Properties

Label 1104.6.a.ba.1.7
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $8$
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] = [1104,6,Mod(1,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4608x^{6} - 3161x^{5} + 6284039x^{4} - 8279002x^{3} - 2677454576x^{2} + 13573192447x + 77901299860 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(34.5562\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +71.1125 q^{5} -188.015 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +71.1125 q^{5} -188.015 q^{7} +81.0000 q^{9} -133.529 q^{11} +661.881 q^{13} -640.012 q^{15} +2237.24 q^{17} +979.973 q^{19} +1692.13 q^{21} +529.000 q^{23} +1931.99 q^{25} -729.000 q^{27} +337.792 q^{29} -423.647 q^{31} +1201.76 q^{33} -13370.2 q^{35} -2257.78 q^{37} -5956.93 q^{39} -11014.8 q^{41} -8574.44 q^{43} +5760.11 q^{45} +1416.94 q^{47} +18542.5 q^{49} -20135.1 q^{51} +1729.01 q^{53} -9495.55 q^{55} -8819.75 q^{57} +50385.7 q^{59} -23301.9 q^{61} -15229.2 q^{63} +47068.0 q^{65} +25758.6 q^{67} -4761.00 q^{69} +19497.0 q^{71} -54323.1 q^{73} -17387.9 q^{75} +25105.3 q^{77} +4710.12 q^{79} +6561.00 q^{81} +31521.3 q^{83} +159095. q^{85} -3040.12 q^{87} -111714. q^{89} -124443. q^{91} +3812.82 q^{93} +69688.3 q^{95} -100692. q^{97} -10815.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) 71.1125 1.27210 0.636050 0.771648i \(-0.280567\pi\)
0.636050 + 0.771648i \(0.280567\pi\)
\(6\) 0 0
\(7\) −188.015 −1.45026 −0.725131 0.688611i \(-0.758221\pi\)
−0.725131 + 0.688611i \(0.758221\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −133.529 −0.332731 −0.166365 0.986064i \(-0.553203\pi\)
−0.166365 + 0.986064i \(0.553203\pi\)
\(12\) 0 0
\(13\) 661.881 1.08623 0.543115 0.839658i \(-0.317245\pi\)
0.543115 + 0.839658i \(0.317245\pi\)
\(14\) 0 0
\(15\) −640.012 −0.734447
\(16\) 0 0
\(17\) 2237.24 1.87754 0.938771 0.344542i \(-0.111966\pi\)
0.938771 + 0.344542i \(0.111966\pi\)
\(18\) 0 0
\(19\) 979.973 0.622773 0.311387 0.950283i \(-0.399207\pi\)
0.311387 + 0.950283i \(0.399207\pi\)
\(20\) 0 0
\(21\) 1692.13 0.837309
\(22\) 0 0
\(23\) 529.000 0.208514
\(24\) 0 0
\(25\) 1931.99 0.618236
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 337.792 0.0745854 0.0372927 0.999304i \(-0.488127\pi\)
0.0372927 + 0.999304i \(0.488127\pi\)
\(30\) 0 0
\(31\) −423.647 −0.0791771 −0.0395886 0.999216i \(-0.512605\pi\)
−0.0395886 + 0.999216i \(0.512605\pi\)
\(32\) 0 0
\(33\) 1201.76 0.192102
\(34\) 0 0
\(35\) −13370.2 −1.84488
\(36\) 0 0
\(37\) −2257.78 −0.271130 −0.135565 0.990768i \(-0.543285\pi\)
−0.135565 + 0.990768i \(0.543285\pi\)
\(38\) 0 0
\(39\) −5956.93 −0.627135
\(40\) 0 0
\(41\) −11014.8 −1.02333 −0.511667 0.859184i \(-0.670972\pi\)
−0.511667 + 0.859184i \(0.670972\pi\)
\(42\) 0 0
\(43\) −8574.44 −0.707187 −0.353594 0.935399i \(-0.615040\pi\)
−0.353594 + 0.935399i \(0.615040\pi\)
\(44\) 0 0
\(45\) 5760.11 0.424033
\(46\) 0 0
\(47\) 1416.94 0.0935634 0.0467817 0.998905i \(-0.485103\pi\)
0.0467817 + 0.998905i \(0.485103\pi\)
\(48\) 0 0
\(49\) 18542.5 1.10326
\(50\) 0 0
\(51\) −20135.1 −1.08400
\(52\) 0 0
\(53\) 1729.01 0.0845490 0.0422745 0.999106i \(-0.486540\pi\)
0.0422745 + 0.999106i \(0.486540\pi\)
\(54\) 0 0
\(55\) −9495.55 −0.423266
\(56\) 0 0
\(57\) −8819.75 −0.359558
\(58\) 0 0
\(59\) 50385.7 1.88442 0.942210 0.335023i \(-0.108744\pi\)
0.942210 + 0.335023i \(0.108744\pi\)
\(60\) 0 0
\(61\) −23301.9 −0.801800 −0.400900 0.916122i \(-0.631303\pi\)
−0.400900 + 0.916122i \(0.631303\pi\)
\(62\) 0 0
\(63\) −15229.2 −0.483421
\(64\) 0 0
\(65\) 47068.0 1.38179
\(66\) 0 0
\(67\) 25758.6 0.701029 0.350514 0.936557i \(-0.386007\pi\)
0.350514 + 0.936557i \(0.386007\pi\)
\(68\) 0 0
\(69\) −4761.00 −0.120386
\(70\) 0 0
\(71\) 19497.0 0.459009 0.229504 0.973308i \(-0.426289\pi\)
0.229504 + 0.973308i \(0.426289\pi\)
\(72\) 0 0
\(73\) −54323.1 −1.19310 −0.596551 0.802575i \(-0.703463\pi\)
−0.596551 + 0.802575i \(0.703463\pi\)
\(74\) 0 0
\(75\) −17387.9 −0.356939
\(76\) 0 0
\(77\) 25105.3 0.482547
\(78\) 0 0
\(79\) 4710.12 0.0849111 0.0424556 0.999098i \(-0.486482\pi\)
0.0424556 + 0.999098i \(0.486482\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 31521.3 0.502237 0.251119 0.967956i \(-0.419202\pi\)
0.251119 + 0.967956i \(0.419202\pi\)
\(84\) 0 0
\(85\) 159095. 2.38842
\(86\) 0 0
\(87\) −3040.12 −0.0430619
\(88\) 0 0
\(89\) −111714. −1.49497 −0.747484 0.664280i \(-0.768738\pi\)
−0.747484 + 0.664280i \(0.768738\pi\)
\(90\) 0 0
\(91\) −124443. −1.57532
\(92\) 0 0
\(93\) 3812.82 0.0457129
\(94\) 0 0
\(95\) 69688.3 0.792229
\(96\) 0 0
\(97\) −100692. −1.08659 −0.543294 0.839543i \(-0.682823\pi\)
−0.543294 + 0.839543i \(0.682823\pi\)
\(98\) 0 0
\(99\) −10815.8 −0.110910
\(100\) 0 0
\(101\) 153987. 1.50204 0.751021 0.660278i \(-0.229562\pi\)
0.751021 + 0.660278i \(0.229562\pi\)
\(102\) 0 0
\(103\) 95768.8 0.889469 0.444735 0.895662i \(-0.353298\pi\)
0.444735 + 0.895662i \(0.353298\pi\)
\(104\) 0 0
\(105\) 120332. 1.06514
\(106\) 0 0
\(107\) −134430. −1.13511 −0.567555 0.823335i \(-0.692111\pi\)
−0.567555 + 0.823335i \(0.692111\pi\)
\(108\) 0 0
\(109\) 75035.9 0.604927 0.302463 0.953161i \(-0.402191\pi\)
0.302463 + 0.953161i \(0.402191\pi\)
\(110\) 0 0
\(111\) 20320.0 0.156537
\(112\) 0 0
\(113\) 176475. 1.30013 0.650066 0.759877i \(-0.274741\pi\)
0.650066 + 0.759877i \(0.274741\pi\)
\(114\) 0 0
\(115\) 37618.5 0.265251
\(116\) 0 0
\(117\) 53612.4 0.362077
\(118\) 0 0
\(119\) −420633. −2.72293
\(120\) 0 0
\(121\) −143221. −0.889290
\(122\) 0 0
\(123\) 99133.4 0.590823
\(124\) 0 0
\(125\) −84838.1 −0.485642
\(126\) 0 0
\(127\) −35819.0 −0.197063 −0.0985314 0.995134i \(-0.531414\pi\)
−0.0985314 + 0.995134i \(0.531414\pi\)
\(128\) 0 0
\(129\) 77169.9 0.408295
\(130\) 0 0
\(131\) 306089. 1.55836 0.779182 0.626798i \(-0.215635\pi\)
0.779182 + 0.626798i \(0.215635\pi\)
\(132\) 0 0
\(133\) −184249. −0.903185
\(134\) 0 0
\(135\) −51841.0 −0.244816
\(136\) 0 0
\(137\) 144889. 0.659530 0.329765 0.944063i \(-0.393031\pi\)
0.329765 + 0.944063i \(0.393031\pi\)
\(138\) 0 0
\(139\) 24005.5 0.105384 0.0526919 0.998611i \(-0.483220\pi\)
0.0526919 + 0.998611i \(0.483220\pi\)
\(140\) 0 0
\(141\) −12752.4 −0.0540188
\(142\) 0 0
\(143\) −88380.1 −0.361422
\(144\) 0 0
\(145\) 24021.2 0.0948800
\(146\) 0 0
\(147\) −166883. −0.636968
\(148\) 0 0
\(149\) 310431. 1.14551 0.572756 0.819726i \(-0.305874\pi\)
0.572756 + 0.819726i \(0.305874\pi\)
\(150\) 0 0
\(151\) −369009. −1.31703 −0.658513 0.752569i \(-0.728814\pi\)
−0.658513 + 0.752569i \(0.728814\pi\)
\(152\) 0 0
\(153\) 181216. 0.625847
\(154\) 0 0
\(155\) −30126.6 −0.100721
\(156\) 0 0
\(157\) −380929. −1.23337 −0.616686 0.787209i \(-0.711525\pi\)
−0.616686 + 0.787209i \(0.711525\pi\)
\(158\) 0 0
\(159\) −15561.1 −0.0488144
\(160\) 0 0
\(161\) −99459.8 −0.302401
\(162\) 0 0
\(163\) −173657. −0.511945 −0.255972 0.966684i \(-0.582396\pi\)
−0.255972 + 0.966684i \(0.582396\pi\)
\(164\) 0 0
\(165\) 85460.0 0.244373
\(166\) 0 0
\(167\) 69591.8 0.193093 0.0965466 0.995328i \(-0.469220\pi\)
0.0965466 + 0.995328i \(0.469220\pi\)
\(168\) 0 0
\(169\) 66793.9 0.179895
\(170\) 0 0
\(171\) 79377.8 0.207591
\(172\) 0 0
\(173\) 680477. 1.72862 0.864308 0.502963i \(-0.167757\pi\)
0.864308 + 0.502963i \(0.167757\pi\)
\(174\) 0 0
\(175\) −363242. −0.896604
\(176\) 0 0
\(177\) −453472. −1.08797
\(178\) 0 0
\(179\) 595916. 1.39012 0.695060 0.718952i \(-0.255378\pi\)
0.695060 + 0.718952i \(0.255378\pi\)
\(180\) 0 0
\(181\) 529495. 1.20134 0.600670 0.799497i \(-0.294901\pi\)
0.600670 + 0.799497i \(0.294901\pi\)
\(182\) 0 0
\(183\) 209717. 0.462920
\(184\) 0 0
\(185\) −160556. −0.344904
\(186\) 0 0
\(187\) −298735. −0.624715
\(188\) 0 0
\(189\) 137063. 0.279103
\(190\) 0 0
\(191\) −51044.2 −0.101242 −0.0506212 0.998718i \(-0.516120\pi\)
−0.0506212 + 0.998718i \(0.516120\pi\)
\(192\) 0 0
\(193\) 662338. 1.27993 0.639965 0.768404i \(-0.278949\pi\)
0.639965 + 0.768404i \(0.278949\pi\)
\(194\) 0 0
\(195\) −423612. −0.797778
\(196\) 0 0
\(197\) −519040. −0.952873 −0.476436 0.879209i \(-0.658072\pi\)
−0.476436 + 0.879209i \(0.658072\pi\)
\(198\) 0 0
\(199\) −292840. −0.524202 −0.262101 0.965041i \(-0.584415\pi\)
−0.262101 + 0.965041i \(0.584415\pi\)
\(200\) 0 0
\(201\) −231828. −0.404739
\(202\) 0 0
\(203\) −63509.8 −0.108168
\(204\) 0 0
\(205\) −783291. −1.30178
\(206\) 0 0
\(207\) 42849.0 0.0695048
\(208\) 0 0
\(209\) −130854. −0.207216
\(210\) 0 0
\(211\) −1.10357e6 −1.70646 −0.853228 0.521539i \(-0.825358\pi\)
−0.853228 + 0.521539i \(0.825358\pi\)
\(212\) 0 0
\(213\) −175473. −0.265009
\(214\) 0 0
\(215\) −609750. −0.899612
\(216\) 0 0
\(217\) 79651.8 0.114828
\(218\) 0 0
\(219\) 488908. 0.688838
\(220\) 0 0
\(221\) 1.48079e6 2.03944
\(222\) 0 0
\(223\) 64393.2 0.0867118 0.0433559 0.999060i \(-0.486195\pi\)
0.0433559 + 0.999060i \(0.486195\pi\)
\(224\) 0 0
\(225\) 156491. 0.206079
\(226\) 0 0
\(227\) 659120. 0.848984 0.424492 0.905432i \(-0.360453\pi\)
0.424492 + 0.905432i \(0.360453\pi\)
\(228\) 0 0
\(229\) 217092. 0.273561 0.136781 0.990601i \(-0.456324\pi\)
0.136781 + 0.990601i \(0.456324\pi\)
\(230\) 0 0
\(231\) −225948. −0.278598
\(232\) 0 0
\(233\) 1.11565e6 1.34629 0.673145 0.739510i \(-0.264943\pi\)
0.673145 + 0.739510i \(0.264943\pi\)
\(234\) 0 0
\(235\) 100762. 0.119022
\(236\) 0 0
\(237\) −42391.1 −0.0490235
\(238\) 0 0
\(239\) −87775.6 −0.0993983 −0.0496992 0.998764i \(-0.515826\pi\)
−0.0496992 + 0.998764i \(0.515826\pi\)
\(240\) 0 0
\(241\) 586801. 0.650801 0.325401 0.945576i \(-0.394501\pi\)
0.325401 + 0.945576i \(0.394501\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) 1.31860e6 1.40346
\(246\) 0 0
\(247\) 648626. 0.676475
\(248\) 0 0
\(249\) −283692. −0.289967
\(250\) 0 0
\(251\) 1.83841e6 1.84186 0.920932 0.389723i \(-0.127429\pi\)
0.920932 + 0.389723i \(0.127429\pi\)
\(252\) 0 0
\(253\) −70636.6 −0.0693791
\(254\) 0 0
\(255\) −1.43186e6 −1.37895
\(256\) 0 0
\(257\) −1.08346e6 −1.02325 −0.511623 0.859210i \(-0.670956\pi\)
−0.511623 + 0.859210i \(0.670956\pi\)
\(258\) 0 0
\(259\) 424495. 0.393209
\(260\) 0 0
\(261\) 27361.1 0.0248618
\(262\) 0 0
\(263\) 1.55245e6 1.38397 0.691987 0.721910i \(-0.256735\pi\)
0.691987 + 0.721910i \(0.256735\pi\)
\(264\) 0 0
\(265\) 122954. 0.107555
\(266\) 0 0
\(267\) 1.00542e6 0.863120
\(268\) 0 0
\(269\) 1.47991e6 1.24697 0.623484 0.781836i \(-0.285717\pi\)
0.623484 + 0.781836i \(0.285717\pi\)
\(270\) 0 0
\(271\) −1.76717e6 −1.46169 −0.730845 0.682544i \(-0.760874\pi\)
−0.730845 + 0.682544i \(0.760874\pi\)
\(272\) 0 0
\(273\) 1.11999e6 0.909511
\(274\) 0 0
\(275\) −257976. −0.205706
\(276\) 0 0
\(277\) 1.32630e6 1.03859 0.519293 0.854597i \(-0.326196\pi\)
0.519293 + 0.854597i \(0.326196\pi\)
\(278\) 0 0
\(279\) −34315.4 −0.0263924
\(280\) 0 0
\(281\) −250126. −0.188970 −0.0944852 0.995526i \(-0.530121\pi\)
−0.0944852 + 0.995526i \(0.530121\pi\)
\(282\) 0 0
\(283\) 256231. 0.190181 0.0950903 0.995469i \(-0.469686\pi\)
0.0950903 + 0.995469i \(0.469686\pi\)
\(284\) 0 0
\(285\) −627195. −0.457394
\(286\) 0 0
\(287\) 2.07095e6 1.48410
\(288\) 0 0
\(289\) 3.58537e6 2.52516
\(290\) 0 0
\(291\) 906226. 0.627342
\(292\) 0 0
\(293\) −746625. −0.508082 −0.254041 0.967193i \(-0.581760\pi\)
−0.254041 + 0.967193i \(0.581760\pi\)
\(294\) 0 0
\(295\) 3.58306e6 2.39717
\(296\) 0 0
\(297\) 97342.4 0.0640340
\(298\) 0 0
\(299\) 350135. 0.226495
\(300\) 0 0
\(301\) 1.61212e6 1.02561
\(302\) 0 0
\(303\) −1.38589e6 −0.867204
\(304\) 0 0
\(305\) −1.65705e6 −1.01997
\(306\) 0 0
\(307\) 2.01926e6 1.22278 0.611389 0.791330i \(-0.290611\pi\)
0.611389 + 0.791330i \(0.290611\pi\)
\(308\) 0 0
\(309\) −861919. −0.513535
\(310\) 0 0
\(311\) 1.92994e6 1.13147 0.565734 0.824588i \(-0.308593\pi\)
0.565734 + 0.824588i \(0.308593\pi\)
\(312\) 0 0
\(313\) 1.26972e6 0.732569 0.366284 0.930503i \(-0.380630\pi\)
0.366284 + 0.930503i \(0.380630\pi\)
\(314\) 0 0
\(315\) −1.08299e6 −0.614959
\(316\) 0 0
\(317\) 557576. 0.311642 0.155821 0.987785i \(-0.450198\pi\)
0.155821 + 0.987785i \(0.450198\pi\)
\(318\) 0 0
\(319\) −45104.8 −0.0248168
\(320\) 0 0
\(321\) 1.20987e6 0.655356
\(322\) 0 0
\(323\) 2.19243e6 1.16928
\(324\) 0 0
\(325\) 1.27875e6 0.671546
\(326\) 0 0
\(327\) −675323. −0.349255
\(328\) 0 0
\(329\) −266405. −0.135691
\(330\) 0 0
\(331\) −1.03458e6 −0.519031 −0.259516 0.965739i \(-0.583563\pi\)
−0.259516 + 0.965739i \(0.583563\pi\)
\(332\) 0 0
\(333\) −182880. −0.0903765
\(334\) 0 0
\(335\) 1.83176e6 0.891778
\(336\) 0 0
\(337\) 2.08776e6 1.00140 0.500698 0.865622i \(-0.333077\pi\)
0.500698 + 0.865622i \(0.333077\pi\)
\(338\) 0 0
\(339\) −1.58828e6 −0.750632
\(340\) 0 0
\(341\) 56569.0 0.0263447
\(342\) 0 0
\(343\) −326302. −0.149756
\(344\) 0 0
\(345\) −338567. −0.153143
\(346\) 0 0
\(347\) 3.07258e6 1.36987 0.684936 0.728603i \(-0.259830\pi\)
0.684936 + 0.728603i \(0.259830\pi\)
\(348\) 0 0
\(349\) 1.06963e6 0.470080 0.235040 0.971986i \(-0.424478\pi\)
0.235040 + 0.971986i \(0.424478\pi\)
\(350\) 0 0
\(351\) −482511. −0.209045
\(352\) 0 0
\(353\) −1.83670e6 −0.784516 −0.392258 0.919855i \(-0.628306\pi\)
−0.392258 + 0.919855i \(0.628306\pi\)
\(354\) 0 0
\(355\) 1.38648e6 0.583905
\(356\) 0 0
\(357\) 3.78570e6 1.57208
\(358\) 0 0
\(359\) −2.05922e6 −0.843272 −0.421636 0.906765i \(-0.638544\pi\)
−0.421636 + 0.906765i \(0.638544\pi\)
\(360\) 0 0
\(361\) −1.51575e6 −0.612154
\(362\) 0 0
\(363\) 1.28899e6 0.513432
\(364\) 0 0
\(365\) −3.86305e6 −1.51774
\(366\) 0 0
\(367\) 4.01460e6 1.55588 0.777941 0.628337i \(-0.216264\pi\)
0.777941 + 0.628337i \(0.216264\pi\)
\(368\) 0 0
\(369\) −892200. −0.341112
\(370\) 0 0
\(371\) −325080. −0.122618
\(372\) 0 0
\(373\) 957150. 0.356212 0.178106 0.984011i \(-0.443003\pi\)
0.178106 + 0.984011i \(0.443003\pi\)
\(374\) 0 0
\(375\) 763543. 0.280385
\(376\) 0 0
\(377\) 223578. 0.0810169
\(378\) 0 0
\(379\) 2.04565e6 0.731534 0.365767 0.930707i \(-0.380807\pi\)
0.365767 + 0.930707i \(0.380807\pi\)
\(380\) 0 0
\(381\) 322371. 0.113774
\(382\) 0 0
\(383\) −262567. −0.0914627 −0.0457313 0.998954i \(-0.514562\pi\)
−0.0457313 + 0.998954i \(0.514562\pi\)
\(384\) 0 0
\(385\) 1.78530e6 0.613847
\(386\) 0 0
\(387\) −694529. −0.235729
\(388\) 0 0
\(389\) −5.57116e6 −1.86669 −0.933343 0.358986i \(-0.883123\pi\)
−0.933343 + 0.358986i \(0.883123\pi\)
\(390\) 0 0
\(391\) 1.18350e6 0.391495
\(392\) 0 0
\(393\) −2.75480e6 −0.899722
\(394\) 0 0
\(395\) 334949. 0.108015
\(396\) 0 0
\(397\) −1.48102e6 −0.471611 −0.235806 0.971800i \(-0.575773\pi\)
−0.235806 + 0.971800i \(0.575773\pi\)
\(398\) 0 0
\(399\) 1.65824e6 0.521454
\(400\) 0 0
\(401\) −1.69121e6 −0.525213 −0.262607 0.964903i \(-0.584582\pi\)
−0.262607 + 0.964903i \(0.584582\pi\)
\(402\) 0 0
\(403\) −280404. −0.0860046
\(404\) 0 0
\(405\) 466569. 0.141344
\(406\) 0 0
\(407\) 301478. 0.0902131
\(408\) 0 0
\(409\) 784184. 0.231798 0.115899 0.993261i \(-0.463025\pi\)
0.115899 + 0.993261i \(0.463025\pi\)
\(410\) 0 0
\(411\) −1.30400e6 −0.380780
\(412\) 0 0
\(413\) −9.47326e6 −2.73290
\(414\) 0 0
\(415\) 2.24156e6 0.638896
\(416\) 0 0
\(417\) −216049. −0.0608433
\(418\) 0 0
\(419\) 1.72033e6 0.478713 0.239357 0.970932i \(-0.423063\pi\)
0.239357 + 0.970932i \(0.423063\pi\)
\(420\) 0 0
\(421\) −1.00599e6 −0.276624 −0.138312 0.990389i \(-0.544168\pi\)
−0.138312 + 0.990389i \(0.544168\pi\)
\(422\) 0 0
\(423\) 114772. 0.0311878
\(424\) 0 0
\(425\) 4.32231e6 1.16076
\(426\) 0 0
\(427\) 4.38109e6 1.16282
\(428\) 0 0
\(429\) 795421. 0.208667
\(430\) 0 0
\(431\) −1.99734e6 −0.517914 −0.258957 0.965889i \(-0.583379\pi\)
−0.258957 + 0.965889i \(0.583379\pi\)
\(432\) 0 0
\(433\) 3.49033e6 0.894637 0.447318 0.894375i \(-0.352379\pi\)
0.447318 + 0.894375i \(0.352379\pi\)
\(434\) 0 0
\(435\) −216191. −0.0547790
\(436\) 0 0
\(437\) 518405. 0.129857
\(438\) 0 0
\(439\) 4.65337e6 1.15241 0.576205 0.817306i \(-0.304533\pi\)
0.576205 + 0.817306i \(0.304533\pi\)
\(440\) 0 0
\(441\) 1.50194e6 0.367754
\(442\) 0 0
\(443\) −433031. −0.104836 −0.0524179 0.998625i \(-0.516693\pi\)
−0.0524179 + 0.998625i \(0.516693\pi\)
\(444\) 0 0
\(445\) −7.94425e6 −1.90175
\(446\) 0 0
\(447\) −2.79388e6 −0.661362
\(448\) 0 0
\(449\) 2.95195e6 0.691025 0.345513 0.938414i \(-0.387705\pi\)
0.345513 + 0.938414i \(0.387705\pi\)
\(450\) 0 0
\(451\) 1.47079e6 0.340495
\(452\) 0 0
\(453\) 3.32108e6 0.760386
\(454\) 0 0
\(455\) −8.84948e6 −2.00396
\(456\) 0 0
\(457\) −7.37465e6 −1.65177 −0.825887 0.563836i \(-0.809325\pi\)
−0.825887 + 0.563836i \(0.809325\pi\)
\(458\) 0 0
\(459\) −1.63095e6 −0.361333
\(460\) 0 0
\(461\) −4.39294e6 −0.962728 −0.481364 0.876521i \(-0.659858\pi\)
−0.481364 + 0.876521i \(0.659858\pi\)
\(462\) 0 0
\(463\) 1.76781e6 0.383252 0.191626 0.981468i \(-0.438624\pi\)
0.191626 + 0.981468i \(0.438624\pi\)
\(464\) 0 0
\(465\) 271139. 0.0581514
\(466\) 0 0
\(467\) −1.89091e6 −0.401216 −0.200608 0.979672i \(-0.564292\pi\)
−0.200608 + 0.979672i \(0.564292\pi\)
\(468\) 0 0
\(469\) −4.84300e6 −1.01668
\(470\) 0 0
\(471\) 3.42836e6 0.712088
\(472\) 0 0
\(473\) 1.14493e6 0.235303
\(474\) 0 0
\(475\) 1.89329e6 0.385021
\(476\) 0 0
\(477\) 140050. 0.0281830
\(478\) 0 0
\(479\) −2.41395e6 −0.480717 −0.240359 0.970684i \(-0.577265\pi\)
−0.240359 + 0.970684i \(0.577265\pi\)
\(480\) 0 0
\(481\) −1.49438e6 −0.294509
\(482\) 0 0
\(483\) 895138. 0.174591
\(484\) 0 0
\(485\) −7.16045e6 −1.38225
\(486\) 0 0
\(487\) −1.64022e6 −0.313386 −0.156693 0.987647i \(-0.550083\pi\)
−0.156693 + 0.987647i \(0.550083\pi\)
\(488\) 0 0
\(489\) 1.56291e6 0.295571
\(490\) 0 0
\(491\) −69046.0 −0.0129251 −0.00646256 0.999979i \(-0.502057\pi\)
−0.00646256 + 0.999979i \(0.502057\pi\)
\(492\) 0 0
\(493\) 755720. 0.140037
\(494\) 0 0
\(495\) −769140. −0.141089
\(496\) 0 0
\(497\) −3.66572e6 −0.665684
\(498\) 0 0
\(499\) −8.65341e6 −1.55574 −0.777868 0.628427i \(-0.783699\pi\)
−0.777868 + 0.628427i \(0.783699\pi\)
\(500\) 0 0
\(501\) −626326. −0.111482
\(502\) 0 0
\(503\) 4.85060e6 0.854822 0.427411 0.904057i \(-0.359426\pi\)
0.427411 + 0.904057i \(0.359426\pi\)
\(504\) 0 0
\(505\) 1.09504e7 1.91075
\(506\) 0 0
\(507\) −601145. −0.103863
\(508\) 0 0
\(509\) 6.59169e6 1.12772 0.563861 0.825870i \(-0.309315\pi\)
0.563861 + 0.825870i \(0.309315\pi\)
\(510\) 0 0
\(511\) 1.02135e7 1.73031
\(512\) 0 0
\(513\) −714400. −0.119853
\(514\) 0 0
\(515\) 6.81036e6 1.13149
\(516\) 0 0
\(517\) −189202. −0.0311314
\(518\) 0 0
\(519\) −6.12430e6 −0.998017
\(520\) 0 0
\(521\) −3.58425e6 −0.578501 −0.289250 0.957253i \(-0.593406\pi\)
−0.289250 + 0.957253i \(0.593406\pi\)
\(522\) 0 0
\(523\) 3.13368e6 0.500958 0.250479 0.968122i \(-0.419412\pi\)
0.250479 + 0.968122i \(0.419412\pi\)
\(524\) 0 0
\(525\) 3.26918e6 0.517655
\(526\) 0 0
\(527\) −947798. −0.148658
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) 4.08125e6 0.628140
\(532\) 0 0
\(533\) −7.29050e6 −1.11158
\(534\) 0 0
\(535\) −9.55968e6 −1.44397
\(536\) 0 0
\(537\) −5.36324e6 −0.802586
\(538\) 0 0
\(539\) −2.47596e6 −0.367089
\(540\) 0 0
\(541\) 2.50065e6 0.367333 0.183667 0.982989i \(-0.441203\pi\)
0.183667 + 0.982989i \(0.441203\pi\)
\(542\) 0 0
\(543\) −4.76546e6 −0.693594
\(544\) 0 0
\(545\) 5.33599e6 0.769527
\(546\) 0 0
\(547\) −1.06868e7 −1.52715 −0.763573 0.645722i \(-0.776556\pi\)
−0.763573 + 0.645722i \(0.776556\pi\)
\(548\) 0 0
\(549\) −1.88745e6 −0.267267
\(550\) 0 0
\(551\) 331027. 0.0464498
\(552\) 0 0
\(553\) −885572. −0.123143
\(554\) 0 0
\(555\) 1.44501e6 0.199130
\(556\) 0 0
\(557\) 604766. 0.0825942 0.0412971 0.999147i \(-0.486851\pi\)
0.0412971 + 0.999147i \(0.486851\pi\)
\(558\) 0 0
\(559\) −5.67526e6 −0.768168
\(560\) 0 0
\(561\) 2.68862e6 0.360680
\(562\) 0 0
\(563\) 6.03544e6 0.802486 0.401243 0.915972i \(-0.368578\pi\)
0.401243 + 0.915972i \(0.368578\pi\)
\(564\) 0 0
\(565\) 1.25496e7 1.65390
\(566\) 0 0
\(567\) −1.23356e6 −0.161140
\(568\) 0 0
\(569\) 1.01774e7 1.31782 0.658910 0.752222i \(-0.271018\pi\)
0.658910 + 0.752222i \(0.271018\pi\)
\(570\) 0 0
\(571\) −5.47052e6 −0.702164 −0.351082 0.936345i \(-0.614186\pi\)
−0.351082 + 0.936345i \(0.614186\pi\)
\(572\) 0 0
\(573\) 459398. 0.0584524
\(574\) 0 0
\(575\) 1.02202e6 0.128911
\(576\) 0 0
\(577\) 7.48524e6 0.935979 0.467989 0.883734i \(-0.344979\pi\)
0.467989 + 0.883734i \(0.344979\pi\)
\(578\) 0 0
\(579\) −5.96104e6 −0.738968
\(580\) 0 0
\(581\) −5.92647e6 −0.728376
\(582\) 0 0
\(583\) −230873. −0.0281320
\(584\) 0 0
\(585\) 3.81251e6 0.460597
\(586\) 0 0
\(587\) 7.99098e6 0.957204 0.478602 0.878032i \(-0.341144\pi\)
0.478602 + 0.878032i \(0.341144\pi\)
\(588\) 0 0
\(589\) −415162. −0.0493094
\(590\) 0 0
\(591\) 4.67136e6 0.550141
\(592\) 0 0
\(593\) 1.24720e7 1.45646 0.728231 0.685331i \(-0.240343\pi\)
0.728231 + 0.685331i \(0.240343\pi\)
\(594\) 0 0
\(595\) −2.99123e7 −3.46383
\(596\) 0 0
\(597\) 2.63556e6 0.302648
\(598\) 0 0
\(599\) 1.12914e7 1.28582 0.642909 0.765942i \(-0.277727\pi\)
0.642909 + 0.765942i \(0.277727\pi\)
\(600\) 0 0
\(601\) 1.73885e7 1.96371 0.981853 0.189642i \(-0.0607326\pi\)
0.981853 + 0.189642i \(0.0607326\pi\)
\(602\) 0 0
\(603\) 2.08645e6 0.233676
\(604\) 0 0
\(605\) −1.01848e7 −1.13127
\(606\) 0 0
\(607\) 1.15433e7 1.27162 0.635811 0.771845i \(-0.280666\pi\)
0.635811 + 0.771845i \(0.280666\pi\)
\(608\) 0 0
\(609\) 571588. 0.0624511
\(610\) 0 0
\(611\) 937844. 0.101631
\(612\) 0 0
\(613\) 5.23860e6 0.563072 0.281536 0.959551i \(-0.409156\pi\)
0.281536 + 0.959551i \(0.409156\pi\)
\(614\) 0 0
\(615\) 7.04962e6 0.751585
\(616\) 0 0
\(617\) 4.57047e6 0.483335 0.241668 0.970359i \(-0.422306\pi\)
0.241668 + 0.970359i \(0.422306\pi\)
\(618\) 0 0
\(619\) 7.60001e6 0.797237 0.398618 0.917117i \(-0.369490\pi\)
0.398618 + 0.917117i \(0.369490\pi\)
\(620\) 0 0
\(621\) −385641. −0.0401286
\(622\) 0 0
\(623\) 2.10038e7 2.16810
\(624\) 0 0
\(625\) −1.20705e7 −1.23602
\(626\) 0 0
\(627\) 1.17769e6 0.119636
\(628\) 0 0
\(629\) −5.05118e6 −0.509057
\(630\) 0 0
\(631\) 893207. 0.0893056 0.0446528 0.999003i \(-0.485782\pi\)
0.0446528 + 0.999003i \(0.485782\pi\)
\(632\) 0 0
\(633\) 9.93216e6 0.985222
\(634\) 0 0
\(635\) −2.54718e6 −0.250683
\(636\) 0 0
\(637\) 1.22729e7 1.19840
\(638\) 0 0
\(639\) 1.57925e6 0.153003
\(640\) 0 0
\(641\) −169721. −0.0163152 −0.00815758 0.999967i \(-0.502597\pi\)
−0.00815758 + 0.999967i \(0.502597\pi\)
\(642\) 0 0
\(643\) 9.10025e6 0.868012 0.434006 0.900910i \(-0.357100\pi\)
0.434006 + 0.900910i \(0.357100\pi\)
\(644\) 0 0
\(645\) 5.48775e6 0.519391
\(646\) 0 0
\(647\) −1.78218e6 −0.167375 −0.0836874 0.996492i \(-0.526670\pi\)
−0.0836874 + 0.996492i \(0.526670\pi\)
\(648\) 0 0
\(649\) −6.72794e6 −0.627004
\(650\) 0 0
\(651\) −716867. −0.0662958
\(652\) 0 0
\(653\) −1.80884e7 −1.66003 −0.830016 0.557739i \(-0.811669\pi\)
−0.830016 + 0.557739i \(0.811669\pi\)
\(654\) 0 0
\(655\) 2.17667e7 1.98239
\(656\) 0 0
\(657\) −4.40017e6 −0.397701
\(658\) 0 0
\(659\) 1.57392e7 1.41179 0.705893 0.708318i \(-0.250546\pi\)
0.705893 + 0.708318i \(0.250546\pi\)
\(660\) 0 0
\(661\) 1.20994e7 1.07711 0.538557 0.842589i \(-0.318970\pi\)
0.538557 + 0.842589i \(0.318970\pi\)
\(662\) 0 0
\(663\) −1.33271e7 −1.17747
\(664\) 0 0
\(665\) −1.31024e7 −1.14894
\(666\) 0 0
\(667\) 178692. 0.0155521
\(668\) 0 0
\(669\) −579539. −0.0500631
\(670\) 0 0
\(671\) 3.11147e6 0.266783
\(672\) 0 0
\(673\) 1.46404e7 1.24599 0.622996 0.782225i \(-0.285915\pi\)
0.622996 + 0.782225i \(0.285915\pi\)
\(674\) 0 0
\(675\) −1.40842e6 −0.118980
\(676\) 0 0
\(677\) −1.88409e7 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(678\) 0 0
\(679\) 1.89315e7 1.57584
\(680\) 0 0
\(681\) −5.93208e6 −0.490161
\(682\) 0 0
\(683\) −2.26555e7 −1.85832 −0.929162 0.369673i \(-0.879470\pi\)
−0.929162 + 0.369673i \(0.879470\pi\)
\(684\) 0 0
\(685\) 1.03034e7 0.838987
\(686\) 0 0
\(687\) −1.95383e6 −0.157941
\(688\) 0 0
\(689\) 1.14440e6 0.0918396
\(690\) 0 0
\(691\) −2.38740e7 −1.90208 −0.951041 0.309065i \(-0.899984\pi\)
−0.951041 + 0.309065i \(0.899984\pi\)
\(692\) 0 0
\(693\) 2.03353e6 0.160849
\(694\) 0 0
\(695\) 1.70709e6 0.134059
\(696\) 0 0
\(697\) −2.46428e7 −1.92135
\(698\) 0 0
\(699\) −1.00409e7 −0.777281
\(700\) 0 0
\(701\) −1.66784e7 −1.28192 −0.640959 0.767575i \(-0.721463\pi\)
−0.640959 + 0.767575i \(0.721463\pi\)
\(702\) 0 0
\(703\) −2.21256e6 −0.168852
\(704\) 0 0
\(705\) −906857. −0.0687173
\(706\) 0 0
\(707\) −2.89519e7 −2.17835
\(708\) 0 0
\(709\) 3.85537e6 0.288038 0.144019 0.989575i \(-0.453997\pi\)
0.144019 + 0.989575i \(0.453997\pi\)
\(710\) 0 0
\(711\) 381520. 0.0283037
\(712\) 0 0
\(713\) −224109. −0.0165096
\(714\) 0 0
\(715\) −6.28493e6 −0.459764
\(716\) 0 0
\(717\) 789980. 0.0573876
\(718\) 0 0
\(719\) 2.14977e7 1.55085 0.775423 0.631442i \(-0.217536\pi\)
0.775423 + 0.631442i \(0.217536\pi\)
\(720\) 0 0
\(721\) −1.80059e7 −1.28996
\(722\) 0 0
\(723\) −5.28121e6 −0.375740
\(724\) 0 0
\(725\) 652609. 0.0461114
\(726\) 0 0
\(727\) 4.96974e6 0.348737 0.174368 0.984680i \(-0.444212\pi\)
0.174368 + 0.984680i \(0.444212\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.91830e7 −1.32777
\(732\) 0 0
\(733\) 2.34633e7 1.61298 0.806489 0.591249i \(-0.201365\pi\)
0.806489 + 0.591249i \(0.201365\pi\)
\(734\) 0 0
\(735\) −1.18674e7 −0.810287
\(736\) 0 0
\(737\) −3.43951e6 −0.233254
\(738\) 0 0
\(739\) −2.29225e7 −1.54401 −0.772007 0.635614i \(-0.780747\pi\)
−0.772007 + 0.635614i \(0.780747\pi\)
\(740\) 0 0
\(741\) −5.83763e6 −0.390563
\(742\) 0 0
\(743\) −1.55951e7 −1.03638 −0.518188 0.855267i \(-0.673393\pi\)
−0.518188 + 0.855267i \(0.673393\pi\)
\(744\) 0 0
\(745\) 2.20755e7 1.45720
\(746\) 0 0
\(747\) 2.55323e6 0.167412
\(748\) 0 0
\(749\) 2.52749e7 1.64621
\(750\) 0 0
\(751\) −2.41940e7 −1.56533 −0.782667 0.622440i \(-0.786141\pi\)
−0.782667 + 0.622440i \(0.786141\pi\)
\(752\) 0 0
\(753\) −1.65457e7 −1.06340
\(754\) 0 0
\(755\) −2.62412e7 −1.67539
\(756\) 0 0
\(757\) 4.45942e6 0.282839 0.141419 0.989950i \(-0.454833\pi\)
0.141419 + 0.989950i \(0.454833\pi\)
\(758\) 0 0
\(759\) 635730. 0.0400560
\(760\) 0 0
\(761\) −8.72756e6 −0.546300 −0.273150 0.961971i \(-0.588066\pi\)
−0.273150 + 0.961971i \(0.588066\pi\)
\(762\) 0 0
\(763\) −1.41078e7 −0.877302
\(764\) 0 0
\(765\) 1.28867e7 0.796140
\(766\) 0 0
\(767\) 3.33494e7 2.04691
\(768\) 0 0
\(769\) −2.27707e7 −1.38855 −0.694275 0.719710i \(-0.744275\pi\)
−0.694275 + 0.719710i \(0.744275\pi\)
\(770\) 0 0
\(771\) 9.75115e6 0.590772
\(772\) 0 0
\(773\) −1.43137e7 −0.861594 −0.430797 0.902449i \(-0.641767\pi\)
−0.430797 + 0.902449i \(0.641767\pi\)
\(774\) 0 0
\(775\) −818480. −0.0489502
\(776\) 0 0
\(777\) −3.82046e6 −0.227019
\(778\) 0 0
\(779\) −1.07942e7 −0.637305
\(780\) 0 0
\(781\) −2.60340e6 −0.152726
\(782\) 0 0
\(783\) −246250. −0.0143540
\(784\) 0 0
\(785\) −2.70888e7 −1.56897
\(786\) 0 0
\(787\) −1.10140e6 −0.0633885 −0.0316942 0.999498i \(-0.510090\pi\)
−0.0316942 + 0.999498i \(0.510090\pi\)
\(788\) 0 0
\(789\) −1.39720e7 −0.799038
\(790\) 0 0
\(791\) −3.31799e7 −1.88553
\(792\) 0 0
\(793\) −1.54231e7 −0.870939
\(794\) 0 0
\(795\) −1.10659e6 −0.0620967
\(796\) 0 0
\(797\) 1.39183e7 0.776142 0.388071 0.921630i \(-0.373142\pi\)
0.388071 + 0.921630i \(0.373142\pi\)
\(798\) 0 0
\(799\) 3.17002e6 0.175669
\(800\) 0 0
\(801\) −9.04882e6 −0.498323
\(802\) 0 0
\(803\) 7.25369e6 0.396981
\(804\) 0 0
\(805\) −7.07283e6 −0.384684
\(806\) 0 0
\(807\) −1.33192e7 −0.719937
\(808\) 0 0
\(809\) 2.39586e7 1.28704 0.643518 0.765431i \(-0.277474\pi\)
0.643518 + 0.765431i \(0.277474\pi\)
\(810\) 0 0
\(811\) −8.68484e6 −0.463671 −0.231835 0.972755i \(-0.574473\pi\)
−0.231835 + 0.972755i \(0.574473\pi\)
\(812\) 0 0
\(813\) 1.59045e7 0.843907
\(814\) 0 0
\(815\) −1.23492e7 −0.651245
\(816\) 0 0
\(817\) −8.40271e6 −0.440417
\(818\) 0 0
\(819\) −1.00799e7 −0.525106
\(820\) 0 0
\(821\) 9.39635e6 0.486521 0.243260 0.969961i \(-0.421783\pi\)
0.243260 + 0.969961i \(0.421783\pi\)
\(822\) 0 0
\(823\) −1.72483e7 −0.887659 −0.443829 0.896111i \(-0.646380\pi\)
−0.443829 + 0.896111i \(0.646380\pi\)
\(824\) 0 0
\(825\) 2.32178e6 0.118764
\(826\) 0 0
\(827\) −3.53805e7 −1.79887 −0.899437 0.437050i \(-0.856023\pi\)
−0.899437 + 0.437050i \(0.856023\pi\)
\(828\) 0 0
\(829\) 3.54998e7 1.79407 0.897035 0.441959i \(-0.145716\pi\)
0.897035 + 0.441959i \(0.145716\pi\)
\(830\) 0 0
\(831\) −1.19367e7 −0.599627
\(832\) 0 0
\(833\) 4.14840e7 2.07142
\(834\) 0 0
\(835\) 4.94885e6 0.245634
\(836\) 0 0
\(837\) 308839. 0.0152376
\(838\) 0 0
\(839\) −2.74841e7 −1.34796 −0.673980 0.738750i \(-0.735417\pi\)
−0.673980 + 0.738750i \(0.735417\pi\)
\(840\) 0 0
\(841\) −2.03970e7 −0.994437
\(842\) 0 0
\(843\) 2.25114e6 0.109102
\(844\) 0 0
\(845\) 4.74988e6 0.228845
\(846\) 0 0
\(847\) 2.69277e7 1.28970
\(848\) 0 0
\(849\) −2.30608e6 −0.109801
\(850\) 0 0
\(851\) −1.19436e6 −0.0565344
\(852\) 0 0
\(853\) −1.40740e7 −0.662286 −0.331143 0.943581i \(-0.607434\pi\)
−0.331143 + 0.943581i \(0.607434\pi\)
\(854\) 0 0
\(855\) 5.64475e6 0.264076
\(856\) 0 0
\(857\) −2.01794e7 −0.938547 −0.469274 0.883053i \(-0.655484\pi\)
−0.469274 + 0.883053i \(0.655484\pi\)
\(858\) 0 0
\(859\) 3.84235e6 0.177670 0.0888350 0.996046i \(-0.471686\pi\)
0.0888350 + 0.996046i \(0.471686\pi\)
\(860\) 0 0
\(861\) −1.86385e7 −0.856848
\(862\) 0 0
\(863\) −3.31172e6 −0.151366 −0.0756828 0.997132i \(-0.524114\pi\)
−0.0756828 + 0.997132i \(0.524114\pi\)
\(864\) 0 0
\(865\) 4.83904e7 2.19897
\(866\) 0 0
\(867\) −3.22683e7 −1.45790
\(868\) 0 0
\(869\) −628936. −0.0282525
\(870\) 0 0
\(871\) 1.70492e7 0.761478
\(872\) 0 0
\(873\) −8.15604e6 −0.362196
\(874\) 0 0
\(875\) 1.59508e7 0.704308
\(876\) 0 0
\(877\) 4.23996e7 1.86150 0.930749 0.365659i \(-0.119156\pi\)
0.930749 + 0.365659i \(0.119156\pi\)
\(878\) 0 0
\(879\) 6.71963e6 0.293341
\(880\) 0 0
\(881\) −7.13977e6 −0.309916 −0.154958 0.987921i \(-0.549524\pi\)
−0.154958 + 0.987921i \(0.549524\pi\)
\(882\) 0 0
\(883\) −1.36386e7 −0.588665 −0.294333 0.955703i \(-0.595097\pi\)
−0.294333 + 0.955703i \(0.595097\pi\)
\(884\) 0 0
\(885\) −3.22475e7 −1.38401
\(886\) 0 0
\(887\) −8.02258e6 −0.342377 −0.171189 0.985238i \(-0.554761\pi\)
−0.171189 + 0.985238i \(0.554761\pi\)
\(888\) 0 0
\(889\) 6.73451e6 0.285793
\(890\) 0 0
\(891\) −876081. −0.0369701
\(892\) 0 0
\(893\) 1.38856e6 0.0582688
\(894\) 0 0
\(895\) 4.23771e7 1.76837
\(896\) 0 0
\(897\) −3.15122e6 −0.130767
\(898\) 0 0
\(899\) −143104. −0.00590546
\(900\) 0 0
\(901\) 3.86821e6 0.158744
\(902\) 0 0
\(903\) −1.45091e7 −0.592135
\(904\) 0 0
\(905\) 3.76537e7 1.52822
\(906\) 0 0
\(907\) 4.74039e6 0.191336 0.0956678 0.995413i \(-0.469501\pi\)
0.0956678 + 0.995413i \(0.469501\pi\)
\(908\) 0 0
\(909\) 1.24730e7 0.500681
\(910\) 0 0
\(911\) 3.59416e7 1.43483 0.717417 0.696644i \(-0.245324\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(912\) 0 0
\(913\) −4.20900e6 −0.167110
\(914\) 0 0
\(915\) 1.49135e7 0.588880
\(916\) 0 0
\(917\) −5.75491e7 −2.26004
\(918\) 0 0
\(919\) 1.99426e7 0.778922 0.389461 0.921043i \(-0.372661\pi\)
0.389461 + 0.921043i \(0.372661\pi\)
\(920\) 0 0
\(921\) −1.81734e7 −0.705971
\(922\) 0 0
\(923\) 1.29047e7 0.498589
\(924\) 0 0
\(925\) −4.36200e6 −0.167622
\(926\) 0 0
\(927\) 7.75727e6 0.296490
\(928\) 0 0
\(929\) −4.68297e6 −0.178026 −0.0890128 0.996030i \(-0.528371\pi\)
−0.0890128 + 0.996030i \(0.528371\pi\)
\(930\) 0 0
\(931\) 1.81712e7 0.687082
\(932\) 0 0
\(933\) −1.73694e7 −0.653253
\(934\) 0 0
\(935\) −2.12438e7 −0.794700
\(936\) 0 0
\(937\) −2.40436e7 −0.894643 −0.447321 0.894373i \(-0.647622\pi\)
−0.447321 + 0.894373i \(0.647622\pi\)
\(938\) 0 0
\(939\) −1.14275e7 −0.422949
\(940\) 0 0
\(941\) 3.66326e7 1.34863 0.674317 0.738442i \(-0.264438\pi\)
0.674317 + 0.738442i \(0.264438\pi\)
\(942\) 0 0
\(943\) −5.82684e6 −0.213380
\(944\) 0 0
\(945\) 9.74687e6 0.355047
\(946\) 0 0
\(947\) 3.79519e7 1.37518 0.687588 0.726101i \(-0.258670\pi\)
0.687588 + 0.726101i \(0.258670\pi\)
\(948\) 0 0
\(949\) −3.59554e7 −1.29598
\(950\) 0 0
\(951\) −5.01819e6 −0.179927
\(952\) 0 0
\(953\) −3.79717e7 −1.35434 −0.677170 0.735827i \(-0.736794\pi\)
−0.677170 + 0.735827i \(0.736794\pi\)
\(954\) 0 0
\(955\) −3.62988e6 −0.128790
\(956\) 0 0
\(957\) 405944. 0.0143280
\(958\) 0 0
\(959\) −2.72413e7 −0.956491
\(960\) 0 0
\(961\) −2.84497e7 −0.993731
\(962\) 0 0
\(963\) −1.08889e7 −0.378370
\(964\) 0 0
\(965\) 4.71005e7 1.62820
\(966\) 0 0
\(967\) 3.35414e7 1.15349 0.576746 0.816924i \(-0.304322\pi\)
0.576746 + 0.816924i \(0.304322\pi\)
\(968\) 0 0
\(969\) −1.97319e7 −0.675086
\(970\) 0 0
\(971\) 3.47181e7 1.18170 0.590851 0.806781i \(-0.298792\pi\)
0.590851 + 0.806781i \(0.298792\pi\)
\(972\) 0 0
\(973\) −4.51338e6 −0.152834
\(974\) 0 0
\(975\) −1.15087e7 −0.387717
\(976\) 0 0
\(977\) −5.46519e6 −0.183176 −0.0915880 0.995797i \(-0.529194\pi\)
−0.0915880 + 0.995797i \(0.529194\pi\)
\(978\) 0 0
\(979\) 1.49170e7 0.497422
\(980\) 0 0
\(981\) 6.07791e6 0.201642
\(982\) 0 0
\(983\) 2.47579e7 0.817203 0.408602 0.912713i \(-0.366017\pi\)
0.408602 + 0.912713i \(0.366017\pi\)
\(984\) 0 0
\(985\) −3.69102e7 −1.21215
\(986\) 0 0
\(987\) 2.39764e6 0.0783415
\(988\) 0 0
\(989\) −4.53588e6 −0.147459
\(990\) 0 0
\(991\) −3.19572e7 −1.03368 −0.516839 0.856083i \(-0.672891\pi\)
−0.516839 + 0.856083i \(0.672891\pi\)
\(992\) 0 0
\(993\) 9.31121e6 0.299663
\(994\) 0 0
\(995\) −2.08246e7 −0.666836
\(996\) 0 0
\(997\) −3.70746e7 −1.18124 −0.590621 0.806949i \(-0.701117\pi\)
−0.590621 + 0.806949i \(0.701117\pi\)
\(998\) 0 0
\(999\) 1.64592e6 0.0521789
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 1104.6.a.ba.1.7 8
4.3 odd 2 552.6.a.i.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.6.a.i.1.7 8 4.3 odd 2
1104.6.a.ba.1.7 8 1.1 even 1 trivial