2.83124
\(218\) 0 0
\(219\) 10.7473 0.726236
\(220\) 0 0
\(221\) 3.69769 0.248734
\(222\) 0 0
\(223\) 13.4188 0.898593 0.449296 0.893383i \(-0.351675\pi\)
0.449296 + 0.893383i \(0.351675\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.4913 −1.22731 −0.613654 0.789575i \(-0.710301\pi\)
−0.613654 + 0.789575i \(0.710301\pi\)
\(228\) 0 0
\(229\) −19.4125 −1.28281 −0.641406 0.767202i \(-0.721648\pi\)
−0.641406 + 0.767202i \(0.721648\pi\)
\(230\) 0 0
\(231\) 15.5649 1.02410
\(232\) 0 0
\(233\) −12.1833 −0.798151 −0.399076 0.916918i \(-0.630669\pi\)
−0.399076 + 0.916918i \(0.630669\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.12492 0.592727
\(238\) 0 0
\(239\) 4.52476 0.292682 0.146341 0.989234i \(-0.453250\pi\)
0.146341 + 0.989234i \(0.453250\pi\)
\(240\) 0 0
\(241\) 9.01177 0.580499 0.290250 0.956951i \(-0.406262\pi\)
0.290250 + 0.956951i \(0.406262\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00680 0.254947
\(248\) 0 0
\(249\) 8.44098 0.534925
\(250\) 0 0
\(251\) 3.16965 0.200066 0.100033 0.994984i \(-0.468105\pi\)
0.100033 + 0.994984i \(0.468105\pi\)
\(252\) 0 0
\(253\) 20.5885 1.29439
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.25241 0.202880 0.101440 0.994842i \(-0.467655\pi\)
0.101440 + 0.994842i \(0.467655\pi\)
\(258\) 0 0
\(259\) −23.9070 −1.48551
\(260\) 0 0
\(261\) −7.02438 −0.434798
\(262\) 0 0
\(263\) −11.0600 −0.681991 −0.340995 0.940065i \(-0.610764\pi\)
−0.340995 + 0.940065i \(0.610764\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.38783 0.207332
\(268\) 0 0
\(269\) 24.7558 1.50939 0.754695 0.656076i \(-0.227785\pi\)
0.754695 + 0.656076i \(0.227785\pi\)
\(270\) 0 0
\(271\) 0.250205 0.0151989 0.00759945 0.999971i \(-0.497581\pi\)
0.00759945 + 0.999971i \(0.497581\pi\)
\(272\) 0 0
\(273\) −2.88735 −0.174750
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.87074 0.532991 0.266496 0.963836i \(-0.414134\pi\)
0.266496 + 0.963836i \(0.414134\pi\)
\(278\) 0 0
\(279\) −10.0920 −0.604192
\(280\) 0 0
\(281\) 19.7379 1.17746 0.588732 0.808328i \(-0.299627\pi\)
0.588732 + 0.808328i \(0.299627\pi\)
\(282\) 0 0
\(283\) −26.4418 −1.57180 −0.785901 0.618352i \(-0.787801\pi\)
−0.785901 + 0.618352i \(0.787801\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −27.2570 −1.60893
\(288\) 0 0
\(289\) 11.0106 0.647683
\(290\) 0 0
\(291\) 10.4351 0.611714
\(292\) 0 0
\(293\) 19.1882 1.12099 0.560494 0.828158i \(-0.310611\pi\)
0.560494 + 0.828158i \(0.310611\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.76632 0.218544
\(298\) 0 0
\(299\) −3.81925 −0.220873
\(300\) 0 0
\(301\) −19.8231 −1.14258
\(302\) 0 0
\(303\) −7.10799 −0.408344
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −9.26979 −0.529055 −0.264527 0.964378i \(-0.585216\pi\)
−0.264527 + 0.964378i \(0.585216\pi\)
\(308\) 0 0
\(309\) −12.2091 −0.694549
\(310\) 0 0
\(311\) −24.8936 −1.41159 −0.705793 0.708418i \(-0.749409\pi\)
−0.705793 + 0.708418i \(0.749409\pi\)
\(312\) 0 0
\(313\) 6.04712 0.341803 0.170902 0.985288i \(-0.445332\pi\)
0.170902 + 0.985288i \(0.445332\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.3864 0.808023 0.404012 0.914754i \(-0.367616\pi\)
0.404012 + 0.914754i \(0.367616\pi\)
\(318\) 0 0
\(319\) 26.4560 1.48125
\(320\) 0 0
\(321\) 4.04303 0.225660
\(322\) 0 0
\(323\) 30.3522 1.68884
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.773815 0.0427921
\(328\) 0 0
\(329\) −39.9785 −2.20409
\(330\) 0 0
\(331\) 1.06688 0.0586411 0.0293206 0.999570i \(-0.490666\pi\)
0.0293206 + 0.999570i \(0.490666\pi\)
\(332\) 0 0
\(333\) −5.78488 −0.317010
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.6020 0.795421 0.397710 0.917511i \(-0.369805\pi\)
0.397710 + 0.917511i \(0.369805\pi\)
\(338\) 0 0
\(339\) 1.13943 0.0618854
\(340\) 0 0
\(341\) 38.0097 2.05834
\(342\) 0 0
\(343\) 12.7239 0.687029
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.1453 −0.866723 −0.433361 0.901220i \(-0.642673\pi\)
−0.433361 + 0.901220i \(0.642673\pi\)
\(348\) 0 0
\(349\) −22.2622 −1.19167 −0.595834 0.803108i \(-0.703178\pi\)
−0.595834 + 0.803108i \(0.703178\pi\)
\(350\) 0 0
\(351\) −0.698665 −0.0372920
\(352\) 0 0
\(353\) 9.52997 0.507229 0.253615 0.967305i \(-0.418381\pi\)
0.253615 + 0.967305i \(0.418381\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −21.8721 −1.15760
\(358\) 0 0
\(359\) −16.7872 −0.885995 −0.442998 0.896523i \(-0.646085\pi\)
−0.442998 + 0.896523i \(0.646085\pi\)
\(360\) 0 0
\(361\) 13.8895 0.731027
\(362\) 0 0
\(363\) −3.18514 −0.167177
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −32.4766 −1.69527 −0.847633 0.530583i \(-0.821973\pi\)
−0.847633 + 0.530583i \(0.821973\pi\)
\(368\) 0 0
\(369\) −6.59552 −0.343349
\(370\) 0 0
\(371\) 14.0361 0.728717
\(372\) 0 0
\(373\) 14.2912 0.739972 0.369986 0.929037i \(-0.379363\pi\)
0.369986 + 0.929037i \(0.379363\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.90769 −0.252759
\(378\) 0 0
\(379\) −3.21569 −0.165179 −0.0825895 0.996584i \(-0.526319\pi\)
−0.0825895 + 0.996584i \(0.526319\pi\)
\(380\) 0 0
\(381\) −0.537868 −0.0275558
\(382\) 0 0
\(383\) 12.3598 0.631556 0.315778 0.948833i \(-0.397734\pi\)
0.315778 + 0.948833i \(0.397734\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.79668 −0.243829
\(388\) 0 0
\(389\) 23.1248 1.17247 0.586236 0.810140i \(-0.300609\pi\)
0.586236 + 0.810140i \(0.300609\pi\)
\(390\) 0 0
\(391\) −28.9314 −1.46313
\(392\) 0 0
\(393\) 6.72242 0.339101
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.12824 −0.106814 −0.0534068 0.998573i \(-0.517008\pi\)
−0.0534068 + 0.998573i \(0.517008\pi\)
\(398\) 0 0
\(399\) −23.7005 −1.18651
\(400\) 0 0
\(401\) −33.8250 −1.68914 −0.844569 0.535447i \(-0.820143\pi\)
−0.844569 + 0.535447i \(0.820143\pi\)
\(402\) 0 0
\(403\) −7.05093 −0.351232
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.7877 1.07998
\(408\) 0 0
\(409\) −34.7663 −1.71908 −0.859542 0.511065i \(-0.829251\pi\)
−0.859542 + 0.511065i \(0.829251\pi\)
\(410\) 0 0
\(411\) −0.183379 −0.00904541
\(412\) 0 0
\(413\) 3.08006 0.151560
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −8.22006 −0.402538
\(418\) 0 0
\(419\) −6.79744 −0.332077 −0.166038 0.986119i \(-0.553098\pi\)
−0.166038 + 0.986119i \(0.553098\pi\)
\(420\) 0 0
\(421\) −10.8246 −0.527560 −0.263780 0.964583i \(-0.584969\pi\)
−0.263780 + 0.964583i \(0.584969\pi\)
\(422\) 0 0
\(423\) −9.67379 −0.470356
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 48.7982 2.36151
\(428\) 0 0
\(429\) 2.63140 0.127045
\(430\) 0 0
\(431\) −10.3869 −0.500319 −0.250159 0.968205i \(-0.580483\pi\)
−0.250159 + 0.968205i \(0.580483\pi\)
\(432\) 0 0
\(433\) 15.6627 0.752701 0.376350 0.926477i \(-0.377179\pi\)
0.376350 + 0.926477i \(0.377179\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −31.3500 −1.49967
\(438\) 0 0
\(439\) 22.0075 1.05036 0.525181 0.850990i \(-0.323997\pi\)
0.525181 + 0.850990i \(0.323997\pi\)
\(440\) 0 0
\(441\) 10.0789 0.479946
\(442\) 0 0
\(443\) −29.4447 −1.39896 −0.699480 0.714652i \(-0.746585\pi\)
−0.699480 + 0.714652i \(0.746585\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.53790 −0.261933
\(448\) 0 0
\(449\) 27.3445 1.29047 0.645234 0.763985i \(-0.276760\pi\)
0.645234 + 0.763985i \(0.276760\pi\)
\(450\) 0 0
\(451\) 24.8408 1.16971
\(452\) 0 0
\(453\) −12.8736 −0.604856
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −33.6394 −1.57359 −0.786793 0.617217i \(-0.788260\pi\)
−0.786793 + 0.617217i \(0.788260\pi\)
\(458\) 0 0
\(459\) −5.29251 −0.247033
\(460\) 0 0
\(461\) 25.4419 1.18495 0.592474 0.805589i \(-0.298151\pi\)
0.592474 + 0.805589i \(0.298151\pi\)
\(462\) 0 0
\(463\) −7.84127 −0.364415 −0.182207 0.983260i \(-0.558324\pi\)
−0.182207 + 0.983260i \(0.558324\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.8007 0.499795 0.249897 0.968272i \(-0.419603\pi\)
0.249897 + 0.968272i \(0.419603\pi\)
\(468\) 0 0
\(469\) 19.8268 0.915517
\(470\) 0 0
\(471\) 14.6395 0.674551
\(472\) 0 0
\(473\) 18.0658 0.830668
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.39638 0.155509
\(478\) 0 0
\(479\) 22.6753 1.03606 0.518030 0.855362i \(-0.326665\pi\)
0.518030 + 0.855362i \(0.326665\pi\)
\(480\) 0 0
\(481\) −4.04170 −0.184286
\(482\) 0 0
\(483\) 22.5912 1.02793
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −4.98587 −0.225931 −0.112966 0.993599i \(-0.536035\pi\)
−0.112966 + 0.993599i \(0.536035\pi\)
\(488\) 0 0
\(489\) 4.44424 0.200975
\(490\) 0 0
\(491\) −26.4530 −1.19381 −0.596904 0.802313i \(-0.703603\pi\)
−0.596904 + 0.802313i \(0.703603\pi\)
\(492\) 0 0
\(493\) −37.1766 −1.67435
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.9871 −0.582553
\(498\) 0 0
\(499\) 38.0875 1.70503 0.852516 0.522702i \(-0.175076\pi\)
0.852516 + 0.522702i \(0.175076\pi\)
\(500\) 0 0
\(501\) 23.7655 1.06176
\(502\) 0 0
\(503\) 2.35598 0.105048 0.0525240 0.998620i \(-0.483273\pi\)
0.0525240 + 0.998620i \(0.483273\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.5119 0.555672
\(508\) 0 0
\(509\) −29.1985 −1.29420 −0.647099 0.762406i \(-0.724018\pi\)
−0.647099 + 0.762406i \(0.724018\pi\)
\(510\) 0 0
\(511\) −44.4150 −1.96480
\(512\) 0 0
\(513\) −5.73494 −0.253204
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 36.4346 1.60239
\(518\) 0 0
\(519\) 15.0868 0.662236
\(520\) 0 0
\(521\) −42.6181 −1.86713 −0.933565 0.358407i \(-0.883320\pi\)
−0.933565 + 0.358407i \(0.883320\pi\)
\(522\) 0 0
\(523\) −20.6750 −0.904055 −0.452028 0.892004i \(-0.649299\pi\)
−0.452028 + 0.892004i \(0.649299\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −53.4120 −2.32666
\(528\) 0 0
\(529\) 6.88254 0.299241
\(530\) 0 0
\(531\) 0.745297 0.0323431
\(532\) 0 0
\(533\) −4.60806 −0.199597
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −16.7163 −0.721363
\(538\) 0 0
\(539\) −37.9602 −1.63506
\(540\) 0 0
\(541\) 11.9549 0.513980 0.256990 0.966414i \(-0.417269\pi\)
0.256990 + 0.966414i \(0.417269\pi\)
\(542\) 0 0
\(543\) −10.3923 −0.445976
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −26.6027 −1.13745 −0.568724 0.822529i \(-0.692563\pi\)
−0.568724 + 0.822529i \(0.692563\pi\)
\(548\) 0 0
\(549\) 11.8079 0.503951
\(550\) 0 0
\(551\) −40.2844 −1.71617
\(552\) 0 0
\(553\) −37.7102 −1.60360
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.3420 1.20089 0.600444 0.799667i \(-0.294990\pi\)
0.600444 + 0.799667i \(0.294990\pi\)
\(558\) 0 0
\(559\) −3.35128 −0.141744
\(560\) 0 0
\(561\) 19.9333 0.841583
\(562\) 0 0
\(563\) −4.05616 −0.170947 −0.0854735 0.996340i \(-0.527240\pi\)
−0.0854735 + 0.996340i \(0.527240\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.13266 0.173555
\(568\) 0 0
\(569\) 1.96688 0.0824558 0.0412279 0.999150i \(-0.486873\pi\)
0.0412279 + 0.999150i \(0.486873\pi\)
\(570\) 0 0
\(571\) 5.15778 0.215846 0.107923 0.994159i \(-0.465580\pi\)
0.107923 + 0.994159i \(0.465580\pi\)
\(572\) 0 0
\(573\) 19.7557 0.825307
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −38.6187 −1.60772 −0.803859 0.594819i \(-0.797224\pi\)
−0.803859 + 0.594819i \(0.797224\pi\)
\(578\) 0 0
\(579\) 2.23549 0.0929039
\(580\) 0 0
\(581\) −34.8837 −1.44722
\(582\) 0 0
\(583\) −12.7918 −0.529784
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −27.1719 −1.12151 −0.560753 0.827983i \(-0.689488\pi\)
−0.560753 + 0.827983i \(0.689488\pi\)
\(588\) 0 0
\(589\) −57.8770 −2.38478
\(590\) 0 0
\(591\) −22.7568 −0.936089
\(592\) 0 0
\(593\) 30.3486 1.24627 0.623135 0.782114i \(-0.285859\pi\)
0.623135 + 0.782114i \(0.285859\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.8171 −0.933842
\(598\) 0 0
\(599\) −26.9205 −1.09994 −0.549971 0.835184i \(-0.685361\pi\)
−0.549971 + 0.835184i \(0.685361\pi\)
\(600\) 0 0
\(601\) −16.9133 −0.689910 −0.344955 0.938619i \(-0.612106\pi\)
−0.344955 + 0.938619i \(0.612106\pi\)
\(602\) 0 0
\(603\) 4.79759 0.195373
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.9715 −0.770028 −0.385014 0.922911i \(-0.625803\pi\)
−0.385014 + 0.922911i \(0.625803\pi\)
\(608\) 0 0
\(609\) 29.0294 1.17633
\(610\) 0 0
\(611\) −6.75874 −0.273430
\(612\) 0 0
\(613\) −5.45816 −0.220453 −0.110227 0.993906i \(-0.535158\pi\)
−0.110227 + 0.993906i \(0.535158\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.8970 −0.478954 −0.239477 0.970902i \(-0.576976\pi\)
−0.239477 + 0.970902i \(0.576976\pi\)
\(618\) 0 0
\(619\) −13.2121 −0.531038 −0.265519 0.964106i \(-0.585543\pi\)
−0.265519 + 0.964106i \(0.585543\pi\)
\(620\) 0 0
\(621\) 5.46649 0.219363
\(622\) 0 0
\(623\) −14.0008 −0.560928
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 21.5996 0.862605
\(628\) 0 0
\(629\) −30.6165 −1.22076
\(630\) 0 0
\(631\) −11.6182 −0.462513 −0.231257 0.972893i \(-0.574284\pi\)
−0.231257 + 0.972893i \(0.574284\pi\)
\(632\) 0 0
\(633\) −20.0966 −0.798768
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.04176 0.279005
\(638\) 0 0
\(639\) −3.14256 −0.124318
\(640\) 0 0
\(641\) −34.8796 −1.37766 −0.688831 0.724922i \(-0.741876\pi\)
−0.688831 + 0.724922i \(0.741876\pi\)
\(642\) 0 0
\(643\) 2.90629 0.114613 0.0573065 0.998357i \(-0.481749\pi\)
0.0573065 + 0.998357i \(0.481749\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.1402 0.477280 0.238640 0.971108i \(-0.423298\pi\)
0.238640 + 0.971108i \(0.423298\pi\)
\(648\) 0 0
\(649\) −2.80702 −0.110185
\(650\) 0 0
\(651\) 41.7068 1.63462
\(652\) 0 0
\(653\) 26.0987 1.02132 0.510661 0.859782i \(-0.329401\pi\)
0.510661 + 0.859782i \(0.329401\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.7473 −0.419293
\(658\) 0 0
\(659\) 21.3268 0.830775 0.415388 0.909644i \(-0.363646\pi\)
0.415388 + 0.909644i \(0.363646\pi\)
\(660\) 0 0
\(661\) −31.6863 −1.23246 −0.616228 0.787568i \(-0.711340\pi\)
−0.616228 + 0.787568i \(0.711340\pi\)
\(662\) 0 0
\(663\) −3.69769 −0.143606
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 38.3987 1.48680
\(668\) 0 0
\(669\) −13.4188 −0.518803
\(670\) 0 0
\(671\) −44.4725 −1.71684
\(672\) 0 0
\(673\) 1.98577 0.0765456 0.0382728 0.999267i \(-0.487814\pi\)
0.0382728 + 0.999267i \(0.487814\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.4359 −1.51564 −0.757822 0.652462i \(-0.773736\pi\)
−0.757822 + 0.652462i \(0.773736\pi\)
\(678\) 0 0
\(679\) −43.1246 −1.65497
\(680\) 0 0
\(681\) 18.4913 0.708587
\(682\) 0 0
\(683\) 12.2965 0.470511 0.235256 0.971934i \(-0.424407\pi\)
0.235256 + 0.971934i \(0.424407\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 19.4125 0.740631
\(688\) 0 0
\(689\) 2.37293 0.0904015
\(690\) 0 0
\(691\) 15.7058 0.597475 0.298737 0.954335i \(-0.403435\pi\)
0.298737 + 0.954335i \(0.403435\pi\)
\(692\) 0 0
\(693\) −15.5649 −0.591262
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −34.9068 −1.32219
\(698\) 0 0
\(699\) 12.1833 0.460813
\(700\) 0 0
\(701\) −26.4222 −0.997953 −0.498976 0.866616i \(-0.666291\pi\)
−0.498976 + 0.866616i \(0.666291\pi\)
\(702\) 0 0
\(703\) −33.1759 −1.25125
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 29.3749 1.10476
\(708\) 0 0
\(709\) 48.7475 1.83075 0.915376 0.402601i \(-0.131894\pi\)
0.915376 + 0.402601i \(0.131894\pi\)
\(710\) 0 0
\(711\) −9.12492 −0.342211
\(712\) 0 0
\(713\) 55.1678 2.06605
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.52476 −0.168980
\(718\) 0 0
\(719\) −34.2535 −1.27744 −0.638721 0.769439i \(-0.720536\pi\)
−0.638721 + 0.769439i \(0.720536\pi\)
\(720\) 0 0
\(721\) 50.4559 1.87907
\(722\) 0 0
\(723\) −9.01177 −0.335151
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.3624 −0.792287 −0.396144 0.918189i \(-0.629652\pi\)
−0.396144 + 0.918189i \(0.629652\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −25.3865 −0.938953
\(732\) 0 0
\(733\) 31.2827 1.15545 0.577727 0.816230i \(-0.303940\pi\)
0.577727 + 0.816230i \(0.303940\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.0692 −0.665589
\(738\) 0 0
\(739\) 1.31491 0.0483699 0.0241849 0.999708i \(-0.492301\pi\)
0.0241849 + 0.999708i \(0.492301\pi\)
\(740\) 0 0
\(741\) −4.00680 −0.147194
\(742\) 0 0
\(743\) 8.76431 0.321531 0.160766 0.986993i \(-0.448604\pi\)
0.160766 + 0.986993i \(0.448604\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.44098 −0.308839
\(748\) 0 0
\(749\) −16.7085 −0.610515
\(750\) 0 0
\(751\) 44.6570 1.62956 0.814779 0.579772i \(-0.196858\pi\)
0.814779 + 0.579772i \(0.196858\pi\)
\(752\) 0 0
\(753\) −3.16965 −0.115508
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.70729 0.280126 0.140063 0.990143i \(-0.455269\pi\)
0.140063 + 0.990143i \(0.455269\pi\)
\(758\) 0 0
\(759\) −20.5885 −0.747317
\(760\) 0 0
\(761\) 39.5914 1.43519 0.717594 0.696461i \(-0.245243\pi\)
0.717594 + 0.696461i \(0.245243\pi\)
\(762\) 0 0
\(763\) −3.19791 −0.115772
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.520713 0.0188019
\(768\) 0 0
\(769\) 29.3363 1.05790 0.528948 0.848655i \(-0.322587\pi\)
0.528948 + 0.848655i \(0.322587\pi\)
\(770\) 0 0
\(771\) −3.25241 −0.117133
\(772\) 0 0
\(773\) −10.9101 −0.392411 −0.196205 0.980563i \(-0.562862\pi\)
−0.196205 + 0.980563i \(0.562862\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 23.9070 0.857657
\(778\) 0 0
\(779\) −37.8249 −1.35522
\(780\) 0 0
\(781\) 11.8359 0.423521
\(782\) 0 0
\(783\) 7.02438 0.251031
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 37.7075 1.34413 0.672063 0.740494i \(-0.265409\pi\)
0.672063 + 0.740494i \(0.265409\pi\)
\(788\) 0 0
\(789\) 11.0600 0.393747
\(790\) 0 0
\(791\) −4.70888 −0.167429
\(792\) 0 0
\(793\) 8.24980 0.292959
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 49.2412 1.74421 0.872106 0.489317i \(-0.162754\pi\)
0.872106 + 0.489317i \(0.162754\pi\)
\(798\) 0 0
\(799\) −51.1986 −1.81128
\(800\) 0 0
\(801\) −3.38783 −0.119703
\(802\) 0 0
\(803\) 40.4778 1.42843
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −24.7558 −0.871447
\(808\) 0 0
\(809\) 26.9783 0.948507 0.474253 0.880388i \(-0.342718\pi\)
0.474253 + 0.880388i \(0.342718\pi\)
\(810\) 0 0
\(811\) 8.17256 0.286977 0.143489 0.989652i \(-0.454168\pi\)
0.143489 + 0.989652i \(0.454168\pi\)
\(812\) 0 0
\(813\) −0.250205 −0.00877509
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −27.5087 −0.962407
\(818\) 0 0
\(819\) 2.88735 0.100892
\(820\) 0 0
\(821\) −43.7884 −1.52823 −0.764113 0.645082i \(-0.776823\pi\)
−0.764113 + 0.645082i \(0.776823\pi\)
\(822\) 0 0
\(823\) −16.4567 −0.573644 −0.286822 0.957984i \(-0.592599\pi\)
−0.286822 + 0.957984i \(0.592599\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.01834 0.139731 0.0698657 0.997556i \(-0.477743\pi\)
0.0698657 + 0.997556i \(0.477743\pi\)
\(828\) 0 0
\(829\) 37.4656 1.30124 0.650618 0.759405i \(-0.274510\pi\)
0.650618 + 0.759405i \(0.274510\pi\)
\(830\) 0 0
\(831\) −8.87074 −0.307723
\(832\) 0 0
\(833\) 53.3425 1.84821
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 10.0920 0.348831
\(838\) 0 0
\(839\) −11.5960 −0.400337 −0.200169 0.979761i \(-0.564149\pi\)
−0.200169 + 0.979761i \(0.564149\pi\)
\(840\) 0 0
\(841\) 20.3419 0.701445
\(842\) 0 0
\(843\) −19.7379 −0.679809
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 13.1631 0.452290
\(848\) 0 0
\(849\) 26.4418 0.907481
\(850\) 0 0
\(851\) 31.6230 1.08402
\(852\) 0 0
\(853\) −1.32197 −0.0452635 −0.0226317 0.999744i \(-0.507205\pi\)
−0.0226317 + 0.999744i \(0.507205\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.5623 0.531598 0.265799 0.964028i \(-0.414364\pi\)
0.265799 + 0.964028i \(0.414364\pi\)
\(858\) 0 0
\(859\) −27.3826 −0.934283 −0.467141 0.884183i \(-0.654716\pi\)
−0.467141 + 0.884183i \(0.654716\pi\)
\(860\) 0 0
\(861\) 27.2570 0.928917
\(862\) 0 0
\(863\) 32.0293 1.09029 0.545145 0.838342i \(-0.316475\pi\)
0.545145 + 0.838342i \(0.316475\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −11.0106 −0.373940
\(868\) 0 0
\(869\) 34.3673 1.16583
\(870\) 0 0
\(871\) 3.35191 0.113575
\(872\) 0 0
\(873\) −10.4351 −0.353173
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.29121 −0.144904 −0.0724520 0.997372i \(-0.523082\pi\)
−0.0724520 + 0.997372i \(0.523082\pi\)
\(878\) 0 0
\(879\) −19.1882 −0.647203
\(880\) 0 0
\(881\) −54.0804 −1.82202 −0.911008 0.412388i \(-0.864695\pi\)
−0.911008 + 0.412388i \(0.864695\pi\)
\(882\) 0 0
\(883\) −31.7020 −1.06686 −0.533428 0.845845i \(-0.679097\pi\)
−0.533428 + 0.845845i \(0.679097\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.38001 −0.180643 −0.0903216 0.995913i \(-0.528789\pi\)
−0.0903216 + 0.995913i \(0.528789\pi\)
\(888\) 0 0
\(889\) 2.22283 0.0745512
\(890\) 0 0
\(891\) −3.76632 −0.126176
\(892\) 0 0
\(893\) −55.4786 −1.85652
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.81925 0.127521
\(898\) 0 0
\(899\) 70.8900 2.36432
\(900\) 0 0
\(901\) 17.9753 0.598846
\(902\) 0 0
\(903\) 19.8231 0.659670
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −45.7593 −1.51941 −0.759706 0.650266i \(-0.774657\pi\)
−0.759706 + 0.650266i \(0.774657\pi\)
\(908\) 0 0
\(909\) 7.10799 0.235757
\(910\) 0 0
\(911\) 2.16774 0.0718206 0.0359103 0.999355i \(-0.488567\pi\)
0.0359103 + 0.999355i \(0.488567\pi\)
\(912\) 0 0
\(913\) 31.7914 1.05214
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −27.7815 −0.917426
\(918\) 0 0
\(919\) −19.6600 −0.648523 −0.324262 0.945967i \(-0.605116\pi\)
−0.324262 + 0.945967i \(0.605116\pi\)
\(920\) 0 0
\(921\) 9.26979 0.305450
\(922\) 0 0
\(923\) −2.19560 −0.0722690
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.2091 0.400998
\(928\) 0 0
\(929\) 43.5995 1.43045 0.715227 0.698892i \(-0.246323\pi\)
0.715227 + 0.698892i \(0.246323\pi\)
\(930\) 0 0
\(931\) 57.8017 1.89438
\(932\) 0 0
\(933\) 24.8936 0.814980
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0735 1.37448 0.687241 0.726429i \(-0.258822\pi\)
0.687241 + 0.726429i \(0.258822\pi\)
\(938\) 0 0
\(939\) −6.04712 −0.197340
\(940\) 0 0
\(941\) 48.9170 1.59465 0.797324 0.603552i \(-0.206248\pi\)
0.797324 + 0.603552i \(0.206248\pi\)
\(942\) 0 0
\(943\) 36.0544 1.17409
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.11340 −0.231155 −0.115577 0.993298i \(-0.536872\pi\)
−0.115577 + 0.993298i \(0.536872\pi\)
\(948\) 0 0
\(949\) −7.50878 −0.243745
\(950\) 0 0
\(951\) −14.3864 −0.466512
\(952\) 0 0
\(953\) −22.2372 −0.720334 −0.360167 0.932888i \(-0.617280\pi\)
−0.360167 + 0.932888i \(0.617280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −26.4560 −0.855202
\(958\) 0 0
\(959\) 0.757842 0.0244720
\(960\) 0 0
\(961\) 70.8485 2.28543
\(962\) 0 0
\(963\) −4.04303 −0.130285
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.9152 0.447482 0.223741 0.974649i \(-0.428173\pi\)
0.223741 + 0.974649i \(0.428173\pi\)
\(968\) 0 0
\(969\) −30.3522 −0.975053
\(970\) 0 0
\(971\) 24.2947 0.779655 0.389827 0.920888i \(-0.372535\pi\)
0.389827 + 0.920888i \(0.372535\pi\)
\(972\) 0 0
\(973\) 33.9707 1.08905
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.3314 1.48227 0.741136 0.671355i \(-0.234287\pi\)
0.741136 + 0.671355i \(0.234287\pi\)
\(978\) 0 0
\(979\) 12.7596 0.407800
\(980\) 0 0
\(981\) −0.773815 −0.0247060
\(982\) 0 0
\(983\) 41.0582 1.30955 0.654777 0.755822i \(-0.272763\pi\)
0.654777 + 0.755822i \(0.272763\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 39.9785 1.27253
\(988\) 0 0
\(989\) 26.2210 0.833780
\(990\) 0 0
\(991\) 20.2765 0.644105 0.322053 0.946722i \(-0.395627\pi\)
0.322053 + 0.946722i \(0.395627\pi\)
\(992\) 0 0
\(993\) −1.06688 −0.0338565
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −24.3634 −0.771596 −0.385798 0.922583i \(-0.626074\pi\)
−0.385798 + 0.922583i \(0.626074\pi\)
\(998\) 0 0
\(999\) 5.78488 0.183026
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 7500.2.a.m.1.12 12
5.2 odd 4 7500.2.d.g.1249.24 24
5.3 odd 4 7500.2.d.g.1249.1 24
5.4 even 2 7500.2.a.n.1.1 12
25.2 odd 20 300.2.o.a.229.5 yes 24
25.9 even 10 1500.2.m.c.901.1 24
25.11 even 5 1500.2.m.d.601.6 24
25.12 odd 20 1500.2.o.c.349.2 24
25.13 odd 20 300.2.o.a.169.5 24
25.14 even 10 1500.2.m.c.601.1 24
25.16 even 5 1500.2.m.d.901.6 24
25.23 odd 20 1500.2.o.c.649.2 24
75.2 even 20 900.2.w.c.829.3 24
75.38 even 20 900.2.w.c.469.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.o.a.169.5 24 25.13 odd 20
300.2.o.a.229.5 yes 24 25.2 odd 20
900.2.w.c.469.3 24 75.38 even 20
900.2.w.c.829.3 24 75.2 even 20
1500.2.m.c.601.1 24 25.14 even 10
1500.2.m.c.901.1 24 25.9 even 10
1500.2.m.d.601.6 24 25.11 even 5
1500.2.m.d.901.6 24 25.16 even 5
1500.2.o.c.349.2 24 25.12 odd 20
1500.2.o.c.649.2 24 25.23 odd 20
7500.2.a.m.1.12 12 1.1 even 1 trivial
7500.2.a.n.1.1 12 5.4 even 2
7500.2.d.g.1249.1 24 5.3 odd 4
7500.2.d.g.1249.24 24 5.2 odd 4