Properties

Label 1008.4.h.a.575.11
Level $1008$
Weight $4$
Character 1008.575
Analytic conductor $59.474$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,4,Mod(575,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.575");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.h (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: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.11
Root \(8.41502 + 0.655192i\) of defining polynomial
Character \(\chi\) \(=\) 1008.575
Dual form 1008.4.h.a.575.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+14.1054i q^{5} -7.00000i q^{7} +O(q^{10})\) \(q+14.1054i q^{5} -7.00000i q^{7} -13.1065 q^{11} +30.2418 q^{13} -78.2552i q^{17} +72.0861i q^{19} -217.553 q^{23} -73.9633 q^{25} -3.49080i q^{29} +305.218i q^{31} +98.7380 q^{35} +191.923 q^{37} +246.011i q^{41} -200.959i q^{43} -338.976 q^{47} -49.0000 q^{49} -423.136i q^{53} -184.873i q^{55} -366.860 q^{59} -730.595 q^{61} +426.573i q^{65} -1024.29i q^{67} -79.3885 q^{71} +127.786 q^{73} +91.7458i q^{77} -413.860i q^{79} -228.695 q^{83} +1103.82 q^{85} -193.891i q^{89} -211.692i q^{91} -1016.81 q^{95} +1004.12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) 14.1054i 1.26163i 0.775934 + 0.630814i \(0.217279\pi\)
−0.775934 + 0.630814i \(0.782721\pi\)
\(6\) 0 0
\(7\) − 7.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −13.1065 −0.359252 −0.179626 0.983735i \(-0.557489\pi\)
−0.179626 + 0.983735i \(0.557489\pi\)
\(12\) 0 0
\(13\) 30.2418 0.645197 0.322598 0.946536i \(-0.395444\pi\)
0.322598 + 0.946536i \(0.395444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 78.2552i − 1.11645i −0.829689 0.558226i \(-0.811482\pi\)
0.829689 0.558226i \(-0.188518\pi\)
\(18\) 0 0
\(19\) 72.0861i 0.870404i 0.900333 + 0.435202i \(0.143323\pi\)
−0.900333 + 0.435202i \(0.856677\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −217.553 −1.97231 −0.986153 0.165840i \(-0.946967\pi\)
−0.986153 + 0.165840i \(0.946967\pi\)
\(24\) 0 0
\(25\) −73.9633 −0.591706
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.49080i − 0.0223526i −0.999938 0.0111763i \(-0.996442\pi\)
0.999938 0.0111763i \(-0.00355760\pi\)
\(30\) 0 0
\(31\) 305.218i 1.76835i 0.467160 + 0.884173i \(0.345277\pi\)
−0.467160 + 0.884173i \(0.654723\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 98.7380 0.476851
\(36\) 0 0
\(37\) 191.923 0.852754 0.426377 0.904546i \(-0.359790\pi\)
0.426377 + 0.904546i \(0.359790\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 246.011i 0.937086i 0.883441 + 0.468543i \(0.155221\pi\)
−0.883441 + 0.468543i \(0.844779\pi\)
\(42\) 0 0
\(43\) − 200.959i − 0.712698i −0.934353 0.356349i \(-0.884021\pi\)
0.934353 0.356349i \(-0.115979\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −338.976 −1.05202 −0.526008 0.850480i \(-0.676312\pi\)
−0.526008 + 0.850480i \(0.676312\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 423.136i − 1.09665i −0.836267 0.548323i \(-0.815266\pi\)
0.836267 0.548323i \(-0.184734\pi\)
\(54\) 0 0
\(55\) − 184.873i − 0.453242i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −366.860 −0.809509 −0.404755 0.914425i \(-0.632643\pi\)
−0.404755 + 0.914425i \(0.632643\pi\)
\(60\) 0 0
\(61\) −730.595 −1.53349 −0.766747 0.641950i \(-0.778126\pi\)
−0.766747 + 0.641950i \(0.778126\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 426.573i 0.813999i
\(66\) 0 0
\(67\) − 1024.29i − 1.86772i −0.357633 0.933862i \(-0.616416\pi\)
0.357633 0.933862i \(-0.383584\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −79.3885 −0.132700 −0.0663499 0.997796i \(-0.521135\pi\)
−0.0663499 + 0.997796i \(0.521135\pi\)
\(72\) 0 0
\(73\) 127.786 0.204880 0.102440 0.994739i \(-0.467335\pi\)
0.102440 + 0.994739i \(0.467335\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 91.7458i 0.135784i
\(78\) 0 0
\(79\) − 413.860i − 0.589403i −0.955589 0.294701i \(-0.904780\pi\)
0.955589 0.294701i \(-0.0952202\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −228.695 −0.302440 −0.151220 0.988500i \(-0.548320\pi\)
−0.151220 + 0.988500i \(0.548320\pi\)
\(84\) 0 0
\(85\) 1103.82 1.40855
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 193.891i − 0.230926i −0.993312 0.115463i \(-0.963165\pi\)
0.993312 0.115463i \(-0.0368352\pi\)
\(90\) 0 0
\(91\) − 211.692i − 0.243861i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1016.81 −1.09813
\(96\) 0 0
\(97\) 1004.12 1.05107 0.525533 0.850773i \(-0.323866\pi\)
0.525533 + 0.850773i \(0.323866\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 218.638i 0.215399i 0.994183 + 0.107700i \(0.0343485\pi\)
−0.994183 + 0.107700i \(0.965652\pi\)
\(102\) 0 0
\(103\) − 965.671i − 0.923790i −0.886935 0.461895i \(-0.847170\pi\)
0.886935 0.461895i \(-0.152830\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 333.264 0.301102 0.150551 0.988602i \(-0.451895\pi\)
0.150551 + 0.988602i \(0.451895\pi\)
\(108\) 0 0
\(109\) −1891.50 −1.66214 −0.831068 0.556170i \(-0.812270\pi\)
−0.831068 + 0.556170i \(0.812270\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 255.337i − 0.212567i −0.994336 0.106284i \(-0.966105\pi\)
0.994336 0.106284i \(-0.0338951\pi\)
\(114\) 0 0
\(115\) − 3068.69i − 2.48832i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −547.787 −0.421979
\(120\) 0 0
\(121\) −1159.22 −0.870938
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 719.895i 0.515115i
\(126\) 0 0
\(127\) 82.2137i 0.0574432i 0.999587 + 0.0287216i \(0.00914363\pi\)
−0.999587 + 0.0287216i \(0.990856\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1925.25 −1.28404 −0.642021 0.766687i \(-0.721904\pi\)
−0.642021 + 0.766687i \(0.721904\pi\)
\(132\) 0 0
\(133\) 504.602 0.328982
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2062.05i − 1.28593i −0.765894 0.642967i \(-0.777703\pi\)
0.765894 0.642967i \(-0.222297\pi\)
\(138\) 0 0
\(139\) − 363.818i − 0.222005i −0.993820 0.111002i \(-0.964594\pi\)
0.993820 0.111002i \(-0.0354061\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −396.365 −0.231788
\(144\) 0 0
\(145\) 49.2392 0.0282007
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 961.387i − 0.528590i −0.964442 0.264295i \(-0.914861\pi\)
0.964442 0.264295i \(-0.0851393\pi\)
\(150\) 0 0
\(151\) 2080.34i 1.12116i 0.828099 + 0.560581i \(0.189422\pi\)
−0.828099 + 0.560581i \(0.810578\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4305.23 −2.23099
\(156\) 0 0
\(157\) 3163.02 1.60788 0.803939 0.594712i \(-0.202734\pi\)
0.803939 + 0.594712i \(0.202734\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1522.87i 0.745461i
\(162\) 0 0
\(163\) − 2325.36i − 1.11740i −0.829369 0.558701i \(-0.811300\pi\)
0.829369 0.558701i \(-0.188700\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4001.48 −1.85415 −0.927077 0.374870i \(-0.877687\pi\)
−0.927077 + 0.374870i \(0.877687\pi\)
\(168\) 0 0
\(169\) −1282.44 −0.583721
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1284.23i − 0.564384i −0.959358 0.282192i \(-0.908938\pi\)
0.959358 0.282192i \(-0.0910616\pi\)
\(174\) 0 0
\(175\) 517.743i 0.223644i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −644.970 −0.269315 −0.134657 0.990892i \(-0.542993\pi\)
−0.134657 + 0.990892i \(0.542993\pi\)
\(180\) 0 0
\(181\) 2374.61 0.975155 0.487578 0.873080i \(-0.337881\pi\)
0.487578 + 0.873080i \(0.337881\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2707.15i 1.07586i
\(186\) 0 0
\(187\) 1025.65i 0.401087i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4428.79 −1.67778 −0.838889 0.544302i \(-0.816795\pi\)
−0.838889 + 0.544302i \(0.816795\pi\)
\(192\) 0 0
\(193\) −3114.28 −1.16151 −0.580753 0.814080i \(-0.697242\pi\)
−0.580753 + 0.814080i \(0.697242\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3892.82i 1.40788i 0.710260 + 0.703939i \(0.248577\pi\)
−0.710260 + 0.703939i \(0.751423\pi\)
\(198\) 0 0
\(199\) 1837.29i 0.654482i 0.944941 + 0.327241i \(0.106119\pi\)
−0.944941 + 0.327241i \(0.893881\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −24.4356 −0.00844848
\(204\) 0 0
\(205\) −3470.10 −1.18225
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 944.799i − 0.312694i
\(210\) 0 0
\(211\) − 1753.48i − 0.572105i −0.958214 0.286053i \(-0.907657\pi\)
0.958214 0.286053i \(-0.0923433\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2834.62 0.899160
\(216\) 0 0
\(217\) 2136.52 0.668372
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2366.58i − 0.720331i
\(222\) 0 0
\(223\) − 2467.93i − 0.741098i −0.928813 0.370549i \(-0.879170\pi\)
0.928813 0.370549i \(-0.120830\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3753.03 −1.09735 −0.548673 0.836037i \(-0.684867\pi\)
−0.548673 + 0.836037i \(0.684867\pi\)
\(228\) 0 0
\(229\) −4115.92 −1.18772 −0.593860 0.804568i \(-0.702397\pi\)
−0.593860 + 0.804568i \(0.702397\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 2719.40i − 0.764608i −0.924037 0.382304i \(-0.875131\pi\)
0.924037 0.382304i \(-0.124869\pi\)
\(234\) 0 0
\(235\) − 4781.41i − 1.32725i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3670.35 0.993370 0.496685 0.867931i \(-0.334550\pi\)
0.496685 + 0.867931i \(0.334550\pi\)
\(240\) 0 0
\(241\) 3043.03 0.813357 0.406678 0.913571i \(-0.366687\pi\)
0.406678 + 0.913571i \(0.366687\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 691.166i − 0.180233i
\(246\) 0 0
\(247\) 2180.01i 0.561582i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1036.93 0.260758 0.130379 0.991464i \(-0.458381\pi\)
0.130379 + 0.991464i \(0.458381\pi\)
\(252\) 0 0
\(253\) 2851.37 0.708554
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 929.028i 0.225491i 0.993624 + 0.112745i \(0.0359645\pi\)
−0.993624 + 0.112745i \(0.964036\pi\)
\(258\) 0 0
\(259\) − 1343.46i − 0.322311i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1676.15 0.392987 0.196494 0.980505i \(-0.437045\pi\)
0.196494 + 0.980505i \(0.437045\pi\)
\(264\) 0 0
\(265\) 5968.52 1.38356
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1717.55i 0.389297i 0.980873 + 0.194649i \(0.0623567\pi\)
−0.980873 + 0.194649i \(0.937643\pi\)
\(270\) 0 0
\(271\) − 3930.25i − 0.880981i −0.897757 0.440490i \(-0.854805\pi\)
0.897757 0.440490i \(-0.145195\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 969.402 0.212571
\(276\) 0 0
\(277\) −4632.58 −1.00486 −0.502428 0.864619i \(-0.667560\pi\)
−0.502428 + 0.864619i \(0.667560\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4487.15i 0.952600i 0.879283 + 0.476300i \(0.158022\pi\)
−0.879283 + 0.476300i \(0.841978\pi\)
\(282\) 0 0
\(283\) 3633.83i 0.763281i 0.924311 + 0.381641i \(0.124641\pi\)
−0.924311 + 0.381641i \(0.875359\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1722.08 0.354185
\(288\) 0 0
\(289\) −1210.88 −0.246464
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 3355.27i − 0.669000i −0.942396 0.334500i \(-0.891433\pi\)
0.942396 0.334500i \(-0.108567\pi\)
\(294\) 0 0
\(295\) − 5174.71i − 1.02130i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6579.20 −1.27253
\(300\) 0 0
\(301\) −1406.72 −0.269375
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 10305.4i − 1.93470i
\(306\) 0 0
\(307\) − 1849.50i − 0.343832i −0.985112 0.171916i \(-0.945004\pi\)
0.985112 0.171916i \(-0.0549957\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1872.43 0.341401 0.170700 0.985323i \(-0.445397\pi\)
0.170700 + 0.985323i \(0.445397\pi\)
\(312\) 0 0
\(313\) −114.858 −0.0207417 −0.0103709 0.999946i \(-0.503301\pi\)
−0.0103709 + 0.999946i \(0.503301\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7436.55i 1.31760i 0.752320 + 0.658798i \(0.228935\pi\)
−0.752320 + 0.658798i \(0.771065\pi\)
\(318\) 0 0
\(319\) 45.7523i 0.00803021i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5641.11 0.971764
\(324\) 0 0
\(325\) −2236.78 −0.381767
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2372.83i 0.397625i
\(330\) 0 0
\(331\) 5161.23i 0.857059i 0.903528 + 0.428530i \(0.140968\pi\)
−0.903528 + 0.428530i \(0.859032\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14448.1 2.35637
\(336\) 0 0
\(337\) −5415.65 −0.875398 −0.437699 0.899122i \(-0.644206\pi\)
−0.437699 + 0.899122i \(0.644206\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4000.35i − 0.635281i
\(342\) 0 0
\(343\) 343.000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6577.28 −1.01754 −0.508770 0.860902i \(-0.669900\pi\)
−0.508770 + 0.860902i \(0.669900\pi\)
\(348\) 0 0
\(349\) 8587.72 1.31716 0.658582 0.752509i \(-0.271157\pi\)
0.658582 + 0.752509i \(0.271157\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 2518.85i − 0.379788i −0.981805 0.189894i \(-0.939186\pi\)
0.981805 0.189894i \(-0.0608144\pi\)
\(354\) 0 0
\(355\) − 1119.81i − 0.167418i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10688.2 1.57132 0.785659 0.618660i \(-0.212324\pi\)
0.785659 + 0.618660i \(0.212324\pi\)
\(360\) 0 0
\(361\) 1662.60 0.242397
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1802.48i 0.258482i
\(366\) 0 0
\(367\) − 451.872i − 0.0642712i −0.999484 0.0321356i \(-0.989769\pi\)
0.999484 0.0321356i \(-0.0102308\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2961.95 −0.414493
\(372\) 0 0
\(373\) 8510.92 1.18144 0.590722 0.806875i \(-0.298843\pi\)
0.590722 + 0.806875i \(0.298843\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 105.568i − 0.0144218i
\(378\) 0 0
\(379\) 11356.2i 1.53912i 0.638574 + 0.769561i \(0.279525\pi\)
−0.638574 + 0.769561i \(0.720475\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10796.9 1.44046 0.720229 0.693736i \(-0.244037\pi\)
0.720229 + 0.693736i \(0.244037\pi\)
\(384\) 0 0
\(385\) −1294.11 −0.171309
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4634.81i 0.604099i 0.953292 + 0.302049i \(0.0976708\pi\)
−0.953292 + 0.302049i \(0.902329\pi\)
\(390\) 0 0
\(391\) 17024.7i 2.20198i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5837.67 0.743608
\(396\) 0 0
\(397\) −8193.89 −1.03587 −0.517934 0.855421i \(-0.673299\pi\)
−0.517934 + 0.855421i \(0.673299\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7505.04i 0.934623i 0.884093 + 0.467311i \(0.154777\pi\)
−0.884093 + 0.467311i \(0.845223\pi\)
\(402\) 0 0
\(403\) 9230.32i 1.14093i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2515.44 −0.306353
\(408\) 0 0
\(409\) 708.588 0.0856661 0.0428330 0.999082i \(-0.486362\pi\)
0.0428330 + 0.999082i \(0.486362\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2568.02i 0.305966i
\(414\) 0 0
\(415\) − 3225.84i − 0.381566i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11975.7 −1.39631 −0.698153 0.715949i \(-0.745994\pi\)
−0.698153 + 0.715949i \(0.745994\pi\)
\(420\) 0 0
\(421\) 1813.85 0.209980 0.104990 0.994473i \(-0.466519\pi\)
0.104990 + 0.994473i \(0.466519\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5788.01i 0.660611i
\(426\) 0 0
\(427\) 5114.16i 0.579606i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9988.53 1.11631 0.558156 0.829736i \(-0.311509\pi\)
0.558156 + 0.829736i \(0.311509\pi\)
\(432\) 0 0
\(433\) −1385.76 −0.153800 −0.0768998 0.997039i \(-0.524502\pi\)
−0.0768998 + 0.997039i \(0.524502\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 15682.6i − 1.71670i
\(438\) 0 0
\(439\) 15869.5i 1.72531i 0.505795 + 0.862653i \(0.331199\pi\)
−0.505795 + 0.862653i \(0.668801\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10484.7 −1.12448 −0.562240 0.826974i \(-0.690060\pi\)
−0.562240 + 0.826974i \(0.690060\pi\)
\(444\) 0 0
\(445\) 2734.92 0.291343
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 2234.73i − 0.234885i −0.993080 0.117442i \(-0.962530\pi\)
0.993080 0.117442i \(-0.0374695\pi\)
\(450\) 0 0
\(451\) − 3224.36i − 0.336650i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2986.01 0.307663
\(456\) 0 0
\(457\) −17478.3 −1.78906 −0.894529 0.447011i \(-0.852489\pi\)
−0.894529 + 0.447011i \(0.852489\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18276.9i − 1.84651i −0.384190 0.923254i \(-0.625519\pi\)
0.384190 0.923254i \(-0.374481\pi\)
\(462\) 0 0
\(463\) 15906.7i 1.59665i 0.602230 + 0.798323i \(0.294279\pi\)
−0.602230 + 0.798323i \(0.705721\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8595.46 0.851714 0.425857 0.904790i \(-0.359973\pi\)
0.425857 + 0.904790i \(0.359973\pi\)
\(468\) 0 0
\(469\) −7170.06 −0.705933
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2633.88i 0.256038i
\(474\) 0 0
\(475\) − 5331.72i − 0.515023i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19485.8 1.85872 0.929360 0.369174i \(-0.120359\pi\)
0.929360 + 0.369174i \(0.120359\pi\)
\(480\) 0 0
\(481\) 5804.08 0.550194
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14163.6i 1.32606i
\(486\) 0 0
\(487\) 10181.7i 0.947386i 0.880690 + 0.473693i \(0.157079\pi\)
−0.880690 + 0.473693i \(0.842921\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3098.15 −0.284760 −0.142380 0.989812i \(-0.545476\pi\)
−0.142380 + 0.989812i \(0.545476\pi\)
\(492\) 0 0
\(493\) −273.173 −0.0249556
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 555.720i 0.0501558i
\(498\) 0 0
\(499\) − 1116.79i − 0.100189i −0.998744 0.0500944i \(-0.984048\pi\)
0.998744 0.0500944i \(-0.0159522\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2573.33 −0.228109 −0.114055 0.993474i \(-0.536384\pi\)
−0.114055 + 0.993474i \(0.536384\pi\)
\(504\) 0 0
\(505\) −3083.99 −0.271754
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16767.6i 1.46014i 0.683373 + 0.730069i \(0.260512\pi\)
−0.683373 + 0.730069i \(0.739488\pi\)
\(510\) 0 0
\(511\) − 894.502i − 0.0774373i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13621.2 1.16548
\(516\) 0 0
\(517\) 4442.80 0.377939
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8773.54i 0.737766i 0.929476 + 0.368883i \(0.120260\pi\)
−0.929476 + 0.368883i \(0.879740\pi\)
\(522\) 0 0
\(523\) − 8809.57i − 0.736550i −0.929717 0.368275i \(-0.879949\pi\)
0.929717 0.368275i \(-0.120051\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23884.9 1.97427
\(528\) 0 0
\(529\) 35162.5 2.88999
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7439.82i 0.604605i
\(534\) 0 0
\(535\) 4700.84i 0.379879i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 642.220 0.0513217
\(540\) 0 0
\(541\) −976.663 −0.0776156 −0.0388078 0.999247i \(-0.512356\pi\)
−0.0388078 + 0.999247i \(0.512356\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 26680.4i − 2.09700i
\(546\) 0 0
\(547\) − 19767.5i − 1.54515i −0.634922 0.772576i \(-0.718968\pi\)
0.634922 0.772576i \(-0.281032\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 251.638 0.0194558
\(552\) 0 0
\(553\) −2897.02 −0.222773
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 17079.5i − 1.29925i −0.760254 0.649626i \(-0.774925\pi\)
0.760254 0.649626i \(-0.225075\pi\)
\(558\) 0 0
\(559\) − 6077.37i − 0.459831i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7784.65 −0.582742 −0.291371 0.956610i \(-0.594111\pi\)
−0.291371 + 0.956610i \(0.594111\pi\)
\(564\) 0 0
\(565\) 3601.64 0.268181
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 8845.46i − 0.651706i −0.945420 0.325853i \(-0.894349\pi\)
0.945420 0.325853i \(-0.105651\pi\)
\(570\) 0 0
\(571\) 1.50330i 0 0.000110177i 1.00000 5.50886e-5i \(1.75352e-5\pi\)
−1.00000 5.50886e-5i \(0.999982\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16091.0 1.16703
\(576\) 0 0
\(577\) 2938.69 0.212026 0.106013 0.994365i \(-0.466191\pi\)
0.106013 + 0.994365i \(0.466191\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1600.86i 0.114311i
\(582\) 0 0
\(583\) 5545.85i 0.393972i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19938.8 1.40198 0.700989 0.713173i \(-0.252742\pi\)
0.700989 + 0.713173i \(0.252742\pi\)
\(588\) 0 0
\(589\) −22001.9 −1.53917
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19762.6i 1.36855i 0.729222 + 0.684277i \(0.239882\pi\)
−0.729222 + 0.684277i \(0.760118\pi\)
\(594\) 0 0
\(595\) − 7726.77i − 0.532381i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10471.7 0.714293 0.357147 0.934048i \(-0.383750\pi\)
0.357147 + 0.934048i \(0.383750\pi\)
\(600\) 0 0
\(601\) −14290.6 −0.969924 −0.484962 0.874535i \(-0.661167\pi\)
−0.484962 + 0.874535i \(0.661167\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 16351.3i − 1.09880i
\(606\) 0 0
\(607\) 26516.3i 1.77308i 0.462648 + 0.886542i \(0.346899\pi\)
−0.462648 + 0.886542i \(0.653101\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10251.2 −0.678757
\(612\) 0 0
\(613\) 25135.2 1.65612 0.828058 0.560642i \(-0.189446\pi\)
0.828058 + 0.560642i \(0.189446\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 4354.81i − 0.284146i −0.989856 0.142073i \(-0.954623\pi\)
0.989856 0.142073i \(-0.0453768\pi\)
\(618\) 0 0
\(619\) − 91.2299i − 0.00592381i −0.999996 0.00296190i \(-0.999057\pi\)
0.999996 0.00296190i \(-0.000942805\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1357.24 −0.0872819
\(624\) 0 0
\(625\) −19399.8 −1.24159
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 15019.0i − 0.952059i
\(630\) 0 0
\(631\) − 1993.10i − 0.125743i −0.998022 0.0628716i \(-0.979974\pi\)
0.998022 0.0628716i \(-0.0200259\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1159.66 −0.0724720
\(636\) 0 0
\(637\) −1481.85 −0.0921710
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27606.0i 1.70105i 0.525937 + 0.850523i \(0.323715\pi\)
−0.525937 + 0.850523i \(0.676285\pi\)
\(642\) 0 0
\(643\) − 3174.43i − 0.194693i −0.995251 0.0973464i \(-0.968965\pi\)
0.995251 0.0973464i \(-0.0310355\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23316.4 −1.41679 −0.708395 0.705816i \(-0.750580\pi\)
−0.708395 + 0.705816i \(0.750580\pi\)
\(648\) 0 0
\(649\) 4808.26 0.290818
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28631.7i 1.71584i 0.513782 + 0.857921i \(0.328244\pi\)
−0.513782 + 0.857921i \(0.671756\pi\)
\(654\) 0 0
\(655\) − 27156.4i − 1.61998i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −538.740 −0.0318457 −0.0159229 0.999873i \(-0.505069\pi\)
−0.0159229 + 0.999873i \(0.505069\pi\)
\(660\) 0 0
\(661\) −15272.0 −0.898656 −0.449328 0.893367i \(-0.648336\pi\)
−0.449328 + 0.893367i \(0.648336\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7117.64i 0.415053i
\(666\) 0 0
\(667\) 759.435i 0.0440861i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9575.57 0.550910
\(672\) 0 0
\(673\) −23556.7 −1.34925 −0.674625 0.738161i \(-0.735695\pi\)
−0.674625 + 0.738161i \(0.735695\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7988.75i 0.453519i 0.973951 + 0.226760i \(0.0728132\pi\)
−0.973951 + 0.226760i \(0.927187\pi\)
\(678\) 0 0
\(679\) − 7028.87i − 0.397266i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18508.0 −1.03688 −0.518439 0.855115i \(-0.673487\pi\)
−0.518439 + 0.855115i \(0.673487\pi\)
\(684\) 0 0
\(685\) 29086.1 1.62237
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 12796.4i − 0.707552i
\(690\) 0 0
\(691\) − 24965.2i − 1.37441i −0.726462 0.687207i \(-0.758837\pi\)
0.726462 0.687207i \(-0.241163\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5131.81 0.280087
\(696\) 0 0
\(697\) 19251.7 1.04621
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19893.7i 1.07186i 0.844262 + 0.535931i \(0.180039\pi\)
−0.844262 + 0.535931i \(0.819961\pi\)
\(702\) 0 0
\(703\) 13835.0i 0.742241i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1530.47 0.0814132
\(708\) 0 0
\(709\) 10872.8 0.575931 0.287966 0.957641i \(-0.407021\pi\)
0.287966 + 0.957641i \(0.407021\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 66401.1i − 3.48772i
\(714\) 0 0
\(715\) − 5590.90i − 0.292430i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32431.5 −1.68218 −0.841092 0.540892i \(-0.818087\pi\)
−0.841092 + 0.540892i \(0.818087\pi\)
\(720\) 0 0
\(721\) −6759.69 −0.349160
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 258.191i 0.0132262i
\(726\) 0 0
\(727\) 20606.8i 1.05126i 0.850714 + 0.525629i \(0.176170\pi\)
−0.850714 + 0.525629i \(0.823830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15726.1 −0.795693
\(732\) 0 0
\(733\) −2770.73 −0.139617 −0.0698085 0.997560i \(-0.522239\pi\)
−0.0698085 + 0.997560i \(0.522239\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13425.0i 0.670983i
\(738\) 0 0
\(739\) − 19617.9i − 0.976532i −0.872695 0.488266i \(-0.837630\pi\)
0.872695 0.488266i \(-0.162370\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2537.18 −0.125276 −0.0626381 0.998036i \(-0.519951\pi\)
−0.0626381 + 0.998036i \(0.519951\pi\)
\(744\) 0 0
\(745\) 13560.8 0.666884
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 2332.85i − 0.113806i
\(750\) 0 0
\(751\) 28280.3i 1.37412i 0.726602 + 0.687059i \(0.241099\pi\)
−0.726602 + 0.687059i \(0.758901\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −29344.1 −1.41449
\(756\) 0 0
\(757\) −30129.0 −1.44658 −0.723288 0.690546i \(-0.757370\pi\)
−0.723288 + 0.690546i \(0.757370\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11243.4i 0.535573i 0.963478 + 0.267787i \(0.0862922\pi\)
−0.963478 + 0.267787i \(0.913708\pi\)
\(762\) 0 0
\(763\) 13240.5i 0.628229i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11094.5 −0.522293
\(768\) 0 0
\(769\) 11539.8 0.541138 0.270569 0.962701i \(-0.412788\pi\)
0.270569 + 0.962701i \(0.412788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 23116.8i − 1.07562i −0.843066 0.537810i \(-0.819252\pi\)
0.843066 0.537810i \(-0.180748\pi\)
\(774\) 0 0
\(775\) − 22574.9i − 1.04634i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17734.0 −0.815643
\(780\) 0 0
\(781\) 1040.51 0.0476726
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 44615.8i 2.02854i
\(786\) 0 0
\(787\) 1401.68i 0.0634875i 0.999496 + 0.0317438i \(0.0101060\pi\)
−0.999496 + 0.0317438i \(0.989894\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1787.36 −0.0803429
\(792\) 0 0
\(793\) −22094.5 −0.989405
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 6424.11i − 0.285513i −0.989758 0.142756i \(-0.954403\pi\)
0.989758 0.142756i \(-0.0455965\pi\)
\(798\) 0 0
\(799\) 26526.7i 1.17453i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1674.83 −0.0736034
\(804\) 0 0
\(805\) −21480.8 −0.940495
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 23279.8i − 1.01171i −0.862618 0.505856i \(-0.831177\pi\)
0.862618 0.505856i \(-0.168823\pi\)
\(810\) 0 0
\(811\) − 24122.3i − 1.04445i −0.852808 0.522225i \(-0.825102\pi\)
0.852808 0.522225i \(-0.174898\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 32800.3 1.40975
\(816\) 0 0
\(817\) 14486.4 0.620336
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25808.0i 1.09708i 0.836123 + 0.548542i \(0.184817\pi\)
−0.836123 + 0.548542i \(0.815183\pi\)
\(822\) 0 0
\(823\) 45232.8i 1.91582i 0.287073 + 0.957909i \(0.407318\pi\)
−0.287073 + 0.957909i \(0.592682\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42035.1 −1.76748 −0.883740 0.467979i \(-0.844982\pi\)
−0.883740 + 0.467979i \(0.844982\pi\)
\(828\) 0 0
\(829\) −24480.4 −1.02562 −0.512810 0.858502i \(-0.671395\pi\)
−0.512810 + 0.858502i \(0.671395\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3834.51i 0.159493i
\(834\) 0 0
\(835\) − 56442.6i − 2.33925i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −26751.5 −1.10079 −0.550396 0.834904i \(-0.685523\pi\)
−0.550396 + 0.834904i \(0.685523\pi\)
\(840\) 0 0
\(841\) 24376.8 0.999500
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 18089.3i − 0.736439i
\(846\) 0 0
\(847\) 8114.53i 0.329184i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −41753.4 −1.68189
\(852\) 0 0
\(853\) 26977.8 1.08289 0.541444 0.840737i \(-0.317878\pi\)
0.541444 + 0.840737i \(0.317878\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25583.7i 1.01975i 0.860250 + 0.509873i \(0.170308\pi\)
−0.860250 + 0.509873i \(0.829692\pi\)
\(858\) 0 0
\(859\) 31259.1i 1.24161i 0.783963 + 0.620807i \(0.213195\pi\)
−0.783963 + 0.620807i \(0.786805\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −45428.8 −1.79190 −0.895952 0.444150i \(-0.853506\pi\)
−0.895952 + 0.444150i \(0.853506\pi\)
\(864\) 0 0
\(865\) 18114.7 0.712043
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5424.27i 0.211744i
\(870\) 0 0
\(871\) − 30976.5i − 1.20505i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5039.27 0.194695
\(876\) 0 0
\(877\) 32177.7 1.23895 0.619477 0.785015i \(-0.287345\pi\)
0.619477 + 0.785015i \(0.287345\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 43816.2i − 1.67560i −0.545977 0.837800i \(-0.683841\pi\)
0.545977 0.837800i \(-0.316159\pi\)
\(882\) 0 0
\(883\) − 30769.3i − 1.17267i −0.810068 0.586336i \(-0.800570\pi\)
0.810068 0.586336i \(-0.199430\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20431.3 0.773411 0.386705 0.922203i \(-0.373613\pi\)
0.386705 + 0.922203i \(0.373613\pi\)
\(888\) 0 0
\(889\) 575.496 0.0217115
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 24435.5i − 0.915679i
\(894\) 0 0
\(895\) − 9097.59i − 0.339775i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1065.45 0.0395271
\(900\) 0 0
\(901\) −33112.6 −1.22435
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33494.8i 1.23028i
\(906\) 0 0
\(907\) − 18439.6i − 0.675057i −0.941315 0.337529i \(-0.890409\pi\)
0.941315 0.337529i \(-0.109591\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11953.8 0.434737 0.217369 0.976090i \(-0.430253\pi\)
0.217369 + 0.976090i \(0.430253\pi\)
\(912\) 0 0
\(913\) 2997.39 0.108652
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13476.7i 0.485322i
\(918\) 0 0
\(919\) − 44910.6i − 1.61204i −0.591889 0.806019i \(-0.701618\pi\)
0.591889 0.806019i \(-0.298382\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2400.85 −0.0856174
\(924\) 0 0
\(925\) −14195.2 −0.504580
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5294.66i 0.186988i 0.995620 + 0.0934942i \(0.0298037\pi\)
−0.995620 + 0.0934942i \(0.970196\pi\)
\(930\) 0 0
\(931\) − 3532.22i − 0.124343i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −14467.3 −0.506023
\(936\) 0 0
\(937\) 52165.8 1.81877 0.909383 0.415960i \(-0.136554\pi\)
0.909383 + 0.415960i \(0.136554\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 29115.8i − 1.00866i −0.863511 0.504330i \(-0.831740\pi\)
0.863511 0.504330i \(-0.168260\pi\)
\(942\) 0 0
\(943\) − 53520.6i − 1.84822i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14121.4 0.484567 0.242284 0.970205i \(-0.422104\pi\)
0.242284 + 0.970205i \(0.422104\pi\)
\(948\) 0 0
\(949\) 3864.48 0.132188
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5587.57i 0.189926i 0.995481 + 0.0949629i \(0.0302732\pi\)
−0.995481 + 0.0949629i \(0.969727\pi\)
\(954\) 0 0
\(955\) − 62470.0i − 2.11673i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14434.4 −0.486037
\(960\) 0 0
\(961\) −63366.8 −2.12705
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 43928.2i − 1.46539i
\(966\) 0 0
\(967\) − 10370.3i − 0.344867i −0.985021 0.172434i \(-0.944837\pi\)
0.985021 0.172434i \(-0.0551630\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12244.2 −0.404671 −0.202335 0.979316i \(-0.564853\pi\)
−0.202335 + 0.979316i \(0.564853\pi\)
\(972\) 0 0
\(973\) −2546.73 −0.0839098
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 34539.2i − 1.13102i −0.824742 0.565510i \(-0.808680\pi\)
0.824742 0.565510i \(-0.191320\pi\)
\(978\) 0 0
\(979\) 2541.24i 0.0829606i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17119.3 −0.555465 −0.277732 0.960659i \(-0.589583\pi\)
−0.277732 + 0.960659i \(0.589583\pi\)
\(984\) 0 0
\(985\) −54909.9 −1.77622
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43719.4i 1.40566i
\(990\) 0 0
\(991\) − 4303.90i − 0.137959i −0.997618 0.0689797i \(-0.978026\pi\)
0.997618 0.0689797i \(-0.0219744\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −25915.7 −0.825713
\(996\) 0 0
\(997\) 10385.4 0.329899 0.164950 0.986302i \(-0.447254\pi\)
0.164950 + 0.986302i \(0.447254\pi\)
\(998\) 0 0
\(999\) 0 0
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 1008.4.h.a.575.11 yes 12
3.2 odd 2 inner 1008.4.h.a.575.1 12
4.3 odd 2 inner 1008.4.h.a.575.12 yes 12
12.11 even 2 inner 1008.4.h.a.575.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.4.h.a.575.1 12 3.2 odd 2 inner
1008.4.h.a.575.2 yes 12 12.11 even 2 inner
1008.4.h.a.575.11 yes 12 1.1 even 1 trivial
1008.4.h.a.575.12 yes 12 4.3 odd 2 inner