Properties

Label 25.34.b.c.24.8
Level $25$
Weight $34$
Character 25.24
Analytic conductor $172.457$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 20265063301 x^{10} + 151877130254993985540 x^{8} + 529361026408998369955851761920 x^{6} + 862054500541443268723145801151954288640 x^{4} + 544606882049987247786598582768552479276511789056 x^{2} + 39700341273686227371481766034965422500156905479886340096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{10}\cdot 5^{36}\cdot 7^{2}\cdot 11^{4} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.8
Root \(38204.6i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.34.b.c.24.5

$q$-expansion

\(f(q)\) \(=\) \(q+76409.2i q^{2} -7.43191e7i q^{3} +2.75157e9 q^{4} +5.67866e12 q^{6} +1.36987e14i q^{7} +8.66595e14i q^{8} +3.57381e13 q^{9} +O(q^{10})\) \(q+76409.2i q^{2} -7.43191e7i q^{3} +2.75157e9 q^{4} +5.67866e12 q^{6} +1.36987e14i q^{7} +8.66595e14i q^{8} +3.57381e13 q^{9} +3.84389e16 q^{11} -2.04494e17i q^{12} -2.57012e18i q^{13} -1.04671e19 q^{14} -4.25800e19 q^{16} +1.44621e20i q^{17} +2.73072e18i q^{18} +2.79512e20 q^{19} +1.01808e22 q^{21} +2.93708e21i q^{22} -3.71118e22i q^{23} +6.44045e22 q^{24} +1.96381e23 q^{26} -4.15800e23i q^{27} +3.76930e23i q^{28} +1.90470e24 q^{29} +5.88152e24 q^{31} +4.19049e24i q^{32} -2.85674e24i q^{33} -1.10503e25 q^{34} +9.83358e22 q^{36} +1.05655e26i q^{37} +2.13573e25i q^{38} -1.91009e26 q^{39} -2.64164e25 q^{41} +7.77905e26i q^{42} -1.20441e27i q^{43} +1.05767e26 q^{44} +2.83569e27 q^{46} -3.53827e27i q^{47} +3.16451e27i q^{48} -1.10345e28 q^{49} +1.07481e28 q^{51} -7.07187e27i q^{52} +3.25098e28i q^{53} +3.17710e28 q^{54} -1.18713e29 q^{56} -2.07731e28i q^{57} +1.45537e29i q^{58} -3.41770e28 q^{59} -6.01152e28 q^{61} +4.49402e29i q^{62} +4.89566e27i q^{63} -6.85952e29 q^{64} +2.18281e29 q^{66} -2.17962e30i q^{67} +3.97934e29i q^{68} -2.75812e30 q^{69} +9.20055e29 q^{71} +3.09704e28i q^{72} +4.72873e30i q^{73} -8.07302e30 q^{74} +7.69097e29 q^{76} +5.26564e30i q^{77} -1.45948e31i q^{78} +2.14123e31 q^{79} -3.07032e31 q^{81} -2.01845e30i q^{82} +6.60838e31i q^{83} +2.80131e31 q^{84} +9.20283e31 q^{86} -1.41556e32i q^{87} +3.33109e31i q^{88} +6.08523e31 q^{89} +3.52074e32 q^{91} -1.02116e32i q^{92} -4.37109e32i q^{93} +2.70356e32 q^{94} +3.11433e32 q^{96} +6.12049e30i q^{97} -8.43141e32i q^{98} +1.37373e30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + O(q^{10}) \) \( 12q - 59041291304q^{4} + 13231518498704q^{6} - 27204205166690876q^{9} + 301521793111780384q^{11} + 14172220017979782912q^{14} + \)\(12\!\cdots\!32\)\(q^{16} - \)\(34\!\cdots\!00\)\(q^{19} - \)\(21\!\cdots\!56\)\(q^{21} - \)\(42\!\cdots\!00\)\(q^{24} - \)\(72\!\cdots\!76\)\(q^{26} - \)\(26\!\cdots\!00\)\(q^{29} + \)\(21\!\cdots\!04\)\(q^{31} - \)\(17\!\cdots\!48\)\(q^{34} + \)\(33\!\cdots\!92\)\(q^{36} - \)\(11\!\cdots\!12\)\(q^{39} + \)\(89\!\cdots\!64\)\(q^{41} - \)\(14\!\cdots\!28\)\(q^{44} + \)\(69\!\cdots\!44\)\(q^{46} + \)\(14\!\cdots\!16\)\(q^{49} - \)\(10\!\cdots\!76\)\(q^{51} + \)\(36\!\cdots\!00\)\(q^{54} - \)\(78\!\cdots\!00\)\(q^{56} + \)\(35\!\cdots\!00\)\(q^{59} - \)\(56\!\cdots\!16\)\(q^{61} - \)\(38\!\cdots\!44\)\(q^{64} + \)\(10\!\cdots\!28\)\(q^{66} - \)\(63\!\cdots\!72\)\(q^{69} + \)\(16\!\cdots\!44\)\(q^{71} - \)\(40\!\cdots\!68\)\(q^{74} + \)\(58\!\cdots\!00\)\(q^{76} + \)\(44\!\cdots\!00\)\(q^{79} - \)\(26\!\cdots\!48\)\(q^{81} + \)\(47\!\cdots\!52\)\(q^{84} - \)\(25\!\cdots\!16\)\(q^{86} + \)\(67\!\cdots\!00\)\(q^{89} + \)\(38\!\cdots\!64\)\(q^{91} - \)\(29\!\cdots\!28\)\(q^{94} + \)\(63\!\cdots\!44\)\(q^{96} - \)\(33\!\cdots\!32\)\(q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 76409.2i 0.824424i 0.911088 + 0.412212i \(0.135244\pi\)
−0.911088 + 0.412212i \(0.864756\pi\)
\(3\) − 7.43191e7i − 0.996780i −0.866953 0.498390i \(-0.833925\pi\)
0.866953 0.498390i \(-0.166075\pi\)
\(4\) 2.75157e9 0.320325
\(5\) 0 0
\(6\) 5.67866e12 0.821770
\(7\) 1.36987e14i 1.55798i 0.627034 + 0.778992i \(0.284269\pi\)
−0.627034 + 0.778992i \(0.715731\pi\)
\(8\) 8.66595e14i 1.08851i
\(9\) 3.57381e13 0.00642880
\(10\) 0 0
\(11\) 3.84389e16 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(12\) − 2.04494e17i − 0.319294i
\(13\) − 2.57012e18i − 1.07124i −0.844458 0.535622i \(-0.820077\pi\)
0.844458 0.535622i \(-0.179923\pi\)
\(14\) −1.04671e19 −1.28444
\(15\) 0 0
\(16\) −4.25800e19 −0.577067
\(17\) 1.44621e20i 0.720814i 0.932795 + 0.360407i \(0.117362\pi\)
−0.932795 + 0.360407i \(0.882638\pi\)
\(18\) 2.73072e18i 0.00530005i
\(19\) 2.79512e20 0.222314 0.111157 0.993803i \(-0.464544\pi\)
0.111157 + 0.993803i \(0.464544\pi\)
\(20\) 0 0
\(21\) 1.01808e22 1.55297
\(22\) 2.93708e21i 0.207942i
\(23\) − 3.71118e22i − 1.26184i −0.775849 0.630918i \(-0.782678\pi\)
0.775849 0.630918i \(-0.217322\pi\)
\(24\) 6.44045e22 1.08500
\(25\) 0 0
\(26\) 1.96381e23 0.883159
\(27\) − 4.15800e23i − 1.00319i
\(28\) 3.76930e23i 0.499061i
\(29\) 1.90470e24 1.41339 0.706693 0.707520i \(-0.250186\pi\)
0.706693 + 0.707520i \(0.250186\pi\)
\(30\) 0 0
\(31\) 5.88152e24 1.45218 0.726092 0.687597i \(-0.241334\pi\)
0.726092 + 0.687597i \(0.241334\pi\)
\(32\) 4.19049e24i 0.612760i
\(33\) − 2.85674e24i − 0.251415i
\(34\) −1.10503e25 −0.594256
\(35\) 0 0
\(36\) 9.83358e22 0.00205930
\(37\) 1.05655e26i 1.40787i 0.710265 + 0.703934i \(0.248575\pi\)
−0.710265 + 0.703934i \(0.751425\pi\)
\(38\) 2.13573e25i 0.183281i
\(39\) −1.91009e26 −1.06779
\(40\) 0 0
\(41\) −2.64164e25 −0.0647052 −0.0323526 0.999477i \(-0.510300\pi\)
−0.0323526 + 0.999477i \(0.510300\pi\)
\(42\) 7.77905e26i 1.28030i
\(43\) − 1.20441e27i − 1.34446i −0.740344 0.672228i \(-0.765338\pi\)
0.740344 0.672228i \(-0.234662\pi\)
\(44\) 1.05767e26 0.0807946
\(45\) 0 0
\(46\) 2.83569e27 1.04029
\(47\) − 3.53827e27i − 0.910282i −0.890419 0.455141i \(-0.849589\pi\)
0.890419 0.455141i \(-0.150411\pi\)
\(48\) 3.16451e27i 0.575209i
\(49\) −1.10345e28 −1.42731
\(50\) 0 0
\(51\) 1.07481e28 0.718493
\(52\) − 7.07187e27i − 0.343146i
\(53\) 3.25098e28i 1.15203i 0.817440 + 0.576014i \(0.195392\pi\)
−0.817440 + 0.576014i \(0.804608\pi\)
\(54\) 3.17710e28 0.827053
\(55\) 0 0
\(56\) −1.18713e29 −1.69588
\(57\) − 2.07731e28i − 0.221598i
\(58\) 1.45537e29i 1.16523i
\(59\) −3.41770e28 −0.206384 −0.103192 0.994661i \(-0.532906\pi\)
−0.103192 + 0.994661i \(0.532906\pi\)
\(60\) 0 0
\(61\) −6.01152e28 −0.209431 −0.104716 0.994502i \(-0.533393\pi\)
−0.104716 + 0.994502i \(0.533393\pi\)
\(62\) 4.49402e29i 1.19722i
\(63\) 4.89566e27i 0.0100160i
\(64\) −6.85952e29 −1.08224
\(65\) 0 0
\(66\) 2.18281e29 0.207272
\(67\) − 2.17962e30i − 1.61490i −0.589934 0.807451i \(-0.700846\pi\)
0.589934 0.807451i \(-0.299154\pi\)
\(68\) 3.97934e29i 0.230895i
\(69\) −2.75812e30 −1.25777
\(70\) 0 0
\(71\) 9.20055e29 0.261849 0.130924 0.991392i \(-0.458205\pi\)
0.130924 + 0.991392i \(0.458205\pi\)
\(72\) 3.09704e28i 0.00699779i
\(73\) 4.72873e30i 0.850977i 0.904964 + 0.425488i \(0.139898\pi\)
−0.904964 + 0.425488i \(0.860102\pi\)
\(74\) −8.07302e30 −1.16068
\(75\) 0 0
\(76\) 7.69097e29 0.0712126
\(77\) 5.26564e30i 0.392965i
\(78\) − 1.45948e31i − 0.880316i
\(79\) 2.14123e31 1.04668 0.523342 0.852122i \(-0.324685\pi\)
0.523342 + 0.852122i \(0.324685\pi\)
\(80\) 0 0
\(81\) −3.07032e31 −0.993530
\(82\) − 2.01845e30i − 0.0533445i
\(83\) 6.60838e31i 1.42990i 0.699174 + 0.714951i \(0.253551\pi\)
−0.699174 + 0.714951i \(0.746449\pi\)
\(84\) 2.80131e31 0.497454
\(85\) 0 0
\(86\) 9.20283e31 1.10840
\(87\) − 1.41556e32i − 1.40884i
\(88\) 3.33109e31i 0.274551i
\(89\) 6.08523e31 0.416238 0.208119 0.978104i \(-0.433266\pi\)
0.208119 + 0.978104i \(0.433266\pi\)
\(90\) 0 0
\(91\) 3.52074e32 1.66898
\(92\) − 1.02116e32i − 0.404198i
\(93\) − 4.37109e32i − 1.44751i
\(94\) 2.70356e32 0.750458
\(95\) 0 0
\(96\) 3.11433e32 0.610787
\(97\) 6.12049e30i 0.0101170i 0.999987 + 0.00505851i \(0.00161018\pi\)
−0.999987 + 0.00505851i \(0.998390\pi\)
\(98\) − 8.43141e32i − 1.17671i
\(99\) 1.37373e30 0.00162152
\(100\) 0 0
\(101\) 1.27681e33 1.08349 0.541743 0.840544i \(-0.317765\pi\)
0.541743 + 0.840544i \(0.317765\pi\)
\(102\) 8.21251e32i 0.592343i
\(103\) 1.91249e33i 1.17432i 0.809473 + 0.587158i \(0.199753\pi\)
−0.809473 + 0.587158i \(0.800247\pi\)
\(104\) 2.22725e33 1.16606
\(105\) 0 0
\(106\) −2.48405e33 −0.949759
\(107\) − 5.25843e33i − 1.72196i −0.508638 0.860981i \(-0.669851\pi\)
0.508638 0.860981i \(-0.330149\pi\)
\(108\) − 1.14410e33i − 0.321346i
\(109\) −1.63289e33 −0.393930 −0.196965 0.980411i \(-0.563109\pi\)
−0.196965 + 0.980411i \(0.563109\pi\)
\(110\) 0 0
\(111\) 7.85219e33 1.40334
\(112\) − 5.83293e33i − 0.899061i
\(113\) 1.10961e34i 1.47699i 0.674259 + 0.738495i \(0.264463\pi\)
−0.674259 + 0.738495i \(0.735537\pi\)
\(114\) 1.58725e33 0.182691
\(115\) 0 0
\(116\) 5.24093e33 0.452743
\(117\) − 9.18512e31i − 0.00688681i
\(118\) − 2.61144e33i − 0.170148i
\(119\) −1.98112e34 −1.12302
\(120\) 0 0
\(121\) −2.17476e34 −0.936382
\(122\) − 4.59335e33i − 0.172660i
\(123\) 1.96324e33i 0.0644969i
\(124\) 1.61834e34 0.465171
\(125\) 0 0
\(126\) −3.74074e32 −0.00825740
\(127\) 9.92625e34i 1.92320i 0.274458 + 0.961599i \(0.411502\pi\)
−0.274458 + 0.961599i \(0.588498\pi\)
\(128\) − 1.64170e34i − 0.279466i
\(129\) −8.95110e34 −1.34013
\(130\) 0 0
\(131\) 5.93400e34 0.689241 0.344621 0.938742i \(-0.388008\pi\)
0.344621 + 0.938742i \(0.388008\pi\)
\(132\) − 7.86052e33i − 0.0805344i
\(133\) 3.82896e34i 0.346361i
\(134\) 1.66543e35 1.33136
\(135\) 0 0
\(136\) −1.25328e35 −0.784611
\(137\) − 7.72864e34i − 0.428759i −0.976750 0.214380i \(-0.931227\pi\)
0.976750 0.214380i \(-0.0687730\pi\)
\(138\) − 2.10746e35i − 1.03694i
\(139\) 3.33723e35 1.45761 0.728806 0.684720i \(-0.240075\pi\)
0.728806 + 0.684720i \(0.240075\pi\)
\(140\) 0 0
\(141\) −2.62961e35 −0.907351
\(142\) 7.03006e34i 0.215875i
\(143\) − 9.87926e34i − 0.270196i
\(144\) −1.52173e33 −0.00370985
\(145\) 0 0
\(146\) −3.61319e35 −0.701566
\(147\) 8.20077e35i 1.42272i
\(148\) 2.90717e35i 0.450975i
\(149\) −9.18769e35 −1.27536 −0.637680 0.770301i \(-0.720106\pi\)
−0.637680 + 0.770301i \(0.720106\pi\)
\(150\) 0 0
\(151\) −1.06733e33 −0.00118899 −0.000594497 1.00000i \(-0.500189\pi\)
−0.000594497 1.00000i \(0.500189\pi\)
\(152\) 2.42224e35i 0.241990i
\(153\) 5.16846e33i 0.00463397i
\(154\) −4.02343e35 −0.323970
\(155\) 0 0
\(156\) −5.25575e35 −0.342041
\(157\) 2.04504e36i 1.19773i 0.800851 + 0.598863i \(0.204381\pi\)
−0.800851 + 0.598863i \(0.795619\pi\)
\(158\) 1.63610e36i 0.862912i
\(159\) 2.41610e36 1.14832
\(160\) 0 0
\(161\) 5.08385e36 1.96592
\(162\) − 2.34601e36i − 0.819090i
\(163\) − 8.53783e35i − 0.269310i −0.990893 0.134655i \(-0.957007\pi\)
0.990893 0.134655i \(-0.0429926\pi\)
\(164\) −7.26865e34 −0.0207267
\(165\) 0 0
\(166\) −5.04941e36 −1.17885
\(167\) 2.30718e36i 0.487819i 0.969798 + 0.243910i \(0.0784301\pi\)
−0.969798 + 0.243910i \(0.921570\pi\)
\(168\) 8.82261e36i 1.69042i
\(169\) −8.49393e35 −0.147563
\(170\) 0 0
\(171\) 9.98922e33 0.00142921
\(172\) − 3.31403e36i − 0.430663i
\(173\) − 7.44289e36i − 0.878984i −0.898247 0.439492i \(-0.855158\pi\)
0.898247 0.439492i \(-0.144842\pi\)
\(174\) 1.08162e37 1.16148
\(175\) 0 0
\(176\) −1.63673e36 −0.145552
\(177\) 2.54000e36i 0.205719i
\(178\) 4.64967e36i 0.343157i
\(179\) 1.14363e37 0.769506 0.384753 0.923020i \(-0.374287\pi\)
0.384753 + 0.923020i \(0.374287\pi\)
\(180\) 0 0
\(181\) 9.37075e36 0.524902 0.262451 0.964945i \(-0.415469\pi\)
0.262451 + 0.964945i \(0.415469\pi\)
\(182\) 2.69017e37i 1.37595i
\(183\) 4.46770e36i 0.208757i
\(184\) 3.21609e37 1.37352
\(185\) 0 0
\(186\) 3.33992e37 1.19336
\(187\) 5.55906e36i 0.181809i
\(188\) − 9.73579e36i − 0.291586i
\(189\) 5.69594e37 1.56295
\(190\) 0 0
\(191\) 1.69883e37 0.391831 0.195915 0.980621i \(-0.437232\pi\)
0.195915 + 0.980621i \(0.437232\pi\)
\(192\) 5.09793e37i 1.07876i
\(193\) − 7.50799e36i − 0.145824i −0.997338 0.0729119i \(-0.976771\pi\)
0.997338 0.0729119i \(-0.0232292\pi\)
\(194\) −4.67662e35 −0.00834072
\(195\) 0 0
\(196\) −3.03623e37 −0.457204
\(197\) 1.41045e38i 1.95284i 0.215886 + 0.976418i \(0.430736\pi\)
−0.215886 + 0.976418i \(0.569264\pi\)
\(198\) 1.04966e35i 0.00133682i
\(199\) −1.95572e37 −0.229208 −0.114604 0.993411i \(-0.536560\pi\)
−0.114604 + 0.993411i \(0.536560\pi\)
\(200\) 0 0
\(201\) −1.61987e38 −1.60970
\(202\) 9.75600e37i 0.893252i
\(203\) 2.60921e38i 2.20203i
\(204\) 2.95741e37 0.230151
\(205\) 0 0
\(206\) −1.46132e38 −0.968134
\(207\) − 1.32631e36i − 0.00811209i
\(208\) 1.09436e38i 0.618179i
\(209\) 1.07441e37 0.0560735
\(210\) 0 0
\(211\) −2.65891e38 −1.18589 −0.592944 0.805244i \(-0.702034\pi\)
−0.592944 + 0.805244i \(0.702034\pi\)
\(212\) 8.94530e37i 0.369023i
\(213\) − 6.83776e37i − 0.261006i
\(214\) 4.01792e38 1.41963
\(215\) 0 0
\(216\) 3.60330e38 1.09198
\(217\) 8.05694e38i 2.26248i
\(218\) − 1.24768e38i − 0.324765i
\(219\) 3.51435e38 0.848237
\(220\) 0 0
\(221\) 3.71693e38 0.772167
\(222\) 5.99979e38i 1.15694i
\(223\) 2.18179e38i 0.390645i 0.980739 + 0.195323i \(0.0625754\pi\)
−0.980739 + 0.195323i \(0.937425\pi\)
\(224\) −5.74044e38 −0.954670
\(225\) 0 0
\(226\) −8.47847e38 −1.21767
\(227\) − 5.43077e38i − 0.725161i −0.931952 0.362581i \(-0.881896\pi\)
0.931952 0.362581i \(-0.118104\pi\)
\(228\) − 5.71586e37i − 0.0709833i
\(229\) 1.10600e39 1.27782 0.638908 0.769284i \(-0.279387\pi\)
0.638908 + 0.769284i \(0.279387\pi\)
\(230\) 0 0
\(231\) 3.91337e38 0.391700
\(232\) 1.65061e39i 1.53848i
\(233\) 5.67177e38i 0.492431i 0.969215 + 0.246216i \(0.0791871\pi\)
−0.969215 + 0.246216i \(0.920813\pi\)
\(234\) 7.01827e36 0.00567765
\(235\) 0 0
\(236\) −9.40404e37 −0.0661099
\(237\) − 1.59134e39i − 1.04332i
\(238\) − 1.51376e39i − 0.925842i
\(239\) 2.69103e39 1.53587 0.767933 0.640530i \(-0.221285\pi\)
0.767933 + 0.640530i \(0.221285\pi\)
\(240\) 0 0
\(241\) 3.85216e39 1.91612 0.958061 0.286565i \(-0.0925134\pi\)
0.958061 + 0.286565i \(0.0925134\pi\)
\(242\) − 1.66172e39i − 0.771976i
\(243\) − 2.96249e37i − 0.0128574i
\(244\) −1.65411e38 −0.0670861
\(245\) 0 0
\(246\) −1.50009e38 −0.0531728
\(247\) − 7.18380e38i − 0.238152i
\(248\) 5.09690e39i 1.58071i
\(249\) 4.91128e39 1.42530
\(250\) 0 0
\(251\) 1.40214e37 0.00356594 0.00178297 0.999998i \(-0.499432\pi\)
0.00178297 + 0.999998i \(0.499432\pi\)
\(252\) 1.34708e37i 0.00320836i
\(253\) − 1.42654e39i − 0.318269i
\(254\) −7.58457e39 −1.58553
\(255\) 0 0
\(256\) −4.63787e39 −0.851843
\(257\) − 3.74219e39i − 0.644510i −0.946653 0.322255i \(-0.895559\pi\)
0.946653 0.322255i \(-0.104441\pi\)
\(258\) − 6.83946e39i − 1.10483i
\(259\) −1.44734e40 −2.19344
\(260\) 0 0
\(261\) 6.80705e37 0.00908637
\(262\) 4.53412e39i 0.568227i
\(263\) 8.59682e39i 1.01174i 0.862610 + 0.505869i \(0.168828\pi\)
−0.862610 + 0.505869i \(0.831172\pi\)
\(264\) 2.47564e39 0.273667
\(265\) 0 0
\(266\) −2.92568e39 −0.285548
\(267\) − 4.52248e39i − 0.414898i
\(268\) − 5.99737e39i − 0.517293i
\(269\) −2.20144e40 −1.78564 −0.892820 0.450414i \(-0.851276\pi\)
−0.892820 + 0.450414i \(0.851276\pi\)
\(270\) 0 0
\(271\) 4.26536e39 0.306170 0.153085 0.988213i \(-0.451079\pi\)
0.153085 + 0.988213i \(0.451079\pi\)
\(272\) − 6.15795e39i − 0.415958i
\(273\) − 2.61658e40i − 1.66361i
\(274\) 5.90539e39 0.353480
\(275\) 0 0
\(276\) −7.58915e39 −0.402896
\(277\) − 2.32580e40i − 1.16321i −0.813473 0.581603i \(-0.802426\pi\)
0.813473 0.581603i \(-0.197574\pi\)
\(278\) 2.54995e40i 1.20169i
\(279\) 2.10194e38 0.00933580
\(280\) 0 0
\(281\) −3.04347e39 −0.120147 −0.0600737 0.998194i \(-0.519134\pi\)
−0.0600737 + 0.998194i \(0.519134\pi\)
\(282\) − 2.00926e40i − 0.748042i
\(283\) − 1.78513e40i − 0.626893i −0.949606 0.313447i \(-0.898516\pi\)
0.949606 0.313447i \(-0.101484\pi\)
\(284\) 2.53160e39 0.0838768
\(285\) 0 0
\(286\) 7.54866e39 0.222756
\(287\) − 3.61871e39i − 0.100810i
\(288\) 1.49760e38i 0.00393931i
\(289\) 1.93394e40 0.480427
\(290\) 0 0
\(291\) 4.54869e38 0.0100844
\(292\) 1.30114e40i 0.272589i
\(293\) − 5.63289e39i − 0.111536i −0.998444 0.0557681i \(-0.982239\pi\)
0.998444 0.0557681i \(-0.0177607\pi\)
\(294\) −6.26614e40 −1.17292
\(295\) 0 0
\(296\) −9.15602e40 −1.53248
\(297\) − 1.59829e40i − 0.253031i
\(298\) − 7.02024e40i − 1.05144i
\(299\) −9.53819e40 −1.35174
\(300\) 0 0
\(301\) 1.64990e41 2.09464
\(302\) − 8.15540e37i 0 0.000980236i
\(303\) − 9.48913e40i − 1.08000i
\(304\) −1.19016e40 −0.128290
\(305\) 0 0
\(306\) −3.94918e38 −0.00382035
\(307\) − 1.51598e41i − 1.38966i −0.719173 0.694832i \(-0.755479\pi\)
0.719173 0.694832i \(-0.244521\pi\)
\(308\) 1.44888e40i 0.125877i
\(309\) 1.42134e41 1.17053
\(310\) 0 0
\(311\) 1.38324e41 1.02412 0.512062 0.858949i \(-0.328882\pi\)
0.512062 + 0.858949i \(0.328882\pi\)
\(312\) − 1.65527e41i − 1.16230i
\(313\) 2.31070e41i 1.53908i 0.638598 + 0.769540i \(0.279515\pi\)
−0.638598 + 0.769540i \(0.720485\pi\)
\(314\) −1.56260e41 −0.987434
\(315\) 0 0
\(316\) 5.89174e40 0.335279
\(317\) − 1.18117e41i − 0.638022i −0.947751 0.319011i \(-0.896649\pi\)
0.947751 0.319011i \(-0.103351\pi\)
\(318\) 1.84612e41i 0.946701i
\(319\) 7.32147e40 0.356494
\(320\) 0 0
\(321\) −3.90801e41 −1.71642
\(322\) 3.88453e41i 1.62075i
\(323\) 4.04232e40i 0.160247i
\(324\) −8.44820e40 −0.318252
\(325\) 0 0
\(326\) 6.52369e40 0.222026
\(327\) 1.21355e41i 0.392662i
\(328\) − 2.28923e40i − 0.0704321i
\(329\) 4.84698e41 1.41820
\(330\) 0 0
\(331\) 2.42794e41 0.642801 0.321400 0.946943i \(-0.395847\pi\)
0.321400 + 0.946943i \(0.395847\pi\)
\(332\) 1.81834e41i 0.458033i
\(333\) 3.77591e39i 0.00905090i
\(334\) −1.76290e41 −0.402170
\(335\) 0 0
\(336\) −4.33498e41 −0.896166
\(337\) 5.79957e41i 1.14157i 0.821100 + 0.570785i \(0.193361\pi\)
−0.821100 + 0.570785i \(0.806639\pi\)
\(338\) − 6.49015e40i − 0.121655i
\(339\) 8.24654e41 1.47223
\(340\) 0 0
\(341\) 2.26079e41 0.366280
\(342\) 7.63268e38i 0.00117827i
\(343\) − 4.52545e41i − 0.665747i
\(344\) 1.04374e42 1.46345
\(345\) 0 0
\(346\) 5.68705e41 0.724656
\(347\) 1.00536e42i 1.22148i 0.791832 + 0.610739i \(0.209128\pi\)
−0.791832 + 0.610739i \(0.790872\pi\)
\(348\) − 3.89501e41i − 0.451285i
\(349\) −1.46484e42 −1.61872 −0.809360 0.587313i \(-0.800186\pi\)
−0.809360 + 0.587313i \(0.800186\pi\)
\(350\) 0 0
\(351\) −1.06866e42 −1.07466
\(352\) 1.61078e41i 0.154554i
\(353\) 7.67079e40i 0.0702356i 0.999383 + 0.0351178i \(0.0111806\pi\)
−0.999383 + 0.0351178i \(0.988819\pi\)
\(354\) −1.94079e41 −0.169600
\(355\) 0 0
\(356\) 1.67439e41 0.133331
\(357\) 1.47235e42i 1.11940i
\(358\) 8.73842e41i 0.634399i
\(359\) 2.27621e42 1.57817 0.789084 0.614285i \(-0.210555\pi\)
0.789084 + 0.614285i \(0.210555\pi\)
\(360\) 0 0
\(361\) −1.50264e42 −0.950577
\(362\) 7.16012e41i 0.432742i
\(363\) 1.61626e42i 0.933367i
\(364\) 9.68757e41 0.534616
\(365\) 0 0
\(366\) −3.41373e41 −0.172104
\(367\) 2.44390e42i 1.17786i 0.808185 + 0.588929i \(0.200450\pi\)
−0.808185 + 0.588929i \(0.799550\pi\)
\(368\) 1.58022e42i 0.728165i
\(369\) −9.44070e38 −0.000415977 0
\(370\) 0 0
\(371\) −4.45343e42 −1.79484
\(372\) − 1.20274e42i − 0.463673i
\(373\) − 2.18298e42i − 0.805108i −0.915396 0.402554i \(-0.868123\pi\)
0.915396 0.402554i \(-0.131877\pi\)
\(374\) −4.24763e41 −0.149887
\(375\) 0 0
\(376\) 3.06625e42 0.990849
\(377\) − 4.89532e42i − 1.51408i
\(378\) 4.35222e42i 1.28853i
\(379\) −3.29619e42 −0.934254 −0.467127 0.884190i \(-0.654711\pi\)
−0.467127 + 0.884190i \(0.654711\pi\)
\(380\) 0 0
\(381\) 7.37710e42 1.91701
\(382\) 1.29806e42i 0.323035i
\(383\) − 2.10779e42i − 0.502399i −0.967935 0.251199i \(-0.919175\pi\)
0.967935 0.251199i \(-0.0808249\pi\)
\(384\) −1.22009e42 −0.278566
\(385\) 0 0
\(386\) 5.73680e41 0.120221
\(387\) − 4.30434e40i − 0.00864323i
\(388\) 1.68410e40i 0.00324073i
\(389\) 4.71038e42 0.868735 0.434368 0.900736i \(-0.356972\pi\)
0.434368 + 0.900736i \(0.356972\pi\)
\(390\) 0 0
\(391\) 5.36714e42 0.909550
\(392\) − 9.56249e42i − 1.55364i
\(393\) − 4.41009e42i − 0.687022i
\(394\) −1.07772e43 −1.60997
\(395\) 0 0
\(396\) 3.77992e39 0.000519412 0
\(397\) 4.19658e42i 0.553162i 0.960991 + 0.276581i \(0.0892014\pi\)
−0.960991 + 0.276581i \(0.910799\pi\)
\(398\) − 1.49435e42i − 0.188965i
\(399\) 2.84565e42 0.345246
\(400\) 0 0
\(401\) 5.75416e42 0.642835 0.321418 0.946938i \(-0.395841\pi\)
0.321418 + 0.946938i \(0.395841\pi\)
\(402\) − 1.23773e43i − 1.32708i
\(403\) − 1.51162e43i − 1.55564i
\(404\) 3.51323e42 0.347068
\(405\) 0 0
\(406\) −1.99367e43 −1.81541
\(407\) 4.06126e42i 0.355102i
\(408\) 9.31423e42i 0.782085i
\(409\) 1.41308e43 1.13954 0.569772 0.821803i \(-0.307032\pi\)
0.569772 + 0.821803i \(0.307032\pi\)
\(410\) 0 0
\(411\) −5.74385e42 −0.427379
\(412\) 5.26235e42i 0.376162i
\(413\) − 4.68182e42i − 0.321543i
\(414\) 1.01342e41 0.00668780
\(415\) 0 0
\(416\) 1.07701e43 0.656415
\(417\) − 2.48020e43i − 1.45292i
\(418\) 8.20950e41i 0.0462283i
\(419\) −2.86224e43 −1.54944 −0.774720 0.632305i \(-0.782109\pi\)
−0.774720 + 0.632305i \(0.782109\pi\)
\(420\) 0 0
\(421\) 1.34194e43 0.671549 0.335774 0.941942i \(-0.391002\pi\)
0.335774 + 0.941942i \(0.391002\pi\)
\(422\) − 2.03165e43i − 0.977674i
\(423\) − 1.26451e41i − 0.00585202i
\(424\) −2.81729e43 −1.25399
\(425\) 0 0
\(426\) 5.22468e42 0.215180
\(427\) − 8.23502e42i − 0.326291i
\(428\) − 1.44689e43i − 0.551587i
\(429\) −7.34217e42 −0.269327
\(430\) 0 0
\(431\) −4.00130e43 −1.35933 −0.679667 0.733521i \(-0.737876\pi\)
−0.679667 + 0.733521i \(0.737876\pi\)
\(432\) 1.77048e43i 0.578907i
\(433\) 1.03509e43i 0.325783i 0.986644 + 0.162891i \(0.0520820\pi\)
−0.986644 + 0.162891i \(0.947918\pi\)
\(434\) −6.15625e43 −1.86524
\(435\) 0 0
\(436\) −4.49300e42 −0.126186
\(437\) − 1.03732e43i − 0.280524i
\(438\) 2.68529e43i 0.699307i
\(439\) −3.15434e42 −0.0791122 −0.0395561 0.999217i \(-0.512594\pi\)
−0.0395561 + 0.999217i \(0.512594\pi\)
\(440\) 0 0
\(441\) −3.94353e41 −0.00917591
\(442\) 2.84007e43i 0.636593i
\(443\) 1.66215e43i 0.358929i 0.983764 + 0.179464i \(0.0574365\pi\)
−0.983764 + 0.179464i \(0.942564\pi\)
\(444\) 2.16058e43 0.449523
\(445\) 0 0
\(446\) −1.66709e43 −0.322057
\(447\) 6.82821e43i 1.27125i
\(448\) − 9.39667e43i − 1.68611i
\(449\) 2.11347e43 0.365537 0.182768 0.983156i \(-0.441494\pi\)
0.182768 + 0.983156i \(0.441494\pi\)
\(450\) 0 0
\(451\) −1.01542e42 −0.0163204
\(452\) 3.05318e43i 0.473117i
\(453\) 7.93231e40i 0.00118517i
\(454\) 4.14961e43 0.597840
\(455\) 0 0
\(456\) 1.80018e43 0.241211
\(457\) − 1.42828e44i − 1.84585i −0.384981 0.922924i \(-0.625792\pi\)
0.384981 0.922924i \(-0.374208\pi\)
\(458\) 8.45082e43i 1.05346i
\(459\) 6.01333e43 0.723112
\(460\) 0 0
\(461\) 3.53458e43 0.395615 0.197807 0.980241i \(-0.436618\pi\)
0.197807 + 0.980241i \(0.436618\pi\)
\(462\) 2.99018e43i 0.322927i
\(463\) 3.07914e43i 0.320880i 0.987046 + 0.160440i \(0.0512914\pi\)
−0.987046 + 0.160440i \(0.948709\pi\)
\(464\) −8.11024e43 −0.815618
\(465\) 0 0
\(466\) −4.33375e43 −0.405972
\(467\) − 3.68923e43i − 0.333586i −0.985992 0.166793i \(-0.946659\pi\)
0.985992 0.166793i \(-0.0533412\pi\)
\(468\) − 2.52735e41i − 0.00220602i
\(469\) 2.98580e44 2.51599
\(470\) 0 0
\(471\) 1.51985e44 1.19387
\(472\) − 2.96176e43i − 0.224650i
\(473\) − 4.62963e43i − 0.339108i
\(474\) 1.21593e44 0.860134
\(475\) 0 0
\(476\) −5.45119e43 −0.359730
\(477\) 1.16184e42i 0.00740615i
\(478\) 2.05619e44i 1.26620i
\(479\) 3.09046e43 0.183860 0.0919300 0.995765i \(-0.470696\pi\)
0.0919300 + 0.995765i \(0.470696\pi\)
\(480\) 0 0
\(481\) 2.71546e44 1.50817
\(482\) 2.94340e44i 1.57970i
\(483\) − 3.77827e44i − 1.95959i
\(484\) −5.98401e43 −0.299946
\(485\) 0 0
\(486\) 2.26361e42 0.0105999
\(487\) 4.31638e44i 1.95385i 0.213578 + 0.976926i \(0.431488\pi\)
−0.213578 + 0.976926i \(0.568512\pi\)
\(488\) − 5.20955e43i − 0.227968i
\(489\) −6.34524e43 −0.268443
\(490\) 0 0
\(491\) −2.93145e44 −1.15941 −0.579706 0.814826i \(-0.696832\pi\)
−0.579706 + 0.814826i \(0.696832\pi\)
\(492\) 5.40199e42i 0.0206600i
\(493\) 2.75460e44i 1.01879i
\(494\) 5.48908e43 0.196338
\(495\) 0 0
\(496\) −2.50435e44 −0.838008
\(497\) 1.26036e44i 0.407956i
\(498\) 3.75267e44i 1.17505i
\(499\) 9.00632e42 0.0272828 0.0136414 0.999907i \(-0.495658\pi\)
0.0136414 + 0.999907i \(0.495658\pi\)
\(500\) 0 0
\(501\) 1.71467e44 0.486249
\(502\) 1.07136e42i 0.00293985i
\(503\) 3.10986e44i 0.825789i 0.910779 + 0.412895i \(0.135482\pi\)
−0.910779 + 0.412895i \(0.864518\pi\)
\(504\) −4.24256e42 −0.0109024
\(505\) 0 0
\(506\) 1.09001e44 0.262389
\(507\) 6.31261e43i 0.147088i
\(508\) 2.73128e44i 0.616048i
\(509\) −7.95788e43 −0.173762 −0.0868808 0.996219i \(-0.527690\pi\)
−0.0868808 + 0.996219i \(0.527690\pi\)
\(510\) 0 0
\(511\) −6.47777e44 −1.32581
\(512\) − 4.95397e44i − 0.981745i
\(513\) − 1.16221e44i − 0.223023i
\(514\) 2.85938e44 0.531350
\(515\) 0 0
\(516\) −2.46296e44 −0.429276
\(517\) − 1.36007e44i − 0.229598i
\(518\) − 1.10590e45i − 1.80832i
\(519\) −5.53149e44 −0.876154
\(520\) 0 0
\(521\) 1.27119e44 0.188968 0.0944841 0.995526i \(-0.469880\pi\)
0.0944841 + 0.995526i \(0.469880\pi\)
\(522\) 5.20121e42i 0.00749102i
\(523\) 1.19591e44i 0.166887i 0.996513 + 0.0834434i \(0.0265918\pi\)
−0.996513 + 0.0834434i \(0.973408\pi\)
\(524\) 1.63278e44 0.220781
\(525\) 0 0
\(526\) −6.56876e44 −0.834101
\(527\) 8.50590e44i 1.04675i
\(528\) 1.21640e44i 0.145083i
\(529\) −5.12284e44 −0.592233
\(530\) 0 0
\(531\) −1.22142e42 −0.00132680
\(532\) 1.05357e44i 0.110948i
\(533\) 6.78932e43i 0.0693151i
\(534\) 3.45559e44 0.342052
\(535\) 0 0
\(536\) 1.88885e45 1.75783
\(537\) − 8.49938e44i − 0.767029i
\(538\) − 1.68210e45i − 1.47212i
\(539\) −4.24156e44 −0.360007
\(540\) 0 0
\(541\) −1.24548e45 −0.994451 −0.497225 0.867621i \(-0.665648\pi\)
−0.497225 + 0.867621i \(0.665648\pi\)
\(542\) 3.25913e44i 0.252414i
\(543\) − 6.96426e44i − 0.523212i
\(544\) −6.06031e44 −0.441686
\(545\) 0 0
\(546\) 1.99931e45 1.37152
\(547\) − 9.41755e44i − 0.626826i −0.949617 0.313413i \(-0.898528\pi\)
0.949617 0.313413i \(-0.101472\pi\)
\(548\) − 2.12659e44i − 0.137342i
\(549\) −2.14840e42 −0.00134639
\(550\) 0 0
\(551\) 5.32388e44 0.314215
\(552\) − 2.39017e45i − 1.36910i
\(553\) 2.93321e45i 1.63072i
\(554\) 1.77713e45 0.958975
\(555\) 0 0
\(556\) 9.18263e44 0.466910
\(557\) − 2.81038e45i − 1.38725i −0.720338 0.693624i \(-0.756013\pi\)
0.720338 0.693624i \(-0.243987\pi\)
\(558\) 1.60608e43i 0.00769666i
\(559\) −3.09549e45 −1.44024
\(560\) 0 0
\(561\) 4.13144e44 0.181223
\(562\) − 2.32549e44i − 0.0990524i
\(563\) − 2.48634e45i − 1.02842i −0.857663 0.514212i \(-0.828084\pi\)
0.857663 0.514212i \(-0.171916\pi\)
\(564\) −7.23555e44 −0.290647
\(565\) 0 0
\(566\) 1.36400e45 0.516826
\(567\) − 4.20595e45i − 1.54790i
\(568\) 7.97315e44i 0.285025i
\(569\) −1.50472e45 −0.522521 −0.261261 0.965268i \(-0.584138\pi\)
−0.261261 + 0.965268i \(0.584138\pi\)
\(570\) 0 0
\(571\) −2.28889e45 −0.750117 −0.375059 0.927001i \(-0.622377\pi\)
−0.375059 + 0.927001i \(0.622377\pi\)
\(572\) − 2.71835e44i − 0.0865507i
\(573\) − 1.26255e45i − 0.390569i
\(574\) 2.76503e44 0.0831099
\(575\) 0 0
\(576\) −2.45146e43 −0.00695751
\(577\) 1.59455e45i 0.439781i 0.975525 + 0.219891i \(0.0705701\pi\)
−0.975525 + 0.219891i \(0.929430\pi\)
\(578\) 1.47771e45i 0.396076i
\(579\) −5.57987e44 −0.145354
\(580\) 0 0
\(581\) −9.05264e45 −2.22776
\(582\) 3.47562e43i 0.00831386i
\(583\) 1.24964e45i 0.290572i
\(584\) −4.09790e45 −0.926295
\(585\) 0 0
\(586\) 4.30405e44 0.0919531
\(587\) 5.55956e45i 1.15481i 0.816457 + 0.577407i \(0.195935\pi\)
−0.816457 + 0.577407i \(0.804065\pi\)
\(588\) 2.25650e45i 0.455732i
\(589\) 1.64396e45 0.322841
\(590\) 0 0
\(591\) 1.04824e46 1.94655
\(592\) − 4.49880e45i − 0.812434i
\(593\) 1.33454e45i 0.234384i 0.993109 + 0.117192i \(0.0373893\pi\)
−0.993109 + 0.117192i \(0.962611\pi\)
\(594\) 1.22124e45 0.208605
\(595\) 0 0
\(596\) −2.52806e45 −0.408530
\(597\) 1.45347e45i 0.228470i
\(598\) − 7.28806e45i − 1.11440i
\(599\) −2.61382e45 −0.388806 −0.194403 0.980922i \(-0.562277\pi\)
−0.194403 + 0.980922i \(0.562277\pi\)
\(600\) 0 0
\(601\) 1.12433e45 0.158295 0.0791474 0.996863i \(-0.474780\pi\)
0.0791474 + 0.996863i \(0.474780\pi\)
\(602\) 1.26067e46i 1.72687i
\(603\) − 7.78954e43i − 0.0103819i
\(604\) −2.93684e42 −0.000380865 0
\(605\) 0 0
\(606\) 7.25056e45 0.890376
\(607\) 1.38809e46i 1.65884i 0.558628 + 0.829418i \(0.311328\pi\)
−0.558628 + 0.829418i \(0.688672\pi\)
\(608\) 1.17129e45i 0.136225i
\(609\) 1.93914e46 2.19494
\(610\) 0 0
\(611\) −9.09378e45 −0.975134
\(612\) 1.42214e43i 0.00148437i
\(613\) − 1.28344e46i − 1.30400i −0.758218 0.652001i \(-0.773930\pi\)
0.758218 0.652001i \(-0.226070\pi\)
\(614\) 1.15835e46 1.14567
\(615\) 0 0
\(616\) −4.56318e45 −0.427746
\(617\) − 1.41494e46i − 1.29131i −0.763628 0.645657i \(-0.776584\pi\)
0.763628 0.645657i \(-0.223416\pi\)
\(618\) 1.08604e46i 0.965017i
\(619\) −3.49373e45 −0.302269 −0.151134 0.988513i \(-0.548293\pi\)
−0.151134 + 0.988513i \(0.548293\pi\)
\(620\) 0 0
\(621\) −1.54311e46 −1.26586
\(622\) 1.05693e46i 0.844312i
\(623\) 8.33600e45i 0.648492i
\(624\) 8.13317e45 0.616189
\(625\) 0 0
\(626\) −1.76558e46 −1.26885
\(627\) − 7.98494e44i − 0.0558929i
\(628\) 5.62707e45i 0.383662i
\(629\) −1.52799e46 −1.01481
\(630\) 0 0
\(631\) 2.21383e46 1.39528 0.697638 0.716451i \(-0.254234\pi\)
0.697638 + 0.716451i \(0.254234\pi\)
\(632\) 1.85558e46i 1.13932i
\(633\) 1.97608e46i 1.18207i
\(634\) 9.02525e45 0.526001
\(635\) 0 0
\(636\) 6.64807e45 0.367835
\(637\) 2.83601e46i 1.52900i
\(638\) 5.59428e45i 0.293902i
\(639\) 3.28810e43 0.00168337
\(640\) 0 0
\(641\) −4.90058e45 −0.238281 −0.119141 0.992877i \(-0.538014\pi\)
−0.119141 + 0.992877i \(0.538014\pi\)
\(642\) − 2.98608e46i − 1.41506i
\(643\) − 2.98051e46i − 1.37661i −0.725423 0.688303i \(-0.758356\pi\)
0.725423 0.688303i \(-0.241644\pi\)
\(644\) 1.39886e46 0.629734
\(645\) 0 0
\(646\) −3.08871e45 −0.132111
\(647\) − 3.18321e45i − 0.132723i −0.997796 0.0663613i \(-0.978861\pi\)
0.997796 0.0663613i \(-0.0211390\pi\)
\(648\) − 2.66073e46i − 1.08146i
\(649\) −1.31372e45 −0.0520555
\(650\) 0 0
\(651\) 5.98785e46 2.25520
\(652\) − 2.34924e45i − 0.0862666i
\(653\) 4.44872e46i 1.59282i 0.604756 + 0.796411i \(0.293271\pi\)
−0.604756 + 0.796411i \(0.706729\pi\)
\(654\) −9.27261e45 −0.323720
\(655\) 0 0
\(656\) 1.12481e45 0.0373392
\(657\) 1.68996e44i 0.00547076i
\(658\) 3.70354e46i 1.16920i
\(659\) 3.64438e46 1.12206 0.561028 0.827797i \(-0.310406\pi\)
0.561028 + 0.827797i \(0.310406\pi\)
\(660\) 0 0
\(661\) −2.16411e46 −0.633804 −0.316902 0.948458i \(-0.602643\pi\)
−0.316902 + 0.948458i \(0.602643\pi\)
\(662\) 1.85517e46i 0.529940i
\(663\) − 2.76238e46i − 0.769681i
\(664\) −5.72679e46 −1.55646
\(665\) 0 0
\(666\) −2.88514e44 −0.00746178
\(667\) − 7.06871e46i − 1.78346i
\(668\) 6.34836e45i 0.156261i
\(669\) 1.62148e46 0.389387
\(670\) 0 0
\(671\) −2.31076e45 −0.0528242
\(672\) 4.26624e46i 0.951596i
\(673\) − 5.34875e46i − 1.16414i −0.813140 0.582068i \(-0.802244\pi\)
0.813140 0.582068i \(-0.197756\pi\)
\(674\) −4.43140e46 −0.941137
\(675\) 0 0
\(676\) −2.33717e45 −0.0472682
\(677\) − 3.90814e46i − 0.771359i −0.922633 0.385679i \(-0.873967\pi\)
0.922633 0.385679i \(-0.126033\pi\)
\(678\) 6.30112e46i 1.21375i
\(679\) −8.38430e44 −0.0157622
\(680\) 0 0
\(681\) −4.03610e46 −0.722826
\(682\) 1.72745e46i 0.301970i
\(683\) − 9.82421e46i − 1.67632i −0.545427 0.838158i \(-0.683632\pi\)
0.545427 0.838158i \(-0.316368\pi\)
\(684\) 2.74860e43 0.000457811 0
\(685\) 0 0
\(686\) 3.45786e46 0.548858
\(687\) − 8.21966e46i − 1.27370i
\(688\) 5.12840e46i 0.775841i
\(689\) 8.35542e46 1.23410
\(690\) 0 0
\(691\) −7.20065e46 −1.01387 −0.506937 0.861983i \(-0.669222\pi\)
−0.506937 + 0.861983i \(0.669222\pi\)
\(692\) − 2.04796e46i − 0.281560i
\(693\) 1.88184e44i 0.00252629i
\(694\) −7.68188e46 −1.00702
\(695\) 0 0
\(696\) 1.22672e47 1.53353
\(697\) − 3.82035e45i − 0.0466404i
\(698\) − 1.11927e47i − 1.33451i
\(699\) 4.21520e46 0.490846
\(700\) 0 0
\(701\) −3.78749e46 −0.420731 −0.210366 0.977623i \(-0.567465\pi\)
−0.210366 + 0.977623i \(0.567465\pi\)
\(702\) − 8.16552e46i − 0.885975i
\(703\) 2.95319e46i 0.312988i
\(704\) −2.63672e46 −0.272970
\(705\) 0 0
\(706\) −5.86119e45 −0.0579039
\(707\) 1.74907e47i 1.68805i
\(708\) 6.98899e45i 0.0658970i
\(709\) −8.09313e46 −0.745510 −0.372755 0.927930i \(-0.621587\pi\)
−0.372755 + 0.927930i \(0.621587\pi\)
\(710\) 0 0
\(711\) 7.65234e44 0.00672892
\(712\) 5.27343e46i 0.453078i
\(713\) − 2.18274e47i − 1.83242i
\(714\) −1.12501e47 −0.922861
\(715\) 0 0
\(716\) 3.14679e46 0.246492
\(717\) − 1.99995e47i − 1.53092i
\(718\) 1.73923e47i 1.30108i
\(719\) −1.77184e47 −1.29538 −0.647692 0.761902i \(-0.724266\pi\)
−0.647692 + 0.761902i \(0.724266\pi\)
\(720\) 0 0
\(721\) −2.61987e47 −1.82956
\(722\) − 1.14816e47i − 0.783678i
\(723\) − 2.86289e47i − 1.90995i
\(724\) 2.57843e46 0.168139
\(725\) 0 0
\(726\) −1.23497e47 −0.769490
\(727\) 1.73206e47i 1.05498i 0.849561 + 0.527490i \(0.176867\pi\)
−0.849561 + 0.527490i \(0.823133\pi\)
\(728\) 3.05106e47i 1.81670i
\(729\) −1.72883e47 −1.00635
\(730\) 0 0
\(731\) 1.74183e47 0.969102
\(732\) 1.22932e46i 0.0668701i
\(733\) 2.26618e47i 1.20525i 0.798024 + 0.602626i \(0.205879\pi\)
−0.798024 + 0.602626i \(0.794121\pi\)
\(734\) −1.86736e47 −0.971054
\(735\) 0 0
\(736\) 1.55517e47 0.773203
\(737\) − 8.37821e46i − 0.407322i
\(738\) − 7.21356e43i 0 0.000342941i
\(739\) −3.80934e46 −0.177099 −0.0885496 0.996072i \(-0.528223\pi\)
−0.0885496 + 0.996072i \(0.528223\pi\)
\(740\) 0 0
\(741\) −5.33893e46 −0.237385
\(742\) − 3.40283e47i − 1.47971i
\(743\) 1.22091e47i 0.519238i 0.965711 + 0.259619i \(0.0835970\pi\)
−0.965711 + 0.259619i \(0.916403\pi\)
\(744\) 3.78797e47 1.57563
\(745\) 0 0
\(746\) 1.66800e47 0.663750
\(747\) 2.36171e45i 0.00919255i
\(748\) 1.52961e46i 0.0582378i
\(749\) 7.20338e47 2.68279
\(750\) 0 0
\(751\) −2.36534e46 −0.0843014 −0.0421507 0.999111i \(-0.513421\pi\)
−0.0421507 + 0.999111i \(0.513421\pi\)
\(752\) 1.50660e47i 0.525294i
\(753\) − 1.04206e45i − 0.00355446i
\(754\) 3.74048e47 1.24824
\(755\) 0 0
\(756\) 1.56728e47 0.500652
\(757\) − 1.36056e47i − 0.425243i −0.977135 0.212621i \(-0.931800\pi\)
0.977135 0.212621i \(-0.0682001\pi\)
\(758\) − 2.51859e47i − 0.770222i
\(759\) −1.06019e47 −0.317245
\(760\) 0 0
\(761\) 2.10711e47 0.603727 0.301863 0.953351i \(-0.402391\pi\)
0.301863 + 0.953351i \(0.402391\pi\)
\(762\) 5.63678e47i 1.58043i
\(763\) − 2.23685e47i − 0.613737i
\(764\) 4.67444e46 0.125513
\(765\) 0 0
\(766\) 1.61055e47 0.414189
\(767\) 8.78390e46i 0.221087i
\(768\) 3.44682e47i 0.849100i
\(769\) −3.42055e47 −0.824728 −0.412364 0.911019i \(-0.635297\pi\)
−0.412364 + 0.911019i \(0.635297\pi\)
\(770\) 0 0
\(771\) −2.78116e47 −0.642435
\(772\) − 2.06588e46i − 0.0467110i
\(773\) − 2.59138e47i − 0.573549i −0.957998 0.286774i \(-0.907417\pi\)
0.957998 0.286774i \(-0.0925830\pi\)
\(774\) 3.28892e45 0.00712569
\(775\) 0 0
\(776\) −5.30399e45 −0.0110125
\(777\) 1.07565e48i 2.18637i
\(778\) 3.59916e47i 0.716206i
\(779\) −7.38369e45 −0.0143849
\(780\) 0 0
\(781\) 3.53659e46 0.0660454
\(782\) 4.10099e47i 0.749855i
\(783\) − 7.91977e47i − 1.41789i
\(784\) 4.69851e47 0.823655
\(785\) 0 0
\(786\) 3.36972e47 0.566398
\(787\) − 3.31448e47i − 0.545546i −0.962078 0.272773i \(-0.912059\pi\)
0.962078 0.272773i \(-0.0879409\pi\)
\(788\) 3.88096e47i 0.625542i
\(789\) 6.38908e47 1.00848
\(790\) 0 0
\(791\) −1.52003e48 −2.30113
\(792\) 1.19047e45i 0.00176503i
\(793\) 1.54503e47i 0.224352i
\(794\) −3.20658e47 −0.456040
\(795\) 0 0
\(796\) −5.38129e46 −0.0734211
\(797\) − 1.21980e48i − 1.63015i −0.579354 0.815076i \(-0.696695\pi\)
0.579354 0.815076i \(-0.303305\pi\)
\(798\) 2.17434e47i 0.284629i
\(799\) 5.11707e47 0.656144
\(800\) 0 0
\(801\) 2.17474e45 0.00267591
\(802\) 4.39671e47i 0.529969i
\(803\) 1.81767e47i 0.214639i
\(804\) −4.45719e47 −0.515628
\(805\) 0 0
\(806\) 1.15502e48 1.28251
\(807\) 1.63609e48i 1.77989i
\(808\) 1.10648e48i 1.17938i
\(809\) 2.89096e47 0.301920 0.150960 0.988540i \(-0.451764\pi\)
0.150960 + 0.988540i \(0.451764\pi\)
\(810\) 0 0
\(811\) −8.79228e47 −0.881571 −0.440786 0.897612i \(-0.645300\pi\)
−0.440786 + 0.897612i \(0.645300\pi\)
\(812\) 7.17941e47i 0.705366i
\(813\) − 3.16998e47i − 0.305184i
\(814\) −3.10318e47 −0.292755
\(815\) 0 0
\(816\) −4.57653e47 −0.414619
\(817\) − 3.36648e47i − 0.298891i
\(818\) 1.07972e48i 0.939467i
\(819\) 1.25825e46 0.0107295
\(820\) 0 0
\(821\) −6.16314e47 −0.504824 −0.252412 0.967620i \(-0.581224\pi\)
−0.252412 + 0.967620i \(0.581224\pi\)
\(822\) − 4.38883e47i − 0.352342i
\(823\) 1.89117e47i 0.148810i 0.997228 + 0.0744052i \(0.0237058\pi\)
−0.997228 + 0.0744052i \(0.976294\pi\)
\(824\) −1.65735e48 −1.27825
\(825\) 0 0
\(826\) 3.57734e47 0.265087
\(827\) − 9.51973e47i − 0.691486i −0.938329 0.345743i \(-0.887627\pi\)
0.938329 0.345743i \(-0.112373\pi\)
\(828\) − 3.64942e45i − 0.00259851i
\(829\) −2.56276e48 −1.78878 −0.894392 0.447284i \(-0.852391\pi\)
−0.894392 + 0.447284i \(0.852391\pi\)
\(830\) 0 0
\(831\) −1.72851e48 −1.15946
\(832\) 1.76298e48i 1.15934i
\(833\) − 1.59582e48i − 1.02883i
\(834\) 1.89510e48 1.19782
\(835\) 0 0
\(836\) 2.95632e46 0.0179617
\(837\) − 2.44554e48i − 1.45682i
\(838\) − 2.18701e48i − 1.27740i
\(839\) 5.48536e47 0.314147 0.157073 0.987587i \(-0.449794\pi\)
0.157073 + 0.987587i \(0.449794\pi\)
\(840\) 0 0
\(841\) 1.81182e48 0.997659
\(842\) 1.02536e48i 0.553641i
\(843\) 2.26188e47i 0.119761i
\(844\) −7.31618e47 −0.379869
\(845\) 0 0
\(846\) 9.66201e45 0.00482454
\(847\) − 2.97915e48i − 1.45887i
\(848\) − 1.38427e48i − 0.664797i
\(849\) −1.32669e48 −0.624875
\(850\) 0 0
\(851\) 3.92105e48 1.77650
\(852\) − 1.88146e47i − 0.0836067i
\(853\) − 6.63073e47i − 0.289003i −0.989505 0.144501i \(-0.953842\pi\)
0.989505 0.144501i \(-0.0461578\pi\)
\(854\) 6.29231e47 0.269002
\(855\) 0 0
\(856\) 4.55693e48 1.87437
\(857\) − 3.61344e48i − 1.45793i −0.684550 0.728966i \(-0.740001\pi\)
0.684550 0.728966i \(-0.259999\pi\)
\(858\) − 5.61009e47i − 0.222039i
\(859\) 2.95644e47 0.114784 0.0573921 0.998352i \(-0.481721\pi\)
0.0573921 + 0.998352i \(0.481721\pi\)
\(860\) 0 0
\(861\) −2.68939e47 −0.100485
\(862\) − 3.05736e48i − 1.12067i
\(863\) − 1.94226e48i − 0.698440i −0.937041 0.349220i \(-0.886447\pi\)
0.937041 0.349220i \(-0.113553\pi\)
\(864\) 1.74241e48 0.614713
\(865\) 0 0
\(866\) −7.90904e47 −0.268583
\(867\) − 1.43728e48i − 0.478881i
\(868\) 2.21692e48i 0.724729i
\(869\) 8.23064e47 0.264002
\(870\) 0 0
\(871\) −5.60188e48 −1.72995
\(872\) − 1.41505e48i − 0.428796i
\(873\) 2.18735e44i 0 6.50403e-5i
\(874\) 7.92608e47 0.231270
\(875\) 0 0
\(876\) 9.66998e47 0.271711
\(877\) 3.36176e48i 0.926985i 0.886101 + 0.463493i \(0.153404\pi\)
−0.886101 + 0.463493i \(0.846596\pi\)
\(878\) − 2.41020e47i − 0.0652220i
\(879\) −4.18631e47 −0.111177
\(880\) 0 0
\(881\) 2.89868e47 0.0741477 0.0370738 0.999313i \(-0.488196\pi\)
0.0370738 + 0.999313i \(0.488196\pi\)
\(882\) − 3.01322e46i − 0.00756484i
\(883\) 2.27915e48i 0.561592i 0.959767 + 0.280796i \(0.0905985\pi\)
−0.959767 + 0.280796i \(0.909402\pi\)
\(884\) 1.02274e48 0.247344
\(885\) 0 0
\(886\) −1.27003e48 −0.295910
\(887\) − 3.67815e48i − 0.841180i −0.907251 0.420590i \(-0.861823\pi\)
0.907251 0.420590i \(-0.138177\pi\)
\(888\) 6.80467e48i 1.52754i
\(889\) −1.35977e49 −2.99631
\(890\) 0 0
\(891\) −1.18020e48 −0.250595
\(892\) 6.00334e47i 0.125133i
\(893\) − 9.88988e47i − 0.202368i
\(894\) −5.21738e48 −1.04805
\(895\) 0 0
\(896\) 2.24892e48 0.435403
\(897\) 7.08870e48i 1.34738i
\(898\) 1.61488e48i 0.301357i
\(899\) 1.12026e49 2.05250
\(900\) 0 0
\(901\) −4.70159e48 −0.830397
\(902\) − 7.75870e46i − 0.0134549i
\(903\) − 1.22619e49i − 2.08790i
\(904\) −9.61586e48 −1.60771
\(905\) 0 0
\(906\) −6.06102e45 −0.000977080 0
\(907\) − 1.60568e48i − 0.254178i −0.991891 0.127089i \(-0.959437\pi\)
0.991891 0.127089i \(-0.0405634\pi\)
\(908\) − 1.49431e48i − 0.232287i
\(909\) 4.56307e46 0.00696551
\(910\) 0 0
\(911\) 7.70098e48 1.13369 0.566843 0.823826i \(-0.308165\pi\)
0.566843 + 0.823826i \(0.308165\pi\)
\(912\) 8.84518e47i 0.127877i
\(913\) 2.54019e48i 0.360660i
\(914\) 1.09134e49 1.52176
\(915\) 0 0
\(916\) 3.04322e48 0.409316
\(917\) 8.12883e48i 1.07383i
\(918\) 4.59474e48i 0.596151i
\(919\) −5.42603e48 −0.691474 −0.345737 0.938331i \(-0.612371\pi\)
−0.345737 + 0.938331i \(0.612371\pi\)
\(920\) 0 0
\(921\) −1.12666e49 −1.38519
\(922\) 2.70074e48i 0.326154i
\(923\) − 2.36465e48i − 0.280504i
\(924\) 1.07679e48 0.125471
\(925\) 0 0
\(926\) −2.35275e48 −0.264542
\(927\) 6.83487e46i 0.00754944i
\(928\) 7.98165e48i 0.866066i
\(929\) 7.82524e48 0.834139 0.417070 0.908875i \(-0.363057\pi\)
0.417070 + 0.908875i \(0.363057\pi\)
\(930\) 0 0
\(931\) −3.08429e48 −0.317311
\(932\) 1.56063e48i 0.157738i
\(933\) − 1.02801e49i − 1.02083i
\(934\) 2.81891e48 0.275016
\(935\) 0 0
\(936\) 7.95978e46 0.00749634
\(937\) 1.34653e49i 1.24599i 0.782227 + 0.622994i \(0.214084\pi\)
−0.782227 + 0.622994i \(0.785916\pi\)
\(938\) 2.28143e49i 2.07424i
\(939\) 1.71729e49 1.53413
\(940\) 0 0
\(941\) −1.57761e49 −1.36073 −0.680365 0.732874i \(-0.738179\pi\)
−0.680365 + 0.732874i \(0.738179\pi\)
\(942\) 1.16131e49i 0.984255i
\(943\) 9.80360e47i 0.0816474i
\(944\) 1.45526e48 0.119097
\(945\) 0 0
\(946\) 3.53747e48 0.279569
\(947\) 7.45162e48i 0.578730i 0.957219 + 0.289365i \(0.0934441\pi\)
−0.957219 + 0.289365i \(0.906556\pi\)
\(948\) − 4.37869e48i − 0.334200i
\(949\) 1.21534e49 0.911604
\(950\) 0 0
\(951\) −8.77837e48 −0.635968
\(952\) − 1.71683e49i − 1.22241i
\(953\) − 2.12970e49i − 1.49034i −0.666874 0.745171i \(-0.732368\pi\)
0.666874 0.745171i \(-0.267632\pi\)
\(954\) −8.87751e46 −0.00610581
\(955\) 0 0
\(956\) 7.40456e48 0.491976
\(957\) − 5.44125e48i − 0.355346i
\(958\) 2.36140e48i 0.151579i
\(959\) 1.05873e49 0.668000
\(960\) 0 0
\(961\) 1.81888e49 1.10884
\(962\) 2.07486e49i 1.24337i
\(963\) − 1.87926e47i − 0.0110701i
\(964\) 1.05995e49 0.613781
\(965\) 0 0
\(966\) 2.88695e49 1.61553
\(967\) 7.81881e47i 0.0430134i 0.999769 + 0.0215067i \(0.00684632\pi\)
−0.999769 + 0.0215067i \(0.993154\pi\)
\(968\) − 1.88464e49i − 1.01926i
\(969\) 3.00422e48 0.159731
\(970\) 0 0
\(971\) 2.07870e49 1.06825 0.534127 0.845404i \(-0.320640\pi\)
0.534127 + 0.845404i \(0.320640\pi\)
\(972\) − 8.15149e46i − 0.00411854i
\(973\) 4.57159e49i 2.27094i
\(974\) −3.29811e49 −1.61080
\(975\) 0 0
\(976\) 2.55971e48 0.120856
\(977\) − 1.16244e49i − 0.539647i −0.962910 0.269824i \(-0.913035\pi\)
0.962910 0.269824i \(-0.0869654\pi\)
\(978\) − 4.84834e48i − 0.221311i
\(979\) 2.33909e48 0.104986
\(980\) 0 0
\(981\) −5.83563e46 −0.00253250
\(982\) − 2.23990e49i − 0.955846i
\(983\) 2.37943e49i 0.998482i 0.866463 + 0.499241i \(0.166388\pi\)
−0.866463 + 0.499241i \(0.833612\pi\)
\(984\) −1.70133e48 −0.0702054
\(985\) 0 0
\(986\) −2.10476e49 −0.839913
\(987\) − 3.60223e49i − 1.41364i
\(988\) − 1.97667e48i − 0.0762861i
\(989\) −4.46980e49 −1.69648
\(990\) 0 0
\(991\) −3.80696e49 −1.39754 −0.698768 0.715348i \(-0.746268\pi\)
−0.698768 + 0.715348i \(0.746268\pi\)
\(992\) 2.46465e49i 0.889840i
\(993\) − 1.80442e49i − 0.640731i
\(994\) −9.63030e48 −0.336329
\(995\) 0 0
\(996\) 1.35137e49 0.456559
\(997\) 6.76901e48i 0.224934i 0.993655 + 0.112467i \(0.0358752\pi\)
−0.993655 + 0.112467i \(0.964125\pi\)
\(998\) 6.88166e47i 0.0224926i
\(999\) 4.39314e49 1.41236
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 25.34.b.c.24.8 12
5.2 odd 4 25.34.a.c.1.3 6
5.3 odd 4 5.34.a.b.1.4 6
5.4 even 2 inner 25.34.b.c.24.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.4 6 5.3 odd 4
25.34.a.c.1.3 6 5.2 odd 4
25.34.b.c.24.5 12 5.4 even 2 inner
25.34.b.c.24.8 12 1.1 even 1 trivial