Properties

Label 177.8.a.a.1.4
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 1.4
Root \(15.0467\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-15.0467 q^{2} -27.0000 q^{3} +98.4026 q^{4} +159.890 q^{5} +406.260 q^{6} +980.332 q^{7} +445.343 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-15.0467 q^{2} -27.0000 q^{3} +98.4026 q^{4} +159.890 q^{5} +406.260 q^{6} +980.332 q^{7} +445.343 q^{8} +729.000 q^{9} -2405.81 q^{10} -3390.91 q^{11} -2656.87 q^{12} -7169.91 q^{13} -14750.7 q^{14} -4317.03 q^{15} -19296.5 q^{16} -12242.5 q^{17} -10969.0 q^{18} +19577.3 q^{19} +15733.6 q^{20} -26469.0 q^{21} +51021.9 q^{22} +103217. q^{23} -12024.3 q^{24} -52560.2 q^{25} +107883. q^{26} -19683.0 q^{27} +96467.2 q^{28} -169076. q^{29} +64956.9 q^{30} -37743.0 q^{31} +233344. q^{32} +91554.5 q^{33} +184209. q^{34} +156745. q^{35} +71735.5 q^{36} -29517.3 q^{37} -294573. q^{38} +193588. q^{39} +71205.8 q^{40} +375081. q^{41} +398270. q^{42} +719695. q^{43} -333674. q^{44} +116560. q^{45} -1.55308e6 q^{46} -317727. q^{47} +521004. q^{48} +137507. q^{49} +790857. q^{50} +330547. q^{51} -705538. q^{52} +290607. q^{53} +296164. q^{54} -542172. q^{55} +436584. q^{56} -528586. q^{57} +2.54404e6 q^{58} +205379. q^{59} -424807. q^{60} -2.85898e6 q^{61} +567906. q^{62} +714662. q^{63} -1.04110e6 q^{64} -1.14640e6 q^{65} -1.37759e6 q^{66} +1.89394e6 q^{67} -1.20469e6 q^{68} -2.78687e6 q^{69} -2.35849e6 q^{70} +3.57204e6 q^{71} +324655. q^{72} -2.76466e6 q^{73} +444137. q^{74} +1.41913e6 q^{75} +1.92645e6 q^{76} -3.32421e6 q^{77} -2.91285e6 q^{78} -4.56707e6 q^{79} -3.08531e6 q^{80} +531441. q^{81} -5.64372e6 q^{82} -661014. q^{83} -2.60461e6 q^{84} -1.95745e6 q^{85} -1.08290e7 q^{86} +4.56506e6 q^{87} -1.51012e6 q^{88} -5.31858e6 q^{89} -1.75384e6 q^{90} -7.02889e6 q^{91} +1.01568e7 q^{92} +1.01906e6 q^{93} +4.78073e6 q^{94} +3.13021e6 q^{95} -6.30028e6 q^{96} -1.84545e6 q^{97} -2.06903e6 q^{98} -2.47197e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.0467 −1.32995 −0.664976 0.746865i \(-0.731558\pi\)
−0.664976 + 0.746865i \(0.731558\pi\)
\(3\) −27.0000 −0.577350
\(4\) 98.4026 0.768770
\(5\) 159.890 0.572039 0.286020 0.958224i \(-0.407668\pi\)
0.286020 + 0.958224i \(0.407668\pi\)
\(6\) 406.260 0.767848
\(7\) 980.332 1.08026 0.540132 0.841580i \(-0.318374\pi\)
0.540132 + 0.841580i \(0.318374\pi\)
\(8\) 445.343 0.307524
\(9\) 729.000 0.333333
\(10\) −2405.81 −0.760785
\(11\) −3390.91 −0.768142 −0.384071 0.923304i \(-0.625478\pi\)
−0.384071 + 0.923304i \(0.625478\pi\)
\(12\) −2656.87 −0.443850
\(13\) −7169.91 −0.905132 −0.452566 0.891731i \(-0.649491\pi\)
−0.452566 + 0.891731i \(0.649491\pi\)
\(14\) −14750.7 −1.43670
\(15\) −4317.03 −0.330267
\(16\) −19296.5 −1.17776
\(17\) −12242.5 −0.604364 −0.302182 0.953250i \(-0.597715\pi\)
−0.302182 + 0.953250i \(0.597715\pi\)
\(18\) −10969.0 −0.443317
\(19\) 19577.3 0.654808 0.327404 0.944884i \(-0.393826\pi\)
0.327404 + 0.944884i \(0.393826\pi\)
\(20\) 15733.6 0.439767
\(21\) −26469.0 −0.623691
\(22\) 51021.9 1.02159
\(23\) 103217. 1.76891 0.884454 0.466628i \(-0.154531\pi\)
0.884454 + 0.466628i \(0.154531\pi\)
\(24\) −12024.3 −0.177549
\(25\) −52560.2 −0.672771
\(26\) 107883. 1.20378
\(27\) −19683.0 −0.192450
\(28\) 96467.2 0.830475
\(29\) −169076. −1.28733 −0.643665 0.765307i \(-0.722587\pi\)
−0.643665 + 0.765307i \(0.722587\pi\)
\(30\) 64956.9 0.439239
\(31\) −37743.0 −0.227546 −0.113773 0.993507i \(-0.536294\pi\)
−0.113773 + 0.993507i \(0.536294\pi\)
\(32\) 233344. 1.25884
\(33\) 91554.5 0.443487
\(34\) 184209. 0.803774
\(35\) 156745. 0.617954
\(36\) 71735.5 0.256257
\(37\) −29517.3 −0.0958011 −0.0479006 0.998852i \(-0.515253\pi\)
−0.0479006 + 0.998852i \(0.515253\pi\)
\(38\) −294573. −0.870863
\(39\) 193588. 0.522578
\(40\) 71205.8 0.175916
\(41\) 375081. 0.849926 0.424963 0.905211i \(-0.360287\pi\)
0.424963 + 0.905211i \(0.360287\pi\)
\(42\) 398270. 0.829478
\(43\) 719695. 1.38041 0.690206 0.723613i \(-0.257520\pi\)
0.690206 + 0.723613i \(0.257520\pi\)
\(44\) −333674. −0.590525
\(45\) 116560. 0.190680
\(46\) −1.55308e6 −2.35256
\(47\) −317727. −0.446387 −0.223193 0.974774i \(-0.571648\pi\)
−0.223193 + 0.974774i \(0.571648\pi\)
\(48\) 521004. 0.679982
\(49\) 137507. 0.166970
\(50\) 790857. 0.894752
\(51\) 330547. 0.348929
\(52\) −705538. −0.695839
\(53\) 290607. 0.268127 0.134063 0.990973i \(-0.457197\pi\)
0.134063 + 0.990973i \(0.457197\pi\)
\(54\) 296164. 0.255949
\(55\) −542172. −0.439407
\(56\) 436584. 0.332208
\(57\) −528586. −0.378054
\(58\) 2.54404e6 1.71209
\(59\) 205379. 0.130189
\(60\) −424807. −0.253900
\(61\) −2.85898e6 −1.61271 −0.806355 0.591432i \(-0.798563\pi\)
−0.806355 + 0.591432i \(0.798563\pi\)
\(62\) 567906. 0.302626
\(63\) 714662. 0.360088
\(64\) −1.04110e6 −0.496436
\(65\) −1.14640e6 −0.517771
\(66\) −1.37759e6 −0.589816
\(67\) 1.89394e6 0.769316 0.384658 0.923059i \(-0.374319\pi\)
0.384658 + 0.923059i \(0.374319\pi\)
\(68\) −1.20469e6 −0.464617
\(69\) −2.78687e6 −1.02128
\(70\) −2.35849e6 −0.821848
\(71\) 3.57204e6 1.18444 0.592218 0.805778i \(-0.298252\pi\)
0.592218 + 0.805778i \(0.298252\pi\)
\(72\) 324655. 0.102508
\(73\) −2.76466e6 −0.831786 −0.415893 0.909413i \(-0.636531\pi\)
−0.415893 + 0.909413i \(0.636531\pi\)
\(74\) 444137. 0.127411
\(75\) 1.41913e6 0.388424
\(76\) 1.92645e6 0.503397
\(77\) −3.32421e6 −0.829796
\(78\) −2.91285e6 −0.695004
\(79\) −4.56707e6 −1.04218 −0.521091 0.853501i \(-0.674475\pi\)
−0.521091 + 0.853501i \(0.674475\pi\)
\(80\) −3.08531e6 −0.673727
\(81\) 531441. 0.111111
\(82\) −5.64372e6 −1.13036
\(83\) −661014. −0.126893 −0.0634464 0.997985i \(-0.520209\pi\)
−0.0634464 + 0.997985i \(0.520209\pi\)
\(84\) −2.60461e6 −0.479475
\(85\) −1.95745e6 −0.345720
\(86\) −1.08290e7 −1.83588
\(87\) 4.56506e6 0.743241
\(88\) −1.51012e6 −0.236222
\(89\) −5.31858e6 −0.799707 −0.399853 0.916579i \(-0.630939\pi\)
−0.399853 + 0.916579i \(0.630939\pi\)
\(90\) −1.75384e6 −0.253595
\(91\) −7.02889e6 −0.977782
\(92\) 1.01568e7 1.35988
\(93\) 1.01906e6 0.131374
\(94\) 4.78073e6 0.593673
\(95\) 3.13021e6 0.374576
\(96\) −6.30028e6 −0.726793
\(97\) −1.84545e6 −0.205305 −0.102653 0.994717i \(-0.532733\pi\)
−0.102653 + 0.994717i \(0.532733\pi\)
\(98\) −2.06903e6 −0.222062
\(99\) −2.47197e6 −0.256047
\(100\) −5.17206e6 −0.517206
\(101\) 4.95032e6 0.478088 0.239044 0.971009i \(-0.423166\pi\)
0.239044 + 0.971009i \(0.423166\pi\)
\(102\) −4.97364e6 −0.464059
\(103\) −2.16194e7 −1.94946 −0.974728 0.223393i \(-0.928287\pi\)
−0.974728 + 0.223393i \(0.928287\pi\)
\(104\) −3.19307e6 −0.278350
\(105\) −4.23212e6 −0.356776
\(106\) −4.37267e6 −0.356595
\(107\) −1.82441e6 −0.143973 −0.0719863 0.997406i \(-0.522934\pi\)
−0.0719863 + 0.997406i \(0.522934\pi\)
\(108\) −1.93686e6 −0.147950
\(109\) 5.23038e6 0.386848 0.193424 0.981115i \(-0.438041\pi\)
0.193424 + 0.981115i \(0.438041\pi\)
\(110\) 8.15788e6 0.584390
\(111\) 796967. 0.0553108
\(112\) −1.89169e7 −1.27229
\(113\) 2.44766e7 1.59579 0.797895 0.602796i \(-0.205947\pi\)
0.797895 + 0.602796i \(0.205947\pi\)
\(114\) 7.95346e6 0.502793
\(115\) 1.65034e7 1.01188
\(116\) −1.66376e7 −0.989661
\(117\) −5.22687e6 −0.301711
\(118\) −3.09027e6 −0.173145
\(119\) −1.20017e7 −0.652872
\(120\) −1.92256e6 −0.101565
\(121\) −7.98892e6 −0.409958
\(122\) 4.30181e7 2.14483
\(123\) −1.01272e7 −0.490705
\(124\) −3.71400e6 −0.174931
\(125\) −2.08952e7 −0.956891
\(126\) −1.07533e7 −0.478899
\(127\) −2.70518e7 −1.17188 −0.585940 0.810355i \(-0.699275\pi\)
−0.585940 + 0.810355i \(0.699275\pi\)
\(128\) −1.42029e7 −0.598606
\(129\) −1.94318e7 −0.796981
\(130\) 1.72495e7 0.688611
\(131\) 1.34945e7 0.524454 0.262227 0.965006i \(-0.415543\pi\)
0.262227 + 0.965006i \(0.415543\pi\)
\(132\) 9.00920e6 0.340940
\(133\) 1.91922e7 0.707366
\(134\) −2.84975e7 −1.02315
\(135\) −3.14711e6 −0.110089
\(136\) −5.45210e6 −0.185857
\(137\) 5.35174e7 1.77817 0.889084 0.457745i \(-0.151343\pi\)
0.889084 + 0.457745i \(0.151343\pi\)
\(138\) 4.19331e7 1.35825
\(139\) 1.33665e7 0.422149 0.211075 0.977470i \(-0.432304\pi\)
0.211075 + 0.977470i \(0.432304\pi\)
\(140\) 1.54241e7 0.475064
\(141\) 8.57863e6 0.257722
\(142\) −5.37473e7 −1.57524
\(143\) 2.43125e7 0.695270
\(144\) −1.40671e7 −0.392588
\(145\) −2.70336e7 −0.736404
\(146\) 4.15990e7 1.10624
\(147\) −3.71269e6 −0.0964003
\(148\) −2.90458e6 −0.0736490
\(149\) −1.60776e7 −0.398170 −0.199085 0.979982i \(-0.563797\pi\)
−0.199085 + 0.979982i \(0.563797\pi\)
\(150\) −2.13531e7 −0.516586
\(151\) −5.18108e7 −1.22462 −0.612310 0.790618i \(-0.709760\pi\)
−0.612310 + 0.790618i \(0.709760\pi\)
\(152\) 8.71859e6 0.201370
\(153\) −8.92477e6 −0.201455
\(154\) 5.00184e7 1.10359
\(155\) −6.03472e6 −0.130166
\(156\) 1.90495e7 0.401743
\(157\) −8.16318e7 −1.68349 −0.841746 0.539874i \(-0.818472\pi\)
−0.841746 + 0.539874i \(0.818472\pi\)
\(158\) 6.87193e7 1.38605
\(159\) −7.84638e6 −0.154803
\(160\) 3.73093e7 0.720107
\(161\) 1.01187e8 1.91089
\(162\) −7.99642e6 −0.147772
\(163\) 8.91769e6 0.161286 0.0806428 0.996743i \(-0.474303\pi\)
0.0806428 + 0.996743i \(0.474303\pi\)
\(164\) 3.69089e7 0.653398
\(165\) 1.46386e7 0.253692
\(166\) 9.94606e6 0.168761
\(167\) −3.21164e7 −0.533604 −0.266802 0.963751i \(-0.585967\pi\)
−0.266802 + 0.963751i \(0.585967\pi\)
\(168\) −1.17878e7 −0.191800
\(169\) −1.13409e7 −0.180736
\(170\) 2.94531e7 0.459790
\(171\) 1.42718e7 0.218269
\(172\) 7.08198e7 1.06122
\(173\) −5.06027e7 −0.743040 −0.371520 0.928425i \(-0.621163\pi\)
−0.371520 + 0.928425i \(0.621163\pi\)
\(174\) −6.86891e7 −0.988474
\(175\) −5.15265e7 −0.726770
\(176\) 6.54325e7 0.904689
\(177\) −5.54523e6 −0.0751646
\(178\) 8.00270e7 1.06357
\(179\) −6.22045e7 −0.810655 −0.405327 0.914172i \(-0.632843\pi\)
−0.405327 + 0.914172i \(0.632843\pi\)
\(180\) 1.14698e7 0.146589
\(181\) 6.29686e7 0.789313 0.394656 0.918829i \(-0.370864\pi\)
0.394656 + 0.918829i \(0.370864\pi\)
\(182\) 1.05761e8 1.30040
\(183\) 7.71924e7 0.931098
\(184\) 4.59671e7 0.543982
\(185\) −4.71952e6 −0.0548020
\(186\) −1.53335e7 −0.174721
\(187\) 4.15131e7 0.464237
\(188\) −3.12651e7 −0.343169
\(189\) −1.92959e7 −0.207897
\(190\) −4.70992e7 −0.498168
\(191\) −7.18839e7 −0.746474 −0.373237 0.927736i \(-0.621752\pi\)
−0.373237 + 0.927736i \(0.621752\pi\)
\(192\) 2.81098e7 0.286618
\(193\) −3.68734e7 −0.369201 −0.184600 0.982814i \(-0.559099\pi\)
−0.184600 + 0.982814i \(0.559099\pi\)
\(194\) 2.77678e7 0.273046
\(195\) 3.09527e7 0.298935
\(196\) 1.35311e7 0.128362
\(197\) 1.75363e8 1.63420 0.817101 0.576495i \(-0.195580\pi\)
0.817101 + 0.576495i \(0.195580\pi\)
\(198\) 3.71950e7 0.340530
\(199\) 8.16489e7 0.734454 0.367227 0.930131i \(-0.380307\pi\)
0.367227 + 0.930131i \(0.380307\pi\)
\(200\) −2.34073e7 −0.206893
\(201\) −5.11364e7 −0.444165
\(202\) −7.44858e7 −0.635834
\(203\) −1.65751e8 −1.39066
\(204\) 3.25267e7 0.268247
\(205\) 5.99716e7 0.486191
\(206\) 3.25300e8 2.59268
\(207\) 7.52454e7 0.589636
\(208\) 1.38354e8 1.06603
\(209\) −6.63846e7 −0.502986
\(210\) 6.36793e7 0.474494
\(211\) 1.04928e7 0.0768955 0.0384478 0.999261i \(-0.487759\pi\)
0.0384478 + 0.999261i \(0.487759\pi\)
\(212\) 2.85965e7 0.206128
\(213\) −9.64450e7 −0.683835
\(214\) 2.74513e7 0.191476
\(215\) 1.15072e8 0.789650
\(216\) −8.76569e6 −0.0591831
\(217\) −3.70006e7 −0.245810
\(218\) −7.86998e7 −0.514489
\(219\) 7.46458e7 0.480232
\(220\) −5.33511e7 −0.337803
\(221\) 8.77775e7 0.547029
\(222\) −1.19917e7 −0.0735607
\(223\) −2.52899e8 −1.52714 −0.763572 0.645723i \(-0.776556\pi\)
−0.763572 + 0.645723i \(0.776556\pi\)
\(224\) 2.28754e8 1.35988
\(225\) −3.83164e7 −0.224257
\(226\) −3.68291e8 −2.12232
\(227\) 457979. 0.00259869 0.00129935 0.999999i \(-0.499586\pi\)
0.00129935 + 0.999999i \(0.499586\pi\)
\(228\) −5.20142e7 −0.290636
\(229\) −1.85147e8 −1.01881 −0.509405 0.860527i \(-0.670135\pi\)
−0.509405 + 0.860527i \(0.670135\pi\)
\(230\) −2.48321e8 −1.34576
\(231\) 8.97538e7 0.479083
\(232\) −7.52970e7 −0.395886
\(233\) −3.11021e8 −1.61081 −0.805404 0.592726i \(-0.798052\pi\)
−0.805404 + 0.592726i \(0.798052\pi\)
\(234\) 7.86470e7 0.401261
\(235\) −5.08013e7 −0.255351
\(236\) 2.02098e7 0.100085
\(237\) 1.23311e8 0.601703
\(238\) 1.80586e8 0.868288
\(239\) −1.44383e8 −0.684107 −0.342053 0.939680i \(-0.611122\pi\)
−0.342053 + 0.939680i \(0.611122\pi\)
\(240\) 8.33034e7 0.388976
\(241\) −9.66758e7 −0.444896 −0.222448 0.974945i \(-0.571405\pi\)
−0.222448 + 0.974945i \(0.571405\pi\)
\(242\) 1.20207e8 0.545224
\(243\) −1.43489e7 −0.0641500
\(244\) −2.81331e8 −1.23980
\(245\) 2.19860e7 0.0955136
\(246\) 1.52380e8 0.652614
\(247\) −1.40367e8 −0.592688
\(248\) −1.68086e7 −0.0699761
\(249\) 1.78474e7 0.0732616
\(250\) 3.14404e8 1.27262
\(251\) 1.46912e8 0.586407 0.293203 0.956050i \(-0.405279\pi\)
0.293203 + 0.956050i \(0.405279\pi\)
\(252\) 7.03246e7 0.276825
\(253\) −3.50000e8 −1.35877
\(254\) 4.07040e8 1.55854
\(255\) 5.28511e7 0.199601
\(256\) 3.46967e8 1.29255
\(257\) −4.93706e8 −1.81427 −0.907136 0.420837i \(-0.861736\pi\)
−0.907136 + 0.420837i \(0.861736\pi\)
\(258\) 2.92384e8 1.05995
\(259\) −2.89367e7 −0.103490
\(260\) −1.12808e8 −0.398047
\(261\) −1.23257e8 −0.429110
\(262\) −2.03048e8 −0.697498
\(263\) −1.94462e8 −0.659157 −0.329578 0.944128i \(-0.606907\pi\)
−0.329578 + 0.944128i \(0.606907\pi\)
\(264\) 4.07731e7 0.136383
\(265\) 4.64651e7 0.153379
\(266\) −2.88779e8 −0.940762
\(267\) 1.43602e8 0.461711
\(268\) 1.86369e8 0.591427
\(269\) −5.97880e7 −0.187275 −0.0936377 0.995606i \(-0.529850\pi\)
−0.0936377 + 0.995606i \(0.529850\pi\)
\(270\) 4.73536e7 0.146413
\(271\) −5.18384e8 −1.58219 −0.791096 0.611693i \(-0.790489\pi\)
−0.791096 + 0.611693i \(0.790489\pi\)
\(272\) 2.36237e8 0.711797
\(273\) 1.89780e8 0.564523
\(274\) −8.05258e8 −2.36488
\(275\) 1.78227e8 0.516783
\(276\) −2.74235e8 −0.785129
\(277\) 2.35392e8 0.665444 0.332722 0.943025i \(-0.392033\pi\)
0.332722 + 0.943025i \(0.392033\pi\)
\(278\) −2.01122e8 −0.561438
\(279\) −2.75146e7 −0.0758488
\(280\) 6.98053e7 0.190036
\(281\) −3.16518e8 −0.850994 −0.425497 0.904960i \(-0.639901\pi\)
−0.425497 + 0.904960i \(0.639901\pi\)
\(282\) −1.29080e8 −0.342757
\(283\) −2.45453e8 −0.643748 −0.321874 0.946782i \(-0.604313\pi\)
−0.321874 + 0.946782i \(0.604313\pi\)
\(284\) 3.51498e8 0.910559
\(285\) −8.45155e7 −0.216262
\(286\) −3.65822e8 −0.924675
\(287\) 3.67703e8 0.918144
\(288\) 1.70108e8 0.419614
\(289\) −2.60460e8 −0.634745
\(290\) 4.06766e8 0.979381
\(291\) 4.98270e7 0.118533
\(292\) −2.72050e8 −0.639452
\(293\) 1.97162e8 0.457916 0.228958 0.973436i \(-0.426468\pi\)
0.228958 + 0.973436i \(0.426468\pi\)
\(294\) 5.58637e7 0.128208
\(295\) 3.28380e7 0.0744732
\(296\) −1.31453e7 −0.0294612
\(297\) 6.67432e7 0.147829
\(298\) 2.41914e8 0.529546
\(299\) −7.40059e8 −1.60110
\(300\) 1.39646e8 0.298609
\(301\) 7.05540e8 1.49121
\(302\) 7.79581e8 1.62868
\(303\) −1.33659e8 −0.276024
\(304\) −3.77772e8 −0.771209
\(305\) −4.57121e8 −0.922534
\(306\) 1.34288e8 0.267925
\(307\) 1.34184e7 0.0264677 0.0132339 0.999912i \(-0.495787\pi\)
0.0132339 + 0.999912i \(0.495787\pi\)
\(308\) −3.27111e8 −0.637922
\(309\) 5.83724e8 1.12552
\(310\) 9.08024e7 0.173114
\(311\) −7.42366e8 −1.39945 −0.699724 0.714413i \(-0.746694\pi\)
−0.699724 + 0.714413i \(0.746694\pi\)
\(312\) 8.62129e7 0.160706
\(313\) 7.47946e8 1.37869 0.689343 0.724435i \(-0.257899\pi\)
0.689343 + 0.724435i \(0.257899\pi\)
\(314\) 1.22829e9 2.23896
\(315\) 1.14267e8 0.205985
\(316\) −4.49412e8 −0.801198
\(317\) −7.65404e8 −1.34953 −0.674766 0.738031i \(-0.735756\pi\)
−0.674766 + 0.738031i \(0.735756\pi\)
\(318\) 1.18062e8 0.205881
\(319\) 5.73323e8 0.988853
\(320\) −1.66462e8 −0.283981
\(321\) 4.92591e7 0.0831226
\(322\) −1.52253e9 −2.54139
\(323\) −2.39674e8 −0.395742
\(324\) 5.22952e7 0.0854189
\(325\) 3.76852e8 0.608947
\(326\) −1.34182e8 −0.214502
\(327\) −1.41220e8 −0.223347
\(328\) 1.67039e8 0.261373
\(329\) −3.11478e8 −0.482216
\(330\) −2.20263e8 −0.337398
\(331\) 3.13126e8 0.474592 0.237296 0.971437i \(-0.423739\pi\)
0.237296 + 0.971437i \(0.423739\pi\)
\(332\) −6.50455e7 −0.0975515
\(333\) −2.15181e7 −0.0319337
\(334\) 4.83245e8 0.709667
\(335\) 3.02822e8 0.440079
\(336\) 5.10757e8 0.734560
\(337\) 1.30356e9 1.85535 0.927675 0.373388i \(-0.121804\pi\)
0.927675 + 0.373388i \(0.121804\pi\)
\(338\) 1.70643e8 0.240369
\(339\) −6.60867e8 −0.921330
\(340\) −1.92618e8 −0.265779
\(341\) 1.27983e8 0.174788
\(342\) −2.14743e8 −0.290288
\(343\) −6.72543e8 −0.899892
\(344\) 3.20511e8 0.424511
\(345\) −4.45592e8 −0.584212
\(346\) 7.61402e8 0.988206
\(347\) −2.10675e8 −0.270683 −0.135341 0.990799i \(-0.543213\pi\)
−0.135341 + 0.990799i \(0.543213\pi\)
\(348\) 4.49214e8 0.571381
\(349\) 6.27337e8 0.789972 0.394986 0.918687i \(-0.370749\pi\)
0.394986 + 0.918687i \(0.370749\pi\)
\(350\) 7.75302e8 0.966569
\(351\) 1.41125e8 0.174193
\(352\) −7.91247e8 −0.966970
\(353\) −8.10514e8 −0.980729 −0.490364 0.871517i \(-0.663136\pi\)
−0.490364 + 0.871517i \(0.663136\pi\)
\(354\) 8.34373e7 0.0999653
\(355\) 5.71133e8 0.677544
\(356\) −5.23362e8 −0.614791
\(357\) 3.24046e8 0.376936
\(358\) 9.35971e8 1.07813
\(359\) 4.90257e8 0.559234 0.279617 0.960112i \(-0.409793\pi\)
0.279617 + 0.960112i \(0.409793\pi\)
\(360\) 5.19091e7 0.0586387
\(361\) −5.10603e8 −0.571226
\(362\) −9.47468e8 −1.04975
\(363\) 2.15701e8 0.236689
\(364\) −6.91661e8 −0.751689
\(365\) −4.42041e8 −0.475815
\(366\) −1.16149e9 −1.23832
\(367\) −6.65868e8 −0.703164 −0.351582 0.936157i \(-0.614356\pi\)
−0.351582 + 0.936157i \(0.614356\pi\)
\(368\) −1.99173e9 −2.08335
\(369\) 2.73434e8 0.283309
\(370\) 7.10131e7 0.0728840
\(371\) 2.84891e8 0.289648
\(372\) 1.00278e8 0.100996
\(373\) −1.50881e9 −1.50541 −0.752704 0.658360i \(-0.771251\pi\)
−0.752704 + 0.658360i \(0.771251\pi\)
\(374\) −6.24635e8 −0.617412
\(375\) 5.64172e8 0.552461
\(376\) −1.41497e8 −0.137275
\(377\) 1.21226e9 1.16520
\(378\) 2.90339e8 0.276493
\(379\) −1.17501e9 −1.10868 −0.554339 0.832291i \(-0.687029\pi\)
−0.554339 + 0.832291i \(0.687029\pi\)
\(380\) 3.08020e8 0.287963
\(381\) 7.30398e8 0.676585
\(382\) 1.08161e9 0.992774
\(383\) 3.09605e8 0.281587 0.140794 0.990039i \(-0.455035\pi\)
0.140794 + 0.990039i \(0.455035\pi\)
\(384\) 3.83478e8 0.345606
\(385\) −5.31508e8 −0.474676
\(386\) 5.54822e8 0.491019
\(387\) 5.24658e8 0.460137
\(388\) −1.81597e8 −0.157833
\(389\) −1.81950e9 −1.56721 −0.783606 0.621258i \(-0.786622\pi\)
−0.783606 + 0.621258i \(0.786622\pi\)
\(390\) −4.65735e8 −0.397570
\(391\) −1.26364e9 −1.06906
\(392\) 6.12379e7 0.0513474
\(393\) −3.64352e8 −0.302794
\(394\) −2.63863e9 −2.17341
\(395\) −7.30229e8 −0.596169
\(396\) −2.43248e8 −0.196842
\(397\) 6.58203e8 0.527950 0.263975 0.964529i \(-0.414966\pi\)
0.263975 + 0.964529i \(0.414966\pi\)
\(398\) −1.22855e9 −0.976788
\(399\) −5.18189e8 −0.408398
\(400\) 1.01423e9 0.792364
\(401\) 7.74197e8 0.599578 0.299789 0.954005i \(-0.403084\pi\)
0.299789 + 0.954005i \(0.403084\pi\)
\(402\) 7.69433e8 0.590717
\(403\) 2.70614e8 0.205960
\(404\) 4.87124e8 0.367540
\(405\) 8.49720e7 0.0635599
\(406\) 2.49400e9 1.84951
\(407\) 1.00090e8 0.0735889
\(408\) 1.47207e8 0.107304
\(409\) 1.08898e9 0.787022 0.393511 0.919320i \(-0.371260\pi\)
0.393511 + 0.919320i \(0.371260\pi\)
\(410\) −9.02373e8 −0.646610
\(411\) −1.44497e9 −1.02663
\(412\) −2.12741e9 −1.49868
\(413\) 2.01340e8 0.140638
\(414\) −1.13219e9 −0.784187
\(415\) −1.05689e8 −0.0725877
\(416\) −1.67305e9 −1.13942
\(417\) −3.60896e8 −0.243728
\(418\) 9.98869e8 0.668946
\(419\) 1.86951e9 1.24159 0.620795 0.783973i \(-0.286810\pi\)
0.620795 + 0.783973i \(0.286810\pi\)
\(420\) −4.16451e8 −0.274278
\(421\) 2.61169e9 1.70583 0.852913 0.522053i \(-0.174834\pi\)
0.852913 + 0.522053i \(0.174834\pi\)
\(422\) −1.57881e8 −0.102267
\(423\) −2.31623e8 −0.148796
\(424\) 1.29420e8 0.0824555
\(425\) 6.43468e8 0.406598
\(426\) 1.45118e9 0.909467
\(427\) −2.80275e9 −1.74215
\(428\) −1.79527e8 −0.110682
\(429\) −6.56438e8 −0.401414
\(430\) −1.73145e9 −1.05020
\(431\) 1.54835e9 0.931534 0.465767 0.884908i \(-0.345779\pi\)
0.465767 + 0.884908i \(0.345779\pi\)
\(432\) 3.79812e8 0.226661
\(433\) −1.73247e9 −1.02555 −0.512776 0.858523i \(-0.671383\pi\)
−0.512776 + 0.858523i \(0.671383\pi\)
\(434\) 5.56736e8 0.326916
\(435\) 7.29908e8 0.425163
\(436\) 5.14683e8 0.297397
\(437\) 2.02071e9 1.15830
\(438\) −1.12317e9 −0.638685
\(439\) −3.14201e9 −1.77248 −0.886239 0.463227i \(-0.846691\pi\)
−0.886239 + 0.463227i \(0.846691\pi\)
\(440\) −2.41452e8 −0.135129
\(441\) 1.00243e8 0.0556568
\(442\) −1.32076e9 −0.727522
\(443\) 1.87980e9 1.02730 0.513651 0.857999i \(-0.328293\pi\)
0.513651 + 0.857999i \(0.328293\pi\)
\(444\) 7.84236e7 0.0425213
\(445\) −8.50388e8 −0.457464
\(446\) 3.80529e9 2.03103
\(447\) 4.34094e8 0.229883
\(448\) −1.02063e9 −0.536282
\(449\) 2.34987e9 1.22513 0.612564 0.790421i \(-0.290138\pi\)
0.612564 + 0.790421i \(0.290138\pi\)
\(450\) 5.76535e8 0.298251
\(451\) −1.27186e9 −0.652864
\(452\) 2.40856e9 1.22680
\(453\) 1.39889e9 0.707035
\(454\) −6.89106e6 −0.00345613
\(455\) −1.12385e9 −0.559330
\(456\) −2.35402e8 −0.116261
\(457\) −1.89675e8 −0.0929615 −0.0464808 0.998919i \(-0.514801\pi\)
−0.0464808 + 0.998919i \(0.514801\pi\)
\(458\) 2.78585e9 1.35497
\(459\) 2.40969e8 0.116310
\(460\) 1.62398e9 0.777907
\(461\) −2.07603e9 −0.986919 −0.493459 0.869769i \(-0.664268\pi\)
−0.493459 + 0.869769i \(0.664268\pi\)
\(462\) −1.35050e9 −0.637157
\(463\) 3.00869e9 1.40878 0.704392 0.709811i \(-0.251220\pi\)
0.704392 + 0.709811i \(0.251220\pi\)
\(464\) 3.26258e9 1.51617
\(465\) 1.62937e8 0.0751511
\(466\) 4.67983e9 2.14230
\(467\) −1.33042e9 −0.604479 −0.302239 0.953232i \(-0.597734\pi\)
−0.302239 + 0.953232i \(0.597734\pi\)
\(468\) −5.14337e8 −0.231946
\(469\) 1.85669e9 0.831064
\(470\) 7.64391e8 0.339604
\(471\) 2.20406e9 0.971964
\(472\) 9.14641e7 0.0400363
\(473\) −2.44042e9 −1.06035
\(474\) −1.85542e9 −0.800236
\(475\) −1.02898e9 −0.440536
\(476\) −1.18100e9 −0.501909
\(477\) 2.11852e8 0.0893756
\(478\) 2.17249e9 0.909829
\(479\) 2.48445e9 1.03290 0.516448 0.856318i \(-0.327254\pi\)
0.516448 + 0.856318i \(0.327254\pi\)
\(480\) −1.00735e9 −0.415754
\(481\) 2.11636e8 0.0867127
\(482\) 1.45465e9 0.591689
\(483\) −2.73205e9 −1.10325
\(484\) −7.86130e8 −0.315163
\(485\) −2.95068e8 −0.117443
\(486\) 2.15903e8 0.0853164
\(487\) −1.39919e9 −0.548940 −0.274470 0.961596i \(-0.588502\pi\)
−0.274470 + 0.961596i \(0.588502\pi\)
\(488\) −1.27323e9 −0.495948
\(489\) −2.40778e8 −0.0931183
\(490\) −3.30816e8 −0.127028
\(491\) 1.98268e9 0.755907 0.377953 0.925825i \(-0.376628\pi\)
0.377953 + 0.925825i \(0.376628\pi\)
\(492\) −9.96540e8 −0.377239
\(493\) 2.06992e9 0.778016
\(494\) 2.11206e9 0.788246
\(495\) −3.95243e8 −0.146469
\(496\) 7.28305e8 0.267996
\(497\) 3.50178e9 1.27950
\(498\) −2.68544e8 −0.0974344
\(499\) 3.50830e9 1.26400 0.631998 0.774970i \(-0.282235\pi\)
0.631998 + 0.774970i \(0.282235\pi\)
\(500\) −2.05615e9 −0.735629
\(501\) 8.67143e8 0.308076
\(502\) −2.21054e9 −0.779893
\(503\) −3.10883e9 −1.08920 −0.544602 0.838694i \(-0.683319\pi\)
−0.544602 + 0.838694i \(0.683319\pi\)
\(504\) 3.18270e8 0.110736
\(505\) 7.91505e8 0.273485
\(506\) 5.26634e9 1.80710
\(507\) 3.06204e8 0.104348
\(508\) −2.66197e9 −0.900906
\(509\) 2.98895e9 1.00463 0.502315 0.864685i \(-0.332482\pi\)
0.502315 + 0.864685i \(0.332482\pi\)
\(510\) −7.95234e8 −0.265460
\(511\) −2.71028e9 −0.898549
\(512\) −3.40274e9 −1.12043
\(513\) −3.85339e8 −0.126018
\(514\) 7.42864e9 2.41289
\(515\) −3.45673e9 −1.11517
\(516\) −1.91214e9 −0.612696
\(517\) 1.07738e9 0.342888
\(518\) 4.35402e8 0.137637
\(519\) 1.36627e9 0.428994
\(520\) −5.10540e8 −0.159227
\(521\) 2.53744e8 0.0786075 0.0393037 0.999227i \(-0.487486\pi\)
0.0393037 + 0.999227i \(0.487486\pi\)
\(522\) 1.85460e9 0.570696
\(523\) 2.79711e9 0.854975 0.427487 0.904021i \(-0.359399\pi\)
0.427487 + 0.904021i \(0.359399\pi\)
\(524\) 1.32789e9 0.403185
\(525\) 1.39121e9 0.419601
\(526\) 2.92600e9 0.876646
\(527\) 4.62067e8 0.137521
\(528\) −1.76668e9 −0.522322
\(529\) 7.24899e9 2.12903
\(530\) −6.99145e8 −0.203987
\(531\) 1.49721e8 0.0433963
\(532\) 1.88856e9 0.543802
\(533\) −2.68929e9 −0.769295
\(534\) −2.16073e9 −0.614053
\(535\) −2.91705e8 −0.0823580
\(536\) 8.43453e8 0.236583
\(537\) 1.67952e9 0.468032
\(538\) 8.99610e8 0.249067
\(539\) −4.66274e8 −0.128257
\(540\) −3.09684e8 −0.0846332
\(541\) −2.58195e9 −0.701064 −0.350532 0.936551i \(-0.613999\pi\)
−0.350532 + 0.936551i \(0.613999\pi\)
\(542\) 7.79996e9 2.10424
\(543\) −1.70015e9 −0.455710
\(544\) −2.85671e9 −0.760798
\(545\) 8.36284e8 0.221292
\(546\) −2.85556e9 −0.750787
\(547\) 1.44486e9 0.377459 0.188730 0.982029i \(-0.439563\pi\)
0.188730 + 0.982029i \(0.439563\pi\)
\(548\) 5.26625e9 1.36700
\(549\) −2.08419e9 −0.537570
\(550\) −2.68172e9 −0.687297
\(551\) −3.31005e9 −0.842955
\(552\) −1.24111e9 −0.314068
\(553\) −4.47725e9 −1.12583
\(554\) −3.54186e9 −0.885008
\(555\) 1.27427e8 0.0316400
\(556\) 1.31530e9 0.324536
\(557\) −4.77372e9 −1.17048 −0.585240 0.810860i \(-0.699000\pi\)
−0.585240 + 0.810860i \(0.699000\pi\)
\(558\) 4.14004e8 0.100875
\(559\) −5.16015e9 −1.24946
\(560\) −3.02463e9 −0.727803
\(561\) −1.12085e9 −0.268027
\(562\) 4.76255e9 1.13178
\(563\) −3.69151e9 −0.871816 −0.435908 0.899991i \(-0.643573\pi\)
−0.435908 + 0.899991i \(0.643573\pi\)
\(564\) 8.44159e8 0.198129
\(565\) 3.91356e9 0.912855
\(566\) 3.69326e9 0.856154
\(567\) 5.20988e8 0.120029
\(568\) 1.59078e9 0.364243
\(569\) −2.15110e9 −0.489516 −0.244758 0.969584i \(-0.578709\pi\)
−0.244758 + 0.969584i \(0.578709\pi\)
\(570\) 1.27168e9 0.287617
\(571\) 5.63827e8 0.126742 0.0633709 0.997990i \(-0.479815\pi\)
0.0633709 + 0.997990i \(0.479815\pi\)
\(572\) 2.39241e9 0.534503
\(573\) 1.94087e9 0.430977
\(574\) −5.53271e9 −1.22109
\(575\) −5.42513e9 −1.19007
\(576\) −7.58964e8 −0.165479
\(577\) −3.12035e9 −0.676221 −0.338110 0.941106i \(-0.609788\pi\)
−0.338110 + 0.941106i \(0.609788\pi\)
\(578\) 3.91906e9 0.844179
\(579\) 9.95581e8 0.213158
\(580\) −2.66018e9 −0.566125
\(581\) −6.48013e8 −0.137078
\(582\) −7.49732e8 −0.157643
\(583\) −9.85420e8 −0.205959
\(584\) −1.23122e9 −0.255795
\(585\) −8.35723e8 −0.172590
\(586\) −2.96663e9 −0.609006
\(587\) −4.80777e9 −0.981094 −0.490547 0.871415i \(-0.663203\pi\)
−0.490547 + 0.871415i \(0.663203\pi\)
\(588\) −3.65339e8 −0.0741097
\(589\) −7.38903e8 −0.148999
\(590\) −4.94103e8 −0.0990457
\(591\) −4.73480e9 −0.943507
\(592\) 5.69579e8 0.112831
\(593\) −4.97973e8 −0.0980651 −0.0490326 0.998797i \(-0.515614\pi\)
−0.0490326 + 0.998797i \(0.515614\pi\)
\(594\) −1.00426e9 −0.196605
\(595\) −1.91895e9 −0.373469
\(596\) −1.58207e9 −0.306101
\(597\) −2.20452e9 −0.424037
\(598\) 1.11354e10 2.12938
\(599\) −1.25174e9 −0.237968 −0.118984 0.992896i \(-0.537964\pi\)
−0.118984 + 0.992896i \(0.537964\pi\)
\(600\) 6.31998e8 0.119450
\(601\) −3.57133e9 −0.671073 −0.335536 0.942027i \(-0.608918\pi\)
−0.335536 + 0.942027i \(0.608918\pi\)
\(602\) −1.06160e10 −1.98324
\(603\) 1.38068e9 0.256439
\(604\) −5.09832e9 −0.941451
\(605\) −1.27735e9 −0.234512
\(606\) 2.01112e9 0.367099
\(607\) −2.73915e9 −0.497113 −0.248556 0.968617i \(-0.579956\pi\)
−0.248556 + 0.968617i \(0.579956\pi\)
\(608\) 4.56823e9 0.824300
\(609\) 4.47528e9 0.802896
\(610\) 6.87816e9 1.22692
\(611\) 2.27807e9 0.404039
\(612\) −8.78220e8 −0.154872
\(613\) 2.74148e9 0.480700 0.240350 0.970686i \(-0.422738\pi\)
0.240350 + 0.970686i \(0.422738\pi\)
\(614\) −2.01903e8 −0.0352008
\(615\) −1.61923e9 −0.280703
\(616\) −1.48042e9 −0.255183
\(617\) −9.78150e9 −1.67652 −0.838258 0.545274i \(-0.816425\pi\)
−0.838258 + 0.545274i \(0.816425\pi\)
\(618\) −8.78311e9 −1.49689
\(619\) 8.34172e9 1.41364 0.706819 0.707394i \(-0.250130\pi\)
0.706819 + 0.707394i \(0.250130\pi\)
\(620\) −5.93832e8 −0.100067
\(621\) −2.03163e9 −0.340426
\(622\) 1.11701e10 1.86120
\(623\) −5.21397e9 −0.863895
\(624\) −3.73556e9 −0.615473
\(625\) 7.65329e8 0.125391
\(626\) −1.12541e10 −1.83358
\(627\) 1.79239e9 0.290399
\(628\) −8.03278e9 −1.29422
\(629\) 3.61365e8 0.0578987
\(630\) −1.71934e9 −0.273949
\(631\) −4.64175e7 −0.00735494 −0.00367747 0.999993i \(-0.501171\pi\)
−0.00367747 + 0.999993i \(0.501171\pi\)
\(632\) −2.03391e9 −0.320496
\(633\) −2.83304e8 −0.0443956
\(634\) 1.15168e10 1.79481
\(635\) −4.32531e9 −0.670361
\(636\) −7.72104e8 −0.119008
\(637\) −9.85914e8 −0.151130
\(638\) −8.62660e9 −1.31513
\(639\) 2.60402e9 0.394812
\(640\) −2.27090e9 −0.342426
\(641\) −9.01725e9 −1.35229 −0.676147 0.736767i \(-0.736352\pi\)
−0.676147 + 0.736767i \(0.736352\pi\)
\(642\) −7.41186e8 −0.110549
\(643\) −6.73082e9 −0.998457 −0.499229 0.866470i \(-0.666383\pi\)
−0.499229 + 0.866470i \(0.666383\pi\)
\(644\) 9.95708e9 1.46903
\(645\) −3.10694e9 −0.455905
\(646\) 3.60630e9 0.526318
\(647\) 1.43491e9 0.208286 0.104143 0.994562i \(-0.466790\pi\)
0.104143 + 0.994562i \(0.466790\pi\)
\(648\) 2.36674e8 0.0341694
\(649\) −6.96421e8 −0.100004
\(650\) −5.67037e9 −0.809869
\(651\) 9.99017e8 0.141919
\(652\) 8.77524e8 0.123992
\(653\) 3.13170e9 0.440133 0.220066 0.975485i \(-0.429373\pi\)
0.220066 + 0.975485i \(0.429373\pi\)
\(654\) 2.12489e9 0.297040
\(655\) 2.15763e9 0.300008
\(656\) −7.23773e9 −1.00101
\(657\) −2.01544e9 −0.277262
\(658\) 4.68671e9 0.641323
\(659\) 8.36231e9 1.13822 0.569111 0.822260i \(-0.307287\pi\)
0.569111 + 0.822260i \(0.307287\pi\)
\(660\) 1.44048e9 0.195031
\(661\) 1.18208e10 1.59200 0.795999 0.605298i \(-0.206946\pi\)
0.795999 + 0.605298i \(0.206946\pi\)
\(662\) −4.71150e9 −0.631185
\(663\) −2.36999e9 −0.315827
\(664\) −2.94378e8 −0.0390227
\(665\) 3.06864e9 0.404641
\(666\) 3.23776e8 0.0424703
\(667\) −1.74516e10 −2.27717
\(668\) −3.16034e9 −0.410219
\(669\) 6.82827e9 0.881697
\(670\) −4.55647e9 −0.585284
\(671\) 9.69452e9 1.23879
\(672\) −6.17637e9 −0.785128
\(673\) 2.34562e9 0.296624 0.148312 0.988941i \(-0.452616\pi\)
0.148312 + 0.988941i \(0.452616\pi\)
\(674\) −1.96142e10 −2.46753
\(675\) 1.03454e9 0.129475
\(676\) −1.11597e9 −0.138944
\(677\) −1.57414e10 −1.94977 −0.974884 0.222713i \(-0.928509\pi\)
−0.974884 + 0.222713i \(0.928509\pi\)
\(678\) 9.94386e9 1.22532
\(679\) −1.80915e9 −0.221784
\(680\) −8.71736e8 −0.106317
\(681\) −1.23654e7 −0.00150036
\(682\) −1.92572e9 −0.232459
\(683\) −6.44477e9 −0.773989 −0.386994 0.922082i \(-0.626487\pi\)
−0.386994 + 0.922082i \(0.626487\pi\)
\(684\) 1.40438e9 0.167799
\(685\) 8.55688e9 1.01718
\(686\) 1.01195e10 1.19681
\(687\) 4.99898e9 0.588211
\(688\) −1.38876e10 −1.62580
\(689\) −2.08362e9 −0.242690
\(690\) 6.70468e9 0.776973
\(691\) 1.59645e10 1.84070 0.920349 0.391097i \(-0.127904\pi\)
0.920349 + 0.391097i \(0.127904\pi\)
\(692\) −4.97943e9 −0.571227
\(693\) −2.42335e9 −0.276599
\(694\) 3.16997e9 0.359995
\(695\) 2.13717e9 0.241486
\(696\) 2.03302e9 0.228565
\(697\) −4.59192e9 −0.513664
\(698\) −9.43933e9 −1.05062
\(699\) 8.39756e9 0.930001
\(700\) −5.07034e9 −0.558719
\(701\) 1.28238e10 1.40606 0.703028 0.711162i \(-0.251831\pi\)
0.703028 + 0.711162i \(0.251831\pi\)
\(702\) −2.12347e9 −0.231668
\(703\) −5.77868e8 −0.0627314
\(704\) 3.53028e9 0.381333
\(705\) 1.37164e9 0.147427
\(706\) 1.21955e10 1.30432
\(707\) 4.85295e9 0.516462
\(708\) −5.45665e8 −0.0577843
\(709\) 4.26207e8 0.0449117 0.0224558 0.999748i \(-0.492851\pi\)
0.0224558 + 0.999748i \(0.492851\pi\)
\(710\) −8.59365e9 −0.901101
\(711\) −3.32940e9 −0.347394
\(712\) −2.36859e9 −0.245929
\(713\) −3.89573e9 −0.402509
\(714\) −4.87581e9 −0.501306
\(715\) 3.88732e9 0.397722
\(716\) −6.12108e9 −0.623207
\(717\) 3.89835e9 0.394969
\(718\) −7.37674e9 −0.743754
\(719\) 9.93256e9 0.996576 0.498288 0.867012i \(-0.333962\pi\)
0.498288 + 0.867012i \(0.333962\pi\)
\(720\) −2.24919e9 −0.224576
\(721\) −2.11942e10 −2.10593
\(722\) 7.68288e9 0.759703
\(723\) 2.61025e9 0.256861
\(724\) 6.19627e9 0.606800
\(725\) 8.88670e9 0.866079
\(726\) −3.24558e9 −0.314785
\(727\) −7.03816e8 −0.0679342 −0.0339671 0.999423i \(-0.510814\pi\)
−0.0339671 + 0.999423i \(0.510814\pi\)
\(728\) −3.13027e9 −0.300692
\(729\) 3.87420e8 0.0370370
\(730\) 6.65125e9 0.632810
\(731\) −8.81085e9 −0.834271
\(732\) 7.59593e9 0.715801
\(733\) 1.24906e10 1.17144 0.585718 0.810515i \(-0.300813\pi\)
0.585718 + 0.810515i \(0.300813\pi\)
\(734\) 1.00191e10 0.935174
\(735\) −5.93622e8 −0.0551448
\(736\) 2.40851e10 2.22678
\(737\) −6.42218e9 −0.590944
\(738\) −4.11427e9 −0.376787
\(739\) −8.92221e9 −0.813237 −0.406618 0.913598i \(-0.633292\pi\)
−0.406618 + 0.913598i \(0.633292\pi\)
\(740\) −4.64413e8 −0.0421302
\(741\) 3.78991e9 0.342189
\(742\) −4.28666e9 −0.385217
\(743\) 4.73069e9 0.423120 0.211560 0.977365i \(-0.432146\pi\)
0.211560 + 0.977365i \(0.432146\pi\)
\(744\) 4.53831e8 0.0404007
\(745\) −2.57064e9 −0.227769
\(746\) 2.27026e10 2.00212
\(747\) −4.81879e8 −0.0422976
\(748\) 4.08500e9 0.356892
\(749\) −1.78853e9 −0.155528
\(750\) −8.48891e9 −0.734746
\(751\) 6.35674e9 0.547640 0.273820 0.961781i \(-0.411713\pi\)
0.273820 + 0.961781i \(0.411713\pi\)
\(752\) 6.13100e9 0.525738
\(753\) −3.96662e9 −0.338562
\(754\) −1.82405e10 −1.54967
\(755\) −8.28403e9 −0.700531
\(756\) −1.89876e9 −0.159825
\(757\) −3.90470e9 −0.327154 −0.163577 0.986531i \(-0.552303\pi\)
−0.163577 + 0.986531i \(0.552303\pi\)
\(758\) 1.76801e10 1.47449
\(759\) 9.45001e9 0.784487
\(760\) 1.39401e9 0.115191
\(761\) 2.13395e10 1.75525 0.877623 0.479351i \(-0.159128\pi\)
0.877623 + 0.479351i \(0.159128\pi\)
\(762\) −1.09901e10 −0.899825
\(763\) 5.12750e9 0.417898
\(764\) −7.07356e9 −0.573867
\(765\) −1.42698e9 −0.115240
\(766\) −4.65853e9 −0.374497
\(767\) −1.47255e9 −0.117838
\(768\) −9.36811e9 −0.746256
\(769\) −1.05289e9 −0.0834910 −0.0417455 0.999128i \(-0.513292\pi\)
−0.0417455 + 0.999128i \(0.513292\pi\)
\(770\) 7.99743e9 0.631296
\(771\) 1.33301e10 1.04747
\(772\) −3.62844e9 −0.283830
\(773\) 5.93967e9 0.462524 0.231262 0.972891i \(-0.425715\pi\)
0.231262 + 0.972891i \(0.425715\pi\)
\(774\) −7.89435e9 −0.611960
\(775\) 1.98378e9 0.153087
\(776\) −8.21856e8 −0.0631364
\(777\) 7.81292e8 0.0597503
\(778\) 2.73774e10 2.08432
\(779\) 7.34305e9 0.556539
\(780\) 3.04583e9 0.229813
\(781\) −1.21124e10 −0.909815
\(782\) 1.90135e10 1.42180
\(783\) 3.32793e9 0.247747
\(784\) −2.65340e9 −0.196651
\(785\) −1.30521e10 −0.963023
\(786\) 5.48228e9 0.402701
\(787\) 1.31636e10 0.962640 0.481320 0.876545i \(-0.340158\pi\)
0.481320 + 0.876545i \(0.340158\pi\)
\(788\) 1.72562e10 1.25633
\(789\) 5.25046e9 0.380564
\(790\) 1.09875e10 0.792875
\(791\) 2.39952e10 1.72388
\(792\) −1.10087e9 −0.0787408
\(793\) 2.04986e10 1.45972
\(794\) −9.90377e9 −0.702148
\(795\) −1.25456e9 −0.0885535
\(796\) 8.03446e9 0.564626
\(797\) −1.29607e10 −0.906825 −0.453412 0.891301i \(-0.649794\pi\)
−0.453412 + 0.891301i \(0.649794\pi\)
\(798\) 7.79703e9 0.543149
\(799\) 3.88977e9 0.269780
\(800\) −1.22646e10 −0.846912
\(801\) −3.87725e9 −0.266569
\(802\) −1.16491e10 −0.797410
\(803\) 9.37471e9 0.638930
\(804\) −5.03195e9 −0.341461
\(805\) 1.61788e10 1.09310
\(806\) −4.07184e9 −0.273916
\(807\) 1.61427e9 0.108123
\(808\) 2.20459e9 0.147024
\(809\) −1.62295e10 −1.07767 −0.538836 0.842411i \(-0.681136\pi\)
−0.538836 + 0.842411i \(0.681136\pi\)
\(810\) −1.27855e9 −0.0845316
\(811\) 8.71515e9 0.573723 0.286861 0.957972i \(-0.407388\pi\)
0.286861 + 0.957972i \(0.407388\pi\)
\(812\) −1.63103e10 −1.06910
\(813\) 1.39964e10 0.913478
\(814\) −1.50603e9 −0.0978696
\(815\) 1.42585e9 0.0922618
\(816\) −6.37839e9 −0.410956
\(817\) 1.40897e10 0.903906
\(818\) −1.63855e10 −1.04670
\(819\) −5.12406e9 −0.325927
\(820\) 5.90136e9 0.373769
\(821\) 2.43096e10 1.53312 0.766561 0.642171i \(-0.221966\pi\)
0.766561 + 0.642171i \(0.221966\pi\)
\(822\) 2.17420e10 1.36536
\(823\) −1.50334e10 −0.940062 −0.470031 0.882650i \(-0.655757\pi\)
−0.470031 + 0.882650i \(0.655757\pi\)
\(824\) −9.62805e9 −0.599506
\(825\) −4.81212e9 −0.298365
\(826\) −3.02949e9 −0.187042
\(827\) −7.68810e9 −0.472661 −0.236331 0.971673i \(-0.575945\pi\)
−0.236331 + 0.971673i \(0.575945\pi\)
\(828\) 7.40434e9 0.453294
\(829\) 4.74776e9 0.289433 0.144716 0.989473i \(-0.453773\pi\)
0.144716 + 0.989473i \(0.453773\pi\)
\(830\) 1.59028e9 0.0965381
\(831\) −6.35557e9 −0.384194
\(832\) 7.46461e9 0.449340
\(833\) −1.68343e9 −0.100911
\(834\) 5.43028e9 0.324146
\(835\) −5.13509e9 −0.305243
\(836\) −6.53242e9 −0.386680
\(837\) 7.42895e8 0.0437913
\(838\) −2.81299e10 −1.65125
\(839\) −1.32986e10 −0.777391 −0.388696 0.921366i \(-0.627074\pi\)
−0.388696 + 0.921366i \(0.627074\pi\)
\(840\) −1.88474e9 −0.109717
\(841\) 1.13370e10 0.657221
\(842\) −3.92973e10 −2.26867
\(843\) 8.54599e9 0.491322
\(844\) 1.03251e9 0.0591150
\(845\) −1.81329e9 −0.103388
\(846\) 3.48516e9 0.197891
\(847\) −7.83179e9 −0.442863
\(848\) −5.60768e9 −0.315790
\(849\) 6.62724e9 0.371668
\(850\) −9.68205e9 −0.540756
\(851\) −3.04670e9 −0.169463
\(852\) −9.49044e9 −0.525712
\(853\) 5.80154e9 0.320053 0.160027 0.987113i \(-0.448842\pi\)
0.160027 + 0.987113i \(0.448842\pi\)
\(854\) 4.21720e10 2.31698
\(855\) 2.28192e9 0.124859
\(856\) −8.12489e8 −0.0442751
\(857\) −2.65557e10 −1.44120 −0.720601 0.693350i \(-0.756134\pi\)
−0.720601 + 0.693350i \(0.756134\pi\)
\(858\) 9.87721e9 0.533861
\(859\) 4.83329e9 0.260176 0.130088 0.991502i \(-0.458474\pi\)
0.130088 + 0.991502i \(0.458474\pi\)
\(860\) 1.13234e10 0.607060
\(861\) −9.92799e9 −0.530091
\(862\) −2.32975e10 −1.23889
\(863\) 9.78741e9 0.518358 0.259179 0.965829i \(-0.416548\pi\)
0.259179 + 0.965829i \(0.416548\pi\)
\(864\) −4.59291e9 −0.242264
\(865\) −8.09085e9 −0.425048
\(866\) 2.60679e10 1.36393
\(867\) 7.03243e9 0.366470
\(868\) −3.64096e9 −0.188972
\(869\) 1.54865e10 0.800543
\(870\) −1.09827e10 −0.565446
\(871\) −1.35794e10 −0.696333
\(872\) 2.32931e9 0.118965
\(873\) −1.34533e9 −0.0684351
\(874\) −3.04050e10 −1.54048
\(875\) −2.04843e10 −1.03369
\(876\) 7.34534e9 0.369188
\(877\) −1.33028e10 −0.665956 −0.332978 0.942935i \(-0.608053\pi\)
−0.332978 + 0.942935i \(0.608053\pi\)
\(878\) 4.72767e10 2.35731
\(879\) −5.32337e9 −0.264378
\(880\) 1.04620e10 0.517518
\(881\) 1.14650e10 0.564885 0.282443 0.959284i \(-0.408855\pi\)
0.282443 + 0.959284i \(0.408855\pi\)
\(882\) −1.50832e9 −0.0740208
\(883\) −1.56517e10 −0.765067 −0.382533 0.923942i \(-0.624948\pi\)
−0.382533 + 0.923942i \(0.624948\pi\)
\(884\) 8.63753e9 0.420540
\(885\) −8.86627e8 −0.0429971
\(886\) −2.82847e10 −1.36626
\(887\) 6.19771e9 0.298194 0.149097 0.988823i \(-0.452363\pi\)
0.149097 + 0.988823i \(0.452363\pi\)
\(888\) 3.54924e8 0.0170094
\(889\) −2.65197e10 −1.26594
\(890\) 1.27955e10 0.608405
\(891\) −1.80207e9 −0.0853491
\(892\) −2.48859e10 −1.17402
\(893\) −6.22022e9 −0.292298
\(894\) −6.53167e9 −0.305734
\(895\) −9.94587e9 −0.463727
\(896\) −1.39235e10 −0.646653
\(897\) 1.99816e10 0.924393
\(898\) −3.53577e10 −1.62936
\(899\) 6.38144e9 0.292927
\(900\) −3.77043e9 −0.172402
\(901\) −3.55775e9 −0.162046
\(902\) 1.91373e10 0.868277
\(903\) −1.90496e10 −0.860950
\(904\) 1.09005e10 0.490745
\(905\) 1.00680e10 0.451518
\(906\) −2.10487e10 −0.940322
\(907\) −2.84672e10 −1.26683 −0.633417 0.773811i \(-0.718348\pi\)
−0.633417 + 0.773811i \(0.718348\pi\)
\(908\) 4.50663e7 0.00199780
\(909\) 3.60878e9 0.159363
\(910\) 1.69102e10 0.743881
\(911\) 1.41611e10 0.620558 0.310279 0.950646i \(-0.399578\pi\)
0.310279 + 0.950646i \(0.399578\pi\)
\(912\) 1.01998e10 0.445258
\(913\) 2.24144e9 0.0974717
\(914\) 2.85398e9 0.123634
\(915\) 1.23423e10 0.532625
\(916\) −1.82190e10 −0.783231
\(917\) 1.32291e10 0.566549
\(918\) −3.62578e9 −0.154686
\(919\) 1.43616e10 0.610379 0.305189 0.952292i \(-0.401280\pi\)
0.305189 + 0.952292i \(0.401280\pi\)
\(920\) 7.34968e9 0.311179
\(921\) −3.62297e8 −0.0152812
\(922\) 3.12374e10 1.31255
\(923\) −2.56112e10 −1.07207
\(924\) 8.83200e9 0.368305
\(925\) 1.55144e9 0.0644522
\(926\) −4.52709e10 −1.87362
\(927\) −1.57606e10 −0.649819
\(928\) −3.94529e10 −1.62055
\(929\) 2.83698e10 1.16092 0.580458 0.814290i \(-0.302873\pi\)
0.580458 + 0.814290i \(0.302873\pi\)
\(930\) −2.45167e9 −0.0999473
\(931\) 2.69201e9 0.109334
\(932\) −3.06053e10 −1.23834
\(933\) 2.00439e10 0.807972
\(934\) 2.00185e10 0.803928
\(935\) 6.63753e9 0.265562
\(936\) −2.32775e9 −0.0927834
\(937\) −4.33315e10 −1.72074 −0.860370 0.509669i \(-0.829768\pi\)
−0.860370 + 0.509669i \(0.829768\pi\)
\(938\) −2.79370e10 −1.10527
\(939\) −2.01945e10 −0.795984
\(940\) −4.99898e9 −0.196306
\(941\) 3.82084e10 1.49484 0.747421 0.664351i \(-0.231292\pi\)
0.747421 + 0.664351i \(0.231292\pi\)
\(942\) −3.31638e10 −1.29266
\(943\) 3.87148e10 1.50344
\(944\) −3.96309e9 −0.153332
\(945\) −3.08521e9 −0.118925
\(946\) 3.67202e10 1.41022
\(947\) 1.03534e10 0.396148 0.198074 0.980187i \(-0.436531\pi\)
0.198074 + 0.980187i \(0.436531\pi\)
\(948\) 1.21341e10 0.462572
\(949\) 1.98224e10 0.752877
\(950\) 1.54828e10 0.585891
\(951\) 2.06659e10 0.779153
\(952\) −5.34487e9 −0.200774
\(953\) 2.92299e10 1.09396 0.546981 0.837145i \(-0.315777\pi\)
0.546981 + 0.837145i \(0.315777\pi\)
\(954\) −3.18767e9 −0.118865
\(955\) −1.14935e10 −0.427013
\(956\) −1.42077e10 −0.525921
\(957\) −1.54797e10 −0.570914
\(958\) −3.73828e10 −1.37370
\(959\) 5.24648e10 1.92089
\(960\) 4.49447e9 0.163957
\(961\) −2.60881e10 −0.948223
\(962\) −3.18443e9 −0.115324
\(963\) −1.33000e9 −0.0479908
\(964\) −9.51315e9 −0.342022
\(965\) −5.89568e9 −0.211197
\(966\) 4.11084e10 1.46727
\(967\) 4.59588e9 0.163447 0.0817233 0.996655i \(-0.473958\pi\)
0.0817233 + 0.996655i \(0.473958\pi\)
\(968\) −3.55781e9 −0.126072
\(969\) 6.47120e9 0.228482
\(970\) 4.43980e9 0.156193
\(971\) −5.52070e10 −1.93520 −0.967602 0.252481i \(-0.918753\pi\)
−0.967602 + 0.252481i \(0.918753\pi\)
\(972\) −1.41197e9 −0.0493166
\(973\) 1.31036e10 0.456033
\(974\) 2.10531e10 0.730063
\(975\) −1.01750e10 −0.351576
\(976\) 5.51681e10 1.89939
\(977\) −4.44629e9 −0.152534 −0.0762671 0.997087i \(-0.524300\pi\)
−0.0762671 + 0.997087i \(0.524300\pi\)
\(978\) 3.62290e9 0.123843
\(979\) 1.80348e10 0.614288
\(980\) 2.16348e9 0.0734280
\(981\) 3.81294e9 0.128949
\(982\) −2.98328e10 −1.00532
\(983\) 3.90750e10 1.31208 0.656041 0.754725i \(-0.272230\pi\)
0.656041 + 0.754725i \(0.272230\pi\)
\(984\) −4.51007e9 −0.150904
\(985\) 2.80387e10 0.934828
\(986\) −3.11454e10 −1.03472
\(987\) 8.40990e9 0.278407
\(988\) −1.38125e10 −0.455641
\(989\) 7.42850e10 2.44182
\(990\) 5.94710e9 0.194797
\(991\) −1.52795e10 −0.498713 −0.249357 0.968412i \(-0.580219\pi\)
−0.249357 + 0.968412i \(0.580219\pi\)
\(992\) −8.80708e9 −0.286445
\(993\) −8.45439e9 −0.274006
\(994\) −5.26902e10 −1.70168
\(995\) 1.30548e10 0.420137
\(996\) 1.75623e9 0.0563214
\(997\) 1.22551e10 0.391637 0.195818 0.980640i \(-0.437264\pi\)
0.195818 + 0.980640i \(0.437264\pi\)
\(998\) −5.27883e10 −1.68105
\(999\) 5.80989e8 0.0184369
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.1.4 16
3.2 odd 2 531.8.a.b.1.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.4 16 1.1 even 1 trivial
531.8.a.b.1.13 16 3.2 odd 2