Properties

Label 4680.2.l.g.2809.6
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: 8.0.57815240704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) 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 \(0.594137 + 0.594137i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.g.2809.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.594137 + 2.15569i) q^{5} -4.92778i q^{7} +O(q^{10})\) \(q+(0.594137 + 2.15569i) q^{5} -4.92778i q^{7} +1.38360 q^{11} -1.00000i q^{13} -0.195329i q^{17} -2.19533i q^{23} +(-4.29400 + 2.56155i) q^{25} +7.49966 q^{29} +(10.6228 - 2.92778i) q^{35} +6.05088i q^{37} +2.11605 q^{41} -3.49966i q^{43} -2.31138i q^{47} -17.2830 q^{49} -6.05088i q^{53} +(0.822051 + 2.98262i) q^{55} +9.43449 q^{59} -3.69498 q^{61} +(2.15569 - 0.594137i) q^{65} -12.6228i q^{67} -13.9716 q^{71} -4.73245i q^{73} -6.81809i q^{77} +8.07153 q^{79} -13.3997i q^{83} +(0.421068 - 0.116052i) q^{85} +3.02770 q^{89} -4.92778 q^{91} +6.01612i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) \) 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
\(5\) 0.594137 + 2.15569i 0.265706 + 0.964054i
\(6\) 0 0
\(7\) 4.92778i 1.86252i −0.364349 0.931262i \(-0.618709\pi\)
0.364349 0.931262i \(-0.381291\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.38360 0.417172 0.208586 0.978004i \(-0.433114\pi\)
0.208586 + 0.978004i \(0.433114\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.195329i 0.0473741i −0.999719 0.0236871i \(-0.992459\pi\)
0.999719 0.0236871i \(-0.00754053\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.19533i 0.457758i −0.973455 0.228879i \(-0.926494\pi\)
0.973455 0.228879i \(-0.0735060\pi\)
\(24\) 0 0
\(25\) −4.29400 + 2.56155i −0.858800 + 0.512311i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.49966 1.39265 0.696326 0.717726i \(-0.254817\pi\)
0.696326 + 0.717726i \(0.254817\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.6228 2.92778i 1.79557 0.494885i
\(36\) 0 0
\(37\) 6.05088i 0.994759i 0.867533 + 0.497379i \(0.165704\pi\)
−0.867533 + 0.497379i \(0.834296\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.11605 0.330472 0.165236 0.986254i \(-0.447161\pi\)
0.165236 + 0.986254i \(0.447161\pi\)
\(42\) 0 0
\(43\) 3.49966i 0.533692i −0.963739 0.266846i \(-0.914018\pi\)
0.963739 0.266846i \(-0.0859816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.31138i 0.337150i −0.985689 0.168575i \(-0.946084\pi\)
0.985689 0.168575i \(-0.0539165\pi\)
\(48\) 0 0
\(49\) −17.2830 −2.46900
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.05088i 0.831153i −0.909558 0.415576i \(-0.863580\pi\)
0.909558 0.415576i \(-0.136420\pi\)
\(54\) 0 0
\(55\) 0.822051 + 2.98262i 0.110845 + 0.402176i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.43449 1.22827 0.614133 0.789203i \(-0.289506\pi\)
0.614133 + 0.789203i \(0.289506\pi\)
\(60\) 0 0
\(61\) −3.69498 −0.473094 −0.236547 0.971620i \(-0.576016\pi\)
−0.236547 + 0.971620i \(0.576016\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.15569 0.594137i 0.267380 0.0736937i
\(66\) 0 0
\(67\) 12.6228i 1.54212i −0.636765 0.771058i \(-0.719728\pi\)
0.636765 0.771058i \(-0.280272\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.9716 −1.65812 −0.829062 0.559156i \(-0.811125\pi\)
−0.829062 + 0.559156i \(0.811125\pi\)
\(72\) 0 0
\(73\) 4.73245i 0.553891i −0.960886 0.276946i \(-0.910678\pi\)
0.960886 0.276946i \(-0.0893222\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.81809i 0.776993i
\(78\) 0 0
\(79\) 8.07153 0.908119 0.454059 0.890971i \(-0.349975\pi\)
0.454059 + 0.890971i \(0.349975\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.3997i 1.47081i −0.677627 0.735406i \(-0.736992\pi\)
0.677627 0.735406i \(-0.263008\pi\)
\(84\) 0 0
\(85\) 0.421068 0.116052i 0.0456712 0.0125876i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.02770 0.320936 0.160468 0.987041i \(-0.448700\pi\)
0.160468 + 0.987041i \(0.448700\pi\)
\(90\) 0 0
\(91\) −4.92778 −0.516571
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.01612i 0.610845i 0.952217 + 0.305422i \(0.0987977\pi\)
−0.952217 + 0.305422i \(0.901202\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.10900 0.309357 0.154678 0.987965i \(-0.450566\pi\)
0.154678 + 0.987965i \(0.450566\pi\)
\(102\) 0 0
\(103\) 16.8690i 1.66215i −0.556161 0.831075i \(-0.687726\pi\)
0.556161 0.831075i \(-0.312274\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.1946i 1.66227i 0.556072 + 0.831134i \(0.312308\pi\)
−0.556072 + 0.831134i \(0.687692\pi\)
\(108\) 0 0
\(109\) −8.76721 −0.839746 −0.419873 0.907583i \(-0.637925\pi\)
−0.419873 + 0.907583i \(0.637925\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.46490i 0.702238i 0.936331 + 0.351119i \(0.114199\pi\)
−0.936331 + 0.351119i \(0.885801\pi\)
\(114\) 0 0
\(115\) 4.73245 1.30433i 0.441303 0.121629i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.962536 −0.0882355
\(120\) 0 0
\(121\) −9.08564 −0.825967
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −8.07314 7.73462i −0.722084 0.691806i
\(126\) 0 0
\(127\) 11.1231i 0.987016i −0.869741 0.493508i \(-0.835714\pi\)
0.869741 0.493508i \(-0.164286\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.82080 0.159084 0.0795418 0.996832i \(-0.474654\pi\)
0.0795418 + 0.996832i \(0.474654\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.70204i 0.743465i 0.928340 + 0.371733i \(0.121236\pi\)
−0.928340 + 0.371733i \(0.878764\pi\)
\(138\) 0 0
\(139\) −6.81809 −0.578303 −0.289151 0.957283i \(-0.593373\pi\)
−0.289151 + 0.957283i \(0.593373\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.38360i 0.115703i
\(144\) 0 0
\(145\) 4.45583 + 16.1669i 0.370036 + 1.34259i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.48537 −0.613225 −0.306613 0.951834i \(-0.599196\pi\)
−0.306613 + 0.951834i \(0.599196\pi\)
\(150\) 0 0
\(151\) 12.8549 1.04611 0.523057 0.852298i \(-0.324791\pi\)
0.523057 + 0.852298i \(0.324791\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.7318i 1.41515i −0.706639 0.707574i \(-0.749790\pi\)
0.706639 0.707574i \(-0.250210\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.8181 −0.852585
\(162\) 0 0
\(163\) 18.1385i 1.42072i −0.703838 0.710360i \(-0.748532\pi\)
0.703838 0.710360i \(-0.251468\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.06517i 0.314572i 0.987553 + 0.157286i \(0.0502745\pi\)
−0.987553 + 0.157286i \(0.949726\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 23.2455i 1.76732i −0.468125 0.883662i \(-0.655070\pi\)
0.468125 0.883662i \(-0.344930\pi\)
\(174\) 0 0
\(175\) 12.6228 + 21.1599i 0.954191 + 1.59954i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.1224 1.05556 0.527779 0.849381i \(-0.323025\pi\)
0.527779 + 0.849381i \(0.323025\pi\)
\(180\) 0 0
\(181\) 3.67433 0.273111 0.136556 0.990632i \(-0.456397\pi\)
0.136556 + 0.990632i \(0.456397\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −13.0438 + 3.59506i −0.959001 + 0.264314i
\(186\) 0 0
\(187\) 0.270257i 0.0197632i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.8549 1.21958 0.609788 0.792565i \(-0.291255\pi\)
0.609788 + 0.792565i \(0.291255\pi\)
\(192\) 0 0
\(193\) 6.53712i 0.470552i −0.971929 0.235276i \(-0.924401\pi\)
0.971929 0.235276i \(-0.0755994\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.7890i 1.83739i −0.394967 0.918695i \(-0.629244\pi\)
0.394967 0.918695i \(-0.370756\pi\)
\(198\) 0 0
\(199\) 16.9646 1.20259 0.601293 0.799029i \(-0.294653\pi\)
0.601293 + 0.799029i \(0.294653\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 36.9566i 2.59385i
\(204\) 0 0
\(205\) 1.25723 + 4.56155i 0.0878085 + 0.318593i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −26.4577 −1.82142 −0.910710 0.413046i \(-0.864465\pi\)
−0.910710 + 0.413046i \(0.864465\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.54417 2.07928i 0.514508 0.141805i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.195329 −0.0131392
\(222\) 0 0
\(223\) 2.02065i 0.135313i 0.997709 + 0.0676564i \(0.0215522\pi\)
−0.997709 + 0.0676564i \(0.978448\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.7904i 1.18079i 0.807115 + 0.590395i \(0.201028\pi\)
−0.807115 + 0.590395i \(0.798972\pi\)
\(228\) 0 0
\(229\) 18.3339 1.21154 0.605768 0.795641i \(-0.292866\pi\)
0.605768 + 0.795641i \(0.292866\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.0361i 1.50914i −0.656217 0.754572i \(-0.727844\pi\)
0.656217 0.754572i \(-0.272156\pi\)
\(234\) 0 0
\(235\) 4.98262 1.37328i 0.325030 0.0895828i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −0.986402 −0.0638051 −0.0319025 0.999491i \(-0.510157\pi\)
−0.0319025 + 0.999491i \(0.510157\pi\)
\(240\) 0 0
\(241\) 1.52031 0.0979316 0.0489658 0.998800i \(-0.484407\pi\)
0.0489658 + 0.998800i \(0.484407\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.2685 37.2568i −0.656028 2.38025i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −11.6763 −0.737005 −0.368502 0.929627i \(-0.620129\pi\)
−0.368502 + 0.929627i \(0.620129\pi\)
\(252\) 0 0
\(253\) 3.03746i 0.190964i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00069i 0.187178i 0.995611 + 0.0935889i \(0.0298339\pi\)
−0.995611 + 0.0935889i \(0.970166\pi\)
\(258\) 0 0
\(259\) 29.8174 1.85276
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 25.8683i 1.59511i 0.603248 + 0.797553i \(0.293873\pi\)
−0.603248 + 0.797553i \(0.706127\pi\)
\(264\) 0 0
\(265\) 13.0438 3.59506i 0.801276 0.220843i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.1090 0.677328 0.338664 0.940907i \(-0.390025\pi\)
0.338664 + 0.940907i \(0.390025\pi\)
\(270\) 0 0
\(271\) −13.9859 −0.849582 −0.424791 0.905291i \(-0.639653\pi\)
−0.424791 + 0.905291i \(0.639653\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.94120 + 3.54417i −0.358268 + 0.213722i
\(276\) 0 0
\(277\) 10.0529i 0.604020i 0.953305 + 0.302010i \(0.0976576\pi\)
−0.953305 + 0.302010i \(0.902342\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.6318 1.17114 0.585568 0.810623i \(-0.300871\pi\)
0.585568 + 0.810623i \(0.300871\pi\)
\(282\) 0 0
\(283\) 25.5667i 1.51978i 0.650051 + 0.759890i \(0.274747\pi\)
−0.650051 + 0.759890i \(0.725253\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.4274i 0.615512i
\(288\) 0 0
\(289\) 16.9618 0.997756
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.71614i 0.392361i 0.980568 + 0.196181i \(0.0628539\pi\)
−0.980568 + 0.196181i \(0.937146\pi\)
\(294\) 0 0
\(295\) 5.60538 + 20.3378i 0.326358 + 1.18411i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.19533 −0.126959
\(300\) 0 0
\(301\) −17.2455 −0.994015
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.19533 7.96524i −0.125704 0.456088i
\(306\) 0 0
\(307\) 14.2689i 0.814368i 0.913346 + 0.407184i \(0.133489\pi\)
−0.913346 + 0.407184i \(0.866511\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.15786 0.405885 0.202943 0.979191i \(-0.434950\pi\)
0.202943 + 0.979191i \(0.434950\pi\)
\(312\) 0 0
\(313\) 21.6568i 1.22412i −0.790813 0.612058i \(-0.790342\pi\)
0.790813 0.612058i \(-0.209658\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.0232i 1.63010i 0.579388 + 0.815052i \(0.303292\pi\)
−0.579388 + 0.815052i \(0.696708\pi\)
\(318\) 0 0
\(319\) 10.3765 0.580975
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.56155 + 4.29400i 0.142089 + 0.238188i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.3900 −0.627949
\(330\) 0 0
\(331\) 21.8683 1.20199 0.600995 0.799253i \(-0.294771\pi\)
0.600995 + 0.799253i \(0.294771\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 27.2108 7.49966i 1.48668 0.409750i
\(336\) 0 0
\(337\) 22.2991i 1.21471i −0.794431 0.607355i \(-0.792231\pi\)
0.794431 0.607355i \(-0.207769\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 50.6723i 2.73605i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.9199i 1.23040i −0.788370 0.615201i \(-0.789075\pi\)
0.788370 0.615201i \(-0.210925\pi\)
\(348\) 0 0
\(349\) −6.75310 −0.361485 −0.180743 0.983530i \(-0.557850\pi\)
−0.180743 + 0.983530i \(0.557850\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.07859i 0.163857i 0.996638 + 0.0819283i \(0.0261079\pi\)
−0.996638 + 0.0819283i \(0.973892\pi\)
\(354\) 0 0
\(355\) −8.30105 30.1185i −0.440574 1.59852i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.0091 −0.686592 −0.343296 0.939227i \(-0.611543\pi\)
−0.343296 + 0.939227i \(0.611543\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 10.2017 2.81173i 0.533981 0.147172i
\(366\) 0 0
\(367\) 27.4041i 1.43048i −0.698878 0.715241i \(-0.746317\pi\)
0.698878 0.715241i \(-0.253683\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.8174 −1.54804
\(372\) 0 0
\(373\) 19.9573i 1.03335i −0.856181 0.516675i \(-0.827169\pi\)
0.856181 0.516675i \(-0.172831\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.49966i 0.386252i
\(378\) 0 0
\(379\) 29.2042 1.50012 0.750060 0.661370i \(-0.230025\pi\)
0.750060 + 0.661370i \(0.230025\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.5764i 0.795918i 0.917403 + 0.397959i \(0.130281\pi\)
−0.917403 + 0.397959i \(0.869719\pi\)
\(384\) 0 0
\(385\) 14.6977 4.05088i 0.749064 0.206452i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.0529 1.01672 0.508361 0.861144i \(-0.330251\pi\)
0.508361 + 0.861144i \(0.330251\pi\)
\(390\) 0 0
\(391\) −0.428810 −0.0216859
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.79560 + 17.3997i 0.241293 + 0.875475i
\(396\) 0 0
\(397\) 30.9057i 1.55112i −0.631277 0.775558i \(-0.717469\pi\)
0.631277 0.775558i \(-0.282531\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.9902 0.598764 0.299382 0.954133i \(-0.403220\pi\)
0.299382 + 0.954133i \(0.403220\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.37202i 0.414986i
\(408\) 0 0
\(409\) −39.2169 −1.93915 −0.969577 0.244788i \(-0.921282\pi\)
−0.969577 + 0.244788i \(0.921282\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 46.4910i 2.28767i
\(414\) 0 0
\(415\) 28.8857 7.96128i 1.41794 0.390804i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −17.8621 −0.872621 −0.436310 0.899796i \(-0.643715\pi\)
−0.436310 + 0.899796i \(0.643715\pi\)
\(420\) 0 0
\(421\) 28.8690 1.40699 0.703494 0.710701i \(-0.251622\pi\)
0.703494 + 0.710701i \(0.251622\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.500344 + 0.838741i 0.0242703 + 0.0406849i
\(426\) 0 0
\(427\) 18.2081i 0.881150i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.0373 −0.820657 −0.410329 0.911938i \(-0.634586\pi\)
−0.410329 + 0.911938i \(0.634586\pi\)
\(432\) 0 0
\(433\) 29.1554i 1.40112i 0.713595 + 0.700558i \(0.247066\pi\)
−0.713595 + 0.700558i \(0.752934\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 23.2087 1.10769 0.553847 0.832619i \(-0.313159\pi\)
0.553847 + 0.832619i \(0.313159\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.9484i 0.710221i −0.934824 0.355111i \(-0.884443\pi\)
0.934824 0.355111i \(-0.115557\pi\)
\(444\) 0 0
\(445\) 1.79887 + 6.52679i 0.0852748 + 0.309400i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −42.2023 −1.99165 −0.995826 0.0912763i \(-0.970905\pi\)
−0.995826 + 0.0912763i \(0.970905\pi\)
\(450\) 0 0
\(451\) 2.92778 0.137864
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.92778 10.6228i −0.137256 0.498003i
\(456\) 0 0
\(457\) 15.1599i 0.709149i −0.935028 0.354575i \(-0.884626\pi\)
0.935028 0.354575i \(-0.115374\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.848501 0.0395186 0.0197593 0.999805i \(-0.493710\pi\)
0.0197593 + 0.999805i \(0.493710\pi\)
\(462\) 0 0
\(463\) 13.8109i 0.641845i −0.947105 0.320922i \(-0.896007\pi\)
0.947105 0.320922i \(-0.103993\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.5165i 0.486644i −0.969946 0.243322i \(-0.921763\pi\)
0.969946 0.243322i \(-0.0782372\pi\)
\(468\) 0 0
\(469\) −62.2022 −2.87223
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.84214i 0.222642i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.41905 −0.293294 −0.146647 0.989189i \(-0.546848\pi\)
−0.146647 + 0.989189i \(0.546848\pi\)
\(480\) 0 0
\(481\) 6.05088 0.275897
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12.9689 + 3.57440i −0.588887 + 0.162305i
\(486\) 0 0
\(487\) 23.5492i 1.06711i 0.845764 + 0.533557i \(0.179145\pi\)
−0.845764 + 0.533557i \(0.820855\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.99346 −0.180222 −0.0901111 0.995932i \(-0.528722\pi\)
−0.0901111 + 0.995932i \(0.528722\pi\)
\(492\) 0 0
\(493\) 1.46490i 0.0659756i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 68.8490i 3.08830i
\(498\) 0 0
\(499\) 5.30231 0.237364 0.118682 0.992932i \(-0.462133\pi\)
0.118682 + 0.992932i \(0.462133\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.8408i 1.55347i 0.629826 + 0.776736i \(0.283126\pi\)
−0.629826 + 0.776736i \(0.716874\pi\)
\(504\) 0 0
\(505\) 1.84717 + 6.70204i 0.0821981 + 0.298237i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 42.2526 1.87281 0.936406 0.350918i \(-0.114130\pi\)
0.936406 + 0.350918i \(0.114130\pi\)
\(510\) 0 0
\(511\) −23.3205 −1.03164
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 36.3643 10.0225i 1.60240 0.441644i
\(516\) 0 0
\(517\) 3.19803i 0.140649i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.65098 0.116141 0.0580707 0.998312i \(-0.481505\pi\)
0.0580707 + 0.998312i \(0.481505\pi\)
\(522\) 0 0
\(523\) 31.6014i 1.38183i 0.722934 + 0.690917i \(0.242793\pi\)
−0.722934 + 0.690917i \(0.757207\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.1805 0.790458
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.11605i 0.0916564i
\(534\) 0 0
\(535\) −37.0663 + 10.2160i −1.60252 + 0.441675i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −23.9128 −1.03000
\(540\) 0 0
\(541\) 1.54852 0.0665761 0.0332881 0.999446i \(-0.489402\pi\)
0.0332881 + 0.999446i \(0.489402\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.20893 18.8994i −0.223126 0.809561i
\(546\) 0 0
\(547\) 33.2376i 1.42114i −0.703628 0.710569i \(-0.748438\pi\)
0.703628 0.710569i \(-0.251562\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 39.7747i 1.69139i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.64594i 0.0697407i −0.999392 0.0348703i \(-0.988898\pi\)
0.999392 0.0348703i \(-0.0111018\pi\)
\(558\) 0 0
\(559\) −3.49966 −0.148020
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.7486i 0.790158i −0.918647 0.395079i \(-0.870717\pi\)
0.918647 0.395079i \(-0.129283\pi\)
\(564\) 0 0
\(565\) −16.0920 + 4.43518i −0.676996 + 0.186589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −41.6543 −1.74624 −0.873120 0.487505i \(-0.837907\pi\)
−0.873120 + 0.487505i \(0.837907\pi\)
\(570\) 0 0
\(571\) −27.3184 −1.14324 −0.571620 0.820518i \(-0.693685\pi\)
−0.571620 + 0.820518i \(0.693685\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.62345 + 9.42674i 0.234514 + 0.393122i
\(576\) 0 0
\(577\) 24.0715i 1.00211i −0.865415 0.501056i \(-0.832945\pi\)
0.865415 0.501056i \(-0.167055\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −66.0309 −2.73942
\(582\) 0 0
\(583\) 8.37202i 0.346734i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.6872i 0.977677i −0.872374 0.488839i \(-0.837421\pi\)
0.872374 0.488839i \(-0.162579\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.8632i 1.59592i 0.602709 + 0.797961i \(0.294088\pi\)
−0.602709 + 0.797961i \(0.705912\pi\)
\(594\) 0 0
\(595\) −0.571878 2.07493i −0.0234447 0.0850638i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.4783 0.755003 0.377502 0.926009i \(-0.376783\pi\)
0.377502 + 0.926009i \(0.376783\pi\)
\(600\) 0 0
\(601\) −2.08702 −0.0851313 −0.0425656 0.999094i \(-0.513553\pi\)
−0.0425656 + 0.999094i \(0.513553\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.39812 19.5858i −0.219465 0.796277i
\(606\) 0 0
\(607\) 3.59120i 0.145762i −0.997341 0.0728812i \(-0.976781\pi\)
0.997341 0.0728812i \(-0.0232194\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.31138 −0.0935084
\(612\) 0 0
\(613\) 24.7895i 1.00124i 0.865667 + 0.500620i \(0.166894\pi\)
−0.865667 + 0.500620i \(0.833106\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.7574i 0.594112i 0.954860 + 0.297056i \(0.0960049\pi\)
−0.954860 + 0.297056i \(0.903995\pi\)
\(618\) 0 0
\(619\) −25.4081 −1.02124 −0.510619 0.859807i \(-0.670584\pi\)
−0.510619 + 0.859807i \(0.670584\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.9199i 0.597751i
\(624\) 0 0
\(625\) 11.8769 21.9986i 0.475076 0.879945i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.18191 0.0471258
\(630\) 0 0
\(631\) 9.46490 0.376792 0.188396 0.982093i \(-0.439671\pi\)
0.188396 + 0.982093i \(0.439671\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 23.9780 6.60865i 0.951537 0.262256i
\(636\) 0 0
\(637\) 17.2830i 0.684777i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −7.89686 −0.311907 −0.155954 0.987764i \(-0.549845\pi\)
−0.155954 + 0.987764i \(0.549845\pi\)
\(642\) 0 0
\(643\) 16.1204i 0.635727i −0.948137 0.317863i \(-0.897035\pi\)
0.948137 0.317863i \(-0.102965\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.99616i 0.117791i −0.998264 0.0588956i \(-0.981242\pi\)
0.998264 0.0588956i \(-0.0187579\pi\)
\(648\) 0 0
\(649\) 13.0536 0.512398
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.5330i 0.842653i 0.906909 + 0.421326i \(0.138435\pi\)
−0.906909 + 0.421326i \(0.861565\pi\)
\(654\) 0 0
\(655\) 1.08180 + 3.92507i 0.0422695 + 0.153365i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.4429 0.679478 0.339739 0.940520i \(-0.389661\pi\)
0.339739 + 0.940520i \(0.389661\pi\)
\(660\) 0 0
\(661\) −22.4642 −0.873756 −0.436878 0.899521i \(-0.643916\pi\)
−0.436878 + 0.899521i \(0.643916\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.4642i 0.637497i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.11239 −0.197362
\(672\) 0 0
\(673\) 46.8122i 1.80448i 0.431237 + 0.902239i \(0.358077\pi\)
−0.431237 + 0.902239i \(0.641923\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.2682i 1.20173i 0.799349 + 0.600867i \(0.205178\pi\)
−0.799349 + 0.600867i \(0.794822\pi\)
\(678\) 0 0
\(679\) 29.6461 1.13771
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.1797i 1.69049i 0.534381 + 0.845244i \(0.320545\pi\)
−0.534381 + 0.845244i \(0.679455\pi\)
\(684\) 0 0
\(685\) −18.7589 + 5.17021i −0.716741 + 0.197543i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.05088 −0.230520
\(690\) 0 0
\(691\) −25.8360 −0.982849 −0.491425 0.870920i \(-0.663524\pi\)
−0.491425 + 0.870920i \(0.663524\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.05088 14.6977i −0.153659 0.557515i
\(696\) 0 0
\(697\) 0.413325i 0.0156558i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.5887 −0.588777 −0.294388 0.955686i \(-0.595116\pi\)
−0.294388 + 0.955686i \(0.595116\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.3205i 0.576185i
\(708\) 0 0
\(709\) 4.05541 0.152304 0.0761521 0.997096i \(-0.475737\pi\)
0.0761521 + 0.997096i \(0.475737\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.98262 0.822051i 0.111544 0.0307430i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.4081 0.500038 0.250019 0.968241i \(-0.419563\pi\)
0.250019 + 0.968241i \(0.419563\pi\)
\(720\) 0 0
\(721\) −83.1265 −3.09579
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −32.2035 + 19.2108i −1.19601 + 0.713470i
\(726\) 0 0
\(727\) 12.9787i 0.481352i −0.970606 0.240676i \(-0.922631\pi\)
0.970606 0.240676i \(-0.0773691\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.683583 −0.0252832
\(732\) 0 0
\(733\) 49.2410i 1.81876i 0.415969 + 0.909379i \(0.363443\pi\)
−0.415969 + 0.909379i \(0.636557\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.4649i 0.643328i
\(738\) 0 0
\(739\) 11.3359 0.416999 0.208500 0.978022i \(-0.433142\pi\)
0.208500 + 0.978022i \(0.433142\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.07790i 0.296349i −0.988961 0.148175i \(-0.952660\pi\)
0.988961 0.148175i \(-0.0473398\pi\)
\(744\) 0 0
\(745\) −4.44734 16.1361i −0.162938 0.591182i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 84.7314 3.09602
\(750\) 0 0
\(751\) 18.2403 0.665598 0.332799 0.942998i \(-0.392007\pi\)
0.332799 + 0.942998i \(0.392007\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.63756 + 27.7111i 0.277959 + 1.00851i
\(756\) 0 0
\(757\) 5.71111i 0.207574i 0.994600 + 0.103787i \(0.0330960\pi\)
−0.994600 + 0.103787i \(0.966904\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.6543 −0.458718 −0.229359 0.973342i \(-0.573663\pi\)
−0.229359 + 0.973342i \(0.573663\pi\)
\(762\) 0 0
\(763\) 43.2028i 1.56405i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.43449i 0.340660i
\(768\) 0 0
\(769\) 21.4222 0.772505 0.386252 0.922393i \(-0.373769\pi\)
0.386252 + 0.922393i \(0.373769\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.0547i 1.29680i 0.761300 + 0.648399i \(0.224561\pi\)
−0.761300 + 0.648399i \(0.775439\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −19.3312 −0.691723
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 38.2242 10.5351i 1.36428 0.376014i
\(786\) 0 0
\(787\) 40.8394i 1.45577i −0.685701 0.727883i \(-0.740504\pi\)
0.685701 0.727883i \(-0.259496\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.7853 1.30794
\(792\) 0 0
\(793\) 3.69498i 0.131213i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.4691i 0.902161i 0.892483 + 0.451080i \(0.148961\pi\)
−0.892483 + 0.451080i \(0.851039\pi\)
\(798\) 0 0
\(799\) −0.451479 −0.0159722
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.54783i 0.231068i
\(804\) 0 0
\(805\) −6.42743 23.3205i −0.226537 0.821938i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.7252 0.552869 0.276435 0.961033i \(-0.410847\pi\)
0.276435 + 0.961033i \(0.410847\pi\)
\(810\) 0 0
\(811\) −49.6815 −1.74455 −0.872277 0.489012i \(-0.837357\pi\)
−0.872277 + 0.489012i \(0.837357\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39.1011 10.7768i 1.36965 0.377494i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.9936 −0.837384 −0.418692 0.908128i \(-0.637511\pi\)
−0.418692 + 0.908128i \(0.637511\pi\)
\(822\) 0 0
\(823\) 23.0457i 0.803321i 0.915789 + 0.401661i \(0.131567\pi\)
−0.915789 + 0.401661i \(0.868433\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 49.1195i 1.70805i −0.520229 0.854027i \(-0.674153\pi\)
0.520229 0.854027i \(-0.325847\pi\)
\(828\) 0 0
\(829\) 5.54739 0.192669 0.0963344 0.995349i \(-0.469288\pi\)
0.0963344 + 0.995349i \(0.469288\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.37586i 0.116967i
\(834\) 0 0
\(835\) −8.76325 + 2.41527i −0.303265 + 0.0835838i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −28.7106 −0.991200 −0.495600 0.868551i \(-0.665052\pi\)
−0.495600 + 0.868551i \(0.665052\pi\)
\(840\) 0 0
\(841\) 27.2448 0.939477
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −0.594137 2.15569i −0.0204390 0.0741580i
\(846\) 0 0
\(847\) 44.7720i 1.53838i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.2837 0.455359
\(852\) 0 0
\(853\) 27.7606i 0.950505i 0.879849 + 0.475253i \(0.157643\pi\)
−0.879849 + 0.475253i \(0.842357\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.8869i 1.22587i 0.790132 + 0.612937i \(0.210012\pi\)
−0.790132 + 0.612937i \(0.789988\pi\)
\(858\) 0 0
\(859\) 26.6107 0.907944 0.453972 0.891016i \(-0.350007\pi\)
0.453972 + 0.891016i \(0.350007\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.0366i 0.682054i −0.940054 0.341027i \(-0.889225\pi\)
0.940054 0.341027i \(-0.110775\pi\)
\(864\) 0 0
\(865\) 50.1101 13.8110i 1.70380 0.469589i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.1678 0.378842
\(870\) 0 0
\(871\) −12.6228 −0.427706
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −38.1145 + 39.7826i −1.28851 + 1.34490i
\(876\) 0 0
\(877\) 2.15855i 0.0728892i 0.999336 + 0.0364446i \(0.0116032\pi\)
−0.999336 + 0.0364446i \(0.988397\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.8421 0.500044 0.250022 0.968240i \(-0.419562\pi\)
0.250022 + 0.968240i \(0.419562\pi\)
\(882\) 0 0
\(883\) 13.0742i 0.439983i −0.975502 0.219992i \(-0.929397\pi\)
0.975502 0.219992i \(-0.0706030\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.4254i 1.32377i 0.749604 + 0.661887i \(0.230244\pi\)
−0.749604 + 0.661887i \(0.769756\pi\)
\(888\) 0 0
\(889\) −54.8122 −1.83834
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.39066 + 30.4436i 0.280469 + 1.01762i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −1.18191 −0.0393751
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.18306 + 7.92072i 0.0725673 + 0.263294i
\(906\) 0 0
\(907\) 49.8397i 1.65490i 0.561539 + 0.827450i \(0.310209\pi\)
−0.561539 + 0.827450i \(0.689791\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.92438 −0.295678 −0.147839 0.989011i \(-0.547232\pi\)
−0.147839 + 0.989011i \(0.547232\pi\)
\(912\) 0 0
\(913\) 18.5399i 0.613581i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.97247i 0.296297i
\(918\) 0 0
\(919\) 21.9876 0.725302 0.362651 0.931925i \(-0.381872\pi\)
0.362651 + 0.931925i \(0.381872\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13.9716i 0.459881i
\(924\) 0 0
\(925\) −15.4997 25.9825i −0.509626 0.854299i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.14289 0.135924 0.0679619 0.997688i \(-0.478350\pi\)
0.0679619 + 0.997688i \(0.478350\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.582591 0.160570i 0.0190528 0.00525120i
\(936\) 0 0
\(937\) 59.2235i 1.93475i 0.253354 + 0.967374i \(0.418466\pi\)
−0.253354 + 0.967374i \(0.581534\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.7615 −0.383415 −0.191707 0.981452i \(-0.561402\pi\)
−0.191707 + 0.981452i \(0.561402\pi\)
\(942\) 0 0
\(943\) 4.64543i 0.151276i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.51629i 0.276742i 0.990380 + 0.138371i \(0.0441867\pi\)
−0.990380 + 0.138371i \(0.955813\pi\)
\(948\) 0 0
\(949\) −4.73245 −0.153622
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 35.4872i 1.14954i 0.818314 + 0.574772i \(0.194909\pi\)
−0.818314 + 0.574772i \(0.805091\pi\)
\(954\) 0 0
\(955\) 10.0141 + 36.3339i 0.324049 + 1.17574i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 42.8817 1.38472
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.0920 3.88395i 0.453638 0.125029i
\(966\) 0 0
\(967\) 8.51445i 0.273806i −0.990584 0.136903i \(-0.956285\pi\)
0.990584 0.136903i \(-0.0437149\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 50.5013 1.62066 0.810331 0.585972i \(-0.199287\pi\)
0.810331 + 0.585972i \(0.199287\pi\)
\(972\) 0 0
\(973\) 33.5980i 1.07710i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39.0540i 1.24945i 0.780845 + 0.624725i \(0.214789\pi\)
−0.780845 + 0.624725i \(0.785211\pi\)
\(978\) 0 0
\(979\) 4.18914 0.133886
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 54.9882i 1.75385i 0.480627 + 0.876925i \(0.340409\pi\)
−0.480627 + 0.876925i \(0.659591\pi\)
\(984\) 0 0
\(985\) 55.5931 15.3222i 1.77134 0.488206i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.68289 −0.244302
\(990\) 0 0
\(991\) −15.0052 −0.476656 −0.238328 0.971185i \(-0.576599\pi\)
−0.238328 + 0.971185i \(0.576599\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.0793 + 36.5703i 0.319535 + 1.15936i
\(996\) 0 0
\(997\) 22.4113i 0.709773i 0.934909 + 0.354887i \(0.115481\pi\)
−0.934909 + 0.354887i \(0.884519\pi\)
\(998\) 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.g.2809.6 8
3.2 odd 2 1560.2.l.d.1249.6 yes 8
5.4 even 2 inner 4680.2.l.g.2809.5 8
12.11 even 2 3120.2.l.n.1249.2 8
15.2 even 4 7800.2.a.by.1.4 4
15.8 even 4 7800.2.a.bt.1.1 4
15.14 odd 2 1560.2.l.d.1249.2 8
60.59 even 2 3120.2.l.n.1249.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.d.1249.2 8 15.14 odd 2
1560.2.l.d.1249.6 yes 8 3.2 odd 2
3120.2.l.n.1249.2 8 12.11 even 2
3120.2.l.n.1249.6 8 60.59 even 2
4680.2.l.g.2809.5 8 5.4 even 2 inner
4680.2.l.g.2809.6 8 1.1 even 1 trivial
7800.2.a.bt.1.1 4 15.8 even 4
7800.2.a.by.1.4 4 15.2 even 4