Properties

Label 1216.2.a.v.1.2
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(1,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.786802\) of defining polynomial
Character \(\chi\) \(=\) 1216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.786802 q^{3} -3.29707 q^{5} +2.08387 q^{7} -2.38094 q^{9} +O(q^{10})\) \(q+0.786802 q^{3} -3.29707 q^{5} +2.08387 q^{7} -2.38094 q^{9} +1.29707 q^{11} -1.21320 q^{13} -2.59414 q^{15} +4.08387 q^{17} -1.00000 q^{19} +1.63959 q^{21} +8.95455 q^{23} +5.87067 q^{25} -4.23374 q^{27} +9.38094 q^{29} +1.02054 q^{33} -6.87067 q^{35} +2.00000 q^{37} -0.954547 q^{39} +3.57360 q^{41} +7.72347 q^{43} +7.85014 q^{45} -9.46482 q^{47} -2.65748 q^{49} +3.21320 q^{51} +11.9751 q^{53} -4.27653 q^{55} -0.786802 q^{57} -7.21320 q^{59} -4.87067 q^{61} -4.96158 q^{63} +4.00000 q^{65} +11.3809 q^{67} +7.04545 q^{69} +9.02054 q^{71} +5.65748 q^{73} +4.61906 q^{75} +2.70293 q^{77} -9.57360 q^{79} +3.81172 q^{81} +10.7619 q^{83} -13.4648 q^{85} +7.38094 q^{87} +11.0205 q^{89} -2.52815 q^{91} +3.29707 q^{95} -8.59414 q^{97} -3.08825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} - q^{5} - 4 q^{7} + 12 q^{9} - 5 q^{11} - 5 q^{13} + 10 q^{15} + 2 q^{17} - 3 q^{19} + 9 q^{21} + 5 q^{23} + 6 q^{25} + q^{27} + 9 q^{29} - 12 q^{33} - 9 q^{35} + 6 q^{37} + 19 q^{39} + 8 q^{41} + 17 q^{43} + 27 q^{45} + q^{47} + 5 q^{49} + 11 q^{51} - q^{53} - 19 q^{55} - q^{57} - 23 q^{59} - 3 q^{61} - 47 q^{63} + 12 q^{65} + 15 q^{67} + 43 q^{69} + 12 q^{71} + 4 q^{73} + 33 q^{75} + 17 q^{77} - 26 q^{79} + 47 q^{81} - 6 q^{83} - 11 q^{85} + 3 q^{87} + 18 q^{89} + 17 q^{91} + q^{95} - 8 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.786802 0.454260 0.227130 0.973864i \(-0.427066\pi\)
0.227130 + 0.973864i \(0.427066\pi\)
\(4\) 0 0
\(5\) −3.29707 −1.47449 −0.737247 0.675623i \(-0.763875\pi\)
−0.737247 + 0.675623i \(0.763875\pi\)
\(6\) 0 0
\(7\) 2.08387 0.787630 0.393815 0.919190i \(-0.371155\pi\)
0.393815 + 0.919190i \(0.371155\pi\)
\(8\) 0 0
\(9\) −2.38094 −0.793648
\(10\) 0 0
\(11\) 1.29707 0.391081 0.195541 0.980696i \(-0.437354\pi\)
0.195541 + 0.980696i \(0.437354\pi\)
\(12\) 0 0
\(13\) −1.21320 −0.336481 −0.168240 0.985746i \(-0.553808\pi\)
−0.168240 + 0.985746i \(0.553808\pi\)
\(14\) 0 0
\(15\) −2.59414 −0.669804
\(16\) 0 0
\(17\) 4.08387 0.990485 0.495242 0.868755i \(-0.335079\pi\)
0.495242 + 0.868755i \(0.335079\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.63959 0.357789
\(22\) 0 0
\(23\) 8.95455 1.86715 0.933576 0.358379i \(-0.116671\pi\)
0.933576 + 0.358379i \(0.116671\pi\)
\(24\) 0 0
\(25\) 5.87067 1.17413
\(26\) 0 0
\(27\) −4.23374 −0.814783
\(28\) 0 0
\(29\) 9.38094 1.74200 0.870999 0.491285i \(-0.163473\pi\)
0.870999 + 0.491285i \(0.163473\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 1.02054 0.177653
\(34\) 0 0
\(35\) −6.87067 −1.16136
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −0.954547 −0.152850
\(40\) 0 0
\(41\) 3.57360 0.558103 0.279052 0.960276i \(-0.409980\pi\)
0.279052 + 0.960276i \(0.409980\pi\)
\(42\) 0 0
\(43\) 7.72347 1.17782 0.588909 0.808199i \(-0.299558\pi\)
0.588909 + 0.808199i \(0.299558\pi\)
\(44\) 0 0
\(45\) 7.85014 1.17023
\(46\) 0 0
\(47\) −9.46482 −1.38059 −0.690293 0.723530i \(-0.742518\pi\)
−0.690293 + 0.723530i \(0.742518\pi\)
\(48\) 0 0
\(49\) −2.65748 −0.379639
\(50\) 0 0
\(51\) 3.21320 0.449938
\(52\) 0 0
\(53\) 11.9751 1.64490 0.822452 0.568834i \(-0.192605\pi\)
0.822452 + 0.568834i \(0.192605\pi\)
\(54\) 0 0
\(55\) −4.27653 −0.576648
\(56\) 0 0
\(57\) −0.786802 −0.104214
\(58\) 0 0
\(59\) −7.21320 −0.939078 −0.469539 0.882912i \(-0.655580\pi\)
−0.469539 + 0.882912i \(0.655580\pi\)
\(60\) 0 0
\(61\) −4.87067 −0.623626 −0.311813 0.950144i \(-0.600936\pi\)
−0.311813 + 0.950144i \(0.600936\pi\)
\(62\) 0 0
\(63\) −4.96158 −0.625100
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 11.3809 1.39040 0.695202 0.718815i \(-0.255315\pi\)
0.695202 + 0.718815i \(0.255315\pi\)
\(68\) 0 0
\(69\) 7.04545 0.848173
\(70\) 0 0
\(71\) 9.02054 1.07054 0.535270 0.844681i \(-0.320210\pi\)
0.535270 + 0.844681i \(0.320210\pi\)
\(72\) 0 0
\(73\) 5.65748 0.662157 0.331079 0.943603i \(-0.392587\pi\)
0.331079 + 0.943603i \(0.392587\pi\)
\(74\) 0 0
\(75\) 4.61906 0.533363
\(76\) 0 0
\(77\) 2.70293 0.308027
\(78\) 0 0
\(79\) −9.57360 −1.07711 −0.538557 0.842589i \(-0.681030\pi\)
−0.538557 + 0.842589i \(0.681030\pi\)
\(80\) 0 0
\(81\) 3.81172 0.423524
\(82\) 0 0
\(83\) 10.7619 1.18127 0.590635 0.806939i \(-0.298877\pi\)
0.590635 + 0.806939i \(0.298877\pi\)
\(84\) 0 0
\(85\) −13.4648 −1.46046
\(86\) 0 0
\(87\) 7.38094 0.791320
\(88\) 0 0
\(89\) 11.0205 1.16817 0.584087 0.811691i \(-0.301453\pi\)
0.584087 + 0.811691i \(0.301453\pi\)
\(90\) 0 0
\(91\) −2.52815 −0.265022
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.29707 0.338272
\(96\) 0 0
\(97\) −8.59414 −0.872603 −0.436301 0.899801i \(-0.643712\pi\)
−0.436301 + 0.899801i \(0.643712\pi\)
\(98\) 0 0
\(99\) −3.08825 −0.310381
\(100\) 0 0
\(101\) −9.14721 −0.910181 −0.455091 0.890445i \(-0.650393\pi\)
−0.455091 + 0.890445i \(0.650393\pi\)
\(102\) 0 0
\(103\) −17.3560 −1.71014 −0.855070 0.518512i \(-0.826486\pi\)
−0.855070 + 0.518512i \(0.826486\pi\)
\(104\) 0 0
\(105\) −5.40586 −0.527558
\(106\) 0 0
\(107\) −0.786802 −0.0760630 −0.0380315 0.999277i \(-0.512109\pi\)
−0.0380315 + 0.999277i \(0.512109\pi\)
\(108\) 0 0
\(109\) 13.5487 1.29773 0.648864 0.760904i \(-0.275244\pi\)
0.648864 + 0.760904i \(0.275244\pi\)
\(110\) 0 0
\(111\) 1.57360 0.149360
\(112\) 0 0
\(113\) −1.14721 −0.107920 −0.0539601 0.998543i \(-0.517184\pi\)
−0.0539601 + 0.998543i \(0.517184\pi\)
\(114\) 0 0
\(115\) −29.5238 −2.75311
\(116\) 0 0
\(117\) 2.88856 0.267047
\(118\) 0 0
\(119\) 8.51027 0.780135
\(120\) 0 0
\(121\) −9.31761 −0.847055
\(122\) 0 0
\(123\) 2.81172 0.253524
\(124\) 0 0
\(125\) −2.87067 −0.256761
\(126\) 0 0
\(127\) 1.02054 0.0905581 0.0452790 0.998974i \(-0.485582\pi\)
0.0452790 + 0.998974i \(0.485582\pi\)
\(128\) 0 0
\(129\) 6.07684 0.535036
\(130\) 0 0
\(131\) −7.72347 −0.674802 −0.337401 0.941361i \(-0.609548\pi\)
−0.337401 + 0.941361i \(0.609548\pi\)
\(132\) 0 0
\(133\) −2.08387 −0.180695
\(134\) 0 0
\(135\) 13.9589 1.20139
\(136\) 0 0
\(137\) 15.2311 1.30128 0.650639 0.759387i \(-0.274501\pi\)
0.650639 + 0.759387i \(0.274501\pi\)
\(138\) 0 0
\(139\) 16.0590 1.36210 0.681051 0.732236i \(-0.261523\pi\)
0.681051 + 0.732236i \(0.261523\pi\)
\(140\) 0 0
\(141\) −7.44693 −0.627145
\(142\) 0 0
\(143\) −1.57360 −0.131591
\(144\) 0 0
\(145\) −30.9296 −2.56857
\(146\) 0 0
\(147\) −2.09091 −0.172455
\(148\) 0 0
\(149\) 2.10879 0.172759 0.0863793 0.996262i \(-0.472470\pi\)
0.0863793 + 0.996262i \(0.472470\pi\)
\(150\) 0 0
\(151\) −17.3560 −1.41241 −0.706207 0.708006i \(-0.749595\pi\)
−0.706207 + 0.708006i \(0.749595\pi\)
\(152\) 0 0
\(153\) −9.72347 −0.786096
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 9.42202 0.747215
\(160\) 0 0
\(161\) 18.6601 1.47062
\(162\) 0 0
\(163\) −0.852793 −0.0667959 −0.0333979 0.999442i \(-0.510633\pi\)
−0.0333979 + 0.999442i \(0.510633\pi\)
\(164\) 0 0
\(165\) −3.36478 −0.261948
\(166\) 0 0
\(167\) 18.5941 1.43886 0.719429 0.694566i \(-0.244404\pi\)
0.719429 + 0.694566i \(0.244404\pi\)
\(168\) 0 0
\(169\) −11.5282 −0.886781
\(170\) 0 0
\(171\) 2.38094 0.182075
\(172\) 0 0
\(173\) −2.85279 −0.216894 −0.108447 0.994102i \(-0.534588\pi\)
−0.108447 + 0.994102i \(0.534588\pi\)
\(174\) 0 0
\(175\) 12.2337 0.924783
\(176\) 0 0
\(177\) −5.67536 −0.426586
\(178\) 0 0
\(179\) −26.0768 −1.94907 −0.974537 0.224226i \(-0.928015\pi\)
−0.974537 + 0.224226i \(0.928015\pi\)
\(180\) 0 0
\(181\) −3.18828 −0.236983 −0.118492 0.992955i \(-0.537806\pi\)
−0.118492 + 0.992955i \(0.537806\pi\)
\(182\) 0 0
\(183\) −3.83226 −0.283288
\(184\) 0 0
\(185\) −6.59414 −0.484811
\(186\) 0 0
\(187\) 5.29707 0.387360
\(188\) 0 0
\(189\) −8.82256 −0.641747
\(190\) 0 0
\(191\) −7.95720 −0.575763 −0.287881 0.957666i \(-0.592951\pi\)
−0.287881 + 0.957666i \(0.592951\pi\)
\(192\) 0 0
\(193\) 20.9296 1.50655 0.753274 0.657707i \(-0.228473\pi\)
0.753274 + 0.657707i \(0.228473\pi\)
\(194\) 0 0
\(195\) 3.14721 0.225376
\(196\) 0 0
\(197\) 3.57360 0.254609 0.127304 0.991864i \(-0.459368\pi\)
0.127304 + 0.991864i \(0.459368\pi\)
\(198\) 0 0
\(199\) 26.4194 1.87282 0.936409 0.350909i \(-0.114127\pi\)
0.936409 + 0.350909i \(0.114127\pi\)
\(200\) 0 0
\(201\) 8.95455 0.631605
\(202\) 0 0
\(203\) 19.5487 1.37205
\(204\) 0 0
\(205\) −11.7824 −0.822920
\(206\) 0 0
\(207\) −21.3203 −1.48186
\(208\) 0 0
\(209\) −1.29707 −0.0897202
\(210\) 0 0
\(211\) −23.5487 −1.62116 −0.810579 0.585629i \(-0.800848\pi\)
−0.810579 + 0.585629i \(0.800848\pi\)
\(212\) 0 0
\(213\) 7.09738 0.486304
\(214\) 0 0
\(215\) −25.4648 −1.73669
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.45131 0.300792
\(220\) 0 0
\(221\) −4.95455 −0.333279
\(222\) 0 0
\(223\) −1.02054 −0.0683402 −0.0341701 0.999416i \(-0.510879\pi\)
−0.0341701 + 0.999416i \(0.510879\pi\)
\(224\) 0 0
\(225\) −13.9777 −0.931849
\(226\) 0 0
\(227\) 3.04545 0.202134 0.101067 0.994880i \(-0.467774\pi\)
0.101067 + 0.994880i \(0.467774\pi\)
\(228\) 0 0
\(229\) 4.01788 0.265509 0.132755 0.991149i \(-0.457618\pi\)
0.132755 + 0.991149i \(0.457618\pi\)
\(230\) 0 0
\(231\) 2.12667 0.139925
\(232\) 0 0
\(233\) 25.2062 1.65131 0.825655 0.564175i \(-0.190806\pi\)
0.825655 + 0.564175i \(0.190806\pi\)
\(234\) 0 0
\(235\) 31.2062 2.03567
\(236\) 0 0
\(237\) −7.53253 −0.489290
\(238\) 0 0
\(239\) 6.93667 0.448696 0.224348 0.974509i \(-0.427975\pi\)
0.224348 + 0.974509i \(0.427975\pi\)
\(240\) 0 0
\(241\) 0.761886 0.0490774 0.0245387 0.999699i \(-0.492188\pi\)
0.0245387 + 0.999699i \(0.492188\pi\)
\(242\) 0 0
\(243\) 15.7003 1.00717
\(244\) 0 0
\(245\) 8.76189 0.559776
\(246\) 0 0
\(247\) 1.21320 0.0771940
\(248\) 0 0
\(249\) 8.46747 0.536604
\(250\) 0 0
\(251\) −6.14986 −0.388176 −0.194088 0.980984i \(-0.562175\pi\)
−0.194088 + 0.980984i \(0.562175\pi\)
\(252\) 0 0
\(253\) 11.6147 0.730209
\(254\) 0 0
\(255\) −10.5941 −0.663431
\(256\) 0 0
\(257\) −23.3560 −1.45691 −0.728454 0.685094i \(-0.759761\pi\)
−0.728454 + 0.685094i \(0.759761\pi\)
\(258\) 0 0
\(259\) 4.16774 0.258971
\(260\) 0 0
\(261\) −22.3355 −1.38253
\(262\) 0 0
\(263\) −12.2765 −0.757003 −0.378502 0.925601i \(-0.623561\pi\)
−0.378502 + 0.925601i \(0.623561\pi\)
\(264\) 0 0
\(265\) −39.4827 −2.42540
\(266\) 0 0
\(267\) 8.67098 0.530655
\(268\) 0 0
\(269\) 7.74135 0.471998 0.235999 0.971753i \(-0.424164\pi\)
0.235999 + 0.971753i \(0.424164\pi\)
\(270\) 0 0
\(271\) 0.954547 0.0579846 0.0289923 0.999580i \(-0.490770\pi\)
0.0289923 + 0.999580i \(0.490770\pi\)
\(272\) 0 0
\(273\) −1.98915 −0.120389
\(274\) 0 0
\(275\) 7.61468 0.459182
\(276\) 0 0
\(277\) −17.0384 −1.02374 −0.511870 0.859063i \(-0.671047\pi\)
−0.511870 + 0.859063i \(0.671047\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.0974 −0.781324 −0.390662 0.920534i \(-0.627754\pi\)
−0.390662 + 0.920534i \(0.627754\pi\)
\(282\) 0 0
\(283\) 1.29707 0.0771028 0.0385514 0.999257i \(-0.487726\pi\)
0.0385514 + 0.999257i \(0.487726\pi\)
\(284\) 0 0
\(285\) 2.59414 0.153664
\(286\) 0 0
\(287\) 7.44693 0.439579
\(288\) 0 0
\(289\) −0.321987 −0.0189404
\(290\) 0 0
\(291\) −6.76189 −0.396389
\(292\) 0 0
\(293\) 21.5487 1.25889 0.629444 0.777046i \(-0.283283\pi\)
0.629444 + 0.777046i \(0.283283\pi\)
\(294\) 0 0
\(295\) 23.7824 1.38467
\(296\) 0 0
\(297\) −5.49145 −0.318646
\(298\) 0 0
\(299\) −10.8636 −0.628260
\(300\) 0 0
\(301\) 16.0947 0.927684
\(302\) 0 0
\(303\) −7.19704 −0.413459
\(304\) 0 0
\(305\) 16.0590 0.919533
\(306\) 0 0
\(307\) −9.18828 −0.524403 −0.262201 0.965013i \(-0.584449\pi\)
−0.262201 + 0.965013i \(0.584449\pi\)
\(308\) 0 0
\(309\) −13.6558 −0.776849
\(310\) 0 0
\(311\) 7.48973 0.424704 0.212352 0.977193i \(-0.431888\pi\)
0.212352 + 0.977193i \(0.431888\pi\)
\(312\) 0 0
\(313\) −9.76626 −0.552022 −0.276011 0.961154i \(-0.589013\pi\)
−0.276011 + 0.961154i \(0.589013\pi\)
\(314\) 0 0
\(315\) 16.3587 0.921707
\(316\) 0 0
\(317\) −13.7164 −0.770392 −0.385196 0.922835i \(-0.625866\pi\)
−0.385196 + 0.922835i \(0.625866\pi\)
\(318\) 0 0
\(319\) 12.1677 0.681263
\(320\) 0 0
\(321\) −0.619057 −0.0345524
\(322\) 0 0
\(323\) −4.08387 −0.227233
\(324\) 0 0
\(325\) −7.12229 −0.395074
\(326\) 0 0
\(327\) 10.6601 0.589507
\(328\) 0 0
\(329\) −19.7235 −1.08739
\(330\) 0 0
\(331\) −14.5282 −0.798539 −0.399270 0.916834i \(-0.630736\pi\)
−0.399270 + 0.916834i \(0.630736\pi\)
\(332\) 0 0
\(333\) −4.76189 −0.260950
\(334\) 0 0
\(335\) −37.5238 −2.05014
\(336\) 0 0
\(337\) 9.31495 0.507418 0.253709 0.967281i \(-0.418349\pi\)
0.253709 + 0.967281i \(0.418349\pi\)
\(338\) 0 0
\(339\) −0.902625 −0.0490238
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1249 −1.08665
\(344\) 0 0
\(345\) −23.2294 −1.25063
\(346\) 0 0
\(347\) −9.29707 −0.499093 −0.249546 0.968363i \(-0.580282\pi\)
−0.249546 + 0.968363i \(0.580282\pi\)
\(348\) 0 0
\(349\) 14.0590 0.752559 0.376279 0.926506i \(-0.377203\pi\)
0.376279 + 0.926506i \(0.377203\pi\)
\(350\) 0 0
\(351\) 5.13636 0.274159
\(352\) 0 0
\(353\) −20.4783 −1.08995 −0.544975 0.838452i \(-0.683461\pi\)
−0.544975 + 0.838452i \(0.683461\pi\)
\(354\) 0 0
\(355\) −29.7413 −1.57851
\(356\) 0 0
\(357\) 6.69589 0.354384
\(358\) 0 0
\(359\) −18.9724 −1.00133 −0.500663 0.865642i \(-0.666910\pi\)
−0.500663 + 0.865642i \(0.666910\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −7.33111 −0.384784
\(364\) 0 0
\(365\) −18.6531 −0.976348
\(366\) 0 0
\(367\) 4.85279 0.253314 0.126657 0.991947i \(-0.459575\pi\)
0.126657 + 0.991947i \(0.459575\pi\)
\(368\) 0 0
\(369\) −8.50855 −0.442937
\(370\) 0 0
\(371\) 24.9545 1.29558
\(372\) 0 0
\(373\) 12.1071 0.626880 0.313440 0.949608i \(-0.398519\pi\)
0.313440 + 0.949608i \(0.398519\pi\)
\(374\) 0 0
\(375\) −2.25865 −0.116636
\(376\) 0 0
\(377\) −11.3809 −0.586148
\(378\) 0 0
\(379\) −2.02492 −0.104013 −0.0520065 0.998647i \(-0.516562\pi\)
−0.0520065 + 0.998647i \(0.516562\pi\)
\(380\) 0 0
\(381\) 0.802961 0.0411369
\(382\) 0 0
\(383\) 1.23811 0.0632647 0.0316323 0.999500i \(-0.489929\pi\)
0.0316323 + 0.999500i \(0.489929\pi\)
\(384\) 0 0
\(385\) −8.91175 −0.454185
\(386\) 0 0
\(387\) −18.3891 −0.934772
\(388\) 0 0
\(389\) −12.8707 −0.652569 −0.326285 0.945272i \(-0.605797\pi\)
−0.326285 + 0.945272i \(0.605797\pi\)
\(390\) 0 0
\(391\) 36.5692 1.84939
\(392\) 0 0
\(393\) −6.07684 −0.306536
\(394\) 0 0
\(395\) 31.5648 1.58820
\(396\) 0 0
\(397\) −17.3739 −0.871971 −0.435986 0.899954i \(-0.643600\pi\)
−0.435986 + 0.899954i \(0.643600\pi\)
\(398\) 0 0
\(399\) −1.63959 −0.0820824
\(400\) 0 0
\(401\) 19.0205 0.949840 0.474920 0.880029i \(-0.342477\pi\)
0.474920 + 0.880029i \(0.342477\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −12.5675 −0.624484
\(406\) 0 0
\(407\) 2.59414 0.128587
\(408\) 0 0
\(409\) 20.4622 1.01179 0.505894 0.862595i \(-0.331163\pi\)
0.505894 + 0.862595i \(0.331163\pi\)
\(410\) 0 0
\(411\) 11.9838 0.591119
\(412\) 0 0
\(413\) −15.0314 −0.739646
\(414\) 0 0
\(415\) −35.4827 −1.74178
\(416\) 0 0
\(417\) 12.6352 0.618749
\(418\) 0 0
\(419\) −7.14721 −0.349164 −0.174582 0.984643i \(-0.555857\pi\)
−0.174582 + 0.984643i \(0.555857\pi\)
\(420\) 0 0
\(421\) −12.3604 −0.602409 −0.301205 0.953560i \(-0.597389\pi\)
−0.301205 + 0.953560i \(0.597389\pi\)
\(422\) 0 0
\(423\) 22.5352 1.09570
\(424\) 0 0
\(425\) 23.9751 1.16296
\(426\) 0 0
\(427\) −10.1499 −0.491186
\(428\) 0 0
\(429\) −1.23811 −0.0597767
\(430\) 0 0
\(431\) 1.90909 0.0919578 0.0459789 0.998942i \(-0.485359\pi\)
0.0459789 + 0.998942i \(0.485359\pi\)
\(432\) 0 0
\(433\) 4.04107 0.194202 0.0971008 0.995275i \(-0.469043\pi\)
0.0971008 + 0.995275i \(0.469043\pi\)
\(434\) 0 0
\(435\) −24.3355 −1.16680
\(436\) 0 0
\(437\) −8.95455 −0.428354
\(438\) 0 0
\(439\) −14.0768 −0.671851 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(440\) 0 0
\(441\) 6.32730 0.301300
\(442\) 0 0
\(443\) −19.6736 −0.934723 −0.467361 0.884066i \(-0.654795\pi\)
−0.467361 + 0.884066i \(0.654795\pi\)
\(444\) 0 0
\(445\) −36.3355 −1.72247
\(446\) 0 0
\(447\) 1.65920 0.0784774
\(448\) 0 0
\(449\) 10.5531 0.498030 0.249015 0.968500i \(-0.419893\pi\)
0.249015 + 0.968500i \(0.419893\pi\)
\(450\) 0 0
\(451\) 4.63522 0.218264
\(452\) 0 0
\(453\) −13.6558 −0.641603
\(454\) 0 0
\(455\) 8.33549 0.390774
\(456\) 0 0
\(457\) 2.72785 0.127603 0.0638016 0.997963i \(-0.479678\pi\)
0.0638016 + 0.997963i \(0.479678\pi\)
\(458\) 0 0
\(459\) −17.2900 −0.807030
\(460\) 0 0
\(461\) 35.1153 1.63548 0.817740 0.575587i \(-0.195227\pi\)
0.817740 + 0.575587i \(0.195227\pi\)
\(462\) 0 0
\(463\) −17.1293 −0.796067 −0.398034 0.917371i \(-0.630307\pi\)
−0.398034 + 0.917371i \(0.630307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.7029 0.680370 0.340185 0.940358i \(-0.389510\pi\)
0.340185 + 0.940358i \(0.389510\pi\)
\(468\) 0 0
\(469\) 23.7164 1.09512
\(470\) 0 0
\(471\) −11.0152 −0.507555
\(472\) 0 0
\(473\) 10.0179 0.460623
\(474\) 0 0
\(475\) −5.87067 −0.269365
\(476\) 0 0
\(477\) −28.5120 −1.30547
\(478\) 0 0
\(479\) −10.0411 −0.458788 −0.229394 0.973334i \(-0.573674\pi\)
−0.229394 + 0.973334i \(0.573674\pi\)
\(480\) 0 0
\(481\) −2.42640 −0.110634
\(482\) 0 0
\(483\) 14.6818 0.668046
\(484\) 0 0
\(485\) 28.3355 1.28665
\(486\) 0 0
\(487\) 23.6504 1.07170 0.535852 0.844312i \(-0.319991\pi\)
0.535852 + 0.844312i \(0.319991\pi\)
\(488\) 0 0
\(489\) −0.670979 −0.0303427
\(490\) 0 0
\(491\) −38.3766 −1.73191 −0.865955 0.500122i \(-0.833289\pi\)
−0.865955 + 0.500122i \(0.833289\pi\)
\(492\) 0 0
\(493\) 38.3106 1.72542
\(494\) 0 0
\(495\) 10.1822 0.457655
\(496\) 0 0
\(497\) 18.7976 0.843190
\(498\) 0 0
\(499\) 10.3176 0.461880 0.230940 0.972968i \(-0.425820\pi\)
0.230940 + 0.972968i \(0.425820\pi\)
\(500\) 0 0
\(501\) 14.6299 0.653616
\(502\) 0 0
\(503\) 20.4372 0.911252 0.455626 0.890171i \(-0.349416\pi\)
0.455626 + 0.890171i \(0.349416\pi\)
\(504\) 0 0
\(505\) 30.1590 1.34206
\(506\) 0 0
\(507\) −9.07037 −0.402829
\(508\) 0 0
\(509\) 15.4059 0.682853 0.341426 0.939909i \(-0.389090\pi\)
0.341426 + 0.939909i \(0.389090\pi\)
\(510\) 0 0
\(511\) 11.7895 0.521535
\(512\) 0 0
\(513\) 4.23374 0.186924
\(514\) 0 0
\(515\) 57.2240 2.52159
\(516\) 0 0
\(517\) −12.2765 −0.539921
\(518\) 0 0
\(519\) −2.24458 −0.0985262
\(520\) 0 0
\(521\) 36.0270 1.57837 0.789186 0.614154i \(-0.210503\pi\)
0.789186 + 0.614154i \(0.210503\pi\)
\(522\) 0 0
\(523\) 44.7370 1.95621 0.978106 0.208109i \(-0.0667310\pi\)
0.978106 + 0.208109i \(0.0667310\pi\)
\(524\) 0 0
\(525\) 9.62553 0.420092
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 57.1839 2.48626
\(530\) 0 0
\(531\) 17.1742 0.745297
\(532\) 0 0
\(533\) −4.33549 −0.187791
\(534\) 0 0
\(535\) 2.59414 0.112154
\(536\) 0 0
\(537\) −20.5173 −0.885387
\(538\) 0 0
\(539\) −3.44693 −0.148470
\(540\) 0 0
\(541\) 37.5059 1.61250 0.806252 0.591572i \(-0.201493\pi\)
0.806252 + 0.591572i \(0.201493\pi\)
\(542\) 0 0
\(543\) −2.50855 −0.107652
\(544\) 0 0
\(545\) −44.6710 −1.91349
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) 11.5968 0.494939
\(550\) 0 0
\(551\) −9.38094 −0.399642
\(552\) 0 0
\(553\) −19.9502 −0.848367
\(554\) 0 0
\(555\) −5.18828 −0.220230
\(556\) 0 0
\(557\) 33.3381 1.41258 0.706291 0.707921i \(-0.250367\pi\)
0.706291 + 0.707921i \(0.250367\pi\)
\(558\) 0 0
\(559\) −9.37010 −0.396313
\(560\) 0 0
\(561\) 4.16774 0.175962
\(562\) 0 0
\(563\) 3.44693 0.145271 0.0726355 0.997359i \(-0.476859\pi\)
0.0726355 + 0.997359i \(0.476859\pi\)
\(564\) 0 0
\(565\) 3.78242 0.159128
\(566\) 0 0
\(567\) 7.94313 0.333580
\(568\) 0 0
\(569\) 27.0205 1.13276 0.566380 0.824144i \(-0.308343\pi\)
0.566380 + 0.824144i \(0.308343\pi\)
\(570\) 0 0
\(571\) −5.70559 −0.238771 −0.119386 0.992848i \(-0.538092\pi\)
−0.119386 + 0.992848i \(0.538092\pi\)
\(572\) 0 0
\(573\) −6.26074 −0.261546
\(574\) 0 0
\(575\) 52.5692 2.19229
\(576\) 0 0
\(577\) 6.34252 0.264043 0.132021 0.991247i \(-0.457853\pi\)
0.132021 + 0.991247i \(0.457853\pi\)
\(578\) 0 0
\(579\) 16.4675 0.684365
\(580\) 0 0
\(581\) 22.4264 0.930404
\(582\) 0 0
\(583\) 15.5325 0.643292
\(584\) 0 0
\(585\) −9.52377 −0.393759
\(586\) 0 0
\(587\) −37.5827 −1.55121 −0.775603 0.631222i \(-0.782554\pi\)
−0.775603 + 0.631222i \(0.782554\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 2.81172 0.115659
\(592\) 0 0
\(593\) −9.48270 −0.389408 −0.194704 0.980862i \(-0.562375\pi\)
−0.194704 + 0.980862i \(0.562375\pi\)
\(594\) 0 0
\(595\) −28.0590 −1.15031
\(596\) 0 0
\(597\) 20.7868 0.850747
\(598\) 0 0
\(599\) −24.6710 −1.00803 −0.504014 0.863695i \(-0.668144\pi\)
−0.504014 + 0.863695i \(0.668144\pi\)
\(600\) 0 0
\(601\) 41.4329 1.69008 0.845041 0.534702i \(-0.179576\pi\)
0.845041 + 0.534702i \(0.179576\pi\)
\(602\) 0 0
\(603\) −27.0974 −1.10349
\(604\) 0 0
\(605\) 30.7208 1.24898
\(606\) 0 0
\(607\) −5.32026 −0.215943 −0.107971 0.994154i \(-0.534436\pi\)
−0.107971 + 0.994154i \(0.534436\pi\)
\(608\) 0 0
\(609\) 15.3809 0.623267
\(610\) 0 0
\(611\) 11.4827 0.464540
\(612\) 0 0
\(613\) 27.2472 1.10051 0.550253 0.834998i \(-0.314531\pi\)
0.550253 + 0.834998i \(0.314531\pi\)
\(614\) 0 0
\(615\) −9.27043 −0.373820
\(616\) 0 0
\(617\) −30.9117 −1.24446 −0.622230 0.782834i \(-0.713773\pi\)
−0.622230 + 0.782834i \(0.713773\pi\)
\(618\) 0 0
\(619\) −24.2857 −0.976123 −0.488061 0.872809i \(-0.662296\pi\)
−0.488061 + 0.872809i \(0.662296\pi\)
\(620\) 0 0
\(621\) −37.9112 −1.52132
\(622\) 0 0
\(623\) 22.9654 0.920089
\(624\) 0 0
\(625\) −19.8886 −0.795542
\(626\) 0 0
\(627\) −1.02054 −0.0407563
\(628\) 0 0
\(629\) 8.16774 0.325669
\(630\) 0 0
\(631\) −24.9117 −0.991721 −0.495861 0.868402i \(-0.665147\pi\)
−0.495861 + 0.868402i \(0.665147\pi\)
\(632\) 0 0
\(633\) −18.5282 −0.736428
\(634\) 0 0
\(635\) −3.36478 −0.133527
\(636\) 0 0
\(637\) 3.22405 0.127741
\(638\) 0 0
\(639\) −21.4774 −0.849632
\(640\) 0 0
\(641\) 4.25865 0.168207 0.0841033 0.996457i \(-0.473197\pi\)
0.0841033 + 0.996457i \(0.473197\pi\)
\(642\) 0 0
\(643\) −13.9323 −0.549436 −0.274718 0.961525i \(-0.588584\pi\)
−0.274718 + 0.961525i \(0.588584\pi\)
\(644\) 0 0
\(645\) −20.0358 −0.788907
\(646\) 0 0
\(647\) −5.58064 −0.219398 −0.109699 0.993965i \(-0.534989\pi\)
−0.109699 + 0.993965i \(0.534989\pi\)
\(648\) 0 0
\(649\) −9.35603 −0.367256
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.77330 0.0693945 0.0346973 0.999398i \(-0.488953\pi\)
0.0346973 + 0.999398i \(0.488953\pi\)
\(654\) 0 0
\(655\) 25.4648 0.994993
\(656\) 0 0
\(657\) −13.4701 −0.525520
\(658\) 0 0
\(659\) −1.12229 −0.0437183 −0.0218591 0.999761i \(-0.506959\pi\)
−0.0218591 + 0.999761i \(0.506959\pi\)
\(660\) 0 0
\(661\) −21.9340 −0.853134 −0.426567 0.904456i \(-0.640277\pi\)
−0.426567 + 0.904456i \(0.640277\pi\)
\(662\) 0 0
\(663\) −3.89825 −0.151395
\(664\) 0 0
\(665\) 6.87067 0.266433
\(666\) 0 0
\(667\) 84.0021 3.25257
\(668\) 0 0
\(669\) −0.802961 −0.0310443
\(670\) 0 0
\(671\) −6.31761 −0.243889
\(672\) 0 0
\(673\) −8.24458 −0.317805 −0.158903 0.987294i \(-0.550796\pi\)
−0.158903 + 0.987294i \(0.550796\pi\)
\(674\) 0 0
\(675\) −24.8549 −0.956665
\(676\) 0 0
\(677\) 5.76626 0.221616 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(678\) 0 0
\(679\) −17.9091 −0.687288
\(680\) 0 0
\(681\) 2.39617 0.0918214
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −50.2179 −1.91873
\(686\) 0 0
\(687\) 3.16128 0.120610
\(688\) 0 0
\(689\) −14.5282 −0.553478
\(690\) 0 0
\(691\) 23.0384 0.876423 0.438211 0.898872i \(-0.355612\pi\)
0.438211 + 0.898872i \(0.355612\pi\)
\(692\) 0 0
\(693\) −6.43552 −0.244465
\(694\) 0 0
\(695\) −52.9475 −2.00841
\(696\) 0 0
\(697\) 14.5941 0.552793
\(698\) 0 0
\(699\) 19.8323 0.750125
\(700\) 0 0
\(701\) 3.90909 0.147644 0.0738222 0.997271i \(-0.476480\pi\)
0.0738222 + 0.997271i \(0.476480\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 24.5531 0.924722
\(706\) 0 0
\(707\) −19.0616 −0.716886
\(708\) 0 0
\(709\) 12.0411 0.452212 0.226106 0.974103i \(-0.427400\pi\)
0.226106 + 0.974103i \(0.427400\pi\)
\(710\) 0 0
\(711\) 22.7942 0.854849
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 5.18828 0.194031
\(716\) 0 0
\(717\) 5.45778 0.203825
\(718\) 0 0
\(719\) −32.2927 −1.20431 −0.602157 0.798378i \(-0.705692\pi\)
−0.602157 + 0.798378i \(0.705692\pi\)
\(720\) 0 0
\(721\) −36.1677 −1.34696
\(722\) 0 0
\(723\) 0.599453 0.0222939
\(724\) 0 0
\(725\) 55.0725 2.04534
\(726\) 0 0
\(727\) −41.1812 −1.52733 −0.763664 0.645614i \(-0.776602\pi\)
−0.763664 + 0.645614i \(0.776602\pi\)
\(728\) 0 0
\(729\) 0.917850 0.0339945
\(730\) 0 0
\(731\) 31.5417 1.16661
\(732\) 0 0
\(733\) −1.27919 −0.0472479 −0.0236240 0.999721i \(-0.507520\pi\)
−0.0236240 + 0.999721i \(0.507520\pi\)
\(734\) 0 0
\(735\) 6.89387 0.254284
\(736\) 0 0
\(737\) 14.7619 0.543761
\(738\) 0 0
\(739\) 30.3534 1.11657 0.558283 0.829650i \(-0.311460\pi\)
0.558283 + 0.829650i \(0.311460\pi\)
\(740\) 0 0
\(741\) 0.954547 0.0350661
\(742\) 0 0
\(743\) 19.4827 0.714751 0.357375 0.933961i \(-0.383672\pi\)
0.357375 + 0.933961i \(0.383672\pi\)
\(744\) 0 0
\(745\) −6.95282 −0.254732
\(746\) 0 0
\(747\) −25.6234 −0.937512
\(748\) 0 0
\(749\) −1.63959 −0.0599095
\(750\) 0 0
\(751\) 48.6710 1.77603 0.888015 0.459815i \(-0.152084\pi\)
0.888015 + 0.459815i \(0.152084\pi\)
\(752\) 0 0
\(753\) −4.83872 −0.176333
\(754\) 0 0
\(755\) 57.2240 2.08260
\(756\) 0 0
\(757\) −41.7235 −1.51647 −0.758233 0.651984i \(-0.773937\pi\)
−0.758233 + 0.651984i \(0.773937\pi\)
\(758\) 0 0
\(759\) 9.13845 0.331705
\(760\) 0 0
\(761\) −9.10441 −0.330035 −0.165017 0.986291i \(-0.552768\pi\)
−0.165017 + 0.986291i \(0.552768\pi\)
\(762\) 0 0
\(763\) 28.2337 1.02213
\(764\) 0 0
\(765\) 32.0590 1.15909
\(766\) 0 0
\(767\) 8.75104 0.315982
\(768\) 0 0
\(769\) 7.14548 0.257673 0.128836 0.991666i \(-0.458876\pi\)
0.128836 + 0.991666i \(0.458876\pi\)
\(770\) 0 0
\(771\) −18.3766 −0.661816
\(772\) 0 0
\(773\) 28.9956 1.04290 0.521450 0.853282i \(-0.325391\pi\)
0.521450 + 0.853282i \(0.325391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.27919 0.117640
\(778\) 0 0
\(779\) −3.57360 −0.128038
\(780\) 0 0
\(781\) 11.7003 0.418669
\(782\) 0 0
\(783\) −39.7164 −1.41935
\(784\) 0 0
\(785\) 46.1590 1.64748
\(786\) 0 0
\(787\) −23.3311 −0.831664 −0.415832 0.909441i \(-0.636510\pi\)
−0.415832 + 0.909441i \(0.636510\pi\)
\(788\) 0 0
\(789\) −9.65920 −0.343877
\(790\) 0 0
\(791\) −2.39063 −0.0850011
\(792\) 0 0
\(793\) 5.90909 0.209838
\(794\) 0 0
\(795\) −31.0651 −1.10176
\(796\) 0 0
\(797\) −19.2543 −0.682021 −0.341011 0.940059i \(-0.610769\pi\)
−0.341011 + 0.940059i \(0.610769\pi\)
\(798\) 0 0
\(799\) −38.6531 −1.36745
\(800\) 0 0
\(801\) −26.2393 −0.927119
\(802\) 0 0
\(803\) 7.33815 0.258958
\(804\) 0 0
\(805\) −61.5238 −2.16843
\(806\) 0 0
\(807\) 6.09091 0.214410
\(808\) 0 0
\(809\) 27.9983 0.984367 0.492184 0.870491i \(-0.336199\pi\)
0.492184 + 0.870491i \(0.336199\pi\)
\(810\) 0 0
\(811\) −47.4631 −1.66665 −0.833327 0.552780i \(-0.813567\pi\)
−0.833327 + 0.552780i \(0.813567\pi\)
\(812\) 0 0
\(813\) 0.751039 0.0263401
\(814\) 0 0
\(815\) 2.81172 0.0984902
\(816\) 0 0
\(817\) −7.72347 −0.270210
\(818\) 0 0
\(819\) 6.01938 0.210334
\(820\) 0 0
\(821\) −2.14455 −0.0748454 −0.0374227 0.999300i \(-0.511915\pi\)
−0.0374227 + 0.999300i \(0.511915\pi\)
\(822\) 0 0
\(823\) 32.8458 1.14493 0.572466 0.819929i \(-0.305987\pi\)
0.572466 + 0.819929i \(0.305987\pi\)
\(824\) 0 0
\(825\) 5.99124 0.208588
\(826\) 0 0
\(827\) 26.6103 0.925331 0.462665 0.886533i \(-0.346893\pi\)
0.462665 + 0.886533i \(0.346893\pi\)
\(828\) 0 0
\(829\) −50.6871 −1.76044 −0.880219 0.474569i \(-0.842604\pi\)
−0.880219 + 0.474569i \(0.842604\pi\)
\(830\) 0 0
\(831\) −13.4059 −0.465044
\(832\) 0 0
\(833\) −10.8528 −0.376027
\(834\) 0 0
\(835\) −61.3062 −2.12159
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 51.2651 1.76987 0.884934 0.465716i \(-0.154203\pi\)
0.884934 + 0.465716i \(0.154203\pi\)
\(840\) 0 0
\(841\) 59.0021 2.03455
\(842\) 0 0
\(843\) −10.3050 −0.354924
\(844\) 0 0
\(845\) 38.0091 1.30755
\(846\) 0 0
\(847\) −19.4167 −0.667166
\(848\) 0 0
\(849\) 1.02054 0.0350248
\(850\) 0 0
\(851\) 17.9091 0.613916
\(852\) 0 0
\(853\) 7.95893 0.272508 0.136254 0.990674i \(-0.456494\pi\)
0.136254 + 0.990674i \(0.456494\pi\)
\(854\) 0 0
\(855\) −7.85014 −0.268469
\(856\) 0 0
\(857\) −4.09091 −0.139743 −0.0698714 0.997556i \(-0.522259\pi\)
−0.0698714 + 0.997556i \(0.522259\pi\)
\(858\) 0 0
\(859\) −13.3328 −0.454910 −0.227455 0.973789i \(-0.573041\pi\)
−0.227455 + 0.973789i \(0.573041\pi\)
\(860\) 0 0
\(861\) 5.85926 0.199683
\(862\) 0 0
\(863\) 20.1677 0.686518 0.343259 0.939241i \(-0.388469\pi\)
0.343259 + 0.939241i \(0.388469\pi\)
\(864\) 0 0
\(865\) 9.40586 0.319809
\(866\) 0 0
\(867\) −0.253340 −0.00860386
\(868\) 0 0
\(869\) −12.4176 −0.421240
\(870\) 0 0
\(871\) −13.8073 −0.467844
\(872\) 0 0
\(873\) 20.4622 0.692539
\(874\) 0 0
\(875\) −5.98212 −0.202233
\(876\) 0 0
\(877\) −23.8572 −0.805599 −0.402800 0.915288i \(-0.631963\pi\)
−0.402800 + 0.915288i \(0.631963\pi\)
\(878\) 0 0
\(879\) 16.9545 0.571863
\(880\) 0 0
\(881\) −47.9182 −1.61441 −0.807203 0.590274i \(-0.799020\pi\)
−0.807203 + 0.590274i \(0.799020\pi\)
\(882\) 0 0
\(883\) −28.5622 −0.961194 −0.480597 0.876942i \(-0.659580\pi\)
−0.480597 + 0.876942i \(0.659580\pi\)
\(884\) 0 0
\(885\) 18.7121 0.628999
\(886\) 0 0
\(887\) −47.3150 −1.58868 −0.794340 0.607473i \(-0.792183\pi\)
−0.794340 + 0.607473i \(0.792183\pi\)
\(888\) 0 0
\(889\) 2.12667 0.0713262
\(890\) 0 0
\(891\) 4.94407 0.165632
\(892\) 0 0
\(893\) 9.46482 0.316728
\(894\) 0 0
\(895\) 85.9772 2.87390
\(896\) 0 0
\(897\) −8.54753 −0.285394
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 48.9047 1.62925
\(902\) 0 0
\(903\) 12.6634 0.421410
\(904\) 0 0
\(905\) 10.5120 0.349430
\(906\) 0 0
\(907\) −15.8982 −0.527893 −0.263946 0.964537i \(-0.585024\pi\)
−0.263946 + 0.964537i \(0.585024\pi\)
\(908\) 0 0
\(909\) 21.7790 0.722363
\(910\) 0 0
\(911\) 20.2997 0.672560 0.336280 0.941762i \(-0.390831\pi\)
0.336280 + 0.941762i \(0.390831\pi\)
\(912\) 0 0
\(913\) 13.9589 0.461973
\(914\) 0 0
\(915\) 12.6352 0.417707
\(916\) 0 0
\(917\) −16.0947 −0.531494
\(918\) 0 0
\(919\) −32.6191 −1.07600 −0.538002 0.842944i \(-0.680821\pi\)
−0.538002 + 0.842944i \(0.680821\pi\)
\(920\) 0 0
\(921\) −7.22936 −0.238215
\(922\) 0 0
\(923\) −10.9437 −0.360216
\(924\) 0 0
\(925\) 11.7413 0.386053
\(926\) 0 0
\(927\) 41.3237 1.35725
\(928\) 0 0
\(929\) 53.4631 1.75407 0.877034 0.480429i \(-0.159519\pi\)
0.877034 + 0.480429i \(0.159519\pi\)
\(930\) 0 0
\(931\) 2.65748 0.0870953
\(932\) 0 0
\(933\) 5.89293 0.192926
\(934\) 0 0
\(935\) −17.4648 −0.571161
\(936\) 0 0
\(937\) −43.0135 −1.40519 −0.702595 0.711590i \(-0.747975\pi\)
−0.702595 + 0.711590i \(0.747975\pi\)
\(938\) 0 0
\(939\) −7.68411 −0.250762
\(940\) 0 0
\(941\) −25.3311 −0.825771 −0.412885 0.910783i \(-0.635479\pi\)
−0.412885 + 0.910783i \(0.635479\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 29.0886 0.946253
\(946\) 0 0
\(947\) 12.6710 0.411751 0.205876 0.978578i \(-0.433996\pi\)
0.205876 + 0.978578i \(0.433996\pi\)
\(948\) 0 0
\(949\) −6.86364 −0.222803
\(950\) 0 0
\(951\) −10.7921 −0.349958
\(952\) 0 0
\(953\) −31.2240 −1.01145 −0.505723 0.862696i \(-0.668774\pi\)
−0.505723 + 0.862696i \(0.668774\pi\)
\(954\) 0 0
\(955\) 26.2355 0.848959
\(956\) 0 0
\(957\) 9.57360 0.309471
\(958\) 0 0
\(959\) 31.7396 1.02493
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 1.87333 0.0603672
\(964\) 0 0
\(965\) −69.0065 −2.22140
\(966\) 0 0
\(967\) −6.29441 −0.202415 −0.101207 0.994865i \(-0.532271\pi\)
−0.101207 + 0.994865i \(0.532271\pi\)
\(968\) 0 0
\(969\) −3.21320 −0.103223
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 33.4648 1.07283
\(974\) 0 0
\(975\) −5.60383 −0.179466
\(976\) 0 0
\(977\) −34.1677 −1.09312 −0.546561 0.837419i \(-0.684064\pi\)
−0.546561 + 0.837419i \(0.684064\pi\)
\(978\) 0 0
\(979\) 14.2944 0.456851
\(980\) 0 0
\(981\) −32.2587 −1.02994
\(982\) 0 0
\(983\) −30.4264 −0.970451 −0.485226 0.874389i \(-0.661263\pi\)
−0.485226 + 0.874389i \(0.661263\pi\)
\(984\) 0 0
\(985\) −11.7824 −0.375419
\(986\) 0 0
\(987\) −15.5185 −0.493958
\(988\) 0 0
\(989\) 69.1601 2.19916
\(990\) 0 0
\(991\) 24.1179 0.766131 0.383065 0.923721i \(-0.374868\pi\)
0.383065 + 0.923721i \(0.374868\pi\)
\(992\) 0 0
\(993\) −11.4308 −0.362745
\(994\) 0 0
\(995\) −87.1065 −2.76146
\(996\) 0 0
\(997\) −15.9323 −0.504581 −0.252290 0.967652i \(-0.581184\pi\)
−0.252290 + 0.967652i \(0.581184\pi\)
\(998\) 0 0
\(999\) −8.46747 −0.267899
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 1216.2.a.v.1.2 3
4.3 odd 2 1216.2.a.u.1.2 3
8.3 odd 2 152.2.a.c.1.2 3
8.5 even 2 304.2.a.g.1.2 3
24.5 odd 2 2736.2.a.bd.1.1 3
24.11 even 2 1368.2.a.n.1.1 3
40.3 even 4 3800.2.d.j.3649.4 6
40.19 odd 2 3800.2.a.r.1.2 3
40.27 even 4 3800.2.d.j.3649.3 6
40.29 even 2 7600.2.a.bv.1.2 3
56.27 even 2 7448.2.a.bf.1.2 3
152.37 odd 2 5776.2.a.bp.1.2 3
152.75 even 2 2888.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.2 3 8.3 odd 2
304.2.a.g.1.2 3 8.5 even 2
1216.2.a.u.1.2 3 4.3 odd 2
1216.2.a.v.1.2 3 1.1 even 1 trivial
1368.2.a.n.1.1 3 24.11 even 2
2736.2.a.bd.1.1 3 24.5 odd 2
2888.2.a.o.1.2 3 152.75 even 2
3800.2.a.r.1.2 3 40.19 odd 2
3800.2.d.j.3649.3 6 40.27 even 4
3800.2.d.j.3649.4 6 40.3 even 4
5776.2.a.bp.1.2 3 152.37 odd 2
7448.2.a.bf.1.2 3 56.27 even 2
7600.2.a.bv.1.2 3 40.29 even 2