Properties

Label 588.6.a.n.1.3
Level $588$
Weight $6$
Character 588.1
Self dual yes
Analytic conductor $94.306$
Analytic rank $0$
Dimension $4$
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] = [588,6,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, 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("588.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 699x^{2} - 686x + 59664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-10.9924\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +46.1154 q^{5} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +46.1154 q^{5} +81.0000 q^{9} -631.165 q^{11} -1079.22 q^{13} -415.038 q^{15} +161.156 q^{17} +1176.86 q^{19} +2163.46 q^{23} -998.372 q^{25} -729.000 q^{27} -4492.01 q^{29} +318.260 q^{31} +5680.48 q^{33} +15186.8 q^{37} +9713.02 q^{39} +20587.2 q^{41} -455.118 q^{43} +3735.35 q^{45} -20762.8 q^{47} -1450.40 q^{51} -19300.1 q^{53} -29106.4 q^{55} -10591.7 q^{57} +6368.31 q^{59} -49145.1 q^{61} -49768.9 q^{65} +34054.0 q^{67} -19471.1 q^{69} +62962.4 q^{71} -8867.76 q^{73} +8985.34 q^{75} +34413.2 q^{79} +6561.00 q^{81} -7041.42 q^{83} +7431.75 q^{85} +40428.1 q^{87} -20242.8 q^{89} -2864.34 q^{93} +54271.1 q^{95} +54066.6 q^{97} -51124.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{3} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{3} + 324 q^{9} + 462 q^{11} - 602 q^{13} - 228 q^{17} - 358 q^{19} + 2148 q^{23} + 5454 q^{25} - 2916 q^{27} - 5532 q^{29} - 830 q^{31} - 4158 q^{33} + 3914 q^{37} + 5418 q^{39} - 8316 q^{41} - 14518 q^{43} - 41700 q^{47} + 2052 q^{51} - 22164 q^{53} + 3892 q^{55} + 3222 q^{57} - 32886 q^{59} - 83732 q^{61} + 93192 q^{65} + 80034 q^{67} - 19332 q^{69} + 44772 q^{71} + 22470 q^{73} - 49086 q^{75} + 75286 q^{79} + 26244 q^{81} - 17418 q^{83} + 139252 q^{85} + 49788 q^{87} - 28944 q^{89} + 7470 q^{93} + 144120 q^{95} - 216678 q^{97} + 37422 q^{99}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) 46.1154 0.824937 0.412469 0.910972i \(-0.364667\pi\)
0.412469 + 0.910972i \(0.364667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −631.165 −1.57276 −0.786378 0.617746i \(-0.788046\pi\)
−0.786378 + 0.617746i \(0.788046\pi\)
\(12\) 0 0
\(13\) −1079.22 −1.77114 −0.885571 0.464503i \(-0.846233\pi\)
−0.885571 + 0.464503i \(0.846233\pi\)
\(14\) 0 0
\(15\) −415.038 −0.476278
\(16\) 0 0
\(17\) 161.156 0.135246 0.0676228 0.997711i \(-0.478459\pi\)
0.0676228 + 0.997711i \(0.478459\pi\)
\(18\) 0 0
\(19\) 1176.86 0.747892 0.373946 0.927450i \(-0.378004\pi\)
0.373946 + 0.927450i \(0.378004\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2163.46 0.852765 0.426382 0.904543i \(-0.359788\pi\)
0.426382 + 0.904543i \(0.359788\pi\)
\(24\) 0 0
\(25\) −998.372 −0.319479
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −4492.01 −0.991850 −0.495925 0.868365i \(-0.665171\pi\)
−0.495925 + 0.868365i \(0.665171\pi\)
\(30\) 0 0
\(31\) 318.260 0.0594809 0.0297405 0.999558i \(-0.490532\pi\)
0.0297405 + 0.999558i \(0.490532\pi\)
\(32\) 0 0
\(33\) 5680.48 0.908031
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 15186.8 1.82374 0.911869 0.410482i \(-0.134639\pi\)
0.911869 + 0.410482i \(0.134639\pi\)
\(38\) 0 0
\(39\) 9713.02 1.02257
\(40\) 0 0
\(41\) 20587.2 1.91266 0.956330 0.292289i \(-0.0944169\pi\)
0.956330 + 0.292289i \(0.0944169\pi\)
\(42\) 0 0
\(43\) −455.118 −0.0375364 −0.0187682 0.999824i \(-0.505974\pi\)
−0.0187682 + 0.999824i \(0.505974\pi\)
\(44\) 0 0
\(45\) 3735.35 0.274979
\(46\) 0 0
\(47\) −20762.8 −1.37101 −0.685504 0.728068i \(-0.740418\pi\)
−0.685504 + 0.728068i \(0.740418\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1450.40 −0.0780841
\(52\) 0 0
\(53\) −19300.1 −0.943776 −0.471888 0.881658i \(-0.656427\pi\)
−0.471888 + 0.881658i \(0.656427\pi\)
\(54\) 0 0
\(55\) −29106.4 −1.29742
\(56\) 0 0
\(57\) −10591.7 −0.431796
\(58\) 0 0
\(59\) 6368.31 0.238174 0.119087 0.992884i \(-0.462003\pi\)
0.119087 + 0.992884i \(0.462003\pi\)
\(60\) 0 0
\(61\) −49145.1 −1.69105 −0.845524 0.533938i \(-0.820712\pi\)
−0.845524 + 0.533938i \(0.820712\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −49768.9 −1.46108
\(66\) 0 0
\(67\) 34054.0 0.926790 0.463395 0.886152i \(-0.346631\pi\)
0.463395 + 0.886152i \(0.346631\pi\)
\(68\) 0 0
\(69\) −19471.1 −0.492344
\(70\) 0 0
\(71\) 62962.4 1.48230 0.741149 0.671341i \(-0.234281\pi\)
0.741149 + 0.671341i \(0.234281\pi\)
\(72\) 0 0
\(73\) −8867.76 −0.194763 −0.0973815 0.995247i \(-0.531047\pi\)
−0.0973815 + 0.995247i \(0.531047\pi\)
\(74\) 0 0
\(75\) 8985.34 0.184451
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 34413.2 0.620380 0.310190 0.950675i \(-0.399607\pi\)
0.310190 + 0.950675i \(0.399607\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −7041.42 −0.112193 −0.0560964 0.998425i \(-0.517865\pi\)
−0.0560964 + 0.998425i \(0.517865\pi\)
\(84\) 0 0
\(85\) 7431.75 0.111569
\(86\) 0 0
\(87\) 40428.1 0.572645
\(88\) 0 0
\(89\) −20242.8 −0.270891 −0.135445 0.990785i \(-0.543247\pi\)
−0.135445 + 0.990785i \(0.543247\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2864.34 −0.0343413
\(94\) 0 0
\(95\) 54271.1 0.616964
\(96\) 0 0
\(97\) 54066.6 0.583444 0.291722 0.956503i \(-0.405772\pi\)
0.291722 + 0.956503i \(0.405772\pi\)
\(98\) 0 0
\(99\) −51124.4 −0.524252
\(100\) 0 0
\(101\) −91486.3 −0.892386 −0.446193 0.894937i \(-0.647220\pi\)
−0.446193 + 0.894937i \(0.647220\pi\)
\(102\) 0 0
\(103\) 75381.4 0.700118 0.350059 0.936728i \(-0.386162\pi\)
0.350059 + 0.936728i \(0.386162\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 152133. 1.28459 0.642293 0.766459i \(-0.277983\pi\)
0.642293 + 0.766459i \(0.277983\pi\)
\(108\) 0 0
\(109\) 76671.9 0.618116 0.309058 0.951043i \(-0.399986\pi\)
0.309058 + 0.951043i \(0.399986\pi\)
\(110\) 0 0
\(111\) −136681. −1.05294
\(112\) 0 0
\(113\) 228515. 1.68352 0.841760 0.539852i \(-0.181520\pi\)
0.841760 + 0.539852i \(0.181520\pi\)
\(114\) 0 0
\(115\) 99768.8 0.703477
\(116\) 0 0
\(117\) −87417.2 −0.590381
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 237318. 1.47356
\(122\) 0 0
\(123\) −185285. −1.10427
\(124\) 0 0
\(125\) −190151. −1.08849
\(126\) 0 0
\(127\) 122111. 0.671809 0.335905 0.941896i \(-0.390958\pi\)
0.335905 + 0.941896i \(0.390958\pi\)
\(128\) 0 0
\(129\) 4096.06 0.0216717
\(130\) 0 0
\(131\) 75795.3 0.385890 0.192945 0.981210i \(-0.438196\pi\)
0.192945 + 0.981210i \(0.438196\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −33618.1 −0.158759
\(136\) 0 0
\(137\) −241611. −1.09981 −0.549903 0.835229i \(-0.685335\pi\)
−0.549903 + 0.835229i \(0.685335\pi\)
\(138\) 0 0
\(139\) 125657. 0.551634 0.275817 0.961210i \(-0.411052\pi\)
0.275817 + 0.961210i \(0.411052\pi\)
\(140\) 0 0
\(141\) 186865. 0.791552
\(142\) 0 0
\(143\) 681169. 2.78557
\(144\) 0 0
\(145\) −207151. −0.818214
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 36046.6 0.133014 0.0665071 0.997786i \(-0.478814\pi\)
0.0665071 + 0.997786i \(0.478814\pi\)
\(150\) 0 0
\(151\) 150668. 0.537748 0.268874 0.963175i \(-0.413349\pi\)
0.268874 + 0.963175i \(0.413349\pi\)
\(152\) 0 0
\(153\) 13053.6 0.0450819
\(154\) 0 0
\(155\) 14676.7 0.0490680
\(156\) 0 0
\(157\) 426414. 1.38065 0.690323 0.723501i \(-0.257468\pi\)
0.690323 + 0.723501i \(0.257468\pi\)
\(158\) 0 0
\(159\) 173701. 0.544889
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −192510. −0.567524 −0.283762 0.958895i \(-0.591583\pi\)
−0.283762 + 0.958895i \(0.591583\pi\)
\(164\) 0 0
\(165\) 261958. 0.749068
\(166\) 0 0
\(167\) 164987. 0.457782 0.228891 0.973452i \(-0.426490\pi\)
0.228891 + 0.973452i \(0.426490\pi\)
\(168\) 0 0
\(169\) 793433. 2.13695
\(170\) 0 0
\(171\) 95325.3 0.249297
\(172\) 0 0
\(173\) −329498. −0.837022 −0.418511 0.908212i \(-0.637448\pi\)
−0.418511 + 0.908212i \(0.637448\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −57314.7 −0.137510
\(178\) 0 0
\(179\) 368348. 0.859262 0.429631 0.903005i \(-0.358644\pi\)
0.429631 + 0.903005i \(0.358644\pi\)
\(180\) 0 0
\(181\) −79607.3 −0.180616 −0.0903080 0.995914i \(-0.528785\pi\)
−0.0903080 + 0.995914i \(0.528785\pi\)
\(182\) 0 0
\(183\) 442306. 0.976327
\(184\) 0 0
\(185\) 700346. 1.50447
\(186\) 0 0
\(187\) −101716. −0.212708
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 565914. 1.12245 0.561225 0.827663i \(-0.310330\pi\)
0.561225 + 0.827663i \(0.310330\pi\)
\(192\) 0 0
\(193\) −38665.8 −0.0747194 −0.0373597 0.999302i \(-0.511895\pi\)
−0.0373597 + 0.999302i \(0.511895\pi\)
\(194\) 0 0
\(195\) 447920. 0.843556
\(196\) 0 0
\(197\) −334957. −0.614927 −0.307463 0.951560i \(-0.599480\pi\)
−0.307463 + 0.951560i \(0.599480\pi\)
\(198\) 0 0
\(199\) −600245. −1.07447 −0.537237 0.843431i \(-0.680532\pi\)
−0.537237 + 0.843431i \(0.680532\pi\)
\(200\) 0 0
\(201\) −306486. −0.535083
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 949387. 1.57782
\(206\) 0 0
\(207\) 175240. 0.284255
\(208\) 0 0
\(209\) −742790. −1.17625
\(210\) 0 0
\(211\) 1.06504e6 1.64687 0.823433 0.567414i \(-0.192056\pi\)
0.823433 + 0.567414i \(0.192056\pi\)
\(212\) 0 0
\(213\) −566662. −0.855805
\(214\) 0 0
\(215\) −20987.9 −0.0309652
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 79809.8 0.112446
\(220\) 0 0
\(221\) −173923. −0.239539
\(222\) 0 0
\(223\) −1.37380e6 −1.84995 −0.924976 0.380027i \(-0.875915\pi\)
−0.924976 + 0.380027i \(0.875915\pi\)
\(224\) 0 0
\(225\) −80868.1 −0.106493
\(226\) 0 0
\(227\) 324880. 0.418464 0.209232 0.977866i \(-0.432904\pi\)
0.209232 + 0.977866i \(0.432904\pi\)
\(228\) 0 0
\(229\) −822392. −1.03631 −0.518155 0.855286i \(-0.673381\pi\)
−0.518155 + 0.855286i \(0.673381\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.13814e6 1.37343 0.686713 0.726928i \(-0.259053\pi\)
0.686713 + 0.726928i \(0.259053\pi\)
\(234\) 0 0
\(235\) −957482. −1.13100
\(236\) 0 0
\(237\) −309719. −0.358176
\(238\) 0 0
\(239\) 483125. 0.547097 0.273549 0.961858i \(-0.411803\pi\)
0.273549 + 0.961858i \(0.411803\pi\)
\(240\) 0 0
\(241\) 1.01481e6 1.12549 0.562747 0.826629i \(-0.309745\pi\)
0.562747 + 0.826629i \(0.309745\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.27009e6 −1.32462
\(248\) 0 0
\(249\) 63372.7 0.0647745
\(250\) 0 0
\(251\) 415812. 0.416593 0.208297 0.978066i \(-0.433208\pi\)
0.208297 + 0.978066i \(0.433208\pi\)
\(252\) 0 0
\(253\) −1.36550e6 −1.34119
\(254\) 0 0
\(255\) −66885.8 −0.0644144
\(256\) 0 0
\(257\) 1.01200e6 0.955753 0.477876 0.878427i \(-0.341407\pi\)
0.477876 + 0.878427i \(0.341407\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −363853. −0.330617
\(262\) 0 0
\(263\) 2.05769e6 1.83439 0.917195 0.398439i \(-0.130448\pi\)
0.917195 + 0.398439i \(0.130448\pi\)
\(264\) 0 0
\(265\) −890030. −0.778556
\(266\) 0 0
\(267\) 182185. 0.156399
\(268\) 0 0
\(269\) −807996. −0.680814 −0.340407 0.940278i \(-0.610565\pi\)
−0.340407 + 0.940278i \(0.610565\pi\)
\(270\) 0 0
\(271\) 196023. 0.162137 0.0810687 0.996709i \(-0.474167\pi\)
0.0810687 + 0.996709i \(0.474167\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 630137. 0.502462
\(276\) 0 0
\(277\) 302584. 0.236945 0.118472 0.992957i \(-0.462200\pi\)
0.118472 + 0.992957i \(0.462200\pi\)
\(278\) 0 0
\(279\) 25779.0 0.0198270
\(280\) 0 0
\(281\) 646014. 0.488063 0.244032 0.969767i \(-0.421530\pi\)
0.244032 + 0.969767i \(0.421530\pi\)
\(282\) 0 0
\(283\) −1.10750e6 −0.822008 −0.411004 0.911634i \(-0.634822\pi\)
−0.411004 + 0.911634i \(0.634822\pi\)
\(284\) 0 0
\(285\) −488440. −0.356204
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.39389e6 −0.981709
\(290\) 0 0
\(291\) −486599. −0.336852
\(292\) 0 0
\(293\) 1.89396e6 1.28885 0.644425 0.764668i \(-0.277097\pi\)
0.644425 + 0.764668i \(0.277097\pi\)
\(294\) 0 0
\(295\) 293677. 0.196478
\(296\) 0 0
\(297\) 460119. 0.302677
\(298\) 0 0
\(299\) −2.33486e6 −1.51037
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 823377. 0.515219
\(304\) 0 0
\(305\) −2.26635e6 −1.39501
\(306\) 0 0
\(307\) 1.97803e6 1.19781 0.598905 0.800820i \(-0.295603\pi\)
0.598905 + 0.800820i \(0.295603\pi\)
\(308\) 0 0
\(309\) −678432. −0.404213
\(310\) 0 0
\(311\) 2.58787e6 1.51719 0.758596 0.651561i \(-0.225885\pi\)
0.758596 + 0.651561i \(0.225885\pi\)
\(312\) 0 0
\(313\) −141357. −0.0815561 −0.0407781 0.999168i \(-0.512984\pi\)
−0.0407781 + 0.999168i \(0.512984\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.17285e6 0.655530 0.327765 0.944759i \(-0.393705\pi\)
0.327765 + 0.944759i \(0.393705\pi\)
\(318\) 0 0
\(319\) 2.83520e6 1.55994
\(320\) 0 0
\(321\) −1.36919e6 −0.741656
\(322\) 0 0
\(323\) 189657. 0.101149
\(324\) 0 0
\(325\) 1.07747e6 0.565843
\(326\) 0 0
\(327\) −690047. −0.356870
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.30661e6 0.655505 0.327752 0.944764i \(-0.393709\pi\)
0.327752 + 0.944764i \(0.393709\pi\)
\(332\) 0 0
\(333\) 1.23013e6 0.607913
\(334\) 0 0
\(335\) 1.57041e6 0.764544
\(336\) 0 0
\(337\) −265059. −0.127136 −0.0635679 0.997978i \(-0.520248\pi\)
−0.0635679 + 0.997978i \(0.520248\pi\)
\(338\) 0 0
\(339\) −2.05663e6 −0.971981
\(340\) 0 0
\(341\) −200874. −0.0935489
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −897919. −0.406153
\(346\) 0 0
\(347\) −3.05583e6 −1.36240 −0.681200 0.732097i \(-0.738542\pi\)
−0.681200 + 0.732097i \(0.738542\pi\)
\(348\) 0 0
\(349\) −1.47164e6 −0.646753 −0.323377 0.946270i \(-0.604818\pi\)
−0.323377 + 0.946270i \(0.604818\pi\)
\(350\) 0 0
\(351\) 786755. 0.340857
\(352\) 0 0
\(353\) −385183. −0.164525 −0.0822623 0.996611i \(-0.526215\pi\)
−0.0822623 + 0.996611i \(0.526215\pi\)
\(354\) 0 0
\(355\) 2.90354e6 1.22280
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.64188e6 −0.672364 −0.336182 0.941797i \(-0.609136\pi\)
−0.336182 + 0.941797i \(0.609136\pi\)
\(360\) 0 0
\(361\) −1.09111e6 −0.440657
\(362\) 0 0
\(363\) −2.13586e6 −0.850760
\(364\) 0 0
\(365\) −408940. −0.160667
\(366\) 0 0
\(367\) −1.12323e6 −0.435315 −0.217658 0.976025i \(-0.569842\pi\)
−0.217658 + 0.976025i \(0.569842\pi\)
\(368\) 0 0
\(369\) 1.66756e6 0.637553
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.22105e6 −1.57090 −0.785450 0.618926i \(-0.787568\pi\)
−0.785450 + 0.618926i \(0.787568\pi\)
\(374\) 0 0
\(375\) 1.71136e6 0.628438
\(376\) 0 0
\(377\) 4.84789e6 1.75671
\(378\) 0 0
\(379\) −4.49923e6 −1.60894 −0.804471 0.593993i \(-0.797551\pi\)
−0.804471 + 0.593993i \(0.797551\pi\)
\(380\) 0 0
\(381\) −1.09900e6 −0.387869
\(382\) 0 0
\(383\) 4.43114e6 1.54354 0.771771 0.635900i \(-0.219371\pi\)
0.771771 + 0.635900i \(0.219371\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −36864.6 −0.0125121
\(388\) 0 0
\(389\) −5.44223e6 −1.82349 −0.911744 0.410759i \(-0.865264\pi\)
−0.911744 + 0.410759i \(0.865264\pi\)
\(390\) 0 0
\(391\) 348654. 0.115333
\(392\) 0 0
\(393\) −682158. −0.222794
\(394\) 0 0
\(395\) 1.58698e6 0.511774
\(396\) 0 0
\(397\) 2.68794e6 0.855939 0.427969 0.903793i \(-0.359229\pi\)
0.427969 + 0.903793i \(0.359229\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 443294. 0.137667 0.0688337 0.997628i \(-0.478072\pi\)
0.0688337 + 0.997628i \(0.478072\pi\)
\(402\) 0 0
\(403\) −343474. −0.105349
\(404\) 0 0
\(405\) 302563. 0.0916597
\(406\) 0 0
\(407\) −9.58539e6 −2.86829
\(408\) 0 0
\(409\) −4.89578e6 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(410\) 0 0
\(411\) 2.17450e6 0.634973
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −324718. −0.0925520
\(416\) 0 0
\(417\) −1.13092e6 −0.318486
\(418\) 0 0
\(419\) −2.23186e6 −0.621058 −0.310529 0.950564i \(-0.600506\pi\)
−0.310529 + 0.950564i \(0.600506\pi\)
\(420\) 0 0
\(421\) 5.48208e6 1.50744 0.753721 0.657195i \(-0.228257\pi\)
0.753721 + 0.657195i \(0.228257\pi\)
\(422\) 0 0
\(423\) −1.68178e6 −0.457003
\(424\) 0 0
\(425\) −160893. −0.0432081
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.13052e6 −1.60825
\(430\) 0 0
\(431\) 4.62366e6 1.19893 0.599463 0.800402i \(-0.295381\pi\)
0.599463 + 0.800402i \(0.295381\pi\)
\(432\) 0 0
\(433\) −3.08314e6 −0.790267 −0.395134 0.918624i \(-0.629302\pi\)
−0.395134 + 0.918624i \(0.629302\pi\)
\(434\) 0 0
\(435\) 1.86436e6 0.472396
\(436\) 0 0
\(437\) 2.54608e6 0.637776
\(438\) 0 0
\(439\) 434470. 0.107597 0.0537983 0.998552i \(-0.482867\pi\)
0.0537983 + 0.998552i \(0.482867\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.72063e6 0.416562 0.208281 0.978069i \(-0.433213\pi\)
0.208281 + 0.978069i \(0.433213\pi\)
\(444\) 0 0
\(445\) −933502. −0.223468
\(446\) 0 0
\(447\) −324419. −0.0767958
\(448\) 0 0
\(449\) −4.06508e6 −0.951598 −0.475799 0.879554i \(-0.657841\pi\)
−0.475799 + 0.879554i \(0.657841\pi\)
\(450\) 0 0
\(451\) −1.29939e7 −3.00815
\(452\) 0 0
\(453\) −1.35601e6 −0.310469
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.26184e6 0.506607 0.253303 0.967387i \(-0.418483\pi\)
0.253303 + 0.967387i \(0.418483\pi\)
\(458\) 0 0
\(459\) −117482. −0.0260280
\(460\) 0 0
\(461\) 7.80980e6 1.71154 0.855771 0.517355i \(-0.173083\pi\)
0.855771 + 0.517355i \(0.173083\pi\)
\(462\) 0 0
\(463\) −525518. −0.113929 −0.0569647 0.998376i \(-0.518142\pi\)
−0.0569647 + 0.998376i \(0.518142\pi\)
\(464\) 0 0
\(465\) −132090. −0.0283294
\(466\) 0 0
\(467\) 5.59376e6 1.18689 0.593447 0.804873i \(-0.297767\pi\)
0.593447 + 0.804873i \(0.297767\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.83773e6 −0.797116
\(472\) 0 0
\(473\) 287255. 0.0590356
\(474\) 0 0
\(475\) −1.17494e6 −0.238936
\(476\) 0 0
\(477\) −1.56330e6 −0.314592
\(478\) 0 0
\(479\) 1.81246e6 0.360936 0.180468 0.983581i \(-0.442239\pi\)
0.180468 + 0.983581i \(0.442239\pi\)
\(480\) 0 0
\(481\) −1.63900e7 −3.23010
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.49330e6 0.481305
\(486\) 0 0
\(487\) 1.44068e6 0.275261 0.137631 0.990484i \(-0.456051\pi\)
0.137631 + 0.990484i \(0.456051\pi\)
\(488\) 0 0
\(489\) 1.73259e6 0.327660
\(490\) 0 0
\(491\) 5.25026e6 0.982827 0.491413 0.870926i \(-0.336480\pi\)
0.491413 + 0.870926i \(0.336480\pi\)
\(492\) 0 0
\(493\) −723913. −0.134143
\(494\) 0 0
\(495\) −2.35762e6 −0.432475
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −8.14160e6 −1.46372 −0.731861 0.681454i \(-0.761348\pi\)
−0.731861 + 0.681454i \(0.761348\pi\)
\(500\) 0 0
\(501\) −1.48488e6 −0.264301
\(502\) 0 0
\(503\) −7.98029e6 −1.40637 −0.703183 0.711009i \(-0.748239\pi\)
−0.703183 + 0.711009i \(0.748239\pi\)
\(504\) 0 0
\(505\) −4.21893e6 −0.736162
\(506\) 0 0
\(507\) −7.14090e6 −1.23377
\(508\) 0 0
\(509\) 8.14283e6 1.39309 0.696547 0.717511i \(-0.254719\pi\)
0.696547 + 0.717511i \(0.254719\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −857928. −0.143932
\(514\) 0 0
\(515\) 3.47624e6 0.577553
\(516\) 0 0
\(517\) 1.31047e7 2.15626
\(518\) 0 0
\(519\) 2.96548e6 0.483255
\(520\) 0 0
\(521\) 5.11967e6 0.826319 0.413160 0.910659i \(-0.364425\pi\)
0.413160 + 0.910659i \(0.364425\pi\)
\(522\) 0 0
\(523\) 4.73473e6 0.756904 0.378452 0.925621i \(-0.376456\pi\)
0.378452 + 0.925621i \(0.376456\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 51289.3 0.00804453
\(528\) 0 0
\(529\) −1.75579e6 −0.272792
\(530\) 0 0
\(531\) 515833. 0.0793912
\(532\) 0 0
\(533\) −2.22182e7 −3.38759
\(534\) 0 0
\(535\) 7.01566e6 1.05970
\(536\) 0 0
\(537\) −3.31513e6 −0.496095
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.20453e6 −1.20520 −0.602602 0.798042i \(-0.705869\pi\)
−0.602602 + 0.798042i \(0.705869\pi\)
\(542\) 0 0
\(543\) 716465. 0.104279
\(544\) 0 0
\(545\) 3.53575e6 0.509907
\(546\) 0 0
\(547\) −1.57733e6 −0.225400 −0.112700 0.993629i \(-0.535950\pi\)
−0.112700 + 0.993629i \(0.535950\pi\)
\(548\) 0 0
\(549\) −3.98076e6 −0.563683
\(550\) 0 0
\(551\) −5.28645e6 −0.741797
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6.30311e6 −0.868605
\(556\) 0 0
\(557\) −2.10411e6 −0.287363 −0.143681 0.989624i \(-0.545894\pi\)
−0.143681 + 0.989624i \(0.545894\pi\)
\(558\) 0 0
\(559\) 491175. 0.0664824
\(560\) 0 0
\(561\) 915442. 0.122807
\(562\) 0 0
\(563\) 7.69167e6 1.02270 0.511351 0.859372i \(-0.329145\pi\)
0.511351 + 0.859372i \(0.329145\pi\)
\(564\) 0 0
\(565\) 1.05381e7 1.38880
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.30139e6 0.945421 0.472710 0.881218i \(-0.343276\pi\)
0.472710 + 0.881218i \(0.343276\pi\)
\(570\) 0 0
\(571\) 3.41934e6 0.438887 0.219443 0.975625i \(-0.429576\pi\)
0.219443 + 0.975625i \(0.429576\pi\)
\(572\) 0 0
\(573\) −5.09323e6 −0.648047
\(574\) 0 0
\(575\) −2.15994e6 −0.272440
\(576\) 0 0
\(577\) −1.09289e7 −1.36659 −0.683295 0.730142i \(-0.739454\pi\)
−0.683295 + 0.730142i \(0.739454\pi\)
\(578\) 0 0
\(579\) 347992. 0.0431393
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.21815e7 1.48433
\(584\) 0 0
\(585\) −4.03128e6 −0.487027
\(586\) 0 0
\(587\) 2.25268e6 0.269839 0.134919 0.990857i \(-0.456922\pi\)
0.134919 + 0.990857i \(0.456922\pi\)
\(588\) 0 0
\(589\) 374546. 0.0444853
\(590\) 0 0
\(591\) 3.01461e6 0.355028
\(592\) 0 0
\(593\) −5.68240e6 −0.663583 −0.331792 0.943353i \(-0.607653\pi\)
−0.331792 + 0.943353i \(0.607653\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.40221e6 0.620348
\(598\) 0 0
\(599\) −1.28026e7 −1.45792 −0.728958 0.684559i \(-0.759995\pi\)
−0.728958 + 0.684559i \(0.759995\pi\)
\(600\) 0 0
\(601\) −9.25870e6 −1.04560 −0.522798 0.852457i \(-0.675112\pi\)
−0.522798 + 0.852457i \(0.675112\pi\)
\(602\) 0 0
\(603\) 2.75838e6 0.308930
\(604\) 0 0
\(605\) 1.09440e7 1.21559
\(606\) 0 0
\(607\) −401320. −0.0442098 −0.0221049 0.999756i \(-0.507037\pi\)
−0.0221049 + 0.999756i \(0.507037\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.24077e7 2.42825
\(612\) 0 0
\(613\) 6.82396e6 0.733475 0.366738 0.930324i \(-0.380475\pi\)
0.366738 + 0.930324i \(0.380475\pi\)
\(614\) 0 0
\(615\) −8.54448e6 −0.910957
\(616\) 0 0
\(617\) −336274. −0.0355615 −0.0177808 0.999842i \(-0.505660\pi\)
−0.0177808 + 0.999842i \(0.505660\pi\)
\(618\) 0 0
\(619\) −9.61151e6 −1.00824 −0.504121 0.863633i \(-0.668183\pi\)
−0.504121 + 0.863633i \(0.668183\pi\)
\(620\) 0 0
\(621\) −1.57716e6 −0.164115
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.64897e6 −0.578454
\(626\) 0 0
\(627\) 6.68511e6 0.679109
\(628\) 0 0
\(629\) 2.44744e6 0.246652
\(630\) 0 0
\(631\) 1.28813e7 1.28791 0.643957 0.765062i \(-0.277292\pi\)
0.643957 + 0.765062i \(0.277292\pi\)
\(632\) 0 0
\(633\) −9.58532e6 −0.950818
\(634\) 0 0
\(635\) 5.63120e6 0.554200
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 5.09996e6 0.494099
\(640\) 0 0
\(641\) −1.49496e7 −1.43709 −0.718545 0.695480i \(-0.755192\pi\)
−0.718545 + 0.695480i \(0.755192\pi\)
\(642\) 0 0
\(643\) −8.03892e6 −0.766780 −0.383390 0.923587i \(-0.625243\pi\)
−0.383390 + 0.923587i \(0.625243\pi\)
\(644\) 0 0
\(645\) 188892. 0.0178778
\(646\) 0 0
\(647\) −6.68775e6 −0.628086 −0.314043 0.949409i \(-0.601684\pi\)
−0.314043 + 0.949409i \(0.601684\pi\)
\(648\) 0 0
\(649\) −4.01945e6 −0.374589
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.04443e6 −0.0958505 −0.0479253 0.998851i \(-0.515261\pi\)
−0.0479253 + 0.998851i \(0.515261\pi\)
\(654\) 0 0
\(655\) 3.49533e6 0.318335
\(656\) 0 0
\(657\) −718288. −0.0649210
\(658\) 0 0
\(659\) 2.10237e7 1.88580 0.942902 0.333071i \(-0.108085\pi\)
0.942902 + 0.333071i \(0.108085\pi\)
\(660\) 0 0
\(661\) 9.46680e6 0.842752 0.421376 0.906886i \(-0.361547\pi\)
0.421376 + 0.906886i \(0.361547\pi\)
\(662\) 0 0
\(663\) 1.56531e6 0.138298
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.71829e6 −0.845815
\(668\) 0 0
\(669\) 1.23642e7 1.06807
\(670\) 0 0
\(671\) 3.10187e7 2.65960
\(672\) 0 0
\(673\) −1.95188e7 −1.66117 −0.830587 0.556889i \(-0.811995\pi\)
−0.830587 + 0.556889i \(0.811995\pi\)
\(674\) 0 0
\(675\) 727813. 0.0614837
\(676\) 0 0
\(677\) −1.70544e7 −1.43010 −0.715049 0.699075i \(-0.753596\pi\)
−0.715049 + 0.699075i \(0.753596\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2.92392e6 −0.241600
\(682\) 0 0
\(683\) 1.09475e7 0.897972 0.448986 0.893539i \(-0.351785\pi\)
0.448986 + 0.893539i \(0.351785\pi\)
\(684\) 0 0
\(685\) −1.11420e7 −0.907270
\(686\) 0 0
\(687\) 7.40153e6 0.598314
\(688\) 0 0
\(689\) 2.08291e7 1.67156
\(690\) 0 0
\(691\) 1.04538e7 0.832874 0.416437 0.909165i \(-0.363279\pi\)
0.416437 + 0.909165i \(0.363279\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.79474e6 0.455063
\(696\) 0 0
\(697\) 3.31774e6 0.258679
\(698\) 0 0
\(699\) −1.02433e7 −0.792948
\(700\) 0 0
\(701\) 8.71564e6 0.669891 0.334946 0.942237i \(-0.391282\pi\)
0.334946 + 0.942237i \(0.391282\pi\)
\(702\) 0 0
\(703\) 1.78727e7 1.36396
\(704\) 0 0
\(705\) 8.61734e6 0.652981
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −938439. −0.0701117 −0.0350559 0.999385i \(-0.511161\pi\)
−0.0350559 + 0.999385i \(0.511161\pi\)
\(710\) 0 0
\(711\) 2.78747e6 0.206793
\(712\) 0 0
\(713\) 688542. 0.0507232
\(714\) 0 0
\(715\) 3.14124e7 2.29792
\(716\) 0 0
\(717\) −4.34812e6 −0.315867
\(718\) 0 0
\(719\) 6.59073e6 0.475457 0.237729 0.971332i \(-0.423597\pi\)
0.237729 + 0.971332i \(0.423597\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −9.13331e6 −0.649804
\(724\) 0 0
\(725\) 4.48470e6 0.316875
\(726\) 0 0
\(727\) −2.32586e7 −1.63210 −0.816052 0.577979i \(-0.803842\pi\)
−0.816052 + 0.577979i \(0.803842\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −73344.8 −0.00507664
\(732\) 0 0
\(733\) −2.02854e7 −1.39452 −0.697259 0.716819i \(-0.745597\pi\)
−0.697259 + 0.716819i \(0.745597\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.14937e7 −1.45761
\(738\) 0 0
\(739\) 593427. 0.0399720 0.0199860 0.999800i \(-0.493638\pi\)
0.0199860 + 0.999800i \(0.493638\pi\)
\(740\) 0 0
\(741\) 1.14308e7 0.764772
\(742\) 0 0
\(743\) 7.70228e6 0.511855 0.255928 0.966696i \(-0.417619\pi\)
0.255928 + 0.966696i \(0.417619\pi\)
\(744\) 0 0
\(745\) 1.66230e6 0.109728
\(746\) 0 0
\(747\) −570355. −0.0373976
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.68333e7 −1.73610 −0.868048 0.496481i \(-0.834625\pi\)
−0.868048 + 0.496481i \(0.834625\pi\)
\(752\) 0 0
\(753\) −3.74230e6 −0.240520
\(754\) 0 0
\(755\) 6.94811e6 0.443608
\(756\) 0 0
\(757\) 2.34943e7 1.49013 0.745063 0.666994i \(-0.232419\pi\)
0.745063 + 0.666994i \(0.232419\pi\)
\(758\) 0 0
\(759\) 1.22895e7 0.774337
\(760\) 0 0
\(761\) −1.26192e7 −0.789895 −0.394947 0.918704i \(-0.629237\pi\)
−0.394947 + 0.918704i \(0.629237\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 601972. 0.0371897
\(766\) 0 0
\(767\) −6.87283e6 −0.421840
\(768\) 0 0
\(769\) 2.73674e6 0.166885 0.0834425 0.996513i \(-0.473408\pi\)
0.0834425 + 0.996513i \(0.473408\pi\)
\(770\) 0 0
\(771\) −9.10796e6 −0.551804
\(772\) 0 0
\(773\) 2.15505e7 1.29720 0.648602 0.761128i \(-0.275354\pi\)
0.648602 + 0.761128i \(0.275354\pi\)
\(774\) 0 0
\(775\) −317742. −0.0190029
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.42282e7 1.43046
\(780\) 0 0
\(781\) −3.97397e7 −2.33129
\(782\) 0 0
\(783\) 3.27468e6 0.190882
\(784\) 0 0
\(785\) 1.96642e7 1.13895
\(786\) 0 0
\(787\) 2.89891e6 0.166839 0.0834195 0.996515i \(-0.473416\pi\)
0.0834195 + 0.996515i \(0.473416\pi\)
\(788\) 0 0
\(789\) −1.85192e7 −1.05909
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 5.30386e7 2.99509
\(794\) 0 0
\(795\) 8.01027e6 0.449499
\(796\) 0 0
\(797\) −1.57797e7 −0.879937 −0.439969 0.898013i \(-0.645010\pi\)
−0.439969 + 0.898013i \(0.645010\pi\)
\(798\) 0 0
\(799\) −3.34603e6 −0.185423
\(800\) 0 0
\(801\) −1.63966e6 −0.0902970
\(802\) 0 0
\(803\) 5.59702e6 0.306315
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 7.27197e6 0.393068
\(808\) 0 0
\(809\) 7.95560e6 0.427367 0.213684 0.976903i \(-0.431454\pi\)
0.213684 + 0.976903i \(0.431454\pi\)
\(810\) 0 0
\(811\) −2.48725e7 −1.32791 −0.663953 0.747774i \(-0.731123\pi\)
−0.663953 + 0.747774i \(0.731123\pi\)
\(812\) 0 0
\(813\) −1.76420e6 −0.0936101
\(814\) 0 0
\(815\) −8.87767e6 −0.468171
\(816\) 0 0
\(817\) −535608. −0.0280732
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.19168e7 1.13480 0.567401 0.823442i \(-0.307949\pi\)
0.567401 + 0.823442i \(0.307949\pi\)
\(822\) 0 0
\(823\) 1.95072e7 1.00391 0.501955 0.864894i \(-0.332614\pi\)
0.501955 + 0.864894i \(0.332614\pi\)
\(824\) 0 0
\(825\) −5.67123e6 −0.290097
\(826\) 0 0
\(827\) 9.90134e6 0.503420 0.251710 0.967803i \(-0.419007\pi\)
0.251710 + 0.967803i \(0.419007\pi\)
\(828\) 0 0
\(829\) 2.39789e7 1.21183 0.605916 0.795529i \(-0.292807\pi\)
0.605916 + 0.795529i \(0.292807\pi\)
\(830\) 0 0
\(831\) −2.72326e6 −0.136800
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.60844e6 0.377641
\(836\) 0 0
\(837\) −232011. −0.0114471
\(838\) 0 0
\(839\) −1.26483e6 −0.0620334 −0.0310167 0.999519i \(-0.509875\pi\)
−0.0310167 + 0.999519i \(0.509875\pi\)
\(840\) 0 0
\(841\) −332952. −0.0162327
\(842\) 0 0
\(843\) −5.81413e6 −0.281784
\(844\) 0 0
\(845\) 3.65895e7 1.76285
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9.96746e6 0.474586
\(850\) 0 0
\(851\) 3.28561e7 1.55522
\(852\) 0 0
\(853\) 2.75502e7 1.29644 0.648219 0.761454i \(-0.275514\pi\)
0.648219 + 0.761454i \(0.275514\pi\)
\(854\) 0 0
\(855\) 4.39596e6 0.205655
\(856\) 0 0
\(857\) 3.59074e7 1.67006 0.835030 0.550204i \(-0.185450\pi\)
0.835030 + 0.550204i \(0.185450\pi\)
\(858\) 0 0
\(859\) −2.34341e7 −1.08359 −0.541795 0.840511i \(-0.682255\pi\)
−0.541795 + 0.840511i \(0.682255\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.35409e7 1.53302 0.766511 0.642231i \(-0.221991\pi\)
0.766511 + 0.642231i \(0.221991\pi\)
\(864\) 0 0
\(865\) −1.51949e7 −0.690491
\(866\) 0 0
\(867\) 1.25450e7 0.566790
\(868\) 0 0
\(869\) −2.17204e7 −0.975705
\(870\) 0 0
\(871\) −3.67520e7 −1.64148
\(872\) 0 0
\(873\) 4.37939e6 0.194481
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.54557e7 −0.678561 −0.339281 0.940685i \(-0.610184\pi\)
−0.339281 + 0.940685i \(0.610184\pi\)
\(878\) 0 0
\(879\) −1.70457e7 −0.744118
\(880\) 0 0
\(881\) 1.02425e7 0.444599 0.222299 0.974978i \(-0.428644\pi\)
0.222299 + 0.974978i \(0.428644\pi\)
\(882\) 0 0
\(883\) −6.85044e6 −0.295676 −0.147838 0.989012i \(-0.547231\pi\)
−0.147838 + 0.989012i \(0.547231\pi\)
\(884\) 0 0
\(885\) −2.64309e6 −0.113437
\(886\) 0 0
\(887\) 6.08759e6 0.259799 0.129899 0.991527i \(-0.458535\pi\)
0.129899 + 0.991527i \(0.458535\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.14107e6 −0.174751
\(892\) 0 0
\(893\) −2.44348e7 −1.02537
\(894\) 0 0
\(895\) 1.69865e7 0.708837
\(896\) 0 0
\(897\) 2.10137e7 0.872011
\(898\) 0 0
\(899\) −1.42963e6 −0.0589962
\(900\) 0 0
\(901\) −3.11031e6 −0.127642
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.67112e6 −0.148997
\(906\) 0 0
\(907\) 4.62504e7 1.86680 0.933398 0.358843i \(-0.116829\pi\)
0.933398 + 0.358843i \(0.116829\pi\)
\(908\) 0 0
\(909\) −7.41039e6 −0.297462
\(910\) 0 0
\(911\) −4.09674e7 −1.63547 −0.817734 0.575596i \(-0.804770\pi\)
−0.817734 + 0.575596i \(0.804770\pi\)
\(912\) 0 0
\(913\) 4.44430e6 0.176452
\(914\) 0 0
\(915\) 2.03971e7 0.805408
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 3.05857e7 1.19462 0.597309 0.802011i \(-0.296237\pi\)
0.597309 + 0.802011i \(0.296237\pi\)
\(920\) 0 0
\(921\) −1.78023e7 −0.691556
\(922\) 0 0
\(923\) −6.79506e7 −2.62536
\(924\) 0 0
\(925\) −1.51621e7 −0.582646
\(926\) 0 0
\(927\) 6.10589e6 0.233373
\(928\) 0 0
\(929\) −3.16851e7 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −2.32908e7 −0.875952
\(934\) 0 0
\(935\) −4.69066e6 −0.175471
\(936\) 0 0
\(937\) 2.43042e7 0.904342 0.452171 0.891931i \(-0.350650\pi\)
0.452171 + 0.891931i \(0.350650\pi\)
\(938\) 0 0
\(939\) 1.27221e6 0.0470865
\(940\) 0 0
\(941\) −2.48459e7 −0.914704 −0.457352 0.889286i \(-0.651202\pi\)
−0.457352 + 0.889286i \(0.651202\pi\)
\(942\) 0 0
\(943\) 4.45396e7 1.63105
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.91617e7 −1.78136 −0.890681 0.454629i \(-0.849772\pi\)
−0.890681 + 0.454629i \(0.849772\pi\)
\(948\) 0 0
\(949\) 9.57030e6 0.344953
\(950\) 0 0
\(951\) −1.05556e7 −0.378471
\(952\) 0 0
\(953\) 1.84741e7 0.658918 0.329459 0.944170i \(-0.393134\pi\)
0.329459 + 0.944170i \(0.393134\pi\)
\(954\) 0 0
\(955\) 2.60973e7 0.925951
\(956\) 0 0
\(957\) −2.55168e7 −0.900631
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.85279e7 −0.996462
\(962\) 0 0
\(963\) 1.23227e7 0.428195
\(964\) 0 0
\(965\) −1.78309e6 −0.0616388
\(966\) 0 0
\(967\) 3.78820e7 1.30277 0.651384 0.758748i \(-0.274189\pi\)
0.651384 + 0.758748i \(0.274189\pi\)
\(968\) 0 0
\(969\) −1.70691e6 −0.0583985
\(970\) 0 0
\(971\) −3.72621e7 −1.26829 −0.634146 0.773213i \(-0.718648\pi\)
−0.634146 + 0.773213i \(0.718648\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −9.69721e6 −0.326689
\(976\) 0 0
\(977\) −3.70590e6 −0.124210 −0.0621052 0.998070i \(-0.519781\pi\)
−0.0621052 + 0.998070i \(0.519781\pi\)
\(978\) 0 0
\(979\) 1.27765e7 0.426045
\(980\) 0 0
\(981\) 6.21043e6 0.206039
\(982\) 0 0
\(983\) 8.28765e6 0.273557 0.136778 0.990602i \(-0.456325\pi\)
0.136778 + 0.990602i \(0.456325\pi\)
\(984\) 0 0
\(985\) −1.54467e7 −0.507276
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −984630. −0.0320097
\(990\) 0 0
\(991\) 1.62498e7 0.525612 0.262806 0.964849i \(-0.415352\pi\)
0.262806 + 0.964849i \(0.415352\pi\)
\(992\) 0 0
\(993\) −1.17595e7 −0.378456
\(994\) 0 0
\(995\) −2.76805e7 −0.886374
\(996\) 0 0
\(997\) −4.35858e7 −1.38870 −0.694348 0.719640i \(-0.744307\pi\)
−0.694348 + 0.719640i \(0.744307\pi\)
\(998\) 0 0
\(999\) −1.10712e7 −0.350978
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 588.6.a.n.1.3 4
7.2 even 3 84.6.i.c.25.2 8
7.3 odd 6 588.6.i.o.373.3 8
7.4 even 3 84.6.i.c.37.2 yes 8
7.5 odd 6 588.6.i.o.361.3 8
7.6 odd 2 588.6.a.p.1.2 4
21.2 odd 6 252.6.k.f.109.3 8
21.11 odd 6 252.6.k.f.37.3 8
28.11 odd 6 336.6.q.i.289.2 8
28.23 odd 6 336.6.q.i.193.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.2 8 7.2 even 3
84.6.i.c.37.2 yes 8 7.4 even 3
252.6.k.f.37.3 8 21.11 odd 6
252.6.k.f.109.3 8 21.2 odd 6
336.6.q.i.193.2 8 28.23 odd 6
336.6.q.i.289.2 8 28.11 odd 6
588.6.a.n.1.3 4 1.1 even 1 trivial
588.6.a.p.1.2 4 7.6 odd 2
588.6.i.o.361.3 8 7.5 odd 6
588.6.i.o.373.3 8 7.3 odd 6