Properties

Label 882.6.a.bb.1.1
Level $882$
Weight $6$
Character 882.1
Self dual yes
Analytic conductor $141.459$
Analytic rank $1$
Dimension $2$
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] = [882,6,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, 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("882.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(141.458529075\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9601}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(49.4923\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -75.4923 q^{5} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} -75.4923 q^{5} -64.0000 q^{8} +301.969 q^{10} +149.462 q^{11} -349.416 q^{13} +256.000 q^{16} -1149.85 q^{17} +2795.20 q^{19} -1207.88 q^{20} -597.847 q^{22} -1813.97 q^{23} +2574.09 q^{25} +1397.66 q^{26} +759.033 q^{29} -9031.74 q^{31} -1024.00 q^{32} +4599.39 q^{34} +7794.89 q^{37} -11180.8 q^{38} +4831.51 q^{40} +7640.49 q^{41} +12188.8 q^{43} +2391.39 q^{44} +7255.88 q^{46} +24598.8 q^{47} -10296.4 q^{50} -5590.65 q^{52} -13596.2 q^{53} -11283.2 q^{55} -3036.13 q^{58} -26358.8 q^{59} -35321.8 q^{61} +36127.0 q^{62} +4096.00 q^{64} +26378.2 q^{65} +54371.9 q^{67} -18397.6 q^{68} +70145.7 q^{71} +44468.8 q^{73} -31179.6 q^{74} +44723.2 q^{76} +61612.5 q^{79} -19326.0 q^{80} -30562.0 q^{82} -87142.0 q^{83} +86804.6 q^{85} -48755.3 q^{86} -9565.55 q^{88} +98569.4 q^{89} -29023.5 q^{92} -98395.2 q^{94} -211016. q^{95} -32342.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 32 q^{4} - 53 q^{5} - 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 32 q^{4} - 53 q^{5} - 128 q^{8} + 212 q^{10} - 191 q^{11} + 379 q^{13} + 512 q^{16} - 340 q^{17} + 1769 q^{19} - 848 q^{20} + 764 q^{22} - 3236 q^{23} - 45 q^{25} - 1516 q^{26} - 4459 q^{29} - 1994 q^{31} - 2048 q^{32} + 1360 q^{34} + 20587 q^{37} - 7076 q^{38} + 3392 q^{40} + 8814 q^{41} + 15853 q^{43} - 3056 q^{44} + 12944 q^{46} + 33912 q^{47} + 180 q^{50} + 6064 q^{52} - 49239 q^{53} - 18941 q^{55} + 17836 q^{58} - 56735 q^{59} - 67508 q^{61} + 7976 q^{62} + 8192 q^{64} + 42762 q^{65} + 75723 q^{67} - 5440 q^{68} + 8992 q^{71} + 3201 q^{73} - 82348 q^{74} + 28304 q^{76} + 26612 q^{79} - 13568 q^{80} - 35256 q^{82} - 949 q^{83} + 105020 q^{85} - 63412 q^{86} + 12224 q^{88} + 176562 q^{89} - 51776 q^{92} - 135648 q^{94} - 234098 q^{95} + 129423 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −75.4923 −1.35045 −0.675224 0.737613i \(-0.735953\pi\)
−0.675224 + 0.737613i \(0.735953\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 301.969 0.954911
\(11\) 149.462 0.372433 0.186217 0.982509i \(-0.440377\pi\)
0.186217 + 0.982509i \(0.440377\pi\)
\(12\) 0 0
\(13\) −349.416 −0.573435 −0.286717 0.958015i \(-0.592564\pi\)
−0.286717 + 0.958015i \(0.592564\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −1149.85 −0.964979 −0.482489 0.875902i \(-0.660267\pi\)
−0.482489 + 0.875902i \(0.660267\pi\)
\(18\) 0 0
\(19\) 2795.20 1.77635 0.888176 0.459503i \(-0.151972\pi\)
0.888176 + 0.459503i \(0.151972\pi\)
\(20\) −1207.88 −0.675224
\(21\) 0 0
\(22\) −597.847 −0.263350
\(23\) −1813.97 −0.715007 −0.357504 0.933912i \(-0.616372\pi\)
−0.357504 + 0.933912i \(0.616372\pi\)
\(24\) 0 0
\(25\) 2574.09 0.823710
\(26\) 1397.66 0.405480
\(27\) 0 0
\(28\) 0 0
\(29\) 759.033 0.167597 0.0837984 0.996483i \(-0.473295\pi\)
0.0837984 + 0.996483i \(0.473295\pi\)
\(30\) 0 0
\(31\) −9031.74 −1.68798 −0.843990 0.536359i \(-0.819799\pi\)
−0.843990 + 0.536359i \(0.819799\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) 4599.39 0.682343
\(35\) 0 0
\(36\) 0 0
\(37\) 7794.89 0.936064 0.468032 0.883711i \(-0.344963\pi\)
0.468032 + 0.883711i \(0.344963\pi\)
\(38\) −11180.8 −1.25607
\(39\) 0 0
\(40\) 4831.51 0.477456
\(41\) 7640.49 0.709842 0.354921 0.934896i \(-0.384508\pi\)
0.354921 + 0.934896i \(0.384508\pi\)
\(42\) 0 0
\(43\) 12188.8 1.00529 0.502645 0.864493i \(-0.332360\pi\)
0.502645 + 0.864493i \(0.332360\pi\)
\(44\) 2391.39 0.186217
\(45\) 0 0
\(46\) 7255.88 0.505586
\(47\) 24598.8 1.62431 0.812156 0.583441i \(-0.198294\pi\)
0.812156 + 0.583441i \(0.198294\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −10296.4 −0.582451
\(51\) 0 0
\(52\) −5590.65 −0.286717
\(53\) −13596.2 −0.664858 −0.332429 0.943128i \(-0.607868\pi\)
−0.332429 + 0.943128i \(0.607868\pi\)
\(54\) 0 0
\(55\) −11283.2 −0.502952
\(56\) 0 0
\(57\) 0 0
\(58\) −3036.13 −0.118509
\(59\) −26358.8 −0.985816 −0.492908 0.870081i \(-0.664066\pi\)
−0.492908 + 0.870081i \(0.664066\pi\)
\(60\) 0 0
\(61\) −35321.8 −1.21540 −0.607698 0.794168i \(-0.707907\pi\)
−0.607698 + 0.794168i \(0.707907\pi\)
\(62\) 36127.0 1.19358
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 26378.2 0.774394
\(66\) 0 0
\(67\) 54371.9 1.47975 0.739874 0.672746i \(-0.234885\pi\)
0.739874 + 0.672746i \(0.234885\pi\)
\(68\) −18397.6 −0.482489
\(69\) 0 0
\(70\) 0 0
\(71\) 70145.7 1.65141 0.825706 0.564101i \(-0.190777\pi\)
0.825706 + 0.564101i \(0.190777\pi\)
\(72\) 0 0
\(73\) 44468.8 0.976671 0.488335 0.872656i \(-0.337604\pi\)
0.488335 + 0.872656i \(0.337604\pi\)
\(74\) −31179.6 −0.661897
\(75\) 0 0
\(76\) 44723.2 0.888176
\(77\) 0 0
\(78\) 0 0
\(79\) 61612.5 1.11071 0.555355 0.831613i \(-0.312582\pi\)
0.555355 + 0.831613i \(0.312582\pi\)
\(80\) −19326.0 −0.337612
\(81\) 0 0
\(82\) −30562.0 −0.501934
\(83\) −87142.0 −1.38846 −0.694228 0.719755i \(-0.744254\pi\)
−0.694228 + 0.719755i \(0.744254\pi\)
\(84\) 0 0
\(85\) 86804.6 1.30315
\(86\) −48755.3 −0.710847
\(87\) 0 0
\(88\) −9565.55 −0.131675
\(89\) 98569.4 1.31907 0.659534 0.751675i \(-0.270754\pi\)
0.659534 + 0.751675i \(0.270754\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −29023.5 −0.357504
\(93\) 0 0
\(94\) −98395.2 −1.14856
\(95\) −211016. −2.39887
\(96\) 0 0
\(97\) −32342.3 −0.349013 −0.174507 0.984656i \(-0.555833\pi\)
−0.174507 + 0.984656i \(0.555833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 41185.5 0.411855
\(101\) 31346.4 0.305763 0.152881 0.988245i \(-0.451145\pi\)
0.152881 + 0.988245i \(0.451145\pi\)
\(102\) 0 0
\(103\) −99332.9 −0.922572 −0.461286 0.887252i \(-0.652612\pi\)
−0.461286 + 0.887252i \(0.652612\pi\)
\(104\) 22362.6 0.202740
\(105\) 0 0
\(106\) 54384.9 0.470125
\(107\) 145268. 1.22662 0.613310 0.789842i \(-0.289837\pi\)
0.613310 + 0.789842i \(0.289837\pi\)
\(108\) 0 0
\(109\) −180851. −1.45799 −0.728994 0.684520i \(-0.760012\pi\)
−0.728994 + 0.684520i \(0.760012\pi\)
\(110\) 45132.9 0.355640
\(111\) 0 0
\(112\) 0 0
\(113\) −197832. −1.45748 −0.728738 0.684793i \(-0.759893\pi\)
−0.728738 + 0.684793i \(0.759893\pi\)
\(114\) 0 0
\(115\) 136941. 0.965580
\(116\) 12144.5 0.0837984
\(117\) 0 0
\(118\) 105435. 0.697077
\(119\) 0 0
\(120\) 0 0
\(121\) −138712. −0.861294
\(122\) 141287. 0.859415
\(123\) 0 0
\(124\) −144508. −0.843990
\(125\) 41589.2 0.238070
\(126\) 0 0
\(127\) −33517.2 −0.184399 −0.0921996 0.995741i \(-0.529390\pi\)
−0.0921996 + 0.995741i \(0.529390\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) −105513. −0.547579
\(131\) −10808.9 −0.0550305 −0.0275153 0.999621i \(-0.508759\pi\)
−0.0275153 + 0.999621i \(0.508759\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −217488. −1.04634
\(135\) 0 0
\(136\) 73590.2 0.341171
\(137\) 18932.0 0.0861778 0.0430889 0.999071i \(-0.486280\pi\)
0.0430889 + 0.999071i \(0.486280\pi\)
\(138\) 0 0
\(139\) 168897. 0.741457 0.370729 0.928741i \(-0.379108\pi\)
0.370729 + 0.928741i \(0.379108\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −280583. −1.16772
\(143\) −52224.3 −0.213566
\(144\) 0 0
\(145\) −57301.2 −0.226331
\(146\) −177875. −0.690611
\(147\) 0 0
\(148\) 124718. 0.468032
\(149\) −264002. −0.974186 −0.487093 0.873350i \(-0.661943\pi\)
−0.487093 + 0.873350i \(0.661943\pi\)
\(150\) 0 0
\(151\) 279175. 0.996400 0.498200 0.867062i \(-0.333994\pi\)
0.498200 + 0.867062i \(0.333994\pi\)
\(152\) −178893. −0.628035
\(153\) 0 0
\(154\) 0 0
\(155\) 681828. 2.27953
\(156\) 0 0
\(157\) −188602. −0.610658 −0.305329 0.952247i \(-0.598766\pi\)
−0.305329 + 0.952247i \(0.598766\pi\)
\(158\) −246450. −0.785391
\(159\) 0 0
\(160\) 77304.2 0.238728
\(161\) 0 0
\(162\) 0 0
\(163\) 89717.4 0.264489 0.132244 0.991217i \(-0.457782\pi\)
0.132244 + 0.991217i \(0.457782\pi\)
\(164\) 122248. 0.354921
\(165\) 0 0
\(166\) 348568. 0.981787
\(167\) 529411. 1.46893 0.734467 0.678645i \(-0.237432\pi\)
0.734467 + 0.678645i \(0.237432\pi\)
\(168\) 0 0
\(169\) −249202. −0.671172
\(170\) −347219. −0.921469
\(171\) 0 0
\(172\) 195021. 0.502645
\(173\) 33838.3 0.0859594 0.0429797 0.999076i \(-0.486315\pi\)
0.0429797 + 0.999076i \(0.486315\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 38262.2 0.0931083
\(177\) 0 0
\(178\) −394278. −0.932722
\(179\) 247744. 0.577923 0.288962 0.957341i \(-0.406690\pi\)
0.288962 + 0.957341i \(0.406690\pi\)
\(180\) 0 0
\(181\) −369470. −0.838268 −0.419134 0.907924i \(-0.637666\pi\)
−0.419134 + 0.907924i \(0.637666\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 116094. 0.252793
\(185\) −588455. −1.26411
\(186\) 0 0
\(187\) −171858. −0.359390
\(188\) 393581. 0.812156
\(189\) 0 0
\(190\) 844065. 1.69626
\(191\) 485225. 0.962408 0.481204 0.876609i \(-0.340200\pi\)
0.481204 + 0.876609i \(0.340200\pi\)
\(192\) 0 0
\(193\) −330817. −0.639285 −0.319643 0.947538i \(-0.603563\pi\)
−0.319643 + 0.947538i \(0.603563\pi\)
\(194\) 129369. 0.246790
\(195\) 0 0
\(196\) 0 0
\(197\) 161963. 0.297337 0.148669 0.988887i \(-0.452501\pi\)
0.148669 + 0.988887i \(0.452501\pi\)
\(198\) 0 0
\(199\) −55396.2 −0.0991625 −0.0495813 0.998770i \(-0.515789\pi\)
−0.0495813 + 0.998770i \(0.515789\pi\)
\(200\) −164742. −0.291226
\(201\) 0 0
\(202\) −125386. −0.216207
\(203\) 0 0
\(204\) 0 0
\(205\) −576799. −0.958605
\(206\) 397332. 0.652357
\(207\) 0 0
\(208\) −89450.4 −0.143359
\(209\) 417776. 0.661572
\(210\) 0 0
\(211\) 481748. 0.744926 0.372463 0.928047i \(-0.378513\pi\)
0.372463 + 0.928047i \(0.378513\pi\)
\(212\) −217540. −0.332429
\(213\) 0 0
\(214\) −581072. −0.867352
\(215\) −920164. −1.35759
\(216\) 0 0
\(217\) 0 0
\(218\) 723403. 1.03095
\(219\) 0 0
\(220\) −180531. −0.251476
\(221\) 401775. 0.553353
\(222\) 0 0
\(223\) −638779. −0.860178 −0.430089 0.902787i \(-0.641518\pi\)
−0.430089 + 0.902787i \(0.641518\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 791330. 1.03059
\(227\) −525682. −0.677109 −0.338554 0.940947i \(-0.609938\pi\)
−0.338554 + 0.940947i \(0.609938\pi\)
\(228\) 0 0
\(229\) −932686. −1.17529 −0.587647 0.809117i \(-0.699946\pi\)
−0.587647 + 0.809117i \(0.699946\pi\)
\(230\) −547763. −0.682768
\(231\) 0 0
\(232\) −48578.1 −0.0592544
\(233\) 707793. 0.854115 0.427058 0.904224i \(-0.359550\pi\)
0.427058 + 0.904224i \(0.359550\pi\)
\(234\) 0 0
\(235\) −1.85702e6 −2.19355
\(236\) −421741. −0.492908
\(237\) 0 0
\(238\) 0 0
\(239\) 500614. 0.566902 0.283451 0.958987i \(-0.408521\pi\)
0.283451 + 0.958987i \(0.408521\pi\)
\(240\) 0 0
\(241\) −1.20822e6 −1.33999 −0.669997 0.742364i \(-0.733705\pi\)
−0.669997 + 0.742364i \(0.733705\pi\)
\(242\) 554849. 0.609027
\(243\) 0 0
\(244\) −565148. −0.607698
\(245\) 0 0
\(246\) 0 0
\(247\) −976688. −1.01862
\(248\) 578032. 0.596791
\(249\) 0 0
\(250\) −166357. −0.168341
\(251\) 97826.7 0.0980106 0.0490053 0.998799i \(-0.484395\pi\)
0.0490053 + 0.998799i \(0.484395\pi\)
\(252\) 0 0
\(253\) −271119. −0.266292
\(254\) 134069. 0.130390
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.10791e6 −1.04634 −0.523170 0.852228i \(-0.675251\pi\)
−0.523170 + 0.852228i \(0.675251\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 422052. 0.387197
\(261\) 0 0
\(262\) 43235.6 0.0389125
\(263\) −1.50685e6 −1.34333 −0.671663 0.740856i \(-0.734420\pi\)
−0.671663 + 0.740856i \(0.734420\pi\)
\(264\) 0 0
\(265\) 1.02641e6 0.897856
\(266\) 0 0
\(267\) 0 0
\(268\) 869951. 0.739874
\(269\) −2.13502e6 −1.79896 −0.899481 0.436960i \(-0.856055\pi\)
−0.899481 + 0.436960i \(0.856055\pi\)
\(270\) 0 0
\(271\) −1.22860e6 −1.01622 −0.508108 0.861293i \(-0.669655\pi\)
−0.508108 + 0.861293i \(0.669655\pi\)
\(272\) −294361. −0.241245
\(273\) 0 0
\(274\) −75728.1 −0.0609369
\(275\) 384729. 0.306777
\(276\) 0 0
\(277\) −1.84682e6 −1.44619 −0.723097 0.690747i \(-0.757282\pi\)
−0.723097 + 0.690747i \(0.757282\pi\)
\(278\) −675590. −0.524289
\(279\) 0 0
\(280\) 0 0
\(281\) 2.28326e6 1.72500 0.862500 0.506056i \(-0.168897\pi\)
0.862500 + 0.506056i \(0.168897\pi\)
\(282\) 0 0
\(283\) −1.33693e6 −0.992296 −0.496148 0.868238i \(-0.665253\pi\)
−0.496148 + 0.868238i \(0.665253\pi\)
\(284\) 1.12233e6 0.825706
\(285\) 0 0
\(286\) 208897. 0.151014
\(287\) 0 0
\(288\) 0 0
\(289\) −97709.0 −0.0688161
\(290\) 229205. 0.160040
\(291\) 0 0
\(292\) 711501. 0.488335
\(293\) −2.23033e6 −1.51775 −0.758875 0.651236i \(-0.774251\pi\)
−0.758875 + 0.651236i \(0.774251\pi\)
\(294\) 0 0
\(295\) 1.98989e6 1.33129
\(296\) −498873. −0.330949
\(297\) 0 0
\(298\) 1.05601e6 0.688853
\(299\) 633830. 0.410010
\(300\) 0 0
\(301\) 0 0
\(302\) −1.11670e6 −0.704561
\(303\) 0 0
\(304\) 715572. 0.444088
\(305\) 2.66652e6 1.64133
\(306\) 0 0
\(307\) −1.77782e6 −1.07657 −0.538284 0.842763i \(-0.680927\pi\)
−0.538284 + 0.842763i \(0.680927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.72731e6 −1.61187
\(311\) 1.40272e6 0.822373 0.411187 0.911551i \(-0.365114\pi\)
0.411187 + 0.911551i \(0.365114\pi\)
\(312\) 0 0
\(313\) −1.21894e6 −0.703271 −0.351636 0.936137i \(-0.614374\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(314\) 754409. 0.431800
\(315\) 0 0
\(316\) 985799. 0.555355
\(317\) −2.68001e6 −1.49792 −0.748961 0.662615i \(-0.769447\pi\)
−0.748961 + 0.662615i \(0.769447\pi\)
\(318\) 0 0
\(319\) 113446. 0.0624186
\(320\) −309217. −0.168806
\(321\) 0 0
\(322\) 0 0
\(323\) −3.21405e6 −1.71414
\(324\) 0 0
\(325\) −899429. −0.472344
\(326\) −358869. −0.187022
\(327\) 0 0
\(328\) −488992. −0.250967
\(329\) 0 0
\(330\) 0 0
\(331\) 142560. 0.0715202 0.0357601 0.999360i \(-0.488615\pi\)
0.0357601 + 0.999360i \(0.488615\pi\)
\(332\) −1.39427e6 −0.694228
\(333\) 0 0
\(334\) −2.11765e6 −1.03869
\(335\) −4.10466e6 −1.99832
\(336\) 0 0
\(337\) −1.21206e6 −0.581367 −0.290683 0.956819i \(-0.593883\pi\)
−0.290683 + 0.956819i \(0.593883\pi\)
\(338\) 996806. 0.474591
\(339\) 0 0
\(340\) 1.38887e6 0.651577
\(341\) −1.34990e6 −0.628660
\(342\) 0 0
\(343\) 0 0
\(344\) −780085. −0.355423
\(345\) 0 0
\(346\) −135353. −0.0607825
\(347\) −3.46853e6 −1.54640 −0.773199 0.634163i \(-0.781345\pi\)
−0.773199 + 0.634163i \(0.781345\pi\)
\(348\) 0 0
\(349\) −1.01692e6 −0.446911 −0.223456 0.974714i \(-0.571734\pi\)
−0.223456 + 0.974714i \(0.571734\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −153049. −0.0658375
\(353\) 1.56385e6 0.667974 0.333987 0.942578i \(-0.391606\pi\)
0.333987 + 0.942578i \(0.391606\pi\)
\(354\) 0 0
\(355\) −5.29547e6 −2.23015
\(356\) 1.57711e6 0.659534
\(357\) 0 0
\(358\) −990975. −0.408653
\(359\) −49553.1 −0.0202925 −0.0101462 0.999949i \(-0.503230\pi\)
−0.0101462 + 0.999949i \(0.503230\pi\)
\(360\) 0 0
\(361\) 5.33705e6 2.15543
\(362\) 1.47788e6 0.592745
\(363\) 0 0
\(364\) 0 0
\(365\) −3.35705e6 −1.31894
\(366\) 0 0
\(367\) −3.54831e6 −1.37517 −0.687585 0.726104i \(-0.741329\pi\)
−0.687585 + 0.726104i \(0.741329\pi\)
\(368\) −464376. −0.178752
\(369\) 0 0
\(370\) 2.35382e6 0.893858
\(371\) 0 0
\(372\) 0 0
\(373\) −2.25573e6 −0.839490 −0.419745 0.907642i \(-0.637880\pi\)
−0.419745 + 0.907642i \(0.637880\pi\)
\(374\) 687432. 0.254127
\(375\) 0 0
\(376\) −1.57432e6 −0.574281
\(377\) −265218. −0.0961059
\(378\) 0 0
\(379\) 4.39503e6 1.57168 0.785840 0.618430i \(-0.212231\pi\)
0.785840 + 0.618430i \(0.212231\pi\)
\(380\) −3.37626e6 −1.19944
\(381\) 0 0
\(382\) −1.94090e6 −0.680525
\(383\) −1.22781e6 −0.427694 −0.213847 0.976867i \(-0.568599\pi\)
−0.213847 + 0.976867i \(0.568599\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.32327e6 0.452043
\(387\) 0 0
\(388\) −517477. −0.174507
\(389\) −2.05424e6 −0.688300 −0.344150 0.938915i \(-0.611833\pi\)
−0.344150 + 0.938915i \(0.611833\pi\)
\(390\) 0 0
\(391\) 2.08579e6 0.689967
\(392\) 0 0
\(393\) 0 0
\(394\) −647851. −0.210249
\(395\) −4.65127e6 −1.49996
\(396\) 0 0
\(397\) 3.80367e6 1.21123 0.605615 0.795758i \(-0.292927\pi\)
0.605615 + 0.795758i \(0.292927\pi\)
\(398\) 221585. 0.0701185
\(399\) 0 0
\(400\) 658968. 0.205928
\(401\) −1.18524e6 −0.368084 −0.184042 0.982918i \(-0.558918\pi\)
−0.184042 + 0.982918i \(0.558918\pi\)
\(402\) 0 0
\(403\) 3.15583e6 0.967947
\(404\) 501543. 0.152881
\(405\) 0 0
\(406\) 0 0
\(407\) 1.16504e6 0.348621
\(408\) 0 0
\(409\) 4.30393e6 1.27220 0.636102 0.771605i \(-0.280546\pi\)
0.636102 + 0.771605i \(0.280546\pi\)
\(410\) 2.30720e6 0.677836
\(411\) 0 0
\(412\) −1.58933e6 −0.461286
\(413\) 0 0
\(414\) 0 0
\(415\) 6.57855e6 1.87504
\(416\) 357802. 0.101370
\(417\) 0 0
\(418\) −1.67110e6 −0.467802
\(419\) −113725. −0.0316461 −0.0158230 0.999875i \(-0.505037\pi\)
−0.0158230 + 0.999875i \(0.505037\pi\)
\(420\) 0 0
\(421\) 443417. 0.121929 0.0609645 0.998140i \(-0.480582\pi\)
0.0609645 + 0.998140i \(0.480582\pi\)
\(422\) −1.92699e6 −0.526743
\(423\) 0 0
\(424\) 870158. 0.235063
\(425\) −2.95981e6 −0.794863
\(426\) 0 0
\(427\) 0 0
\(428\) 2.32429e6 0.613310
\(429\) 0 0
\(430\) 3.68065e6 0.959962
\(431\) −4.63310e6 −1.20138 −0.600688 0.799484i \(-0.705107\pi\)
−0.600688 + 0.799484i \(0.705107\pi\)
\(432\) 0 0
\(433\) −6.57955e6 −1.68646 −0.843231 0.537552i \(-0.819349\pi\)
−0.843231 + 0.537552i \(0.819349\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.89361e6 −0.728994
\(437\) −5.07041e6 −1.27010
\(438\) 0 0
\(439\) 1.82070e6 0.450896 0.225448 0.974255i \(-0.427615\pi\)
0.225448 + 0.974255i \(0.427615\pi\)
\(440\) 722126. 0.177820
\(441\) 0 0
\(442\) −1.60710e6 −0.391279
\(443\) 2.06990e6 0.501119 0.250559 0.968101i \(-0.419385\pi\)
0.250559 + 0.968101i \(0.419385\pi\)
\(444\) 0 0
\(445\) −7.44123e6 −1.78133
\(446\) 2.55511e6 0.608238
\(447\) 0 0
\(448\) 0 0
\(449\) −5.72581e6 −1.34036 −0.670180 0.742199i \(-0.733783\pi\)
−0.670180 + 0.742199i \(0.733783\pi\)
\(450\) 0 0
\(451\) 1.14196e6 0.264369
\(452\) −3.16532e6 −0.728738
\(453\) 0 0
\(454\) 2.10273e6 0.478788
\(455\) 0 0
\(456\) 0 0
\(457\) 318676. 0.0713771 0.0356886 0.999363i \(-0.488638\pi\)
0.0356886 + 0.999363i \(0.488638\pi\)
\(458\) 3.73074e6 0.831059
\(459\) 0 0
\(460\) 2.19105e6 0.482790
\(461\) −3.42470e6 −0.750535 −0.375267 0.926917i \(-0.622449\pi\)
−0.375267 + 0.926917i \(0.622449\pi\)
\(462\) 0 0
\(463\) 3.82945e6 0.830201 0.415101 0.909775i \(-0.363746\pi\)
0.415101 + 0.909775i \(0.363746\pi\)
\(464\) 194312. 0.0418992
\(465\) 0 0
\(466\) −2.83117e6 −0.603951
\(467\) 2.84462e6 0.603576 0.301788 0.953375i \(-0.402417\pi\)
0.301788 + 0.953375i \(0.402417\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 7.42809e6 1.55107
\(471\) 0 0
\(472\) 1.68696e6 0.348539
\(473\) 1.82176e6 0.374403
\(474\) 0 0
\(475\) 7.19511e6 1.46320
\(476\) 0 0
\(477\) 0 0
\(478\) −2.00246e6 −0.400860
\(479\) 1.30365e6 0.259611 0.129806 0.991539i \(-0.458565\pi\)
0.129806 + 0.991539i \(0.458565\pi\)
\(480\) 0 0
\(481\) −2.72366e6 −0.536772
\(482\) 4.83287e6 0.947519
\(483\) 0 0
\(484\) −2.21940e6 −0.430647
\(485\) 2.44160e6 0.471324
\(486\) 0 0
\(487\) 6.23624e6 1.19152 0.595759 0.803164i \(-0.296852\pi\)
0.595759 + 0.803164i \(0.296852\pi\)
\(488\) 2.26059e6 0.429707
\(489\) 0 0
\(490\) 0 0
\(491\) 3.93928e6 0.737417 0.368709 0.929545i \(-0.379800\pi\)
0.368709 + 0.929545i \(0.379800\pi\)
\(492\) 0 0
\(493\) −872772. −0.161727
\(494\) 3.90675e6 0.720275
\(495\) 0 0
\(496\) −2.31213e6 −0.421995
\(497\) 0 0
\(498\) 0 0
\(499\) 7.97571e6 1.43390 0.716948 0.697126i \(-0.245538\pi\)
0.716948 + 0.697126i \(0.245538\pi\)
\(500\) 665427. 0.119035
\(501\) 0 0
\(502\) −391307. −0.0693040
\(503\) −2.70777e6 −0.477190 −0.238595 0.971119i \(-0.576687\pi\)
−0.238595 + 0.971119i \(0.576687\pi\)
\(504\) 0 0
\(505\) −2.36641e6 −0.412917
\(506\) 1.08448e6 0.188297
\(507\) 0 0
\(508\) −536276. −0.0921996
\(509\) 3.90049e6 0.667306 0.333653 0.942696i \(-0.391719\pi\)
0.333653 + 0.942696i \(0.391719\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) 4.43165e6 0.739874
\(515\) 7.49887e6 1.24589
\(516\) 0 0
\(517\) 3.67658e6 0.604947
\(518\) 0 0
\(519\) 0 0
\(520\) −1.68821e6 −0.273790
\(521\) 7.00063e6 1.12991 0.564954 0.825122i \(-0.308894\pi\)
0.564954 + 0.825122i \(0.308894\pi\)
\(522\) 0 0
\(523\) 2.11208e6 0.337641 0.168821 0.985647i \(-0.446004\pi\)
0.168821 + 0.985647i \(0.446004\pi\)
\(524\) −172943. −0.0275153
\(525\) 0 0
\(526\) 6.02741e6 0.949875
\(527\) 1.03851e7 1.62887
\(528\) 0 0
\(529\) −3.14586e6 −0.488765
\(530\) −4.10564e6 −0.634880
\(531\) 0 0
\(532\) 0 0
\(533\) −2.66971e6 −0.407048
\(534\) 0 0
\(535\) −1.09666e7 −1.65649
\(536\) −3.47980e6 −0.523170
\(537\) 0 0
\(538\) 8.54009e6 1.27206
\(539\) 0 0
\(540\) 0 0
\(541\) 5.70567e6 0.838134 0.419067 0.907955i \(-0.362357\pi\)
0.419067 + 0.907955i \(0.362357\pi\)
\(542\) 4.91439e6 0.718574
\(543\) 0 0
\(544\) 1.17744e6 0.170586
\(545\) 1.36528e7 1.96894
\(546\) 0 0
\(547\) 2.28054e6 0.325888 0.162944 0.986635i \(-0.447901\pi\)
0.162944 + 0.986635i \(0.447901\pi\)
\(548\) 302912. 0.0430889
\(549\) 0 0
\(550\) −1.53891e6 −0.216924
\(551\) 2.12165e6 0.297711
\(552\) 0 0
\(553\) 0 0
\(554\) 7.38730e6 1.02261
\(555\) 0 0
\(556\) 2.70236e6 0.370729
\(557\) 6.73941e6 0.920415 0.460208 0.887811i \(-0.347775\pi\)
0.460208 + 0.887811i \(0.347775\pi\)
\(558\) 0 0
\(559\) −4.25897e6 −0.576468
\(560\) 0 0
\(561\) 0 0
\(562\) −9.13304e6 −1.21976
\(563\) −9.66820e6 −1.28551 −0.642754 0.766073i \(-0.722208\pi\)
−0.642754 + 0.766073i \(0.722208\pi\)
\(564\) 0 0
\(565\) 1.49348e7 1.96825
\(566\) 5.34770e6 0.701659
\(567\) 0 0
\(568\) −4.48933e6 −0.583862
\(569\) 1.39587e7 1.80744 0.903719 0.428125i \(-0.140826\pi\)
0.903719 + 0.428125i \(0.140826\pi\)
\(570\) 0 0
\(571\) −2.35841e6 −0.302711 −0.151355 0.988479i \(-0.548364\pi\)
−0.151355 + 0.988479i \(0.548364\pi\)
\(572\) −835589. −0.106783
\(573\) 0 0
\(574\) 0 0
\(575\) −4.66933e6 −0.588959
\(576\) 0 0
\(577\) −5.01098e6 −0.626589 −0.313295 0.949656i \(-0.601433\pi\)
−0.313295 + 0.949656i \(0.601433\pi\)
\(578\) 390836. 0.0486603
\(579\) 0 0
\(580\) −916819. −0.113165
\(581\) 0 0
\(582\) 0 0
\(583\) −2.03211e6 −0.247615
\(584\) −2.84600e6 −0.345305
\(585\) 0 0
\(586\) 8.92132e6 1.07321
\(587\) 3.95106e6 0.473280 0.236640 0.971597i \(-0.423954\pi\)
0.236640 + 0.971597i \(0.423954\pi\)
\(588\) 0 0
\(589\) −2.52455e7 −2.99845
\(590\) −7.95955e6 −0.941367
\(591\) 0 0
\(592\) 1.99549e6 0.234016
\(593\) −2.53680e6 −0.296244 −0.148122 0.988969i \(-0.547323\pi\)
−0.148122 + 0.988969i \(0.547323\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.22403e6 −0.487093
\(597\) 0 0
\(598\) −2.53532e6 −0.289921
\(599\) −7.17714e6 −0.817305 −0.408652 0.912690i \(-0.634001\pi\)
−0.408652 + 0.912690i \(0.634001\pi\)
\(600\) 0 0
\(601\) −1.12527e7 −1.27078 −0.635392 0.772190i \(-0.719162\pi\)
−0.635392 + 0.772190i \(0.719162\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 4.46680e6 0.498200
\(605\) 1.04717e7 1.16313
\(606\) 0 0
\(607\) 1.40370e7 1.54633 0.773167 0.634202i \(-0.218671\pi\)
0.773167 + 0.634202i \(0.218671\pi\)
\(608\) −2.86229e6 −0.314018
\(609\) 0 0
\(610\) −1.06661e7 −1.16059
\(611\) −8.59521e6 −0.931437
\(612\) 0 0
\(613\) −7.62872e6 −0.819975 −0.409988 0.912091i \(-0.634467\pi\)
−0.409988 + 0.912091i \(0.634467\pi\)
\(614\) 7.11128e6 0.761249
\(615\) 0 0
\(616\) 0 0
\(617\) 4.41080e6 0.466450 0.233225 0.972423i \(-0.425072\pi\)
0.233225 + 0.972423i \(0.425072\pi\)
\(618\) 0 0
\(619\) 9.04355e6 0.948664 0.474332 0.880346i \(-0.342690\pi\)
0.474332 + 0.880346i \(0.342690\pi\)
\(620\) 1.09092e7 1.13977
\(621\) 0 0
\(622\) −5.61087e6 −0.581506
\(623\) 0 0
\(624\) 0 0
\(625\) −1.11837e7 −1.14521
\(626\) 4.87578e6 0.497288
\(627\) 0 0
\(628\) −3.01764e6 −0.305329
\(629\) −8.96293e6 −0.903282
\(630\) 0 0
\(631\) −7.33039e6 −0.732916 −0.366458 0.930435i \(-0.619430\pi\)
−0.366458 + 0.930435i \(0.619430\pi\)
\(632\) −3.94320e6 −0.392695
\(633\) 0 0
\(634\) 1.07201e7 1.05919
\(635\) 2.53030e6 0.249022
\(636\) 0 0
\(637\) 0 0
\(638\) −453786. −0.0441366
\(639\) 0 0
\(640\) 1.23687e6 0.119364
\(641\) −3.68824e6 −0.354547 −0.177273 0.984162i \(-0.556728\pi\)
−0.177273 + 0.984162i \(0.556728\pi\)
\(642\) 0 0
\(643\) 1.17584e7 1.12155 0.560776 0.827968i \(-0.310503\pi\)
0.560776 + 0.827968i \(0.310503\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.28562e7 1.21208
\(647\) −9.94050e6 −0.933572 −0.466786 0.884370i \(-0.654588\pi\)
−0.466786 + 0.884370i \(0.654588\pi\)
\(648\) 0 0
\(649\) −3.93963e6 −0.367151
\(650\) 3.59772e6 0.333998
\(651\) 0 0
\(652\) 1.43548e6 0.132244
\(653\) 1.02620e6 0.0941778 0.0470889 0.998891i \(-0.485006\pi\)
0.0470889 + 0.998891i \(0.485006\pi\)
\(654\) 0 0
\(655\) 815990. 0.0743159
\(656\) 1.95597e6 0.177461
\(657\) 0 0
\(658\) 0 0
\(659\) −1.00207e7 −0.898846 −0.449423 0.893319i \(-0.648370\pi\)
−0.449423 + 0.893319i \(0.648370\pi\)
\(660\) 0 0
\(661\) −2.62826e6 −0.233972 −0.116986 0.993134i \(-0.537323\pi\)
−0.116986 + 0.993134i \(0.537323\pi\)
\(662\) −570241. −0.0505724
\(663\) 0 0
\(664\) 5.57709e6 0.490893
\(665\) 0 0
\(666\) 0 0
\(667\) −1.37686e6 −0.119833
\(668\) 8.47058e6 0.734467
\(669\) 0 0
\(670\) 1.64187e7 1.41303
\(671\) −5.27925e6 −0.452654
\(672\) 0 0
\(673\) −1.50220e7 −1.27847 −0.639233 0.769013i \(-0.720748\pi\)
−0.639233 + 0.769013i \(0.720748\pi\)
\(674\) 4.84825e6 0.411088
\(675\) 0 0
\(676\) −3.98723e6 −0.335586
\(677\) −6.03310e6 −0.505905 −0.252953 0.967479i \(-0.581402\pi\)
−0.252953 + 0.967479i \(0.581402\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.55550e6 −0.460734
\(681\) 0 0
\(682\) 5.39960e6 0.444530
\(683\) −1.10720e7 −0.908187 −0.454093 0.890954i \(-0.650037\pi\)
−0.454093 + 0.890954i \(0.650037\pi\)
\(684\) 0 0
\(685\) −1.42922e6 −0.116379
\(686\) 0 0
\(687\) 0 0
\(688\) 3.12034e6 0.251322
\(689\) 4.75073e6 0.381253
\(690\) 0 0
\(691\) 1.05548e7 0.840919 0.420460 0.907311i \(-0.361869\pi\)
0.420460 + 0.907311i \(0.361869\pi\)
\(692\) 541413. 0.0429797
\(693\) 0 0
\(694\) 1.38741e7 1.09347
\(695\) −1.27505e7 −1.00130
\(696\) 0 0
\(697\) −8.78540e6 −0.684983
\(698\) 4.06766e6 0.316014
\(699\) 0 0
\(700\) 0 0
\(701\) 7.20675e6 0.553917 0.276958 0.960882i \(-0.410674\pi\)
0.276958 + 0.960882i \(0.410674\pi\)
\(702\) 0 0
\(703\) 2.17883e7 1.66278
\(704\) 612195. 0.0465541
\(705\) 0 0
\(706\) −6.25542e6 −0.472329
\(707\) 0 0
\(708\) 0 0
\(709\) 2.69373e6 0.201252 0.100626 0.994924i \(-0.467916\pi\)
0.100626 + 0.994924i \(0.467916\pi\)
\(710\) 2.11819e7 1.57695
\(711\) 0 0
\(712\) −6.30844e6 −0.466361
\(713\) 1.63833e7 1.20692
\(714\) 0 0
\(715\) 3.94253e6 0.288410
\(716\) 3.96390e6 0.288962
\(717\) 0 0
\(718\) 198213. 0.0143489
\(719\) 6.92550e6 0.499608 0.249804 0.968296i \(-0.419634\pi\)
0.249804 + 0.968296i \(0.419634\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.13482e7 −1.52412
\(723\) 0 0
\(724\) −5.91152e6 −0.419134
\(725\) 1.95382e6 0.138051
\(726\) 0 0
\(727\) −4.11366e6 −0.288664 −0.144332 0.989529i \(-0.546103\pi\)
−0.144332 + 0.989529i \(0.546103\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.34282e7 0.932634
\(731\) −1.40153e7 −0.970083
\(732\) 0 0
\(733\) −7.92038e6 −0.544485 −0.272243 0.962229i \(-0.587765\pi\)
−0.272243 + 0.962229i \(0.587765\pi\)
\(734\) 1.41932e7 0.972392
\(735\) 0 0
\(736\) 1.85750e6 0.126397
\(737\) 8.12652e6 0.551107
\(738\) 0 0
\(739\) 1.60410e7 1.08049 0.540245 0.841508i \(-0.318332\pi\)
0.540245 + 0.841508i \(0.318332\pi\)
\(740\) −9.41527e6 −0.632053
\(741\) 0 0
\(742\) 0 0
\(743\) −1.53453e7 −1.01977 −0.509887 0.860241i \(-0.670313\pi\)
−0.509887 + 0.860241i \(0.670313\pi\)
\(744\) 0 0
\(745\) 1.99301e7 1.31559
\(746\) 9.02293e6 0.593609
\(747\) 0 0
\(748\) −2.74973e6 −0.179695
\(749\) 0 0
\(750\) 0 0
\(751\) 2.24976e7 1.45558 0.727790 0.685800i \(-0.240547\pi\)
0.727790 + 0.685800i \(0.240547\pi\)
\(752\) 6.29729e6 0.406078
\(753\) 0 0
\(754\) 1.06087e6 0.0679571
\(755\) −2.10756e7 −1.34559
\(756\) 0 0
\(757\) 2.30349e7 1.46099 0.730494 0.682919i \(-0.239290\pi\)
0.730494 + 0.682919i \(0.239290\pi\)
\(758\) −1.75801e7 −1.11135
\(759\) 0 0
\(760\) 1.35050e7 0.848129
\(761\) 5.40735e6 0.338472 0.169236 0.985576i \(-0.445870\pi\)
0.169236 + 0.985576i \(0.445870\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 7.76359e6 0.481204
\(765\) 0 0
\(766\) 4.91123e6 0.302426
\(767\) 9.21019e6 0.565301
\(768\) 0 0
\(769\) 7.93100e6 0.483629 0.241814 0.970323i \(-0.422258\pi\)
0.241814 + 0.970323i \(0.422258\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.29307e6 −0.319643
\(773\) 269579. 0.0162270 0.00811349 0.999967i \(-0.497417\pi\)
0.00811349 + 0.999967i \(0.497417\pi\)
\(774\) 0 0
\(775\) −2.32486e7 −1.39041
\(776\) 2.06991e6 0.123395
\(777\) 0 0
\(778\) 8.21698e6 0.486702
\(779\) 2.13567e7 1.26093
\(780\) 0 0
\(781\) 1.04841e7 0.615041
\(782\) −8.34315e6 −0.487880
\(783\) 0 0
\(784\) 0 0
\(785\) 1.42380e7 0.824662
\(786\) 0 0
\(787\) 8.98621e6 0.517177 0.258589 0.965988i \(-0.416743\pi\)
0.258589 + 0.965988i \(0.416743\pi\)
\(788\) 2.59140e6 0.148669
\(789\) 0 0
\(790\) 1.86051e7 1.06063
\(791\) 0 0
\(792\) 0 0
\(793\) 1.23420e7 0.696950
\(794\) −1.52147e7 −0.856469
\(795\) 0 0
\(796\) −886340. −0.0495813
\(797\) −2.65558e7 −1.48086 −0.740429 0.672135i \(-0.765378\pi\)
−0.740429 + 0.672135i \(0.765378\pi\)
\(798\) 0 0
\(799\) −2.82849e7 −1.56743
\(800\) −2.63587e6 −0.145613
\(801\) 0 0
\(802\) 4.74098e6 0.260275
\(803\) 6.64638e6 0.363745
\(804\) 0 0
\(805\) 0 0
\(806\) −1.26233e7 −0.684442
\(807\) 0 0
\(808\) −2.00617e6 −0.108103
\(809\) −2.52557e7 −1.35671 −0.678356 0.734733i \(-0.737307\pi\)
−0.678356 + 0.734733i \(0.737307\pi\)
\(810\) 0 0
\(811\) 4.76866e6 0.254592 0.127296 0.991865i \(-0.459370\pi\)
0.127296 + 0.991865i \(0.459370\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −4.66015e6 −0.246513
\(815\) −6.77297e6 −0.357179
\(816\) 0 0
\(817\) 3.40702e7 1.78575
\(818\) −1.72157e7 −0.899585
\(819\) 0 0
\(820\) −9.22878e6 −0.479303
\(821\) 5.95534e6 0.308353 0.154177 0.988043i \(-0.450727\pi\)
0.154177 + 0.988043i \(0.450727\pi\)
\(822\) 0 0
\(823\) 1.06659e7 0.548907 0.274453 0.961600i \(-0.411503\pi\)
0.274453 + 0.961600i \(0.411503\pi\)
\(824\) 6.35731e6 0.326178
\(825\) 0 0
\(826\) 0 0
\(827\) 8.52356e6 0.433369 0.216684 0.976242i \(-0.430476\pi\)
0.216684 + 0.976242i \(0.430476\pi\)
\(828\) 0 0
\(829\) −3.87458e6 −0.195811 −0.0979057 0.995196i \(-0.531214\pi\)
−0.0979057 + 0.995196i \(0.531214\pi\)
\(830\) −2.63142e7 −1.32585
\(831\) 0 0
\(832\) −1.43121e6 −0.0716794
\(833\) 0 0
\(834\) 0 0
\(835\) −3.99665e7 −1.98372
\(836\) 6.68441e6 0.330786
\(837\) 0 0
\(838\) 454899. 0.0223772
\(839\) 2.13769e7 1.04843 0.524215 0.851586i \(-0.324359\pi\)
0.524215 + 0.851586i \(0.324359\pi\)
\(840\) 0 0
\(841\) −1.99350e7 −0.971911
\(842\) −1.77367e6 −0.0862168
\(843\) 0 0
\(844\) 7.70796e6 0.372463
\(845\) 1.88128e7 0.906383
\(846\) 0 0
\(847\) 0 0
\(848\) −3.48063e6 −0.166214
\(849\) 0 0
\(850\) 1.18393e7 0.562053
\(851\) −1.41397e7 −0.669293
\(852\) 0 0
\(853\) 85930.0 0.00404364 0.00202182 0.999998i \(-0.499356\pi\)
0.00202182 + 0.999998i \(0.499356\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.29715e6 −0.433676
\(857\) −1.85664e7 −0.863528 −0.431764 0.901987i \(-0.642109\pi\)
−0.431764 + 0.901987i \(0.642109\pi\)
\(858\) 0 0
\(859\) −3.92215e7 −1.81360 −0.906799 0.421564i \(-0.861481\pi\)
−0.906799 + 0.421564i \(0.861481\pi\)
\(860\) −1.47226e7 −0.678795
\(861\) 0 0
\(862\) 1.85324e7 0.849501
\(863\) −1.83642e7 −0.839352 −0.419676 0.907674i \(-0.637856\pi\)
−0.419676 + 0.907674i \(0.637856\pi\)
\(864\) 0 0
\(865\) −2.55453e6 −0.116084
\(866\) 2.63182e7 1.19251
\(867\) 0 0
\(868\) 0 0
\(869\) 9.20870e6 0.413665
\(870\) 0 0
\(871\) −1.89984e7 −0.848539
\(872\) 1.15744e7 0.515477
\(873\) 0 0
\(874\) 2.02816e7 0.898100
\(875\) 0 0
\(876\) 0 0
\(877\) −9.42578e6 −0.413826 −0.206913 0.978359i \(-0.566342\pi\)
−0.206913 + 0.978359i \(0.566342\pi\)
\(878\) −7.28278e6 −0.318831
\(879\) 0 0
\(880\) −2.88850e6 −0.125738
\(881\) 1.32820e7 0.576534 0.288267 0.957550i \(-0.406921\pi\)
0.288267 + 0.957550i \(0.406921\pi\)
\(882\) 0 0
\(883\) 1.88897e7 0.815309 0.407655 0.913136i \(-0.366347\pi\)
0.407655 + 0.913136i \(0.366347\pi\)
\(884\) 6.42840e6 0.276676
\(885\) 0 0
\(886\) −8.27961e6 −0.354345
\(887\) 4.17083e7 1.77997 0.889987 0.455985i \(-0.150713\pi\)
0.889987 + 0.455985i \(0.150713\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.97649e7 1.25959
\(891\) 0 0
\(892\) −1.02205e7 −0.430089
\(893\) 6.87586e7 2.88535
\(894\) 0 0
\(895\) −1.87028e7 −0.780455
\(896\) 0 0
\(897\) 0 0
\(898\) 2.29033e7 0.947777
\(899\) −6.85539e6 −0.282900
\(900\) 0 0
\(901\) 1.56336e7 0.641573
\(902\) −4.56785e6 −0.186937
\(903\) 0 0
\(904\) 1.26613e7 0.515295
\(905\) 2.78922e7 1.13204
\(906\) 0 0
\(907\) 4.60801e7 1.85993 0.929963 0.367654i \(-0.119839\pi\)
0.929963 + 0.367654i \(0.119839\pi\)
\(908\) −8.41091e6 −0.338554
\(909\) 0 0
\(910\) 0 0
\(911\) −2.25527e7 −0.900334 −0.450167 0.892944i \(-0.648636\pi\)
−0.450167 + 0.892944i \(0.648636\pi\)
\(912\) 0 0
\(913\) −1.30244e7 −0.517107
\(914\) −1.27470e6 −0.0504712
\(915\) 0 0
\(916\) −1.49230e7 −0.587647
\(917\) 0 0
\(918\) 0 0
\(919\) −3.25898e7 −1.27290 −0.636448 0.771319i \(-0.719597\pi\)
−0.636448 + 0.771319i \(0.719597\pi\)
\(920\) −8.76421e6 −0.341384
\(921\) 0 0
\(922\) 1.36988e7 0.530708
\(923\) −2.45100e7 −0.946977
\(924\) 0 0
\(925\) 2.00648e7 0.771046
\(926\) −1.53178e7 −0.587041
\(927\) 0 0
\(928\) −777250. −0.0296272
\(929\) −4.56149e7 −1.73407 −0.867036 0.498246i \(-0.833978\pi\)
−0.867036 + 0.498246i \(0.833978\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.13247e7 0.427058
\(933\) 0 0
\(934\) −1.13785e7 −0.426792
\(935\) 1.29740e7 0.485338
\(936\) 0 0
\(937\) −3.60574e6 −0.134167 −0.0670835 0.997747i \(-0.521369\pi\)
−0.0670835 + 0.997747i \(0.521369\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.97123e7 −1.09677
\(941\) −4.40826e6 −0.162290 −0.0811452 0.996702i \(-0.525858\pi\)
−0.0811452 + 0.996702i \(0.525858\pi\)
\(942\) 0 0
\(943\) −1.38596e7 −0.507542
\(944\) −6.74786e6 −0.246454
\(945\) 0 0
\(946\) −7.28706e6 −0.264743
\(947\) −7.36278e6 −0.266788 −0.133394 0.991063i \(-0.542588\pi\)
−0.133394 + 0.991063i \(0.542588\pi\)
\(948\) 0 0
\(949\) −1.55381e7 −0.560057
\(950\) −2.87804e7 −1.03464
\(951\) 0 0
\(952\) 0 0
\(953\) −3.73903e7 −1.33360 −0.666802 0.745235i \(-0.732337\pi\)
−0.666802 + 0.745235i \(0.732337\pi\)
\(954\) 0 0
\(955\) −3.66307e7 −1.29968
\(956\) 8.00982e6 0.283451
\(957\) 0 0
\(958\) −5.21462e6 −0.183573
\(959\) 0 0
\(960\) 0 0
\(961\) 5.29433e7 1.84928
\(962\) 1.08946e7 0.379555
\(963\) 0 0
\(964\) −1.93315e7 −0.669997
\(965\) 2.49742e7 0.863322
\(966\) 0 0
\(967\) 1.30730e7 0.449582 0.224791 0.974407i \(-0.427830\pi\)
0.224791 + 0.974407i \(0.427830\pi\)
\(968\) 8.87758e6 0.304513
\(969\) 0 0
\(970\) −9.76640e6 −0.333277
\(971\) −1.52454e7 −0.518908 −0.259454 0.965755i \(-0.583543\pi\)
−0.259454 + 0.965755i \(0.583543\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.49450e7 −0.842530
\(975\) 0 0
\(976\) −9.04237e6 −0.303849
\(977\) −3.08690e6 −0.103463 −0.0517316 0.998661i \(-0.516474\pi\)
−0.0517316 + 0.998661i \(0.516474\pi\)
\(978\) 0 0
\(979\) 1.47324e7 0.491264
\(980\) 0 0
\(981\) 0 0
\(982\) −1.57571e7 −0.521433
\(983\) 1.70201e7 0.561796 0.280898 0.959738i \(-0.409368\pi\)
0.280898 + 0.959738i \(0.409368\pi\)
\(984\) 0 0
\(985\) −1.22269e7 −0.401539
\(986\) 3.49109e6 0.114359
\(987\) 0 0
\(988\) −1.56270e7 −0.509311
\(989\) −2.21102e7 −0.718789
\(990\) 0 0
\(991\) −4.34672e7 −1.40597 −0.702987 0.711203i \(-0.748151\pi\)
−0.702987 + 0.711203i \(0.748151\pi\)
\(992\) 9.24851e6 0.298396
\(993\) 0 0
\(994\) 0 0
\(995\) 4.18199e6 0.133914
\(996\) 0 0
\(997\) 2.46672e7 0.785926 0.392963 0.919554i \(-0.371450\pi\)
0.392963 + 0.919554i \(0.371450\pi\)
\(998\) −3.19028e7 −1.01392
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.6.a.bb.1.1 2
3.2 odd 2 294.6.a.w.1.2 2
7.3 odd 6 126.6.g.h.37.1 4
7.5 odd 6 126.6.g.h.109.1 4
7.6 odd 2 882.6.a.bh.1.2 2
21.2 odd 6 294.6.e.s.67.1 4
21.5 even 6 42.6.e.c.25.2 4
21.11 odd 6 294.6.e.s.79.1 4
21.17 even 6 42.6.e.c.37.2 yes 4
21.20 even 2 294.6.a.r.1.1 2
84.47 odd 6 336.6.q.f.193.2 4
84.59 odd 6 336.6.q.f.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.c.25.2 4 21.5 even 6
42.6.e.c.37.2 yes 4 21.17 even 6
126.6.g.h.37.1 4 7.3 odd 6
126.6.g.h.109.1 4 7.5 odd 6
294.6.a.r.1.1 2 21.20 even 2
294.6.a.w.1.2 2 3.2 odd 2
294.6.e.s.67.1 4 21.2 odd 6
294.6.e.s.79.1 4 21.11 odd 6
336.6.q.f.193.2 4 84.47 odd 6
336.6.q.f.289.2 4 84.59 odd 6
882.6.a.bb.1.1 2 1.1 even 1 trivial
882.6.a.bh.1.2 2 7.6 odd 2