Properties

Label 3800.2.d.j.3649.3
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.59105344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 21x^{4} + 116x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.3
Root \(-3.29707i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.j.3649.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +O(q^{10})\) \(q-0.786802i q^{3} -2.08387i q^{7} +2.38094 q^{9} +1.29707 q^{11} -1.21320i q^{13} +4.08387i q^{17} +1.00000 q^{19} -1.63959 q^{21} +8.95455i q^{23} -4.23374i q^{27} +9.38094 q^{29} -1.02054i q^{33} -2.00000i q^{37} -0.954547 q^{39} +3.57360 q^{41} -7.72347i q^{43} +9.46482i q^{47} +2.65748 q^{49} +3.21320 q^{51} +11.9751i q^{53} -0.786802i q^{57} +7.21320 q^{59} +4.87067 q^{61} -4.96158i q^{63} +11.3809i q^{67} +7.04545 q^{69} -9.02054 q^{71} -5.65748i q^{73} -2.70293i q^{77} -9.57360 q^{79} +3.81172 q^{81} -10.7619i q^{83} -7.38094i q^{87} -11.0205 q^{89} -2.52815 q^{91} -8.59414i q^{97} +3.08825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 24 q^{9} - 10 q^{11} + 6 q^{19} - 18 q^{21} + 18 q^{29} + 38 q^{39} + 16 q^{41} - 10 q^{49} + 22 q^{51} + 46 q^{59} + 6 q^{61} + 86 q^{69} - 24 q^{71} - 52 q^{79} + 94 q^{81} - 36 q^{89} + 34 q^{91} + 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 0.786802i − 0.454260i −0.973864 0.227130i \(-0.927066\pi\)
0.973864 0.227130i \(-0.0729343\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.08387i − 0.787630i −0.919190 0.393815i \(-0.871155\pi\)
0.919190 0.393815i \(-0.128845\pi\)
\(8\) 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
\(13\) − 1.21320i − 0.336481i −0.985746 0.168240i \(-0.946192\pi\)
0.985746 0.168240i \(-0.0538085\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.08387i 0.990485i 0.868755 + 0.495242i \(0.164921\pi\)
−0.868755 + 0.495242i \(0.835079\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.63959 −0.357789
\(22\) 0 0
\(23\) 8.95455i 1.86715i 0.358379 + 0.933576i \(0.383329\pi\)
−0.358379 + 0.933576i \(0.616671\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.23374i − 0.814783i
\(28\) 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
\(33\) − 1.02054i − 0.177653i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 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
\(43\) − 7.72347i − 1.17782i −0.808199 0.588909i \(-0.799558\pi\)
0.808199 0.588909i \(-0.200442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.46482i 1.38059i 0.723530 + 0.690293i \(0.242518\pi\)
−0.723530 + 0.690293i \(0.757482\pi\)
\(48\) 0 0
\(49\) 2.65748 0.379639
\(50\) 0 0
\(51\) 3.21320 0.449938
\(52\) 0 0
\(53\) 11.9751i 1.64490i 0.568834 + 0.822452i \(0.307395\pi\)
−0.568834 + 0.822452i \(0.692605\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 0.786802i − 0.104214i
\(58\) 0 0
\(59\) 7.21320 0.939078 0.469539 0.882912i \(-0.344420\pi\)
0.469539 + 0.882912i \(0.344420\pi\)
\(60\) 0 0
\(61\) 4.87067 0.623626 0.311813 0.950144i \(-0.399064\pi\)
0.311813 + 0.950144i \(0.399064\pi\)
\(62\) 0 0
\(63\) − 4.96158i − 0.625100i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.3809i 1.39040i 0.718815 + 0.695202i \(0.244685\pi\)
−0.718815 + 0.695202i \(0.755315\pi\)
\(68\) 0 0
\(69\) 7.04545 0.848173
\(70\) 0 0
\(71\) −9.02054 −1.07054 −0.535270 0.844681i \(-0.679790\pi\)
−0.535270 + 0.844681i \(0.679790\pi\)
\(72\) 0 0
\(73\) − 5.65748i − 0.662157i −0.943603 0.331079i \(-0.892587\pi\)
0.943603 0.331079i \(-0.107413\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.70293i − 0.308027i
\(78\) 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
\(83\) − 10.7619i − 1.18127i −0.806939 0.590635i \(-0.798877\pi\)
0.806939 0.590635i \(-0.201123\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.38094i − 0.791320i
\(88\) 0 0
\(89\) −11.0205 −1.16817 −0.584087 0.811691i \(-0.698547\pi\)
−0.584087 + 0.811691i \(0.698547\pi\)
\(90\) 0 0
\(91\) −2.52815 −0.265022
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.59414i − 0.872603i −0.899801 0.436301i \(-0.856288\pi\)
0.899801 0.436301i \(-0.143712\pi\)
\(98\) 0 0
\(99\) 3.08825 0.310381
\(100\) 0 0
\(101\) 9.14721 0.910181 0.455091 0.890445i \(-0.349607\pi\)
0.455091 + 0.890445i \(0.349607\pi\)
\(102\) 0 0
\(103\) − 17.3560i − 1.71014i −0.518512 0.855070i \(-0.673514\pi\)
0.518512 0.855070i \(-0.326486\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 0.786802i − 0.0760630i −0.999277 0.0380315i \(-0.987891\pi\)
0.999277 0.0380315i \(-0.0121087\pi\)
\(108\) 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
\(113\) 1.14721i 0.107920i 0.998543 + 0.0539601i \(0.0171844\pi\)
−0.998543 + 0.0539601i \(0.982816\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.88856i − 0.267047i
\(118\) 0 0
\(119\) 8.51027 0.780135
\(120\) 0 0
\(121\) −9.31761 −0.847055
\(122\) 0 0
\(123\) − 2.81172i − 0.253524i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1.02054i − 0.0905581i −0.998974 0.0452790i \(-0.985582\pi\)
0.998974 0.0452790i \(-0.0144177\pi\)
\(128\) 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
\(133\) − 2.08387i − 0.180695i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.2311i 1.30128i 0.759387 + 0.650639i \(0.225499\pi\)
−0.759387 + 0.650639i \(0.774501\pi\)
\(138\) 0 0
\(139\) −16.0590 −1.36210 −0.681051 0.732236i \(-0.738477\pi\)
−0.681051 + 0.732236i \(0.738477\pi\)
\(140\) 0 0
\(141\) 7.44693 0.627145
\(142\) 0 0
\(143\) − 1.57360i − 0.131591i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.09091i − 0.172455i
\(148\) 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 \(-0.250405\pi\)
0.706207 + 0.708006i \(0.250405\pi\)
\(152\) 0 0
\(153\) 9.72347i 0.786096i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 0 0
\(159\) 9.42202 0.747215
\(160\) 0 0
\(161\) 18.6601 1.47062
\(162\) 0 0
\(163\) 0.852793i 0.0667959i 0.999442 + 0.0333979i \(0.0106329\pi\)
−0.999442 + 0.0333979i \(0.989367\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 18.5941i − 1.43886i −0.694566 0.719429i \(-0.744404\pi\)
0.694566 0.719429i \(-0.255596\pi\)
\(168\) 0 0
\(169\) 11.5282 0.886781
\(170\) 0 0
\(171\) 2.38094 0.182075
\(172\) 0 0
\(173\) − 2.85279i − 0.216894i −0.994102 0.108447i \(-0.965412\pi\)
0.994102 0.108447i \(-0.0345877\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 5.67536i − 0.426586i
\(178\) 0 0
\(179\) 26.0768 1.94907 0.974537 0.224226i \(-0.0719854\pi\)
0.974537 + 0.224226i \(0.0719854\pi\)
\(180\) 0 0
\(181\) 3.18828 0.236983 0.118492 0.992955i \(-0.462194\pi\)
0.118492 + 0.992955i \(0.462194\pi\)
\(182\) 0 0
\(183\) − 3.83226i − 0.283288i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.29707i 0.387360i
\(188\) 0 0
\(189\) −8.82256 −0.641747
\(190\) 0 0
\(191\) 7.95720 0.575763 0.287881 0.957666i \(-0.407049\pi\)
0.287881 + 0.957666i \(0.407049\pi\)
\(192\) 0 0
\(193\) − 20.9296i − 1.50655i −0.657707 0.753274i \(-0.728473\pi\)
0.657707 0.753274i \(-0.271527\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.57360i − 0.254609i −0.991864 0.127304i \(-0.959368\pi\)
0.991864 0.127304i \(-0.0406325\pi\)
\(198\) 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
\(203\) − 19.5487i − 1.37205i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 21.3203i 1.48186i
\(208\) 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
\(213\) 7.09738i 0.486304i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.45131 −0.300792
\(220\) 0 0
\(221\) 4.95455 0.333279
\(222\) 0 0
\(223\) − 1.02054i − 0.0683402i −0.999416 0.0341701i \(-0.989121\pi\)
0.999416 0.0341701i \(-0.0108788\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.04545i 0.202134i 0.994880 + 0.101067i \(0.0322256\pi\)
−0.994880 + 0.101067i \(0.967774\pi\)
\(228\) 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
\(233\) − 25.2062i − 1.65131i −0.564175 0.825655i \(-0.690806\pi\)
0.564175 0.825655i \(-0.309194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.53253i 0.489290i
\(238\) 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
\(243\) − 15.7003i − 1.00717i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.21320i − 0.0771940i
\(248\) 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
\(253\) 11.6147i 0.730209i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 23.3560i − 1.45691i −0.685094 0.728454i \(-0.740239\pi\)
0.685094 0.728454i \(-0.259761\pi\)
\(258\) 0 0
\(259\) −4.16774 −0.258971
\(260\) 0 0
\(261\) 22.3355 1.38253
\(262\) 0 0
\(263\) − 12.2765i − 0.757003i −0.925601 0.378502i \(-0.876439\pi\)
0.925601 0.378502i \(-0.123561\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.67098i 0.530655i
\(268\) 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 \(-0.509230\pi\)
−0.0289923 + 0.999580i \(0.509230\pi\)
\(272\) 0 0
\(273\) 1.98915i 0.120389i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.0384i 1.02374i 0.859063 + 0.511870i \(0.171047\pi\)
−0.859063 + 0.511870i \(0.828953\pi\)
\(278\) 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
\(283\) − 1.29707i − 0.0771028i −0.999257 0.0385514i \(-0.987726\pi\)
0.999257 0.0385514i \(-0.0122743\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 7.44693i − 0.439579i
\(288\) 0 0
\(289\) 0.321987 0.0189404
\(290\) 0 0
\(291\) −6.76189 −0.396389
\(292\) 0 0
\(293\) 21.5487i 1.25889i 0.777046 + 0.629444i \(0.216717\pi\)
−0.777046 + 0.629444i \(0.783283\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.49145i − 0.318646i
\(298\) 0 0
\(299\) 10.8636 0.628260
\(300\) 0 0
\(301\) −16.0947 −0.927684
\(302\) 0 0
\(303\) − 7.19704i − 0.413459i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 9.18828i − 0.524403i −0.965013 0.262201i \(-0.915551\pi\)
0.965013 0.262201i \(-0.0844485\pi\)
\(308\) 0 0
\(309\) −13.6558 −0.776849
\(310\) 0 0
\(311\) −7.48973 −0.424704 −0.212352 0.977193i \(-0.568112\pi\)
−0.212352 + 0.977193i \(0.568112\pi\)
\(312\) 0 0
\(313\) 9.76626i 0.552022i 0.961154 + 0.276011i \(0.0890126\pi\)
−0.961154 + 0.276011i \(0.910987\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.7164i 0.770392i 0.922835 + 0.385196i \(0.125866\pi\)
−0.922835 + 0.385196i \(0.874134\pi\)
\(318\) 0 0
\(319\) 12.1677 0.681263
\(320\) 0 0
\(321\) −0.619057 −0.0345524
\(322\) 0 0
\(323\) 4.08387i 0.227233i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10.6601i − 0.589507i
\(328\) 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
\(333\) − 4.76189i − 0.260950i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.31495i 0.507418i 0.967281 + 0.253709i \(0.0816505\pi\)
−0.967281 + 0.253709i \(0.918349\pi\)
\(338\) 0 0
\(339\) 0.902625 0.0490238
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.1249i − 1.08665i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.29707i − 0.499093i −0.968363 0.249546i \(-0.919718\pi\)
0.968363 0.249546i \(-0.0802815\pi\)
\(348\) 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
\(353\) 20.4783i 1.08995i 0.838452 + 0.544975i \(0.183461\pi\)
−0.838452 + 0.544975i \(0.816539\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.69589i − 0.354384i
\(358\) 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
\(363\) 7.33111i 0.384784i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.85279i − 0.253314i −0.991947 0.126657i \(-0.959575\pi\)
0.991947 0.126657i \(-0.0404247\pi\)
\(368\) 0 0
\(369\) 8.50855 0.442937
\(370\) 0 0
\(371\) 24.9545 1.29558
\(372\) 0 0
\(373\) 12.1071i 0.626880i 0.949608 + 0.313440i \(0.101481\pi\)
−0.949608 + 0.313440i \(0.898519\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.3809i − 0.586148i
\(378\) 0 0
\(379\) 2.02492 0.104013 0.0520065 0.998647i \(-0.483438\pi\)
0.0520065 + 0.998647i \(0.483438\pi\)
\(380\) 0 0
\(381\) −0.802961 −0.0411369
\(382\) 0 0
\(383\) 1.23811i 0.0632647i 0.999500 + 0.0316323i \(0.0100706\pi\)
−0.999500 + 0.0316323i \(0.989929\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 18.3891i − 0.934772i
\(388\) 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
\(393\) 6.07684i 0.306536i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 17.3739i 0.871971i 0.899954 + 0.435986i \(0.143600\pi\)
−0.899954 + 0.435986i \(0.856400\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.59414i − 0.128587i
\(408\) 0 0
\(409\) −20.4622 −1.01179 −0.505894 0.862595i \(-0.668837\pi\)
−0.505894 + 0.862595i \(0.668837\pi\)
\(410\) 0 0
\(411\) 11.9838 0.591119
\(412\) 0 0
\(413\) − 15.0314i − 0.739646i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.6352i 0.618749i
\(418\) 0 0
\(419\) 7.14721 0.349164 0.174582 0.984643i \(-0.444143\pi\)
0.174582 + 0.984643i \(0.444143\pi\)
\(420\) 0 0
\(421\) 12.3604 0.602409 0.301205 0.953560i \(-0.402611\pi\)
0.301205 + 0.953560i \(0.402611\pi\)
\(422\) 0 0
\(423\) 22.5352i 1.09570i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 10.1499i − 0.491186i
\(428\) 0 0
\(429\) −1.23811 −0.0597767
\(430\) 0 0
\(431\) −1.90909 −0.0919578 −0.0459789 0.998942i \(-0.514641\pi\)
−0.0459789 + 0.998942i \(0.514641\pi\)
\(432\) 0 0
\(433\) − 4.04107i − 0.194202i −0.995275 0.0971008i \(-0.969043\pi\)
0.995275 0.0971008i \(-0.0309569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.95455i 0.428354i
\(438\) 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
\(443\) 19.6736i 0.934723i 0.884066 + 0.467361i \(0.154795\pi\)
−0.884066 + 0.467361i \(0.845205\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1.65920i − 0.0784774i
\(448\) 0 0
\(449\) −10.5531 −0.498030 −0.249015 0.968500i \(-0.580107\pi\)
−0.249015 + 0.968500i \(0.580107\pi\)
\(450\) 0 0
\(451\) 4.63522 0.218264
\(452\) 0 0
\(453\) − 13.6558i − 0.641603i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.72785i 0.127603i 0.997963 + 0.0638016i \(0.0203225\pi\)
−0.997963 + 0.0638016i \(0.979678\pi\)
\(458\) 0 0
\(459\) 17.2900 0.807030
\(460\) 0 0
\(461\) −35.1153 −1.63548 −0.817740 0.575587i \(-0.804773\pi\)
−0.817740 + 0.575587i \(0.804773\pi\)
\(462\) 0 0
\(463\) − 17.1293i − 0.796067i −0.917371 0.398034i \(-0.869693\pi\)
0.917371 0.398034i \(-0.130307\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.7029i 0.680370i 0.940358 + 0.340185i \(0.110490\pi\)
−0.940358 + 0.340185i \(0.889510\pi\)
\(468\) 0 0
\(469\) 23.7164 1.09512
\(470\) 0 0
\(471\) 11.0152 0.507555
\(472\) 0 0
\(473\) − 10.0179i − 0.460623i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 28.5120i 1.30547i
\(478\) 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
\(483\) − 14.6818i − 0.668046i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 23.6504i − 1.07170i −0.844312 0.535852i \(-0.819991\pi\)
0.844312 0.535852i \(-0.180009\pi\)
\(488\) 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
\(493\) 38.3106i 1.72542i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.7976i 0.843190i
\(498\) 0 0
\(499\) −10.3176 −0.461880 −0.230940 0.972968i \(-0.574180\pi\)
−0.230940 + 0.972968i \(0.574180\pi\)
\(500\) 0 0
\(501\) −14.6299 −0.653616
\(502\) 0 0
\(503\) 20.4372i 0.911252i 0.890171 + 0.455626i \(0.150584\pi\)
−0.890171 + 0.455626i \(0.849416\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.07037i − 0.402829i
\(508\) 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
\(513\) − 4.23374i − 0.186924i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.2765i 0.539921i
\(518\) 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
\(523\) − 44.7370i − 1.95621i −0.208109 0.978106i \(-0.566731\pi\)
0.208109 0.978106i \(-0.433269\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −57.1839 −2.48626
\(530\) 0 0
\(531\) 17.1742 0.745297
\(532\) 0 0
\(533\) − 4.33549i − 0.187791i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 20.5173i − 0.885387i
\(538\) 0 0
\(539\) 3.44693 0.148470
\(540\) 0 0
\(541\) −37.5059 −1.61250 −0.806252 0.591572i \(-0.798507\pi\)
−0.806252 + 0.591572i \(0.798507\pi\)
\(542\) 0 0
\(543\) − 2.50855i − 0.107652i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 0 0
\(549\) 11.5968 0.494939
\(550\) 0 0
\(551\) 9.38094 0.399642
\(552\) 0 0
\(553\) 19.9502i 0.848367i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 33.3381i − 1.41258i −0.707921 0.706291i \(-0.750367\pi\)
0.707921 0.706291i \(-0.249633\pi\)
\(558\) 0 0
\(559\) −9.37010 −0.396313
\(560\) 0 0
\(561\) 4.16774 0.175962
\(562\) 0 0
\(563\) − 3.44693i − 0.145271i −0.997359 0.0726355i \(-0.976859\pi\)
0.997359 0.0726355i \(-0.0231410\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 7.94313i − 0.333580i
\(568\) 0 0
\(569\) −27.0205 −1.13276 −0.566380 0.824144i \(-0.691657\pi\)
−0.566380 + 0.824144i \(0.691657\pi\)
\(570\) 0 0
\(571\) −5.70559 −0.238771 −0.119386 0.992848i \(-0.538092\pi\)
−0.119386 + 0.992848i \(0.538092\pi\)
\(572\) 0 0
\(573\) − 6.26074i − 0.261546i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.34252i 0.264043i 0.991247 + 0.132021i \(0.0421467\pi\)
−0.991247 + 0.132021i \(0.957853\pi\)
\(578\) 0 0
\(579\) −16.4675 −0.684365
\(580\) 0 0
\(581\) −22.4264 −0.930404
\(582\) 0 0
\(583\) 15.5325i 0.643292i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 37.5827i − 1.55121i −0.631222 0.775603i \(-0.717446\pi\)
0.631222 0.775603i \(-0.282554\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.81172 −0.115659
\(592\) 0 0
\(593\) 9.48270i 0.389408i 0.980862 + 0.194704i \(0.0623746\pi\)
−0.980862 + 0.194704i \(0.937625\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 20.7868i − 0.850747i
\(598\) 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
\(603\) 27.0974i 1.10349i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 5.32026i 0.215943i 0.994154 + 0.107971i \(0.0344355\pi\)
−0.994154 + 0.107971i \(0.965564\pi\)
\(608\) 0 0
\(609\) −15.3809 −0.623267
\(610\) 0 0
\(611\) 11.4827 0.464540
\(612\) 0 0
\(613\) 27.2472i 1.10051i 0.834998 + 0.550253i \(0.185469\pi\)
−0.834998 + 0.550253i \(0.814531\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 30.9117i − 1.24446i −0.782834 0.622230i \(-0.786227\pi\)
0.782834 0.622230i \(-0.213773\pi\)
\(618\) 0 0
\(619\) 24.2857 0.976123 0.488061 0.872809i \(-0.337704\pi\)
0.488061 + 0.872809i \(0.337704\pi\)
\(620\) 0 0
\(621\) 37.9112 1.52132
\(622\) 0 0
\(623\) 22.9654i 0.920089i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.02054i − 0.0407563i
\(628\) 0 0
\(629\) 8.16774 0.325669
\(630\) 0 0
\(631\) 24.9117 0.991721 0.495861 0.868402i \(-0.334853\pi\)
0.495861 + 0.868402i \(0.334853\pi\)
\(632\) 0 0
\(633\) 18.5282i 0.736428i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.22405i − 0.127741i
\(638\) 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
\(643\) 13.9323i 0.549436i 0.961525 + 0.274718i \(0.0885845\pi\)
−0.961525 + 0.274718i \(0.911416\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.58064i 0.219398i 0.993965 + 0.109699i \(0.0349886\pi\)
−0.993965 + 0.109699i \(0.965011\pi\)
\(648\) 0 0
\(649\) 9.35603 0.367256
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.77330i 0.0693945i 0.999398 + 0.0346973i \(0.0110467\pi\)
−0.999398 + 0.0346973i \(0.988953\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 13.4701i − 0.525520i
\(658\) 0 0
\(659\) 1.12229 0.0437183 0.0218591 0.999761i \(-0.493041\pi\)
0.0218591 + 0.999761i \(0.493041\pi\)
\(660\) 0 0
\(661\) 21.9340 0.853134 0.426567 0.904456i \(-0.359723\pi\)
0.426567 + 0.904456i \(0.359723\pi\)
\(662\) 0 0
\(663\) − 3.89825i − 0.151395i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 84.0021i 3.25257i
\(668\) 0 0
\(669\) −0.802961 −0.0310443
\(670\) 0 0
\(671\) 6.31761 0.243889
\(672\) 0 0
\(673\) 8.24458i 0.317805i 0.987294 + 0.158903i \(0.0507956\pi\)
−0.987294 + 0.158903i \(0.949204\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 5.76626i − 0.221616i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353438\pi\)
\(678\) 0 0
\(679\) −17.9091 −0.687288
\(680\) 0 0
\(681\) 2.39617 0.0918214
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 3.16128i − 0.120610i
\(688\) 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
\(693\) − 6.43552i − 0.244465i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 14.5941i 0.552793i
\(698\) 0 0
\(699\) −19.8323 −0.750125
\(700\) 0 0
\(701\) −3.90909 −0.147644 −0.0738222 0.997271i \(-0.523520\pi\)
−0.0738222 + 0.997271i \(0.523520\pi\)
\(702\) 0 0
\(703\) − 2.00000i − 0.0754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 19.0616i − 0.716886i
\(708\) 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
\(715\) 0 0
\(716\) 0 0
\(717\) − 5.45778i − 0.203825i
\(718\) 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
\(723\) − 0.599453i − 0.0222939i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 41.1812i 1.52733i 0.645614 + 0.763664i \(0.276602\pi\)
−0.645614 + 0.763664i \(0.723398\pi\)
\(728\) 0 0
\(729\) −0.917850 −0.0339945
\(730\) 0 0
\(731\) 31.5417 1.16661
\(732\) 0 0
\(733\) − 1.27919i − 0.0472479i −0.999721 0.0236240i \(-0.992480\pi\)
0.999721 0.0236240i \(-0.00752044\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 14.7619i 0.543761i
\(738\) 0 0
\(739\) −30.3534 −1.11657 −0.558283 0.829650i \(-0.688540\pi\)
−0.558283 + 0.829650i \(0.688540\pi\)
\(740\) 0 0
\(741\) −0.954547 −0.0350661
\(742\) 0 0
\(743\) 19.4827i 0.714751i 0.933961 + 0.357375i \(0.116328\pi\)
−0.933961 + 0.357375i \(0.883672\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 25.6234i − 0.937512i
\(748\) 0 0
\(749\) −1.63959 −0.0599095
\(750\) 0 0
\(751\) −48.6710 −1.77603 −0.888015 0.459815i \(-0.847916\pi\)
−0.888015 + 0.459815i \(0.847916\pi\)
\(752\) 0 0
\(753\) 4.83872i 0.176333i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 41.7235i 1.51647i 0.651984 + 0.758233i \(0.273937\pi\)
−0.651984 + 0.758233i \(0.726063\pi\)
\(758\) 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
\(763\) − 28.2337i − 1.02213i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 8.75104i − 0.315982i
\(768\) 0 0
\(769\) −7.14548 −0.257673 −0.128836 0.991666i \(-0.541124\pi\)
−0.128836 + 0.991666i \(0.541124\pi\)
\(770\) 0 0
\(771\) −18.3766 −0.661816
\(772\) 0 0
\(773\) 28.9956i 1.04290i 0.853282 + 0.521450i \(0.174609\pi\)
−0.853282 + 0.521450i \(0.825391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3.27919i 0.117640i
\(778\) 0 0
\(779\) 3.57360 0.128038
\(780\) 0 0
\(781\) −11.7003 −0.418669
\(782\) 0 0
\(783\) − 39.7164i − 1.41935i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 23.3311i − 0.831664i −0.909441 0.415832i \(-0.863490\pi\)
0.909441 0.415832i \(-0.136510\pi\)
\(788\) 0 0
\(789\) −9.65920 −0.343877
\(790\) 0 0
\(791\) 2.39063 0.0850011
\(792\) 0 0
\(793\) − 5.90909i − 0.209838i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.2543i 0.682021i 0.940059 + 0.341011i \(0.110769\pi\)
−0.940059 + 0.341011i \(0.889231\pi\)
\(798\) 0 0
\(799\) −38.6531 −1.36745
\(800\) 0 0
\(801\) −26.2393 −0.927119
\(802\) 0 0
\(803\) − 7.33815i − 0.258958i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.09091i − 0.214410i
\(808\) 0 0
\(809\) −27.9983 −0.984367 −0.492184 0.870491i \(-0.663801\pi\)
−0.492184 + 0.870491i \(0.663801\pi\)
\(810\) 0 0
\(811\) −47.4631 −1.66665 −0.833327 0.552780i \(-0.813567\pi\)
−0.833327 + 0.552780i \(0.813567\pi\)
\(812\) 0 0
\(813\) 0.751039i 0.0263401i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 7.72347i − 0.270210i
\(818\) 0 0
\(819\) −6.01938 −0.210334
\(820\) 0 0
\(821\) 2.14455 0.0748454 0.0374227 0.999300i \(-0.488085\pi\)
0.0374227 + 0.999300i \(0.488085\pi\)
\(822\) 0 0
\(823\) 32.8458i 1.14493i 0.819929 + 0.572466i \(0.194013\pi\)
−0.819929 + 0.572466i \(0.805987\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.6103i 0.925331i 0.886533 + 0.462665i \(0.153107\pi\)
−0.886533 + 0.462665i \(0.846893\pi\)
\(828\) 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
\(833\) 10.8528i 0.376027i
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(843\) 10.3050i 0.354924i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 19.4167i 0.667166i
\(848\) 0 0
\(849\) −1.02054 −0.0350248
\(850\) 0 0
\(851\) 17.9091 0.613916
\(852\) 0 0
\(853\) 7.95893i 0.272508i 0.990674 + 0.136254i \(0.0435064\pi\)
−0.990674 + 0.136254i \(0.956494\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.09091i − 0.139743i −0.997556 0.0698714i \(-0.977741\pi\)
0.997556 0.0698714i \(-0.0222589\pi\)
\(858\) 0 0
\(859\) 13.3328 0.454910 0.227455 0.973789i \(-0.426959\pi\)
0.227455 + 0.973789i \(0.426959\pi\)
\(860\) 0 0
\(861\) −5.85926 −0.199683
\(862\) 0 0
\(863\) 20.1677i 0.686518i 0.939241 + 0.343259i \(0.111531\pi\)
−0.939241 + 0.343259i \(0.888469\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 0.253340i − 0.00860386i
\(868\) 0 0
\(869\) −12.4176 −0.421240
\(870\) 0 0
\(871\) 13.8073 0.467844
\(872\) 0 0
\(873\) − 20.4622i − 0.692539i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.8572i 0.805599i 0.915288 + 0.402800i \(0.131963\pi\)
−0.915288 + 0.402800i \(0.868037\pi\)
\(878\) 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
\(883\) 28.5622i 0.961194i 0.876942 + 0.480597i \(0.159580\pi\)
−0.876942 + 0.480597i \(0.840420\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.3150i 1.58868i 0.607473 + 0.794340i \(0.292183\pi\)
−0.607473 + 0.794340i \(0.707817\pi\)
\(888\) 0 0
\(889\) −2.12667 −0.0713262
\(890\) 0 0
\(891\) 4.94407 0.165632
\(892\) 0 0
\(893\) 9.46482i 0.316728i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 8.54753i − 0.285394i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −48.9047 −1.62925
\(902\) 0 0
\(903\) 12.6634i 0.421410i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 15.8982i − 0.527893i −0.964537 0.263946i \(-0.914976\pi\)
0.964537 0.263946i \(-0.0850242\pi\)
\(908\) 0 0
\(909\) 21.7790 0.722363
\(910\) 0 0
\(911\) −20.2997 −0.672560 −0.336280 0.941762i \(-0.609169\pi\)
−0.336280 + 0.941762i \(0.609169\pi\)
\(912\) 0 0
\(913\) − 13.9589i − 0.461973i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0947i 0.531494i
\(918\) 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
\(923\) 10.9437i 0.360216i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 41.3237i − 1.35725i
\(928\) 0 0
\(929\) −53.4631 −1.75407 −0.877034 0.480429i \(-0.840481\pi\)
−0.877034 + 0.480429i \(0.840481\pi\)
\(930\) 0 0
\(931\) 2.65748 0.0870953
\(932\) 0 0
\(933\) 5.89293i 0.192926i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 43.0135i − 1.40519i −0.711590 0.702595i \(-0.752025\pi\)
0.711590 0.702595i \(-0.247975\pi\)
\(938\) 0 0
\(939\) 7.68411 0.250762
\(940\) 0 0
\(941\) 25.3311 0.825771 0.412885 0.910783i \(-0.364521\pi\)
0.412885 + 0.910783i \(0.364521\pi\)
\(942\) 0 0
\(943\) 32.0000i 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.6710i 0.411751i 0.978578 + 0.205876i \(0.0660043\pi\)
−0.978578 + 0.205876i \(0.933996\pi\)
\(948\) 0 0
\(949\) −6.86364 −0.222803
\(950\) 0 0
\(951\) 10.7921 0.349958
\(952\) 0 0
\(953\) 31.2240i 1.01145i 0.862696 + 0.505723i \(0.168774\pi\)
−0.862696 + 0.505723i \(0.831226\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 9.57360i − 0.309471i
\(958\) 0 0
\(959\) 31.7396 1.02493
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 1.87333i − 0.0603672i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.29441i 0.202415i 0.994865 + 0.101207i \(0.0322706\pi\)
−0.994865 + 0.101207i \(0.967729\pi\)
\(968\) 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
\(973\) 33.4648i 1.07283i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 34.1677i − 1.09312i −0.837419 0.546561i \(-0.815936\pi\)
0.837419 0.546561i \(-0.184064\pi\)
\(978\) 0 0
\(979\) −14.2944 −0.456851
\(980\) 0 0
\(981\) 32.2587 1.02994
\(982\) 0 0
\(983\) − 30.4264i − 0.970451i −0.874389 0.485226i \(-0.838737\pi\)
0.874389 0.485226i \(-0.161263\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 15.5185i − 0.493958i
\(988\) 0 0
\(989\) 69.1601 2.19916
\(990\) 0 0
\(991\) −24.1179 −0.766131 −0.383065 0.923721i \(-0.625132\pi\)
−0.383065 + 0.923721i \(0.625132\pi\)
\(992\) 0 0
\(993\) 11.4308i 0.362745i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15.9323i 0.504581i 0.967652 + 0.252290i \(0.0811838\pi\)
−0.967652 + 0.252290i \(0.918816\pi\)
\(998\) 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 3800.2.d.j.3649.3 6
5.2 odd 4 3800.2.a.r.1.2 3
5.3 odd 4 152.2.a.c.1.2 3
5.4 even 2 inner 3800.2.d.j.3649.4 6
15.8 even 4 1368.2.a.n.1.1 3
20.3 even 4 304.2.a.g.1.2 3
20.7 even 4 7600.2.a.bv.1.2 3
35.13 even 4 7448.2.a.bf.1.2 3
40.3 even 4 1216.2.a.v.1.2 3
40.13 odd 4 1216.2.a.u.1.2 3
60.23 odd 4 2736.2.a.bd.1.1 3
95.18 even 4 2888.2.a.o.1.2 3
380.303 odd 4 5776.2.a.bp.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.c.1.2 3 5.3 odd 4
304.2.a.g.1.2 3 20.3 even 4
1216.2.a.u.1.2 3 40.13 odd 4
1216.2.a.v.1.2 3 40.3 even 4
1368.2.a.n.1.1 3 15.8 even 4
2736.2.a.bd.1.1 3 60.23 odd 4
2888.2.a.o.1.2 3 95.18 even 4
3800.2.a.r.1.2 3 5.2 odd 4
3800.2.d.j.3649.3 6 1.1 even 1 trivial
3800.2.d.j.3649.4 6 5.4 even 2 inner
5776.2.a.bp.1.2 3 380.303 odd 4
7448.2.a.bf.1.2 3 35.13 even 4
7600.2.a.bv.1.2 3 20.7 even 4