Properties

Label 177.12.a.b.1.15
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.16187 q^{2} +243.000 q^{3} -2043.33 q^{4} +654.400 q^{5} -525.335 q^{6} +79533.7 q^{7} +8844.92 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-2.16187 q^{2} +243.000 q^{3} -2043.33 q^{4} +654.400 q^{5} -525.335 q^{6} +79533.7 q^{7} +8844.92 q^{8} +59049.0 q^{9} -1414.73 q^{10} -361945. q^{11} -496528. q^{12} -608677. q^{13} -171942. q^{14} +159019. q^{15} +4.16561e6 q^{16} +1.14080e6 q^{17} -127656. q^{18} -6.87969e6 q^{19} -1.33715e6 q^{20} +1.93267e7 q^{21} +782479. q^{22} -2.10594e7 q^{23} +2.14932e6 q^{24} -4.83999e7 q^{25} +1.31588e6 q^{26} +1.43489e7 q^{27} -1.62513e8 q^{28} -1.41194e8 q^{29} -343779. q^{30} +8.71043e7 q^{31} -2.71199e7 q^{32} -8.79527e7 q^{33} -2.46625e6 q^{34} +5.20469e7 q^{35} -1.20656e8 q^{36} +5.08594e8 q^{37} +1.48730e7 q^{38} -1.47909e8 q^{39} +5.78812e6 q^{40} -2.45057e8 q^{41} -4.17818e7 q^{42} +1.63401e8 q^{43} +7.39572e8 q^{44} +3.86417e7 q^{45} +4.55276e7 q^{46} +1.77966e9 q^{47} +1.01224e9 q^{48} +4.34828e9 q^{49} +1.04634e8 q^{50} +2.77213e8 q^{51} +1.24373e9 q^{52} -4.11758e9 q^{53} -3.10205e7 q^{54} -2.36857e8 q^{55} +7.03469e8 q^{56} -1.67176e9 q^{57} +3.05244e8 q^{58} -7.14924e8 q^{59} -3.24928e8 q^{60} -2.50892e9 q^{61} -1.88308e8 q^{62} +4.69639e9 q^{63} -8.47254e9 q^{64} -3.98318e8 q^{65} +1.90142e8 q^{66} -3.37618e9 q^{67} -2.33102e9 q^{68} -5.11743e9 q^{69} -1.12519e8 q^{70} -6.16327e9 q^{71} +5.22284e8 q^{72} +2.20149e9 q^{73} -1.09951e9 q^{74} -1.17612e10 q^{75} +1.40574e10 q^{76} -2.87868e10 q^{77} +3.19759e8 q^{78} +2.69116e10 q^{79} +2.72598e9 q^{80} +3.48678e9 q^{81} +5.29781e8 q^{82} -1.69513e10 q^{83} -3.94907e10 q^{84} +7.46537e8 q^{85} -3.53251e8 q^{86} -3.43102e10 q^{87} -3.20138e9 q^{88} -5.65822e10 q^{89} -8.35384e7 q^{90} -4.84103e10 q^{91} +4.30312e10 q^{92} +2.11663e10 q^{93} -3.84740e9 q^{94} -4.50207e9 q^{95} -6.59014e9 q^{96} -2.14261e9 q^{97} -9.40043e9 q^{98} -2.13725e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} + O(q^{10}) \) \( 27q - 128q^{2} + 6561q^{3} + 26142q^{4} - 17188q^{5} - 31104q^{6} - 126579q^{7} - 355797q^{8} + 1594323q^{9} - 383719q^{10} - 1816556q^{11} + 6352506q^{12} - 3951804q^{13} - 6207867q^{14} - 4176684q^{15} + 28295194q^{16} - 17723275q^{17} - 7558272q^{18} - 19573013q^{19} - 48468099q^{20} - 30758697q^{21} - 1729910q^{22} - 88593797q^{23} - 86458671q^{24} + 345714963q^{25} - 6676346q^{26} + 387420489q^{27} + 126954286q^{28} - 276632427q^{29} - 93243717q^{30} - 357680917q^{31} - 859842334q^{32} - 441423108q^{33} + 232730000q^{34} - 510315139q^{35} + 1543658958q^{36} - 660238257q^{37} - 2067286961q^{38} - 960288372q^{39} - 3388951110q^{40} - 1671147569q^{41} - 1508511681q^{42} - 1883107790q^{43} - 3895687630q^{44} - 1014934212q^{45} - 1720344243q^{46} - 5818572501q^{47} + 6875732142q^{48} - 18858180q^{49} - 21474519647q^{50} - 4306755825q^{51} - 42214560062q^{52} - 11444513368q^{53} - 1836660096q^{54} - 24401486484q^{55} - 50583585764q^{56} - 4756242159q^{57} - 45017395090q^{58} - 19302956073q^{59} - 11777748057q^{60} + 408637955q^{61} - 28543084070q^{62} - 7474363371q^{63} + 33067284293q^{64} - 21656714730q^{65} - 420368130q^{66} - 49803132690q^{67} - 16500749319q^{68} - 21528292671q^{69} - 45808890782q^{70} - 34127492216q^{71} - 21009457053q^{72} - 55734362153q^{73} - 40367816298q^{74} + 84008736009q^{75} - 14840406404q^{76} - 99723443615q^{77} - 1622352078q^{78} - 76484916442q^{79} + 93882788915q^{80} + 94143178827q^{81} + 52951239205q^{82} - 140433865655q^{83} + 30849891498q^{84} + 34329063335q^{85} + 175223869508q^{86} - 67221679761q^{87} + 268823645069q^{88} - 1191878597q^{89} - 22658223231q^{90} + 201632581559q^{91} - 206501888812q^{92} - 86916462831q^{93} + 319770144384q^{94} - 81387074885q^{95} - 208941687162q^{96} - 144896178730q^{97} + 135739195260q^{98} - 107265815244q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.16187 −0.0477711 −0.0238855 0.999715i \(-0.507604\pi\)
−0.0238855 + 0.999715i \(0.507604\pi\)
\(3\) 243.000 0.577350
\(4\) −2043.33 −0.997718
\(5\) 654.400 0.0936502 0.0468251 0.998903i \(-0.485090\pi\)
0.0468251 + 0.998903i \(0.485090\pi\)
\(6\) −525.335 −0.0275806
\(7\) 79533.7 1.78859 0.894297 0.447473i \(-0.147676\pi\)
0.894297 + 0.447473i \(0.147676\pi\)
\(8\) 8844.92 0.0954331
\(9\) 59049.0 0.333333
\(10\) −1414.73 −0.00447377
\(11\) −361945. −0.677615 −0.338808 0.940856i \(-0.610024\pi\)
−0.338808 + 0.940856i \(0.610024\pi\)
\(12\) −496528. −0.576033
\(13\) −608677. −0.454672 −0.227336 0.973816i \(-0.573002\pi\)
−0.227336 + 0.973816i \(0.573002\pi\)
\(14\) −171942. −0.0854431
\(15\) 159019. 0.0540689
\(16\) 4.16561e6 0.993159
\(17\) 1.14080e6 0.194867 0.0974336 0.995242i \(-0.468937\pi\)
0.0974336 + 0.995242i \(0.468937\pi\)
\(18\) −127656. −0.0159237
\(19\) −6.87969e6 −0.637418 −0.318709 0.947853i \(-0.603249\pi\)
−0.318709 + 0.947853i \(0.603249\pi\)
\(20\) −1.33715e6 −0.0934364
\(21\) 1.93267e7 1.03265
\(22\) 782479. 0.0323704
\(23\) −2.10594e7 −0.682248 −0.341124 0.940018i \(-0.610808\pi\)
−0.341124 + 0.940018i \(0.610808\pi\)
\(24\) 2.14932e6 0.0550983
\(25\) −4.83999e7 −0.991230
\(26\) 1.31588e6 0.0217202
\(27\) 1.43489e7 0.192450
\(28\) −1.62513e8 −1.78451
\(29\) −1.41194e8 −1.27829 −0.639143 0.769088i \(-0.720711\pi\)
−0.639143 + 0.769088i \(0.720711\pi\)
\(30\) −343779. −0.00258293
\(31\) 8.71043e7 0.546450 0.273225 0.961950i \(-0.411910\pi\)
0.273225 + 0.961950i \(0.411910\pi\)
\(32\) −2.71199e7 −0.142877
\(33\) −8.79527e7 −0.391221
\(34\) −2.46625e6 −0.00930901
\(35\) 5.20469e7 0.167502
\(36\) −1.20656e8 −0.332573
\(37\) 5.08594e8 1.20576 0.602881 0.797831i \(-0.294019\pi\)
0.602881 + 0.797831i \(0.294019\pi\)
\(38\) 1.48730e7 0.0304501
\(39\) −1.47909e8 −0.262505
\(40\) 5.78812e6 0.00893733
\(41\) −2.45057e8 −0.330336 −0.165168 0.986265i \(-0.552817\pi\)
−0.165168 + 0.986265i \(0.552817\pi\)
\(42\) −4.17818e7 −0.0493306
\(43\) 1.63401e8 0.169503 0.0847514 0.996402i \(-0.472990\pi\)
0.0847514 + 0.996402i \(0.472990\pi\)
\(44\) 7.39572e8 0.676069
\(45\) 3.86417e7 0.0312167
\(46\) 4.55276e7 0.0325917
\(47\) 1.77966e9 1.13188 0.565938 0.824448i \(-0.308514\pi\)
0.565938 + 0.824448i \(0.308514\pi\)
\(48\) 1.01224e9 0.573401
\(49\) 4.34828e9 2.19907
\(50\) 1.04634e8 0.0473521
\(51\) 2.77213e8 0.112507
\(52\) 1.24373e9 0.453635
\(53\) −4.11758e9 −1.35246 −0.676231 0.736690i \(-0.736388\pi\)
−0.676231 + 0.736690i \(0.736388\pi\)
\(54\) −3.10205e7 −0.00919355
\(55\) −2.36857e8 −0.0634588
\(56\) 7.03469e8 0.170691
\(57\) −1.67176e9 −0.368013
\(58\) 3.05244e8 0.0610651
\(59\) −7.14924e8 −0.130189
\(60\) −3.24928e8 −0.0539456
\(61\) −2.50892e9 −0.380340 −0.190170 0.981751i \(-0.560904\pi\)
−0.190170 + 0.981751i \(0.560904\pi\)
\(62\) −1.88308e8 −0.0261045
\(63\) 4.69639e9 0.596198
\(64\) −8.47254e9 −0.986334
\(65\) −3.98318e8 −0.0425801
\(66\) 1.90142e8 0.0186891
\(67\) −3.37618e9 −0.305502 −0.152751 0.988265i \(-0.548813\pi\)
−0.152751 + 0.988265i \(0.548813\pi\)
\(68\) −2.33102e9 −0.194423
\(69\) −5.11743e9 −0.393896
\(70\) −1.12519e8 −0.00800176
\(71\) −6.16327e9 −0.405406 −0.202703 0.979240i \(-0.564973\pi\)
−0.202703 + 0.979240i \(0.564973\pi\)
\(72\) 5.22284e8 0.0318110
\(73\) 2.20149e9 0.124292 0.0621458 0.998067i \(-0.480206\pi\)
0.0621458 + 0.998067i \(0.480206\pi\)
\(74\) −1.09951e9 −0.0576005
\(75\) −1.17612e10 −0.572287
\(76\) 1.40574e10 0.635963
\(77\) −2.87868e10 −1.21198
\(78\) 3.19759e8 0.0125402
\(79\) 2.69116e10 0.983991 0.491995 0.870598i \(-0.336268\pi\)
0.491995 + 0.870598i \(0.336268\pi\)
\(80\) 2.72598e9 0.0930095
\(81\) 3.48678e9 0.111111
\(82\) 5.29781e8 0.0157805
\(83\) −1.69513e10 −0.472361 −0.236180 0.971709i \(-0.575896\pi\)
−0.236180 + 0.971709i \(0.575896\pi\)
\(84\) −3.94907e10 −1.03029
\(85\) 7.46537e8 0.0182493
\(86\) −3.53251e8 −0.00809733
\(87\) −3.43102e10 −0.738019
\(88\) −3.20138e9 −0.0646669
\(89\) −5.65822e10 −1.07408 −0.537038 0.843558i \(-0.680457\pi\)
−0.537038 + 0.843558i \(0.680457\pi\)
\(90\) −8.35384e7 −0.00149126
\(91\) −4.84103e10 −0.813224
\(92\) 4.30312e10 0.680691
\(93\) 2.11663e10 0.315493
\(94\) −3.84740e9 −0.0540710
\(95\) −4.50207e9 −0.0596942
\(96\) −6.59014e9 −0.0824903
\(97\) −2.14261e9 −0.0253336 −0.0126668 0.999920i \(-0.504032\pi\)
−0.0126668 + 0.999920i \(0.504032\pi\)
\(98\) −9.40043e9 −0.105052
\(99\) −2.13725e10 −0.225872
\(100\) 9.88968e10 0.988968
\(101\) 9.24270e10 0.875047 0.437524 0.899207i \(-0.355856\pi\)
0.437524 + 0.899207i \(0.355856\pi\)
\(102\) −5.99300e8 −0.00537456
\(103\) −1.07966e11 −0.917665 −0.458832 0.888523i \(-0.651732\pi\)
−0.458832 + 0.888523i \(0.651732\pi\)
\(104\) −5.38370e9 −0.0433908
\(105\) 1.26474e10 0.0967074
\(106\) 8.90168e9 0.0646085
\(107\) 5.38904e10 0.371450 0.185725 0.982602i \(-0.440537\pi\)
0.185725 + 0.982602i \(0.440537\pi\)
\(108\) −2.93195e10 −0.192011
\(109\) −8.67119e10 −0.539800 −0.269900 0.962888i \(-0.586991\pi\)
−0.269900 + 0.962888i \(0.586991\pi\)
\(110\) 5.12055e8 0.00303149
\(111\) 1.23588e11 0.696147
\(112\) 3.31306e11 1.77636
\(113\) −2.77274e11 −1.41572 −0.707861 0.706352i \(-0.750340\pi\)
−0.707861 + 0.706352i \(0.750340\pi\)
\(114\) 3.61414e9 0.0175804
\(115\) −1.37813e10 −0.0638926
\(116\) 2.88506e11 1.27537
\(117\) −3.59418e10 −0.151557
\(118\) 1.54557e9 0.00621926
\(119\) 9.07317e10 0.348538
\(120\) 1.40651e9 0.00515997
\(121\) −1.54307e11 −0.540838
\(122\) 5.42396e9 0.0181693
\(123\) −5.95488e10 −0.190719
\(124\) −1.77982e11 −0.545203
\(125\) −6.36260e10 −0.186479
\(126\) −1.01530e10 −0.0284810
\(127\) −2.23170e11 −0.599399 −0.299700 0.954034i \(-0.596886\pi\)
−0.299700 + 0.954034i \(0.596886\pi\)
\(128\) 7.38581e10 0.189996
\(129\) 3.97063e10 0.0978625
\(130\) 8.61113e8 0.00203410
\(131\) −8.35848e10 −0.189293 −0.0946467 0.995511i \(-0.530172\pi\)
−0.0946467 + 0.995511i \(0.530172\pi\)
\(132\) 1.79716e11 0.390329
\(133\) −5.47167e11 −1.14008
\(134\) 7.29887e9 0.0145942
\(135\) 9.38993e9 0.0180230
\(136\) 1.00902e10 0.0185968
\(137\) 4.52915e10 0.0801777 0.0400889 0.999196i \(-0.487236\pi\)
0.0400889 + 0.999196i \(0.487236\pi\)
\(138\) 1.10632e10 0.0188168
\(139\) −4.20037e11 −0.686604 −0.343302 0.939225i \(-0.611545\pi\)
−0.343302 + 0.939225i \(0.611545\pi\)
\(140\) −1.06349e11 −0.167120
\(141\) 4.32458e11 0.653489
\(142\) 1.33242e10 0.0193667
\(143\) 2.20308e11 0.308093
\(144\) 2.45975e11 0.331053
\(145\) −9.23976e10 −0.119712
\(146\) −4.75934e9 −0.00593754
\(147\) 1.05663e12 1.26963
\(148\) −1.03922e12 −1.20301
\(149\) 3.64336e11 0.406422 0.203211 0.979135i \(-0.434862\pi\)
0.203211 + 0.979135i \(0.434862\pi\)
\(150\) 2.54261e10 0.0273387
\(151\) −1.39489e12 −1.44599 −0.722996 0.690852i \(-0.757236\pi\)
−0.722996 + 0.690852i \(0.757236\pi\)
\(152\) −6.08503e10 −0.0608307
\(153\) 6.73628e10 0.0649557
\(154\) 6.22335e10 0.0578975
\(155\) 5.70011e10 0.0511751
\(156\) 3.02225e11 0.261906
\(157\) 4.10340e11 0.343317 0.171659 0.985156i \(-0.445087\pi\)
0.171659 + 0.985156i \(0.445087\pi\)
\(158\) −5.81795e10 −0.0470063
\(159\) −1.00057e12 −0.780844
\(160\) −1.77473e10 −0.0133805
\(161\) −1.67493e12 −1.22027
\(162\) −7.53798e9 −0.00530790
\(163\) −2.54325e12 −1.73124 −0.865620 0.500701i \(-0.833076\pi\)
−0.865620 + 0.500701i \(0.833076\pi\)
\(164\) 5.00731e11 0.329582
\(165\) −5.75563e10 −0.0366379
\(166\) 3.66465e10 0.0225652
\(167\) −1.38079e11 −0.0822598 −0.0411299 0.999154i \(-0.513096\pi\)
−0.0411299 + 0.999154i \(0.513096\pi\)
\(168\) 1.70943e11 0.0985486
\(169\) −1.42167e12 −0.793273
\(170\) −1.61392e9 −0.000871791 0
\(171\) −4.06239e11 −0.212473
\(172\) −3.33881e11 −0.169116
\(173\) 1.91015e11 0.0937159 0.0468580 0.998902i \(-0.485079\pi\)
0.0468580 + 0.998902i \(0.485079\pi\)
\(174\) 7.41743e10 0.0352559
\(175\) −3.84942e12 −1.77291
\(176\) −1.50772e12 −0.672980
\(177\) −1.73727e11 −0.0751646
\(178\) 1.22323e11 0.0513097
\(179\) 3.55183e12 1.44464 0.722321 0.691558i \(-0.243075\pi\)
0.722321 + 0.691558i \(0.243075\pi\)
\(180\) −7.89576e10 −0.0311455
\(181\) 3.50928e12 1.34272 0.671360 0.741132i \(-0.265711\pi\)
0.671360 + 0.741132i \(0.265711\pi\)
\(182\) 1.04657e11 0.0388486
\(183\) −6.09667e11 −0.219589
\(184\) −1.86268e11 −0.0651090
\(185\) 3.32824e11 0.112920
\(186\) −4.57589e10 −0.0150714
\(187\) −4.12906e11 −0.132045
\(188\) −3.63643e12 −1.12929
\(189\) 1.14122e12 0.344215
\(190\) 9.73290e9 0.00285166
\(191\) −4.78607e12 −1.36237 −0.681186 0.732111i \(-0.738535\pi\)
−0.681186 + 0.732111i \(0.738535\pi\)
\(192\) −2.05883e12 −0.569460
\(193\) −2.49448e12 −0.670524 −0.335262 0.942125i \(-0.608825\pi\)
−0.335262 + 0.942125i \(0.608825\pi\)
\(194\) 4.63204e9 0.00121022
\(195\) −9.67914e10 −0.0245836
\(196\) −8.88496e12 −2.19405
\(197\) −7.83320e12 −1.88094 −0.940470 0.339877i \(-0.889615\pi\)
−0.940470 + 0.339877i \(0.889615\pi\)
\(198\) 4.62046e10 0.0107901
\(199\) 1.97236e12 0.448016 0.224008 0.974587i \(-0.428086\pi\)
0.224008 + 0.974587i \(0.428086\pi\)
\(200\) −4.28093e11 −0.0945961
\(201\) −8.20412e11 −0.176382
\(202\) −1.99815e11 −0.0418019
\(203\) −1.12297e13 −2.28634
\(204\) −5.66437e11 −0.112250
\(205\) −1.60365e11 −0.0309360
\(206\) 2.33410e11 0.0438378
\(207\) −1.24353e12 −0.227416
\(208\) −2.53551e12 −0.451562
\(209\) 2.49007e12 0.431924
\(210\) −2.73420e10 −0.00461982
\(211\) −1.10288e12 −0.181540 −0.0907702 0.995872i \(-0.528933\pi\)
−0.0907702 + 0.995872i \(0.528933\pi\)
\(212\) 8.41356e12 1.34937
\(213\) −1.49768e12 −0.234061
\(214\) −1.16504e11 −0.0177446
\(215\) 1.06929e11 0.0158740
\(216\) 1.26915e11 0.0183661
\(217\) 6.92773e12 0.977377
\(218\) 1.87460e11 0.0257868
\(219\) 5.34963e11 0.0717597
\(220\) 4.83976e11 0.0633139
\(221\) −6.94376e11 −0.0886007
\(222\) −2.67182e11 −0.0332557
\(223\) −6.53558e12 −0.793610 −0.396805 0.917903i \(-0.629881\pi\)
−0.396805 + 0.917903i \(0.629881\pi\)
\(224\) −2.15695e12 −0.255550
\(225\) −2.85796e12 −0.330410
\(226\) 5.99431e11 0.0676305
\(227\) 1.24917e13 1.37556 0.687778 0.725921i \(-0.258586\pi\)
0.687778 + 0.725921i \(0.258586\pi\)
\(228\) 3.41596e12 0.367173
\(229\) 6.05763e12 0.635635 0.317817 0.948152i \(-0.397050\pi\)
0.317817 + 0.948152i \(0.397050\pi\)
\(230\) 2.97933e10 0.00305222
\(231\) −6.99520e12 −0.699736
\(232\) −1.24885e12 −0.121991
\(233\) −8.58130e12 −0.818644 −0.409322 0.912390i \(-0.634235\pi\)
−0.409322 + 0.912390i \(0.634235\pi\)
\(234\) 7.77015e10 0.00724006
\(235\) 1.16461e12 0.106000
\(236\) 1.46082e12 0.129892
\(237\) 6.53953e12 0.568107
\(238\) −1.96150e11 −0.0166501
\(239\) −1.96699e12 −0.163160 −0.0815799 0.996667i \(-0.525997\pi\)
−0.0815799 + 0.996667i \(0.525997\pi\)
\(240\) 6.62412e11 0.0536991
\(241\) 1.96077e13 1.55358 0.776789 0.629761i \(-0.216847\pi\)
0.776789 + 0.629761i \(0.216847\pi\)
\(242\) 3.33593e11 0.0258364
\(243\) 8.47289e11 0.0641500
\(244\) 5.12654e12 0.379472
\(245\) 2.84552e12 0.205943
\(246\) 1.28737e11 0.00911087
\(247\) 4.18751e12 0.289816
\(248\) 7.70431e11 0.0521494
\(249\) −4.11917e12 −0.272718
\(250\) 1.37551e11 0.00890830
\(251\) −4.19882e12 −0.266025 −0.133012 0.991114i \(-0.542465\pi\)
−0.133012 + 0.991114i \(0.542465\pi\)
\(252\) −9.59625e12 −0.594838
\(253\) 7.62234e12 0.462302
\(254\) 4.82466e11 0.0286339
\(255\) 1.81409e11 0.0105363
\(256\) 1.71921e13 0.977257
\(257\) −2.94651e13 −1.63937 −0.819684 0.572816i \(-0.805851\pi\)
−0.819684 + 0.572816i \(0.805851\pi\)
\(258\) −8.58400e10 −0.00467500
\(259\) 4.04503e13 2.15662
\(260\) 8.13895e11 0.0424830
\(261\) −8.33738e12 −0.426095
\(262\) 1.80700e11 0.00904274
\(263\) −1.13059e13 −0.554049 −0.277024 0.960863i \(-0.589348\pi\)
−0.277024 + 0.960863i \(0.589348\pi\)
\(264\) −7.77935e11 −0.0373355
\(265\) −2.69455e12 −0.126658
\(266\) 1.18290e12 0.0544629
\(267\) −1.37495e13 −0.620118
\(268\) 6.89864e12 0.304805
\(269\) −3.82578e13 −1.65608 −0.828042 0.560665i \(-0.810545\pi\)
−0.828042 + 0.560665i \(0.810545\pi\)
\(270\) −2.02998e10 −0.000860977 0
\(271\) −9.60782e12 −0.399295 −0.199647 0.979868i \(-0.563980\pi\)
−0.199647 + 0.979868i \(0.563980\pi\)
\(272\) 4.75211e12 0.193534
\(273\) −1.17637e13 −0.469515
\(274\) −9.79144e10 −0.00383017
\(275\) 1.75181e13 0.671672
\(276\) 1.04566e13 0.392997
\(277\) −2.31076e13 −0.851366 −0.425683 0.904872i \(-0.639966\pi\)
−0.425683 + 0.904872i \(0.639966\pi\)
\(278\) 9.08066e11 0.0327998
\(279\) 5.14342e12 0.182150
\(280\) 4.60351e11 0.0159853
\(281\) −3.08669e13 −1.05101 −0.525506 0.850790i \(-0.676124\pi\)
−0.525506 + 0.850790i \(0.676124\pi\)
\(282\) −9.34918e11 −0.0312179
\(283\) −2.49874e13 −0.818268 −0.409134 0.912474i \(-0.634169\pi\)
−0.409134 + 0.912474i \(0.634169\pi\)
\(284\) 1.25936e13 0.404481
\(285\) −1.09400e12 −0.0344645
\(286\) −4.76277e11 −0.0147179
\(287\) −1.94903e13 −0.590837
\(288\) −1.60140e12 −0.0476258
\(289\) −3.29705e13 −0.962027
\(290\) 1.99752e11 0.00571875
\(291\) −5.20653e11 −0.0146264
\(292\) −4.49837e12 −0.124008
\(293\) −2.28908e13 −0.619283 −0.309642 0.950853i \(-0.600209\pi\)
−0.309642 + 0.950853i \(0.600209\pi\)
\(294\) −2.28430e12 −0.0606518
\(295\) −4.67847e11 −0.0121922
\(296\) 4.49847e12 0.115070
\(297\) −5.19352e12 −0.130407
\(298\) −7.87647e11 −0.0194152
\(299\) 1.28184e13 0.310199
\(300\) 2.40319e13 0.570981
\(301\) 1.29958e13 0.303172
\(302\) 3.01557e12 0.0690766
\(303\) 2.24598e13 0.505209
\(304\) −2.86581e13 −0.633057
\(305\) −1.64184e12 −0.0356189
\(306\) −1.45630e11 −0.00310300
\(307\) 9.03824e12 0.189157 0.0945786 0.995517i \(-0.469850\pi\)
0.0945786 + 0.995517i \(0.469850\pi\)
\(308\) 5.88209e13 1.20921
\(309\) −2.62358e13 −0.529814
\(310\) −1.23229e11 −0.00244469
\(311\) 2.75466e13 0.536891 0.268445 0.963295i \(-0.413490\pi\)
0.268445 + 0.963295i \(0.413490\pi\)
\(312\) −1.30824e12 −0.0250517
\(313\) −5.33653e13 −1.00407 −0.502036 0.864847i \(-0.667415\pi\)
−0.502036 + 0.864847i \(0.667415\pi\)
\(314\) −8.87103e11 −0.0164006
\(315\) 3.07332e12 0.0558341
\(316\) −5.49892e13 −0.981745
\(317\) 2.18062e13 0.382607 0.191304 0.981531i \(-0.438728\pi\)
0.191304 + 0.981531i \(0.438728\pi\)
\(318\) 2.16311e12 0.0373017
\(319\) 5.11046e13 0.866186
\(320\) −5.54443e12 −0.0923703
\(321\) 1.30954e13 0.214457
\(322\) 3.62098e12 0.0582934
\(323\) −7.84832e12 −0.124212
\(324\) −7.12464e12 −0.110858
\(325\) 2.94599e13 0.450685
\(326\) 5.49818e12 0.0827032
\(327\) −2.10710e13 −0.311654
\(328\) −2.16751e12 −0.0315250
\(329\) 1.41543e14 2.02447
\(330\) 1.24429e11 0.00175023
\(331\) 5.38836e13 0.745423 0.372712 0.927947i \(-0.378428\pi\)
0.372712 + 0.927947i \(0.378428\pi\)
\(332\) 3.46371e13 0.471283
\(333\) 3.00320e13 0.401921
\(334\) 2.98509e11 0.00392964
\(335\) −2.20937e12 −0.0286103
\(336\) 8.05075e13 1.02558
\(337\) 6.64623e13 0.832935 0.416467 0.909151i \(-0.363268\pi\)
0.416467 + 0.909151i \(0.363268\pi\)
\(338\) 3.07347e12 0.0378955
\(339\) −6.73776e13 −0.817367
\(340\) −1.52542e12 −0.0182077
\(341\) −3.15270e13 −0.370283
\(342\) 8.78236e11 0.0101500
\(343\) 1.88571e14 2.14465
\(344\) 1.44527e12 0.0161762
\(345\) −3.34885e12 −0.0368884
\(346\) −4.12949e11 −0.00447691
\(347\) −7.03218e12 −0.0750374 −0.0375187 0.999296i \(-0.511945\pi\)
−0.0375187 + 0.999296i \(0.511945\pi\)
\(348\) 7.01069e13 0.736335
\(349\) −8.02464e13 −0.829632 −0.414816 0.909905i \(-0.636154\pi\)
−0.414816 + 0.909905i \(0.636154\pi\)
\(350\) 8.32195e12 0.0846937
\(351\) −8.73385e12 −0.0875017
\(352\) 9.81592e12 0.0968159
\(353\) 4.65152e13 0.451683 0.225842 0.974164i \(-0.427487\pi\)
0.225842 + 0.974164i \(0.427487\pi\)
\(354\) 3.75575e11 0.00359069
\(355\) −4.03325e12 −0.0379664
\(356\) 1.15616e14 1.07162
\(357\) 2.20478e13 0.201229
\(358\) −7.67860e12 −0.0690121
\(359\) 3.18415e13 0.281822 0.140911 0.990022i \(-0.454997\pi\)
0.140911 + 0.990022i \(0.454997\pi\)
\(360\) 3.41783e11 0.00297911
\(361\) −6.91601e13 −0.593699
\(362\) −7.58660e12 −0.0641431
\(363\) −3.74967e13 −0.312253
\(364\) 9.89181e13 0.811369
\(365\) 1.44066e12 0.0116399
\(366\) 1.31802e12 0.0104900
\(367\) 1.63532e14 1.28215 0.641074 0.767479i \(-0.278489\pi\)
0.641074 + 0.767479i \(0.278489\pi\)
\(368\) −8.77251e13 −0.677581
\(369\) −1.44704e13 −0.110112
\(370\) −7.19523e11 −0.00539430
\(371\) −3.27486e14 −2.41900
\(372\) −4.32497e13 −0.314773
\(373\) 1.16018e14 0.832010 0.416005 0.909362i \(-0.363430\pi\)
0.416005 + 0.909362i \(0.363430\pi\)
\(374\) 8.92649e11 0.00630793
\(375\) −1.54611e13 −0.107664
\(376\) 1.57410e13 0.108019
\(377\) 8.59417e13 0.581201
\(378\) −2.46717e12 −0.0164435
\(379\) 2.33930e14 1.53664 0.768318 0.640069i \(-0.221094\pi\)
0.768318 + 0.640069i \(0.221094\pi\)
\(380\) 9.19920e12 0.0595580
\(381\) −5.42304e13 −0.346063
\(382\) 1.03469e13 0.0650819
\(383\) 3.07371e14 1.90577 0.952884 0.303336i \(-0.0981005\pi\)
0.952884 + 0.303336i \(0.0981005\pi\)
\(384\) 1.79475e13 0.109694
\(385\) −1.88381e13 −0.113502
\(386\) 5.39274e12 0.0320316
\(387\) 9.64864e12 0.0565010
\(388\) 4.37804e12 0.0252758
\(389\) −1.43081e14 −0.814442 −0.407221 0.913330i \(-0.633502\pi\)
−0.407221 + 0.913330i \(0.633502\pi\)
\(390\) 2.09251e11 0.00117439
\(391\) −2.40244e13 −0.132948
\(392\) 3.84602e13 0.209864
\(393\) −2.03111e13 −0.109289
\(394\) 1.69344e13 0.0898545
\(395\) 1.76110e13 0.0921509
\(396\) 4.36710e13 0.225356
\(397\) 1.73602e14 0.883502 0.441751 0.897138i \(-0.354358\pi\)
0.441751 + 0.897138i \(0.354358\pi\)
\(398\) −4.26398e12 −0.0214022
\(399\) −1.32962e14 −0.658226
\(400\) −2.01615e14 −0.984449
\(401\) 2.06844e14 0.996206 0.498103 0.867118i \(-0.334030\pi\)
0.498103 + 0.867118i \(0.334030\pi\)
\(402\) 1.77362e12 0.00842594
\(403\) −5.30184e13 −0.248456
\(404\) −1.88859e14 −0.873050
\(405\) 2.28175e12 0.0104056
\(406\) 2.42772e13 0.109221
\(407\) −1.84083e14 −0.817043
\(408\) 2.45193e12 0.0107369
\(409\) 4.82766e13 0.208573 0.104287 0.994547i \(-0.466744\pi\)
0.104287 + 0.994547i \(0.466744\pi\)
\(410\) 3.46689e11 0.00147784
\(411\) 1.10058e13 0.0462906
\(412\) 2.20611e14 0.915570
\(413\) −5.68606e13 −0.232855
\(414\) 2.68836e12 0.0108639
\(415\) −1.10929e13 −0.0442366
\(416\) 1.65073e13 0.0649624
\(417\) −1.02069e14 −0.396411
\(418\) −5.38321e12 −0.0206335
\(419\) 9.62691e13 0.364175 0.182087 0.983282i \(-0.441715\pi\)
0.182087 + 0.983282i \(0.441715\pi\)
\(420\) −2.58427e13 −0.0964867
\(421\) 3.57648e14 1.31797 0.658984 0.752157i \(-0.270987\pi\)
0.658984 + 0.752157i \(0.270987\pi\)
\(422\) 2.38428e12 0.00867238
\(423\) 1.05087e14 0.377292
\(424\) −3.64197e13 −0.129070
\(425\) −5.52144e13 −0.193158
\(426\) 3.23778e12 0.0111814
\(427\) −1.99543e14 −0.680274
\(428\) −1.10116e14 −0.370603
\(429\) 5.35348e13 0.177877
\(430\) −2.31168e11 −0.000758316 0
\(431\) −1.62300e14 −0.525648 −0.262824 0.964844i \(-0.584654\pi\)
−0.262824 + 0.964844i \(0.584654\pi\)
\(432\) 5.97720e13 0.191134
\(433\) 1.89586e14 0.598581 0.299290 0.954162i \(-0.403250\pi\)
0.299290 + 0.954162i \(0.403250\pi\)
\(434\) −1.49769e13 −0.0466903
\(435\) −2.24526e13 −0.0691156
\(436\) 1.77181e14 0.538569
\(437\) 1.44882e14 0.434877
\(438\) −1.15652e12 −0.00342804
\(439\) 5.87391e14 1.71938 0.859691 0.510815i \(-0.170656\pi\)
0.859691 + 0.510815i \(0.170656\pi\)
\(440\) −2.09498e12 −0.00605607
\(441\) 2.56762e14 0.733024
\(442\) 1.50115e12 0.00423255
\(443\) 9.54123e13 0.265695 0.132848 0.991136i \(-0.457588\pi\)
0.132848 + 0.991136i \(0.457588\pi\)
\(444\) −2.52531e14 −0.694558
\(445\) −3.70274e13 −0.100587
\(446\) 1.41291e13 0.0379116
\(447\) 8.85335e13 0.234648
\(448\) −6.73852e14 −1.76415
\(449\) −2.82045e14 −0.729396 −0.364698 0.931126i \(-0.618828\pi\)
−0.364698 + 0.931126i \(0.618828\pi\)
\(450\) 6.17855e12 0.0157840
\(451\) 8.86971e13 0.223840
\(452\) 5.66562e14 1.41249
\(453\) −3.38958e14 −0.834844
\(454\) −2.70054e13 −0.0657117
\(455\) −3.16797e13 −0.0761586
\(456\) −1.47866e13 −0.0351206
\(457\) −5.40931e14 −1.26941 −0.634706 0.772754i \(-0.718879\pi\)
−0.634706 + 0.772754i \(0.718879\pi\)
\(458\) −1.30958e13 −0.0303649
\(459\) 1.63692e13 0.0375022
\(460\) 2.81596e13 0.0637468
\(461\) 3.87690e14 0.867221 0.433610 0.901100i \(-0.357239\pi\)
0.433610 + 0.901100i \(0.357239\pi\)
\(462\) 1.51227e13 0.0334271
\(463\) −6.36506e14 −1.39029 −0.695147 0.718867i \(-0.744661\pi\)
−0.695147 + 0.718867i \(0.744661\pi\)
\(464\) −5.88160e14 −1.26954
\(465\) 1.38513e13 0.0295460
\(466\) 1.85517e13 0.0391075
\(467\) −5.78961e14 −1.20616 −0.603082 0.797679i \(-0.706061\pi\)
−0.603082 + 0.797679i \(0.706061\pi\)
\(468\) 7.34408e13 0.151212
\(469\) −2.68520e14 −0.546419
\(470\) −2.51774e12 −0.00506375
\(471\) 9.97126e13 0.198214
\(472\) −6.32345e12 −0.0124243
\(473\) −5.91421e13 −0.114858
\(474\) −1.41376e13 −0.0271391
\(475\) 3.32976e14 0.631827
\(476\) −1.85394e14 −0.347743
\(477\) −2.43139e14 −0.450820
\(478\) 4.25237e12 0.00779432
\(479\) −3.77517e14 −0.684055 −0.342027 0.939690i \(-0.611114\pi\)
−0.342027 + 0.939690i \(0.611114\pi\)
\(480\) −4.31259e12 −0.00772523
\(481\) −3.09569e14 −0.548227
\(482\) −4.23894e13 −0.0742161
\(483\) −4.07008e14 −0.704520
\(484\) 3.15300e14 0.539603
\(485\) −1.40212e12 −0.00237250
\(486\) −1.83173e12 −0.00306452
\(487\) 5.88866e14 0.974109 0.487054 0.873372i \(-0.338071\pi\)
0.487054 + 0.873372i \(0.338071\pi\)
\(488\) −2.21912e13 −0.0362970
\(489\) −6.18010e14 −0.999532
\(490\) −6.15164e12 −0.00983813
\(491\) 7.81722e14 1.23624 0.618122 0.786082i \(-0.287894\pi\)
0.618122 + 0.786082i \(0.287894\pi\)
\(492\) 1.21678e14 0.190284
\(493\) −1.61074e14 −0.249096
\(494\) −9.05286e12 −0.0138448
\(495\) −1.39862e13 −0.0211529
\(496\) 3.62843e14 0.542711
\(497\) −4.90188e14 −0.725108
\(498\) 8.90511e12 0.0130280
\(499\) 3.04135e14 0.440062 0.220031 0.975493i \(-0.429384\pi\)
0.220031 + 0.975493i \(0.429384\pi\)
\(500\) 1.30009e14 0.186053
\(501\) −3.35532e13 −0.0474927
\(502\) 9.07731e12 0.0127083
\(503\) 5.32879e14 0.737912 0.368956 0.929447i \(-0.379715\pi\)
0.368956 + 0.929447i \(0.379715\pi\)
\(504\) 4.15392e13 0.0568970
\(505\) 6.04843e13 0.0819483
\(506\) −1.64785e13 −0.0220846
\(507\) −3.45466e14 −0.457996
\(508\) 4.56010e14 0.598031
\(509\) −1.28977e15 −1.67326 −0.836631 0.547767i \(-0.815478\pi\)
−0.836631 + 0.547767i \(0.815478\pi\)
\(510\) −3.92182e11 −0.000503329 0
\(511\) 1.75093e14 0.222307
\(512\) −1.88429e14 −0.236680
\(513\) −9.87160e13 −0.122671
\(514\) 6.36998e13 0.0783143
\(515\) −7.06533e13 −0.0859394
\(516\) −8.11330e13 −0.0976392
\(517\) −6.44140e14 −0.766977
\(518\) −8.74485e13 −0.103024
\(519\) 4.64166e13 0.0541069
\(520\) −3.52310e12 −0.00406355
\(521\) −1.32247e15 −1.50930 −0.754652 0.656125i \(-0.772194\pi\)
−0.754652 + 0.656125i \(0.772194\pi\)
\(522\) 1.80243e13 0.0203550
\(523\) 4.00732e14 0.447811 0.223906 0.974611i \(-0.428119\pi\)
0.223906 + 0.974611i \(0.428119\pi\)
\(524\) 1.70791e14 0.188861
\(525\) −9.35409e14 −1.02359
\(526\) 2.44419e13 0.0264675
\(527\) 9.93682e13 0.106485
\(528\) −3.66377e14 −0.388545
\(529\) −5.09313e14 −0.534538
\(530\) 5.82526e12 0.00605060
\(531\) −4.22156e13 −0.0433963
\(532\) 1.11804e15 1.13748
\(533\) 1.49160e14 0.150194
\(534\) 2.97246e13 0.0296237
\(535\) 3.52659e13 0.0347864
\(536\) −2.98621e13 −0.0291550
\(537\) 8.63094e14 0.834065
\(538\) 8.27085e13 0.0791129
\(539\) −1.57384e15 −1.49012
\(540\) −1.91867e13 −0.0179819
\(541\) −7.68108e14 −0.712586 −0.356293 0.934374i \(-0.615959\pi\)
−0.356293 + 0.934374i \(0.615959\pi\)
\(542\) 2.07709e13 0.0190747
\(543\) 8.52754e14 0.775219
\(544\) −3.09383e13 −0.0278421
\(545\) −5.67443e13 −0.0505524
\(546\) 2.54316e13 0.0224292
\(547\) 1.79943e14 0.157111 0.0785553 0.996910i \(-0.474969\pi\)
0.0785553 + 0.996910i \(0.474969\pi\)
\(548\) −9.25453e13 −0.0799947
\(549\) −1.48149e14 −0.126780
\(550\) −3.78719e13 −0.0320865
\(551\) 9.71373e14 0.814802
\(552\) −4.52632e13 −0.0375907
\(553\) 2.14038e15 1.75996
\(554\) 4.99557e13 0.0406707
\(555\) 8.08762e13 0.0651943
\(556\) 8.58273e14 0.685037
\(557\) −2.08144e14 −0.164498 −0.0822491 0.996612i \(-0.526210\pi\)
−0.0822491 + 0.996612i \(0.526210\pi\)
\(558\) −1.11194e13 −0.00870150
\(559\) −9.94582e13 −0.0770682
\(560\) 2.16807e14 0.166356
\(561\) −1.00336e14 −0.0762362
\(562\) 6.67302e13 0.0502080
\(563\) −1.07254e14 −0.0799131 −0.0399566 0.999201i \(-0.512722\pi\)
−0.0399566 + 0.999201i \(0.512722\pi\)
\(564\) −8.83652e14 −0.651998
\(565\) −1.81448e14 −0.132583
\(566\) 5.40196e13 0.0390895
\(567\) 2.77317e14 0.198733
\(568\) −5.45137e13 −0.0386892
\(569\) −2.74553e15 −1.92978 −0.964892 0.262648i \(-0.915404\pi\)
−0.964892 + 0.262648i \(0.915404\pi\)
\(570\) 2.36509e12 0.00164641
\(571\) 2.88426e14 0.198855 0.0994275 0.995045i \(-0.468299\pi\)
0.0994275 + 0.995045i \(0.468299\pi\)
\(572\) −4.50161e14 −0.307390
\(573\) −1.16301e15 −0.786565
\(574\) 4.21355e13 0.0282249
\(575\) 1.01927e15 0.676264
\(576\) −5.00295e14 −0.328778
\(577\) −7.87983e13 −0.0512920 −0.0256460 0.999671i \(-0.508164\pi\)
−0.0256460 + 0.999671i \(0.508164\pi\)
\(578\) 7.12779e13 0.0459570
\(579\) −6.06158e14 −0.387127
\(580\) 1.88798e14 0.119438
\(581\) −1.34820e15 −0.844862
\(582\) 1.12559e12 0.000698718 0
\(583\) 1.49034e15 0.916448
\(584\) 1.94720e13 0.0118615
\(585\) −2.35203e13 −0.0141934
\(586\) 4.94870e13 0.0295838
\(587\) 4.26298e14 0.252466 0.126233 0.992001i \(-0.459711\pi\)
0.126233 + 0.992001i \(0.459711\pi\)
\(588\) −2.15904e15 −1.26674
\(589\) −5.99250e14 −0.348317
\(590\) 1.01142e12 0.000582435 0
\(591\) −1.90347e15 −1.08596
\(592\) 2.11860e15 1.19751
\(593\) −2.68295e15 −1.50249 −0.751245 0.660023i \(-0.770546\pi\)
−0.751245 + 0.660023i \(0.770546\pi\)
\(594\) 1.12277e13 0.00622969
\(595\) 5.93748e13 0.0326407
\(596\) −7.44456e14 −0.405494
\(597\) 4.79283e14 0.258662
\(598\) −2.77116e13 −0.0148185
\(599\) −1.34478e15 −0.712533 −0.356266 0.934384i \(-0.615950\pi\)
−0.356266 + 0.934384i \(0.615950\pi\)
\(600\) −1.04027e14 −0.0546151
\(601\) −1.99759e15 −1.03919 −0.519597 0.854412i \(-0.673918\pi\)
−0.519597 + 0.854412i \(0.673918\pi\)
\(602\) −2.80954e13 −0.0144828
\(603\) −1.99360e14 −0.101834
\(604\) 2.85021e15 1.44269
\(605\) −1.00979e14 −0.0506495
\(606\) −4.85551e13 −0.0241344
\(607\) −1.08277e15 −0.533331 −0.266666 0.963789i \(-0.585922\pi\)
−0.266666 + 0.963789i \(0.585922\pi\)
\(608\) 1.86577e14 0.0910725
\(609\) −2.72882e15 −1.32002
\(610\) 3.54944e12 0.00170155
\(611\) −1.08324e15 −0.514633
\(612\) −1.37644e14 −0.0648075
\(613\) 2.68608e15 1.25339 0.626694 0.779265i \(-0.284407\pi\)
0.626694 + 0.779265i \(0.284407\pi\)
\(614\) −1.95395e13 −0.00903624
\(615\) −3.89687e13 −0.0178609
\(616\) −2.54617e14 −0.115663
\(617\) 1.09022e15 0.490847 0.245423 0.969416i \(-0.421073\pi\)
0.245423 + 0.969416i \(0.421073\pi\)
\(618\) 5.67185e13 0.0253098
\(619\) 1.99913e15 0.884182 0.442091 0.896970i \(-0.354237\pi\)
0.442091 + 0.896970i \(0.354237\pi\)
\(620\) −1.16472e14 −0.0510583
\(621\) −3.02179e14 −0.131299
\(622\) −5.95522e13 −0.0256479
\(623\) −4.50019e15 −1.92109
\(624\) −6.16129e14 −0.260709
\(625\) 2.32164e15 0.973766
\(626\) 1.15369e14 0.0479656
\(627\) 6.05087e14 0.249371
\(628\) −8.38459e14 −0.342534
\(629\) 5.80202e14 0.234964
\(630\) −6.64411e12 −0.00266725
\(631\) −1.60230e14 −0.0637650 −0.0318825 0.999492i \(-0.510150\pi\)
−0.0318825 + 0.999492i \(0.510150\pi\)
\(632\) 2.38031e14 0.0939053
\(633\) −2.67999e14 −0.104812
\(634\) −4.71421e13 −0.0182776
\(635\) −1.46043e14 −0.0561338
\(636\) 2.04450e15 0.779062
\(637\) −2.64670e15 −0.999857
\(638\) −1.10482e14 −0.0413786
\(639\) −3.63935e14 −0.135135
\(640\) 4.83328e13 0.0177931
\(641\) 3.08509e15 1.12603 0.563013 0.826448i \(-0.309642\pi\)
0.563013 + 0.826448i \(0.309642\pi\)
\(642\) −2.83105e13 −0.0102448
\(643\) 1.17294e15 0.420837 0.210419 0.977611i \(-0.432517\pi\)
0.210419 + 0.977611i \(0.432517\pi\)
\(644\) 3.42243e15 1.21748
\(645\) 2.59838e13 0.00916484
\(646\) 1.69671e13 0.00593373
\(647\) 1.06206e15 0.368277 0.184139 0.982900i \(-0.441050\pi\)
0.184139 + 0.982900i \(0.441050\pi\)
\(648\) 3.08403e13 0.0106037
\(649\) 2.58763e14 0.0882180
\(650\) −6.36885e13 −0.0215297
\(651\) 1.68344e15 0.564289
\(652\) 5.19669e15 1.72729
\(653\) 5.17742e15 1.70644 0.853219 0.521552i \(-0.174647\pi\)
0.853219 + 0.521552i \(0.174647\pi\)
\(654\) 4.55528e13 0.0148880
\(655\) −5.46979e13 −0.0177273
\(656\) −1.02081e15 −0.328076
\(657\) 1.29996e14 0.0414305
\(658\) −3.05998e14 −0.0967110
\(659\) −5.76292e15 −1.80623 −0.903114 0.429400i \(-0.858725\pi\)
−0.903114 + 0.429400i \(0.858725\pi\)
\(660\) 1.17606e14 0.0365543
\(661\) 1.35104e15 0.416447 0.208224 0.978081i \(-0.433232\pi\)
0.208224 + 0.978081i \(0.433232\pi\)
\(662\) −1.16489e14 −0.0356097
\(663\) −1.68733e14 −0.0511537
\(664\) −1.49933e14 −0.0450788
\(665\) −3.58066e14 −0.106769
\(666\) −6.49252e13 −0.0192002
\(667\) 2.97346e15 0.872108
\(668\) 2.82141e14 0.0820720
\(669\) −1.58815e15 −0.458191
\(670\) 4.77638e12 0.00136675
\(671\) 9.08091e14 0.257724
\(672\) −5.24138e14 −0.147542
\(673\) −3.33582e14 −0.0931364 −0.0465682 0.998915i \(-0.514828\pi\)
−0.0465682 + 0.998915i \(0.514828\pi\)
\(674\) −1.43683e14 −0.0397902
\(675\) −6.94485e14 −0.190762
\(676\) 2.90494e15 0.791463
\(677\) 1.21173e15 0.327469 0.163734 0.986504i \(-0.447646\pi\)
0.163734 + 0.986504i \(0.447646\pi\)
\(678\) 1.45662e14 0.0390465
\(679\) −1.70409e14 −0.0453116
\(680\) 6.60306e12 0.00174159
\(681\) 3.03547e15 0.794177
\(682\) 6.81573e13 0.0176888
\(683\) 3.59538e15 0.925617 0.462808 0.886458i \(-0.346842\pi\)
0.462808 + 0.886458i \(0.346842\pi\)
\(684\) 8.30078e14 0.211988
\(685\) 2.96388e13 0.00750866
\(686\) −4.07666e14 −0.102452
\(687\) 1.47200e15 0.366984
\(688\) 6.80663e14 0.168343
\(689\) 2.50628e15 0.614927
\(690\) 7.23977e12 0.00176220
\(691\) −1.63600e15 −0.395051 −0.197525 0.980298i \(-0.563290\pi\)
−0.197525 + 0.980298i \(0.563290\pi\)
\(692\) −3.90305e14 −0.0935020
\(693\) −1.69983e15 −0.403993
\(694\) 1.52027e13 0.00358462
\(695\) −2.74872e14 −0.0643005
\(696\) −3.03471e14 −0.0704314
\(697\) −2.79560e14 −0.0643716
\(698\) 1.73482e14 0.0396324
\(699\) −2.08525e15 −0.472645
\(700\) 7.86562e15 1.76886
\(701\) −3.60539e15 −0.804458 −0.402229 0.915539i \(-0.631764\pi\)
−0.402229 + 0.915539i \(0.631764\pi\)
\(702\) 1.88815e13 0.00418005
\(703\) −3.49897e15 −0.768574
\(704\) 3.06660e15 0.668355
\(705\) 2.83000e14 0.0611994
\(706\) −1.00560e14 −0.0215774
\(707\) 7.35106e15 1.56510
\(708\) 3.54980e14 0.0749931
\(709\) 3.80021e15 0.796624 0.398312 0.917250i \(-0.369596\pi\)
0.398312 + 0.917250i \(0.369596\pi\)
\(710\) 8.71937e12 0.00181369
\(711\) 1.58910e15 0.327997
\(712\) −5.00465e14 −0.102502
\(713\) −1.83436e15 −0.372814
\(714\) −4.76645e13 −0.00961291
\(715\) 1.44169e14 0.0288529
\(716\) −7.25755e15 −1.44135
\(717\) −4.77978e14 −0.0942003
\(718\) −6.88373e13 −0.0134629
\(719\) −3.94798e15 −0.766243 −0.383121 0.923698i \(-0.625151\pi\)
−0.383121 + 0.923698i \(0.625151\pi\)
\(720\) 1.60966e14 0.0310032
\(721\) −8.58697e15 −1.64133
\(722\) 1.49515e14 0.0283616
\(723\) 4.76467e15 0.896959
\(724\) −7.17059e15 −1.33966
\(725\) 6.83379e15 1.26708
\(726\) 8.10630e13 0.0149166
\(727\) 3.22833e15 0.589574 0.294787 0.955563i \(-0.404751\pi\)
0.294787 + 0.955563i \(0.404751\pi\)
\(728\) −4.28186e14 −0.0776085
\(729\) 2.05891e14 0.0370370
\(730\) −3.11452e12 −0.000556051 0
\(731\) 1.86407e14 0.0330306
\(732\) 1.24575e15 0.219088
\(733\) −3.11347e14 −0.0543466 −0.0271733 0.999631i \(-0.508651\pi\)
−0.0271733 + 0.999631i \(0.508651\pi\)
\(734\) −3.53534e14 −0.0612496
\(735\) 6.91461e14 0.118901
\(736\) 5.71128e14 0.0974778
\(737\) 1.22199e15 0.207013
\(738\) 3.12830e13 0.00526016
\(739\) −8.06786e15 −1.34652 −0.673262 0.739404i \(-0.735107\pi\)
−0.673262 + 0.739404i \(0.735107\pi\)
\(740\) −6.80068e14 −0.112662
\(741\) 1.01756e15 0.167325
\(742\) 7.07983e14 0.115558
\(743\) −5.25372e15 −0.851194 −0.425597 0.904913i \(-0.639936\pi\)
−0.425597 + 0.904913i \(0.639936\pi\)
\(744\) 1.87215e14 0.0301085
\(745\) 2.38421e14 0.0380615
\(746\) −2.50817e14 −0.0397460
\(747\) −1.00096e15 −0.157454
\(748\) 8.43701e14 0.131744
\(749\) 4.28610e15 0.664374
\(750\) 3.34250e13 0.00514321
\(751\) 9.17937e15 1.40215 0.701073 0.713090i \(-0.252705\pi\)
0.701073 + 0.713090i \(0.252705\pi\)
\(752\) 7.41338e15 1.12413
\(753\) −1.02031e15 −0.153589
\(754\) −1.85795e14 −0.0277646
\(755\) −9.12815e14 −0.135417
\(756\) −2.33189e15 −0.343430
\(757\) 4.74779e15 0.694168 0.347084 0.937834i \(-0.387172\pi\)
0.347084 + 0.937834i \(0.387172\pi\)
\(758\) −5.05727e14 −0.0734067
\(759\) 1.85223e15 0.266910
\(760\) −3.98205e13 −0.00569681
\(761\) −1.64969e14 −0.0234307 −0.0117154 0.999931i \(-0.503729\pi\)
−0.0117154 + 0.999931i \(0.503729\pi\)
\(762\) 1.17239e14 0.0165318
\(763\) −6.89652e15 −0.965484
\(764\) 9.77950e15 1.35926
\(765\) 4.40823e13 0.00608311
\(766\) −6.64497e14 −0.0910405
\(767\) 4.35158e14 0.0591933
\(768\) 4.17768e15 0.564220
\(769\) −7.50627e15 −1.00654 −0.503268 0.864130i \(-0.667869\pi\)
−0.503268 + 0.864130i \(0.667869\pi\)
\(770\) 4.07256e13 0.00542211
\(771\) −7.16003e15 −0.946489
\(772\) 5.09703e15 0.668993
\(773\) −9.39637e15 −1.22454 −0.612270 0.790649i \(-0.709743\pi\)
−0.612270 + 0.790649i \(0.709743\pi\)
\(774\) −2.08591e13 −0.00269911
\(775\) −4.21584e15 −0.541657
\(776\) −1.89512e13 −0.00241767
\(777\) 9.82943e15 1.24512
\(778\) 3.09324e14 0.0389068
\(779\) 1.68591e15 0.210562
\(780\) 1.97776e14 0.0245275
\(781\) 2.23077e15 0.274710
\(782\) 5.19377e13 0.00635106
\(783\) −2.02598e15 −0.246006
\(784\) 1.81132e16 2.18403
\(785\) 2.68527e14 0.0321517
\(786\) 4.39100e13 0.00522083
\(787\) 9.95129e15 1.17495 0.587474 0.809243i \(-0.300123\pi\)
0.587474 + 0.809243i \(0.300123\pi\)
\(788\) 1.60058e16 1.87665
\(789\) −2.74733e15 −0.319880
\(790\) −3.80727e13 −0.00440214
\(791\) −2.20526e16 −2.53215
\(792\) −1.89038e14 −0.0215556
\(793\) 1.52712e15 0.172930
\(794\) −3.75306e14 −0.0422058
\(795\) −6.54775e14 −0.0731261
\(796\) −4.03017e15 −0.446994
\(797\) 1.29552e15 0.142700 0.0713498 0.997451i \(-0.477269\pi\)
0.0713498 + 0.997451i \(0.477269\pi\)
\(798\) 2.87446e14 0.0314442
\(799\) 2.03023e15 0.220566
\(800\) 1.31260e15 0.141624
\(801\) −3.34112e15 −0.358025
\(802\) −4.47171e14 −0.0475898
\(803\) −7.96820e14 −0.0842218
\(804\) 1.67637e15 0.175979
\(805\) −1.09607e15 −0.114278
\(806\) 1.14619e14 0.0118690
\(807\) −9.29665e15 −0.956141
\(808\) 8.17510e14 0.0835085
\(809\) −1.76699e16 −1.79274 −0.896372 0.443303i \(-0.853807\pi\)
−0.896372 + 0.443303i \(0.853807\pi\)
\(810\) −4.93286e12 −0.000497085 0
\(811\) 5.13792e15 0.514248 0.257124 0.966378i \(-0.417225\pi\)
0.257124 + 0.966378i \(0.417225\pi\)
\(812\) 2.29459e16 2.28112
\(813\) −2.33470e15 −0.230533
\(814\) 3.97964e14 0.0390310
\(815\) −1.66430e15 −0.162131
\(816\) 1.15476e15 0.111737
\(817\) −1.12415e15 −0.108044
\(818\) −1.04368e14 −0.00996377
\(819\) −2.85858e15 −0.271075
\(820\) 3.27678e14 0.0308654
\(821\) 1.65798e16 1.55128 0.775641 0.631174i \(-0.217427\pi\)
0.775641 + 0.631174i \(0.217427\pi\)
\(822\) −2.37932e13 −0.00221135
\(823\) 8.54594e15 0.788970 0.394485 0.918902i \(-0.370923\pi\)
0.394485 + 0.918902i \(0.370923\pi\)
\(824\) −9.54955e14 −0.0875756
\(825\) 4.25690e15 0.387790
\(826\) 1.22925e14 0.0111237
\(827\) −9.25100e15 −0.831588 −0.415794 0.909459i \(-0.636496\pi\)
−0.415794 + 0.909459i \(0.636496\pi\)
\(828\) 2.54095e15 0.226897
\(829\) 1.22466e16 1.08634 0.543168 0.839624i \(-0.317225\pi\)
0.543168 + 0.839624i \(0.317225\pi\)
\(830\) 2.39815e13 0.00211323
\(831\) −5.61515e15 −0.491536
\(832\) 5.15704e15 0.448459
\(833\) 4.96050e15 0.428527
\(834\) 2.20660e14 0.0189370
\(835\) −9.03590e13 −0.00770364
\(836\) −5.08803e15 −0.430938
\(837\) 1.24985e15 0.105164
\(838\) −2.08121e14 −0.0173970
\(839\) 1.38137e16 1.14715 0.573575 0.819153i \(-0.305556\pi\)
0.573575 + 0.819153i \(0.305556\pi\)
\(840\) 1.11865e14 0.00922909
\(841\) 7.73531e15 0.634015
\(842\) −7.73190e14 −0.0629607
\(843\) −7.50065e15 −0.606802
\(844\) 2.25354e15 0.181126
\(845\) −9.30343e14 −0.0742901
\(846\) −2.27185e14 −0.0180237
\(847\) −1.22726e16 −0.967339
\(848\) −1.71522e16 −1.34321
\(849\) −6.07194e15 −0.472427
\(850\) 1.19366e14 0.00922737
\(851\) −1.07107e16 −0.822629
\(852\) 3.06024e15 0.233527
\(853\) 1.29744e16 0.983712 0.491856 0.870677i \(-0.336319\pi\)
0.491856 + 0.870677i \(0.336319\pi\)
\(854\) 4.31387e14 0.0324974
\(855\) −2.65843e14 −0.0198981
\(856\) 4.76657e14 0.0354487
\(857\) 6.64958e15 0.491360 0.245680 0.969351i \(-0.420989\pi\)
0.245680 + 0.969351i \(0.420989\pi\)
\(858\) −1.15735e14 −0.00849740
\(859\) 1.88846e15 0.137767 0.0688836 0.997625i \(-0.478056\pi\)
0.0688836 + 0.997625i \(0.478056\pi\)
\(860\) −2.18492e14 −0.0158377
\(861\) −4.73613e15 −0.341120
\(862\) 3.50873e14 0.0251108
\(863\) −1.84854e16 −1.31453 −0.657264 0.753661i \(-0.728286\pi\)
−0.657264 + 0.753661i \(0.728286\pi\)
\(864\) −3.89141e14 −0.0274968
\(865\) 1.25000e14 0.00877651
\(866\) −4.09860e14 −0.0285948
\(867\) −8.01183e15 −0.555426
\(868\) −1.41556e16 −0.975147
\(869\) −9.74054e15 −0.666767
\(870\) 4.85397e13 0.00330172
\(871\) 2.05500e15 0.138903
\(872\) −7.66960e14 −0.0515148
\(873\) −1.26519e14 −0.00844455
\(874\) −3.13216e14 −0.0207745
\(875\) −5.06041e15 −0.333535
\(876\) −1.09310e15 −0.0715960
\(877\) −1.83116e16 −1.19187 −0.595934 0.803033i \(-0.703218\pi\)
−0.595934 + 0.803033i \(0.703218\pi\)
\(878\) −1.26986e15 −0.0821367
\(879\) −5.56247e15 −0.357543
\(880\) −9.86654e14 −0.0630246
\(881\) −2.42101e16 −1.53684 −0.768419 0.639947i \(-0.778956\pi\)
−0.768419 + 0.639947i \(0.778956\pi\)
\(882\) −5.55086e14 −0.0350173
\(883\) −8.16078e14 −0.0511620 −0.0255810 0.999673i \(-0.508144\pi\)
−0.0255810 + 0.999673i \(0.508144\pi\)
\(884\) 1.41884e15 0.0883985
\(885\) −1.13687e14 −0.00703918
\(886\) −2.06269e14 −0.0126925
\(887\) 3.04467e16 1.86192 0.930959 0.365124i \(-0.118974\pi\)
0.930959 + 0.365124i \(0.118974\pi\)
\(888\) 1.09313e15 0.0664355
\(889\) −1.77496e16 −1.07208
\(890\) 8.00485e13 0.00480516
\(891\) −1.26203e15 −0.0752906
\(892\) 1.33543e16 0.791799
\(893\) −1.22435e16 −0.721478
\(894\) −1.91398e14 −0.0112094
\(895\) 2.32432e15 0.135291
\(896\) 5.87421e15 0.339825
\(897\) 3.11486e15 0.179094
\(898\) 6.09745e14 0.0348440
\(899\) −1.22986e16 −0.698519
\(900\) 5.83975e15 0.329656
\(901\) −4.69732e15 −0.263550
\(902\) −1.91752e14 −0.0106931
\(903\) 3.15799e15 0.175036
\(904\) −2.45247e15 −0.135107
\(905\) 2.29647e15 0.125746
\(906\) 7.32783e14 0.0398814
\(907\) −6.06387e15 −0.328027 −0.164013 0.986458i \(-0.552444\pi\)
−0.164013 + 0.986458i \(0.552444\pi\)
\(908\) −2.55245e16 −1.37242
\(909\) 5.45772e15 0.291682
\(910\) 6.84875e13 0.00363818
\(911\) −2.44461e16 −1.29080 −0.645400 0.763845i \(-0.723309\pi\)
−0.645400 + 0.763845i \(0.723309\pi\)
\(912\) −6.96392e15 −0.365496
\(913\) 6.13545e15 0.320079
\(914\) 1.16942e15 0.0606412
\(915\) −3.98966e14 −0.0205646
\(916\) −1.23777e16 −0.634184
\(917\) −6.64781e15 −0.338569
\(918\) −3.53880e13 −0.00179152
\(919\) −4.61347e15 −0.232162 −0.116081 0.993240i \(-0.537033\pi\)
−0.116081 + 0.993240i \(0.537033\pi\)
\(920\) −1.21894e14 −0.00609747
\(921\) 2.19629e15 0.109210
\(922\) −8.38136e14 −0.0414281
\(923\) 3.75144e15 0.184327
\(924\) 1.42935e16 0.698139
\(925\) −2.46159e16 −1.19519
\(926\) 1.37604e15 0.0664159
\(927\) −6.37531e15 −0.305888
\(928\) 3.82918e15 0.182638
\(929\) −1.59191e16 −0.754802 −0.377401 0.926050i \(-0.623182\pi\)
−0.377401 + 0.926050i \(0.623182\pi\)
\(930\) −2.99446e13 −0.00141144
\(931\) −2.99148e16 −1.40173
\(932\) 1.75344e16 0.816776
\(933\) 6.69383e15 0.309974
\(934\) 1.25164e15 0.0576197
\(935\) −2.70206e14 −0.0123660
\(936\) −3.17902e14 −0.0144636
\(937\) 9.86284e15 0.446102 0.223051 0.974807i \(-0.428398\pi\)
0.223051 + 0.974807i \(0.428398\pi\)
\(938\) 5.80506e14 0.0261030
\(939\) −1.29678e16 −0.579701
\(940\) −2.37968e15 −0.105759
\(941\) 4.42260e16 1.95404 0.977022 0.213138i \(-0.0683685\pi\)
0.977022 + 0.213138i \(0.0683685\pi\)
\(942\) −2.15566e14 −0.00946891
\(943\) 5.16074e15 0.225371
\(944\) −2.97810e15 −0.129298
\(945\) 7.46816e14 0.0322358
\(946\) 1.27858e14 0.00548687
\(947\) −1.88298e16 −0.803381 −0.401690 0.915776i \(-0.631577\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(948\) −1.33624e16 −0.566811
\(949\) −1.34000e15 −0.0565119
\(950\) −7.19852e14 −0.0301831
\(951\) 5.29890e15 0.220898
\(952\) 8.02515e14 0.0332621
\(953\) −1.95109e15 −0.0804018 −0.0402009 0.999192i \(-0.512800\pi\)
−0.0402009 + 0.999192i \(0.512800\pi\)
\(954\) 5.25635e14 0.0215362
\(955\) −3.13200e15 −0.127586
\(956\) 4.01920e15 0.162787
\(957\) 1.24184e16 0.500093
\(958\) 8.16142e14 0.0326780
\(959\) 3.60220e15 0.143405
\(960\) −1.34730e15 −0.0533300
\(961\) −1.78213e16 −0.701393
\(962\) 6.69249e14 0.0261894
\(963\) 3.18218e15 0.123817
\(964\) −4.00650e16 −1.55003
\(965\) −1.63239e15 −0.0627946
\(966\) 8.79899e14 0.0336557
\(967\) 3.19445e16 1.21493 0.607464 0.794347i \(-0.292187\pi\)
0.607464 + 0.794347i \(0.292187\pi\)
\(968\) −1.36484e15 −0.0516138
\(969\) −1.90714e15 −0.0717137
\(970\) 3.03121e12 0.000113337 0
\(971\) 1.80781e16 0.672120 0.336060 0.941841i \(-0.390905\pi\)
0.336060 + 0.941841i \(0.390905\pi\)
\(972\) −1.73129e15 −0.0640036
\(973\) −3.34071e16 −1.22806
\(974\) −1.27305e15 −0.0465342
\(975\) 7.15876e15 0.260203
\(976\) −1.04512e16 −0.377738
\(977\) 1.30976e16 0.470731 0.235366 0.971907i \(-0.424371\pi\)
0.235366 + 0.971907i \(0.424371\pi\)
\(978\) 1.33606e15 0.0477487
\(979\) 2.04797e16 0.727810
\(980\) −5.81432e15 −0.205473
\(981\) −5.12025e15 −0.179933
\(982\) −1.68998e15 −0.0590567
\(983\) −4.30264e16 −1.49517 −0.747586 0.664165i \(-0.768787\pi\)
−0.747586 + 0.664165i \(0.768787\pi\)
\(984\) −5.26704e14 −0.0182009
\(985\) −5.12605e15 −0.176150
\(986\) 3.48221e14 0.0118996
\(987\) 3.43950e16 1.16883
\(988\) −8.55645e15 −0.289155
\(989\) −3.44111e15 −0.115643
\(990\) 3.02363e13 0.00101050
\(991\) −1.77981e16 −0.591520 −0.295760 0.955262i \(-0.595573\pi\)
−0.295760 + 0.955262i \(0.595573\pi\)
\(992\) −2.36226e15 −0.0780753
\(993\) 1.30937e16 0.430370
\(994\) 1.05972e15 0.0346392
\(995\) 1.29071e15 0.0419568
\(996\) 8.41680e15 0.272095
\(997\) −2.02344e16 −0.650529 −0.325264 0.945623i \(-0.605453\pi\)
−0.325264 + 0.945623i \(0.605453\pi\)
\(998\) −6.57501e14 −0.0210222
\(999\) 7.29777e15 0.232049
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 177.12.a.b.1.15 27
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.b.1.15 27 1.1 even 1 trivial