Properties

Label 4536.2.a.x.1.3
Level $4536$
Weight $2$
Character 4536.1
Self dual yes
Analytic conductor $36.220$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4536,2,Mod(1,4536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4536.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4536.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: \(36.2201423569\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.519120\) of defining polynomial
Character \(\chi\) \(=\) 4536.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.936586 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+0.936586 q^{5} +1.00000 q^{7} +4.97483 q^{11} +1.24431 q^{13} +5.22446 q^{17} +5.18622 q^{19} -2.00532 q^{23} -4.12281 q^{25} +6.87849 q^{29} -5.73051 q^{31} +0.936586 q^{35} +9.73051 q^{37} -11.4741 q^{41} +9.60369 q^{43} -1.96951 q^{47} +1.00000 q^{49} -7.63418 q^{53} +4.65935 q^{55} +4.87849 q^{59} -3.04356 q^{61} +1.16541 q^{65} -1.14798 q^{67} -8.83749 q^{71} +6.10698 q^{73} +4.97483 q^{77} -12.1083 q^{79} -0.862663 q^{83} +4.89316 q^{85} -10.8480 q^{89} +1.24431 q^{91} +4.85734 q^{95} +7.57043 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 4 q^{7} + 6 q^{11} + 3 q^{13} - 8 q^{17} - 2 q^{19} + 5 q^{23} + 14 q^{25} - q^{29} - 11 q^{31} - 4 q^{35} + 27 q^{37} - 2 q^{41} + 11 q^{43} - 7 q^{47} + 4 q^{49} - 4 q^{53} + 6 q^{55} - 9 q^{59} + 7 q^{61} + 9 q^{65} + 12 q^{67} - 12 q^{71} + 13 q^{73} + 6 q^{77} + 22 q^{79} + 6 q^{83} + 11 q^{85} - 14 q^{89} + 3 q^{91} + 23 q^{95} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.936586 0.418854 0.209427 0.977824i \(-0.432840\pi\)
0.209427 + 0.977824i \(0.432840\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.97483 1.49997 0.749983 0.661457i \(-0.230061\pi\)
0.749983 + 0.661457i \(0.230061\pi\)
\(12\) 0 0
\(13\) 1.24431 0.345110 0.172555 0.985000i \(-0.444798\pi\)
0.172555 + 0.985000i \(0.444798\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.22446 1.26712 0.633559 0.773694i \(-0.281593\pi\)
0.633559 + 0.773694i \(0.281593\pi\)
\(18\) 0 0
\(19\) 5.18622 1.18980 0.594900 0.803800i \(-0.297192\pi\)
0.594900 + 0.803800i \(0.297192\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00532 −0.418138 −0.209069 0.977901i \(-0.567043\pi\)
−0.209069 + 0.977901i \(0.567043\pi\)
\(24\) 0 0
\(25\) −4.12281 −0.824561
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.87849 1.27730 0.638652 0.769496i \(-0.279492\pi\)
0.638652 + 0.769496i \(0.279492\pi\)
\(30\) 0 0
\(31\) −5.73051 −1.02923 −0.514615 0.857421i \(-0.672065\pi\)
−0.514615 + 0.857421i \(0.672065\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.936586 0.158312
\(36\) 0 0
\(37\) 9.73051 1.59969 0.799843 0.600209i \(-0.204916\pi\)
0.799843 + 0.600209i \(0.204916\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.4741 −1.79195 −0.895976 0.444102i \(-0.853523\pi\)
−0.895976 + 0.444102i \(0.853523\pi\)
\(42\) 0 0
\(43\) 9.60369 1.46455 0.732274 0.681010i \(-0.238459\pi\)
0.732274 + 0.681010i \(0.238459\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.96951 −0.287282 −0.143641 0.989630i \(-0.545881\pi\)
−0.143641 + 0.989630i \(0.545881\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.63418 −1.04864 −0.524318 0.851523i \(-0.675680\pi\)
−0.524318 + 0.851523i \(0.675680\pi\)
\(54\) 0 0
\(55\) 4.65935 0.628267
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.87849 0.635126 0.317563 0.948237i \(-0.397136\pi\)
0.317563 + 0.948237i \(0.397136\pi\)
\(60\) 0 0
\(61\) −3.04356 −0.389688 −0.194844 0.980834i \(-0.562420\pi\)
−0.194844 + 0.980834i \(0.562420\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.16541 0.144551
\(66\) 0 0
\(67\) −1.14798 −0.140248 −0.0701240 0.997538i \(-0.522340\pi\)
−0.0701240 + 0.997538i \(0.522340\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.83749 −1.04882 −0.524409 0.851467i \(-0.675714\pi\)
−0.524409 + 0.851467i \(0.675714\pi\)
\(72\) 0 0
\(73\) 6.10698 0.714768 0.357384 0.933958i \(-0.383669\pi\)
0.357384 + 0.933958i \(0.383669\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.97483 0.566934
\(78\) 0 0
\(79\) −12.1083 −1.36229 −0.681144 0.732150i \(-0.738517\pi\)
−0.681144 + 0.732150i \(0.738517\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.862663 −0.0946896 −0.0473448 0.998879i \(-0.515076\pi\)
−0.0473448 + 0.998879i \(0.515076\pi\)
\(84\) 0 0
\(85\) 4.89316 0.530737
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.8480 −1.14989 −0.574943 0.818194i \(-0.694976\pi\)
−0.574943 + 0.818194i \(0.694976\pi\)
\(90\) 0 0
\(91\) 1.24431 0.130439
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.85734 0.498353
\(96\) 0 0
\(97\) 7.57043 0.768661 0.384330 0.923196i \(-0.374432\pi\)
0.384330 + 0.923196i \(0.374432\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.05939 0.503428 0.251714 0.967802i \(-0.419006\pi\)
0.251714 + 0.967802i \(0.419006\pi\)
\(102\) 0 0
\(103\) 0.238992 0.0235485 0.0117743 0.999931i \(-0.496252\pi\)
0.0117743 + 0.999931i \(0.496252\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.25496 0.894710 0.447355 0.894356i \(-0.352366\pi\)
0.447355 + 0.894356i \(0.352366\pi\)
\(108\) 0 0
\(109\) −10.5453 −1.01005 −0.505026 0.863104i \(-0.668517\pi\)
−0.505026 + 0.863104i \(0.668517\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.1281 −0.952774 −0.476387 0.879236i \(-0.658054\pi\)
−0.476387 + 0.879236i \(0.658054\pi\)
\(114\) 0 0
\(115\) −1.87816 −0.175139
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.22446 0.478926
\(120\) 0 0
\(121\) 13.7489 1.24990
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.54429 −0.764225
\(126\) 0 0
\(127\) 6.52720 0.579196 0.289598 0.957148i \(-0.406478\pi\)
0.289598 + 0.957148i \(0.406478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.53495 −0.308850 −0.154425 0.988005i \(-0.549353\pi\)
−0.154425 + 0.988005i \(0.549353\pi\)
\(132\) 0 0
\(133\) 5.18622 0.449702
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.97206 0.424792 0.212396 0.977184i \(-0.431873\pi\)
0.212396 + 0.977184i \(0.431873\pi\)
\(138\) 0 0
\(139\) 9.75133 0.827097 0.413548 0.910482i \(-0.364289\pi\)
0.413548 + 0.910482i \(0.364289\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.19024 0.517654
\(144\) 0 0
\(145\) 6.44230 0.535004
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.7822 −1.04716 −0.523578 0.851978i \(-0.675403\pi\)
−0.523578 + 0.851978i \(0.675403\pi\)
\(150\) 0 0
\(151\) 22.9563 1.86816 0.934078 0.357070i \(-0.116224\pi\)
0.934078 + 0.357070i \(0.116224\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.36712 −0.431097
\(156\) 0 0
\(157\) 16.5098 1.31762 0.658812 0.752308i \(-0.271059\pi\)
0.658812 + 0.752308i \(0.271059\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00532 −0.158041
\(162\) 0 0
\(163\) 17.2245 1.34912 0.674562 0.738218i \(-0.264333\pi\)
0.674562 + 0.738218i \(0.264333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.1281 1.01589 0.507943 0.861391i \(-0.330406\pi\)
0.507943 + 0.861391i \(0.330406\pi\)
\(168\) 0 0
\(169\) −11.4517 −0.880899
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −18.9882 −1.44365 −0.721824 0.692076i \(-0.756696\pi\)
−0.721824 + 0.692076i \(0.756696\pi\)
\(174\) 0 0
\(175\) −4.12281 −0.311655
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.61835 0.569422 0.284711 0.958613i \(-0.408102\pi\)
0.284711 + 0.958613i \(0.408102\pi\)
\(180\) 0 0
\(181\) −9.27737 −0.689581 −0.344791 0.938680i \(-0.612050\pi\)
−0.344791 + 0.938680i \(0.612050\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.11346 0.670035
\(186\) 0 0
\(187\) 25.9908 1.90063
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.60239 0.694804 0.347402 0.937716i \(-0.387064\pi\)
0.347402 + 0.937716i \(0.387064\pi\)
\(192\) 0 0
\(193\) −15.0831 −1.08571 −0.542853 0.839828i \(-0.682656\pi\)
−0.542853 + 0.839828i \(0.682656\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.0712 −0.788788 −0.394394 0.918942i \(-0.629045\pi\)
−0.394394 + 0.918942i \(0.629045\pi\)
\(198\) 0 0
\(199\) 18.3368 1.29986 0.649929 0.759995i \(-0.274799\pi\)
0.649929 + 0.759995i \(0.274799\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.87849 0.482776
\(204\) 0 0
\(205\) −10.7465 −0.750567
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 25.8005 1.78466
\(210\) 0 0
\(211\) 2.61822 0.180245 0.0901227 0.995931i \(-0.471274\pi\)
0.0901227 + 0.995931i \(0.471274\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.99468 0.613432
\(216\) 0 0
\(217\) −5.73051 −0.389013
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.50087 0.437296
\(222\) 0 0
\(223\) −25.0884 −1.68005 −0.840023 0.542551i \(-0.817458\pi\)
−0.840023 + 0.542551i \(0.817458\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −25.5562 −1.69623 −0.848113 0.529815i \(-0.822262\pi\)
−0.848113 + 0.529815i \(0.822262\pi\)
\(228\) 0 0
\(229\) 5.47313 0.361675 0.180837 0.983513i \(-0.442119\pi\)
0.180837 + 0.983513i \(0.442119\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.6774 0.699500 0.349750 0.936843i \(-0.386267\pi\)
0.349750 + 0.936843i \(0.386267\pi\)
\(234\) 0 0
\(235\) −1.84461 −0.120329
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) 20.3368 1.31001 0.655003 0.755626i \(-0.272667\pi\)
0.655003 + 0.755626i \(0.272667\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.936586 0.0598363
\(246\) 0 0
\(247\) 6.45328 0.410612
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.8151 1.25072 0.625358 0.780338i \(-0.284953\pi\)
0.625358 + 0.780338i \(0.284953\pi\)
\(252\) 0 0
\(253\) −9.97613 −0.627194
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.93901 −0.120952 −0.0604761 0.998170i \(-0.519262\pi\)
−0.0604761 + 0.998170i \(0.519262\pi\)
\(258\) 0 0
\(259\) 9.73051 0.604625
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.3554 −0.823526 −0.411763 0.911291i \(-0.635087\pi\)
−0.411763 + 0.911291i \(0.635087\pi\)
\(264\) 0 0
\(265\) −7.15007 −0.439225
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.15040 0.253055 0.126527 0.991963i \(-0.459617\pi\)
0.126527 + 0.991963i \(0.459617\pi\)
\(270\) 0 0
\(271\) −16.6050 −1.00868 −0.504341 0.863505i \(-0.668264\pi\)
−0.504341 + 0.863505i \(0.668264\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.5102 −1.23681
\(276\) 0 0
\(277\) 17.1083 1.02794 0.513968 0.857809i \(-0.328175\pi\)
0.513968 + 0.857809i \(0.328175\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.9801 −0.893641 −0.446820 0.894624i \(-0.647444\pi\)
−0.446820 + 0.894624i \(0.647444\pi\)
\(282\) 0 0
\(283\) −18.0116 −1.07068 −0.535339 0.844637i \(-0.679816\pi\)
−0.535339 + 0.844637i \(0.679816\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.4741 −0.677294
\(288\) 0 0
\(289\) 10.2950 0.605588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −27.6724 −1.61664 −0.808320 0.588743i \(-0.799623\pi\)
−0.808320 + 0.588743i \(0.799623\pi\)
\(294\) 0 0
\(295\) 4.56913 0.266025
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.49525 −0.144304
\(300\) 0 0
\(301\) 9.60369 0.553547
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.85056 −0.163222
\(306\) 0 0
\(307\) −6.10040 −0.348168 −0.174084 0.984731i \(-0.555696\pi\)
−0.174084 + 0.984731i \(0.555696\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.6553 1.51149 0.755743 0.654869i \(-0.227276\pi\)
0.755743 + 0.654869i \(0.227276\pi\)
\(312\) 0 0
\(313\) 24.1362 1.36426 0.682130 0.731231i \(-0.261054\pi\)
0.682130 + 0.731231i \(0.261054\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.6009 −0.988566 −0.494283 0.869301i \(-0.664569\pi\)
−0.494283 + 0.869301i \(0.664569\pi\)
\(318\) 0 0
\(319\) 34.2193 1.91591
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 27.0952 1.50762
\(324\) 0 0
\(325\) −5.13006 −0.284565
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.96951 −0.108582
\(330\) 0 0
\(331\) 24.2995 1.33562 0.667810 0.744332i \(-0.267232\pi\)
0.667810 + 0.744332i \(0.267232\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.07518 −0.0587435
\(336\) 0 0
\(337\) 22.9443 1.24986 0.624929 0.780682i \(-0.285128\pi\)
0.624929 + 0.780682i \(0.285128\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −28.5083 −1.54381
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.7386 −0.683844 −0.341922 0.939728i \(-0.611078\pi\)
−0.341922 + 0.939728i \(0.611078\pi\)
\(348\) 0 0
\(349\) −7.98936 −0.427660 −0.213830 0.976871i \(-0.568594\pi\)
−0.213830 + 0.976871i \(0.568594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.9563 −1.16861 −0.584307 0.811532i \(-0.698634\pi\)
−0.584307 + 0.811532i \(0.698634\pi\)
\(354\) 0 0
\(355\) −8.27707 −0.439301
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.4673 1.08022 0.540112 0.841593i \(-0.318382\pi\)
0.540112 + 0.841593i \(0.318382\pi\)
\(360\) 0 0
\(361\) 7.89688 0.415625
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.71971 0.299383
\(366\) 0 0
\(367\) −19.8835 −1.03791 −0.518955 0.854802i \(-0.673679\pi\)
−0.518955 + 0.854802i \(0.673679\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.63418 −0.396347
\(372\) 0 0
\(373\) 34.2876 1.77534 0.887672 0.460477i \(-0.152322\pi\)
0.887672 + 0.460477i \(0.152322\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.55900 0.440811
\(378\) 0 0
\(379\) 18.6248 0.956694 0.478347 0.878171i \(-0.341236\pi\)
0.478347 + 0.878171i \(0.341236\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.8720 1.67968 0.839842 0.542832i \(-0.182648\pi\)
0.839842 + 0.542832i \(0.182648\pi\)
\(384\) 0 0
\(385\) 4.65935 0.237463
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −21.6500 −1.09770 −0.548850 0.835921i \(-0.684934\pi\)
−0.548850 + 0.835921i \(0.684934\pi\)
\(390\) 0 0
\(391\) −10.4767 −0.529831
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.3404 −0.570600
\(396\) 0 0
\(397\) 16.2721 0.816673 0.408336 0.912832i \(-0.366109\pi\)
0.408336 + 0.912832i \(0.366109\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.14266 −0.456563 −0.228281 0.973595i \(-0.573311\pi\)
−0.228281 + 0.973595i \(0.573311\pi\)
\(402\) 0 0
\(403\) −7.13055 −0.355198
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 48.4076 2.39948
\(408\) 0 0
\(409\) −22.1953 −1.09749 −0.548743 0.835991i \(-0.684893\pi\)
−0.548743 + 0.835991i \(0.684893\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.87849 0.240055
\(414\) 0 0
\(415\) −0.807958 −0.0396611
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.9201 1.36399 0.681994 0.731358i \(-0.261113\pi\)
0.681994 + 0.731358i \(0.261113\pi\)
\(420\) 0 0
\(421\) 24.0883 1.17399 0.586996 0.809590i \(-0.300311\pi\)
0.586996 + 0.809590i \(0.300311\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −21.5394 −1.04482
\(426\) 0 0
\(427\) −3.04356 −0.147288
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.1623 −1.26020 −0.630098 0.776516i \(-0.716985\pi\)
−0.630098 + 0.776516i \(0.716985\pi\)
\(432\) 0 0
\(433\) −18.5630 −0.892082 −0.446041 0.895013i \(-0.647166\pi\)
−0.446041 + 0.895013i \(0.647166\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.4000 −0.497501
\(438\) 0 0
\(439\) 7.70790 0.367878 0.183939 0.982938i \(-0.441115\pi\)
0.183939 + 0.982938i \(0.441115\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.0408 1.56982 0.784909 0.619611i \(-0.212710\pi\)
0.784909 + 0.619611i \(0.212710\pi\)
\(444\) 0 0
\(445\) −10.1601 −0.481634
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.9283 0.940477 0.470238 0.882540i \(-0.344168\pi\)
0.470238 + 0.882540i \(0.344168\pi\)
\(450\) 0 0
\(451\) −57.0816 −2.68787
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.16541 0.0546351
\(456\) 0 0
\(457\) −33.8653 −1.58415 −0.792075 0.610424i \(-0.790999\pi\)
−0.792075 + 0.610424i \(0.790999\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14.5070 −0.675659 −0.337830 0.941207i \(-0.609693\pi\)
−0.337830 + 0.941207i \(0.609693\pi\)
\(462\) 0 0
\(463\) 25.0697 1.16509 0.582544 0.812799i \(-0.302057\pi\)
0.582544 + 0.812799i \(0.302057\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.102955 −0.00476419 −0.00238209 0.999997i \(-0.500758\pi\)
−0.00238209 + 0.999997i \(0.500758\pi\)
\(468\) 0 0
\(469\) −1.14798 −0.0530088
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 47.7767 2.19677
\(474\) 0 0
\(475\) −21.3818 −0.981064
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.25336 0.240032 0.120016 0.992772i \(-0.461705\pi\)
0.120016 + 0.992772i \(0.461705\pi\)
\(480\) 0 0
\(481\) 12.1078 0.552068
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.09036 0.321957
\(486\) 0 0
\(487\) −30.7319 −1.39260 −0.696299 0.717752i \(-0.745171\pi\)
−0.696299 + 0.717752i \(0.745171\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.71841 −0.212939 −0.106469 0.994316i \(-0.533955\pi\)
−0.106469 + 0.994316i \(0.533955\pi\)
\(492\) 0 0
\(493\) 35.9364 1.61850
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.83749 −0.396416
\(498\) 0 0
\(499\) 24.7132 1.10632 0.553158 0.833076i \(-0.313423\pi\)
0.553158 + 0.833076i \(0.313423\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.4083 −0.865371 −0.432686 0.901545i \(-0.642434\pi\)
−0.432686 + 0.901545i \(0.642434\pi\)
\(504\) 0 0
\(505\) 4.73856 0.210863
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.45701 0.241878 0.120939 0.992660i \(-0.461410\pi\)
0.120939 + 0.992660i \(0.461410\pi\)
\(510\) 0 0
\(511\) 6.10698 0.270157
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.223836 0.00986340
\(516\) 0 0
\(517\) −9.79795 −0.430913
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −24.7683 −1.08512 −0.542558 0.840018i \(-0.682544\pi\)
−0.542558 + 0.840018i \(0.682544\pi\)
\(522\) 0 0
\(523\) −32.7530 −1.43219 −0.716094 0.698004i \(-0.754072\pi\)
−0.716094 + 0.698004i \(0.754072\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −29.9388 −1.30416
\(528\) 0 0
\(529\) −18.9787 −0.825160
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.2774 −0.618421
\(534\) 0 0
\(535\) 8.66806 0.374753
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.97483 0.214281
\(540\) 0 0
\(541\) 37.8575 1.62762 0.813811 0.581130i \(-0.197389\pi\)
0.813811 + 0.581130i \(0.197389\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.87654 −0.423064
\(546\) 0 0
\(547\) 3.49671 0.149509 0.0747543 0.997202i \(-0.476183\pi\)
0.0747543 + 0.997202i \(0.476183\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 35.6734 1.51974
\(552\) 0 0
\(553\) −12.1083 −0.514896
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −34.1226 −1.44582 −0.722911 0.690941i \(-0.757197\pi\)
−0.722911 + 0.690941i \(0.757197\pi\)
\(558\) 0 0
\(559\) 11.9500 0.505431
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.3014 0.644878 0.322439 0.946590i \(-0.395497\pi\)
0.322439 + 0.946590i \(0.395497\pi\)
\(564\) 0 0
\(565\) −9.48586 −0.399073
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.9907 1.38304 0.691520 0.722357i \(-0.256941\pi\)
0.691520 + 0.722357i \(0.256941\pi\)
\(570\) 0 0
\(571\) −21.4300 −0.896819 −0.448410 0.893828i \(-0.648009\pi\)
−0.448410 + 0.893828i \(0.648009\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.26755 0.344781
\(576\) 0 0
\(577\) −38.4188 −1.59939 −0.799697 0.600404i \(-0.795007\pi\)
−0.799697 + 0.600404i \(0.795007\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.862663 −0.0357893
\(582\) 0 0
\(583\) −37.9787 −1.57292
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.8754 −0.696524 −0.348262 0.937397i \(-0.613228\pi\)
−0.348262 + 0.937397i \(0.613228\pi\)
\(588\) 0 0
\(589\) −29.7197 −1.22458
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.7817 0.894467 0.447233 0.894417i \(-0.352409\pi\)
0.447233 + 0.894417i \(0.352409\pi\)
\(594\) 0 0
\(595\) 4.89316 0.200600
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −26.7555 −1.09320 −0.546601 0.837393i \(-0.684078\pi\)
−0.546601 + 0.837393i \(0.684078\pi\)
\(600\) 0 0
\(601\) −25.1229 −1.02479 −0.512393 0.858751i \(-0.671241\pi\)
−0.512393 + 0.858751i \(0.671241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.8770 0.523526
\(606\) 0 0
\(607\) 8.63950 0.350667 0.175333 0.984509i \(-0.443900\pi\)
0.175333 + 0.984509i \(0.443900\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.45068 −0.0991440
\(612\) 0 0
\(613\) 46.1789 1.86515 0.932575 0.360977i \(-0.117557\pi\)
0.932575 + 0.360977i \(0.117557\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.5244 −0.463957 −0.231978 0.972721i \(-0.574520\pi\)
−0.231978 + 0.972721i \(0.574520\pi\)
\(618\) 0 0
\(619\) 39.2650 1.57819 0.789096 0.614270i \(-0.210549\pi\)
0.789096 + 0.614270i \(0.210549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.8480 −0.434616
\(624\) 0 0
\(625\) 12.6116 0.504463
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 50.8367 2.02699
\(630\) 0 0
\(631\) 22.8387 0.909193 0.454596 0.890698i \(-0.349784\pi\)
0.454596 + 0.890698i \(0.349784\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.11329 0.242598
\(636\) 0 0
\(637\) 1.24431 0.0493015
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0980 −0.714827 −0.357413 0.933946i \(-0.616341\pi\)
−0.357413 + 0.933946i \(0.616341\pi\)
\(642\) 0 0
\(643\) −30.0833 −1.18637 −0.593184 0.805067i \(-0.702129\pi\)
−0.593184 + 0.805067i \(0.702129\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.1323 −0.988054 −0.494027 0.869447i \(-0.664476\pi\)
−0.494027 + 0.869447i \(0.664476\pi\)
\(648\) 0 0
\(649\) 24.2697 0.952668
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.4977 0.410806 0.205403 0.978678i \(-0.434150\pi\)
0.205403 + 0.978678i \(0.434150\pi\)
\(654\) 0 0
\(655\) −3.31079 −0.129363
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.08277 −0.314860 −0.157430 0.987530i \(-0.550321\pi\)
−0.157430 + 0.987530i \(0.550321\pi\)
\(660\) 0 0
\(661\) −26.9059 −1.04652 −0.523260 0.852173i \(-0.675284\pi\)
−0.523260 + 0.852173i \(0.675284\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.85734 0.188360
\(666\) 0 0
\(667\) −13.7936 −0.534090
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.1412 −0.584519
\(672\) 0 0
\(673\) −12.2232 −0.471170 −0.235585 0.971854i \(-0.575701\pi\)
−0.235585 + 0.971854i \(0.575701\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.68030 0.256745 0.128372 0.991726i \(-0.459025\pi\)
0.128372 + 0.991726i \(0.459025\pi\)
\(678\) 0 0
\(679\) 7.57043 0.290526
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.2125 0.735146 0.367573 0.929995i \(-0.380189\pi\)
0.367573 + 0.929995i \(0.380189\pi\)
\(684\) 0 0
\(685\) 4.65677 0.177926
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.49931 −0.361895
\(690\) 0 0
\(691\) −11.4541 −0.435735 −0.217867 0.975978i \(-0.569910\pi\)
−0.217867 + 0.975978i \(0.569910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.13296 0.346433
\(696\) 0 0
\(697\) −59.9460 −2.27062
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.29596 −0.162256 −0.0811281 0.996704i \(-0.525852\pi\)
−0.0811281 + 0.996704i \(0.525852\pi\)
\(702\) 0 0
\(703\) 50.4646 1.90331
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.05939 0.190278
\(708\) 0 0
\(709\) 12.8901 0.484099 0.242049 0.970264i \(-0.422180\pi\)
0.242049 + 0.970264i \(0.422180\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.4915 0.430361
\(714\) 0 0
\(715\) 5.79769 0.216821
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 5.46489 0.203806 0.101903 0.994794i \(-0.467507\pi\)
0.101903 + 0.994794i \(0.467507\pi\)
\(720\) 0 0
\(721\) 0.238992 0.00890051
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −28.3587 −1.05322
\(726\) 0 0
\(727\) −49.8599 −1.84920 −0.924601 0.380936i \(-0.875602\pi\)
−0.924601 + 0.380936i \(0.875602\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 50.1741 1.85576
\(732\) 0 0
\(733\) −35.2246 −1.30105 −0.650525 0.759485i \(-0.725451\pi\)
−0.650525 + 0.759485i \(0.725451\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.71100 −0.210367
\(738\) 0 0
\(739\) −35.7209 −1.31401 −0.657007 0.753885i \(-0.728178\pi\)
−0.657007 + 0.753885i \(0.728178\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.0609 0.479156 0.239578 0.970877i \(-0.422991\pi\)
0.239578 + 0.970877i \(0.422991\pi\)
\(744\) 0 0
\(745\) −11.9716 −0.438605
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.25496 0.338169
\(750\) 0 0
\(751\) −2.22446 −0.0811717 −0.0405859 0.999176i \(-0.512922\pi\)
−0.0405859 + 0.999176i \(0.512922\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.5005 0.782484
\(756\) 0 0
\(757\) 4.27335 0.155317 0.0776587 0.996980i \(-0.475256\pi\)
0.0776587 + 0.996980i \(0.475256\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 50.4411 1.82849 0.914245 0.405162i \(-0.132785\pi\)
0.914245 + 0.405162i \(0.132785\pi\)
\(762\) 0 0
\(763\) −10.5453 −0.381764
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.07037 0.219188
\(768\) 0 0
\(769\) −2.06986 −0.0746411 −0.0373205 0.999303i \(-0.511882\pi\)
−0.0373205 + 0.999303i \(0.511882\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.3739 −1.02054 −0.510269 0.860015i \(-0.670454\pi\)
−0.510269 + 0.860015i \(0.670454\pi\)
\(774\) 0 0
\(775\) 23.6258 0.848664
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −59.5072 −2.13207
\(780\) 0 0
\(781\) −43.9650 −1.57319
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.4628 0.551892
\(786\) 0 0
\(787\) −3.45072 −0.123005 −0.0615025 0.998107i \(-0.519589\pi\)
−0.0615025 + 0.998107i \(0.519589\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.1281 −0.360115
\(792\) 0 0
\(793\) −3.78714 −0.134485
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.4460 −0.795079 −0.397540 0.917585i \(-0.630136\pi\)
−0.397540 + 0.917585i \(0.630136\pi\)
\(798\) 0 0
\(799\) −10.2896 −0.364020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 30.3812 1.07213
\(804\) 0 0
\(805\) −1.87816 −0.0661963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.2366 0.641164 0.320582 0.947221i \(-0.396122\pi\)
0.320582 + 0.947221i \(0.396122\pi\)
\(810\) 0 0
\(811\) 16.8280 0.590910 0.295455 0.955357i \(-0.404529\pi\)
0.295455 + 0.955357i \(0.404529\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.1322 0.565086
\(816\) 0 0
\(817\) 49.8068 1.74252
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37.2298 −1.29933 −0.649665 0.760221i \(-0.725091\pi\)
−0.649665 + 0.760221i \(0.725091\pi\)
\(822\) 0 0
\(823\) −12.2749 −0.427878 −0.213939 0.976847i \(-0.568629\pi\)
−0.213939 + 0.976847i \(0.568629\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.60355 −0.299175 −0.149587 0.988749i \(-0.547795\pi\)
−0.149587 + 0.988749i \(0.547795\pi\)
\(828\) 0 0
\(829\) −44.2887 −1.53821 −0.769105 0.639123i \(-0.779298\pi\)
−0.769105 + 0.639123i \(0.779298\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.22446 0.181017
\(834\) 0 0
\(835\) 12.2956 0.425508
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 47.2307 1.63059 0.815293 0.579048i \(-0.196576\pi\)
0.815293 + 0.579048i \(0.196576\pi\)
\(840\) 0 0
\(841\) 18.3137 0.631506
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.7255 −0.368968
\(846\) 0 0
\(847\) 13.7489 0.472418
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −19.5128 −0.668890
\(852\) 0 0
\(853\) −23.9650 −0.820546 −0.410273 0.911963i \(-0.634567\pi\)
−0.410273 + 0.911963i \(0.634567\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.0181 0.444688 0.222344 0.974968i \(-0.428629\pi\)
0.222344 + 0.974968i \(0.428629\pi\)
\(858\) 0 0
\(859\) −0.0947371 −0.00323239 −0.00161619 0.999999i \(-0.500514\pi\)
−0.00161619 + 0.999999i \(0.500514\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.8729 −0.880723 −0.440361 0.897821i \(-0.645150\pi\)
−0.440361 + 0.897821i \(0.645150\pi\)
\(864\) 0 0
\(865\) −17.7841 −0.604678
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −60.2366 −2.04339
\(870\) 0 0
\(871\) −1.42845 −0.0484010
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −8.54429 −0.288850
\(876\) 0 0
\(877\) −53.8083 −1.81698 −0.908489 0.417908i \(-0.862763\pi\)
−0.908489 + 0.417908i \(0.862763\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −42.8689 −1.44429 −0.722144 0.691743i \(-0.756843\pi\)
−0.722144 + 0.691743i \(0.756843\pi\)
\(882\) 0 0
\(883\) 34.5967 1.16427 0.582136 0.813092i \(-0.302217\pi\)
0.582136 + 0.813092i \(0.302217\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.4555 −0.686830 −0.343415 0.939184i \(-0.611584\pi\)
−0.343415 + 0.939184i \(0.611584\pi\)
\(888\) 0 0
\(889\) 6.52720 0.218915
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.2143 −0.341808
\(894\) 0 0
\(895\) 7.13524 0.238505
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −39.4173 −1.31464
\(900\) 0 0
\(901\) −39.8845 −1.32874
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.68905 −0.288834
\(906\) 0 0
\(907\) −7.84912 −0.260626 −0.130313 0.991473i \(-0.541598\pi\)
−0.130313 + 0.991473i \(0.541598\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7.74379 −0.256563 −0.128282 0.991738i \(-0.540946\pi\)
−0.128282 + 0.991738i \(0.540946\pi\)
\(912\) 0 0
\(913\) −4.29160 −0.142031
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.53495 −0.116734
\(918\) 0 0
\(919\) 19.3702 0.638963 0.319482 0.947592i \(-0.396491\pi\)
0.319482 + 0.947592i \(0.396491\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.9966 −0.361958
\(924\) 0 0
\(925\) −40.1170 −1.31904
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.55979 −0.116793 −0.0583964 0.998293i \(-0.518599\pi\)
−0.0583964 + 0.998293i \(0.518599\pi\)
\(930\) 0 0
\(931\) 5.18622 0.169972
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.3426 0.796089
\(936\) 0 0
\(937\) −34.2230 −1.11802 −0.559008 0.829162i \(-0.688818\pi\)
−0.559008 + 0.829162i \(0.688818\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.0407 −0.392514 −0.196257 0.980552i \(-0.562879\pi\)
−0.196257 + 0.980552i \(0.562879\pi\)
\(942\) 0 0
\(943\) 23.0093 0.749284
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.7247 −0.738453 −0.369227 0.929339i \(-0.620377\pi\)
−0.369227 + 0.929339i \(0.620377\pi\)
\(948\) 0 0
\(949\) 7.59899 0.246674
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.74488 −0.218488 −0.109244 0.994015i \(-0.534843\pi\)
−0.109244 + 0.994015i \(0.534843\pi\)
\(954\) 0 0
\(955\) 8.99346 0.291022
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.97206 0.160556
\(960\) 0 0
\(961\) 1.83879 0.0593158
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.1266 −0.454752
\(966\) 0 0
\(967\) −25.6534 −0.824958 −0.412479 0.910967i \(-0.635337\pi\)
−0.412479 + 0.910967i \(0.635337\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.9298 1.31350 0.656750 0.754108i \(-0.271931\pi\)
0.656750 + 0.754108i \(0.271931\pi\)
\(972\) 0 0
\(973\) 9.75133 0.312613
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.4730 1.42282 0.711408 0.702779i \(-0.248058\pi\)
0.711408 + 0.702779i \(0.248058\pi\)
\(978\) 0 0
\(979\) −53.9669 −1.72479
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −56.9100 −1.81515 −0.907573 0.419893i \(-0.862067\pi\)
−0.907573 + 0.419893i \(0.862067\pi\)
\(984\) 0 0
\(985\) −10.3691 −0.330387
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −19.2585 −0.612384
\(990\) 0 0
\(991\) −44.7331 −1.42099 −0.710496 0.703701i \(-0.751530\pi\)
−0.710496 + 0.703701i \(0.751530\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.1740 0.544451
\(996\) 0 0
\(997\) −18.7336 −0.593300 −0.296650 0.954986i \(-0.595869\pi\)
−0.296650 + 0.954986i \(0.595869\pi\)
\(998\) 0 0
\(999\) 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 4536.2.a.x.1.3 4
3.2 odd 2 4536.2.a.ba.1.2 4
4.3 odd 2 9072.2.a.ce.1.3 4
9.2 odd 6 1512.2.r.d.1009.3 8
9.4 even 3 504.2.r.d.169.2 8
9.5 odd 6 1512.2.r.d.505.3 8
9.7 even 3 504.2.r.d.337.2 yes 8
12.11 even 2 9072.2.a.cl.1.2 4
36.7 odd 6 1008.2.r.m.337.3 8
36.11 even 6 3024.2.r.l.1009.3 8
36.23 even 6 3024.2.r.l.2017.3 8
36.31 odd 6 1008.2.r.m.673.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.d.169.2 8 9.4 even 3
504.2.r.d.337.2 yes 8 9.7 even 3
1008.2.r.m.337.3 8 36.7 odd 6
1008.2.r.m.673.3 8 36.31 odd 6
1512.2.r.d.505.3 8 9.5 odd 6
1512.2.r.d.1009.3 8 9.2 odd 6
3024.2.r.l.1009.3 8 36.11 even 6
3024.2.r.l.2017.3 8 36.23 even 6
4536.2.a.x.1.3 4 1.1 even 1 trivial
4536.2.a.ba.1.2 4 3.2 odd 2
9072.2.a.ce.1.3 4 4.3 odd 2
9072.2.a.cl.1.2 4 12.11 even 2