Properties

Label 3.28.a.b.1.1
Level $3$
Weight $28$
Character 3.1
Self dual yes
Analytic conductor $13.856$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.8556672451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{30001}) \)
Defining polynomial: \( x^{2} - x - 7500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(87.1040\) of defining polynomial
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q-121.102 q^{2} -1.59432e6 q^{3} -1.34203e8 q^{4} -1.07803e9 q^{5} +1.93076e8 q^{6} -1.13904e10 q^{7} +3.25063e10 q^{8} +2.54187e12 q^{9} +O(q^{10})\) \(q-121.102 q^{2} -1.59432e6 q^{3} -1.34203e8 q^{4} -1.07803e9 q^{5} +1.93076e8 q^{6} -1.13904e10 q^{7} +3.25063e10 q^{8} +2.54187e12 q^{9} +1.30551e11 q^{10} -5.05749e13 q^{11} +2.13963e14 q^{12} +1.21304e15 q^{13} +1.37940e12 q^{14} +1.71872e15 q^{15} +1.80085e16 q^{16} +7.00452e16 q^{17} -3.07825e14 q^{18} +2.51027e16 q^{19} +1.44674e17 q^{20} +1.81600e16 q^{21} +6.12472e15 q^{22} -3.40370e18 q^{23} -5.18255e16 q^{24} -6.28844e18 q^{25} -1.46901e17 q^{26} -4.05256e18 q^{27} +1.52863e18 q^{28} +5.74556e19 q^{29} -2.08141e17 q^{30} +1.08167e20 q^{31} -6.54378e18 q^{32} +8.06328e19 q^{33} -8.48261e18 q^{34} +1.22791e19 q^{35} -3.41126e20 q^{36} -1.90073e21 q^{37} -3.03999e18 q^{38} -1.93398e21 q^{39} -3.50426e19 q^{40} +6.66148e21 q^{41} -2.19921e18 q^{42} +1.03265e21 q^{43} +6.78731e21 q^{44} -2.74020e21 q^{45} +4.12194e20 q^{46} +6.43133e22 q^{47} -2.87114e22 q^{48} -6.55826e22 q^{49} +7.61542e20 q^{50} -1.11675e23 q^{51} -1.62794e23 q^{52} +2.13714e23 q^{53} +4.90772e20 q^{54} +5.45211e22 q^{55} -3.70259e20 q^{56} -4.00218e22 q^{57} -6.95799e21 q^{58} +1.07026e24 q^{59} -2.30658e23 q^{60} +8.16628e23 q^{61} -1.30992e22 q^{62} -2.89528e22 q^{63} -2.41627e24 q^{64} -1.30769e24 q^{65} -9.76479e21 q^{66} +1.68630e24 q^{67} -9.40028e24 q^{68} +5.42659e24 q^{69} -1.48703e21 q^{70} -3.18547e24 q^{71} +8.26266e22 q^{72} +2.09853e25 q^{73} +2.30183e23 q^{74} +1.00258e25 q^{75} -3.36886e24 q^{76} +5.76068e23 q^{77} +2.34208e23 q^{78} -8.88148e22 q^{79} -1.94136e25 q^{80} +6.46108e24 q^{81} -8.06718e23 q^{82} -6.50402e25 q^{83} -2.43712e24 q^{84} -7.55105e25 q^{85} -1.25056e23 q^{86} -9.16028e25 q^{87} -1.64400e24 q^{88} +3.38617e26 q^{89} +3.31843e23 q^{90} -1.38170e25 q^{91} +4.56787e26 q^{92} -1.72453e26 q^{93} -7.78847e24 q^{94} -2.70614e25 q^{95} +1.04329e25 q^{96} -4.16124e26 q^{97} +7.94218e24 q^{98} -1.28555e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 21582 q^{2} - 3188646 q^{3} + 202603844 q^{4} - 1771946100 q^{5} - 34408678986 q^{6} + 369665199904 q^{7} + 4429319872824 q^{8} + 5083731656658 q^{9} - 14929666656300 q^{10} + 75762335668248 q^{11} - 323015968377612 q^{12} - 103021079177588 q^{13} + 82\!\cdots\!36 q^{14}+ \cdots + 19\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −121.102 −0.0104531 −0.00522656 0.999986i \(-0.501664\pi\)
−0.00522656 + 0.999986i \(0.501664\pi\)
\(3\) −1.59432e6 −0.577350
\(4\) −1.34203e8 −0.999891
\(5\) −1.07803e9 −0.394943 −0.197471 0.980309i \(-0.563273\pi\)
−0.197471 + 0.980309i \(0.563273\pi\)
\(6\) 1.93076e8 0.00603512
\(7\) −1.13904e10 −0.0444340 −0.0222170 0.999753i \(-0.507072\pi\)
−0.0222170 + 0.999753i \(0.507072\pi\)
\(8\) 3.25063e10 0.0209051
\(9\) 2.54187e12 0.333333
\(10\) 1.30551e11 0.00412839
\(11\) −5.05749e13 −0.441707 −0.220853 0.975307i \(-0.570884\pi\)
−0.220853 + 0.975307i \(0.570884\pi\)
\(12\) 2.13963e14 0.577287
\(13\) 1.21304e15 1.11081 0.555405 0.831580i \(-0.312563\pi\)
0.555405 + 0.831580i \(0.312563\pi\)
\(14\) 1.37940e12 0.000464474 0
\(15\) 1.71872e15 0.228020
\(16\) 1.80085e16 0.999672
\(17\) 7.00452e16 1.71521 0.857606 0.514307i \(-0.171951\pi\)
0.857606 + 0.514307i \(0.171951\pi\)
\(18\) −3.07825e14 −0.00348438
\(19\) 2.51027e16 0.136945 0.0684726 0.997653i \(-0.478187\pi\)
0.0684726 + 0.997653i \(0.478187\pi\)
\(20\) 1.44674e17 0.394900
\(21\) 1.81600e16 0.0256540
\(22\) 6.12472e15 0.00461722
\(23\) −3.40370e18 −1.40807 −0.704037 0.710164i \(-0.748621\pi\)
−0.704037 + 0.710164i \(0.748621\pi\)
\(24\) −5.18255e16 −0.0120696
\(25\) −6.28844e18 −0.844020
\(26\) −1.46901e17 −0.0116114
\(27\) −4.05256e18 −0.192450
\(28\) 1.52863e18 0.0444291
\(29\) 5.74556e19 1.03982 0.519912 0.854220i \(-0.325965\pi\)
0.519912 + 0.854220i \(0.325965\pi\)
\(30\) −2.08141e17 −0.00238353
\(31\) 1.08167e20 0.795632 0.397816 0.917465i \(-0.369768\pi\)
0.397816 + 0.917465i \(0.369768\pi\)
\(32\) −6.54378e18 −0.0313548
\(33\) 8.06328e19 0.255020
\(34\) −8.48261e18 −0.0179293
\(35\) 1.22791e19 0.0175489
\(36\) −3.41126e20 −0.333297
\(37\) −1.90073e21 −1.28292 −0.641458 0.767159i \(-0.721670\pi\)
−0.641458 + 0.767159i \(0.721670\pi\)
\(38\) −3.03999e18 −0.00143151
\(39\) −1.93398e21 −0.641326
\(40\) −3.50426e19 −0.00825632
\(41\) 6.66148e21 1.12458 0.562288 0.826942i \(-0.309921\pi\)
0.562288 + 0.826942i \(0.309921\pi\)
\(42\) −2.19921e18 −0.000268164 0
\(43\) 1.03265e21 0.0916490 0.0458245 0.998950i \(-0.485408\pi\)
0.0458245 + 0.998950i \(0.485408\pi\)
\(44\) 6.78731e21 0.441659
\(45\) −2.74020e21 −0.131648
\(46\) 4.12194e20 0.0147188
\(47\) 6.43133e22 1.71783 0.858914 0.512120i \(-0.171140\pi\)
0.858914 + 0.512120i \(0.171140\pi\)
\(48\) −2.87114e22 −0.577161
\(49\) −6.55826e22 −0.998026
\(50\) 7.61542e20 0.00882265
\(51\) −1.11675e23 −0.990278
\(52\) −1.62794e23 −1.11069
\(53\) 2.13714e23 1.12748 0.563740 0.825953i \(-0.309362\pi\)
0.563740 + 0.825953i \(0.309362\pi\)
\(54\) 4.90772e20 0.00201171
\(55\) 5.45211e22 0.174449
\(56\) −3.70259e20 −0.000928897 0
\(57\) −4.00218e22 −0.0790653
\(58\) −6.95799e21 −0.0108694
\(59\) 1.07026e24 1.32736 0.663679 0.748017i \(-0.268994\pi\)
0.663679 + 0.748017i \(0.268994\pi\)
\(60\) −2.30658e23 −0.227995
\(61\) 8.16628e23 0.645761 0.322880 0.946440i \(-0.395349\pi\)
0.322880 + 0.946440i \(0.395349\pi\)
\(62\) −1.30992e22 −0.00831684
\(63\) −2.89528e22 −0.0148113
\(64\) −2.41627e24 −0.999344
\(65\) −1.30769e24 −0.438706
\(66\) −9.76479e21 −0.00266575
\(67\) 1.68630e24 0.375773 0.187886 0.982191i \(-0.439836\pi\)
0.187886 + 0.982191i \(0.439836\pi\)
\(68\) −9.40028e24 −1.71502
\(69\) 5.42659e24 0.812951
\(70\) −1.48703e21 −0.000183441 0
\(71\) −3.18547e24 −0.324478 −0.162239 0.986751i \(-0.551872\pi\)
−0.162239 + 0.986751i \(0.551872\pi\)
\(72\) 8.26266e22 0.00696837
\(73\) 2.09853e25 1.46912 0.734559 0.678545i \(-0.237389\pi\)
0.734559 + 0.678545i \(0.237389\pi\)
\(74\) 2.30183e23 0.0134105
\(75\) 1.00258e25 0.487295
\(76\) −3.36886e24 −0.136930
\(77\) 5.76068e23 0.0196268
\(78\) 2.34208e23 0.00670386
\(79\) −8.88148e22 −0.00214052 −0.00107026 0.999999i \(-0.500341\pi\)
−0.00107026 + 0.999999i \(0.500341\pi\)
\(80\) −1.94136e25 −0.394813
\(81\) 6.46108e24 0.111111
\(82\) −8.06718e23 −0.0117553
\(83\) −6.50402e25 −0.804689 −0.402344 0.915488i \(-0.631805\pi\)
−0.402344 + 0.915488i \(0.631805\pi\)
\(84\) −2.43712e24 −0.0256512
\(85\) −7.55105e25 −0.677411
\(86\) −1.25056e23 −0.000958019 0
\(87\) −9.16028e25 −0.600342
\(88\) −1.64400e24 −0.00923393
\(89\) 3.38617e26 1.63284 0.816420 0.577459i \(-0.195956\pi\)
0.816420 + 0.577459i \(0.195956\pi\)
\(90\) 3.31843e23 0.00137613
\(91\) −1.38170e25 −0.0493577
\(92\) 4.56787e26 1.40792
\(93\) −1.72453e26 −0.459358
\(94\) −7.78847e24 −0.0179567
\(95\) −2.70614e25 −0.0540855
\(96\) 1.04329e25 0.0181027
\(97\) −4.16124e26 −0.627775 −0.313888 0.949460i \(-0.601632\pi\)
−0.313888 + 0.949460i \(0.601632\pi\)
\(98\) 7.94218e24 0.0104325
\(99\) −1.28555e26 −0.147236
\(100\) 8.43928e26 0.843928
\(101\) −9.90798e26 −0.866256 −0.433128 0.901332i \(-0.642590\pi\)
−0.433128 + 0.901332i \(0.642590\pi\)
\(102\) 1.35240e25 0.0103515
\(103\) −2.01221e27 −1.35012 −0.675059 0.737764i \(-0.735882\pi\)
−0.675059 + 0.737764i \(0.735882\pi\)
\(104\) 3.94314e25 0.0232216
\(105\) −1.95769e25 −0.0101318
\(106\) −2.58812e25 −0.0117857
\(107\) 2.23219e25 0.00895470 0.00447735 0.999990i \(-0.498575\pi\)
0.00447735 + 0.999990i \(0.498575\pi\)
\(108\) 5.43865e26 0.192429
\(109\) 5.40319e27 1.68808 0.844039 0.536281i \(-0.180171\pi\)
0.844039 + 0.536281i \(0.180171\pi\)
\(110\) −6.60261e24 −0.00182354
\(111\) 3.03038e27 0.740691
\(112\) −2.05124e26 −0.0444194
\(113\) −3.71413e27 −0.713342 −0.356671 0.934230i \(-0.616088\pi\)
−0.356671 + 0.934230i \(0.616088\pi\)
\(114\) 4.84672e24 0.000826480 0
\(115\) 3.66927e27 0.556108
\(116\) −7.71072e27 −1.03971
\(117\) 3.08338e27 0.370270
\(118\) −1.29611e26 −0.0138750
\(119\) −7.97842e26 −0.0762137
\(120\) 5.58693e25 0.00476679
\(121\) −1.05522e28 −0.804895
\(122\) −9.88953e25 −0.00675022
\(123\) −1.06205e28 −0.649274
\(124\) −1.45164e28 −0.795545
\(125\) 1.48110e28 0.728282
\(126\) 3.50625e24 0.000154825 0
\(127\) 2.13925e28 0.849007 0.424503 0.905426i \(-0.360449\pi\)
0.424503 + 0.905426i \(0.360449\pi\)
\(128\) 1.17091e27 0.0418011
\(129\) −1.64637e27 −0.0529136
\(130\) 1.58364e26 0.00458585
\(131\) 5.14435e28 1.34328 0.671642 0.740876i \(-0.265589\pi\)
0.671642 + 0.740876i \(0.265589\pi\)
\(132\) −1.08212e28 −0.254992
\(133\) −2.85930e26 −0.00608502
\(134\) −2.04214e26 −0.00392800
\(135\) 4.36876e27 0.0760068
\(136\) 2.27691e27 0.0358567
\(137\) 3.71968e28 0.530612 0.265306 0.964164i \(-0.414527\pi\)
0.265306 + 0.964164i \(0.414527\pi\)
\(138\) −6.57171e26 −0.00849788
\(139\) −1.00529e29 −1.17921 −0.589605 0.807692i \(-0.700717\pi\)
−0.589605 + 0.807692i \(0.700717\pi\)
\(140\) −1.64790e27 −0.0175470
\(141\) −1.02536e29 −0.991788
\(142\) 3.85766e26 0.00339181
\(143\) −6.13494e28 −0.490652
\(144\) 4.57752e28 0.333224
\(145\) −6.19387e28 −0.410671
\(146\) −2.54136e27 −0.0153569
\(147\) 1.04560e29 0.576210
\(148\) 2.55084e29 1.28277
\(149\) −2.06193e29 −0.946801 −0.473400 0.880847i \(-0.656974\pi\)
−0.473400 + 0.880847i \(0.656974\pi\)
\(150\) −1.21414e27 −0.00509376
\(151\) 8.73857e28 0.335160 0.167580 0.985859i \(-0.446405\pi\)
0.167580 + 0.985859i \(0.446405\pi\)
\(152\) 8.15995e26 0.00286285
\(153\) 1.78045e29 0.571737
\(154\) −6.97630e25 −0.000205161 0
\(155\) −1.16607e29 −0.314229
\(156\) 2.59545e29 0.641256
\(157\) 3.66647e29 0.831004 0.415502 0.909592i \(-0.363606\pi\)
0.415502 + 0.909592i \(0.363606\pi\)
\(158\) 1.07556e25 2.23751e−5 0
\(159\) −3.40729e29 −0.650950
\(160\) 7.05437e27 0.0123834
\(161\) 3.87694e28 0.0625663
\(162\) −7.82450e26 −0.00116146
\(163\) −9.33235e29 −1.27485 −0.637424 0.770513i \(-0.720000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(164\) −8.93991e29 −1.12445
\(165\) −8.69242e28 −0.100718
\(166\) 7.87650e27 0.00841151
\(167\) 3.46815e29 0.341527 0.170764 0.985312i \(-0.445377\pi\)
0.170764 + 0.985312i \(0.445377\pi\)
\(168\) 5.90313e26 0.000536299 0
\(169\) 2.78930e29 0.233897
\(170\) 9.14447e27 0.00708106
\(171\) 6.38077e28 0.0456484
\(172\) −1.38584e29 −0.0916390
\(173\) 2.65803e30 1.62532 0.812659 0.582740i \(-0.198019\pi\)
0.812659 + 0.582740i \(0.198019\pi\)
\(174\) 1.10933e28 0.00627545
\(175\) 7.16278e28 0.0375032
\(176\) −9.10778e29 −0.441562
\(177\) −1.70634e30 −0.766351
\(178\) −4.10072e28 −0.0170683
\(179\) 4.30907e30 1.66291 0.831454 0.555594i \(-0.187509\pi\)
0.831454 + 0.555594i \(0.187509\pi\)
\(180\) 3.67743e29 0.131633
\(181\) −5.56401e29 −0.184811 −0.0924053 0.995721i \(-0.529456\pi\)
−0.0924053 + 0.995721i \(0.529456\pi\)
\(182\) 1.67326e27 0.000515942 0
\(183\) −1.30197e30 −0.372830
\(184\) −1.10642e29 −0.0294359
\(185\) 2.04904e30 0.506678
\(186\) 2.08844e28 0.00480173
\(187\) −3.54253e30 −0.757621
\(188\) −8.63104e30 −1.71764
\(189\) 4.61602e28 0.00855132
\(190\) 3.27718e27 0.000565363 0
\(191\) −4.19469e30 −0.674139 −0.337070 0.941480i \(-0.609436\pi\)
−0.337070 + 0.941480i \(0.609436\pi\)
\(192\) 3.85231e30 0.576972
\(193\) 4.85628e28 0.00678078 0.00339039 0.999994i \(-0.498921\pi\)
0.00339039 + 0.999994i \(0.498921\pi\)
\(194\) 5.03934e28 0.00656221
\(195\) 2.08488e30 0.253287
\(196\) 8.80139e30 0.997917
\(197\) −2.40097e29 −0.0254151 −0.0127075 0.999919i \(-0.504045\pi\)
−0.0127075 + 0.999919i \(0.504045\pi\)
\(198\) 1.55682e28 0.00153907
\(199\) 7.40632e30 0.684047 0.342024 0.939691i \(-0.388888\pi\)
0.342024 + 0.939691i \(0.388888\pi\)
\(200\) −2.04414e29 −0.0176443
\(201\) −2.68851e30 −0.216953
\(202\) 1.19988e29 0.00905509
\(203\) −6.54442e29 −0.0462035
\(204\) 1.49871e31 0.990170
\(205\) −7.18125e30 −0.444143
\(206\) 2.43683e29 0.0141129
\(207\) −8.65174e30 −0.469358
\(208\) 2.18450e31 1.11045
\(209\) −1.26957e30 −0.0604896
\(210\) 2.37080e27 0.000105909 0
\(211\) −3.11843e31 −1.30654 −0.653271 0.757124i \(-0.726604\pi\)
−0.653271 + 0.757124i \(0.726604\pi\)
\(212\) −2.86811e31 −1.12736
\(213\) 5.07866e30 0.187337
\(214\) −2.70323e27 −9.36046e−5 0
\(215\) −1.11322e30 −0.0361961
\(216\) −1.31733e29 −0.00402319
\(217\) −1.23207e30 −0.0353531
\(218\) −6.54337e29 −0.0176457
\(219\) −3.34574e31 −0.848196
\(220\) −7.31690e30 −0.174430
\(221\) 8.49675e31 1.90527
\(222\) −3.66985e29 −0.00774254
\(223\) −2.26940e31 −0.450605 −0.225302 0.974289i \(-0.572337\pi\)
−0.225302 + 0.974289i \(0.572337\pi\)
\(224\) 7.45362e28 0.00139322
\(225\) −1.59844e31 −0.281340
\(226\) 4.49788e29 0.00745665
\(227\) 8.41908e31 1.31497 0.657484 0.753468i \(-0.271621\pi\)
0.657484 + 0.753468i \(0.271621\pi\)
\(228\) 5.37105e30 0.0790567
\(229\) 4.09888e31 0.568703 0.284352 0.958720i \(-0.408222\pi\)
0.284352 + 0.958720i \(0.408222\pi\)
\(230\) −4.44356e29 −0.00581307
\(231\) −9.18439e29 −0.0113315
\(232\) 1.86767e30 0.0217376
\(233\) 6.99046e31 0.767717 0.383858 0.923392i \(-0.374595\pi\)
0.383858 + 0.923392i \(0.374595\pi\)
\(234\) −3.73404e29 −0.00387048
\(235\) −6.93314e31 −0.678444
\(236\) −1.43633e32 −1.32721
\(237\) 1.41599e29 0.00123583
\(238\) 9.66202e28 0.000796671 0
\(239\) −1.89074e32 −1.47320 −0.736598 0.676330i \(-0.763569\pi\)
−0.736598 + 0.676330i \(0.763569\pi\)
\(240\) 3.09516e31 0.227946
\(241\) 1.01767e32 0.708563 0.354281 0.935139i \(-0.384725\pi\)
0.354281 + 0.935139i \(0.384725\pi\)
\(242\) 1.27789e30 0.00841367
\(243\) −1.03011e31 −0.0641500
\(244\) −1.09594e32 −0.645690
\(245\) 7.06998e31 0.394163
\(246\) 1.28617e30 0.00678694
\(247\) 3.04506e31 0.152120
\(248\) 3.51611e30 0.0166328
\(249\) 1.03695e32 0.464587
\(250\) −1.79364e30 −0.00761283
\(251\) −2.25120e32 −0.905356 −0.452678 0.891674i \(-0.649531\pi\)
−0.452678 + 0.891674i \(0.649531\pi\)
\(252\) 3.88556e30 0.0148097
\(253\) 1.72142e32 0.621955
\(254\) −2.59067e30 −0.00887478
\(255\) 1.20388e32 0.391103
\(256\) 3.24164e32 0.998908
\(257\) 1.28310e32 0.375113 0.187557 0.982254i \(-0.439943\pi\)
0.187557 + 0.982254i \(0.439943\pi\)
\(258\) 1.99379e29 0.000553113 0
\(259\) 2.16501e31 0.0570050
\(260\) 1.75496e32 0.438658
\(261\) 1.46045e32 0.346608
\(262\) −6.22990e30 −0.0140415
\(263\) −6.17345e32 −1.32168 −0.660838 0.750528i \(-0.729799\pi\)
−0.660838 + 0.750528i \(0.729799\pi\)
\(264\) 2.62107e30 0.00533121
\(265\) −2.30389e32 −0.445290
\(266\) 3.46266e28 6.36075e−5 0
\(267\) −5.39865e32 −0.942720
\(268\) −2.26307e32 −0.375732
\(269\) −1.70730e31 −0.0269559 −0.0134779 0.999909i \(-0.504290\pi\)
−0.0134779 + 0.999909i \(0.504290\pi\)
\(270\) −5.29065e29 −0.000794508 0
\(271\) −1.06709e33 −1.52446 −0.762229 0.647307i \(-0.775895\pi\)
−0.762229 + 0.647307i \(0.775895\pi\)
\(272\) 1.26141e33 1.71465
\(273\) 2.20287e31 0.0284967
\(274\) −4.50460e30 −0.00554656
\(275\) 3.18037e32 0.372809
\(276\) −7.28265e32 −0.812863
\(277\) 5.90772e32 0.627976 0.313988 0.949427i \(-0.398335\pi\)
0.313988 + 0.949427i \(0.398335\pi\)
\(278\) 1.21742e31 0.0123264
\(279\) 2.74946e32 0.265211
\(280\) 3.99149e29 0.000366861 0
\(281\) 3.49531e32 0.306161 0.153081 0.988214i \(-0.451081\pi\)
0.153081 + 0.988214i \(0.451081\pi\)
\(282\) 1.24173e31 0.0103673
\(283\) −1.04098e32 −0.0828565 −0.0414282 0.999141i \(-0.513191\pi\)
−0.0414282 + 0.999141i \(0.513191\pi\)
\(284\) 4.27499e32 0.324443
\(285\) 4.31446e31 0.0312263
\(286\) 7.42953e30 0.00512885
\(287\) −7.58768e31 −0.0499693
\(288\) −1.66334e31 −0.0104516
\(289\) 3.23862e33 1.94195
\(290\) 7.50089e30 0.00429279
\(291\) 6.63436e32 0.362446
\(292\) −2.81629e33 −1.46896
\(293\) −4.57358e32 −0.227794 −0.113897 0.993493i \(-0.536333\pi\)
−0.113897 + 0.993493i \(0.536333\pi\)
\(294\) −1.26624e31 −0.00602320
\(295\) −1.15377e33 −0.524231
\(296\) −6.17858e31 −0.0268195
\(297\) 2.04958e32 0.0850065
\(298\) 2.49703e31 0.00989703
\(299\) −4.12882e33 −1.56410
\(300\) −1.34549e33 −0.487242
\(301\) −1.17623e31 −0.00407233
\(302\) −1.05826e31 −0.00350346
\(303\) 1.57965e33 0.500133
\(304\) 4.52062e32 0.136900
\(305\) −8.80347e32 −0.255039
\(306\) −2.15617e31 −0.00597644
\(307\) 2.50586e33 0.664642 0.332321 0.943166i \(-0.392168\pi\)
0.332321 + 0.943166i \(0.392168\pi\)
\(308\) −7.73101e31 −0.0196246
\(309\) 3.20812e33 0.779491
\(310\) 1.41213e31 0.00328468
\(311\) −4.09104e33 −0.911104 −0.455552 0.890209i \(-0.650558\pi\)
−0.455552 + 0.890209i \(0.650558\pi\)
\(312\) −6.28664e31 −0.0134070
\(313\) 5.57324e33 1.13831 0.569154 0.822231i \(-0.307271\pi\)
0.569154 + 0.822231i \(0.307271\pi\)
\(314\) −4.44017e31 −0.00868659
\(315\) 3.12119e31 0.00584962
\(316\) 1.19192e31 0.00214029
\(317\) 6.22580e33 1.07126 0.535631 0.844452i \(-0.320074\pi\)
0.535631 + 0.844452i \(0.320074\pi\)
\(318\) 4.12629e31 0.00680447
\(319\) −2.90581e33 −0.459297
\(320\) 2.60480e33 0.394684
\(321\) −3.55884e31 −0.00517000
\(322\) −4.69505e30 −0.000654013 0
\(323\) 1.75832e33 0.234890
\(324\) −8.67097e32 −0.111099
\(325\) −7.62812e33 −0.937545
\(326\) 1.13017e32 0.0133261
\(327\) −8.61443e33 −0.974613
\(328\) 2.16540e32 0.0235094
\(329\) −7.32553e32 −0.0763299
\(330\) 1.05267e31 0.00105282
\(331\) −6.69049e33 −0.642363 −0.321181 0.947018i \(-0.604080\pi\)
−0.321181 + 0.947018i \(0.604080\pi\)
\(332\) 8.72860e33 0.804601
\(333\) −4.83141e33 −0.427638
\(334\) −4.20000e31 −0.00357003
\(335\) −1.81788e33 −0.148409
\(336\) 3.27033e32 0.0256455
\(337\) −8.45330e33 −0.636829 −0.318414 0.947952i \(-0.603150\pi\)
−0.318414 + 0.947952i \(0.603150\pi\)
\(338\) −3.37790e31 −0.00244496
\(339\) 5.92152e33 0.411848
\(340\) 1.01337e34 0.677337
\(341\) −5.47054e33 −0.351436
\(342\) −7.72724e30 −0.000477168 0
\(343\) 1.49550e33 0.0887802
\(344\) 3.35675e31 0.00191593
\(345\) −5.85001e33 −0.321069
\(346\) −3.21893e32 −0.0169896
\(347\) 3.54455e34 1.79934 0.899671 0.436568i \(-0.143806\pi\)
0.899671 + 0.436568i \(0.143806\pi\)
\(348\) 1.22934e34 0.600277
\(349\) 1.62742e34 0.764462 0.382231 0.924067i \(-0.375156\pi\)
0.382231 + 0.924067i \(0.375156\pi\)
\(350\) −8.67426e30 −0.000392025 0
\(351\) −4.91591e33 −0.213775
\(352\) 3.30951e32 0.0138496
\(353\) −6.68547e33 −0.269261 −0.134630 0.990896i \(-0.542985\pi\)
−0.134630 + 0.990896i \(0.542985\pi\)
\(354\) 2.06642e32 0.00801076
\(355\) 3.43402e33 0.128150
\(356\) −4.54434e34 −1.63266
\(357\) 1.27202e33 0.0440020
\(358\) −5.21837e32 −0.0173826
\(359\) −7.80470e33 −0.250369 −0.125185 0.992133i \(-0.539952\pi\)
−0.125185 + 0.992133i \(0.539952\pi\)
\(360\) −8.90736e31 −0.00275211
\(361\) −3.29705e34 −0.981246
\(362\) 6.73812e31 0.00193185
\(363\) 1.68236e34 0.464706
\(364\) 1.85428e33 0.0493523
\(365\) −2.26227e34 −0.580218
\(366\) 1.57671e32 0.00389724
\(367\) 6.61615e34 1.57621 0.788104 0.615542i \(-0.211063\pi\)
0.788104 + 0.615542i \(0.211063\pi\)
\(368\) −6.12955e34 −1.40761
\(369\) 1.69326e34 0.374858
\(370\) −2.48143e32 −0.00529637
\(371\) −2.43428e33 −0.0500984
\(372\) 2.31438e34 0.459308
\(373\) −6.79473e34 −1.30048 −0.650239 0.759729i \(-0.725331\pi\)
−0.650239 + 0.759729i \(0.725331\pi\)
\(374\) 4.29007e32 0.00791951
\(375\) −2.36136e34 −0.420474
\(376\) 2.09059e33 0.0359114
\(377\) 6.96959e34 1.15505
\(378\) −5.59009e30 −8.93880e−5 0
\(379\) 7.98595e34 1.23224 0.616122 0.787650i \(-0.288703\pi\)
0.616122 + 0.787650i \(0.288703\pi\)
\(380\) 3.63172e33 0.0540796
\(381\) −3.41065e34 −0.490174
\(382\) 5.07985e32 0.00704686
\(383\) 1.19033e35 1.59399 0.796994 0.603987i \(-0.206422\pi\)
0.796994 + 0.603987i \(0.206422\pi\)
\(384\) −1.86680e33 −0.0241339
\(385\) −6.21016e32 −0.00775146
\(386\) −5.88105e30 −7.08804e−5 0
\(387\) 2.62485e33 0.0305497
\(388\) 5.58451e34 0.627707
\(389\) 1.48935e34 0.161688 0.0808441 0.996727i \(-0.474238\pi\)
0.0808441 + 0.996727i \(0.474238\pi\)
\(390\) −2.52483e32 −0.00264764
\(391\) −2.38413e35 −2.41514
\(392\) −2.13185e33 −0.0208638
\(393\) −8.20175e34 −0.775545
\(394\) 2.90762e31 0.000265667 0
\(395\) 9.57447e31 0.000845384 0
\(396\) 1.72524e34 0.147220
\(397\) 4.07374e34 0.335986 0.167993 0.985788i \(-0.446271\pi\)
0.167993 + 0.985788i \(0.446271\pi\)
\(398\) −8.96920e32 −0.00715043
\(399\) 4.55864e32 0.00351319
\(400\) −1.13245e35 −0.843744
\(401\) −1.64933e35 −1.18812 −0.594058 0.804422i \(-0.702475\pi\)
−0.594058 + 0.804422i \(0.702475\pi\)
\(402\) 3.25584e32 0.00226783
\(403\) 1.31211e35 0.883796
\(404\) 1.32968e35 0.866162
\(405\) −6.96521e33 −0.0438825
\(406\) 7.92542e31 0.000482971 0
\(407\) 9.61295e34 0.566672
\(408\) −3.63013e33 −0.0207019
\(409\) −2.13655e35 −1.17882 −0.589411 0.807833i \(-0.700640\pi\)
−0.589411 + 0.807833i \(0.700640\pi\)
\(410\) 8.69663e32 0.00464268
\(411\) −5.93036e34 −0.306349
\(412\) 2.70045e35 1.34997
\(413\) −1.21907e34 −0.0589798
\(414\) 1.04774e33 0.00490626
\(415\) 7.01151e34 0.317806
\(416\) −7.93786e33 −0.0348292
\(417\) 1.60276e35 0.680817
\(418\) 1.53747e32 0.000632306 0
\(419\) 6.25908e34 0.249242 0.124621 0.992204i \(-0.460228\pi\)
0.124621 + 0.992204i \(0.460228\pi\)
\(420\) 2.62728e33 0.0101307
\(421\) 1.62873e35 0.608191 0.304096 0.952641i \(-0.401646\pi\)
0.304096 + 0.952641i \(0.401646\pi\)
\(422\) 3.77648e33 0.0136574
\(423\) 1.63476e35 0.572609
\(424\) 6.94704e33 0.0235701
\(425\) −4.40475e35 −1.44767
\(426\) −6.15036e32 −0.00195826
\(427\) −9.30171e33 −0.0286937
\(428\) −2.99567e33 −0.00895372
\(429\) 9.78107e34 0.283278
\(430\) 1.34813e32 0.000378363 0
\(431\) −6.21025e35 −1.68914 −0.844571 0.535444i \(-0.820144\pi\)
−0.844571 + 0.535444i \(0.820144\pi\)
\(432\) −7.29804e34 −0.192387
\(433\) 1.04792e35 0.267757 0.133878 0.990998i \(-0.457257\pi\)
0.133878 + 0.990998i \(0.457257\pi\)
\(434\) 1.49206e32 0.000369550 0
\(435\) 9.87502e34 0.237101
\(436\) −7.25125e35 −1.68789
\(437\) −8.54420e34 −0.192829
\(438\) 4.05175e33 0.00886630
\(439\) 6.93788e35 1.47216 0.736081 0.676893i \(-0.236674\pi\)
0.736081 + 0.676893i \(0.236674\pi\)
\(440\) 1.77228e33 0.00364687
\(441\) −1.66702e35 −0.332675
\(442\) −1.02897e34 −0.0199161
\(443\) 5.15844e35 0.968430 0.484215 0.874949i \(-0.339105\pi\)
0.484215 + 0.874949i \(0.339105\pi\)
\(444\) −4.06687e35 −0.740610
\(445\) −3.65038e35 −0.644878
\(446\) 2.74829e33 0.00471023
\(447\) 3.28738e35 0.546636
\(448\) 2.75222e34 0.0444048
\(449\) 9.16926e35 1.43552 0.717759 0.696292i \(-0.245168\pi\)
0.717759 + 0.696292i \(0.245168\pi\)
\(450\) 1.93574e33 0.00294088
\(451\) −3.36904e35 −0.496732
\(452\) 4.98447e35 0.713264
\(453\) −1.39321e35 −0.193504
\(454\) −1.01957e34 −0.0137455
\(455\) 1.48951e34 0.0194934
\(456\) −1.30096e33 −0.00165287
\(457\) 1.55254e36 1.91502 0.957510 0.288400i \(-0.0931232\pi\)
0.957510 + 0.288400i \(0.0931232\pi\)
\(458\) −4.96382e33 −0.00594473
\(459\) −2.83862e35 −0.330093
\(460\) −4.92428e35 −0.556048
\(461\) −5.06840e35 −0.555787 −0.277893 0.960612i \(-0.589636\pi\)
−0.277893 + 0.960612i \(0.589636\pi\)
\(462\) 1.11225e32 0.000118450 0
\(463\) −1.45476e36 −1.50469 −0.752347 0.658767i \(-0.771078\pi\)
−0.752347 + 0.658767i \(0.771078\pi\)
\(464\) 1.03469e36 1.03948
\(465\) 1.85909e35 0.181420
\(466\) −8.46559e33 −0.00802504
\(467\) −6.89399e35 −0.634881 −0.317441 0.948278i \(-0.602823\pi\)
−0.317441 + 0.948278i \(0.602823\pi\)
\(468\) −4.13799e35 −0.370229
\(469\) −1.92076e34 −0.0166971
\(470\) 8.39617e33 0.00709186
\(471\) −5.84554e35 −0.479780
\(472\) 3.47903e34 0.0277486
\(473\) −5.22261e34 −0.0404820
\(474\) −1.71480e31 −1.29183e−5 0
\(475\) −1.57857e35 −0.115585
\(476\) 1.07073e35 0.0762053
\(477\) 5.43232e35 0.375826
\(478\) 2.28973e34 0.0153995
\(479\) −6.34595e35 −0.414923 −0.207461 0.978243i \(-0.566520\pi\)
−0.207461 + 0.978243i \(0.566520\pi\)
\(480\) −1.12469e34 −0.00714953
\(481\) −2.30566e36 −1.42507
\(482\) −1.23242e34 −0.00740670
\(483\) −6.18110e34 −0.0361226
\(484\) 1.41613e36 0.804807
\(485\) 4.48592e35 0.247935
\(486\) 1.24748e33 0.000670568 0
\(487\) 2.48287e36 1.29812 0.649058 0.760739i \(-0.275163\pi\)
0.649058 + 0.760739i \(0.275163\pi\)
\(488\) 2.65456e34 0.0134997
\(489\) 1.48788e36 0.736034
\(490\) −8.56188e33 −0.00412024
\(491\) −2.37415e36 −1.11150 −0.555748 0.831351i \(-0.687568\pi\)
−0.555748 + 0.831351i \(0.687568\pi\)
\(492\) 1.42531e36 0.649203
\(493\) 4.02449e36 1.78352
\(494\) −3.68762e33 −0.00159013
\(495\) 1.38585e35 0.0581496
\(496\) 1.94793e36 0.795371
\(497\) 3.62837e34 0.0144178
\(498\) −1.25577e34 −0.00485639
\(499\) −5.00732e36 −1.88472 −0.942362 0.334595i \(-0.891400\pi\)
−0.942362 + 0.334595i \(0.891400\pi\)
\(500\) −1.98768e36 −0.728203
\(501\) −5.52935e35 −0.197181
\(502\) 2.72625e34 0.00946380
\(503\) −1.61560e36 −0.545966 −0.272983 0.962019i \(-0.588010\pi\)
−0.272983 + 0.962019i \(0.588010\pi\)
\(504\) −9.41149e32 −0.000309632 0
\(505\) 1.06811e36 0.342122
\(506\) −2.08467e34 −0.00650138
\(507\) −4.44705e35 −0.135041
\(508\) −2.87094e36 −0.848914
\(509\) 1.80638e36 0.520139 0.260069 0.965590i \(-0.416255\pi\)
0.260069 + 0.965590i \(0.416255\pi\)
\(510\) −1.45792e34 −0.00408825
\(511\) −2.39031e35 −0.0652787
\(512\) −1.96413e35 −0.0522428
\(513\) −1.01730e35 −0.0263551
\(514\) −1.55386e34 −0.00392111
\(515\) 2.16922e36 0.533219
\(516\) 2.20948e35 0.0529078
\(517\) −3.25264e36 −0.758776
\(518\) −2.62187e33 −0.000595880 0
\(519\) −4.23776e36 −0.938377
\(520\) −4.25081e34 −0.00917120
\(521\) 4.59028e36 0.965006 0.482503 0.875894i \(-0.339728\pi\)
0.482503 + 0.875894i \(0.339728\pi\)
\(522\) −1.76863e34 −0.00362314
\(523\) 5.82915e35 0.116368 0.0581838 0.998306i \(-0.481469\pi\)
0.0581838 + 0.998306i \(0.481469\pi\)
\(524\) −6.90387e36 −1.34314
\(525\) −1.14198e35 −0.0216525
\(526\) 7.47617e34 0.0138157
\(527\) 7.57658e36 1.36468
\(528\) 1.45207e36 0.254936
\(529\) 5.74195e36 0.982670
\(530\) 2.79006e34 0.00465467
\(531\) 2.72046e36 0.442453
\(532\) 3.83726e34 0.00608435
\(533\) 8.08063e36 1.24919
\(534\) 6.53787e34 0.00985437
\(535\) −2.40636e34 −0.00353659
\(536\) 5.48154e34 0.00785558
\(537\) −6.87006e36 −0.960080
\(538\) 2.06757e33 0.000281773 0
\(539\) 3.31684e36 0.440835
\(540\) −5.86301e35 −0.0759985
\(541\) −6.32438e36 −0.799567 −0.399783 0.916610i \(-0.630915\pi\)
−0.399783 + 0.916610i \(0.630915\pi\)
\(542\) 1.29227e35 0.0159354
\(543\) 8.87082e35 0.106700
\(544\) −4.58360e35 −0.0537801
\(545\) −5.82478e36 −0.666695
\(546\) −2.66772e33 −0.000297879 0
\(547\) −9.05384e36 −0.986289 −0.493144 0.869948i \(-0.664153\pi\)
−0.493144 + 0.869948i \(0.664153\pi\)
\(548\) −4.99192e36 −0.530554
\(549\) 2.07576e36 0.215254
\(550\) −3.85150e34 −0.00389702
\(551\) 1.44229e36 0.142399
\(552\) 1.76398e35 0.0169948
\(553\) 1.01164e33 9.51118e−5 0
\(554\) −7.15436e34 −0.00656431
\(555\) −3.26683e36 −0.292531
\(556\) 1.34913e37 1.17908
\(557\) −1.74285e37 −1.48668 −0.743338 0.668917i \(-0.766758\pi\)
−0.743338 + 0.668917i \(0.766758\pi\)
\(558\) −3.32965e34 −0.00277228
\(559\) 1.25264e36 0.101805
\(560\) 2.21129e35 0.0175431
\(561\) 5.64794e36 0.437413
\(562\) −4.23289e34 −0.00320034
\(563\) −7.81650e36 −0.576964 −0.288482 0.957485i \(-0.593151\pi\)
−0.288482 + 0.957485i \(0.593151\pi\)
\(564\) 1.37607e37 0.991680
\(565\) 4.00393e36 0.281729
\(566\) 1.26065e34 0.000866109 0
\(567\) −7.35942e34 −0.00493711
\(568\) −1.03548e35 −0.00678325
\(569\) −1.42845e37 −0.913797 −0.456899 0.889519i \(-0.651040\pi\)
−0.456899 + 0.889519i \(0.651040\pi\)
\(570\) −5.22489e33 −0.000326412 0
\(571\) 2.26975e37 1.38481 0.692406 0.721508i \(-0.256551\pi\)
0.692406 + 0.721508i \(0.256551\pi\)
\(572\) 8.23327e36 0.490598
\(573\) 6.68769e36 0.389214
\(574\) 9.18883e33 0.000522336 0
\(575\) 2.14040e37 1.18844
\(576\) −6.14183e36 −0.333115
\(577\) −1.49672e37 −0.792987 −0.396494 0.918038i \(-0.629773\pi\)
−0.396494 + 0.918038i \(0.629773\pi\)
\(578\) −3.92203e35 −0.0202995
\(579\) −7.74247e34 −0.00391489
\(580\) 8.31236e36 0.410626
\(581\) 7.40834e35 0.0357555
\(582\) −8.03434e34 −0.00378870
\(583\) −1.08086e37 −0.498015
\(584\) 6.82154e35 0.0307121
\(585\) −3.32397e36 −0.146235
\(586\) 5.53869e34 0.00238116
\(587\) −7.65028e36 −0.321413 −0.160706 0.987002i \(-0.551377\pi\)
−0.160706 + 0.987002i \(0.551377\pi\)
\(588\) −1.40323e37 −0.576147
\(589\) 2.71529e36 0.108958
\(590\) 1.39724e35 0.00547985
\(591\) 3.82791e35 0.0146734
\(592\) −3.42293e37 −1.28249
\(593\) −1.64866e37 −0.603799 −0.301900 0.953340i \(-0.597621\pi\)
−0.301900 + 0.953340i \(0.597621\pi\)
\(594\) −2.48208e34 −0.000888584 0
\(595\) 8.60094e35 0.0301000
\(596\) 2.76717e37 0.946698
\(597\) −1.18081e37 −0.394935
\(598\) 5.00008e35 0.0163497
\(599\) −5.74500e36 −0.183665 −0.0918327 0.995774i \(-0.529273\pi\)
−0.0918327 + 0.995774i \(0.529273\pi\)
\(600\) 3.25902e35 0.0101870
\(601\) −5.09841e36 −0.155822 −0.0779112 0.996960i \(-0.524825\pi\)
−0.0779112 + 0.996960i \(0.524825\pi\)
\(602\) 1.42443e33 4.25686e−5 0
\(603\) 4.28635e36 0.125258
\(604\) −1.17274e37 −0.335123
\(605\) 1.13755e37 0.317888
\(606\) −1.91299e35 −0.00522796
\(607\) 1.21769e37 0.325453 0.162727 0.986671i \(-0.447971\pi\)
0.162727 + 0.986671i \(0.447971\pi\)
\(608\) −1.64267e35 −0.00429389
\(609\) 1.04339e36 0.0266756
\(610\) 1.06612e35 0.00266595
\(611\) 7.80145e37 1.90818
\(612\) −2.38942e37 −0.571675
\(613\) 2.55443e37 0.597829 0.298915 0.954280i \(-0.403375\pi\)
0.298915 + 0.954280i \(0.403375\pi\)
\(614\) −3.03464e35 −0.00694759
\(615\) 1.14492e37 0.256426
\(616\) 1.87258e34 0.000410300 0
\(617\) 1.30077e37 0.278838 0.139419 0.990234i \(-0.455477\pi\)
0.139419 + 0.990234i \(0.455477\pi\)
\(618\) −3.88509e35 −0.00814812
\(619\) −2.04833e37 −0.420316 −0.210158 0.977667i \(-0.567398\pi\)
−0.210158 + 0.977667i \(0.567398\pi\)
\(620\) 1.56490e37 0.314195
\(621\) 1.37937e37 0.270984
\(622\) 4.95433e35 0.00952389
\(623\) −3.85698e36 −0.0725535
\(624\) −3.48280e37 −0.641116
\(625\) 3.08859e37 0.556390
\(626\) −6.74931e35 −0.0118989
\(627\) 2.02410e36 0.0349237
\(628\) −4.92052e37 −0.830913
\(629\) −1.33137e38 −2.20047
\(630\) −3.77982e33 −6.11469e−5 0
\(631\) 7.16988e37 1.13531 0.567656 0.823266i \(-0.307850\pi\)
0.567656 + 0.823266i \(0.307850\pi\)
\(632\) −2.88704e33 −4.47478e−5 0
\(633\) 4.97179e37 0.754332
\(634\) −7.53957e35 −0.0111980
\(635\) −2.30617e37 −0.335309
\(636\) 4.57269e37 0.650879
\(637\) −7.95543e37 −1.10862
\(638\) 3.51900e35 0.00480109
\(639\) −8.09703e36 −0.108159
\(640\) −1.26227e36 −0.0165090
\(641\) −1.00079e38 −1.28161 −0.640807 0.767702i \(-0.721400\pi\)
−0.640807 + 0.767702i \(0.721400\pi\)
\(642\) 4.30982e33 5.40426e−5 0
\(643\) 7.58172e37 0.930936 0.465468 0.885065i \(-0.345886\pi\)
0.465468 + 0.885065i \(0.345886\pi\)
\(644\) −5.20298e36 −0.0625594
\(645\) 1.77483e36 0.0208978
\(646\) −2.12936e35 −0.00245534
\(647\) −7.13094e37 −0.805265 −0.402632 0.915362i \(-0.631905\pi\)
−0.402632 + 0.915362i \(0.631905\pi\)
\(648\) 2.10026e35 0.00232279
\(649\) −5.41285e37 −0.586303
\(650\) 9.23781e35 0.00980028
\(651\) 1.96431e36 0.0204111
\(652\) 1.25243e38 1.27471
\(653\) 2.47590e37 0.246835 0.123417 0.992355i \(-0.460615\pi\)
0.123417 + 0.992355i \(0.460615\pi\)
\(654\) 1.04322e36 0.0101878
\(655\) −5.54574e37 −0.530520
\(656\) 1.19963e38 1.12421
\(657\) 5.33418e37 0.489706
\(658\) 8.87136e34 0.000797886 0
\(659\) 1.51836e38 1.33789 0.668946 0.743311i \(-0.266746\pi\)
0.668946 + 0.743311i \(0.266746\pi\)
\(660\) 1.16655e37 0.100707
\(661\) 1.92148e38 1.62523 0.812617 0.582798i \(-0.198042\pi\)
0.812617 + 0.582798i \(0.198042\pi\)
\(662\) 8.10232e35 0.00671470
\(663\) −1.35466e38 −1.10001
\(664\) −2.11422e36 −0.0168221
\(665\) 3.08239e35 0.00240323
\(666\) 5.85093e35 0.00447016
\(667\) −1.95562e38 −1.46415
\(668\) −4.65436e37 −0.341490
\(669\) 3.61816e37 0.260157
\(670\) 2.20149e35 0.00155134
\(671\) −4.13009e37 −0.285237
\(672\) −1.18835e35 −0.000804375 0
\(673\) −9.37859e37 −0.622206 −0.311103 0.950376i \(-0.600698\pi\)
−0.311103 + 0.950376i \(0.600698\pi\)
\(674\) 1.02371e36 0.00665685
\(675\) 2.54843e37 0.162432
\(676\) −3.74333e37 −0.233872
\(677\) −2.93781e37 −0.179919 −0.0899595 0.995945i \(-0.528674\pi\)
−0.0899595 + 0.995945i \(0.528674\pi\)
\(678\) −7.17107e35 −0.00430510
\(679\) 4.73981e36 0.0278945
\(680\) −2.45457e36 −0.0141613
\(681\) −1.34227e38 −0.759197
\(682\) 6.62494e35 0.00367361
\(683\) 1.68599e38 0.916591 0.458295 0.888800i \(-0.348460\pi\)
0.458295 + 0.888800i \(0.348460\pi\)
\(684\) −8.56319e36 −0.0456434
\(685\) −4.00991e37 −0.209562
\(686\) −1.81108e35 −0.000928031 0
\(687\) −6.53493e37 −0.328341
\(688\) 1.85964e37 0.0916190
\(689\) 2.59243e38 1.25241
\(690\) 7.08448e35 0.00335618
\(691\) −3.72820e38 −1.73199 −0.865996 0.500051i \(-0.833314\pi\)
−0.865996 + 0.500051i \(0.833314\pi\)
\(692\) −3.56716e38 −1.62514
\(693\) 1.46429e36 0.00654226
\(694\) −4.29252e36 −0.0188088
\(695\) 1.08373e38 0.465720
\(696\) −2.97767e36 −0.0125502
\(697\) 4.66604e38 1.92888
\(698\) −1.97084e36 −0.00799102
\(699\) −1.11451e38 −0.443241
\(700\) −9.61267e36 −0.0374991
\(701\) 3.38534e38 1.29541 0.647707 0.761890i \(-0.275728\pi\)
0.647707 + 0.761890i \(0.275728\pi\)
\(702\) 5.95326e35 0.00223462
\(703\) −4.77135e37 −0.175689
\(704\) 1.22203e38 0.441417
\(705\) 1.10537e38 0.391700
\(706\) 8.09623e35 0.00281462
\(707\) 1.12856e37 0.0384912
\(708\) 2.28997e38 0.766267
\(709\) 2.34891e38 0.771155 0.385578 0.922675i \(-0.374002\pi\)
0.385578 + 0.922675i \(0.374002\pi\)
\(710\) −4.15866e35 −0.00133957
\(711\) −2.25755e35 −0.000713507 0
\(712\) 1.10072e37 0.0341347
\(713\) −3.68168e38 −1.12031
\(714\) −1.54044e35 −0.000459958 0
\(715\) 6.61362e37 0.193779
\(716\) −5.78291e38 −1.66273
\(717\) 3.01445e38 0.850551
\(718\) 9.45165e35 0.00261714
\(719\) −1.57531e38 −0.428082 −0.214041 0.976825i \(-0.568663\pi\)
−0.214041 + 0.976825i \(0.568663\pi\)
\(720\) −4.93468e37 −0.131604
\(721\) 2.29199e37 0.0599911
\(722\) 3.99279e36 0.0102571
\(723\) −1.62250e38 −0.409089
\(724\) 7.46707e37 0.184790
\(725\) −3.61306e38 −0.877632
\(726\) −2.03737e36 −0.00485763
\(727\) −2.38862e38 −0.559027 −0.279513 0.960142i \(-0.590173\pi\)
−0.279513 + 0.960142i \(0.590173\pi\)
\(728\) −4.49139e35 −0.00103183
\(729\) 1.64232e37 0.0370370
\(730\) 2.73965e36 0.00606509
\(731\) 7.23320e37 0.157198
\(732\) 1.74728e38 0.372789
\(733\) −1.39015e38 −0.291177 −0.145589 0.989345i \(-0.546508\pi\)
−0.145589 + 0.989345i \(0.546508\pi\)
\(734\) −8.01229e36 −0.0164763
\(735\) −1.12718e38 −0.227570
\(736\) 2.22731e37 0.0441499
\(737\) −8.52846e37 −0.165981
\(738\) −2.05057e36 −0.00391844
\(739\) 3.01753e37 0.0566176 0.0283088 0.999599i \(-0.490988\pi\)
0.0283088 + 0.999599i \(0.490988\pi\)
\(740\) −2.74987e38 −0.506623
\(741\) −4.85480e37 −0.0878265
\(742\) 2.94797e35 0.000523685 0
\(743\) 8.48049e38 1.47935 0.739677 0.672962i \(-0.234978\pi\)
0.739677 + 0.672962i \(0.234978\pi\)
\(744\) −5.60582e36 −0.00960294
\(745\) 2.22281e38 0.373932
\(746\) 8.22856e36 0.0135941
\(747\) −1.65324e38 −0.268230
\(748\) 4.75418e38 0.757538
\(749\) −2.54256e35 −0.000397893 0
\(750\) 2.85965e36 0.00439527
\(751\) −1.00379e39 −1.51532 −0.757658 0.652651i \(-0.773657\pi\)
−0.757658 + 0.652651i \(0.773657\pi\)
\(752\) 1.15819e39 1.71726
\(753\) 3.58914e38 0.522707
\(754\) −8.44031e36 −0.0120738
\(755\) −9.42041e37 −0.132369
\(756\) −6.19484e36 −0.00855038
\(757\) 1.01436e39 1.37530 0.687652 0.726040i \(-0.258641\pi\)
0.687652 + 0.726040i \(0.258641\pi\)
\(758\) −9.67114e36 −0.0128808
\(759\) −2.74450e38 −0.359086
\(760\) −8.79664e35 −0.00113066
\(761\) −1.18512e39 −1.49648 −0.748240 0.663428i \(-0.769101\pi\)
−0.748240 + 0.663428i \(0.769101\pi\)
\(762\) 4.13037e36 0.00512385
\(763\) −6.15445e37 −0.0750080
\(764\) 5.62940e38 0.674065
\(765\) −1.91938e38 −0.225804
\(766\) −1.44152e37 −0.0166622
\(767\) 1.29827e39 1.47444
\(768\) −5.16822e38 −0.576720
\(769\) −1.38650e39 −1.52024 −0.760122 0.649780i \(-0.774861\pi\)
−0.760122 + 0.649780i \(0.774861\pi\)
\(770\) 7.52063e34 8.10269e−5 0
\(771\) −2.04567e38 −0.216572
\(772\) −6.51727e36 −0.00678004
\(773\) −3.84501e38 −0.393074 −0.196537 0.980496i \(-0.562970\pi\)
−0.196537 + 0.980496i \(0.562970\pi\)
\(774\) −3.17875e35 −0.000319340 0
\(775\) −6.80202e38 −0.671530
\(776\) −1.35266e37 −0.0131237
\(777\) −3.45172e37 −0.0329118
\(778\) −1.80364e36 −0.00169015
\(779\) 1.67221e38 0.154005
\(780\) −2.79797e38 −0.253259
\(781\) 1.61105e38 0.143324
\(782\) 2.88722e37 0.0252458
\(783\) −2.32842e38 −0.200114
\(784\) −1.18104e39 −0.997698
\(785\) −3.95255e38 −0.328199
\(786\) 9.93248e36 0.00810687
\(787\) 8.31177e38 0.666860 0.333430 0.942775i \(-0.391794\pi\)
0.333430 + 0.942775i \(0.391794\pi\)
\(788\) 3.22217e37 0.0254123
\(789\) 9.84247e38 0.763070
\(790\) −1.15949e34 −8.83690e−6 0
\(791\) 4.23054e37 0.0316966
\(792\) −4.17883e36 −0.00307798
\(793\) 9.90602e38 0.717317
\(794\) −4.93337e36 −0.00351210
\(795\) 3.67315e38 0.257088
\(796\) −9.93951e38 −0.683973
\(797\) −5.86889e38 −0.397071 −0.198536 0.980094i \(-0.563619\pi\)
−0.198536 + 0.980094i \(0.563619\pi\)
\(798\) −5.52060e34 −3.67238e−5 0
\(799\) 4.50484e39 2.94644
\(800\) 4.11502e37 0.0264641
\(801\) 8.60719e38 0.544280
\(802\) 1.99737e37 0.0124195
\(803\) −1.06133e39 −0.648919
\(804\) 3.60806e38 0.216929
\(805\) −4.17945e37 −0.0247101
\(806\) −1.58899e37 −0.00923843
\(807\) 2.72198e37 0.0155630
\(808\) −3.22072e37 −0.0181092
\(809\) −2.02827e39 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(810\) 8.43501e35 0.000458710 0
\(811\) −1.61063e39 −0.861421 −0.430710 0.902490i \(-0.641737\pi\)
−0.430710 + 0.902490i \(0.641737\pi\)
\(812\) 8.78281e37 0.0461984
\(813\) 1.70129e39 0.880147
\(814\) −1.16415e37 −0.00592350
\(815\) 1.00605e39 0.503492
\(816\) −2.01109e39 −0.989954
\(817\) 2.59222e37 0.0125509
\(818\) 2.58740e37 0.0123224
\(819\) −3.51209e37 −0.0164526
\(820\) 9.63745e38 0.444094
\(821\) −2.30031e39 −1.04268 −0.521342 0.853348i \(-0.674569\pi\)
−0.521342 + 0.853348i \(0.674569\pi\)
\(822\) 7.18179e36 0.00320231
\(823\) −2.08776e39 −0.915764 −0.457882 0.889013i \(-0.651392\pi\)
−0.457882 + 0.889013i \(0.651392\pi\)
\(824\) −6.54096e37 −0.0282244
\(825\) −5.07054e38 −0.215242
\(826\) 1.47632e36 0.000616523 0
\(827\) 3.43290e39 1.41038 0.705192 0.709017i \(-0.250861\pi\)
0.705192 + 0.709017i \(0.250861\pi\)
\(828\) 1.16109e39 0.469306
\(829\) 8.03456e38 0.319504 0.159752 0.987157i \(-0.448931\pi\)
0.159752 + 0.987157i \(0.448931\pi\)
\(830\) −8.49107e36 −0.00332207
\(831\) −9.41881e38 −0.362562
\(832\) −2.93103e39 −1.11008
\(833\) −4.59375e39 −1.71183
\(834\) −1.94097e37 −0.00711667
\(835\) −3.73875e38 −0.134884
\(836\) 1.70380e38 0.0604830
\(837\) −4.38353e38 −0.153119
\(838\) −7.57987e36 −0.00260536
\(839\) 2.54669e39 0.861369 0.430685 0.902502i \(-0.358272\pi\)
0.430685 + 0.902502i \(0.358272\pi\)
\(840\) −6.36372e35 −0.000211807 0
\(841\) 2.48015e38 0.0812329
\(842\) −1.97242e37 −0.00635750
\(843\) −5.57266e38 −0.176762
\(844\) 4.18503e39 1.30640
\(845\) −3.00694e38 −0.0923760
\(846\) −1.97972e37 −0.00598556
\(847\) 1.20193e38 0.0357647
\(848\) 3.84867e39 1.12711
\(849\) 1.65966e38 0.0478372
\(850\) 5.33424e37 0.0151327
\(851\) 6.46952e39 1.80644
\(852\) −6.81572e38 −0.187317
\(853\) 1.61849e39 0.437823 0.218912 0.975745i \(-0.429749\pi\)
0.218912 + 0.975745i \(0.429749\pi\)
\(854\) 1.12646e36 0.000299939 0
\(855\) −6.87864e37 −0.0180285
\(856\) 7.25603e35 0.000187199 0
\(857\) 1.79678e38 0.0456304 0.0228152 0.999740i \(-0.492737\pi\)
0.0228152 + 0.999740i \(0.492737\pi\)
\(858\) −1.18451e37 −0.00296114
\(859\) −6.83644e39 −1.68237 −0.841187 0.540744i \(-0.818143\pi\)
−0.841187 + 0.540744i \(0.818143\pi\)
\(860\) 1.49398e38 0.0361922
\(861\) 1.20972e38 0.0288498
\(862\) 7.52074e37 0.0176568
\(863\) −8.93182e37 −0.0206440 −0.0103220 0.999947i \(-0.503286\pi\)
−0.0103220 + 0.999947i \(0.503286\pi\)
\(864\) 2.65190e37 0.00603424
\(865\) −2.86543e39 −0.641907
\(866\) −1.26905e37 −0.00279890
\(867\) −5.16340e39 −1.12119
\(868\) 1.65347e38 0.0353492
\(869\) 4.49180e36 0.000945483 0
\(870\) −1.19588e37 −0.00247845
\(871\) 2.04555e39 0.417412
\(872\) 1.75638e38 0.0352895
\(873\) −1.05773e39 −0.209258
\(874\) 1.03472e37 0.00201566
\(875\) −1.68703e38 −0.0323605
\(876\) 4.49008e39 0.848103
\(877\) −4.74244e39 −0.882078 −0.441039 0.897488i \(-0.645390\pi\)
−0.441039 + 0.897488i \(0.645390\pi\)
\(878\) −8.40191e37 −0.0153887
\(879\) 7.29176e38 0.131517
\(880\) 9.81843e38 0.174392
\(881\) −4.04555e39 −0.707624 −0.353812 0.935317i \(-0.615115\pi\)
−0.353812 + 0.935317i \(0.615115\pi\)
\(882\) 2.01880e37 0.00347750
\(883\) 7.15911e39 1.21448 0.607239 0.794519i \(-0.292277\pi\)
0.607239 + 0.794519i \(0.292277\pi\)
\(884\) −1.14029e40 −1.90507
\(885\) 1.83948e39 0.302665
\(886\) −6.24697e37 −0.0101231
\(887\) −3.43934e39 −0.548916 −0.274458 0.961599i \(-0.588498\pi\)
−0.274458 + 0.961599i \(0.588498\pi\)
\(888\) 9.85065e37 0.0154842
\(889\) −2.43669e38 −0.0377247
\(890\) 4.42068e37 0.00674099
\(891\) −3.26769e38 −0.0490785
\(892\) 3.04561e39 0.450556
\(893\) 1.61444e39 0.235248
\(894\) −3.98108e37 −0.00571405
\(895\) −4.64529e39 −0.656753
\(896\) −1.33371e37 −0.00185739
\(897\) 6.58267e39 0.903034
\(898\) −1.11041e38 −0.0150056
\(899\) 6.21481e39 0.827317
\(900\) 2.14515e39 0.281309
\(901\) 1.49696e40 1.93387
\(902\) 4.07997e37 0.00519241
\(903\) 1.87528e37 0.00235116
\(904\) −1.20732e38 −0.0149125
\(905\) 5.99814e38 0.0729896
\(906\) 1.68720e37 0.00202273
\(907\) 2.71200e38 0.0320325 0.0160163 0.999872i \(-0.494902\pi\)
0.0160163 + 0.999872i \(0.494902\pi\)
\(908\) −1.12987e40 −1.31482
\(909\) −2.51848e39 −0.288752
\(910\) −1.80382e36 −0.000203767 0
\(911\) 1.18591e40 1.31994 0.659970 0.751292i \(-0.270569\pi\)
0.659970 + 0.751292i \(0.270569\pi\)
\(912\) −7.20733e38 −0.0790394
\(913\) 3.28941e39 0.355436
\(914\) −1.88015e38 −0.0200179
\(915\) 1.40356e39 0.147247
\(916\) −5.50082e39 −0.568641
\(917\) −5.85961e38 −0.0596874
\(918\) 3.43762e37 0.00345050
\(919\) −1.70613e40 −1.68753 −0.843766 0.536711i \(-0.819667\pi\)
−0.843766 + 0.536711i \(0.819667\pi\)
\(920\) 1.19274e38 0.0116255
\(921\) −3.99514e39 −0.383731
\(922\) 6.13793e37 0.00580971
\(923\) −3.86409e39 −0.360433
\(924\) 1.23257e38 0.0113303
\(925\) 1.19526e40 1.08281
\(926\) 1.76174e38 0.0157287
\(927\) −5.11478e39 −0.450039
\(928\) −3.75977e38 −0.0326035
\(929\) −8.89239e39 −0.759987 −0.379994 0.924989i \(-0.624074\pi\)
−0.379994 + 0.924989i \(0.624074\pi\)
\(930\) −2.25140e37 −0.00189641
\(931\) −1.64630e39 −0.136675
\(932\) −9.38141e39 −0.767633
\(933\) 6.52243e39 0.526026
\(934\) 8.34875e37 0.00663649
\(935\) 3.81894e39 0.299217
\(936\) 1.00229e38 0.00774053
\(937\) −1.80674e40 −1.37534 −0.687670 0.726023i \(-0.741367\pi\)
−0.687670 + 0.726023i \(0.741367\pi\)
\(938\) 2.32608e36 0.000174537 0
\(939\) −8.88555e39 −0.657202
\(940\) 9.30449e39 0.678370
\(941\) 2.62451e40 1.88620 0.943101 0.332507i \(-0.107895\pi\)
0.943101 + 0.332507i \(0.107895\pi\)
\(942\) 7.07906e37 0.00501520
\(943\) −2.26737e40 −1.58348
\(944\) 1.92738e40 1.32692
\(945\) −4.97619e37 −0.00337728
\(946\) 6.32468e36 0.000423163 0
\(947\) −6.71744e39 −0.443077 −0.221539 0.975152i \(-0.571108\pi\)
−0.221539 + 0.975152i \(0.571108\pi\)
\(948\) −1.90031e37 −0.00123570
\(949\) 2.54560e40 1.63191
\(950\) 1.91168e37 0.00120822
\(951\) −9.92594e39 −0.618493
\(952\) −2.59349e37 −0.00159325
\(953\) 1.95322e39 0.118303 0.0591516 0.998249i \(-0.481160\pi\)
0.0591516 + 0.998249i \(0.481160\pi\)
\(954\) −6.57865e37 −0.00392856
\(955\) 4.52199e39 0.266246
\(956\) 2.53743e40 1.47304
\(957\) 4.63281e39 0.265175
\(958\) 7.68507e37 0.00433724
\(959\) −4.23686e38 −0.0235772
\(960\) −4.15289e39 −0.227871
\(961\) −6.78259e39 −0.366969
\(962\) 2.79220e38 0.0148965
\(963\) 5.67394e37 0.00298490
\(964\) −1.36575e40 −0.708486
\(965\) −5.23519e37 −0.00267802
\(966\) 7.48543e36 0.000377595 0
\(967\) 1.48699e40 0.739690 0.369845 0.929093i \(-0.379411\pi\)
0.369845 + 0.929093i \(0.379411\pi\)
\(968\) −3.43012e38 −0.0168264
\(969\) −2.80333e39 −0.135614
\(970\) −5.43254e37 −0.00259170
\(971\) −1.17711e40 −0.553804 −0.276902 0.960898i \(-0.589308\pi\)
−0.276902 + 0.960898i \(0.589308\pi\)
\(972\) 1.38243e39 0.0641430
\(973\) 1.14506e39 0.0523970
\(974\) −3.00681e38 −0.0135694
\(975\) 1.21617e40 0.541292
\(976\) 1.47062e40 0.645549
\(977\) −3.52154e40 −1.52460 −0.762300 0.647224i \(-0.775930\pi\)
−0.762300 + 0.647224i \(0.775930\pi\)
\(978\) −1.80185e38 −0.00769385
\(979\) −1.71255e40 −0.721236
\(980\) −9.48813e39 −0.394120
\(981\) 1.37342e40 0.562693
\(982\) 2.87514e38 0.0116186
\(983\) −3.23914e40 −1.29109 −0.645547 0.763721i \(-0.723371\pi\)
−0.645547 + 0.763721i \(0.723371\pi\)
\(984\) −3.45234e38 −0.0135731
\(985\) 2.58830e38 0.0100375
\(986\) −4.87374e38 −0.0186433
\(987\) 1.16793e39 0.0440691
\(988\) −4.08656e39 −0.152103
\(989\) −3.51482e39 −0.129049
\(990\) −1.67830e37 −0.000607845 0
\(991\) 5.97175e38 0.0213357 0.0106679 0.999943i \(-0.496604\pi\)
0.0106679 + 0.999943i \(0.496604\pi\)
\(992\) −7.07822e38 −0.0249469
\(993\) 1.06668e40 0.370868
\(994\) −4.39403e36 −0.000150712 0
\(995\) −7.98421e39 −0.270160
\(996\) −1.39162e40 −0.464536
\(997\) −1.51823e40 −0.499980 −0.249990 0.968248i \(-0.580427\pi\)
−0.249990 + 0.968248i \(0.580427\pi\)
\(998\) 6.06396e38 0.0197013
\(999\) 7.70283e39 0.246897
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 3.28.a.b.1.1 2
3.2 odd 2 9.28.a.b.1.2 2
4.3 odd 2 48.28.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.28.a.b.1.1 2 1.1 even 1 trivial
9.28.a.b.1.2 2 3.2 odd 2
48.28.a.e.1.1 2 4.3 odd 2