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Properties

Label 7360.2.a.i.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
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] = [7360,2,Mod(1,7360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7360.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.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: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +6.00000 q^{11} +1.00000 q^{13} -1.00000 q^{15} -2.00000 q^{19} +4.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -9.00000 q^{29} +5.00000 q^{31} -6.00000 q^{33} -4.00000 q^{35} -2.00000 q^{37} -1.00000 q^{39} -9.00000 q^{41} +4.00000 q^{43} -2.00000 q^{45} -3.00000 q^{47} +9.00000 q^{49} +6.00000 q^{53} +6.00000 q^{55} +2.00000 q^{57} -2.00000 q^{61} +8.00000 q^{63} +1.00000 q^{65} +10.0000 q^{67} -1.00000 q^{69} -3.00000 q^{71} -7.00000 q^{73} -1.00000 q^{75} -24.0000 q^{77} -10.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +9.00000 q^{87} -4.00000 q^{91} -5.00000 q^{93} -2.00000 q^{95} +8.00000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 8.00000 1.00791
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −24.0000 −2.73505
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −20.0000 −1.91565 −0.957826 − 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 − 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −23.0000 −1.95083 −0.975417 − 0.220366i \(-0.929275\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −9.00000 −0.747409
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 − 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −19.0000 −1.36765 −0.683825 − 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −21.0000 −1.49619 −0.748094 − 0.663593i \(-0.769031\pi\)
−0.748094 + 0.663593i \(0.769031\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 − 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) 36.0000 2.52670
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 − 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 3.00000 0.205557
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 − 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 − 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) 24.0000 1.57908
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 − 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) −3.00000 −0.194054 −0.0970269 − 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 − 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 − 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.0000 1.64622 0.823110 − 0.567883i \(-0.192237\pi\)
0.823110 + 0.567883i \(0.192237\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) 31.0000 1.86261 0.931305 − 0.364241i \(-0.118672\pi\)
0.931305 + 0.364241i \(0.118672\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 − 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −26.0000 −1.54554 −0.772770 − 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 36.0000 2.12501
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) −24.0000 −1.40209 −0.701047 − 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 30.0000 1.74078
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 − 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 − 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) 32.0000 1.80875 0.904373 − 0.426742i \(-0.140339\pi\)
0.904373 + 0.426742i \(0.140339\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) −54.0000 −3.02342
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 0 0
\(327\) 20.0000 1.10600
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −35.0000 −1.92377 −0.961887 − 0.273447i \(-0.911836\pi\)
−0.961887 + 0.273447i \(0.911836\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 − 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 36.0000 1.93258 0.966291 − 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 − 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 − 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −7.00000 −0.366397
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 − 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 − 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −11.0000 −0.563547
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) −23.0000 −1.15434 −0.577168 − 0.816625i \(-0.695842\pi\)
−0.577168 + 0.816625i \(0.695842\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 5.00000 0.249068
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 − 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 23.0000 1.12631
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 − 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 11.0000 0.525001 0.262501 − 0.964932i \(-0.415453\pi\)
0.262501 + 0.964932i \(0.415453\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −54.0000 −2.54276
\(452\) 0 0
\(453\) −5.00000 −0.234920
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 − 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) −5.00000 −0.231869
\(466\) 0 0
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 24.0000 1.10352
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 − 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 − 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 3.00000 0.132973 0.0664863 − 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 0 0
\(513\) −10.0000 −0.441511
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 − 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.00000 −0.389833
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 15.0000 0.647298
\(538\) 0 0
\(539\) 54.0000 2.32594
\(540\) 0 0
\(541\) 7.00000 0.300954 0.150477 − 0.988614i \(-0.451919\pi\)
0.150477 + 0.988614i \(0.451919\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 − 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 40.0000 1.70097
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) 24.0000 1.01691 0.508456 − 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 − 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 − 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −43.0000 −1.79011 −0.895057 − 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −33.0000 −1.36206 −0.681028 − 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −10.0000 −0.412043
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 − 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −14.0000 −0.572982
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 − 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) −20.0000 −0.814463
\(604\) 0 0
\(605\) 25.0000 1.01639
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) −36.0000 −1.45879
\(610\) 0 0
\(611\) −3.00000 −0.121367
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 − 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 11.0000 0.436522
\(636\) 0 0
\(637\) 9.00000 0.356593
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 21.0000 0.825595 0.412798 − 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 0 0
\(653\) 9.00000 0.352197 0.176099 − 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 0 0
\(655\) 15.0000 0.586098
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 − 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −42.0000 −1.61419 −0.807096 − 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 0 0
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) −3.00000 −0.114792 −0.0573959 − 0.998351i \(-0.518280\pi\)
−0.0573959 + 0.998351i \(0.518280\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 48.0000 1.82337
\(694\) 0 0
\(695\) −23.0000 −0.872440
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 − 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) −24.0000 −0.902613
\(708\) 0 0
\(709\) −32.0000 −1.20179 −0.600893 − 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) 3.00000 0.112037
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 22.0000 0.818189
\(724\) 0 0
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) −10.0000 −0.370879 −0.185440 − 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 40.0000 1.47743 0.738717 − 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) 0 0
\(735\) −9.00000 −0.331970
\(736\) 0 0
\(737\) 60.0000 2.21013
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 − 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) −24.0000 −0.878114
\(748\) 0 0
\(749\) −72.0000 −2.63082
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 − 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 − 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 80.0000 2.89619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 0 0
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) −45.0000 −1.60817
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 6.00000 0.212531 0.106265 − 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −42.0000 −1.48215
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) −27.0000 −0.950445
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 19.0000 0.667180 0.333590 − 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 1.00000 0.0350285
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 − 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 − 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −31.0000 −1.07538
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 25.0000 0.864126
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 12.0000 0.413302
\(844\) 0 0
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) −100.000 −3.43604
\(848\) 0 0
\(849\) 26.0000 0.892318
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 − 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 39.0000 1.33221 0.666107 − 0.745856i \(-0.267959\pi\)
0.666107 + 0.745856i \(0.267959\pi\)
\(858\) 0 0
\(859\) −5.00000 −0.170598 −0.0852989 − 0.996355i \(-0.527185\pi\)
−0.0852989 + 0.996355i \(0.527185\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 0 0
\(863\) −45.0000 −1.53182 −0.765909 − 0.642949i \(-0.777711\pi\)
−0.765909 + 0.642949i \(0.777711\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 − 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 − 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.00000 −0.100730 −0.0503651 − 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) 0 0
\(889\) −44.0000 −1.47571
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 0 0
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 0 0
\(899\) −45.0000 −1.50083
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 − 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 0 0
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 − 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 72.0000 2.38285
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) −60.0000 −1.98137
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 − 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 32.0000 1.05444
\(922\) 0 0
\(923\) −3.00000 −0.0987462
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 39.0000 1.27955 0.639774 − 0.768563i \(-0.279028\pi\)
0.639774 + 0.768563i \(0.279028\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) −21.0000 −0.687509
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) 12.0000 0.391189 0.195594 − 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) 0 0
\(945\) −20.0000 −0.650600
\(946\) 0 0
\(947\) −33.0000 −1.07236 −0.536178 − 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 − 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) 54.0000 1.74557
\(958\) 0 0
\(959\) 48.0000 1.55000
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) −19.0000 −0.611632
\(966\) 0 0
\(967\) 23.0000 0.739630 0.369815 − 0.929105i \(-0.379421\pi\)
0.369815 + 0.929105i \(0.379421\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 − 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) 92.0000 2.94938
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 40.0000 1.27710
\(982\) 0 0
\(983\) 54.0000 1.72233 0.861166 − 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 0 0
\(985\) −21.0000 −0.669116
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 − 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 35.0000 1.11069
\(994\) 0 0
\(995\) 14.0000 0.443830
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 − 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 7360.2.a.i.1.1 1
4.3 odd 2 7360.2.a.v.1.1 1
8.3 odd 2 1840.2.a.c.1.1 1
8.5 even 2 460.2.a.c.1.1 ✓ 1
24.5 odd 2 4140.2.a.f.1.1 1
40.13 odd 4 2300.2.c.d.1749.2 2
40.19 odd 2 9200.2.a.y.1.1 1
40.29 even 2 2300.2.a.d.1.1 1
40.37 odd 4 2300.2.c.d.1749.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.c.1.1 ✓ 1 8.5 even 2
1840.2.a.c.1.1 1 8.3 odd 2
2300.2.a.d.1.1 1 40.29 even 2
2300.2.c.d.1749.1 2 40.37 odd 4
2300.2.c.d.1749.2 2 40.13 odd 4
4140.2.a.f.1.1 1 24.5 odd 2
7360.2.a.i.1.1 1 1.1 even 1 trivial
7360.2.a.v.1.1 1 4.3 odd 2
9200.2.a.y.1.1 1 40.19 odd 2