Properties

Label 177.8.a.a.1.5
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 1.5
Root \(14.7989\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-14.7989 q^{2} -27.0000 q^{3} +91.0084 q^{4} -296.536 q^{5} +399.571 q^{6} -1410.76 q^{7} +547.437 q^{8} +729.000 q^{9} +4388.42 q^{10} -7538.45 q^{11} -2457.23 q^{12} +7066.47 q^{13} +20877.7 q^{14} +8006.47 q^{15} -19750.5 q^{16} +14832.8 q^{17} -10788.4 q^{18} -24023.8 q^{19} -26987.3 q^{20} +38090.5 q^{21} +111561. q^{22} +41590.5 q^{23} -14780.8 q^{24} +9808.63 q^{25} -104576. q^{26} -19683.0 q^{27} -128391. q^{28} +15791.6 q^{29} -118487. q^{30} +108246. q^{31} +222215. q^{32} +203538. q^{33} -219509. q^{34} +418341. q^{35} +66345.1 q^{36} +331221. q^{37} +355527. q^{38} -190795. q^{39} -162335. q^{40} +224396. q^{41} -563699. q^{42} -134226. q^{43} -686063. q^{44} -216175. q^{45} -615494. q^{46} +816259. q^{47} +533265. q^{48} +1.16670e6 q^{49} -145157. q^{50} -400485. q^{51} +643108. q^{52} +1.21786e6 q^{53} +291287. q^{54} +2.23542e6 q^{55} -772301. q^{56} +648644. q^{57} -233699. q^{58} +205379. q^{59} +728656. q^{60} +1.88156e6 q^{61} -1.60193e6 q^{62} -1.02844e6 q^{63} -760476. q^{64} -2.09546e6 q^{65} -3.01215e6 q^{66} -2.09166e6 q^{67} +1.34991e6 q^{68} -1.12294e6 q^{69} -6.19100e6 q^{70} -1.80830e6 q^{71} +399081. q^{72} -4.93066e6 q^{73} -4.90172e6 q^{74} -264833. q^{75} -2.18637e6 q^{76} +1.06349e7 q^{77} +2.82356e6 q^{78} -8.26089e6 q^{79} +5.85675e6 q^{80} +531441. q^{81} -3.32083e6 q^{82} -2.95933e6 q^{83} +3.46655e6 q^{84} -4.39846e6 q^{85} +1.98640e6 q^{86} -426373. q^{87} -4.12683e6 q^{88} +3.82628e6 q^{89} +3.19916e6 q^{90} -9.96908e6 q^{91} +3.78508e6 q^{92} -2.92265e6 q^{93} -1.20798e7 q^{94} +7.12393e6 q^{95} -5.99981e6 q^{96} -6.24098e6 q^{97} -1.72659e7 q^{98} -5.49553e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.7989 −1.30805 −0.654027 0.756472i \(-0.726922\pi\)
−0.654027 + 0.756472i \(0.726922\pi\)
\(3\) −27.0000 −0.577350
\(4\) 91.0084 0.711003
\(5\) −296.536 −1.06092 −0.530460 0.847710i \(-0.677981\pi\)
−0.530460 + 0.847710i \(0.677981\pi\)
\(6\) 399.571 0.755205
\(7\) −1410.76 −1.55457 −0.777284 0.629150i \(-0.783403\pi\)
−0.777284 + 0.629150i \(0.783403\pi\)
\(8\) 547.437 0.378024
\(9\) 729.000 0.333333
\(10\) 4388.42 1.38774
\(11\) −7538.45 −1.70769 −0.853843 0.520531i \(-0.825734\pi\)
−0.853843 + 0.520531i \(0.825734\pi\)
\(12\) −2457.23 −0.410498
\(13\) 7066.47 0.892073 0.446037 0.895015i \(-0.352835\pi\)
0.446037 + 0.895015i \(0.352835\pi\)
\(14\) 20877.7 2.03346
\(15\) 8006.47 0.612522
\(16\) −19750.5 −1.20548
\(17\) 14832.8 0.732237 0.366118 0.930568i \(-0.380687\pi\)
0.366118 + 0.930568i \(0.380687\pi\)
\(18\) −10788.4 −0.436018
\(19\) −24023.8 −0.803535 −0.401767 0.915742i \(-0.631604\pi\)
−0.401767 + 0.915742i \(0.631604\pi\)
\(20\) −26987.3 −0.754317
\(21\) 38090.5 0.897530
\(22\) 111561. 2.23374
\(23\) 41590.5 0.712765 0.356382 0.934340i \(-0.384010\pi\)
0.356382 + 0.934340i \(0.384010\pi\)
\(24\) −14780.8 −0.218252
\(25\) 9808.63 0.125550
\(26\) −104576. −1.16688
\(27\) −19683.0 −0.192450
\(28\) −128391. −1.10530
\(29\) 15791.6 0.120235 0.0601177 0.998191i \(-0.480852\pi\)
0.0601177 + 0.998191i \(0.480852\pi\)
\(30\) −118487. −0.801212
\(31\) 108246. 0.652601 0.326300 0.945266i \(-0.394198\pi\)
0.326300 + 0.945266i \(0.394198\pi\)
\(32\) 222215. 1.19881
\(33\) 203538. 0.985933
\(34\) −219509. −0.957805
\(35\) 418341. 1.64927
\(36\) 66345.1 0.237001
\(37\) 331221. 1.07501 0.537504 0.843261i \(-0.319367\pi\)
0.537504 + 0.843261i \(0.319367\pi\)
\(38\) 355527. 1.05107
\(39\) −190795. −0.515039
\(40\) −162335. −0.401053
\(41\) 224396. 0.508478 0.254239 0.967141i \(-0.418175\pi\)
0.254239 + 0.967141i \(0.418175\pi\)
\(42\) −563699. −1.17402
\(43\) −134226. −0.257453 −0.128726 0.991680i \(-0.541089\pi\)
−0.128726 + 0.991680i \(0.541089\pi\)
\(44\) −686063. −1.21417
\(45\) −216175. −0.353640
\(46\) −615494. −0.932334
\(47\) 816259. 1.14679 0.573397 0.819278i \(-0.305625\pi\)
0.573397 + 0.819278i \(0.305625\pi\)
\(48\) 533265. 0.695983
\(49\) 1.16670e6 1.41668
\(50\) −145157. −0.164227
\(51\) −400485. −0.422757
\(52\) 643108. 0.634267
\(53\) 1.21786e6 1.12365 0.561825 0.827256i \(-0.310100\pi\)
0.561825 + 0.827256i \(0.310100\pi\)
\(54\) 291287. 0.251735
\(55\) 2.23542e6 1.81172
\(56\) −772301. −0.587663
\(57\) 648644. 0.463921
\(58\) −233699. −0.157274
\(59\) 205379. 0.130189
\(60\) 728656. 0.435505
\(61\) 1.88156e6 1.06136 0.530681 0.847571i \(-0.321936\pi\)
0.530681 + 0.847571i \(0.321936\pi\)
\(62\) −1.60193e6 −0.853637
\(63\) −1.02844e6 −0.518189
\(64\) −760476. −0.362623
\(65\) −2.09546e6 −0.946418
\(66\) −3.01215e6 −1.28965
\(67\) −2.09166e6 −0.849627 −0.424814 0.905281i \(-0.639660\pi\)
−0.424814 + 0.905281i \(0.639660\pi\)
\(68\) 1.34991e6 0.520622
\(69\) −1.12294e6 −0.411515
\(70\) −6.19100e6 −2.15734
\(71\) −1.80830e6 −0.599605 −0.299802 0.954001i \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(72\) 399081. 0.126008
\(73\) −4.93066e6 −1.48346 −0.741729 0.670699i \(-0.765994\pi\)
−0.741729 + 0.670699i \(0.765994\pi\)
\(74\) −4.90172e6 −1.40617
\(75\) −264833. −0.0724866
\(76\) −2.18637e6 −0.571316
\(77\) 1.06349e7 2.65471
\(78\) 2.82356e6 0.673698
\(79\) −8.26089e6 −1.88509 −0.942545 0.334078i \(-0.891575\pi\)
−0.942545 + 0.334078i \(0.891575\pi\)
\(80\) 5.85675e6 1.27892
\(81\) 531441. 0.111111
\(82\) −3.32083e6 −0.665116
\(83\) −2.95933e6 −0.568093 −0.284047 0.958810i \(-0.591677\pi\)
−0.284047 + 0.958810i \(0.591677\pi\)
\(84\) 3.46655e6 0.638147
\(85\) −4.39846e6 −0.776844
\(86\) 1.98640e6 0.336762
\(87\) −426373. −0.0694180
\(88\) −4.12683e6 −0.645545
\(89\) 3.82628e6 0.575323 0.287661 0.957732i \(-0.407122\pi\)
0.287661 + 0.957732i \(0.407122\pi\)
\(90\) 3.19916e6 0.462580
\(91\) −9.96908e6 −1.38679
\(92\) 3.78508e6 0.506778
\(93\) −2.92265e6 −0.376779
\(94\) −1.20798e7 −1.50007
\(95\) 7.12393e6 0.852486
\(96\) −5.99981e6 −0.692131
\(97\) −6.24098e6 −0.694308 −0.347154 0.937808i \(-0.612852\pi\)
−0.347154 + 0.937808i \(0.612852\pi\)
\(98\) −1.72659e7 −1.85310
\(99\) −5.49553e6 −0.569229
\(100\) 892667. 0.0892667
\(101\) 1.50527e7 1.45375 0.726873 0.686772i \(-0.240973\pi\)
0.726873 + 0.686772i \(0.240973\pi\)
\(102\) 5.92675e6 0.552989
\(103\) 1.06416e7 0.959572 0.479786 0.877386i \(-0.340714\pi\)
0.479786 + 0.877386i \(0.340714\pi\)
\(104\) 3.86844e6 0.337225
\(105\) −1.12952e7 −0.952207
\(106\) −1.80230e7 −1.46979
\(107\) −1.71933e7 −1.35680 −0.678402 0.734691i \(-0.737327\pi\)
−0.678402 + 0.734691i \(0.737327\pi\)
\(108\) −1.79132e6 −0.136833
\(109\) −2.61258e7 −1.93231 −0.966154 0.257968i \(-0.916947\pi\)
−0.966154 + 0.257968i \(0.916947\pi\)
\(110\) −3.30819e7 −2.36982
\(111\) −8.94296e6 −0.620656
\(112\) 2.78633e7 1.87400
\(113\) 1.49158e7 0.972463 0.486231 0.873830i \(-0.338371\pi\)
0.486231 + 0.873830i \(0.338371\pi\)
\(114\) −9.59923e6 −0.606833
\(115\) −1.23331e7 −0.756186
\(116\) 1.43717e6 0.0854878
\(117\) 5.15145e6 0.297358
\(118\) −3.03939e6 −0.170294
\(119\) −2.09255e7 −1.13831
\(120\) 4.38304e6 0.231548
\(121\) 3.73411e7 1.91619
\(122\) −2.78451e7 −1.38832
\(123\) −6.05870e6 −0.293570
\(124\) 9.85133e6 0.464001
\(125\) 2.02583e7 0.927721
\(126\) 1.52199e7 0.677819
\(127\) −1.99083e7 −0.862426 −0.431213 0.902250i \(-0.641914\pi\)
−0.431213 + 0.902250i \(0.641914\pi\)
\(128\) −1.71893e7 −0.724475
\(129\) 3.62410e6 0.148640
\(130\) 3.10106e7 1.23797
\(131\) 2.18462e7 0.849038 0.424519 0.905419i \(-0.360443\pi\)
0.424519 + 0.905419i \(0.360443\pi\)
\(132\) 1.85237e7 0.701001
\(133\) 3.38918e7 1.24915
\(134\) 3.09543e7 1.11136
\(135\) 5.83672e6 0.204174
\(136\) 8.12001e6 0.276803
\(137\) −1.96904e7 −0.654233 −0.327117 0.944984i \(-0.606077\pi\)
−0.327117 + 0.944984i \(0.606077\pi\)
\(138\) 1.66183e7 0.538284
\(139\) 2.42247e6 0.0765078 0.0382539 0.999268i \(-0.487820\pi\)
0.0382539 + 0.999268i \(0.487820\pi\)
\(140\) 3.80725e7 1.17264
\(141\) −2.20390e7 −0.662102
\(142\) 2.67608e7 0.784315
\(143\) −5.32702e7 −1.52338
\(144\) −1.43981e7 −0.401826
\(145\) −4.68277e6 −0.127560
\(146\) 7.29686e7 1.94044
\(147\) −3.15009e7 −0.817922
\(148\) 3.01439e7 0.764334
\(149\) 2.09513e7 0.518871 0.259436 0.965760i \(-0.416463\pi\)
0.259436 + 0.965760i \(0.416463\pi\)
\(150\) 3.91924e6 0.0948163
\(151\) 4.77834e6 0.112943 0.0564713 0.998404i \(-0.482015\pi\)
0.0564713 + 0.998404i \(0.482015\pi\)
\(152\) −1.31515e7 −0.303755
\(153\) 1.08131e7 0.244079
\(154\) −1.57386e8 −3.47251
\(155\) −3.20990e7 −0.692357
\(156\) −1.73639e7 −0.366194
\(157\) −6.17818e7 −1.27412 −0.637062 0.770813i \(-0.719850\pi\)
−0.637062 + 0.770813i \(0.719850\pi\)
\(158\) 1.22252e8 2.46580
\(159\) −3.28821e7 −0.648739
\(160\) −6.58948e7 −1.27184
\(161\) −5.86741e7 −1.10804
\(162\) −7.86476e6 −0.145339
\(163\) 8.27535e7 1.49668 0.748342 0.663313i \(-0.230850\pi\)
0.748342 + 0.663313i \(0.230850\pi\)
\(164\) 2.04219e7 0.361529
\(165\) −6.03564e7 −1.04600
\(166\) 4.37949e7 0.743096
\(167\) −1.39381e7 −0.231578 −0.115789 0.993274i \(-0.536940\pi\)
−0.115789 + 0.993274i \(0.536940\pi\)
\(168\) 2.08521e7 0.339288
\(169\) −1.28136e7 −0.204205
\(170\) 6.50925e7 1.01615
\(171\) −1.75134e7 −0.267845
\(172\) −1.22157e7 −0.183049
\(173\) 9.03037e7 1.32600 0.663001 0.748618i \(-0.269282\pi\)
0.663001 + 0.748618i \(0.269282\pi\)
\(174\) 6.30986e6 0.0908024
\(175\) −1.38376e7 −0.195177
\(176\) 1.48889e8 2.05858
\(177\) −5.54523e6 −0.0751646
\(178\) −5.66248e7 −0.752553
\(179\) 9.61633e7 1.25321 0.626605 0.779337i \(-0.284444\pi\)
0.626605 + 0.779337i \(0.284444\pi\)
\(180\) −1.96737e7 −0.251439
\(181\) 1.27453e8 1.59763 0.798816 0.601575i \(-0.205460\pi\)
0.798816 + 0.601575i \(0.205460\pi\)
\(182\) 1.47532e8 1.81399
\(183\) −5.08022e7 −0.612778
\(184\) 2.27681e7 0.269442
\(185\) −9.82189e7 −1.14050
\(186\) 4.32522e7 0.492847
\(187\) −1.11816e8 −1.25043
\(188\) 7.42864e7 0.815374
\(189\) 2.77680e7 0.299177
\(190\) −1.05427e8 −1.11510
\(191\) 8.72604e6 0.0906151 0.0453075 0.998973i \(-0.485573\pi\)
0.0453075 + 0.998973i \(0.485573\pi\)
\(192\) 2.05329e7 0.209361
\(193\) 6.92616e7 0.693492 0.346746 0.937959i \(-0.387287\pi\)
0.346746 + 0.937959i \(0.387287\pi\)
\(194\) 9.23599e7 0.908191
\(195\) 5.65775e7 0.546415
\(196\) 1.06179e8 1.00726
\(197\) 1.99367e7 0.185790 0.0928949 0.995676i \(-0.470388\pi\)
0.0928949 + 0.995676i \(0.470388\pi\)
\(198\) 8.13280e7 0.744581
\(199\) −1.15995e8 −1.04341 −0.521705 0.853126i \(-0.674704\pi\)
−0.521705 + 0.853126i \(0.674704\pi\)
\(200\) 5.36960e6 0.0474610
\(201\) 5.64747e7 0.490533
\(202\) −2.22763e8 −1.90158
\(203\) −2.22781e7 −0.186914
\(204\) −3.64475e7 −0.300582
\(205\) −6.65416e7 −0.539454
\(206\) −1.57485e8 −1.25517
\(207\) 3.03194e7 0.237588
\(208\) −1.39567e8 −1.07537
\(209\) 1.81103e8 1.37218
\(210\) 1.67157e8 1.24554
\(211\) 6.67529e7 0.489195 0.244597 0.969625i \(-0.421344\pi\)
0.244597 + 0.969625i \(0.421344\pi\)
\(212\) 1.10835e8 0.798918
\(213\) 4.88240e7 0.346182
\(214\) 2.54443e8 1.77477
\(215\) 3.98029e7 0.273136
\(216\) −1.07752e7 −0.0727507
\(217\) −1.52710e8 −1.01451
\(218\) 3.86633e8 2.52756
\(219\) 1.33128e8 0.856475
\(220\) 2.03442e8 1.28814
\(221\) 1.04815e8 0.653209
\(222\) 1.32346e8 0.811851
\(223\) 1.76404e8 1.06522 0.532611 0.846360i \(-0.321211\pi\)
0.532611 + 0.846360i \(0.321211\pi\)
\(224\) −3.13492e8 −1.86362
\(225\) 7.15049e6 0.0418501
\(226\) −2.20738e8 −1.27203
\(227\) −1.70858e8 −0.969491 −0.484745 0.874655i \(-0.661088\pi\)
−0.484745 + 0.874655i \(0.661088\pi\)
\(228\) 5.90320e7 0.329849
\(229\) −1.06631e8 −0.586757 −0.293379 0.955996i \(-0.594780\pi\)
−0.293379 + 0.955996i \(0.594780\pi\)
\(230\) 1.82516e8 0.989132
\(231\) −2.87143e8 −1.53270
\(232\) 8.64489e6 0.0454518
\(233\) −2.36647e8 −1.22562 −0.612809 0.790231i \(-0.709960\pi\)
−0.612809 + 0.790231i \(0.709960\pi\)
\(234\) −7.62360e7 −0.388960
\(235\) −2.42050e8 −1.21666
\(236\) 1.86912e7 0.0925647
\(237\) 2.23044e8 1.08836
\(238\) 3.09675e8 1.48897
\(239\) 4.70420e6 0.0222891 0.0111446 0.999938i \(-0.496453\pi\)
0.0111446 + 0.999938i \(0.496453\pi\)
\(240\) −1.58132e8 −0.738382
\(241\) −1.57232e8 −0.723573 −0.361786 0.932261i \(-0.617833\pi\)
−0.361786 + 0.932261i \(0.617833\pi\)
\(242\) −5.52609e8 −2.50648
\(243\) −1.43489e7 −0.0641500
\(244\) 1.71238e8 0.754632
\(245\) −3.45968e8 −1.50299
\(246\) 8.96623e7 0.384005
\(247\) −1.69764e8 −0.716812
\(248\) 5.92581e7 0.246699
\(249\) 7.99018e7 0.327989
\(250\) −2.99801e8 −1.21351
\(251\) 1.41210e7 0.0563647 0.0281823 0.999603i \(-0.491028\pi\)
0.0281823 + 0.999603i \(0.491028\pi\)
\(252\) −9.35970e7 −0.368434
\(253\) −3.13528e8 −1.21718
\(254\) 2.94622e8 1.12810
\(255\) 1.18758e8 0.448511
\(256\) 3.51724e8 1.31027
\(257\) −3.85019e8 −1.41487 −0.707434 0.706780i \(-0.750147\pi\)
−0.707434 + 0.706780i \(0.750147\pi\)
\(258\) −5.36328e7 −0.194429
\(259\) −4.67273e8 −1.67117
\(260\) −1.90705e8 −0.672906
\(261\) 1.15121e7 0.0400785
\(262\) −3.23301e8 −1.11059
\(263\) 1.74319e8 0.590882 0.295441 0.955361i \(-0.404533\pi\)
0.295441 + 0.955361i \(0.404533\pi\)
\(264\) 1.11424e8 0.372706
\(265\) −3.61138e8 −1.19210
\(266\) −5.01563e8 −1.63395
\(267\) −1.03310e8 −0.332163
\(268\) −1.90358e8 −0.604087
\(269\) 2.39784e7 0.0751083 0.0375541 0.999295i \(-0.488043\pi\)
0.0375541 + 0.999295i \(0.488043\pi\)
\(270\) −8.63772e7 −0.267071
\(271\) 5.61954e8 1.71518 0.857588 0.514338i \(-0.171962\pi\)
0.857588 + 0.514338i \(0.171962\pi\)
\(272\) −2.92956e8 −0.882695
\(273\) 2.69165e8 0.800663
\(274\) 2.91397e8 0.855772
\(275\) −7.39419e7 −0.214401
\(276\) −1.02197e8 −0.292588
\(277\) 6.23564e7 0.176280 0.0881398 0.996108i \(-0.471908\pi\)
0.0881398 + 0.996108i \(0.471908\pi\)
\(278\) −3.58499e7 −0.100076
\(279\) 7.89117e7 0.217534
\(280\) 2.29015e8 0.623463
\(281\) 5.33351e8 1.43397 0.716986 0.697087i \(-0.245521\pi\)
0.716986 + 0.697087i \(0.245521\pi\)
\(282\) 3.26153e8 0.866064
\(283\) −7.38516e8 −1.93690 −0.968450 0.249207i \(-0.919830\pi\)
−0.968450 + 0.249207i \(0.919830\pi\)
\(284\) −1.64570e8 −0.426321
\(285\) −1.92346e8 −0.492183
\(286\) 7.88343e8 1.99266
\(287\) −3.16569e8 −0.790464
\(288\) 1.61995e8 0.399602
\(289\) −1.90327e8 −0.463829
\(290\) 6.93001e7 0.166856
\(291\) 1.68507e8 0.400859
\(292\) −4.48732e8 −1.05474
\(293\) 4.24434e8 0.985764 0.492882 0.870096i \(-0.335943\pi\)
0.492882 + 0.870096i \(0.335943\pi\)
\(294\) 4.66179e8 1.06988
\(295\) −6.09023e7 −0.138120
\(296\) 1.81322e8 0.406378
\(297\) 1.48379e8 0.328644
\(298\) −3.10057e8 −0.678711
\(299\) 2.93898e8 0.635839
\(300\) −2.41020e7 −0.0515382
\(301\) 1.89361e8 0.400227
\(302\) −7.07143e7 −0.147735
\(303\) −4.06422e8 −0.839320
\(304\) 4.74484e8 0.968643
\(305\) −5.57951e8 −1.12602
\(306\) −1.60022e8 −0.319268
\(307\) 5.81873e8 1.14774 0.573871 0.818946i \(-0.305441\pi\)
0.573871 + 0.818946i \(0.305441\pi\)
\(308\) 9.67869e8 1.88751
\(309\) −2.87324e8 −0.554009
\(310\) 4.75031e8 0.905640
\(311\) 4.40429e8 0.830260 0.415130 0.909762i \(-0.363736\pi\)
0.415130 + 0.909762i \(0.363736\pi\)
\(312\) −1.04448e8 −0.194697
\(313\) 1.33282e8 0.245678 0.122839 0.992427i \(-0.460800\pi\)
0.122839 + 0.992427i \(0.460800\pi\)
\(314\) 9.14304e8 1.66662
\(315\) 3.04971e8 0.549757
\(316\) −7.51811e8 −1.34031
\(317\) −5.31810e8 −0.937668 −0.468834 0.883286i \(-0.655326\pi\)
−0.468834 + 0.883286i \(0.655326\pi\)
\(318\) 4.86620e8 0.848585
\(319\) −1.19044e8 −0.205324
\(320\) 2.25509e8 0.384714
\(321\) 4.64220e8 0.783351
\(322\) 8.68314e8 1.44938
\(323\) −3.56340e8 −0.588378
\(324\) 4.83656e7 0.0790003
\(325\) 6.93123e7 0.112000
\(326\) −1.22466e9 −1.95774
\(327\) 7.05395e8 1.11562
\(328\) 1.22843e8 0.192217
\(329\) −1.15154e9 −1.78277
\(330\) 8.93211e8 1.36822
\(331\) −1.41095e8 −0.213853 −0.106926 0.994267i \(-0.534101\pi\)
−0.106926 + 0.994267i \(0.534101\pi\)
\(332\) −2.69323e8 −0.403916
\(333\) 2.41460e8 0.358336
\(334\) 2.06269e8 0.302916
\(335\) 6.20251e8 0.901386
\(336\) −7.52308e8 −1.08195
\(337\) −1.15891e9 −1.64948 −0.824739 0.565513i \(-0.808678\pi\)
−0.824739 + 0.565513i \(0.808678\pi\)
\(338\) 1.89627e8 0.267111
\(339\) −4.02728e8 −0.561452
\(340\) −4.00296e8 −0.552339
\(341\) −8.16011e8 −1.11444
\(342\) 2.59179e8 0.350355
\(343\) −4.84110e8 −0.647760
\(344\) −7.34802e7 −0.0973231
\(345\) 3.32993e8 0.436584
\(346\) −1.33640e9 −1.73448
\(347\) 4.17462e8 0.536369 0.268184 0.963368i \(-0.413576\pi\)
0.268184 + 0.963368i \(0.413576\pi\)
\(348\) −3.88035e7 −0.0493564
\(349\) −9.78142e7 −0.123172 −0.0615861 0.998102i \(-0.519616\pi\)
−0.0615861 + 0.998102i \(0.519616\pi\)
\(350\) 2.04782e8 0.255301
\(351\) −1.39089e8 −0.171680
\(352\) −1.67516e9 −2.04718
\(353\) −1.27204e9 −1.53918 −0.769590 0.638539i \(-0.779539\pi\)
−0.769590 + 0.638539i \(0.779539\pi\)
\(354\) 8.20635e7 0.0983193
\(355\) 5.36225e8 0.636133
\(356\) 3.48223e8 0.409056
\(357\) 5.64988e8 0.657205
\(358\) −1.42311e9 −1.63927
\(359\) −2.70429e8 −0.308477 −0.154238 0.988034i \(-0.549292\pi\)
−0.154238 + 0.988034i \(0.549292\pi\)
\(360\) −1.18342e8 −0.133684
\(361\) −3.16727e8 −0.354332
\(362\) −1.88618e9 −2.08979
\(363\) −1.00821e9 −1.10631
\(364\) −9.07270e8 −0.986011
\(365\) 1.46212e9 1.57383
\(366\) 7.51818e8 0.801546
\(367\) 1.26226e9 1.33296 0.666481 0.745522i \(-0.267800\pi\)
0.666481 + 0.745522i \(0.267800\pi\)
\(368\) −8.21434e8 −0.859222
\(369\) 1.63585e8 0.169493
\(370\) 1.45354e9 1.49183
\(371\) −1.71810e9 −1.74679
\(372\) −2.65986e8 −0.267891
\(373\) −4.77047e7 −0.0475971 −0.0237985 0.999717i \(-0.507576\pi\)
−0.0237985 + 0.999717i \(0.507576\pi\)
\(374\) 1.65476e9 1.63563
\(375\) −5.46973e8 −0.535620
\(376\) 4.46850e8 0.433515
\(377\) 1.11591e8 0.107259
\(378\) −4.10936e8 −0.391339
\(379\) 1.65053e9 1.55735 0.778674 0.627429i \(-0.215893\pi\)
0.778674 + 0.627429i \(0.215893\pi\)
\(380\) 6.48338e8 0.606120
\(381\) 5.37525e8 0.497922
\(382\) −1.29136e8 −0.118529
\(383\) 4.84522e7 0.0440674 0.0220337 0.999757i \(-0.492986\pi\)
0.0220337 + 0.999757i \(0.492986\pi\)
\(384\) 4.64111e8 0.418276
\(385\) −3.15364e9 −2.81644
\(386\) −1.02500e9 −0.907125
\(387\) −9.78508e7 −0.0858175
\(388\) −5.67982e8 −0.493655
\(389\) 2.13932e9 1.84269 0.921345 0.388745i \(-0.127091\pi\)
0.921345 + 0.388745i \(0.127091\pi\)
\(390\) −8.37286e8 −0.714740
\(391\) 6.16902e8 0.521913
\(392\) 6.38693e8 0.535539
\(393\) −5.89848e8 −0.490192
\(394\) −2.95042e8 −0.243023
\(395\) 2.44965e9 1.99993
\(396\) −5.00140e8 −0.404723
\(397\) 1.67380e9 1.34257 0.671284 0.741200i \(-0.265743\pi\)
0.671284 + 0.741200i \(0.265743\pi\)
\(398\) 1.71661e9 1.36484
\(399\) −9.15080e8 −0.721197
\(400\) −1.93726e8 −0.151348
\(401\) −5.90271e8 −0.457137 −0.228568 0.973528i \(-0.573404\pi\)
−0.228568 + 0.973528i \(0.573404\pi\)
\(402\) −8.35765e8 −0.641643
\(403\) 7.64920e8 0.582168
\(404\) 1.36992e9 1.03362
\(405\) −1.57591e8 −0.117880
\(406\) 3.29692e8 0.244494
\(407\) −2.49689e9 −1.83578
\(408\) −2.19240e8 −0.159812
\(409\) −2.01375e8 −0.145537 −0.0727685 0.997349i \(-0.523183\pi\)
−0.0727685 + 0.997349i \(0.523183\pi\)
\(410\) 9.84745e8 0.705635
\(411\) 5.31641e8 0.377722
\(412\) 9.68476e8 0.682258
\(413\) −2.89740e8 −0.202388
\(414\) −4.48695e8 −0.310778
\(415\) 8.77547e8 0.602701
\(416\) 1.57028e9 1.06942
\(417\) −6.54066e7 −0.0441718
\(418\) −2.68012e9 −1.79489
\(419\) 1.53310e9 1.01817 0.509086 0.860716i \(-0.329983\pi\)
0.509086 + 0.860716i \(0.329983\pi\)
\(420\) −1.02796e9 −0.677022
\(421\) 1.10242e9 0.720044 0.360022 0.932944i \(-0.382769\pi\)
0.360022 + 0.932944i \(0.382769\pi\)
\(422\) −9.87872e8 −0.639892
\(423\) 5.95053e8 0.382265
\(424\) 6.66699e8 0.424766
\(425\) 1.45489e8 0.0919326
\(426\) −7.22543e8 −0.452825
\(427\) −2.65443e9 −1.64996
\(428\) −1.56474e9 −0.964691
\(429\) 1.43830e9 0.879524
\(430\) −5.89040e8 −0.357277
\(431\) −2.15146e9 −1.29438 −0.647192 0.762327i \(-0.724057\pi\)
−0.647192 + 0.762327i \(0.724057\pi\)
\(432\) 3.88750e8 0.231994
\(433\) 4.65835e8 0.275756 0.137878 0.990449i \(-0.455972\pi\)
0.137878 + 0.990449i \(0.455972\pi\)
\(434\) 2.25994e9 1.32704
\(435\) 1.26435e8 0.0736469
\(436\) −2.37766e9 −1.37388
\(437\) −9.99162e8 −0.572732
\(438\) −1.97015e9 −1.12032
\(439\) 2.25997e9 1.27490 0.637451 0.770491i \(-0.279989\pi\)
0.637451 + 0.770491i \(0.279989\pi\)
\(440\) 1.22375e9 0.684872
\(441\) 8.50523e8 0.472227
\(442\) −1.55116e9 −0.854432
\(443\) −1.51398e9 −0.827383 −0.413691 0.910417i \(-0.635761\pi\)
−0.413691 + 0.910417i \(0.635761\pi\)
\(444\) −8.13885e8 −0.441288
\(445\) −1.13463e9 −0.610371
\(446\) −2.61058e9 −1.39337
\(447\) −5.65686e8 −0.299570
\(448\) 1.07285e9 0.563723
\(449\) −2.99785e9 −1.56296 −0.781480 0.623930i \(-0.785535\pi\)
−0.781480 + 0.623930i \(0.785535\pi\)
\(450\) −1.05820e8 −0.0547422
\(451\) −1.69160e9 −0.868321
\(452\) 1.35747e9 0.691424
\(453\) −1.29015e8 −0.0652074
\(454\) 2.52851e9 1.26815
\(455\) 2.95619e9 1.47127
\(456\) 3.55091e8 0.175373
\(457\) 2.58428e9 1.26658 0.633289 0.773915i \(-0.281704\pi\)
0.633289 + 0.773915i \(0.281704\pi\)
\(458\) 1.57802e9 0.767510
\(459\) −2.91954e8 −0.140919
\(460\) −1.12241e9 −0.537651
\(461\) −1.75068e9 −0.832247 −0.416124 0.909308i \(-0.636612\pi\)
−0.416124 + 0.909308i \(0.636612\pi\)
\(462\) 4.24942e9 2.00485
\(463\) −3.28077e9 −1.53618 −0.768091 0.640341i \(-0.778793\pi\)
−0.768091 + 0.640341i \(0.778793\pi\)
\(464\) −3.11892e8 −0.144941
\(465\) 8.66672e8 0.399733
\(466\) 3.50212e9 1.60317
\(467\) 2.15325e9 0.978329 0.489164 0.872192i \(-0.337302\pi\)
0.489164 + 0.872192i \(0.337302\pi\)
\(468\) 4.68825e8 0.211422
\(469\) 2.95082e9 1.32080
\(470\) 3.58208e9 1.59145
\(471\) 1.66811e9 0.735616
\(472\) 1.12432e8 0.0492145
\(473\) 1.01186e9 0.439648
\(474\) −3.30082e9 −1.42363
\(475\) −2.35641e8 −0.100884
\(476\) −1.90439e9 −0.809343
\(477\) 8.87818e8 0.374550
\(478\) −6.96171e7 −0.0291554
\(479\) −4.22659e9 −1.75718 −0.878589 0.477578i \(-0.841515\pi\)
−0.878589 + 0.477578i \(0.841515\pi\)
\(480\) 1.77916e9 0.734295
\(481\) 2.34056e9 0.958986
\(482\) 2.32687e9 0.946471
\(483\) 1.58420e9 0.639728
\(484\) 3.39836e9 1.36242
\(485\) 1.85068e9 0.736605
\(486\) 2.12348e8 0.0839116
\(487\) −3.71716e9 −1.45835 −0.729173 0.684330i \(-0.760095\pi\)
−0.729173 + 0.684330i \(0.760095\pi\)
\(488\) 1.03004e9 0.401220
\(489\) −2.23435e9 −0.864111
\(490\) 5.11996e9 1.96599
\(491\) 3.64965e9 1.39144 0.695722 0.718311i \(-0.255085\pi\)
0.695722 + 0.718311i \(0.255085\pi\)
\(492\) −5.51392e8 −0.208729
\(493\) 2.34233e8 0.0880409
\(494\) 2.51232e9 0.937628
\(495\) 1.62962e9 0.603906
\(496\) −2.13793e9 −0.786696
\(497\) 2.55107e9 0.932127
\(498\) −1.18246e9 −0.429027
\(499\) −8.31205e8 −0.299472 −0.149736 0.988726i \(-0.547842\pi\)
−0.149736 + 0.988726i \(0.547842\pi\)
\(500\) 1.84367e9 0.659612
\(501\) 3.76330e8 0.133702
\(502\) −2.08976e8 −0.0737280
\(503\) −3.38642e9 −1.18646 −0.593230 0.805033i \(-0.702147\pi\)
−0.593230 + 0.805033i \(0.702147\pi\)
\(504\) −5.63007e8 −0.195888
\(505\) −4.46366e9 −1.54231
\(506\) 4.63988e9 1.59213
\(507\) 3.45966e8 0.117898
\(508\) −1.81183e9 −0.613188
\(509\) −2.52186e9 −0.847633 −0.423817 0.905748i \(-0.639310\pi\)
−0.423817 + 0.905748i \(0.639310\pi\)
\(510\) −1.75750e9 −0.586677
\(511\) 6.95598e9 2.30614
\(512\) −3.00491e9 −0.989434
\(513\) 4.72861e8 0.154640
\(514\) 5.69786e9 1.85072
\(515\) −3.15562e9 −1.01803
\(516\) 3.29824e8 0.105684
\(517\) −6.15333e9 −1.95836
\(518\) 6.91514e9 2.18598
\(519\) −2.43820e9 −0.765568
\(520\) −1.14713e9 −0.357768
\(521\) 4.55029e9 1.40964 0.704818 0.709388i \(-0.251029\pi\)
0.704818 + 0.709388i \(0.251029\pi\)
\(522\) −1.70366e8 −0.0524248
\(523\) −1.39010e9 −0.424903 −0.212451 0.977172i \(-0.568145\pi\)
−0.212451 + 0.977172i \(0.568145\pi\)
\(524\) 1.98819e9 0.603668
\(525\) 3.73616e8 0.112685
\(526\) −2.57974e9 −0.772905
\(527\) 1.60560e9 0.477858
\(528\) −4.01999e9 −1.18852
\(529\) −1.67506e9 −0.491966
\(530\) 5.34446e9 1.55933
\(531\) 1.49721e8 0.0433963
\(532\) 3.08444e9 0.888149
\(533\) 1.58569e9 0.453600
\(534\) 1.52887e9 0.434487
\(535\) 5.09844e9 1.43946
\(536\) −1.14505e9 −0.321179
\(537\) −2.59641e9 −0.723541
\(538\) −3.54855e8 −0.0982456
\(539\) −8.79510e9 −2.41925
\(540\) 5.31190e8 0.145168
\(541\) 3.45589e9 0.938360 0.469180 0.883103i \(-0.344550\pi\)
0.469180 + 0.883103i \(0.344550\pi\)
\(542\) −8.31632e9 −2.24354
\(543\) −3.44124e9 −0.922393
\(544\) 3.29607e9 0.877809
\(545\) 7.74723e9 2.05002
\(546\) −3.98336e9 −1.04731
\(547\) 5.69696e9 1.48829 0.744145 0.668018i \(-0.232857\pi\)
0.744145 + 0.668018i \(0.232857\pi\)
\(548\) −1.79199e9 −0.465162
\(549\) 1.37166e9 0.353788
\(550\) 1.09426e9 0.280447
\(551\) −3.79374e8 −0.0966134
\(552\) −6.14740e8 −0.155562
\(553\) 1.16541e10 2.93050
\(554\) −9.22809e8 −0.230583
\(555\) 2.65191e9 0.658466
\(556\) 2.20465e8 0.0543973
\(557\) −7.56400e9 −1.85464 −0.927318 0.374275i \(-0.877892\pi\)
−0.927318 + 0.374275i \(0.877892\pi\)
\(558\) −1.16781e9 −0.284546
\(559\) −9.48504e8 −0.229667
\(560\) −8.26246e9 −1.98816
\(561\) 3.01904e9 0.721936
\(562\) −7.89302e9 −1.87571
\(563\) −5.67154e9 −1.33943 −0.669717 0.742616i \(-0.733585\pi\)
−0.669717 + 0.742616i \(0.733585\pi\)
\(564\) −2.00573e9 −0.470756
\(565\) −4.42308e9 −1.03170
\(566\) 1.09292e10 2.53357
\(567\) −7.49735e8 −0.172730
\(568\) −9.89927e8 −0.226665
\(569\) 1.36290e9 0.310149 0.155074 0.987903i \(-0.450438\pi\)
0.155074 + 0.987903i \(0.450438\pi\)
\(570\) 2.84652e9 0.643801
\(571\) −2.64928e9 −0.595528 −0.297764 0.954640i \(-0.596241\pi\)
−0.297764 + 0.954640i \(0.596241\pi\)
\(572\) −4.84804e9 −1.08313
\(573\) −2.35603e8 −0.0523166
\(574\) 4.68489e9 1.03397
\(575\) 4.07945e8 0.0894880
\(576\) −5.54387e8 −0.120874
\(577\) 7.60073e9 1.64718 0.823588 0.567188i \(-0.191969\pi\)
0.823588 + 0.567188i \(0.191969\pi\)
\(578\) 2.81664e9 0.606713
\(579\) −1.87006e9 −0.400388
\(580\) −4.26172e8 −0.0906957
\(581\) 4.17490e9 0.883139
\(582\) −2.49372e9 −0.524344
\(583\) −9.18076e9 −1.91884
\(584\) −2.69923e9 −0.560782
\(585\) −1.52759e9 −0.315473
\(586\) −6.28116e9 −1.28943
\(587\) −3.30015e9 −0.673443 −0.336721 0.941604i \(-0.609318\pi\)
−0.336721 + 0.941604i \(0.609318\pi\)
\(588\) −2.86684e9 −0.581545
\(589\) −2.60049e9 −0.524388
\(590\) 9.01289e8 0.180668
\(591\) −5.38292e8 −0.107266
\(592\) −6.54179e9 −1.29590
\(593\) 5.80009e9 1.14220 0.571102 0.820879i \(-0.306516\pi\)
0.571102 + 0.820879i \(0.306516\pi\)
\(594\) −2.19586e9 −0.429884
\(595\) 6.20516e9 1.20766
\(596\) 1.90675e9 0.368919
\(597\) 3.13188e9 0.602413
\(598\) −4.34937e9 −0.831711
\(599\) 3.53769e9 0.672551 0.336276 0.941764i \(-0.390833\pi\)
0.336276 + 0.941764i \(0.390833\pi\)
\(600\) −1.44979e8 −0.0274016
\(601\) −7.08661e9 −1.33161 −0.665806 0.746125i \(-0.731912\pi\)
−0.665806 + 0.746125i \(0.731912\pi\)
\(602\) −2.80233e9 −0.523519
\(603\) −1.52482e9 −0.283209
\(604\) 4.34869e8 0.0803025
\(605\) −1.10730e10 −2.03292
\(606\) 6.01461e9 1.09788
\(607\) −9.13138e8 −0.165720 −0.0828602 0.996561i \(-0.526406\pi\)
−0.0828602 + 0.996561i \(0.526406\pi\)
\(608\) −5.33846e9 −0.963282
\(609\) 6.01509e8 0.107915
\(610\) 8.25707e9 1.47289
\(611\) 5.76807e9 1.02302
\(612\) 9.84083e8 0.173541
\(613\) 6.24314e8 0.109469 0.0547346 0.998501i \(-0.482569\pi\)
0.0547346 + 0.998501i \(0.482569\pi\)
\(614\) −8.61110e9 −1.50131
\(615\) 1.79662e9 0.311454
\(616\) 5.82196e9 1.00354
\(617\) −1.00319e10 −1.71943 −0.859714 0.510775i \(-0.829358\pi\)
−0.859714 + 0.510775i \(0.829358\pi\)
\(618\) 4.25208e9 0.724673
\(619\) −1.02470e10 −1.73652 −0.868258 0.496114i \(-0.834760\pi\)
−0.868258 + 0.496114i \(0.834760\pi\)
\(620\) −2.92128e9 −0.492268
\(621\) −8.18625e8 −0.137172
\(622\) −6.51787e9 −1.08602
\(623\) −5.39796e9 −0.894378
\(624\) 3.76830e9 0.620868
\(625\) −6.77361e9 −1.10979
\(626\) −1.97243e9 −0.321360
\(627\) −4.88977e9 −0.792231
\(628\) −5.62266e9 −0.905906
\(629\) 4.91293e9 0.787160
\(630\) −4.51324e9 −0.719112
\(631\) 3.58757e9 0.568457 0.284228 0.958757i \(-0.408263\pi\)
0.284228 + 0.958757i \(0.408263\pi\)
\(632\) −4.52232e9 −0.712609
\(633\) −1.80233e9 −0.282437
\(634\) 7.87022e9 1.22652
\(635\) 5.90354e9 0.914965
\(636\) −2.99255e9 −0.461255
\(637\) 8.24444e9 1.26378
\(638\) 1.76173e9 0.268575
\(639\) −1.31825e9 −0.199868
\(640\) 5.09725e9 0.768609
\(641\) −2.92622e9 −0.438837 −0.219419 0.975631i \(-0.570416\pi\)
−0.219419 + 0.975631i \(0.570416\pi\)
\(642\) −6.86996e9 −1.02466
\(643\) 5.85273e9 0.868200 0.434100 0.900865i \(-0.357066\pi\)
0.434100 + 0.900865i \(0.357066\pi\)
\(644\) −5.33984e9 −0.787821
\(645\) −1.07468e9 −0.157695
\(646\) 5.27346e9 0.769629
\(647\) −7.40954e9 −1.07554 −0.537770 0.843092i \(-0.680733\pi\)
−0.537770 + 0.843092i \(0.680733\pi\)
\(648\) 2.90930e8 0.0420026
\(649\) −1.54824e9 −0.222322
\(650\) −1.02575e9 −0.146502
\(651\) 4.12316e9 0.585729
\(652\) 7.53127e9 1.06415
\(653\) −4.89548e9 −0.688018 −0.344009 0.938966i \(-0.611785\pi\)
−0.344009 + 0.938966i \(0.611785\pi\)
\(654\) −1.04391e10 −1.45929
\(655\) −6.47820e9 −0.900761
\(656\) −4.43195e9 −0.612959
\(657\) −3.59445e9 −0.494486
\(658\) 1.70416e10 2.33196
\(659\) −1.26032e10 −1.71547 −0.857736 0.514091i \(-0.828129\pi\)
−0.857736 + 0.514091i \(0.828129\pi\)
\(660\) −5.49294e9 −0.743706
\(661\) 3.60198e8 0.0485106 0.0242553 0.999706i \(-0.492279\pi\)
0.0242553 + 0.999706i \(0.492279\pi\)
\(662\) 2.08806e9 0.279731
\(663\) −2.83002e9 −0.377130
\(664\) −1.62004e9 −0.214753
\(665\) −1.00502e10 −1.32525
\(666\) −3.57335e9 −0.468723
\(667\) 6.56779e8 0.0856996
\(668\) −1.26849e9 −0.164652
\(669\) −4.76290e9 −0.615007
\(670\) −9.17906e9 −1.17906
\(671\) −1.41841e10 −1.81247
\(672\) 8.46428e9 1.07596
\(673\) 1.40565e9 0.177756 0.0888782 0.996043i \(-0.471672\pi\)
0.0888782 + 0.996043i \(0.471672\pi\)
\(674\) 1.71507e10 2.15761
\(675\) −1.93063e8 −0.0241622
\(676\) −1.16614e9 −0.145190
\(677\) 5.12306e9 0.634555 0.317278 0.948333i \(-0.397231\pi\)
0.317278 + 0.948333i \(0.397231\pi\)
\(678\) 5.95994e9 0.734409
\(679\) 8.80452e9 1.07935
\(680\) −2.40788e9 −0.293665
\(681\) 4.61315e9 0.559736
\(682\) 1.20761e10 1.45774
\(683\) −1.70981e9 −0.205341 −0.102671 0.994715i \(-0.532739\pi\)
−0.102671 + 0.994715i \(0.532739\pi\)
\(684\) −1.59386e9 −0.190439
\(685\) 5.83891e9 0.694089
\(686\) 7.16431e9 0.847305
\(687\) 2.87903e9 0.338764
\(688\) 2.65104e9 0.310353
\(689\) 8.60594e9 1.00238
\(690\) −4.92794e9 −0.571076
\(691\) −1.38466e10 −1.59651 −0.798253 0.602323i \(-0.794242\pi\)
−0.798253 + 0.602323i \(0.794242\pi\)
\(692\) 8.21839e9 0.942791
\(693\) 7.75287e9 0.884905
\(694\) −6.17799e9 −0.701599
\(695\) −7.18348e8 −0.0811686
\(696\) −2.33412e8 −0.0262416
\(697\) 3.32842e9 0.372326
\(698\) 1.44755e9 0.161116
\(699\) 6.38947e9 0.707611
\(700\) −1.25934e9 −0.138771
\(701\) −1.16579e10 −1.27823 −0.639115 0.769111i \(-0.720699\pi\)
−0.639115 + 0.769111i \(0.720699\pi\)
\(702\) 2.05837e9 0.224566
\(703\) −7.95720e9 −0.863806
\(704\) 5.73282e9 0.619247
\(705\) 6.53535e9 0.702437
\(706\) 1.88248e10 2.01333
\(707\) −2.12357e10 −2.25995
\(708\) −5.04663e8 −0.0534423
\(709\) −1.12659e10 −1.18714 −0.593572 0.804781i \(-0.702283\pi\)
−0.593572 + 0.804781i \(0.702283\pi\)
\(710\) −7.93555e9 −0.832095
\(711\) −6.02219e9 −0.628364
\(712\) 2.09464e9 0.217486
\(713\) 4.50202e9 0.465151
\(714\) −8.36122e9 −0.859659
\(715\) 1.57965e10 1.61618
\(716\) 8.75167e9 0.891036
\(717\) −1.27013e8 −0.0128686
\(718\) 4.00206e9 0.403504
\(719\) −9.57915e9 −0.961116 −0.480558 0.876963i \(-0.659566\pi\)
−0.480558 + 0.876963i \(0.659566\pi\)
\(720\) 4.26957e9 0.426305
\(721\) −1.50128e10 −1.49172
\(722\) 4.68722e9 0.463485
\(723\) 4.24527e9 0.417755
\(724\) 1.15993e10 1.13592
\(725\) 1.54894e8 0.0150956
\(726\) 1.49204e10 1.44712
\(727\) 1.06396e10 1.02697 0.513484 0.858099i \(-0.328355\pi\)
0.513484 + 0.858099i \(0.328355\pi\)
\(728\) −5.45744e9 −0.524239
\(729\) 3.87420e8 0.0370370
\(730\) −2.16378e10 −2.05865
\(731\) −1.99095e9 −0.188516
\(732\) −4.62342e9 −0.435687
\(733\) −1.85210e10 −1.73700 −0.868500 0.495689i \(-0.834916\pi\)
−0.868500 + 0.495689i \(0.834916\pi\)
\(734\) −1.86801e10 −1.74359
\(735\) 9.34114e9 0.867749
\(736\) 9.24203e9 0.854467
\(737\) 1.57679e10 1.45090
\(738\) −2.42088e9 −0.221705
\(739\) 2.17412e9 0.198166 0.0990828 0.995079i \(-0.468409\pi\)
0.0990828 + 0.995079i \(0.468409\pi\)
\(740\) −8.93875e9 −0.810897
\(741\) 4.58362e9 0.413852
\(742\) 2.54261e10 2.28489
\(743\) 1.86262e10 1.66595 0.832977 0.553307i \(-0.186634\pi\)
0.832977 + 0.553307i \(0.186634\pi\)
\(744\) −1.59997e9 −0.142431
\(745\) −6.21282e9 −0.550481
\(746\) 7.05979e8 0.0622595
\(747\) −2.15735e9 −0.189364
\(748\) −1.01762e10 −0.889060
\(749\) 2.42557e10 2.10924
\(750\) 8.09462e9 0.700619
\(751\) −2.24532e9 −0.193437 −0.0967185 0.995312i \(-0.530835\pi\)
−0.0967185 + 0.995312i \(0.530835\pi\)
\(752\) −1.61216e10 −1.38243
\(753\) −3.81267e8 −0.0325422
\(754\) −1.65142e9 −0.140300
\(755\) −1.41695e9 −0.119823
\(756\) 2.52712e9 0.212716
\(757\) 1.91646e10 1.60570 0.802852 0.596179i \(-0.203315\pi\)
0.802852 + 0.596179i \(0.203315\pi\)
\(758\) −2.44260e10 −2.03709
\(759\) 8.46525e9 0.702738
\(760\) 3.89990e9 0.322260
\(761\) 1.33554e10 1.09853 0.549265 0.835648i \(-0.314908\pi\)
0.549265 + 0.835648i \(0.314908\pi\)
\(762\) −7.95480e9 −0.651309
\(763\) 3.68572e10 3.00390
\(764\) 7.94143e8 0.0644276
\(765\) −3.20647e9 −0.258948
\(766\) −7.17041e8 −0.0576426
\(767\) 1.45130e9 0.116138
\(768\) −9.49655e9 −0.756488
\(769\) −5.20020e9 −0.412361 −0.206181 0.978514i \(-0.566103\pi\)
−0.206181 + 0.978514i \(0.566103\pi\)
\(770\) 4.66706e10 3.68405
\(771\) 1.03955e10 0.816874
\(772\) 6.30338e9 0.493075
\(773\) 2.37920e10 1.85269 0.926347 0.376672i \(-0.122932\pi\)
0.926347 + 0.376672i \(0.122932\pi\)
\(774\) 1.44809e9 0.112254
\(775\) 1.06175e9 0.0819343
\(776\) −3.41654e9 −0.262465
\(777\) 1.26164e10 0.964852
\(778\) −3.16597e10 −2.41034
\(779\) −5.39086e9 −0.408580
\(780\) 5.14902e9 0.388502
\(781\) 1.36318e10 1.02394
\(782\) −9.12950e9 −0.682690
\(783\) −3.10826e8 −0.0231393
\(784\) −2.30429e10 −1.70778
\(785\) 1.83205e10 1.35174
\(786\) 8.72913e9 0.641198
\(787\) 5.45988e9 0.399275 0.199637 0.979870i \(-0.436024\pi\)
0.199637 + 0.979870i \(0.436024\pi\)
\(788\) 1.81441e9 0.132097
\(789\) −4.70662e9 −0.341146
\(790\) −3.62522e10 −2.61601
\(791\) −2.10427e10 −1.51176
\(792\) −3.00846e9 −0.215182
\(793\) 1.32960e10 0.946814
\(794\) −2.47704e10 −1.75615
\(795\) 9.75074e9 0.688260
\(796\) −1.05566e10 −0.741868
\(797\) −4.12638e9 −0.288712 −0.144356 0.989526i \(-0.546111\pi\)
−0.144356 + 0.989526i \(0.546111\pi\)
\(798\) 1.35422e10 0.943364
\(799\) 1.21074e10 0.839725
\(800\) 2.17963e9 0.150511
\(801\) 2.78936e9 0.191774
\(802\) 8.73538e9 0.597959
\(803\) 3.71696e10 2.53328
\(804\) 5.13967e9 0.348770
\(805\) 1.73990e10 1.17554
\(806\) −1.13200e10 −0.761507
\(807\) −6.47418e8 −0.0433638
\(808\) 8.24037e9 0.549550
\(809\) 4.36775e9 0.290027 0.145013 0.989430i \(-0.453678\pi\)
0.145013 + 0.989430i \(0.453678\pi\)
\(810\) 2.33218e9 0.154193
\(811\) −1.81321e10 −1.19364 −0.596822 0.802374i \(-0.703570\pi\)
−0.596822 + 0.802374i \(0.703570\pi\)
\(812\) −2.02750e9 −0.132897
\(813\) −1.51728e10 −0.990257
\(814\) 3.69514e10 2.40129
\(815\) −2.45394e10 −1.58786
\(816\) 7.90980e9 0.509624
\(817\) 3.22462e9 0.206872
\(818\) 2.98013e9 0.190370
\(819\) −7.26746e9 −0.462263
\(820\) −6.05584e9 −0.383554
\(821\) 1.18439e10 0.746951 0.373475 0.927640i \(-0.378166\pi\)
0.373475 + 0.927640i \(0.378166\pi\)
\(822\) −7.86772e9 −0.494080
\(823\) 7.82921e8 0.0489574 0.0244787 0.999700i \(-0.492207\pi\)
0.0244787 + 0.999700i \(0.492207\pi\)
\(824\) 5.82561e9 0.362741
\(825\) 1.99643e9 0.123784
\(826\) 4.28785e9 0.264734
\(827\) −1.09138e10 −0.670974 −0.335487 0.942045i \(-0.608901\pi\)
−0.335487 + 0.942045i \(0.608901\pi\)
\(828\) 2.75932e9 0.168926
\(829\) −2.91147e10 −1.77489 −0.887446 0.460911i \(-0.847523\pi\)
−0.887446 + 0.460911i \(0.847523\pi\)
\(830\) −1.29868e10 −0.788365
\(831\) −1.68362e9 −0.101775
\(832\) −5.37388e9 −0.323487
\(833\) 1.73054e10 1.03735
\(834\) 9.67947e8 0.0577791
\(835\) 4.13316e9 0.245685
\(836\) 1.64819e10 0.975628
\(837\) −2.13061e9 −0.125593
\(838\) −2.26882e10 −1.33182
\(839\) −3.47596e9 −0.203193 −0.101596 0.994826i \(-0.532395\pi\)
−0.101596 + 0.994826i \(0.532395\pi\)
\(840\) −6.18341e9 −0.359957
\(841\) −1.70005e10 −0.985543
\(842\) −1.63146e10 −0.941855
\(843\) −1.44005e10 −0.827905
\(844\) 6.07507e9 0.347819
\(845\) 3.79969e9 0.216645
\(846\) −8.80614e9 −0.500022
\(847\) −5.26793e10 −2.97885
\(848\) −2.40533e10 −1.35453
\(849\) 1.99399e10 1.11827
\(850\) −2.15309e9 −0.120253
\(851\) 1.37756e10 0.766228
\(852\) 4.44339e9 0.246136
\(853\) −2.90613e10 −1.60322 −0.801610 0.597847i \(-0.796023\pi\)
−0.801610 + 0.597847i \(0.796023\pi\)
\(854\) 3.92827e10 2.15824
\(855\) 5.19335e9 0.284162
\(856\) −9.41226e9 −0.512904
\(857\) 3.15674e10 1.71319 0.856597 0.515987i \(-0.172575\pi\)
0.856597 + 0.515987i \(0.172575\pi\)
\(858\) −2.12852e10 −1.15046
\(859\) 4.24010e9 0.228244 0.114122 0.993467i \(-0.463594\pi\)
0.114122 + 0.993467i \(0.463594\pi\)
\(860\) 3.62239e9 0.194201
\(861\) 8.54737e9 0.456374
\(862\) 3.18393e10 1.69312
\(863\) −2.94013e9 −0.155715 −0.0778573 0.996965i \(-0.524808\pi\)
−0.0778573 + 0.996965i \(0.524808\pi\)
\(864\) −4.37386e9 −0.230710
\(865\) −2.67783e10 −1.40678
\(866\) −6.89387e9 −0.360703
\(867\) 5.13883e9 0.267792
\(868\) −1.38979e10 −0.721321
\(869\) 6.22744e10 3.21914
\(870\) −1.87110e9 −0.0963341
\(871\) −1.47806e10 −0.757930
\(872\) −1.43022e10 −0.730458
\(873\) −4.54968e9 −0.231436
\(874\) 1.47865e10 0.749163
\(875\) −2.85795e10 −1.44220
\(876\) 1.21158e10 0.608956
\(877\) 2.78040e10 1.39190 0.695952 0.718088i \(-0.254983\pi\)
0.695952 + 0.718088i \(0.254983\pi\)
\(878\) −3.34451e10 −1.66764
\(879\) −1.14597e10 −0.569131
\(880\) −4.41508e10 −2.18398
\(881\) 2.13975e10 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(882\) −1.25868e10 −0.617698
\(883\) 2.73260e10 1.33572 0.667858 0.744289i \(-0.267211\pi\)
0.667858 + 0.744289i \(0.267211\pi\)
\(884\) 9.53908e9 0.464433
\(885\) 1.64436e9 0.0797436
\(886\) 2.24053e10 1.08226
\(887\) 2.35865e10 1.13483 0.567416 0.823431i \(-0.307943\pi\)
0.567416 + 0.823431i \(0.307943\pi\)
\(888\) −4.89571e9 −0.234623
\(889\) 2.80859e10 1.34070
\(890\) 1.67913e10 0.798398
\(891\) −4.00624e9 −0.189743
\(892\) 1.60542e10 0.757376
\(893\) −1.96097e10 −0.921489
\(894\) 8.37154e9 0.391854
\(895\) −2.85159e10 −1.32955
\(896\) 2.42500e10 1.12625
\(897\) −7.93524e9 −0.367102
\(898\) 4.43650e10 2.04444
\(899\) 1.70938e9 0.0784658
\(900\) 6.50754e8 0.0297556
\(901\) 1.80642e10 0.822777
\(902\) 2.50339e10 1.13581
\(903\) −5.11274e9 −0.231071
\(904\) 8.16547e9 0.367614
\(905\) −3.77946e10 −1.69496
\(906\) 1.90929e9 0.0852948
\(907\) −1.48021e9 −0.0658716 −0.0329358 0.999457i \(-0.510486\pi\)
−0.0329358 + 0.999457i \(0.510486\pi\)
\(908\) −1.55495e10 −0.689311
\(909\) 1.09734e10 0.484582
\(910\) −4.37485e10 −1.92450
\(911\) 1.06953e10 0.468681 0.234341 0.972155i \(-0.424707\pi\)
0.234341 + 0.972155i \(0.424707\pi\)
\(912\) −1.28111e10 −0.559247
\(913\) 2.23087e10 0.970124
\(914\) −3.82445e10 −1.65675
\(915\) 1.50647e10 0.650108
\(916\) −9.70430e9 −0.417186
\(917\) −3.08198e10 −1.31989
\(918\) 4.32060e9 0.184330
\(919\) −3.15248e9 −0.133983 −0.0669913 0.997754i \(-0.521340\pi\)
−0.0669913 + 0.997754i \(0.521340\pi\)
\(920\) −6.75157e9 −0.285856
\(921\) −1.57106e10 −0.662649
\(922\) 2.59081e10 1.08862
\(923\) −1.27783e10 −0.534892
\(924\) −2.61325e10 −1.08975
\(925\) 3.24882e9 0.134968
\(926\) 4.85519e10 2.00941
\(927\) 7.75774e9 0.319857
\(928\) 3.50913e9 0.144139
\(929\) 4.53386e10 1.85529 0.927647 0.373458i \(-0.121828\pi\)
0.927647 + 0.373458i \(0.121828\pi\)
\(930\) −1.28258e10 −0.522871
\(931\) −2.80286e10 −1.13835
\(932\) −2.15368e10 −0.871418
\(933\) −1.18916e10 −0.479351
\(934\) −3.18658e10 −1.27971
\(935\) 3.31576e10 1.32661
\(936\) 2.82009e9 0.112408
\(937\) 4.84194e9 0.192279 0.0961393 0.995368i \(-0.469351\pi\)
0.0961393 + 0.995368i \(0.469351\pi\)
\(938\) −4.36690e10 −1.72768
\(939\) −3.59861e9 −0.141842
\(940\) −2.20286e10 −0.865046
\(941\) −8.67653e9 −0.339455 −0.169728 0.985491i \(-0.554289\pi\)
−0.169728 + 0.985491i \(0.554289\pi\)
\(942\) −2.46862e10 −0.962224
\(943\) 9.33275e9 0.362425
\(944\) −4.05635e9 −0.156940
\(945\) −8.23420e9 −0.317402
\(946\) −1.49744e10 −0.575083
\(947\) 1.85330e10 0.709121 0.354561 0.935033i \(-0.384630\pi\)
0.354561 + 0.935033i \(0.384630\pi\)
\(948\) 2.02989e10 0.773825
\(949\) −3.48424e10 −1.32335
\(950\) 3.48723e9 0.131962
\(951\) 1.43589e10 0.541363
\(952\) −1.14554e10 −0.430309
\(953\) 1.33156e10 0.498352 0.249176 0.968458i \(-0.419840\pi\)
0.249176 + 0.968458i \(0.419840\pi\)
\(954\) −1.31388e10 −0.489931
\(955\) −2.58759e9 −0.0961353
\(956\) 4.28122e8 0.0158476
\(957\) 3.21419e9 0.118544
\(958\) 6.25490e10 2.29848
\(959\) 2.77784e10 1.01705
\(960\) −6.08873e9 −0.222115
\(961\) −1.57953e10 −0.574112
\(962\) −3.46378e10 −1.25440
\(963\) −1.25339e10 −0.452268
\(964\) −1.43095e10 −0.514462
\(965\) −2.05385e10 −0.735740
\(966\) −2.34445e10 −0.836798
\(967\) 1.13617e10 0.404064 0.202032 0.979379i \(-0.435246\pi\)
0.202032 + 0.979379i \(0.435246\pi\)
\(968\) 2.04419e10 0.724365
\(969\) 9.62119e9 0.339700
\(970\) −2.73880e10 −0.963518
\(971\) 1.97070e10 0.690802 0.345401 0.938455i \(-0.387743\pi\)
0.345401 + 0.938455i \(0.387743\pi\)
\(972\) −1.30587e9 −0.0456109
\(973\) −3.41751e9 −0.118937
\(974\) 5.50100e10 1.90759
\(975\) −1.87143e9 −0.0646633
\(976\) −3.71619e10 −1.27945
\(977\) −4.83923e10 −1.66014 −0.830072 0.557657i \(-0.811701\pi\)
−0.830072 + 0.557657i \(0.811701\pi\)
\(978\) 3.30659e10 1.13030
\(979\) −2.88442e10 −0.982470
\(980\) −3.14860e10 −1.06863
\(981\) −1.90457e10 −0.644102
\(982\) −5.40109e10 −1.82008
\(983\) 2.12104e10 0.712214 0.356107 0.934445i \(-0.384104\pi\)
0.356107 + 0.934445i \(0.384104\pi\)
\(984\) −3.31675e9 −0.110976
\(985\) −5.91196e9 −0.197108
\(986\) −3.46640e9 −0.115162
\(987\) 3.10917e10 1.02928
\(988\) −1.54499e10 −0.509655
\(989\) −5.58252e9 −0.183503
\(990\) −2.41167e10 −0.789941
\(991\) 3.64631e10 1.19013 0.595067 0.803676i \(-0.297125\pi\)
0.595067 + 0.803676i \(0.297125\pi\)
\(992\) 2.40540e10 0.782342
\(993\) 3.80958e9 0.123468
\(994\) −3.77531e10 −1.21927
\(995\) 3.43968e10 1.10697
\(996\) 7.27173e9 0.233201
\(997\) −3.41980e10 −1.09287 −0.546434 0.837502i \(-0.684015\pi\)
−0.546434 + 0.837502i \(0.684015\pi\)
\(998\) 1.23009e10 0.391725
\(999\) −6.51942e9 −0.206885
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.1.5 16
3.2 odd 2 531.8.a.b.1.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.5 16 1.1 even 1 trivial
531.8.a.b.1.12 16 3.2 odd 2