Properties

Label 6525.2.a.p.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
Dimension $2$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})\) \(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} -0.828427 q^{11} +2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843 q^{17} -4.82843 q^{19} -0.343146 q^{22} -3.17157 q^{23} +0.828427 q^{26} -8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} +4.41421 q^{32} +1.17157 q^{34} +8.48528 q^{37} -2.00000 q^{38} +6.00000 q^{41} +6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} -11.6569 q^{47} +16.3137 q^{49} -3.65685 q^{52} -3.65685 q^{53} -7.65685 q^{56} -0.414214 q^{58} -3.65685 q^{61} +2.68629 q^{62} -4.17157 q^{64} -6.48528 q^{67} -5.17157 q^{68} +15.3137 q^{71} -8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} -4.00000 q^{77} -2.48528 q^{79} +2.48528 q^{82} +7.17157 q^{83} +2.48528 q^{86} +1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} +5.79899 q^{92} -4.82843 q^{94} +12.4853 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8} + 4 q^{11} + 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 12 q^{22} - 12 q^{23} - 4 q^{26} - 12 q^{28} - 2 q^{29} - 4 q^{31} + 6 q^{32} + 8 q^{34} - 4 q^{38} + 12 q^{41} + 12 q^{43} + 20 q^{44} + 20 q^{46} - 12 q^{47} + 10 q^{49} + 4 q^{52} + 4 q^{53} - 4 q^{56} + 2 q^{58} + 4 q^{61} + 28 q^{62} - 14 q^{64} + 4 q^{67} - 16 q^{68} + 8 q^{71} + 24 q^{74} + 12 q^{76} - 8 q^{77} + 12 q^{79} - 12 q^{82} + 20 q^{83} - 12 q^{86} - 20 q^{88} + 4 q^{89} + 8 q^{91} - 28 q^{92} - 4 q^{94} + 8 q^{97} + 22 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) −1.58579 −0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) −4.82843 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.343146 −0.0731589
\(23\) −3.17157 −0.661319 −0.330659 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.828427 0.162468
\(27\) 0 0
\(28\) −8.82843 −1.66842
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) 1.17157 0.200923
\(35\) 0 0
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 1.51472 0.228352
\(45\) 0 0
\(46\) −1.31371 −0.193696
\(47\) −11.6569 −1.70033 −0.850163 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) −3.65685 −0.507114
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.65685 −1.02319
\(57\) 0 0
\(58\) −0.414214 −0.0543889
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 2.68629 0.341159
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) −6.48528 −0.792303 −0.396152 0.918185i \(-0.629655\pi\)
−0.396152 + 0.918185i \(0.629655\pi\)
\(68\) −5.17157 −0.627145
\(69\) 0 0
\(70\) 0 0
\(71\) 15.3137 1.81740 0.908701 0.417447i \(-0.137075\pi\)
0.908701 + 0.417447i \(0.137075\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 3.51472 0.408578
\(75\) 0 0
\(76\) 8.82843 1.01269
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −2.48528 −0.279616 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.48528 0.274453
\(83\) 7.17157 0.787182 0.393591 0.919286i \(-0.371233\pi\)
0.393591 + 0.919286i \(0.371233\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.48528 0.267995
\(87\) 0 0
\(88\) 1.31371 0.140042
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 5.79899 0.604586
\(93\) 0 0
\(94\) −4.82843 −0.498014
\(95\) 0 0
\(96\) 0 0
\(97\) 12.4853 1.26769 0.633844 0.773461i \(-0.281476\pi\)
0.633844 + 0.773461i \(0.281476\pi\)
\(98\) 6.75736 0.682596
\(99\) 0 0
\(100\) 0 0
\(101\) −15.6569 −1.55792 −0.778958 0.627077i \(-0.784251\pi\)
−0.778958 + 0.627077i \(0.784251\pi\)
\(102\) 0 0
\(103\) −16.1421 −1.59053 −0.795266 0.606261i \(-0.792669\pi\)
−0.795266 + 0.606261i \(0.792669\pi\)
\(104\) −3.17157 −0.310998
\(105\) 0 0
\(106\) −1.51472 −0.147122
\(107\) 20.1421 1.94721 0.973607 0.228232i \(-0.0732943\pi\)
0.973607 + 0.228232i \(0.0732943\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 14.4853 1.36873
\(113\) −2.82843 −0.266076 −0.133038 0.991111i \(-0.542473\pi\)
−0.133038 + 0.991111i \(0.542473\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.82843 0.169765
\(117\) 0 0
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) −1.51472 −0.137136
\(123\) 0 0
\(124\) −11.8579 −1.06487
\(125\) 0 0
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −10.5563 −0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) 12.1421 1.06086 0.530432 0.847728i \(-0.322030\pi\)
0.530432 + 0.847728i \(0.322030\pi\)
\(132\) 0 0
\(133\) −23.3137 −2.02155
\(134\) −2.68629 −0.232060
\(135\) 0 0
\(136\) −4.48528 −0.384610
\(137\) −5.17157 −0.441837 −0.220919 0.975292i \(-0.570906\pi\)
−0.220919 + 0.975292i \(0.570906\pi\)
\(138\) 0 0
\(139\) 21.6569 1.83691 0.918455 0.395525i \(-0.129437\pi\)
0.918455 + 0.395525i \(0.129437\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.34315 0.532305
\(143\) −1.65685 −0.138553
\(144\) 0 0
\(145\) 0 0
\(146\) −3.51472 −0.290880
\(147\) 0 0
\(148\) −15.5147 −1.27530
\(149\) −9.31371 −0.763009 −0.381504 0.924367i \(-0.624594\pi\)
−0.381504 + 0.924367i \(0.624594\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 7.65685 0.621053
\(153\) 0 0
\(154\) −1.65685 −0.133513
\(155\) 0 0
\(156\) 0 0
\(157\) −0.485281 −0.0387297 −0.0193648 0.999812i \(-0.506164\pi\)
−0.0193648 + 0.999812i \(0.506164\pi\)
\(158\) −1.02944 −0.0818976
\(159\) 0 0
\(160\) 0 0
\(161\) −15.3137 −1.20689
\(162\) 0 0
\(163\) 8.34315 0.653486 0.326743 0.945113i \(-0.394049\pi\)
0.326743 + 0.945113i \(0.394049\pi\)
\(164\) −10.9706 −0.856657
\(165\) 0 0
\(166\) 2.97056 0.230560
\(167\) −2.48528 −0.192317 −0.0961584 0.995366i \(-0.530656\pi\)
−0.0961584 + 0.995366i \(0.530656\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −10.9706 −0.836498
\(173\) 17.3137 1.31634 0.658168 0.752871i \(-0.271331\pi\)
0.658168 + 0.752871i \(0.271331\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.48528 −0.187335
\(177\) 0 0
\(178\) 3.17157 0.237719
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 5.02944 0.370775
\(185\) 0 0
\(186\) 0 0
\(187\) −2.34315 −0.171348
\(188\) 21.3137 1.55446
\(189\) 0 0
\(190\) 0 0
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) 0 0
\(193\) −4.48528 −0.322858 −0.161429 0.986884i \(-0.551610\pi\)
−0.161429 + 0.986884i \(0.551610\pi\)
\(194\) 5.17157 0.371297
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) −19.6569 −1.40049 −0.700246 0.713901i \(-0.746927\pi\)
−0.700246 + 0.713901i \(0.746927\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.48528 −0.456303
\(203\) −4.82843 −0.338889
\(204\) 0 0
\(205\) 0 0
\(206\) −6.68629 −0.465856
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −0.828427 −0.0570313 −0.0285156 0.999593i \(-0.509078\pi\)
−0.0285156 + 0.999593i \(0.509078\pi\)
\(212\) 6.68629 0.459216
\(213\) 0 0
\(214\) 8.34315 0.570326
\(215\) 0 0
\(216\) 0 0
\(217\) 31.3137 2.12571
\(218\) 0.828427 0.0561082
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) 0 0
\(223\) 17.7990 1.19191 0.595954 0.803018i \(-0.296774\pi\)
0.595954 + 0.803018i \(0.296774\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) −1.17157 −0.0779319
\(227\) 20.1421 1.33688 0.668440 0.743766i \(-0.266962\pi\)
0.668440 + 0.743766i \(0.266962\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.58579 0.104112
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 5.65685 0.366679
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −4.27208 −0.274620
\(243\) 0 0
\(244\) 6.68629 0.428046
\(245\) 0 0
\(246\) 0 0
\(247\) −9.65685 −0.614451
\(248\) −10.2843 −0.653052
\(249\) 0 0
\(250\) 0 0
\(251\) −8.82843 −0.557245 −0.278623 0.960401i \(-0.589878\pi\)
−0.278623 + 0.960401i \(0.589878\pi\)
\(252\) 0 0
\(253\) 2.62742 0.165184
\(254\) −2.48528 −0.155940
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 6.68629 0.417079 0.208540 0.978014i \(-0.433129\pi\)
0.208540 + 0.978014i \(0.433129\pi\)
\(258\) 0 0
\(259\) 40.9706 2.54579
\(260\) 0 0
\(261\) 0 0
\(262\) 5.02944 0.310720
\(263\) −19.6569 −1.21209 −0.606047 0.795429i \(-0.707246\pi\)
−0.606047 + 0.795429i \(0.707246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −9.65685 −0.592100
\(267\) 0 0
\(268\) 11.8579 0.724334
\(269\) 21.3137 1.29952 0.649760 0.760140i \(-0.274869\pi\)
0.649760 + 0.760140i \(0.274869\pi\)
\(270\) 0 0
\(271\) −9.79899 −0.595246 −0.297623 0.954683i \(-0.596194\pi\)
−0.297623 + 0.954683i \(0.596194\pi\)
\(272\) 8.48528 0.514496
\(273\) 0 0
\(274\) −2.14214 −0.129411
\(275\) 0 0
\(276\) 0 0
\(277\) 3.65685 0.219719 0.109860 0.993947i \(-0.464960\pi\)
0.109860 + 0.993947i \(0.464960\pi\)
\(278\) 8.97056 0.538019
\(279\) 0 0
\(280\) 0 0
\(281\) 29.3137 1.74871 0.874355 0.485288i \(-0.161285\pi\)
0.874355 + 0.485288i \(0.161285\pi\)
\(282\) 0 0
\(283\) −4.82843 −0.287020 −0.143510 0.989649i \(-0.545839\pi\)
−0.143510 + 0.989649i \(0.545839\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −0.686292 −0.0405813
\(287\) 28.9706 1.71008
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 15.5147 0.907930
\(293\) 8.48528 0.495715 0.247858 0.968796i \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −13.4558 −0.782105
\(297\) 0 0
\(298\) −3.85786 −0.223480
\(299\) −6.34315 −0.366834
\(300\) 0 0
\(301\) 28.9706 1.66984
\(302\) −4.97056 −0.286024
\(303\) 0 0
\(304\) −14.4853 −0.830788
\(305\) 0 0
\(306\) 0 0
\(307\) 22.9706 1.31100 0.655500 0.755195i \(-0.272458\pi\)
0.655500 + 0.755195i \(0.272458\pi\)
\(308\) 7.31371 0.416737
\(309\) 0 0
\(310\) 0 0
\(311\) −14.4853 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −0.201010 −0.0113437
\(315\) 0 0
\(316\) 4.54416 0.255629
\(317\) 2.82843 0.158860 0.0794301 0.996840i \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 0 0
\(319\) 0.828427 0.0463830
\(320\) 0 0
\(321\) 0 0
\(322\) −6.34315 −0.353490
\(323\) −13.6569 −0.759888
\(324\) 0 0
\(325\) 0 0
\(326\) 3.45584 0.191402
\(327\) 0 0
\(328\) −9.51472 −0.525362
\(329\) −56.2843 −3.10305
\(330\) 0 0
\(331\) 21.7990 1.19818 0.599090 0.800681i \(-0.295529\pi\)
0.599090 + 0.800681i \(0.295529\pi\)
\(332\) −13.1127 −0.719653
\(333\) 0 0
\(334\) −1.02944 −0.0563283
\(335\) 0 0
\(336\) 0 0
\(337\) 1.17157 0.0638196 0.0319098 0.999491i \(-0.489841\pi\)
0.0319098 + 0.999491i \(0.489841\pi\)
\(338\) −3.72792 −0.202772
\(339\) 0 0
\(340\) 0 0
\(341\) −5.37258 −0.290942
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) −9.51472 −0.512999
\(345\) 0 0
\(346\) 7.17157 0.385546
\(347\) −8.14214 −0.437093 −0.218546 0.975827i \(-0.570131\pi\)
−0.218546 + 0.975827i \(0.570131\pi\)
\(348\) 0 0
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.65685 −0.194911
\(353\) −4.34315 −0.231162 −0.115581 0.993298i \(-0.536873\pi\)
−0.115581 + 0.993298i \(0.536873\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 9.65685 0.510381
\(359\) 3.85786 0.203610 0.101805 0.994804i \(-0.467538\pi\)
0.101805 + 0.994804i \(0.467538\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) −2.48528 −0.130623
\(363\) 0 0
\(364\) −17.6569 −0.925471
\(365\) 0 0
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −9.51472 −0.495989
\(369\) 0 0
\(370\) 0 0
\(371\) −17.6569 −0.916698
\(372\) 0 0
\(373\) 6.97056 0.360922 0.180461 0.983582i \(-0.442241\pi\)
0.180461 + 0.983582i \(0.442241\pi\)
\(374\) −0.970563 −0.0501866
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 22.4853 1.15499 0.577496 0.816394i \(-0.304030\pi\)
0.577496 + 0.816394i \(0.304030\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 8.62742 0.441417
\(383\) 2.48528 0.126992 0.0634960 0.997982i \(-0.479775\pi\)
0.0634960 + 0.997982i \(0.479775\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.85786 −0.0945628
\(387\) 0 0
\(388\) −22.8284 −1.15894
\(389\) 29.3137 1.48626 0.743132 0.669145i \(-0.233339\pi\)
0.743132 + 0.669145i \(0.233339\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) −25.8701 −1.30664
\(393\) 0 0
\(394\) −8.14214 −0.410195
\(395\) 0 0
\(396\) 0 0
\(397\) 19.6569 0.986549 0.493275 0.869874i \(-0.335800\pi\)
0.493275 + 0.869874i \(0.335800\pi\)
\(398\) 4.97056 0.249152
\(399\) 0 0
\(400\) 0 0
\(401\) 6.68629 0.333897 0.166949 0.985966i \(-0.446609\pi\)
0.166949 + 0.985966i \(0.446609\pi\)
\(402\) 0 0
\(403\) 12.9706 0.646110
\(404\) 28.6274 1.42427
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −7.02944 −0.348436
\(408\) 0 0
\(409\) −2.97056 −0.146885 −0.0734424 0.997299i \(-0.523399\pi\)
−0.0734424 + 0.997299i \(0.523399\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 29.5147 1.45409
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 8.82843 0.432849
\(417\) 0 0
\(418\) 1.65685 0.0810394
\(419\) −28.9706 −1.41530 −0.707652 0.706561i \(-0.750246\pi\)
−0.707652 + 0.706561i \(0.750246\pi\)
\(420\) 0 0
\(421\) 18.9706 0.924569 0.462284 0.886732i \(-0.347030\pi\)
0.462284 + 0.886732i \(0.347030\pi\)
\(422\) −0.343146 −0.0167041
\(423\) 0 0
\(424\) 5.79899 0.281624
\(425\) 0 0
\(426\) 0 0
\(427\) −17.6569 −0.854475
\(428\) −36.8284 −1.78017
\(429\) 0 0
\(430\) 0 0
\(431\) −3.31371 −0.159616 −0.0798079 0.996810i \(-0.525431\pi\)
−0.0798079 + 0.996810i \(0.525431\pi\)
\(432\) 0 0
\(433\) 29.1716 1.40190 0.700948 0.713212i \(-0.252760\pi\)
0.700948 + 0.713212i \(0.252760\pi\)
\(434\) 12.9706 0.622607
\(435\) 0 0
\(436\) −3.65685 −0.175132
\(437\) 15.3137 0.732554
\(438\) 0 0
\(439\) −10.3431 −0.493651 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.34315 0.111452
\(443\) −7.65685 −0.363788 −0.181894 0.983318i \(-0.558223\pi\)
−0.181894 + 0.983318i \(0.558223\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.37258 0.349102
\(447\) 0 0
\(448\) −20.1421 −0.951626
\(449\) −11.6569 −0.550121 −0.275060 0.961427i \(-0.588698\pi\)
−0.275060 + 0.961427i \(0.588698\pi\)
\(450\) 0 0
\(451\) −4.97056 −0.234055
\(452\) 5.17157 0.243250
\(453\) 0 0
\(454\) 8.34315 0.391563
\(455\) 0 0
\(456\) 0 0
\(457\) −19.6569 −0.919509 −0.459754 0.888046i \(-0.652063\pi\)
−0.459754 + 0.888046i \(0.652063\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 0 0
\(460\) 0 0
\(461\) 35.6569 1.66071 0.830353 0.557238i \(-0.188139\pi\)
0.830353 + 0.557238i \(0.188139\pi\)
\(462\) 0 0
\(463\) −21.7990 −1.01308 −0.506542 0.862215i \(-0.669077\pi\)
−0.506542 + 0.862215i \(0.669077\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 7.45584 0.345385
\(467\) 10.9706 0.507657 0.253829 0.967249i \(-0.418310\pi\)
0.253829 + 0.967249i \(0.418310\pi\)
\(468\) 0 0
\(469\) −31.3137 −1.44593
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.97056 −0.228547
\(474\) 0 0
\(475\) 0 0
\(476\) −24.9706 −1.14452
\(477\) 0 0
\(478\) 0.284271 0.0130023
\(479\) −7.17157 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(480\) 0 0
\(481\) 16.9706 0.773791
\(482\) 4.14214 0.188669
\(483\) 0 0
\(484\) 18.8579 0.857176
\(485\) 0 0
\(486\) 0 0
\(487\) −9.79899 −0.444035 −0.222017 0.975043i \(-0.571264\pi\)
−0.222017 + 0.975043i \(0.571264\pi\)
\(488\) 5.79899 0.262508
\(489\) 0 0
\(490\) 0 0
\(491\) 7.45584 0.336478 0.168239 0.985746i \(-0.446192\pi\)
0.168239 + 0.985746i \(0.446192\pi\)
\(492\) 0 0
\(493\) −2.82843 −0.127386
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) 73.9411 3.31671
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.65685 −0.163213
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.08831 0.0483814
\(507\) 0 0
\(508\) 10.9706 0.486740
\(509\) −0.627417 −0.0278098 −0.0139049 0.999903i \(-0.504426\pi\)
−0.0139049 + 0.999903i \(0.504426\pi\)
\(510\) 0 0
\(511\) −40.9706 −1.81243
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) 2.76955 0.122160
\(515\) 0 0
\(516\) 0 0
\(517\) 9.65685 0.424708
\(518\) 16.9706 0.745644
\(519\) 0 0
\(520\) 0 0
\(521\) 21.3137 0.933771 0.466885 0.884318i \(-0.345376\pi\)
0.466885 + 0.884318i \(0.345376\pi\)
\(522\) 0 0
\(523\) −2.48528 −0.108674 −0.0543369 0.998523i \(-0.517304\pi\)
−0.0543369 + 0.998523i \(0.517304\pi\)
\(524\) −22.2010 −0.969856
\(525\) 0 0
\(526\) −8.14214 −0.355014
\(527\) 18.3431 0.799040
\(528\) 0 0
\(529\) −12.9411 −0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) 42.6274 1.84813
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 10.2843 0.444213
\(537\) 0 0
\(538\) 8.82843 0.380621
\(539\) −13.5147 −0.582120
\(540\) 0 0
\(541\) −5.02944 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(542\) −4.05887 −0.174344
\(543\) 0 0
\(544\) 12.4853 0.535302
\(545\) 0 0
\(546\) 0 0
\(547\) 2.48528 0.106263 0.0531315 0.998588i \(-0.483080\pi\)
0.0531315 + 0.998588i \(0.483080\pi\)
\(548\) 9.45584 0.403934
\(549\) 0 0
\(550\) 0 0
\(551\) 4.82843 0.205698
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 1.51472 0.0643542
\(555\) 0 0
\(556\) −39.5980 −1.67933
\(557\) −27.9411 −1.18390 −0.591952 0.805973i \(-0.701642\pi\)
−0.591952 + 0.805973i \(0.701642\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 12.1421 0.512185
\(563\) 7.65685 0.322698 0.161349 0.986897i \(-0.448416\pi\)
0.161349 + 0.986897i \(0.448416\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) −24.2843 −1.01895
\(569\) −27.6569 −1.15944 −0.579718 0.814817i \(-0.696837\pi\)
−0.579718 + 0.814817i \(0.696837\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 3.02944 0.126667
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) −23.7990 −0.990765 −0.495382 0.868675i \(-0.664972\pi\)
−0.495382 + 0.868675i \(0.664972\pi\)
\(578\) −3.72792 −0.155061
\(579\) 0 0
\(580\) 0 0
\(581\) 34.6274 1.43659
\(582\) 0 0
\(583\) 3.02944 0.125466
\(584\) 13.4558 0.556807
\(585\) 0 0
\(586\) 3.51472 0.145192
\(587\) −29.7990 −1.22994 −0.614968 0.788552i \(-0.710831\pi\)
−0.614968 + 0.788552i \(0.710831\pi\)
\(588\) 0 0
\(589\) −31.3137 −1.29026
\(590\) 0 0
\(591\) 0 0
\(592\) 25.4558 1.04623
\(593\) −7.65685 −0.314429 −0.157215 0.987564i \(-0.550251\pi\)
−0.157215 + 0.987564i \(0.550251\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17.0294 0.697553
\(597\) 0 0
\(598\) −2.62742 −0.107443
\(599\) 37.7990 1.54442 0.772212 0.635364i \(-0.219150\pi\)
0.772212 + 0.635364i \(0.219150\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) 21.9411 0.892772
\(605\) 0 0
\(606\) 0 0
\(607\) −9.02944 −0.366494 −0.183247 0.983067i \(-0.558661\pi\)
−0.183247 + 0.983067i \(0.558661\pi\)
\(608\) −21.3137 −0.864385
\(609\) 0 0
\(610\) 0 0
\(611\) −23.3137 −0.943172
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 9.51472 0.383983
\(615\) 0 0
\(616\) 6.34315 0.255573
\(617\) −9.17157 −0.369234 −0.184617 0.982811i \(-0.559104\pi\)
−0.184617 + 0.982811i \(0.559104\pi\)
\(618\) 0 0
\(619\) −9.79899 −0.393855 −0.196927 0.980418i \(-0.563096\pi\)
−0.196927 + 0.980418i \(0.563096\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 36.9706 1.48119
\(624\) 0 0
\(625\) 0 0
\(626\) 2.48528 0.0993318
\(627\) 0 0
\(628\) 0.887302 0.0354072
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 36.9706 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(632\) 3.94113 0.156770
\(633\) 0 0
\(634\) 1.17157 0.0465291
\(635\) 0 0
\(636\) 0 0
\(637\) 32.6274 1.29275
\(638\) 0.343146 0.0135853
\(639\) 0 0
\(640\) 0 0
\(641\) −0.627417 −0.0247815 −0.0123907 0.999923i \(-0.503944\pi\)
−0.0123907 + 0.999923i \(0.503944\pi\)
\(642\) 0 0
\(643\) −19.4558 −0.767264 −0.383632 0.923486i \(-0.625327\pi\)
−0.383632 + 0.923486i \(0.625327\pi\)
\(644\) 28.0000 1.10335
\(645\) 0 0
\(646\) −5.65685 −0.222566
\(647\) −41.1127 −1.61631 −0.808153 0.588972i \(-0.799533\pi\)
−0.808153 + 0.588972i \(0.799533\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −15.2548 −0.597425
\(653\) −17.1716 −0.671976 −0.335988 0.941866i \(-0.609070\pi\)
−0.335988 + 0.941866i \(0.609070\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 18.0000 0.702782
\(657\) 0 0
\(658\) −23.3137 −0.908863
\(659\) −1.79899 −0.0700787 −0.0350393 0.999386i \(-0.511156\pi\)
−0.0350393 + 0.999386i \(0.511156\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 9.02944 0.350939
\(663\) 0 0
\(664\) −11.3726 −0.441342
\(665\) 0 0
\(666\) 0 0
\(667\) 3.17157 0.122804
\(668\) 4.54416 0.175819
\(669\) 0 0
\(670\) 0 0
\(671\) 3.02944 0.116950
\(672\) 0 0
\(673\) −22.9706 −0.885450 −0.442725 0.896657i \(-0.645988\pi\)
−0.442725 + 0.896657i \(0.645988\pi\)
\(674\) 0.485281 0.0186923
\(675\) 0 0
\(676\) 16.4558 0.632917
\(677\) −36.7696 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) 0 0
\(679\) 60.2843 2.31350
\(680\) 0 0
\(681\) 0 0
\(682\) −2.22540 −0.0852148
\(683\) 11.8579 0.453729 0.226864 0.973926i \(-0.427153\pi\)
0.226864 + 0.973926i \(0.427153\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18.6274 0.711198
\(687\) 0 0
\(688\) 18.0000 0.686244
\(689\) −7.31371 −0.278630
\(690\) 0 0
\(691\) −44.9706 −1.71076 −0.855380 0.518000i \(-0.826677\pi\)
−0.855380 + 0.518000i \(0.826677\pi\)
\(692\) −31.6569 −1.20341
\(693\) 0 0
\(694\) −3.37258 −0.128022
\(695\) 0 0
\(696\) 0 0
\(697\) 16.9706 0.642806
\(698\) 8.54416 0.323401
\(699\) 0 0
\(700\) 0 0
\(701\) −6.68629 −0.252538 −0.126269 0.991996i \(-0.540300\pi\)
−0.126269 + 0.991996i \(0.540300\pi\)
\(702\) 0 0
\(703\) −40.9706 −1.54523
\(704\) 3.45584 0.130247
\(705\) 0 0
\(706\) −1.79899 −0.0677059
\(707\) −75.5980 −2.84315
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.1421 −0.455046
\(713\) −20.5685 −0.770298
\(714\) 0 0
\(715\) 0 0
\(716\) −42.6274 −1.59306
\(717\) 0 0
\(718\) 1.59798 0.0596361
\(719\) 34.6274 1.29138 0.645692 0.763598i \(-0.276569\pi\)
0.645692 + 0.763598i \(0.276569\pi\)
\(720\) 0 0
\(721\) −77.9411 −2.90268
\(722\) 1.78680 0.0664977
\(723\) 0 0
\(724\) 10.9706 0.407718
\(725\) 0 0
\(726\) 0 0
\(727\) 23.9411 0.887927 0.443964 0.896045i \(-0.353572\pi\)
0.443964 + 0.896045i \(0.353572\pi\)
\(728\) −15.3137 −0.567564
\(729\) 0 0
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) 22.8284 0.843187 0.421594 0.906785i \(-0.361471\pi\)
0.421594 + 0.906785i \(0.361471\pi\)
\(734\) −7.45584 −0.275200
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) 5.37258 0.197902
\(738\) 0 0
\(739\) 14.4853 0.532850 0.266425 0.963856i \(-0.414158\pi\)
0.266425 + 0.963856i \(0.414158\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −7.31371 −0.268495
\(743\) −52.6274 −1.93071 −0.965356 0.260935i \(-0.915969\pi\)
−0.965356 + 0.260935i \(0.915969\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.88730 0.105712
\(747\) 0 0
\(748\) 4.28427 0.156648
\(749\) 97.2548 3.55361
\(750\) 0 0
\(751\) 16.1421 0.589035 0.294517 0.955646i \(-0.404841\pi\)
0.294517 + 0.955646i \(0.404841\pi\)
\(752\) −34.9706 −1.27525
\(753\) 0 0
\(754\) −0.828427 −0.0301695
\(755\) 0 0
\(756\) 0 0
\(757\) −19.5147 −0.709275 −0.354637 0.935004i \(-0.615396\pi\)
−0.354637 + 0.935004i \(0.615396\pi\)
\(758\) 9.31371 0.338289
\(759\) 0 0
\(760\) 0 0
\(761\) −8.62742 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(762\) 0 0
\(763\) 9.65685 0.349602
\(764\) −38.0833 −1.37780
\(765\) 0 0
\(766\) 1.02944 0.0371951
\(767\) 0 0
\(768\) 0 0
\(769\) −15.6569 −0.564601 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.20101 0.295161
\(773\) −8.48528 −0.305194 −0.152597 0.988288i \(-0.548764\pi\)
−0.152597 + 0.988288i \(0.548764\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −19.7990 −0.710742
\(777\) 0 0
\(778\) 12.1421 0.435317
\(779\) −28.9706 −1.03798
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) −3.71573 −0.132874
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) 0 0
\(786\) 0 0
\(787\) −17.7990 −0.634465 −0.317233 0.948348i \(-0.602754\pi\)
−0.317233 + 0.948348i \(0.602754\pi\)
\(788\) 35.9411 1.28035
\(789\) 0 0
\(790\) 0 0
\(791\) −13.6569 −0.485582
\(792\) 0 0
\(793\) −7.31371 −0.259717
\(794\) 8.14214 0.288954
\(795\) 0 0
\(796\) −21.9411 −0.777683
\(797\) −5.85786 −0.207496 −0.103748 0.994604i \(-0.533084\pi\)
−0.103748 + 0.994604i \(0.533084\pi\)
\(798\) 0 0
\(799\) −32.9706 −1.16641
\(800\) 0 0
\(801\) 0 0
\(802\) 2.76955 0.0977963
\(803\) 7.02944 0.248063
\(804\) 0 0
\(805\) 0 0
\(806\) 5.37258 0.189241
\(807\) 0 0
\(808\) 24.8284 0.873461
\(809\) −42.2843 −1.48664 −0.743318 0.668938i \(-0.766749\pi\)
−0.743318 + 0.668938i \(0.766749\pi\)
\(810\) 0 0
\(811\) 37.6569 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(812\) 8.82843 0.309817
\(813\) 0 0
\(814\) −2.91169 −0.102055
\(815\) 0 0
\(816\) 0 0
\(817\) −28.9706 −1.01355
\(818\) −1.23045 −0.0430216
\(819\) 0 0
\(820\) 0 0
\(821\) 22.6863 0.791757 0.395879 0.918303i \(-0.370440\pi\)
0.395879 + 0.918303i \(0.370440\pi\)
\(822\) 0 0
\(823\) 30.9706 1.07957 0.539783 0.841804i \(-0.318506\pi\)
0.539783 + 0.841804i \(0.318506\pi\)
\(824\) 25.5980 0.891748
\(825\) 0 0
\(826\) 0 0
\(827\) 17.3137 0.602057 0.301028 0.953615i \(-0.402670\pi\)
0.301028 + 0.953615i \(0.402670\pi\)
\(828\) 0 0
\(829\) 20.6274 0.716420 0.358210 0.933641i \(-0.383387\pi\)
0.358210 + 0.933641i \(0.383387\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.34315 −0.289247
\(833\) 46.1421 1.59873
\(834\) 0 0
\(835\) 0 0
\(836\) −7.31371 −0.252950
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −2.48528 −0.0858014 −0.0429007 0.999079i \(-0.513660\pi\)
−0.0429007 + 0.999079i \(0.513660\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.85786 0.270800
\(843\) 0 0
\(844\) 1.51472 0.0521388
\(845\) 0 0
\(846\) 0 0
\(847\) −49.7990 −1.71111
\(848\) −10.9706 −0.376731
\(849\) 0 0
\(850\) 0 0
\(851\) −26.9117 −0.922521
\(852\) 0 0
\(853\) −51.1127 −1.75007 −0.875033 0.484064i \(-0.839160\pi\)
−0.875033 + 0.484064i \(0.839160\pi\)
\(854\) −7.31371 −0.250270
\(855\) 0 0
\(856\) −31.9411 −1.09173
\(857\) 3.37258 0.115205 0.0576026 0.998340i \(-0.481654\pi\)
0.0576026 + 0.998340i \(0.481654\pi\)
\(858\) 0 0
\(859\) −56.4264 −1.92524 −0.962622 0.270848i \(-0.912696\pi\)
−0.962622 + 0.270848i \(0.912696\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.37258 −0.0467504
\(863\) −36.1421 −1.23029 −0.615146 0.788413i \(-0.710903\pi\)
−0.615146 + 0.788413i \(0.710903\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 12.0833 0.410606
\(867\) 0 0
\(868\) −57.2548 −1.94336
\(869\) 2.05887 0.0698425
\(870\) 0 0
\(871\) −12.9706 −0.439491
\(872\) −3.17157 −0.107403
\(873\) 0 0
\(874\) 6.34315 0.214560
\(875\) 0 0
\(876\) 0 0
\(877\) −38.2843 −1.29277 −0.646384 0.763012i \(-0.723720\pi\)
−0.646384 + 0.763012i \(0.723720\pi\)
\(878\) −4.28427 −0.144587
\(879\) 0 0
\(880\) 0 0
\(881\) −29.3137 −0.987604 −0.493802 0.869574i \(-0.664393\pi\)
−0.493802 + 0.869574i \(0.664393\pi\)
\(882\) 0 0
\(883\) 14.4853 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(884\) −10.3431 −0.347878
\(885\) 0 0
\(886\) −3.17157 −0.106551
\(887\) −6.68629 −0.224504 −0.112252 0.993680i \(-0.535806\pi\)
−0.112252 + 0.993680i \(0.535806\pi\)
\(888\) 0 0
\(889\) −28.9706 −0.971641
\(890\) 0 0
\(891\) 0 0
\(892\) −32.5442 −1.08966
\(893\) 56.2843 1.88348
\(894\) 0 0
\(895\) 0 0
\(896\) −50.9706 −1.70281
\(897\) 0 0
\(898\) −4.82843 −0.161127
\(899\) −6.48528 −0.216296
\(900\) 0 0
\(901\) −10.3431 −0.344580
\(902\) −2.05887 −0.0685530
\(903\) 0 0
\(904\) 4.48528 0.149178
\(905\) 0 0
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −36.8284 −1.22219
\(909\) 0 0
\(910\) 0 0
\(911\) −32.1421 −1.06492 −0.532458 0.846456i \(-0.678732\pi\)
−0.532458 + 0.846456i \(0.678732\pi\)
\(912\) 0 0
\(913\) −5.94113 −0.196623
\(914\) −8.14214 −0.269318
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) 58.6274 1.93605
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 14.7696 0.486409
\(923\) 30.6274 1.00811
\(924\) 0 0
\(925\) 0 0
\(926\) −9.02944 −0.296726
\(927\) 0 0
\(928\) −4.41421 −0.144904
\(929\) 4.62742 0.151821 0.0759103 0.997115i \(-0.475814\pi\)
0.0759103 + 0.997115i \(0.475814\pi\)
\(930\) 0 0
\(931\) −78.7696 −2.58157
\(932\) −32.9117 −1.07806
\(933\) 0 0
\(934\) 4.54416 0.148689
\(935\) 0 0
\(936\) 0 0
\(937\) −19.6569 −0.642161 −0.321081 0.947052i \(-0.604046\pi\)
−0.321081 + 0.947052i \(0.604046\pi\)
\(938\) −12.9706 −0.423504
\(939\) 0 0
\(940\) 0 0
\(941\) 27.9411 0.910855 0.455427 0.890273i \(-0.349486\pi\)
0.455427 + 0.890273i \(0.349486\pi\)
\(942\) 0 0
\(943\) −19.0294 −0.619684
\(944\) 0 0
\(945\) 0 0
\(946\) −2.05887 −0.0669398
\(947\) 44.9117 1.45943 0.729717 0.683749i \(-0.239652\pi\)
0.729717 + 0.683749i \(0.239652\pi\)
\(948\) 0 0
\(949\) −16.9706 −0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) −21.6569 −0.701903
\(953\) 29.3137 0.949564 0.474782 0.880103i \(-0.342527\pi\)
0.474782 + 0.880103i \(0.342527\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.25483 −0.0405842
\(957\) 0 0
\(958\) −2.97056 −0.0959745
\(959\) −24.9706 −0.806342
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) 7.02944 0.226638
\(963\) 0 0
\(964\) −18.2843 −0.588897
\(965\) 0 0
\(966\) 0 0
\(967\) −14.9706 −0.481421 −0.240710 0.970597i \(-0.577380\pi\)
−0.240710 + 0.970597i \(0.577380\pi\)
\(968\) 16.3553 0.525681
\(969\) 0 0
\(970\) 0 0
\(971\) −28.1421 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(972\) 0 0
\(973\) 104.569 3.35231
\(974\) −4.05887 −0.130055
\(975\) 0 0
\(976\) −10.9706 −0.351159
\(977\) −2.68629 −0.0859421 −0.0429710 0.999076i \(-0.513682\pi\)
−0.0429710 + 0.999076i \(0.513682\pi\)
\(978\) 0 0
\(979\) −6.34315 −0.202728
\(980\) 0 0
\(981\) 0 0
\(982\) 3.08831 0.0985520
\(983\) −9.31371 −0.297061 −0.148531 0.988908i \(-0.547454\pi\)
−0.148531 + 0.988908i \(0.547454\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.17157 −0.0373105
\(987\) 0 0
\(988\) 17.6569 0.561739
\(989\) −19.0294 −0.605101
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 28.6274 0.908921
\(993\) 0 0
\(994\) 30.6274 0.971443
\(995\) 0 0
\(996\) 0 0
\(997\) −6.82843 −0.216258 −0.108129 0.994137i \(-0.534486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(998\) 14.9117 0.472021
\(999\) 0 0
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 6525.2.a.p.1.2 2
3.2 odd 2 725.2.a.c.1.1 2
5.4 even 2 1305.2.a.n.1.1 2
15.2 even 4 725.2.b.c.349.2 4
15.8 even 4 725.2.b.c.349.3 4
15.14 odd 2 145.2.a.b.1.2 2
60.59 even 2 2320.2.a.k.1.2 2
105.104 even 2 7105.2.a.e.1.2 2
120.29 odd 2 9280.2.a.be.1.1 2
120.59 even 2 9280.2.a.w.1.2 2
435.434 odd 2 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 15.14 odd 2
725.2.a.c.1.1 2 3.2 odd 2
725.2.b.c.349.2 4 15.2 even 4
725.2.b.c.349.3 4 15.8 even 4
1305.2.a.n.1.1 2 5.4 even 2
2320.2.a.k.1.2 2 60.59 even 2
4205.2.a.d.1.1 2 435.434 odd 2
6525.2.a.p.1.2 2 1.1 even 1 trivial
7105.2.a.e.1.2 2 105.104 even 2
9280.2.a.w.1.2 2 120.59 even 2
9280.2.a.be.1.1 2 120.29 odd 2