Properties

Label 25.36.a.a.1.3
Level $25$
Weight $36$
Character 25.1
Self dual yes
Analytic conductor $193.988$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(193.987826584\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 12422194 x - 2645665785\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-213.765\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q+165109. q^{2} +3.45913e8 q^{3} -7.09870e9 q^{4} +5.71135e13 q^{6} +1.25160e14 q^{7} -6.84517e15 q^{8} +6.96245e16 q^{9} +O(q^{10})\) \(q+165109. q^{2} +3.45913e8 q^{3} -7.09870e9 q^{4} +5.71135e13 q^{6} +1.25160e14 q^{7} -6.84517e15 q^{8} +6.96245e16 q^{9} -1.70842e18 q^{11} -2.45554e18 q^{12} +4.94986e19 q^{13} +2.06651e19 q^{14} -8.86291e20 q^{16} -1.32045e21 q^{17} +1.14956e22 q^{18} +3.94388e21 q^{19} +4.32945e22 q^{21} -2.82076e23 q^{22} +3.48881e23 q^{23} -2.36784e24 q^{24} +8.17268e24 q^{26} +6.77748e24 q^{27} -8.88473e23 q^{28} +3.21628e25 q^{29} +3.41044e25 q^{31} +8.88635e25 q^{32} -5.90965e26 q^{33} -2.18018e26 q^{34} -4.94244e26 q^{36} +4.03624e27 q^{37} +6.51171e26 q^{38} +1.71222e28 q^{39} +8.65857e27 q^{41} +7.14832e27 q^{42} -9.89389e26 q^{43} +1.21276e28 q^{44} +5.76034e28 q^{46} -1.95852e29 q^{47} -3.06580e29 q^{48} -3.63154e29 q^{49} -4.56760e29 q^{51} -3.51376e29 q^{52} +9.96909e29 q^{53} +1.11902e30 q^{54} -8.56741e29 q^{56} +1.36424e30 q^{57} +5.31037e30 q^{58} +3.91953e30 q^{59} +7.64909e29 q^{61} +5.63096e30 q^{62} +8.71420e30 q^{63} +4.51249e31 q^{64} -9.75738e31 q^{66} +1.64301e32 q^{67} +9.37346e30 q^{68} +1.20682e32 q^{69} +7.65930e31 q^{71} -4.76592e32 q^{72} +7.08063e32 q^{73} +6.66421e32 q^{74} -2.79964e31 q^{76} -2.13826e32 q^{77} +2.82704e33 q^{78} +2.34599e33 q^{79} -1.13900e33 q^{81} +1.42961e33 q^{82} -5.18971e33 q^{83} -3.07335e32 q^{84} -1.63357e32 q^{86} +1.11255e34 q^{87} +1.16944e34 q^{88} +1.44578e34 q^{89} +6.19525e33 q^{91} -2.47660e33 q^{92} +1.17972e34 q^{93} -3.23369e34 q^{94} +3.07391e34 q^{96} +2.87399e34 q^{97} -5.99600e34 q^{98} -1.18948e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} + O(q^{10}) \) \( 3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} - 1157945428549987044q^{11} + 22548891526776486144q^{12} + 62139610550998650558q^{13} + \)\(13\!\cdots\!28\)\(q^{14} + \)\(21\!\cdots\!48\)\(q^{16} + \)\(39\!\cdots\!94\)\(q^{17} + \)\(23\!\cdots\!08\)\(q^{18} - \)\(32\!\cdots\!40\)\(q^{19} + \)\(22\!\cdots\!76\)\(q^{21} - \)\(22\!\cdots\!12\)\(q^{22} + \)\(51\!\cdots\!08\)\(q^{23} - \)\(37\!\cdots\!20\)\(q^{24} + \)\(42\!\cdots\!36\)\(q^{26} - \)\(27\!\cdots\!00\)\(q^{27} - \)\(10\!\cdots\!08\)\(q^{28} - \)\(38\!\cdots\!10\)\(q^{29} + \)\(10\!\cdots\!56\)\(q^{31} + \)\(92\!\cdots\!44\)\(q^{32} - \)\(11\!\cdots\!84\)\(q^{33} - \)\(55\!\cdots\!12\)\(q^{34} - \)\(60\!\cdots\!52\)\(q^{36} - \)\(24\!\cdots\!06\)\(q^{37} + \)\(12\!\cdots\!00\)\(q^{38} + \)\(18\!\cdots\!12\)\(q^{39} + \)\(23\!\cdots\!06\)\(q^{41} + \)\(33\!\cdots\!88\)\(q^{42} + \)\(47\!\cdots\!08\)\(q^{43} - \)\(80\!\cdots\!12\)\(q^{44} + \)\(31\!\cdots\!56\)\(q^{46} - \)\(16\!\cdots\!56\)\(q^{47} - \)\(44\!\cdots\!92\)\(q^{48} - \)\(59\!\cdots\!21\)\(q^{49} - \)\(18\!\cdots\!04\)\(q^{51} + \)\(51\!\cdots\!44\)\(q^{52} + \)\(16\!\cdots\!58\)\(q^{53} + \)\(44\!\cdots\!60\)\(q^{54} + \)\(56\!\cdots\!40\)\(q^{56} - \)\(40\!\cdots\!00\)\(q^{57} + \)\(23\!\cdots\!00\)\(q^{58} + \)\(43\!\cdots\!80\)\(q^{59} + \)\(23\!\cdots\!06\)\(q^{61} - \)\(29\!\cdots\!12\)\(q^{62} - \)\(45\!\cdots\!92\)\(q^{63} + \)\(93\!\cdots\!84\)\(q^{64} - \)\(75\!\cdots\!68\)\(q^{66} + \)\(18\!\cdots\!44\)\(q^{67} - \)\(21\!\cdots\!08\)\(q^{68} - \)\(32\!\cdots\!48\)\(q^{69} + \)\(34\!\cdots\!56\)\(q^{71} - \)\(31\!\cdots\!00\)\(q^{72} + \)\(28\!\cdots\!58\)\(q^{73} + \)\(21\!\cdots\!08\)\(q^{74} - \)\(27\!\cdots\!20\)\(q^{76} - \)\(69\!\cdots\!12\)\(q^{77} + \)\(22\!\cdots\!16\)\(q^{78} - \)\(42\!\cdots\!60\)\(q^{79} + \)\(18\!\cdots\!63\)\(q^{81} + \)\(40\!\cdots\!88\)\(q^{82} - \)\(14\!\cdots\!92\)\(q^{83} - \)\(13\!\cdots\!52\)\(q^{84} + \)\(14\!\cdots\!96\)\(q^{86} + \)\(78\!\cdots\!00\)\(q^{87} + \)\(18\!\cdots\!00\)\(q^{88} + \)\(30\!\cdots\!70\)\(q^{89} + \)\(10\!\cdots\!96\)\(q^{91} - \)\(83\!\cdots\!56\)\(q^{92} + \)\(40\!\cdots\!16\)\(q^{93} + \)\(19\!\cdots\!68\)\(q^{94} - \)\(32\!\cdots\!84\)\(q^{96} + \)\(10\!\cdots\!94\)\(q^{97} + \)\(13\!\cdots\!92\)\(q^{98} + \)\(30\!\cdots\!52\)\(q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 165109. 0.890730 0.445365 0.895349i \(-0.353074\pi\)
0.445365 + 0.895349i \(0.353074\pi\)
\(3\) 3.45913e8 1.54648 0.773242 0.634111i \(-0.218634\pi\)
0.773242 + 0.634111i \(0.218634\pi\)
\(4\) −7.09870e9 −0.206599
\(5\) 0 0
\(6\) 5.71135e13 1.37750
\(7\) 1.25160e14 0.203353 0.101676 0.994818i \(-0.467579\pi\)
0.101676 + 0.994818i \(0.467579\pi\)
\(8\) −6.84517e15 −1.07475
\(9\) 6.96245e16 1.39161
\(10\) 0 0
\(11\) −1.70842e18 −1.01911 −0.509557 0.860437i \(-0.670191\pi\)
−0.509557 + 0.860437i \(0.670191\pi\)
\(12\) −2.45554e18 −0.319503
\(13\) 4.94986e19 1.58703 0.793514 0.608552i \(-0.208249\pi\)
0.793514 + 0.608552i \(0.208249\pi\)
\(14\) 2.06651e19 0.181132
\(15\) 0 0
\(16\) −8.86291e20 −0.750717
\(17\) −1.32045e21 −0.387137 −0.193569 0.981087i \(-0.562006\pi\)
−0.193569 + 0.981087i \(0.562006\pi\)
\(18\) 1.14956e22 1.23955
\(19\) 3.94388e21 0.165096 0.0825479 0.996587i \(-0.473694\pi\)
0.0825479 + 0.996587i \(0.473694\pi\)
\(20\) 0 0
\(21\) 4.32945e22 0.314482
\(22\) −2.82076e23 −0.907756
\(23\) 3.48881e23 0.515750 0.257875 0.966178i \(-0.416978\pi\)
0.257875 + 0.966178i \(0.416978\pi\)
\(24\) −2.36784e24 −1.66209
\(25\) 0 0
\(26\) 8.17268e24 1.41361
\(27\) 6.77748e24 0.605623
\(28\) −8.88473e23 −0.0420125
\(29\) 3.21628e25 0.822979 0.411489 0.911415i \(-0.365009\pi\)
0.411489 + 0.911415i \(0.365009\pi\)
\(30\) 0 0
\(31\) 3.41044e25 0.271632 0.135816 0.990734i \(-0.456634\pi\)
0.135816 + 0.990734i \(0.456634\pi\)
\(32\) 8.88635e25 0.406068
\(33\) −5.90965e26 −1.57604
\(34\) −2.18018e26 −0.344835
\(35\) 0 0
\(36\) −4.94244e26 −0.287506
\(37\) 4.03624e27 1.45361 0.726804 0.686845i \(-0.241005\pi\)
0.726804 + 0.686845i \(0.241005\pi\)
\(38\) 6.51171e26 0.147056
\(39\) 1.71222e28 2.45431
\(40\) 0 0
\(41\) 8.65857e27 0.517283 0.258642 0.965973i \(-0.416725\pi\)
0.258642 + 0.965973i \(0.416725\pi\)
\(42\) 7.14832e27 0.280118
\(43\) −9.89389e26 −0.0256844 −0.0128422 0.999918i \(-0.504088\pi\)
−0.0128422 + 0.999918i \(0.504088\pi\)
\(44\) 1.21276e28 0.210548
\(45\) 0 0
\(46\) 5.76034e28 0.459395
\(47\) −1.95852e29 −1.07205 −0.536025 0.844202i \(-0.680075\pi\)
−0.536025 + 0.844202i \(0.680075\pi\)
\(48\) −3.06580e29 −1.16097
\(49\) −3.63154e29 −0.958648
\(50\) 0 0
\(51\) −4.56760e29 −0.598702
\(52\) −3.51376e29 −0.327879
\(53\) 9.96909e29 0.666543 0.333271 0.942831i \(-0.391847\pi\)
0.333271 + 0.942831i \(0.391847\pi\)
\(54\) 1.11902e30 0.539447
\(55\) 0 0
\(56\) −8.56741e29 −0.218554
\(57\) 1.36424e30 0.255318
\(58\) 5.31037e30 0.733052
\(59\) 3.91953e30 0.401166 0.200583 0.979677i \(-0.435716\pi\)
0.200583 + 0.979677i \(0.435716\pi\)
\(60\) 0 0
\(61\) 7.64909e29 0.0436855 0.0218428 0.999761i \(-0.493047\pi\)
0.0218428 + 0.999761i \(0.493047\pi\)
\(62\) 5.63096e30 0.241951
\(63\) 8.71420e30 0.282988
\(64\) 4.51249e31 1.11241
\(65\) 0 0
\(66\) −9.75738e31 −1.40383
\(67\) 1.64301e32 1.81690 0.908451 0.417992i \(-0.137266\pi\)
0.908451 + 0.417992i \(0.137266\pi\)
\(68\) 9.37346e30 0.0799823
\(69\) 1.20682e32 0.797600
\(70\) 0 0
\(71\) 7.65930e31 0.307021 0.153510 0.988147i \(-0.450942\pi\)
0.153510 + 0.988147i \(0.450942\pi\)
\(72\) −4.76592e32 −1.49564
\(73\) 7.08063e32 1.74551 0.872754 0.488160i \(-0.162332\pi\)
0.872754 + 0.488160i \(0.162332\pi\)
\(74\) 6.66421e32 1.29477
\(75\) 0 0
\(76\) −2.79964e31 −0.0341087
\(77\) −2.13826e32 −0.207239
\(78\) 2.82704e33 2.18613
\(79\) 2.34599e33 1.45162 0.725809 0.687896i \(-0.241466\pi\)
0.725809 + 0.687896i \(0.241466\pi\)
\(80\) 0 0
\(81\) −1.13900e33 −0.455027
\(82\) 1.42961e33 0.460760
\(83\) −5.18971e33 −1.35293 −0.676467 0.736473i \(-0.736490\pi\)
−0.676467 + 0.736473i \(0.736490\pi\)
\(84\) −3.07335e32 −0.0649717
\(85\) 0 0
\(86\) −1.63357e32 −0.0228779
\(87\) 1.11255e34 1.27272
\(88\) 1.16944e34 1.09530
\(89\) 1.44578e34 1.11116 0.555582 0.831461i \(-0.312495\pi\)
0.555582 + 0.831461i \(0.312495\pi\)
\(90\) 0 0
\(91\) 6.19525e33 0.322726
\(92\) −2.47660e33 −0.106554
\(93\) 1.17972e34 0.420075
\(94\) −3.23369e34 −0.954907
\(95\) 0 0
\(96\) 3.07391e34 0.627978
\(97\) 2.87399e34 0.489756 0.244878 0.969554i \(-0.421252\pi\)
0.244878 + 0.969554i \(0.421252\pi\)
\(98\) −5.99600e34 −0.853897
\(99\) −1.18948e35 −1.41821
\(100\) 0 0
\(101\) 1.13198e35 0.951079 0.475539 0.879694i \(-0.342253\pi\)
0.475539 + 0.879694i \(0.342253\pi\)
\(102\) −7.54153e34 −0.533282
\(103\) −2.53635e35 −1.51202 −0.756011 0.654559i \(-0.772855\pi\)
−0.756011 + 0.654559i \(0.772855\pi\)
\(104\) −3.38826e35 −1.70567
\(105\) 0 0
\(106\) 1.64599e35 0.593710
\(107\) 3.63939e35 1.11381 0.556907 0.830575i \(-0.311988\pi\)
0.556907 + 0.830575i \(0.311988\pi\)
\(108\) −4.81113e34 −0.125121
\(109\) −4.65371e34 −0.103000 −0.0514998 0.998673i \(-0.516400\pi\)
−0.0514998 + 0.998673i \(0.516400\pi\)
\(110\) 0 0
\(111\) 1.39619e36 2.24798
\(112\) −1.10928e35 −0.152660
\(113\) 2.18346e35 0.257201 0.128601 0.991696i \(-0.458951\pi\)
0.128601 + 0.991696i \(0.458951\pi\)
\(114\) 2.25249e35 0.227419
\(115\) 0 0
\(116\) −2.28314e35 −0.170027
\(117\) 3.44632e36 2.20853
\(118\) 6.47150e35 0.357330
\(119\) −1.65267e35 −0.0787254
\(120\) 0 0
\(121\) 1.08456e35 0.0385932
\(122\) 1.26293e35 0.0389120
\(123\) 2.99511e36 0.799970
\(124\) −2.42097e35 −0.0561191
\(125\) 0 0
\(126\) 1.43880e36 0.252066
\(127\) −4.19558e36 −0.640070 −0.320035 0.947406i \(-0.603695\pi\)
−0.320035 + 0.947406i \(0.603695\pi\)
\(128\) 4.39721e36 0.584793
\(129\) −3.42243e35 −0.0397205
\(130\) 0 0
\(131\) 2.40631e36 0.213355 0.106678 0.994294i \(-0.465979\pi\)
0.106678 + 0.994294i \(0.465979\pi\)
\(132\) 4.19509e36 0.325610
\(133\) 4.93616e35 0.0335727
\(134\) 2.71276e37 1.61837
\(135\) 0 0
\(136\) 9.03869e36 0.416078
\(137\) 2.14718e37 0.869477 0.434739 0.900557i \(-0.356841\pi\)
0.434739 + 0.900557i \(0.356841\pi\)
\(138\) 1.99258e37 0.710446
\(139\) 2.63209e37 0.827068 0.413534 0.910489i \(-0.364294\pi\)
0.413534 + 0.910489i \(0.364294\pi\)
\(140\) 0 0
\(141\) −6.77477e37 −1.65791
\(142\) 1.26462e37 0.273473
\(143\) −8.45645e37 −1.61736
\(144\) −6.17076e37 −1.04471
\(145\) 0 0
\(146\) 1.16908e38 1.55478
\(147\) −1.25620e38 −1.48253
\(148\) −2.86521e37 −0.300314
\(149\) −1.74389e38 −1.62465 −0.812324 0.583206i \(-0.801798\pi\)
−0.812324 + 0.583206i \(0.801798\pi\)
\(150\) 0 0
\(151\) −1.40947e38 −1.03982 −0.519912 0.854220i \(-0.674035\pi\)
−0.519912 + 0.854220i \(0.674035\pi\)
\(152\) −2.69965e37 −0.177437
\(153\) −9.19355e37 −0.538745
\(154\) −3.53046e37 −0.184595
\(155\) 0 0
\(156\) −1.21546e38 −0.507059
\(157\) −1.49924e38 −0.559277 −0.279639 0.960105i \(-0.590215\pi\)
−0.279639 + 0.960105i \(0.590215\pi\)
\(158\) 3.87345e38 1.29300
\(159\) 3.44844e38 1.03080
\(160\) 0 0
\(161\) 4.36659e37 0.104879
\(162\) −1.88060e38 −0.405306
\(163\) −8.60787e38 −1.66576 −0.832880 0.553453i \(-0.813310\pi\)
−0.832880 + 0.553453i \(0.813310\pi\)
\(164\) −6.14646e37 −0.106870
\(165\) 0 0
\(166\) −8.56869e38 −1.20510
\(167\) 9.23344e38 1.16903 0.584515 0.811383i \(-0.301285\pi\)
0.584515 + 0.811383i \(0.301285\pi\)
\(168\) −2.96358e38 −0.337991
\(169\) 1.47733e39 1.51866
\(170\) 0 0
\(171\) 2.74591e38 0.229749
\(172\) 7.02337e36 0.00530638
\(173\) 1.33161e39 0.909012 0.454506 0.890744i \(-0.349816\pi\)
0.454506 + 0.890744i \(0.349816\pi\)
\(174\) 1.83693e39 1.13365
\(175\) 0 0
\(176\) 1.51416e39 0.765066
\(177\) 1.35582e39 0.620396
\(178\) 2.38712e39 0.989748
\(179\) 6.27202e38 0.235765 0.117883 0.993028i \(-0.462389\pi\)
0.117883 + 0.993028i \(0.462389\pi\)
\(180\) 0 0
\(181\) −2.39965e39 −0.742629 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(182\) 1.02289e39 0.287462
\(183\) 2.64592e38 0.0675590
\(184\) −2.38815e39 −0.554305
\(185\) 0 0
\(186\) 1.94782e39 0.374174
\(187\) 2.25588e39 0.394537
\(188\) 1.39029e39 0.221485
\(189\) 8.48269e38 0.123155
\(190\) 0 0
\(191\) 8.14518e39 0.983596 0.491798 0.870709i \(-0.336340\pi\)
0.491798 + 0.870709i \(0.336340\pi\)
\(192\) 1.56093e40 1.72033
\(193\) −1.38549e40 −1.39428 −0.697139 0.716936i \(-0.745544\pi\)
−0.697139 + 0.716936i \(0.745544\pi\)
\(194\) 4.74522e39 0.436240
\(195\) 0 0
\(196\) 2.57792e39 0.198056
\(197\) 1.15539e40 0.812026 0.406013 0.913867i \(-0.366919\pi\)
0.406013 + 0.913867i \(0.366919\pi\)
\(198\) −1.96394e40 −1.26324
\(199\) −4.76691e39 −0.280742 −0.140371 0.990099i \(-0.544830\pi\)
−0.140371 + 0.990099i \(0.544830\pi\)
\(200\) 0 0
\(201\) 5.68340e40 2.80981
\(202\) 1.86901e40 0.847155
\(203\) 4.02549e39 0.167355
\(204\) 3.24241e39 0.123691
\(205\) 0 0
\(206\) −4.18775e40 −1.34680
\(207\) 2.42906e40 0.717725
\(208\) −4.38702e40 −1.19141
\(209\) −6.73781e39 −0.168251
\(210\) 0 0
\(211\) −1.87354e40 −0.396022 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(212\) −7.07676e39 −0.137707
\(213\) 2.64945e40 0.474803
\(214\) 6.00896e40 0.992107
\(215\) 0 0
\(216\) −4.63930e40 −0.650896
\(217\) 4.26851e39 0.0552371
\(218\) −7.68370e39 −0.0917448
\(219\) 2.44929e41 2.69940
\(220\) 0 0
\(221\) −6.53603e40 −0.614398
\(222\) 2.30524e41 2.00234
\(223\) 7.12100e40 0.571750 0.285875 0.958267i \(-0.407716\pi\)
0.285875 + 0.958267i \(0.407716\pi\)
\(224\) 1.11222e40 0.0825750
\(225\) 0 0
\(226\) 3.60510e40 0.229097
\(227\) 1.04005e41 0.611787 0.305893 0.952066i \(-0.401045\pi\)
0.305893 + 0.952066i \(0.401045\pi\)
\(228\) −9.68434e39 −0.0527485
\(229\) 1.82859e41 0.922564 0.461282 0.887254i \(-0.347390\pi\)
0.461282 + 0.887254i \(0.347390\pi\)
\(230\) 0 0
\(231\) −7.39652e40 −0.320493
\(232\) −2.20160e41 −0.884501
\(233\) −1.57847e41 −0.588177 −0.294088 0.955778i \(-0.595016\pi\)
−0.294088 + 0.955778i \(0.595016\pi\)
\(234\) 5.69019e41 1.96720
\(235\) 0 0
\(236\) −2.78236e40 −0.0828806
\(237\) 8.11510e41 2.24490
\(238\) −2.72871e40 −0.0701231
\(239\) 5.00984e41 1.19636 0.598178 0.801363i \(-0.295892\pi\)
0.598178 + 0.801363i \(0.295892\pi\)
\(240\) 0 0
\(241\) −3.33164e41 −0.687638 −0.343819 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(242\) 1.79072e40 0.0343762
\(243\) −7.33084e41 −1.30931
\(244\) −5.42986e39 −0.00902540
\(245\) 0 0
\(246\) 4.94521e41 0.712558
\(247\) 1.95217e41 0.262011
\(248\) −2.33451e41 −0.291938
\(249\) −1.79519e42 −2.09229
\(250\) 0 0
\(251\) 1.52760e42 1.54782 0.773909 0.633297i \(-0.218299\pi\)
0.773909 + 0.633297i \(0.218299\pi\)
\(252\) −6.18595e40 −0.0584652
\(253\) −5.96035e41 −0.525608
\(254\) −6.92730e41 −0.570130
\(255\) 0 0
\(256\) −8.24460e41 −0.591521
\(257\) 2.72482e42 1.82603 0.913016 0.407925i \(-0.133747\pi\)
0.913016 + 0.407925i \(0.133747\pi\)
\(258\) −5.65074e40 −0.0353802
\(259\) 5.05176e41 0.295595
\(260\) 0 0
\(261\) 2.23932e42 1.14527
\(262\) 3.97303e41 0.190042
\(263\) −4.22055e40 −0.0188861 −0.00944307 0.999955i \(-0.503006\pi\)
−0.00944307 + 0.999955i \(0.503006\pi\)
\(264\) 4.04526e42 1.69386
\(265\) 0 0
\(266\) 8.15005e40 0.0299042
\(267\) 5.00116e42 1.71840
\(268\) −1.16632e42 −0.375371
\(269\) −1.30618e42 −0.393858 −0.196929 0.980418i \(-0.563097\pi\)
−0.196929 + 0.980418i \(0.563097\pi\)
\(270\) 0 0
\(271\) −4.11918e41 −0.109106 −0.0545529 0.998511i \(-0.517373\pi\)
−0.0545529 + 0.998511i \(0.517373\pi\)
\(272\) 1.17030e42 0.290631
\(273\) 2.14302e42 0.499091
\(274\) 3.54519e42 0.774470
\(275\) 0 0
\(276\) −8.56689e41 −0.164784
\(277\) −7.38232e42 −1.33290 −0.666449 0.745551i \(-0.732187\pi\)
−0.666449 + 0.745551i \(0.732187\pi\)
\(278\) 4.34582e42 0.736694
\(279\) 2.37451e42 0.378007
\(280\) 0 0
\(281\) 4.08223e42 0.573505 0.286752 0.958005i \(-0.407424\pi\)
0.286752 + 0.958005i \(0.407424\pi\)
\(282\) −1.11858e43 −1.47675
\(283\) 1.06503e43 1.32159 0.660796 0.750565i \(-0.270219\pi\)
0.660796 + 0.750565i \(0.270219\pi\)
\(284\) −5.43711e41 −0.0634303
\(285\) 0 0
\(286\) −1.39624e43 −1.44063
\(287\) 1.08371e42 0.105191
\(288\) 6.18708e42 0.565089
\(289\) −9.88997e42 −0.850125
\(290\) 0 0
\(291\) 9.94151e42 0.757399
\(292\) −5.02633e42 −0.360621
\(293\) −1.90474e43 −1.28722 −0.643610 0.765353i \(-0.722564\pi\)
−0.643610 + 0.765353i \(0.722564\pi\)
\(294\) −2.07410e43 −1.32054
\(295\) 0 0
\(296\) −2.76288e43 −1.56227
\(297\) −1.15788e43 −0.617199
\(298\) −2.87932e43 −1.44712
\(299\) 1.72691e43 0.818510
\(300\) 0 0
\(301\) −1.23832e41 −0.00522298
\(302\) −2.32717e43 −0.926203
\(303\) 3.91569e43 1.47083
\(304\) −3.49543e42 −0.123940
\(305\) 0 0
\(306\) −1.51794e43 −0.479877
\(307\) 1.39752e43 0.417288 0.208644 0.977992i \(-0.433095\pi\)
0.208644 + 0.977992i \(0.433095\pi\)
\(308\) 1.51789e42 0.0428156
\(309\) −8.77359e43 −2.33832
\(310\) 0 0
\(311\) 2.22377e43 0.529399 0.264700 0.964331i \(-0.414727\pi\)
0.264700 + 0.964331i \(0.414727\pi\)
\(312\) −1.17205e44 −2.63778
\(313\) −3.63171e43 −0.772833 −0.386417 0.922324i \(-0.626287\pi\)
−0.386417 + 0.922324i \(0.626287\pi\)
\(314\) −2.47538e43 −0.498165
\(315\) 0 0
\(316\) −1.66535e43 −0.299903
\(317\) −5.29385e43 −0.902058 −0.451029 0.892509i \(-0.648943\pi\)
−0.451029 + 0.892509i \(0.648943\pi\)
\(318\) 5.69370e43 0.918163
\(319\) −5.49476e43 −0.838709
\(320\) 0 0
\(321\) 1.25891e44 1.72249
\(322\) 7.20964e42 0.0934191
\(323\) −5.20769e42 −0.0639147
\(324\) 8.08544e42 0.0940083
\(325\) 0 0
\(326\) −1.42124e44 −1.48374
\(327\) −1.60978e43 −0.159287
\(328\) −5.92693e43 −0.555953
\(329\) −2.45128e43 −0.218004
\(330\) 0 0
\(331\) 6.25551e43 0.500349 0.250174 0.968201i \(-0.419512\pi\)
0.250174 + 0.968201i \(0.419512\pi\)
\(332\) 3.68402e43 0.279515
\(333\) 2.81022e44 2.02286
\(334\) 1.52453e44 1.04129
\(335\) 0 0
\(336\) −3.83715e43 −0.236087
\(337\) 1.04185e44 0.608529 0.304264 0.952588i \(-0.401589\pi\)
0.304264 + 0.952588i \(0.401589\pi\)
\(338\) 2.43920e44 1.35271
\(339\) 7.55289e43 0.397757
\(340\) 0 0
\(341\) −5.82647e43 −0.276824
\(342\) 4.53375e43 0.204645
\(343\) −9.28652e43 −0.398296
\(344\) 6.77253e42 0.0276044
\(345\) 0 0
\(346\) 2.19861e44 0.809685
\(347\) 3.07657e44 1.07721 0.538606 0.842557i \(-0.318951\pi\)
0.538606 + 0.842557i \(0.318951\pi\)
\(348\) −7.89769e43 −0.262944
\(349\) −2.62767e44 −0.832005 −0.416003 0.909363i \(-0.636569\pi\)
−0.416003 + 0.909363i \(0.636569\pi\)
\(350\) 0 0
\(351\) 3.35476e44 0.961140
\(352\) −1.51816e44 −0.413830
\(353\) −7.06671e44 −1.83299 −0.916494 0.400048i \(-0.868994\pi\)
−0.916494 + 0.400048i \(0.868994\pi\)
\(354\) 2.23858e44 0.552606
\(355\) 0 0
\(356\) −1.02632e44 −0.229566
\(357\) −5.71681e43 −0.121748
\(358\) 1.03557e44 0.210003
\(359\) −5.35788e44 −1.03476 −0.517380 0.855756i \(-0.673093\pi\)
−0.517380 + 0.855756i \(0.673093\pi\)
\(360\) 0 0
\(361\) −5.55104e44 −0.972743
\(362\) −3.96204e44 −0.661483
\(363\) 3.75165e43 0.0596838
\(364\) −4.39782e43 −0.0666750
\(365\) 0 0
\(366\) 4.36866e43 0.0601768
\(367\) 3.00056e44 0.394045 0.197022 0.980399i \(-0.436873\pi\)
0.197022 + 0.980399i \(0.436873\pi\)
\(368\) −3.09210e44 −0.387183
\(369\) 6.02849e44 0.719858
\(370\) 0 0
\(371\) 1.24773e44 0.135543
\(372\) −8.37447e43 −0.0867872
\(373\) −1.08201e45 −1.06986 −0.534932 0.844895i \(-0.679663\pi\)
−0.534932 + 0.844895i \(0.679663\pi\)
\(374\) 3.72466e44 0.351426
\(375\) 0 0
\(376\) 1.34064e45 1.15219
\(377\) 1.59201e45 1.30609
\(378\) 1.40057e44 0.109698
\(379\) 1.21847e45 0.911229 0.455614 0.890177i \(-0.349420\pi\)
0.455614 + 0.890177i \(0.349420\pi\)
\(380\) 0 0
\(381\) −1.45131e45 −0.989858
\(382\) 1.34484e45 0.876119
\(383\) 9.53201e43 0.0593207 0.0296603 0.999560i \(-0.490557\pi\)
0.0296603 + 0.999560i \(0.490557\pi\)
\(384\) 1.52105e45 0.904373
\(385\) 0 0
\(386\) −2.28757e45 −1.24193
\(387\) −6.88857e43 −0.0357427
\(388\) −2.04016e44 −0.101183
\(389\) 1.90493e45 0.903152 0.451576 0.892233i \(-0.350862\pi\)
0.451576 + 0.892233i \(0.350862\pi\)
\(390\) 0 0
\(391\) −4.60678e44 −0.199666
\(392\) 2.48585e45 1.03031
\(393\) 8.32373e44 0.329950
\(394\) 1.90766e45 0.723296
\(395\) 0 0
\(396\) 8.44376e44 0.293002
\(397\) −3.96399e45 −1.31613 −0.658065 0.752961i \(-0.728625\pi\)
−0.658065 + 0.752961i \(0.728625\pi\)
\(398\) −7.87060e44 −0.250066
\(399\) 1.70748e44 0.0519196
\(400\) 0 0
\(401\) −5.53039e44 −0.154074 −0.0770369 0.997028i \(-0.524546\pi\)
−0.0770369 + 0.997028i \(0.524546\pi\)
\(402\) 9.38381e45 2.50278
\(403\) 1.68812e45 0.431088
\(404\) −8.03562e44 −0.196492
\(405\) 0 0
\(406\) 6.64646e44 0.149068
\(407\) −6.89560e45 −1.48139
\(408\) 3.12660e45 0.643457
\(409\) 3.35495e45 0.661498 0.330749 0.943719i \(-0.392699\pi\)
0.330749 + 0.943719i \(0.392699\pi\)
\(410\) 0 0
\(411\) 7.42738e45 1.34463
\(412\) 1.80048e45 0.312383
\(413\) 4.90568e44 0.0815781
\(414\) 4.01061e45 0.639299
\(415\) 0 0
\(416\) 4.39862e45 0.644441
\(417\) 9.10474e45 1.27905
\(418\) −1.11247e45 −0.149867
\(419\) −7.69138e45 −0.993708 −0.496854 0.867834i \(-0.665512\pi\)
−0.496854 + 0.867834i \(0.665512\pi\)
\(420\) 0 0
\(421\) −2.77229e45 −0.329535 −0.164767 0.986332i \(-0.552687\pi\)
−0.164767 + 0.986332i \(0.552687\pi\)
\(422\) −3.09338e45 −0.352749
\(423\) −1.36361e46 −1.49188
\(424\) −6.82401e45 −0.716370
\(425\) 0 0
\(426\) 4.37449e45 0.422921
\(427\) 9.57360e43 0.00888357
\(428\) −2.58349e45 −0.230113
\(429\) −2.92520e46 −2.50122
\(430\) 0 0
\(431\) 6.18484e45 0.487502 0.243751 0.969838i \(-0.421622\pi\)
0.243751 + 0.969838i \(0.421622\pi\)
\(432\) −6.00681e45 −0.454651
\(433\) −2.21783e46 −1.61209 −0.806047 0.591852i \(-0.798397\pi\)
−0.806047 + 0.591852i \(0.798397\pi\)
\(434\) 7.04770e44 0.0492014
\(435\) 0 0
\(436\) 3.30353e44 0.0212796
\(437\) 1.37594e45 0.0851482
\(438\) 4.04399e46 2.40444
\(439\) 1.91766e45 0.109557 0.0547787 0.998499i \(-0.482555\pi\)
0.0547787 + 0.998499i \(0.482555\pi\)
\(440\) 0 0
\(441\) −2.52844e46 −1.33407
\(442\) −1.07916e46 −0.547263
\(443\) 3.00864e46 1.46658 0.733290 0.679916i \(-0.237984\pi\)
0.733290 + 0.679916i \(0.237984\pi\)
\(444\) −9.91114e45 −0.464431
\(445\) 0 0
\(446\) 1.17574e46 0.509275
\(447\) −6.03235e46 −2.51249
\(448\) 5.64783e45 0.226212
\(449\) 5.68788e45 0.219099 0.109549 0.993981i \(-0.465059\pi\)
0.109549 + 0.993981i \(0.465059\pi\)
\(450\) 0 0
\(451\) −1.47925e46 −0.527171
\(452\) −1.54997e45 −0.0531376
\(453\) −4.87555e46 −1.60807
\(454\) 1.71721e46 0.544937
\(455\) 0 0
\(456\) −9.33846e45 −0.274404
\(457\) −4.67626e46 −1.32241 −0.661204 0.750206i \(-0.729954\pi\)
−0.661204 + 0.750206i \(0.729954\pi\)
\(458\) 3.01918e46 0.821755
\(459\) −8.94930e45 −0.234459
\(460\) 0 0
\(461\) −3.72978e46 −0.905560 −0.452780 0.891622i \(-0.649568\pi\)
−0.452780 + 0.891622i \(0.649568\pi\)
\(462\) −1.22123e46 −0.285472
\(463\) 3.67649e46 0.827497 0.413748 0.910391i \(-0.364219\pi\)
0.413748 + 0.910391i \(0.364219\pi\)
\(464\) −2.85056e46 −0.617825
\(465\) 0 0
\(466\) −2.60620e46 −0.523907
\(467\) 2.37743e46 0.460323 0.230161 0.973152i \(-0.426075\pi\)
0.230161 + 0.973152i \(0.426075\pi\)
\(468\) −2.44644e46 −0.456280
\(469\) 2.05639e46 0.369472
\(470\) 0 0
\(471\) −5.18607e46 −0.864914
\(472\) −2.68298e46 −0.431155
\(473\) 1.69029e45 0.0261753
\(474\) 1.33988e47 1.99960
\(475\) 0 0
\(476\) 1.17318e45 0.0162646
\(477\) 6.94094e46 0.927569
\(478\) 8.27170e46 1.06563
\(479\) −8.80315e46 −1.09337 −0.546684 0.837339i \(-0.684110\pi\)
−0.546684 + 0.837339i \(0.684110\pi\)
\(480\) 0 0
\(481\) 1.99788e47 2.30691
\(482\) −5.50084e46 −0.612500
\(483\) 1.51046e46 0.162194
\(484\) −7.69900e44 −0.00797334
\(485\) 0 0
\(486\) −1.21039e47 −1.16625
\(487\) −7.13443e46 −0.663136 −0.331568 0.943431i \(-0.607578\pi\)
−0.331568 + 0.943431i \(0.607578\pi\)
\(488\) −5.23593e45 −0.0469512
\(489\) −2.97758e47 −2.57607
\(490\) 0 0
\(491\) 8.49115e46 0.683975 0.341988 0.939704i \(-0.388900\pi\)
0.341988 + 0.939704i \(0.388900\pi\)
\(492\) −2.12614e46 −0.165273
\(493\) −4.24693e46 −0.318606
\(494\) 3.22321e46 0.233382
\(495\) 0 0
\(496\) −3.02265e46 −0.203919
\(497\) 9.58638e45 0.0624335
\(498\) −2.96402e47 −1.86367
\(499\) −1.75876e47 −1.06770 −0.533848 0.845581i \(-0.679255\pi\)
−0.533848 + 0.845581i \(0.679255\pi\)
\(500\) 0 0
\(501\) 3.19397e47 1.80789
\(502\) 2.52221e47 1.37869
\(503\) −1.24051e47 −0.654880 −0.327440 0.944872i \(-0.606186\pi\)
−0.327440 + 0.944872i \(0.606186\pi\)
\(504\) −5.96502e46 −0.304143
\(505\) 0 0
\(506\) −9.84108e46 −0.468175
\(507\) 5.11027e47 2.34858
\(508\) 2.97832e46 0.132238
\(509\) 3.71512e47 1.59372 0.796859 0.604166i \(-0.206493\pi\)
0.796859 + 0.604166i \(0.206493\pi\)
\(510\) 0 0
\(511\) 8.86212e46 0.354954
\(512\) −2.87213e47 −1.11168
\(513\) 2.67296e46 0.0999857
\(514\) 4.49893e47 1.62650
\(515\) 0 0
\(516\) 2.42948e45 0.00820623
\(517\) 3.34597e47 1.09254
\(518\) 8.34092e46 0.263295
\(519\) 4.60621e47 1.40577
\(520\) 0 0
\(521\) 7.89659e45 0.0225309 0.0112655 0.999937i \(-0.496414\pi\)
0.0112655 + 0.999937i \(0.496414\pi\)
\(522\) 3.69732e47 1.02012
\(523\) −1.18122e47 −0.315175 −0.157588 0.987505i \(-0.550372\pi\)
−0.157588 + 0.987505i \(0.550372\pi\)
\(524\) −1.70816e46 −0.0440790
\(525\) 0 0
\(526\) −6.96851e45 −0.0168225
\(527\) −4.50331e46 −0.105159
\(528\) 5.23767e47 1.18316
\(529\) −3.35870e47 −0.734001
\(530\) 0 0
\(531\) 2.72895e47 0.558267
\(532\) −3.50403e45 −0.00693609
\(533\) 4.28587e47 0.820943
\(534\) 8.25738e47 1.53063
\(535\) 0 0
\(536\) −1.12467e48 −1.95272
\(537\) 2.16958e47 0.364607
\(538\) −2.15663e47 −0.350821
\(539\) 6.20419e47 0.976971
\(540\) 0 0
\(541\) −6.62658e47 −0.977997 −0.488998 0.872285i \(-0.662638\pi\)
−0.488998 + 0.872285i \(0.662638\pi\)
\(542\) −6.80114e46 −0.0971839
\(543\) −8.30070e47 −1.14846
\(544\) −1.17340e47 −0.157204
\(545\) 0 0
\(546\) 3.53832e47 0.444555
\(547\) 1.12424e47 0.136798 0.0683990 0.997658i \(-0.478211\pi\)
0.0683990 + 0.997658i \(0.478211\pi\)
\(548\) −1.52422e47 −0.179633
\(549\) 5.32564e46 0.0607933
\(550\) 0 0
\(551\) 1.26846e47 0.135870
\(552\) −8.26092e47 −0.857224
\(553\) 2.93624e47 0.295190
\(554\) −1.21889e48 −1.18725
\(555\) 0 0
\(556\) −1.86844e47 −0.170872
\(557\) −6.68090e47 −0.592064 −0.296032 0.955178i \(-0.595664\pi\)
−0.296032 + 0.955178i \(0.595664\pi\)
\(558\) 3.92053e47 0.336702
\(559\) −4.89734e46 −0.0407618
\(560\) 0 0
\(561\) 7.80339e47 0.610145
\(562\) 6.74014e47 0.510838
\(563\) −2.47733e48 −1.82007 −0.910033 0.414536i \(-0.863944\pi\)
−0.910033 + 0.414536i \(0.863944\pi\)
\(564\) 4.80921e47 0.342523
\(565\) 0 0
\(566\) 1.75846e48 1.17718
\(567\) −1.42558e47 −0.0925309
\(568\) −5.24292e47 −0.329972
\(569\) 8.59661e47 0.524641 0.262321 0.964981i \(-0.415512\pi\)
0.262321 + 0.964981i \(0.415512\pi\)
\(570\) 0 0
\(571\) −3.39185e48 −1.94673 −0.973364 0.229266i \(-0.926367\pi\)
−0.973364 + 0.229266i \(0.926367\pi\)
\(572\) 6.00298e47 0.334146
\(573\) 2.81753e48 1.52112
\(574\) 1.78930e47 0.0936968
\(575\) 0 0
\(576\) 3.14180e48 1.54805
\(577\) −2.12534e48 −1.01590 −0.507951 0.861386i \(-0.669597\pi\)
−0.507951 + 0.861386i \(0.669597\pi\)
\(578\) −1.63292e48 −0.757232
\(579\) −4.79258e48 −2.15623
\(580\) 0 0
\(581\) −6.49544e47 −0.275123
\(582\) 1.64143e48 0.674639
\(583\) −1.70314e48 −0.679283
\(584\) −4.84681e48 −1.87599
\(585\) 0 0
\(586\) −3.14491e48 −1.14657
\(587\) −1.06269e48 −0.376044 −0.188022 0.982165i \(-0.560208\pi\)
−0.188022 + 0.982165i \(0.560208\pi\)
\(588\) 8.91737e47 0.306290
\(589\) 1.34504e47 0.0448453
\(590\) 0 0
\(591\) 3.99665e48 1.25578
\(592\) −3.57728e48 −1.09125
\(593\) 3.98260e48 1.17953 0.589766 0.807574i \(-0.299220\pi\)
0.589766 + 0.807574i \(0.299220\pi\)
\(594\) −1.91176e48 −0.549758
\(595\) 0 0
\(596\) 1.23794e48 0.335651
\(597\) −1.64894e48 −0.434163
\(598\) 2.85129e48 0.729072
\(599\) 5.14535e48 1.27775 0.638874 0.769311i \(-0.279401\pi\)
0.638874 + 0.769311i \(0.279401\pi\)
\(600\) 0 0
\(601\) −1.83878e48 −0.430751 −0.215375 0.976531i \(-0.569098\pi\)
−0.215375 + 0.976531i \(0.569098\pi\)
\(602\) −2.04458e46 −0.00465227
\(603\) 1.14394e49 2.52842
\(604\) 1.00054e48 0.214827
\(605\) 0 0
\(606\) 6.46516e48 1.31011
\(607\) −2.06846e48 −0.407236 −0.203618 0.979050i \(-0.565270\pi\)
−0.203618 + 0.979050i \(0.565270\pi\)
\(608\) 3.50467e47 0.0670401
\(609\) 1.39247e48 0.258812
\(610\) 0 0
\(611\) −9.69439e48 −1.70137
\(612\) 6.52623e47 0.111304
\(613\) −1.02050e48 −0.169143 −0.0845717 0.996417i \(-0.526952\pi\)
−0.0845717 + 0.996417i \(0.526952\pi\)
\(614\) 2.30743e48 0.371691
\(615\) 0 0
\(616\) 1.46367e48 0.222732
\(617\) 3.46313e48 0.512246 0.256123 0.966644i \(-0.417555\pi\)
0.256123 + 0.966644i \(0.417555\pi\)
\(618\) −1.44860e49 −2.08281
\(619\) 6.56428e48 0.917488 0.458744 0.888568i \(-0.348299\pi\)
0.458744 + 0.888568i \(0.348299\pi\)
\(620\) 0 0
\(621\) 2.36453e48 0.312350
\(622\) 3.67165e48 0.471552
\(623\) 1.80954e48 0.225958
\(624\) −1.51753e49 −1.84249
\(625\) 0 0
\(626\) −5.99629e48 −0.688386
\(627\) −2.33070e48 −0.260198
\(628\) 1.06427e48 0.115546
\(629\) −5.32965e48 −0.562746
\(630\) 0 0
\(631\) 5.18692e48 0.518079 0.259039 0.965867i \(-0.416594\pi\)
0.259039 + 0.965867i \(0.416594\pi\)
\(632\) −1.60587e49 −1.56013
\(633\) −6.48082e48 −0.612442
\(634\) −8.74063e48 −0.803490
\(635\) 0 0
\(636\) −2.44795e48 −0.212962
\(637\) −1.79756e49 −1.52140
\(638\) −9.07235e48 −0.747064
\(639\) 5.33275e48 0.427254
\(640\) 0 0
\(641\) −6.15569e48 −0.466940 −0.233470 0.972364i \(-0.575008\pi\)
−0.233470 + 0.972364i \(0.575008\pi\)
\(642\) 2.07858e49 1.53428
\(643\) 6.50853e48 0.467510 0.233755 0.972296i \(-0.424899\pi\)
0.233755 + 0.972296i \(0.424899\pi\)
\(644\) −3.09971e47 −0.0216680
\(645\) 0 0
\(646\) −8.59837e47 −0.0569308
\(647\) −2.43533e49 −1.56940 −0.784698 0.619878i \(-0.787182\pi\)
−0.784698 + 0.619878i \(0.787182\pi\)
\(648\) 7.79667e48 0.489042
\(649\) −6.69620e48 −0.408833
\(650\) 0 0
\(651\) 1.47654e48 0.0854233
\(652\) 6.11047e48 0.344145
\(653\) −4.33343e48 −0.237602 −0.118801 0.992918i \(-0.537905\pi\)
−0.118801 + 0.992918i \(0.537905\pi\)
\(654\) −2.65789e48 −0.141882
\(655\) 0 0
\(656\) −7.67401e48 −0.388334
\(657\) 4.92986e49 2.42907
\(658\) −4.04729e48 −0.194183
\(659\) 2.26798e49 1.05960 0.529802 0.848121i \(-0.322266\pi\)
0.529802 + 0.848121i \(0.322266\pi\)
\(660\) 0 0
\(661\) −2.84364e49 −1.25994 −0.629969 0.776620i \(-0.716932\pi\)
−0.629969 + 0.776620i \(0.716932\pi\)
\(662\) 1.03284e49 0.445676
\(663\) −2.26090e49 −0.950156
\(664\) 3.55245e49 1.45407
\(665\) 0 0
\(666\) 4.63992e49 1.80182
\(667\) 1.12210e49 0.424452
\(668\) −6.55455e48 −0.241521
\(669\) 2.46325e49 0.884202
\(670\) 0 0
\(671\) −1.30679e48 −0.0445205
\(672\) 3.84730e48 0.127701
\(673\) −1.14434e49 −0.370077 −0.185039 0.982731i \(-0.559241\pi\)
−0.185039 + 0.982731i \(0.559241\pi\)
\(674\) 1.72019e49 0.542035
\(675\) 0 0
\(676\) −1.04871e49 −0.313753
\(677\) 3.00594e49 0.876351 0.438175 0.898890i \(-0.355625\pi\)
0.438175 + 0.898890i \(0.355625\pi\)
\(678\) 1.24705e49 0.354295
\(679\) 3.59708e48 0.0995931
\(680\) 0 0
\(681\) 3.59766e49 0.946119
\(682\) −9.62004e48 −0.246576
\(683\) 7.78736e49 1.94549 0.972743 0.231885i \(-0.0744894\pi\)
0.972743 + 0.231885i \(0.0744894\pi\)
\(684\) −1.94924e48 −0.0474661
\(685\) 0 0
\(686\) −1.53329e49 −0.354774
\(687\) 6.32535e49 1.42673
\(688\) 8.76886e47 0.0192817
\(689\) 4.93456e49 1.05782
\(690\) 0 0
\(691\) 9.61614e49 1.95945 0.979726 0.200340i \(-0.0642046\pi\)
0.979726 + 0.200340i \(0.0642046\pi\)
\(692\) −9.45269e48 −0.187801
\(693\) −1.48875e49 −0.288397
\(694\) 5.07971e49 0.959506
\(695\) 0 0
\(696\) −7.61562e49 −1.36787
\(697\) −1.14332e49 −0.200260
\(698\) −4.33853e49 −0.741093
\(699\) −5.46014e49 −0.909606
\(700\) 0 0
\(701\) 1.19741e49 0.189748 0.0948742 0.995489i \(-0.469755\pi\)
0.0948742 + 0.995489i \(0.469755\pi\)
\(702\) 5.53901e49 0.856116
\(703\) 1.59185e49 0.239984
\(704\) −7.70923e49 −1.13368
\(705\) 0 0
\(706\) −1.16678e50 −1.63270
\(707\) 1.41679e49 0.193404
\(708\) −9.62454e48 −0.128173
\(709\) −4.80032e49 −0.623680 −0.311840 0.950135i \(-0.600945\pi\)
−0.311840 + 0.950135i \(0.600945\pi\)
\(710\) 0 0
\(711\) 1.63339e50 2.02009
\(712\) −9.89664e49 −1.19423
\(713\) 1.18984e49 0.140094
\(714\) −9.43898e48 −0.108444
\(715\) 0 0
\(716\) −4.45232e48 −0.0487089
\(717\) 1.73297e50 1.85015
\(718\) −8.84635e49 −0.921693
\(719\) −1.45387e49 −0.147832 −0.0739162 0.997264i \(-0.523550\pi\)
−0.0739162 + 0.997264i \(0.523550\pi\)
\(720\) 0 0
\(721\) −3.17450e49 −0.307474
\(722\) −9.16528e49 −0.866452
\(723\) −1.15246e50 −1.06342
\(724\) 1.70344e49 0.153427
\(725\) 0 0
\(726\) 6.19432e48 0.0531622
\(727\) −1.25503e50 −1.05148 −0.525742 0.850644i \(-0.676212\pi\)
−0.525742 + 0.850644i \(0.676212\pi\)
\(728\) −4.24075e49 −0.346851
\(729\) −1.96598e50 −1.56981
\(730\) 0 0
\(731\) 1.30644e48 0.00994338
\(732\) −1.87826e48 −0.0139576
\(733\) 1.40440e50 1.01899 0.509495 0.860473i \(-0.329832\pi\)
0.509495 + 0.860473i \(0.329832\pi\)
\(734\) 4.95419e49 0.350988
\(735\) 0 0
\(736\) 3.10027e49 0.209430
\(737\) −2.80695e50 −1.85163
\(738\) 9.95358e49 0.641199
\(739\) 2.93572e49 0.184687 0.0923435 0.995727i \(-0.470564\pi\)
0.0923435 + 0.995727i \(0.470564\pi\)
\(740\) 0 0
\(741\) 6.75281e49 0.405197
\(742\) 2.06012e49 0.120732
\(743\) −1.10795e50 −0.634185 −0.317093 0.948395i \(-0.602707\pi\)
−0.317093 + 0.948395i \(0.602707\pi\)
\(744\) −8.07537e49 −0.451478
\(745\) 0 0
\(746\) −1.78650e50 −0.952960
\(747\) −3.61331e50 −1.88276
\(748\) −1.60138e49 −0.0815111
\(749\) 4.55506e49 0.226497
\(750\) 0 0
\(751\) −4.13601e50 −1.96283 −0.981416 0.191892i \(-0.938538\pi\)
−0.981416 + 0.191892i \(0.938538\pi\)
\(752\) 1.73582e50 0.804806
\(753\) 5.28418e50 2.39368
\(754\) 2.62856e50 1.16337
\(755\) 0 0
\(756\) −6.02161e48 −0.0254437
\(757\) −3.15399e50 −1.30221 −0.651106 0.758987i \(-0.725695\pi\)
−0.651106 + 0.758987i \(0.725695\pi\)
\(758\) 2.01180e50 0.811659
\(759\) −2.06176e50 −0.812845
\(760\) 0 0
\(761\) −3.65841e50 −1.37740 −0.688702 0.725045i \(-0.741819\pi\)
−0.688702 + 0.725045i \(0.741819\pi\)
\(762\) −2.39624e50 −0.881696
\(763\) −5.82458e48 −0.0209452
\(764\) −5.78202e49 −0.203210
\(765\) 0 0
\(766\) 1.57382e49 0.0528387
\(767\) 1.94011e50 0.636661
\(768\) −2.85192e50 −0.914778
\(769\) −2.07264e50 −0.649849 −0.324925 0.945740i \(-0.605339\pi\)
−0.324925 + 0.945740i \(0.605339\pi\)
\(770\) 0 0
\(771\) 9.42552e50 2.82393
\(772\) 9.83516e49 0.288057
\(773\) −2.04692e50 −0.586084 −0.293042 0.956100i \(-0.594668\pi\)
−0.293042 + 0.956100i \(0.594668\pi\)
\(774\) −1.13737e49 −0.0318371
\(775\) 0 0
\(776\) −1.96729e50 −0.526367
\(777\) 1.74747e50 0.457133
\(778\) 3.14521e50 0.804465
\(779\) 3.41484e49 0.0854013
\(780\) 0 0
\(781\) −1.30853e50 −0.312889
\(782\) −7.60622e49 −0.177849
\(783\) 2.17983e50 0.498415
\(784\) 3.21860e50 0.719673
\(785\) 0 0
\(786\) 1.37432e50 0.293897
\(787\) 2.74398e50 0.573882 0.286941 0.957948i \(-0.407362\pi\)
0.286941 + 0.957948i \(0.407362\pi\)
\(788\) −8.20177e49 −0.167764
\(789\) −1.45994e49 −0.0292071
\(790\) 0 0
\(791\) 2.73282e49 0.0523025
\(792\) 8.14219e50 1.52423
\(793\) 3.78619e49 0.0693301
\(794\) −6.54491e50 −1.17232
\(795\) 0 0
\(796\) 3.38388e49 0.0580012
\(797\) −3.73868e50 −0.626899 −0.313450 0.949605i \(-0.601485\pi\)
−0.313450 + 0.949605i \(0.601485\pi\)
\(798\) 2.81921e49 0.0462463
\(799\) 2.58612e50 0.415030
\(800\) 0 0
\(801\) 1.00662e51 1.54631
\(802\) −9.13117e49 −0.137238
\(803\) −1.20967e51 −1.77887
\(804\) −4.03447e50 −0.580505
\(805\) 0 0
\(806\) 2.78725e50 0.383983
\(807\) −4.51827e50 −0.609095
\(808\) −7.74862e50 −1.02218
\(809\) −3.87407e50 −0.500114 −0.250057 0.968231i \(-0.580449\pi\)
−0.250057 + 0.968231i \(0.580449\pi\)
\(810\) 0 0
\(811\) −9.21816e50 −1.13967 −0.569835 0.821759i \(-0.692993\pi\)
−0.569835 + 0.821759i \(0.692993\pi\)
\(812\) −2.85758e49 −0.0345754
\(813\) −1.42488e50 −0.168730
\(814\) −1.13853e51 −1.31952
\(815\) 0 0
\(816\) 4.04823e50 0.449456
\(817\) −3.90203e48 −0.00424038
\(818\) 5.53933e50 0.589216
\(819\) 4.31341e50 0.449110
\(820\) 0 0
\(821\) 1.64737e51 1.64357 0.821783 0.569801i \(-0.192980\pi\)
0.821783 + 0.569801i \(0.192980\pi\)
\(822\) 1.22633e51 1.19771
\(823\) 6.02163e50 0.575727 0.287864 0.957671i \(-0.407055\pi\)
0.287864 + 0.957671i \(0.407055\pi\)
\(824\) 1.73618e51 1.62505
\(825\) 0 0
\(826\) 8.09973e49 0.0726641
\(827\) 4.30476e50 0.378097 0.189048 0.981968i \(-0.439460\pi\)
0.189048 + 0.981968i \(0.439460\pi\)
\(828\) −1.72432e50 −0.148282
\(829\) 8.88751e50 0.748300 0.374150 0.927368i \(-0.377935\pi\)
0.374150 + 0.927368i \(0.377935\pi\)
\(830\) 0 0
\(831\) −2.55364e51 −2.06131
\(832\) 2.23362e51 1.76543
\(833\) 4.79525e50 0.371128
\(834\) 1.50328e51 1.13929
\(835\) 0 0
\(836\) 4.78297e49 0.0347606
\(837\) 2.31142e50 0.164507
\(838\) −1.26992e51 −0.885126
\(839\) 3.76912e50 0.257280 0.128640 0.991691i \(-0.458939\pi\)
0.128640 + 0.991691i \(0.458939\pi\)
\(840\) 0 0
\(841\) −4.92875e50 −0.322706
\(842\) −4.57730e50 −0.293527
\(843\) 1.41210e51 0.886916
\(844\) 1.32997e50 0.0818179
\(845\) 0 0
\(846\) −2.25144e51 −1.32886
\(847\) 1.35744e49 0.00784804
\(848\) −8.83551e50 −0.500385
\(849\) 3.68407e51 2.04382
\(850\) 0 0
\(851\) 1.40817e51 0.749699
\(852\) −1.88077e50 −0.0980940
\(853\) 3.00248e51 1.53417 0.767083 0.641547i \(-0.221707\pi\)
0.767083 + 0.641547i \(0.221707\pi\)
\(854\) 1.58069e49 0.00791286
\(855\) 0 0
\(856\) −2.49122e51 −1.19708
\(857\) 2.46088e50 0.115858 0.0579291 0.998321i \(-0.481550\pi\)
0.0579291 + 0.998321i \(0.481550\pi\)
\(858\) −4.82977e51 −2.22792
\(859\) 2.64506e51 1.19551 0.597757 0.801677i \(-0.296059\pi\)
0.597757 + 0.801677i \(0.296059\pi\)
\(860\) 0 0
\(861\) 3.74868e50 0.162676
\(862\) 1.02117e51 0.434233
\(863\) 9.93861e50 0.414130 0.207065 0.978327i \(-0.433609\pi\)
0.207065 + 0.978327i \(0.433609\pi\)
\(864\) 6.02270e50 0.245924
\(865\) 0 0
\(866\) −3.66184e51 −1.43594
\(867\) −3.42107e51 −1.31470
\(868\) −3.03009e49 −0.0114120
\(869\) −4.00794e51 −1.47936
\(870\) 0 0
\(871\) 8.13268e51 2.88347
\(872\) 3.18554e50 0.110699
\(873\) 2.00100e51 0.681550
\(874\) 2.27181e50 0.0758441
\(875\) 0 0
\(876\) −1.73867e51 −0.557695
\(877\) 2.79832e50 0.0879842 0.0439921 0.999032i \(-0.485992\pi\)
0.0439921 + 0.999032i \(0.485992\pi\)
\(878\) 3.16623e50 0.0975861
\(879\) −6.58876e51 −1.99067
\(880\) 0 0
\(881\) −5.95050e50 −0.172773 −0.0863863 0.996262i \(-0.527532\pi\)
−0.0863863 + 0.996262i \(0.527532\pi\)
\(882\) −4.17469e51 −1.18829
\(883\) 3.85350e51 1.07533 0.537667 0.843157i \(-0.319306\pi\)
0.537667 + 0.843157i \(0.319306\pi\)
\(884\) 4.63973e50 0.126934
\(885\) 0 0
\(886\) 4.96755e51 1.30633
\(887\) 4.56338e50 0.117658 0.0588292 0.998268i \(-0.481263\pi\)
0.0588292 + 0.998268i \(0.481263\pi\)
\(888\) −9.55716e51 −2.41603
\(889\) −5.25119e50 −0.130160
\(890\) 0 0
\(891\) 1.94590e51 0.463724
\(892\) −5.05498e50 −0.118123
\(893\) −7.72416e50 −0.176991
\(894\) −9.95996e51 −2.23795
\(895\) 0 0
\(896\) 5.50354e50 0.118919
\(897\) 5.97362e51 1.26581
\(898\) 9.39122e50 0.195158
\(899\) 1.09689e51 0.223548
\(900\) 0 0
\(901\) −1.31637e51 −0.258044
\(902\) −2.44237e51 −0.469567
\(903\) −4.28351e49 −0.00807726
\(904\) −1.49462e51 −0.276428
\(905\) 0 0
\(906\) −8.04998e51 −1.43236
\(907\) 4.12286e51 0.719566 0.359783 0.933036i \(-0.382851\pi\)
0.359783 + 0.933036i \(0.382851\pi\)
\(908\) −7.38298e50 −0.126395
\(909\) 7.88139e51 1.32353
\(910\) 0 0
\(911\) 6.95561e51 1.12399 0.561996 0.827140i \(-0.310034\pi\)
0.561996 + 0.827140i \(0.310034\pi\)
\(912\) −1.20911e51 −0.191672
\(913\) 8.86621e51 1.37879
\(914\) −7.72094e51 −1.17791
\(915\) 0 0
\(916\) −1.29806e51 −0.190601
\(917\) 3.01173e50 0.0433863
\(918\) −1.47761e51 −0.208840
\(919\) 1.23335e51 0.171027 0.0855136 0.996337i \(-0.472747\pi\)
0.0855136 + 0.996337i \(0.472747\pi\)
\(920\) 0 0
\(921\) 4.83420e51 0.645329
\(922\) −6.15821e51 −0.806610
\(923\) 3.79125e51 0.487251
\(924\) 5.25057e50 0.0662136
\(925\) 0 0
\(926\) 6.07023e51 0.737076
\(927\) −1.76592e52 −2.10415
\(928\) 2.85810e51 0.334185
\(929\) −9.97208e51 −1.14422 −0.572112 0.820176i \(-0.693876\pi\)
−0.572112 + 0.820176i \(0.693876\pi\)
\(930\) 0 0
\(931\) −1.43224e51 −0.158269
\(932\) 1.12051e51 0.121517
\(933\) 7.69232e51 0.818707
\(934\) 3.92536e51 0.410024
\(935\) 0 0
\(936\) −2.35906e52 −2.37363
\(937\) −6.01334e51 −0.593844 −0.296922 0.954902i \(-0.595960\pi\)
−0.296922 + 0.954902i \(0.595960\pi\)
\(938\) 3.39529e51 0.329100
\(939\) −1.25626e52 −1.19517
\(940\) 0 0
\(941\) 1.38035e52 1.26523 0.632617 0.774465i \(-0.281981\pi\)
0.632617 + 0.774465i \(0.281981\pi\)
\(942\) −8.56268e51 −0.770405
\(943\) 3.02081e51 0.266789
\(944\) −3.47384e51 −0.301162
\(945\) 0 0
\(946\) 2.79083e50 0.0233151
\(947\) 9.01648e51 0.739456 0.369728 0.929140i \(-0.379451\pi\)
0.369728 + 0.929140i \(0.379451\pi\)
\(948\) −5.76067e51 −0.463796
\(949\) 3.50482e52 2.77017
\(950\) 0 0
\(951\) −1.83121e52 −1.39502
\(952\) 1.13128e51 0.0846105
\(953\) −4.32450e51 −0.317549 −0.158774 0.987315i \(-0.550754\pi\)
−0.158774 + 0.987315i \(0.550754\pi\)
\(954\) 1.14601e52 0.826214
\(955\) 0 0
\(956\) −3.55633e51 −0.247166
\(957\) −1.90071e52 −1.29705
\(958\) −1.45348e52 −0.973897
\(959\) 2.68741e51 0.176810
\(960\) 0 0
\(961\) −1.46006e52 −0.926216
\(962\) 3.29869e52 2.05484
\(963\) 2.53391e52 1.55000
\(964\) 2.36503e51 0.142066
\(965\) 0 0
\(966\) 2.49391e51 0.144471
\(967\) 1.91085e51 0.108709 0.0543543 0.998522i \(-0.482690\pi\)
0.0543543 + 0.998522i \(0.482690\pi\)
\(968\) −7.42403e50 −0.0414783
\(969\) −1.80141e51 −0.0988431
\(970\) 0 0
\(971\) 2.54698e52 1.34800 0.674000 0.738732i \(-0.264575\pi\)
0.674000 + 0.738732i \(0.264575\pi\)
\(972\) 5.20394e51 0.270504
\(973\) 3.29432e51 0.168186
\(974\) −1.17796e52 −0.590675
\(975\) 0 0
\(976\) −6.77932e50 −0.0327955
\(977\) −2.34465e52 −1.11410 −0.557048 0.830480i \(-0.688066\pi\)
−0.557048 + 0.830480i \(0.688066\pi\)
\(978\) −4.91625e52 −2.29459
\(979\) −2.47001e52 −1.13240
\(980\) 0 0
\(981\) −3.24012e51 −0.143335
\(982\) 1.40197e52 0.609238
\(983\) −1.06892e52 −0.456308 −0.228154 0.973625i \(-0.573269\pi\)
−0.228154 + 0.973625i \(0.573269\pi\)
\(984\) −2.05021e52 −0.859772
\(985\) 0 0
\(986\) −7.01207e51 −0.283792
\(987\) −8.47931e51 −0.337140
\(988\) −1.38579e51 −0.0541314
\(989\) −3.45178e50 −0.0132467
\(990\) 0 0
\(991\) 1.97227e52 0.730597 0.365298 0.930891i \(-0.380967\pi\)
0.365298 + 0.930891i \(0.380967\pi\)
\(992\) 3.03064e51 0.110301
\(993\) 2.16387e52 0.773782
\(994\) 1.58280e51 0.0556114
\(995\) 0 0
\(996\) 1.27435e52 0.432266
\(997\) 1.44359e52 0.481149 0.240575 0.970631i \(-0.422664\pi\)
0.240575 + 0.970631i \(0.422664\pi\)
\(998\) −2.90388e52 −0.951029
\(999\) 2.73555e52 0.880338
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 25.36.a.a.1.3 3
5.2 odd 4 25.36.b.a.24.5 6
5.3 odd 4 25.36.b.a.24.2 6
5.4 even 2 1.36.a.a.1.1 3
15.14 odd 2 9.36.a.b.1.3 3
20.19 odd 2 16.36.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.36.a.a.1.1 3 5.4 even 2
9.36.a.b.1.3 3 15.14 odd 2
16.36.a.d.1.3 3 20.19 odd 2
25.36.a.a.1.3 3 1.1 even 1 trivial
25.36.b.a.24.2 6 5.3 odd 4
25.36.b.a.24.5 6 5.2 odd 4