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Properties

Label 1800.4.a.r.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,4,Mod(1,1800)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.1");
 
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.a (trivial)

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: yes
Analytic conductor: \(106.203438010\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{7} +34.0000 q^{11} +68.0000 q^{13} -38.0000 q^{17} +4.00000 q^{19} +152.000 q^{23} +46.0000 q^{29} -260.000 q^{31} +312.000 q^{37} -48.0000 q^{41} +200.000 q^{43} +104.000 q^{47} -339.000 q^{49} -414.000 q^{53} +2.00000 q^{59} -38.0000 q^{61} +244.000 q^{67} -708.000 q^{71} +378.000 q^{73} -68.0000 q^{77} -852.000 q^{79} +844.000 q^{83} +1380.00 q^{89} -136.000 q^{91} -514.000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(7\) −2.00000 −0.107990 −0.0539949 − 0.998541i \(-0.517195\pi\)
−0.0539949 + 0.998541i \(0.517195\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 34.0000 0.931944 0.465972 − 0.884799i \(-0.345705\pi\)
0.465972 + 0.884799i \(0.345705\pi\)
\(12\) 0 0
\(13\) 68.0000 1.45075 0.725377 − 0.688352i \(-0.241665\pi\)
0.725377 + 0.688352i \(0.241665\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −38.0000 −0.542138 −0.271069 − 0.962560i \(-0.587377\pi\)
−0.271069 + 0.962560i \(0.587377\pi\)
\(18\) 0 0
\(19\) 4.00000 0.0482980 0.0241490 − 0.999708i \(-0.492312\pi\)
0.0241490 + 0.999708i \(0.492312\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 152.000 1.37801 0.689004 − 0.724757i \(-0.258048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 46.0000 0.294551 0.147276 − 0.989095i \(-0.452950\pi\)
0.147276 + 0.989095i \(0.452950\pi\)
\(30\) 0 0
\(31\) −260.000 −1.50637 −0.753184 − 0.657810i \(-0.771483\pi\)
−0.753184 + 0.657810i \(0.771483\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 312.000 1.38628 0.693142 − 0.720801i \(-0.256226\pi\)
0.693142 + 0.720801i \(0.256226\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −48.0000 −0.182838 −0.0914188 − 0.995813i \(-0.529140\pi\)
−0.0914188 + 0.995813i \(0.529140\pi\)
\(42\) 0 0
\(43\) 200.000 0.709296 0.354648 − 0.935000i \(-0.384601\pi\)
0.354648 + 0.935000i \(0.384601\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 104.000 0.322765 0.161383 − 0.986892i \(-0.448405\pi\)
0.161383 + 0.986892i \(0.448405\pi\)
\(48\) 0 0
\(49\) −339.000 −0.988338
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 − 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 0.00441318 0.00220659 − 0.999998i \(-0.499298\pi\)
0.00220659 + 0.999998i \(0.499298\pi\)
\(60\) 0 0
\(61\) −38.0000 −0.0797607 −0.0398803 − 0.999204i \(-0.512698\pi\)
−0.0398803 + 0.999204i \(0.512698\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 244.000 0.444916 0.222458 − 0.974942i \(-0.428592\pi\)
0.222458 + 0.974942i \(0.428592\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −708.000 −1.18344 −0.591719 − 0.806144i \(-0.701551\pi\)
−0.591719 + 0.806144i \(0.701551\pi\)
\(72\) 0 0
\(73\) 378.000 0.606049 0.303024 − 0.952983i \(-0.402004\pi\)
0.303024 + 0.952983i \(0.402004\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −68.0000 −0.100641
\(78\) 0 0
\(79\) −852.000 −1.21339 −0.606693 − 0.794936i \(-0.707504\pi\)
−0.606693 + 0.794936i \(0.707504\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 844.000 1.11616 0.558079 − 0.829788i \(-0.311539\pi\)
0.558079 + 0.829788i \(0.311539\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1380.00 1.64359 0.821796 − 0.569782i \(-0.192972\pi\)
0.821796 + 0.569782i \(0.192972\pi\)
\(90\) 0 0
\(91\) −136.000 −0.156667
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −514.000 −0.538029 −0.269014 − 0.963136i \(-0.586698\pi\)
−0.269014 + 0.963136i \(0.586698\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 702.000 0.691600 0.345800 − 0.938308i \(-0.387608\pi\)
0.345800 + 0.938308i \(0.387608\pi\)
\(102\) 0 0
\(103\) −898.000 −0.859054 −0.429527 − 0.903054i \(-0.641320\pi\)
−0.429527 + 0.903054i \(0.641320\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 876.000 0.791459 0.395730 − 0.918367i \(-0.370492\pi\)
0.395730 + 0.918367i \(0.370492\pi\)
\(108\) 0 0
\(109\) 602.000 0.529001 0.264501 − 0.964386i \(-0.414793\pi\)
0.264501 + 0.964386i \(0.414793\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1350.00 1.12387 0.561935 − 0.827181i \(-0.310057\pi\)
0.561935 + 0.827181i \(0.310057\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 76.0000 0.0585455
\(120\) 0 0
\(121\) −175.000 −0.131480
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 366.000 0.255726 0.127863 − 0.991792i \(-0.459188\pi\)
0.127863 + 0.991792i \(0.459188\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −498.000 −0.332141 −0.166070 − 0.986114i \(-0.553108\pi\)
−0.166070 + 0.986114i \(0.553108\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.00521570
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2026.00 −1.26345 −0.631726 − 0.775192i \(-0.717653\pi\)
−0.631726 + 0.775192i \(0.717653\pi\)
\(138\) 0 0
\(139\) 2460.00 1.50111 0.750556 − 0.660807i \(-0.229786\pi\)
0.750556 + 0.660807i \(0.229786\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2312.00 1.35202
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3362.00 1.84850 0.924248 − 0.381794i \(-0.124694\pi\)
0.924248 + 0.381794i \(0.124694\pi\)
\(150\) 0 0
\(151\) 2096.00 1.12960 0.564802 − 0.825227i \(-0.308953\pi\)
0.564802 + 0.825227i \(0.308953\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2092.00 −1.06344 −0.531719 − 0.846921i \(-0.678454\pi\)
−0.531719 + 0.846921i \(0.678454\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −304.000 −0.148811
\(162\) 0 0
\(163\) −244.000 −0.117249 −0.0586244 − 0.998280i \(-0.518671\pi\)
−0.0586244 + 0.998280i \(0.518671\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2064.00 0.956390 0.478195 − 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(168\) 0 0
\(169\) 2427.00 1.10469
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1258.00 0.552855 0.276428 − 0.961035i \(-0.410849\pi\)
0.276428 + 0.961035i \(0.410849\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3986.00 1.66440 0.832200 − 0.554475i \(-0.187081\pi\)
0.832200 + 0.554475i \(0.187081\pi\)
\(180\) 0 0
\(181\) 2570.00 1.05540 0.527698 − 0.849432i \(-0.323055\pi\)
0.527698 + 0.849432i \(0.323055\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1292.00 −0.505243
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4684.00 −1.77446 −0.887231 − 0.461325i \(-0.847374\pi\)
−0.887231 + 0.461325i \(0.847374\pi\)
\(192\) 0 0
\(193\) −214.000 −0.0798138 −0.0399069 − 0.999203i \(-0.512706\pi\)
−0.0399069 + 0.999203i \(0.512706\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3014.00 1.09004 0.545022 − 0.838422i \(-0.316521\pi\)
0.545022 + 0.838422i \(0.316521\pi\)
\(198\) 0 0
\(199\) −1792.00 −0.638349 −0.319175 − 0.947696i \(-0.603406\pi\)
−0.319175 + 0.947696i \(0.603406\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −92.0000 −0.0318085
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 136.000 0.0450111
\(210\) 0 0
\(211\) −4540.00 −1.48126 −0.740631 − 0.671911i \(-0.765474\pi\)
−0.740631 + 0.671911i \(0.765474\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 520.000 0.162672
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2584.00 −0.786510
\(222\) 0 0
\(223\) −6506.00 −1.95369 −0.976847 − 0.213937i \(-0.931371\pi\)
−0.976847 + 0.213937i \(0.931371\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3696.00 −1.08067 −0.540335 − 0.841450i \(-0.681703\pi\)
−0.540335 + 0.841450i \(0.681703\pi\)
\(228\) 0 0
\(229\) −3386.00 −0.977088 −0.488544 − 0.872539i \(-0.662472\pi\)
−0.488544 + 0.872539i \(0.662472\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3306.00 0.929542 0.464771 − 0.885431i \(-0.346137\pi\)
0.464771 + 0.885431i \(0.346137\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4188.00 1.13347 0.566735 − 0.823900i \(-0.308206\pi\)
0.566735 + 0.823900i \(0.308206\pi\)
\(240\) 0 0
\(241\) 5462.00 1.45991 0.729955 − 0.683495i \(-0.239541\pi\)
0.729955 + 0.683495i \(0.239541\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 272.000 0.0700686
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3366.00 0.846454 0.423227 − 0.906024i \(-0.360897\pi\)
0.423227 + 0.906024i \(0.360897\pi\)
\(252\) 0 0
\(253\) 5168.00 1.28423
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1158.00 0.281066 0.140533 − 0.990076i \(-0.455118\pi\)
0.140533 + 0.990076i \(0.455118\pi\)
\(258\) 0 0
\(259\) −624.000 −0.149705
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8304.00 1.94695 0.973473 − 0.228804i \(-0.0734814\pi\)
0.973473 + 0.228804i \(0.0734814\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7478.00 1.69495 0.847475 − 0.530835i \(-0.178122\pi\)
0.847475 + 0.530835i \(0.178122\pi\)
\(270\) 0 0
\(271\) −6792.00 −1.52245 −0.761226 − 0.648486i \(-0.775402\pi\)
−0.761226 + 0.648486i \(0.775402\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2296.00 0.498026 0.249013 − 0.968500i \(-0.419894\pi\)
0.249013 + 0.968500i \(0.419894\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3980.00 −0.844936 −0.422468 − 0.906378i \(-0.638836\pi\)
−0.422468 + 0.906378i \(0.638836\pi\)
\(282\) 0 0
\(283\) 1972.00 0.414216 0.207108 − 0.978318i \(-0.433595\pi\)
0.207108 + 0.978318i \(0.433595\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 96.0000 0.0197446
\(288\) 0 0
\(289\) −3469.00 −0.706086
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9254.00 1.84513 0.922567 − 0.385836i \(-0.126087\pi\)
0.922567 + 0.385836i \(0.126087\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10336.0 1.99915
\(300\) 0 0
\(301\) −400.000 −0.0765967
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5888.00 1.09461 0.547306 − 0.836933i \(-0.315653\pi\)
0.547306 + 0.836933i \(0.315653\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4604.00 0.839450 0.419725 − 0.907651i \(-0.362127\pi\)
0.419725 + 0.907651i \(0.362127\pi\)
\(312\) 0 0
\(313\) 8026.00 1.44938 0.724691 − 0.689074i \(-0.241983\pi\)
0.724691 + 0.689074i \(0.241983\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2838.00 0.502833 0.251416 − 0.967879i \(-0.419104\pi\)
0.251416 + 0.967879i \(0.419104\pi\)
\(318\) 0 0
\(319\) 1564.00 0.274505
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −152.000 −0.0261842
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −208.000 −0.0348554
\(330\) 0 0
\(331\) −1020.00 −0.169378 −0.0846892 − 0.996407i \(-0.526990\pi\)
−0.0846892 + 0.996407i \(0.526990\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −814.000 −0.131577 −0.0657884 − 0.997834i \(-0.520956\pi\)
−0.0657884 + 0.997834i \(0.520956\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8840.00 −1.40385
\(342\) 0 0
\(343\) 1364.00 0.214720
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4544.00 −0.702982 −0.351491 − 0.936191i \(-0.614325\pi\)
−0.351491 + 0.936191i \(0.614325\pi\)
\(348\) 0 0
\(349\) 6978.00 1.07027 0.535134 − 0.844767i \(-0.320261\pi\)
0.535134 + 0.844767i \(0.320261\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2818.00 0.424892 0.212446 − 0.977173i \(-0.431857\pi\)
0.212446 + 0.977173i \(0.431857\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −744.000 −0.109378 −0.0546892 − 0.998503i \(-0.517417\pi\)
−0.0546892 + 0.998503i \(0.517417\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6454.00 0.917973 0.458986 − 0.888443i \(-0.348213\pi\)
0.458986 + 0.888443i \(0.348213\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 828.000 0.115870
\(372\) 0 0
\(373\) 5900.00 0.819009 0.409505 − 0.912308i \(-0.365702\pi\)
0.409505 + 0.912308i \(0.365702\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3128.00 0.427321
\(378\) 0 0
\(379\) −11876.0 −1.60958 −0.804788 − 0.593563i \(-0.797721\pi\)
−0.804788 + 0.593563i \(0.797721\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −552.000 −0.0736446 −0.0368223 − 0.999322i \(-0.511724\pi\)
−0.0368223 + 0.999322i \(0.511724\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1722.00 0.224444 0.112222 − 0.993683i \(-0.464203\pi\)
0.112222 + 0.993683i \(0.464203\pi\)
\(390\) 0 0
\(391\) −5776.00 −0.747071
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −4576.00 −0.578496 −0.289248 − 0.957254i \(-0.593405\pi\)
−0.289248 + 0.957254i \(0.593405\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2892.00 −0.360149 −0.180074 − 0.983653i \(-0.557634\pi\)
−0.180074 + 0.983653i \(0.557634\pi\)
\(402\) 0 0
\(403\) −17680.0 −2.18537
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10608.0 1.29194
\(408\) 0 0
\(409\) −230.000 −0.0278063 −0.0139031 − 0.999903i \(-0.504426\pi\)
−0.0139031 + 0.999903i \(0.504426\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.000476579 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15438.0 1.79999 0.899995 − 0.435901i \(-0.143570\pi\)
0.899995 + 0.435901i \(0.143570\pi\)
\(420\) 0 0
\(421\) 12294.0 1.42321 0.711607 − 0.702578i \(-0.247968\pi\)
0.711607 + 0.702578i \(0.247968\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 76.0000 0.00861334
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17488.0 −1.95445 −0.977224 − 0.212209i \(-0.931934\pi\)
−0.977224 + 0.212209i \(0.931934\pi\)
\(432\) 0 0
\(433\) 8698.00 0.965356 0.482678 − 0.875798i \(-0.339664\pi\)
0.482678 + 0.875798i \(0.339664\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 608.000 0.0665551
\(438\) 0 0
\(439\) 8536.00 0.928021 0.464010 − 0.885830i \(-0.346410\pi\)
0.464010 + 0.885830i \(0.346410\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8712.00 −0.934356 −0.467178 − 0.884163i \(-0.654729\pi\)
−0.467178 + 0.884163i \(0.654729\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5484.00 0.576405 0.288203 − 0.957569i \(-0.406942\pi\)
0.288203 + 0.957569i \(0.406942\pi\)
\(450\) 0 0
\(451\) −1632.00 −0.170394
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19402.0 −1.98597 −0.992984 − 0.118250i \(-0.962272\pi\)
−0.992984 + 0.118250i \(0.962272\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13578.0 1.37178 0.685890 − 0.727705i \(-0.259413\pi\)
0.685890 + 0.727705i \(0.259413\pi\)
\(462\) 0 0
\(463\) −6222.00 −0.624537 −0.312269 − 0.949994i \(-0.601089\pi\)
−0.312269 + 0.949994i \(0.601089\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15260.0 −1.51210 −0.756048 − 0.654516i \(-0.772872\pi\)
−0.756048 + 0.654516i \(0.772872\pi\)
\(468\) 0 0
\(469\) −488.000 −0.0480464
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6800.00 0.661024
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9812.00 −0.935953 −0.467977 − 0.883741i \(-0.655017\pi\)
−0.467977 + 0.883741i \(0.655017\pi\)
\(480\) 0 0
\(481\) 21216.0 2.01116
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7226.00 0.672364 0.336182 − 0.941797i \(-0.390864\pi\)
0.336182 + 0.941797i \(0.390864\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6750.00 0.620414 0.310207 − 0.950669i \(-0.399602\pi\)
0.310207 + 0.950669i \(0.399602\pi\)
\(492\) 0 0
\(493\) −1748.00 −0.159688
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1416.00 0.127799
\(498\) 0 0
\(499\) −4156.00 −0.372842 −0.186421 − 0.982470i \(-0.559689\pi\)
−0.186421 + 0.982470i \(0.559689\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14088.0 1.24881 0.624406 − 0.781100i \(-0.285341\pi\)
0.624406 + 0.781100i \(0.285341\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16970.0 −1.47776 −0.738882 − 0.673835i \(-0.764646\pi\)
−0.738882 + 0.673835i \(0.764646\pi\)
\(510\) 0 0
\(511\) −756.000 −0.0654471
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3536.00 0.300799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8500.00 0.714763 0.357382 − 0.933958i \(-0.383670\pi\)
0.357382 + 0.933958i \(0.383670\pi\)
\(522\) 0 0
\(523\) −20620.0 −1.72400 −0.861998 − 0.506912i \(-0.830787\pi\)
−0.861998 + 0.506912i \(0.830787\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9880.00 0.816660
\(528\) 0 0
\(529\) 10937.0 0.898907
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3264.00 −0.265252
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −11526.0 −0.921076
\(540\) 0 0
\(541\) 5314.00 0.422304 0.211152 − 0.977453i \(-0.432278\pi\)
0.211152 + 0.977453i \(0.432278\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −24104.0 −1.88412 −0.942059 − 0.335447i \(-0.891113\pi\)
−0.942059 + 0.335447i \(0.891113\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 184.000 0.0142262
\(552\) 0 0
\(553\) 1704.00 0.131033
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −23582.0 −1.79390 −0.896949 − 0.442134i \(-0.854222\pi\)
−0.896949 + 0.442134i \(0.854222\pi\)
\(558\) 0 0
\(559\) 13600.0 1.02901
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2680.00 −0.200619 −0.100310 − 0.994956i \(-0.531983\pi\)
−0.100310 + 0.994956i \(0.531983\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25004.0 1.84222 0.921109 − 0.389304i \(-0.127285\pi\)
0.921109 + 0.389304i \(0.127285\pi\)
\(570\) 0 0
\(571\) −11180.0 −0.819384 −0.409692 − 0.912224i \(-0.634364\pi\)
−0.409692 + 0.912224i \(0.634364\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15862.0 1.14444 0.572222 − 0.820099i \(-0.306082\pi\)
0.572222 + 0.820099i \(0.306082\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1688.00 −0.120534
\(582\) 0 0
\(583\) −14076.0 −0.999946
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15036.0 1.05724 0.528622 − 0.848857i \(-0.322709\pi\)
0.528622 + 0.848857i \(0.322709\pi\)
\(588\) 0 0
\(589\) −1040.00 −0.0727546
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12786.0 −0.885427 −0.442713 − 0.896663i \(-0.645984\pi\)
−0.442713 + 0.896663i \(0.645984\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13464.0 −0.918404 −0.459202 − 0.888332i \(-0.651865\pi\)
−0.459202 + 0.888332i \(0.651865\pi\)
\(600\) 0 0
\(601\) 8518.00 0.578131 0.289065 − 0.957309i \(-0.406656\pi\)
0.289065 + 0.957309i \(0.406656\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11082.0 0.741029 0.370514 − 0.928827i \(-0.379181\pi\)
0.370514 + 0.928827i \(0.379181\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7072.00 0.468253
\(612\) 0 0
\(613\) 26568.0 1.75052 0.875262 − 0.483649i \(-0.160689\pi\)
0.875262 + 0.483649i \(0.160689\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3282.00 0.214146 0.107073 − 0.994251i \(-0.465852\pi\)
0.107073 + 0.994251i \(0.465852\pi\)
\(618\) 0 0
\(619\) −2308.00 −0.149865 −0.0749324 − 0.997189i \(-0.523874\pi\)
−0.0749324 + 0.997189i \(0.523874\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2760.00 −0.177491
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −11856.0 −0.751558
\(630\) 0 0
\(631\) −24572.0 −1.55023 −0.775116 − 0.631819i \(-0.782308\pi\)
−0.775116 + 0.631819i \(0.782308\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −23052.0 −1.43384
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2136.00 −0.131618 −0.0658088 − 0.997832i \(-0.520963\pi\)
−0.0658088 + 0.997832i \(0.520963\pi\)
\(642\) 0 0
\(643\) 5508.00 0.337814 0.168907 − 0.985632i \(-0.445976\pi\)
0.168907 + 0.985632i \(0.445976\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4536.00 −0.275624 −0.137812 − 0.990458i \(-0.544007\pi\)
−0.137812 + 0.990458i \(0.544007\pi\)
\(648\) 0 0
\(649\) 68.0000 0.00411284
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27914.0 −1.67283 −0.836416 − 0.548095i \(-0.815353\pi\)
−0.836416 + 0.548095i \(0.815353\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22842.0 −1.35022 −0.675112 − 0.737715i \(-0.735905\pi\)
−0.675112 + 0.737715i \(0.735905\pi\)
\(660\) 0 0
\(661\) 16458.0 0.968445 0.484222 − 0.874945i \(-0.339103\pi\)
0.484222 + 0.874945i \(0.339103\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6992.00 0.405894
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1292.00 −0.0743325
\(672\) 0 0
\(673\) 16050.0 0.919290 0.459645 − 0.888103i \(-0.347977\pi\)
0.459645 + 0.888103i \(0.347977\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5314.00 −0.301674 −0.150837 − 0.988559i \(-0.548197\pi\)
−0.150837 + 0.988559i \(0.548197\pi\)
\(678\) 0 0
\(679\) 1028.00 0.0581016
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15876.0 −0.889426 −0.444713 − 0.895673i \(-0.646694\pi\)
−0.444713 + 0.895673i \(0.646694\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28152.0 −1.55661
\(690\) 0 0
\(691\) −13372.0 −0.736172 −0.368086 − 0.929792i \(-0.619987\pi\)
−0.368086 + 0.929792i \(0.619987\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1824.00 0.0991233
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3230.00 0.174031 0.0870153 − 0.996207i \(-0.472267\pi\)
0.0870153 + 0.996207i \(0.472267\pi\)
\(702\) 0 0
\(703\) 1248.00 0.0669548
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1404.00 −0.0746858
\(708\) 0 0
\(709\) −6154.00 −0.325978 −0.162989 − 0.986628i \(-0.552113\pi\)
−0.162989 + 0.986628i \(0.552113\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −39520.0 −2.07579
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20264.0 −1.05107 −0.525535 − 0.850772i \(-0.676135\pi\)
−0.525535 + 0.850772i \(0.676135\pi\)
\(720\) 0 0
\(721\) 1796.00 0.0927691
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25354.0 1.29344 0.646718 − 0.762729i \(-0.276141\pi\)
0.646718 + 0.762729i \(0.276141\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7600.00 −0.384536
\(732\) 0 0
\(733\) 13344.0 0.672404 0.336202 − 0.941790i \(-0.390858\pi\)
0.336202 + 0.941790i \(0.390858\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8296.00 0.414636
\(738\) 0 0
\(739\) −28452.0 −1.41627 −0.708135 − 0.706077i \(-0.750463\pi\)
−0.708135 + 0.706077i \(0.750463\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5784.00 −0.285591 −0.142796 − 0.989752i \(-0.545609\pi\)
−0.142796 + 0.989752i \(0.545609\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1752.00 −0.0854695
\(750\) 0 0
\(751\) 852.000 0.0413980 0.0206990 − 0.999786i \(-0.493411\pi\)
0.0206990 + 0.999786i \(0.493411\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 5704.00 0.273864 0.136932 − 0.990580i \(-0.456276\pi\)
0.136932 + 0.990580i \(0.456276\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24828.0 1.18267 0.591337 − 0.806425i \(-0.298600\pi\)
0.591337 + 0.806425i \(0.298600\pi\)
\(762\) 0 0
\(763\) −1204.00 −0.0571268
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 136.000 0.00640245
\(768\) 0 0
\(769\) −13298.0 −0.623587 −0.311793 − 0.950150i \(-0.600930\pi\)
−0.311793 + 0.950150i \(0.600930\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −642.000 −0.0298721 −0.0149361 − 0.999888i \(-0.504754\pi\)
−0.0149361 + 0.999888i \(0.504754\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −192.000 −0.00883070
\(780\) 0 0
\(781\) −24072.0 −1.10290
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20236.0 0.916564 0.458282 − 0.888807i \(-0.348465\pi\)
0.458282 + 0.888807i \(0.348465\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2700.00 −0.121367
\(792\) 0 0
\(793\) −2584.00 −0.115713
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11562.0 0.513861 0.256930 − 0.966430i \(-0.417289\pi\)
0.256930 + 0.966430i \(0.417289\pi\)
\(798\) 0 0
\(799\) −3952.00 −0.174983
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12852.0 0.564804
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18984.0 −0.825021 −0.412510 − 0.910953i \(-0.635348\pi\)
−0.412510 + 0.910953i \(0.635348\pi\)
\(810\) 0 0
\(811\) −2332.00 −0.100971 −0.0504856 − 0.998725i \(-0.516077\pi\)
−0.0504856 + 0.998725i \(0.516077\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 800.000 0.0342576
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19126.0 0.813035 0.406518 − 0.913643i \(-0.366743\pi\)
0.406518 + 0.913643i \(0.366743\pi\)
\(822\) 0 0
\(823\) −37102.0 −1.57144 −0.785720 − 0.618583i \(-0.787707\pi\)
−0.785720 + 0.618583i \(0.787707\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11304.0 −0.475307 −0.237653 − 0.971350i \(-0.576378\pi\)
−0.237653 + 0.971350i \(0.576378\pi\)
\(828\) 0 0
\(829\) −974.000 −0.0408063 −0.0204031 − 0.999792i \(-0.506495\pi\)
−0.0204031 + 0.999792i \(0.506495\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12882.0 0.535816
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16480.0 −0.678132 −0.339066 − 0.940763i \(-0.610111\pi\)
−0.339066 + 0.940763i \(0.610111\pi\)
\(840\) 0 0
\(841\) −22273.0 −0.913240
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 350.000 0.0141985
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 47424.0 1.91031
\(852\) 0 0
\(853\) −11192.0 −0.449246 −0.224623 − 0.974446i \(-0.572115\pi\)
−0.224623 + 0.974446i \(0.572115\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −34278.0 −1.36629 −0.683147 − 0.730281i \(-0.739389\pi\)
−0.683147 + 0.730281i \(0.739389\pi\)
\(858\) 0 0
\(859\) −14020.0 −0.556876 −0.278438 − 0.960454i \(-0.589817\pi\)
−0.278438 + 0.960454i \(0.589817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30528.0 −1.20415 −0.602077 − 0.798438i \(-0.705660\pi\)
−0.602077 + 0.798438i \(0.705660\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −28968.0 −1.13081
\(870\) 0 0
\(871\) 16592.0 0.645463
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2932.00 −0.112892 −0.0564462 − 0.998406i \(-0.517977\pi\)
−0.0564462 + 0.998406i \(0.517977\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7116.00 −0.272127 −0.136064 − 0.990700i \(-0.543445\pi\)
−0.136064 + 0.990700i \(0.543445\pi\)
\(882\) 0 0
\(883\) 35140.0 1.33925 0.669624 − 0.742701i \(-0.266455\pi\)
0.669624 + 0.742701i \(0.266455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20296.0 −0.768290 −0.384145 − 0.923273i \(-0.625504\pi\)
−0.384145 + 0.923273i \(0.625504\pi\)
\(888\) 0 0
\(889\) −732.000 −0.0276159
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 416.000 0.0155889
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11960.0 −0.443702
\(900\) 0 0
\(901\) 15732.0 0.581697
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19512.0 −0.714317 −0.357158 − 0.934044i \(-0.616254\pi\)
−0.357158 + 0.934044i \(0.616254\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 16720.0 0.608077 0.304039 − 0.952660i \(-0.401665\pi\)
0.304039 + 0.952660i \(0.401665\pi\)
\(912\) 0 0
\(913\) 28696.0 1.04020
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 996.000 0.0358678
\(918\) 0 0
\(919\) 7340.00 0.263465 0.131732 − 0.991285i \(-0.457946\pi\)
0.131732 + 0.991285i \(0.457946\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −48144.0 −1.71688
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −48932.0 −1.72810 −0.864051 − 0.503404i \(-0.832081\pi\)
−0.864051 + 0.503404i \(0.832081\pi\)
\(930\) 0 0
\(931\) −1356.00 −0.0477348
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −30298.0 −1.05634 −0.528171 − 0.849138i \(-0.677122\pi\)
−0.528171 + 0.849138i \(0.677122\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8414.00 −0.291486 −0.145743 − 0.989322i \(-0.546557\pi\)
−0.145743 + 0.989322i \(0.546557\pi\)
\(942\) 0 0
\(943\) −7296.00 −0.251952
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23912.0 0.820523 0.410262 − 0.911968i \(-0.365437\pi\)
0.410262 + 0.911968i \(0.365437\pi\)
\(948\) 0 0
\(949\) 25704.0 0.879228
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22866.0 −0.777232 −0.388616 − 0.921400i \(-0.627047\pi\)
−0.388616 + 0.921400i \(0.627047\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4052.00 0.136440
\(960\) 0 0
\(961\) 37809.0 1.26914
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −738.000 −0.0245424 −0.0122712 − 0.999925i \(-0.503906\pi\)
−0.0122712 + 0.999925i \(0.503906\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −44098.0 −1.45744 −0.728719 − 0.684813i \(-0.759884\pi\)
−0.728719 + 0.684813i \(0.759884\pi\)
\(972\) 0 0
\(973\) −4920.00 −0.162105
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 34426.0 1.12731 0.563657 − 0.826009i \(-0.309394\pi\)
0.563657 + 0.826009i \(0.309394\pi\)
\(978\) 0 0
\(979\) 46920.0 1.53174
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −30216.0 −0.980408 −0.490204 − 0.871608i \(-0.663078\pi\)
−0.490204 + 0.871608i \(0.663078\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30400.0 0.977415
\(990\) 0 0
\(991\) 4592.00 0.147194 0.0735972 − 0.997288i \(-0.476552\pi\)
0.0735972 + 0.997288i \(0.476552\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 24276.0 0.771142 0.385571 − 0.922678i \(-0.374004\pi\)
0.385571 + 0.922678i \(0.374004\pi\)
\(998\) 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.a.r.1.1 1
3.2 odd 2 1800.4.a.q.1.1 1
5.2 odd 4 1800.4.f.t.649.1 2
5.3 odd 4 1800.4.f.t.649.2 2
5.4 even 2 360.4.a.d.1.1 ✓ 1
15.2 even 4 1800.4.f.f.649.1 2
15.8 even 4 1800.4.f.f.649.2 2
15.14 odd 2 360.4.a.k.1.1 yes 1
20.19 odd 2 720.4.a.g.1.1 1
60.59 even 2 720.4.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.4.a.d.1.1 ✓ 1 5.4 even 2
360.4.a.k.1.1 yes 1 15.14 odd 2
720.4.a.g.1.1 1 20.19 odd 2
720.4.a.x.1.1 1 60.59 even 2
1800.4.a.q.1.1 1 3.2 odd 2
1800.4.a.r.1.1 1 1.1 even 1 trivial
1800.4.f.f.649.1 2 15.2 even 4
1800.4.f.f.649.2 2 15.8 even 4
1800.4.f.t.649.1 2 5.2 odd 4
1800.4.f.t.649.2 2 5.3 odd 4