Properties

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

$q$-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 \(-0.756131\pi\)
0.720595 0.693357i \(-0.243869\pi\)
\(3\) − 388445.i − 1.26600i −0.774150 0.633002i \(-0.781823\pi\)
0.774150 0.633002i \(-0.218177\pi\)
\(4\) −7.74247e6 −0.922974
\(5\) 0 0
\(6\) −1.56013e9 −1.75558
\(7\) 3.81217e9i 0.728693i 0.931263 + 0.364346i \(0.118708\pi\)
−0.931263 + 0.364346i \(0.881292\pi\)
\(8\) − 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 \(0.0896273\pi\)
−0.960620 + 0.277867i \(0.910373\pi\)
\(14\) 1.53110e13 1.01049
\(15\) 0 0
\(16\) −7.53715e13 −1.07109
\(17\) 2.34190e14i 1.65731i 0.559756 + 0.828657i \(0.310895\pi\)
−0.559756 + 0.828657i \(0.689105\pi\)
\(18\) 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 \(0.128420\pi\)
−0.919715 + 0.392587i \(0.871580\pi\)
\(24\) −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 \(-0.806005\pi\)
0.819961 0.572419i \(-0.193995\pi\)
\(38\) − 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 \(-0.994285\pi\)
0.999839 0.0179522i \(-0.00571465\pi\)
\(44\) −1.95265e18 −0.245990
\(45\) 0 0
\(46\) 1.44101e19 1.08881
\(47\) − 8.67836e18i − 0.512050i −0.966670 0.256025i \(-0.917587\pi\)
0.966670 0.256025i \(-0.0824129\pi\)
\(48\) 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 \(0.191386\pi\)
−0.824626 + 0.565679i \(0.808614\pi\)
\(54\) −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 \(0.264537\pi\)
−0.674088 + 0.738652i \(0.735463\pi\)
\(68\) − 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 \(0.225990\pi\)
−0.758382 + 0.651810i \(0.774010\pi\)
\(74\) −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 \(0.106652\pi\)
−0.944392 + 0.328823i \(0.893348\pi\)
\(84\) −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 \(-0.701095\pi\)
0.590564 0.806991i \(-0.298905\pi\)
\(98\) − 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 \(-0.983974\pi\)
0.998733 0.0503264i \(-0.0160262\pi\)
\(104\) 9.31907e21 0.0593594
\(105\) 0 0
\(106\) 3.06623e23 1.56887
\(107\) 6.02971e22i 0.276938i 0.990367 + 0.138469i \(0.0442182\pi\)
−0.990367 + 0.138469i \(0.955782\pi\)
\(108\) 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 \(-0.858145\pi\)
0.902331 0.431044i \(-0.141855\pi\)
\(114\) −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 \(0.266443\pi\)
−0.669653 + 0.742674i \(0.733557\pi\)
\(128\) 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 \(0.202534\pi\)
−0.804312 + 0.594207i \(0.797466\pi\)
\(138\) − 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 \(0.137708\pi\)
−0.907870 + 0.419252i \(0.862292\pi\)
\(158\) 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 \(0.157173\pi\)
−0.880550 + 0.473953i \(0.842827\pi\)
\(164\) −1.08452e25 −0.366888
\(165\) 0 0
\(166\) 3.09900e25 0.911967
\(167\) 1.77408e25i 0.487230i 0.969872 + 0.243615i \(0.0783334\pi\)
−0.969872 + 0.243615i \(0.921667\pi\)
\(168\) − 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 \(-0.598356\pi\)
0.304100 0.952640i \(-0.401644\pi\)
\(174\) −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 \(0.108177\pi\)
−0.942805 + 0.333344i \(0.891823\pi\)
\(194\) −4.56673e26 −2.23813
\(195\) 0 0
\(196\) −9.93833e25 −0.432881
\(197\) − 3.89967e26i − 1.60201i −0.598655 0.801007i \(-0.704298\pi\)
0.598655 0.801007i \(-0.295702\pi\)
\(198\) 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 \(-0.759644\pi\)
0.728203 0.685361i \(-0.240356\pi\)
\(224\) −1.07102e27 −1.00449
\(225\) 0 0
\(226\) −1.41184e27 −1.19547
\(227\) − 2.14958e27i − 1.73004i −0.501734 0.865022i \(-0.667304\pi\)
0.501734 0.865022i \(-0.332696\pi\)
\(228\) 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 \(-0.773381\pi\)
0.757094 0.653306i \(-0.226619\pi\)
\(234\) −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 \(-0.875454\pi\)
0.924424 0.381366i \(-0.124546\pi\)
\(258\) 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 \(-0.952327\pi\)
0.988805 0.149210i \(-0.0476731\pi\)
\(264\) −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 \(0.177268\pi\)
−0.848896 + 0.528560i \(0.822732\pi\)
\(278\) − 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 \(0.240360\pi\)
−0.728193 + 0.685372i \(0.759640\pi\)
\(284\) −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 \(0.0828026\pi\)
−0.966356 + 0.257208i \(0.917197\pi\)
\(294\) −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 \(-0.799919\pi\)
0.808867 0.587992i \(-0.200081\pi\)
\(308\) − 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 \(0.0362317\pi\)
−0.993529 + 0.113579i \(0.963768\pi\)
\(314\) 6.02814e28 1.16277
\(315\) 0 0
\(316\) 7.92439e28 1.42090
\(317\) 1.37896e28i 0.238436i 0.992868 + 0.119218i \(0.0380387\pi\)
−0.992868 + 0.119218i \(0.961961\pi\)
\(318\) − 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 \(-0.752219\pi\)
0.712019 0.702160i \(-0.247781\pi\)
\(338\) − 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 \(0.146774\pi\)
−0.895561 + 0.444938i \(0.853226\pi\)
\(348\) 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 \(-0.979445\pi\)
0.997916 0.0645320i \(-0.0205555\pi\)
\(354\) −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 \(0.0840447\pi\)
−0.965345 + 0.260977i \(0.915955\pi\)
\(368\) − 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 \(-0.955979\pi\)
0.990452 0.137856i \(-0.0440211\pi\)
\(374\) 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 \(-0.905914\pi\)
0.956633 0.291295i \(-0.0940862\pi\)
\(384\) 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 \(0.00346461\pi\)
−0.999941 + 0.0108842i \(0.996535\pi\)
\(398\) − 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 \(0.222324\pi\)
−0.765839 + 0.643032i \(0.777676\pi\)
\(434\) 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 \(0.0478302\pi\)
−0.988732 + 0.149698i \(0.952170\pi\)
\(444\) 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 \(0.0889291\pi\)
−0.961227 + 0.275759i \(0.911071\pi\)
\(458\) − 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 \(0.323979\pi\)
−0.525232 + 0.850959i \(0.676021\pi\)
\(464\) −1.55207e30 −0.335703
\(465\) 0 0
\(466\) −8.80180e30 −1.81190
\(467\) − 2.87884e30i − 0.578194i −0.957300 0.289097i \(-0.906645\pi\)
0.957300 0.289097i \(-0.0933551\pi\)
\(468\) 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 \(0.0169675\pi\)
−0.998580 + 0.0532798i \(0.983032\pi\)
\(488\) − 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 \(-0.909497\pi\)
0.959852 0.280507i \(-0.0905027\pi\)
\(504\) −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 \(0.383269\pi\)
−0.358557 + 0.933508i \(0.616731\pi\)
\(524\) −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 \(-0.946210\pi\)
0.985756 0.168184i \(-0.0537901\pi\)
\(548\) − 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 \(0.0577090\pi\)
−0.983610 + 0.180307i \(0.942291\pi\)
\(558\) 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 \(-0.882236\pi\)
0.932339 0.361585i \(-0.117764\pi\)
\(564\) 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 \(-0.955006\pi\)
0.990026 0.140884i \(-0.0449944\pi\)
\(578\) 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 \(0.201306\pi\)
−0.806599 + 0.591099i \(0.798694\pi\)
\(588\) 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 \(0.0115089\pi\)
−0.999346 + 0.0361485i \(0.988491\pi\)
\(594\) −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 \(-0.974301\pi\)
0.996743 0.0806481i \(-0.0256990\pi\)
\(608\) 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 \(-0.972876\pi\)
0.996371 0.0851110i \(-0.0271245\pi\)
\(614\) −1.88940e32 −1.63075
\(615\) 0 0
\(616\) 2.49502e30 0.0207442
\(617\) 5.25618e31i 0.428934i 0.976731 + 0.214467i \(0.0688015\pi\)
−0.976731 + 0.214467i \(0.931199\pi\)
\(618\) 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 \(-0.969947\pi\)
0.995546 0.0942729i \(-0.0300526\pi\)
\(644\) 1.05898e32 0.528080
\(645\) 0 0
\(646\) 5.86067e32 2.82016
\(647\) − 3.77554e32i − 1.78476i −0.451286 0.892379i \(-0.649035\pi\)
0.451286 0.892379i \(-0.350965\pi\)
\(648\) 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 \(0.0583801\pi\)
−0.983228 + 0.182380i \(0.941620\pi\)
\(654\) −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 \(-0.703053\pi\)
0.595519 0.803341i \(-0.296947\pi\)
\(674\) −6.59237e32 −1.94739
\(675\) 0 0
\(676\) −2.23437e32 −0.637924
\(677\) 1.10049e32i 0.308899i 0.988001 + 0.154449i \(0.0493604\pi\)
−0.988001 + 0.154449i \(0.950640\pi\)
\(678\) 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 \(0.123039\pi\)
−0.926220 + 0.376983i \(0.876961\pi\)
\(684\) 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 \(0.0925900\pi\)
−0.957992 + 0.286795i \(0.907410\pi\)
\(728\) 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 \(0.336490\pi\)
−0.491387 + 0.870941i \(0.663510\pi\)
\(734\) 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 \(-0.981608\pi\)
0.998331 0.0577487i \(-0.0183922\pi\)
\(744\) −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 \(-0.792439\pi\)
0.794828 0.606835i \(-0.207561\pi\)
\(758\) 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 \(0.0829822\pi\)
−0.966211 + 0.257753i \(0.917018\pi\)
\(774\) 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 \(0.110583\pi\)
−0.940259 + 0.340460i \(0.889417\pi\)
\(788\) 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 \(0.109871\pi\)
−0.941018 + 0.338357i \(0.890129\pi\)
\(798\) − 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 \(-0.667170\pi\)
0.501370 0.865233i \(-0.332830\pi\)
\(824\) −3.66954e31 −0.0107510
\(825\) 0 0
\(826\) 1.55961e31 0.00444372
\(827\) − 3.61128e33i − 1.01472i −0.861733 0.507362i \(-0.830621\pi\)
0.861733 0.507362i \(-0.169379\pi\)
\(828\) 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 \(0.354140\pi\)
−0.442364 + 0.896836i \(0.645860\pi\)
\(854\) 4.39943e33 0.854339
\(855\) 0 0
\(856\) 1.56478e32 0.0295805
\(857\) 6.17423e33i 1.15160i 0.817589 + 0.575802i \(0.195310\pi\)
−0.817589 + 0.575802i \(0.804690\pi\)
\(858\) − 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 \(0.148363\pi\)
−0.893329 + 0.449403i \(0.851637\pi\)
\(864\) 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 \(-0.984653\pi\)
0.998838 0.0481954i \(-0.0153470\pi\)
\(878\) 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 \(-0.968458\pi\)
0.995094 0.0989294i \(-0.0315418\pi\)
\(884\) 6.51121e33 0.850082
\(885\) 0 0
\(886\) 3.26378e33 0.415177
\(887\) − 6.44904e33i − 0.809792i −0.914363 0.404896i \(-0.867308\pi\)
0.914363 0.404896i \(-0.132692\pi\)
\(888\) 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 \(0.101013\pi\)
−0.950069 + 0.312041i \(0.898987\pi\)
\(908\) 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 \(-0.818484\pi\)
0.841766 0.539843i \(-0.181516\pi\)
\(938\) 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 \(-0.663778\pi\)
0.492119 0.870528i \(-0.336222\pi\)
\(948\) − 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 \(-0.817425\pi\)
0.839965 0.542641i \(-0.182575\pi\)
\(954\) −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 \(-0.843262\pi\)
0.881198 0.472748i \(-0.156738\pi\)
\(968\) 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 \(-0.997579\pi\)
0.999971 0.00760725i \(-0.00242149\pi\)
\(978\) − 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 \(0.0507078\pi\)
−0.987338 + 0.158630i \(0.949292\pi\)
\(984\) −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 \(-0.975767\pi\)
0.997103 0.0760573i \(-0.0242332\pi\)
\(998\) 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.24.2 4
5.2 odd 4 25.24.a.a.1.2 2
5.3 odd 4 1.24.a.a.1.1 2
5.4 even 2 inner 25.24.b.a.24.3 4
15.8 even 4 9.24.a.b.1.2 2
20.3 even 4 16.24.a.b.1.1 2
35.13 even 4 49.24.a.b.1.1 2
40.3 even 4 64.24.a.g.1.2 2
40.13 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 5.3 odd 4
9.24.a.b.1.2 2 15.8 even 4
16.24.a.b.1.1 2 20.3 even 4
25.24.a.a.1.2 2 5.2 odd 4
25.24.b.a.24.2 4 1.1 even 1 trivial
25.24.b.a.24.3 4 5.4 even 2 inner
49.24.a.b.1.1 2 35.13 even 4
64.24.a.d.1.1 2 40.13 odd 4
64.24.a.g.1.2 2 40.3 even 4