Properties

Label 810.3.b.c.809.15
Level $810$
Weight $3$
Character 810.809
Analytic conductor $22.071$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [810,3,Mod(809,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.b (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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 809.15
Character \(\chi\) \(=\) 810.809
Dual form 810.3.b.c.809.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +2.00000 q^{4} +(-4.21457 - 2.69024i) q^{5} +7.64672i q^{7} +2.82843 q^{8} +O(q^{10})\) \(q+1.41421 q^{2} +2.00000 q^{4} +(-4.21457 - 2.69024i) q^{5} +7.64672i q^{7} +2.82843 q^{8} +(-5.96030 - 3.80458i) q^{10} -11.3743i q^{11} +14.3829i q^{13} +10.8141i q^{14} +4.00000 q^{16} -17.3175 q^{17} -31.8326 q^{19} +(-8.42914 - 5.38048i) q^{20} -16.0857i q^{22} +16.5149 q^{23} +(10.5252 + 22.6764i) q^{25} +20.3405i q^{26} +15.2934i q^{28} +8.49209i q^{29} -45.8192 q^{31} +5.65685 q^{32} -24.4907 q^{34} +(20.5715 - 32.2276i) q^{35} -31.8348i q^{37} -45.0180 q^{38} +(-11.9206 - 7.60915i) q^{40} +68.3775i q^{41} +31.3825i q^{43} -22.7486i q^{44} +23.3555 q^{46} -46.0552 q^{47} -9.47230 q^{49} +(14.8849 + 32.0693i) q^{50} +28.7659i q^{52} -53.6946 q^{53} +(-30.5996 + 47.9378i) q^{55} +21.6282i q^{56} +12.0096i q^{58} +64.7702i q^{59} -96.2754 q^{61} -64.7981 q^{62} +8.00000 q^{64} +(38.6935 - 60.6179i) q^{65} -1.72446i q^{67} -34.6351 q^{68} +(29.0925 - 45.5768i) q^{70} +26.5730i q^{71} -51.4318i q^{73} -45.0212i q^{74} -63.6651 q^{76} +86.9762 q^{77} +59.2176 q^{79} +(-16.8583 - 10.7610i) q^{80} +96.7004i q^{82} +6.92825 q^{83} +(72.9859 + 46.5883i) q^{85} +44.3815i q^{86} -32.1714i q^{88} -115.787i q^{89} -109.982 q^{91} +33.0297 q^{92} -65.1318 q^{94} +(134.161 + 85.6373i) q^{95} +158.654i q^{97} -13.3959 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-1\) \(-1\)

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
\(5\) −4.21457 2.69024i −0.842914 0.538048i
\(6\) 0 0
\(7\) 7.64672i 1.09239i 0.837659 + 0.546194i \(0.183924\pi\)
−0.837659 + 0.546194i \(0.816076\pi\)
\(8\) 2.82843 0.353553
\(9\) 0 0
\(10\) −5.96030 3.80458i −0.596030 0.380458i
\(11\) 11.3743i 1.03403i −0.855977 0.517014i \(-0.827043\pi\)
0.855977 0.517014i \(-0.172957\pi\)
\(12\) 0 0
\(13\) 14.3829i 1.10638i 0.833055 + 0.553190i \(0.186589\pi\)
−0.833055 + 0.553190i \(0.813411\pi\)
\(14\) 10.8141i 0.772435i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) −17.3175 −1.01868 −0.509339 0.860566i \(-0.670110\pi\)
−0.509339 + 0.860566i \(0.670110\pi\)
\(18\) 0 0
\(19\) −31.8326 −1.67540 −0.837699 0.546132i \(-0.816100\pi\)
−0.837699 + 0.546132i \(0.816100\pi\)
\(20\) −8.42914 5.38048i −0.421457 0.269024i
\(21\) 0 0
\(22\) 16.0857i 0.731169i
\(23\) 16.5149 0.718037 0.359019 0.933330i \(-0.383111\pi\)
0.359019 + 0.933330i \(0.383111\pi\)
\(24\) 0 0
\(25\) 10.5252 + 22.6764i 0.421008 + 0.907057i
\(26\) 20.3405i 0.782328i
\(27\) 0 0
\(28\) 15.2934i 0.546194i
\(29\) 8.49209i 0.292831i 0.989223 + 0.146415i \(0.0467736\pi\)
−0.989223 + 0.146415i \(0.953226\pi\)
\(30\) 0 0
\(31\) −45.8192 −1.47804 −0.739019 0.673684i \(-0.764711\pi\)
−0.739019 + 0.673684i \(0.764711\pi\)
\(32\) 5.65685 0.176777
\(33\) 0 0
\(34\) −24.4907 −0.720314
\(35\) 20.5715 32.2276i 0.587758 0.920789i
\(36\) 0 0
\(37\) 31.8348i 0.860400i −0.902734 0.430200i \(-0.858443\pi\)
0.902734 0.430200i \(-0.141557\pi\)
\(38\) −45.0180 −1.18469
\(39\) 0 0
\(40\) −11.9206 7.60915i −0.298015 0.190229i
\(41\) 68.3775i 1.66774i 0.551958 + 0.833872i \(0.313881\pi\)
−0.551958 + 0.833872i \(0.686119\pi\)
\(42\) 0 0
\(43\) 31.3825i 0.729825i 0.931042 + 0.364913i \(0.118901\pi\)
−0.931042 + 0.364913i \(0.881099\pi\)
\(44\) 22.7486i 0.517014i
\(45\) 0 0
\(46\) 23.3555 0.507729
\(47\) −46.0552 −0.979897 −0.489948 0.871751i \(-0.662984\pi\)
−0.489948 + 0.871751i \(0.662984\pi\)
\(48\) 0 0
\(49\) −9.47230 −0.193312
\(50\) 14.8849 + 32.0693i 0.297698 + 0.641386i
\(51\) 0 0
\(52\) 28.7659i 0.553190i
\(53\) −53.6946 −1.01311 −0.506553 0.862209i \(-0.669080\pi\)
−0.506553 + 0.862209i \(0.669080\pi\)
\(54\) 0 0
\(55\) −30.5996 + 47.9378i −0.556357 + 0.871597i
\(56\) 21.6282i 0.386218i
\(57\) 0 0
\(58\) 12.0096i 0.207063i
\(59\) 64.7702i 1.09780i 0.835888 + 0.548900i \(0.184953\pi\)
−0.835888 + 0.548900i \(0.815047\pi\)
\(60\) 0 0
\(61\) −96.2754 −1.57828 −0.789142 0.614210i \(-0.789475\pi\)
−0.789142 + 0.614210i \(0.789475\pi\)
\(62\) −64.7981 −1.04513
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) 38.6935 60.6179i 0.595285 0.932583i
\(66\) 0 0
\(67\) 1.72446i 0.0257382i −0.999917 0.0128691i \(-0.995904\pi\)
0.999917 0.0128691i \(-0.00409648\pi\)
\(68\) −34.6351 −0.509339
\(69\) 0 0
\(70\) 29.0925 45.5768i 0.415607 0.651096i
\(71\) 26.5730i 0.374268i 0.982334 + 0.187134i \(0.0599199\pi\)
−0.982334 + 0.187134i \(0.940080\pi\)
\(72\) 0 0
\(73\) 51.4318i 0.704545i −0.935898 0.352272i \(-0.885409\pi\)
0.935898 0.352272i \(-0.114591\pi\)
\(74\) 45.0212i 0.608395i
\(75\) 0 0
\(76\) −63.6651 −0.837699
\(77\) 86.9762 1.12956
\(78\) 0 0
\(79\) 59.2176 0.749590 0.374795 0.927108i \(-0.377713\pi\)
0.374795 + 0.927108i \(0.377713\pi\)
\(80\) −16.8583 10.7610i −0.210729 0.134512i
\(81\) 0 0
\(82\) 96.7004i 1.17927i
\(83\) 6.92825 0.0834729 0.0417365 0.999129i \(-0.486711\pi\)
0.0417365 + 0.999129i \(0.486711\pi\)
\(84\) 0 0
\(85\) 72.9859 + 46.5883i 0.858658 + 0.548098i
\(86\) 44.3815i 0.516064i
\(87\) 0 0
\(88\) 32.1714i 0.365584i
\(89\) 115.787i 1.30097i −0.759518 0.650486i \(-0.774565\pi\)
0.759518 0.650486i \(-0.225435\pi\)
\(90\) 0 0
\(91\) −109.982 −1.20860
\(92\) 33.0297 0.359019
\(93\) 0 0
\(94\) −65.1318 −0.692892
\(95\) 134.161 + 85.6373i 1.41222 + 0.901445i
\(96\) 0 0
\(97\) 158.654i 1.63561i 0.575498 + 0.817803i \(0.304808\pi\)
−0.575498 + 0.817803i \(0.695192\pi\)
\(98\) −13.3959 −0.136692
\(99\) 0 0
\(100\) 21.0504 + 45.3528i 0.210504 + 0.453528i
\(101\) 108.879i 1.07801i −0.842303 0.539004i \(-0.818801\pi\)
0.842303 0.539004i \(-0.181199\pi\)
\(102\) 0 0
\(103\) 134.361i 1.30447i 0.758016 + 0.652236i \(0.226169\pi\)
−0.758016 + 0.652236i \(0.773831\pi\)
\(104\) 40.6811i 0.391164i
\(105\) 0 0
\(106\) −75.9356 −0.716374
\(107\) 125.758 1.17531 0.587655 0.809111i \(-0.300051\pi\)
0.587655 + 0.809111i \(0.300051\pi\)
\(108\) 0 0
\(109\) −63.2406 −0.580189 −0.290094 0.956998i \(-0.593687\pi\)
−0.290094 + 0.956998i \(0.593687\pi\)
\(110\) −43.2744 + 67.7943i −0.393404 + 0.616312i
\(111\) 0 0
\(112\) 30.5869i 0.273097i
\(113\) 104.017 0.920509 0.460254 0.887787i \(-0.347758\pi\)
0.460254 + 0.887787i \(0.347758\pi\)
\(114\) 0 0
\(115\) −69.6030 44.4289i −0.605244 0.386339i
\(116\) 16.9842i 0.146415i
\(117\) 0 0
\(118\) 91.5989i 0.776262i
\(119\) 132.422i 1.11279i
\(120\) 0 0
\(121\) −8.37500 −0.0692149
\(122\) −136.154 −1.11602
\(123\) 0 0
\(124\) −91.6384 −0.739019
\(125\) 16.6458 123.887i 0.133167 0.991094i
\(126\) 0 0
\(127\) 142.712i 1.12371i −0.827235 0.561857i \(-0.810087\pi\)
0.827235 0.561857i \(-0.189913\pi\)
\(128\) 11.3137 0.0883883
\(129\) 0 0
\(130\) 54.7209 85.7266i 0.420930 0.659435i
\(131\) 206.387i 1.57547i −0.616014 0.787735i \(-0.711253\pi\)
0.616014 0.787735i \(-0.288747\pi\)
\(132\) 0 0
\(133\) 243.415i 1.83019i
\(134\) 2.43876i 0.0181997i
\(135\) 0 0
\(136\) −48.9814 −0.360157
\(137\) −164.121 −1.19796 −0.598981 0.800763i \(-0.704428\pi\)
−0.598981 + 0.800763i \(0.704428\pi\)
\(138\) 0 0
\(139\) 30.5580 0.219842 0.109921 0.993940i \(-0.464940\pi\)
0.109921 + 0.993940i \(0.464940\pi\)
\(140\) 41.1430 64.4553i 0.293879 0.460395i
\(141\) 0 0
\(142\) 37.5799i 0.264647i
\(143\) 163.596 1.14403
\(144\) 0 0
\(145\) 22.8458 35.7905i 0.157557 0.246831i
\(146\) 72.7355i 0.498188i
\(147\) 0 0
\(148\) 63.6696i 0.430200i
\(149\) 77.9229i 0.522973i −0.965207 0.261486i \(-0.915787\pi\)
0.965207 0.261486i \(-0.0842127\pi\)
\(150\) 0 0
\(151\) 43.0430 0.285053 0.142527 0.989791i \(-0.454477\pi\)
0.142527 + 0.989791i \(0.454477\pi\)
\(152\) −90.0361 −0.592343
\(153\) 0 0
\(154\) 123.003 0.798720
\(155\) 193.108 + 123.265i 1.24586 + 0.795256i
\(156\) 0 0
\(157\) 59.3282i 0.377887i −0.981988 0.188943i \(-0.939494\pi\)
0.981988 0.188943i \(-0.0605063\pi\)
\(158\) 83.7463 0.530040
\(159\) 0 0
\(160\) −23.8412 15.2183i −0.149008 0.0951144i
\(161\) 126.284i 0.784375i
\(162\) 0 0
\(163\) 79.1245i 0.485426i 0.970098 + 0.242713i \(0.0780374\pi\)
−0.970098 + 0.242713i \(0.921963\pi\)
\(164\) 136.755i 0.833872i
\(165\) 0 0
\(166\) 9.79803 0.0590243
\(167\) −101.516 −0.607878 −0.303939 0.952691i \(-0.598302\pi\)
−0.303939 + 0.952691i \(0.598302\pi\)
\(168\) 0 0
\(169\) −37.8686 −0.224075
\(170\) 103.218 + 65.8858i 0.607163 + 0.387564i
\(171\) 0 0
\(172\) 62.7650i 0.364913i
\(173\) 181.350 1.04826 0.524132 0.851637i \(-0.324390\pi\)
0.524132 + 0.851637i \(0.324390\pi\)
\(174\) 0 0
\(175\) −173.400 + 80.4833i −0.990858 + 0.459904i
\(176\) 45.4973i 0.258507i
\(177\) 0 0
\(178\) 163.747i 0.919926i
\(179\) 90.4200i 0.505140i 0.967579 + 0.252570i \(0.0812758\pi\)
−0.967579 + 0.252570i \(0.918724\pi\)
\(180\) 0 0
\(181\) −160.779 −0.888283 −0.444141 0.895957i \(-0.646491\pi\)
−0.444141 + 0.895957i \(0.646491\pi\)
\(182\) −155.538 −0.854606
\(183\) 0 0
\(184\) 46.7111 0.253865
\(185\) −85.6433 + 134.170i −0.462937 + 0.725243i
\(186\) 0 0
\(187\) 196.975i 1.05334i
\(188\) −92.1103 −0.489948
\(189\) 0 0
\(190\) 189.732 + 121.109i 0.998588 + 0.637418i
\(191\) 170.467i 0.892497i −0.894909 0.446249i \(-0.852760\pi\)
0.894909 0.446249i \(-0.147240\pi\)
\(192\) 0 0
\(193\) 214.247i 1.11009i 0.831821 + 0.555044i \(0.187299\pi\)
−0.831821 + 0.555044i \(0.812701\pi\)
\(194\) 224.370i 1.15655i
\(195\) 0 0
\(196\) −18.9446 −0.0966561
\(197\) −209.813 −1.06504 −0.532522 0.846416i \(-0.678755\pi\)
−0.532522 + 0.846416i \(0.678755\pi\)
\(198\) 0 0
\(199\) 327.667 1.64657 0.823285 0.567628i \(-0.192139\pi\)
0.823285 + 0.567628i \(0.192139\pi\)
\(200\) 29.7698 + 64.1386i 0.148849 + 0.320693i
\(201\) 0 0
\(202\) 153.978i 0.762267i
\(203\) −64.9366 −0.319885
\(204\) 0 0
\(205\) 183.952 288.182i 0.897327 1.40577i
\(206\) 190.014i 0.922400i
\(207\) 0 0
\(208\) 57.5317i 0.276595i
\(209\) 362.074i 1.73241i
\(210\) 0 0
\(211\) 102.013 0.483473 0.241737 0.970342i \(-0.422283\pi\)
0.241737 + 0.970342i \(0.422283\pi\)
\(212\) −107.389 −0.506553
\(213\) 0 0
\(214\) 177.849 0.831070
\(215\) 84.4265 132.264i 0.392681 0.615180i
\(216\) 0 0
\(217\) 350.366i 1.61459i
\(218\) −89.4357 −0.410255
\(219\) 0 0
\(220\) −61.1993 + 95.8757i −0.278179 + 0.435799i
\(221\) 249.077i 1.12704i
\(222\) 0 0
\(223\) 377.504i 1.69284i 0.532514 + 0.846421i \(0.321247\pi\)
−0.532514 + 0.846421i \(0.678753\pi\)
\(224\) 43.2564i 0.193109i
\(225\) 0 0
\(226\) 147.103 0.650898
\(227\) 269.454 1.18702 0.593512 0.804825i \(-0.297741\pi\)
0.593512 + 0.804825i \(0.297741\pi\)
\(228\) 0 0
\(229\) 423.862 1.85093 0.925463 0.378837i \(-0.123676\pi\)
0.925463 + 0.378837i \(0.123676\pi\)
\(230\) −98.4335 62.8320i −0.427972 0.273183i
\(231\) 0 0
\(232\) 24.0193i 0.103531i
\(233\) 70.8761 0.304189 0.152095 0.988366i \(-0.451398\pi\)
0.152095 + 0.988366i \(0.451398\pi\)
\(234\) 0 0
\(235\) 194.103 + 123.899i 0.825969 + 0.527232i
\(236\) 129.540i 0.548900i
\(237\) 0 0
\(238\) 187.273i 0.786863i
\(239\) 352.073i 1.47311i −0.676377 0.736555i \(-0.736451\pi\)
0.676377 0.736555i \(-0.263549\pi\)
\(240\) 0 0
\(241\) 87.5764 0.363388 0.181694 0.983355i \(-0.441842\pi\)
0.181694 + 0.983355i \(0.441842\pi\)
\(242\) −11.8440 −0.0489423
\(243\) 0 0
\(244\) −192.551 −0.789142
\(245\) 39.9217 + 25.4828i 0.162946 + 0.104011i
\(246\) 0 0
\(247\) 457.846i 1.85363i
\(248\) −129.596 −0.522565
\(249\) 0 0
\(250\) 23.5407 175.202i 0.0941630 0.700809i
\(251\) 2.67686i 0.0106648i −0.999986 0.00533238i \(-0.998303\pi\)
0.999986 0.00533238i \(-0.00169736\pi\)
\(252\) 0 0
\(253\) 187.845i 0.742471i
\(254\) 201.825i 0.794585i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) −390.653 −1.52005 −0.760025 0.649894i \(-0.774813\pi\)
−0.760025 + 0.649894i \(0.774813\pi\)
\(258\) 0 0
\(259\) 243.432 0.939891
\(260\) 77.3871 121.236i 0.297643 0.466291i
\(261\) 0 0
\(262\) 291.875i 1.11403i
\(263\) −0.831141 −0.00316023 −0.00158012 0.999999i \(-0.500503\pi\)
−0.00158012 + 0.999999i \(0.500503\pi\)
\(264\) 0 0
\(265\) 226.300 + 144.451i 0.853961 + 0.545100i
\(266\) 344.240i 1.29414i
\(267\) 0 0
\(268\) 3.44892i 0.0128691i
\(269\) 126.427i 0.469989i 0.971997 + 0.234994i \(0.0755072\pi\)
−0.971997 + 0.234994i \(0.924493\pi\)
\(270\) 0 0
\(271\) −359.481 −1.32650 −0.663249 0.748399i \(-0.730823\pi\)
−0.663249 + 0.748399i \(0.730823\pi\)
\(272\) −69.2701 −0.254669
\(273\) 0 0
\(274\) −232.102 −0.847087
\(275\) 257.929 119.717i 0.937923 0.435335i
\(276\) 0 0
\(277\) 424.766i 1.53345i 0.641975 + 0.766726i \(0.278115\pi\)
−0.641975 + 0.766726i \(0.721885\pi\)
\(278\) 43.2156 0.155452
\(279\) 0 0
\(280\) 58.1850 91.1535i 0.207804 0.325548i
\(281\) 372.569i 1.32587i 0.748677 + 0.662935i \(0.230689\pi\)
−0.748677 + 0.662935i \(0.769311\pi\)
\(282\) 0 0
\(283\) 453.237i 1.60154i −0.598970 0.800771i \(-0.704423\pi\)
0.598970 0.800771i \(-0.295577\pi\)
\(284\) 53.1461i 0.187134i
\(285\) 0 0
\(286\) 231.360 0.808950
\(287\) −522.864 −1.82182
\(288\) 0 0
\(289\) 10.8967 0.0377048
\(290\) 32.3088 50.6154i 0.111410 0.174536i
\(291\) 0 0
\(292\) 102.864i 0.352272i
\(293\) −105.139 −0.358835 −0.179418 0.983773i \(-0.557421\pi\)
−0.179418 + 0.983773i \(0.557421\pi\)
\(294\) 0 0
\(295\) 174.247 272.978i 0.590669 0.925351i
\(296\) 90.0424i 0.304197i
\(297\) 0 0
\(298\) 110.200i 0.369798i
\(299\) 237.532i 0.794421i
\(300\) 0 0
\(301\) −239.973 −0.797253
\(302\) 60.8720 0.201563
\(303\) 0 0
\(304\) −127.330 −0.418850
\(305\) 405.759 + 259.004i 1.33036 + 0.849193i
\(306\) 0 0
\(307\) 488.983i 1.59278i −0.604785 0.796389i \(-0.706741\pi\)
0.604785 0.796389i \(-0.293259\pi\)
\(308\) 173.952 0.564780
\(309\) 0 0
\(310\) 273.096 + 174.323i 0.880956 + 0.562331i
\(311\) 57.9226i 0.186246i 0.995655 + 0.0931231i \(0.0296850\pi\)
−0.995655 + 0.0931231i \(0.970315\pi\)
\(312\) 0 0
\(313\) 456.139i 1.45731i −0.684880 0.728656i \(-0.740145\pi\)
0.684880 0.728656i \(-0.259855\pi\)
\(314\) 83.9028i 0.267206i
\(315\) 0 0
\(316\) 118.435 0.374795
\(317\) −483.080 −1.52391 −0.761955 0.647629i \(-0.775761\pi\)
−0.761955 + 0.647629i \(0.775761\pi\)
\(318\) 0 0
\(319\) 96.5917 0.302795
\(320\) −33.7166 21.5219i −0.105364 0.0672560i
\(321\) 0 0
\(322\) 178.593i 0.554637i
\(323\) 551.261 1.70669
\(324\) 0 0
\(325\) −326.153 + 151.383i −1.00355 + 0.465795i
\(326\) 111.899i 0.343248i
\(327\) 0 0
\(328\) 193.401i 0.589637i
\(329\) 352.171i 1.07043i
\(330\) 0 0
\(331\) −85.4008 −0.258008 −0.129004 0.991644i \(-0.541178\pi\)
−0.129004 + 0.991644i \(0.541178\pi\)
\(332\) 13.8565 0.0417365
\(333\) 0 0
\(334\) −143.565 −0.429835
\(335\) −4.63922 + 7.26786i −0.0138484 + 0.0216951i
\(336\) 0 0
\(337\) 498.720i 1.47988i 0.672672 + 0.739941i \(0.265147\pi\)
−0.672672 + 0.739941i \(0.734853\pi\)
\(338\) −53.5543 −0.158445
\(339\) 0 0
\(340\) 145.972 + 93.1766i 0.429329 + 0.274049i
\(341\) 521.162i 1.52833i
\(342\) 0 0
\(343\) 302.257i 0.881216i
\(344\) 88.7631i 0.258032i
\(345\) 0 0
\(346\) 256.467 0.741235
\(347\) −494.070 −1.42383 −0.711917 0.702264i \(-0.752173\pi\)
−0.711917 + 0.702264i \(0.752173\pi\)
\(348\) 0 0
\(349\) 371.772 1.06525 0.532624 0.846352i \(-0.321206\pi\)
0.532624 + 0.846352i \(0.321206\pi\)
\(350\) −245.225 + 113.821i −0.700643 + 0.325202i
\(351\) 0 0
\(352\) 64.3428i 0.182792i
\(353\) 295.348 0.836679 0.418340 0.908291i \(-0.362612\pi\)
0.418340 + 0.908291i \(0.362612\pi\)
\(354\) 0 0
\(355\) 71.4879 111.994i 0.201374 0.315476i
\(356\) 231.573i 0.650486i
\(357\) 0 0
\(358\) 127.873i 0.357188i
\(359\) 251.070i 0.699359i −0.936869 0.349679i \(-0.886291\pi\)
0.936869 0.349679i \(-0.113709\pi\)
\(360\) 0 0
\(361\) 652.312 1.80696
\(362\) −227.376 −0.628111
\(363\) 0 0
\(364\) −219.964 −0.604298
\(365\) −138.364 + 216.763i −0.379079 + 0.593871i
\(366\) 0 0
\(367\) 632.976i 1.72473i 0.506286 + 0.862366i \(0.331018\pi\)
−0.506286 + 0.862366i \(0.668982\pi\)
\(368\) 66.0594 0.179509
\(369\) 0 0
\(370\) −121.118 + 189.745i −0.327346 + 0.512824i
\(371\) 410.587i 1.10670i
\(372\) 0 0
\(373\) 183.030i 0.490696i −0.969435 0.245348i \(-0.921098\pi\)
0.969435 0.245348i \(-0.0789022\pi\)
\(374\) 278.565i 0.744825i
\(375\) 0 0
\(376\) −130.264 −0.346446
\(377\) −122.141 −0.323982
\(378\) 0 0
\(379\) 274.778 0.725009 0.362504 0.931982i \(-0.381922\pi\)
0.362504 + 0.931982i \(0.381922\pi\)
\(380\) 268.321 + 171.275i 0.706108 + 0.450723i
\(381\) 0 0
\(382\) 241.077i 0.631091i
\(383\) 374.185 0.976985 0.488492 0.872568i \(-0.337547\pi\)
0.488492 + 0.872568i \(0.337547\pi\)
\(384\) 0 0
\(385\) −366.567 233.987i −0.952123 0.607758i
\(386\) 302.991i 0.784951i
\(387\) 0 0
\(388\) 317.308i 0.817803i
\(389\) 241.583i 0.621037i −0.950567 0.310518i \(-0.899497\pi\)
0.950567 0.310518i \(-0.100503\pi\)
\(390\) 0 0
\(391\) −285.996 −0.731449
\(392\) −26.7917 −0.0683462
\(393\) 0 0
\(394\) −296.721 −0.753099
\(395\) −249.577 159.310i −0.631840 0.403315i
\(396\) 0 0
\(397\) 267.415i 0.673590i 0.941578 + 0.336795i \(0.109343\pi\)
−0.941578 + 0.336795i \(0.890657\pi\)
\(398\) 463.392 1.16430
\(399\) 0 0
\(400\) 42.1008 + 90.7057i 0.105252 + 0.226764i
\(401\) 688.894i 1.71794i 0.512026 + 0.858970i \(0.328895\pi\)
−0.512026 + 0.858970i \(0.671105\pi\)
\(402\) 0 0
\(403\) 659.014i 1.63527i
\(404\) 217.758i 0.539004i
\(405\) 0 0
\(406\) −91.8343 −0.226193
\(407\) −362.099 −0.889678
\(408\) 0 0
\(409\) −253.327 −0.619382 −0.309691 0.950837i \(-0.600226\pi\)
−0.309691 + 0.950837i \(0.600226\pi\)
\(410\) 260.147 407.551i 0.634506 0.994026i
\(411\) 0 0
\(412\) 268.721i 0.652236i
\(413\) −495.279 −1.19922
\(414\) 0 0
\(415\) −29.1996 18.6387i −0.0703605 0.0449125i
\(416\) 81.3621i 0.195582i
\(417\) 0 0
\(418\) 512.049i 1.22500i
\(419\) 515.612i 1.23058i −0.788302 0.615289i \(-0.789039\pi\)
0.788302 0.615289i \(-0.210961\pi\)
\(420\) 0 0
\(421\) −517.082 −1.22822 −0.614112 0.789219i \(-0.710486\pi\)
−0.614112 + 0.789219i \(0.710486\pi\)
\(422\) 144.268 0.341867
\(423\) 0 0
\(424\) −151.871 −0.358187
\(425\) −182.271 392.699i −0.428872 0.923999i
\(426\) 0 0
\(427\) 736.190i 1.72410i
\(428\) 251.516 0.587655
\(429\) 0 0
\(430\) 119.397 187.049i 0.277668 0.434998i
\(431\) 290.795i 0.674698i −0.941380 0.337349i \(-0.890470\pi\)
0.941380 0.337349i \(-0.109530\pi\)
\(432\) 0 0
\(433\) 358.909i 0.828889i 0.910075 + 0.414444i \(0.136024\pi\)
−0.910075 + 0.414444i \(0.863976\pi\)
\(434\) 495.493i 1.14169i
\(435\) 0 0
\(436\) −126.481 −0.290094
\(437\) −525.710 −1.20300
\(438\) 0 0
\(439\) −203.637 −0.463866 −0.231933 0.972732i \(-0.574505\pi\)
−0.231933 + 0.972732i \(0.574505\pi\)
\(440\) −86.5489 + 135.589i −0.196702 + 0.308156i
\(441\) 0 0
\(442\) 352.248i 0.796940i
\(443\) 21.2268 0.0479160 0.0239580 0.999713i \(-0.492373\pi\)
0.0239580 + 0.999713i \(0.492373\pi\)
\(444\) 0 0
\(445\) −311.494 + 487.990i −0.699986 + 1.09661i
\(446\) 533.871i 1.19702i
\(447\) 0 0
\(448\) 61.1737i 0.136549i
\(449\) 650.346i 1.44843i 0.689573 + 0.724216i \(0.257798\pi\)
−0.689573 + 0.724216i \(0.742202\pi\)
\(450\) 0 0
\(451\) 777.747 1.72450
\(452\) 208.035 0.460254
\(453\) 0 0
\(454\) 381.066 0.839353
\(455\) 463.528 + 295.879i 1.01874 + 0.650283i
\(456\) 0 0
\(457\) 154.972i 0.339107i 0.985521 + 0.169553i \(0.0542325\pi\)
−0.985521 + 0.169553i \(0.945767\pi\)
\(458\) 599.432 1.30880
\(459\) 0 0
\(460\) −139.206 88.8579i −0.302622 0.193169i
\(461\) 351.985i 0.763525i 0.924261 + 0.381762i \(0.124683\pi\)
−0.924261 + 0.381762i \(0.875317\pi\)
\(462\) 0 0
\(463\) 100.306i 0.216644i 0.994116 + 0.108322i \(0.0345477\pi\)
−0.994116 + 0.108322i \(0.965452\pi\)
\(464\) 33.9684i 0.0732077i
\(465\) 0 0
\(466\) 100.234 0.215094
\(467\) 31.5687 0.0675989 0.0337994 0.999429i \(-0.489239\pi\)
0.0337994 + 0.999429i \(0.489239\pi\)
\(468\) 0 0
\(469\) 13.1865 0.0281161
\(470\) 274.503 + 175.220i 0.584048 + 0.372809i
\(471\) 0 0
\(472\) 183.198i 0.388131i
\(473\) 356.954 0.754660
\(474\) 0 0
\(475\) −335.044 721.849i −0.705356 1.51968i
\(476\) 264.844i 0.556396i
\(477\) 0 0
\(478\) 497.907i 1.04165i
\(479\) 634.355i 1.32433i 0.749357 + 0.662166i \(0.230362\pi\)
−0.749357 + 0.662166i \(0.769638\pi\)
\(480\) 0 0
\(481\) 457.878 0.951929
\(482\) 123.852 0.256954
\(483\) 0 0
\(484\) −16.7500 −0.0346075
\(485\) 426.817 668.658i 0.880035 1.37868i
\(486\) 0 0
\(487\) 639.112i 1.31235i 0.754611 + 0.656173i \(0.227826\pi\)
−0.754611 + 0.656173i \(0.772174\pi\)
\(488\) −272.308 −0.558008
\(489\) 0 0
\(490\) 56.4578 + 36.0381i 0.115220 + 0.0735471i
\(491\) 367.869i 0.749224i 0.927182 + 0.374612i \(0.122224\pi\)
−0.927182 + 0.374612i \(0.877776\pi\)
\(492\) 0 0
\(493\) 147.062i 0.298300i
\(494\) 647.491i 1.31071i
\(495\) 0 0
\(496\) −183.277 −0.369510
\(497\) −203.196 −0.408846
\(498\) 0 0
\(499\) −494.543 −0.991069 −0.495534 0.868588i \(-0.665028\pi\)
−0.495534 + 0.868588i \(0.665028\pi\)
\(500\) 33.2916 247.773i 0.0665833 0.495547i
\(501\) 0 0
\(502\) 3.78565i 0.00754113i
\(503\) −176.118 −0.350135 −0.175067 0.984556i \(-0.556014\pi\)
−0.175067 + 0.984556i \(0.556014\pi\)
\(504\) 0 0
\(505\) −292.910 + 458.878i −0.580020 + 0.908668i
\(506\) 265.653i 0.525006i
\(507\) 0 0
\(508\) 285.423i 0.561857i
\(509\) 624.613i 1.22714i 0.789641 + 0.613569i \(0.210267\pi\)
−0.789641 + 0.613569i \(0.789733\pi\)
\(510\) 0 0
\(511\) 393.284 0.769636
\(512\) 22.6274 0.0441942
\(513\) 0 0
\(514\) −552.467 −1.07484
\(515\) 361.462 566.272i 0.701868 1.09956i
\(516\) 0 0
\(517\) 523.846i 1.01324i
\(518\) 344.265 0.664603
\(519\) 0 0
\(520\) 109.442 171.453i 0.210465 0.329718i
\(521\) 908.634i 1.74402i −0.489489 0.872009i \(-0.662817\pi\)
0.489489 0.872009i \(-0.337183\pi\)
\(522\) 0 0
\(523\) 534.875i 1.02271i 0.859371 + 0.511353i \(0.170855\pi\)
−0.859371 + 0.511353i \(0.829145\pi\)
\(524\) 412.773i 0.787735i
\(525\) 0 0
\(526\) −1.17541 −0.00223462
\(527\) 793.475 1.50565
\(528\) 0 0
\(529\) −256.260 −0.484423
\(530\) 320.036 + 204.285i 0.603841 + 0.385444i
\(531\) 0 0
\(532\) 486.829i 0.915093i
\(533\) −983.469 −1.84516
\(534\) 0 0
\(535\) −530.017 338.320i −0.990686 0.632374i
\(536\) 4.87751i 0.00909984i
\(537\) 0 0
\(538\) 178.795i 0.332332i
\(539\) 107.741i 0.199890i
\(540\) 0 0
\(541\) −996.576 −1.84210 −0.921050 0.389444i \(-0.872667\pi\)
−0.921050 + 0.389444i \(0.872667\pi\)
\(542\) −508.383 −0.937975
\(543\) 0 0
\(544\) −97.9627 −0.180079
\(545\) 266.532 + 170.132i 0.489049 + 0.312169i
\(546\) 0 0
\(547\) 632.447i 1.15621i 0.815963 + 0.578105i \(0.196208\pi\)
−0.815963 + 0.578105i \(0.803792\pi\)
\(548\) −328.242 −0.598981
\(549\) 0 0
\(550\) 364.766 169.305i 0.663211 0.307828i
\(551\) 270.325i 0.490608i
\(552\) 0 0
\(553\) 452.820i 0.818843i
\(554\) 600.710i 1.08431i
\(555\) 0 0
\(556\) 61.1161 0.109921
\(557\) −673.973 −1.21001 −0.605003 0.796223i \(-0.706828\pi\)
−0.605003 + 0.796223i \(0.706828\pi\)
\(558\) 0 0
\(559\) −451.372 −0.807463
\(560\) 82.2861 128.911i 0.146939 0.230197i
\(561\) 0 0
\(562\) 526.893i 0.937531i
\(563\) 336.483 0.597661 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(564\) 0 0
\(565\) −438.389 279.832i −0.775910 0.495278i
\(566\) 640.973i 1.13246i
\(567\) 0 0
\(568\) 75.1599i 0.132324i
\(569\) 410.792i 0.721954i −0.932575 0.360977i \(-0.882443\pi\)
0.932575 0.360977i \(-0.117557\pi\)
\(570\) 0 0
\(571\) −444.795 −0.778976 −0.389488 0.921032i \(-0.627348\pi\)
−0.389488 + 0.921032i \(0.627348\pi\)
\(572\) 327.192 0.572014
\(573\) 0 0
\(574\) −739.441 −1.28822
\(575\) 173.822 + 374.498i 0.302300 + 0.651301i
\(576\) 0 0
\(577\) 365.707i 0.633808i −0.948458 0.316904i \(-0.897357\pi\)
0.948458 0.316904i \(-0.102643\pi\)
\(578\) 15.4102 0.0266613
\(579\) 0 0
\(580\) 45.6916 71.5811i 0.0787786 0.123416i
\(581\) 52.9784i 0.0911848i
\(582\) 0 0
\(583\) 610.739i 1.04758i
\(584\) 145.471i 0.249094i
\(585\) 0 0
\(586\) −148.689 −0.253735
\(587\) 387.993 0.660976 0.330488 0.943810i \(-0.392787\pi\)
0.330488 + 0.943810i \(0.392787\pi\)
\(588\) 0 0
\(589\) 1458.54 2.47630
\(590\) 246.423 386.050i 0.417666 0.654322i
\(591\) 0 0
\(592\) 127.339i 0.215100i
\(593\) −657.706 −1.10912 −0.554558 0.832145i \(-0.687113\pi\)
−0.554558 + 0.832145i \(0.687113\pi\)
\(594\) 0 0
\(595\) −356.248 + 558.103i −0.598736 + 0.937988i
\(596\) 155.846i 0.261486i
\(597\) 0 0
\(598\) 335.921i 0.561741i
\(599\) 534.335i 0.892045i 0.895022 + 0.446022i \(0.147160\pi\)
−0.895022 + 0.446022i \(0.852840\pi\)
\(600\) 0 0
\(601\) −900.532 −1.49839 −0.749195 0.662350i \(-0.769559\pi\)
−0.749195 + 0.662350i \(0.769559\pi\)
\(602\) −339.373 −0.563743
\(603\) 0 0
\(604\) 86.0861 0.142527
\(605\) 35.2970 + 22.5308i 0.0583422 + 0.0372410i
\(606\) 0 0
\(607\) 870.321i 1.43381i −0.697172 0.716904i \(-0.745559\pi\)
0.697172 0.716904i \(-0.254441\pi\)
\(608\) −180.072 −0.296171
\(609\) 0 0
\(610\) 573.830 + 366.287i 0.940705 + 0.600470i
\(611\) 662.408i 1.08414i
\(612\) 0 0
\(613\) 252.101i 0.411258i −0.978630 0.205629i \(-0.934076\pi\)
0.978630 0.205629i \(-0.0659240\pi\)
\(614\) 691.526i 1.12626i
\(615\) 0 0
\(616\) 246.006 0.399360
\(617\) −724.936 −1.17494 −0.587468 0.809247i \(-0.699875\pi\)
−0.587468 + 0.809247i \(0.699875\pi\)
\(618\) 0 0
\(619\) 293.289 0.473811 0.236906 0.971533i \(-0.423867\pi\)
0.236906 + 0.971533i \(0.423867\pi\)
\(620\) 386.216 + 246.529i 0.622930 + 0.397628i
\(621\) 0 0
\(622\) 81.9149i 0.131696i
\(623\) 885.387 1.42117
\(624\) 0 0
\(625\) −403.440 + 477.348i −0.645504 + 0.763757i
\(626\) 645.078i 1.03048i
\(627\) 0 0
\(628\) 118.656i 0.188943i
\(629\) 551.300i 0.876471i
\(630\) 0 0
\(631\) 397.922 0.630622 0.315311 0.948988i \(-0.397891\pi\)
0.315311 + 0.948988i \(0.397891\pi\)
\(632\) 167.493 0.265020
\(633\) 0 0
\(634\) −683.178 −1.07757
\(635\) −383.929 + 601.468i −0.604612 + 0.947194i
\(636\) 0 0
\(637\) 136.239i 0.213877i
\(638\) 136.601 0.214109
\(639\) 0 0
\(640\) −47.6824 30.4366i −0.0745038 0.0475572i
\(641\) 265.282i 0.413857i 0.978356 + 0.206928i \(0.0663467\pi\)
−0.978356 + 0.206928i \(0.933653\pi\)
\(642\) 0 0
\(643\) 251.091i 0.390499i 0.980754 + 0.195250i \(0.0625517\pi\)
−0.980754 + 0.195250i \(0.937448\pi\)
\(644\) 252.569i 0.392188i
\(645\) 0 0
\(646\) 779.601 1.20681
\(647\) −415.167 −0.641680 −0.320840 0.947133i \(-0.603965\pi\)
−0.320840 + 0.947133i \(0.603965\pi\)
\(648\) 0 0
\(649\) 736.716 1.13516
\(650\) −461.250 + 214.088i −0.709616 + 0.329367i
\(651\) 0 0
\(652\) 158.249i 0.242713i
\(653\) −847.279 −1.29752 −0.648758 0.760994i \(-0.724711\pi\)
−0.648758 + 0.760994i \(0.724711\pi\)
\(654\) 0 0
\(655\) −555.230 + 869.831i −0.847679 + 1.32799i
\(656\) 273.510i 0.416936i
\(657\) 0 0
\(658\) 498.045i 0.756907i
\(659\) 143.448i 0.217676i 0.994059 + 0.108838i \(0.0347130\pi\)
−0.994059 + 0.108838i \(0.965287\pi\)
\(660\) 0 0
\(661\) −303.035 −0.458449 −0.229224 0.973374i \(-0.573619\pi\)
−0.229224 + 0.973374i \(0.573619\pi\)
\(662\) −120.775 −0.182439
\(663\) 0 0
\(664\) 19.5961 0.0295121
\(665\) −654.844 + 1025.89i −0.984728 + 1.54269i
\(666\) 0 0
\(667\) 140.246i 0.210263i
\(668\) −203.031 −0.303939
\(669\) 0 0
\(670\) −6.56084 + 10.2783i −0.00979230 + 0.0153408i
\(671\) 1095.07i 1.63199i
\(672\) 0 0
\(673\) 278.524i 0.413854i −0.978356 0.206927i \(-0.933654\pi\)
0.978356 0.206927i \(-0.0663462\pi\)
\(674\) 705.297i 1.04643i
\(675\) 0 0
\(676\) −75.7373 −0.112037
\(677\) −813.121 −1.20106 −0.600532 0.799600i \(-0.705045\pi\)
−0.600532 + 0.799600i \(0.705045\pi\)
\(678\) 0 0
\(679\) −1213.18 −1.78672
\(680\) 206.435 + 131.772i 0.303581 + 0.193782i
\(681\) 0 0
\(682\) 737.034i 1.08070i
\(683\) 1081.42 1.58334 0.791668 0.610952i \(-0.209213\pi\)
0.791668 + 0.610952i \(0.209213\pi\)
\(684\) 0 0
\(685\) 691.699 + 441.525i 1.00978 + 0.644561i
\(686\) 427.456i 0.623114i
\(687\) 0 0
\(688\) 125.530i 0.182456i
\(689\) 772.285i 1.12088i
\(690\) 0 0
\(691\) 1059.73 1.53361 0.766807 0.641878i \(-0.221844\pi\)
0.766807 + 0.641878i \(0.221844\pi\)
\(692\) 362.700 0.524132
\(693\) 0 0
\(694\) −698.721 −1.00680
\(695\) −128.789 82.2085i −0.185308 0.118286i
\(696\) 0 0
\(697\) 1184.13i 1.69889i
\(698\) 525.765 0.753245
\(699\) 0 0
\(700\) −346.800 + 160.967i −0.495429 + 0.229952i
\(701\) 716.640i 1.02231i −0.859488 0.511155i \(-0.829218\pi\)
0.859488 0.511155i \(-0.170782\pi\)
\(702\) 0 0
\(703\) 1013.38i 1.44151i
\(704\) 90.9945i 0.129254i
\(705\) 0 0
\(706\) 417.685 0.591622
\(707\) 832.566 1.17760
\(708\) 0 0
\(709\) −146.474 −0.206592 −0.103296 0.994651i \(-0.532939\pi\)
−0.103296 + 0.994651i \(0.532939\pi\)
\(710\) 101.099 158.383i 0.142393 0.223075i
\(711\) 0 0
\(712\) 327.494i 0.459963i
\(713\) −756.697 −1.06129
\(714\) 0 0
\(715\) −689.487 440.113i −0.964317 0.615542i
\(716\) 180.840i 0.252570i
\(717\) 0 0
\(718\) 355.066i 0.494521i
\(719\) 329.272i 0.457958i 0.973431 + 0.228979i \(0.0735388\pi\)
−0.973431 + 0.228979i \(0.926461\pi\)
\(720\) 0 0
\(721\) −1027.42 −1.42499
\(722\) 922.509 1.27771
\(723\) 0 0
\(724\) −321.558 −0.444141
\(725\) −192.570 + 89.3810i −0.265614 + 0.123284i
\(726\) 0 0
\(727\) 332.250i 0.457015i 0.973542 + 0.228507i \(0.0733845\pi\)
−0.973542 + 0.228507i \(0.926615\pi\)
\(728\) −311.077 −0.427303
\(729\) 0 0
\(730\) −195.676 + 306.549i −0.268049 + 0.419930i
\(731\) 543.467i 0.743457i
\(732\) 0 0
\(733\) 496.851i 0.677833i 0.940817 + 0.338916i \(0.110060\pi\)
−0.940817 + 0.338916i \(0.889940\pi\)
\(734\) 895.164i 1.21957i
\(735\) 0 0
\(736\) 93.4221 0.126932
\(737\) −19.6146 −0.0266141
\(738\) 0 0
\(739\) 493.117 0.667276 0.333638 0.942701i \(-0.391724\pi\)
0.333638 + 0.942701i \(0.391724\pi\)
\(740\) −171.287 + 268.340i −0.231468 + 0.362622i
\(741\) 0 0
\(742\) 580.658i 0.782558i
\(743\) −816.847 −1.09939 −0.549695 0.835365i \(-0.685256\pi\)
−0.549695 + 0.835365i \(0.685256\pi\)
\(744\) 0 0
\(745\) −209.632 + 328.412i −0.281385 + 0.440821i
\(746\) 258.843i 0.346975i
\(747\) 0 0
\(748\) 393.950i 0.526671i
\(749\) 961.638i 1.28390i
\(750\) 0 0
\(751\) 583.577 0.777067 0.388533 0.921435i \(-0.372982\pi\)
0.388533 + 0.921435i \(0.372982\pi\)
\(752\) −184.221 −0.244974
\(753\) 0 0
\(754\) −172.734 −0.229090
\(755\) −181.408 115.796i −0.240275 0.153372i
\(756\) 0 0
\(757\) 633.635i 0.837034i −0.908209 0.418517i \(-0.862550\pi\)
0.908209 0.418517i \(-0.137450\pi\)
\(758\) 388.595 0.512658
\(759\) 0 0
\(760\) 379.463 + 242.219i 0.499294 + 0.318709i
\(761\) 745.689i 0.979880i 0.871756 + 0.489940i \(0.162981\pi\)
−0.871756 + 0.489940i \(0.837019\pi\)
\(762\) 0 0
\(763\) 483.583i 0.633791i
\(764\) 340.934i 0.446249i
\(765\) 0 0
\(766\) 529.178 0.690832
\(767\) −931.585 −1.21458
\(768\) 0 0
\(769\) −431.418 −0.561012 −0.280506 0.959852i \(-0.590502\pi\)
−0.280506 + 0.959852i \(0.590502\pi\)
\(770\) −518.404 330.907i −0.673252 0.429750i
\(771\) 0 0
\(772\) 428.494i 0.555044i
\(773\) −248.851 −0.321929 −0.160964 0.986960i \(-0.551460\pi\)
−0.160964 + 0.986960i \(0.551460\pi\)
\(774\) 0 0
\(775\) −482.256 1039.02i −0.622266 1.34066i
\(776\) 448.741i 0.578274i
\(777\) 0 0
\(778\) 341.650i 0.439139i
\(779\) 2176.63i 2.79414i
\(780\) 0 0
\(781\) 302.250 0.387004
\(782\) −404.460 −0.517212
\(783\) 0 0
\(784\) −37.8892 −0.0483280
\(785\) −159.607 + 250.043i −0.203321 + 0.318526i
\(786\) 0 0
\(787\) 929.437i 1.18099i 0.807042 + 0.590494i \(0.201067\pi\)
−0.807042 + 0.590494i \(0.798933\pi\)
\(788\) −419.627 −0.532522
\(789\) 0 0
\(790\) −352.955 225.298i −0.446778 0.285187i
\(791\) 795.392i 1.00555i
\(792\) 0 0
\(793\) 1384.72i 1.74618i
\(794\) 378.182i 0.476300i
\(795\) 0 0
\(796\) 655.335 0.823285
\(797\) 584.059 0.732821 0.366411 0.930453i \(-0.380587\pi\)
0.366411 + 0.930453i \(0.380587\pi\)
\(798\) 0 0
\(799\) 797.561 0.998199
\(800\) 59.5396 + 128.277i 0.0744244 + 0.160347i
\(801\) 0 0
\(802\) 974.243i 1.21477i
\(803\) −585.001 −0.728519
\(804\) 0 0
\(805\) 339.736 532.235i 0.422032 0.661161i
\(806\) 931.987i 1.15631i
\(807\) 0 0
\(808\) 307.956i 0.381134i
\(809\) 363.295i 0.449067i 0.974466 + 0.224533i \(0.0720858\pi\)
−0.974466 + 0.224533i \(0.927914\pi\)
\(810\) 0 0
\(811\) 69.8507 0.0861290 0.0430645 0.999072i \(-0.486288\pi\)
0.0430645 + 0.999072i \(0.486288\pi\)
\(812\) −129.873 −0.159942
\(813\) 0 0
\(814\) −512.085 −0.629097
\(815\) 212.864 333.476i 0.261183 0.409173i
\(816\) 0 0
\(817\) 998.985i 1.22275i
\(818\) −358.259 −0.437969
\(819\) 0 0
\(820\) 367.904 576.364i 0.448663 0.702883i
\(821\) 183.472i 0.223474i −0.993738 0.111737i \(-0.964359\pi\)
0.993738 0.111737i \(-0.0356414\pi\)
\(822\) 0 0
\(823\) 534.931i 0.649977i −0.945718 0.324988i \(-0.894640\pi\)
0.945718 0.324988i \(-0.105360\pi\)
\(824\) 380.029i 0.461200i
\(825\) 0 0
\(826\) −700.431 −0.847979
\(827\) 640.632 0.774646 0.387323 0.921944i \(-0.373400\pi\)
0.387323 + 0.921944i \(0.373400\pi\)
\(828\) 0 0
\(829\) 910.462 1.09827 0.549133 0.835735i \(-0.314958\pi\)
0.549133 + 0.835735i \(0.314958\pi\)
\(830\) −41.2945 26.3591i −0.0497524 0.0317579i
\(831\) 0 0
\(832\) 115.063i 0.138297i
\(833\) 164.037 0.196923
\(834\) 0 0
\(835\) 427.845 + 273.102i 0.512389 + 0.327068i
\(836\) 724.147i 0.866205i
\(837\) 0 0
\(838\) 729.186i 0.870150i
\(839\) 41.5328i 0.0495027i −0.999694 0.0247513i \(-0.992121\pi\)
0.999694 0.0247513i \(-0.00787940\pi\)
\(840\) 0 0
\(841\) 768.884 0.914250
\(842\) −731.265 −0.868486
\(843\) 0 0
\(844\) 204.026 0.241737
\(845\) 159.600 + 101.876i 0.188876 + 0.120563i
\(846\) 0 0
\(847\) 64.0413i 0.0756096i
\(848\) −214.778 −0.253276
\(849\) 0 0
\(850\) −257.769 555.361i −0.303258 0.653366i
\(851\) 525.747i 0.617799i
\(852\) 0 0
\(853\) 426.015i 0.499431i 0.968319 + 0.249716i \(0.0803371\pi\)
−0.968319 + 0.249716i \(0.919663\pi\)
\(854\) 1041.13i 1.21912i
\(855\) 0 0
\(856\) 355.698 0.415535
\(857\) 662.580 0.773139 0.386570 0.922260i \(-0.373660\pi\)
0.386570 + 0.922260i \(0.373660\pi\)
\(858\) 0 0
\(859\) 1402.18 1.63234 0.816168 0.577815i \(-0.196094\pi\)
0.816168 + 0.577815i \(0.196094\pi\)
\(860\) 168.853 264.527i 0.196341 0.307590i
\(861\) 0 0
\(862\) 411.246i 0.477083i
\(863\) −54.6082 −0.0632771 −0.0316386 0.999499i \(-0.510073\pi\)
−0.0316386 + 0.999499i \(0.510073\pi\)
\(864\) 0 0
\(865\) −764.312 487.875i −0.883597 0.564017i
\(866\) 507.574i 0.586113i
\(867\) 0 0
\(868\) 700.733i 0.807296i
\(869\) 673.559i 0.775097i
\(870\) 0 0
\(871\) 24.8028 0.0284762
\(872\) −178.871 −0.205128
\(873\) 0 0
\(874\) −743.467 −0.850648
\(875\) 947.327 + 127.286i 1.08266 + 0.145470i
\(876\) 0 0
\(877\) 875.174i 0.997918i 0.866625 + 0.498959i \(0.166284\pi\)
−0.866625 + 0.498959i \(0.833716\pi\)
\(878\) −287.987 −0.328003
\(879\) 0 0
\(880\) −122.399 + 191.751i −0.139089 + 0.217899i
\(881\) 1103.84i 1.25294i −0.779445 0.626471i \(-0.784499\pi\)
0.779445 0.626471i \(-0.215501\pi\)
\(882\) 0 0
\(883\) 455.458i 0.515808i −0.966171 0.257904i \(-0.916968\pi\)
0.966171 0.257904i \(-0.0830318\pi\)
\(884\) 498.153i 0.563522i
\(885\) 0 0
\(886\) 30.0192 0.0338817
\(887\) −951.037 −1.07219 −0.536097 0.844156i \(-0.680102\pi\)
−0.536097 + 0.844156i \(0.680102\pi\)
\(888\) 0 0
\(889\) 1091.28 1.22753
\(890\) −440.518 + 690.123i −0.494965 + 0.775419i
\(891\) 0 0
\(892\) 755.008i 0.846421i
\(893\) 1466.05 1.64172
\(894\) 0 0
\(895\) 243.252 381.082i 0.271790 0.425789i
\(896\) 86.5127i 0.0965544i
\(897\) 0 0
\(898\) 919.728i 1.02420i
\(899\) 389.101i 0.432815i
\(900\) 0 0
\(901\) 929.857 1.03203
\(902\) 1099.90 1.21940
\(903\) 0 0
\(904\) 294.206 0.325449
\(905\) 677.615 + 432.535i 0.748746 + 0.477939i
\(906\) 0 0
\(907\) 1349.31i 1.48766i −0.668370 0.743829i \(-0.733008\pi\)
0.668370 0.743829i \(-0.266992\pi\)
\(908\) 538.909 0.593512
\(909\) 0 0
\(910\) 655.527 + 418.436i 0.720360 + 0.459819i
\(911\) 409.175i 0.449149i 0.974457 + 0.224575i \(0.0720993\pi\)
−0.974457 + 0.224575i \(0.927901\pi\)
\(912\) 0 0
\(913\) 78.8041i 0.0863134i
\(914\) 219.163i 0.239785i
\(915\) 0 0
\(916\) 847.725 0.925463
\(917\) 1578.18 1.72103
\(918\) 0 0
\(919\) −247.618 −0.269443 −0.134721 0.990884i \(-0.543014\pi\)
−0.134721 + 0.990884i \(0.543014\pi\)
\(920\) −196.867 125.664i −0.213986 0.136591i
\(921\) 0 0
\(922\) 497.782i 0.539893i
\(923\) −382.198 −0.414082
\(924\) 0 0
\(925\) 721.899 335.068i 0.780432 0.362236i
\(926\) 141.854i 0.153190i
\(927\) 0 0
\(928\) 48.0385i 0.0517657i
\(929\) 274.384i 0.295354i 0.989036 + 0.147677i \(0.0471796\pi\)
−0.989036 + 0.147677i \(0.952820\pi\)
\(930\) 0 0
\(931\) 301.528 0.323875
\(932\) 141.752 0.152095
\(933\) 0 0
\(934\) 44.6449 0.0477996
\(935\) 529.910 830.165i 0.566749 0.887877i
\(936\) 0 0
\(937\) 313.547i 0.334629i −0.985904 0.167315i \(-0.946490\pi\)
0.985904 0.167315i \(-0.0535095\pi\)
\(938\) 18.6485 0.0198811
\(939\) 0 0
\(940\) 388.205 + 247.799i 0.412984 + 0.263616i
\(941\) 508.607i 0.540496i −0.962791 0.270248i \(-0.912894\pi\)
0.962791 0.270248i \(-0.0871058\pi\)
\(942\) 0 0
\(943\) 1129.24i 1.19750i
\(944\) 259.081i 0.274450i
\(945\) 0 0
\(946\) 504.810 0.533625
\(947\) 1008.92 1.06539 0.532695 0.846307i \(-0.321179\pi\)
0.532695 + 0.846307i \(0.321179\pi\)
\(948\) 0 0
\(949\) 739.739 0.779494
\(950\) −473.824 1020.85i −0.498762 1.07458i
\(951\) 0 0
\(952\) 374.547i 0.393431i
\(953\) −918.564 −0.963865 −0.481933 0.876208i \(-0.660065\pi\)
−0.481933 + 0.876208i \(0.660065\pi\)
\(954\) 0 0
\(955\) −458.597 + 718.445i −0.480206 + 0.752298i
\(956\) 704.147i 0.736555i
\(957\) 0 0
\(958\) 897.113i 0.936444i
\(959\) 1254.99i 1.30864i
\(960\) 0 0
\(961\) 1138.40 1.18460
\(962\) 647.537 0.673115
\(963\) 0 0
\(964\) 175.153 0.181694
\(965\) 576.376 902.959i 0.597281 0.935709i
\(966\) 0 0
\(967\) 30.2629i 0.0312957i −0.999878 0.0156479i \(-0.995019\pi\)
0.999878 0.0156479i \(-0.00498107\pi\)
\(968\) −23.6881 −0.0244712
\(969\) 0 0
\(970\) 603.610 945.625i 0.622279 0.974871i
\(971\) 1003.74i 1.03372i 0.856071 + 0.516858i \(0.172899\pi\)
−0.856071 + 0.516858i \(0.827101\pi\)
\(972\) 0 0
\(973\) 233.669i 0.240153i
\(974\) 903.841i 0.927968i
\(975\) 0 0
\(976\) −385.101 −0.394571
\(977\) −784.493 −0.802961 −0.401481 0.915868i \(-0.631504\pi\)
−0.401481 + 0.915868i \(0.631504\pi\)
\(978\) 0 0
\(979\) −1316.99 −1.34524
\(980\) 79.8433 + 50.9655i 0.0814728 + 0.0520056i
\(981\) 0 0
\(982\) 520.245i 0.529781i
\(983\) −53.2332 −0.0541539 −0.0270769 0.999633i \(-0.508620\pi\)
−0.0270769 + 0.999633i \(0.508620\pi\)
\(984\) 0 0
\(985\) 884.274 + 564.449i 0.897740 + 0.573045i
\(986\) 207.977i 0.210930i
\(987\) 0 0
\(988\) 915.691i 0.926813i
\(989\) 518.277i 0.524042i
\(990\) 0 0
\(991\) 1202.09 1.21300 0.606501 0.795082i \(-0.292572\pi\)
0.606501 + 0.795082i \(0.292572\pi\)
\(992\) −259.192 −0.261283
\(993\) 0 0
\(994\) −287.363 −0.289098
\(995\) −1380.98 881.505i −1.38792 0.885934i
\(996\) 0 0
\(997\) 1336.97i 1.34099i −0.741914 0.670495i \(-0.766082\pi\)
0.741914 0.670495i \(-0.233918\pi\)
\(998\) −699.390 −0.700791
\(999\) 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 810.3.b.c.809.15 yes 24
3.2 odd 2 inner 810.3.b.c.809.10 yes 24
5.4 even 2 inner 810.3.b.c.809.9 24
9.2 odd 6 810.3.j.g.539.7 24
9.4 even 3 810.3.j.h.269.1 24
9.5 odd 6 810.3.j.h.269.7 24
9.7 even 3 810.3.j.g.539.1 24
15.14 odd 2 inner 810.3.b.c.809.16 yes 24
45.4 even 6 810.3.j.g.269.7 24
45.14 odd 6 810.3.j.g.269.1 24
45.29 odd 6 810.3.j.h.539.1 24
45.34 even 6 810.3.j.h.539.7 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
810.3.b.c.809.9 24 5.4 even 2 inner
810.3.b.c.809.10 yes 24 3.2 odd 2 inner
810.3.b.c.809.15 yes 24 1.1 even 1 trivial
810.3.b.c.809.16 yes 24 15.14 odd 2 inner
810.3.j.g.269.1 24 45.14 odd 6
810.3.j.g.269.7 24 45.4 even 6
810.3.j.g.539.1 24 9.7 even 3
810.3.j.g.539.7 24 9.2 odd 6
810.3.j.h.269.1 24 9.4 even 3
810.3.j.h.269.7 24 9.5 odd 6
810.3.j.h.539.1 24 45.29 odd 6
810.3.j.h.539.7 24 45.34 even 6