Properties

Label 1200.3.l.s.401.2
Level $1200$
Weight $3$
Character 1200.401
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
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] = [1200,3,Mod(401,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.2
Root \(0.500000 - 1.65831i\) of defining polynomial
Character \(\chi\) \(=\) 1200.401
Dual form 1200.3.l.s.401.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.50000 + 1.65831i) q^{3} +(3.50000 + 8.29156i) q^{9} +O(q^{10})\) \(q+(2.50000 + 1.65831i) q^{3} +(3.50000 + 8.29156i) q^{9} +16.5831i q^{11} -10.0000 q^{13} -3.31662i q^{17} -7.00000 q^{19} +19.8997i q^{23} +(-5.00000 + 26.5330i) q^{27} -33.1662i q^{29} -42.0000 q^{31} +(-27.5000 + 41.4578i) q^{33} +40.0000 q^{37} +(-25.0000 - 16.5831i) q^{39} +16.5831i q^{41} -50.0000 q^{43} -46.4327i q^{47} -49.0000 q^{49} +(5.50000 - 8.29156i) q^{51} +46.4327i q^{53} +(-17.5000 - 11.6082i) q^{57} +66.3325i q^{59} -8.00000 q^{61} +45.0000 q^{67} +(-33.0000 + 49.7494i) q^{69} +33.1662i q^{71} +35.0000 q^{73} -12.0000 q^{79} +(-56.5000 + 58.0409i) q^{81} +69.6491i q^{83} +(55.0000 - 82.9156i) q^{87} +149.248i q^{89} +(-105.000 - 69.6491i) q^{93} +70.0000 q^{97} +(-137.500 + 58.0409i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} + 7 q^{9} - 20 q^{13} - 14 q^{19} - 10 q^{27} - 84 q^{31} - 55 q^{33} + 80 q^{37} - 50 q^{39} - 100 q^{43} - 98 q^{49} + 11 q^{51} - 35 q^{57} - 16 q^{61} + 90 q^{67} - 66 q^{69} + 70 q^{73} - 24 q^{79} - 113 q^{81} + 110 q^{87} - 210 q^{93} + 140 q^{97} - 275 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 2.50000 + 1.65831i 0.833333 + 0.552771i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 3.50000 + 8.29156i 0.388889 + 0.921285i
\(10\) 0 0
\(11\) 16.5831i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.769231 −0.384615 0.923077i \(-0.625666\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.31662i 0.195096i −0.995231 0.0975478i \(-0.968900\pi\)
0.995231 0.0975478i \(-0.0310999\pi\)
\(18\) 0 0
\(19\) −7.00000 −0.368421 −0.184211 0.982887i \(-0.558973\pi\)
−0.184211 + 0.982887i \(0.558973\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 19.8997i 0.865206i 0.901584 + 0.432603i \(0.142405\pi\)
−0.901584 + 0.432603i \(0.857595\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 + 26.5330i −0.185185 + 0.982704i
\(28\) 0 0
\(29\) 33.1662i 1.14366i −0.820371 0.571832i \(-0.806233\pi\)
0.820371 0.571832i \(-0.193767\pi\)
\(30\) 0 0
\(31\) −42.0000 −1.35484 −0.677419 0.735597i \(-0.736902\pi\)
−0.677419 + 0.735597i \(0.736902\pi\)
\(32\) 0 0
\(33\) −27.5000 + 41.4578i −0.833333 + 1.25630i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 40.0000 1.08108 0.540541 0.841318i \(-0.318220\pi\)
0.540541 + 0.841318i \(0.318220\pi\)
\(38\) 0 0
\(39\) −25.0000 16.5831i −0.641026 0.425208i
\(40\) 0 0
\(41\) 16.5831i 0.404466i 0.979337 + 0.202233i \(0.0648199\pi\)
−0.979337 + 0.202233i \(0.935180\pi\)
\(42\) 0 0
\(43\) −50.0000 −1.16279 −0.581395 0.813621i \(-0.697493\pi\)
−0.581395 + 0.813621i \(0.697493\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 46.4327i 0.987931i −0.869482 0.493965i \(-0.835547\pi\)
0.869482 0.493965i \(-0.164453\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 0 0
\(51\) 5.50000 8.29156i 0.107843 0.162580i
\(52\) 0 0
\(53\) 46.4327i 0.876090i 0.898953 + 0.438045i \(0.144329\pi\)
−0.898953 + 0.438045i \(0.855671\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −17.5000 11.6082i −0.307018 0.203652i
\(58\) 0 0
\(59\) 66.3325i 1.12428i 0.827042 + 0.562140i \(0.190022\pi\)
−0.827042 + 0.562140i \(0.809978\pi\)
\(60\) 0 0
\(61\) −8.00000 −0.131148 −0.0655738 0.997848i \(-0.520888\pi\)
−0.0655738 + 0.997848i \(0.520888\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 45.0000 0.671642 0.335821 0.941926i \(-0.390986\pi\)
0.335821 + 0.941926i \(0.390986\pi\)
\(68\) 0 0
\(69\) −33.0000 + 49.7494i −0.478261 + 0.721005i
\(70\) 0 0
\(71\) 33.1662i 0.467130i 0.972341 + 0.233565i \(0.0750392\pi\)
−0.972341 + 0.233565i \(0.924961\pi\)
\(72\) 0 0
\(73\) 35.0000 0.479452 0.239726 0.970841i \(-0.422942\pi\)
0.239726 + 0.970841i \(0.422942\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −0.151899 −0.0759494 0.997112i \(-0.524199\pi\)
−0.0759494 + 0.997112i \(0.524199\pi\)
\(80\) 0 0
\(81\) −56.5000 + 58.0409i −0.697531 + 0.716555i
\(82\) 0 0
\(83\) 69.6491i 0.839146i 0.907722 + 0.419573i \(0.137820\pi\)
−0.907722 + 0.419573i \(0.862180\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 55.0000 82.9156i 0.632184 0.953053i
\(88\) 0 0
\(89\) 149.248i 1.67695i 0.544944 + 0.838473i \(0.316551\pi\)
−0.544944 + 0.838473i \(0.683449\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −105.000 69.6491i −1.12903 0.748915i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 70.0000 0.721649 0.360825 0.932634i \(-0.382495\pi\)
0.360825 + 0.932634i \(0.382495\pi\)
\(98\) 0 0
\(99\) −137.500 + 58.0409i −1.38889 + 0.586272i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −70.0000 −0.679612 −0.339806 0.940496i \(-0.610361\pi\)
−0.339806 + 0.940496i \(0.610361\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 69.6491i 0.650926i 0.945555 + 0.325463i \(0.105520\pi\)
−0.945555 + 0.325463i \(0.894480\pi\)
\(108\) 0 0
\(109\) −88.0000 −0.807339 −0.403670 0.914905i \(-0.632266\pi\)
−0.403670 + 0.914905i \(0.632266\pi\)
\(110\) 0 0
\(111\) 100.000 + 66.3325i 0.900901 + 0.597590i
\(112\) 0 0
\(113\) 102.815i 0.909871i −0.890525 0.454935i \(-0.849662\pi\)
0.890525 0.454935i \(-0.150338\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −35.0000 82.9156i −0.299145 0.708681i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −154.000 −1.27273
\(122\) 0 0
\(123\) −27.5000 + 41.4578i −0.223577 + 0.337055i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 190.000 1.49606 0.748031 0.663663i \(-0.230999\pi\)
0.748031 + 0.663663i \(0.230999\pi\)
\(128\) 0 0
\(129\) −125.000 82.9156i −0.968992 0.642757i
\(130\) 0 0
\(131\) 198.997i 1.51906i −0.650469 0.759532i \(-0.725428\pi\)
0.650469 0.759532i \(-0.274572\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 69.6491i 0.508388i −0.967153 0.254194i \(-0.918190\pi\)
0.967153 0.254194i \(-0.0818101\pi\)
\(138\) 0 0
\(139\) −77.0000 −0.553957 −0.276978 0.960876i \(-0.589333\pi\)
−0.276978 + 0.960876i \(0.589333\pi\)
\(140\) 0 0
\(141\) 77.0000 116.082i 0.546099 0.823276i
\(142\) 0 0
\(143\) 165.831i 1.15966i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −122.500 81.2573i −0.833333 0.552771i
\(148\) 0 0
\(149\) 165.831i 1.11296i 0.830861 + 0.556481i \(0.187849\pi\)
−0.830861 + 0.556481i \(0.812151\pi\)
\(150\) 0 0
\(151\) −172.000 −1.13907 −0.569536 0.821966i \(-0.692877\pi\)
−0.569536 + 0.821966i \(0.692877\pi\)
\(152\) 0 0
\(153\) 27.5000 11.6082i 0.179739 0.0758705i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 250.000 1.59236 0.796178 0.605062i \(-0.206852\pi\)
0.796178 + 0.605062i \(0.206852\pi\)
\(158\) 0 0
\(159\) −77.0000 + 116.082i −0.484277 + 0.730075i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 35.0000 0.214724 0.107362 0.994220i \(-0.465760\pi\)
0.107362 + 0.994220i \(0.465760\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 179.098i 1.07244i −0.844078 0.536221i \(-0.819851\pi\)
0.844078 0.536221i \(-0.180149\pi\)
\(168\) 0 0
\(169\) −69.0000 −0.408284
\(170\) 0 0
\(171\) −24.5000 58.0409i −0.143275 0.339421i
\(172\) 0 0
\(173\) 278.596i 1.61038i 0.593014 + 0.805192i \(0.297938\pi\)
−0.593014 + 0.805192i \(0.702062\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −110.000 + 165.831i −0.621469 + 0.936900i
\(178\) 0 0
\(179\) 116.082i 0.648502i 0.945971 + 0.324251i \(0.105112\pi\)
−0.945971 + 0.324251i \(0.894888\pi\)
\(180\) 0 0
\(181\) 182.000 1.00552 0.502762 0.864425i \(-0.332317\pi\)
0.502762 + 0.864425i \(0.332317\pi\)
\(182\) 0 0
\(183\) −20.0000 13.2665i −0.109290 0.0724945i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 55.0000 0.294118
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 232.164i 1.21552i 0.794122 + 0.607758i \(0.207931\pi\)
−0.794122 + 0.607758i \(0.792069\pi\)
\(192\) 0 0
\(193\) 25.0000 0.129534 0.0647668 0.997900i \(-0.479370\pi\)
0.0647668 + 0.997900i \(0.479370\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 218.897i 1.11115i −0.831465 0.555577i \(-0.812498\pi\)
0.831465 0.555577i \(-0.187502\pi\)
\(198\) 0 0
\(199\) 68.0000 0.341709 0.170854 0.985296i \(-0.445347\pi\)
0.170854 + 0.985296i \(0.445347\pi\)
\(200\) 0 0
\(201\) 112.500 + 74.6241i 0.559701 + 0.371264i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −165.000 + 69.6491i −0.797101 + 0.336469i
\(208\) 0 0
\(209\) 116.082i 0.555416i
\(210\) 0 0
\(211\) −77.0000 −0.364929 −0.182464 0.983212i \(-0.558407\pi\)
−0.182464 + 0.983212i \(0.558407\pi\)
\(212\) 0 0
\(213\) −55.0000 + 82.9156i −0.258216 + 0.389275i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 87.5000 + 58.0409i 0.399543 + 0.265027i
\(220\) 0 0
\(221\) 33.1662i 0.150074i
\(222\) 0 0
\(223\) 140.000 0.627803 0.313901 0.949456i \(-0.398364\pi\)
0.313901 + 0.949456i \(0.398364\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 185.731i 0.818198i 0.912490 + 0.409099i \(0.134157\pi\)
−0.912490 + 0.409099i \(0.865843\pi\)
\(228\) 0 0
\(229\) 372.000 1.62445 0.812227 0.583341i \(-0.198255\pi\)
0.812227 + 0.583341i \(0.198255\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 119.398i 0.512440i −0.966619 0.256220i \(-0.917523\pi\)
0.966619 0.256220i \(-0.0824771\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −30.0000 19.8997i −0.126582 0.0839652i
\(238\) 0 0
\(239\) 232.164i 0.971396i −0.874127 0.485698i \(-0.838565\pi\)
0.874127 0.485698i \(-0.161435\pi\)
\(240\) 0 0
\(241\) −413.000 −1.71369 −0.856846 0.515572i \(-0.827580\pi\)
−0.856846 + 0.515572i \(0.827580\pi\)
\(242\) 0 0
\(243\) −237.500 + 51.4077i −0.977366 + 0.211554i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 70.0000 0.283401
\(248\) 0 0
\(249\) −115.500 + 174.123i −0.463855 + 0.699288i
\(250\) 0 0
\(251\) 248.747i 0.991023i 0.868601 + 0.495512i \(0.165019\pi\)
−0.868601 + 0.495512i \(0.834981\pi\)
\(252\) 0 0
\(253\) −330.000 −1.30435
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 278.596i 1.08403i 0.840368 + 0.542017i \(0.182339\pi\)
−0.840368 + 0.542017i \(0.817661\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 275.000 116.082i 1.05364 0.444758i
\(262\) 0 0
\(263\) 285.230i 1.08452i 0.840210 + 0.542262i \(0.182432\pi\)
−0.840210 + 0.542262i \(0.817568\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −247.500 + 373.120i −0.926966 + 1.39745i
\(268\) 0 0
\(269\) 464.327i 1.72612i −0.505098 0.863062i \(-0.668544\pi\)
0.505098 0.863062i \(-0.331456\pi\)
\(270\) 0 0
\(271\) −22.0000 −0.0811808 −0.0405904 0.999176i \(-0.512924\pi\)
−0.0405904 + 0.999176i \(0.512924\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 210.000 0.758123 0.379061 0.925372i \(-0.376247\pi\)
0.379061 + 0.925372i \(0.376247\pi\)
\(278\) 0 0
\(279\) −147.000 348.246i −0.526882 1.24819i
\(280\) 0 0
\(281\) 198.997i 0.708176i 0.935212 + 0.354088i \(0.115209\pi\)
−0.935212 + 0.354088i \(0.884791\pi\)
\(282\) 0 0
\(283\) 345.000 1.21908 0.609541 0.792755i \(-0.291354\pi\)
0.609541 + 0.792755i \(0.291354\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 278.000 0.961938
\(290\) 0 0
\(291\) 175.000 + 116.082i 0.601375 + 0.398907i
\(292\) 0 0
\(293\) 318.396i 1.08668i −0.839514 0.543338i \(-0.817160\pi\)
0.839514 0.543338i \(-0.182840\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −440.000 82.9156i −1.48148 0.279177i
\(298\) 0 0
\(299\) 198.997i 0.665543i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 325.000 1.05863 0.529316 0.848425i \(-0.322449\pi\)
0.529316 + 0.848425i \(0.322449\pi\)
\(308\) 0 0
\(309\) −175.000 116.082i −0.566343 0.375669i
\(310\) 0 0
\(311\) 397.995i 1.27973i −0.768489 0.639863i \(-0.778991\pi\)
0.768489 0.639863i \(-0.221009\pi\)
\(312\) 0 0
\(313\) 490.000 1.56550 0.782748 0.622339i \(-0.213818\pi\)
0.782748 + 0.622339i \(0.213818\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 212.264i 0.669602i 0.942289 + 0.334801i \(0.108669\pi\)
−0.942289 + 0.334801i \(0.891331\pi\)
\(318\) 0 0
\(319\) 550.000 1.72414
\(320\) 0 0
\(321\) −115.500 + 174.123i −0.359813 + 0.542439i
\(322\) 0 0
\(323\) 23.2164i 0.0718773i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −220.000 145.931i −0.672783 0.446274i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 243.000 0.734139 0.367069 0.930194i \(-0.380361\pi\)
0.367069 + 0.930194i \(0.380361\pi\)
\(332\) 0 0
\(333\) 140.000 + 331.662i 0.420420 + 0.995983i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −385.000 −1.14243 −0.571217 0.820799i \(-0.693528\pi\)
−0.571217 + 0.820799i \(0.693528\pi\)
\(338\) 0 0
\(339\) 170.500 257.038i 0.502950 0.758225i
\(340\) 0 0
\(341\) 696.491i 2.04250i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 295.180i 0.850662i −0.905038 0.425331i \(-0.860158\pi\)
0.905038 0.425331i \(-0.139842\pi\)
\(348\) 0 0
\(349\) 532.000 1.52436 0.762178 0.647368i \(-0.224130\pi\)
0.762178 + 0.647368i \(0.224130\pi\)
\(350\) 0 0
\(351\) 50.0000 265.330i 0.142450 0.755926i
\(352\) 0 0
\(353\) 278.596i 0.789225i 0.918848 + 0.394613i \(0.129121\pi\)
−0.918848 + 0.394613i \(0.870879\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 397.995i 1.10862i 0.832310 + 0.554311i \(0.187018\pi\)
−0.832310 + 0.554311i \(0.812982\pi\)
\(360\) 0 0
\(361\) −312.000 −0.864266
\(362\) 0 0
\(363\) −385.000 255.380i −1.06061 0.703526i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −180.000 −0.490463 −0.245232 0.969465i \(-0.578864\pi\)
−0.245232 + 0.969465i \(0.578864\pi\)
\(368\) 0 0
\(369\) −137.500 + 58.0409i −0.372629 + 0.157293i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 110.000 0.294906 0.147453 0.989069i \(-0.452892\pi\)
0.147453 + 0.989069i \(0.452892\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 331.662i 0.879741i
\(378\) 0 0
\(379\) 533.000 1.40633 0.703166 0.711025i \(-0.251769\pi\)
0.703166 + 0.711025i \(0.251769\pi\)
\(380\) 0 0
\(381\) 475.000 + 315.079i 1.24672 + 0.826980i
\(382\) 0 0
\(383\) 79.5990i 0.207830i −0.994586 0.103915i \(-0.966863\pi\)
0.994586 0.103915i \(-0.0331370\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −175.000 414.578i −0.452196 1.07126i
\(388\) 0 0
\(389\) 99.4987i 0.255781i −0.991788 0.127890i \(-0.959179\pi\)
0.991788 0.127890i \(-0.0408206\pi\)
\(390\) 0 0
\(391\) 66.0000 0.168798
\(392\) 0 0
\(393\) 330.000 497.494i 0.839695 1.26589i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 20.0000 0.0503778 0.0251889 0.999683i \(-0.491981\pi\)
0.0251889 + 0.999683i \(0.491981\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 746.241i 1.86095i 0.366357 + 0.930475i \(0.380605\pi\)
−0.366357 + 0.930475i \(0.619395\pi\)
\(402\) 0 0
\(403\) 420.000 1.04218
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 663.325i 1.62979i
\(408\) 0 0
\(409\) 77.0000 0.188264 0.0941320 0.995560i \(-0.469992\pi\)
0.0941320 + 0.995560i \(0.469992\pi\)
\(410\) 0 0
\(411\) 115.500 174.123i 0.281022 0.423656i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −192.500 127.690i −0.461631 0.306211i
\(418\) 0 0
\(419\) 116.082i 0.277045i −0.990359 0.138523i \(-0.955765\pi\)
0.990359 0.138523i \(-0.0442353\pi\)
\(420\) 0 0
\(421\) 412.000 0.978622 0.489311 0.872109i \(-0.337248\pi\)
0.489311 + 0.872109i \(0.337248\pi\)
\(422\) 0 0
\(423\) 385.000 162.515i 0.910165 0.384195i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 275.000 414.578i 0.641026 0.966383i
\(430\) 0 0
\(431\) 198.997i 0.461711i −0.972988 0.230856i \(-0.925848\pi\)
0.972988 0.230856i \(-0.0741525\pi\)
\(432\) 0 0
\(433\) 455.000 1.05081 0.525404 0.850853i \(-0.323914\pi\)
0.525404 + 0.850853i \(0.323914\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 139.298i 0.318760i
\(438\) 0 0
\(439\) −22.0000 −0.0501139 −0.0250569 0.999686i \(-0.507977\pi\)
−0.0250569 + 0.999686i \(0.507977\pi\)
\(440\) 0 0
\(441\) −171.500 406.287i −0.388889 0.921285i
\(442\) 0 0
\(443\) 527.343i 1.19039i −0.803581 0.595196i \(-0.797075\pi\)
0.803581 0.595196i \(-0.202925\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −275.000 + 414.578i −0.615213 + 0.927468i
\(448\) 0 0
\(449\) 82.9156i 0.184667i 0.995728 + 0.0923337i \(0.0294326\pi\)
−0.995728 + 0.0923337i \(0.970567\pi\)
\(450\) 0 0
\(451\) −275.000 −0.609756
\(452\) 0 0
\(453\) −430.000 285.230i −0.949227 0.629646i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 275.000 0.601751 0.300875 0.953663i \(-0.402721\pi\)
0.300875 + 0.953663i \(0.402721\pi\)
\(458\) 0 0
\(459\) 88.0000 + 16.5831i 0.191721 + 0.0361288i
\(460\) 0 0
\(461\) 596.992i 1.29499i −0.762068 0.647497i \(-0.775816\pi\)
0.762068 0.647497i \(-0.224184\pi\)
\(462\) 0 0
\(463\) −240.000 −0.518359 −0.259179 0.965829i \(-0.583452\pi\)
−0.259179 + 0.965829i \(0.583452\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.2665i 0.0284079i −0.999899 0.0142040i \(-0.995479\pi\)
0.999899 0.0142040i \(-0.00452141\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 625.000 + 414.578i 1.32696 + 0.880208i
\(472\) 0 0
\(473\) 829.156i 1.75297i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −385.000 + 162.515i −0.807128 + 0.340701i
\(478\) 0 0
\(479\) 298.496i 0.623165i −0.950219 0.311583i \(-0.899141\pi\)
0.950219 0.311583i \(-0.100859\pi\)
\(480\) 0 0
\(481\) −400.000 −0.831601
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 410.000 0.841889 0.420945 0.907086i \(-0.361699\pi\)
0.420945 + 0.907086i \(0.361699\pi\)
\(488\) 0 0
\(489\) 87.5000 + 58.0409i 0.178937 + 0.118693i
\(490\) 0 0
\(491\) 265.330i 0.540387i −0.962806 0.270193i \(-0.912912\pi\)
0.962806 0.270193i \(-0.0870877\pi\)
\(492\) 0 0
\(493\) −110.000 −0.223124
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −322.000 −0.645291 −0.322645 0.946520i \(-0.604572\pi\)
−0.322645 + 0.946520i \(0.604572\pi\)
\(500\) 0 0
\(501\) 297.000 447.744i 0.592814 0.893701i
\(502\) 0 0
\(503\) 411.261i 0.817617i −0.912620 0.408809i \(-0.865944\pi\)
0.912620 0.408809i \(-0.134056\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −172.500 114.424i −0.340237 0.225687i
\(508\) 0 0
\(509\) 431.161i 0.847075i 0.905879 + 0.423538i \(0.139212\pi\)
−0.905879 + 0.423538i \(0.860788\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 35.0000 185.731i 0.0682261 0.362049i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 770.000 1.48936
\(518\) 0 0
\(519\) −462.000 + 696.491i −0.890173 + 1.34199i
\(520\) 0 0
\(521\) 281.913i 0.541100i −0.962706 0.270550i \(-0.912794\pi\)
0.962706 0.270550i \(-0.0872055\pi\)
\(522\) 0 0
\(523\) 1015.00 1.94073 0.970363 0.241651i \(-0.0776888\pi\)
0.970363 + 0.241651i \(0.0776888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 139.298i 0.264323i
\(528\) 0 0
\(529\) 133.000 0.251418
\(530\) 0 0
\(531\) −550.000 + 232.164i −1.03578 + 0.437220i
\(532\) 0 0
\(533\) 165.831i 0.311128i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −192.500 + 290.205i −0.358473 + 0.540418i
\(538\) 0 0
\(539\) 812.573i 1.50756i
\(540\) 0 0
\(541\) 912.000 1.68577 0.842884 0.538096i \(-0.180856\pi\)
0.842884 + 0.538096i \(0.180856\pi\)
\(542\) 0 0
\(543\) 455.000 + 301.813i 0.837937 + 0.555825i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 55.0000 0.100548 0.0502742 0.998735i \(-0.483990\pi\)
0.0502742 + 0.998735i \(0.483990\pi\)
\(548\) 0 0
\(549\) −28.0000 66.3325i −0.0510018 0.120824i
\(550\) 0 0
\(551\) 232.164i 0.421350i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1014.89i 1.82206i −0.412341 0.911030i \(-0.635289\pi\)
0.412341 0.911030i \(-0.364711\pi\)
\(558\) 0 0
\(559\) 500.000 0.894454
\(560\) 0 0
\(561\) 137.500 + 91.2072i 0.245098 + 0.162580i
\(562\) 0 0
\(563\) 119.398i 0.212075i 0.994362 + 0.106038i \(0.0338164\pi\)
−0.994362 + 0.106038i \(0.966184\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 49.7494i 0.0874330i −0.999044 0.0437165i \(-0.986080\pi\)
0.999044 0.0437165i \(-0.0139198\pi\)
\(570\) 0 0
\(571\) −242.000 −0.423818 −0.211909 0.977289i \(-0.567968\pi\)
−0.211909 + 0.977289i \(0.567968\pi\)
\(572\) 0 0
\(573\) −385.000 + 580.409i −0.671902 + 1.01293i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −665.000 −1.15251 −0.576256 0.817269i \(-0.695487\pi\)
−0.576256 + 0.817269i \(0.695487\pi\)
\(578\) 0 0
\(579\) 62.5000 + 41.4578i 0.107945 + 0.0716024i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −770.000 −1.32075
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 626.842i 1.06787i −0.845524 0.533937i \(-0.820712\pi\)
0.845524 0.533937i \(-0.179288\pi\)
\(588\) 0 0
\(589\) 294.000 0.499151
\(590\) 0 0
\(591\) 363.000 547.243i 0.614213 0.925961i
\(592\) 0 0
\(593\) 859.006i 1.44858i 0.689497 + 0.724288i \(0.257831\pi\)
−0.689497 + 0.724288i \(0.742169\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 170.000 + 112.765i 0.284757 + 0.188887i
\(598\) 0 0
\(599\) 331.662i 0.553694i 0.960914 + 0.276847i \(0.0892895\pi\)
−0.960914 + 0.276847i \(0.910711\pi\)
\(600\) 0 0
\(601\) −343.000 −0.570715 −0.285358 0.958421i \(-0.592112\pi\)
−0.285358 + 0.958421i \(0.592112\pi\)
\(602\) 0 0
\(603\) 157.500 + 373.120i 0.261194 + 0.618773i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1100.00 −1.81219 −0.906096 0.423073i \(-0.860951\pi\)
−0.906096 + 0.423073i \(0.860951\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 464.327i 0.759947i
\(612\) 0 0
\(613\) 290.000 0.473083 0.236542 0.971621i \(-0.423986\pi\)
0.236542 + 0.971621i \(0.423986\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 79.5990i 0.129010i 0.997917 + 0.0645049i \(0.0205468\pi\)
−0.997917 + 0.0645049i \(0.979453\pi\)
\(618\) 0 0
\(619\) 58.0000 0.0936995 0.0468498 0.998902i \(-0.485082\pi\)
0.0468498 + 0.998902i \(0.485082\pi\)
\(620\) 0 0
\(621\) −528.000 99.4987i −0.850242 0.160223i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 192.500 290.205i 0.307018 0.462846i
\(628\) 0 0
\(629\) 132.665i 0.210914i
\(630\) 0 0
\(631\) −862.000 −1.36609 −0.683043 0.730378i \(-0.739344\pi\)
−0.683043 + 0.730378i \(0.739344\pi\)
\(632\) 0 0
\(633\) −192.500 127.690i −0.304107 0.201722i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 490.000 0.769231
\(638\) 0 0
\(639\) −275.000 + 116.082i −0.430360 + 0.181662i
\(640\) 0 0
\(641\) 596.992i 0.931345i 0.884957 + 0.465673i \(0.154188\pi\)
−0.884957 + 0.465673i \(0.845812\pi\)
\(642\) 0 0
\(643\) −1050.00 −1.63297 −0.816485 0.577366i \(-0.804080\pi\)
−0.816485 + 0.577366i \(0.804080\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 252.063i 0.389588i 0.980844 + 0.194794i \(0.0624038\pi\)
−0.980844 + 0.194794i \(0.937596\pi\)
\(648\) 0 0
\(649\) −1100.00 −1.69492
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1207.25i 1.84878i 0.381452 + 0.924389i \(0.375424\pi\)
−0.381452 + 0.924389i \(0.624576\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 122.500 + 290.205i 0.186454 + 0.441712i
\(658\) 0 0
\(659\) 812.573i 1.23304i 0.787339 + 0.616520i \(0.211458\pi\)
−0.787339 + 0.616520i \(0.788542\pi\)
\(660\) 0 0
\(661\) −98.0000 −0.148260 −0.0741301 0.997249i \(-0.523618\pi\)
−0.0741301 + 0.997249i \(0.523618\pi\)
\(662\) 0 0
\(663\) −55.0000 + 82.9156i −0.0829563 + 0.125061i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 660.000 0.989505
\(668\) 0 0
\(669\) 350.000 + 232.164i 0.523169 + 0.347031i
\(670\) 0 0
\(671\) 132.665i 0.197712i
\(672\) 0 0
\(673\) 210.000 0.312036 0.156018 0.987754i \(-0.450134\pi\)
0.156018 + 0.987754i \(0.450134\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 79.5990i 0.117576i 0.998270 + 0.0587880i \(0.0187236\pi\)
−0.998270 + 0.0587880i \(0.981276\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −308.000 + 464.327i −0.452276 + 0.681832i
\(682\) 0 0
\(683\) 169.148i 0.247654i 0.992304 + 0.123827i \(0.0395168\pi\)
−0.992304 + 0.123827i \(0.960483\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 930.000 + 616.892i 1.35371 + 0.897951i
\(688\) 0 0
\(689\) 464.327i 0.673915i
\(690\) 0 0
\(691\) 713.000 1.03184 0.515919 0.856637i \(-0.327451\pi\)
0.515919 + 0.856637i \(0.327451\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 55.0000 0.0789096
\(698\) 0 0
\(699\) 198.000 298.496i 0.283262 0.427033i
\(700\) 0 0
\(701\) 1160.82i 1.65595i 0.560767 + 0.827973i \(0.310506\pi\)
−0.560767 + 0.827973i \(0.689494\pi\)
\(702\) 0 0
\(703\) −280.000 −0.398293
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −248.000 −0.349788 −0.174894 0.984587i \(-0.555958\pi\)
−0.174894 + 0.984587i \(0.555958\pi\)
\(710\) 0 0
\(711\) −42.0000 99.4987i −0.0590717 0.139942i
\(712\) 0 0
\(713\) 835.789i 1.17222i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 385.000 580.409i 0.536960 0.809497i
\(718\) 0 0
\(719\) 198.997i 0.276770i −0.990379 0.138385i \(-0.955809\pi\)
0.990379 0.138385i \(-0.0441911\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1032.50 684.883i −1.42808 0.947279i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −10.0000 −0.0137552 −0.00687758 0.999976i \(-0.502189\pi\)
−0.00687758 + 0.999976i \(0.502189\pi\)
\(728\) 0 0
\(729\) −679.000 265.330i −0.931413 0.363964i
\(730\) 0 0
\(731\) 165.831i 0.226855i
\(732\) 0 0
\(733\) −770.000 −1.05048 −0.525239 0.850955i \(-0.676024\pi\)
−0.525239 + 0.850955i \(0.676024\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 746.241i 1.01254i
\(738\) 0 0
\(739\) −802.000 −1.08525 −0.542625 0.839975i \(-0.682570\pi\)
−0.542625 + 0.839975i \(0.682570\pi\)
\(740\) 0 0
\(741\) 175.000 + 116.082i 0.236167 + 0.156656i
\(742\) 0 0
\(743\) 1346.55i 1.81231i 0.422941 + 0.906157i \(0.360998\pi\)
−0.422941 + 0.906157i \(0.639002\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −577.500 + 243.772i −0.773092 + 0.326335i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −322.000 −0.428762 −0.214381 0.976750i \(-0.568773\pi\)
−0.214381 + 0.976750i \(0.568773\pi\)
\(752\) 0 0
\(753\) −412.500 + 621.867i −0.547809 + 0.825853i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −400.000 −0.528402 −0.264201 0.964468i \(-0.585108\pi\)
−0.264201 + 0.964468i \(0.585108\pi\)
\(758\) 0 0
\(759\) −825.000 547.243i −1.08696 0.721005i
\(760\) 0 0
\(761\) 348.246i 0.457616i −0.973472 0.228808i \(-0.926517\pi\)
0.973472 0.228808i \(-0.0734828\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 663.325i 0.864830i
\(768\) 0 0
\(769\) −193.000 −0.250975 −0.125488 0.992095i \(-0.540050\pi\)
−0.125488 + 0.992095i \(0.540050\pi\)
\(770\) 0 0
\(771\) −462.000 + 696.491i −0.599222 + 0.903361i
\(772\) 0 0
\(773\) 417.895i 0.540614i −0.962774 0.270307i \(-0.912875\pi\)
0.962774 0.270307i \(-0.0871252\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 116.082i 0.149014i
\(780\) 0 0
\(781\) −550.000 −0.704225
\(782\) 0 0
\(783\) 880.000 + 165.831i 1.12388 + 0.211790i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 910.000 1.15629 0.578145 0.815934i \(-0.303777\pi\)
0.578145 + 0.815934i \(0.303777\pi\)
\(788\) 0 0
\(789\) −473.000 + 713.074i −0.599493 + 0.903770i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 80.0000 0.100883
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1107.75i 1.38990i 0.719057 + 0.694951i \(0.244574\pi\)
−0.719057 + 0.694951i \(0.755426\pi\)
\(798\) 0 0
\(799\) −154.000 −0.192741
\(800\) 0 0
\(801\) −1237.50 + 522.368i −1.54494 + 0.652145i
\(802\) 0 0
\(803\) 580.409i 0.722801i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 770.000 1160.82i 0.954151 1.43844i
\(808\) 0 0
\(809\) 1260.32i 1.55787i 0.627104 + 0.778935i \(0.284240\pi\)
−0.627104 + 0.778935i \(0.715760\pi\)
\(810\) 0 0
\(811\) 858.000 1.05795 0.528977 0.848636i \(-0.322576\pi\)
0.528977 + 0.848636i \(0.322576\pi\)
\(812\) 0 0
\(813\) −55.0000 36.4829i −0.0676507 0.0448744i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 350.000 0.428397
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 696.491i 0.848345i −0.905581 0.424172i \(-0.860565\pi\)
0.905581 0.424172i \(-0.139435\pi\)
\(822\) 0 0
\(823\) −1060.00 −1.28797 −0.643985 0.765038i \(-0.722720\pi\)
−0.643985 + 0.765038i \(0.722720\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 500.810i 0.605575i 0.953058 + 0.302787i \(0.0979172\pi\)
−0.953058 + 0.302787i \(0.902083\pi\)
\(828\) 0 0
\(829\) −1038.00 −1.25211 −0.626055 0.779779i \(-0.715332\pi\)
−0.626055 + 0.779779i \(0.715332\pi\)
\(830\) 0 0
\(831\) 525.000 + 348.246i 0.631769 + 0.419068i
\(832\) 0 0
\(833\) 162.515i 0.195096i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 210.000 1114.39i 0.250896 1.33140i
\(838\) 0 0
\(839\) 928.655i 1.10686i 0.832896 + 0.553430i \(0.186681\pi\)
−0.832896 + 0.553430i \(0.813319\pi\)
\(840\) 0 0
\(841\) −259.000 −0.307967
\(842\) 0 0
\(843\) −330.000 + 497.494i −0.391459 + 0.590147i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 862.500 + 572.118i 1.01590 + 0.673873i
\(850\) 0 0
\(851\) 795.990i 0.935358i
\(852\) 0 0
\(853\) −630.000 −0.738570 −0.369285 0.929316i \(-0.620397\pi\)
−0.369285 + 0.929316i \(0.620397\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1296.80i 1.51319i −0.653886 0.756593i \(-0.726862\pi\)
0.653886 0.756593i \(-0.273138\pi\)
\(858\) 0 0
\(859\) −307.000 −0.357392 −0.178696 0.983904i \(-0.557188\pi\)
−0.178696 + 0.983904i \(0.557188\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 484.227i 0.561098i 0.959840 + 0.280549i \(0.0905165\pi\)
−0.959840 + 0.280549i \(0.909484\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 695.000 + 461.011i 0.801615 + 0.531731i
\(868\) 0 0
\(869\) 198.997i 0.228996i
\(870\) 0 0
\(871\) −450.000 −0.516648
\(872\) 0 0
\(873\) 245.000 + 580.409i 0.280641 + 0.664845i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −840.000 −0.957811 −0.478905 0.877867i \(-0.658966\pi\)
−0.478905 + 0.877867i \(0.658966\pi\)
\(878\) 0 0
\(879\) 528.000 795.990i 0.600683 0.905563i
\(880\) 0 0
\(881\) 464.327i 0.527046i −0.964653 0.263523i \(-0.915116\pi\)
0.964653 0.263523i \(-0.0848845\pi\)
\(882\) 0 0
\(883\) 995.000 1.12684 0.563420 0.826171i \(-0.309485\pi\)
0.563420 + 0.826171i \(0.309485\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 477.594i 0.538437i −0.963079 0.269219i \(-0.913235\pi\)
0.963079 0.269219i \(-0.0867654\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −962.500 936.947i −1.08025 1.05157i
\(892\) 0 0
\(893\) 325.029i 0.363975i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 330.000 497.494i 0.367893 0.554620i
\(898\) 0 0
\(899\) 1392.98i 1.54948i
\(900\) 0 0
\(901\) 154.000 0.170921
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1050.00 −1.15766 −0.578831 0.815447i \(-0.696491\pi\)
−0.578831 + 0.815447i \(0.696491\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 729.657i 0.800941i −0.916310 0.400471i \(-0.868847\pi\)
0.916310 0.400471i \(-0.131153\pi\)
\(912\) 0 0
\(913\) −1155.00 −1.26506
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1342.00 −1.46028 −0.730141 0.683296i \(-0.760546\pi\)
−0.730141 + 0.683296i \(0.760546\pi\)
\(920\) 0 0
\(921\) 812.500 + 538.952i 0.882193 + 0.585181i
\(922\) 0 0
\(923\) 331.662i 0.359331i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −245.000 580.409i −0.264293 0.626116i
\(928\) 0 0
\(929\) 464.327i 0.499814i 0.968270 + 0.249907i \(0.0804001\pi\)
−0.968270 + 0.249907i \(0.919600\pi\)
\(930\) 0 0
\(931\) 343.000 0.368421
\(932\) 0 0
\(933\) 660.000 994.987i 0.707395 1.06644i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 165.000 0.176094 0.0880470 0.996116i \(-0.471937\pi\)
0.0880470 + 0.996116i \(0.471937\pi\)
\(938\) 0 0
\(939\) 1225.00 + 812.573i 1.30458 + 0.865360i
\(940\) 0 0
\(941\) 232.164i 0.246720i −0.992362 0.123360i \(-0.960633\pi\)
0.992362 0.123360i \(-0.0393670\pi\)
\(942\) 0 0
\(943\) −330.000 −0.349947
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 278.596i 0.294188i −0.989122 0.147094i \(-0.953008\pi\)
0.989122 0.147094i \(-0.0469921\pi\)
\(948\) 0 0
\(949\) −350.000 −0.368809
\(950\) 0 0
\(951\) −352.000 + 530.660i −0.370137 + 0.558002i
\(952\) 0 0
\(953\) 195.681i 0.205331i 0.994716 + 0.102666i \(0.0327372\pi\)
−0.994716 + 0.102666i \(0.967263\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1375.00 + 912.072i 1.43678 + 0.953053i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 803.000 0.835588
\(962\) 0 0
\(963\) −577.500 + 243.772i −0.599688 + 0.253138i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −280.000 −0.289555 −0.144778 0.989464i \(-0.546247\pi\)
−0.144778 + 0.989464i \(0.546247\pi\)
\(968\) 0 0
\(969\) −38.5000 + 58.0409i −0.0397317 + 0.0598978i
\(970\) 0 0
\(971\) 1044.74i 1.07594i −0.842964 0.537970i \(-0.819192\pi\)
0.842964 0.537970i \(-0.180808\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1097.80i 1.12365i −0.827257 0.561823i \(-0.810100\pi\)
0.827257 0.561823i \(-0.189900\pi\)
\(978\) 0 0
\(979\) −2475.00 −2.52809
\(980\) 0 0
\(981\) −308.000 729.657i −0.313965 0.743789i
\(982\) 0 0
\(983\) 1671.58i 1.70049i −0.526389 0.850244i \(-0.676455\pi\)
0.526389 0.850244i \(-0.323545\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 994.987i 1.00605i
\(990\) 0 0
\(991\) −452.000 −0.456105 −0.228052 0.973649i \(-0.573236\pi\)
−0.228052 + 0.973649i \(0.573236\pi\)
\(992\) 0 0
\(993\) 607.500 + 402.970i 0.611782 + 0.405811i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 420.000 0.421264 0.210632 0.977565i \(-0.432448\pi\)
0.210632 + 0.977565i \(0.432448\pi\)
\(998\) 0 0
\(999\) −200.000 + 1061.32i −0.200200 + 1.06238i
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 1200.3.l.s.401.2 2
3.2 odd 2 inner 1200.3.l.s.401.1 2
4.3 odd 2 75.3.c.c.26.1 2
5.2 odd 4 1200.3.c.d.449.3 4
5.3 odd 4 1200.3.c.d.449.2 4
5.4 even 2 1200.3.l.f.401.1 2
12.11 even 2 75.3.c.c.26.2 yes 2
15.2 even 4 1200.3.c.d.449.1 4
15.8 even 4 1200.3.c.d.449.4 4
15.14 odd 2 1200.3.l.f.401.2 2
20.3 even 4 75.3.d.c.74.1 4
20.7 even 4 75.3.d.c.74.4 4
20.19 odd 2 75.3.c.f.26.2 yes 2
60.23 odd 4 75.3.d.c.74.3 4
60.47 odd 4 75.3.d.c.74.2 4
60.59 even 2 75.3.c.f.26.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.c.c.26.1 2 4.3 odd 2
75.3.c.c.26.2 yes 2 12.11 even 2
75.3.c.f.26.1 yes 2 60.59 even 2
75.3.c.f.26.2 yes 2 20.19 odd 2
75.3.d.c.74.1 4 20.3 even 4
75.3.d.c.74.2 4 60.47 odd 4
75.3.d.c.74.3 4 60.23 odd 4
75.3.d.c.74.4 4 20.7 even 4
1200.3.c.d.449.1 4 15.2 even 4
1200.3.c.d.449.2 4 5.3 odd 4
1200.3.c.d.449.3 4 5.2 odd 4
1200.3.c.d.449.4 4 15.8 even 4
1200.3.l.f.401.1 2 5.4 even 2
1200.3.l.f.401.2 2 15.14 odd 2
1200.3.l.s.401.1 2 3.2 odd 2 inner
1200.3.l.s.401.2 2 1.1 even 1 trivial