Properties

Label 9.76.a.c.1.6
Level $9$
Weight $76$
Character 9.1
Self dual yes
Analytic conductor $320.606$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 76 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(320.605553540\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 38457853073924058692 x^{4} - 10276556354621685339901678086 x^{3} + 371187556674475060057870954681799784505 x^{2} + 52686123927652036687598761277591247931691204025 x - 675344021115865838575279495800656435684060652010336995750\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{58}\cdot 3^{36}\cdot 5^{7}\cdot 7^{3}\cdot 11\cdot 13\cdot 19 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-4.40594e9\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.26741e11 q^{2} +6.89808e22 q^{4} +4.33451e25 q^{5} +4.24638e31 q^{7} +1.01949e34 q^{8} +O(q^{10})\) \(q+3.26741e11 q^{2} +6.89808e22 q^{4} +4.33451e25 q^{5} +4.24638e31 q^{7} +1.01949e34 q^{8} +1.41626e37 q^{10} -1.56603e38 q^{11} -1.88327e41 q^{13} +1.38747e43 q^{14} +7.25086e44 q^{16} -9.86838e45 q^{17} -1.40944e47 q^{19} +2.98998e48 q^{20} -5.11687e49 q^{22} -8.97012e50 q^{23} -2.45910e52 q^{25} -6.15343e52 q^{26} +2.92919e54 q^{28} -7.48449e54 q^{29} +9.83848e55 q^{31} -1.48239e56 q^{32} -3.22441e57 q^{34} +1.84060e57 q^{35} -4.42368e58 q^{37} -4.60523e58 q^{38} +4.41901e59 q^{40} -3.18294e60 q^{41} -2.58944e61 q^{43} -1.08026e61 q^{44} -2.93091e62 q^{46} +4.72886e61 q^{47} -6.08688e62 q^{49} -8.03489e63 q^{50} -1.29910e64 q^{52} +7.75824e64 q^{53} -6.78797e63 q^{55} +4.32917e65 q^{56} -2.44549e66 q^{58} +8.17255e64 q^{59} -6.09863e66 q^{61} +3.21464e67 q^{62} -7.58287e67 q^{64} -8.16306e66 q^{65} +4.31126e68 q^{67} -6.80729e68 q^{68} +6.01399e68 q^{70} -2.46110e69 q^{71} -4.09634e69 q^{73} -1.44540e70 q^{74} -9.72245e69 q^{76} -6.64997e69 q^{77} +9.26377e69 q^{79} +3.14289e70 q^{80} -1.04000e72 q^{82} +2.78206e69 q^{83} -4.27746e71 q^{85} -8.46075e72 q^{86} -1.59656e72 q^{88} -1.96684e73 q^{89} -7.99710e72 q^{91} -6.18767e73 q^{92} +1.54511e73 q^{94} -6.10924e72 q^{95} +2.01886e74 q^{97} -1.98883e74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 57080822040q^{2} + \)\(17\!\cdots\!28\)\(q^{4} + \)\(38\!\cdots\!40\)\(q^{5} + \)\(19\!\cdots\!00\)\(q^{7} - \)\(44\!\cdots\!20\)\(q^{8} + O(q^{10}) \) \( 6q + 57080822040q^{2} + \)\(17\!\cdots\!28\)\(q^{4} + \)\(38\!\cdots\!40\)\(q^{5} + \)\(19\!\cdots\!00\)\(q^{7} - \)\(44\!\cdots\!20\)\(q^{8} + \)\(13\!\cdots\!60\)\(q^{10} + \)\(94\!\cdots\!88\)\(q^{11} + \)\(53\!\cdots\!20\)\(q^{13} - \)\(82\!\cdots\!76\)\(q^{14} + \)\(26\!\cdots\!16\)\(q^{16} - \)\(18\!\cdots\!80\)\(q^{17} + \)\(10\!\cdots\!80\)\(q^{19} - \)\(92\!\cdots\!80\)\(q^{20} + \)\(15\!\cdots\!20\)\(q^{22} - \)\(15\!\cdots\!80\)\(q^{23} + \)\(19\!\cdots\!50\)\(q^{25} - \)\(11\!\cdots\!52\)\(q^{26} + \)\(14\!\cdots\!20\)\(q^{28} - \)\(14\!\cdots\!20\)\(q^{29} - \)\(41\!\cdots\!88\)\(q^{31} - \)\(11\!\cdots\!60\)\(q^{32} + \)\(30\!\cdots\!56\)\(q^{34} - \)\(27\!\cdots\!60\)\(q^{35} + \)\(98\!\cdots\!40\)\(q^{37} - \)\(12\!\cdots\!80\)\(q^{38} + \)\(88\!\cdots\!00\)\(q^{40} - \)\(50\!\cdots\!12\)\(q^{41} + \)\(27\!\cdots\!00\)\(q^{43} + \)\(86\!\cdots\!44\)\(q^{44} - \)\(82\!\cdots\!88\)\(q^{46} + \)\(13\!\cdots\!80\)\(q^{47} - \)\(57\!\cdots\!42\)\(q^{49} - \)\(31\!\cdots\!00\)\(q^{50} + \)\(41\!\cdots\!00\)\(q^{52} - \)\(64\!\cdots\!60\)\(q^{53} + \)\(43\!\cdots\!20\)\(q^{55} - \)\(28\!\cdots\!20\)\(q^{56} - \)\(17\!\cdots\!80\)\(q^{58} + \)\(24\!\cdots\!60\)\(q^{59} - \)\(25\!\cdots\!88\)\(q^{61} + \)\(29\!\cdots\!80\)\(q^{62} + \)\(47\!\cdots\!48\)\(q^{64} - \)\(12\!\cdots\!20\)\(q^{65} + \)\(95\!\cdots\!80\)\(q^{67} - \)\(12\!\cdots\!60\)\(q^{68} - \)\(34\!\cdots\!40\)\(q^{70} + \)\(25\!\cdots\!88\)\(q^{71} - \)\(30\!\cdots\!20\)\(q^{73} + \)\(24\!\cdots\!84\)\(q^{74} + \)\(10\!\cdots\!40\)\(q^{76} - \)\(15\!\cdots\!00\)\(q^{77} + \)\(11\!\cdots\!20\)\(q^{79} - \)\(12\!\cdots\!60\)\(q^{80} - \)\(25\!\cdots\!80\)\(q^{82} + \)\(79\!\cdots\!60\)\(q^{83} - \)\(36\!\cdots\!40\)\(q^{85} - \)\(72\!\cdots\!32\)\(q^{86} - \)\(48\!\cdots\!60\)\(q^{88} - \)\(53\!\cdots\!60\)\(q^{89} + \)\(34\!\cdots\!32\)\(q^{91} - \)\(18\!\cdots\!80\)\(q^{92} - \)\(29\!\cdots\!04\)\(q^{94} - \)\(19\!\cdots\!00\)\(q^{95} - \)\(74\!\cdots\!80\)\(q^{97} + \)\(16\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.26741e11 1.68104 0.840522 0.541778i \(-0.182249\pi\)
0.840522 + 0.541778i \(0.182249\pi\)
\(3\) 0 0
\(4\) 6.89808e22 1.82591
\(5\) 4.33451e25 0.266419 0.133209 0.991088i \(-0.457472\pi\)
0.133209 + 0.991088i \(0.457472\pi\)
\(6\) 0 0
\(7\) 4.24638e31 0.864655 0.432327 0.901717i \(-0.357692\pi\)
0.432327 + 0.901717i \(0.357692\pi\)
\(8\) 1.01949e34 1.38839
\(9\) 0 0
\(10\) 1.41626e37 0.447861
\(11\) −1.56603e38 −0.138859 −0.0694296 0.997587i \(-0.522118\pi\)
−0.0694296 + 0.997587i \(0.522118\pi\)
\(12\) 0 0
\(13\) −1.88327e41 −0.317715 −0.158857 0.987302i \(-0.550781\pi\)
−0.158857 + 0.987302i \(0.550781\pi\)
\(14\) 1.38747e43 1.45352
\(15\) 0 0
\(16\) 7.25086e44 0.508031
\(17\) −9.86838e45 −0.711888 −0.355944 0.934507i \(-0.615841\pi\)
−0.355944 + 0.934507i \(0.615841\pi\)
\(18\) 0 0
\(19\) −1.40944e47 −0.156959 −0.0784797 0.996916i \(-0.525007\pi\)
−0.0784797 + 0.996916i \(0.525007\pi\)
\(20\) 2.98998e48 0.486456
\(21\) 0 0
\(22\) −5.11687e49 −0.233428
\(23\) −8.97012e50 −0.772689 −0.386344 0.922355i \(-0.626262\pi\)
−0.386344 + 0.922355i \(0.626262\pi\)
\(24\) 0 0
\(25\) −2.45910e52 −0.929021
\(26\) −6.15343e52 −0.534093
\(27\) 0 0
\(28\) 2.92919e54 1.57878
\(29\) −7.48449e54 −1.08202 −0.541012 0.841015i \(-0.681959\pi\)
−0.541012 + 0.841015i \(0.681959\pi\)
\(30\) 0 0
\(31\) 9.83848e55 1.16645 0.583223 0.812312i \(-0.301791\pi\)
0.583223 + 0.812312i \(0.301791\pi\)
\(32\) −1.48239e56 −0.534365
\(33\) 0 0
\(34\) −3.22441e57 −1.19671
\(35\) 1.84060e57 0.230360
\(36\) 0 0
\(37\) −4.42368e58 −0.689000 −0.344500 0.938786i \(-0.611952\pi\)
−0.344500 + 0.938786i \(0.611952\pi\)
\(38\) −4.60523e58 −0.263856
\(39\) 0 0
\(40\) 4.41901e59 0.369892
\(41\) −3.18294e60 −1.05544 −0.527721 0.849418i \(-0.676953\pi\)
−0.527721 + 0.849418i \(0.676953\pi\)
\(42\) 0 0
\(43\) −2.58944e61 −1.43926 −0.719628 0.694360i \(-0.755688\pi\)
−0.719628 + 0.694360i \(0.755688\pi\)
\(44\) −1.08026e61 −0.253544
\(45\) 0 0
\(46\) −2.93091e62 −1.29892
\(47\) 4.72886e61 0.0935593 0.0467797 0.998905i \(-0.485104\pi\)
0.0467797 + 0.998905i \(0.485104\pi\)
\(48\) 0 0
\(49\) −6.08688e62 −0.252372
\(50\) −8.03489e63 −1.56173
\(51\) 0 0
\(52\) −1.29910e64 −0.580118
\(53\) 7.75824e64 1.69597 0.847985 0.530020i \(-0.177816\pi\)
0.847985 + 0.530020i \(0.177816\pi\)
\(54\) 0 0
\(55\) −6.78797e63 −0.0369947
\(56\) 4.32917e65 1.20048
\(57\) 0 0
\(58\) −2.44549e66 −1.81893
\(59\) 8.17255e64 0.0320190 0.0160095 0.999872i \(-0.494904\pi\)
0.0160095 + 0.999872i \(0.494904\pi\)
\(60\) 0 0
\(61\) −6.09863e66 −0.684484 −0.342242 0.939612i \(-0.611186\pi\)
−0.342242 + 0.939612i \(0.611186\pi\)
\(62\) 3.21464e67 1.96085
\(63\) 0 0
\(64\) −7.58287e67 −1.40632
\(65\) −8.16306e66 −0.0846451
\(66\) 0 0
\(67\) 4.31126e68 1.43483 0.717413 0.696648i \(-0.245326\pi\)
0.717413 + 0.696648i \(0.245326\pi\)
\(68\) −6.80729e68 −1.29984
\(69\) 0 0
\(70\) 6.01399e68 0.387245
\(71\) −2.46110e69 −0.930981 −0.465491 0.885053i \(-0.654122\pi\)
−0.465491 + 0.885053i \(0.654122\pi\)
\(72\) 0 0
\(73\) −4.09634e69 −0.546749 −0.273375 0.961908i \(-0.588140\pi\)
−0.273375 + 0.961908i \(0.588140\pi\)
\(74\) −1.44540e70 −1.15824
\(75\) 0 0
\(76\) −9.72245e69 −0.286593
\(77\) −6.64997e69 −0.120065
\(78\) 0 0
\(79\) 9.26377e69 0.0639399 0.0319699 0.999489i \(-0.489822\pi\)
0.0319699 + 0.999489i \(0.489822\pi\)
\(80\) 3.14289e70 0.135349
\(81\) 0 0
\(82\) −1.04000e72 −1.77424
\(83\) 2.78206e69 0.00301257 0.00150629 0.999999i \(-0.499521\pi\)
0.00150629 + 0.999999i \(0.499521\pi\)
\(84\) 0 0
\(85\) −4.27746e71 −0.189660
\(86\) −8.46075e72 −2.41945
\(87\) 0 0
\(88\) −1.59656e72 −0.192790
\(89\) −1.96684e73 −1.55469 −0.777346 0.629073i \(-0.783435\pi\)
−0.777346 + 0.629073i \(0.783435\pi\)
\(90\) 0 0
\(91\) −7.99710e72 −0.274714
\(92\) −6.18767e73 −1.41086
\(93\) 0 0
\(94\) 1.54511e73 0.157277
\(95\) −6.10924e72 −0.0418169
\(96\) 0 0
\(97\) 2.01886e74 0.632653 0.316326 0.948650i \(-0.397551\pi\)
0.316326 + 0.948650i \(0.397551\pi\)
\(98\) −1.98883e74 −0.424249
\(99\) 0 0
\(100\) −1.69631e75 −1.69631
\(101\) 1.56337e75 1.07649 0.538247 0.842787i \(-0.319087\pi\)
0.538247 + 0.842787i \(0.319087\pi\)
\(102\) 0 0
\(103\) −5.07128e74 −0.167387 −0.0836935 0.996492i \(-0.526672\pi\)
−0.0836935 + 0.996492i \(0.526672\pi\)
\(104\) −1.91999e75 −0.441111
\(105\) 0 0
\(106\) 2.53494e76 2.85100
\(107\) 8.29288e75 0.655864 0.327932 0.944701i \(-0.393648\pi\)
0.327932 + 0.944701i \(0.393648\pi\)
\(108\) 0 0
\(109\) 4.14553e76 1.63714 0.818570 0.574406i \(-0.194767\pi\)
0.818570 + 0.574406i \(0.194767\pi\)
\(110\) −2.21791e75 −0.0621897
\(111\) 0 0
\(112\) 3.07899e76 0.439271
\(113\) 1.53804e76 0.157227 0.0786134 0.996905i \(-0.474951\pi\)
0.0786134 + 0.996905i \(0.474951\pi\)
\(114\) 0 0
\(115\) −3.88811e76 −0.205859
\(116\) −5.16286e77 −1.97568
\(117\) 0 0
\(118\) 2.67031e76 0.0538253
\(119\) −4.19049e77 −0.615537
\(120\) 0 0
\(121\) −1.24737e78 −0.980718
\(122\) −1.99267e78 −1.15065
\(123\) 0 0
\(124\) 6.78667e78 2.12982
\(125\) −2.21323e78 −0.513927
\(126\) 0 0
\(127\) 1.95181e78 0.249918 0.124959 0.992162i \(-0.460120\pi\)
0.124959 + 0.992162i \(0.460120\pi\)
\(128\) −1.91760e79 −1.82972
\(129\) 0 0
\(130\) −2.66721e78 −0.142292
\(131\) 2.48929e78 0.0996328 0.0498164 0.998758i \(-0.484136\pi\)
0.0498164 + 0.998758i \(0.484136\pi\)
\(132\) 0 0
\(133\) −5.98503e78 −0.135716
\(134\) 1.40867e80 2.41201
\(135\) 0 0
\(136\) −1.00608e80 −0.988375
\(137\) 1.62508e80 1.21298 0.606491 0.795090i \(-0.292577\pi\)
0.606491 + 0.795090i \(0.292577\pi\)
\(138\) 0 0
\(139\) 2.33640e80 1.01273 0.506364 0.862320i \(-0.330989\pi\)
0.506364 + 0.862320i \(0.330989\pi\)
\(140\) 1.26966e80 0.420616
\(141\) 0 0
\(142\) −8.04142e80 −1.56502
\(143\) 2.94926e79 0.0441176
\(144\) 0 0
\(145\) −3.24416e80 −0.288272
\(146\) −1.33844e81 −0.919110
\(147\) 0 0
\(148\) −3.05149e81 −1.25805
\(149\) −3.44890e81 −1.10457 −0.552287 0.833654i \(-0.686245\pi\)
−0.552287 + 0.833654i \(0.686245\pi\)
\(150\) 0 0
\(151\) −2.98374e81 −0.579597 −0.289798 0.957088i \(-0.593588\pi\)
−0.289798 + 0.957088i \(0.593588\pi\)
\(152\) −1.43692e81 −0.217920
\(153\) 0 0
\(154\) −2.17282e81 −0.201835
\(155\) 4.26450e81 0.310763
\(156\) 0 0
\(157\) −2.36792e82 −1.06692 −0.533458 0.845826i \(-0.679108\pi\)
−0.533458 + 0.845826i \(0.679108\pi\)
\(158\) 3.02685e81 0.107486
\(159\) 0 0
\(160\) −6.42542e81 −0.142365
\(161\) −3.80906e82 −0.668109
\(162\) 0 0
\(163\) 5.73833e82 0.633506 0.316753 0.948508i \(-0.397407\pi\)
0.316753 + 0.948508i \(0.397407\pi\)
\(164\) −2.19562e83 −1.92714
\(165\) 0 0
\(166\) 9.09012e80 0.00506426
\(167\) 7.47244e82 0.332349 0.166175 0.986096i \(-0.446858\pi\)
0.166175 + 0.986096i \(0.446858\pi\)
\(168\) 0 0
\(169\) −3.15892e83 −0.899057
\(170\) −1.39762e83 −0.318827
\(171\) 0 0
\(172\) −1.78621e84 −2.62795
\(173\) −7.78810e83 −0.921940 −0.460970 0.887416i \(-0.652498\pi\)
−0.460970 + 0.887416i \(0.652498\pi\)
\(174\) 0 0
\(175\) −1.04423e84 −0.803283
\(176\) −1.13551e83 −0.0705448
\(177\) 0 0
\(178\) −6.42648e84 −2.61351
\(179\) −3.91485e84 −1.29041 −0.645203 0.764011i \(-0.723227\pi\)
−0.645203 + 0.764011i \(0.723227\pi\)
\(180\) 0 0
\(181\) −4.40838e84 −0.957928 −0.478964 0.877835i \(-0.658987\pi\)
−0.478964 + 0.877835i \(0.658987\pi\)
\(182\) −2.61298e84 −0.461806
\(183\) 0 0
\(184\) −9.14499e84 −1.07279
\(185\) −1.91745e84 −0.183562
\(186\) 0 0
\(187\) 1.54542e84 0.0988521
\(188\) 3.26201e84 0.170831
\(189\) 0 0
\(190\) −1.99614e84 −0.0702960
\(191\) 2.54828e85 0.737047 0.368524 0.929618i \(-0.379863\pi\)
0.368524 + 0.929618i \(0.379863\pi\)
\(192\) 0 0
\(193\) −6.92114e85 −1.35450 −0.677250 0.735753i \(-0.736828\pi\)
−0.677250 + 0.735753i \(0.736828\pi\)
\(194\) 6.59643e85 1.06352
\(195\) 0 0
\(196\) −4.19878e85 −0.460808
\(197\) −1.15851e86 −1.05055 −0.525276 0.850932i \(-0.676038\pi\)
−0.525276 + 0.850932i \(0.676038\pi\)
\(198\) 0 0
\(199\) 2.78135e86 1.72689 0.863445 0.504444i \(-0.168302\pi\)
0.863445 + 0.504444i \(0.168302\pi\)
\(200\) −2.50704e86 −1.28984
\(201\) 0 0
\(202\) 5.10818e86 1.80963
\(203\) −3.17820e86 −0.935578
\(204\) 0 0
\(205\) −1.37965e86 −0.281189
\(206\) −1.65700e86 −0.281385
\(207\) 0 0
\(208\) −1.36553e86 −0.161409
\(209\) 2.20723e85 0.0217953
\(210\) 0 0
\(211\) 6.21488e86 0.429379 0.214689 0.976682i \(-0.431126\pi\)
0.214689 + 0.976682i \(0.431126\pi\)
\(212\) 5.35170e87 3.09668
\(213\) 0 0
\(214\) 2.70963e87 1.10254
\(215\) −1.12239e87 −0.383445
\(216\) 0 0
\(217\) 4.17780e87 1.00857
\(218\) 1.35451e88 2.75211
\(219\) 0 0
\(220\) −4.68240e86 −0.0675489
\(221\) 1.85849e87 0.226177
\(222\) 0 0
\(223\) 1.14012e88 0.989737 0.494869 0.868968i \(-0.335216\pi\)
0.494869 + 0.868968i \(0.335216\pi\)
\(224\) −6.29479e87 −0.462041
\(225\) 0 0
\(226\) 5.02541e87 0.264305
\(227\) −2.58517e88 −1.15218 −0.576090 0.817386i \(-0.695422\pi\)
−0.576090 + 0.817386i \(0.695422\pi\)
\(228\) 0 0
\(229\) 6.20331e87 0.198973 0.0994863 0.995039i \(-0.468280\pi\)
0.0994863 + 0.995039i \(0.468280\pi\)
\(230\) −1.27040e88 −0.346057
\(231\) 0 0
\(232\) −7.63040e88 −1.50227
\(233\) −4.08817e88 −0.684986 −0.342493 0.939520i \(-0.611271\pi\)
−0.342493 + 0.939520i \(0.611271\pi\)
\(234\) 0 0
\(235\) 2.04973e87 0.0249259
\(236\) 5.63749e87 0.0584637
\(237\) 0 0
\(238\) −1.36921e89 −1.03474
\(239\) −5.69769e88 −0.367941 −0.183970 0.982932i \(-0.558895\pi\)
−0.183970 + 0.982932i \(0.558895\pi\)
\(240\) 0 0
\(241\) 1.40344e89 0.663064 0.331532 0.943444i \(-0.392435\pi\)
0.331532 + 0.943444i \(0.392435\pi\)
\(242\) −4.07567e89 −1.64863
\(243\) 0 0
\(244\) −4.20688e89 −1.24980
\(245\) −2.63836e88 −0.0672366
\(246\) 0 0
\(247\) 2.65436e88 0.0498683
\(248\) 1.00303e90 1.61948
\(249\) 0 0
\(250\) −7.23154e89 −0.863934
\(251\) 1.01537e89 0.104439 0.0522193 0.998636i \(-0.483371\pi\)
0.0522193 + 0.998636i \(0.483371\pi\)
\(252\) 0 0
\(253\) 1.40475e89 0.107295
\(254\) 6.37736e89 0.420123
\(255\) 0 0
\(256\) −3.40088e90 −1.66952
\(257\) −1.81313e90 −0.769018 −0.384509 0.923121i \(-0.625629\pi\)
−0.384509 + 0.923121i \(0.625629\pi\)
\(258\) 0 0
\(259\) −1.87846e90 −0.595747
\(260\) −5.63095e89 −0.154554
\(261\) 0 0
\(262\) 8.13354e89 0.167487
\(263\) −3.47473e90 −0.620270 −0.310135 0.950693i \(-0.600374\pi\)
−0.310135 + 0.950693i \(0.600374\pi\)
\(264\) 0 0
\(265\) 3.36281e90 0.451838
\(266\) −1.95556e90 −0.228144
\(267\) 0 0
\(268\) 2.97395e91 2.61986
\(269\) 1.34158e91 1.02779 0.513897 0.857852i \(-0.328201\pi\)
0.513897 + 0.857852i \(0.328201\pi\)
\(270\) 0 0
\(271\) 2.32034e91 1.34649 0.673246 0.739418i \(-0.264899\pi\)
0.673246 + 0.739418i \(0.264899\pi\)
\(272\) −7.15542e90 −0.361661
\(273\) 0 0
\(274\) 5.30982e91 2.03908
\(275\) 3.85102e90 0.129003
\(276\) 0 0
\(277\) −5.73199e91 −1.46324 −0.731618 0.681715i \(-0.761235\pi\)
−0.731618 + 0.681715i \(0.761235\pi\)
\(278\) 7.63398e91 1.70244
\(279\) 0 0
\(280\) 1.87648e91 0.319829
\(281\) 6.46798e91 0.964454 0.482227 0.876046i \(-0.339828\pi\)
0.482227 + 0.876046i \(0.339828\pi\)
\(282\) 0 0
\(283\) −7.40884e91 −0.846757 −0.423379 0.905953i \(-0.639156\pi\)
−0.423379 + 0.905953i \(0.639156\pi\)
\(284\) −1.69769e92 −1.69989
\(285\) 0 0
\(286\) 9.63646e90 0.0741637
\(287\) −1.35160e92 −0.912593
\(288\) 0 0
\(289\) −9.47777e91 −0.493216
\(290\) −1.06000e92 −0.484597
\(291\) 0 0
\(292\) −2.82569e92 −0.998314
\(293\) 1.11299e92 0.345903 0.172951 0.984930i \(-0.444670\pi\)
0.172951 + 0.984930i \(0.444670\pi\)
\(294\) 0 0
\(295\) 3.54240e90 0.00853045
\(296\) −4.50992e92 −0.956598
\(297\) 0 0
\(298\) −1.12690e93 −1.85684
\(299\) 1.68932e92 0.245495
\(300\) 0 0
\(301\) −1.09957e93 −1.24446
\(302\) −9.74911e92 −0.974327
\(303\) 0 0
\(304\) −1.02197e92 −0.0797402
\(305\) −2.64345e92 −0.182359
\(306\) 0 0
\(307\) 5.54198e92 0.299210 0.149605 0.988746i \(-0.452200\pi\)
0.149605 + 0.988746i \(0.452200\pi\)
\(308\) −4.58720e92 −0.219228
\(309\) 0 0
\(310\) 1.39339e93 0.522406
\(311\) 2.52122e93 0.837715 0.418858 0.908052i \(-0.362431\pi\)
0.418858 + 0.908052i \(0.362431\pi\)
\(312\) 0 0
\(313\) 2.94579e93 0.769643 0.384822 0.922991i \(-0.374263\pi\)
0.384822 + 0.922991i \(0.374263\pi\)
\(314\) −7.73697e93 −1.79353
\(315\) 0 0
\(316\) 6.39022e92 0.116748
\(317\) −1.03547e94 −1.68041 −0.840203 0.542272i \(-0.817564\pi\)
−0.840203 + 0.542272i \(0.817564\pi\)
\(318\) 0 0
\(319\) 1.17209e93 0.150249
\(320\) −3.28680e93 −0.374670
\(321\) 0 0
\(322\) −1.24458e94 −1.12312
\(323\) 1.39089e93 0.111737
\(324\) 0 0
\(325\) 4.63115e93 0.295164
\(326\) 1.87495e94 1.06495
\(327\) 0 0
\(328\) −3.24499e94 −1.46536
\(329\) 2.00806e93 0.0808965
\(330\) 0 0
\(331\) 4.31190e94 1.38395 0.691973 0.721924i \(-0.256742\pi\)
0.691973 + 0.721924i \(0.256742\pi\)
\(332\) 1.91908e92 0.00550068
\(333\) 0 0
\(334\) 2.44155e94 0.558694
\(335\) 1.86872e94 0.382265
\(336\) 0 0
\(337\) 4.32493e94 0.707714 0.353857 0.935300i \(-0.384870\pi\)
0.353857 + 0.935300i \(0.384870\pi\)
\(338\) −1.03215e95 −1.51135
\(339\) 0 0
\(340\) −2.95063e94 −0.346302
\(341\) −1.54074e94 −0.161972
\(342\) 0 0
\(343\) −1.28264e95 −1.08287
\(344\) −2.63992e95 −1.99824
\(345\) 0 0
\(346\) −2.54469e95 −1.54982
\(347\) −3.28354e95 −1.79468 −0.897339 0.441343i \(-0.854502\pi\)
−0.897339 + 0.441343i \(0.854502\pi\)
\(348\) 0 0
\(349\) −2.28464e95 −1.00662 −0.503309 0.864106i \(-0.667884\pi\)
−0.503309 + 0.864106i \(0.667884\pi\)
\(350\) −3.41192e95 −1.35035
\(351\) 0 0
\(352\) 2.32147e94 0.0742014
\(353\) 2.28907e95 0.657820 0.328910 0.944361i \(-0.393319\pi\)
0.328910 + 0.944361i \(0.393319\pi\)
\(354\) 0 0
\(355\) −1.06676e95 −0.248031
\(356\) −1.35674e96 −2.83872
\(357\) 0 0
\(358\) −1.27914e96 −2.16923
\(359\) 3.43961e95 0.525372 0.262686 0.964881i \(-0.415392\pi\)
0.262686 + 0.964881i \(0.415392\pi\)
\(360\) 0 0
\(361\) −7.86478e95 −0.975364
\(362\) −1.44040e96 −1.61032
\(363\) 0 0
\(364\) −5.51647e95 −0.501602
\(365\) −1.77556e95 −0.145664
\(366\) 0 0
\(367\) 1.95272e96 1.30516 0.652579 0.757720i \(-0.273687\pi\)
0.652579 + 0.757720i \(0.273687\pi\)
\(368\) −6.50411e95 −0.392550
\(369\) 0 0
\(370\) −6.26509e95 −0.308576
\(371\) 3.29445e96 1.46643
\(372\) 0 0
\(373\) −3.14216e94 −0.0114326 −0.00571632 0.999984i \(-0.501820\pi\)
−0.00571632 + 0.999984i \(0.501820\pi\)
\(374\) 5.04952e95 0.166175
\(375\) 0 0
\(376\) 4.82105e95 0.129896
\(377\) 1.40953e96 0.343775
\(378\) 0 0
\(379\) −1.17713e96 −0.235427 −0.117713 0.993048i \(-0.537556\pi\)
−0.117713 + 0.993048i \(0.537556\pi\)
\(380\) −4.21420e95 −0.0763538
\(381\) 0 0
\(382\) 8.32627e96 1.23901
\(383\) −1.08925e97 −1.46951 −0.734755 0.678333i \(-0.762703\pi\)
−0.734755 + 0.678333i \(0.762703\pi\)
\(384\) 0 0
\(385\) −2.88243e95 −0.0319876
\(386\) −2.26142e97 −2.27697
\(387\) 0 0
\(388\) 1.39262e97 1.15517
\(389\) −1.78980e96 −0.134801 −0.0674007 0.997726i \(-0.521471\pi\)
−0.0674007 + 0.997726i \(0.521471\pi\)
\(390\) 0 0
\(391\) 8.85206e96 0.550067
\(392\) −6.20554e96 −0.350390
\(393\) 0 0
\(394\) −3.78534e97 −1.76602
\(395\) 4.01539e95 0.0170348
\(396\) 0 0
\(397\) 2.74242e97 0.962698 0.481349 0.876529i \(-0.340147\pi\)
0.481349 + 0.876529i \(0.340147\pi\)
\(398\) 9.08780e97 2.90298
\(399\) 0 0
\(400\) −1.78306e97 −0.471971
\(401\) −5.08366e97 −1.22536 −0.612678 0.790333i \(-0.709908\pi\)
−0.612678 + 0.790333i \(0.709908\pi\)
\(402\) 0 0
\(403\) −1.85285e97 −0.370597
\(404\) 1.07843e98 1.96558
\(405\) 0 0
\(406\) −1.03845e98 −1.57275
\(407\) 6.92762e96 0.0956740
\(408\) 0 0
\(409\) −3.24764e97 −0.373201 −0.186601 0.982436i \(-0.559747\pi\)
−0.186601 + 0.982436i \(0.559747\pi\)
\(410\) −4.50788e97 −0.472692
\(411\) 0 0
\(412\) −3.49821e97 −0.305633
\(413\) 3.47038e96 0.0276853
\(414\) 0 0
\(415\) 1.20588e95 0.000802605 0
\(416\) 2.79174e97 0.169776
\(417\) 0 0
\(418\) 7.21193e96 0.0366388
\(419\) −3.50672e98 −1.62883 −0.814413 0.580286i \(-0.802941\pi\)
−0.814413 + 0.580286i \(0.802941\pi\)
\(420\) 0 0
\(421\) 1.68894e98 0.656199 0.328099 0.944643i \(-0.393592\pi\)
0.328099 + 0.944643i \(0.393592\pi\)
\(422\) 2.03066e98 0.721804
\(423\) 0 0
\(424\) 7.90948e98 2.35466
\(425\) 2.42673e98 0.661359
\(426\) 0 0
\(427\) −2.58971e98 −0.591842
\(428\) 5.72050e98 1.19755
\(429\) 0 0
\(430\) −3.66732e98 −0.644587
\(431\) 9.95538e98 1.60384 0.801919 0.597433i \(-0.203813\pi\)
0.801919 + 0.597433i \(0.203813\pi\)
\(432\) 0 0
\(433\) 6.16347e98 0.834698 0.417349 0.908746i \(-0.362959\pi\)
0.417349 + 0.908746i \(0.362959\pi\)
\(434\) 1.36506e99 1.69545
\(435\) 0 0
\(436\) 2.85962e99 2.98927
\(437\) 1.26429e98 0.121281
\(438\) 0 0
\(439\) 1.65566e99 1.33829 0.669147 0.743130i \(-0.266660\pi\)
0.669147 + 0.743130i \(0.266660\pi\)
\(440\) −6.92030e97 −0.0513629
\(441\) 0 0
\(442\) 6.07244e98 0.380214
\(443\) 6.77161e97 0.0389540 0.0194770 0.999810i \(-0.493800\pi\)
0.0194770 + 0.999810i \(0.493800\pi\)
\(444\) 0 0
\(445\) −8.52529e98 −0.414199
\(446\) 3.72526e99 1.66379
\(447\) 0 0
\(448\) −3.21998e99 −1.21598
\(449\) −2.65101e99 −0.920819 −0.460409 0.887707i \(-0.652297\pi\)
−0.460409 + 0.887707i \(0.652297\pi\)
\(450\) 0 0
\(451\) 4.98459e98 0.146558
\(452\) 1.06095e99 0.287081
\(453\) 0 0
\(454\) −8.44683e99 −1.93686
\(455\) −3.46635e98 −0.0731888
\(456\) 0 0
\(457\) −4.84378e99 −0.867615 −0.433807 0.901006i \(-0.642830\pi\)
−0.433807 + 0.901006i \(0.642830\pi\)
\(458\) 2.02688e99 0.334482
\(459\) 0 0
\(460\) −2.68205e99 −0.375879
\(461\) −1.31364e100 −1.69704 −0.848521 0.529162i \(-0.822507\pi\)
−0.848521 + 0.529162i \(0.822507\pi\)
\(462\) 0 0
\(463\) 7.77164e99 0.853543 0.426772 0.904359i \(-0.359651\pi\)
0.426772 + 0.904359i \(0.359651\pi\)
\(464\) −5.42690e99 −0.549702
\(465\) 0 0
\(466\) −1.33577e100 −1.15149
\(467\) −7.55574e99 −0.601027 −0.300513 0.953778i \(-0.597158\pi\)
−0.300513 + 0.953778i \(0.597158\pi\)
\(468\) 0 0
\(469\) 1.83073e100 1.24063
\(470\) 6.69731e98 0.0419016
\(471\) 0 0
\(472\) 8.33187e98 0.0444547
\(473\) 4.05514e99 0.199854
\(474\) 0 0
\(475\) 3.46596e99 0.145819
\(476\) −2.89064e100 −1.12391
\(477\) 0 0
\(478\) −1.86167e100 −0.618524
\(479\) −3.90184e100 −1.19864 −0.599319 0.800510i \(-0.704562\pi\)
−0.599319 + 0.800510i \(0.704562\pi\)
\(480\) 0 0
\(481\) 8.33100e99 0.218905
\(482\) 4.58562e100 1.11464
\(483\) 0 0
\(484\) −8.60447e100 −1.79070
\(485\) 8.75075e99 0.168550
\(486\) 0 0
\(487\) 1.26161e100 0.208251 0.104126 0.994564i \(-0.466796\pi\)
0.104126 + 0.994564i \(0.466796\pi\)
\(488\) −6.21752e100 −0.950328
\(489\) 0 0
\(490\) −8.62061e99 −0.113028
\(491\) −6.80665e99 −0.0826756 −0.0413378 0.999145i \(-0.513162\pi\)
−0.0413378 + 0.999145i \(0.513162\pi\)
\(492\) 0 0
\(493\) 7.38598e100 0.770280
\(494\) 8.67290e99 0.0838308
\(495\) 0 0
\(496\) 7.13374e100 0.592590
\(497\) −1.04508e101 −0.804977
\(498\) 0 0
\(499\) −2.15058e100 −0.142490 −0.0712451 0.997459i \(-0.522697\pi\)
−0.0712451 + 0.997459i \(0.522697\pi\)
\(500\) −1.52671e101 −0.938383
\(501\) 0 0
\(502\) 3.31764e100 0.175566
\(503\) 1.06452e101 0.522824 0.261412 0.965227i \(-0.415812\pi\)
0.261412 + 0.965227i \(0.415812\pi\)
\(504\) 0 0
\(505\) 6.77645e100 0.286798
\(506\) 4.58989e100 0.180367
\(507\) 0 0
\(508\) 1.34637e101 0.456327
\(509\) 4.78996e101 1.50805 0.754023 0.656848i \(-0.228111\pi\)
0.754023 + 0.656848i \(0.228111\pi\)
\(510\) 0 0
\(511\) −1.73946e101 −0.472749
\(512\) −3.86756e101 −0.976817
\(513\) 0 0
\(514\) −5.92423e101 −1.29275
\(515\) −2.19815e100 −0.0445950
\(516\) 0 0
\(517\) −7.40554e99 −0.0129916
\(518\) −6.13772e101 −1.00148
\(519\) 0 0
\(520\) −8.32219e100 −0.117520
\(521\) −8.84088e101 −1.16166 −0.580832 0.814023i \(-0.697273\pi\)
−0.580832 + 0.814023i \(0.697273\pi\)
\(522\) 0 0
\(523\) −1.09779e100 −0.0124942 −0.00624709 0.999980i \(-0.501989\pi\)
−0.00624709 + 0.999980i \(0.501989\pi\)
\(524\) 1.71713e101 0.181920
\(525\) 0 0
\(526\) −1.13534e102 −1.04270
\(527\) −9.70899e101 −0.830378
\(528\) 0 0
\(529\) −5.43052e101 −0.402952
\(530\) 1.09877e102 0.759559
\(531\) 0 0
\(532\) −4.12853e101 −0.247804
\(533\) 5.99435e101 0.335330
\(534\) 0 0
\(535\) 3.59456e101 0.174734
\(536\) 4.39531e102 1.99209
\(537\) 0 0
\(538\) 4.38350e102 1.72777
\(539\) 9.53224e100 0.0350442
\(540\) 0 0
\(541\) 4.54819e101 0.145526 0.0727632 0.997349i \(-0.476818\pi\)
0.0727632 + 0.997349i \(0.476818\pi\)
\(542\) 7.58152e102 2.26351
\(543\) 0 0
\(544\) 1.46288e102 0.380408
\(545\) 1.79688e102 0.436165
\(546\) 0 0
\(547\) 3.65619e102 0.773577 0.386789 0.922168i \(-0.373584\pi\)
0.386789 + 0.922168i \(0.373584\pi\)
\(548\) 1.12100e103 2.21479
\(549\) 0 0
\(550\) 1.25829e102 0.216860
\(551\) 1.05490e102 0.169834
\(552\) 0 0
\(553\) 3.93375e101 0.0552859
\(554\) −1.87288e103 −2.45976
\(555\) 0 0
\(556\) 1.61167e103 1.84915
\(557\) 4.10459e102 0.440251 0.220126 0.975472i \(-0.429353\pi\)
0.220126 + 0.975472i \(0.429353\pi\)
\(558\) 0 0
\(559\) 4.87661e102 0.457273
\(560\) 1.33459e102 0.117030
\(561\) 0 0
\(562\) 2.11335e103 1.62129
\(563\) 2.03919e102 0.146350 0.0731751 0.997319i \(-0.476687\pi\)
0.0731751 + 0.997319i \(0.476687\pi\)
\(564\) 0 0
\(565\) 6.66665e101 0.0418881
\(566\) −2.42077e103 −1.42344
\(567\) 0 0
\(568\) −2.50908e103 −1.29256
\(569\) 1.21675e103 0.586803 0.293401 0.955989i \(-0.405213\pi\)
0.293401 + 0.955989i \(0.405213\pi\)
\(570\) 0 0
\(571\) −3.95470e103 −1.67209 −0.836044 0.548663i \(-0.815137\pi\)
−0.836044 + 0.548663i \(0.815137\pi\)
\(572\) 2.03443e102 0.0805547
\(573\) 0 0
\(574\) −4.41623e103 −1.53411
\(575\) 2.20584e103 0.717844
\(576\) 0 0
\(577\) −2.30970e103 −0.659877 −0.329938 0.944002i \(-0.607028\pi\)
−0.329938 + 0.944002i \(0.607028\pi\)
\(578\) −3.09678e103 −0.829118
\(579\) 0 0
\(580\) −2.23785e103 −0.526357
\(581\) 1.18137e101 0.00260483
\(582\) 0 0
\(583\) −1.21496e103 −0.235501
\(584\) −4.17619e103 −0.759099
\(585\) 0 0
\(586\) 3.63658e103 0.581478
\(587\) 4.27950e103 0.641896 0.320948 0.947097i \(-0.395999\pi\)
0.320948 + 0.947097i \(0.395999\pi\)
\(588\) 0 0
\(589\) −1.38668e103 −0.183085
\(590\) 1.15745e102 0.0143401
\(591\) 0 0
\(592\) −3.20755e103 −0.350033
\(593\) −5.73284e103 −0.587245 −0.293622 0.955922i \(-0.594861\pi\)
−0.293622 + 0.955922i \(0.594861\pi\)
\(594\) 0 0
\(595\) −1.81637e103 −0.163990
\(596\) −2.37908e104 −2.01685
\(597\) 0 0
\(598\) 5.51970e103 0.412687
\(599\) 8.09810e103 0.568692 0.284346 0.958722i \(-0.408224\pi\)
0.284346 + 0.958722i \(0.408224\pi\)
\(600\) 0 0
\(601\) 1.48364e103 0.0919469 0.0459735 0.998943i \(-0.485361\pi\)
0.0459735 + 0.998943i \(0.485361\pi\)
\(602\) −3.59276e104 −2.09199
\(603\) 0 0
\(604\) −2.05821e104 −1.05829
\(605\) −5.40674e103 −0.261282
\(606\) 0 0
\(607\) 3.19159e104 1.36280 0.681400 0.731912i \(-0.261372\pi\)
0.681400 + 0.731912i \(0.261372\pi\)
\(608\) 2.08934e103 0.0838735
\(609\) 0 0
\(610\) −8.63725e103 −0.306554
\(611\) −8.90574e102 −0.0297252
\(612\) 0 0
\(613\) 1.90887e104 0.563650 0.281825 0.959466i \(-0.409060\pi\)
0.281825 + 0.959466i \(0.409060\pi\)
\(614\) 1.81079e104 0.502985
\(615\) 0 0
\(616\) −6.77961e103 −0.166697
\(617\) 4.16335e104 0.963273 0.481636 0.876371i \(-0.340043\pi\)
0.481636 + 0.876371i \(0.340043\pi\)
\(618\) 0 0
\(619\) 5.63359e104 1.15448 0.577239 0.816575i \(-0.304130\pi\)
0.577239 + 0.816575i \(0.304130\pi\)
\(620\) 2.94168e104 0.567424
\(621\) 0 0
\(622\) 8.23787e104 1.40824
\(623\) −8.35197e104 −1.34427
\(624\) 0 0
\(625\) 5.54985e104 0.792101
\(626\) 9.62511e104 1.29380
\(627\) 0 0
\(628\) −1.63341e105 −1.94809
\(629\) 4.36546e104 0.490490
\(630\) 0 0
\(631\) −4.77847e104 −0.476637 −0.238319 0.971187i \(-0.576596\pi\)
−0.238319 + 0.971187i \(0.576596\pi\)
\(632\) 9.44436e103 0.0887733
\(633\) 0 0
\(634\) −3.38330e105 −2.82484
\(635\) 8.46013e103 0.0665828
\(636\) 0 0
\(637\) 1.14632e104 0.0801824
\(638\) 3.82971e104 0.252575
\(639\) 0 0
\(640\) −8.31187e104 −0.487472
\(641\) 1.19877e104 0.0663069 0.0331534 0.999450i \(-0.489445\pi\)
0.0331534 + 0.999450i \(0.489445\pi\)
\(642\) 0 0
\(643\) 2.25954e105 1.11201 0.556006 0.831178i \(-0.312333\pi\)
0.556006 + 0.831178i \(0.312333\pi\)
\(644\) −2.62752e105 −1.21991
\(645\) 0 0
\(646\) 4.54462e104 0.187836
\(647\) −3.62715e103 −0.0141467 −0.00707334 0.999975i \(-0.502252\pi\)
−0.00707334 + 0.999975i \(0.502252\pi\)
\(648\) 0 0
\(649\) −1.27985e103 −0.00444613
\(650\) 1.51319e105 0.496183
\(651\) 0 0
\(652\) 3.95835e105 1.15672
\(653\) −3.74683e105 −1.03376 −0.516880 0.856058i \(-0.672907\pi\)
−0.516880 + 0.856058i \(0.672907\pi\)
\(654\) 0 0
\(655\) 1.07899e104 0.0265440
\(656\) −2.30791e105 −0.536197
\(657\) 0 0
\(658\) 6.56115e104 0.135991
\(659\) 8.83130e105 1.72911 0.864553 0.502542i \(-0.167602\pi\)
0.864553 + 0.502542i \(0.167602\pi\)
\(660\) 0 0
\(661\) 3.25607e105 0.569036 0.284518 0.958671i \(-0.408166\pi\)
0.284518 + 0.958671i \(0.408166\pi\)
\(662\) 1.40888e106 2.32647
\(663\) 0 0
\(664\) 2.83629e103 0.00418261
\(665\) −2.59422e104 −0.0361572
\(666\) 0 0
\(667\) 6.71368e105 0.836068
\(668\) 5.15455e105 0.606839
\(669\) 0 0
\(670\) 6.10588e105 0.642603
\(671\) 9.55064e104 0.0950469
\(672\) 0 0
\(673\) 1.25330e106 1.11555 0.557777 0.829991i \(-0.311654\pi\)
0.557777 + 0.829991i \(0.311654\pi\)
\(674\) 1.41313e106 1.18970
\(675\) 0 0
\(676\) −2.17905e106 −1.64160
\(677\) 1.53726e106 1.09565 0.547824 0.836594i \(-0.315456\pi\)
0.547824 + 0.836594i \(0.315456\pi\)
\(678\) 0 0
\(679\) 8.57284e105 0.547026
\(680\) −4.36084e105 −0.263322
\(681\) 0 0
\(682\) −5.03422e105 −0.272281
\(683\) −1.70977e106 −0.875311 −0.437655 0.899143i \(-0.644191\pi\)
−0.437655 + 0.899143i \(0.644191\pi\)
\(684\) 0 0
\(685\) 7.04394e105 0.323161
\(686\) −4.19092e106 −1.82035
\(687\) 0 0
\(688\) −1.87756e106 −0.731186
\(689\) −1.46109e106 −0.538835
\(690\) 0 0
\(691\) 4.55165e106 1.50571 0.752856 0.658185i \(-0.228676\pi\)
0.752856 + 0.658185i \(0.228676\pi\)
\(692\) −5.37230e106 −1.68338
\(693\) 0 0
\(694\) −1.07287e107 −3.01693
\(695\) 1.01271e106 0.269809
\(696\) 0 0
\(697\) 3.14105e106 0.751356
\(698\) −7.46487e106 −1.69217
\(699\) 0 0
\(700\) −7.20317e106 −1.46672
\(701\) 6.86314e106 1.32464 0.662319 0.749222i \(-0.269573\pi\)
0.662319 + 0.749222i \(0.269573\pi\)
\(702\) 0 0
\(703\) 6.23492e105 0.108145
\(704\) 1.18750e106 0.195281
\(705\) 0 0
\(706\) 7.47932e106 1.10582
\(707\) 6.63868e106 0.930795
\(708\) 0 0
\(709\) −4.47111e105 −0.0563873 −0.0281936 0.999602i \(-0.508976\pi\)
−0.0281936 + 0.999602i \(0.508976\pi\)
\(710\) −3.48556e106 −0.416950
\(711\) 0 0
\(712\) −2.00519e107 −2.15851
\(713\) −8.82524e106 −0.901299
\(714\) 0 0
\(715\) 1.27836e105 0.0117538
\(716\) −2.70049e107 −2.35616
\(717\) 0 0
\(718\) 1.12386e107 0.883174
\(719\) 4.46581e106 0.333094 0.166547 0.986034i \(-0.446738\pi\)
0.166547 + 0.986034i \(0.446738\pi\)
\(720\) 0 0
\(721\) −2.15346e106 −0.144732
\(722\) −2.56975e107 −1.63963
\(723\) 0 0
\(724\) −3.04094e107 −1.74909
\(725\) 1.84051e107 1.00522
\(726\) 0 0
\(727\) 7.07466e106 0.348470 0.174235 0.984704i \(-0.444255\pi\)
0.174235 + 0.984704i \(0.444255\pi\)
\(728\) −8.15300e106 −0.381409
\(729\) 0 0
\(730\) −5.80148e106 −0.244868
\(731\) 2.55535e107 1.02459
\(732\) 0 0
\(733\) 4.53195e107 1.64016 0.820080 0.572249i \(-0.193929\pi\)
0.820080 + 0.572249i \(0.193929\pi\)
\(734\) 6.38035e107 2.19403
\(735\) 0 0
\(736\) 1.32972e107 0.412897
\(737\) −6.75157e106 −0.199239
\(738\) 0 0
\(739\) −3.01848e107 −0.804679 −0.402339 0.915491i \(-0.631803\pi\)
−0.402339 + 0.915491i \(0.631803\pi\)
\(740\) −1.32267e107 −0.335168
\(741\) 0 0
\(742\) 1.07643e108 2.46513
\(743\) 6.62438e107 1.44233 0.721165 0.692764i \(-0.243607\pi\)
0.721165 + 0.692764i \(0.243607\pi\)
\(744\) 0 0
\(745\) −1.49493e107 −0.294279
\(746\) −1.02667e106 −0.0192188
\(747\) 0 0
\(748\) 1.06604e107 0.180495
\(749\) 3.52148e107 0.567096
\(750\) 0 0
\(751\) 1.02618e108 1.49529 0.747645 0.664099i \(-0.231185\pi\)
0.747645 + 0.664099i \(0.231185\pi\)
\(752\) 3.42883e106 0.0475310
\(753\) 0 0
\(754\) 4.60553e107 0.577901
\(755\) −1.29331e107 −0.154415
\(756\) 0 0
\(757\) −8.55357e107 −0.924808 −0.462404 0.886669i \(-0.653013\pi\)
−0.462404 + 0.886669i \(0.653013\pi\)
\(758\) −3.84618e107 −0.395762
\(759\) 0 0
\(760\) −6.22833e106 −0.0580580
\(761\) −1.22517e108 −1.08711 −0.543553 0.839375i \(-0.682921\pi\)
−0.543553 + 0.839375i \(0.682921\pi\)
\(762\) 0 0
\(763\) 1.76035e108 1.41556
\(764\) 1.75782e108 1.34578
\(765\) 0 0
\(766\) −3.55902e108 −2.47031
\(767\) −1.53911e106 −0.0101729
\(768\) 0 0
\(769\) 1.12833e108 0.676393 0.338197 0.941075i \(-0.390183\pi\)
0.338197 + 0.941075i \(0.390183\pi\)
\(770\) −9.41810e106 −0.0537726
\(771\) 0 0
\(772\) −4.77426e108 −2.47319
\(773\) −2.94129e107 −0.145147 −0.0725734 0.997363i \(-0.523121\pi\)
−0.0725734 + 0.997363i \(0.523121\pi\)
\(774\) 0 0
\(775\) −2.41938e108 −1.08365
\(776\) 2.05821e108 0.878367
\(777\) 0 0
\(778\) −5.84801e107 −0.226607
\(779\) 4.48618e107 0.165662
\(780\) 0 0
\(781\) 3.85416e107 0.129275
\(782\) 2.89233e108 0.924687
\(783\) 0 0
\(784\) −4.41351e107 −0.128213
\(785\) −1.02638e108 −0.284246
\(786\) 0 0
\(787\) 7.86662e108 1.98032 0.990161 0.139936i \(-0.0446896\pi\)
0.990161 + 0.139936i \(0.0446896\pi\)
\(788\) −7.99152e108 −1.91821
\(789\) 0 0
\(790\) 1.31199e107 0.0286362
\(791\) 6.53111e107 0.135947
\(792\) 0 0
\(793\) 1.14854e108 0.217471
\(794\) 8.96061e108 1.61834
\(795\) 0 0
\(796\) 1.91860e109 3.15314
\(797\) −9.68204e107 −0.151803 −0.0759014 0.997115i \(-0.524183\pi\)
−0.0759014 + 0.997115i \(0.524183\pi\)
\(798\) 0 0
\(799\) −4.66662e107 −0.0666037
\(800\) 3.64534e108 0.496436
\(801\) 0 0
\(802\) −1.66104e109 −2.05988
\(803\) 6.41499e107 0.0759212
\(804\) 0 0
\(805\) −1.65104e108 −0.177997
\(806\) −6.05404e108 −0.622990
\(807\) 0 0
\(808\) 1.59385e109 1.49459
\(809\) −1.25823e109 −1.12640 −0.563199 0.826321i \(-0.690430\pi\)
−0.563199 + 0.826321i \(0.690430\pi\)
\(810\) 0 0
\(811\) −1.83325e109 −1.49603 −0.748014 0.663683i \(-0.768992\pi\)
−0.748014 + 0.663683i \(0.768992\pi\)
\(812\) −2.19235e109 −1.70828
\(813\) 0 0
\(814\) 2.26354e108 0.160832
\(815\) 2.48728e108 0.168778
\(816\) 0 0
\(817\) 3.64966e108 0.225905
\(818\) −1.06114e109 −0.627368
\(819\) 0 0
\(820\) −9.51694e108 −0.513426
\(821\) −5.30578e108 −0.273452 −0.136726 0.990609i \(-0.543658\pi\)
−0.136726 + 0.990609i \(0.543658\pi\)
\(822\) 0 0
\(823\) −4.35738e107 −0.0204989 −0.0102495 0.999947i \(-0.503263\pi\)
−0.0102495 + 0.999947i \(0.503263\pi\)
\(824\) −5.17014e108 −0.232398
\(825\) 0 0
\(826\) 1.13392e108 0.0465403
\(827\) −9.43799e107 −0.0370188 −0.0185094 0.999829i \(-0.505892\pi\)
−0.0185094 + 0.999829i \(0.505892\pi\)
\(828\) 0 0
\(829\) −4.56308e109 −1.63480 −0.817398 0.576073i \(-0.804584\pi\)
−0.817398 + 0.576073i \(0.804584\pi\)
\(830\) 3.94012e106 0.00134921
\(831\) 0 0
\(832\) 1.42806e109 0.446809
\(833\) 6.00676e108 0.179661
\(834\) 0 0
\(835\) 3.23893e108 0.0885440
\(836\) 1.52257e108 0.0397961
\(837\) 0 0
\(838\) −1.14579e110 −2.73813
\(839\) −7.28934e108 −0.166577 −0.0832884 0.996525i \(-0.526542\pi\)
−0.0832884 + 0.996525i \(0.526542\pi\)
\(840\) 0 0
\(841\) 8.17112e108 0.170778
\(842\) 5.51846e109 1.10310
\(843\) 0 0
\(844\) 4.28707e109 0.784006
\(845\) −1.36924e109 −0.239526
\(846\) 0 0
\(847\) −5.29682e109 −0.847983
\(848\) 5.62539e109 0.861605
\(849\) 0 0
\(850\) 7.92913e109 1.11177
\(851\) 3.96810e109 0.532382
\(852\) 0 0
\(853\) −1.01292e110 −1.24448 −0.622240 0.782827i \(-0.713777\pi\)
−0.622240 + 0.782827i \(0.713777\pi\)
\(854\) −8.46165e109 −0.994913
\(855\) 0 0
\(856\) 8.45455e109 0.910593
\(857\) −1.65427e110 −1.70540 −0.852698 0.522405i \(-0.825035\pi\)
−0.852698 + 0.522405i \(0.825035\pi\)
\(858\) 0 0
\(859\) 5.68954e109 0.537444 0.268722 0.963218i \(-0.413399\pi\)
0.268722 + 0.963218i \(0.413399\pi\)
\(860\) −7.74236e109 −0.700134
\(861\) 0 0
\(862\) 3.25283e110 2.69612
\(863\) 9.44371e108 0.0749442 0.0374721 0.999298i \(-0.488069\pi\)
0.0374721 + 0.999298i \(0.488069\pi\)
\(864\) 0 0
\(865\) −3.37576e109 −0.245622
\(866\) 2.01386e110 1.40316
\(867\) 0 0
\(868\) 2.88188e110 1.84156
\(869\) −1.45073e108 −0.00887864
\(870\) 0 0
\(871\) −8.11929e109 −0.455866
\(872\) 4.22634e110 2.27298
\(873\) 0 0
\(874\) 4.13095e109 0.203878
\(875\) −9.39823e109 −0.444370
\(876\) 0 0
\(877\) −1.65254e110 −0.717246 −0.358623 0.933482i \(-0.616754\pi\)
−0.358623 + 0.933482i \(0.616754\pi\)
\(878\) 5.40972e110 2.24973
\(879\) 0 0
\(880\) −4.92186e108 −0.0187944
\(881\) 3.05928e110 1.11950 0.559748 0.828663i \(-0.310898\pi\)
0.559748 + 0.828663i \(0.310898\pi\)
\(882\) 0 0
\(883\) −1.03341e110 −0.347334 −0.173667 0.984804i \(-0.555562\pi\)
−0.173667 + 0.984804i \(0.555562\pi\)
\(884\) 1.28200e110 0.412979
\(885\) 0 0
\(886\) 2.21256e109 0.0654834
\(887\) 4.02103e110 1.14078 0.570390 0.821374i \(-0.306792\pi\)
0.570390 + 0.821374i \(0.306792\pi\)
\(888\) 0 0
\(889\) 8.28813e109 0.216093
\(890\) −2.78556e110 −0.696286
\(891\) 0 0
\(892\) 7.86467e110 1.80717
\(893\) −6.66506e108 −0.0146850
\(894\) 0 0
\(895\) −1.69689e110 −0.343788
\(896\) −8.14288e110 −1.58208
\(897\) 0 0
\(898\) −8.66195e110 −1.54794
\(899\) −7.36360e110 −1.26212
\(900\) 0 0
\(901\) −7.65613e110 −1.20734
\(902\) 1.62867e110 0.246370
\(903\) 0 0
\(904\) 1.56802e110 0.218291
\(905\) −1.91082e110 −0.255210
\(906\) 0 0
\(907\) −1.12894e111 −1.38803 −0.694013 0.719963i \(-0.744159\pi\)
−0.694013 + 0.719963i \(0.744159\pi\)
\(908\) −1.78328e111 −2.10377
\(909\) 0 0
\(910\) −1.13260e110 −0.123034
\(911\) 1.25185e110 0.130501 0.0652506 0.997869i \(-0.479215\pi\)
0.0652506 + 0.997869i \(0.479215\pi\)
\(912\) 0 0
\(913\) −4.35678e107 −0.000418323 0
\(914\) −1.58266e111 −1.45850
\(915\) 0 0
\(916\) 4.27910e110 0.363306
\(917\) 1.05705e110 0.0861480
\(918\) 0 0
\(919\) 1.73300e111 1.30157 0.650785 0.759262i \(-0.274440\pi\)
0.650785 + 0.759262i \(0.274440\pi\)
\(920\) −3.96390e110 −0.285811
\(921\) 0 0
\(922\) −4.29221e111 −2.85280
\(923\) 4.63492e110 0.295787
\(924\) 0 0
\(925\) 1.08783e111 0.640095
\(926\) 2.53932e111 1.43484
\(927\) 0 0
\(928\) 1.10949e111 0.578196
\(929\) −6.51740e110 −0.326201 −0.163100 0.986609i \(-0.552149\pi\)
−0.163100 + 0.986609i \(0.552149\pi\)
\(930\) 0 0
\(931\) 8.57910e109 0.0396122
\(932\) −2.82005e111 −1.25072
\(933\) 0 0
\(934\) −2.46877e111 −1.01035
\(935\) 6.69863e109 0.0263360
\(936\) 0 0
\(937\) 5.15863e111 1.87197 0.935987 0.352035i \(-0.114510\pi\)
0.935987 + 0.352035i \(0.114510\pi\)
\(938\) 5.98174e111 2.08555
\(939\) 0 0
\(940\) 1.41392e110 0.0455125
\(941\) −5.35193e110 −0.165539 −0.0827694 0.996569i \(-0.526376\pi\)
−0.0827694 + 0.996569i \(0.526376\pi\)
\(942\) 0 0
\(943\) 2.85514e111 0.815528
\(944\) 5.92580e109 0.0162666
\(945\) 0 0
\(946\) 1.32498e111 0.335963
\(947\) −6.35901e111 −1.54976 −0.774881 0.632107i \(-0.782190\pi\)
−0.774881 + 0.632107i \(0.782190\pi\)
\(948\) 0 0
\(949\) 7.71452e110 0.173710
\(950\) 1.13247e111 0.245127
\(951\) 0 0
\(952\) −4.27219e111 −0.854603
\(953\) 2.74523e111 0.527952 0.263976 0.964529i \(-0.414966\pi\)
0.263976 + 0.964529i \(0.414966\pi\)
\(954\) 0 0
\(955\) 1.10455e111 0.196363
\(956\) −3.93032e111 −0.671825
\(957\) 0 0
\(958\) −1.27489e112 −2.01496
\(959\) 6.90073e111 1.04881
\(960\) 0 0
\(961\) 2.56535e111 0.360595
\(962\) 2.72208e111 0.367990
\(963\) 0 0
\(964\) 9.68106e111 1.21069
\(965\) −2.99997e111 −0.360864
\(966\) 0 0
\(967\) −6.60617e111 −0.735287 −0.367643 0.929967i \(-0.619835\pi\)
−0.367643 + 0.929967i \(0.619835\pi\)
\(968\) −1.27169e112 −1.36162
\(969\) 0 0
\(970\) 2.85923e111 0.283341
\(971\) −1.03596e112 −0.987695 −0.493847 0.869549i \(-0.664410\pi\)
−0.493847 + 0.869549i \(0.664410\pi\)
\(972\) 0 0
\(973\) 9.92126e111 0.875659
\(974\) 4.12219e111 0.350080
\(975\) 0 0
\(976\) −4.42203e111 −0.347739
\(977\) 2.05932e112 1.55839 0.779197 0.626779i \(-0.215627\pi\)
0.779197 + 0.626779i \(0.215627\pi\)
\(978\) 0 0
\(979\) 3.08014e111 0.215883
\(980\) −1.81996e111 −0.122768
\(981\) 0 0
\(982\) −2.22401e111 −0.138981
\(983\) 9.74562e111 0.586209 0.293105 0.956080i \(-0.405312\pi\)
0.293105 + 0.956080i \(0.405312\pi\)
\(984\) 0 0
\(985\) −5.02158e111 −0.279886
\(986\) 2.41330e112 1.29487
\(987\) 0 0
\(988\) 1.83100e111 0.0910550
\(989\) 2.32276e112 1.11210
\(990\) 0 0
\(991\) 2.65057e112 1.17646 0.588230 0.808694i \(-0.299825\pi\)
0.588230 + 0.808694i \(0.299825\pi\)
\(992\) −1.45844e112 −0.623307
\(993\) 0 0
\(994\) −3.41470e112 −1.35320
\(995\) 1.20558e112 0.460075
\(996\) 0 0
\(997\) −2.01612e112 −0.713587 −0.356793 0.934183i \(-0.616130\pi\)
−0.356793 + 0.934183i \(0.616130\pi\)
\(998\) −7.02683e111 −0.239532
\(999\) 0 0
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 9.76.a.c.1.6 6
3.2 odd 2 1.76.a.a.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.76.a.a.1.1 6 3.2 odd 2
9.76.a.c.1.6 6 1.1 even 1 trivial