Properties

Label 6960.2.a.co.1.4
Level $6960$
Weight $2$
Character 6960.1
Self dual yes
Analytic conductor $55.576$
Analytic rank $1$
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] = [6960,2,Mod(1,6960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6960 = 2^{4} \cdot 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6960.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: \(55.5758798068\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 6960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.74301 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +2.74301 q^{7} +1.00000 q^{9} -2.74301 q^{11} -5.14744 q^{13} -1.00000 q^{15} -3.72913 q^{17} +0.404431 q^{19} +2.74301 q^{21} +5.45825 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{29} -1.45825 q^{31} -2.74301 q^{33} -2.74301 q^{35} +6.76702 q^{37} -5.14744 q^{39} +9.78090 q^{41} -4.43220 q^{43} -1.00000 q^{45} -2.60569 q^{47} +0.524103 q^{49} -3.72913 q^{51} -6.43220 q^{53} +2.74301 q^{55} +0.404431 q^{57} +9.91822 q^{59} -13.0816 q^{61} +2.74301 q^{63} +5.14744 q^{65} -12.4961 q^{67} +5.45825 q^{69} +11.3487 q^{71} -10.7670 q^{73} +1.00000 q^{75} -7.52410 q^{77} -14.1576 q^{79} +1.00000 q^{81} -1.62334 q^{83} +3.72913 q^{85} -1.00000 q^{87} -8.87281 q^{89} -14.1195 q^{91} -1.45825 q^{93} -0.404431 q^{95} +7.82084 q^{97} -2.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{5} - 2 q^{7} + 4 q^{9} + 2 q^{11} - 8 q^{13} - 4 q^{15} - 10 q^{17} + 2 q^{19} - 2 q^{21} + 12 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 16 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 4 q^{45} + 12 q^{47} + 6 q^{49} - 10 q^{51} - 10 q^{53} - 2 q^{55} + 2 q^{57} - 2 q^{59} - 26 q^{61} - 2 q^{63} + 8 q^{65} - 2 q^{67} + 12 q^{69} + 10 q^{71} + 4 q^{75} - 34 q^{77} - 22 q^{79} + 4 q^{81} + 10 q^{83} + 10 q^{85} - 4 q^{87} - 4 q^{89} + 8 q^{91} + 4 q^{93} - 2 q^{95} - 22 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.74301 1.03676 0.518380 0.855150i \(-0.326535\pi\)
0.518380 + 0.855150i \(0.326535\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.74301 −0.827049 −0.413524 0.910493i \(-0.635702\pi\)
−0.413524 + 0.910493i \(0.635702\pi\)
\(12\) 0 0
\(13\) −5.14744 −1.42764 −0.713822 0.700328i \(-0.753037\pi\)
−0.713822 + 0.700328i \(0.753037\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −3.72913 −0.904446 −0.452223 0.891905i \(-0.649369\pi\)
−0.452223 + 0.891905i \(0.649369\pi\)
\(18\) 0 0
\(19\) 0.404431 0.0927827 0.0463914 0.998923i \(-0.485228\pi\)
0.0463914 + 0.998923i \(0.485228\pi\)
\(20\) 0 0
\(21\) 2.74301 0.598574
\(22\) 0 0
\(23\) 5.45825 1.13812 0.569062 0.822295i \(-0.307306\pi\)
0.569062 + 0.822295i \(0.307306\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −1.45825 −0.261910 −0.130955 0.991388i \(-0.541804\pi\)
−0.130955 + 0.991388i \(0.541804\pi\)
\(32\) 0 0
\(33\) −2.74301 −0.477497
\(34\) 0 0
\(35\) −2.74301 −0.463653
\(36\) 0 0
\(37\) 6.76702 1.11249 0.556245 0.831018i \(-0.312241\pi\)
0.556245 + 0.831018i \(0.312241\pi\)
\(38\) 0 0
\(39\) −5.14744 −0.824250
\(40\) 0 0
\(41\) 9.78090 1.52752 0.763760 0.645500i \(-0.223351\pi\)
0.763760 + 0.645500i \(0.223351\pi\)
\(42\) 0 0
\(43\) −4.43220 −0.675904 −0.337952 0.941163i \(-0.609734\pi\)
−0.337952 + 0.941163i \(0.609734\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −2.60569 −0.380079 −0.190040 0.981776i \(-0.560862\pi\)
−0.190040 + 0.981776i \(0.560862\pi\)
\(48\) 0 0
\(49\) 0.524103 0.0748719
\(50\) 0 0
\(51\) −3.72913 −0.522182
\(52\) 0 0
\(53\) −6.43220 −0.883530 −0.441765 0.897131i \(-0.645648\pi\)
−0.441765 + 0.897131i \(0.645648\pi\)
\(54\) 0 0
\(55\) 2.74301 0.369867
\(56\) 0 0
\(57\) 0.404431 0.0535681
\(58\) 0 0
\(59\) 9.91822 1.29124 0.645621 0.763658i \(-0.276599\pi\)
0.645621 + 0.763658i \(0.276599\pi\)
\(60\) 0 0
\(61\) −13.0816 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(62\) 0 0
\(63\) 2.74301 0.345587
\(64\) 0 0
\(65\) 5.14744 0.638461
\(66\) 0 0
\(67\) −12.4961 −1.52665 −0.763323 0.646017i \(-0.776434\pi\)
−0.763323 + 0.646017i \(0.776434\pi\)
\(68\) 0 0
\(69\) 5.45825 0.657096
\(70\) 0 0
\(71\) 11.3487 1.34684 0.673422 0.739259i \(-0.264824\pi\)
0.673422 + 0.739259i \(0.264824\pi\)
\(72\) 0 0
\(73\) −10.7670 −1.26018 −0.630092 0.776520i \(-0.716983\pi\)
−0.630092 + 0.776520i \(0.716983\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −7.52410 −0.857451
\(78\) 0 0
\(79\) −14.1576 −1.59285 −0.796425 0.604737i \(-0.793278\pi\)
−0.796425 + 0.604737i \(0.793278\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.62334 −0.178184 −0.0890922 0.996023i \(-0.528397\pi\)
−0.0890922 + 0.996023i \(0.528397\pi\)
\(84\) 0 0
\(85\) 3.72913 0.404481
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −8.87281 −0.940516 −0.470258 0.882529i \(-0.655839\pi\)
−0.470258 + 0.882529i \(0.655839\pi\)
\(90\) 0 0
\(91\) −14.1195 −1.48012
\(92\) 0 0
\(93\) −1.45825 −0.151214
\(94\) 0 0
\(95\) −0.404431 −0.0414937
\(96\) 0 0
\(97\) 7.82084 0.794086 0.397043 0.917800i \(-0.370036\pi\)
0.397043 + 0.917800i \(0.370036\pi\)
\(98\) 0 0
\(99\) −2.74301 −0.275683
\(100\) 0 0
\(101\) −4.88033 −0.485611 −0.242805 0.970075i \(-0.578068\pi\)
−0.242805 + 0.970075i \(0.578068\pi\)
\(102\) 0 0
\(103\) 0.294881 0.0290555 0.0145277 0.999894i \(-0.495376\pi\)
0.0145277 + 0.999894i \(0.495376\pi\)
\(104\) 0 0
\(105\) −2.74301 −0.267690
\(106\) 0 0
\(107\) 13.7809 1.33225 0.666125 0.745840i \(-0.267952\pi\)
0.666125 + 0.745840i \(0.267952\pi\)
\(108\) 0 0
\(109\) −6.20126 −0.593973 −0.296987 0.954882i \(-0.595982\pi\)
−0.296987 + 0.954882i \(0.595982\pi\)
\(110\) 0 0
\(111\) 6.76702 0.642297
\(112\) 0 0
\(113\) −10.5658 −0.993943 −0.496971 0.867767i \(-0.665555\pi\)
−0.496971 + 0.867767i \(0.665555\pi\)
\(114\) 0 0
\(115\) −5.45825 −0.508985
\(116\) 0 0
\(117\) −5.14744 −0.475881
\(118\) 0 0
\(119\) −10.2290 −0.937694
\(120\) 0 0
\(121\) −3.47590 −0.315991
\(122\) 0 0
\(123\) 9.78090 0.881914
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.54175 −0.757958 −0.378979 0.925405i \(-0.623725\pi\)
−0.378979 + 0.925405i \(0.623725\pi\)
\(128\) 0 0
\(129\) −4.43220 −0.390233
\(130\) 0 0
\(131\) −13.3050 −1.16246 −0.581232 0.813738i \(-0.697429\pi\)
−0.581232 + 0.813738i \(0.697429\pi\)
\(132\) 0 0
\(133\) 1.10936 0.0961934
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −18.2253 −1.55709 −0.778545 0.627589i \(-0.784042\pi\)
−0.778545 + 0.627589i \(0.784042\pi\)
\(138\) 0 0
\(139\) 5.66142 0.480195 0.240098 0.970749i \(-0.422821\pi\)
0.240098 + 0.970749i \(0.422821\pi\)
\(140\) 0 0
\(141\) −2.60569 −0.219439
\(142\) 0 0
\(143\) 14.1195 1.18073
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) 0.524103 0.0432273
\(148\) 0 0
\(149\) −11.9182 −0.976378 −0.488189 0.872738i \(-0.662342\pi\)
−0.488189 + 0.872738i \(0.662342\pi\)
\(150\) 0 0
\(151\) −7.19114 −0.585207 −0.292603 0.956234i \(-0.594522\pi\)
−0.292603 + 0.956234i \(0.594522\pi\)
\(152\) 0 0
\(153\) −3.72913 −0.301482
\(154\) 0 0
\(155\) 1.45825 0.117130
\(156\) 0 0
\(157\) −4.31457 −0.344340 −0.172170 0.985067i \(-0.555078\pi\)
−0.172170 + 0.985067i \(0.555078\pi\)
\(158\) 0 0
\(159\) −6.43220 −0.510106
\(160\) 0 0
\(161\) 14.9720 1.17996
\(162\) 0 0
\(163\) −21.6436 −1.69526 −0.847628 0.530591i \(-0.821970\pi\)
−0.847628 + 0.530591i \(0.821970\pi\)
\(164\) 0 0
\(165\) 2.74301 0.213543
\(166\) 0 0
\(167\) 16.4303 1.27141 0.635707 0.771930i \(-0.280709\pi\)
0.635707 + 0.771930i \(0.280709\pi\)
\(168\) 0 0
\(169\) 13.4961 1.03816
\(170\) 0 0
\(171\) 0.404431 0.0309276
\(172\) 0 0
\(173\) −4.64939 −0.353487 −0.176743 0.984257i \(-0.556556\pi\)
−0.176743 + 0.984257i \(0.556556\pi\)
\(174\) 0 0
\(175\) 2.74301 0.207352
\(176\) 0 0
\(177\) 9.91822 0.745499
\(178\) 0 0
\(179\) −7.62334 −0.569795 −0.284897 0.958558i \(-0.591960\pi\)
−0.284897 + 0.958558i \(0.591960\pi\)
\(180\) 0 0
\(181\) −21.3050 −1.58359 −0.791794 0.610788i \(-0.790853\pi\)
−0.791794 + 0.610788i \(0.790853\pi\)
\(182\) 0 0
\(183\) −13.0816 −0.967019
\(184\) 0 0
\(185\) −6.76702 −0.497521
\(186\) 0 0
\(187\) 10.2290 0.748021
\(188\) 0 0
\(189\) 2.74301 0.199525
\(190\) 0 0
\(191\) −3.07597 −0.222570 −0.111285 0.993789i \(-0.535497\pi\)
−0.111285 + 0.993789i \(0.535497\pi\)
\(192\) 0 0
\(193\) 6.77454 0.487642 0.243821 0.969820i \(-0.421599\pi\)
0.243821 + 0.969820i \(0.421599\pi\)
\(194\) 0 0
\(195\) 5.14744 0.368616
\(196\) 0 0
\(197\) −13.6455 −0.972201 −0.486100 0.873903i \(-0.661581\pi\)
−0.486100 + 0.873903i \(0.661581\pi\)
\(198\) 0 0
\(199\) 6.33858 0.449330 0.224665 0.974436i \(-0.427871\pi\)
0.224665 + 0.974436i \(0.427871\pi\)
\(200\) 0 0
\(201\) −12.4961 −0.881410
\(202\) 0 0
\(203\) −2.74301 −0.192522
\(204\) 0 0
\(205\) −9.78090 −0.683128
\(206\) 0 0
\(207\) 5.45825 0.379375
\(208\) 0 0
\(209\) −1.10936 −0.0767358
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 11.3487 0.777600
\(214\) 0 0
\(215\) 4.43220 0.302273
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) −10.7670 −0.727568
\(220\) 0 0
\(221\) 19.1955 1.29123
\(222\) 0 0
\(223\) 2.20126 0.147407 0.0737037 0.997280i \(-0.476518\pi\)
0.0737037 + 0.997280i \(0.476518\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.1853 −0.808769 −0.404384 0.914589i \(-0.632514\pi\)
−0.404384 + 0.914589i \(0.632514\pi\)
\(228\) 0 0
\(229\) −3.16337 −0.209041 −0.104521 0.994523i \(-0.533331\pi\)
−0.104521 + 0.994523i \(0.533331\pi\)
\(230\) 0 0
\(231\) −7.52410 −0.495050
\(232\) 0 0
\(233\) 23.8087 1.55976 0.779879 0.625930i \(-0.215281\pi\)
0.779879 + 0.625930i \(0.215281\pi\)
\(234\) 0 0
\(235\) 2.60569 0.169977
\(236\) 0 0
\(237\) −14.1576 −0.919633
\(238\) 0 0
\(239\) 6.02025 0.389417 0.194709 0.980861i \(-0.437624\pi\)
0.194709 + 0.980861i \(0.437624\pi\)
\(240\) 0 0
\(241\) 24.5517 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.524103 −0.0334837
\(246\) 0 0
\(247\) −2.08178 −0.132461
\(248\) 0 0
\(249\) −1.62334 −0.102875
\(250\) 0 0
\(251\) 27.5162 1.73681 0.868403 0.495858i \(-0.165146\pi\)
0.868403 + 0.495858i \(0.165146\pi\)
\(252\) 0 0
\(253\) −14.9720 −0.941284
\(254\) 0 0
\(255\) 3.72913 0.233527
\(256\) 0 0
\(257\) 15.6436 0.975820 0.487910 0.872894i \(-0.337759\pi\)
0.487910 + 0.872894i \(0.337759\pi\)
\(258\) 0 0
\(259\) 18.5620 1.15339
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 18.6175 1.14801 0.574003 0.818853i \(-0.305390\pi\)
0.574003 + 0.818853i \(0.305390\pi\)
\(264\) 0 0
\(265\) 6.43220 0.395127
\(266\) 0 0
\(267\) −8.87281 −0.543007
\(268\) 0 0
\(269\) 10.9006 0.664620 0.332310 0.943170i \(-0.392172\pi\)
0.332310 + 0.943170i \(0.392172\pi\)
\(270\) 0 0
\(271\) −17.7809 −1.08011 −0.540056 0.841629i \(-0.681597\pi\)
−0.540056 + 0.841629i \(0.681597\pi\)
\(272\) 0 0
\(273\) −14.1195 −0.854550
\(274\) 0 0
\(275\) −2.74301 −0.165410
\(276\) 0 0
\(277\) −20.6612 −1.24141 −0.620706 0.784043i \(-0.713154\pi\)
−0.620706 + 0.784043i \(0.713154\pi\)
\(278\) 0 0
\(279\) −1.45825 −0.0873033
\(280\) 0 0
\(281\) 28.2652 1.68616 0.843080 0.537787i \(-0.180740\pi\)
0.843080 + 0.537787i \(0.180740\pi\)
\(282\) 0 0
\(283\) 3.21139 0.190897 0.0954485 0.995434i \(-0.469571\pi\)
0.0954485 + 0.995434i \(0.469571\pi\)
\(284\) 0 0
\(285\) −0.404431 −0.0239564
\(286\) 0 0
\(287\) 26.8291 1.58367
\(288\) 0 0
\(289\) −3.09362 −0.181978
\(290\) 0 0
\(291\) 7.82084 0.458466
\(292\) 0 0
\(293\) −3.99624 −0.233463 −0.116731 0.993164i \(-0.537242\pi\)
−0.116731 + 0.993164i \(0.537242\pi\)
\(294\) 0 0
\(295\) −9.91822 −0.577461
\(296\) 0 0
\(297\) −2.74301 −0.159166
\(298\) 0 0
\(299\) −28.0960 −1.62484
\(300\) 0 0
\(301\) −12.1576 −0.700750
\(302\) 0 0
\(303\) −4.88033 −0.280367
\(304\) 0 0
\(305\) 13.0816 0.749050
\(306\) 0 0
\(307\) 12.0555 0.688046 0.344023 0.938961i \(-0.388210\pi\)
0.344023 + 0.938961i \(0.388210\pi\)
\(308\) 0 0
\(309\) 0.294881 0.0167752
\(310\) 0 0
\(311\) −15.6873 −0.889544 −0.444772 0.895644i \(-0.646715\pi\)
−0.444772 + 0.895644i \(0.646715\pi\)
\(312\) 0 0
\(313\) −33.8726 −1.91459 −0.957297 0.289107i \(-0.906641\pi\)
−0.957297 + 0.289107i \(0.906641\pi\)
\(314\) 0 0
\(315\) −2.74301 −0.154551
\(316\) 0 0
\(317\) 1.75689 0.0986770 0.0493385 0.998782i \(-0.484289\pi\)
0.0493385 + 0.998782i \(0.484289\pi\)
\(318\) 0 0
\(319\) 2.74301 0.153579
\(320\) 0 0
\(321\) 13.7809 0.769175
\(322\) 0 0
\(323\) −1.50817 −0.0839170
\(324\) 0 0
\(325\) −5.14744 −0.285529
\(326\) 0 0
\(327\) −6.20126 −0.342931
\(328\) 0 0
\(329\) −7.14744 −0.394051
\(330\) 0 0
\(331\) 22.4505 1.23399 0.616997 0.786966i \(-0.288349\pi\)
0.616997 + 0.786966i \(0.288349\pi\)
\(332\) 0 0
\(333\) 6.76702 0.370830
\(334\) 0 0
\(335\) 12.4961 0.682737
\(336\) 0 0
\(337\) 15.3886 0.838273 0.419136 0.907923i \(-0.362333\pi\)
0.419136 + 0.907923i \(0.362333\pi\)
\(338\) 0 0
\(339\) −10.5658 −0.573853
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −17.7634 −0.959136
\(344\) 0 0
\(345\) −5.45825 −0.293862
\(346\) 0 0
\(347\) 12.3504 0.663005 0.331503 0.943454i \(-0.392444\pi\)
0.331503 + 0.943454i \(0.392444\pi\)
\(348\) 0 0
\(349\) −1.94446 −0.104085 −0.0520424 0.998645i \(-0.516573\pi\)
−0.0520424 + 0.998645i \(0.516573\pi\)
\(350\) 0 0
\(351\) −5.14744 −0.274750
\(352\) 0 0
\(353\) 6.72517 0.357945 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(354\) 0 0
\(355\) −11.3487 −0.602327
\(356\) 0 0
\(357\) −10.2290 −0.541378
\(358\) 0 0
\(359\) −15.2114 −0.802826 −0.401413 0.915897i \(-0.631481\pi\)
−0.401413 + 0.915897i \(0.631481\pi\)
\(360\) 0 0
\(361\) −18.8364 −0.991391
\(362\) 0 0
\(363\) −3.47590 −0.182437
\(364\) 0 0
\(365\) 10.7670 0.563571
\(366\) 0 0
\(367\) −13.2189 −0.690021 −0.345011 0.938599i \(-0.612125\pi\)
−0.345011 + 0.938599i \(0.612125\pi\)
\(368\) 0 0
\(369\) 9.78090 0.509173
\(370\) 0 0
\(371\) −17.6436 −0.916009
\(372\) 0 0
\(373\) 26.8050 1.38791 0.693956 0.720017i \(-0.255866\pi\)
0.693956 + 0.720017i \(0.255866\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 5.14744 0.265107
\(378\) 0 0
\(379\) 24.4228 1.25451 0.627257 0.778813i \(-0.284178\pi\)
0.627257 + 0.778813i \(0.284178\pi\)
\(380\) 0 0
\(381\) −8.54175 −0.437607
\(382\) 0 0
\(383\) −27.2372 −1.39176 −0.695879 0.718159i \(-0.744985\pi\)
−0.695879 + 0.718159i \(0.744985\pi\)
\(384\) 0 0
\(385\) 7.52410 0.383464
\(386\) 0 0
\(387\) −4.43220 −0.225301
\(388\) 0 0
\(389\) −12.1994 −0.618532 −0.309266 0.950976i \(-0.600083\pi\)
−0.309266 + 0.950976i \(0.600083\pi\)
\(390\) 0 0
\(391\) −20.3545 −1.02937
\(392\) 0 0
\(393\) −13.3050 −0.671149
\(394\) 0 0
\(395\) 14.1576 0.712344
\(396\) 0 0
\(397\) −17.7254 −0.889611 −0.444805 0.895627i \(-0.646727\pi\)
−0.444805 + 0.895627i \(0.646727\pi\)
\(398\) 0 0
\(399\) 1.10936 0.0555373
\(400\) 0 0
\(401\) −2.48773 −0.124231 −0.0621157 0.998069i \(-0.519785\pi\)
−0.0621157 + 0.998069i \(0.519785\pi\)
\(402\) 0 0
\(403\) 7.50627 0.373914
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −18.5620 −0.920084
\(408\) 0 0
\(409\) 37.7194 1.86510 0.932551 0.361038i \(-0.117577\pi\)
0.932551 + 0.361038i \(0.117577\pi\)
\(410\) 0 0
\(411\) −18.2253 −0.898986
\(412\) 0 0
\(413\) 27.2058 1.33871
\(414\) 0 0
\(415\) 1.62334 0.0796865
\(416\) 0 0
\(417\) 5.66142 0.277241
\(418\) 0 0
\(419\) 2.29488 0.112112 0.0560561 0.998428i \(-0.482147\pi\)
0.0560561 + 0.998428i \(0.482147\pi\)
\(420\) 0 0
\(421\) 35.9662 1.75289 0.876443 0.481505i \(-0.159910\pi\)
0.876443 + 0.481505i \(0.159910\pi\)
\(422\) 0 0
\(423\) −2.60569 −0.126693
\(424\) 0 0
\(425\) −3.72913 −0.180889
\(426\) 0 0
\(427\) −35.8829 −1.73650
\(428\) 0 0
\(429\) 14.1195 0.681695
\(430\) 0 0
\(431\) −22.6974 −1.09330 −0.546648 0.837363i \(-0.684096\pi\)
−0.546648 + 0.837363i \(0.684096\pi\)
\(432\) 0 0
\(433\) −11.4108 −0.548368 −0.274184 0.961677i \(-0.588408\pi\)
−0.274184 + 0.961677i \(0.588408\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) 2.20748 0.105598
\(438\) 0 0
\(439\) 7.36654 0.351586 0.175793 0.984427i \(-0.443751\pi\)
0.175793 + 0.984427i \(0.443751\pi\)
\(440\) 0 0
\(441\) 0.524103 0.0249573
\(442\) 0 0
\(443\) −36.5423 −1.73617 −0.868087 0.496411i \(-0.834651\pi\)
−0.868087 + 0.496411i \(0.834651\pi\)
\(444\) 0 0
\(445\) 8.87281 0.420611
\(446\) 0 0
\(447\) −11.9182 −0.563712
\(448\) 0 0
\(449\) −39.6182 −1.86970 −0.934850 0.355043i \(-0.884466\pi\)
−0.934850 + 0.355043i \(0.884466\pi\)
\(450\) 0 0
\(451\) −26.8291 −1.26333
\(452\) 0 0
\(453\) −7.19114 −0.337869
\(454\) 0 0
\(455\) 14.1195 0.661931
\(456\) 0 0
\(457\) 28.5220 1.33420 0.667102 0.744967i \(-0.267535\pi\)
0.667102 + 0.744967i \(0.267535\pi\)
\(458\) 0 0
\(459\) −3.72913 −0.174061
\(460\) 0 0
\(461\) −22.6696 −1.05583 −0.527915 0.849297i \(-0.677026\pi\)
−0.527915 + 0.849297i \(0.677026\pi\)
\(462\) 0 0
\(463\) −28.5517 −1.32691 −0.663455 0.748217i \(-0.730910\pi\)
−0.663455 + 0.748217i \(0.730910\pi\)
\(464\) 0 0
\(465\) 1.45825 0.0676248
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −34.2770 −1.58277
\(470\) 0 0
\(471\) −4.31457 −0.198805
\(472\) 0 0
\(473\) 12.1576 0.559005
\(474\) 0 0
\(475\) 0.404431 0.0185565
\(476\) 0 0
\(477\) −6.43220 −0.294510
\(478\) 0 0
\(479\) 4.43410 0.202599 0.101300 0.994856i \(-0.467700\pi\)
0.101300 + 0.994856i \(0.467700\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) 0 0
\(483\) 14.9720 0.681251
\(484\) 0 0
\(485\) −7.82084 −0.355126
\(486\) 0 0
\(487\) −14.1035 −0.639093 −0.319546 0.947571i \(-0.603531\pi\)
−0.319546 + 0.947571i \(0.603531\pi\)
\(488\) 0 0
\(489\) −21.6436 −0.978757
\(490\) 0 0
\(491\) −39.6376 −1.78882 −0.894410 0.447249i \(-0.852404\pi\)
−0.894410 + 0.447249i \(0.852404\pi\)
\(492\) 0 0
\(493\) 3.72913 0.167951
\(494\) 0 0
\(495\) 2.74301 0.123289
\(496\) 0 0
\(497\) 31.1296 1.39635
\(498\) 0 0
\(499\) −6.33858 −0.283754 −0.141877 0.989884i \(-0.545314\pi\)
−0.141877 + 0.989884i \(0.545314\pi\)
\(500\) 0 0
\(501\) 16.4303 0.734051
\(502\) 0 0
\(503\) 19.0638 0.850011 0.425005 0.905191i \(-0.360272\pi\)
0.425005 + 0.905191i \(0.360272\pi\)
\(504\) 0 0
\(505\) 4.88033 0.217172
\(506\) 0 0
\(507\) 13.4961 0.599385
\(508\) 0 0
\(509\) −12.8145 −0.567992 −0.283996 0.958826i \(-0.591660\pi\)
−0.283996 + 0.958826i \(0.591660\pi\)
\(510\) 0 0
\(511\) −29.5340 −1.30651
\(512\) 0 0
\(513\) 0.404431 0.0178560
\(514\) 0 0
\(515\) −0.294881 −0.0129940
\(516\) 0 0
\(517\) 7.14744 0.314344
\(518\) 0 0
\(519\) −4.64939 −0.204086
\(520\) 0 0
\(521\) −35.4800 −1.55441 −0.777204 0.629249i \(-0.783363\pi\)
−0.777204 + 0.629249i \(0.783363\pi\)
\(522\) 0 0
\(523\) 14.9227 0.652525 0.326263 0.945279i \(-0.394211\pi\)
0.326263 + 0.945279i \(0.394211\pi\)
\(524\) 0 0
\(525\) 2.74301 0.119715
\(526\) 0 0
\(527\) 5.43801 0.236883
\(528\) 0 0
\(529\) 6.79252 0.295327
\(530\) 0 0
\(531\) 9.91822 0.430414
\(532\) 0 0
\(533\) −50.3466 −2.18075
\(534\) 0 0
\(535\) −13.7809 −0.595800
\(536\) 0 0
\(537\) −7.62334 −0.328971
\(538\) 0 0
\(539\) −1.43762 −0.0619227
\(540\) 0 0
\(541\) −22.8644 −0.983017 −0.491509 0.870873i \(-0.663554\pi\)
−0.491509 + 0.870873i \(0.663554\pi\)
\(542\) 0 0
\(543\) −21.3050 −0.914285
\(544\) 0 0
\(545\) 6.20126 0.265633
\(546\) 0 0
\(547\) 35.3290 1.51056 0.755279 0.655404i \(-0.227502\pi\)
0.755279 + 0.655404i \(0.227502\pi\)
\(548\) 0 0
\(549\) −13.0816 −0.558309
\(550\) 0 0
\(551\) −0.404431 −0.0172293
\(552\) 0 0
\(553\) −38.8343 −1.65140
\(554\) 0 0
\(555\) −6.76702 −0.287244
\(556\) 0 0
\(557\) 32.7994 1.38976 0.694878 0.719127i \(-0.255458\pi\)
0.694878 + 0.719127i \(0.255458\pi\)
\(558\) 0 0
\(559\) 22.8145 0.964950
\(560\) 0 0
\(561\) 10.2290 0.431870
\(562\) 0 0
\(563\) −16.8169 −0.708747 −0.354374 0.935104i \(-0.615306\pi\)
−0.354374 + 0.935104i \(0.615306\pi\)
\(564\) 0 0
\(565\) 10.5658 0.444505
\(566\) 0 0
\(567\) 2.74301 0.115196
\(568\) 0 0
\(569\) 8.88033 0.372283 0.186141 0.982523i \(-0.440402\pi\)
0.186141 + 0.982523i \(0.440402\pi\)
\(570\) 0 0
\(571\) −25.1240 −1.05141 −0.525703 0.850668i \(-0.676198\pi\)
−0.525703 + 0.850668i \(0.676198\pi\)
\(572\) 0 0
\(573\) −3.07597 −0.128501
\(574\) 0 0
\(575\) 5.45825 0.227625
\(576\) 0 0
\(577\) −13.9026 −0.578774 −0.289387 0.957212i \(-0.593451\pi\)
−0.289387 + 0.957212i \(0.593451\pi\)
\(578\) 0 0
\(579\) 6.77454 0.281540
\(580\) 0 0
\(581\) −4.45283 −0.184735
\(582\) 0 0
\(583\) 17.6436 0.730723
\(584\) 0 0
\(585\) 5.14744 0.212820
\(586\) 0 0
\(587\) 20.9942 0.866523 0.433262 0.901268i \(-0.357363\pi\)
0.433262 + 0.901268i \(0.357363\pi\)
\(588\) 0 0
\(589\) −0.589762 −0.0243007
\(590\) 0 0
\(591\) −13.6455 −0.561300
\(592\) 0 0
\(593\) 3.54003 0.145372 0.0726859 0.997355i \(-0.476843\pi\)
0.0726859 + 0.997355i \(0.476843\pi\)
\(594\) 0 0
\(595\) 10.2290 0.419349
\(596\) 0 0
\(597\) 6.33858 0.259421
\(598\) 0 0
\(599\) 32.4886 1.32745 0.663725 0.747977i \(-0.268975\pi\)
0.663725 + 0.747977i \(0.268975\pi\)
\(600\) 0 0
\(601\) −12.5620 −0.512414 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(602\) 0 0
\(603\) −12.4961 −0.508882
\(604\) 0 0
\(605\) 3.47590 0.141315
\(606\) 0 0
\(607\) 0.369141 0.0149830 0.00749149 0.999972i \(-0.497615\pi\)
0.00749149 + 0.999972i \(0.497615\pi\)
\(608\) 0 0
\(609\) −2.74301 −0.111152
\(610\) 0 0
\(611\) 13.4126 0.542618
\(612\) 0 0
\(613\) 44.7017 1.80549 0.902743 0.430181i \(-0.141550\pi\)
0.902743 + 0.430181i \(0.141550\pi\)
\(614\) 0 0
\(615\) −9.78090 −0.394404
\(616\) 0 0
\(617\) −5.91014 −0.237933 −0.118967 0.992898i \(-0.537958\pi\)
−0.118967 + 0.992898i \(0.537958\pi\)
\(618\) 0 0
\(619\) 44.2187 1.77730 0.888650 0.458586i \(-0.151644\pi\)
0.888650 + 0.458586i \(0.151644\pi\)
\(620\) 0 0
\(621\) 5.45825 0.219032
\(622\) 0 0
\(623\) −24.3382 −0.975089
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.10936 −0.0443034
\(628\) 0 0
\(629\) −25.2351 −1.00619
\(630\) 0 0
\(631\) 29.5702 1.17717 0.588586 0.808435i \(-0.299685\pi\)
0.588586 + 0.808435i \(0.299685\pi\)
\(632\) 0 0
\(633\) −2.00000 −0.0794929
\(634\) 0 0
\(635\) 8.54175 0.338969
\(636\) 0 0
\(637\) −2.69779 −0.106890
\(638\) 0 0
\(639\) 11.3487 0.448948
\(640\) 0 0
\(641\) 26.6335 1.05196 0.525979 0.850497i \(-0.323699\pi\)
0.525979 + 0.850497i \(0.323699\pi\)
\(642\) 0 0
\(643\) −8.16597 −0.322035 −0.161017 0.986952i \(-0.551477\pi\)
−0.161017 + 0.986952i \(0.551477\pi\)
\(644\) 0 0
\(645\) 4.43220 0.174518
\(646\) 0 0
\(647\) −20.8381 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(648\) 0 0
\(649\) −27.2058 −1.06792
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 0 0
\(653\) 13.4267 0.525428 0.262714 0.964874i \(-0.415382\pi\)
0.262714 + 0.964874i \(0.415382\pi\)
\(654\) 0 0
\(655\) 13.3050 0.519870
\(656\) 0 0
\(657\) −10.7670 −0.420061
\(658\) 0 0
\(659\) 42.9744 1.67405 0.837023 0.547167i \(-0.184294\pi\)
0.837023 + 0.547167i \(0.184294\pi\)
\(660\) 0 0
\(661\) −10.0178 −0.389649 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(662\) 0 0
\(663\) 19.1955 0.745490
\(664\) 0 0
\(665\) −1.10936 −0.0430190
\(666\) 0 0
\(667\) −5.45825 −0.211344
\(668\) 0 0
\(669\) 2.20126 0.0851057
\(670\) 0 0
\(671\) 35.8829 1.38525
\(672\) 0 0
\(673\) 23.0878 0.889970 0.444985 0.895538i \(-0.353209\pi\)
0.444985 + 0.895538i \(0.353209\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 4.95555 0.190457 0.0952287 0.995455i \(-0.469642\pi\)
0.0952287 + 0.995455i \(0.469642\pi\)
\(678\) 0 0
\(679\) 21.4526 0.823277
\(680\) 0 0
\(681\) −12.1853 −0.466943
\(682\) 0 0
\(683\) 36.8010 1.40815 0.704075 0.710126i \(-0.251362\pi\)
0.704075 + 0.710126i \(0.251362\pi\)
\(684\) 0 0
\(685\) 18.2253 0.696352
\(686\) 0 0
\(687\) −3.16337 −0.120690
\(688\) 0 0
\(689\) 33.1094 1.26137
\(690\) 0 0
\(691\) −15.8246 −0.601996 −0.300998 0.953625i \(-0.597320\pi\)
−0.300998 + 0.953625i \(0.597320\pi\)
\(692\) 0 0
\(693\) −7.52410 −0.285817
\(694\) 0 0
\(695\) −5.66142 −0.214750
\(696\) 0 0
\(697\) −36.4742 −1.38156
\(698\) 0 0
\(699\) 23.8087 0.900527
\(700\) 0 0
\(701\) −42.6156 −1.60957 −0.804785 0.593567i \(-0.797719\pi\)
−0.804785 + 0.593567i \(0.797719\pi\)
\(702\) 0 0
\(703\) 2.73679 0.103220
\(704\) 0 0
\(705\) 2.60569 0.0981361
\(706\) 0 0
\(707\) −13.3868 −0.503462
\(708\) 0 0
\(709\) 35.1795 1.32119 0.660597 0.750740i \(-0.270303\pi\)
0.660597 + 0.750740i \(0.270303\pi\)
\(710\) 0 0
\(711\) −14.1576 −0.530950
\(712\) 0 0
\(713\) −7.95951 −0.298086
\(714\) 0 0
\(715\) −14.1195 −0.528039
\(716\) 0 0
\(717\) 6.02025 0.224830
\(718\) 0 0
\(719\) 29.4563 1.09854 0.549268 0.835646i \(-0.314907\pi\)
0.549268 + 0.835646i \(0.314907\pi\)
\(720\) 0 0
\(721\) 0.808861 0.0301236
\(722\) 0 0
\(723\) 24.5517 0.913087
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −46.3391 −1.71862 −0.859311 0.511454i \(-0.829107\pi\)
−0.859311 + 0.511454i \(0.829107\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.5282 0.611319
\(732\) 0 0
\(733\) −3.73344 −0.137898 −0.0689489 0.997620i \(-0.521965\pi\)
−0.0689489 + 0.997620i \(0.521965\pi\)
\(734\) 0 0
\(735\) −0.524103 −0.0193318
\(736\) 0 0
\(737\) 34.2770 1.26261
\(738\) 0 0
\(739\) 6.48021 0.238378 0.119189 0.992872i \(-0.461970\pi\)
0.119189 + 0.992872i \(0.461970\pi\)
\(740\) 0 0
\(741\) −2.08178 −0.0764762
\(742\) 0 0
\(743\) 51.0638 1.87335 0.936674 0.350203i \(-0.113888\pi\)
0.936674 + 0.350203i \(0.113888\pi\)
\(744\) 0 0
\(745\) 11.9182 0.436650
\(746\) 0 0
\(747\) −1.62334 −0.0593948
\(748\) 0 0
\(749\) 37.8011 1.38122
\(750\) 0 0
\(751\) −24.6494 −0.899469 −0.449735 0.893162i \(-0.648481\pi\)
−0.449735 + 0.893162i \(0.648481\pi\)
\(752\) 0 0
\(753\) 27.5162 1.00275
\(754\) 0 0
\(755\) 7.19114 0.261712
\(756\) 0 0
\(757\) −38.6833 −1.40597 −0.702985 0.711205i \(-0.748150\pi\)
−0.702985 + 0.711205i \(0.748150\pi\)
\(758\) 0 0
\(759\) −14.9720 −0.543451
\(760\) 0 0
\(761\) −3.23725 −0.117350 −0.0586750 0.998277i \(-0.518688\pi\)
−0.0586750 + 0.998277i \(0.518688\pi\)
\(762\) 0 0
\(763\) −17.0101 −0.615808
\(764\) 0 0
\(765\) 3.72913 0.134827
\(766\) 0 0
\(767\) −51.0534 −1.84343
\(768\) 0 0
\(769\) 46.3166 1.67022 0.835111 0.550082i \(-0.185404\pi\)
0.835111 + 0.550082i \(0.185404\pi\)
\(770\) 0 0
\(771\) 15.6436 0.563390
\(772\) 0 0
\(773\) −27.0341 −0.972350 −0.486175 0.873861i \(-0.661608\pi\)
−0.486175 + 0.873861i \(0.661608\pi\)
\(774\) 0 0
\(775\) −1.45825 −0.0523820
\(776\) 0 0
\(777\) 18.5620 0.665908
\(778\) 0 0
\(779\) 3.95569 0.141727
\(780\) 0 0
\(781\) −31.1296 −1.11390
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 4.31457 0.153994
\(786\) 0 0
\(787\) 39.1870 1.39687 0.698434 0.715675i \(-0.253881\pi\)
0.698434 + 0.715675i \(0.253881\pi\)
\(788\) 0 0
\(789\) 18.6175 0.662802
\(790\) 0 0
\(791\) −28.9820 −1.03048
\(792\) 0 0
\(793\) 67.3367 2.39120
\(794\) 0 0
\(795\) 6.43220 0.228127
\(796\) 0 0
\(797\) −53.2933 −1.88775 −0.943873 0.330307i \(-0.892848\pi\)
−0.943873 + 0.330307i \(0.892848\pi\)
\(798\) 0 0
\(799\) 9.71696 0.343761
\(800\) 0 0
\(801\) −8.87281 −0.313505
\(802\) 0 0
\(803\) 29.5340 1.04223
\(804\) 0 0
\(805\) −14.9720 −0.527695
\(806\) 0 0
\(807\) 10.9006 0.383718
\(808\) 0 0
\(809\) 0.613214 0.0215595 0.0107797 0.999942i \(-0.496569\pi\)
0.0107797 + 0.999942i \(0.496569\pi\)
\(810\) 0 0
\(811\) 0.230936 0.00810927 0.00405463 0.999992i \(-0.498709\pi\)
0.00405463 + 0.999992i \(0.498709\pi\)
\(812\) 0 0
\(813\) −17.7809 −0.623603
\(814\) 0 0
\(815\) 21.6436 0.758142
\(816\) 0 0
\(817\) −1.79252 −0.0627122
\(818\) 0 0
\(819\) −14.1195 −0.493375
\(820\) 0 0
\(821\) 25.7254 0.897821 0.448911 0.893577i \(-0.351812\pi\)
0.448911 + 0.893577i \(0.351812\pi\)
\(822\) 0 0
\(823\) 38.2258 1.33247 0.666234 0.745743i \(-0.267905\pi\)
0.666234 + 0.745743i \(0.267905\pi\)
\(824\) 0 0
\(825\) −2.74301 −0.0954993
\(826\) 0 0
\(827\) −24.9199 −0.866551 −0.433275 0.901262i \(-0.642642\pi\)
−0.433275 + 0.901262i \(0.642642\pi\)
\(828\) 0 0
\(829\) −1.65301 −0.0574115 −0.0287057 0.999588i \(-0.509139\pi\)
−0.0287057 + 0.999588i \(0.509139\pi\)
\(830\) 0 0
\(831\) −20.6612 −0.716730
\(832\) 0 0
\(833\) −1.95445 −0.0677176
\(834\) 0 0
\(835\) −16.4303 −0.568594
\(836\) 0 0
\(837\) −1.45825 −0.0504046
\(838\) 0 0
\(839\) 31.4757 1.08666 0.543331 0.839519i \(-0.317163\pi\)
0.543331 + 0.839519i \(0.317163\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 28.2652 0.973505
\(844\) 0 0
\(845\) −13.4961 −0.464281
\(846\) 0 0
\(847\) −9.53442 −0.327607
\(848\) 0 0
\(849\) 3.21139 0.110214
\(850\) 0 0
\(851\) 36.9361 1.26615
\(852\) 0 0
\(853\) −30.3753 −1.04003 −0.520015 0.854157i \(-0.674074\pi\)
−0.520015 + 0.854157i \(0.674074\pi\)
\(854\) 0 0
\(855\) −0.404431 −0.0138312
\(856\) 0 0
\(857\) −6.53423 −0.223205 −0.111602 0.993753i \(-0.535598\pi\)
−0.111602 + 0.993753i \(0.535598\pi\)
\(858\) 0 0
\(859\) −27.0220 −0.921977 −0.460989 0.887406i \(-0.652505\pi\)
−0.460989 + 0.887406i \(0.652505\pi\)
\(860\) 0 0
\(861\) 26.8291 0.914334
\(862\) 0 0
\(863\) −31.9587 −1.08789 −0.543944 0.839122i \(-0.683069\pi\)
−0.543944 + 0.839122i \(0.683069\pi\)
\(864\) 0 0
\(865\) 4.64939 0.158084
\(866\) 0 0
\(867\) −3.09362 −0.105065
\(868\) 0 0
\(869\) 38.8343 1.31736
\(870\) 0 0
\(871\) 64.3232 2.17951
\(872\) 0 0
\(873\) 7.82084 0.264695
\(874\) 0 0
\(875\) −2.74301 −0.0927307
\(876\) 0 0
\(877\) −50.0679 −1.69067 −0.845336 0.534235i \(-0.820600\pi\)
−0.845336 + 0.534235i \(0.820600\pi\)
\(878\) 0 0
\(879\) −3.99624 −0.134790
\(880\) 0 0
\(881\) −13.6817 −0.460947 −0.230474 0.973079i \(-0.574028\pi\)
−0.230474 + 0.973079i \(0.574028\pi\)
\(882\) 0 0
\(883\) −29.5693 −0.995087 −0.497543 0.867439i \(-0.665764\pi\)
−0.497543 + 0.867439i \(0.665764\pi\)
\(884\) 0 0
\(885\) −9.91822 −0.333397
\(886\) 0 0
\(887\) −4.35130 −0.146102 −0.0730512 0.997328i \(-0.523274\pi\)
−0.0730512 + 0.997328i \(0.523274\pi\)
\(888\) 0 0
\(889\) −23.4301 −0.785820
\(890\) 0 0
\(891\) −2.74301 −0.0918943
\(892\) 0 0
\(893\) −1.05382 −0.0352648
\(894\) 0 0
\(895\) 7.62334 0.254820
\(896\) 0 0
\(897\) −28.0960 −0.938099
\(898\) 0 0
\(899\) 1.45825 0.0486354
\(900\) 0 0
\(901\) 23.9865 0.799105
\(902\) 0 0
\(903\) −12.1576 −0.404578
\(904\) 0 0
\(905\) 21.3050 0.708202
\(906\) 0 0
\(907\) 52.3965 1.73980 0.869899 0.493230i \(-0.164184\pi\)
0.869899 + 0.493230i \(0.164184\pi\)
\(908\) 0 0
\(909\) −4.88033 −0.161870
\(910\) 0 0
\(911\) −7.85085 −0.260110 −0.130055 0.991507i \(-0.541515\pi\)
−0.130055 + 0.991507i \(0.541515\pi\)
\(912\) 0 0
\(913\) 4.45283 0.147367
\(914\) 0 0
\(915\) 13.0816 0.432464
\(916\) 0 0
\(917\) −36.4958 −1.20520
\(918\) 0 0
\(919\) −13.4979 −0.445253 −0.222627 0.974904i \(-0.571463\pi\)
−0.222627 + 0.974904i \(0.571463\pi\)
\(920\) 0 0
\(921\) 12.0555 0.397243
\(922\) 0 0
\(923\) −58.4168 −1.92281
\(924\) 0 0
\(925\) 6.76702 0.222498
\(926\) 0 0
\(927\) 0.294881 0.00968516
\(928\) 0 0
\(929\) 11.6177 0.381165 0.190583 0.981671i \(-0.438962\pi\)
0.190583 + 0.981671i \(0.438962\pi\)
\(930\) 0 0
\(931\) 0.211963 0.00694682
\(932\) 0 0
\(933\) −15.6873 −0.513579
\(934\) 0 0
\(935\) −10.2290 −0.334525
\(936\) 0 0
\(937\) −42.6740 −1.39410 −0.697049 0.717024i \(-0.745504\pi\)
−0.697049 + 0.717024i \(0.745504\pi\)
\(938\) 0 0
\(939\) −33.8726 −1.10539
\(940\) 0 0
\(941\) −5.91479 −0.192817 −0.0964083 0.995342i \(-0.530735\pi\)
−0.0964083 + 0.995342i \(0.530735\pi\)
\(942\) 0 0
\(943\) 53.3866 1.73851
\(944\) 0 0
\(945\) −2.74301 −0.0892301
\(946\) 0 0
\(947\) 45.0915 1.46528 0.732639 0.680618i \(-0.238289\pi\)
0.732639 + 0.680618i \(0.238289\pi\)
\(948\) 0 0
\(949\) 55.4226 1.79909
\(950\) 0 0
\(951\) 1.75689 0.0569712
\(952\) 0 0
\(953\) 30.3166 0.982053 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(954\) 0 0
\(955\) 3.07597 0.0995362
\(956\) 0 0
\(957\) 2.74301 0.0886689
\(958\) 0 0
\(959\) −49.9921 −1.61433
\(960\) 0 0
\(961\) −28.8735 −0.931403
\(962\) 0 0
\(963\) 13.7809 0.444083
\(964\) 0 0
\(965\) −6.77454 −0.218080
\(966\) 0 0
\(967\) −5.99809 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(968\) 0 0
\(969\) −1.50817 −0.0484495
\(970\) 0 0
\(971\) −28.5140 −0.915057 −0.457529 0.889195i \(-0.651265\pi\)
−0.457529 + 0.889195i \(0.651265\pi\)
\(972\) 0 0
\(973\) 15.5293 0.497848
\(974\) 0 0
\(975\) −5.14744 −0.164850
\(976\) 0 0
\(977\) −5.53813 −0.177180 −0.0885902 0.996068i \(-0.528236\pi\)
−0.0885902 + 0.996068i \(0.528236\pi\)
\(978\) 0 0
\(979\) 24.3382 0.777852
\(980\) 0 0
\(981\) −6.20126 −0.197991
\(982\) 0 0
\(983\) −7.37086 −0.235094 −0.117547 0.993067i \(-0.537503\pi\)
−0.117547 + 0.993067i \(0.537503\pi\)
\(984\) 0 0
\(985\) 13.6455 0.434781
\(986\) 0 0
\(987\) −7.14744 −0.227506
\(988\) 0 0
\(989\) −24.1921 −0.769263
\(990\) 0 0
\(991\) −3.68167 −0.116952 −0.0584760 0.998289i \(-0.518624\pi\)
−0.0584760 + 0.998289i \(0.518624\pi\)
\(992\) 0 0
\(993\) 22.4505 0.712446
\(994\) 0 0
\(995\) −6.33858 −0.200946
\(996\) 0 0
\(997\) 36.8747 1.16783 0.583916 0.811814i \(-0.301520\pi\)
0.583916 + 0.811814i \(0.301520\pi\)
\(998\) 0 0
\(999\) 6.76702 0.214099
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 6960.2.a.co.1.4 4
4.3 odd 2 435.2.a.j.1.4 4
12.11 even 2 1305.2.a.r.1.1 4
20.3 even 4 2175.2.c.n.349.3 8
20.7 even 4 2175.2.c.n.349.6 8
20.19 odd 2 2175.2.a.v.1.1 4
60.59 even 2 6525.2.a.bi.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 4.3 odd 2
1305.2.a.r.1.1 4 12.11 even 2
2175.2.a.v.1.1 4 20.19 odd 2
2175.2.c.n.349.3 8 20.3 even 4
2175.2.c.n.349.6 8 20.7 even 4
6525.2.a.bi.1.4 4 60.59 even 2
6960.2.a.co.1.4 4 1.1 even 1 trivial