Properties

Label 4680.2.l.e.2809.6
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4680,2,Mod(2809,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4680.2809");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (of order \(2\), degree \(1\), 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: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2809.6
Root \(-1.75233 - 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.e.2809.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.75233 + 1.38900i) q^{5} +O(q^{10})\) \(q+(1.75233 + 1.38900i) q^{5} -1.50466 q^{11} +1.00000i q^{13} -2.72666i q^{17} -0.726656 q^{19} +4.72666i q^{23} +(1.14134 + 4.86799i) q^{25} -7.55602 q^{29} -3.00933 q^{31} +5.00933i q^{37} -5.78734 q^{41} +2.72666i q^{43} -10.2313i q^{47} +7.00000 q^{49} +7.55602i q^{53} +(-2.63667 - 2.08998i) q^{55} -12.5140 q^{59} +6.28267 q^{61} +(-1.38900 + 1.75233i) q^{65} +12.5653i q^{67} -4.77801 q^{71} +12.0187i q^{73} -5.27334 q^{79} -7.78734i q^{83} +(3.78734 - 4.77801i) q^{85} +1.78734 q^{89} +(-1.27334 - 1.00933i) q^{95} +6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(5\) 1.75233 + 1.38900i 0.783667 + 0.621181i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50466 −0.453673 −0.226837 0.973933i \(-0.572838\pi\)
−0.226837 + 0.973933i \(0.572838\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.72666i 0.661311i −0.943751 0.330656i \(-0.892730\pi\)
0.943751 0.330656i \(-0.107270\pi\)
\(18\) 0 0
\(19\) −0.726656 −0.166706 −0.0833532 0.996520i \(-0.526563\pi\)
−0.0833532 + 0.996520i \(0.526563\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.72666i 0.985576i 0.870149 + 0.492788i \(0.164022\pi\)
−0.870149 + 0.492788i \(0.835978\pi\)
\(24\) 0 0
\(25\) 1.14134 + 4.86799i 0.228267 + 0.973599i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.55602 −1.40312 −0.701558 0.712612i \(-0.747512\pi\)
−0.701558 + 0.712612i \(0.747512\pi\)
\(30\) 0 0
\(31\) −3.00933 −0.540491 −0.270246 0.962791i \(-0.587105\pi\)
−0.270246 + 0.962791i \(0.587105\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00933i 0.823529i 0.911290 + 0.411764i \(0.135087\pi\)
−0.911290 + 0.411764i \(0.864913\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.78734 −0.903830 −0.451915 0.892061i \(-0.649259\pi\)
−0.451915 + 0.892061i \(0.649259\pi\)
\(42\) 0 0
\(43\) 2.72666i 0.415811i 0.978149 + 0.207906i \(0.0666647\pi\)
−0.978149 + 0.207906i \(0.933335\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.2313i 1.49239i −0.665727 0.746196i \(-0.731878\pi\)
0.665727 0.746196i \(-0.268122\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.55602i 1.03790i 0.854805 + 0.518949i \(0.173677\pi\)
−0.854805 + 0.518949i \(0.826323\pi\)
\(54\) 0 0
\(55\) −2.63667 2.08998i −0.355529 0.281813i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.5140 −1.62918 −0.814592 0.580035i \(-0.803039\pi\)
−0.814592 + 0.580035i \(0.803039\pi\)
\(60\) 0 0
\(61\) 6.28267 0.804414 0.402207 0.915549i \(-0.368243\pi\)
0.402207 + 0.915549i \(0.368243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.38900 + 1.75233i −0.172285 + 0.217350i
\(66\) 0 0
\(67\) 12.5653i 1.53510i 0.640988 + 0.767551i \(0.278525\pi\)
−0.640988 + 0.767551i \(0.721475\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.77801 −0.567045 −0.283523 0.958966i \(-0.591503\pi\)
−0.283523 + 0.958966i \(0.591503\pi\)
\(72\) 0 0
\(73\) 12.0187i 1.40668i 0.710855 + 0.703339i \(0.248308\pi\)
−0.710855 + 0.703339i \(0.751692\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.27334 −0.593297 −0.296649 0.954987i \(-0.595869\pi\)
−0.296649 + 0.954987i \(0.595869\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.78734i 0.854771i −0.904069 0.427386i \(-0.859435\pi\)
0.904069 0.427386i \(-0.140565\pi\)
\(84\) 0 0
\(85\) 3.78734 4.77801i 0.410794 0.518248i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.78734 0.189457 0.0947286 0.995503i \(-0.469802\pi\)
0.0947286 + 0.995503i \(0.469802\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.27334 1.00933i −0.130642 0.103555i
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.99067 0.297583 0.148791 0.988869i \(-0.452462\pi\)
0.148791 + 0.988869i \(0.452462\pi\)
\(102\) 0 0
\(103\) 0.443984i 0.0437471i −0.999761 0.0218735i \(-0.993037\pi\)
0.999761 0.0218735i \(-0.00696312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.00933i 0.677617i 0.940855 + 0.338809i \(0.110024\pi\)
−0.940855 + 0.338809i \(0.889976\pi\)
\(108\) 0 0
\(109\) 13.8387 1.32551 0.662753 0.748838i \(-0.269388\pi\)
0.662753 + 0.748838i \(0.269388\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.28267i 0.402880i 0.979501 + 0.201440i \(0.0645621\pi\)
−0.979501 + 0.201440i \(0.935438\pi\)
\(114\) 0 0
\(115\) −6.56534 + 8.28267i −0.612222 + 0.772363i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.73599 −0.794180
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.76166 + 10.1157i −0.425896 + 0.904772i
\(126\) 0 0
\(127\) 5.71733i 0.507331i 0.967292 + 0.253665i \(0.0816362\pi\)
−0.967292 + 0.253665i \(0.918364\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.55602 −0.485431 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.67531i 0.570310i −0.958481 0.285155i \(-0.907955\pi\)
0.958481 0.285155i \(-0.0920450\pi\)
\(138\) 0 0
\(139\) −19.4720 −1.65159 −0.825795 0.563970i \(-0.809273\pi\)
−0.825795 + 0.563970i \(0.809273\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.50466i 0.125826i
\(144\) 0 0
\(145\) −13.2406 10.4953i −1.09958 0.871590i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.5140 −1.18903 −0.594516 0.804084i \(-0.702656\pi\)
−0.594516 + 0.804084i \(0.702656\pi\)
\(150\) 0 0
\(151\) −4.46264 −0.363165 −0.181582 0.983376i \(-0.558122\pi\)
−0.181582 + 0.983376i \(0.558122\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.27334 4.17997i −0.423565 0.335743i
\(156\) 0 0
\(157\) 8.30133i 0.662518i −0.943540 0.331259i \(-0.892527\pi\)
0.943540 0.331259i \(-0.107473\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0093i 0.862317i −0.902276 0.431159i \(-0.858105\pi\)
0.902276 0.431159i \(-0.141895\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.76868i 0.446394i −0.974773 0.223197i \(-0.928351\pi\)
0.974773 0.223197i \(-0.0716493\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.90663i 0.373044i −0.982451 0.186522i \(-0.940278\pi\)
0.982451 0.186522i \(-0.0597215\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.45331 0.706574 0.353287 0.935515i \(-0.385064\pi\)
0.353287 + 0.935515i \(0.385064\pi\)
\(180\) 0 0
\(181\) 17.4720 1.29868 0.649341 0.760498i \(-0.275045\pi\)
0.649341 + 0.760498i \(0.275045\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.95798 + 8.77801i −0.511561 + 0.645372i
\(186\) 0 0
\(187\) 4.10270i 0.300019i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.1120 0.804038 0.402019 0.915631i \(-0.368309\pi\)
0.402019 + 0.915631i \(0.368309\pi\)
\(192\) 0 0
\(193\) 6.10270i 0.439282i 0.975581 + 0.219641i \(0.0704886\pi\)
−0.975581 + 0.219641i \(0.929511\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.3620i 1.52198i 0.648764 + 0.760990i \(0.275286\pi\)
−0.648764 + 0.760990i \(0.724714\pi\)
\(198\) 0 0
\(199\) −8.38538 −0.594423 −0.297212 0.954812i \(-0.596057\pi\)
−0.297212 + 0.954812i \(0.596057\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −10.1413 8.03863i −0.708302 0.561443i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.09337 0.0756303
\(210\) 0 0
\(211\) −1.27334 −0.0876606 −0.0438303 0.999039i \(-0.513956\pi\)
−0.0438303 + 0.999039i \(0.513956\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.78734 + 4.77801i −0.258294 + 0.325857i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.72666 0.183415
\(222\) 0 0
\(223\) 16.4626i 1.10242i −0.834367 0.551210i \(-0.814166\pi\)
0.834367 0.551210i \(-0.185834\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.2500i 0.813060i −0.913638 0.406530i \(-0.866739\pi\)
0.913638 0.406530i \(-0.133261\pi\)
\(228\) 0 0
\(229\) 1.27334 0.0841449 0.0420725 0.999115i \(-0.486604\pi\)
0.0420725 + 0.999115i \(0.486604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.3013i 1.19896i 0.800390 + 0.599480i \(0.204626\pi\)
−0.800390 + 0.599480i \(0.795374\pi\)
\(234\) 0 0
\(235\) 14.2113 17.9287i 0.927046 1.16954i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.23132 0.403071 0.201535 0.979481i \(-0.435407\pi\)
0.201535 + 0.979481i \(0.435407\pi\)
\(240\) 0 0
\(241\) 19.5560 1.25971 0.629857 0.776711i \(-0.283114\pi\)
0.629857 + 0.776711i \(0.283114\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.2663 + 9.72303i 0.783667 + 0.621181i
\(246\) 0 0
\(247\) 0.726656i 0.0462360i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −26.4813 −1.67148 −0.835742 0.549122i \(-0.814962\pi\)
−0.835742 + 0.549122i \(0.814962\pi\)
\(252\) 0 0
\(253\) 7.11203i 0.447130i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.8294i 0.925030i −0.886611 0.462515i \(-0.846947\pi\)
0.886611 0.462515i \(-0.153053\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.9507i 1.41520i 0.706612 + 0.707601i \(0.250223\pi\)
−0.706612 + 0.707601i \(0.749777\pi\)
\(264\) 0 0
\(265\) −10.4953 + 13.2406i −0.644723 + 0.813367i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −27.9160 −1.70207 −0.851033 0.525112i \(-0.824023\pi\)
−0.851033 + 0.525112i \(0.824023\pi\)
\(270\) 0 0
\(271\) −5.65872 −0.343743 −0.171871 0.985119i \(-0.554981\pi\)
−0.171871 + 0.985119i \(0.554981\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.71733 7.32469i −0.103559 0.441696i
\(276\) 0 0
\(277\) 7.73599i 0.464810i 0.972619 + 0.232405i \(0.0746595\pi\)
−0.972619 + 0.232405i \(0.925340\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.11929 −0.305391 −0.152696 0.988273i \(-0.548795\pi\)
−0.152696 + 0.988273i \(0.548795\pi\)
\(282\) 0 0
\(283\) 6.90663i 0.410556i −0.978704 0.205278i \(-0.934190\pi\)
0.978704 0.205278i \(-0.0658099\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.56534 0.562667
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26.3527i 1.53954i 0.638321 + 0.769770i \(0.279629\pi\)
−0.638321 + 0.769770i \(0.720371\pi\)
\(294\) 0 0
\(295\) −21.9287 17.3820i −1.27674 1.01202i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.72666 −0.273350
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.0093 + 8.72666i 0.630392 + 0.499687i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.8973 0.674634 0.337317 0.941391i \(-0.390481\pi\)
0.337317 + 0.941391i \(0.390481\pi\)
\(312\) 0 0
\(313\) 17.7360i 1.00250i −0.865303 0.501249i \(-0.832874\pi\)
0.865303 0.501249i \(-0.167126\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.32469i 0.0744023i 0.999308 + 0.0372011i \(0.0118442\pi\)
−0.999308 + 0.0372011i \(0.988156\pi\)
\(318\) 0 0
\(319\) 11.3693 0.636557
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.98134i 0.110245i
\(324\) 0 0
\(325\) −4.86799 + 1.14134i −0.270028 + 0.0633099i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.73599 −0.205348 −0.102674 0.994715i \(-0.532740\pi\)
−0.102674 + 0.994715i \(0.532740\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −17.4533 + 22.0187i −0.953576 + 1.20301i
\(336\) 0 0
\(337\) 23.3947i 1.27439i 0.770702 + 0.637195i \(0.219906\pi\)
−0.770702 + 0.637195i \(0.780094\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.52803 0.245207
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 30.5840i 1.64184i 0.571047 + 0.820918i \(0.306538\pi\)
−0.571047 + 0.820918i \(0.693462\pi\)
\(348\) 0 0
\(349\) −1.37605 −0.0736581 −0.0368290 0.999322i \(-0.511726\pi\)
−0.0368290 + 0.999322i \(0.511726\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.9087i 1.05963i −0.848112 0.529817i \(-0.822261\pi\)
0.848112 0.529817i \(-0.177739\pi\)
\(354\) 0 0
\(355\) −8.37266 6.63667i −0.444374 0.352238i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.6940 1.40885 0.704427 0.709777i \(-0.251204\pi\)
0.704427 + 0.709777i \(0.251204\pi\)
\(360\) 0 0
\(361\) −18.4720 −0.972209
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.6940 + 21.0607i −0.873802 + 1.10237i
\(366\) 0 0
\(367\) 8.12136i 0.423932i −0.977277 0.211966i \(-0.932013\pi\)
0.977277 0.211966i \(-0.0679865\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 9.47197i 0.490440i 0.969467 + 0.245220i \(0.0788602\pi\)
−0.969467 + 0.245220i \(0.921140\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.55602i 0.389155i
\(378\) 0 0
\(379\) −16.7267 −0.859191 −0.429595 0.903022i \(-0.641344\pi\)
−0.429595 + 0.903022i \(0.641344\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.6846i 1.00584i −0.864334 0.502919i \(-0.832259\pi\)
0.864334 0.502919i \(-0.167741\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 12.8880 0.651773
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.24065 7.32469i −0.464948 0.368545i
\(396\) 0 0
\(397\) 3.35061i 0.168162i 0.996459 + 0.0840812i \(0.0267955\pi\)
−0.996459 + 0.0840812i \(0.973205\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.3713 1.81630 0.908149 0.418647i \(-0.137496\pi\)
0.908149 + 0.418647i \(0.137496\pi\)
\(402\) 0 0
\(403\) 3.00933i 0.149905i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.53736i 0.373613i
\(408\) 0 0
\(409\) −35.5933 −1.75998 −0.879988 0.474995i \(-0.842450\pi\)
−0.879988 + 0.474995i \(0.842450\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.8166 13.6460i 0.530968 0.669856i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −13.9160 −0.679839 −0.339919 0.940455i \(-0.610400\pi\)
−0.339919 + 0.940455i \(0.610400\pi\)
\(420\) 0 0
\(421\) 4.70800 0.229454 0.114727 0.993397i \(-0.463401\pi\)
0.114727 + 0.993397i \(0.463401\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.2733 3.11203i 0.643852 0.150956i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.89004 −0.380050 −0.190025 0.981779i \(-0.560857\pi\)
−0.190025 + 0.981779i \(0.560857\pi\)
\(432\) 0 0
\(433\) 2.30133i 0.110595i −0.998470 0.0552974i \(-0.982389\pi\)
0.998470 0.0552974i \(-0.0176107\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.43466i 0.164302i
\(438\) 0 0
\(439\) −41.1307 −1.96306 −0.981530 0.191307i \(-0.938728\pi\)
−0.981530 + 0.191307i \(0.938728\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 29.5933i 1.40602i 0.711179 + 0.703011i \(0.248161\pi\)
−0.711179 + 0.703011i \(0.751839\pi\)
\(444\) 0 0
\(445\) 3.13201 + 2.48262i 0.148471 + 0.117687i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11.4461 −0.540173 −0.270086 0.962836i \(-0.587052\pi\)
−0.270086 + 0.962836i \(0.587052\pi\)
\(450\) 0 0
\(451\) 8.70800 0.410044
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.5467i 0.774021i 0.922075 + 0.387011i \(0.126492\pi\)
−0.922075 + 0.387011i \(0.873508\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.50466 −0.163228 −0.0816142 0.996664i \(-0.526008\pi\)
−0.0816142 + 0.996664i \(0.526008\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.44398i 0.298192i −0.988823 0.149096i \(-0.952364\pi\)
0.988823 0.149096i \(-0.0476363\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.10270i 0.188642i
\(474\) 0 0
\(475\) −0.829359 3.53736i −0.0380536 0.162305i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.2313 1.01577 0.507887 0.861423i \(-0.330427\pi\)
0.507887 + 0.861423i \(0.330427\pi\)
\(480\) 0 0
\(481\) −5.00933 −0.228406
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.33402 + 10.5140i −0.378429 + 0.477416i
\(486\) 0 0
\(487\) 13.9160i 0.630592i −0.948993 0.315296i \(-0.897896\pi\)
0.948993 0.315296i \(-0.102104\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.1400 1.26994 0.634971 0.772536i \(-0.281012\pi\)
0.634971 + 0.772536i \(0.281012\pi\)
\(492\) 0 0
\(493\) 20.6027i 0.927897i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −38.8480 −1.73908 −0.869538 0.493866i \(-0.835583\pi\)
−0.869538 + 0.493866i \(0.835583\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.19863i 0.365559i 0.983154 + 0.182779i \(0.0585094\pi\)
−0.983154 + 0.182779i \(0.941491\pi\)
\(504\) 0 0
\(505\) 5.24065 + 4.15405i 0.233206 + 0.184853i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.0700 0.712291 0.356145 0.934431i \(-0.384091\pi\)
0.356145 + 0.934431i \(0.384091\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.616696 0.778008i 0.0271749 0.0342831i
\(516\) 0 0
\(517\) 15.3947i 0.677058i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.0280 1.00887 0.504437 0.863448i \(-0.331700\pi\)
0.504437 + 0.863448i \(0.331700\pi\)
\(522\) 0 0
\(523\) 5.47875i 0.239569i −0.992800 0.119784i \(-0.961780\pi\)
0.992800 0.119784i \(-0.0382204\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.20541i 0.357433i
\(528\) 0 0
\(529\) 0.658719 0.0286399
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.78734i 0.250677i
\(534\) 0 0
\(535\) −9.73599 + 12.2827i −0.420923 + 0.531026i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.5327 −0.453673
\(540\) 0 0
\(541\) 17.8387 0.766945 0.383473 0.923552i \(-0.374728\pi\)
0.383473 + 0.923552i \(0.374728\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.2500 + 19.2220i 1.03875 + 0.823380i
\(546\) 0 0
\(547\) 29.1053i 1.24445i 0.782838 + 0.622225i \(0.213771\pi\)
−0.782838 + 0.622225i \(0.786229\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.49063 0.233909
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 44.0114i 1.86482i −0.361399 0.932411i \(-0.617701\pi\)
0.361399 0.932411i \(-0.382299\pi\)
\(558\) 0 0
\(559\) −2.72666 −0.115325
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.43466i 0.313333i 0.987652 + 0.156667i \(0.0500748\pi\)
−0.987652 + 0.156667i \(0.949925\pi\)
\(564\) 0 0
\(565\) −5.94865 + 7.50466i −0.250262 + 0.315724i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.48130 0.355555 0.177777 0.984071i \(-0.443109\pi\)
0.177777 + 0.984071i \(0.443109\pi\)
\(570\) 0 0
\(571\) 40.7826 1.70670 0.853350 0.521339i \(-0.174567\pi\)
0.853350 + 0.521339i \(0.174567\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −23.0093 + 5.39470i −0.959555 + 0.224975i
\(576\) 0 0
\(577\) 1.57467i 0.0655545i 0.999463 + 0.0327773i \(0.0104352\pi\)
−0.999463 + 0.0327773i \(0.989565\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.3693i 0.470867i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.8247i 1.14845i −0.818699 0.574223i \(-0.805304\pi\)
0.818699 0.574223i \(-0.194696\pi\)
\(588\) 0 0
\(589\) 2.18675 0.0901034
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 32.6940i 1.34258i 0.741195 + 0.671290i \(0.234260\pi\)
−0.741195 + 0.671290i \(0.765740\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.1400 −1.31321 −0.656603 0.754237i \(-0.728007\pi\)
−0.656603 + 0.754237i \(0.728007\pi\)
\(600\) 0 0
\(601\) 40.8667 1.66699 0.833493 0.552530i \(-0.186337\pi\)
0.833493 + 0.552530i \(0.186337\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −15.3083 12.1343i −0.622373 0.493330i
\(606\) 0 0
\(607\) 14.9907i 0.608453i 0.952600 + 0.304226i \(0.0983979\pi\)
−0.952600 + 0.304226i \(0.901602\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10.2313 0.413915
\(612\) 0 0
\(613\) 13.5747i 0.548276i −0.961690 0.274138i \(-0.911608\pi\)
0.961690 0.274138i \(-0.0883925\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.69396i 0.188972i 0.995526 + 0.0944859i \(0.0301207\pi\)
−0.995526 + 0.0944859i \(0.969879\pi\)
\(618\) 0 0
\(619\) 20.8667 0.838702 0.419351 0.907824i \(-0.362258\pi\)
0.419351 + 0.907824i \(0.362258\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −22.3947 + 11.1120i −0.895788 + 0.444481i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.6587 0.544609
\(630\) 0 0
\(631\) 45.9533 1.82937 0.914685 0.404167i \(-0.132438\pi\)
0.914685 + 0.404167i \(0.132438\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.94139 + 10.0187i −0.315144 + 0.397578i
\(636\) 0 0
\(637\) 7.00000i 0.277350i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.0187 −0.790689 −0.395345 0.918533i \(-0.629375\pi\)
−0.395345 + 0.918533i \(0.629375\pi\)
\(642\) 0 0
\(643\) 26.5840i 1.04837i 0.851604 + 0.524185i \(0.175630\pi\)
−0.851604 + 0.524185i \(0.824370\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.39470i 0.369344i −0.982800 0.184672i \(-0.940878\pi\)
0.982800 0.184672i \(-0.0591223\pi\)
\(648\) 0 0
\(649\) 18.8294 0.739117
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000i 0.0782660i 0.999234 + 0.0391330i \(0.0124596\pi\)
−0.999234 + 0.0391330i \(0.987540\pi\)
\(654\) 0 0
\(655\) −9.73599 7.71733i −0.380416 0.301541i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 47.6774 1.85725 0.928623 0.371024i \(-0.120993\pi\)
0.928623 + 0.371024i \(0.120993\pi\)
\(660\) 0 0
\(661\) −22.0959 −0.859432 −0.429716 0.902964i \(-0.641386\pi\)
−0.429716 + 0.902964i \(0.641386\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7147i 1.38288i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.45331 −0.364941
\(672\) 0 0
\(673\) 48.6027i 1.87349i 0.350006 + 0.936747i \(0.386180\pi\)
−0.350006 + 0.936747i \(0.613820\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.4253i 0.400678i −0.979727 0.200339i \(-0.935796\pi\)
0.979727 0.200339i \(-0.0642043\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.13795i 0.349654i −0.984599 0.174827i \(-0.944063\pi\)
0.984599 0.174827i \(-0.0559366\pi\)
\(684\) 0 0
\(685\) 9.27203 11.6974i 0.354266 0.446933i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.55602 −0.287861
\(690\) 0 0
\(691\) 33.7173 1.28267 0.641334 0.767262i \(-0.278381\pi\)
0.641334 + 0.767262i \(0.278381\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34.1214 27.0466i −1.29430 1.02594i
\(696\) 0 0
\(697\) 15.7801i 0.597713i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.9813 0.754685 0.377342 0.926074i \(-0.376838\pi\)
0.377342 + 0.926074i \(0.376838\pi\)
\(702\) 0 0
\(703\) 3.64006i 0.137288i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 43.3293 1.62727 0.813633 0.581378i \(-0.197486\pi\)
0.813633 + 0.581378i \(0.197486\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.2241i 0.532695i
\(714\) 0 0
\(715\) 2.08998 2.63667i 0.0781610 0.0986059i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.6867 −1.29360 −0.646798 0.762661i \(-0.723892\pi\)
−0.646798 + 0.762661i \(0.723892\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.62395 36.7826i −0.320286 1.36607i
\(726\) 0 0
\(727\) 5.39470i 0.200078i −0.994983 0.100039i \(-0.968103\pi\)
0.994983 0.100039i \(-0.0318968\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.43466 0.274981
\(732\) 0 0
\(733\) 9.11203i 0.336561i 0.985739 + 0.168280i \(0.0538214\pi\)
−0.985739 + 0.168280i \(0.946179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18.9066i 0.696435i
\(738\) 0 0
\(739\) 4.82936 0.177651 0.0888254 0.996047i \(-0.471689\pi\)
0.0888254 + 0.996047i \(0.471689\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 50.4087i 1.84931i −0.380801 0.924657i \(-0.624352\pi\)
0.380801 0.924657i \(-0.375648\pi\)
\(744\) 0 0
\(745\) −25.4333 20.1600i −0.931805 0.738605i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.88797 0.178365 0.0891823 0.996015i \(-0.471575\pi\)
0.0891823 + 0.996015i \(0.471575\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −7.82003 6.19863i −0.284600 0.225591i
\(756\) 0 0
\(757\) 32.2241i 1.17120i −0.810599 0.585602i \(-0.800858\pi\)
0.810599 0.585602i \(-0.199142\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.4740 −1.32218 −0.661091 0.750305i \(-0.729906\pi\)
−0.661091 + 0.750305i \(0.729906\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.5140i 0.451854i
\(768\) 0 0
\(769\) −13.1120 −0.472832 −0.236416 0.971652i \(-0.575973\pi\)
−0.236416 + 0.971652i \(0.575973\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.5913i 0.452876i 0.974026 + 0.226438i \(0.0727081\pi\)
−0.974026 + 0.226438i \(0.927292\pi\)
\(774\) 0 0
\(775\) −3.43466 14.6494i −0.123376 0.526222i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.20541 0.150674
\(780\) 0 0
\(781\) 7.18930 0.257253
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.5306 14.5467i 0.411544 0.519194i
\(786\) 0 0
\(787\) 36.9253i 1.31624i −0.752911 0.658122i \(-0.771351\pi\)
0.752911 0.658122i \(-0.228649\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.28267i 0.223104i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.0466i 0.887198i −0.896225 0.443599i \(-0.853702\pi\)
0.896225 0.443599i \(-0.146298\pi\)
\(798\) 0 0
\(799\) −27.8973 −0.986935
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 18.0840i 0.638172i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 11.2334 0.394945 0.197473 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(810\) 0 0
\(811\) 31.8387 1.11801 0.559004 0.829165i \(-0.311184\pi\)
0.559004 + 0.829165i \(0.311184\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.2920 19.2920i 0.535655 0.675769i
\(816\) 0 0
\(817\) 1.98134i 0.0693184i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32.1073 1.12055 0.560277 0.828306i \(-0.310695\pi\)
0.560277 + 0.828306i \(0.310695\pi\)
\(822\) 0 0
\(823\) 46.7054i 1.62805i −0.580832 0.814023i \(-0.697273\pi\)
0.580832 0.814023i \(-0.302727\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.6940i 0.928240i 0.885772 + 0.464120i \(0.153629\pi\)
−0.885772 + 0.464120i \(0.846371\pi\)
\(828\) 0 0
\(829\) 28.8294 1.00129 0.500643 0.865654i \(-0.333097\pi\)
0.500643 + 0.865654i \(0.333097\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 19.0866i 0.661311i
\(834\) 0 0
\(835\) 8.01272 10.1086i 0.277292 0.349824i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.68463 0.265303 0.132652 0.991163i \(-0.457651\pi\)
0.132652 + 0.991163i \(0.457651\pi\)
\(840\) 0 0
\(841\) 28.0934 0.968737
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.75233 1.38900i −0.0602821 0.0477832i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −23.6774 −0.811650
\(852\) 0 0
\(853\) 18.4626i 0.632149i 0.948734 + 0.316074i \(0.102365\pi\)
−0.948734 + 0.316074i \(0.897635\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.2827i 1.10276i −0.834256 0.551378i \(-0.814102\pi\)
0.834256 0.551378i \(-0.185898\pi\)
\(858\) 0 0
\(859\) 18.9694 0.647227 0.323613 0.946189i \(-0.395102\pi\)
0.323613 + 0.946189i \(0.395102\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.8807i 0.574626i −0.957837 0.287313i \(-0.907238\pi\)
0.957837 0.287313i \(-0.0927620\pi\)
\(864\) 0 0
\(865\) 6.81532 8.59804i 0.231728 0.292342i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.93461 0.269163
\(870\) 0 0
\(871\) −12.5653 −0.425760
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 28.1214i 0.949591i −0.880096 0.474795i \(-0.842522\pi\)
0.880096 0.474795i \(-0.157478\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 49.4066 1.66455 0.832275 0.554363i \(-0.187038\pi\)
0.832275 + 0.554363i \(0.187038\pi\)
\(882\) 0 0
\(883\) 5.65872i 0.190431i −0.995457 0.0952155i \(-0.969646\pi\)
0.995457 0.0952155i \(-0.0303540\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.2640i 0.949013i 0.880252 + 0.474506i \(0.157373\pi\)
−0.880252 + 0.474506i \(0.842627\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.43466i 0.248791i
\(894\) 0 0
\(895\) 16.5653 + 13.1307i 0.553718 + 0.438911i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22.7385 0.758373
\(900\) 0 0
\(901\) 20.6027 0.686374
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 30.6167 + 24.2686i 1.01773 + 0.806717i
\(906\) 0 0
\(907\) 14.5840i 0.484254i 0.970245 + 0.242127i \(0.0778450\pi\)
−0.970245 + 0.242127i \(0.922155\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −29.1680 −0.966379 −0.483190 0.875516i \(-0.660522\pi\)
−0.483190 + 0.875516i \(0.660522\pi\)
\(912\) 0 0
\(913\) 11.7173i 0.387787i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.36672 −0.0780708 −0.0390354 0.999238i \(-0.512429\pi\)
−0.0390354 + 0.999238i \(0.512429\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.77801i 0.157270i
\(924\) 0 0
\(925\) −24.3854 + 5.71733i −0.801786 + 0.187985i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 31.1753 1.02283 0.511414 0.859335i \(-0.329122\pi\)
0.511414 + 0.859335i \(0.329122\pi\)
\(930\) 0 0
\(931\) −5.08660 −0.166706
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.69867 + 7.18930i −0.186366 + 0.235115i
\(936\) 0 0
\(937\) 6.46942i 0.211347i −0.994401 0.105673i \(-0.966300\pi\)
0.994401 0.105673i \(-0.0336998\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 19.4020 0.632486 0.316243 0.948678i \(-0.397579\pi\)
0.316243 + 0.948678i \(0.397579\pi\)
\(942\) 0 0
\(943\) 27.3548i 0.890793i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50.0632i 1.62684i 0.581679 + 0.813418i \(0.302396\pi\)
−0.581679 + 0.813418i \(0.697604\pi\)
\(948\) 0 0
\(949\) −12.0187 −0.390142
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.30133i 0.204120i −0.994778 0.102060i \(-0.967457\pi\)
0.994778 0.102060i \(-0.0325434\pi\)
\(954\) 0 0
\(955\) 19.4720 + 15.4347i 0.630098 + 0.499454i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21.9439 −0.707869
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.47668 + 10.6940i −0.272874 + 0.344251i
\(966\) 0 0
\(967\) 37.2334i 1.19735i 0.800994 + 0.598673i \(0.204305\pi\)
−0.800994 + 0.598673i \(0.795695\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −13.4533 −0.431737 −0.215869 0.976422i \(-0.569258\pi\)
−0.215869 + 0.976422i \(0.569258\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 37.4647i 1.19860i −0.800524 0.599301i \(-0.795445\pi\)
0.800524 0.599301i \(-0.204555\pi\)
\(978\) 0 0
\(979\) −2.68934 −0.0859517
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 29.2406i 0.932632i −0.884618 0.466316i \(-0.845581\pi\)
0.884618 0.466316i \(-0.154419\pi\)
\(984\) 0 0
\(985\) −29.6719 + 37.4333i −0.945426 + 1.19273i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.8880 −0.409814
\(990\) 0 0
\(991\) 3.11203 0.0988569 0.0494285 0.998778i \(-0.484260\pi\)
0.0494285 + 0.998778i \(0.484260\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −14.6940 11.6473i −0.465830 0.369245i
\(996\) 0 0
\(997\) 42.3200i 1.34029i 0.742231 + 0.670144i \(0.233768\pi\)
−0.742231 + 0.670144i \(0.766232\pi\)
\(998\) 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 4680.2.l.e.2809.6 6
3.2 odd 2 1560.2.l.c.1249.4 yes 6
5.4 even 2 inner 4680.2.l.e.2809.5 6
12.11 even 2 3120.2.l.m.1249.1 6
15.2 even 4 7800.2.a.bp.1.3 3
15.8 even 4 7800.2.a.bj.1.3 3
15.14 odd 2 1560.2.l.c.1249.1 6
60.59 even 2 3120.2.l.m.1249.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.c.1249.1 6 15.14 odd 2
1560.2.l.c.1249.4 yes 6 3.2 odd 2
3120.2.l.m.1249.1 6 12.11 even 2
3120.2.l.m.1249.4 6 60.59 even 2
4680.2.l.e.2809.5 6 5.4 even 2 inner
4680.2.l.e.2809.6 6 1.1 even 1 trivial
7800.2.a.bj.1.3 3 15.8 even 4
7800.2.a.bp.1.3 3 15.2 even 4