Properties

Label 25.34.a.c.1.2
Level $25$
Weight $34$
Character 25.1
Self dual yes
Analytic conductor $172.457$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.457072203\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 9680197848 x^{4} + 8052423720422 x^{3} + 24239866893261762265 x^{2} - 69081627028404093368325 x - 10572274201725134136583265250\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{5}\cdot 5^{6}\cdot 7\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(54181.8\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q-132922. q^{2} +1.17526e8 q^{3} +9.07819e9 q^{4} -1.56217e13 q^{6} +1.34853e13 q^{7} -6.49000e13 q^{8} +8.25322e15 q^{9} +O(q^{10})\) \(q-132922. q^{2} +1.17526e8 q^{3} +9.07819e9 q^{4} -1.56217e13 q^{6} +1.34853e13 q^{7} -6.49000e13 q^{8} +8.25322e15 q^{9} -7.96470e16 q^{11} +1.06692e18 q^{12} +1.16823e18 q^{13} -1.79249e18 q^{14} -6.93545e19 q^{16} -3.27833e20 q^{17} -1.09703e21 q^{18} +2.21358e21 q^{19} +1.58487e21 q^{21} +1.05868e22 q^{22} -3.93052e22 q^{23} -7.62742e21 q^{24} -1.55283e23 q^{26} +3.16633e23 q^{27} +1.22422e23 q^{28} +1.59428e24 q^{29} +2.07133e24 q^{31} +9.77619e24 q^{32} -9.36057e24 q^{33} +4.35761e25 q^{34} +7.49243e25 q^{36} +4.76387e25 q^{37} -2.94233e26 q^{38} +1.37297e26 q^{39} -6.39281e26 q^{41} -2.10663e26 q^{42} +2.16469e26 q^{43} -7.23051e26 q^{44} +5.22451e27 q^{46} -7.45743e27 q^{47} -8.15093e27 q^{48} -7.54914e27 q^{49} -3.85288e28 q^{51} +1.06055e28 q^{52} -1.70592e28 q^{53} -4.20873e28 q^{54} -8.75196e26 q^{56} +2.60153e29 q^{57} -2.11913e29 q^{58} -5.91397e28 q^{59} +2.18117e29 q^{61} -2.75325e29 q^{62} +1.11297e29 q^{63} -7.03715e29 q^{64} +1.24422e30 q^{66} -2.21631e30 q^{67} -2.97613e30 q^{68} -4.61937e30 q^{69} +2.87498e30 q^{71} -5.35634e29 q^{72} +3.41825e30 q^{73} -6.33220e30 q^{74} +2.00953e31 q^{76} -1.07406e30 q^{77} -1.82498e31 q^{78} +5.87385e30 q^{79} -8.66769e30 q^{81} +8.49743e31 q^{82} +2.11211e31 q^{83} +1.43877e31 q^{84} -2.87734e31 q^{86} +1.87368e32 q^{87} +5.16909e30 q^{88} -2.67672e32 q^{89} +1.57540e31 q^{91} -3.56821e32 q^{92} +2.43435e32 q^{93} +9.91253e32 q^{94} +1.14895e33 q^{96} -1.07066e31 q^{97} +1.00344e33 q^{98} -6.57344e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + O(q^{10}) \) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + 150760896555890192q^{11} - 466351581756413200q^{12} + 1403095636752804700q^{13} - 7086110008989891456q^{14} + 64511520005858668816q^{16} - \)\(13\!\cdots\!00\)\(q^{17} - \)\(67\!\cdots\!50\)\(q^{18} + \)\(17\!\cdots\!00\)\(q^{19} - \)\(10\!\cdots\!28\)\(q^{21} - \)\(24\!\cdots\!00\)\(q^{22} - \)\(68\!\cdots\!00\)\(q^{23} + \)\(21\!\cdots\!00\)\(q^{24} - \)\(36\!\cdots\!88\)\(q^{26} + \)\(34\!\cdots\!00\)\(q^{27} + \)\(17\!\cdots\!00\)\(q^{28} + \)\(13\!\cdots\!00\)\(q^{29} + \)\(10\!\cdots\!52\)\(q^{31} - \)\(26\!\cdots\!00\)\(q^{32} - \)\(34\!\cdots\!00\)\(q^{33} + \)\(88\!\cdots\!24\)\(q^{34} + \)\(16\!\cdots\!96\)\(q^{36} - \)\(27\!\cdots\!00\)\(q^{37} - \)\(83\!\cdots\!00\)\(q^{38} + \)\(57\!\cdots\!56\)\(q^{39} + \)\(44\!\cdots\!32\)\(q^{41} - \)\(14\!\cdots\!00\)\(q^{42} - \)\(22\!\cdots\!00\)\(q^{43} + \)\(71\!\cdots\!64\)\(q^{44} + \)\(34\!\cdots\!72\)\(q^{46} - \)\(38\!\cdots\!00\)\(q^{47} - \)\(24\!\cdots\!00\)\(q^{48} - \)\(71\!\cdots\!58\)\(q^{49} - \)\(54\!\cdots\!88\)\(q^{51} + \)\(90\!\cdots\!00\)\(q^{52} - \)\(47\!\cdots\!00\)\(q^{53} - \)\(18\!\cdots\!00\)\(q^{54} - \)\(39\!\cdots\!00\)\(q^{56} + \)\(24\!\cdots\!00\)\(q^{57} + \)\(44\!\cdots\!00\)\(q^{58} - \)\(17\!\cdots\!00\)\(q^{59} - \)\(28\!\cdots\!08\)\(q^{61} - \)\(34\!\cdots\!00\)\(q^{62} + \)\(55\!\cdots\!00\)\(q^{63} + \)\(19\!\cdots\!72\)\(q^{64} + \)\(50\!\cdots\!64\)\(q^{66} - \)\(33\!\cdots\!00\)\(q^{67} - \)\(38\!\cdots\!00\)\(q^{68} + \)\(31\!\cdots\!36\)\(q^{69} + \)\(84\!\cdots\!72\)\(q^{71} - \)\(87\!\cdots\!00\)\(q^{72} + \)\(17\!\cdots\!00\)\(q^{73} + \)\(20\!\cdots\!84\)\(q^{74} + \)\(29\!\cdots\!00\)\(q^{76} + \)\(11\!\cdots\!00\)\(q^{77} + \)\(57\!\cdots\!00\)\(q^{78} - \)\(22\!\cdots\!00\)\(q^{79} - \)\(13\!\cdots\!74\)\(q^{81} + \)\(15\!\cdots\!00\)\(q^{82} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(23\!\cdots\!76\)\(q^{84} - \)\(12\!\cdots\!08\)\(q^{86} + \)\(17\!\cdots\!00\)\(q^{87} - \)\(49\!\cdots\!00\)\(q^{88} - \)\(33\!\cdots\!00\)\(q^{89} + \)\(19\!\cdots\!32\)\(q^{91} + \)\(41\!\cdots\!00\)\(q^{92} - \)\(21\!\cdots\!00\)\(q^{93} + \)\(14\!\cdots\!64\)\(q^{94} + \)\(31\!\cdots\!72\)\(q^{96} - \)\(25\!\cdots\!00\)\(q^{97} - \)\(69\!\cdots\!50\)\(q^{98} + \)\(16\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −132922. −1.43417 −0.717085 0.696986i \(-0.754524\pi\)
−0.717085 + 0.696986i \(0.754524\pi\)
\(3\) 1.17526e8 1.57627 0.788137 0.615499i \(-0.211046\pi\)
0.788137 + 0.615499i \(0.211046\pi\)
\(4\) 9.07819e9 1.05684
\(5\) 0 0
\(6\) −1.56217e13 −2.26064
\(7\) 1.34853e13 0.153371 0.0766854 0.997055i \(-0.475566\pi\)
0.0766854 + 0.997055i \(0.475566\pi\)
\(8\) −6.49000e13 −0.0815192
\(9\) 8.25322e15 1.48464
\(10\) 0 0
\(11\) −7.96470e16 −0.522625 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(12\) 1.06692e18 1.66587
\(13\) 1.16823e18 0.486928 0.243464 0.969910i \(-0.421716\pi\)
0.243464 + 0.969910i \(0.421716\pi\)
\(14\) −1.79249e18 −0.219960
\(15\) 0 0
\(16\) −6.93545e19 −0.939928
\(17\) −3.27833e20 −1.63398 −0.816988 0.576654i \(-0.804358\pi\)
−0.816988 + 0.576654i \(0.804358\pi\)
\(18\) −1.09703e21 −2.12923
\(19\) 2.21358e21 1.76060 0.880301 0.474416i \(-0.157341\pi\)
0.880301 + 0.474416i \(0.157341\pi\)
\(20\) 0 0
\(21\) 1.58487e21 0.241755
\(22\) 1.05868e22 0.749533
\(23\) −3.93052e22 −1.33641 −0.668207 0.743975i \(-0.732938\pi\)
−0.668207 + 0.743975i \(0.732938\pi\)
\(24\) −7.62742e21 −0.128497
\(25\) 0 0
\(26\) −1.55283e23 −0.698337
\(27\) 3.16633e23 0.763930
\(28\) 1.22422e23 0.162089
\(29\) 1.59428e24 1.18303 0.591516 0.806293i \(-0.298530\pi\)
0.591516 + 0.806293i \(0.298530\pi\)
\(30\) 0 0
\(31\) 2.07133e24 0.511425 0.255712 0.966753i \(-0.417690\pi\)
0.255712 + 0.966753i \(0.417690\pi\)
\(32\) 9.77619e24 1.42954
\(33\) −9.36057e24 −0.823801
\(34\) 4.35761e25 2.34340
\(35\) 0 0
\(36\) 7.49243e25 1.56903
\(37\) 4.76387e25 0.634792 0.317396 0.948293i \(-0.397192\pi\)
0.317396 + 0.948293i \(0.397192\pi\)
\(38\) −2.94233e26 −2.52500
\(39\) 1.37297e26 0.767532
\(40\) 0 0
\(41\) −6.39281e26 −1.56588 −0.782940 0.622097i \(-0.786281\pi\)
−0.782940 + 0.622097i \(0.786281\pi\)
\(42\) −2.10663e26 −0.346717
\(43\) 2.16469e26 0.241639 0.120819 0.992674i \(-0.461448\pi\)
0.120819 + 0.992674i \(0.461448\pi\)
\(44\) −7.23051e26 −0.552331
\(45\) 0 0
\(46\) 5.22451e27 1.91664
\(47\) −7.45743e27 −1.91856 −0.959278 0.282462i \(-0.908849\pi\)
−0.959278 + 0.282462i \(0.908849\pi\)
\(48\) −8.15093e27 −1.48159
\(49\) −7.54914e27 −0.976477
\(50\) 0 0
\(51\) −3.85288e28 −2.57560
\(52\) 1.06055e28 0.514605
\(53\) −1.70592e28 −0.604513 −0.302257 0.953227i \(-0.597740\pi\)
−0.302257 + 0.953227i \(0.597740\pi\)
\(54\) −4.20873e28 −1.09560
\(55\) 0 0
\(56\) −8.75196e26 −0.0125027
\(57\) 2.60153e29 2.77519
\(58\) −2.11913e29 −1.69667
\(59\) −5.91397e28 −0.357126 −0.178563 0.983928i \(-0.557145\pi\)
−0.178563 + 0.983928i \(0.557145\pi\)
\(60\) 0 0
\(61\) 2.18117e29 0.759884 0.379942 0.925010i \(-0.375944\pi\)
0.379942 + 0.925010i \(0.375944\pi\)
\(62\) −2.75325e29 −0.733470
\(63\) 1.11297e29 0.227701
\(64\) −7.03715e29 −1.11027
\(65\) 0 0
\(66\) 1.24422e30 1.18147
\(67\) −2.21631e30 −1.64209 −0.821044 0.570865i \(-0.806608\pi\)
−0.821044 + 0.570865i \(0.806608\pi\)
\(68\) −2.97613e30 −1.72685
\(69\) −4.61937e30 −2.10656
\(70\) 0 0
\(71\) 2.87498e30 0.818223 0.409112 0.912484i \(-0.365839\pi\)
0.409112 + 0.912484i \(0.365839\pi\)
\(72\) −5.35634e29 −0.121027
\(73\) 3.41825e30 0.615143 0.307571 0.951525i \(-0.400484\pi\)
0.307571 + 0.951525i \(0.400484\pi\)
\(74\) −6.33220e30 −0.910398
\(75\) 0 0
\(76\) 2.00953e31 1.86068
\(77\) −1.07406e30 −0.0801555
\(78\) −1.82498e31 −1.10077
\(79\) 5.87385e30 0.287128 0.143564 0.989641i \(-0.454144\pi\)
0.143564 + 0.989641i \(0.454144\pi\)
\(80\) 0 0
\(81\) −8.66769e30 −0.280479
\(82\) 8.49743e31 2.24574
\(83\) 2.11211e31 0.457013 0.228507 0.973542i \(-0.426616\pi\)
0.228507 + 0.973542i \(0.426616\pi\)
\(84\) 1.43877e31 0.255496
\(85\) 0 0
\(86\) −2.87734e31 −0.346551
\(87\) 1.87368e32 1.86478
\(88\) 5.16909e30 0.0426040
\(89\) −2.67672e32 −1.83092 −0.915458 0.402413i \(-0.868172\pi\)
−0.915458 + 0.402413i \(0.868172\pi\)
\(90\) 0 0
\(91\) 1.57540e31 0.0746806
\(92\) −3.56821e32 −1.41238
\(93\) 2.43435e32 0.806146
\(94\) 9.91253e32 2.75153
\(95\) 0 0
\(96\) 1.14895e33 2.25334
\(97\) −1.07066e31 −0.0176977 −0.00884885 0.999961i \(-0.502817\pi\)
−0.00884885 + 0.999961i \(0.502817\pi\)
\(98\) 1.00344e33 1.40043
\(99\) −6.57344e32 −0.775911
\(100\) 0 0
\(101\) 1.55093e33 1.31610 0.658051 0.752974i \(-0.271381\pi\)
0.658051 + 0.752974i \(0.271381\pi\)
\(102\) 5.12131e33 3.69384
\(103\) −6.99665e32 −0.429612 −0.214806 0.976657i \(-0.568912\pi\)
−0.214806 + 0.976657i \(0.568912\pi\)
\(104\) −7.58184e31 −0.0396940
\(105\) 0 0
\(106\) 2.26753e33 0.866974
\(107\) 6.35208e32 0.208010 0.104005 0.994577i \(-0.466834\pi\)
0.104005 + 0.994577i \(0.466834\pi\)
\(108\) 2.87445e33 0.807352
\(109\) 2.09396e33 0.505163 0.252581 0.967576i \(-0.418720\pi\)
0.252581 + 0.967576i \(0.418720\pi\)
\(110\) 0 0
\(111\) 5.59876e33 1.00061
\(112\) −9.35266e32 −0.144158
\(113\) −7.28662e33 −0.969910 −0.484955 0.874539i \(-0.661164\pi\)
−0.484955 + 0.874539i \(0.661164\pi\)
\(114\) −3.45799e34 −3.98009
\(115\) 0 0
\(116\) 1.44731e34 1.25028
\(117\) 9.64169e33 0.722914
\(118\) 7.86094e33 0.512179
\(119\) −4.42093e33 −0.250604
\(120\) 0 0
\(121\) −1.68815e34 −0.726863
\(122\) −2.89924e34 −1.08980
\(123\) −7.51320e34 −2.46826
\(124\) 1.88040e34 0.540495
\(125\) 0 0
\(126\) −1.47938e34 −0.326562
\(127\) 4.47086e34 0.866222 0.433111 0.901341i \(-0.357416\pi\)
0.433111 + 0.901341i \(0.357416\pi\)
\(128\) 9.56208e33 0.162775
\(129\) 2.54407e34 0.380889
\(130\) 0 0
\(131\) −2.63964e34 −0.306597 −0.153298 0.988180i \(-0.548990\pi\)
−0.153298 + 0.988180i \(0.548990\pi\)
\(132\) −8.49770e34 −0.870626
\(133\) 2.98508e34 0.270025
\(134\) 2.94595e35 2.35503
\(135\) 0 0
\(136\) 2.12764e34 0.133201
\(137\) −1.03805e35 −0.575874 −0.287937 0.957649i \(-0.592969\pi\)
−0.287937 + 0.957649i \(0.592969\pi\)
\(138\) 6.14014e35 3.02116
\(139\) −1.77157e34 −0.0773774 −0.0386887 0.999251i \(-0.512318\pi\)
−0.0386887 + 0.999251i \(0.512318\pi\)
\(140\) 0 0
\(141\) −8.76440e35 −3.02417
\(142\) −3.82147e35 −1.17347
\(143\) −9.30464e34 −0.254481
\(144\) −5.72398e35 −1.39546
\(145\) 0 0
\(146\) −4.54358e35 −0.882219
\(147\) −8.87218e35 −1.53920
\(148\) 4.32473e35 0.670874
\(149\) 9.39618e35 1.30430 0.652151 0.758089i \(-0.273867\pi\)
0.652151 + 0.758089i \(0.273867\pi\)
\(150\) 0 0
\(151\) 4.46650e35 0.497563 0.248781 0.968560i \(-0.419970\pi\)
0.248781 + 0.968560i \(0.419970\pi\)
\(152\) −1.43661e35 −0.143523
\(153\) −2.70568e36 −2.42587
\(154\) 1.42766e35 0.114956
\(155\) 0 0
\(156\) 1.24641e36 0.811159
\(157\) 7.30629e35 0.427910 0.213955 0.976843i \(-0.431365\pi\)
0.213955 + 0.976843i \(0.431365\pi\)
\(158\) −7.80761e35 −0.411791
\(159\) −2.00489e36 −0.952879
\(160\) 0 0
\(161\) −5.30043e35 −0.204967
\(162\) 1.15212e36 0.402254
\(163\) 2.77089e35 0.0874026 0.0437013 0.999045i \(-0.486085\pi\)
0.0437013 + 0.999045i \(0.486085\pi\)
\(164\) −5.80352e36 −1.65489
\(165\) 0 0
\(166\) −2.80745e36 −0.655434
\(167\) −5.22919e36 −1.10564 −0.552818 0.833302i \(-0.686448\pi\)
−0.552818 + 0.833302i \(0.686448\pi\)
\(168\) −1.02858e35 −0.0197077
\(169\) −4.39136e36 −0.762901
\(170\) 0 0
\(171\) 1.82692e37 2.61386
\(172\) 1.96515e36 0.255374
\(173\) 5.38814e36 0.636324 0.318162 0.948036i \(-0.396934\pi\)
0.318162 + 0.948036i \(0.396934\pi\)
\(174\) −2.49053e37 −2.67441
\(175\) 0 0
\(176\) 5.52388e36 0.491230
\(177\) −6.95044e36 −0.562928
\(178\) 3.55794e37 2.62584
\(179\) 2.52793e37 1.70094 0.850472 0.526020i \(-0.176316\pi\)
0.850472 + 0.526020i \(0.176316\pi\)
\(180\) 0 0
\(181\) 1.65835e36 0.0928924 0.0464462 0.998921i \(-0.485210\pi\)
0.0464462 + 0.998921i \(0.485210\pi\)
\(182\) −2.09404e36 −0.107105
\(183\) 2.56343e37 1.19779
\(184\) 2.55091e36 0.108943
\(185\) 0 0
\(186\) −3.23577e37 −1.15615
\(187\) 2.61109e37 0.853957
\(188\) −6.77000e37 −2.02761
\(189\) 4.26989e36 0.117165
\(190\) 0 0
\(191\) 6.04480e37 1.39422 0.697110 0.716964i \(-0.254469\pi\)
0.697110 + 0.716964i \(0.254469\pi\)
\(192\) −8.27046e37 −1.75009
\(193\) −6.49082e36 −0.126068 −0.0630339 0.998011i \(-0.520078\pi\)
−0.0630339 + 0.998011i \(0.520078\pi\)
\(194\) 1.42313e36 0.0253815
\(195\) 0 0
\(196\) −6.85326e37 −1.03198
\(197\) −9.24040e37 −1.27938 −0.639688 0.768635i \(-0.720936\pi\)
−0.639688 + 0.768635i \(0.720936\pi\)
\(198\) 8.73752e37 1.11279
\(199\) −1.09203e38 −1.27985 −0.639925 0.768437i \(-0.721035\pi\)
−0.639925 + 0.768437i \(0.721035\pi\)
\(200\) 0 0
\(201\) −2.60473e38 −2.58838
\(202\) −2.06152e38 −1.88751
\(203\) 2.14993e37 0.181443
\(204\) −3.49772e38 −2.72200
\(205\) 0 0
\(206\) 9.30006e37 0.616136
\(207\) −3.24395e38 −1.98410
\(208\) −8.10223e37 −0.457677
\(209\) −1.76305e38 −0.920134
\(210\) 0 0
\(211\) −6.11672e37 −0.272808 −0.136404 0.990653i \(-0.543555\pi\)
−0.136404 + 0.990653i \(0.543555\pi\)
\(212\) −1.54866e38 −0.638874
\(213\) 3.37884e38 1.28974
\(214\) −8.44328e37 −0.298321
\(215\) 0 0
\(216\) −2.05495e37 −0.0622750
\(217\) 2.79325e37 0.0784377
\(218\) −2.78332e38 −0.724489
\(219\) 4.01731e38 0.969634
\(220\) 0 0
\(221\) −3.82986e38 −0.795629
\(222\) −7.44196e38 −1.43504
\(223\) −9.45280e38 −1.69251 −0.846253 0.532781i \(-0.821147\pi\)
−0.846253 + 0.532781i \(0.821147\pi\)
\(224\) 1.31835e38 0.219249
\(225\) 0 0
\(226\) 9.68548e38 1.39102
\(227\) −4.90469e38 −0.654914 −0.327457 0.944866i \(-0.606192\pi\)
−0.327457 + 0.944866i \(0.606192\pi\)
\(228\) 2.36172e39 2.93294
\(229\) −4.57241e38 −0.528275 −0.264137 0.964485i \(-0.585087\pi\)
−0.264137 + 0.964485i \(0.585087\pi\)
\(230\) 0 0
\(231\) −1.26230e38 −0.126347
\(232\) −1.03468e38 −0.0964398
\(233\) 8.69571e38 0.754975 0.377487 0.926015i \(-0.376788\pi\)
0.377487 + 0.926015i \(0.376788\pi\)
\(234\) −1.28159e39 −1.03678
\(235\) 0 0
\(236\) −5.36882e38 −0.377425
\(237\) 6.90328e38 0.452593
\(238\) 5.87637e38 0.359409
\(239\) −1.38618e39 −0.791140 −0.395570 0.918436i \(-0.629453\pi\)
−0.395570 + 0.918436i \(0.629453\pi\)
\(240\) 0 0
\(241\) −2.94732e39 −1.46604 −0.733019 0.680208i \(-0.761890\pi\)
−0.733019 + 0.680208i \(0.761890\pi\)
\(242\) 2.24392e39 1.04244
\(243\) −2.77886e39 −1.20604
\(244\) 1.98011e39 0.803076
\(245\) 0 0
\(246\) 9.98665e39 3.53990
\(247\) 2.58598e39 0.857286
\(248\) −1.34430e38 −0.0416910
\(249\) 2.48228e39 0.720379
\(250\) 0 0
\(251\) −5.17168e39 −1.31527 −0.657636 0.753336i \(-0.728444\pi\)
−0.657636 + 0.753336i \(0.728444\pi\)
\(252\) 1.01038e39 0.240644
\(253\) 3.13055e39 0.698444
\(254\) −5.94273e39 −1.24231
\(255\) 0 0
\(256\) 4.77386e39 0.876820
\(257\) 4.09624e39 0.705488 0.352744 0.935720i \(-0.385249\pi\)
0.352744 + 0.935720i \(0.385249\pi\)
\(258\) −3.38162e39 −0.546260
\(259\) 6.42422e38 0.0973585
\(260\) 0 0
\(261\) 1.31579e40 1.75638
\(262\) 3.50865e39 0.439712
\(263\) −2.59968e39 −0.305950 −0.152975 0.988230i \(-0.548885\pi\)
−0.152975 + 0.988230i \(0.548885\pi\)
\(264\) 6.07501e38 0.0671556
\(265\) 0 0
\(266\) −3.96781e39 −0.387262
\(267\) −3.14584e40 −2.88603
\(268\) −2.01201e40 −1.73543
\(269\) 7.20422e39 0.584351 0.292176 0.956365i \(-0.405621\pi\)
0.292176 + 0.956365i \(0.405621\pi\)
\(270\) 0 0
\(271\) −1.89743e40 −1.36199 −0.680993 0.732290i \(-0.738451\pi\)
−0.680993 + 0.732290i \(0.738451\pi\)
\(272\) 2.27367e40 1.53582
\(273\) 1.85150e39 0.117717
\(274\) 1.37979e40 0.825901
\(275\) 0 0
\(276\) −4.19356e40 −2.22629
\(277\) −3.08340e40 −1.54211 −0.771053 0.636771i \(-0.780270\pi\)
−0.771053 + 0.636771i \(0.780270\pi\)
\(278\) 2.35480e39 0.110972
\(279\) 1.70952e40 0.759283
\(280\) 0 0
\(281\) −1.56575e40 −0.618112 −0.309056 0.951044i \(-0.600013\pi\)
−0.309056 + 0.951044i \(0.600013\pi\)
\(282\) 1.16498e41 4.33718
\(283\) −7.24188e39 −0.254317 −0.127158 0.991882i \(-0.540586\pi\)
−0.127158 + 0.991882i \(0.540586\pi\)
\(284\) 2.60996e40 0.864732
\(285\) 0 0
\(286\) 1.23679e40 0.364968
\(287\) −8.62090e39 −0.240160
\(288\) 8.06850e40 2.12235
\(289\) 6.72202e40 1.66988
\(290\) 0 0
\(291\) −1.25830e39 −0.0278964
\(292\) 3.10315e40 0.650108
\(293\) −7.67091e40 −1.51891 −0.759453 0.650562i \(-0.774533\pi\)
−0.759453 + 0.650562i \(0.774533\pi\)
\(294\) 1.17930e41 2.20747
\(295\) 0 0
\(296\) −3.09175e39 −0.0517477
\(297\) −2.52188e40 −0.399249
\(298\) −1.24895e41 −1.87059
\(299\) −4.59177e40 −0.650737
\(300\) 0 0
\(301\) 2.91915e39 0.0370604
\(302\) −5.93694e40 −0.713589
\(303\) 1.82274e41 2.07454
\(304\) −1.53522e41 −1.65484
\(305\) 0 0
\(306\) 3.59643e41 3.47911
\(307\) 1.29298e41 1.18525 0.592625 0.805478i \(-0.298091\pi\)
0.592625 + 0.805478i \(0.298091\pi\)
\(308\) −9.75056e39 −0.0847116
\(309\) −8.22286e40 −0.677186
\(310\) 0 0
\(311\) −5.39990e40 −0.399797 −0.199898 0.979817i \(-0.564061\pi\)
−0.199898 + 0.979817i \(0.564061\pi\)
\(312\) −8.91061e39 −0.0625686
\(313\) −1.08588e41 −0.723270 −0.361635 0.932320i \(-0.617781\pi\)
−0.361635 + 0.932320i \(0.617781\pi\)
\(314\) −9.71163e40 −0.613696
\(315\) 0 0
\(316\) 5.33240e40 0.303449
\(317\) 1.38129e41 0.746117 0.373059 0.927808i \(-0.378309\pi\)
0.373059 + 0.927808i \(0.378309\pi\)
\(318\) 2.66493e41 1.36659
\(319\) −1.26979e41 −0.618282
\(320\) 0 0
\(321\) 7.46533e40 0.327881
\(322\) 7.04541e40 0.293957
\(323\) −7.25686e41 −2.87678
\(324\) −7.86869e40 −0.296422
\(325\) 0 0
\(326\) −3.68311e40 −0.125350
\(327\) 2.46094e41 0.796275
\(328\) 4.14894e40 0.127649
\(329\) −1.00566e41 −0.294251
\(330\) 0 0
\(331\) −4.29457e41 −1.13699 −0.568496 0.822686i \(-0.692474\pi\)
−0.568496 + 0.822686i \(0.692474\pi\)
\(332\) 1.91742e41 0.482990
\(333\) 3.93172e41 0.942439
\(334\) 6.95072e41 1.58567
\(335\) 0 0
\(336\) −1.09918e41 −0.227232
\(337\) −6.97107e41 −1.37216 −0.686082 0.727524i \(-0.740671\pi\)
−0.686082 + 0.727524i \(0.740671\pi\)
\(338\) 5.83706e41 1.09413
\(339\) −8.56364e41 −1.52885
\(340\) 0 0
\(341\) −1.64975e41 −0.267284
\(342\) −2.42837e42 −3.74872
\(343\) −2.06057e41 −0.303134
\(344\) −1.40489e40 −0.0196982
\(345\) 0 0
\(346\) −7.16200e41 −0.912596
\(347\) −7.56830e41 −0.919522 −0.459761 0.888043i \(-0.652065\pi\)
−0.459761 + 0.888043i \(0.652065\pi\)
\(348\) 1.70096e42 1.97078
\(349\) 4.06051e41 0.448705 0.224353 0.974508i \(-0.427973\pi\)
0.224353 + 0.974508i \(0.427973\pi\)
\(350\) 0 0
\(351\) 3.69901e41 0.371979
\(352\) −7.78644e41 −0.747111
\(353\) 1.97994e42 1.81288 0.906439 0.422336i \(-0.138790\pi\)
0.906439 + 0.422336i \(0.138790\pi\)
\(354\) 9.23863e41 0.807335
\(355\) 0 0
\(356\) −2.42998e42 −1.93499
\(357\) −5.19573e41 −0.395022
\(358\) −3.36016e42 −2.43944
\(359\) −6.70289e41 −0.464733 −0.232366 0.972628i \(-0.574647\pi\)
−0.232366 + 0.972628i \(0.574647\pi\)
\(360\) 0 0
\(361\) 3.31917e42 2.09972
\(362\) −2.20431e41 −0.133223
\(363\) −1.98401e42 −1.14574
\(364\) 1.43018e41 0.0789255
\(365\) 0 0
\(366\) −3.40736e42 −1.71783
\(367\) 1.36571e42 0.658215 0.329108 0.944292i \(-0.393252\pi\)
0.329108 + 0.944292i \(0.393252\pi\)
\(368\) 2.72599e42 1.25613
\(369\) −5.27613e42 −2.32477
\(370\) 0 0
\(371\) −2.30048e41 −0.0927147
\(372\) 2.20995e42 0.851968
\(373\) −3.32617e41 −0.122673 −0.0613364 0.998117i \(-0.519536\pi\)
−0.0613364 + 0.998117i \(0.519536\pi\)
\(374\) −3.47071e42 −1.22472
\(375\) 0 0
\(376\) 4.83988e41 0.156399
\(377\) 1.86249e42 0.576051
\(378\) −5.67560e41 −0.168034
\(379\) 1.32387e42 0.375230 0.187615 0.982243i \(-0.439924\pi\)
0.187615 + 0.982243i \(0.439924\pi\)
\(380\) 0 0
\(381\) 5.25440e42 1.36540
\(382\) −8.03484e42 −1.99955
\(383\) −5.38010e42 −1.28236 −0.641181 0.767390i \(-0.721555\pi\)
−0.641181 + 0.767390i \(0.721555\pi\)
\(384\) 1.12379e42 0.256578
\(385\) 0 0
\(386\) 8.62770e41 0.180803
\(387\) 1.78657e42 0.358747
\(388\) −9.71963e40 −0.0187036
\(389\) −6.93155e42 −1.27839 −0.639193 0.769046i \(-0.720731\pi\)
−0.639193 + 0.769046i \(0.720731\pi\)
\(390\) 0 0
\(391\) 1.28856e43 2.18367
\(392\) 4.89939e41 0.0796017
\(393\) −3.10225e42 −0.483281
\(394\) 1.22825e43 1.83484
\(395\) 0 0
\(396\) −5.96750e42 −0.820015
\(397\) 9.33530e42 1.23051 0.615254 0.788329i \(-0.289053\pi\)
0.615254 + 0.788329i \(0.289053\pi\)
\(398\) 1.45154e43 1.83552
\(399\) 3.50824e42 0.425634
\(400\) 0 0
\(401\) 3.40931e42 0.380877 0.190438 0.981699i \(-0.439009\pi\)
0.190438 + 0.981699i \(0.439009\pi\)
\(402\) 3.46225e43 3.71218
\(403\) 2.41980e42 0.249027
\(404\) 1.40796e43 1.39091
\(405\) 0 0
\(406\) −2.85772e42 −0.260219
\(407\) −3.79428e42 −0.331758
\(408\) 2.50052e42 0.209961
\(409\) 3.12051e42 0.251646 0.125823 0.992053i \(-0.459843\pi\)
0.125823 + 0.992053i \(0.459843\pi\)
\(410\) 0 0
\(411\) −1.21997e43 −0.907736
\(412\) −6.35170e42 −0.454031
\(413\) −7.97517e41 −0.0547727
\(414\) 4.31190e43 2.84553
\(415\) 0 0
\(416\) 1.14209e43 0.696081
\(417\) −2.08205e42 −0.121968
\(418\) 2.34347e43 1.31963
\(419\) −3.20466e43 −1.73480 −0.867402 0.497608i \(-0.834212\pi\)
−0.867402 + 0.497608i \(0.834212\pi\)
\(420\) 0 0
\(421\) 5.44061e42 0.272266 0.136133 0.990691i \(-0.456533\pi\)
0.136133 + 0.990691i \(0.456533\pi\)
\(422\) 8.13043e42 0.391253
\(423\) −6.15478e43 −2.84837
\(424\) 1.10714e42 0.0492794
\(425\) 0 0
\(426\) −4.49120e43 −1.84971
\(427\) 2.94137e42 0.116544
\(428\) 5.76654e42 0.219833
\(429\) −1.09353e43 −0.401132
\(430\) 0 0
\(431\) −3.50771e43 −1.19165 −0.595826 0.803114i \(-0.703175\pi\)
−0.595826 + 0.803114i \(0.703175\pi\)
\(432\) −2.19599e43 −0.718039
\(433\) −4.30567e43 −1.35516 −0.677580 0.735449i \(-0.736971\pi\)
−0.677580 + 0.735449i \(0.736971\pi\)
\(434\) −3.71284e42 −0.112493
\(435\) 0 0
\(436\) 1.90094e43 0.533876
\(437\) −8.70054e43 −2.35289
\(438\) −5.33988e43 −1.39062
\(439\) −1.81689e43 −0.455685 −0.227843 0.973698i \(-0.573167\pi\)
−0.227843 + 0.973698i \(0.573167\pi\)
\(440\) 0 0
\(441\) −6.23047e43 −1.44972
\(442\) 5.09071e43 1.14107
\(443\) 2.81575e43 0.608040 0.304020 0.952666i \(-0.401671\pi\)
0.304020 + 0.952666i \(0.401671\pi\)
\(444\) 5.08267e43 1.05748
\(445\) 0 0
\(446\) 1.25648e44 2.42734
\(447\) 1.10429e44 2.05594
\(448\) −9.48981e42 −0.170283
\(449\) 2.21288e43 0.382731 0.191365 0.981519i \(-0.438708\pi\)
0.191365 + 0.981519i \(0.438708\pi\)
\(450\) 0 0
\(451\) 5.09169e43 0.818368
\(452\) −6.61493e43 −1.02504
\(453\) 5.24928e43 0.784295
\(454\) 6.51938e43 0.939258
\(455\) 0 0
\(456\) −1.68839e43 −0.226231
\(457\) 6.39404e43 0.826339 0.413169 0.910654i \(-0.364422\pi\)
0.413169 + 0.910654i \(0.364422\pi\)
\(458\) 6.07772e43 0.757636
\(459\) −1.03803e44 −1.24824
\(460\) 0 0
\(461\) 8.79013e42 0.0983853 0.0491926 0.998789i \(-0.484335\pi\)
0.0491926 + 0.998789i \(0.484335\pi\)
\(462\) 1.67787e43 0.181203
\(463\) 1.18763e44 1.23764 0.618821 0.785532i \(-0.287611\pi\)
0.618821 + 0.785532i \(0.287611\pi\)
\(464\) −1.10570e44 −1.11196
\(465\) 0 0
\(466\) −1.15585e44 −1.08276
\(467\) −1.78679e44 −1.61564 −0.807819 0.589431i \(-0.799352\pi\)
−0.807819 + 0.589431i \(0.799352\pi\)
\(468\) 8.75291e43 0.764005
\(469\) −2.98876e43 −0.251849
\(470\) 0 0
\(471\) 8.58677e43 0.674504
\(472\) 3.83817e42 0.0291126
\(473\) −1.72411e43 −0.126287
\(474\) −9.17595e43 −0.649095
\(475\) 0 0
\(476\) −4.01341e43 −0.264849
\(477\) −1.40793e44 −0.897486
\(478\) 1.84253e44 1.13463
\(479\) 1.06259e44 0.632165 0.316082 0.948732i \(-0.397632\pi\)
0.316082 + 0.948732i \(0.397632\pi\)
\(480\) 0 0
\(481\) 5.56531e43 0.309098
\(482\) 3.91762e44 2.10255
\(483\) −6.22936e43 −0.323084
\(484\) −1.53254e44 −0.768178
\(485\) 0 0
\(486\) 3.69370e44 1.72967
\(487\) −1.85504e44 −0.839700 −0.419850 0.907593i \(-0.637917\pi\)
−0.419850 + 0.907593i \(0.637917\pi\)
\(488\) −1.41558e43 −0.0619451
\(489\) 3.25651e43 0.137771
\(490\) 0 0
\(491\) 8.88210e43 0.351294 0.175647 0.984453i \(-0.443798\pi\)
0.175647 + 0.984453i \(0.443798\pi\)
\(492\) −6.82062e44 −2.60855
\(493\) −5.22657e44 −1.93305
\(494\) −3.43733e44 −1.22949
\(495\) 0 0
\(496\) −1.43656e44 −0.480703
\(497\) 3.87700e43 0.125492
\(498\) −3.29948e44 −1.03314
\(499\) 2.77853e44 0.841699 0.420849 0.907131i \(-0.361732\pi\)
0.420849 + 0.907131i \(0.361732\pi\)
\(500\) 0 0
\(501\) −6.14564e44 −1.74279
\(502\) 6.87428e44 1.88632
\(503\) −8.80760e41 −0.00233876 −0.00116938 0.999999i \(-0.500372\pi\)
−0.00116938 + 0.999999i \(0.500372\pi\)
\(504\) −7.22319e42 −0.0185620
\(505\) 0 0
\(506\) −4.16117e44 −1.00169
\(507\) −5.16097e44 −1.20254
\(508\) 4.05873e44 0.915459
\(509\) −8.68803e44 −1.89705 −0.948523 0.316709i \(-0.897422\pi\)
−0.948523 + 0.316709i \(0.897422\pi\)
\(510\) 0 0
\(511\) 4.60961e43 0.0943450
\(512\) −7.16687e44 −1.42028
\(513\) 7.00892e44 1.34498
\(514\) −5.44479e44 −1.01179
\(515\) 0 0
\(516\) 2.30956e44 0.402539
\(517\) 5.93962e44 1.00269
\(518\) −8.53917e43 −0.139629
\(519\) 6.33245e44 1.00302
\(520\) 0 0
\(521\) −2.56437e44 −0.381205 −0.190602 0.981667i \(-0.561044\pi\)
−0.190602 + 0.981667i \(0.561044\pi\)
\(522\) −1.74897e45 −2.51894
\(523\) 6.37079e43 0.0889028 0.0444514 0.999012i \(-0.485846\pi\)
0.0444514 + 0.999012i \(0.485846\pi\)
\(524\) −2.39631e44 −0.324024
\(525\) 0 0
\(526\) 3.45554e44 0.438784
\(527\) −6.79052e44 −0.835657
\(528\) 6.49197e44 0.774314
\(529\) 6.79897e44 0.786004
\(530\) 0 0
\(531\) −4.88093e44 −0.530204
\(532\) 2.70991e44 0.285373
\(533\) −7.46830e44 −0.762471
\(534\) 4.18149e45 4.13905
\(535\) 0 0
\(536\) 1.43839e44 0.133862
\(537\) 2.97097e45 2.68116
\(538\) −9.57595e44 −0.838059
\(539\) 6.01267e44 0.510332
\(540\) 0 0
\(541\) −1.56706e45 −1.25121 −0.625605 0.780140i \(-0.715148\pi\)
−0.625605 + 0.780140i \(0.715148\pi\)
\(542\) 2.52209e45 1.95332
\(543\) 1.94899e44 0.146424
\(544\) −3.20496e45 −2.33583
\(545\) 0 0
\(546\) −2.46104e44 −0.168826
\(547\) 2.05395e45 1.36710 0.683548 0.729905i \(-0.260436\pi\)
0.683548 + 0.729905i \(0.260436\pi\)
\(548\) −9.42359e44 −0.608607
\(549\) 1.80017e45 1.12816
\(550\) 0 0
\(551\) 3.52906e45 2.08285
\(552\) 2.99797e44 0.171725
\(553\) 7.92107e43 0.0440371
\(554\) 4.09851e45 2.21164
\(555\) 0 0
\(556\) −1.60827e44 −0.0817756
\(557\) −3.02985e44 −0.149558 −0.0747792 0.997200i \(-0.523825\pi\)
−0.0747792 + 0.997200i \(0.523825\pi\)
\(558\) −2.27231e45 −1.08894
\(559\) 2.52887e44 0.117661
\(560\) 0 0
\(561\) 3.06871e45 1.34607
\(562\) 2.08122e45 0.886478
\(563\) 2.25447e45 0.932514 0.466257 0.884649i \(-0.345602\pi\)
0.466257 + 0.884649i \(0.345602\pi\)
\(564\) −7.95649e45 −3.19607
\(565\) 0 0
\(566\) 9.62602e44 0.364733
\(567\) −1.16886e44 −0.0430173
\(568\) −1.86586e44 −0.0667009
\(569\) −3.27232e45 −1.13633 −0.568163 0.822916i \(-0.692346\pi\)
−0.568163 + 0.822916i \(0.692346\pi\)
\(570\) 0 0
\(571\) 3.07381e45 1.00735 0.503675 0.863893i \(-0.331981\pi\)
0.503675 + 0.863893i \(0.331981\pi\)
\(572\) −8.44693e44 −0.268946
\(573\) 7.10419e45 2.19768
\(574\) 1.14590e45 0.344431
\(575\) 0 0
\(576\) −5.80792e45 −1.64835
\(577\) 6.81899e43 0.0188070 0.00940349 0.999956i \(-0.497007\pi\)
0.00940349 + 0.999956i \(0.497007\pi\)
\(578\) −8.93501e45 −2.39489
\(579\) −7.62838e44 −0.198718
\(580\) 0 0
\(581\) 2.84825e44 0.0700926
\(582\) 1.67255e44 0.0400082
\(583\) 1.35871e45 0.315934
\(584\) −2.21844e44 −0.0501460
\(585\) 0 0
\(586\) 1.01963e46 2.17837
\(587\) 8.96797e45 1.86280 0.931398 0.364002i \(-0.118590\pi\)
0.931398 + 0.364002i \(0.118590\pi\)
\(588\) −8.05433e45 −1.62669
\(589\) 4.58506e45 0.900416
\(590\) 0 0
\(591\) −1.08598e46 −2.01665
\(592\) −3.30395e45 −0.596659
\(593\) 3.75910e45 0.660209 0.330104 0.943944i \(-0.392916\pi\)
0.330104 + 0.943944i \(0.392916\pi\)
\(594\) 3.35213e45 0.572590
\(595\) 0 0
\(596\) 8.53004e45 1.37844
\(597\) −1.28342e46 −2.01740
\(598\) 6.10345e45 0.933268
\(599\) 8.80446e45 1.30966 0.654832 0.755774i \(-0.272739\pi\)
0.654832 + 0.755774i \(0.272739\pi\)
\(600\) 0 0
\(601\) −4.53048e45 −0.637846 −0.318923 0.947781i \(-0.603321\pi\)
−0.318923 + 0.947781i \(0.603321\pi\)
\(602\) −3.88018e44 −0.0531508
\(603\) −1.82917e46 −2.43791
\(604\) 4.05477e45 0.525844
\(605\) 0 0
\(606\) −2.42281e46 −2.97524
\(607\) −1.02463e46 −1.22448 −0.612242 0.790671i \(-0.709732\pi\)
−0.612242 + 0.790671i \(0.709732\pi\)
\(608\) 2.16404e46 2.51684
\(609\) 2.52672e45 0.286003
\(610\) 0 0
\(611\) −8.71203e45 −0.934199
\(612\) −2.45627e46 −2.56376
\(613\) 4.85425e45 0.493201 0.246601 0.969117i \(-0.420686\pi\)
0.246601 + 0.969117i \(0.420686\pi\)
\(614\) −1.71865e46 −1.69985
\(615\) 0 0
\(616\) 6.97068e43 0.00653421
\(617\) 1.62668e46 1.48455 0.742277 0.670093i \(-0.233746\pi\)
0.742277 + 0.670093i \(0.233746\pi\)
\(618\) 1.09300e46 0.971200
\(619\) −2.50157e45 −0.216430 −0.108215 0.994128i \(-0.534514\pi\)
−0.108215 + 0.994128i \(0.534514\pi\)
\(620\) 0 0
\(621\) −1.24453e46 −1.02093
\(622\) 7.17763e45 0.573376
\(623\) −3.60964e45 −0.280809
\(624\) −9.52220e45 −0.721425
\(625\) 0 0
\(626\) 1.44337e46 1.03729
\(627\) −2.07204e46 −1.45038
\(628\) 6.63279e45 0.452233
\(629\) −1.56175e46 −1.03723
\(630\) 0 0
\(631\) 6.94339e45 0.437610 0.218805 0.975769i \(-0.429784\pi\)
0.218805 + 0.975769i \(0.429784\pi\)
\(632\) −3.81213e44 −0.0234065
\(633\) −7.18871e45 −0.430021
\(634\) −1.83603e46 −1.07006
\(635\) 0 0
\(636\) −1.82008e46 −1.00704
\(637\) −8.81916e45 −0.475474
\(638\) 1.68783e46 0.886721
\(639\) 2.37278e46 1.21477
\(640\) 0 0
\(641\) 4.34529e45 0.211282 0.105641 0.994404i \(-0.466311\pi\)
0.105641 + 0.994404i \(0.466311\pi\)
\(642\) −9.92303e45 −0.470236
\(643\) −2.56764e46 −1.18591 −0.592956 0.805235i \(-0.702039\pi\)
−0.592956 + 0.805235i \(0.702039\pi\)
\(644\) −4.81183e45 −0.216618
\(645\) 0 0
\(646\) 9.64592e46 4.12579
\(647\) 1.55928e46 0.650136 0.325068 0.945691i \(-0.394613\pi\)
0.325068 + 0.945691i \(0.394613\pi\)
\(648\) 5.62533e44 0.0228644
\(649\) 4.71030e45 0.186643
\(650\) 0 0
\(651\) 3.28279e45 0.123639
\(652\) 2.51547e45 0.0923706
\(653\) −1.54838e45 −0.0554384 −0.0277192 0.999616i \(-0.508824\pi\)
−0.0277192 + 0.999616i \(0.508824\pi\)
\(654\) −3.27112e46 −1.14199
\(655\) 0 0
\(656\) 4.43370e46 1.47182
\(657\) 2.82115e46 0.913267
\(658\) 1.33673e46 0.422005
\(659\) −1.05260e46 −0.324080 −0.162040 0.986784i \(-0.551807\pi\)
−0.162040 + 0.986784i \(0.551807\pi\)
\(660\) 0 0
\(661\) −1.48865e46 −0.435981 −0.217990 0.975951i \(-0.569950\pi\)
−0.217990 + 0.975951i \(0.569950\pi\)
\(662\) 5.70840e46 1.63064
\(663\) −4.50107e46 −1.25413
\(664\) −1.37076e45 −0.0372554
\(665\) 0 0
\(666\) −5.22611e46 −1.35162
\(667\) −6.26634e46 −1.58102
\(668\) −4.74716e46 −1.16848
\(669\) −1.11095e47 −2.66786
\(670\) 0 0
\(671\) −1.73724e46 −0.397134
\(672\) 1.54940e46 0.345597
\(673\) 2.78172e46 0.605432 0.302716 0.953081i \(-0.402107\pi\)
0.302716 + 0.953081i \(0.402107\pi\)
\(674\) 9.26605e46 1.96791
\(675\) 0 0
\(676\) −3.98656e46 −0.806265
\(677\) 9.09948e45 0.179599 0.0897995 0.995960i \(-0.471377\pi\)
0.0897995 + 0.995960i \(0.471377\pi\)
\(678\) 1.13829e47 2.19262
\(679\) −1.44381e44 −0.00271431
\(680\) 0 0
\(681\) −5.76426e46 −1.03233
\(682\) 2.19288e46 0.383330
\(683\) −2.07397e46 −0.353883 −0.176942 0.984221i \(-0.556620\pi\)
−0.176942 + 0.984221i \(0.556620\pi\)
\(684\) 1.65851e47 2.76244
\(685\) 0 0
\(686\) 2.73894e46 0.434746
\(687\) −5.37376e46 −0.832707
\(688\) −1.50131e46 −0.227123
\(689\) −1.99291e46 −0.294354
\(690\) 0 0
\(691\) 7.80876e46 1.09950 0.549749 0.835330i \(-0.314723\pi\)
0.549749 + 0.835330i \(0.314723\pi\)
\(692\) 4.89146e46 0.672493
\(693\) −8.86448e45 −0.119002
\(694\) 1.00599e47 1.31875
\(695\) 0 0
\(696\) −1.21602e46 −0.152016
\(697\) 2.09578e47 2.55861
\(698\) −5.39729e46 −0.643519
\(699\) 1.02197e47 1.19005
\(700\) 0 0
\(701\) −7.28838e46 −0.809624 −0.404812 0.914400i \(-0.632663\pi\)
−0.404812 + 0.914400i \(0.632663\pi\)
\(702\) −4.91678e46 −0.533480
\(703\) 1.05452e47 1.11761
\(704\) 5.60488e46 0.580253
\(705\) 0 0
\(706\) −2.63176e47 −2.59997
\(707\) 2.09147e46 0.201852
\(708\) −6.30974e46 −0.594926
\(709\) 4.67334e46 0.430491 0.215246 0.976560i \(-0.430945\pi\)
0.215246 + 0.976560i \(0.430945\pi\)
\(710\) 0 0
\(711\) 4.84782e46 0.426283
\(712\) 1.73719e46 0.149255
\(713\) −8.14142e46 −0.683476
\(714\) 6.90624e46 0.566528
\(715\) 0 0
\(716\) 2.29490e47 1.79763
\(717\) −1.62911e47 −1.24705
\(718\) 8.90958e46 0.666506
\(719\) −1.27931e47 −0.935299 −0.467649 0.883914i \(-0.654899\pi\)
−0.467649 + 0.883914i \(0.654899\pi\)
\(720\) 0 0
\(721\) −9.43520e45 −0.0658899
\(722\) −4.41189e47 −3.01135
\(723\) −3.46385e47 −2.31088
\(724\) 1.50548e46 0.0981725
\(725\) 0 0
\(726\) 2.63718e47 1.64318
\(727\) 1.68652e47 1.02724 0.513621 0.858017i \(-0.328304\pi\)
0.513621 + 0.858017i \(0.328304\pi\)
\(728\) −1.02243e45 −0.00608790
\(729\) −2.78403e47 −1.62057
\(730\) 0 0
\(731\) −7.09659e46 −0.394832
\(732\) 2.32713e47 1.26587
\(733\) −3.19951e47 −1.70164 −0.850821 0.525456i \(-0.823895\pi\)
−0.850821 + 0.525456i \(0.823895\pi\)
\(734\) −1.81532e47 −0.943992
\(735\) 0 0
\(736\) −3.84255e47 −1.91045
\(737\) 1.76523e47 0.858196
\(738\) 7.01311e47 3.33412
\(739\) −1.33356e47 −0.619980 −0.309990 0.950740i \(-0.600326\pi\)
−0.309990 + 0.950740i \(0.600326\pi\)
\(740\) 0 0
\(741\) 3.03919e47 1.35132
\(742\) 3.05783e46 0.132969
\(743\) −3.22840e46 −0.137300 −0.0686502 0.997641i \(-0.521869\pi\)
−0.0686502 + 0.997641i \(0.521869\pi\)
\(744\) −1.57989e46 −0.0657164
\(745\) 0 0
\(746\) 4.42119e46 0.175934
\(747\) 1.74317e47 0.678502
\(748\) 2.37040e47 0.902497
\(749\) 8.56597e45 0.0319027
\(750\) 0 0
\(751\) 2.39672e46 0.0854196 0.0427098 0.999088i \(-0.486401\pi\)
0.0427098 + 0.999088i \(0.486401\pi\)
\(752\) 5.17206e47 1.80331
\(753\) −6.07806e47 −2.07323
\(754\) −2.47565e47 −0.826154
\(755\) 0 0
\(756\) 3.87628e46 0.123824
\(757\) 6.92584e46 0.216466 0.108233 0.994126i \(-0.465481\pi\)
0.108233 + 0.994126i \(0.465481\pi\)
\(758\) −1.75970e47 −0.538143
\(759\) 3.67919e47 1.10094
\(760\) 0 0
\(761\) −4.97287e47 −1.42482 −0.712411 0.701763i \(-0.752397\pi\)
−0.712411 + 0.701763i \(0.752397\pi\)
\(762\) −6.98423e47 −1.95822
\(763\) 2.82377e46 0.0774773
\(764\) 5.48759e47 1.47347
\(765\) 0 0
\(766\) 7.15131e47 1.83912
\(767\) −6.90891e46 −0.173895
\(768\) 5.61051e47 1.38211
\(769\) 2.89778e47 0.698683 0.349342 0.936995i \(-0.386405\pi\)
0.349342 + 0.936995i \(0.386405\pi\)
\(770\) 0 0
\(771\) 4.81414e47 1.11204
\(772\) −5.89249e46 −0.133234
\(773\) 5.10656e47 1.13023 0.565115 0.825012i \(-0.308832\pi\)
0.565115 + 0.825012i \(0.308832\pi\)
\(774\) −2.37473e47 −0.514504
\(775\) 0 0
\(776\) 6.94857e44 0.00144270
\(777\) 7.55010e46 0.153464
\(778\) 9.21352e47 1.83342
\(779\) −1.41510e48 −2.75689
\(780\) 0 0
\(781\) −2.28983e47 −0.427624
\(782\) −1.71277e48 −3.13175
\(783\) 5.04799e47 0.903753
\(784\) 5.23567e47 0.917819
\(785\) 0 0
\(786\) 4.12356e47 0.693107
\(787\) −3.90010e47 −0.641938 −0.320969 0.947090i \(-0.604009\pi\)
−0.320969 + 0.947090i \(0.604009\pi\)
\(788\) −8.38862e47 −1.35210
\(789\) −3.05529e47 −0.482261
\(790\) 0 0
\(791\) −9.82622e46 −0.148756
\(792\) 4.26617e46 0.0632517
\(793\) 2.54812e47 0.370009
\(794\) −1.24086e48 −1.76476
\(795\) 0 0
\(796\) −9.91367e47 −1.35260
\(797\) −6.38986e47 −0.853943 −0.426972 0.904265i \(-0.640420\pi\)
−0.426972 + 0.904265i \(0.640420\pi\)
\(798\) −4.66320e47 −0.610431
\(799\) 2.44479e48 3.13488
\(800\) 0 0
\(801\) −2.20916e48 −2.71826
\(802\) −4.53171e47 −0.546242
\(803\) −2.72253e47 −0.321489
\(804\) −2.36463e48 −2.73551
\(805\) 0 0
\(806\) −3.21644e47 −0.357147
\(807\) 8.46680e47 0.921098
\(808\) −1.00655e47 −0.107288
\(809\) −3.00339e47 −0.313661 −0.156831 0.987626i \(-0.550128\pi\)
−0.156831 + 0.987626i \(0.550128\pi\)
\(810\) 0 0
\(811\) −1.57205e48 −1.57624 −0.788121 0.615521i \(-0.788946\pi\)
−0.788121 + 0.615521i \(0.788946\pi\)
\(812\) 1.95175e47 0.191756
\(813\) −2.22997e48 −2.14686
\(814\) 5.04341e47 0.475797
\(815\) 0 0
\(816\) 2.67215e48 2.42088
\(817\) 4.79172e47 0.425430
\(818\) −4.14782e47 −0.360903
\(819\) 1.30021e47 0.110874
\(820\) 0 0
\(821\) 2.23751e48 1.83275 0.916374 0.400324i \(-0.131102\pi\)
0.916374 + 0.400324i \(0.131102\pi\)
\(822\) 1.62160e48 1.30185
\(823\) 5.02472e47 0.395380 0.197690 0.980265i \(-0.436656\pi\)
0.197690 + 0.980265i \(0.436656\pi\)
\(824\) 4.54083e46 0.0350216
\(825\) 0 0
\(826\) 1.06007e47 0.0785533
\(827\) −1.52395e48 −1.10695 −0.553475 0.832865i \(-0.686699\pi\)
−0.553475 + 0.832865i \(0.686699\pi\)
\(828\) −2.94492e48 −2.09688
\(829\) 7.14872e47 0.498975 0.249487 0.968378i \(-0.419738\pi\)
0.249487 + 0.968378i \(0.419738\pi\)
\(830\) 0 0
\(831\) −3.62379e48 −2.43078
\(832\) −8.22104e47 −0.540620
\(833\) 2.47486e48 1.59554
\(834\) 2.76749e47 0.174923
\(835\) 0 0
\(836\) −1.60053e48 −0.972436
\(837\) 6.55851e47 0.390693
\(838\) 4.25968e48 2.48800
\(839\) 2.41800e48 1.38479 0.692394 0.721519i \(-0.256556\pi\)
0.692394 + 0.721519i \(0.256556\pi\)
\(840\) 0 0
\(841\) 7.25637e47 0.399563
\(842\) −7.23174e47 −0.390475
\(843\) −1.84015e48 −0.974315
\(844\) −5.55287e47 −0.288315
\(845\) 0 0
\(846\) 8.18103e48 4.08505
\(847\) −2.27652e47 −0.111480
\(848\) 1.18313e48 0.568199
\(849\) −8.51107e47 −0.400873
\(850\) 0 0
\(851\) −1.87245e48 −0.848345
\(852\) 3.06737e48 1.36306
\(853\) −5.58595e47 −0.243466 −0.121733 0.992563i \(-0.538845\pi\)
−0.121733 + 0.992563i \(0.538845\pi\)
\(854\) −3.90972e47 −0.167144
\(855\) 0 0
\(856\) −4.12250e46 −0.0169568
\(857\) 7.99378e47 0.322529 0.161264 0.986911i \(-0.448443\pi\)
0.161264 + 0.986911i \(0.448443\pi\)
\(858\) 1.45354e48 0.575290
\(859\) 3.21299e48 1.24745 0.623723 0.781646i \(-0.285619\pi\)
0.623723 + 0.781646i \(0.285619\pi\)
\(860\) 0 0
\(861\) −1.01318e48 −0.378559
\(862\) 4.66251e48 1.70903
\(863\) −2.15382e48 −0.774519 −0.387259 0.921971i \(-0.626578\pi\)
−0.387259 + 0.921971i \(0.626578\pi\)
\(864\) 3.09546e48 1.09206
\(865\) 0 0
\(866\) 5.72316e48 1.94353
\(867\) 7.90010e48 2.63219
\(868\) 2.53577e47 0.0828962
\(869\) −4.67835e47 −0.150060
\(870\) 0 0
\(871\) −2.58917e48 −0.799578
\(872\) −1.35898e47 −0.0411805
\(873\) −8.83637e46 −0.0262748
\(874\) 1.15649e49 3.37445
\(875\) 0 0
\(876\) 3.64700e48 1.02475
\(877\) −4.36089e48 −1.20249 −0.601245 0.799064i \(-0.705329\pi\)
−0.601245 + 0.799064i \(0.705329\pi\)
\(878\) 2.41504e48 0.653530
\(879\) −9.01528e48 −2.39421
\(880\) 0 0
\(881\) −4.71885e48 −1.20707 −0.603537 0.797335i \(-0.706242\pi\)
−0.603537 + 0.797335i \(0.706242\pi\)
\(882\) 8.28163e48 2.07914
\(883\) 2.47214e48 0.609146 0.304573 0.952489i \(-0.401486\pi\)
0.304573 + 0.952489i \(0.401486\pi\)
\(884\) −3.47682e48 −0.840853
\(885\) 0 0
\(886\) −3.74274e48 −0.872033
\(887\) 6.65983e48 1.52308 0.761542 0.648116i \(-0.224443\pi\)
0.761542 + 0.648116i \(0.224443\pi\)
\(888\) −3.63360e47 −0.0815686
\(889\) 6.02908e47 0.132853
\(890\) 0 0
\(891\) 6.90355e47 0.146585
\(892\) −8.58144e48 −1.78871
\(893\) −1.65076e49 −3.37781
\(894\) −1.46784e49 −2.94856
\(895\) 0 0
\(896\) 1.28948e47 0.0249650
\(897\) −5.39651e48 −1.02574
\(898\) −2.94139e48 −0.548901
\(899\) 3.30227e48 0.605032
\(900\) 0 0
\(901\) 5.59256e48 0.987760
\(902\) −6.76795e48 −1.17368
\(903\) 3.43075e47 0.0584173
\(904\) 4.72902e47 0.0790663
\(905\) 0 0
\(906\) −6.97743e48 −1.12481
\(907\) −4.30056e48 −0.680777 −0.340388 0.940285i \(-0.610559\pi\)
−0.340388 + 0.940285i \(0.610559\pi\)
\(908\) −4.45257e48 −0.692140
\(909\) 1.28002e49 1.95394
\(910\) 0 0
\(911\) −2.43257e48 −0.358106 −0.179053 0.983839i \(-0.557303\pi\)
−0.179053 + 0.983839i \(0.557303\pi\)
\(912\) −1.80427e49 −2.60848
\(913\) −1.68224e48 −0.238847
\(914\) −8.49906e48 −1.18511
\(915\) 0 0
\(916\) −4.15093e48 −0.558303
\(917\) −3.55963e47 −0.0470230
\(918\) 1.37976e49 1.79019
\(919\) 1.59035e48 0.202669 0.101334 0.994852i \(-0.467689\pi\)
0.101334 + 0.994852i \(0.467689\pi\)
\(920\) 0 0
\(921\) 1.51959e49 1.86828
\(922\) −1.16840e48 −0.141101
\(923\) 3.35865e48 0.398416
\(924\) −1.14594e48 −0.133529
\(925\) 0 0
\(926\) −1.57862e49 −1.77499
\(927\) −5.77449e48 −0.637820
\(928\) 1.55859e49 1.69119
\(929\) −1.48456e49 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(930\) 0 0
\(931\) −1.67106e49 −1.71919
\(932\) 7.89414e48 0.797888
\(933\) −6.34627e48 −0.630190
\(934\) 2.37502e49 2.31710
\(935\) 0 0
\(936\) −6.25746e47 −0.0589314
\(937\) 1.00831e49 0.933020 0.466510 0.884516i \(-0.345511\pi\)
0.466510 + 0.884516i \(0.345511\pi\)
\(938\) 3.97271e48 0.361193
\(939\) −1.27619e49 −1.14007
\(940\) 0 0
\(941\) 9.04832e48 0.780440 0.390220 0.920722i \(-0.372399\pi\)
0.390220 + 0.920722i \(0.372399\pi\)
\(942\) −1.14137e49 −0.967353
\(943\) 2.51271e49 2.09266
\(944\) 4.10161e48 0.335673
\(945\) 0 0
\(946\) 2.29172e48 0.181116
\(947\) 9.36347e48 0.727214 0.363607 0.931552i \(-0.381545\pi\)
0.363607 + 0.931552i \(0.381545\pi\)
\(948\) 6.26693e48 0.478319
\(949\) 3.99331e48 0.299530
\(950\) 0 0
\(951\) 1.62337e49 1.17609
\(952\) 2.86919e47 0.0204291
\(953\) −1.92193e49 −1.34494 −0.672472 0.740123i \(-0.734767\pi\)
−0.672472 + 0.740123i \(0.734767\pi\)
\(954\) 1.87144e49 1.28715
\(955\) 0 0
\(956\) −1.25840e49 −0.836109
\(957\) −1.49233e49 −0.974582
\(958\) −1.41241e49 −0.906631
\(959\) −1.39984e48 −0.0883223
\(960\) 0 0
\(961\) −1.21131e49 −0.738444
\(962\) −7.39750e48 −0.443298
\(963\) 5.24251e48 0.308820
\(964\) −2.67563e49 −1.54937
\(965\) 0 0
\(966\) 8.28017e48 0.463358
\(967\) 1.11182e49 0.611642 0.305821 0.952089i \(-0.401069\pi\)
0.305821 + 0.952089i \(0.401069\pi\)
\(968\) 1.09561e48 0.0592533
\(969\) −8.52867e49 −4.53460
\(970\) 0 0
\(971\) 1.55536e49 0.799308 0.399654 0.916666i \(-0.369130\pi\)
0.399654 + 0.916666i \(0.369130\pi\)
\(972\) −2.52270e49 −1.27459
\(973\) −2.38902e47 −0.0118674
\(974\) 2.46574e49 1.20427
\(975\) 0 0
\(976\) −1.51274e49 −0.714236
\(977\) 2.43707e49 1.13138 0.565690 0.824618i \(-0.308610\pi\)
0.565690 + 0.824618i \(0.308610\pi\)
\(978\) −4.32860e48 −0.197586
\(979\) 2.13193e49 0.956883
\(980\) 0 0
\(981\) 1.72819e49 0.749986
\(982\) −1.18062e49 −0.503815
\(983\) −1.27767e48 −0.0536148 −0.0268074 0.999641i \(-0.508534\pi\)
−0.0268074 + 0.999641i \(0.508534\pi\)
\(984\) 4.87607e48 0.201210
\(985\) 0 0
\(986\) 6.94723e49 2.77231
\(987\) −1.18191e49 −0.463820
\(988\) 2.34760e49 0.906015
\(989\) −8.50838e48 −0.322930
\(990\) 0 0
\(991\) −8.47131e48 −0.310982 −0.155491 0.987837i \(-0.549696\pi\)
−0.155491 + 0.987837i \(0.549696\pi\)
\(992\) 2.02497e49 0.731100
\(993\) −5.04722e49 −1.79221
\(994\) −5.15336e48 −0.179976
\(995\) 0 0
\(996\) 2.25346e49 0.761326
\(997\) 4.23201e49 1.40629 0.703147 0.711045i \(-0.251778\pi\)
0.703147 + 0.711045i \(0.251778\pi\)
\(998\) −3.69327e49 −1.20714
\(999\) 1.50840e49 0.484936
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.a.c.1.2 6
5.2 odd 4 25.34.b.c.24.2 12
5.3 odd 4 25.34.b.c.24.11 12
5.4 even 2 5.34.a.b.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.5 6 5.4 even 2
25.34.a.c.1.2 6 1.1 even 1 trivial
25.34.b.c.24.2 12 5.2 odd 4
25.34.b.c.24.11 12 5.3 odd 4