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Properties

Label 72.6.a.a.1.1
Level $72$
Weight $6$
Character 72.1
Self dual yes
Analytic conductor $11.548$
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] = [72,6,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, 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("72.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(11.5476350265\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-94.0000 q^{5} +144.000 q^{7} +O(q^{10})\) \(q-94.0000 q^{5} +144.000 q^{7} +380.000 q^{11} +814.000 q^{13} +862.000 q^{17} -1156.00 q^{19} +488.000 q^{23} +5711.00 q^{25} +5466.00 q^{29} +9560.00 q^{31} -13536.0 q^{35} -10506.0 q^{37} +5190.00 q^{41} -17084.0 q^{43} -3168.00 q^{47} +3929.00 q^{49} +24770.0 q^{53} -35720.0 q^{55} -17380.0 q^{59} +4366.00 q^{61} -76516.0 q^{65} -5284.00 q^{67} -8360.00 q^{71} +39466.0 q^{73} +54720.0 q^{77} +42376.0 q^{79} +61828.0 q^{83} -81028.0 q^{85} +63078.0 q^{89} +117216. q^{91} +108664. q^{95} -16318.0 q^{97} +O(q^{100})\)

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
\(5\) −94.0000 −1.68152 −0.840762 − 0.541406i \(-0.817892\pi\)
−0.840762 + 0.541406i \(0.817892\pi\)
\(6\) 0 0
\(7\) 144.000 1.11075 0.555376 − 0.831599i \(-0.312574\pi\)
0.555376 + 0.831599i \(0.312574\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 380.000 0.946895 0.473448 − 0.880822i \(-0.343009\pi\)
0.473448 + 0.880822i \(0.343009\pi\)
\(12\) 0 0
\(13\) 814.000 1.33588 0.667938 − 0.744217i \(-0.267177\pi\)
0.667938 + 0.744217i \(0.267177\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 862.000 0.723411 0.361705 − 0.932292i \(-0.382195\pi\)
0.361705 + 0.932292i \(0.382195\pi\)
\(18\) 0 0
\(19\) −1156.00 −0.734639 −0.367319 − 0.930095i \(-0.619724\pi\)
−0.367319 + 0.930095i \(0.619724\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 488.000 0.192354 0.0961768 − 0.995364i \(-0.469339\pi\)
0.0961768 + 0.995364i \(0.469339\pi\)
\(24\) 0 0
\(25\) 5711.00 1.82752
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5466.00 1.20691 0.603455 − 0.797397i \(-0.293790\pi\)
0.603455 + 0.797397i \(0.293790\pi\)
\(30\) 0 0
\(31\) 9560.00 1.78671 0.893354 − 0.449353i \(-0.148346\pi\)
0.893354 + 0.449353i \(0.148346\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −13536.0 −1.86776
\(36\) 0 0
\(37\) −10506.0 −1.26163 −0.630817 − 0.775932i \(-0.717280\pi\)
−0.630817 + 0.775932i \(0.717280\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5190.00 0.482178 0.241089 − 0.970503i \(-0.422495\pi\)
0.241089 + 0.970503i \(0.422495\pi\)
\(42\) 0 0
\(43\) −17084.0 −1.40902 −0.704512 − 0.709692i \(-0.748834\pi\)
−0.704512 + 0.709692i \(0.748834\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3168.00 −0.209190 −0.104595 − 0.994515i \(-0.533355\pi\)
−0.104595 + 0.994515i \(0.533355\pi\)
\(48\) 0 0
\(49\) 3929.00 0.233772
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 24770.0 1.21126 0.605629 − 0.795747i \(-0.292922\pi\)
0.605629 + 0.795747i \(0.292922\pi\)
\(54\) 0 0
\(55\) −35720.0 −1.59223
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −17380.0 −0.650010 −0.325005 − 0.945712i \(-0.605366\pi\)
−0.325005 + 0.945712i \(0.605366\pi\)
\(60\) 0 0
\(61\) 4366.00 0.150231 0.0751154 − 0.997175i \(-0.476067\pi\)
0.0751154 + 0.997175i \(0.476067\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −76516.0 −2.24631
\(66\) 0 0
\(67\) −5284.00 −0.143806 −0.0719028 − 0.997412i \(-0.522907\pi\)
−0.0719028 + 0.997412i \(0.522907\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8360.00 −0.196816 −0.0984080 − 0.995146i \(-0.531375\pi\)
−0.0984080 + 0.995146i \(0.531375\pi\)
\(72\) 0 0
\(73\) 39466.0 0.866794 0.433397 − 0.901203i \(-0.357315\pi\)
0.433397 + 0.901203i \(0.357315\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 54720.0 1.05177
\(78\) 0 0
\(79\) 42376.0 0.763928 0.381964 − 0.924177i \(-0.375248\pi\)
0.381964 + 0.924177i \(0.375248\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 61828.0 0.985122 0.492561 − 0.870278i \(-0.336061\pi\)
0.492561 + 0.870278i \(0.336061\pi\)
\(84\) 0 0
\(85\) −81028.0 −1.21643
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 63078.0 0.844117 0.422059 − 0.906568i \(-0.361308\pi\)
0.422059 + 0.906568i \(0.361308\pi\)
\(90\) 0 0
\(91\) 117216. 1.48383
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 108664. 1.23531
\(96\) 0 0
\(97\) −16318.0 −0.176091 −0.0880456 − 0.996116i \(-0.528062\pi\)
−0.0880456 + 0.996116i \(0.528062\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −134766. −1.31455 −0.657275 − 0.753651i \(-0.728291\pi\)
−0.657275 + 0.753651i \(0.728291\pi\)
\(102\) 0 0
\(103\) 45184.0 0.419654 0.209827 − 0.977739i \(-0.432710\pi\)
0.209827 + 0.977739i \(0.432710\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −143124. −1.20852 −0.604259 − 0.796788i \(-0.706531\pi\)
−0.604259 + 0.796788i \(0.706531\pi\)
\(108\) 0 0
\(109\) 153070. 1.23402 0.617012 − 0.786953i \(-0.288343\pi\)
0.617012 + 0.786953i \(0.288343\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 64494.0 0.475142 0.237571 − 0.971370i \(-0.423649\pi\)
0.237571 + 0.971370i \(0.423649\pi\)
\(114\) 0 0
\(115\) −45872.0 −0.323447
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 124128. 0.803530
\(120\) 0 0
\(121\) −16651.0 −0.103390
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −243084. −1.39149
\(126\) 0 0
\(127\) 33256.0 0.182962 0.0914810 − 0.995807i \(-0.470840\pi\)
0.0914810 + 0.995807i \(0.470840\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 149284. 0.760038 0.380019 − 0.924979i \(-0.375918\pi\)
0.380019 + 0.924979i \(0.375918\pi\)
\(132\) 0 0
\(133\) −166464. −0.816002
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −148602. −0.676431 −0.338215 − 0.941069i \(-0.609823\pi\)
−0.338215 + 0.941069i \(0.609823\pi\)
\(138\) 0 0
\(139\) −324012. −1.42241 −0.711204 − 0.702986i \(-0.751850\pi\)
−0.711204 + 0.702986i \(0.751850\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 309320. 1.26493
\(144\) 0 0
\(145\) −513804. −2.02945
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 59986.0 0.221352 0.110676 − 0.993857i \(-0.464698\pi\)
0.110676 + 0.993857i \(0.464698\pi\)
\(150\) 0 0
\(151\) 91520.0 0.326643 0.163322 − 0.986573i \(-0.447779\pi\)
0.163322 + 0.986573i \(0.447779\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −898640. −3.00439
\(156\) 0 0
\(157\) 345550. 1.11882 0.559412 − 0.828890i \(-0.311027\pi\)
0.559412 + 0.828890i \(0.311027\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 70272.0 0.213657
\(162\) 0 0
\(163\) −634164. −1.86953 −0.934765 − 0.355266i \(-0.884390\pi\)
−0.934765 + 0.355266i \(0.884390\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 415848. 1.15383 0.576917 − 0.816803i \(-0.304256\pi\)
0.576917 + 0.816803i \(0.304256\pi\)
\(168\) 0 0
\(169\) 291303. 0.784564
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 324282. 0.823773 0.411887 − 0.911235i \(-0.364870\pi\)
0.411887 + 0.911235i \(0.364870\pi\)
\(174\) 0 0
\(175\) 822384. 2.02992
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −606828. −1.41558 −0.707788 − 0.706425i \(-0.750307\pi\)
−0.707788 + 0.706425i \(0.750307\pi\)
\(180\) 0 0
\(181\) 463254. 1.05105 0.525524 − 0.850779i \(-0.323869\pi\)
0.525524 + 0.850779i \(0.323869\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 987564. 2.12147
\(186\) 0 0
\(187\) 327560. 0.684994
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −955008. −1.89419 −0.947095 − 0.320953i \(-0.895997\pi\)
−0.947095 + 0.320953i \(0.895997\pi\)
\(192\) 0 0
\(193\) −256126. −0.494949 −0.247474 − 0.968894i \(-0.579601\pi\)
−0.247474 + 0.968894i \(0.579601\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −111150. −0.204053 −0.102027 − 0.994782i \(-0.532533\pi\)
−0.102027 + 0.994782i \(0.532533\pi\)
\(198\) 0 0
\(199\) −516800. −0.925102 −0.462551 − 0.886593i \(-0.653066\pi\)
−0.462551 + 0.886593i \(0.653066\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 787104. 1.34058
\(204\) 0 0
\(205\) −487860. −0.810794
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −439280. −0.695626
\(210\) 0 0
\(211\) 400972. 0.620023 0.310012 − 0.950733i \(-0.399667\pi\)
0.310012 + 0.950733i \(0.399667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.60590e6 2.36931
\(216\) 0 0
\(217\) 1.37664e6 1.98459
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 701668. 0.966387
\(222\) 0 0
\(223\) −206168. −0.277625 −0.138813 − 0.990319i \(-0.544329\pi\)
−0.138813 + 0.990319i \(0.544329\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −363468. −0.468168 −0.234084 − 0.972216i \(-0.575209\pi\)
−0.234084 + 0.972216i \(0.575209\pi\)
\(228\) 0 0
\(229\) −589274. −0.742555 −0.371277 − 0.928522i \(-0.621080\pi\)
−0.371277 + 0.928522i \(0.621080\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −730090. −0.881022 −0.440511 − 0.897747i \(-0.645203\pi\)
−0.440511 + 0.897747i \(0.645203\pi\)
\(234\) 0 0
\(235\) 297792. 0.351758
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −472496. −0.535061 −0.267531 − 0.963549i \(-0.586208\pi\)
−0.267531 + 0.963549i \(0.586208\pi\)
\(240\) 0 0
\(241\) 993042. 1.10135 0.550675 − 0.834720i \(-0.314371\pi\)
0.550675 + 0.834720i \(0.314371\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −369326. −0.393092
\(246\) 0 0
\(247\) −940984. −0.981386
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.93653e6 −1.94017 −0.970086 − 0.242760i \(-0.921947\pi\)
−0.970086 + 0.242760i \(0.921947\pi\)
\(252\) 0 0
\(253\) 185440. 0.182139
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −735746. −0.694856 −0.347428 − 0.937707i \(-0.612945\pi\)
−0.347428 + 0.937707i \(0.612945\pi\)
\(258\) 0 0
\(259\) −1.51286e6 −1.40136
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 542984. 0.484058 0.242029 − 0.970269i \(-0.422187\pi\)
0.242029 + 0.970269i \(0.422187\pi\)
\(264\) 0 0
\(265\) −2.32838e6 −2.03676
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −438614. −0.369574 −0.184787 − 0.982779i \(-0.559160\pi\)
−0.184787 + 0.982779i \(0.559160\pi\)
\(270\) 0 0
\(271\) 644408. 0.533013 0.266506 − 0.963833i \(-0.414131\pi\)
0.266506 + 0.963833i \(0.414131\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.17018e6 1.73047
\(276\) 0 0
\(277\) −1.41953e6 −1.11159 −0.555796 − 0.831319i \(-0.687586\pi\)
−0.555796 + 0.831319i \(0.687586\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −645242. −0.487480 −0.243740 − 0.969841i \(-0.578374\pi\)
−0.243740 + 0.969841i \(0.578374\pi\)
\(282\) 0 0
\(283\) 1.93696e6 1.43766 0.718829 − 0.695187i \(-0.244679\pi\)
0.718829 + 0.695187i \(0.244679\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 747360. 0.535581
\(288\) 0 0
\(289\) −676813. −0.476677
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.65429e6 1.12575 0.562876 − 0.826541i \(-0.309695\pi\)
0.562876 + 0.826541i \(0.309695\pi\)
\(294\) 0 0
\(295\) 1.63372e6 1.09301
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 397232. 0.256960
\(300\) 0 0
\(301\) −2.46010e6 −1.56508
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −410404. −0.252617
\(306\) 0 0
\(307\) −2.78236e6 −1.68487 −0.842436 − 0.538797i \(-0.818879\pi\)
−0.842436 + 0.538797i \(0.818879\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00510e6 −0.589259 −0.294630 − 0.955611i \(-0.595196\pi\)
−0.294630 + 0.955611i \(0.595196\pi\)
\(312\) 0 0
\(313\) 535386. 0.308892 0.154446 − 0.988001i \(-0.450641\pi\)
0.154446 + 0.988001i \(0.450641\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.96266e6 −1.65590 −0.827950 − 0.560802i \(-0.810493\pi\)
−0.827950 + 0.560802i \(0.810493\pi\)
\(318\) 0 0
\(319\) 2.07708e6 1.14282
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −996472. −0.531446
\(324\) 0 0
\(325\) 4.64875e6 2.44134
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −456192. −0.232358
\(330\) 0 0
\(331\) 1.55935e6 0.782300 0.391150 − 0.920327i \(-0.372077\pi\)
0.391150 + 0.920327i \(0.372077\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 496696. 0.241812
\(336\) 0 0
\(337\) 3.61557e6 1.73421 0.867106 − 0.498124i \(-0.165978\pi\)
0.867106 + 0.498124i \(0.165978\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.63280e6 1.69183
\(342\) 0 0
\(343\) −1.85443e6 −0.851090
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.76488e6 0.786847 0.393424 − 0.919357i \(-0.371291\pi\)
0.393424 + 0.919357i \(0.371291\pi\)
\(348\) 0 0
\(349\) 553134. 0.243090 0.121545 − 0.992586i \(-0.461215\pi\)
0.121545 + 0.992586i \(0.461215\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.17408e6 0.501488 0.250744 − 0.968053i \(-0.419325\pi\)
0.250744 + 0.968053i \(0.419325\pi\)
\(354\) 0 0
\(355\) 785840. 0.330951
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.18975e6 0.487215 0.243607 − 0.969874i \(-0.421669\pi\)
0.243607 + 0.969874i \(0.421669\pi\)
\(360\) 0 0
\(361\) −1.13976e6 −0.460306
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.70980e6 −1.45753
\(366\) 0 0
\(367\) −1.35327e6 −0.524469 −0.262235 − 0.965004i \(-0.584459\pi\)
−0.262235 + 0.965004i \(0.584459\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.56688e6 1.34541
\(372\) 0 0
\(373\) 2.79306e6 1.03946 0.519731 − 0.854330i \(-0.326032\pi\)
0.519731 + 0.854330i \(0.326032\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.44932e6 1.61228
\(378\) 0 0
\(379\) −3.37304e6 −1.20621 −0.603105 − 0.797662i \(-0.706070\pi\)
−0.603105 + 0.797662i \(0.706070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.72781e6 −1.64688 −0.823442 − 0.567401i \(-0.807949\pi\)
−0.823442 + 0.567401i \(0.807949\pi\)
\(384\) 0 0
\(385\) −5.14368e6 −1.76857
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −188254. −0.0630769 −0.0315384 − 0.999503i \(-0.510041\pi\)
−0.0315384 + 0.999503i \(0.510041\pi\)
\(390\) 0 0
\(391\) 420656. 0.139151
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.98334e6 −1.28456
\(396\) 0 0
\(397\) −524578. −0.167045 −0.0835226 − 0.996506i \(-0.526617\pi\)
−0.0835226 + 0.996506i \(0.526617\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.08653e6 0.958537 0.479269 − 0.877668i \(-0.340902\pi\)
0.479269 + 0.877668i \(0.340902\pi\)
\(402\) 0 0
\(403\) 7.78184e6 2.38682
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.99228e6 −1.19463
\(408\) 0 0
\(409\) −4.60196e6 −1.36030 −0.680150 − 0.733073i \(-0.738085\pi\)
−0.680150 + 0.733073i \(0.738085\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.50272e6 −0.722000
\(414\) 0 0
\(415\) −5.81183e6 −1.65651
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.09492e6 −0.582953 −0.291476 − 0.956578i \(-0.594146\pi\)
−0.291476 + 0.956578i \(0.594146\pi\)
\(420\) 0 0
\(421\) 987206. 0.271458 0.135729 − 0.990746i \(-0.456662\pi\)
0.135729 + 0.990746i \(0.456662\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.92288e6 1.32205
\(426\) 0 0
\(427\) 628704. 0.166869
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.25933e6 −0.845152 −0.422576 − 0.906327i \(-0.638874\pi\)
−0.422576 + 0.906327i \(0.638874\pi\)
\(432\) 0 0
\(433\) 7.31461e6 1.87487 0.937436 − 0.348159i \(-0.113193\pi\)
0.937436 + 0.348159i \(0.113193\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −564128. −0.141310
\(438\) 0 0
\(439\) −7.31419e6 −1.81136 −0.905681 − 0.423961i \(-0.860639\pi\)
−0.905681 + 0.423961i \(0.860639\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.07403e6 1.71261 0.856303 − 0.516474i \(-0.172756\pi\)
0.856303 + 0.516474i \(0.172756\pi\)
\(444\) 0 0
\(445\) −5.92933e6 −1.41940
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.48079e6 −1.75118 −0.875591 − 0.483053i \(-0.839528\pi\)
−0.875591 + 0.483053i \(0.839528\pi\)
\(450\) 0 0
\(451\) 1.97220e6 0.456572
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.10183e7 −2.49509
\(456\) 0 0
\(457\) 2.97108e6 0.665463 0.332732 − 0.943022i \(-0.392030\pi\)
0.332732 + 0.943022i \(0.392030\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.29078e6 1.81695 0.908475 − 0.417939i \(-0.137247\pi\)
0.908475 + 0.417939i \(0.137247\pi\)
\(462\) 0 0
\(463\) 955064. 0.207052 0.103526 − 0.994627i \(-0.466987\pi\)
0.103526 + 0.994627i \(0.466987\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.19018e6 1.73781 0.868903 − 0.494983i \(-0.164826\pi\)
0.868903 + 0.494983i \(0.164826\pi\)
\(468\) 0 0
\(469\) −760896. −0.159732
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.49192e6 −1.33420
\(474\) 0 0
\(475\) −6.60192e6 −1.34257
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.39774e6 0.676631 0.338315 − 0.941033i \(-0.390143\pi\)
0.338315 + 0.941033i \(0.390143\pi\)
\(480\) 0 0
\(481\) −8.55188e6 −1.68538
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.53389e6 0.296101
\(486\) 0 0
\(487\) 1.94814e6 0.372219 0.186110 − 0.982529i \(-0.440412\pi\)
0.186110 + 0.982529i \(0.440412\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.92574e6 0.734882 0.367441 − 0.930047i \(-0.380234\pi\)
0.367441 + 0.930047i \(0.380234\pi\)
\(492\) 0 0
\(493\) 4.71169e6 0.873091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.20384e6 −0.218614
\(498\) 0 0
\(499\) 2.07102e6 0.372334 0.186167 − 0.982518i \(-0.440393\pi\)
0.186167 + 0.982518i \(0.440393\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.16500e6 1.61515 0.807574 − 0.589766i \(-0.200780\pi\)
0.807574 + 0.589766i \(0.200780\pi\)
\(504\) 0 0
\(505\) 1.26680e7 2.21045
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.91517e6 −1.18307 −0.591533 − 0.806281i \(-0.701477\pi\)
−0.591533 + 0.806281i \(0.701477\pi\)
\(510\) 0 0
\(511\) 5.68310e6 0.962794
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.24730e6 −0.705658
\(516\) 0 0
\(517\) −1.20384e6 −0.198081
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.26077e6 0.364891 0.182445 − 0.983216i \(-0.441599\pi\)
0.182445 + 0.983216i \(0.441599\pi\)
\(522\) 0 0
\(523\) 8.25524e6 1.31970 0.659850 − 0.751397i \(-0.270620\pi\)
0.659850 + 0.751397i \(0.270620\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.24072e6 1.29252
\(528\) 0 0
\(529\) −6.19820e6 −0.963000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.22466e6 0.644130
\(534\) 0 0
\(535\) 1.34537e7 2.03215
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.49302e6 0.221357
\(540\) 0 0
\(541\) 1.90467e6 0.279786 0.139893 − 0.990167i \(-0.455324\pi\)
0.139893 + 0.990167i \(0.455324\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.43886e7 −2.07504
\(546\) 0 0
\(547\) 1.57060e6 0.224439 0.112220 − 0.993683i \(-0.464204\pi\)
0.112220 + 0.993683i \(0.464204\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.31870e6 −0.886642
\(552\) 0 0
\(553\) 6.10214e6 0.848535
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.36849e6 −0.323469 −0.161735 − 0.986834i \(-0.551709\pi\)
−0.161735 + 0.986834i \(0.551709\pi\)
\(558\) 0 0
\(559\) −1.39064e7 −1.88228
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.64976e6 −0.884168 −0.442084 − 0.896974i \(-0.645761\pi\)
−0.442084 + 0.896974i \(0.645761\pi\)
\(564\) 0 0
\(565\) −6.06244e6 −0.798962
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.94565e6 1.15833 0.579164 − 0.815211i \(-0.303379\pi\)
0.579164 + 0.815211i \(0.303379\pi\)
\(570\) 0 0
\(571\) −4.19526e6 −0.538479 −0.269239 − 0.963073i \(-0.586772\pi\)
−0.269239 + 0.963073i \(0.586772\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.78697e6 0.351530
\(576\) 0 0
\(577\) −6.24845e6 −0.781326 −0.390663 − 0.920534i \(-0.627754\pi\)
−0.390663 + 0.920534i \(0.627754\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.90323e6 1.09423
\(582\) 0 0
\(583\) 9.41260e6 1.14693
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.55284e6 0.545365 0.272683 − 0.962104i \(-0.412089\pi\)
0.272683 + 0.962104i \(0.412089\pi\)
\(588\) 0 0
\(589\) −1.10514e7 −1.31259
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −210802. −0.0246172 −0.0123086 − 0.999924i \(-0.503918\pi\)
−0.0123086 + 0.999924i \(0.503918\pi\)
\(594\) 0 0
\(595\) −1.16680e7 −1.35116
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.05945e6 −0.348398 −0.174199 − 0.984710i \(-0.555734\pi\)
−0.174199 + 0.984710i \(0.555734\pi\)
\(600\) 0 0
\(601\) 7.37094e6 0.832409 0.416204 − 0.909271i \(-0.363360\pi\)
0.416204 + 0.909271i \(0.363360\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.56519e6 0.173852
\(606\) 0 0
\(607\) 2.12257e6 0.233824 0.116912 − 0.993142i \(-0.462700\pi\)
0.116912 + 0.993142i \(0.462700\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.57875e6 −0.279452
\(612\) 0 0
\(613\) −1.09689e7 −1.17899 −0.589496 − 0.807771i \(-0.700674\pi\)
−0.589496 + 0.807771i \(0.700674\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.28953e6 0.770879 0.385440 − 0.922733i \(-0.374050\pi\)
0.385440 + 0.922733i \(0.374050\pi\)
\(618\) 0 0
\(619\) −2.23675e6 −0.234634 −0.117317 − 0.993095i \(-0.537429\pi\)
−0.117317 + 0.993095i \(0.537429\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.08323e6 0.937606
\(624\) 0 0
\(625\) 5.00302e6 0.512309
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.05617e6 −0.912679
\(630\) 0 0
\(631\) −7.06160e6 −0.706041 −0.353020 − 0.935616i \(-0.614845\pi\)
−0.353020 + 0.935616i \(0.614845\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.12606e6 −0.307655
\(636\) 0 0
\(637\) 3.19821e6 0.312290
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.80189e7 −1.73214 −0.866070 − 0.499922i \(-0.833362\pi\)
−0.866070 + 0.499922i \(0.833362\pi\)
\(642\) 0 0
\(643\) 1.07252e6 0.102301 0.0511505 − 0.998691i \(-0.483711\pi\)
0.0511505 + 0.998691i \(0.483711\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.03054e7 −1.90701 −0.953503 − 0.301385i \(-0.902551\pi\)
−0.953503 + 0.301385i \(0.902551\pi\)
\(648\) 0 0
\(649\) −6.60440e6 −0.615491
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.97292e7 −1.81061 −0.905306 − 0.424759i \(-0.860359\pi\)
−0.905306 + 0.424759i \(0.860359\pi\)
\(654\) 0 0
\(655\) −1.40327e7 −1.27802
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.29511e7 −1.16170 −0.580848 − 0.814012i \(-0.697279\pi\)
−0.580848 + 0.814012i \(0.697279\pi\)
\(660\) 0 0
\(661\) 2.01888e7 1.79725 0.898623 − 0.438721i \(-0.144568\pi\)
0.898623 + 0.438721i \(0.144568\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.56476e7 1.37213
\(666\) 0 0
\(667\) 2.66741e6 0.232153
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.65908e6 0.142253
\(672\) 0 0
\(673\) −1.83989e7 −1.56587 −0.782934 − 0.622105i \(-0.786278\pi\)
−0.782934 + 0.622105i \(0.786278\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.72791e6 0.396458 0.198229 − 0.980156i \(-0.436481\pi\)
0.198229 + 0.980156i \(0.436481\pi\)
\(678\) 0 0
\(679\) −2.34979e6 −0.195594
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.38084e7 1.13264 0.566321 − 0.824184i \(-0.308366\pi\)
0.566321 + 0.824184i \(0.308366\pi\)
\(684\) 0 0
\(685\) 1.39686e7 1.13743
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.01628e7 1.61809
\(690\) 0 0
\(691\) 1.28964e7 1.02748 0.513742 − 0.857945i \(-0.328259\pi\)
0.513742 + 0.857945i \(0.328259\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.04571e7 2.39181
\(696\) 0 0
\(697\) 4.47378e6 0.348813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.81078e6 0.677204 0.338602 − 0.940930i \(-0.390046\pi\)
0.338602 + 0.940930i \(0.390046\pi\)
\(702\) 0 0
\(703\) 1.21449e7 0.926845
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.94063e7 −1.46014
\(708\) 0 0
\(709\) 1.83369e7 1.36997 0.684984 − 0.728558i \(-0.259809\pi\)
0.684984 + 0.728558i \(0.259809\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.66528e6 0.343680
\(714\) 0 0
\(715\) −2.90761e7 −2.12702
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.26238e7 −1.63209 −0.816044 − 0.577990i \(-0.803837\pi\)
−0.816044 + 0.577990i \(0.803837\pi\)
\(720\) 0 0
\(721\) 6.50650e6 0.466132
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.12163e7 2.20565
\(726\) 0 0
\(727\) −1.25658e7 −0.881769 −0.440885 − 0.897564i \(-0.645335\pi\)
−0.440885 + 0.897564i \(0.645335\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.47264e7 −1.01930
\(732\) 0 0
\(733\) 833534. 0.0573012 0.0286506 − 0.999589i \(-0.490879\pi\)
0.0286506 + 0.999589i \(0.490879\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.00792e6 −0.136169
\(738\) 0 0
\(739\) 8.84188e6 0.595571 0.297786 − 0.954633i \(-0.403752\pi\)
0.297786 + 0.954633i \(0.403752\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.62140e7 1.74205 0.871025 − 0.491239i \(-0.163456\pi\)
0.871025 + 0.491239i \(0.163456\pi\)
\(744\) 0 0
\(745\) −5.63868e6 −0.372209
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.06099e7 −1.34236
\(750\) 0 0
\(751\) 6.17143e6 0.399288 0.199644 − 0.979869i \(-0.436021\pi\)
0.199644 + 0.979869i \(0.436021\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.60288e6 −0.549258
\(756\) 0 0
\(757\) −6.70734e6 −0.425413 −0.212706 − 0.977116i \(-0.568228\pi\)
−0.212706 + 0.977116i \(0.568228\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.08114e7 0.676734 0.338367 − 0.941014i \(-0.390125\pi\)
0.338367 + 0.941014i \(0.390125\pi\)
\(762\) 0 0
\(763\) 2.20421e7 1.37070
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.41473e7 −0.868332
\(768\) 0 0
\(769\) −1.33048e7 −0.811322 −0.405661 − 0.914024i \(-0.632959\pi\)
−0.405661 + 0.914024i \(0.632959\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.29053e7 0.776818 0.388409 − 0.921487i \(-0.373025\pi\)
0.388409 + 0.921487i \(0.373025\pi\)
\(774\) 0 0
\(775\) 5.45972e7 3.26525
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.99964e6 −0.354227
\(780\) 0 0
\(781\) −3.17680e6 −0.186364
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.24817e7 −1.88133
\(786\) 0 0
\(787\) 6.77705e6 0.390035 0.195018 − 0.980800i \(-0.437524\pi\)
0.195018 + 0.980800i \(0.437524\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.28714e6 0.527765
\(792\) 0 0
\(793\) 3.55392e6 0.200690
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.49551e7 −0.833956 −0.416978 − 0.908916i \(-0.636911\pi\)
−0.416978 + 0.908916i \(0.636911\pi\)
\(798\) 0 0
\(799\) −2.73082e6 −0.151330
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.49971e7 0.820763
\(804\) 0 0
\(805\) −6.60557e6 −0.359270
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.65841e6 0.465122 0.232561 − 0.972582i \(-0.425290\pi\)
0.232561 + 0.972582i \(0.425290\pi\)
\(810\) 0 0
\(811\) −1.98323e7 −1.05881 −0.529407 − 0.848368i \(-0.677586\pi\)
−0.529407 + 0.848368i \(0.677586\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.96114e7 3.14366
\(816\) 0 0
\(817\) 1.97491e7 1.03512
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.02228e7 −1.56486 −0.782431 − 0.622737i \(-0.786021\pi\)
−0.782431 + 0.622737i \(0.786021\pi\)
\(822\) 0 0
\(823\) −2.64692e7 −1.36220 −0.681100 − 0.732190i \(-0.738498\pi\)
−0.681100 + 0.732190i \(0.738498\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.81882e6 −0.448380 −0.224190 − 0.974545i \(-0.571974\pi\)
−0.224190 + 0.974545i \(0.571974\pi\)
\(828\) 0 0
\(829\) −2.31053e7 −1.16768 −0.583841 − 0.811868i \(-0.698451\pi\)
−0.583841 + 0.811868i \(0.698451\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.38680e6 0.169113
\(834\) 0 0
\(835\) −3.90897e7 −1.94020
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.65084e7 −1.30010 −0.650052 − 0.759890i \(-0.725253\pi\)
−0.650052 + 0.759890i \(0.725253\pi\)
\(840\) 0 0
\(841\) 9.36601e6 0.456630
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.73825e7 −1.31926
\(846\) 0 0
\(847\) −2.39774e6 −0.114840
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.12693e6 −0.242680
\(852\) 0 0
\(853\) −7.84201e6 −0.369024 −0.184512 − 0.982830i \(-0.559070\pi\)
−0.184512 + 0.982830i \(0.559070\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.26563e7 1.05375 0.526875 − 0.849943i \(-0.323364\pi\)
0.526875 + 0.849943i \(0.323364\pi\)
\(858\) 0 0
\(859\) −2.67391e7 −1.23641 −0.618207 − 0.786016i \(-0.712140\pi\)
−0.618207 + 0.786016i \(0.712140\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.29957e7 0.593983 0.296992 − 0.954880i \(-0.404017\pi\)
0.296992 + 0.954880i \(0.404017\pi\)
\(864\) 0 0
\(865\) −3.04825e7 −1.38519
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.61029e7 0.723359
\(870\) 0 0
\(871\) −4.30118e6 −0.192106
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.50041e7 −1.54561
\(876\) 0 0
\(877\) −3.16845e6 −0.139107 −0.0695533 − 0.997578i \(-0.522157\pi\)
−0.0695533 + 0.997578i \(0.522157\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.85811e7 −1.67469 −0.837346 − 0.546673i \(-0.815894\pi\)
−0.837346 + 0.546673i \(0.815894\pi\)
\(882\) 0 0
\(883\) −2.31224e7 −0.998000 −0.499000 − 0.866602i \(-0.666299\pi\)
−0.499000 + 0.866602i \(0.666299\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.62602e7 1.12070 0.560350 − 0.828256i \(-0.310667\pi\)
0.560350 + 0.828256i \(0.310667\pi\)
\(888\) 0 0
\(889\) 4.78886e6 0.203225
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.66221e6 0.153679
\(894\) 0 0
\(895\) 5.70418e7 2.38032
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.22550e7 2.15639
\(900\) 0 0
\(901\) 2.13517e7 0.876236
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.35459e7 −1.76736
\(906\) 0 0
\(907\) 3.13213e7 1.26422 0.632109 − 0.774879i \(-0.282189\pi\)
0.632109 + 0.774879i \(0.282189\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.49877e6 −0.179596 −0.0897982 − 0.995960i \(-0.528622\pi\)
−0.0897982 + 0.995960i \(0.528622\pi\)
\(912\) 0 0
\(913\) 2.34946e7 0.932807
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.14969e7 0.844214
\(918\) 0 0
\(919\) −3.03824e6 −0.118668 −0.0593340 − 0.998238i \(-0.518898\pi\)
−0.0593340 + 0.998238i \(0.518898\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.80504e6 −0.262922
\(924\) 0 0
\(925\) −5.99998e7 −2.30566
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 193966. 0.00737371 0.00368686 − 0.999993i \(-0.498826\pi\)
0.00368686 + 0.999993i \(0.498826\pi\)
\(930\) 0 0
\(931\) −4.54192e6 −0.171738
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.07906e7 −1.15183
\(936\) 0 0
\(937\) −3.23213e7 −1.20265 −0.601326 − 0.799003i \(-0.705361\pi\)
−0.601326 + 0.799003i \(0.705361\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.50288e7 1.65774 0.828870 − 0.559442i \(-0.188984\pi\)
0.828870 + 0.559442i \(0.188984\pi\)
\(942\) 0 0
\(943\) 2.53272e6 0.0927487
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.04495e7 −0.378635 −0.189318 − 0.981916i \(-0.560628\pi\)
−0.189318 + 0.981916i \(0.560628\pi\)
\(948\) 0 0
\(949\) 3.21253e7 1.15793
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4.31343e7 −1.53847 −0.769237 − 0.638963i \(-0.779364\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(954\) 0 0
\(955\) 8.97708e7 3.18512
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.13987e7 −0.751347
\(960\) 0 0
\(961\) 6.27644e7 2.19233
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.40758e7 0.832268
\(966\) 0 0
\(967\) −1.43982e7 −0.495157 −0.247579 − 0.968868i \(-0.579635\pi\)
−0.247579 + 0.968868i \(0.579635\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.00272e6 −0.102204 −0.0511019 − 0.998693i \(-0.516273\pi\)
−0.0511019 + 0.998693i \(0.516273\pi\)
\(972\) 0 0
\(973\) −4.66577e7 −1.57994
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.75164e7 1.25743 0.628716 − 0.777635i \(-0.283581\pi\)
0.628716 + 0.777635i \(0.283581\pi\)
\(978\) 0 0
\(979\) 2.39696e7 0.799291
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.57244e6 0.0849105 0.0424553 − 0.999098i \(-0.486482\pi\)
0.0424553 + 0.999098i \(0.486482\pi\)
\(984\) 0 0
\(985\) 1.04481e7 0.343121
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −8.33699e6 −0.271031
\(990\) 0 0
\(991\) −5.41887e7 −1.75277 −0.876385 − 0.481611i \(-0.840052\pi\)
−0.876385 + 0.481611i \(0.840052\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.85792e7 1.55558
\(996\) 0 0
\(997\) 3.52674e7 1.12366 0.561830 − 0.827253i \(-0.310097\pi\)
0.561830 + 0.827253i \(0.310097\pi\)
\(998\) 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 72.6.a.a.1.1 1
3.2 odd 2 24.6.a.b.1.1 ✓ 1
4.3 odd 2 144.6.a.b.1.1 1
8.3 odd 2 576.6.a.bf.1.1 1
8.5 even 2 576.6.a.bg.1.1 1
12.11 even 2 48.6.a.e.1.1 1
15.2 even 4 600.6.f.b.49.2 2
15.8 even 4 600.6.f.b.49.1 2
15.14 odd 2 600.6.a.d.1.1 1
24.5 odd 2 192.6.a.i.1.1 1
24.11 even 2 192.6.a.a.1.1 1
48.5 odd 4 768.6.d.d.385.2 2
48.11 even 4 768.6.d.o.385.1 2
48.29 odd 4 768.6.d.d.385.1 2
48.35 even 4 768.6.d.o.385.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.6.a.b.1.1 ✓ 1 3.2 odd 2
48.6.a.e.1.1 1 12.11 even 2
72.6.a.a.1.1 1 1.1 even 1 trivial
144.6.a.b.1.1 1 4.3 odd 2
192.6.a.a.1.1 1 24.11 even 2
192.6.a.i.1.1 1 24.5 odd 2
576.6.a.bf.1.1 1 8.3 odd 2
576.6.a.bg.1.1 1 8.5 even 2
600.6.a.d.1.1 1 15.14 odd 2
600.6.f.b.49.1 2 15.8 even 4
600.6.f.b.49.2 2 15.2 even 4
768.6.d.d.385.1 2 48.29 odd 4
768.6.d.d.385.2 2 48.5 odd 4
768.6.d.o.385.1 2 48.11 even 4
768.6.d.o.385.2 2 48.35 even 4