Properties

Label 1800.4.f.l.649.2
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
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] = [1800,4,Mod(649,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (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: \(106.203438010\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.l.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000i q^{7} +O(q^{10})\) \(q+5.00000i q^{7} -14.0000 q^{11} -1.00000i q^{13} -46.0000i q^{17} -19.0000 q^{19} -46.0000i q^{23} +14.0000 q^{29} +133.000 q^{31} +258.000i q^{37} -84.0000 q^{41} +167.000i q^{43} -410.000i q^{47} +318.000 q^{49} +456.000i q^{53} -194.000 q^{59} -17.0000 q^{61} +653.000i q^{67} -828.000 q^{71} -570.000i q^{73} -70.0000i q^{77} +552.000 q^{79} +142.000i q^{83} -1104.00 q^{89} +5.00000 q^{91} +841.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 28 q^{11} - 38 q^{19} + 28 q^{29} + 266 q^{31} - 168 q^{41} + 636 q^{49} - 388 q^{59} - 34 q^{61} - 1656 q^{71} + 1104 q^{79} - 2208 q^{89} + 10 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000i 0.269975i 0.990847 + 0.134987i \(0.0430994\pi\)
−0.990847 + 0.134987i \(0.956901\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −14.0000 −0.383742 −0.191871 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.0213346i −0.999943 0.0106673i \(-0.996604\pi\)
0.999943 0.0106673i \(-0.00339558\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 46.0000i − 0.656273i −0.944630 0.328136i \(-0.893579\pi\)
0.944630 0.328136i \(-0.106421\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 46.0000i − 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
\(27\) 0 0
\(28\) 0 0
\(29\) 14.0000 0.0896460 0.0448230 0.998995i \(-0.485728\pi\)
0.0448230 + 0.998995i \(0.485728\pi\)
\(30\) 0 0
\(31\) 133.000 0.770565 0.385282 0.922799i \(-0.374104\pi\)
0.385282 + 0.922799i \(0.374104\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 258.000i 1.14635i 0.819433 + 0.573175i \(0.194288\pi\)
−0.819433 + 0.573175i \(0.805712\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −84.0000 −0.319966 −0.159983 0.987120i \(-0.551144\pi\)
−0.159983 + 0.987120i \(0.551144\pi\)
\(42\) 0 0
\(43\) 167.000i 0.592262i 0.955147 + 0.296131i \(0.0956965\pi\)
−0.955147 + 0.296131i \(0.904304\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 410.000i − 1.27244i −0.771508 0.636220i \(-0.780497\pi\)
0.771508 0.636220i \(-0.219503\pi\)
\(48\) 0 0
\(49\) 318.000 0.927114
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 456.000i 1.18182i 0.806738 + 0.590910i \(0.201231\pi\)
−0.806738 + 0.590910i \(0.798769\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −194.000 −0.428079 −0.214039 0.976825i \(-0.568662\pi\)
−0.214039 + 0.976825i \(0.568662\pi\)
\(60\) 0 0
\(61\) −17.0000 −0.0356824 −0.0178412 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 653.000i 1.19070i 0.803468 + 0.595348i \(0.202986\pi\)
−0.803468 + 0.595348i \(0.797014\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −828.000 −1.38402 −0.692011 0.721887i \(-0.743275\pi\)
−0.692011 + 0.721887i \(0.743275\pi\)
\(72\) 0 0
\(73\) − 570.000i − 0.913883i −0.889497 0.456941i \(-0.848945\pi\)
0.889497 0.456941i \(-0.151055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 70.0000i − 0.103601i
\(78\) 0 0
\(79\) 552.000 0.786137 0.393069 0.919509i \(-0.371413\pi\)
0.393069 + 0.919509i \(0.371413\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 142.000i 0.187789i 0.995582 + 0.0938947i \(0.0299317\pi\)
−0.995582 + 0.0938947i \(0.970068\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1104.00 −1.31487 −0.657437 0.753510i \(-0.728359\pi\)
−0.657437 + 0.753510i \(0.728359\pi\)
\(90\) 0 0
\(91\) 5.00000 0.00575981
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 841.000i 0.880316i 0.897920 + 0.440158i \(0.145077\pi\)
−0.897920 + 0.440158i \(0.854923\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −552.000 −0.543822 −0.271911 0.962322i \(-0.587656\pi\)
−0.271911 + 0.962322i \(0.587656\pi\)
\(102\) 0 0
\(103\) 308.000i 0.294642i 0.989089 + 0.147321i \(0.0470651\pi\)
−0.989089 + 0.147321i \(0.952935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 984.000i − 0.889036i −0.895770 0.444518i \(-0.853375\pi\)
0.895770 0.444518i \(-0.146625\pi\)
\(108\) 0 0
\(109\) 1843.00 1.61952 0.809759 0.586763i \(-0.199598\pi\)
0.809759 + 0.586763i \(0.199598\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 876.000i 0.729267i 0.931151 + 0.364633i \(0.118806\pi\)
−0.931151 + 0.364633i \(0.881194\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 230.000 0.177177
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2376.00i 1.66013i 0.557670 + 0.830063i \(0.311695\pi\)
−0.557670 + 0.830063i \(0.688305\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1056.00 −0.704299 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(132\) 0 0
\(133\) − 95.0000i − 0.0619364i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 778.000i 0.485175i 0.970129 + 0.242588i \(0.0779962\pi\)
−0.970129 + 0.242588i \(0.922004\pi\)
\(138\) 0 0
\(139\) −1692.00 −1.03247 −0.516236 0.856446i \(-0.672667\pi\)
−0.516236 + 0.856446i \(0.672667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.0000i 0.00818698i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −494.000 −0.271611 −0.135806 0.990736i \(-0.543362\pi\)
−0.135806 + 0.990736i \(0.543362\pi\)
\(150\) 0 0
\(151\) −841.000 −0.453242 −0.226621 0.973983i \(-0.572768\pi\)
−0.226621 + 0.973983i \(0.572768\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.0000i 0.00965838i 0.999988 + 0.00482919i \(0.00153718\pi\)
−0.999988 + 0.00482919i \(0.998463\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 230.000 0.112587
\(162\) 0 0
\(163\) 2261.00i 1.08647i 0.839580 + 0.543237i \(0.182801\pi\)
−0.839580 + 0.543237i \(0.817199\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 2112.00i − 0.978632i −0.872107 0.489316i \(-0.837247\pi\)
0.872107 0.489316i \(-0.162753\pi\)
\(168\) 0 0
\(169\) 2196.00 0.999545
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 562.000i 0.246983i 0.992346 + 0.123492i \(0.0394092\pi\)
−0.992346 + 0.123492i \(0.960591\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3718.00 1.55249 0.776247 0.630429i \(-0.217121\pi\)
0.776247 + 0.630429i \(0.217121\pi\)
\(180\) 0 0
\(181\) −1639.00 −0.673071 −0.336536 0.941671i \(-0.609255\pi\)
−0.336536 + 0.941671i \(0.609255\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 644.000i 0.251839i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2410.00 −0.912992 −0.456496 0.889725i \(-0.650896\pi\)
−0.456496 + 0.889725i \(0.650896\pi\)
\(192\) 0 0
\(193\) 2621.00i 0.977532i 0.872415 + 0.488766i \(0.162553\pi\)
−0.872415 + 0.488766i \(0.837447\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4954.00i 1.79166i 0.444392 + 0.895832i \(0.353420\pi\)
−0.444392 + 0.895832i \(0.646580\pi\)
\(198\) 0 0
\(199\) −1739.00 −0.619470 −0.309735 0.950823i \(-0.600240\pi\)
−0.309735 + 0.950823i \(0.600240\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 70.0000i 0.0242022i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 266.000 0.0880364
\(210\) 0 0
\(211\) −4525.00 −1.47637 −0.738184 0.674599i \(-0.764317\pi\)
−0.738184 + 0.674599i \(0.764317\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 665.000i 0.208033i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −46.0000 −0.0140013
\(222\) 0 0
\(223\) 3211.00i 0.964235i 0.876106 + 0.482118i \(0.160132\pi\)
−0.876106 + 0.482118i \(0.839868\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2484.00i 0.726295i 0.931732 + 0.363147i \(0.118298\pi\)
−0.931732 + 0.363147i \(0.881702\pi\)
\(228\) 0 0
\(229\) 1847.00 0.532983 0.266492 0.963837i \(-0.414136\pi\)
0.266492 + 0.963837i \(0.414136\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1020.00i − 0.286792i −0.989665 0.143396i \(-0.954198\pi\)
0.989665 0.143396i \(-0.0458022\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1176.00 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(240\) 0 0
\(241\) −6967.00 −1.86217 −0.931087 0.364797i \(-0.881138\pi\)
−0.931087 + 0.364797i \(0.881138\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 19.0000i 0.00489450i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1380.00 −0.347031 −0.173516 0.984831i \(-0.555513\pi\)
−0.173516 + 0.984831i \(0.555513\pi\)
\(252\) 0 0
\(253\) 644.000i 0.160031i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6924.00i 1.68057i 0.542143 + 0.840286i \(0.317613\pi\)
−0.542143 + 0.840286i \(0.682387\pi\)
\(258\) 0 0
\(259\) −1290.00 −0.309485
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1884.00i 0.441720i 0.975305 + 0.220860i \(0.0708864\pi\)
−0.975305 + 0.220860i \(0.929114\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3610.00 0.818236 0.409118 0.912481i \(-0.365836\pi\)
0.409118 + 0.912481i \(0.365836\pi\)
\(270\) 0 0
\(271\) −6072.00 −1.36106 −0.680531 0.732719i \(-0.738251\pi\)
−0.680531 + 0.732719i \(0.738251\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 2803.00i − 0.608000i −0.952672 0.304000i \(-0.901678\pi\)
0.952672 0.304000i \(-0.0983223\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6694.00 1.42111 0.710553 0.703644i \(-0.248445\pi\)
0.710553 + 0.703644i \(0.248445\pi\)
\(282\) 0 0
\(283\) 6481.00i 1.36133i 0.732596 + 0.680663i \(0.238308\pi\)
−0.732596 + 0.680663i \(0.761692\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 420.000i − 0.0863826i
\(288\) 0 0
\(289\) 2797.00 0.569306
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3014.00i 0.600955i 0.953789 + 0.300477i \(0.0971460\pi\)
−0.953789 + 0.300477i \(0.902854\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −46.0000 −0.00889715
\(300\) 0 0
\(301\) −835.000 −0.159896
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5369.00i − 0.998127i −0.866565 0.499064i \(-0.833677\pi\)
0.866565 0.499064i \(-0.166323\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4846.00 −0.883574 −0.441787 0.897120i \(-0.645655\pi\)
−0.441787 + 0.897120i \(0.645655\pi\)
\(312\) 0 0
\(313\) 757.000i 0.136703i 0.997661 + 0.0683517i \(0.0217740\pi\)
−0.997661 + 0.0683517i \(0.978226\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7632.00i 1.35223i 0.736798 + 0.676113i \(0.236337\pi\)
−0.736798 + 0.676113i \(0.763663\pi\)
\(318\) 0 0
\(319\) −196.000 −0.0344009
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 874.000i 0.150559i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2050.00 0.343526
\(330\) 0 0
\(331\) −6780.00 −1.12587 −0.562934 0.826502i \(-0.690328\pi\)
−0.562934 + 0.826502i \(0.690328\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7849.00i 1.26873i 0.773033 + 0.634365i \(0.218739\pi\)
−0.773033 + 0.634365i \(0.781261\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1862.00 −0.295698
\(342\) 0 0
\(343\) 3305.00i 0.520272i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 634.000i − 0.0980833i −0.998797 0.0490416i \(-0.984383\pi\)
0.998797 0.0490416i \(-0.0156167\pi\)
\(348\) 0 0
\(349\) −930.000 −0.142641 −0.0713206 0.997453i \(-0.522721\pi\)
−0.0713206 + 0.997453i \(0.522721\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 4286.00i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4236.00 −0.622751 −0.311375 0.950287i \(-0.600790\pi\)
−0.311375 + 0.950287i \(0.600790\pi\)
\(360\) 0 0
\(361\) −6498.00 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1451.00i 0.206380i 0.994662 + 0.103190i \(0.0329050\pi\)
−0.994662 + 0.103190i \(0.967095\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2280.00 −0.319061
\(372\) 0 0
\(373\) − 3115.00i − 0.432409i −0.976348 0.216205i \(-0.930632\pi\)
0.976348 0.216205i \(-0.0693678\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14.0000i − 0.00191256i
\(378\) 0 0
\(379\) 1415.00 0.191777 0.0958887 0.995392i \(-0.469431\pi\)
0.0958887 + 0.995392i \(0.469431\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 180.000i − 0.0240145i −0.999928 0.0120073i \(-0.996178\pi\)
0.999928 0.0120073i \(-0.00382213\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12372.0 1.61256 0.806279 0.591535i \(-0.201478\pi\)
0.806279 + 0.591535i \(0.201478\pi\)
\(390\) 0 0
\(391\) −2116.00 −0.273685
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5767.00i 0.729062i 0.931191 + 0.364531i \(0.118771\pi\)
−0.931191 + 0.364531i \(0.881229\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3120.00 0.388542 0.194271 0.980948i \(-0.437766\pi\)
0.194271 + 0.980948i \(0.437766\pi\)
\(402\) 0 0
\(403\) − 133.000i − 0.0164397i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 3612.00i − 0.439902i
\(408\) 0 0
\(409\) −1501.00 −0.181466 −0.0907331 0.995875i \(-0.528921\pi\)
−0.0907331 + 0.995875i \(0.528921\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 970.000i − 0.115570i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9072.00 −1.05775 −0.528874 0.848701i \(-0.677386\pi\)
−0.528874 + 0.848701i \(0.677386\pi\)
\(420\) 0 0
\(421\) 7350.00 0.850872 0.425436 0.904989i \(-0.360121\pi\)
0.425436 + 0.904989i \(0.360121\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 85.0000i − 0.00963334i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5962.00 −0.666310 −0.333155 0.942872i \(-0.608113\pi\)
−0.333155 + 0.942872i \(0.608113\pi\)
\(432\) 0 0
\(433\) 10093.0i 1.12018i 0.828431 + 0.560091i \(0.189234\pi\)
−0.828431 + 0.560091i \(0.810766\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 874.000i 0.0956730i
\(438\) 0 0
\(439\) 2555.00 0.277776 0.138888 0.990308i \(-0.455647\pi\)
0.138888 + 0.990308i \(0.455647\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6240.00i 0.669236i 0.942354 + 0.334618i \(0.108607\pi\)
−0.942354 + 0.334618i \(0.891393\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3324.00 0.349375 0.174687 0.984624i \(-0.444109\pi\)
0.174687 + 0.984624i \(0.444109\pi\)
\(450\) 0 0
\(451\) 1176.00 0.122784
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16774.0i 1.71697i 0.512840 + 0.858484i \(0.328593\pi\)
−0.512840 + 0.858484i \(0.671407\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14304.0 −1.44513 −0.722564 0.691304i \(-0.757036\pi\)
−0.722564 + 0.691304i \(0.757036\pi\)
\(462\) 0 0
\(463\) − 6936.00i − 0.696206i −0.937456 0.348103i \(-0.886826\pi\)
0.937456 0.348103i \(-0.113174\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 15622.0i − 1.54797i −0.633207 0.773983i \(-0.718262\pi\)
0.633207 0.773983i \(-0.281738\pi\)
\(468\) 0 0
\(469\) −3265.00 −0.321458
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 2338.00i − 0.227276i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −13354.0 −1.27382 −0.636910 0.770938i \(-0.719788\pi\)
−0.636910 + 0.770938i \(0.719788\pi\)
\(480\) 0 0
\(481\) 258.000 0.0244569
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 461.000i − 0.0428951i −0.999770 0.0214475i \(-0.993173\pi\)
0.999770 0.0214475i \(-0.00682749\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3768.00 −0.346329 −0.173164 0.984893i \(-0.555399\pi\)
−0.173164 + 0.984893i \(0.555399\pi\)
\(492\) 0 0
\(493\) − 644.000i − 0.0588323i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 4140.00i − 0.373651i
\(498\) 0 0
\(499\) 14317.0 1.28440 0.642201 0.766536i \(-0.278021\pi\)
0.642201 + 0.766536i \(0.278021\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9228.00i 0.818004i 0.912533 + 0.409002i \(0.134123\pi\)
−0.912533 + 0.409002i \(0.865877\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4574.00 0.398308 0.199154 0.979968i \(-0.436181\pi\)
0.199154 + 0.979968i \(0.436181\pi\)
\(510\) 0 0
\(511\) 2850.00 0.246725
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5740.00i 0.488288i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8494.00 0.714259 0.357129 0.934055i \(-0.383755\pi\)
0.357129 + 0.934055i \(0.383755\pi\)
\(522\) 0 0
\(523\) − 8263.00i − 0.690852i −0.938446 0.345426i \(-0.887734\pi\)
0.938446 0.345426i \(-0.112266\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 6118.00i − 0.505701i
\(528\) 0 0
\(529\) 10051.0 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 84.0000i 0.00682635i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4452.00 −0.355772
\(540\) 0 0
\(541\) 21157.0 1.68135 0.840675 0.541540i \(-0.182158\pi\)
0.840675 + 0.541540i \(0.182158\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 4048.00i − 0.316417i −0.987406 0.158208i \(-0.949428\pi\)
0.987406 0.158208i \(-0.0505718\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −266.000 −0.0205662
\(552\) 0 0
\(553\) 2760.00i 0.212237i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6758.00i 0.514086i 0.966400 + 0.257043i \(0.0827481\pi\)
−0.966400 + 0.257043i \(0.917252\pi\)
\(558\) 0 0
\(559\) 167.000 0.0126357
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24506.0i 1.83447i 0.398350 + 0.917233i \(0.369583\pi\)
−0.398350 + 0.917233i \(0.630417\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1430.00 −0.105358 −0.0526790 0.998611i \(-0.516776\pi\)
−0.0526790 + 0.998611i \(0.516776\pi\)
\(570\) 0 0
\(571\) 3691.00 0.270514 0.135257 0.990811i \(-0.456814\pi\)
0.135257 + 0.990811i \(0.456814\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 4571.00i 0.329798i 0.986310 + 0.164899i \(0.0527298\pi\)
−0.986310 + 0.164899i \(0.947270\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −710.000 −0.0506984
\(582\) 0 0
\(583\) − 6384.00i − 0.453513i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 14808.0i − 1.04121i −0.853797 0.520606i \(-0.825706\pi\)
0.853797 0.520606i \(-0.174294\pi\)
\(588\) 0 0
\(589\) −2527.00 −0.176780
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 24588.0i − 1.70271i −0.524588 0.851356i \(-0.675781\pi\)
0.524588 0.851356i \(-0.324219\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27564.0 −1.88019 −0.940096 0.340911i \(-0.889265\pi\)
−0.940096 + 0.340911i \(0.889265\pi\)
\(600\) 0 0
\(601\) 10987.0 0.745706 0.372853 0.927891i \(-0.378380\pi\)
0.372853 + 0.927891i \(0.378380\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 13200.0i − 0.882655i −0.897346 0.441327i \(-0.854508\pi\)
0.897346 0.441327i \(-0.145492\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −410.000 −0.0271470
\(612\) 0 0
\(613\) − 21066.0i − 1.38801i −0.719972 0.694003i \(-0.755845\pi\)
0.719972 0.694003i \(-0.244155\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12336.0i − 0.804909i −0.915440 0.402454i \(-0.868157\pi\)
0.915440 0.402454i \(-0.131843\pi\)
\(618\) 0 0
\(619\) 1441.00 0.0935681 0.0467841 0.998905i \(-0.485103\pi\)
0.0467841 + 0.998905i \(0.485103\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 5520.00i − 0.354983i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11868.0 0.752318
\(630\) 0 0
\(631\) −9839.00 −0.620736 −0.310368 0.950616i \(-0.600452\pi\)
−0.310368 + 0.950616i \(0.600452\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 318.000i − 0.0197796i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21564.0 1.32875 0.664373 0.747401i \(-0.268698\pi\)
0.664373 + 0.747401i \(0.268698\pi\)
\(642\) 0 0
\(643\) − 8604.00i − 0.527696i −0.964564 0.263848i \(-0.915008\pi\)
0.964564 0.263848i \(-0.0849918\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 3444.00i − 0.209270i −0.994511 0.104635i \(-0.966633\pi\)
0.994511 0.104635i \(-0.0333674\pi\)
\(648\) 0 0
\(649\) 2716.00 0.164272
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 3518.00i − 0.210827i −0.994428 0.105413i \(-0.966383\pi\)
0.994428 0.105413i \(-0.0336166\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12612.0 −0.745514 −0.372757 0.927929i \(-0.621588\pi\)
−0.372757 + 0.927929i \(0.621588\pi\)
\(660\) 0 0
\(661\) 27090.0 1.59407 0.797034 0.603935i \(-0.206401\pi\)
0.797034 + 0.603935i \(0.206401\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 644.000i − 0.0373850i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 238.000 0.0136928
\(672\) 0 0
\(673\) 5442.00i 0.311699i 0.987781 + 0.155850i \(0.0498115\pi\)
−0.987781 + 0.155850i \(0.950188\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15226.0i 0.864376i 0.901783 + 0.432188i \(0.142258\pi\)
−0.901783 + 0.432188i \(0.857742\pi\)
\(678\) 0 0
\(679\) −4205.00 −0.237663
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 552.000i − 0.0309249i −0.999880 0.0154624i \(-0.995078\pi\)
0.999880 0.0154624i \(-0.00492204\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 456.000 0.0252137
\(690\) 0 0
\(691\) 9776.00 0.538201 0.269100 0.963112i \(-0.413274\pi\)
0.269100 + 0.963112i \(0.413274\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3864.00i 0.209985i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −13066.0 −0.703989 −0.351994 0.936002i \(-0.614496\pi\)
−0.351994 + 0.936002i \(0.614496\pi\)
\(702\) 0 0
\(703\) − 4902.00i − 0.262991i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 2760.00i − 0.146818i
\(708\) 0 0
\(709\) −28985.0 −1.53534 −0.767669 0.640847i \(-0.778583\pi\)
−0.767669 + 0.640847i \(0.778583\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 6118.00i − 0.321348i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15722.0 0.815482 0.407741 0.913098i \(-0.366317\pi\)
0.407741 + 0.913098i \(0.366317\pi\)
\(720\) 0 0
\(721\) −1540.00 −0.0795459
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 32611.0i − 1.66365i −0.555036 0.831826i \(-0.687296\pi\)
0.555036 0.831826i \(-0.312704\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7682.00 0.388685
\(732\) 0 0
\(733\) − 8358.00i − 0.421159i −0.977577 0.210580i \(-0.932465\pi\)
0.977577 0.210580i \(-0.0675351\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9142.00i − 0.456920i
\(738\) 0 0
\(739\) −20604.0 −1.02562 −0.512808 0.858503i \(-0.671395\pi\)
−0.512808 + 0.858503i \(0.671395\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 19476.0i − 0.961649i −0.876817 0.480824i \(-0.840337\pi\)
0.876817 0.480824i \(-0.159663\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4920.00 0.240017
\(750\) 0 0
\(751\) 3864.00 0.187749 0.0938744 0.995584i \(-0.470075\pi\)
0.0938744 + 0.995584i \(0.470075\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 18871.0i − 0.906048i −0.891498 0.453024i \(-0.850345\pi\)
0.891498 0.453024i \(-0.149655\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36372.0 1.73257 0.866284 0.499552i \(-0.166502\pi\)
0.866284 + 0.499552i \(0.166502\pi\)
\(762\) 0 0
\(763\) 9215.00i 0.437229i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 194.000i 0.00913290i
\(768\) 0 0
\(769\) −4603.00 −0.215850 −0.107925 0.994159i \(-0.534421\pi\)
−0.107925 + 0.994159i \(0.534421\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 36.0000i − 0.00167507i −1.00000 0.000837536i \(-0.999733\pi\)
1.00000 0.000837536i \(-0.000266596\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1596.00 0.0734052
\(780\) 0 0
\(781\) 11592.0 0.531107
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20281.0i − 0.918602i −0.888281 0.459301i \(-0.848100\pi\)
0.888281 0.459301i \(-0.151900\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4380.00 −0.196884
\(792\) 0 0
\(793\) 17.0000i 0 0.000761271i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37524.0i 1.66771i 0.551980 + 0.833857i \(0.313872\pi\)
−0.551980 + 0.833857i \(0.686128\pi\)
\(798\) 0 0
\(799\) −18860.0 −0.835067
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 7980.00i 0.350695i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31224.0 1.35696 0.678478 0.734621i \(-0.262640\pi\)
0.678478 + 0.734621i \(0.262640\pi\)
\(810\) 0 0
\(811\) 32579.0 1.41061 0.705304 0.708905i \(-0.250810\pi\)
0.705304 + 0.708905i \(0.250810\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3173.00i − 0.135874i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19810.0 0.842112 0.421056 0.907035i \(-0.361660\pi\)
0.421056 + 0.907035i \(0.361660\pi\)
\(822\) 0 0
\(823\) − 10273.0i − 0.435108i −0.976048 0.217554i \(-0.930192\pi\)
0.976048 0.217554i \(-0.0698079\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 16656.0i − 0.700346i −0.936685 0.350173i \(-0.886123\pi\)
0.936685 0.350173i \(-0.113877\pi\)
\(828\) 0 0
\(829\) 4790.00 0.200680 0.100340 0.994953i \(-0.468007\pi\)
0.100340 + 0.994953i \(0.468007\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 14628.0i − 0.608440i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7414.00 0.305077 0.152539 0.988298i \(-0.451255\pi\)
0.152539 + 0.988298i \(0.451255\pi\)
\(840\) 0 0
\(841\) −24193.0 −0.991964
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5675.00i − 0.230219i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11868.0 0.478061
\(852\) 0 0
\(853\) − 30155.0i − 1.21042i −0.796066 0.605210i \(-0.793089\pi\)
0.796066 0.605210i \(-0.206911\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8244.00i 0.328599i 0.986410 + 0.164300i \(0.0525364\pi\)
−0.986410 + 0.164300i \(0.947464\pi\)
\(858\) 0 0
\(859\) −17552.0 −0.697167 −0.348584 0.937278i \(-0.613337\pi\)
−0.348584 + 0.937278i \(0.613337\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 34104.0i − 1.34521i −0.740003 0.672604i \(-0.765176\pi\)
0.740003 0.672604i \(-0.234824\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7728.00 −0.301674
\(870\) 0 0
\(871\) 653.000 0.0254031
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 46229.0i − 1.77998i −0.455980 0.889990i \(-0.650711\pi\)
0.455980 0.889990i \(-0.349289\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22440.0 0.858142 0.429071 0.903271i \(-0.358841\pi\)
0.429071 + 0.903271i \(0.358841\pi\)
\(882\) 0 0
\(883\) 17143.0i 0.653350i 0.945137 + 0.326675i \(0.105928\pi\)
−0.945137 + 0.326675i \(0.894072\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23626.0i 0.894344i 0.894448 + 0.447172i \(0.147569\pi\)
−0.894448 + 0.447172i \(0.852431\pi\)
\(888\) 0 0
\(889\) −11880.0 −0.448192
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7790.00i 0.291918i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1862.00 0.0690781
\(900\) 0 0
\(901\) 20976.0 0.775596
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 11268.0i − 0.412511i −0.978498 0.206256i \(-0.933872\pi\)
0.978498 0.206256i \(-0.0661279\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −10046.0 −0.365355 −0.182678 0.983173i \(-0.558476\pi\)
−0.182678 + 0.983173i \(0.558476\pi\)
\(912\) 0 0
\(913\) − 1988.00i − 0.0720626i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5280.00i − 0.190143i
\(918\) 0 0
\(919\) −15359.0 −0.551302 −0.275651 0.961258i \(-0.588893\pi\)
−0.275651 + 0.961258i \(0.588893\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 828.000i 0.0295276i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −39790.0 −1.40524 −0.702620 0.711565i \(-0.747986\pi\)
−0.702620 + 0.711565i \(0.747986\pi\)
\(930\) 0 0
\(931\) −6042.00 −0.212694
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 17009.0i − 0.593020i −0.955030 0.296510i \(-0.904177\pi\)
0.955030 0.296510i \(-0.0958228\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 11674.0 0.404422 0.202211 0.979342i \(-0.435187\pi\)
0.202211 + 0.979342i \(0.435187\pi\)
\(942\) 0 0
\(943\) 3864.00i 0.133435i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6026.00i − 0.206778i −0.994641 0.103389i \(-0.967031\pi\)
0.994641 0.103389i \(-0.0329686\pi\)
\(948\) 0 0
\(949\) −570.000 −0.0194973
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 14088.0i − 0.478862i −0.970913 0.239431i \(-0.923039\pi\)
0.970913 0.239431i \(-0.0769608\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3890.00 −0.130985
\(960\) 0 0
\(961\) −12102.0 −0.406230
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 11208.0i − 0.372725i −0.982481 0.186362i \(-0.940330\pi\)
0.982481 0.186362i \(-0.0596699\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 26054.0 0.861084 0.430542 0.902571i \(-0.358322\pi\)
0.430542 + 0.902571i \(0.358322\pi\)
\(972\) 0 0
\(973\) − 8460.00i − 0.278741i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26870.0i 0.879885i 0.898026 + 0.439942i \(0.145001\pi\)
−0.898026 + 0.439942i \(0.854999\pi\)
\(978\) 0 0
\(979\) 15456.0 0.504572
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 23388.0i − 0.758862i −0.925220 0.379431i \(-0.876120\pi\)
0.925220 0.379431i \(-0.123880\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7682.00 0.246990
\(990\) 0 0
\(991\) 17345.0 0.555986 0.277993 0.960583i \(-0.410331\pi\)
0.277993 + 0.960583i \(0.410331\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 25998.0i − 0.825842i −0.910767 0.412921i \(-0.864508\pi\)
0.910767 0.412921i \(-0.135492\pi\)
\(998\) 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 1800.4.f.l.649.2 2
3.2 odd 2 600.4.f.e.49.1 2
5.2 odd 4 1800.4.a.m.1.1 1
5.3 odd 4 1800.4.a.v.1.1 1
5.4 even 2 inner 1800.4.f.l.649.1 2
12.11 even 2 1200.4.f.i.49.2 2
15.2 even 4 600.4.a.e.1.1 1
15.8 even 4 600.4.a.n.1.1 yes 1
15.14 odd 2 600.4.f.e.49.2 2
60.23 odd 4 1200.4.a.g.1.1 1
60.47 odd 4 1200.4.a.bd.1.1 1
60.59 even 2 1200.4.f.i.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.e.1.1 1 15.2 even 4
600.4.a.n.1.1 yes 1 15.8 even 4
600.4.f.e.49.1 2 3.2 odd 2
600.4.f.e.49.2 2 15.14 odd 2
1200.4.a.g.1.1 1 60.23 odd 4
1200.4.a.bd.1.1 1 60.47 odd 4
1200.4.f.i.49.1 2 60.59 even 2
1200.4.f.i.49.2 2 12.11 even 2
1800.4.a.m.1.1 1 5.2 odd 4
1800.4.a.v.1.1 1 5.3 odd 4
1800.4.f.l.649.1 2 5.4 even 2 inner
1800.4.f.l.649.2 2 1.1 even 1 trivial