Properties

Label 91.10.a.a.1.1
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 1.1
Root \(-37.6810\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-39.6810 q^{2} -201.837 q^{3} +1062.58 q^{4} +764.411 q^{5} +8009.09 q^{6} +2401.00 q^{7} -21847.5 q^{8} +21055.2 q^{9} +O(q^{10})\) \(q-39.6810 q^{2} -201.837 q^{3} +1062.58 q^{4} +764.411 q^{5} +8009.09 q^{6} +2401.00 q^{7} -21847.5 q^{8} +21055.2 q^{9} -30332.6 q^{10} -87028.8 q^{11} -214468. q^{12} +28561.0 q^{13} -95274.0 q^{14} -154286. q^{15} +322890. q^{16} -46185.1 q^{17} -835490. q^{18} +504454. q^{19} +812247. q^{20} -484611. q^{21} +3.45339e6 q^{22} -599831. q^{23} +4.40964e6 q^{24} -1.36880e6 q^{25} -1.13333e6 q^{26} -276959. q^{27} +2.55125e6 q^{28} +2.58941e6 q^{29} +6.12224e6 q^{30} +3.96298e6 q^{31} -1.62665e6 q^{32} +1.75656e7 q^{33} +1.83267e6 q^{34} +1.83535e6 q^{35} +2.23728e7 q^{36} -6.19845e6 q^{37} -2.00172e7 q^{38} -5.76467e6 q^{39} -1.67005e7 q^{40} -1.66091e7 q^{41} +1.92298e7 q^{42} +3.31849e7 q^{43} -9.24750e7 q^{44} +1.60948e7 q^{45} +2.38019e7 q^{46} +2.64372e7 q^{47} -6.51711e7 q^{48} +5.76480e6 q^{49} +5.43153e7 q^{50} +9.32186e6 q^{51} +3.03483e7 q^{52} -3.90874e7 q^{53} +1.09900e7 q^{54} -6.65258e7 q^{55} -5.24559e7 q^{56} -1.01817e8 q^{57} -1.02750e8 q^{58} +4.81308e7 q^{59} -1.63942e8 q^{60} +1.43492e8 q^{61} -1.57255e8 q^{62} +5.05535e7 q^{63} -1.00773e8 q^{64} +2.18323e7 q^{65} -6.97022e8 q^{66} -1.77854e7 q^{67} -4.90753e7 q^{68} +1.21068e8 q^{69} -7.28285e7 q^{70} -8.31277e7 q^{71} -4.60003e8 q^{72} +2.29798e8 q^{73} +2.45960e8 q^{74} +2.76275e8 q^{75} +5.36022e8 q^{76} -2.08956e8 q^{77} +2.28748e8 q^{78} -2.99993e8 q^{79} +2.46820e8 q^{80} -3.58529e8 q^{81} +6.59064e8 q^{82} +8.34408e8 q^{83} -5.14937e8 q^{84} -3.53044e7 q^{85} -1.31681e9 q^{86} -5.22638e8 q^{87} +1.90136e9 q^{88} -2.13503e8 q^{89} -6.38658e8 q^{90} +6.85750e7 q^{91} -6.37368e8 q^{92} -7.99875e8 q^{93} -1.04906e9 q^{94} +3.85610e8 q^{95} +3.28318e8 q^{96} -1.36917e9 q^{97} -2.28753e8 q^{98} -1.83241e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.6810 −1.75367 −0.876834 0.480794i \(-0.840349\pi\)
−0.876834 + 0.480794i \(0.840349\pi\)
\(3\) −201.837 −1.43865 −0.719325 0.694674i \(-0.755549\pi\)
−0.719325 + 0.694674i \(0.755549\pi\)
\(4\) 1062.58 2.07535
\(5\) 764.411 0.546968 0.273484 0.961877i \(-0.411824\pi\)
0.273484 + 0.961877i \(0.411824\pi\)
\(6\) 8009.09 2.52291
\(7\) 2401.00 0.377964
\(8\) −21847.5 −1.88581
\(9\) 21055.2 1.06971
\(10\) −30332.6 −0.959200
\(11\) −87028.8 −1.79224 −0.896120 0.443812i \(-0.853626\pi\)
−0.896120 + 0.443812i \(0.853626\pi\)
\(12\) −214468. −2.98570
\(13\) 28561.0 0.277350
\(14\) −95274.0 −0.662824
\(15\) −154286. −0.786896
\(16\) 322890. 1.23173
\(17\) −46185.1 −0.134116 −0.0670581 0.997749i \(-0.521361\pi\)
−0.0670581 + 0.997749i \(0.521361\pi\)
\(18\) −835490. −1.87592
\(19\) 504454. 0.888035 0.444017 0.896018i \(-0.353553\pi\)
0.444017 + 0.896018i \(0.353553\pi\)
\(20\) 812247. 1.13515
\(21\) −484611. −0.543759
\(22\) 3.45339e6 3.14299
\(23\) −599831. −0.446945 −0.223472 0.974710i \(-0.571739\pi\)
−0.223472 + 0.974710i \(0.571739\pi\)
\(24\) 4.40964e6 2.71301
\(25\) −1.36880e6 −0.700826
\(26\) −1.13333e6 −0.486380
\(27\) −276959. −0.100295
\(28\) 2.55125e6 0.784408
\(29\) 2.58941e6 0.679844 0.339922 0.940454i \(-0.389599\pi\)
0.339922 + 0.940454i \(0.389599\pi\)
\(30\) 6.12224e6 1.37995
\(31\) 3.96298e6 0.770715 0.385357 0.922767i \(-0.374078\pi\)
0.385357 + 0.922767i \(0.374078\pi\)
\(32\) −1.62665e6 −0.274232
\(33\) 1.75656e7 2.57841
\(34\) 1.83267e6 0.235195
\(35\) 1.83535e6 0.206734
\(36\) 2.23728e7 2.22003
\(37\) −6.19845e6 −0.543720 −0.271860 0.962337i \(-0.587639\pi\)
−0.271860 + 0.962337i \(0.587639\pi\)
\(38\) −2.00172e7 −1.55732
\(39\) −5.76467e6 −0.399010
\(40\) −1.67005e7 −1.03148
\(41\) −1.66091e7 −0.917948 −0.458974 0.888450i \(-0.651783\pi\)
−0.458974 + 0.888450i \(0.651783\pi\)
\(42\) 1.92298e7 0.953572
\(43\) 3.31849e7 1.48024 0.740121 0.672474i \(-0.234768\pi\)
0.740121 + 0.672474i \(0.234768\pi\)
\(44\) −9.24750e7 −3.71952
\(45\) 1.60948e7 0.585100
\(46\) 2.38019e7 0.783792
\(47\) 2.64372e7 0.790271 0.395135 0.918623i \(-0.370698\pi\)
0.395135 + 0.918623i \(0.370698\pi\)
\(48\) −6.51711e7 −1.77202
\(49\) 5.76480e6 0.142857
\(50\) 5.43153e7 1.22902
\(51\) 9.32186e6 0.192946
\(52\) 3.03483e7 0.575598
\(53\) −3.90874e7 −0.680448 −0.340224 0.940344i \(-0.610503\pi\)
−0.340224 + 0.940344i \(0.610503\pi\)
\(54\) 1.09900e7 0.175884
\(55\) −6.65258e7 −0.980298
\(56\) −5.24559e7 −0.712767
\(57\) −1.01817e8 −1.27757
\(58\) −1.02750e8 −1.19222
\(59\) 4.81308e7 0.517117 0.258558 0.965996i \(-0.416753\pi\)
0.258558 + 0.965996i \(0.416753\pi\)
\(60\) −1.63942e8 −1.63308
\(61\) 1.43492e8 1.32692 0.663458 0.748214i \(-0.269088\pi\)
0.663458 + 0.748214i \(0.269088\pi\)
\(62\) −1.57255e8 −1.35158
\(63\) 5.05535e7 0.404314
\(64\) −1.00773e8 −0.750814
\(65\) 2.18323e7 0.151702
\(66\) −6.97022e8 −4.52167
\(67\) −1.77854e7 −0.107827 −0.0539133 0.998546i \(-0.517169\pi\)
−0.0539133 + 0.998546i \(0.517169\pi\)
\(68\) −4.90753e7 −0.278338
\(69\) 1.21068e8 0.642997
\(70\) −7.28285e7 −0.362544
\(71\) −8.31277e7 −0.388225 −0.194112 0.980979i \(-0.562183\pi\)
−0.194112 + 0.980979i \(0.562183\pi\)
\(72\) −4.60003e8 −2.01727
\(73\) 2.29798e8 0.947095 0.473547 0.880768i \(-0.342973\pi\)
0.473547 + 0.880768i \(0.342973\pi\)
\(74\) 2.45960e8 0.953504
\(75\) 2.76275e8 1.00824
\(76\) 5.36022e8 1.84298
\(77\) −2.08956e8 −0.677403
\(78\) 2.28748e8 0.699730
\(79\) −2.99993e8 −0.866540 −0.433270 0.901264i \(-0.642640\pi\)
−0.433270 + 0.901264i \(0.642640\pi\)
\(80\) 2.46820e8 0.673715
\(81\) −3.58529e8 −0.925425
\(82\) 6.59064e8 1.60978
\(83\) 8.34408e8 1.92987 0.964933 0.262497i \(-0.0845461\pi\)
0.964933 + 0.262497i \(0.0845461\pi\)
\(84\) −5.14937e8 −1.12849
\(85\) −3.53044e7 −0.0733573
\(86\) −1.31681e9 −2.59585
\(87\) −5.22638e8 −0.978057
\(88\) 1.90136e9 3.37982
\(89\) −2.13503e8 −0.360702 −0.180351 0.983602i \(-0.557723\pi\)
−0.180351 + 0.983602i \(0.557723\pi\)
\(90\) −6.38658e8 −1.02607
\(91\) 6.85750e7 0.104828
\(92\) −6.37368e8 −0.927566
\(93\) −7.99875e8 −1.10879
\(94\) −1.04906e9 −1.38587
\(95\) 3.85610e8 0.485727
\(96\) 3.28318e8 0.394524
\(97\) −1.36917e9 −1.57031 −0.785155 0.619300i \(-0.787417\pi\)
−0.785155 + 0.619300i \(0.787417\pi\)
\(98\) −2.28753e8 −0.250524
\(99\) −1.83241e9 −1.91719
\(100\) −1.45446e9 −1.45446
\(101\) −1.06790e9 −1.02114 −0.510571 0.859836i \(-0.670566\pi\)
−0.510571 + 0.859836i \(0.670566\pi\)
\(102\) −3.69900e8 −0.338364
\(103\) 1.98007e9 1.73346 0.866728 0.498782i \(-0.166219\pi\)
0.866728 + 0.498782i \(0.166219\pi\)
\(104\) −6.23987e8 −0.523028
\(105\) −3.70442e8 −0.297419
\(106\) 1.55102e9 1.19328
\(107\) 2.70735e9 1.99672 0.998362 0.0572160i \(-0.0182224\pi\)
0.998362 + 0.0572160i \(0.0182224\pi\)
\(108\) −2.94291e8 −0.208147
\(109\) −8.51469e8 −0.577763 −0.288882 0.957365i \(-0.593283\pi\)
−0.288882 + 0.957365i \(0.593283\pi\)
\(110\) 2.63981e9 1.71912
\(111\) 1.25108e9 0.782223
\(112\) 7.75258e8 0.465549
\(113\) −1.97004e9 −1.13664 −0.568318 0.822809i \(-0.692406\pi\)
−0.568318 + 0.822809i \(0.692406\pi\)
\(114\) 4.04021e9 2.24044
\(115\) −4.58517e8 −0.244464
\(116\) 2.75145e9 1.41091
\(117\) 6.01357e8 0.296685
\(118\) −1.90988e9 −0.906851
\(119\) −1.10890e8 −0.0506912
\(120\) 3.37077e9 1.48393
\(121\) 5.21607e9 2.21212
\(122\) −5.69390e9 −2.32697
\(123\) 3.35233e9 1.32061
\(124\) 4.21097e9 1.59950
\(125\) −2.53932e9 −0.930297
\(126\) −2.00601e9 −0.709033
\(127\) 1.62686e9 0.554923 0.277462 0.960737i \(-0.410507\pi\)
0.277462 + 0.960737i \(0.410507\pi\)
\(128\) 4.83159e9 1.59091
\(129\) −6.69794e9 −2.12955
\(130\) −8.66328e8 −0.266034
\(131\) −2.20493e9 −0.654146 −0.327073 0.944999i \(-0.606062\pi\)
−0.327073 + 0.944999i \(0.606062\pi\)
\(132\) 1.86649e10 5.35109
\(133\) 1.21119e9 0.335646
\(134\) 7.05740e8 0.189092
\(135\) −2.11711e8 −0.0548581
\(136\) 1.00903e9 0.252917
\(137\) 4.32814e9 1.04969 0.524843 0.851199i \(-0.324124\pi\)
0.524843 + 0.851199i \(0.324124\pi\)
\(138\) −4.80410e9 −1.12760
\(139\) −2.38823e9 −0.542637 −0.271318 0.962490i \(-0.587460\pi\)
−0.271318 + 0.962490i \(0.587460\pi\)
\(140\) 1.95021e9 0.429046
\(141\) −5.33602e9 −1.13692
\(142\) 3.29859e9 0.680817
\(143\) −2.48563e9 −0.497078
\(144\) 6.79850e9 1.31760
\(145\) 1.97937e9 0.371853
\(146\) −9.11861e9 −1.66089
\(147\) −1.16355e9 −0.205521
\(148\) −6.58634e9 −1.12841
\(149\) −6.32030e9 −1.05051 −0.525254 0.850945i \(-0.676030\pi\)
−0.525254 + 0.850945i \(0.676030\pi\)
\(150\) −1.09628e10 −1.76812
\(151\) −7.50622e9 −1.17497 −0.587483 0.809237i \(-0.699881\pi\)
−0.587483 + 0.809237i \(0.699881\pi\)
\(152\) −1.10211e10 −1.67466
\(153\) −9.72436e8 −0.143466
\(154\) 8.29159e9 1.18794
\(155\) 3.02934e9 0.421556
\(156\) −6.12541e9 −0.828085
\(157\) −9.47459e8 −0.124455 −0.0622274 0.998062i \(-0.519820\pi\)
−0.0622274 + 0.998062i \(0.519820\pi\)
\(158\) 1.19040e10 1.51962
\(159\) 7.88928e9 0.978927
\(160\) −1.24343e9 −0.149996
\(161\) −1.44019e9 −0.168929
\(162\) 1.42268e10 1.62289
\(163\) −1.13452e10 −1.25883 −0.629417 0.777067i \(-0.716706\pi\)
−0.629417 + 0.777067i \(0.716706\pi\)
\(164\) −1.76485e10 −1.90506
\(165\) 1.34274e10 1.41031
\(166\) −3.31101e10 −3.38434
\(167\) −1.09555e10 −1.08995 −0.544975 0.838453i \(-0.683461\pi\)
−0.544975 + 0.838453i \(0.683461\pi\)
\(168\) 1.05875e10 1.02542
\(169\) 8.15731e8 0.0769231
\(170\) 1.40091e9 0.128644
\(171\) 1.06214e10 0.949944
\(172\) 3.52616e10 3.07202
\(173\) −1.16926e10 −0.992436 −0.496218 0.868198i \(-0.665278\pi\)
−0.496218 + 0.868198i \(0.665278\pi\)
\(174\) 2.07388e10 1.71519
\(175\) −3.28649e9 −0.264887
\(176\) −2.81007e10 −2.20755
\(177\) −9.71457e9 −0.743950
\(178\) 8.47201e9 0.632552
\(179\) −2.27115e10 −1.65351 −0.826756 0.562561i \(-0.809816\pi\)
−0.826756 + 0.562561i \(0.809816\pi\)
\(180\) 1.71020e10 1.21429
\(181\) −1.64962e10 −1.14243 −0.571216 0.820800i \(-0.693528\pi\)
−0.571216 + 0.820800i \(0.693528\pi\)
\(182\) −2.72112e9 −0.183834
\(183\) −2.89620e10 −1.90897
\(184\) 1.31048e10 0.842850
\(185\) −4.73816e9 −0.297397
\(186\) 3.17398e10 1.94445
\(187\) 4.01943e9 0.240369
\(188\) 2.80917e10 1.64009
\(189\) −6.64979e8 −0.0379079
\(190\) −1.53014e10 −0.851803
\(191\) −7.64027e9 −0.415392 −0.207696 0.978193i \(-0.566597\pi\)
−0.207696 + 0.978193i \(0.566597\pi\)
\(192\) 2.03396e10 1.08016
\(193\) 1.38071e10 0.716301 0.358151 0.933664i \(-0.383407\pi\)
0.358151 + 0.933664i \(0.383407\pi\)
\(194\) 5.43301e10 2.75380
\(195\) −4.40658e9 −0.218246
\(196\) 6.12556e9 0.296478
\(197\) −2.81607e10 −1.33213 −0.666064 0.745895i \(-0.732022\pi\)
−0.666064 + 0.745895i \(0.732022\pi\)
\(198\) 7.27118e10 3.36211
\(199\) −2.65123e9 −0.119842 −0.0599209 0.998203i \(-0.519085\pi\)
−0.0599209 + 0.998203i \(0.519085\pi\)
\(200\) 2.99049e10 1.32162
\(201\) 3.58974e9 0.155125
\(202\) 4.23755e10 1.79074
\(203\) 6.21716e9 0.256957
\(204\) 9.90521e9 0.400431
\(205\) −1.26962e10 −0.502088
\(206\) −7.85710e10 −3.03990
\(207\) −1.26296e10 −0.478103
\(208\) 9.22205e9 0.341619
\(209\) −4.39020e10 −1.59157
\(210\) 1.46995e10 0.521573
\(211\) 5.04101e10 1.75084 0.875421 0.483362i \(-0.160584\pi\)
0.875421 + 0.483362i \(0.160584\pi\)
\(212\) −4.15334e10 −1.41217
\(213\) 1.67783e10 0.558520
\(214\) −1.07430e11 −3.50159
\(215\) 2.53669e10 0.809645
\(216\) 6.05087e9 0.189137
\(217\) 9.51510e9 0.291303
\(218\) 3.37871e10 1.01320
\(219\) −4.63817e10 −1.36254
\(220\) −7.06889e10 −2.03446
\(221\) −1.31909e9 −0.0371972
\(222\) −4.96439e10 −1.37176
\(223\) 6.29895e8 0.0170568 0.00852838 0.999964i \(-0.497285\pi\)
0.00852838 + 0.999964i \(0.497285\pi\)
\(224\) −3.90558e9 −0.103650
\(225\) −2.88204e10 −0.749684
\(226\) 7.81729e10 1.99328
\(227\) −5.51412e10 −1.37835 −0.689175 0.724595i \(-0.742027\pi\)
−0.689175 + 0.724595i \(0.742027\pi\)
\(228\) −1.08189e11 −2.65141
\(229\) 2.46673e10 0.592737 0.296369 0.955074i \(-0.404224\pi\)
0.296369 + 0.955074i \(0.404224\pi\)
\(230\) 1.81944e10 0.428709
\(231\) 4.21751e10 0.974546
\(232\) −5.65720e10 −1.28205
\(233\) 2.87752e10 0.639613 0.319806 0.947483i \(-0.396382\pi\)
0.319806 + 0.947483i \(0.396382\pi\)
\(234\) −2.38624e10 −0.520288
\(235\) 2.02089e10 0.432253
\(236\) 5.11427e10 1.07320
\(237\) 6.05496e10 1.24665
\(238\) 4.40024e9 0.0888955
\(239\) −2.14647e10 −0.425534 −0.212767 0.977103i \(-0.568248\pi\)
−0.212767 + 0.977103i \(0.568248\pi\)
\(240\) −4.98175e10 −0.969240
\(241\) 5.95005e10 1.13617 0.568086 0.822969i \(-0.307684\pi\)
0.568086 + 0.822969i \(0.307684\pi\)
\(242\) −2.06979e11 −3.87933
\(243\) 7.78158e10 1.43166
\(244\) 1.52471e11 2.75381
\(245\) 4.40668e9 0.0781383
\(246\) −1.33024e11 −2.31590
\(247\) 1.44077e10 0.246297
\(248\) −8.65811e10 −1.45342
\(249\) −1.68414e11 −2.77640
\(250\) 1.00763e11 1.63143
\(251\) −9.95398e10 −1.58294 −0.791471 0.611207i \(-0.790684\pi\)
−0.791471 + 0.611207i \(0.790684\pi\)
\(252\) 5.37171e10 0.839093
\(253\) 5.22026e10 0.801032
\(254\) −6.45553e10 −0.973151
\(255\) 7.12573e9 0.105536
\(256\) −1.40127e11 −2.03911
\(257\) 4.89105e9 0.0699364 0.0349682 0.999388i \(-0.488867\pi\)
0.0349682 + 0.999388i \(0.488867\pi\)
\(258\) 2.65781e11 3.73452
\(259\) −1.48825e10 −0.205507
\(260\) 2.31986e10 0.314834
\(261\) 5.45204e10 0.727239
\(262\) 8.74939e10 1.14716
\(263\) −1.03933e10 −0.133953 −0.0669767 0.997755i \(-0.521335\pi\)
−0.0669767 + 0.997755i \(0.521335\pi\)
\(264\) −3.83765e11 −4.86237
\(265\) −2.98788e10 −0.372183
\(266\) −4.80613e10 −0.588611
\(267\) 4.30928e10 0.518925
\(268\) −1.88983e10 −0.223778
\(269\) −1.23860e11 −1.44227 −0.721135 0.692795i \(-0.756379\pi\)
−0.721135 + 0.692795i \(0.756379\pi\)
\(270\) 8.40089e9 0.0962029
\(271\) 1.04040e11 1.17176 0.585880 0.810398i \(-0.300749\pi\)
0.585880 + 0.810398i \(0.300749\pi\)
\(272\) −1.49127e10 −0.165194
\(273\) −1.38410e10 −0.150812
\(274\) −1.71745e11 −1.84080
\(275\) 1.19125e11 1.25605
\(276\) 1.28644e11 1.33444
\(277\) −1.11720e11 −1.14017 −0.570087 0.821584i \(-0.693090\pi\)
−0.570087 + 0.821584i \(0.693090\pi\)
\(278\) 9.47672e10 0.951605
\(279\) 8.34412e10 0.824445
\(280\) −4.00978e10 −0.389861
\(281\) 8.46126e10 0.809574 0.404787 0.914411i \(-0.367346\pi\)
0.404787 + 0.914411i \(0.367346\pi\)
\(282\) 2.11738e11 1.99378
\(283\) −9.70546e9 −0.0899450 −0.0449725 0.998988i \(-0.514320\pi\)
−0.0449725 + 0.998988i \(0.514320\pi\)
\(284\) −8.83298e10 −0.805702
\(285\) −7.78304e10 −0.698791
\(286\) 9.86322e10 0.871709
\(287\) −3.98784e10 −0.346952
\(288\) −3.42494e10 −0.293350
\(289\) −1.16455e11 −0.982013
\(290\) −7.85433e10 −0.652106
\(291\) 2.76350e11 2.25913
\(292\) 2.44178e11 1.96555
\(293\) 1.53609e11 1.21762 0.608809 0.793317i \(-0.291648\pi\)
0.608809 + 0.793317i \(0.291648\pi\)
\(294\) 4.61708e10 0.360416
\(295\) 3.67917e10 0.282846
\(296\) 1.35421e11 1.02535
\(297\) 2.41035e10 0.179753
\(298\) 2.50796e11 1.84224
\(299\) −1.71318e10 −0.123960
\(300\) 2.93564e11 2.09246
\(301\) 7.96770e10 0.559479
\(302\) 2.97854e11 2.06050
\(303\) 2.15543e11 1.46907
\(304\) 1.62883e11 1.09382
\(305\) 1.09687e11 0.725780
\(306\) 3.85872e10 0.251592
\(307\) 1.44131e11 0.926048 0.463024 0.886346i \(-0.346764\pi\)
0.463024 + 0.886346i \(0.346764\pi\)
\(308\) −2.22033e11 −1.40585
\(309\) −3.99651e11 −2.49384
\(310\) −1.20207e11 −0.739270
\(311\) −2.59353e11 −1.57206 −0.786030 0.618188i \(-0.787867\pi\)
−0.786030 + 0.618188i \(0.787867\pi\)
\(312\) 1.25944e11 0.752455
\(313\) 1.49010e11 0.877538 0.438769 0.898600i \(-0.355415\pi\)
0.438769 + 0.898600i \(0.355415\pi\)
\(314\) 3.75961e10 0.218252
\(315\) 3.86437e10 0.221147
\(316\) −3.18766e11 −1.79837
\(317\) 3.08871e11 1.71795 0.858976 0.512016i \(-0.171101\pi\)
0.858976 + 0.512016i \(0.171101\pi\)
\(318\) −3.13054e11 −1.71671
\(319\) −2.25353e11 −1.21844
\(320\) −7.70316e10 −0.410671
\(321\) −5.46444e11 −2.87259
\(322\) 5.71483e10 0.296246
\(323\) −2.32982e10 −0.119100
\(324\) −3.80965e11 −1.92058
\(325\) −3.90943e10 −0.194374
\(326\) 4.50189e11 2.20758
\(327\) 1.71858e11 0.831199
\(328\) 3.62867e11 1.73107
\(329\) 6.34758e10 0.298694
\(330\) −5.32811e11 −2.47321
\(331\) 2.79598e11 1.28029 0.640144 0.768255i \(-0.278875\pi\)
0.640144 + 0.768255i \(0.278875\pi\)
\(332\) 8.86624e11 4.00515
\(333\) −1.30510e11 −0.581625
\(334\) 4.34723e11 1.91141
\(335\) −1.35953e10 −0.0589777
\(336\) −1.56476e11 −0.669762
\(337\) −1.24066e11 −0.523984 −0.261992 0.965070i \(-0.584379\pi\)
−0.261992 + 0.965070i \(0.584379\pi\)
\(338\) −3.23690e10 −0.134897
\(339\) 3.97626e11 1.63522
\(340\) −3.75137e10 −0.152242
\(341\) −3.44893e11 −1.38131
\(342\) −4.21466e11 −1.66589
\(343\) 1.38413e10 0.0539949
\(344\) −7.25007e11 −2.79145
\(345\) 9.25458e10 0.351699
\(346\) 4.63972e11 1.74040
\(347\) −4.24443e11 −1.57158 −0.785791 0.618492i \(-0.787744\pi\)
−0.785791 + 0.618492i \(0.787744\pi\)
\(348\) −5.55344e11 −2.02981
\(349\) −2.73427e11 −0.986568 −0.493284 0.869868i \(-0.664204\pi\)
−0.493284 + 0.869868i \(0.664204\pi\)
\(350\) 1.30411e11 0.464524
\(351\) −7.91024e9 −0.0278168
\(352\) 1.41565e11 0.491490
\(353\) 1.64625e11 0.564301 0.282151 0.959370i \(-0.408952\pi\)
0.282151 + 0.959370i \(0.408952\pi\)
\(354\) 3.85484e11 1.30464
\(355\) −6.35437e10 −0.212347
\(356\) −2.26864e11 −0.748584
\(357\) 2.23818e10 0.0729269
\(358\) 9.01214e11 2.89971
\(359\) 3.73577e11 1.18701 0.593507 0.804829i \(-0.297743\pi\)
0.593507 + 0.804829i \(0.297743\pi\)
\(360\) −3.51632e11 −1.10338
\(361\) −6.82143e10 −0.211394
\(362\) 6.54585e11 2.00344
\(363\) −1.05280e12 −3.18247
\(364\) 7.28663e10 0.217556
\(365\) 1.75660e11 0.518030
\(366\) 1.14924e12 3.34769
\(367\) −2.58080e11 −0.742604 −0.371302 0.928512i \(-0.621088\pi\)
−0.371302 + 0.928512i \(0.621088\pi\)
\(368\) −1.93679e11 −0.550513
\(369\) −3.49707e11 −0.981942
\(370\) 1.88015e11 0.521536
\(371\) −9.38488e10 −0.257185
\(372\) −8.49931e11 −2.30112
\(373\) −5.05371e11 −1.35182 −0.675912 0.736982i \(-0.736250\pi\)
−0.675912 + 0.736982i \(0.736250\pi\)
\(374\) −1.59495e11 −0.421526
\(375\) 5.12528e11 1.33837
\(376\) −5.77588e11 −1.49030
\(377\) 7.39560e10 0.188555
\(378\) 2.63870e10 0.0664779
\(379\) −7.39437e10 −0.184088 −0.0920438 0.995755i \(-0.529340\pi\)
−0.0920438 + 0.995755i \(0.529340\pi\)
\(380\) 4.09741e11 1.00805
\(381\) −3.28360e11 −0.798341
\(382\) 3.03173e11 0.728460
\(383\) −1.19857e10 −0.0284623 −0.0142312 0.999899i \(-0.504530\pi\)
−0.0142312 + 0.999899i \(0.504530\pi\)
\(384\) −9.75195e11 −2.28876
\(385\) −1.59728e11 −0.370518
\(386\) −5.47880e11 −1.25615
\(387\) 6.98715e11 1.58344
\(388\) −1.45485e12 −3.25894
\(389\) −2.82555e11 −0.625647 −0.312823 0.949811i \(-0.601275\pi\)
−0.312823 + 0.949811i \(0.601275\pi\)
\(390\) 1.74857e11 0.382730
\(391\) 2.77032e10 0.0599425
\(392\) −1.25947e11 −0.269401
\(393\) 4.45038e11 0.941088
\(394\) 1.11745e12 2.33611
\(395\) −2.29318e11 −0.473970
\(396\) −1.94708e12 −3.97883
\(397\) −5.51391e11 −1.11404 −0.557022 0.830497i \(-0.688056\pi\)
−0.557022 + 0.830497i \(0.688056\pi\)
\(398\) 1.05203e11 0.210163
\(399\) −2.44464e11 −0.482877
\(400\) −4.41971e11 −0.863226
\(401\) −8.56723e11 −1.65459 −0.827296 0.561767i \(-0.810122\pi\)
−0.827296 + 0.561767i \(0.810122\pi\)
\(402\) −1.42444e11 −0.272037
\(403\) 1.13187e11 0.213758
\(404\) −1.13473e12 −2.11923
\(405\) −2.74063e11 −0.506178
\(406\) −2.46703e11 −0.450617
\(407\) 5.39444e11 0.974476
\(408\) −2.03659e11 −0.363859
\(409\) −4.41170e11 −0.779563 −0.389782 0.920907i \(-0.627450\pi\)
−0.389782 + 0.920907i \(0.627450\pi\)
\(410\) 5.03796e11 0.880496
\(411\) −8.73580e11 −1.51013
\(412\) 2.10398e12 3.59753
\(413\) 1.15562e11 0.195452
\(414\) 5.01153e11 0.838434
\(415\) 6.37831e11 1.05557
\(416\) −4.64587e10 −0.0760584
\(417\) 4.82033e11 0.780665
\(418\) 1.74207e12 2.79109
\(419\) −1.04353e12 −1.65402 −0.827009 0.562189i \(-0.809959\pi\)
−0.827009 + 0.562189i \(0.809959\pi\)
\(420\) −3.93624e11 −0.617248
\(421\) −4.64177e11 −0.720135 −0.360068 0.932926i \(-0.617246\pi\)
−0.360068 + 0.932926i \(0.617246\pi\)
\(422\) −2.00032e12 −3.07039
\(423\) 5.56641e11 0.845364
\(424\) 8.53962e11 1.28319
\(425\) 6.32182e10 0.0939922
\(426\) −6.65777e11 −0.979458
\(427\) 3.44524e11 0.501527
\(428\) 2.87678e12 4.14390
\(429\) 5.01692e11 0.715121
\(430\) −1.00658e12 −1.41985
\(431\) 1.03602e12 1.44617 0.723086 0.690758i \(-0.242723\pi\)
0.723086 + 0.690758i \(0.242723\pi\)
\(432\) −8.94273e10 −0.123536
\(433\) 2.95265e11 0.403661 0.201830 0.979420i \(-0.435311\pi\)
0.201830 + 0.979420i \(0.435311\pi\)
\(434\) −3.77568e11 −0.510848
\(435\) −3.99510e11 −0.534966
\(436\) −9.04753e11 −1.19906
\(437\) −3.02587e11 −0.396902
\(438\) 1.84047e12 2.38944
\(439\) 1.60114e11 0.205750 0.102875 0.994694i \(-0.467196\pi\)
0.102875 + 0.994694i \(0.467196\pi\)
\(440\) 1.45342e12 1.84865
\(441\) 1.21379e11 0.152816
\(442\) 5.23428e10 0.0652314
\(443\) −6.60319e11 −0.814586 −0.407293 0.913297i \(-0.633527\pi\)
−0.407293 + 0.913297i \(0.633527\pi\)
\(444\) 1.32937e12 1.62339
\(445\) −1.63204e11 −0.197293
\(446\) −2.49948e10 −0.0299119
\(447\) 1.27567e12 1.51131
\(448\) −2.41955e11 −0.283781
\(449\) 1.17755e12 1.36732 0.683661 0.729800i \(-0.260387\pi\)
0.683661 + 0.729800i \(0.260387\pi\)
\(450\) 1.14362e12 1.31470
\(451\) 1.44547e12 1.64518
\(452\) −2.09332e12 −2.35892
\(453\) 1.51503e12 1.69036
\(454\) 2.18805e12 2.41717
\(455\) 5.24195e10 0.0573378
\(456\) 2.22446e12 2.40925
\(457\) 4.73878e11 0.508210 0.254105 0.967177i \(-0.418219\pi\)
0.254105 + 0.967177i \(0.418219\pi\)
\(458\) −9.78823e11 −1.03946
\(459\) 1.27914e10 0.0134512
\(460\) −4.87211e11 −0.507349
\(461\) −7.86748e11 −0.811300 −0.405650 0.914028i \(-0.632955\pi\)
−0.405650 + 0.914028i \(0.632955\pi\)
\(462\) −1.67355e12 −1.70903
\(463\) −1.98673e11 −0.200921 −0.100461 0.994941i \(-0.532032\pi\)
−0.100461 + 0.994941i \(0.532032\pi\)
\(464\) 8.36092e11 0.837381
\(465\) −6.11433e11 −0.606472
\(466\) −1.14183e12 −1.12167
\(467\) −4.02028e11 −0.391138 −0.195569 0.980690i \(-0.562655\pi\)
−0.195569 + 0.980690i \(0.562655\pi\)
\(468\) 6.38990e11 0.615726
\(469\) −4.27026e10 −0.0407546
\(470\) −8.01910e11 −0.758028
\(471\) 1.91232e11 0.179047
\(472\) −1.05154e12 −0.975182
\(473\) −2.88804e12 −2.65295
\(474\) −2.40267e12 −2.18621
\(475\) −6.90496e11 −0.622358
\(476\) −1.17830e11 −0.105202
\(477\) −8.22992e11 −0.727885
\(478\) 8.51740e11 0.746245
\(479\) −4.03884e11 −0.350548 −0.175274 0.984520i \(-0.556081\pi\)
−0.175274 + 0.984520i \(0.556081\pi\)
\(480\) 2.50970e11 0.215792
\(481\) −1.77034e11 −0.150801
\(482\) −2.36104e12 −1.99247
\(483\) 2.90685e11 0.243030
\(484\) 5.54249e12 4.59093
\(485\) −1.04661e12 −0.858909
\(486\) −3.08780e12 −2.51065
\(487\) −1.32903e12 −1.07067 −0.535335 0.844640i \(-0.679815\pi\)
−0.535335 + 0.844640i \(0.679815\pi\)
\(488\) −3.13494e12 −2.50230
\(489\) 2.28989e12 1.81102
\(490\) −1.74861e11 −0.137029
\(491\) −1.73030e12 −1.34355 −0.671777 0.740753i \(-0.734469\pi\)
−0.671777 + 0.740753i \(0.734469\pi\)
\(492\) 3.56211e12 2.74072
\(493\) −1.19592e11 −0.0911781
\(494\) −5.71711e11 −0.431922
\(495\) −1.40071e12 −1.04864
\(496\) 1.27960e12 0.949309
\(497\) −1.99590e11 −0.146735
\(498\) 6.68285e12 4.86888
\(499\) −1.01113e12 −0.730051 −0.365026 0.930998i \(-0.618940\pi\)
−0.365026 + 0.930998i \(0.618940\pi\)
\(500\) −2.69822e12 −1.93069
\(501\) 2.21122e12 1.56806
\(502\) 3.94984e12 2.77595
\(503\) −1.40816e12 −0.980835 −0.490418 0.871488i \(-0.663156\pi\)
−0.490418 + 0.871488i \(0.663156\pi\)
\(504\) −1.10447e12 −0.762458
\(505\) −8.16318e11 −0.558532
\(506\) −2.07145e12 −1.40474
\(507\) −1.64645e11 −0.110665
\(508\) 1.72867e12 1.15166
\(509\) 5.65749e11 0.373589 0.186794 0.982399i \(-0.440190\pi\)
0.186794 + 0.982399i \(0.440190\pi\)
\(510\) −2.82756e11 −0.185074
\(511\) 5.51745e11 0.357968
\(512\) 3.08659e12 1.98502
\(513\) −1.39713e11 −0.0890654
\(514\) −1.94082e11 −0.122645
\(515\) 1.51359e12 0.948145
\(516\) −7.11709e12 −4.41956
\(517\) −2.30080e12 −1.41635
\(518\) 5.90551e11 0.360391
\(519\) 2.35999e12 1.42777
\(520\) −4.76982e11 −0.286080
\(521\) 2.40433e12 1.42963 0.714816 0.699312i \(-0.246510\pi\)
0.714816 + 0.699312i \(0.246510\pi\)
\(522\) −2.16342e12 −1.27533
\(523\) −2.71200e12 −1.58501 −0.792505 0.609865i \(-0.791224\pi\)
−0.792505 + 0.609865i \(0.791224\pi\)
\(524\) −2.34292e12 −1.35758
\(525\) 6.63336e11 0.381080
\(526\) 4.12417e11 0.234910
\(527\) −1.83030e11 −0.103365
\(528\) 5.67176e12 3.17589
\(529\) −1.44136e12 −0.800241
\(530\) 1.18562e12 0.652686
\(531\) 1.01340e12 0.553168
\(532\) 1.28699e12 0.696582
\(533\) −4.74372e11 −0.254593
\(534\) −1.70997e12 −0.910021
\(535\) 2.06953e12 1.09214
\(536\) 3.88566e11 0.203340
\(537\) 4.58402e12 2.37882
\(538\) 4.91489e12 2.52926
\(539\) −5.01704e11 −0.256034
\(540\) −2.24959e11 −0.113850
\(541\) −1.85773e12 −0.932382 −0.466191 0.884684i \(-0.654374\pi\)
−0.466191 + 0.884684i \(0.654374\pi\)
\(542\) −4.12841e12 −2.05488
\(543\) 3.32954e12 1.64356
\(544\) 7.51268e10 0.0367790
\(545\) −6.50873e11 −0.316018
\(546\) 5.49223e11 0.264473
\(547\) 3.11372e11 0.148709 0.0743543 0.997232i \(-0.476310\pi\)
0.0743543 + 0.997232i \(0.476310\pi\)
\(548\) 4.59899e12 2.17846
\(549\) 3.02125e12 1.41942
\(550\) −4.72700e12 −2.20269
\(551\) 1.30623e12 0.603725
\(552\) −2.64504e12 −1.21257
\(553\) −7.20282e11 −0.327521
\(554\) 4.43315e12 1.99949
\(555\) 9.56337e11 0.427851
\(556\) −2.53768e12 −1.12616
\(557\) 1.34798e12 0.593382 0.296691 0.954974i \(-0.404117\pi\)
0.296691 + 0.954974i \(0.404117\pi\)
\(558\) −3.31103e12 −1.44580
\(559\) 9.47794e11 0.410545
\(560\) 5.92616e11 0.254640
\(561\) −8.11271e11 −0.345806
\(562\) −3.35751e12 −1.41972
\(563\) 2.32125e12 0.973718 0.486859 0.873481i \(-0.338143\pi\)
0.486859 + 0.873481i \(0.338143\pi\)
\(564\) −5.66994e12 −2.35951
\(565\) −1.50592e12 −0.621703
\(566\) 3.85122e11 0.157734
\(567\) −8.60827e11 −0.349778
\(568\) 1.81613e12 0.732116
\(569\) 1.44089e12 0.576268 0.288134 0.957590i \(-0.406965\pi\)
0.288134 + 0.957590i \(0.406965\pi\)
\(570\) 3.08838e12 1.22545
\(571\) −1.18856e12 −0.467906 −0.233953 0.972248i \(-0.575166\pi\)
−0.233953 + 0.972248i \(0.575166\pi\)
\(572\) −2.64118e12 −1.03161
\(573\) 1.54209e12 0.597604
\(574\) 1.58241e12 0.608438
\(575\) 8.21049e11 0.313230
\(576\) −2.12178e12 −0.803156
\(577\) 7.32215e11 0.275009 0.137505 0.990501i \(-0.456092\pi\)
0.137505 + 0.990501i \(0.456092\pi\)
\(578\) 4.62104e12 1.72212
\(579\) −2.78679e12 −1.03051
\(580\) 2.10324e12 0.771724
\(581\) 2.00341e12 0.729421
\(582\) −1.09658e13 −3.96176
\(583\) 3.40173e12 1.21953
\(584\) −5.02051e12 −1.78604
\(585\) 4.59684e11 0.162277
\(586\) −6.09533e12 −2.13530
\(587\) 1.83141e12 0.636669 0.318334 0.947978i \(-0.396877\pi\)
0.318334 + 0.947978i \(0.396877\pi\)
\(588\) −1.23636e12 −0.426529
\(589\) 1.99914e12 0.684422
\(590\) −1.45993e12 −0.496019
\(591\) 5.68388e12 1.91647
\(592\) −2.00141e12 −0.669714
\(593\) 1.13605e12 0.377270 0.188635 0.982047i \(-0.439594\pi\)
0.188635 + 0.982047i \(0.439594\pi\)
\(594\) −9.56448e11 −0.315226
\(595\) −8.47658e10 −0.0277265
\(596\) −6.71582e12 −2.18017
\(597\) 5.35116e11 0.172410
\(598\) 6.79805e11 0.217385
\(599\) 5.79013e12 1.83767 0.918836 0.394640i \(-0.129131\pi\)
0.918836 + 0.394640i \(0.129131\pi\)
\(600\) −6.03591e12 −1.90135
\(601\) −4.77276e11 −0.149222 −0.0746112 0.997213i \(-0.523772\pi\)
−0.0746112 + 0.997213i \(0.523772\pi\)
\(602\) −3.16166e12 −0.981139
\(603\) −3.74474e11 −0.115344
\(604\) −7.97595e12 −2.43846
\(605\) 3.98722e12 1.20996
\(606\) −8.55294e12 −2.57625
\(607\) −5.55565e12 −1.66106 −0.830531 0.556972i \(-0.811963\pi\)
−0.830531 + 0.556972i \(0.811963\pi\)
\(608\) −8.20568e11 −0.243528
\(609\) −1.25485e12 −0.369671
\(610\) −4.35248e12 −1.27278
\(611\) 7.55074e11 0.219182
\(612\) −1.03329e12 −0.297742
\(613\) 3.19190e12 0.913014 0.456507 0.889720i \(-0.349100\pi\)
0.456507 + 0.889720i \(0.349100\pi\)
\(614\) −5.71924e12 −1.62398
\(615\) 2.56256e12 0.722329
\(616\) 4.56517e12 1.27745
\(617\) −2.54892e12 −0.708064 −0.354032 0.935233i \(-0.615190\pi\)
−0.354032 + 0.935233i \(0.615190\pi\)
\(618\) 1.58585e13 4.37336
\(619\) −3.56626e12 −0.976348 −0.488174 0.872746i \(-0.662337\pi\)
−0.488174 + 0.872746i \(0.662337\pi\)
\(620\) 3.21891e12 0.874877
\(621\) 1.66129e11 0.0448263
\(622\) 1.02914e13 2.75687
\(623\) −5.12621e11 −0.136333
\(624\) −1.86135e12 −0.491471
\(625\) 7.32357e11 0.191983
\(626\) −5.91286e12 −1.53891
\(627\) 8.86105e12 2.28971
\(628\) −1.00675e12 −0.258287
\(629\) 2.86276e11 0.0729217
\(630\) −1.53342e12 −0.387818
\(631\) 6.93158e12 1.74060 0.870302 0.492518i \(-0.163924\pi\)
0.870302 + 0.492518i \(0.163924\pi\)
\(632\) 6.55409e12 1.63413
\(633\) −1.01746e13 −2.51885
\(634\) −1.22563e13 −3.01272
\(635\) 1.24359e12 0.303525
\(636\) 8.38298e12 2.03162
\(637\) 1.64648e11 0.0396214
\(638\) 8.94222e12 2.13674
\(639\) −1.75027e12 −0.415290
\(640\) 3.69332e12 0.870177
\(641\) 6.87093e12 1.60751 0.803756 0.594959i \(-0.202832\pi\)
0.803756 + 0.594959i \(0.202832\pi\)
\(642\) 2.16834e13 5.03756
\(643\) 2.96454e12 0.683924 0.341962 0.939714i \(-0.388909\pi\)
0.341962 + 0.939714i \(0.388909\pi\)
\(644\) −1.53032e12 −0.350587
\(645\) −5.11998e12 −1.16480
\(646\) 9.24496e11 0.208862
\(647\) −3.09024e12 −0.693302 −0.346651 0.937994i \(-0.612681\pi\)
−0.346651 + 0.937994i \(0.612681\pi\)
\(648\) 7.83296e12 1.74517
\(649\) −4.18877e12 −0.926798
\(650\) 1.55130e12 0.340868
\(651\) −1.92050e12 −0.419083
\(652\) −1.20552e13 −2.61252
\(653\) −1.34253e12 −0.288944 −0.144472 0.989509i \(-0.546148\pi\)
−0.144472 + 0.989509i \(0.546148\pi\)
\(654\) −6.81949e12 −1.45765
\(655\) −1.68548e12 −0.357797
\(656\) −5.36290e12 −1.13066
\(657\) 4.83844e12 1.01312
\(658\) −2.51878e12 −0.523810
\(659\) 5.64054e12 1.16503 0.582514 0.812821i \(-0.302069\pi\)
0.582514 + 0.812821i \(0.302069\pi\)
\(660\) 1.42676e13 2.92688
\(661\) −7.62722e12 −1.55403 −0.777015 0.629482i \(-0.783267\pi\)
−0.777015 + 0.629482i \(0.783267\pi\)
\(662\) −1.10947e13 −2.24520
\(663\) 2.66242e11 0.0535137
\(664\) −1.82297e13 −3.63935
\(665\) 9.25849e11 0.183587
\(666\) 5.17874e12 1.01998
\(667\) −1.55321e12 −0.303852
\(668\) −1.16410e13 −2.26203
\(669\) −1.27136e11 −0.0245387
\(670\) 5.39475e11 0.103427
\(671\) −1.24879e13 −2.37815
\(672\) 7.88291e11 0.149116
\(673\) −6.65404e12 −1.25031 −0.625154 0.780501i \(-0.714964\pi\)
−0.625154 + 0.780501i \(0.714964\pi\)
\(674\) 4.92305e12 0.918893
\(675\) 3.79102e11 0.0702893
\(676\) 8.66778e11 0.159642
\(677\) 3.62414e11 0.0663065 0.0331532 0.999450i \(-0.489445\pi\)
0.0331532 + 0.999450i \(0.489445\pi\)
\(678\) −1.57782e13 −2.86763
\(679\) −3.28738e12 −0.593521
\(680\) 7.71312e11 0.138338
\(681\) 1.11295e13 1.98296
\(682\) 1.36857e13 2.42235
\(683\) 7.87971e12 1.38553 0.692767 0.721162i \(-0.256392\pi\)
0.692767 + 0.721162i \(0.256392\pi\)
\(684\) 1.12860e13 1.97147
\(685\) 3.30848e12 0.574144
\(686\) −5.49236e11 −0.0946891
\(687\) −4.97878e12 −0.852741
\(688\) 1.07151e13 1.82325
\(689\) −1.11637e12 −0.188722
\(690\) −3.67231e12 −0.616763
\(691\) −3.59867e11 −0.0600469 −0.0300235 0.999549i \(-0.509558\pi\)
−0.0300235 + 0.999549i \(0.509558\pi\)
\(692\) −1.24243e13 −2.05965
\(693\) −4.39961e12 −0.724628
\(694\) 1.68423e13 2.75603
\(695\) −1.82559e12 −0.296805
\(696\) 1.14183e13 1.84443
\(697\) 7.67091e11 0.123112
\(698\) 1.08499e13 1.73011
\(699\) −5.80791e12 −0.920179
\(700\) −3.49216e12 −0.549734
\(701\) −9.48905e12 −1.48420 −0.742099 0.670291i \(-0.766169\pi\)
−0.742099 + 0.670291i \(0.766169\pi\)
\(702\) 3.13886e11 0.0487815
\(703\) −3.12683e12 −0.482842
\(704\) 8.77011e12 1.34564
\(705\) −4.07891e12 −0.621861
\(706\) −6.53250e12 −0.989596
\(707\) −2.56404e12 −0.385955
\(708\) −1.03225e13 −1.54396
\(709\) −1.69854e11 −0.0252446 −0.0126223 0.999920i \(-0.504018\pi\)
−0.0126223 + 0.999920i \(0.504018\pi\)
\(710\) 2.52148e12 0.372385
\(711\) −6.31640e12 −0.926951
\(712\) 4.66451e12 0.680215
\(713\) −2.37712e12 −0.344467
\(714\) −8.88131e11 −0.127889
\(715\) −1.90004e12 −0.271886
\(716\) −2.41328e13 −3.43161
\(717\) 4.33237e12 0.612195
\(718\) −1.48239e13 −2.08163
\(719\) 9.92019e12 1.38433 0.692166 0.721739i \(-0.256657\pi\)
0.692166 + 0.721739i \(0.256657\pi\)
\(720\) 5.19685e12 0.720682
\(721\) 4.75414e12 0.655185
\(722\) 2.70681e12 0.370715
\(723\) −1.20094e13 −1.63455
\(724\) −1.75285e13 −2.37094
\(725\) −3.54438e12 −0.476452
\(726\) 4.17760e13 5.58100
\(727\) −9.34395e11 −0.124058 −0.0620291 0.998074i \(-0.519757\pi\)
−0.0620291 + 0.998074i \(0.519757\pi\)
\(728\) −1.49819e12 −0.197686
\(729\) −8.64918e12 −1.13423
\(730\) −6.97036e12 −0.908453
\(731\) −1.53265e12 −0.198524
\(732\) −3.07744e13 −3.96177
\(733\) −4.07900e12 −0.521899 −0.260949 0.965352i \(-0.584036\pi\)
−0.260949 + 0.965352i \(0.584036\pi\)
\(734\) 1.02409e13 1.30228
\(735\) −8.89431e11 −0.112414
\(736\) 9.75714e11 0.122567
\(737\) 1.54784e12 0.193251
\(738\) 1.38767e13 1.72200
\(739\) 3.37654e12 0.416458 0.208229 0.978080i \(-0.433230\pi\)
0.208229 + 0.978080i \(0.433230\pi\)
\(740\) −5.03467e12 −0.617203
\(741\) −2.90801e12 −0.354335
\(742\) 3.72401e12 0.451017
\(743\) −8.79058e12 −1.05820 −0.529100 0.848560i \(-0.677470\pi\)
−0.529100 + 0.848560i \(0.677470\pi\)
\(744\) 1.74753e13 2.09096
\(745\) −4.83131e12 −0.574594
\(746\) 2.00536e13 2.37065
\(747\) 1.75686e13 2.06441
\(748\) 4.27097e12 0.498849
\(749\) 6.50036e12 0.754691
\(750\) −2.03376e13 −2.34706
\(751\) 1.49673e13 1.71697 0.858484 0.512840i \(-0.171406\pi\)
0.858484 + 0.512840i \(0.171406\pi\)
\(752\) 8.53631e12 0.973397
\(753\) 2.00908e13 2.27730
\(754\) −2.93465e12 −0.330662
\(755\) −5.73784e12 −0.642668
\(756\) −7.06593e11 −0.0786722
\(757\) −1.55150e13 −1.71720 −0.858598 0.512650i \(-0.828664\pi\)
−0.858598 + 0.512650i \(0.828664\pi\)
\(758\) 2.93416e12 0.322828
\(759\) −1.05364e13 −1.15240
\(760\) −8.42461e12 −0.915986
\(761\) −1.49750e13 −1.61859 −0.809294 0.587404i \(-0.800150\pi\)
−0.809294 + 0.587404i \(0.800150\pi\)
\(762\) 1.30297e13 1.40002
\(763\) −2.04438e12 −0.218374
\(764\) −8.11839e12 −0.862084
\(765\) −7.43340e11 −0.0784714
\(766\) 4.75606e11 0.0499135
\(767\) 1.37466e12 0.143422
\(768\) 2.82828e13 2.93357
\(769\) 1.66756e13 1.71954 0.859769 0.510683i \(-0.170607\pi\)
0.859769 + 0.510683i \(0.170607\pi\)
\(770\) 6.33818e12 0.649765
\(771\) −9.87195e11 −0.100614
\(772\) 1.46712e13 1.48658
\(773\) 3.21304e12 0.323674 0.161837 0.986817i \(-0.448258\pi\)
0.161837 + 0.986817i \(0.448258\pi\)
\(774\) −2.77257e13 −2.77682
\(775\) −5.42452e12 −0.540137
\(776\) 2.99130e13 2.96130
\(777\) 3.00383e12 0.295652
\(778\) 1.12120e13 1.09718
\(779\) −8.37851e12 −0.815170
\(780\) −4.68233e12 −0.452936
\(781\) 7.23451e12 0.695792
\(782\) −1.09929e12 −0.105119
\(783\) −7.17160e11 −0.0681849
\(784\) 1.86139e12 0.175961
\(785\) −7.24248e11 −0.0680728
\(786\) −1.76595e13 −1.65036
\(787\) −1.41868e11 −0.0131826 −0.00659128 0.999978i \(-0.502098\pi\)
−0.00659128 + 0.999978i \(0.502098\pi\)
\(788\) −2.99230e13 −2.76463
\(789\) 2.09776e12 0.192712
\(790\) 9.09955e12 0.831185
\(791\) −4.73006e12 −0.429608
\(792\) 4.00336e13 3.61544
\(793\) 4.09827e12 0.368020
\(794\) 2.18797e13 1.95366
\(795\) 6.03065e12 0.535442
\(796\) −2.81714e12 −0.248714
\(797\) 9.50720e12 0.834623 0.417311 0.908764i \(-0.362972\pi\)
0.417311 + 0.908764i \(0.362972\pi\)
\(798\) 9.70055e12 0.846805
\(799\) −1.22101e12 −0.105988
\(800\) 2.22656e12 0.192189
\(801\) −4.49535e12 −0.385849
\(802\) 3.39956e13 2.90160
\(803\) −1.99991e13 −1.69742
\(804\) 3.81439e12 0.321938
\(805\) −1.10090e12 −0.0923988
\(806\) −4.49135e12 −0.374860
\(807\) 2.49996e13 2.07492
\(808\) 2.33310e13 1.92567
\(809\) 1.98831e13 1.63198 0.815990 0.578066i \(-0.196192\pi\)
0.815990 + 0.578066i \(0.196192\pi\)
\(810\) 1.08751e13 0.887668
\(811\) −4.24777e12 −0.344800 −0.172400 0.985027i \(-0.555152\pi\)
−0.172400 + 0.985027i \(0.555152\pi\)
\(812\) 6.60623e12 0.533275
\(813\) −2.09991e13 −1.68575
\(814\) −2.14056e13 −1.70891
\(815\) −8.67241e12 −0.688542
\(816\) 3.00993e12 0.237657
\(817\) 1.67402e13 1.31451
\(818\) 1.75061e13 1.36709
\(819\) 1.44386e12 0.112137
\(820\) −1.34907e13 −1.04201
\(821\) −1.55168e13 −1.19195 −0.595974 0.803003i \(-0.703234\pi\)
−0.595974 + 0.803003i \(0.703234\pi\)
\(822\) 3.46645e13 2.64827
\(823\) −1.21234e13 −0.921136 −0.460568 0.887624i \(-0.652354\pi\)
−0.460568 + 0.887624i \(0.652354\pi\)
\(824\) −4.32595e13 −3.26896
\(825\) −2.40439e13 −1.80701
\(826\) −4.58561e12 −0.342758
\(827\) 1.41174e13 1.04950 0.524748 0.851257i \(-0.324159\pi\)
0.524748 + 0.851257i \(0.324159\pi\)
\(828\) −1.34199e13 −0.992231
\(829\) 8.72820e12 0.641844 0.320922 0.947106i \(-0.396007\pi\)
0.320922 + 0.947106i \(0.396007\pi\)
\(830\) −2.53097e13 −1.85113
\(831\) 2.25492e13 1.64031
\(832\) −2.87816e12 −0.208238
\(833\) −2.66248e11 −0.0191595
\(834\) −1.91275e13 −1.36903
\(835\) −8.37447e12 −0.596167
\(836\) −4.66494e13 −3.30307
\(837\) −1.09758e12 −0.0772988
\(838\) 4.14081e13 2.90060
\(839\) 1.74217e13 1.21384 0.606922 0.794762i \(-0.292404\pi\)
0.606922 + 0.794762i \(0.292404\pi\)
\(840\) 8.09323e12 0.560874
\(841\) −7.80213e12 −0.537813
\(842\) 1.84190e13 1.26288
\(843\) −1.70780e13 −1.16469
\(844\) 5.35647e13 3.63361
\(845\) 6.23554e11 0.0420745
\(846\) −2.20881e13 −1.48249
\(847\) 1.25238e13 0.836104
\(848\) −1.26209e13 −0.838126
\(849\) 1.95892e12 0.129399
\(850\) −2.50856e12 −0.164831
\(851\) 3.71802e12 0.243013
\(852\) 1.78282e13 1.15912
\(853\) 1.67487e12 0.108320 0.0541602 0.998532i \(-0.482752\pi\)
0.0541602 + 0.998532i \(0.482752\pi\)
\(854\) −1.36710e13 −0.879511
\(855\) 8.11909e12 0.519589
\(856\) −5.91489e13 −3.76543
\(857\) 2.04783e13 1.29682 0.648412 0.761290i \(-0.275434\pi\)
0.648412 + 0.761290i \(0.275434\pi\)
\(858\) −1.99076e13 −1.25408
\(859\) −2.22613e13 −1.39502 −0.697510 0.716575i \(-0.745709\pi\)
−0.697510 + 0.716575i \(0.745709\pi\)
\(860\) 2.69543e13 1.68030
\(861\) 8.04894e12 0.499142
\(862\) −4.11102e13 −2.53610
\(863\) −1.70230e13 −1.04469 −0.522345 0.852734i \(-0.674943\pi\)
−0.522345 + 0.852734i \(0.674943\pi\)
\(864\) 4.50515e11 0.0275041
\(865\) −8.93793e12 −0.542831
\(866\) −1.17164e13 −0.707887
\(867\) 2.35049e13 1.41277
\(868\) 1.01105e13 0.604555
\(869\) 2.61080e13 1.55305
\(870\) 1.58529e13 0.938153
\(871\) −5.07968e11 −0.0299057
\(872\) 1.86025e13 1.08955
\(873\) −2.88282e13 −1.67978
\(874\) 1.20069e13 0.696035
\(875\) −6.09690e12 −0.351619
\(876\) −4.92843e13 −2.82774
\(877\) −5.12343e12 −0.292458 −0.146229 0.989251i \(-0.546714\pi\)
−0.146229 + 0.989251i \(0.546714\pi\)
\(878\) −6.35349e12 −0.360817
\(879\) −3.10039e13 −1.75173
\(880\) −2.14805e13 −1.20746
\(881\) −2.66945e13 −1.49290 −0.746450 0.665441i \(-0.768243\pi\)
−0.746450 + 0.665441i \(0.768243\pi\)
\(882\) −4.81644e12 −0.267989
\(883\) −5.33946e12 −0.295579 −0.147790 0.989019i \(-0.547216\pi\)
−0.147790 + 0.989019i \(0.547216\pi\)
\(884\) −1.40164e12 −0.0771971
\(885\) −7.42593e12 −0.406917
\(886\) 2.62021e13 1.42851
\(887\) −3.02361e13 −1.64010 −0.820048 0.572295i \(-0.806053\pi\)
−0.820048 + 0.572295i \(0.806053\pi\)
\(888\) −2.73329e13 −1.47512
\(889\) 3.90609e12 0.209741
\(890\) 6.47610e12 0.345986
\(891\) 3.12023e13 1.65858
\(892\) 6.69313e11 0.0353987
\(893\) 1.33364e13 0.701788
\(894\) −5.06198e13 −2.65034
\(895\) −1.73609e13 −0.904418
\(896\) 1.16007e13 0.601307
\(897\) 3.45783e12 0.178335
\(898\) −4.67263e13 −2.39783
\(899\) 1.02617e13 0.523966
\(900\) −3.06239e13 −1.55586
\(901\) 1.80525e12 0.0912592
\(902\) −5.73576e13 −2.88510
\(903\) −1.60818e13 −0.804894
\(904\) 4.30404e13 2.14347
\(905\) −1.26099e13 −0.624873
\(906\) −6.01180e13 −2.96434
\(907\) 2.31549e13 1.13608 0.568040 0.823001i \(-0.307702\pi\)
0.568040 + 0.823001i \(0.307702\pi\)
\(908\) −5.85918e13 −2.86056
\(909\) −2.24849e13 −1.09233
\(910\) −2.08005e12 −0.100551
\(911\) 7.59486e11 0.0365332 0.0182666 0.999833i \(-0.494185\pi\)
0.0182666 + 0.999833i \(0.494185\pi\)
\(912\) −3.28758e13 −1.57362
\(913\) −7.26176e13 −3.45878
\(914\) −1.88039e13 −0.891232
\(915\) −2.21389e13 −1.04414
\(916\) 2.62110e13 1.23014
\(917\) −5.29405e12 −0.247244
\(918\) −5.07575e11 −0.0235889
\(919\) 2.94594e13 1.36240 0.681199 0.732098i \(-0.261459\pi\)
0.681199 + 0.732098i \(0.261459\pi\)
\(920\) 1.00175e13 0.461012
\(921\) −2.90909e13 −1.33226
\(922\) 3.12189e13 1.42275
\(923\) −2.37421e12 −0.107674
\(924\) 4.48144e13 2.02252
\(925\) 8.48444e12 0.381053
\(926\) 7.88355e12 0.352349
\(927\) 4.16907e13 1.85430
\(928\) −4.21205e12 −0.186435
\(929\) −8.83235e12 −0.389050 −0.194525 0.980898i \(-0.562317\pi\)
−0.194525 + 0.980898i \(0.562317\pi\)
\(930\) 2.42623e13 1.06355
\(931\) 2.90807e12 0.126862
\(932\) 3.05760e13 1.32742
\(933\) 5.23470e13 2.26165
\(934\) 1.59529e13 0.685926
\(935\) 3.07250e12 0.131474
\(936\) −1.31382e13 −0.559491
\(937\) −2.01513e13 −0.854033 −0.427017 0.904244i \(-0.640435\pi\)
−0.427017 + 0.904244i \(0.640435\pi\)
\(938\) 1.69448e12 0.0714701
\(939\) −3.00757e13 −1.26247
\(940\) 2.14736e13 0.897076
\(941\) 1.48474e13 0.617302 0.308651 0.951175i \(-0.400122\pi\)
0.308651 + 0.951175i \(0.400122\pi\)
\(942\) −7.58828e12 −0.313989
\(943\) 9.96264e12 0.410272
\(944\) 1.55409e13 0.636946
\(945\) −5.08318e11 −0.0207344
\(946\) 1.14600e14 4.65239
\(947\) 1.67021e12 0.0674832 0.0337416 0.999431i \(-0.489258\pi\)
0.0337416 + 0.999431i \(0.489258\pi\)
\(948\) 6.43387e13 2.58723
\(949\) 6.56326e12 0.262677
\(950\) 2.73996e13 1.09141
\(951\) −6.23417e13 −2.47153
\(952\) 2.42268e12 0.0955937
\(953\) 1.48456e13 0.583014 0.291507 0.956569i \(-0.405843\pi\)
0.291507 + 0.956569i \(0.405843\pi\)
\(954\) 3.26571e13 1.27647
\(955\) −5.84030e12 −0.227206
\(956\) −2.28079e13 −0.883132
\(957\) 4.54846e13 1.75291
\(958\) 1.60265e13 0.614744
\(959\) 1.03919e13 0.396744
\(960\) 1.55478e13 0.590812
\(961\) −1.07345e13 −0.405999
\(962\) 7.02487e12 0.264454
\(963\) 5.70039e13 2.13592
\(964\) 6.32240e13 2.35795
\(965\) 1.05543e13 0.391794
\(966\) −1.15346e13 −0.426194
\(967\) 3.82621e13 1.40718 0.703590 0.710607i \(-0.251579\pi\)
0.703590 + 0.710607i \(0.251579\pi\)
\(968\) −1.13958e14 −4.17164
\(969\) 4.70244e12 0.171343
\(970\) 4.15305e13 1.50624
\(971\) −4.68778e12 −0.169231 −0.0846156 0.996414i \(-0.526966\pi\)
−0.0846156 + 0.996414i \(0.526966\pi\)
\(972\) 8.26854e13 2.97119
\(973\) −5.73414e12 −0.205097
\(974\) 5.27374e13 1.87760
\(975\) 7.89068e12 0.279636
\(976\) 4.63320e13 1.63440
\(977\) −2.05720e13 −0.722355 −0.361177 0.932497i \(-0.617625\pi\)
−0.361177 + 0.932497i \(0.617625\pi\)
\(978\) −9.08649e13 −3.17593
\(979\) 1.85809e13 0.646465
\(980\) 4.68244e12 0.162164
\(981\) −1.79278e13 −0.618042
\(982\) 6.86600e13 2.35615
\(983\) −1.66354e13 −0.568255 −0.284128 0.958787i \(-0.591704\pi\)
−0.284128 + 0.958787i \(0.591704\pi\)
\(984\) −7.32400e13 −2.49041
\(985\) −2.15264e13 −0.728632
\(986\) 4.74552e12 0.159896
\(987\) −1.28118e13 −0.429717
\(988\) 1.53093e13 0.511151
\(989\) −1.99053e13 −0.661586
\(990\) 5.55817e13 1.83896
\(991\) 2.83299e13 0.933067 0.466534 0.884503i \(-0.345503\pi\)
0.466534 + 0.884503i \(0.345503\pi\)
\(992\) −6.44636e12 −0.211355
\(993\) −5.64332e13 −1.84189
\(994\) 7.91991e12 0.257325
\(995\) −2.02663e12 −0.0655496
\(996\) −1.78954e14 −5.76200
\(997\) 1.89205e13 0.606463 0.303232 0.952917i \(-0.401934\pi\)
0.303232 + 0.952917i \(0.401934\pi\)
\(998\) 4.01225e13 1.28027
\(999\) 1.71672e12 0.0545324
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.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.1 12 1.1 even 1 trivial