Properties

Label 3100.3.d.e.1301.11
Level $3100$
Weight $3$
Character 3100.1301
Analytic conductor $84.469$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3100,3,Mod(1301,3100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3100.1301");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3100.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.4688819517\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 114 x^{18} + 5280 x^{16} + 128422 x^{14} + 1776819 x^{12} + 14249420 x^{10} + 65297060 x^{8} + \cdots + 20793600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{23}\cdot 3 \)
Twist minimal: no (minimal twist has level 620)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1301.11
Root \(0.693074i\) of defining polynomial
Character \(\chi\) \(=\) 3100.1301
Dual form 3100.3.d.e.1301.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.693074i q^{3} +9.01983 q^{7} +8.51965 q^{9} +O(q^{10})\) \(q+0.693074i q^{3} +9.01983 q^{7} +8.51965 q^{9} -7.32398i q^{11} +1.20530i q^{13} -2.61928i q^{17} -15.5460 q^{19} +6.25141i q^{21} +39.1974i q^{23} +12.1424i q^{27} +37.8559i q^{29} +(11.3467 - 28.8488i) q^{31} +5.07606 q^{33} +72.7141i q^{37} -0.835364 q^{39} -54.6523 q^{41} +53.8375i q^{43} -5.08458 q^{47} +32.3573 q^{49} +1.81536 q^{51} -34.0833i q^{53} -10.7745i q^{57} +23.0023 q^{59} +1.49235i q^{61} +76.8458 q^{63} +96.1191 q^{67} -27.1667 q^{69} +5.02664 q^{71} +76.6699i q^{73} -66.0611i q^{77} +50.4131i q^{79} +68.2612 q^{81} +113.505i q^{83} -26.2369 q^{87} -145.171i q^{89} +10.8716i q^{91} +(19.9943 + 7.86413i) q^{93} +17.3814 q^{97} -62.3978i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 12 q^{7} - 48 q^{9} + 60 q^{19} + 8 q^{31} - 68 q^{33} + 28 q^{39} - 80 q^{41} + 48 q^{47} + 84 q^{49} + 344 q^{51} + 160 q^{59} + 232 q^{63} + 180 q^{67} - 140 q^{69} - 108 q^{71} + 336 q^{81} + 236 q^{87} + 332 q^{93} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times\).

\(n\) \(1551\) \(1801\) \(2977\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.693074i 0.231025i 0.993306 + 0.115512i \(0.0368510\pi\)
−0.993306 + 0.115512i \(0.963149\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 9.01983 1.28855 0.644274 0.764795i \(-0.277160\pi\)
0.644274 + 0.764795i \(0.277160\pi\)
\(8\) 0 0
\(9\) 8.51965 0.946628
\(10\) 0 0
\(11\) 7.32398i 0.665817i −0.942959 0.332908i \(-0.891970\pi\)
0.942959 0.332908i \(-0.108030\pi\)
\(12\) 0 0
\(13\) 1.20530i 0.0927156i 0.998925 + 0.0463578i \(0.0147614\pi\)
−0.998925 + 0.0463578i \(0.985239\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.61928i 0.154075i −0.997028 0.0770377i \(-0.975454\pi\)
0.997028 0.0770377i \(-0.0245462\pi\)
\(18\) 0 0
\(19\) −15.5460 −0.818211 −0.409105 0.912487i \(-0.634159\pi\)
−0.409105 + 0.912487i \(0.634159\pi\)
\(20\) 0 0
\(21\) 6.25141i 0.297686i
\(22\) 0 0
\(23\) 39.1974i 1.70424i 0.523350 + 0.852118i \(0.324682\pi\)
−0.523350 + 0.852118i \(0.675318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 12.1424i 0.449719i
\(28\) 0 0
\(29\) 37.8559i 1.30538i 0.757627 + 0.652688i \(0.226359\pi\)
−0.757627 + 0.652688i \(0.773641\pi\)
\(30\) 0 0
\(31\) 11.3467 28.8488i 0.366024 0.930605i
\(32\) 0 0
\(33\) 5.07606 0.153820
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 72.7141i 1.96525i 0.185613 + 0.982623i \(0.440573\pi\)
−0.185613 + 0.982623i \(0.559427\pi\)
\(38\) 0 0
\(39\) −0.835364 −0.0214196
\(40\) 0 0
\(41\) −54.6523 −1.33298 −0.666491 0.745513i \(-0.732205\pi\)
−0.666491 + 0.745513i \(0.732205\pi\)
\(42\) 0 0
\(43\) 53.8375i 1.25204i 0.779809 + 0.626018i \(0.215316\pi\)
−0.779809 + 0.626018i \(0.784684\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.08458 −0.108183 −0.0540913 0.998536i \(-0.517226\pi\)
−0.0540913 + 0.998536i \(0.517226\pi\)
\(48\) 0 0
\(49\) 32.3573 0.660354
\(50\) 0 0
\(51\) 1.81536 0.0355952
\(52\) 0 0
\(53\) 34.0833i 0.643081i −0.946896 0.321540i \(-0.895799\pi\)
0.946896 0.321540i \(-0.104201\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.7745i 0.189027i
\(58\) 0 0
\(59\) 23.0023 0.389870 0.194935 0.980816i \(-0.437550\pi\)
0.194935 + 0.980816i \(0.437550\pi\)
\(60\) 0 0
\(61\) 1.49235i 0.0244647i 0.999925 + 0.0122323i \(0.00389377\pi\)
−0.999925 + 0.0122323i \(0.996106\pi\)
\(62\) 0 0
\(63\) 76.8458 1.21977
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 96.1191 1.43461 0.717306 0.696758i \(-0.245375\pi\)
0.717306 + 0.696758i \(0.245375\pi\)
\(68\) 0 0
\(69\) −27.1667 −0.393721
\(70\) 0 0
\(71\) 5.02664 0.0707978 0.0353989 0.999373i \(-0.488730\pi\)
0.0353989 + 0.999373i \(0.488730\pi\)
\(72\) 0 0
\(73\) 76.6699i 1.05027i 0.851018 + 0.525136i \(0.175986\pi\)
−0.851018 + 0.525136i \(0.824014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 66.0611i 0.857936i
\(78\) 0 0
\(79\) 50.4131i 0.638141i 0.947731 + 0.319070i \(0.103371\pi\)
−0.947731 + 0.319070i \(0.896629\pi\)
\(80\) 0 0
\(81\) 68.2612 0.842731
\(82\) 0 0
\(83\) 113.505i 1.36753i 0.729703 + 0.683764i \(0.239658\pi\)
−0.729703 + 0.683764i \(0.760342\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −26.2369 −0.301574
\(88\) 0 0
\(89\) 145.171i 1.63114i −0.578661 0.815568i \(-0.696424\pi\)
0.578661 0.815568i \(-0.303576\pi\)
\(90\) 0 0
\(91\) 10.8716i 0.119468i
\(92\) 0 0
\(93\) 19.9943 + 7.86413i 0.214993 + 0.0845606i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.3814 0.179190 0.0895949 0.995978i \(-0.471443\pi\)
0.0895949 + 0.995978i \(0.471443\pi\)
\(98\) 0 0
\(99\) 62.3978i 0.630280i
\(100\) 0 0
\(101\) 123.598 1.22374 0.611869 0.790959i \(-0.290418\pi\)
0.611869 + 0.790959i \(0.290418\pi\)
\(102\) 0 0
\(103\) −34.2866 −0.332879 −0.166440 0.986052i \(-0.553227\pi\)
−0.166440 + 0.986052i \(0.553227\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −122.332 −1.14329 −0.571646 0.820501i \(-0.693695\pi\)
−0.571646 + 0.820501i \(0.693695\pi\)
\(108\) 0 0
\(109\) −193.649 −1.77659 −0.888297 0.459269i \(-0.848111\pi\)
−0.888297 + 0.459269i \(0.848111\pi\)
\(110\) 0 0
\(111\) −50.3963 −0.454020
\(112\) 0 0
\(113\) 189.325 1.67544 0.837721 0.546098i \(-0.183888\pi\)
0.837721 + 0.546098i \(0.183888\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.2688i 0.0877672i
\(118\) 0 0
\(119\) 23.6255i 0.198533i
\(120\) 0 0
\(121\) 67.3593 0.556688
\(122\) 0 0
\(123\) 37.8781i 0.307952i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 190.990i 1.50386i −0.659244 0.751929i \(-0.729124\pi\)
0.659244 0.751929i \(-0.270876\pi\)
\(128\) 0 0
\(129\) −37.3134 −0.289251
\(130\) 0 0
\(131\) 61.2275 0.467385 0.233693 0.972311i \(-0.424919\pi\)
0.233693 + 0.972311i \(0.424919\pi\)
\(132\) 0 0
\(133\) −140.222 −1.05430
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 66.3248i 0.484123i −0.970261 0.242061i \(-0.922176\pi\)
0.970261 0.242061i \(-0.0778235\pi\)
\(138\) 0 0
\(139\) 81.0932i 0.583404i −0.956509 0.291702i \(-0.905778\pi\)
0.956509 0.291702i \(-0.0942216\pi\)
\(140\) 0 0
\(141\) 3.52399i 0.0249929i
\(142\) 0 0
\(143\) 8.82762 0.0617316
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 22.4260i 0.152558i
\(148\) 0 0
\(149\) 115.602 0.775850 0.387925 0.921691i \(-0.373192\pi\)
0.387925 + 0.921691i \(0.373192\pi\)
\(150\) 0 0
\(151\) 203.042i 1.34465i 0.740258 + 0.672323i \(0.234703\pi\)
−0.740258 + 0.672323i \(0.765297\pi\)
\(152\) 0 0
\(153\) 22.3154i 0.145852i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −144.573 −0.920847 −0.460423 0.887699i \(-0.652302\pi\)
−0.460423 + 0.887699i \(0.652302\pi\)
\(158\) 0 0
\(159\) 23.6222 0.148567
\(160\) 0 0
\(161\) 353.554i 2.19599i
\(162\) 0 0
\(163\) 49.2890 0.302386 0.151193 0.988504i \(-0.451688\pi\)
0.151193 + 0.988504i \(0.451688\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 225.149i 1.34820i −0.738642 0.674098i \(-0.764532\pi\)
0.738642 0.674098i \(-0.235468\pi\)
\(168\) 0 0
\(169\) 167.547 0.991404
\(170\) 0 0
\(171\) −132.446 −0.774541
\(172\) 0 0
\(173\) −83.1425 −0.480592 −0.240296 0.970700i \(-0.577245\pi\)
−0.240296 + 0.970700i \(0.577245\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.9423i 0.0900696i
\(178\) 0 0
\(179\) 221.904i 1.23969i 0.784725 + 0.619844i \(0.212804\pi\)
−0.784725 + 0.619844i \(0.787196\pi\)
\(180\) 0 0
\(181\) 310.184i 1.71372i 0.515547 + 0.856862i \(0.327589\pi\)
−0.515547 + 0.856862i \(0.672411\pi\)
\(182\) 0 0
\(183\) −1.03431 −0.00565195
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −19.1836 −0.102586
\(188\) 0 0
\(189\) 109.523i 0.579484i
\(190\) 0 0
\(191\) −102.675 −0.537564 −0.268782 0.963201i \(-0.586621\pi\)
−0.268782 + 0.963201i \(0.586621\pi\)
\(192\) 0 0
\(193\) −231.065 −1.19723 −0.598613 0.801039i \(-0.704281\pi\)
−0.598613 + 0.801039i \(0.704281\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 67.7244i 0.343778i −0.985116 0.171889i \(-0.945013\pi\)
0.985116 0.171889i \(-0.0549871\pi\)
\(198\) 0 0
\(199\) 67.6929i 0.340165i −0.985430 0.170083i \(-0.945597\pi\)
0.985430 0.170083i \(-0.0544034\pi\)
\(200\) 0 0
\(201\) 66.6176i 0.331431i
\(202\) 0 0
\(203\) 341.454i 1.68204i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 333.948i 1.61328i
\(208\) 0 0
\(209\) 113.859i 0.544778i
\(210\) 0 0
\(211\) 415.648 1.96990 0.984949 0.172847i \(-0.0552965\pi\)
0.984949 + 0.172847i \(0.0552965\pi\)
\(212\) 0 0
\(213\) 3.48384i 0.0163560i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 102.346 260.211i 0.471639 1.19913i
\(218\) 0 0
\(219\) −53.1379 −0.242639
\(220\) 0 0
\(221\) 3.15703 0.0142852
\(222\) 0 0
\(223\) 8.69524i 0.0389921i −0.999810 0.0194961i \(-0.993794\pi\)
0.999810 0.0194961i \(-0.00620618\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 352.159 1.55136 0.775680 0.631126i \(-0.217407\pi\)
0.775680 + 0.631126i \(0.217407\pi\)
\(228\) 0 0
\(229\) 74.0055i 0.323168i 0.986859 + 0.161584i \(0.0516603\pi\)
−0.986859 + 0.161584i \(0.948340\pi\)
\(230\) 0 0
\(231\) 45.7852 0.198204
\(232\) 0 0
\(233\) 268.408 1.15197 0.575983 0.817461i \(-0.304619\pi\)
0.575983 + 0.817461i \(0.304619\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −34.9400 −0.147426
\(238\) 0 0
\(239\) 157.032i 0.657036i −0.944498 0.328518i \(-0.893451\pi\)
0.944498 0.328518i \(-0.106549\pi\)
\(240\) 0 0
\(241\) 10.6119i 0.0440327i 0.999758 + 0.0220164i \(0.00700859\pi\)
−0.999758 + 0.0220164i \(0.992991\pi\)
\(242\) 0 0
\(243\) 156.592i 0.644411i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.7377i 0.0758609i
\(248\) 0 0
\(249\) −78.6673 −0.315933
\(250\) 0 0
\(251\) 404.851i 1.61295i −0.591265 0.806477i \(-0.701371\pi\)
0.591265 0.806477i \(-0.298629\pi\)
\(252\) 0 0
\(253\) 287.081 1.13471
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −271.328 −1.05575 −0.527875 0.849322i \(-0.677011\pi\)
−0.527875 + 0.849322i \(0.677011\pi\)
\(258\) 0 0
\(259\) 655.869i 2.53231i
\(260\) 0 0
\(261\) 322.519i 1.23570i
\(262\) 0 0
\(263\) 120.787i 0.459267i 0.973277 + 0.229633i \(0.0737527\pi\)
−0.973277 + 0.229633i \(0.926247\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 100.614 0.376833
\(268\) 0 0
\(269\) 280.569i 1.04301i 0.853249 + 0.521503i \(0.174629\pi\)
−0.853249 + 0.521503i \(0.825371\pi\)
\(270\) 0 0
\(271\) 244.579i 0.902504i 0.892397 + 0.451252i \(0.149022\pi\)
−0.892397 + 0.451252i \(0.850978\pi\)
\(272\) 0 0
\(273\) −7.53485 −0.0276002
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 98.5407i 0.355743i −0.984054 0.177871i \(-0.943079\pi\)
0.984054 0.177871i \(-0.0569211\pi\)
\(278\) 0 0
\(279\) 96.6703 245.781i 0.346488 0.880937i
\(280\) 0 0
\(281\) 206.677 0.735504 0.367752 0.929924i \(-0.380128\pi\)
0.367752 + 0.929924i \(0.380128\pi\)
\(282\) 0 0
\(283\) 389.808 1.37741 0.688706 0.725041i \(-0.258179\pi\)
0.688706 + 0.725041i \(0.258179\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −492.954 −1.71761
\(288\) 0 0
\(289\) 282.139 0.976261
\(290\) 0 0
\(291\) 12.0466i 0.0413973i
\(292\) 0 0
\(293\) −103.473 −0.353150 −0.176575 0.984287i \(-0.556502\pi\)
−0.176575 + 0.984287i \(0.556502\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 88.9308 0.299430
\(298\) 0 0
\(299\) −47.2448 −0.158009
\(300\) 0 0
\(301\) 485.605i 1.61331i
\(302\) 0 0
\(303\) 85.6622i 0.282714i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −230.191 −0.749807 −0.374903 0.927064i \(-0.622324\pi\)
−0.374903 + 0.927064i \(0.622324\pi\)
\(308\) 0 0
\(309\) 23.7631i 0.0769033i
\(310\) 0 0
\(311\) −209.481 −0.673572 −0.336786 0.941581i \(-0.609340\pi\)
−0.336786 + 0.941581i \(0.609340\pi\)
\(312\) 0 0
\(313\) 296.621i 0.947671i 0.880613 + 0.473835i \(0.157131\pi\)
−0.880613 + 0.473835i \(0.842869\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −59.4210 −0.187448 −0.0937239 0.995598i \(-0.529877\pi\)
−0.0937239 + 0.995598i \(0.529877\pi\)
\(318\) 0 0
\(319\) 277.256 0.869141
\(320\) 0 0
\(321\) 84.7852i 0.264128i
\(322\) 0 0
\(323\) 40.7194i 0.126066i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 134.213i 0.410437i
\(328\) 0 0
\(329\) −45.8621 −0.139398
\(330\) 0 0
\(331\) 286.896i 0.866757i 0.901212 + 0.433378i \(0.142679\pi\)
−0.901212 + 0.433378i \(0.857321\pi\)
\(332\) 0 0
\(333\) 619.499i 1.86036i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 496.446i 1.47313i −0.676364 0.736567i \(-0.736445\pi\)
0.676364 0.736567i \(-0.263555\pi\)
\(338\) 0 0
\(339\) 131.216i 0.387068i
\(340\) 0 0
\(341\) −211.288 83.1034i −0.619613 0.243705i
\(342\) 0 0
\(343\) −150.114 −0.437650
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 31.3455i 0.0903329i 0.998979 + 0.0451664i \(0.0143818\pi\)
−0.998979 + 0.0451664i \(0.985618\pi\)
\(348\) 0 0
\(349\) 364.479 1.04435 0.522176 0.852838i \(-0.325120\pi\)
0.522176 + 0.852838i \(0.325120\pi\)
\(350\) 0 0
\(351\) −14.6353 −0.0416960
\(352\) 0 0
\(353\) 537.862i 1.52369i 0.647760 + 0.761845i \(0.275706\pi\)
−0.647760 + 0.761845i \(0.724294\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.3742 0.0458661
\(358\) 0 0
\(359\) −530.782 −1.47850 −0.739251 0.673430i \(-0.764820\pi\)
−0.739251 + 0.673430i \(0.764820\pi\)
\(360\) 0 0
\(361\) −119.322 −0.330531
\(362\) 0 0
\(363\) 46.6850i 0.128609i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 515.419i 1.40441i 0.711974 + 0.702206i \(0.247801\pi\)
−0.711974 + 0.702206i \(0.752199\pi\)
\(368\) 0 0
\(369\) −465.618 −1.26184
\(370\) 0 0
\(371\) 307.425i 0.828640i
\(372\) 0 0
\(373\) 434.313 1.16438 0.582189 0.813053i \(-0.302196\pi\)
0.582189 + 0.813053i \(0.302196\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −45.6278 −0.121029
\(378\) 0 0
\(379\) 485.580 1.28121 0.640607 0.767869i \(-0.278683\pi\)
0.640607 + 0.767869i \(0.278683\pi\)
\(380\) 0 0
\(381\) 132.370 0.347428
\(382\) 0 0
\(383\) 465.117i 1.21440i 0.794548 + 0.607202i \(0.207708\pi\)
−0.794548 + 0.607202i \(0.792292\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 458.677i 1.18521i
\(388\) 0 0
\(389\) 400.345i 1.02917i −0.857441 0.514583i \(-0.827947\pi\)
0.857441 0.514583i \(-0.172053\pi\)
\(390\) 0 0
\(391\) 102.669 0.262581
\(392\) 0 0
\(393\) 42.4352i 0.107978i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.7892 0.0901490 0.0450745 0.998984i \(-0.485647\pi\)
0.0450745 + 0.998984i \(0.485647\pi\)
\(398\) 0 0
\(399\) 97.1845i 0.243570i
\(400\) 0 0
\(401\) 369.186i 0.920664i −0.887747 0.460332i \(-0.847730\pi\)
0.887747 0.460332i \(-0.152270\pi\)
\(402\) 0 0
\(403\) 34.7715 + 13.6763i 0.0862817 + 0.0339362i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 532.557 1.30849
\(408\) 0 0
\(409\) 687.009i 1.67973i −0.542796 0.839864i \(-0.682634\pi\)
0.542796 0.839864i \(-0.317366\pi\)
\(410\) 0 0
\(411\) 45.9680 0.111844
\(412\) 0 0
\(413\) 207.477 0.502366
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 56.2036 0.134781
\(418\) 0 0
\(419\) −123.910 −0.295729 −0.147864 0.989008i \(-0.547240\pi\)
−0.147864 + 0.989008i \(0.547240\pi\)
\(420\) 0 0
\(421\) 69.4701 0.165012 0.0825060 0.996591i \(-0.473708\pi\)
0.0825060 + 0.996591i \(0.473708\pi\)
\(422\) 0 0
\(423\) −43.3189 −0.102409
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 13.4607i 0.0315239i
\(428\) 0 0
\(429\) 6.11820i 0.0142615i
\(430\) 0 0
\(431\) −92.6617 −0.214992 −0.107496 0.994205i \(-0.534283\pi\)
−0.107496 + 0.994205i \(0.534283\pi\)
\(432\) 0 0
\(433\) 226.729i 0.523622i −0.965119 0.261811i \(-0.915680\pi\)
0.965119 0.261811i \(-0.0843198\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 609.363i 1.39442i
\(438\) 0 0
\(439\) −699.079 −1.59244 −0.796218 0.605010i \(-0.793169\pi\)
−0.796218 + 0.605010i \(0.793169\pi\)
\(440\) 0 0
\(441\) 275.673 0.625109
\(442\) 0 0
\(443\) 165.572 0.373751 0.186876 0.982384i \(-0.440164\pi\)
0.186876 + 0.982384i \(0.440164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 80.1205i 0.179240i
\(448\) 0 0
\(449\) 199.826i 0.445046i −0.974927 0.222523i \(-0.928571\pi\)
0.974927 0.222523i \(-0.0714293\pi\)
\(450\) 0 0
\(451\) 400.272i 0.887522i
\(452\) 0 0
\(453\) −140.723 −0.310646
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 91.6074i 0.200454i 0.994965 + 0.100227i \(0.0319569\pi\)
−0.994965 + 0.100227i \(0.968043\pi\)
\(458\) 0 0
\(459\) 31.8044 0.0692906
\(460\) 0 0
\(461\) 510.100i 1.10651i −0.833013 0.553254i \(-0.813386\pi\)
0.833013 0.553254i \(-0.186614\pi\)
\(462\) 0 0
\(463\) 35.1149i 0.0758422i −0.999281 0.0379211i \(-0.987926\pi\)
0.999281 0.0379211i \(-0.0120736\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −584.719 −1.25207 −0.626037 0.779793i \(-0.715324\pi\)
−0.626037 + 0.779793i \(0.715324\pi\)
\(468\) 0 0
\(469\) 866.978 1.84857
\(470\) 0 0
\(471\) 100.200i 0.212738i
\(472\) 0 0
\(473\) 394.305 0.833626
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 290.377i 0.608758i
\(478\) 0 0
\(479\) −354.617 −0.740327 −0.370163 0.928967i \(-0.620698\pi\)
−0.370163 + 0.928967i \(0.620698\pi\)
\(480\) 0 0
\(481\) −87.6426 −0.182209
\(482\) 0 0
\(483\) −245.039 −0.507327
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 100.571i 0.206511i 0.994655 + 0.103255i \(0.0329259\pi\)
−0.994655 + 0.103255i \(0.967074\pi\)
\(488\) 0 0
\(489\) 34.1609i 0.0698587i
\(490\) 0 0
\(491\) 751.241i 1.53002i 0.644017 + 0.765011i \(0.277266\pi\)
−0.644017 + 0.765011i \(0.722734\pi\)
\(492\) 0 0
\(493\) 99.1553 0.201126
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 45.3395 0.0912263
\(498\) 0 0
\(499\) 47.1472i 0.0944833i 0.998883 + 0.0472416i \(0.0150431\pi\)
−0.998883 + 0.0472416i \(0.984957\pi\)
\(500\) 0 0
\(501\) 156.045 0.311467
\(502\) 0 0
\(503\) −254.863 −0.506686 −0.253343 0.967377i \(-0.581530\pi\)
−0.253343 + 0.967377i \(0.581530\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 116.123i 0.229039i
\(508\) 0 0
\(509\) 126.136i 0.247811i −0.992294 0.123905i \(-0.960458\pi\)
0.992294 0.123905i \(-0.0395419\pi\)
\(510\) 0 0
\(511\) 691.549i 1.35333i
\(512\) 0 0
\(513\) 188.766i 0.367965i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 37.2394i 0.0720298i
\(518\) 0 0
\(519\) 57.6239i 0.111029i
\(520\) 0 0
\(521\) −814.952 −1.56421 −0.782103 0.623149i \(-0.785853\pi\)
−0.782103 + 0.623149i \(0.785853\pi\)
\(522\) 0 0
\(523\) 24.1549i 0.0461853i −0.999733 0.0230926i \(-0.992649\pi\)
0.999733 0.0230926i \(-0.00735127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −75.5630 29.7203i −0.143383 0.0563953i
\(528\) 0 0
\(529\) −1007.44 −1.90442
\(530\) 0 0
\(531\) 195.972 0.369062
\(532\) 0 0
\(533\) 65.8726i 0.123588i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −153.796 −0.286398
\(538\) 0 0
\(539\) 236.985i 0.439675i
\(540\) 0 0
\(541\) 241.017 0.445503 0.222752 0.974875i \(-0.428496\pi\)
0.222752 + 0.974875i \(0.428496\pi\)
\(542\) 0 0
\(543\) −214.980 −0.395912
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 640.236 1.17045 0.585225 0.810871i \(-0.301006\pi\)
0.585225 + 0.810871i \(0.301006\pi\)
\(548\) 0 0
\(549\) 12.7143i 0.0231589i
\(550\) 0 0
\(551\) 588.508i 1.06807i
\(552\) 0 0
\(553\) 454.718i 0.822274i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 290.241i 0.521079i −0.965463 0.260539i \(-0.916100\pi\)
0.965463 0.260539i \(-0.0839004\pi\)
\(558\) 0 0
\(559\) −64.8906 −0.116083
\(560\) 0 0
\(561\) 13.2956i 0.0236999i
\(562\) 0 0
\(563\) −546.321 −0.970375 −0.485187 0.874410i \(-0.661249\pi\)
−0.485187 + 0.874410i \(0.661249\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 615.705 1.08590
\(568\) 0 0
\(569\) 799.608i 1.40529i −0.711543 0.702643i \(-0.752003\pi\)
0.711543 0.702643i \(-0.247997\pi\)
\(570\) 0 0
\(571\) 343.911i 0.602296i −0.953577 0.301148i \(-0.902630\pi\)
0.953577 0.301148i \(-0.0973697\pi\)
\(572\) 0 0
\(573\) 71.1612i 0.124191i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −388.763 −0.673767 −0.336883 0.941546i \(-0.609373\pi\)
−0.336883 + 0.941546i \(0.609373\pi\)
\(578\) 0 0
\(579\) 160.145i 0.276589i
\(580\) 0 0
\(581\) 1023.79i 1.76213i
\(582\) 0 0
\(583\) −249.625 −0.428174
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 501.375i 0.854131i 0.904221 + 0.427065i \(0.140453\pi\)
−0.904221 + 0.427065i \(0.859547\pi\)
\(588\) 0 0
\(589\) −176.397 + 448.483i −0.299485 + 0.761431i
\(590\) 0 0
\(591\) 46.9380 0.0794213
\(592\) 0 0
\(593\) −914.823 −1.54270 −0.771352 0.636409i \(-0.780419\pi\)
−0.771352 + 0.636409i \(0.780419\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 46.9162 0.0785865
\(598\) 0 0
\(599\) 891.875 1.48894 0.744470 0.667656i \(-0.232702\pi\)
0.744470 + 0.667656i \(0.232702\pi\)
\(600\) 0 0
\(601\) 532.088i 0.885338i −0.896685 0.442669i \(-0.854032\pi\)
0.896685 0.442669i \(-0.145968\pi\)
\(602\) 0 0
\(603\) 818.901 1.35804
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 777.101 1.28023 0.640116 0.768278i \(-0.278886\pi\)
0.640116 + 0.768278i \(0.278886\pi\)
\(608\) 0 0
\(609\) −236.653 −0.388592
\(610\) 0 0
\(611\) 6.12847i 0.0100302i
\(612\) 0 0
\(613\) 253.403i 0.413381i −0.978406 0.206691i \(-0.933731\pi\)
0.978406 0.206691i \(-0.0662693\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −590.743 −0.957444 −0.478722 0.877966i \(-0.658900\pi\)
−0.478722 + 0.877966i \(0.658900\pi\)
\(618\) 0 0
\(619\) 905.297i 1.46252i 0.682101 + 0.731258i \(0.261066\pi\)
−0.682101 + 0.731258i \(0.738934\pi\)
\(620\) 0 0
\(621\) −475.951 −0.766427
\(622\) 0 0
\(623\) 1309.42i 2.10180i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −78.9125 −0.125857
\(628\) 0 0
\(629\) 190.459 0.302796
\(630\) 0 0
\(631\) 384.376i 0.609154i 0.952488 + 0.304577i \(0.0985151\pi\)
−0.952488 + 0.304577i \(0.901485\pi\)
\(632\) 0 0
\(633\) 288.075i 0.455095i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 39.0004i 0.0612251i
\(638\) 0 0
\(639\) 42.8252 0.0670191
\(640\) 0 0
\(641\) 1141.36i 1.78059i −0.455379 0.890297i \(-0.650496\pi\)
0.455379 0.890297i \(-0.349504\pi\)
\(642\) 0 0
\(643\) 1011.71i 1.57342i −0.617324 0.786709i \(-0.711783\pi\)
0.617324 0.786709i \(-0.288217\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 230.511i 0.356276i 0.984005 + 0.178138i \(0.0570074\pi\)
−0.984005 + 0.178138i \(0.942993\pi\)
\(648\) 0 0
\(649\) 168.469i 0.259582i
\(650\) 0 0
\(651\) 180.345 + 70.9332i 0.277028 + 0.108960i
\(652\) 0 0
\(653\) −462.269 −0.707916 −0.353958 0.935261i \(-0.615164\pi\)
−0.353958 + 0.935261i \(0.615164\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 653.200i 0.994217i
\(658\) 0 0
\(659\) 911.200 1.38270 0.691350 0.722520i \(-0.257016\pi\)
0.691350 + 0.722520i \(0.257016\pi\)
\(660\) 0 0
\(661\) 122.411 0.185191 0.0925954 0.995704i \(-0.470484\pi\)
0.0925954 + 0.995704i \(0.470484\pi\)
\(662\) 0 0
\(663\) 2.18805i 0.00330023i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1483.85 −2.22467
\(668\) 0 0
\(669\) 6.02645 0.00900814
\(670\) 0 0
\(671\) 10.9299 0.0162890
\(672\) 0 0
\(673\) 381.492i 0.566853i 0.958994 + 0.283427i \(0.0914712\pi\)
−0.958994 + 0.283427i \(0.908529\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 882.565i 1.30364i −0.758373 0.651821i \(-0.774005\pi\)
0.758373 0.651821i \(-0.225995\pi\)
\(678\) 0 0
\(679\) 156.777 0.230895
\(680\) 0 0
\(681\) 244.072i 0.358403i
\(682\) 0 0
\(683\) 1084.00 1.58712 0.793561 0.608490i \(-0.208225\pi\)
0.793561 + 0.608490i \(0.208225\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −51.2913 −0.0746599
\(688\) 0 0
\(689\) 41.0807 0.0596236
\(690\) 0 0
\(691\) −6.47645 −0.00937258 −0.00468629 0.999989i \(-0.501492\pi\)
−0.00468629 + 0.999989i \(0.501492\pi\)
\(692\) 0 0
\(693\) 562.817i 0.812146i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 143.150i 0.205380i
\(698\) 0 0
\(699\) 186.027i 0.266133i
\(700\) 0 0
\(701\) 780.180 1.11295 0.556476 0.830864i \(-0.312153\pi\)
0.556476 + 0.830864i \(0.312153\pi\)
\(702\) 0 0
\(703\) 1130.41i 1.60799i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1114.83 1.57684
\(708\) 0 0
\(709\) 1369.94i 1.93221i 0.258149 + 0.966105i \(0.416887\pi\)
−0.258149 + 0.966105i \(0.583113\pi\)
\(710\) 0 0
\(711\) 429.502i 0.604081i
\(712\) 0 0
\(713\) 1130.80 + 444.763i 1.58597 + 0.623791i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 108.835 0.151792
\(718\) 0 0
\(719\) 1101.50i 1.53198i 0.642851 + 0.765992i \(0.277752\pi\)
−0.642851 + 0.765992i \(0.722248\pi\)
\(720\) 0 0
\(721\) −309.259 −0.428931
\(722\) 0 0
\(723\) −7.35482 −0.0101726
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 556.384 0.765314 0.382657 0.923890i \(-0.375009\pi\)
0.382657 + 0.923890i \(0.375009\pi\)
\(728\) 0 0
\(729\) 505.821 0.693857
\(730\) 0 0
\(731\) 141.016 0.192908
\(732\) 0 0
\(733\) −482.566 −0.658343 −0.329172 0.944270i \(-0.606769\pi\)
−0.329172 + 0.944270i \(0.606769\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 703.974i 0.955189i
\(738\) 0 0
\(739\) 386.972i 0.523643i 0.965116 + 0.261822i \(0.0843232\pi\)
−0.965116 + 0.261822i \(0.915677\pi\)
\(740\) 0 0
\(741\) 12.9866 0.0175257
\(742\) 0 0
\(743\) 230.629i 0.310402i 0.987883 + 0.155201i \(0.0496026\pi\)
−0.987883 + 0.155201i \(0.950397\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 967.022i 1.29454i
\(748\) 0 0
\(749\) −1103.42 −1.47318
\(750\) 0 0
\(751\) 736.091 0.980148 0.490074 0.871681i \(-0.336970\pi\)
0.490074 + 0.871681i \(0.336970\pi\)
\(752\) 0 0
\(753\) 280.592 0.372632
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 151.962i 0.200743i 0.994950 + 0.100371i \(0.0320031\pi\)
−0.994950 + 0.100371i \(0.967997\pi\)
\(758\) 0 0
\(759\) 198.969i 0.262146i
\(760\) 0 0
\(761\) 400.905i 0.526813i −0.964685 0.263407i \(-0.915154\pi\)
0.964685 0.263407i \(-0.0848461\pi\)
\(762\) 0 0
\(763\) −1746.68 −2.28923
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.7248i 0.0361470i
\(768\) 0 0
\(769\) −806.936 −1.04933 −0.524666 0.851308i \(-0.675810\pi\)
−0.524666 + 0.851308i \(0.675810\pi\)
\(770\) 0 0
\(771\) 188.050i 0.243904i
\(772\) 0 0
\(773\) 478.951i 0.619601i −0.950802 0.309800i \(-0.899738\pi\)
0.950802 0.309800i \(-0.100262\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −454.566 −0.585027
\(778\) 0 0
\(779\) 849.624 1.09066
\(780\) 0 0
\(781\) 36.8150i 0.0471383i
\(782\) 0 0
\(783\) −459.662 −0.587052
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 85.3203i 0.108412i −0.998530 0.0542060i \(-0.982737\pi\)
0.998530 0.0542060i \(-0.0172628\pi\)
\(788\) 0 0
\(789\) −83.7144 −0.106102
\(790\) 0 0
\(791\) 1707.68 2.15889
\(792\) 0 0
\(793\) −1.79873 −0.00226826
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 176.296i 0.221200i −0.993865 0.110600i \(-0.964723\pi\)
0.993865 0.110600i \(-0.0352772\pi\)
\(798\) 0 0
\(799\) 13.3180i 0.0166683i
\(800\) 0 0
\(801\) 1236.81i 1.54408i
\(802\) 0 0
\(803\) 561.529 0.699289
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −194.455 −0.240960
\(808\) 0 0
\(809\) 708.863i 0.876222i −0.898921 0.438111i \(-0.855648\pi\)
0.898921 0.438111i \(-0.144352\pi\)
\(810\) 0 0
\(811\) −361.574 −0.445837 −0.222919 0.974837i \(-0.571558\pi\)
−0.222919 + 0.974837i \(0.571558\pi\)
\(812\) 0 0
\(813\) −169.511 −0.208501
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 836.959i 1.02443i
\(818\) 0 0
\(819\) 92.6225i 0.113092i
\(820\) 0 0
\(821\) 1155.39i 1.40730i −0.710547 0.703650i \(-0.751552\pi\)
0.710547 0.703650i \(-0.248448\pi\)
\(822\) 0 0
\(823\) 578.491i 0.702905i −0.936206 0.351453i \(-0.885688\pi\)
0.936206 0.351453i \(-0.114312\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 824.460i 0.996929i −0.866910 0.498465i \(-0.833897\pi\)
0.866910 0.498465i \(-0.166103\pi\)
\(828\) 0 0
\(829\) 745.335i 0.899077i −0.893261 0.449538i \(-0.851588\pi\)
0.893261 0.449538i \(-0.148412\pi\)
\(830\) 0 0
\(831\) 68.2960 0.0821853
\(832\) 0 0
\(833\) 84.7530i 0.101744i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 350.294 + 137.777i 0.418511 + 0.164608i
\(838\) 0 0
\(839\) −1102.25 −1.31377 −0.656884 0.753991i \(-0.728126\pi\)
−0.656884 + 0.753991i \(0.728126\pi\)
\(840\) 0 0
\(841\) −592.069 −0.704006
\(842\) 0 0
\(843\) 143.242i 0.169920i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 607.569 0.717319
\(848\) 0 0
\(849\) 270.165i 0.318216i
\(850\) 0 0
\(851\) −2850.21 −3.34924
\(852\) 0 0
\(853\) −733.709 −0.860151 −0.430075 0.902793i \(-0.641513\pi\)
−0.430075 + 0.902793i \(0.641513\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1050.67 1.22599 0.612995 0.790087i \(-0.289964\pi\)
0.612995 + 0.790087i \(0.289964\pi\)
\(858\) 0 0
\(859\) 1488.78i 1.73316i 0.499040 + 0.866579i \(0.333686\pi\)
−0.499040 + 0.866579i \(0.666314\pi\)
\(860\) 0 0
\(861\) 341.654i 0.396810i
\(862\) 0 0
\(863\) 268.208i 0.310786i −0.987853 0.155393i \(-0.950336\pi\)
0.987853 0.155393i \(-0.0496644\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 195.543i 0.225540i
\(868\) 0 0
\(869\) 369.225 0.424885
\(870\) 0 0
\(871\) 115.853i 0.133011i
\(872\) 0 0
\(873\) 148.084 0.169626
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1595.55 −1.81933 −0.909664 0.415344i \(-0.863661\pi\)
−0.909664 + 0.415344i \(0.863661\pi\)
\(878\) 0 0
\(879\) 71.7144i 0.0815864i
\(880\) 0 0
\(881\) 134.904i 0.153126i −0.997065 0.0765630i \(-0.975605\pi\)
0.997065 0.0765630i \(-0.0243946\pi\)
\(882\) 0 0
\(883\) 721.579i 0.817191i 0.912716 + 0.408595i \(0.133981\pi\)
−0.912716 + 0.408595i \(0.866019\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −357.058 −0.402545 −0.201273 0.979535i \(-0.564508\pi\)
−0.201273 + 0.979535i \(0.564508\pi\)
\(888\) 0 0
\(889\) 1722.70i 1.93779i
\(890\) 0 0
\(891\) 499.944i 0.561105i
\(892\) 0 0
\(893\) 79.0450 0.0885162
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 32.7441i 0.0365041i
\(898\) 0 0
\(899\) 1092.10 + 429.541i 1.21479 + 0.477799i
\(900\) 0 0
\(901\) −89.2737 −0.0990829
\(902\) 0 0
\(903\) −336.561 −0.372714
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −695.351 −0.766650 −0.383325 0.923614i \(-0.625221\pi\)
−0.383325 + 0.923614i \(0.625221\pi\)
\(908\) 0 0
\(909\) 1053.01 1.15842
\(910\) 0 0
\(911\) 700.661i 0.769112i 0.923102 + 0.384556i \(0.125645\pi\)
−0.923102 + 0.384556i \(0.874355\pi\)
\(912\) 0 0
\(913\) 831.308 0.910523
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 552.261 0.602248
\(918\) 0 0
\(919\) 1217.24 1.32452 0.662261 0.749273i \(-0.269597\pi\)
0.662261 + 0.749273i \(0.269597\pi\)
\(920\) 0 0
\(921\) 159.539i 0.173224i
\(922\) 0 0
\(923\) 6.05863i 0.00656406i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −292.110 −0.315113
\(928\) 0 0
\(929\) 1412.58i 1.52054i −0.649610 0.760268i \(-0.725068\pi\)
0.649610 0.760268i \(-0.274932\pi\)
\(930\) 0 0
\(931\) −503.027 −0.540309
\(932\) 0 0
\(933\) 145.186i 0.155612i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 768.384 0.820047 0.410023 0.912075i \(-0.365521\pi\)
0.410023 + 0.912075i \(0.365521\pi\)
\(938\) 0 0
\(939\) −205.580 −0.218935
\(940\) 0 0
\(941\) 1009.58i 1.07289i 0.843937 + 0.536443i \(0.180232\pi\)
−0.843937 + 0.536443i \(0.819768\pi\)
\(942\) 0 0
\(943\) 2142.23i 2.27172i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 759.057i 0.801539i 0.916179 + 0.400769i \(0.131257\pi\)
−0.916179 + 0.400769i \(0.868743\pi\)
\(948\) 0 0
\(949\) −92.4105 −0.0973767
\(950\) 0 0
\(951\) 41.1831i 0.0433051i
\(952\) 0 0
\(953\) 122.027i 0.128045i −0.997948 0.0640227i \(-0.979607\pi\)
0.997948 0.0640227i \(-0.0203930\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 192.159i 0.200793i
\(958\) 0 0
\(959\) 598.239i 0.623815i
\(960\) 0 0
\(961\) −703.503 654.679i −0.732053 0.681248i
\(962\) 0 0
\(963\) −1042.23 −1.08227
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 294.473i 0.304523i 0.988340 + 0.152261i \(0.0486555\pi\)
−0.988340 + 0.152261i \(0.951344\pi\)
\(968\) 0 0
\(969\) −28.2215 −0.0291244
\(970\) 0 0
\(971\) 27.5432 0.0283658 0.0141829 0.999899i \(-0.495485\pi\)
0.0141829 + 0.999899i \(0.495485\pi\)
\(972\) 0 0
\(973\) 731.447i 0.751744i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −353.842 −0.362172 −0.181086 0.983467i \(-0.557961\pi\)
−0.181086 + 0.983467i \(0.557961\pi\)
\(978\) 0 0
\(979\) −1063.23 −1.08604
\(980\) 0 0
\(981\) −1649.82 −1.68177
\(982\) 0 0
\(983\) 514.232i 0.523125i 0.965186 + 0.261563i \(0.0842378\pi\)
−0.965186 + 0.261563i \(0.915762\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 31.7858i 0.0322045i
\(988\) 0 0
\(989\) −2110.29 −2.13376
\(990\) 0 0
\(991\) 1484.21i 1.49769i −0.662746 0.748845i \(-0.730609\pi\)
0.662746 0.748845i \(-0.269391\pi\)
\(992\) 0 0
\(993\) −198.840 −0.200242
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −876.530 −0.879168 −0.439584 0.898202i \(-0.644874\pi\)
−0.439584 + 0.898202i \(0.644874\pi\)
\(998\) 0 0
\(999\) −882.925 −0.883808
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 3100.3.d.e.1301.11 20
5.2 odd 4 3100.3.f.c.1549.21 40
5.3 odd 4 3100.3.f.c.1549.20 40
5.4 even 2 620.3.d.a.61.10 20
31.30 odd 2 inner 3100.3.d.e.1301.10 20
155.92 even 4 3100.3.f.c.1549.19 40
155.123 even 4 3100.3.f.c.1549.22 40
155.154 odd 2 620.3.d.a.61.11 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
620.3.d.a.61.10 20 5.4 even 2
620.3.d.a.61.11 yes 20 155.154 odd 2
3100.3.d.e.1301.10 20 31.30 odd 2 inner
3100.3.d.e.1301.11 20 1.1 even 1 trivial
3100.3.f.c.1549.19 40 155.92 even 4
3100.3.f.c.1549.20 40 5.3 odd 4
3100.3.f.c.1549.21 40 5.2 odd 4
3100.3.f.c.1549.22 40 155.123 even 4