Properties

Label 9075.2.a.r.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

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: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +4.00000 q^{21} -4.00000 q^{23} -3.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -4.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +2.00000 q^{34} -1.00000 q^{36} -8.00000 q^{37} -8.00000 q^{38} +2.00000 q^{39} -12.0000 q^{41} +4.00000 q^{42} -8.00000 q^{43} -4.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +2.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} -12.0000 q^{56} -8.00000 q^{57} -4.00000 q^{58} -8.00000 q^{59} -8.00000 q^{62} +4.00000 q^{63} +7.00000 q^{64} +4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -12.0000 q^{71} -3.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} +8.00000 q^{76} +2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -8.00000 q^{86} -4.00000 q^{87} +6.00000 q^{89} +8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} +5.00000 q^{96} +8.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −8.00000 −1.29777
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 4.00000 0.617213
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) −2.00000 −0.277350
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) −8.00000 −1.05963
\(58\) −4.00000 −0.525226
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 2.00000 0.198030
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −4.00000 −0.377964
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −8.00000 −0.749269
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 2.00000 0.184900
\(118\) −8.00000 −0.736460
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −3.00000 −0.265165
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −32.0000 −2.77475
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −4.00000 −0.340503
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 9.00000 0.742307
\(148\) 8.00000 0.657596
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 24.0000 1.94666
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −8.00000 −0.636446
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −12.0000 −0.925820
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 8.00000 0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 6.00000 0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 7.00000 0.505181
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 4.00000 0.281439
\(203\) −16.0000 −1.12298
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) −4.00000 −0.278019
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −4.00000 −0.274721
\(213\) −12.0000 −0.822226
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) −32.0000 −2.17230
\(218\) 8.00000 0.541828
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −8.00000 −0.536925
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 20.0000 1.33631
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 8.00000 0.529813
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12.0000 0.787839
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) −8.00000 −0.519656
\(238\) 8.00000 0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) −16.0000 −1.01806
\(248\) 24.0000 1.52400
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −4.00000 −0.251976
\(253\) 0 0
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −4.00000 −0.249513 −0.124757 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) −8.00000 −0.498058
\(259\) −32.0000 −1.98838
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) −8.00000 −0.494242
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −32.0000 −1.96205
\(267\) 6.00000 0.367194
\(268\) −4.00000 −0.244339
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −2.00000 −0.121268
\(273\) 8.00000 0.484182
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −16.0000 −0.959616
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 4.00000 0.238197
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −48.0000 −2.83335
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −2.00000 −0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 8.00000 0.458831
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 24.0000 1.36975 0.684876 0.728659i \(-0.259856\pi\)
0.684876 + 0.728659i \(0.259856\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) −6.00000 −0.339683
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −20.0000 −1.12331 −0.561656 0.827371i \(-0.689836\pi\)
−0.561656 + 0.827371i \(0.689836\pi\)
\(318\) 4.00000 0.224309
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −16.0000 −0.891645
\(323\) −16.0000 −0.890264
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 8.00000 0.442401
\(328\) 36.0000 1.98777
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) −8.00000 −0.438397
\(334\) 24.0000 1.31322
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.00000 −0.489535
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) 8.00000 0.431959
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 4.00000 0.214423
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 8.00000 0.423405
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) −36.0000 −1.87918 −0.939592 0.342296i \(-0.888796\pi\)
−0.939592 + 0.342296i \(0.888796\pi\)
\(368\) 4.00000 0.208514
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 16.0000 0.830679
\(372\) 8.00000 0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) −8.00000 −0.412021
\(378\) 4.00000 0.205738
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 12.0000 0.613973
\(383\) 36.0000 1.83951 0.919757 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −8.00000 −0.406663
\(388\) −8.00000 −0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −27.0000 −1.36371
\(393\) −8.00000 −0.403547
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) −16.0000 −0.802008
\(399\) −32.0000 −1.60200
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 4.00000 0.199502
\(403\) −16.0000 −0.797017
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −16.0000 −0.794067
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) 40.0000 1.97787 0.988936 0.148340i \(-0.0473931\pi\)
0.988936 + 0.148340i \(0.0473931\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) −12.0000 −0.591198
\(413\) −32.0000 −1.57462
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 16.0000 0.778868
\(423\) 4.00000 0.194487
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 32.0000 1.53077
\(438\) 2.00000 0.0955637
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 4.00000 0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) −20.0000 −0.945968
\(448\) 28.0000 1.32288
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 24.0000 1.12390
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −22.0000 −1.02799
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) −8.00000 −0.368621
\(472\) 24.0000 1.10469
\(473\) 0 0
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 4.00000 0.183147
\(478\) 24.0000 1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) −8.00000 −0.364390
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 12.0000 0.541002
\(493\) −8.00000 −0.360302
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −48.0000 −2.15309
\(498\) 4.00000 0.179244
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −12.0000 −0.534522
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 4.00000 0.177471
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −11.0000 −0.486136
\(513\) −8.00000 −0.353209
\(514\) −4.00000 −0.176432
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −32.0000 −1.40600
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −4.00000 −0.175075
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 32.0000 1.38738
\(533\) −24.0000 −1.03956
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) −26.0000 −1.12094
\(539\) 0 0
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −16.0000 −0.687259
\(543\) −10.0000 −0.429141
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −4.00000 −0.170872
\(549\) 0 0
\(550\) 0 0
\(551\) 32.0000 1.36325
\(552\) 12.0000 0.510754
\(553\) −32.0000 −1.36078
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) −8.00000 −0.338667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 4.00000 0.167984
\(568\) 36.0000 1.51053
\(569\) −44.0000 −1.84458 −0.922288 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) −48.0000 −2.00348
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −13.0000 −0.540729
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 16.0000 0.663792
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −9.00000 −0.371154
\(589\) 64.0000 2.63707
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 8.00000 0.328798
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) −16.0000 −0.654836
\(598\) −8.00000 −0.327144
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) −32.0000 −1.30422
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) 20.0000 0.811775 0.405887 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) −40.0000 −1.62221
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) −2.00000 −0.0808452
\(613\) −42.0000 −1.69636 −0.848182 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) 24.0000 0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 12.0000 0.482711
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 28.0000 1.12270
\(623\) 24.0000 0.961540
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −16.0000 −0.639489
\(627\) 0 0
\(628\) 8.00000 0.319235
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 24.0000 0.954669
\(633\) 16.0000 0.635943
\(634\) −20.0000 −0.794301
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 12.0000 0.473602
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) −20.0000 −0.783260
\(653\) −28.0000 −1.09572 −0.547862 0.836569i \(-0.684558\pi\)
−0.547862 + 0.836569i \(0.684558\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 12.0000 0.468521
\(657\) 2.00000 0.0780274
\(658\) 16.0000 0.623745
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 6.00000 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(662\) 4.00000 0.155464
\(663\) 4.00000 0.155347
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 16.0000 0.619522
\(668\) −24.0000 −0.928588
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 0 0
\(672\) 20.0000 0.771517
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −12.0000 −0.460857
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 8.00000 0.305888
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) −22.0000 −0.839352
\(688\) 8.00000 0.304997
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −14.0000 −0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 12.0000 0.454859
\(697\) −24.0000 −0.909065
\(698\) −16.0000 −0.605609
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 2.00000 0.0754851
\(703\) 64.0000 2.41381
\(704\) 0 0
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 16.0000 0.601742
\(708\) 8.00000 0.300658
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 32.0000 1.19841
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) −24.0000 −0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 45.0000 1.67473
\(723\) −8.00000 −0.297523
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) −24.0000 −0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 16.0000 0.587378
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −4.00000 −0.144905
\(763\) 32.0000 1.15848
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) −16.0000 −0.577727
\(768\) −17.0000 −0.613435
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) 6.00000 0.215945
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −24.0000 −0.861550
\(777\) −32.0000 −1.14799
\(778\) 18.0000 0.645331
\(779\) 96.0000 3.43956
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) −4.00000 −0.142948
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 0 0
\(794\) −8.00000 −0.283909
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) −32.0000 −1.13279
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) −2.00000 −0.0706225
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −26.0000 −0.915243
\(808\) −12.0000 −0.422159
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 16.0000 0.561490
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) 64.0000 2.23908
\(818\) 40.0000 1.39857
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) −20.0000 −0.698005 −0.349002 0.937122i \(-0.613479\pi\)
−0.349002 + 0.937122i \(0.613479\pi\)
\(822\) 4.00000 0.139516
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) −36.0000 −1.25412
\(825\) 0 0
\(826\) −32.0000 −1.11342
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 4.00000 0.139010
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 14.0000 0.485363
\(833\) 18.0000 0.623663
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 8.00000 0.276355
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −10.0000 −0.344623
\(843\) −12.0000 −0.413302
\(844\) −16.0000 −0.550743
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 12.0000 0.411113
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) −48.0000 −1.63584
\(862\) 24.0000 0.817443
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) −13.0000 −0.441503
\(868\) 32.0000 1.08615
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −24.0000 −0.812743
\(873\) 8.00000 0.270759
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 9.00000 0.303046
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 24.0000 0.805387
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −32.0000 −1.07084
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) −8.00000 −0.267112
\(898\) −30.0000 −1.00111
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) −32.0000 −1.06489
\(904\) 36.0000 1.19734
\(905\) 0 0
\(906\) 0 0
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) 12.0000 0.398234
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −32.0000 −1.05673
\(918\) 2.00000 0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 12.0000 0.395199
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) −28.0000 −0.920137
\(927\) 12.0000 0.394132
\(928\) −20.0000 −0.656532
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −72.0000 −2.35970
\(932\) −6.00000 −0.196537
\(933\) 28.0000 0.916679
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 16.0000 0.522419
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 4.00000 0.130396 0.0651981 0.997872i \(-0.479232\pi\)
0.0651981 + 0.997872i \(0.479232\pi\)
\(942\) −8.00000 −0.260654
\(943\) 48.0000 1.56310
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) 0 0
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 8.00000 0.259828
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −20.0000 −0.648544
\(952\) −24.0000 −0.777844
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −16.0000 −0.515861
\(963\) 12.0000 0.386695
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) 0 0
\(969\) −16.0000 −0.513994
\(970\) 0 0
\(971\) 16.0000 0.513464 0.256732 0.966483i \(-0.417354\pi\)
0.256732 + 0.966483i \(0.417354\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −64.0000 −2.05175
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) 0 0
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 36.0000 1.14764
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 16.0000 0.509286
\(988\) 16.0000 0.509028
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) −40.0000 −1.27000
\(993\) 4.00000 0.126936
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −12.0000 −0.379853
\(999\) −8.00000 −0.253109
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 9075.2.a.r.1.1 1
5.2 odd 4 1815.2.c.b.364.2 yes 2
5.3 odd 4 1815.2.c.b.364.1 yes 2
5.4 even 2 9075.2.a.c.1.1 1
11.10 odd 2 9075.2.a.f.1.1 1
55.32 even 4 1815.2.c.a.364.1 2
55.43 even 4 1815.2.c.a.364.2 yes 2
55.54 odd 2 9075.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.c.a.364.1 2 55.32 even 4
1815.2.c.a.364.2 yes 2 55.43 even 4
1815.2.c.b.364.1 yes 2 5.3 odd 4
1815.2.c.b.364.2 yes 2 5.2 odd 4
9075.2.a.c.1.1 1 5.4 even 2
9075.2.a.f.1.1 1 11.10 odd 2
9075.2.a.o.1.1 1 55.54 odd 2
9075.2.a.r.1.1 1 1.1 even 1 trivial