Properties

Label 25.34.a.c.1.4
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 1.4
Root \(-21408.2\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q+18258.4 q^{2} +3.67420e7 q^{3} -8.25656e9 q^{4} +6.70852e11 q^{6} +1.51681e13 q^{7} -3.07591e14 q^{8} -4.20908e15 q^{9} +O(q^{10})\) \(q+18258.4 q^{2} +3.67420e7 q^{3} -8.25656e9 q^{4} +6.70852e11 q^{6} +1.51681e13 q^{7} -3.07591e14 q^{8} -4.20908e15 q^{9} -2.61480e17 q^{11} -3.03363e17 q^{12} +3.70714e17 q^{13} +2.76946e17 q^{14} +6.53072e19 q^{16} +3.71730e20 q^{17} -7.68513e19 q^{18} +1.42255e21 q^{19} +5.57308e20 q^{21} -4.77422e21 q^{22} +3.26990e22 q^{23} -1.13015e22 q^{24} +6.76865e21 q^{26} -3.58901e23 q^{27} -1.25237e23 q^{28} +6.41343e23 q^{29} -1.44554e23 q^{31} +3.83459e24 q^{32} -9.60732e24 q^{33} +6.78720e24 q^{34} +3.47526e25 q^{36} -1.34909e26 q^{37} +2.59736e25 q^{38} +1.36208e25 q^{39} +6.37865e26 q^{41} +1.01756e25 q^{42} -3.37044e26 q^{43} +2.15893e27 q^{44} +5.97033e26 q^{46} -3.12828e27 q^{47} +2.39952e27 q^{48} -7.50092e27 q^{49} +1.36581e28 q^{51} -3.06082e27 q^{52} +4.06250e28 q^{53} -6.55298e27 q^{54} -4.66557e27 q^{56} +5.22675e28 q^{57} +1.17099e28 q^{58} -8.77955e28 q^{59} -1.84902e28 q^{61} -2.63933e27 q^{62} -6.38439e28 q^{63} -4.90971e29 q^{64} -1.75415e29 q^{66} -7.69531e29 q^{67} -3.06921e30 q^{68} +1.20143e30 q^{69} +2.72988e30 q^{71} +1.29468e30 q^{72} -4.44452e30 q^{73} -2.46323e30 q^{74} -1.17454e31 q^{76} -3.96617e30 q^{77} +2.48694e29 q^{78} -1.16758e31 q^{79} +1.02118e31 q^{81} +1.16464e31 q^{82} +4.84088e31 q^{83} -4.60145e30 q^{84} -6.15390e30 q^{86} +2.35643e31 q^{87} +8.04290e31 q^{88} +1.18101e32 q^{89} +5.62303e30 q^{91} -2.69981e32 q^{92} -5.31122e30 q^{93} -5.71176e31 q^{94} +1.40891e32 q^{96} +2.71846e31 q^{97} -1.36955e32 q^{98} +1.10059e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + O(q^{10}) \) \( 6q - 147350q^{2} - 26513900q^{3} + 29520645652q^{4} + 6615759249352q^{6} - 19113832847500q^{7} - 3069820115785800q^{8} + 13602102583345438q^{9} + 150760896555890192q^{11} - 466351581756413200q^{12} + 1403095636752804700q^{13} - 7086110008989891456q^{14} + 64511520005858668816q^{16} - \)\(13\!\cdots\!00\)\(q^{17} - \)\(67\!\cdots\!50\)\(q^{18} + \)\(17\!\cdots\!00\)\(q^{19} - \)\(10\!\cdots\!28\)\(q^{21} - \)\(24\!\cdots\!00\)\(q^{22} - \)\(68\!\cdots\!00\)\(q^{23} + \)\(21\!\cdots\!00\)\(q^{24} - \)\(36\!\cdots\!88\)\(q^{26} + \)\(34\!\cdots\!00\)\(q^{27} + \)\(17\!\cdots\!00\)\(q^{28} + \)\(13\!\cdots\!00\)\(q^{29} + \)\(10\!\cdots\!52\)\(q^{31} - \)\(26\!\cdots\!00\)\(q^{32} - \)\(34\!\cdots\!00\)\(q^{33} + \)\(88\!\cdots\!24\)\(q^{34} + \)\(16\!\cdots\!96\)\(q^{36} - \)\(27\!\cdots\!00\)\(q^{37} - \)\(83\!\cdots\!00\)\(q^{38} + \)\(57\!\cdots\!56\)\(q^{39} + \)\(44\!\cdots\!32\)\(q^{41} - \)\(14\!\cdots\!00\)\(q^{42} - \)\(22\!\cdots\!00\)\(q^{43} + \)\(71\!\cdots\!64\)\(q^{44} + \)\(34\!\cdots\!72\)\(q^{46} - \)\(38\!\cdots\!00\)\(q^{47} - \)\(24\!\cdots\!00\)\(q^{48} - \)\(71\!\cdots\!58\)\(q^{49} - \)\(54\!\cdots\!88\)\(q^{51} + \)\(90\!\cdots\!00\)\(q^{52} - \)\(47\!\cdots\!00\)\(q^{53} - \)\(18\!\cdots\!00\)\(q^{54} - \)\(39\!\cdots\!00\)\(q^{56} + \)\(24\!\cdots\!00\)\(q^{57} + \)\(44\!\cdots\!00\)\(q^{58} - \)\(17\!\cdots\!00\)\(q^{59} - \)\(28\!\cdots\!08\)\(q^{61} - \)\(34\!\cdots\!00\)\(q^{62} + \)\(55\!\cdots\!00\)\(q^{63} + \)\(19\!\cdots\!72\)\(q^{64} + \)\(50\!\cdots\!64\)\(q^{66} - \)\(33\!\cdots\!00\)\(q^{67} - \)\(38\!\cdots\!00\)\(q^{68} + \)\(31\!\cdots\!36\)\(q^{69} + \)\(84\!\cdots\!72\)\(q^{71} - \)\(87\!\cdots\!00\)\(q^{72} + \)\(17\!\cdots\!00\)\(q^{73} + \)\(20\!\cdots\!84\)\(q^{74} + \)\(29\!\cdots\!00\)\(q^{76} + \)\(11\!\cdots\!00\)\(q^{77} + \)\(57\!\cdots\!00\)\(q^{78} - \)\(22\!\cdots\!00\)\(q^{79} - \)\(13\!\cdots\!74\)\(q^{81} + \)\(15\!\cdots\!00\)\(q^{82} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(23\!\cdots\!76\)\(q^{84} - \)\(12\!\cdots\!08\)\(q^{86} + \)\(17\!\cdots\!00\)\(q^{87} - \)\(49\!\cdots\!00\)\(q^{88} - \)\(33\!\cdots\!00\)\(q^{89} + \)\(19\!\cdots\!32\)\(q^{91} + \)\(41\!\cdots\!00\)\(q^{92} - \)\(21\!\cdots\!00\)\(q^{93} + \)\(14\!\cdots\!64\)\(q^{94} + \)\(31\!\cdots\!72\)\(q^{96} - \)\(25\!\cdots\!00\)\(q^{97} - \)\(69\!\cdots\!50\)\(q^{98} + \)\(16\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18258.4 0.197001 0.0985006 0.995137i \(-0.468595\pi\)
0.0985006 + 0.995137i \(0.468595\pi\)
\(3\) 3.67420e7 0.492791 0.246395 0.969169i \(-0.420754\pi\)
0.246395 + 0.969169i \(0.420754\pi\)
\(4\) −8.25656e9 −0.961191
\(5\) 0 0
\(6\) 6.70852e11 0.0970803
\(7\) 1.51681e13 0.172510 0.0862550 0.996273i \(-0.472510\pi\)
0.0862550 + 0.996273i \(0.472510\pi\)
\(8\) −3.07591e14 −0.386357
\(9\) −4.20908e15 −0.757157
\(10\) 0 0
\(11\) −2.61480e17 −1.71577 −0.857887 0.513839i \(-0.828223\pi\)
−0.857887 + 0.513839i \(0.828223\pi\)
\(12\) −3.03363e17 −0.473666
\(13\) 3.70714e17 0.154516 0.0772580 0.997011i \(-0.475383\pi\)
0.0772580 + 0.997011i \(0.475383\pi\)
\(14\) 2.76946e17 0.0339847
\(15\) 0 0
\(16\) 6.53072e19 0.885078
\(17\) 3.71730e20 1.85276 0.926382 0.376585i \(-0.122902\pi\)
0.926382 + 0.376585i \(0.122902\pi\)
\(18\) −7.68513e19 −0.149161
\(19\) 1.42255e21 1.13145 0.565723 0.824595i \(-0.308597\pi\)
0.565723 + 0.824595i \(0.308597\pi\)
\(20\) 0 0
\(21\) 5.57308e20 0.0850113
\(22\) −4.77422e21 −0.338009
\(23\) 3.26990e22 1.11180 0.555898 0.831250i \(-0.312374\pi\)
0.555898 + 0.831250i \(0.312374\pi\)
\(24\) −1.13015e22 −0.190393
\(25\) 0 0
\(26\) 6.76865e21 0.0304398
\(27\) −3.58901e23 −0.865911
\(28\) −1.25237e23 −0.165815
\(29\) 6.41343e23 0.475909 0.237954 0.971276i \(-0.423523\pi\)
0.237954 + 0.971276i \(0.423523\pi\)
\(30\) 0 0
\(31\) −1.44554e23 −0.0356914 −0.0178457 0.999841i \(-0.505681\pi\)
−0.0178457 + 0.999841i \(0.505681\pi\)
\(32\) 3.83459e24 0.560718
\(33\) −9.60732e24 −0.845517
\(34\) 6.78720e24 0.364997
\(35\) 0 0
\(36\) 3.47526e25 0.727772
\(37\) −1.34909e26 −1.79768 −0.898842 0.438273i \(-0.855590\pi\)
−0.898842 + 0.438273i \(0.855590\pi\)
\(38\) 2.59736e25 0.222896
\(39\) 1.36208e25 0.0761440
\(40\) 0 0
\(41\) 6.37865e26 1.56241 0.781205 0.624275i \(-0.214605\pi\)
0.781205 + 0.624275i \(0.214605\pi\)
\(42\) 1.01756e25 0.0167473
\(43\) −3.37044e26 −0.376233 −0.188117 0.982147i \(-0.560238\pi\)
−0.188117 + 0.982147i \(0.560238\pi\)
\(44\) 2.15893e27 1.64919
\(45\) 0 0
\(46\) 5.97033e26 0.219025
\(47\) −3.12828e27 −0.804807 −0.402403 0.915462i \(-0.631825\pi\)
−0.402403 + 0.915462i \(0.631825\pi\)
\(48\) 2.39952e27 0.436158
\(49\) −7.50092e27 −0.970240
\(50\) 0 0
\(51\) 1.36581e28 0.913025
\(52\) −3.06082e27 −0.148519
\(53\) 4.06250e28 1.43960 0.719799 0.694182i \(-0.244234\pi\)
0.719799 + 0.694182i \(0.244234\pi\)
\(54\) −6.55298e27 −0.170585
\(55\) 0 0
\(56\) −4.66557e27 −0.0666504
\(57\) 5.22675e28 0.557566
\(58\) 1.17099e28 0.0937545
\(59\) −8.77955e28 −0.530169 −0.265084 0.964225i \(-0.585400\pi\)
−0.265084 + 0.964225i \(0.585400\pi\)
\(60\) 0 0
\(61\) −1.84902e28 −0.0644168 −0.0322084 0.999481i \(-0.510254\pi\)
−0.0322084 + 0.999481i \(0.510254\pi\)
\(62\) −2.63933e27 −0.00703124
\(63\) −6.38439e28 −0.130617
\(64\) −4.90971e29 −0.774616
\(65\) 0 0
\(66\) −1.75415e29 −0.166568
\(67\) −7.69531e29 −0.570154 −0.285077 0.958505i \(-0.592019\pi\)
−0.285077 + 0.958505i \(0.592019\pi\)
\(68\) −3.06921e30 −1.78086
\(69\) 1.20143e30 0.547883
\(70\) 0 0
\(71\) 2.72988e30 0.776929 0.388464 0.921464i \(-0.373006\pi\)
0.388464 + 0.921464i \(0.373006\pi\)
\(72\) 1.29468e30 0.292533
\(73\) −4.44452e30 −0.799831 −0.399915 0.916552i \(-0.630960\pi\)
−0.399915 + 0.916552i \(0.630960\pi\)
\(74\) −2.46323e30 −0.354146
\(75\) 0 0
\(76\) −1.17454e31 −1.08754
\(77\) −3.96617e30 −0.295988
\(78\) 2.48694e29 0.0150005
\(79\) −1.16758e31 −0.570743 −0.285371 0.958417i \(-0.592117\pi\)
−0.285371 + 0.958417i \(0.592117\pi\)
\(80\) 0 0
\(81\) 1.02118e31 0.330445
\(82\) 1.16464e31 0.307796
\(83\) 4.84088e31 1.04746 0.523728 0.851886i \(-0.324541\pi\)
0.523728 + 0.851886i \(0.324541\pi\)
\(84\) −4.60145e30 −0.0817121
\(85\) 0 0
\(86\) −6.15390e30 −0.0741184
\(87\) 2.35643e31 0.234523
\(88\) 8.04290e31 0.662901
\(89\) 1.18101e32 0.807831 0.403915 0.914796i \(-0.367649\pi\)
0.403915 + 0.914796i \(0.367649\pi\)
\(90\) 0 0
\(91\) 5.62303e30 0.0266555
\(92\) −2.69981e32 −1.06865
\(93\) −5.31122e30 −0.0175884
\(94\) −5.71176e31 −0.158548
\(95\) 0 0
\(96\) 1.40891e32 0.276317
\(97\) 2.71846e31 0.0449355 0.0224678 0.999748i \(-0.492848\pi\)
0.0224678 + 0.999748i \(0.492848\pi\)
\(98\) −1.36955e32 −0.191138
\(99\) 1.10059e33 1.29911
\(100\) 0 0
\(101\) −1.30031e33 −1.10343 −0.551714 0.834033i \(-0.686026\pi\)
−0.551714 + 0.834033i \(0.686026\pi\)
\(102\) 2.49376e32 0.179867
\(103\) −1.73233e33 −1.06369 −0.531846 0.846841i \(-0.678502\pi\)
−0.531846 + 0.846841i \(0.678502\pi\)
\(104\) −1.14028e32 −0.0596983
\(105\) 0 0
\(106\) 7.41749e32 0.283603
\(107\) −3.72062e32 −0.121838 −0.0609191 0.998143i \(-0.519403\pi\)
−0.0609191 + 0.998143i \(0.519403\pi\)
\(108\) 2.96329e33 0.832305
\(109\) −5.92448e33 −1.42927 −0.714633 0.699500i \(-0.753406\pi\)
−0.714633 + 0.699500i \(0.753406\pi\)
\(110\) 0 0
\(111\) −4.95684e33 −0.885882
\(112\) 9.90588e32 0.152685
\(113\) −2.74196e33 −0.364978 −0.182489 0.983208i \(-0.558415\pi\)
−0.182489 + 0.983208i \(0.558415\pi\)
\(114\) 9.54322e32 0.109841
\(115\) 0 0
\(116\) −5.29529e33 −0.457439
\(117\) −1.56036e33 −0.116993
\(118\) −1.60301e33 −0.104444
\(119\) 5.63844e33 0.319620
\(120\) 0 0
\(121\) 4.51469e34 1.94388
\(122\) −3.37602e32 −0.0126902
\(123\) 2.34364e34 0.769941
\(124\) 1.19352e33 0.0343062
\(125\) 0 0
\(126\) −1.16569e33 −0.0257317
\(127\) −7.72774e34 −1.49724 −0.748619 0.663000i \(-0.769283\pi\)
−0.748619 + 0.663000i \(0.769283\pi\)
\(128\) −4.19033e34 −0.713318
\(129\) −1.23837e34 −0.185404
\(130\) 0 0
\(131\) −1.77088e34 −0.205690 −0.102845 0.994697i \(-0.532795\pi\)
−0.102845 + 0.994697i \(0.532795\pi\)
\(132\) 7.93235e34 0.812703
\(133\) 2.15774e34 0.195186
\(134\) −1.40504e34 −0.112321
\(135\) 0 0
\(136\) −1.14341e35 −0.715828
\(137\) 1.26744e35 0.703133 0.351567 0.936163i \(-0.385649\pi\)
0.351567 + 0.936163i \(0.385649\pi\)
\(138\) 2.19362e34 0.107934
\(139\) 1.04160e35 0.454941 0.227471 0.973785i \(-0.426954\pi\)
0.227471 + 0.973785i \(0.426954\pi\)
\(140\) 0 0
\(141\) −1.14940e35 −0.396601
\(142\) 4.98434e34 0.153056
\(143\) −9.69344e34 −0.265114
\(144\) −2.74884e35 −0.670143
\(145\) 0 0
\(146\) −8.11501e34 −0.157568
\(147\) −2.75599e35 −0.478125
\(148\) 1.11389e36 1.72792
\(149\) −3.09463e35 −0.429572 −0.214786 0.976661i \(-0.568905\pi\)
−0.214786 + 0.976661i \(0.568905\pi\)
\(150\) 0 0
\(151\) 1.12720e36 1.25569 0.627844 0.778340i \(-0.283938\pi\)
0.627844 + 0.778340i \(0.283938\pi\)
\(152\) −4.37564e35 −0.437142
\(153\) −1.56464e36 −1.40283
\(154\) −7.24160e34 −0.0583100
\(155\) 0 0
\(156\) −1.12461e35 −0.0731889
\(157\) −1.58631e36 −0.929061 −0.464530 0.885557i \(-0.653777\pi\)
−0.464530 + 0.885557i \(0.653777\pi\)
\(158\) −2.13182e35 −0.112437
\(159\) 1.49264e36 0.709421
\(160\) 0 0
\(161\) 4.95982e35 0.191796
\(162\) 1.86451e35 0.0650979
\(163\) 3.56599e36 1.12483 0.562413 0.826857i \(-0.309873\pi\)
0.562413 + 0.826857i \(0.309873\pi\)
\(164\) −5.26657e36 −1.50177
\(165\) 0 0
\(166\) 8.83869e35 0.206350
\(167\) −6.77798e36 −1.43311 −0.716553 0.697532i \(-0.754281\pi\)
−0.716553 + 0.697532i \(0.754281\pi\)
\(168\) −1.71423e35 −0.0328447
\(169\) −5.61870e36 −0.976125
\(170\) 0 0
\(171\) −5.98764e36 −0.856683
\(172\) 2.78283e36 0.361632
\(173\) −1.72200e36 −0.203364 −0.101682 0.994817i \(-0.532422\pi\)
−0.101682 + 0.994817i \(0.532422\pi\)
\(174\) 4.30247e35 0.0462014
\(175\) 0 0
\(176\) −1.70766e37 −1.51859
\(177\) −3.22578e36 −0.261262
\(178\) 2.15635e36 0.159144
\(179\) −6.69163e36 −0.450253 −0.225127 0.974330i \(-0.572280\pi\)
−0.225127 + 0.974330i \(0.572280\pi\)
\(180\) 0 0
\(181\) 1.34636e37 0.754163 0.377082 0.926180i \(-0.376928\pi\)
0.377082 + 0.926180i \(0.376928\pi\)
\(182\) 1.02668e35 0.00525117
\(183\) −6.79367e35 −0.0317440
\(184\) −1.00579e37 −0.429550
\(185\) 0 0
\(186\) −9.69745e34 −0.00346493
\(187\) −9.72001e37 −3.17892
\(188\) 2.58289e37 0.773573
\(189\) −5.44386e36 −0.149378
\(190\) 0 0
\(191\) −6.65942e36 −0.153598 −0.0767990 0.997047i \(-0.524470\pi\)
−0.0767990 + 0.997047i \(0.524470\pi\)
\(192\) −1.80393e37 −0.381723
\(193\) 5.05982e37 0.982742 0.491371 0.870950i \(-0.336496\pi\)
0.491371 + 0.870950i \(0.336496\pi\)
\(194\) 4.96349e35 0.00885235
\(195\) 0 0
\(196\) 6.19318e37 0.932586
\(197\) −2.33016e37 −0.322621 −0.161311 0.986904i \(-0.551572\pi\)
−0.161311 + 0.986904i \(0.551572\pi\)
\(198\) 2.00951e37 0.255926
\(199\) −1.24079e38 −1.45420 −0.727100 0.686531i \(-0.759132\pi\)
−0.727100 + 0.686531i \(0.759132\pi\)
\(200\) 0 0
\(201\) −2.82742e37 −0.280967
\(202\) −2.37416e37 −0.217377
\(203\) 9.72798e36 0.0820990
\(204\) −1.12769e38 −0.877591
\(205\) 0 0
\(206\) −3.16296e37 −0.209549
\(207\) −1.37633e38 −0.841805
\(208\) 2.42103e37 0.136759
\(209\) −3.71970e38 −1.94130
\(210\) 0 0
\(211\) −2.15498e38 −0.961133 −0.480567 0.876958i \(-0.659569\pi\)
−0.480567 + 0.876958i \(0.659569\pi\)
\(212\) −3.35423e38 −1.38373
\(213\) 1.00301e38 0.382863
\(214\) −6.79328e36 −0.0240023
\(215\) 0 0
\(216\) 1.10395e38 0.334550
\(217\) −2.19262e36 −0.00615711
\(218\) −1.08172e38 −0.281567
\(219\) −1.63301e38 −0.394149
\(220\) 0 0
\(221\) 1.37805e38 0.286282
\(222\) −9.05042e37 −0.174520
\(223\) 1.94520e37 0.0348285 0.0174142 0.999848i \(-0.494457\pi\)
0.0174142 + 0.999848i \(0.494457\pi\)
\(224\) 5.81636e37 0.0967294
\(225\) 0 0
\(226\) −5.00639e37 −0.0719011
\(227\) −1.37345e39 −1.83395 −0.916974 0.398947i \(-0.869376\pi\)
−0.916974 + 0.398947i \(0.869376\pi\)
\(228\) −4.31550e38 −0.535927
\(229\) −3.69693e38 −0.427126 −0.213563 0.976929i \(-0.568507\pi\)
−0.213563 + 0.976929i \(0.568507\pi\)
\(230\) 0 0
\(231\) −1.45725e38 −0.145860
\(232\) −1.97271e38 −0.183871
\(233\) 2.02428e39 1.75751 0.878754 0.477275i \(-0.158375\pi\)
0.878754 + 0.477275i \(0.158375\pi\)
\(234\) −2.84898e37 −0.0230477
\(235\) 0 0
\(236\) 7.24889e38 0.509593
\(237\) −4.28994e38 −0.281257
\(238\) 1.02949e38 0.0629655
\(239\) −6.31944e38 −0.360673 −0.180336 0.983605i \(-0.557719\pi\)
−0.180336 + 0.983605i \(0.557719\pi\)
\(240\) 0 0
\(241\) −1.82829e39 −0.909420 −0.454710 0.890640i \(-0.650257\pi\)
−0.454710 + 0.890640i \(0.650257\pi\)
\(242\) 8.24311e38 0.382946
\(243\) 2.37036e39 1.02875
\(244\) 1.52665e38 0.0619168
\(245\) 0 0
\(246\) 4.27913e38 0.151679
\(247\) 5.27360e38 0.174826
\(248\) 4.44636e37 0.0137896
\(249\) 1.77864e39 0.516177
\(250\) 0 0
\(251\) −2.17903e39 −0.554174 −0.277087 0.960845i \(-0.589369\pi\)
−0.277087 + 0.960845i \(0.589369\pi\)
\(252\) 5.27131e38 0.125548
\(253\) −8.55015e39 −1.90759
\(254\) −1.41096e39 −0.294958
\(255\) 0 0
\(256\) 3.45232e39 0.634091
\(257\) 1.48926e39 0.256492 0.128246 0.991742i \(-0.459065\pi\)
0.128246 + 0.991742i \(0.459065\pi\)
\(258\) −2.26107e38 −0.0365249
\(259\) −2.04632e39 −0.310118
\(260\) 0 0
\(261\) −2.69947e39 −0.360338
\(262\) −3.23335e38 −0.0405211
\(263\) −1.17319e40 −1.38069 −0.690346 0.723479i \(-0.742542\pi\)
−0.690346 + 0.723479i \(0.742542\pi\)
\(264\) 2.95512e39 0.326671
\(265\) 0 0
\(266\) 3.93970e38 0.0384518
\(267\) 4.33929e39 0.398091
\(268\) 6.35369e39 0.548027
\(269\) 2.30936e40 1.87318 0.936590 0.350428i \(-0.113964\pi\)
0.936590 + 0.350428i \(0.113964\pi\)
\(270\) 0 0
\(271\) 2.16176e40 1.55172 0.775861 0.630903i \(-0.217316\pi\)
0.775861 + 0.630903i \(0.217316\pi\)
\(272\) 2.42766e40 1.63984
\(273\) 2.06602e38 0.0131356
\(274\) 2.31414e39 0.138518
\(275\) 0 0
\(276\) −9.91967e39 −0.526620
\(277\) 1.86721e40 0.933847 0.466924 0.884298i \(-0.345362\pi\)
0.466924 + 0.884298i \(0.345362\pi\)
\(278\) 1.90179e39 0.0896239
\(279\) 6.08441e38 0.0270240
\(280\) 0 0
\(281\) −6.80173e39 −0.268513 −0.134256 0.990947i \(-0.542865\pi\)
−0.134256 + 0.990947i \(0.542865\pi\)
\(282\) −2.09862e39 −0.0781309
\(283\) 3.60211e40 1.26497 0.632485 0.774573i \(-0.282035\pi\)
0.632485 + 0.774573i \(0.282035\pi\)
\(284\) −2.25395e40 −0.746777
\(285\) 0 0
\(286\) −1.76987e39 −0.0522278
\(287\) 9.67521e39 0.269531
\(288\) −1.61401e40 −0.424552
\(289\) 9.79285e40 2.43274
\(290\) 0 0
\(291\) 9.98819e38 0.0221438
\(292\) 3.66965e40 0.768790
\(293\) −6.02024e40 −1.19206 −0.596030 0.802962i \(-0.703256\pi\)
−0.596030 + 0.802962i \(0.703256\pi\)
\(294\) −5.03201e39 −0.0941912
\(295\) 0 0
\(296\) 4.14968e40 0.694547
\(297\) 9.38457e40 1.48571
\(298\) −5.65032e39 −0.0846261
\(299\) 1.21220e40 0.171790
\(300\) 0 0
\(301\) −5.11233e39 −0.0649040
\(302\) 2.05809e40 0.247372
\(303\) −4.77760e40 −0.543759
\(304\) 9.29029e40 1.00142
\(305\) 0 0
\(306\) −2.85679e40 −0.276360
\(307\) −2.05983e41 −1.88820 −0.944098 0.329664i \(-0.893064\pi\)
−0.944098 + 0.329664i \(0.893064\pi\)
\(308\) 3.27469e40 0.284501
\(309\) −6.36493e40 −0.524178
\(310\) 0 0
\(311\) −1.19843e41 −0.887294 −0.443647 0.896202i \(-0.646316\pi\)
−0.443647 + 0.896202i \(0.646316\pi\)
\(312\) −4.18962e39 −0.0294188
\(313\) −2.13282e41 −1.42060 −0.710302 0.703897i \(-0.751442\pi\)
−0.710302 + 0.703897i \(0.751442\pi\)
\(314\) −2.89636e40 −0.183026
\(315\) 0 0
\(316\) 9.64022e40 0.548593
\(317\) 2.31538e41 1.25068 0.625338 0.780354i \(-0.284961\pi\)
0.625338 + 0.780354i \(0.284961\pi\)
\(318\) 2.72534e40 0.139757
\(319\) −1.67699e41 −0.816552
\(320\) 0 0
\(321\) −1.36703e40 −0.0600407
\(322\) 9.05586e39 0.0377840
\(323\) 5.28805e41 2.09630
\(324\) −8.43142e40 −0.317620
\(325\) 0 0
\(326\) 6.51094e40 0.221592
\(327\) −2.17677e41 −0.704329
\(328\) −1.96201e41 −0.603647
\(329\) −4.74502e40 −0.138837
\(330\) 0 0
\(331\) −3.27790e40 −0.0867830 −0.0433915 0.999058i \(-0.513816\pi\)
−0.0433915 + 0.999058i \(0.513816\pi\)
\(332\) −3.99690e41 −1.00680
\(333\) 5.67844e41 1.36113
\(334\) −1.23755e41 −0.282324
\(335\) 0 0
\(336\) 3.63962e40 0.0752416
\(337\) −5.05149e41 −0.994321 −0.497160 0.867659i \(-0.665624\pi\)
−0.497160 + 0.867659i \(0.665624\pi\)
\(338\) −1.02589e41 −0.192298
\(339\) −1.00745e41 −0.179858
\(340\) 0 0
\(341\) 3.77981e40 0.0612383
\(342\) −1.09325e41 −0.168767
\(343\) −2.31040e41 −0.339886
\(344\) 1.03672e41 0.145360
\(345\) 0 0
\(346\) −3.14411e40 −0.0400629
\(347\) −5.54748e40 −0.0673999 −0.0337000 0.999432i \(-0.510729\pi\)
−0.0337000 + 0.999432i \(0.510729\pi\)
\(348\) −1.94560e41 −0.225422
\(349\) −4.36140e41 −0.481955 −0.240978 0.970531i \(-0.577468\pi\)
−0.240978 + 0.970531i \(0.577468\pi\)
\(350\) 0 0
\(351\) −1.33050e41 −0.133797
\(352\) −1.00267e42 −0.962065
\(353\) −2.19124e41 −0.200635 −0.100318 0.994955i \(-0.531986\pi\)
−0.100318 + 0.994955i \(0.531986\pi\)
\(354\) −5.88978e40 −0.0514689
\(355\) 0 0
\(356\) −9.75113e41 −0.776479
\(357\) 2.07168e41 0.157506
\(358\) −1.22179e41 −0.0887004
\(359\) −1.83249e42 −1.27053 −0.635264 0.772295i \(-0.719109\pi\)
−0.635264 + 0.772295i \(0.719109\pi\)
\(360\) 0 0
\(361\) 4.42885e41 0.280170
\(362\) 2.45825e41 0.148571
\(363\) 1.65879e42 0.957925
\(364\) −4.64269e40 −0.0256211
\(365\) 0 0
\(366\) −1.24042e40 −0.00625360
\(367\) −2.46022e42 −1.18572 −0.592862 0.805304i \(-0.702002\pi\)
−0.592862 + 0.805304i \(0.702002\pi\)
\(368\) 2.13548e42 0.984026
\(369\) −2.68483e42 −1.18299
\(370\) 0 0
\(371\) 6.16205e41 0.248345
\(372\) 4.38524e40 0.0169058
\(373\) −9.17264e41 −0.338297 −0.169149 0.985591i \(-0.554102\pi\)
−0.169149 + 0.985591i \(0.554102\pi\)
\(374\) −1.77472e42 −0.626251
\(375\) 0 0
\(376\) 9.62231e41 0.310942
\(377\) 2.37755e41 0.0735355
\(378\) −9.93964e40 −0.0294277
\(379\) 5.49346e42 1.55704 0.778518 0.627622i \(-0.215972\pi\)
0.778518 + 0.627622i \(0.215972\pi\)
\(380\) 0 0
\(381\) −2.83933e42 −0.737825
\(382\) −1.21591e41 −0.0302590
\(383\) −2.70725e42 −0.645281 −0.322640 0.946522i \(-0.604570\pi\)
−0.322640 + 0.946522i \(0.604570\pi\)
\(384\) −1.53961e42 −0.351517
\(385\) 0 0
\(386\) 9.23844e41 0.193601
\(387\) 1.41865e42 0.284868
\(388\) −2.24452e41 −0.0431916
\(389\) 5.33079e42 0.983157 0.491579 0.870833i \(-0.336420\pi\)
0.491579 + 0.870833i \(0.336420\pi\)
\(390\) 0 0
\(391\) 1.21552e43 2.05990
\(392\) 2.30721e42 0.374859
\(393\) −6.50657e41 −0.101362
\(394\) −4.25451e41 −0.0635568
\(395\) 0 0
\(396\) −9.08712e42 −1.24869
\(397\) −8.08784e42 −1.06608 −0.533039 0.846091i \(-0.678950\pi\)
−0.533039 + 0.846091i \(0.678950\pi\)
\(398\) −2.26550e42 −0.286479
\(399\) 7.92799e41 0.0961857
\(400\) 0 0
\(401\) −1.16113e43 −1.29717 −0.648587 0.761140i \(-0.724640\pi\)
−0.648587 + 0.761140i \(0.724640\pi\)
\(402\) −5.16242e41 −0.0553507
\(403\) −5.35882e40 −0.00551488
\(404\) 1.07361e43 1.06061
\(405\) 0 0
\(406\) 1.77618e41 0.0161736
\(407\) 3.52761e43 3.08442
\(408\) −4.20111e42 −0.352753
\(409\) 4.40488e42 0.355222 0.177611 0.984101i \(-0.443163\pi\)
0.177611 + 0.984101i \(0.443163\pi\)
\(410\) 0 0
\(411\) 4.65683e42 0.346498
\(412\) 1.43031e43 1.02241
\(413\) −1.33169e42 −0.0914594
\(414\) −2.51296e42 −0.165836
\(415\) 0 0
\(416\) 1.42154e42 0.0866399
\(417\) 3.82704e42 0.224191
\(418\) −6.79158e42 −0.382439
\(419\) 2.34926e43 1.27174 0.635871 0.771796i \(-0.280641\pi\)
0.635871 + 0.771796i \(0.280641\pi\)
\(420\) 0 0
\(421\) −4.10149e42 −0.205252 −0.102626 0.994720i \(-0.532724\pi\)
−0.102626 + 0.994720i \(0.532724\pi\)
\(422\) −3.93466e42 −0.189344
\(423\) 1.31672e43 0.609365
\(424\) −1.24959e43 −0.556199
\(425\) 0 0
\(426\) 1.83135e42 0.0754245
\(427\) −2.80461e41 −0.0111125
\(428\) 3.07196e42 0.117110
\(429\) −3.56157e42 −0.130646
\(430\) 0 0
\(431\) 1.92923e43 0.655404 0.327702 0.944781i \(-0.393726\pi\)
0.327702 + 0.944781i \(0.393726\pi\)
\(432\) −2.34389e43 −0.766398
\(433\) −1.20050e43 −0.377843 −0.188922 0.981992i \(-0.560499\pi\)
−0.188922 + 0.981992i \(0.560499\pi\)
\(434\) −4.00338e40 −0.00121296
\(435\) 0 0
\(436\) 4.89159e43 1.37380
\(437\) 4.65160e43 1.25794
\(438\) −2.98162e42 −0.0776478
\(439\) 3.55254e43 0.890993 0.445496 0.895284i \(-0.353027\pi\)
0.445496 + 0.895284i \(0.353027\pi\)
\(440\) 0 0
\(441\) 3.15720e43 0.734625
\(442\) 2.51611e42 0.0563978
\(443\) −4.50074e43 −0.971901 −0.485951 0.873986i \(-0.661526\pi\)
−0.485951 + 0.873986i \(0.661526\pi\)
\(444\) 4.09265e43 0.851501
\(445\) 0 0
\(446\) 3.55163e41 0.00686124
\(447\) −1.13703e43 −0.211689
\(448\) −7.44711e42 −0.133629
\(449\) −4.48522e43 −0.775745 −0.387872 0.921713i \(-0.626790\pi\)
−0.387872 + 0.921713i \(0.626790\pi\)
\(450\) 0 0
\(451\) −1.66789e44 −2.68074
\(452\) 2.26392e43 0.350814
\(453\) 4.14156e43 0.618791
\(454\) −2.50771e43 −0.361290
\(455\) 0 0
\(456\) −1.60770e43 −0.215419
\(457\) −5.99465e43 −0.774723 −0.387362 0.921928i \(-0.626614\pi\)
−0.387362 + 0.921928i \(0.626614\pi\)
\(458\) −6.75002e42 −0.0841442
\(459\) −1.33414e44 −1.60433
\(460\) 0 0
\(461\) 3.64117e43 0.407546 0.203773 0.979018i \(-0.434680\pi\)
0.203773 + 0.979018i \(0.434680\pi\)
\(462\) −2.66071e42 −0.0287346
\(463\) 4.93497e43 0.514278 0.257139 0.966374i \(-0.417220\pi\)
0.257139 + 0.966374i \(0.417220\pi\)
\(464\) 4.18844e43 0.421216
\(465\) 0 0
\(466\) 3.69601e43 0.346231
\(467\) −6.19636e43 −0.560284 −0.280142 0.959959i \(-0.590381\pi\)
−0.280142 + 0.959959i \(0.590381\pi\)
\(468\) 1.28833e43 0.112452
\(469\) −1.16723e43 −0.0983572
\(470\) 0 0
\(471\) −5.82843e43 −0.457832
\(472\) 2.70051e43 0.204834
\(473\) 8.81305e43 0.645531
\(474\) −7.83275e42 −0.0554079
\(475\) 0 0
\(476\) −4.65542e43 −0.307216
\(477\) −1.70994e44 −1.09000
\(478\) −1.15383e43 −0.0710530
\(479\) 1.65648e44 0.985483 0.492741 0.870176i \(-0.335995\pi\)
0.492741 + 0.870176i \(0.335995\pi\)
\(480\) 0 0
\(481\) −5.00127e43 −0.277771
\(482\) −3.33818e43 −0.179157
\(483\) 1.82234e43 0.0945152
\(484\) −3.72758e44 −1.86844
\(485\) 0 0
\(486\) 4.32790e43 0.202665
\(487\) −3.41088e44 −1.54397 −0.771984 0.635642i \(-0.780736\pi\)
−0.771984 + 0.635642i \(0.780736\pi\)
\(488\) 5.68741e42 0.0248879
\(489\) 1.31022e44 0.554303
\(490\) 0 0
\(491\) 4.35160e44 1.72109 0.860547 0.509371i \(-0.170122\pi\)
0.860547 + 0.509371i \(0.170122\pi\)
\(492\) −1.93505e44 −0.740060
\(493\) 2.38406e44 0.881747
\(494\) 9.62876e42 0.0344410
\(495\) 0 0
\(496\) −9.44044e42 −0.0315896
\(497\) 4.14072e43 0.134028
\(498\) 3.24751e43 0.101687
\(499\) −6.19626e44 −1.87703 −0.938514 0.345242i \(-0.887797\pi\)
−0.938514 + 0.345242i \(0.887797\pi\)
\(500\) 0 0
\(501\) −2.49037e44 −0.706222
\(502\) −3.97856e43 −0.109173
\(503\) −5.28443e44 −1.40322 −0.701611 0.712560i \(-0.747536\pi\)
−0.701611 + 0.712560i \(0.747536\pi\)
\(504\) 1.96378e43 0.0504648
\(505\) 0 0
\(506\) −1.56112e44 −0.375797
\(507\) −2.06443e44 −0.481025
\(508\) 6.38046e44 1.43913
\(509\) −4.72823e43 −0.103242 −0.0516209 0.998667i \(-0.516439\pi\)
−0.0516209 + 0.998667i \(0.516439\pi\)
\(510\) 0 0
\(511\) −6.74151e43 −0.137979
\(512\) 4.22980e44 0.838235
\(513\) −5.10556e44 −0.979731
\(514\) 2.71916e43 0.0505293
\(515\) 0 0
\(516\) 1.02247e44 0.178209
\(517\) 8.17985e44 1.38087
\(518\) −3.73626e43 −0.0610937
\(519\) −6.32699e43 −0.100216
\(520\) 0 0
\(521\) −4.38738e44 −0.652203 −0.326102 0.945335i \(-0.605735\pi\)
−0.326102 + 0.945335i \(0.605735\pi\)
\(522\) −4.92881e43 −0.0709869
\(523\) 1.40016e44 0.195389 0.0976944 0.995216i \(-0.468853\pi\)
0.0976944 + 0.995216i \(0.468853\pi\)
\(524\) 1.46214e44 0.197707
\(525\) 0 0
\(526\) −2.14205e44 −0.271998
\(527\) −5.37351e43 −0.0661277
\(528\) −6.27428e44 −0.748349
\(529\) 2.04220e44 0.236091
\(530\) 0 0
\(531\) 3.69539e44 0.401421
\(532\) −1.78156e44 −0.187611
\(533\) 2.36465e44 0.241417
\(534\) 7.92286e43 0.0784244
\(535\) 0 0
\(536\) 2.36701e44 0.220283
\(537\) −2.45864e44 −0.221881
\(538\) 4.21653e44 0.369018
\(539\) 1.96134e45 1.66471
\(540\) 0 0
\(541\) 1.56960e45 1.25324 0.626622 0.779324i \(-0.284437\pi\)
0.626622 + 0.779324i \(0.284437\pi\)
\(542\) 3.94704e44 0.305691
\(543\) 4.94681e44 0.371645
\(544\) 1.42543e45 1.03888
\(545\) 0 0
\(546\) 3.77222e42 0.00258773
\(547\) 2.11683e45 1.40895 0.704474 0.709730i \(-0.251183\pi\)
0.704474 + 0.709730i \(0.251183\pi\)
\(548\) −1.04647e45 −0.675845
\(549\) 7.78267e43 0.0487736
\(550\) 0 0
\(551\) 9.12345e44 0.538465
\(552\) −3.69548e44 −0.211678
\(553\) −1.77100e44 −0.0984588
\(554\) 3.40922e44 0.183969
\(555\) 0 0
\(556\) −8.60001e44 −0.437285
\(557\) 1.56733e45 0.773661 0.386830 0.922151i \(-0.373570\pi\)
0.386830 + 0.922151i \(0.373570\pi\)
\(558\) 1.11092e43 0.00532375
\(559\) −1.24947e44 −0.0581341
\(560\) 0 0
\(561\) −3.57133e45 −1.56654
\(562\) −1.24189e44 −0.0528974
\(563\) −1.29413e44 −0.0535291 −0.0267646 0.999642i \(-0.508520\pi\)
−0.0267646 + 0.999642i \(0.508520\pi\)
\(564\) 9.49006e44 0.381209
\(565\) 0 0
\(566\) 6.57688e44 0.249200
\(567\) 1.54894e44 0.0570050
\(568\) −8.39687e44 −0.300172
\(569\) −2.93499e45 −1.01919 −0.509593 0.860415i \(-0.670204\pi\)
−0.509593 + 0.860415i \(0.670204\pi\)
\(570\) 0 0
\(571\) −9.20277e44 −0.301594 −0.150797 0.988565i \(-0.548184\pi\)
−0.150797 + 0.988565i \(0.548184\pi\)
\(572\) 8.00345e44 0.254825
\(573\) −2.44681e44 −0.0756917
\(574\) 1.76654e44 0.0530979
\(575\) 0 0
\(576\) 2.06654e45 0.586506
\(577\) 6.78370e45 1.87097 0.935483 0.353371i \(-0.114965\pi\)
0.935483 + 0.353371i \(0.114965\pi\)
\(578\) 1.78802e45 0.479251
\(579\) 1.85908e45 0.484286
\(580\) 0 0
\(581\) 7.34270e44 0.180697
\(582\) 1.82369e43 0.00436236
\(583\) −1.06226e46 −2.47003
\(584\) 1.36709e45 0.309020
\(585\) 0 0
\(586\) −1.09920e45 −0.234837
\(587\) −1.92291e45 −0.399421 −0.199710 0.979855i \(-0.564000\pi\)
−0.199710 + 0.979855i \(0.564000\pi\)
\(588\) 2.27550e45 0.459570
\(589\) −2.05636e44 −0.0403828
\(590\) 0 0
\(591\) −8.56148e44 −0.158985
\(592\) −8.81055e45 −1.59109
\(593\) 3.84061e45 0.674524 0.337262 0.941411i \(-0.390499\pi\)
0.337262 + 0.941411i \(0.390499\pi\)
\(594\) 1.71348e45 0.292686
\(595\) 0 0
\(596\) 2.55510e45 0.412900
\(597\) −4.55893e45 −0.716616
\(598\) 2.21328e44 0.0338429
\(599\) −1.48464e45 −0.220840 −0.110420 0.993885i \(-0.535220\pi\)
−0.110420 + 0.993885i \(0.535220\pi\)
\(600\) 0 0
\(601\) 5.01345e45 0.705843 0.352921 0.935653i \(-0.385188\pi\)
0.352921 + 0.935653i \(0.385188\pi\)
\(602\) −9.33431e43 −0.0127862
\(603\) 3.23902e45 0.431696
\(604\) −9.30680e45 −1.20695
\(605\) 0 0
\(606\) −8.72316e44 −0.107121
\(607\) −3.29284e45 −0.393511 −0.196756 0.980453i \(-0.563041\pi\)
−0.196756 + 0.980453i \(0.563041\pi\)
\(608\) 5.45491e45 0.634422
\(609\) 3.57426e44 0.0404576
\(610\) 0 0
\(611\) −1.15970e45 −0.124355
\(612\) 1.29186e46 1.34839
\(613\) 1.00399e46 1.02008 0.510038 0.860152i \(-0.329631\pi\)
0.510038 + 0.860152i \(0.329631\pi\)
\(614\) −3.76092e45 −0.371977
\(615\) 0 0
\(616\) 1.21996e45 0.114357
\(617\) −1.43698e46 −1.31144 −0.655718 0.755006i \(-0.727634\pi\)
−0.655718 + 0.755006i \(0.727634\pi\)
\(618\) −1.16214e45 −0.103264
\(619\) 1.61224e46 1.39487 0.697435 0.716648i \(-0.254325\pi\)
0.697435 + 0.716648i \(0.254325\pi\)
\(620\) 0 0
\(621\) −1.17357e46 −0.962716
\(622\) −2.18815e45 −0.174798
\(623\) 1.79138e45 0.139359
\(624\) 8.89535e44 0.0673934
\(625\) 0 0
\(626\) −3.89420e45 −0.279861
\(627\) −1.36669e46 −0.956657
\(628\) 1.30975e46 0.893004
\(629\) −5.01498e46 −3.33068
\(630\) 0 0
\(631\) 2.47683e46 1.56103 0.780516 0.625136i \(-0.214957\pi\)
0.780516 + 0.625136i \(0.214957\pi\)
\(632\) 3.59138e45 0.220510
\(633\) −7.91785e45 −0.473638
\(634\) 4.22752e45 0.246384
\(635\) 0 0
\(636\) −1.23241e46 −0.681889
\(637\) −2.78069e45 −0.149918
\(638\) −3.06192e45 −0.160862
\(639\) −1.14903e46 −0.588257
\(640\) 0 0
\(641\) −3.57575e46 −1.73864 −0.869321 0.494249i \(-0.835443\pi\)
−0.869321 + 0.494249i \(0.835443\pi\)
\(642\) −2.49599e44 −0.0118281
\(643\) −1.88292e46 −0.869663 −0.434831 0.900512i \(-0.643192\pi\)
−0.434831 + 0.900512i \(0.643192\pi\)
\(644\) −4.09511e45 −0.184352
\(645\) 0 0
\(646\) 9.65515e45 0.412974
\(647\) −2.33501e46 −0.973573 −0.486787 0.873521i \(-0.661831\pi\)
−0.486787 + 0.873521i \(0.661831\pi\)
\(648\) −3.14105e45 −0.127669
\(649\) 2.29568e46 0.909649
\(650\) 0 0
\(651\) −8.05612e43 −0.00303417
\(652\) −2.94428e46 −1.08117
\(653\) −2.35876e46 −0.844531 −0.422266 0.906472i \(-0.638765\pi\)
−0.422266 + 0.906472i \(0.638765\pi\)
\(654\) −3.97445e45 −0.138754
\(655\) 0 0
\(656\) 4.16572e46 1.38285
\(657\) 1.87074e46 0.605598
\(658\) −8.66366e44 −0.0273511
\(659\) 2.73242e46 0.841276 0.420638 0.907228i \(-0.361806\pi\)
0.420638 + 0.907228i \(0.361806\pi\)
\(660\) 0 0
\(661\) −2.72690e46 −0.798629 −0.399315 0.916814i \(-0.630752\pi\)
−0.399315 + 0.916814i \(0.630752\pi\)
\(662\) −5.98494e44 −0.0170963
\(663\) 5.06325e45 0.141077
\(664\) −1.48901e46 −0.404692
\(665\) 0 0
\(666\) 1.03679e46 0.268144
\(667\) 2.09713e46 0.529114
\(668\) 5.59628e46 1.37749
\(669\) 7.14706e44 0.0171631
\(670\) 0 0
\(671\) 4.83482e45 0.110525
\(672\) 2.13705e45 0.0476674
\(673\) 1.12643e46 0.245163 0.122581 0.992458i \(-0.460883\pi\)
0.122581 + 0.992458i \(0.460883\pi\)
\(674\) −9.22324e45 −0.195882
\(675\) 0 0
\(676\) 4.63912e46 0.938242
\(677\) 4.69606e46 0.926874 0.463437 0.886130i \(-0.346616\pi\)
0.463437 + 0.886130i \(0.346616\pi\)
\(678\) −1.83945e45 −0.0354322
\(679\) 4.12340e44 0.00775183
\(680\) 0 0
\(681\) −5.04634e46 −0.903753
\(682\) 6.90134e44 0.0120640
\(683\) 1.23577e46 0.210860 0.105430 0.994427i \(-0.466378\pi\)
0.105430 + 0.994427i \(0.466378\pi\)
\(684\) 4.94373e46 0.823435
\(685\) 0 0
\(686\) −4.21842e45 −0.0669579
\(687\) −1.35833e46 −0.210484
\(688\) −2.20114e46 −0.332996
\(689\) 1.50602e46 0.222441
\(690\) 0 0
\(691\) −3.45406e46 −0.486342 −0.243171 0.969983i \(-0.578188\pi\)
−0.243171 + 0.969983i \(0.578188\pi\)
\(692\) 1.42178e46 0.195471
\(693\) 1.66939e46 0.224109
\(694\) −1.01288e45 −0.0132779
\(695\) 0 0
\(696\) −7.24815e45 −0.0906097
\(697\) 2.37113e47 2.89478
\(698\) −7.96324e45 −0.0949457
\(699\) 7.43761e46 0.866084
\(700\) 0 0
\(701\) 4.58600e46 0.509433 0.254716 0.967016i \(-0.418018\pi\)
0.254716 + 0.967016i \(0.418018\pi\)
\(702\) −2.42928e45 −0.0263582
\(703\) −1.91915e47 −2.03398
\(704\) 1.28379e47 1.32907
\(705\) 0 0
\(706\) −4.00086e45 −0.0395254
\(707\) −1.97233e46 −0.190352
\(708\) 2.66339e46 0.251123
\(709\) 1.01825e46 0.0937978 0.0468989 0.998900i \(-0.485066\pi\)
0.0468989 + 0.998900i \(0.485066\pi\)
\(710\) 0 0
\(711\) 4.91445e46 0.432142
\(712\) −3.63269e46 −0.312111
\(713\) −4.72678e45 −0.0396815
\(714\) 3.78256e45 0.0310288
\(715\) 0 0
\(716\) 5.52498e46 0.432779
\(717\) −2.32189e46 −0.177736
\(718\) −3.34585e46 −0.250295
\(719\) −2.37751e46 −0.173818 −0.0869092 0.996216i \(-0.527699\pi\)
−0.0869092 + 0.996216i \(0.527699\pi\)
\(720\) 0 0
\(721\) −2.62762e46 −0.183498
\(722\) 8.08638e45 0.0551938
\(723\) −6.71752e46 −0.448154
\(724\) −1.11163e47 −0.724895
\(725\) 0 0
\(726\) 3.02869e46 0.188712
\(727\) 7.60235e46 0.463053 0.231526 0.972829i \(-0.425628\pi\)
0.231526 + 0.972829i \(0.425628\pi\)
\(728\) −1.72959e45 −0.0102985
\(729\) 3.03238e46 0.176514
\(730\) 0 0
\(731\) −1.25289e47 −0.697072
\(732\) 5.60924e45 0.0305120
\(733\) 9.46431e46 0.503354 0.251677 0.967811i \(-0.419018\pi\)
0.251677 + 0.967811i \(0.419018\pi\)
\(734\) −4.49197e46 −0.233589
\(735\) 0 0
\(736\) 1.25387e47 0.623404
\(737\) 2.01217e47 0.978255
\(738\) −4.90207e46 −0.233050
\(739\) 1.36554e46 0.0634851 0.0317425 0.999496i \(-0.489894\pi\)
0.0317425 + 0.999496i \(0.489894\pi\)
\(740\) 0 0
\(741\) 1.93763e46 0.0861529
\(742\) 1.12509e46 0.0489243
\(743\) −2.14591e47 −0.912632 −0.456316 0.889818i \(-0.650831\pi\)
−0.456316 + 0.889818i \(0.650831\pi\)
\(744\) 1.63368e45 0.00679538
\(745\) 0 0
\(746\) −1.67478e46 −0.0666449
\(747\) −2.03757e47 −0.793089
\(748\) 8.02539e47 3.05555
\(749\) −5.64349e45 −0.0210183
\(750\) 0 0
\(751\) −3.00124e47 −1.06965 −0.534825 0.844963i \(-0.679622\pi\)
−0.534825 + 0.844963i \(0.679622\pi\)
\(752\) −2.04300e47 −0.712317
\(753\) −8.00619e46 −0.273092
\(754\) 4.34103e45 0.0144866
\(755\) 0 0
\(756\) 4.49476e46 0.143581
\(757\) −3.75997e47 −1.17517 −0.587587 0.809161i \(-0.699922\pi\)
−0.587587 + 0.809161i \(0.699922\pi\)
\(758\) 1.00302e47 0.306738
\(759\) −3.14150e47 −0.940043
\(760\) 0 0
\(761\) 2.05125e47 0.587723 0.293861 0.955848i \(-0.405060\pi\)
0.293861 + 0.955848i \(0.405060\pi\)
\(762\) −5.18417e46 −0.145352
\(763\) −8.98632e46 −0.246563
\(764\) 5.49839e46 0.147637
\(765\) 0 0
\(766\) −4.94302e46 −0.127121
\(767\) −3.25470e46 −0.0819195
\(768\) 1.26845e47 0.312474
\(769\) −3.91739e47 −0.944522 −0.472261 0.881459i \(-0.656562\pi\)
−0.472261 + 0.881459i \(0.656562\pi\)
\(770\) 0 0
\(771\) 5.47185e46 0.126397
\(772\) −4.17767e47 −0.944602
\(773\) −6.18683e47 −1.36933 −0.684663 0.728860i \(-0.740051\pi\)
−0.684663 + 0.728860i \(0.740051\pi\)
\(774\) 2.59023e46 0.0561193
\(775\) 0 0
\(776\) −8.36174e45 −0.0173611
\(777\) −7.51860e46 −0.152823
\(778\) 9.73319e46 0.193683
\(779\) 9.07396e47 1.76778
\(780\) 0 0
\(781\) −7.13811e47 −1.33303
\(782\) 2.21935e47 0.405802
\(783\) −2.30179e47 −0.412095
\(784\) −4.89864e47 −0.858738
\(785\) 0 0
\(786\) −1.18800e46 −0.0199684
\(787\) −2.50651e45 −0.00412559 −0.00206280 0.999998i \(-0.500657\pi\)
−0.00206280 + 0.999998i \(0.500657\pi\)
\(788\) 1.92391e47 0.310101
\(789\) −4.31052e47 −0.680392
\(790\) 0 0
\(791\) −4.15904e46 −0.0629624
\(792\) −3.38532e47 −0.501920
\(793\) −6.85457e45 −0.00995342
\(794\) −1.47671e47 −0.210019
\(795\) 0 0
\(796\) 1.02447e48 1.39776
\(797\) −7.37335e47 −0.985378 −0.492689 0.870206i \(-0.663986\pi\)
−0.492689 + 0.870206i \(0.663986\pi\)
\(798\) 1.44753e46 0.0189487
\(799\) −1.16288e48 −1.49112
\(800\) 0 0
\(801\) −4.97099e47 −0.611655
\(802\) −2.12004e47 −0.255545
\(803\) 1.16216e48 1.37233
\(804\) 2.33447e47 0.270062
\(805\) 0 0
\(806\) −9.78438e44 −0.00108644
\(807\) 8.48506e47 0.923085
\(808\) 3.99963e47 0.426317
\(809\) −9.92771e47 −1.03681 −0.518404 0.855136i \(-0.673474\pi\)
−0.518404 + 0.855136i \(0.673474\pi\)
\(810\) 0 0
\(811\) 1.07185e47 0.107470 0.0537351 0.998555i \(-0.482887\pi\)
0.0537351 + 0.998555i \(0.482887\pi\)
\(812\) −8.03197e46 −0.0789128
\(813\) 7.94274e47 0.764674
\(814\) 6.44087e47 0.607634
\(815\) 0 0
\(816\) 8.91973e47 0.808098
\(817\) −4.79463e47 −0.425688
\(818\) 8.04262e46 0.0699791
\(819\) −2.36678e46 −0.0201824
\(820\) 0 0
\(821\) −1.20939e48 −0.990610 −0.495305 0.868719i \(-0.664944\pi\)
−0.495305 + 0.868719i \(0.664944\pi\)
\(822\) 8.50264e46 0.0682604
\(823\) −2.45743e48 −1.93367 −0.966837 0.255396i \(-0.917794\pi\)
−0.966837 + 0.255396i \(0.917794\pi\)
\(824\) 5.32848e47 0.410965
\(825\) 0 0
\(826\) −2.43146e46 −0.0180176
\(827\) 1.80057e48 1.30789 0.653943 0.756544i \(-0.273114\pi\)
0.653943 + 0.756544i \(0.273114\pi\)
\(828\) 1.13637e48 0.809135
\(829\) 6.57735e47 0.459094 0.229547 0.973298i \(-0.426276\pi\)
0.229547 + 0.973298i \(0.426276\pi\)
\(830\) 0 0
\(831\) 6.86049e47 0.460191
\(832\) −1.82010e47 −0.119691
\(833\) −2.78832e48 −1.79763
\(834\) 6.98757e46 0.0441658
\(835\) 0 0
\(836\) 3.07119e48 1.86596
\(837\) 5.18807e46 0.0309055
\(838\) 4.28937e47 0.250534
\(839\) 6.26140e47 0.358591 0.179295 0.983795i \(-0.442618\pi\)
0.179295 + 0.983795i \(0.442618\pi\)
\(840\) 0 0
\(841\) −1.40475e48 −0.773511
\(842\) −7.48868e46 −0.0404348
\(843\) −2.49910e47 −0.132321
\(844\) 1.77928e48 0.923832
\(845\) 0 0
\(846\) 2.40413e47 0.120046
\(847\) 6.84793e47 0.335338
\(848\) 2.65311e48 1.27416
\(849\) 1.32349e48 0.623365
\(850\) 0 0
\(851\) −4.41140e48 −1.99866
\(852\) −8.28145e47 −0.368005
\(853\) −3.16331e48 −1.37874 −0.689370 0.724410i \(-0.742112\pi\)
−0.689370 + 0.724410i \(0.742112\pi\)
\(854\) −5.12079e45 −0.00218918
\(855\) 0 0
\(856\) 1.14443e47 0.0470730
\(857\) 3.13666e48 1.26556 0.632782 0.774330i \(-0.281913\pi\)
0.632782 + 0.774330i \(0.281913\pi\)
\(858\) −6.50286e46 −0.0257374
\(859\) 2.20270e48 0.855202 0.427601 0.903968i \(-0.359359\pi\)
0.427601 + 0.903968i \(0.359359\pi\)
\(860\) 0 0
\(861\) 3.55487e47 0.132822
\(862\) 3.52247e47 0.129115
\(863\) −5.00806e47 −0.180091 −0.0900453 0.995938i \(-0.528701\pi\)
−0.0900453 + 0.995938i \(0.528701\pi\)
\(864\) −1.37624e48 −0.485532
\(865\) 0 0
\(866\) −2.19192e47 −0.0744356
\(867\) 3.59809e48 1.19883
\(868\) 1.81035e46 0.00591816
\(869\) 3.05300e48 0.979266
\(870\) 0 0
\(871\) −2.85276e47 −0.0880979
\(872\) 1.82232e48 0.552207
\(873\) −1.14422e47 −0.0340233
\(874\) 8.49310e47 0.247815
\(875\) 0 0
\(876\) 1.34830e48 0.378852
\(877\) 1.77120e48 0.488400 0.244200 0.969725i \(-0.421475\pi\)
0.244200 + 0.969725i \(0.421475\pi\)
\(878\) 6.48638e47 0.175527
\(879\) −2.21196e48 −0.587436
\(880\) 0 0
\(881\) 1.59935e48 0.409111 0.204555 0.978855i \(-0.434425\pi\)
0.204555 + 0.978855i \(0.434425\pi\)
\(882\) 5.76455e47 0.144722
\(883\) 2.38704e48 0.588177 0.294089 0.955778i \(-0.404984\pi\)
0.294089 + 0.955778i \(0.404984\pi\)
\(884\) −1.13780e48 −0.275171
\(885\) 0 0
\(886\) −8.21764e47 −0.191466
\(887\) −5.13361e48 −1.17404 −0.587020 0.809572i \(-0.699699\pi\)
−0.587020 + 0.809572i \(0.699699\pi\)
\(888\) 1.52468e48 0.342266
\(889\) −1.17215e48 −0.258289
\(890\) 0 0
\(891\) −2.67018e48 −0.566968
\(892\) −1.60607e47 −0.0334768
\(893\) −4.45015e48 −0.910595
\(894\) −2.07604e47 −0.0417030
\(895\) 0 0
\(896\) −6.35594e47 −0.123054
\(897\) 4.45386e47 0.0846566
\(898\) −8.18931e47 −0.152823
\(899\) −9.27089e46 −0.0169858
\(900\) 0 0
\(901\) 1.51015e49 2.66724
\(902\) −3.04531e48 −0.528109
\(903\) −1.87837e47 −0.0319841
\(904\) 8.43402e47 0.141012
\(905\) 0 0
\(906\) 7.56185e47 0.121903
\(907\) −5.18589e48 −0.820924 −0.410462 0.911878i \(-0.634632\pi\)
−0.410462 + 0.911878i \(0.634632\pi\)
\(908\) 1.13400e49 1.76277
\(909\) 5.47311e48 0.835469
\(910\) 0 0
\(911\) −6.63747e48 −0.977123 −0.488562 0.872529i \(-0.662478\pi\)
−0.488562 + 0.872529i \(0.662478\pi\)
\(912\) 3.41344e48 0.493489
\(913\) −1.26580e49 −1.79720
\(914\) −1.09453e48 −0.152621
\(915\) 0 0
\(916\) 3.05239e48 0.410549
\(917\) −2.68609e47 −0.0354835
\(918\) −2.43594e48 −0.316054
\(919\) −7.10984e48 −0.906054 −0.453027 0.891497i \(-0.649656\pi\)
−0.453027 + 0.891497i \(0.649656\pi\)
\(920\) 0 0
\(921\) −7.56822e48 −0.930486
\(922\) 6.64821e47 0.0802869
\(923\) 1.01200e48 0.120048
\(924\) 1.20319e48 0.140199
\(925\) 0 0
\(926\) 9.01048e47 0.101313
\(927\) 7.29151e48 0.805382
\(928\) 2.45929e48 0.266851
\(929\) −1.17316e49 −1.25054 −0.625272 0.780407i \(-0.715012\pi\)
−0.625272 + 0.780407i \(0.715012\pi\)
\(930\) 0 0
\(931\) −1.06705e49 −1.09777
\(932\) −1.67136e49 −1.68930
\(933\) −4.40329e48 −0.437250
\(934\) −1.13136e48 −0.110376
\(935\) 0 0
\(936\) 4.79954e47 0.0452010
\(937\) −7.60807e48 −0.703998 −0.351999 0.936000i \(-0.614498\pi\)
−0.351999 + 0.936000i \(0.614498\pi\)
\(938\) −2.13119e47 −0.0193765
\(939\) −7.83643e48 −0.700061
\(940\) 0 0
\(941\) −5.90432e48 −0.509262 −0.254631 0.967038i \(-0.581954\pi\)
−0.254631 + 0.967038i \(0.581954\pi\)
\(942\) −1.06418e48 −0.0901935
\(943\) 2.08575e49 1.73708
\(944\) −5.73368e48 −0.469241
\(945\) 0 0
\(946\) 1.60912e48 0.127170
\(947\) 7.50195e47 0.0582639 0.0291319 0.999576i \(-0.490726\pi\)
0.0291319 + 0.999576i \(0.490726\pi\)
\(948\) 3.54201e48 0.270341
\(949\) −1.64765e48 −0.123587
\(950\) 0 0
\(951\) 8.50718e48 0.616321
\(952\) −1.73433e48 −0.123487
\(953\) 1.06031e49 0.741993 0.370997 0.928634i \(-0.379016\pi\)
0.370997 + 0.928634i \(0.379016\pi\)
\(954\) −3.12208e48 −0.214732
\(955\) 0 0
\(956\) 5.21769e48 0.346675
\(957\) −6.16159e48 −0.402389
\(958\) 3.02447e48 0.194141
\(959\) 1.92247e48 0.121297
\(960\) 0 0
\(961\) −1.63826e49 −0.998726
\(962\) −9.13154e47 −0.0547212
\(963\) 1.56604e48 0.0922507
\(964\) 1.50954e49 0.874126
\(965\) 0 0
\(966\) 3.32731e47 0.0186196
\(967\) 3.31961e48 0.182621 0.0913104 0.995822i \(-0.470894\pi\)
0.0913104 + 0.995822i \(0.470894\pi\)
\(968\) −1.38868e49 −0.751030
\(969\) 1.94294e49 1.03304
\(970\) 0 0
\(971\) 1.67736e49 0.862003 0.431001 0.902351i \(-0.358160\pi\)
0.431001 + 0.902351i \(0.358160\pi\)
\(972\) −1.95710e49 −0.988826
\(973\) 1.57991e48 0.0784819
\(974\) −6.22773e48 −0.304164
\(975\) 0 0
\(976\) −1.20754e48 −0.0570139
\(977\) −2.50785e49 −1.16424 −0.582118 0.813104i \(-0.697776\pi\)
−0.582118 + 0.813104i \(0.697776\pi\)
\(978\) 2.39225e48 0.109198
\(979\) −3.08812e49 −1.38605
\(980\) 0 0
\(981\) 2.49366e49 1.08218
\(982\) 7.94535e48 0.339058
\(983\) −1.24304e49 −0.521616 −0.260808 0.965391i \(-0.583989\pi\)
−0.260808 + 0.965391i \(0.583989\pi\)
\(984\) −7.20883e48 −0.297472
\(985\) 0 0
\(986\) 4.35293e48 0.173705
\(987\) −1.74342e48 −0.0684177
\(988\) −4.35418e48 −0.168042
\(989\) −1.10210e49 −0.418295
\(990\) 0 0
\(991\) −2.48644e49 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(992\) −5.54307e47 −0.0200128
\(993\) −1.20437e48 −0.0427658
\(994\) 7.56031e47 0.0264037
\(995\) 0 0
\(996\) −1.46854e49 −0.496144
\(997\) 2.01041e49 0.668058 0.334029 0.942563i \(-0.391592\pi\)
0.334029 + 0.942563i \(0.391592\pi\)
\(998\) −1.13134e49 −0.369777
\(999\) 4.84191e49 1.55663
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.1.4 6
5.2 odd 4 25.34.b.c.24.7 12
5.3 odd 4 25.34.b.c.24.6 12
5.4 even 2 5.34.a.b.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.34.a.b.1.3 6 5.4 even 2
25.34.a.c.1.4 6 1.1 even 1 trivial
25.34.b.c.24.6 12 5.3 odd 4
25.34.b.c.24.7 12 5.2 odd 4