Properties

Label 3700.2.d.c.149.2
Level $3700$
Weight $2$
Character 3700.149
Analytic conductor $29.545$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3700,2,Mod(149,3700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3700.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.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: \(29.5446487479\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3700.149
Dual form 3700.2.d.c.149.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.00000i q^{7} +2.00000 q^{9} -3.00000 q^{11} +6.00000i q^{13} -1.00000 q^{21} -2.00000i q^{23} +5.00000i q^{27} +6.00000 q^{29} -3.00000i q^{33} -1.00000i q^{37} -6.00000 q^{39} -9.00000 q^{41} +10.0000i q^{43} +1.00000i q^{47} +6.00000 q^{49} -1.00000i q^{53} -12.0000 q^{61} +2.00000i q^{63} +2.00000 q^{69} -5.00000 q^{71} -3.00000i q^{73} -3.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} -11.0000i q^{83} +6.00000i q^{87} -6.00000 q^{91} +8.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 6 q^{11} - 2 q^{21} + 12 q^{29} - 12 q^{39} - 18 q^{41} + 12 q^{49} - 24 q^{61} + 4 q^{69} - 10 q^{71} - 32 q^{79} + 2 q^{81} - 12 q^{91} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) − 3.00000i − 0.522233i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000i 0.145865i 0.997337 + 0.0729325i \(0.0232358\pi\)
−0.997337 + 0.0729325i \(0.976764\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.00000i − 0.137361i −0.997639 0.0686803i \(-0.978121\pi\)
0.997639 0.0686803i \(-0.0218788\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) − 3.00000i − 0.351123i −0.984468 0.175562i \(-0.943826\pi\)
0.984468 0.175562i \(-0.0561742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.00000i − 0.341882i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 11.0000i − 1.20741i −0.797209 0.603703i \(-0.793691\pi\)
0.797209 0.603703i \(-0.206309\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 20.0000i 1.93347i 0.255774 + 0.966736i \(0.417670\pi\)
−0.255774 + 0.966736i \(0.582330\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 0 0
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 12.0000i 1.10940i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1.00000i − 0.0887357i −0.999015 0.0443678i \(-0.985873\pi\)
0.999015 0.0443678i \(-0.0141274\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 0 0
\(143\) − 18.0000i − 1.50524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.0000i 1.19713i 0.801074 + 0.598565i \(0.204262\pi\)
−0.801074 + 0.598565i \(0.795738\pi\)
\(158\) 0 0
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 2.00000i 0.156652i 0.996928 + 0.0783260i \(0.0249575\pi\)
−0.996928 + 0.0783260i \(0.975042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0000i 1.14043i 0.821496 + 0.570214i \(0.193140\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) − 12.0000i − 0.887066i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000i 0.213741i 0.994273 + 0.106871i \(0.0340831\pi\)
−0.994273 + 0.106871i \(0.965917\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 4.00000i − 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −27.0000 −1.85876 −0.929378 0.369129i \(-0.879656\pi\)
−0.929378 + 0.369129i \(0.879656\pi\)
\(212\) 0 0
\(213\) − 5.00000i − 0.342594i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.00000 0.202721
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 17.0000i − 1.13840i −0.822198 0.569202i \(-0.807252\pi\)
0.822198 0.569202i \(-0.192748\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) −3.00000 −0.198246 −0.0991228 0.995075i \(-0.531604\pi\)
−0.0991228 + 0.995075i \(0.531604\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) 2.00000i 0.131024i 0.997852 + 0.0655122i \(0.0208681\pi\)
−0.997852 + 0.0655122i \(0.979132\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 11.0000 0.697097
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 6.00000i 0.377217i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0000i 1.24757i 0.781598 + 0.623783i \(0.214405\pi\)
−0.781598 + 0.623783i \(0.785595\pi\)
\(258\) 0 0
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) 12.0000 0.742781
\(262\) 0 0
\(263\) − 5.00000i − 0.308313i −0.988046 0.154157i \(-0.950734\pi\)
0.988046 0.154157i \(-0.0492660\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 0 0
\(273\) − 6.00000i − 0.363137i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 8.00000i − 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 9.00000i − 0.531253i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 15.0000i − 0.870388i
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 0 0
\(303\) − 1.00000i − 0.0574485i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 11.0000i − 0.627803i −0.949456 0.313902i \(-0.898364\pi\)
0.949456 0.313902i \(-0.101636\pi\)
\(308\) 0 0
\(309\) −10.0000 −0.568880
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) −1.00000 −0.0551318
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.00000i 0.163420i 0.996656 + 0.0817102i \(0.0260382\pi\)
−0.996656 + 0.0817102i \(0.973962\pi\)
\(338\) 0 0
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 34.0000i − 1.82522i −0.408836 0.912608i \(-0.634065\pi\)
0.408836 0.912608i \(-0.365935\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) −30.0000 −1.60128
\(352\) 0 0
\(353\) − 4.00000i − 0.212899i −0.994318 0.106449i \(-0.966052\pi\)
0.994318 0.106449i \(-0.0339482\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.0000 −1.00278 −0.501391 0.865221i \(-0.667178\pi\)
−0.501391 + 0.865221i \(0.667178\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 2.00000i − 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 1.00000 0.0519174
\(372\) 0 0
\(373\) 11.0000i 0.569558i 0.958593 + 0.284779i \(0.0919203\pi\)
−0.958593 + 0.284779i \(0.908080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36.0000i 1.85409i
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 1.00000 0.0512316
\(382\) 0 0
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.0000i 1.01666i
\(388\) 0 0
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 21.0000i − 1.05396i −0.849878 0.526980i \(-0.823324\pi\)
0.849878 0.526980i \(-0.176676\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.00000i 0.148704i
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.0000i 0.587643i
\(418\) 0 0
\(419\) 15.0000 0.732798 0.366399 0.930458i \(-0.380591\pi\)
0.366399 + 0.930458i \(0.380591\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 0 0
\(423\) 2.00000i 0.0972433i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 12.0000i − 0.580721i
\(428\) 0 0
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) 26.0000 1.25238 0.626188 0.779672i \(-0.284614\pi\)
0.626188 + 0.779672i \(0.284614\pi\)
\(432\) 0 0
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) 0 0
\(443\) 13.0000i 0.617649i 0.951119 + 0.308824i \(0.0999355\pi\)
−0.951119 + 0.308824i \(0.900064\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.00000i 0.425685i
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 27.0000 1.27138
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 30.0000i − 1.40334i −0.712502 0.701670i \(-0.752438\pi\)
0.712502 0.701670i \(-0.247562\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) − 6.00000i − 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 22.0000i − 1.01804i −0.860755 0.509019i \(-0.830008\pi\)
0.860755 0.509019i \(-0.169992\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −15.0000 −0.691164
\(472\) 0 0
\(473\) − 30.0000i − 1.37940i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 5.00000i − 0.224281i
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 23.0000i − 1.02147i
\(508\) 0 0
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.00000i − 0.131940i
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) 7.00000 0.306676 0.153338 0.988174i \(-0.450998\pi\)
0.153338 + 0.988174i \(0.450998\pi\)
\(522\) 0 0
\(523\) 26.0000i 1.13690i 0.822718 + 0.568450i \(0.192457\pi\)
−0.822718 + 0.568450i \(0.807543\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 54.0000i − 2.33900i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 18.0000i − 0.776757i
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 17.0000i 0.729540i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 40.0000i − 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) 0 0
\(549\) −24.0000 −1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 16.0000i − 0.680389i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) 0 0
\(559\) −60.0000 −2.53773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.0000i 1.60151i 0.598993 + 0.800755i \(0.295568\pi\)
−0.598993 + 0.800755i \(0.704432\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) 17.0000 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(572\) 0 0
\(573\) − 12.0000i − 0.501307i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 24.0000i − 0.999133i −0.866276 0.499567i \(-0.833493\pi\)
0.866276 0.499567i \(-0.166507\pi\)
\(578\) 0 0
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 11.0000 0.456357
\(582\) 0 0
\(583\) 3.00000i 0.124247i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.0000i 0.825488i 0.910847 + 0.412744i \(0.135430\pi\)
−0.910847 + 0.412744i \(0.864570\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −3.00000 −0.123404
\(592\) 0 0
\(593\) 33.0000i 1.35515i 0.735455 + 0.677574i \(0.236969\pi\)
−0.735455 + 0.677574i \(0.763031\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 14.0000i − 0.572982i
\(598\) 0 0
\(599\) 21.0000 0.858037 0.429018 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) − 23.0000i − 0.928961i −0.885583 0.464481i \(-0.846241\pi\)
0.885583 0.464481i \(-0.153759\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 3.00000i − 0.120775i −0.998175 0.0603877i \(-0.980766\pi\)
0.998175 0.0603877i \(-0.0192337\pi\)
\(618\) 0 0
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 0 0
\(621\) 10.0000 0.401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) − 27.0000i − 1.07315i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 36.0000i 1.42637i
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) 15.0000 0.592464 0.296232 0.955116i \(-0.404270\pi\)
0.296232 + 0.955116i \(0.404270\pi\)
\(642\) 0 0
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) 17.0000 0.657258
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) − 47.0000i − 1.81172i −0.423581 0.905858i \(-0.639227\pi\)
0.423581 0.905858i \(-0.360773\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.0000i 1.42203i 0.703179 + 0.711013i \(0.251763\pi\)
−0.703179 + 0.711013i \(0.748237\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) − 26.0000i − 0.994862i −0.867503 0.497431i \(-0.834277\pi\)
0.867503 0.497431i \(-0.165723\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 3.00000i − 0.114457i
\(688\) 0 0
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.00000i − 0.0376089i
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) −32.0000 −1.20009
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 30.0000i − 1.12037i
\(718\) 0 0
\(719\) 3.00000 0.111881 0.0559406 0.998434i \(-0.482184\pi\)
0.0559406 + 0.998434i \(0.482184\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) − 6.00000i − 0.223142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 17.0000i 0.627909i 0.949438 + 0.313955i \(0.101654\pi\)
−0.949438 + 0.313955i \(0.898346\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −33.0000 −1.21392 −0.606962 0.794731i \(-0.707612\pi\)
−0.606962 + 0.794731i \(0.707612\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.0000i 1.65089i 0.564483 + 0.825445i \(0.309076\pi\)
−0.564483 + 0.825445i \(0.690924\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 22.0000i − 0.804938i
\(748\) 0 0
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 11.0000 0.401396 0.200698 0.979653i \(-0.435679\pi\)
0.200698 + 0.979653i \(0.435679\pi\)
\(752\) 0 0
\(753\) 6.00000i 0.218652i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) 0 0
\(773\) − 31.0000i − 1.11499i −0.830179 0.557496i \(-0.811762\pi\)
0.830179 0.557496i \(-0.188238\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.00000i 0.0358748i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 15.0000 0.536742
\(782\) 0 0
\(783\) 30.0000i 1.07211i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) 0 0
\(789\) 5.00000 0.178005
\(790\) 0 0
\(791\) 14.0000 0.497783
\(792\) 0 0
\(793\) − 72.0000i − 2.55679i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0000i 0.991811i 0.868377 + 0.495905i \(0.165164\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.00000i 0.317603i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.00000i − 0.0704033i
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 41.0000 1.43970 0.719852 0.694127i \(-0.244209\pi\)
0.719852 + 0.694127i \(0.244209\pi\)
\(812\) 0 0
\(813\) − 13.0000i − 0.455930i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −19.0000 −0.663105 −0.331552 0.943437i \(-0.607572\pi\)
−0.331552 + 0.943437i \(0.607572\pi\)
\(822\) 0 0
\(823\) − 24.0000i − 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.00000i − 0.0695468i −0.999395 0.0347734i \(-0.988929\pi\)
0.999395 0.0347734i \(-0.0110710\pi\)
\(828\) 0 0
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 0 0
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.0000i 0.577350i
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 33.0000i − 1.10803i −0.832506 0.554016i \(-0.813095\pi\)
0.832506 0.554016i \(-0.186905\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000i 0.400668i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 10.0000i − 0.332779i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 33.0000i 1.09214i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) 11.0000 0.362462
\(922\) 0 0
\(923\) − 30.0000i − 0.987462i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 20.0000i 0.656886i
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 16.0000i 0.523816i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.00000i 0.0326686i 0.999867 + 0.0163343i \(0.00519960\pi\)
−0.999867 + 0.0163343i \(0.994800\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 18.0000i 0.586161i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 18.0000 0.584305
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) − 9.00000i − 0.291539i −0.989319 0.145769i \(-0.953434\pi\)
0.989319 0.145769i \(-0.0465657\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 18.0000i − 0.581857i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 40.0000i 1.28898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.00000i 0.192947i 0.995336 + 0.0964735i \(0.0307563\pi\)
−0.995336 + 0.0964735i \(0.969244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 12.0000i 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 52.0000i − 1.66363i −0.555055 0.831814i \(-0.687303\pi\)
0.555055 0.831814i \(-0.312697\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) − 39.0000i − 1.24391i −0.783054 0.621953i \(-0.786339\pi\)
0.783054 0.621953i \(-0.213661\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 1.00000i − 0.0318304i
\(988\) 0 0
\(989\) 20.0000 0.635963
\(990\) 0 0
\(991\) −30.0000 −0.952981 −0.476491 0.879180i \(-0.658091\pi\)
−0.476491 + 0.879180i \(0.658091\pi\)
\(992\) 0 0
\(993\) 6.00000i 0.190404i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 42.0000i 1.33015i 0.746775 + 0.665077i \(0.231601\pi\)
−0.746775 + 0.665077i \(0.768399\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 3700.2.d.c.149.2 2
5.2 odd 4 3700.2.a.d.1.1 1
5.3 odd 4 740.2.a.a.1.1 1
5.4 even 2 inner 3700.2.d.c.149.1 2
15.8 even 4 6660.2.a.a.1.1 1
20.3 even 4 2960.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
740.2.a.a.1.1 1 5.3 odd 4
2960.2.a.h.1.1 1 20.3 even 4
3700.2.a.d.1.1 1 5.2 odd 4
3700.2.d.c.149.1 2 5.4 even 2 inner
3700.2.d.c.149.2 2 1.1 even 1 trivial
6660.2.a.a.1.1 1 15.8 even 4