Properties

Label 91.10.a.a.1.4
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-22.2119\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-24.2119 q^{2} +6.29566 q^{3} +74.2177 q^{4} +2011.41 q^{5} -152.430 q^{6} +2401.00 q^{7} +10599.6 q^{8} -19643.4 q^{9} +O(q^{10})\) \(q-24.2119 q^{2} +6.29566 q^{3} +74.2177 q^{4} +2011.41 q^{5} -152.430 q^{6} +2401.00 q^{7} +10599.6 q^{8} -19643.4 q^{9} -48700.2 q^{10} -30042.8 q^{11} +467.250 q^{12} +28561.0 q^{13} -58132.8 q^{14} +12663.2 q^{15} -294635. q^{16} -576543. q^{17} +475604. q^{18} +282669. q^{19} +149282. q^{20} +15115.9 q^{21} +727393. q^{22} +1.75925e6 q^{23} +66731.3 q^{24} +2.09265e6 q^{25} -691517. q^{26} -247586. q^{27} +178197. q^{28} -28745.8 q^{29} -306600. q^{30} -6.18815e6 q^{31} +1.70671e6 q^{32} -189139. q^{33} +1.39592e7 q^{34} +4.82940e6 q^{35} -1.45788e6 q^{36} -4.71914e6 q^{37} -6.84397e6 q^{38} +179810. q^{39} +2.13201e7 q^{40} +3.07403e7 q^{41} -365985. q^{42} +1.13424e7 q^{43} -2.22970e6 q^{44} -3.95109e7 q^{45} -4.25948e7 q^{46} -3.74184e7 q^{47} -1.85492e6 q^{48} +5.76480e6 q^{49} -5.06672e7 q^{50} -3.62972e6 q^{51} +2.11973e6 q^{52} +3.09065e7 q^{53} +5.99453e6 q^{54} -6.04284e7 q^{55} +2.54495e7 q^{56} +1.77959e6 q^{57} +695991. q^{58} -3.11616e7 q^{59} +939831. q^{60} -1.96023e8 q^{61} +1.49827e8 q^{62} -4.71637e7 q^{63} +1.09530e8 q^{64} +5.74479e7 q^{65} +4.57943e6 q^{66} -2.53265e8 q^{67} -4.27897e7 q^{68} +1.10756e7 q^{69} -1.16929e8 q^{70} -3.89974e8 q^{71} -2.08211e8 q^{72} -1.45469e8 q^{73} +1.14260e8 q^{74} +1.31746e7 q^{75} +2.09791e7 q^{76} -7.21327e7 q^{77} -4.35356e6 q^{78} -1.55125e8 q^{79} -5.92633e8 q^{80} +3.85082e8 q^{81} -7.44283e8 q^{82} -2.55020e8 q^{83} +1.12187e6 q^{84} -1.15967e9 q^{85} -2.74622e8 q^{86} -180974. q^{87} -3.18440e8 q^{88} +8.80255e8 q^{89} +9.56635e8 q^{90} +6.85750e7 q^{91} +1.30567e8 q^{92} -3.89585e7 q^{93} +9.05973e8 q^{94} +5.68564e8 q^{95} +1.07449e7 q^{96} +1.53722e8 q^{97} -1.39577e8 q^{98} +5.90141e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −24.2119 −1.07003 −0.535013 0.844844i \(-0.679693\pi\)
−0.535013 + 0.844844i \(0.679693\pi\)
\(3\) 6.29566 0.0448741 0.0224371 0.999748i \(-0.492857\pi\)
0.0224371 + 0.999748i \(0.492857\pi\)
\(4\) 74.2177 0.144956
\(5\) 2011.41 1.43925 0.719625 0.694363i \(-0.244314\pi\)
0.719625 + 0.694363i \(0.244314\pi\)
\(6\) −152.430 −0.0480165
\(7\) 2401.00 0.377964
\(8\) 10599.6 0.914919
\(9\) −19643.4 −0.997986
\(10\) −48700.2 −1.54003
\(11\) −30042.8 −0.618690 −0.309345 0.950950i \(-0.600110\pi\)
−0.309345 + 0.950950i \(0.600110\pi\)
\(12\) 467.250 0.00650479
\(13\) 28561.0 0.277350
\(14\) −58132.8 −0.404432
\(15\) 12663.2 0.0645850
\(16\) −294635. −1.12394
\(17\) −576543. −1.67422 −0.837108 0.547037i \(-0.815756\pi\)
−0.837108 + 0.547037i \(0.815756\pi\)
\(18\) 475604. 1.06787
\(19\) 282669. 0.497608 0.248804 0.968554i \(-0.419962\pi\)
0.248804 + 0.968554i \(0.419962\pi\)
\(20\) 149282. 0.208628
\(21\) 15115.9 0.0169608
\(22\) 727393. 0.662014
\(23\) 1.75925e6 1.31085 0.655423 0.755262i \(-0.272491\pi\)
0.655423 + 0.755262i \(0.272491\pi\)
\(24\) 66731.3 0.0410562
\(25\) 2.09265e6 1.07144
\(26\) −691517. −0.296772
\(27\) −247586. −0.0896579
\(28\) 178197. 0.0547884
\(29\) −28745.8 −0.00754715 −0.00377358 0.999993i \(-0.501201\pi\)
−0.00377358 + 0.999993i \(0.501201\pi\)
\(30\) −306600. −0.0691077
\(31\) −6.18815e6 −1.20347 −0.601733 0.798698i \(-0.705523\pi\)
−0.601733 + 0.798698i \(0.705523\pi\)
\(32\) 1.70671e6 0.287731
\(33\) −189139. −0.0277632
\(34\) 1.39592e7 1.79146
\(35\) 4.82940e6 0.543985
\(36\) −1.45788e6 −0.144664
\(37\) −4.71914e6 −0.413957 −0.206979 0.978345i \(-0.566363\pi\)
−0.206979 + 0.978345i \(0.566363\pi\)
\(38\) −6.84397e6 −0.532454
\(39\) 179810. 0.0124458
\(40\) 2.13201e7 1.31680
\(41\) 3.07403e7 1.69895 0.849476 0.527628i \(-0.176918\pi\)
0.849476 + 0.527628i \(0.176918\pi\)
\(42\) −365985. −0.0181485
\(43\) 1.13424e7 0.505939 0.252969 0.967474i \(-0.418593\pi\)
0.252969 + 0.967474i \(0.418593\pi\)
\(44\) −2.22970e6 −0.0896830
\(45\) −3.95109e7 −1.43635
\(46\) −4.25948e7 −1.40264
\(47\) −3.74184e7 −1.11852 −0.559262 0.828991i \(-0.688915\pi\)
−0.559262 + 0.828991i \(0.688915\pi\)
\(48\) −1.85492e6 −0.0504360
\(49\) 5.76480e6 0.142857
\(50\) −5.06672e7 −1.14647
\(51\) −3.62972e6 −0.0751290
\(52\) 2.11973e6 0.0402037
\(53\) 3.09065e7 0.538033 0.269016 0.963136i \(-0.413301\pi\)
0.269016 + 0.963136i \(0.413301\pi\)
\(54\) 5.99453e6 0.0959363
\(55\) −6.04284e7 −0.890449
\(56\) 2.54495e7 0.345807
\(57\) 1.77959e6 0.0223297
\(58\) 695991. 0.00807565
\(59\) −3.11616e7 −0.334800 −0.167400 0.985889i \(-0.553537\pi\)
−0.167400 + 0.985889i \(0.553537\pi\)
\(60\) 939831. 0.00936201
\(61\) −1.96023e8 −1.81268 −0.906342 0.422545i \(-0.861137\pi\)
−0.906342 + 0.422545i \(0.861137\pi\)
\(62\) 1.49827e8 1.28774
\(63\) −4.71637e7 −0.377203
\(64\) 1.09530e8 0.816065
\(65\) 5.74479e7 0.399176
\(66\) 4.57943e6 0.0297073
\(67\) −2.53265e8 −1.53546 −0.767731 0.640773i \(-0.778614\pi\)
−0.767731 + 0.640773i \(0.778614\pi\)
\(68\) −4.27897e7 −0.242688
\(69\) 1.10756e7 0.0588230
\(70\) −1.16929e8 −0.582078
\(71\) −3.89974e8 −1.82127 −0.910633 0.413216i \(-0.864406\pi\)
−0.910633 + 0.413216i \(0.864406\pi\)
\(72\) −2.08211e8 −0.913077
\(73\) −1.45469e8 −0.599540 −0.299770 0.954012i \(-0.596910\pi\)
−0.299770 + 0.954012i \(0.596910\pi\)
\(74\) 1.14260e8 0.442945
\(75\) 1.31746e7 0.0480798
\(76\) 2.09791e7 0.0721315
\(77\) −7.21327e7 −0.233843
\(78\) −4.35356e6 −0.0133174
\(79\) −1.55125e8 −0.448085 −0.224042 0.974579i \(-0.571925\pi\)
−0.224042 + 0.974579i \(0.571925\pi\)
\(80\) −5.92633e8 −1.61764
\(81\) 3.85082e8 0.993963
\(82\) −7.44283e8 −1.81792
\(83\) −2.55020e8 −0.589825 −0.294913 0.955524i \(-0.595291\pi\)
−0.294913 + 0.955524i \(0.595291\pi\)
\(84\) 1.12187e6 0.00245858
\(85\) −1.15967e9 −2.40961
\(86\) −2.74622e8 −0.541368
\(87\) −180974. −0.000338672 0
\(88\) −3.18440e8 −0.566051
\(89\) 8.80255e8 1.48715 0.743573 0.668655i \(-0.233130\pi\)
0.743573 + 0.668655i \(0.233130\pi\)
\(90\) 9.56635e8 1.53693
\(91\) 6.85750e7 0.104828
\(92\) 1.30567e8 0.190015
\(93\) −3.89585e7 −0.0540044
\(94\) 9.05973e8 1.19685
\(95\) 5.68564e8 0.716182
\(96\) 1.07449e7 0.0129117
\(97\) 1.53722e8 0.176304 0.0881522 0.996107i \(-0.471904\pi\)
0.0881522 + 0.996107i \(0.471904\pi\)
\(98\) −1.39577e8 −0.152861
\(99\) 5.90141e8 0.617444
\(100\) 1.55312e8 0.155312
\(101\) −8.47811e7 −0.0810687 −0.0405343 0.999178i \(-0.512906\pi\)
−0.0405343 + 0.999178i \(0.512906\pi\)
\(102\) 8.78826e7 0.0803900
\(103\) −7.58258e8 −0.663818 −0.331909 0.943311i \(-0.607693\pi\)
−0.331909 + 0.943311i \(0.607693\pi\)
\(104\) 3.02734e8 0.253753
\(105\) 3.04043e7 0.0244108
\(106\) −7.48306e8 −0.575709
\(107\) 3.33641e8 0.246066 0.123033 0.992403i \(-0.460738\pi\)
0.123033 + 0.992403i \(0.460738\pi\)
\(108\) −1.83752e7 −0.0129965
\(109\) 1.41941e9 0.963135 0.481568 0.876409i \(-0.340068\pi\)
0.481568 + 0.876409i \(0.340068\pi\)
\(110\) 1.46309e9 0.952803
\(111\) −2.97101e7 −0.0185760
\(112\) −7.07419e8 −0.424811
\(113\) −1.52248e9 −0.878412 −0.439206 0.898386i \(-0.644740\pi\)
−0.439206 + 0.898386i \(0.644740\pi\)
\(114\) −4.30873e7 −0.0238934
\(115\) 3.53857e9 1.88663
\(116\) −2.13345e6 −0.00109401
\(117\) −5.61034e8 −0.276792
\(118\) 7.54482e8 0.358245
\(119\) −1.38428e9 −0.632794
\(120\) 1.34224e8 0.0590901
\(121\) −1.45538e9 −0.617223
\(122\) 4.74609e9 1.93962
\(123\) 1.93531e8 0.0762390
\(124\) −4.59270e8 −0.174450
\(125\) 2.80647e8 0.102817
\(126\) 1.14192e9 0.403618
\(127\) 1.69398e8 0.0577820 0.0288910 0.999583i \(-0.490802\pi\)
0.0288910 + 0.999583i \(0.490802\pi\)
\(128\) −3.52578e9 −1.16094
\(129\) 7.14081e7 0.0227036
\(130\) −1.39093e9 −0.427129
\(131\) 1.11398e9 0.330487 0.165244 0.986253i \(-0.447159\pi\)
0.165244 + 0.986253i \(0.447159\pi\)
\(132\) −1.40375e7 −0.00402445
\(133\) 6.78689e8 0.188078
\(134\) 6.13204e9 1.64298
\(135\) −4.97997e8 −0.129040
\(136\) −6.11110e9 −1.53177
\(137\) −3.16000e9 −0.766380 −0.383190 0.923669i \(-0.625175\pi\)
−0.383190 + 0.923669i \(0.625175\pi\)
\(138\) −2.68162e8 −0.0629422
\(139\) 1.38977e9 0.315773 0.157887 0.987457i \(-0.449532\pi\)
0.157887 + 0.987457i \(0.449532\pi\)
\(140\) 3.58427e8 0.0788541
\(141\) −2.35574e8 −0.0501928
\(142\) 9.44203e9 1.94880
\(143\) −8.58051e8 −0.171594
\(144\) 5.78763e9 1.12168
\(145\) −5.78196e7 −0.0108622
\(146\) 3.52209e9 0.641523
\(147\) 3.62933e7 0.00641059
\(148\) −3.50244e8 −0.0600057
\(149\) −9.59418e9 −1.59466 −0.797332 0.603540i \(-0.793756\pi\)
−0.797332 + 0.603540i \(0.793756\pi\)
\(150\) −3.18983e8 −0.0514467
\(151\) 4.88287e9 0.764326 0.382163 0.924095i \(-0.375179\pi\)
0.382163 + 0.924095i \(0.375179\pi\)
\(152\) 2.99617e9 0.455271
\(153\) 1.13252e10 1.67085
\(154\) 1.74647e9 0.250218
\(155\) −1.24469e10 −1.73209
\(156\) 1.33451e7 0.00180410
\(157\) −1.28335e10 −1.68576 −0.842882 0.538098i \(-0.819143\pi\)
−0.842882 + 0.538098i \(0.819143\pi\)
\(158\) 3.75588e9 0.479463
\(159\) 1.94577e8 0.0241437
\(160\) 3.43291e9 0.414116
\(161\) 4.22395e9 0.495453
\(162\) −9.32357e9 −1.06357
\(163\) −4.22651e9 −0.468962 −0.234481 0.972121i \(-0.575339\pi\)
−0.234481 + 0.972121i \(0.575339\pi\)
\(164\) 2.28148e9 0.246274
\(165\) −3.80437e8 −0.0399581
\(166\) 6.17454e9 0.631129
\(167\) −1.66408e10 −1.65558 −0.827791 0.561037i \(-0.810403\pi\)
−0.827791 + 0.561037i \(0.810403\pi\)
\(168\) 1.60222e8 0.0155178
\(169\) 8.15731e8 0.0769231
\(170\) 2.80777e10 2.57835
\(171\) −5.55258e9 −0.496606
\(172\) 8.41809e8 0.0733391
\(173\) 8.89469e9 0.754959 0.377480 0.926018i \(-0.376791\pi\)
0.377480 + 0.926018i \(0.376791\pi\)
\(174\) 4.38173e6 0.000362388 0
\(175\) 5.02446e9 0.404965
\(176\) 8.85166e9 0.695373
\(177\) −1.96183e8 −0.0150238
\(178\) −2.13127e10 −1.59129
\(179\) −2.13424e10 −1.55383 −0.776916 0.629604i \(-0.783217\pi\)
−0.776916 + 0.629604i \(0.783217\pi\)
\(180\) −2.93241e9 −0.208208
\(181\) −1.27540e9 −0.0883268 −0.0441634 0.999024i \(-0.514062\pi\)
−0.0441634 + 0.999024i \(0.514062\pi\)
\(182\) −1.66033e9 −0.112169
\(183\) −1.23409e9 −0.0813426
\(184\) 1.86472e10 1.19932
\(185\) −9.49214e9 −0.595788
\(186\) 9.43262e8 0.0577862
\(187\) 1.73210e10 1.03582
\(188\) −2.77711e9 −0.162137
\(189\) −5.94453e8 −0.0338875
\(190\) −1.37660e10 −0.766334
\(191\) 6.83481e9 0.371601 0.185800 0.982588i \(-0.440512\pi\)
0.185800 + 0.982588i \(0.440512\pi\)
\(192\) 6.89566e8 0.0366202
\(193\) −3.40888e10 −1.76850 −0.884248 0.467018i \(-0.845329\pi\)
−0.884248 + 0.467018i \(0.845329\pi\)
\(194\) −3.72190e9 −0.188650
\(195\) 3.61673e8 0.0179127
\(196\) 4.27850e8 0.0207081
\(197\) 2.78745e8 0.0131859 0.00659295 0.999978i \(-0.497901\pi\)
0.00659295 + 0.999978i \(0.497901\pi\)
\(198\) −1.42885e10 −0.660681
\(199\) 1.97894e10 0.894528 0.447264 0.894402i \(-0.352399\pi\)
0.447264 + 0.894402i \(0.352399\pi\)
\(200\) 2.21812e10 0.980279
\(201\) −1.59447e9 −0.0689025
\(202\) 2.05272e9 0.0867456
\(203\) −6.90186e7 −0.00285256
\(204\) −2.69390e8 −0.0108904
\(205\) 6.18314e10 2.44521
\(206\) 1.83589e10 0.710303
\(207\) −3.45575e10 −1.30821
\(208\) −8.41508e9 −0.311726
\(209\) −8.49217e9 −0.307865
\(210\) −7.36146e8 −0.0261203
\(211\) 1.12980e10 0.392403 0.196201 0.980564i \(-0.437139\pi\)
0.196201 + 0.980564i \(0.437139\pi\)
\(212\) 2.29381e9 0.0779913
\(213\) −2.45515e9 −0.0817277
\(214\) −8.07809e9 −0.263297
\(215\) 2.28143e10 0.728172
\(216\) −2.62430e9 −0.0820297
\(217\) −1.48578e10 −0.454867
\(218\) −3.43665e10 −1.03058
\(219\) −9.15825e8 −0.0269038
\(220\) −4.48485e9 −0.129076
\(221\) −1.64666e10 −0.464344
\(222\) 7.19340e8 0.0198768
\(223\) 3.00947e10 0.814927 0.407464 0.913221i \(-0.366413\pi\)
0.407464 + 0.913221i \(0.366413\pi\)
\(224\) 4.09782e9 0.108752
\(225\) −4.11067e10 −1.06928
\(226\) 3.68621e10 0.939924
\(227\) 4.82417e9 0.120589 0.0602943 0.998181i \(-0.480796\pi\)
0.0602943 + 0.998181i \(0.480796\pi\)
\(228\) 1.32077e8 0.00323684
\(229\) 5.42517e9 0.130363 0.0651814 0.997873i \(-0.479237\pi\)
0.0651814 + 0.997873i \(0.479237\pi\)
\(230\) −8.56756e10 −2.01875
\(231\) −4.54123e8 −0.0104935
\(232\) −3.04693e8 −0.00690504
\(233\) −7.74224e10 −1.72094 −0.860469 0.509504i \(-0.829829\pi\)
−0.860469 + 0.509504i \(0.829829\pi\)
\(234\) 1.35837e10 0.296174
\(235\) −7.52639e10 −1.60983
\(236\) −2.31274e9 −0.0485314
\(237\) −9.76616e8 −0.0201074
\(238\) 3.35161e10 0.677107
\(239\) −2.04264e10 −0.404950 −0.202475 0.979287i \(-0.564898\pi\)
−0.202475 + 0.979287i \(0.564898\pi\)
\(240\) −3.73102e9 −0.0725900
\(241\) −6.20364e10 −1.18460 −0.592298 0.805719i \(-0.701779\pi\)
−0.592298 + 0.805719i \(0.701779\pi\)
\(242\) 3.52376e10 0.660445
\(243\) 7.29757e9 0.134261
\(244\) −1.45484e10 −0.262760
\(245\) 1.15954e10 0.205607
\(246\) −4.68575e9 −0.0815777
\(247\) 8.07332e9 0.138012
\(248\) −6.55917e10 −1.10107
\(249\) −1.60552e9 −0.0264679
\(250\) −6.79500e9 −0.110017
\(251\) 4.99797e10 0.794807 0.397404 0.917644i \(-0.369911\pi\)
0.397404 + 0.917644i \(0.369911\pi\)
\(252\) −3.50038e9 −0.0546780
\(253\) −5.28526e10 −0.811006
\(254\) −4.10146e9 −0.0618283
\(255\) −7.30087e9 −0.108129
\(256\) 2.92864e10 0.426173
\(257\) −6.64363e10 −0.949962 −0.474981 0.879996i \(-0.657545\pi\)
−0.474981 + 0.879996i \(0.657545\pi\)
\(258\) −1.72893e9 −0.0242934
\(259\) −1.13307e10 −0.156461
\(260\) 4.26365e9 0.0578631
\(261\) 5.64664e8 0.00753196
\(262\) −2.69715e10 −0.353630
\(263\) 3.84218e10 0.495196 0.247598 0.968863i \(-0.420359\pi\)
0.247598 + 0.968863i \(0.420359\pi\)
\(264\) −2.00479e9 −0.0254010
\(265\) 6.21657e10 0.774363
\(266\) −1.64324e10 −0.201249
\(267\) 5.54179e9 0.0667344
\(268\) −1.87968e10 −0.222575
\(269\) −1.40719e11 −1.63857 −0.819287 0.573384i \(-0.805630\pi\)
−0.819287 + 0.573384i \(0.805630\pi\)
\(270\) 1.20575e10 0.138076
\(271\) 1.54656e11 1.74182 0.870912 0.491440i \(-0.163529\pi\)
0.870912 + 0.491440i \(0.163529\pi\)
\(272\) 1.69870e11 1.88173
\(273\) 4.31725e8 0.00470409
\(274\) 7.65097e10 0.820047
\(275\) −6.28691e10 −0.662888
\(276\) 8.22007e8 0.00852677
\(277\) 3.62631e10 0.370089 0.185044 0.982730i \(-0.440757\pi\)
0.185044 + 0.982730i \(0.440757\pi\)
\(278\) −3.36489e10 −0.337885
\(279\) 1.21556e11 1.20104
\(280\) 5.11895e10 0.497702
\(281\) −1.13196e11 −1.08306 −0.541531 0.840680i \(-0.682155\pi\)
−0.541531 + 0.840680i \(0.682155\pi\)
\(282\) 5.70370e9 0.0537076
\(283\) 1.31351e11 1.21729 0.608647 0.793441i \(-0.291713\pi\)
0.608647 + 0.793441i \(0.291713\pi\)
\(284\) −2.89430e10 −0.264004
\(285\) 3.57949e9 0.0321380
\(286\) 2.07751e10 0.183610
\(287\) 7.38075e10 0.642143
\(288\) −3.35256e10 −0.287151
\(289\) 2.13814e11 1.80300
\(290\) 1.39992e9 0.0116229
\(291\) 9.67782e8 0.00791150
\(292\) −1.07964e10 −0.0869071
\(293\) 1.04899e11 0.831508 0.415754 0.909477i \(-0.363518\pi\)
0.415754 + 0.909477i \(0.363518\pi\)
\(294\) −8.78730e8 −0.00685950
\(295\) −6.26787e10 −0.481860
\(296\) −5.00208e10 −0.378737
\(297\) 7.43816e9 0.0554704
\(298\) 2.32294e11 1.70633
\(299\) 5.02458e10 0.363563
\(300\) 9.77791e8 0.00696948
\(301\) 2.72332e10 0.191227
\(302\) −1.18224e11 −0.817849
\(303\) −5.33754e8 −0.00363789
\(304\) −8.32843e10 −0.559284
\(305\) −3.94282e11 −2.60890
\(306\) −2.74206e11 −1.78785
\(307\) 1.47584e11 0.948236 0.474118 0.880461i \(-0.342767\pi\)
0.474118 + 0.880461i \(0.342767\pi\)
\(308\) −5.35352e9 −0.0338970
\(309\) −4.77374e9 −0.0297883
\(310\) 3.01364e11 1.85338
\(311\) 1.19975e10 0.0727227 0.0363613 0.999339i \(-0.488423\pi\)
0.0363613 + 0.999339i \(0.488423\pi\)
\(312\) 1.90591e9 0.0113869
\(313\) 1.15683e11 0.681269 0.340635 0.940196i \(-0.389358\pi\)
0.340635 + 0.940196i \(0.389358\pi\)
\(314\) 3.10724e11 1.80381
\(315\) −9.48657e10 −0.542890
\(316\) −1.15130e10 −0.0649528
\(317\) 4.25724e10 0.236789 0.118395 0.992967i \(-0.462225\pi\)
0.118395 + 0.992967i \(0.462225\pi\)
\(318\) −4.71109e9 −0.0258344
\(319\) 8.63603e8 0.00466935
\(320\) 2.20311e11 1.17452
\(321\) 2.10049e9 0.0110420
\(322\) −1.02270e11 −0.530148
\(323\) −1.62971e11 −0.833104
\(324\) 2.85799e10 0.144081
\(325\) 5.97682e10 0.297163
\(326\) 1.02332e11 0.501802
\(327\) 8.93610e9 0.0432198
\(328\) 3.25834e11 1.55440
\(329\) −8.98417e10 −0.422762
\(330\) 9.21111e9 0.0427562
\(331\) 1.41353e11 0.647259 0.323629 0.946184i \(-0.395097\pi\)
0.323629 + 0.946184i \(0.395097\pi\)
\(332\) −1.89270e10 −0.0854990
\(333\) 9.26999e10 0.413124
\(334\) 4.02907e11 1.77152
\(335\) −5.09421e11 −2.20991
\(336\) −4.45367e9 −0.0190630
\(337\) −2.51597e11 −1.06260 −0.531302 0.847182i \(-0.678297\pi\)
−0.531302 + 0.847182i \(0.678297\pi\)
\(338\) −1.97504e10 −0.0823097
\(339\) −9.58501e9 −0.0394180
\(340\) −8.60677e10 −0.349289
\(341\) 1.85909e11 0.744571
\(342\) 1.34439e11 0.531382
\(343\) 1.38413e10 0.0539949
\(344\) 1.20225e11 0.462893
\(345\) 2.22776e10 0.0846610
\(346\) −2.15358e11 −0.807826
\(347\) 4.63528e11 1.71630 0.858151 0.513398i \(-0.171614\pi\)
0.858151 + 0.513398i \(0.171614\pi\)
\(348\) −1.34315e7 −4.90927e−5 0
\(349\) −1.00354e11 −0.362092 −0.181046 0.983475i \(-0.557948\pi\)
−0.181046 + 0.983475i \(0.557948\pi\)
\(350\) −1.21652e11 −0.433324
\(351\) −7.07129e9 −0.0248666
\(352\) −5.12744e10 −0.178016
\(353\) 1.87907e11 0.644104 0.322052 0.946722i \(-0.395627\pi\)
0.322052 + 0.946722i \(0.395627\pi\)
\(354\) 4.74996e9 0.0160759
\(355\) −7.84399e11 −2.62126
\(356\) 6.53305e10 0.215571
\(357\) −8.71496e9 −0.0283961
\(358\) 5.16740e11 1.66264
\(359\) 4.90223e11 1.55764 0.778822 0.627244i \(-0.215817\pi\)
0.778822 + 0.627244i \(0.215817\pi\)
\(360\) −4.18798e11 −1.31415
\(361\) −2.42786e11 −0.752386
\(362\) 3.08799e10 0.0945120
\(363\) −9.16258e9 −0.0276973
\(364\) 5.08947e9 0.0151956
\(365\) −2.92598e11 −0.862887
\(366\) 2.98798e10 0.0870387
\(367\) −3.42258e9 −0.00984820 −0.00492410 0.999988i \(-0.501567\pi\)
−0.00492410 + 0.999988i \(0.501567\pi\)
\(368\) −5.18336e11 −1.47332
\(369\) −6.03843e11 −1.69553
\(370\) 2.29823e11 0.637508
\(371\) 7.42065e10 0.203357
\(372\) −2.89141e9 −0.00782829
\(373\) 1.85283e11 0.495616 0.247808 0.968809i \(-0.420290\pi\)
0.247808 + 0.968809i \(0.420290\pi\)
\(374\) −4.19374e11 −1.10836
\(375\) 1.76686e9 0.00461382
\(376\) −3.96619e11 −1.02336
\(377\) −8.21008e8 −0.00209320
\(378\) 1.43929e10 0.0362605
\(379\) −1.06955e11 −0.266272 −0.133136 0.991098i \(-0.542505\pi\)
−0.133136 + 0.991098i \(0.542505\pi\)
\(380\) 4.21975e10 0.103815
\(381\) 1.06648e9 0.00259292
\(382\) −1.65484e11 −0.397623
\(383\) −6.69645e11 −1.59019 −0.795097 0.606482i \(-0.792580\pi\)
−0.795097 + 0.606482i \(0.792580\pi\)
\(384\) −2.21971e10 −0.0520962
\(385\) −1.45089e11 −0.336558
\(386\) 8.25356e11 1.89234
\(387\) −2.22803e11 −0.504920
\(388\) 1.14089e10 0.0255564
\(389\) −4.35983e11 −0.965376 −0.482688 0.875792i \(-0.660339\pi\)
−0.482688 + 0.875792i \(0.660339\pi\)
\(390\) −8.75680e9 −0.0191670
\(391\) −1.01428e12 −2.19464
\(392\) 6.11043e10 0.130703
\(393\) 7.01322e9 0.0148303
\(394\) −6.74897e9 −0.0141093
\(395\) −3.12021e11 −0.644906
\(396\) 4.37989e10 0.0895024
\(397\) 9.25523e11 1.86995 0.934975 0.354713i \(-0.115421\pi\)
0.934975 + 0.354713i \(0.115421\pi\)
\(398\) −4.79140e11 −0.957168
\(399\) 4.27280e9 0.00843984
\(400\) −6.16569e11 −1.20424
\(401\) 1.01599e12 1.96218 0.981088 0.193562i \(-0.0620041\pi\)
0.981088 + 0.193562i \(0.0620041\pi\)
\(402\) 3.86053e10 0.0737275
\(403\) −1.76740e11 −0.333781
\(404\) −6.29226e9 −0.0117514
\(405\) 7.74558e11 1.43056
\(406\) 1.67107e9 0.00305231
\(407\) 1.41776e11 0.256111
\(408\) −3.84734e10 −0.0687370
\(409\) −5.69331e11 −1.00603 −0.503014 0.864279i \(-0.667775\pi\)
−0.503014 + 0.864279i \(0.667775\pi\)
\(410\) −1.49706e12 −2.61644
\(411\) −1.98943e10 −0.0343906
\(412\) −5.62761e10 −0.0962247
\(413\) −7.48189e10 −0.126542
\(414\) 8.36704e11 1.39981
\(415\) −5.12951e11 −0.848906
\(416\) 4.87455e10 0.0798021
\(417\) 8.74950e9 0.0141700
\(418\) 2.05612e11 0.329424
\(419\) −9.48916e11 −1.50406 −0.752029 0.659130i \(-0.770925\pi\)
−0.752029 + 0.659130i \(0.770925\pi\)
\(420\) 2.25653e9 0.00353851
\(421\) 4.56911e11 0.708863 0.354431 0.935082i \(-0.384674\pi\)
0.354431 + 0.935082i \(0.384674\pi\)
\(422\) −2.73547e11 −0.419881
\(423\) 7.35024e11 1.11627
\(424\) 3.27595e11 0.492256
\(425\) −1.20650e12 −1.79382
\(426\) 5.94439e10 0.0874508
\(427\) −4.70651e11 −0.685130
\(428\) 2.47620e10 0.0356689
\(429\) −5.40200e9 −0.00770011
\(430\) −5.52378e11 −0.779163
\(431\) 1.20269e11 0.167882 0.0839410 0.996471i \(-0.473249\pi\)
0.0839410 + 0.996471i \(0.473249\pi\)
\(432\) 7.29474e10 0.100770
\(433\) 1.29563e12 1.77128 0.885639 0.464374i \(-0.153721\pi\)
0.885639 + 0.464374i \(0.153721\pi\)
\(434\) 3.59735e11 0.486720
\(435\) −3.64013e8 −0.000487433 0
\(436\) 1.05345e11 0.139613
\(437\) 4.97285e11 0.652287
\(438\) 2.21739e10 0.0287878
\(439\) 1.55250e11 0.199499 0.0997493 0.995013i \(-0.468196\pi\)
0.0997493 + 0.995013i \(0.468196\pi\)
\(440\) −6.40514e11 −0.814688
\(441\) −1.13240e11 −0.142569
\(442\) 3.98689e11 0.496860
\(443\) 1.44640e12 1.78432 0.892160 0.451720i \(-0.149190\pi\)
0.892160 + 0.451720i \(0.149190\pi\)
\(444\) −2.20502e9 −0.00269270
\(445\) 1.77056e12 2.14037
\(446\) −7.28652e11 −0.871994
\(447\) −6.04017e10 −0.0715592
\(448\) 2.62982e11 0.308443
\(449\) 6.38236e11 0.741093 0.370546 0.928814i \(-0.379170\pi\)
0.370546 + 0.928814i \(0.379170\pi\)
\(450\) 9.95273e11 1.14416
\(451\) −9.23524e11 −1.05112
\(452\) −1.12995e11 −0.127331
\(453\) 3.07409e10 0.0342985
\(454\) −1.16802e11 −0.129033
\(455\) 1.37932e11 0.150874
\(456\) 1.88629e10 0.0204299
\(457\) −8.32269e10 −0.0892567 −0.0446284 0.999004i \(-0.514210\pi\)
−0.0446284 + 0.999004i \(0.514210\pi\)
\(458\) −1.31354e11 −0.139492
\(459\) 1.42744e11 0.150107
\(460\) 2.62624e11 0.273479
\(461\) −1.79458e12 −1.85058 −0.925292 0.379254i \(-0.876181\pi\)
−0.925292 + 0.379254i \(0.876181\pi\)
\(462\) 1.09952e10 0.0112283
\(463\) 1.55403e12 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(464\) 8.46952e9 0.00848258
\(465\) −7.83617e10 −0.0777258
\(466\) 1.87455e12 1.84145
\(467\) 1.55205e11 0.151001 0.0755003 0.997146i \(-0.475945\pi\)
0.0755003 + 0.997146i \(0.475945\pi\)
\(468\) −4.16386e10 −0.0401227
\(469\) −6.08090e11 −0.580350
\(470\) 1.82228e12 1.72257
\(471\) −8.07955e10 −0.0756472
\(472\) −3.30299e11 −0.306315
\(473\) −3.40758e11 −0.313019
\(474\) 2.36458e10 0.0215155
\(475\) 5.91529e11 0.533156
\(476\) −1.02738e11 −0.0917276
\(477\) −6.07108e11 −0.536949
\(478\) 4.94562e11 0.433307
\(479\) 1.67162e11 0.145086 0.0725432 0.997365i \(-0.476888\pi\)
0.0725432 + 0.997365i \(0.476888\pi\)
\(480\) 2.16124e10 0.0185831
\(481\) −1.34783e11 −0.114811
\(482\) 1.50202e12 1.26755
\(483\) 2.65926e10 0.0222330
\(484\) −1.08015e11 −0.0894704
\(485\) 3.09198e11 0.253746
\(486\) −1.76688e11 −0.143663
\(487\) −2.29324e12 −1.84743 −0.923717 0.383076i \(-0.874865\pi\)
−0.923717 + 0.383076i \(0.874865\pi\)
\(488\) −2.07775e12 −1.65846
\(489\) −2.66087e10 −0.0210443
\(490\) −2.80747e11 −0.220005
\(491\) −8.75896e10 −0.0680121 −0.0340060 0.999422i \(-0.510827\pi\)
−0.0340060 + 0.999422i \(0.510827\pi\)
\(492\) 1.43634e10 0.0110513
\(493\) 1.65732e10 0.0126356
\(494\) −1.95471e11 −0.147676
\(495\) 1.18702e12 0.888655
\(496\) 1.82325e12 1.35263
\(497\) −9.36328e11 −0.688374
\(498\) 3.88728e10 0.0283213
\(499\) −4.13751e11 −0.298735 −0.149368 0.988782i \(-0.547724\pi\)
−0.149368 + 0.988782i \(0.547724\pi\)
\(500\) 2.08290e10 0.0149040
\(501\) −1.04765e11 −0.0742928
\(502\) −1.21010e12 −0.850465
\(503\) −1.87574e12 −1.30652 −0.653260 0.757134i \(-0.726599\pi\)
−0.653260 + 0.757134i \(0.726599\pi\)
\(504\) −4.99914e11 −0.345111
\(505\) −1.70530e11 −0.116678
\(506\) 1.27966e12 0.867798
\(507\) 5.13557e9 0.00345186
\(508\) 1.25724e10 0.00837587
\(509\) −8.08683e11 −0.534009 −0.267004 0.963695i \(-0.586034\pi\)
−0.267004 + 0.963695i \(0.586034\pi\)
\(510\) 1.76768e11 0.115701
\(511\) −3.49271e11 −0.226605
\(512\) 1.09612e12 0.704925
\(513\) −6.99848e10 −0.0446145
\(514\) 1.60855e12 1.01648
\(515\) −1.52517e12 −0.955400
\(516\) 5.29974e9 0.00329103
\(517\) 1.12415e12 0.692019
\(518\) 2.74337e11 0.167418
\(519\) 5.59980e10 0.0338781
\(520\) 6.08923e11 0.365214
\(521\) −2.14661e12 −1.27639 −0.638195 0.769874i \(-0.720319\pi\)
−0.638195 + 0.769874i \(0.720319\pi\)
\(522\) −1.36716e10 −0.00805939
\(523\) 6.45639e11 0.377339 0.188670 0.982041i \(-0.439582\pi\)
0.188670 + 0.982041i \(0.439582\pi\)
\(524\) 8.26767e10 0.0479063
\(525\) 3.16323e10 0.0181725
\(526\) −9.30266e11 −0.529872
\(527\) 3.56774e12 2.01486
\(528\) 5.57271e10 0.0312042
\(529\) 1.29379e12 0.718315
\(530\) −1.50515e12 −0.828589
\(531\) 6.12118e11 0.334126
\(532\) 5.03707e10 0.0272631
\(533\) 8.77974e11 0.471204
\(534\) −1.34177e11 −0.0714075
\(535\) 6.71089e11 0.354151
\(536\) −2.68450e12 −1.40482
\(537\) −1.34364e11 −0.0697269
\(538\) 3.40707e12 1.75332
\(539\) −1.73191e11 −0.0883842
\(540\) −3.69601e10 −0.0187052
\(541\) 1.62332e12 0.814733 0.407366 0.913265i \(-0.366447\pi\)
0.407366 + 0.913265i \(0.366447\pi\)
\(542\) −3.74451e12 −1.86380
\(543\) −8.02948e9 −0.00396359
\(544\) −9.83995e11 −0.481723
\(545\) 2.85501e12 1.38619
\(546\) −1.04529e10 −0.00503350
\(547\) 1.50699e12 0.719725 0.359863 0.933005i \(-0.382824\pi\)
0.359863 + 0.933005i \(0.382824\pi\)
\(548\) −2.34528e11 −0.111092
\(549\) 3.85055e12 1.80903
\(550\) 1.52218e12 0.709307
\(551\) −8.12555e9 −0.00375552
\(552\) 1.17397e11 0.0538183
\(553\) −3.72456e11 −0.169360
\(554\) −8.77999e11 −0.396005
\(555\) −5.97593e10 −0.0267354
\(556\) 1.03145e11 0.0457733
\(557\) 3.20939e12 1.41278 0.706388 0.707824i \(-0.250323\pi\)
0.706388 + 0.707824i \(0.250323\pi\)
\(558\) −2.94311e12 −1.28515
\(559\) 3.23951e11 0.140322
\(560\) −1.42291e12 −0.611409
\(561\) 1.09047e11 0.0464815
\(562\) 2.74070e12 1.15891
\(563\) 2.95439e12 1.23931 0.619654 0.784875i \(-0.287273\pi\)
0.619654 + 0.784875i \(0.287273\pi\)
\(564\) −1.74838e10 −0.00727576
\(565\) −3.06233e12 −1.26425
\(566\) −3.18027e12 −1.30254
\(567\) 9.24581e11 0.375683
\(568\) −4.13355e12 −1.66631
\(569\) −1.52799e12 −0.611103 −0.305551 0.952176i \(-0.598841\pi\)
−0.305551 + 0.952176i \(0.598841\pi\)
\(570\) −8.66664e10 −0.0343885
\(571\) −2.58983e12 −1.01955 −0.509776 0.860307i \(-0.670272\pi\)
−0.509776 + 0.860307i \(0.670272\pi\)
\(572\) −6.36826e10 −0.0248736
\(573\) 4.30297e10 0.0166753
\(574\) −1.78702e12 −0.687110
\(575\) 3.68149e12 1.40449
\(576\) −2.15154e12 −0.814421
\(577\) 9.80075e11 0.368102 0.184051 0.982917i \(-0.441079\pi\)
0.184051 + 0.982917i \(0.441079\pi\)
\(578\) −5.17685e12 −1.92926
\(579\) −2.14612e11 −0.0793597
\(580\) −4.29124e9 −0.00157455
\(581\) −6.12304e11 −0.222933
\(582\) −2.34319e10 −0.00846551
\(583\) −9.28517e11 −0.332875
\(584\) −1.54191e12 −0.548530
\(585\) −1.12847e12 −0.398372
\(586\) −2.53980e12 −0.889735
\(587\) −5.81451e11 −0.202135 −0.101067 0.994880i \(-0.532226\pi\)
−0.101067 + 0.994880i \(0.532226\pi\)
\(588\) 2.69360e9 0.000929256 0
\(589\) −1.74920e12 −0.598854
\(590\) 1.51757e12 0.515603
\(591\) 1.75489e9 0.000591706 0
\(592\) 1.39043e12 0.465265
\(593\) −1.04203e12 −0.346046 −0.173023 0.984918i \(-0.555353\pi\)
−0.173023 + 0.984918i \(0.555353\pi\)
\(594\) −1.80092e11 −0.0593548
\(595\) −2.78436e12 −0.910749
\(596\) −7.12057e11 −0.231157
\(597\) 1.24587e11 0.0401411
\(598\) −1.21655e12 −0.389022
\(599\) 5.43809e12 1.72594 0.862970 0.505256i \(-0.168602\pi\)
0.862970 + 0.505256i \(0.168602\pi\)
\(600\) 1.39645e11 0.0439892
\(601\) 3.95819e12 1.23755 0.618774 0.785569i \(-0.287630\pi\)
0.618774 + 0.785569i \(0.287630\pi\)
\(602\) −6.59368e11 −0.204618
\(603\) 4.97498e12 1.53237
\(604\) 3.62395e11 0.110794
\(605\) −2.92737e12 −0.888338
\(606\) 1.29232e10 0.00389263
\(607\) 3.05579e12 0.913640 0.456820 0.889559i \(-0.348988\pi\)
0.456820 + 0.889559i \(0.348988\pi\)
\(608\) 4.82436e11 0.143177
\(609\) −4.34518e8 −0.000128006 0
\(610\) 9.54634e12 2.79160
\(611\) −1.06871e12 −0.310223
\(612\) 8.40533e11 0.242200
\(613\) −1.18677e12 −0.339466 −0.169733 0.985490i \(-0.554290\pi\)
−0.169733 + 0.985490i \(0.554290\pi\)
\(614\) −3.57329e12 −1.01464
\(615\) 3.89270e11 0.109727
\(616\) −7.64574e11 −0.213947
\(617\) 6.94626e11 0.192960 0.0964801 0.995335i \(-0.469242\pi\)
0.0964801 + 0.995335i \(0.469242\pi\)
\(618\) 1.15581e11 0.0318742
\(619\) 2.54412e12 0.696515 0.348257 0.937399i \(-0.386774\pi\)
0.348257 + 0.937399i \(0.386774\pi\)
\(620\) −9.23782e11 −0.251077
\(621\) −4.35564e11 −0.117528
\(622\) −2.90483e11 −0.0778152
\(623\) 2.11349e12 0.562088
\(624\) −5.29785e10 −0.0139884
\(625\) −3.52272e12 −0.923459
\(626\) −2.80090e12 −0.728976
\(627\) −5.34638e10 −0.0138152
\(628\) −9.52473e11 −0.244362
\(629\) 2.72079e12 0.693054
\(630\) 2.29688e12 0.580906
\(631\) −4.10571e12 −1.03099 −0.515497 0.856891i \(-0.672393\pi\)
−0.515497 + 0.856891i \(0.672393\pi\)
\(632\) −1.64426e12 −0.409962
\(633\) 7.11287e10 0.0176087
\(634\) −1.03076e12 −0.253371
\(635\) 3.40730e11 0.0831627
\(636\) 1.44411e10 0.00349979
\(637\) 1.64648e11 0.0396214
\(638\) −2.09095e10 −0.00499632
\(639\) 7.66041e12 1.81760
\(640\) −7.09179e12 −1.67088
\(641\) 2.93476e12 0.686612 0.343306 0.939224i \(-0.388453\pi\)
0.343306 + 0.939224i \(0.388453\pi\)
\(642\) −5.08569e10 −0.0118152
\(643\) −6.69969e12 −1.54563 −0.772815 0.634632i \(-0.781152\pi\)
−0.772815 + 0.634632i \(0.781152\pi\)
\(644\) 3.13492e11 0.0718191
\(645\) 1.43631e11 0.0326761
\(646\) 3.94584e12 0.891443
\(647\) −6.23009e12 −1.39774 −0.698868 0.715250i \(-0.746313\pi\)
−0.698868 + 0.715250i \(0.746313\pi\)
\(648\) 4.08169e12 0.909396
\(649\) 9.36180e11 0.207137
\(650\) −1.44710e12 −0.317973
\(651\) −9.35395e10 −0.0204118
\(652\) −3.13682e11 −0.0679791
\(653\) −4.42425e12 −0.952206 −0.476103 0.879390i \(-0.657951\pi\)
−0.476103 + 0.879390i \(0.657951\pi\)
\(654\) −2.16360e11 −0.0462464
\(655\) 2.24066e12 0.475654
\(656\) −9.05718e12 −1.90953
\(657\) 2.85750e12 0.598332
\(658\) 2.17524e12 0.452367
\(659\) −1.19733e12 −0.247304 −0.123652 0.992326i \(-0.539461\pi\)
−0.123652 + 0.992326i \(0.539461\pi\)
\(660\) −2.82351e10 −0.00579218
\(661\) −8.59424e12 −1.75106 −0.875530 0.483164i \(-0.839487\pi\)
−0.875530 + 0.483164i \(0.839487\pi\)
\(662\) −3.42242e12 −0.692584
\(663\) −1.03668e11 −0.0208370
\(664\) −2.70310e12 −0.539643
\(665\) 1.36512e12 0.270691
\(666\) −2.24444e12 −0.442053
\(667\) −5.05709e10 −0.00989315
\(668\) −1.23504e12 −0.239987
\(669\) 1.89466e11 0.0365691
\(670\) 1.23341e13 2.36466
\(671\) 5.88907e12 1.12149
\(672\) 2.57985e10 0.00488015
\(673\) −4.63034e11 −0.0870051 −0.0435026 0.999053i \(-0.513852\pi\)
−0.0435026 + 0.999053i \(0.513852\pi\)
\(674\) 6.09166e12 1.13701
\(675\) −5.18111e11 −0.0960629
\(676\) 6.05416e10 0.0111505
\(677\) 4.72455e12 0.864393 0.432196 0.901780i \(-0.357739\pi\)
0.432196 + 0.901780i \(0.357739\pi\)
\(678\) 2.32072e11 0.0421783
\(679\) 3.69086e11 0.0666368
\(680\) −1.22919e13 −2.20460
\(681\) 3.03714e10 0.00541131
\(682\) −4.50122e12 −0.796711
\(683\) −1.89286e12 −0.332832 −0.166416 0.986056i \(-0.553219\pi\)
−0.166416 + 0.986056i \(0.553219\pi\)
\(684\) −4.12099e11 −0.0719862
\(685\) −6.35606e12 −1.10301
\(686\) −3.35124e11 −0.0577760
\(687\) 3.41551e10 0.00584992
\(688\) −3.34188e12 −0.568647
\(689\) 8.82721e11 0.149223
\(690\) −5.39385e11 −0.0905895
\(691\) 9.50737e12 1.58639 0.793193 0.608970i \(-0.208417\pi\)
0.793193 + 0.608970i \(0.208417\pi\)
\(692\) 6.60143e11 0.109436
\(693\) 1.41693e12 0.233372
\(694\) −1.12229e13 −1.83649
\(695\) 2.79539e12 0.454476
\(696\) −1.91824e9 −0.000309857 0
\(697\) −1.77231e13 −2.84441
\(698\) 2.42976e12 0.387448
\(699\) −4.87425e11 −0.0772255
\(700\) 3.72904e11 0.0587023
\(701\) 3.69030e12 0.577205 0.288603 0.957449i \(-0.406809\pi\)
0.288603 + 0.957449i \(0.406809\pi\)
\(702\) 1.71210e11 0.0266079
\(703\) −1.33396e12 −0.205988
\(704\) −3.29060e12 −0.504891
\(705\) −4.73836e11 −0.0722399
\(706\) −4.54958e12 −0.689208
\(707\) −2.03560e11 −0.0306411
\(708\) −1.45602e10 −0.00217780
\(709\) 9.02589e12 1.34147 0.670737 0.741695i \(-0.265978\pi\)
0.670737 + 0.741695i \(0.265978\pi\)
\(710\) 1.89918e13 2.80481
\(711\) 3.04718e12 0.447183
\(712\) 9.33031e12 1.36062
\(713\) −1.08865e13 −1.57756
\(714\) 2.11006e11 0.0303846
\(715\) −1.72589e12 −0.246966
\(716\) −1.58398e12 −0.225238
\(717\) −1.28598e11 −0.0181718
\(718\) −1.18692e13 −1.66672
\(719\) −5.57596e12 −0.778108 −0.389054 0.921215i \(-0.627198\pi\)
−0.389054 + 0.921215i \(0.627198\pi\)
\(720\) 1.16413e13 1.61438
\(721\) −1.82058e12 −0.250900
\(722\) 5.87831e12 0.805073
\(723\) −3.90560e11 −0.0531577
\(724\) −9.46571e10 −0.0128035
\(725\) −6.01549e10 −0.00808631
\(726\) 2.21844e11 0.0296369
\(727\) 5.66814e12 0.752551 0.376276 0.926508i \(-0.377205\pi\)
0.376276 + 0.926508i \(0.377205\pi\)
\(728\) 7.26864e11 0.0959096
\(729\) −7.53362e12 −0.987938
\(730\) 7.08437e12 0.923312
\(731\) −6.53940e12 −0.847051
\(732\) −9.15915e10 −0.0117911
\(733\) −2.25858e12 −0.288980 −0.144490 0.989506i \(-0.546154\pi\)
−0.144490 + 0.989506i \(0.546154\pi\)
\(734\) 8.28673e10 0.0105378
\(735\) 7.30007e10 0.00922643
\(736\) 3.00253e12 0.377170
\(737\) 7.60879e12 0.949974
\(738\) 1.46202e13 1.81426
\(739\) 1.34627e13 1.66047 0.830235 0.557413i \(-0.188206\pi\)
0.830235 + 0.557413i \(0.188206\pi\)
\(740\) −7.04485e11 −0.0863632
\(741\) 5.08269e10 0.00619315
\(742\) −1.79668e12 −0.217598
\(743\) −1.23724e13 −1.48938 −0.744688 0.667413i \(-0.767402\pi\)
−0.744688 + 0.667413i \(0.767402\pi\)
\(744\) −4.12943e11 −0.0494097
\(745\) −1.92978e13 −2.29512
\(746\) −4.48605e12 −0.530322
\(747\) 5.00946e12 0.588638
\(748\) 1.28552e12 0.150149
\(749\) 8.01071e11 0.0930043
\(750\) −4.27790e10 −0.00493691
\(751\) −4.20252e12 −0.482092 −0.241046 0.970514i \(-0.577490\pi\)
−0.241046 + 0.970514i \(0.577490\pi\)
\(752\) 1.10248e13 1.25716
\(753\) 3.14655e11 0.0356663
\(754\) 1.98782e10 0.00223978
\(755\) 9.82146e12 1.10006
\(756\) −4.41189e10 −0.00491221
\(757\) 1.32766e13 1.46946 0.734728 0.678361i \(-0.237310\pi\)
0.734728 + 0.678361i \(0.237310\pi\)
\(758\) 2.58959e12 0.284918
\(759\) −3.32742e11 −0.0363932
\(760\) 6.02653e12 0.655249
\(761\) 3.70152e11 0.0400083 0.0200041 0.999800i \(-0.493632\pi\)
0.0200041 + 0.999800i \(0.493632\pi\)
\(762\) −2.58214e10 −0.00277449
\(763\) 3.40799e12 0.364031
\(764\) 5.07264e11 0.0538659
\(765\) 2.27797e13 2.40476
\(766\) 1.62134e13 1.70155
\(767\) −8.90005e11 −0.0928568
\(768\) 1.84377e11 0.0191241
\(769\) −1.57793e13 −1.62712 −0.813558 0.581484i \(-0.802472\pi\)
−0.813558 + 0.581484i \(0.802472\pi\)
\(770\) 3.51287e12 0.360126
\(771\) −4.18261e11 −0.0426287
\(772\) −2.52999e12 −0.256355
\(773\) 3.76505e12 0.379283 0.189642 0.981853i \(-0.439267\pi\)
0.189642 + 0.981853i \(0.439267\pi\)
\(774\) 5.39450e12 0.540278
\(775\) −1.29497e13 −1.28944
\(776\) 1.62938e12 0.161304
\(777\) −7.13341e10 −0.00702105
\(778\) 1.05560e13 1.03298
\(779\) 8.68934e12 0.845412
\(780\) 2.68425e10 0.00259656
\(781\) 1.17159e13 1.12680
\(782\) 2.45577e13 2.34832
\(783\) 7.11704e9 0.000676662 0
\(784\) −1.69851e12 −0.160563
\(785\) −2.58135e13 −2.42624
\(786\) −1.69804e11 −0.0158688
\(787\) 1.18260e13 1.09889 0.549444 0.835531i \(-0.314840\pi\)
0.549444 + 0.835531i \(0.314840\pi\)
\(788\) 2.06878e10 0.00191138
\(789\) 2.41891e11 0.0222215
\(790\) 7.55462e12 0.690066
\(791\) −3.65547e12 −0.332008
\(792\) 6.25523e12 0.564911
\(793\) −5.59861e12 −0.502748
\(794\) −2.24087e13 −2.00090
\(795\) 3.91375e11 0.0347489
\(796\) 1.46872e12 0.129668
\(797\) −6.28239e12 −0.551521 −0.275761 0.961226i \(-0.588930\pi\)
−0.275761 + 0.961226i \(0.588930\pi\)
\(798\) −1.03453e11 −0.00903085
\(799\) 2.15733e13 1.87265
\(800\) 3.57156e12 0.308285
\(801\) −1.72912e13 −1.48415
\(802\) −2.45990e13 −2.09958
\(803\) 4.37029e12 0.370929
\(804\) −1.18338e11 −0.00998785
\(805\) 8.49610e12 0.713080
\(806\) 4.27921e12 0.357155
\(807\) −8.85917e11 −0.0735296
\(808\) −8.98642e11 −0.0741713
\(809\) 3.18104e12 0.261097 0.130548 0.991442i \(-0.458326\pi\)
0.130548 + 0.991442i \(0.458326\pi\)
\(810\) −1.87535e13 −1.53074
\(811\) 1.45353e13 1.17986 0.589929 0.807455i \(-0.299156\pi\)
0.589929 + 0.807455i \(0.299156\pi\)
\(812\) −5.12240e9 −0.000413496 0
\(813\) 9.73660e11 0.0781628
\(814\) −3.43267e12 −0.274046
\(815\) −8.50126e12 −0.674954
\(816\) 1.06944e12 0.0844408
\(817\) 3.20616e12 0.251759
\(818\) 1.37846e13 1.07648
\(819\) −1.34704e12 −0.104617
\(820\) 4.58899e12 0.354449
\(821\) −2.29827e13 −1.76545 −0.882726 0.469888i \(-0.844294\pi\)
−0.882726 + 0.469888i \(0.844294\pi\)
\(822\) 4.81679e11 0.0367989
\(823\) 1.19282e13 0.906309 0.453154 0.891432i \(-0.350299\pi\)
0.453154 + 0.891432i \(0.350299\pi\)
\(824\) −8.03719e12 −0.607340
\(825\) −3.95803e11 −0.0297465
\(826\) 1.81151e12 0.135404
\(827\) −1.61714e13 −1.20219 −0.601094 0.799178i \(-0.705268\pi\)
−0.601094 + 0.799178i \(0.705268\pi\)
\(828\) −2.56478e12 −0.189633
\(829\) −1.82619e13 −1.34292 −0.671462 0.741039i \(-0.734333\pi\)
−0.671462 + 0.741039i \(0.734333\pi\)
\(830\) 1.24195e13 0.908352
\(831\) 2.28300e11 0.0166074
\(832\) 3.12830e12 0.226336
\(833\) −3.32366e12 −0.239174
\(834\) −2.11842e11 −0.0151623
\(835\) −3.34715e13 −2.38279
\(836\) −6.30269e11 −0.0446270
\(837\) 1.53210e12 0.107900
\(838\) 2.29751e13 1.60938
\(839\) 1.51573e13 1.05607 0.528035 0.849222i \(-0.322929\pi\)
0.528035 + 0.849222i \(0.322929\pi\)
\(840\) 3.22272e11 0.0223340
\(841\) −1.45063e13 −0.999943
\(842\) −1.10627e13 −0.758502
\(843\) −7.12646e11 −0.0486015
\(844\) 8.38514e11 0.0568813
\(845\) 1.64077e12 0.110711
\(846\) −1.77964e13 −1.19444
\(847\) −3.49437e12 −0.233288
\(848\) −9.10615e12 −0.604718
\(849\) 8.26943e11 0.0546250
\(850\) 2.92118e13 1.91943
\(851\) −8.30214e12 −0.542634
\(852\) −1.82215e11 −0.0118470
\(853\) 2.16876e13 1.40262 0.701312 0.712854i \(-0.252598\pi\)
0.701312 + 0.712854i \(0.252598\pi\)
\(854\) 1.13954e13 0.733107
\(855\) −1.11685e13 −0.714740
\(856\) 3.53644e12 0.225131
\(857\) 1.19059e13 0.753961 0.376981 0.926221i \(-0.376962\pi\)
0.376981 + 0.926221i \(0.376962\pi\)
\(858\) 1.30793e11 0.00823932
\(859\) 2.06615e13 1.29477 0.647386 0.762162i \(-0.275862\pi\)
0.647386 + 0.762162i \(0.275862\pi\)
\(860\) 1.69322e12 0.105553
\(861\) 4.64667e11 0.0288156
\(862\) −2.91193e12 −0.179638
\(863\) −2.69412e13 −1.65336 −0.826682 0.562670i \(-0.809774\pi\)
−0.826682 + 0.562670i \(0.809774\pi\)
\(864\) −4.22558e11 −0.0257973
\(865\) 1.78909e13 1.08657
\(866\) −3.13698e13 −1.89531
\(867\) 1.34610e12 0.0809081
\(868\) −1.10271e12 −0.0659359
\(869\) 4.66039e12 0.277226
\(870\) 8.81346e9 0.000521566 0
\(871\) −7.23351e12 −0.425860
\(872\) 1.50451e13 0.881191
\(873\) −3.01962e12 −0.175949
\(874\) −1.20402e13 −0.697964
\(875\) 6.73833e11 0.0388612
\(876\) −6.79704e10 −0.00389988
\(877\) 2.18247e12 0.124580 0.0622902 0.998058i \(-0.480160\pi\)
0.0622902 + 0.998058i \(0.480160\pi\)
\(878\) −3.75889e12 −0.213469
\(879\) 6.60408e11 0.0373132
\(880\) 1.78043e13 1.00081
\(881\) −1.17134e13 −0.655077 −0.327538 0.944838i \(-0.606219\pi\)
−0.327538 + 0.944838i \(0.606219\pi\)
\(882\) 2.74176e12 0.152553
\(883\) 1.38647e13 0.767517 0.383759 0.923433i \(-0.374629\pi\)
0.383759 + 0.923433i \(0.374629\pi\)
\(884\) −1.22212e12 −0.0673097
\(885\) −3.94604e11 −0.0216231
\(886\) −3.50202e13 −1.90927
\(887\) −2.07690e13 −1.12657 −0.563286 0.826262i \(-0.690463\pi\)
−0.563286 + 0.826262i \(0.690463\pi\)
\(888\) −3.14914e11 −0.0169955
\(889\) 4.06726e11 0.0218395
\(890\) −4.28686e13 −2.29026
\(891\) −1.15689e13 −0.614955
\(892\) 2.23356e12 0.118129
\(893\) −1.05770e13 −0.556587
\(894\) 1.46244e12 0.0765702
\(895\) −4.29283e13 −2.23635
\(896\) −8.46540e12 −0.438795
\(897\) 3.16331e11 0.0163146
\(898\) −1.54529e13 −0.792989
\(899\) 1.77883e11 0.00908274
\(900\) −3.05085e12 −0.154999
\(901\) −1.78189e13 −0.900783
\(902\) 2.23603e13 1.12473
\(903\) 1.71451e11 0.00858114
\(904\) −1.61376e13 −0.803676
\(905\) −2.56535e12 −0.127124
\(906\) −7.44297e11 −0.0367003
\(907\) 1.25577e13 0.616138 0.308069 0.951364i \(-0.400317\pi\)
0.308069 + 0.951364i \(0.400317\pi\)
\(908\) 3.58039e11 0.0174801
\(909\) 1.66539e12 0.0809054
\(910\) −3.33961e12 −0.161439
\(911\) −1.91027e12 −0.0918888 −0.0459444 0.998944i \(-0.514630\pi\)
−0.0459444 + 0.998944i \(0.514630\pi\)
\(912\) −5.24330e11 −0.0250974
\(913\) 7.66152e12 0.364919
\(914\) 2.01509e12 0.0955071
\(915\) −2.48227e12 −0.117072
\(916\) 4.02644e11 0.0188969
\(917\) 2.67466e12 0.124913
\(918\) −3.45610e12 −0.160618
\(919\) −6.24982e12 −0.289033 −0.144517 0.989502i \(-0.546163\pi\)
−0.144517 + 0.989502i \(0.546163\pi\)
\(920\) 3.75073e13 1.72612
\(921\) 9.29139e11 0.0425513
\(922\) 4.34503e13 1.98017
\(923\) −1.11381e13 −0.505128
\(924\) −3.37040e10 −0.00152110
\(925\) −9.87553e12 −0.443529
\(926\) −3.76261e13 −1.68167
\(927\) 1.48947e13 0.662482
\(928\) −4.90609e10 −0.00217155
\(929\) −3.02983e13 −1.33459 −0.667293 0.744795i \(-0.732547\pi\)
−0.667293 + 0.744795i \(0.732547\pi\)
\(930\) 1.89729e12 0.0831687
\(931\) 1.62953e12 0.0710869
\(932\) −5.74611e12 −0.249461
\(933\) 7.55324e10 0.00326337
\(934\) −3.75780e12 −0.161575
\(935\) 3.48396e13 1.49080
\(936\) −5.94671e12 −0.253242
\(937\) 6.44264e12 0.273046 0.136523 0.990637i \(-0.456407\pi\)
0.136523 + 0.990637i \(0.456407\pi\)
\(938\) 1.47230e13 0.620990
\(939\) 7.28299e11 0.0305714
\(940\) −5.58591e12 −0.233356
\(941\) −3.76681e12 −0.156610 −0.0783052 0.996929i \(-0.524951\pi\)
−0.0783052 + 0.996929i \(0.524951\pi\)
\(942\) 1.95621e12 0.0809445
\(943\) 5.40798e13 2.22706
\(944\) 9.18129e12 0.376296
\(945\) −1.19569e12 −0.0487725
\(946\) 8.25041e12 0.334939
\(947\) −3.85590e13 −1.55794 −0.778969 0.627062i \(-0.784257\pi\)
−0.778969 + 0.627062i \(0.784257\pi\)
\(948\) −7.24822e10 −0.00291470
\(949\) −4.15474e12 −0.166282
\(950\) −1.43220e13 −0.570491
\(951\) 2.68022e11 0.0106257
\(952\) −1.46728e13 −0.578956
\(953\) −3.84061e13 −1.50828 −0.754140 0.656713i \(-0.771946\pi\)
−0.754140 + 0.656713i \(0.771946\pi\)
\(954\) 1.46993e13 0.574550
\(955\) 1.37476e13 0.534826
\(956\) −1.51600e12 −0.0587000
\(957\) 5.43695e9 0.000209533 0
\(958\) −4.04731e12 −0.155246
\(959\) −7.58716e12 −0.289665
\(960\) 1.38700e12 0.0527056
\(961\) 1.18536e13 0.448329
\(962\) 3.26337e12 0.122851
\(963\) −6.55383e12 −0.245571
\(964\) −4.60420e12 −0.171715
\(965\) −6.85667e13 −2.54531
\(966\) −6.43858e11 −0.0237899
\(967\) 1.79148e13 0.658859 0.329430 0.944180i \(-0.393144\pi\)
0.329430 + 0.944180i \(0.393144\pi\)
\(968\) −1.54264e13 −0.564709
\(969\) −1.02601e12 −0.0373848
\(970\) −7.48628e12 −0.271515
\(971\) 3.22379e13 1.16381 0.581903 0.813258i \(-0.302308\pi\)
0.581903 + 0.813258i \(0.302308\pi\)
\(972\) 5.41609e11 0.0194620
\(973\) 3.33683e12 0.119351
\(974\) 5.55237e13 1.97680
\(975\) 3.76281e11 0.0133349
\(976\) 5.77552e13 2.03736
\(977\) 7.62010e12 0.267569 0.133784 0.991010i \(-0.457287\pi\)
0.133784 + 0.991010i \(0.457287\pi\)
\(978\) 6.44248e11 0.0225179
\(979\) −2.64453e13 −0.920082
\(980\) 8.60583e11 0.0298041
\(981\) −2.78819e13 −0.961196
\(982\) 2.12071e12 0.0727747
\(983\) 2.22466e13 0.759928 0.379964 0.925001i \(-0.375936\pi\)
0.379964 + 0.925001i \(0.375936\pi\)
\(984\) 2.05134e12 0.0697525
\(985\) 5.60672e11 0.0189778
\(986\) −4.01269e11 −0.0135204
\(987\) −5.65613e11 −0.0189711
\(988\) 5.99183e11 0.0200057
\(989\) 1.99541e13 0.663207
\(990\) −2.87400e13 −0.950885
\(991\) 1.18711e13 0.390985 0.195493 0.980705i \(-0.437369\pi\)
0.195493 + 0.980705i \(0.437369\pi\)
\(992\) −1.05614e13 −0.346274
\(993\) 8.89908e11 0.0290452
\(994\) 2.26703e13 0.736578
\(995\) 3.98046e13 1.28745
\(996\) −1.19158e11 −0.00383669
\(997\) −1.63112e13 −0.522828 −0.261414 0.965227i \(-0.584189\pi\)
−0.261414 + 0.965227i \(0.584189\pi\)
\(998\) 1.00177e13 0.319654
\(999\) 1.16839e12 0.0371145
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 91.10.a.a.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.4 12 1.1 even 1 trivial