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Properties

Label 252.6.a.a.1.1
Level $252$
Weight $6$
Character 252.1
Self dual yes
Analytic conductor $40.417$
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] = [252,6,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(40.4167225929\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 252.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{5} -49.0000 q^{7} +O(q^{10})\) \(q-16.0000 q^{5} -49.0000 q^{7} -8.00000 q^{11} +684.000 q^{13} +2218.00 q^{17} -2698.00 q^{19} -3344.00 q^{23} -2869.00 q^{25} +3254.00 q^{29} +4788.00 q^{31} +784.000 q^{35} -11470.0 q^{37} -13350.0 q^{41} -928.000 q^{43} -1212.00 q^{47} +2401.00 q^{49} -13110.0 q^{53} +128.000 q^{55} -34702.0 q^{59} -1032.00 q^{61} -10944.0 q^{65} +10108.0 q^{67} -62720.0 q^{71} -18926.0 q^{73} +392.000 q^{77} +11400.0 q^{79} -88958.0 q^{83} -35488.0 q^{85} -19722.0 q^{89} -33516.0 q^{91} +43168.0 q^{95} +17062.0 q^{97} +O(q^{100})\)

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
\(5\) −16.0000 −0.286217 −0.143108 − 0.989707i \(-0.545710\pi\)
−0.143108 + 0.989707i \(0.545710\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.00000 −0.0199346 −0.00996732 − 0.999950i \(-0.503173\pi\)
−0.00996732 + 0.999950i \(0.503173\pi\)
\(12\) 0 0
\(13\) 684.000 1.12253 0.561265 − 0.827636i \(-0.310315\pi\)
0.561265 + 0.827636i \(0.310315\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2218.00 1.86140 0.930699 − 0.365786i \(-0.119200\pi\)
0.930699 + 0.365786i \(0.119200\pi\)
\(18\) 0 0
\(19\) −2698.00 −1.71458 −0.857290 − 0.514833i \(-0.827854\pi\)
−0.857290 + 0.514833i \(0.827854\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3344.00 −1.31809 −0.659047 − 0.752101i \(-0.729040\pi\)
−0.659047 + 0.752101i \(0.729040\pi\)
\(24\) 0 0
\(25\) −2869.00 −0.918080
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3254.00 0.718493 0.359247 − 0.933243i \(-0.383034\pi\)
0.359247 + 0.933243i \(0.383034\pi\)
\(30\) 0 0
\(31\) 4788.00 0.894849 0.447425 − 0.894322i \(-0.352341\pi\)
0.447425 + 0.894322i \(0.352341\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 784.000 0.108180
\(36\) 0 0
\(37\) −11470.0 −1.37740 −0.688698 − 0.725048i \(-0.741818\pi\)
−0.688698 + 0.725048i \(0.741818\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −13350.0 −1.24029 −0.620143 − 0.784489i \(-0.712925\pi\)
−0.620143 + 0.784489i \(0.712925\pi\)
\(42\) 0 0
\(43\) −928.000 −0.0765380 −0.0382690 − 0.999267i \(-0.512184\pi\)
−0.0382690 + 0.999267i \(0.512184\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1212.00 −0.0800310 −0.0400155 − 0.999199i \(-0.512741\pi\)
−0.0400155 + 0.999199i \(0.512741\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13110.0 −0.641081 −0.320541 − 0.947235i \(-0.603865\pi\)
−0.320541 + 0.947235i \(0.603865\pi\)
\(54\) 0 0
\(55\) 128.000 0.00570563
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −34702.0 −1.29785 −0.648925 − 0.760852i \(-0.724781\pi\)
−0.648925 + 0.760852i \(0.724781\pi\)
\(60\) 0 0
\(61\) −1032.00 −0.0355104 −0.0177552 − 0.999842i \(-0.505652\pi\)
−0.0177552 + 0.999842i \(0.505652\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10944.0 −0.321287
\(66\) 0 0
\(67\) 10108.0 0.275092 0.137546 − 0.990495i \(-0.456078\pi\)
0.137546 + 0.990495i \(0.456078\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −62720.0 −1.47659 −0.738295 − 0.674477i \(-0.764369\pi\)
−0.738295 + 0.674477i \(0.764369\pi\)
\(72\) 0 0
\(73\) −18926.0 −0.415673 −0.207836 − 0.978164i \(-0.566642\pi\)
−0.207836 + 0.978164i \(0.566642\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 392.000 0.00753458
\(78\) 0 0
\(79\) 11400.0 0.205512 0.102756 − 0.994707i \(-0.467234\pi\)
0.102756 + 0.994707i \(0.467234\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −88958.0 −1.41739 −0.708696 − 0.705514i \(-0.750716\pi\)
−0.708696 + 0.705514i \(0.750716\pi\)
\(84\) 0 0
\(85\) −35488.0 −0.532763
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −19722.0 −0.263922 −0.131961 − 0.991255i \(-0.542127\pi\)
−0.131961 + 0.991255i \(0.542127\pi\)
\(90\) 0 0
\(91\) −33516.0 −0.424276
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 43168.0 0.490742
\(96\) 0 0
\(97\) 17062.0 0.184120 0.0920599 − 0.995753i \(-0.470655\pi\)
0.0920599 + 0.995753i \(0.470655\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −45904.0 −0.447762 −0.223881 − 0.974617i \(-0.571873\pi\)
−0.223881 + 0.974617i \(0.571873\pi\)
\(102\) 0 0
\(103\) −136012. −1.26324 −0.631618 − 0.775280i \(-0.717609\pi\)
−0.631618 + 0.775280i \(0.717609\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 69156.0 0.583943 0.291971 − 0.956427i \(-0.405689\pi\)
0.291971 + 0.956427i \(0.405689\pi\)
\(108\) 0 0
\(109\) −146414. −1.18037 −0.590183 − 0.807270i \(-0.700944\pi\)
−0.590183 + 0.807270i \(0.700944\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 80186.0 0.590748 0.295374 − 0.955382i \(-0.404556\pi\)
0.295374 + 0.955382i \(0.404556\pi\)
\(114\) 0 0
\(115\) 53504.0 0.377261
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −108682. −0.703542
\(120\) 0 0
\(121\) −160987. −0.999603
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 95904.0 0.548987
\(126\) 0 0
\(127\) 274800. 1.51185 0.755923 − 0.654661i \(-0.227189\pi\)
0.755923 + 0.654661i \(0.227189\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −180742. −0.920197 −0.460099 − 0.887868i \(-0.652186\pi\)
−0.460099 + 0.887868i \(0.652186\pi\)
\(132\) 0 0
\(133\) 132202. 0.648051
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 209678. 0.954446 0.477223 − 0.878782i \(-0.341643\pi\)
0.477223 + 0.878782i \(0.341643\pi\)
\(138\) 0 0
\(139\) 17242.0 0.0756921 0.0378461 − 0.999284i \(-0.487950\pi\)
0.0378461 + 0.999284i \(0.487950\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5472.00 −0.0223772
\(144\) 0 0
\(145\) −52064.0 −0.205645
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −59358.0 −0.219035 −0.109518 − 0.993985i \(-0.534931\pi\)
−0.109518 + 0.993985i \(0.534931\pi\)
\(150\) 0 0
\(151\) −336344. −1.20044 −0.600221 − 0.799834i \(-0.704921\pi\)
−0.600221 + 0.799834i \(0.704921\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −76608.0 −0.256121
\(156\) 0 0
\(157\) 464588. 1.50425 0.752123 − 0.659023i \(-0.229030\pi\)
0.752123 + 0.659023i \(0.229030\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 163856. 0.498193
\(162\) 0 0
\(163\) 314792. 0.928014 0.464007 − 0.885831i \(-0.346411\pi\)
0.464007 + 0.885831i \(0.346411\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −285724. −0.792785 −0.396393 − 0.918081i \(-0.629738\pi\)
−0.396393 + 0.918081i \(0.629738\pi\)
\(168\) 0 0
\(169\) 96563.0 0.260072
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 709148. 1.80145 0.900724 − 0.434392i \(-0.143037\pi\)
0.900724 + 0.434392i \(0.143037\pi\)
\(174\) 0 0
\(175\) 140581. 0.347002
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 617148. 1.43965 0.719825 − 0.694156i \(-0.244222\pi\)
0.719825 + 0.694156i \(0.244222\pi\)
\(180\) 0 0
\(181\) 237828. 0.539593 0.269797 − 0.962917i \(-0.413044\pi\)
0.269797 + 0.962917i \(0.413044\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 183520. 0.394234
\(186\) 0 0
\(187\) −17744.0 −0.0371063
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 133512. 0.264812 0.132406 − 0.991196i \(-0.457730\pi\)
0.132406 + 0.991196i \(0.457730\pi\)
\(192\) 0 0
\(193\) 270446. 0.522622 0.261311 − 0.965255i \(-0.415845\pi\)
0.261311 + 0.965255i \(0.415845\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −875102. −1.60655 −0.803273 − 0.595611i \(-0.796910\pi\)
−0.803273 + 0.595611i \(0.796910\pi\)
\(198\) 0 0
\(199\) −347620. −0.622260 −0.311130 − 0.950367i \(-0.600708\pi\)
−0.311130 + 0.950367i \(0.600708\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −159446. −0.271565
\(204\) 0 0
\(205\) 213600. 0.354990
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21584.0 0.0341795
\(210\) 0 0
\(211\) −425380. −0.657765 −0.328883 − 0.944371i \(-0.606672\pi\)
−0.328883 + 0.944371i \(0.606672\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14848.0 0.0219064
\(216\) 0 0
\(217\) −234612. −0.338221
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.51711e6 2.08947
\(222\) 0 0
\(223\) 481592. 0.648511 0.324255 − 0.945970i \(-0.394886\pi\)
0.324255 + 0.945970i \(0.394886\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6042.00 0.00778245 0.00389122 − 0.999992i \(-0.498761\pi\)
0.00389122 + 0.999992i \(0.498761\pi\)
\(228\) 0 0
\(229\) 1804.00 0.00227325 0.00113663 − 0.999999i \(-0.499638\pi\)
0.00113663 + 0.999999i \(0.499638\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.61153e6 1.94468 0.972339 − 0.233576i \(-0.0750427\pi\)
0.972339 + 0.233576i \(0.0750427\pi\)
\(234\) 0 0
\(235\) 19392.0 0.0229062
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 987096. 1.11780 0.558901 − 0.829235i \(-0.311223\pi\)
0.558901 + 0.829235i \(0.311223\pi\)
\(240\) 0 0
\(241\) 893510. 0.990962 0.495481 − 0.868619i \(-0.334992\pi\)
0.495481 + 0.868619i \(0.334992\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −38416.0 −0.0408881
\(246\) 0 0
\(247\) −1.84543e6 −1.92467
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −365946. −0.366634 −0.183317 − 0.983054i \(-0.558683\pi\)
−0.183317 + 0.983054i \(0.558683\pi\)
\(252\) 0 0
\(253\) 26752.0 0.0262757
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.40459e6 −1.32653 −0.663266 − 0.748383i \(-0.730830\pi\)
−0.663266 + 0.748383i \(0.730830\pi\)
\(258\) 0 0
\(259\) 562030. 0.520607
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.09968e6 −0.980341 −0.490170 − 0.871627i \(-0.663065\pi\)
−0.490170 + 0.871627i \(0.663065\pi\)
\(264\) 0 0
\(265\) 209760. 0.183488
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −814948. −0.686672 −0.343336 − 0.939213i \(-0.611557\pi\)
−0.343336 + 0.939213i \(0.611557\pi\)
\(270\) 0 0
\(271\) −1.69906e6 −1.40535 −0.702675 − 0.711511i \(-0.748011\pi\)
−0.702675 + 0.711511i \(0.748011\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22952.0 0.0183016
\(276\) 0 0
\(277\) −1.36508e6 −1.06895 −0.534477 − 0.845183i \(-0.679492\pi\)
−0.534477 + 0.845183i \(0.679492\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 715846. 0.540821 0.270411 − 0.962745i \(-0.412841\pi\)
0.270411 + 0.962745i \(0.412841\pi\)
\(282\) 0 0
\(283\) 217726. 0.161601 0.0808005 − 0.996730i \(-0.474252\pi\)
0.0808005 + 0.996730i \(0.474252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 654150. 0.468784
\(288\) 0 0
\(289\) 3.49967e6 2.46480
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.50708e6 −1.02557 −0.512787 − 0.858516i \(-0.671387\pi\)
−0.512787 + 0.858516i \(0.671387\pi\)
\(294\) 0 0
\(295\) 555232. 0.371466
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.28730e6 −1.47960
\(300\) 0 0
\(301\) 45472.0 0.0289286
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16512.0 0.0101637
\(306\) 0 0
\(307\) 12502.0 0.00757066 0.00378533 − 0.999993i \(-0.498795\pi\)
0.00378533 + 0.999993i \(0.498795\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 647432. 0.379571 0.189786 − 0.981826i \(-0.439221\pi\)
0.189786 + 0.981826i \(0.439221\pi\)
\(312\) 0 0
\(313\) −935978. −0.540014 −0.270007 − 0.962858i \(-0.587026\pi\)
−0.270007 + 0.962858i \(0.587026\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −705942. −0.394567 −0.197284 − 0.980346i \(-0.563212\pi\)
−0.197284 + 0.980346i \(0.563212\pi\)
\(318\) 0 0
\(319\) −26032.0 −0.0143229
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.98416e6 −3.19152
\(324\) 0 0
\(325\) −1.96240e6 −1.03057
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 59388.0 0.0302489
\(330\) 0 0
\(331\) −1.14304e6 −0.573445 −0.286722 − 0.958014i \(-0.592566\pi\)
−0.286722 + 0.958014i \(0.592566\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −161728. −0.0787360
\(336\) 0 0
\(337\) −2.36402e6 −1.13390 −0.566952 − 0.823751i \(-0.691877\pi\)
−0.566952 + 0.823751i \(0.691877\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −38304.0 −0.0178385
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −726240. −0.323785 −0.161892 − 0.986808i \(-0.551760\pi\)
−0.161892 + 0.986808i \(0.551760\pi\)
\(348\) 0 0
\(349\) 136180. 0.0598480 0.0299240 − 0.999552i \(-0.490473\pi\)
0.0299240 + 0.999552i \(0.490473\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.16907e6 0.499349 0.249674 − 0.968330i \(-0.419676\pi\)
0.249674 + 0.968330i \(0.419676\pi\)
\(354\) 0 0
\(355\) 1.00352e6 0.422625
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4280.00 0.00175270 0.000876350 − 1.00000i \(-0.499721\pi\)
0.000876350 1.00000i \(0.499721\pi\)
\(360\) 0 0
\(361\) 4.80310e6 1.93979
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 302816. 0.118973
\(366\) 0 0
\(367\) 2.44796e6 0.948722 0.474361 − 0.880330i \(-0.342679\pi\)
0.474361 + 0.880330i \(0.342679\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 642390. 0.242306
\(372\) 0 0
\(373\) −904514. −0.336623 −0.168311 − 0.985734i \(-0.553831\pi\)
−0.168311 + 0.985734i \(0.553831\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.22574e6 0.806530
\(378\) 0 0
\(379\) −4.23034e6 −1.51279 −0.756393 − 0.654117i \(-0.773040\pi\)
−0.756393 + 0.654117i \(0.773040\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.55400e6 −1.58634 −0.793169 − 0.609002i \(-0.791570\pi\)
−0.793169 + 0.609002i \(0.791570\pi\)
\(384\) 0 0
\(385\) −6272.00 −0.00215652
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.98541e6 1.33536 0.667680 − 0.744448i \(-0.267287\pi\)
0.667680 + 0.744448i \(0.267287\pi\)
\(390\) 0 0
\(391\) −7.41699e6 −2.45350
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −182400. −0.0588210
\(396\) 0 0
\(397\) 552420. 0.175911 0.0879555 − 0.996124i \(-0.471967\pi\)
0.0879555 + 0.996124i \(0.471967\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38190.0 −0.0118601 −0.00593006 − 0.999982i \(-0.501888\pi\)
−0.00593006 + 0.999982i \(0.501888\pi\)
\(402\) 0 0
\(403\) 3.27499e6 1.00449
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 91760.0 0.0274579
\(408\) 0 0
\(409\) −3.92475e6 −1.16012 −0.580062 − 0.814573i \(-0.696972\pi\)
−0.580062 + 0.814573i \(0.696972\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.70040e6 0.490541
\(414\) 0 0
\(415\) 1.42333e6 0.405681
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −598386. −0.166512 −0.0832562 − 0.996528i \(-0.526532\pi\)
−0.0832562 + 0.996528i \(0.526532\pi\)
\(420\) 0 0
\(421\) 4.61597e6 1.26928 0.634641 − 0.772807i \(-0.281148\pi\)
0.634641 + 0.772807i \(0.281148\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.36344e6 −1.70891
\(426\) 0 0
\(427\) 50568.0 0.0134217
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 61560.0 0.0159627 0.00798133 − 0.999968i \(-0.497459\pi\)
0.00798133 + 0.999968i \(0.497459\pi\)
\(432\) 0 0
\(433\) −3.79727e6 −0.973310 −0.486655 − 0.873594i \(-0.661783\pi\)
−0.486655 + 0.873594i \(0.661783\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.02211e6 2.25998
\(438\) 0 0
\(439\) −2.28852e6 −0.566752 −0.283376 − 0.959009i \(-0.591455\pi\)
−0.283376 + 0.959009i \(0.591455\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.75976e6 1.15233 0.576163 − 0.817335i \(-0.304549\pi\)
0.576163 + 0.817335i \(0.304549\pi\)
\(444\) 0 0
\(445\) 315552. 0.0755389
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4.36715e6 1.02231 0.511155 − 0.859489i \(-0.329218\pi\)
0.511155 + 0.859489i \(0.329218\pi\)
\(450\) 0 0
\(451\) 106800. 0.0247246
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 536256. 0.121435
\(456\) 0 0
\(457\) 5.44994e6 1.22068 0.610339 − 0.792140i \(-0.291033\pi\)
0.610339 + 0.792140i \(0.291033\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.66966e6 −0.365911 −0.182956 − 0.983121i \(-0.558566\pi\)
−0.182956 + 0.983121i \(0.558566\pi\)
\(462\) 0 0
\(463\) 70768.0 0.0153421 0.00767104 − 0.999971i \(-0.497558\pi\)
0.00767104 + 0.999971i \(0.497558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.66083e6 −1.20112 −0.600562 − 0.799578i \(-0.705056\pi\)
−0.600562 + 0.799578i \(0.705056\pi\)
\(468\) 0 0
\(469\) −495292. −0.103975
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7424.00 0.00152576
\(474\) 0 0
\(475\) 7.74056e6 1.57412
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.44948e6 0.288652 0.144326 − 0.989530i \(-0.453899\pi\)
0.144326 + 0.989530i \(0.453899\pi\)
\(480\) 0 0
\(481\) −7.84548e6 −1.54617
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −272992. −0.0526982
\(486\) 0 0
\(487\) 4.07504e6 0.778591 0.389296 − 0.921113i \(-0.372718\pi\)
0.389296 + 0.921113i \(0.372718\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −986100. −0.184594 −0.0922969 − 0.995732i \(-0.529421\pi\)
−0.0922969 + 0.995732i \(0.529421\pi\)
\(492\) 0 0
\(493\) 7.21737e6 1.33740
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.07328e6 0.558099
\(498\) 0 0
\(499\) 5.98342e6 1.07572 0.537859 − 0.843035i \(-0.319233\pi\)
0.537859 + 0.843035i \(0.319233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.49373e6 0.615700 0.307850 − 0.951435i \(-0.400391\pi\)
0.307850 + 0.951435i \(0.400391\pi\)
\(504\) 0 0
\(505\) 734464. 0.128157
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.15711e6 −0.369043 −0.184522 − 0.982828i \(-0.559074\pi\)
−0.184522 + 0.982828i \(0.559074\pi\)
\(510\) 0 0
\(511\) 927374. 0.157110
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.17619e6 0.361559
\(516\) 0 0
\(517\) 9696.00 0.00159539
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.65817e6 1.07463 0.537317 − 0.843380i \(-0.319438\pi\)
0.537317 + 0.843380i \(0.319438\pi\)
\(522\) 0 0
\(523\) 5.95223e6 0.951537 0.475768 − 0.879571i \(-0.342170\pi\)
0.475768 + 0.879571i \(0.342170\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.06198e7 1.66567
\(528\) 0 0
\(529\) 4.74599e6 0.737374
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.13140e6 −1.39226
\(534\) 0 0
\(535\) −1.10650e6 −0.167134
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −19208.0 −0.00284780
\(540\) 0 0
\(541\) −6.39681e6 −0.939659 −0.469830 − 0.882757i \(-0.655685\pi\)
−0.469830 + 0.882757i \(0.655685\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.34262e6 0.337840
\(546\) 0 0
\(547\) −5.51851e6 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.77929e6 −1.23191
\(552\) 0 0
\(553\) −558600. −0.0776762
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.02159e6 −0.276093 −0.138046 − 0.990426i \(-0.544082\pi\)
−0.138046 + 0.990426i \(0.544082\pi\)
\(558\) 0 0
\(559\) −634752. −0.0859161
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.14678e6 −1.08322 −0.541608 − 0.840631i \(-0.682184\pi\)
−0.541608 + 0.840631i \(0.682184\pi\)
\(564\) 0 0
\(565\) −1.28298e6 −0.169082
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.19824e7 1.55154 0.775772 − 0.631013i \(-0.217361\pi\)
0.775772 + 0.631013i \(0.217361\pi\)
\(570\) 0 0
\(571\) 1.39582e6 0.179159 0.0895793 − 0.995980i \(-0.471448\pi\)
0.0895793 + 0.995980i \(0.471448\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.59394e6 1.21012
\(576\) 0 0
\(577\) 1.96784e6 0.246065 0.123033 − 0.992403i \(-0.460738\pi\)
0.123033 + 0.992403i \(0.460738\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.35894e6 0.535724
\(582\) 0 0
\(583\) 104880. 0.0127797
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.18897e6 0.381993 0.190997 − 0.981591i \(-0.438828\pi\)
0.190997 + 0.981591i \(0.438828\pi\)
\(588\) 0 0
\(589\) −1.29180e7 −1.53429
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.67500e6 −0.195604 −0.0978022 − 0.995206i \(-0.531181\pi\)
−0.0978022 + 0.995206i \(0.531181\pi\)
\(594\) 0 0
\(595\) 1.73891e6 0.201366
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.00635e7 1.14599 0.572994 − 0.819559i \(-0.305782\pi\)
0.572994 + 0.819559i \(0.305782\pi\)
\(600\) 0 0
\(601\) 1.72798e6 0.195143 0.0975713 − 0.995229i \(-0.468893\pi\)
0.0975713 + 0.995229i \(0.468893\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.57579e6 0.286103
\(606\) 0 0
\(607\) −1.69523e7 −1.86748 −0.933740 − 0.357953i \(-0.883475\pi\)
−0.933740 + 0.357953i \(0.883475\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −829008. −0.0898371
\(612\) 0 0
\(613\) 1.01942e7 1.09572 0.547861 − 0.836569i \(-0.315442\pi\)
0.547861 + 0.836569i \(0.315442\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.57452e7 −1.66508 −0.832540 − 0.553965i \(-0.813114\pi\)
−0.832540 + 0.553965i \(0.813114\pi\)
\(618\) 0 0
\(619\) −332690. −0.0348990 −0.0174495 − 0.999848i \(-0.505555\pi\)
−0.0174495 + 0.999848i \(0.505555\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 966378. 0.0997532
\(624\) 0 0
\(625\) 7.43116e6 0.760951
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.54405e7 −2.56388
\(630\) 0 0
\(631\) 3.59720e6 0.359659 0.179830 − 0.983698i \(-0.442445\pi\)
0.179830 + 0.983698i \(0.442445\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.39680e6 −0.432715
\(636\) 0 0
\(637\) 1.64228e6 0.160361
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.46389e7 1.40723 0.703614 − 0.710583i \(-0.251569\pi\)
0.703614 + 0.710583i \(0.251569\pi\)
\(642\) 0 0
\(643\) −1.38386e7 −1.31997 −0.659987 − 0.751277i \(-0.729438\pi\)
−0.659987 + 0.751277i \(0.729438\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.40358e7 1.31819 0.659093 − 0.752061i \(-0.270940\pi\)
0.659093 + 0.752061i \(0.270940\pi\)
\(648\) 0 0
\(649\) 277616. 0.0258722
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.61063e7 −1.47813 −0.739064 − 0.673635i \(-0.764732\pi\)
−0.739064 + 0.673635i \(0.764732\pi\)
\(654\) 0 0
\(655\) 2.89187e6 0.263376
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.80075e6 −0.430622 −0.215311 − 0.976546i \(-0.569076\pi\)
−0.215311 + 0.976546i \(0.569076\pi\)
\(660\) 0 0
\(661\) −1.76565e7 −1.57181 −0.785905 − 0.618347i \(-0.787803\pi\)
−0.785905 + 0.618347i \(0.787803\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.11523e6 −0.185483
\(666\) 0 0
\(667\) −1.08814e7 −0.947042
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8256.00 0.000707886 0
\(672\) 0 0
\(673\) 6.59225e6 0.561043 0.280521 − 0.959848i \(-0.409493\pi\)
0.280521 + 0.959848i \(0.409493\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.77178e6 −0.819411 −0.409706 − 0.912218i \(-0.634369\pi\)
−0.409706 + 0.912218i \(0.634369\pi\)
\(678\) 0 0
\(679\) −836038. −0.0695908
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.88663e7 1.54752 0.773758 − 0.633481i \(-0.218374\pi\)
0.773758 + 0.633481i \(0.218374\pi\)
\(684\) 0 0
\(685\) −3.35485e6 −0.273178
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.96724e6 −0.719632
\(690\) 0 0
\(691\) 8.67018e6 0.690769 0.345385 − 0.938461i \(-0.387748\pi\)
0.345385 + 0.938461i \(0.387748\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −275872. −0.0216643
\(696\) 0 0
\(697\) −2.96103e7 −2.30866
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −7.93482e6 −0.609877 −0.304938 − 0.952372i \(-0.598636\pi\)
−0.304938 + 0.952372i \(0.598636\pi\)
\(702\) 0 0
\(703\) 3.09461e7 2.36166
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.24930e6 0.169238
\(708\) 0 0
\(709\) 2.62600e7 1.96191 0.980956 − 0.194228i \(-0.0622202\pi\)
0.980956 + 0.194228i \(0.0622202\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.60111e7 −1.17950
\(714\) 0 0
\(715\) 87552.0 0.00640473
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.20763e7 −1.59259 −0.796295 − 0.604909i \(-0.793210\pi\)
−0.796295 + 0.604909i \(0.793210\pi\)
\(720\) 0 0
\(721\) 6.66459e6 0.477458
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −9.33573e6 −0.659634
\(726\) 0 0
\(727\) 8.49245e6 0.595933 0.297966 − 0.954576i \(-0.403692\pi\)
0.297966 + 0.954576i \(0.403692\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.05830e6 −0.142468
\(732\) 0 0
\(733\) −1.90713e7 −1.31105 −0.655526 − 0.755172i \(-0.727553\pi\)
−0.655526 + 0.755172i \(0.727553\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −80864.0 −0.00548386
\(738\) 0 0
\(739\) −1.46832e7 −0.989032 −0.494516 − 0.869169i \(-0.664655\pi\)
−0.494516 + 0.869169i \(0.664655\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.64265e7 −1.09162 −0.545812 − 0.837908i \(-0.683779\pi\)
−0.545812 + 0.837908i \(0.683779\pi\)
\(744\) 0 0
\(745\) 949728. 0.0626915
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.38864e6 −0.220710
\(750\) 0 0
\(751\) −2.44357e7 −1.58097 −0.790486 − 0.612479i \(-0.790172\pi\)
−0.790486 + 0.612479i \(0.790172\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.38150e6 0.343587
\(756\) 0 0
\(757\) −295566. −0.0187463 −0.00937313 − 0.999956i \(-0.502984\pi\)
−0.00937313 + 0.999956i \(0.502984\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 473842. 0.0296601 0.0148300 − 0.999890i \(-0.495279\pi\)
0.0148300 + 0.999890i \(0.495279\pi\)
\(762\) 0 0
\(763\) 7.17429e6 0.446136
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.37362e7 −1.45687
\(768\) 0 0
\(769\) 2.33241e7 1.42229 0.711145 − 0.703045i \(-0.248177\pi\)
0.711145 + 0.703045i \(0.248177\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.55583e7 −0.936511 −0.468255 − 0.883593i \(-0.655117\pi\)
−0.468255 + 0.883593i \(0.655117\pi\)
\(774\) 0 0
\(775\) −1.37368e7 −0.821543
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.60183e7 2.12657
\(780\) 0 0
\(781\) 501760. 0.0294353
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.43341e6 −0.430540
\(786\) 0 0
\(787\) 6.66843e6 0.383784 0.191892 − 0.981416i \(-0.438538\pi\)
0.191892 + 0.981416i \(0.438538\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.92911e6 −0.223282
\(792\) 0 0
\(793\) −705888. −0.0398614
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.22461e7 0.682892 0.341446 − 0.939901i \(-0.389083\pi\)
0.341446 + 0.939901i \(0.389083\pi\)
\(798\) 0 0
\(799\) −2.68822e6 −0.148969
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 151408. 0.00828629
\(804\) 0 0
\(805\) −2.62170e6 −0.142591
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.91495e7 1.56588 0.782941 − 0.622095i \(-0.213718\pi\)
0.782941 + 0.622095i \(0.213718\pi\)
\(810\) 0 0
\(811\) 7.58849e6 0.405138 0.202569 − 0.979268i \(-0.435071\pi\)
0.202569 + 0.979268i \(0.435071\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.03667e6 −0.265613
\(816\) 0 0
\(817\) 2.50374e6 0.131230
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.98849e6 0.310070 0.155035 − 0.987909i \(-0.450451\pi\)
0.155035 + 0.987909i \(0.450451\pi\)
\(822\) 0 0
\(823\) 817960. 0.0420952 0.0210476 − 0.999778i \(-0.493300\pi\)
0.0210476 + 0.999778i \(0.493300\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.51963e6 −0.128107 −0.0640535 − 0.997946i \(-0.520403\pi\)
−0.0640535 + 0.997946i \(0.520403\pi\)
\(828\) 0 0
\(829\) −1.61006e7 −0.813684 −0.406842 − 0.913499i \(-0.633370\pi\)
−0.406842 + 0.913499i \(0.633370\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.32542e6 0.265914
\(834\) 0 0
\(835\) 4.57158e6 0.226908
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.58167e7 1.26618 0.633091 − 0.774077i \(-0.281786\pi\)
0.633091 + 0.774077i \(0.281786\pi\)
\(840\) 0 0
\(841\) −9.92263e6 −0.483768
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.54501e6 −0.0744370
\(846\) 0 0
\(847\) 7.88836e6 0.377814
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.83557e7 1.81554
\(852\) 0 0
\(853\) −1.54270e7 −0.725954 −0.362977 − 0.931798i \(-0.618240\pi\)
−0.362977 + 0.931798i \(0.618240\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.60517e6 −0.167677 −0.0838384 − 0.996479i \(-0.526718\pi\)
−0.0838384 + 0.996479i \(0.526718\pi\)
\(858\) 0 0
\(859\) 4.06995e6 0.188194 0.0940970 − 0.995563i \(-0.470004\pi\)
0.0940970 + 0.995563i \(0.470004\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.25111e6 −0.331419 −0.165710 − 0.986175i \(-0.552991\pi\)
−0.165710 + 0.986175i \(0.552991\pi\)
\(864\) 0 0
\(865\) −1.13464e7 −0.515604
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −91200.0 −0.00409681
\(870\) 0 0
\(871\) 6.91387e6 0.308799
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.69930e6 −0.207497
\(876\) 0 0
\(877\) 2.37414e6 0.104233 0.0521167 − 0.998641i \(-0.483403\pi\)
0.0521167 + 0.998641i \(0.483403\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.03558e7 −1.31766 −0.658828 − 0.752293i \(-0.728948\pi\)
−0.658828 + 0.752293i \(0.728948\pi\)
\(882\) 0 0
\(883\) 1.53338e6 0.0661832 0.0330916 − 0.999452i \(-0.489465\pi\)
0.0330916 + 0.999452i \(0.489465\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.92379e7 1.24778 0.623888 − 0.781514i \(-0.285552\pi\)
0.623888 + 0.781514i \(0.285552\pi\)
\(888\) 0 0
\(889\) −1.34652e7 −0.571424
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.26998e6 0.137220
\(894\) 0 0
\(895\) −9.87437e6 −0.412052
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.55802e7 0.642943
\(900\) 0 0
\(901\) −2.90780e7 −1.19331
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.80525e6 −0.154441
\(906\) 0 0
\(907\) 4.48227e7 1.80917 0.904587 − 0.426289i \(-0.140179\pi\)
0.904587 + 0.426289i \(0.140179\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.62906e7 1.44877 0.724384 − 0.689397i \(-0.242124\pi\)
0.724384 + 0.689397i \(0.242124\pi\)
\(912\) 0 0
\(913\) 711664. 0.0282552
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.85636e6 0.347802
\(918\) 0 0
\(919\) 3.25350e7 1.27076 0.635378 − 0.772201i \(-0.280844\pi\)
0.635378 + 0.772201i \(0.280844\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.29005e7 −1.65752
\(924\) 0 0
\(925\) 3.29074e7 1.26456
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.46676e7 1.69806 0.849030 − 0.528344i \(-0.177187\pi\)
0.849030 + 0.528344i \(0.177187\pi\)
\(930\) 0 0
\(931\) −6.47790e6 −0.244940
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 283904. 0.0106204
\(936\) 0 0
\(937\) 1.56680e7 0.582995 0.291498 − 0.956572i \(-0.405846\pi\)
0.291498 + 0.956572i \(0.405846\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.01175e7 0.740627 0.370313 − 0.928907i \(-0.379250\pi\)
0.370313 + 0.928907i \(0.379250\pi\)
\(942\) 0 0
\(943\) 4.46424e7 1.63481
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 8.84518e6 0.320503 0.160251 − 0.987076i \(-0.448769\pi\)
0.160251 + 0.987076i \(0.448769\pi\)
\(948\) 0 0
\(949\) −1.29454e7 −0.466605
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.14364e7 1.12124 0.560622 − 0.828072i \(-0.310562\pi\)
0.560622 + 0.828072i \(0.310562\pi\)
\(954\) 0 0
\(955\) −2.13619e6 −0.0757935
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.02742e7 −0.360747
\(960\) 0 0
\(961\) −5.70421e6 −0.199245
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.32714e6 −0.149583
\(966\) 0 0
\(967\) −9.52158e6 −0.327449 −0.163724 − 0.986506i \(-0.552351\pi\)
−0.163724 + 0.986506i \(0.552351\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.06520e7 −0.362564 −0.181282 − 0.983431i \(-0.558025\pi\)
−0.181282 + 0.983431i \(0.558025\pi\)
\(972\) 0 0
\(973\) −844858. −0.0286089
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.72931e7 −0.914779 −0.457389 − 0.889266i \(-0.651215\pi\)
−0.457389 + 0.889266i \(0.651215\pi\)
\(978\) 0 0
\(979\) 157776. 0.00526119
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.04764e7 −0.345802 −0.172901 − 0.984939i \(-0.555314\pi\)
−0.172901 + 0.984939i \(0.555314\pi\)
\(984\) 0 0
\(985\) 1.40016e7 0.459820
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.10323e6 0.100884
\(990\) 0 0
\(991\) 1.88230e6 0.0608843 0.0304422 − 0.999537i \(-0.490308\pi\)
0.0304422 + 0.999537i \(0.490308\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.56192e6 0.178101
\(996\) 0 0
\(997\) 2.71518e7 0.865090 0.432545 − 0.901612i \(-0.357616\pi\)
0.432545 + 0.901612i \(0.357616\pi\)
\(998\) 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 252.6.a.a.1.1 1
3.2 odd 2 28.6.a.b.1.1 ✓ 1
4.3 odd 2 1008.6.a.l.1.1 1
12.11 even 2 112.6.a.b.1.1 1
15.2 even 4 700.6.e.b.449.1 2
15.8 even 4 700.6.e.b.449.2 2
15.14 odd 2 700.6.a.b.1.1 1
21.2 odd 6 196.6.e.a.165.1 2
21.5 even 6 196.6.e.i.165.1 2
21.11 odd 6 196.6.e.a.177.1 2
21.17 even 6 196.6.e.i.177.1 2
21.20 even 2 196.6.a.a.1.1 1
24.5 odd 2 448.6.a.b.1.1 1
24.11 even 2 448.6.a.o.1.1 1
84.83 odd 2 784.6.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.6.a.b.1.1 ✓ 1 3.2 odd 2
112.6.a.b.1.1 1 12.11 even 2
196.6.a.a.1.1 1 21.20 even 2
196.6.e.a.165.1 2 21.2 odd 6
196.6.e.a.177.1 2 21.11 odd 6
196.6.e.i.165.1 2 21.5 even 6
196.6.e.i.177.1 2 21.17 even 6
252.6.a.a.1.1 1 1.1 even 1 trivial
448.6.a.b.1.1 1 24.5 odd 2
448.6.a.o.1.1 1 24.11 even 2
700.6.a.b.1.1 1 15.14 odd 2
700.6.e.b.449.1 2 15.2 even 4
700.6.e.b.449.2 2 15.8 even 4
784.6.a.m.1.1 1 84.83 odd 2
1008.6.a.l.1.1 1 4.3 odd 2