Properties

Label 441.6.a.y.1.1
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,6,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, 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("441.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.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: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{19}, \sqrt{69})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 71x^{2} + 72x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.705587\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.71780 q^{2} +44.0000 q^{4} -99.6795 q^{5} -104.614 q^{8} +O(q^{10})\) \(q-8.71780 q^{2} +44.0000 q^{4} -99.6795 q^{5} -104.614 q^{8} +868.986 q^{10} -374.865 q^{11} +868.986 q^{13} -496.000 q^{16} +1096.47 q^{17} -868.986 q^{19} -4385.90 q^{20} +3268.00 q^{22} -2972.77 q^{23} +6811.00 q^{25} -7575.64 q^{26} -4515.82 q^{29} -9558.84 q^{31} +7671.66 q^{32} -9558.84 q^{34} -5466.00 q^{37} +7575.64 q^{38} +10427.8 q^{40} -9868.27 q^{41} +12540.0 q^{43} -16494.1 q^{44} +25916.0 q^{46} +9967.95 q^{47} -59376.9 q^{50} +38235.4 q^{52} +15151.5 q^{53} +37366.4 q^{55} +39368.0 q^{58} -42662.8 q^{59} -47794.2 q^{61} +83332.1 q^{62} -51008.0 q^{64} -86620.0 q^{65} -29996.0 q^{67} +48244.9 q^{68} -61469.2 q^{71} -48663.2 q^{73} +47651.5 q^{74} -38235.4 q^{76} +80168.0 q^{79} +49441.0 q^{80} +86029.6 q^{82} -30701.3 q^{83} -109296. q^{85} -109321. q^{86} +39216.0 q^{88} +20833.0 q^{89} -130802. q^{92} -86898.6 q^{94} +86620.0 q^{95} -133824. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 176 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.71780 −1.54110 −0.770552 0.637377i \(-0.780019\pi\)
−0.770552 + 0.637377i \(0.780019\pi\)
\(3\) 0 0
\(4\) 44.0000 1.37500
\(5\) −99.6795 −1.78312 −0.891560 0.452902i \(-0.850389\pi\)
−0.891560 + 0.452902i \(0.850389\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −104.614 −0.577914
\(9\) 0 0
\(10\) 868.986 2.74797
\(11\) −374.865 −0.934100 −0.467050 0.884231i \(-0.654683\pi\)
−0.467050 + 0.884231i \(0.654683\pi\)
\(12\) 0 0
\(13\) 868.986 1.42611 0.713057 0.701106i \(-0.247310\pi\)
0.713057 + 0.701106i \(0.247310\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −496.000 −0.484375
\(17\) 1096.47 0.920187 0.460094 0.887870i \(-0.347816\pi\)
0.460094 + 0.887870i \(0.347816\pi\)
\(18\) 0 0
\(19\) −868.986 −0.552241 −0.276120 0.961123i \(-0.589049\pi\)
−0.276120 + 0.961123i \(0.589049\pi\)
\(20\) −4385.90 −2.45179
\(21\) 0 0
\(22\) 3268.00 1.43955
\(23\) −2972.77 −1.17177 −0.585884 0.810395i \(-0.699253\pi\)
−0.585884 + 0.810395i \(0.699253\pi\)
\(24\) 0 0
\(25\) 6811.00 2.17952
\(26\) −7575.64 −2.19779
\(27\) 0 0
\(28\) 0 0
\(29\) −4515.82 −0.997107 −0.498553 0.866859i \(-0.666135\pi\)
−0.498553 + 0.866859i \(0.666135\pi\)
\(30\) 0 0
\(31\) −9558.84 −1.78649 −0.893246 0.449568i \(-0.851578\pi\)
−0.893246 + 0.449568i \(0.851578\pi\)
\(32\) 7671.66 1.32439
\(33\) 0 0
\(34\) −9558.84 −1.41810
\(35\) 0 0
\(36\) 0 0
\(37\) −5466.00 −0.656395 −0.328198 0.944609i \(-0.606441\pi\)
−0.328198 + 0.944609i \(0.606441\pi\)
\(38\) 7575.64 0.851060
\(39\) 0 0
\(40\) 10427.8 1.03049
\(41\) −9868.27 −0.916814 −0.458407 0.888742i \(-0.651580\pi\)
−0.458407 + 0.888742i \(0.651580\pi\)
\(42\) 0 0
\(43\) 12540.0 1.03425 0.517126 0.855909i \(-0.327002\pi\)
0.517126 + 0.855909i \(0.327002\pi\)
\(44\) −16494.1 −1.28439
\(45\) 0 0
\(46\) 25916.0 1.80582
\(47\) 9967.95 0.658205 0.329102 0.944294i \(-0.393254\pi\)
0.329102 + 0.944294i \(0.393254\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −59376.9 −3.35887
\(51\) 0 0
\(52\) 38235.4 1.96091
\(53\) 15151.5 0.740912 0.370456 0.928850i \(-0.379201\pi\)
0.370456 + 0.928850i \(0.379201\pi\)
\(54\) 0 0
\(55\) 37366.4 1.66561
\(56\) 0 0
\(57\) 0 0
\(58\) 39368.0 1.53664
\(59\) −42662.8 −1.59558 −0.797792 0.602933i \(-0.793999\pi\)
−0.797792 + 0.602933i \(0.793999\pi\)
\(60\) 0 0
\(61\) −47794.2 −1.64456 −0.822282 0.569080i \(-0.807299\pi\)
−0.822282 + 0.569080i \(0.807299\pi\)
\(62\) 83332.1 2.75317
\(63\) 0 0
\(64\) −51008.0 −1.55664
\(65\) −86620.0 −2.54293
\(66\) 0 0
\(67\) −29996.0 −0.816350 −0.408175 0.912904i \(-0.633835\pi\)
−0.408175 + 0.912904i \(0.633835\pi\)
\(68\) 48244.9 1.26526
\(69\) 0 0
\(70\) 0 0
\(71\) −61469.2 −1.44714 −0.723572 0.690249i \(-0.757501\pi\)
−0.723572 + 0.690249i \(0.757501\pi\)
\(72\) 0 0
\(73\) −48663.2 −1.06879 −0.534396 0.845234i \(-0.679461\pi\)
−0.534396 + 0.845234i \(0.679461\pi\)
\(74\) 47651.5 1.01157
\(75\) 0 0
\(76\) −38235.4 −0.759331
\(77\) 0 0
\(78\) 0 0
\(79\) 80168.0 1.44522 0.722609 0.691257i \(-0.242943\pi\)
0.722609 + 0.691257i \(0.242943\pi\)
\(80\) 49441.0 0.863699
\(81\) 0 0
\(82\) 86029.6 1.41291
\(83\) −30701.3 −0.489172 −0.244586 0.969628i \(-0.578652\pi\)
−0.244586 + 0.969628i \(0.578652\pi\)
\(84\) 0 0
\(85\) −109296. −1.64080
\(86\) −109321. −1.59389
\(87\) 0 0
\(88\) 39216.0 0.539829
\(89\) 20833.0 0.278790 0.139395 0.990237i \(-0.455484\pi\)
0.139395 + 0.990237i \(0.455484\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −130802. −1.61118
\(93\) 0 0
\(94\) −86898.6 −1.01436
\(95\) 86620.0 0.984712
\(96\) 0 0
\(97\) −133824. −1.44412 −0.722061 0.691829i \(-0.756805\pi\)
−0.722061 + 0.691829i \(0.756805\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 299684. 2.99684
\(101\) −106358. −1.03745 −0.518725 0.854941i \(-0.673593\pi\)
−0.518725 + 0.854941i \(0.673593\pi\)
\(102\) 0 0
\(103\) 47794.2 0.443897 0.221949 0.975058i \(-0.428758\pi\)
0.221949 + 0.975058i \(0.428758\pi\)
\(104\) −90907.7 −0.824171
\(105\) 0 0
\(106\) −132088. −1.14182
\(107\) 102983. 0.869576 0.434788 0.900533i \(-0.356823\pi\)
0.434788 + 0.900533i \(0.356823\pi\)
\(108\) 0 0
\(109\) 73854.0 0.595399 0.297699 0.954660i \(-0.403781\pi\)
0.297699 + 0.954660i \(0.403781\pi\)
\(110\) −325753. −2.56688
\(111\) 0 0
\(112\) 0 0
\(113\) 29535.9 0.217598 0.108799 0.994064i \(-0.465300\pi\)
0.108799 + 0.994064i \(0.465300\pi\)
\(114\) 0 0
\(115\) 296324. 2.08940
\(116\) −198696. −1.37102
\(117\) 0 0
\(118\) 371926. 2.45896
\(119\) 0 0
\(120\) 0 0
\(121\) −20527.0 −0.127457
\(122\) 416660. 2.53444
\(123\) 0 0
\(124\) −420589. −2.45643
\(125\) −367419. −2.10323
\(126\) 0 0
\(127\) 175120. 0.963444 0.481722 0.876324i \(-0.340012\pi\)
0.481722 + 0.876324i \(0.340012\pi\)
\(128\) 199184. 1.07456
\(129\) 0 0
\(130\) 755136. 3.91892
\(131\) 144735. 0.736876 0.368438 0.929652i \(-0.379893\pi\)
0.368438 + 0.929652i \(0.379893\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 261499. 1.25808
\(135\) 0 0
\(136\) −114706. −0.531789
\(137\) 172229. 0.783979 0.391989 0.919970i \(-0.371787\pi\)
0.391989 + 0.919970i \(0.371787\pi\)
\(138\) 0 0
\(139\) −19117.7 −0.0839263 −0.0419632 0.999119i \(-0.513361\pi\)
−0.0419632 + 0.999119i \(0.513361\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 535876. 2.23020
\(143\) −325753. −1.33213
\(144\) 0 0
\(145\) 450135. 1.77796
\(146\) 424236. 1.64712
\(147\) 0 0
\(148\) −240504. −0.902543
\(149\) 9118.82 0.0336491 0.0168245 0.999858i \(-0.494644\pi\)
0.0168245 + 0.999858i \(0.494644\pi\)
\(150\) 0 0
\(151\) 310200. 1.10713 0.553566 0.832805i \(-0.313267\pi\)
0.553566 + 0.832805i \(0.313267\pi\)
\(152\) 90907.7 0.319148
\(153\) 0 0
\(154\) 0 0
\(155\) 952820. 3.18553
\(156\) 0 0
\(157\) −258089. −0.835641 −0.417821 0.908530i \(-0.637206\pi\)
−0.417821 + 0.908530i \(0.637206\pi\)
\(158\) −698888. −2.22723
\(159\) 0 0
\(160\) −764707. −2.36154
\(161\) 0 0
\(162\) 0 0
\(163\) 126556. 0.373090 0.186545 0.982446i \(-0.440271\pi\)
0.186545 + 0.982446i \(0.440271\pi\)
\(164\) −434204. −1.26062
\(165\) 0 0
\(166\) 267648. 0.753864
\(167\) 342100. 0.949209 0.474605 0.880199i \(-0.342591\pi\)
0.474605 + 0.880199i \(0.342591\pi\)
\(168\) 0 0
\(169\) 383843. 1.03380
\(170\) 952820. 2.52865
\(171\) 0 0
\(172\) 551760. 1.42210
\(173\) 187497. 0.476299 0.238149 0.971229i \(-0.423459\pi\)
0.238149 + 0.971229i \(0.423459\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 185933. 0.452455
\(177\) 0 0
\(178\) −181618. −0.429644
\(179\) 127829. 0.298193 0.149096 0.988823i \(-0.452364\pi\)
0.149096 + 0.988823i \(0.452364\pi\)
\(180\) 0 0
\(181\) 219853. 0.498812 0.249406 0.968399i \(-0.419765\pi\)
0.249406 + 0.968399i \(0.419765\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 310992. 0.677181
\(185\) 544848. 1.17043
\(186\) 0 0
\(187\) −411030. −0.859547
\(188\) 438590. 0.905032
\(189\) 0 0
\(190\) −755136. −1.51754
\(191\) 116130. 0.230335 0.115168 0.993346i \(-0.463259\pi\)
0.115168 + 0.993346i \(0.463259\pi\)
\(192\) 0 0
\(193\) −306306. −0.591919 −0.295959 0.955200i \(-0.595639\pi\)
−0.295959 + 0.955200i \(0.595639\pi\)
\(194\) 1.16665e6 2.22554
\(195\) 0 0
\(196\) 0 0
\(197\) 360202. 0.661273 0.330636 0.943758i \(-0.392737\pi\)
0.330636 + 0.943758i \(0.392737\pi\)
\(198\) 0 0
\(199\) 229412. 0.410661 0.205331 0.978693i \(-0.434173\pi\)
0.205331 + 0.978693i \(0.434173\pi\)
\(200\) −712523. −1.25957
\(201\) 0 0
\(202\) 927208. 1.59882
\(203\) 0 0
\(204\) 0 0
\(205\) 983664. 1.63479
\(206\) −416660. −0.684091
\(207\) 0 0
\(208\) −431017. −0.690774
\(209\) 325753. 0.515848
\(210\) 0 0
\(211\) −1.21031e6 −1.87150 −0.935750 0.352664i \(-0.885276\pi\)
−0.935750 + 0.352664i \(0.885276\pi\)
\(212\) 666667. 1.01875
\(213\) 0 0
\(214\) −897788. −1.34011
\(215\) −1.24998e6 −1.84420
\(216\) 0 0
\(217\) 0 0
\(218\) −643844. −0.917571
\(219\) 0 0
\(220\) 1.64412e6 2.29022
\(221\) 952820. 1.31229
\(222\) 0 0
\(223\) 745590. 1.00401 0.502005 0.864865i \(-0.332596\pi\)
0.502005 + 0.864865i \(0.332596\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −257488. −0.335340
\(227\) −140349. −0.180777 −0.0903886 0.995907i \(-0.528811\pi\)
−0.0903886 + 0.995907i \(0.528811\pi\)
\(228\) 0 0
\(229\) −774266. −0.975667 −0.487833 0.872937i \(-0.662213\pi\)
−0.487833 + 0.872937i \(0.662213\pi\)
\(230\) −2.58329e6 −3.21999
\(231\) 0 0
\(232\) 472416. 0.576242
\(233\) −1.03550e6 −1.24957 −0.624785 0.780797i \(-0.714813\pi\)
−0.624785 + 0.780797i \(0.714813\pi\)
\(234\) 0 0
\(235\) −993600. −1.17366
\(236\) −1.87716e6 −2.19393
\(237\) 0 0
\(238\) 0 0
\(239\) 1.11622e6 1.26402 0.632011 0.774960i \(-0.282230\pi\)
0.632011 + 0.774960i \(0.282230\pi\)
\(240\) 0 0
\(241\) −1.44425e6 −1.60177 −0.800887 0.598816i \(-0.795638\pi\)
−0.800887 + 0.598816i \(0.795638\pi\)
\(242\) 178950. 0.196424
\(243\) 0 0
\(244\) −2.10295e6 −2.26128
\(245\) 0 0
\(246\) 0 0
\(247\) −755136. −0.787558
\(248\) 999985. 1.03244
\(249\) 0 0
\(250\) 3.20308e6 3.24129
\(251\) 216105. 0.216511 0.108256 0.994123i \(-0.465473\pi\)
0.108256 + 0.994123i \(0.465473\pi\)
\(252\) 0 0
\(253\) 1.11439e6 1.09455
\(254\) −1.52666e6 −1.48477
\(255\) 0 0
\(256\) −104192. −0.0993652
\(257\) 1.28517e6 1.21374 0.606872 0.794800i \(-0.292424\pi\)
0.606872 + 0.794800i \(0.292424\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −3.81128e6 −3.49653
\(261\) 0 0
\(262\) −1.26177e6 −1.13560
\(263\) 1.47794e6 1.31755 0.658774 0.752341i \(-0.271075\pi\)
0.658774 + 0.752341i \(0.271075\pi\)
\(264\) 0 0
\(265\) −1.51030e6 −1.32114
\(266\) 0 0
\(267\) 0 0
\(268\) −1.31982e6 −1.12248
\(269\) 2.28176e6 1.92260 0.961302 0.275497i \(-0.0888423\pi\)
0.961302 + 0.275497i \(0.0888423\pi\)
\(270\) 0 0
\(271\) −1.09058e6 −0.902055 −0.451028 0.892510i \(-0.648942\pi\)
−0.451028 + 0.892510i \(0.648942\pi\)
\(272\) −543851. −0.445716
\(273\) 0 0
\(274\) −1.50146e6 −1.20819
\(275\) −2.55321e6 −2.03589
\(276\) 0 0
\(277\) −256234. −0.200649 −0.100325 0.994955i \(-0.531988\pi\)
−0.100325 + 0.994955i \(0.531988\pi\)
\(278\) 166664. 0.129339
\(279\) 0 0
\(280\) 0 0
\(281\) −889041. −0.671670 −0.335835 0.941921i \(-0.609018\pi\)
−0.335835 + 0.941921i \(0.609018\pi\)
\(282\) 0 0
\(283\) −496191. −0.368284 −0.184142 0.982900i \(-0.558951\pi\)
−0.184142 + 0.982900i \(0.558951\pi\)
\(284\) −2.70464e6 −1.98982
\(285\) 0 0
\(286\) 2.83984e6 2.05296
\(287\) 0 0
\(288\) 0 0
\(289\) −217601. −0.153256
\(290\) −3.92418e6 −2.74002
\(291\) 0 0
\(292\) −2.14118e6 −1.46959
\(293\) 731348. 0.497686 0.248843 0.968544i \(-0.419950\pi\)
0.248843 + 0.968544i \(0.419950\pi\)
\(294\) 0 0
\(295\) 4.25261e6 2.84512
\(296\) 571818. 0.379340
\(297\) 0 0
\(298\) −79496.0 −0.0518567
\(299\) −2.58329e6 −1.67107
\(300\) 0 0
\(301\) 0 0
\(302\) −2.70426e6 −1.70621
\(303\) 0 0
\(304\) 431017. 0.267492
\(305\) 4.76410e6 2.93246
\(306\) 0 0
\(307\) 66911.9 0.0405189 0.0202594 0.999795i \(-0.493551\pi\)
0.0202594 + 0.999795i \(0.493551\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.30650e6 −4.90923
\(311\) 302627. 0.177422 0.0887108 0.996057i \(-0.471725\pi\)
0.0887108 + 0.996057i \(0.471725\pi\)
\(312\) 0 0
\(313\) 1.01324e6 0.584589 0.292294 0.956328i \(-0.405581\pi\)
0.292294 + 0.956328i \(0.405581\pi\)
\(314\) 2.24997e6 1.28781
\(315\) 0 0
\(316\) 3.52739e6 1.98717
\(317\) −638474. −0.356858 −0.178429 0.983953i \(-0.557101\pi\)
−0.178429 + 0.983953i \(0.557101\pi\)
\(318\) 0 0
\(319\) 1.69282e6 0.931398
\(320\) 5.08445e6 2.77568
\(321\) 0 0
\(322\) 0 0
\(323\) −952820. −0.508165
\(324\) 0 0
\(325\) 5.91866e6 3.10824
\(326\) −1.10329e6 −0.574970
\(327\) 0 0
\(328\) 1.03235e6 0.529840
\(329\) 0 0
\(330\) 0 0
\(331\) 2.63905e6 1.32397 0.661985 0.749517i \(-0.269714\pi\)
0.661985 + 0.749517i \(0.269714\pi\)
\(332\) −1.35086e6 −0.672611
\(333\) 0 0
\(334\) −2.98236e6 −1.46283
\(335\) 2.98999e6 1.45565
\(336\) 0 0
\(337\) −455070. −0.218275 −0.109137 0.994027i \(-0.534809\pi\)
−0.109137 + 0.994027i \(0.534809\pi\)
\(338\) −3.34627e6 −1.59319
\(339\) 0 0
\(340\) −4.80902e6 −2.25611
\(341\) 3.58328e6 1.66876
\(342\) 0 0
\(343\) 0 0
\(344\) −1.31185e6 −0.597709
\(345\) 0 0
\(346\) −1.63456e6 −0.734025
\(347\) 1.30756e6 0.582957 0.291479 0.956577i \(-0.405853\pi\)
0.291479 + 0.956577i \(0.405853\pi\)
\(348\) 0 0
\(349\) 1.05234e6 0.462480 0.231240 0.972897i \(-0.425722\pi\)
0.231240 + 0.972897i \(0.425722\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.87584e6 −1.23711
\(353\) 1.17333e6 0.501167 0.250583 0.968095i \(-0.419378\pi\)
0.250583 + 0.968095i \(0.419378\pi\)
\(354\) 0 0
\(355\) 6.12722e6 2.58043
\(356\) 916653. 0.383336
\(357\) 0 0
\(358\) −1.11439e6 −0.459546
\(359\) 2.98746e6 1.22339 0.611696 0.791093i \(-0.290488\pi\)
0.611696 + 0.791093i \(0.290488\pi\)
\(360\) 0 0
\(361\) −1.72096e6 −0.695030
\(362\) −1.91664e6 −0.768721
\(363\) 0 0
\(364\) 0 0
\(365\) 4.85072e6 1.90579
\(366\) 0 0
\(367\) −2.98236e6 −1.15583 −0.577916 0.816096i \(-0.696134\pi\)
−0.577916 + 0.816096i \(0.696134\pi\)
\(368\) 1.47449e6 0.567575
\(369\) 0 0
\(370\) −4.74988e6 −1.80376
\(371\) 0 0
\(372\) 0 0
\(373\) −1.58662e6 −0.590473 −0.295237 0.955424i \(-0.595398\pi\)
−0.295237 + 0.955424i \(0.595398\pi\)
\(374\) 3.58328e6 1.32465
\(375\) 0 0
\(376\) −1.04278e6 −0.380386
\(377\) −3.92418e6 −1.42199
\(378\) 0 0
\(379\) 1.27168e6 0.454756 0.227378 0.973807i \(-0.426985\pi\)
0.227378 + 0.973807i \(0.426985\pi\)
\(380\) 3.81128e6 1.35398
\(381\) 0 0
\(382\) −1.01240e6 −0.354970
\(383\) −5.13589e6 −1.78903 −0.894517 0.447035i \(-0.852480\pi\)
−0.894517 + 0.447035i \(0.852480\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.67031e6 0.912208
\(387\) 0 0
\(388\) −5.88825e6 −1.98567
\(389\) −4.92655e6 −1.65070 −0.825351 0.564619i \(-0.809023\pi\)
−0.825351 + 0.564619i \(0.809023\pi\)
\(390\) 0 0
\(391\) −3.25957e6 −1.07825
\(392\) 0 0
\(393\) 0 0
\(394\) −3.14017e6 −1.01909
\(395\) −7.99111e6 −2.57700
\(396\) 0 0
\(397\) −219853. −0.0700095 −0.0350047 0.999387i \(-0.511145\pi\)
−0.0350047 + 0.999387i \(0.511145\pi\)
\(398\) −1.99997e6 −0.632872
\(399\) 0 0
\(400\) −3.37826e6 −1.05570
\(401\) −3.18796e6 −0.990038 −0.495019 0.868882i \(-0.664839\pi\)
−0.495019 + 0.868882i \(0.664839\pi\)
\(402\) 0 0
\(403\) −8.30650e6 −2.54774
\(404\) −4.67975e6 −1.42649
\(405\) 0 0
\(406\) 0 0
\(407\) 2.04901e6 0.613139
\(408\) 0 0
\(409\) −2.54265e6 −0.751586 −0.375793 0.926704i \(-0.622630\pi\)
−0.375793 + 0.926704i \(0.622630\pi\)
\(410\) −8.57538e6 −2.51938
\(411\) 0 0
\(412\) 2.10295e6 0.610358
\(413\) 0 0
\(414\) 0 0
\(415\) 3.06029e6 0.872252
\(416\) 6.66656e6 1.88873
\(417\) 0 0
\(418\) −2.83984e6 −0.794976
\(419\) −3.57132e6 −0.993787 −0.496893 0.867812i \(-0.665526\pi\)
−0.496893 + 0.867812i \(0.665526\pi\)
\(420\) 0 0
\(421\) 991750. 0.272707 0.136354 0.990660i \(-0.456462\pi\)
0.136354 + 0.990660i \(0.456462\pi\)
\(422\) 1.05512e7 2.88417
\(423\) 0 0
\(424\) −1.58506e6 −0.428184
\(425\) 7.46809e6 2.00557
\(426\) 0 0
\(427\) 0 0
\(428\) 4.53127e6 1.19567
\(429\) 0 0
\(430\) 1.08971e7 2.84210
\(431\) 3.40602e6 0.883189 0.441595 0.897215i \(-0.354413\pi\)
0.441595 + 0.897215i \(0.354413\pi\)
\(432\) 0 0
\(433\) 7.74266e6 1.98459 0.992294 0.123902i \(-0.0395409\pi\)
0.992294 + 0.123902i \(0.0395409\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.24958e6 0.818673
\(437\) 2.58329e6 0.647098
\(438\) 0 0
\(439\) −6.31753e6 −1.56454 −0.782268 0.622942i \(-0.785937\pi\)
−0.782268 + 0.622942i \(0.785937\pi\)
\(440\) −3.90903e6 −0.962581
\(441\) 0 0
\(442\) −8.30650e6 −2.02238
\(443\) 485712. 0.117590 0.0587949 0.998270i \(-0.481274\pi\)
0.0587949 + 0.998270i \(0.481274\pi\)
\(444\) 0 0
\(445\) −2.07662e6 −0.497116
\(446\) −6.49990e6 −1.54728
\(447\) 0 0
\(448\) 0 0
\(449\) 5.18374e6 1.21347 0.606733 0.794906i \(-0.292480\pi\)
0.606733 + 0.794906i \(0.292480\pi\)
\(450\) 0 0
\(451\) 3.69927e6 0.856396
\(452\) 1.29958e6 0.299197
\(453\) 0 0
\(454\) 1.22353e6 0.278596
\(455\) 0 0
\(456\) 0 0
\(457\) 6.55690e6 1.46862 0.734308 0.678817i \(-0.237507\pi\)
0.734308 + 0.678817i \(0.237507\pi\)
\(458\) 6.74990e6 1.50360
\(459\) 0 0
\(460\) 1.30383e7 2.87293
\(461\) −4.15893e6 −0.911442 −0.455721 0.890123i \(-0.650619\pi\)
−0.455721 + 0.890123i \(0.650619\pi\)
\(462\) 0 0
\(463\) 5.62854e6 1.22023 0.610117 0.792311i \(-0.291123\pi\)
0.610117 + 0.792311i \(0.291123\pi\)
\(464\) 2.23985e6 0.482973
\(465\) 0 0
\(466\) 9.02728e6 1.92572
\(467\) −4.20727e6 −0.892706 −0.446353 0.894857i \(-0.647277\pi\)
−0.446353 + 0.894857i \(0.647277\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.66200e6 1.80873
\(471\) 0 0
\(472\) 4.46311e6 0.922110
\(473\) −4.70081e6 −0.966095
\(474\) 0 0
\(475\) −5.91866e6 −1.20362
\(476\) 0 0
\(477\) 0 0
\(478\) −9.73096e6 −1.94799
\(479\) −7.10954e6 −1.41580 −0.707901 0.706311i \(-0.750358\pi\)
−0.707901 + 0.706311i \(0.750358\pi\)
\(480\) 0 0
\(481\) −4.74988e6 −0.936094
\(482\) 1.25907e7 2.46850
\(483\) 0 0
\(484\) −903188. −0.175253
\(485\) 1.33395e7 2.57504
\(486\) 0 0
\(487\) 7.03615e6 1.34435 0.672176 0.740392i \(-0.265360\pi\)
0.672176 + 0.740392i \(0.265360\pi\)
\(488\) 4.99992e6 0.950416
\(489\) 0 0
\(490\) 0 0
\(491\) 1.08668e6 0.203422 0.101711 0.994814i \(-0.467568\pi\)
0.101711 + 0.994814i \(0.467568\pi\)
\(492\) 0 0
\(493\) −4.95148e6 −0.917525
\(494\) 6.58312e6 1.21371
\(495\) 0 0
\(496\) 4.74119e6 0.865332
\(497\) 0 0
\(498\) 0 0
\(499\) 3.05896e6 0.549948 0.274974 0.961452i \(-0.411331\pi\)
0.274974 + 0.961452i \(0.411331\pi\)
\(500\) −1.61664e7 −2.89194
\(501\) 0 0
\(502\) −1.88396e6 −0.333666
\(503\) 6.23236e6 1.09833 0.549165 0.835714i \(-0.314946\pi\)
0.549165 + 0.835714i \(0.314946\pi\)
\(504\) 0 0
\(505\) 1.06017e7 1.84990
\(506\) −9.71501e6 −1.68681
\(507\) 0 0
\(508\) 7.70528e6 1.32474
\(509\) 6.14574e6 1.05143 0.525714 0.850661i \(-0.323798\pi\)
0.525714 + 0.850661i \(0.323798\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −5.46557e6 −0.921426
\(513\) 0 0
\(514\) −1.12038e7 −1.87050
\(515\) −4.76410e6 −0.791522
\(516\) 0 0
\(517\) −3.73664e6 −0.614829
\(518\) 0 0
\(519\) 0 0
\(520\) 9.06163e6 1.46960
\(521\) 6.78050e6 1.09438 0.547189 0.837009i \(-0.315698\pi\)
0.547189 + 0.837009i \(0.315698\pi\)
\(522\) 0 0
\(523\) −1.90134e6 −0.303952 −0.151976 0.988384i \(-0.548564\pi\)
−0.151976 + 0.988384i \(0.548564\pi\)
\(524\) 6.36832e6 1.01320
\(525\) 0 0
\(526\) −1.28844e7 −2.03048
\(527\) −1.04810e7 −1.64391
\(528\) 0 0
\(529\) 2.40101e6 0.373040
\(530\) 1.31665e7 2.03601
\(531\) 0 0
\(532\) 0 0
\(533\) −8.57538e6 −1.30748
\(534\) 0 0
\(535\) −1.02653e7 −1.55056
\(536\) 3.13799e6 0.471780
\(537\) 0 0
\(538\) −1.98919e7 −2.96293
\(539\) 0 0
\(540\) 0 0
\(541\) 9.04950e6 1.32933 0.664663 0.747143i \(-0.268575\pi\)
0.664663 + 0.747143i \(0.268575\pi\)
\(542\) 9.50743e6 1.39016
\(543\) 0 0
\(544\) 8.41178e6 1.21868
\(545\) −7.36173e6 −1.06167
\(546\) 0 0
\(547\) 1.91721e6 0.273969 0.136985 0.990573i \(-0.456259\pi\)
0.136985 + 0.990573i \(0.456259\pi\)
\(548\) 7.57807e6 1.07797
\(549\) 0 0
\(550\) 2.22583e7 3.13752
\(551\) 3.92418e6 0.550643
\(552\) 0 0
\(553\) 0 0
\(554\) 2.23380e6 0.309221
\(555\) 0 0
\(556\) −841178. −0.115399
\(557\) −2.62380e6 −0.358337 −0.179169 0.983818i \(-0.557341\pi\)
−0.179169 + 0.983818i \(0.557341\pi\)
\(558\) 0 0
\(559\) 1.08971e7 1.47496
\(560\) 0 0
\(561\) 0 0
\(562\) 7.75048e6 1.03511
\(563\) −1.07498e7 −1.42932 −0.714662 0.699470i \(-0.753420\pi\)
−0.714662 + 0.699470i \(0.753420\pi\)
\(564\) 0 0
\(565\) −2.94412e6 −0.388003
\(566\) 4.32569e6 0.567563
\(567\) 0 0
\(568\) 6.43051e6 0.836324
\(569\) −914671. −0.118436 −0.0592181 0.998245i \(-0.518861\pi\)
−0.0592181 + 0.998245i \(0.518861\pi\)
\(570\) 0 0
\(571\) 5.24616e6 0.673367 0.336683 0.941618i \(-0.390695\pi\)
0.336683 + 0.941618i \(0.390695\pi\)
\(572\) −1.43331e7 −1.83168
\(573\) 0 0
\(574\) 0 0
\(575\) −2.02475e7 −2.55389
\(576\) 0 0
\(577\) 1.15471e7 1.44389 0.721943 0.691953i \(-0.243249\pi\)
0.721943 + 0.691953i \(0.243249\pi\)
\(578\) 1.89700e6 0.236183
\(579\) 0 0
\(580\) 1.98059e7 2.44470
\(581\) 0 0
\(582\) 0 0
\(583\) −5.67978e6 −0.692087
\(584\) 5.09083e6 0.617670
\(585\) 0 0
\(586\) −6.37575e6 −0.766985
\(587\) 5.01428e6 0.600638 0.300319 0.953839i \(-0.402907\pi\)
0.300319 + 0.953839i \(0.402907\pi\)
\(588\) 0 0
\(589\) 8.30650e6 0.986574
\(590\) −3.70734e7 −4.38462
\(591\) 0 0
\(592\) 2.71114e6 0.317941
\(593\) −4.82778e6 −0.563781 −0.281890 0.959447i \(-0.590961\pi\)
−0.281890 + 0.959447i \(0.590961\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 401228. 0.0462674
\(597\) 0 0
\(598\) 2.25206e7 2.57530
\(599\) −9.24522e6 −1.05281 −0.526405 0.850234i \(-0.676460\pi\)
−0.526405 + 0.850234i \(0.676460\pi\)
\(600\) 0 0
\(601\) −3.60455e6 −0.407066 −0.203533 0.979068i \(-0.565242\pi\)
−0.203533 + 0.979068i \(0.565242\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.36488e7 1.52231
\(605\) 2.04612e6 0.227270
\(606\) 0 0
\(607\) 7.34988e6 0.809671 0.404835 0.914390i \(-0.367329\pi\)
0.404835 + 0.914390i \(0.367329\pi\)
\(608\) −6.66656e6 −0.731380
\(609\) 0 0
\(610\) −4.15325e7 −4.51922
\(611\) 8.66200e6 0.938675
\(612\) 0 0
\(613\) −1.52374e7 −1.63780 −0.818898 0.573940i \(-0.805414\pi\)
−0.818898 + 0.573940i \(0.805414\pi\)
\(614\) −583324. −0.0624438
\(615\) 0 0
\(616\) 0 0
\(617\) −1.13748e7 −1.20290 −0.601450 0.798910i \(-0.705410\pi\)
−0.601450 + 0.798910i \(0.705410\pi\)
\(618\) 0 0
\(619\) 1.50074e7 1.57427 0.787133 0.616783i \(-0.211564\pi\)
0.787133 + 0.616783i \(0.211564\pi\)
\(620\) 4.19241e7 4.38010
\(621\) 0 0
\(622\) −2.63824e6 −0.273425
\(623\) 0 0
\(624\) 0 0
\(625\) 1.53397e7 1.57079
\(626\) −8.83320e6 −0.900911
\(627\) 0 0
\(628\) −1.13559e7 −1.14901
\(629\) −5.99333e6 −0.604006
\(630\) 0 0
\(631\) −4.06654e6 −0.406586 −0.203293 0.979118i \(-0.565164\pi\)
−0.203293 + 0.979118i \(0.565164\pi\)
\(632\) −8.38666e6 −0.835211
\(633\) 0 0
\(634\) 5.56609e6 0.549955
\(635\) −1.74559e7 −1.71794
\(636\) 0 0
\(637\) 0 0
\(638\) −1.47577e7 −1.43538
\(639\) 0 0
\(640\) −1.98546e7 −1.91607
\(641\) 9.79786e6 0.941860 0.470930 0.882171i \(-0.343918\pi\)
0.470930 + 0.882171i \(0.343918\pi\)
\(642\) 0 0
\(643\) −678678. −0.0647346 −0.0323673 0.999476i \(-0.510305\pi\)
−0.0323673 + 0.999476i \(0.510305\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.30650e6 0.783135
\(647\) 1.51545e7 1.42325 0.711623 0.702561i \(-0.247960\pi\)
0.711623 + 0.702561i \(0.247960\pi\)
\(648\) 0 0
\(649\) 1.59928e7 1.49044
\(650\) −5.15977e7 −4.79013
\(651\) 0 0
\(652\) 5.56846e6 0.512999
\(653\) −6.64999e6 −0.610292 −0.305146 0.952306i \(-0.598705\pi\)
−0.305146 + 0.952306i \(0.598705\pi\)
\(654\) 0 0
\(655\) −1.44271e7 −1.31394
\(656\) 4.89466e6 0.444082
\(657\) 0 0
\(658\) 0 0
\(659\) −9.20368e6 −0.825559 −0.412780 0.910831i \(-0.635442\pi\)
−0.412780 + 0.910831i \(0.635442\pi\)
\(660\) 0 0
\(661\) −1.24361e7 −1.10708 −0.553540 0.832823i \(-0.686723\pi\)
−0.553540 + 0.832823i \(0.686723\pi\)
\(662\) −2.30067e7 −2.04037
\(663\) 0 0
\(664\) 3.21177e6 0.282699
\(665\) 0 0
\(666\) 0 0
\(667\) 1.34245e7 1.16838
\(668\) 1.50524e7 1.30516
\(669\) 0 0
\(670\) −2.60661e7 −2.24331
\(671\) 1.79164e7 1.53619
\(672\) 0 0
\(673\) −9.53297e6 −0.811317 −0.405659 0.914025i \(-0.632958\pi\)
−0.405659 + 0.914025i \(0.632958\pi\)
\(674\) 3.96721e6 0.336384
\(675\) 0 0
\(676\) 1.68891e7 1.42148
\(677\) 6.45494e6 0.541279 0.270639 0.962681i \(-0.412765\pi\)
0.270639 + 0.962681i \(0.412765\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 1.14338e7 0.948244
\(681\) 0 0
\(682\) −3.12383e7 −2.57174
\(683\) −4.39884e6 −0.360816 −0.180408 0.983592i \(-0.557742\pi\)
−0.180408 + 0.983592i \(0.557742\pi\)
\(684\) 0 0
\(685\) −1.71677e7 −1.39793
\(686\) 0 0
\(687\) 0 0
\(688\) −6.21984e6 −0.500966
\(689\) 1.31665e7 1.05663
\(690\) 0 0
\(691\) 2.30559e7 1.83691 0.918454 0.395528i \(-0.129438\pi\)
0.918454 + 0.395528i \(0.129438\pi\)
\(692\) 8.24987e6 0.654911
\(693\) 0 0
\(694\) −1.13990e7 −0.898397
\(695\) 1.90564e6 0.149651
\(696\) 0 0
\(697\) −1.08203e7 −0.843641
\(698\) −9.17410e6 −0.712730
\(699\) 0 0
\(700\) 0 0
\(701\) −1.69755e7 −1.30475 −0.652374 0.757898i \(-0.726227\pi\)
−0.652374 + 0.757898i \(0.726227\pi\)
\(702\) 0 0
\(703\) 4.74988e6 0.362488
\(704\) 1.91211e7 1.45406
\(705\) 0 0
\(706\) −1.02288e7 −0.772350
\(707\) 0 0
\(708\) 0 0
\(709\) 3.61369e6 0.269982 0.134991 0.990847i \(-0.456899\pi\)
0.134991 + 0.990847i \(0.456899\pi\)
\(710\) −5.34158e7 −3.97671
\(711\) 0 0
\(712\) −2.17942e6 −0.161117
\(713\) 2.84162e7 2.09335
\(714\) 0 0
\(715\) 3.24708e7 2.37536
\(716\) 5.62448e6 0.410015
\(717\) 0 0
\(718\) −2.60441e7 −1.88537
\(719\) −5.66778e6 −0.408875 −0.204437 0.978880i \(-0.565536\pi\)
−0.204437 + 0.978880i \(0.565536\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.50030e7 1.07111
\(723\) 0 0
\(724\) 9.67355e6 0.685866
\(725\) −3.07572e7 −2.17321
\(726\) 0 0
\(727\) 1.72919e7 1.21341 0.606705 0.794927i \(-0.292491\pi\)
0.606705 + 0.794927i \(0.292491\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −4.22876e7 −2.93701
\(731\) 1.37498e7 0.951705
\(732\) 0 0
\(733\) 3.37427e6 0.231964 0.115982 0.993251i \(-0.462999\pi\)
0.115982 + 0.993251i \(0.462999\pi\)
\(734\) 2.59996e7 1.78126
\(735\) 0 0
\(736\) −2.28061e7 −1.55187
\(737\) 1.12445e7 0.762553
\(738\) 0 0
\(739\) 2.16148e7 1.45593 0.727964 0.685616i \(-0.240467\pi\)
0.727964 + 0.685616i \(0.240467\pi\)
\(740\) 2.39733e7 1.60934
\(741\) 0 0
\(742\) 0 0
\(743\) 1.51151e7 1.00447 0.502237 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(744\) 0 0
\(745\) −908959. −0.0600003
\(746\) 1.38318e7 0.909981
\(747\) 0 0
\(748\) −1.80853e7 −1.18188
\(749\) 0 0
\(750\) 0 0
\(751\) 2.45624e7 1.58917 0.794587 0.607151i \(-0.207688\pi\)
0.794587 + 0.607151i \(0.207688\pi\)
\(752\) −4.94410e6 −0.318818
\(753\) 0 0
\(754\) 3.42102e7 2.19143
\(755\) −3.09206e7 −1.97415
\(756\) 0 0
\(757\) 2.02916e6 0.128699 0.0643496 0.997927i \(-0.479503\pi\)
0.0643496 + 0.997927i \(0.479503\pi\)
\(758\) −1.10862e7 −0.700826
\(759\) 0 0
\(760\) −9.06163e6 −0.569079
\(761\) 5.94179e6 0.371926 0.185963 0.982557i \(-0.440460\pi\)
0.185963 + 0.982557i \(0.440460\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 5.10971e6 0.316711
\(765\) 0 0
\(766\) 4.47736e7 2.75709
\(767\) −3.70734e7 −2.27548
\(768\) 0 0
\(769\) 1.07928e6 0.0658140 0.0329070 0.999458i \(-0.489523\pi\)
0.0329070 + 0.999458i \(0.489523\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.34775e7 −0.813889
\(773\) 1.84656e6 0.111151 0.0555757 0.998454i \(-0.482301\pi\)
0.0555757 + 0.998454i \(0.482301\pi\)
\(774\) 0 0
\(775\) −6.51053e7 −3.89369
\(776\) 1.39998e7 0.834578
\(777\) 0 0
\(778\) 4.29487e7 2.54390
\(779\) 8.57538e6 0.506302
\(780\) 0 0
\(781\) 2.30427e7 1.35178
\(782\) 2.84162e7 1.66169
\(783\) 0 0
\(784\) 0 0
\(785\) 2.57262e7 1.49005
\(786\) 0 0
\(787\) 1.32764e7 0.764086 0.382043 0.924145i \(-0.375221\pi\)
0.382043 + 0.924145i \(0.375221\pi\)
\(788\) 1.58489e7 0.909250
\(789\) 0 0
\(790\) 6.96648e7 3.97142
\(791\) 0 0
\(792\) 0 0
\(793\) −4.15325e7 −2.34534
\(794\) 1.91664e6 0.107892
\(795\) 0 0
\(796\) 1.00941e7 0.564659
\(797\) 8.63793e6 0.481686 0.240843 0.970564i \(-0.422576\pi\)
0.240843 + 0.970564i \(0.422576\pi\)
\(798\) 0 0
\(799\) 1.09296e7 0.605672
\(800\) 5.22517e7 2.88653
\(801\) 0 0
\(802\) 2.77920e7 1.52575
\(803\) 1.82421e7 0.998360
\(804\) 0 0
\(805\) 0 0
\(806\) 7.24144e7 3.92633
\(807\) 0 0
\(808\) 1.11265e7 0.599556
\(809\) 2.39614e6 0.128718 0.0643592 0.997927i \(-0.479500\pi\)
0.0643592 + 0.997927i \(0.479500\pi\)
\(810\) 0 0
\(811\) −6.79547e6 −0.362800 −0.181400 0.983409i \(-0.558063\pi\)
−0.181400 + 0.983409i \(0.558063\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.78629e7 −0.944910
\(815\) −1.26150e7 −0.665265
\(816\) 0 0
\(817\) −1.08971e7 −0.571156
\(818\) 2.21663e7 1.15827
\(819\) 0 0
\(820\) 4.32812e7 2.24784
\(821\) 2.64758e6 0.137085 0.0685426 0.997648i \(-0.478165\pi\)
0.0685426 + 0.997648i \(0.478165\pi\)
\(822\) 0 0
\(823\) −2.40725e7 −1.23886 −0.619430 0.785052i \(-0.712636\pi\)
−0.619430 + 0.785052i \(0.712636\pi\)
\(824\) −4.99992e6 −0.256534
\(825\) 0 0
\(826\) 0 0
\(827\) 1.55007e6 0.0788110 0.0394055 0.999223i \(-0.487454\pi\)
0.0394055 + 0.999223i \(0.487454\pi\)
\(828\) 0 0
\(829\) −2.64111e7 −1.33475 −0.667375 0.744722i \(-0.732582\pi\)
−0.667375 + 0.744722i \(0.732582\pi\)
\(830\) −2.66790e7 −1.34423
\(831\) 0 0
\(832\) −4.43252e7 −2.21995
\(833\) 0 0
\(834\) 0 0
\(835\) −3.41004e7 −1.69255
\(836\) 1.43331e7 0.709292
\(837\) 0 0
\(838\) 3.11340e7 1.53153
\(839\) 1.47573e7 0.723775 0.361887 0.932222i \(-0.382132\pi\)
0.361887 + 0.932222i \(0.382132\pi\)
\(840\) 0 0
\(841\) −118525. −0.00577856
\(842\) −8.64588e6 −0.420270
\(843\) 0 0
\(844\) −5.32536e7 −2.57331
\(845\) −3.82613e7 −1.84339
\(846\) 0 0
\(847\) 0 0
\(848\) −7.51516e6 −0.358879
\(849\) 0 0
\(850\) −6.51053e7 −3.09079
\(851\) 1.62492e7 0.769143
\(852\) 0 0
\(853\) 6.12722e6 0.288331 0.144165 0.989554i \(-0.453950\pi\)
0.144165 + 0.989554i \(0.453950\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −1.07735e7 −0.502540
\(857\) −3.88119e7 −1.80515 −0.902574 0.430534i \(-0.858325\pi\)
−0.902574 + 0.430534i \(0.858325\pi\)
\(858\) 0 0
\(859\) −2.09339e6 −0.0967980 −0.0483990 0.998828i \(-0.515412\pi\)
−0.0483990 + 0.998828i \(0.515412\pi\)
\(860\) −5.49992e7 −2.53577
\(861\) 0 0
\(862\) −2.96930e7 −1.36109
\(863\) 4.08889e7 1.86887 0.934434 0.356137i \(-0.115906\pi\)
0.934434 + 0.356137i \(0.115906\pi\)
\(864\) 0 0
\(865\) −1.86896e7 −0.849298
\(866\) −6.74990e7 −3.05846
\(867\) 0 0
\(868\) 0 0
\(869\) −3.00522e7 −1.34998
\(870\) 0 0
\(871\) −2.60661e7 −1.16421
\(872\) −7.72613e6 −0.344089
\(873\) 0 0
\(874\) −2.25206e7 −0.997245
\(875\) 0 0
\(876\) 0 0
\(877\) 1.65515e7 0.726670 0.363335 0.931659i \(-0.381638\pi\)
0.363335 + 0.931659i \(0.381638\pi\)
\(878\) 5.50749e7 2.41111
\(879\) 0 0
\(880\) −1.85337e7 −0.806782
\(881\) −1.48157e7 −0.643104 −0.321552 0.946892i \(-0.604205\pi\)
−0.321552 + 0.946892i \(0.604205\pi\)
\(882\) 0 0
\(883\) 2.52632e7 1.09040 0.545202 0.838305i \(-0.316453\pi\)
0.545202 + 0.838305i \(0.316453\pi\)
\(884\) 4.19241e7 1.80440
\(885\) 0 0
\(886\) −4.23434e6 −0.181218
\(887\) 3.23328e7 1.37986 0.689930 0.723877i \(-0.257641\pi\)
0.689930 + 0.723877i \(0.257641\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.81036e7 0.766107
\(891\) 0 0
\(892\) 3.28059e7 1.38051
\(893\) −8.66200e6 −0.363488
\(894\) 0 0
\(895\) −1.27419e7 −0.531714
\(896\) 0 0
\(897\) 0 0
\(898\) −4.51908e7 −1.87008
\(899\) 4.31660e7 1.78132
\(900\) 0 0
\(901\) 1.66133e7 0.681778
\(902\) −3.22495e7 −1.31980
\(903\) 0 0
\(904\) −3.08986e6 −0.125753
\(905\) −2.19149e7 −0.889442
\(906\) 0 0
\(907\) 1.10034e7 0.444129 0.222065 0.975032i \(-0.428720\pi\)
0.222065 + 0.975032i \(0.428720\pi\)
\(908\) −6.17534e6 −0.248569
\(909\) 0 0
\(910\) 0 0
\(911\) 4.12930e7 1.64847 0.824234 0.566249i \(-0.191606\pi\)
0.824234 + 0.566249i \(0.191606\pi\)
\(912\) 0 0
\(913\) 1.15088e7 0.456935
\(914\) −5.71617e7 −2.26329
\(915\) 0 0
\(916\) −3.40677e7 −1.34154
\(917\) 0 0
\(918\) 0 0
\(919\) 3.78206e6 0.147720 0.0738601 0.997269i \(-0.476468\pi\)
0.0738601 + 0.997269i \(0.476468\pi\)
\(920\) −3.09995e7 −1.20750
\(921\) 0 0
\(922\) 3.62567e7 1.40463
\(923\) −5.34158e7 −2.06379
\(924\) 0 0
\(925\) −3.72289e7 −1.43063
\(926\) −4.90684e7 −1.88051
\(927\) 0 0
\(928\) −3.46438e7 −1.32055
\(929\) −1.47979e7 −0.562550 −0.281275 0.959627i \(-0.590757\pi\)
−0.281275 + 0.959627i \(0.590757\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −4.55620e7 −1.71816
\(933\) 0 0
\(934\) 3.66781e7 1.37575
\(935\) 4.09713e7 1.53268
\(936\) 0 0
\(937\) 3.07412e7 1.14386 0.571929 0.820303i \(-0.306195\pi\)
0.571929 + 0.820303i \(0.306195\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.37184e7 −1.61378
\(941\) −2.56871e7 −0.945673 −0.472837 0.881150i \(-0.656770\pi\)
−0.472837 + 0.881150i \(0.656770\pi\)
\(942\) 0 0
\(943\) 2.93361e7 1.07429
\(944\) 2.11608e7 0.772861
\(945\) 0 0
\(946\) 4.09807e7 1.48885
\(947\) 6.25960e6 0.226815 0.113407 0.993549i \(-0.463823\pi\)
0.113407 + 0.993549i \(0.463823\pi\)
\(948\) 0 0
\(949\) −4.22876e7 −1.52422
\(950\) 5.15977e7 1.85490
\(951\) 0 0
\(952\) 0 0
\(953\) 2.65561e7 0.947180 0.473590 0.880746i \(-0.342958\pi\)
0.473590 + 0.880746i \(0.342958\pi\)
\(954\) 0 0
\(955\) −1.15758e7 −0.410715
\(956\) 4.91136e7 1.73803
\(957\) 0 0
\(958\) 6.19795e7 2.18190
\(959\) 0 0
\(960\) 0 0
\(961\) 6.27423e7 2.19155
\(962\) 4.14085e7 1.44262
\(963\) 0 0
\(964\) −6.35472e7 −2.20244
\(965\) 3.05324e7 1.05546
\(966\) 0 0
\(967\) −3.33142e7 −1.14568 −0.572841 0.819667i \(-0.694159\pi\)
−0.572841 + 0.819667i \(0.694159\pi\)
\(968\) 2.14740e6 0.0736589
\(969\) 0 0
\(970\) −1.16291e8 −3.96841
\(971\) −4.04061e6 −0.137530 −0.0687652 0.997633i \(-0.521906\pi\)
−0.0687652 + 0.997633i \(0.521906\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −6.13398e7 −2.07178
\(975\) 0 0
\(976\) 2.37059e7 0.796586
\(977\) 1.10096e7 0.369008 0.184504 0.982832i \(-0.440932\pi\)
0.184504 + 0.982832i \(0.440932\pi\)
\(978\) 0 0
\(979\) −7.80957e6 −0.260418
\(980\) 0 0
\(981\) 0 0
\(982\) −9.47348e6 −0.313495
\(983\) −2.74119e7 −0.904804 −0.452402 0.891814i \(-0.649433\pi\)
−0.452402 + 0.891814i \(0.649433\pi\)
\(984\) 0 0
\(985\) −3.59047e7 −1.17913
\(986\) 4.31660e7 1.41400
\(987\) 0 0
\(988\) −3.32260e7 −1.08289
\(989\) −3.72785e7 −1.21190
\(990\) 0 0
\(991\) 2.29977e7 0.743875 0.371938 0.928258i \(-0.378694\pi\)
0.371938 + 0.928258i \(0.378694\pi\)
\(992\) −7.33322e7 −2.36600
\(993\) 0 0
\(994\) 0 0
\(995\) −2.28677e7 −0.732259
\(996\) 0 0
\(997\) 5.23347e7 1.66744 0.833722 0.552184i \(-0.186205\pi\)
0.833722 + 0.552184i \(0.186205\pi\)
\(998\) −2.66674e7 −0.847527
\(999\) 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 441.6.a.y.1.1 4
3.2 odd 2 inner 441.6.a.y.1.4 yes 4
7.6 odd 2 inner 441.6.a.y.1.2 yes 4
21.20 even 2 inner 441.6.a.y.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.6.a.y.1.1 4 1.1 even 1 trivial
441.6.a.y.1.2 yes 4 7.6 odd 2 inner
441.6.a.y.1.3 yes 4 21.20 even 2 inner
441.6.a.y.1.4 yes 4 3.2 odd 2 inner