Properties

Label 6525.2.a.p.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} +3.82843 q^{4} -0.828427 q^{7} -4.41421 q^{8} +4.82843 q^{11} +2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} +0.828427 q^{19} -11.6569 q^{22} -8.82843 q^{23} -4.82843 q^{26} -3.17157 q^{28} -1.00000 q^{29} -10.4853 q^{31} +1.58579 q^{32} +6.82843 q^{34} -8.48528 q^{37} -2.00000 q^{38} +6.00000 q^{41} +6.00000 q^{43} +18.4853 q^{44} +21.3137 q^{46} -0.343146 q^{47} -6.31371 q^{49} +7.65685 q^{52} +7.65685 q^{53} +3.65685 q^{56} +2.41421 q^{58} +7.65685 q^{61} +25.3137 q^{62} -9.82843 q^{64} +10.4853 q^{67} -10.8284 q^{68} -7.31371 q^{71} +8.48528 q^{73} +20.4853 q^{74} +3.17157 q^{76} -4.00000 q^{77} +14.4853 q^{79} -14.4853 q^{82} +12.8284 q^{83} -14.4853 q^{86} -21.3137 q^{88} -3.65685 q^{89} -1.65685 q^{91} -33.7990 q^{92} +0.828427 q^{94} -4.48528 q^{97} +15.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) 0 0
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) −4.41421 −1.56066
\(9\) 0 0
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 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 \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 0.828427 0.190054 0.0950271 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −11.6569 −2.48525
\(23\) −8.82843 −1.84085 −0.920427 0.390914i \(-0.872159\pi\)
−0.920427 + 0.390914i \(0.872159\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.82843 −0.946932
\(27\) 0 0
\(28\) −3.17157 −0.599371
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −10.4853 −1.88321 −0.941606 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) 1.58579 0.280330
\(33\) 0 0
\(34\) 6.82843 1.17107
\(35\) 0 0
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −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 \(0.348747\pi\)
\(44\) 18.4853 2.78676
\(45\) 0 0
\(46\) 21.3137 3.14253
\(47\) −0.343146 −0.0500530 −0.0250265 0.999687i \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) 0 0
\(52\) 7.65685 1.06181
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.65685 0.488668
\(57\) 0 0
\(58\) 2.41421 0.317002
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 25.3137 3.21484
\(63\) 0 0
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 0 0
\(67\) 10.4853 1.28098 0.640490 0.767966i \(-0.278731\pi\)
0.640490 + 0.767966i \(0.278731\pi\)
\(68\) −10.8284 −1.31314
\(69\) 0 0
\(70\) 0 0
\(71\) −7.31371 −0.867978 −0.433989 0.900918i \(-0.642894\pi\)
−0.433989 + 0.900918i \(0.642894\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 20.4853 2.38137
\(75\) 0 0
\(76\) 3.17157 0.363804
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 14.4853 1.62972 0.814861 0.579657i \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.4853 −1.59963
\(83\) 12.8284 1.40810 0.704051 0.710149i \(-0.251372\pi\)
0.704051 + 0.710149i \(0.251372\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −14.4853 −1.56199
\(87\) 0 0
\(88\) −21.3137 −2.27205
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −1.65685 −0.173686
\(92\) −33.7990 −3.52379
\(93\) 0 0
\(94\) 0.828427 0.0854457
\(95\) 0 0
\(96\) 0 0
\(97\) −4.48528 −0.455411 −0.227706 0.973730i \(-0.573122\pi\)
−0.227706 + 0.973730i \(0.573122\pi\)
\(98\) 15.2426 1.53974
\(99\) 0 0
\(100\) 0 0
\(101\) −4.34315 −0.432159 −0.216080 0.976376i \(-0.569327\pi\)
−0.216080 + 0.976376i \(0.569327\pi\)
\(102\) 0 0
\(103\) 12.1421 1.19640 0.598200 0.801347i \(-0.295883\pi\)
0.598200 + 0.801347i \(0.295883\pi\)
\(104\) −8.82843 −0.865699
\(105\) 0 0
\(106\) −18.4853 −1.79545
\(107\) −8.14214 −0.787130 −0.393565 0.919297i \(-0.628758\pi\)
−0.393565 + 0.919297i \(0.628758\pi\)
\(108\) 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
\(112\) −2.48528 −0.234837
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −3.82843 −0.355461
\(117\) 0 0
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) −18.4853 −1.67358
\(123\) 0 0
\(124\) −40.1421 −3.60487
\(125\) 0 0
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 20.5563 1.81694
\(129\) 0 0
\(130\) 0 0
\(131\) −16.1421 −1.41034 −0.705172 0.709036i \(-0.749130\pi\)
−0.705172 + 0.709036i \(0.749130\pi\)
\(132\) 0 0
\(133\) −0.686292 −0.0595090
\(134\) −25.3137 −2.18677
\(135\) 0 0
\(136\) 12.4853 1.07060
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) 0 0
\(139\) 10.3431 0.877294 0.438647 0.898659i \(-0.355458\pi\)
0.438647 + 0.898659i \(0.355458\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 17.6569 1.48173
\(143\) 9.65685 0.807547
\(144\) 0 0
\(145\) 0 0
\(146\) −20.4853 −1.69537
\(147\) 0 0
\(148\) −32.4853 −2.67027
\(149\) 13.3137 1.09070 0.545351 0.838208i \(-0.316396\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −3.65685 −0.296610
\(153\) 0 0
\(154\) 9.65685 0.778171
\(155\) 0 0
\(156\) 0 0
\(157\) 16.4853 1.31567 0.657834 0.753163i \(-0.271473\pi\)
0.657834 + 0.753163i \(0.271473\pi\)
\(158\) −34.9706 −2.78211
\(159\) 0 0
\(160\) 0 0
\(161\) 7.31371 0.576401
\(162\) 0 0
\(163\) 19.6569 1.53964 0.769822 0.638259i \(-0.220345\pi\)
0.769822 + 0.638259i \(0.220345\pi\)
\(164\) 22.9706 1.79370
\(165\) 0 0
\(166\) −30.9706 −2.40378
\(167\) 14.4853 1.12090 0.560452 0.828187i \(-0.310627\pi\)
0.560452 + 0.828187i \(0.310627\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 22.9706 1.75149
\(173\) −5.31371 −0.403994 −0.201997 0.979386i \(-0.564743\pi\)
−0.201997 + 0.979386i \(0.564743\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14.4853 1.09187
\(177\) 0 0
\(178\) 8.82843 0.661719
\(179\) 0.686292 0.0512958 0.0256479 0.999671i \(-0.491835\pi\)
0.0256479 + 0.999671i \(0.491835\pi\)
\(180\) 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
\(184\) 38.9706 2.87295
\(185\) 0 0
\(186\) 0 0
\(187\) −13.6569 −0.998688
\(188\) −1.31371 −0.0958120
\(189\) 0 0
\(190\) 0 0
\(191\) 15.1716 1.09778 0.548888 0.835896i \(-0.315051\pi\)
0.548888 + 0.835896i \(0.315051\pi\)
\(192\) 0 0
\(193\) 12.4853 0.898710 0.449355 0.893353i \(-0.351654\pi\)
0.449355 + 0.893353i \(0.351654\pi\)
\(194\) 10.8284 0.777436
\(195\) 0 0
\(196\) −24.1716 −1.72654
\(197\) −8.34315 −0.594425 −0.297212 0.954811i \(-0.596057\pi\)
−0.297212 + 0.954811i \(0.596057\pi\)
\(198\) 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
\(202\) 10.4853 0.737742
\(203\) 0.828427 0.0581442
\(204\) 0 0
\(205\) 0 0
\(206\) −29.3137 −2.04238
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 4.82843 0.332403 0.166201 0.986092i \(-0.446850\pi\)
0.166201 + 0.986092i \(0.446850\pi\)
\(212\) 29.3137 2.01327
\(213\) 0 0
\(214\) 19.6569 1.34371
\(215\) 0 0
\(216\) 0 0
\(217\) 8.68629 0.589664
\(218\) −4.82843 −0.327022
\(219\) 0 0
\(220\) 0 0
\(221\) −5.65685 −0.380521
\(222\) 0 0
\(223\) −21.7990 −1.45977 −0.729884 0.683571i \(-0.760426\pi\)
−0.729884 + 0.683571i \(0.760426\pi\)
\(224\) −1.31371 −0.0877758
\(225\) 0 0
\(226\) −6.82843 −0.454220
\(227\) −8.14214 −0.540413 −0.270206 0.962802i \(-0.587092\pi\)
−0.270206 + 0.962802i \(0.587092\pi\)
\(228\) 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
\(232\) 4.41421 0.289807
\(233\) 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
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −29.7279 −1.91098
\(243\) 0 0
\(244\) 29.3137 1.87662
\(245\) 0 0
\(246\) 0 0
\(247\) 1.65685 0.105423
\(248\) 46.2843 2.93905
\(249\) 0 0
\(250\) 0 0
\(251\) −3.17157 −0.200188 −0.100094 0.994978i \(-0.531914\pi\)
−0.100094 + 0.994978i \(0.531914\pi\)
\(252\) 0 0
\(253\) −42.6274 −2.67996
\(254\) 14.4853 0.908887
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 29.3137 1.82854 0.914269 0.405107i \(-0.132766\pi\)
0.914269 + 0.405107i \(0.132766\pi\)
\(258\) 0 0
\(259\) 7.02944 0.436788
\(260\) 0 0
\(261\) 0 0
\(262\) 38.9706 2.40761
\(263\) −8.34315 −0.514460 −0.257230 0.966350i \(-0.582810\pi\)
−0.257230 + 0.966350i \(0.582810\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.65685 0.101588
\(267\) 0 0
\(268\) 40.1421 2.45207
\(269\) −1.31371 −0.0800982 −0.0400491 0.999198i \(-0.512751\pi\)
−0.0400491 + 0.999198i \(0.512751\pi\)
\(270\) 0 0
\(271\) 29.7990 1.81016 0.905080 0.425242i \(-0.139811\pi\)
0.905080 + 0.425242i \(0.139811\pi\)
\(272\) −8.48528 −0.514496
\(273\) 0 0
\(274\) 26.1421 1.57930
\(275\) 0 0
\(276\) 0 0
\(277\) −7.65685 −0.460056 −0.230028 0.973184i \(-0.573882\pi\)
−0.230028 + 0.973184i \(0.573882\pi\)
\(278\) −24.9706 −1.49763
\(279\) 0 0
\(280\) 0 0
\(281\) 6.68629 0.398871 0.199435 0.979911i \(-0.436089\pi\)
0.199435 + 0.979911i \(0.436089\pi\)
\(282\) 0 0
\(283\) 0.828427 0.0492449 0.0246224 0.999697i \(-0.492162\pi\)
0.0246224 + 0.999697i \(0.492162\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −23.3137 −1.37857
\(287\) −4.97056 −0.293403
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 32.4853 1.90106
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 37.4558 2.17708
\(297\) 0 0
\(298\) −32.1421 −1.86194
\(299\) −17.6569 −1.02112
\(300\) 0 0
\(301\) −4.97056 −0.286498
\(302\) 28.9706 1.66707
\(303\) 0 0
\(304\) 2.48528 0.142541
\(305\) 0 0
\(306\) 0 0
\(307\) −10.9706 −0.626123 −0.313062 0.949733i \(-0.601355\pi\)
−0.313062 + 0.949733i \(0.601355\pi\)
\(308\) −15.3137 −0.872580
\(309\) 0 0
\(310\) 0 0
\(311\) 2.48528 0.140927 0.0704637 0.997514i \(-0.477552\pi\)
0.0704637 + 0.997514i \(0.477552\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −39.7990 −2.24599
\(315\) 0 0
\(316\) 55.4558 3.11963
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) 0 0
\(319\) −4.82843 −0.270340
\(320\) 0 0
\(321\) 0 0
\(322\) −17.6569 −0.983978
\(323\) −2.34315 −0.130376
\(324\) 0 0
\(325\) 0 0
\(326\) −47.4558 −2.62834
\(327\) 0 0
\(328\) −26.4853 −1.46241
\(329\) 0.284271 0.0156724
\(330\) 0 0
\(331\) −17.7990 −0.978321 −0.489160 0.872194i \(-0.662697\pi\)
−0.489160 + 0.872194i \(0.662697\pi\)
\(332\) 49.1127 2.69541
\(333\) 0 0
\(334\) −34.9706 −1.91350
\(335\) 0 0
\(336\) 0 0
\(337\) 6.82843 0.371968 0.185984 0.982553i \(-0.440453\pi\)
0.185984 + 0.982553i \(0.440453\pi\)
\(338\) 21.7279 1.18184
\(339\) 0 0
\(340\) 0 0
\(341\) −50.6274 −2.74163
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) −26.4853 −1.42799
\(345\) 0 0
\(346\) 12.8284 0.689661
\(347\) 20.1421 1.08129 0.540643 0.841252i \(-0.318181\pi\)
0.540643 + 0.841252i \(0.318181\pi\)
\(348\) 0 0
\(349\) −24.6274 −1.31828 −0.659138 0.752022i \(-0.729079\pi\)
−0.659138 + 0.752022i \(0.729079\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.65685 0.408112
\(353\) −15.6569 −0.833330 −0.416665 0.909060i \(-0.636801\pi\)
−0.416665 + 0.909060i \(0.636801\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −1.65685 −0.0875675
\(359\) 32.1421 1.69640 0.848199 0.529678i \(-0.177687\pi\)
0.848199 + 0.529678i \(0.177687\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) 14.4853 0.761329
\(363\) 0 0
\(364\) −6.34315 −0.332471
\(365\) 0 0
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) −26.4853 −1.38064
\(369\) 0 0
\(370\) 0 0
\(371\) −6.34315 −0.329320
\(372\) 0 0
\(373\) −26.9706 −1.39648 −0.698241 0.715862i \(-0.746034\pi\)
−0.698241 + 0.715862i \(0.746034\pi\)
\(374\) 32.9706 1.70487
\(375\) 0 0
\(376\) 1.51472 0.0781156
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 5.51472 0.283272 0.141636 0.989919i \(-0.454764\pi\)
0.141636 + 0.989919i \(0.454764\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −36.6274 −1.87402
\(383\) −14.4853 −0.740163 −0.370082 0.928999i \(-0.620670\pi\)
−0.370082 + 0.928999i \(0.620670\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −30.1421 −1.53419
\(387\) 0 0
\(388\) −17.1716 −0.871755
\(389\) 6.68629 0.339008 0.169504 0.985529i \(-0.445783\pi\)
0.169504 + 0.985529i \(0.445783\pi\)
\(390\) 0 0
\(391\) 24.9706 1.26282
\(392\) 27.8701 1.40765
\(393\) 0 0
\(394\) 20.1421 1.01475
\(395\) 0 0
\(396\) 0 0
\(397\) 8.34315 0.418730 0.209365 0.977838i \(-0.432860\pi\)
0.209365 + 0.977838i \(0.432860\pi\)
\(398\) −28.9706 −1.45216
\(399\) 0 0
\(400\) 0 0
\(401\) 29.3137 1.46386 0.731928 0.681382i \(-0.238621\pi\)
0.731928 + 0.681382i \(0.238621\pi\)
\(402\) 0 0
\(403\) −20.9706 −1.04462
\(404\) −16.6274 −0.827245
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −40.9706 −2.03084
\(408\) 0 0
\(409\) 30.9706 1.53140 0.765698 0.643200i \(-0.222394\pi\)
0.765698 + 0.643200i \(0.222394\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 46.4853 2.29017
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 3.17157 0.155499
\(417\) 0 0
\(418\) −9.65685 −0.472332
\(419\) 4.97056 0.242828 0.121414 0.992602i \(-0.461257\pi\)
0.121414 + 0.992602i \(0.461257\pi\)
\(420\) 0 0
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) −11.6569 −0.567447
\(423\) 0 0
\(424\) −33.7990 −1.64142
\(425\) 0 0
\(426\) 0 0
\(427\) −6.34315 −0.306966
\(428\) −31.1716 −1.50673
\(429\) 0 0
\(430\) 0 0
\(431\) 19.3137 0.930309 0.465154 0.885230i \(-0.345999\pi\)
0.465154 + 0.885230i \(0.345999\pi\)
\(432\) 0 0
\(433\) 34.8284 1.67375 0.836874 0.547396i \(-0.184381\pi\)
0.836874 + 0.547396i \(0.184381\pi\)
\(434\) −20.9706 −1.00662
\(435\) 0 0
\(436\) 7.65685 0.366697
\(437\) −7.31371 −0.349862
\(438\) 0 0
\(439\) −21.6569 −1.03363 −0.516813 0.856099i \(-0.672882\pi\)
−0.516813 + 0.856099i \(0.672882\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 13.6569 0.649590
\(443\) 3.65685 0.173742 0.0868712 0.996220i \(-0.472313\pi\)
0.0868712 + 0.996220i \(0.472313\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 52.6274 2.49198
\(447\) 0 0
\(448\) 8.14214 0.384680
\(449\) −0.343146 −0.0161940 −0.00809702 0.999967i \(-0.502577\pi\)
−0.00809702 + 0.999967i \(0.502577\pi\)
\(450\) 0 0
\(451\) 28.9706 1.36417
\(452\) 10.8284 0.509326
\(453\) 0 0
\(454\) 19.6569 0.922542
\(455\) 0 0
\(456\) 0 0
\(457\) −8.34315 −0.390276 −0.195138 0.980776i \(-0.562515\pi\)
−0.195138 + 0.980776i \(0.562515\pi\)
\(458\) 4.82843 0.225618
\(459\) 0 0
\(460\) 0 0
\(461\) 24.3431 1.13377 0.566887 0.823796i \(-0.308148\pi\)
0.566887 + 0.823796i \(0.308148\pi\)
\(462\) 0 0
\(463\) 17.7990 0.827189 0.413595 0.910461i \(-0.364273\pi\)
0.413595 + 0.910461i \(0.364273\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −43.4558 −2.01305
\(467\) −22.9706 −1.06295 −0.531475 0.847074i \(-0.678362\pi\)
−0.531475 + 0.847074i \(0.678362\pi\)
\(468\) 0 0
\(469\) −8.68629 −0.401096
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.9706 1.33207
\(474\) 0 0
\(475\) 0 0
\(476\) 8.97056 0.411165
\(477\) 0 0
\(478\) −56.2843 −2.57438
\(479\) −12.8284 −0.586146 −0.293073 0.956090i \(-0.594678\pi\)
−0.293073 + 0.956090i \(0.594678\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) −24.1421 −1.09964
\(483\) 0 0
\(484\) 47.1421 2.14282
\(485\) 0 0
\(486\) 0 0
\(487\) 29.7990 1.35032 0.675161 0.737671i \(-0.264074\pi\)
0.675161 + 0.737671i \(0.264074\pi\)
\(488\) −33.7990 −1.53001
\(489\) 0 0
\(490\) 0 0
\(491\) −43.4558 −1.96113 −0.980567 0.196183i \(-0.937145\pi\)
−0.980567 + 0.196183i \(0.937145\pi\)
\(492\) 0 0
\(493\) 2.82843 0.127386
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −31.4558 −1.41241
\(497\) 6.05887 0.271778
\(498\) 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
\(502\) 7.65685 0.341742
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 102.912 4.57498
\(507\) 0 0
\(508\) −22.9706 −1.01915
\(509\) 44.6274 1.97808 0.989038 0.147663i \(-0.0471751\pi\)
0.989038 + 0.147663i \(0.0471751\pi\)
\(510\) 0 0
\(511\) −7.02944 −0.310964
\(512\) 31.2426 1.38074
\(513\) 0 0
\(514\) −70.7696 −3.12151
\(515\) 0 0
\(516\) 0 0
\(517\) −1.65685 −0.0728684
\(518\) −16.9706 −0.745644
\(519\) 0 0
\(520\) 0 0
\(521\) −1.31371 −0.0575546 −0.0287773 0.999586i \(-0.509161\pi\)
−0.0287773 + 0.999586i \(0.509161\pi\)
\(522\) 0 0
\(523\) 14.4853 0.633397 0.316699 0.948526i \(-0.397426\pi\)
0.316699 + 0.948526i \(0.397426\pi\)
\(524\) −61.7990 −2.69970
\(525\) 0 0
\(526\) 20.1421 0.878239
\(527\) 29.6569 1.29187
\(528\) 0 0
\(529\) 54.9411 2.38874
\(530\) 0 0
\(531\) 0 0
\(532\) −2.62742 −0.113913
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) −46.2843 −1.99918
\(537\) 0 0
\(538\) 3.17157 0.136736
\(539\) −30.4853 −1.31309
\(540\) 0 0
\(541\) −38.9706 −1.67548 −0.837738 0.546073i \(-0.816122\pi\)
−0.837738 + 0.546073i \(0.816122\pi\)
\(542\) −71.9411 −3.09014
\(543\) 0 0
\(544\) −4.48528 −0.192305
\(545\) 0 0
\(546\) 0 0
\(547\) −14.4853 −0.619346 −0.309673 0.950843i \(-0.600220\pi\)
−0.309673 + 0.950843i \(0.600220\pi\)
\(548\) −41.4558 −1.77091
\(549\) 0 0
\(550\) 0 0
\(551\) −0.828427 −0.0352922
\(552\) 0 0
\(553\) −12.0000 −0.510292
\(554\) 18.4853 0.785364
\(555\) 0 0
\(556\) 39.5980 1.67933
\(557\) 39.9411 1.69236 0.846180 0.532897i \(-0.178897\pi\)
0.846180 + 0.532897i \(0.178897\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) −16.1421 −0.680915
\(563\) −3.65685 −0.154118 −0.0770590 0.997027i \(-0.524553\pi\)
−0.0770590 + 0.997027i \(0.524553\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) 32.2843 1.35462
\(569\) −16.3431 −0.685140 −0.342570 0.939492i \(-0.611297\pi\)
−0.342570 + 0.939492i \(0.611297\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 36.9706 1.54582
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 15.7990 0.657721 0.328860 0.944379i \(-0.393335\pi\)
0.328860 + 0.944379i \(0.393335\pi\)
\(578\) 21.7279 0.903762
\(579\) 0 0
\(580\) 0 0
\(581\) −10.6274 −0.440900
\(582\) 0 0
\(583\) 36.9706 1.53116
\(584\) −37.4558 −1.54993
\(585\) 0 0
\(586\) 20.4853 0.846239
\(587\) 9.79899 0.404448 0.202224 0.979339i \(-0.435183\pi\)
0.202224 + 0.979339i \(0.435183\pi\)
\(588\) 0 0
\(589\) −8.68629 −0.357912
\(590\) 0 0
\(591\) 0 0
\(592\) −25.4558 −1.04623
\(593\) 3.65685 0.150169 0.0750845 0.997177i \(-0.476077\pi\)
0.0750845 + 0.997177i \(0.476077\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 50.9706 2.08784
\(597\) 0 0
\(598\) 42.6274 1.74316
\(599\) −1.79899 −0.0735047 −0.0367524 0.999324i \(-0.511701\pi\)
−0.0367524 + 0.999324i \(0.511701\pi\)
\(600\) 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
\(604\) −45.9411 −1.86932
\(605\) 0 0
\(606\) 0 0
\(607\) −42.9706 −1.74412 −0.872061 0.489398i \(-0.837217\pi\)
−0.872061 + 0.489398i \(0.837217\pi\)
\(608\) 1.31371 0.0532779
\(609\) 0 0
\(610\) 0 0
\(611\) −0.686292 −0.0277644
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 26.4853 1.06886
\(615\) 0 0
\(616\) 17.6569 0.711415
\(617\) −14.8284 −0.596970 −0.298485 0.954414i \(-0.596481\pi\)
−0.298485 + 0.954414i \(0.596481\pi\)
\(618\) 0 0
\(619\) 29.7990 1.19772 0.598861 0.800853i \(-0.295620\pi\)
0.598861 + 0.800853i \(0.295620\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.00000 −0.240578
\(623\) 3.02944 0.121372
\(624\) 0 0
\(625\) 0 0
\(626\) −14.4853 −0.578948
\(627\) 0 0
\(628\) 63.1127 2.51847
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 3.02944 0.120600 0.0603000 0.998180i \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) −63.9411 −2.54344
\(633\) 0 0
\(634\) 6.82843 0.271191
\(635\) 0 0
\(636\) 0 0
\(637\) −12.6274 −0.500316
\(638\) 11.6569 0.461499
\(639\) 0 0
\(640\) 0 0
\(641\) 44.6274 1.76268 0.881338 0.472485i \(-0.156643\pi\)
0.881338 + 0.472485i \(0.156643\pi\)
\(642\) 0 0
\(643\) 31.4558 1.24050 0.620249 0.784405i \(-0.287032\pi\)
0.620249 + 0.784405i \(0.287032\pi\)
\(644\) 28.0000 1.10335
\(645\) 0 0
\(646\) 5.65685 0.222566
\(647\) 21.1127 0.830026 0.415013 0.909816i \(-0.363777\pi\)
0.415013 + 0.909816i \(0.363777\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 75.2548 2.94721
\(653\) −22.8284 −0.893345 −0.446673 0.894697i \(-0.647391\pi\)
−0.446673 + 0.894697i \(0.647391\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 18.0000 0.702782
\(657\) 0 0
\(658\) −0.686292 −0.0267544
\(659\) 37.7990 1.47244 0.736220 0.676743i \(-0.236609\pi\)
0.736220 + 0.676743i \(0.236609\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 42.9706 1.67010
\(663\) 0 0
\(664\) −56.6274 −2.19757
\(665\) 0 0
\(666\) 0 0
\(667\) 8.82843 0.341838
\(668\) 55.4558 2.14565
\(669\) 0 0
\(670\) 0 0
\(671\) 36.9706 1.42723
\(672\) 0 0
\(673\) 10.9706 0.422884 0.211442 0.977391i \(-0.432184\pi\)
0.211442 + 0.977391i \(0.432184\pi\)
\(674\) −16.4853 −0.634989
\(675\) 0 0
\(676\) −34.4558 −1.32522
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) 0 0
\(679\) 3.71573 0.142597
\(680\) 0 0
\(681\) 0 0
\(682\) 122.225 4.68025
\(683\) 40.1421 1.53600 0.767998 0.640452i \(-0.221253\pi\)
0.767998 + 0.640452i \(0.221253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −26.6274 −1.01664
\(687\) 0 0
\(688\) 18.0000 0.686244
\(689\) 15.3137 0.583406
\(690\) 0 0
\(691\) −11.0294 −0.419580 −0.209790 0.977747i \(-0.567278\pi\)
−0.209790 + 0.977747i \(0.567278\pi\)
\(692\) −20.3431 −0.773330
\(693\) 0 0
\(694\) −48.6274 −1.84587
\(695\) 0 0
\(696\) 0 0
\(697\) −16.9706 −0.642806
\(698\) 59.4558 2.25044
\(699\) 0 0
\(700\) 0 0
\(701\) −29.3137 −1.10716 −0.553582 0.832795i \(-0.686739\pi\)
−0.553582 + 0.832795i \(0.686739\pi\)
\(702\) 0 0
\(703\) −7.02944 −0.265120
\(704\) −47.4558 −1.78856
\(705\) 0 0
\(706\) 37.7990 1.42258
\(707\) 3.59798 0.135316
\(708\) 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
\(712\) 16.1421 0.604952
\(713\) 92.5685 3.46672
\(714\) 0 0
\(715\) 0 0
\(716\) 2.62742 0.0981912
\(717\) 0 0
\(718\) −77.5980 −2.89593
\(719\) −10.6274 −0.396336 −0.198168 0.980168i \(-0.563499\pi\)
−0.198168 + 0.980168i \(0.563499\pi\)
\(720\) 0 0
\(721\) −10.0589 −0.374612
\(722\) 44.2132 1.64545
\(723\) 0 0
\(724\) −22.9706 −0.853694
\(725\) 0 0
\(726\) 0 0
\(727\) −43.9411 −1.62969 −0.814843 0.579682i \(-0.803177\pi\)
−0.814843 + 0.579682i \(0.803177\pi\)
\(728\) 7.31371 0.271064
\(729\) 0 0
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) 0 0
\(733\) 17.1716 0.634247 0.317123 0.948384i \(-0.397283\pi\)
0.317123 + 0.948384i \(0.397283\pi\)
\(734\) 43.4558 1.60398
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) 50.6274 1.86488
\(738\) 0 0
\(739\) −2.48528 −0.0914226 −0.0457113 0.998955i \(-0.514555\pi\)
−0.0457113 + 0.998955i \(0.514555\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 15.3137 0.562184
\(743\) −7.37258 −0.270474 −0.135237 0.990813i \(-0.543180\pi\)
−0.135237 + 0.990813i \(0.543180\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 65.1127 2.38395
\(747\) 0 0
\(748\) −52.2843 −1.91170
\(749\) 6.74517 0.246463
\(750\) 0 0
\(751\) −12.1421 −0.443073 −0.221536 0.975152i \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) −1.02944 −0.0375397
\(753\) 0 0
\(754\) 4.82843 0.175841
\(755\) 0 0
\(756\) 0 0
\(757\) −36.4853 −1.32608 −0.663040 0.748584i \(-0.730734\pi\)
−0.663040 + 0.748584i \(0.730734\pi\)
\(758\) −13.3137 −0.483576
\(759\) 0 0
\(760\) 0 0
\(761\) 36.6274 1.32774 0.663871 0.747847i \(-0.268912\pi\)
0.663871 + 0.747847i \(0.268912\pi\)
\(762\) 0 0
\(763\) −1.65685 −0.0599822
\(764\) 58.0833 2.10138
\(765\) 0 0
\(766\) 34.9706 1.26354
\(767\) 0 0
\(768\) 0 0
\(769\) −4.34315 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 47.7990 1.72032
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 19.7990 0.710742
\(777\) 0 0
\(778\) −16.1421 −0.578724
\(779\) 4.97056 0.178089
\(780\) 0 0
\(781\) −35.3137 −1.26362
\(782\) −60.2843 −2.15576
\(783\) 0 0
\(784\) −18.9411 −0.676469
\(785\) 0 0
\(786\) 0 0
\(787\) 21.7990 0.777050 0.388525 0.921438i \(-0.372985\pi\)
0.388525 + 0.921438i \(0.372985\pi\)
\(788\) −31.9411 −1.13786
\(789\) 0 0
\(790\) 0 0
\(791\) −2.34315 −0.0833127
\(792\) 0 0
\(793\) 15.3137 0.543806
\(794\) −20.1421 −0.714818
\(795\) 0 0
\(796\) 45.9411 1.62834
\(797\) −34.1421 −1.20938 −0.604688 0.796462i \(-0.706702\pi\)
−0.604688 + 0.796462i \(0.706702\pi\)
\(798\) 0 0
\(799\) 0.970563 0.0343360
\(800\) 0 0
\(801\) 0 0
\(802\) −70.7696 −2.49896
\(803\) 40.9706 1.44582
\(804\) 0 0
\(805\) 0 0
\(806\) 50.6274 1.78327
\(807\) 0 0
\(808\) 19.1716 0.674454
\(809\) 14.2843 0.502208 0.251104 0.967960i \(-0.419206\pi\)
0.251104 + 0.967960i \(0.419206\pi\)
\(810\) 0 0
\(811\) 26.3431 0.925033 0.462516 0.886611i \(-0.346947\pi\)
0.462516 + 0.886611i \(0.346947\pi\)
\(812\) 3.17157 0.111300
\(813\) 0 0
\(814\) 98.9117 3.46685
\(815\) 0 0
\(816\) 0 0
\(817\) 4.97056 0.173898
\(818\) −74.7696 −2.61426
\(819\) 0 0
\(820\) 0 0
\(821\) 45.3137 1.58146 0.790730 0.612165i \(-0.209701\pi\)
0.790730 + 0.612165i \(0.209701\pi\)
\(822\) 0 0
\(823\) −2.97056 −0.103547 −0.0517737 0.998659i \(-0.516487\pi\)
−0.0517737 + 0.998659i \(0.516487\pi\)
\(824\) −53.5980 −1.86717
\(825\) 0 0
\(826\) 0 0
\(827\) −5.31371 −0.184776 −0.0923879 0.995723i \(-0.529450\pi\)
−0.0923879 + 0.995723i \(0.529450\pi\)
\(828\) 0 0
\(829\) −24.6274 −0.855346 −0.427673 0.903934i \(-0.640666\pi\)
−0.427673 + 0.903934i \(0.640666\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −19.6569 −0.681479
\(833\) 17.8579 0.618738
\(834\) 0 0
\(835\) 0 0
\(836\) 15.3137 0.529636
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 14.4853 0.500087 0.250044 0.968235i \(-0.419555\pi\)
0.250044 + 0.968235i \(0.419555\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 36.1421 1.24554
\(843\) 0 0
\(844\) 18.4853 0.636290
\(845\) 0 0
\(846\) 0 0
\(847\) −10.2010 −0.350511
\(848\) 22.9706 0.788812
\(849\) 0 0
\(850\) 0 0
\(851\) 74.9117 2.56794
\(852\) 0 0
\(853\) 11.1127 0.380492 0.190246 0.981736i \(-0.439072\pi\)
0.190246 + 0.981736i \(0.439072\pi\)
\(854\) 15.3137 0.524024
\(855\) 0 0
\(856\) 35.9411 1.22844
\(857\) 48.6274 1.66108 0.830540 0.556958i \(-0.188032\pi\)
0.830540 + 0.556958i \(0.188032\pi\)
\(858\) 0 0
\(859\) 28.4264 0.969896 0.484948 0.874543i \(-0.338838\pi\)
0.484948 + 0.874543i \(0.338838\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −46.6274 −1.58814
\(863\) −7.85786 −0.267485 −0.133742 0.991016i \(-0.542699\pi\)
−0.133742 + 0.991016i \(0.542699\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −84.0833 −2.85727
\(867\) 0 0
\(868\) 33.2548 1.12874
\(869\) 69.9411 2.37259
\(870\) 0 0
\(871\) 20.9706 0.710560
\(872\) −8.82843 −0.298968
\(873\) 0 0
\(874\) 17.6569 0.597252
\(875\) 0 0
\(876\) 0 0
\(877\) 18.2843 0.617416 0.308708 0.951157i \(-0.400103\pi\)
0.308708 + 0.951157i \(0.400103\pi\)
\(878\) 52.2843 1.76451
\(879\) 0 0
\(880\) 0 0
\(881\) −6.68629 −0.225267 −0.112633 0.993637i \(-0.535929\pi\)
−0.112633 + 0.993637i \(0.535929\pi\)
\(882\) 0 0
\(883\) −2.48528 −0.0836364 −0.0418182 0.999125i \(-0.513315\pi\)
−0.0418182 + 0.999125i \(0.513315\pi\)
\(884\) −21.6569 −0.728399
\(885\) 0 0
\(886\) −8.82843 −0.296597
\(887\) −29.3137 −0.984258 −0.492129 0.870522i \(-0.663781\pi\)
−0.492129 + 0.870522i \(0.663781\pi\)
\(888\) 0 0
\(889\) 4.97056 0.166707
\(890\) 0 0
\(891\) 0 0
\(892\) −83.4558 −2.79431
\(893\) −0.284271 −0.00951277
\(894\) 0 0
\(895\) 0 0
\(896\) −17.0294 −0.568914
\(897\) 0 0
\(898\) 0.828427 0.0276450
\(899\) 10.4853 0.349704
\(900\) 0 0
\(901\) −21.6569 −0.721494
\(902\) −69.9411 −2.32878
\(903\) 0 0
\(904\) −12.4853 −0.415254
\(905\) 0 0
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) −31.1716 −1.03446
\(909\) 0 0
\(910\) 0 0
\(911\) −3.85786 −0.127817 −0.0639084 0.997956i \(-0.520357\pi\)
−0.0639084 + 0.997956i \(0.520357\pi\)
\(912\) 0 0
\(913\) 61.9411 2.04995
\(914\) 20.1421 0.666243
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) 13.3726 0.441602
\(918\) 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
\(922\) −58.7696 −1.93547
\(923\) −14.6274 −0.481467
\(924\) 0 0
\(925\) 0 0
\(926\) −42.9706 −1.41210
\(927\) 0 0
\(928\) −1.58579 −0.0520560
\(929\) −40.6274 −1.33294 −0.666471 0.745531i \(-0.732196\pi\)
−0.666471 + 0.745531i \(0.732196\pi\)
\(930\) 0 0
\(931\) −5.23045 −0.171421
\(932\) 68.9117 2.25728
\(933\) 0 0
\(934\) 55.4558 1.81457
\(935\) 0 0
\(936\) 0 0
\(937\) −8.34315 −0.272559 −0.136279 0.990670i \(-0.543514\pi\)
−0.136279 + 0.990670i \(0.543514\pi\)
\(938\) 20.9706 0.684713
\(939\) 0 0
\(940\) 0 0
\(941\) −39.9411 −1.30204 −0.651022 0.759059i \(-0.725659\pi\)
−0.651022 + 0.759059i \(0.725659\pi\)
\(942\) 0 0
\(943\) −52.9706 −1.72496
\(944\) 0 0
\(945\) 0 0
\(946\) −69.9411 −2.27398
\(947\) −56.9117 −1.84938 −0.924691 0.380719i \(-0.875676\pi\)
−0.924691 + 0.380719i \(0.875676\pi\)
\(948\) 0 0
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) −10.3431 −0.335223
\(953\) 6.68629 0.216590 0.108295 0.994119i \(-0.465461\pi\)
0.108295 + 0.994119i \(0.465461\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 89.2548 2.88671
\(957\) 0 0
\(958\) 30.9706 1.00061
\(959\) 8.97056 0.289675
\(960\) 0 0
\(961\) 78.9411 2.54649
\(962\) 40.9706 1.32094
\(963\) 0 0
\(964\) 38.2843 1.23305
\(965\) 0 0
\(966\) 0 0
\(967\) 18.9706 0.610052 0.305026 0.952344i \(-0.401335\pi\)
0.305026 + 0.952344i \(0.401335\pi\)
\(968\) −54.3553 −1.74705
\(969\) 0 0
\(970\) 0 0
\(971\) 0.142136 0.00456135 0.00228067 0.999997i \(-0.499274\pi\)
0.00228067 + 0.999997i \(0.499274\pi\)
\(972\) 0 0
\(973\) −8.56854 −0.274695
\(974\) −71.9411 −2.30514
\(975\) 0 0
\(976\) 22.9706 0.735270
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) 0 0
\(979\) −17.6569 −0.564316
\(980\) 0 0
\(981\) 0 0
\(982\) 104.912 3.34787
\(983\) 13.3137 0.424641 0.212321 0.977200i \(-0.431898\pi\)
0.212321 + 0.977200i \(0.431898\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.82843 −0.217461
\(987\) 0 0
\(988\) 6.34315 0.201802
\(989\) −52.9706 −1.68437
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −16.6274 −0.527921
\(993\) 0 0
\(994\) −14.6274 −0.463953
\(995\) 0 0
\(996\) 0 0
\(997\) −1.17157 −0.0371041 −0.0185520 0.999828i \(-0.505906\pi\)
−0.0185520 + 0.999828i \(0.505906\pi\)
\(998\) −86.9117 −2.75114
\(999\) 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.1.1 2
3.2 odd 2 725.2.a.c.1.2 2
5.4 even 2 1305.2.a.n.1.2 2
15.2 even 4 725.2.b.c.349.4 4
15.8 even 4 725.2.b.c.349.1 4
15.14 odd 2 145.2.a.b.1.1 2
60.59 even 2 2320.2.a.k.1.1 2
105.104 even 2 7105.2.a.e.1.1 2
120.29 odd 2 9280.2.a.be.1.2 2
120.59 even 2 9280.2.a.w.1.1 2
435.434 odd 2 4205.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 15.14 odd 2
725.2.a.c.1.2 2 3.2 odd 2
725.2.b.c.349.1 4 15.8 even 4
725.2.b.c.349.4 4 15.2 even 4
1305.2.a.n.1.2 2 5.4 even 2
2320.2.a.k.1.1 2 60.59 even 2
4205.2.a.d.1.2 2 435.434 odd 2
6525.2.a.p.1.1 2 1.1 even 1 trivial
7105.2.a.e.1.1 2 105.104 even 2
9280.2.a.w.1.1 2 120.59 even 2
9280.2.a.be.1.2 2 120.29 odd 2