Properties

Label 16.36.a.d.1.3
Level $16$
Weight $36$
Character 16.1
Self dual yes
Analytic conductor $124.152$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(124.152209014\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 12422194 x - 2645665785\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{3}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-213.765\) of defining polynomial
Character \(\chi\) \(=\) 16.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +O(q^{10})\) \(q+3.45913e8 q^{3} -2.05014e12 q^{5} +1.25160e14 q^{7} +6.96245e16 q^{9} +1.70842e18 q^{11} -4.94986e19 q^{13} -7.09169e20 q^{15} +1.32045e21 q^{17} -3.94388e21 q^{19} +4.32945e22 q^{21} +3.48881e23 q^{23} +1.29267e24 q^{25} +6.77748e24 q^{27} +3.21628e25 q^{29} -3.41044e25 q^{31} +5.90965e26 q^{33} -2.56595e26 q^{35} -4.03624e27 q^{37} -1.71222e28 q^{39} +8.65857e27 q^{41} -9.89389e26 q^{43} -1.42740e29 q^{45} -1.95852e29 q^{47} -3.63154e29 q^{49} +4.56760e29 q^{51} -9.96909e29 q^{53} -3.50249e30 q^{55} -1.36424e30 q^{57} -3.91953e30 q^{59} +7.64909e29 q^{61} +8.71420e30 q^{63} +1.01479e32 q^{65} +1.64301e32 q^{67} +1.20682e32 q^{69} -7.65930e31 q^{71} -7.08063e32 q^{73} +4.47152e32 q^{75} +2.13826e32 q^{77} -2.34599e33 q^{79} -1.13900e33 q^{81} -5.18971e33 q^{83} -2.70710e33 q^{85} +1.11255e34 q^{87} +1.44578e34 q^{89} -6.19525e33 q^{91} -1.17972e34 q^{93} +8.08549e33 q^{95} -2.87399e34 q^{97} +1.18948e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 104875308q^{3} + 892652054010q^{5} - 878422149346056q^{7} + 150091978876243551q^{9} + O(q^{10}) \) \( 3q + 104875308q^{3} + 892652054010q^{5} - 878422149346056q^{7} + 150091978876243551q^{9} + 1157945428549987044q^{11} - 62139610550998650558q^{13} - \)\(70\!\cdots\!20\)\(q^{15} - \)\(39\!\cdots\!94\)\(q^{17} + \)\(32\!\cdots\!40\)\(q^{19} + \)\(22\!\cdots\!76\)\(q^{21} + \)\(51\!\cdots\!08\)\(q^{23} + \)\(64\!\cdots\!25\)\(q^{25} - \)\(27\!\cdots\!00\)\(q^{27} - \)\(38\!\cdots\!10\)\(q^{29} - \)\(10\!\cdots\!56\)\(q^{31} + \)\(11\!\cdots\!84\)\(q^{33} - \)\(15\!\cdots\!60\)\(q^{35} + \)\(24\!\cdots\!06\)\(q^{37} - \)\(18\!\cdots\!12\)\(q^{39} + \)\(23\!\cdots\!06\)\(q^{41} + \)\(47\!\cdots\!08\)\(q^{43} - \)\(11\!\cdots\!30\)\(q^{45} - \)\(16\!\cdots\!56\)\(q^{47} - \)\(59\!\cdots\!21\)\(q^{49} + \)\(18\!\cdots\!04\)\(q^{51} - \)\(16\!\cdots\!58\)\(q^{53} - \)\(30\!\cdots\!20\)\(q^{55} + \)\(40\!\cdots\!00\)\(q^{57} - \)\(43\!\cdots\!80\)\(q^{59} + \)\(23\!\cdots\!06\)\(q^{61} - \)\(45\!\cdots\!92\)\(q^{63} + \)\(75\!\cdots\!20\)\(q^{65} + \)\(18\!\cdots\!44\)\(q^{67} - \)\(32\!\cdots\!48\)\(q^{69} - \)\(34\!\cdots\!56\)\(q^{71} - \)\(28\!\cdots\!58\)\(q^{73} + \)\(15\!\cdots\!00\)\(q^{75} + \)\(69\!\cdots\!12\)\(q^{77} + \)\(42\!\cdots\!60\)\(q^{79} + \)\(18\!\cdots\!63\)\(q^{81} - \)\(14\!\cdots\!92\)\(q^{83} - \)\(87\!\cdots\!40\)\(q^{85} + \)\(78\!\cdots\!00\)\(q^{87} + \)\(30\!\cdots\!70\)\(q^{89} - \)\(10\!\cdots\!96\)\(q^{91} - \)\(40\!\cdots\!16\)\(q^{93} + \)\(84\!\cdots\!00\)\(q^{95} - \)\(10\!\cdots\!94\)\(q^{97} - \)\(30\!\cdots\!52\)\(q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.45913e8 1.54648 0.773242 0.634111i \(-0.218634\pi\)
0.773242 + 0.634111i \(0.218634\pi\)
\(4\) 0 0
\(5\) −2.05014e12 −1.20173 −0.600866 0.799350i \(-0.705177\pi\)
−0.600866 + 0.799350i \(0.705177\pi\)
\(6\) 0 0
\(7\) 1.25160e14 0.203353 0.101676 0.994818i \(-0.467579\pi\)
0.101676 + 0.994818i \(0.467579\pi\)
\(8\) 0 0
\(9\) 6.96245e16 1.39161
\(10\) 0 0
\(11\) 1.70842e18 1.01911 0.509557 0.860437i \(-0.329809\pi\)
0.509557 + 0.860437i \(0.329809\pi\)
\(12\) 0 0
\(13\) −4.94986e19 −1.58703 −0.793514 0.608552i \(-0.791751\pi\)
−0.793514 + 0.608552i \(0.791751\pi\)
\(14\) 0 0
\(15\) −7.09169e20 −1.85846
\(16\) 0 0
\(17\) 1.32045e21 0.387137 0.193569 0.981087i \(-0.437994\pi\)
0.193569 + 0.981087i \(0.437994\pi\)
\(18\) 0 0
\(19\) −3.94388e21 −0.165096 −0.0825479 0.996587i \(-0.526306\pi\)
−0.0825479 + 0.996587i \(0.526306\pi\)
\(20\) 0 0
\(21\) 4.32945e22 0.314482
\(22\) 0 0
\(23\) 3.48881e23 0.515750 0.257875 0.966178i \(-0.416978\pi\)
0.257875 + 0.966178i \(0.416978\pi\)
\(24\) 0 0
\(25\) 1.29267e24 0.444159
\(26\) 0 0
\(27\) 6.77748e24 0.605623
\(28\) 0 0
\(29\) 3.21628e25 0.822979 0.411489 0.911415i \(-0.365009\pi\)
0.411489 + 0.911415i \(0.365009\pi\)
\(30\) 0 0
\(31\) −3.41044e25 −0.271632 −0.135816 0.990734i \(-0.543366\pi\)
−0.135816 + 0.990734i \(0.543366\pi\)
\(32\) 0 0
\(33\) 5.90965e26 1.57604
\(34\) 0 0
\(35\) −2.56595e26 −0.244375
\(36\) 0 0
\(37\) −4.03624e27 −1.45361 −0.726804 0.686845i \(-0.758995\pi\)
−0.726804 + 0.686845i \(0.758995\pi\)
\(38\) 0 0
\(39\) −1.71222e28 −2.45431
\(40\) 0 0
\(41\) 8.65857e27 0.517283 0.258642 0.965973i \(-0.416725\pi\)
0.258642 + 0.965973i \(0.416725\pi\)
\(42\) 0 0
\(43\) −9.89389e26 −0.0256844 −0.0128422 0.999918i \(-0.504088\pi\)
−0.0128422 + 0.999918i \(0.504088\pi\)
\(44\) 0 0
\(45\) −1.42740e29 −1.67234
\(46\) 0 0
\(47\) −1.95852e29 −1.07205 −0.536025 0.844202i \(-0.680075\pi\)
−0.536025 + 0.844202i \(0.680075\pi\)
\(48\) 0 0
\(49\) −3.63154e29 −0.958648
\(50\) 0 0
\(51\) 4.56760e29 0.598702
\(52\) 0 0
\(53\) −9.96909e29 −0.666543 −0.333271 0.942831i \(-0.608153\pi\)
−0.333271 + 0.942831i \(0.608153\pi\)
\(54\) 0 0
\(55\) −3.50249e30 −1.22470
\(56\) 0 0
\(57\) −1.36424e30 −0.255318
\(58\) 0 0
\(59\) −3.91953e30 −0.401166 −0.200583 0.979677i \(-0.564284\pi\)
−0.200583 + 0.979677i \(0.564284\pi\)
\(60\) 0 0
\(61\) 7.64909e29 0.0436855 0.0218428 0.999761i \(-0.493047\pi\)
0.0218428 + 0.999761i \(0.493047\pi\)
\(62\) 0 0
\(63\) 8.71420e30 0.282988
\(64\) 0 0
\(65\) 1.01479e32 1.90718
\(66\) 0 0
\(67\) 1.64301e32 1.81690 0.908451 0.417992i \(-0.137266\pi\)
0.908451 + 0.417992i \(0.137266\pi\)
\(68\) 0 0
\(69\) 1.20682e32 0.797600
\(70\) 0 0
\(71\) −7.65930e31 −0.307021 −0.153510 0.988147i \(-0.549058\pi\)
−0.153510 + 0.988147i \(0.549058\pi\)
\(72\) 0 0
\(73\) −7.08063e32 −1.74551 −0.872754 0.488160i \(-0.837668\pi\)
−0.872754 + 0.488160i \(0.837668\pi\)
\(74\) 0 0
\(75\) 4.47152e32 0.686884
\(76\) 0 0
\(77\) 2.13826e32 0.207239
\(78\) 0 0
\(79\) −2.34599e33 −1.45162 −0.725809 0.687896i \(-0.758534\pi\)
−0.725809 + 0.687896i \(0.758534\pi\)
\(80\) 0 0
\(81\) −1.13900e33 −0.455027
\(82\) 0 0
\(83\) −5.18971e33 −1.35293 −0.676467 0.736473i \(-0.736490\pi\)
−0.676467 + 0.736473i \(0.736490\pi\)
\(84\) 0 0
\(85\) −2.70710e33 −0.465235
\(86\) 0 0
\(87\) 1.11255e34 1.27272
\(88\) 0 0
\(89\) 1.44578e34 1.11116 0.555582 0.831461i \(-0.312495\pi\)
0.555582 + 0.831461i \(0.312495\pi\)
\(90\) 0 0
\(91\) −6.19525e33 −0.322726
\(92\) 0 0
\(93\) −1.17972e34 −0.420075
\(94\) 0 0
\(95\) 8.08549e33 0.198401
\(96\) 0 0
\(97\) −2.87399e34 −0.489756 −0.244878 0.969554i \(-0.578748\pi\)
−0.244878 + 0.969554i \(0.578748\pi\)
\(98\) 0 0
\(99\) 1.18948e35 1.41821
\(100\) 0 0
\(101\) 1.13198e35 0.951079 0.475539 0.879694i \(-0.342253\pi\)
0.475539 + 0.879694i \(0.342253\pi\)
\(102\) 0 0
\(103\) −2.53635e35 −1.51202 −0.756011 0.654559i \(-0.772855\pi\)
−0.756011 + 0.654559i \(0.772855\pi\)
\(104\) 0 0
\(105\) −8.87596e34 −0.377922
\(106\) 0 0
\(107\) 3.63939e35 1.11381 0.556907 0.830575i \(-0.311988\pi\)
0.556907 + 0.830575i \(0.311988\pi\)
\(108\) 0 0
\(109\) −4.65371e34 −0.103000 −0.0514998 0.998673i \(-0.516400\pi\)
−0.0514998 + 0.998673i \(0.516400\pi\)
\(110\) 0 0
\(111\) −1.39619e36 −2.24798
\(112\) 0 0
\(113\) −2.18346e35 −0.257201 −0.128601 0.991696i \(-0.541049\pi\)
−0.128601 + 0.991696i \(0.541049\pi\)
\(114\) 0 0
\(115\) −7.15252e35 −0.619794
\(116\) 0 0
\(117\) −3.44632e36 −2.20853
\(118\) 0 0
\(119\) 1.65267e35 0.0787254
\(120\) 0 0
\(121\) 1.08456e35 0.0385932
\(122\) 0 0
\(123\) 2.99511e36 0.799970
\(124\) 0 0
\(125\) 3.31653e36 0.667972
\(126\) 0 0
\(127\) −4.19558e36 −0.640070 −0.320035 0.947406i \(-0.603695\pi\)
−0.320035 + 0.947406i \(0.603695\pi\)
\(128\) 0 0
\(129\) −3.42243e35 −0.0397205
\(130\) 0 0
\(131\) −2.40631e36 −0.213355 −0.106678 0.994294i \(-0.534021\pi\)
−0.106678 + 0.994294i \(0.534021\pi\)
\(132\) 0 0
\(133\) −4.93616e35 −0.0335727
\(134\) 0 0
\(135\) −1.38947e37 −0.727796
\(136\) 0 0
\(137\) −2.14718e37 −0.869477 −0.434739 0.900557i \(-0.643159\pi\)
−0.434739 + 0.900557i \(0.643159\pi\)
\(138\) 0 0
\(139\) −2.63209e37 −0.827068 −0.413534 0.910489i \(-0.635706\pi\)
−0.413534 + 0.910489i \(0.635706\pi\)
\(140\) 0 0
\(141\) −6.77477e37 −1.65791
\(142\) 0 0
\(143\) −8.45645e37 −1.61736
\(144\) 0 0
\(145\) −6.59381e37 −0.989000
\(146\) 0 0
\(147\) −1.25620e38 −1.48253
\(148\) 0 0
\(149\) −1.74389e38 −1.62465 −0.812324 0.583206i \(-0.801798\pi\)
−0.812324 + 0.583206i \(0.801798\pi\)
\(150\) 0 0
\(151\) 1.40947e38 1.03982 0.519912 0.854220i \(-0.325965\pi\)
0.519912 + 0.854220i \(0.325965\pi\)
\(152\) 0 0
\(153\) 9.19355e37 0.538745
\(154\) 0 0
\(155\) 6.99187e37 0.326429
\(156\) 0 0
\(157\) 1.49924e38 0.559277 0.279639 0.960105i \(-0.409785\pi\)
0.279639 + 0.960105i \(0.409785\pi\)
\(158\) 0 0
\(159\) −3.44844e38 −1.03080
\(160\) 0 0
\(161\) 4.36659e37 0.104879
\(162\) 0 0
\(163\) −8.60787e38 −1.66576 −0.832880 0.553453i \(-0.813310\pi\)
−0.832880 + 0.553453i \(0.813310\pi\)
\(164\) 0 0
\(165\) −1.21156e39 −1.89398
\(166\) 0 0
\(167\) 9.23344e38 1.16903 0.584515 0.811383i \(-0.301285\pi\)
0.584515 + 0.811383i \(0.301285\pi\)
\(168\) 0 0
\(169\) 1.47733e39 1.51866
\(170\) 0 0
\(171\) −2.74591e38 −0.229749
\(172\) 0 0
\(173\) −1.33161e39 −0.909012 −0.454506 0.890744i \(-0.650184\pi\)
−0.454506 + 0.890744i \(0.650184\pi\)
\(174\) 0 0
\(175\) 1.61791e38 0.0903208
\(176\) 0 0
\(177\) −1.35582e39 −0.620396
\(178\) 0 0
\(179\) −6.27202e38 −0.235765 −0.117883 0.993028i \(-0.537611\pi\)
−0.117883 + 0.993028i \(0.537611\pi\)
\(180\) 0 0
\(181\) −2.39965e39 −0.742629 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(182\) 0 0
\(183\) 2.64592e38 0.0675590
\(184\) 0 0
\(185\) 8.27484e39 1.74685
\(186\) 0 0
\(187\) 2.25588e39 0.394537
\(188\) 0 0
\(189\) 8.48269e38 0.123155
\(190\) 0 0
\(191\) −8.14518e39 −0.983596 −0.491798 0.870709i \(-0.663660\pi\)
−0.491798 + 0.870709i \(0.663660\pi\)
\(192\) 0 0
\(193\) 1.38549e40 1.39428 0.697139 0.716936i \(-0.254456\pi\)
0.697139 + 0.716936i \(0.254456\pi\)
\(194\) 0 0
\(195\) 3.51029e40 2.94942
\(196\) 0 0
\(197\) −1.15539e40 −0.812026 −0.406013 0.913867i \(-0.633081\pi\)
−0.406013 + 0.913867i \(0.633081\pi\)
\(198\) 0 0
\(199\) 4.76691e39 0.280742 0.140371 0.990099i \(-0.455170\pi\)
0.140371 + 0.990099i \(0.455170\pi\)
\(200\) 0 0
\(201\) 5.68340e40 2.80981
\(202\) 0 0
\(203\) 4.02549e39 0.167355
\(204\) 0 0
\(205\) −1.77512e40 −0.621636
\(206\) 0 0
\(207\) 2.42906e40 0.717725
\(208\) 0 0
\(209\) −6.73781e39 −0.168251
\(210\) 0 0
\(211\) 1.87354e40 0.396022 0.198011 0.980200i \(-0.436552\pi\)
0.198011 + 0.980200i \(0.436552\pi\)
\(212\) 0 0
\(213\) −2.64945e40 −0.474803
\(214\) 0 0
\(215\) 2.02838e39 0.0308657
\(216\) 0 0
\(217\) −4.26851e39 −0.0552371
\(218\) 0 0
\(219\) −2.44929e41 −2.69940
\(220\) 0 0
\(221\) −6.53603e40 −0.614398
\(222\) 0 0
\(223\) 7.12100e40 0.571750 0.285875 0.958267i \(-0.407716\pi\)
0.285875 + 0.958267i \(0.407716\pi\)
\(224\) 0 0
\(225\) 9.00017e40 0.618097
\(226\) 0 0
\(227\) 1.04005e41 0.611787 0.305893 0.952066i \(-0.401045\pi\)
0.305893 + 0.952066i \(0.401045\pi\)
\(228\) 0 0
\(229\) 1.82859e41 0.922564 0.461282 0.887254i \(-0.347390\pi\)
0.461282 + 0.887254i \(0.347390\pi\)
\(230\) 0 0
\(231\) 7.39652e40 0.320493
\(232\) 0 0
\(233\) 1.57847e41 0.588177 0.294088 0.955778i \(-0.404984\pi\)
0.294088 + 0.955778i \(0.404984\pi\)
\(234\) 0 0
\(235\) 4.01523e41 1.28832
\(236\) 0 0
\(237\) −8.11510e41 −2.24490
\(238\) 0 0
\(239\) −5.00984e41 −1.19636 −0.598178 0.801363i \(-0.704108\pi\)
−0.598178 + 0.801363i \(0.704108\pi\)
\(240\) 0 0
\(241\) −3.33164e41 −0.687638 −0.343819 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(242\) 0 0
\(243\) −7.33084e41 −1.30931
\(244\) 0 0
\(245\) 7.44514e41 1.15204
\(246\) 0 0
\(247\) 1.95217e41 0.262011
\(248\) 0 0
\(249\) −1.79519e42 −2.09229
\(250\) 0 0
\(251\) −1.52760e42 −1.54782 −0.773909 0.633297i \(-0.781701\pi\)
−0.773909 + 0.633297i \(0.781701\pi\)
\(252\) 0 0
\(253\) 5.96035e41 0.525608
\(254\) 0 0
\(255\) −9.36421e41 −0.719479
\(256\) 0 0
\(257\) −2.72482e42 −1.82603 −0.913016 0.407925i \(-0.866253\pi\)
−0.913016 + 0.407925i \(0.866253\pi\)
\(258\) 0 0
\(259\) −5.05176e41 −0.295595
\(260\) 0 0
\(261\) 2.23932e42 1.14527
\(262\) 0 0
\(263\) −4.22055e40 −0.0188861 −0.00944307 0.999955i \(-0.503006\pi\)
−0.00944307 + 0.999955i \(0.503006\pi\)
\(264\) 0 0
\(265\) 2.04380e42 0.801005
\(266\) 0 0
\(267\) 5.00116e42 1.71840
\(268\) 0 0
\(269\) −1.30618e42 −0.393858 −0.196929 0.980418i \(-0.563097\pi\)
−0.196929 + 0.980418i \(0.563097\pi\)
\(270\) 0 0
\(271\) 4.11918e41 0.109106 0.0545529 0.998511i \(-0.482627\pi\)
0.0545529 + 0.998511i \(0.482627\pi\)
\(272\) 0 0
\(273\) −2.14302e42 −0.499091
\(274\) 0 0
\(275\) 2.20843e42 0.452648
\(276\) 0 0
\(277\) 7.38232e42 1.33290 0.666449 0.745551i \(-0.267813\pi\)
0.666449 + 0.745551i \(0.267813\pi\)
\(278\) 0 0
\(279\) −2.37451e42 −0.378007
\(280\) 0 0
\(281\) 4.08223e42 0.573505 0.286752 0.958005i \(-0.407424\pi\)
0.286752 + 0.958005i \(0.407424\pi\)
\(282\) 0 0
\(283\) 1.06503e43 1.32159 0.660796 0.750565i \(-0.270219\pi\)
0.660796 + 0.750565i \(0.270219\pi\)
\(284\) 0 0
\(285\) 2.79688e42 0.306824
\(286\) 0 0
\(287\) 1.08371e42 0.105191
\(288\) 0 0
\(289\) −9.88997e42 −0.850125
\(290\) 0 0
\(291\) −9.94151e42 −0.757399
\(292\) 0 0
\(293\) 1.90474e43 1.28722 0.643610 0.765353i \(-0.277436\pi\)
0.643610 + 0.765353i \(0.277436\pi\)
\(294\) 0 0
\(295\) 8.03556e42 0.482093
\(296\) 0 0
\(297\) 1.15788e43 0.617199
\(298\) 0 0
\(299\) −1.72691e43 −0.818510
\(300\) 0 0
\(301\) −1.23832e41 −0.00522298
\(302\) 0 0
\(303\) 3.91569e43 1.47083
\(304\) 0 0
\(305\) −1.56817e42 −0.0524983
\(306\) 0 0
\(307\) 1.39752e43 0.417288 0.208644 0.977992i \(-0.433095\pi\)
0.208644 + 0.977992i \(0.433095\pi\)
\(308\) 0 0
\(309\) −8.77359e43 −2.33832
\(310\) 0 0
\(311\) −2.22377e43 −0.529399 −0.264700 0.964331i \(-0.585273\pi\)
−0.264700 + 0.964331i \(0.585273\pi\)
\(312\) 0 0
\(313\) 3.63171e43 0.772833 0.386417 0.922324i \(-0.373713\pi\)
0.386417 + 0.922324i \(0.373713\pi\)
\(314\) 0 0
\(315\) −1.78653e43 −0.340076
\(316\) 0 0
\(317\) 5.29385e43 0.902058 0.451029 0.892509i \(-0.351057\pi\)
0.451029 + 0.892509i \(0.351057\pi\)
\(318\) 0 0
\(319\) 5.49476e43 0.838709
\(320\) 0 0
\(321\) 1.25891e44 1.72249
\(322\) 0 0
\(323\) −5.20769e42 −0.0639147
\(324\) 0 0
\(325\) −6.39855e43 −0.704892
\(326\) 0 0
\(327\) −1.60978e43 −0.159287
\(328\) 0 0
\(329\) −2.45128e43 −0.218004
\(330\) 0 0
\(331\) −6.25551e43 −0.500349 −0.250174 0.968201i \(-0.580488\pi\)
−0.250174 + 0.968201i \(0.580488\pi\)
\(332\) 0 0
\(333\) −2.81022e44 −2.02286
\(334\) 0 0
\(335\) −3.36840e44 −2.18343
\(336\) 0 0
\(337\) −1.04185e44 −0.608529 −0.304264 0.952588i \(-0.598411\pi\)
−0.304264 + 0.952588i \(0.598411\pi\)
\(338\) 0 0
\(339\) −7.55289e43 −0.397757
\(340\) 0 0
\(341\) −5.82647e43 −0.276824
\(342\) 0 0
\(343\) −9.28652e43 −0.398296
\(344\) 0 0
\(345\) −2.47415e44 −0.958501
\(346\) 0 0
\(347\) 3.07657e44 1.07721 0.538606 0.842557i \(-0.318951\pi\)
0.538606 + 0.842557i \(0.318951\pi\)
\(348\) 0 0
\(349\) −2.62767e44 −0.832005 −0.416003 0.909363i \(-0.636569\pi\)
−0.416003 + 0.909363i \(0.636569\pi\)
\(350\) 0 0
\(351\) −3.35476e44 −0.961140
\(352\) 0 0
\(353\) 7.06671e44 1.83299 0.916494 0.400048i \(-0.131006\pi\)
0.916494 + 0.400048i \(0.131006\pi\)
\(354\) 0 0
\(355\) 1.57026e44 0.368957
\(356\) 0 0
\(357\) 5.71681e43 0.121748
\(358\) 0 0
\(359\) 5.35788e44 1.03476 0.517380 0.855756i \(-0.326907\pi\)
0.517380 + 0.855756i \(0.326907\pi\)
\(360\) 0 0
\(361\) −5.55104e44 −0.972743
\(362\) 0 0
\(363\) 3.75165e43 0.0596838
\(364\) 0 0
\(365\) 1.45163e45 2.09763
\(366\) 0 0
\(367\) 3.00056e44 0.394045 0.197022 0.980399i \(-0.436873\pi\)
0.197022 + 0.980399i \(0.436873\pi\)
\(368\) 0 0
\(369\) 6.02849e44 0.719858
\(370\) 0 0
\(371\) −1.24773e44 −0.135543
\(372\) 0 0
\(373\) 1.08201e45 1.06986 0.534932 0.844895i \(-0.320337\pi\)
0.534932 + 0.844895i \(0.320337\pi\)
\(374\) 0 0
\(375\) 1.14723e45 1.03301
\(376\) 0 0
\(377\) −1.59201e45 −1.30609
\(378\) 0 0
\(379\) −1.21847e45 −0.911229 −0.455614 0.890177i \(-0.650580\pi\)
−0.455614 + 0.890177i \(0.650580\pi\)
\(380\) 0 0
\(381\) −1.45131e45 −0.989858
\(382\) 0 0
\(383\) 9.53201e43 0.0593207 0.0296603 0.999560i \(-0.490557\pi\)
0.0296603 + 0.999560i \(0.490557\pi\)
\(384\) 0 0
\(385\) −4.38372e44 −0.249046
\(386\) 0 0
\(387\) −6.88857e43 −0.0357427
\(388\) 0 0
\(389\) 1.90493e45 0.903152 0.451576 0.892233i \(-0.350862\pi\)
0.451576 + 0.892233i \(0.350862\pi\)
\(390\) 0 0
\(391\) 4.60678e44 0.199666
\(392\) 0 0
\(393\) −8.32373e44 −0.329950
\(394\) 0 0
\(395\) 4.80960e45 1.74446
\(396\) 0 0
\(397\) 3.96399e45 1.31613 0.658065 0.752961i \(-0.271375\pi\)
0.658065 + 0.752961i \(0.271375\pi\)
\(398\) 0 0
\(399\) −1.70748e44 −0.0519196
\(400\) 0 0
\(401\) −5.53039e44 −0.154074 −0.0770369 0.997028i \(-0.524546\pi\)
−0.0770369 + 0.997028i \(0.524546\pi\)
\(402\) 0 0
\(403\) 1.68812e45 0.431088
\(404\) 0 0
\(405\) 2.33511e45 0.546820
\(406\) 0 0
\(407\) −6.89560e45 −1.48139
\(408\) 0 0
\(409\) 3.35495e45 0.661498 0.330749 0.943719i \(-0.392699\pi\)
0.330749 + 0.943719i \(0.392699\pi\)
\(410\) 0 0
\(411\) −7.42738e45 −1.34463
\(412\) 0 0
\(413\) −4.90568e44 −0.0815781
\(414\) 0 0
\(415\) 1.06396e46 1.62586
\(416\) 0 0
\(417\) −9.10474e45 −1.27905
\(418\) 0 0
\(419\) 7.69138e45 0.993708 0.496854 0.867834i \(-0.334488\pi\)
0.496854 + 0.867834i \(0.334488\pi\)
\(420\) 0 0
\(421\) −2.77229e45 −0.329535 −0.164767 0.986332i \(-0.552687\pi\)
−0.164767 + 0.986332i \(0.552687\pi\)
\(422\) 0 0
\(423\) −1.36361e46 −1.49188
\(424\) 0 0
\(425\) 1.70691e45 0.171950
\(426\) 0 0
\(427\) 9.57360e43 0.00888357
\(428\) 0 0
\(429\) −2.92520e46 −2.50122
\(430\) 0 0
\(431\) −6.18484e45 −0.487502 −0.243751 0.969838i \(-0.578378\pi\)
−0.243751 + 0.969838i \(0.578378\pi\)
\(432\) 0 0
\(433\) 2.21783e46 1.61209 0.806047 0.591852i \(-0.201603\pi\)
0.806047 + 0.591852i \(0.201603\pi\)
\(434\) 0 0
\(435\) −2.28089e46 −1.52947
\(436\) 0 0
\(437\) −1.37594e45 −0.0851482
\(438\) 0 0
\(439\) −1.91766e45 −0.109557 −0.0547787 0.998499i \(-0.517445\pi\)
−0.0547787 + 0.998499i \(0.517445\pi\)
\(440\) 0 0
\(441\) −2.52844e46 −1.33407
\(442\) 0 0
\(443\) 3.00864e46 1.46658 0.733290 0.679916i \(-0.237984\pi\)
0.733290 + 0.679916i \(0.237984\pi\)
\(444\) 0 0
\(445\) −2.96405e46 −1.33532
\(446\) 0 0
\(447\) −6.03235e46 −2.51249
\(448\) 0 0
\(449\) 5.68788e45 0.219099 0.109549 0.993981i \(-0.465059\pi\)
0.109549 + 0.993981i \(0.465059\pi\)
\(450\) 0 0
\(451\) 1.47925e46 0.527171
\(452\) 0 0
\(453\) 4.87555e46 1.60807
\(454\) 0 0
\(455\) 1.27011e46 0.387830
\(456\) 0 0
\(457\) 4.67626e46 1.32241 0.661204 0.750206i \(-0.270046\pi\)
0.661204 + 0.750206i \(0.270046\pi\)
\(458\) 0 0
\(459\) 8.94930e45 0.234459
\(460\) 0 0
\(461\) −3.72978e46 −0.905560 −0.452780 0.891622i \(-0.649568\pi\)
−0.452780 + 0.891622i \(0.649568\pi\)
\(462\) 0 0
\(463\) 3.67649e46 0.827497 0.413748 0.910391i \(-0.364219\pi\)
0.413748 + 0.910391i \(0.364219\pi\)
\(464\) 0 0
\(465\) 2.41858e46 0.504817
\(466\) 0 0
\(467\) 2.37743e46 0.460323 0.230161 0.973152i \(-0.426075\pi\)
0.230161 + 0.973152i \(0.426075\pi\)
\(468\) 0 0
\(469\) 2.05639e46 0.369472
\(470\) 0 0
\(471\) 5.18607e46 0.864914
\(472\) 0 0
\(473\) −1.69029e45 −0.0261753
\(474\) 0 0
\(475\) −5.09814e45 −0.0733287
\(476\) 0 0
\(477\) −6.94094e46 −0.927569
\(478\) 0 0
\(479\) 8.80315e46 1.09337 0.546684 0.837339i \(-0.315890\pi\)
0.546684 + 0.837339i \(0.315890\pi\)
\(480\) 0 0
\(481\) 1.99788e47 2.30691
\(482\) 0 0
\(483\) 1.51046e46 0.162194
\(484\) 0 0
\(485\) 5.89206e46 0.588555
\(486\) 0 0
\(487\) −7.13443e46 −0.663136 −0.331568 0.943431i \(-0.607578\pi\)
−0.331568 + 0.943431i \(0.607578\pi\)
\(488\) 0 0
\(489\) −2.97758e47 −2.57607
\(490\) 0 0
\(491\) −8.49115e46 −0.683975 −0.341988 0.939704i \(-0.611100\pi\)
−0.341988 + 0.939704i \(0.611100\pi\)
\(492\) 0 0
\(493\) 4.24693e46 0.318606
\(494\) 0 0
\(495\) −2.43859e47 −1.70431
\(496\) 0 0
\(497\) −9.58638e45 −0.0624335
\(498\) 0 0
\(499\) 1.75876e47 1.06770 0.533848 0.845581i \(-0.320745\pi\)
0.533848 + 0.845581i \(0.320745\pi\)
\(500\) 0 0
\(501\) 3.19397e47 1.80789
\(502\) 0 0
\(503\) −1.24051e47 −0.654880 −0.327440 0.944872i \(-0.606186\pi\)
−0.327440 + 0.944872i \(0.606186\pi\)
\(504\) 0 0
\(505\) −2.32072e47 −1.14294
\(506\) 0 0
\(507\) 5.11027e47 2.34858
\(508\) 0 0
\(509\) 3.71512e47 1.59372 0.796859 0.604166i \(-0.206493\pi\)
0.796859 + 0.604166i \(0.206493\pi\)
\(510\) 0 0
\(511\) −8.86212e46 −0.354954
\(512\) 0 0
\(513\) −2.67296e46 −0.0999857
\(514\) 0 0
\(515\) 5.19987e47 1.81705
\(516\) 0 0
\(517\) −3.34597e47 −1.09254
\(518\) 0 0
\(519\) −4.60621e47 −1.40577
\(520\) 0 0
\(521\) 7.89659e45 0.0225309 0.0112655 0.999937i \(-0.496414\pi\)
0.0112655 + 0.999937i \(0.496414\pi\)
\(522\) 0 0
\(523\) −1.18122e47 −0.315175 −0.157588 0.987505i \(-0.550372\pi\)
−0.157588 + 0.987505i \(0.550372\pi\)
\(524\) 0 0
\(525\) 5.59656e46 0.139680
\(526\) 0 0
\(527\) −4.50331e46 −0.105159
\(528\) 0 0
\(529\) −3.35870e47 −0.734001
\(530\) 0 0
\(531\) −2.72895e47 −0.558267
\(532\) 0 0
\(533\) −4.28587e47 −0.820943
\(534\) 0 0
\(535\) −7.46124e47 −1.33850
\(536\) 0 0
\(537\) −2.16958e47 −0.364607
\(538\) 0 0
\(539\) −6.20419e47 −0.976971
\(540\) 0 0
\(541\) −6.62658e47 −0.977997 −0.488998 0.872285i \(-0.662638\pi\)
−0.488998 + 0.872285i \(0.662638\pi\)
\(542\) 0 0
\(543\) −8.30070e47 −1.14846
\(544\) 0 0
\(545\) 9.54073e46 0.123778
\(546\) 0 0
\(547\) 1.12424e47 0.136798 0.0683990 0.997658i \(-0.478211\pi\)
0.0683990 + 0.997658i \(0.478211\pi\)
\(548\) 0 0
\(549\) 5.32564e46 0.0607933
\(550\) 0 0
\(551\) −1.26846e47 −0.135870
\(552\) 0 0
\(553\) −2.93624e47 −0.295190
\(554\) 0 0
\(555\) 2.86238e48 2.70147
\(556\) 0 0
\(557\) 6.68090e47 0.592064 0.296032 0.955178i \(-0.404336\pi\)
0.296032 + 0.955178i \(0.404336\pi\)
\(558\) 0 0
\(559\) 4.89734e46 0.0407618
\(560\) 0 0
\(561\) 7.80339e47 0.610145
\(562\) 0 0
\(563\) −2.47733e48 −1.82007 −0.910033 0.414536i \(-0.863944\pi\)
−0.910033 + 0.414536i \(0.863944\pi\)
\(564\) 0 0
\(565\) 4.47639e47 0.309087
\(566\) 0 0
\(567\) −1.42558e47 −0.0925309
\(568\) 0 0
\(569\) 8.59661e47 0.524641 0.262321 0.964981i \(-0.415512\pi\)
0.262321 + 0.964981i \(0.415512\pi\)
\(570\) 0 0
\(571\) 3.39185e48 1.94673 0.973364 0.229266i \(-0.0736327\pi\)
0.973364 + 0.229266i \(0.0736327\pi\)
\(572\) 0 0
\(573\) −2.81753e48 −1.52112
\(574\) 0 0
\(575\) 4.50988e47 0.229075
\(576\) 0 0
\(577\) 2.12534e48 1.01590 0.507951 0.861386i \(-0.330403\pi\)
0.507951 + 0.861386i \(0.330403\pi\)
\(578\) 0 0
\(579\) 4.79258e48 2.15623
\(580\) 0 0
\(581\) −6.49544e47 −0.275123
\(582\) 0 0
\(583\) −1.70314e48 −0.679283
\(584\) 0 0
\(585\) 7.06542e48 2.65406
\(586\) 0 0
\(587\) −1.06269e48 −0.376044 −0.188022 0.982165i \(-0.560208\pi\)
−0.188022 + 0.982165i \(0.560208\pi\)
\(588\) 0 0
\(589\) 1.34504e47 0.0448453
\(590\) 0 0
\(591\) −3.99665e48 −1.25578
\(592\) 0 0
\(593\) −3.98260e48 −1.17953 −0.589766 0.807574i \(-0.700780\pi\)
−0.589766 + 0.807574i \(0.700780\pi\)
\(594\) 0 0
\(595\) −3.38820e47 −0.0946068
\(596\) 0 0
\(597\) 1.64894e48 0.434163
\(598\) 0 0
\(599\) −5.14535e48 −1.27775 −0.638874 0.769311i \(-0.720599\pi\)
−0.638874 + 0.769311i \(0.720599\pi\)
\(600\) 0 0
\(601\) −1.83878e48 −0.430751 −0.215375 0.976531i \(-0.569098\pi\)
−0.215375 + 0.976531i \(0.569098\pi\)
\(602\) 0 0
\(603\) 1.14394e49 2.52842
\(604\) 0 0
\(605\) −2.22350e47 −0.0463787
\(606\) 0 0
\(607\) −2.06846e48 −0.407236 −0.203618 0.979050i \(-0.565270\pi\)
−0.203618 + 0.979050i \(0.565270\pi\)
\(608\) 0 0
\(609\) 1.39247e48 0.258812
\(610\) 0 0
\(611\) 9.69439e48 1.70137
\(612\) 0 0
\(613\) 1.02050e48 0.169143 0.0845717 0.996417i \(-0.473048\pi\)
0.0845717 + 0.996417i \(0.473048\pi\)
\(614\) 0 0
\(615\) −6.14039e48 −0.961350
\(616\) 0 0
\(617\) −3.46313e48 −0.512246 −0.256123 0.966644i \(-0.582445\pi\)
−0.256123 + 0.966644i \(0.582445\pi\)
\(618\) 0 0
\(619\) −6.56428e48 −0.917488 −0.458744 0.888568i \(-0.651701\pi\)
−0.458744 + 0.888568i \(0.651701\pi\)
\(620\) 0 0
\(621\) 2.36453e48 0.312350
\(622\) 0 0
\(623\) 1.80954e48 0.225958
\(624\) 0 0
\(625\) −1.05615e49 −1.24688
\(626\) 0 0
\(627\) −2.33070e48 −0.260198
\(628\) 0 0
\(629\) −5.32965e48 −0.562746
\(630\) 0 0
\(631\) −5.18692e48 −0.518079 −0.259039 0.965867i \(-0.583406\pi\)
−0.259039 + 0.965867i \(0.583406\pi\)
\(632\) 0 0
\(633\) 6.48082e48 0.612442
\(634\) 0 0
\(635\) 8.60152e48 0.769192
\(636\) 0 0
\(637\) 1.79756e49 1.52140
\(638\) 0 0
\(639\) −5.33275e48 −0.427254
\(640\) 0 0
\(641\) −6.15569e48 −0.466940 −0.233470 0.972364i \(-0.575008\pi\)
−0.233470 + 0.972364i \(0.575008\pi\)
\(642\) 0 0
\(643\) 6.50853e48 0.467510 0.233755 0.972296i \(-0.424899\pi\)
0.233755 + 0.972296i \(0.424899\pi\)
\(644\) 0 0
\(645\) 7.01644e47 0.0477333
\(646\) 0 0
\(647\) −2.43533e49 −1.56940 −0.784698 0.619878i \(-0.787182\pi\)
−0.784698 + 0.619878i \(0.787182\pi\)
\(648\) 0 0
\(649\) −6.69620e48 −0.408833
\(650\) 0 0
\(651\) −1.47654e48 −0.0854233
\(652\) 0 0
\(653\) 4.33343e48 0.237602 0.118801 0.992918i \(-0.462095\pi\)
0.118801 + 0.992918i \(0.462095\pi\)
\(654\) 0 0
\(655\) 4.93325e48 0.256396
\(656\) 0 0
\(657\) −4.92986e49 −2.42907
\(658\) 0 0
\(659\) −2.26798e49 −1.05960 −0.529802 0.848121i \(-0.677734\pi\)
−0.529802 + 0.848121i \(0.677734\pi\)
\(660\) 0 0
\(661\) −2.84364e49 −1.25994 −0.629969 0.776620i \(-0.716932\pi\)
−0.629969 + 0.776620i \(0.716932\pi\)
\(662\) 0 0
\(663\) −2.26090e49 −0.950156
\(664\) 0 0
\(665\) 1.01198e48 0.0403453
\(666\) 0 0
\(667\) 1.12210e49 0.424452
\(668\) 0 0
\(669\) 2.46325e49 0.884202
\(670\) 0 0
\(671\) 1.30679e48 0.0445205
\(672\) 0 0
\(673\) 1.14434e49 0.370077 0.185039 0.982731i \(-0.440759\pi\)
0.185039 + 0.982731i \(0.440759\pi\)
\(674\) 0 0
\(675\) 8.76105e48 0.268993
\(676\) 0 0
\(677\) −3.00594e49 −0.876351 −0.438175 0.898890i \(-0.644375\pi\)
−0.438175 + 0.898890i \(0.644375\pi\)
\(678\) 0 0
\(679\) −3.59708e48 −0.0995931
\(680\) 0 0
\(681\) 3.59766e49 0.946119
\(682\) 0 0
\(683\) 7.78736e49 1.94549 0.972743 0.231885i \(-0.0744894\pi\)
0.972743 + 0.231885i \(0.0744894\pi\)
\(684\) 0 0
\(685\) 4.40201e49 1.04488
\(686\) 0 0
\(687\) 6.32535e49 1.42673
\(688\) 0 0
\(689\) 4.93456e49 1.05782
\(690\) 0 0
\(691\) −9.61614e49 −1.95945 −0.979726 0.200340i \(-0.935795\pi\)
−0.979726 + 0.200340i \(0.935795\pi\)
\(692\) 0 0
\(693\) 1.48875e49 0.288397
\(694\) 0 0
\(695\) 5.39613e49 0.993913
\(696\) 0 0
\(697\) 1.14332e49 0.200260
\(698\) 0 0
\(699\) 5.46014e49 0.909606
\(700\) 0 0
\(701\) 1.19741e49 0.189748 0.0948742 0.995489i \(-0.469755\pi\)
0.0948742 + 0.995489i \(0.469755\pi\)
\(702\) 0 0
\(703\) 1.59185e49 0.239984
\(704\) 0 0
\(705\) 1.38892e50 1.99236
\(706\) 0 0
\(707\) 1.41679e49 0.193404
\(708\) 0 0
\(709\) −4.80032e49 −0.623680 −0.311840 0.950135i \(-0.600945\pi\)
−0.311840 + 0.950135i \(0.600945\pi\)
\(710\) 0 0
\(711\) −1.63339e50 −2.02009
\(712\) 0 0
\(713\) −1.18984e49 −0.140094
\(714\) 0 0
\(715\) 1.73369e50 1.94363
\(716\) 0 0
\(717\) −1.73297e50 −1.85015
\(718\) 0 0
\(719\) 1.45387e49 0.147832 0.0739162 0.997264i \(-0.476450\pi\)
0.0739162 + 0.997264i \(0.476450\pi\)
\(720\) 0 0
\(721\) −3.17450e49 −0.307474
\(722\) 0 0
\(723\) −1.15246e50 −1.06342
\(724\) 0 0
\(725\) 4.15759e49 0.365533
\(726\) 0 0
\(727\) −1.25503e50 −1.05148 −0.525742 0.850644i \(-0.676212\pi\)
−0.525742 + 0.850644i \(0.676212\pi\)
\(728\) 0 0
\(729\) −1.96598e50 −1.56981
\(730\) 0 0
\(731\) −1.30644e48 −0.00994338
\(732\) 0 0
\(733\) −1.40440e50 −1.01899 −0.509495 0.860473i \(-0.670168\pi\)
−0.509495 + 0.860473i \(0.670168\pi\)
\(734\) 0 0
\(735\) 2.57537e50 1.78161
\(736\) 0 0
\(737\) 2.80695e50 1.85163
\(738\) 0 0
\(739\) −2.93572e49 −0.184687 −0.0923435 0.995727i \(-0.529436\pi\)
−0.0923435 + 0.995727i \(0.529436\pi\)
\(740\) 0 0
\(741\) 6.75281e49 0.405197
\(742\) 0 0
\(743\) −1.10795e50 −0.634185 −0.317093 0.948395i \(-0.602707\pi\)
−0.317093 + 0.948395i \(0.602707\pi\)
\(744\) 0 0
\(745\) 3.57521e50 1.95239
\(746\) 0 0
\(747\) −3.61331e50 −1.88276
\(748\) 0 0
\(749\) 4.55506e49 0.226497
\(750\) 0 0
\(751\) 4.13601e50 1.96283 0.981416 0.191892i \(-0.0614622\pi\)
0.981416 + 0.191892i \(0.0614622\pi\)
\(752\) 0 0
\(753\) −5.28418e50 −2.39368
\(754\) 0 0
\(755\) −2.88961e50 −1.24959
\(756\) 0 0
\(757\) 3.15399e50 1.30221 0.651106 0.758987i \(-0.274305\pi\)
0.651106 + 0.758987i \(0.274305\pi\)
\(758\) 0 0
\(759\) 2.06176e50 0.812845
\(760\) 0 0
\(761\) −3.65841e50 −1.37740 −0.688702 0.725045i \(-0.741819\pi\)
−0.688702 + 0.725045i \(0.741819\pi\)
\(762\) 0 0
\(763\) −5.82458e48 −0.0209452
\(764\) 0 0
\(765\) −1.88480e50 −0.647427
\(766\) 0 0
\(767\) 1.94011e50 0.636661
\(768\) 0 0
\(769\) −2.07264e50 −0.649849 −0.324925 0.945740i \(-0.605339\pi\)
−0.324925 + 0.945740i \(0.605339\pi\)
\(770\) 0 0
\(771\) −9.42552e50 −2.82393
\(772\) 0 0
\(773\) 2.04692e50 0.586084 0.293042 0.956100i \(-0.405332\pi\)
0.293042 + 0.956100i \(0.405332\pi\)
\(774\) 0 0
\(775\) −4.40859e49 −0.120648
\(776\) 0 0
\(777\) −1.74747e50 −0.457133
\(778\) 0 0
\(779\) −3.41484e49 −0.0854013
\(780\) 0 0
\(781\) −1.30853e50 −0.312889
\(782\) 0 0
\(783\) 2.17983e50 0.498415
\(784\) 0 0
\(785\) −3.07365e50 −0.672101
\(786\) 0 0
\(787\) 2.74398e50 0.573882 0.286941 0.957948i \(-0.407362\pi\)
0.286941 + 0.957948i \(0.407362\pi\)
\(788\) 0 0
\(789\) −1.45994e49 −0.0292071
\(790\) 0 0
\(791\) −2.73282e49 −0.0523025
\(792\) 0 0
\(793\) −3.78619e49 −0.0693301
\(794\) 0 0
\(795\) 7.06978e50 1.23874
\(796\) 0 0
\(797\) 3.73868e50 0.626899 0.313450 0.949605i \(-0.398515\pi\)
0.313450 + 0.949605i \(0.398515\pi\)
\(798\) 0 0
\(799\) −2.58612e50 −0.415030
\(800\) 0 0
\(801\) 1.00662e51 1.54631
\(802\) 0 0
\(803\) −1.20967e51 −1.77887
\(804\) 0 0
\(805\) −8.95210e49 −0.126037
\(806\) 0 0
\(807\) −4.51827e50 −0.609095
\(808\) 0 0
\(809\) −3.87407e50 −0.500114 −0.250057 0.968231i \(-0.580449\pi\)
−0.250057 + 0.968231i \(0.580449\pi\)
\(810\) 0 0
\(811\) 9.21816e50 1.13967 0.569835 0.821759i \(-0.307007\pi\)
0.569835 + 0.821759i \(0.307007\pi\)
\(812\) 0 0
\(813\) 1.42488e50 0.168730
\(814\) 0 0
\(815\) 1.76473e51 2.00180
\(816\) 0 0
\(817\) 3.90203e48 0.00424038
\(818\) 0 0
\(819\) −4.31341e50 −0.449110
\(820\) 0 0
\(821\) 1.64737e51 1.64357 0.821783 0.569801i \(-0.192980\pi\)
0.821783 + 0.569801i \(0.192980\pi\)
\(822\) 0 0
\(823\) 6.02163e50 0.575727 0.287864 0.957671i \(-0.407055\pi\)
0.287864 + 0.957671i \(0.407055\pi\)
\(824\) 0 0
\(825\) 7.63924e50 0.700013
\(826\) 0 0
\(827\) 4.30476e50 0.378097 0.189048 0.981968i \(-0.439460\pi\)
0.189048 + 0.981968i \(0.439460\pi\)
\(828\) 0 0
\(829\) 8.88751e50 0.748300 0.374150 0.927368i \(-0.377935\pi\)
0.374150 + 0.927368i \(0.377935\pi\)
\(830\) 0 0
\(831\) 2.55364e51 2.06131
\(832\) 0 0
\(833\) −4.79525e50 −0.371128
\(834\) 0 0
\(835\) −1.89298e51 −1.40486
\(836\) 0 0
\(837\) −2.31142e50 −0.164507
\(838\) 0 0
\(839\) −3.76912e50 −0.257280 −0.128640 0.991691i \(-0.541061\pi\)
−0.128640 + 0.991691i \(0.541061\pi\)
\(840\) 0 0
\(841\) −4.92875e50 −0.322706
\(842\) 0 0
\(843\) 1.41210e51 0.886916
\(844\) 0 0
\(845\) −3.02872e51 −1.82502
\(846\) 0 0
\(847\) 1.35744e49 0.00784804
\(848\) 0 0
\(849\) 3.68407e51 2.04382
\(850\) 0 0
\(851\) −1.40817e51 −0.749699
\(852\) 0 0
\(853\) −3.00248e51 −1.53417 −0.767083 0.641547i \(-0.778293\pi\)
−0.767083 + 0.641547i \(0.778293\pi\)
\(854\) 0 0
\(855\) 5.62949e50 0.276097
\(856\) 0 0
\(857\) −2.46088e50 −0.115858 −0.0579291 0.998321i \(-0.518450\pi\)
−0.0579291 + 0.998321i \(0.518450\pi\)
\(858\) 0 0
\(859\) −2.64506e51 −1.19551 −0.597757 0.801677i \(-0.703941\pi\)
−0.597757 + 0.801677i \(0.703941\pi\)
\(860\) 0 0
\(861\) 3.74868e50 0.162676
\(862\) 0 0
\(863\) 9.93861e50 0.414130 0.207065 0.978327i \(-0.433609\pi\)
0.207065 + 0.978327i \(0.433609\pi\)
\(864\) 0 0
\(865\) 2.72998e51 1.09239
\(866\) 0 0
\(867\) −3.42107e51 −1.31470
\(868\) 0 0
\(869\) −4.00794e51 −1.47936
\(870\) 0 0
\(871\) −8.13268e51 −2.88347
\(872\) 0 0
\(873\) −2.00100e51 −0.681550
\(874\) 0 0
\(875\) 4.15096e50 0.135834
\(876\) 0 0
\(877\) −2.79832e50 −0.0879842 −0.0439921 0.999032i \(-0.514008\pi\)
−0.0439921 + 0.999032i \(0.514008\pi\)
\(878\) 0 0
\(879\) 6.58876e51 1.99067
\(880\) 0 0
\(881\) −5.95050e50 −0.172773 −0.0863863 0.996262i \(-0.527532\pi\)
−0.0863863 + 0.996262i \(0.527532\pi\)
\(882\) 0 0
\(883\) 3.85350e51 1.07533 0.537667 0.843157i \(-0.319306\pi\)
0.537667 + 0.843157i \(0.319306\pi\)
\(884\) 0 0
\(885\) 2.77961e51 0.745550
\(886\) 0 0
\(887\) 4.56338e50 0.117658 0.0588292 0.998268i \(-0.481263\pi\)
0.0588292 + 0.998268i \(0.481263\pi\)
\(888\) 0 0
\(889\) −5.25119e50 −0.130160
\(890\) 0 0
\(891\) −1.94590e51 −0.463724
\(892\) 0 0
\(893\) 7.72416e50 0.176991
\(894\) 0 0
\(895\) 1.28585e51 0.283326
\(896\) 0 0
\(897\) −5.97362e51 −1.26581
\(898\) 0 0
\(899\) −1.09689e51 −0.223548
\(900\) 0 0
\(901\) −1.31637e51 −0.258044
\(902\) 0 0
\(903\) −4.28351e49 −0.00807726
\(904\) 0 0
\(905\) 4.91960e51 0.892441
\(906\) 0 0
\(907\) 4.12286e51 0.719566 0.359783 0.933036i \(-0.382851\pi\)
0.359783 + 0.933036i \(0.382851\pi\)
\(908\) 0 0
\(909\) 7.88139e51 1.32353
\(910\) 0 0
\(911\) −6.95561e51 −1.12399 −0.561996 0.827140i \(-0.689966\pi\)
−0.561996 + 0.827140i \(0.689966\pi\)
\(912\) 0 0
\(913\) −8.86621e51 −1.37879
\(914\) 0 0
\(915\) −5.42450e50 −0.0811877
\(916\) 0 0
\(917\) −3.01173e50 −0.0433863
\(918\) 0 0
\(919\) −1.23335e51 −0.171027 −0.0855136 0.996337i \(-0.527253\pi\)
−0.0855136 + 0.996337i \(0.527253\pi\)
\(920\) 0 0
\(921\) 4.83420e51 0.645329
\(922\) 0 0
\(923\) 3.79125e51 0.487251
\(924\) 0 0
\(925\) −5.21754e51 −0.645632
\(926\) 0 0
\(927\) −1.76592e52 −2.10415
\(928\) 0 0
\(929\) −9.97208e51 −1.14422 −0.572112 0.820176i \(-0.693876\pi\)
−0.572112 + 0.820176i \(0.693876\pi\)
\(930\) 0 0
\(931\) 1.43224e51 0.158269
\(932\) 0 0
\(933\) −7.69232e51 −0.818707
\(934\) 0 0
\(935\) −4.62486e51 −0.474128
\(936\) 0 0
\(937\) 6.01334e51 0.593844 0.296922 0.954902i \(-0.404040\pi\)
0.296922 + 0.954902i \(0.404040\pi\)
\(938\) 0 0
\(939\) 1.25626e52 1.19517
\(940\) 0 0
\(941\) 1.38035e52 1.26523 0.632617 0.774465i \(-0.281981\pi\)
0.632617 + 0.774465i \(0.281981\pi\)
\(942\) 0 0
\(943\) 3.02081e51 0.266789
\(944\) 0 0
\(945\) −1.73907e51 −0.147999
\(946\) 0 0
\(947\) 9.01648e51 0.739456 0.369728 0.929140i \(-0.379451\pi\)
0.369728 + 0.929140i \(0.379451\pi\)
\(948\) 0 0
\(949\) 3.50482e52 2.77017
\(950\) 0 0
\(951\) 1.83121e52 1.39502
\(952\) 0 0
\(953\) 4.32450e51 0.317549 0.158774 0.987315i \(-0.449246\pi\)
0.158774 + 0.987315i \(0.449246\pi\)
\(954\) 0 0
\(955\) 1.66987e52 1.18202
\(956\) 0 0
\(957\) 1.90071e52 1.29705
\(958\) 0 0
\(959\) −2.68741e51 −0.176810
\(960\) 0 0
\(961\) −1.46006e52 −0.926216
\(962\) 0 0
\(963\) 2.53391e52 1.55000
\(964\) 0 0
\(965\) −2.84044e52 −1.67555
\(966\) 0 0
\(967\) 1.91085e51 0.108709 0.0543543 0.998522i \(-0.482690\pi\)
0.0543543 + 0.998522i \(0.482690\pi\)
\(968\) 0 0
\(969\) −1.80141e51 −0.0988431
\(970\) 0 0
\(971\) −2.54698e52 −1.34800 −0.674000 0.738732i \(-0.735425\pi\)
−0.674000 + 0.738732i \(0.735425\pi\)
\(972\) 0 0
\(973\) −3.29432e51 −0.168186
\(974\) 0 0
\(975\) −2.21334e52 −1.09010
\(976\) 0 0
\(977\) 2.34465e52 1.11410 0.557048 0.830480i \(-0.311934\pi\)
0.557048 + 0.830480i \(0.311934\pi\)
\(978\) 0 0
\(979\) 2.47001e52 1.13240
\(980\) 0 0
\(981\) −3.24012e51 −0.143335
\(982\) 0 0
\(983\) −1.06892e52 −0.456308 −0.228154 0.973625i \(-0.573269\pi\)
−0.228154 + 0.973625i \(0.573269\pi\)
\(984\) 0 0
\(985\) 2.36871e52 0.975837
\(986\) 0 0
\(987\) −8.47931e51 −0.337140
\(988\) 0 0
\(989\) −3.45178e50 −0.0132467
\(990\) 0 0
\(991\) −1.97227e52 −0.730597 −0.365298 0.930891i \(-0.619033\pi\)
−0.365298 + 0.930891i \(0.619033\pi\)
\(992\) 0 0
\(993\) −2.16387e52 −0.773782
\(994\) 0 0
\(995\) −9.77280e51 −0.337377
\(996\) 0 0
\(997\) −1.44359e52 −0.481149 −0.240575 0.970631i \(-0.577336\pi\)
−0.240575 + 0.970631i \(0.577336\pi\)
\(998\) 0 0
\(999\) −2.73555e52 −0.880338
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 16.36.a.d.1.3 3
4.3 odd 2 1.36.a.a.1.1 3
12.11 even 2 9.36.a.b.1.3 3
20.3 even 4 25.36.b.a.24.5 6
20.7 even 4 25.36.b.a.24.2 6
20.19 odd 2 25.36.a.a.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.36.a.a.1.1 3 4.3 odd 2
9.36.a.b.1.3 3 12.11 even 2
16.36.a.d.1.3 3 1.1 even 1 trivial
25.36.a.a.1.3 3 20.19 odd 2
25.36.b.a.24.2 6 20.7 even 4
25.36.b.a.24.5 6 20.3 even 4