Properties

Label 25.24.b.a.24.3
Level $25$
Weight $24$
Character 25.24
Analytic conductor $83.801$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(83.8010093363\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \(x^{4} + 72085 x^{2} + 1299025764\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 24.3
Root \(189.348i\) of defining polynomial
Character \(\chi\) \(=\) 25.24
Dual form 25.24.b.a.24.2

$q$-expansion

\(f(q)\) \(=\) \(q+4016.35i q^{2} +388445. i q^{3} -7.74247e6 q^{4} -1.56013e9 q^{6} -3.81217e9i q^{7} +2.59512e9i q^{8} -5.67462e10 q^{9} +O(q^{10})\) \(q+4016.35i q^{2} +388445. i q^{3} -7.74247e6 q^{4} -1.56013e9 q^{6} -3.81217e9i q^{7} +2.59512e9i q^{8} -5.67462e10 q^{9} +2.52200e11 q^{11} -3.00752e12i q^{12} -3.59099e12i q^{13} +1.53110e13 q^{14} -7.53715e13 q^{16} -2.34190e14i q^{17} -2.27913e14i q^{18} +6.23086e14 q^{19} +1.48082e15 q^{21} +1.01292e15i q^{22} -3.58786e15i q^{23} -1.00806e15 q^{24} +1.44227e16 q^{26} +1.45267e16i q^{27} +2.95156e16i q^{28} +2.05923e16 q^{29} +1.36357e17 q^{31} -2.80949e17i q^{32} +9.79656e16i q^{33} +9.40588e17 q^{34} +4.39356e17 q^{36} +1.23898e18i q^{37} +2.50253e18i q^{38} +1.39490e18 q^{39} +1.40074e18 q^{41} +5.94748e18i q^{42} +2.18793e17i q^{43} -1.95265e18 q^{44} +1.44101e19 q^{46} +8.67836e18i q^{47} -2.92777e19i q^{48} +1.28361e19 q^{49} +9.09698e19 q^{51} +2.78032e19i q^{52} -7.63436e19i q^{53} -5.83441e19 q^{54} +9.89304e18 q^{56} +2.42034e20i q^{57} +8.27059e19i q^{58} +1.01862e18 q^{59} +2.87337e20 q^{61} +5.47658e20i q^{62} +2.16326e20i q^{63} +4.96127e20 q^{64} -3.93464e20 q^{66} -1.47683e21i q^{67} +1.81321e21i q^{68} +1.39369e21 q^{69} +7.64346e20 q^{71} -1.47263e20i q^{72} -3.49433e21i q^{73} -4.97617e21 q^{74} -4.82422e21 q^{76} -9.61427e20i q^{77} +5.60242e21i q^{78} -1.02350e22 q^{79} -1.09851e22 q^{81} +5.62587e21i q^{82} -7.71597e21i q^{83} -1.14652e22 q^{84} -8.78750e20 q^{86} +7.99897e21i q^{87} +6.54489e20i q^{88} -4.58518e21 q^{89} -1.36895e22 q^{91} +2.77789e22i q^{92} +5.29672e22i q^{93} -3.48553e22 q^{94} +1.09133e23 q^{96} +1.13703e23i q^{97} +5.15544e22i q^{98} -1.43114e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 50653312q^{4} - 3619346112q^{6} + 69998788332q^{9} + O(q^{10}) \) \( 4q - 50653312q^{4} - 3619346112q^{6} + 69998788332q^{9} + 1713603936528q^{11} + 83332069059456q^{14} + 31913172803584q^{16} - 8521201959920q^{19} + 3468063275445888q^{21} + 2573244630282240q^{24} + 110051709317470368q^{26} - 41636867203246680q^{29} + 275428034354000768q^{31} + 1678967311922650656q^{34} - 2347832600155149696q^{36} + 3570022947330058464q^{39} - 4588870954336629912q^{41} - 25168177680066077184q^{44} - 17626412036100442752q^{46} + 26923963408752401628q^{49} + 179996725690156584288q^{51} - 209462446834079598720q^{54} - 464913424108576235520q^{56} - 561745979942681543760q^{59} - 360905785033004447272q^{61} + 1787380508938705829888q^{64} - 1088676281827303766784q^{66} + 2341121344344808447104q^{69} + 6110067020388286657248q^{71} - 13430688566297615514144q^{74} + 12414308589026161180160q^{76} - 12489833629119279961280q^{79} - 5587057161859667951196q^{81} - 31835503814380804952064q^{84} - 21832577625837998486592q^{86} - 12790186172346140009640q^{89} - 109780356325409120292032q^{91} - 318801036013069863654144q^{94} + 211184243339168788512768q^{96} + 82316490264624271963824q^{99} + O(q^{100}) \)

Character values

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

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4016.35i 1.38671i 0.720595 + 0.693357i \(0.243869\pi\)
−0.720595 + 0.693357i \(0.756131\pi\)
\(3\) 388445.i 1.26600i 0.774150 + 0.633002i \(0.218177\pi\)
−0.774150 + 0.633002i \(0.781823\pi\)
\(4\) −7.74247e6 −0.922974
\(5\) 0 0
\(6\) −1.56013e9 −1.75558
\(7\) − 3.81217e9i − 0.728693i −0.931263 0.364346i \(-0.881292\pi\)
0.931263 0.364346i \(-0.118708\pi\)
\(8\) 2.59512e9i 0.106813i
\(9\) −5.67462e10 −0.602765
\(10\) 0 0
\(11\) 2.52200e11 0.266519 0.133260 0.991081i \(-0.457456\pi\)
0.133260 + 0.991081i \(0.457456\pi\)
\(12\) − 3.00752e12i − 1.16849i
\(13\) − 3.59099e12i − 0.555733i −0.960620 0.277867i \(-0.910373\pi\)
0.960620 0.277867i \(-0.0896273\pi\)
\(14\) 1.53110e13 1.01049
\(15\) 0 0
\(16\) −7.53715e13 −1.07109
\(17\) − 2.34190e14i − 1.65731i −0.559756 0.828657i \(-0.689105\pi\)
0.559756 0.828657i \(-0.310895\pi\)
\(18\) − 2.27913e14i − 0.835863i
\(19\) 6.23086e14 1.22710 0.613552 0.789654i \(-0.289740\pi\)
0.613552 + 0.789654i \(0.289740\pi\)
\(20\) 0 0
\(21\) 1.48082e15 0.922528
\(22\) 1.01292e15i 0.369586i
\(23\) − 3.58786e15i − 0.785173i −0.919715 0.392587i \(-0.871580\pi\)
0.919715 0.392587i \(-0.128420\pi\)
\(24\) −1.00806e15 −0.135225
\(25\) 0 0
\(26\) 1.44227e16 0.770642
\(27\) 1.45267e16i 0.502901i
\(28\) 2.95156e16i 0.672565i
\(29\) 2.05923e16 0.313421 0.156710 0.987645i \(-0.449911\pi\)
0.156710 + 0.987645i \(0.449911\pi\)
\(30\) 0 0
\(31\) 1.36357e17 0.963870 0.481935 0.876207i \(-0.339934\pi\)
0.481935 + 0.876207i \(0.339934\pi\)
\(32\) − 2.80949e17i − 1.37849i
\(33\) 9.79656e16i 0.337414i
\(34\) 9.40588e17 2.29822
\(35\) 0 0
\(36\) 4.39356e17 0.556337
\(37\) 1.23898e18i 1.14484i 0.819961 + 0.572419i \(0.193995\pi\)
−0.819961 + 0.572419i \(0.806005\pi\)
\(38\) 2.50253e18i 1.70164i
\(39\) 1.39490e18 0.703560
\(40\) 0 0
\(41\) 1.40074e18 0.397506 0.198753 0.980050i \(-0.436311\pi\)
0.198753 + 0.980050i \(0.436311\pi\)
\(42\) 5.94748e18i 1.27928i
\(43\) 2.18793e17i 0.0359043i 0.999839 + 0.0179522i \(0.00571465\pi\)
−0.999839 + 0.0179522i \(0.994285\pi\)
\(44\) −1.95265e18 −0.245990
\(45\) 0 0
\(46\) 1.44101e19 1.08881
\(47\) 8.67836e18i 0.512050i 0.966670 + 0.256025i \(0.0824129\pi\)
−0.966670 + 0.256025i \(0.917587\pi\)
\(48\) − 2.92777e19i − 1.35601i
\(49\) 1.28361e19 0.469007
\(50\) 0 0
\(51\) 9.09698e19 2.09817
\(52\) 2.78032e19i 0.512927i
\(53\) − 7.63436e19i − 1.13136i −0.824626 0.565679i \(-0.808614\pi\)
0.824626 0.565679i \(-0.191386\pi\)
\(54\) −5.83441e19 −0.697379
\(55\) 0 0
\(56\) 9.89304e18 0.0778337
\(57\) 2.42034e20i 1.55352i
\(58\) 8.27059e19i 0.434625i
\(59\) 1.01862e18 0.00439760 0.00219880 0.999998i \(-0.499300\pi\)
0.00219880 + 0.999998i \(0.499300\pi\)
\(60\) 0 0
\(61\) 2.87337e20 0.845472 0.422736 0.906253i \(-0.361070\pi\)
0.422736 + 0.906253i \(0.361070\pi\)
\(62\) 5.47658e20i 1.33661i
\(63\) 2.16326e20i 0.439231i
\(64\) 4.96127e20 0.840472
\(65\) 0 0
\(66\) −3.93464e20 −0.467897
\(67\) − 1.47683e21i − 1.47730i −0.674088 0.738652i \(-0.735463\pi\)
0.674088 0.738652i \(-0.264537\pi\)
\(68\) 1.81321e21i 1.52966i
\(69\) 1.39369e21 0.994032
\(70\) 0 0
\(71\) 7.64346e20 0.392481 0.196241 0.980556i \(-0.437127\pi\)
0.196241 + 0.980556i \(0.437127\pi\)
\(72\) − 1.47263e20i − 0.0643830i
\(73\) − 3.49433e21i − 1.30362i −0.758382 0.651810i \(-0.774010\pi\)
0.758382 0.651810i \(-0.225990\pi\)
\(74\) −4.97617e21 −1.58756
\(75\) 0 0
\(76\) −4.82422e21 −1.13259
\(77\) − 9.61427e20i − 0.194211i
\(78\) 5.60242e21i 0.975636i
\(79\) −1.02350e22 −1.53948 −0.769742 0.638356i \(-0.779615\pi\)
−0.769742 + 0.638356i \(0.779615\pi\)
\(80\) 0 0
\(81\) −1.09851e22 −1.23944
\(82\) 5.62587e21i 0.551227i
\(83\) − 7.71597e21i − 0.657646i −0.944392 0.328823i \(-0.893348\pi\)
0.944392 0.328823i \(-0.106652\pi\)
\(84\) −1.14652e22 −0.851469
\(85\) 0 0
\(86\) −8.78750e20 −0.0497890
\(87\) 7.99897e21i 0.396792i
\(88\) 6.54489e20i 0.0284676i
\(89\) −4.58518e21 −0.175134 −0.0875672 0.996159i \(-0.527909\pi\)
−0.0875672 + 0.996159i \(0.527909\pi\)
\(90\) 0 0
\(91\) −1.36895e22 −0.404959
\(92\) 2.77789e22i 0.724695i
\(93\) 5.29672e22i 1.22026i
\(94\) −3.48553e22 −0.710067
\(95\) 0 0
\(96\) 1.09133e23 1.74517
\(97\) 1.13703e23i 1.61398i 0.590564 + 0.806991i \(0.298905\pi\)
−0.590564 + 0.806991i \(0.701095\pi\)
\(98\) 5.15544e22i 0.650378i
\(99\) −1.43114e22 −0.160648
\(100\) 0 0
\(101\) 1.36243e23 1.21512 0.607558 0.794275i \(-0.292149\pi\)
0.607558 + 0.794275i \(0.292149\pi\)
\(102\) 3.65366e23i 2.90956i
\(103\) 1.41401e22i 0.100653i 0.998733 + 0.0503264i \(0.0160262\pi\)
−0.998733 + 0.0503264i \(0.983974\pi\)
\(104\) 9.31907e21 0.0593594
\(105\) 0 0
\(106\) 3.06623e23 1.56887
\(107\) − 6.02971e22i − 0.276938i −0.990367 0.138469i \(-0.955782\pi\)
0.990367 0.138469i \(-0.0442182\pi\)
\(108\) − 1.12472e23i − 0.464164i
\(109\) 5.58169e22 0.207186 0.103593 0.994620i \(-0.466966\pi\)
0.103593 + 0.994620i \(0.466966\pi\)
\(110\) 0 0
\(111\) −4.81275e23 −1.44937
\(112\) 2.87329e23i 0.780498i
\(113\) 3.51523e23i 0.862089i 0.902331 + 0.431044i \(0.141855\pi\)
−0.902331 + 0.431044i \(0.858145\pi\)
\(114\) −9.72095e23 −2.15428
\(115\) 0 0
\(116\) −1.59435e23 −0.289279
\(117\) 2.03775e23i 0.334976i
\(118\) 4.09115e21i 0.00609821i
\(119\) −8.92770e23 −1.20767
\(120\) 0 0
\(121\) −8.31826e23 −0.928968
\(122\) 1.15405e24i 1.17243i
\(123\) 5.44111e23i 0.503244i
\(124\) −1.05574e24 −0.889627
\(125\) 0 0
\(126\) −8.68842e23 −0.609087
\(127\) − 2.32044e24i − 1.48535i −0.669653 0.742674i \(-0.733557\pi\)
0.669653 0.742674i \(-0.266443\pi\)
\(128\) − 3.64148e23i − 0.212992i
\(129\) −8.49891e22 −0.0454550
\(130\) 0 0
\(131\) 8.70825e23 0.390221 0.195111 0.980781i \(-0.437493\pi\)
0.195111 + 0.980781i \(0.437493\pi\)
\(132\) − 7.58496e23i − 0.311425i
\(133\) − 2.37531e24i − 0.894182i
\(134\) 5.93146e24 2.04860
\(135\) 0 0
\(136\) 6.07751e23 0.177022
\(137\) − 4.43869e24i − 1.18841i −0.804312 0.594207i \(-0.797466\pi\)
0.804312 0.594207i \(-0.202534\pi\)
\(138\) 5.59753e24i 1.37844i
\(139\) 4.97229e23 0.112690 0.0563452 0.998411i \(-0.482055\pi\)
0.0563452 + 0.998411i \(0.482055\pi\)
\(140\) 0 0
\(141\) −3.37106e24 −0.648257
\(142\) 3.06988e24i 0.544259i
\(143\) − 9.05647e23i − 0.148114i
\(144\) 4.27705e24 0.645617
\(145\) 0 0
\(146\) 1.40345e25 1.80775
\(147\) 4.98613e24i 0.593764i
\(148\) − 9.59276e24i − 1.05666i
\(149\) −1.10598e24 −0.112748 −0.0563738 0.998410i \(-0.517954\pi\)
−0.0563738 + 0.998410i \(0.517954\pi\)
\(150\) 0 0
\(151\) 3.76304e24 0.329082 0.164541 0.986370i \(-0.447386\pi\)
0.164541 + 0.986370i \(0.447386\pi\)
\(152\) 1.61698e24i 0.131070i
\(153\) 1.32894e25i 0.998972i
\(154\) 3.86143e24 0.269315
\(155\) 0 0
\(156\) −1.08000e25 −0.649368
\(157\) − 1.50090e25i − 0.838504i −0.907870 0.419252i \(-0.862292\pi\)
0.907870 0.419252i \(-0.137708\pi\)
\(158\) − 4.11072e25i − 2.13482i
\(159\) 2.96553e25 1.43230
\(160\) 0 0
\(161\) −1.36775e25 −0.572150
\(162\) − 4.41199e25i − 1.71875i
\(163\) − 2.61170e25i − 0.947907i −0.880550 0.473953i \(-0.842827\pi\)
0.880550 0.473953i \(-0.157173\pi\)
\(164\) −1.08452e25 −0.366888
\(165\) 0 0
\(166\) 3.09900e25 0.911967
\(167\) − 1.77408e25i − 0.487230i −0.969872 0.243615i \(-0.921667\pi\)
0.969872 0.243615i \(-0.0783334\pi\)
\(168\) 3.84290e24i 0.0985377i
\(169\) 2.88587e25 0.691161
\(170\) 0 0
\(171\) −3.53578e25 −0.739656
\(172\) − 1.69400e24i − 0.0331387i
\(173\) 1.04109e26i 1.90528i 0.304100 + 0.952640i \(0.401644\pi\)
−0.304100 + 0.952640i \(0.598356\pi\)
\(174\) −3.21267e25 −0.550237
\(175\) 0 0
\(176\) −1.90087e25 −0.285467
\(177\) 3.95679e23i 0.00556737i
\(178\) − 1.84157e25i − 0.242861i
\(179\) 1.00142e25 0.123824 0.0619122 0.998082i \(-0.480280\pi\)
0.0619122 + 0.998082i \(0.480280\pi\)
\(180\) 0 0
\(181\) −5.17169e25 −0.562768 −0.281384 0.959595i \(-0.590793\pi\)
−0.281384 + 0.959595i \(0.590793\pi\)
\(182\) − 5.49817e25i − 0.561562i
\(183\) 1.11615e26i 1.07037i
\(184\) 9.31094e24 0.0838665
\(185\) 0 0
\(186\) −2.12735e26 −1.69216
\(187\) − 5.90625e25i − 0.441706i
\(188\) − 6.71919e25i − 0.472609i
\(189\) 5.53780e25 0.366460
\(190\) 0 0
\(191\) 3.10126e26 1.81825 0.909127 0.416520i \(-0.136750\pi\)
0.909127 + 0.416520i \(0.136750\pi\)
\(192\) 1.92718e26i 1.06404i
\(193\) − 1.28183e26i − 0.666687i −0.942805 0.333344i \(-0.891823\pi\)
0.942805 0.333344i \(-0.108177\pi\)
\(194\) −4.56673e26 −2.23813
\(195\) 0 0
\(196\) −9.93833e25 −0.432881
\(197\) 3.89967e26i 1.60201i 0.598655 + 0.801007i \(0.295702\pi\)
−0.598655 + 0.801007i \(0.704298\pi\)
\(198\) − 5.74795e25i − 0.222773i
\(199\) 1.25611e26 0.459426 0.229713 0.973258i \(-0.426221\pi\)
0.229713 + 0.973258i \(0.426221\pi\)
\(200\) 0 0
\(201\) 5.73666e26 1.87027
\(202\) 5.47199e26i 1.68502i
\(203\) − 7.85013e25i − 0.228387i
\(204\) −7.04330e26 −1.93655
\(205\) 0 0
\(206\) −5.67918e25 −0.139577
\(207\) 2.03598e26i 0.473275i
\(208\) 2.70658e26i 0.595242i
\(209\) 1.57142e26 0.327047
\(210\) 0 0
\(211\) 5.88286e26 1.09734 0.548669 0.836040i \(-0.315135\pi\)
0.548669 + 0.836040i \(0.315135\pi\)
\(212\) 5.91088e26i 1.04421i
\(213\) 2.96906e26i 0.496883i
\(214\) 2.42174e26 0.384034
\(215\) 0 0
\(216\) −3.76984e25 −0.0537162
\(217\) − 5.19817e26i − 0.702365i
\(218\) 2.24180e26i 0.287308i
\(219\) 1.35735e27 1.65039
\(220\) 0 0
\(221\) −8.40974e26 −0.921025
\(222\) − 1.93297e27i − 2.00986i
\(223\) 1.38821e27i 1.37072i 0.728203 + 0.685361i \(0.240356\pi\)
−0.728203 + 0.685361i \(0.759644\pi\)
\(224\) −1.07102e27 −1.00449
\(225\) 0 0
\(226\) −1.41184e27 −1.19547
\(227\) 2.14958e27i 1.73004i 0.501734 + 0.865022i \(0.332696\pi\)
−0.501734 + 0.865022i \(0.667304\pi\)
\(228\) − 1.87394e27i − 1.43386i
\(229\) 6.39851e26 0.465554 0.232777 0.972530i \(-0.425219\pi\)
0.232777 + 0.972530i \(0.425219\pi\)
\(230\) 0 0
\(231\) 3.73461e26 0.245871
\(232\) 5.34395e25i 0.0334773i
\(233\) 2.19149e27i 1.30661i 0.757094 + 0.653306i \(0.226619\pi\)
−0.757094 + 0.653306i \(0.773381\pi\)
\(234\) −8.18434e26 −0.464516
\(235\) 0 0
\(236\) −7.88666e24 −0.00405887
\(237\) − 3.97572e27i − 1.94899i
\(238\) − 3.58568e27i − 1.67470i
\(239\) −1.09944e27 −0.489322 −0.244661 0.969609i \(-0.578677\pi\)
−0.244661 + 0.969609i \(0.578677\pi\)
\(240\) 0 0
\(241\) −1.44651e27 −0.584961 −0.292480 0.956272i \(-0.594481\pi\)
−0.292480 + 0.956272i \(0.594481\pi\)
\(242\) − 3.34090e27i − 1.28821i
\(243\) − 2.89951e27i − 1.06623i
\(244\) −2.22470e27 −0.780349
\(245\) 0 0
\(246\) −2.18534e27 −0.697855
\(247\) − 2.23750e27i − 0.681942i
\(248\) 3.53864e26i 0.102954i
\(249\) 2.99723e27 0.832582
\(250\) 0 0
\(251\) 2.24453e26 0.0568693 0.0284346 0.999596i \(-0.490948\pi\)
0.0284346 + 0.999596i \(0.490948\pi\)
\(252\) − 1.67490e27i − 0.405399i
\(253\) − 9.04857e26i − 0.209264i
\(254\) 9.31972e27 2.05975
\(255\) 0 0
\(256\) 5.62436e27 1.13583
\(257\) 3.95005e27i 0.762732i 0.924424 + 0.381366i \(0.124546\pi\)
−0.924424 + 0.381366i \(0.875454\pi\)
\(258\) − 3.41346e26i − 0.0630330i
\(259\) 4.72320e27 0.834236
\(260\) 0 0
\(261\) −1.16853e27 −0.188919
\(262\) 3.49754e27i 0.541125i
\(263\) 2.01521e27i 0.298420i 0.988805 + 0.149210i \(0.0476731\pi\)
−0.988805 + 0.149210i \(0.952327\pi\)
\(264\) −2.54233e26 −0.0360401
\(265\) 0 0
\(266\) 9.54007e27 1.23997
\(267\) − 1.78109e27i − 0.221721i
\(268\) 1.14343e28i 1.36351i
\(269\) −5.72063e27 −0.653571 −0.326785 0.945099i \(-0.605965\pi\)
−0.326785 + 0.945099i \(0.605965\pi\)
\(270\) 0 0
\(271\) 5.18050e27 0.543531 0.271766 0.962363i \(-0.412392\pi\)
0.271766 + 0.962363i \(0.412392\pi\)
\(272\) 1.76512e28i 1.77514i
\(273\) − 5.31761e27i − 0.512679i
\(274\) 1.78273e28 1.64799
\(275\) 0 0
\(276\) −1.07906e28 −0.917466
\(277\) − 1.29611e28i − 1.05712i −0.848896 0.528560i \(-0.822732\pi\)
0.848896 0.528560i \(-0.177268\pi\)
\(278\) 1.99705e27i 0.156269i
\(279\) −7.73776e27 −0.580987
\(280\) 0 0
\(281\) 1.24154e28 0.858691 0.429346 0.903140i \(-0.358744\pi\)
0.429346 + 0.903140i \(0.358744\pi\)
\(282\) − 1.35394e28i − 0.898947i
\(283\) − 2.15031e28i − 1.37074i −0.728193 0.685372i \(-0.759640\pi\)
0.728193 0.685372i \(-0.240360\pi\)
\(284\) −5.91792e27 −0.362250
\(285\) 0 0
\(286\) 3.63740e27 0.205391
\(287\) − 5.33986e27i − 0.289660i
\(288\) 1.59428e28i 0.830903i
\(289\) −3.48772e28 −1.74669
\(290\) 0 0
\(291\) −4.41675e28 −2.04331
\(292\) 2.70547e28i 1.20321i
\(293\) − 1.20307e28i − 0.514416i −0.966356 0.257208i \(-0.917197\pi\)
0.966356 0.257208i \(-0.0828026\pi\)
\(294\) −2.00260e28 −0.823380
\(295\) 0 0
\(296\) −3.21530e27 −0.122283
\(297\) 3.66362e27i 0.134033i
\(298\) − 4.44202e27i − 0.156349i
\(299\) −1.28840e28 −0.436347
\(300\) 0 0
\(301\) 8.34076e26 0.0261632
\(302\) 1.51137e28i 0.456342i
\(303\) 5.29228e28i 1.53834i
\(304\) −4.69629e28 −1.31434
\(305\) 0 0
\(306\) −5.33748e28 −1.38529
\(307\) 4.70428e28i 1.17598i 0.808867 + 0.587992i \(0.200081\pi\)
−0.808867 + 0.587992i \(0.799919\pi\)
\(308\) 7.44382e27i 0.179251i
\(309\) −5.49267e27 −0.127427
\(310\) 0 0
\(311\) −8.99672e26 −0.0193794 −0.00968968 0.999953i \(-0.503084\pi\)
−0.00968968 + 0.999953i \(0.503084\pi\)
\(312\) 3.61994e27i 0.0751492i
\(313\) − 1.13525e28i − 0.227159i −0.993529 0.113579i \(-0.963768\pi\)
0.993529 0.113579i \(-0.0362317\pi\)
\(314\) 6.02814e28 1.16277
\(315\) 0 0
\(316\) 7.92439e28 1.42090
\(317\) − 1.37896e28i − 0.238436i −0.992868 0.119218i \(-0.961961\pi\)
0.992868 0.119218i \(-0.0380387\pi\)
\(318\) 1.19106e29i 1.98619i
\(319\) 5.19337e27 0.0835326
\(320\) 0 0
\(321\) 2.34221e28 0.350605
\(322\) − 5.49338e28i − 0.793408i
\(323\) − 1.45920e29i − 2.03370i
\(324\) 8.50516e28 1.14397
\(325\) 0 0
\(326\) 1.04895e29 1.31448
\(327\) 2.16818e28i 0.262298i
\(328\) 3.63510e27i 0.0424587i
\(329\) 3.30834e28 0.373127
\(330\) 0 0
\(331\) 1.62668e29 1.71112 0.855559 0.517706i \(-0.173214\pi\)
0.855559 + 0.517706i \(0.173214\pi\)
\(332\) 5.97406e28i 0.606990i
\(333\) − 7.03074e28i − 0.690069i
\(334\) 7.12534e28 0.675649
\(335\) 0 0
\(336\) −1.11611e29 −0.988113
\(337\) 1.64138e29i 1.40432i 0.712019 + 0.702160i \(0.247781\pi\)
−0.712019 + 0.702160i \(0.752219\pi\)
\(338\) 1.15907e29i 0.958442i
\(339\) −1.36547e29 −1.09141
\(340\) 0 0
\(341\) 3.43892e28 0.256890
\(342\) − 1.42009e29i − 1.02569i
\(343\) − 1.53268e29i − 1.07045i
\(344\) −5.67795e26 −0.00383504
\(345\) 0 0
\(346\) −4.18139e29 −2.64208
\(347\) − 1.45586e29i − 0.889876i −0.895561 0.444938i \(-0.853226\pi\)
0.895561 0.444938i \(-0.146774\pi\)
\(348\) − 6.19318e28i − 0.366229i
\(349\) −2.22110e28 −0.127080 −0.0635398 0.997979i \(-0.520239\pi\)
−0.0635398 + 0.997979i \(0.520239\pi\)
\(350\) 0 0
\(351\) 5.21651e28 0.279479
\(352\) − 7.08552e28i − 0.367393i
\(353\) 2.57166e28i 0.129064i 0.997916 + 0.0645320i \(0.0205555\pi\)
−0.997916 + 0.0645320i \(0.979445\pi\)
\(354\) −1.58919e27 −0.00772035
\(355\) 0 0
\(356\) 3.55006e28 0.161644
\(357\) − 3.46792e29i − 1.52892i
\(358\) 4.02205e28i 0.171709i
\(359\) 3.34157e29 1.38154 0.690771 0.723074i \(-0.257271\pi\)
0.690771 + 0.723074i \(0.257271\pi\)
\(360\) 0 0
\(361\) 1.30406e29 0.505785
\(362\) − 2.07713e29i − 0.780398i
\(363\) − 3.23118e29i − 1.17608i
\(364\) 1.05990e29 0.373766
\(365\) 0 0
\(366\) −4.48284e29 −1.48430
\(367\) − 1.62664e29i − 0.521954i −0.965345 0.260977i \(-0.915955\pi\)
0.965345 0.260977i \(-0.0840447\pi\)
\(368\) 2.70422e29i 0.840994i
\(369\) −7.94868e28 −0.239603
\(370\) 0 0
\(371\) −2.91034e29 −0.824412
\(372\) − 4.10097e29i − 1.12627i
\(373\) 1.03540e29i 0.275712i 0.990452 + 0.137856i \(0.0440211\pi\)
−0.990452 + 0.137856i \(0.955979\pi\)
\(374\) 2.37216e29 0.612520
\(375\) 0 0
\(376\) −2.25214e28 −0.0546935
\(377\) − 7.39468e28i − 0.174178i
\(378\) 2.22418e29i 0.508175i
\(379\) −6.62210e29 −1.46772 −0.733862 0.679298i \(-0.762284\pi\)
−0.733862 + 0.679298i \(0.762284\pi\)
\(380\) 0 0
\(381\) 9.01364e29 1.88046
\(382\) 1.24557e30i 2.52140i
\(383\) 2.96585e29i 0.582591i 0.956633 + 0.291295i \(0.0940862\pi\)
−0.956633 + 0.291295i \(0.905914\pi\)
\(384\) 1.41451e29 0.269648
\(385\) 0 0
\(386\) 5.14829e29 0.924504
\(387\) − 1.24157e28i − 0.0216419i
\(388\) − 8.80345e29i − 1.48966i
\(389\) 1.06609e30 1.75134 0.875672 0.482906i \(-0.160419\pi\)
0.875672 + 0.482906i \(0.160419\pi\)
\(390\) 0 0
\(391\) −8.40240e29 −1.30128
\(392\) 3.33113e28i 0.0500959i
\(393\) 3.38267e29i 0.494021i
\(394\) −1.56624e30 −2.22153
\(395\) 0 0
\(396\) 1.10805e29 0.148274
\(397\) − 1.67462e28i − 0.0217683i −0.999941 0.0108842i \(-0.996535\pi\)
0.999941 0.0108842i \(-0.00346461\pi\)
\(398\) 5.04496e29i 0.637093i
\(399\) 9.22676e29 1.13204
\(400\) 0 0
\(401\) 3.87121e29 0.448421 0.224211 0.974541i \(-0.428020\pi\)
0.224211 + 0.974541i \(0.428020\pi\)
\(402\) 2.30404e30i 2.59353i
\(403\) − 4.89658e29i − 0.535654i
\(404\) −1.05486e30 −1.12152
\(405\) 0 0
\(406\) 3.15289e29 0.316708
\(407\) 3.12470e29i 0.305121i
\(408\) 2.36078e29i 0.224111i
\(409\) −5.00511e29 −0.461950 −0.230975 0.972960i \(-0.574192\pi\)
−0.230975 + 0.972960i \(0.574192\pi\)
\(410\) 0 0
\(411\) 1.72418e30 1.50454
\(412\) − 1.09480e29i − 0.0928999i
\(413\) − 3.88316e27i − 0.00320450i
\(414\) −8.17719e29 −0.656297
\(415\) 0 0
\(416\) −1.00889e30 −0.766070
\(417\) 1.93146e29i 0.142666i
\(418\) 6.31137e29i 0.453520i
\(419\) −1.70760e30 −1.19378 −0.596891 0.802322i \(-0.703598\pi\)
−0.596891 + 0.802322i \(0.703598\pi\)
\(420\) 0 0
\(421\) −5.97218e29 −0.395265 −0.197633 0.980276i \(-0.563325\pi\)
−0.197633 + 0.980276i \(0.563325\pi\)
\(422\) 2.36276e30i 1.52169i
\(423\) − 4.92464e29i − 0.308646i
\(424\) 1.98121e29 0.120843
\(425\) 0 0
\(426\) −1.19248e30 −0.689034
\(427\) − 1.09538e30i − 0.616089i
\(428\) 4.66848e29i 0.255607i
\(429\) 3.51794e29 0.187512
\(430\) 0 0
\(431\) 1.91854e30 0.969353 0.484676 0.874694i \(-0.338937\pi\)
0.484676 + 0.874694i \(0.338937\pi\)
\(432\) − 1.09490e30i − 0.538653i
\(433\) − 2.68456e30i − 1.28606i −0.765839 0.643032i \(-0.777676\pi\)
0.765839 0.643032i \(-0.222324\pi\)
\(434\) 2.08777e30 0.973980
\(435\) 0 0
\(436\) −4.32160e29 −0.191228
\(437\) − 2.23555e30i − 0.963489i
\(438\) 5.45161e30i 2.28862i
\(439\) −2.11168e29 −0.0863549 −0.0431775 0.999067i \(-0.513748\pi\)
−0.0431775 + 0.999067i \(0.513748\pi\)
\(440\) 0 0
\(441\) −7.28401e29 −0.282701
\(442\) − 3.37765e30i − 1.27720i
\(443\) − 8.12623e29i − 0.299396i −0.988732 0.149698i \(-0.952170\pi\)
0.988732 0.149698i \(-0.0478302\pi\)
\(444\) 3.72626e30 1.33773
\(445\) 0 0
\(446\) −5.57554e30 −1.90080
\(447\) − 4.29614e29i − 0.142739i
\(448\) − 1.89132e30i − 0.612446i
\(449\) 3.76596e30 1.18862 0.594310 0.804236i \(-0.297425\pi\)
0.594310 + 0.804236i \(0.297425\pi\)
\(450\) 0 0
\(451\) 3.53267e29 0.105943
\(452\) − 2.72166e30i − 0.795686i
\(453\) 1.46173e30i 0.416619i
\(454\) −8.63348e30 −2.39907
\(455\) 0 0
\(456\) −6.28109e29 −0.165936
\(457\) − 2.14090e30i − 0.551518i −0.961227 0.275759i \(-0.911071\pi\)
0.961227 0.275759i \(-0.0889291\pi\)
\(458\) 2.56986e30i 0.645591i
\(459\) 3.40199e30 0.833465
\(460\) 0 0
\(461\) 4.76379e30 1.11018 0.555088 0.831792i \(-0.312685\pi\)
0.555088 + 0.831792i \(0.312685\pi\)
\(462\) 1.49995e30i 0.340953i
\(463\) − 7.67574e30i − 1.70192i −0.525232 0.850959i \(-0.676021\pi\)
0.525232 0.850959i \(-0.323979\pi\)
\(464\) −1.55207e30 −0.335703
\(465\) 0 0
\(466\) −8.80180e30 −1.81190
\(467\) 2.87884e30i 0.578194i 0.957300 + 0.289097i \(0.0933551\pi\)
−0.957300 + 0.289097i \(0.906645\pi\)
\(468\) − 1.57772e30i − 0.309175i
\(469\) −5.62992e30 −1.07650
\(470\) 0 0
\(471\) 5.83016e30 1.06155
\(472\) 2.64345e27i 0 0.000469719i
\(473\) 5.51796e28i 0.00956919i
\(474\) 1.59679e31 2.70269
\(475\) 0 0
\(476\) 6.91225e30 1.11465
\(477\) 4.33221e30i 0.681943i
\(478\) − 4.41573e30i − 0.678549i
\(479\) −6.82143e30 −1.02333 −0.511666 0.859184i \(-0.670972\pi\)
−0.511666 + 0.859184i \(0.670972\pi\)
\(480\) 0 0
\(481\) 4.44917e30 0.636224
\(482\) − 5.80971e30i − 0.811173i
\(483\) − 5.31297e30i − 0.724344i
\(484\) 6.44038e30 0.857413
\(485\) 0 0
\(486\) 1.16455e31 1.47856
\(487\) − 8.59361e29i − 0.106560i −0.998580 0.0532798i \(-0.983032\pi\)
0.998580 0.0532798i \(-0.0169675\pi\)
\(488\) 7.45676e29i 0.0903071i
\(489\) 1.01450e31 1.20005
\(490\) 0 0
\(491\) −7.15140e30 −0.807149 −0.403575 0.914947i \(-0.632232\pi\)
−0.403575 + 0.914947i \(0.632232\pi\)
\(492\) − 4.21276e30i − 0.464481i
\(493\) − 4.82250e30i − 0.519437i
\(494\) 8.98658e30 0.945659
\(495\) 0 0
\(496\) −1.02774e31 −1.03239
\(497\) − 2.91382e30i − 0.285998i
\(498\) 1.20379e31i 1.15455i
\(499\) −1.77913e31 −1.66744 −0.833722 0.552185i \(-0.813794\pi\)
−0.833722 + 0.552185i \(0.813794\pi\)
\(500\) 0 0
\(501\) 6.89133e30 0.616835
\(502\) 9.01483e29i 0.0788614i
\(503\) 6.56155e30i 0.561015i 0.959852 + 0.280507i \(0.0905027\pi\)
−0.959852 + 0.280507i \(0.909497\pi\)
\(504\) −5.61393e29 −0.0469154
\(505\) 0 0
\(506\) 3.63422e30 0.290189
\(507\) 1.12100e31i 0.875012i
\(508\) 1.79660e31i 1.37094i
\(509\) 1.19826e31 0.893917 0.446959 0.894555i \(-0.352507\pi\)
0.446959 + 0.894555i \(0.352507\pi\)
\(510\) 0 0
\(511\) −1.33210e31 −0.949939
\(512\) 1.95347e31i 1.36208i
\(513\) 9.05135e30i 0.617112i
\(514\) −1.58648e31 −1.05769
\(515\) 0 0
\(516\) 6.58025e29 0.0419538
\(517\) 2.18868e30i 0.136471i
\(518\) 1.89700e31i 1.15685i
\(519\) −4.04407e31 −2.41209
\(520\) 0 0
\(521\) −2.26240e31 −1.29103 −0.645514 0.763749i \(-0.723357\pi\)
−0.645514 + 0.763749i \(0.723357\pi\)
\(522\) − 4.69325e30i − 0.261977i
\(523\) − 3.41914e31i − 1.86702i −0.358557 0.933508i \(-0.616731\pi\)
0.358557 0.933508i \(-0.383269\pi\)
\(524\) −6.74233e30 −0.360164
\(525\) 0 0
\(526\) −8.09378e30 −0.413824
\(527\) − 3.19334e31i − 1.59744i
\(528\) − 7.38381e30i − 0.361402i
\(529\) 8.00772e30 0.383503
\(530\) 0 0
\(531\) −5.78030e28 −0.00265072
\(532\) 1.83907e31i 0.825307i
\(533\) − 5.03006e30i − 0.220907i
\(534\) 7.15348e30 0.307463
\(535\) 0 0
\(536\) 3.83255e30 0.157795
\(537\) 3.88996e30i 0.156762i
\(538\) − 2.29761e31i − 0.906315i
\(539\) 3.23726e30 0.124999
\(540\) 0 0
\(541\) −2.75024e30 −0.101766 −0.0508829 0.998705i \(-0.516204\pi\)
−0.0508829 + 0.998705i \(0.516204\pi\)
\(542\) 2.08067e31i 0.753722i
\(543\) − 2.00892e31i − 0.712466i
\(544\) −6.57953e31 −2.28459
\(545\) 0 0
\(546\) 2.13574e31 0.710939
\(547\) 1.03197e31i 0.336367i 0.985756 + 0.168184i \(0.0537901\pi\)
−0.985756 + 0.168184i \(0.946210\pi\)
\(548\) 3.43664e31i 1.09688i
\(549\) −1.63053e31 −0.509621
\(550\) 0 0
\(551\) 1.28308e31 0.384600
\(552\) 3.61679e30i 0.106175i
\(553\) 3.90174e31i 1.12181i
\(554\) 5.20563e31 1.46592
\(555\) 0 0
\(556\) −3.84978e30 −0.104010
\(557\) − 1.36262e31i − 0.360613i −0.983610 0.180307i \(-0.942291\pi\)
0.983610 0.180307i \(-0.0577090\pi\)
\(558\) − 3.10775e31i − 0.805663i
\(559\) 7.85685e29 0.0199532
\(560\) 0 0
\(561\) 2.29425e31 0.559202
\(562\) 4.98645e31i 1.19076i
\(563\) 3.09091e31i 0.723169i 0.932339 + 0.361585i \(0.117764\pi\)
−0.932339 + 0.361585i \(0.882236\pi\)
\(564\) 2.61004e31 0.598325
\(565\) 0 0
\(566\) 8.63638e31 1.90083
\(567\) 4.18769e31i 0.903171i
\(568\) 1.98357e30i 0.0419220i
\(569\) 6.93051e31 1.43540 0.717702 0.696350i \(-0.245194\pi\)
0.717702 + 0.696350i \(0.245194\pi\)
\(570\) 0 0
\(571\) −6.24961e30 −0.124319 −0.0621595 0.998066i \(-0.519799\pi\)
−0.0621595 + 0.998066i \(0.519799\pi\)
\(572\) 7.01195e30i 0.136705i
\(573\) 1.20467e32i 2.30192i
\(574\) 2.14468e31 0.401675
\(575\) 0 0
\(576\) −2.81534e31 −0.506607
\(577\) 1.59740e31i 0.281768i 0.990026 + 0.140884i \(0.0449944\pi\)
−0.990026 + 0.140884i \(0.955006\pi\)
\(578\) − 1.40079e32i − 2.42216i
\(579\) 4.97921e31 0.844029
\(580\) 0 0
\(581\) −2.94146e31 −0.479222
\(582\) − 1.77392e32i − 2.83348i
\(583\) − 1.92538e31i − 0.301529i
\(584\) 9.06822e30 0.139243
\(585\) 0 0
\(586\) 4.83197e31 0.713348
\(587\) − 8.16635e31i − 1.18220i −0.806599 0.591099i \(-0.798694\pi\)
0.806599 0.591099i \(-0.201306\pi\)
\(588\) − 3.86049e31i − 0.548029i
\(589\) 8.49622e31 1.18277
\(590\) 0 0
\(591\) −1.51481e32 −2.02816
\(592\) − 9.33837e31i − 1.22623i
\(593\) − 5.61370e30i − 0.0722970i −0.999346 0.0361485i \(-0.988491\pi\)
0.999346 0.0361485i \(-0.0115089\pi\)
\(594\) −1.47144e31 −0.185865
\(595\) 0 0
\(596\) 8.56305e30 0.104063
\(597\) 4.87928e31i 0.581635i
\(598\) − 5.17466e31i − 0.605088i
\(599\) −1.60231e32 −1.83797 −0.918985 0.394293i \(-0.870990\pi\)
−0.918985 + 0.394293i \(0.870990\pi\)
\(600\) 0 0
\(601\) 8.47641e31 0.935741 0.467870 0.883797i \(-0.345021\pi\)
0.467870 + 0.883797i \(0.345021\pi\)
\(602\) 3.34994e30i 0.0362809i
\(603\) 8.38044e31i 0.890467i
\(604\) −2.91352e31 −0.303734
\(605\) 0 0
\(606\) −2.12557e32 −2.13324
\(607\) 1.63793e31i 0.161296i 0.996743 + 0.0806481i \(0.0256990\pi\)
−0.996743 + 0.0806481i \(0.974301\pi\)
\(608\) − 1.75055e32i − 1.69155i
\(609\) 3.04934e31 0.289139
\(610\) 0 0
\(611\) 3.11639e31 0.284563
\(612\) − 1.02893e32i − 0.922025i
\(613\) 1.93558e31i 0.170222i 0.996371 + 0.0851110i \(0.0271245\pi\)
−0.996371 + 0.0851110i \(0.972876\pi\)
\(614\) −1.88940e32 −1.63075
\(615\) 0 0
\(616\) 2.49502e30 0.0207442
\(617\) − 5.25618e31i − 0.428934i −0.976731 0.214467i \(-0.931199\pi\)
0.976731 0.214467i \(-0.0688015\pi\)
\(618\) − 2.20605e31i − 0.176704i
\(619\) 9.42009e31 0.740650 0.370325 0.928902i \(-0.379246\pi\)
0.370325 + 0.928902i \(0.379246\pi\)
\(620\) 0 0
\(621\) 5.21196e31 0.394864
\(622\) − 3.61340e30i − 0.0268736i
\(623\) 1.74795e31i 0.127619i
\(624\) −1.05136e32 −0.753578
\(625\) 0 0
\(626\) 4.55954e31 0.315004
\(627\) 6.10410e31i 0.414042i
\(628\) 1.16207e32i 0.773918i
\(629\) 2.90156e32 1.89736
\(630\) 0 0
\(631\) −2.22784e32 −1.40458 −0.702288 0.711892i \(-0.747838\pi\)
−0.702288 + 0.711892i \(0.747838\pi\)
\(632\) − 2.65610e31i − 0.164436i
\(633\) 2.28517e32i 1.38923i
\(634\) 5.53840e31 0.330642
\(635\) 0 0
\(636\) −2.29605e32 −1.32198
\(637\) − 4.60944e31i − 0.260642i
\(638\) 2.08584e31i 0.115836i
\(639\) −4.33738e31 −0.236574
\(640\) 0 0
\(641\) 1.38602e32 0.729295 0.364648 0.931146i \(-0.381189\pi\)
0.364648 + 0.931146i \(0.381189\pi\)
\(642\) 9.40714e31i 0.486188i
\(643\) 3.71401e31i 0.188546i 0.995546 + 0.0942729i \(0.0300526\pi\)
−0.995546 + 0.0942729i \(0.969947\pi\)
\(644\) 1.05898e32 0.528080
\(645\) 0 0
\(646\) 5.86067e32 2.82016
\(647\) 3.77554e32i 1.78476i 0.451286 + 0.892379i \(0.350965\pi\)
−0.451286 + 0.892379i \(0.649035\pi\)
\(648\) − 2.85076e31i − 0.132388i
\(649\) 2.56896e29 0.00117204
\(650\) 0 0
\(651\) 2.01920e32 0.889197
\(652\) 2.02210e32i 0.874893i
\(653\) − 8.58042e31i − 0.364760i −0.983228 0.182380i \(-0.941620\pi\)
0.983228 0.182380i \(-0.0583801\pi\)
\(654\) −8.70816e31 −0.363733
\(655\) 0 0
\(656\) −1.05576e32 −0.425766
\(657\) 1.98290e32i 0.785777i
\(658\) 1.32874e32i 0.517421i
\(659\) −1.46687e32 −0.561319 −0.280660 0.959807i \(-0.590553\pi\)
−0.280660 + 0.959807i \(0.590553\pi\)
\(660\) 0 0
\(661\) 2.02066e32 0.746752 0.373376 0.927680i \(-0.378200\pi\)
0.373376 + 0.927680i \(0.378200\pi\)
\(662\) 6.53331e32i 2.37283i
\(663\) − 3.26672e32i − 1.16602i
\(664\) 2.00239e31 0.0702450
\(665\) 0 0
\(666\) 2.82379e32 0.956927
\(667\) − 7.38823e31i − 0.246090i
\(668\) 1.37358e32i 0.449701i
\(669\) −5.39243e32 −1.73534
\(670\) 0 0
\(671\) 7.24664e31 0.225334
\(672\) − 4.16034e32i − 1.27169i
\(673\) 5.34692e32i 1.60668i 0.595519 + 0.803341i \(0.296947\pi\)
−0.595519 + 0.803341i \(0.703053\pi\)
\(674\) −6.59237e32 −1.94739
\(675\) 0 0
\(676\) −2.23437e32 −0.637924
\(677\) − 1.10049e32i − 0.308899i −0.988001 0.154449i \(-0.950640\pi\)
0.988001 0.154449i \(-0.0493604\pi\)
\(678\) − 5.48422e32i − 1.51347i
\(679\) 4.33457e32 1.17610
\(680\) 0 0
\(681\) −8.34995e32 −2.19024
\(682\) 1.38119e32i 0.356233i
\(683\) − 2.97297e32i − 0.753967i −0.926220 0.376983i \(-0.876961\pi\)
0.926220 0.376983i \(-0.123039\pi\)
\(684\) 2.73756e32 0.682683
\(685\) 0 0
\(686\) 6.15577e32 1.48441
\(687\) 2.48547e32i 0.589394i
\(688\) − 1.64908e31i − 0.0384568i
\(689\) −2.74149e32 −0.628733
\(690\) 0 0
\(691\) −7.18838e32 −1.59453 −0.797266 0.603628i \(-0.793721\pi\)
−0.797266 + 0.603628i \(0.793721\pi\)
\(692\) − 8.06062e32i − 1.75852i
\(693\) 5.45574e31i 0.117063i
\(694\) 5.84723e32 1.23400
\(695\) 0 0
\(696\) −2.07583e31 −0.0423824
\(697\) − 3.28039e32i − 0.658793i
\(698\) − 8.92072e31i − 0.176223i
\(699\) −8.51274e32 −1.65418
\(700\) 0 0
\(701\) −1.02832e33 −1.93362 −0.966810 0.255497i \(-0.917761\pi\)
−0.966810 + 0.255497i \(0.917761\pi\)
\(702\) 2.09513e32i 0.387557i
\(703\) 7.71990e32i 1.40484i
\(704\) 1.25123e32 0.224002
\(705\) 0 0
\(706\) −1.03287e32 −0.178975
\(707\) − 5.19380e32i − 0.885446i
\(708\) − 3.06353e30i − 0.00513854i
\(709\) 2.50942e32 0.414134 0.207067 0.978327i \(-0.433608\pi\)
0.207067 + 0.978327i \(0.433608\pi\)
\(710\) 0 0
\(711\) 5.80796e32 0.927947
\(712\) − 1.18991e31i − 0.0187066i
\(713\) − 4.89231e32i − 0.756805i
\(714\) 1.39284e33 2.12017
\(715\) 0 0
\(716\) −7.75346e31 −0.114287
\(717\) − 4.27071e32i − 0.619483i
\(718\) 1.34209e33i 1.91580i
\(719\) −7.09247e32 −0.996358 −0.498179 0.867074i \(-0.665998\pi\)
−0.498179 + 0.867074i \(0.665998\pi\)
\(720\) 0 0
\(721\) 5.39046e31 0.0733450
\(722\) 5.23757e32i 0.701378i
\(723\) − 5.61891e32i − 0.740562i
\(724\) 4.00417e32 0.519420
\(725\) 0 0
\(726\) 1.29776e33 1.63088
\(727\) − 4.63712e32i − 0.573591i −0.957992 0.286795i \(-0.907410\pi\)
0.957992 0.286795i \(-0.0925900\pi\)
\(728\) − 3.55259e31i − 0.0432547i
\(729\) 9.21300e31 0.110417
\(730\) 0 0
\(731\) 5.12391e31 0.0595047
\(732\) − 8.64174e32i − 0.987924i
\(733\) − 1.54780e33i − 1.74188i −0.491387 0.870941i \(-0.663510\pi\)
0.491387 0.870941i \(-0.336490\pi\)
\(734\) 6.53317e32 0.723801
\(735\) 0 0
\(736\) −1.00801e33 −1.08235
\(737\) − 3.72455e32i − 0.393730i
\(738\) − 3.19247e32i − 0.332260i
\(739\) 1.49249e33 1.52932 0.764662 0.644431i \(-0.222906\pi\)
0.764662 + 0.644431i \(0.222906\pi\)
\(740\) 0 0
\(741\) 8.69144e32 0.863341
\(742\) − 1.16890e33i − 1.14322i
\(743\) 1.19934e32i 0.115497i 0.998331 + 0.0577487i \(0.0183922\pi\)
−0.998331 + 0.0577487i \(0.981608\pi\)
\(744\) −1.37456e32 −0.130340
\(745\) 0 0
\(746\) −4.15852e32 −0.382334
\(747\) 4.37852e32i 0.396406i
\(748\) 4.57290e32i 0.407683i
\(749\) −2.29863e32 −0.201803
\(750\) 0 0
\(751\) 1.42812e33 1.21592 0.607959 0.793969i \(-0.291989\pi\)
0.607959 + 0.793969i \(0.291989\pi\)
\(752\) − 6.54101e32i − 0.548453i
\(753\) 8.71877e31i 0.0719967i
\(754\) 2.96996e32 0.241535
\(755\) 0 0
\(756\) −4.28763e32 −0.338233
\(757\) 1.56208e33i 1.21367i 0.794828 + 0.606835i \(0.207561\pi\)
−0.794828 + 0.606835i \(0.792439\pi\)
\(758\) − 2.65967e33i − 2.03531i
\(759\) 3.51487e32 0.264929
\(760\) 0 0
\(761\) 1.92115e33 1.40488 0.702439 0.711744i \(-0.252094\pi\)
0.702439 + 0.711744i \(0.252094\pi\)
\(762\) 3.62020e33i 2.60765i
\(763\) − 2.12783e32i − 0.150975i
\(764\) −2.40114e33 −1.67820
\(765\) 0 0
\(766\) −1.19119e33 −0.807886
\(767\) − 3.65787e30i − 0.00244389i
\(768\) 2.18475e33i 1.43797i
\(769\) −1.22615e33 −0.795044 −0.397522 0.917593i \(-0.630130\pi\)
−0.397522 + 0.917593i \(0.630130\pi\)
\(770\) 0 0
\(771\) −1.53438e33 −0.965621
\(772\) 9.92455e32i 0.615335i
\(773\) − 8.43914e32i − 0.515507i −0.966211 0.257753i \(-0.917018\pi\)
0.966211 0.257753i \(-0.0829822\pi\)
\(774\) 4.98658e31 0.0300111
\(775\) 0 0
\(776\) −2.95074e32 −0.172394
\(777\) 1.83470e33i 1.05615i
\(778\) 4.28177e33i 2.42861i
\(779\) 8.72782e32 0.487781
\(780\) 0 0
\(781\) 1.92768e32 0.104604
\(782\) − 3.37470e33i − 1.80450i
\(783\) 2.99137e32i 0.157620i
\(784\) −9.67477e32 −0.502350
\(785\) 0 0
\(786\) −1.35860e33 −0.685066
\(787\) − 1.37027e33i − 0.680919i −0.940259 0.340460i \(-0.889417\pi\)
0.940259 0.340460i \(-0.110583\pi\)
\(788\) − 3.01931e33i − 1.47862i
\(789\) −7.82797e32 −0.377801
\(790\) 0 0
\(791\) 1.34007e33 0.628198
\(792\) − 3.71398e31i − 0.0171593i
\(793\) − 1.03183e33i − 0.469857i
\(794\) 6.72586e31 0.0301865
\(795\) 0 0
\(796\) −9.72536e32 −0.424039
\(797\) − 1.57462e33i − 0.676714i −0.941018 0.338357i \(-0.890129\pi\)
0.941018 0.338357i \(-0.109871\pi\)
\(798\) 3.70579e33i 1.56981i
\(799\) 2.03238e33 0.848628
\(800\) 0 0
\(801\) 2.60192e32 0.105565
\(802\) 1.55481e33i 0.621832i
\(803\) − 8.81269e32i − 0.347440i
\(804\) −4.44159e33 −1.72621
\(805\) 0 0
\(806\) 1.96664e33 0.742799
\(807\) − 2.22215e33i − 0.827423i
\(808\) 3.53567e32i 0.129790i
\(809\) 2.56942e33 0.929882 0.464941 0.885342i \(-0.346076\pi\)
0.464941 + 0.885342i \(0.346076\pi\)
\(810\) 0 0
\(811\) 1.25770e33 0.442423 0.221212 0.975226i \(-0.428999\pi\)
0.221212 + 0.975226i \(0.428999\pi\)
\(812\) 6.07794e32i 0.210796i
\(813\) 2.01234e33i 0.688112i
\(814\) −1.25499e33 −0.423116
\(815\) 0 0
\(816\) −6.85652e33 −2.24733
\(817\) 1.36327e32i 0.0440583i
\(818\) − 2.01023e33i − 0.640593i
\(819\) 7.76826e32 0.244095
\(820\) 0 0
\(821\) −2.38117e32 −0.0727519 −0.0363759 0.999338i \(-0.511581\pi\)
−0.0363759 + 0.999338i \(0.511581\pi\)
\(822\) 6.92493e33i 2.08636i
\(823\) 5.82453e33i 1.73047i 0.501370 + 0.865233i \(0.332830\pi\)
−0.501370 + 0.865233i \(0.667170\pi\)
\(824\) −3.66954e31 −0.0107510
\(825\) 0 0
\(826\) 1.55961e31 0.00444372
\(827\) 3.61128e33i 1.01472i 0.861733 + 0.507362i \(0.169379\pi\)
−0.861733 + 0.507362i \(0.830621\pi\)
\(828\) − 1.57635e33i − 0.436821i
\(829\) −6.27935e33 −1.71608 −0.858040 0.513584i \(-0.828318\pi\)
−0.858040 + 0.513584i \(0.828318\pi\)
\(830\) 0 0
\(831\) 5.03467e33 1.33832
\(832\) − 1.78159e33i − 0.467078i
\(833\) − 3.00609e33i − 0.777292i
\(834\) −7.75743e32 −0.197837
\(835\) 0 0
\(836\) −1.21667e33 −0.301856
\(837\) 1.98081e33i 0.484731i
\(838\) − 6.85832e33i − 1.65543i
\(839\) −7.51696e32 −0.178970 −0.0894849 0.995988i \(-0.528522\pi\)
−0.0894849 + 0.995988i \(0.528522\pi\)
\(840\) 0 0
\(841\) −3.89268e33 −0.901767
\(842\) − 2.39864e33i − 0.548120i
\(843\) 4.82269e33i 1.08711i
\(844\) −4.55478e33 −1.01281
\(845\) 0 0
\(846\) 1.97791e33 0.428003
\(847\) 3.17106e33i 0.676932i
\(848\) 5.75412e33i 1.21179i
\(849\) 8.35275e33 1.73537
\(850\) 0 0
\(851\) 4.44528e33 0.898896
\(852\) − 2.29879e33i − 0.458610i
\(853\) − 9.11290e33i − 1.79367i −0.442364 0.896836i \(-0.645860\pi\)
0.442364 0.896836i \(-0.354140\pi\)
\(854\) 4.39943e33 0.854339
\(855\) 0 0
\(856\) 1.56478e32 0.0295805
\(857\) − 6.17423e33i − 1.15160i −0.817589 0.575802i \(-0.804690\pi\)
0.817589 0.575802i \(-0.195310\pi\)
\(858\) 1.41293e33i 0.260026i
\(859\) 1.86175e33 0.338065 0.169033 0.985610i \(-0.445936\pi\)
0.169033 + 0.985610i \(0.445936\pi\)
\(860\) 0 0
\(861\) 2.07424e33 0.366710
\(862\) 7.70552e33i 1.34421i
\(863\) − 5.22144e33i − 0.898806i −0.893329 0.449403i \(-0.851637\pi\)
0.893329 0.449403i \(-0.148363\pi\)
\(864\) 4.08125e33 0.693242
\(865\) 0 0
\(866\) 1.07821e34 1.78340
\(867\) − 1.35479e34i − 2.21132i
\(868\) 4.02466e33i 0.648265i
\(869\) −2.58126e33 −0.410302
\(870\) 0 0
\(871\) −5.30328e33 −0.820986
\(872\) 1.44852e32i 0.0221301i
\(873\) − 6.45224e33i − 0.972852i
\(874\) 8.97874e33 1.33608
\(875\) 0 0
\(876\) −1.05093e34 −1.52327
\(877\) 6.73800e32i 0.0963908i 0.998838 + 0.0481954i \(0.0153470\pi\)
−0.998838 + 0.0481954i \(0.984653\pi\)
\(878\) − 8.48127e32i − 0.119750i
\(879\) 4.67328e33 0.651253
\(880\) 0 0
\(881\) 1.36978e34 1.85963 0.929815 0.368028i \(-0.119967\pi\)
0.929815 + 0.368028i \(0.119967\pi\)
\(882\) − 2.92552e33i − 0.392025i
\(883\) 1.49590e33i 0.197859i 0.995094 + 0.0989294i \(0.0315418\pi\)
−0.995094 + 0.0989294i \(0.968458\pi\)
\(884\) 6.51121e33 0.850082
\(885\) 0 0
\(886\) 3.26378e33 0.415177
\(887\) 6.44904e33i 0.809792i 0.914363 + 0.404896i \(0.132692\pi\)
−0.914363 + 0.404896i \(0.867308\pi\)
\(888\) − 1.24897e33i − 0.154811i
\(889\) −8.84592e33 −1.08236
\(890\) 0 0
\(891\) −2.77043e33 −0.330334
\(892\) − 1.07482e34i − 1.26514i
\(893\) 5.40736e33i 0.628339i
\(894\) 1.72548e33 0.197938
\(895\) 0 0
\(896\) −1.38819e33 −0.155206
\(897\) − 5.00472e33i − 0.552417i
\(898\) 1.51254e34i 1.64827i
\(899\) 2.80791e33 0.302097
\(900\) 0 0
\(901\) −1.78789e34 −1.87502
\(902\) 1.41884e33i 0.146913i
\(903\) 3.23993e32i 0.0331227i
\(904\) −9.12246e32 −0.0920820
\(905\) 0 0
\(906\) −5.87083e33 −0.577731
\(907\) − 6.42282e33i − 0.624083i −0.950069 0.312041i \(-0.898987\pi\)
0.950069 0.312041i \(-0.101013\pi\)
\(908\) − 1.66431e34i − 1.59679i
\(909\) −7.73126e33 −0.732430
\(910\) 0 0
\(911\) −1.48932e34 −1.37571 −0.687855 0.725848i \(-0.741447\pi\)
−0.687855 + 0.725848i \(0.741447\pi\)
\(912\) − 1.82425e34i − 1.66396i
\(913\) − 1.94596e33i − 0.175275i
\(914\) 8.59859e33 0.764797
\(915\) 0 0
\(916\) −4.95402e33 −0.429695
\(917\) − 3.31973e33i − 0.284351i
\(918\) 1.36636e34i 1.15578i
\(919\) −3.10701e33 −0.259546 −0.129773 0.991544i \(-0.541425\pi\)
−0.129773 + 0.991544i \(0.541425\pi\)
\(920\) 0 0
\(921\) −1.82735e34 −1.48880
\(922\) 1.91330e34i 1.53950i
\(923\) − 2.74476e33i − 0.218115i
\(924\) −2.89151e33 −0.226933
\(925\) 0 0
\(926\) 3.08285e34 2.36007
\(927\) − 8.02400e32i − 0.0606700i
\(928\) − 5.78538e33i − 0.432046i
\(929\) −1.72846e34 −1.27490 −0.637452 0.770490i \(-0.720012\pi\)
−0.637452 + 0.770490i \(0.720012\pi\)
\(930\) 0 0
\(931\) 7.99801e33 0.575520
\(932\) − 1.69676e34i − 1.20597i
\(933\) − 3.49473e32i − 0.0245343i
\(934\) −1.15624e34 −0.801790
\(935\) 0 0
\(936\) −5.28822e32 −0.0357797
\(937\) 1.61548e34i 1.07969i 0.841766 + 0.539843i \(0.181516\pi\)
−0.841766 + 0.539843i \(0.818484\pi\)
\(938\) − 2.26117e34i − 1.49280i
\(939\) 4.40980e33 0.287584
\(940\) 0 0
\(941\) 7.69275e33 0.489554 0.244777 0.969579i \(-0.421285\pi\)
0.244777 + 0.969579i \(0.421285\pi\)
\(942\) 2.34160e34i 1.47206i
\(943\) − 5.02567e33i − 0.312111i
\(944\) −7.67751e31 −0.00471023
\(945\) 0 0
\(946\) −2.21620e32 −0.0132697
\(947\) 2.94332e34i 1.74106i 0.492119 + 0.870528i \(0.336222\pi\)
−0.492119 + 0.870528i \(0.663778\pi\)
\(948\) 3.07819e34i 1.79887i
\(949\) −1.25481e34 −0.724465
\(950\) 0 0
\(951\) 5.35651e33 0.301860
\(952\) − 2.31685e33i − 0.128995i
\(953\) 1.97292e34i 1.08528i 0.839965 + 0.542641i \(0.182575\pi\)
−0.839965 + 0.542641i \(0.817425\pi\)
\(954\) −1.73997e34 −0.945659
\(955\) 0 0
\(956\) 8.51236e33 0.451632
\(957\) 2.01734e33i 0.105753i
\(958\) − 2.73973e34i − 1.41907i
\(959\) −1.69210e34 −0.865989
\(960\) 0 0
\(961\) −1.42003e33 −0.0709542
\(962\) 1.78694e34i 0.882261i
\(963\) 3.42163e33i 0.166929i
\(964\) 1.11996e34 0.539904
\(965\) 0 0
\(966\) 2.13387e34 1.00446
\(967\) 2.03266e34i 0.945496i 0.881198 + 0.472748i \(0.156738\pi\)
−0.881198 + 0.472748i \(0.843262\pi\)
\(968\) − 2.15869e33i − 0.0992255i
\(969\) 5.66820e34 2.57467
\(970\) 0 0
\(971\) 3.40470e34 1.51028 0.755141 0.655563i \(-0.227569\pi\)
0.755141 + 0.655563i \(0.227569\pi\)
\(972\) 2.24494e34i 0.984106i
\(973\) − 1.89552e33i − 0.0821166i
\(974\) 3.45150e33 0.147768
\(975\) 0 0
\(976\) −2.16570e34 −0.905579
\(977\) 3.68167e32i 0.0152145i 0.999971 + 0.00760725i \(0.00242149\pi\)
−0.999971 + 0.00760725i \(0.997579\pi\)
\(978\) 4.07459e34i 1.66413i
\(979\) −1.15638e33 −0.0466767
\(980\) 0 0
\(981\) −3.16740e33 −0.124885
\(982\) − 2.87225e34i − 1.11928i
\(983\) − 8.23724e33i − 0.317261i −0.987338 0.158630i \(-0.949292\pi\)
0.987338 0.158630i \(-0.0507078\pi\)
\(984\) −1.41203e33 −0.0537529
\(985\) 0 0
\(986\) 1.93689e34 0.720310
\(987\) 1.28511e34i 0.472380i
\(988\) 1.73238e34i 0.629415i
\(989\) 7.85000e32 0.0281911
\(990\) 0 0
\(991\) 2.70349e33 0.0948590 0.0474295 0.998875i \(-0.484897\pi\)
0.0474295 + 0.998875i \(0.484897\pi\)
\(992\) − 3.83094e34i − 1.32868i
\(993\) 6.31875e34i 2.16628i
\(994\) 1.17029e34 0.396598
\(995\) 0 0
\(996\) −2.32059e34 −0.768452
\(997\) 4.64692e33i 0.152115i 0.997103 + 0.0760573i \(0.0242332\pi\)
−0.997103 + 0.0760573i \(0.975767\pi\)
\(998\) − 7.14561e34i − 2.31227i
\(999\) −1.79982e34 −0.575740
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 25.24.b.a.24.3 4
5.2 odd 4 1.24.a.a.1.1 2
5.3 odd 4 25.24.a.a.1.2 2
5.4 even 2 inner 25.24.b.a.24.2 4
15.2 even 4 9.24.a.b.1.2 2
20.7 even 4 16.24.a.b.1.1 2
35.27 even 4 49.24.a.b.1.1 2
40.27 even 4 64.24.a.g.1.2 2
40.37 odd 4 64.24.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.24.a.a.1.1 2 5.2 odd 4
9.24.a.b.1.2 2 15.2 even 4
16.24.a.b.1.1 2 20.7 even 4
25.24.a.a.1.2 2 5.3 odd 4
25.24.b.a.24.2 4 5.4 even 2 inner
25.24.b.a.24.3 4 1.1 even 1 trivial
49.24.a.b.1.1 2 35.27 even 4
64.24.a.d.1.1 2 40.37 odd 4
64.24.a.g.1.2 2 40.27 even 4