Properties

Label 225.6.a.c.1.1
Level $225$
Weight $6$
Character 225.1
Self dual yes
Analytic conductor $36.086$
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] = [225,6,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("225.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -28.0000 q^{4} +132.000 q^{7} +120.000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} -28.0000 q^{4} +132.000 q^{7} +120.000 q^{8} -472.000 q^{11} +686.000 q^{13} -264.000 q^{14} +656.000 q^{16} -1562.00 q^{17} -2180.00 q^{19} +944.000 q^{22} +264.000 q^{23} -1372.00 q^{26} -3696.00 q^{28} -170.000 q^{29} +7272.00 q^{31} -5152.00 q^{32} +3124.00 q^{34} +142.000 q^{37} +4360.00 q^{38} +16198.0 q^{41} +10316.0 q^{43} +13216.0 q^{44} -528.000 q^{46} +18568.0 q^{47} +617.000 q^{49} -19208.0 q^{52} +21514.0 q^{53} +15840.0 q^{56} +340.000 q^{58} -34600.0 q^{59} -35738.0 q^{61} -14544.0 q^{62} -10688.0 q^{64} +5772.00 q^{67} +43736.0 q^{68} +69088.0 q^{71} +70526.0 q^{73} -284.000 q^{74} +61040.0 q^{76} -62304.0 q^{77} +47640.0 q^{79} -32396.0 q^{82} +74004.0 q^{83} -20632.0 q^{86} -56640.0 q^{88} +90030.0 q^{89} +90552.0 q^{91} -7392.00 q^{92} -37136.0 q^{94} +33502.0 q^{97} -1234.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −28.0000 −0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 132.000 1.01819 0.509095 0.860710i \(-0.329980\pi\)
0.509095 + 0.860710i \(0.329980\pi\)
\(8\) 120.000 0.662913
\(9\) 0 0
\(10\) 0 0
\(11\) −472.000 −1.17614 −0.588072 0.808809i \(-0.700113\pi\)
−0.588072 + 0.808809i \(0.700113\pi\)
\(12\) 0 0
\(13\) 686.000 1.12581 0.562906 0.826521i \(-0.309683\pi\)
0.562906 + 0.826521i \(0.309683\pi\)
\(14\) −264.000 −0.359985
\(15\) 0 0
\(16\) 656.000 0.640625
\(17\) −1562.00 −1.31087 −0.655434 0.755253i \(-0.727514\pi\)
−0.655434 + 0.755253i \(0.727514\pi\)
\(18\) 0 0
\(19\) −2180.00 −1.38539 −0.692696 0.721230i \(-0.743577\pi\)
−0.692696 + 0.721230i \(0.743577\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 944.000 0.415829
\(23\) 264.000 0.104060 0.0520301 0.998646i \(-0.483431\pi\)
0.0520301 + 0.998646i \(0.483431\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1372.00 −0.398035
\(27\) 0 0
\(28\) −3696.00 −0.890916
\(29\) −170.000 −0.0375365 −0.0187683 0.999824i \(-0.505974\pi\)
−0.0187683 + 0.999824i \(0.505974\pi\)
\(30\) 0 0
\(31\) 7272.00 1.35909 0.679547 0.733632i \(-0.262176\pi\)
0.679547 + 0.733632i \(0.262176\pi\)
\(32\) −5152.00 −0.889408
\(33\) 0 0
\(34\) 3124.00 0.463462
\(35\) 0 0
\(36\) 0 0
\(37\) 142.000 0.0170523 0.00852617 0.999964i \(-0.497286\pi\)
0.00852617 + 0.999964i \(0.497286\pi\)
\(38\) 4360.00 0.489810
\(39\) 0 0
\(40\) 0 0
\(41\) 16198.0 1.50488 0.752440 0.658661i \(-0.228877\pi\)
0.752440 + 0.658661i \(0.228877\pi\)
\(42\) 0 0
\(43\) 10316.0 0.850825 0.425412 0.905000i \(-0.360129\pi\)
0.425412 + 0.905000i \(0.360129\pi\)
\(44\) 13216.0 1.02913
\(45\) 0 0
\(46\) −528.000 −0.0367908
\(47\) 18568.0 1.22608 0.613042 0.790050i \(-0.289945\pi\)
0.613042 + 0.790050i \(0.289945\pi\)
\(48\) 0 0
\(49\) 617.000 0.0367109
\(50\) 0 0
\(51\) 0 0
\(52\) −19208.0 −0.985085
\(53\) 21514.0 1.05204 0.526019 0.850473i \(-0.323684\pi\)
0.526019 + 0.850473i \(0.323684\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 15840.0 0.674971
\(57\) 0 0
\(58\) 340.000 0.0132712
\(59\) −34600.0 −1.29404 −0.647018 0.762475i \(-0.723984\pi\)
−0.647018 + 0.762475i \(0.723984\pi\)
\(60\) 0 0
\(61\) −35738.0 −1.22972 −0.614859 0.788637i \(-0.710787\pi\)
−0.614859 + 0.788637i \(0.710787\pi\)
\(62\) −14544.0 −0.480512
\(63\) 0 0
\(64\) −10688.0 −0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 5772.00 0.157087 0.0785433 0.996911i \(-0.474973\pi\)
0.0785433 + 0.996911i \(0.474973\pi\)
\(68\) 43736.0 1.14701
\(69\) 0 0
\(70\) 0 0
\(71\) 69088.0 1.62651 0.813255 0.581907i \(-0.197693\pi\)
0.813255 + 0.581907i \(0.197693\pi\)
\(72\) 0 0
\(73\) 70526.0 1.54897 0.774483 0.632594i \(-0.218010\pi\)
0.774483 + 0.632594i \(0.218010\pi\)
\(74\) −284.000 −0.00602891
\(75\) 0 0
\(76\) 61040.0 1.21222
\(77\) −62304.0 −1.19754
\(78\) 0 0
\(79\) 47640.0 0.858824 0.429412 0.903109i \(-0.358721\pi\)
0.429412 + 0.903109i \(0.358721\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −32396.0 −0.532055
\(83\) 74004.0 1.17913 0.589563 0.807723i \(-0.299300\pi\)
0.589563 + 0.807723i \(0.299300\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −20632.0 −0.300812
\(87\) 0 0
\(88\) −56640.0 −0.779680
\(89\) 90030.0 1.20479 0.602396 0.798197i \(-0.294213\pi\)
0.602396 + 0.798197i \(0.294213\pi\)
\(90\) 0 0
\(91\) 90552.0 1.14629
\(92\) −7392.00 −0.0910526
\(93\) 0 0
\(94\) −37136.0 −0.433486
\(95\) 0 0
\(96\) 0 0
\(97\) 33502.0 0.361528 0.180764 0.983527i \(-0.442143\pi\)
0.180764 + 0.983527i \(0.442143\pi\)
\(98\) −1234.00 −0.0129793
\(99\) 0 0
\(100\) 0 0
\(101\) −78882.0 −0.769440 −0.384720 0.923033i \(-0.625702\pi\)
−0.384720 + 0.923033i \(0.625702\pi\)
\(102\) 0 0
\(103\) 82036.0 0.761924 0.380962 0.924591i \(-0.375593\pi\)
0.380962 + 0.924591i \(0.375593\pi\)
\(104\) 82320.0 0.746315
\(105\) 0 0
\(106\) −43028.0 −0.371952
\(107\) −41652.0 −0.351703 −0.175852 0.984417i \(-0.556268\pi\)
−0.175852 + 0.984417i \(0.556268\pi\)
\(108\) 0 0
\(109\) 104870. 0.845444 0.422722 0.906259i \(-0.361075\pi\)
0.422722 + 0.906259i \(0.361075\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 86592.0 0.652278
\(113\) −153746. −1.13268 −0.566341 0.824171i \(-0.691641\pi\)
−0.566341 + 0.824171i \(0.691641\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4760.00 0.0328444
\(117\) 0 0
\(118\) 69200.0 0.457511
\(119\) −206184. −1.33471
\(120\) 0 0
\(121\) 61733.0 0.383313
\(122\) 71476.0 0.434771
\(123\) 0 0
\(124\) −203616. −1.18921
\(125\) 0 0
\(126\) 0 0
\(127\) 59372.0 0.326642 0.163321 0.986573i \(-0.447779\pi\)
0.163321 + 0.986573i \(0.447779\pi\)
\(128\) 186240. 1.00473
\(129\) 0 0
\(130\) 0 0
\(131\) 98808.0 0.503053 0.251527 0.967850i \(-0.419067\pi\)
0.251527 + 0.967850i \(0.419067\pi\)
\(132\) 0 0
\(133\) −287760. −1.41059
\(134\) −11544.0 −0.0555385
\(135\) 0 0
\(136\) −187440. −0.868990
\(137\) 306918. 1.39708 0.698539 0.715572i \(-0.253834\pi\)
0.698539 + 0.715572i \(0.253834\pi\)
\(138\) 0 0
\(139\) −36820.0 −0.161639 −0.0808196 0.996729i \(-0.525754\pi\)
−0.0808196 + 0.996729i \(0.525754\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −138176. −0.575058
\(143\) −323792. −1.32412
\(144\) 0 0
\(145\) 0 0
\(146\) −141052. −0.547642
\(147\) 0 0
\(148\) −3976.00 −0.0149208
\(149\) 138070. 0.509488 0.254744 0.967009i \(-0.418009\pi\)
0.254744 + 0.967009i \(0.418009\pi\)
\(150\) 0 0
\(151\) −362408. −1.29347 −0.646734 0.762716i \(-0.723865\pi\)
−0.646734 + 0.762716i \(0.723865\pi\)
\(152\) −261600. −0.918393
\(153\) 0 0
\(154\) 124608. 0.423393
\(155\) 0 0
\(156\) 0 0
\(157\) −246098. −0.796818 −0.398409 0.917208i \(-0.630437\pi\)
−0.398409 + 0.917208i \(0.630437\pi\)
\(158\) −95280.0 −0.303640
\(159\) 0 0
\(160\) 0 0
\(161\) 34848.0 0.105953
\(162\) 0 0
\(163\) −170084. −0.501412 −0.250706 0.968063i \(-0.580663\pi\)
−0.250706 + 0.968063i \(0.580663\pi\)
\(164\) −453544. −1.31677
\(165\) 0 0
\(166\) −148008. −0.416884
\(167\) 274008. 0.760277 0.380139 0.924929i \(-0.375876\pi\)
0.380139 + 0.924929i \(0.375876\pi\)
\(168\) 0 0
\(169\) 99303.0 0.267452
\(170\) 0 0
\(171\) 0 0
\(172\) −288848. −0.744472
\(173\) −281886. −0.716075 −0.358037 0.933707i \(-0.616554\pi\)
−0.358037 + 0.933707i \(0.616554\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −309632. −0.753467
\(177\) 0 0
\(178\) −180060. −0.425958
\(179\) 110240. 0.257162 0.128581 0.991699i \(-0.458958\pi\)
0.128581 + 0.991699i \(0.458958\pi\)
\(180\) 0 0
\(181\) 693182. 1.57272 0.786359 0.617770i \(-0.211964\pi\)
0.786359 + 0.617770i \(0.211964\pi\)
\(182\) −181104. −0.405275
\(183\) 0 0
\(184\) 31680.0 0.0689828
\(185\) 0 0
\(186\) 0 0
\(187\) 737264. 1.54177
\(188\) −519904. −1.07282
\(189\) 0 0
\(190\) 0 0
\(191\) −374792. −0.743373 −0.371687 0.928358i \(-0.621220\pi\)
−0.371687 + 0.928358i \(0.621220\pi\)
\(192\) 0 0
\(193\) −247754. −0.478771 −0.239385 0.970925i \(-0.576946\pi\)
−0.239385 + 0.970925i \(0.576946\pi\)
\(194\) −67004.0 −0.127819
\(195\) 0 0
\(196\) −17276.0 −0.0321220
\(197\) 373578. 0.685829 0.342914 0.939367i \(-0.388586\pi\)
0.342914 + 0.939367i \(0.388586\pi\)
\(198\) 0 0
\(199\) −482840. −0.864312 −0.432156 0.901799i \(-0.642247\pi\)
−0.432156 + 0.901799i \(0.642247\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 157764. 0.272038
\(203\) −22440.0 −0.0382193
\(204\) 0 0
\(205\) 0 0
\(206\) −164072. −0.269381
\(207\) 0 0
\(208\) 450016. 0.721223
\(209\) 1.02896e6 1.62942
\(210\) 0 0
\(211\) −708748. −1.09594 −0.547969 0.836499i \(-0.684599\pi\)
−0.547969 + 0.836499i \(0.684599\pi\)
\(212\) −602392. −0.920533
\(213\) 0 0
\(214\) 83304.0 0.124346
\(215\) 0 0
\(216\) 0 0
\(217\) 959904. 1.38382
\(218\) −209740. −0.298910
\(219\) 0 0
\(220\) 0 0
\(221\) −1.07153e6 −1.47579
\(222\) 0 0
\(223\) 211036. 0.284181 0.142090 0.989854i \(-0.454618\pi\)
0.142090 + 0.989854i \(0.454618\pi\)
\(224\) −680064. −0.905586
\(225\) 0 0
\(226\) 307492. 0.400463
\(227\) 920828. 1.18608 0.593040 0.805173i \(-0.297928\pi\)
0.593040 + 0.805173i \(0.297928\pi\)
\(228\) 0 0
\(229\) −71130.0 −0.0896322 −0.0448161 0.998995i \(-0.514270\pi\)
−0.0448161 + 0.998995i \(0.514270\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −20400.0 −0.0248834
\(233\) 1.00049e6 1.20733 0.603663 0.797239i \(-0.293707\pi\)
0.603663 + 0.797239i \(0.293707\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 968800. 1.13228
\(237\) 0 0
\(238\) 412368. 0.471892
\(239\) 550520. 0.623417 0.311708 0.950178i \(-0.399099\pi\)
0.311708 + 0.950178i \(0.399099\pi\)
\(240\) 0 0
\(241\) 543682. 0.602979 0.301490 0.953469i \(-0.402516\pi\)
0.301490 + 0.953469i \(0.402516\pi\)
\(242\) −123466. −0.135522
\(243\) 0 0
\(244\) 1.00066e6 1.07600
\(245\) 0 0
\(246\) 0 0
\(247\) −1.49548e6 −1.55969
\(248\) 872640. 0.900961
\(249\) 0 0
\(250\) 0 0
\(251\) 659568. 0.660808 0.330404 0.943840i \(-0.392815\pi\)
0.330404 + 0.943840i \(0.392815\pi\)
\(252\) 0 0
\(253\) −124608. −0.122390
\(254\) −118744. −0.115485
\(255\) 0 0
\(256\) −30464.0 −0.0290527
\(257\) −1.54600e6 −1.46008 −0.730041 0.683403i \(-0.760499\pi\)
−0.730041 + 0.683403i \(0.760499\pi\)
\(258\) 0 0
\(259\) 18744.0 0.0173625
\(260\) 0 0
\(261\) 0 0
\(262\) −197616. −0.177856
\(263\) −96856.0 −0.0863450 −0.0431725 0.999068i \(-0.513747\pi\)
−0.0431725 + 0.999068i \(0.513747\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 575520. 0.498719
\(267\) 0 0
\(268\) −161616. −0.137451
\(269\) −83530.0 −0.0703820 −0.0351910 0.999381i \(-0.511204\pi\)
−0.0351910 + 0.999381i \(0.511204\pi\)
\(270\) 0 0
\(271\) −857888. −0.709590 −0.354795 0.934944i \(-0.615449\pi\)
−0.354795 + 0.934944i \(0.615449\pi\)
\(272\) −1.02467e6 −0.839774
\(273\) 0 0
\(274\) −613836. −0.493942
\(275\) 0 0
\(276\) 0 0
\(277\) −647538. −0.507068 −0.253534 0.967327i \(-0.581593\pi\)
−0.253534 + 0.967327i \(0.581593\pi\)
\(278\) 73640.0 0.0571481
\(279\) 0 0
\(280\) 0 0
\(281\) 2.21480e6 1.67328 0.836639 0.547754i \(-0.184517\pi\)
0.836639 + 0.547754i \(0.184517\pi\)
\(282\) 0 0
\(283\) 156276. 0.115992 0.0579958 0.998317i \(-0.481529\pi\)
0.0579958 + 0.998317i \(0.481529\pi\)
\(284\) −1.93446e6 −1.42320
\(285\) 0 0
\(286\) 647584. 0.468146
\(287\) 2.13814e6 1.53225
\(288\) 0 0
\(289\) 1.01999e6 0.718373
\(290\) 0 0
\(291\) 0 0
\(292\) −1.97473e6 −1.35535
\(293\) −56406.0 −0.0383845 −0.0191923 0.999816i \(-0.506109\pi\)
−0.0191923 + 0.999816i \(0.506109\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17040.0 0.0113042
\(297\) 0 0
\(298\) −276140. −0.180131
\(299\) 181104. 0.117152
\(300\) 0 0
\(301\) 1.36171e6 0.866301
\(302\) 724816. 0.457310
\(303\) 0 0
\(304\) −1.43008e6 −0.887516
\(305\) 0 0
\(306\) 0 0
\(307\) 1.76061e6 1.06615 0.533074 0.846068i \(-0.321037\pi\)
0.533074 + 0.846068i \(0.321037\pi\)
\(308\) 1.74451e6 1.04785
\(309\) 0 0
\(310\) 0 0
\(311\) 1.71389e6 1.00480 0.502402 0.864634i \(-0.332450\pi\)
0.502402 + 0.864634i \(0.332450\pi\)
\(312\) 0 0
\(313\) 1.42877e6 0.824328 0.412164 0.911110i \(-0.364773\pi\)
0.412164 + 0.911110i \(0.364773\pi\)
\(314\) 492196. 0.281718
\(315\) 0 0
\(316\) −1.33392e6 −0.751471
\(317\) −744382. −0.416052 −0.208026 0.978123i \(-0.566704\pi\)
−0.208026 + 0.978123i \(0.566704\pi\)
\(318\) 0 0
\(319\) 80240.0 0.0441483
\(320\) 0 0
\(321\) 0 0
\(322\) −69696.0 −0.0374600
\(323\) 3.40516e6 1.81606
\(324\) 0 0
\(325\) 0 0
\(326\) 340168. 0.177276
\(327\) 0 0
\(328\) 1.94376e6 0.997604
\(329\) 2.45098e6 1.24839
\(330\) 0 0
\(331\) −136908. −0.0686845 −0.0343423 0.999410i \(-0.510934\pi\)
−0.0343423 + 0.999410i \(0.510934\pi\)
\(332\) −2.07211e6 −1.03173
\(333\) 0 0
\(334\) −548016. −0.268799
\(335\) 0 0
\(336\) 0 0
\(337\) 2.43594e6 1.16840 0.584201 0.811609i \(-0.301408\pi\)
0.584201 + 0.811609i \(0.301408\pi\)
\(338\) −198606. −0.0945585
\(339\) 0 0
\(340\) 0 0
\(341\) −3.43238e6 −1.59849
\(342\) 0 0
\(343\) −2.13708e6 −0.980811
\(344\) 1.23792e6 0.564023
\(345\) 0 0
\(346\) 563772. 0.253171
\(347\) 1.16127e6 0.517736 0.258868 0.965913i \(-0.416650\pi\)
0.258868 + 0.965913i \(0.416650\pi\)
\(348\) 0 0
\(349\) 725830. 0.318986 0.159493 0.987199i \(-0.449014\pi\)
0.159493 + 0.987199i \(0.449014\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.43174e6 1.04607
\(353\) −3.94867e6 −1.68661 −0.843303 0.537438i \(-0.819392\pi\)
−0.843303 + 0.537438i \(0.819392\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.52084e6 −1.05419
\(357\) 0 0
\(358\) −220480. −0.0909205
\(359\) −1.21188e6 −0.496276 −0.248138 0.968725i \(-0.579819\pi\)
−0.248138 + 0.968725i \(0.579819\pi\)
\(360\) 0 0
\(361\) 2.27630e6 0.919309
\(362\) −1.38636e6 −0.556040
\(363\) 0 0
\(364\) −2.53546e6 −1.00300
\(365\) 0 0
\(366\) 0 0
\(367\) −3.92999e6 −1.52309 −0.761546 0.648111i \(-0.775559\pi\)
−0.761546 + 0.648111i \(0.775559\pi\)
\(368\) 173184. 0.0666635
\(369\) 0 0
\(370\) 0 0
\(371\) 2.83985e6 1.07117
\(372\) 0 0
\(373\) −1.08519e6 −0.403864 −0.201932 0.979400i \(-0.564722\pi\)
−0.201932 + 0.979400i \(0.564722\pi\)
\(374\) −1.47453e6 −0.545097
\(375\) 0 0
\(376\) 2.22816e6 0.812787
\(377\) −116620. −0.0422590
\(378\) 0 0
\(379\) 4.06654e6 1.45421 0.727105 0.686526i \(-0.240865\pi\)
0.727105 + 0.686526i \(0.240865\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 749584. 0.262822
\(383\) −1.53454e6 −0.534540 −0.267270 0.963622i \(-0.586122\pi\)
−0.267270 + 0.963622i \(0.586122\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 495508. 0.169271
\(387\) 0 0
\(388\) −938056. −0.316337
\(389\) −1.90737e6 −0.639088 −0.319544 0.947571i \(-0.603530\pi\)
−0.319544 + 0.947571i \(0.603530\pi\)
\(390\) 0 0
\(391\) −412368. −0.136409
\(392\) 74040.0 0.0243361
\(393\) 0 0
\(394\) −747156. −0.242477
\(395\) 0 0
\(396\) 0 0
\(397\) 1.71162e6 0.545044 0.272522 0.962150i \(-0.412142\pi\)
0.272522 + 0.962150i \(0.412142\pi\)
\(398\) 965680. 0.305580
\(399\) 0 0
\(400\) 0 0
\(401\) −2.46268e6 −0.764799 −0.382400 0.923997i \(-0.624902\pi\)
−0.382400 + 0.923997i \(0.624902\pi\)
\(402\) 0 0
\(403\) 4.98859e6 1.53008
\(404\) 2.20870e6 0.673260
\(405\) 0 0
\(406\) 44880.0 0.0135126
\(407\) −67024.0 −0.0200560
\(408\) 0 0
\(409\) −3.69703e6 −1.09281 −0.546405 0.837521i \(-0.684004\pi\)
−0.546405 + 0.837521i \(0.684004\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.29701e6 −0.666683
\(413\) −4.56720e6 −1.31757
\(414\) 0 0
\(415\) 0 0
\(416\) −3.53427e6 −1.00131
\(417\) 0 0
\(418\) −2.05792e6 −0.576087
\(419\) 1.93060e6 0.537226 0.268613 0.963248i \(-0.413435\pi\)
0.268613 + 0.963248i \(0.413435\pi\)
\(420\) 0 0
\(421\) −4.83910e6 −1.33064 −0.665318 0.746560i \(-0.731704\pi\)
−0.665318 + 0.746560i \(0.731704\pi\)
\(422\) 1.41750e6 0.387472
\(423\) 0 0
\(424\) 2.58168e6 0.697409
\(425\) 0 0
\(426\) 0 0
\(427\) −4.71742e6 −1.25209
\(428\) 1.16626e6 0.307740
\(429\) 0 0
\(430\) 0 0
\(431\) 2.74325e6 0.711331 0.355666 0.934613i \(-0.384254\pi\)
0.355666 + 0.934613i \(0.384254\pi\)
\(432\) 0 0
\(433\) −6.05823e6 −1.55284 −0.776419 0.630217i \(-0.782966\pi\)
−0.776419 + 0.630217i \(0.782966\pi\)
\(434\) −1.91981e6 −0.489253
\(435\) 0 0
\(436\) −2.93636e6 −0.739764
\(437\) −575520. −0.144164
\(438\) 0 0
\(439\) 5.21044e6 1.29037 0.645183 0.764028i \(-0.276781\pi\)
0.645183 + 0.764028i \(0.276781\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.14306e6 0.521770
\(443\) 1.17248e6 0.283856 0.141928 0.989877i \(-0.454670\pi\)
0.141928 + 0.989877i \(0.454670\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −422072. −0.100473
\(447\) 0 0
\(448\) −1.41082e6 −0.332105
\(449\) −4.14361e6 −0.969981 −0.484990 0.874520i \(-0.661177\pi\)
−0.484990 + 0.874520i \(0.661177\pi\)
\(450\) 0 0
\(451\) −7.64546e6 −1.76995
\(452\) 4.30489e6 0.991096
\(453\) 0 0
\(454\) −1.84166e6 −0.419343
\(455\) 0 0
\(456\) 0 0
\(457\) 5.41338e6 1.21249 0.606245 0.795278i \(-0.292675\pi\)
0.606245 + 0.795278i \(0.292675\pi\)
\(458\) 142260. 0.0316898
\(459\) 0 0
\(460\) 0 0
\(461\) −5.36680e6 −1.17615 −0.588076 0.808806i \(-0.700114\pi\)
−0.588076 + 0.808806i \(0.700114\pi\)
\(462\) 0 0
\(463\) 5.33284e6 1.15613 0.578064 0.815992i \(-0.303808\pi\)
0.578064 + 0.815992i \(0.303808\pi\)
\(464\) −111520. −0.0240468
\(465\) 0 0
\(466\) −2.00099e6 −0.426854
\(467\) 5.56831e6 1.18149 0.590746 0.806857i \(-0.298833\pi\)
0.590746 + 0.806857i \(0.298833\pi\)
\(468\) 0 0
\(469\) 761904. 0.159944
\(470\) 0 0
\(471\) 0 0
\(472\) −4.15200e6 −0.857832
\(473\) −4.86915e6 −1.00069
\(474\) 0 0
\(475\) 0 0
\(476\) 5.77315e6 1.16787
\(477\) 0 0
\(478\) −1.10104e6 −0.220411
\(479\) 1.57092e6 0.312835 0.156417 0.987691i \(-0.450005\pi\)
0.156417 + 0.987691i \(0.450005\pi\)
\(480\) 0 0
\(481\) 97412.0 0.0191977
\(482\) −1.08736e6 −0.213185
\(483\) 0 0
\(484\) −1.72852e6 −0.335399
\(485\) 0 0
\(486\) 0 0
\(487\) 7.74497e6 1.47978 0.739891 0.672727i \(-0.234877\pi\)
0.739891 + 0.672727i \(0.234877\pi\)
\(488\) −4.28856e6 −0.815196
\(489\) 0 0
\(490\) 0 0
\(491\) −1.97715e6 −0.370115 −0.185057 0.982728i \(-0.559247\pi\)
−0.185057 + 0.982728i \(0.559247\pi\)
\(492\) 0 0
\(493\) 265540. 0.0492054
\(494\) 2.99096e6 0.551434
\(495\) 0 0
\(496\) 4.77043e6 0.870670
\(497\) 9.11962e6 1.65610
\(498\) 0 0
\(499\) −6.55994e6 −1.17937 −0.589683 0.807635i \(-0.700747\pi\)
−0.589683 + 0.807635i \(0.700747\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.31914e6 −0.233631
\(503\) −5.92850e6 −1.04478 −0.522390 0.852707i \(-0.674959\pi\)
−0.522390 + 0.852707i \(0.674959\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 249216. 0.0432713
\(507\) 0 0
\(508\) −1.66242e6 −0.285812
\(509\) 1.03055e6 0.176309 0.0881545 0.996107i \(-0.471903\pi\)
0.0881545 + 0.996107i \(0.471903\pi\)
\(510\) 0 0
\(511\) 9.30943e6 1.57714
\(512\) −5.89875e6 −0.994455
\(513\) 0 0
\(514\) 3.09200e6 0.516217
\(515\) 0 0
\(516\) 0 0
\(517\) −8.76410e6 −1.44205
\(518\) −37488.0 −0.00613858
\(519\) 0 0
\(520\) 0 0
\(521\) −8.48280e6 −1.36913 −0.684566 0.728951i \(-0.740008\pi\)
−0.684566 + 0.728951i \(0.740008\pi\)
\(522\) 0 0
\(523\) −8.93676e6 −1.42865 −0.714325 0.699814i \(-0.753266\pi\)
−0.714325 + 0.699814i \(0.753266\pi\)
\(524\) −2.76662e6 −0.440172
\(525\) 0 0
\(526\) 193712. 0.0305276
\(527\) −1.13589e7 −1.78159
\(528\) 0 0
\(529\) −6.36665e6 −0.989171
\(530\) 0 0
\(531\) 0 0
\(532\) 8.05728e6 1.23427
\(533\) 1.11118e7 1.69421
\(534\) 0 0
\(535\) 0 0
\(536\) 692640. 0.104135
\(537\) 0 0
\(538\) 167060. 0.0248838
\(539\) −291224. −0.0431773
\(540\) 0 0
\(541\) 4.14394e6 0.608724 0.304362 0.952556i \(-0.401557\pi\)
0.304362 + 0.952556i \(0.401557\pi\)
\(542\) 1.71578e6 0.250878
\(543\) 0 0
\(544\) 8.04742e6 1.16590
\(545\) 0 0
\(546\) 0 0
\(547\) −2.63115e6 −0.375991 −0.187995 0.982170i \(-0.560199\pi\)
−0.187995 + 0.982170i \(0.560199\pi\)
\(548\) −8.59370e6 −1.22244
\(549\) 0 0
\(550\) 0 0
\(551\) 370600. 0.0520028
\(552\) 0 0
\(553\) 6.28848e6 0.874446
\(554\) 1.29508e6 0.179275
\(555\) 0 0
\(556\) 1.03096e6 0.141434
\(557\) −1.24917e7 −1.70602 −0.853012 0.521892i \(-0.825227\pi\)
−0.853012 + 0.521892i \(0.825227\pi\)
\(558\) 0 0
\(559\) 7.07678e6 0.957869
\(560\) 0 0
\(561\) 0 0
\(562\) −4.42960e6 −0.591593
\(563\) 7.63892e6 1.01569 0.507845 0.861448i \(-0.330442\pi\)
0.507845 + 0.861448i \(0.330442\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −312552. −0.0410092
\(567\) 0 0
\(568\) 8.29056e6 1.07823
\(569\) −455250. −0.0589480 −0.0294740 0.999566i \(-0.509383\pi\)
−0.0294740 + 0.999566i \(0.509383\pi\)
\(570\) 0 0
\(571\) −259228. −0.0332730 −0.0166365 0.999862i \(-0.505296\pi\)
−0.0166365 + 0.999862i \(0.505296\pi\)
\(572\) 9.06618e6 1.15860
\(573\) 0 0
\(574\) −4.27627e6 −0.541733
\(575\) 0 0
\(576\) 0 0
\(577\) 1.33636e7 1.67103 0.835513 0.549470i \(-0.185170\pi\)
0.835513 + 0.549470i \(0.185170\pi\)
\(578\) −2.03997e6 −0.253983
\(579\) 0 0
\(580\) 0 0
\(581\) 9.76853e6 1.20057
\(582\) 0 0
\(583\) −1.01546e7 −1.23735
\(584\) 8.46312e6 1.02683
\(585\) 0 0
\(586\) 112812. 0.0135710
\(587\) −1.22332e7 −1.46536 −0.732679 0.680574i \(-0.761730\pi\)
−0.732679 + 0.680574i \(0.761730\pi\)
\(588\) 0 0
\(589\) −1.58530e7 −1.88288
\(590\) 0 0
\(591\) 0 0
\(592\) 93152.0 0.0109242
\(593\) 1.17588e7 1.37318 0.686589 0.727046i \(-0.259107\pi\)
0.686589 + 0.727046i \(0.259107\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.86596e6 −0.445802
\(597\) 0 0
\(598\) −362208. −0.0414195
\(599\) −8.58404e6 −0.977518 −0.488759 0.872419i \(-0.662550\pi\)
−0.488759 + 0.872419i \(0.662550\pi\)
\(600\) 0 0
\(601\) −1.59372e6 −0.179980 −0.0899902 0.995943i \(-0.528684\pi\)
−0.0899902 + 0.995943i \(0.528684\pi\)
\(602\) −2.72342e6 −0.306284
\(603\) 0 0
\(604\) 1.01474e7 1.13178
\(605\) 0 0
\(606\) 0 0
\(607\) −1.52801e7 −1.68327 −0.841637 0.540044i \(-0.818408\pi\)
−0.841637 + 0.540044i \(0.818408\pi\)
\(608\) 1.12314e7 1.23218
\(609\) 0 0
\(610\) 0 0
\(611\) 1.27376e7 1.38034
\(612\) 0 0
\(613\) 1.67004e7 1.79504 0.897521 0.440971i \(-0.145366\pi\)
0.897521 + 0.440971i \(0.145366\pi\)
\(614\) −3.52122e6 −0.376940
\(615\) 0 0
\(616\) −7.47648e6 −0.793863
\(617\) 1.03986e7 1.09966 0.549832 0.835275i \(-0.314692\pi\)
0.549832 + 0.835275i \(0.314692\pi\)
\(618\) 0 0
\(619\) −1.22631e7 −1.28640 −0.643199 0.765699i \(-0.722393\pi\)
−0.643199 + 0.765699i \(0.722393\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −3.42778e6 −0.355252
\(623\) 1.18840e7 1.22671
\(624\) 0 0
\(625\) 0 0
\(626\) −2.85753e6 −0.291444
\(627\) 0 0
\(628\) 6.89074e6 0.697215
\(629\) −221804. −0.0223534
\(630\) 0 0
\(631\) 1.22509e7 1.22488 0.612442 0.790515i \(-0.290187\pi\)
0.612442 + 0.790515i \(0.290187\pi\)
\(632\) 5.71680e6 0.569325
\(633\) 0 0
\(634\) 1.48876e6 0.147097
\(635\) 0 0
\(636\) 0 0
\(637\) 423262. 0.0413296
\(638\) −160480. −0.0156088
\(639\) 0 0
\(640\) 0 0
\(641\) 6.85904e6 0.659353 0.329677 0.944094i \(-0.393060\pi\)
0.329677 + 0.944094i \(0.393060\pi\)
\(642\) 0 0
\(643\) 3.31120e6 0.315833 0.157916 0.987452i \(-0.449522\pi\)
0.157916 + 0.987452i \(0.449522\pi\)
\(644\) −975744. −0.0927089
\(645\) 0 0
\(646\) −6.81032e6 −0.642076
\(647\) 9.91821e6 0.931478 0.465739 0.884922i \(-0.345789\pi\)
0.465739 + 0.884922i \(0.345789\pi\)
\(648\) 0 0
\(649\) 1.63312e7 1.52197
\(650\) 0 0
\(651\) 0 0
\(652\) 4.76235e6 0.438735
\(653\) −1.39102e7 −1.27659 −0.638294 0.769793i \(-0.720359\pi\)
−0.638294 + 0.769793i \(0.720359\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.06259e7 0.964063
\(657\) 0 0
\(658\) −4.90195e6 −0.441372
\(659\) 3.27884e6 0.294108 0.147054 0.989128i \(-0.453021\pi\)
0.147054 + 0.989128i \(0.453021\pi\)
\(660\) 0 0
\(661\) −7.00330e6 −0.623446 −0.311723 0.950173i \(-0.600906\pi\)
−0.311723 + 0.950173i \(0.600906\pi\)
\(662\) 273816. 0.0242836
\(663\) 0 0
\(664\) 8.88048e6 0.781657
\(665\) 0 0
\(666\) 0 0
\(667\) −44880.0 −0.00390605
\(668\) −7.67222e6 −0.665243
\(669\) 0 0
\(670\) 0 0
\(671\) 1.68683e7 1.44633
\(672\) 0 0
\(673\) −1.09747e7 −0.934019 −0.467009 0.884252i \(-0.654669\pi\)
−0.467009 + 0.884252i \(0.654669\pi\)
\(674\) −4.87188e6 −0.413092
\(675\) 0 0
\(676\) −2.78048e6 −0.234020
\(677\) −8.90482e6 −0.746713 −0.373356 0.927688i \(-0.621793\pi\)
−0.373356 + 0.927688i \(0.621793\pi\)
\(678\) 0 0
\(679\) 4.42226e6 0.368104
\(680\) 0 0
\(681\) 0 0
\(682\) 6.86477e6 0.565152
\(683\) 1.61956e7 1.32845 0.664227 0.747531i \(-0.268761\pi\)
0.664227 + 0.747531i \(0.268761\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 4.27416e6 0.346769
\(687\) 0 0
\(688\) 6.76730e6 0.545060
\(689\) 1.47586e7 1.18440
\(690\) 0 0
\(691\) −1.51015e6 −0.120316 −0.0601581 0.998189i \(-0.519160\pi\)
−0.0601581 + 0.998189i \(0.519160\pi\)
\(692\) 7.89281e6 0.626565
\(693\) 0 0
\(694\) −2.32254e6 −0.183047
\(695\) 0 0
\(696\) 0 0
\(697\) −2.53013e7 −1.97270
\(698\) −1.45166e6 −0.112779
\(699\) 0 0
\(700\) 0 0
\(701\) −1.80508e7 −1.38740 −0.693698 0.720266i \(-0.744020\pi\)
−0.693698 + 0.720266i \(0.744020\pi\)
\(702\) 0 0
\(703\) −309560. −0.0236242
\(704\) 5.04474e6 0.383625
\(705\) 0 0
\(706\) 7.89733e6 0.596305
\(707\) −1.04124e7 −0.783436
\(708\) 0 0
\(709\) 1.57918e7 1.17982 0.589912 0.807468i \(-0.299163\pi\)
0.589912 + 0.807468i \(0.299163\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.08036e7 0.798672
\(713\) 1.91981e6 0.141428
\(714\) 0 0
\(715\) 0 0
\(716\) −3.08672e6 −0.225017
\(717\) 0 0
\(718\) 2.42376e6 0.175460
\(719\) −897600. −0.0647531 −0.0323766 0.999476i \(-0.510308\pi\)
−0.0323766 + 0.999476i \(0.510308\pi\)
\(720\) 0 0
\(721\) 1.08288e7 0.775783
\(722\) −4.55260e6 −0.325025
\(723\) 0 0
\(724\) −1.94091e7 −1.37613
\(725\) 0 0
\(726\) 0 0
\(727\) −8.33311e6 −0.584751 −0.292376 0.956304i \(-0.594446\pi\)
−0.292376 + 0.956304i \(0.594446\pi\)
\(728\) 1.08662e7 0.759890
\(729\) 0 0
\(730\) 0 0
\(731\) −1.61136e7 −1.11532
\(732\) 0 0
\(733\) 6.05633e6 0.416341 0.208171 0.978093i \(-0.433249\pi\)
0.208171 + 0.978093i \(0.433249\pi\)
\(734\) 7.85998e6 0.538494
\(735\) 0 0
\(736\) −1.36013e6 −0.0925519
\(737\) −2.72438e6 −0.184756
\(738\) 0 0
\(739\) 5.19646e6 0.350023 0.175011 0.984566i \(-0.444004\pi\)
0.175011 + 0.984566i \(0.444004\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5.67970e6 −0.378717
\(743\) 2.67222e6 0.177583 0.0887914 0.996050i \(-0.471700\pi\)
0.0887914 + 0.996050i \(0.471700\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.17039e6 0.142788
\(747\) 0 0
\(748\) −2.06434e7 −1.34905
\(749\) −5.49806e6 −0.358101
\(750\) 0 0
\(751\) 1.49981e7 0.970366 0.485183 0.874413i \(-0.338753\pi\)
0.485183 + 0.874413i \(0.338753\pi\)
\(752\) 1.21806e7 0.785461
\(753\) 0 0
\(754\) 233240. 0.0149408
\(755\) 0 0
\(756\) 0 0
\(757\) 2.56846e7 1.62905 0.814523 0.580131i \(-0.196999\pi\)
0.814523 + 0.580131i \(0.196999\pi\)
\(758\) −8.13308e6 −0.514141
\(759\) 0 0
\(760\) 0 0
\(761\) −6.07960e6 −0.380552 −0.190276 0.981731i \(-0.560938\pi\)
−0.190276 + 0.981731i \(0.560938\pi\)
\(762\) 0 0
\(763\) 1.38428e7 0.860823
\(764\) 1.04942e7 0.650452
\(765\) 0 0
\(766\) 3.06907e6 0.188988
\(767\) −2.37356e7 −1.45684
\(768\) 0 0
\(769\) −2.82501e7 −1.72268 −0.861339 0.508030i \(-0.830374\pi\)
−0.861339 + 0.508030i \(0.830374\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.93711e6 0.418924
\(773\) −2.11430e7 −1.27268 −0.636339 0.771409i \(-0.719552\pi\)
−0.636339 + 0.771409i \(0.719552\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4.02024e6 0.239661
\(777\) 0 0
\(778\) 3.81474e6 0.225952
\(779\) −3.53116e7 −2.08485
\(780\) 0 0
\(781\) −3.26095e7 −1.91301
\(782\) 824736. 0.0482279
\(783\) 0 0
\(784\) 404752. 0.0235179
\(785\) 0 0
\(786\) 0 0
\(787\) 2.42307e7 1.39454 0.697268 0.716811i \(-0.254399\pi\)
0.697268 + 0.716811i \(0.254399\pi\)
\(788\) −1.04602e7 −0.600100
\(789\) 0 0
\(790\) 0 0
\(791\) −2.02945e7 −1.15328
\(792\) 0 0
\(793\) −2.45163e7 −1.38443
\(794\) −3.42324e6 −0.192702
\(795\) 0 0
\(796\) 1.35195e7 0.756273
\(797\) 1.44322e6 0.0804797 0.0402398 0.999190i \(-0.487188\pi\)
0.0402398 + 0.999190i \(0.487188\pi\)
\(798\) 0 0
\(799\) −2.90032e7 −1.60723
\(800\) 0 0
\(801\) 0 0
\(802\) 4.92536e6 0.270397
\(803\) −3.32883e7 −1.82181
\(804\) 0 0
\(805\) 0 0
\(806\) −9.97718e6 −0.540967
\(807\) 0 0
\(808\) −9.46584e6 −0.510071
\(809\) −2.99186e7 −1.60720 −0.803599 0.595171i \(-0.797084\pi\)
−0.803599 + 0.595171i \(0.797084\pi\)
\(810\) 0 0
\(811\) −5.30183e6 −0.283057 −0.141528 0.989934i \(-0.545202\pi\)
−0.141528 + 0.989934i \(0.545202\pi\)
\(812\) 628320. 0.0334419
\(813\) 0 0
\(814\) 134048. 0.00709087
\(815\) 0 0
\(816\) 0 0
\(817\) −2.24889e7 −1.17873
\(818\) 7.39406e6 0.386367
\(819\) 0 0
\(820\) 0 0
\(821\) −4.55008e6 −0.235592 −0.117796 0.993038i \(-0.537583\pi\)
−0.117796 + 0.993038i \(0.537583\pi\)
\(822\) 0 0
\(823\) 2.42775e7 1.24941 0.624705 0.780861i \(-0.285219\pi\)
0.624705 + 0.780861i \(0.285219\pi\)
\(824\) 9.84432e6 0.505089
\(825\) 0 0
\(826\) 9.13440e6 0.465833
\(827\) 3.63999e6 0.185070 0.0925350 0.995709i \(-0.470503\pi\)
0.0925350 + 0.995709i \(0.470503\pi\)
\(828\) 0 0
\(829\) 1.98674e7 1.00405 0.502023 0.864854i \(-0.332589\pi\)
0.502023 + 0.864854i \(0.332589\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −7.33197e6 −0.367208
\(833\) −963754. −0.0481231
\(834\) 0 0
\(835\) 0 0
\(836\) −2.88109e7 −1.42574
\(837\) 0 0
\(838\) −3.86120e6 −0.189938
\(839\) 3.58666e7 1.75908 0.879539 0.475826i \(-0.157851\pi\)
0.879539 + 0.475826i \(0.157851\pi\)
\(840\) 0 0
\(841\) −2.04822e7 −0.998591
\(842\) 9.67820e6 0.470451
\(843\) 0 0
\(844\) 1.98449e7 0.958945
\(845\) 0 0
\(846\) 0 0
\(847\) 8.14876e6 0.390286
\(848\) 1.41132e7 0.673962
\(849\) 0 0
\(850\) 0 0
\(851\) 37488.0 0.00177447
\(852\) 0 0
\(853\) −2.25791e7 −1.06251 −0.531256 0.847212i \(-0.678280\pi\)
−0.531256 + 0.847212i \(0.678280\pi\)
\(854\) 9.43483e6 0.442680
\(855\) 0 0
\(856\) −4.99824e6 −0.233149
\(857\) 1.78039e7 0.828063 0.414032 0.910262i \(-0.364120\pi\)
0.414032 + 0.910262i \(0.364120\pi\)
\(858\) 0 0
\(859\) −302900. −0.0140061 −0.00700304 0.999975i \(-0.502229\pi\)
−0.00700304 + 0.999975i \(0.502229\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.48650e6 −0.251494
\(863\) −4.37306e6 −0.199875 −0.0999374 0.994994i \(-0.531864\pi\)
−0.0999374 + 0.994994i \(0.531864\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.21165e7 0.549011
\(867\) 0 0
\(868\) −2.68773e7 −1.21084
\(869\) −2.24861e7 −1.01010
\(870\) 0 0
\(871\) 3.95959e6 0.176850
\(872\) 1.25844e7 0.560456
\(873\) 0 0
\(874\) 1.15104e6 0.0509697
\(875\) 0 0
\(876\) 0 0
\(877\) −2.24189e7 −0.984272 −0.492136 0.870518i \(-0.663784\pi\)
−0.492136 + 0.870518i \(0.663784\pi\)
\(878\) −1.04209e7 −0.456213
\(879\) 0 0
\(880\) 0 0
\(881\) 1.40470e7 0.609739 0.304869 0.952394i \(-0.401387\pi\)
0.304869 + 0.952394i \(0.401387\pi\)
\(882\) 0 0
\(883\) 2.41189e7 1.04101 0.520505 0.853858i \(-0.325744\pi\)
0.520505 + 0.853858i \(0.325744\pi\)
\(884\) 3.00029e7 1.29132
\(885\) 0 0
\(886\) −2.34497e6 −0.100358
\(887\) 2.05454e7 0.876809 0.438404 0.898778i \(-0.355544\pi\)
0.438404 + 0.898778i \(0.355544\pi\)
\(888\) 0 0
\(889\) 7.83710e6 0.332584
\(890\) 0 0
\(891\) 0 0
\(892\) −5.90901e6 −0.248658
\(893\) −4.04782e7 −1.69861
\(894\) 0 0
\(895\) 0 0
\(896\) 2.45837e7 1.02300
\(897\) 0 0
\(898\) 8.28722e6 0.342940
\(899\) −1.23624e6 −0.0510157
\(900\) 0 0
\(901\) −3.36049e7 −1.37908
\(902\) 1.52909e7 0.625773
\(903\) 0 0
\(904\) −1.84495e7 −0.750869
\(905\) 0 0
\(906\) 0 0
\(907\) 7.37425e6 0.297646 0.148823 0.988864i \(-0.452452\pi\)
0.148823 + 0.988864i \(0.452452\pi\)
\(908\) −2.57832e7 −1.03782
\(909\) 0 0
\(910\) 0 0
\(911\) 2.23806e7 0.893462 0.446731 0.894668i \(-0.352588\pi\)
0.446731 + 0.894668i \(0.352588\pi\)
\(912\) 0 0
\(913\) −3.49299e7 −1.38682
\(914\) −1.08268e7 −0.428680
\(915\) 0 0
\(916\) 1.99164e6 0.0784282
\(917\) 1.30427e7 0.512204
\(918\) 0 0
\(919\) 3.04744e7 1.19027 0.595136 0.803625i \(-0.297098\pi\)
0.595136 + 0.803625i \(0.297098\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.07336e7 0.415832
\(923\) 4.73944e7 1.83114
\(924\) 0 0
\(925\) 0 0
\(926\) −1.06657e7 −0.408753
\(927\) 0 0
\(928\) 875840. 0.0333853
\(929\) −1.87929e7 −0.714421 −0.357211 0.934024i \(-0.616272\pi\)
−0.357211 + 0.934024i \(0.616272\pi\)
\(930\) 0 0
\(931\) −1.34506e6 −0.0508590
\(932\) −2.80138e7 −1.05641
\(933\) 0 0
\(934\) −1.11366e7 −0.417721
\(935\) 0 0
\(936\) 0 0
\(937\) 9.77386e6 0.363678 0.181839 0.983328i \(-0.441795\pi\)
0.181839 + 0.983328i \(0.441795\pi\)
\(938\) −1.52381e6 −0.0565488
\(939\) 0 0
\(940\) 0 0
\(941\) −1.24022e7 −0.456590 −0.228295 0.973592i \(-0.573315\pi\)
−0.228295 + 0.973592i \(0.573315\pi\)
\(942\) 0 0
\(943\) 4.27627e6 0.156598
\(944\) −2.26976e7 −0.828991
\(945\) 0 0
\(946\) 9.73830e6 0.353798
\(947\) −1.77052e7 −0.641542 −0.320771 0.947157i \(-0.603942\pi\)
−0.320771 + 0.947157i \(0.603942\pi\)
\(948\) 0 0
\(949\) 4.83808e7 1.74384
\(950\) 0 0
\(951\) 0 0
\(952\) −2.47421e7 −0.884797
\(953\) 2.37213e6 0.0846071 0.0423036 0.999105i \(-0.486530\pi\)
0.0423036 + 0.999105i \(0.486530\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.54146e7 −0.545490
\(957\) 0 0
\(958\) −3.14184e6 −0.110604
\(959\) 4.05132e7 1.42249
\(960\) 0 0
\(961\) 2.42528e7 0.847138
\(962\) −194824. −0.00678742
\(963\) 0 0
\(964\) −1.52231e7 −0.527607
\(965\) 0 0
\(966\) 0 0
\(967\) −567548. −0.0195180 −0.00975902 0.999952i \(-0.503106\pi\)
−0.00975902 + 0.999952i \(0.503106\pi\)
\(968\) 7.40796e6 0.254103
\(969\) 0 0
\(970\) 0 0
\(971\) 2.33139e7 0.793536 0.396768 0.917919i \(-0.370132\pi\)
0.396768 + 0.917919i \(0.370132\pi\)
\(972\) 0 0
\(973\) −4.86024e6 −0.164579
\(974\) −1.54899e7 −0.523182
\(975\) 0 0
\(976\) −2.34441e7 −0.787788
\(977\) 3.94860e6 0.132345 0.0661723 0.997808i \(-0.478921\pi\)
0.0661723 + 0.997808i \(0.478921\pi\)
\(978\) 0 0
\(979\) −4.24942e7 −1.41701
\(980\) 0 0
\(981\) 0 0
\(982\) 3.95430e6 0.130855
\(983\) 5.01417e7 1.65506 0.827532 0.561418i \(-0.189744\pi\)
0.827532 + 0.561418i \(0.189744\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −531080. −0.0173967
\(987\) 0 0
\(988\) 4.18734e7 1.36473
\(989\) 2.72342e6 0.0885369
\(990\) 0 0
\(991\) 3.48675e6 0.112781 0.0563906 0.998409i \(-0.482041\pi\)
0.0563906 + 0.998409i \(0.482041\pi\)
\(992\) −3.74653e7 −1.20879
\(993\) 0 0
\(994\) −1.82392e7 −0.585518
\(995\) 0 0
\(996\) 0 0
\(997\) 1.32373e7 0.421756 0.210878 0.977512i \(-0.432368\pi\)
0.210878 + 0.977512i \(0.432368\pi\)
\(998\) 1.31199e7 0.416969
\(999\) 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 225.6.a.c.1.1 1
3.2 odd 2 75.6.a.c.1.1 1
5.2 odd 4 225.6.b.d.199.1 2
5.3 odd 4 225.6.b.d.199.2 2
5.4 even 2 45.6.a.c.1.1 1
15.2 even 4 75.6.b.d.49.2 2
15.8 even 4 75.6.b.d.49.1 2
15.14 odd 2 15.6.a.a.1.1 1
20.19 odd 2 720.6.a.w.1.1 1
60.59 even 2 240.6.a.k.1.1 1
105.104 even 2 735.6.a.a.1.1 1
120.29 odd 2 960.6.a.v.1.1 1
120.59 even 2 960.6.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.6.a.a.1.1 1 15.14 odd 2
45.6.a.c.1.1 1 5.4 even 2
75.6.a.c.1.1 1 3.2 odd 2
75.6.b.d.49.1 2 15.8 even 4
75.6.b.d.49.2 2 15.2 even 4
225.6.a.c.1.1 1 1.1 even 1 trivial
225.6.b.d.199.1 2 5.2 odd 4
225.6.b.d.199.2 2 5.3 odd 4
240.6.a.k.1.1 1 60.59 even 2
720.6.a.w.1.1 1 20.19 odd 2
735.6.a.a.1.1 1 105.104 even 2
960.6.a.m.1.1 1 120.59 even 2
960.6.a.v.1.1 1 120.29 odd 2