Properties

Label 7600.2.a.cf.1.4
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $4$
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] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.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: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.28734\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.04306 q^{3} -0.574672 q^{7} +6.26020 q^{9} +O(q^{10})\) \(q+3.04306 q^{3} -0.574672 q^{7} +6.26020 q^{9} -2.57467 q^{11} +0.468387 q^{13} +4.08612 q^{17} -1.00000 q^{19} -1.74876 q^{21} +1.51145 q^{23} +9.92099 q^{27} -4.08612 q^{29} +9.92099 q^{31} -7.83488 q^{33} +8.30326 q^{37} +1.42533 q^{39} -1.83488 q^{41} -0.574672 q^{43} +7.09508 q^{47} -6.66975 q^{49} +12.4343 q^{51} -4.30326 q^{53} -3.04306 q^{57} +2.68553 q^{59} +12.4095 q^{61} -3.59756 q^{63} -2.70570 q^{67} +4.59942 q^{69} +7.40058 q^{71} -12.0861 q^{73} +1.47959 q^{77} +6.68553 q^{79} +11.4095 q^{81} -6.66079 q^{83} -12.4343 q^{87} +14.6065 q^{89} -0.269169 q^{91} +30.1902 q^{93} -17.4526 q^{97} -16.1180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} - 12 q^{47} + 20 q^{49} + 36 q^{51} + 10 q^{53} - 2 q^{57} + 20 q^{61} + 20 q^{63} - 18 q^{67} + 28 q^{69} + 20 q^{71} - 28 q^{73} + 40 q^{77} + 16 q^{79} + 16 q^{81} - 36 q^{87} + 4 q^{89} + 36 q^{91} + 40 q^{93} - 30 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 3.04306 1.75691 0.878455 0.477825i \(-0.158575\pi\)
0.878455 + 0.477825i \(0.158575\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.574672 −0.217205 −0.108603 0.994085i \(-0.534638\pi\)
−0.108603 + 0.994085i \(0.534638\pi\)
\(8\) 0 0
\(9\) 6.26020 2.08673
\(10\) 0 0
\(11\) −2.57467 −0.776293 −0.388146 0.921598i \(-0.626884\pi\)
−0.388146 + 0.921598i \(0.626884\pi\)
\(12\) 0 0
\(13\) 0.468387 0.129907 0.0649536 0.997888i \(-0.479310\pi\)
0.0649536 + 0.997888i \(0.479310\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.08612 0.991029 0.495514 0.868600i \(-0.334980\pi\)
0.495514 + 0.868600i \(0.334980\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.74876 −0.381611
\(22\) 0 0
\(23\) 1.51145 0.315158 0.157579 0.987506i \(-0.449631\pi\)
0.157579 + 0.987506i \(0.449631\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.92099 1.90930
\(28\) 0 0
\(29\) −4.08612 −0.758773 −0.379386 0.925238i \(-0.623865\pi\)
−0.379386 + 0.925238i \(0.623865\pi\)
\(30\) 0 0
\(31\) 9.92099 1.78186 0.890931 0.454138i \(-0.150053\pi\)
0.890931 + 0.454138i \(0.150053\pi\)
\(32\) 0 0
\(33\) −7.83488 −1.36388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.30326 1.36505 0.682524 0.730863i \(-0.260882\pi\)
0.682524 + 0.730863i \(0.260882\pi\)
\(38\) 0 0
\(39\) 1.42533 0.228235
\(40\) 0 0
\(41\) −1.83488 −0.286560 −0.143280 0.989682i \(-0.545765\pi\)
−0.143280 + 0.989682i \(0.545765\pi\)
\(42\) 0 0
\(43\) −0.574672 −0.0876366 −0.0438183 0.999040i \(-0.513952\pi\)
−0.0438183 + 0.999040i \(0.513952\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.09508 1.03492 0.517462 0.855706i \(-0.326877\pi\)
0.517462 + 0.855706i \(0.326877\pi\)
\(48\) 0 0
\(49\) −6.66975 −0.952822
\(50\) 0 0
\(51\) 12.4343 1.74115
\(52\) 0 0
\(53\) −4.30326 −0.591099 −0.295549 0.955327i \(-0.595503\pi\)
−0.295549 + 0.955327i \(0.595503\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.04306 −0.403063
\(58\) 0 0
\(59\) 2.68553 0.349627 0.174813 0.984602i \(-0.444068\pi\)
0.174813 + 0.984602i \(0.444068\pi\)
\(60\) 0 0
\(61\) 12.4095 1.58888 0.794440 0.607343i \(-0.207765\pi\)
0.794440 + 0.607343i \(0.207765\pi\)
\(62\) 0 0
\(63\) −3.59756 −0.453250
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.70570 −0.330554 −0.165277 0.986247i \(-0.552852\pi\)
−0.165277 + 0.986247i \(0.552852\pi\)
\(68\) 0 0
\(69\) 4.59942 0.553705
\(70\) 0 0
\(71\) 7.40058 0.878288 0.439144 0.898417i \(-0.355282\pi\)
0.439144 + 0.898417i \(0.355282\pi\)
\(72\) 0 0
\(73\) −12.0861 −1.41457 −0.707286 0.706927i \(-0.750081\pi\)
−0.707286 + 0.706927i \(0.750081\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.47959 0.168615
\(78\) 0 0
\(79\) 6.68553 0.752181 0.376091 0.926583i \(-0.377268\pi\)
0.376091 + 0.926583i \(0.377268\pi\)
\(80\) 0 0
\(81\) 11.4095 1.26773
\(82\) 0 0
\(83\) −6.66079 −0.731117 −0.365558 0.930788i \(-0.619122\pi\)
−0.365558 + 0.930788i \(0.619122\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.4343 −1.33310
\(88\) 0 0
\(89\) 14.6065 1.54829 0.774144 0.633009i \(-0.218180\pi\)
0.774144 + 0.633009i \(0.218180\pi\)
\(90\) 0 0
\(91\) −0.269169 −0.0282165
\(92\) 0 0
\(93\) 30.1902 3.13057
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.4526 −1.77204 −0.886022 0.463643i \(-0.846542\pi\)
−0.886022 + 0.463643i \(0.846542\pi\)
\(98\) 0 0
\(99\) −16.1180 −1.61992
\(100\) 0 0
\(101\) 14.2831 1.42122 0.710611 0.703586i \(-0.248419\pi\)
0.710611 + 0.703586i \(0.248419\pi\)
\(102\) 0 0
\(103\) 4.79182 0.472152 0.236076 0.971735i \(-0.424139\pi\)
0.236076 + 0.971735i \(0.424139\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.22611 0.891922 0.445961 0.895052i \(-0.352862\pi\)
0.445961 + 0.895052i \(0.352862\pi\)
\(108\) 0 0
\(109\) 4.89810 0.469153 0.234577 0.972098i \(-0.424630\pi\)
0.234577 + 0.972098i \(0.424630\pi\)
\(110\) 0 0
\(111\) 25.2673 2.39827
\(112\) 0 0
\(113\) −1.61773 −0.152183 −0.0760916 0.997101i \(-0.524244\pi\)
−0.0760916 + 0.997101i \(0.524244\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.93220 0.271082
\(118\) 0 0
\(119\) −2.34818 −0.215257
\(120\) 0 0
\(121\) −4.37107 −0.397370
\(122\) 0 0
\(123\) −5.58364 −0.503459
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −13.1292 −1.16503 −0.582513 0.812821i \(-0.697930\pi\)
−0.582513 + 0.812821i \(0.697930\pi\)
\(128\) 0 0
\(129\) −1.74876 −0.153970
\(130\) 0 0
\(131\) 8.17223 0.714011 0.357006 0.934102i \(-0.383798\pi\)
0.357006 + 0.934102i \(0.383798\pi\)
\(132\) 0 0
\(133\) 0.574672 0.0498304
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.6065 1.24792 0.623960 0.781456i \(-0.285523\pi\)
0.623960 + 0.781456i \(0.285523\pi\)
\(138\) 0 0
\(139\) 2.07219 0.175761 0.0878804 0.996131i \(-0.471991\pi\)
0.0878804 + 0.996131i \(0.471991\pi\)
\(140\) 0 0
\(141\) 21.5907 1.81827
\(142\) 0 0
\(143\) −1.20594 −0.100846
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −20.2964 −1.67402
\(148\) 0 0
\(149\) 8.91203 0.730102 0.365051 0.930988i \(-0.381052\pi\)
0.365051 + 0.930988i \(0.381052\pi\)
\(150\) 0 0
\(151\) −11.4572 −0.932372 −0.466186 0.884687i \(-0.654372\pi\)
−0.466186 + 0.884687i \(0.654372\pi\)
\(152\) 0 0
\(153\) 25.5799 2.06801
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.60653 0.527258 0.263629 0.964624i \(-0.415081\pi\)
0.263629 + 0.964624i \(0.415081\pi\)
\(158\) 0 0
\(159\) −13.0951 −1.03851
\(160\) 0 0
\(161\) −0.868585 −0.0684541
\(162\) 0 0
\(163\) 20.2444 1.58567 0.792833 0.609439i \(-0.208605\pi\)
0.792833 + 0.609439i \(0.208605\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.89372 −0.456069 −0.228035 0.973653i \(-0.573230\pi\)
−0.228035 + 0.973653i \(0.573230\pi\)
\(168\) 0 0
\(169\) −12.7806 −0.983124
\(170\) 0 0
\(171\) −6.26020 −0.478730
\(172\) 0 0
\(173\) 3.53161 0.268504 0.134252 0.990947i \(-0.457137\pi\)
0.134252 + 0.990947i \(0.457137\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.17223 0.614263
\(178\) 0 0
\(179\) −7.18801 −0.537257 −0.268629 0.963244i \(-0.586570\pi\)
−0.268629 + 0.963244i \(0.586570\pi\)
\(180\) 0 0
\(181\) 15.5433 1.15532 0.577662 0.816276i \(-0.303965\pi\)
0.577662 + 0.816276i \(0.303965\pi\)
\(182\) 0 0
\(183\) 37.7630 2.79152
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.5204 −0.769329
\(188\) 0 0
\(189\) −5.70131 −0.414710
\(190\) 0 0
\(191\) −13.3216 −0.963915 −0.481958 0.876194i \(-0.660074\pi\)
−0.481958 + 0.876194i \(0.660074\pi\)
\(192\) 0 0
\(193\) −18.9959 −1.36736 −0.683678 0.729784i \(-0.739621\pi\)
−0.683678 + 0.729784i \(0.739621\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.17223 −0.154765 −0.0773826 0.997001i \(-0.524656\pi\)
−0.0773826 + 0.997001i \(0.524656\pi\)
\(198\) 0 0
\(199\) 1.87355 0.132812 0.0664061 0.997793i \(-0.478847\pi\)
0.0664061 + 0.997793i \(0.478847\pi\)
\(200\) 0 0
\(201\) −8.23361 −0.580754
\(202\) 0 0
\(203\) 2.34818 0.164810
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.46196 0.657651
\(208\) 0 0
\(209\) 2.57467 0.178094
\(210\) 0 0
\(211\) 17.1090 1.17783 0.588916 0.808194i \(-0.299555\pi\)
0.588916 + 0.808194i \(0.299555\pi\)
\(212\) 0 0
\(213\) 22.5204 1.54307
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.70131 −0.387030
\(218\) 0 0
\(219\) −36.7788 −2.48528
\(220\) 0 0
\(221\) 1.91388 0.128742
\(222\) 0 0
\(223\) 5.12918 0.343475 0.171737 0.985143i \(-0.445062\pi\)
0.171737 + 0.985143i \(0.445062\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.31223 −0.485330 −0.242665 0.970110i \(-0.578022\pi\)
−0.242665 + 0.970110i \(0.578022\pi\)
\(228\) 0 0
\(229\) −8.40955 −0.555719 −0.277859 0.960622i \(-0.589625\pi\)
−0.277859 + 0.960622i \(0.589625\pi\)
\(230\) 0 0
\(231\) 4.50248 0.296242
\(232\) 0 0
\(233\) 14.1722 0.928454 0.464227 0.885716i \(-0.346332\pi\)
0.464227 + 0.885716i \(0.346332\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 20.3445 1.32152
\(238\) 0 0
\(239\) 14.1902 0.917885 0.458943 0.888466i \(-0.348228\pi\)
0.458943 + 0.888466i \(0.348228\pi\)
\(240\) 0 0
\(241\) 27.8807 1.79595 0.897976 0.440045i \(-0.145038\pi\)
0.897976 + 0.440045i \(0.145038\pi\)
\(242\) 0 0
\(243\) 4.95694 0.317988
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.468387 −0.0298027
\(248\) 0 0
\(249\) −20.2692 −1.28451
\(250\) 0 0
\(251\) 26.1902 1.65311 0.826554 0.562857i \(-0.190298\pi\)
0.826554 + 0.562857i \(0.190298\pi\)
\(252\) 0 0
\(253\) −3.89148 −0.244655
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.01831 −0.562547 −0.281273 0.959628i \(-0.590757\pi\)
−0.281273 + 0.959628i \(0.590757\pi\)
\(258\) 0 0
\(259\) −4.77165 −0.296496
\(260\) 0 0
\(261\) −25.5799 −1.58336
\(262\) 0 0
\(263\) −9.00896 −0.555517 −0.277758 0.960651i \(-0.589591\pi\)
−0.277758 + 0.960651i \(0.589591\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 44.4485 2.72020
\(268\) 0 0
\(269\) −30.6136 −1.86655 −0.933273 0.359167i \(-0.883061\pi\)
−0.933273 + 0.359167i \(0.883061\pi\)
\(270\) 0 0
\(271\) −24.1180 −1.46506 −0.732531 0.680733i \(-0.761661\pi\)
−0.732531 + 0.680733i \(0.761661\pi\)
\(272\) 0 0
\(273\) −0.819096 −0.0495739
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.56075 −0.274029 −0.137014 0.990569i \(-0.543751\pi\)
−0.137014 + 0.990569i \(0.543751\pi\)
\(278\) 0 0
\(279\) 62.1074 3.71828
\(280\) 0 0
\(281\) −6.11563 −0.364828 −0.182414 0.983222i \(-0.558391\pi\)
−0.182414 + 0.983222i \(0.558391\pi\)
\(282\) 0 0
\(283\) −19.0547 −1.13269 −0.566344 0.824169i \(-0.691642\pi\)
−0.566344 + 0.824169i \(0.691642\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.05445 0.0622423
\(288\) 0 0
\(289\) −0.303649 −0.0178617
\(290\) 0 0
\(291\) −53.1093 −3.11332
\(292\) 0 0
\(293\) 6.43887 0.376163 0.188081 0.982153i \(-0.439773\pi\)
0.188081 + 0.982153i \(0.439773\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −25.5433 −1.48217
\(298\) 0 0
\(299\) 0.707941 0.0409413
\(300\) 0 0
\(301\) 0.330247 0.0190351
\(302\) 0 0
\(303\) 43.4643 2.49696
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.77389 −0.386606 −0.193303 0.981139i \(-0.561920\pi\)
−0.193303 + 0.981139i \(0.561920\pi\)
\(308\) 0 0
\(309\) 14.5818 0.829529
\(310\) 0 0
\(311\) −20.6205 −1.16928 −0.584639 0.811293i \(-0.698764\pi\)
−0.584639 + 0.811293i \(0.698764\pi\)
\(312\) 0 0
\(313\) −19.3711 −1.09492 −0.547459 0.836833i \(-0.684405\pi\)
−0.547459 + 0.836833i \(0.684405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.37360 −0.189480 −0.0947401 0.995502i \(-0.530202\pi\)
−0.0947401 + 0.995502i \(0.530202\pi\)
\(318\) 0 0
\(319\) 10.5204 0.589030
\(320\) 0 0
\(321\) 28.0756 1.56703
\(322\) 0 0
\(323\) −4.08612 −0.227358
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 14.9052 0.824260
\(328\) 0 0
\(329\) −4.07734 −0.224791
\(330\) 0 0
\(331\) 32.7788 1.80168 0.900842 0.434148i \(-0.142950\pi\)
0.900842 + 0.434148i \(0.142950\pi\)
\(332\) 0 0
\(333\) 51.9801 2.84849
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.74915 −0.476596 −0.238298 0.971192i \(-0.576590\pi\)
−0.238298 + 0.971192i \(0.576590\pi\)
\(338\) 0 0
\(339\) −4.92285 −0.267372
\(340\) 0 0
\(341\) −25.5433 −1.38325
\(342\) 0 0
\(343\) 7.85562 0.424164
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.5028 −0.993281 −0.496640 0.867956i \(-0.665433\pi\)
−0.496640 + 0.867956i \(0.665433\pi\)
\(348\) 0 0
\(349\) 3.54330 0.189668 0.0948342 0.995493i \(-0.469768\pi\)
0.0948342 + 0.995493i \(0.469768\pi\)
\(350\) 0 0
\(351\) 4.64686 0.248031
\(352\) 0 0
\(353\) 3.41140 0.181571 0.0907853 0.995870i \(-0.471062\pi\)
0.0907853 + 0.995870i \(0.471062\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −7.14564 −0.378187
\(358\) 0 0
\(359\) 1.70609 0.0900438 0.0450219 0.998986i \(-0.485664\pi\)
0.0450219 + 0.998986i \(0.485664\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −13.3014 −0.698143
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 37.6155 1.96351 0.981756 0.190144i \(-0.0608953\pi\)
0.981756 + 0.190144i \(0.0608953\pi\)
\(368\) 0 0
\(369\) −11.4867 −0.597974
\(370\) 0 0
\(371\) 2.47296 0.128390
\(372\) 0 0
\(373\) −13.4031 −0.693987 −0.346994 0.937868i \(-0.612797\pi\)
−0.346994 + 0.937868i \(0.612797\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.91388 −0.0985700
\(378\) 0 0
\(379\) 9.37107 0.481359 0.240680 0.970605i \(-0.422630\pi\)
0.240680 + 0.970605i \(0.422630\pi\)
\(380\) 0 0
\(381\) −39.9528 −2.04685
\(382\) 0 0
\(383\) −2.09917 −0.107263 −0.0536314 0.998561i \(-0.517080\pi\)
−0.0536314 + 0.998561i \(0.517080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.59756 −0.182874
\(388\) 0 0
\(389\) 1.07238 0.0543718 0.0271859 0.999630i \(-0.491345\pi\)
0.0271859 + 0.999630i \(0.491345\pi\)
\(390\) 0 0
\(391\) 6.17594 0.312331
\(392\) 0 0
\(393\) 24.8686 1.25445
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.5341 1.18114 0.590572 0.806985i \(-0.298902\pi\)
0.590572 + 0.806985i \(0.298902\pi\)
\(398\) 0 0
\(399\) 1.74876 0.0875475
\(400\) 0 0
\(401\) 18.6469 0.931180 0.465590 0.885001i \(-0.345842\pi\)
0.465590 + 0.885001i \(0.345842\pi\)
\(402\) 0 0
\(403\) 4.64686 0.231477
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.3782 −1.05968
\(408\) 0 0
\(409\) −6.88017 −0.340203 −0.170101 0.985427i \(-0.554410\pi\)
−0.170101 + 0.985427i \(0.554410\pi\)
\(410\) 0 0
\(411\) 44.4485 2.19248
\(412\) 0 0
\(413\) −1.54330 −0.0759408
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.30580 0.308796
\(418\) 0 0
\(419\) −13.4796 −0.658521 −0.329261 0.944239i \(-0.606799\pi\)
−0.329261 + 0.944239i \(0.606799\pi\)
\(420\) 0 0
\(421\) −5.83488 −0.284374 −0.142187 0.989840i \(-0.545414\pi\)
−0.142187 + 0.989840i \(0.545414\pi\)
\(422\) 0 0
\(423\) 44.4167 2.15961
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −7.13142 −0.345113
\(428\) 0 0
\(429\) −3.66975 −0.177177
\(430\) 0 0
\(431\) −29.2039 −1.40670 −0.703351 0.710843i \(-0.748314\pi\)
−0.703351 + 0.710843i \(0.748314\pi\)
\(432\) 0 0
\(433\) 12.5229 0.601814 0.300907 0.953653i \(-0.402711\pi\)
0.300907 + 0.953653i \(0.402711\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.51145 −0.0723022
\(438\) 0 0
\(439\) −15.6769 −0.748216 −0.374108 0.927385i \(-0.622051\pi\)
−0.374108 + 0.927385i \(0.622051\pi\)
\(440\) 0 0
\(441\) −41.7540 −1.98829
\(442\) 0 0
\(443\) −29.8281 −1.41717 −0.708587 0.705624i \(-0.750667\pi\)
−0.708587 + 0.705624i \(0.750667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 27.1198 1.28272
\(448\) 0 0
\(449\) 29.9668 1.41422 0.707110 0.707104i \(-0.249999\pi\)
0.707110 + 0.707104i \(0.249999\pi\)
\(450\) 0 0
\(451\) 4.72420 0.222454
\(452\) 0 0
\(453\) −34.8649 −1.63809
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.6698 −0.826556 −0.413278 0.910605i \(-0.635616\pi\)
−0.413278 + 0.910605i \(0.635616\pi\)
\(458\) 0 0
\(459\) 40.5383 1.89217
\(460\) 0 0
\(461\) 22.3445 1.04069 0.520343 0.853957i \(-0.325804\pi\)
0.520343 + 0.853957i \(0.325804\pi\)
\(462\) 0 0
\(463\) −6.83302 −0.317557 −0.158779 0.987314i \(-0.550756\pi\)
−0.158779 + 0.987314i \(0.550756\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.00896 0.416885 0.208443 0.978035i \(-0.433161\pi\)
0.208443 + 0.978035i \(0.433161\pi\)
\(468\) 0 0
\(469\) 1.55489 0.0717981
\(470\) 0 0
\(471\) 20.1040 0.926345
\(472\) 0 0
\(473\) 1.47959 0.0680317
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −26.9393 −1.23347
\(478\) 0 0
\(479\) −9.26731 −0.423434 −0.211717 0.977331i \(-0.567906\pi\)
−0.211717 + 0.977331i \(0.567906\pi\)
\(480\) 0 0
\(481\) 3.88914 0.177329
\(482\) 0 0
\(483\) −2.64315 −0.120268
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −38.7694 −1.75681 −0.878405 0.477917i \(-0.841392\pi\)
−0.878405 + 0.477917i \(0.841392\pi\)
\(488\) 0 0
\(489\) 61.6050 2.78587
\(490\) 0 0
\(491\) −21.6877 −0.978751 −0.489376 0.872073i \(-0.662775\pi\)
−0.489376 + 0.872073i \(0.662775\pi\)
\(492\) 0 0
\(493\) −16.6964 −0.751966
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.25291 −0.190769
\(498\) 0 0
\(499\) 27.8372 1.24616 0.623082 0.782156i \(-0.285880\pi\)
0.623082 + 0.782156i \(0.285880\pi\)
\(500\) 0 0
\(501\) −17.9349 −0.801273
\(502\) 0 0
\(503\) −41.2449 −1.83902 −0.919510 0.393067i \(-0.871414\pi\)
−0.919510 + 0.393067i \(0.871414\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −38.8922 −1.72726
\(508\) 0 0
\(509\) 0.416364 0.0184550 0.00922751 0.999957i \(-0.497063\pi\)
0.00922751 + 0.999957i \(0.497063\pi\)
\(510\) 0 0
\(511\) 6.94555 0.307253
\(512\) 0 0
\(513\) −9.92099 −0.438023
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −18.2675 −0.803404
\(518\) 0 0
\(519\) 10.7469 0.471737
\(520\) 0 0
\(521\) 24.0458 1.05346 0.526732 0.850031i \(-0.323417\pi\)
0.526732 + 0.850031i \(0.323417\pi\)
\(522\) 0 0
\(523\) −5.19736 −0.227265 −0.113632 0.993523i \(-0.536249\pi\)
−0.113632 + 0.993523i \(0.536249\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.5383 1.76588
\(528\) 0 0
\(529\) −20.7155 −0.900675
\(530\) 0 0
\(531\) 16.8120 0.729578
\(532\) 0 0
\(533\) −0.859432 −0.0372261
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −21.8735 −0.943913
\(538\) 0 0
\(539\) 17.1724 0.739669
\(540\) 0 0
\(541\) −1.93863 −0.0833481 −0.0416741 0.999131i \(-0.513269\pi\)
−0.0416741 + 0.999131i \(0.513269\pi\)
\(542\) 0 0
\(543\) 47.2992 2.02980
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −12.9075 −0.551883 −0.275941 0.961174i \(-0.588990\pi\)
−0.275941 + 0.961174i \(0.588990\pi\)
\(548\) 0 0
\(549\) 77.6863 3.31557
\(550\) 0 0
\(551\) 4.08612 0.174074
\(552\) 0 0
\(553\) −3.84199 −0.163378
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 34.1040 1.44503 0.722517 0.691353i \(-0.242985\pi\)
0.722517 + 0.691353i \(0.242985\pi\)
\(558\) 0 0
\(559\) −0.269169 −0.0113846
\(560\) 0 0
\(561\) −32.0142 −1.35164
\(562\) 0 0
\(563\) −14.4911 −0.610727 −0.305363 0.952236i \(-0.598778\pi\)
−0.305363 + 0.952236i \(0.598778\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −6.55674 −0.275357
\(568\) 0 0
\(569\) −39.2110 −1.64381 −0.821906 0.569624i \(-0.807089\pi\)
−0.821906 + 0.569624i \(0.807089\pi\)
\(570\) 0 0
\(571\) −21.9915 −0.920316 −0.460158 0.887837i \(-0.652207\pi\)
−0.460158 + 0.887837i \(0.652207\pi\)
\(572\) 0 0
\(573\) −40.5383 −1.69351
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.8735 −0.993869 −0.496934 0.867788i \(-0.665541\pi\)
−0.496934 + 0.867788i \(0.665541\pi\)
\(578\) 0 0
\(579\) −57.8057 −2.40232
\(580\) 0 0
\(581\) 3.82777 0.158803
\(582\) 0 0
\(583\) 11.0795 0.458866
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.8281 0.735843 0.367921 0.929857i \(-0.380070\pi\)
0.367921 + 0.929857i \(0.380070\pi\)
\(588\) 0 0
\(589\) −9.92099 −0.408787
\(590\) 0 0
\(591\) −6.61023 −0.271909
\(592\) 0 0
\(593\) 21.2446 0.872412 0.436206 0.899847i \(-0.356322\pi\)
0.436206 + 0.899847i \(0.356322\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.70131 0.233339
\(598\) 0 0
\(599\) −24.3374 −0.994397 −0.497199 0.867637i \(-0.665638\pi\)
−0.497199 + 0.867637i \(0.665638\pi\)
\(600\) 0 0
\(601\) 15.7891 0.644051 0.322025 0.946731i \(-0.395636\pi\)
0.322025 + 0.946731i \(0.395636\pi\)
\(602\) 0 0
\(603\) −16.9382 −0.689779
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.81471 0.317189 0.158595 0.987344i \(-0.449304\pi\)
0.158595 + 0.987344i \(0.449304\pi\)
\(608\) 0 0
\(609\) 7.14564 0.289556
\(610\) 0 0
\(611\) 3.32324 0.134444
\(612\) 0 0
\(613\) −12.9547 −0.523235 −0.261618 0.965172i \(-0.584256\pi\)
−0.261618 + 0.965172i \(0.584256\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.8873 −0.760373 −0.380187 0.924910i \(-0.624140\pi\)
−0.380187 + 0.924910i \(0.624140\pi\)
\(618\) 0 0
\(619\) −29.2673 −1.17635 −0.588176 0.808733i \(-0.700154\pi\)
−0.588176 + 0.808733i \(0.700154\pi\)
\(620\) 0 0
\(621\) 14.9950 0.601730
\(622\) 0 0
\(623\) −8.39396 −0.336297
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.83488 0.312895
\(628\) 0 0
\(629\) 33.9281 1.35280
\(630\) 0 0
\(631\) 5.25309 0.209122 0.104561 0.994518i \(-0.466656\pi\)
0.104561 + 0.994518i \(0.466656\pi\)
\(632\) 0 0
\(633\) 52.0637 2.06935
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.12402 −0.123778
\(638\) 0 0
\(639\) 46.3292 1.83275
\(640\) 0 0
\(641\) −13.0021 −0.513554 −0.256777 0.966471i \(-0.582661\pi\)
−0.256777 + 0.966471i \(0.582661\pi\)
\(642\) 0 0
\(643\) −17.3534 −0.684353 −0.342176 0.939636i \(-0.611164\pi\)
−0.342176 + 0.939636i \(0.611164\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.4848 0.490830 0.245415 0.969418i \(-0.421076\pi\)
0.245415 + 0.969418i \(0.421076\pi\)
\(648\) 0 0
\(649\) −6.91437 −0.271413
\(650\) 0 0
\(651\) −17.3494 −0.679978
\(652\) 0 0
\(653\) 28.3532 1.10955 0.554774 0.832001i \(-0.312805\pi\)
0.554774 + 0.832001i \(0.312805\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −75.6616 −2.95184
\(658\) 0 0
\(659\) −45.2202 −1.76153 −0.880764 0.473556i \(-0.842970\pi\)
−0.880764 + 0.473556i \(0.842970\pi\)
\(660\) 0 0
\(661\) −14.6086 −0.568207 −0.284104 0.958794i \(-0.591696\pi\)
−0.284104 + 0.958794i \(0.591696\pi\)
\(662\) 0 0
\(663\) 5.82406 0.226188
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.17594 −0.239133
\(668\) 0 0
\(669\) 15.6084 0.603455
\(670\) 0 0
\(671\) −31.9505 −1.23344
\(672\) 0 0
\(673\) 0.440534 0.0169813 0.00849067 0.999964i \(-0.497297\pi\)
0.00849067 + 0.999964i \(0.497297\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.8057 −1.26083 −0.630414 0.776259i \(-0.717115\pi\)
−0.630414 + 0.776259i \(0.717115\pi\)
\(678\) 0 0
\(679\) 10.0295 0.384898
\(680\) 0 0
\(681\) −22.2515 −0.852681
\(682\) 0 0
\(683\) −39.6092 −1.51561 −0.757803 0.652484i \(-0.773727\pi\)
−0.757803 + 0.652484i \(0.773727\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −25.5907 −0.976348
\(688\) 0 0
\(689\) −2.01559 −0.0767879
\(690\) 0 0
\(691\) 19.8962 0.756889 0.378444 0.925624i \(-0.376459\pi\)
0.378444 + 0.925624i \(0.376459\pi\)
\(692\) 0 0
\(693\) 9.26254 0.351855
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7.49752 −0.283989
\(698\) 0 0
\(699\) 43.1269 1.63121
\(700\) 0 0
\(701\) −14.1251 −0.533497 −0.266748 0.963766i \(-0.585949\pi\)
−0.266748 + 0.963766i \(0.585949\pi\)
\(702\) 0 0
\(703\) −8.30326 −0.313163
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.20809 −0.308697
\(708\) 0 0
\(709\) −41.1815 −1.54660 −0.773302 0.634038i \(-0.781396\pi\)
−0.773302 + 0.634038i \(0.781396\pi\)
\(710\) 0 0
\(711\) 41.8528 1.56960
\(712\) 0 0
\(713\) 14.9950 0.561569
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 43.1815 1.61264
\(718\) 0 0
\(719\) 18.0227 0.672133 0.336067 0.941838i \(-0.390903\pi\)
0.336067 + 0.941838i \(0.390903\pi\)
\(720\) 0 0
\(721\) −2.75372 −0.102554
\(722\) 0 0
\(723\) 84.8425 3.15533
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −21.1266 −0.783544 −0.391772 0.920062i \(-0.628138\pi\)
−0.391772 + 0.920062i \(0.628138\pi\)
\(728\) 0 0
\(729\) −19.1444 −0.709051
\(730\) 0 0
\(731\) −2.34818 −0.0868504
\(732\) 0 0
\(733\) 27.6660 1.02187 0.510934 0.859620i \(-0.329300\pi\)
0.510934 + 0.859620i \(0.329300\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.96629 0.256607
\(738\) 0 0
\(739\) 14.2987 0.525986 0.262993 0.964798i \(-0.415290\pi\)
0.262993 + 0.964798i \(0.415290\pi\)
\(740\) 0 0
\(741\) −1.42533 −0.0523607
\(742\) 0 0
\(743\) 49.9438 1.83226 0.916130 0.400881i \(-0.131296\pi\)
0.916130 + 0.400881i \(0.131296\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −41.6979 −1.52565
\(748\) 0 0
\(749\) −5.30198 −0.193730
\(750\) 0 0
\(751\) −9.69216 −0.353672 −0.176836 0.984240i \(-0.556586\pi\)
−0.176836 + 0.984240i \(0.556586\pi\)
\(752\) 0 0
\(753\) 79.6982 2.90436
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −46.6889 −1.69694 −0.848469 0.529245i \(-0.822475\pi\)
−0.848469 + 0.529245i \(0.822475\pi\)
\(758\) 0 0
\(759\) −11.8420 −0.429837
\(760\) 0 0
\(761\) −38.7361 −1.40418 −0.702091 0.712087i \(-0.747750\pi\)
−0.702091 + 0.712087i \(0.747750\pi\)
\(762\) 0 0
\(763\) −2.81480 −0.101903
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.25787 0.0454190
\(768\) 0 0
\(769\) −5.38666 −0.194248 −0.0971239 0.995272i \(-0.530964\pi\)
−0.0971239 + 0.995272i \(0.530964\pi\)
\(770\) 0 0
\(771\) −27.4433 −0.988345
\(772\) 0 0
\(773\) −7.20137 −0.259015 −0.129508 0.991578i \(-0.541340\pi\)
−0.129508 + 0.991578i \(0.541340\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −14.5204 −0.520917
\(778\) 0 0
\(779\) 1.83488 0.0657413
\(780\) 0 0
\(781\) −19.0541 −0.681808
\(782\) 0 0
\(783\) −40.5383 −1.44872
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −33.5231 −1.19497 −0.597485 0.801880i \(-0.703833\pi\)
−0.597485 + 0.801880i \(0.703833\pi\)
\(788\) 0 0
\(789\) −27.4148 −0.975993
\(790\) 0 0
\(791\) 0.929664 0.0330550
\(792\) 0 0
\(793\) 5.81247 0.206407
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.4979 0.371855 0.185927 0.982563i \(-0.440471\pi\)
0.185927 + 0.982563i \(0.440471\pi\)
\(798\) 0 0
\(799\) 28.9913 1.02564
\(800\) 0 0
\(801\) 91.4398 3.23087
\(802\) 0 0
\(803\) 31.1178 1.09812
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −93.1591 −3.27936
\(808\) 0 0
\(809\) −13.0724 −0.459600 −0.229800 0.973238i \(-0.573807\pi\)
−0.229800 + 0.973238i \(0.573807\pi\)
\(810\) 0 0
\(811\) 10.0790 0.353922 0.176961 0.984218i \(-0.443373\pi\)
0.176961 + 0.984218i \(0.443373\pi\)
\(812\) 0 0
\(813\) −73.3924 −2.57398
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.574672 0.0201052
\(818\) 0 0
\(819\) −1.68505 −0.0588804
\(820\) 0 0
\(821\) 40.2987 1.40643 0.703217 0.710975i \(-0.251746\pi\)
0.703217 + 0.710975i \(0.251746\pi\)
\(822\) 0 0
\(823\) −5.07715 −0.176978 −0.0884892 0.996077i \(-0.528204\pi\)
−0.0884892 + 0.996077i \(0.528204\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42.8077 −1.48857 −0.744285 0.667862i \(-0.767209\pi\)
−0.744285 + 0.667862i \(0.767209\pi\)
\(828\) 0 0
\(829\) 24.2018 0.840562 0.420281 0.907394i \(-0.361932\pi\)
0.420281 + 0.907394i \(0.361932\pi\)
\(830\) 0 0
\(831\) −13.8786 −0.481444
\(832\) 0 0
\(833\) −27.2534 −0.944274
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 98.4261 3.40210
\(838\) 0 0
\(839\) 8.63975 0.298277 0.149139 0.988816i \(-0.452350\pi\)
0.149139 + 0.988816i \(0.452350\pi\)
\(840\) 0 0
\(841\) −12.3036 −0.424264
\(842\) 0 0
\(843\) −18.6102 −0.640971
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.51193 0.0863109
\(848\) 0 0
\(849\) −57.9847 −1.99003
\(850\) 0 0
\(851\) 12.5499 0.430206
\(852\) 0 0
\(853\) 13.0229 0.445895 0.222948 0.974830i \(-0.428432\pi\)
0.222948 + 0.974830i \(0.428432\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.82367 0.0964548 0.0482274 0.998836i \(-0.484643\pi\)
0.0482274 + 0.998836i \(0.484643\pi\)
\(858\) 0 0
\(859\) −24.1227 −0.823057 −0.411529 0.911397i \(-0.635005\pi\)
−0.411529 + 0.911397i \(0.635005\pi\)
\(860\) 0 0
\(861\) 3.20876 0.109354
\(862\) 0 0
\(863\) 32.7307 1.11417 0.557084 0.830456i \(-0.311920\pi\)
0.557084 + 0.830456i \(0.311920\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.924022 −0.0313814
\(868\) 0 0
\(869\) −17.2131 −0.583913
\(870\) 0 0
\(871\) −1.26731 −0.0429413
\(872\) 0 0
\(873\) −109.257 −3.69779
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 49.5072 1.67174 0.835869 0.548929i \(-0.184964\pi\)
0.835869 + 0.548929i \(0.184964\pi\)
\(878\) 0 0
\(879\) 19.5939 0.660884
\(880\) 0 0
\(881\) 40.3152 1.35826 0.679128 0.734020i \(-0.262358\pi\)
0.679128 + 0.734020i \(0.262358\pi\)
\(882\) 0 0
\(883\) −29.3347 −0.987192 −0.493596 0.869691i \(-0.664318\pi\)
−0.493596 + 0.869691i \(0.664318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 43.3214 1.45459 0.727295 0.686325i \(-0.240777\pi\)
0.727295 + 0.686325i \(0.240777\pi\)
\(888\) 0 0
\(889\) 7.54496 0.253050
\(890\) 0 0
\(891\) −29.3758 −0.984128
\(892\) 0 0
\(893\) −7.09508 −0.237428
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.15431 0.0719302
\(898\) 0 0
\(899\) −40.5383 −1.35203
\(900\) 0 0
\(901\) −17.5836 −0.585796
\(902\) 0 0
\(903\) 1.00496 0.0334431
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0731 0.467288 0.233644 0.972322i \(-0.424935\pi\)
0.233644 + 0.972322i \(0.424935\pi\)
\(908\) 0 0
\(909\) 89.4151 2.96571
\(910\) 0 0
\(911\) −30.0725 −0.996346 −0.498173 0.867078i \(-0.665996\pi\)
−0.498173 + 0.867078i \(0.665996\pi\)
\(912\) 0 0
\(913\) 17.1493 0.567560
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −4.69635 −0.155087
\(918\) 0 0
\(919\) −29.6518 −0.978123 −0.489062 0.872249i \(-0.662661\pi\)
−0.489062 + 0.872249i \(0.662661\pi\)
\(920\) 0 0
\(921\) −20.6133 −0.679233
\(922\) 0 0
\(923\) 3.46634 0.114096
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 29.9978 0.985256
\(928\) 0 0
\(929\) 58.3803 1.91540 0.957698 0.287775i \(-0.0929154\pi\)
0.957698 + 0.287775i \(0.0929154\pi\)
\(930\) 0 0
\(931\) 6.66975 0.218592
\(932\) 0 0
\(933\) −62.7492 −2.05432
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.15850 0.103184 0.0515918 0.998668i \(-0.483571\pi\)
0.0515918 + 0.998668i \(0.483571\pi\)
\(938\) 0 0
\(939\) −58.9473 −1.92367
\(940\) 0 0
\(941\) −9.66975 −0.315225 −0.157612 0.987501i \(-0.550380\pi\)
−0.157612 + 0.987501i \(0.550380\pi\)
\(942\) 0 0
\(943\) −2.77331 −0.0903116
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.3714 1.01943 0.509716 0.860343i \(-0.329750\pi\)
0.509716 + 0.860343i \(0.329750\pi\)
\(948\) 0 0
\(949\) −5.66098 −0.183763
\(950\) 0 0
\(951\) −10.2661 −0.332900
\(952\) 0 0
\(953\) 13.1224 0.425075 0.212537 0.977153i \(-0.431827\pi\)
0.212537 + 0.977153i \(0.431827\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.0142 1.03487
\(958\) 0 0
\(959\) −8.39396 −0.271055
\(960\) 0 0
\(961\) 67.4261 2.17504
\(962\) 0 0
\(963\) 57.7573 1.86120
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −44.4400 −1.42910 −0.714548 0.699587i \(-0.753367\pi\)
−0.714548 + 0.699587i \(0.753367\pi\)
\(968\) 0 0
\(969\) −12.4343 −0.399447
\(970\) 0 0
\(971\) 5.69927 0.182898 0.0914491 0.995810i \(-0.470850\pi\)
0.0914491 + 0.995810i \(0.470850\pi\)
\(972\) 0 0
\(973\) −1.19083 −0.0381762
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.9164 0.765154 0.382577 0.923924i \(-0.375037\pi\)
0.382577 + 0.923924i \(0.375037\pi\)
\(978\) 0 0
\(979\) −37.6070 −1.20193
\(980\) 0 0
\(981\) 30.6631 0.978998
\(982\) 0 0
\(983\) −45.5302 −1.45219 −0.726095 0.687595i \(-0.758667\pi\)
−0.726095 + 0.687595i \(0.758667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −12.4076 −0.394938
\(988\) 0 0
\(989\) −0.868585 −0.0276194
\(990\) 0 0
\(991\) −27.5521 −0.875220 −0.437610 0.899165i \(-0.644175\pi\)
−0.437610 + 0.899165i \(0.644175\pi\)
\(992\) 0 0
\(993\) 99.7477 3.16540
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −16.8557 −0.533826 −0.266913 0.963721i \(-0.586004\pi\)
−0.266913 + 0.963721i \(0.586004\pi\)
\(998\) 0 0
\(999\) 82.3766 2.60628
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 7600.2.a.cf.1.4 4
4.3 odd 2 475.2.a.i.1.4 4
5.4 even 2 1520.2.a.t.1.1 4
12.11 even 2 4275.2.a.bo.1.1 4
20.3 even 4 475.2.b.e.324.1 8
20.7 even 4 475.2.b.e.324.8 8
20.19 odd 2 95.2.a.b.1.1 4
40.19 odd 2 6080.2.a.cc.1.1 4
40.29 even 2 6080.2.a.ch.1.4 4
60.59 even 2 855.2.a.m.1.4 4
76.75 even 2 9025.2.a.bf.1.1 4
140.139 even 2 4655.2.a.y.1.1 4
380.379 even 2 1805.2.a.p.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.a.b.1.1 4 20.19 odd 2
475.2.a.i.1.4 4 4.3 odd 2
475.2.b.e.324.1 8 20.3 even 4
475.2.b.e.324.8 8 20.7 even 4
855.2.a.m.1.4 4 60.59 even 2
1520.2.a.t.1.1 4 5.4 even 2
1805.2.a.p.1.4 4 380.379 even 2
4275.2.a.bo.1.1 4 12.11 even 2
4655.2.a.y.1.1 4 140.139 even 2
6080.2.a.cc.1.1 4 40.19 odd 2
6080.2.a.ch.1.4 4 40.29 even 2
7600.2.a.cf.1.4 4 1.1 even 1 trivial
9025.2.a.bf.1.1 4 76.75 even 2