Properties

Label 546.8.a.p.1.4
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 1.4
Root \(-14.4733\) 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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +7.47333 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +7.47333 q^{5} -216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} -59.7866 q^{10} +5391.75 q^{11} +1728.00 q^{12} -2197.00 q^{13} -2744.00 q^{14} +201.780 q^{15} +4096.00 q^{16} +23811.5 q^{17} -5832.00 q^{18} +46865.1 q^{19} +478.293 q^{20} +9261.00 q^{21} -43134.0 q^{22} +73930.3 q^{23} -13824.0 q^{24} -78069.1 q^{25} +17576.0 q^{26} +19683.0 q^{27} +21952.0 q^{28} -175803. q^{29} -1614.24 q^{30} +277477. q^{31} -32768.0 q^{32} +145577. q^{33} -190492. q^{34} +2563.35 q^{35} +46656.0 q^{36} +241049. q^{37} -374921. q^{38} -59319.0 q^{39} -3826.34 q^{40} -78727.1 q^{41} -74088.0 q^{42} -199540. q^{43} +345072. q^{44} +5448.06 q^{45} -591442. q^{46} +142125. q^{47} +110592. q^{48} +117649. q^{49} +624553. q^{50} +642910. q^{51} -140608. q^{52} -854767. q^{53} -157464. q^{54} +40294.3 q^{55} -175616. q^{56} +1.26536e6 q^{57} +1.40642e6 q^{58} -533880. q^{59} +12913.9 q^{60} +3.35232e6 q^{61} -2.21981e6 q^{62} +250047. q^{63} +262144. q^{64} -16418.9 q^{65} -1.16462e6 q^{66} -4.13096e6 q^{67} +1.52393e6 q^{68} +1.99612e6 q^{69} -20506.8 q^{70} -2.83222e6 q^{71} -373248. q^{72} +3.21360e6 q^{73} -1.92839e6 q^{74} -2.10787e6 q^{75} +2.99937e6 q^{76} +1.84937e6 q^{77} +474552. q^{78} -8.08645e6 q^{79} +30610.7 q^{80} +531441. q^{81} +629817. q^{82} +6.07145e6 q^{83} +592704. q^{84} +177951. q^{85} +1.59632e6 q^{86} -4.74667e6 q^{87} -2.76057e6 q^{88} +7.87841e6 q^{89} -43584.4 q^{90} -753571. q^{91} +4.73154e6 q^{92} +7.49187e6 q^{93} -1.13700e6 q^{94} +350238. q^{95} -884736. q^{96} +1.58135e6 q^{97} -941192. q^{98} +3.93058e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} + 162 q^{3} + 384 q^{4} - 43 q^{5} - 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) 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
\(5\) 7.47333 0.0267374 0.0133687 0.999911i \(-0.495744\pi\)
0.0133687 + 0.999911i \(0.495744\pi\)
\(6\) −216.000 −0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) −59.7866 −0.0189062
\(11\) 5391.75 1.22139 0.610696 0.791865i \(-0.290890\pi\)
0.610696 + 0.791865i \(0.290890\pi\)
\(12\) 1728.00 0.288675
\(13\) −2197.00 −0.277350
\(14\) −2744.00 −0.267261
\(15\) 201.780 0.0154368
\(16\) 4096.00 0.250000
\(17\) 23811.5 1.17548 0.587740 0.809050i \(-0.300018\pi\)
0.587740 + 0.809050i \(0.300018\pi\)
\(18\) −5832.00 −0.235702
\(19\) 46865.1 1.56752 0.783758 0.621066i \(-0.213300\pi\)
0.783758 + 0.621066i \(0.213300\pi\)
\(20\) 478.293 0.0133687
\(21\) 9261.00 0.218218
\(22\) −43134.0 −0.863654
\(23\) 73930.3 1.26699 0.633497 0.773745i \(-0.281619\pi\)
0.633497 + 0.773745i \(0.281619\pi\)
\(24\) −13824.0 −0.204124
\(25\) −78069.1 −0.999285
\(26\) 17576.0 0.196116
\(27\) 19683.0 0.192450
\(28\) 21952.0 0.188982
\(29\) −175803. −1.33854 −0.669272 0.743018i \(-0.733394\pi\)
−0.669272 + 0.743018i \(0.733394\pi\)
\(30\) −1614.24 −0.0109155
\(31\) 277477. 1.67286 0.836432 0.548071i \(-0.184638\pi\)
0.836432 + 0.548071i \(0.184638\pi\)
\(32\) −32768.0 −0.176777
\(33\) 145577. 0.705171
\(34\) −190492. −0.831190
\(35\) 2563.35 0.0101058
\(36\) 46656.0 0.166667
\(37\) 241049. 0.782345 0.391173 0.920317i \(-0.372070\pi\)
0.391173 + 0.920317i \(0.372070\pi\)
\(38\) −374921. −1.10840
\(39\) −59319.0 −0.160128
\(40\) −3826.34 −0.00945309
\(41\) −78727.1 −0.178394 −0.0891972 0.996014i \(-0.528430\pi\)
−0.0891972 + 0.996014i \(0.528430\pi\)
\(42\) −74088.0 −0.154303
\(43\) −199540. −0.382729 −0.191364 0.981519i \(-0.561291\pi\)
−0.191364 + 0.981519i \(0.561291\pi\)
\(44\) 345072. 0.610696
\(45\) 5448.06 0.00891246
\(46\) −591442. −0.895901
\(47\) 142125. 0.199677 0.0998383 0.995004i \(-0.468167\pi\)
0.0998383 + 0.995004i \(0.468167\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) 624553. 0.706601
\(51\) 642910. 0.678664
\(52\) −140608. −0.138675
\(53\) −854767. −0.788646 −0.394323 0.918972i \(-0.629021\pi\)
−0.394323 + 0.918972i \(0.629021\pi\)
\(54\) −157464. −0.136083
\(55\) 40294.3 0.0326568
\(56\) −175616. −0.133631
\(57\) 1.26536e6 0.905006
\(58\) 1.40642e6 0.946493
\(59\) −533880. −0.338424 −0.169212 0.985580i \(-0.554122\pi\)
−0.169212 + 0.985580i \(0.554122\pi\)
\(60\) 12913.9 0.00771842
\(61\) 3.35232e6 1.89100 0.945499 0.325624i \(-0.105574\pi\)
0.945499 + 0.325624i \(0.105574\pi\)
\(62\) −2.21981e6 −1.18289
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −16418.9 −0.00741562
\(66\) −1.16462e6 −0.498631
\(67\) −4.13096e6 −1.67799 −0.838994 0.544141i \(-0.816856\pi\)
−0.838994 + 0.544141i \(0.816856\pi\)
\(68\) 1.52393e6 0.587740
\(69\) 1.99612e6 0.731500
\(70\) −20506.8 −0.00714587
\(71\) −2.83222e6 −0.939122 −0.469561 0.882900i \(-0.655588\pi\)
−0.469561 + 0.882900i \(0.655588\pi\)
\(72\) −373248. −0.117851
\(73\) 3.21360e6 0.966857 0.483429 0.875384i \(-0.339391\pi\)
0.483429 + 0.875384i \(0.339391\pi\)
\(74\) −1.92839e6 −0.553202
\(75\) −2.10787e6 −0.576938
\(76\) 2.99937e6 0.783758
\(77\) 1.84937e6 0.461643
\(78\) 474552. 0.113228
\(79\) −8.08645e6 −1.84528 −0.922641 0.385659i \(-0.873974\pi\)
−0.922641 + 0.385659i \(0.873974\pi\)
\(80\) 30610.7 0.00668435
\(81\) 531441. 0.111111
\(82\) 629817. 0.126144
\(83\) 6.07145e6 1.16552 0.582760 0.812645i \(-0.301973\pi\)
0.582760 + 0.812645i \(0.301973\pi\)
\(84\) 592704. 0.109109
\(85\) 177951. 0.0314293
\(86\) 1.59632e6 0.270630
\(87\) −4.74667e6 −0.772809
\(88\) −2.76057e6 −0.431827
\(89\) 7.87841e6 1.18460 0.592302 0.805716i \(-0.298219\pi\)
0.592302 + 0.805716i \(0.298219\pi\)
\(90\) −43584.4 −0.00630206
\(91\) −753571. −0.104828
\(92\) 4.73154e6 0.633497
\(93\) 7.49187e6 0.965828
\(94\) −1.13700e6 −0.141193
\(95\) 350238. 0.0419113
\(96\) −884736. −0.102062
\(97\) 1.58135e6 0.175924 0.0879622 0.996124i \(-0.471965\pi\)
0.0879622 + 0.996124i \(0.471965\pi\)
\(98\) −941192. −0.101015
\(99\) 3.93058e6 0.407131
\(100\) −4.99643e6 −0.499643
\(101\) 9.21450e6 0.889912 0.444956 0.895553i \(-0.353219\pi\)
0.444956 + 0.895553i \(0.353219\pi\)
\(102\) −5.14328e6 −0.479888
\(103\) 1.38313e7 1.24719 0.623596 0.781747i \(-0.285671\pi\)
0.623596 + 0.781747i \(0.285671\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 69210.5 0.00583458
\(106\) 6.83813e6 0.557657
\(107\) 1.47082e7 1.16069 0.580343 0.814372i \(-0.302918\pi\)
0.580343 + 0.814372i \(0.302918\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −6.12182e6 −0.452781 −0.226390 0.974037i \(-0.572692\pi\)
−0.226390 + 0.974037i \(0.572692\pi\)
\(110\) −322354. −0.0230919
\(111\) 6.50831e6 0.451687
\(112\) 1.40493e6 0.0944911
\(113\) −1.98529e7 −1.29434 −0.647172 0.762344i \(-0.724049\pi\)
−0.647172 + 0.762344i \(0.724049\pi\)
\(114\) −1.01229e7 −0.639936
\(115\) 552505. 0.0338761
\(116\) −1.12514e7 −0.669272
\(117\) −1.60161e6 −0.0924500
\(118\) 4.27104e6 0.239302
\(119\) 8.16734e6 0.444290
\(120\) −103311. −0.00545775
\(121\) 9.58375e6 0.491798
\(122\) −2.68186e7 −1.33714
\(123\) −2.12563e6 −0.102996
\(124\) 1.77585e7 0.836432
\(125\) −1.16729e6 −0.0534557
\(126\) −2.00038e6 −0.0890871
\(127\) −3.96590e7 −1.71802 −0.859010 0.511958i \(-0.828920\pi\)
−0.859010 + 0.511958i \(0.828920\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −5.38759e6 −0.220968
\(130\) 131351. 0.00524363
\(131\) −3.36490e7 −1.30774 −0.653872 0.756605i \(-0.726857\pi\)
−0.653872 + 0.756605i \(0.726857\pi\)
\(132\) 9.31694e6 0.352585
\(133\) 1.60747e7 0.592466
\(134\) 3.30476e7 1.18652
\(135\) 147097. 0.00514561
\(136\) −1.21915e7 −0.415595
\(137\) 948259. 0.0315069 0.0157534 0.999876i \(-0.494985\pi\)
0.0157534 + 0.999876i \(0.494985\pi\)
\(138\) −1.59689e7 −0.517248
\(139\) −5.20308e7 −1.64327 −0.821634 0.570016i \(-0.806937\pi\)
−0.821634 + 0.570016i \(0.806937\pi\)
\(140\) 164054. 0.00505289
\(141\) 3.83737e6 0.115283
\(142\) 2.26577e7 0.664060
\(143\) −1.18457e7 −0.338753
\(144\) 2.98598e6 0.0833333
\(145\) −1.31383e6 −0.0357892
\(146\) −2.57088e7 −0.683671
\(147\) 3.17652e6 0.0824786
\(148\) 1.54271e7 0.391173
\(149\) 1.33560e7 0.330768 0.165384 0.986229i \(-0.447114\pi\)
0.165384 + 0.986229i \(0.447114\pi\)
\(150\) 1.68629e7 0.407956
\(151\) 5.22301e6 0.123453 0.0617265 0.998093i \(-0.480339\pi\)
0.0617265 + 0.998093i \(0.480339\pi\)
\(152\) −2.39949e7 −0.554201
\(153\) 1.73586e7 0.391827
\(154\) −1.47949e7 −0.326431
\(155\) 2.07367e6 0.0447280
\(156\) −3.79642e6 −0.0800641
\(157\) −7.00155e7 −1.44393 −0.721964 0.691931i \(-0.756760\pi\)
−0.721964 + 0.691931i \(0.756760\pi\)
\(158\) 6.46916e7 1.30481
\(159\) −2.30787e7 −0.455325
\(160\) −244886. −0.00472655
\(161\) 2.53581e7 0.478879
\(162\) −4.25153e6 −0.0785674
\(163\) −6.95049e7 −1.25707 −0.628534 0.777782i \(-0.716345\pi\)
−0.628534 + 0.777782i \(0.716345\pi\)
\(164\) −5.03854e6 −0.0891972
\(165\) 1.08795e6 0.0188544
\(166\) −4.85716e7 −0.824146
\(167\) 5.45990e7 0.907145 0.453572 0.891219i \(-0.350149\pi\)
0.453572 + 0.891219i \(0.350149\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −1.42361e6 −0.0222238
\(171\) 3.41647e7 0.522506
\(172\) −1.27706e7 −0.191364
\(173\) 2.84648e7 0.417971 0.208986 0.977919i \(-0.432984\pi\)
0.208986 + 0.977919i \(0.432984\pi\)
\(174\) 3.79734e7 0.546458
\(175\) −2.67777e7 −0.377694
\(176\) 2.20846e7 0.305348
\(177\) −1.44148e7 −0.195389
\(178\) −6.30273e7 −0.837642
\(179\) 9.25538e7 1.20617 0.603085 0.797677i \(-0.293938\pi\)
0.603085 + 0.797677i \(0.293938\pi\)
\(180\) 348676. 0.00445623
\(181\) 3.43985e7 0.431185 0.215593 0.976483i \(-0.430832\pi\)
0.215593 + 0.976483i \(0.430832\pi\)
\(182\) 6.02857e6 0.0741249
\(183\) 9.05127e7 1.09177
\(184\) −3.78523e7 −0.447950
\(185\) 1.80143e6 0.0209179
\(186\) −5.99349e7 −0.682944
\(187\) 1.28385e8 1.43572
\(188\) 9.09598e6 0.0998383
\(189\) 6.75127e6 0.0727393
\(190\) −2.80191e6 −0.0296358
\(191\) 1.08204e7 0.112364 0.0561822 0.998421i \(-0.482107\pi\)
0.0561822 + 0.998421i \(0.482107\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −2.69909e7 −0.270250 −0.135125 0.990829i \(-0.543144\pi\)
−0.135125 + 0.990829i \(0.543144\pi\)
\(194\) −1.26508e7 −0.124397
\(195\) −443310. −0.00428141
\(196\) 7.52954e6 0.0714286
\(197\) −3.57337e7 −0.333001 −0.166501 0.986041i \(-0.553247\pi\)
−0.166501 + 0.986041i \(0.553247\pi\)
\(198\) −3.14447e7 −0.287885
\(199\) 2.84954e7 0.256324 0.128162 0.991753i \(-0.459092\pi\)
0.128162 + 0.991753i \(0.459092\pi\)
\(200\) 3.99714e7 0.353301
\(201\) −1.11536e8 −0.968787
\(202\) −7.37160e7 −0.629263
\(203\) −6.03003e7 −0.505922
\(204\) 4.11462e7 0.339332
\(205\) −588354. −0.00476980
\(206\) −1.10650e8 −0.881897
\(207\) 5.38952e7 0.422332
\(208\) −8.99891e6 −0.0693375
\(209\) 2.52685e8 1.91455
\(210\) −553684. −0.00412567
\(211\) 9.59044e7 0.702829 0.351415 0.936220i \(-0.385701\pi\)
0.351415 + 0.936220i \(0.385701\pi\)
\(212\) −5.47051e7 −0.394323
\(213\) −7.64698e7 −0.542203
\(214\) −1.17665e8 −0.820729
\(215\) −1.49123e6 −0.0102332
\(216\) −1.00777e7 −0.0680414
\(217\) 9.51745e7 0.632283
\(218\) 4.89746e7 0.320164
\(219\) 8.67673e7 0.558215
\(220\) 2.57883e6 0.0163284
\(221\) −5.23138e7 −0.326019
\(222\) −5.20665e7 −0.319391
\(223\) 6.07312e7 0.366729 0.183364 0.983045i \(-0.441301\pi\)
0.183364 + 0.983045i \(0.441301\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −5.69124e7 −0.333095
\(226\) 1.58823e8 0.915240
\(227\) −1.60653e8 −0.911589 −0.455794 0.890085i \(-0.650645\pi\)
−0.455794 + 0.890085i \(0.650645\pi\)
\(228\) 8.09829e7 0.452503
\(229\) −1.06088e8 −0.583770 −0.291885 0.956453i \(-0.594282\pi\)
−0.291885 + 0.956453i \(0.594282\pi\)
\(230\) −4.42004e6 −0.0239540
\(231\) 4.99330e7 0.266530
\(232\) 9.00110e7 0.473247
\(233\) 2.39951e8 1.24273 0.621364 0.783522i \(-0.286579\pi\)
0.621364 + 0.783522i \(0.286579\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) 1.06214e6 0.00533883
\(236\) −3.41683e7 −0.169212
\(237\) −2.18334e8 −1.06537
\(238\) −6.53387e7 −0.314160
\(239\) −5.80559e7 −0.275077 −0.137538 0.990496i \(-0.543919\pi\)
−0.137538 + 0.990496i \(0.543919\pi\)
\(240\) 826490. 0.00385921
\(241\) 1.29424e7 0.0595600 0.0297800 0.999556i \(-0.490519\pi\)
0.0297800 + 0.999556i \(0.490519\pi\)
\(242\) −7.66700e7 −0.347754
\(243\) 1.43489e7 0.0641500
\(244\) 2.14549e8 0.945499
\(245\) 879229. 0.00381963
\(246\) 1.70051e7 0.0728292
\(247\) −1.02963e8 −0.434751
\(248\) −1.42068e8 −0.591447
\(249\) 1.63929e8 0.672913
\(250\) 9.33832e6 0.0377989
\(251\) −1.10147e8 −0.439656 −0.219828 0.975539i \(-0.570550\pi\)
−0.219828 + 0.975539i \(0.570550\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 3.98613e8 1.54750
\(254\) 3.17272e8 1.21482
\(255\) 4.80468e6 0.0181457
\(256\) 1.67772e7 0.0625000
\(257\) 4.12507e8 1.51588 0.757940 0.652324i \(-0.226206\pi\)
0.757940 + 0.652324i \(0.226206\pi\)
\(258\) 4.31007e7 0.156248
\(259\) 8.26797e7 0.295699
\(260\) −1.05081e6 −0.00370781
\(261\) −1.28160e8 −0.446181
\(262\) 2.69192e8 0.924715
\(263\) 4.55717e8 1.54472 0.772360 0.635185i \(-0.219076\pi\)
0.772360 + 0.635185i \(0.219076\pi\)
\(264\) −7.45355e7 −0.249316
\(265\) −6.38795e6 −0.0210863
\(266\) −1.28598e8 −0.418936
\(267\) 2.12717e8 0.683932
\(268\) −2.64381e8 −0.838994
\(269\) 2.59372e8 0.812439 0.406219 0.913776i \(-0.366847\pi\)
0.406219 + 0.913776i \(0.366847\pi\)
\(270\) −1.17678e6 −0.00363850
\(271\) 3.38421e8 1.03292 0.516458 0.856313i \(-0.327250\pi\)
0.516458 + 0.856313i \(0.327250\pi\)
\(272\) 9.75318e7 0.293870
\(273\) −2.03464e7 −0.0605228
\(274\) −7.58607e6 −0.0222787
\(275\) −4.20929e8 −1.22052
\(276\) 1.27751e8 0.365750
\(277\) 3.61274e8 1.02131 0.510654 0.859786i \(-0.329403\pi\)
0.510654 + 0.859786i \(0.329403\pi\)
\(278\) 4.16246e8 1.16197
\(279\) 2.02280e8 0.557621
\(280\) −1.31244e6 −0.00357293
\(281\) 4.75931e8 1.27959 0.639797 0.768544i \(-0.279019\pi\)
0.639797 + 0.768544i \(0.279019\pi\)
\(282\) −3.06989e7 −0.0815176
\(283\) 4.36716e8 1.14537 0.572686 0.819775i \(-0.305901\pi\)
0.572686 + 0.819775i \(0.305901\pi\)
\(284\) −1.81262e8 −0.469561
\(285\) 9.45644e6 0.0241975
\(286\) 9.47653e7 0.239535
\(287\) −2.70034e7 −0.0674267
\(288\) −2.38879e7 −0.0589256
\(289\) 1.56648e8 0.381753
\(290\) 1.05106e7 0.0253068
\(291\) 4.26964e7 0.101570
\(292\) 2.05671e8 0.483429
\(293\) 1.06844e8 0.248149 0.124075 0.992273i \(-0.460404\pi\)
0.124075 + 0.992273i \(0.460404\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −3.98986e6 −0.00904858
\(296\) −1.23417e8 −0.276601
\(297\) 1.06126e8 0.235057
\(298\) −1.06848e8 −0.233889
\(299\) −1.62425e8 −0.351401
\(300\) −1.34903e8 −0.288469
\(301\) −6.84423e7 −0.144658
\(302\) −4.17841e7 −0.0872944
\(303\) 2.48791e8 0.513791
\(304\) 1.91960e8 0.391879
\(305\) 2.50530e7 0.0505604
\(306\) −1.38869e8 −0.277063
\(307\) −7.83514e8 −1.54548 −0.772738 0.634725i \(-0.781114\pi\)
−0.772738 + 0.634725i \(0.781114\pi\)
\(308\) 1.18360e8 0.230821
\(309\) 3.73445e8 0.720066
\(310\) −1.65894e7 −0.0316275
\(311\) −3.68900e8 −0.695421 −0.347711 0.937602i \(-0.613041\pi\)
−0.347711 + 0.937602i \(0.613041\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 7.75507e8 1.42949 0.714744 0.699387i \(-0.246543\pi\)
0.714744 + 0.699387i \(0.246543\pi\)
\(314\) 5.60124e8 1.02101
\(315\) 1.86868e6 0.00336859
\(316\) −5.17533e8 −0.922641
\(317\) 8.70639e8 1.53508 0.767540 0.641001i \(-0.221481\pi\)
0.767540 + 0.641001i \(0.221481\pi\)
\(318\) 1.84630e8 0.321963
\(319\) −9.47883e8 −1.63489
\(320\) 1.95909e6 0.00334217
\(321\) 3.97120e8 0.670123
\(322\) −2.02865e8 −0.338619
\(323\) 1.11593e9 1.84258
\(324\) 3.40122e7 0.0555556
\(325\) 1.71518e8 0.277152
\(326\) 5.56039e8 0.888882
\(327\) −1.65289e8 −0.261413
\(328\) 4.03083e7 0.0630719
\(329\) 4.87488e7 0.0754706
\(330\) −8.70356e6 −0.0133321
\(331\) 8.64912e8 1.31091 0.655457 0.755233i \(-0.272476\pi\)
0.655457 + 0.755233i \(0.272476\pi\)
\(332\) 3.88573e8 0.582760
\(333\) 1.75724e8 0.260782
\(334\) −4.36792e8 −0.641448
\(335\) −3.08720e7 −0.0448650
\(336\) 3.79331e7 0.0545545
\(337\) −1.27542e9 −1.81530 −0.907651 0.419727i \(-0.862126\pi\)
−0.907651 + 0.419727i \(0.862126\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −5.36029e8 −0.747290
\(340\) 1.13889e7 0.0157146
\(341\) 1.49608e9 2.04322
\(342\) −2.73317e8 −0.369467
\(343\) 4.03536e7 0.0539949
\(344\) 1.02165e8 0.135315
\(345\) 1.49176e7 0.0195584
\(346\) −2.27718e8 −0.295550
\(347\) −5.04685e8 −0.648436 −0.324218 0.945982i \(-0.605101\pi\)
−0.324218 + 0.945982i \(0.605101\pi\)
\(348\) −3.03787e8 −0.386404
\(349\) −4.12949e8 −0.520005 −0.260003 0.965608i \(-0.583723\pi\)
−0.260003 + 0.965608i \(0.583723\pi\)
\(350\) 2.14222e8 0.267070
\(351\) −4.32436e7 −0.0533761
\(352\) −1.76677e8 −0.215914
\(353\) −9.13045e8 −1.10479 −0.552396 0.833582i \(-0.686287\pi\)
−0.552396 + 0.833582i \(0.686287\pi\)
\(354\) 1.15318e8 0.138161
\(355\) −2.11661e7 −0.0251097
\(356\) 5.04218e8 0.592302
\(357\) 2.20518e8 0.256511
\(358\) −7.40430e8 −0.852891
\(359\) 1.03015e9 1.17508 0.587542 0.809193i \(-0.300096\pi\)
0.587542 + 0.809193i \(0.300096\pi\)
\(360\) −2.78940e6 −0.00315103
\(361\) 1.30247e9 1.45711
\(362\) −2.75188e8 −0.304894
\(363\) 2.58761e8 0.283940
\(364\) −4.82285e7 −0.0524142
\(365\) 2.40163e7 0.0258512
\(366\) −7.24101e8 −0.771997
\(367\) 1.00109e9 1.05716 0.528581 0.848883i \(-0.322724\pi\)
0.528581 + 0.848883i \(0.322724\pi\)
\(368\) 3.02818e8 0.316749
\(369\) −5.73921e7 −0.0594648
\(370\) −1.44115e7 −0.0147912
\(371\) −2.93185e8 −0.298080
\(372\) 4.79480e8 0.482914
\(373\) 1.08945e9 1.08700 0.543498 0.839410i \(-0.317099\pi\)
0.543498 + 0.839410i \(0.317099\pi\)
\(374\) −1.02708e9 −1.01521
\(375\) −3.15168e7 −0.0308626
\(376\) −7.27678e7 −0.0705963
\(377\) 3.86239e8 0.371245
\(378\) −5.40102e7 −0.0514344
\(379\) 1.89668e8 0.178960 0.0894801 0.995989i \(-0.471479\pi\)
0.0894801 + 0.995989i \(0.471479\pi\)
\(380\) 2.24153e7 0.0209556
\(381\) −1.07079e9 −0.991900
\(382\) −8.65636e7 −0.0794536
\(383\) 7.33219e8 0.666865 0.333432 0.942774i \(-0.391793\pi\)
0.333432 + 0.942774i \(0.391793\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 1.38209e7 0.0123431
\(386\) 2.15927e8 0.191096
\(387\) −1.45465e8 −0.127576
\(388\) 1.01206e8 0.0879622
\(389\) −1.17464e9 −1.01177 −0.505885 0.862601i \(-0.668834\pi\)
−0.505885 + 0.862601i \(0.668834\pi\)
\(390\) 3.54648e6 0.00302741
\(391\) 1.76039e9 1.48933
\(392\) −6.02363e7 −0.0505076
\(393\) −9.08524e8 −0.755027
\(394\) 2.85869e8 0.235467
\(395\) −6.04327e7 −0.0493380
\(396\) 2.51557e8 0.203565
\(397\) −1.96244e9 −1.57409 −0.787044 0.616897i \(-0.788389\pi\)
−0.787044 + 0.616897i \(0.788389\pi\)
\(398\) −2.27964e8 −0.181249
\(399\) 4.34018e8 0.342060
\(400\) −3.19771e8 −0.249821
\(401\) 1.98515e8 0.153741 0.0768703 0.997041i \(-0.475507\pi\)
0.0768703 + 0.997041i \(0.475507\pi\)
\(402\) 8.92286e8 0.685036
\(403\) −6.09616e8 −0.463969
\(404\) 5.89728e8 0.444956
\(405\) 3.97163e6 0.00297082
\(406\) 4.82403e8 0.357741
\(407\) 1.29967e9 0.955550
\(408\) −3.29170e8 −0.239944
\(409\) 1.84368e9 1.33246 0.666231 0.745745i \(-0.267907\pi\)
0.666231 + 0.745745i \(0.267907\pi\)
\(410\) 4.70683e6 0.00337276
\(411\) 2.56030e7 0.0181905
\(412\) 8.85204e8 0.623596
\(413\) −1.83121e8 −0.127912
\(414\) −4.31161e8 −0.298634
\(415\) 4.53740e7 0.0311629
\(416\) 7.19913e7 0.0490290
\(417\) −1.40483e9 −0.948741
\(418\) −2.02148e9 −1.35379
\(419\) 2.40877e9 1.59973 0.799863 0.600183i \(-0.204906\pi\)
0.799863 + 0.600183i \(0.204906\pi\)
\(420\) 4.42947e6 0.00291729
\(421\) −1.10193e9 −0.719727 −0.359864 0.933005i \(-0.617177\pi\)
−0.359864 + 0.933005i \(0.617177\pi\)
\(422\) −7.67235e8 −0.496975
\(423\) 1.03609e8 0.0665588
\(424\) 4.37641e8 0.278828
\(425\) −1.85894e9 −1.17464
\(426\) 6.11759e8 0.383395
\(427\) 1.14985e9 0.714730
\(428\) 9.41322e8 0.580343
\(429\) −3.19833e8 −0.195579
\(430\) 1.19298e7 0.00723594
\(431\) −1.52987e8 −0.0920414 −0.0460207 0.998940i \(-0.514654\pi\)
−0.0460207 + 0.998940i \(0.514654\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −9.21111e8 −0.545261 −0.272630 0.962119i \(-0.587894\pi\)
−0.272630 + 0.962119i \(0.587894\pi\)
\(434\) −7.61396e8 −0.447092
\(435\) −3.54734e7 −0.0206629
\(436\) −3.91797e8 −0.226390
\(437\) 3.46475e9 1.98604
\(438\) −6.94139e8 −0.394718
\(439\) −1.42647e9 −0.804708 −0.402354 0.915484i \(-0.631808\pi\)
−0.402354 + 0.915484i \(0.631808\pi\)
\(440\) −2.06307e7 −0.0115459
\(441\) 8.57661e7 0.0476190
\(442\) 4.18511e8 0.230531
\(443\) −1.42706e9 −0.779884 −0.389942 0.920840i \(-0.627505\pi\)
−0.389942 + 0.920840i \(0.627505\pi\)
\(444\) 4.16532e8 0.225844
\(445\) 5.88779e7 0.0316732
\(446\) −4.85850e8 −0.259316
\(447\) 3.60612e8 0.190969
\(448\) 8.99154e7 0.0472456
\(449\) 2.50667e9 1.30688 0.653439 0.756979i \(-0.273325\pi\)
0.653439 + 0.756979i \(0.273325\pi\)
\(450\) 4.55299e8 0.235534
\(451\) −4.24477e8 −0.217889
\(452\) −1.27059e9 −0.647172
\(453\) 1.41021e8 0.0712756
\(454\) 1.28523e9 0.644591
\(455\) −5.63168e6 −0.00280284
\(456\) −6.47863e8 −0.319968
\(457\) −2.47735e9 −1.21417 −0.607087 0.794635i \(-0.707662\pi\)
−0.607087 + 0.794635i \(0.707662\pi\)
\(458\) 8.48703e8 0.412788
\(459\) 4.68681e8 0.226221
\(460\) 3.53603e7 0.0169381
\(461\) 3.93003e9 1.86828 0.934141 0.356904i \(-0.116168\pi\)
0.934141 + 0.356904i \(0.116168\pi\)
\(462\) −3.99464e8 −0.188465
\(463\) −1.11168e9 −0.520532 −0.260266 0.965537i \(-0.583810\pi\)
−0.260266 + 0.965537i \(0.583810\pi\)
\(464\) −7.20088e8 −0.334636
\(465\) 5.59892e7 0.0258237
\(466\) −1.91960e9 −0.878741
\(467\) −1.34613e9 −0.611616 −0.305808 0.952093i \(-0.598927\pi\)
−0.305808 + 0.952093i \(0.598927\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −1.41692e9 −0.634220
\(470\) −8.49715e6 −0.00377512
\(471\) −1.89042e9 −0.833652
\(472\) 2.73346e8 0.119651
\(473\) −1.07587e9 −0.467462
\(474\) 1.74667e9 0.753333
\(475\) −3.65872e9 −1.56640
\(476\) 5.22710e8 0.222145
\(477\) −6.23125e8 −0.262882
\(478\) 4.64447e8 0.194509
\(479\) 4.82168e8 0.200458 0.100229 0.994964i \(-0.468042\pi\)
0.100229 + 0.994964i \(0.468042\pi\)
\(480\) −6.61192e6 −0.00272887
\(481\) −5.29584e8 −0.216984
\(482\) −1.03539e8 −0.0421153
\(483\) 6.84668e8 0.276481
\(484\) 6.13360e8 0.245899
\(485\) 1.18179e7 0.00470376
\(486\) −1.14791e8 −0.0453609
\(487\) −2.23691e9 −0.877600 −0.438800 0.898585i \(-0.644596\pi\)
−0.438800 + 0.898585i \(0.644596\pi\)
\(488\) −1.71639e9 −0.668569
\(489\) −1.87663e9 −0.725769
\(490\) −7.03384e6 −0.00270088
\(491\) −1.00077e9 −0.381549 −0.190774 0.981634i \(-0.561100\pi\)
−0.190774 + 0.981634i \(0.561100\pi\)
\(492\) −1.36041e8 −0.0514980
\(493\) −4.18612e9 −1.57343
\(494\) 8.23701e8 0.307415
\(495\) 2.93745e7 0.0108856
\(496\) 1.13654e9 0.418216
\(497\) −9.71450e8 −0.354955
\(498\) −1.31143e9 −0.475821
\(499\) 1.35572e9 0.488447 0.244224 0.969719i \(-0.421467\pi\)
0.244224 + 0.969719i \(0.421467\pi\)
\(500\) −7.47066e7 −0.0267278
\(501\) 1.47417e9 0.523740
\(502\) 8.81172e8 0.310884
\(503\) 5.92942e8 0.207742 0.103871 0.994591i \(-0.466877\pi\)
0.103871 + 0.994591i \(0.466877\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 6.88630e7 0.0237939
\(506\) −3.18891e9 −1.09425
\(507\) 1.30324e8 0.0444116
\(508\) −2.53817e9 −0.859010
\(509\) 7.46490e8 0.250906 0.125453 0.992100i \(-0.459962\pi\)
0.125453 + 0.992100i \(0.459962\pi\)
\(510\) −3.84374e7 −0.0128309
\(511\) 1.10227e9 0.365438
\(512\) −1.34218e8 −0.0441942
\(513\) 9.22446e8 0.301669
\(514\) −3.30005e9 −1.07189
\(515\) 1.03366e8 0.0333466
\(516\) −3.44806e8 −0.110484
\(517\) 7.66300e8 0.243883
\(518\) −6.61437e8 −0.209091
\(519\) 7.68549e8 0.241316
\(520\) 8.40648e6 0.00262182
\(521\) −1.06134e8 −0.0328792 −0.0164396 0.999865i \(-0.505233\pi\)
−0.0164396 + 0.999865i \(0.505233\pi\)
\(522\) 1.02528e9 0.315498
\(523\) −1.91251e9 −0.584587 −0.292293 0.956329i \(-0.594418\pi\)
−0.292293 + 0.956329i \(0.594418\pi\)
\(524\) −2.15354e9 −0.653872
\(525\) −7.22998e8 −0.218062
\(526\) −3.64573e9 −1.09228
\(527\) 6.60713e9 1.96642
\(528\) 5.96284e8 0.176293
\(529\) 2.06086e9 0.605276
\(530\) 5.11036e7 0.0149103
\(531\) −3.89198e8 −0.112808
\(532\) 1.02878e9 0.296233
\(533\) 1.72964e8 0.0494777
\(534\) −1.70174e9 −0.483613
\(535\) 1.09919e8 0.0310337
\(536\) 2.11505e9 0.593258
\(537\) 2.49895e9 0.696382
\(538\) −2.07498e9 −0.574481
\(539\) 6.34333e8 0.174485
\(540\) 9.41424e6 0.00257281
\(541\) −5.18332e9 −1.40740 −0.703700 0.710498i \(-0.748470\pi\)
−0.703700 + 0.710498i \(0.748470\pi\)
\(542\) −2.70737e9 −0.730381
\(543\) 9.28758e8 0.248945
\(544\) −7.80255e8 −0.207797
\(545\) −4.57504e7 −0.0121062
\(546\) 1.62771e8 0.0427960
\(547\) 2.21651e9 0.579047 0.289523 0.957171i \(-0.406503\pi\)
0.289523 + 0.957171i \(0.406503\pi\)
\(548\) 6.06886e7 0.0157534
\(549\) 2.44384e9 0.630333
\(550\) 3.36743e9 0.863037
\(551\) −8.23902e9 −2.09819
\(552\) −1.02201e9 −0.258624
\(553\) −2.77365e9 −0.697451
\(554\) −2.89019e9 −0.722174
\(555\) 4.86387e7 0.0120769
\(556\) −3.32997e9 −0.821634
\(557\) −2.37436e9 −0.582175 −0.291087 0.956697i \(-0.594017\pi\)
−0.291087 + 0.956697i \(0.594017\pi\)
\(558\) −1.61824e9 −0.394298
\(559\) 4.38390e8 0.106150
\(560\) 1.04995e7 0.00252645
\(561\) 3.46641e9 0.828914
\(562\) −3.80745e9 −0.904809
\(563\) −5.53237e9 −1.30657 −0.653283 0.757114i \(-0.726609\pi\)
−0.653283 + 0.757114i \(0.726609\pi\)
\(564\) 2.45591e8 0.0576416
\(565\) −1.48367e8 −0.0346074
\(566\) −3.49373e9 −0.809901
\(567\) 1.82284e8 0.0419961
\(568\) 1.45009e9 0.332030
\(569\) −8.34861e8 −0.189986 −0.0949930 0.995478i \(-0.530283\pi\)
−0.0949930 + 0.995478i \(0.530283\pi\)
\(570\) −7.56515e7 −0.0171102
\(571\) 5.84940e9 1.31488 0.657438 0.753509i \(-0.271640\pi\)
0.657438 + 0.753509i \(0.271640\pi\)
\(572\) −7.58123e8 −0.169377
\(573\) 2.92152e8 0.0648736
\(574\) 2.16027e8 0.0476779
\(575\) −5.77167e9 −1.26609
\(576\) 1.91103e8 0.0416667
\(577\) 2.46903e9 0.535071 0.267535 0.963548i \(-0.413791\pi\)
0.267535 + 0.963548i \(0.413791\pi\)
\(578\) −1.25318e9 −0.269940
\(579\) −7.28754e8 −0.156029
\(580\) −8.40852e7 −0.0178946
\(581\) 2.08251e9 0.440525
\(582\) −3.41571e8 −0.0718208
\(583\) −4.60868e9 −0.963246
\(584\) −1.64537e9 −0.341836
\(585\) −1.19694e7 −0.00247187
\(586\) −8.54751e8 −0.175468
\(587\) 5.50894e9 1.12418 0.562089 0.827077i \(-0.309998\pi\)
0.562089 + 0.827077i \(0.309998\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) 1.30040e10 2.62224
\(590\) 3.19189e7 0.00639831
\(591\) −9.64809e8 −0.192258
\(592\) 9.87335e8 0.195586
\(593\) −5.66367e9 −1.11534 −0.557669 0.830063i \(-0.688304\pi\)
−0.557669 + 0.830063i \(0.688304\pi\)
\(594\) −8.49006e8 −0.166210
\(595\) 6.10372e7 0.0118791
\(596\) 8.54783e8 0.165384
\(597\) 7.69377e8 0.147989
\(598\) 1.29940e9 0.248478
\(599\) 4.65297e9 0.884578 0.442289 0.896872i \(-0.354166\pi\)
0.442289 + 0.896872i \(0.354166\pi\)
\(600\) 1.07923e9 0.203978
\(601\) −4.89509e9 −0.919814 −0.459907 0.887967i \(-0.652117\pi\)
−0.459907 + 0.887967i \(0.652117\pi\)
\(602\) 5.47538e8 0.102289
\(603\) −3.01147e9 −0.559329
\(604\) 3.34273e8 0.0617265
\(605\) 7.16225e7 0.0131494
\(606\) −1.99033e9 −0.363305
\(607\) 8.11357e8 0.147249 0.0736244 0.997286i \(-0.476543\pi\)
0.0736244 + 0.997286i \(0.476543\pi\)
\(608\) −1.53568e9 −0.277100
\(609\) −1.62811e9 −0.292094
\(610\) −2.00424e8 −0.0357516
\(611\) −3.12248e8 −0.0553803
\(612\) 1.11095e9 0.195913
\(613\) 2.27379e9 0.398693 0.199347 0.979929i \(-0.436118\pi\)
0.199347 + 0.979929i \(0.436118\pi\)
\(614\) 6.26811e9 1.09282
\(615\) −1.58856e7 −0.00275384
\(616\) −9.46877e8 −0.163215
\(617\) −8.69457e9 −1.49022 −0.745109 0.666943i \(-0.767603\pi\)
−0.745109 + 0.666943i \(0.767603\pi\)
\(618\) −2.98756e9 −0.509164
\(619\) −9.22069e9 −1.56259 −0.781297 0.624160i \(-0.785442\pi\)
−0.781297 + 0.624160i \(0.785442\pi\)
\(620\) 1.32715e8 0.0223640
\(621\) 1.45517e9 0.243833
\(622\) 2.95120e9 0.491737
\(623\) 2.70229e9 0.447739
\(624\) −2.42971e8 −0.0400320
\(625\) 6.09043e9 0.997856
\(626\) −6.20405e9 −1.01080
\(627\) 6.82249e9 1.10537
\(628\) −4.48099e9 −0.721964
\(629\) 5.73972e9 0.919631
\(630\) −1.49495e7 −0.00238196
\(631\) −1.06468e9 −0.168701 −0.0843503 0.996436i \(-0.526881\pi\)
−0.0843503 + 0.996436i \(0.526881\pi\)
\(632\) 4.14026e9 0.652406
\(633\) 2.58942e9 0.405779
\(634\) −6.96511e9 −1.08547
\(635\) −2.96384e8 −0.0459354
\(636\) −1.47704e9 −0.227662
\(637\) −2.58475e8 −0.0396214
\(638\) 7.58307e9 1.15604
\(639\) −2.06469e9 −0.313041
\(640\) −1.56727e7 −0.00236327
\(641\) −3.49397e9 −0.523981 −0.261991 0.965070i \(-0.584379\pi\)
−0.261991 + 0.965070i \(0.584379\pi\)
\(642\) −3.17696e9 −0.473848
\(643\) −1.19059e10 −1.76613 −0.883066 0.469249i \(-0.844525\pi\)
−0.883066 + 0.469249i \(0.844525\pi\)
\(644\) 1.62292e9 0.239439
\(645\) −4.02632e7 −0.00590812
\(646\) −8.92742e9 −1.30290
\(647\) 8.84544e9 1.28397 0.641984 0.766718i \(-0.278111\pi\)
0.641984 + 0.766718i \(0.278111\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −2.87854e9 −0.413349
\(650\) −1.37214e9 −0.195976
\(651\) 2.56971e9 0.365049
\(652\) −4.44831e9 −0.628534
\(653\) 4.23921e9 0.595785 0.297892 0.954599i \(-0.403716\pi\)
0.297892 + 0.954599i \(0.403716\pi\)
\(654\) 1.32231e9 0.184847
\(655\) −2.51470e8 −0.0349657
\(656\) −3.22466e8 −0.0445986
\(657\) 2.34272e9 0.322286
\(658\) −3.89990e8 −0.0533658
\(659\) −2.08660e9 −0.284014 −0.142007 0.989866i \(-0.545356\pi\)
−0.142007 + 0.989866i \(0.545356\pi\)
\(660\) 6.96285e7 0.00942721
\(661\) −5.01321e9 −0.675167 −0.337583 0.941296i \(-0.609609\pi\)
−0.337583 + 0.941296i \(0.609609\pi\)
\(662\) −6.91930e9 −0.926956
\(663\) −1.41247e9 −0.188227
\(664\) −3.10858e9 −0.412073
\(665\) 1.20132e8 0.0158410
\(666\) −1.40580e9 −0.184401
\(667\) −1.29971e10 −1.69593
\(668\) 3.49433e9 0.453572
\(669\) 1.63974e9 0.211731
\(670\) 2.46976e8 0.0317243
\(671\) 1.80749e10 2.30965
\(672\) −3.03464e8 −0.0385758
\(673\) −3.78775e9 −0.478993 −0.239496 0.970897i \(-0.576982\pi\)
−0.239496 + 0.970897i \(0.576982\pi\)
\(674\) 1.02034e10 1.28361
\(675\) −1.53664e9 −0.192313
\(676\) 3.08916e8 0.0384615
\(677\) 1.42684e9 0.176732 0.0883662 0.996088i \(-0.471835\pi\)
0.0883662 + 0.996088i \(0.471835\pi\)
\(678\) 4.28823e9 0.528414
\(679\) 5.42402e8 0.0664932
\(680\) −9.11109e7 −0.0111119
\(681\) −4.33764e9 −0.526306
\(682\) −1.19687e10 −1.44478
\(683\) −2.34845e9 −0.282039 −0.141020 0.990007i \(-0.545038\pi\)
−0.141020 + 0.990007i \(0.545038\pi\)
\(684\) 2.18654e9 0.261253
\(685\) 7.08665e6 0.000842411 0
\(686\) −3.22829e8 −0.0381802
\(687\) −2.86437e9 −0.337040
\(688\) −8.17317e8 −0.0956821
\(689\) 1.87792e9 0.218731
\(690\) −1.19341e8 −0.0138299
\(691\) 7.74789e8 0.0893327 0.0446663 0.999002i \(-0.485778\pi\)
0.0446663 + 0.999002i \(0.485778\pi\)
\(692\) 1.82175e9 0.208986
\(693\) 1.34819e9 0.153881
\(694\) 4.03748e9 0.458513
\(695\) −3.88843e8 −0.0439367
\(696\) 2.43030e9 0.273229
\(697\) −1.87461e9 −0.209699
\(698\) 3.30360e9 0.367699
\(699\) 6.47866e9 0.717489
\(700\) −1.71377e9 −0.188847
\(701\) 1.22722e10 1.34558 0.672789 0.739834i \(-0.265096\pi\)
0.672789 + 0.739834i \(0.265096\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 1.12968e10 1.22634
\(704\) 1.41341e9 0.152674
\(705\) 2.86779e7 0.00308237
\(706\) 7.30436e9 0.781206
\(707\) 3.16057e9 0.336355
\(708\) −9.22544e8 −0.0976946
\(709\) 4.42611e9 0.466402 0.233201 0.972428i \(-0.425080\pi\)
0.233201 + 0.972428i \(0.425080\pi\)
\(710\) 1.69329e8 0.0177552
\(711\) −5.89502e9 −0.615094
\(712\) −4.03375e9 −0.418821
\(713\) 2.05139e10 2.11951
\(714\) −1.76415e9 −0.181380
\(715\) −8.85265e7 −0.00905737
\(716\) 5.92344e9 0.603085
\(717\) −1.56751e9 −0.158816
\(718\) −8.24119e9 −0.830910
\(719\) −2.30699e9 −0.231470 −0.115735 0.993280i \(-0.536922\pi\)
−0.115735 + 0.993280i \(0.536922\pi\)
\(720\) 2.23152e7 0.00222812
\(721\) 4.74414e9 0.471394
\(722\) −1.04197e10 −1.03033
\(723\) 3.49444e8 0.0343870
\(724\) 2.20150e9 0.215593
\(725\) 1.37248e10 1.33759
\(726\) −2.07009e9 −0.200776
\(727\) 1.45573e10 1.40511 0.702555 0.711629i \(-0.252042\pi\)
0.702555 + 0.711629i \(0.252042\pi\)
\(728\) 3.85828e8 0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −1.92131e8 −0.0182796
\(731\) −4.75135e9 −0.449890
\(732\) 5.79281e9 0.545884
\(733\) 1.18974e10 1.11580 0.557902 0.829907i \(-0.311606\pi\)
0.557902 + 0.829907i \(0.311606\pi\)
\(734\) −8.00872e9 −0.747527
\(735\) 2.37392e7 0.00220526
\(736\) −2.42255e9 −0.223975
\(737\) −2.22731e10 −2.04948
\(738\) 4.59137e8 0.0420479
\(739\) −6.61920e8 −0.0603324 −0.0301662 0.999545i \(-0.509604\pi\)
−0.0301662 + 0.999545i \(0.509604\pi\)
\(740\) 1.15292e8 0.0104589
\(741\) −2.77999e9 −0.251004
\(742\) 2.34548e9 0.210774
\(743\) −6.56840e9 −0.587488 −0.293744 0.955884i \(-0.594901\pi\)
−0.293744 + 0.955884i \(0.594901\pi\)
\(744\) −3.83584e9 −0.341472
\(745\) 9.98136e7 0.00884388
\(746\) −8.71564e9 −0.768623
\(747\) 4.42609e9 0.388506
\(748\) 8.21667e9 0.717861
\(749\) 5.04490e9 0.438698
\(750\) 2.52135e8 0.0218232
\(751\) −1.77094e10 −1.52568 −0.762840 0.646587i \(-0.776196\pi\)
−0.762840 + 0.646587i \(0.776196\pi\)
\(752\) 5.82143e8 0.0499191
\(753\) −2.97396e9 −0.253835
\(754\) −3.08991e9 −0.262510
\(755\) 3.90332e7 0.00330081
\(756\) 4.32081e8 0.0363696
\(757\) −1.73394e10 −1.45277 −0.726386 0.687286i \(-0.758802\pi\)
−0.726386 + 0.687286i \(0.758802\pi\)
\(758\) −1.51734e9 −0.126544
\(759\) 1.07626e10 0.893448
\(760\) −1.79322e8 −0.0148179
\(761\) −1.34566e10 −1.10685 −0.553424 0.832900i \(-0.686679\pi\)
−0.553424 + 0.832900i \(0.686679\pi\)
\(762\) 8.56634e9 0.701379
\(763\) −2.09978e9 −0.171135
\(764\) 6.92509e8 0.0561822
\(765\) 1.29726e8 0.0104764
\(766\) −5.86575e9 −0.471545
\(767\) 1.17293e9 0.0938620
\(768\) 4.52985e8 0.0360844
\(769\) 7.37377e9 0.584720 0.292360 0.956308i \(-0.405560\pi\)
0.292360 + 0.956308i \(0.405560\pi\)
\(770\) −1.10567e8 −0.00872790
\(771\) 1.11377e10 0.875194
\(772\) −1.72742e9 −0.135125
\(773\) 1.75212e10 1.36438 0.682189 0.731176i \(-0.261028\pi\)
0.682189 + 0.731176i \(0.261028\pi\)
\(774\) 1.16372e9 0.0902100
\(775\) −2.16624e10 −1.67167
\(776\) −8.09649e8 −0.0621986
\(777\) 2.23235e9 0.170722
\(778\) 9.39713e9 0.715429
\(779\) −3.68956e9 −0.279636
\(780\) −2.83719e7 −0.00214070
\(781\) −1.52706e10 −1.14704
\(782\) −1.40831e10 −1.05311
\(783\) −3.46032e9 −0.257603
\(784\) 4.81890e8 0.0357143
\(785\) −5.23249e8 −0.0386069
\(786\) 7.26819e9 0.533884
\(787\) −2.08015e10 −1.52119 −0.760594 0.649228i \(-0.775092\pi\)
−0.760594 + 0.649228i \(0.775092\pi\)
\(788\) −2.28695e9 −0.166501
\(789\) 1.23044e10 0.891845
\(790\) 4.83461e8 0.0348873
\(791\) −6.80955e9 −0.489216
\(792\) −2.01246e9 −0.143942
\(793\) −7.36505e9 −0.524469
\(794\) 1.56995e10 1.11305
\(795\) −1.72475e8 −0.0121742
\(796\) 1.82371e9 0.128162
\(797\) 2.06060e10 1.44175 0.720874 0.693066i \(-0.243741\pi\)
0.720874 + 0.693066i \(0.243741\pi\)
\(798\) −3.47214e9 −0.241873
\(799\) 3.38420e9 0.234716
\(800\) 2.55817e9 0.176650
\(801\) 5.74336e9 0.394868
\(802\) −1.58812e9 −0.108711
\(803\) 1.73269e10 1.18091
\(804\) −7.13829e9 −0.484393
\(805\) 1.89509e8 0.0128040
\(806\) 4.87693e9 0.328075
\(807\) 7.00305e9 0.469062
\(808\) −4.71782e9 −0.314631
\(809\) −4.06471e9 −0.269904 −0.134952 0.990852i \(-0.543088\pi\)
−0.134952 + 0.990852i \(0.543088\pi\)
\(810\) −3.17731e7 −0.00210069
\(811\) −6.91651e9 −0.455317 −0.227659 0.973741i \(-0.573107\pi\)
−0.227659 + 0.973741i \(0.573107\pi\)
\(812\) −3.85922e9 −0.252961
\(813\) 9.13737e9 0.596354
\(814\) −1.03974e10 −0.675676
\(815\) −5.19433e8 −0.0336107
\(816\) 2.63336e9 0.169666
\(817\) −9.35148e9 −0.599933
\(818\) −1.47495e10 −0.942193
\(819\) −5.49353e8 −0.0349428
\(820\) −3.76546e7 −0.00238490
\(821\) −1.65831e10 −1.04584 −0.522920 0.852382i \(-0.675157\pi\)
−0.522920 + 0.852382i \(0.675157\pi\)
\(822\) −2.04824e8 −0.0128626
\(823\) −3.10529e9 −0.194179 −0.0970896 0.995276i \(-0.530953\pi\)
−0.0970896 + 0.995276i \(0.530953\pi\)
\(824\) −7.08163e9 −0.440949
\(825\) −1.13651e10 −0.704667
\(826\) 1.46497e9 0.0904477
\(827\) −1.22275e10 −0.751741 −0.375871 0.926672i \(-0.622656\pi\)
−0.375871 + 0.926672i \(0.622656\pi\)
\(828\) 3.44929e9 0.211166
\(829\) 3.00252e10 1.83039 0.915197 0.403007i \(-0.132035\pi\)
0.915197 + 0.403007i \(0.132035\pi\)
\(830\) −3.62992e8 −0.0220355
\(831\) 9.75439e9 0.589653
\(832\) −5.75930e8 −0.0346688
\(833\) 2.80140e9 0.167926
\(834\) 1.12386e10 0.670861
\(835\) 4.08036e8 0.0242547
\(836\) 1.61718e10 0.957276
\(837\) 5.46157e9 0.321943
\(838\) −1.92701e10 −1.13118
\(839\) −2.44073e10 −1.42676 −0.713382 0.700775i \(-0.752838\pi\)
−0.713382 + 0.700775i \(0.752838\pi\)
\(840\) −3.54358e7 −0.00206283
\(841\) 1.36567e10 0.791699
\(842\) 8.81546e9 0.508924
\(843\) 1.28501e10 0.738774
\(844\) 6.13788e9 0.351415
\(845\) 3.60723e7 0.00205672
\(846\) −8.28871e8 −0.0470642
\(847\) 3.28723e9 0.185882
\(848\) −3.50112e9 −0.197161
\(849\) 1.17913e10 0.661281
\(850\) 1.48715e10 0.830596
\(851\) 1.78208e10 0.991227
\(852\) −4.89407e9 −0.271101
\(853\) 2.26002e10 1.24678 0.623391 0.781911i \(-0.285755\pi\)
0.623391 + 0.781911i \(0.285755\pi\)
\(854\) −9.19877e9 −0.505391
\(855\) 2.55324e8 0.0139704
\(856\) −7.53058e9 −0.410365
\(857\) 8.14738e8 0.0442166 0.0221083 0.999756i \(-0.492962\pi\)
0.0221083 + 0.999756i \(0.492962\pi\)
\(858\) 2.55866e9 0.138295
\(859\) 2.14476e10 1.15452 0.577262 0.816559i \(-0.304121\pi\)
0.577262 + 0.816559i \(0.304121\pi\)
\(860\) −9.54387e7 −0.00511658
\(861\) −7.29092e8 −0.0389288
\(862\) 1.22389e9 0.0650831
\(863\) 2.51368e10 1.33129 0.665645 0.746269i \(-0.268157\pi\)
0.665645 + 0.746269i \(0.268157\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 2.12727e8 0.0111755
\(866\) 7.36889e9 0.385558
\(867\) 4.22950e9 0.220405
\(868\) 6.09117e9 0.316141
\(869\) −4.36001e10 −2.25381
\(870\) 2.83788e8 0.0146109
\(871\) 9.07571e9 0.465390
\(872\) 3.13437e9 0.160082
\(873\) 1.15280e9 0.0586414
\(874\) −2.77180e10 −1.40434
\(875\) −4.00380e8 −0.0202043
\(876\) 5.55311e9 0.279108
\(877\) −8.07486e9 −0.404237 −0.202119 0.979361i \(-0.564783\pi\)
−0.202119 + 0.979361i \(0.564783\pi\)
\(878\) 1.14118e10 0.569014
\(879\) 2.88479e9 0.143269
\(880\) 1.65045e8 0.00816421
\(881\) −2.80996e10 −1.38447 −0.692235 0.721672i \(-0.743374\pi\)
−0.692235 + 0.721672i \(0.743374\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −2.39587e10 −1.17112 −0.585559 0.810630i \(-0.699125\pi\)
−0.585559 + 0.810630i \(0.699125\pi\)
\(884\) −3.34809e9 −0.163010
\(885\) −1.07726e8 −0.00522420
\(886\) 1.14165e10 0.551461
\(887\) 1.33775e10 0.643637 0.321819 0.946801i \(-0.395706\pi\)
0.321819 + 0.946801i \(0.395706\pi\)
\(888\) −3.33226e9 −0.159696
\(889\) −1.36030e10 −0.649351
\(890\) −4.71023e8 −0.0223964
\(891\) 2.86539e9 0.135710
\(892\) 3.88680e9 0.183364
\(893\) 6.66069e9 0.312996
\(894\) −2.88489e9 −0.135036
\(895\) 6.91684e8 0.0322498
\(896\) −7.19323e8 −0.0334077
\(897\) −4.38547e9 −0.202882
\(898\) −2.00534e10 −0.924103
\(899\) −4.87811e10 −2.23920
\(900\) −3.64239e9 −0.166548
\(901\) −2.03533e10 −0.927038
\(902\) 3.39581e9 0.154071
\(903\) −1.84794e9 −0.0835182
\(904\) 1.01647e10 0.457620
\(905\) 2.57071e8 0.0115288
\(906\) −1.12817e9 −0.0503994
\(907\) −2.81472e10 −1.25259 −0.626296 0.779585i \(-0.715430\pi\)
−0.626296 + 0.779585i \(0.715430\pi\)
\(908\) −1.02818e10 −0.455794
\(909\) 6.71737e9 0.296637
\(910\) 4.50535e7 0.00198191
\(911\) −1.72184e10 −0.754534 −0.377267 0.926105i \(-0.623136\pi\)
−0.377267 + 0.926105i \(0.623136\pi\)
\(912\) 5.18291e9 0.226252
\(913\) 3.27357e10 1.42356
\(914\) 1.98188e10 0.858551
\(915\) 6.76431e8 0.0291910
\(916\) −6.78962e9 −0.291885
\(917\) −1.15416e10 −0.494281
\(918\) −3.74945e9 −0.159963
\(919\) 2.64594e10 1.12454 0.562272 0.826953i \(-0.309928\pi\)
0.562272 + 0.826953i \(0.309928\pi\)
\(920\) −2.82883e8 −0.0119770
\(921\) −2.11549e10 −0.892282
\(922\) −3.14402e10 −1.32107
\(923\) 6.22238e9 0.260466
\(924\) 3.19571e9 0.133265
\(925\) −1.88185e10 −0.781786
\(926\) 8.89346e9 0.368071
\(927\) 1.00830e10 0.415730
\(928\) 5.76070e9 0.236623
\(929\) 3.96644e8 0.0162310 0.00811551 0.999967i \(-0.497417\pi\)
0.00811551 + 0.999967i \(0.497417\pi\)
\(930\) −4.47913e8 −0.0182601
\(931\) 5.51363e9 0.223931
\(932\) 1.53568e10 0.621364
\(933\) −9.96031e9 −0.401502
\(934\) 1.07691e10 0.432478
\(935\) 9.59467e8 0.0383874
\(936\) 8.20026e8 0.0326860
\(937\) 1.37233e10 0.544967 0.272483 0.962160i \(-0.412155\pi\)
0.272483 + 0.962160i \(0.412155\pi\)
\(938\) 1.13353e10 0.448461
\(939\) 2.09387e10 0.825315
\(940\) 6.79772e7 0.00266941
\(941\) 9.81482e9 0.383989 0.191994 0.981396i \(-0.438504\pi\)
0.191994 + 0.981396i \(0.438504\pi\)
\(942\) 1.51234e10 0.589481
\(943\) −5.82032e9 −0.226025
\(944\) −2.18677e9 −0.0846060
\(945\) 5.04544e7 0.00194486
\(946\) 8.60696e9 0.330545
\(947\) 3.35417e10 1.28340 0.641698 0.766958i \(-0.278230\pi\)
0.641698 + 0.766958i \(0.278230\pi\)
\(948\) −1.39734e10 −0.532687
\(949\) −7.06029e9 −0.268158
\(950\) 2.92698e10 1.10761
\(951\) 2.35073e10 0.886279
\(952\) −4.18168e9 −0.157080
\(953\) 2.09755e10 0.785032 0.392516 0.919745i \(-0.371605\pi\)
0.392516 + 0.919745i \(0.371605\pi\)
\(954\) 4.98500e9 0.185886
\(955\) 8.08647e7 0.00300433
\(956\) −3.71558e9 −0.137538
\(957\) −2.55929e10 −0.943902
\(958\) −3.85735e9 −0.141746
\(959\) 3.25253e8 0.0119085
\(960\) 5.28954e7 0.00192960
\(961\) 4.94807e10 1.79847
\(962\) 4.23667e9 0.153431
\(963\) 1.07222e10 0.386896
\(964\) 8.28313e8 0.0297800
\(965\) −2.01712e8 −0.00722579
\(966\) −5.47734e9 −0.195502
\(967\) −3.47197e10 −1.23476 −0.617381 0.786664i \(-0.711806\pi\)
−0.617381 + 0.786664i \(0.711806\pi\)
\(968\) −4.90688e9 −0.173877
\(969\) 3.01301e10 1.06382
\(970\) −9.45433e7 −0.00332606
\(971\) 4.45891e9 0.156301 0.0781504 0.996942i \(-0.475099\pi\)
0.0781504 + 0.996942i \(0.475099\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.78465e10 −0.621097
\(974\) 1.78953e10 0.620557
\(975\) 4.63098e9 0.160014
\(976\) 1.37311e10 0.472750
\(977\) 1.71692e10 0.589007 0.294503 0.955650i \(-0.404846\pi\)
0.294503 + 0.955650i \(0.404846\pi\)
\(978\) 1.50131e10 0.513196
\(979\) 4.24784e10 1.44687
\(980\) 5.62707e7 0.00190981
\(981\) −4.46281e9 −0.150927
\(982\) 8.00618e9 0.269796
\(983\) 3.75095e10 1.25952 0.629758 0.776791i \(-0.283154\pi\)
0.629758 + 0.776791i \(0.283154\pi\)
\(984\) 1.08832e9 0.0364146
\(985\) −2.67049e8 −0.00890358
\(986\) 3.34890e10 1.11258
\(987\) 1.31622e9 0.0435730
\(988\) −6.58961e9 −0.217375
\(989\) −1.47521e10 −0.484915
\(990\) −2.34996e8 −0.00769729
\(991\) 1.97803e10 0.645616 0.322808 0.946464i \(-0.395373\pi\)
0.322808 + 0.946464i \(0.395373\pi\)
\(992\) −9.09235e9 −0.295723
\(993\) 2.33526e10 0.756856
\(994\) 7.77160e9 0.250991
\(995\) 2.12956e8 0.00685344
\(996\) 1.04915e10 0.336456
\(997\) 1.83361e9 0.0585967 0.0292984 0.999571i \(-0.490673\pi\)
0.0292984 + 0.999571i \(0.490673\pi\)
\(998\) −1.08458e10 −0.345384
\(999\) 4.74456e9 0.150562
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.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.p.1.4 6 1.1 even 1 trivial