Properties

Label 546.8.a.n.1.2
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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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - x^{5} - 309949x^{4} - 14548431x^{3} + 25221499020x^{2} + 1862570808000x - 308009568384000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(348.888\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -378.888 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +3031.11 q^{10} +5295.06 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +10230.0 q^{15} +4096.00 q^{16} -1617.93 q^{17} -5832.00 q^{18} +42012.9 q^{19} -24248.8 q^{20} -9261.00 q^{21} -42360.5 q^{22} +101886. q^{23} +13824.0 q^{24} +65431.2 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +65637.5 q^{29} -81839.8 q^{30} +117455. q^{31} -32768.0 q^{32} -142967. q^{33} +12943.4 q^{34} -129959. q^{35} +46656.0 q^{36} +170252. q^{37} -336103. q^{38} -59319.0 q^{39} +193991. q^{40} +305044. q^{41} +74088.0 q^{42} +388832. q^{43} +338884. q^{44} -276209. q^{45} -815087. q^{46} -165989. q^{47} -110592. q^{48} +117649. q^{49} -523450. q^{50} +43684.0 q^{51} +140608. q^{52} +124874. q^{53} +157464. q^{54} -2.00624e6 q^{55} -175616. q^{56} -1.13435e6 q^{57} -525100. q^{58} -1.79173e6 q^{59} +654719. q^{60} +121728. q^{61} -939639. q^{62} +250047. q^{63} +262144. q^{64} -832417. q^{65} +1.14373e6 q^{66} -154439. q^{67} -103547. q^{68} -2.75092e6 q^{69} +1.03967e6 q^{70} -349494. q^{71} -373248. q^{72} +742316. q^{73} -1.36202e6 q^{74} -1.76664e6 q^{75} +2.68883e6 q^{76} +1.81621e6 q^{77} +474552. q^{78} +4.09788e6 q^{79} -1.55193e6 q^{80} +531441. q^{81} -2.44035e6 q^{82} -2.21258e6 q^{83} -592704. q^{84} +613013. q^{85} -3.11065e6 q^{86} -1.77221e6 q^{87} -2.71107e6 q^{88} +585039. q^{89} +2.20968e6 q^{90} +753571. q^{91} +6.52070e6 q^{92} -3.17128e6 q^{93} +1.32792e6 q^{94} -1.59182e7 q^{95} +884736. q^{96} +8.57271e6 q^{97} -941192. q^{98} +3.86010e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99}+O(q^{100}) \) 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
\(5\) −378.888 −1.35555 −0.677776 0.735269i \(-0.737056\pi\)
−0.677776 + 0.735269i \(0.737056\pi\)
\(6\) 216.000 0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 3031.11 0.958520
\(11\) 5295.06 1.19949 0.599745 0.800191i \(-0.295269\pi\)
0.599745 + 0.800191i \(0.295269\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) 10230.0 0.782628
\(16\) 4096.00 0.250000
\(17\) −1617.93 −0.0798707 −0.0399354 0.999202i \(-0.512715\pi\)
−0.0399354 + 0.999202i \(0.512715\pi\)
\(18\) −5832.00 −0.235702
\(19\) 42012.9 1.40522 0.702611 0.711574i \(-0.252017\pi\)
0.702611 + 0.711574i \(0.252017\pi\)
\(20\) −24248.8 −0.677776
\(21\) −9261.00 −0.218218
\(22\) −42360.5 −0.848167
\(23\) 101886. 1.74609 0.873045 0.487639i \(-0.162142\pi\)
0.873045 + 0.487639i \(0.162142\pi\)
\(24\) 13824.0 0.204124
\(25\) 65431.2 0.837520
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 65637.5 0.499757 0.249879 0.968277i \(-0.419609\pi\)
0.249879 + 0.968277i \(0.419609\pi\)
\(30\) −81839.8 −0.553402
\(31\) 117455. 0.708117 0.354058 0.935223i \(-0.384801\pi\)
0.354058 + 0.935223i \(0.384801\pi\)
\(32\) −32768.0 −0.176777
\(33\) −142967. −0.692526
\(34\) 12943.4 0.0564771
\(35\) −129959. −0.512350
\(36\) 46656.0 0.166667
\(37\) 170252. 0.552568 0.276284 0.961076i \(-0.410897\pi\)
0.276284 + 0.961076i \(0.410897\pi\)
\(38\) −336103. −0.993643
\(39\) −59319.0 −0.160128
\(40\) 193991. 0.479260
\(41\) 305044. 0.691225 0.345612 0.938377i \(-0.387671\pi\)
0.345612 + 0.938377i \(0.387671\pi\)
\(42\) 74088.0 0.154303
\(43\) 388832. 0.745800 0.372900 0.927872i \(-0.378364\pi\)
0.372900 + 0.927872i \(0.378364\pi\)
\(44\) 338884. 0.599745
\(45\) −276209. −0.451851
\(46\) −815087. −1.23467
\(47\) −165989. −0.233205 −0.116603 0.993179i \(-0.537200\pi\)
−0.116603 + 0.993179i \(0.537200\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −523450. −0.592216
\(51\) 43684.0 0.0461134
\(52\) 140608. 0.138675
\(53\) 124874. 0.115214 0.0576070 0.998339i \(-0.481653\pi\)
0.0576070 + 0.998339i \(0.481653\pi\)
\(54\) 157464. 0.136083
\(55\) −2.00624e6 −1.62597
\(56\) −175616. −0.133631
\(57\) −1.13435e6 −0.811306
\(58\) −525100. −0.353382
\(59\) −1.79173e6 −1.13577 −0.567886 0.823107i \(-0.692239\pi\)
−0.567886 + 0.823107i \(0.692239\pi\)
\(60\) 654719. 0.391314
\(61\) 121728. 0.0686651 0.0343326 0.999410i \(-0.489069\pi\)
0.0343326 + 0.999410i \(0.489069\pi\)
\(62\) −939639. −0.500714
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −832417. −0.375962
\(66\) 1.14373e6 0.489690
\(67\) −154439. −0.0627328 −0.0313664 0.999508i \(-0.509986\pi\)
−0.0313664 + 0.999508i \(0.509986\pi\)
\(68\) −103547. −0.0399354
\(69\) −2.75092e6 −1.00811
\(70\) 1.03967e6 0.362286
\(71\) −349494. −0.115887 −0.0579437 0.998320i \(-0.518454\pi\)
−0.0579437 + 0.998320i \(0.518454\pi\)
\(72\) −373248. −0.117851
\(73\) 742316. 0.223336 0.111668 0.993746i \(-0.464381\pi\)
0.111668 + 0.993746i \(0.464381\pi\)
\(74\) −1.36202e6 −0.390725
\(75\) −1.76664e6 −0.483542
\(76\) 2.68883e6 0.702611
\(77\) 1.81621e6 0.453364
\(78\) 474552. 0.113228
\(79\) 4.09788e6 0.935114 0.467557 0.883963i \(-0.345134\pi\)
0.467557 + 0.883963i \(0.345134\pi\)
\(80\) −1.55193e6 −0.338888
\(81\) 531441. 0.111111
\(82\) −2.44035e6 −0.488770
\(83\) −2.21258e6 −0.424742 −0.212371 0.977189i \(-0.568119\pi\)
−0.212371 + 0.977189i \(0.568119\pi\)
\(84\) −592704. −0.109109
\(85\) 613013. 0.108269
\(86\) −3.11065e6 −0.527360
\(87\) −1.77221e6 −0.288535
\(88\) −2.71107e6 −0.424084
\(89\) 585039. 0.0879669 0.0439835 0.999032i \(-0.485995\pi\)
0.0439835 + 0.999032i \(0.485995\pi\)
\(90\) 2.20968e6 0.319507
\(91\) 753571. 0.104828
\(92\) 6.52070e6 0.873045
\(93\) −3.17128e6 −0.408832
\(94\) 1.32792e6 0.164901
\(95\) −1.59182e7 −1.90485
\(96\) 884736. 0.102062
\(97\) 8.57271e6 0.953711 0.476856 0.878982i \(-0.341776\pi\)
0.476856 + 0.878982i \(0.341776\pi\)
\(98\) −941192. −0.101015
\(99\) 3.86010e6 0.399830
\(100\) 4.18760e6 0.418760
\(101\) −6.25098e6 −0.603703 −0.301852 0.953355i \(-0.597605\pi\)
−0.301852 + 0.953355i \(0.597605\pi\)
\(102\) −349472. −0.0326071
\(103\) −1.07333e7 −0.967840 −0.483920 0.875112i \(-0.660787\pi\)
−0.483920 + 0.875112i \(0.660787\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 3.50888e6 0.295806
\(106\) −998990. −0.0814686
\(107\) −1.82392e6 −0.143934 −0.0719669 0.997407i \(-0.522928\pi\)
−0.0719669 + 0.997407i \(0.522928\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 1.27114e7 0.940157 0.470079 0.882624i \(-0.344226\pi\)
0.470079 + 0.882624i \(0.344226\pi\)
\(110\) 1.60499e7 1.14973
\(111\) −4.59680e6 −0.319025
\(112\) 1.40493e6 0.0944911
\(113\) 7.44600e6 0.485454 0.242727 0.970095i \(-0.421958\pi\)
0.242727 + 0.970095i \(0.421958\pi\)
\(114\) 9.07479e6 0.573680
\(115\) −3.86034e7 −2.36692
\(116\) 4.20080e6 0.249879
\(117\) 1.60161e6 0.0924500
\(118\) 1.43339e7 0.803112
\(119\) −554949. −0.0301883
\(120\) −5.23775e6 −0.276701
\(121\) 8.55049e6 0.438776
\(122\) −973824. −0.0485536
\(123\) −8.23619e6 −0.399079
\(124\) 7.51711e6 0.354058
\(125\) 4.80951e6 0.220250
\(126\) −2.00038e6 −0.0890871
\(127\) −5.83896e6 −0.252943 −0.126471 0.991970i \(-0.540365\pi\)
−0.126471 + 0.991970i \(0.540365\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) −1.04985e7 −0.430588
\(130\) 6.65934e6 0.265846
\(131\) −1.48836e7 −0.578441 −0.289220 0.957263i \(-0.593396\pi\)
−0.289220 + 0.957263i \(0.593396\pi\)
\(132\) −9.14986e6 −0.346263
\(133\) 1.44104e7 0.531124
\(134\) 1.23551e6 0.0443588
\(135\) 7.45766e6 0.260876
\(136\) 828378. 0.0282386
\(137\) −4.91113e7 −1.63177 −0.815885 0.578214i \(-0.803750\pi\)
−0.815885 + 0.578214i \(0.803750\pi\)
\(138\) 2.20074e7 0.712839
\(139\) −2.44767e6 −0.0773038 −0.0386519 0.999253i \(-0.512306\pi\)
−0.0386519 + 0.999253i \(0.512306\pi\)
\(140\) −8.31735e6 −0.256175
\(141\) 4.48171e6 0.134641
\(142\) 2.79596e6 0.0819447
\(143\) 1.16332e7 0.332679
\(144\) 2.98598e6 0.0833333
\(145\) −2.48693e7 −0.677447
\(146\) −5.93853e6 −0.157922
\(147\) −3.17652e6 −0.0824786
\(148\) 1.08961e7 0.276284
\(149\) −5.05865e7 −1.25280 −0.626401 0.779501i \(-0.715473\pi\)
−0.626401 + 0.779501i \(0.715473\pi\)
\(150\) 1.41331e7 0.341916
\(151\) 3.76324e7 0.889493 0.444747 0.895656i \(-0.353294\pi\)
0.444747 + 0.895656i \(0.353294\pi\)
\(152\) −2.15106e7 −0.496821
\(153\) −1.17947e6 −0.0266236
\(154\) −1.45296e7 −0.320577
\(155\) −4.45023e7 −0.959889
\(156\) −3.79642e6 −0.0800641
\(157\) 5.73898e7 1.18355 0.591774 0.806104i \(-0.298428\pi\)
0.591774 + 0.806104i \(0.298428\pi\)
\(158\) −3.27831e7 −0.661226
\(159\) −3.37159e6 −0.0665189
\(160\) 1.24154e7 0.239630
\(161\) 3.49469e7 0.659960
\(162\) −4.25153e6 −0.0785674
\(163\) −9.76916e6 −0.176685 −0.0883427 0.996090i \(-0.528157\pi\)
−0.0883427 + 0.996090i \(0.528157\pi\)
\(164\) 1.95228e7 0.345612
\(165\) 5.41684e7 0.938754
\(166\) 1.77006e7 0.300338
\(167\) −1.14275e8 −1.89865 −0.949324 0.314300i \(-0.898230\pi\)
−0.949324 + 0.314300i \(0.898230\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −4.90411e6 −0.0765577
\(171\) 3.06274e7 0.468408
\(172\) 2.48852e7 0.372900
\(173\) 1.54944e7 0.227517 0.113759 0.993508i \(-0.463711\pi\)
0.113759 + 0.993508i \(0.463711\pi\)
\(174\) 1.41777e7 0.204025
\(175\) 2.24429e7 0.316553
\(176\) 2.16886e7 0.299872
\(177\) 4.83768e7 0.655738
\(178\) −4.68031e6 −0.0622020
\(179\) −1.59073e7 −0.207306 −0.103653 0.994614i \(-0.533053\pi\)
−0.103653 + 0.994614i \(0.533053\pi\)
\(180\) −1.76774e7 −0.225925
\(181\) 3.88537e6 0.0487032 0.0243516 0.999703i \(-0.492248\pi\)
0.0243516 + 0.999703i \(0.492248\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) −3.28666e6 −0.0396438
\(184\) −5.21656e7 −0.617336
\(185\) −6.45064e7 −0.749035
\(186\) 2.53702e7 0.289088
\(187\) −8.56702e6 −0.0958041
\(188\) −1.06233e7 −0.116603
\(189\) −6.75127e6 −0.0727393
\(190\) 1.27346e8 1.34693
\(191\) −5.63071e7 −0.584718 −0.292359 0.956309i \(-0.594440\pi\)
−0.292359 + 0.956309i \(0.594440\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 1.35067e8 1.35238 0.676188 0.736729i \(-0.263631\pi\)
0.676188 + 0.736729i \(0.263631\pi\)
\(194\) −6.85817e7 −0.674376
\(195\) 2.24753e7 0.217062
\(196\) 7.52954e6 0.0714286
\(197\) 7.06656e7 0.658531 0.329265 0.944237i \(-0.393199\pi\)
0.329265 + 0.944237i \(0.393199\pi\)
\(198\) −3.08808e7 −0.282722
\(199\) 7.79675e7 0.701339 0.350670 0.936499i \(-0.385954\pi\)
0.350670 + 0.936499i \(0.385954\pi\)
\(200\) −3.35008e7 −0.296108
\(201\) 4.16985e6 0.0362188
\(202\) 5.00079e7 0.426883
\(203\) 2.25137e7 0.188891
\(204\) 2.79578e6 0.0230567
\(205\) −1.15578e8 −0.936991
\(206\) 8.58665e7 0.684366
\(207\) 7.42748e7 0.582030
\(208\) 8.99891e6 0.0693375
\(209\) 2.22461e8 1.68555
\(210\) −2.80711e7 −0.209166
\(211\) 2.39403e8 1.75445 0.877226 0.480077i \(-0.159391\pi\)
0.877226 + 0.480077i \(0.159391\pi\)
\(212\) 7.99192e6 0.0576070
\(213\) 9.43635e6 0.0669076
\(214\) 1.45914e7 0.101777
\(215\) −1.47324e8 −1.01097
\(216\) 1.00777e7 0.0680414
\(217\) 4.02870e7 0.267643
\(218\) −1.01691e8 −0.664792
\(219\) −2.00425e7 −0.128943
\(220\) −1.28399e8 −0.812985
\(221\) −3.55458e6 −0.0221521
\(222\) 3.67744e7 0.225585
\(223\) −2.41348e8 −1.45740 −0.728698 0.684835i \(-0.759874\pi\)
−0.728698 + 0.684835i \(0.759874\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) 4.76994e7 0.279173
\(226\) −5.95680e7 −0.343268
\(227\) 9.31249e7 0.528415 0.264208 0.964466i \(-0.414890\pi\)
0.264208 + 0.964466i \(0.414890\pi\)
\(228\) −7.25983e7 −0.405653
\(229\) −2.07153e8 −1.13990 −0.569950 0.821680i \(-0.693037\pi\)
−0.569950 + 0.821680i \(0.693037\pi\)
\(230\) 3.08827e8 1.67366
\(231\) −4.90376e7 −0.261750
\(232\) −3.36064e7 −0.176691
\(233\) −3.44213e8 −1.78271 −0.891356 0.453304i \(-0.850245\pi\)
−0.891356 + 0.453304i \(0.850245\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 6.28914e7 0.316121
\(236\) −1.14671e8 −0.567886
\(237\) −1.10643e8 −0.539888
\(238\) 4.43959e6 0.0213463
\(239\) 2.99906e8 1.42100 0.710498 0.703699i \(-0.248470\pi\)
0.710498 + 0.703699i \(0.248470\pi\)
\(240\) 4.19020e7 0.195657
\(241\) 2.57421e8 1.18463 0.592317 0.805705i \(-0.298213\pi\)
0.592317 + 0.805705i \(0.298213\pi\)
\(242\) −6.84040e7 −0.310261
\(243\) −1.43489e7 −0.0641500
\(244\) 7.79060e6 0.0343326
\(245\) −4.45758e7 −0.193650
\(246\) 6.58895e7 0.282191
\(247\) 9.23024e7 0.389739
\(248\) −6.01369e7 −0.250357
\(249\) 5.97396e7 0.245225
\(250\) −3.84761e7 −0.155740
\(251\) 2.04128e8 0.814789 0.407394 0.913252i \(-0.366437\pi\)
0.407394 + 0.913252i \(0.366437\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) 5.39492e8 2.09442
\(254\) 4.67117e7 0.178858
\(255\) −1.65514e7 −0.0625091
\(256\) 1.67772e7 0.0625000
\(257\) 4.09448e8 1.50464 0.752320 0.658798i \(-0.228935\pi\)
0.752320 + 0.658798i \(0.228935\pi\)
\(258\) 8.39877e7 0.304471
\(259\) 5.83964e7 0.208851
\(260\) −5.32747e7 −0.187981
\(261\) 4.78498e7 0.166586
\(262\) 1.19069e8 0.409019
\(263\) 1.00893e8 0.341992 0.170996 0.985272i \(-0.445301\pi\)
0.170996 + 0.985272i \(0.445301\pi\)
\(264\) 7.31989e7 0.244845
\(265\) −4.73132e7 −0.156179
\(266\) −1.15283e8 −0.375562
\(267\) −1.57960e7 −0.0507877
\(268\) −9.88409e6 −0.0313664
\(269\) 4.56374e7 0.142951 0.0714756 0.997442i \(-0.477229\pi\)
0.0714756 + 0.997442i \(0.477229\pi\)
\(270\) −5.96612e7 −0.184467
\(271\) 1.81284e8 0.553309 0.276654 0.960969i \(-0.410774\pi\)
0.276654 + 0.960969i \(0.410774\pi\)
\(272\) −6.62703e6 −0.0199677
\(273\) −2.03464e7 −0.0605228
\(274\) 3.92890e8 1.15384
\(275\) 3.46462e8 1.00460
\(276\) −1.76059e8 −0.504053
\(277\) 1.89089e8 0.534548 0.267274 0.963621i \(-0.413877\pi\)
0.267274 + 0.963621i \(0.413877\pi\)
\(278\) 1.95813e7 0.0546620
\(279\) 8.56246e7 0.236039
\(280\) 6.65388e7 0.181143
\(281\) 6.97181e8 1.87445 0.937224 0.348727i \(-0.113386\pi\)
0.937224 + 0.348727i \(0.113386\pi\)
\(282\) −3.58537e7 −0.0952056
\(283\) −1.76668e8 −0.463345 −0.231673 0.972794i \(-0.574420\pi\)
−0.231673 + 0.972794i \(0.574420\pi\)
\(284\) −2.23676e7 −0.0579437
\(285\) 4.29791e8 1.09977
\(286\) −9.30660e7 −0.235239
\(287\) 1.04630e8 0.261258
\(288\) −2.38879e7 −0.0589256
\(289\) −4.07721e8 −0.993621
\(290\) 1.98954e8 0.479027
\(291\) −2.31463e8 −0.550626
\(292\) 4.75082e7 0.111668
\(293\) 1.56661e7 0.0363853 0.0181926 0.999835i \(-0.494209\pi\)
0.0181926 + 0.999835i \(0.494209\pi\)
\(294\) 2.54122e7 0.0583212
\(295\) 6.78866e8 1.53960
\(296\) −8.71690e7 −0.195362
\(297\) −1.04223e8 −0.230842
\(298\) 4.04692e8 0.885865
\(299\) 2.23843e8 0.484278
\(300\) −1.13065e8 −0.241771
\(301\) 1.33369e8 0.281886
\(302\) −3.01059e8 −0.628967
\(303\) 1.68777e8 0.348548
\(304\) 1.72085e8 0.351306
\(305\) −4.61213e7 −0.0930791
\(306\) 9.43575e6 0.0188257
\(307\) 3.01774e8 0.595248 0.297624 0.954683i \(-0.403806\pi\)
0.297624 + 0.954683i \(0.403806\pi\)
\(308\) 1.16237e8 0.226682
\(309\) 2.89799e8 0.558783
\(310\) 3.56018e8 0.678744
\(311\) 8.63103e8 1.62705 0.813526 0.581528i \(-0.197545\pi\)
0.813526 + 0.581528i \(0.197545\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 3.80020e8 0.700490 0.350245 0.936658i \(-0.386098\pi\)
0.350245 + 0.936658i \(0.386098\pi\)
\(314\) −4.59119e8 −0.836895
\(315\) −9.47399e7 −0.170783
\(316\) 2.62264e8 0.467557
\(317\) 1.43755e8 0.253464 0.126732 0.991937i \(-0.459551\pi\)
0.126732 + 0.991937i \(0.459551\pi\)
\(318\) 2.69727e7 0.0470359
\(319\) 3.47555e8 0.599454
\(320\) −9.93233e7 −0.169444
\(321\) 4.92459e7 0.0831002
\(322\) −2.79575e8 −0.466662
\(323\) −6.79738e7 −0.112236
\(324\) 3.40122e7 0.0555556
\(325\) 1.43752e8 0.232286
\(326\) 7.81533e7 0.124935
\(327\) −3.43208e8 −0.542800
\(328\) −1.56183e8 −0.244385
\(329\) −5.69344e7 −0.0881432
\(330\) −4.33347e8 −0.663800
\(331\) 3.40991e8 0.516827 0.258413 0.966034i \(-0.416800\pi\)
0.258413 + 0.966034i \(0.416800\pi\)
\(332\) −1.41605e8 −0.212371
\(333\) 1.24114e8 0.184189
\(334\) 9.14202e8 1.34255
\(335\) 5.85151e7 0.0850376
\(336\) −3.79331e7 −0.0545545
\(337\) −5.50524e7 −0.0783558 −0.0391779 0.999232i \(-0.512474\pi\)
−0.0391779 + 0.999232i \(0.512474\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) −2.01042e8 −0.280277
\(340\) 3.92328e7 0.0541344
\(341\) 6.21931e8 0.849379
\(342\) −2.45019e8 −0.331214
\(343\) 4.03536e7 0.0539949
\(344\) −1.99082e8 −0.263680
\(345\) 1.04229e9 1.36654
\(346\) −1.23955e8 −0.160879
\(347\) −6.65870e8 −0.855532 −0.427766 0.903889i \(-0.640699\pi\)
−0.427766 + 0.903889i \(0.640699\pi\)
\(348\) −1.13422e8 −0.144268
\(349\) −8.81952e8 −1.11060 −0.555298 0.831652i \(-0.687396\pi\)
−0.555298 + 0.831652i \(0.687396\pi\)
\(350\) −1.79543e8 −0.223837
\(351\) −4.32436e7 −0.0533761
\(352\) −1.73509e8 −0.212042
\(353\) 6.08276e8 0.736019 0.368010 0.929822i \(-0.380039\pi\)
0.368010 + 0.929822i \(0.380039\pi\)
\(354\) −3.87014e8 −0.463677
\(355\) 1.32419e8 0.157091
\(356\) 3.74425e7 0.0439835
\(357\) 1.49836e7 0.0174292
\(358\) 1.27259e8 0.146587
\(359\) −9.63926e8 −1.09955 −0.549773 0.835314i \(-0.685286\pi\)
−0.549773 + 0.835314i \(0.685286\pi\)
\(360\) 1.41419e8 0.159753
\(361\) 8.71213e8 0.974651
\(362\) −3.10829e7 −0.0344383
\(363\) −2.30863e8 −0.253327
\(364\) 4.82285e7 0.0524142
\(365\) −2.81255e8 −0.302743
\(366\) 2.62933e7 0.0280324
\(367\) −1.50372e9 −1.58795 −0.793975 0.607950i \(-0.791992\pi\)
−0.793975 + 0.607950i \(0.791992\pi\)
\(368\) 4.17325e8 0.436523
\(369\) 2.22377e8 0.230408
\(370\) 5.16051e8 0.529648
\(371\) 4.28317e7 0.0435468
\(372\) −2.02962e8 −0.204416
\(373\) 1.13197e8 0.112941 0.0564707 0.998404i \(-0.482015\pi\)
0.0564707 + 0.998404i \(0.482015\pi\)
\(374\) 6.85361e7 0.0677437
\(375\) −1.29857e8 −0.127161
\(376\) 8.49866e7 0.0824504
\(377\) 1.44206e8 0.138608
\(378\) 5.40102e7 0.0514344
\(379\) −6.91870e8 −0.652811 −0.326405 0.945230i \(-0.605837\pi\)
−0.326405 + 0.945230i \(0.605837\pi\)
\(380\) −1.01876e9 −0.952426
\(381\) 1.57652e8 0.146037
\(382\) 4.50457e8 0.413458
\(383\) −5.86351e8 −0.533289 −0.266644 0.963795i \(-0.585915\pi\)
−0.266644 + 0.963795i \(0.585915\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −6.88139e8 −0.614559
\(386\) −1.08053e9 −0.956274
\(387\) 2.83458e8 0.248600
\(388\) 5.48653e8 0.476856
\(389\) 6.66861e8 0.574396 0.287198 0.957871i \(-0.407276\pi\)
0.287198 + 0.957871i \(0.407276\pi\)
\(390\) −1.79802e8 −0.153486
\(391\) −1.64844e8 −0.139462
\(392\) −6.02363e7 −0.0505076
\(393\) 4.01858e8 0.333963
\(394\) −5.65325e8 −0.465652
\(395\) −1.55264e9 −1.26760
\(396\) 2.47046e8 0.199915
\(397\) −1.64583e9 −1.32014 −0.660069 0.751205i \(-0.729473\pi\)
−0.660069 + 0.751205i \(0.729473\pi\)
\(398\) −6.23740e8 −0.495922
\(399\) −3.89082e8 −0.306645
\(400\) 2.68006e8 0.209380
\(401\) 2.12678e7 0.0164709 0.00823544 0.999966i \(-0.497379\pi\)
0.00823544 + 0.999966i \(0.497379\pi\)
\(402\) −3.33588e7 −0.0256106
\(403\) 2.58048e8 0.196396
\(404\) −4.00063e8 −0.301852
\(405\) −2.01357e8 −0.150617
\(406\) −1.80109e8 −0.133566
\(407\) 9.01494e8 0.662800
\(408\) −2.23662e7 −0.0163035
\(409\) −1.10363e9 −0.797611 −0.398805 0.917036i \(-0.630575\pi\)
−0.398805 + 0.917036i \(0.630575\pi\)
\(410\) 9.24621e8 0.662553
\(411\) 1.32600e9 0.942103
\(412\) −6.86932e8 −0.483920
\(413\) −6.14564e8 −0.429281
\(414\) −5.94199e8 −0.411558
\(415\) 8.38320e8 0.575760
\(416\) −7.19913e7 −0.0490290
\(417\) 6.60870e7 0.0446314
\(418\) −1.77969e9 −1.19186
\(419\) −1.44045e9 −0.956638 −0.478319 0.878186i \(-0.658754\pi\)
−0.478319 + 0.878186i \(0.658754\pi\)
\(420\) 2.24569e8 0.147903
\(421\) −1.45953e8 −0.0953291 −0.0476646 0.998863i \(-0.515178\pi\)
−0.0476646 + 0.998863i \(0.515178\pi\)
\(422\) −1.91523e9 −1.24059
\(423\) −1.21006e8 −0.0777350
\(424\) −6.39353e7 −0.0407343
\(425\) −1.05863e8 −0.0668933
\(426\) −7.54908e7 −0.0473108
\(427\) 4.17527e7 0.0259530
\(428\) −1.16731e8 −0.0719669
\(429\) −3.14098e8 −0.192072
\(430\) 1.17859e9 0.714864
\(431\) 7.55565e8 0.454571 0.227285 0.973828i \(-0.427015\pi\)
0.227285 + 0.973828i \(0.427015\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 1.90514e9 1.12777 0.563883 0.825855i \(-0.309307\pi\)
0.563883 + 0.825855i \(0.309307\pi\)
\(434\) −3.22296e8 −0.189252
\(435\) 6.71471e8 0.391124
\(436\) 8.13530e8 0.470079
\(437\) 4.28053e9 2.45365
\(438\) 1.60340e8 0.0911765
\(439\) 8.25574e6 0.00465726 0.00232863 0.999997i \(-0.499259\pi\)
0.00232863 + 0.999997i \(0.499259\pi\)
\(440\) 1.02719e9 0.574867
\(441\) 8.57661e7 0.0476190
\(442\) 2.84367e7 0.0156639
\(443\) 1.87904e9 1.02689 0.513443 0.858123i \(-0.328370\pi\)
0.513443 + 0.858123i \(0.328370\pi\)
\(444\) −2.94195e8 −0.159513
\(445\) −2.21664e8 −0.119244
\(446\) 1.93079e9 1.03053
\(447\) 1.36584e9 0.723306
\(448\) 8.99154e7 0.0472456
\(449\) 2.21500e9 1.15481 0.577406 0.816457i \(-0.304065\pi\)
0.577406 + 0.816457i \(0.304065\pi\)
\(450\) −3.81595e8 −0.197405
\(451\) 1.61523e9 0.829117
\(452\) 4.76544e8 0.242727
\(453\) −1.01607e9 −0.513549
\(454\) −7.44999e8 −0.373646
\(455\) −2.85519e8 −0.142100
\(456\) 5.80787e8 0.286840
\(457\) 1.02114e9 0.500469 0.250234 0.968185i \(-0.419492\pi\)
0.250234 + 0.968185i \(0.419492\pi\)
\(458\) 1.65722e9 0.806031
\(459\) 3.18456e7 0.0153711
\(460\) −2.47062e9 −1.18346
\(461\) −2.27846e9 −1.08315 −0.541574 0.840653i \(-0.682171\pi\)
−0.541574 + 0.840653i \(0.682171\pi\)
\(462\) 3.92300e8 0.185085
\(463\) −1.31776e9 −0.617026 −0.308513 0.951220i \(-0.599831\pi\)
−0.308513 + 0.951220i \(0.599831\pi\)
\(464\) 2.68851e8 0.124939
\(465\) 1.20156e9 0.554192
\(466\) 2.75370e9 1.26057
\(467\) −4.21868e9 −1.91676 −0.958379 0.285498i \(-0.907841\pi\)
−0.958379 + 0.285498i \(0.907841\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) −5.29726e7 −0.0237108
\(470\) −5.03131e8 −0.223532
\(471\) −1.54953e9 −0.683322
\(472\) 9.17367e8 0.401556
\(473\) 2.05889e9 0.894579
\(474\) 8.85143e8 0.381759
\(475\) 2.74896e9 1.17690
\(476\) −3.55167e7 −0.0150941
\(477\) 9.10329e7 0.0384047
\(478\) −2.39925e9 −1.00480
\(479\) 2.47664e9 1.02965 0.514823 0.857296i \(-0.327858\pi\)
0.514823 + 0.857296i \(0.327858\pi\)
\(480\) −3.35216e8 −0.138350
\(481\) 3.74043e8 0.153255
\(482\) −2.05937e9 −0.837663
\(483\) −9.43566e8 −0.381028
\(484\) 5.47232e8 0.219388
\(485\) −3.24810e9 −1.29280
\(486\) 1.14791e8 0.0453609
\(487\) 1.36666e9 0.536178 0.268089 0.963394i \(-0.413608\pi\)
0.268089 + 0.963394i \(0.413608\pi\)
\(488\) −6.23248e7 −0.0242768
\(489\) 2.63767e8 0.102009
\(490\) 3.56607e8 0.136931
\(491\) −2.79999e9 −1.06751 −0.533754 0.845640i \(-0.679219\pi\)
−0.533754 + 0.845640i \(0.679219\pi\)
\(492\) −5.27116e8 −0.199539
\(493\) −1.06197e8 −0.0399160
\(494\) −7.38419e8 −0.275587
\(495\) −1.46255e9 −0.541990
\(496\) 4.81095e8 0.177029
\(497\) −1.19877e8 −0.0438013
\(498\) −4.77917e8 −0.173400
\(499\) 3.58384e9 1.29121 0.645604 0.763672i \(-0.276606\pi\)
0.645604 + 0.763672i \(0.276606\pi\)
\(500\) 3.07809e8 0.110125
\(501\) 3.08543e9 1.09618
\(502\) −1.63303e9 −0.576143
\(503\) −3.08646e9 −1.08136 −0.540682 0.841227i \(-0.681834\pi\)
−0.540682 + 0.841227i \(0.681834\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 2.36842e9 0.818351
\(506\) −4.31594e9 −1.48098
\(507\) −1.30324e8 −0.0444116
\(508\) −3.73694e8 −0.126471
\(509\) −8.26000e8 −0.277631 −0.138815 0.990318i \(-0.544329\pi\)
−0.138815 + 0.990318i \(0.544329\pi\)
\(510\) 1.32411e8 0.0442006
\(511\) 2.54614e8 0.0844131
\(512\) −1.34218e8 −0.0441942
\(513\) −8.26940e8 −0.270435
\(514\) −3.27558e9 −1.06394
\(515\) 4.06673e9 1.31196
\(516\) −6.71901e8 −0.215294
\(517\) −8.78924e8 −0.279727
\(518\) −4.67171e8 −0.147680
\(519\) −4.18350e8 −0.131357
\(520\) 4.26198e8 0.132923
\(521\) −6.38043e9 −1.97660 −0.988298 0.152539i \(-0.951255\pi\)
−0.988298 + 0.152539i \(0.951255\pi\)
\(522\) −3.82798e8 −0.117794
\(523\) −1.94369e9 −0.594117 −0.297058 0.954859i \(-0.596006\pi\)
−0.297058 + 0.954859i \(0.596006\pi\)
\(524\) −9.52551e8 −0.289220
\(525\) −6.05959e8 −0.182762
\(526\) −8.07145e8 −0.241825
\(527\) −1.90033e8 −0.0565578
\(528\) −5.85591e8 −0.173131
\(529\) 6.97592e9 2.04883
\(530\) 3.78505e8 0.110435
\(531\) −1.30617e9 −0.378591
\(532\) 9.22268e8 0.265562
\(533\) 6.70182e8 0.191711
\(534\) 1.26368e8 0.0359124
\(535\) 6.91062e8 0.195110
\(536\) 7.90727e7 0.0221794
\(537\) 4.29498e8 0.119688
\(538\) −3.65099e8 −0.101082
\(539\) 6.22959e8 0.171356
\(540\) 4.77290e8 0.130438
\(541\) 6.03178e9 1.63778 0.818889 0.573952i \(-0.194591\pi\)
0.818889 + 0.573952i \(0.194591\pi\)
\(542\) −1.45027e9 −0.391248
\(543\) −1.04905e8 −0.0281188
\(544\) 5.30162e7 0.0141193
\(545\) −4.81620e9 −1.27443
\(546\) 1.62771e8 0.0427960
\(547\) 3.94623e9 1.03092 0.515462 0.856913i \(-0.327620\pi\)
0.515462 + 0.856913i \(0.327620\pi\)
\(548\) −3.14312e9 −0.815885
\(549\) 8.87398e7 0.0228884
\(550\) −2.77170e9 −0.710357
\(551\) 2.75762e9 0.702270
\(552\) 1.40847e9 0.356419
\(553\) 1.40557e9 0.353440
\(554\) −1.51271e9 −0.377983
\(555\) 1.74167e9 0.432455
\(556\) −1.56651e8 −0.0386519
\(557\) 5.88604e9 1.44321 0.721606 0.692304i \(-0.243404\pi\)
0.721606 + 0.692304i \(0.243404\pi\)
\(558\) −6.84997e8 −0.166905
\(559\) 8.54263e8 0.206848
\(560\) −5.32311e8 −0.128088
\(561\) 2.31309e8 0.0553125
\(562\) −5.57745e9 −1.32544
\(563\) −3.36222e9 −0.794049 −0.397024 0.917808i \(-0.629957\pi\)
−0.397024 + 0.917808i \(0.629957\pi\)
\(564\) 2.86830e8 0.0673205
\(565\) −2.82120e9 −0.658059
\(566\) 1.41334e9 0.327634
\(567\) 1.82284e8 0.0419961
\(568\) 1.78941e8 0.0409724
\(569\) −3.24680e9 −0.738861 −0.369430 0.929258i \(-0.620447\pi\)
−0.369430 + 0.929258i \(0.620447\pi\)
\(570\) −3.43833e9 −0.777653
\(571\) −5.21927e9 −1.17323 −0.586616 0.809865i \(-0.699540\pi\)
−0.586616 + 0.809865i \(0.699540\pi\)
\(572\) 7.44528e8 0.166339
\(573\) 1.52029e9 0.337587
\(574\) −8.37041e8 −0.184738
\(575\) 6.66652e9 1.46239
\(576\) 1.91103e8 0.0416667
\(577\) −6.94575e9 −1.50523 −0.752616 0.658460i \(-0.771208\pi\)
−0.752616 + 0.658460i \(0.771208\pi\)
\(578\) 3.26177e9 0.702596
\(579\) −3.64680e9 −0.780794
\(580\) −1.59163e9 −0.338723
\(581\) −7.58915e8 −0.160537
\(582\) 1.85170e9 0.389351
\(583\) 6.61214e8 0.138198
\(584\) −3.80066e8 −0.0789612
\(585\) −6.06832e8 −0.125321
\(586\) −1.25329e8 −0.0257283
\(587\) −6.75547e9 −1.37855 −0.689275 0.724500i \(-0.742071\pi\)
−0.689275 + 0.724500i \(0.742071\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 4.93462e9 0.995062
\(590\) −5.43093e9 −1.08866
\(591\) −1.90797e9 −0.380203
\(592\) 6.97352e8 0.138142
\(593\) −3.66897e8 −0.0722524 −0.0361262 0.999347i \(-0.511502\pi\)
−0.0361262 + 0.999347i \(0.511502\pi\)
\(594\) 8.33781e8 0.163230
\(595\) 2.10264e8 0.0409218
\(596\) −3.23754e9 −0.626401
\(597\) −2.10512e9 −0.404918
\(598\) −1.79075e9 −0.342437
\(599\) −3.63967e9 −0.691940 −0.345970 0.938246i \(-0.612450\pi\)
−0.345970 + 0.938246i \(0.612450\pi\)
\(600\) 9.04522e8 0.170958
\(601\) −1.32849e9 −0.249631 −0.124815 0.992180i \(-0.539834\pi\)
−0.124815 + 0.992180i \(0.539834\pi\)
\(602\) −1.06695e9 −0.199323
\(603\) −1.12586e8 −0.0209109
\(604\) 2.40847e9 0.444747
\(605\) −3.23968e9 −0.594783
\(606\) −1.35021e9 −0.246461
\(607\) 2.37205e9 0.430490 0.215245 0.976560i \(-0.430945\pi\)
0.215245 + 0.976560i \(0.430945\pi\)
\(608\) −1.37668e9 −0.248411
\(609\) −6.07869e8 −0.109056
\(610\) 3.68971e8 0.0658169
\(611\) −3.64679e8 −0.0646794
\(612\) −7.54860e7 −0.0133118
\(613\) −9.60638e8 −0.168441 −0.0842206 0.996447i \(-0.526840\pi\)
−0.0842206 + 0.996447i \(0.526840\pi\)
\(614\) −2.41419e9 −0.420904
\(615\) 3.12060e9 0.540972
\(616\) −9.29897e8 −0.160289
\(617\) 3.07342e8 0.0526773 0.0263386 0.999653i \(-0.491615\pi\)
0.0263386 + 0.999653i \(0.491615\pi\)
\(618\) −2.31840e9 −0.395119
\(619\) 3.18881e9 0.540395 0.270197 0.962805i \(-0.412911\pi\)
0.270197 + 0.962805i \(0.412911\pi\)
\(620\) −2.84814e9 −0.479945
\(621\) −2.00542e9 −0.336035
\(622\) −6.90483e9 −1.15050
\(623\) 2.00668e8 0.0332484
\(624\) −2.42971e8 −0.0400320
\(625\) −6.93408e9 −1.13608
\(626\) −3.04016e9 −0.495321
\(627\) −6.00644e9 −0.973153
\(628\) 3.67295e9 0.591774
\(629\) −2.75455e8 −0.0441340
\(630\) 7.57919e8 0.120762
\(631\) −1.04712e10 −1.65918 −0.829589 0.558375i \(-0.811425\pi\)
−0.829589 + 0.558375i \(0.811425\pi\)
\(632\) −2.09812e9 −0.330613
\(633\) −6.46389e9 −1.01293
\(634\) −1.15004e9 −0.179226
\(635\) 2.21231e9 0.342877
\(636\) −2.15782e8 −0.0332594
\(637\) 2.58475e8 0.0396214
\(638\) −2.78044e9 −0.423878
\(639\) −2.54781e8 −0.0386291
\(640\) 7.94586e8 0.119815
\(641\) −3.75555e9 −0.563211 −0.281605 0.959530i \(-0.590867\pi\)
−0.281605 + 0.959530i \(0.590867\pi\)
\(642\) −3.93967e8 −0.0587607
\(643\) 8.85818e9 1.31403 0.657016 0.753876i \(-0.271818\pi\)
0.657016 + 0.753876i \(0.271818\pi\)
\(644\) 2.23660e9 0.329980
\(645\) 3.97774e9 0.583684
\(646\) 5.43790e8 0.0793629
\(647\) −8.35966e9 −1.21345 −0.606727 0.794910i \(-0.707518\pi\)
−0.606727 + 0.794910i \(0.707518\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −9.48733e9 −1.36235
\(650\) −1.15002e9 −0.164251
\(651\) −1.08775e9 −0.154524
\(652\) −6.25226e8 −0.0883427
\(653\) −2.32417e9 −0.326642 −0.163321 0.986573i \(-0.552221\pi\)
−0.163321 + 0.986573i \(0.552221\pi\)
\(654\) 2.74566e9 0.383818
\(655\) 5.63923e9 0.784106
\(656\) 1.24946e9 0.172806
\(657\) 5.41148e8 0.0744453
\(658\) 4.55475e8 0.0623267
\(659\) −1.82416e9 −0.248293 −0.124147 0.992264i \(-0.539619\pi\)
−0.124147 + 0.992264i \(0.539619\pi\)
\(660\) 3.46678e9 0.469377
\(661\) 8.34870e9 1.12438 0.562191 0.827007i \(-0.309959\pi\)
0.562191 + 0.827007i \(0.309959\pi\)
\(662\) −2.72793e9 −0.365452
\(663\) 9.59738e7 0.0127895
\(664\) 1.13284e9 0.150169
\(665\) −5.45994e9 −0.719966
\(666\) −9.92909e8 −0.130242
\(667\) 6.68754e9 0.872622
\(668\) −7.31361e9 −0.949324
\(669\) 6.51641e9 0.841428
\(670\) −4.68121e8 −0.0601307
\(671\) 6.44557e8 0.0823631
\(672\) 3.03464e8 0.0385758
\(673\) −1.11689e10 −1.41240 −0.706200 0.708012i \(-0.749592\pi\)
−0.706200 + 0.708012i \(0.749592\pi\)
\(674\) 4.40419e8 0.0554059
\(675\) −1.28788e9 −0.161181
\(676\) 3.08916e8 0.0384615
\(677\) −1.38490e10 −1.71537 −0.857687 0.514173i \(-0.828099\pi\)
−0.857687 + 0.514173i \(0.828099\pi\)
\(678\) 1.60834e9 0.198186
\(679\) 2.94044e9 0.360469
\(680\) −3.13863e8 −0.0382788
\(681\) −2.51437e9 −0.305081
\(682\) −4.97544e9 −0.600602
\(683\) −1.46785e10 −1.76283 −0.881414 0.472345i \(-0.843408\pi\)
−0.881414 + 0.472345i \(0.843408\pi\)
\(684\) 1.96015e9 0.234204
\(685\) 1.86077e10 2.21195
\(686\) −3.22829e8 −0.0381802
\(687\) 5.59312e9 0.658121
\(688\) 1.59266e9 0.186450
\(689\) 2.74348e8 0.0319546
\(690\) −8.33833e9 −0.966289
\(691\) −4.89134e8 −0.0563969 −0.0281984 0.999602i \(-0.508977\pi\)
−0.0281984 + 0.999602i \(0.508977\pi\)
\(692\) 9.91644e8 0.113759
\(693\) 1.32401e9 0.151121
\(694\) 5.32696e9 0.604953
\(695\) 9.27392e8 0.104789
\(696\) 9.07373e8 0.102013
\(697\) −4.93539e8 −0.0552086
\(698\) 7.05562e9 0.785310
\(699\) 9.29375e9 1.02925
\(700\) 1.43635e9 0.158276
\(701\) −5.28855e9 −0.579860 −0.289930 0.957048i \(-0.593632\pi\)
−0.289930 + 0.957048i \(0.593632\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 7.15278e9 0.776482
\(704\) 1.38807e9 0.149936
\(705\) −1.69807e9 −0.182513
\(706\) −4.86621e9 −0.520444
\(707\) −2.14409e9 −0.228178
\(708\) 3.09611e9 0.327869
\(709\) 1.15419e10 1.21623 0.608116 0.793849i \(-0.291926\pi\)
0.608116 + 0.793849i \(0.291926\pi\)
\(710\) −1.05935e9 −0.111080
\(711\) 2.98736e9 0.311705
\(712\) −2.99540e8 −0.0311010
\(713\) 1.19670e10 1.23644
\(714\) −1.19869e8 −0.0123243
\(715\) −4.40770e9 −0.450963
\(716\) −1.01807e9 −0.103653
\(717\) −8.09747e9 −0.820412
\(718\) 7.71141e9 0.777496
\(719\) 1.19175e10 1.19574 0.597868 0.801594i \(-0.296015\pi\)
0.597868 + 0.801594i \(0.296015\pi\)
\(720\) −1.13135e9 −0.112963
\(721\) −3.68153e9 −0.365809
\(722\) −6.96971e9 −0.689183
\(723\) −6.95037e9 −0.683949
\(724\) 2.48664e8 0.0243516
\(725\) 4.29475e9 0.418557
\(726\) 1.84691e9 0.179129
\(727\) 3.61469e9 0.348900 0.174450 0.984666i \(-0.444185\pi\)
0.174450 + 0.984666i \(0.444185\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 2.25004e9 0.214072
\(731\) −6.29101e8 −0.0595676
\(732\) −2.10346e8 −0.0198219
\(733\) −2.25776e9 −0.211745 −0.105873 0.994380i \(-0.533764\pi\)
−0.105873 + 0.994380i \(0.533764\pi\)
\(734\) 1.20298e10 1.12285
\(735\) 1.20355e9 0.111804
\(736\) −3.33860e9 −0.308668
\(737\) −8.17764e8 −0.0752474
\(738\) −1.77902e9 −0.162923
\(739\) 2.04054e10 1.85990 0.929949 0.367688i \(-0.119851\pi\)
0.929949 + 0.367688i \(0.119851\pi\)
\(740\) −4.12841e9 −0.374517
\(741\) −2.49216e9 −0.225016
\(742\) −3.42653e8 −0.0307923
\(743\) −5.54813e9 −0.496233 −0.248116 0.968730i \(-0.579812\pi\)
−0.248116 + 0.968730i \(0.579812\pi\)
\(744\) 1.62370e9 0.144544
\(745\) 1.91666e10 1.69824
\(746\) −9.05574e8 −0.0798616
\(747\) −1.61297e9 −0.141581
\(748\) −5.48289e8 −0.0479020
\(749\) −6.25605e8 −0.0544019
\(750\) 1.03885e9 0.0899167
\(751\) −9.72081e9 −0.837457 −0.418729 0.908111i \(-0.637524\pi\)
−0.418729 + 0.908111i \(0.637524\pi\)
\(752\) −6.79893e8 −0.0583013
\(753\) −5.51146e9 −0.470419
\(754\) −1.15365e9 −0.0980105
\(755\) −1.42585e10 −1.20575
\(756\) −4.32081e8 −0.0363696
\(757\) −1.93064e9 −0.161758 −0.0808791 0.996724i \(-0.525773\pi\)
−0.0808791 + 0.996724i \(0.525773\pi\)
\(758\) 5.53496e9 0.461607
\(759\) −1.45663e10 −1.20921
\(760\) 8.15012e9 0.673467
\(761\) −2.22943e9 −0.183378 −0.0916892 0.995788i \(-0.529227\pi\)
−0.0916892 + 0.995788i \(0.529227\pi\)
\(762\) −1.26122e9 −0.103264
\(763\) 4.36001e9 0.355346
\(764\) −3.60366e9 −0.292359
\(765\) 4.46887e8 0.0360896
\(766\) 4.69081e9 0.377092
\(767\) −3.93644e9 −0.315006
\(768\) −4.52985e8 −0.0360844
\(769\) 1.03220e10 0.818506 0.409253 0.912421i \(-0.365789\pi\)
0.409253 + 0.912421i \(0.365789\pi\)
\(770\) 5.50511e9 0.434559
\(771\) −1.10551e10 −0.868704
\(772\) 8.64426e9 0.676188
\(773\) 1.74584e10 1.35949 0.679745 0.733449i \(-0.262091\pi\)
0.679745 + 0.733449i \(0.262091\pi\)
\(774\) −2.26767e9 −0.175787
\(775\) 7.68522e9 0.593062
\(776\) −4.38923e9 −0.337188
\(777\) −1.57670e9 −0.120580
\(778\) −5.33489e9 −0.406159
\(779\) 1.28158e10 0.971325
\(780\) 1.43842e9 0.108531
\(781\) −1.85059e9 −0.139006
\(782\) 1.31875e9 0.0986142
\(783\) −1.29194e9 −0.0961783
\(784\) 4.81890e8 0.0357143
\(785\) −2.17443e10 −1.60436
\(786\) −3.21486e9 −0.236147
\(787\) −9.17018e9 −0.670604 −0.335302 0.942111i \(-0.608838\pi\)
−0.335302 + 0.942111i \(0.608838\pi\)
\(788\) 4.52260e9 0.329265
\(789\) −2.72411e9 −0.197449
\(790\) 1.24211e10 0.896325
\(791\) 2.55398e9 0.183485
\(792\) −1.97637e9 −0.141361
\(793\) 2.67437e8 0.0190443
\(794\) 1.31667e10 0.933478
\(795\) 1.27746e9 0.0901698
\(796\) 4.98992e9 0.350670
\(797\) 1.55129e10 1.08540 0.542698 0.839928i \(-0.317403\pi\)
0.542698 + 0.839928i \(0.317403\pi\)
\(798\) 3.11265e9 0.216831
\(799\) 2.68559e8 0.0186263
\(800\) −2.14405e9 −0.148054
\(801\) 4.26493e8 0.0293223
\(802\) −1.70142e8 −0.0116467
\(803\) 3.93061e9 0.267889
\(804\) 2.66870e8 0.0181094
\(805\) −1.32410e10 −0.894610
\(806\) −2.06439e9 −0.138873
\(807\) −1.23221e9 −0.0825329
\(808\) 3.20050e9 0.213441
\(809\) 1.88511e10 1.25175 0.625875 0.779924i \(-0.284742\pi\)
0.625875 + 0.779924i \(0.284742\pi\)
\(810\) 1.61085e9 0.106502
\(811\) 1.37909e10 0.907859 0.453930 0.891038i \(-0.350022\pi\)
0.453930 + 0.891038i \(0.350022\pi\)
\(812\) 1.44087e9 0.0944453
\(813\) −4.89467e9 −0.319453
\(814\) −7.21195e9 −0.468670
\(815\) 3.70142e9 0.239506
\(816\) 1.78930e8 0.0115283
\(817\) 1.63360e10 1.04801
\(818\) 8.82902e9 0.563996
\(819\) 5.49353e8 0.0349428
\(820\) −7.39697e9 −0.468495
\(821\) 2.18934e10 1.38074 0.690369 0.723457i \(-0.257448\pi\)
0.690369 + 0.723457i \(0.257448\pi\)
\(822\) −1.06080e10 −0.666168
\(823\) 1.28762e10 0.805171 0.402586 0.915382i \(-0.368112\pi\)
0.402586 + 0.915382i \(0.368112\pi\)
\(824\) 5.49546e9 0.342183
\(825\) −9.35449e9 −0.580004
\(826\) 4.91651e9 0.303548
\(827\) −4.75080e9 −0.292077 −0.146039 0.989279i \(-0.546652\pi\)
−0.146039 + 0.989279i \(0.546652\pi\)
\(828\) 4.75359e9 0.291015
\(829\) −2.70326e10 −1.64796 −0.823980 0.566619i \(-0.808251\pi\)
−0.823980 + 0.566619i \(0.808251\pi\)
\(830\) −6.70656e9 −0.407124
\(831\) −5.10540e9 −0.308622
\(832\) 5.75930e8 0.0346688
\(833\) −1.90347e8 −0.0114101
\(834\) −5.28696e8 −0.0315591
\(835\) 4.32975e10 2.57371
\(836\) 1.42375e10 0.842775
\(837\) −2.31186e9 −0.136277
\(838\) 1.15236e10 0.676445
\(839\) 2.00634e10 1.17284 0.586418 0.810008i \(-0.300537\pi\)
0.586418 + 0.810008i \(0.300537\pi\)
\(840\) −1.79655e9 −0.104583
\(841\) −1.29416e10 −0.750243
\(842\) 1.16762e9 0.0674079
\(843\) −1.88239e10 −1.08221
\(844\) 1.53218e10 0.877226
\(845\) −1.82882e9 −0.104273
\(846\) 9.68050e8 0.0549670
\(847\) 2.93282e9 0.165842
\(848\) 5.11483e8 0.0288035
\(849\) 4.77003e9 0.267512
\(850\) 8.46904e8 0.0473007
\(851\) 1.73463e10 0.964834
\(852\) 6.03926e8 0.0334538
\(853\) 1.10456e10 0.609353 0.304677 0.952456i \(-0.401452\pi\)
0.304677 + 0.952456i \(0.401452\pi\)
\(854\) −3.34022e8 −0.0183515
\(855\) −1.16044e10 −0.634951
\(856\) 9.33848e8 0.0508883
\(857\) 3.08171e10 1.67247 0.836237 0.548368i \(-0.184751\pi\)
0.836237 + 0.548368i \(0.184751\pi\)
\(858\) 2.51278e9 0.135815
\(859\) −1.22323e10 −0.658465 −0.329232 0.944249i \(-0.606790\pi\)
−0.329232 + 0.944249i \(0.606790\pi\)
\(860\) −9.42872e9 −0.505485
\(861\) −2.82501e9 −0.150838
\(862\) −6.04452e9 −0.321430
\(863\) 4.41026e9 0.233575 0.116788 0.993157i \(-0.462740\pi\)
0.116788 + 0.993157i \(0.462740\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −5.87066e9 −0.308411
\(866\) −1.52411e10 −0.797451
\(867\) 1.10085e10 0.573667
\(868\) 2.57837e9 0.133822
\(869\) 2.16985e10 1.12166
\(870\) −5.37176e9 −0.276567
\(871\) −3.39302e8 −0.0173990
\(872\) −6.50824e9 −0.332396
\(873\) 6.24950e9 0.317904
\(874\) −3.42442e10 −1.73499
\(875\) 1.64966e9 0.0832467
\(876\) −1.28272e9 −0.0644716
\(877\) −2.81168e10 −1.40756 −0.703782 0.710416i \(-0.748507\pi\)
−0.703782 + 0.710416i \(0.748507\pi\)
\(878\) −6.60459e7 −0.00329318
\(879\) −4.22986e8 −0.0210070
\(880\) −8.21754e9 −0.406493
\(881\) 1.66508e10 0.820388 0.410194 0.911998i \(-0.365461\pi\)
0.410194 + 0.911998i \(0.365461\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) −9.07838e8 −0.0443758 −0.0221879 0.999754i \(-0.507063\pi\)
−0.0221879 + 0.999754i \(0.507063\pi\)
\(884\) −2.27493e8 −0.0110761
\(885\) −1.83294e10 −0.888887
\(886\) −1.50323e10 −0.726119
\(887\) 1.76337e10 0.848421 0.424211 0.905564i \(-0.360552\pi\)
0.424211 + 0.905564i \(0.360552\pi\)
\(888\) 2.35356e9 0.112793
\(889\) −2.00276e9 −0.0956034
\(890\) 1.77331e9 0.0843180
\(891\) 2.81401e9 0.133277
\(892\) −1.54463e10 −0.728698
\(893\) −6.97370e9 −0.327705
\(894\) −1.09267e10 −0.511455
\(895\) 6.02710e9 0.281014
\(896\) −7.19323e8 −0.0334077
\(897\) −6.04377e9 −0.279598
\(898\) −1.77200e10 −0.816576
\(899\) 7.70945e9 0.353887
\(900\) 3.05276e9 0.139587
\(901\) −2.02036e8 −0.00920223
\(902\) −1.29218e10 −0.586274
\(903\) −3.60097e9 −0.162747
\(904\) −3.81235e9 −0.171634
\(905\) −1.47212e9 −0.0660196
\(906\) 8.12860e9 0.363134
\(907\) 1.18339e10 0.526628 0.263314 0.964710i \(-0.415184\pi\)
0.263314 + 0.964710i \(0.415184\pi\)
\(908\) 5.95999e9 0.264208
\(909\) −4.55697e9 −0.201234
\(910\) 2.28415e9 0.100480
\(911\) 2.18365e10 0.956906 0.478453 0.878113i \(-0.341198\pi\)
0.478453 + 0.878113i \(0.341198\pi\)
\(912\) −4.64629e9 −0.202826
\(913\) −1.17157e10 −0.509474
\(914\) −8.16909e9 −0.353885
\(915\) 1.24528e9 0.0537393
\(916\) −1.32578e10 −0.569950
\(917\) −5.10508e9 −0.218630
\(918\) −2.54765e8 −0.0108690
\(919\) 6.10212e9 0.259344 0.129672 0.991557i \(-0.458608\pi\)
0.129672 + 0.991557i \(0.458608\pi\)
\(920\) 1.97649e10 0.836831
\(921\) −8.14790e9 −0.343666
\(922\) 1.82277e10 0.765901
\(923\) −7.67839e8 −0.0321414
\(924\) −3.13840e9 −0.130875
\(925\) 1.11398e10 0.462787
\(926\) 1.05421e10 0.436303
\(927\) −7.82459e9 −0.322613
\(928\) −2.15081e9 −0.0883455
\(929\) −2.88458e10 −1.18039 −0.590197 0.807259i \(-0.700950\pi\)
−0.590197 + 0.807259i \(0.700950\pi\)
\(930\) −9.61249e9 −0.391873
\(931\) 4.94278e9 0.200746
\(932\) −2.20296e10 −0.891356
\(933\) −2.33038e10 −0.939379
\(934\) 3.37494e10 1.35535
\(935\) 3.24594e9 0.129867
\(936\) −8.20026e8 −0.0326860
\(937\) −5.53047e9 −0.219621 −0.109810 0.993953i \(-0.535024\pi\)
−0.109810 + 0.993953i \(0.535024\pi\)
\(938\) 4.23780e8 0.0167661
\(939\) −1.02606e10 −0.404428
\(940\) 4.02505e9 0.158061
\(941\) −2.60273e10 −1.01828 −0.509139 0.860685i \(-0.670036\pi\)
−0.509139 + 0.860685i \(0.670036\pi\)
\(942\) 1.23962e10 0.483182
\(943\) 3.10797e10 1.20694
\(944\) −7.33894e9 −0.283943
\(945\) 2.55798e9 0.0986019
\(946\) −1.64711e10 −0.632563
\(947\) 2.09596e10 0.801972 0.400986 0.916084i \(-0.368668\pi\)
0.400986 + 0.916084i \(0.368668\pi\)
\(948\) −7.08114e9 −0.269944
\(949\) 1.63087e9 0.0619423
\(950\) −2.19917e10 −0.832196
\(951\) −3.88139e9 −0.146338
\(952\) 2.84134e8 0.0106732
\(953\) −4.15816e10 −1.55624 −0.778119 0.628117i \(-0.783826\pi\)
−0.778119 + 0.628117i \(0.783826\pi\)
\(954\) −7.28264e8 −0.0271562
\(955\) 2.13341e10 0.792616
\(956\) 1.91940e10 0.710498
\(957\) −9.38398e9 −0.346095
\(958\) −1.98131e10 −0.728070
\(959\) −1.68452e10 −0.616751
\(960\) 2.68173e9 0.0978285
\(961\) −1.37170e10 −0.498570
\(962\) −2.99235e9 −0.108368
\(963\) −1.32964e9 −0.0479779
\(964\) 1.64749e10 0.592317
\(965\) −5.11751e10 −1.83321
\(966\) 7.54853e9 0.269428
\(967\) 2.40353e10 0.854784 0.427392 0.904066i \(-0.359432\pi\)
0.427392 + 0.904066i \(0.359432\pi\)
\(968\) −4.37785e9 −0.155131
\(969\) 1.83529e9 0.0647996
\(970\) 2.59848e10 0.914151
\(971\) 2.42256e10 0.849194 0.424597 0.905382i \(-0.360416\pi\)
0.424597 + 0.905382i \(0.360416\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −8.39550e8 −0.0292181
\(974\) −1.09333e10 −0.379135
\(975\) −3.88132e9 −0.134111
\(976\) 4.98598e8 0.0171663
\(977\) −5.17070e10 −1.77386 −0.886928 0.461908i \(-0.847165\pi\)
−0.886928 + 0.461908i \(0.847165\pi\)
\(978\) −2.11014e9 −0.0721315
\(979\) 3.09781e9 0.105515
\(980\) −2.85285e9 −0.0968251
\(981\) 9.26661e9 0.313386
\(982\) 2.23999e10 0.754842
\(983\) 3.76673e10 1.26482 0.632408 0.774636i \(-0.282067\pi\)
0.632408 + 0.774636i \(0.282067\pi\)
\(984\) 4.21693e9 0.141096
\(985\) −2.67744e10 −0.892672
\(986\) 8.49574e8 0.0282249
\(987\) 1.53723e9 0.0508895
\(988\) 5.90735e9 0.194869
\(989\) 3.96165e10 1.30223
\(990\) 1.17004e10 0.383245
\(991\) 4.35196e10 1.42045 0.710227 0.703973i \(-0.248592\pi\)
0.710227 + 0.703973i \(0.248592\pi\)
\(992\) −3.84876e9 −0.125179
\(993\) −9.20675e9 −0.298390
\(994\) 9.59013e8 0.0309722
\(995\) −2.95410e10 −0.950701
\(996\) 3.82334e9 0.122613
\(997\) 1.93909e10 0.619676 0.309838 0.950789i \(-0.399725\pi\)
0.309838 + 0.950789i \(0.399725\pi\)
\(998\) −2.86707e10 −0.913022
\(999\) −3.35107e9 −0.106342
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.n.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.n.1.2 6 1.1 even 1 trivial