Properties

Label 392.6.a.j.1.3
Level $392$
Weight $6$
Character 392.1
Self dual yes
Analytic conductor $62.870$
Analytic rank $0$
Dimension $5$
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] = [392,6,Mod(1,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, 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("392.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 392.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: \(62.8704573667\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 7 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.638887\) of defining polynomial
Character \(\chi\) \(=\) 392.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.27777 q^{3} -50.5110 q^{5} -237.812 q^{9} +O(q^{10})\) \(q-2.27777 q^{3} -50.5110 q^{5} -237.812 q^{9} +192.028 q^{11} +656.798 q^{13} +115.053 q^{15} -2189.37 q^{17} +279.330 q^{19} -3713.60 q^{23} -573.636 q^{25} +1095.18 q^{27} -5276.51 q^{29} +6485.80 q^{31} -437.396 q^{33} +2950.08 q^{37} -1496.04 q^{39} +2394.37 q^{41} +17261.5 q^{43} +12012.1 q^{45} -7720.77 q^{47} +4986.89 q^{51} -8497.26 q^{53} -9699.52 q^{55} -636.251 q^{57} +8935.01 q^{59} +22389.6 q^{61} -33175.6 q^{65} +52695.0 q^{67} +8458.73 q^{69} -25479.0 q^{71} +22908.1 q^{73} +1306.61 q^{75} -45919.1 q^{79} +55293.7 q^{81} +97180.7 q^{83} +110587. q^{85} +12018.7 q^{87} +59131.4 q^{89} -14773.2 q^{93} -14109.3 q^{95} +73491.5 q^{97} -45666.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 81 q^{5} + 390 q^{9} + 361 q^{11} + 342 q^{13} - 1049 q^{15} + 1809 q^{17} + 1277 q^{19} + 911 q^{23} + 3940 q^{25} - 4751 q^{27} + 5442 q^{29} + 2187 q^{31} - 5553 q^{33} - 8181 q^{37} - 3422 q^{39} + 16578 q^{41} + 6332 q^{43} + 41310 q^{45} + 16101 q^{47} - 67865 q^{51} + 16047 q^{53} + 45629 q^{55} + 22347 q^{57} + 71027 q^{59} + 31093 q^{61} - 64370 q^{65} + 47981 q^{67} + 137249 q^{69} + 22512 q^{71} + 123333 q^{73} - 45460 q^{75} - 212481 q^{79} + 52917 q^{81} + 87460 q^{83} + 222141 q^{85} + 318070 q^{87} + 129045 q^{89} - 252835 q^{93} + 300417 q^{95} + 328274 q^{97} + 249798 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.27777 −0.146119 −0.0730596 0.997328i \(-0.523276\pi\)
−0.0730596 + 0.997328i \(0.523276\pi\)
\(4\) 0 0
\(5\) −50.5110 −0.903569 −0.451784 0.892127i \(-0.649212\pi\)
−0.451784 + 0.892127i \(0.649212\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −237.812 −0.978649
\(10\) 0 0
\(11\) 192.028 0.478500 0.239250 0.970958i \(-0.423098\pi\)
0.239250 + 0.970958i \(0.423098\pi\)
\(12\) 0 0
\(13\) 656.798 1.07789 0.538944 0.842342i \(-0.318823\pi\)
0.538944 + 0.842342i \(0.318823\pi\)
\(14\) 0 0
\(15\) 115.053 0.132029
\(16\) 0 0
\(17\) −2189.37 −1.83737 −0.918685 0.394991i \(-0.870748\pi\)
−0.918685 + 0.394991i \(0.870748\pi\)
\(18\) 0 0
\(19\) 279.330 0.177515 0.0887573 0.996053i \(-0.471710\pi\)
0.0887573 + 0.996053i \(0.471710\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3713.60 −1.46378 −0.731889 0.681424i \(-0.761361\pi\)
−0.731889 + 0.681424i \(0.761361\pi\)
\(24\) 0 0
\(25\) −573.636 −0.183564
\(26\) 0 0
\(27\) 1095.18 0.289119
\(28\) 0 0
\(29\) −5276.51 −1.16507 −0.582535 0.812806i \(-0.697939\pi\)
−0.582535 + 0.812806i \(0.697939\pi\)
\(30\) 0 0
\(31\) 6485.80 1.21216 0.606079 0.795405i \(-0.292742\pi\)
0.606079 + 0.795405i \(0.292742\pi\)
\(32\) 0 0
\(33\) −437.396 −0.0699181
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2950.08 0.354266 0.177133 0.984187i \(-0.443318\pi\)
0.177133 + 0.984187i \(0.443318\pi\)
\(38\) 0 0
\(39\) −1496.04 −0.157500
\(40\) 0 0
\(41\) 2394.37 0.222449 0.111225 0.993795i \(-0.464523\pi\)
0.111225 + 0.993795i \(0.464523\pi\)
\(42\) 0 0
\(43\) 17261.5 1.42366 0.711832 0.702350i \(-0.247866\pi\)
0.711832 + 0.702350i \(0.247866\pi\)
\(44\) 0 0
\(45\) 12012.1 0.884277
\(46\) 0 0
\(47\) −7720.77 −0.509819 −0.254909 0.966965i \(-0.582046\pi\)
−0.254909 + 0.966965i \(0.582046\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4986.89 0.268475
\(52\) 0 0
\(53\) −8497.26 −0.415517 −0.207759 0.978180i \(-0.566617\pi\)
−0.207759 + 0.978180i \(0.566617\pi\)
\(54\) 0 0
\(55\) −9699.52 −0.432358
\(56\) 0 0
\(57\) −636.251 −0.0259383
\(58\) 0 0
\(59\) 8935.01 0.334168 0.167084 0.985943i \(-0.446565\pi\)
0.167084 + 0.985943i \(0.446565\pi\)
\(60\) 0 0
\(61\) 22389.6 0.770410 0.385205 0.922831i \(-0.374131\pi\)
0.385205 + 0.922831i \(0.374131\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −33175.6 −0.973946
\(66\) 0 0
\(67\) 52695.0 1.43411 0.717055 0.697017i \(-0.245490\pi\)
0.717055 + 0.697017i \(0.245490\pi\)
\(68\) 0 0
\(69\) 8458.73 0.213886
\(70\) 0 0
\(71\) −25479.0 −0.599842 −0.299921 0.953964i \(-0.596960\pi\)
−0.299921 + 0.953964i \(0.596960\pi\)
\(72\) 0 0
\(73\) 22908.1 0.503133 0.251566 0.967840i \(-0.419054\pi\)
0.251566 + 0.967840i \(0.419054\pi\)
\(74\) 0 0
\(75\) 1306.61 0.0268222
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −45919.1 −0.827800 −0.413900 0.910322i \(-0.635834\pi\)
−0.413900 + 0.910322i \(0.635834\pi\)
\(80\) 0 0
\(81\) 55293.7 0.936403
\(82\) 0 0
\(83\) 97180.7 1.54841 0.774203 0.632937i \(-0.218151\pi\)
0.774203 + 0.632937i \(0.218151\pi\)
\(84\) 0 0
\(85\) 110587. 1.66019
\(86\) 0 0
\(87\) 12018.7 0.170239
\(88\) 0 0
\(89\) 59131.4 0.791304 0.395652 0.918401i \(-0.370519\pi\)
0.395652 + 0.918401i \(0.370519\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −14773.2 −0.177120
\(94\) 0 0
\(95\) −14109.3 −0.160397
\(96\) 0 0
\(97\) 73491.5 0.793063 0.396531 0.918021i \(-0.370214\pi\)
0.396531 + 0.918021i \(0.370214\pi\)
\(98\) 0 0
\(99\) −45666.5 −0.468284
\(100\) 0 0
\(101\) 178860. 1.74465 0.872326 0.488925i \(-0.162611\pi\)
0.872326 + 0.488925i \(0.162611\pi\)
\(102\) 0 0
\(103\) −8711.96 −0.0809138 −0.0404569 0.999181i \(-0.512881\pi\)
−0.0404569 + 0.999181i \(0.512881\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −172719. −1.45841 −0.729206 0.684294i \(-0.760110\pi\)
−0.729206 + 0.684294i \(0.760110\pi\)
\(108\) 0 0
\(109\) −166306. −1.34073 −0.670367 0.742030i \(-0.733863\pi\)
−0.670367 + 0.742030i \(0.733863\pi\)
\(110\) 0 0
\(111\) −6719.61 −0.0517651
\(112\) 0 0
\(113\) 29604.7 0.218104 0.109052 0.994036i \(-0.465218\pi\)
0.109052 + 0.994036i \(0.465218\pi\)
\(114\) 0 0
\(115\) 187578. 1.32262
\(116\) 0 0
\(117\) −156194. −1.05487
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −124176. −0.771037
\(122\) 0 0
\(123\) −5453.82 −0.0325041
\(124\) 0 0
\(125\) 186822. 1.06943
\(126\) 0 0
\(127\) 333136. 1.83279 0.916395 0.400275i \(-0.131085\pi\)
0.916395 + 0.400275i \(0.131085\pi\)
\(128\) 0 0
\(129\) −39317.8 −0.208025
\(130\) 0 0
\(131\) 58228.0 0.296452 0.148226 0.988954i \(-0.452644\pi\)
0.148226 + 0.988954i \(0.452644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −55318.7 −0.261239
\(136\) 0 0
\(137\) 327685. 1.49161 0.745805 0.666164i \(-0.232065\pi\)
0.745805 + 0.666164i \(0.232065\pi\)
\(138\) 0 0
\(139\) −165074. −0.724674 −0.362337 0.932047i \(-0.618021\pi\)
−0.362337 + 0.932047i \(0.618021\pi\)
\(140\) 0 0
\(141\) 17586.2 0.0744943
\(142\) 0 0
\(143\) 126124. 0.515770
\(144\) 0 0
\(145\) 266522. 1.05272
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 117988. 0.435383 0.217692 0.976018i \(-0.430147\pi\)
0.217692 + 0.976018i \(0.430147\pi\)
\(150\) 0 0
\(151\) 415301. 1.48225 0.741123 0.671369i \(-0.234294\pi\)
0.741123 + 0.671369i \(0.234294\pi\)
\(152\) 0 0
\(153\) 520658. 1.79814
\(154\) 0 0
\(155\) −327604. −1.09527
\(156\) 0 0
\(157\) 453131. 1.46715 0.733575 0.679608i \(-0.237850\pi\)
0.733575 + 0.679608i \(0.237850\pi\)
\(158\) 0 0
\(159\) 19354.8 0.0607151
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −432334. −1.27453 −0.637265 0.770645i \(-0.719934\pi\)
−0.637265 + 0.770645i \(0.719934\pi\)
\(164\) 0 0
\(165\) 22093.3 0.0631758
\(166\) 0 0
\(167\) −343081. −0.951932 −0.475966 0.879464i \(-0.657901\pi\)
−0.475966 + 0.879464i \(0.657901\pi\)
\(168\) 0 0
\(169\) 60091.2 0.161843
\(170\) 0 0
\(171\) −66428.0 −0.173724
\(172\) 0 0
\(173\) 443505. 1.12663 0.563317 0.826241i \(-0.309525\pi\)
0.563317 + 0.826241i \(0.309525\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20351.9 −0.0488284
\(178\) 0 0
\(179\) −95299.6 −0.222310 −0.111155 0.993803i \(-0.535455\pi\)
−0.111155 + 0.993803i \(0.535455\pi\)
\(180\) 0 0
\(181\) 78562.3 0.178245 0.0891226 0.996021i \(-0.471594\pi\)
0.0891226 + 0.996021i \(0.471594\pi\)
\(182\) 0 0
\(183\) −50998.4 −0.112572
\(184\) 0 0
\(185\) −149012. −0.320104
\(186\) 0 0
\(187\) −420420. −0.879182
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 294761. 0.584638 0.292319 0.956321i \(-0.405573\pi\)
0.292319 + 0.956321i \(0.405573\pi\)
\(192\) 0 0
\(193\) −874656. −1.69022 −0.845111 0.534590i \(-0.820466\pi\)
−0.845111 + 0.534590i \(0.820466\pi\)
\(194\) 0 0
\(195\) 75566.4 0.142312
\(196\) 0 0
\(197\) 338078. 0.620656 0.310328 0.950630i \(-0.399561\pi\)
0.310328 + 0.950630i \(0.399561\pi\)
\(198\) 0 0
\(199\) 353044. 0.631970 0.315985 0.948764i \(-0.397665\pi\)
0.315985 + 0.948764i \(0.397665\pi\)
\(200\) 0 0
\(201\) −120027. −0.209551
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −120942. −0.200998
\(206\) 0 0
\(207\) 883137. 1.43253
\(208\) 0 0
\(209\) 53639.2 0.0849408
\(210\) 0 0
\(211\) 351876. 0.544106 0.272053 0.962282i \(-0.412297\pi\)
0.272053 + 0.962282i \(0.412297\pi\)
\(212\) 0 0
\(213\) 58035.4 0.0876484
\(214\) 0 0
\(215\) −871896. −1.28638
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −52179.5 −0.0735174
\(220\) 0 0
\(221\) −1.43797e6 −1.98048
\(222\) 0 0
\(223\) −434832. −0.585544 −0.292772 0.956182i \(-0.594578\pi\)
−0.292772 + 0.956182i \(0.594578\pi\)
\(224\) 0 0
\(225\) 136417. 0.179644
\(226\) 0 0
\(227\) −1.07056e6 −1.37894 −0.689472 0.724312i \(-0.742157\pi\)
−0.689472 + 0.724312i \(0.742157\pi\)
\(228\) 0 0
\(229\) −131866. −0.166167 −0.0830836 0.996543i \(-0.526477\pi\)
−0.0830836 + 0.996543i \(0.526477\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 62663.0 0.0756174 0.0378087 0.999285i \(-0.487962\pi\)
0.0378087 + 0.999285i \(0.487962\pi\)
\(234\) 0 0
\(235\) 389984. 0.460656
\(236\) 0 0
\(237\) 104593. 0.120957
\(238\) 0 0
\(239\) −1.29337e6 −1.46462 −0.732312 0.680969i \(-0.761559\pi\)
−0.732312 + 0.680969i \(0.761559\pi\)
\(240\) 0 0
\(241\) −116282. −0.128965 −0.0644823 0.997919i \(-0.520540\pi\)
−0.0644823 + 0.997919i \(0.520540\pi\)
\(242\) 0 0
\(243\) −392075. −0.425945
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 183464. 0.191341
\(248\) 0 0
\(249\) −221356. −0.226252
\(250\) 0 0
\(251\) −360974. −0.361653 −0.180827 0.983515i \(-0.557877\pi\)
−0.180827 + 0.983515i \(0.557877\pi\)
\(252\) 0 0
\(253\) −713114. −0.700419
\(254\) 0 0
\(255\) −251893. −0.242586
\(256\) 0 0
\(257\) 1.93335e6 1.82591 0.912953 0.408064i \(-0.133796\pi\)
0.912953 + 0.408064i \(0.133796\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.25482e6 1.14020
\(262\) 0 0
\(263\) 214788. 0.191479 0.0957396 0.995406i \(-0.469478\pi\)
0.0957396 + 0.995406i \(0.469478\pi\)
\(264\) 0 0
\(265\) 429205. 0.375449
\(266\) 0 0
\(267\) −134688. −0.115625
\(268\) 0 0
\(269\) −1.90275e6 −1.60325 −0.801625 0.597827i \(-0.796031\pi\)
−0.801625 + 0.597827i \(0.796031\pi\)
\(270\) 0 0
\(271\) 1.28492e6 1.06281 0.531403 0.847119i \(-0.321665\pi\)
0.531403 + 0.847119i \(0.321665\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −110154. −0.0878352
\(276\) 0 0
\(277\) 1.42670e6 1.11720 0.558602 0.829436i \(-0.311338\pi\)
0.558602 + 0.829436i \(0.311338\pi\)
\(278\) 0 0
\(279\) −1.54240e6 −1.18628
\(280\) 0 0
\(281\) 492636. 0.372186 0.186093 0.982532i \(-0.440417\pi\)
0.186093 + 0.982532i \(0.440417\pi\)
\(282\) 0 0
\(283\) −2.01296e6 −1.49406 −0.747031 0.664789i \(-0.768522\pi\)
−0.747031 + 0.664789i \(0.768522\pi\)
\(284\) 0 0
\(285\) 32137.7 0.0234370
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.37348e6 2.37593
\(290\) 0 0
\(291\) −167397. −0.115882
\(292\) 0 0
\(293\) 58017.9 0.0394814 0.0197407 0.999805i \(-0.493716\pi\)
0.0197407 + 0.999805i \(0.493716\pi\)
\(294\) 0 0
\(295\) −451317. −0.301944
\(296\) 0 0
\(297\) 210305. 0.138343
\(298\) 0 0
\(299\) −2.43909e6 −1.57779
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −407402. −0.254927
\(304\) 0 0
\(305\) −1.13092e6 −0.696118
\(306\) 0 0
\(307\) −3.08036e6 −1.86533 −0.932665 0.360744i \(-0.882523\pi\)
−0.932665 + 0.360744i \(0.882523\pi\)
\(308\) 0 0
\(309\) 19843.9 0.0118231
\(310\) 0 0
\(311\) 561853. 0.329399 0.164699 0.986344i \(-0.447335\pi\)
0.164699 + 0.986344i \(0.447335\pi\)
\(312\) 0 0
\(313\) −2.97128e6 −1.71428 −0.857141 0.515082i \(-0.827761\pi\)
−0.857141 + 0.515082i \(0.827761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −750487. −0.419465 −0.209732 0.977759i \(-0.567259\pi\)
−0.209732 + 0.977759i \(0.567259\pi\)
\(318\) 0 0
\(319\) −1.01324e6 −0.557487
\(320\) 0 0
\(321\) 393414. 0.213102
\(322\) 0 0
\(323\) −611557. −0.326160
\(324\) 0 0
\(325\) −376763. −0.197861
\(326\) 0 0
\(327\) 378808. 0.195907
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.68677e6 1.34791 0.673954 0.738773i \(-0.264595\pi\)
0.673954 + 0.738773i \(0.264595\pi\)
\(332\) 0 0
\(333\) −701564. −0.346702
\(334\) 0 0
\(335\) −2.66168e6 −1.29582
\(336\) 0 0
\(337\) 49365.1 0.0236780 0.0118390 0.999930i \(-0.496231\pi\)
0.0118390 + 0.999930i \(0.496231\pi\)
\(338\) 0 0
\(339\) −67432.8 −0.0318692
\(340\) 0 0
\(341\) 1.24545e6 0.580018
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −427259. −0.193261
\(346\) 0 0
\(347\) −1.21898e6 −0.543468 −0.271734 0.962372i \(-0.587597\pi\)
−0.271734 + 0.962372i \(0.587597\pi\)
\(348\) 0 0
\(349\) 3.25801e6 1.43182 0.715910 0.698193i \(-0.246012\pi\)
0.715910 + 0.698193i \(0.246012\pi\)
\(350\) 0 0
\(351\) 719313. 0.311638
\(352\) 0 0
\(353\) 635449. 0.271421 0.135711 0.990749i \(-0.456668\pi\)
0.135711 + 0.990749i \(0.456668\pi\)
\(354\) 0 0
\(355\) 1.28697e6 0.541998
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 943982. 0.386569 0.193285 0.981143i \(-0.438086\pi\)
0.193285 + 0.981143i \(0.438086\pi\)
\(360\) 0 0
\(361\) −2.39807e6 −0.968489
\(362\) 0 0
\(363\) 282846. 0.112663
\(364\) 0 0
\(365\) −1.15711e6 −0.454615
\(366\) 0 0
\(367\) −4.01448e6 −1.55584 −0.777918 0.628366i \(-0.783724\pi\)
−0.777918 + 0.628366i \(0.783724\pi\)
\(368\) 0 0
\(369\) −569408. −0.217700
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 167016. 0.0621566 0.0310783 0.999517i \(-0.490106\pi\)
0.0310783 + 0.999517i \(0.490106\pi\)
\(374\) 0 0
\(375\) −425538. −0.156264
\(376\) 0 0
\(377\) −3.46561e6 −1.25582
\(378\) 0 0
\(379\) −2.51875e6 −0.900716 −0.450358 0.892848i \(-0.648704\pi\)
−0.450358 + 0.892848i \(0.648704\pi\)
\(380\) 0 0
\(381\) −758809. −0.267806
\(382\) 0 0
\(383\) −1.14410e6 −0.398535 −0.199267 0.979945i \(-0.563856\pi\)
−0.199267 + 0.979945i \(0.563856\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.10499e6 −1.39327
\(388\) 0 0
\(389\) 2.43472e6 0.815785 0.407892 0.913030i \(-0.366264\pi\)
0.407892 + 0.913030i \(0.366264\pi\)
\(390\) 0 0
\(391\) 8.13044e6 2.68950
\(392\) 0 0
\(393\) −132630. −0.0433173
\(394\) 0 0
\(395\) 2.31942e6 0.747974
\(396\) 0 0
\(397\) 2.75120e6 0.876084 0.438042 0.898954i \(-0.355672\pi\)
0.438042 + 0.898954i \(0.355672\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −110452. −0.0343016 −0.0171508 0.999853i \(-0.505460\pi\)
−0.0171508 + 0.999853i \(0.505460\pi\)
\(402\) 0 0
\(403\) 4.25986e6 1.30657
\(404\) 0 0
\(405\) −2.79294e6 −0.846105
\(406\) 0 0
\(407\) 566497. 0.169516
\(408\) 0 0
\(409\) 170695. 0.0504560 0.0252280 0.999682i \(-0.491969\pi\)
0.0252280 + 0.999682i \(0.491969\pi\)
\(410\) 0 0
\(411\) −746393. −0.217953
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.90870e6 −1.39909
\(416\) 0 0
\(417\) 376002. 0.105889
\(418\) 0 0
\(419\) −2852.47 −0.000793754 0 −0.000396877 1.00000i \(-0.500126\pi\)
−0.000396877 1.00000i \(0.500126\pi\)
\(420\) 0 0
\(421\) −730555. −0.200885 −0.100443 0.994943i \(-0.532026\pi\)
−0.100443 + 0.994943i \(0.532026\pi\)
\(422\) 0 0
\(423\) 1.83609e6 0.498934
\(424\) 0 0
\(425\) 1.25590e6 0.337274
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −287281. −0.0753639
\(430\) 0 0
\(431\) 3.69410e6 0.957889 0.478945 0.877845i \(-0.341019\pi\)
0.478945 + 0.877845i \(0.341019\pi\)
\(432\) 0 0
\(433\) 674656. 0.172927 0.0864635 0.996255i \(-0.472443\pi\)
0.0864635 + 0.996255i \(0.472443\pi\)
\(434\) 0 0
\(435\) −607077. −0.153823
\(436\) 0 0
\(437\) −1.03732e6 −0.259842
\(438\) 0 0
\(439\) −3.95626e6 −0.979769 −0.489884 0.871787i \(-0.662961\pi\)
−0.489884 + 0.871787i \(0.662961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.29412e6 1.76589 0.882945 0.469477i \(-0.155557\pi\)
0.882945 + 0.469477i \(0.155557\pi\)
\(444\) 0 0
\(445\) −2.98679e6 −0.714997
\(446\) 0 0
\(447\) −268750. −0.0636179
\(448\) 0 0
\(449\) 5.49637e6 1.28665 0.643325 0.765593i \(-0.277554\pi\)
0.643325 + 0.765593i \(0.277554\pi\)
\(450\) 0 0
\(451\) 459785. 0.106442
\(452\) 0 0
\(453\) −945960. −0.216585
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.73755e6 −0.837137 −0.418568 0.908185i \(-0.637468\pi\)
−0.418568 + 0.908185i \(0.637468\pi\)
\(458\) 0 0
\(459\) −2.39775e6 −0.531218
\(460\) 0 0
\(461\) −414493. −0.0908374 −0.0454187 0.998968i \(-0.514462\pi\)
−0.0454187 + 0.998968i \(0.514462\pi\)
\(462\) 0 0
\(463\) 6.90895e6 1.49782 0.748910 0.662672i \(-0.230577\pi\)
0.748910 + 0.662672i \(0.230577\pi\)
\(464\) 0 0
\(465\) 746208. 0.160040
\(466\) 0 0
\(467\) 5.30705e6 1.12606 0.563029 0.826437i \(-0.309636\pi\)
0.563029 + 0.826437i \(0.309636\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.03213e6 −0.214379
\(472\) 0 0
\(473\) 3.31469e6 0.681223
\(474\) 0 0
\(475\) −160234. −0.0325852
\(476\) 0 0
\(477\) 2.02075e6 0.406646
\(478\) 0 0
\(479\) 9.03751e6 1.79974 0.899870 0.436157i \(-0.143661\pi\)
0.899870 + 0.436157i \(0.143661\pi\)
\(480\) 0 0
\(481\) 1.93761e6 0.381859
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.71213e6 −0.716587
\(486\) 0 0
\(487\) −4.45282e6 −0.850770 −0.425385 0.905012i \(-0.639861\pi\)
−0.425385 + 0.905012i \(0.639861\pi\)
\(488\) 0 0
\(489\) 984758. 0.186233
\(490\) 0 0
\(491\) 2.27049e6 0.425027 0.212513 0.977158i \(-0.431835\pi\)
0.212513 + 0.977158i \(0.431835\pi\)
\(492\) 0 0
\(493\) 1.15522e7 2.14067
\(494\) 0 0
\(495\) 2.30666e6 0.423127
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.47102e6 1.34316 0.671582 0.740931i \(-0.265615\pi\)
0.671582 + 0.740931i \(0.265615\pi\)
\(500\) 0 0
\(501\) 781461. 0.139095
\(502\) 0 0
\(503\) 6.64380e6 1.17084 0.585418 0.810731i \(-0.300930\pi\)
0.585418 + 0.810731i \(0.300930\pi\)
\(504\) 0 0
\(505\) −9.03438e6 −1.57641
\(506\) 0 0
\(507\) −136874. −0.0236484
\(508\) 0 0
\(509\) −8.66213e6 −1.48194 −0.740970 0.671539i \(-0.765634\pi\)
−0.740970 + 0.671539i \(0.765634\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 305917. 0.0513228
\(514\) 0 0
\(515\) 440050. 0.0731112
\(516\) 0 0
\(517\) −1.48260e6 −0.243948
\(518\) 0 0
\(519\) −1.01020e6 −0.164623
\(520\) 0 0
\(521\) 2.48870e6 0.401679 0.200839 0.979624i \(-0.435633\pi\)
0.200839 + 0.979624i \(0.435633\pi\)
\(522\) 0 0
\(523\) 8.36398e6 1.33708 0.668542 0.743674i \(-0.266919\pi\)
0.668542 + 0.743674i \(0.266919\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.41998e7 −2.22718
\(528\) 0 0
\(529\) 7.35447e6 1.14265
\(530\) 0 0
\(531\) −2.12485e6 −0.327033
\(532\) 0 0
\(533\) 1.57262e6 0.239775
\(534\) 0 0
\(535\) 8.72421e6 1.31778
\(536\) 0 0
\(537\) 217071. 0.0324837
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.12558e6 −0.752922 −0.376461 0.926433i \(-0.622859\pi\)
−0.376461 + 0.926433i \(0.622859\pi\)
\(542\) 0 0
\(543\) −178947. −0.0260450
\(544\) 0 0
\(545\) 8.40031e6 1.21145
\(546\) 0 0
\(547\) 8.27719e6 1.18281 0.591404 0.806375i \(-0.298574\pi\)
0.591404 + 0.806375i \(0.298574\pi\)
\(548\) 0 0
\(549\) −5.32451e6 −0.753961
\(550\) 0 0
\(551\) −1.47389e6 −0.206817
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 339415. 0.0467733
\(556\) 0 0
\(557\) 1.14592e7 1.56501 0.782503 0.622646i \(-0.213942\pi\)
0.782503 + 0.622646i \(0.213942\pi\)
\(558\) 0 0
\(559\) 1.13373e7 1.53455
\(560\) 0 0
\(561\) 957621. 0.128465
\(562\) 0 0
\(563\) −9.23070e6 −1.22734 −0.613668 0.789564i \(-0.710307\pi\)
−0.613668 + 0.789564i \(0.710307\pi\)
\(564\) 0 0
\(565\) −1.49536e6 −0.197072
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.29747e6 −1.07440 −0.537199 0.843456i \(-0.680518\pi\)
−0.537199 + 0.843456i \(0.680518\pi\)
\(570\) 0 0
\(571\) −1.00735e7 −1.29298 −0.646490 0.762922i \(-0.723764\pi\)
−0.646490 + 0.762922i \(0.723764\pi\)
\(572\) 0 0
\(573\) −671399. −0.0854268
\(574\) 0 0
\(575\) 2.13025e6 0.268696
\(576\) 0 0
\(577\) −6.42515e6 −0.803422 −0.401711 0.915767i \(-0.631584\pi\)
−0.401711 + 0.915767i \(0.631584\pi\)
\(578\) 0 0
\(579\) 1.99227e6 0.246974
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.63171e6 −0.198825
\(584\) 0 0
\(585\) 7.88954e6 0.953152
\(586\) 0 0
\(587\) 3.97124e6 0.475698 0.237849 0.971302i \(-0.423558\pi\)
0.237849 + 0.971302i \(0.423558\pi\)
\(588\) 0 0
\(589\) 1.81168e6 0.215176
\(590\) 0 0
\(591\) −770064. −0.0906897
\(592\) 0 0
\(593\) −4.89322e6 −0.571423 −0.285712 0.958316i \(-0.592230\pi\)
−0.285712 + 0.958316i \(0.592230\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −804155. −0.0923430
\(598\) 0 0
\(599\) −3.97986e6 −0.453212 −0.226606 0.973987i \(-0.572763\pi\)
−0.226606 + 0.973987i \(0.572763\pi\)
\(600\) 0 0
\(601\) −6.77069e6 −0.764622 −0.382311 0.924034i \(-0.624872\pi\)
−0.382311 + 0.924034i \(0.624872\pi\)
\(602\) 0 0
\(603\) −1.25315e7 −1.40349
\(604\) 0 0
\(605\) 6.27227e6 0.696685
\(606\) 0 0
\(607\) 1.52249e7 1.67719 0.838596 0.544754i \(-0.183377\pi\)
0.838596 + 0.544754i \(0.183377\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.07099e6 −0.549528
\(612\) 0 0
\(613\) −1.16036e7 −1.24721 −0.623606 0.781739i \(-0.714333\pi\)
−0.623606 + 0.781739i \(0.714333\pi\)
\(614\) 0 0
\(615\) 275478. 0.0293697
\(616\) 0 0
\(617\) −4.51891e6 −0.477882 −0.238941 0.971034i \(-0.576800\pi\)
−0.238941 + 0.971034i \(0.576800\pi\)
\(618\) 0 0
\(619\) 6.87077e6 0.720741 0.360370 0.932809i \(-0.382650\pi\)
0.360370 + 0.932809i \(0.382650\pi\)
\(620\) 0 0
\(621\) −4.06706e6 −0.423206
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.64395e6 −0.782741
\(626\) 0 0
\(627\) −122178. −0.0124115
\(628\) 0 0
\(629\) −6.45882e6 −0.650918
\(630\) 0 0
\(631\) 3.17504e6 0.317450 0.158725 0.987323i \(-0.449262\pi\)
0.158725 + 0.987323i \(0.449262\pi\)
\(632\) 0 0
\(633\) −801494. −0.0795044
\(634\) 0 0
\(635\) −1.68271e7 −1.65605
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.05921e6 0.587034
\(640\) 0 0
\(641\) 3.71197e6 0.356829 0.178414 0.983955i \(-0.442903\pi\)
0.178414 + 0.983955i \(0.442903\pi\)
\(642\) 0 0
\(643\) 2.23112e6 0.212811 0.106406 0.994323i \(-0.466066\pi\)
0.106406 + 0.994323i \(0.466066\pi\)
\(644\) 0 0
\(645\) 1.98598e6 0.187964
\(646\) 0 0
\(647\) 7.06649e6 0.663656 0.331828 0.943340i \(-0.392335\pi\)
0.331828 + 0.943340i \(0.392335\pi\)
\(648\) 0 0
\(649\) 1.71577e6 0.159900
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.83924e7 1.68793 0.843966 0.536397i \(-0.180215\pi\)
0.843966 + 0.536397i \(0.180215\pi\)
\(654\) 0 0
\(655\) −2.94116e6 −0.267864
\(656\) 0 0
\(657\) −5.44782e6 −0.492390
\(658\) 0 0
\(659\) 4.02375e6 0.360926 0.180463 0.983582i \(-0.442240\pi\)
0.180463 + 0.983582i \(0.442240\pi\)
\(660\) 0 0
\(661\) 2.52922e6 0.225156 0.112578 0.993643i \(-0.464089\pi\)
0.112578 + 0.993643i \(0.464089\pi\)
\(662\) 0 0
\(663\) 3.27538e6 0.289386
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.95949e7 1.70540
\(668\) 0 0
\(669\) 990449. 0.0855592
\(670\) 0 0
\(671\) 4.29942e6 0.368641
\(672\) 0 0
\(673\) 1.31377e7 1.11811 0.559053 0.829132i \(-0.311165\pi\)
0.559053 + 0.829132i \(0.311165\pi\)
\(674\) 0 0
\(675\) −628235. −0.0530716
\(676\) 0 0
\(677\) −2.00815e6 −0.168393 −0.0841965 0.996449i \(-0.526832\pi\)
−0.0841965 + 0.996449i \(0.526832\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 2.43849e6 0.201490
\(682\) 0 0
\(683\) 3.35825e6 0.275462 0.137731 0.990470i \(-0.456019\pi\)
0.137731 + 0.990470i \(0.456019\pi\)
\(684\) 0 0
\(685\) −1.65517e7 −1.34777
\(686\) 0 0
\(687\) 300362. 0.0242802
\(688\) 0 0
\(689\) −5.58099e6 −0.447881
\(690\) 0 0
\(691\) 1.79752e6 0.143211 0.0716057 0.997433i \(-0.477188\pi\)
0.0716057 + 0.997433i \(0.477188\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.33808e6 0.654793
\(696\) 0 0
\(697\) −5.24215e6 −0.408722
\(698\) 0 0
\(699\) −142732. −0.0110492
\(700\) 0 0
\(701\) −1.74200e7 −1.33891 −0.669457 0.742851i \(-0.733473\pi\)
−0.669457 + 0.742851i \(0.733473\pi\)
\(702\) 0 0
\(703\) 824047. 0.0628874
\(704\) 0 0
\(705\) −888295. −0.0673107
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 173533. 0.0129648 0.00648240 0.999979i \(-0.497937\pi\)
0.00648240 + 0.999979i \(0.497937\pi\)
\(710\) 0 0
\(711\) 1.09201e7 0.810125
\(712\) 0 0
\(713\) −2.40856e7 −1.77433
\(714\) 0 0
\(715\) −6.37063e6 −0.466034
\(716\) 0 0
\(717\) 2.94599e6 0.214010
\(718\) 0 0
\(719\) 1.44188e7 1.04018 0.520089 0.854112i \(-0.325899\pi\)
0.520089 + 0.854112i \(0.325899\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 264865. 0.0188442
\(724\) 0 0
\(725\) 3.02680e6 0.213864
\(726\) 0 0
\(727\) 1.47752e7 1.03681 0.518403 0.855136i \(-0.326527\pi\)
0.518403 + 0.855136i \(0.326527\pi\)
\(728\) 0 0
\(729\) −1.25433e7 −0.874165
\(730\) 0 0
\(731\) −3.77918e7 −2.61580
\(732\) 0 0
\(733\) 8.73446e6 0.600449 0.300225 0.953869i \(-0.402938\pi\)
0.300225 + 0.953869i \(0.402938\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.01189e7 0.686222
\(738\) 0 0
\(739\) −2.56930e7 −1.73063 −0.865314 0.501229i \(-0.832881\pi\)
−0.865314 + 0.501229i \(0.832881\pi\)
\(740\) 0 0
\(741\) −417889. −0.0279586
\(742\) 0 0
\(743\) −3.71107e6 −0.246619 −0.123310 0.992368i \(-0.539351\pi\)
−0.123310 + 0.992368i \(0.539351\pi\)
\(744\) 0 0
\(745\) −5.95969e6 −0.393399
\(746\) 0 0
\(747\) −2.31107e7 −1.51535
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −876028. −0.0566785 −0.0283392 0.999598i \(-0.509022\pi\)
−0.0283392 + 0.999598i \(0.509022\pi\)
\(752\) 0 0
\(753\) 822218. 0.0528445
\(754\) 0 0
\(755\) −2.09773e7 −1.33931
\(756\) 0 0
\(757\) 1.30192e7 0.825743 0.412871 0.910789i \(-0.364526\pi\)
0.412871 + 0.910789i \(0.364526\pi\)
\(758\) 0 0
\(759\) 1.62431e6 0.102345
\(760\) 0 0
\(761\) −7.80270e6 −0.488409 −0.244204 0.969724i \(-0.578527\pi\)
−0.244204 + 0.969724i \(0.578527\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.62990e7 −1.62474
\(766\) 0 0
\(767\) 5.86850e6 0.360196
\(768\) 0 0
\(769\) 1.99560e7 1.21691 0.608454 0.793589i \(-0.291790\pi\)
0.608454 + 0.793589i \(0.291790\pi\)
\(770\) 0 0
\(771\) −4.40374e6 −0.266800
\(772\) 0 0
\(773\) −2.74909e7 −1.65478 −0.827390 0.561628i \(-0.810175\pi\)
−0.827390 + 0.561628i \(0.810175\pi\)
\(774\) 0 0
\(775\) −3.72049e6 −0.222508
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 668819. 0.0394880
\(780\) 0 0
\(781\) −4.89268e6 −0.287024
\(782\) 0 0
\(783\) −5.77873e6 −0.336844
\(784\) 0 0
\(785\) −2.28881e7 −1.32567
\(786\) 0 0
\(787\) −2.84767e7 −1.63890 −0.819451 0.573149i \(-0.805721\pi\)
−0.819451 + 0.573149i \(0.805721\pi\)
\(788\) 0 0
\(789\) −489239. −0.0279788
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.47055e7 0.830415
\(794\) 0 0
\(795\) −977632. −0.0548603
\(796\) 0 0
\(797\) 1.32245e7 0.737450 0.368725 0.929538i \(-0.379794\pi\)
0.368725 + 0.929538i \(0.379794\pi\)
\(798\) 0 0
\(799\) 1.69036e7 0.936726
\(800\) 0 0
\(801\) −1.40621e7 −0.774409
\(802\) 0 0
\(803\) 4.39900e6 0.240749
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.33403e6 0.234266
\(808\) 0 0
\(809\) −1.13419e7 −0.609276 −0.304638 0.952468i \(-0.598536\pi\)
−0.304638 + 0.952468i \(0.598536\pi\)
\(810\) 0 0
\(811\) −6.47233e6 −0.345548 −0.172774 0.984961i \(-0.555273\pi\)
−0.172774 + 0.984961i \(0.555273\pi\)
\(812\) 0 0
\(813\) −2.92677e6 −0.155296
\(814\) 0 0
\(815\) 2.18376e7 1.15163
\(816\) 0 0
\(817\) 4.82166e6 0.252721
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.83476e7 0.949996 0.474998 0.879987i \(-0.342449\pi\)
0.474998 + 0.879987i \(0.342449\pi\)
\(822\) 0 0
\(823\) 1.00423e7 0.516814 0.258407 0.966036i \(-0.416803\pi\)
0.258407 + 0.966036i \(0.416803\pi\)
\(824\) 0 0
\(825\) 250906. 0.0128344
\(826\) 0 0
\(827\) −334853. −0.0170251 −0.00851256 0.999964i \(-0.502710\pi\)
−0.00851256 + 0.999964i \(0.502710\pi\)
\(828\) 0 0
\(829\) −3.12894e7 −1.58129 −0.790643 0.612277i \(-0.790254\pi\)
−0.790643 + 0.612277i \(0.790254\pi\)
\(830\) 0 0
\(831\) −3.24969e6 −0.163245
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.73294e7 0.860136
\(836\) 0 0
\(837\) 7.10312e6 0.350457
\(838\) 0 0
\(839\) −2.42772e7 −1.19068 −0.595339 0.803475i \(-0.702982\pi\)
−0.595339 + 0.803475i \(0.702982\pi\)
\(840\) 0 0
\(841\) 7.33046e6 0.357389
\(842\) 0 0
\(843\) −1.12211e6 −0.0543836
\(844\) 0 0
\(845\) −3.03527e6 −0.146236
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.58506e6 0.218311
\(850\) 0 0
\(851\) −1.09554e7 −0.518567
\(852\) 0 0
\(853\) 3.17995e7 1.49640 0.748200 0.663474i \(-0.230919\pi\)
0.748200 + 0.663474i \(0.230919\pi\)
\(854\) 0 0
\(855\) 3.35535e6 0.156972
\(856\) 0 0
\(857\) 3.42848e7 1.59459 0.797296 0.603588i \(-0.206263\pi\)
0.797296 + 0.603588i \(0.206263\pi\)
\(858\) 0 0
\(859\) 7.63560e6 0.353070 0.176535 0.984294i \(-0.443511\pi\)
0.176535 + 0.984294i \(0.443511\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.60447e7 −1.64746 −0.823729 0.566983i \(-0.808110\pi\)
−0.823729 + 0.566983i \(0.808110\pi\)
\(864\) 0 0
\(865\) −2.24019e7 −1.01799
\(866\) 0 0
\(867\) −7.68402e6 −0.347169
\(868\) 0 0
\(869\) −8.81773e6 −0.396102
\(870\) 0 0
\(871\) 3.46100e7 1.54581
\(872\) 0 0
\(873\) −1.74771e7 −0.776130
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.25755e7 1.43018 0.715092 0.699030i \(-0.246385\pi\)
0.715092 + 0.699030i \(0.246385\pi\)
\(878\) 0 0
\(879\) −132152. −0.00576900
\(880\) 0 0
\(881\) −3.83192e7 −1.66332 −0.831661 0.555284i \(-0.812610\pi\)
−0.831661 + 0.555284i \(0.812610\pi\)
\(882\) 0 0
\(883\) −3.50607e7 −1.51328 −0.756640 0.653832i \(-0.773160\pi\)
−0.756640 + 0.653832i \(0.773160\pi\)
\(884\) 0 0
\(885\) 1.02800e6 0.0441198
\(886\) 0 0
\(887\) −1.96135e7 −0.837041 −0.418521 0.908207i \(-0.637451\pi\)
−0.418521 + 0.908207i \(0.637451\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.06179e7 0.448069
\(892\) 0 0
\(893\) −2.15664e6 −0.0905002
\(894\) 0 0
\(895\) 4.81368e6 0.200872
\(896\) 0 0
\(897\) 5.55568e6 0.230545
\(898\) 0 0
\(899\) −3.42224e7 −1.41225
\(900\) 0 0
\(901\) 1.86036e7 0.763459
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.96826e6 −0.161057
\(906\) 0 0
\(907\) −2.41242e7 −0.973721 −0.486860 0.873480i \(-0.661858\pi\)
−0.486860 + 0.873480i \(0.661858\pi\)
\(908\) 0 0
\(909\) −4.25349e7 −1.70740
\(910\) 0 0
\(911\) −346019. −0.0138135 −0.00690676 0.999976i \(-0.502199\pi\)
−0.00690676 + 0.999976i \(0.502199\pi\)
\(912\) 0 0
\(913\) 1.86614e7 0.740913
\(914\) 0 0
\(915\) 2.57598e6 0.101716
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −6.54640e6 −0.255690 −0.127845 0.991794i \(-0.540806\pi\)
−0.127845 + 0.991794i \(0.540806\pi\)
\(920\) 0 0
\(921\) 7.01636e6 0.272560
\(922\) 0 0
\(923\) −1.67346e7 −0.646562
\(924\) 0 0
\(925\) −1.69227e6 −0.0650303
\(926\) 0 0
\(927\) 2.07181e6 0.0791863
\(928\) 0 0
\(929\) 1.84856e7 0.702740 0.351370 0.936237i \(-0.385716\pi\)
0.351370 + 0.936237i \(0.385716\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1.27977e6 −0.0481315
\(934\) 0 0
\(935\) 2.12358e7 0.794402
\(936\) 0 0
\(937\) 4.28083e7 1.59286 0.796432 0.604729i \(-0.206718\pi\)
0.796432 + 0.604729i \(0.206718\pi\)
\(938\) 0 0
\(939\) 6.76789e6 0.250490
\(940\) 0 0
\(941\) 1.84060e7 0.677619 0.338810 0.940855i \(-0.389976\pi\)
0.338810 + 0.940855i \(0.389976\pi\)
\(942\) 0 0
\(943\) −8.89171e6 −0.325616
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.71884e7 0.622818 0.311409 0.950276i \(-0.399199\pi\)
0.311409 + 0.950276i \(0.399199\pi\)
\(948\) 0 0
\(949\) 1.50460e7 0.542321
\(950\) 0 0
\(951\) 1.70944e6 0.0612918
\(952\) 0 0
\(953\) 3.40220e7 1.21347 0.606733 0.794905i \(-0.292480\pi\)
0.606733 + 0.794905i \(0.292480\pi\)
\(954\) 0 0
\(955\) −1.48887e7 −0.528260
\(956\) 0 0
\(957\) 2.30792e6 0.0814595
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.34364e7 0.469327
\(962\) 0 0
\(963\) 4.10746e7 1.42727
\(964\) 0 0
\(965\) 4.41798e7 1.52723
\(966\) 0 0
\(967\) −1.22029e6 −0.0419658 −0.0209829 0.999780i \(-0.506680\pi\)
−0.0209829 + 0.999780i \(0.506680\pi\)
\(968\) 0 0
\(969\) 1.39299e6 0.0476582
\(970\) 0 0
\(971\) 1.86907e7 0.636176 0.318088 0.948061i \(-0.396959\pi\)
0.318088 + 0.948061i \(0.396959\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 858181. 0.0289113
\(976\) 0 0
\(977\) −2.74507e7 −0.920062 −0.460031 0.887903i \(-0.652162\pi\)
−0.460031 + 0.887903i \(0.652162\pi\)
\(978\) 0 0
\(979\) 1.13549e7 0.378639
\(980\) 0 0
\(981\) 3.95496e7 1.31211
\(982\) 0 0
\(983\) −1.11626e7 −0.368453 −0.184227 0.982884i \(-0.558978\pi\)
−0.184227 + 0.982884i \(0.558978\pi\)
\(984\) 0 0
\(985\) −1.70766e7 −0.560805
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.41023e7 −2.08393
\(990\) 0 0
\(991\) 4.48808e7 1.45170 0.725850 0.687853i \(-0.241447\pi\)
0.725850 + 0.687853i \(0.241447\pi\)
\(992\) 0 0
\(993\) −6.11985e6 −0.196955
\(994\) 0 0
\(995\) −1.78326e7 −0.571028
\(996\) 0 0
\(997\) 2.06308e6 0.0657321 0.0328661 0.999460i \(-0.489537\pi\)
0.0328661 + 0.999460i \(0.489537\pi\)
\(998\) 0 0
\(999\) 3.23087e6 0.102425
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 392.6.a.j.1.3 5
4.3 odd 2 784.6.a.bl.1.3 5
7.2 even 3 392.6.i.o.361.3 10
7.3 odd 6 56.6.i.b.9.3 10
7.4 even 3 392.6.i.o.177.3 10
7.5 odd 6 56.6.i.b.25.3 yes 10
7.6 odd 2 392.6.a.k.1.3 5
21.5 even 6 504.6.s.b.361.5 10
21.17 even 6 504.6.s.b.289.5 10
28.3 even 6 112.6.i.f.65.3 10
28.19 even 6 112.6.i.f.81.3 10
28.27 even 2 784.6.a.bk.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.i.b.9.3 10 7.3 odd 6
56.6.i.b.25.3 yes 10 7.5 odd 6
112.6.i.f.65.3 10 28.3 even 6
112.6.i.f.81.3 10 28.19 even 6
392.6.a.j.1.3 5 1.1 even 1 trivial
392.6.a.k.1.3 5 7.6 odd 2
392.6.i.o.177.3 10 7.4 even 3
392.6.i.o.361.3 10 7.2 even 3
504.6.s.b.289.5 10 21.17 even 6
504.6.s.b.361.5 10 21.5 even 6
784.6.a.bk.1.3 5 28.27 even 2
784.6.a.bl.1.3 5 4.3 odd 2