Properties

Label 900.3.b.b.449.3
Level $900$
Weight $3$
Character 900.449
Analytic conductor $24.523$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(449,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(0.437016 + 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 900.449
Dual form 900.3.b.b.449.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.48683i q^{7} +O(q^{10})\) \(q-5.48683i q^{7} -9.17377i q^{11} +11.4868i q^{13} -16.9706 q^{17} -26.9737 q^{19} +4.93113 q^{23} -20.5247i q^{29} +20.9737 q^{31} +62.4605i q^{37} +40.9377i q^{41} +1.02633i q^{43} -86.2298 q^{47} +18.8947 q^{49} -96.0920 q^{53} -112.374i q^{59} -66.9210 q^{61} +76.0000i q^{67} +24.0789i q^{71} -18.9210i q^{73} -50.3349 q^{77} -106.921 q^{79} +45.1804 q^{83} -115.928i q^{89} +63.0263 q^{91} +87.0263i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{19} + 16 q^{31} - 456 q^{49} - 80 q^{61} - 400 q^{79} + 656 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 0
\(6\) 0 0
\(7\) − 5.48683i − 0.783833i −0.920001 0.391917i \(-0.871812\pi\)
0.920001 0.391917i \(-0.128188\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 9.17377i − 0.833979i −0.908911 0.416989i \(-0.863085\pi\)
0.908911 0.416989i \(-0.136915\pi\)
\(12\) 0 0
\(13\) 11.4868i 0.883603i 0.897113 + 0.441801i \(0.145660\pi\)
−0.897113 + 0.441801i \(0.854340\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −16.9706 −0.998268 −0.499134 0.866525i \(-0.666349\pi\)
−0.499134 + 0.866525i \(0.666349\pi\)
\(18\) 0 0
\(19\) −26.9737 −1.41967 −0.709833 0.704370i \(-0.751230\pi\)
−0.709833 + 0.704370i \(0.751230\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.93113 0.214397 0.107198 0.994238i \(-0.465812\pi\)
0.107198 + 0.994238i \(0.465812\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 20.5247i − 0.707749i −0.935293 0.353874i \(-0.884864\pi\)
0.935293 0.353874i \(-0.115136\pi\)
\(30\) 0 0
\(31\) 20.9737 0.676570 0.338285 0.941044i \(-0.390153\pi\)
0.338285 + 0.941044i \(0.390153\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 62.4605i 1.68812i 0.536247 + 0.844061i \(0.319841\pi\)
−0.536247 + 0.844061i \(0.680159\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 40.9377i 0.998481i 0.866464 + 0.499240i \(0.166388\pi\)
−0.866464 + 0.499240i \(0.833612\pi\)
\(42\) 0 0
\(43\) 1.02633i 0.0238682i 0.999929 + 0.0119341i \(0.00379884\pi\)
−0.999929 + 0.0119341i \(0.996201\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −86.2298 −1.83468 −0.917338 0.398109i \(-0.869667\pi\)
−0.917338 + 0.398109i \(0.869667\pi\)
\(48\) 0 0
\(49\) 18.8947 0.385605
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −96.0920 −1.81306 −0.906529 0.422144i \(-0.861278\pi\)
−0.906529 + 0.422144i \(0.861278\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 112.374i − 1.90465i −0.305092 0.952323i \(-0.598687\pi\)
0.305092 0.952323i \(-0.401313\pi\)
\(60\) 0 0
\(61\) −66.9210 −1.09707 −0.548533 0.836129i \(-0.684813\pi\)
−0.548533 + 0.836129i \(0.684813\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 76.0000i 1.13433i 0.823605 + 0.567164i \(0.191960\pi\)
−0.823605 + 0.567164i \(0.808040\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 24.0789i 0.339139i 0.985518 + 0.169570i \(0.0542377\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(72\) 0 0
\(73\) − 18.9210i − 0.259192i −0.991567 0.129596i \(-0.958632\pi\)
0.991567 0.129596i \(-0.0413680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −50.3349 −0.653700
\(78\) 0 0
\(79\) −106.921 −1.35343 −0.676715 0.736245i \(-0.736597\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 45.1804 0.544342 0.272171 0.962249i \(-0.412258\pi\)
0.272171 + 0.962249i \(0.412258\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 115.928i − 1.30256i −0.758835 0.651282i \(-0.774231\pi\)
0.758835 0.651282i \(-0.225769\pi\)
\(90\) 0 0
\(91\) 63.0263 0.692597
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 87.0263i 0.897179i 0.893738 + 0.448589i \(0.148073\pi\)
−0.893738 + 0.448589i \(0.851927\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 129.233i − 1.27953i −0.768569 0.639767i \(-0.779031\pi\)
0.768569 0.639767i \(-0.220969\pi\)
\(102\) 0 0
\(103\) 114.302i 1.10973i 0.831939 + 0.554866i \(0.187231\pi\)
−0.831939 + 0.554866i \(0.812769\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −93.3381 −0.872319 −0.436159 0.899869i \(-0.643662\pi\)
−0.436159 + 0.899869i \(0.643662\pi\)
\(108\) 0 0
\(109\) −120.868 −1.10888 −0.554442 0.832222i \(-0.687068\pi\)
−0.554442 + 0.832222i \(0.687068\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −110.309 −0.976183 −0.488091 0.872793i \(-0.662307\pi\)
−0.488091 + 0.872793i \(0.662307\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 93.1146i 0.782476i
\(120\) 0 0
\(121\) 36.8420 0.304479
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 105.381i − 0.829776i −0.909873 0.414888i \(-0.863821\pi\)
0.909873 0.414888i \(-0.136179\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 140.584i 1.07316i 0.843850 + 0.536580i \(0.180284\pi\)
−0.843850 + 0.536580i \(0.819716\pi\)
\(132\) 0 0
\(133\) 148.000i 1.11278i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −266.951 −1.94855 −0.974274 0.225365i \(-0.927643\pi\)
−0.974274 + 0.225365i \(0.927643\pi\)
\(138\) 0 0
\(139\) 15.8420 0.113971 0.0569856 0.998375i \(-0.481851\pi\)
0.0569856 + 0.998375i \(0.481851\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 105.378 0.736906
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 41.6262i − 0.279370i −0.990196 0.139685i \(-0.955391\pi\)
0.990196 0.139685i \(-0.0446091\pi\)
\(150\) 0 0
\(151\) 103.842 0.687695 0.343848 0.939025i \(-0.388270\pi\)
0.343848 + 0.939025i \(0.388270\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.5132i 0.0797017i 0.999206 + 0.0398509i \(0.0126883\pi\)
−0.999206 + 0.0398509i \(0.987312\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 27.0563i − 0.168051i
\(162\) 0 0
\(163\) − 290.868i − 1.78447i −0.451573 0.892234i \(-0.649137\pi\)
0.451573 0.892234i \(-0.350863\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 174.637 1.04573 0.522865 0.852416i \(-0.324863\pi\)
0.522865 + 0.852416i \(0.324863\pi\)
\(168\) 0 0
\(169\) 37.0527 0.219247
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 43.5799 0.251907 0.125954 0.992036i \(-0.459801\pi\)
0.125954 + 0.992036i \(0.459801\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 88.2952i − 0.493270i −0.969109 0.246635i \(-0.920675\pi\)
0.969109 0.246635i \(-0.0793248\pi\)
\(180\) 0 0
\(181\) −56.8683 −0.314190 −0.157095 0.987584i \(-0.550213\pi\)
−0.157095 + 0.987584i \(0.550213\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 155.684i 0.832535i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 56.6430i 0.296560i 0.988945 + 0.148280i \(0.0473737\pi\)
−0.988945 + 0.148280i \(0.952626\pi\)
\(192\) 0 0
\(193\) 110.000i 0.569948i 0.958535 + 0.284974i \(0.0919850\pi\)
−0.958535 + 0.284974i \(0.908015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 245.049 1.24391 0.621953 0.783055i \(-0.286339\pi\)
0.621953 + 0.783055i \(0.286339\pi\)
\(198\) 0 0
\(199\) 169.895 0.853742 0.426871 0.904313i \(-0.359616\pi\)
0.426871 + 0.904313i \(0.359616\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −112.616 −0.554757
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 247.450i 1.18397i
\(210\) 0 0
\(211\) −265.579 −1.25867 −0.629333 0.777136i \(-0.716672\pi\)
−0.629333 + 0.777136i \(0.716672\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 115.079i − 0.530318i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 194.938i − 0.882072i
\(222\) 0 0
\(223\) − 187.329i − 0.840040i −0.907515 0.420020i \(-0.862023\pi\)
0.907515 0.420020i \(-0.137977\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 421.063 1.85490 0.927452 0.373942i \(-0.121994\pi\)
0.927452 + 0.373942i \(0.121994\pi\)
\(228\) 0 0
\(229\) 102.105 0.445875 0.222937 0.974833i \(-0.428435\pi\)
0.222937 + 0.974833i \(0.428435\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −99.0694 −0.425191 −0.212595 0.977140i \(-0.568192\pi\)
−0.212595 + 0.977140i \(0.568192\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 77.9680i 0.326226i 0.986607 + 0.163113i \(0.0521535\pi\)
−0.986607 + 0.163113i \(0.947847\pi\)
\(240\) 0 0
\(241\) −257.947 −1.07032 −0.535160 0.844750i \(-0.679749\pi\)
−0.535160 + 0.844750i \(0.679749\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 309.842i − 1.25442i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 274.971i 1.09550i 0.836641 + 0.547752i \(0.184516\pi\)
−0.836641 + 0.547752i \(0.815484\pi\)
\(252\) 0 0
\(253\) − 45.2370i − 0.178802i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 131.634 0.512193 0.256096 0.966651i \(-0.417564\pi\)
0.256096 + 0.966651i \(0.417564\pi\)
\(258\) 0 0
\(259\) 342.710 1.32321
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 458.782 1.74442 0.872209 0.489133i \(-0.162687\pi\)
0.872209 + 0.489133i \(0.162687\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 301.916i 1.12236i 0.827692 + 0.561182i \(0.189653\pi\)
−0.827692 + 0.561182i \(0.810347\pi\)
\(270\) 0 0
\(271\) 475.842 1.75587 0.877937 0.478776i \(-0.158919\pi\)
0.877937 + 0.478776i \(0.158919\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 322.039i − 1.16260i −0.813691 0.581298i \(-0.802545\pi\)
0.813691 0.581298i \(-0.197455\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 352.139i − 1.25316i −0.779355 0.626582i \(-0.784453\pi\)
0.779355 0.626582i \(-0.215547\pi\)
\(282\) 0 0
\(283\) − 281.631i − 0.995164i −0.867417 0.497582i \(-0.834221\pi\)
0.867417 0.497582i \(-0.165779\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 224.618 0.782642
\(288\) 0 0
\(289\) −1.00000 −0.00346021
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −64.3281 −0.219550 −0.109775 0.993956i \(-0.535013\pi\)
−0.109775 + 0.993956i \(0.535013\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 56.6430i 0.189442i
\(300\) 0 0
\(301\) 5.63132 0.0187087
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 230.158i − 0.749700i −0.927085 0.374850i \(-0.877694\pi\)
0.927085 0.374850i \(-0.122306\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.48528i 0.0272839i 0.999907 + 0.0136419i \(0.00434250\pi\)
−0.999907 + 0.0136419i \(0.995658\pi\)
\(312\) 0 0
\(313\) 605.579i 1.93476i 0.253337 + 0.967378i \(0.418472\pi\)
−0.253337 + 0.967378i \(0.581528\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −87.9601 −0.277477 −0.138738 0.990329i \(-0.544305\pi\)
−0.138738 + 0.990329i \(0.544305\pi\)
\(318\) 0 0
\(319\) −188.289 −0.590248
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 457.758 1.41721
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 473.128i 1.43808i
\(330\) 0 0
\(331\) −49.2370 −0.148752 −0.0743761 0.997230i \(-0.523697\pi\)
−0.0743761 + 0.997230i \(0.523697\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.71033i 0.00804251i 0.999992 + 0.00402125i \(0.00128001\pi\)
−0.999992 + 0.00402125i \(0.998720\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 192.408i − 0.564245i
\(342\) 0 0
\(343\) − 372.527i − 1.08608i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 74.9906 0.216111 0.108056 0.994145i \(-0.465538\pi\)
0.108056 + 0.994145i \(0.465538\pi\)
\(348\) 0 0
\(349\) 150.921 0.432438 0.216219 0.976345i \(-0.430627\pi\)
0.216219 + 0.976345i \(0.430627\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −292.407 −0.828349 −0.414174 0.910198i \(-0.635930\pi\)
−0.414174 + 0.910198i \(0.635930\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 576.776i 1.60662i 0.595563 + 0.803309i \(0.296929\pi\)
−0.595563 + 0.803309i \(0.703071\pi\)
\(360\) 0 0
\(361\) 366.579 1.01545
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 243.540i − 0.663595i −0.943351 0.331798i \(-0.892345\pi\)
0.943351 0.331798i \(-0.107655\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 527.241i 1.42113i
\(372\) 0 0
\(373\) 115.908i 0.310746i 0.987856 + 0.155373i \(0.0496579\pi\)
−0.987856 + 0.155373i \(0.950342\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 235.764 0.625369
\(378\) 0 0
\(379\) −30.2107 −0.0797115 −0.0398558 0.999205i \(-0.512690\pi\)
−0.0398558 + 0.999205i \(0.512690\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 651.319 1.70057 0.850286 0.526320i \(-0.176429\pi\)
0.850286 + 0.526320i \(0.176429\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 205.600i − 0.528536i −0.964449 0.264268i \(-0.914870\pi\)
0.964449 0.264268i \(-0.0851303\pi\)
\(390\) 0 0
\(391\) −83.6840 −0.214026
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 118.355i − 0.298124i −0.988828 0.149062i \(-0.952375\pi\)
0.988828 0.149062i \(-0.0476254\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 170.971i − 0.426361i −0.977013 0.213181i \(-0.931618\pi\)
0.977013 0.213181i \(-0.0683823\pi\)
\(402\) 0 0
\(403\) 240.921i 0.597819i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 572.998 1.40786
\(408\) 0 0
\(409\) 335.947 0.821387 0.410694 0.911773i \(-0.365287\pi\)
0.410694 + 0.911773i \(0.365287\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −616.578 −1.49292
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 407.535i − 0.972637i −0.873781 0.486319i \(-0.838339\pi\)
0.873781 0.486319i \(-0.161661\pi\)
\(420\) 0 0
\(421\) −771.210 −1.83185 −0.915926 0.401346i \(-0.868542\pi\)
−0.915926 + 0.401346i \(0.868542\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 367.184i 0.859916i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 436.210i − 1.01209i −0.862508 0.506044i \(-0.831107\pi\)
0.862508 0.506044i \(-0.168893\pi\)
\(432\) 0 0
\(433\) − 838.500i − 1.93649i −0.250006 0.968244i \(-0.580432\pi\)
0.250006 0.968244i \(-0.419568\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −133.011 −0.304372
\(438\) 0 0
\(439\) −50.0000 −0.113895 −0.0569476 0.998377i \(-0.518137\pi\)
−0.0569476 + 0.998377i \(0.518137\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −492.853 −1.11253 −0.556267 0.831003i \(-0.687767\pi\)
−0.556267 + 0.831003i \(0.687767\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 483.102i − 1.07595i −0.842960 0.537976i \(-0.819189\pi\)
0.842960 0.537976i \(-0.180811\pi\)
\(450\) 0 0
\(451\) 375.553 0.832712
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 546.921i 1.19676i 0.801211 + 0.598382i \(0.204189\pi\)
−0.801211 + 0.598382i \(0.795811\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 130.386i 0.282834i 0.989950 + 0.141417i \(0.0451658\pi\)
−0.989950 + 0.141417i \(0.954834\pi\)
\(462\) 0 0
\(463\) 24.7765i 0.0535130i 0.999642 + 0.0267565i \(0.00851787\pi\)
−0.999642 + 0.0267565i \(0.991482\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −671.491 −1.43788 −0.718941 0.695071i \(-0.755373\pi\)
−0.718941 + 0.695071i \(0.755373\pi\)
\(468\) 0 0
\(469\) 416.999 0.889124
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.41535 0.0199056
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 538.257i − 1.12371i −0.827236 0.561855i \(-0.810088\pi\)
0.827236 0.561855i \(-0.189912\pi\)
\(480\) 0 0
\(481\) −717.473 −1.49163
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 608.250i − 1.24897i −0.781036 0.624486i \(-0.785308\pi\)
0.781036 0.624486i \(-0.214692\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 143.561i 0.292386i 0.989256 + 0.146193i \(0.0467020\pi\)
−0.989256 + 0.146193i \(0.953298\pi\)
\(492\) 0 0
\(493\) 348.316i 0.706523i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 132.117 0.265828
\(498\) 0 0
\(499\) −616.605 −1.23568 −0.617841 0.786303i \(-0.711992\pi\)
−0.617841 + 0.786303i \(0.711992\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 374.729 0.744989 0.372494 0.928034i \(-0.378503\pi\)
0.372494 + 0.928034i \(0.378503\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 173.036i 0.339953i 0.985448 + 0.169977i \(0.0543693\pi\)
−0.985448 + 0.169977i \(0.945631\pi\)
\(510\) 0 0
\(511\) −103.816 −0.203163
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 791.052i 1.53008i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 455.116i 0.873543i 0.899572 + 0.436772i \(0.143878\pi\)
−0.899572 + 0.436772i \(0.856122\pi\)
\(522\) 0 0
\(523\) − 295.395i − 0.564809i −0.959295 0.282404i \(-0.908868\pi\)
0.959295 0.282404i \(-0.0911320\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −355.935 −0.675398
\(528\) 0 0
\(529\) −504.684 −0.954034
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −470.245 −0.882260
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 173.335i − 0.321587i
\(540\) 0 0
\(541\) 906.394 1.67541 0.837703 0.546127i \(-0.183898\pi\)
0.837703 + 0.546127i \(0.183898\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 672.921i 1.23020i 0.788448 + 0.615101i \(0.210885\pi\)
−0.788448 + 0.615101i \(0.789115\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 553.627i 1.00477i
\(552\) 0 0
\(553\) 586.658i 1.06086i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −870.336 −1.56254 −0.781271 0.624192i \(-0.785428\pi\)
−0.781271 + 0.624192i \(0.785428\pi\)
\(558\) 0 0
\(559\) −11.7893 −0.0210900
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.6320 0.0419751 0.0209875 0.999780i \(-0.493319\pi\)
0.0209875 + 0.999780i \(0.493319\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 295.273i − 0.518933i −0.965752 0.259466i \(-0.916453\pi\)
0.965752 0.259466i \(-0.0835467\pi\)
\(570\) 0 0
\(571\) 893.920 1.56553 0.782767 0.622314i \(-0.213807\pi\)
0.782767 + 0.622314i \(0.213807\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 264.763i 0.458861i 0.973325 + 0.229431i \(0.0736864\pi\)
−0.973325 + 0.229431i \(0.926314\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 247.897i − 0.426673i
\(582\) 0 0
\(583\) 881.526i 1.51205i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −791.885 −1.34904 −0.674519 0.738258i \(-0.735649\pi\)
−0.674519 + 0.738258i \(0.735649\pi\)
\(588\) 0 0
\(589\) −565.737 −0.960504
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.82388 −0.00307567 −0.00153784 0.999999i \(-0.500490\pi\)
−0.00153784 + 0.999999i \(0.500490\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 758.204i 1.26578i 0.774241 + 0.632891i \(0.218132\pi\)
−0.774241 + 0.632891i \(0.781868\pi\)
\(600\) 0 0
\(601\) −283.579 −0.471845 −0.235922 0.971772i \(-0.575811\pi\)
−0.235922 + 0.971772i \(0.575811\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 167.828i − 0.276488i −0.990398 0.138244i \(-0.955854\pi\)
0.990398 0.138244i \(-0.0441459\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 990.507i − 1.62112i
\(612\) 0 0
\(613\) − 395.698i − 0.645510i −0.946483 0.322755i \(-0.895391\pi\)
0.946483 0.322755i \(-0.104609\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −855.190 −1.38604 −0.693022 0.720916i \(-0.743721\pi\)
−0.693022 + 0.720916i \(0.743721\pi\)
\(618\) 0 0
\(619\) −308.158 −0.497832 −0.248916 0.968525i \(-0.580074\pi\)
−0.248916 + 0.968525i \(0.580074\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −636.079 −1.02099
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1059.99i − 1.68520i
\(630\) 0 0
\(631\) 344.974 0.546709 0.273355 0.961913i \(-0.411867\pi\)
0.273355 + 0.961913i \(0.411867\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 217.040i 0.340722i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 202.641i − 0.316133i −0.987428 0.158066i \(-0.949474\pi\)
0.987428 0.158066i \(-0.0505260\pi\)
\(642\) 0 0
\(643\) 724.605i 1.12691i 0.826146 + 0.563456i \(0.190529\pi\)
−0.826146 + 0.563456i \(0.809471\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −125.679 −0.194249 −0.0971243 0.995272i \(-0.530964\pi\)
−0.0971243 + 0.995272i \(0.530964\pi\)
\(648\) 0 0
\(649\) −1030.89 −1.58843
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 551.226 0.844144 0.422072 0.906562i \(-0.361303\pi\)
0.422072 + 0.906562i \(0.361303\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 801.765i − 1.21664i −0.793692 0.608320i \(-0.791844\pi\)
0.793692 0.608320i \(-0.208156\pi\)
\(660\) 0 0
\(661\) −529.079 −0.800422 −0.400211 0.916423i \(-0.631063\pi\)
−0.400211 + 0.916423i \(0.631063\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 101.210i − 0.151739i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 613.918i 0.914929i
\(672\) 0 0
\(673\) 413.395i 0.614257i 0.951668 + 0.307129i \(0.0993682\pi\)
−0.951668 + 0.307129i \(0.900632\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −359.936 −0.531663 −0.265832 0.964019i \(-0.585647\pi\)
−0.265832 + 0.964019i \(0.585647\pi\)
\(678\) 0 0
\(679\) 477.499 0.703239
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −21.5484 −0.0315496 −0.0157748 0.999876i \(-0.505021\pi\)
−0.0157748 + 0.999876i \(0.505021\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1103.79i − 1.60202i
\(690\) 0 0
\(691\) −384.921 −0.557049 −0.278525 0.960429i \(-0.589845\pi\)
−0.278525 + 0.960429i \(0.589845\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 694.736i − 0.996752i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 439.057i 0.626330i 0.949699 + 0.313165i \(0.101389\pi\)
−0.949699 + 0.313165i \(0.898611\pi\)
\(702\) 0 0
\(703\) − 1684.79i − 2.39657i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −709.080 −1.00294
\(708\) 0 0
\(709\) 126.421 0.178309 0.0891547 0.996018i \(-0.471583\pi\)
0.0891547 + 0.996018i \(0.471583\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 103.424 0.145054
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 571.715i 0.795153i 0.917569 + 0.397576i \(0.130149\pi\)
−0.917569 + 0.397576i \(0.869851\pi\)
\(720\) 0 0
\(721\) 627.159 0.869846
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 841.038i 1.15686i 0.815731 + 0.578431i \(0.196335\pi\)
−0.815731 + 0.578431i \(0.803665\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 17.4175i − 0.0238269i
\(732\) 0 0
\(733\) 494.749i 0.674965i 0.941332 + 0.337483i \(0.109575\pi\)
−0.941332 + 0.337483i \(0.890425\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 697.206 0.946006
\(738\) 0 0
\(739\) −1263.97 −1.71038 −0.855191 0.518312i \(-0.826561\pi\)
−0.855191 + 0.518312i \(0.826561\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −956.343 −1.28714 −0.643568 0.765389i \(-0.722547\pi\)
−0.643568 + 0.765389i \(0.722547\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 512.131i 0.683752i
\(750\) 0 0
\(751\) −1121.66 −1.49355 −0.746776 0.665076i \(-0.768399\pi\)
−0.746776 + 0.665076i \(0.768399\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 466.433i 0.616160i 0.951360 + 0.308080i \(0.0996865\pi\)
−0.951360 + 0.308080i \(0.900313\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 1223.41i − 1.60763i −0.594880 0.803814i \(-0.702801\pi\)
0.594880 0.803814i \(-0.297199\pi\)
\(762\) 0 0
\(763\) 663.184i 0.869180i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1290.82 1.68295
\(768\) 0 0
\(769\) 1290.63 1.67832 0.839162 0.543882i \(-0.183046\pi\)
0.839162 + 0.543882i \(0.183046\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −329.455 −0.426204 −0.213102 0.977030i \(-0.568357\pi\)
−0.213102 + 0.977030i \(0.568357\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1104.24i − 1.41751i
\(780\) 0 0
\(781\) 220.894 0.282835
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1506.13i 1.91376i 0.290480 + 0.956881i \(0.406185\pi\)
−0.290480 + 0.956881i \(0.593815\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 605.245i 0.765165i
\(792\) 0 0
\(793\) − 768.710i − 0.969370i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 250.874 0.314773 0.157387 0.987537i \(-0.449693\pi\)
0.157387 + 0.987537i \(0.449693\pi\)
\(798\) 0 0
\(799\) 1463.37 1.83150
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −173.577 −0.216160
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1097.02i 1.35602i 0.735053 + 0.678010i \(0.237157\pi\)
−0.735053 + 0.678010i \(0.762843\pi\)
\(810\) 0 0
\(811\) −221.473 −0.273087 −0.136543 0.990634i \(-0.543599\pi\)
−0.136543 + 0.990634i \(0.543599\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 27.6840i − 0.0338849i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1003.92i 1.22281i 0.791319 + 0.611403i \(0.209395\pi\)
−0.791319 + 0.611403i \(0.790605\pi\)
\(822\) 0 0
\(823\) 335.013i 0.407063i 0.979068 + 0.203531i \(0.0652419\pi\)
−0.979068 + 0.203531i \(0.934758\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1004.72 −1.21490 −0.607451 0.794357i \(-0.707808\pi\)
−0.607451 + 0.794357i \(0.707808\pi\)
\(828\) 0 0
\(829\) −676.763 −0.816361 −0.408180 0.912901i \(-0.633837\pi\)
−0.408180 + 0.912901i \(0.633837\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −320.653 −0.384938
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 621.286i − 0.740507i −0.928931 0.370254i \(-0.879271\pi\)
0.928931 0.370254i \(-0.120729\pi\)
\(840\) 0 0
\(841\) 419.736 0.499092
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 202.146i − 0.238661i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 308.001i 0.361928i
\(852\) 0 0
\(853\) − 544.591i − 0.638443i −0.947680 0.319221i \(-0.896579\pi\)
0.947680 0.319221i \(-0.103421\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −362.149 −0.422578 −0.211289 0.977424i \(-0.567766\pi\)
−0.211289 + 0.977424i \(0.567766\pi\)
\(858\) 0 0
\(859\) 21.6840 0.0252433 0.0126216 0.999920i \(-0.495982\pi\)
0.0126216 + 0.999920i \(0.495982\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 220.300 0.255273 0.127636 0.991821i \(-0.459261\pi\)
0.127636 + 0.991821i \(0.459261\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 980.868i 1.12873i
\(870\) 0 0
\(871\) −872.999 −1.00230
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1136.41i − 1.29579i −0.761730 0.647895i \(-0.775650\pi\)
0.761730 0.647895i \(-0.224350\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 711.758i 0.807898i 0.914782 + 0.403949i \(0.132363\pi\)
−0.914782 + 0.403949i \(0.867637\pi\)
\(882\) 0 0
\(883\) − 536.394i − 0.607468i −0.952757 0.303734i \(-0.901767\pi\)
0.952757 0.303734i \(-0.0982334\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1463.95 1.65045 0.825227 0.564801i \(-0.191047\pi\)
0.825227 + 0.564801i \(0.191047\pi\)
\(888\) 0 0
\(889\) −578.211 −0.650406
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2325.93 2.60463
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 430.479i − 0.478842i
\(900\) 0 0
\(901\) 1630.74 1.80992
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 855.657i − 0.943392i −0.881761 0.471696i \(-0.843642\pi\)
0.881761 0.471696i \(-0.156358\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1581.02i − 1.73547i −0.497024 0.867737i \(-0.665574\pi\)
0.497024 0.867737i \(-0.334426\pi\)
\(912\) 0 0
\(913\) − 414.474i − 0.453969i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 771.360 0.841178
\(918\) 0 0
\(919\) 1489.08 1.62033 0.810163 0.586205i \(-0.199379\pi\)
0.810163 + 0.586205i \(0.199379\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −276.590 −0.299664
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 736.990i 0.793316i 0.917966 + 0.396658i \(0.129830\pi\)
−0.917966 + 0.396658i \(0.870170\pi\)
\(930\) 0 0
\(931\) −509.658 −0.547431
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 1550.05i − 1.65427i −0.562003 0.827135i \(-0.689969\pi\)
0.562003 0.827135i \(-0.310031\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 628.711i 0.668131i 0.942550 + 0.334065i \(0.108421\pi\)
−0.942550 + 0.334065i \(0.891579\pi\)
\(942\) 0 0
\(943\) 201.869i 0.214071i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1498.73 −1.58261 −0.791304 0.611423i \(-0.790598\pi\)
−0.791304 + 0.611423i \(0.790598\pi\)
\(948\) 0 0
\(949\) 217.342 0.229022
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 224.748 0.235832 0.117916 0.993024i \(-0.462379\pi\)
0.117916 + 0.993024i \(0.462379\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1464.72i 1.52734i
\(960\) 0 0
\(961\) −521.105 −0.542253
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 233.986i − 0.241972i −0.992654 0.120986i \(-0.961394\pi\)
0.992654 0.120986i \(-0.0386055\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 691.940i − 0.712606i −0.934371 0.356303i \(-0.884037\pi\)
0.934371 0.356303i \(-0.115963\pi\)
\(972\) 0 0
\(973\) − 86.9224i − 0.0893344i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 395.571 0.404883 0.202442 0.979294i \(-0.435112\pi\)
0.202442 + 0.979294i \(0.435112\pi\)
\(978\) 0 0
\(979\) −1063.50 −1.08631
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.82388 0.00185542 0.000927709 1.00000i \(-0.499705\pi\)
0.000927709 1.00000i \(0.499705\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.06098i 0.00511727i
\(990\) 0 0
\(991\) 506.316 0.510914 0.255457 0.966820i \(-0.417774\pi\)
0.255457 + 0.966820i \(0.417774\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 295.670i 0.296560i 0.988945 + 0.148280i \(0.0473737\pi\)
−0.988945 + 0.148280i \(0.952626\pi\)
\(998\) 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 900.3.b.b.449.3 8
3.2 odd 2 inner 900.3.b.b.449.4 8
4.3 odd 2 3600.3.c.k.449.6 8
5.2 odd 4 180.3.g.a.161.2 4
5.3 odd 4 900.3.g.d.701.1 4
5.4 even 2 inner 900.3.b.b.449.5 8
12.11 even 2 3600.3.c.k.449.5 8
15.2 even 4 180.3.g.a.161.4 yes 4
15.8 even 4 900.3.g.d.701.2 4
15.14 odd 2 inner 900.3.b.b.449.6 8
20.3 even 4 3600.3.l.n.1601.4 4
20.7 even 4 720.3.l.c.161.1 4
20.19 odd 2 3600.3.c.k.449.4 8
40.27 even 4 2880.3.l.f.1601.3 4
40.37 odd 4 2880.3.l.b.1601.4 4
45.2 even 12 1620.3.o.f.701.1 8
45.7 odd 12 1620.3.o.f.701.3 8
45.22 odd 12 1620.3.o.f.1241.1 8
45.32 even 12 1620.3.o.f.1241.3 8
60.23 odd 4 3600.3.l.n.1601.3 4
60.47 odd 4 720.3.l.c.161.3 4
60.59 even 2 3600.3.c.k.449.3 8
120.77 even 4 2880.3.l.b.1601.2 4
120.107 odd 4 2880.3.l.f.1601.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.g.a.161.2 4 5.2 odd 4
180.3.g.a.161.4 yes 4 15.2 even 4
720.3.l.c.161.1 4 20.7 even 4
720.3.l.c.161.3 4 60.47 odd 4
900.3.b.b.449.3 8 1.1 even 1 trivial
900.3.b.b.449.4 8 3.2 odd 2 inner
900.3.b.b.449.5 8 5.4 even 2 inner
900.3.b.b.449.6 8 15.14 odd 2 inner
900.3.g.d.701.1 4 5.3 odd 4
900.3.g.d.701.2 4 15.8 even 4
1620.3.o.f.701.1 8 45.2 even 12
1620.3.o.f.701.3 8 45.7 odd 12
1620.3.o.f.1241.1 8 45.22 odd 12
1620.3.o.f.1241.3 8 45.32 even 12
2880.3.l.b.1601.2 4 120.77 even 4
2880.3.l.b.1601.4 4 40.37 odd 4
2880.3.l.f.1601.1 4 120.107 odd 4
2880.3.l.f.1601.3 4 40.27 even 4
3600.3.c.k.449.3 8 60.59 even 2
3600.3.c.k.449.4 8 20.19 odd 2
3600.3.c.k.449.5 8 12.11 even 2
3600.3.c.k.449.6 8 4.3 odd 2
3600.3.l.n.1601.3 4 60.23 odd 4
3600.3.l.n.1601.4 4 20.3 even 4