Properties

Label 2475.2.c.k.199.4
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(199,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2475.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (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: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.k.199.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.30278i q^{2} -3.30278 q^{4} -4.30278i q^{7} -3.00000i q^{8} +O(q^{10})\) \(q+2.30278i q^{2} -3.30278 q^{4} -4.30278i q^{7} -3.00000i q^{8} +1.00000 q^{11} +5.00000i q^{13} +9.90833 q^{14} +0.302776 q^{16} -3.90833i q^{17} +1.00000 q^{19} +2.30278i q^{22} +3.69722i q^{23} -11.5139 q^{26} +14.2111i q^{28} -9.90833 q^{29} -4.21110 q^{31} -5.30278i q^{32} +9.00000 q^{34} -9.60555i q^{37} +2.30278i q^{38} -1.60555 q^{41} -7.21110i q^{43} -3.30278 q^{44} -8.51388 q^{46} -3.00000i q^{47} -11.5139 q^{49} -16.5139i q^{52} -2.30278i q^{53} -12.9083 q^{56} -22.8167i q^{58} +0.211103 q^{59} +2.90833 q^{61} -9.69722i q^{62} +12.8167 q^{64} +4.00000i q^{67} +12.9083i q^{68} -4.60555 q^{71} +2.90833i q^{73} +22.1194 q^{74} -3.30278 q^{76} -4.30278i q^{77} +0.0916731 q^{79} -3.69722i q^{82} -14.5139i q^{83} +16.6056 q^{86} -3.00000i q^{88} +5.30278 q^{89} +21.5139 q^{91} -12.2111i q^{92} +6.90833 q^{94} -11.6972i q^{97} -26.5139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30278i 1.62831i 0.580649 + 0.814154i \(0.302799\pi\)
−0.580649 + 0.814154i \(0.697201\pi\)
\(3\) 0 0
\(4\) −3.30278 −1.65139
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.30278i − 1.62630i −0.582057 0.813148i \(-0.697752\pi\)
0.582057 0.813148i \(-0.302248\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 9.90833 2.64811
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) − 3.90833i − 0.947909i −0.880549 0.473954i \(-0.842826\pi\)
0.880549 0.473954i \(-0.157174\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.30278i 0.490953i
\(23\) 3.69722i 0.770925i 0.922724 + 0.385462i \(0.125958\pi\)
−0.922724 + 0.385462i \(0.874042\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −11.5139 −2.25806
\(27\) 0 0
\(28\) 14.2111i 2.68565i
\(29\) −9.90833 −1.83993 −0.919965 0.392000i \(-0.871783\pi\)
−0.919965 + 0.392000i \(0.871783\pi\)
\(30\) 0 0
\(31\) −4.21110 −0.756336 −0.378168 0.925737i \(-0.623446\pi\)
−0.378168 + 0.925737i \(0.623446\pi\)
\(32\) − 5.30278i − 0.937407i
\(33\) 0 0
\(34\) 9.00000 1.54349
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.60555i − 1.57914i −0.613659 0.789571i \(-0.710303\pi\)
0.613659 0.789571i \(-0.289697\pi\)
\(38\) 2.30278i 0.373560i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.60555 −0.250745 −0.125372 0.992110i \(-0.540013\pi\)
−0.125372 + 0.992110i \(0.540013\pi\)
\(42\) 0 0
\(43\) − 7.21110i − 1.09968i −0.835269 0.549841i \(-0.814688\pi\)
0.835269 0.549841i \(-0.185312\pi\)
\(44\) −3.30278 −0.497912
\(45\) 0 0
\(46\) −8.51388 −1.25530
\(47\) − 3.00000i − 0.437595i −0.975770 0.218797i \(-0.929787\pi\)
0.975770 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −11.5139 −1.64484
\(50\) 0 0
\(51\) 0 0
\(52\) − 16.5139i − 2.29006i
\(53\) − 2.30278i − 0.316311i −0.987414 0.158155i \(-0.949445\pi\)
0.987414 0.158155i \(-0.0505547\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.9083 −1.72495
\(57\) 0 0
\(58\) − 22.8167i − 2.99597i
\(59\) 0.211103 0.0274832 0.0137416 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) − 9.69722i − 1.23155i
\(63\) 0 0
\(64\) 12.8167 1.60208
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 12.9083i 1.56536i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.60555 −0.546578 −0.273289 0.961932i \(-0.588112\pi\)
−0.273289 + 0.961932i \(0.588112\pi\)
\(72\) 0 0
\(73\) 2.90833i 0.340394i 0.985410 + 0.170197i \(0.0544404\pi\)
−0.985410 + 0.170197i \(0.945560\pi\)
\(74\) 22.1194 2.57133
\(75\) 0 0
\(76\) −3.30278 −0.378854
\(77\) − 4.30278i − 0.490347i
\(78\) 0 0
\(79\) 0.0916731 0.0103140 0.00515701 0.999987i \(-0.498358\pi\)
0.00515701 + 0.999987i \(0.498358\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 3.69722i − 0.408290i
\(83\) − 14.5139i − 1.59311i −0.604569 0.796553i \(-0.706655\pi\)
0.604569 0.796553i \(-0.293345\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 16.6056 1.79062
\(87\) 0 0
\(88\) − 3.00000i − 0.319801i
\(89\) 5.30278 0.562093 0.281047 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) − 12.2111i − 1.27310i
\(93\) 0 0
\(94\) 6.90833 0.712540
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.6972i − 1.18767i −0.804586 0.593837i \(-0.797613\pi\)
0.804586 0.593837i \(-0.202387\pi\)
\(98\) − 26.5139i − 2.67831i
\(99\) 0 0
\(100\) 0 0
\(101\) −17.5139 −1.74270 −0.871348 0.490666i \(-0.836754\pi\)
−0.871348 + 0.490666i \(0.836754\pi\)
\(102\) 0 0
\(103\) − 7.90833i − 0.779231i −0.920978 0.389615i \(-0.872608\pi\)
0.920978 0.389615i \(-0.127392\pi\)
\(104\) 15.0000 1.47087
\(105\) 0 0
\(106\) 5.30278 0.515051
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) 0 0
\(109\) 6.51388 0.623916 0.311958 0.950096i \(-0.399015\pi\)
0.311958 + 0.950096i \(0.399015\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.30278i − 0.123101i
\(113\) − 10.8167i − 1.01755i −0.860901 0.508773i \(-0.830099\pi\)
0.860901 0.508773i \(-0.169901\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 32.7250 3.03844
\(117\) 0 0
\(118\) 0.486122i 0.0447511i
\(119\) −16.8167 −1.54158
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 6.69722i 0.606338i
\(123\) 0 0
\(124\) 13.9083 1.24900
\(125\) 0 0
\(126\) 0 0
\(127\) 17.1194i 1.51910i 0.650447 + 0.759552i \(0.274582\pi\)
−0.650447 + 0.759552i \(0.725418\pi\)
\(128\) 18.9083i 1.67128i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.908327 −0.0793609 −0.0396804 0.999212i \(-0.512634\pi\)
−0.0396804 + 0.999212i \(0.512634\pi\)
\(132\) 0 0
\(133\) − 4.30278i − 0.373098i
\(134\) −9.21110 −0.795718
\(135\) 0 0
\(136\) −11.7250 −1.00541
\(137\) − 2.09167i − 0.178704i −0.996000 0.0893518i \(-0.971520\pi\)
0.996000 0.0893518i \(-0.0284796\pi\)
\(138\) 0 0
\(139\) −8.21110 −0.696457 −0.348228 0.937410i \(-0.613217\pi\)
−0.348228 + 0.937410i \(0.613217\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 10.6056i − 0.889998i
\(143\) 5.00000i 0.418121i
\(144\) 0 0
\(145\) 0 0
\(146\) −6.69722 −0.554266
\(147\) 0 0
\(148\) 31.7250i 2.60778i
\(149\) 2.78890 0.228475 0.114238 0.993453i \(-0.463557\pi\)
0.114238 + 0.993453i \(0.463557\pi\)
\(150\) 0 0
\(151\) −20.8167 −1.69404 −0.847018 0.531565i \(-0.821604\pi\)
−0.847018 + 0.531565i \(0.821604\pi\)
\(152\) − 3.00000i − 0.243332i
\(153\) 0 0
\(154\) 9.90833 0.798436
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.78890i − 0.382196i −0.981571 0.191098i \(-0.938795\pi\)
0.981571 0.191098i \(-0.0612048\pi\)
\(158\) 0.211103i 0.0167944i
\(159\) 0 0
\(160\) 0 0
\(161\) 15.9083 1.25375
\(162\) 0 0
\(163\) 5.69722i 0.446241i 0.974791 + 0.223121i \(0.0716243\pi\)
−0.974791 + 0.223121i \(0.928376\pi\)
\(164\) 5.30278 0.414077
\(165\) 0 0
\(166\) 33.4222 2.59407
\(167\) − 15.4222i − 1.19341i −0.802462 0.596703i \(-0.796477\pi\)
0.802462 0.596703i \(-0.203523\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 23.8167i 1.81600i
\(173\) − 16.8167i − 1.27855i −0.768980 0.639273i \(-0.779235\pi\)
0.768980 0.639273i \(-0.220765\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.302776 0.0228226
\(177\) 0 0
\(178\) 12.2111i 0.915261i
\(179\) −5.51388 −0.412127 −0.206063 0.978539i \(-0.566065\pi\)
−0.206063 + 0.978539i \(0.566065\pi\)
\(180\) 0 0
\(181\) −9.09167 −0.675779 −0.337889 0.941186i \(-0.609713\pi\)
−0.337889 + 0.941186i \(0.609713\pi\)
\(182\) 49.5416i 3.67227i
\(183\) 0 0
\(184\) 11.0917 0.817689
\(185\) 0 0
\(186\) 0 0
\(187\) − 3.90833i − 0.285805i
\(188\) 9.90833i 0.722639i
\(189\) 0 0
\(190\) 0 0
\(191\) −6.69722 −0.484594 −0.242297 0.970202i \(-0.577901\pi\)
−0.242297 + 0.970202i \(0.577901\pi\)
\(192\) 0 0
\(193\) − 1.21110i − 0.0871771i −0.999050 0.0435885i \(-0.986121\pi\)
0.999050 0.0435885i \(-0.0138791\pi\)
\(194\) 26.9361 1.93390
\(195\) 0 0
\(196\) 38.0278 2.71627
\(197\) 9.69722i 0.690899i 0.938437 + 0.345449i \(0.112273\pi\)
−0.938437 + 0.345449i \(0.887727\pi\)
\(198\) 0 0
\(199\) 24.5139 1.73774 0.868871 0.495038i \(-0.164846\pi\)
0.868871 + 0.495038i \(0.164846\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 40.3305i − 2.83765i
\(203\) 42.6333i 2.99227i
\(204\) 0 0
\(205\) 0 0
\(206\) 18.2111 1.26883
\(207\) 0 0
\(208\) 1.51388i 0.104969i
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 25.2389 1.73751 0.868757 0.495238i \(-0.164919\pi\)
0.868757 + 0.495238i \(0.164919\pi\)
\(212\) 7.60555i 0.522351i
\(213\) 0 0
\(214\) 6.90833 0.472244
\(215\) 0 0
\(216\) 0 0
\(217\) 18.1194i 1.23003i
\(218\) 15.0000i 1.01593i
\(219\) 0 0
\(220\) 0 0
\(221\) 19.5416 1.31451
\(222\) 0 0
\(223\) 20.6333i 1.38171i 0.722994 + 0.690854i \(0.242765\pi\)
−0.722994 + 0.690854i \(0.757235\pi\)
\(224\) −22.8167 −1.52450
\(225\) 0 0
\(226\) 24.9083 1.65688
\(227\) 5.30278i 0.351958i 0.984394 + 0.175979i \(0.0563090\pi\)
−0.984394 + 0.175979i \(0.943691\pi\)
\(228\) 0 0
\(229\) −13.7250 −0.906972 −0.453486 0.891263i \(-0.649820\pi\)
−0.453486 + 0.891263i \(0.649820\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 29.7250i 1.95154i
\(233\) − 5.09167i − 0.333567i −0.985994 0.166783i \(-0.946662\pi\)
0.985994 0.166783i \(-0.0533380\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.697224 −0.0453854
\(237\) 0 0
\(238\) − 38.7250i − 2.51017i
\(239\) −4.11943 −0.266464 −0.133232 0.991085i \(-0.542535\pi\)
−0.133232 + 0.991085i \(0.542535\pi\)
\(240\) 0 0
\(241\) −24.9361 −1.60627 −0.803137 0.595794i \(-0.796837\pi\)
−0.803137 + 0.595794i \(0.796837\pi\)
\(242\) 2.30278i 0.148028i
\(243\) 0 0
\(244\) −9.60555 −0.614932
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000i 0.318142i
\(248\) 12.6333i 0.802216i
\(249\) 0 0
\(250\) 0 0
\(251\) 3.90833 0.246691 0.123346 0.992364i \(-0.460638\pi\)
0.123346 + 0.992364i \(0.460638\pi\)
\(252\) 0 0
\(253\) 3.69722i 0.232443i
\(254\) −39.4222 −2.47357
\(255\) 0 0
\(256\) −17.9083 −1.11927
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) −41.3305 −2.56815
\(260\) 0 0
\(261\) 0 0
\(262\) − 2.09167i − 0.129224i
\(263\) 1.18335i 0.0729683i 0.999334 + 0.0364841i \(0.0116158\pi\)
−0.999334 + 0.0364841i \(0.988384\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 9.90833 0.607519
\(267\) 0 0
\(268\) − 13.2111i − 0.806997i
\(269\) −23.7250 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(270\) 0 0
\(271\) 14.2111 0.863263 0.431632 0.902050i \(-0.357938\pi\)
0.431632 + 0.902050i \(0.357938\pi\)
\(272\) − 1.18335i − 0.0717509i
\(273\) 0 0
\(274\) 4.81665 0.290985
\(275\) 0 0
\(276\) 0 0
\(277\) − 21.6056i − 1.29815i −0.760724 0.649076i \(-0.775156\pi\)
0.760724 0.649076i \(-0.224844\pi\)
\(278\) − 18.9083i − 1.13405i
\(279\) 0 0
\(280\) 0 0
\(281\) 22.8167 1.36113 0.680564 0.732689i \(-0.261735\pi\)
0.680564 + 0.732689i \(0.261735\pi\)
\(282\) 0 0
\(283\) 2.69722i 0.160333i 0.996781 + 0.0801667i \(0.0255453\pi\)
−0.996781 + 0.0801667i \(0.974455\pi\)
\(284\) 15.2111 0.902613
\(285\) 0 0
\(286\) −11.5139 −0.680830
\(287\) 6.90833i 0.407786i
\(288\) 0 0
\(289\) 1.72498 0.101469
\(290\) 0 0
\(291\) 0 0
\(292\) − 9.60555i − 0.562122i
\(293\) − 15.2111i − 0.888642i −0.895868 0.444321i \(-0.853445\pi\)
0.895868 0.444321i \(-0.146555\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −28.8167 −1.67493
\(297\) 0 0
\(298\) 6.42221i 0.372028i
\(299\) −18.4861 −1.06908
\(300\) 0 0
\(301\) −31.0278 −1.78841
\(302\) − 47.9361i − 2.75841i
\(303\) 0 0
\(304\) 0.302776 0.0173654
\(305\) 0 0
\(306\) 0 0
\(307\) 6.09167i 0.347670i 0.984775 + 0.173835i \(0.0556160\pi\)
−0.984775 + 0.173835i \(0.944384\pi\)
\(308\) 14.2111i 0.809753i
\(309\) 0 0
\(310\) 0 0
\(311\) 16.8167 0.953585 0.476792 0.879016i \(-0.341799\pi\)
0.476792 + 0.879016i \(0.341799\pi\)
\(312\) 0 0
\(313\) 21.8167i 1.23315i 0.787296 + 0.616575i \(0.211480\pi\)
−0.787296 + 0.616575i \(0.788520\pi\)
\(314\) 11.0278 0.622332
\(315\) 0 0
\(316\) −0.302776 −0.0170325
\(317\) 9.90833i 0.556507i 0.960508 + 0.278254i \(0.0897555\pi\)
−0.960508 + 0.278254i \(0.910244\pi\)
\(318\) 0 0
\(319\) −9.90833 −0.554760
\(320\) 0 0
\(321\) 0 0
\(322\) 36.6333i 2.04149i
\(323\) − 3.90833i − 0.217465i
\(324\) 0 0
\(325\) 0 0
\(326\) −13.1194 −0.726618
\(327\) 0 0
\(328\) 4.81665i 0.265955i
\(329\) −12.9083 −0.711659
\(330\) 0 0
\(331\) −14.3944 −0.791190 −0.395595 0.918425i \(-0.629462\pi\)
−0.395595 + 0.918425i \(0.629462\pi\)
\(332\) 47.9361i 2.63083i
\(333\) 0 0
\(334\) 35.5139 1.94323
\(335\) 0 0
\(336\) 0 0
\(337\) − 26.8444i − 1.46231i −0.682212 0.731154i \(-0.738982\pi\)
0.682212 0.731154i \(-0.261018\pi\)
\(338\) − 27.6333i − 1.50305i
\(339\) 0 0
\(340\) 0 0
\(341\) −4.21110 −0.228044
\(342\) 0 0
\(343\) 19.4222i 1.04870i
\(344\) −21.6333 −1.16639
\(345\) 0 0
\(346\) 38.7250 2.08187
\(347\) 5.51388i 0.296000i 0.988987 + 0.148000i \(0.0472836\pi\)
−0.988987 + 0.148000i \(0.952716\pi\)
\(348\) 0 0
\(349\) 26.8167 1.43546 0.717731 0.696320i \(-0.245181\pi\)
0.717731 + 0.696320i \(0.245181\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 5.30278i − 0.282639i
\(353\) 24.6333i 1.31110i 0.755152 + 0.655549i \(0.227563\pi\)
−0.755152 + 0.655549i \(0.772437\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −17.5139 −0.928234
\(357\) 0 0
\(358\) − 12.6972i − 0.671069i
\(359\) 15.2111 0.802811 0.401406 0.915900i \(-0.368522\pi\)
0.401406 + 0.915900i \(0.368522\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 20.9361i − 1.10038i
\(363\) 0 0
\(364\) −71.0555 −3.72432
\(365\) 0 0
\(366\) 0 0
\(367\) 24.3028i 1.26859i 0.773089 + 0.634297i \(0.218710\pi\)
−0.773089 + 0.634297i \(0.781290\pi\)
\(368\) 1.11943i 0.0583543i
\(369\) 0 0
\(370\) 0 0
\(371\) −9.90833 −0.514415
\(372\) 0 0
\(373\) − 1.42221i − 0.0736390i −0.999322 0.0368195i \(-0.988277\pi\)
0.999322 0.0368195i \(-0.0117227\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) − 49.5416i − 2.55152i
\(378\) 0 0
\(379\) −24.8167 −1.27475 −0.637373 0.770555i \(-0.719979\pi\)
−0.637373 + 0.770555i \(0.719979\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 15.4222i − 0.789069i
\(383\) 21.6333i 1.10541i 0.833377 + 0.552705i \(0.186404\pi\)
−0.833377 + 0.552705i \(0.813596\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.78890 0.141951
\(387\) 0 0
\(388\) 38.6333i 1.96131i
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 14.4500 0.730766
\(392\) 34.5416i 1.74462i
\(393\) 0 0
\(394\) −22.3305 −1.12500
\(395\) 0 0
\(396\) 0 0
\(397\) − 25.3028i − 1.26991i −0.772549 0.634955i \(-0.781019\pi\)
0.772549 0.634955i \(-0.218981\pi\)
\(398\) 56.4500i 2.82958i
\(399\) 0 0
\(400\) 0 0
\(401\) 27.2111 1.35886 0.679429 0.733741i \(-0.262228\pi\)
0.679429 + 0.733741i \(0.262228\pi\)
\(402\) 0 0
\(403\) − 21.0555i − 1.04885i
\(404\) 57.8444 2.87787
\(405\) 0 0
\(406\) −98.1749 −4.87234
\(407\) − 9.60555i − 0.476129i
\(408\) 0 0
\(409\) −8.21110 −0.406013 −0.203006 0.979177i \(-0.565071\pi\)
−0.203006 + 0.979177i \(0.565071\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 26.1194i 1.28681i
\(413\) − 0.908327i − 0.0446958i
\(414\) 0 0
\(415\) 0 0
\(416\) 26.5139 1.29995
\(417\) 0 0
\(418\) 2.30278i 0.112632i
\(419\) 13.6056 0.664675 0.332337 0.943161i \(-0.392163\pi\)
0.332337 + 0.943161i \(0.392163\pi\)
\(420\) 0 0
\(421\) 4.30278 0.209704 0.104852 0.994488i \(-0.466563\pi\)
0.104852 + 0.994488i \(0.466563\pi\)
\(422\) 58.1194i 2.82921i
\(423\) 0 0
\(424\) −6.90833 −0.335498
\(425\) 0 0
\(426\) 0 0
\(427\) − 12.5139i − 0.605589i
\(428\) 9.90833i 0.478937i
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) −41.7250 −2.00286
\(435\) 0 0
\(436\) −21.5139 −1.03033
\(437\) 3.69722i 0.176862i
\(438\) 0 0
\(439\) −20.6972 −0.987825 −0.493912 0.869512i \(-0.664434\pi\)
−0.493912 + 0.869512i \(0.664434\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 45.0000i 2.14043i
\(443\) 1.39445i 0.0662523i 0.999451 + 0.0331261i \(0.0105463\pi\)
−0.999451 + 0.0331261i \(0.989454\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −47.5139 −2.24985
\(447\) 0 0
\(448\) − 55.1472i − 2.60546i
\(449\) 41.5139 1.95916 0.979581 0.201052i \(-0.0644361\pi\)
0.979581 + 0.201052i \(0.0644361\pi\)
\(450\) 0 0
\(451\) −1.60555 −0.0756025
\(452\) 35.7250i 1.68036i
\(453\) 0 0
\(454\) −12.2111 −0.573095
\(455\) 0 0
\(456\) 0 0
\(457\) 24.3028i 1.13684i 0.822740 + 0.568418i \(0.192444\pi\)
−0.822740 + 0.568418i \(0.807556\pi\)
\(458\) − 31.6056i − 1.47683i
\(459\) 0 0
\(460\) 0 0
\(461\) −17.7889 −0.828512 −0.414256 0.910161i \(-0.635958\pi\)
−0.414256 + 0.910161i \(0.635958\pi\)
\(462\) 0 0
\(463\) 26.2111i 1.21813i 0.793119 + 0.609067i \(0.208456\pi\)
−0.793119 + 0.609067i \(0.791544\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 11.7250 0.543149
\(467\) 24.6333i 1.13989i 0.821682 + 0.569947i \(0.193036\pi\)
−0.821682 + 0.569947i \(0.806964\pi\)
\(468\) 0 0
\(469\) 17.2111 0.794735
\(470\) 0 0
\(471\) 0 0
\(472\) − 0.633308i − 0.0291503i
\(473\) − 7.21110i − 0.331567i
\(474\) 0 0
\(475\) 0 0
\(476\) 55.5416 2.54575
\(477\) 0 0
\(478\) − 9.48612i − 0.433885i
\(479\) −13.1833 −0.602362 −0.301181 0.953567i \(-0.597381\pi\)
−0.301181 + 0.953567i \(0.597381\pi\)
\(480\) 0 0
\(481\) 48.0278 2.18988
\(482\) − 57.4222i − 2.61551i
\(483\) 0 0
\(484\) −3.30278 −0.150126
\(485\) 0 0
\(486\) 0 0
\(487\) 10.2111i 0.462709i 0.972869 + 0.231355i \(0.0743158\pi\)
−0.972869 + 0.231355i \(0.925684\pi\)
\(488\) − 8.72498i − 0.394961i
\(489\) 0 0
\(490\) 0 0
\(491\) 24.2111 1.09263 0.546316 0.837579i \(-0.316030\pi\)
0.546316 + 0.837579i \(0.316030\pi\)
\(492\) 0 0
\(493\) 38.7250i 1.74409i
\(494\) −11.5139 −0.518034
\(495\) 0 0
\(496\) −1.27502 −0.0572501
\(497\) 19.8167i 0.888898i
\(498\) 0 0
\(499\) 21.5139 0.963093 0.481547 0.876420i \(-0.340075\pi\)
0.481547 + 0.876420i \(0.340075\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 9.00000i 0.401690i
\(503\) − 16.6056i − 0.740405i −0.928951 0.370202i \(-0.879288\pi\)
0.928951 0.370202i \(-0.120712\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.51388 −0.378488
\(507\) 0 0
\(508\) − 56.5416i − 2.50863i
\(509\) −26.3028 −1.16585 −0.582925 0.812526i \(-0.698092\pi\)
−0.582925 + 0.812526i \(0.698092\pi\)
\(510\) 0 0
\(511\) 12.5139 0.553581
\(512\) − 3.42221i − 0.151242i
\(513\) 0 0
\(514\) 41.4500 1.82828
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.00000i − 0.131940i
\(518\) − 95.1749i − 4.18175i
\(519\) 0 0
\(520\) 0 0
\(521\) −23.4500 −1.02736 −0.513681 0.857981i \(-0.671718\pi\)
−0.513681 + 0.857981i \(0.671718\pi\)
\(522\) 0 0
\(523\) − 3.57779i − 0.156446i −0.996936 0.0782230i \(-0.975075\pi\)
0.996936 0.0782230i \(-0.0249246\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) −2.72498 −0.118815
\(527\) 16.4584i 0.716938i
\(528\) 0 0
\(529\) 9.33053 0.405675
\(530\) 0 0
\(531\) 0 0
\(532\) 14.2111i 0.616129i
\(533\) − 8.02776i − 0.347721i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) − 54.6333i − 2.35541i
\(539\) −11.5139 −0.495938
\(540\) 0 0
\(541\) −6.72498 −0.289130 −0.144565 0.989495i \(-0.546178\pi\)
−0.144565 + 0.989495i \(0.546178\pi\)
\(542\) 32.7250i 1.40566i
\(543\) 0 0
\(544\) −20.7250 −0.888576
\(545\) 0 0
\(546\) 0 0
\(547\) − 18.1194i − 0.774731i −0.921926 0.387365i \(-0.873385\pi\)
0.921926 0.387365i \(-0.126615\pi\)
\(548\) 6.90833i 0.295109i
\(549\) 0 0
\(550\) 0 0
\(551\) −9.90833 −0.422109
\(552\) 0 0
\(553\) − 0.394449i − 0.0167737i
\(554\) 49.7527 2.11379
\(555\) 0 0
\(556\) 27.1194 1.15012
\(557\) − 9.42221i − 0.399232i −0.979874 0.199616i \(-0.936031\pi\)
0.979874 0.199616i \(-0.0639694\pi\)
\(558\) 0 0
\(559\) 36.0555 1.52499
\(560\) 0 0
\(561\) 0 0
\(562\) 52.5416i 2.21634i
\(563\) − 18.9083i − 0.796891i −0.917192 0.398445i \(-0.869550\pi\)
0.917192 0.398445i \(-0.130450\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.21110 −0.261072
\(567\) 0 0
\(568\) 13.8167i 0.579734i
\(569\) 15.1472 0.635003 0.317502 0.948258i \(-0.397156\pi\)
0.317502 + 0.948258i \(0.397156\pi\)
\(570\) 0 0
\(571\) −17.3305 −0.725260 −0.362630 0.931933i \(-0.618121\pi\)
−0.362630 + 0.931933i \(0.618121\pi\)
\(572\) − 16.5139i − 0.690480i
\(573\) 0 0
\(574\) −15.9083 −0.664001
\(575\) 0 0
\(576\) 0 0
\(577\) 31.3583i 1.30546i 0.757589 + 0.652731i \(0.226377\pi\)
−0.757589 + 0.652731i \(0.773623\pi\)
\(578\) 3.97224i 0.165224i
\(579\) 0 0
\(580\) 0 0
\(581\) −62.4500 −2.59086
\(582\) 0 0
\(583\) − 2.30278i − 0.0953712i
\(584\) 8.72498 0.361042
\(585\) 0 0
\(586\) 35.0278 1.44698
\(587\) − 37.5416i − 1.54951i −0.632262 0.774755i \(-0.717873\pi\)
0.632262 0.774755i \(-0.282127\pi\)
\(588\) 0 0
\(589\) −4.21110 −0.173515
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.90833i − 0.119531i
\(593\) 13.6056i 0.558713i 0.960187 + 0.279357i \(0.0901211\pi\)
−0.960187 + 0.279357i \(0.909879\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −9.21110 −0.377301
\(597\) 0 0
\(598\) − 42.5694i − 1.74079i
\(599\) −14.0917 −0.575770 −0.287885 0.957665i \(-0.592952\pi\)
−0.287885 + 0.957665i \(0.592952\pi\)
\(600\) 0 0
\(601\) 8.90833 0.363378 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(602\) − 71.4500i − 2.91208i
\(603\) 0 0
\(604\) 68.7527 2.79751
\(605\) 0 0
\(606\) 0 0
\(607\) 7.21110i 0.292690i 0.989234 + 0.146345i \(0.0467509\pi\)
−0.989234 + 0.146345i \(0.953249\pi\)
\(608\) − 5.30278i − 0.215056i
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) − 41.1194i − 1.66080i −0.557169 0.830399i \(-0.688112\pi\)
0.557169 0.830399i \(-0.311888\pi\)
\(614\) −14.0278 −0.566114
\(615\) 0 0
\(616\) −12.9083 −0.520091
\(617\) 10.6056i 0.426963i 0.976947 + 0.213482i \(0.0684804\pi\)
−0.976947 + 0.213482i \(0.931520\pi\)
\(618\) 0 0
\(619\) −17.4222 −0.700258 −0.350129 0.936702i \(-0.613862\pi\)
−0.350129 + 0.936702i \(0.613862\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 38.7250i 1.55273i
\(623\) − 22.8167i − 0.914130i
\(624\) 0 0
\(625\) 0 0
\(626\) −50.2389 −2.00795
\(627\) 0 0
\(628\) 15.8167i 0.631153i
\(629\) −37.5416 −1.49688
\(630\) 0 0
\(631\) −39.9361 −1.58983 −0.794915 0.606721i \(-0.792485\pi\)
−0.794915 + 0.606721i \(0.792485\pi\)
\(632\) − 0.275019i − 0.0109397i
\(633\) 0 0
\(634\) −22.8167 −0.906165
\(635\) 0 0
\(636\) 0 0
\(637\) − 57.5694i − 2.28098i
\(638\) − 22.8167i − 0.903320i
\(639\) 0 0
\(640\) 0 0
\(641\) −42.2111 −1.66724 −0.833619 0.552340i \(-0.813735\pi\)
−0.833619 + 0.552340i \(0.813735\pi\)
\(642\) 0 0
\(643\) − 22.0000i − 0.867595i −0.901010 0.433798i \(-0.857173\pi\)
0.901010 0.433798i \(-0.142827\pi\)
\(644\) −52.5416 −2.07043
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) 17.2389i 0.677729i 0.940835 + 0.338865i \(0.110043\pi\)
−0.940835 + 0.338865i \(0.889957\pi\)
\(648\) 0 0
\(649\) 0.211103 0.00828650
\(650\) 0 0
\(651\) 0 0
\(652\) − 18.8167i − 0.736917i
\(653\) 19.1194i 0.748201i 0.927388 + 0.374101i \(0.122049\pi\)
−0.927388 + 0.374101i \(0.877951\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.486122 −0.0189799
\(657\) 0 0
\(658\) − 29.7250i − 1.15880i
\(659\) −20.0917 −0.782660 −0.391330 0.920250i \(-0.627985\pi\)
−0.391330 + 0.920250i \(0.627985\pi\)
\(660\) 0 0
\(661\) 12.8167 0.498510 0.249255 0.968438i \(-0.419814\pi\)
0.249255 + 0.968438i \(0.419814\pi\)
\(662\) − 33.1472i − 1.28830i
\(663\) 0 0
\(664\) −43.5416 −1.68974
\(665\) 0 0
\(666\) 0 0
\(667\) − 36.6333i − 1.41845i
\(668\) 50.9361i 1.97078i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.90833 0.112275
\(672\) 0 0
\(673\) − 6.02776i − 0.232353i −0.993229 0.116176i \(-0.962936\pi\)
0.993229 0.116176i \(-0.0370638\pi\)
\(674\) 61.8167 2.38109
\(675\) 0 0
\(676\) 39.6333 1.52436
\(677\) − 26.2389i − 1.00844i −0.863575 0.504221i \(-0.831780\pi\)
0.863575 0.504221i \(-0.168220\pi\)
\(678\) 0 0
\(679\) −50.3305 −1.93151
\(680\) 0 0
\(681\) 0 0
\(682\) − 9.69722i − 0.371326i
\(683\) 9.84441i 0.376686i 0.982103 + 0.188343i \(0.0603116\pi\)
−0.982103 + 0.188343i \(0.939688\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −44.7250 −1.70761
\(687\) 0 0
\(688\) − 2.18335i − 0.0832393i
\(689\) 11.5139 0.438644
\(690\) 0 0
\(691\) −26.5416 −1.00969 −0.504846 0.863210i \(-0.668451\pi\)
−0.504846 + 0.863210i \(0.668451\pi\)
\(692\) 55.5416i 2.11138i
\(693\) 0 0
\(694\) −12.6972 −0.481980
\(695\) 0 0
\(696\) 0 0
\(697\) 6.27502i 0.237683i
\(698\) 61.7527i 2.33738i
\(699\) 0 0
\(700\) 0 0
\(701\) 26.7889 1.01180 0.505901 0.862591i \(-0.331160\pi\)
0.505901 + 0.862591i \(0.331160\pi\)
\(702\) 0 0
\(703\) − 9.60555i − 0.362280i
\(704\) 12.8167 0.483046
\(705\) 0 0
\(706\) −56.7250 −2.13487
\(707\) 75.3583i 2.83414i
\(708\) 0 0
\(709\) −11.6333 −0.436898 −0.218449 0.975848i \(-0.570100\pi\)
−0.218449 + 0.975848i \(0.570100\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 15.9083i − 0.596190i
\(713\) − 15.5694i − 0.583078i
\(714\) 0 0
\(715\) 0 0
\(716\) 18.2111 0.680581
\(717\) 0 0
\(718\) 35.0278i 1.30722i
\(719\) 28.8167 1.07468 0.537340 0.843366i \(-0.319429\pi\)
0.537340 + 0.843366i \(0.319429\pi\)
\(720\) 0 0
\(721\) −34.0278 −1.26726
\(722\) − 41.4500i − 1.54261i
\(723\) 0 0
\(724\) 30.0278 1.11597
\(725\) 0 0
\(726\) 0 0
\(727\) − 0.330532i − 0.0122588i −0.999981 0.00612938i \(-0.998049\pi\)
0.999981 0.00612938i \(-0.00195105\pi\)
\(728\) − 64.5416i − 2.39207i
\(729\) 0 0
\(730\) 0 0
\(731\) −28.1833 −1.04240
\(732\) 0 0
\(733\) 12.3944i 0.457799i 0.973450 + 0.228900i \(0.0735128\pi\)
−0.973450 + 0.228900i \(0.926487\pi\)
\(734\) −55.9638 −2.06566
\(735\) 0 0
\(736\) 19.6056 0.722670
\(737\) 4.00000i 0.147342i
\(738\) 0 0
\(739\) −9.88057 −0.363463 −0.181731 0.983348i \(-0.558170\pi\)
−0.181731 + 0.983348i \(0.558170\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 22.8167i − 0.837626i
\(743\) − 44.3028i − 1.62531i −0.582744 0.812656i \(-0.698021\pi\)
0.582744 0.812656i \(-0.301979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 3.27502 0.119907
\(747\) 0 0
\(748\) 12.9083i 0.471975i
\(749\) −12.9083 −0.471660
\(750\) 0 0
\(751\) −5.66947 −0.206882 −0.103441 0.994636i \(-0.532985\pi\)
−0.103441 + 0.994636i \(0.532985\pi\)
\(752\) − 0.908327i − 0.0331233i
\(753\) 0 0
\(754\) 114.083 4.15467
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0555i 0.837967i 0.907994 + 0.418983i \(0.137613\pi\)
−0.907994 + 0.418983i \(0.862387\pi\)
\(758\) − 57.1472i − 2.07568i
\(759\) 0 0
\(760\) 0 0
\(761\) 42.4222 1.53780 0.768902 0.639367i \(-0.220803\pi\)
0.768902 + 0.639367i \(0.220803\pi\)
\(762\) 0 0
\(763\) − 28.0278i − 1.01467i
\(764\) 22.1194 0.800253
\(765\) 0 0
\(766\) −49.8167 −1.79995
\(767\) 1.05551i 0.0381124i
\(768\) 0 0
\(769\) 26.8167 0.967033 0.483517 0.875335i \(-0.339359\pi\)
0.483517 + 0.875335i \(0.339359\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) 22.1194i 0.795581i 0.917476 + 0.397790i \(0.130223\pi\)
−0.917476 + 0.397790i \(0.869777\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −35.0917 −1.25972
\(777\) 0 0
\(778\) 27.6333i 0.990702i
\(779\) −1.60555 −0.0575248
\(780\) 0 0
\(781\) −4.60555 −0.164800
\(782\) 33.2750i 1.18991i
\(783\) 0 0
\(784\) −3.48612 −0.124504
\(785\) 0 0
\(786\) 0 0
\(787\) 4.21110i 0.150110i 0.997179 + 0.0750548i \(0.0239132\pi\)
−0.997179 + 0.0750548i \(0.976087\pi\)
\(788\) − 32.0278i − 1.14094i
\(789\) 0 0
\(790\) 0 0
\(791\) −46.5416 −1.65483
\(792\) 0 0
\(793\) 14.5416i 0.516389i
\(794\) 58.2666 2.06780
\(795\) 0 0
\(796\) −80.9638 −2.86969
\(797\) − 14.5139i − 0.514108i −0.966397 0.257054i \(-0.917248\pi\)
0.966397 0.257054i \(-0.0827518\pi\)
\(798\) 0 0
\(799\) −11.7250 −0.414800
\(800\) 0 0
\(801\) 0 0
\(802\) 62.6611i 2.21264i
\(803\) 2.90833i 0.102633i
\(804\) 0 0
\(805\) 0 0
\(806\) 48.4861 1.70785
\(807\) 0 0
\(808\) 52.5416i 1.84841i
\(809\) −3.63331 −0.127740 −0.0638701 0.997958i \(-0.520344\pi\)
−0.0638701 + 0.997958i \(0.520344\pi\)
\(810\) 0 0
\(811\) −54.8722 −1.92682 −0.963411 0.268028i \(-0.913628\pi\)
−0.963411 + 0.268028i \(0.913628\pi\)
\(812\) − 140.808i − 4.94140i
\(813\) 0 0
\(814\) 22.1194 0.775286
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.21110i − 0.252285i
\(818\) − 18.9083i − 0.661114i
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) − 10.4222i − 0.363295i −0.983364 0.181648i \(-0.941857\pi\)
0.983364 0.181648i \(-0.0581430\pi\)
\(824\) −23.7250 −0.826499
\(825\) 0 0
\(826\) 2.09167 0.0727786
\(827\) − 7.81665i − 0.271812i −0.990722 0.135906i \(-0.956606\pi\)
0.990722 0.135906i \(-0.0433945\pi\)
\(828\) 0 0
\(829\) 38.7527 1.34594 0.672969 0.739671i \(-0.265019\pi\)
0.672969 + 0.739671i \(0.265019\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 64.0833i 2.22169i
\(833\) 45.0000i 1.55916i
\(834\) 0 0
\(835\) 0 0
\(836\) −3.30278 −0.114229
\(837\) 0 0
\(838\) 31.3305i 1.08230i
\(839\) 16.1194 0.556505 0.278252 0.960508i \(-0.410245\pi\)
0.278252 + 0.960508i \(0.410245\pi\)
\(840\) 0 0
\(841\) 69.1749 2.38534
\(842\) 9.90833i 0.341463i
\(843\) 0 0
\(844\) −83.3583 −2.86931
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.30278i − 0.147845i
\(848\) − 0.697224i − 0.0239428i
\(849\) 0 0
\(850\) 0 0
\(851\) 35.5139 1.21740
\(852\) 0 0
\(853\) 19.7250i 0.675370i 0.941259 + 0.337685i \(0.109644\pi\)
−0.941259 + 0.337685i \(0.890356\pi\)
\(854\) 28.8167 0.986086
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) 0 0
\(859\) −48.6056 −1.65840 −0.829200 0.558952i \(-0.811204\pi\)
−0.829200 + 0.558952i \(0.811204\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 75.9916i − 2.58828i
\(863\) 19.6056i 0.667381i 0.942683 + 0.333690i \(0.108294\pi\)
−0.942683 + 0.333690i \(0.891706\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −11.5139 −0.391258
\(867\) 0 0
\(868\) − 59.8444i − 2.03125i
\(869\) 0.0916731 0.00310980
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) − 19.5416i − 0.661763i
\(873\) 0 0
\(874\) −8.51388 −0.287986
\(875\) 0 0
\(876\) 0 0
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) − 47.6611i − 1.60848i
\(879\) 0 0
\(880\) 0 0
\(881\) 34.5416 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(882\) 0 0
\(883\) − 12.4500i − 0.418975i −0.977811 0.209487i \(-0.932821\pi\)
0.977811 0.209487i \(-0.0671795\pi\)
\(884\) −64.5416 −2.17077
\(885\) 0 0
\(886\) −3.21110 −0.107879
\(887\) 47.2389i 1.58613i 0.609140 + 0.793063i \(0.291515\pi\)
−0.609140 + 0.793063i \(0.708485\pi\)
\(888\) 0 0
\(889\) 73.6611 2.47051
\(890\) 0 0
\(891\) 0 0
\(892\) − 68.1472i − 2.28174i
\(893\) − 3.00000i − 0.100391i
\(894\) 0 0
\(895\) 0 0
\(896\) 81.3583 2.71799
\(897\) 0 0
\(898\) 95.5971i 3.19012i
\(899\) 41.7250 1.39161
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) − 3.69722i − 0.123104i
\(903\) 0 0
\(904\) −32.4500 −1.07927
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 17.5139i − 0.581218i
\(909\) 0 0
\(910\) 0 0
\(911\) 39.2111 1.29912 0.649561 0.760310i \(-0.274953\pi\)
0.649561 + 0.760310i \(0.274953\pi\)
\(912\) 0 0
\(913\) − 14.5139i − 0.480339i
\(914\) −55.9638 −1.85112
\(915\) 0 0
\(916\) 45.3305 1.49776
\(917\) 3.90833i 0.129064i
\(918\) 0 0
\(919\) −41.2111 −1.35943 −0.679714 0.733477i \(-0.737896\pi\)
−0.679714 + 0.733477i \(0.737896\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 40.9638i − 1.34907i
\(923\) − 23.0278i − 0.757968i
\(924\) 0 0
\(925\) 0 0
\(926\) −60.3583 −1.98350
\(927\) 0 0
\(928\) 52.5416i 1.72476i
\(929\) 46.3944 1.52215 0.761076 0.648662i \(-0.224671\pi\)
0.761076 + 0.648662i \(0.224671\pi\)
\(930\) 0 0
\(931\) −11.5139 −0.377352
\(932\) 16.8167i 0.550848i
\(933\) 0 0
\(934\) −56.7250 −1.85610
\(935\) 0 0
\(936\) 0 0
\(937\) − 5.21110i − 0.170239i −0.996371 0.0851196i \(-0.972873\pi\)
0.996371 0.0851196i \(-0.0271273\pi\)
\(938\) 39.6333i 1.29407i
\(939\) 0 0
\(940\) 0 0
\(941\) −52.3944 −1.70801 −0.854005 0.520265i \(-0.825833\pi\)
−0.854005 + 0.520265i \(0.825833\pi\)
\(942\) 0 0
\(943\) − 5.93608i − 0.193305i
\(944\) 0.0639167 0.00208031
\(945\) 0 0
\(946\) 16.6056 0.539893
\(947\) − 36.6333i − 1.19042i −0.803569 0.595211i \(-0.797068\pi\)
0.803569 0.595211i \(-0.202932\pi\)
\(948\) 0 0
\(949\) −14.5416 −0.472041
\(950\) 0 0
\(951\) 0 0
\(952\) 50.4500i 1.63509i
\(953\) 49.2666i 1.59590i 0.602722 + 0.797951i \(0.294083\pi\)
−0.602722 + 0.797951i \(0.705917\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 13.6056 0.440035
\(957\) 0 0
\(958\) − 30.3583i − 0.980832i
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −13.2666 −0.427955
\(962\) 110.597i 3.56580i
\(963\) 0 0
\(964\) 82.3583 2.65258
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.09167i − 0.131579i −0.997834 0.0657897i \(-0.979043\pi\)
0.997834 0.0657897i \(-0.0209566\pi\)
\(968\) − 3.00000i − 0.0964237i
\(969\) 0 0
\(970\) 0 0
\(971\) 30.3583 0.974244 0.487122 0.873334i \(-0.338047\pi\)
0.487122 + 0.873334i \(0.338047\pi\)
\(972\) 0 0
\(973\) 35.3305i 1.13264i
\(974\) −23.5139 −0.753433
\(975\) 0 0
\(976\) 0.880571 0.0281864
\(977\) − 15.9722i − 0.510997i −0.966809 0.255499i \(-0.917760\pi\)
0.966809 0.255499i \(-0.0822396\pi\)
\(978\) 0 0
\(979\) 5.30278 0.169477
\(980\) 0 0
\(981\) 0 0
\(982\) 55.7527i 1.77914i
\(983\) 48.8444i 1.55789i 0.627089 + 0.778947i \(0.284246\pi\)
−0.627089 + 0.778947i \(0.715754\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −89.1749 −2.83991
\(987\) 0 0
\(988\) − 16.5139i − 0.525376i
\(989\) 26.6611 0.847773
\(990\) 0 0
\(991\) −6.09167 −0.193508 −0.0967542 0.995308i \(-0.530846\pi\)
−0.0967542 + 0.995308i \(0.530846\pi\)
\(992\) 22.3305i 0.708995i
\(993\) 0 0
\(994\) −45.6333 −1.44740
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.2750i − 0.452094i −0.974116 0.226047i \(-0.927420\pi\)
0.974116 0.226047i \(-0.0725804\pi\)
\(998\) 49.5416i 1.56821i
\(999\) 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 2475.2.c.k.199.4 4
3.2 odd 2 275.2.b.c.199.1 4
5.2 odd 4 2475.2.a.o.1.1 2
5.3 odd 4 2475.2.a.t.1.2 2
5.4 even 2 inner 2475.2.c.k.199.1 4
12.11 even 2 4400.2.b.y.4049.3 4
15.2 even 4 275.2.a.f.1.2 yes 2
15.8 even 4 275.2.a.e.1.1 2
15.14 odd 2 275.2.b.c.199.4 4
60.23 odd 4 4400.2.a.bs.1.1 2
60.47 odd 4 4400.2.a.bh.1.2 2
60.59 even 2 4400.2.b.y.4049.2 4
165.32 odd 4 3025.2.a.h.1.1 2
165.98 odd 4 3025.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 15.8 even 4
275.2.a.f.1.2 yes 2 15.2 even 4
275.2.b.c.199.1 4 3.2 odd 2
275.2.b.c.199.4 4 15.14 odd 2
2475.2.a.o.1.1 2 5.2 odd 4
2475.2.a.t.1.2 2 5.3 odd 4
2475.2.c.k.199.1 4 5.4 even 2 inner
2475.2.c.k.199.4 4 1.1 even 1 trivial
3025.2.a.h.1.1 2 165.32 odd 4
3025.2.a.n.1.2 2 165.98 odd 4
4400.2.a.bh.1.2 2 60.47 odd 4
4400.2.a.bs.1.1 2 60.23 odd 4
4400.2.b.y.4049.2 4 60.59 even 2
4400.2.b.y.4049.3 4 12.11 even 2