Properties

Label 4680.2.l.f.2809.4
Level $4680$
Weight $2$
Character 4680.2809
Analytic conductor $37.370$
Analytic rank $0$
Dimension $8$
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] = [4680,2,Mod(2809,4680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4680.2809");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.l (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: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2809.4
Root \(1.69230i\) of defining polynomial
Character \(\chi\) \(=\) 4680.2809
Dual form 4680.2.l.f.2809.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.489528 + 2.18183i) q^{5} -2.82843i q^{7} +O(q^{10})\) \(q+(-0.489528 + 2.18183i) q^{5} -2.82843i q^{7} +2.97906 q^{11} +1.00000i q^{13} -1.44383i q^{17} -4.17113 q^{19} +6.21302i q^{23} +(-4.52072 - 2.13613i) q^{25} -0.828427 q^{29} +1.65685 q^{31} +(6.17113 + 1.38459i) q^{35} -0.0418875i q^{37} +4.36365 q^{41} +8.99956i q^{43} +4.40554i q^{47} -1.00000 q^{49} -6.72730i q^{53} +(-1.45833 + 6.49978i) q^{55} -2.97906 q^{59} +6.27226 q^{61} +(-2.18183 - 0.489528i) q^{65} +0.727302i q^{67} +8.32176 q^{71} +6.87031i q^{73} -8.42604i q^{77} -6.11190 q^{79} +14.7059i q^{83} +(3.15019 + 0.706797i) q^{85} +14.7478 q^{89} +2.82843 q^{91} +(2.04189 - 9.10069i) q^{95} -6.78654i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{11} + 24 q^{19} - 4 q^{25} + 16 q^{29} - 32 q^{31} - 8 q^{35} + 8 q^{41} - 8 q^{49} - 36 q^{55} - 16 q^{59} + 24 q^{61} - 4 q^{65} + 24 q^{71} + 24 q^{79} - 40 q^{85} - 8 q^{89} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

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
\(5\) −0.489528 + 2.18183i −0.218924 + 0.975742i
\(6\) 0 0
\(7\) 2.82843i 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.97906 0.898219 0.449110 0.893477i \(-0.351741\pi\)
0.449110 + 0.893477i \(0.351741\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.44383i 0.350181i −0.984552 0.175090i \(-0.943978\pi\)
0.984552 0.175090i \(-0.0560218\pi\)
\(18\) 0 0
\(19\) −4.17113 −0.956924 −0.478462 0.878108i \(-0.658806\pi\)
−0.478462 + 0.878108i \(0.658806\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.21302i 1.29550i 0.761851 + 0.647752i \(0.224291\pi\)
−0.761851 + 0.647752i \(0.775709\pi\)
\(24\) 0 0
\(25\) −4.52072 2.13613i −0.904145 0.427226i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 1.65685 0.297580 0.148790 0.988869i \(-0.452462\pi\)
0.148790 + 0.988869i \(0.452462\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.17113 + 1.38459i 1.04311 + 0.234039i
\(36\) 0 0
\(37\) 0.0418875i 0.00688626i −0.999994 0.00344313i \(-0.998904\pi\)
0.999994 0.00344313i \(-0.00109599\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.36365 0.681488 0.340744 0.940156i \(-0.389321\pi\)
0.340744 + 0.940156i \(0.389321\pi\)
\(42\) 0 0
\(43\) 8.99956i 1.37242i 0.727403 + 0.686210i \(0.240727\pi\)
−0.727403 + 0.686210i \(0.759273\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.40554i 0.642614i 0.946975 + 0.321307i \(0.104122\pi\)
−0.946975 + 0.321307i \(0.895878\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.72730i 0.924066i −0.886863 0.462033i \(-0.847120\pi\)
0.886863 0.462033i \(-0.152880\pi\)
\(54\) 0 0
\(55\) −1.45833 + 6.49978i −0.196641 + 0.876430i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.97906 −0.387840 −0.193920 0.981017i \(-0.562120\pi\)
−0.193920 + 0.981017i \(0.562120\pi\)
\(60\) 0 0
\(61\) 6.27226 0.803081 0.401540 0.915841i \(-0.368475\pi\)
0.401540 + 0.915841i \(0.368475\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.18183 0.489528i −0.270622 0.0607185i
\(66\) 0 0
\(67\) 0.727302i 0.0888540i 0.999013 + 0.0444270i \(0.0141462\pi\)
−0.999013 + 0.0444270i \(0.985854\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.32176 0.987612 0.493806 0.869572i \(-0.335605\pi\)
0.493806 + 0.869572i \(0.335605\pi\)
\(72\) 0 0
\(73\) 6.87031i 0.804110i 0.915616 + 0.402055i \(0.131704\pi\)
−0.915616 + 0.402055i \(0.868296\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.42604i 0.960237i
\(78\) 0 0
\(79\) −6.11190 −0.687642 −0.343821 0.939035i \(-0.611721\pi\)
−0.343821 + 0.939035i \(0.611721\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.7059i 1.61418i 0.590425 + 0.807092i \(0.298960\pi\)
−0.590425 + 0.807092i \(0.701040\pi\)
\(84\) 0 0
\(85\) 3.15019 + 0.706797i 0.341686 + 0.0766629i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.7478 1.56326 0.781632 0.623740i \(-0.214387\pi\)
0.781632 + 0.623740i \(0.214387\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.04189 9.10069i 0.209493 0.933711i
\(96\) 0 0
\(97\) 6.78654i 0.689069i −0.938774 0.344534i \(-0.888037\pi\)
0.938774 0.344534i \(-0.111963\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.1421 −1.20819 −0.604094 0.796913i \(-0.706465\pi\)
−0.604094 + 0.796913i \(0.706465\pi\)
\(102\) 0 0
\(103\) 2.81108i 0.276984i −0.990364 0.138492i \(-0.955775\pi\)
0.990364 0.138492i \(-0.0442255\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.3842i 1.77726i 0.458621 + 0.888632i \(0.348343\pi\)
−0.458621 + 0.888632i \(0.651657\pi\)
\(108\) 0 0
\(109\) −1.44383 −0.138294 −0.0691470 0.997606i \(-0.522028\pi\)
−0.0691470 + 0.997606i \(0.522028\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.40194i 0.320028i −0.987115 0.160014i \(-0.948846\pi\)
0.987115 0.160014i \(-0.0511539\pi\)
\(114\) 0 0
\(115\) −13.5557 3.04145i −1.26408 0.283617i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.08378 −0.374359
\(120\) 0 0
\(121\) −2.12522 −0.193202
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.87368 8.81774i 0.614801 0.788682i
\(126\) 0 0
\(127\) 15.8106i 1.40297i −0.712686 0.701484i \(-0.752521\pi\)
0.712686 0.701484i \(-0.247479\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.1707 1.67495 0.837476 0.546475i \(-0.184030\pi\)
0.837476 + 0.546475i \(0.184030\pi\)
\(132\) 0 0
\(133\) 11.7977i 1.02299i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.67824i 0.143381i 0.997427 + 0.0716907i \(0.0228395\pi\)
−0.997427 + 0.0716907i \(0.977161\pi\)
\(138\) 0 0
\(139\) 12.3423 1.04686 0.523429 0.852069i \(-0.324653\pi\)
0.523429 + 0.852069i \(0.324653\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.97906i 0.249121i
\(144\) 0 0
\(145\) 0.405538 1.80748i 0.0336781 0.150103i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.677798 0.0555274 0.0277637 0.999615i \(-0.491161\pi\)
0.0277637 + 0.999615i \(0.491161\pi\)
\(150\) 0 0
\(151\) −17.9991 −1.46475 −0.732374 0.680903i \(-0.761588\pi\)
−0.732374 + 0.680903i \(0.761588\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.811077 + 3.61497i −0.0651473 + 0.290361i
\(156\) 0 0
\(157\) 12.8392i 1.02468i 0.858783 + 0.512340i \(0.171221\pi\)
−0.858783 + 0.512340i \(0.828779\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.5731 1.38495
\(162\) 0 0
\(163\) 15.3137i 1.19946i 0.800202 + 0.599731i \(0.204726\pi\)
−0.800202 + 0.599731i \(0.795274\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.3922i 1.19109i 0.803324 + 0.595543i \(0.203063\pi\)
−0.803324 + 0.595543i \(0.796937\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.9706i 1.13819i 0.822272 + 0.569095i \(0.192706\pi\)
−0.822272 + 0.569095i \(0.807294\pi\)
\(174\) 0 0
\(175\) −6.04189 + 12.7865i −0.456724 + 0.966572i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.63950 −0.571003 −0.285502 0.958378i \(-0.592160\pi\)
−0.285502 + 0.958378i \(0.592160\pi\)
\(180\) 0 0
\(181\) 15.0286 1.11706 0.558532 0.829483i \(-0.311365\pi\)
0.558532 + 0.829483i \(0.311365\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0913912 + 0.0205051i 0.00671922 + 0.00150757i
\(186\) 0 0
\(187\) 4.30126i 0.314539i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.8877 0.787804 0.393902 0.919152i \(-0.371125\pi\)
0.393902 + 0.919152i \(0.371125\pi\)
\(192\) 0 0
\(193\) 4.94690i 0.356086i 0.984023 + 0.178043i \(0.0569766\pi\)
−0.984023 + 0.178043i \(0.943023\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.3628i 1.87827i 0.343549 + 0.939135i \(0.388371\pi\)
−0.343549 + 0.939135i \(0.611629\pi\)
\(198\) 0 0
\(199\) −6.11190 −0.433261 −0.216630 0.976254i \(-0.569507\pi\)
−0.216630 + 0.976254i \(0.569507\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.34315i 0.164457i
\(204\) 0 0
\(205\) −2.13613 + 9.52072i −0.149194 + 0.664956i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −12.4260 −0.859527
\(210\) 0 0
\(211\) 9.68585 0.666802 0.333401 0.942785i \(-0.391804\pi\)
0.333401 + 0.942785i \(0.391804\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −19.6355 4.40554i −1.33913 0.300455i
\(216\) 0 0
\(217\) 4.68629i 0.318126i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.44383 0.0971227
\(222\) 0 0
\(223\) 9.21258i 0.616920i 0.951237 + 0.308460i \(0.0998136\pi\)
−0.951237 + 0.308460i \(0.900186\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.3218i 1.34880i 0.738365 + 0.674401i \(0.235598\pi\)
−0.738365 + 0.674401i \(0.764402\pi\)
\(228\) 0 0
\(229\) 16.4715 1.08847 0.544234 0.838933i \(-0.316820\pi\)
0.544234 + 0.838933i \(0.316820\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.37339i 0.155486i −0.996973 0.0777428i \(-0.975229\pi\)
0.996973 0.0777428i \(-0.0247713\pi\)
\(234\) 0 0
\(235\) −9.61212 2.15663i −0.627025 0.140683i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.6363 −0.882062 −0.441031 0.897492i \(-0.645387\pi\)
−0.441031 + 0.897492i \(0.645387\pi\)
\(240\) 0 0
\(241\) 1.07045 0.0689536 0.0344768 0.999405i \(-0.489024\pi\)
0.0344768 + 0.999405i \(0.489024\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.489528 2.18183i 0.0312748 0.139392i
\(246\) 0 0
\(247\) 4.17113i 0.265403i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −14.6262 −0.923196 −0.461598 0.887089i \(-0.652724\pi\)
−0.461598 + 0.887089i \(0.652724\pi\)
\(252\) 0 0
\(253\) 18.5089i 1.16365i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.9403i 1.55573i 0.628429 + 0.777867i \(0.283698\pi\)
−0.628429 + 0.777867i \(0.716302\pi\)
\(258\) 0 0
\(259\) −0.118476 −0.00736173
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.01691i 0.556007i 0.960580 + 0.278003i \(0.0896726\pi\)
−0.960580 + 0.278003i \(0.910327\pi\)
\(264\) 0 0
\(265\) 14.6778 + 3.29320i 0.901650 + 0.202300i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.28391 0.505079 0.252539 0.967587i \(-0.418734\pi\)
0.252539 + 0.967587i \(0.418734\pi\)
\(270\) 0 0
\(271\) −19.9572 −1.21232 −0.606158 0.795344i \(-0.707290\pi\)
−0.606158 + 0.795344i \(0.707290\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.4675 6.36365i −0.812120 0.383743i
\(276\) 0 0
\(277\) 22.6145i 1.35878i 0.733780 + 0.679388i \(0.237755\pi\)
−0.733780 + 0.679388i \(0.762245\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.89572 0.113089 0.0565446 0.998400i \(-0.481992\pi\)
0.0565446 + 0.998400i \(0.481992\pi\)
\(282\) 0 0
\(283\) 11.2299i 0.667550i −0.942653 0.333775i \(-0.891677\pi\)
0.942653 0.333775i \(-0.108323\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.3423i 0.728541i
\(288\) 0 0
\(289\) 14.9153 0.877373
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 27.6774i 1.61693i −0.588545 0.808464i \(-0.700299\pi\)
0.588545 0.808464i \(-0.299701\pi\)
\(294\) 0 0
\(295\) 1.45833 6.49978i 0.0849074 0.378432i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.21302 −0.359308
\(300\) 0 0
\(301\) 25.4546 1.46718
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.07045 + 13.6850i −0.175813 + 0.783599i
\(306\) 0 0
\(307\) 24.9858i 1.42601i −0.701157 0.713007i \(-0.747333\pi\)
0.701157 0.713007i \(-0.252667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.3842 −1.04247 −0.521235 0.853413i \(-0.674528\pi\)
−0.521235 + 0.853413i \(0.674528\pi\)
\(312\) 0 0
\(313\) 8.61453i 0.486922i −0.969911 0.243461i \(-0.921717\pi\)
0.969911 0.243461i \(-0.0782828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.0491i 0.845240i 0.906307 + 0.422620i \(0.138889\pi\)
−0.906307 + 0.422620i \(0.861111\pi\)
\(318\) 0 0
\(319\) −2.46793 −0.138178
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.02242i 0.335096i
\(324\) 0 0
\(325\) 2.13613 4.52072i 0.118491 0.250765i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.4607 0.686983
\(330\) 0 0
\(331\) 22.6738 1.24626 0.623131 0.782117i \(-0.285860\pi\)
0.623131 + 0.782117i \(0.285860\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.58685 0.356035i −0.0866986 0.0194523i
\(336\) 0 0
\(337\) 27.9282i 1.52135i −0.649134 0.760674i \(-0.724869\pi\)
0.649134 0.760674i \(-0.275131\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.93586 0.267292
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.7398i 1.05969i 0.848096 + 0.529843i \(0.177749\pi\)
−0.848096 + 0.529843i \(0.822251\pi\)
\(348\) 0 0
\(349\) −30.7157 −1.64417 −0.822086 0.569364i \(-0.807190\pi\)
−0.822086 + 0.569364i \(0.807190\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 17.0072i 0.905201i 0.891713 + 0.452600i \(0.149504\pi\)
−0.891713 + 0.452600i \(0.850496\pi\)
\(354\) 0 0
\(355\) −4.07374 + 18.1566i −0.216212 + 0.963654i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.90203 −0.417053 −0.208527 0.978017i \(-0.566867\pi\)
−0.208527 + 0.978017i \(0.566867\pi\)
\(360\) 0 0
\(361\) −1.60164 −0.0842968
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −14.9898 3.36321i −0.784603 0.176039i
\(366\) 0 0
\(367\) 34.1810i 1.78424i 0.451803 + 0.892118i \(0.350781\pi\)
−0.451803 + 0.892118i \(0.649219\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.0277 −0.987868
\(372\) 0 0
\(373\) 18.5098i 0.958402i 0.877705 + 0.479201i \(0.159074\pi\)
−0.877705 + 0.479201i \(0.840926\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.828427i 0.0426662i
\(378\) 0 0
\(379\) 12.7994 0.657462 0.328731 0.944424i \(-0.393379\pi\)
0.328731 + 0.944424i \(0.393379\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.664032i 0.0339304i 0.999856 + 0.0169652i \(0.00540046\pi\)
−0.999856 + 0.0169652i \(0.994600\pi\)
\(384\) 0 0
\(385\) 18.3842 + 4.12479i 0.936943 + 0.210219i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.32580 0.117923 0.0589613 0.998260i \(-0.481221\pi\)
0.0589613 + 0.998260i \(0.481221\pi\)
\(390\) 0 0
\(391\) 8.97056 0.453661
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.99195 13.3351i 0.150541 0.670961i
\(396\) 0 0
\(397\) 18.2433i 0.915603i −0.889054 0.457802i \(-0.848637\pi\)
0.889054 0.457802i \(-0.151363\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −16.9919 −0.848537 −0.424269 0.905536i \(-0.639469\pi\)
−0.424269 + 0.905536i \(0.639469\pi\)
\(402\) 0 0
\(403\) 1.65685i 0.0825338i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.124785i 0.00618538i
\(408\) 0 0
\(409\) 10.8111 0.534573 0.267287 0.963617i \(-0.413873\pi\)
0.267287 + 0.963617i \(0.413873\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.42604i 0.414618i
\(414\) 0 0
\(415\) −32.0857 7.19896i −1.57503 0.353383i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5.08780 −0.248555 −0.124278 0.992247i \(-0.539661\pi\)
−0.124278 + 0.992247i \(0.539661\pi\)
\(420\) 0 0
\(421\) 12.0945 0.589452 0.294726 0.955582i \(-0.404772\pi\)
0.294726 + 0.955582i \(0.404772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.08421 + 6.52717i −0.149606 + 0.316614i
\(426\) 0 0
\(427\) 17.7406i 0.858529i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.48931 0.408916 0.204458 0.978875i \(-0.434457\pi\)
0.204458 + 0.978875i \(0.434457\pi\)
\(432\) 0 0
\(433\) 13.8453i 0.665365i −0.943039 0.332682i \(-0.892046\pi\)
0.943039 0.332682i \(-0.107954\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.9153i 1.23970i
\(438\) 0 0
\(439\) −0.602516 −0.0287565 −0.0143783 0.999897i \(-0.504577\pi\)
−0.0143783 + 0.999897i \(0.504577\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.6721i 0.839626i −0.907611 0.419813i \(-0.862096\pi\)
0.907611 0.419813i \(-0.137904\pi\)
\(444\) 0 0
\(445\) −7.21947 + 32.1771i −0.342236 + 1.52534i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.33509 0.157393 0.0786963 0.996899i \(-0.474924\pi\)
0.0786963 + 0.996899i \(0.474924\pi\)
\(450\) 0 0
\(451\) 12.9996 0.612125
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.38459 + 6.17113i −0.0649108 + 0.289307i
\(456\) 0 0
\(457\) 38.9266i 1.82091i −0.413611 0.910454i \(-0.635733\pi\)
0.413611 0.910454i \(-0.364267\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −28.3918 −1.32234 −0.661168 0.750238i \(-0.729939\pi\)
−0.661168 + 0.750238i \(0.729939\pi\)
\(462\) 0 0
\(463\) 10.2259i 0.475238i −0.971358 0.237619i \(-0.923633\pi\)
0.971358 0.237619i \(-0.0763670\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.61584i 0.259870i 0.991522 + 0.129935i \(0.0414769\pi\)
−0.991522 + 0.129935i \(0.958523\pi\)
\(468\) 0 0
\(469\) 2.05712 0.0949890
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 26.8102i 1.23273i
\(474\) 0 0
\(475\) 18.8565 + 8.91008i 0.865198 + 0.408823i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.95006 −0.408939 −0.204469 0.978873i \(-0.565547\pi\)
−0.204469 + 0.978873i \(0.565547\pi\)
\(480\) 0 0
\(481\) 0.0418875 0.00190991
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.8070 + 3.32220i 0.672353 + 0.150853i
\(486\) 0 0
\(487\) 18.3596i 0.831954i 0.909375 + 0.415977i \(0.136560\pi\)
−0.909375 + 0.415977i \(0.863440\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.7981 1.52529 0.762644 0.646819i \(-0.223901\pi\)
0.762644 + 0.646819i \(0.223901\pi\)
\(492\) 0 0
\(493\) 1.19611i 0.0538701i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 23.5375i 1.05580i
\(498\) 0 0
\(499\) −5.72898 −0.256464 −0.128232 0.991744i \(-0.540930\pi\)
−0.128232 + 0.991744i \(0.540930\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.2121i 0.901215i −0.892722 0.450607i \(-0.851207\pi\)
0.892722 0.450607i \(-0.148793\pi\)
\(504\) 0 0
\(505\) 5.94392 26.4920i 0.264501 1.17888i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.02708 0.267146 0.133573 0.991039i \(-0.457355\pi\)
0.133573 + 0.991039i \(0.457355\pi\)
\(510\) 0 0
\(511\) 19.4322 0.859629
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.13328 + 1.37610i 0.270265 + 0.0606383i
\(516\) 0 0
\(517\) 13.1243i 0.577208i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.95899 0.0858249 0.0429125 0.999079i \(-0.486336\pi\)
0.0429125 + 0.999079i \(0.486336\pi\)
\(522\) 0 0
\(523\) 21.6859i 0.948256i 0.880456 + 0.474128i \(0.157237\pi\)
−0.880456 + 0.474128i \(0.842763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.39222i 0.104207i
\(528\) 0 0
\(529\) −15.6016 −0.678332
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.36365i 0.189011i
\(534\) 0 0
\(535\) −40.1110 8.99956i −1.73415 0.389085i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.97906 −0.128317
\(540\) 0 0
\(541\) 39.3245 1.69069 0.845346 0.534220i \(-0.179394\pi\)
0.845346 + 0.534220i \(0.179394\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.706797 3.15019i 0.0302758 0.134939i
\(546\) 0 0
\(547\) 14.1466i 0.604865i 0.953171 + 0.302432i \(0.0977987\pi\)
−0.953171 + 0.302432i \(0.902201\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.45548 0.147208
\(552\) 0 0
\(553\) 17.2871i 0.735120i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.7603i 1.26098i 0.776196 + 0.630491i \(0.217147\pi\)
−0.776196 + 0.630491i \(0.782853\pi\)
\(558\) 0 0
\(559\) −8.99956 −0.380641
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 28.6498i 1.20745i −0.797194 0.603723i \(-0.793683\pi\)
0.797194 0.603723i \(-0.206317\pi\)
\(564\) 0 0
\(565\) 7.42245 + 1.66535i 0.312265 + 0.0700617i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 39.7326 1.66568 0.832838 0.553517i \(-0.186715\pi\)
0.832838 + 0.553517i \(0.186715\pi\)
\(570\) 0 0
\(571\) 30.5726 1.27943 0.639713 0.768614i \(-0.279053\pi\)
0.639713 + 0.768614i \(0.279053\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.2718 28.0874i 0.553473 1.17132i
\(576\) 0 0
\(577\) 35.3116i 1.47004i −0.678045 0.735020i \(-0.737173\pi\)
0.678045 0.735020i \(-0.262827\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 41.5946 1.72564
\(582\) 0 0
\(583\) 20.0410i 0.830014i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.5517i 0.806985i −0.914983 0.403492i \(-0.867796\pi\)
0.914983 0.403492i \(-0.132204\pi\)
\(588\) 0 0
\(589\) −6.91096 −0.284761
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.66491i 0.191565i 0.995402 + 0.0957824i \(0.0305353\pi\)
−0.995402 + 0.0957824i \(0.969465\pi\)
\(594\) 0 0
\(595\) 1.99912 8.91008i 0.0819560 0.365278i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −19.6497 −0.802864 −0.401432 0.915889i \(-0.631487\pi\)
−0.401432 + 0.915889i \(0.631487\pi\)
\(600\) 0 0
\(601\) 44.2714 1.80587 0.902934 0.429780i \(-0.141409\pi\)
0.902934 + 0.429780i \(0.141409\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.04036 4.63687i 0.0422965 0.188515i
\(606\) 0 0
\(607\) 34.2648i 1.39077i 0.718640 + 0.695383i \(0.244765\pi\)
−0.718640 + 0.695383i \(0.755235\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.40554 −0.178229
\(612\) 0 0
\(613\) 33.7203i 1.36195i −0.732307 0.680975i \(-0.761556\pi\)
0.732307 0.680975i \(-0.238444\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.0901i 1.41267i 0.707876 + 0.706337i \(0.249653\pi\)
−0.707876 + 0.706337i \(0.750347\pi\)
\(618\) 0 0
\(619\) −2.63907 −0.106073 −0.0530365 0.998593i \(-0.516890\pi\)
−0.0530365 + 0.998593i \(0.516890\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 41.7131i 1.67120i
\(624\) 0 0
\(625\) 15.8739 + 19.3137i 0.634956 + 0.772548i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.0604785 −0.00241144
\(630\) 0 0
\(631\) −28.7683 −1.14525 −0.572624 0.819818i \(-0.694075\pi\)
−0.572624 + 0.819818i \(0.694075\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.4961 + 7.73975i 1.36893 + 0.307143i
\(636\) 0 0
\(637\) 1.00000i 0.0396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −35.1944 −1.39009 −0.695047 0.718965i \(-0.744616\pi\)
−0.695047 + 0.718965i \(0.744616\pi\)
\(642\) 0 0
\(643\) 27.2709i 1.07546i 0.843117 + 0.537731i \(0.180718\pi\)
−0.843117 + 0.537731i \(0.819282\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.22913i 0.166264i 0.996539 + 0.0831322i \(0.0264924\pi\)
−0.996539 + 0.0831322i \(0.973508\pi\)
\(648\) 0 0
\(649\) −8.87478 −0.348365
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.1400i 0.866406i −0.901296 0.433203i \(-0.857383\pi\)
0.901296 0.433203i \(-0.142617\pi\)
\(654\) 0 0
\(655\) −9.38459 + 41.8271i −0.366686 + 1.63432i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 27.0932 1.05540 0.527701 0.849430i \(-0.323054\pi\)
0.527701 + 0.849430i \(0.323054\pi\)
\(660\) 0 0
\(661\) 19.0597 0.741335 0.370668 0.928766i \(-0.379129\pi\)
0.370668 + 0.928766i \(0.379129\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −25.7406 5.77533i −0.998179 0.223958i
\(666\) 0 0
\(667\) 5.14704i 0.199294i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.6854 0.721342
\(672\) 0 0
\(673\) 23.6213i 0.910533i −0.890355 0.455267i \(-0.849544\pi\)
0.890355 0.455267i \(-0.150456\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.8111i 0.722968i −0.932378 0.361484i \(-0.882270\pi\)
0.932378 0.361484i \(-0.117730\pi\)
\(678\) 0 0
\(679\) −19.1952 −0.736645
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.7827i 1.29266i −0.763059 0.646329i \(-0.776303\pi\)
0.763059 0.646329i \(-0.223697\pi\)
\(684\) 0 0
\(685\) −3.66162 0.821544i −0.139903 0.0313896i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.72730 0.256290
\(690\) 0 0
\(691\) −47.3236 −1.80027 −0.900137 0.435606i \(-0.856534\pi\)
−0.900137 + 0.435606i \(0.856534\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.04189 + 26.9287i −0.229182 + 1.02146i
\(696\) 0 0
\(697\) 6.30038i 0.238644i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.29636 0.124502 0.0622509 0.998061i \(-0.480172\pi\)
0.0622509 + 0.998061i \(0.480172\pi\)
\(702\) 0 0
\(703\) 0.174718i 0.00658963i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 34.3431i 1.29161i
\(708\) 0 0
\(709\) −46.8404 −1.75913 −0.879565 0.475779i \(-0.842166\pi\)
−0.879565 + 0.475779i \(0.842166\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.2941i 0.385516i
\(714\) 0 0
\(715\) −6.49978 1.45833i −0.243078 0.0545385i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0152 −0.746442 −0.373221 0.927742i \(-0.621747\pi\)
−0.373221 + 0.927742i \(0.621747\pi\)
\(720\) 0 0
\(721\) −7.95093 −0.296108
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.74509 + 1.76963i 0.139089 + 0.0657223i
\(726\) 0 0
\(727\) 10.8383i 0.401971i −0.979594 0.200986i \(-0.935586\pi\)
0.979594 0.200986i \(-0.0644144\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.9939 0.480595
\(732\) 0 0
\(733\) 21.1952i 0.782864i −0.920207 0.391432i \(-0.871980\pi\)
0.920207 0.391432i \(-0.128020\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.16667i 0.0798104i
\(738\) 0 0
\(739\) −7.07403 −0.260222 −0.130111 0.991499i \(-0.541533\pi\)
−0.130111 + 0.991499i \(0.541533\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.3913i 0.784772i 0.919801 + 0.392386i \(0.128350\pi\)
−0.919801 + 0.392386i \(0.871650\pi\)
\(744\) 0 0
\(745\) −0.331801 + 1.47884i −0.0121563 + 0.0541804i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 51.9982 1.89997
\(750\) 0 0
\(751\) 18.2247 0.665028 0.332514 0.943098i \(-0.392103\pi\)
0.332514 + 0.943098i \(0.392103\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.81108 39.2709i 0.320668 1.42922i
\(756\) 0 0
\(757\) 39.3708i 1.43096i −0.698635 0.715479i \(-0.746209\pi\)
0.698635 0.715479i \(-0.253791\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.8610 0.502462 0.251231 0.967927i \(-0.419165\pi\)
0.251231 + 0.967927i \(0.419165\pi\)
\(762\) 0 0
\(763\) 4.08378i 0.147843i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.97906i 0.107567i
\(768\) 0 0
\(769\) 12.6274 0.455356 0.227678 0.973736i \(-0.426887\pi\)
0.227678 + 0.973736i \(0.426887\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.7825i 1.25104i −0.780208 0.625520i \(-0.784887\pi\)
0.780208 0.625520i \(-0.215113\pi\)
\(774\) 0 0
\(775\) −7.49018 3.53926i −0.269055 0.127134i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.2014 −0.652132
\(780\) 0 0
\(781\) 24.7910 0.887092
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −28.0129 6.28515i −0.999823 0.224327i
\(786\) 0 0
\(787\) 36.7835i 1.31119i 0.755112 + 0.655596i \(0.227582\pi\)
−0.755112 + 0.655596i \(0.772418\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.62215 −0.342124
\(792\) 0 0
\(793\) 6.27226i 0.222734i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.4413i 1.50335i −0.659535 0.751674i \(-0.729247\pi\)
0.659535 0.751674i \(-0.270753\pi\)
\(798\) 0 0
\(799\) 6.36086 0.225031
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.4671i 0.722267i
\(804\) 0 0
\(805\) −8.60252 + 38.3414i −0.303199 + 1.35136i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 55.3771 1.94696 0.973478 0.228780i \(-0.0734735\pi\)
0.973478 + 0.228780i \(0.0734735\pi\)
\(810\) 0 0
\(811\) −17.1159 −0.601021 −0.300511 0.953778i \(-0.597157\pi\)
−0.300511 + 0.953778i \(0.597157\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33.4118 7.49649i −1.17037 0.262591i
\(816\) 0 0
\(817\) 37.5384i 1.31330i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.5460 −0.542559 −0.271279 0.962501i \(-0.587447\pi\)
−0.271279 + 0.962501i \(0.587447\pi\)
\(822\) 0 0
\(823\) 0.0998847i 0.00348176i 0.999998 + 0.00174088i \(0.000554140\pi\)
−0.999998 + 0.00174088i \(0.999446\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.53311i 0.331499i −0.986168 0.165749i \(-0.946996\pi\)
0.986168 0.165749i \(-0.0530042\pi\)
\(828\) 0 0
\(829\) −50.6361 −1.75866 −0.879332 0.476210i \(-0.842010\pi\)
−0.879332 + 0.476210i \(0.842010\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.44383i 0.0500258i
\(834\) 0 0
\(835\) −33.5831 7.53492i −1.16219 0.260757i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −43.2310 −1.49250 −0.746249 0.665666i \(-0.768147\pi\)
−0.746249 + 0.665666i \(0.768147\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.489528 2.18183i 0.0168403 0.0750571i
\(846\) 0 0
\(847\) 6.01104i 0.206542i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0.260248 0.00892119
\(852\) 0 0
\(853\) 31.6979i 1.08531i −0.839954 0.542657i \(-0.817418\pi\)
0.839954 0.542657i \(-0.182582\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.8912i 0.986906i −0.869772 0.493453i \(-0.835734\pi\)
0.869772 0.493453i \(-0.164266\pi\)
\(858\) 0 0
\(859\) −49.7099 −1.69608 −0.848040 0.529933i \(-0.822217\pi\)
−0.848040 + 0.529933i \(0.822217\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.56502i 0.155395i 0.996977 + 0.0776976i \(0.0247569\pi\)
−0.996977 + 0.0776976i \(0.975243\pi\)
\(864\) 0 0
\(865\) −32.6632 7.32851i −1.11058 0.249177i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −18.2077 −0.617653
\(870\) 0 0
\(871\) −0.727302 −0.0246437
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24.9403 19.4417i −0.843137 0.657250i
\(876\) 0 0
\(877\) 13.5312i 0.456916i −0.973554 0.228458i \(-0.926632\pi\)
0.973554 0.228458i \(-0.0733683\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 28.0634 0.945481 0.472740 0.881202i \(-0.343265\pi\)
0.472740 + 0.881202i \(0.343265\pi\)
\(882\) 0 0
\(883\) 47.3700i 1.59413i −0.603896 0.797063i \(-0.706386\pi\)
0.603896 0.797063i \(-0.293614\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.606101i 0.0203509i −0.999948 0.0101754i \(-0.996761\pi\)
0.999948 0.0101754i \(-0.00323900\pi\)
\(888\) 0 0
\(889\) −44.7192 −1.49984
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.3761i 0.614932i
\(894\) 0 0
\(895\) 3.73975 16.6681i 0.125006 0.557152i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.37258 −0.0457782
\(900\) 0 0
\(901\) −9.71310 −0.323590
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.35690 + 32.7897i −0.244552 + 1.08997i
\(906\) 0 0
\(907\) 11.8582i 0.393746i 0.980429 + 0.196873i \(0.0630787\pi\)
−0.980429 + 0.196873i \(0.936921\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 33.5642 1.11203 0.556015 0.831172i \(-0.312330\pi\)
0.556015 + 0.831172i \(0.312330\pi\)
\(912\) 0 0
\(913\) 43.8098i 1.44989i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 54.2229i 1.79060i
\(918\) 0 0
\(919\) 19.2242 0.634149 0.317074 0.948401i \(-0.397299\pi\)
0.317074 + 0.948401i \(0.397299\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.32176i 0.273914i
\(924\) 0 0
\(925\) −0.0894772 + 0.189362i −0.00294199 + 0.00622618i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −36.0349 −1.18227 −0.591133 0.806574i \(-0.701319\pi\)
−0.591133 + 0.806574i \(0.701319\pi\)
\(930\) 0 0
\(931\) 4.17113 0.136703
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.38459 + 2.10559i 0.306909 + 0.0688601i
\(936\) 0 0
\(937\) 41.5851i 1.35853i 0.733895 + 0.679263i \(0.237700\pi\)
−0.733895 + 0.679263i \(0.762300\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 56.4933 1.84163 0.920814 0.390002i \(-0.127526\pi\)
0.920814 + 0.390002i \(0.127526\pi\)
\(942\) 0 0
\(943\) 27.1115i 0.882871i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.3922i 0.370197i 0.982720 + 0.185099i \(0.0592604\pi\)
−0.982720 + 0.185099i \(0.940740\pi\)
\(948\) 0 0
\(949\) −6.87031 −0.223020
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 56.7986i 1.83989i 0.392053 + 0.919943i \(0.371765\pi\)
−0.392053 + 0.919943i \(0.628235\pi\)
\(954\) 0 0
\(955\) −5.32982 + 23.7550i −0.172469 + 0.768693i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.74677 0.153281
\(960\) 0 0
\(961\) −28.2548 −0.911446
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −10.7933 2.42165i −0.347448 0.0779556i
\(966\) 0 0
\(967\) 25.7224i 0.827177i −0.910464 0.413588i \(-0.864275\pi\)
0.910464 0.413588i \(-0.135725\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 32.8703 1.05486 0.527429 0.849599i \(-0.323156\pi\)
0.527429 + 0.849599i \(0.323156\pi\)
\(972\) 0 0
\(973\) 34.9092i 1.11914i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 30.1453i 0.964433i −0.876052 0.482217i \(-0.839832\pi\)
0.876052 0.482217i \(-0.160168\pi\)
\(978\) 0 0
\(979\) 43.9345 1.40415
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 32.4965i 1.03648i −0.855236 0.518238i \(-0.826588\pi\)
0.855236 0.518238i \(-0.173412\pi\)
\(984\) 0 0
\(985\) −57.5190 12.9053i −1.83271 0.411198i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −55.9145 −1.77798
\(990\) 0 0
\(991\) 37.4893 1.19089 0.595443 0.803397i \(-0.296976\pi\)
0.595443 + 0.803397i \(0.296976\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.99195 13.3351i 0.0948510 0.422751i
\(996\) 0 0
\(997\) 18.8659i 0.597488i −0.954333 0.298744i \(-0.903432\pi\)
0.954333 0.298744i \(-0.0965676\pi\)
\(998\) 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 4680.2.l.f.2809.4 8
3.2 odd 2 1560.2.l.e.1249.3 8
5.4 even 2 inner 4680.2.l.f.2809.3 8
12.11 even 2 3120.2.l.o.1249.7 8
15.2 even 4 7800.2.a.bv.1.3 4
15.8 even 4 7800.2.a.bw.1.1 4
15.14 odd 2 1560.2.l.e.1249.7 yes 8
60.59 even 2 3120.2.l.o.1249.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.l.e.1249.3 8 3.2 odd 2
1560.2.l.e.1249.7 yes 8 15.14 odd 2
3120.2.l.o.1249.3 8 60.59 even 2
3120.2.l.o.1249.7 8 12.11 even 2
4680.2.l.f.2809.3 8 5.4 even 2 inner
4680.2.l.f.2809.4 8 1.1 even 1 trivial
7800.2.a.bv.1.3 4 15.2 even 4
7800.2.a.bw.1.1 4 15.8 even 4