Properties

Label 7600.2.a.bw.1.1
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) 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)
 
\(f(q)\) \(=\) \(q-0.801938 q^{3} +1.69202 q^{7} -2.35690 q^{9} +O(q^{10})\) \(q-0.801938 q^{3} +1.69202 q^{7} -2.35690 q^{9} +0.911854 q^{11} -1.55496 q^{13} -5.29590 q^{17} +1.00000 q^{19} -1.35690 q^{21} +4.24698 q^{23} +4.29590 q^{27} +5.00969 q^{29} -1.82908 q^{31} -0.731250 q^{33} -6.29590 q^{37} +1.24698 q^{39} +4.18060 q^{41} +7.31767 q^{43} -2.04892 q^{47} -4.13706 q^{49} +4.24698 q^{51} +2.70171 q^{53} -0.801938 q^{57} -9.87800 q^{59} +0.542877 q^{61} -3.98792 q^{63} -13.9976 q^{67} -3.40581 q^{69} +12.8780 q^{71} +2.80731 q^{73} +1.54288 q^{77} -1.59419 q^{79} +3.62565 q^{81} +12.2349 q^{83} -4.01746 q^{87} +2.91723 q^{89} -2.63102 q^{91} +1.46681 q^{93} -1.55496 q^{97} -2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53} + 2 q^{57} - 10 q^{59} - 17 q^{61} + 7 q^{63} - q^{67} + 3 q^{69} + 19 q^{71} + q^{73} - 14 q^{77} - 18 q^{79} - q^{81} + 13 q^{83} - 28 q^{87} + 2 q^{89} + 7 q^{91} + q^{93} - 5 q^{97} - 20 q^{99}+O(q^{100}) \) 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
\(3\) −0.801938 −0.462999 −0.231499 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.69202 0.639524 0.319762 0.947498i \(-0.396397\pi\)
0.319762 + 0.947498i \(0.396397\pi\)
\(8\) 0 0
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) 0.911854 0.274934 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(12\) 0 0
\(13\) −1.55496 −0.431268 −0.215634 0.976474i \(-0.569182\pi\)
−0.215634 + 0.976474i \(0.569182\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.29590 −1.28444 −0.642222 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.35690 −0.296099
\(22\) 0 0
\(23\) 4.24698 0.885556 0.442778 0.896631i \(-0.353993\pi\)
0.442778 + 0.896631i \(0.353993\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.29590 0.826746
\(28\) 0 0
\(29\) 5.00969 0.930276 0.465138 0.885238i \(-0.346005\pi\)
0.465138 + 0.885238i \(0.346005\pi\)
\(30\) 0 0
\(31\) −1.82908 −0.328513 −0.164257 0.986418i \(-0.552523\pi\)
−0.164257 + 0.986418i \(0.552523\pi\)
\(32\) 0 0
\(33\) −0.731250 −0.127294
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.29590 −1.03504 −0.517520 0.855671i \(-0.673145\pi\)
−0.517520 + 0.855671i \(0.673145\pi\)
\(38\) 0 0
\(39\) 1.24698 0.199677
\(40\) 0 0
\(41\) 4.18060 0.652901 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(42\) 0 0
\(43\) 7.31767 1.11593 0.557967 0.829863i \(-0.311582\pi\)
0.557967 + 0.829863i \(0.311582\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.04892 −0.298865 −0.149433 0.988772i \(-0.547745\pi\)
−0.149433 + 0.988772i \(0.547745\pi\)
\(48\) 0 0
\(49\) −4.13706 −0.591009
\(50\) 0 0
\(51\) 4.24698 0.594696
\(52\) 0 0
\(53\) 2.70171 0.371108 0.185554 0.982634i \(-0.440592\pi\)
0.185554 + 0.982634i \(0.440592\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.801938 −0.106219
\(58\) 0 0
\(59\) −9.87800 −1.28601 −0.643003 0.765864i \(-0.722312\pi\)
−0.643003 + 0.765864i \(0.722312\pi\)
\(60\) 0 0
\(61\) 0.542877 0.0695082 0.0347541 0.999396i \(-0.488935\pi\)
0.0347541 + 0.999396i \(0.488935\pi\)
\(62\) 0 0
\(63\) −3.98792 −0.502430
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.9976 −1.71008 −0.855040 0.518562i \(-0.826467\pi\)
−0.855040 + 0.518562i \(0.826467\pi\)
\(68\) 0 0
\(69\) −3.40581 −0.410012
\(70\) 0 0
\(71\) 12.8780 1.52834 0.764169 0.645016i \(-0.223149\pi\)
0.764169 + 0.645016i \(0.223149\pi\)
\(72\) 0 0
\(73\) 2.80731 0.328571 0.164286 0.986413i \(-0.447468\pi\)
0.164286 + 0.986413i \(0.447468\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.54288 0.175827
\(78\) 0 0
\(79\) −1.59419 −0.179360 −0.0896800 0.995971i \(-0.528584\pi\)
−0.0896800 + 0.995971i \(0.528584\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) 0 0
\(83\) 12.2349 1.34295 0.671477 0.741025i \(-0.265660\pi\)
0.671477 + 0.741025i \(0.265660\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −4.01746 −0.430717
\(88\) 0 0
\(89\) 2.91723 0.309226 0.154613 0.987975i \(-0.450587\pi\)
0.154613 + 0.987975i \(0.450587\pi\)
\(90\) 0 0
\(91\) −2.63102 −0.275806
\(92\) 0 0
\(93\) 1.46681 0.152101
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.55496 −0.157882 −0.0789410 0.996879i \(-0.525154\pi\)
−0.0789410 + 0.996879i \(0.525154\pi\)
\(98\) 0 0
\(99\) −2.14914 −0.215997
\(100\) 0 0
\(101\) −16.6015 −1.65191 −0.825955 0.563737i \(-0.809363\pi\)
−0.825955 + 0.563737i \(0.809363\pi\)
\(102\) 0 0
\(103\) −4.84548 −0.477439 −0.238720 0.971089i \(-0.576728\pi\)
−0.238720 + 0.971089i \(0.576728\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.46681 −0.431823 −0.215912 0.976413i \(-0.569272\pi\)
−0.215912 + 0.976413i \(0.569272\pi\)
\(108\) 0 0
\(109\) 18.8267 1.80327 0.901635 0.432498i \(-0.142368\pi\)
0.901635 + 0.432498i \(0.142368\pi\)
\(110\) 0 0
\(111\) 5.04892 0.479222
\(112\) 0 0
\(113\) −20.0368 −1.88491 −0.942453 0.334337i \(-0.891488\pi\)
−0.942453 + 0.334337i \(0.891488\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.66487 0.338818
\(118\) 0 0
\(119\) −8.96077 −0.821433
\(120\) 0 0
\(121\) −10.1685 −0.924411
\(122\) 0 0
\(123\) −3.35258 −0.302292
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −17.8702 −1.58573 −0.792863 0.609399i \(-0.791411\pi\)
−0.792863 + 0.609399i \(0.791411\pi\)
\(128\) 0 0
\(129\) −5.86831 −0.516676
\(130\) 0 0
\(131\) −7.44265 −0.650267 −0.325134 0.945668i \(-0.605409\pi\)
−0.325134 + 0.945668i \(0.605409\pi\)
\(132\) 0 0
\(133\) 1.69202 0.146717
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.68664 −0.485843 −0.242921 0.970046i \(-0.578106\pi\)
−0.242921 + 0.970046i \(0.578106\pi\)
\(138\) 0 0
\(139\) −3.61596 −0.306701 −0.153351 0.988172i \(-0.549006\pi\)
−0.153351 + 0.988172i \(0.549006\pi\)
\(140\) 0 0
\(141\) 1.64310 0.138374
\(142\) 0 0
\(143\) −1.41789 −0.118570
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.31767 0.273637
\(148\) 0 0
\(149\) 3.29052 0.269570 0.134785 0.990875i \(-0.456966\pi\)
0.134785 + 0.990875i \(0.456966\pi\)
\(150\) 0 0
\(151\) 10.2131 0.831133 0.415566 0.909563i \(-0.363583\pi\)
0.415566 + 0.909563i \(0.363583\pi\)
\(152\) 0 0
\(153\) 12.4819 1.00910
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.3448 −1.14484 −0.572420 0.819960i \(-0.693995\pi\)
−0.572420 + 0.819960i \(0.693995\pi\)
\(158\) 0 0
\(159\) −2.16660 −0.171823
\(160\) 0 0
\(161\) 7.18598 0.566335
\(162\) 0 0
\(163\) −19.5308 −1.52977 −0.764885 0.644167i \(-0.777204\pi\)
−0.764885 + 0.644167i \(0.777204\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.8823 0.919481 0.459741 0.888053i \(-0.347942\pi\)
0.459741 + 0.888053i \(0.347942\pi\)
\(168\) 0 0
\(169\) −10.5821 −0.814008
\(170\) 0 0
\(171\) −2.35690 −0.180236
\(172\) 0 0
\(173\) −15.4722 −1.17633 −0.588164 0.808741i \(-0.700149\pi\)
−0.588164 + 0.808741i \(0.700149\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.92154 0.595420
\(178\) 0 0
\(179\) −2.16421 −0.161761 −0.0808803 0.996724i \(-0.525773\pi\)
−0.0808803 + 0.996724i \(0.525773\pi\)
\(180\) 0 0
\(181\) 16.8974 1.25597 0.627986 0.778225i \(-0.283879\pi\)
0.627986 + 0.778225i \(0.283879\pi\)
\(182\) 0 0
\(183\) −0.435353 −0.0321822
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.82908 −0.353138
\(188\) 0 0
\(189\) 7.26875 0.528724
\(190\) 0 0
\(191\) −5.92394 −0.428641 −0.214320 0.976763i \(-0.568754\pi\)
−0.214320 + 0.976763i \(0.568754\pi\)
\(192\) 0 0
\(193\) 4.43535 0.319264 0.159632 0.987177i \(-0.448969\pi\)
0.159632 + 0.987177i \(0.448969\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.4722 −1.17359 −0.586797 0.809734i \(-0.699612\pi\)
−0.586797 + 0.809734i \(0.699612\pi\)
\(198\) 0 0
\(199\) 24.4131 1.73060 0.865300 0.501255i \(-0.167128\pi\)
0.865300 + 0.501255i \(0.167128\pi\)
\(200\) 0 0
\(201\) 11.2252 0.791765
\(202\) 0 0
\(203\) 8.47650 0.594934
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −10.0097 −0.695721
\(208\) 0 0
\(209\) 0.911854 0.0630743
\(210\) 0 0
\(211\) 4.34050 0.298813 0.149406 0.988776i \(-0.452264\pi\)
0.149406 + 0.988776i \(0.452264\pi\)
\(212\) 0 0
\(213\) −10.3274 −0.707619
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.09485 −0.210092
\(218\) 0 0
\(219\) −2.25129 −0.152128
\(220\) 0 0
\(221\) 8.23490 0.553939
\(222\) 0 0
\(223\) 26.2379 1.75702 0.878509 0.477725i \(-0.158539\pi\)
0.878509 + 0.477725i \(0.158539\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.6853 0.974699 0.487349 0.873207i \(-0.337964\pi\)
0.487349 + 0.873207i \(0.337964\pi\)
\(228\) 0 0
\(229\) −21.6407 −1.43006 −0.715029 0.699095i \(-0.753587\pi\)
−0.715029 + 0.699095i \(0.753587\pi\)
\(230\) 0 0
\(231\) −1.23729 −0.0814078
\(232\) 0 0
\(233\) −27.1183 −1.77658 −0.888289 0.459286i \(-0.848105\pi\)
−0.888289 + 0.459286i \(0.848105\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.27844 0.0830435
\(238\) 0 0
\(239\) 11.5308 0.745865 0.372933 0.927858i \(-0.378352\pi\)
0.372933 + 0.927858i \(0.378352\pi\)
\(240\) 0 0
\(241\) 11.8194 0.761354 0.380677 0.924708i \(-0.375691\pi\)
0.380677 + 0.924708i \(0.375691\pi\)
\(242\) 0 0
\(243\) −15.7952 −1.01326
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.55496 −0.0989396
\(248\) 0 0
\(249\) −9.81163 −0.621787
\(250\) 0 0
\(251\) 9.66487 0.610041 0.305021 0.952346i \(-0.401337\pi\)
0.305021 + 0.952346i \(0.401337\pi\)
\(252\) 0 0
\(253\) 3.87263 0.243470
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.1860 −0.884897 −0.442449 0.896794i \(-0.645890\pi\)
−0.442449 + 0.896794i \(0.645890\pi\)
\(258\) 0 0
\(259\) −10.6528 −0.661932
\(260\) 0 0
\(261\) −11.8073 −0.730854
\(262\) 0 0
\(263\) −21.7942 −1.34389 −0.671943 0.740603i \(-0.734540\pi\)
−0.671943 + 0.740603i \(0.734540\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.33944 −0.143171
\(268\) 0 0
\(269\) −24.7265 −1.50760 −0.753800 0.657104i \(-0.771781\pi\)
−0.753800 + 0.657104i \(0.771781\pi\)
\(270\) 0 0
\(271\) 13.2295 0.803636 0.401818 0.915720i \(-0.368378\pi\)
0.401818 + 0.915720i \(0.368378\pi\)
\(272\) 0 0
\(273\) 2.10992 0.127698
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.560335 0.0336673 0.0168336 0.999858i \(-0.494641\pi\)
0.0168336 + 0.999858i \(0.494641\pi\)
\(278\) 0 0
\(279\) 4.31096 0.258091
\(280\) 0 0
\(281\) 27.6039 1.64671 0.823355 0.567527i \(-0.192100\pi\)
0.823355 + 0.567527i \(0.192100\pi\)
\(282\) 0 0
\(283\) −15.9608 −0.948769 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.07367 0.417546
\(288\) 0 0
\(289\) 11.0465 0.649796
\(290\) 0 0
\(291\) 1.24698 0.0730992
\(292\) 0 0
\(293\) −25.3327 −1.47995 −0.739977 0.672632i \(-0.765164\pi\)
−0.739977 + 0.672632i \(0.765164\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.91723 0.227301
\(298\) 0 0
\(299\) −6.60388 −0.381912
\(300\) 0 0
\(301\) 12.3817 0.713666
\(302\) 0 0
\(303\) 13.3134 0.764832
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −11.5574 −0.659613 −0.329806 0.944049i \(-0.606983\pi\)
−0.329806 + 0.944049i \(0.606983\pi\)
\(308\) 0 0
\(309\) 3.88577 0.221054
\(310\) 0 0
\(311\) 18.3424 1.04010 0.520052 0.854135i \(-0.325913\pi\)
0.520052 + 0.854135i \(0.325913\pi\)
\(312\) 0 0
\(313\) 18.9119 1.06896 0.534481 0.845181i \(-0.320507\pi\)
0.534481 + 0.845181i \(0.320507\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.2784 −1.13895 −0.569475 0.822008i \(-0.692854\pi\)
−0.569475 + 0.822008i \(0.692854\pi\)
\(318\) 0 0
\(319\) 4.56810 0.255765
\(320\) 0 0
\(321\) 3.58211 0.199934
\(322\) 0 0
\(323\) −5.29590 −0.294672
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −15.0978 −0.834912
\(328\) 0 0
\(329\) −3.46681 −0.191132
\(330\) 0 0
\(331\) 4.77479 0.262446 0.131223 0.991353i \(-0.458110\pi\)
0.131223 + 0.991353i \(0.458110\pi\)
\(332\) 0 0
\(333\) 14.8388 0.813160
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −24.3967 −1.32897 −0.664487 0.747300i \(-0.731350\pi\)
−0.664487 + 0.747300i \(0.731350\pi\)
\(338\) 0 0
\(339\) 16.0683 0.872710
\(340\) 0 0
\(341\) −1.66786 −0.0903196
\(342\) 0 0
\(343\) −18.8442 −1.01749
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.99761 −0.536700 −0.268350 0.963322i \(-0.586478\pi\)
−0.268350 + 0.963322i \(0.586478\pi\)
\(348\) 0 0
\(349\) 21.9584 1.17541 0.587703 0.809077i \(-0.300033\pi\)
0.587703 + 0.809077i \(0.300033\pi\)
\(350\) 0 0
\(351\) −6.67994 −0.356549
\(352\) 0 0
\(353\) 36.8786 1.96285 0.981425 0.191848i \(-0.0614479\pi\)
0.981425 + 0.191848i \(0.0614479\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.18598 0.380322
\(358\) 0 0
\(359\) −16.5187 −0.871824 −0.435912 0.899989i \(-0.643574\pi\)
−0.435912 + 0.899989i \(0.643574\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 8.15452 0.428001
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.83148 −0.356600 −0.178300 0.983976i \(-0.557060\pi\)
−0.178300 + 0.983976i \(0.557060\pi\)
\(368\) 0 0
\(369\) −9.85325 −0.512940
\(370\) 0 0
\(371\) 4.57135 0.237333
\(372\) 0 0
\(373\) 4.76271 0.246604 0.123302 0.992369i \(-0.460652\pi\)
0.123302 + 0.992369i \(0.460652\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.78986 −0.401198
\(378\) 0 0
\(379\) −14.3773 −0.738514 −0.369257 0.929327i \(-0.620388\pi\)
−0.369257 + 0.929327i \(0.620388\pi\)
\(380\) 0 0
\(381\) 14.3308 0.734190
\(382\) 0 0
\(383\) −30.6708 −1.56721 −0.783603 0.621261i \(-0.786621\pi\)
−0.783603 + 0.621261i \(0.786621\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.2470 −0.876713
\(388\) 0 0
\(389\) −12.9215 −0.655148 −0.327574 0.944825i \(-0.606231\pi\)
−0.327574 + 0.944825i \(0.606231\pi\)
\(390\) 0 0
\(391\) −22.4916 −1.13745
\(392\) 0 0
\(393\) 5.96854 0.301073
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 29.8471 1.49798 0.748992 0.662579i \(-0.230538\pi\)
0.748992 + 0.662579i \(0.230538\pi\)
\(398\) 0 0
\(399\) −1.35690 −0.0679298
\(400\) 0 0
\(401\) −28.7101 −1.43371 −0.716856 0.697221i \(-0.754420\pi\)
−0.716856 + 0.697221i \(0.754420\pi\)
\(402\) 0 0
\(403\) 2.84415 0.141677
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.74094 −0.284568
\(408\) 0 0
\(409\) −13.6203 −0.673479 −0.336739 0.941598i \(-0.609324\pi\)
−0.336739 + 0.941598i \(0.609324\pi\)
\(410\) 0 0
\(411\) 4.56033 0.224945
\(412\) 0 0
\(413\) −16.7138 −0.822432
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.89977 0.142002
\(418\) 0 0
\(419\) 2.13946 0.104519 0.0522596 0.998634i \(-0.483358\pi\)
0.0522596 + 0.998634i \(0.483358\pi\)
\(420\) 0 0
\(421\) −15.0562 −0.733795 −0.366897 0.930261i \(-0.619580\pi\)
−0.366897 + 0.930261i \(0.619580\pi\)
\(422\) 0 0
\(423\) 4.82908 0.234798
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.918559 0.0444522
\(428\) 0 0
\(429\) 1.13706 0.0548979
\(430\) 0 0
\(431\) −4.37435 −0.210705 −0.105353 0.994435i \(-0.533597\pi\)
−0.105353 + 0.994435i \(0.533597\pi\)
\(432\) 0 0
\(433\) 33.1564 1.59340 0.796698 0.604377i \(-0.206578\pi\)
0.796698 + 0.604377i \(0.206578\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.24698 0.203161
\(438\) 0 0
\(439\) −32.2368 −1.53858 −0.769290 0.638900i \(-0.779390\pi\)
−0.769290 + 0.638900i \(0.779390\pi\)
\(440\) 0 0
\(441\) 9.75063 0.464316
\(442\) 0 0
\(443\) 21.7362 1.03272 0.516358 0.856373i \(-0.327287\pi\)
0.516358 + 0.856373i \(0.327287\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −2.63879 −0.124811
\(448\) 0 0
\(449\) −24.9584 −1.17786 −0.588929 0.808185i \(-0.700450\pi\)
−0.588929 + 0.808185i \(0.700450\pi\)
\(450\) 0 0
\(451\) 3.81210 0.179505
\(452\) 0 0
\(453\) −8.19029 −0.384814
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13.1347 −0.614414 −0.307207 0.951643i \(-0.599394\pi\)
−0.307207 + 0.951643i \(0.599394\pi\)
\(458\) 0 0
\(459\) −22.7506 −1.06191
\(460\) 0 0
\(461\) −35.3726 −1.64746 −0.823732 0.566979i \(-0.808112\pi\)
−0.823732 + 0.566979i \(0.808112\pi\)
\(462\) 0 0
\(463\) 14.6963 0.682997 0.341498 0.939882i \(-0.389066\pi\)
0.341498 + 0.939882i \(0.389066\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −33.2121 −1.53687 −0.768435 0.639927i \(-0.778964\pi\)
−0.768435 + 0.639927i \(0.778964\pi\)
\(468\) 0 0
\(469\) −23.6843 −1.09364
\(470\) 0 0
\(471\) 11.5036 0.530060
\(472\) 0 0
\(473\) 6.67264 0.306809
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.36765 −0.291555
\(478\) 0 0
\(479\) −24.6219 −1.12500 −0.562502 0.826796i \(-0.690161\pi\)
−0.562502 + 0.826796i \(0.690161\pi\)
\(480\) 0 0
\(481\) 9.78986 0.446379
\(482\) 0 0
\(483\) −5.76271 −0.262212
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −29.5646 −1.33970 −0.669851 0.742496i \(-0.733642\pi\)
−0.669851 + 0.742496i \(0.733642\pi\)
\(488\) 0 0
\(489\) 15.6625 0.708282
\(490\) 0 0
\(491\) −36.2978 −1.63810 −0.819049 0.573724i \(-0.805498\pi\)
−0.819049 + 0.573724i \(0.805498\pi\)
\(492\) 0 0
\(493\) −26.5308 −1.19489
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.7899 0.977409
\(498\) 0 0
\(499\) −7.84415 −0.351152 −0.175576 0.984466i \(-0.556179\pi\)
−0.175576 + 0.984466i \(0.556179\pi\)
\(500\) 0 0
\(501\) −9.52888 −0.425719
\(502\) 0 0
\(503\) −20.4166 −0.910330 −0.455165 0.890407i \(-0.650420\pi\)
−0.455165 + 0.890407i \(0.650420\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.48619 0.376885
\(508\) 0 0
\(509\) −14.7530 −0.653916 −0.326958 0.945039i \(-0.606024\pi\)
−0.326958 + 0.945039i \(0.606024\pi\)
\(510\) 0 0
\(511\) 4.75004 0.210129
\(512\) 0 0
\(513\) 4.29590 0.189668
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.86831 −0.0821683
\(518\) 0 0
\(519\) 12.4077 0.544639
\(520\) 0 0
\(521\) 37.1487 1.62751 0.813756 0.581206i \(-0.197419\pi\)
0.813756 + 0.581206i \(0.197419\pi\)
\(522\) 0 0
\(523\) −16.3623 −0.715472 −0.357736 0.933823i \(-0.616451\pi\)
−0.357736 + 0.933823i \(0.616451\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.68664 0.421957
\(528\) 0 0
\(529\) −4.96316 −0.215790
\(530\) 0 0
\(531\) 23.2814 1.01033
\(532\) 0 0
\(533\) −6.50066 −0.281575
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.73556 0.0748950
\(538\) 0 0
\(539\) −3.77240 −0.162489
\(540\) 0 0
\(541\) −2.08947 −0.0898335 −0.0449168 0.998991i \(-0.514302\pi\)
−0.0449168 + 0.998991i \(0.514302\pi\)
\(542\) 0 0
\(543\) −13.5506 −0.581514
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.86054 −0.0795511 −0.0397756 0.999209i \(-0.512664\pi\)
−0.0397756 + 0.999209i \(0.512664\pi\)
\(548\) 0 0
\(549\) −1.27950 −0.0546079
\(550\) 0 0
\(551\) 5.00969 0.213420
\(552\) 0 0
\(553\) −2.69740 −0.114705
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.80061 −0.415265 −0.207633 0.978207i \(-0.566576\pi\)
−0.207633 + 0.978207i \(0.566576\pi\)
\(558\) 0 0
\(559\) −11.3787 −0.481266
\(560\) 0 0
\(561\) 3.87263 0.163502
\(562\) 0 0
\(563\) −38.0170 −1.60222 −0.801112 0.598514i \(-0.795758\pi\)
−0.801112 + 0.598514i \(0.795758\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 6.13467 0.257632
\(568\) 0 0
\(569\) −7.32975 −0.307279 −0.153640 0.988127i \(-0.549099\pi\)
−0.153640 + 0.988127i \(0.549099\pi\)
\(570\) 0 0
\(571\) −37.8775 −1.58513 −0.792563 0.609791i \(-0.791254\pi\)
−0.792563 + 0.609791i \(0.791254\pi\)
\(572\) 0 0
\(573\) 4.75063 0.198460
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 28.6993 1.19477 0.597384 0.801955i \(-0.296207\pi\)
0.597384 + 0.801955i \(0.296207\pi\)
\(578\) 0 0
\(579\) −3.55688 −0.147819
\(580\) 0 0
\(581\) 20.7017 0.858852
\(582\) 0 0
\(583\) 2.46357 0.102030
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.72348 −0.153684 −0.0768422 0.997043i \(-0.524484\pi\)
−0.0768422 + 0.997043i \(0.524484\pi\)
\(588\) 0 0
\(589\) −1.82908 −0.0753661
\(590\) 0 0
\(591\) 13.2097 0.543373
\(592\) 0 0
\(593\) 27.7399 1.13914 0.569570 0.821943i \(-0.307110\pi\)
0.569570 + 0.821943i \(0.307110\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.5778 −0.801266
\(598\) 0 0
\(599\) 23.5579 0.962551 0.481276 0.876569i \(-0.340174\pi\)
0.481276 + 0.876569i \(0.340174\pi\)
\(600\) 0 0
\(601\) −7.36898 −0.300587 −0.150293 0.988641i \(-0.548022\pi\)
−0.150293 + 0.988641i \(0.548022\pi\)
\(602\) 0 0
\(603\) 32.9909 1.34349
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.4198 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(608\) 0 0
\(609\) −6.79763 −0.275454
\(610\) 0 0
\(611\) 3.18598 0.128891
\(612\) 0 0
\(613\) −6.48129 −0.261777 −0.130888 0.991397i \(-0.541783\pi\)
−0.130888 + 0.991397i \(0.541783\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4650 0.984924 0.492462 0.870334i \(-0.336097\pi\)
0.492462 + 0.870334i \(0.336097\pi\)
\(618\) 0 0
\(619\) 22.2457 0.894128 0.447064 0.894502i \(-0.352470\pi\)
0.447064 + 0.894502i \(0.352470\pi\)
\(620\) 0 0
\(621\) 18.2446 0.732130
\(622\) 0 0
\(623\) 4.93602 0.197757
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.731250 −0.0292033
\(628\) 0 0
\(629\) 33.3424 1.32945
\(630\) 0 0
\(631\) 15.5888 0.620581 0.310290 0.950642i \(-0.399574\pi\)
0.310290 + 0.950642i \(0.399574\pi\)
\(632\) 0 0
\(633\) −3.48081 −0.138350
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.43296 0.254883
\(638\) 0 0
\(639\) −30.3521 −1.20071
\(640\) 0 0
\(641\) 8.17496 0.322892 0.161446 0.986882i \(-0.448384\pi\)
0.161446 + 0.986882i \(0.448384\pi\)
\(642\) 0 0
\(643\) 11.1836 0.441038 0.220519 0.975383i \(-0.429225\pi\)
0.220519 + 0.975383i \(0.429225\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.2121 0.637362 0.318681 0.947862i \(-0.396760\pi\)
0.318681 + 0.947862i \(0.396760\pi\)
\(648\) 0 0
\(649\) −9.00730 −0.353567
\(650\) 0 0
\(651\) 2.48188 0.0972725
\(652\) 0 0
\(653\) −24.7952 −0.970312 −0.485156 0.874427i \(-0.661237\pi\)
−0.485156 + 0.874427i \(0.661237\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.61655 −0.258136
\(658\) 0 0
\(659\) 20.2868 0.790262 0.395131 0.918625i \(-0.370699\pi\)
0.395131 + 0.918625i \(0.370699\pi\)
\(660\) 0 0
\(661\) 20.4222 0.794332 0.397166 0.917747i \(-0.369994\pi\)
0.397166 + 0.917747i \(0.369994\pi\)
\(662\) 0 0
\(663\) −6.60388 −0.256473
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 21.2760 0.823812
\(668\) 0 0
\(669\) −21.0411 −0.813498
\(670\) 0 0
\(671\) 0.495024 0.0191102
\(672\) 0 0
\(673\) −28.7254 −1.10728 −0.553641 0.832755i \(-0.686762\pi\)
−0.553641 + 0.832755i \(0.686762\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −44.0062 −1.69130 −0.845648 0.533740i \(-0.820786\pi\)
−0.845648 + 0.533740i \(0.820786\pi\)
\(678\) 0 0
\(679\) −2.63102 −0.100969
\(680\) 0 0
\(681\) −11.7767 −0.451284
\(682\) 0 0
\(683\) 8.48427 0.324642 0.162321 0.986738i \(-0.448102\pi\)
0.162321 + 0.986738i \(0.448102\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 17.3545 0.662116
\(688\) 0 0
\(689\) −4.20105 −0.160047
\(690\) 0 0
\(691\) −20.2586 −0.770673 −0.385336 0.922776i \(-0.625915\pi\)
−0.385336 + 0.922776i \(0.625915\pi\)
\(692\) 0 0
\(693\) −3.63640 −0.138135
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −22.1400 −0.838614
\(698\) 0 0
\(699\) 21.7472 0.822553
\(700\) 0 0
\(701\) −23.4101 −0.884188 −0.442094 0.896969i \(-0.645764\pi\)
−0.442094 + 0.896969i \(0.645764\pi\)
\(702\) 0 0
\(703\) −6.29590 −0.237454
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −28.0901 −1.05644
\(708\) 0 0
\(709\) 30.3472 1.13971 0.569857 0.821744i \(-0.306999\pi\)
0.569857 + 0.821744i \(0.306999\pi\)
\(710\) 0 0
\(711\) 3.75733 0.140911
\(712\) 0 0
\(713\) −7.76809 −0.290917
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.24698 −0.345335
\(718\) 0 0
\(719\) 38.3230 1.42921 0.714604 0.699529i \(-0.246607\pi\)
0.714604 + 0.699529i \(0.246607\pi\)
\(720\) 0 0
\(721\) −8.19865 −0.305334
\(722\) 0 0
\(723\) −9.47842 −0.352506
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.69069 0.0997923 0.0498961 0.998754i \(-0.484111\pi\)
0.0498961 + 0.998754i \(0.484111\pi\)
\(728\) 0 0
\(729\) 1.78986 0.0662910
\(730\) 0 0
\(731\) −38.7536 −1.43335
\(732\) 0 0
\(733\) −18.9651 −0.700491 −0.350246 0.936658i \(-0.613902\pi\)
−0.350246 + 0.936658i \(0.613902\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.7638 −0.470160
\(738\) 0 0
\(739\) −29.8278 −1.09723 −0.548616 0.836075i \(-0.684845\pi\)
−0.548616 + 0.836075i \(0.684845\pi\)
\(740\) 0 0
\(741\) 1.24698 0.0458089
\(742\) 0 0
\(743\) 8.78448 0.322271 0.161136 0.986932i \(-0.448484\pi\)
0.161136 + 0.986932i \(0.448484\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −28.8364 −1.05507
\(748\) 0 0
\(749\) −7.55794 −0.276161
\(750\) 0 0
\(751\) 18.1142 0.660998 0.330499 0.943806i \(-0.392783\pi\)
0.330499 + 0.943806i \(0.392783\pi\)
\(752\) 0 0
\(753\) −7.75063 −0.282449
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.222816 0.00809840 0.00404920 0.999992i \(-0.498711\pi\)
0.00404920 + 0.999992i \(0.498711\pi\)
\(758\) 0 0
\(759\) −3.10560 −0.112726
\(760\) 0 0
\(761\) −6.07798 −0.220327 −0.110163 0.993913i \(-0.535137\pi\)
−0.110163 + 0.993913i \(0.535137\pi\)
\(762\) 0 0
\(763\) 31.8552 1.15323
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.3599 0.554613
\(768\) 0 0
\(769\) −14.6267 −0.527453 −0.263726 0.964598i \(-0.584952\pi\)
−0.263726 + 0.964598i \(0.584952\pi\)
\(770\) 0 0
\(771\) 11.3763 0.409706
\(772\) 0 0
\(773\) 4.05728 0.145930 0.0729651 0.997334i \(-0.476754\pi\)
0.0729651 + 0.997334i \(0.476754\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.54288 0.306474
\(778\) 0 0
\(779\) 4.18060 0.149786
\(780\) 0 0
\(781\) 11.7429 0.420192
\(782\) 0 0
\(783\) 21.5211 0.769102
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 19.5657 0.697442 0.348721 0.937227i \(-0.386616\pi\)
0.348721 + 0.937227i \(0.386616\pi\)
\(788\) 0 0
\(789\) 17.4776 0.622218
\(790\) 0 0
\(791\) −33.9028 −1.20544
\(792\) 0 0
\(793\) −0.844150 −0.0299767
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −38.9051 −1.37809 −0.689046 0.724718i \(-0.741970\pi\)
−0.689046 + 0.724718i \(0.741970\pi\)
\(798\) 0 0
\(799\) 10.8509 0.383876
\(800\) 0 0
\(801\) −6.87561 −0.242938
\(802\) 0 0
\(803\) 2.55986 0.0903355
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.8291 0.698017
\(808\) 0 0
\(809\) −22.5730 −0.793625 −0.396812 0.917900i \(-0.629884\pi\)
−0.396812 + 0.917900i \(0.629884\pi\)
\(810\) 0 0
\(811\) 46.3605 1.62794 0.813968 0.580909i \(-0.197303\pi\)
0.813968 + 0.580909i \(0.197303\pi\)
\(812\) 0 0
\(813\) −10.6093 −0.372083
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.31767 0.256013
\(818\) 0 0
\(819\) 6.20105 0.216682
\(820\) 0 0
\(821\) −17.8194 −0.621901 −0.310951 0.950426i \(-0.600647\pi\)
−0.310951 + 0.950426i \(0.600647\pi\)
\(822\) 0 0
\(823\) −32.5394 −1.13425 −0.567126 0.823631i \(-0.691945\pi\)
−0.567126 + 0.823631i \(0.691945\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.8256 1.38487 0.692436 0.721479i \(-0.256537\pi\)
0.692436 + 0.721479i \(0.256537\pi\)
\(828\) 0 0
\(829\) 38.7525 1.34593 0.672966 0.739674i \(-0.265020\pi\)
0.672966 + 0.739674i \(0.265020\pi\)
\(830\) 0 0
\(831\) −0.449354 −0.0155879
\(832\) 0 0
\(833\) 21.9095 0.759118
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −7.85756 −0.271597
\(838\) 0 0
\(839\) −2.76749 −0.0955445 −0.0477723 0.998858i \(-0.515212\pi\)
−0.0477723 + 0.998858i \(0.515212\pi\)
\(840\) 0 0
\(841\) −3.90302 −0.134587
\(842\) 0 0
\(843\) −22.1366 −0.762425
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −17.2054 −0.591183
\(848\) 0 0
\(849\) 12.7995 0.439279
\(850\) 0 0
\(851\) −26.7385 −0.916586
\(852\) 0 0
\(853\) −24.1390 −0.826503 −0.413252 0.910617i \(-0.635607\pi\)
−0.413252 + 0.910617i \(0.635607\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.16229 0.108022 0.0540109 0.998540i \(-0.482799\pi\)
0.0540109 + 0.998540i \(0.482799\pi\)
\(858\) 0 0
\(859\) −4.86426 −0.165967 −0.0829833 0.996551i \(-0.526445\pi\)
−0.0829833 + 0.996551i \(0.526445\pi\)
\(860\) 0 0
\(861\) −5.67264 −0.193323
\(862\) 0 0
\(863\) 4.54958 0.154870 0.0774348 0.996997i \(-0.475327\pi\)
0.0774348 + 0.996997i \(0.475327\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.85862 −0.300855
\(868\) 0 0
\(869\) −1.45367 −0.0493122
\(870\) 0 0
\(871\) 21.7657 0.737502
\(872\) 0 0
\(873\) 3.66487 0.124037
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18.6595 0.630086 0.315043 0.949077i \(-0.397981\pi\)
0.315043 + 0.949077i \(0.397981\pi\)
\(878\) 0 0
\(879\) 20.3153 0.685217
\(880\) 0 0
\(881\) 19.4306 0.654632 0.327316 0.944915i \(-0.393856\pi\)
0.327316 + 0.944915i \(0.393856\pi\)
\(882\) 0 0
\(883\) 25.6437 0.862979 0.431490 0.902118i \(-0.357988\pi\)
0.431490 + 0.902118i \(0.357988\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −31.0062 −1.04109 −0.520544 0.853835i \(-0.674271\pi\)
−0.520544 + 0.853835i \(0.674271\pi\)
\(888\) 0 0
\(889\) −30.2368 −1.01411
\(890\) 0 0
\(891\) 3.30606 0.110757
\(892\) 0 0
\(893\) −2.04892 −0.0685644
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.29590 0.176825
\(898\) 0 0
\(899\) −9.16315 −0.305608
\(900\) 0 0
\(901\) −14.3080 −0.476668
\(902\) 0 0
\(903\) −9.92931 −0.330427
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −17.7676 −0.589964 −0.294982 0.955503i \(-0.595314\pi\)
−0.294982 + 0.955503i \(0.595314\pi\)
\(908\) 0 0
\(909\) 39.1280 1.29779
\(910\) 0 0
\(911\) −41.7313 −1.38262 −0.691309 0.722559i \(-0.742966\pi\)
−0.691309 + 0.722559i \(0.742966\pi\)
\(912\) 0 0
\(913\) 11.1564 0.369224
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.5931 −0.415862
\(918\) 0 0
\(919\) −30.4088 −1.00309 −0.501547 0.865130i \(-0.667236\pi\)
−0.501547 + 0.865130i \(0.667236\pi\)
\(920\) 0 0
\(921\) 9.26828 0.305400
\(922\) 0 0
\(923\) −20.0248 −0.659123
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 11.4203 0.375091
\(928\) 0 0
\(929\) −28.8219 −0.945616 −0.472808 0.881165i \(-0.656760\pi\)
−0.472808 + 0.881165i \(0.656760\pi\)
\(930\) 0 0
\(931\) −4.13706 −0.135587
\(932\) 0 0
\(933\) −14.7095 −0.481567
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 50.4601 1.64846 0.824230 0.566255i \(-0.191608\pi\)
0.824230 + 0.566255i \(0.191608\pi\)
\(938\) 0 0
\(939\) −15.1661 −0.494928
\(940\) 0 0
\(941\) 1.10082 0.0358857 0.0179428 0.999839i \(-0.494288\pi\)
0.0179428 + 0.999839i \(0.494288\pi\)
\(942\) 0 0
\(943\) 17.7549 0.578180
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.9353 −0.452836 −0.226418 0.974030i \(-0.572701\pi\)
−0.226418 + 0.974030i \(0.572701\pi\)
\(948\) 0 0
\(949\) −4.36526 −0.141702
\(950\) 0 0
\(951\) 16.2620 0.527333
\(952\) 0 0
\(953\) 41.3400 1.33913 0.669567 0.742751i \(-0.266480\pi\)
0.669567 + 0.742751i \(0.266480\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −3.66334 −0.118419
\(958\) 0 0
\(959\) −9.62192 −0.310708
\(960\) 0 0
\(961\) −27.6544 −0.892079
\(962\) 0 0
\(963\) 10.5278 0.339254
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.26875 −0.169432 −0.0847158 0.996405i \(-0.526998\pi\)
−0.0847158 + 0.996405i \(0.526998\pi\)
\(968\) 0 0
\(969\) 4.24698 0.136433
\(970\) 0 0
\(971\) 5.15346 0.165382 0.0826911 0.996575i \(-0.473649\pi\)
0.0826911 + 0.996575i \(0.473649\pi\)
\(972\) 0 0
\(973\) −6.11828 −0.196143
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.77612 −0.152802 −0.0764008 0.997077i \(-0.524343\pi\)
−0.0764008 + 0.997077i \(0.524343\pi\)
\(978\) 0 0
\(979\) 2.66009 0.0850168
\(980\) 0 0
\(981\) −44.3726 −1.41671
\(982\) 0 0
\(983\) −28.9758 −0.924186 −0.462093 0.886832i \(-0.652901\pi\)
−0.462093 + 0.886832i \(0.652901\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.78017 0.0884937
\(988\) 0 0
\(989\) 31.0780 0.988222
\(990\) 0 0
\(991\) 38.5042 1.22313 0.611564 0.791195i \(-0.290541\pi\)
0.611564 + 0.791195i \(0.290541\pi\)
\(992\) 0 0
\(993\) −3.82908 −0.121512
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −10.1491 −0.321427 −0.160713 0.987001i \(-0.551379\pi\)
−0.160713 + 0.987001i \(0.551379\pi\)
\(998\) 0 0
\(999\) −27.0465 −0.855714
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.bw.1.1 3
4.3 odd 2 475.2.a.d.1.3 3
5.4 even 2 7600.2.a.bn.1.3 3
12.11 even 2 4275.2.a.bn.1.1 3
20.3 even 4 475.2.b.c.324.3 6
20.7 even 4 475.2.b.c.324.4 6
20.19 odd 2 475.2.a.h.1.1 yes 3
60.59 even 2 4275.2.a.z.1.3 3
76.75 even 2 9025.2.a.be.1.1 3
380.379 even 2 9025.2.a.w.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 3 4.3 odd 2
475.2.a.h.1.1 yes 3 20.19 odd 2
475.2.b.c.324.3 6 20.3 even 4
475.2.b.c.324.4 6 20.7 even 4
4275.2.a.z.1.3 3 60.59 even 2
4275.2.a.bn.1.1 3 12.11 even 2
7600.2.a.bn.1.3 3 5.4 even 2
7600.2.a.bw.1.1 3 1.1 even 1 trivial
9025.2.a.w.1.3 3 380.379 even 2
9025.2.a.be.1.1 3 76.75 even 2