Properties

Label 546.8.a.n.1.3
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: \(x^{6} - x^{5} - 309949 x^{4} - 14548431 x^{3} + 25221499020 x^{2} + 1862570808000 x - 308009568384000\)
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 1.3
Root \(83.4678\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -113.468 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -113.468 q^{5} +216.000 q^{6} +343.000 q^{7} -512.000 q^{8} +729.000 q^{9} +907.742 q^{10} -3827.20 q^{11} -1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} +3063.63 q^{15} +4096.00 q^{16} -36941.0 q^{17} -5832.00 q^{18} -56827.3 q^{19} -7261.94 q^{20} -9261.00 q^{21} +30617.6 q^{22} +42283.4 q^{23} +13824.0 q^{24} -65250.1 q^{25} -17576.0 q^{26} -19683.0 q^{27} +21952.0 q^{28} +158520. q^{29} -24509.0 q^{30} +98296.8 q^{31} -32768.0 q^{32} +103335. q^{33} +295528. q^{34} -38919.5 q^{35} +46656.0 q^{36} +352941. q^{37} +454619. q^{38} -59319.0 q^{39} +58095.5 q^{40} -786907. q^{41} +74088.0 q^{42} -744062. q^{43} -244941. q^{44} -82718.0 q^{45} -338267. q^{46} -498725. q^{47} -110592. q^{48} +117649. q^{49} +522000. q^{50} +997408. q^{51} +140608. q^{52} -664787. q^{53} +157464. q^{54} +434264. q^{55} -175616. q^{56} +1.53434e6 q^{57} -1.26816e6 q^{58} +1.39438e6 q^{59} +196072. q^{60} -1.33040e6 q^{61} -786374. q^{62} +250047. q^{63} +262144. q^{64} -249289. q^{65} -826676. q^{66} +2.54161e6 q^{67} -2.36423e6 q^{68} -1.14165e6 q^{69} +311356. q^{70} -5.37482e6 q^{71} -373248. q^{72} +3.59276e6 q^{73} -2.82353e6 q^{74} +1.76175e6 q^{75} -3.63695e6 q^{76} -1.31273e6 q^{77} +474552. q^{78} +5.53769e6 q^{79} -464764. q^{80} +531441. q^{81} +6.29525e6 q^{82} -5.11360e6 q^{83} -592704. q^{84} +4.19162e6 q^{85} +5.95249e6 q^{86} -4.28003e6 q^{87} +1.95953e6 q^{88} -7.66025e6 q^{89} +661744. q^{90} +753571. q^{91} +2.70614e6 q^{92} -2.65401e6 q^{93} +3.98980e6 q^{94} +6.44807e6 q^{95} +884736. q^{96} -1.56564e7 q^{97} -941192. q^{98} -2.79003e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + O(q^{10}) \) \( 6 q - 48 q^{2} - 162 q^{3} + 384 q^{4} - 181 q^{5} + 1296 q^{6} + 2058 q^{7} - 3072 q^{8} + 4374 q^{9} + 1448 q^{10} - 6130 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} + 4887 q^{15} + 24576 q^{16} - 34610 q^{17} - 34992 q^{18} - 4085 q^{19} - 11584 q^{20} - 55566 q^{21} + 49040 q^{22} + 1515 q^{23} + 82944 q^{24} + 156609 q^{25} - 105456 q^{26} - 118098 q^{27} + 131712 q^{28} - 59395 q^{29} - 39096 q^{30} + 478241 q^{31} - 196608 q^{32} + 165510 q^{33} + 276880 q^{34} - 62083 q^{35} + 279936 q^{36} + 574310 q^{37} + 32680 q^{38} - 355914 q^{39} + 92672 q^{40} + 201552 q^{41} + 444528 q^{42} + 728605 q^{43} - 392320 q^{44} - 131949 q^{45} - 12120 q^{46} + 227615 q^{47} - 663552 q^{48} + 705894 q^{49} - 1252872 q^{50} + 934470 q^{51} + 843648 q^{52} + 26321 q^{53} + 944784 q^{54} + 2115010 q^{55} - 1053696 q^{56} + 110295 q^{57} + 475160 q^{58} + 478280 q^{59} + 312768 q^{60} - 501406 q^{61} - 3825928 q^{62} + 1500282 q^{63} + 1572864 q^{64} - 397657 q^{65} - 1324080 q^{66} - 3156366 q^{67} - 2215040 q^{68} - 40905 q^{69} + 496664 q^{70} - 2003644 q^{71} - 2239488 q^{72} + 3659111 q^{73} - 4594480 q^{74} - 4228443 q^{75} - 261440 q^{76} - 2102590 q^{77} + 2847312 q^{78} + 1131065 q^{79} - 741376 q^{80} + 3188646 q^{81} - 1612416 q^{82} - 9629297 q^{83} - 3556224 q^{84} + 895068 q^{85} - 5828840 q^{86} + 1603665 q^{87} + 3138560 q^{88} - 21977377 q^{89} + 1055592 q^{90} + 4521426 q^{91} + 96960 q^{92} - 12912507 q^{93} - 1820920 q^{94} - 19325507 q^{95} + 5308416 q^{96} - 26386649 q^{97} - 5647152 q^{98} - 4468770 q^{99} + O(q^{100}) \)

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
\(5\) −113.468 −0.405955 −0.202977 0.979183i \(-0.565062\pi\)
−0.202977 + 0.979183i \(0.565062\pi\)
\(6\) 216.000 0.408248
\(7\) 343.000 0.377964
\(8\) −512.000 −0.353553
\(9\) 729.000 0.333333
\(10\) 907.742 0.287053
\(11\) −3827.20 −0.866976 −0.433488 0.901159i \(-0.642717\pi\)
−0.433488 + 0.901159i \(0.642717\pi\)
\(12\) −1728.00 −0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) 3063.63 0.234378
\(16\) 4096.00 0.250000
\(17\) −36941.0 −1.82363 −0.911817 0.410596i \(-0.865321\pi\)
−0.911817 + 0.410596i \(0.865321\pi\)
\(18\) −5832.00 −0.235702
\(19\) −56827.3 −1.90073 −0.950364 0.311142i \(-0.899289\pi\)
−0.950364 + 0.311142i \(0.899289\pi\)
\(20\) −7261.94 −0.202977
\(21\) −9261.00 −0.218218
\(22\) 30617.6 0.613045
\(23\) 42283.4 0.724641 0.362320 0.932054i \(-0.381985\pi\)
0.362320 + 0.932054i \(0.381985\pi\)
\(24\) 13824.0 0.204124
\(25\) −65250.1 −0.835201
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) 21952.0 0.188982
\(29\) 158520. 1.20695 0.603476 0.797381i \(-0.293782\pi\)
0.603476 + 0.797381i \(0.293782\pi\)
\(30\) −24509.0 −0.165730
\(31\) 98296.8 0.592616 0.296308 0.955092i \(-0.404245\pi\)
0.296308 + 0.955092i \(0.404245\pi\)
\(32\) −32768.0 −0.176777
\(33\) 103335. 0.500549
\(34\) 295528. 1.28950
\(35\) −38919.5 −0.153436
\(36\) 46656.0 0.166667
\(37\) 352941. 1.14550 0.572751 0.819729i \(-0.305876\pi\)
0.572751 + 0.819729i \(0.305876\pi\)
\(38\) 454619. 1.34402
\(39\) −59319.0 −0.160128
\(40\) 58095.5 0.143527
\(41\) −786907. −1.78312 −0.891558 0.452906i \(-0.850387\pi\)
−0.891558 + 0.452906i \(0.850387\pi\)
\(42\) 74088.0 0.154303
\(43\) −744062. −1.42715 −0.713575 0.700579i \(-0.752925\pi\)
−0.713575 + 0.700579i \(0.752925\pi\)
\(44\) −244941. −0.433488
\(45\) −82718.0 −0.135318
\(46\) −338267. −0.512398
\(47\) −498725. −0.700678 −0.350339 0.936623i \(-0.613934\pi\)
−0.350339 + 0.936623i \(0.613934\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) 522000. 0.590576
\(51\) 997408. 1.05288
\(52\) 140608. 0.138675
\(53\) −664787. −0.613362 −0.306681 0.951812i \(-0.599218\pi\)
−0.306681 + 0.951812i \(0.599218\pi\)
\(54\) 157464. 0.136083
\(55\) 434264. 0.351953
\(56\) −175616. −0.133631
\(57\) 1.53434e6 1.09739
\(58\) −1.26816e6 −0.853445
\(59\) 1.39438e6 0.883894 0.441947 0.897041i \(-0.354288\pi\)
0.441947 + 0.897041i \(0.354288\pi\)
\(60\) 196072. 0.117189
\(61\) −1.33040e6 −0.750462 −0.375231 0.926931i \(-0.622437\pi\)
−0.375231 + 0.926931i \(0.622437\pi\)
\(62\) −786374. −0.419043
\(63\) 250047. 0.125988
\(64\) 262144. 0.125000
\(65\) −249289. −0.112592
\(66\) −826676. −0.353942
\(67\) 2.54161e6 1.03240 0.516198 0.856469i \(-0.327347\pi\)
0.516198 + 0.856469i \(0.327347\pi\)
\(68\) −2.36423e6 −0.911817
\(69\) −1.14165e6 −0.418372
\(70\) 311356. 0.108496
\(71\) −5.37482e6 −1.78221 −0.891107 0.453792i \(-0.850071\pi\)
−0.891107 + 0.453792i \(0.850071\pi\)
\(72\) −373248. −0.117851
\(73\) 3.59276e6 1.08093 0.540466 0.841366i \(-0.318248\pi\)
0.540466 + 0.841366i \(0.318248\pi\)
\(74\) −2.82353e6 −0.809992
\(75\) 1.76175e6 0.482203
\(76\) −3.63695e6 −0.950364
\(77\) −1.31273e6 −0.327686
\(78\) 474552. 0.113228
\(79\) 5.53769e6 1.26367 0.631835 0.775103i \(-0.282302\pi\)
0.631835 + 0.775103i \(0.282302\pi\)
\(80\) −464764. −0.101489
\(81\) 531441. 0.111111
\(82\) 6.29525e6 1.26085
\(83\) −5.11360e6 −0.981643 −0.490822 0.871260i \(-0.663303\pi\)
−0.490822 + 0.871260i \(0.663303\pi\)
\(84\) −592704. −0.109109
\(85\) 4.19162e6 0.740313
\(86\) 5.95249e6 1.00915
\(87\) −4.28003e6 −0.696835
\(88\) 1.95953e6 0.306522
\(89\) −7.66025e6 −1.15180 −0.575901 0.817519i \(-0.695349\pi\)
−0.575901 + 0.817519i \(0.695349\pi\)
\(90\) 661744. 0.0956844
\(91\) 753571. 0.104828
\(92\) 2.70614e6 0.362320
\(93\) −2.65401e6 −0.342147
\(94\) 3.98980e6 0.495454
\(95\) 6.44807e6 0.771609
\(96\) 884736. 0.102062
\(97\) −1.56564e7 −1.74177 −0.870884 0.491488i \(-0.836453\pi\)
−0.870884 + 0.491488i \(0.836453\pi\)
\(98\) −941192. −0.101015
\(99\) −2.79003e6 −0.288992
\(100\) −4.17600e6 −0.417600
\(101\) −103574. −0.0100029 −0.00500147 0.999987i \(-0.501592\pi\)
−0.00500147 + 0.999987i \(0.501592\pi\)
\(102\) −7.97926e6 −0.744496
\(103\) −1.64565e7 −1.48391 −0.741954 0.670451i \(-0.766101\pi\)
−0.741954 + 0.670451i \(0.766101\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) 1.05083e6 0.0885866
\(106\) 5.31829e6 0.433712
\(107\) −7.14120e6 −0.563544 −0.281772 0.959481i \(-0.590922\pi\)
−0.281772 + 0.959481i \(0.590922\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) 5.81181e6 0.429851 0.214926 0.976630i \(-0.431049\pi\)
0.214926 + 0.976630i \(0.431049\pi\)
\(110\) −3.47411e6 −0.248868
\(111\) −9.52940e6 −0.661356
\(112\) 1.40493e6 0.0944911
\(113\) −1.89753e7 −1.23713 −0.618564 0.785735i \(-0.712285\pi\)
−0.618564 + 0.785735i \(0.712285\pi\)
\(114\) −1.22747e7 −0.775969
\(115\) −4.79781e6 −0.294171
\(116\) 1.01453e7 0.603476
\(117\) 1.60161e6 0.0924500
\(118\) −1.11551e7 −0.625008
\(119\) −1.26708e7 −0.689269
\(120\) −1.56858e6 −0.0828652
\(121\) −4.83968e6 −0.248352
\(122\) 1.06432e7 0.530657
\(123\) 2.12465e7 1.02948
\(124\) 6.29099e6 0.296308
\(125\) 1.62685e7 0.745008
\(126\) −2.00038e6 −0.0890871
\(127\) 3.95319e7 1.71252 0.856258 0.516548i \(-0.172783\pi\)
0.856258 + 0.516548i \(0.172783\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 2.00897e7 0.823965
\(130\) 1.99431e6 0.0796143
\(131\) −2.42781e6 −0.0943551 −0.0471775 0.998887i \(-0.515023\pi\)
−0.0471775 + 0.998887i \(0.515023\pi\)
\(132\) 6.61341e6 0.250274
\(133\) −1.94918e7 −0.718407
\(134\) −2.03329e7 −0.730015
\(135\) 2.23339e6 0.0781260
\(136\) 1.89138e7 0.644752
\(137\) 4.16766e7 1.38475 0.692374 0.721539i \(-0.256565\pi\)
0.692374 + 0.721539i \(0.256565\pi\)
\(138\) 9.13322e6 0.295833
\(139\) −5.96582e7 −1.88416 −0.942082 0.335383i \(-0.891134\pi\)
−0.942082 + 0.335383i \(0.891134\pi\)
\(140\) −2.49084e6 −0.0767182
\(141\) 1.34656e7 0.404537
\(142\) 4.29986e7 1.26022
\(143\) −8.40837e6 −0.240456
\(144\) 2.98598e6 0.0833333
\(145\) −1.79869e7 −0.489968
\(146\) −2.87421e7 −0.764335
\(147\) −3.17652e6 −0.0824786
\(148\) 2.25882e7 0.572751
\(149\) 1.78622e7 0.442366 0.221183 0.975232i \(-0.429008\pi\)
0.221183 + 0.975232i \(0.429008\pi\)
\(150\) −1.40940e7 −0.340969
\(151\) −3.78116e7 −0.893729 −0.446865 0.894602i \(-0.647459\pi\)
−0.446865 + 0.894602i \(0.647459\pi\)
\(152\) 2.90956e7 0.672009
\(153\) −2.69300e7 −0.607878
\(154\) 1.05018e7 0.231709
\(155\) −1.11535e7 −0.240575
\(156\) −3.79642e6 −0.0800641
\(157\) −1.14825e7 −0.236804 −0.118402 0.992966i \(-0.537777\pi\)
−0.118402 + 0.992966i \(0.537777\pi\)
\(158\) −4.43015e7 −0.893549
\(159\) 1.79492e7 0.354125
\(160\) 3.71811e6 0.0717633
\(161\) 1.45032e7 0.273888
\(162\) −4.25153e6 −0.0785674
\(163\) 3.82479e7 0.691752 0.345876 0.938280i \(-0.387582\pi\)
0.345876 + 0.938280i \(0.387582\pi\)
\(164\) −5.03620e7 −0.891558
\(165\) −1.17251e7 −0.203200
\(166\) 4.09088e7 0.694127
\(167\) 4.21224e7 0.699851 0.349926 0.936778i \(-0.386207\pi\)
0.349926 + 0.936778i \(0.386207\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −3.35329e7 −0.523480
\(171\) −4.14271e7 −0.633576
\(172\) −4.76199e7 −0.713575
\(173\) −2.41841e7 −0.355114 −0.177557 0.984110i \(-0.556819\pi\)
−0.177557 + 0.984110i \(0.556819\pi\)
\(174\) 3.42403e7 0.492736
\(175\) −2.23808e7 −0.315676
\(176\) −1.56762e7 −0.216744
\(177\) −3.76484e7 −0.510317
\(178\) 6.12820e7 0.814447
\(179\) −3.53619e7 −0.460840 −0.230420 0.973091i \(-0.574010\pi\)
−0.230420 + 0.973091i \(0.574010\pi\)
\(180\) −5.29395e6 −0.0676591
\(181\) 2.96542e7 0.371716 0.185858 0.982577i \(-0.440494\pi\)
0.185858 + 0.982577i \(0.440494\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 3.59209e7 0.433280
\(184\) −2.16491e7 −0.256199
\(185\) −4.00474e7 −0.465022
\(186\) 2.12321e7 0.241934
\(187\) 1.41381e8 1.58105
\(188\) −3.19184e7 −0.350339
\(189\) −6.75127e6 −0.0727393
\(190\) −5.15846e7 −0.545610
\(191\) −1.47397e8 −1.53064 −0.765319 0.643651i \(-0.777419\pi\)
−0.765319 + 0.643651i \(0.777419\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −9.04774e7 −0.905919 −0.452960 0.891531i \(-0.649632\pi\)
−0.452960 + 0.891531i \(0.649632\pi\)
\(194\) 1.25251e8 1.23162
\(195\) 6.73080e6 0.0650048
\(196\) 7.52954e6 0.0714286
\(197\) −5.91676e7 −0.551382 −0.275691 0.961246i \(-0.588907\pi\)
−0.275691 + 0.961246i \(0.588907\pi\)
\(198\) 2.23203e7 0.204348
\(199\) 4.33390e7 0.389846 0.194923 0.980819i \(-0.437554\pi\)
0.194923 + 0.980819i \(0.437554\pi\)
\(200\) 3.34080e7 0.295288
\(201\) −6.86234e7 −0.596055
\(202\) 828596. 0.00707315
\(203\) 5.43723e7 0.456185
\(204\) 6.38341e7 0.526438
\(205\) 8.92886e7 0.723865
\(206\) 1.31652e8 1.04928
\(207\) 3.08246e7 0.241547
\(208\) 8.99891e6 0.0693375
\(209\) 2.17490e8 1.64789
\(210\) −8.40660e6 −0.0626402
\(211\) 2.26404e8 1.65918 0.829592 0.558369i \(-0.188573\pi\)
0.829592 + 0.558369i \(0.188573\pi\)
\(212\) −4.25464e7 −0.306681
\(213\) 1.45120e8 1.02896
\(214\) 5.71296e7 0.398486
\(215\) 8.44270e7 0.579358
\(216\) 1.00777e7 0.0680414
\(217\) 3.37158e7 0.223988
\(218\) −4.64945e7 −0.303951
\(219\) −9.70046e7 −0.624077
\(220\) 2.77929e7 0.175977
\(221\) −8.11595e7 −0.505785
\(222\) 7.62352e7 0.467649
\(223\) 1.76572e8 1.06624 0.533121 0.846039i \(-0.321019\pi\)
0.533121 + 0.846039i \(0.321019\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −4.75673e7 −0.278400
\(226\) 1.51803e8 0.874781
\(227\) 5.10006e7 0.289391 0.144695 0.989476i \(-0.453780\pi\)
0.144695 + 0.989476i \(0.453780\pi\)
\(228\) 9.81977e7 0.548693
\(229\) −3.09360e8 −1.70232 −0.851158 0.524909i \(-0.824099\pi\)
−0.851158 + 0.524909i \(0.824099\pi\)
\(230\) 3.83825e7 0.208011
\(231\) 3.54437e7 0.189190
\(232\) −8.11621e7 −0.426722
\(233\) 1.46068e8 0.756501 0.378250 0.925703i \(-0.376526\pi\)
0.378250 + 0.925703i \(0.376526\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 5.65892e7 0.284443
\(236\) 8.92406e7 0.441947
\(237\) −1.49518e8 −0.729580
\(238\) 1.01366e8 0.487387
\(239\) 9.61072e7 0.455369 0.227684 0.973735i \(-0.426885\pi\)
0.227684 + 0.973735i \(0.426885\pi\)
\(240\) 1.25486e7 0.0585945
\(241\) 3.91257e8 1.80054 0.900270 0.435333i \(-0.143369\pi\)
0.900270 + 0.435333i \(0.143369\pi\)
\(242\) 3.87175e7 0.175612
\(243\) −1.43489e7 −0.0641500
\(244\) −8.51458e7 −0.375231
\(245\) −1.33494e7 −0.0579935
\(246\) −1.69972e8 −0.727954
\(247\) −1.24850e8 −0.527167
\(248\) −5.03280e7 −0.209521
\(249\) 1.38067e8 0.566752
\(250\) −1.30148e8 −0.526800
\(251\) 1.97659e8 0.788967 0.394483 0.918903i \(-0.370924\pi\)
0.394483 + 0.918903i \(0.370924\pi\)
\(252\) 1.60030e7 0.0629941
\(253\) −1.61827e8 −0.628246
\(254\) −3.16255e8 −1.21093
\(255\) −1.13174e8 −0.427420
\(256\) 1.67772e7 0.0625000
\(257\) −1.30951e8 −0.481219 −0.240610 0.970622i \(-0.577347\pi\)
−0.240610 + 0.970622i \(0.577347\pi\)
\(258\) −1.60717e8 −0.582631
\(259\) 1.21059e8 0.432959
\(260\) −1.59545e7 −0.0562958
\(261\) 1.15561e8 0.402318
\(262\) 1.94225e7 0.0667191
\(263\) −3.42123e8 −1.15968 −0.579839 0.814731i \(-0.696884\pi\)
−0.579839 + 0.814731i \(0.696884\pi\)
\(264\) −5.29073e7 −0.176971
\(265\) 7.54319e7 0.248997
\(266\) 1.55934e8 0.507991
\(267\) 2.06827e8 0.664993
\(268\) 1.62663e8 0.516198
\(269\) 1.20771e8 0.378295 0.189148 0.981949i \(-0.439428\pi\)
0.189148 + 0.981949i \(0.439428\pi\)
\(270\) −1.78671e7 −0.0552434
\(271\) 5.29161e8 1.61508 0.807542 0.589809i \(-0.200797\pi\)
0.807542 + 0.589809i \(0.200797\pi\)
\(272\) −1.51310e8 −0.455909
\(273\) −2.03464e7 −0.0605228
\(274\) −3.33413e8 −0.979165
\(275\) 2.49725e8 0.724099
\(276\) −7.30658e7 −0.209186
\(277\) 3.39180e8 0.958850 0.479425 0.877583i \(-0.340845\pi\)
0.479425 + 0.877583i \(0.340845\pi\)
\(278\) 4.77266e8 1.33230
\(279\) 7.16584e7 0.197539
\(280\) 1.99268e7 0.0542480
\(281\) 6.33478e8 1.70317 0.851587 0.524213i \(-0.175640\pi\)
0.851587 + 0.524213i \(0.175640\pi\)
\(282\) −1.07725e8 −0.286051
\(283\) 5.96829e8 1.56530 0.782650 0.622463i \(-0.213868\pi\)
0.782650 + 0.622463i \(0.213868\pi\)
\(284\) −3.43989e8 −0.891107
\(285\) −1.74098e8 −0.445489
\(286\) 6.72669e7 0.170028
\(287\) −2.69909e8 −0.673955
\(288\) −2.38879e7 −0.0589256
\(289\) 9.54302e8 2.32564
\(290\) 1.43895e8 0.346460
\(291\) 4.22722e8 1.00561
\(292\) 2.29937e8 0.540466
\(293\) 1.70024e8 0.394887 0.197443 0.980314i \(-0.436736\pi\)
0.197443 + 0.980314i \(0.436736\pi\)
\(294\) 2.54122e7 0.0583212
\(295\) −1.58218e8 −0.358821
\(296\) −1.80706e8 −0.404996
\(297\) 7.53309e7 0.166850
\(298\) −1.42897e8 −0.312800
\(299\) 9.28967e7 0.200979
\(300\) 1.12752e8 0.241102
\(301\) −2.55213e8 −0.539412
\(302\) 3.02493e8 0.631962
\(303\) 2.79651e6 0.00577520
\(304\) −2.32765e8 −0.475182
\(305\) 1.50958e8 0.304654
\(306\) 2.15440e8 0.429835
\(307\) 4.12442e8 0.813540 0.406770 0.913531i \(-0.366655\pi\)
0.406770 + 0.913531i \(0.366655\pi\)
\(308\) −8.40148e7 −0.163843
\(309\) 4.44325e8 0.856735
\(310\) 8.92282e7 0.170112
\(311\) −7.47583e8 −1.40928 −0.704641 0.709564i \(-0.748892\pi\)
−0.704641 + 0.709564i \(0.748892\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 2.27958e8 0.420194 0.210097 0.977681i \(-0.432622\pi\)
0.210097 + 0.977681i \(0.432622\pi\)
\(314\) 9.18604e7 0.167446
\(315\) −2.83723e7 −0.0511455
\(316\) 3.54412e8 0.631835
\(317\) −1.72903e8 −0.304856 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(318\) −1.43594e8 −0.250404
\(319\) −6.06687e8 −1.04640
\(320\) −2.97449e7 −0.0507443
\(321\) 1.92812e8 0.325362
\(322\) −1.16026e8 −0.193668
\(323\) 2.09926e9 3.46623
\(324\) 3.40122e7 0.0555556
\(325\) −1.43354e8 −0.231643
\(326\) −3.05983e8 −0.489143
\(327\) −1.56919e8 −0.248175
\(328\) 4.02896e8 0.630427
\(329\) −1.71063e8 −0.264831
\(330\) 9.38011e7 0.143684
\(331\) −6.78468e8 −1.02833 −0.514164 0.857692i \(-0.671898\pi\)
−0.514164 + 0.857692i \(0.671898\pi\)
\(332\) −3.27271e8 −0.490822
\(333\) 2.57294e8 0.381834
\(334\) −3.36979e8 −0.494869
\(335\) −2.88391e8 −0.419106
\(336\) −3.79331e7 −0.0545545
\(337\) −5.33477e8 −0.759295 −0.379648 0.925131i \(-0.623955\pi\)
−0.379648 + 0.925131i \(0.623955\pi\)
\(338\) −3.86145e7 −0.0543928
\(339\) 5.12333e8 0.714256
\(340\) 2.68264e8 0.370157
\(341\) −3.76202e8 −0.513784
\(342\) 3.31417e8 0.448006
\(343\) 4.03536e7 0.0539949
\(344\) 3.80960e8 0.504573
\(345\) 1.29541e8 0.169840
\(346\) 1.93473e8 0.251104
\(347\) 5.83712e8 0.749972 0.374986 0.927030i \(-0.377647\pi\)
0.374986 + 0.927030i \(0.377647\pi\)
\(348\) −2.73922e8 −0.348417
\(349\) −1.00692e9 −1.26796 −0.633978 0.773351i \(-0.718579\pi\)
−0.633978 + 0.773351i \(0.718579\pi\)
\(350\) 1.79046e8 0.223217
\(351\) −4.32436e7 −0.0533761
\(352\) 1.25410e8 0.153261
\(353\) −1.61837e8 −0.195825 −0.0979123 0.995195i \(-0.531216\pi\)
−0.0979123 + 0.995195i \(0.531216\pi\)
\(354\) 3.01187e8 0.360848
\(355\) 6.09869e8 0.723498
\(356\) −4.90256e8 −0.575901
\(357\) 3.42111e8 0.397950
\(358\) 2.82895e8 0.325863
\(359\) −1.36320e9 −1.55500 −0.777500 0.628883i \(-0.783512\pi\)
−0.777500 + 0.628883i \(0.783512\pi\)
\(360\) 4.23516e7 0.0478422
\(361\) 2.33548e9 2.61276
\(362\) −2.37233e8 −0.262843
\(363\) 1.30671e8 0.143386
\(364\) 4.82285e7 0.0524142
\(365\) −4.07663e8 −0.438810
\(366\) −2.87367e8 −0.306375
\(367\) 9.23714e8 0.975452 0.487726 0.872997i \(-0.337826\pi\)
0.487726 + 0.872997i \(0.337826\pi\)
\(368\) 1.73193e8 0.181160
\(369\) −5.73655e8 −0.594372
\(370\) 3.20379e8 0.328820
\(371\) −2.28022e8 −0.231829
\(372\) −1.69857e8 −0.171074
\(373\) −7.84082e7 −0.0782313 −0.0391157 0.999235i \(-0.512454\pi\)
−0.0391157 + 0.999235i \(0.512454\pi\)
\(374\) −1.13105e9 −1.11797
\(375\) −4.39248e8 −0.430131
\(376\) 2.55347e8 0.247727
\(377\) 3.48268e8 0.334749
\(378\) 5.40102e7 0.0514344
\(379\) 1.54173e8 0.145469 0.0727346 0.997351i \(-0.476827\pi\)
0.0727346 + 0.997351i \(0.476827\pi\)
\(380\) 4.12677e8 0.385805
\(381\) −1.06736e9 −0.988722
\(382\) 1.17918e9 1.08232
\(383\) 5.09223e8 0.463140 0.231570 0.972818i \(-0.425614\pi\)
0.231570 + 0.972818i \(0.425614\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 1.48953e8 0.133026
\(386\) 7.23819e8 0.640582
\(387\) −5.42421e8 −0.475716
\(388\) −1.00201e9 −0.870884
\(389\) 2.44111e7 0.0210263 0.0105132 0.999945i \(-0.496653\pi\)
0.0105132 + 0.999945i \(0.496653\pi\)
\(390\) −5.38464e7 −0.0459653
\(391\) −1.56199e9 −1.32148
\(392\) −6.02363e7 −0.0505076
\(393\) 6.55509e7 0.0544759
\(394\) 4.73341e8 0.389886
\(395\) −6.28349e8 −0.512993
\(396\) −1.78562e8 −0.144496
\(397\) 5.09478e8 0.408657 0.204329 0.978902i \(-0.434499\pi\)
0.204329 + 0.978902i \(0.434499\pi\)
\(398\) −3.46712e8 −0.275663
\(399\) 5.26278e8 0.414773
\(400\) −2.67264e8 −0.208800
\(401\) 9.01074e8 0.697839 0.348919 0.937153i \(-0.386549\pi\)
0.348919 + 0.937153i \(0.386549\pi\)
\(402\) 5.48987e8 0.421474
\(403\) 2.15958e8 0.164362
\(404\) −6.62877e6 −0.00500147
\(405\) −6.03014e7 −0.0451061
\(406\) −4.34978e8 −0.322572
\(407\) −1.35078e9 −0.993123
\(408\) −5.10673e8 −0.372248
\(409\) 1.63738e9 1.18337 0.591683 0.806171i \(-0.298464\pi\)
0.591683 + 0.806171i \(0.298464\pi\)
\(410\) −7.14308e8 −0.511850
\(411\) −1.12527e9 −0.799485
\(412\) −1.05322e9 −0.741954
\(413\) 4.78274e8 0.334081
\(414\) −2.46597e8 −0.170799
\(415\) 5.80229e8 0.398503
\(416\) −7.19913e7 −0.0490290
\(417\) 1.61077e9 1.08782
\(418\) −1.73992e9 −1.16523
\(419\) 1.35298e8 0.0898548 0.0449274 0.998990i \(-0.485694\pi\)
0.0449274 + 0.998990i \(0.485694\pi\)
\(420\) 6.72528e7 0.0442933
\(421\) −1.92665e9 −1.25839 −0.629196 0.777247i \(-0.716616\pi\)
−0.629196 + 0.777247i \(0.716616\pi\)
\(422\) −1.81123e9 −1.17322
\(423\) −3.63570e8 −0.233559
\(424\) 3.40371e8 0.216856
\(425\) 2.41041e9 1.52310
\(426\) −1.16096e9 −0.727586
\(427\) −4.56328e8 −0.283648
\(428\) −4.57037e8 −0.281772
\(429\) 2.27026e8 0.138827
\(430\) −6.75416e8 −0.409668
\(431\) 2.85171e9 1.71567 0.857837 0.513922i \(-0.171808\pi\)
0.857837 + 0.513922i \(0.171808\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 1.71853e8 0.101730 0.0508650 0.998706i \(-0.483802\pi\)
0.0508650 + 0.998706i \(0.483802\pi\)
\(434\) −2.69726e8 −0.158383
\(435\) 4.85646e8 0.282883
\(436\) 3.71956e8 0.214926
\(437\) −2.40286e9 −1.37734
\(438\) 7.76037e8 0.441289
\(439\) 9.19463e8 0.518691 0.259345 0.965785i \(-0.416493\pi\)
0.259345 + 0.965785i \(0.416493\pi\)
\(440\) −2.22343e8 −0.124434
\(441\) 8.57661e7 0.0476190
\(442\) 6.49276e8 0.357644
\(443\) −2.59886e9 −1.42027 −0.710134 0.704067i \(-0.751366\pi\)
−0.710134 + 0.704067i \(0.751366\pi\)
\(444\) −6.09882e8 −0.330678
\(445\) 8.69192e8 0.467579
\(446\) −1.41258e9 −0.753947
\(447\) −4.82278e8 −0.255400
\(448\) 8.99154e7 0.0472456
\(449\) 5.80704e8 0.302756 0.151378 0.988476i \(-0.451629\pi\)
0.151378 + 0.988476i \(0.451629\pi\)
\(450\) 3.80538e8 0.196859
\(451\) 3.01165e9 1.54592
\(452\) −1.21442e9 −0.618564
\(453\) 1.02091e9 0.515995
\(454\) −4.08005e8 −0.204630
\(455\) −8.55060e7 −0.0425556
\(456\) −7.85581e8 −0.387984
\(457\) −2.67969e9 −1.31334 −0.656672 0.754176i \(-0.728036\pi\)
−0.656672 + 0.754176i \(0.728036\pi\)
\(458\) 2.47488e9 1.20372
\(459\) 7.27110e8 0.350959
\(460\) −3.07060e8 −0.147086
\(461\) 2.97997e9 1.41664 0.708319 0.705892i \(-0.249454\pi\)
0.708319 + 0.705892i \(0.249454\pi\)
\(462\) −2.83550e8 −0.133777
\(463\) 2.53656e9 1.18771 0.593856 0.804571i \(-0.297605\pi\)
0.593856 + 0.804571i \(0.297605\pi\)
\(464\) 6.49297e8 0.301738
\(465\) 3.01145e8 0.138896
\(466\) −1.16854e9 −0.534927
\(467\) −1.87235e9 −0.850703 −0.425351 0.905028i \(-0.639849\pi\)
−0.425351 + 0.905028i \(0.639849\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 8.71771e8 0.390209
\(470\) −4.52714e8 −0.201132
\(471\) 3.10029e8 0.136719
\(472\) −7.13925e8 −0.312504
\(473\) 2.84768e9 1.23730
\(474\) 1.19614e9 0.515891
\(475\) 3.70799e9 1.58749
\(476\) −8.10930e8 −0.344635
\(477\) −4.84630e8 −0.204454
\(478\) −7.68858e8 −0.321994
\(479\) −2.75021e9 −1.14338 −0.571691 0.820469i \(-0.693713\pi\)
−0.571691 + 0.820469i \(0.693713\pi\)
\(480\) −1.00389e8 −0.0414326
\(481\) 7.75411e8 0.317705
\(482\) −3.13006e9 −1.27317
\(483\) −3.91587e8 −0.158130
\(484\) −3.09740e8 −0.124176
\(485\) 1.77650e9 0.707079
\(486\) 1.14791e8 0.0453609
\(487\) 3.55069e9 1.39303 0.696516 0.717541i \(-0.254732\pi\)
0.696516 + 0.717541i \(0.254732\pi\)
\(488\) 6.81167e8 0.265329
\(489\) −1.03269e9 −0.399383
\(490\) 1.06795e8 0.0410076
\(491\) −1.46470e9 −0.558425 −0.279213 0.960229i \(-0.590073\pi\)
−0.279213 + 0.960229i \(0.590073\pi\)
\(492\) 1.35977e9 0.514741
\(493\) −5.85588e9 −2.20104
\(494\) 9.98797e8 0.372763
\(495\) 3.16579e8 0.117318
\(496\) 4.02624e8 0.148154
\(497\) −1.84356e9 −0.673614
\(498\) −1.10454e9 −0.400754
\(499\) −1.64336e9 −0.592081 −0.296040 0.955175i \(-0.595666\pi\)
−0.296040 + 0.955175i \(0.595666\pi\)
\(500\) 1.04118e9 0.372504
\(501\) −1.13731e9 −0.404059
\(502\) −1.58127e9 −0.557884
\(503\) 2.63997e9 0.924934 0.462467 0.886636i \(-0.346964\pi\)
0.462467 + 0.886636i \(0.346964\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 1.17524e7 0.00406074
\(506\) 1.29462e9 0.444237
\(507\) −1.30324e8 −0.0444116
\(508\) 2.53004e9 0.856258
\(509\) 1.19059e9 0.400177 0.200088 0.979778i \(-0.435877\pi\)
0.200088 + 0.979778i \(0.435877\pi\)
\(510\) 9.05389e8 0.302232
\(511\) 1.23232e9 0.408554
\(512\) −1.34218e8 −0.0441942
\(513\) 1.11853e9 0.365795
\(514\) 1.04761e9 0.340273
\(515\) 1.86728e9 0.602400
\(516\) 1.28574e9 0.411982
\(517\) 1.90872e9 0.607471
\(518\) −9.68470e8 −0.306148
\(519\) 6.52970e8 0.205025
\(520\) 1.27636e8 0.0398071
\(521\) −3.82843e8 −0.118601 −0.0593005 0.998240i \(-0.518887\pi\)
−0.0593005 + 0.998240i \(0.518887\pi\)
\(522\) −9.24487e8 −0.284482
\(523\) −5.99389e9 −1.83212 −0.916058 0.401046i \(-0.868647\pi\)
−0.916058 + 0.401046i \(0.868647\pi\)
\(524\) −1.55380e8 −0.0471775
\(525\) 6.04281e8 0.182256
\(526\) 2.73698e9 0.820016
\(527\) −3.63119e9 −1.08072
\(528\) 4.23258e8 0.125137
\(529\) −1.61694e9 −0.474896
\(530\) −6.03455e8 −0.176068
\(531\) 1.01651e9 0.294631
\(532\) −1.24747e9 −0.359204
\(533\) −1.72883e9 −0.494548
\(534\) −1.65461e9 −0.470221
\(535\) 8.10296e8 0.228773
\(536\) −1.30130e9 −0.365007
\(537\) 9.54772e8 0.266066
\(538\) −9.66171e8 −0.267495
\(539\) −4.50267e8 −0.123854
\(540\) 1.42937e8 0.0390630
\(541\) 3.40349e9 0.924132 0.462066 0.886846i \(-0.347108\pi\)
0.462066 + 0.886846i \(0.347108\pi\)
\(542\) −4.23329e9 −1.14204
\(543\) −8.00662e8 −0.214610
\(544\) 1.21048e9 0.322376
\(545\) −6.59453e8 −0.174500
\(546\) 1.62771e8 0.0427960
\(547\) −2.88099e8 −0.0752638 −0.0376319 0.999292i \(-0.511981\pi\)
−0.0376319 + 0.999292i \(0.511981\pi\)
\(548\) 2.66730e9 0.692374
\(549\) −9.69864e8 −0.250154
\(550\) −1.99780e9 −0.512015
\(551\) −9.00826e9 −2.29409
\(552\) 5.84526e8 0.147917
\(553\) 1.89943e9 0.477622
\(554\) −2.71344e9 −0.678009
\(555\) 1.08128e9 0.268481
\(556\) −3.81813e9 −0.942082
\(557\) −8.55398e8 −0.209737 −0.104868 0.994486i \(-0.533442\pi\)
−0.104868 + 0.994486i \(0.533442\pi\)
\(558\) −5.73267e8 −0.139681
\(559\) −1.63470e9 −0.395820
\(560\) −1.59414e8 −0.0383591
\(561\) −3.81728e9 −0.912819
\(562\) −5.06782e9 −1.20433
\(563\) 9.35132e8 0.220848 0.110424 0.993885i \(-0.464779\pi\)
0.110424 + 0.993885i \(0.464779\pi\)
\(564\) 8.61796e8 0.202268
\(565\) 2.15309e9 0.502218
\(566\) −4.77463e9 −1.10683
\(567\) 1.82284e8 0.0419961
\(568\) 2.75191e9 0.630108
\(569\) 1.19108e9 0.271049 0.135524 0.990774i \(-0.456728\pi\)
0.135524 + 0.990774i \(0.456728\pi\)
\(570\) 1.39278e9 0.315008
\(571\) −4.44856e9 −0.999983 −0.499992 0.866030i \(-0.666664\pi\)
−0.499992 + 0.866030i \(0.666664\pi\)
\(572\) −5.38135e8 −0.120228
\(573\) 3.97973e9 0.883714
\(574\) 2.15927e9 0.476558
\(575\) −2.75900e9 −0.605221
\(576\) 1.91103e8 0.0416667
\(577\) 7.29923e9 1.58184 0.790918 0.611922i \(-0.209603\pi\)
0.790918 + 0.611922i \(0.209603\pi\)
\(578\) −7.63441e9 −1.64448
\(579\) 2.44289e9 0.523033
\(580\) −1.15116e9 −0.244984
\(581\) −1.75397e9 −0.371026
\(582\) −3.38178e9 −0.711074
\(583\) 2.54427e9 0.531770
\(584\) −1.83949e9 −0.382167
\(585\) −1.81731e8 −0.0375305
\(586\) −1.36019e9 −0.279227
\(587\) 2.04418e9 0.417145 0.208572 0.978007i \(-0.433118\pi\)
0.208572 + 0.978007i \(0.433118\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) −5.58595e9 −1.12640
\(590\) 1.26574e9 0.253725
\(591\) 1.59753e9 0.318340
\(592\) 1.44565e9 0.286376
\(593\) −6.01899e9 −1.18531 −0.592655 0.805456i \(-0.701920\pi\)
−0.592655 + 0.805456i \(0.701920\pi\)
\(594\) −6.02647e8 −0.117981
\(595\) 1.43772e9 0.279812
\(596\) 1.14318e9 0.221183
\(597\) −1.17015e9 −0.225078
\(598\) −7.43174e8 −0.142114
\(599\) −4.49081e9 −0.853750 −0.426875 0.904311i \(-0.640386\pi\)
−0.426875 + 0.904311i \(0.640386\pi\)
\(600\) −9.02017e8 −0.170485
\(601\) −6.69638e8 −0.125829 −0.0629143 0.998019i \(-0.520039\pi\)
−0.0629143 + 0.998019i \(0.520039\pi\)
\(602\) 2.04171e9 0.381422
\(603\) 1.85283e9 0.344132
\(604\) −2.41994e9 −0.446865
\(605\) 5.49148e8 0.100820
\(606\) −2.23721e7 −0.00408369
\(607\) 8.39969e9 1.52441 0.762207 0.647333i \(-0.224116\pi\)
0.762207 + 0.647333i \(0.224116\pi\)
\(608\) 1.86212e9 0.336004
\(609\) −1.46805e9 −0.263379
\(610\) −1.20766e9 −0.215423
\(611\) −1.09570e9 −0.194333
\(612\) −1.72352e9 −0.303939
\(613\) −3.69061e9 −0.647122 −0.323561 0.946207i \(-0.604880\pi\)
−0.323561 + 0.946207i \(0.604880\pi\)
\(614\) −3.29954e9 −0.575259
\(615\) −2.41079e9 −0.417923
\(616\) 6.72118e8 0.115855
\(617\) −8.35635e9 −1.43225 −0.716125 0.697972i \(-0.754086\pi\)
−0.716125 + 0.697972i \(0.754086\pi\)
\(618\) −3.55460e9 −0.605803
\(619\) 5.90457e9 1.00062 0.500312 0.865845i \(-0.333219\pi\)
0.500312 + 0.865845i \(0.333219\pi\)
\(620\) −7.13825e8 −0.120288
\(621\) −8.32265e8 −0.139457
\(622\) 5.98066e9 0.996513
\(623\) −2.62747e9 −0.435340
\(624\) −2.42971e8 −0.0400320
\(625\) 3.25172e9 0.532761
\(626\) −1.82366e9 −0.297122
\(627\) −5.87223e9 −0.951407
\(628\) −7.34883e8 −0.118402
\(629\) −1.30380e10 −2.08898
\(630\) 2.26978e8 0.0361653
\(631\) 1.02737e9 0.162789 0.0813944 0.996682i \(-0.474063\pi\)
0.0813944 + 0.996682i \(0.474063\pi\)
\(632\) −2.83530e9 −0.446775
\(633\) −6.11290e9 −0.957931
\(634\) 1.38322e9 0.215565
\(635\) −4.48560e9 −0.695204
\(636\) 1.14875e9 0.177062
\(637\) 2.58475e8 0.0396214
\(638\) 4.85350e9 0.739916
\(639\) −3.91825e9 −0.594072
\(640\) 2.37959e8 0.0358817
\(641\) 9.46369e9 1.41924 0.709622 0.704582i \(-0.248866\pi\)
0.709622 + 0.704582i \(0.248866\pi\)
\(642\) −1.54250e9 −0.230066
\(643\) −1.09823e10 −1.62913 −0.814563 0.580075i \(-0.803023\pi\)
−0.814563 + 0.580075i \(0.803023\pi\)
\(644\) 9.28206e8 0.136944
\(645\) −2.27953e9 −0.334492
\(646\) −1.67941e10 −2.45100
\(647\) −3.36347e9 −0.488228 −0.244114 0.969747i \(-0.578497\pi\)
−0.244114 + 0.969747i \(0.578497\pi\)
\(648\) −2.72098e8 −0.0392837
\(649\) −5.33659e9 −0.766315
\(650\) 1.14684e9 0.163796
\(651\) −9.10327e8 −0.129319
\(652\) 2.44786e9 0.345876
\(653\) 6.26787e8 0.0880895 0.0440447 0.999030i \(-0.485976\pi\)
0.0440447 + 0.999030i \(0.485976\pi\)
\(654\) 1.25535e9 0.175486
\(655\) 2.75478e8 0.0383039
\(656\) −3.22317e9 −0.445779
\(657\) 2.61912e9 0.360311
\(658\) 1.36850e9 0.187264
\(659\) 6.73142e9 0.916236 0.458118 0.888891i \(-0.348524\pi\)
0.458118 + 0.888891i \(0.348524\pi\)
\(660\) −7.50409e8 −0.101600
\(661\) 8.92439e8 0.120191 0.0600957 0.998193i \(-0.480859\pi\)
0.0600957 + 0.998193i \(0.480859\pi\)
\(662\) 5.42774e9 0.727137
\(663\) 2.19131e9 0.292015
\(664\) 2.61816e9 0.347063
\(665\) 2.21169e9 0.291641
\(666\) −2.05835e9 −0.269997
\(667\) 6.70276e9 0.874607
\(668\) 2.69583e9 0.349926
\(669\) −4.76746e9 −0.615595
\(670\) 2.30712e9 0.296353
\(671\) 5.09173e9 0.650633
\(672\) 3.03464e8 0.0385758
\(673\) 7.99740e9 1.01134 0.505669 0.862728i \(-0.331246\pi\)
0.505669 + 0.862728i \(0.331246\pi\)
\(674\) 4.26781e9 0.536903
\(675\) 1.28432e9 0.160734
\(676\) 3.08916e8 0.0384615
\(677\) 6.29402e8 0.0779592 0.0389796 0.999240i \(-0.487589\pi\)
0.0389796 + 0.999240i \(0.487589\pi\)
\(678\) −4.09867e9 −0.505055
\(679\) −5.37014e9 −0.658327
\(680\) −2.14611e9 −0.261740
\(681\) −1.37702e9 −0.167080
\(682\) 3.00961e9 0.363300
\(683\) 6.50089e9 0.780729 0.390364 0.920660i \(-0.372349\pi\)
0.390364 + 0.920660i \(0.372349\pi\)
\(684\) −2.65134e9 −0.316788
\(685\) −4.72896e9 −0.562145
\(686\) −3.22829e8 −0.0381802
\(687\) 8.35272e9 0.982833
\(688\) −3.04768e9 −0.356787
\(689\) −1.46054e9 −0.170116
\(690\) −1.03633e9 −0.120095
\(691\) 7.64042e9 0.880935 0.440468 0.897768i \(-0.354813\pi\)
0.440468 + 0.897768i \(0.354813\pi\)
\(692\) −1.54778e9 −0.177557
\(693\) −9.56981e8 −0.109229
\(694\) −4.66969e9 −0.530310
\(695\) 6.76929e9 0.764885
\(696\) 2.19138e9 0.246368
\(697\) 2.90692e10 3.25175
\(698\) 8.05532e9 0.896580
\(699\) −3.94384e9 −0.436766
\(700\) −1.43237e9 −0.157838
\(701\) 7.48269e7 0.00820436 0.00410218 0.999992i \(-0.498694\pi\)
0.00410218 + 0.999992i \(0.498694\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) −2.00567e10 −2.17729
\(704\) −1.00328e9 −0.108372
\(705\) −1.52791e9 −0.164224
\(706\) 1.29470e9 0.138469
\(707\) −3.55261e7 −0.00378076
\(708\) −2.40950e9 −0.255158
\(709\) 4.53424e9 0.477796 0.238898 0.971045i \(-0.423214\pi\)
0.238898 + 0.971045i \(0.423214\pi\)
\(710\) −4.87896e9 −0.511591
\(711\) 4.03697e9 0.421223
\(712\) 3.92205e9 0.407224
\(713\) 4.15633e9 0.429434
\(714\) −2.73689e9 −0.281393
\(715\) 9.54079e8 0.0976142
\(716\) −2.26316e9 −0.230420
\(717\) −2.59489e9 −0.262907
\(718\) 1.09056e10 1.09955
\(719\) −1.88677e9 −0.189307 −0.0946536 0.995510i \(-0.530174\pi\)
−0.0946536 + 0.995510i \(0.530174\pi\)
\(720\) −3.38813e8 −0.0338296
\(721\) −5.64458e9 −0.560865
\(722\) −1.86838e10 −1.84750
\(723\) −1.05639e10 −1.03954
\(724\) 1.89787e9 0.185858
\(725\) −1.03434e10 −1.00805
\(726\) −1.04537e9 −0.101389
\(727\) 1.04020e10 1.00403 0.502014 0.864859i \(-0.332592\pi\)
0.502014 + 0.864859i \(0.332592\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 3.26130e9 0.310285
\(731\) 2.74864e10 2.60260
\(732\) 2.29894e9 0.216640
\(733\) −1.80922e10 −1.69679 −0.848394 0.529366i \(-0.822430\pi\)
−0.848394 + 0.529366i \(0.822430\pi\)
\(734\) −7.38971e9 −0.689749
\(735\) 3.60433e8 0.0334826
\(736\) −1.38554e9 −0.128100
\(737\) −9.72725e9 −0.895064
\(738\) 4.58924e9 0.420285
\(739\) 2.67865e9 0.244152 0.122076 0.992521i \(-0.461045\pi\)
0.122076 + 0.992521i \(0.461045\pi\)
\(740\) −2.56303e9 −0.232511
\(741\) 3.37094e9 0.304360
\(742\) 1.82418e9 0.163928
\(743\) 2.77097e8 0.0247840 0.0123920 0.999923i \(-0.496055\pi\)
0.0123920 + 0.999923i \(0.496055\pi\)
\(744\) 1.35885e9 0.120967
\(745\) −2.02678e9 −0.179581
\(746\) 6.27266e8 0.0553179
\(747\) −3.72782e9 −0.327214
\(748\) 9.04838e9 0.790524
\(749\) −2.44943e9 −0.213000
\(750\) 3.51399e9 0.304148
\(751\) 1.25214e10 1.07873 0.539367 0.842071i \(-0.318663\pi\)
0.539367 + 0.842071i \(0.318663\pi\)
\(752\) −2.04278e9 −0.175169
\(753\) −5.33679e9 −0.455510
\(754\) −2.78614e9 −0.236703
\(755\) 4.29040e9 0.362814
\(756\) −4.32081e8 −0.0363696
\(757\) −9.37050e9 −0.785104 −0.392552 0.919730i \(-0.628408\pi\)
−0.392552 + 0.919730i \(0.628408\pi\)
\(758\) −1.23338e9 −0.102862
\(759\) 4.36934e9 0.362718
\(760\) −3.30141e9 −0.272805
\(761\) −1.28866e10 −1.05997 −0.529984 0.848008i \(-0.677802\pi\)
−0.529984 + 0.848008i \(0.677802\pi\)
\(762\) 8.53889e9 0.699132
\(763\) 1.99345e9 0.162469
\(764\) −9.43342e9 −0.765319
\(765\) 3.05569e9 0.246771
\(766\) −4.07378e9 −0.327489
\(767\) 3.06346e9 0.245148
\(768\) −4.52985e8 −0.0360844
\(769\) 2.35638e9 0.186855 0.0934273 0.995626i \(-0.470218\pi\)
0.0934273 + 0.995626i \(0.470218\pi\)
\(770\) −1.19162e9 −0.0940634
\(771\) 3.53568e9 0.277832
\(772\) −5.79055e9 −0.452960
\(773\) −1.60854e10 −1.25258 −0.626288 0.779592i \(-0.715426\pi\)
−0.626288 + 0.779592i \(0.715426\pi\)
\(774\) 4.33937e9 0.336382
\(775\) −6.41387e9 −0.494953
\(776\) 8.01607e9 0.615808
\(777\) −3.26859e9 −0.249969
\(778\) −1.95289e8 −0.0148679
\(779\) 4.47178e10 3.38922
\(780\) 4.30771e8 0.0325024
\(781\) 2.05705e10 1.54514
\(782\) 1.24960e10 0.934428
\(783\) −3.12014e9 −0.232278
\(784\) 4.81890e8 0.0357143
\(785\) 1.30290e9 0.0961318
\(786\) −5.24407e8 −0.0385203
\(787\) −1.42841e10 −1.04458 −0.522288 0.852769i \(-0.674921\pi\)
−0.522288 + 0.852769i \(0.674921\pi\)
\(788\) −3.78673e9 −0.275691
\(789\) 9.23732e9 0.669540
\(790\) 5.02679e9 0.362740
\(791\) −6.50853e9 −0.467590
\(792\) 1.42850e9 0.102174
\(793\) −2.92290e9 −0.208141
\(794\) −4.07583e9 −0.288964
\(795\) −2.03666e9 −0.143759
\(796\) 2.77370e9 0.194923
\(797\) 4.63692e9 0.324433 0.162217 0.986755i \(-0.448136\pi\)
0.162217 + 0.986755i \(0.448136\pi\)
\(798\) −4.21022e9 −0.293289
\(799\) 1.84234e10 1.27778
\(800\) 2.13811e9 0.147644
\(801\) −5.58432e9 −0.383934
\(802\) −7.20859e9 −0.493447
\(803\) −1.37502e10 −0.937143
\(804\) −4.39190e9 −0.298027
\(805\) −1.64565e9 −0.111186
\(806\) −1.72766e9 −0.116222
\(807\) −3.26083e9 −0.218409
\(808\) 5.30301e7 0.00353658
\(809\) −1.37884e10 −0.915577 −0.457789 0.889061i \(-0.651358\pi\)
−0.457789 + 0.889061i \(0.651358\pi\)
\(810\) 4.82411e8 0.0318948
\(811\) 2.64001e10 1.73793 0.868967 0.494871i \(-0.164785\pi\)
0.868967 + 0.494871i \(0.164785\pi\)
\(812\) 3.47983e9 0.228093
\(813\) −1.42873e10 −0.932470
\(814\) 1.08062e10 0.702244
\(815\) −4.33990e9 −0.280820
\(816\) 4.08538e9 0.263219
\(817\) 4.22831e10 2.71262
\(818\) −1.30991e10 −0.836766
\(819\) 5.49353e8 0.0349428
\(820\) 5.71447e9 0.361932
\(821\) −2.32509e10 −1.46635 −0.733177 0.680037i \(-0.761964\pi\)
−0.733177 + 0.680037i \(0.761964\pi\)
\(822\) 9.00215e9 0.565321
\(823\) 1.26784e10 0.792800 0.396400 0.918078i \(-0.370259\pi\)
0.396400 + 0.918078i \(0.370259\pi\)
\(824\) 8.42573e9 0.524641
\(825\) −6.74258e9 −0.418059
\(826\) −3.82619e9 −0.236231
\(827\) 1.11464e10 0.685279 0.342639 0.939467i \(-0.388679\pi\)
0.342639 + 0.939467i \(0.388679\pi\)
\(828\) 1.97278e9 0.120773
\(829\) 1.52385e9 0.0928968 0.0464484 0.998921i \(-0.485210\pi\)
0.0464484 + 0.998921i \(0.485210\pi\)
\(830\) −4.64183e9 −0.281784
\(831\) −9.15785e9 −0.553592
\(832\) 5.75930e8 0.0346688
\(833\) −4.34608e9 −0.260519
\(834\) −1.28862e10 −0.769207
\(835\) −4.77954e9 −0.284108
\(836\) 1.39193e10 0.823943
\(837\) −1.93478e9 −0.114049
\(838\) −1.08238e9 −0.0635369
\(839\) 1.97275e10 1.15320 0.576602 0.817025i \(-0.304378\pi\)
0.576602 + 0.817025i \(0.304378\pi\)
\(840\) −5.38022e8 −0.0313201
\(841\) 7.87863e9 0.456736
\(842\) 1.54132e10 0.889818
\(843\) −1.71039e10 −0.983328
\(844\) 1.44898e10 0.829592
\(845\) −5.47687e8 −0.0312273
\(846\) 2.90856e9 0.165151
\(847\) −1.66001e9 −0.0938683
\(848\) −2.72297e9 −0.153341
\(849\) −1.61144e10 −0.903726
\(850\) −1.92832e10 −1.07700
\(851\) 1.49235e10 0.830078
\(852\) 9.28770e9 0.514481
\(853\) 1.83654e9 0.101316 0.0506582 0.998716i \(-0.483868\pi\)
0.0506582 + 0.998716i \(0.483868\pi\)
\(854\) 3.65063e9 0.200570
\(855\) 4.70065e9 0.257203
\(856\) 3.65629e9 0.199243
\(857\) 1.24123e10 0.673627 0.336813 0.941571i \(-0.390651\pi\)
0.336813 + 0.941571i \(0.390651\pi\)
\(858\) −1.81621e9 −0.0981657
\(859\) 5.30403e9 0.285516 0.142758 0.989758i \(-0.454403\pi\)
0.142758 + 0.989758i \(0.454403\pi\)
\(860\) 5.40333e9 0.289679
\(861\) 7.28754e9 0.389108
\(862\) −2.28137e10 −1.21316
\(863\) 2.55569e10 1.35354 0.676770 0.736195i \(-0.263379\pi\)
0.676770 + 0.736195i \(0.263379\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 2.74411e9 0.144160
\(866\) −1.37482e9 −0.0719340
\(867\) −2.57661e10 −1.34271
\(868\) 2.15781e9 0.111994
\(869\) −2.11939e10 −1.09557
\(870\) −3.88517e9 −0.200029
\(871\) 5.58391e9 0.286335
\(872\) −2.97565e9 −0.151975
\(873\) −1.14135e10 −0.580589
\(874\) 1.92228e10 0.973930
\(875\) 5.58008e9 0.281587
\(876\) −6.20829e9 −0.312038
\(877\) 2.75111e10 1.37724 0.688621 0.725122i \(-0.258217\pi\)
0.688621 + 0.725122i \(0.258217\pi\)
\(878\) −7.35570e9 −0.366770
\(879\) −4.59064e9 −0.227988
\(880\) 1.77875e9 0.0879883
\(881\) −2.66910e10 −1.31507 −0.657536 0.753424i \(-0.728401\pi\)
−0.657536 + 0.753424i \(0.728401\pi\)
\(882\) −6.86129e8 −0.0336718
\(883\) 7.52996e9 0.368070 0.184035 0.982920i \(-0.441084\pi\)
0.184035 + 0.982920i \(0.441084\pi\)
\(884\) −5.19421e9 −0.252893
\(885\) 4.27188e9 0.207165
\(886\) 2.07909e10 1.00428
\(887\) 2.61087e9 0.125618 0.0628092 0.998026i \(-0.479994\pi\)
0.0628092 + 0.998026i \(0.479994\pi\)
\(888\) 4.87905e9 0.233825
\(889\) 1.35594e10 0.647270
\(890\) −6.95353e9 −0.330629
\(891\) −2.03393e9 −0.0963307
\(892\) 1.13006e10 0.533121
\(893\) 2.83412e10 1.33180
\(894\) 3.85823e9 0.180595
\(895\) 4.01244e9 0.187080
\(896\) −7.19323e8 −0.0334077
\(897\) −2.50821e9 −0.116035
\(898\) −4.64563e9 −0.214081
\(899\) 1.55820e10 0.715260
\(900\) −3.04431e9 −0.139200
\(901\) 2.45579e10 1.11855
\(902\) −2.40932e10 −1.09313
\(903\) 6.89076e9 0.311429
\(904\) 9.71536e9 0.437391
\(905\) −3.36479e9 −0.150900
\(906\) −8.16731e9 −0.364863
\(907\) −9.59738e9 −0.427098 −0.213549 0.976932i \(-0.568502\pi\)
−0.213549 + 0.976932i \(0.568502\pi\)
\(908\) 3.26404e9 0.144695
\(909\) −7.55058e7 −0.00333432
\(910\) 6.84048e8 0.0300914
\(911\) 2.70205e10 1.18408 0.592038 0.805910i \(-0.298324\pi\)
0.592038 + 0.805910i \(0.298324\pi\)
\(912\) 6.28465e9 0.274346
\(913\) 1.95708e10 0.851061
\(914\) 2.14375e10 0.928674
\(915\) −4.07586e9 −0.175892
\(916\) −1.97991e10 −0.851158
\(917\) −8.32739e8 −0.0356629
\(918\) −5.81688e9 −0.248165
\(919\) 5.19047e9 0.220599 0.110299 0.993898i \(-0.464819\pi\)
0.110299 + 0.993898i \(0.464819\pi\)
\(920\) 2.45648e9 0.104005
\(921\) −1.11359e10 −0.469697
\(922\) −2.38398e10 −1.00171
\(923\) −1.18085e10 −0.494297
\(924\) 2.26840e9 0.0945949
\(925\) −2.30294e10 −0.956724
\(926\) −2.02925e10 −0.839840
\(927\) −1.19968e10 −0.494636
\(928\) −5.19437e9 −0.213361
\(929\) 3.23414e10 1.32344 0.661719 0.749752i \(-0.269827\pi\)
0.661719 + 0.749752i \(0.269827\pi\)
\(930\) −2.40916e9 −0.0982144
\(931\) −6.68568e9 −0.271532
\(932\) 9.34835e9 0.378250
\(933\) 2.01847e10 0.813649
\(934\) 1.49788e10 0.601538
\(935\) −1.60422e10 −0.641834
\(936\) −8.20026e8 −0.0326860
\(937\) −7.25331e9 −0.288037 −0.144018 0.989575i \(-0.546002\pi\)
−0.144018 + 0.989575i \(0.546002\pi\)
\(938\) −6.97417e9 −0.275920
\(939\) −6.15486e9 −0.242599
\(940\) 3.62171e9 0.142222
\(941\) −2.51889e10 −0.985473 −0.492737 0.870179i \(-0.664003\pi\)
−0.492737 + 0.870179i \(0.664003\pi\)
\(942\) −2.48023e9 −0.0966749
\(943\) −3.32731e10 −1.29212
\(944\) 5.71140e9 0.220974
\(945\) 7.66052e8 0.0295289
\(946\) −2.27814e10 −0.874906
\(947\) −2.22847e10 −0.852671 −0.426335 0.904565i \(-0.640196\pi\)
−0.426335 + 0.904565i \(0.640196\pi\)
\(948\) −9.56912e9 −0.364790
\(949\) 7.89330e9 0.299797
\(950\) −2.96639e10 −1.12252
\(951\) 4.66837e9 0.176008
\(952\) 6.48744e9 0.243693
\(953\) −2.43375e10 −0.910857 −0.455429 0.890272i \(-0.650514\pi\)
−0.455429 + 0.890272i \(0.650514\pi\)
\(954\) 3.87704e9 0.144571
\(955\) 1.67248e10 0.621370
\(956\) 6.15086e9 0.227684
\(957\) 1.63806e10 0.604139
\(958\) 2.20017e10 0.808494
\(959\) 1.42951e10 0.523385
\(960\) 8.03112e8 0.0292973
\(961\) −1.78504e10 −0.648806
\(962\) −6.20329e9 −0.224651
\(963\) −5.20593e9 −0.187848
\(964\) 2.50405e10 0.900270
\(965\) 1.02663e10 0.367762
\(966\) 3.13269e9 0.111814
\(967\) −3.42527e10 −1.21815 −0.609076 0.793112i \(-0.708460\pi\)
−0.609076 + 0.793112i \(0.708460\pi\)
\(968\) 2.47792e9 0.0878058
\(969\) −5.66801e10 −2.00123
\(970\) −1.42120e10 −0.499980
\(971\) 1.06157e10 0.372117 0.186059 0.982539i \(-0.440429\pi\)
0.186059 + 0.982539i \(0.440429\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −2.04628e10 −0.712147
\(974\) −2.84055e10 −0.985023
\(975\) 3.87057e9 0.133739
\(976\) −5.44933e9 −0.187616
\(977\) 3.29891e10 1.13172 0.565860 0.824501i \(-0.308544\pi\)
0.565860 + 0.824501i \(0.308544\pi\)
\(978\) 8.26154e9 0.282407
\(979\) 2.93173e10 0.998585
\(980\) −8.54360e8 −0.0289968
\(981\) 4.23681e9 0.143284
\(982\) 1.17176e10 0.394866
\(983\) 1.51536e10 0.508837 0.254418 0.967094i \(-0.418116\pi\)
0.254418 + 0.967094i \(0.418116\pi\)
\(984\) −1.08782e10 −0.363977
\(985\) 6.71362e9 0.223836
\(986\) 4.68471e10 1.55637
\(987\) 4.61869e9 0.152900
\(988\) −7.99038e9 −0.263583
\(989\) −3.14615e10 −1.03417
\(990\) −2.53263e9 −0.0829561
\(991\) −8.21417e9 −0.268106 −0.134053 0.990974i \(-0.542799\pi\)
−0.134053 + 0.990974i \(0.542799\pi\)
\(992\) −3.22099e9 −0.104761
\(993\) 1.83186e10 0.593705
\(994\) 1.47485e10 0.476317
\(995\) −4.91758e9 −0.158260
\(996\) 8.83630e9 0.283376
\(997\) 9.33193e9 0.298221 0.149111 0.988821i \(-0.452359\pi\)
0.149111 + 0.988821i \(0.452359\pi\)
\(998\) 1.31469e10 0.418664
\(999\) −6.94693e9 −0.220452
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.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.n.1.3 6 1.1 even 1 trivial