Properties

Label 245.6.a.h.1.2
Level $245$
Weight $6$
Character 245.1
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $6$
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] = [245,6,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.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: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.22733\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.22733 q^{2} +15.9209 q^{3} +20.2343 q^{4} +25.0000 q^{5} -115.066 q^{6} +85.0347 q^{8} +10.4759 q^{9} +O(q^{10})\) \(q-7.22733 q^{2} +15.9209 q^{3} +20.2343 q^{4} +25.0000 q^{5} -115.066 q^{6} +85.0347 q^{8} +10.4759 q^{9} -180.683 q^{10} -387.066 q^{11} +322.149 q^{12} +920.823 q^{13} +398.023 q^{15} -1262.07 q^{16} -678.473 q^{17} -75.7130 q^{18} -2764.19 q^{19} +505.857 q^{20} +2797.46 q^{22} +2065.73 q^{23} +1353.83 q^{24} +625.000 q^{25} -6655.09 q^{26} -3702.00 q^{27} -2921.75 q^{29} -2876.64 q^{30} -4255.06 q^{31} +6400.29 q^{32} -6162.46 q^{33} +4903.55 q^{34} +211.973 q^{36} -154.621 q^{37} +19977.7 q^{38} +14660.4 q^{39} +2125.87 q^{40} +7600.35 q^{41} +13533.2 q^{43} -7832.01 q^{44} +261.898 q^{45} -14929.7 q^{46} -15002.5 q^{47} -20093.3 q^{48} -4517.08 q^{50} -10801.9 q^{51} +18632.2 q^{52} +18078.9 q^{53} +26755.6 q^{54} -9676.66 q^{55} -44008.4 q^{57} +21116.4 q^{58} -6479.69 q^{59} +8053.71 q^{60} -4006.33 q^{61} +30752.7 q^{62} -5870.74 q^{64} +23020.6 q^{65} +44538.1 q^{66} -33620.3 q^{67} -13728.4 q^{68} +32888.3 q^{69} -61404.3 q^{71} +890.817 q^{72} +42664.3 q^{73} +1117.50 q^{74} +9950.58 q^{75} -55931.4 q^{76} -105955. q^{78} -107893. q^{79} -31551.8 q^{80} -61484.9 q^{81} -54930.2 q^{82} -99024.8 q^{83} -16961.8 q^{85} -97808.7 q^{86} -46516.9 q^{87} -32914.1 q^{88} -7661.84 q^{89} -1892.82 q^{90} +41798.5 q^{92} -67744.5 q^{93} +108428. q^{94} -69104.7 q^{95} +101899. q^{96} -87087.4 q^{97} -4054.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 5 q^{2} - 20 q^{3} + 31 q^{4} + 150 q^{5} - 96 q^{6} - 135 q^{8} + 378 q^{9} - 125 q^{10} - 924 q^{11} - 370 q^{12} + 150 q^{13} - 500 q^{15} + 435 q^{16} - 1540 q^{17} + 195 q^{18} - 92 q^{19} + 775 q^{20} - 6855 q^{22} - 3920 q^{23} + 7200 q^{24} + 3750 q^{25} - 2635 q^{26} - 2060 q^{27} + 1264 q^{29} - 2400 q^{30} + 7160 q^{31} + 9105 q^{32} - 4460 q^{33} - 2166 q^{34} - 26375 q^{36} - 14170 q^{37} + 46215 q^{38} - 15376 q^{39} - 3375 q^{40} - 4098 q^{41} - 24460 q^{43} - 27873 q^{44} + 9450 q^{45} + 6815 q^{46} - 42940 q^{47} - 11610 q^{48} - 3125 q^{50} - 42008 q^{51} + 36115 q^{52} - 2450 q^{53} - 19566 q^{54} - 23100 q^{55} - 97100 q^{57} - 36110 q^{58} + 64600 q^{59} - 9250 q^{60} - 73620 q^{61} - 111440 q^{62} - 157997 q^{64} + 3750 q^{65} + 139138 q^{66} - 142620 q^{67} - 124330 q^{68} - 17344 q^{69} - 154256 q^{71} - 117495 q^{72} + 5120 q^{73} + 2785 q^{74} - 12500 q^{75} - 7775 q^{76} - 214090 q^{78} - 222504 q^{79} + 10875 q^{80} - 43986 q^{81} - 31665 q^{82} - 179580 q^{83} - 38500 q^{85} - 207160 q^{86} + 209300 q^{87} - 45145 q^{88} - 41648 q^{89} + 4875 q^{90} - 292185 q^{92} - 198520 q^{93} + 333699 q^{94} - 2300 q^{95} - 61824 q^{96} + 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.22733 −1.27762 −0.638812 0.769363i \(-0.720574\pi\)
−0.638812 + 0.769363i \(0.720574\pi\)
\(3\) 15.9209 1.02133 0.510664 0.859780i \(-0.329400\pi\)
0.510664 + 0.859780i \(0.329400\pi\)
\(4\) 20.2343 0.632321
\(5\) 25.0000 0.447214
\(6\) −115.066 −1.30487
\(7\) 0 0
\(8\) 85.0347 0.469755
\(9\) 10.4759 0.0431108
\(10\) −180.683 −0.571371
\(11\) −387.066 −0.964504 −0.482252 0.876033i \(-0.660181\pi\)
−0.482252 + 0.876033i \(0.660181\pi\)
\(12\) 322.149 0.645808
\(13\) 920.823 1.51118 0.755592 0.655042i \(-0.227349\pi\)
0.755592 + 0.655042i \(0.227349\pi\)
\(14\) 0 0
\(15\) 398.023 0.456752
\(16\) −1262.07 −1.23249
\(17\) −678.473 −0.569390 −0.284695 0.958618i \(-0.591892\pi\)
−0.284695 + 0.958618i \(0.591892\pi\)
\(18\) −75.7130 −0.0550794
\(19\) −2764.19 −1.75664 −0.878322 0.478070i \(-0.841337\pi\)
−0.878322 + 0.478070i \(0.841337\pi\)
\(20\) 505.857 0.282783
\(21\) 0 0
\(22\) 2797.46 1.23227
\(23\) 2065.73 0.814241 0.407121 0.913374i \(-0.366533\pi\)
0.407121 + 0.913374i \(0.366533\pi\)
\(24\) 1353.83 0.479774
\(25\) 625.000 0.200000
\(26\) −6655.09 −1.93073
\(27\) −3702.00 −0.977298
\(28\) 0 0
\(29\) −2921.75 −0.645131 −0.322565 0.946547i \(-0.604545\pi\)
−0.322565 + 0.946547i \(0.604545\pi\)
\(30\) −2876.64 −0.583557
\(31\) −4255.06 −0.795246 −0.397623 0.917549i \(-0.630165\pi\)
−0.397623 + 0.917549i \(0.630165\pi\)
\(32\) 6400.29 1.10490
\(33\) −6162.46 −0.985074
\(34\) 4903.55 0.727467
\(35\) 0 0
\(36\) 211.973 0.0272599
\(37\) −154.621 −0.0185680 −0.00928399 0.999957i \(-0.502955\pi\)
−0.00928399 + 0.999957i \(0.502955\pi\)
\(38\) 19977.7 2.24433
\(39\) 14660.4 1.54342
\(40\) 2125.87 0.210081
\(41\) 7600.35 0.706112 0.353056 0.935602i \(-0.385142\pi\)
0.353056 + 0.935602i \(0.385142\pi\)
\(42\) 0 0
\(43\) 13533.2 1.11616 0.558082 0.829786i \(-0.311537\pi\)
0.558082 + 0.829786i \(0.311537\pi\)
\(44\) −7832.01 −0.609876
\(45\) 261.898 0.0192797
\(46\) −14929.7 −1.04029
\(47\) −15002.5 −0.990650 −0.495325 0.868708i \(-0.664951\pi\)
−0.495325 + 0.868708i \(0.664951\pi\)
\(48\) −20093.3 −1.25878
\(49\) 0 0
\(50\) −4517.08 −0.255525
\(51\) −10801.9 −0.581534
\(52\) 18632.2 0.955555
\(53\) 18078.9 0.884060 0.442030 0.897000i \(-0.354259\pi\)
0.442030 + 0.897000i \(0.354259\pi\)
\(54\) 26755.6 1.24862
\(55\) −9676.66 −0.431339
\(56\) 0 0
\(57\) −44008.4 −1.79411
\(58\) 21116.4 0.824234
\(59\) −6479.69 −0.242339 −0.121170 0.992632i \(-0.538665\pi\)
−0.121170 + 0.992632i \(0.538665\pi\)
\(60\) 8053.71 0.288814
\(61\) −4006.33 −0.137855 −0.0689274 0.997622i \(-0.521958\pi\)
−0.0689274 + 0.997622i \(0.521958\pi\)
\(62\) 30752.7 1.01602
\(63\) 0 0
\(64\) −5870.74 −0.179161
\(65\) 23020.6 0.675822
\(66\) 44538.1 1.25855
\(67\) −33620.3 −0.914986 −0.457493 0.889213i \(-0.651252\pi\)
−0.457493 + 0.889213i \(0.651252\pi\)
\(68\) −13728.4 −0.360038
\(69\) 32888.3 0.831608
\(70\) 0 0
\(71\) −61404.3 −1.44562 −0.722808 0.691049i \(-0.757149\pi\)
−0.722808 + 0.691049i \(0.757149\pi\)
\(72\) 890.817 0.0202515
\(73\) 42664.3 0.937039 0.468520 0.883453i \(-0.344788\pi\)
0.468520 + 0.883453i \(0.344788\pi\)
\(74\) 1117.50 0.0237229
\(75\) 9950.58 0.204266
\(76\) −55931.4 −1.11076
\(77\) 0 0
\(78\) −105955. −1.97190
\(79\) −107893. −1.94502 −0.972510 0.232862i \(-0.925191\pi\)
−0.972510 + 0.232862i \(0.925191\pi\)
\(80\) −31551.8 −0.551187
\(81\) −61484.9 −1.04125
\(82\) −54930.2 −0.902145
\(83\) −99024.8 −1.57779 −0.788894 0.614529i \(-0.789346\pi\)
−0.788894 + 0.614529i \(0.789346\pi\)
\(84\) 0 0
\(85\) −16961.8 −0.254639
\(86\) −97808.7 −1.42604
\(87\) −46516.9 −0.658890
\(88\) −32914.1 −0.453080
\(89\) −7661.84 −0.102532 −0.0512659 0.998685i \(-0.516326\pi\)
−0.0512659 + 0.998685i \(0.516326\pi\)
\(90\) −1892.82 −0.0246323
\(91\) 0 0
\(92\) 41798.5 0.514862
\(93\) −67744.5 −0.812207
\(94\) 108428. 1.26568
\(95\) −69104.7 −0.785595
\(96\) 101899. 1.12847
\(97\) −87087.4 −0.939779 −0.469890 0.882725i \(-0.655706\pi\)
−0.469890 + 0.882725i \(0.655706\pi\)
\(98\) 0 0
\(99\) −4054.88 −0.0415805
\(100\) 12646.4 0.126464
\(101\) −72757.1 −0.709695 −0.354848 0.934924i \(-0.615467\pi\)
−0.354848 + 0.934924i \(0.615467\pi\)
\(102\) 78069.0 0.742982
\(103\) −49822.1 −0.462732 −0.231366 0.972867i \(-0.574319\pi\)
−0.231366 + 0.972867i \(0.574319\pi\)
\(104\) 78301.9 0.709886
\(105\) 0 0
\(106\) −130662. −1.12950
\(107\) −79308.8 −0.669672 −0.334836 0.942276i \(-0.608681\pi\)
−0.334836 + 0.942276i \(0.608681\pi\)
\(108\) −74907.3 −0.617966
\(109\) −183107. −1.47618 −0.738089 0.674703i \(-0.764272\pi\)
−0.738089 + 0.674703i \(0.764272\pi\)
\(110\) 69936.4 0.551089
\(111\) −2461.71 −0.0189640
\(112\) 0 0
\(113\) 137042. 1.00962 0.504809 0.863231i \(-0.331563\pi\)
0.504809 + 0.863231i \(0.331563\pi\)
\(114\) 318064. 2.29220
\(115\) 51643.2 0.364140
\(116\) −59119.5 −0.407930
\(117\) 9646.47 0.0651484
\(118\) 46830.8 0.309618
\(119\) 0 0
\(120\) 33845.8 0.214561
\(121\) −11230.6 −0.0697329
\(122\) 28955.0 0.176126
\(123\) 121005. 0.721172
\(124\) −86098.1 −0.502851
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −43001.7 −0.236579 −0.118290 0.992979i \(-0.537741\pi\)
−0.118290 + 0.992979i \(0.537741\pi\)
\(128\) −162380. −0.876005
\(129\) 215461. 1.13997
\(130\) −166377. −0.863447
\(131\) 21205.7 0.107963 0.0539815 0.998542i \(-0.482809\pi\)
0.0539815 + 0.998542i \(0.482809\pi\)
\(132\) −124693. −0.622884
\(133\) 0 0
\(134\) 242985. 1.16901
\(135\) −92550.0 −0.437061
\(136\) −57693.7 −0.267474
\(137\) 285197. 1.29820 0.649102 0.760701i \(-0.275145\pi\)
0.649102 + 0.760701i \(0.275145\pi\)
\(138\) −237694. −1.06248
\(139\) 282350. 1.23951 0.619756 0.784794i \(-0.287231\pi\)
0.619756 + 0.784794i \(0.287231\pi\)
\(140\) 0 0
\(141\) −238855. −1.01178
\(142\) 443789. 1.84695
\(143\) −356420. −1.45754
\(144\) −13221.4 −0.0531337
\(145\) −73043.7 −0.288511
\(146\) −308349. −1.19718
\(147\) 0 0
\(148\) −3128.65 −0.0117409
\(149\) 69436.1 0.256224 0.128112 0.991760i \(-0.459108\pi\)
0.128112 + 0.991760i \(0.459108\pi\)
\(150\) −71916.1 −0.260974
\(151\) 258978. 0.924315 0.462157 0.886798i \(-0.347075\pi\)
0.462157 + 0.886798i \(0.347075\pi\)
\(152\) −235052. −0.825192
\(153\) −7107.63 −0.0245469
\(154\) 0 0
\(155\) −106376. −0.355645
\(156\) 296642. 0.975935
\(157\) 102995. 0.333477 0.166738 0.986001i \(-0.446676\pi\)
0.166738 + 0.986001i \(0.446676\pi\)
\(158\) 779775. 2.48500
\(159\) 287832. 0.902915
\(160\) 160007. 0.494128
\(161\) 0 0
\(162\) 444372. 1.33033
\(163\) −581424. −1.71405 −0.857026 0.515272i \(-0.827691\pi\)
−0.857026 + 0.515272i \(0.827691\pi\)
\(164\) 153788. 0.446490
\(165\) −154061. −0.440539
\(166\) 715685. 2.01582
\(167\) 475920. 1.32051 0.660257 0.751040i \(-0.270447\pi\)
0.660257 + 0.751040i \(0.270447\pi\)
\(168\) 0 0
\(169\) 476621. 1.28368
\(170\) 122589. 0.325333
\(171\) −28957.4 −0.0757304
\(172\) 273834. 0.705775
\(173\) −285658. −0.725657 −0.362828 0.931856i \(-0.618189\pi\)
−0.362828 + 0.931856i \(0.618189\pi\)
\(174\) 336193. 0.841813
\(175\) 0 0
\(176\) 488505. 1.18874
\(177\) −103163. −0.247508
\(178\) 55374.7 0.130997
\(179\) −450892. −1.05182 −0.525909 0.850541i \(-0.676275\pi\)
−0.525909 + 0.850541i \(0.676275\pi\)
\(180\) 5299.32 0.0121910
\(181\) −151606. −0.343969 −0.171985 0.985100i \(-0.555018\pi\)
−0.171985 + 0.985100i \(0.555018\pi\)
\(182\) 0 0
\(183\) −63784.4 −0.140795
\(184\) 175658. 0.382494
\(185\) −3865.53 −0.00830385
\(186\) 489612. 1.03769
\(187\) 262614. 0.549179
\(188\) −303566. −0.626409
\(189\) 0 0
\(190\) 499443. 1.00369
\(191\) 514986. 1.02144 0.510719 0.859748i \(-0.329379\pi\)
0.510719 + 0.859748i \(0.329379\pi\)
\(192\) −93467.7 −0.182982
\(193\) −769968. −1.48792 −0.743959 0.668225i \(-0.767054\pi\)
−0.743959 + 0.668225i \(0.767054\pi\)
\(194\) 629409. 1.20068
\(195\) 366509. 0.690236
\(196\) 0 0
\(197\) −246901. −0.453269 −0.226635 0.973980i \(-0.572772\pi\)
−0.226635 + 0.973980i \(0.572772\pi\)
\(198\) 29306.0 0.0531243
\(199\) −890639. −1.59430 −0.797148 0.603783i \(-0.793659\pi\)
−0.797148 + 0.603783i \(0.793659\pi\)
\(200\) 53146.7 0.0939509
\(201\) −535266. −0.934501
\(202\) 525839. 0.906723
\(203\) 0 0
\(204\) −218569. −0.367717
\(205\) 190009. 0.315783
\(206\) 360081. 0.591197
\(207\) 21640.4 0.0351026
\(208\) −1.16214e6 −1.86252
\(209\) 1.06992e6 1.69429
\(210\) 0 0
\(211\) −538746. −0.833063 −0.416531 0.909121i \(-0.636754\pi\)
−0.416531 + 0.909121i \(0.636754\pi\)
\(212\) 365813. 0.559010
\(213\) −977614. −1.47645
\(214\) 573191. 0.855588
\(215\) 338329. 0.499164
\(216\) −314798. −0.459090
\(217\) 0 0
\(218\) 1.32338e6 1.88600
\(219\) 679256. 0.957025
\(220\) −195800. −0.272745
\(221\) −624753. −0.860454
\(222\) 17791.6 0.0242288
\(223\) −1.34117e6 −1.80602 −0.903011 0.429618i \(-0.858648\pi\)
−0.903011 + 0.429618i \(0.858648\pi\)
\(224\) 0 0
\(225\) 6547.46 0.00862216
\(226\) −990447. −1.28991
\(227\) 581025. 0.748394 0.374197 0.927349i \(-0.377918\pi\)
0.374197 + 0.927349i \(0.377918\pi\)
\(228\) −890479. −1.13445
\(229\) 998618. 1.25838 0.629188 0.777253i \(-0.283387\pi\)
0.629188 + 0.777253i \(0.283387\pi\)
\(230\) −373242. −0.465234
\(231\) 0 0
\(232\) −248450. −0.303053
\(233\) 1.37979e6 1.66503 0.832517 0.553999i \(-0.186899\pi\)
0.832517 + 0.553999i \(0.186899\pi\)
\(234\) −69718.2 −0.0832351
\(235\) −375064. −0.443032
\(236\) −131112. −0.153236
\(237\) −1.71775e6 −1.98650
\(238\) 0 0
\(239\) 508394. 0.575713 0.287856 0.957674i \(-0.407057\pi\)
0.287856 + 0.957674i \(0.407057\pi\)
\(240\) −502333. −0.562942
\(241\) −185871. −0.206143 −0.103072 0.994674i \(-0.532867\pi\)
−0.103072 + 0.994674i \(0.532867\pi\)
\(242\) 81166.9 0.0890924
\(243\) −79311.0 −0.0861624
\(244\) −81065.1 −0.0871685
\(245\) 0 0
\(246\) −874540. −0.921386
\(247\) −2.54533e6 −2.65461
\(248\) −361828. −0.373570
\(249\) −1.57657e6 −1.61144
\(250\) −112927. −0.114274
\(251\) 421837. 0.422630 0.211315 0.977418i \(-0.432225\pi\)
0.211315 + 0.977418i \(0.432225\pi\)
\(252\) 0 0
\(253\) −799573. −0.785339
\(254\) 310788. 0.302259
\(255\) −270048. −0.260070
\(256\) 1.36143e6 1.29836
\(257\) 1.94672e6 1.83853 0.919265 0.393639i \(-0.128784\pi\)
0.919265 + 0.393639i \(0.128784\pi\)
\(258\) −1.55720e6 −1.45645
\(259\) 0 0
\(260\) 465805. 0.427337
\(261\) −30608.0 −0.0278121
\(262\) −153261. −0.137936
\(263\) 1.63031e6 1.45339 0.726693 0.686962i \(-0.241056\pi\)
0.726693 + 0.686962i \(0.241056\pi\)
\(264\) −524023. −0.462743
\(265\) 451972. 0.395363
\(266\) 0 0
\(267\) −121984. −0.104719
\(268\) −680282. −0.578565
\(269\) −1.37145e6 −1.15558 −0.577788 0.816187i \(-0.696084\pi\)
−0.577788 + 0.816187i \(0.696084\pi\)
\(270\) 668889. 0.558399
\(271\) 761737. 0.630060 0.315030 0.949082i \(-0.397985\pi\)
0.315030 + 0.949082i \(0.397985\pi\)
\(272\) 856281. 0.701769
\(273\) 0 0
\(274\) −2.06121e6 −1.65862
\(275\) −241917. −0.192901
\(276\) 665471. 0.525843
\(277\) −2.06875e6 −1.61997 −0.809987 0.586448i \(-0.800526\pi\)
−0.809987 + 0.586448i \(0.800526\pi\)
\(278\) −2.04064e6 −1.58363
\(279\) −44575.7 −0.0342837
\(280\) 0 0
\(281\) 431842. 0.326257 0.163128 0.986605i \(-0.447842\pi\)
0.163128 + 0.986605i \(0.447842\pi\)
\(282\) 1.72628e6 1.29267
\(283\) 1.43347e6 1.06395 0.531977 0.846759i \(-0.321449\pi\)
0.531977 + 0.846759i \(0.321449\pi\)
\(284\) −1.24247e6 −0.914094
\(285\) −1.10021e6 −0.802350
\(286\) 2.57596e6 1.86219
\(287\) 0 0
\(288\) 67049.0 0.0476333
\(289\) −959531. −0.675794
\(290\) 527911. 0.368609
\(291\) −1.38651e6 −0.959823
\(292\) 863282. 0.592510
\(293\) 138921. 0.0945363 0.0472682 0.998882i \(-0.484948\pi\)
0.0472682 + 0.998882i \(0.484948\pi\)
\(294\) 0 0
\(295\) −161992. −0.108377
\(296\) −13148.2 −0.00872240
\(297\) 1.43292e6 0.942607
\(298\) −501838. −0.327358
\(299\) 1.90217e6 1.23047
\(300\) 201343. 0.129162
\(301\) 0 0
\(302\) −1.87172e6 −1.18093
\(303\) −1.15836e6 −0.724832
\(304\) 3.48860e6 2.16505
\(305\) −100158. −0.0616505
\(306\) 51369.2 0.0313617
\(307\) 631297. 0.382285 0.191143 0.981562i \(-0.438781\pi\)
0.191143 + 0.981562i \(0.438781\pi\)
\(308\) 0 0
\(309\) −793215. −0.472601
\(310\) 768818. 0.454380
\(311\) 2.20444e6 1.29240 0.646199 0.763169i \(-0.276358\pi\)
0.646199 + 0.763169i \(0.276358\pi\)
\(312\) 1.24664e6 0.725027
\(313\) −441934. −0.254974 −0.127487 0.991840i \(-0.540691\pi\)
−0.127487 + 0.991840i \(0.540691\pi\)
\(314\) −744376. −0.426058
\(315\) 0 0
\(316\) −2.18313e6 −1.22988
\(317\) 1.33132e6 0.744103 0.372052 0.928212i \(-0.378654\pi\)
0.372052 + 0.928212i \(0.378654\pi\)
\(318\) −2.08026e6 −1.15359
\(319\) 1.13091e6 0.622231
\(320\) −146769. −0.0801232
\(321\) −1.26267e6 −0.683954
\(322\) 0 0
\(323\) 1.87543e6 1.00022
\(324\) −1.24410e6 −0.658406
\(325\) 575514. 0.302237
\(326\) 4.20215e6 2.18991
\(327\) −2.91523e6 −1.50766
\(328\) 646293. 0.331700
\(329\) 0 0
\(330\) 1.11345e6 0.562843
\(331\) 1.50280e6 0.753931 0.376966 0.926227i \(-0.376967\pi\)
0.376966 + 0.926227i \(0.376967\pi\)
\(332\) −2.00370e6 −0.997670
\(333\) −1619.80 −0.000800481 0
\(334\) −3.43963e6 −1.68712
\(335\) −840507. −0.409194
\(336\) 0 0
\(337\) −2.34434e6 −1.12447 −0.562233 0.826979i \(-0.690058\pi\)
−0.562233 + 0.826979i \(0.690058\pi\)
\(338\) −3.44470e6 −1.64006
\(339\) 2.18183e6 1.03115
\(340\) −343210. −0.161014
\(341\) 1.64699e6 0.767017
\(342\) 209285. 0.0967549
\(343\) 0 0
\(344\) 1.15079e6 0.524324
\(345\) 822207. 0.371906
\(346\) 2.06454e6 0.927116
\(347\) 279935. 0.124805 0.0624026 0.998051i \(-0.480124\pi\)
0.0624026 + 0.998051i \(0.480124\pi\)
\(348\) −941237. −0.416630
\(349\) −1.16659e6 −0.512691 −0.256346 0.966585i \(-0.582519\pi\)
−0.256346 + 0.966585i \(0.582519\pi\)
\(350\) 0 0
\(351\) −3.40888e6 −1.47688
\(352\) −2.47734e6 −1.06568
\(353\) −1.55443e6 −0.663949 −0.331975 0.943288i \(-0.607715\pi\)
−0.331975 + 0.943288i \(0.607715\pi\)
\(354\) 745590. 0.316222
\(355\) −1.53511e6 −0.646499
\(356\) −155032. −0.0648330
\(357\) 0 0
\(358\) 3.25875e6 1.34383
\(359\) 856494. 0.350742 0.175371 0.984502i \(-0.443887\pi\)
0.175371 + 0.984502i \(0.443887\pi\)
\(360\) 22270.4 0.00905675
\(361\) 5.16464e6 2.08580
\(362\) 1.09571e6 0.439463
\(363\) −178801. −0.0712202
\(364\) 0 0
\(365\) 1.06661e6 0.419057
\(366\) 460991. 0.179883
\(367\) 909272. 0.352394 0.176197 0.984355i \(-0.443620\pi\)
0.176197 + 0.984355i \(0.443620\pi\)
\(368\) −2.60709e6 −1.00355
\(369\) 79620.7 0.0304411
\(370\) 27937.5 0.0106092
\(371\) 0 0
\(372\) −1.37076e6 −0.513576
\(373\) −133835. −0.0498080 −0.0249040 0.999690i \(-0.507928\pi\)
−0.0249040 + 0.999690i \(0.507928\pi\)
\(374\) −1.89800e6 −0.701644
\(375\) 248764. 0.0913504
\(376\) −1.27574e6 −0.465363
\(377\) −2.69041e6 −0.974912
\(378\) 0 0
\(379\) 1.07715e6 0.385192 0.192596 0.981278i \(-0.438309\pi\)
0.192596 + 0.981278i \(0.438309\pi\)
\(380\) −1.39828e6 −0.496749
\(381\) −684627. −0.241625
\(382\) −3.72198e6 −1.30501
\(383\) −461084. −0.160614 −0.0803069 0.996770i \(-0.525590\pi\)
−0.0803069 + 0.996770i \(0.525590\pi\)
\(384\) −2.58523e6 −0.894688
\(385\) 0 0
\(386\) 5.56481e6 1.90100
\(387\) 141773. 0.0481188
\(388\) −1.76215e6 −0.594243
\(389\) 235175. 0.0787983 0.0393992 0.999224i \(-0.487456\pi\)
0.0393992 + 0.999224i \(0.487456\pi\)
\(390\) −2.64888e6 −0.881862
\(391\) −1.40154e6 −0.463621
\(392\) 0 0
\(393\) 337615. 0.110266
\(394\) 1.78443e6 0.579108
\(395\) −2.69732e6 −0.869839
\(396\) −82047.6 −0.0262923
\(397\) 3.60978e6 1.14949 0.574744 0.818334i \(-0.305102\pi\)
0.574744 + 0.818334i \(0.305102\pi\)
\(398\) 6.43694e6 2.03691
\(399\) 0 0
\(400\) −788794. −0.246498
\(401\) 4.78064e6 1.48465 0.742327 0.670038i \(-0.233722\pi\)
0.742327 + 0.670038i \(0.233722\pi\)
\(402\) 3.86854e6 1.19394
\(403\) −3.91815e6 −1.20176
\(404\) −1.47219e6 −0.448756
\(405\) −1.53712e6 −0.465662
\(406\) 0 0
\(407\) 59848.7 0.0179089
\(408\) −918538. −0.273179
\(409\) −332478. −0.0982778 −0.0491389 0.998792i \(-0.515648\pi\)
−0.0491389 + 0.998792i \(0.515648\pi\)
\(410\) −1.37325e6 −0.403452
\(411\) 4.54060e6 1.32589
\(412\) −1.00812e6 −0.292595
\(413\) 0 0
\(414\) −156402. −0.0448479
\(415\) −2.47562e6 −0.705609
\(416\) 5.89353e6 1.66972
\(417\) 4.49528e6 1.26595
\(418\) −7.73270e6 −2.16466
\(419\) 3.01019e6 0.837643 0.418822 0.908068i \(-0.362443\pi\)
0.418822 + 0.908068i \(0.362443\pi\)
\(420\) 0 0
\(421\) −2.57143e6 −0.707083 −0.353541 0.935419i \(-0.615023\pi\)
−0.353541 + 0.935419i \(0.615023\pi\)
\(422\) 3.89369e6 1.06434
\(423\) −157166. −0.0427078
\(424\) 1.53733e6 0.415291
\(425\) −424046. −0.113878
\(426\) 7.06554e6 1.88634
\(427\) 0 0
\(428\) −1.60476e6 −0.423448
\(429\) −5.67453e6 −1.48863
\(430\) −2.44522e6 −0.637744
\(431\) −5.76915e6 −1.49596 −0.747978 0.663723i \(-0.768975\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(432\) 4.67218e6 1.20451
\(433\) 1.93748e6 0.496611 0.248306 0.968682i \(-0.420126\pi\)
0.248306 + 0.968682i \(0.420126\pi\)
\(434\) 0 0
\(435\) −1.16292e6 −0.294665
\(436\) −3.70504e6 −0.933419
\(437\) −5.71006e6 −1.43033
\(438\) −4.90921e6 −1.22272
\(439\) −4.48026e6 −1.10954 −0.554769 0.832004i \(-0.687193\pi\)
−0.554769 + 0.832004i \(0.687193\pi\)
\(440\) −822852. −0.202624
\(441\) 0 0
\(442\) 4.51530e6 1.09934
\(443\) 2.62761e6 0.636139 0.318069 0.948067i \(-0.396965\pi\)
0.318069 + 0.948067i \(0.396965\pi\)
\(444\) −49811.0 −0.0119913
\(445\) −191546. −0.0458536
\(446\) 9.69310e6 2.30742
\(447\) 1.10549e6 0.261689
\(448\) 0 0
\(449\) −6.11083e6 −1.43049 −0.715244 0.698875i \(-0.753684\pi\)
−0.715244 + 0.698875i \(0.753684\pi\)
\(450\) −47320.6 −0.0110159
\(451\) −2.94184e6 −0.681048
\(452\) 2.77294e6 0.638403
\(453\) 4.12316e6 0.944029
\(454\) −4.19926e6 −0.956166
\(455\) 0 0
\(456\) −3.74224e6 −0.842791
\(457\) 3.93604e6 0.881595 0.440797 0.897607i \(-0.354696\pi\)
0.440797 + 0.897607i \(0.354696\pi\)
\(458\) −7.21734e6 −1.60773
\(459\) 2.51171e6 0.556464
\(460\) 1.04496e6 0.230253
\(461\) −3.61769e6 −0.792827 −0.396414 0.918072i \(-0.629745\pi\)
−0.396414 + 0.918072i \(0.629745\pi\)
\(462\) 0 0
\(463\) 6.68308e6 1.44885 0.724426 0.689352i \(-0.242105\pi\)
0.724426 + 0.689352i \(0.242105\pi\)
\(464\) 3.68745e6 0.795118
\(465\) −1.69361e6 −0.363230
\(466\) −9.97219e6 −2.12729
\(467\) 2.36830e6 0.502510 0.251255 0.967921i \(-0.419157\pi\)
0.251255 + 0.967921i \(0.419157\pi\)
\(468\) 195189. 0.0411947
\(469\) 0 0
\(470\) 2.71071e6 0.566028
\(471\) 1.63977e6 0.340589
\(472\) −550998. −0.113840
\(473\) −5.23824e6 −1.07654
\(474\) 1.24147e7 2.53800
\(475\) −1.72762e6 −0.351329
\(476\) 0 0
\(477\) 189393. 0.0381125
\(478\) −3.67433e6 −0.735544
\(479\) 2.26976e6 0.452004 0.226002 0.974127i \(-0.427434\pi\)
0.226002 + 0.974127i \(0.427434\pi\)
\(480\) 2.54746e6 0.504667
\(481\) −142379. −0.0280597
\(482\) 1.34335e6 0.263373
\(483\) 0 0
\(484\) −227242. −0.0440936
\(485\) −2.17718e6 −0.420282
\(486\) 573207. 0.110083
\(487\) −1.69003e6 −0.322904 −0.161452 0.986881i \(-0.551618\pi\)
−0.161452 + 0.986881i \(0.551618\pi\)
\(488\) −340677. −0.0647579
\(489\) −9.25681e6 −1.75061
\(490\) 0 0
\(491\) 3.12583e6 0.585143 0.292572 0.956244i \(-0.405489\pi\)
0.292572 + 0.956244i \(0.405489\pi\)
\(492\) 2.44844e6 0.456013
\(493\) 1.98233e6 0.367331
\(494\) 1.83959e7 3.39160
\(495\) −101372. −0.0185954
\(496\) 5.37018e6 0.980133
\(497\) 0 0
\(498\) 1.13944e7 2.05881
\(499\) −9.40680e6 −1.69118 −0.845591 0.533831i \(-0.820752\pi\)
−0.845591 + 0.533831i \(0.820752\pi\)
\(500\) 316161. 0.0565565
\(501\) 7.57709e6 1.34868
\(502\) −3.04875e6 −0.539961
\(503\) −7.23245e6 −1.27458 −0.637288 0.770626i \(-0.719944\pi\)
−0.637288 + 0.770626i \(0.719944\pi\)
\(504\) 0 0
\(505\) −1.81893e6 −0.317385
\(506\) 5.77878e6 1.00337
\(507\) 7.58825e6 1.31106
\(508\) −870109. −0.149594
\(509\) −4.10701e6 −0.702638 −0.351319 0.936256i \(-0.614267\pi\)
−0.351319 + 0.936256i \(0.614267\pi\)
\(510\) 1.95173e6 0.332272
\(511\) 0 0
\(512\) −4.64339e6 −0.782816
\(513\) 1.02330e7 1.71676
\(514\) −1.40696e7 −2.34895
\(515\) −1.24555e6 −0.206940
\(516\) 4.35969e6 0.720828
\(517\) 5.80698e6 0.955486
\(518\) 0 0
\(519\) −4.54794e6 −0.741134
\(520\) 1.95755e6 0.317471
\(521\) −6.80125e6 −1.09773 −0.548863 0.835912i \(-0.684939\pi\)
−0.548863 + 0.835912i \(0.684939\pi\)
\(522\) 221214. 0.0355334
\(523\) 3.31515e6 0.529968 0.264984 0.964253i \(-0.414633\pi\)
0.264984 + 0.964253i \(0.414633\pi\)
\(524\) 429083. 0.0682674
\(525\) 0 0
\(526\) −1.17828e7 −1.85688
\(527\) 2.88694e6 0.452805
\(528\) 7.77746e6 1.21410
\(529\) −2.16912e6 −0.337011
\(530\) −3.26655e6 −0.505126
\(531\) −67880.7 −0.0104475
\(532\) 0 0
\(533\) 6.99857e6 1.06707
\(534\) 881616. 0.133791
\(535\) −1.98272e6 −0.299486
\(536\) −2.85889e6 −0.429819
\(537\) −7.17863e6 −1.07425
\(538\) 9.91190e6 1.47639
\(539\) 0 0
\(540\) −1.87268e6 −0.276363
\(541\) 3.47955e6 0.511129 0.255564 0.966792i \(-0.417739\pi\)
0.255564 + 0.966792i \(0.417739\pi\)
\(542\) −5.50532e6 −0.804979
\(543\) −2.41371e6 −0.351305
\(544\) −4.34242e6 −0.629122
\(545\) −4.57768e6 −0.660167
\(546\) 0 0
\(547\) 2.55022e6 0.364426 0.182213 0.983259i \(-0.441674\pi\)
0.182213 + 0.983259i \(0.441674\pi\)
\(548\) 5.77075e6 0.820882
\(549\) −41970.0 −0.00594303
\(550\) 1.74841e6 0.246454
\(551\) 8.07626e6 1.13326
\(552\) 2.79664e6 0.390652
\(553\) 0 0
\(554\) 1.49515e7 2.06972
\(555\) −61542.8 −0.00848096
\(556\) 5.71315e6 0.783771
\(557\) 3.34227e6 0.456461 0.228230 0.973607i \(-0.426706\pi\)
0.228230 + 0.973607i \(0.426706\pi\)
\(558\) 322163. 0.0438016
\(559\) 1.24616e7 1.68673
\(560\) 0 0
\(561\) 4.18106e6 0.560892
\(562\) −3.12107e6 −0.416833
\(563\) 6.30412e6 0.838212 0.419106 0.907937i \(-0.362344\pi\)
0.419106 + 0.907937i \(0.362344\pi\)
\(564\) −4.83305e6 −0.639770
\(565\) 3.42605e6 0.451515
\(566\) −1.03602e7 −1.35933
\(567\) 0 0
\(568\) −5.22150e6 −0.679085
\(569\) 3.52056e6 0.455860 0.227930 0.973678i \(-0.426804\pi\)
0.227930 + 0.973678i \(0.426804\pi\)
\(570\) 7.95159e6 1.02510
\(571\) −7.51318e6 −0.964348 −0.482174 0.876076i \(-0.660153\pi\)
−0.482174 + 0.876076i \(0.660153\pi\)
\(572\) −7.21190e6 −0.921636
\(573\) 8.19906e6 1.04322
\(574\) 0 0
\(575\) 1.29108e6 0.162848
\(576\) −61501.5 −0.00772377
\(577\) 6.67495e6 0.834658 0.417329 0.908756i \(-0.362966\pi\)
0.417329 + 0.908756i \(0.362966\pi\)
\(578\) 6.93485e6 0.863411
\(579\) −1.22586e7 −1.51965
\(580\) −1.47799e6 −0.182432
\(581\) 0 0
\(582\) 1.00208e7 1.22629
\(583\) −6.99772e6 −0.852679
\(584\) 3.62795e6 0.440179
\(585\) 241162. 0.0291353
\(586\) −1.00403e6 −0.120782
\(587\) 1.44246e7 1.72785 0.863927 0.503617i \(-0.167998\pi\)
0.863927 + 0.503617i \(0.167998\pi\)
\(588\) 0 0
\(589\) 1.17618e7 1.39696
\(590\) 1.17077e6 0.138466
\(591\) −3.93089e6 −0.462937
\(592\) 195143. 0.0228849
\(593\) −8.63292e6 −1.00814 −0.504070 0.863663i \(-0.668165\pi\)
−0.504070 + 0.863663i \(0.668165\pi\)
\(594\) −1.03562e7 −1.20430
\(595\) 0 0
\(596\) 1.40499e6 0.162016
\(597\) −1.41798e7 −1.62830
\(598\) −1.37476e7 −1.57208
\(599\) 5.75996e6 0.655922 0.327961 0.944691i \(-0.393638\pi\)
0.327961 + 0.944691i \(0.393638\pi\)
\(600\) 846144. 0.0959547
\(601\) 6.23001e6 0.703562 0.351781 0.936082i \(-0.385576\pi\)
0.351781 + 0.936082i \(0.385576\pi\)
\(602\) 0 0
\(603\) −352204. −0.0394458
\(604\) 5.24023e6 0.584464
\(605\) −280764. −0.0311855
\(606\) 8.37185e6 0.926062
\(607\) −2.79863e6 −0.308301 −0.154150 0.988047i \(-0.549264\pi\)
−0.154150 + 0.988047i \(0.549264\pi\)
\(608\) −1.76916e7 −1.94092
\(609\) 0 0
\(610\) 723876. 0.0787661
\(611\) −1.38147e7 −1.49706
\(612\) −143818. −0.0155215
\(613\) 1.24192e7 1.33488 0.667438 0.744665i \(-0.267391\pi\)
0.667438 + 0.744665i \(0.267391\pi\)
\(614\) −4.56259e6 −0.488417
\(615\) 3.02511e6 0.322518
\(616\) 0 0
\(617\) 5.63980e6 0.596418 0.298209 0.954501i \(-0.403611\pi\)
0.298209 + 0.954501i \(0.403611\pi\)
\(618\) 5.73282e6 0.603806
\(619\) −5.79714e6 −0.608117 −0.304059 0.952653i \(-0.598342\pi\)
−0.304059 + 0.952653i \(0.598342\pi\)
\(620\) −2.15245e6 −0.224882
\(621\) −7.64732e6 −0.795756
\(622\) −1.59322e7 −1.65120
\(623\) 0 0
\(624\) −1.85024e7 −1.90225
\(625\) 390625. 0.0400000
\(626\) 3.19400e6 0.325761
\(627\) 1.70342e7 1.73042
\(628\) 2.08402e6 0.210864
\(629\) 104906. 0.0105724
\(630\) 0 0
\(631\) −3.41197e6 −0.341140 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(632\) −9.17461e6 −0.913682
\(633\) −8.57733e6 −0.850830
\(634\) −9.62186e6 −0.950684
\(635\) −1.07504e6 −0.105801
\(636\) 5.82408e6 0.570932
\(637\) 0 0
\(638\) −8.17346e6 −0.794977
\(639\) −643267. −0.0623217
\(640\) −4.05949e6 −0.391761
\(641\) −1.18486e7 −1.13900 −0.569499 0.821992i \(-0.692863\pi\)
−0.569499 + 0.821992i \(0.692863\pi\)
\(642\) 9.12573e6 0.873836
\(643\) 6.05156e6 0.577218 0.288609 0.957447i \(-0.406807\pi\)
0.288609 + 0.957447i \(0.406807\pi\)
\(644\) 0 0
\(645\) 5.38651e6 0.509810
\(646\) −1.35543e7 −1.27790
\(647\) 9.82251e6 0.922490 0.461245 0.887273i \(-0.347403\pi\)
0.461245 + 0.887273i \(0.347403\pi\)
\(648\) −5.22835e6 −0.489133
\(649\) 2.50807e6 0.233737
\(650\) −4.15943e6 −0.386145
\(651\) 0 0
\(652\) −1.17647e7 −1.08383
\(653\) −957185. −0.0878442 −0.0439221 0.999035i \(-0.513985\pi\)
−0.0439221 + 0.999035i \(0.513985\pi\)
\(654\) 2.10694e7 1.92622
\(655\) 530144. 0.0482826
\(656\) −9.59217e6 −0.870277
\(657\) 446949. 0.0403965
\(658\) 0 0
\(659\) −1.54679e7 −1.38745 −0.693725 0.720240i \(-0.744032\pi\)
−0.693725 + 0.720240i \(0.744032\pi\)
\(660\) −3.11732e6 −0.278562
\(661\) 3.95036e6 0.351668 0.175834 0.984420i \(-0.443738\pi\)
0.175834 + 0.984420i \(0.443738\pi\)
\(662\) −1.08612e7 −0.963240
\(663\) −9.94665e6 −0.878806
\(664\) −8.42055e6 −0.741174
\(665\) 0 0
\(666\) 11706.8 0.00102271
\(667\) −6.03553e6 −0.525292
\(668\) 9.62990e6 0.834989
\(669\) −2.13527e7 −1.84454
\(670\) 6.07462e6 0.522796
\(671\) 1.55071e6 0.132961
\(672\) 0 0
\(673\) −1.64772e7 −1.40231 −0.701157 0.713007i \(-0.747333\pi\)
−0.701157 + 0.713007i \(0.747333\pi\)
\(674\) 1.69433e7 1.43664
\(675\) −2.31375e6 −0.195460
\(676\) 9.64409e6 0.811698
\(677\) 9.00324e6 0.754966 0.377483 0.926017i \(-0.376790\pi\)
0.377483 + 0.926017i \(0.376790\pi\)
\(678\) −1.57688e7 −1.31742
\(679\) 0 0
\(680\) −1.44234e6 −0.119618
\(681\) 9.25046e6 0.764356
\(682\) −1.19033e7 −0.979959
\(683\) 9.38382e6 0.769711 0.384856 0.922977i \(-0.374251\pi\)
0.384856 + 0.922977i \(0.374251\pi\)
\(684\) −585933. −0.0478859
\(685\) 7.12992e6 0.580575
\(686\) 0 0
\(687\) 1.58989e7 1.28522
\(688\) −1.70798e7 −1.37566
\(689\) 1.66474e7 1.33598
\(690\) −5.94236e6 −0.475156
\(691\) 5.05803e6 0.402983 0.201491 0.979490i \(-0.435421\pi\)
0.201491 + 0.979490i \(0.435421\pi\)
\(692\) −5.78009e6 −0.458848
\(693\) 0 0
\(694\) −2.02318e6 −0.159454
\(695\) 7.05875e6 0.554327
\(696\) −3.95555e6 −0.309517
\(697\) −5.15663e6 −0.402054
\(698\) 8.43135e6 0.655026
\(699\) 2.19675e7 1.70055
\(700\) 0 0
\(701\) −2.25374e7 −1.73224 −0.866121 0.499835i \(-0.833394\pi\)
−0.866121 + 0.499835i \(0.833394\pi\)
\(702\) 2.46371e7 1.88689
\(703\) 427402. 0.0326173
\(704\) 2.27237e6 0.172801
\(705\) −5.97136e6 −0.452481
\(706\) 1.12344e7 0.848277
\(707\) 0 0
\(708\) −2.08742e6 −0.156505
\(709\) −2.37037e6 −0.177093 −0.0885463 0.996072i \(-0.528222\pi\)
−0.0885463 + 0.996072i \(0.528222\pi\)
\(710\) 1.10947e7 0.825982
\(711\) −1.13028e6 −0.0838514
\(712\) −651523. −0.0481648
\(713\) −8.78979e6 −0.647522
\(714\) 0 0
\(715\) −8.91049e6 −0.651833
\(716\) −9.12349e6 −0.665087
\(717\) 8.09411e6 0.587992
\(718\) −6.19016e6 −0.448117
\(719\) 1.25461e7 0.905077 0.452539 0.891745i \(-0.350518\pi\)
0.452539 + 0.891745i \(0.350518\pi\)
\(720\) −330534. −0.0237621
\(721\) 0 0
\(722\) −3.73266e7 −2.66486
\(723\) −2.95924e6 −0.210540
\(724\) −3.06764e6 −0.217499
\(725\) −1.82609e6 −0.129026
\(726\) 1.29225e6 0.0909926
\(727\) −5.18009e6 −0.363498 −0.181749 0.983345i \(-0.558176\pi\)
−0.181749 + 0.983345i \(0.558176\pi\)
\(728\) 0 0
\(729\) 1.36781e7 0.953252
\(730\) −7.70873e6 −0.535397
\(731\) −9.18189e6 −0.635534
\(732\) −1.29063e6 −0.0890276
\(733\) 1.54379e7 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(734\) −6.57161e6 −0.450227
\(735\) 0 0
\(736\) 1.32213e7 0.899659
\(737\) 1.30133e7 0.882507
\(738\) −575445. −0.0388922
\(739\) −1.64732e7 −1.10960 −0.554801 0.831983i \(-0.687206\pi\)
−0.554801 + 0.831983i \(0.687206\pi\)
\(740\) −78216.2 −0.00525070
\(741\) −4.05240e7 −2.71123
\(742\) 0 0
\(743\) −2.59403e7 −1.72386 −0.861931 0.507026i \(-0.830745\pi\)
−0.861931 + 0.507026i \(0.830745\pi\)
\(744\) −5.76063e6 −0.381538
\(745\) 1.73590e6 0.114587
\(746\) 967273. 0.0636358
\(747\) −1.03738e6 −0.0680198
\(748\) 5.31381e6 0.347258
\(749\) 0 0
\(750\) −1.79790e6 −0.116711
\(751\) 1.18608e7 0.767386 0.383693 0.923461i \(-0.374652\pi\)
0.383693 + 0.923461i \(0.374652\pi\)
\(752\) 1.89343e7 1.22097
\(753\) 6.71603e6 0.431643
\(754\) 1.94445e7 1.24557
\(755\) 6.47444e6 0.413366
\(756\) 0 0
\(757\) −6.25104e6 −0.396472 −0.198236 0.980154i \(-0.563521\pi\)
−0.198236 + 0.980154i \(0.563521\pi\)
\(758\) −7.78491e6 −0.492131
\(759\) −1.27300e7 −0.802088
\(760\) −5.87630e6 −0.369037
\(761\) −2.20531e7 −1.38041 −0.690206 0.723613i \(-0.742480\pi\)
−0.690206 + 0.723613i \(0.742480\pi\)
\(762\) 4.94803e6 0.308706
\(763\) 0 0
\(764\) 1.04204e7 0.645877
\(765\) −177691. −0.0109777
\(766\) 3.33240e6 0.205204
\(767\) −5.96664e6 −0.366220
\(768\) 2.16753e7 1.32606
\(769\) 1.37289e7 0.837181 0.418591 0.908175i \(-0.362524\pi\)
0.418591 + 0.908175i \(0.362524\pi\)
\(770\) 0 0
\(771\) 3.09936e7 1.87774
\(772\) −1.55797e7 −0.940843
\(773\) 1.50716e7 0.907213 0.453606 0.891202i \(-0.350137\pi\)
0.453606 + 0.891202i \(0.350137\pi\)
\(774\) −1.02464e6 −0.0614777
\(775\) −2.65941e6 −0.159049
\(776\) −7.40545e6 −0.441466
\(777\) 0 0
\(778\) −1.69969e6 −0.100675
\(779\) −2.10088e7 −1.24039
\(780\) 7.41604e6 0.436451
\(781\) 2.37675e7 1.39430
\(782\) 1.01294e7 0.592333
\(783\) 1.08163e7 0.630485
\(784\) 0 0
\(785\) 2.57487e6 0.149135
\(786\) −2.44006e6 −0.140878
\(787\) −1.93056e7 −1.11108 −0.555541 0.831489i \(-0.687489\pi\)
−0.555541 + 0.831489i \(0.687489\pi\)
\(788\) −4.99586e6 −0.286612
\(789\) 2.59561e7 1.48438
\(790\) 1.94944e7 1.11133
\(791\) 0 0
\(792\) −344806. −0.0195327
\(793\) −3.68911e6 −0.208324
\(794\) −2.60891e7 −1.46861
\(795\) 7.19581e6 0.403796
\(796\) −1.80215e7 −1.00811
\(797\) −2.05347e7 −1.14510 −0.572549 0.819871i \(-0.694045\pi\)
−0.572549 + 0.819871i \(0.694045\pi\)
\(798\) 0 0
\(799\) 1.01788e7 0.564067
\(800\) 4.00018e6 0.220981
\(801\) −80264.9 −0.00442023
\(802\) −3.45513e7 −1.89683
\(803\) −1.65139e7 −0.903778
\(804\) −1.08307e7 −0.590905
\(805\) 0 0
\(806\) 2.83178e7 1.53540
\(807\) −2.18347e7 −1.18022
\(808\) −6.18688e6 −0.333383
\(809\) 7.34708e6 0.394678 0.197339 0.980335i \(-0.436770\pi\)
0.197339 + 0.980335i \(0.436770\pi\)
\(810\) 1.11093e7 0.594941
\(811\) −2.36201e7 −1.26104 −0.630522 0.776171i \(-0.717159\pi\)
−0.630522 + 0.776171i \(0.717159\pi\)
\(812\) 0 0
\(813\) 1.21276e7 0.643498
\(814\) −432546. −0.0228808
\(815\) −1.45356e7 −0.766548
\(816\) 1.36328e7 0.716736
\(817\) −3.74082e7 −1.96070
\(818\) 2.40293e6 0.125562
\(819\) 0 0
\(820\) 3.84469e6 0.199676
\(821\) 1.84495e7 0.955272 0.477636 0.878558i \(-0.341494\pi\)
0.477636 + 0.878558i \(0.341494\pi\)
\(822\) −3.28164e7 −1.69399
\(823\) −1.01309e7 −0.521374 −0.260687 0.965423i \(-0.583949\pi\)
−0.260687 + 0.965423i \(0.583949\pi\)
\(824\) −4.23661e6 −0.217370
\(825\) −3.85154e6 −0.197015
\(826\) 0 0
\(827\) −9.10559e6 −0.462961 −0.231480 0.972840i \(-0.574357\pi\)
−0.231480 + 0.972840i \(0.574357\pi\)
\(828\) 437878. 0.0221961
\(829\) 2.40432e7 1.21508 0.607540 0.794289i \(-0.292156\pi\)
0.607540 + 0.794289i \(0.292156\pi\)
\(830\) 1.78921e7 0.901502
\(831\) −3.29364e7 −1.65452
\(832\) −5.40591e6 −0.270745
\(833\) 0 0
\(834\) −3.24888e7 −1.61741
\(835\) 1.18980e7 0.590552
\(836\) 2.16492e7 1.07134
\(837\) 1.57522e7 0.777192
\(838\) −2.17557e7 −1.07019
\(839\) 3.70147e7 1.81539 0.907694 0.419633i \(-0.137841\pi\)
0.907694 + 0.419633i \(0.137841\pi\)
\(840\) 0 0
\(841\) −1.19745e7 −0.583806
\(842\) 1.85846e7 0.903385
\(843\) 6.87533e6 0.333215
\(844\) −1.09011e7 −0.526763
\(845\) 1.19155e7 0.574079
\(846\) 1.13589e6 0.0545644
\(847\) 0 0
\(848\) −2.28168e7 −1.08960
\(849\) 2.28222e7 1.08665
\(850\) 3.06472e6 0.145493
\(851\) −319405. −0.0151188
\(852\) −1.97813e7 −0.933590
\(853\) 79427.6 0.00373765 0.00186883 0.999998i \(-0.499405\pi\)
0.00186883 + 0.999998i \(0.499405\pi\)
\(854\) 0 0
\(855\) −723936. −0.0338676
\(856\) −6.74400e6 −0.314581
\(857\) −1.28759e7 −0.598859 −0.299430 0.954118i \(-0.596796\pi\)
−0.299430 + 0.954118i \(0.596796\pi\)
\(858\) 4.10117e7 1.90191
\(859\) −3.82897e6 −0.177051 −0.0885256 0.996074i \(-0.528216\pi\)
−0.0885256 + 0.996074i \(0.528216\pi\)
\(860\) 6.84585e6 0.315632
\(861\) 0 0
\(862\) 4.16956e7 1.91127
\(863\) −1.27055e7 −0.580718 −0.290359 0.956918i \(-0.593775\pi\)
−0.290359 + 0.956918i \(0.593775\pi\)
\(864\) −2.36939e7 −1.07982
\(865\) −7.14145e6 −0.324524
\(866\) −1.40028e7 −0.634482
\(867\) −1.52766e7 −0.690208
\(868\) 0 0
\(869\) 4.17616e7 1.87598
\(870\) 8.40483e6 0.376470
\(871\) −3.09583e7 −1.38271
\(872\) −1.55705e7 −0.693442
\(873\) −912321. −0.0405147
\(874\) 4.12685e7 1.82743
\(875\) 0 0
\(876\) 1.37443e7 0.605147
\(877\) −1.88584e7 −0.827953 −0.413977 0.910288i \(-0.635860\pi\)
−0.413977 + 0.910288i \(0.635860\pi\)
\(878\) 3.23803e7 1.41757
\(879\) 2.21175e6 0.0965526
\(880\) 1.22126e7 0.531622
\(881\) 3.82921e7 1.66215 0.831073 0.556163i \(-0.187727\pi\)
0.831073 + 0.556163i \(0.187727\pi\)
\(882\) 0 0
\(883\) 3.02825e7 1.30704 0.653521 0.756909i \(-0.273291\pi\)
0.653521 + 0.756909i \(0.273291\pi\)
\(884\) −1.26414e7 −0.544084
\(885\) −2.57907e6 −0.110689
\(886\) −1.89906e7 −0.812746
\(887\) 2.80881e7 1.19871 0.599354 0.800484i \(-0.295424\pi\)
0.599354 + 0.800484i \(0.295424\pi\)
\(888\) −209331. −0.00890843
\(889\) 0 0
\(890\) 1.38437e6 0.0585836
\(891\) 2.37987e7 1.00429
\(892\) −2.71377e7 −1.14199
\(893\) 4.14699e7 1.74022
\(894\) −7.98972e6 −0.334340
\(895\) −1.12723e7 −0.470387
\(896\) 0 0
\(897\) 3.02843e7 1.25671
\(898\) 4.41650e7 1.82762
\(899\) 1.24322e7 0.513037
\(900\) 132483. 0.00545198
\(901\) −1.22660e7 −0.503375
\(902\) 2.12616e7 0.870122
\(903\) 0 0
\(904\) 1.16533e7 0.474273
\(905\) −3.79015e6 −0.153828
\(906\) −2.97995e7 −1.20611
\(907\) 1.50416e7 0.607121 0.303561 0.952812i \(-0.401824\pi\)
0.303561 + 0.952812i \(0.401824\pi\)
\(908\) 1.17566e7 0.473226
\(909\) −762198. −0.0305955
\(910\) 0 0
\(911\) 1.53071e7 0.611078 0.305539 0.952179i \(-0.401163\pi\)
0.305539 + 0.952179i \(0.401163\pi\)
\(912\) 5.55418e7 2.21122
\(913\) 3.83292e7 1.52178
\(914\) −2.84471e7 −1.12635
\(915\) −1.59461e6 −0.0629654
\(916\) 2.02063e7 0.795698
\(917\) 0 0
\(918\) −1.81529e7 −0.710951
\(919\) 2.42413e7 0.946820 0.473410 0.880842i \(-0.343023\pi\)
0.473410 + 0.880842i \(0.343023\pi\)
\(920\) 4.39146e6 0.171056
\(921\) 1.00508e7 0.390439
\(922\) 2.61462e7 1.01293
\(923\) −5.65425e7 −2.18459
\(924\) 0 0
\(925\) −96638.2 −0.00371360
\(926\) −4.83008e7 −1.85109
\(927\) −521933. −0.0199488
\(928\) −1.87000e7 −0.712808
\(929\) 2.10562e7 0.800461 0.400231 0.916414i \(-0.368930\pi\)
0.400231 + 0.916414i \(0.368930\pi\)
\(930\) 1.22403e7 0.464071
\(931\) 0 0
\(932\) 2.79191e7 1.05284
\(933\) 3.50967e7 1.31996
\(934\) −1.71165e7 −0.642019
\(935\) 6.56535e6 0.245600
\(936\) 820285. 0.0306038
\(937\) 1.43242e7 0.532993 0.266497 0.963836i \(-0.414134\pi\)
0.266497 + 0.963836i \(0.414134\pi\)
\(938\) 0 0
\(939\) −7.03600e6 −0.260413
\(940\) −7.58915e6 −0.280139
\(941\) 1.12722e7 0.414988 0.207494 0.978236i \(-0.433469\pi\)
0.207494 + 0.978236i \(0.433469\pi\)
\(942\) −1.18512e7 −0.435145
\(943\) 1.57002e7 0.574946
\(944\) 8.17782e6 0.298681
\(945\) 0 0
\(946\) 3.78584e7 1.37542
\(947\) −2.86149e7 −1.03685 −0.518427 0.855122i \(-0.673482\pi\)
−0.518427 + 0.855122i \(0.673482\pi\)
\(948\) −3.47574e7 −1.25611
\(949\) 3.92863e7 1.41604
\(950\) 1.24861e7 0.448866
\(951\) 2.11958e7 0.759974
\(952\) 0 0
\(953\) −2.67535e7 −0.954219 −0.477109 0.878844i \(-0.658315\pi\)
−0.477109 + 0.878844i \(0.658315\pi\)
\(954\) −1.36881e6 −0.0486935
\(955\) 1.28747e7 0.456801
\(956\) 1.02870e7 0.364036
\(957\) 1.80051e7 0.635502
\(958\) −1.64043e7 −0.577490
\(959\) 0 0
\(960\) −2.33669e6 −0.0818320
\(961\) −1.05236e7 −0.367585
\(962\) 1.02902e6 0.0358497
\(963\) −830833. −0.0288701
\(964\) −3.76096e6 −0.130349
\(965\) −1.92492e7 −0.665417
\(966\) 0 0
\(967\) −1.79212e7 −0.616311 −0.308155 0.951336i \(-0.599712\pi\)
−0.308155 + 0.951336i \(0.599712\pi\)
\(968\) −954987. −0.0327574
\(969\) 2.98585e7 1.02155
\(970\) 1.57352e7 0.536962
\(971\) −3.38258e7 −1.15133 −0.575666 0.817685i \(-0.695257\pi\)
−0.575666 + 0.817685i \(0.695257\pi\)
\(972\) −1.60480e6 −0.0544823
\(973\) 0 0
\(974\) 1.22144e7 0.412549
\(975\) 9.16272e6 0.308683
\(976\) 5.05627e6 0.169905
\(977\) 3.62165e7 1.21386 0.606931 0.794754i \(-0.292400\pi\)
0.606931 + 0.794754i \(0.292400\pi\)
\(978\) 6.69020e7 2.23662
\(979\) 2.96564e6 0.0988922
\(980\) 0 0
\(981\) −1.91822e6 −0.0636393
\(982\) −2.25914e7 −0.747593
\(983\) −1.07386e7 −0.354456 −0.177228 0.984170i \(-0.556713\pi\)
−0.177228 + 0.984170i \(0.556713\pi\)
\(984\) 1.02896e7 0.338774
\(985\) −6.17251e6 −0.202708
\(986\) −1.43269e7 −0.469311
\(987\) 0 0
\(988\) −5.15029e7 −1.67857
\(989\) 2.79558e7 0.908828
\(990\) 732649. 0.0237579
\(991\) 5.29006e6 0.171110 0.0855552 0.996333i \(-0.472734\pi\)
0.0855552 + 0.996333i \(0.472734\pi\)
\(992\) −2.72336e7 −0.878670
\(993\) 2.39260e7 0.770011
\(994\) 0 0
\(995\) −2.22660e7 −0.712991
\(996\) −3.19007e7 −1.01895
\(997\) 1.70145e7 0.542103 0.271051 0.962565i \(-0.412629\pi\)
0.271051 + 0.962565i \(0.412629\pi\)
\(998\) 6.79860e7 2.16069
\(999\) 572407. 0.0181464
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 245.6.a.h.1.2 6
7.3 odd 6 35.6.e.a.16.5 yes 12
7.5 odd 6 35.6.e.a.11.5 12
7.6 odd 2 245.6.a.i.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.6.e.a.11.5 12 7.5 odd 6
35.6.e.a.16.5 yes 12 7.3 odd 6
245.6.a.h.1.2 6 1.1 even 1 trivial
245.6.a.i.1.2 6 7.6 odd 2