Properties

Label 9.36.a.a.1.1
Level $9$
Weight $36$
Character 9.1
Self dual yes
Analytic conductor $69.836$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.8356175703\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2196841}) \)
Defining polynomial: \(x^{2} - x - 549210\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(741.587\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-218549. q^{2} +1.34041e10 q^{4} +9.68425e11 q^{5} -5.65365e14 q^{7} +4.57985e15 q^{8} +O(q^{10})\) \(q-218549. q^{2} +1.34041e10 q^{4} +9.68425e11 q^{5} -5.65365e14 q^{7} +4.57985e15 q^{8} -2.11649e17 q^{10} +1.62372e18 q^{11} -2.52538e19 q^{13} +1.23560e20 q^{14} -1.46148e21 q^{16} -4.37227e21 q^{17} +1.72036e22 q^{19} +1.29808e22 q^{20} -3.54863e23 q^{22} +5.13356e23 q^{23} -1.97254e24 q^{25} +5.51920e24 q^{26} -7.57819e24 q^{28} +3.39397e25 q^{29} -9.89761e25 q^{31} +1.62044e26 q^{32} +9.55557e26 q^{34} -5.47514e26 q^{35} +4.58147e27 q^{37} -3.75982e27 q^{38} +4.43524e27 q^{40} +2.87989e28 q^{41} +6.85354e28 q^{43} +2.17644e28 q^{44} -1.12194e29 q^{46} +4.79363e26 q^{47} -5.91811e28 q^{49} +4.31096e29 q^{50} -3.38503e29 q^{52} -2.32896e30 q^{53} +1.57245e30 q^{55} -2.58929e30 q^{56} -7.41751e30 q^{58} -5.75836e30 q^{59} -2.67580e31 q^{61} +2.16312e31 q^{62} +1.48016e31 q^{64} -2.44564e31 q^{65} +7.66448e31 q^{67} -5.86062e31 q^{68} +1.19659e32 q^{70} +5.08152e31 q^{71} -2.74698e32 q^{73} -1.00128e33 q^{74} +2.30597e32 q^{76} -9.17994e32 q^{77} -2.13670e33 q^{79} -1.41534e33 q^{80} -6.29399e33 q^{82} +2.22116e33 q^{83} -4.23422e33 q^{85} -1.49784e34 q^{86} +7.43639e33 q^{88} -2.07592e34 q^{89} +1.42776e34 q^{91} +6.88105e33 q^{92} -1.04765e32 q^{94} +1.66604e34 q^{95} +1.55170e34 q^{97} +1.29340e34 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} + O(q^{10}) \) \( 2q + 60912q^{2} + 57142939904q^{4} + 1333779496740q^{5} - 1201512782952944q^{7} + 7200956480385024q^{8} - 109546268366103840q^{10} + 1474443852221320632q^{11} + 30005213658205678828q^{13} - 54218555116626208896q^{14} - \)\(22\!\cdots\!20\)\(q^{16} - \)\(61\!\cdots\!32\)\(q^{17} - \)\(16\!\cdots\!64\)\(q^{19} + \)\(28\!\cdots\!20\)\(q^{20} - \)\(39\!\cdots\!16\)\(q^{22} - \)\(49\!\cdots\!72\)\(q^{23} - \)\(47\!\cdots\!50\)\(q^{25} + \)\(20\!\cdots\!56\)\(q^{26} - \)\(35\!\cdots\!72\)\(q^{28} + \)\(78\!\cdots\!48\)\(q^{29} - \)\(12\!\cdots\!48\)\(q^{31} - \)\(14\!\cdots\!76\)\(q^{32} + \)\(44\!\cdots\!88\)\(q^{34} - \)\(77\!\cdots\!60\)\(q^{35} + \)\(16\!\cdots\!16\)\(q^{37} - \)\(13\!\cdots\!12\)\(q^{38} + \)\(53\!\cdots\!00\)\(q^{40} + \)\(32\!\cdots\!88\)\(q^{41} + \)\(10\!\cdots\!20\)\(q^{43} + \)\(15\!\cdots\!68\)\(q^{44} - \)\(39\!\cdots\!48\)\(q^{46} + \)\(52\!\cdots\!20\)\(q^{47} - \)\(33\!\cdots\!50\)\(q^{49} - \)\(34\!\cdots\!00\)\(q^{50} + \)\(20\!\cdots\!12\)\(q^{52} + \)\(31\!\cdots\!68\)\(q^{53} + \)\(15\!\cdots\!20\)\(q^{55} - \)\(42\!\cdots\!20\)\(q^{56} + \)\(51\!\cdots\!24\)\(q^{58} - \)\(82\!\cdots\!64\)\(q^{59} - \)\(40\!\cdots\!40\)\(q^{61} + \)\(15\!\cdots\!68\)\(q^{62} - \)\(44\!\cdots\!64\)\(q^{64} - \)\(42\!\cdots\!20\)\(q^{65} + \)\(96\!\cdots\!12\)\(q^{67} - \)\(13\!\cdots\!04\)\(q^{68} + \)\(54\!\cdots\!60\)\(q^{70} - \)\(44\!\cdots\!24\)\(q^{71} - \)\(13\!\cdots\!08\)\(q^{73} - \)\(18\!\cdots\!76\)\(q^{74} - \)\(12\!\cdots\!64\)\(q^{76} - \)\(82\!\cdots\!12\)\(q^{77} - \)\(10\!\cdots\!80\)\(q^{79} - \)\(16\!\cdots\!60\)\(q^{80} - \)\(51\!\cdots\!48\)\(q^{82} + \)\(55\!\cdots\!44\)\(q^{83} - \)\(48\!\cdots\!40\)\(q^{85} - \)\(31\!\cdots\!88\)\(q^{86} + \)\(70\!\cdots\!36\)\(q^{88} - \)\(24\!\cdots\!36\)\(q^{89} - \)\(20\!\cdots\!28\)\(q^{91} - \)\(37\!\cdots\!36\)\(q^{92} + \)\(14\!\cdots\!04\)\(q^{94} + \)\(45\!\cdots\!00\)\(q^{95} + \)\(83\!\cdots\!52\)\(q^{97} + \)\(20\!\cdots\!08\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −218549. −1.17903 −0.589515 0.807758i \(-0.700681\pi\)
−0.589515 + 0.807758i \(0.700681\pi\)
\(3\) 0 0
\(4\) 1.34041e10 0.390109
\(5\) 9.68425e11 0.567664 0.283832 0.958874i \(-0.408394\pi\)
0.283832 + 0.958874i \(0.408394\pi\)
\(6\) 0 0
\(7\) −5.65365e14 −0.918572 −0.459286 0.888288i \(-0.651895\pi\)
−0.459286 + 0.888288i \(0.651895\pi\)
\(8\) 4.57985e15 0.719079
\(9\) 0 0
\(10\) −2.11649e17 −0.669292
\(11\) 1.62372e18 0.968588 0.484294 0.874905i \(-0.339077\pi\)
0.484294 + 0.874905i \(0.339077\pi\)
\(12\) 0 0
\(13\) −2.52538e19 −0.809688 −0.404844 0.914386i \(-0.632674\pi\)
−0.404844 + 0.914386i \(0.632674\pi\)
\(14\) 1.23560e20 1.08302
\(15\) 0 0
\(16\) −1.46148e21 −1.23792
\(17\) −4.37227e21 −1.28189 −0.640946 0.767586i \(-0.721458\pi\)
−0.640946 + 0.767586i \(0.721458\pi\)
\(18\) 0 0
\(19\) 1.72036e22 0.720162 0.360081 0.932921i \(-0.382749\pi\)
0.360081 + 0.932921i \(0.382749\pi\)
\(20\) 1.29808e22 0.221451
\(21\) 0 0
\(22\) −3.54863e23 −1.14199
\(23\) 5.13356e23 0.758895 0.379447 0.925213i \(-0.376114\pi\)
0.379447 + 0.925213i \(0.376114\pi\)
\(24\) 0 0
\(25\) −1.97254e24 −0.677758
\(26\) 5.51920e24 0.954646
\(27\) 0 0
\(28\) −7.57819e24 −0.358344
\(29\) 3.39397e25 0.868447 0.434224 0.900805i \(-0.357023\pi\)
0.434224 + 0.900805i \(0.357023\pi\)
\(30\) 0 0
\(31\) −9.89761e25 −0.788317 −0.394158 0.919043i \(-0.628964\pi\)
−0.394158 + 0.919043i \(0.628964\pi\)
\(32\) 1.62044e26 0.740470
\(33\) 0 0
\(34\) 9.55557e26 1.51139
\(35\) −5.47514e26 −0.521440
\(36\) 0 0
\(37\) 4.58147e27 1.64996 0.824982 0.565160i \(-0.191185\pi\)
0.824982 + 0.565160i \(0.191185\pi\)
\(38\) −3.75982e27 −0.849092
\(39\) 0 0
\(40\) 4.43524e27 0.408195
\(41\) 2.87989e28 1.72052 0.860259 0.509858i \(-0.170302\pi\)
0.860259 + 0.509858i \(0.170302\pi\)
\(42\) 0 0
\(43\) 6.85354e28 1.77917 0.889583 0.456773i \(-0.150995\pi\)
0.889583 + 0.456773i \(0.150995\pi\)
\(44\) 2.17644e28 0.377855
\(45\) 0 0
\(46\) −1.12194e29 −0.894759
\(47\) 4.79363e26 0.00262393 0.00131197 0.999999i \(-0.499582\pi\)
0.00131197 + 0.999999i \(0.499582\pi\)
\(48\) 0 0
\(49\) −5.91811e28 −0.156225
\(50\) 4.31096e29 0.799096
\(51\) 0 0
\(52\) −3.38503e29 −0.315867
\(53\) −2.32896e30 −1.55717 −0.778583 0.627541i \(-0.784061\pi\)
−0.778583 + 0.627541i \(0.784061\pi\)
\(54\) 0 0
\(55\) 1.57245e30 0.549832
\(56\) −2.58929e30 −0.660526
\(57\) 0 0
\(58\) −7.41751e30 −1.02392
\(59\) −5.75836e30 −0.589371 −0.294685 0.955594i \(-0.595215\pi\)
−0.294685 + 0.955594i \(0.595215\pi\)
\(60\) 0 0
\(61\) −2.67580e31 −1.52820 −0.764102 0.645095i \(-0.776818\pi\)
−0.764102 + 0.645095i \(0.776818\pi\)
\(62\) 2.16312e31 0.929448
\(63\) 0 0
\(64\) 1.48016e31 0.364889
\(65\) −2.44564e31 −0.459630
\(66\) 0 0
\(67\) 7.66448e31 0.847566 0.423783 0.905764i \(-0.360702\pi\)
0.423783 + 0.905764i \(0.360702\pi\)
\(68\) −5.86062e31 −0.500078
\(69\) 0 0
\(70\) 1.19659e32 0.614793
\(71\) 5.08152e31 0.203691 0.101846 0.994800i \(-0.467525\pi\)
0.101846 + 0.994800i \(0.467525\pi\)
\(72\) 0 0
\(73\) −2.74698e32 −0.677181 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(74\) −1.00128e33 −1.94535
\(75\) 0 0
\(76\) 2.30597e32 0.280942
\(77\) −9.17994e32 −0.889718
\(78\) 0 0
\(79\) −2.13670e33 −1.32212 −0.661059 0.750334i \(-0.729893\pi\)
−0.661059 + 0.750334i \(0.729893\pi\)
\(80\) −1.41534e33 −0.702724
\(81\) 0 0
\(82\) −6.29399e33 −2.02854
\(83\) 2.22116e33 0.579046 0.289523 0.957171i \(-0.406503\pi\)
0.289523 + 0.957171i \(0.406503\pi\)
\(84\) 0 0
\(85\) −4.23422e33 −0.727683
\(86\) −1.49784e34 −2.09769
\(87\) 0 0
\(88\) 7.43639e33 0.696491
\(89\) −2.07592e34 −1.59545 −0.797727 0.603018i \(-0.793965\pi\)
−0.797727 + 0.603018i \(0.793965\pi\)
\(90\) 0 0
\(91\) 1.42776e34 0.743757
\(92\) 6.88105e33 0.296052
\(93\) 0 0
\(94\) −1.04765e32 −0.00309369
\(95\) 1.66604e34 0.408810
\(96\) 0 0
\(97\) 1.55170e34 0.264426 0.132213 0.991221i \(-0.457792\pi\)
0.132213 + 0.991221i \(0.457792\pi\)
\(98\) 1.29340e34 0.184194
\(99\) 0 0
\(100\) −2.64400e34 −0.264400
\(101\) 1.03897e35 0.872929 0.436464 0.899722i \(-0.356230\pi\)
0.436464 + 0.899722i \(0.356230\pi\)
\(102\) 0 0
\(103\) −1.94833e35 −1.16148 −0.580740 0.814089i \(-0.697237\pi\)
−0.580740 + 0.814089i \(0.697237\pi\)
\(104\) −1.15659e35 −0.582229
\(105\) 0 0
\(106\) 5.08993e35 1.83594
\(107\) 3.61557e35 1.10652 0.553262 0.833007i \(-0.313383\pi\)
0.553262 + 0.833007i \(0.313383\pi\)
\(108\) 0 0
\(109\) −7.33560e35 −1.62357 −0.811786 0.583954i \(-0.801505\pi\)
−0.811786 + 0.583954i \(0.801505\pi\)
\(110\) −3.43658e35 −0.648268
\(111\) 0 0
\(112\) 8.26271e35 1.13712
\(113\) 5.95677e35 0.701678 0.350839 0.936436i \(-0.385896\pi\)
0.350839 + 0.936436i \(0.385896\pi\)
\(114\) 0 0
\(115\) 4.97147e35 0.430797
\(116\) 4.54930e35 0.338790
\(117\) 0 0
\(118\) 1.25849e36 0.694885
\(119\) 2.47193e36 1.17751
\(120\) 0 0
\(121\) −1.73778e35 −0.0618372
\(122\) 5.84794e36 1.80180
\(123\) 0 0
\(124\) −1.32668e36 −0.307530
\(125\) −4.72874e36 −0.952402
\(126\) 0 0
\(127\) 1.88168e36 0.287065 0.143533 0.989646i \(-0.454154\pi\)
0.143533 + 0.989646i \(0.454154\pi\)
\(128\) −8.80267e36 −1.17068
\(129\) 0 0
\(130\) 5.34493e36 0.541918
\(131\) −6.22324e36 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(132\) 0 0
\(133\) −9.72629e36 −0.661521
\(134\) −1.67507e37 −0.999305
\(135\) 0 0
\(136\) −2.00244e37 −0.921781
\(137\) −6.91179e36 −0.279886 −0.139943 0.990160i \(-0.544692\pi\)
−0.139943 + 0.990160i \(0.544692\pi\)
\(138\) 0 0
\(139\) −4.10165e37 −1.28884 −0.644421 0.764671i \(-0.722901\pi\)
−0.644421 + 0.764671i \(0.722901\pi\)
\(140\) −7.33891e36 −0.203419
\(141\) 0 0
\(142\) −1.11056e37 −0.240158
\(143\) −4.10051e37 −0.784254
\(144\) 0 0
\(145\) 3.28681e37 0.492986
\(146\) 6.00350e37 0.798416
\(147\) 0 0
\(148\) 6.14102e37 0.643666
\(149\) −3.38483e37 −0.315339 −0.157670 0.987492i \(-0.550398\pi\)
−0.157670 + 0.987492i \(0.550398\pi\)
\(150\) 0 0
\(151\) −4.14774e37 −0.305995 −0.152998 0.988227i \(-0.548893\pi\)
−0.152998 + 0.988227i \(0.548893\pi\)
\(152\) 7.87897e37 0.517853
\(153\) 0 0
\(154\) 2.00627e38 1.04900
\(155\) −9.58510e37 −0.447499
\(156\) 0 0
\(157\) 2.79097e38 1.04115 0.520573 0.853817i \(-0.325718\pi\)
0.520573 + 0.853817i \(0.325718\pi\)
\(158\) 4.66975e38 1.55881
\(159\) 0 0
\(160\) 1.56927e38 0.420338
\(161\) −2.90234e38 −0.697099
\(162\) 0 0
\(163\) −9.06631e38 −1.75448 −0.877238 0.480056i \(-0.840616\pi\)
−0.877238 + 0.480056i \(0.840616\pi\)
\(164\) 3.86023e38 0.671190
\(165\) 0 0
\(166\) −4.85433e38 −0.682712
\(167\) 2.20471e38 0.279134 0.139567 0.990213i \(-0.455429\pi\)
0.139567 + 0.990213i \(0.455429\pi\)
\(168\) 0 0
\(169\) −3.35033e38 −0.344405
\(170\) 9.25386e38 0.857959
\(171\) 0 0
\(172\) 9.18652e38 0.694070
\(173\) 1.20919e39 0.825442 0.412721 0.910858i \(-0.364579\pi\)
0.412721 + 0.910858i \(0.364579\pi\)
\(174\) 0 0
\(175\) 1.11520e39 0.622570
\(176\) −2.37304e39 −1.19904
\(177\) 0 0
\(178\) 4.53690e39 1.88109
\(179\) −2.04025e39 −0.766930 −0.383465 0.923555i \(-0.625269\pi\)
−0.383465 + 0.923555i \(0.625269\pi\)
\(180\) 0 0
\(181\) −4.15321e39 −1.28531 −0.642656 0.766155i \(-0.722168\pi\)
−0.642656 + 0.766155i \(0.722168\pi\)
\(182\) −3.12036e39 −0.876911
\(183\) 0 0
\(184\) 2.35109e39 0.545705
\(185\) 4.43681e39 0.936624
\(186\) 0 0
\(187\) −7.09935e39 −1.24162
\(188\) 6.42542e36 0.00102362
\(189\) 0 0
\(190\) −3.64111e39 −0.481998
\(191\) −6.52044e39 −0.787396 −0.393698 0.919240i \(-0.628804\pi\)
−0.393698 + 0.919240i \(0.628804\pi\)
\(192\) 0 0
\(193\) −1.97397e39 −0.198649 −0.0993246 0.995055i \(-0.531668\pi\)
−0.0993246 + 0.995055i \(0.531668\pi\)
\(194\) −3.39124e39 −0.311765
\(195\) 0 0
\(196\) −7.93267e38 −0.0609450
\(197\) −6.10473e39 −0.429050 −0.214525 0.976719i \(-0.568820\pi\)
−0.214525 + 0.976719i \(0.568820\pi\)
\(198\) 0 0
\(199\) −8.83517e39 −0.520338 −0.260169 0.965563i \(-0.583778\pi\)
−0.260169 + 0.965563i \(0.583778\pi\)
\(200\) −9.03392e39 −0.487361
\(201\) 0 0
\(202\) −2.27066e40 −1.02921
\(203\) −1.91883e40 −0.797732
\(204\) 0 0
\(205\) 2.78896e40 0.976675
\(206\) 4.25807e40 1.36942
\(207\) 0 0
\(208\) 3.69080e40 1.00233
\(209\) 2.79337e40 0.697540
\(210\) 0 0
\(211\) 7.88511e40 1.66673 0.833364 0.552724i \(-0.186412\pi\)
0.833364 + 0.552724i \(0.186412\pi\)
\(212\) −3.12176e40 −0.607465
\(213\) 0 0
\(214\) −7.90181e40 −1.30462
\(215\) 6.63714e40 1.00997
\(216\) 0 0
\(217\) 5.59576e40 0.724126
\(218\) 1.60319e41 1.91424
\(219\) 0 0
\(220\) 2.10772e40 0.214495
\(221\) 1.10416e41 1.03793
\(222\) 0 0
\(223\) −2.84802e40 −0.228670 −0.114335 0.993442i \(-0.536474\pi\)
−0.114335 + 0.993442i \(0.536474\pi\)
\(224\) −9.16138e40 −0.680175
\(225\) 0 0
\(226\) −1.30185e41 −0.827299
\(227\) −2.11879e41 −1.24634 −0.623168 0.782088i \(-0.714155\pi\)
−0.623168 + 0.782088i \(0.714155\pi\)
\(228\) 0 0
\(229\) 2.44651e40 0.123432 0.0617159 0.998094i \(-0.480343\pi\)
0.0617159 + 0.998094i \(0.480343\pi\)
\(230\) −1.08651e41 −0.507922
\(231\) 0 0
\(232\) 1.55439e41 0.624482
\(233\) −3.49375e41 −1.30186 −0.650929 0.759139i \(-0.725620\pi\)
−0.650929 + 0.759139i \(0.725620\pi\)
\(234\) 0 0
\(235\) 4.64228e38 0.00148951
\(236\) −7.71854e40 −0.229919
\(237\) 0 0
\(238\) −5.40239e41 −1.38832
\(239\) 3.32939e41 0.795063 0.397531 0.917589i \(-0.369867\pi\)
0.397531 + 0.917589i \(0.369867\pi\)
\(240\) 0 0
\(241\) 6.93771e40 0.143192 0.0715959 0.997434i \(-0.477191\pi\)
0.0715959 + 0.997434i \(0.477191\pi\)
\(242\) 3.79790e40 0.0729078
\(243\) 0 0
\(244\) −3.58666e41 −0.596167
\(245\) −5.73125e40 −0.0886835
\(246\) 0 0
\(247\) −4.34455e41 −0.583106
\(248\) −4.53296e41 −0.566862
\(249\) 0 0
\(250\) 1.03346e42 1.12291
\(251\) −3.37146e40 −0.0341608 −0.0170804 0.999854i \(-0.505437\pi\)
−0.0170804 + 0.999854i \(0.505437\pi\)
\(252\) 0 0
\(253\) 8.33546e41 0.735056
\(254\) −4.11240e41 −0.338458
\(255\) 0 0
\(256\) 1.41524e42 1.01538
\(257\) 2.45764e41 0.164698 0.0823489 0.996604i \(-0.473758\pi\)
0.0823489 + 0.996604i \(0.473758\pi\)
\(258\) 0 0
\(259\) −2.59020e42 −1.51561
\(260\) −3.27815e41 −0.179306
\(261\) 0 0
\(262\) 1.36009e42 0.650569
\(263\) 1.00792e42 0.451026 0.225513 0.974240i \(-0.427594\pi\)
0.225513 + 0.974240i \(0.427594\pi\)
\(264\) 0 0
\(265\) −2.25543e42 −0.883947
\(266\) 2.12567e42 0.779952
\(267\) 0 0
\(268\) 1.02735e42 0.330644
\(269\) −3.53460e42 −1.06580 −0.532900 0.846178i \(-0.678898\pi\)
−0.532900 + 0.846178i \(0.678898\pi\)
\(270\) 0 0
\(271\) 2.05300e40 0.00543785 0.00271893 0.999996i \(-0.499135\pi\)
0.00271893 + 0.999996i \(0.499135\pi\)
\(272\) 6.39000e42 1.58688
\(273\) 0 0
\(274\) 1.51057e42 0.329993
\(275\) −3.20285e42 −0.656468
\(276\) 0 0
\(277\) −3.99065e42 −0.720522 −0.360261 0.932851i \(-0.617312\pi\)
−0.360261 + 0.932851i \(0.617312\pi\)
\(278\) 8.96412e42 1.51958
\(279\) 0 0
\(280\) −2.50753e42 −0.374956
\(281\) 6.62219e42 0.930338 0.465169 0.885222i \(-0.345994\pi\)
0.465169 + 0.885222i \(0.345994\pi\)
\(282\) 0 0
\(283\) 9.02584e42 1.12002 0.560009 0.828487i \(-0.310798\pi\)
0.560009 + 0.828487i \(0.310798\pi\)
\(284\) 6.81130e41 0.0794619
\(285\) 0 0
\(286\) 8.96163e42 0.924658
\(287\) −1.62819e43 −1.58042
\(288\) 0 0
\(289\) 7.48323e42 0.643246
\(290\) −7.18330e42 −0.581245
\(291\) 0 0
\(292\) −3.68206e42 −0.264175
\(293\) −1.83072e43 −1.23719 −0.618597 0.785708i \(-0.712299\pi\)
−0.618597 + 0.785708i \(0.712299\pi\)
\(294\) 0 0
\(295\) −5.57654e42 −0.334564
\(296\) 2.09824e43 1.18645
\(297\) 0 0
\(298\) 7.39753e42 0.371794
\(299\) −1.29642e43 −0.614468
\(300\) 0 0
\(301\) −3.87475e43 −1.63429
\(302\) 9.06486e42 0.360778
\(303\) 0 0
\(304\) −2.51427e43 −0.891506
\(305\) −2.59131e43 −0.867506
\(306\) 0 0
\(307\) −5.94182e43 −1.77418 −0.887091 0.461595i \(-0.847277\pi\)
−0.887091 + 0.461595i \(0.847277\pi\)
\(308\) −1.23049e43 −0.347087
\(309\) 0 0
\(310\) 2.09482e43 0.527614
\(311\) −9.43273e42 −0.224559 −0.112280 0.993677i \(-0.535815\pi\)
−0.112280 + 0.993677i \(0.535815\pi\)
\(312\) 0 0
\(313\) 5.95209e43 1.26661 0.633306 0.773902i \(-0.281697\pi\)
0.633306 + 0.773902i \(0.281697\pi\)
\(314\) −6.09965e43 −1.22754
\(315\) 0 0
\(316\) −2.86405e43 −0.515770
\(317\) 1.09193e44 1.86062 0.930311 0.366772i \(-0.119537\pi\)
0.930311 + 0.366772i \(0.119537\pi\)
\(318\) 0 0
\(319\) 5.51086e43 0.841168
\(320\) 1.43343e43 0.207134
\(321\) 0 0
\(322\) 6.34303e43 0.821901
\(323\) −7.52186e43 −0.923169
\(324\) 0 0
\(325\) 4.98140e43 0.548773
\(326\) 1.98144e44 2.06858
\(327\) 0 0
\(328\) 1.31895e44 1.23719
\(329\) −2.71015e41 −0.00241027
\(330\) 0 0
\(331\) −6.64486e43 −0.531491 −0.265746 0.964043i \(-0.585618\pi\)
−0.265746 + 0.964043i \(0.585618\pi\)
\(332\) 2.97726e43 0.225891
\(333\) 0 0
\(334\) −4.81837e43 −0.329107
\(335\) 7.42248e43 0.481132
\(336\) 0 0
\(337\) −1.32087e44 −0.771504 −0.385752 0.922602i \(-0.626058\pi\)
−0.385752 + 0.922602i \(0.626058\pi\)
\(338\) 7.32211e43 0.406064
\(339\) 0 0
\(340\) −5.67557e43 −0.283876
\(341\) −1.60709e44 −0.763554
\(342\) 0 0
\(343\) 2.47630e44 1.06208
\(344\) 3.13882e44 1.27936
\(345\) 0 0
\(346\) −2.64267e44 −0.973220
\(347\) −1.51937e44 −0.531983 −0.265991 0.963975i \(-0.585699\pi\)
−0.265991 + 0.963975i \(0.585699\pi\)
\(348\) 0 0
\(349\) −4.35219e44 −1.37804 −0.689022 0.724741i \(-0.741960\pi\)
−0.689022 + 0.724741i \(0.741960\pi\)
\(350\) −2.43727e44 −0.734028
\(351\) 0 0
\(352\) 2.63113e44 0.717210
\(353\) −3.37356e44 −0.875045 −0.437523 0.899207i \(-0.644144\pi\)
−0.437523 + 0.899207i \(0.644144\pi\)
\(354\) 0 0
\(355\) 4.92107e43 0.115628
\(356\) −2.78257e44 −0.622402
\(357\) 0 0
\(358\) 4.45896e44 0.904233
\(359\) −8.65151e44 −1.67086 −0.835428 0.549600i \(-0.814780\pi\)
−0.835428 + 0.549600i \(0.814780\pi\)
\(360\) 0 0
\(361\) −2.74696e44 −0.481367
\(362\) 9.07681e44 1.51542
\(363\) 0 0
\(364\) 1.91378e44 0.290147
\(365\) −2.66024e44 −0.384411
\(366\) 0 0
\(367\) 2.11463e44 0.277701 0.138851 0.990313i \(-0.455659\pi\)
0.138851 + 0.990313i \(0.455659\pi\)
\(368\) −7.50261e44 −0.939454
\(369\) 0 0
\(370\) −9.69661e44 −1.10431
\(371\) 1.31671e45 1.43037
\(372\) 0 0
\(373\) −8.06525e44 −0.797468 −0.398734 0.917067i \(-0.630550\pi\)
−0.398734 + 0.917067i \(0.630550\pi\)
\(374\) 1.55156e45 1.46391
\(375\) 0 0
\(376\) 2.19541e42 0.00188681
\(377\) −8.57107e44 −0.703172
\(378\) 0 0
\(379\) 1.62962e45 1.21871 0.609356 0.792897i \(-0.291428\pi\)
0.609356 + 0.792897i \(0.291428\pi\)
\(380\) 2.23316e44 0.159480
\(381\) 0 0
\(382\) 1.42504e45 0.928362
\(383\) −3.08699e45 −1.92113 −0.960565 0.278055i \(-0.910310\pi\)
−0.960565 + 0.278055i \(0.910310\pi\)
\(384\) 0 0
\(385\) −8.89009e44 −0.505060
\(386\) 4.31409e44 0.234213
\(387\) 0 0
\(388\) 2.07991e44 0.103155
\(389\) 2.76724e45 1.31199 0.655993 0.754767i \(-0.272250\pi\)
0.655993 + 0.754767i \(0.272250\pi\)
\(390\) 0 0
\(391\) −2.24453e45 −0.972821
\(392\) −2.71040e44 −0.112338
\(393\) 0 0
\(394\) 1.33418e45 0.505862
\(395\) −2.06924e45 −0.750518
\(396\) 0 0
\(397\) −3.58547e45 −1.19045 −0.595226 0.803558i \(-0.702938\pi\)
−0.595226 + 0.803558i \(0.702938\pi\)
\(398\) 1.93092e45 0.613494
\(399\) 0 0
\(400\) 2.88283e45 0.839013
\(401\) 4.26941e45 1.18944 0.594718 0.803934i \(-0.297264\pi\)
0.594718 + 0.803934i \(0.297264\pi\)
\(402\) 0 0
\(403\) 2.49952e45 0.638291
\(404\) 1.39264e45 0.340538
\(405\) 0 0
\(406\) 4.19360e45 0.940549
\(407\) 7.43902e45 1.59813
\(408\) 0 0
\(409\) −8.56530e43 −0.0168883 −0.00844413 0.999964i \(-0.502688\pi\)
−0.00844413 + 0.999964i \(0.502688\pi\)
\(410\) −6.09526e45 −1.15153
\(411\) 0 0
\(412\) −2.61156e45 −0.453104
\(413\) 3.25557e45 0.541379
\(414\) 0 0
\(415\) 2.15103e45 0.328704
\(416\) −4.09221e45 −0.599550
\(417\) 0 0
\(418\) −6.10490e45 −0.822420
\(419\) −5.67787e45 −0.733567 −0.366783 0.930306i \(-0.619541\pi\)
−0.366783 + 0.930306i \(0.619541\pi\)
\(420\) 0 0
\(421\) 4.40748e45 0.523907 0.261953 0.965081i \(-0.415633\pi\)
0.261953 + 0.965081i \(0.415633\pi\)
\(422\) −1.72329e46 −1.96512
\(423\) 0 0
\(424\) −1.06663e46 −1.11973
\(425\) 8.62447e45 0.868812
\(426\) 0 0
\(427\) 1.51280e46 1.40377
\(428\) 4.84633e45 0.431666
\(429\) 0 0
\(430\) −1.45054e46 −1.19078
\(431\) −2.36863e46 −1.86700 −0.933500 0.358578i \(-0.883262\pi\)
−0.933500 + 0.358578i \(0.883262\pi\)
\(432\) 0 0
\(433\) 6.19663e45 0.450420 0.225210 0.974310i \(-0.427693\pi\)
0.225210 + 0.974310i \(0.427693\pi\)
\(434\) −1.22295e46 −0.853765
\(435\) 0 0
\(436\) −9.83268e45 −0.633371
\(437\) 8.83155e45 0.546527
\(438\) 0 0
\(439\) −4.03275e45 −0.230394 −0.115197 0.993343i \(-0.536750\pi\)
−0.115197 + 0.993343i \(0.536750\pi\)
\(440\) 7.20159e45 0.395373
\(441\) 0 0
\(442\) −2.41314e46 −1.22375
\(443\) −9.72231e44 −0.0473919 −0.0236959 0.999719i \(-0.507543\pi\)
−0.0236959 + 0.999719i \(0.507543\pi\)
\(444\) 0 0
\(445\) −2.01037e46 −0.905681
\(446\) 6.22433e45 0.269608
\(447\) 0 0
\(448\) −8.36833e45 −0.335177
\(449\) 3.45806e46 1.33205 0.666027 0.745928i \(-0.267994\pi\)
0.666027 + 0.745928i \(0.267994\pi\)
\(450\) 0 0
\(451\) 4.67614e46 1.66647
\(452\) 7.98448e45 0.273731
\(453\) 0 0
\(454\) 4.63060e46 1.46947
\(455\) 1.38268e46 0.422204
\(456\) 0 0
\(457\) 3.54182e46 1.00160 0.500799 0.865564i \(-0.333040\pi\)
0.500799 + 0.865564i \(0.333040\pi\)
\(458\) −5.34684e45 −0.145530
\(459\) 0 0
\(460\) 6.66379e45 0.168058
\(461\) −7.06935e45 −0.171638 −0.0858189 0.996311i \(-0.527351\pi\)
−0.0858189 + 0.996311i \(0.527351\pi\)
\(462\) 0 0
\(463\) −6.22902e44 −0.0140201 −0.00701007 0.999975i \(-0.502231\pi\)
−0.00701007 + 0.999975i \(0.502231\pi\)
\(464\) −4.96023e46 −1.07507
\(465\) 0 0
\(466\) 7.63557e46 1.53493
\(467\) 6.28266e46 1.21646 0.608231 0.793760i \(-0.291880\pi\)
0.608231 + 0.793760i \(0.291880\pi\)
\(468\) 0 0
\(469\) −4.33323e46 −0.778551
\(470\) −1.01457e44 −0.00175618
\(471\) 0 0
\(472\) −2.63724e46 −0.423804
\(473\) 1.11282e47 1.72328
\(474\) 0 0
\(475\) −3.39346e46 −0.488095
\(476\) 3.31339e46 0.459358
\(477\) 0 0
\(478\) −7.27636e46 −0.937402
\(479\) −1.10146e47 −1.36804 −0.684020 0.729463i \(-0.739770\pi\)
−0.684020 + 0.729463i \(0.739770\pi\)
\(480\) 0 0
\(481\) −1.15699e47 −1.33596
\(482\) −1.51623e46 −0.168827
\(483\) 0 0
\(484\) −2.32932e45 −0.0241233
\(485\) 1.50271e46 0.150105
\(486\) 0 0
\(487\) 8.13246e46 0.755901 0.377951 0.925826i \(-0.376629\pi\)
0.377951 + 0.925826i \(0.376629\pi\)
\(488\) −1.22548e47 −1.09890
\(489\) 0 0
\(490\) 1.25256e46 0.104560
\(491\) −1.83224e47 −1.47590 −0.737950 0.674855i \(-0.764206\pi\)
−0.737950 + 0.674855i \(0.764206\pi\)
\(492\) 0 0
\(493\) −1.48394e47 −1.11326
\(494\) 9.49498e46 0.687499
\(495\) 0 0
\(496\) 1.44652e47 0.975876
\(497\) −2.87291e46 −0.187105
\(498\) 0 0
\(499\) 9.12339e45 0.0553856 0.0276928 0.999616i \(-0.491184\pi\)
0.0276928 + 0.999616i \(0.491184\pi\)
\(500\) −6.33843e46 −0.371541
\(501\) 0 0
\(502\) 7.36830e45 0.0402766
\(503\) 1.87342e47 0.989001 0.494500 0.869177i \(-0.335351\pi\)
0.494500 + 0.869177i \(0.335351\pi\)
\(504\) 0 0
\(505\) 1.00616e47 0.495530
\(506\) −1.82171e47 −0.866653
\(507\) 0 0
\(508\) 2.52221e46 0.111987
\(509\) −1.22247e47 −0.524418 −0.262209 0.965011i \(-0.584451\pi\)
−0.262209 + 0.965011i \(0.584451\pi\)
\(510\) 0 0
\(511\) 1.55304e47 0.622039
\(512\) −6.84151e45 −0.0264806
\(513\) 0 0
\(514\) −5.37115e46 −0.194184
\(515\) −1.88681e47 −0.659329
\(516\) 0 0
\(517\) 7.78352e44 0.00254151
\(518\) 5.66086e47 1.78695
\(519\) 0 0
\(520\) −1.12007e47 −0.330510
\(521\) 6.28279e46 0.179264 0.0896318 0.995975i \(-0.471431\pi\)
0.0896318 + 0.995975i \(0.471431\pi\)
\(522\) 0 0
\(523\) −2.46925e47 −0.658849 −0.329425 0.944182i \(-0.606855\pi\)
−0.329425 + 0.944182i \(0.606855\pi\)
\(524\) −8.34167e46 −0.215256
\(525\) 0 0
\(526\) −2.20281e47 −0.531773
\(527\) 4.32751e47 1.01054
\(528\) 0 0
\(529\) −1.94053e47 −0.424079
\(530\) 4.92922e47 1.04220
\(531\) 0 0
\(532\) −1.30372e47 −0.258065
\(533\) −7.27282e47 −1.39308
\(534\) 0 0
\(535\) 3.50141e47 0.628133
\(536\) 3.51022e47 0.609467
\(537\) 0 0
\(538\) 7.72485e47 1.25661
\(539\) −9.60935e46 −0.151318
\(540\) 0 0
\(541\) 1.20408e47 0.177706 0.0888529 0.996045i \(-0.471680\pi\)
0.0888529 + 0.996045i \(0.471680\pi\)
\(542\) −4.48683e45 −0.00641139
\(543\) 0 0
\(544\) −7.08499e47 −0.949202
\(545\) −7.10398e47 −0.921643
\(546\) 0 0
\(547\) 8.44104e47 1.02711 0.513556 0.858056i \(-0.328328\pi\)
0.513556 + 0.858056i \(0.328328\pi\)
\(548\) −9.26461e46 −0.109186
\(549\) 0 0
\(550\) 6.99980e47 0.773995
\(551\) 5.83884e47 0.625423
\(552\) 0 0
\(553\) 1.20802e48 1.21446
\(554\) 8.72153e47 0.849517
\(555\) 0 0
\(556\) −5.49787e47 −0.502789
\(557\) 1.28303e48 1.13703 0.568514 0.822674i \(-0.307519\pi\)
0.568514 + 0.822674i \(0.307519\pi\)
\(558\) 0 0
\(559\) −1.73078e48 −1.44057
\(560\) 8.00182e47 0.645503
\(561\) 0 0
\(562\) −1.44727e48 −1.09690
\(563\) −2.05838e48 −1.51227 −0.756133 0.654418i \(-0.772914\pi\)
−0.756133 + 0.654418i \(0.772914\pi\)
\(564\) 0 0
\(565\) 5.76868e47 0.398317
\(566\) −1.97259e48 −1.32053
\(567\) 0 0
\(568\) 2.32726e47 0.146470
\(569\) 4.38677e47 0.267720 0.133860 0.991000i \(-0.457263\pi\)
0.133860 + 0.991000i \(0.457263\pi\)
\(570\) 0 0
\(571\) −2.36363e48 −1.35658 −0.678292 0.734792i \(-0.737280\pi\)
−0.678292 + 0.734792i \(0.737280\pi\)
\(572\) −5.49634e47 −0.305945
\(573\) 0 0
\(574\) 3.55840e48 1.86336
\(575\) −1.01261e48 −0.514347
\(576\) 0 0
\(577\) −1.45741e48 −0.696632 −0.348316 0.937377i \(-0.613246\pi\)
−0.348316 + 0.937377i \(0.613246\pi\)
\(578\) −1.63546e48 −0.758406
\(579\) 0 0
\(580\) 4.40566e47 0.192318
\(581\) −1.25577e48 −0.531896
\(582\) 0 0
\(583\) −3.78159e48 −1.50825
\(584\) −1.25807e48 −0.486946
\(585\) 0 0
\(586\) 4.00102e48 1.45869
\(587\) 2.31218e48 0.818190 0.409095 0.912492i \(-0.365844\pi\)
0.409095 + 0.912492i \(0.365844\pi\)
\(588\) 0 0
\(589\) −1.70274e48 −0.567716
\(590\) 1.21875e48 0.394461
\(591\) 0 0
\(592\) −6.69573e48 −2.04253
\(593\) −4.31701e48 −1.27857 −0.639287 0.768968i \(-0.720770\pi\)
−0.639287 + 0.768968i \(0.720770\pi\)
\(594\) 0 0
\(595\) 2.39388e48 0.668429
\(596\) −4.53705e47 −0.123017
\(597\) 0 0
\(598\) 2.83331e48 0.724476
\(599\) 2.66626e48 0.662114 0.331057 0.943611i \(-0.392595\pi\)
0.331057 + 0.943611i \(0.392595\pi\)
\(600\) 0 0
\(601\) 7.51112e48 1.75955 0.879775 0.475391i \(-0.157693\pi\)
0.879775 + 0.475391i \(0.157693\pi\)
\(602\) 8.46824e48 1.92688
\(603\) 0 0
\(604\) −5.55966e47 −0.119372
\(605\) −1.68291e47 −0.0351027
\(606\) 0 0
\(607\) −4.81557e48 −0.948081 −0.474040 0.880503i \(-0.657205\pi\)
−0.474040 + 0.880503i \(0.657205\pi\)
\(608\) 2.78773e48 0.533258
\(609\) 0 0
\(610\) 5.66329e48 1.02281
\(611\) −1.21057e46 −0.00212457
\(612\) 0 0
\(613\) 1.36361e48 0.226012 0.113006 0.993594i \(-0.463952\pi\)
0.113006 + 0.993594i \(0.463952\pi\)
\(614\) 1.29858e49 2.09181
\(615\) 0 0
\(616\) −4.20428e48 −0.639777
\(617\) −1.76559e48 −0.261156 −0.130578 0.991438i \(-0.541683\pi\)
−0.130578 + 0.991438i \(0.541683\pi\)
\(618\) 0 0
\(619\) 7.44879e48 1.04112 0.520559 0.853826i \(-0.325724\pi\)
0.520559 + 0.853826i \(0.325724\pi\)
\(620\) −1.28479e48 −0.174573
\(621\) 0 0
\(622\) 2.06152e48 0.264762
\(623\) 1.17365e49 1.46554
\(624\) 0 0
\(625\) 1.16140e48 0.137114
\(626\) −1.30083e49 −1.49337
\(627\) 0 0
\(628\) 3.74104e48 0.406161
\(629\) −2.00314e49 −2.11507
\(630\) 0 0
\(631\) 4.61699e48 0.461153 0.230576 0.973054i \(-0.425939\pi\)
0.230576 + 0.973054i \(0.425939\pi\)
\(632\) −9.78578e48 −0.950706
\(633\) 0 0
\(634\) −2.38641e49 −2.19373
\(635\) 1.82227e48 0.162956
\(636\) 0 0
\(637\) 1.49455e48 0.126494
\(638\) −1.20440e49 −0.991761
\(639\) 0 0
\(640\) −8.52472e48 −0.664555
\(641\) 2.29073e49 1.73763 0.868816 0.495134i \(-0.164881\pi\)
0.868816 + 0.495134i \(0.164881\pi\)
\(642\) 0 0
\(643\) −1.75961e49 −1.26394 −0.631969 0.774994i \(-0.717753\pi\)
−0.631969 + 0.774994i \(0.717753\pi\)
\(644\) −3.89031e48 −0.271945
\(645\) 0 0
\(646\) 1.64390e49 1.08844
\(647\) 1.34530e49 0.866950 0.433475 0.901166i \(-0.357287\pi\)
0.433475 + 0.901166i \(0.357287\pi\)
\(648\) 0 0
\(649\) −9.34996e48 −0.570857
\(650\) −1.08868e49 −0.647019
\(651\) 0 0
\(652\) −1.21525e49 −0.684438
\(653\) 1.55134e49 0.850601 0.425300 0.905052i \(-0.360168\pi\)
0.425300 + 0.905052i \(0.360168\pi\)
\(654\) 0 0
\(655\) −6.02675e48 −0.313228
\(656\) −4.20892e49 −2.12987
\(657\) 0 0
\(658\) 5.92302e46 0.00284178
\(659\) 1.67026e46 0.000780348 0 0.000390174 1.00000i \(-0.499876\pi\)
0.000390174 1.00000i \(0.499876\pi\)
\(660\) 0 0
\(661\) 3.86727e49 1.71348 0.856740 0.515748i \(-0.172486\pi\)
0.856740 + 0.515748i \(0.172486\pi\)
\(662\) 1.45223e49 0.626644
\(663\) 0 0
\(664\) 1.01726e49 0.416380
\(665\) −9.41918e48 −0.375521
\(666\) 0 0
\(667\) 1.74232e49 0.659060
\(668\) 2.95520e48 0.108893
\(669\) 0 0
\(670\) −1.62218e49 −0.567269
\(671\) −4.34475e49 −1.48020
\(672\) 0 0
\(673\) 1.92251e49 0.621734 0.310867 0.950453i \(-0.399381\pi\)
0.310867 + 0.950453i \(0.399381\pi\)
\(674\) 2.88676e49 0.909626
\(675\) 0 0
\(676\) −4.49080e48 −0.134356
\(677\) 2.27583e49 0.663495 0.331747 0.943368i \(-0.392362\pi\)
0.331747 + 0.943368i \(0.392362\pi\)
\(678\) 0 0
\(679\) −8.77279e48 −0.242894
\(680\) −1.93921e49 −0.523261
\(681\) 0 0
\(682\) 3.51229e49 0.900253
\(683\) 7.47926e49 1.86851 0.934257 0.356599i \(-0.116064\pi\)
0.934257 + 0.356599i \(0.116064\pi\)
\(684\) 0 0
\(685\) −6.69355e48 −0.158881
\(686\) −5.41193e49 −1.25222
\(687\) 0 0
\(688\) −1.00163e50 −2.20247
\(689\) 5.88151e49 1.26082
\(690\) 0 0
\(691\) 5.04033e49 1.02705 0.513527 0.858073i \(-0.328339\pi\)
0.513527 + 0.858073i \(0.328339\pi\)
\(692\) 1.62080e49 0.322013
\(693\) 0 0
\(694\) 3.32057e49 0.627223
\(695\) −3.97214e49 −0.731628
\(696\) 0 0
\(697\) −1.25917e50 −2.20552
\(698\) 9.51168e49 1.62475
\(699\) 0 0
\(700\) 1.49482e49 0.242870
\(701\) −3.72125e48 −0.0589690 −0.0294845 0.999565i \(-0.509387\pi\)
−0.0294845 + 0.999565i \(0.509387\pi\)
\(702\) 0 0
\(703\) 7.88175e49 1.18824
\(704\) 2.40337e49 0.353427
\(705\) 0 0
\(706\) 7.37288e49 1.03170
\(707\) −5.87397e49 −0.801848
\(708\) 0 0
\(709\) −5.15574e49 −0.669857 −0.334928 0.942244i \(-0.608712\pi\)
−0.334928 + 0.942244i \(0.608712\pi\)
\(710\) −1.07550e49 −0.136329
\(711\) 0 0
\(712\) −9.50738e49 −1.14726
\(713\) −5.08100e49 −0.598249
\(714\) 0 0
\(715\) −3.97103e49 −0.445193
\(716\) −2.73477e49 −0.299187
\(717\) 0 0
\(718\) 1.89078e50 1.96999
\(719\) 3.36048e49 0.341700 0.170850 0.985297i \(-0.445349\pi\)
0.170850 + 0.985297i \(0.445349\pi\)
\(720\) 0 0
\(721\) 1.10152e50 1.06690
\(722\) 6.00346e49 0.567546
\(723\) 0 0
\(724\) −5.56699e49 −0.501412
\(725\) −6.69473e49 −0.588597
\(726\) 0 0
\(727\) 1.31541e50 1.10207 0.551033 0.834484i \(-0.314234\pi\)
0.551033 + 0.834484i \(0.314234\pi\)
\(728\) 6.53893e49 0.534820
\(729\) 0 0
\(730\) 5.81394e49 0.453232
\(731\) −2.99655e50 −2.28070
\(732\) 0 0
\(733\) −2.25359e50 −1.63514 −0.817570 0.575829i \(-0.804680\pi\)
−0.817570 + 0.575829i \(0.804680\pi\)
\(734\) −4.62151e49 −0.327418
\(735\) 0 0
\(736\) 8.31861e49 0.561939
\(737\) 1.24450e50 0.820943
\(738\) 0 0
\(739\) −1.93424e50 −1.21684 −0.608419 0.793616i \(-0.708196\pi\)
−0.608419 + 0.793616i \(0.708196\pi\)
\(740\) 5.94712e49 0.365386
\(741\) 0 0
\(742\) −2.87767e50 −1.68645
\(743\) −8.43115e49 −0.482595 −0.241297 0.970451i \(-0.577573\pi\)
−0.241297 + 0.970451i \(0.577573\pi\)
\(744\) 0 0
\(745\) −3.27796e49 −0.179006
\(746\) 1.76266e50 0.940238
\(747\) 0 0
\(748\) −9.51601e49 −0.484370
\(749\) −2.04412e50 −1.01642
\(750\) 0 0
\(751\) −2.50076e50 −1.18679 −0.593394 0.804912i \(-0.702212\pi\)
−0.593394 + 0.804912i \(0.702212\pi\)
\(752\) −7.00581e47 −0.00324823
\(753\) 0 0
\(754\) 1.87320e50 0.829060
\(755\) −4.01678e49 −0.173702
\(756\) 0 0
\(757\) −8.14773e49 −0.336402 −0.168201 0.985753i \(-0.553796\pi\)
−0.168201 + 0.985753i \(0.553796\pi\)
\(758\) −3.56153e50 −1.43690
\(759\) 0 0
\(760\) 7.63019e49 0.293966
\(761\) 3.09181e50 1.16408 0.582038 0.813161i \(-0.302255\pi\)
0.582038 + 0.813161i \(0.302255\pi\)
\(762\) 0 0
\(763\) 4.14729e50 1.49137
\(764\) −8.74004e49 −0.307171
\(765\) 0 0
\(766\) 6.74660e50 2.26507
\(767\) 1.45420e50 0.477206
\(768\) 0 0
\(769\) −9.48513e49 −0.297394 −0.148697 0.988883i \(-0.547508\pi\)
−0.148697 + 0.988883i \(0.547508\pi\)
\(770\) 1.94292e50 0.595481
\(771\) 0 0
\(772\) −2.64592e49 −0.0774950
\(773\) −1.05155e50 −0.301086 −0.150543 0.988603i \(-0.548102\pi\)
−0.150543 + 0.988603i \(0.548102\pi\)
\(774\) 0 0
\(775\) 1.95234e50 0.534288
\(776\) 7.10657e49 0.190143
\(777\) 0 0
\(778\) −6.04779e50 −1.54687
\(779\) 4.95444e50 1.23905
\(780\) 0 0
\(781\) 8.25096e49 0.197293
\(782\) 4.90541e50 1.14698
\(783\) 0 0
\(784\) 8.64922e49 0.193395
\(785\) 2.70285e50 0.591021
\(786\) 0 0
\(787\) −1.75648e50 −0.367354 −0.183677 0.982987i \(-0.558800\pi\)
−0.183677 + 0.982987i \(0.558800\pi\)
\(788\) −8.18282e49 −0.167376
\(789\) 0 0
\(790\) 4.52230e50 0.884882
\(791\) −3.36775e50 −0.644542
\(792\) 0 0
\(793\) 6.75741e50 1.23737
\(794\) 7.83602e50 1.40358
\(795\) 0 0
\(796\) −1.18427e50 −0.202989
\(797\) −5.53388e49 −0.0927917 −0.0463959 0.998923i \(-0.514774\pi\)
−0.0463959 + 0.998923i \(0.514774\pi\)
\(798\) 0 0
\(799\) −2.09591e48 −0.00336359
\(800\) −3.19637e50 −0.501859
\(801\) 0 0
\(802\) −9.33077e50 −1.40238
\(803\) −4.46032e50 −0.655909
\(804\) 0 0
\(805\) −2.81069e50 −0.395718
\(806\) −5.46269e50 −0.752563
\(807\) 0 0
\(808\) 4.75832e50 0.627704
\(809\) −4.98563e50 −0.643607 −0.321803 0.946807i \(-0.604289\pi\)
−0.321803 + 0.946807i \(0.604289\pi\)
\(810\) 0 0
\(811\) −1.07303e51 −1.32662 −0.663309 0.748346i \(-0.730848\pi\)
−0.663309 + 0.748346i \(0.730848\pi\)
\(812\) −2.57202e50 −0.311203
\(813\) 0 0
\(814\) −1.62579e51 −1.88425
\(815\) −8.78004e50 −0.995952
\(816\) 0 0
\(817\) 1.17905e51 1.28129
\(818\) 1.87194e49 0.0199117
\(819\) 0 0
\(820\) 3.73834e50 0.381010
\(821\) 2.21488e50 0.220976 0.110488 0.993877i \(-0.464759\pi\)
0.110488 + 0.993877i \(0.464759\pi\)
\(822\) 0 0
\(823\) 1.24351e51 1.18892 0.594459 0.804126i \(-0.297366\pi\)
0.594459 + 0.804126i \(0.297366\pi\)
\(824\) −8.92306e50 −0.835195
\(825\) 0 0
\(826\) −7.11504e50 −0.638302
\(827\) −1.45573e49 −0.0127860 −0.00639299 0.999980i \(-0.502035\pi\)
−0.00639299 + 0.999980i \(0.502035\pi\)
\(828\) 0 0
\(829\) −9.35859e50 −0.787964 −0.393982 0.919118i \(-0.628903\pi\)
−0.393982 + 0.919118i \(0.628903\pi\)
\(830\) −4.70106e50 −0.387551
\(831\) 0 0
\(832\) −3.73797e50 −0.295446
\(833\) 2.58756e50 0.200264
\(834\) 0 0
\(835\) 2.13510e50 0.158454
\(836\) 3.74426e50 0.272117
\(837\) 0 0
\(838\) 1.24089e51 0.864897
\(839\) 7.95070e50 0.542714 0.271357 0.962479i \(-0.412528\pi\)
0.271357 + 0.962479i \(0.412528\pi\)
\(840\) 0 0
\(841\) −3.75414e50 −0.245799
\(842\) −9.63252e50 −0.617701
\(843\) 0 0
\(844\) 1.05692e51 0.650206
\(845\) −3.24454e50 −0.195506
\(846\) 0 0
\(847\) 9.82478e49 0.0568019
\(848\) 3.40374e51 1.92765
\(849\) 0 0
\(850\) −1.88487e51 −1.02435
\(851\) 2.35192e51 1.25215
\(852\) 0 0
\(853\) −2.54840e50 −0.130214 −0.0651072 0.997878i \(-0.520739\pi\)
−0.0651072 + 0.997878i \(0.520739\pi\)
\(854\) −3.30622e51 −1.65508
\(855\) 0 0
\(856\) 1.65588e51 0.795678
\(857\) 2.64469e51 1.24512 0.622558 0.782573i \(-0.286093\pi\)
0.622558 + 0.782573i \(0.286093\pi\)
\(858\) 0 0
\(859\) −1.92463e51 −0.869894 −0.434947 0.900456i \(-0.643233\pi\)
−0.434947 + 0.900456i \(0.643233\pi\)
\(860\) 8.89646e50 0.393998
\(861\) 0 0
\(862\) 5.17661e51 2.20125
\(863\) 1.08280e51 0.451190 0.225595 0.974221i \(-0.427567\pi\)
0.225595 + 0.974221i \(0.427567\pi\)
\(864\) 0 0
\(865\) 1.17101e51 0.468573
\(866\) −1.35427e51 −0.531058
\(867\) 0 0
\(868\) 7.50059e50 0.282488
\(869\) −3.46941e51 −1.28059
\(870\) 0 0
\(871\) −1.93557e51 −0.686264
\(872\) −3.35959e51 −1.16748
\(873\) 0 0
\(874\) −1.93013e51 −0.644371
\(875\) 2.67347e51 0.874850
\(876\) 0 0
\(877\) 1.72464e51 0.542258 0.271129 0.962543i \(-0.412603\pi\)
0.271129 + 0.962543i \(0.412603\pi\)
\(878\) 8.81354e50 0.271642
\(879\) 0 0
\(880\) −2.29811e51 −0.680650
\(881\) −8.76279e50 −0.254428 −0.127214 0.991875i \(-0.540603\pi\)
−0.127214 + 0.991875i \(0.540603\pi\)
\(882\) 0 0
\(883\) −2.14016e51 −0.597221 −0.298610 0.954375i \(-0.596523\pi\)
−0.298610 + 0.954375i \(0.596523\pi\)
\(884\) 1.48003e51 0.404907
\(885\) 0 0
\(886\) 2.12480e50 0.0558764
\(887\) 3.78615e51 0.976189 0.488095 0.872791i \(-0.337692\pi\)
0.488095 + 0.872791i \(0.337692\pi\)
\(888\) 0 0
\(889\) −1.06384e51 −0.263690
\(890\) 4.39365e51 1.06782
\(891\) 0 0
\(892\) −3.81751e50 −0.0892061
\(893\) 8.24675e48 0.00188965
\(894\) 0 0
\(895\) −1.97583e51 −0.435358
\(896\) 4.97672e51 1.07536
\(897\) 0 0
\(898\) −7.55757e51 −1.57053
\(899\) −3.35922e51 −0.684612
\(900\) 0 0
\(901\) 1.01829e52 1.99612
\(902\) −1.02197e52 −1.96482
\(903\) 0 0
\(904\) 2.72811e51 0.504562
\(905\) −4.02207e51 −0.729625
\(906\) 0 0
\(907\) 5.56638e51 0.971506 0.485753 0.874096i \(-0.338545\pi\)
0.485753 + 0.874096i \(0.338545\pi\)
\(908\) −2.84004e51 −0.486208
\(909\) 0 0
\(910\) −3.02184e51 −0.497790
\(911\) 4.53769e51 0.733268 0.366634 0.930365i \(-0.380510\pi\)
0.366634 + 0.930365i \(0.380510\pi\)
\(912\) 0 0
\(913\) 3.60654e51 0.560857
\(914\) −7.74063e51 −1.18091
\(915\) 0 0
\(916\) 3.27932e50 0.0481519
\(917\) 3.51840e51 0.506853
\(918\) 0 0
\(919\) −9.66145e50 −0.133974 −0.0669870 0.997754i \(-0.521339\pi\)
−0.0669870 + 0.997754i \(0.521339\pi\)
\(920\) 2.27686e51 0.309777
\(921\) 0 0
\(922\) 1.54500e51 0.202366
\(923\) −1.28328e51 −0.164926
\(924\) 0 0
\(925\) −9.03710e51 −1.11828
\(926\) 1.36135e50 0.0165302
\(927\) 0 0
\(928\) 5.49972e51 0.643059
\(929\) −1.12588e52 −1.29187 −0.645935 0.763392i \(-0.723532\pi\)
−0.645935 + 0.763392i \(0.723532\pi\)
\(930\) 0 0
\(931\) −1.01812e51 −0.112508
\(932\) −4.68305e51 −0.507867
\(933\) 0 0
\(934\) −1.37307e52 −1.43424
\(935\) −6.87519e51 −0.704825
\(936\) 0 0
\(937\) −9.93521e51 −0.981147 −0.490574 0.871400i \(-0.663213\pi\)
−0.490574 + 0.871400i \(0.663213\pi\)
\(938\) 9.47024e51 0.917934
\(939\) 0 0
\(940\) 6.22254e48 0.000581072 0
\(941\) 7.86741e51 0.721129 0.360565 0.932734i \(-0.382584\pi\)
0.360565 + 0.932734i \(0.382584\pi\)
\(942\) 0 0
\(943\) 1.47841e52 1.30569
\(944\) 8.41574e51 0.729596
\(945\) 0 0
\(946\) −2.43206e52 −2.03180
\(947\) −1.76717e52 −1.44928 −0.724641 0.689127i \(-0.757994\pi\)
−0.724641 + 0.689127i \(0.757994\pi\)
\(948\) 0 0
\(949\) 6.93715e51 0.548305
\(950\) 7.41639e51 0.575479
\(951\) 0 0
\(952\) 1.13211e52 0.846722
\(953\) 2.15077e52 1.57931 0.789656 0.613550i \(-0.210259\pi\)
0.789656 + 0.613550i \(0.210259\pi\)
\(954\) 0 0
\(955\) −6.31456e51 −0.446976
\(956\) 4.46273e51 0.310162
\(957\) 0 0
\(958\) 2.40724e52 1.61296
\(959\) 3.90769e51 0.257095
\(960\) 0 0
\(961\) −5.96747e51 −0.378557
\(962\) 2.52860e52 1.57513
\(963\) 0 0
\(964\) 9.29935e50 0.0558605
\(965\) −1.91164e51 −0.112766
\(966\) 0 0
\(967\) 5.62647e51 0.320091 0.160045 0.987110i \(-0.448836\pi\)
0.160045 + 0.987110i \(0.448836\pi\)
\(968\) −7.95875e50 −0.0444658
\(969\) 0 0
\(970\) −3.28416e51 −0.176978
\(971\) −3.42787e51 −0.181421 −0.0907104 0.995877i \(-0.528914\pi\)
−0.0907104 + 0.995877i \(0.528914\pi\)
\(972\) 0 0
\(973\) 2.31893e52 1.18389
\(974\) −1.77734e52 −0.891230
\(975\) 0 0
\(976\) 3.91064e52 1.89180
\(977\) −1.18458e52 −0.562872 −0.281436 0.959580i \(-0.590811\pi\)
−0.281436 + 0.959580i \(0.590811\pi\)
\(978\) 0 0
\(979\) −3.37070e52 −1.54534
\(980\) −7.68220e50 −0.0345963
\(981\) 0 0
\(982\) 4.00436e52 1.74013
\(983\) −2.75159e52 −1.17462 −0.587311 0.809362i \(-0.699813\pi\)
−0.587311 + 0.809362i \(0.699813\pi\)
\(984\) 0 0
\(985\) −5.91197e51 −0.243556
\(986\) 3.24314e52 1.31256
\(987\) 0 0
\(988\) −5.82346e51 −0.227475
\(989\) 3.51830e52 1.35020
\(990\) 0 0
\(991\) −3.30440e52 −1.22406 −0.612031 0.790834i \(-0.709647\pi\)
−0.612031 + 0.790834i \(0.709647\pi\)
\(992\) −1.60384e52 −0.583725
\(993\) 0 0
\(994\) 6.27873e51 0.220602
\(995\) −8.55620e51 −0.295377
\(996\) 0 0
\(997\) −4.71253e52 −1.57068 −0.785342 0.619062i \(-0.787513\pi\)
−0.785342 + 0.619062i \(0.787513\pi\)
\(998\) −1.99391e51 −0.0653012
\(999\) 0 0
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 9.36.a.a.1.1 2
3.2 odd 2 3.36.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.36.a.a.1.2 2 3.2 odd 2
9.36.a.a.1.1 2 1.1 even 1 trivial