Properties

Label 588.6.f.c.293.20
Level $588$
Weight $6$
Character 588.293
Analytic conductor $94.306$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,6,Mod(293,588)]
 
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, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.293");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 293.20
Character \(\chi\) \(=\) 588.293
Dual form 588.6.f.c.293.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.39153 + 15.5262i) q^{3} -69.8984 q^{5} +(-239.127 - 43.2103i) q^{9} +465.771i q^{11} +285.857i q^{13} +(97.2655 - 1085.26i) q^{15} +898.214 q^{17} +2096.30i q^{19} +4901.65i q^{23} +1760.78 q^{25} +(1003.65 - 3652.62i) q^{27} +5878.95i q^{29} +3854.71i q^{31} +(-7231.66 - 648.133i) q^{33} -3618.34 q^{37} +(-4438.28 - 397.778i) q^{39} +17142.4 q^{41} -18567.3 q^{43} +(16714.6 + 3020.33i) q^{45} +2422.15 q^{47} +(-1249.89 + 13945.9i) q^{51} +7618.31i q^{53} -32556.6i q^{55} +(-32547.6 - 2917.06i) q^{57} -10644.1 q^{59} +15624.4i q^{61} -19980.9i q^{65} -2973.20 q^{67} +(-76104.2 - 6820.79i) q^{69} +16268.3i q^{71} +14677.7i q^{73} +(-2450.17 + 27338.3i) q^{75} -86199.9 q^{79} +(55314.7 + 20665.5i) q^{81} +2747.03 q^{83} -62783.7 q^{85} +(-91277.9 - 8180.72i) q^{87} +101804. q^{89} +(-59849.1 - 5363.94i) q^{93} -146528. i q^{95} -23406.7i q^{97} +(20126.1 - 111378. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 440 q^{9} - 1224 q^{15} + 19896 q^{25} + 26224 q^{37} - 17384 q^{39} - 21584 q^{43} - 52776 q^{51} + 34880 q^{57} + 89776 q^{67} - 288240 q^{79} + 72504 q^{81} - 50640 q^{85} + 263376 q^{93} - 87056 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −1.39153 + 15.5262i −0.0892665 + 0.996008i
\(4\) 0 0
\(5\) −69.8984 −1.25038 −0.625190 0.780473i \(-0.714978\pi\)
−0.625190 + 0.780473i \(0.714978\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −239.127 43.2103i −0.984063 0.177820i
\(10\) 0 0
\(11\) 465.771i 1.16062i 0.814395 + 0.580310i \(0.197069\pi\)
−0.814395 + 0.580310i \(0.802931\pi\)
\(12\) 0 0
\(13\) 285.857i 0.469127i 0.972101 + 0.234563i \(0.0753660\pi\)
−0.972101 + 0.234563i \(0.924634\pi\)
\(14\) 0 0
\(15\) 97.2655 1085.26i 0.111617 1.24539i
\(16\) 0 0
\(17\) 898.214 0.753803 0.376901 0.926253i \(-0.376990\pi\)
0.376901 + 0.926253i \(0.376990\pi\)
\(18\) 0 0
\(19\) 2096.30i 1.33220i 0.745863 + 0.666100i \(0.232037\pi\)
−0.745863 + 0.666100i \(0.767963\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4901.65i 1.93207i 0.258409 + 0.966036i \(0.416802\pi\)
−0.258409 + 0.966036i \(0.583198\pi\)
\(24\) 0 0
\(25\) 1760.78 0.563450
\(26\) 0 0
\(27\) 1003.65 3652.62i 0.264954 0.964261i
\(28\) 0 0
\(29\) 5878.95i 1.29809i 0.760750 + 0.649045i \(0.224831\pi\)
−0.760750 + 0.649045i \(0.775169\pi\)
\(30\) 0 0
\(31\) 3854.71i 0.720423i 0.932871 + 0.360211i \(0.117295\pi\)
−0.932871 + 0.360211i \(0.882705\pi\)
\(32\) 0 0
\(33\) −7231.66 648.133i −1.15599 0.103605i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3618.34 −0.434515 −0.217257 0.976114i \(-0.569711\pi\)
−0.217257 + 0.976114i \(0.569711\pi\)
\(38\) 0 0
\(39\) −4438.28 397.778i −0.467254 0.0418773i
\(40\) 0 0
\(41\) 17142.4 1.59262 0.796308 0.604891i \(-0.206783\pi\)
0.796308 + 0.604891i \(0.206783\pi\)
\(42\) 0 0
\(43\) −18567.3 −1.53136 −0.765680 0.643221i \(-0.777598\pi\)
−0.765680 + 0.643221i \(0.777598\pi\)
\(44\) 0 0
\(45\) 16714.6 + 3020.33i 1.23045 + 0.222343i
\(46\) 0 0
\(47\) 2422.15 0.159940 0.0799698 0.996797i \(-0.474518\pi\)
0.0799698 + 0.996797i \(0.474518\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1249.89 + 13945.9i −0.0672894 + 0.750793i
\(52\) 0 0
\(53\) 7618.31i 0.372537i 0.982499 + 0.186268i \(0.0596394\pi\)
−0.982499 + 0.186268i \(0.940361\pi\)
\(54\) 0 0
\(55\) 32556.6i 1.45122i
\(56\) 0 0
\(57\) −32547.6 2917.06i −1.32688 0.118921i
\(58\) 0 0
\(59\) −10644.1 −0.398089 −0.199044 0.979990i \(-0.563784\pi\)
−0.199044 + 0.979990i \(0.563784\pi\)
\(60\) 0 0
\(61\) 15624.4i 0.537624i 0.963193 + 0.268812i \(0.0866310\pi\)
−0.963193 + 0.268812i \(0.913369\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 19980.9i 0.586586i
\(66\) 0 0
\(67\) −2973.20 −0.0809165 −0.0404582 0.999181i \(-0.512882\pi\)
−0.0404582 + 0.999181i \(0.512882\pi\)
\(68\) 0 0
\(69\) −76104.2 6820.79i −1.92436 0.172469i
\(70\) 0 0
\(71\) 16268.3i 0.382997i 0.981493 + 0.191498i \(0.0613347\pi\)
−0.981493 + 0.191498i \(0.938665\pi\)
\(72\) 0 0
\(73\) 14677.7i 0.322368i 0.986924 + 0.161184i \(0.0515312\pi\)
−0.986924 + 0.161184i \(0.948469\pi\)
\(74\) 0 0
\(75\) −2450.17 + 27338.3i −0.0502972 + 0.561200i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −86199.9 −1.55396 −0.776978 0.629527i \(-0.783249\pi\)
−0.776978 + 0.629527i \(0.783249\pi\)
\(80\) 0 0
\(81\) 55314.7 + 20665.5i 0.936760 + 0.349973i
\(82\) 0 0
\(83\) 2747.03 0.0437691 0.0218846 0.999761i \(-0.493033\pi\)
0.0218846 + 0.999761i \(0.493033\pi\)
\(84\) 0 0
\(85\) −62783.7 −0.942540
\(86\) 0 0
\(87\) −91277.9 8180.72i −1.29291 0.115876i
\(88\) 0 0
\(89\) 101804. 1.36235 0.681173 0.732122i \(-0.261470\pi\)
0.681173 + 0.732122i \(0.261470\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −59849.1 5363.94i −0.717547 0.0643097i
\(94\) 0 0
\(95\) 146528.i 1.66576i
\(96\) 0 0
\(97\) 23406.7i 0.252587i −0.991993 0.126294i \(-0.959692\pi\)
0.991993 0.126294i \(-0.0403082\pi\)
\(98\) 0 0
\(99\) 20126.1 111378.i 0.206382 1.14212i
\(100\) 0 0
\(101\) 62202.4 0.606741 0.303371 0.952873i \(-0.401888\pi\)
0.303371 + 0.952873i \(0.401888\pi\)
\(102\) 0 0
\(103\) 38730.7i 0.359718i −0.983692 0.179859i \(-0.942436\pi\)
0.983692 0.179859i \(-0.0575641\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 188678.i 1.59317i −0.604525 0.796586i \(-0.706637\pi\)
0.604525 0.796586i \(-0.293363\pi\)
\(108\) 0 0
\(109\) 158294. 1.27614 0.638071 0.769978i \(-0.279733\pi\)
0.638071 + 0.769978i \(0.279733\pi\)
\(110\) 0 0
\(111\) 5035.02 56179.1i 0.0387877 0.432780i
\(112\) 0 0
\(113\) 168309.i 1.23997i 0.784615 + 0.619984i \(0.212861\pi\)
−0.784615 + 0.619984i \(0.787139\pi\)
\(114\) 0 0
\(115\) 342618.i 2.41582i
\(116\) 0 0
\(117\) 12352.0 68356.2i 0.0834203 0.461650i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −55891.3 −0.347041
\(122\) 0 0
\(123\) −23854.1 + 266156.i −0.142167 + 1.58626i
\(124\) 0 0
\(125\) 95356.7 0.545854
\(126\) 0 0
\(127\) −120698. −0.664036 −0.332018 0.943273i \(-0.607730\pi\)
−0.332018 + 0.943273i \(0.607730\pi\)
\(128\) 0 0
\(129\) 25836.9 288280.i 0.136699 1.52525i
\(130\) 0 0
\(131\) −242876. −1.23653 −0.618267 0.785968i \(-0.712165\pi\)
−0.618267 + 0.785968i \(0.712165\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −70153.2 + 255312.i −0.331294 + 1.20569i
\(136\) 0 0
\(137\) 208676.i 0.949887i −0.880016 0.474944i \(-0.842468\pi\)
0.880016 0.474944i \(-0.157532\pi\)
\(138\) 0 0
\(139\) 202179.i 0.887563i 0.896135 + 0.443782i \(0.146363\pi\)
−0.896135 + 0.443782i \(0.853637\pi\)
\(140\) 0 0
\(141\) −3370.48 + 37606.8i −0.0142772 + 0.159301i
\(142\) 0 0
\(143\) −133144. −0.544478
\(144\) 0 0
\(145\) 410929.i 1.62310i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 501543.i 1.85073i 0.379081 + 0.925364i \(0.376240\pi\)
−0.379081 + 0.925364i \(0.623760\pi\)
\(150\) 0 0
\(151\) 352128. 1.25678 0.628389 0.777899i \(-0.283715\pi\)
0.628389 + 0.777899i \(0.283715\pi\)
\(152\) 0 0
\(153\) −214788. 38812.1i −0.741789 0.134041i
\(154\) 0 0
\(155\) 269438.i 0.900802i
\(156\) 0 0
\(157\) 547248.i 1.77188i −0.463797 0.885941i \(-0.653513\pi\)
0.463797 0.885941i \(-0.346487\pi\)
\(158\) 0 0
\(159\) −118284. 10601.1i −0.371050 0.0332551i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −629293. −1.85517 −0.927585 0.373611i \(-0.878119\pi\)
−0.927585 + 0.373611i \(0.878119\pi\)
\(164\) 0 0
\(165\) 505481. + 45303.4i 1.44542 + 0.129545i
\(166\) 0 0
\(167\) 588586. 1.63312 0.816562 0.577258i \(-0.195877\pi\)
0.816562 + 0.577258i \(0.195877\pi\)
\(168\) 0 0
\(169\) 289579. 0.779920
\(170\) 0 0
\(171\) 90581.8 501282.i 0.236892 1.31097i
\(172\) 0 0
\(173\) −161240. −0.409598 −0.204799 0.978804i \(-0.565654\pi\)
−0.204799 + 0.978804i \(0.565654\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14811.6 165263.i 0.0355360 0.396499i
\(178\) 0 0
\(179\) 330630.i 0.771275i 0.922650 + 0.385638i \(0.126018\pi\)
−0.922650 + 0.385638i \(0.873982\pi\)
\(180\) 0 0
\(181\) 421555.i 0.956439i −0.878240 0.478220i \(-0.841282\pi\)
0.878240 0.478220i \(-0.158718\pi\)
\(182\) 0 0
\(183\) −242588. 21741.8i −0.535477 0.0479918i
\(184\) 0 0
\(185\) 252916. 0.543309
\(186\) 0 0
\(187\) 418362.i 0.874879i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 398002.i 0.789408i −0.918808 0.394704i \(-0.870847\pi\)
0.918808 0.394704i \(-0.129153\pi\)
\(192\) 0 0
\(193\) 519175. 1.00328 0.501638 0.865078i \(-0.332731\pi\)
0.501638 + 0.865078i \(0.332731\pi\)
\(194\) 0 0
\(195\) 310228. + 27804.0i 0.584245 + 0.0523626i
\(196\) 0 0
\(197\) 580266.i 1.06527i 0.846344 + 0.532637i \(0.178799\pi\)
−0.846344 + 0.532637i \(0.821201\pi\)
\(198\) 0 0
\(199\) 844112.i 1.51101i −0.655142 0.755505i \(-0.727391\pi\)
0.655142 0.755505i \(-0.272609\pi\)
\(200\) 0 0
\(201\) 4137.29 46162.5i 0.00722313 0.0805934i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.19822e6 −1.99138
\(206\) 0 0
\(207\) 211802. 1.17212e6i 0.343562 1.90128i
\(208\) 0 0
\(209\) −976395. −1.54618
\(210\) 0 0
\(211\) −1.04382e6 −1.61406 −0.807032 0.590508i \(-0.798927\pi\)
−0.807032 + 0.590508i \(0.798927\pi\)
\(212\) 0 0
\(213\) −252585. 22637.7i −0.381468 0.0341888i
\(214\) 0 0
\(215\) 1.29782e6 1.91478
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −227890. 20424.5i −0.321081 0.0287767i
\(220\) 0 0
\(221\) 256761.i 0.353629i
\(222\) 0 0
\(223\) 185286.i 0.249506i 0.992188 + 0.124753i \(0.0398138\pi\)
−0.992188 + 0.124753i \(0.960186\pi\)
\(224\) 0 0
\(225\) −421051. 76083.9i −0.554470 0.100193i
\(226\) 0 0
\(227\) −877852. −1.13072 −0.565362 0.824843i \(-0.691264\pi\)
−0.565362 + 0.824843i \(0.691264\pi\)
\(228\) 0 0
\(229\) 1.22612e6i 1.54505i −0.634983 0.772526i \(-0.718993\pi\)
0.634983 0.772526i \(-0.281007\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 75659.1i 0.0913001i −0.998957 0.0456501i \(-0.985464\pi\)
0.998957 0.0456501i \(-0.0145359\pi\)
\(234\) 0 0
\(235\) −169304. −0.199985
\(236\) 0 0
\(237\) 119950. 1.33836e6i 0.138716 1.54775i
\(238\) 0 0
\(239\) 658418.i 0.745601i −0.927911 0.372801i \(-0.878397\pi\)
0.927911 0.372801i \(-0.121603\pi\)
\(240\) 0 0
\(241\) 567326.i 0.629202i 0.949224 + 0.314601i \(0.101871\pi\)
−0.949224 + 0.314601i \(0.898129\pi\)
\(242\) 0 0
\(243\) −397830. + 830072.i −0.432197 + 0.901779i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −599241. −0.624970
\(248\) 0 0
\(249\) −3822.57 + 42651.0i −0.00390712 + 0.0435944i
\(250\) 0 0
\(251\) −1.83665e6 −1.84010 −0.920050 0.391800i \(-0.871852\pi\)
−0.920050 + 0.391800i \(0.871852\pi\)
\(252\) 0 0
\(253\) −2.28305e6 −2.24240
\(254\) 0 0
\(255\) 87365.3 974794.i 0.0841373 0.938777i
\(256\) 0 0
\(257\) −902411. −0.852259 −0.426129 0.904662i \(-0.640123\pi\)
−0.426129 + 0.904662i \(0.640123\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 254031. 1.40582e6i 0.230827 1.27740i
\(262\) 0 0
\(263\) 933055.i 0.831798i 0.909411 + 0.415899i \(0.136533\pi\)
−0.909411 + 0.415899i \(0.863467\pi\)
\(264\) 0 0
\(265\) 532508.i 0.465813i
\(266\) 0 0
\(267\) −141662. + 1.58062e6i −0.121612 + 1.35691i
\(268\) 0 0
\(269\) −533435. −0.449470 −0.224735 0.974420i \(-0.572152\pi\)
−0.224735 + 0.974420i \(0.572152\pi\)
\(270\) 0 0
\(271\) 798687.i 0.660622i 0.943872 + 0.330311i \(0.107154\pi\)
−0.943872 + 0.330311i \(0.892846\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 820120.i 0.653951i
\(276\) 0 0
\(277\) 1.18032e6 0.924271 0.462136 0.886809i \(-0.347083\pi\)
0.462136 + 0.886809i \(0.347083\pi\)
\(278\) 0 0
\(279\) 166563. 921766.i 0.128106 0.708941i
\(280\) 0 0
\(281\) 1.56254e6i 1.18050i −0.807221 0.590250i \(-0.799029\pi\)
0.807221 0.590250i \(-0.200971\pi\)
\(282\) 0 0
\(283\) 1.08464e6i 0.805044i 0.915410 + 0.402522i \(0.131866\pi\)
−0.915410 + 0.402522i \(0.868134\pi\)
\(284\) 0 0
\(285\) 2.27502e6 + 203898.i 1.65911 + 0.148696i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −613068. −0.431782
\(290\) 0 0
\(291\) 363418. + 32571.1i 0.251579 + 0.0225476i
\(292\) 0 0
\(293\) 1.39243e6 0.947552 0.473776 0.880645i \(-0.342891\pi\)
0.473776 + 0.880645i \(0.342891\pi\)
\(294\) 0 0
\(295\) 744007. 0.497762
\(296\) 0 0
\(297\) 1.70128e6 + 467469.i 1.11914 + 0.307512i
\(298\) 0 0
\(299\) −1.40117e6 −0.906386
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −86556.3 + 965768.i −0.0541617 + 0.604319i
\(304\) 0 0
\(305\) 1.09212e6i 0.672234i
\(306\) 0 0
\(307\) 1.19400e6i 0.723034i 0.932366 + 0.361517i \(0.117741\pi\)
−0.932366 + 0.361517i \(0.882259\pi\)
\(308\) 0 0
\(309\) 601341. + 53894.8i 0.358282 + 0.0321108i
\(310\) 0 0
\(311\) 2.62556e6 1.53929 0.769646 0.638470i \(-0.220433\pi\)
0.769646 + 0.638470i \(0.220433\pi\)
\(312\) 0 0
\(313\) 887758.i 0.512193i −0.966651 0.256097i \(-0.917563\pi\)
0.966651 0.256097i \(-0.0824365\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.81415e6i 1.01397i 0.861955 + 0.506986i \(0.169240\pi\)
−0.861955 + 0.506986i \(0.830760\pi\)
\(318\) 0 0
\(319\) −2.73824e6 −1.50659
\(320\) 0 0
\(321\) 2.92946e6 + 262551.i 1.58681 + 0.142217i
\(322\) 0 0
\(323\) 1.88293e6i 1.00422i
\(324\) 0 0
\(325\) 503331.i 0.264329i
\(326\) 0 0
\(327\) −220271. + 2.45771e6i −0.113917 + 1.27105i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.55409e6 −1.28135 −0.640673 0.767814i \(-0.721345\pi\)
−0.640673 + 0.767814i \(0.721345\pi\)
\(332\) 0 0
\(333\) 865243. + 156350.i 0.427590 + 0.0772656i
\(334\) 0 0
\(335\) 207822. 0.101176
\(336\) 0 0
\(337\) 1.68914e6 0.810196 0.405098 0.914273i \(-0.367237\pi\)
0.405098 + 0.914273i \(0.367237\pi\)
\(338\) 0 0
\(339\) −2.61320e6 234206.i −1.23502 0.110688i
\(340\) 0 0
\(341\) −1.79541e6 −0.836138
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 5.31956e6 + 476762.i 2.40618 + 0.215652i
\(346\) 0 0
\(347\) 169943.i 0.0757668i 0.999282 + 0.0378834i \(0.0120615\pi\)
−0.999282 + 0.0378834i \(0.987938\pi\)
\(348\) 0 0
\(349\) 2.36988e6i 1.04151i 0.853706 + 0.520755i \(0.174350\pi\)
−0.853706 + 0.520755i \(0.825650\pi\)
\(350\) 0 0
\(351\) 1.04412e6 + 286899.i 0.452361 + 0.124297i
\(352\) 0 0
\(353\) −2.02957e6 −0.866897 −0.433448 0.901178i \(-0.642703\pi\)
−0.433448 + 0.901178i \(0.642703\pi\)
\(354\) 0 0
\(355\) 1.13712e6i 0.478892i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 352368.i 0.144298i −0.997394 0.0721489i \(-0.977014\pi\)
0.997394 0.0721489i \(-0.0229857\pi\)
\(360\) 0 0
\(361\) −1.91837e6 −0.774755
\(362\) 0 0
\(363\) 77774.3 867781.i 0.0309791 0.345655i
\(364\) 0 0
\(365\) 1.02595e6i 0.403082i
\(366\) 0 0
\(367\) 1.80805e6i 0.700723i 0.936615 + 0.350362i \(0.113941\pi\)
−0.936615 + 0.350362i \(0.886059\pi\)
\(368\) 0 0
\(369\) −4.09921e6 740728.i −1.56723 0.283200i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.63970e6 1.35454 0.677272 0.735733i \(-0.263162\pi\)
0.677272 + 0.735733i \(0.263162\pi\)
\(374\) 0 0
\(375\) −132692. + 1.48053e6i −0.0487265 + 0.543675i
\(376\) 0 0
\(377\) −1.68054e6 −0.608968
\(378\) 0 0
\(379\) 4.81458e6 1.72171 0.860856 0.508849i \(-0.169929\pi\)
0.860856 + 0.508849i \(0.169929\pi\)
\(380\) 0 0
\(381\) 167955. 1.87399e6i 0.0592762 0.661385i
\(382\) 0 0
\(383\) 3.70544e6 1.29075 0.645376 0.763865i \(-0.276701\pi\)
0.645376 + 0.763865i \(0.276701\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.43995e6 + 802299.i 1.50696 + 0.272307i
\(388\) 0 0
\(389\) 1.09914e6i 0.368282i 0.982900 + 0.184141i \(0.0589503\pi\)
−0.982900 + 0.184141i \(0.941050\pi\)
\(390\) 0 0
\(391\) 4.40274e6i 1.45640i
\(392\) 0 0
\(393\) 337969. 3.77095e6i 0.110381 1.23160i
\(394\) 0 0
\(395\) 6.02523e6 1.94304
\(396\) 0 0
\(397\) 5.54292e6i 1.76507i −0.470246 0.882535i \(-0.655835\pi\)
0.470246 0.882535i \(-0.344165\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.50177e6i 1.08749i −0.839250 0.543746i \(-0.817005\pi\)
0.839250 0.543746i \(-0.182995\pi\)
\(402\) 0 0
\(403\) −1.10189e6 −0.337970
\(404\) 0 0
\(405\) −3.86641e6 1.44449e6i −1.17131 0.437599i
\(406\) 0 0
\(407\) 1.68532e6i 0.504307i
\(408\) 0 0
\(409\) 4.96495e6i 1.46760i 0.679367 + 0.733799i \(0.262254\pi\)
−0.679367 + 0.733799i \(0.737746\pi\)
\(410\) 0 0
\(411\) 3.23996e6 + 290379.i 0.946095 + 0.0847932i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −192013. −0.0547281
\(416\) 0 0
\(417\) −3.13908e6 281338.i −0.884020 0.0792297i
\(418\) 0 0
\(419\) 2.93856e6 0.817711 0.408855 0.912599i \(-0.365928\pi\)
0.408855 + 0.912599i \(0.365928\pi\)
\(420\) 0 0
\(421\) −90444.5 −0.0248701 −0.0124350 0.999923i \(-0.503958\pi\)
−0.0124350 + 0.999923i \(0.503958\pi\)
\(422\) 0 0
\(423\) −579201. 104662.i −0.157391 0.0284405i
\(424\) 0 0
\(425\) 1.58156e6 0.424730
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 185273. 2.06722e6i 0.0486037 0.542305i
\(430\) 0 0
\(431\) 295281.i 0.0765672i 0.999267 + 0.0382836i \(0.0121890\pi\)
−0.999267 + 0.0382836i \(0.987811\pi\)
\(432\) 0 0
\(433\) 7.22505e6i 1.85191i 0.377628 + 0.925957i \(0.376740\pi\)
−0.377628 + 0.925957i \(0.623260\pi\)
\(434\) 0 0
\(435\) 6.38017e6 + 571819.i 1.61663 + 0.144889i
\(436\) 0 0
\(437\) −1.02753e7 −2.57390
\(438\) 0 0
\(439\) 1.59769e6i 0.395668i 0.980236 + 0.197834i \(0.0633906\pi\)
−0.980236 + 0.197834i \(0.936609\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.69173e6i 1.37795i −0.724783 0.688977i \(-0.758060\pi\)
0.724783 0.688977i \(-0.241940\pi\)
\(444\) 0 0
\(445\) −7.11590e6 −1.70345
\(446\) 0 0
\(447\) −7.78707e6 697911.i −1.84334 0.165208i
\(448\) 0 0
\(449\) 7.13493e6i 1.67022i 0.550082 + 0.835111i \(0.314597\pi\)
−0.550082 + 0.835111i \(0.685403\pi\)
\(450\) 0 0
\(451\) 7.98441e6i 1.84842i
\(452\) 0 0
\(453\) −489997. + 5.46723e6i −0.112188 + 1.25176i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.94203e6 0.434976 0.217488 0.976063i \(-0.430214\pi\)
0.217488 + 0.976063i \(0.430214\pi\)
\(458\) 0 0
\(459\) 901489. 3.28083e6i 0.199723 0.726862i
\(460\) 0 0
\(461\) 6.10948e6 1.33891 0.669456 0.742852i \(-0.266527\pi\)
0.669456 + 0.742852i \(0.266527\pi\)
\(462\) 0 0
\(463\) 4.08866e6 0.886398 0.443199 0.896423i \(-0.353844\pi\)
0.443199 + 0.896423i \(0.353844\pi\)
\(464\) 0 0
\(465\) 4.18335e6 + 374930.i 0.897206 + 0.0804115i
\(466\) 0 0
\(467\) −444247. −0.0942611 −0.0471306 0.998889i \(-0.515008\pi\)
−0.0471306 + 0.998889i \(0.515008\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 8.49669e6 + 761511.i 1.76481 + 0.158170i
\(472\) 0 0
\(473\) 8.64810e6i 1.77733i
\(474\) 0 0
\(475\) 3.69112e6i 0.750627i
\(476\) 0 0
\(477\) 329190. 1.82175e6i 0.0662446 0.366600i
\(478\) 0 0
\(479\) −1.50806e6 −0.300317 −0.150159 0.988662i \(-0.547978\pi\)
−0.150159 + 0.988662i \(0.547978\pi\)
\(480\) 0 0
\(481\) 1.03433e6i 0.203843i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.63609e6i 0.315830i
\(486\) 0 0
\(487\) 2.85810e6 0.546078 0.273039 0.962003i \(-0.411971\pi\)
0.273039 + 0.962003i \(0.411971\pi\)
\(488\) 0 0
\(489\) 875679. 9.77054e6i 0.165605 1.84776i
\(490\) 0 0
\(491\) 4.95391e6i 0.927352i −0.886005 0.463676i \(-0.846530\pi\)
0.886005 0.463676i \(-0.153470\pi\)
\(492\) 0 0
\(493\) 5.28056e6i 0.978503i
\(494\) 0 0
\(495\) −1.40678e6 + 7.78517e6i −0.258056 + 1.42809i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −7.98813e6 −1.43613 −0.718065 0.695976i \(-0.754972\pi\)
−0.718065 + 0.695976i \(0.754972\pi\)
\(500\) 0 0
\(501\) −819034. + 9.13853e6i −0.145783 + 1.62660i
\(502\) 0 0
\(503\) 2.57392e6 0.453602 0.226801 0.973941i \(-0.427173\pi\)
0.226801 + 0.973941i \(0.427173\pi\)
\(504\) 0 0
\(505\) −4.34784e6 −0.758657
\(506\) 0 0
\(507\) −402957. + 4.49607e6i −0.0696208 + 0.776807i
\(508\) 0 0
\(509\) 9.29786e6 1.59070 0.795350 0.606150i \(-0.207287\pi\)
0.795350 + 0.606150i \(0.207287\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 7.65697e6 + 2.10394e6i 1.28459 + 0.352972i
\(514\) 0 0
\(515\) 2.70721e6i 0.449784i
\(516\) 0 0
\(517\) 1.12816e6i 0.185629i
\(518\) 0 0
\(519\) 224370. 2.50345e6i 0.0365634 0.407963i
\(520\) 0 0
\(521\) 1.09866e7 1.77325 0.886624 0.462490i \(-0.153044\pi\)
0.886624 + 0.462490i \(0.153044\pi\)
\(522\) 0 0
\(523\) 278694.i 0.0445527i −0.999752 0.0222763i \(-0.992909\pi\)
0.999752 0.0222763i \(-0.00709137\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.46235e6i 0.543057i
\(528\) 0 0
\(529\) −1.75899e7 −2.73290
\(530\) 0 0
\(531\) 2.54530e6 + 459936.i 0.391744 + 0.0707883i
\(532\) 0 0
\(533\) 4.90026e6i 0.747139i
\(534\) 0 0
\(535\) 1.31883e7i 1.99207i
\(536\) 0 0
\(537\) −5.13343e6 460080.i −0.768196 0.0688491i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.24150e6 −0.182370 −0.0911849 0.995834i \(-0.529065\pi\)
−0.0911849 + 0.995834i \(0.529065\pi\)
\(542\) 0 0
\(543\) 6.54515e6 + 586605.i 0.952621 + 0.0853780i
\(544\) 0 0
\(545\) −1.10645e7 −1.59566
\(546\) 0 0
\(547\) −5.28542e6 −0.755286 −0.377643 0.925951i \(-0.623265\pi\)
−0.377643 + 0.925951i \(0.623265\pi\)
\(548\) 0 0
\(549\) 675135. 3.73622e6i 0.0956004 0.529056i
\(550\) 0 0
\(551\) −1.23240e7 −1.72931
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −351939. + 3.92683e6i −0.0484993 + 0.541140i
\(556\) 0 0
\(557\) 7.68636e6i 1.04974i 0.851182 + 0.524871i \(0.175887\pi\)
−0.851182 + 0.524871i \(0.824113\pi\)
\(558\) 0 0
\(559\) 5.30759e6i 0.718402i
\(560\) 0 0
\(561\) −6.49558e6 582162.i −0.871386 0.0780974i
\(562\) 0 0
\(563\) 4.56529e6 0.607012 0.303506 0.952830i \(-0.401843\pi\)
0.303506 + 0.952830i \(0.401843\pi\)
\(564\) 0 0
\(565\) 1.17645e7i 1.55043i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.68731e6i 0.606936i 0.952842 + 0.303468i \(0.0981446\pi\)
−0.952842 + 0.303468i \(0.901855\pi\)
\(570\) 0 0
\(571\) 5.13561e6 0.659177 0.329589 0.944125i \(-0.393090\pi\)
0.329589 + 0.944125i \(0.393090\pi\)
\(572\) 0 0
\(573\) 6.17947e6 + 553831.i 0.786257 + 0.0704678i
\(574\) 0 0
\(575\) 8.63073e6i 1.08862i
\(576\) 0 0
\(577\) 667514.i 0.0834682i −0.999129 0.0417341i \(-0.986712\pi\)
0.999129 0.0417341i \(-0.0132882\pi\)
\(578\) 0 0
\(579\) −722447. + 8.06083e6i −0.0895590 + 0.999271i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3.54839e6 −0.432374
\(584\) 0 0
\(585\) −863382. + 4.77798e6i −0.104307 + 0.577238i
\(586\) 0 0
\(587\) 9.60922e6 1.15105 0.575523 0.817785i \(-0.304798\pi\)
0.575523 + 0.817785i \(0.304798\pi\)
\(588\) 0 0
\(589\) −8.08062e6 −0.959747
\(590\) 0 0
\(591\) −9.00934e6 807456.i −1.06102 0.0950934i
\(592\) 0 0
\(593\) −1.00253e7 −1.17074 −0.585371 0.810766i \(-0.699051\pi\)
−0.585371 + 0.810766i \(0.699051\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.31059e7 + 1.17461e6i 1.50498 + 0.134883i
\(598\) 0 0
\(599\) 4.96759e6i 0.565690i −0.959166 0.282845i \(-0.908722\pi\)
0.959166 0.282845i \(-0.0912782\pi\)
\(600\) 0 0
\(601\) 2.91804e6i 0.329538i 0.986332 + 0.164769i \(0.0526878\pi\)
−0.986332 + 0.164769i \(0.947312\pi\)
\(602\) 0 0
\(603\) 710973. + 128473.i 0.0796269 + 0.0143886i
\(604\) 0 0
\(605\) 3.90671e6 0.433933
\(606\) 0 0
\(607\) 5.53707e6i 0.609969i 0.952357 + 0.304985i \(0.0986513\pi\)
−0.952357 + 0.304985i \(0.901349\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 692387.i 0.0750319i
\(612\) 0 0
\(613\) −5.39229e6 −0.579591 −0.289796 0.957089i \(-0.593587\pi\)
−0.289796 + 0.957089i \(0.593587\pi\)
\(614\) 0 0
\(615\) 1.66736e6 1.86039e7i 0.177763 1.98343i
\(616\) 0 0
\(617\) 1.18727e7i 1.25556i −0.778392 0.627778i \(-0.783965\pi\)
0.778392 0.627778i \(-0.216035\pi\)
\(618\) 0 0
\(619\) 6.71087e6i 0.703967i 0.936006 + 0.351983i \(0.114493\pi\)
−0.936006 + 0.351983i \(0.885507\pi\)
\(620\) 0 0
\(621\) 1.79039e7 + 4.91953e6i 1.86302 + 0.511911i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.21677e7 −1.24597
\(626\) 0 0
\(627\) 1.35868e6 1.51597e7i 0.138022 1.54001i
\(628\) 0 0
\(629\) −3.25004e6 −0.327539
\(630\) 0 0
\(631\) 1.01133e7 1.01116 0.505580 0.862780i \(-0.331279\pi\)
0.505580 + 0.862780i \(0.331279\pi\)
\(632\) 0 0
\(633\) 1.45251e6 1.62066e7i 0.144082 1.60762i
\(634\) 0 0
\(635\) 8.43661e6 0.830297
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 702957. 3.89019e6i 0.0681046 0.376893i
\(640\) 0 0
\(641\) 8.43591e6i 0.810937i −0.914109 0.405468i \(-0.867108\pi\)
0.914109 0.405468i \(-0.132892\pi\)
\(642\) 0 0
\(643\) 1.51051e7i 1.44077i 0.693573 + 0.720386i \(0.256035\pi\)
−0.693573 + 0.720386i \(0.743965\pi\)
\(644\) 0 0
\(645\) −1.80596e6 + 2.01503e7i −0.170926 + 1.90714i
\(646\) 0 0
\(647\) −1.08522e7 −1.01919 −0.509597 0.860413i \(-0.670205\pi\)
−0.509597 + 0.860413i \(0.670205\pi\)
\(648\) 0 0
\(649\) 4.95772e6i 0.462030i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 590041.i 0.0541501i 0.999633 + 0.0270750i \(0.00861931\pi\)
−0.999633 + 0.0270750i \(0.991381\pi\)
\(654\) 0 0
\(655\) 1.69766e7 1.54614
\(656\) 0 0
\(657\) 634229. 3.50984e6i 0.0573235 0.317230i
\(658\) 0 0
\(659\) 1.38679e7i 1.24394i −0.783042 0.621969i \(-0.786333\pi\)
0.783042 0.621969i \(-0.213667\pi\)
\(660\) 0 0
\(661\) 1.62028e7i 1.44240i −0.692727 0.721200i \(-0.743591\pi\)
0.692727 0.721200i \(-0.256409\pi\)
\(662\) 0 0
\(663\) −3.98652e6 357290.i −0.352217 0.0315672i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.88166e7 −2.50800
\(668\) 0 0
\(669\) −2.87679e6 257831.i −0.248510 0.0222725i
\(670\) 0 0
\(671\) −7.27738e6 −0.623977
\(672\) 0 0
\(673\) −1.09799e7 −0.934456 −0.467228 0.884137i \(-0.654747\pi\)
−0.467228 + 0.884137i \(0.654747\pi\)
\(674\) 0 0
\(675\) 1.76720e6 6.43145e6i 0.149288 0.543312i
\(676\) 0 0
\(677\) −190993. −0.0160157 −0.00800785 0.999968i \(-0.502549\pi\)
−0.00800785 + 0.999968i \(0.502549\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.22156e6 1.36297e7i 0.100936 1.12621i
\(682\) 0 0
\(683\) 1.35316e7i 1.10993i 0.831873 + 0.554967i \(0.187269\pi\)
−0.831873 + 0.554967i \(0.812731\pi\)
\(684\) 0 0
\(685\) 1.45861e7i 1.18772i
\(686\) 0 0
\(687\) 1.90370e7 + 1.70618e6i 1.53888 + 0.137922i
\(688\) 0 0
\(689\) −2.17775e6 −0.174767
\(690\) 0 0
\(691\) 9.88908e6i 0.787881i −0.919136 0.393941i \(-0.871112\pi\)
0.919136 0.393941i \(-0.128888\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.41320e7i 1.10979i
\(696\) 0 0
\(697\) 1.53975e7 1.20052
\(698\) 0 0
\(699\) 1.17470e6 + 105282.i 0.0909357 + 0.00815005i
\(700\) 0 0
\(701\) 1.46554e6i 0.112643i 0.998413 + 0.0563213i \(0.0179371\pi\)
−0.998413 + 0.0563213i \(0.982063\pi\)
\(702\) 0 0
\(703\) 7.58512e6i 0.578861i
\(704\) 0 0
\(705\) 235591. 2.62865e6i 0.0178520 0.199187i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.66337e6 0.124272 0.0621361 0.998068i \(-0.480209\pi\)
0.0621361 + 0.998068i \(0.480209\pi\)
\(710\) 0 0
\(711\) 2.06127e7 + 3.72473e6i 1.52919 + 0.276325i
\(712\) 0 0
\(713\) −1.88945e7 −1.39191
\(714\) 0 0
\(715\) 9.30652e6 0.680805
\(716\) 0 0
\(717\) 1.02227e7 + 916207.i 0.742625 + 0.0665573i
\(718\) 0 0
\(719\) 7.28664e6 0.525660 0.262830 0.964842i \(-0.415344\pi\)
0.262830 + 0.964842i \(0.415344\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −8.80843e6 789449.i −0.626690 0.0561667i
\(724\) 0 0
\(725\) 1.03515e7i 0.731408i
\(726\) 0 0
\(727\) 2.37534e6i 0.166682i −0.996521 0.0833411i \(-0.973441\pi\)
0.996521 0.0833411i \(-0.0265591\pi\)
\(728\) 0 0
\(729\) −1.23343e7 7.33187e6i −0.859598 0.510970i
\(730\) 0 0
\(731\) −1.66774e7 −1.15434
\(732\) 0 0
\(733\) 2.12018e7i 1.45751i −0.684774 0.728755i \(-0.740099\pi\)
0.684774 0.728755i \(-0.259901\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.38483e6i 0.0939133i
\(738\) 0 0
\(739\) 8248.53 0.000555604 0.000277802 1.00000i \(-0.499912\pi\)
0.000277802 1.00000i \(0.499912\pi\)
\(740\) 0 0
\(741\) 833861. 9.30395e6i 0.0557889 0.622475i
\(742\) 0 0
\(743\) 2.10286e7i 1.39746i −0.715386 0.698730i \(-0.753749\pi\)
0.715386 0.698730i \(-0.246251\pi\)
\(744\) 0 0
\(745\) 3.50570e7i 2.31411i
\(746\) 0 0
\(747\) −656890. 118700.i −0.0430716 0.00778305i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.24333e7 −0.804426 −0.402213 0.915546i \(-0.631759\pi\)
−0.402213 + 0.915546i \(0.631759\pi\)
\(752\) 0 0
\(753\) 2.55575e6 2.85162e7i 0.164259 1.83275i
\(754\) 0 0
\(755\) −2.46132e7 −1.57145
\(756\) 0 0
\(757\) −1.56314e7 −0.991421 −0.495710 0.868488i \(-0.665092\pi\)
−0.495710 + 0.868488i \(0.665092\pi\)
\(758\) 0 0
\(759\) 3.17692e6 3.54471e7i 0.200172 2.23345i
\(760\) 0 0
\(761\) 3.54527e6 0.221915 0.110958 0.993825i \(-0.464608\pi\)
0.110958 + 0.993825i \(0.464608\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.50133e7 + 2.71291e6i 0.927518 + 0.167603i
\(766\) 0 0
\(767\) 3.04269e6i 0.186754i
\(768\) 0 0
\(769\) 4.01821e6i 0.245028i −0.992467 0.122514i \(-0.960904\pi\)
0.992467 0.122514i \(-0.0390957\pi\)
\(770\) 0 0
\(771\) 1.25573e6 1.40110e7i 0.0760782 0.848857i
\(772\) 0 0
\(773\) −2.96681e7 −1.78584 −0.892918 0.450219i \(-0.851346\pi\)
−0.892918 + 0.450219i \(0.851346\pi\)
\(774\) 0 0
\(775\) 6.78729e6i 0.405922i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.59355e7i 2.12168i
\(780\) 0 0
\(781\) −7.57728e6 −0.444514
\(782\) 0 0
\(783\) 2.14735e7 + 5.90038e6i 1.25170 + 0.343934i
\(784\) 0 0
\(785\) 3.82517e7i 2.21553i
\(786\) 0 0
\(787\) 6.48183e6i 0.373045i −0.982451 0.186522i \(-0.940278\pi\)
0.982451 0.186522i \(-0.0597217\pi\)
\(788\) 0 0
\(789\) −1.44868e7 1.29837e6i −0.828477 0.0742517i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.46634e6 −0.252214
\(794\) 0 0
\(795\) 8.26783e6 + 740999.i 0.463953 + 0.0415815i
\(796\) 0 0
\(797\) 3.52053e7 1.96319 0.981594 0.190979i \(-0.0611663\pi\)
0.981594 + 0.190979i \(0.0611663\pi\)
\(798\) 0 0
\(799\) 2.17561e6 0.120563
\(800\) 0 0
\(801\) −2.43440e7 4.39896e6i −1.34064 0.242253i
\(802\) 0 0
\(803\) −6.83645e6 −0.374147
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 742290. 8.28224e6i 0.0401227 0.447676i
\(808\) 0 0
\(809\) 1.62772e7i 0.874397i 0.899365 + 0.437198i \(0.144029\pi\)
−0.899365 + 0.437198i \(0.855971\pi\)
\(810\) 0 0
\(811\) 823682.i 0.0439752i −0.999758 0.0219876i \(-0.993001\pi\)
0.999758 0.0219876i \(-0.00699943\pi\)
\(812\) 0 0
\(813\) −1.24006e7 1.11139e6i −0.657985 0.0589715i
\(814\) 0 0
\(815\) 4.39865e7 2.31967
\(816\) 0 0
\(817\) 3.89226e7i 2.04008i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.33763e6i 0.224592i 0.993675 + 0.112296i \(0.0358205\pi\)
−0.993675 + 0.112296i \(0.964179\pi\)
\(822\) 0 0
\(823\) 1.01908e7 0.524454 0.262227 0.965006i \(-0.415543\pi\)
0.262227 + 0.965006i \(0.415543\pi\)
\(824\) 0 0
\(825\) −1.27334e7 1.14122e6i −0.651341 0.0583760i
\(826\) 0 0
\(827\) 1.90780e7i 0.969996i 0.874515 + 0.484998i \(0.161180\pi\)
−0.874515 + 0.484998i \(0.838820\pi\)
\(828\) 0 0
\(829\) 5.64200e6i 0.285133i −0.989785 0.142566i \(-0.954465\pi\)
0.989785 0.142566i \(-0.0455354\pi\)
\(830\) 0 0
\(831\) −1.64244e6 + 1.83259e7i −0.0825065 + 0.920581i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −4.11412e7 −2.04202
\(836\) 0 0
\(837\) 1.40798e7 + 3.86876e6i 0.694675 + 0.190879i
\(838\) 0 0
\(839\) 1.11858e7 0.548608 0.274304 0.961643i \(-0.411552\pi\)
0.274304 + 0.961643i \(0.411552\pi\)
\(840\) 0 0
\(841\) −1.40509e7 −0.685037
\(842\) 0 0
\(843\) 2.42604e7 + 2.17432e6i 1.17579 + 0.105379i
\(844\) 0 0
\(845\) −2.02411e7 −0.975196
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.68404e7 1.50931e6i −0.801830 0.0718635i
\(850\) 0 0
\(851\) 1.77358e7i 0.839514i
\(852\) 0 0
\(853\) 8.52648e6i 0.401233i −0.979670 0.200617i \(-0.935705\pi\)
0.979670 0.200617i \(-0.0642946\pi\)
\(854\) 0 0
\(855\) −6.33152e6 + 3.50388e7i −0.296205 + 1.63921i
\(856\) 0 0
\(857\) 2.71350e7 1.26205 0.631027 0.775761i \(-0.282634\pi\)
0.631027 + 0.775761i \(0.282634\pi\)
\(858\) 0 0
\(859\) 2.17238e7i 1.00451i 0.864720 + 0.502254i \(0.167496\pi\)
−0.864720 + 0.502254i \(0.832504\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.77634e7i 0.811895i −0.913896 0.405948i \(-0.866942\pi\)
0.913896 0.405948i \(-0.133058\pi\)
\(864\) 0 0
\(865\) 1.12704e7 0.512153
\(866\) 0 0
\(867\) 853101. 9.51863e6i 0.0385437 0.430058i
\(868\) 0 0
\(869\) 4.01494e7i 1.80355i
\(870\) 0 0
\(871\) 849909.i 0.0379601i
\(872\) 0 0
\(873\) −1.01141e6 + 5.59719e6i −0.0449152 + 0.248562i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −621725. −0.0272960 −0.0136480 0.999907i \(-0.504344\pi\)
−0.0136480 + 0.999907i \(0.504344\pi\)
\(878\) 0 0
\(879\) −1.93760e6 + 2.16191e7i −0.0845847 + 0.943769i
\(880\) 0 0
\(881\) 3.18993e7 1.38465 0.692327 0.721584i \(-0.256586\pi\)
0.692327 + 0.721584i \(0.256586\pi\)
\(882\) 0 0
\(883\) −1.60937e7 −0.694633 −0.347316 0.937748i \(-0.612907\pi\)
−0.347316 + 0.937748i \(0.612907\pi\)
\(884\) 0 0
\(885\) −1.03531e6 + 1.15516e7i −0.0444335 + 0.495775i
\(886\) 0 0
\(887\) 2.19434e7 0.936471 0.468235 0.883604i \(-0.344890\pi\)
0.468235 + 0.883604i \(0.344890\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.62541e6 + 2.57640e7i −0.406186 + 1.08722i
\(892\) 0 0
\(893\) 5.07754e6i 0.213071i
\(894\) 0 0
\(895\) 2.31105e7i 0.964387i
\(896\) 0 0
\(897\) 1.94977e6 2.17549e7i 0.0809100 0.902768i
\(898\) 0 0
\(899\) −2.26616e7 −0.935173
\(900\) 0 0
\(901\) 6.84288e6i 0.280819i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.94660e7i 1.19591i
\(906\) 0 0
\(907\) 2.41122e7 0.973238 0.486619 0.873614i \(-0.338230\pi\)
0.486619 + 0.873614i \(0.338230\pi\)
\(908\) 0 0
\(909\) −1.48743e7 2.68779e6i −0.597071 0.107891i
\(910\) 0 0
\(911\) 1.16994e7i 0.467056i 0.972350 + 0.233528i \(0.0750271\pi\)
−0.972350 + 0.233528i \(0.924973\pi\)
\(912\) 0 0
\(913\) 1.27949e6i 0.0507994i
\(914\) 0 0
\(915\) 1.69565e7 + 1.51971e6i 0.669550 + 0.0600080i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.35972e7 0.531080 0.265540 0.964100i \(-0.414450\pi\)
0.265540 + 0.964100i \(0.414450\pi\)
\(920\) 0 0
\(921\) −1.85383e7 1.66148e6i −0.720147 0.0645427i
\(922\) 0 0
\(923\) −4.65039e6 −0.179674
\(924\) 0 0
\(925\) −6.37110e6 −0.244827
\(926\) 0 0
\(927\) −1.67357e6 + 9.26156e6i −0.0639651 + 0.353985i
\(928\) 0 0
\(929\) −1.13421e7 −0.431177 −0.215589 0.976484i \(-0.569167\pi\)
−0.215589 + 0.976484i \(0.569167\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.65354e6 + 4.07651e7i −0.137407 + 1.53315i
\(934\) 0 0
\(935\) 2.92428e7i 1.09393i
\(936\) 0 0
\(937\) 2.45048e7i 0.911806i −0.890029 0.455903i \(-0.849316\pi\)
0.890029 0.455903i \(-0.150684\pi\)
\(938\) 0 0
\(939\) 1.37835e7 + 1.23534e6i 0.510149 + 0.0457217i
\(940\) 0 0
\(941\) 5.41453e6 0.199336 0.0996682 0.995021i \(-0.468222\pi\)
0.0996682 + 0.995021i \(0.468222\pi\)
\(942\) 0 0
\(943\) 8.40260e7i 3.07705i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00820e7i 1.09001i −0.838432 0.545006i \(-0.816527\pi\)
0.838432 0.545006i \(-0.183473\pi\)
\(948\) 0 0
\(949\) −4.19573e6 −0.151231
\(950\) 0 0
\(951\) −2.81669e7 2.52444e6i −1.00992 0.0905137i
\(952\) 0 0
\(953\) 2.03196e7i 0.724742i −0.932034 0.362371i \(-0.881967\pi\)
0.932034 0.362371i \(-0.118033\pi\)
\(954\) 0 0
\(955\) 2.78197e7i 0.987060i
\(956\) 0 0
\(957\) 3.81034e6 4.25146e7i 0.134488 1.50058i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.37704e7 0.480991
\(962\) 0 0
\(963\) −8.15286e6 + 4.51182e7i −0.283299 + 1.56778i
\(964\) 0 0
\(965\) −3.62895e7 −1.25448
\(966\) 0 0
\(967\) −2.69494e7 −0.926792 −0.463396 0.886151i \(-0.653369\pi\)
−0.463396 + 0.886151i \(0.653369\pi\)
\(968\) 0 0
\(969\) −2.92347e7 2.62014e6i −1.00021 0.0896428i
\(970\) 0 0
\(971\) −3.42537e7 −1.16590 −0.582948 0.812510i \(-0.698101\pi\)
−0.582948 + 0.812510i \(0.698101\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7.81483e6 700399.i −0.263274 0.0235958i
\(976\) 0 0
\(977\) 2.50876e7i 0.840858i −0.907325 0.420429i \(-0.861880\pi\)
0.907325 0.420429i \(-0.138120\pi\)
\(978\) 0 0
\(979\) 4.74171e7i 1.58117i
\(980\) 0 0
\(981\) −3.78525e7 6.83995e6i −1.25580 0.226924i
\(982\) 0 0
\(983\) −2.36116e7 −0.779367 −0.389684 0.920949i \(-0.627416\pi\)
−0.389684 + 0.920949i \(0.627416\pi\)
\(984\) 0 0
\(985\) 4.05596e7i 1.33200i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.10105e7i 2.95870i
\(990\) 0 0
\(991\) −2.17500e7 −0.703519 −0.351760 0.936090i \(-0.614417\pi\)
−0.351760 + 0.936090i \(0.614417\pi\)
\(992\) 0 0
\(993\) 3.55409e6 3.96554e7i 0.114381 1.27623i
\(994\) 0 0
\(995\) 5.90021e7i 1.88934i
\(996\) 0 0
\(997\) 4.35621e7i 1.38794i 0.720004 + 0.693970i \(0.244140\pi\)
−0.720004 + 0.693970i \(0.755860\pi\)
\(998\) 0 0
\(999\) −3.63153e6 + 1.32164e7i −0.115127 + 0.418986i
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.f.c.293.20 yes 40
3.2 odd 2 inner 588.6.f.c.293.22 yes 40
7.6 odd 2 inner 588.6.f.c.293.21 yes 40
21.20 even 2 inner 588.6.f.c.293.19 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.f.c.293.19 40 21.20 even 2 inner
588.6.f.c.293.20 yes 40 1.1 even 1 trivial
588.6.f.c.293.21 yes 40 7.6 odd 2 inner
588.6.f.c.293.22 yes 40 3.2 odd 2 inner