Properties

Label 177.8.a.a.1.7
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(4.55626\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.55626 q^{2} -27.0000 q^{3} -107.241 q^{4} -540.445 q^{5} +123.019 q^{6} -1238.10 q^{7} +1071.82 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-4.55626 q^{2} -27.0000 q^{3} -107.241 q^{4} -540.445 q^{5} +123.019 q^{6} -1238.10 q^{7} +1071.82 q^{8} +729.000 q^{9} +2462.40 q^{10} +3521.59 q^{11} +2895.49 q^{12} -4832.75 q^{13} +5641.10 q^{14} +14592.0 q^{15} +8843.32 q^{16} +2243.59 q^{17} -3321.51 q^{18} +53136.6 q^{19} +57957.6 q^{20} +33428.7 q^{21} -16045.3 q^{22} -56390.0 q^{23} -28939.0 q^{24} +213955. q^{25} +22019.3 q^{26} -19683.0 q^{27} +132774. q^{28} -137347. q^{29} -66484.9 q^{30} -121430. q^{31} -177485. q^{32} -95082.8 q^{33} -10222.4 q^{34} +669124. q^{35} -78178.3 q^{36} -476697. q^{37} -242104. q^{38} +130484. q^{39} -579257. q^{40} +140564. q^{41} -152310. q^{42} +696919. q^{43} -377657. q^{44} -393984. q^{45} +256927. q^{46} +783363. q^{47} -238770. q^{48} +709348. q^{49} -974835. q^{50} -60577.0 q^{51} +518267. q^{52} +1.05434e6 q^{53} +89680.8 q^{54} -1.90322e6 q^{55} -1.32702e6 q^{56} -1.43469e6 q^{57} +625790. q^{58} +205379. q^{59} -1.56485e6 q^{60} +378219. q^{61} +553265. q^{62} -902575. q^{63} -323278. q^{64} +2.61184e6 q^{65} +433222. q^{66} -1.12190e6 q^{67} -240604. q^{68} +1.52253e6 q^{69} -3.04870e6 q^{70} +3.82821e6 q^{71} +781354. q^{72} -1.68355e6 q^{73} +2.17195e6 q^{74} -5.77679e6 q^{75} -5.69840e6 q^{76} -4.36008e6 q^{77} -594520. q^{78} +5.56662e6 q^{79} -4.77932e6 q^{80} +531441. q^{81} -640447. q^{82} -1.80553e6 q^{83} -3.58491e6 q^{84} -1.21254e6 q^{85} -3.17534e6 q^{86} +3.70838e6 q^{87} +3.77449e6 q^{88} +8.26085e6 q^{89} +1.79509e6 q^{90} +5.98343e6 q^{91} +6.04729e6 q^{92} +3.27860e6 q^{93} -3.56920e6 q^{94} -2.87174e7 q^{95} +4.79209e6 q^{96} -9.36799e6 q^{97} -3.23197e6 q^{98} +2.56724e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.55626 −0.402720 −0.201360 0.979517i \(-0.564536\pi\)
−0.201360 + 0.979517i \(0.564536\pi\)
\(3\) −27.0000 −0.577350
\(4\) −107.241 −0.837817
\(5\) −540.445 −1.93355 −0.966777 0.255623i \(-0.917719\pi\)
−0.966777 + 0.255623i \(0.917719\pi\)
\(6\) 123.019 0.232510
\(7\) −1238.10 −1.36431 −0.682154 0.731208i \(-0.738957\pi\)
−0.682154 + 0.731208i \(0.738957\pi\)
\(8\) 1071.82 0.740125
\(9\) 729.000 0.333333
\(10\) 2462.40 0.778680
\(11\) 3521.59 0.797745 0.398872 0.917006i \(-0.369402\pi\)
0.398872 + 0.917006i \(0.369402\pi\)
\(12\) 2895.49 0.483714
\(13\) −4832.75 −0.610089 −0.305044 0.952338i \(-0.598671\pi\)
−0.305044 + 0.952338i \(0.598671\pi\)
\(14\) 5641.10 0.549434
\(15\) 14592.0 1.11634
\(16\) 8843.32 0.539753
\(17\) 2243.59 0.110757 0.0553787 0.998465i \(-0.482363\pi\)
0.0553787 + 0.998465i \(0.482363\pi\)
\(18\) −3321.51 −0.134240
\(19\) 53136.6 1.77728 0.888641 0.458603i \(-0.151650\pi\)
0.888641 + 0.458603i \(0.151650\pi\)
\(20\) 57957.6 1.61996
\(21\) 33428.7 0.787684
\(22\) −16045.3 −0.321268
\(23\) −56390.0 −0.966394 −0.483197 0.875512i \(-0.660525\pi\)
−0.483197 + 0.875512i \(0.660525\pi\)
\(24\) −28939.0 −0.427312
\(25\) 213955. 2.73863
\(26\) 22019.3 0.245695
\(27\) −19683.0 −0.192450
\(28\) 132774. 1.14304
\(29\) −137347. −1.04575 −0.522874 0.852410i \(-0.675140\pi\)
−0.522874 + 0.852410i \(0.675140\pi\)
\(30\) −66484.9 −0.449571
\(31\) −121430. −0.732081 −0.366041 0.930599i \(-0.619287\pi\)
−0.366041 + 0.930599i \(0.619287\pi\)
\(32\) −177485. −0.957495
\(33\) −95082.8 −0.460578
\(34\) −10222.4 −0.0446042
\(35\) 669124. 2.63796
\(36\) −78178.3 −0.279272
\(37\) −476697. −1.54716 −0.773581 0.633697i \(-0.781537\pi\)
−0.773581 + 0.633697i \(0.781537\pi\)
\(38\) −242104. −0.715747
\(39\) 130484. 0.352235
\(40\) −579257. −1.43107
\(41\) 140564. 0.318516 0.159258 0.987237i \(-0.449090\pi\)
0.159258 + 0.987237i \(0.449090\pi\)
\(42\) −152310. −0.317216
\(43\) 696919. 1.33673 0.668363 0.743835i \(-0.266995\pi\)
0.668363 + 0.743835i \(0.266995\pi\)
\(44\) −377657. −0.668364
\(45\) −393984. −0.644518
\(46\) 256927. 0.389186
\(47\) 783363. 1.10058 0.550289 0.834974i \(-0.314518\pi\)
0.550289 + 0.834974i \(0.314518\pi\)
\(48\) −238770. −0.311627
\(49\) 709348. 0.861337
\(50\) −974835. −1.10290
\(51\) −60577.0 −0.0639458
\(52\) 518267. 0.511142
\(53\) 1.05434e6 0.972781 0.486390 0.873742i \(-0.338313\pi\)
0.486390 + 0.873742i \(0.338313\pi\)
\(54\) 89680.8 0.0775035
\(55\) −1.90322e6 −1.54248
\(56\) −1.32702e6 −1.00976
\(57\) −1.43469e6 −1.02611
\(58\) 625790. 0.421144
\(59\) 205379. 0.130189
\(60\) −1.56485e6 −0.935286
\(61\) 378219. 0.213348 0.106674 0.994294i \(-0.465980\pi\)
0.106674 + 0.994294i \(0.465980\pi\)
\(62\) 553265. 0.294824
\(63\) −902575. −0.454769
\(64\) −323278. −0.154151
\(65\) 2.61184e6 1.17964
\(66\) 433222. 0.185484
\(67\) −1.12190e6 −0.455714 −0.227857 0.973695i \(-0.573172\pi\)
−0.227857 + 0.973695i \(0.573172\pi\)
\(68\) −240604. −0.0927943
\(69\) 1.52253e6 0.557948
\(70\) −3.04870e6 −1.06236
\(71\) 3.82821e6 1.26938 0.634689 0.772767i \(-0.281128\pi\)
0.634689 + 0.772767i \(0.281128\pi\)
\(72\) 781354. 0.246708
\(73\) −1.68355e6 −0.506518 −0.253259 0.967398i \(-0.581503\pi\)
−0.253259 + 0.967398i \(0.581503\pi\)
\(74\) 2.17195e6 0.623073
\(75\) −5.77679e6 −1.58115
\(76\) −5.69840e6 −1.48904
\(77\) −4.36008e6 −1.08837
\(78\) −594520. −0.141852
\(79\) 5.56662e6 1.27027 0.635136 0.772401i \(-0.280944\pi\)
0.635136 + 0.772401i \(0.280944\pi\)
\(80\) −4.77932e6 −1.04364
\(81\) 531441. 0.111111
\(82\) −640447. −0.128273
\(83\) −1.80553e6 −0.346602 −0.173301 0.984869i \(-0.555443\pi\)
−0.173301 + 0.984869i \(0.555443\pi\)
\(84\) −3.58491e6 −0.659935
\(85\) −1.21254e6 −0.214155
\(86\) −3.17534e6 −0.538327
\(87\) 3.70838e6 0.603763
\(88\) 3.77449e6 0.590431
\(89\) 8.26085e6 1.24211 0.621055 0.783767i \(-0.286704\pi\)
0.621055 + 0.783767i \(0.286704\pi\)
\(90\) 1.79509e6 0.259560
\(91\) 5.98343e6 0.832349
\(92\) 6.04729e6 0.809661
\(93\) 3.27860e6 0.422667
\(94\) −3.56920e6 −0.443225
\(95\) −2.87174e7 −3.43647
\(96\) 4.79209e6 0.552810
\(97\) −9.36799e6 −1.04219 −0.521093 0.853500i \(-0.674476\pi\)
−0.521093 + 0.853500i \(0.674476\pi\)
\(98\) −3.23197e6 −0.346878
\(99\) 2.56724e6 0.265915
\(100\) −2.29447e7 −2.29447
\(101\) 5.17821e6 0.500097 0.250049 0.968233i \(-0.419553\pi\)
0.250049 + 0.968233i \(0.419553\pi\)
\(102\) 276004. 0.0257522
\(103\) −1.63595e7 −1.47516 −0.737579 0.675261i \(-0.764031\pi\)
−0.737579 + 0.675261i \(0.764031\pi\)
\(104\) −5.17982e6 −0.451542
\(105\) −1.80664e7 −1.52303
\(106\) −4.80384e6 −0.391758
\(107\) −1.15281e7 −0.909734 −0.454867 0.890559i \(-0.650313\pi\)
−0.454867 + 0.890559i \(0.650313\pi\)
\(108\) 2.11082e6 0.161238
\(109\) −1.66988e6 −0.123507 −0.0617535 0.998091i \(-0.519669\pi\)
−0.0617535 + 0.998091i \(0.519669\pi\)
\(110\) 8.67157e6 0.621188
\(111\) 1.28708e7 0.893255
\(112\) −1.09489e7 −0.736390
\(113\) −2.82765e6 −0.184353 −0.0921767 0.995743i \(-0.529382\pi\)
−0.0921767 + 0.995743i \(0.529382\pi\)
\(114\) 6.53681e6 0.413237
\(115\) 3.04756e7 1.86857
\(116\) 1.47292e7 0.876145
\(117\) −3.52308e6 −0.203363
\(118\) −935759. −0.0524297
\(119\) −2.77779e6 −0.151107
\(120\) 1.56399e7 0.826230
\(121\) −7.08560e6 −0.363603
\(122\) −1.72326e6 −0.0859195
\(123\) −3.79524e6 −0.183895
\(124\) 1.30222e7 0.613350
\(125\) −7.34087e7 −3.36173
\(126\) 4.11236e6 0.183145
\(127\) 4.15154e7 1.79844 0.899221 0.437496i \(-0.144134\pi\)
0.899221 + 0.437496i \(0.144134\pi\)
\(128\) 2.41910e7 1.01957
\(129\) −1.88168e7 −0.771760
\(130\) −1.19002e7 −0.475064
\(131\) 4.63759e7 1.80237 0.901183 0.433438i \(-0.142700\pi\)
0.901183 + 0.433438i \(0.142700\pi\)
\(132\) 1.01967e7 0.385880
\(133\) −6.57885e7 −2.42476
\(134\) 5.11167e6 0.183525
\(135\) 1.06376e7 0.372112
\(136\) 2.40472e6 0.0819743
\(137\) −3.20418e6 −0.106462 −0.0532310 0.998582i \(-0.516952\pi\)
−0.0532310 + 0.998582i \(0.516952\pi\)
\(138\) −6.93703e6 −0.224697
\(139\) 3.74591e7 1.18306 0.591529 0.806284i \(-0.298525\pi\)
0.591529 + 0.806284i \(0.298525\pi\)
\(140\) −7.17572e7 −2.21013
\(141\) −2.11508e7 −0.635419
\(142\) −1.74423e7 −0.511204
\(143\) −1.70190e7 −0.486695
\(144\) 6.44678e6 0.179918
\(145\) 7.42286e7 2.02201
\(146\) 7.67067e6 0.203985
\(147\) −1.91524e7 −0.497293
\(148\) 5.11212e7 1.29624
\(149\) 2.39531e7 0.593212 0.296606 0.955000i \(-0.404145\pi\)
0.296606 + 0.955000i \(0.404145\pi\)
\(150\) 2.63205e7 0.636760
\(151\) −6.59208e7 −1.55813 −0.779064 0.626945i \(-0.784305\pi\)
−0.779064 + 0.626945i \(0.784305\pi\)
\(152\) 5.69527e7 1.31541
\(153\) 1.63558e6 0.0369191
\(154\) 1.98656e7 0.438308
\(155\) 6.56261e7 1.41552
\(156\) −1.39932e7 −0.295108
\(157\) −6.45045e7 −1.33027 −0.665137 0.746721i \(-0.731627\pi\)
−0.665137 + 0.746721i \(0.731627\pi\)
\(158\) −2.53629e7 −0.511564
\(159\) −2.84672e7 −0.561635
\(160\) 9.59207e7 1.85137
\(161\) 6.98164e7 1.31846
\(162\) −2.42138e6 −0.0447467
\(163\) −3.88407e7 −0.702474 −0.351237 0.936287i \(-0.614239\pi\)
−0.351237 + 0.936287i \(0.614239\pi\)
\(164\) −1.50742e7 −0.266858
\(165\) 5.13870e7 0.890552
\(166\) 8.22644e6 0.139583
\(167\) 2.62525e7 0.436178 0.218089 0.975929i \(-0.430018\pi\)
0.218089 + 0.975929i \(0.430018\pi\)
\(168\) 3.58294e7 0.582985
\(169\) −3.93930e7 −0.627792
\(170\) 5.52463e6 0.0862446
\(171\) 3.87366e7 0.592428
\(172\) −7.47380e7 −1.11993
\(173\) −9.81294e6 −0.144091 −0.0720457 0.997401i \(-0.522953\pi\)
−0.0720457 + 0.997401i \(0.522953\pi\)
\(174\) −1.68963e7 −0.243147
\(175\) −2.64898e8 −3.73633
\(176\) 3.11425e7 0.430585
\(177\) −5.54523e6 −0.0751646
\(178\) −3.76386e7 −0.500222
\(179\) −3.88790e7 −0.506675 −0.253337 0.967378i \(-0.581528\pi\)
−0.253337 + 0.967378i \(0.581528\pi\)
\(180\) 4.22511e7 0.539988
\(181\) −5.75554e7 −0.721458 −0.360729 0.932671i \(-0.617472\pi\)
−0.360729 + 0.932671i \(0.617472\pi\)
\(182\) −2.72621e7 −0.335204
\(183\) −1.02119e7 −0.123177
\(184\) −6.04397e7 −0.715253
\(185\) 2.57628e8 2.99152
\(186\) −1.49382e7 −0.170217
\(187\) 7.90100e6 0.0883561
\(188\) −8.40083e7 −0.922082
\(189\) 2.43695e7 0.262561
\(190\) 1.30844e8 1.38394
\(191\) 1.56708e8 1.62733 0.813663 0.581337i \(-0.197470\pi\)
0.813663 + 0.581337i \(0.197470\pi\)
\(192\) 8.72850e6 0.0889991
\(193\) 5.44941e7 0.545631 0.272816 0.962066i \(-0.412045\pi\)
0.272816 + 0.962066i \(0.412045\pi\)
\(194\) 4.26830e7 0.419709
\(195\) −7.05196e7 −0.681065
\(196\) −7.60709e7 −0.721642
\(197\) 1.46168e8 1.36213 0.681066 0.732222i \(-0.261517\pi\)
0.681066 + 0.732222i \(0.261517\pi\)
\(198\) −1.16970e7 −0.107089
\(199\) 5.06585e7 0.455687 0.227843 0.973698i \(-0.426833\pi\)
0.227843 + 0.973698i \(0.426833\pi\)
\(200\) 2.29321e8 2.02693
\(201\) 3.02913e7 0.263107
\(202\) −2.35932e7 −0.201399
\(203\) 1.70050e8 1.42672
\(204\) 6.49631e6 0.0535748
\(205\) −7.59672e7 −0.615868
\(206\) 7.45379e7 0.594076
\(207\) −4.11083e7 −0.322131
\(208\) −4.27376e7 −0.329297
\(209\) 1.87125e8 1.41782
\(210\) 8.23149e7 0.613354
\(211\) 3.36247e7 0.246416 0.123208 0.992381i \(-0.460682\pi\)
0.123208 + 0.992381i \(0.460682\pi\)
\(212\) −1.13068e8 −0.815012
\(213\) −1.03362e8 −0.732876
\(214\) 5.25250e7 0.366368
\(215\) −3.76646e8 −2.58463
\(216\) −2.10966e7 −0.142437
\(217\) 1.50342e8 0.998784
\(218\) 7.60839e6 0.0497388
\(219\) 4.54558e7 0.292438
\(220\) 2.04103e8 1.29232
\(221\) −1.08427e7 −0.0675718
\(222\) −5.86427e7 −0.359732
\(223\) 8.68957e7 0.524725 0.262362 0.964969i \(-0.415498\pi\)
0.262362 + 0.964969i \(0.415498\pi\)
\(224\) 2.19744e8 1.30632
\(225\) 1.55973e8 0.912876
\(226\) 1.28835e7 0.0742428
\(227\) 3.64696e7 0.206938 0.103469 0.994633i \(-0.467006\pi\)
0.103469 + 0.994633i \(0.467006\pi\)
\(228\) 1.53857e8 0.859696
\(229\) 2.19546e8 1.20809 0.604047 0.796949i \(-0.293554\pi\)
0.604047 + 0.796949i \(0.293554\pi\)
\(230\) −1.38855e8 −0.752512
\(231\) 1.17722e8 0.628371
\(232\) −1.47211e8 −0.773985
\(233\) 1.19989e8 0.621435 0.310718 0.950502i \(-0.399431\pi\)
0.310718 + 0.950502i \(0.399431\pi\)
\(234\) 1.60520e7 0.0818983
\(235\) −4.23364e8 −2.12803
\(236\) −2.20250e7 −0.109074
\(237\) −1.50299e8 −0.733391
\(238\) 1.26563e7 0.0608539
\(239\) −2.23797e8 −1.06038 −0.530189 0.847879i \(-0.677879\pi\)
−0.530189 + 0.847879i \(0.677879\pi\)
\(240\) 1.29042e8 0.602547
\(241\) −1.78627e8 −0.822030 −0.411015 0.911629i \(-0.634826\pi\)
−0.411015 + 0.911629i \(0.634826\pi\)
\(242\) 3.22838e7 0.146430
\(243\) −1.43489e7 −0.0641500
\(244\) −4.05604e7 −0.178746
\(245\) −3.83363e8 −1.66544
\(246\) 1.72921e7 0.0740584
\(247\) −2.56796e8 −1.08430
\(248\) −1.30150e8 −0.541832
\(249\) 4.87492e7 0.200111
\(250\) 3.34469e8 1.35384
\(251\) −3.44628e8 −1.37560 −0.687800 0.725900i \(-0.741423\pi\)
−0.687800 + 0.725900i \(0.741423\pi\)
\(252\) 9.67926e7 0.381013
\(253\) −1.98582e8 −0.770936
\(254\) −1.89155e8 −0.724268
\(255\) 3.27385e7 0.123643
\(256\) −6.88409e7 −0.256452
\(257\) −1.65611e7 −0.0608588 −0.0304294 0.999537i \(-0.509687\pi\)
−0.0304294 + 0.999537i \(0.509687\pi\)
\(258\) 8.57342e7 0.310803
\(259\) 5.90198e8 2.11081
\(260\) −2.80095e8 −0.988321
\(261\) −1.00126e8 −0.348583
\(262\) −2.11301e8 −0.725849
\(263\) 3.89298e8 1.31958 0.659792 0.751448i \(-0.270644\pi\)
0.659792 + 0.751448i \(0.270644\pi\)
\(264\) −1.01911e8 −0.340886
\(265\) −5.69812e8 −1.88092
\(266\) 2.99749e8 0.976500
\(267\) −2.23043e8 −0.717132
\(268\) 1.20313e8 0.381805
\(269\) −2.69260e8 −0.843410 −0.421705 0.906733i \(-0.638568\pi\)
−0.421705 + 0.906733i \(0.638568\pi\)
\(270\) −4.84675e7 −0.149857
\(271\) −4.95264e8 −1.51163 −0.755813 0.654787i \(-0.772758\pi\)
−0.755813 + 0.654787i \(0.772758\pi\)
\(272\) 1.98408e7 0.0597816
\(273\) −1.61553e8 −0.480557
\(274\) 1.45990e7 0.0428743
\(275\) 7.53462e8 2.18473
\(276\) −1.63277e8 −0.467458
\(277\) 3.16965e8 0.896051 0.448025 0.894021i \(-0.352127\pi\)
0.448025 + 0.894021i \(0.352127\pi\)
\(278\) −1.70673e8 −0.476441
\(279\) −8.85223e7 −0.244027
\(280\) 7.17178e8 1.95242
\(281\) −4.50536e8 −1.21132 −0.605658 0.795725i \(-0.707090\pi\)
−0.605658 + 0.795725i \(0.707090\pi\)
\(282\) 9.63685e7 0.255896
\(283\) 4.67571e8 1.22629 0.613147 0.789969i \(-0.289903\pi\)
0.613147 + 0.789969i \(0.289903\pi\)
\(284\) −4.10539e8 −1.06351
\(285\) 7.75370e8 1.98405
\(286\) 7.75428e7 0.196002
\(287\) −1.74033e8 −0.434554
\(288\) −1.29386e8 −0.319165
\(289\) −4.05305e8 −0.987733
\(290\) −3.38205e8 −0.814304
\(291\) 2.52936e8 0.601706
\(292\) 1.80544e8 0.424369
\(293\) −1.60508e7 −0.0372786 −0.0186393 0.999826i \(-0.505933\pi\)
−0.0186393 + 0.999826i \(0.505933\pi\)
\(294\) 8.72632e7 0.200270
\(295\) −1.10996e8 −0.251727
\(296\) −5.10931e8 −1.14509
\(297\) −6.93154e7 −0.153526
\(298\) −1.09136e8 −0.238898
\(299\) 2.72519e8 0.589586
\(300\) 6.19506e8 1.32471
\(301\) −8.62855e8 −1.82371
\(302\) 3.00352e8 0.627489
\(303\) −1.39812e8 −0.288731
\(304\) 4.69904e8 0.959294
\(305\) −2.04406e8 −0.412520
\(306\) −7.45211e6 −0.0148681
\(307\) −6.56164e7 −0.129428 −0.0647140 0.997904i \(-0.520614\pi\)
−0.0647140 + 0.997904i \(0.520614\pi\)
\(308\) 4.67577e8 0.911854
\(309\) 4.41705e8 0.851683
\(310\) −2.99009e8 −0.570057
\(311\) 4.44680e8 0.838275 0.419138 0.907923i \(-0.362332\pi\)
0.419138 + 0.907923i \(0.362332\pi\)
\(312\) 1.39855e8 0.260698
\(313\) 2.94757e8 0.543324 0.271662 0.962393i \(-0.412427\pi\)
0.271662 + 0.962393i \(0.412427\pi\)
\(314\) 2.93899e8 0.535728
\(315\) 4.87792e8 0.879321
\(316\) −5.96967e8 −1.06425
\(317\) −3.97400e8 −0.700682 −0.350341 0.936622i \(-0.613934\pi\)
−0.350341 + 0.936622i \(0.613934\pi\)
\(318\) 1.29704e8 0.226182
\(319\) −4.83680e8 −0.834240
\(320\) 1.74714e8 0.298059
\(321\) 3.11259e8 0.525235
\(322\) −3.18101e8 −0.530970
\(323\) 1.19217e8 0.196847
\(324\) −5.69920e7 −0.0930907
\(325\) −1.03399e9 −1.67081
\(326\) 1.76968e8 0.282900
\(327\) 4.50867e7 0.0713069
\(328\) 1.50659e8 0.235742
\(329\) −9.69882e8 −1.50153
\(330\) −2.34132e8 −0.358643
\(331\) −1.15349e9 −1.74830 −0.874150 0.485656i \(-0.838581\pi\)
−0.874150 + 0.485656i \(0.838581\pi\)
\(332\) 1.93626e8 0.290389
\(333\) −3.47512e8 −0.515721
\(334\) −1.19613e8 −0.175658
\(335\) 6.06325e8 0.881148
\(336\) 2.95621e8 0.425155
\(337\) −1.00153e9 −1.42547 −0.712735 0.701434i \(-0.752544\pi\)
−0.712735 + 0.701434i \(0.752544\pi\)
\(338\) 1.79485e8 0.252824
\(339\) 7.63466e7 0.106437
\(340\) 1.30033e8 0.179423
\(341\) −4.27625e8 −0.584014
\(342\) −1.76494e8 −0.238582
\(343\) 1.41385e8 0.189179
\(344\) 7.46969e8 0.989346
\(345\) −8.22842e8 −1.07882
\(346\) 4.47103e7 0.0580285
\(347\) −2.60064e8 −0.334139 −0.167069 0.985945i \(-0.553430\pi\)
−0.167069 + 0.985945i \(0.553430\pi\)
\(348\) −3.97688e8 −0.505843
\(349\) −2.45544e8 −0.309201 −0.154600 0.987977i \(-0.549409\pi\)
−0.154600 + 0.987977i \(0.549409\pi\)
\(350\) 1.20694e9 1.50470
\(351\) 9.51231e7 0.117412
\(352\) −6.25028e8 −0.763836
\(353\) −8.28455e8 −1.00244 −0.501219 0.865320i \(-0.667115\pi\)
−0.501219 + 0.865320i \(0.667115\pi\)
\(354\) 2.52655e7 0.0302703
\(355\) −2.06893e9 −2.45441
\(356\) −8.85898e8 −1.04066
\(357\) 7.50003e7 0.0872418
\(358\) 1.77143e8 0.204048
\(359\) 8.66890e8 0.988857 0.494428 0.869218i \(-0.335377\pi\)
0.494428 + 0.869218i \(0.335377\pi\)
\(360\) −4.22278e8 −0.477024
\(361\) 1.92963e9 2.15873
\(362\) 2.62237e8 0.290546
\(363\) 1.91311e8 0.209927
\(364\) −6.41666e8 −0.697356
\(365\) 9.09863e8 0.979380
\(366\) 4.65281e7 0.0496056
\(367\) −1.45949e9 −1.54124 −0.770620 0.637295i \(-0.780053\pi\)
−0.770620 + 0.637295i \(0.780053\pi\)
\(368\) −4.98674e8 −0.521615
\(369\) 1.02471e8 0.106172
\(370\) −1.17382e9 −1.20475
\(371\) −1.30538e9 −1.32717
\(372\) −3.51599e8 −0.354118
\(373\) −1.35964e9 −1.35657 −0.678287 0.734798i \(-0.737277\pi\)
−0.678287 + 0.734798i \(0.737277\pi\)
\(374\) −3.59990e7 −0.0355828
\(375\) 1.98204e9 1.94090
\(376\) 8.39622e8 0.814566
\(377\) 6.63766e8 0.637999
\(378\) −1.11034e8 −0.105739
\(379\) 1.81549e9 1.71300 0.856500 0.516147i \(-0.172634\pi\)
0.856500 + 0.516147i \(0.172634\pi\)
\(380\) 3.07967e9 2.87913
\(381\) −1.12092e9 −1.03833
\(382\) −7.14002e8 −0.655357
\(383\) −5.43039e8 −0.493896 −0.246948 0.969029i \(-0.579428\pi\)
−0.246948 + 0.969029i \(0.579428\pi\)
\(384\) −6.53157e8 −0.588652
\(385\) 2.35638e9 2.10442
\(386\) −2.48289e8 −0.219737
\(387\) 5.08054e8 0.445576
\(388\) 1.00463e9 0.873161
\(389\) 1.90428e9 1.64024 0.820121 0.572190i \(-0.193906\pi\)
0.820121 + 0.572190i \(0.193906\pi\)
\(390\) 3.21305e8 0.274278
\(391\) −1.26516e8 −0.107035
\(392\) 7.60291e8 0.637498
\(393\) −1.25215e9 −1.04060
\(394\) −6.65977e8 −0.548558
\(395\) −3.00845e9 −2.45614
\(396\) −2.75312e8 −0.222788
\(397\) −2.75416e8 −0.220913 −0.110457 0.993881i \(-0.535231\pi\)
−0.110457 + 0.993881i \(0.535231\pi\)
\(398\) −2.30813e8 −0.183514
\(399\) 1.77629e9 1.39994
\(400\) 1.89207e9 1.47818
\(401\) 1.17696e9 0.911501 0.455751 0.890108i \(-0.349371\pi\)
0.455751 + 0.890108i \(0.349371\pi\)
\(402\) −1.38015e8 −0.105958
\(403\) 5.86840e8 0.446635
\(404\) −5.55314e8 −0.418990
\(405\) −2.87214e8 −0.214839
\(406\) −7.74790e8 −0.574570
\(407\) −1.67873e9 −1.23424
\(408\) −6.49274e7 −0.0473279
\(409\) −1.75837e9 −1.27080 −0.635402 0.772181i \(-0.719166\pi\)
−0.635402 + 0.772181i \(0.719166\pi\)
\(410\) 3.46126e8 0.248022
\(411\) 8.65127e7 0.0614658
\(412\) 1.75440e9 1.23591
\(413\) −2.54280e8 −0.177618
\(414\) 1.87300e8 0.129729
\(415\) 9.75787e8 0.670173
\(416\) 8.57741e8 0.584157
\(417\) −1.01140e9 −0.683039
\(418\) −8.52591e8 −0.570984
\(419\) 2.47153e9 1.64141 0.820705 0.571353i \(-0.193581\pi\)
0.820705 + 0.571353i \(0.193581\pi\)
\(420\) 1.93745e9 1.27602
\(421\) −1.28349e9 −0.838313 −0.419157 0.907914i \(-0.637674\pi\)
−0.419157 + 0.907914i \(0.637674\pi\)
\(422\) −1.53203e8 −0.0992367
\(423\) 5.71072e8 0.366859
\(424\) 1.13006e9 0.719980
\(425\) 4.80028e8 0.303323
\(426\) 4.70942e8 0.295144
\(427\) −4.68272e8 −0.291072
\(428\) 1.23628e9 0.762191
\(429\) 4.59512e8 0.280994
\(430\) 1.71610e9 1.04088
\(431\) 1.78192e9 1.07206 0.536030 0.844199i \(-0.319923\pi\)
0.536030 + 0.844199i \(0.319923\pi\)
\(432\) −1.74063e8 −0.103876
\(433\) 8.17222e8 0.483763 0.241881 0.970306i \(-0.422236\pi\)
0.241881 + 0.970306i \(0.422236\pi\)
\(434\) −6.84998e8 −0.402230
\(435\) −2.00417e9 −1.16741
\(436\) 1.79079e8 0.103476
\(437\) −2.99637e9 −1.71756
\(438\) −2.07108e8 −0.117771
\(439\) −2.53136e9 −1.42800 −0.714001 0.700145i \(-0.753119\pi\)
−0.714001 + 0.700145i \(0.753119\pi\)
\(440\) −2.03990e9 −1.14163
\(441\) 5.17115e8 0.287112
\(442\) 4.94022e7 0.0272125
\(443\) −7.10880e8 −0.388493 −0.194247 0.980953i \(-0.562226\pi\)
−0.194247 + 0.980953i \(0.562226\pi\)
\(444\) −1.38027e9 −0.748384
\(445\) −4.46453e9 −2.40169
\(446\) −3.95919e8 −0.211317
\(447\) −6.46734e8 −0.342491
\(448\) 4.00250e8 0.210309
\(449\) 1.22783e9 0.640139 0.320070 0.947394i \(-0.396294\pi\)
0.320070 + 0.947394i \(0.396294\pi\)
\(450\) −7.10655e8 −0.367633
\(451\) 4.95009e8 0.254095
\(452\) 3.03239e8 0.154454
\(453\) 1.77986e9 0.899585
\(454\) −1.66165e8 −0.0833382
\(455\) −3.23371e9 −1.60939
\(456\) −1.53772e9 −0.759454
\(457\) 3.07883e9 1.50897 0.754483 0.656319i \(-0.227888\pi\)
0.754483 + 0.656319i \(0.227888\pi\)
\(458\) −1.00031e9 −0.486524
\(459\) −4.41606e7 −0.0213153
\(460\) −3.26822e9 −1.56552
\(461\) 1.77409e9 0.843379 0.421689 0.906740i \(-0.361437\pi\)
0.421689 + 0.906740i \(0.361437\pi\)
\(462\) −5.36372e8 −0.253057
\(463\) 3.91361e9 1.83250 0.916249 0.400608i \(-0.131201\pi\)
0.916249 + 0.400608i \(0.131201\pi\)
\(464\) −1.21461e9 −0.564446
\(465\) −1.77190e9 −0.817250
\(466\) −5.46701e8 −0.250264
\(467\) −3.15577e9 −1.43383 −0.716914 0.697162i \(-0.754446\pi\)
−0.716914 + 0.697162i \(0.754446\pi\)
\(468\) 3.77817e8 0.170381
\(469\) 1.38902e9 0.621735
\(470\) 1.92896e9 0.856998
\(471\) 1.74162e9 0.768034
\(472\) 2.20129e8 0.0963561
\(473\) 2.45426e9 1.06637
\(474\) 6.84799e8 0.295351
\(475\) 1.13689e10 4.86732
\(476\) 2.97892e8 0.126600
\(477\) 7.68614e8 0.324260
\(478\) 1.01967e9 0.427036
\(479\) −3.08907e9 −1.28426 −0.642130 0.766596i \(-0.721949\pi\)
−0.642130 + 0.766596i \(0.721949\pi\)
\(480\) −2.58986e9 −1.06889
\(481\) 2.30376e9 0.943907
\(482\) 8.13870e8 0.331048
\(483\) −1.88504e9 −0.761213
\(484\) 7.59864e8 0.304633
\(485\) 5.06288e9 2.01512
\(486\) 6.53773e7 0.0258345
\(487\) −8.99853e8 −0.353037 −0.176518 0.984297i \(-0.556484\pi\)
−0.176518 + 0.984297i \(0.556484\pi\)
\(488\) 4.05381e8 0.157904
\(489\) 1.04870e9 0.405574
\(490\) 1.74670e9 0.670706
\(491\) −8.98595e8 −0.342593 −0.171297 0.985220i \(-0.554796\pi\)
−0.171297 + 0.985220i \(0.554796\pi\)
\(492\) 4.07003e8 0.154071
\(493\) −3.08151e8 −0.115824
\(494\) 1.17003e9 0.436669
\(495\) −1.38745e9 −0.514161
\(496\) −1.07384e9 −0.395143
\(497\) −4.73970e9 −1.73182
\(498\) −2.22114e8 −0.0805885
\(499\) −2.94063e8 −0.105947 −0.0529734 0.998596i \(-0.516870\pi\)
−0.0529734 + 0.998596i \(0.516870\pi\)
\(500\) 7.87239e9 2.81651
\(501\) −7.08819e8 −0.251827
\(502\) 1.57021e9 0.553982
\(503\) −3.01250e9 −1.05545 −0.527727 0.849414i \(-0.676956\pi\)
−0.527727 + 0.849414i \(0.676956\pi\)
\(504\) −9.67394e8 −0.336586
\(505\) −2.79853e9 −0.966965
\(506\) 9.04791e8 0.310471
\(507\) 1.06361e9 0.362456
\(508\) −4.45213e9 −1.50676
\(509\) 7.20984e8 0.242333 0.121167 0.992632i \(-0.461336\pi\)
0.121167 + 0.992632i \(0.461336\pi\)
\(510\) −1.49165e8 −0.0497933
\(511\) 2.08440e9 0.691047
\(512\) −2.78279e9 −0.916296
\(513\) −1.04589e9 −0.342038
\(514\) 7.54567e7 0.0245091
\(515\) 8.84138e9 2.85230
\(516\) 2.01792e9 0.646593
\(517\) 2.75868e9 0.877980
\(518\) −2.68909e9 −0.850064
\(519\) 2.64949e8 0.0831912
\(520\) 2.79941e9 0.873081
\(521\) −3.99688e9 −1.23820 −0.619098 0.785314i \(-0.712502\pi\)
−0.619098 + 0.785314i \(0.712502\pi\)
\(522\) 4.56201e8 0.140381
\(523\) 2.17294e9 0.664190 0.332095 0.943246i \(-0.392245\pi\)
0.332095 + 0.943246i \(0.392245\pi\)
\(524\) −4.97338e9 −1.51005
\(525\) 7.15225e9 2.15717
\(526\) −1.77374e9 −0.531423
\(527\) −2.72439e8 −0.0810834
\(528\) −8.40848e8 −0.248599
\(529\) −2.24997e8 −0.0660819
\(530\) 2.59621e9 0.757485
\(531\) 1.49721e8 0.0433963
\(532\) 7.05519e9 2.03151
\(533\) −6.79313e8 −0.194323
\(534\) 1.01624e9 0.288804
\(535\) 6.23030e9 1.75902
\(536\) −1.20247e9 −0.337286
\(537\) 1.04973e9 0.292529
\(538\) 1.22682e9 0.339658
\(539\) 2.49803e9 0.687127
\(540\) −1.14078e9 −0.311762
\(541\) −4.30836e9 −1.16983 −0.584914 0.811096i \(-0.698872\pi\)
−0.584914 + 0.811096i \(0.698872\pi\)
\(542\) 2.25655e9 0.608762
\(543\) 1.55400e9 0.416534
\(544\) −3.98204e8 −0.106050
\(545\) 9.02476e8 0.238808
\(546\) 7.36075e8 0.193530
\(547\) 7.88729e8 0.206050 0.103025 0.994679i \(-0.467148\pi\)
0.103025 + 0.994679i \(0.467148\pi\)
\(548\) 3.43617e8 0.0891956
\(549\) 2.75721e8 0.0711160
\(550\) −3.43297e9 −0.879833
\(551\) −7.29818e9 −1.85859
\(552\) 1.63187e9 0.412952
\(553\) −6.89203e9 −1.73304
\(554\) −1.44418e9 −0.360858
\(555\) −6.95596e9 −1.72716
\(556\) −4.01714e9 −0.991186
\(557\) 5.29917e9 1.29932 0.649658 0.760226i \(-0.274912\pi\)
0.649658 + 0.760226i \(0.274912\pi\)
\(558\) 4.03330e8 0.0982746
\(559\) −3.36804e9 −0.815522
\(560\) 5.91728e9 1.42385
\(561\) −2.13327e8 −0.0510124
\(562\) 2.05276e9 0.487821
\(563\) −5.56156e9 −1.31346 −0.656731 0.754125i \(-0.728061\pi\)
−0.656731 + 0.754125i \(0.728061\pi\)
\(564\) 2.26822e9 0.532365
\(565\) 1.52819e9 0.356457
\(566\) −2.13037e9 −0.493853
\(567\) −6.57977e8 −0.151590
\(568\) 4.10314e9 0.939500
\(569\) 3.31033e9 0.753319 0.376659 0.926352i \(-0.377073\pi\)
0.376659 + 0.926352i \(0.377073\pi\)
\(570\) −3.53278e9 −0.799015
\(571\) −3.86639e8 −0.0869118 −0.0434559 0.999055i \(-0.513837\pi\)
−0.0434559 + 0.999055i \(0.513837\pi\)
\(572\) 1.82512e9 0.407761
\(573\) −4.23112e9 −0.939537
\(574\) 7.92938e8 0.175004
\(575\) −1.20649e10 −2.64659
\(576\) −2.35670e8 −0.0513836
\(577\) 3.94872e9 0.855739 0.427870 0.903840i \(-0.359264\pi\)
0.427870 + 0.903840i \(0.359264\pi\)
\(578\) 1.84667e9 0.397780
\(579\) −1.47134e9 −0.315020
\(580\) −7.96031e9 −1.69407
\(581\) 2.23542e9 0.472871
\(582\) −1.15244e9 −0.242319
\(583\) 3.71295e9 0.776031
\(584\) −1.80445e9 −0.374887
\(585\) 1.90403e9 0.393213
\(586\) 7.31315e7 0.0150128
\(587\) −1.13309e9 −0.231223 −0.115612 0.993294i \(-0.536883\pi\)
−0.115612 + 0.993294i \(0.536883\pi\)
\(588\) 2.05391e9 0.416640
\(589\) −6.45237e9 −1.30112
\(590\) 5.05726e8 0.101376
\(591\) −3.94652e9 −0.786427
\(592\) −4.21558e9 −0.835086
\(593\) 6.61359e9 1.30240 0.651202 0.758904i \(-0.274265\pi\)
0.651202 + 0.758904i \(0.274265\pi\)
\(594\) 3.15819e8 0.0618280
\(595\) 1.50124e9 0.292174
\(596\) −2.56874e9 −0.497003
\(597\) −1.36778e9 −0.263091
\(598\) −1.24167e9 −0.237438
\(599\) −2.36330e9 −0.449289 −0.224644 0.974441i \(-0.572122\pi\)
−0.224644 + 0.974441i \(0.572122\pi\)
\(600\) −6.19166e9 −1.17025
\(601\) −4.40437e9 −0.827605 −0.413803 0.910367i \(-0.635800\pi\)
−0.413803 + 0.910367i \(0.635800\pi\)
\(602\) 3.93139e9 0.734443
\(603\) −8.17866e8 −0.151905
\(604\) 7.06938e9 1.30543
\(605\) 3.82937e9 0.703047
\(606\) 6.37018e8 0.116278
\(607\) −2.71725e9 −0.493138 −0.246569 0.969125i \(-0.579303\pi\)
−0.246569 + 0.969125i \(0.579303\pi\)
\(608\) −9.43095e9 −1.70174
\(609\) −4.59134e9 −0.823719
\(610\) 9.31327e8 0.166130
\(611\) −3.78580e9 −0.671450
\(612\) −1.75400e8 −0.0309314
\(613\) −1.02118e10 −1.79056 −0.895282 0.445500i \(-0.853026\pi\)
−0.895282 + 0.445500i \(0.853026\pi\)
\(614\) 2.98965e8 0.0521233
\(615\) 2.05112e9 0.355572
\(616\) −4.67320e9 −0.805530
\(617\) 3.23817e9 0.555011 0.277506 0.960724i \(-0.410492\pi\)
0.277506 + 0.960724i \(0.410492\pi\)
\(618\) −2.01252e9 −0.342990
\(619\) −1.72870e9 −0.292956 −0.146478 0.989214i \(-0.546794\pi\)
−0.146478 + 0.989214i \(0.546794\pi\)
\(620\) −7.03777e9 −1.18594
\(621\) 1.10992e9 0.185983
\(622\) −2.02608e9 −0.337590
\(623\) −1.02278e10 −1.69462
\(624\) 1.15391e9 0.190120
\(625\) 2.29581e10 3.76145
\(626\) −1.34299e9 −0.218807
\(627\) −5.05238e9 −0.818578
\(628\) 6.91750e9 1.11453
\(629\) −1.06951e9 −0.171360
\(630\) −2.22250e9 −0.354120
\(631\) 7.57564e9 1.20037 0.600187 0.799859i \(-0.295093\pi\)
0.600187 + 0.799859i \(0.295093\pi\)
\(632\) 5.96639e9 0.940160
\(633\) −9.07866e8 −0.142268
\(634\) 1.81066e9 0.282179
\(635\) −2.24368e10 −3.47738
\(636\) 3.05283e9 0.470547
\(637\) −3.42811e9 −0.525492
\(638\) 2.20377e9 0.335965
\(639\) 2.79076e9 0.423126
\(640\) −1.30739e10 −1.97140
\(641\) −5.56603e9 −0.834723 −0.417362 0.908740i \(-0.637045\pi\)
−0.417362 + 0.908740i \(0.637045\pi\)
\(642\) −1.41817e9 −0.211523
\(643\) −7.76877e9 −1.15243 −0.576214 0.817299i \(-0.695471\pi\)
−0.576214 + 0.817299i \(0.695471\pi\)
\(644\) −7.48715e9 −1.10463
\(645\) 1.01694e10 1.49224
\(646\) −5.43183e8 −0.0792743
\(647\) 6.50327e9 0.943989 0.471994 0.881602i \(-0.343534\pi\)
0.471994 + 0.881602i \(0.343534\pi\)
\(648\) 5.69607e8 0.0822362
\(649\) 7.23260e8 0.103858
\(650\) 4.71114e9 0.672867
\(651\) −4.05924e9 −0.576648
\(652\) 4.16530e9 0.588545
\(653\) −9.96747e9 −1.40084 −0.700420 0.713731i \(-0.747004\pi\)
−0.700420 + 0.713731i \(0.747004\pi\)
\(654\) −2.05427e8 −0.0287167
\(655\) −2.50636e10 −3.48497
\(656\) 1.24306e9 0.171920
\(657\) −1.22731e9 −0.168839
\(658\) 4.41903e9 0.604695
\(659\) 1.28708e10 1.75189 0.875946 0.482410i \(-0.160238\pi\)
0.875946 + 0.482410i \(0.160238\pi\)
\(660\) −5.51077e9 −0.746119
\(661\) −4.36910e9 −0.588420 −0.294210 0.955741i \(-0.595056\pi\)
−0.294210 + 0.955741i \(0.595056\pi\)
\(662\) 5.25560e9 0.704075
\(663\) 2.92754e8 0.0390126
\(664\) −1.93519e9 −0.256529
\(665\) 3.55550e10 4.68841
\(666\) 1.58335e9 0.207691
\(667\) 7.74501e9 1.01061
\(668\) −2.81534e9 −0.365437
\(669\) −2.34618e9 −0.302950
\(670\) −2.76257e9 −0.354856
\(671\) 1.33193e9 0.170197
\(672\) −5.93309e9 −0.754203
\(673\) −2.50718e9 −0.317053 −0.158527 0.987355i \(-0.550674\pi\)
−0.158527 + 0.987355i \(0.550674\pi\)
\(674\) 4.56321e9 0.574065
\(675\) −4.21128e9 −0.527049
\(676\) 4.22453e9 0.525974
\(677\) 4.46850e9 0.553479 0.276739 0.960945i \(-0.410746\pi\)
0.276739 + 0.960945i \(0.410746\pi\)
\(678\) −3.47855e8 −0.0428641
\(679\) 1.15985e10 1.42186
\(680\) −1.29962e9 −0.158502
\(681\) −9.84680e8 −0.119476
\(682\) 1.94837e9 0.235194
\(683\) −3.36244e9 −0.403815 −0.201908 0.979405i \(-0.564714\pi\)
−0.201908 + 0.979405i \(0.564714\pi\)
\(684\) −4.15413e9 −0.496346
\(685\) 1.73168e9 0.205850
\(686\) −6.44185e8 −0.0761862
\(687\) −5.92773e9 −0.697493
\(688\) 6.16308e9 0.721503
\(689\) −5.09536e9 −0.593483
\(690\) 3.74908e9 0.434463
\(691\) 1.73704e9 0.200279 0.100140 0.994973i \(-0.468071\pi\)
0.100140 + 0.994973i \(0.468071\pi\)
\(692\) 1.05235e9 0.120722
\(693\) −3.17849e9 −0.362790
\(694\) 1.18492e9 0.134564
\(695\) −2.02446e10 −2.28751
\(696\) 3.97470e9 0.446860
\(697\) 3.15369e8 0.0352780
\(698\) 1.11876e9 0.124521
\(699\) −3.23971e9 −0.358786
\(700\) 2.84078e10 3.13036
\(701\) 7.43794e9 0.815529 0.407764 0.913087i \(-0.366308\pi\)
0.407764 + 0.913087i \(0.366308\pi\)
\(702\) −4.33405e8 −0.0472840
\(703\) −2.53301e10 −2.74975
\(704\) −1.13845e9 −0.122973
\(705\) 1.14308e10 1.22862
\(706\) 3.77465e9 0.403702
\(707\) −6.41114e9 −0.682287
\(708\) 5.94674e8 0.0629742
\(709\) −3.13976e9 −0.330853 −0.165426 0.986222i \(-0.552900\pi\)
−0.165426 + 0.986222i \(0.552900\pi\)
\(710\) 9.42659e9 0.988441
\(711\) 4.05806e9 0.423424
\(712\) 8.85412e9 0.919317
\(713\) 6.84742e9 0.707479
\(714\) −3.41721e8 −0.0351340
\(715\) 9.19780e9 0.941051
\(716\) 4.16940e9 0.424501
\(717\) 6.04251e9 0.612210
\(718\) −3.94977e9 −0.398232
\(719\) −1.62139e10 −1.62680 −0.813402 0.581701i \(-0.802387\pi\)
−0.813402 + 0.581701i \(0.802387\pi\)
\(720\) −3.48413e9 −0.347881
\(721\) 2.02546e10 2.01257
\(722\) −8.79190e9 −0.869365
\(723\) 4.82293e9 0.474599
\(724\) 6.17227e9 0.604449
\(725\) −2.93862e10 −2.86392
\(726\) −8.71663e8 −0.0845416
\(727\) 1.44572e10 1.39545 0.697725 0.716366i \(-0.254196\pi\)
0.697725 + 0.716366i \(0.254196\pi\)
\(728\) 6.41314e9 0.616043
\(729\) 3.87420e8 0.0370370
\(730\) −4.14557e9 −0.394416
\(731\) 1.56360e9 0.148052
\(732\) 1.09513e9 0.103199
\(733\) 8.79960e9 0.825275 0.412638 0.910895i \(-0.364608\pi\)
0.412638 + 0.910895i \(0.364608\pi\)
\(734\) 6.64982e9 0.620688
\(735\) 1.03508e10 0.961543
\(736\) 1.00084e10 0.925318
\(737\) −3.95087e9 −0.363544
\(738\) −4.66886e8 −0.0427576
\(739\) 1.06478e10 0.970523 0.485262 0.874369i \(-0.338724\pi\)
0.485262 + 0.874369i \(0.338724\pi\)
\(740\) −2.76282e10 −2.50635
\(741\) 6.93350e9 0.626021
\(742\) 5.94764e9 0.534479
\(743\) −6.58260e9 −0.588758 −0.294379 0.955689i \(-0.595113\pi\)
−0.294379 + 0.955689i \(0.595113\pi\)
\(744\) 3.51406e9 0.312827
\(745\) −1.29453e10 −1.14701
\(746\) 6.19487e9 0.546319
\(747\) −1.31623e9 −0.115534
\(748\) −8.47307e8 −0.0740262
\(749\) 1.42729e10 1.24116
\(750\) −9.03066e9 −0.781637
\(751\) −8.17849e9 −0.704585 −0.352293 0.935890i \(-0.614598\pi\)
−0.352293 + 0.935890i \(0.614598\pi\)
\(752\) 6.92753e9 0.594040
\(753\) 9.30495e9 0.794203
\(754\) −3.02429e9 −0.256935
\(755\) 3.56265e10 3.01272
\(756\) −2.61340e9 −0.219978
\(757\) 1.76129e10 1.47569 0.737844 0.674972i \(-0.235844\pi\)
0.737844 + 0.674972i \(0.235844\pi\)
\(758\) −8.27185e9 −0.689860
\(759\) 5.36172e9 0.445100
\(760\) −3.07798e10 −2.54342
\(761\) −1.93708e10 −1.59331 −0.796657 0.604431i \(-0.793400\pi\)
−0.796657 + 0.604431i \(0.793400\pi\)
\(762\) 5.10718e9 0.418156
\(763\) 2.06748e9 0.168502
\(764\) −1.68055e10 −1.36340
\(765\) −8.83939e8 −0.0713851
\(766\) 2.47423e9 0.198902
\(767\) −9.92546e8 −0.0794268
\(768\) 1.85870e9 0.148063
\(769\) 9.07869e9 0.719915 0.359957 0.932969i \(-0.382791\pi\)
0.359957 + 0.932969i \(0.382791\pi\)
\(770\) −1.07363e10 −0.847492
\(771\) 4.47150e8 0.0351369
\(772\) −5.84398e9 −0.457139
\(773\) −1.79639e10 −1.39886 −0.699428 0.714703i \(-0.746562\pi\)
−0.699428 + 0.714703i \(0.746562\pi\)
\(774\) −2.31482e9 −0.179442
\(775\) −2.59805e10 −2.00490
\(776\) −1.00408e10 −0.771348
\(777\) −1.59353e10 −1.21867
\(778\) −8.67641e9 −0.660558
\(779\) 7.46912e9 0.566093
\(780\) 7.56256e9 0.570607
\(781\) 1.34814e10 1.01264
\(782\) 5.76440e8 0.0431052
\(783\) 2.70341e9 0.201254
\(784\) 6.27299e9 0.464909
\(785\) 3.48611e10 2.57216
\(786\) 5.70512e9 0.419069
\(787\) −9.95896e9 −0.728287 −0.364143 0.931343i \(-0.618638\pi\)
−0.364143 + 0.931343i \(0.618638\pi\)
\(788\) −1.56751e10 −1.14122
\(789\) −1.05110e10 −0.761862
\(790\) 1.37073e10 0.989135
\(791\) 3.50092e9 0.251515
\(792\) 2.75161e9 0.196810
\(793\) −1.82784e9 −0.130161
\(794\) 1.25486e9 0.0889662
\(795\) 1.53849e10 1.08595
\(796\) −5.43264e9 −0.381782
\(797\) 2.78445e10 1.94821 0.974105 0.226095i \(-0.0725959\pi\)
0.974105 + 0.226095i \(0.0725959\pi\)
\(798\) −8.09323e9 −0.563783
\(799\) 1.75755e9 0.121897
\(800\) −3.79738e10 −2.62222
\(801\) 6.02216e9 0.414037
\(802\) −5.36254e9 −0.367080
\(803\) −5.92875e9 −0.404072
\(804\) −3.24846e9 −0.220435
\(805\) −3.77319e10 −2.54931
\(806\) −2.67380e9 −0.179869
\(807\) 7.27002e9 0.486943
\(808\) 5.55009e9 0.370135
\(809\) 6.81852e9 0.452762 0.226381 0.974039i \(-0.427311\pi\)
0.226381 + 0.974039i \(0.427311\pi\)
\(810\) 1.30862e9 0.0865201
\(811\) 3.94141e9 0.259465 0.129732 0.991549i \(-0.458588\pi\)
0.129732 + 0.991549i \(0.458588\pi\)
\(812\) −1.82362e10 −1.19533
\(813\) 1.33721e10 0.872738
\(814\) 7.64872e9 0.497053
\(815\) 2.09912e10 1.35827
\(816\) −5.35701e8 −0.0345149
\(817\) 3.70319e10 2.37574
\(818\) 8.01159e9 0.511778
\(819\) 4.36192e9 0.277450
\(820\) 8.14676e9 0.515985
\(821\) 2.49119e10 1.57111 0.785553 0.618795i \(-0.212379\pi\)
0.785553 + 0.618795i \(0.212379\pi\)
\(822\) −3.94174e8 −0.0247535
\(823\) −2.21681e9 −0.138621 −0.0693105 0.997595i \(-0.522080\pi\)
−0.0693105 + 0.997595i \(0.522080\pi\)
\(824\) −1.75343e10 −1.09180
\(825\) −2.03435e10 −1.26135
\(826\) 1.15856e9 0.0715302
\(827\) −1.70597e10 −1.04882 −0.524410 0.851466i \(-0.675714\pi\)
−0.524410 + 0.851466i \(0.675714\pi\)
\(828\) 4.40847e9 0.269887
\(829\) −5.39778e9 −0.329060 −0.164530 0.986372i \(-0.552611\pi\)
−0.164530 + 0.986372i \(0.552611\pi\)
\(830\) −4.44594e9 −0.269892
\(831\) −8.55806e9 −0.517335
\(832\) 1.56232e9 0.0940457
\(833\) 1.59149e9 0.0953994
\(834\) 4.60818e9 0.275073
\(835\) −1.41880e10 −0.843373
\(836\) −2.00674e10 −1.18787
\(837\) 2.39010e9 0.140889
\(838\) −1.12609e10 −0.661028
\(839\) 1.48286e10 0.866830 0.433415 0.901194i \(-0.357308\pi\)
0.433415 + 0.901194i \(0.357308\pi\)
\(840\) −1.93638e10 −1.12723
\(841\) 1.61441e9 0.0935895
\(842\) 5.84793e9 0.337606
\(843\) 1.21645e10 0.699353
\(844\) −3.60593e9 −0.206452
\(845\) 2.12897e10 1.21387
\(846\) −2.60195e9 −0.147742
\(847\) 8.77268e9 0.496067
\(848\) 9.32386e9 0.525062
\(849\) −1.26244e10 −0.708001
\(850\) −2.18713e9 −0.122154
\(851\) 2.68809e10 1.49517
\(852\) 1.10846e10 0.614016
\(853\) 1.65358e10 0.912227 0.456114 0.889922i \(-0.349241\pi\)
0.456114 + 0.889922i \(0.349241\pi\)
\(854\) 2.13357e9 0.117221
\(855\) −2.09350e10 −1.14549
\(856\) −1.23560e10 −0.673318
\(857\) 1.32332e10 0.718176 0.359088 0.933304i \(-0.383088\pi\)
0.359088 + 0.933304i \(0.383088\pi\)
\(858\) −2.09365e9 −0.113162
\(859\) −4.23318e9 −0.227872 −0.113936 0.993488i \(-0.536346\pi\)
−0.113936 + 0.993488i \(0.536346\pi\)
\(860\) 4.03917e10 2.16545
\(861\) 4.69888e9 0.250890
\(862\) −8.11891e9 −0.431740
\(863\) 3.99379e8 0.0211518 0.0105759 0.999944i \(-0.496634\pi\)
0.0105759 + 0.999944i \(0.496634\pi\)
\(864\) 3.49344e9 0.184270
\(865\) 5.30335e9 0.278608
\(866\) −3.72347e9 −0.194821
\(867\) 1.09432e10 0.570268
\(868\) −1.61228e10 −0.836798
\(869\) 1.96033e10 1.01335
\(870\) 9.13152e9 0.470139
\(871\) 5.42187e9 0.278026
\(872\) −1.78980e9 −0.0914108
\(873\) −6.82926e9 −0.347395
\(874\) 1.36522e10 0.691694
\(875\) 9.08873e10 4.58643
\(876\) −4.87470e9 −0.245010
\(877\) 9.41245e9 0.471199 0.235599 0.971850i \(-0.424295\pi\)
0.235599 + 0.971850i \(0.424295\pi\)
\(878\) 1.15335e10 0.575085
\(879\) 4.33371e8 0.0215228
\(880\) −1.68308e10 −0.832560
\(881\) −6.45929e8 −0.0318250 −0.0159125 0.999873i \(-0.505065\pi\)
−0.0159125 + 0.999873i \(0.505065\pi\)
\(882\) −2.35611e9 −0.115626
\(883\) 3.76342e10 1.83959 0.919793 0.392403i \(-0.128356\pi\)
0.919793 + 0.392403i \(0.128356\pi\)
\(884\) 1.16278e9 0.0566128
\(885\) 2.99689e9 0.145335
\(886\) 3.23895e9 0.156454
\(887\) −1.95731e10 −0.941732 −0.470866 0.882205i \(-0.656059\pi\)
−0.470866 + 0.882205i \(0.656059\pi\)
\(888\) 1.37951e10 0.661121
\(889\) −5.14002e10 −2.45363
\(890\) 2.03416e10 0.967207
\(891\) 1.87152e9 0.0886383
\(892\) −9.31874e9 −0.439623
\(893\) 4.16253e10 1.95604
\(894\) 2.94668e9 0.137928
\(895\) 2.10119e10 0.979683
\(896\) −2.99509e10 −1.39101
\(897\) −7.35801e9 −0.340398
\(898\) −5.59429e9 −0.257797
\(899\) 1.66781e10 0.765573
\(900\) −1.67267e10 −0.764823
\(901\) 2.36551e9 0.107743
\(902\) −2.25539e9 −0.102329
\(903\) 2.32971e10 1.05292
\(904\) −3.03072e9 −0.136445
\(905\) 3.11055e10 1.39498
\(906\) −8.10950e9 −0.362281
\(907\) −2.68155e10 −1.19333 −0.596665 0.802491i \(-0.703508\pi\)
−0.596665 + 0.802491i \(0.703508\pi\)
\(908\) −3.91102e9 −0.173376
\(909\) 3.77491e9 0.166699
\(910\) 1.47336e10 0.648134
\(911\) −3.43001e10 −1.50307 −0.751537 0.659690i \(-0.770687\pi\)
−0.751537 + 0.659690i \(0.770687\pi\)
\(912\) −1.26874e10 −0.553849
\(913\) −6.35832e9 −0.276500
\(914\) −1.40280e10 −0.607691
\(915\) 5.51897e9 0.238168
\(916\) −2.35442e10 −1.01216
\(917\) −5.74180e10 −2.45898
\(918\) 2.01207e8 0.00858408
\(919\) 1.59456e10 0.677698 0.338849 0.940841i \(-0.389962\pi\)
0.338849 + 0.940841i \(0.389962\pi\)
\(920\) 3.26643e10 1.38298
\(921\) 1.77164e9 0.0747253
\(922\) −8.08321e9 −0.339645
\(923\) −1.85008e10 −0.774434
\(924\) −1.26246e10 −0.526459
\(925\) −1.01992e11 −4.23710
\(926\) −1.78314e10 −0.737984
\(927\) −1.19260e10 −0.491720
\(928\) 2.43771e10 1.00130
\(929\) 1.42285e9 0.0582242 0.0291121 0.999576i \(-0.490732\pi\)
0.0291121 + 0.999576i \(0.490732\pi\)
\(930\) 8.07325e9 0.329123
\(931\) 3.76924e10 1.53084
\(932\) −1.28677e10 −0.520649
\(933\) −1.20064e10 −0.483978
\(934\) 1.43785e10 0.577431
\(935\) −4.27005e9 −0.170841
\(936\) −3.77609e9 −0.150514
\(937\) −2.31807e10 −0.920532 −0.460266 0.887781i \(-0.652246\pi\)
−0.460266 + 0.887781i \(0.652246\pi\)
\(938\) −6.32875e9 −0.250385
\(939\) −7.95843e9 −0.313688
\(940\) 4.54018e10 1.78290
\(941\) 3.37421e10 1.32010 0.660052 0.751220i \(-0.270534\pi\)
0.660052 + 0.751220i \(0.270534\pi\)
\(942\) −7.93527e9 −0.309303
\(943\) −7.92642e9 −0.307812
\(944\) 1.81623e9 0.0702699
\(945\) −1.31704e10 −0.507676
\(946\) −1.11822e10 −0.429447
\(947\) 2.26076e10 0.865029 0.432514 0.901627i \(-0.357626\pi\)
0.432514 + 0.901627i \(0.357626\pi\)
\(948\) 1.61181e10 0.614448
\(949\) 8.13617e9 0.309021
\(950\) −5.17995e10 −1.96017
\(951\) 1.07298e10 0.404539
\(952\) −2.97728e9 −0.111838
\(953\) 1.78597e10 0.668420 0.334210 0.942499i \(-0.391531\pi\)
0.334210 + 0.942499i \(0.391531\pi\)
\(954\) −3.50200e9 −0.130586
\(955\) −8.46920e10 −3.14652
\(956\) 2.40001e10 0.888403
\(957\) 1.30594e10 0.481649
\(958\) 1.40746e10 0.517197
\(959\) 3.96709e9 0.145247
\(960\) −4.71727e9 −0.172084
\(961\) −1.27674e10 −0.464057
\(962\) −1.04965e10 −0.380130
\(963\) −8.40399e9 −0.303245
\(964\) 1.91561e10 0.688710
\(965\) −2.94511e10 −1.05501
\(966\) 8.58874e9 0.306556
\(967\) −1.61338e8 −0.00573778 −0.00286889 0.999996i \(-0.500913\pi\)
−0.00286889 + 0.999996i \(0.500913\pi\)
\(968\) −7.59446e9 −0.269112
\(969\) −3.21886e9 −0.113650
\(970\) −2.30678e10 −0.811530
\(971\) −4.94599e10 −1.73375 −0.866874 0.498527i \(-0.833875\pi\)
−0.866874 + 0.498527i \(0.833875\pi\)
\(972\) 1.53878e9 0.0537460
\(973\) −4.63782e10 −1.61406
\(974\) 4.09996e9 0.142175
\(975\) 2.79178e10 0.964640
\(976\) 3.34471e9 0.115155
\(977\) −1.43208e10 −0.491290 −0.245645 0.969360i \(-0.579000\pi\)
−0.245645 + 0.969360i \(0.579000\pi\)
\(978\) −4.77814e9 −0.163333
\(979\) 2.90913e10 0.990886
\(980\) 4.11121e10 1.39533
\(981\) −1.21734e9 −0.0411690
\(982\) 4.09423e9 0.137969
\(983\) −5.16305e10 −1.73368 −0.866840 0.498587i \(-0.833852\pi\)
−0.866840 + 0.498587i \(0.833852\pi\)
\(984\) −4.06780e9 −0.136106
\(985\) −7.89955e10 −2.63375
\(986\) 1.40402e9 0.0466448
\(987\) 2.61868e10 0.866907
\(988\) 2.75390e10 0.908445
\(989\) −3.92992e10 −1.29181
\(990\) 6.32157e9 0.207063
\(991\) −4.70999e10 −1.53731 −0.768656 0.639662i \(-0.779074\pi\)
−0.768656 + 0.639662i \(0.779074\pi\)
\(992\) 2.15520e10 0.700964
\(993\) 3.11442e10 1.00938
\(994\) 2.15953e10 0.697440
\(995\) −2.73781e10 −0.881094
\(996\) −5.22789e9 −0.167656
\(997\) −5.20068e10 −1.66198 −0.830992 0.556284i \(-0.812227\pi\)
−0.830992 + 0.556284i \(0.812227\pi\)
\(998\) 1.33982e9 0.0426669
\(999\) 9.38282e9 0.297752
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 177.8.a.a.1.7 16
3.2 odd 2 531.8.a.b.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.7 16 1.1 even 1 trivial
531.8.a.b.1.10 16 3.2 odd 2