Properties

Label 2070.3.c.a.91.5
Level $2070$
Weight $3$
Character 2070.91
Analytic conductor $56.403$
Analytic rank $0$
Dimension $16$
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] = [2070,3,Mod(91,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2070.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2070.c (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: \(56.4034147226\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 78x^{14} + 2165x^{12} + 28310x^{10} + 184804x^{8} + 569634x^{6} + 696037x^{4} + 285578x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 91.5
Root \(-2.26343i\) of defining polynomial
Character \(\chi\) \(=\) 2070.91
Dual form 2070.3.c.a.91.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -10.1866i q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -10.1866i q^{7} -2.82843 q^{8} -3.16228i q^{10} +13.0237i q^{11} +23.2154 q^{13} +14.4060i q^{14} +4.00000 q^{16} -28.2228i q^{17} +11.6665i q^{19} +4.47214i q^{20} -18.4183i q^{22} +(17.3999 - 15.0414i) q^{23} -5.00000 q^{25} -32.8315 q^{26} -20.3731i q^{28} -42.4794 q^{29} +18.7683 q^{31} -5.65685 q^{32} +39.9131i q^{34} +22.7778 q^{35} +1.14094i q^{37} -16.4989i q^{38} -6.32456i q^{40} +72.8198 q^{41} +4.96573i q^{43} +26.0474i q^{44} +(-24.6071 + 21.2718i) q^{46} +0.813360 q^{47} -54.7661 q^{49} +7.07107 q^{50} +46.4308 q^{52} +26.7286i q^{53} -29.1219 q^{55} +28.8120i q^{56} +60.0750 q^{58} -94.0845 q^{59} +74.5293i q^{61} -26.5423 q^{62} +8.00000 q^{64} +51.9112i q^{65} -80.0906i q^{67} -56.4456i q^{68} -32.2127 q^{70} +83.5303 q^{71} -8.98897 q^{73} -1.61354i q^{74} +23.3330i q^{76} +132.667 q^{77} +80.2841i q^{79} +8.94427i q^{80} -102.983 q^{82} -94.6451i q^{83} +63.1081 q^{85} -7.02261i q^{86} -36.8367i q^{88} -136.812i q^{89} -236.485i q^{91} +(34.7997 - 30.0829i) q^{92} -1.15026 q^{94} -26.0871 q^{95} +2.32666i q^{97} +77.4509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} + 24 q^{13} + 64 q^{16} - 4 q^{23} - 80 q^{25} - 96 q^{26} + 108 q^{29} - 116 q^{31} - 60 q^{35} + 156 q^{41} - 124 q^{46} + 128 q^{47} - 28 q^{49} + 48 q^{52} + 160 q^{58} - 204 q^{59} - 64 q^{62} + 128 q^{64} - 120 q^{70} - 236 q^{71} - 112 q^{73} + 936 q^{77} - 64 q^{82} + 60 q^{85} - 8 q^{92} - 216 q^{94} + 160 q^{95} - 256 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(1891\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 2.23607i 0.447214i
\(6\) 0 0
\(7\) 10.1866i 1.45522i −0.685989 0.727612i \(-0.740630\pi\)
0.685989 0.727612i \(-0.259370\pi\)
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 3.16228i 0.316228i
\(11\) 13.0237i 1.18397i 0.805947 + 0.591987i \(0.201657\pi\)
−0.805947 + 0.591987i \(0.798343\pi\)
\(12\) 0 0
\(13\) 23.2154 1.78580 0.892900 0.450255i \(-0.148667\pi\)
0.892900 + 0.450255i \(0.148667\pi\)
\(14\) 14.4060i 1.02900i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 28.2228i 1.66016i −0.557641 0.830082i \(-0.688293\pi\)
0.557641 0.830082i \(-0.311707\pi\)
\(18\) 0 0
\(19\) 11.6665i 0.614027i 0.951705 + 0.307013i \(0.0993297\pi\)
−0.951705 + 0.307013i \(0.900670\pi\)
\(20\) 4.47214i 0.223607i
\(21\) 0 0
\(22\) 18.4183i 0.837197i
\(23\) 17.3999 15.0414i 0.756516 0.653976i
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) −32.8315 −1.26275
\(27\) 0 0
\(28\) 20.3731i 0.727612i
\(29\) −42.4794 −1.46481 −0.732404 0.680870i \(-0.761602\pi\)
−0.732404 + 0.680870i \(0.761602\pi\)
\(30\) 0 0
\(31\) 18.7683 0.605428 0.302714 0.953082i \(-0.402107\pi\)
0.302714 + 0.953082i \(0.402107\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 39.9131i 1.17391i
\(35\) 22.7778 0.650796
\(36\) 0 0
\(37\) 1.14094i 0.0308363i 0.999881 + 0.0154182i \(0.00490795\pi\)
−0.999881 + 0.0154182i \(0.995092\pi\)
\(38\) 16.4989i 0.434183i
\(39\) 0 0
\(40\) 6.32456i 0.158114i
\(41\) 72.8198 1.77609 0.888046 0.459755i \(-0.152063\pi\)
0.888046 + 0.459755i \(0.152063\pi\)
\(42\) 0 0
\(43\) 4.96573i 0.115482i 0.998332 + 0.0577411i \(0.0183898\pi\)
−0.998332 + 0.0577411i \(0.981610\pi\)
\(44\) 26.0474i 0.591987i
\(45\) 0 0
\(46\) −24.6071 + 21.2718i −0.534937 + 0.462431i
\(47\) 0.813360 0.0173055 0.00865277 0.999963i \(-0.497246\pi\)
0.00865277 + 0.999963i \(0.497246\pi\)
\(48\) 0 0
\(49\) −54.7661 −1.11768
\(50\) 7.07107 0.141421
\(51\) 0 0
\(52\) 46.4308 0.892900
\(53\) 26.7286i 0.504313i 0.967686 + 0.252157i \(0.0811398\pi\)
−0.967686 + 0.252157i \(0.918860\pi\)
\(54\) 0 0
\(55\) −29.1219 −0.529490
\(56\) 28.8120i 0.514499i
\(57\) 0 0
\(58\) 60.0750 1.03578
\(59\) −94.0845 −1.59465 −0.797326 0.603549i \(-0.793753\pi\)
−0.797326 + 0.603549i \(0.793753\pi\)
\(60\) 0 0
\(61\) 74.5293i 1.22179i 0.791711 + 0.610896i \(0.209191\pi\)
−0.791711 + 0.610896i \(0.790809\pi\)
\(62\) −26.5423 −0.428102
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 51.9112i 0.798634i
\(66\) 0 0
\(67\) 80.0906i 1.19538i −0.801727 0.597691i \(-0.796085\pi\)
0.801727 0.597691i \(-0.203915\pi\)
\(68\) 56.4456i 0.830082i
\(69\) 0 0
\(70\) −32.2127 −0.460182
\(71\) 83.5303 1.17648 0.588241 0.808686i \(-0.299821\pi\)
0.588241 + 0.808686i \(0.299821\pi\)
\(72\) 0 0
\(73\) −8.98897 −0.123137 −0.0615683 0.998103i \(-0.519610\pi\)
−0.0615683 + 0.998103i \(0.519610\pi\)
\(74\) 1.61354i 0.0218046i
\(75\) 0 0
\(76\) 23.3330i 0.307013i
\(77\) 132.667 1.72295
\(78\) 0 0
\(79\) 80.2841i 1.01626i 0.861282 + 0.508128i \(0.169662\pi\)
−0.861282 + 0.508128i \(0.830338\pi\)
\(80\) 8.94427i 0.111803i
\(81\) 0 0
\(82\) −102.983 −1.25589
\(83\) 94.6451i 1.14030i −0.821540 0.570151i \(-0.806885\pi\)
0.821540 0.570151i \(-0.193115\pi\)
\(84\) 0 0
\(85\) 63.1081 0.742448
\(86\) 7.02261i 0.0816582i
\(87\) 0 0
\(88\) 36.8367i 0.418598i
\(89\) 136.812i 1.53722i −0.639720 0.768608i \(-0.720950\pi\)
0.639720 0.768608i \(-0.279050\pi\)
\(90\) 0 0
\(91\) 236.485i 2.59874i
\(92\) 34.7997 30.0829i 0.378258 0.326988i
\(93\) 0 0
\(94\) −1.15026 −0.0122369
\(95\) −26.0871 −0.274601
\(96\) 0 0
\(97\) 2.32666i 0.0239862i 0.999928 + 0.0119931i \(0.00381761\pi\)
−0.999928 + 0.0119931i \(0.996182\pi\)
\(98\) 77.4509 0.790316
\(99\) 0 0
\(100\) −10.0000 −0.100000
\(101\) −12.9918 −0.128632 −0.0643161 0.997930i \(-0.520487\pi\)
−0.0643161 + 0.997930i \(0.520487\pi\)
\(102\) 0 0
\(103\) 90.3165i 0.876859i −0.898766 0.438430i \(-0.855535\pi\)
0.898766 0.438430i \(-0.144465\pi\)
\(104\) −65.6631 −0.631376
\(105\) 0 0
\(106\) 37.7999i 0.356603i
\(107\) 185.173i 1.73059i 0.501264 + 0.865294i \(0.332869\pi\)
−0.501264 + 0.865294i \(0.667131\pi\)
\(108\) 0 0
\(109\) 13.8248i 0.126833i −0.997987 0.0634167i \(-0.979800\pi\)
0.997987 0.0634167i \(-0.0201997\pi\)
\(110\) 41.1846 0.374406
\(111\) 0 0
\(112\) 40.7463i 0.363806i
\(113\) 178.530i 1.57991i −0.613164 0.789956i \(-0.710103\pi\)
0.613164 0.789956i \(-0.289897\pi\)
\(114\) 0 0
\(115\) 33.6337 + 38.9073i 0.292467 + 0.338324i
\(116\) −84.9589 −0.732404
\(117\) 0 0
\(118\) 133.056 1.12759
\(119\) −287.493 −2.41591
\(120\) 0 0
\(121\) −48.6174 −0.401797
\(122\) 105.400i 0.863938i
\(123\) 0 0
\(124\) 37.5365 0.302714
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −112.959 −0.889442 −0.444721 0.895669i \(-0.646697\pi\)
−0.444721 + 0.895669i \(0.646697\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 73.4135i 0.564720i
\(131\) 12.1419 0.0926860 0.0463430 0.998926i \(-0.485243\pi\)
0.0463430 + 0.998926i \(0.485243\pi\)
\(132\) 0 0
\(133\) 118.842 0.893546
\(134\) 113.265i 0.845263i
\(135\) 0 0
\(136\) 79.8261i 0.586957i
\(137\) 170.543i 1.24484i 0.782685 + 0.622418i \(0.213850\pi\)
−0.782685 + 0.622418i \(0.786150\pi\)
\(138\) 0 0
\(139\) 38.5478 0.277322 0.138661 0.990340i \(-0.455720\pi\)
0.138661 + 0.990340i \(0.455720\pi\)
\(140\) 45.5557 0.325398
\(141\) 0 0
\(142\) −118.130 −0.831899
\(143\) 302.351i 2.11434i
\(144\) 0 0
\(145\) 94.9869i 0.655082i
\(146\) 12.7123 0.0870707
\(147\) 0 0
\(148\) 2.28189i 0.0154182i
\(149\) 81.4125i 0.546393i −0.961958 0.273196i \(-0.911919\pi\)
0.961958 0.273196i \(-0.0880808\pi\)
\(150\) 0 0
\(151\) 159.680 1.05748 0.528742 0.848782i \(-0.322664\pi\)
0.528742 + 0.848782i \(0.322664\pi\)
\(152\) 32.9979i 0.217091i
\(153\) 0 0
\(154\) −187.619 −1.21831
\(155\) 41.9671i 0.270755i
\(156\) 0 0
\(157\) 38.5953i 0.245830i −0.992417 0.122915i \(-0.960776\pi\)
0.992417 0.122915i \(-0.0392242\pi\)
\(158\) 113.539i 0.718601i
\(159\) 0 0
\(160\) 12.6491i 0.0790569i
\(161\) −153.221 177.245i −0.951681 1.10090i
\(162\) 0 0
\(163\) 150.291 0.922028 0.461014 0.887393i \(-0.347486\pi\)
0.461014 + 0.887393i \(0.347486\pi\)
\(164\) 145.640 0.888046
\(165\) 0 0
\(166\) 133.848i 0.806316i
\(167\) −27.7604 −0.166230 −0.0831149 0.996540i \(-0.526487\pi\)
−0.0831149 + 0.996540i \(0.526487\pi\)
\(168\) 0 0
\(169\) 369.955 2.18908
\(170\) −89.2483 −0.524990
\(171\) 0 0
\(172\) 9.93146i 0.0577411i
\(173\) 24.0064 0.138765 0.0693826 0.997590i \(-0.477897\pi\)
0.0693826 + 0.997590i \(0.477897\pi\)
\(174\) 0 0
\(175\) 50.9328i 0.291045i
\(176\) 52.0949i 0.295994i
\(177\) 0 0
\(178\) 193.482i 1.08698i
\(179\) 201.178 1.12390 0.561949 0.827172i \(-0.310052\pi\)
0.561949 + 0.827172i \(0.310052\pi\)
\(180\) 0 0
\(181\) 202.358i 1.11800i −0.829168 0.558999i \(-0.811186\pi\)
0.829168 0.558999i \(-0.188814\pi\)
\(182\) 334.441i 1.83759i
\(183\) 0 0
\(184\) −49.2142 + 42.5436i −0.267469 + 0.231215i
\(185\) −2.55123 −0.0137904
\(186\) 0 0
\(187\) 367.566 1.96559
\(188\) 1.62672 0.00865277
\(189\) 0 0
\(190\) 36.8927 0.194172
\(191\) 111.302i 0.582735i −0.956611 0.291368i \(-0.905890\pi\)
0.956611 0.291368i \(-0.0941103\pi\)
\(192\) 0 0
\(193\) −164.081 −0.850163 −0.425081 0.905155i \(-0.639754\pi\)
−0.425081 + 0.905155i \(0.639754\pi\)
\(194\) 3.29040i 0.0169608i
\(195\) 0 0
\(196\) −109.532 −0.558838
\(197\) 329.562 1.67290 0.836451 0.548042i \(-0.184627\pi\)
0.836451 + 0.548042i \(0.184627\pi\)
\(198\) 0 0
\(199\) 337.886i 1.69792i −0.528455 0.848961i \(-0.677229\pi\)
0.528455 0.848961i \(-0.322771\pi\)
\(200\) 14.1421 0.0707107
\(201\) 0 0
\(202\) 18.3732 0.0909566
\(203\) 432.720i 2.13162i
\(204\) 0 0
\(205\) 162.830i 0.794292i
\(206\) 127.727i 0.620033i
\(207\) 0 0
\(208\) 92.8616 0.446450
\(209\) −151.941 −0.726992
\(210\) 0 0
\(211\) 111.656 0.529173 0.264587 0.964362i \(-0.414765\pi\)
0.264587 + 0.964362i \(0.414765\pi\)
\(212\) 53.4572i 0.252157i
\(213\) 0 0
\(214\) 261.874i 1.22371i
\(215\) −11.1037 −0.0516452
\(216\) 0 0
\(217\) 191.184i 0.881032i
\(218\) 19.5513i 0.0896847i
\(219\) 0 0
\(220\) −58.2439 −0.264745
\(221\) 655.204i 2.96472i
\(222\) 0 0
\(223\) −363.545 −1.63025 −0.815123 0.579288i \(-0.803331\pi\)
−0.815123 + 0.579288i \(0.803331\pi\)
\(224\) 57.6239i 0.257250i
\(225\) 0 0
\(226\) 252.480i 1.11717i
\(227\) 11.3368i 0.0499418i −0.999688 0.0249709i \(-0.992051\pi\)
0.999688 0.0249709i \(-0.00794931\pi\)
\(228\) 0 0
\(229\) 112.509i 0.491306i −0.969358 0.245653i \(-0.920998\pi\)
0.969358 0.245653i \(-0.0790024\pi\)
\(230\) −47.5652 55.0232i −0.206805 0.239231i
\(231\) 0 0
\(232\) 120.150 0.517888
\(233\) 381.634 1.63791 0.818957 0.573855i \(-0.194553\pi\)
0.818957 + 0.573855i \(0.194553\pi\)
\(234\) 0 0
\(235\) 1.81873i 0.00773927i
\(236\) −188.169 −0.797326
\(237\) 0 0
\(238\) 406.577 1.70831
\(239\) −139.296 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(240\) 0 0
\(241\) 82.3347i 0.341638i −0.985302 0.170819i \(-0.945359\pi\)
0.985302 0.170819i \(-0.0546413\pi\)
\(242\) 68.7554 0.284113
\(243\) 0 0
\(244\) 149.059i 0.610896i
\(245\) 122.461i 0.499840i
\(246\) 0 0
\(247\) 270.843i 1.09653i
\(248\) −53.0846 −0.214051
\(249\) 0 0
\(250\) 15.8114i 0.0632456i
\(251\) 323.924i 1.29054i 0.763957 + 0.645268i \(0.223254\pi\)
−0.763957 + 0.645268i \(0.776746\pi\)
\(252\) 0 0
\(253\) 195.896 + 226.611i 0.774291 + 0.895695i
\(254\) 159.748 0.628931
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 250.796 0.975858 0.487929 0.872883i \(-0.337752\pi\)
0.487929 + 0.872883i \(0.337752\pi\)
\(258\) 0 0
\(259\) 11.6223 0.0448737
\(260\) 103.822i 0.399317i
\(261\) 0 0
\(262\) −17.1712 −0.0655389
\(263\) 276.878i 1.05277i −0.850247 0.526384i \(-0.823548\pi\)
0.850247 0.526384i \(-0.176452\pi\)
\(264\) 0 0
\(265\) −59.7670 −0.225536
\(266\) −168.067 −0.631833
\(267\) 0 0
\(268\) 160.181i 0.597691i
\(269\) 78.1002 0.290335 0.145168 0.989407i \(-0.453628\pi\)
0.145168 + 0.989407i \(0.453628\pi\)
\(270\) 0 0
\(271\) 331.551 1.22343 0.611717 0.791076i \(-0.290479\pi\)
0.611717 + 0.791076i \(0.290479\pi\)
\(272\) 112.891i 0.415041i
\(273\) 0 0
\(274\) 241.184i 0.880232i
\(275\) 65.1186i 0.236795i
\(276\) 0 0
\(277\) −355.974 −1.28511 −0.642553 0.766241i \(-0.722125\pi\)
−0.642553 + 0.766241i \(0.722125\pi\)
\(278\) −54.5148 −0.196097
\(279\) 0 0
\(280\) −64.4255 −0.230091
\(281\) 391.437i 1.39301i −0.717550 0.696507i \(-0.754736\pi\)
0.717550 0.696507i \(-0.245264\pi\)
\(282\) 0 0
\(283\) 133.616i 0.472143i 0.971736 + 0.236071i \(0.0758599\pi\)
−0.971736 + 0.236071i \(0.924140\pi\)
\(284\) 167.061 0.588241
\(285\) 0 0
\(286\) 427.589i 1.49507i
\(287\) 741.783i 2.58461i
\(288\) 0 0
\(289\) −507.527 −1.75615
\(290\) 134.332i 0.463213i
\(291\) 0 0
\(292\) −17.9779 −0.0615683
\(293\) 533.454i 1.82066i −0.413880 0.910331i \(-0.635827\pi\)
0.413880 0.910331i \(-0.364173\pi\)
\(294\) 0 0
\(295\) 210.379i 0.713150i
\(296\) 3.22708i 0.0109023i
\(297\) 0 0
\(298\) 115.135i 0.386358i
\(299\) 403.945 349.193i 1.35099 1.16787i
\(300\) 0 0
\(301\) 50.5837 0.168052
\(302\) −225.822 −0.747755
\(303\) 0 0
\(304\) 46.6660i 0.153507i
\(305\) −166.653 −0.546402
\(306\) 0 0
\(307\) 282.656 0.920705 0.460352 0.887736i \(-0.347723\pi\)
0.460352 + 0.887736i \(0.347723\pi\)
\(308\) 265.334 0.861474
\(309\) 0 0
\(310\) 59.3504i 0.191453i
\(311\) 193.206 0.621241 0.310620 0.950534i \(-0.399463\pi\)
0.310620 + 0.950534i \(0.399463\pi\)
\(312\) 0 0
\(313\) 399.279i 1.27565i −0.770181 0.637826i \(-0.779834\pi\)
0.770181 0.637826i \(-0.220166\pi\)
\(314\) 54.5820i 0.173828i
\(315\) 0 0
\(316\) 160.568i 0.508128i
\(317\) 36.5297 0.115236 0.0576178 0.998339i \(-0.481650\pi\)
0.0576178 + 0.998339i \(0.481650\pi\)
\(318\) 0 0
\(319\) 553.241i 1.73430i
\(320\) 17.8885i 0.0559017i
\(321\) 0 0
\(322\) 216.687 + 250.662i 0.672940 + 0.778453i
\(323\) 329.262 1.01939
\(324\) 0 0
\(325\) −116.077 −0.357160
\(326\) −212.543 −0.651973
\(327\) 0 0
\(328\) −205.965 −0.627943
\(329\) 8.28534i 0.0251834i
\(330\) 0 0
\(331\) 85.9406 0.259639 0.129820 0.991538i \(-0.458560\pi\)
0.129820 + 0.991538i \(0.458560\pi\)
\(332\) 189.290i 0.570151i
\(333\) 0 0
\(334\) 39.2591 0.117542
\(335\) 179.088 0.534591
\(336\) 0 0
\(337\) 548.713i 1.62823i −0.580706 0.814114i \(-0.697223\pi\)
0.580706 0.814114i \(-0.302777\pi\)
\(338\) −523.195 −1.54791
\(339\) 0 0
\(340\) 126.216 0.371224
\(341\) 244.433i 0.716811i
\(342\) 0 0
\(343\) 58.7366i 0.171244i
\(344\) 14.0452i 0.0408291i
\(345\) 0 0
\(346\) −33.9502 −0.0981219
\(347\) −62.0036 −0.178685 −0.0893424 0.996001i \(-0.528477\pi\)
−0.0893424 + 0.996001i \(0.528477\pi\)
\(348\) 0 0
\(349\) −131.393 −0.376485 −0.188242 0.982123i \(-0.560279\pi\)
−0.188242 + 0.982123i \(0.560279\pi\)
\(350\) 72.0299i 0.205800i
\(351\) 0 0
\(352\) 73.6733i 0.209299i
\(353\) 176.990 0.501389 0.250694 0.968066i \(-0.419341\pi\)
0.250694 + 0.968066i \(0.419341\pi\)
\(354\) 0 0
\(355\) 186.779i 0.526139i
\(356\) 273.624i 0.768608i
\(357\) 0 0
\(358\) −284.508 −0.794716
\(359\) 187.851i 0.523260i −0.965168 0.261630i \(-0.915740\pi\)
0.965168 0.261630i \(-0.0842601\pi\)
\(360\) 0 0
\(361\) 224.893 0.622971
\(362\) 286.177i 0.790544i
\(363\) 0 0
\(364\) 472.970i 1.29937i
\(365\) 20.0999i 0.0550684i
\(366\) 0 0
\(367\) 128.510i 0.350162i 0.984554 + 0.175081i \(0.0560188\pi\)
−0.984554 + 0.175081i \(0.943981\pi\)
\(368\) 69.5994 60.1658i 0.189129 0.163494i
\(369\) 0 0
\(370\) 3.60798 0.00975130
\(371\) 272.273 0.733888
\(372\) 0 0
\(373\) 79.2928i 0.212581i 0.994335 + 0.106291i \(0.0338974\pi\)
−0.994335 + 0.106291i \(0.966103\pi\)
\(374\) −519.817 −1.38988
\(375\) 0 0
\(376\) −2.30053 −0.00611843
\(377\) −986.177 −2.61585
\(378\) 0 0
\(379\) 402.569i 1.06219i 0.847313 + 0.531094i \(0.178219\pi\)
−0.847313 + 0.531094i \(0.821781\pi\)
\(380\) −52.1742 −0.137301
\(381\) 0 0
\(382\) 157.405i 0.412056i
\(383\) 370.198i 0.966575i 0.875462 + 0.483288i \(0.160557\pi\)
−0.875462 + 0.483288i \(0.839443\pi\)
\(384\) 0 0
\(385\) 296.652i 0.770526i
\(386\) 232.046 0.601156
\(387\) 0 0
\(388\) 4.65332i 0.0119931i
\(389\) 12.0598i 0.0310021i −0.999880 0.0155011i \(-0.995066\pi\)
0.999880 0.0155011i \(-0.00493434\pi\)
\(390\) 0 0
\(391\) −424.512 491.073i −1.08571 1.25594i
\(392\) 154.902 0.395158
\(393\) 0 0
\(394\) −466.071 −1.18292
\(395\) −179.521 −0.454483
\(396\) 0 0
\(397\) 233.543 0.588270 0.294135 0.955764i \(-0.404968\pi\)
0.294135 + 0.955764i \(0.404968\pi\)
\(398\) 477.844i 1.20061i
\(399\) 0 0
\(400\) −20.0000 −0.0500000
\(401\) 114.072i 0.284469i −0.989833 0.142235i \(-0.954571\pi\)
0.989833 0.142235i \(-0.0454287\pi\)
\(402\) 0 0
\(403\) 435.713 1.08117
\(404\) −25.9837 −0.0643161
\(405\) 0 0
\(406\) 611.958i 1.50729i
\(407\) −14.8593 −0.0365094
\(408\) 0 0
\(409\) 125.952 0.307950 0.153975 0.988075i \(-0.450792\pi\)
0.153975 + 0.988075i \(0.450792\pi\)
\(410\) 230.276i 0.561650i
\(411\) 0 0
\(412\) 180.633i 0.438430i
\(413\) 958.397i 2.32057i
\(414\) 0 0
\(415\) 211.633 0.509959
\(416\) −131.326 −0.315688
\(417\) 0 0
\(418\) 214.878 0.514061
\(419\) 135.868i 0.324268i 0.986769 + 0.162134i \(0.0518377\pi\)
−0.986769 + 0.162134i \(0.948162\pi\)
\(420\) 0 0
\(421\) 194.111i 0.461070i −0.973064 0.230535i \(-0.925952\pi\)
0.973064 0.230535i \(-0.0740477\pi\)
\(422\) −157.905 −0.374182
\(423\) 0 0
\(424\) 75.5999i 0.178302i
\(425\) 141.114i 0.332033i
\(426\) 0 0
\(427\) 759.198 1.77798
\(428\) 370.346i 0.865294i
\(429\) 0 0
\(430\) 15.7030 0.0365187
\(431\) 421.699i 0.978420i 0.872166 + 0.489210i \(0.162715\pi\)
−0.872166 + 0.489210i \(0.837285\pi\)
\(432\) 0 0
\(433\) 3.78505i 0.00874144i −0.999990 0.00437072i \(-0.998609\pi\)
0.999990 0.00437072i \(-0.00139125\pi\)
\(434\) 270.375i 0.622984i
\(435\) 0 0
\(436\) 27.6497i 0.0634167i
\(437\) 175.481 + 202.996i 0.401559 + 0.464521i
\(438\) 0 0
\(439\) −6.17744 −0.0140716 −0.00703581 0.999975i \(-0.502240\pi\)
−0.00703581 + 0.999975i \(0.502240\pi\)
\(440\) 82.3693 0.187203
\(441\) 0 0
\(442\) 926.598i 2.09638i
\(443\) 406.821 0.918331 0.459166 0.888351i \(-0.348148\pi\)
0.459166 + 0.888351i \(0.348148\pi\)
\(444\) 0 0
\(445\) 305.921 0.687464
\(446\) 514.130 1.15276
\(447\) 0 0
\(448\) 81.4925i 0.181903i
\(449\) 289.981 0.645837 0.322918 0.946427i \(-0.395336\pi\)
0.322918 + 0.946427i \(0.395336\pi\)
\(450\) 0 0
\(451\) 948.385i 2.10285i
\(452\) 357.060i 0.789956i
\(453\) 0 0
\(454\) 16.0326i 0.0353142i
\(455\) 528.797 1.16219
\(456\) 0 0
\(457\) 224.430i 0.491094i −0.969385 0.245547i \(-0.921032\pi\)
0.969385 0.245547i \(-0.0789676\pi\)
\(458\) 159.112i 0.347406i
\(459\) 0 0
\(460\) 67.2674 + 77.8145i 0.146233 + 0.169162i
\(461\) −534.660 −1.15978 −0.579891 0.814694i \(-0.696905\pi\)
−0.579891 + 0.814694i \(0.696905\pi\)
\(462\) 0 0
\(463\) 166.226 0.359019 0.179509 0.983756i \(-0.442549\pi\)
0.179509 + 0.983756i \(0.442549\pi\)
\(464\) −169.918 −0.366202
\(465\) 0 0
\(466\) −539.712 −1.15818
\(467\) 113.755i 0.243587i −0.992555 0.121794i \(-0.961135\pi\)
0.992555 0.121794i \(-0.0388646\pi\)
\(468\) 0 0
\(469\) −815.848 −1.73955
\(470\) 2.57207i 0.00547249i
\(471\) 0 0
\(472\) 266.111 0.563795
\(473\) −64.6723 −0.136728
\(474\) 0 0
\(475\) 58.3326i 0.122805i
\(476\) −574.987 −1.20796
\(477\) 0 0
\(478\) 196.994 0.412121
\(479\) 156.314i 0.326334i 0.986598 + 0.163167i \(0.0521710\pi\)
−0.986598 + 0.163167i \(0.947829\pi\)
\(480\) 0 0
\(481\) 26.4875i 0.0550675i
\(482\) 116.439i 0.241574i
\(483\) 0 0
\(484\) −97.2348 −0.200898
\(485\) −5.20257 −0.0107270
\(486\) 0 0
\(487\) −207.919 −0.426938 −0.213469 0.976950i \(-0.568476\pi\)
−0.213469 + 0.976950i \(0.568476\pi\)
\(488\) 210.801i 0.431969i
\(489\) 0 0
\(490\) 173.186i 0.353440i
\(491\) 463.345 0.943676 0.471838 0.881685i \(-0.343591\pi\)
0.471838 + 0.881685i \(0.343591\pi\)
\(492\) 0 0
\(493\) 1198.89i 2.43182i
\(494\) 383.029i 0.775363i
\(495\) 0 0
\(496\) 75.0730 0.151357
\(497\) 850.886i 1.71204i
\(498\) 0 0
\(499\) −50.7777 −0.101759 −0.0508794 0.998705i \(-0.516202\pi\)
−0.0508794 + 0.998705i \(0.516202\pi\)
\(500\) 22.3607i 0.0447214i
\(501\) 0 0
\(502\) 458.098i 0.912546i
\(503\) 91.1857i 0.181284i 0.995884 + 0.0906419i \(0.0288919\pi\)
−0.995884 + 0.0906419i \(0.971108\pi\)
\(504\) 0 0
\(505\) 29.0506i 0.0575260i
\(506\) −277.038 320.476i −0.547506 0.633352i
\(507\) 0 0
\(508\) −225.918 −0.444721
\(509\) −317.681 −0.624128 −0.312064 0.950061i \(-0.601020\pi\)
−0.312064 + 0.950061i \(0.601020\pi\)
\(510\) 0 0
\(511\) 91.5667i 0.179191i
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) −354.678 −0.690036
\(515\) 201.954 0.392143
\(516\) 0 0
\(517\) 10.5930i 0.0204893i
\(518\) −16.4364 −0.0317305
\(519\) 0 0
\(520\) 146.827i 0.282360i
\(521\) 522.346i 1.00258i 0.865279 + 0.501291i \(0.167142\pi\)
−0.865279 + 0.501291i \(0.832858\pi\)
\(522\) 0 0
\(523\) 245.926i 0.470221i 0.971969 + 0.235111i \(0.0755452\pi\)
−0.971969 + 0.235111i \(0.924455\pi\)
\(524\) 24.2837 0.0463430
\(525\) 0 0
\(526\) 391.564i 0.744419i
\(527\) 529.693i 1.00511i
\(528\) 0 0
\(529\) 76.5101 523.438i 0.144632 0.989486i
\(530\) 84.5232 0.159478
\(531\) 0 0
\(532\) 237.683 0.446773
\(533\) 1690.54 3.17174
\(534\) 0 0
\(535\) −414.059 −0.773943
\(536\) 226.530i 0.422631i
\(537\) 0 0
\(538\) −110.450 −0.205298
\(539\) 713.258i 1.32330i
\(540\) 0 0
\(541\) −307.138 −0.567723 −0.283861 0.958865i \(-0.591616\pi\)
−0.283861 + 0.958865i \(0.591616\pi\)
\(542\) −468.884 −0.865099
\(543\) 0 0
\(544\) 159.652i 0.293478i
\(545\) 30.9133 0.0567216
\(546\) 0 0
\(547\) −712.547 −1.30265 −0.651323 0.758801i \(-0.725786\pi\)
−0.651323 + 0.758801i \(0.725786\pi\)
\(548\) 341.085i 0.622418i
\(549\) 0 0
\(550\) 92.0916i 0.167439i
\(551\) 495.587i 0.899432i
\(552\) 0 0
\(553\) 817.820 1.47888
\(554\) 503.424 0.908707
\(555\) 0 0
\(556\) 77.0956 0.138661
\(557\) 1.17109i 0.00210250i 0.999999 + 0.00105125i \(0.000334623\pi\)
−0.999999 + 0.00105125i \(0.999665\pi\)
\(558\) 0 0
\(559\) 115.281i 0.206228i
\(560\) 91.1114 0.162699
\(561\) 0 0
\(562\) 553.575i 0.985010i
\(563\) 900.058i 1.59868i 0.600878 + 0.799341i \(0.294818\pi\)
−0.600878 + 0.799341i \(0.705182\pi\)
\(564\) 0 0
\(565\) 399.205 0.706558
\(566\) 188.962i 0.333855i
\(567\) 0 0
\(568\) −236.259 −0.415949
\(569\) 589.836i 1.03662i −0.855193 0.518309i \(-0.826562\pi\)
0.855193 0.518309i \(-0.173438\pi\)
\(570\) 0 0
\(571\) 117.755i 0.206225i −0.994670 0.103113i \(-0.967120\pi\)
0.994670 0.103113i \(-0.0328802\pi\)
\(572\) 604.702i 1.05717i
\(573\) 0 0
\(574\) 1049.04i 1.82760i
\(575\) −86.9993 + 75.2072i −0.151303 + 0.130795i
\(576\) 0 0
\(577\) −403.533 −0.699365 −0.349682 0.936868i \(-0.613710\pi\)
−0.349682 + 0.936868i \(0.613710\pi\)
\(578\) 717.751 1.24178
\(579\) 0 0
\(580\) 189.974i 0.327541i
\(581\) −964.109 −1.65940
\(582\) 0 0
\(583\) −348.106 −0.597094
\(584\) 25.4246 0.0435354
\(585\) 0 0
\(586\) 754.418i 1.28740i
\(587\) −607.731 −1.03532 −0.517659 0.855587i \(-0.673196\pi\)
−0.517659 + 0.855587i \(0.673196\pi\)
\(588\) 0 0
\(589\) 218.960i 0.371749i
\(590\) 297.521i 0.504273i
\(591\) 0 0
\(592\) 4.56377i 0.00770908i
\(593\) −428.555 −0.722690 −0.361345 0.932432i \(-0.617682\pi\)
−0.361345 + 0.932432i \(0.617682\pi\)
\(594\) 0 0
\(595\) 642.855i 1.08043i
\(596\) 162.825i 0.273196i
\(597\) 0 0
\(598\) −571.264 + 493.834i −0.955291 + 0.825809i
\(599\) 162.171 0.270736 0.135368 0.990795i \(-0.456778\pi\)
0.135368 + 0.990795i \(0.456778\pi\)
\(600\) 0 0
\(601\) 957.407 1.59302 0.796512 0.604623i \(-0.206676\pi\)
0.796512 + 0.604623i \(0.206676\pi\)
\(602\) −71.5362 −0.118831
\(603\) 0 0
\(604\) 319.360 0.528742
\(605\) 108.712i 0.179689i
\(606\) 0 0
\(607\) −989.893 −1.63080 −0.815398 0.578901i \(-0.803482\pi\)
−0.815398 + 0.578901i \(0.803482\pi\)
\(608\) 65.9958i 0.108546i
\(609\) 0 0
\(610\) 235.682 0.386365
\(611\) 18.8825 0.0309042
\(612\) 0 0
\(613\) 134.181i 0.218892i 0.993993 + 0.109446i \(0.0349076\pi\)
−0.993993 + 0.109446i \(0.965092\pi\)
\(614\) −399.736 −0.651037
\(615\) 0 0
\(616\) −375.239 −0.609154
\(617\) 151.269i 0.245168i 0.992458 + 0.122584i \(0.0391181\pi\)
−0.992458 + 0.122584i \(0.960882\pi\)
\(618\) 0 0
\(619\) 115.852i 0.187159i 0.995612 + 0.0935797i \(0.0298310\pi\)
−0.995612 + 0.0935797i \(0.970169\pi\)
\(620\) 83.9342i 0.135378i
\(621\) 0 0
\(622\) −273.234 −0.439284
\(623\) −1393.65 −2.23699
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 564.665i 0.902022i
\(627\) 0 0
\(628\) 77.1906i 0.122915i
\(629\) 32.2006 0.0511934
\(630\) 0 0
\(631\) 730.187i 1.15719i 0.815615 + 0.578595i \(0.196399\pi\)
−0.815615 + 0.578595i \(0.803601\pi\)
\(632\) 227.078i 0.359300i
\(633\) 0 0
\(634\) −51.6608 −0.0814839
\(635\) 252.584i 0.397771i
\(636\) 0 0
\(637\) −1271.42 −1.99594
\(638\) 782.400i 1.22633i
\(639\) 0 0
\(640\) 25.2982i 0.0395285i
\(641\) 725.551i 1.13190i 0.824438 + 0.565952i \(0.191491\pi\)
−0.824438 + 0.565952i \(0.808509\pi\)
\(642\) 0 0
\(643\) 505.908i 0.786792i 0.919369 + 0.393396i \(0.128700\pi\)
−0.919369 + 0.393396i \(0.871300\pi\)
\(644\) −306.441 354.490i −0.475840 0.550450i
\(645\) 0 0
\(646\) −465.646 −0.720815
\(647\) 324.943 0.502230 0.251115 0.967957i \(-0.419203\pi\)
0.251115 + 0.967957i \(0.419203\pi\)
\(648\) 0 0
\(649\) 1225.33i 1.88803i
\(650\) 164.158 0.252550
\(651\) 0 0
\(652\) 300.581 0.461014
\(653\) 146.801 0.224810 0.112405 0.993662i \(-0.464145\pi\)
0.112405 + 0.993662i \(0.464145\pi\)
\(654\) 0 0
\(655\) 27.1500i 0.0414504i
\(656\) 291.279 0.444023
\(657\) 0 0
\(658\) 11.7172i 0.0178074i
\(659\) 366.667i 0.556399i −0.960523 0.278200i \(-0.910262\pi\)
0.960523 0.278200i \(-0.0897377\pi\)
\(660\) 0 0
\(661\) 1241.87i 1.87878i 0.342852 + 0.939389i \(0.388607\pi\)
−0.342852 + 0.939389i \(0.611393\pi\)
\(662\) −121.538 −0.183593
\(663\) 0 0
\(664\) 267.697i 0.403158i
\(665\) 265.738i 0.399606i
\(666\) 0 0
\(667\) −739.136 + 638.952i −1.10815 + 0.957949i
\(668\) −55.5208 −0.0831149
\(669\) 0 0
\(670\) −253.269 −0.378013
\(671\) −970.649 −1.44657
\(672\) 0 0
\(673\) 812.966 1.20797 0.603987 0.796994i \(-0.293578\pi\)
0.603987 + 0.796994i \(0.293578\pi\)
\(674\) 775.997i 1.15133i
\(675\) 0 0
\(676\) 739.910 1.09454
\(677\) 1053.03i 1.55544i 0.628610 + 0.777721i \(0.283624\pi\)
−0.628610 + 0.777721i \(0.716376\pi\)
\(678\) 0 0
\(679\) 23.7007 0.0349053
\(680\) −178.497 −0.262495
\(681\) 0 0
\(682\) 345.680i 0.506862i
\(683\) −947.718 −1.38758 −0.693791 0.720177i \(-0.744061\pi\)
−0.693791 + 0.720177i \(0.744061\pi\)
\(684\) 0 0
\(685\) −381.345 −0.556708
\(686\) 83.0661i 0.121088i
\(687\) 0 0
\(688\) 19.8629i 0.0288705i
\(689\) 620.515i 0.900602i
\(690\) 0 0
\(691\) −600.541 −0.869090 −0.434545 0.900650i \(-0.643091\pi\)
−0.434545 + 0.900650i \(0.643091\pi\)
\(692\) 48.0128 0.0693826
\(693\) 0 0
\(694\) 87.6863 0.126349
\(695\) 86.1955i 0.124022i
\(696\) 0 0
\(697\) 2055.18i 2.94861i
\(698\) 185.818 0.266215
\(699\) 0 0
\(700\) 101.866i 0.145522i
\(701\) 907.376i 1.29440i −0.762319 0.647201i \(-0.775939\pi\)
0.762319 0.647201i \(-0.224061\pi\)
\(702\) 0 0
\(703\) −13.3108 −0.0189343
\(704\) 104.190i 0.147997i
\(705\) 0 0
\(706\) −250.302 −0.354536
\(707\) 132.342i 0.187188i
\(708\) 0 0
\(709\) 657.300i 0.927081i 0.886076 + 0.463540i \(0.153421\pi\)
−0.886076 + 0.463540i \(0.846579\pi\)
\(710\) 264.146i 0.372036i
\(711\) 0 0
\(712\) 386.963i 0.543488i
\(713\) 326.565 282.302i 0.458015 0.395935i
\(714\) 0 0
\(715\) −676.077 −0.945563
\(716\) 402.356 0.561949
\(717\) 0 0
\(718\) 265.661i 0.370001i
\(719\) −1095.72 −1.52395 −0.761975 0.647606i \(-0.775770\pi\)
−0.761975 + 0.647606i \(0.775770\pi\)
\(720\) 0 0
\(721\) −920.015 −1.27603
\(722\) −318.046 −0.440507
\(723\) 0 0
\(724\) 404.715i 0.558999i
\(725\) 212.397 0.292962
\(726\) 0 0
\(727\) 814.942i 1.12097i 0.828166 + 0.560483i \(0.189384\pi\)
−0.828166 + 0.560483i \(0.810616\pi\)
\(728\) 668.881i 0.918793i
\(729\) 0 0
\(730\) 28.4256i 0.0389392i
\(731\) 140.147 0.191719
\(732\) 0 0
\(733\) 742.293i 1.01268i −0.862335 0.506339i \(-0.830999\pi\)
0.862335 0.506339i \(-0.169001\pi\)
\(734\) 181.740i 0.247602i
\(735\) 0 0
\(736\) −98.4285 + 85.0872i −0.133734 + 0.115608i
\(737\) 1043.08 1.41530
\(738\) 0 0
\(739\) −288.013 −0.389733 −0.194866 0.980830i \(-0.562427\pi\)
−0.194866 + 0.980830i \(0.562427\pi\)
\(740\) −5.10245 −0.00689521
\(741\) 0 0
\(742\) −385.052 −0.518937
\(743\) 199.651i 0.268709i −0.990933 0.134354i \(-0.957104\pi\)
0.990933 0.134354i \(-0.0428961\pi\)
\(744\) 0 0
\(745\) 182.044 0.244354
\(746\) 112.137i 0.150318i
\(747\) 0 0
\(748\) 735.132 0.982797
\(749\) 1886.28 2.51839
\(750\) 0 0
\(751\) 610.667i 0.813138i 0.913620 + 0.406569i \(0.133275\pi\)
−0.913620 + 0.406569i \(0.866725\pi\)
\(752\) 3.25344 0.00432638
\(753\) 0 0
\(754\) 1394.67 1.84969
\(755\) 357.056i 0.472922i
\(756\) 0 0
\(757\) 241.235i 0.318672i 0.987224 + 0.159336i \(0.0509354\pi\)
−0.987224 + 0.159336i \(0.949065\pi\)
\(758\) 569.319i 0.751080i
\(759\) 0 0
\(760\) 73.7855 0.0970862
\(761\) −568.126 −0.746552 −0.373276 0.927720i \(-0.621765\pi\)
−0.373276 + 0.927720i \(0.621765\pi\)
\(762\) 0 0
\(763\) −140.828 −0.184571
\(764\) 222.605i 0.291368i
\(765\) 0 0
\(766\) 523.539i 0.683472i
\(767\) −2184.21 −2.84773
\(768\) 0 0
\(769\) 425.966i 0.553922i −0.960881 0.276961i \(-0.910673\pi\)
0.960881 0.276961i \(-0.0893273\pi\)
\(770\) 419.530i 0.544844i
\(771\) 0 0
\(772\) −328.163 −0.425081
\(773\) 849.417i 1.09886i −0.835540 0.549429i \(-0.814845\pi\)
0.835540 0.549429i \(-0.185155\pi\)
\(774\) 0 0
\(775\) −93.8413 −0.121086
\(776\) 6.58079i 0.00848040i
\(777\) 0 0
\(778\) 17.0552i 0.0219218i
\(779\) 849.553i 1.09057i
\(780\) 0 0
\(781\) 1087.87i 1.39293i
\(782\) 600.350 + 694.482i 0.767711 + 0.888084i
\(783\) 0 0
\(784\) −219.064 −0.279419
\(785\) 86.3017 0.109938
\(786\) 0 0
\(787\) 1448.81i 1.84093i −0.390828 0.920464i \(-0.627811\pi\)
0.390828 0.920464i \(-0.372189\pi\)
\(788\) 659.123 0.836451
\(789\) 0 0
\(790\) 253.881 0.321368
\(791\) −1818.61 −2.29912
\(792\) 0 0
\(793\) 1730.23i 2.18188i
\(794\) −330.280 −0.415970
\(795\) 0 0
\(796\) 675.773i 0.848961i
\(797\) 405.786i 0.509141i −0.967054 0.254571i \(-0.918066\pi\)
0.967054 0.254571i \(-0.0819342\pi\)
\(798\) 0 0
\(799\) 22.9553i 0.0287300i
\(800\) 28.2843 0.0353553
\(801\) 0 0
\(802\) 161.322i 0.201150i
\(803\) 117.070i 0.145791i
\(804\) 0 0
\(805\) 396.331 342.612i 0.492337 0.425605i
\(806\) −616.191 −0.764504
\(807\) 0 0
\(808\) 36.7465 0.0454783
\(809\) −755.183 −0.933477 −0.466739 0.884395i \(-0.654571\pi\)
−0.466739 + 0.884395i \(0.654571\pi\)
\(810\) 0 0
\(811\) −1132.66 −1.39662 −0.698311 0.715795i \(-0.746065\pi\)
−0.698311 + 0.715795i \(0.746065\pi\)
\(812\) 865.439i 1.06581i
\(813\) 0 0
\(814\) 21.0143 0.0258161
\(815\) 336.060i 0.412344i
\(816\) 0 0
\(817\) −57.9328 −0.0709091
\(818\) −178.122 −0.217754
\(819\) 0 0
\(820\) 325.660i 0.397146i
\(821\) −167.684 −0.204244 −0.102122 0.994772i \(-0.532563\pi\)
−0.102122 + 0.994772i \(0.532563\pi\)
\(822\) 0 0
\(823\) −403.176 −0.489886 −0.244943 0.969538i \(-0.578769\pi\)
−0.244943 + 0.969538i \(0.578769\pi\)
\(824\) 255.454i 0.310016i
\(825\) 0 0
\(826\) 1355.38i 1.64089i
\(827\) 901.234i 1.08976i 0.838513 + 0.544881i \(0.183425\pi\)
−0.838513 + 0.544881i \(0.816575\pi\)
\(828\) 0 0
\(829\) −455.150 −0.549034 −0.274517 0.961582i \(-0.588518\pi\)
−0.274517 + 0.961582i \(0.588518\pi\)
\(830\) −299.294 −0.360595
\(831\) 0 0
\(832\) 185.723 0.223225
\(833\) 1545.65i 1.85553i
\(834\) 0 0
\(835\) 62.0741i 0.0743402i
\(836\) −303.883 −0.363496
\(837\) 0 0
\(838\) 192.147i 0.229292i
\(839\) 150.136i 0.178946i −0.995989 0.0894730i \(-0.971482\pi\)
0.995989 0.0894730i \(-0.0285183\pi\)
\(840\) 0 0
\(841\) 963.503 1.14566
\(842\) 274.514i 0.326026i
\(843\) 0 0
\(844\) 223.311 0.264587
\(845\) 827.244i 0.978987i
\(846\) 0 0
\(847\) 495.244i 0.584704i
\(848\) 106.914i 0.126078i
\(849\) 0 0
\(850\) 199.565i 0.234783i
\(851\) 17.1614 + 19.8523i 0.0201662 + 0.0233281i
\(852\) 0 0
\(853\) 582.826 0.683266 0.341633 0.939833i \(-0.389020\pi\)
0.341633 + 0.939833i \(0.389020\pi\)
\(854\) −1073.67 −1.25722
\(855\) 0 0
\(856\) 523.748i 0.611856i
\(857\) 435.647 0.508339 0.254170 0.967160i \(-0.418198\pi\)
0.254170 + 0.967160i \(0.418198\pi\)
\(858\) 0 0
\(859\) −427.848 −0.498076 −0.249038 0.968494i \(-0.580114\pi\)
−0.249038 + 0.968494i \(0.580114\pi\)
\(860\) −22.2074 −0.0258226
\(861\) 0 0
\(862\) 596.373i 0.691848i
\(863\) 698.719 0.809640 0.404820 0.914396i \(-0.367334\pi\)
0.404820 + 0.914396i \(0.367334\pi\)
\(864\) 0 0
\(865\) 53.6799i 0.0620577i
\(866\) 5.35286i 0.00618113i
\(867\) 0 0
\(868\) 382.368i 0.440516i
\(869\) −1045.60 −1.20322
\(870\) 0 0
\(871\) 1859.34i 2.13471i
\(872\) 39.1025i 0.0448424i
\(873\) 0 0
\(874\) −248.168 287.079i −0.283945 0.328466i
\(875\) −113.889 −0.130159
\(876\) 0 0
\(877\) 779.272 0.888566 0.444283 0.895887i \(-0.353459\pi\)
0.444283 + 0.895887i \(0.353459\pi\)
\(878\) 8.73622 0.00995014
\(879\) 0 0
\(880\) −116.488 −0.132372
\(881\) 1612.85i 1.83070i −0.402659 0.915350i \(-0.631914\pi\)
0.402659 0.915350i \(-0.368086\pi\)
\(882\) 0 0
\(883\) −336.489 −0.381075 −0.190537 0.981680i \(-0.561023\pi\)
−0.190537 + 0.981680i \(0.561023\pi\)
\(884\) 1310.41i 1.48236i
\(885\) 0 0
\(886\) −575.331 −0.649358
\(887\) 12.5177 0.0141124 0.00705619 0.999975i \(-0.497754\pi\)
0.00705619 + 0.999975i \(0.497754\pi\)
\(888\) 0 0
\(889\) 1150.67i 1.29434i
\(890\) −432.638 −0.486110
\(891\) 0 0
\(892\) −727.090 −0.815123
\(893\) 9.48907i 0.0106261i
\(894\) 0 0
\(895\) 449.847i 0.502623i
\(896\) 115.248i 0.128625i
\(897\) 0 0
\(898\) −410.095 −0.456676
\(899\) −797.265 −0.886835
\(900\) 0 0
\(901\) 754.356 0.837243
\(902\) 1341.22i 1.48694i
\(903\) 0 0
\(904\) 504.959i 0.558583i
\(905\) 452.485 0.499984
\(906\) 0 0
\(907\) 1320.74i 1.45616i 0.685492 + 0.728080i \(0.259587\pi\)
−0.685492 + 0.728080i \(0.740413\pi\)
\(908\) 22.6736i 0.0249709i
\(909\) 0 0
\(910\) −747.832 −0.821793
\(911\) 1388.64i 1.52430i 0.647398 + 0.762152i \(0.275857\pi\)
−0.647398 + 0.762152i \(0.724143\pi\)
\(912\) 0 0
\(913\) 1232.63 1.35009
\(914\) 317.392i 0.347256i
\(915\) 0 0
\(916\) 225.018i 0.245653i
\(917\) 123.684i 0.134879i
\(918\) 0 0
\(919\) 814.108i 0.885863i −0.896555 0.442932i \(-0.853938\pi\)
0.896555 0.442932i \(-0.146062\pi\)
\(920\) −95.1304 110.046i −0.103403 0.119616i
\(921\) 0 0
\(922\) 756.123 0.820090
\(923\) 1939.19 2.10096
\(924\) 0 0
\(925\) 5.70472i 0.00616726i
\(926\) −235.078 −0.253864
\(927\) 0 0
\(928\) 240.300 0.258944
\(929\) −951.041 −1.02373 −0.511863 0.859067i \(-0.671044\pi\)
−0.511863 + 0.859067i \(0.671044\pi\)
\(930\) 0 0
\(931\) 638.929i 0.686283i
\(932\) 763.268 0.818957
\(933\) 0 0
\(934\) 160.874i 0.172242i
\(935\) 821.903i 0.879040i
\(936\) 0 0
\(937\) 25.4596i 0.0271713i −0.999908 0.0135857i \(-0.995675\pi\)
0.999908 0.0135857i \(-0.00432459\pi\)
\(938\) 1153.78 1.23005
\(939\) 0 0
\(940\) 3.63746i 0.00386963i
\(941\) 1304.69i 1.38650i −0.720698 0.693249i \(-0.756179\pi\)
0.720698 0.693249i \(-0.243821\pi\)
\(942\) 0 0
\(943\) 1267.05 1095.31i 1.34364 1.16152i
\(944\) −376.338 −0.398663
\(945\) 0 0
\(946\) 91.4605 0.0966813
\(947\) 25.1178 0.0265236 0.0132618 0.999912i \(-0.495779\pi\)
0.0132618 + 0.999912i \(0.495779\pi\)
\(948\) 0 0
\(949\) −208.683 −0.219897
\(950\) 82.4947i 0.0868365i
\(951\) 0 0
\(952\) 813.154 0.854153
\(953\) 663.560i 0.696286i 0.937442 + 0.348143i \(0.113188\pi\)
−0.937442 + 0.348143i \(0.886812\pi\)
\(954\) 0 0
\(955\) 248.880 0.260607
\(956\) −278.592 −0.291414
\(957\) 0 0
\(958\) 221.061i 0.230753i
\(959\) 1737.24 1.81152
\(960\) 0 0
\(961\) −608.753 −0.633457
\(962\) 37.4589i 0.0389386i
\(963\) 0 0
\(964\) 164.669i 0.170819i
\(965\) 366.897i 0.380204i
\(966\) 0 0
\(967\) −661.312 −0.683880 −0.341940 0.939722i \(-0.611084\pi\)
−0.341940 + 0.939722i \(0.611084\pi\)
\(968\) 137.511 0.142057
\(969\) 0 0
\(970\) 7.35755 0.00758510
\(971\) 1198.53i 1.23433i 0.786834 + 0.617165i \(0.211719\pi\)
−0.786834 + 0.617165i \(0.788281\pi\)
\(972\) 0 0
\(973\) 392.670i 0.403566i
\(974\) 294.042 0.301891
\(975\) 0 0
\(976\) 298.117i 0.305448i
\(977\) 1156.31i 1.18353i −0.806110 0.591765i \(-0.798431\pi\)
0.806110 0.591765i \(-0.201569\pi\)
\(978\) 0 0
\(979\) 1781.80 1.82002
\(980\) 244.921i 0.249920i
\(981\) 0 0
\(982\) −655.268 −0.667279
\(983\) 534.831i 0.544080i −0.962286 0.272040i \(-0.912302\pi\)
0.962286 0.272040i \(-0.0876983\pi\)
\(984\) 0 0
\(985\) 736.922i 0.748144i
\(986\) 1695.48i 1.71956i
\(987\) 0 0
\(988\) 541.685i 0.548265i
\(989\) 74.6918 + 86.4030i 0.0755225 + 0.0873640i
\(990\) 0 0
\(991\) −217.485 −0.219461 −0.109730 0.993961i \(-0.534999\pi\)
−0.109730 + 0.993961i \(0.534999\pi\)
\(992\) −106.169 −0.107025
\(993\) 0 0
\(994\) 1203.33i 1.21060i
\(995\) 755.537 0.759334
\(996\) 0 0
\(997\) −248.055 −0.248801 −0.124400 0.992232i \(-0.539701\pi\)
−0.124400 + 0.992232i \(0.539701\pi\)
\(998\) 71.8105 0.0719544
\(999\) 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 2070.3.c.a.91.5 16
3.2 odd 2 230.3.d.a.91.13 16
12.11 even 2 1840.3.k.d.321.7 16
15.2 even 4 1150.3.c.c.1149.32 32
15.8 even 4 1150.3.c.c.1149.1 32
15.14 odd 2 1150.3.d.b.551.4 16
23.22 odd 2 inner 2070.3.c.a.91.4 16
69.68 even 2 230.3.d.a.91.14 yes 16
276.275 odd 2 1840.3.k.d.321.8 16
345.68 odd 4 1150.3.c.c.1149.31 32
345.137 odd 4 1150.3.c.c.1149.2 32
345.344 even 2 1150.3.d.b.551.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.3.d.a.91.13 16 3.2 odd 2
230.3.d.a.91.14 yes 16 69.68 even 2
1150.3.c.c.1149.1 32 15.8 even 4
1150.3.c.c.1149.2 32 345.137 odd 4
1150.3.c.c.1149.31 32 345.68 odd 4
1150.3.c.c.1149.32 32 15.2 even 4
1150.3.d.b.551.3 16 345.344 even 2
1150.3.d.b.551.4 16 15.14 odd 2
1840.3.k.d.321.7 16 12.11 even 2
1840.3.k.d.321.8 16 276.275 odd 2
2070.3.c.a.91.4 16 23.22 odd 2 inner
2070.3.c.a.91.5 16 1.1 even 1 trivial