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Properties

Label 900.4.a.h.1.1
Level $900$
Weight $4$
Character 900.1
Self dual yes
Analytic conductor $53.102$
Analytic rank $1$
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] = [900,4,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("900.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(53.1017190052\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.00000 q^{7} +O(q^{10})\) \(q-7.00000 q^{7} +54.0000 q^{11} -55.0000 q^{13} -18.0000 q^{17} -25.0000 q^{19} +18.0000 q^{23} +54.0000 q^{29} -271.000 q^{31} +314.000 q^{37} +360.000 q^{41} -163.000 q^{43} -522.000 q^{47} -294.000 q^{49} +36.0000 q^{53} -126.000 q^{59} +47.0000 q^{61} -343.000 q^{67} +1080.00 q^{71} -1054.00 q^{73} -378.000 q^{77} -568.000 q^{79} -1422.00 q^{83} -1440.00 q^{89} +385.000 q^{91} -439.000 q^{97} +O(q^{100})\)

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
\(7\) −7.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 54.0000 1.48015 0.740073 − 0.672526i \(-0.234791\pi\)
0.740073 + 0.672526i \(0.234791\pi\)
\(12\) 0 0
\(13\) −55.0000 −1.17340 −0.586702 − 0.809803i \(-0.699574\pi\)
−0.586702 + 0.809803i \(0.699574\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −18.0000 −0.256802 −0.128401 − 0.991722i \(-0.540985\pi\)
−0.128401 + 0.991722i \(0.540985\pi\)
\(18\) 0 0
\(19\) −25.0000 −0.301863 −0.150931 − 0.988544i \(-0.548227\pi\)
−0.150931 + 0.988544i \(0.548227\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 18.0000 0.163185 0.0815926 − 0.996666i \(-0.473999\pi\)
0.0815926 + 0.996666i \(0.473999\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 54.0000 0.345778 0.172889 − 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) −271.000 −1.57010 −0.785049 − 0.619434i \(-0.787362\pi\)
−0.785049 + 0.619434i \(0.787362\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 314.000 1.39517 0.697585 − 0.716502i \(-0.254258\pi\)
0.697585 + 0.716502i \(0.254258\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 360.000 1.37128 0.685641 − 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 0 0
\(43\) −163.000 −0.578076 −0.289038 − 0.957318i \(-0.593335\pi\)
−0.289038 + 0.957318i \(0.593335\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −522.000 −1.62003 −0.810016 − 0.586407i \(-0.800542\pi\)
−0.810016 + 0.586407i \(0.800542\pi\)
\(48\) 0 0
\(49\) −294.000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 36.0000 0.0933015 0.0466508 − 0.998911i \(-0.485145\pi\)
0.0466508 + 0.998911i \(0.485145\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −126.000 −0.278031 −0.139015 − 0.990290i \(-0.544394\pi\)
−0.139015 + 0.990290i \(0.544394\pi\)
\(60\) 0 0
\(61\) 47.0000 0.0986514 0.0493257 − 0.998783i \(-0.484293\pi\)
0.0493257 + 0.998783i \(0.484293\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −343.000 −0.625435 −0.312717 − 0.949846i \(-0.601239\pi\)
−0.312717 + 0.949846i \(0.601239\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1080.00 1.80525 0.902623 − 0.430433i \(-0.141639\pi\)
0.902623 + 0.430433i \(0.141639\pi\)
\(72\) 0 0
\(73\) −1054.00 −1.68988 −0.844941 − 0.534860i \(-0.820364\pi\)
−0.844941 + 0.534860i \(0.820364\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −378.000 −0.559443
\(78\) 0 0
\(79\) −568.000 −0.808924 −0.404462 − 0.914555i \(-0.632541\pi\)
−0.404462 + 0.914555i \(0.632541\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1422.00 −1.88054 −0.940270 − 0.340430i \(-0.889427\pi\)
−0.940270 + 0.340430i \(0.889427\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1440.00 −1.71505 −0.857526 − 0.514440i \(-0.828000\pi\)
−0.857526 + 0.514440i \(0.828000\pi\)
\(90\) 0 0
\(91\) 385.000 0.443505
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −439.000 −0.459523 −0.229761 − 0.973247i \(-0.573795\pi\)
−0.229761 + 0.973247i \(0.573795\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −828.000 −0.815733 −0.407867 − 0.913041i \(-0.633727\pi\)
−0.407867 + 0.913041i \(0.633727\pi\)
\(102\) 0 0
\(103\) 548.000 0.524233 0.262117 − 0.965036i \(-0.415579\pi\)
0.262117 + 0.965036i \(0.415579\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1476.00 1.33355 0.666777 − 0.745257i \(-0.267673\pi\)
0.666777 + 0.745257i \(0.267673\pi\)
\(108\) 0 0
\(109\) 1277.00 1.12215 0.561075 − 0.827765i \(-0.310388\pi\)
0.561075 + 0.827765i \(0.310388\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1836.00 −1.52846 −0.764232 − 0.644942i \(-0.776882\pi\)
−0.764232 + 0.644942i \(0.776882\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 126.000 0.0970622
\(120\) 0 0
\(121\) 1585.00 1.19083
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −592.000 −0.413634 −0.206817 − 0.978380i \(-0.566310\pi\)
−0.206817 + 0.978380i \(0.566310\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 468.000 0.312132 0.156066 − 0.987747i \(-0.450119\pi\)
0.156066 + 0.987747i \(0.450119\pi\)
\(132\) 0 0
\(133\) 175.000 0.114093
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2574.00 −1.60519 −0.802597 − 0.596521i \(-0.796549\pi\)
−0.802597 + 0.596521i \(0.796549\pi\)
\(138\) 0 0
\(139\) −1756.00 −1.07153 −0.535763 − 0.844369i \(-0.679976\pi\)
−0.535763 + 0.844369i \(0.679976\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2970.00 −1.73681
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2682.00 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 3395.00 1.82968 0.914838 − 0.403820i \(-0.132318\pi\)
0.914838 + 0.403820i \(0.132318\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1549.00 −0.787412 −0.393706 − 0.919236i \(-0.628807\pi\)
−0.393706 + 0.919236i \(0.628807\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −126.000 −0.0616782
\(162\) 0 0
\(163\) −505.000 −0.242667 −0.121333 − 0.992612i \(-0.538717\pi\)
−0.121333 + 0.992612i \(0.538717\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1476.00 0.683930 0.341965 − 0.939713i \(-0.388908\pi\)
0.341965 + 0.939713i \(0.388908\pi\)
\(168\) 0 0
\(169\) 828.000 0.376878
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2358.00 −1.03627 −0.518137 − 0.855298i \(-0.673374\pi\)
−0.518137 + 0.855298i \(0.673374\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1566.00 −0.653901 −0.326951 − 0.945041i \(-0.606021\pi\)
−0.326951 + 0.945041i \(0.606021\pi\)
\(180\) 0 0
\(181\) 305.000 0.125251 0.0626256 − 0.998037i \(-0.480053\pi\)
0.0626256 + 0.998037i \(0.480053\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −972.000 −0.380105
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1746.00 0.661446 0.330723 − 0.943728i \(-0.392707\pi\)
0.330723 + 0.943728i \(0.392707\pi\)
\(192\) 0 0
\(193\) −3877.00 −1.44597 −0.722986 − 0.690863i \(-0.757231\pi\)
−0.722986 + 0.690863i \(0.757231\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2142.00 −0.774676 −0.387338 − 0.921938i \(-0.626605\pi\)
−0.387338 + 0.921938i \(0.626605\pi\)
\(198\) 0 0
\(199\) −4033.00 −1.43664 −0.718321 − 0.695712i \(-0.755089\pi\)
−0.718321 + 0.695712i \(0.755089\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −378.000 −0.130692
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1350.00 −0.446801
\(210\) 0 0
\(211\) −4105.00 −1.33934 −0.669668 − 0.742661i \(-0.733564\pi\)
−0.669668 + 0.742661i \(0.733564\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1897.00 0.593441
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 990.000 0.301333
\(222\) 0 0
\(223\) 1385.00 0.415903 0.207952 − 0.978139i \(-0.433320\pi\)
0.207952 + 0.978139i \(0.433320\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2520.00 0.736821 0.368410 − 0.929663i \(-0.379902\pi\)
0.368410 + 0.929663i \(0.379902\pi\)
\(228\) 0 0
\(229\) 5129.00 1.48006 0.740030 − 0.672574i \(-0.234811\pi\)
0.740030 + 0.672574i \(0.234811\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3240.00 0.910985 0.455492 − 0.890240i \(-0.349463\pi\)
0.455492 + 0.890240i \(0.349463\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2988.00 0.808693 0.404347 − 0.914606i \(-0.367499\pi\)
0.404347 + 0.914606i \(0.367499\pi\)
\(240\) 0 0
\(241\) −2647.00 −0.707503 −0.353752 − 0.935339i \(-0.615094\pi\)
−0.353752 + 0.935339i \(0.615094\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1375.00 0.354207
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4212.00 1.05920 0.529600 − 0.848248i \(-0.322342\pi\)
0.529600 + 0.848248i \(0.322342\pi\)
\(252\) 0 0
\(253\) 972.000 0.241538
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5724.00 −1.38931 −0.694656 − 0.719342i \(-0.744444\pi\)
−0.694656 + 0.719342i \(0.744444\pi\)
\(258\) 0 0
\(259\) −2198.00 −0.527325
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4608.00 1.08039 0.540193 − 0.841541i \(-0.318351\pi\)
0.540193 + 0.841541i \(0.318351\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6426.00 1.45651 0.728253 − 0.685308i \(-0.240333\pi\)
0.728253 + 0.685308i \(0.240333\pi\)
\(270\) 0 0
\(271\) −3376.00 −0.756743 −0.378372 − 0.925654i \(-0.623516\pi\)
−0.378372 + 0.925654i \(0.623516\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5381.00 1.16719 0.583597 − 0.812043i \(-0.301645\pi\)
0.583597 + 0.812043i \(0.301645\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3474.00 0.737514 0.368757 − 0.929526i \(-0.379783\pi\)
0.368757 + 0.929526i \(0.379783\pi\)
\(282\) 0 0
\(283\) −2269.00 −0.476601 −0.238300 − 0.971191i \(-0.576590\pi\)
−0.238300 + 0.971191i \(0.576590\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2520.00 −0.518296
\(288\) 0 0
\(289\) −4589.00 −0.934053
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1674.00 0.333775 0.166888 − 0.985976i \(-0.446628\pi\)
0.166888 + 0.985976i \(0.446628\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −990.000 −0.191482
\(300\) 0 0
\(301\) 1141.00 0.218492
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 539.000 0.100203 0.0501016 − 0.998744i \(-0.484046\pi\)
0.0501016 + 0.998744i \(0.484046\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1494.00 −0.272402 −0.136201 − 0.990681i \(-0.543489\pi\)
−0.136201 + 0.990681i \(0.543489\pi\)
\(312\) 0 0
\(313\) −3997.00 −0.721801 −0.360901 − 0.932604i \(-0.617531\pi\)
−0.360901 + 0.932604i \(0.617531\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3672.00 0.650600 0.325300 − 0.945611i \(-0.394535\pi\)
0.325300 + 0.945611i \(0.394535\pi\)
\(318\) 0 0
\(319\) 2916.00 0.511801
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 450.000 0.0775191
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3654.00 0.612315
\(330\) 0 0
\(331\) 1052.00 0.174692 0.0873461 − 0.996178i \(-0.472161\pi\)
0.0873461 + 0.996178i \(0.472161\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3871.00 −0.625718 −0.312859 − 0.949800i \(-0.601287\pi\)
−0.312859 + 0.949800i \(0.601287\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14634.0 −2.32398
\(342\) 0 0
\(343\) 4459.00 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7686.00 1.18907 0.594533 − 0.804071i \(-0.297337\pi\)
0.594533 + 0.804071i \(0.297337\pi\)
\(348\) 0 0
\(349\) −46.0000 −0.00705537 −0.00352768 − 0.999994i \(-0.501123\pi\)
−0.00352768 + 0.999994i \(0.501123\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6714.00 1.01232 0.506162 − 0.862439i \(-0.331064\pi\)
0.506162 + 0.862439i \(0.331064\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1296.00 −0.190530 −0.0952650 − 0.995452i \(-0.530370\pi\)
−0.0952650 + 0.995452i \(0.530370\pi\)
\(360\) 0 0
\(361\) −6234.00 −0.908879
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5903.00 0.839602 0.419801 − 0.907616i \(-0.362100\pi\)
0.419801 + 0.907616i \(0.362100\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −252.000 −0.0352647
\(372\) 0 0
\(373\) 8867.00 1.23087 0.615437 − 0.788186i \(-0.288980\pi\)
0.615437 + 0.788186i \(0.288980\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2970.00 −0.405737
\(378\) 0 0
\(379\) 8837.00 1.19769 0.598847 − 0.800863i \(-0.295626\pi\)
0.598847 + 0.800863i \(0.295626\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10044.0 1.34001 0.670006 − 0.742356i \(-0.266292\pi\)
0.670006 + 0.742356i \(0.266292\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2736.00 0.356609 0.178304 − 0.983975i \(-0.442939\pi\)
0.178304 + 0.983975i \(0.442939\pi\)
\(390\) 0 0
\(391\) −324.000 −0.0419064
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12241.0 −1.54750 −0.773751 − 0.633490i \(-0.781622\pi\)
−0.773751 + 0.633490i \(0.781622\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9036.00 −1.12528 −0.562639 − 0.826703i \(-0.690214\pi\)
−0.562639 + 0.826703i \(0.690214\pi\)
\(402\) 0 0
\(403\) 14905.0 1.84236
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16956.0 2.06506
\(408\) 0 0
\(409\) 8549.00 1.03355 0.516774 − 0.856122i \(-0.327133\pi\)
0.516774 + 0.856122i \(0.327133\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 882.000 0.105086
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1548.00 −0.180489 −0.0902443 − 0.995920i \(-0.528765\pi\)
−0.0902443 + 0.995920i \(0.528765\pi\)
\(420\) 0 0
\(421\) 6110.00 0.707323 0.353662 − 0.935373i \(-0.384936\pi\)
0.353662 + 0.935373i \(0.384936\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −329.000 −0.0372867
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5958.00 −0.665863 −0.332931 − 0.942951i \(-0.608038\pi\)
−0.332931 + 0.942951i \(0.608038\pi\)
\(432\) 0 0
\(433\) 7163.00 0.794993 0.397496 − 0.917604i \(-0.369879\pi\)
0.397496 + 0.917604i \(0.369879\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −450.000 −0.0492595
\(438\) 0 0
\(439\) 17.0000 0.00184821 0.000924107 − 1.00000i \(-0.499706\pi\)
0.000924107 1.00000i \(0.499706\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9432.00 1.01158 0.505788 − 0.862658i \(-0.331202\pi\)
0.505788 + 0.862658i \(0.331202\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15228.0 1.60057 0.800283 − 0.599623i \(-0.204683\pi\)
0.800283 + 0.599623i \(0.204683\pi\)
\(450\) 0 0
\(451\) 19440.0 2.02970
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −250.000 −0.0255897 −0.0127949 − 0.999918i \(-0.504073\pi\)
−0.0127949 + 0.999918i \(0.504073\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16956.0 −1.71306 −0.856529 − 0.516099i \(-0.827384\pi\)
−0.856529 + 0.516099i \(0.827384\pi\)
\(462\) 0 0
\(463\) −4384.00 −0.440047 −0.220023 − 0.975495i \(-0.570613\pi\)
−0.220023 + 0.975495i \(0.570613\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5166.00 0.511893 0.255946 − 0.966691i \(-0.417613\pi\)
0.255946 + 0.966691i \(0.417613\pi\)
\(468\) 0 0
\(469\) 2401.00 0.236392
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8802.00 −0.855637
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6966.00 −0.664477 −0.332239 − 0.943195i \(-0.607804\pi\)
−0.332239 + 0.943195i \(0.607804\pi\)
\(480\) 0 0
\(481\) −17270.0 −1.63710
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 18431.0 1.71497 0.857483 − 0.514512i \(-0.172027\pi\)
0.857483 + 0.514512i \(0.172027\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1224.00 −0.112502 −0.0562509 − 0.998417i \(-0.517915\pi\)
−0.0562509 + 0.998417i \(0.517915\pi\)
\(492\) 0 0
\(493\) −972.000 −0.0887965
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7560.00 −0.682319
\(498\) 0 0
\(499\) −2449.00 −0.219704 −0.109852 − 0.993948i \(-0.535038\pi\)
−0.109852 + 0.993948i \(0.535038\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3312.00 0.293588 0.146794 − 0.989167i \(-0.453105\pi\)
0.146794 + 0.989167i \(0.453105\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9162.00 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 7378.00 0.638715
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −28188.0 −2.39789
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5418.00 0.455599 0.227799 − 0.973708i \(-0.426847\pi\)
0.227799 + 0.973708i \(0.426847\pi\)
\(522\) 0 0
\(523\) −6829.00 −0.570959 −0.285479 − 0.958385i \(-0.592153\pi\)
−0.285479 + 0.958385i \(0.592153\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4878.00 0.403205
\(528\) 0 0
\(529\) −11843.0 −0.973371
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −19800.0 −1.60907
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −15876.0 −1.26870
\(540\) 0 0
\(541\) 3053.00 0.242622 0.121311 − 0.992615i \(-0.461290\pi\)
0.121311 + 0.992615i \(0.461290\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20456.0 1.59897 0.799484 − 0.600687i \(-0.205106\pi\)
0.799484 + 0.600687i \(0.205106\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1350.00 −0.104377
\(552\) 0 0
\(553\) 3976.00 0.305745
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10854.0 0.825671 0.412835 − 0.910806i \(-0.364538\pi\)
0.412835 + 0.910806i \(0.364538\pi\)
\(558\) 0 0
\(559\) 8965.00 0.678317
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24930.0 1.86621 0.933103 − 0.359609i \(-0.117090\pi\)
0.933103 + 0.359609i \(0.117090\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24786.0 1.82616 0.913078 − 0.407784i \(-0.133698\pi\)
0.913078 + 0.407784i \(0.133698\pi\)
\(570\) 0 0
\(571\) −14785.0 −1.08360 −0.541798 − 0.840509i \(-0.682256\pi\)
−0.541798 + 0.840509i \(0.682256\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15851.0 1.14365 0.571825 − 0.820376i \(-0.306236\pi\)
0.571825 + 0.820376i \(0.306236\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9954.00 0.710777
\(582\) 0 0
\(583\) 1944.00 0.138100
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23148.0 −1.62763 −0.813816 − 0.581122i \(-0.802614\pi\)
−0.813816 + 0.581122i \(0.802614\pi\)
\(588\) 0 0
\(589\) 6775.00 0.473954
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21888.0 1.51574 0.757869 − 0.652407i \(-0.226241\pi\)
0.757869 + 0.652407i \(0.226241\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10764.0 −0.734232 −0.367116 − 0.930175i \(-0.619655\pi\)
−0.367116 + 0.930175i \(0.619655\pi\)
\(600\) 0 0
\(601\) −25597.0 −1.73731 −0.868655 − 0.495417i \(-0.835015\pi\)
−0.868655 + 0.495417i \(0.835015\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −24976.0 −1.67009 −0.835045 − 0.550182i \(-0.814558\pi\)
−0.835045 + 0.550182i \(0.814558\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28710.0 1.90095
\(612\) 0 0
\(613\) −2134.00 −0.140606 −0.0703030 − 0.997526i \(-0.522397\pi\)
−0.0703030 + 0.997526i \(0.522397\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13932.0 −0.909046 −0.454523 − 0.890735i \(-0.650190\pi\)
−0.454523 + 0.890735i \(0.650190\pi\)
\(618\) 0 0
\(619\) −10429.0 −0.677184 −0.338592 − 0.940933i \(-0.609951\pi\)
−0.338592 + 0.940933i \(0.609951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10080.0 0.648229
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5652.00 −0.358283
\(630\) 0 0
\(631\) −6283.00 −0.396390 −0.198195 − 0.980163i \(-0.563508\pi\)
−0.198195 + 0.980163i \(0.563508\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 16170.0 1.00578
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20916.0 1.28882 0.644409 − 0.764681i \(-0.277103\pi\)
0.644409 + 0.764681i \(0.277103\pi\)
\(642\) 0 0
\(643\) −23452.0 −1.43835 −0.719173 − 0.694831i \(-0.755479\pi\)
−0.719173 + 0.694831i \(0.755479\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11916.0 0.724059 0.362030 − 0.932167i \(-0.382084\pi\)
0.362030 + 0.932167i \(0.382084\pi\)
\(648\) 0 0
\(649\) −6804.00 −0.411526
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31842.0 −1.90823 −0.954115 − 0.299441i \(-0.903200\pi\)
−0.954115 + 0.299441i \(0.903200\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3672.00 −0.217057 −0.108529 − 0.994093i \(-0.534614\pi\)
−0.108529 + 0.994093i \(0.534614\pi\)
\(660\) 0 0
\(661\) 5138.00 0.302337 0.151169 − 0.988508i \(-0.451696\pi\)
0.151169 + 0.988508i \(0.451696\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 972.000 0.0564258
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2538.00 0.146018
\(672\) 0 0
\(673\) −5050.00 −0.289247 −0.144623 − 0.989487i \(-0.546197\pi\)
−0.144623 + 0.989487i \(0.546197\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28458.0 −1.61555 −0.807777 − 0.589488i \(-0.799329\pi\)
−0.807777 + 0.589488i \(0.799329\pi\)
\(678\) 0 0
\(679\) 3073.00 0.173683
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24408.0 −1.36742 −0.683709 − 0.729755i \(-0.739634\pi\)
−0.683709 + 0.729755i \(0.739634\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1980.00 −0.109480
\(690\) 0 0
\(691\) 16328.0 0.898909 0.449455 − 0.893303i \(-0.351618\pi\)
0.449455 + 0.893303i \(0.351618\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6480.00 −0.352148
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6246.00 −0.336531 −0.168265 − 0.985742i \(-0.553817\pi\)
−0.168265 + 0.985742i \(0.553817\pi\)
\(702\) 0 0
\(703\) −7850.00 −0.421150
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5796.00 0.308318
\(708\) 0 0
\(709\) −30679.0 −1.62507 −0.812535 − 0.582913i \(-0.801913\pi\)
−0.812535 + 0.582913i \(0.801913\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4878.00 −0.256217
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15462.0 −0.801996 −0.400998 − 0.916079i \(-0.631337\pi\)
−0.400998 + 0.916079i \(0.631337\pi\)
\(720\) 0 0
\(721\) −3836.00 −0.198142
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 26801.0 1.36725 0.683627 − 0.729831i \(-0.260401\pi\)
0.683627 + 0.729831i \(0.260401\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2934.00 0.148451
\(732\) 0 0
\(733\) −22858.0 −1.15181 −0.575907 − 0.817515i \(-0.695351\pi\)
−0.575907 + 0.817515i \(0.695351\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18522.0 −0.925735
\(738\) 0 0
\(739\) −16900.0 −0.841240 −0.420620 − 0.907237i \(-0.638187\pi\)
−0.420620 + 0.907237i \(0.638187\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8028.00 −0.396391 −0.198196 − 0.980162i \(-0.563508\pi\)
−0.198196 + 0.980162i \(0.563508\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10332.0 −0.504036
\(750\) 0 0
\(751\) −12448.0 −0.604839 −0.302419 − 0.953175i \(-0.597794\pi\)
−0.302419 + 0.953175i \(0.597794\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −30103.0 −1.44533 −0.722663 − 0.691200i \(-0.757082\pi\)
−0.722663 + 0.691200i \(0.757082\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17748.0 0.845420 0.422710 − 0.906265i \(-0.361079\pi\)
0.422710 + 0.906265i \(0.361079\pi\)
\(762\) 0 0
\(763\) −8939.00 −0.424133
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6930.00 0.326242
\(768\) 0 0
\(769\) 13283.0 0.622883 0.311442 − 0.950265i \(-0.399188\pi\)
0.311442 + 0.950265i \(0.399188\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26424.0 −1.22950 −0.614751 − 0.788721i \(-0.710744\pi\)
−0.614751 + 0.788721i \(0.710744\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9000.00 −0.413939
\(780\) 0 0
\(781\) 58320.0 2.67203
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −9709.00 −0.439757 −0.219878 − 0.975527i \(-0.570566\pi\)
−0.219878 + 0.975527i \(0.570566\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12852.0 0.577705
\(792\) 0 0
\(793\) −2585.00 −0.115758
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7884.00 0.350396 0.175198 − 0.984533i \(-0.443943\pi\)
0.175198 + 0.984533i \(0.443943\pi\)
\(798\) 0 0
\(799\) 9396.00 0.416028
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −56916.0 −2.50127
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1476.00 0.0641451 0.0320726 − 0.999486i \(-0.489789\pi\)
0.0320726 + 0.999486i \(0.489789\pi\)
\(810\) 0 0
\(811\) 21455.0 0.928960 0.464480 − 0.885583i \(-0.346241\pi\)
0.464480 + 0.885583i \(0.346241\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4075.00 0.174500
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31086.0 −1.32145 −0.660724 − 0.750629i \(-0.729751\pi\)
−0.660724 + 0.750629i \(0.729751\pi\)
\(822\) 0 0
\(823\) 23381.0 0.990292 0.495146 − 0.868810i \(-0.335115\pi\)
0.495146 + 0.868810i \(0.335115\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3636.00 −0.152885 −0.0764426 − 0.997074i \(-0.524356\pi\)
−0.0764426 + 0.997074i \(0.524356\pi\)
\(828\) 0 0
\(829\) 4058.00 0.170012 0.0850061 − 0.996380i \(-0.472909\pi\)
0.0850061 + 0.996380i \(0.472909\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5292.00 0.220116
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −306.000 −0.0125915 −0.00629576 − 0.999980i \(-0.502004\pi\)
−0.00629576 + 0.999980i \(0.502004\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −11095.0 −0.450093
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5652.00 0.227671
\(852\) 0 0
\(853\) 42299.0 1.69788 0.848939 − 0.528491i \(-0.177242\pi\)
0.848939 + 0.528491i \(0.177242\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11484.0 0.457743 0.228872 − 0.973457i \(-0.426496\pi\)
0.228872 + 0.973457i \(0.426496\pi\)
\(858\) 0 0
\(859\) 33560.0 1.33301 0.666503 − 0.745502i \(-0.267790\pi\)
0.666503 + 0.745502i \(0.267790\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14976.0 0.590717 0.295359 − 0.955386i \(-0.404561\pi\)
0.295359 + 0.955386i \(0.404561\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −30672.0 −1.19733
\(870\) 0 0
\(871\) 18865.0 0.733888
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −41893.0 −1.61303 −0.806514 − 0.591215i \(-0.798649\pi\)
−0.806514 + 0.591215i \(0.798649\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 720.000 0.0275340 0.0137670 − 0.999905i \(-0.495618\pi\)
0.0137670 + 0.999905i \(0.495618\pi\)
\(882\) 0 0
\(883\) 17309.0 0.659676 0.329838 − 0.944037i \(-0.393006\pi\)
0.329838 + 0.944037i \(0.393006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7002.00 0.265055 0.132528 − 0.991179i \(-0.457691\pi\)
0.132528 + 0.991179i \(0.457691\pi\)
\(888\) 0 0
\(889\) 4144.00 0.156339
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 13050.0 0.489028
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −14634.0 −0.542905
\(900\) 0 0
\(901\) −648.000 −0.0239601
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1484.00 0.0543279 0.0271640 − 0.999631i \(-0.491352\pi\)
0.0271640 + 0.999631i \(0.491352\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18882.0 0.686705 0.343353 − 0.939207i \(-0.388437\pi\)
0.343353 + 0.939207i \(0.388437\pi\)
\(912\) 0 0
\(913\) −76788.0 −2.78347
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3276.00 −0.117975
\(918\) 0 0
\(919\) −50653.0 −1.81816 −0.909080 − 0.416622i \(-0.863214\pi\)
−0.909080 + 0.416622i \(0.863214\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −59400.0 −2.11828
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35262.0 −1.24533 −0.622663 − 0.782490i \(-0.713949\pi\)
−0.622663 + 0.782490i \(0.713949\pi\)
\(930\) 0 0
\(931\) 7350.00 0.258740
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20279.0 0.707029 0.353514 − 0.935429i \(-0.384987\pi\)
0.353514 + 0.935429i \(0.384987\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −42390.0 −1.46852 −0.734259 − 0.678870i \(-0.762470\pi\)
−0.734259 + 0.678870i \(0.762470\pi\)
\(942\) 0 0
\(943\) 6480.00 0.223773
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42426.0 1.45582 0.727909 − 0.685674i \(-0.240492\pi\)
0.727909 + 0.685674i \(0.240492\pi\)
\(948\) 0 0
\(949\) 57970.0 1.98291
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −48168.0 −1.63727 −0.818633 − 0.574317i \(-0.805268\pi\)
−0.818633 + 0.574317i \(0.805268\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18018.0 0.606707
\(960\) 0 0
\(961\) 43650.0 1.46521
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −51400.0 −1.70932 −0.854660 − 0.519188i \(-0.826234\pi\)
−0.854660 + 0.519188i \(0.826234\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33858.0 −1.11901 −0.559503 − 0.828828i \(-0.689008\pi\)
−0.559503 + 0.828828i \(0.689008\pi\)
\(972\) 0 0
\(973\) 12292.0 0.404998
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −47106.0 −1.54253 −0.771266 − 0.636513i \(-0.780376\pi\)
−0.771266 + 0.636513i \(0.780376\pi\)
\(978\) 0 0
\(979\) −77760.0 −2.53853
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20844.0 0.676318 0.338159 − 0.941089i \(-0.390196\pi\)
0.338159 + 0.941089i \(0.390196\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2934.00 −0.0943334
\(990\) 0 0
\(991\) 4133.00 0.132481 0.0662407 − 0.997804i \(-0.478899\pi\)
0.0662407 + 0.997804i \(0.478899\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33122.0 1.05214 0.526070 − 0.850441i \(-0.323665\pi\)
0.526070 + 0.850441i \(0.323665\pi\)
\(998\) 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 900.4.a.h.1.1 1
3.2 odd 2 300.4.a.g.1.1 yes 1
5.2 odd 4 900.4.d.j.649.1 2
5.3 odd 4 900.4.d.j.649.2 2
5.4 even 2 900.4.a.k.1.1 1
12.11 even 2 1200.4.a.m.1.1 1
15.2 even 4 300.4.d.a.49.1 2
15.8 even 4 300.4.d.a.49.2 2
15.14 odd 2 300.4.a.c.1.1 ✓ 1
60.23 odd 4 1200.4.f.s.49.1 2
60.47 odd 4 1200.4.f.s.49.2 2
60.59 even 2 1200.4.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.4.a.c.1.1 ✓ 1 15.14 odd 2
300.4.a.g.1.1 yes 1 3.2 odd 2
300.4.d.a.49.1 2 15.2 even 4
300.4.d.a.49.2 2 15.8 even 4
900.4.a.h.1.1 1 1.1 even 1 trivial
900.4.a.k.1.1 1 5.4 even 2
900.4.d.j.649.1 2 5.2 odd 4
900.4.d.j.649.2 2 5.3 odd 4
1200.4.a.m.1.1 1 12.11 even 2
1200.4.a.y.1.1 1 60.59 even 2
1200.4.f.s.49.1 2 60.23 odd 4
1200.4.f.s.49.2 2 60.47 odd 4