Properties

Label 2160.3.c.l.1889.2
Level $2160$
Weight $3$
Character 2160.1889
Analytic conductor $58.856$
Analytic rank $0$
Dimension $4$
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] = [2160,3,Mod(1889,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2160.1889");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2160.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.8557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-11}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.2
Root \(3.83776i\) of defining polynomial
Character \(\chi\) \(=\) 2160.1889
Dual form 2160.3.c.l.1889.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.86421 + 4.09833i) q^{5} +7.15439i q^{7} +O(q^{10})\) \(q+(-2.86421 + 4.09833i) q^{5} +7.15439i q^{7} -5.06984i q^{11} -3.12682i q^{13} +8.72842 q^{17} +20.1852 q^{19} +14.7284 q^{23} +(-8.59262 - 23.4769i) q^{25} -39.7995i q^{29} +39.3705 q^{31} +(-29.3210 - 20.4917i) q^{35} -34.8712i q^{37} -13.2665i q^{41} +66.6156i q^{43} +16.9137 q^{47} -2.18525 q^{49} -4.62950 q^{53} +(20.7779 + 14.5211i) q^{55} +25.7738i q^{59} -12.1852 q^{61} +(12.8148 + 8.95587i) q^{65} +106.415i q^{67} -101.487i q^{71} -23.2646i q^{73} +36.2716 q^{77} +66.5557 q^{79} +144.284 q^{83} +(-25.0000 + 35.7719i) q^{85} +154.553i q^{89} +22.3705 q^{91} +(-57.8148 + 82.7258i) q^{95} +175.733i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{5} + 6 q^{17} - 6 q^{19} + 30 q^{23} + 9 q^{25} - 16 q^{31} - 45 q^{35} - 48 q^{47} + 78 q^{49} - 192 q^{53} - 47 q^{55} + 38 q^{61} + 138 q^{65} + 174 q^{77} + 6 q^{79} + 288 q^{83} - 100 q^{85} - 84 q^{91} - 318 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −2.86421 + 4.09833i −0.572842 + 0.819666i
\(6\) 0 0
\(7\) 7.15439i 1.02206i 0.859564 + 0.511028i \(0.170735\pi\)
−0.859564 + 0.511028i \(0.829265\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.06984i 0.460894i −0.973085 0.230447i \(-0.925981\pi\)
0.973085 0.230447i \(-0.0740189\pi\)
\(12\) 0 0
\(13\) 3.12682i 0.240525i −0.992742 0.120262i \(-0.961626\pi\)
0.992742 0.120262i \(-0.0383736\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.72842 0.513436 0.256718 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(18\) 0 0
\(19\) 20.1852 1.06238 0.531191 0.847252i \(-0.321745\pi\)
0.531191 + 0.847252i \(0.321745\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 14.7284 0.640366 0.320183 0.947356i \(-0.396256\pi\)
0.320183 + 0.947356i \(0.396256\pi\)
\(24\) 0 0
\(25\) −8.59262 23.4769i −0.343705 0.939078i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 39.7995i 1.37240i −0.727415 0.686198i \(-0.759278\pi\)
0.727415 0.686198i \(-0.240722\pi\)
\(30\) 0 0
\(31\) 39.3705 1.27002 0.635008 0.772506i \(-0.280997\pi\)
0.635008 + 0.772506i \(0.280997\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −29.3210 20.4917i −0.837744 0.585476i
\(36\) 0 0
\(37\) 34.8712i 0.942465i −0.882009 0.471232i \(-0.843809\pi\)
0.882009 0.471232i \(-0.156191\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 13.2665i 0.323573i −0.986826 0.161787i \(-0.948274\pi\)
0.986826 0.161787i \(-0.0517256\pi\)
\(42\) 0 0
\(43\) 66.6156i 1.54920i 0.632452 + 0.774600i \(0.282049\pi\)
−0.632452 + 0.774600i \(0.717951\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.9137 0.359865 0.179933 0.983679i \(-0.442412\pi\)
0.179933 + 0.983679i \(0.442412\pi\)
\(48\) 0 0
\(49\) −2.18525 −0.0445969
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.62950 −0.0873491 −0.0436746 0.999046i \(-0.513906\pi\)
−0.0436746 + 0.999046i \(0.513906\pi\)
\(54\) 0 0
\(55\) 20.7779 + 14.5211i 0.377780 + 0.264019i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 25.7738i 0.436844i 0.975854 + 0.218422i \(0.0700909\pi\)
−0.975854 + 0.218422i \(0.929909\pi\)
\(60\) 0 0
\(61\) −12.1852 −0.199758 −0.0998791 0.995000i \(-0.531846\pi\)
−0.0998791 + 0.995000i \(0.531846\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.8148 + 8.95587i 0.197150 + 0.137783i
\(66\) 0 0
\(67\) 106.415i 1.58828i 0.607732 + 0.794142i \(0.292080\pi\)
−0.607732 + 0.794142i \(0.707920\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 101.487i 1.42939i −0.699436 0.714695i \(-0.746565\pi\)
0.699436 0.714695i \(-0.253435\pi\)
\(72\) 0 0
\(73\) 23.2646i 0.318694i −0.987223 0.159347i \(-0.949061\pi\)
0.987223 0.159347i \(-0.0509388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 36.2716 0.471060
\(78\) 0 0
\(79\) 66.5557 0.842478 0.421239 0.906950i \(-0.361595\pi\)
0.421239 + 0.906950i \(0.361595\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 144.284 1.73836 0.869182 0.494493i \(-0.164646\pi\)
0.869182 + 0.494493i \(0.164646\pi\)
\(84\) 0 0
\(85\) −25.0000 + 35.7719i −0.294118 + 0.420846i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 154.553i 1.73655i 0.496085 + 0.868274i \(0.334770\pi\)
−0.496085 + 0.868274i \(0.665230\pi\)
\(90\) 0 0
\(91\) 22.3705 0.245830
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −57.8148 + 82.7258i −0.608576 + 0.870798i
\(96\) 0 0
\(97\) 175.733i 1.81168i 0.423621 + 0.905839i \(0.360759\pi\)
−0.423621 + 0.905839i \(0.639241\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 149.483i 1.48003i −0.672591 0.740014i \(-0.734819\pi\)
0.672591 0.740014i \(-0.265181\pi\)
\(102\) 0 0
\(103\) 28.6175i 0.277840i 0.990304 + 0.138920i \(0.0443631\pi\)
−0.990304 + 0.138920i \(0.955637\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −156.297 −1.46072 −0.730359 0.683064i \(-0.760647\pi\)
−0.730359 + 0.683064i \(0.760647\pi\)
\(108\) 0 0
\(109\) 8.55575 0.0784931 0.0392465 0.999230i \(-0.487504\pi\)
0.0392465 + 0.999230i \(0.487504\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −184.370 −1.63160 −0.815799 0.578336i \(-0.803702\pi\)
−0.815799 + 0.578336i \(0.803702\pi\)
\(114\) 0 0
\(115\) −42.1852 + 60.3619i −0.366828 + 0.524886i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 62.4465i 0.524760i
\(120\) 0 0
\(121\) 95.2967 0.787576
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 120.827 + 32.0274i 0.966619 + 0.256219i
\(126\) 0 0
\(127\) 40.7002i 0.320474i 0.987079 + 0.160237i \(0.0512259\pi\)
−0.987079 + 0.160237i \(0.948774\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 80.0236i 0.610867i 0.952213 + 0.305434i \(0.0988014\pi\)
−0.952213 + 0.305434i \(0.901199\pi\)
\(132\) 0 0
\(133\) 144.413i 1.08581i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.0738 0.0954289 0.0477144 0.998861i \(-0.484806\pi\)
0.0477144 + 0.998861i \(0.484806\pi\)
\(138\) 0 0
\(139\) −144.370 −1.03864 −0.519318 0.854581i \(-0.673814\pi\)
−0.519318 + 0.854581i \(0.673814\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −15.8525 −0.110857
\(144\) 0 0
\(145\) 163.111 + 113.994i 1.12491 + 0.786166i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 54.2498i 0.364093i 0.983290 + 0.182046i \(0.0582721\pi\)
−0.983290 + 0.182046i \(0.941728\pi\)
\(150\) 0 0
\(151\) −21.1852 −0.140300 −0.0701498 0.997536i \(-0.522348\pi\)
−0.0701498 + 0.997536i \(0.522348\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −112.765 + 161.353i −0.727518 + 1.04099i
\(156\) 0 0
\(157\) 105.939i 0.674770i 0.941367 + 0.337385i \(0.109542\pi\)
−0.941367 + 0.337385i \(0.890458\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 105.373i 0.654489i
\(162\) 0 0
\(163\) 154.746i 0.949361i −0.880158 0.474680i \(-0.842564\pi\)
0.880158 0.474680i \(-0.157436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 151.667 0.908187 0.454094 0.890954i \(-0.349963\pi\)
0.454094 + 0.890954i \(0.349963\pi\)
\(168\) 0 0
\(169\) 159.223 0.942148
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −311.198 −1.79883 −0.899416 0.437094i \(-0.856008\pi\)
−0.899416 + 0.437094i \(0.856008\pi\)
\(174\) 0 0
\(175\) 167.963 61.4750i 0.959789 0.351285i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 211.170i 1.17972i −0.807505 0.589861i \(-0.799183\pi\)
0.807505 0.589861i \(-0.200817\pi\)
\(180\) 0 0
\(181\) 149.297 0.824844 0.412422 0.910993i \(-0.364683\pi\)
0.412422 + 0.910993i \(0.364683\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 142.914 + 99.8783i 0.772506 + 0.539883i
\(186\) 0 0
\(187\) 44.2517i 0.236640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.1654i 0.126520i −0.997997 0.0632602i \(-0.979850\pi\)
0.997997 0.0632602i \(-0.0201498\pi\)
\(192\) 0 0
\(193\) 332.653i 1.72359i 0.507255 + 0.861796i \(0.330660\pi\)
−0.507255 + 0.861796i \(0.669340\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 279.568 1.41913 0.709564 0.704641i \(-0.248892\pi\)
0.709564 + 0.704641i \(0.248892\pi\)
\(198\) 0 0
\(199\) 161.185 0.809976 0.404988 0.914322i \(-0.367276\pi\)
0.404988 + 0.914322i \(0.367276\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 284.741 1.40266
\(204\) 0 0
\(205\) 54.3705 + 37.9980i 0.265222 + 0.185356i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 102.336i 0.489646i
\(210\) 0 0
\(211\) 92.1852 0.436897 0.218448 0.975848i \(-0.429900\pi\)
0.218448 + 0.975848i \(0.429900\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −273.013 190.801i −1.26983 0.887446i
\(216\) 0 0
\(217\) 281.672i 1.29803i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 27.2922i 0.123494i
\(222\) 0 0
\(223\) 197.672i 0.886422i 0.896417 + 0.443211i \(0.146161\pi\)
−0.896417 + 0.443211i \(0.853839\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −32.7284 −0.144178 −0.0720890 0.997398i \(-0.522967\pi\)
−0.0720890 + 0.997398i \(0.522967\pi\)
\(228\) 0 0
\(229\) 350.926 1.53243 0.766215 0.642585i \(-0.222138\pi\)
0.766215 + 0.642585i \(0.222138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −399.827 −1.71600 −0.857999 0.513652i \(-0.828292\pi\)
−0.857999 + 0.513652i \(0.828292\pi\)
\(234\) 0 0
\(235\) −48.4443 + 69.3178i −0.206146 + 0.294969i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 363.690i 1.52172i 0.648919 + 0.760858i \(0.275221\pi\)
−0.648919 + 0.760858i \(0.724779\pi\)
\(240\) 0 0
\(241\) 227.667 0.944677 0.472339 0.881417i \(-0.343410\pi\)
0.472339 + 0.881417i \(0.343410\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.25901 8.95587i 0.0255470 0.0365546i
\(246\) 0 0
\(247\) 63.1157i 0.255529i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 227.988i 0.908319i −0.890920 0.454160i \(-0.849940\pi\)
0.890920 0.454160i \(-0.150060\pi\)
\(252\) 0 0
\(253\) 74.6707i 0.295141i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 456.038 1.77447 0.887233 0.461322i \(-0.152625\pi\)
0.887233 + 0.461322i \(0.152625\pi\)
\(258\) 0 0
\(259\) 249.482 0.963251
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 186.543 0.709290 0.354645 0.935001i \(-0.384602\pi\)
0.354645 + 0.935001i \(0.384602\pi\)
\(264\) 0 0
\(265\) 13.2599 18.9732i 0.0500372 0.0715971i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 337.157i 1.25337i −0.779272 0.626686i \(-0.784411\pi\)
0.779272 0.626686i \(-0.215589\pi\)
\(270\) 0 0
\(271\) −58.2590 −0.214978 −0.107489 0.994206i \(-0.534281\pi\)
−0.107489 + 0.994206i \(0.534281\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −119.024 + 43.5632i −0.432816 + 0.158412i
\(276\) 0 0
\(277\) 332.229i 1.19938i −0.800232 0.599691i \(-0.795290\pi\)
0.800232 0.599691i \(-0.204710\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 112.296i 0.399629i −0.979834 0.199814i \(-0.935966\pi\)
0.979834 0.199814i \(-0.0640339\pi\)
\(282\) 0 0
\(283\) 325.975i 1.15185i 0.817501 + 0.575927i \(0.195359\pi\)
−0.817501 + 0.575927i \(0.804641\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 94.9137 0.330710
\(288\) 0 0
\(289\) −212.815 −0.736383
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −71.9623 −0.245605 −0.122803 0.992431i \(-0.539188\pi\)
−0.122803 + 0.992431i \(0.539188\pi\)
\(294\) 0 0
\(295\) −105.630 73.8215i −0.358066 0.250242i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 46.0531i 0.154024i
\(300\) 0 0
\(301\) −476.593 −1.58337
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 34.9011 49.9392i 0.114430 0.163735i
\(306\) 0 0
\(307\) 245.900i 0.800977i 0.916302 + 0.400488i \(0.131159\pi\)
−0.916302 + 0.400488i \(0.868841\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 69.3693i 0.223053i −0.993761 0.111526i \(-0.964426\pi\)
0.993761 0.111526i \(-0.0355739\pi\)
\(312\) 0 0
\(313\) 218.659i 0.698592i 0.937013 + 0.349296i \(0.113579\pi\)
−0.937013 + 0.349296i \(0.886421\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 74.4317 0.234800 0.117400 0.993085i \(-0.462544\pi\)
0.117400 + 0.993085i \(0.462544\pi\)
\(318\) 0 0
\(319\) −201.777 −0.632530
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 176.185 0.545465
\(324\) 0 0
\(325\) −73.4082 + 26.8676i −0.225871 + 0.0826696i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 121.007i 0.367802i
\(330\) 0 0
\(331\) 220.223 0.665326 0.332663 0.943046i \(-0.392053\pi\)
0.332663 + 0.943046i \(0.392053\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −436.124 304.795i −1.30186 0.909835i
\(336\) 0 0
\(337\) 24.6415i 0.0731203i −0.999331 0.0365601i \(-0.988360\pi\)
0.999331 0.0365601i \(-0.0116400\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 199.602i 0.585343i
\(342\) 0 0
\(343\) 334.931i 0.976475i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 147.556 0.425233 0.212616 0.977136i \(-0.431802\pi\)
0.212616 + 0.977136i \(0.431802\pi\)
\(348\) 0 0
\(349\) −503.149 −1.44169 −0.720844 0.693097i \(-0.756246\pi\)
−0.720844 + 0.693097i \(0.756246\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −236.716 −0.670583 −0.335292 0.942114i \(-0.608835\pi\)
−0.335292 + 0.942114i \(0.608835\pi\)
\(354\) 0 0
\(355\) 415.926 + 290.679i 1.17162 + 0.818815i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 195.871i 0.545601i −0.962071 0.272800i \(-0.912050\pi\)
0.962071 0.272800i \(-0.0879499\pi\)
\(360\) 0 0
\(361\) 46.4443 0.128654
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 95.3462 + 66.6348i 0.261222 + 0.182561i
\(366\) 0 0
\(367\) 339.756i 0.925766i 0.886419 + 0.462883i \(0.153185\pi\)
−0.886419 + 0.462883i \(0.846815\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 33.1213i 0.0892756i
\(372\) 0 0
\(373\) 540.980i 1.45035i −0.688566 0.725174i \(-0.741759\pi\)
0.688566 0.725174i \(-0.258241\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −124.446 −0.330095
\(378\) 0 0
\(379\) 432.926 1.14229 0.571143 0.820851i \(-0.306500\pi\)
0.571143 + 0.820851i \(0.306500\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 478.977 1.25059 0.625296 0.780388i \(-0.284978\pi\)
0.625296 + 0.780388i \(0.284978\pi\)
\(384\) 0 0
\(385\) −103.889 + 148.653i −0.269843 + 0.386112i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 339.190i 0.871954i 0.899958 + 0.435977i \(0.143597\pi\)
−0.899958 + 0.435977i \(0.856403\pi\)
\(390\) 0 0
\(391\) 128.556 0.328787
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −190.630 + 272.767i −0.482606 + 0.690550i
\(396\) 0 0
\(397\) 113.042i 0.284740i −0.989814 0.142370i \(-0.954528\pi\)
0.989814 0.142370i \(-0.0454723\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 477.774i 1.19146i 0.803186 + 0.595728i \(0.203136\pi\)
−0.803186 + 0.595728i \(0.796864\pi\)
\(402\) 0 0
\(403\) 123.105i 0.305470i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −176.791 −0.434377
\(408\) 0 0
\(409\) 677.370 1.65616 0.828081 0.560608i \(-0.189433\pi\)
0.828081 + 0.560608i \(0.189433\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −184.396 −0.446479
\(414\) 0 0
\(415\) −413.260 + 591.324i −0.995807 + 1.42488i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 344.170i 0.821408i −0.911769 0.410704i \(-0.865283\pi\)
0.911769 0.410704i \(-0.134717\pi\)
\(420\) 0 0
\(421\) 29.5918 0.0702892 0.0351446 0.999382i \(-0.488811\pi\)
0.0351446 + 0.999382i \(0.488811\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −75.0000 204.917i −0.176471 0.482157i
\(426\) 0 0
\(427\) 87.1780i 0.204164i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 724.253i 1.68040i 0.542276 + 0.840201i \(0.317563\pi\)
−0.542276 + 0.840201i \(0.682437\pi\)
\(432\) 0 0
\(433\) 96.6100i 0.223118i −0.993758 0.111559i \(-0.964416\pi\)
0.993758 0.111559i \(-0.0355844\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 297.297 0.680313
\(438\) 0 0
\(439\) −101.223 −0.230576 −0.115288 0.993332i \(-0.536779\pi\)
−0.115288 + 0.993332i \(0.536779\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −377.520 −0.852189 −0.426094 0.904679i \(-0.640111\pi\)
−0.426094 + 0.904679i \(0.640111\pi\)
\(444\) 0 0
\(445\) −633.408 442.671i −1.42339 0.994767i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 654.974i 1.45874i −0.684120 0.729369i \(-0.739814\pi\)
0.684120 0.729369i \(-0.260186\pi\)
\(450\) 0 0
\(451\) −67.2590 −0.149133
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −64.0738 + 91.6817i −0.140821 + 0.201498i
\(456\) 0 0
\(457\) 321.574i 0.703664i −0.936063 0.351832i \(-0.885559\pi\)
0.936063 0.351832i \(-0.114441\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 465.511i 1.00979i 0.863182 + 0.504893i \(0.168468\pi\)
−0.863182 + 0.504893i \(0.831532\pi\)
\(462\) 0 0
\(463\) 250.880i 0.541857i −0.962600 0.270928i \(-0.912669\pi\)
0.962600 0.270928i \(-0.0873307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 246.259 0.527321 0.263661 0.964616i \(-0.415070\pi\)
0.263661 + 0.964616i \(0.415070\pi\)
\(468\) 0 0
\(469\) −761.334 −1.62331
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 337.730 0.714017
\(474\) 0 0
\(475\) −173.444 473.888i −0.365146 0.997659i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 175.012i 0.365370i −0.983172 0.182685i \(-0.941521\pi\)
0.983172 0.182685i \(-0.0584788\pi\)
\(480\) 0 0
\(481\) −109.036 −0.226686
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −720.211 503.335i −1.48497 1.03781i
\(486\) 0 0
\(487\) 28.9906i 0.0595289i −0.999557 0.0297645i \(-0.990524\pi\)
0.999557 0.0297645i \(-0.00947573\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 974.373i 1.98447i 0.124388 + 0.992234i \(0.460303\pi\)
−0.124388 + 0.992234i \(0.539697\pi\)
\(492\) 0 0
\(493\) 347.387i 0.704638i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 726.075 1.46092
\(498\) 0 0
\(499\) −626.185 −1.25488 −0.627440 0.778665i \(-0.715897\pi\)
−0.627440 + 0.778665i \(0.715897\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 71.5197 0.142186 0.0710932 0.997470i \(-0.477351\pi\)
0.0710932 + 0.997470i \(0.477351\pi\)
\(504\) 0 0
\(505\) 612.630 + 428.150i 1.21313 + 0.847822i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 932.296i 1.83162i 0.401608 + 0.915812i \(0.368451\pi\)
−0.401608 + 0.915812i \(0.631549\pi\)
\(510\) 0 0
\(511\) 166.444 0.325723
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −117.284 81.9666i −0.227736 0.159158i
\(516\) 0 0
\(517\) 85.7495i 0.165860i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 373.070i 0.716066i −0.933709 0.358033i \(-0.883448\pi\)
0.933709 0.358033i \(-0.116552\pi\)
\(522\) 0 0
\(523\) 828.480i 1.58409i −0.610461 0.792046i \(-0.709016\pi\)
0.610461 0.792046i \(-0.290984\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 343.642 0.652072
\(528\) 0 0
\(529\) −312.074 −0.589931
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −41.4820 −0.0778274
\(534\) 0 0
\(535\) 447.666 640.556i 0.836760 1.19730i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 11.0789i 0.0205545i
\(540\) 0 0
\(541\) −50.5935 −0.0935184 −0.0467592 0.998906i \(-0.514889\pi\)
−0.0467592 + 0.998906i \(0.514889\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −24.5054 + 35.0643i −0.0449641 + 0.0643381i
\(546\) 0 0
\(547\) 485.070i 0.886782i −0.896328 0.443391i \(-0.853775\pi\)
0.896328 0.443391i \(-0.146225\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 803.363i 1.45801i
\(552\) 0 0
\(553\) 476.166i 0.861059i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −198.372 −0.356144 −0.178072 0.984017i \(-0.556986\pi\)
−0.178072 + 0.984017i \(0.556986\pi\)
\(558\) 0 0
\(559\) 208.295 0.372621
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −660.840 −1.17378 −0.586892 0.809666i \(-0.699649\pi\)
−0.586892 + 0.809666i \(0.699649\pi\)
\(564\) 0 0
\(565\) 528.075 755.611i 0.934647 1.33736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 162.994i 0.286457i −0.989690 0.143229i \(-0.954252\pi\)
0.989690 0.143229i \(-0.0457484\pi\)
\(570\) 0 0
\(571\) −262.631 −0.459950 −0.229975 0.973197i \(-0.573864\pi\)
−0.229975 + 0.973197i \(0.573864\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −126.556 345.778i −0.220097 0.601353i
\(576\) 0 0
\(577\) 532.551i 0.922966i 0.887149 + 0.461483i \(0.152682\pi\)
−0.887149 + 0.461483i \(0.847318\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1032.26i 1.77670i
\(582\) 0 0
\(583\) 23.4708i 0.0402587i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 927.593 1.58023 0.790114 0.612960i \(-0.210021\pi\)
0.790114 + 0.612960i \(0.210021\pi\)
\(588\) 0 0
\(589\) 794.703 1.34924
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −890.336 −1.50141 −0.750705 0.660638i \(-0.770286\pi\)
−0.750705 + 0.660638i \(0.770286\pi\)
\(594\) 0 0
\(595\) −255.926 178.860i −0.430128 0.300604i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1036.76i 1.73081i 0.501073 + 0.865405i \(0.332939\pi\)
−0.501073 + 0.865405i \(0.667061\pi\)
\(600\) 0 0
\(601\) 865.816 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −272.950 + 390.558i −0.451157 + 0.645550i
\(606\) 0 0
\(607\) 708.606i 1.16739i −0.811973 0.583695i \(-0.801606\pi\)
0.811973 0.583695i \(-0.198394\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 52.8860i 0.0865565i
\(612\) 0 0
\(613\) 677.814i 1.10573i −0.833270 0.552866i \(-0.813534\pi\)
0.833270 0.552866i \(-0.186466\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 474.063 0.768335 0.384168 0.923263i \(-0.374488\pi\)
0.384168 + 0.923263i \(0.374488\pi\)
\(618\) 0 0
\(619\) −323.187 −0.522111 −0.261056 0.965324i \(-0.584071\pi\)
−0.261056 + 0.965324i \(0.584071\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1105.73 −1.77485
\(624\) 0 0
\(625\) −477.334 + 403.457i −0.763734 + 0.645531i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 304.370i 0.483896i
\(630\) 0 0
\(631\) −977.669 −1.54940 −0.774698 0.632331i \(-0.782098\pi\)
−0.774698 + 0.632331i \(0.782098\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −166.803 116.574i −0.262682 0.183581i
\(636\) 0 0
\(637\) 6.83288i 0.0107267i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 245.231i 0.382575i −0.981534 0.191288i \(-0.938734\pi\)
0.981534 0.191288i \(-0.0612663\pi\)
\(642\) 0 0
\(643\) 244.471i 0.380204i −0.981764 0.190102i \(-0.939118\pi\)
0.981764 0.190102i \(-0.0608819\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 866.779 1.33969 0.669844 0.742501i \(-0.266361\pi\)
0.669844 + 0.742501i \(0.266361\pi\)
\(648\) 0 0
\(649\) 130.669 0.201339
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 641.198 0.981926 0.490963 0.871180i \(-0.336645\pi\)
0.490963 + 0.871180i \(0.336645\pi\)
\(654\) 0 0
\(655\) −327.963 229.204i −0.500707 0.349930i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 771.490i 1.17070i −0.810781 0.585349i \(-0.800958\pi\)
0.810781 0.585349i \(-0.199042\pi\)
\(660\) 0 0
\(661\) −796.075 −1.20435 −0.602175 0.798364i \(-0.705699\pi\)
−0.602175 + 0.798364i \(0.705699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −591.852 413.629i −0.890004 0.621999i
\(666\) 0 0
\(667\) 586.184i 0.878836i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 61.7772i 0.0920674i
\(672\) 0 0
\(673\) 698.325i 1.03763i 0.854887 + 0.518815i \(0.173627\pi\)
−0.854887 + 0.518815i \(0.826373\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 433.655 0.640553 0.320277 0.947324i \(-0.396224\pi\)
0.320277 + 0.947324i \(0.396224\pi\)
\(678\) 0 0
\(679\) −1257.26 −1.85164
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1112.83 −1.62933 −0.814663 0.579935i \(-0.803078\pi\)
−0.814663 + 0.579935i \(0.803078\pi\)
\(684\) 0 0
\(685\) −37.4460 + 53.5806i −0.0546656 + 0.0782198i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.4756i 0.0210096i
\(690\) 0 0
\(691\) −497.743 −0.720322 −0.360161 0.932890i \(-0.617278\pi\)
−0.360161 + 0.932890i \(0.617278\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 413.507 591.678i 0.594974 0.851335i
\(696\) 0 0
\(697\) 115.796i 0.166134i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 660.713i 0.942529i −0.881992 0.471264i \(-0.843798\pi\)
0.881992 0.471264i \(-0.156202\pi\)
\(702\) 0 0
\(703\) 703.884i 1.00126i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1069.46 1.51267
\(708\) 0 0
\(709\) −1013.56 −1.42956 −0.714780 0.699350i \(-0.753473\pi\)
−0.714780 + 0.699350i \(0.753473\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 579.865 0.813275
\(714\) 0 0
\(715\) 45.4048 64.9687i 0.0635032 0.0908653i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1067.57i 1.48480i 0.669955 + 0.742402i \(0.266313\pi\)
−0.669955 + 0.742402i \(0.733687\pi\)
\(720\) 0 0
\(721\) −204.741 −0.283968
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −934.370 + 341.982i −1.28879 + 0.471699i
\(726\) 0 0
\(727\) 878.561i 1.20847i −0.796804 0.604237i \(-0.793478\pi\)
0.796804 0.604237i \(-0.206522\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 581.448i 0.795415i
\(732\) 0 0
\(733\) 76.9483i 0.104977i −0.998622 0.0524886i \(-0.983285\pi\)
0.998622 0.0524886i \(-0.0167153\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 539.507 0.732031
\(738\) 0 0
\(739\) 613.815 0.830602 0.415301 0.909684i \(-0.363676\pi\)
0.415301 + 0.909684i \(0.363676\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −331.036 −0.445540 −0.222770 0.974871i \(-0.571510\pi\)
−0.222770 + 0.974871i \(0.571510\pi\)
\(744\) 0 0
\(745\) −222.334 155.383i −0.298434 0.208567i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1118.21i 1.49293i
\(750\) 0 0
\(751\) 496.852 0.661588 0.330794 0.943703i \(-0.392683\pi\)
0.330794 + 0.943703i \(0.392683\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 60.6790 86.8241i 0.0803695 0.114999i
\(756\) 0 0
\(757\) 1234.26i 1.63046i −0.579135 0.815232i \(-0.696610\pi\)
0.579135 0.815232i \(-0.303390\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 69.3693i 0.0911555i 0.998961 + 0.0455777i \(0.0145129\pi\)
−0.998961 + 0.0455777i \(0.985487\pi\)
\(762\) 0 0
\(763\) 61.2111i 0.0802243i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 80.5901 0.105072
\(768\) 0 0
\(769\) 1110.85 1.44454 0.722271 0.691610i \(-0.243098\pi\)
0.722271 + 0.691610i \(0.243098\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 733.074 0.948349 0.474174 0.880431i \(-0.342747\pi\)
0.474174 + 0.880431i \(0.342747\pi\)
\(774\) 0 0
\(775\) −338.296 924.299i −0.436511 1.19264i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 267.788i 0.343758i
\(780\) 0 0
\(781\) −514.521 −0.658798
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −434.173 303.431i −0.553086 0.386536i
\(786\) 0 0
\(787\) 1004.84i 1.27680i 0.769703 + 0.638402i \(0.220404\pi\)
−0.769703 + 0.638402i \(0.779596\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1319.06i 1.66758i
\(792\) 0 0
\(793\) 38.1011i 0.0480468i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 621.101 0.779298 0.389649 0.920963i \(-0.372596\pi\)
0.389649 + 0.920963i \(0.372596\pi\)
\(798\) 0 0
\(799\) 147.630 0.184768
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −117.948 −0.146884
\(804\) 0 0
\(805\) −431.852 301.810i −0.536463 0.374919i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 276.049i 0.341222i 0.985338 + 0.170611i \(0.0545742\pi\)
−0.985338 + 0.170611i \(0.945426\pi\)
\(810\) 0 0
\(811\) −1024.08 −1.26273 −0.631366 0.775485i \(-0.717505\pi\)
−0.631366 + 0.775485i \(0.717505\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 634.200 + 443.224i 0.778159 + 0.543833i
\(816\) 0 0
\(817\) 1344.65i 1.64584i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.43362i 0.00783632i 0.999992 + 0.00391816i \(0.00124719\pi\)
−0.999992 + 0.00391816i \(0.998753\pi\)
\(822\) 0 0
\(823\) 282.727i 0.343532i −0.985138 0.171766i \(-0.945053\pi\)
0.985138 0.171766i \(-0.0549474\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 223.567 0.270334 0.135167 0.990823i \(-0.456843\pi\)
0.135167 + 0.990823i \(0.456843\pi\)
\(828\) 0 0
\(829\) −1355.63 −1.63526 −0.817631 0.575742i \(-0.804713\pi\)
−0.817631 + 0.575742i \(0.804713\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −19.0738 −0.0228977
\(834\) 0 0
\(835\) −434.407 + 621.582i −0.520247 + 0.744410i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 643.715i 0.767241i −0.923491 0.383620i \(-0.874677\pi\)
0.923491 0.383620i \(-0.125323\pi\)
\(840\) 0 0
\(841\) −743.000 −0.883472
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −456.048 + 652.548i −0.539701 + 0.772247i
\(846\) 0 0
\(847\) 681.790i 0.804947i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 513.597i 0.603522i
\(852\) 0 0
\(853\) 964.735i 1.13099i −0.824751 0.565495i \(-0.808685\pi\)
0.824751 0.565495i \(-0.191315\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −455.077 −0.531012 −0.265506 0.964109i \(-0.585539\pi\)
−0.265506 + 0.964109i \(0.585539\pi\)
\(858\) 0 0
\(859\) −1086.78 −1.26517 −0.632584 0.774492i \(-0.718006\pi\)
−0.632584 + 0.774492i \(0.718006\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1273.67 −1.47586 −0.737930 0.674877i \(-0.764197\pi\)
−0.737930 + 0.674877i \(0.764197\pi\)
\(864\) 0 0
\(865\) 891.335 1275.39i 1.03045 1.47444i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 337.427i 0.388293i
\(870\) 0 0
\(871\) 332.741 0.382022
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −229.137 + 864.445i −0.261870 + 0.987938i
\(876\) 0 0
\(877\) 1169.03i 1.33299i −0.745509 0.666496i \(-0.767793\pi\)
0.745509 0.666496i \(-0.232207\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 547.812i 0.621808i −0.950441 0.310904i \(-0.899368\pi\)
0.950441 0.310904i \(-0.100632\pi\)
\(882\) 0 0
\(883\) 494.077i 0.559544i 0.960066 + 0.279772i \(0.0902589\pi\)
−0.960066 + 0.279772i \(0.909741\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 919.667 1.03683 0.518414 0.855129i \(-0.326522\pi\)
0.518414 + 0.855129i \(0.326522\pi\)
\(888\) 0 0
\(889\) −291.185 −0.327542
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 341.407 0.382314
\(894\) 0 0
\(895\) 865.445 + 604.835i 0.966978 + 0.675794i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1566.93i 1.74297i
\(900\) 0 0
\(901\) −40.4082 −0.0448482
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −427.617 + 611.867i −0.472505 + 0.676097i
\(906\) 0 0
\(907\) 142.135i 0.156709i 0.996926 + 0.0783547i \(0.0249667\pi\)
−0.996926 + 0.0783547i \(0.975033\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.6341i 0.0171615i 0.999963 + 0.00858074i \(0.00273137\pi\)
−0.999963 + 0.00858074i \(0.997269\pi\)
\(912\) 0 0
\(913\) 731.497i 0.801202i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −572.520 −0.624340
\(918\) 0 0
\(919\) 1549.48 1.68605 0.843027 0.537871i \(-0.180771\pi\)
0.843027 + 0.537871i \(0.180771\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −317.331 −0.343804
\(924\) 0 0
\(925\) −818.669 + 299.635i −0.885048 + 0.323930i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 982.480i 1.05757i 0.848757 + 0.528784i \(0.177352\pi\)
−0.848757 + 0.528784i \(0.822648\pi\)
\(930\) 0 0
\(931\) −44.1098 −0.0473789
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 181.358 + 126.746i 0.193966 + 0.135557i
\(936\) 0 0
\(937\) 329.526i 0.351682i 0.984419 + 0.175841i \(0.0562645\pi\)
−0.984419 + 0.175841i \(0.943735\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 939.888i 0.998819i −0.866366 0.499409i \(-0.833550\pi\)
0.866366 0.499409i \(-0.166450\pi\)
\(942\) 0 0
\(943\) 195.395i 0.207205i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1456.78 1.53831 0.769155 0.639062i \(-0.220677\pi\)
0.769155 + 0.639062i \(0.220677\pi\)
\(948\) 0 0
\(949\) −72.7444 −0.0766538
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1391.14 1.45975 0.729874 0.683582i \(-0.239579\pi\)
0.729874 + 0.683582i \(0.239579\pi\)
\(954\) 0 0
\(955\) 99.0377 + 69.2147i 0.103704 + 0.0724761i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 93.5347i 0.0975336i
\(960\) 0 0
\(961\) 589.036 0.612941
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1363.32 952.788i −1.41277 0.987345i
\(966\) 0 0
\(967\) 11.9796i 0.0123884i −0.999981 0.00619421i \(-0.998028\pi\)
0.999981 0.00619421i \(-0.00197169\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 504.732i 0.519806i −0.965635 0.259903i \(-0.916309\pi\)
0.965635 0.259903i \(-0.0836906\pi\)
\(972\) 0 0
\(973\) 1032.88i 1.06154i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 529.554 0.542021 0.271010 0.962576i \(-0.412642\pi\)
0.271010 + 0.962576i \(0.412642\pi\)
\(978\) 0 0
\(979\) 783.557 0.800365
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1110.66 −1.12986 −0.564932 0.825137i \(-0.691098\pi\)
−0.564932 + 0.825137i \(0.691098\pi\)
\(984\) 0 0
\(985\) −800.742 + 1145.76i −0.812936 + 1.16321i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 981.142i 0.992054i
\(990\) 0 0
\(991\) 1048.11 1.05763 0.528817 0.848736i \(-0.322636\pi\)
0.528817 + 0.848736i \(0.322636\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −461.668 + 660.590i −0.463988 + 0.663910i
\(996\) 0 0
\(997\) 1596.81i 1.60161i −0.598925 0.800805i \(-0.704405\pi\)
0.598925 0.800805i \(-0.295595\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2160.3.c.l.1889.2 4
3.2 odd 2 2160.3.c.h.1889.3 4
4.3 odd 2 540.3.b.b.269.2 yes 4
5.4 even 2 2160.3.c.h.1889.4 4
12.11 even 2 540.3.b.a.269.3 4
15.14 odd 2 inner 2160.3.c.l.1889.1 4
20.3 even 4 2700.3.g.s.701.2 8
20.7 even 4 2700.3.g.s.701.8 8
20.19 odd 2 540.3.b.a.269.4 yes 4
36.7 odd 6 1620.3.t.a.1349.2 8
36.11 even 6 1620.3.t.d.1349.3 8
36.23 even 6 1620.3.t.d.269.1 8
36.31 odd 6 1620.3.t.a.269.4 8
60.23 odd 4 2700.3.g.s.701.1 8
60.47 odd 4 2700.3.g.s.701.7 8
60.59 even 2 540.3.b.b.269.1 yes 4
180.59 even 6 1620.3.t.a.269.2 8
180.79 odd 6 1620.3.t.d.1349.1 8
180.119 even 6 1620.3.t.a.1349.4 8
180.139 odd 6 1620.3.t.d.269.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
540.3.b.a.269.3 4 12.11 even 2
540.3.b.a.269.4 yes 4 20.19 odd 2
540.3.b.b.269.1 yes 4 60.59 even 2
540.3.b.b.269.2 yes 4 4.3 odd 2
1620.3.t.a.269.2 8 180.59 even 6
1620.3.t.a.269.4 8 36.31 odd 6
1620.3.t.a.1349.2 8 36.7 odd 6
1620.3.t.a.1349.4 8 180.119 even 6
1620.3.t.d.269.1 8 36.23 even 6
1620.3.t.d.269.3 8 180.139 odd 6
1620.3.t.d.1349.1 8 180.79 odd 6
1620.3.t.d.1349.3 8 36.11 even 6
2160.3.c.h.1889.3 4 3.2 odd 2
2160.3.c.h.1889.4 4 5.4 even 2
2160.3.c.l.1889.1 4 15.14 odd 2 inner
2160.3.c.l.1889.2 4 1.1 even 1 trivial
2700.3.g.s.701.1 8 60.23 odd 4
2700.3.g.s.701.2 8 20.3 even 4
2700.3.g.s.701.7 8 60.47 odd 4
2700.3.g.s.701.8 8 20.7 even 4