Properties

Label 8550.2.a.cb.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
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] = [8550,2,Mod(1,8550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8550, 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("8550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.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: \(68.2720937282\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.41421 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +4.41421 q^{7} +1.00000 q^{8} +1.41421 q^{11} +5.82843 q^{13} +4.41421 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} +1.41421 q^{22} -0.757359 q^{23} +5.82843 q^{26} +4.41421 q^{28} +0.171573 q^{29} +6.24264 q^{31} +1.00000 q^{32} -1.00000 q^{34} -8.48528 q^{37} -1.00000 q^{38} +4.24264 q^{41} +1.75736 q^{43} +1.41421 q^{44} -0.757359 q^{46} +12.4853 q^{49} +5.82843 q^{52} -5.48528 q^{53} +4.41421 q^{56} +0.171573 q^{58} -6.89949 q^{59} +14.2426 q^{61} +6.24264 q^{62} +1.00000 q^{64} +4.75736 q^{67} -1.00000 q^{68} +13.4142 q^{71} +11.4853 q^{73} -8.48528 q^{74} -1.00000 q^{76} +6.24264 q^{77} -6.48528 q^{79} +4.24264 q^{82} -14.4853 q^{83} +1.75736 q^{86} +1.41421 q^{88} -7.07107 q^{89} +25.7279 q^{91} -0.757359 q^{92} -0.343146 q^{97} +12.4853 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 6 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 6 q^{7} + 2 q^{8} + 6 q^{13} + 6 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 10 q^{23} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{38} + 12 q^{43} - 10 q^{46} + 8 q^{49} + 6 q^{52} + 6 q^{53} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 20 q^{61} + 4 q^{62} + 2 q^{64} + 18 q^{67} - 2 q^{68} + 24 q^{71} + 6 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} - 12 q^{83} + 12 q^{86} + 26 q^{91} - 10 q^{92} - 12 q^{97} + 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 4.41421 1.66842 0.834208 0.551450i \(-0.185925\pi\)
0.834208 + 0.551450i \(0.185925\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 5.82843 1.61651 0.808257 0.588829i \(-0.200411\pi\)
0.808257 + 0.588829i \(0.200411\pi\)
\(14\) 4.41421 1.17975
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 1.41421 0.301511
\(23\) −0.757359 −0.157920 −0.0789602 0.996878i \(-0.525160\pi\)
−0.0789602 + 0.996878i \(0.525160\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.82843 1.14305
\(27\) 0 0
\(28\) 4.41421 0.834208
\(29\) 0.171573 0.0318603 0.0159301 0.999873i \(-0.494929\pi\)
0.0159301 + 0.999873i \(0.494929\pi\)
\(30\) 0 0
\(31\) 6.24264 1.12121 0.560606 0.828083i \(-0.310568\pi\)
0.560606 + 0.828083i \(0.310568\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264 0.662589 0.331295 0.943527i \(-0.392515\pi\)
0.331295 + 0.943527i \(0.392515\pi\)
\(42\) 0 0
\(43\) 1.75736 0.267995 0.133997 0.990982i \(-0.457219\pi\)
0.133997 + 0.990982i \(0.457219\pi\)
\(44\) 1.41421 0.213201
\(45\) 0 0
\(46\) −0.757359 −0.111667
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 12.4853 1.78361
\(50\) 0 0
\(51\) 0 0
\(52\) 5.82843 0.808257
\(53\) −5.48528 −0.753461 −0.376731 0.926323i \(-0.622952\pi\)
−0.376731 + 0.926323i \(0.622952\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.41421 0.589874
\(57\) 0 0
\(58\) 0.171573 0.0225286
\(59\) −6.89949 −0.898238 −0.449119 0.893472i \(-0.648262\pi\)
−0.449119 + 0.893472i \(0.648262\pi\)
\(60\) 0 0
\(61\) 14.2426 1.82358 0.911792 0.410653i \(-0.134699\pi\)
0.911792 + 0.410653i \(0.134699\pi\)
\(62\) 6.24264 0.792816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.75736 0.581204 0.290602 0.956844i \(-0.406144\pi\)
0.290602 + 0.956844i \(0.406144\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 13.4142 1.59197 0.795987 0.605314i \(-0.206952\pi\)
0.795987 + 0.605314i \(0.206952\pi\)
\(72\) 0 0
\(73\) 11.4853 1.34425 0.672125 0.740437i \(-0.265382\pi\)
0.672125 + 0.740437i \(0.265382\pi\)
\(74\) −8.48528 −0.986394
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 6.24264 0.711415
\(78\) 0 0
\(79\) −6.48528 −0.729651 −0.364826 0.931076i \(-0.618871\pi\)
−0.364826 + 0.931076i \(0.618871\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.24264 0.468521
\(83\) −14.4853 −1.58997 −0.794983 0.606632i \(-0.792520\pi\)
−0.794983 + 0.606632i \(0.792520\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.75736 0.189501
\(87\) 0 0
\(88\) 1.41421 0.150756
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 25.7279 2.69702
\(92\) −0.757359 −0.0789602
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 12.4853 1.26120
\(99\) 0 0
\(100\) 0 0
\(101\) −13.0711 −1.30062 −0.650310 0.759669i \(-0.725361\pi\)
−0.650310 + 0.759669i \(0.725361\pi\)
\(102\) 0 0
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) 5.82843 0.571524
\(105\) 0 0
\(106\) −5.48528 −0.532778
\(107\) −19.7279 −1.90717 −0.953585 0.301124i \(-0.902638\pi\)
−0.953585 + 0.301124i \(0.902638\pi\)
\(108\) 0 0
\(109\) −17.9706 −1.72127 −0.860634 0.509224i \(-0.829932\pi\)
−0.860634 + 0.509224i \(0.829932\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.41421 0.417104
\(113\) 10.2426 0.963547 0.481773 0.876296i \(-0.339993\pi\)
0.481773 + 0.876296i \(0.339993\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.171573 0.0159301
\(117\) 0 0
\(118\) −6.89949 −0.635150
\(119\) −4.41421 −0.404650
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 14.2426 1.28947
\(123\) 0 0
\(124\) 6.24264 0.560606
\(125\) 0 0
\(126\) 0 0
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 16.9706 1.48272 0.741362 0.671105i \(-0.234180\pi\)
0.741362 + 0.671105i \(0.234180\pi\)
\(132\) 0 0
\(133\) −4.41421 −0.382761
\(134\) 4.75736 0.410973
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 13.4142 1.12570
\(143\) 8.24264 0.689284
\(144\) 0 0
\(145\) 0 0
\(146\) 11.4853 0.950529
\(147\) 0 0
\(148\) −8.48528 −0.697486
\(149\) −17.6569 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(150\) 0 0
\(151\) −10.4853 −0.853280 −0.426640 0.904422i \(-0.640303\pi\)
−0.426640 + 0.904422i \(0.640303\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 6.24264 0.503046
\(155\) 0 0
\(156\) 0 0
\(157\) 11.6569 0.930318 0.465159 0.885227i \(-0.345997\pi\)
0.465159 + 0.885227i \(0.345997\pi\)
\(158\) −6.48528 −0.515941
\(159\) 0 0
\(160\) 0 0
\(161\) −3.34315 −0.263477
\(162\) 0 0
\(163\) −10.2426 −0.802266 −0.401133 0.916020i \(-0.631383\pi\)
−0.401133 + 0.916020i \(0.631383\pi\)
\(164\) 4.24264 0.331295
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) 18.2426 1.41166 0.705829 0.708382i \(-0.250575\pi\)
0.705829 + 0.708382i \(0.250575\pi\)
\(168\) 0 0
\(169\) 20.9706 1.61312
\(170\) 0 0
\(171\) 0 0
\(172\) 1.75736 0.133997
\(173\) −0.485281 −0.0368953 −0.0184476 0.999830i \(-0.505872\pi\)
−0.0184476 + 0.999830i \(0.505872\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421 0.106600
\(177\) 0 0
\(178\) −7.07107 −0.529999
\(179\) 11.6569 0.871274 0.435637 0.900122i \(-0.356523\pi\)
0.435637 + 0.900122i \(0.356523\pi\)
\(180\) 0 0
\(181\) −8.48528 −0.630706 −0.315353 0.948974i \(-0.602123\pi\)
−0.315353 + 0.948974i \(0.602123\pi\)
\(182\) 25.7279 1.90708
\(183\) 0 0
\(184\) −0.757359 −0.0558333
\(185\) 0 0
\(186\) 0 0
\(187\) −1.41421 −0.103418
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −12.5563 −0.908546 −0.454273 0.890863i \(-0.650101\pi\)
−0.454273 + 0.890863i \(0.650101\pi\)
\(192\) 0 0
\(193\) −0.343146 −0.0247002 −0.0123501 0.999924i \(-0.503931\pi\)
−0.0123501 + 0.999924i \(0.503931\pi\)
\(194\) −0.343146 −0.0246364
\(195\) 0 0
\(196\) 12.4853 0.891806
\(197\) 11.7574 0.837677 0.418839 0.908061i \(-0.362437\pi\)
0.418839 + 0.908061i \(0.362437\pi\)
\(198\) 0 0
\(199\) 9.24264 0.655193 0.327597 0.944818i \(-0.393761\pi\)
0.327597 + 0.944818i \(0.393761\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −13.0711 −0.919677
\(203\) 0.757359 0.0531562
\(204\) 0 0
\(205\) 0 0
\(206\) 4.24264 0.295599
\(207\) 0 0
\(208\) 5.82843 0.404129
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) −19.7279 −1.35813 −0.679063 0.734080i \(-0.737614\pi\)
−0.679063 + 0.734080i \(0.737614\pi\)
\(212\) −5.48528 −0.376731
\(213\) 0 0
\(214\) −19.7279 −1.34857
\(215\) 0 0
\(216\) 0 0
\(217\) 27.5563 1.87065
\(218\) −17.9706 −1.21712
\(219\) 0 0
\(220\) 0 0
\(221\) −5.82843 −0.392062
\(222\) 0 0
\(223\) −15.1716 −1.01596 −0.507982 0.861368i \(-0.669608\pi\)
−0.507982 + 0.861368i \(0.669608\pi\)
\(224\) 4.41421 0.294937
\(225\) 0 0
\(226\) 10.2426 0.681330
\(227\) 16.7574 1.11223 0.556113 0.831107i \(-0.312292\pi\)
0.556113 + 0.831107i \(0.312292\pi\)
\(228\) 0 0
\(229\) 14.9706 0.989283 0.494641 0.869097i \(-0.335299\pi\)
0.494641 + 0.869097i \(0.335299\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.171573 0.0112643
\(233\) 24.9706 1.63588 0.817938 0.575306i \(-0.195117\pi\)
0.817938 + 0.575306i \(0.195117\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.89949 −0.449119
\(237\) 0 0
\(238\) −4.41421 −0.286131
\(239\) 6.89949 0.446291 0.223146 0.974785i \(-0.428367\pi\)
0.223146 + 0.974785i \(0.428367\pi\)
\(240\) 0 0
\(241\) 8.97056 0.577845 0.288922 0.957353i \(-0.406703\pi\)
0.288922 + 0.957353i \(0.406703\pi\)
\(242\) −9.00000 −0.578542
\(243\) 0 0
\(244\) 14.2426 0.911792
\(245\) 0 0
\(246\) 0 0
\(247\) −5.82843 −0.370854
\(248\) 6.24264 0.396408
\(249\) 0 0
\(250\) 0 0
\(251\) −3.55635 −0.224475 −0.112237 0.993681i \(-0.535802\pi\)
−0.112237 + 0.993681i \(0.535802\pi\)
\(252\) 0 0
\(253\) −1.07107 −0.0673375
\(254\) −2.48528 −0.155940
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.7279 −1.29297 −0.646486 0.762926i \(-0.723762\pi\)
−0.646486 + 0.762926i \(0.723762\pi\)
\(258\) 0 0
\(259\) −37.4558 −2.32739
\(260\) 0 0
\(261\) 0 0
\(262\) 16.9706 1.04844
\(263\) 26.9706 1.66308 0.831538 0.555468i \(-0.187461\pi\)
0.831538 + 0.555468i \(0.187461\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.41421 −0.270653
\(267\) 0 0
\(268\) 4.75736 0.290602
\(269\) 16.6274 1.01379 0.506896 0.862007i \(-0.330793\pi\)
0.506896 + 0.862007i \(0.330793\pi\)
\(270\) 0 0
\(271\) −27.2426 −1.65487 −0.827436 0.561560i \(-0.810202\pi\)
−0.827436 + 0.561560i \(0.810202\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −13.0000 −0.785359
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −4.24264 −0.253095 −0.126547 0.991961i \(-0.540390\pi\)
−0.126547 + 0.991961i \(0.540390\pi\)
\(282\) 0 0
\(283\) 32.1421 1.91065 0.955326 0.295555i \(-0.0955045\pi\)
0.955326 + 0.295555i \(0.0955045\pi\)
\(284\) 13.4142 0.795987
\(285\) 0 0
\(286\) 8.24264 0.487398
\(287\) 18.7279 1.10547
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 11.4853 0.672125
\(293\) 5.48528 0.320454 0.160227 0.987080i \(-0.448777\pi\)
0.160227 + 0.987080i \(0.448777\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.48528 −0.493197
\(297\) 0 0
\(298\) −17.6569 −1.02283
\(299\) −4.41421 −0.255281
\(300\) 0 0
\(301\) 7.75736 0.447127
\(302\) −10.4853 −0.603360
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −17.6569 −1.00773 −0.503865 0.863782i \(-0.668089\pi\)
−0.503865 + 0.863782i \(0.668089\pi\)
\(308\) 6.24264 0.355707
\(309\) 0 0
\(310\) 0 0
\(311\) 4.75736 0.269765 0.134883 0.990862i \(-0.456934\pi\)
0.134883 + 0.990862i \(0.456934\pi\)
\(312\) 0 0
\(313\) −7.97056 −0.450523 −0.225261 0.974298i \(-0.572324\pi\)
−0.225261 + 0.974298i \(0.572324\pi\)
\(314\) 11.6569 0.657834
\(315\) 0 0
\(316\) −6.48528 −0.364826
\(317\) 9.48528 0.532746 0.266373 0.963870i \(-0.414175\pi\)
0.266373 + 0.963870i \(0.414175\pi\)
\(318\) 0 0
\(319\) 0.242641 0.0135853
\(320\) 0 0
\(321\) 0 0
\(322\) −3.34315 −0.186306
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 0 0
\(326\) −10.2426 −0.567287
\(327\) 0 0
\(328\) 4.24264 0.234261
\(329\) 0 0
\(330\) 0 0
\(331\) −19.2426 −1.05767 −0.528836 0.848724i \(-0.677371\pi\)
−0.528836 + 0.848724i \(0.677371\pi\)
\(332\) −14.4853 −0.794983
\(333\) 0 0
\(334\) 18.2426 0.998193
\(335\) 0 0
\(336\) 0 0
\(337\) 14.1005 0.768103 0.384052 0.923312i \(-0.374528\pi\)
0.384052 + 0.923312i \(0.374528\pi\)
\(338\) 20.9706 1.14065
\(339\) 0 0
\(340\) 0 0
\(341\) 8.82843 0.478086
\(342\) 0 0
\(343\) 24.2132 1.30739
\(344\) 1.75736 0.0947505
\(345\) 0 0
\(346\) −0.485281 −0.0260889
\(347\) 5.51472 0.296046 0.148023 0.988984i \(-0.452709\pi\)
0.148023 + 0.988984i \(0.452709\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421 0.0753778
\(353\) 2.51472 0.133845 0.0669225 0.997758i \(-0.478682\pi\)
0.0669225 + 0.997758i \(0.478682\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.07107 −0.374766
\(357\) 0 0
\(358\) 11.6569 0.616084
\(359\) 22.7574 1.20109 0.600544 0.799592i \(-0.294951\pi\)
0.600544 + 0.799592i \(0.294951\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −8.48528 −0.445976
\(363\) 0 0
\(364\) 25.7279 1.34851
\(365\) 0 0
\(366\) 0 0
\(367\) −25.4558 −1.32878 −0.664392 0.747384i \(-0.731309\pi\)
−0.664392 + 0.747384i \(0.731309\pi\)
\(368\) −0.757359 −0.0394801
\(369\) 0 0
\(370\) 0 0
\(371\) −24.2132 −1.25709
\(372\) 0 0
\(373\) 9.00000 0.466002 0.233001 0.972476i \(-0.425145\pi\)
0.233001 + 0.972476i \(0.425145\pi\)
\(374\) −1.41421 −0.0731272
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) 11.2426 0.577496 0.288748 0.957405i \(-0.406761\pi\)
0.288748 + 0.957405i \(0.406761\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −12.5563 −0.642439
\(383\) −12.2426 −0.625570 −0.312785 0.949824i \(-0.601262\pi\)
−0.312785 + 0.949824i \(0.601262\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.343146 −0.0174657
\(387\) 0 0
\(388\) −0.343146 −0.0174206
\(389\) −22.9289 −1.16254 −0.581272 0.813710i \(-0.697445\pi\)
−0.581272 + 0.813710i \(0.697445\pi\)
\(390\) 0 0
\(391\) 0.757359 0.0383013
\(392\) 12.4853 0.630602
\(393\) 0 0
\(394\) 11.7574 0.592327
\(395\) 0 0
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) 9.24264 0.463292
\(399\) 0 0
\(400\) 0 0
\(401\) 25.4142 1.26913 0.634563 0.772871i \(-0.281180\pi\)
0.634563 + 0.772871i \(0.281180\pi\)
\(402\) 0 0
\(403\) 36.3848 1.81245
\(404\) −13.0711 −0.650310
\(405\) 0 0
\(406\) 0.757359 0.0375871
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) 25.2132 1.24671 0.623356 0.781938i \(-0.285769\pi\)
0.623356 + 0.781938i \(0.285769\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.24264 0.209020
\(413\) −30.4558 −1.49863
\(414\) 0 0
\(415\) 0 0
\(416\) 5.82843 0.285762
\(417\) 0 0
\(418\) −1.41421 −0.0691714
\(419\) −19.4142 −0.948446 −0.474223 0.880405i \(-0.657271\pi\)
−0.474223 + 0.880405i \(0.657271\pi\)
\(420\) 0 0
\(421\) 19.4853 0.949655 0.474827 0.880079i \(-0.342511\pi\)
0.474827 + 0.880079i \(0.342511\pi\)
\(422\) −19.7279 −0.960340
\(423\) 0 0
\(424\) −5.48528 −0.266389
\(425\) 0 0
\(426\) 0 0
\(427\) 62.8701 3.04250
\(428\) −19.7279 −0.953585
\(429\) 0 0
\(430\) 0 0
\(431\) 6.38478 0.307544 0.153772 0.988106i \(-0.450858\pi\)
0.153772 + 0.988106i \(0.450858\pi\)
\(432\) 0 0
\(433\) −9.55635 −0.459249 −0.229624 0.973279i \(-0.573750\pi\)
−0.229624 + 0.973279i \(0.573750\pi\)
\(434\) 27.5563 1.32275
\(435\) 0 0
\(436\) −17.9706 −0.860634
\(437\) 0.757359 0.0362294
\(438\) 0 0
\(439\) −5.75736 −0.274784 −0.137392 0.990517i \(-0.543872\pi\)
−0.137392 + 0.990517i \(0.543872\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.82843 −0.277230
\(443\) −4.24264 −0.201574 −0.100787 0.994908i \(-0.532136\pi\)
−0.100787 + 0.994908i \(0.532136\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −15.1716 −0.718395
\(447\) 0 0
\(448\) 4.41421 0.208552
\(449\) −17.3137 −0.817084 −0.408542 0.912739i \(-0.633963\pi\)
−0.408542 + 0.912739i \(0.633963\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 10.2426 0.481773
\(453\) 0 0
\(454\) 16.7574 0.786462
\(455\) 0 0
\(456\) 0 0
\(457\) −3.00000 −0.140334 −0.0701670 0.997535i \(-0.522353\pi\)
−0.0701670 + 0.997535i \(0.522353\pi\)
\(458\) 14.9706 0.699528
\(459\) 0 0
\(460\) 0 0
\(461\) 3.55635 0.165636 0.0828178 0.996565i \(-0.473608\pi\)
0.0828178 + 0.996565i \(0.473608\pi\)
\(462\) 0 0
\(463\) 14.1421 0.657241 0.328620 0.944462i \(-0.393416\pi\)
0.328620 + 0.944462i \(0.393416\pi\)
\(464\) 0.171573 0.00796507
\(465\) 0 0
\(466\) 24.9706 1.15674
\(467\) −0.727922 −0.0336842 −0.0168421 0.999858i \(-0.505361\pi\)
−0.0168421 + 0.999858i \(0.505361\pi\)
\(468\) 0 0
\(469\) 21.0000 0.969690
\(470\) 0 0
\(471\) 0 0
\(472\) −6.89949 −0.317575
\(473\) 2.48528 0.114273
\(474\) 0 0
\(475\) 0 0
\(476\) −4.41421 −0.202325
\(477\) 0 0
\(478\) 6.89949 0.315576
\(479\) 31.1127 1.42158 0.710788 0.703407i \(-0.248339\pi\)
0.710788 + 0.703407i \(0.248339\pi\)
\(480\) 0 0
\(481\) −49.4558 −2.25499
\(482\) 8.97056 0.408598
\(483\) 0 0
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) 37.7990 1.71284 0.856418 0.516283i \(-0.172685\pi\)
0.856418 + 0.516283i \(0.172685\pi\)
\(488\) 14.2426 0.644734
\(489\) 0 0
\(490\) 0 0
\(491\) 33.5563 1.51438 0.757188 0.653197i \(-0.226572\pi\)
0.757188 + 0.653197i \(0.226572\pi\)
\(492\) 0 0
\(493\) −0.171573 −0.00772725
\(494\) −5.82843 −0.262233
\(495\) 0 0
\(496\) 6.24264 0.280303
\(497\) 59.2132 2.65608
\(498\) 0 0
\(499\) −25.7574 −1.15306 −0.576529 0.817077i \(-0.695593\pi\)
−0.576529 + 0.817077i \(0.695593\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.55635 −0.158728
\(503\) −14.2721 −0.636361 −0.318180 0.948030i \(-0.603072\pi\)
−0.318180 + 0.948030i \(0.603072\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.07107 −0.0476148
\(507\) 0 0
\(508\) −2.48528 −0.110267
\(509\) −28.9706 −1.28410 −0.642049 0.766664i \(-0.721915\pi\)
−0.642049 + 0.766664i \(0.721915\pi\)
\(510\) 0 0
\(511\) 50.6985 2.24277
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −20.7279 −0.914269
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −37.4558 −1.64572
\(519\) 0 0
\(520\) 0 0
\(521\) −23.3137 −1.02139 −0.510696 0.859761i \(-0.670612\pi\)
−0.510696 + 0.859761i \(0.670612\pi\)
\(522\) 0 0
\(523\) −2.27208 −0.0993510 −0.0496755 0.998765i \(-0.515819\pi\)
−0.0496755 + 0.998765i \(0.515819\pi\)
\(524\) 16.9706 0.741362
\(525\) 0 0
\(526\) 26.9706 1.17597
\(527\) −6.24264 −0.271934
\(528\) 0 0
\(529\) −22.4264 −0.975061
\(530\) 0 0
\(531\) 0 0
\(532\) −4.41421 −0.191380
\(533\) 24.7279 1.07109
\(534\) 0 0
\(535\) 0 0
\(536\) 4.75736 0.205487
\(537\) 0 0
\(538\) 16.6274 0.716859
\(539\) 17.6569 0.760535
\(540\) 0 0
\(541\) 9.75736 0.419502 0.209751 0.977755i \(-0.432735\pi\)
0.209751 + 0.977755i \(0.432735\pi\)
\(542\) −27.2426 −1.17017
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) 0 0
\(547\) −17.3137 −0.740281 −0.370140 0.928976i \(-0.620690\pi\)
−0.370140 + 0.928976i \(0.620690\pi\)
\(548\) −13.0000 −0.555332
\(549\) 0 0
\(550\) 0 0
\(551\) −0.171573 −0.00730925
\(552\) 0 0
\(553\) −28.6274 −1.21736
\(554\) 0 0
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) 10.2426 0.433218
\(560\) 0 0
\(561\) 0 0
\(562\) −4.24264 −0.178965
\(563\) −4.97056 −0.209484 −0.104742 0.994499i \(-0.533402\pi\)
−0.104742 + 0.994499i \(0.533402\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.1421 1.35103
\(567\) 0 0
\(568\) 13.4142 0.562848
\(569\) 28.2843 1.18574 0.592869 0.805299i \(-0.297995\pi\)
0.592869 + 0.805299i \(0.297995\pi\)
\(570\) 0 0
\(571\) −2.24264 −0.0938516 −0.0469258 0.998898i \(-0.514942\pi\)
−0.0469258 + 0.998898i \(0.514942\pi\)
\(572\) 8.24264 0.344642
\(573\) 0 0
\(574\) 18.7279 0.781688
\(575\) 0 0
\(576\) 0 0
\(577\) −20.3137 −0.845671 −0.422835 0.906207i \(-0.638965\pi\)
−0.422835 + 0.906207i \(0.638965\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 0 0
\(581\) −63.9411 −2.65272
\(582\) 0 0
\(583\) −7.75736 −0.321277
\(584\) 11.4853 0.475264
\(585\) 0 0
\(586\) 5.48528 0.226595
\(587\) −30.2426 −1.24825 −0.624124 0.781326i \(-0.714544\pi\)
−0.624124 + 0.781326i \(0.714544\pi\)
\(588\) 0 0
\(589\) −6.24264 −0.257224
\(590\) 0 0
\(591\) 0 0
\(592\) −8.48528 −0.348743
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17.6569 −0.723253
\(597\) 0 0
\(598\) −4.41421 −0.180511
\(599\) 15.2132 0.621595 0.310797 0.950476i \(-0.399404\pi\)
0.310797 + 0.950476i \(0.399404\pi\)
\(600\) 0 0
\(601\) −20.2426 −0.825715 −0.412857 0.910796i \(-0.635469\pi\)
−0.412857 + 0.910796i \(0.635469\pi\)
\(602\) 7.75736 0.316166
\(603\) 0 0
\(604\) −10.4853 −0.426640
\(605\) 0 0
\(606\) 0 0
\(607\) 3.17157 0.128730 0.0643651 0.997926i \(-0.479498\pi\)
0.0643651 + 0.997926i \(0.479498\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −11.6985 −0.472497 −0.236249 0.971693i \(-0.575918\pi\)
−0.236249 + 0.971693i \(0.575918\pi\)
\(614\) −17.6569 −0.712573
\(615\) 0 0
\(616\) 6.24264 0.251523
\(617\) 4.48528 0.180571 0.0902853 0.995916i \(-0.471222\pi\)
0.0902853 + 0.995916i \(0.471222\pi\)
\(618\) 0 0
\(619\) 7.75736 0.311795 0.155897 0.987773i \(-0.450173\pi\)
0.155897 + 0.987773i \(0.450173\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 4.75736 0.190753
\(623\) −31.2132 −1.25053
\(624\) 0 0
\(625\) 0 0
\(626\) −7.97056 −0.318568
\(627\) 0 0
\(628\) 11.6569 0.465159
\(629\) 8.48528 0.338330
\(630\) 0 0
\(631\) −38.9706 −1.55139 −0.775697 0.631106i \(-0.782601\pi\)
−0.775697 + 0.631106i \(0.782601\pi\)
\(632\) −6.48528 −0.257971
\(633\) 0 0
\(634\) 9.48528 0.376709
\(635\) 0 0
\(636\) 0 0
\(637\) 72.7696 2.88323
\(638\) 0.242641 0.00960624
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0416 0.712602 0.356301 0.934371i \(-0.384038\pi\)
0.356301 + 0.934371i \(0.384038\pi\)
\(642\) 0 0
\(643\) −14.4853 −0.571244 −0.285622 0.958342i \(-0.592200\pi\)
−0.285622 + 0.958342i \(0.592200\pi\)
\(644\) −3.34315 −0.131738
\(645\) 0 0
\(646\) 1.00000 0.0393445
\(647\) 18.7574 0.737428 0.368714 0.929543i \(-0.379798\pi\)
0.368714 + 0.929543i \(0.379798\pi\)
\(648\) 0 0
\(649\) −9.75736 −0.383010
\(650\) 0 0
\(651\) 0 0
\(652\) −10.2426 −0.401133
\(653\) 28.9706 1.13371 0.566853 0.823819i \(-0.308161\pi\)
0.566853 + 0.823819i \(0.308161\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.24264 0.165647
\(657\) 0 0
\(658\) 0 0
\(659\) −18.8995 −0.736220 −0.368110 0.929782i \(-0.619995\pi\)
−0.368110 + 0.929782i \(0.619995\pi\)
\(660\) 0 0
\(661\) −18.4558 −0.717849 −0.358925 0.933367i \(-0.616856\pi\)
−0.358925 + 0.933367i \(0.616856\pi\)
\(662\) −19.2426 −0.747886
\(663\) 0 0
\(664\) −14.4853 −0.562138
\(665\) 0 0
\(666\) 0 0
\(667\) −0.129942 −0.00503139
\(668\) 18.2426 0.705829
\(669\) 0 0
\(670\) 0 0
\(671\) 20.1421 0.777579
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 14.1005 0.543131
\(675\) 0 0
\(676\) 20.9706 0.806560
\(677\) 40.9411 1.57350 0.786748 0.617275i \(-0.211763\pi\)
0.786748 + 0.617275i \(0.211763\pi\)
\(678\) 0 0
\(679\) −1.51472 −0.0581296
\(680\) 0 0
\(681\) 0 0
\(682\) 8.82843 0.338058
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 24.2132 0.924464
\(687\) 0 0
\(688\) 1.75736 0.0669987
\(689\) −31.9706 −1.21798
\(690\) 0 0
\(691\) 22.4853 0.855380 0.427690 0.903925i \(-0.359327\pi\)
0.427690 + 0.903925i \(0.359327\pi\)
\(692\) −0.485281 −0.0184476
\(693\) 0 0
\(694\) 5.51472 0.209336
\(695\) 0 0
\(696\) 0 0
\(697\) −4.24264 −0.160701
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 0 0
\(701\) −16.9706 −0.640969 −0.320485 0.947254i \(-0.603846\pi\)
−0.320485 + 0.947254i \(0.603846\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 1.41421 0.0533002
\(705\) 0 0
\(706\) 2.51472 0.0946427
\(707\) −57.6985 −2.16997
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7.07107 −0.264999
\(713\) −4.72792 −0.177062
\(714\) 0 0
\(715\) 0 0
\(716\) 11.6569 0.435637
\(717\) 0 0
\(718\) 22.7574 0.849297
\(719\) 5.10051 0.190217 0.0951084 0.995467i \(-0.469680\pi\)
0.0951084 + 0.995467i \(0.469680\pi\)
\(720\) 0 0
\(721\) 18.7279 0.697464
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −8.48528 −0.315353
\(725\) 0 0
\(726\) 0 0
\(727\) 9.72792 0.360789 0.180394 0.983594i \(-0.442263\pi\)
0.180394 + 0.983594i \(0.442263\pi\)
\(728\) 25.7279 0.953540
\(729\) 0 0
\(730\) 0 0
\(731\) −1.75736 −0.0649983
\(732\) 0 0
\(733\) 12.0000 0.443230 0.221615 0.975134i \(-0.428867\pi\)
0.221615 + 0.975134i \(0.428867\pi\)
\(734\) −25.4558 −0.939592
\(735\) 0 0
\(736\) −0.757359 −0.0279166
\(737\) 6.72792 0.247826
\(738\) 0 0
\(739\) 1.27208 0.0467941 0.0233971 0.999726i \(-0.492552\pi\)
0.0233971 + 0.999726i \(0.492552\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −24.2132 −0.888895
\(743\) −6.72792 −0.246824 −0.123412 0.992356i \(-0.539384\pi\)
−0.123412 + 0.992356i \(0.539384\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 9.00000 0.329513
\(747\) 0 0
\(748\) −1.41421 −0.0517088
\(749\) −87.0833 −3.18195
\(750\) 0 0
\(751\) −26.7279 −0.975316 −0.487658 0.873035i \(-0.662149\pi\)
−0.487658 + 0.873035i \(0.662149\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) 0 0
\(757\) 12.3431 0.448619 0.224310 0.974518i \(-0.427987\pi\)
0.224310 + 0.974518i \(0.427987\pi\)
\(758\) 11.2426 0.408351
\(759\) 0 0
\(760\) 0 0
\(761\) −43.9706 −1.59393 −0.796966 0.604024i \(-0.793563\pi\)
−0.796966 + 0.604024i \(0.793563\pi\)
\(762\) 0 0
\(763\) −79.3259 −2.87179
\(764\) −12.5563 −0.454273
\(765\) 0 0
\(766\) −12.2426 −0.442345
\(767\) −40.2132 −1.45201
\(768\) 0 0
\(769\) −36.4558 −1.31463 −0.657316 0.753615i \(-0.728308\pi\)
−0.657316 + 0.753615i \(0.728308\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.343146 −0.0123501
\(773\) −13.9706 −0.502486 −0.251243 0.967924i \(-0.580839\pi\)
−0.251243 + 0.967924i \(0.580839\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.343146 −0.0123182
\(777\) 0 0
\(778\) −22.9289 −0.822042
\(779\) −4.24264 −0.152008
\(780\) 0 0
\(781\) 18.9706 0.678820
\(782\) 0.757359 0.0270831
\(783\) 0 0
\(784\) 12.4853 0.445903
\(785\) 0 0
\(786\) 0 0
\(787\) −41.1838 −1.46804 −0.734021 0.679126i \(-0.762359\pi\)
−0.734021 + 0.679126i \(0.762359\pi\)
\(788\) 11.7574 0.418839
\(789\) 0 0
\(790\) 0 0
\(791\) 45.2132 1.60760
\(792\) 0 0
\(793\) 83.0122 2.94785
\(794\) 24.0000 0.851728
\(795\) 0 0
\(796\) 9.24264 0.327597
\(797\) 47.4853 1.68201 0.841007 0.541023i \(-0.181963\pi\)
0.841007 + 0.541023i \(0.181963\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 25.4142 0.897407
\(803\) 16.2426 0.573190
\(804\) 0 0
\(805\) 0 0
\(806\) 36.3848 1.28160
\(807\) 0 0
\(808\) −13.0711 −0.459839
\(809\) 28.7990 1.01252 0.506259 0.862381i \(-0.331028\pi\)
0.506259 + 0.862381i \(0.331028\pi\)
\(810\) 0 0
\(811\) 52.6985 1.85049 0.925247 0.379365i \(-0.123858\pi\)
0.925247 + 0.379365i \(0.123858\pi\)
\(812\) 0.757359 0.0265781
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) −1.75736 −0.0614822
\(818\) 25.2132 0.881559
\(819\) 0 0
\(820\) 0 0
\(821\) −6.34315 −0.221377 −0.110689 0.993855i \(-0.535306\pi\)
−0.110689 + 0.993855i \(0.535306\pi\)
\(822\) 0 0
\(823\) −15.7279 −0.548241 −0.274120 0.961695i \(-0.588387\pi\)
−0.274120 + 0.961695i \(0.588387\pi\)
\(824\) 4.24264 0.147799
\(825\) 0 0
\(826\) −30.4558 −1.05969
\(827\) 20.6985 0.719757 0.359878 0.932999i \(-0.382818\pi\)
0.359878 + 0.932999i \(0.382818\pi\)
\(828\) 0 0
\(829\) 10.4558 0.363146 0.181573 0.983377i \(-0.441881\pi\)
0.181573 + 0.983377i \(0.441881\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.82843 0.202064
\(833\) −12.4853 −0.432589
\(834\) 0 0
\(835\) 0 0
\(836\) −1.41421 −0.0489116
\(837\) 0 0
\(838\) −19.4142 −0.670653
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.9706 −0.998985
\(842\) 19.4853 0.671507
\(843\) 0 0
\(844\) −19.7279 −0.679063
\(845\) 0 0
\(846\) 0 0
\(847\) −39.7279 −1.36507
\(848\) −5.48528 −0.188365
\(849\) 0 0
\(850\) 0 0
\(851\) 6.42641 0.220294
\(852\) 0 0
\(853\) −27.9411 −0.956686 −0.478343 0.878173i \(-0.658762\pi\)
−0.478343 + 0.878173i \(0.658762\pi\)
\(854\) 62.8701 2.15137
\(855\) 0 0
\(856\) −19.7279 −0.674286
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 4.72792 0.161315 0.0806573 0.996742i \(-0.474298\pi\)
0.0806573 + 0.996742i \(0.474298\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.38478 0.217466
\(863\) −0.727922 −0.0247788 −0.0123894 0.999923i \(-0.503944\pi\)
−0.0123894 + 0.999923i \(0.503944\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −9.55635 −0.324738
\(867\) 0 0
\(868\) 27.5563 0.935323
\(869\) −9.17157 −0.311124
\(870\) 0 0
\(871\) 27.7279 0.939525
\(872\) −17.9706 −0.608560
\(873\) 0 0
\(874\) 0.757359 0.0256181
\(875\) 0 0
\(876\) 0 0
\(877\) −18.8579 −0.636785 −0.318392 0.947959i \(-0.603143\pi\)
−0.318392 + 0.947959i \(0.603143\pi\)
\(878\) −5.75736 −0.194301
\(879\) 0 0
\(880\) 0 0
\(881\) 39.5980 1.33409 0.667045 0.745018i \(-0.267559\pi\)
0.667045 + 0.745018i \(0.267559\pi\)
\(882\) 0 0
\(883\) 14.4853 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(884\) −5.82843 −0.196031
\(885\) 0 0
\(886\) −4.24264 −0.142534
\(887\) 45.2132 1.51811 0.759055 0.651026i \(-0.225661\pi\)
0.759055 + 0.651026i \(0.225661\pi\)
\(888\) 0 0
\(889\) −10.9706 −0.367941
\(890\) 0 0
\(891\) 0 0
\(892\) −15.1716 −0.507982
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 4.41421 0.147469
\(897\) 0 0
\(898\) −17.3137 −0.577766
\(899\) 1.07107 0.0357221
\(900\) 0 0
\(901\) 5.48528 0.182741
\(902\) 6.00000 0.199778
\(903\) 0 0
\(904\) 10.2426 0.340665
\(905\) 0 0
\(906\) 0 0
\(907\) 38.6985 1.28496 0.642481 0.766302i \(-0.277905\pi\)
0.642481 + 0.766302i \(0.277905\pi\)
\(908\) 16.7574 0.556113
\(909\) 0 0
\(910\) 0 0
\(911\) −40.2843 −1.33468 −0.667339 0.744754i \(-0.732567\pi\)
−0.667339 + 0.744754i \(0.732567\pi\)
\(912\) 0 0
\(913\) −20.4853 −0.677964
\(914\) −3.00000 −0.0992312
\(915\) 0 0
\(916\) 14.9706 0.494641
\(917\) 74.9117 2.47380
\(918\) 0 0
\(919\) −6.21320 −0.204955 −0.102477 0.994735i \(-0.532677\pi\)
−0.102477 + 0.994735i \(0.532677\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.55635 0.117122
\(923\) 78.1838 2.57345
\(924\) 0 0
\(925\) 0 0
\(926\) 14.1421 0.464739
\(927\) 0 0
\(928\) 0.171573 0.00563216
\(929\) 12.1716 0.399336 0.199668 0.979864i \(-0.436014\pi\)
0.199668 + 0.979864i \(0.436014\pi\)
\(930\) 0 0
\(931\) −12.4853 −0.409189
\(932\) 24.9706 0.817938
\(933\) 0 0
\(934\) −0.727922 −0.0238183
\(935\) 0 0
\(936\) 0 0
\(937\) 27.0000 0.882052 0.441026 0.897494i \(-0.354615\pi\)
0.441026 + 0.897494i \(0.354615\pi\)
\(938\) 21.0000 0.685674
\(939\) 0 0
\(940\) 0 0
\(941\) −53.1421 −1.73238 −0.866192 0.499711i \(-0.833439\pi\)
−0.866192 + 0.499711i \(0.833439\pi\)
\(942\) 0 0
\(943\) −3.21320 −0.104636
\(944\) −6.89949 −0.224559
\(945\) 0 0
\(946\) 2.48528 0.0808035
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 66.9411 2.17300
\(950\) 0 0
\(951\) 0 0
\(952\) −4.41421 −0.143065
\(953\) −18.7279 −0.606657 −0.303328 0.952886i \(-0.598098\pi\)
−0.303328 + 0.952886i \(0.598098\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.89949 0.223146
\(957\) 0 0
\(958\) 31.1127 1.00521
\(959\) −57.3848 −1.85305
\(960\) 0 0
\(961\) 7.97056 0.257115
\(962\) −49.4558 −1.59452
\(963\) 0 0
\(964\) 8.97056 0.288922
\(965\) 0 0
\(966\) 0 0
\(967\) 43.1127 1.38641 0.693205 0.720740i \(-0.256198\pi\)
0.693205 + 0.720740i \(0.256198\pi\)
\(968\) −9.00000 −0.289271
\(969\) 0 0
\(970\) 0 0
\(971\) −41.6569 −1.33683 −0.668416 0.743788i \(-0.733027\pi\)
−0.668416 + 0.743788i \(0.733027\pi\)
\(972\) 0 0
\(973\) −52.9706 −1.69816
\(974\) 37.7990 1.21116
\(975\) 0 0
\(976\) 14.2426 0.455896
\(977\) −0.242641 −0.00776276 −0.00388138 0.999992i \(-0.501235\pi\)
−0.00388138 + 0.999992i \(0.501235\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 33.5563 1.07083
\(983\) 51.4558 1.64119 0.820593 0.571513i \(-0.193643\pi\)
0.820593 + 0.571513i \(0.193643\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.171573 −0.00546399
\(987\) 0 0
\(988\) −5.82843 −0.185427
\(989\) −1.33095 −0.0423218
\(990\) 0 0
\(991\) 13.7574 0.437017 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(992\) 6.24264 0.198204
\(993\) 0 0
\(994\) 59.2132 1.87813
\(995\) 0 0
\(996\) 0 0
\(997\) 48.7279 1.54323 0.771614 0.636091i \(-0.219450\pi\)
0.771614 + 0.636091i \(0.219450\pi\)
\(998\) −25.7574 −0.815335
\(999\) 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 8550.2.a.cb.1.2 2
3.2 odd 2 950.2.a.f.1.1 2
5.2 odd 4 1710.2.d.c.1369.3 4
5.3 odd 4 1710.2.d.c.1369.1 4
5.4 even 2 8550.2.a.bn.1.1 2
12.11 even 2 7600.2.a.v.1.2 2
15.2 even 4 190.2.b.a.39.2 4
15.8 even 4 190.2.b.a.39.3 yes 4
15.14 odd 2 950.2.a.g.1.2 2
60.23 odd 4 1520.2.d.e.609.3 4
60.47 odd 4 1520.2.d.e.609.2 4
60.59 even 2 7600.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.a.39.2 4 15.2 even 4
190.2.b.a.39.3 yes 4 15.8 even 4
950.2.a.f.1.1 2 3.2 odd 2
950.2.a.g.1.2 2 15.14 odd 2
1520.2.d.e.609.2 4 60.47 odd 4
1520.2.d.e.609.3 4 60.23 odd 4
1710.2.d.c.1369.1 4 5.3 odd 4
1710.2.d.c.1369.3 4 5.2 odd 4
7600.2.a.v.1.2 2 12.11 even 2
7600.2.a.bg.1.1 2 60.59 even 2
8550.2.a.bn.1.1 2 5.4 even 2
8550.2.a.cb.1.2 2 1.1 even 1 trivial