Properties

Label 546.8.a.p.1.1
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,8,Mod(1,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.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: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 302081x^{4} - 2628147x^{3} + 19116974952x^{2} - 78725393748x - 5138711063280 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(472.635\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -479.635 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -479.635 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3837.08 q^{10} +6141.65 q^{11} +1728.00 q^{12} -2197.00 q^{13} -2744.00 q^{14} -12950.1 q^{15} +4096.00 q^{16} -28277.0 q^{17} -5832.00 q^{18} -42997.9 q^{19} -30696.6 q^{20} +9261.00 q^{21} -49133.2 q^{22} +91517.2 q^{23} -13824.0 q^{24} +151925. q^{25} +17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} +38280.3 q^{29} +103601. q^{30} +15397.8 q^{31} -32768.0 q^{32} +165825. q^{33} +226216. q^{34} -164515. q^{35} +46656.0 q^{36} -380722. q^{37} +343983. q^{38} -59319.0 q^{39} +245573. q^{40} +82935.4 q^{41} -74088.0 q^{42} +254960. q^{43} +393066. q^{44} -349654. q^{45} -732137. q^{46} +800846. q^{47} +110592. q^{48} +117649. q^{49} -1.21540e6 q^{50} -763479. q^{51} -140608. q^{52} -1.98040e6 q^{53} -157464. q^{54} -2.94575e6 q^{55} -175616. q^{56} -1.16094e6 q^{57} -306242. q^{58} -2.20647e6 q^{59} -828809. q^{60} -1.21835e6 q^{61} -123183. q^{62} +250047. q^{63} +262144. q^{64} +1.05376e6 q^{65} -1.32660e6 q^{66} +1.98455e6 q^{67} -1.80973e6 q^{68} +2.47096e6 q^{69} +1.31612e6 q^{70} +3.12075e6 q^{71} -373248. q^{72} -883562. q^{73} +3.04577e6 q^{74} +4.10197e6 q^{75} -2.75187e6 q^{76} +2.10659e6 q^{77} +474552. q^{78} -4.21311e6 q^{79} -1.96459e6 q^{80} +531441. q^{81} -663483. q^{82} +1.90022e6 q^{83} +592704. q^{84} +1.35626e7 q^{85} -2.03968e6 q^{86} +1.03357e6 q^{87} -3.14453e6 q^{88} -1.19875e6 q^{89} +2.79723e6 q^{90} -753571. q^{91} +5.85710e6 q^{92} +415742. q^{93} -6.40677e6 q^{94} +2.06233e7 q^{95} -884736. q^{96} -9.55500e6 q^{97} -941192. q^{98} +4.47726e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} + 162 q^{3} + 384 q^{4} - 43 q^{5} - 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{2} + 162 q^{3} + 384 q^{4} - 43 q^{5} - 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 344 q^{10} + 7370 q^{11} + 10368 q^{12} - 13182 q^{13} - 16464 q^{14} - 1161 q^{15} + 24576 q^{16} + 7950 q^{17} - 34992 q^{18} - 57145 q^{19} - 2752 q^{20} + 55566 q^{21} - 58960 q^{22} + 31769 q^{23} - 82944 q^{24} + 135721 q^{25} + 105456 q^{26} + 118098 q^{27} + 131712 q^{28} - 36455 q^{29} + 9288 q^{30} + 215069 q^{31} - 196608 q^{32} + 198990 q^{33} - 63600 q^{34} - 14749 q^{35} + 279936 q^{36} + 133074 q^{37} + 457160 q^{38} - 355914 q^{39} + 22016 q^{40} + 516452 q^{41} - 444528 q^{42} - 3085 q^{43} + 471680 q^{44} - 31347 q^{45} - 254152 q^{46} + 1463947 q^{47} + 663552 q^{48} + 705894 q^{49} - 1085768 q^{50} + 214650 q^{51} - 843648 q^{52} - 1344571 q^{53} - 944784 q^{54} - 1568062 q^{55} - 1053696 q^{56} - 1542915 q^{57} + 291640 q^{58} + 1810408 q^{59} - 74304 q^{60} + 4047390 q^{61} - 1720552 q^{62} + 1500282 q^{63} + 1572864 q^{64} + 94471 q^{65} - 1591920 q^{66} + 2393614 q^{67} + 508800 q^{68} + 857763 q^{69} + 117992 q^{70} + 10341084 q^{71} - 2239488 q^{72} + 5180001 q^{73} - 1064592 q^{74} + 3664467 q^{75} - 3657280 q^{76} + 2527910 q^{77} + 2847312 q^{78} + 4624979 q^{79} - 176128 q^{80} + 3188646 q^{81} - 4131616 q^{82} + 11892699 q^{83} + 3556224 q^{84} + 750368 q^{85} + 24680 q^{86} - 984285 q^{87} - 3773440 q^{88} + 9781713 q^{89} + 250776 q^{90} - 4521426 q^{91} + 2033216 q^{92} + 5806863 q^{93} - 11711576 q^{94} + 26244263 q^{95} - 5308416 q^{96} + 5202537 q^{97} - 5647152 q^{98} + 5372730 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −479.635 −1.71599 −0.857997 0.513654i \(-0.828291\pi\)
−0.857997 + 0.513654i \(0.828291\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 3837.08 1.21339
\(11\) 6141.65 1.39127 0.695634 0.718397i \(-0.255124\pi\)
0.695634 + 0.718397i \(0.255124\pi\)
\(12\) 1728.00 0.288675
\(13\) −2197.00 −0.277350
\(14\) −2744.00 −0.267261
\(15\) −12950.1 −0.990730
\(16\) 4096.00 0.250000
\(17\) −28277.0 −1.39593 −0.697963 0.716134i \(-0.745910\pi\)
−0.697963 + 0.716134i \(0.745910\pi\)
\(18\) −5832.00 −0.235702
\(19\) −42997.9 −1.43817 −0.719084 0.694923i \(-0.755438\pi\)
−0.719084 + 0.694923i \(0.755438\pi\)
\(20\) −30696.6 −0.857997
\(21\) 9261.00 0.218218
\(22\) −49133.2 −0.983775
\(23\) 91517.2 1.56839 0.784197 0.620512i \(-0.213075\pi\)
0.784197 + 0.620512i \(0.213075\pi\)
\(24\) −13824.0 −0.204124
\(25\) 151925. 1.94464
\(26\) 17576.0 0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) 38280.3 0.291462 0.145731 0.989324i \(-0.453447\pi\)
0.145731 + 0.989324i \(0.453447\pi\)
\(30\) 103601. 0.700552
\(31\) 15397.8 0.0928312 0.0464156 0.998922i \(-0.485220\pi\)
0.0464156 + 0.998922i \(0.485220\pi\)
\(32\) −32768.0 −0.176777
\(33\) 165825. 0.803249
\(34\) 226216. 0.987069
\(35\) −164515. −0.648585
\(36\) 46656.0 0.166667
\(37\) −380722. −1.23567 −0.617834 0.786309i \(-0.711989\pi\)
−0.617834 + 0.786309i \(0.711989\pi\)
\(38\) 343983. 1.01694
\(39\) −59319.0 −0.160128
\(40\) 245573. 0.606696
\(41\) 82935.4 0.187930 0.0939650 0.995575i \(-0.470046\pi\)
0.0939650 + 0.995575i \(0.470046\pi\)
\(42\) −74088.0 −0.154303
\(43\) 254960. 0.489026 0.244513 0.969646i \(-0.421372\pi\)
0.244513 + 0.969646i \(0.421372\pi\)
\(44\) 393066. 0.695634
\(45\) −349654. −0.571998
\(46\) −732137. −1.10902
\(47\) 800846. 1.12514 0.562570 0.826750i \(-0.309813\pi\)
0.562570 + 0.826750i \(0.309813\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) −1.21540e6 −1.37507
\(51\) −763479. −0.805938
\(52\) −140608. −0.138675
\(53\) −1.98040e6 −1.82720 −0.913601 0.406611i \(-0.866711\pi\)
−0.913601 + 0.406611i \(0.866711\pi\)
\(54\) −157464. −0.136083
\(55\) −2.94575e6 −2.38741
\(56\) −175616. −0.133631
\(57\) −1.16094e6 −0.830327
\(58\) −306242. −0.206095
\(59\) −2.20647e6 −1.39867 −0.699336 0.714793i \(-0.746521\pi\)
−0.699336 + 0.714793i \(0.746521\pi\)
\(60\) −828809. −0.495365
\(61\) −1.21835e6 −0.687252 −0.343626 0.939107i \(-0.611655\pi\)
−0.343626 + 0.939107i \(0.611655\pi\)
\(62\) −123183. −0.0656416
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) 1.05376e6 0.475931
\(66\) −1.32660e6 −0.567983
\(67\) 1.98455e6 0.806122 0.403061 0.915173i \(-0.367946\pi\)
0.403061 + 0.915173i \(0.367946\pi\)
\(68\) −1.80973e6 −0.697963
\(69\) 2.47096e6 0.905513
\(70\) 1.31612e6 0.458619
\(71\) 3.12075e6 1.03480 0.517398 0.855745i \(-0.326901\pi\)
0.517398 + 0.855745i \(0.326901\pi\)
\(72\) −373248. −0.117851
\(73\) −883562. −0.265832 −0.132916 0.991127i \(-0.542434\pi\)
−0.132916 + 0.991127i \(0.542434\pi\)
\(74\) 3.04577e6 0.873749
\(75\) 4.10197e6 1.12274
\(76\) −2.75187e6 −0.719084
\(77\) 2.10659e6 0.525850
\(78\) 474552. 0.113228
\(79\) −4.21311e6 −0.961409 −0.480705 0.876883i \(-0.659619\pi\)
−0.480705 + 0.876883i \(0.659619\pi\)
\(80\) −1.96459e6 −0.428999
\(81\) 531441. 0.111111
\(82\) −663483. −0.132887
\(83\) 1.90022e6 0.364780 0.182390 0.983226i \(-0.441617\pi\)
0.182390 + 0.983226i \(0.441617\pi\)
\(84\) 592704. 0.109109
\(85\) 1.35626e7 2.39540
\(86\) −2.03968e6 −0.345794
\(87\) 1.03357e6 0.168276
\(88\) −3.14453e6 −0.491887
\(89\) −1.19875e6 −0.180244 −0.0901222 0.995931i \(-0.528726\pi\)
−0.0901222 + 0.995931i \(0.528726\pi\)
\(90\) 2.79723e6 0.404464
\(91\) −753571. −0.104828
\(92\) 5.85710e6 0.784197
\(93\) 415742. 0.0535961
\(94\) −6.40677e6 −0.795594
\(95\) 2.06233e7 2.46789
\(96\) −884736. −0.102062
\(97\) −9.55500e6 −1.06299 −0.531496 0.847061i \(-0.678370\pi\)
−0.531496 + 0.847061i \(0.678370\pi\)
\(98\) −941192. −0.101015
\(99\) 4.47726e6 0.463756
\(100\) 9.72318e6 0.972318
\(101\) −3.60039e6 −0.347716 −0.173858 0.984771i \(-0.555623\pi\)
−0.173858 + 0.984771i \(0.555623\pi\)
\(102\) 6.10784e6 0.569884
\(103\) −372595. −0.0335974 −0.0167987 0.999859i \(-0.505347\pi\)
−0.0167987 + 0.999859i \(0.505347\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) −4.44190e6 −0.374461
\(106\) 1.58432e7 1.29203
\(107\) −3.08080e6 −0.243119 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −2.69326e7 −1.99198 −0.995990 0.0894624i \(-0.971485\pi\)
−0.995990 + 0.0894624i \(0.971485\pi\)
\(110\) 2.35660e7 1.68815
\(111\) −1.02795e7 −0.713413
\(112\) 1.40493e6 0.0944911
\(113\) 1.57681e7 1.02803 0.514013 0.857782i \(-0.328158\pi\)
0.514013 + 0.857782i \(0.328158\pi\)
\(114\) 9.28755e6 0.587130
\(115\) −4.38948e7 −2.69136
\(116\) 2.44994e6 0.145731
\(117\) −1.60161e6 −0.0924500
\(118\) 1.76518e7 0.989011
\(119\) −9.69902e6 −0.527610
\(120\) 6.63047e6 0.350276
\(121\) 1.82327e7 0.935626
\(122\) 9.74677e6 0.485961
\(123\) 2.23926e6 0.108501
\(124\) 985462. 0.0464156
\(125\) −3.53969e7 −1.62099
\(126\) −2.00038e6 −0.0890871
\(127\) 2.51985e7 1.09160 0.545799 0.837916i \(-0.316226\pi\)
0.545799 + 0.837916i \(0.316226\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 6.88391e6 0.282339
\(130\) −8.43007e6 −0.336534
\(131\) 2.79431e7 1.08599 0.542994 0.839736i \(-0.317290\pi\)
0.542994 + 0.839736i \(0.317290\pi\)
\(132\) 1.06128e7 0.401624
\(133\) −1.47483e7 −0.543577
\(134\) −1.58764e7 −0.570014
\(135\) −9.44066e6 −0.330243
\(136\) 1.44778e7 0.493534
\(137\) 48254.4 0.00160330 0.000801651 1.00000i \(-0.499745\pi\)
0.000801651 1.00000i \(0.499745\pi\)
\(138\) −1.97677e7 −0.640294
\(139\) 2.07595e7 0.655639 0.327819 0.944740i \(-0.393686\pi\)
0.327819 + 0.944740i \(0.393686\pi\)
\(140\) −1.05289e7 −0.324292
\(141\) 2.16228e7 0.649600
\(142\) −2.49660e7 −0.731711
\(143\) −1.34932e7 −0.385868
\(144\) 2.98598e6 0.0833333
\(145\) −1.83606e7 −0.500147
\(146\) 7.06850e6 0.187971
\(147\) 3.17652e6 0.0824786
\(148\) −2.43662e7 −0.617834
\(149\) 7.41183e7 1.83558 0.917791 0.397065i \(-0.129971\pi\)
0.917791 + 0.397065i \(0.129971\pi\)
\(150\) −3.28157e7 −0.793895
\(151\) −6.13143e7 −1.44925 −0.724624 0.689144i \(-0.757987\pi\)
−0.724624 + 0.689144i \(0.757987\pi\)
\(152\) 2.20149e7 0.508469
\(153\) −2.06139e7 −0.465309
\(154\) −1.68527e7 −0.371832
\(155\) −7.38534e6 −0.159298
\(156\) −3.79642e6 −0.0800641
\(157\) 1.74448e7 0.359763 0.179882 0.983688i \(-0.442429\pi\)
0.179882 + 0.983688i \(0.442429\pi\)
\(158\) 3.37049e7 0.679819
\(159\) −5.34707e7 −1.05494
\(160\) 1.57167e7 0.303348
\(161\) 3.13904e7 0.592797
\(162\) −4.25153e6 −0.0785674
\(163\) −6.00021e7 −1.08520 −0.542600 0.839991i \(-0.682560\pi\)
−0.542600 + 0.839991i \(0.682560\pi\)
\(164\) 5.30786e6 0.0939650
\(165\) −7.95353e7 −1.37837
\(166\) −1.52018e7 −0.257938
\(167\) 5.97842e7 0.993297 0.496648 0.867952i \(-0.334564\pi\)
0.496648 + 0.867952i \(0.334564\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −1.08501e8 −1.69380
\(171\) −3.13455e7 −0.479389
\(172\) 1.63174e7 0.244513
\(173\) 4.45329e7 0.653913 0.326957 0.945039i \(-0.393977\pi\)
0.326957 + 0.945039i \(0.393977\pi\)
\(174\) −8.26854e6 −0.118989
\(175\) 5.21102e7 0.735004
\(176\) 2.51562e7 0.347817
\(177\) −5.95747e7 −0.807524
\(178\) 9.58996e6 0.127452
\(179\) 6.47574e6 0.0843925 0.0421962 0.999109i \(-0.486565\pi\)
0.0421962 + 0.999109i \(0.486565\pi\)
\(180\) −2.23779e7 −0.285999
\(181\) 1.02887e8 1.28969 0.644846 0.764313i \(-0.276922\pi\)
0.644846 + 0.764313i \(0.276922\pi\)
\(182\) 6.02857e6 0.0741249
\(183\) −3.28953e7 −0.396785
\(184\) −4.68568e7 −0.554511
\(185\) 1.82607e8 2.12040
\(186\) −3.32593e6 −0.0378982
\(187\) −1.73668e8 −1.94211
\(188\) 5.12542e7 0.562570
\(189\) 6.75127e6 0.0727393
\(190\) −1.64986e8 −1.74506
\(191\) 1.61049e8 1.67241 0.836204 0.548418i \(-0.184770\pi\)
0.836204 + 0.548418i \(0.184770\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 7.30667e7 0.731592 0.365796 0.930695i \(-0.380797\pi\)
0.365796 + 0.930695i \(0.380797\pi\)
\(194\) 7.64400e7 0.751648
\(195\) 2.84515e7 0.274779
\(196\) 7.52954e6 0.0714286
\(197\) 8.95011e7 0.834059 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(198\) −3.58181e7 −0.327925
\(199\) −9.67078e7 −0.869913 −0.434956 0.900452i \(-0.643236\pi\)
−0.434956 + 0.900452i \(0.643236\pi\)
\(200\) −7.77855e7 −0.687533
\(201\) 5.35829e7 0.465415
\(202\) 2.88031e7 0.245872
\(203\) 1.31301e7 0.110162
\(204\) −4.88627e7 −0.402969
\(205\) −3.97787e7 −0.322487
\(206\) 2.98076e6 0.0237570
\(207\) 6.67160e7 0.522798
\(208\) −8.99891e6 −0.0693375
\(209\) −2.64078e8 −2.00088
\(210\) 3.55352e7 0.264784
\(211\) 1.09331e8 0.801225 0.400613 0.916248i \(-0.368797\pi\)
0.400613 + 0.916248i \(0.368797\pi\)
\(212\) −1.26745e8 −0.913601
\(213\) 8.42603e7 0.597440
\(214\) 2.46464e7 0.171911
\(215\) −1.22288e8 −0.839166
\(216\) −1.00777e7 −0.0680414
\(217\) 5.28146e6 0.0350869
\(218\) 2.15461e8 1.40854
\(219\) −2.38562e7 −0.153478
\(220\) −1.88528e8 −1.19370
\(221\) 6.21246e7 0.387160
\(222\) 8.22359e7 0.504459
\(223\) 1.56478e8 0.944902 0.472451 0.881357i \(-0.343369\pi\)
0.472451 + 0.881357i \(0.343369\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 1.10753e8 0.648212
\(226\) −1.26145e8 −0.726924
\(227\) −2.98940e8 −1.69626 −0.848131 0.529787i \(-0.822272\pi\)
−0.848131 + 0.529787i \(0.822272\pi\)
\(228\) −7.43004e7 −0.415163
\(229\) −2.43669e8 −1.34084 −0.670418 0.741984i \(-0.733885\pi\)
−0.670418 + 0.741984i \(0.733885\pi\)
\(230\) 3.51159e8 1.90308
\(231\) 5.68778e7 0.303599
\(232\) −1.95995e7 −0.103047
\(233\) 1.42543e8 0.738244 0.369122 0.929381i \(-0.379658\pi\)
0.369122 + 0.929381i \(0.379658\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −3.84114e8 −1.93073
\(236\) −1.41214e8 −0.699336
\(237\) −1.13754e8 −0.555070
\(238\) 7.75921e7 0.373077
\(239\) 1.58648e8 0.751696 0.375848 0.926681i \(-0.377351\pi\)
0.375848 + 0.926681i \(0.377351\pi\)
\(240\) −5.30438e7 −0.247682
\(241\) 2.93080e8 1.34873 0.674367 0.738396i \(-0.264416\pi\)
0.674367 + 0.738396i \(0.264416\pi\)
\(242\) −1.45862e8 −0.661587
\(243\) 1.43489e7 0.0641500
\(244\) −7.79741e7 −0.343626
\(245\) −5.64286e7 −0.245142
\(246\) −1.79140e7 −0.0767221
\(247\) 9.44664e7 0.398876
\(248\) −7.88369e6 −0.0328208
\(249\) 5.13060e7 0.210606
\(250\) 2.83176e8 1.14621
\(251\) −2.85090e7 −0.113795 −0.0568976 0.998380i \(-0.518121\pi\)
−0.0568976 + 0.998380i \(0.518121\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 5.62067e8 2.18206
\(254\) −2.01588e8 −0.771876
\(255\) 3.66191e8 1.38299
\(256\) 1.67772e7 0.0625000
\(257\) −1.43945e8 −0.528969 −0.264484 0.964390i \(-0.585202\pi\)
−0.264484 + 0.964390i \(0.585202\pi\)
\(258\) −5.50713e7 −0.199644
\(259\) −1.30588e8 −0.467038
\(260\) 6.74405e7 0.237966
\(261\) 2.79063e7 0.0971540
\(262\) −2.23545e8 −0.767910
\(263\) 5.25151e8 1.78008 0.890040 0.455883i \(-0.150677\pi\)
0.890040 + 0.455883i \(0.150677\pi\)
\(264\) −8.49022e7 −0.283991
\(265\) 9.49868e8 3.13547
\(266\) 1.17986e8 0.384367
\(267\) −3.23661e7 −0.104064
\(268\) 1.27011e8 0.403061
\(269\) 2.56559e8 0.803627 0.401814 0.915722i \(-0.368380\pi\)
0.401814 + 0.915722i \(0.368380\pi\)
\(270\) 7.55252e7 0.233517
\(271\) 2.69629e8 0.822950 0.411475 0.911421i \(-0.365014\pi\)
0.411475 + 0.911421i \(0.365014\pi\)
\(272\) −1.15823e8 −0.348981
\(273\) −2.03464e7 −0.0605228
\(274\) −386035. −0.00113371
\(275\) 9.33069e8 2.70551
\(276\) 1.58142e8 0.452756
\(277\) 1.38687e8 0.392065 0.196032 0.980597i \(-0.437194\pi\)
0.196032 + 0.980597i \(0.437194\pi\)
\(278\) −1.66076e8 −0.463607
\(279\) 1.12250e7 0.0309437
\(280\) 8.42316e7 0.229309
\(281\) 1.99903e8 0.537460 0.268730 0.963215i \(-0.413396\pi\)
0.268730 + 0.963215i \(0.413396\pi\)
\(282\) −1.72983e8 −0.459337
\(283\) −2.76398e8 −0.724908 −0.362454 0.932002i \(-0.618061\pi\)
−0.362454 + 0.932002i \(0.618061\pi\)
\(284\) 1.99728e8 0.517398
\(285\) 5.56829e8 1.42484
\(286\) 1.07946e8 0.272850
\(287\) 2.84468e7 0.0710309
\(288\) −2.38879e7 −0.0589256
\(289\) 3.89251e8 0.948609
\(290\) 1.46884e8 0.353658
\(291\) −2.57985e8 −0.613718
\(292\) −5.65480e7 −0.132916
\(293\) −4.16733e8 −0.967880 −0.483940 0.875101i \(-0.660795\pi\)
−0.483940 + 0.875101i \(0.660795\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 1.05830e9 2.40011
\(296\) 1.94929e8 0.436874
\(297\) 1.20886e8 0.267750
\(298\) −5.92947e8 −1.29795
\(299\) −2.01063e8 −0.434994
\(300\) 2.62526e8 0.561368
\(301\) 8.74512e7 0.184834
\(302\) 4.90515e8 1.02477
\(303\) −9.72105e7 −0.200754
\(304\) −1.76119e8 −0.359542
\(305\) 5.84361e8 1.17932
\(306\) 1.64912e8 0.329023
\(307\) 3.42872e6 0.00676313 0.00338157 0.999994i \(-0.498924\pi\)
0.00338157 + 0.999994i \(0.498924\pi\)
\(308\) 1.34822e8 0.262925
\(309\) −1.00601e7 −0.0193975
\(310\) 5.90827e7 0.112641
\(311\) 5.83328e8 1.09964 0.549821 0.835282i \(-0.314696\pi\)
0.549821 + 0.835282i \(0.314696\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 6.80001e8 1.25344 0.626721 0.779244i \(-0.284396\pi\)
0.626721 + 0.779244i \(0.284396\pi\)
\(314\) −1.39558e8 −0.254391
\(315\) −1.19931e8 −0.216195
\(316\) −2.69639e8 −0.480705
\(317\) −2.78709e8 −0.491409 −0.245705 0.969345i \(-0.579019\pi\)
−0.245705 + 0.969345i \(0.579019\pi\)
\(318\) 4.27766e8 0.745952
\(319\) 2.35104e8 0.405502
\(320\) −1.25733e8 −0.214499
\(321\) −8.31815e7 −0.140365
\(322\) −2.51123e8 −0.419171
\(323\) 1.21585e9 2.00758
\(324\) 3.40122e7 0.0555556
\(325\) −3.33779e8 −0.539345
\(326\) 4.80017e8 0.767352
\(327\) −7.27179e8 −1.15007
\(328\) −4.24629e7 −0.0664433
\(329\) 2.74690e8 0.425263
\(330\) 6.36282e8 0.974655
\(331\) 8.50726e6 0.0128941 0.00644706 0.999979i \(-0.497948\pi\)
0.00644706 + 0.999979i \(0.497948\pi\)
\(332\) 1.21614e8 0.182390
\(333\) −2.77546e8 −0.411889
\(334\) −4.78274e8 −0.702367
\(335\) −9.51861e8 −1.38330
\(336\) 3.79331e7 0.0545545
\(337\) 1.17226e9 1.66847 0.834236 0.551408i \(-0.185909\pi\)
0.834236 + 0.551408i \(0.185909\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 4.25738e8 0.593531
\(340\) 8.68009e8 1.19770
\(341\) 9.45681e7 0.129153
\(342\) 2.50764e8 0.338980
\(343\) 4.03536e7 0.0539949
\(344\) −1.30539e8 −0.172897
\(345\) −1.18516e9 −1.55385
\(346\) −3.56264e8 −0.462386
\(347\) −3.54944e8 −0.456044 −0.228022 0.973656i \(-0.573226\pi\)
−0.228022 + 0.973656i \(0.573226\pi\)
\(348\) 6.61483e7 0.0841379
\(349\) 4.68755e8 0.590278 0.295139 0.955454i \(-0.404634\pi\)
0.295139 + 0.955454i \(0.404634\pi\)
\(350\) −4.16882e8 −0.519726
\(351\) −4.32436e7 −0.0533761
\(352\) −2.01250e8 −0.245944
\(353\) 4.99694e8 0.604634 0.302317 0.953207i \(-0.402240\pi\)
0.302317 + 0.953207i \(0.402240\pi\)
\(354\) 4.76598e8 0.571006
\(355\) −1.49682e9 −1.77570
\(356\) −7.67197e7 −0.0901222
\(357\) −2.61873e8 −0.304616
\(358\) −5.18059e7 −0.0596745
\(359\) 2.02205e8 0.230654 0.115327 0.993328i \(-0.463208\pi\)
0.115327 + 0.993328i \(0.463208\pi\)
\(360\) 1.79023e8 0.202232
\(361\) 9.54949e8 1.06833
\(362\) −8.23097e8 −0.911950
\(363\) 4.92283e8 0.540184
\(364\) −4.82285e7 −0.0524142
\(365\) 4.23787e8 0.456166
\(366\) 2.63163e8 0.280570
\(367\) −8.99197e8 −0.949563 −0.474782 0.880104i \(-0.657473\pi\)
−0.474782 + 0.880104i \(0.657473\pi\)
\(368\) 3.74854e8 0.392099
\(369\) 6.04599e7 0.0626434
\(370\) −1.46086e9 −1.49935
\(371\) −6.79276e8 −0.690618
\(372\) 2.66075e7 0.0267981
\(373\) −2.21113e8 −0.220614 −0.110307 0.993898i \(-0.535183\pi\)
−0.110307 + 0.993898i \(0.535183\pi\)
\(374\) 1.38934e9 1.37328
\(375\) −9.55718e8 −0.935880
\(376\) −4.10033e8 −0.397797
\(377\) −8.41018e7 −0.0808370
\(378\) −5.40102e7 −0.0514344
\(379\) −3.08848e8 −0.291412 −0.145706 0.989328i \(-0.546545\pi\)
−0.145706 + 0.989328i \(0.546545\pi\)
\(380\) 1.31989e9 1.23394
\(381\) 6.80361e8 0.630234
\(382\) −1.28840e9 −1.18257
\(383\) −7.66524e8 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −1.01039e9 −0.902355
\(386\) −5.84534e8 −0.517314
\(387\) 1.85866e8 0.163009
\(388\) −6.11520e8 −0.531496
\(389\) −6.89048e7 −0.0593507 −0.0296754 0.999560i \(-0.509447\pi\)
−0.0296754 + 0.999560i \(0.509447\pi\)
\(390\) −2.27612e8 −0.194298
\(391\) −2.58783e9 −2.18936
\(392\) −6.02363e7 −0.0505076
\(393\) 7.54464e8 0.626996
\(394\) −7.16009e8 −0.589769
\(395\) 2.02076e9 1.64977
\(396\) 2.86545e8 0.231878
\(397\) 2.18830e9 1.75525 0.877627 0.479343i \(-0.159125\pi\)
0.877627 + 0.479343i \(0.159125\pi\)
\(398\) 7.73662e8 0.615121
\(399\) −3.98204e8 −0.313834
\(400\) 6.22284e8 0.486159
\(401\) 6.73081e8 0.521269 0.260635 0.965438i \(-0.416068\pi\)
0.260635 + 0.965438i \(0.416068\pi\)
\(402\) −4.28663e8 −0.329098
\(403\) −3.38291e7 −0.0257467
\(404\) −2.30425e8 −0.173858
\(405\) −2.54898e8 −0.190666
\(406\) −1.05041e8 −0.0778965
\(407\) −2.33826e9 −1.71914
\(408\) 3.90901e8 0.284942
\(409\) 2.44101e9 1.76416 0.882078 0.471103i \(-0.156144\pi\)
0.882078 + 0.471103i \(0.156144\pi\)
\(410\) 3.18230e8 0.228033
\(411\) 1.30287e6 0.000925667 0
\(412\) −2.38461e7 −0.0167987
\(413\) −7.56819e8 −0.528649
\(414\) −5.33728e8 −0.369674
\(415\) −9.11412e8 −0.625960
\(416\) 7.19913e7 0.0490290
\(417\) 5.60506e8 0.378533
\(418\) 2.11263e9 1.41483
\(419\) 1.37884e9 0.915725 0.457863 0.889023i \(-0.348615\pi\)
0.457863 + 0.889023i \(0.348615\pi\)
\(420\) −2.84282e8 −0.187230
\(421\) 1.78764e9 1.16760 0.583800 0.811898i \(-0.301565\pi\)
0.583800 + 0.811898i \(0.301565\pi\)
\(422\) −8.74648e8 −0.566552
\(423\) 5.83817e8 0.375047
\(424\) 1.01396e9 0.646014
\(425\) −4.29598e9 −2.71457
\(426\) −6.74082e8 −0.422454
\(427\) −4.17893e8 −0.259757
\(428\) −1.97171e8 −0.121560
\(429\) −3.64317e8 −0.222781
\(430\) 9.78301e8 0.593380
\(431\) −2.91772e8 −0.175539 −0.0877695 0.996141i \(-0.527974\pi\)
−0.0877695 + 0.996141i \(0.527974\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −1.15485e9 −0.683623 −0.341811 0.939769i \(-0.611040\pi\)
−0.341811 + 0.939769i \(0.611040\pi\)
\(434\) −4.22517e7 −0.0248102
\(435\) −4.95735e8 −0.288760
\(436\) −1.72368e9 −0.995990
\(437\) −3.93505e9 −2.25562
\(438\) 1.90849e8 0.108525
\(439\) 2.62647e9 1.48166 0.740828 0.671695i \(-0.234433\pi\)
0.740828 + 0.671695i \(0.234433\pi\)
\(440\) 1.50822e9 0.844076
\(441\) 8.57661e7 0.0476190
\(442\) −4.96997e8 −0.273764
\(443\) 2.50764e9 1.37042 0.685209 0.728347i \(-0.259711\pi\)
0.685209 + 0.728347i \(0.259711\pi\)
\(444\) −6.57887e8 −0.356706
\(445\) 5.74960e8 0.309299
\(446\) −1.25183e9 −0.668146
\(447\) 2.00119e9 1.05977
\(448\) 8.99154e7 0.0472456
\(449\) −2.66780e9 −1.39088 −0.695442 0.718583i \(-0.744791\pi\)
−0.695442 + 0.718583i \(0.744791\pi\)
\(450\) −8.86025e8 −0.458355
\(451\) 5.09360e8 0.261461
\(452\) 1.00916e9 0.514013
\(453\) −1.65549e9 −0.836724
\(454\) 2.39152e9 1.19944
\(455\) 3.61439e8 0.179885
\(456\) 5.94403e8 0.293565
\(457\) −9.89279e8 −0.484855 −0.242428 0.970169i \(-0.577944\pi\)
−0.242428 + 0.970169i \(0.577944\pi\)
\(458\) 1.94935e9 0.948114
\(459\) −5.56576e8 −0.268646
\(460\) −2.80927e9 −1.34568
\(461\) 1.54225e9 0.733165 0.366583 0.930386i \(-0.380528\pi\)
0.366583 + 0.930386i \(0.380528\pi\)
\(462\) −4.55023e8 −0.214677
\(463\) 2.35562e9 1.10299 0.551496 0.834178i \(-0.314057\pi\)
0.551496 + 0.834178i \(0.314057\pi\)
\(464\) 1.56796e8 0.0728655
\(465\) −1.99404e8 −0.0919706
\(466\) −1.14034e9 −0.522017
\(467\) 2.80161e9 1.27291 0.636455 0.771314i \(-0.280400\pi\)
0.636455 + 0.771314i \(0.280400\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) 6.80701e8 0.304685
\(470\) 3.07291e9 1.36524
\(471\) 4.71009e8 0.207709
\(472\) 1.12971e9 0.494505
\(473\) 1.56587e9 0.680366
\(474\) 9.10032e8 0.392494
\(475\) −6.53245e9 −2.79672
\(476\) −6.20737e8 −0.263805
\(477\) −1.44371e9 −0.609068
\(478\) −1.26919e9 −0.531530
\(479\) −3.51487e9 −1.46129 −0.730643 0.682760i \(-0.760780\pi\)
−0.730643 + 0.682760i \(0.760780\pi\)
\(480\) 4.24350e8 0.175138
\(481\) 8.36445e8 0.342712
\(482\) −2.34464e9 −0.953699
\(483\) 8.47541e8 0.342252
\(484\) 1.16689e9 0.467813
\(485\) 4.58291e9 1.82409
\(486\) −1.14791e8 −0.0453609
\(487\) 1.17215e9 0.459866 0.229933 0.973206i \(-0.426149\pi\)
0.229933 + 0.973206i \(0.426149\pi\)
\(488\) 6.23793e8 0.242980
\(489\) −1.62006e9 −0.626540
\(490\) 4.51429e8 0.173342
\(491\) 3.21207e9 1.22461 0.612307 0.790620i \(-0.290242\pi\)
0.612307 + 0.790620i \(0.290242\pi\)
\(492\) 1.43312e8 0.0542507
\(493\) −1.08245e9 −0.406859
\(494\) −7.55731e8 −0.282048
\(495\) −2.14745e9 −0.795803
\(496\) 6.30695e7 0.0232078
\(497\) 1.07042e9 0.391116
\(498\) −4.10448e8 −0.148921
\(499\) −5.25523e9 −1.89339 −0.946695 0.322132i \(-0.895601\pi\)
−0.946695 + 0.322132i \(0.895601\pi\)
\(500\) −2.26540e9 −0.810496
\(501\) 1.61417e9 0.573480
\(502\) 2.28072e8 0.0804653
\(503\) −4.58055e9 −1.60483 −0.802416 0.596765i \(-0.796453\pi\)
−0.802416 + 0.596765i \(0.796453\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 1.72687e9 0.596679
\(506\) −4.49653e9 −1.54295
\(507\) 1.30324e8 0.0444116
\(508\) 1.61271e9 0.545799
\(509\) 2.78712e9 0.936793 0.468397 0.883518i \(-0.344832\pi\)
0.468397 + 0.883518i \(0.344832\pi\)
\(510\) −2.92953e9 −0.977918
\(511\) −3.03062e8 −0.100475
\(512\) −1.34218e8 −0.0441942
\(513\) −8.46328e8 −0.276776
\(514\) 1.15156e9 0.374037
\(515\) 1.78709e8 0.0576530
\(516\) 4.40570e8 0.141170
\(517\) 4.91852e9 1.56537
\(518\) 1.04470e9 0.330246
\(519\) 1.20239e9 0.377537
\(520\) −5.39524e8 −0.168267
\(521\) −3.50734e8 −0.108654 −0.0543270 0.998523i \(-0.517301\pi\)
−0.0543270 + 0.998523i \(0.517301\pi\)
\(522\) −2.23251e8 −0.0686983
\(523\) −5.56937e9 −1.70235 −0.851177 0.524879i \(-0.824111\pi\)
−0.851177 + 0.524879i \(0.824111\pi\)
\(524\) 1.78836e9 0.542994
\(525\) 1.40698e9 0.424355
\(526\) −4.20121e9 −1.25871
\(527\) −4.35405e8 −0.129585
\(528\) 6.79217e8 0.200812
\(529\) 4.97057e9 1.45986
\(530\) −7.59894e9 −2.21711
\(531\) −1.60852e9 −0.466224
\(532\) −9.43890e8 −0.271788
\(533\) −1.82209e8 −0.0521224
\(534\) 2.58929e8 0.0735845
\(535\) 1.47766e9 0.417192
\(536\) −1.01609e9 −0.285007
\(537\) 1.74845e8 0.0487240
\(538\) −2.05247e9 −0.568250
\(539\) 7.22559e8 0.198753
\(540\) −6.04202e8 −0.165122
\(541\) −2.30163e9 −0.624950 −0.312475 0.949926i \(-0.601158\pi\)
−0.312475 + 0.949926i \(0.601158\pi\)
\(542\) −2.15703e9 −0.581914
\(543\) 2.77795e9 0.744604
\(544\) 9.26581e8 0.246767
\(545\) 1.29178e10 3.41823
\(546\) 1.62771e8 0.0427960
\(547\) −3.53474e9 −0.923425 −0.461712 0.887030i \(-0.652765\pi\)
−0.461712 + 0.887030i \(0.652765\pi\)
\(548\) 3.08828e6 0.000801651 0
\(549\) −8.88174e8 −0.229084
\(550\) −7.46455e9 −1.91308
\(551\) −1.64597e9 −0.419172
\(552\) −1.26513e9 −0.320147
\(553\) −1.44510e9 −0.363378
\(554\) −1.10950e9 −0.277232
\(555\) 4.93040e9 1.22421
\(556\) 1.32861e9 0.327819
\(557\) −1.42066e9 −0.348335 −0.174167 0.984716i \(-0.555723\pi\)
−0.174167 + 0.984716i \(0.555723\pi\)
\(558\) −8.98002e7 −0.0218805
\(559\) −5.60147e8 −0.135631
\(560\) −6.73853e8 −0.162146
\(561\) −4.68902e9 −1.12128
\(562\) −1.59922e9 −0.380042
\(563\) −7.14098e8 −0.168647 −0.0843235 0.996438i \(-0.526873\pi\)
−0.0843235 + 0.996438i \(0.526873\pi\)
\(564\) 1.38386e9 0.324800
\(565\) −7.56292e9 −1.76409
\(566\) 2.21119e9 0.512587
\(567\) 1.82284e8 0.0419961
\(568\) −1.59782e9 −0.365856
\(569\) 6.03946e9 1.37437 0.687187 0.726481i \(-0.258845\pi\)
0.687187 + 0.726481i \(0.258845\pi\)
\(570\) −4.45463e9 −1.00751
\(571\) −8.20167e9 −1.84364 −0.921820 0.387619i \(-0.873298\pi\)
−0.921820 + 0.387619i \(0.873298\pi\)
\(572\) −8.63565e8 −0.192934
\(573\) 4.34833e9 0.965565
\(574\) −2.27575e8 −0.0502264
\(575\) 1.39037e10 3.04996
\(576\) 1.91103e8 0.0416667
\(577\) −6.95530e9 −1.50730 −0.753651 0.657275i \(-0.771709\pi\)
−0.753651 + 0.657275i \(0.771709\pi\)
\(578\) −3.11401e9 −0.670768
\(579\) 1.97280e9 0.422385
\(580\) −1.17508e9 −0.250074
\(581\) 6.51776e8 0.137874
\(582\) 2.06388e9 0.433964
\(583\) −1.21629e10 −2.54213
\(584\) 4.52384e8 0.0939857
\(585\) 7.68190e8 0.158644
\(586\) 3.33387e9 0.684394
\(587\) 7.08547e9 1.44589 0.722945 0.690905i \(-0.242788\pi\)
0.722945 + 0.690905i \(0.242788\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) −6.62075e8 −0.133507
\(590\) −8.46640e9 −1.69714
\(591\) 2.41653e9 0.481544
\(592\) −1.55944e9 −0.308917
\(593\) −6.49353e9 −1.27876 −0.639381 0.768890i \(-0.720809\pi\)
−0.639381 + 0.768890i \(0.720809\pi\)
\(594\) −9.67089e8 −0.189328
\(595\) 4.65199e9 0.905376
\(596\) 4.74357e9 0.917791
\(597\) −2.61111e9 −0.502244
\(598\) 1.60851e9 0.307587
\(599\) −4.35434e9 −0.827806 −0.413903 0.910321i \(-0.635835\pi\)
−0.413903 + 0.910321i \(0.635835\pi\)
\(600\) −2.10021e9 −0.396947
\(601\) 3.14701e9 0.591341 0.295670 0.955290i \(-0.404457\pi\)
0.295670 + 0.955290i \(0.404457\pi\)
\(602\) −6.99610e8 −0.130698
\(603\) 1.44674e9 0.268707
\(604\) −3.92412e9 −0.724624
\(605\) −8.74504e9 −1.60553
\(606\) 7.77684e8 0.141954
\(607\) −1.93569e9 −0.351298 −0.175649 0.984453i \(-0.556202\pi\)
−0.175649 + 0.984453i \(0.556202\pi\)
\(608\) 1.40896e9 0.254235
\(609\) 3.54514e8 0.0636022
\(610\) −4.67489e9 −0.833906
\(611\) −1.75946e9 −0.312058
\(612\) −1.31929e9 −0.232654
\(613\) −3.11478e9 −0.546154 −0.273077 0.961992i \(-0.588041\pi\)
−0.273077 + 0.961992i \(0.588041\pi\)
\(614\) −2.74298e7 −0.00478226
\(615\) −1.07403e9 −0.186188
\(616\) −1.07857e9 −0.185916
\(617\) 2.70867e9 0.464257 0.232128 0.972685i \(-0.425431\pi\)
0.232128 + 0.972685i \(0.425431\pi\)
\(618\) 8.04804e7 0.0137161
\(619\) −4.55504e9 −0.771925 −0.385963 0.922514i \(-0.626131\pi\)
−0.385963 + 0.922514i \(0.626131\pi\)
\(620\) −4.72662e8 −0.0796489
\(621\) 1.80133e9 0.301838
\(622\) −4.66663e9 −0.777565
\(623\) −4.11170e8 −0.0681260
\(624\) −2.42971e8 −0.0400320
\(625\) 5.10849e9 0.836976
\(626\) −5.44001e9 −0.886318
\(627\) −7.13011e9 −1.15521
\(628\) 1.11647e9 0.179882
\(629\) 1.07657e10 1.72490
\(630\) 9.59450e8 0.152873
\(631\) 6.79597e9 1.07683 0.538417 0.842679i \(-0.319023\pi\)
0.538417 + 0.842679i \(0.319023\pi\)
\(632\) 2.15711e9 0.339909
\(633\) 2.95194e9 0.462588
\(634\) 2.22967e9 0.347479
\(635\) −1.20861e10 −1.87317
\(636\) −3.42213e9 −0.527468
\(637\) −2.58475e8 −0.0396214
\(638\) −1.88083e9 −0.286733
\(639\) 2.27503e9 0.344932
\(640\) 1.00587e9 0.151674
\(641\) 1.28737e10 1.93063 0.965316 0.261083i \(-0.0840796\pi\)
0.965316 + 0.261083i \(0.0840796\pi\)
\(642\) 6.65452e8 0.0992531
\(643\) 9.44890e9 1.40166 0.700830 0.713328i \(-0.252813\pi\)
0.700830 + 0.713328i \(0.252813\pi\)
\(644\) 2.00899e9 0.296399
\(645\) −3.30177e9 −0.484493
\(646\) −9.72682e9 −1.41957
\(647\) 3.05647e9 0.443666 0.221833 0.975085i \(-0.428796\pi\)
0.221833 + 0.975085i \(0.428796\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −1.35514e10 −1.94593
\(650\) 2.67023e9 0.381375
\(651\) 1.42599e8 0.0202574
\(652\) −3.84013e9 −0.542600
\(653\) 7.10654e9 0.998762 0.499381 0.866382i \(-0.333561\pi\)
0.499381 + 0.866382i \(0.333561\pi\)
\(654\) 5.81744e9 0.813223
\(655\) −1.34025e10 −1.86355
\(656\) 3.39703e8 0.0469825
\(657\) −6.44117e8 −0.0886106
\(658\) −2.19752e9 −0.300706
\(659\) 3.63623e9 0.494940 0.247470 0.968896i \(-0.420401\pi\)
0.247470 + 0.968896i \(0.420401\pi\)
\(660\) −5.09026e9 −0.689185
\(661\) 6.86469e9 0.924519 0.462259 0.886745i \(-0.347039\pi\)
0.462259 + 0.886745i \(0.347039\pi\)
\(662\) −6.80581e7 −0.00911752
\(663\) 1.67736e9 0.223527
\(664\) −9.72913e8 −0.128969
\(665\) 7.07379e9 0.932774
\(666\) 2.22037e9 0.291250
\(667\) 3.50330e9 0.457127
\(668\) 3.82619e9 0.496648
\(669\) 4.22491e9 0.545539
\(670\) 7.61489e9 0.978141
\(671\) −7.48266e9 −0.956152
\(672\) −3.03464e8 −0.0385758
\(673\) −1.37598e10 −1.74004 −0.870020 0.493016i \(-0.835895\pi\)
−0.870020 + 0.493016i \(0.835895\pi\)
\(674\) −9.37807e9 −1.17979
\(675\) 2.99033e9 0.374246
\(676\) 3.08916e8 0.0384615
\(677\) 1.29093e10 1.59897 0.799487 0.600683i \(-0.205105\pi\)
0.799487 + 0.600683i \(0.205105\pi\)
\(678\) −3.40590e9 −0.419690
\(679\) −3.27737e9 −0.401773
\(680\) −6.94408e9 −0.846902
\(681\) −8.07137e9 −0.979337
\(682\) −7.56545e8 −0.0913250
\(683\) −4.67362e9 −0.561282 −0.280641 0.959813i \(-0.590547\pi\)
−0.280641 + 0.959813i \(0.590547\pi\)
\(684\) −2.00611e9 −0.239695
\(685\) −2.31445e7 −0.00275126
\(686\) −3.22829e8 −0.0381802
\(687\) −6.57906e9 −0.774132
\(688\) 1.04432e9 0.122257
\(689\) 4.35093e9 0.506775
\(690\) 9.48129e9 1.09874
\(691\) 1.39132e10 1.60418 0.802092 0.597200i \(-0.203720\pi\)
0.802092 + 0.597200i \(0.203720\pi\)
\(692\) 2.85011e9 0.326957
\(693\) 1.53570e9 0.175283
\(694\) 2.83955e9 0.322472
\(695\) −9.95697e9 −1.12507
\(696\) −5.29187e8 −0.0594944
\(697\) −2.34516e9 −0.262336
\(698\) −3.75004e9 −0.417390
\(699\) 3.84866e9 0.426225
\(700\) 3.33505e9 0.367502
\(701\) 3.95020e9 0.433117 0.216559 0.976270i \(-0.430517\pi\)
0.216559 + 0.976270i \(0.430517\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 1.63702e10 1.77710
\(704\) 1.61000e9 0.173908
\(705\) −1.03711e10 −1.11471
\(706\) −3.99755e9 −0.427541
\(707\) −1.23493e9 −0.131424
\(708\) −3.81278e9 −0.403762
\(709\) 9.43852e9 0.994585 0.497293 0.867583i \(-0.334328\pi\)
0.497293 + 0.867583i \(0.334328\pi\)
\(710\) 1.19746e10 1.25561
\(711\) −3.07136e9 −0.320470
\(712\) 6.13758e8 0.0637260
\(713\) 1.40917e9 0.145596
\(714\) 2.09499e9 0.215396
\(715\) 6.47181e9 0.662148
\(716\) 4.14447e8 0.0421962
\(717\) 4.28350e9 0.433992
\(718\) −1.61764e9 −0.163097
\(719\) −1.77940e10 −1.78535 −0.892673 0.450704i \(-0.851173\pi\)
−0.892673 + 0.450704i \(0.851173\pi\)
\(720\) −1.43218e9 −0.143000
\(721\) −1.27800e8 −0.0126986
\(722\) −7.63959e9 −0.755422
\(723\) 7.91316e9 0.778692
\(724\) 6.58477e9 0.644846
\(725\) 5.81572e9 0.566788
\(726\) −3.93826e9 −0.381968
\(727\) −1.27477e10 −1.23045 −0.615224 0.788353i \(-0.710934\pi\)
−0.615224 + 0.788353i \(0.710934\pi\)
\(728\) 3.85828e8 0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −3.39030e9 −0.322558
\(731\) −7.20950e9 −0.682644
\(732\) −2.10530e9 −0.198393
\(733\) −1.26461e10 −1.18602 −0.593012 0.805194i \(-0.702061\pi\)
−0.593012 + 0.805194i \(0.702061\pi\)
\(734\) 7.19358e9 0.671442
\(735\) −1.52357e9 −0.141533
\(736\) −2.99884e9 −0.277256
\(737\) 1.21884e10 1.12153
\(738\) −4.83679e8 −0.0442955
\(739\) −1.25576e10 −1.14459 −0.572297 0.820046i \(-0.693948\pi\)
−0.572297 + 0.820046i \(0.693948\pi\)
\(740\) 1.16869e10 1.06020
\(741\) 2.55059e9 0.230291
\(742\) 5.43421e9 0.488341
\(743\) −1.09171e10 −0.976445 −0.488222 0.872719i \(-0.662354\pi\)
−0.488222 + 0.872719i \(0.662354\pi\)
\(744\) −2.12860e8 −0.0189491
\(745\) −3.55497e10 −3.14985
\(746\) 1.76890e9 0.155997
\(747\) 1.38526e9 0.121593
\(748\) −1.11147e10 −0.971053
\(749\) −1.05671e9 −0.0918905
\(750\) 7.64574e9 0.661767
\(751\) 1.05692e10 0.910548 0.455274 0.890351i \(-0.349541\pi\)
0.455274 + 0.890351i \(0.349541\pi\)
\(752\) 3.28027e9 0.281285
\(753\) −7.69742e8 −0.0656996
\(754\) 6.72814e8 0.0571604
\(755\) 2.94085e10 2.48690
\(756\) 4.32081e8 0.0363696
\(757\) 1.52485e10 1.27759 0.638794 0.769378i \(-0.279434\pi\)
0.638794 + 0.769378i \(0.279434\pi\)
\(758\) 2.47079e9 0.206060
\(759\) 1.51758e10 1.25981
\(760\) −1.05591e10 −0.872531
\(761\) −3.85782e9 −0.317319 −0.158659 0.987333i \(-0.550717\pi\)
−0.158659 + 0.987333i \(0.550717\pi\)
\(762\) −5.44288e9 −0.445643
\(763\) −9.23787e9 −0.752898
\(764\) 1.03072e10 0.836204
\(765\) 9.88717e9 0.798467
\(766\) 6.13219e9 0.492964
\(767\) 4.84762e9 0.387922
\(768\) 4.52985e8 0.0360844
\(769\) 2.02192e10 1.60333 0.801663 0.597777i \(-0.203949\pi\)
0.801663 + 0.597777i \(0.203949\pi\)
\(770\) 8.08314e9 0.638062
\(771\) −3.88651e9 −0.305400
\(772\) 4.67627e9 0.365796
\(773\) −8.17755e9 −0.636788 −0.318394 0.947958i \(-0.603144\pi\)
−0.318394 + 0.947958i \(0.603144\pi\)
\(774\) −1.48693e9 −0.115265
\(775\) 2.33931e9 0.180523
\(776\) 4.89216e9 0.375824
\(777\) −3.52586e9 −0.269645
\(778\) 5.51238e8 0.0419673
\(779\) −3.56605e9 −0.270275
\(780\) 1.82089e9 0.137390
\(781\) 1.91666e10 1.43968
\(782\) 2.07027e10 1.54811
\(783\) 7.53471e8 0.0560919
\(784\) 4.81890e8 0.0357143
\(785\) −8.36713e9 −0.617352
\(786\) −6.03571e9 −0.443353
\(787\) −8.75406e9 −0.640174 −0.320087 0.947388i \(-0.603712\pi\)
−0.320087 + 0.947388i \(0.603712\pi\)
\(788\) 5.72807e9 0.417029
\(789\) 1.41791e10 1.02773
\(790\) −1.61660e10 −1.16657
\(791\) 5.40845e9 0.388557
\(792\) −2.29236e9 −0.163962
\(793\) 2.67671e9 0.190610
\(794\) −1.75064e10 −1.24115
\(795\) 2.56464e10 1.81026
\(796\) −6.18930e9 −0.434956
\(797\) −7.25916e9 −0.507905 −0.253952 0.967217i \(-0.581731\pi\)
−0.253952 + 0.967217i \(0.581731\pi\)
\(798\) 3.18563e9 0.221914
\(799\) −2.26455e10 −1.57061
\(800\) −4.97827e9 −0.343766
\(801\) −8.73885e8 −0.0600815
\(802\) −5.38465e9 −0.368593
\(803\) −5.42653e9 −0.369843
\(804\) 3.42931e9 0.232707
\(805\) −1.50559e10 −1.01724
\(806\) 2.70632e8 0.0182057
\(807\) 6.92710e9 0.463974
\(808\) 1.84340e9 0.122936
\(809\) 1.05605e9 0.0701235 0.0350618 0.999385i \(-0.488837\pi\)
0.0350618 + 0.999385i \(0.488837\pi\)
\(810\) 2.03918e9 0.134821
\(811\) 2.15233e10 1.41689 0.708446 0.705765i \(-0.249397\pi\)
0.708446 + 0.705765i \(0.249397\pi\)
\(812\) 8.40329e8 0.0550812
\(813\) 7.27998e9 0.475131
\(814\) 1.87061e10 1.21562
\(815\) 2.87791e10 1.86220
\(816\) −3.12721e9 −0.201485
\(817\) −1.09627e10 −0.703302
\(818\) −1.95280e10 −1.24745
\(819\) −5.49353e8 −0.0349428
\(820\) −2.54584e9 −0.161243
\(821\) −2.08774e10 −1.31666 −0.658331 0.752728i \(-0.728737\pi\)
−0.658331 + 0.752728i \(0.728737\pi\)
\(822\) −1.04230e7 −0.000654545 0
\(823\) 7.12682e9 0.445653 0.222826 0.974858i \(-0.428472\pi\)
0.222826 + 0.974858i \(0.428472\pi\)
\(824\) 1.90768e8 0.0118785
\(825\) 2.51929e10 1.56203
\(826\) 6.05455e9 0.373811
\(827\) −2.41290e10 −1.48344 −0.741720 0.670709i \(-0.765990\pi\)
−0.741720 + 0.670709i \(0.765990\pi\)
\(828\) 4.26983e9 0.261399
\(829\) 1.23027e10 0.749996 0.374998 0.927026i \(-0.377643\pi\)
0.374998 + 0.927026i \(0.377643\pi\)
\(830\) 7.29130e9 0.442621
\(831\) 3.74456e9 0.226359
\(832\) −5.75930e8 −0.0346688
\(833\) −3.32676e9 −0.199418
\(834\) −4.48405e9 −0.267663
\(835\) −2.86746e10 −1.70449
\(836\) −1.69010e10 −1.00044
\(837\) 3.03076e8 0.0178654
\(838\) −1.10307e10 −0.647516
\(839\) 1.47506e10 0.862269 0.431134 0.902288i \(-0.358113\pi\)
0.431134 + 0.902288i \(0.358113\pi\)
\(840\) 2.27425e9 0.132392
\(841\) −1.57845e10 −0.915050
\(842\) −1.43012e10 −0.825617
\(843\) 5.39737e9 0.310303
\(844\) 6.99718e9 0.400613
\(845\) −2.31511e9 −0.132000
\(846\) −4.67054e9 −0.265198
\(847\) 6.25382e9 0.353633
\(848\) −8.11171e9 −0.456801
\(849\) −7.46275e9 −0.418526
\(850\) 3.43678e10 1.91949
\(851\) −3.48426e10 −1.93801
\(852\) 5.39266e9 0.298720
\(853\) 1.07693e10 0.594109 0.297055 0.954860i \(-0.403996\pi\)
0.297055 + 0.954860i \(0.403996\pi\)
\(854\) 3.34314e9 0.183676
\(855\) 1.50344e10 0.822630
\(856\) 1.57737e9 0.0859557
\(857\) −1.90321e10 −1.03289 −0.516444 0.856321i \(-0.672744\pi\)
−0.516444 + 0.856321i \(0.672744\pi\)
\(858\) 2.91453e9 0.157530
\(859\) 2.90696e9 0.156482 0.0782408 0.996934i \(-0.475070\pi\)
0.0782408 + 0.996934i \(0.475070\pi\)
\(860\) −7.82641e9 −0.419583
\(861\) 7.68064e8 0.0410097
\(862\) 2.33418e9 0.124125
\(863\) 6.55988e9 0.347423 0.173711 0.984797i \(-0.444424\pi\)
0.173711 + 0.984797i \(0.444424\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −2.13596e10 −1.12211
\(866\) 9.23877e9 0.483394
\(867\) 1.05098e10 0.547680
\(868\) 3.38013e8 0.0175434
\(869\) −2.58755e10 −1.33758
\(870\) 3.96588e9 0.204184
\(871\) −4.36006e9 −0.223578
\(872\) 1.37895e10 0.704271
\(873\) −6.96560e9 −0.354330
\(874\) 3.14804e10 1.59496
\(875\) −1.21412e10 −0.612677
\(876\) −1.52680e9 −0.0767390
\(877\) 1.55910e10 0.780502 0.390251 0.920709i \(-0.372388\pi\)
0.390251 + 0.920709i \(0.372388\pi\)
\(878\) −2.10118e10 −1.04769
\(879\) −1.12518e10 −0.558806
\(880\) −1.20658e10 −0.596852
\(881\) 2.06150e10 1.01571 0.507854 0.861443i \(-0.330439\pi\)
0.507854 + 0.861443i \(0.330439\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −2.09284e10 −1.02299 −0.511497 0.859285i \(-0.670909\pi\)
−0.511497 + 0.859285i \(0.670909\pi\)
\(884\) 3.97597e9 0.193580
\(885\) 2.85741e10 1.38571
\(886\) −2.00612e10 −0.969031
\(887\) 1.89135e10 0.909995 0.454998 0.890493i \(-0.349640\pi\)
0.454998 + 0.890493i \(0.349640\pi\)
\(888\) 5.26310e9 0.252229
\(889\) 8.64310e9 0.412585
\(890\) −4.59968e9 −0.218707
\(891\) 3.26392e9 0.154585
\(892\) 1.00146e10 0.472451
\(893\) −3.44347e10 −1.61814
\(894\) −1.60096e10 −0.749373
\(895\) −3.10599e9 −0.144817
\(896\) −7.19323e8 −0.0334077
\(897\) −5.42871e9 −0.251144
\(898\) 2.13424e10 0.983503
\(899\) 5.89433e8 0.0270568
\(900\) 7.08820e9 0.324106
\(901\) 5.59997e10 2.55064
\(902\) −4.07488e9 −0.184881
\(903\) 2.36118e9 0.106714
\(904\) −8.07325e9 −0.363462
\(905\) −4.93483e10 −2.21310
\(906\) 1.32439e10 0.591653
\(907\) 5.08875e9 0.226457 0.113228 0.993569i \(-0.463881\pi\)
0.113228 + 0.993569i \(0.463881\pi\)
\(908\) −1.91321e10 −0.848131
\(909\) −2.62468e9 −0.115905
\(910\) −2.89151e9 −0.127198
\(911\) 8.88416e9 0.389316 0.194658 0.980871i \(-0.437640\pi\)
0.194658 + 0.980871i \(0.437640\pi\)
\(912\) −4.75523e9 −0.207582
\(913\) 1.16705e10 0.507506
\(914\) 7.91424e9 0.342845
\(915\) 1.57778e10 0.680881
\(916\) −1.55948e10 −0.670418
\(917\) 9.58449e9 0.410465
\(918\) 4.45261e9 0.189961
\(919\) −1.06207e10 −0.451386 −0.225693 0.974198i \(-0.572465\pi\)
−0.225693 + 0.974198i \(0.572465\pi\)
\(920\) 2.24742e10 0.951538
\(921\) 9.25755e7 0.00390470
\(922\) −1.23380e10 −0.518426
\(923\) −6.85629e9 −0.287001
\(924\) 3.64018e9 0.151800
\(925\) −5.78410e10 −2.40292
\(926\) −1.88450e10 −0.779933
\(927\) −2.71621e8 −0.0111991
\(928\) −1.25437e9 −0.0515237
\(929\) 2.32764e10 0.952492 0.476246 0.879312i \(-0.341997\pi\)
0.476246 + 0.879312i \(0.341997\pi\)
\(930\) 1.59523e9 0.0650330
\(931\) −5.05866e9 −0.205453
\(932\) 9.12275e9 0.369122
\(933\) 1.57499e10 0.634879
\(934\) −2.24128e10 −0.900084
\(935\) 8.32970e10 3.33264
\(936\) 8.20026e8 0.0326860
\(937\) 3.85180e9 0.152959 0.0764796 0.997071i \(-0.475632\pi\)
0.0764796 + 0.997071i \(0.475632\pi\)
\(938\) −5.44561e9 −0.215445
\(939\) 1.83600e10 0.723675
\(940\) −2.45833e10 −0.965367
\(941\) −3.57401e10 −1.39827 −0.699137 0.714988i \(-0.746432\pi\)
−0.699137 + 0.714988i \(0.746432\pi\)
\(942\) −3.76807e9 −0.146873
\(943\) 7.59001e9 0.294748
\(944\) −9.03770e9 −0.349668
\(945\) −3.23815e9 −0.124820
\(946\) −1.25270e10 −0.481092
\(947\) 4.60758e10 1.76298 0.881491 0.472201i \(-0.156540\pi\)
0.881491 + 0.472201i \(0.156540\pi\)
\(948\) −7.28026e9 −0.277535
\(949\) 1.94119e9 0.0737285
\(950\) 5.22596e10 1.97758
\(951\) −7.52514e9 −0.283715
\(952\) 4.96590e9 0.186538
\(953\) −3.69639e10 −1.38342 −0.691709 0.722177i \(-0.743142\pi\)
−0.691709 + 0.722177i \(0.743142\pi\)
\(954\) 1.15497e10 0.430676
\(955\) −7.72449e10 −2.86984
\(956\) 1.01535e10 0.375848
\(957\) 6.34781e9 0.234117
\(958\) 2.81190e10 1.03329
\(959\) 1.65513e7 0.000605991 0
\(960\) −3.39480e9 −0.123841
\(961\) −2.72755e10 −0.991382
\(962\) −6.69156e9 −0.242334
\(963\) −2.24590e9 −0.0810398
\(964\) 1.87571e10 0.674367
\(965\) −3.50453e10 −1.25541
\(966\) −6.78033e9 −0.242008
\(967\) 1.06965e10 0.380408 0.190204 0.981745i \(-0.439085\pi\)
0.190204 + 0.981745i \(0.439085\pi\)
\(968\) −9.33514e9 −0.330794
\(969\) 3.28280e10 1.15907
\(970\) −3.66633e10 −1.28982
\(971\) −9.50978e9 −0.333352 −0.166676 0.986012i \(-0.553303\pi\)
−0.166676 + 0.986012i \(0.553303\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 7.12050e9 0.247808
\(974\) −9.37720e9 −0.325175
\(975\) −9.01202e9 −0.311391
\(976\) −4.99035e9 −0.171813
\(977\) −1.79177e10 −0.614682 −0.307341 0.951599i \(-0.599439\pi\)
−0.307341 + 0.951599i \(0.599439\pi\)
\(978\) 1.29604e10 0.443031
\(979\) −7.36228e9 −0.250768
\(980\) −3.61143e9 −0.122571
\(981\) −1.96338e10 −0.663993
\(982\) −2.56965e10 −0.865933
\(983\) −4.35337e10 −1.46180 −0.730901 0.682483i \(-0.760900\pi\)
−0.730901 + 0.682483i \(0.760900\pi\)
\(984\) −1.14650e9 −0.0383611
\(985\) −4.29279e10 −1.43124
\(986\) 8.65962e9 0.287693
\(987\) 7.41664e9 0.245526
\(988\) 6.04585e9 0.199438
\(989\) 2.33332e10 0.766986
\(990\) 1.71796e10 0.562717
\(991\) −4.10553e10 −1.34002 −0.670010 0.742352i \(-0.733710\pi\)
−0.670010 + 0.742352i \(0.733710\pi\)
\(992\) −5.04556e8 −0.0164104
\(993\) 2.29696e8 0.00744443
\(994\) −8.56334e9 −0.276561
\(995\) 4.63844e10 1.49277
\(996\) 3.28358e9 0.105303
\(997\) 3.76132e10 1.20201 0.601004 0.799246i \(-0.294768\pi\)
0.601004 + 0.799246i \(0.294768\pi\)
\(998\) 4.20419e10 1.33883
\(999\) −7.49374e9 −0.237804
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 546.8.a.p.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.p.1.1 6 1.1 even 1 trivial