Properties

Label 9.38.a.c.1.1
Level 9
Weight 38
Character 9.1
Self dual yes
Analytic conductor 78.043
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 38 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0426343121\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{10}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-101317.\)
Character \(\chi\) = 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-717293. q^{2} +3.77070e11 q^{4} +9.63758e12 q^{5} +1.74370e15 q^{7} -1.71886e17 q^{8} +O(q^{10})\) \(q-717293. q^{2} +3.77070e11 q^{4} +9.63758e12 q^{5} +1.74370e15 q^{7} -1.71886e17 q^{8} -6.91297e18 q^{10} -2.16917e19 q^{11} +4.06617e20 q^{13} -1.25074e21 q^{14} +7.14682e22 q^{16} -4.16602e21 q^{17} -4.03822e23 q^{19} +3.63404e24 q^{20} +1.55593e25 q^{22} +3.07818e25 q^{23} +2.01235e25 q^{25} -2.91664e26 q^{26} +6.57497e26 q^{28} +3.72833e26 q^{29} -1.69414e27 q^{31} -2.76399e28 q^{32} +2.98826e27 q^{34} +1.68050e28 q^{35} +3.17528e28 q^{37} +2.89659e29 q^{38} -1.65656e30 q^{40} +1.14320e29 q^{41} +7.17875e29 q^{43} -8.17928e30 q^{44} -2.20796e31 q^{46} -3.03496e29 q^{47} -1.55216e31 q^{49} -1.44344e31 q^{50} +1.53323e32 q^{52} +6.59860e31 q^{53} -2.09055e32 q^{55} -2.99717e32 q^{56} -2.67431e32 q^{58} +8.50758e32 q^{59} +6.05924e32 q^{61} +1.21519e33 q^{62} +1.00034e34 q^{64} +3.91881e33 q^{65} -2.62114e33 q^{67} -1.57088e33 q^{68} -1.20541e34 q^{70} -1.38853e34 q^{71} -1.07813e34 q^{73} -2.27761e34 q^{74} -1.52269e35 q^{76} -3.78238e34 q^{77} +2.43825e35 q^{79} +6.88781e35 q^{80} -8.20011e34 q^{82} +3.57499e34 q^{83} -4.01504e34 q^{85} -5.14927e35 q^{86} +3.72849e36 q^{88} -1.28259e36 q^{89} +7.09018e35 q^{91} +1.16069e37 q^{92} +2.17695e35 q^{94} -3.89187e36 q^{95} -8.70371e36 q^{97} +1.11336e37 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} + O(q^{10}) \) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} - 21511023001649316q^{10} - 20953708852195292976q^{11} + 51830892788989874168q^{13} + \)\(18\!\cdots\!12\)\(q^{14} + \)\(55\!\cdots\!40\)\(q^{16} - \)\(81\!\cdots\!28\)\(q^{17} - \)\(54\!\cdots\!32\)\(q^{19} + \)\(35\!\cdots\!76\)\(q^{20} + \)\(23\!\cdots\!76\)\(q^{22} + \)\(61\!\cdots\!88\)\(q^{23} + \)\(95\!\cdots\!16\)\(q^{25} - \)\(42\!\cdots\!88\)\(q^{26} + \)\(24\!\cdots\!48\)\(q^{28} - \)\(41\!\cdots\!36\)\(q^{29} + \)\(89\!\cdots\!64\)\(q^{31} - \)\(36\!\cdots\!84\)\(q^{32} + \)\(31\!\cdots\!24\)\(q^{34} - \)\(42\!\cdots\!40\)\(q^{35} + \)\(55\!\cdots\!24\)\(q^{37} + \)\(73\!\cdots\!68\)\(q^{38} - \)\(26\!\cdots\!52\)\(q^{40} + \)\(86\!\cdots\!76\)\(q^{41} - \)\(50\!\cdots\!80\)\(q^{43} - \)\(28\!\cdots\!36\)\(q^{44} - \)\(14\!\cdots\!96\)\(q^{46} - \)\(42\!\cdots\!20\)\(q^{47} + \)\(40\!\cdots\!20\)\(q^{49} - \)\(10\!\cdots\!14\)\(q^{50} + \)\(18\!\cdots\!68\)\(q^{52} + \)\(12\!\cdots\!88\)\(q^{53} - \)\(32\!\cdots\!24\)\(q^{55} - \)\(49\!\cdots\!80\)\(q^{56} - \)\(75\!\cdots\!56\)\(q^{58} + \)\(13\!\cdots\!88\)\(q^{59} - \)\(12\!\cdots\!60\)\(q^{61} + \)\(37\!\cdots\!92\)\(q^{62} + \)\(85\!\cdots\!28\)\(q^{64} - \)\(15\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!48\)\(q^{67} - \)\(63\!\cdots\!84\)\(q^{68} + \)\(82\!\cdots\!60\)\(q^{70} - \)\(10\!\cdots\!88\)\(q^{71} - \)\(19\!\cdots\!48\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} - \)\(95\!\cdots\!68\)\(q^{76} + \)\(25\!\cdots\!92\)\(q^{77} + \)\(42\!\cdots\!20\)\(q^{79} + \)\(95\!\cdots\!36\)\(q^{80} + \)\(33\!\cdots\!48\)\(q^{82} + \)\(46\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!12\)\(q^{85} + \)\(36\!\cdots\!04\)\(q^{86} + \)\(42\!\cdots\!56\)\(q^{88} + \)\(31\!\cdots\!52\)\(q^{89} + \)\(26\!\cdots\!24\)\(q^{91} + \)\(16\!\cdots\!16\)\(q^{92} - \)\(57\!\cdots\!48\)\(q^{94} + \)\(89\!\cdots\!56\)\(q^{95} + \)\(44\!\cdots\!48\)\(q^{97} + \)\(43\!\cdots\!78\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −717293. −1.93482 −0.967412 0.253207i \(-0.918515\pi\)
−0.967412 + 0.253207i \(0.918515\pi\)
\(3\) 0 0
\(4\) 3.77070e11 2.74355
\(5\) 9.63758e12 1.12986 0.564928 0.825140i \(-0.308904\pi\)
0.564928 + 0.825140i \(0.308904\pi\)
\(6\) 0 0
\(7\) 1.74370e15 0.404723 0.202361 0.979311i \(-0.435138\pi\)
0.202361 + 0.979311i \(0.435138\pi\)
\(8\) −1.71886e17 −3.37345
\(9\) 0 0
\(10\) −6.91297e18 −2.18607
\(11\) −2.16917e19 −1.17633 −0.588164 0.808742i \(-0.700149\pi\)
−0.588164 + 0.808742i \(0.700149\pi\)
\(12\) 0 0
\(13\) 4.06617e20 1.00285 0.501423 0.865202i \(-0.332810\pi\)
0.501423 + 0.865202i \(0.332810\pi\)
\(14\) −1.25074e21 −0.783068
\(15\) 0 0
\(16\) 7.14682e22 3.78350
\(17\) −4.16602e21 −0.0718483 −0.0359241 0.999355i \(-0.511437\pi\)
−0.0359241 + 0.999355i \(0.511437\pi\)
\(18\) 0 0
\(19\) −4.03822e23 −0.889710 −0.444855 0.895603i \(-0.646745\pi\)
−0.444855 + 0.895603i \(0.646745\pi\)
\(20\) 3.63404e24 3.09981
\(21\) 0 0
\(22\) 1.55593e25 2.27599
\(23\) 3.07818e25 1.97847 0.989234 0.146344i \(-0.0467506\pi\)
0.989234 + 0.146344i \(0.0467506\pi\)
\(24\) 0 0
\(25\) 2.01235e25 0.276575
\(26\) −2.91664e26 −1.94033
\(27\) 0 0
\(28\) 6.57497e26 1.11038
\(29\) 3.72833e26 0.328967 0.164483 0.986380i \(-0.447404\pi\)
0.164483 + 0.986380i \(0.447404\pi\)
\(30\) 0 0
\(31\) −1.69414e27 −0.435269 −0.217635 0.976030i \(-0.569834\pi\)
−0.217635 + 0.976030i \(0.569834\pi\)
\(32\) −2.76399e28 −3.94695
\(33\) 0 0
\(34\) 2.98826e27 0.139014
\(35\) 1.68050e28 0.457279
\(36\) 0 0
\(37\) 3.17528e28 0.309066 0.154533 0.987988i \(-0.450613\pi\)
0.154533 + 0.987988i \(0.450613\pi\)
\(38\) 2.89659e29 1.72143
\(39\) 0 0
\(40\) −1.65656e30 −3.81152
\(41\) 1.14320e29 0.166580 0.0832898 0.996525i \(-0.473457\pi\)
0.0832898 + 0.996525i \(0.473457\pi\)
\(42\) 0 0
\(43\) 7.17875e29 0.433394 0.216697 0.976239i \(-0.430472\pi\)
0.216697 + 0.976239i \(0.430472\pi\)
\(44\) −8.17928e30 −3.22731
\(45\) 0 0
\(46\) −2.20796e31 −3.82799
\(47\) −3.03496e29 −0.0353462 −0.0176731 0.999844i \(-0.505626\pi\)
−0.0176731 + 0.999844i \(0.505626\pi\)
\(48\) 0 0
\(49\) −1.55216e31 −0.836199
\(50\) −1.44344e31 −0.535123
\(51\) 0 0
\(52\) 1.53323e32 2.75135
\(53\) 6.59860e31 0.832431 0.416216 0.909266i \(-0.363356\pi\)
0.416216 + 0.909266i \(0.363356\pi\)
\(54\) 0 0
\(55\) −2.09055e32 −1.32908
\(56\) −2.99717e32 −1.36531
\(57\) 0 0
\(58\) −2.67431e32 −0.636492
\(59\) 8.50758e32 1.47586 0.737928 0.674879i \(-0.235804\pi\)
0.737928 + 0.674879i \(0.235804\pi\)
\(60\) 0 0
\(61\) 6.05924e32 0.567304 0.283652 0.958927i \(-0.408454\pi\)
0.283652 + 0.958927i \(0.408454\pi\)
\(62\) 1.21519e33 0.842170
\(63\) 0 0
\(64\) 1.00034e34 3.85315
\(65\) 3.91881e33 1.13307
\(66\) 0 0
\(67\) −2.62114e33 −0.432619 −0.216310 0.976325i \(-0.569402\pi\)
−0.216310 + 0.976325i \(0.569402\pi\)
\(68\) −1.57088e33 −0.197119
\(69\) 0 0
\(70\) −1.20541e34 −0.884754
\(71\) −1.38853e34 −0.783925 −0.391963 0.919981i \(-0.628204\pi\)
−0.391963 + 0.919981i \(0.628204\pi\)
\(72\) 0 0
\(73\) −1.07813e34 −0.364079 −0.182040 0.983291i \(-0.558270\pi\)
−0.182040 + 0.983291i \(0.558270\pi\)
\(74\) −2.27761e34 −0.597987
\(75\) 0 0
\(76\) −1.52269e35 −2.44096
\(77\) −3.78238e34 −0.476087
\(78\) 0 0
\(79\) 2.43825e35 1.90976 0.954878 0.297000i \(-0.0959860\pi\)
0.954878 + 0.297000i \(0.0959860\pi\)
\(80\) 6.88781e35 4.27481
\(81\) 0 0
\(82\) −8.20011e34 −0.322302
\(83\) 3.57499e34 0.112287 0.0561436 0.998423i \(-0.482120\pi\)
0.0561436 + 0.998423i \(0.482120\pi\)
\(84\) 0 0
\(85\) −4.01504e34 −0.0811782
\(86\) −5.14927e35 −0.838541
\(87\) 0 0
\(88\) 3.72849e36 3.96829
\(89\) −1.28259e36 −1.10757 −0.553785 0.832659i \(-0.686817\pi\)
−0.553785 + 0.832659i \(0.686817\pi\)
\(90\) 0 0
\(91\) 7.09018e35 0.405875
\(92\) 1.16069e37 5.42802
\(93\) 0 0
\(94\) 2.17695e35 0.0683886
\(95\) −3.89187e36 −1.00524
\(96\) 0 0
\(97\) −8.70371e36 −1.52907 −0.764535 0.644582i \(-0.777031\pi\)
−0.764535 + 0.644582i \(0.777031\pi\)
\(98\) 1.11336e37 1.61790
\(99\) 0 0
\(100\) 7.58795e36 0.758795
\(101\) 1.10433e37 0.918653 0.459327 0.888267i \(-0.348091\pi\)
0.459327 + 0.888267i \(0.348091\pi\)
\(102\) 0 0
\(103\) −2.97966e36 −0.172456 −0.0862280 0.996275i \(-0.527481\pi\)
−0.0862280 + 0.996275i \(0.527481\pi\)
\(104\) −6.98917e37 −3.38305
\(105\) 0 0
\(106\) −4.73313e37 −1.61061
\(107\) −7.82998e36 −0.223955 −0.111978 0.993711i \(-0.535718\pi\)
−0.111978 + 0.993711i \(0.535718\pi\)
\(108\) 0 0
\(109\) 1.17745e37 0.239085 0.119543 0.992829i \(-0.461857\pi\)
0.119543 + 0.992829i \(0.461857\pi\)
\(110\) 1.49954e38 2.57154
\(111\) 0 0
\(112\) 1.24619e38 1.53127
\(113\) −1.31720e38 −1.37310 −0.686548 0.727085i \(-0.740875\pi\)
−0.686548 + 0.727085i \(0.740875\pi\)
\(114\) 0 0
\(115\) 2.96662e38 2.23538
\(116\) 1.40584e38 0.902535
\(117\) 0 0
\(118\) −6.10243e38 −2.85552
\(119\) −7.26429e36 −0.0290786
\(120\) 0 0
\(121\) 1.30490e38 0.383748
\(122\) −4.34625e38 −1.09763
\(123\) 0 0
\(124\) −6.38810e38 −1.19418
\(125\) −5.07285e38 −0.817367
\(126\) 0 0
\(127\) 1.23543e39 1.48405 0.742026 0.670371i \(-0.233865\pi\)
0.742026 + 0.670371i \(0.233865\pi\)
\(128\) −3.37654e39 −3.50823
\(129\) 0 0
\(130\) −2.81093e39 −2.19229
\(131\) 2.21862e39 1.50163 0.750816 0.660511i \(-0.229660\pi\)
0.750816 + 0.660511i \(0.229660\pi\)
\(132\) 0 0
\(133\) −7.04144e38 −0.360086
\(134\) 1.88012e39 0.837042
\(135\) 0 0
\(136\) 7.16079e38 0.242377
\(137\) −1.86892e39 −0.552408 −0.276204 0.961099i \(-0.589077\pi\)
−0.276204 + 0.961099i \(0.589077\pi\)
\(138\) 0 0
\(139\) −3.65432e39 −0.826101 −0.413050 0.910708i \(-0.635537\pi\)
−0.413050 + 0.910708i \(0.635537\pi\)
\(140\) 6.33668e39 1.25456
\(141\) 0 0
\(142\) 9.95980e39 1.51676
\(143\) −8.82022e39 −1.17968
\(144\) 0 0
\(145\) 3.59321e39 0.371685
\(146\) 7.73331e39 0.704430
\(147\) 0 0
\(148\) 1.19730e40 0.847935
\(149\) 2.36586e40 1.47925 0.739627 0.673017i \(-0.235002\pi\)
0.739627 + 0.673017i \(0.235002\pi\)
\(150\) 0 0
\(151\) 3.03874e40 1.48464 0.742319 0.670047i \(-0.233726\pi\)
0.742319 + 0.670047i \(0.233726\pi\)
\(152\) 6.94112e40 3.00140
\(153\) 0 0
\(154\) 2.71307e40 0.921145
\(155\) −1.63274e40 −0.491792
\(156\) 0 0
\(157\) −1.39003e40 −0.330280 −0.165140 0.986270i \(-0.552808\pi\)
−0.165140 + 0.986270i \(0.552808\pi\)
\(158\) −1.74894e41 −3.69504
\(159\) 0 0
\(160\) −2.66382e41 −4.45948
\(161\) 5.36742e40 0.800731
\(162\) 0 0
\(163\) 8.33630e40 0.989698 0.494849 0.868979i \(-0.335223\pi\)
0.494849 + 0.868979i \(0.335223\pi\)
\(164\) 4.31067e40 0.457018
\(165\) 0 0
\(166\) −2.56432e40 −0.217256
\(167\) 5.26775e40 0.399365 0.199683 0.979861i \(-0.436009\pi\)
0.199683 + 0.979861i \(0.436009\pi\)
\(168\) 0 0
\(169\) 9.36904e38 0.00569890
\(170\) 2.87996e40 0.157066
\(171\) 0 0
\(172\) 2.70689e41 1.18904
\(173\) 3.80189e41 1.50019 0.750096 0.661329i \(-0.230007\pi\)
0.750096 + 0.661329i \(0.230007\pi\)
\(174\) 0 0
\(175\) 3.50892e40 0.111936
\(176\) −1.55027e42 −4.45063
\(177\) 0 0
\(178\) 9.19990e41 2.14295
\(179\) 5.74115e41 1.20564 0.602820 0.797877i \(-0.294044\pi\)
0.602820 + 0.797877i \(0.294044\pi\)
\(180\) 0 0
\(181\) 9.26735e41 1.58453 0.792267 0.610174i \(-0.208900\pi\)
0.792267 + 0.610174i \(0.208900\pi\)
\(182\) −5.08574e41 −0.785296
\(183\) 0 0
\(184\) −5.29095e42 −6.67427
\(185\) 3.06021e41 0.349200
\(186\) 0 0
\(187\) 9.03680e40 0.0845171
\(188\) −1.14439e41 −0.0969738
\(189\) 0 0
\(190\) 2.79161e42 1.94497
\(191\) −1.34788e42 −0.852183 −0.426092 0.904680i \(-0.640110\pi\)
−0.426092 + 0.904680i \(0.640110\pi\)
\(192\) 0 0
\(193\) 2.58423e42 1.34748 0.673738 0.738970i \(-0.264688\pi\)
0.673738 + 0.738970i \(0.264688\pi\)
\(194\) 6.24311e42 2.95848
\(195\) 0 0
\(196\) −5.85274e42 −2.29415
\(197\) −3.24438e42 −1.15746 −0.578730 0.815519i \(-0.696451\pi\)
−0.578730 + 0.815519i \(0.696451\pi\)
\(198\) 0 0
\(199\) 4.99334e42 1.47778 0.738890 0.673826i \(-0.235350\pi\)
0.738890 + 0.673826i \(0.235350\pi\)
\(200\) −3.45893e42 −0.933012
\(201\) 0 0
\(202\) −7.92125e42 −1.77743
\(203\) 6.50109e41 0.133140
\(204\) 0 0
\(205\) 1.10177e42 0.188211
\(206\) 2.13729e42 0.333672
\(207\) 0 0
\(208\) 2.90602e43 3.79426
\(209\) 8.75958e42 1.04659
\(210\) 0 0
\(211\) 4.78475e42 0.479329 0.239665 0.970856i \(-0.422963\pi\)
0.239665 + 0.970856i \(0.422963\pi\)
\(212\) 2.48814e43 2.28381
\(213\) 0 0
\(214\) 5.61639e42 0.433314
\(215\) 6.91858e42 0.489672
\(216\) 0 0
\(217\) −2.95407e42 −0.176164
\(218\) −8.44577e42 −0.462588
\(219\) 0 0
\(220\) −7.88285e43 −3.64640
\(221\) −1.69398e42 −0.0720527
\(222\) 0 0
\(223\) −2.35490e43 −0.847875 −0.423938 0.905691i \(-0.639352\pi\)
−0.423938 + 0.905691i \(0.639352\pi\)
\(224\) −4.81956e43 −1.59742
\(225\) 0 0
\(226\) 9.44818e43 2.65670
\(227\) −3.01075e43 −0.780183 −0.390092 0.920776i \(-0.627557\pi\)
−0.390092 + 0.920776i \(0.627557\pi\)
\(228\) 0 0
\(229\) −5.09965e43 −1.12353 −0.561764 0.827297i \(-0.689877\pi\)
−0.561764 + 0.827297i \(0.689877\pi\)
\(230\) −2.12794e44 −4.32507
\(231\) 0 0
\(232\) −6.40847e43 −1.10975
\(233\) −1.15459e43 −0.184647 −0.0923237 0.995729i \(-0.529429\pi\)
−0.0923237 + 0.995729i \(0.529429\pi\)
\(234\) 0 0
\(235\) −2.92497e42 −0.0399361
\(236\) 3.20795e44 4.04908
\(237\) 0 0
\(238\) 5.21062e42 0.0562621
\(239\) 9.64335e43 0.963533 0.481767 0.876300i \(-0.339995\pi\)
0.481767 + 0.876300i \(0.339995\pi\)
\(240\) 0 0
\(241\) 2.88872e43 0.247394 0.123697 0.992320i \(-0.460525\pi\)
0.123697 + 0.992320i \(0.460525\pi\)
\(242\) −9.35993e43 −0.742486
\(243\) 0 0
\(244\) 2.28476e44 1.55643
\(245\) −1.49591e44 −0.944785
\(246\) 0 0
\(247\) −1.64201e44 −0.892241
\(248\) 2.91198e44 1.46836
\(249\) 0 0
\(250\) 3.63872e44 1.58146
\(251\) −3.22802e43 −0.130308 −0.0651542 0.997875i \(-0.520754\pi\)
−0.0651542 + 0.997875i \(0.520754\pi\)
\(252\) 0 0
\(253\) −6.67709e44 −2.32733
\(254\) −8.86165e44 −2.87138
\(255\) 0 0
\(256\) 1.04712e45 2.93465
\(257\) 2.79016e44 0.727555 0.363777 0.931486i \(-0.381487\pi\)
0.363777 + 0.931486i \(0.381487\pi\)
\(258\) 0 0
\(259\) 5.53674e43 0.125086
\(260\) 1.47767e45 3.10863
\(261\) 0 0
\(262\) −1.59140e45 −2.90539
\(263\) −1.50410e43 −0.0255916 −0.0127958 0.999918i \(-0.504073\pi\)
−0.0127958 + 0.999918i \(0.504073\pi\)
\(264\) 0 0
\(265\) 6.35946e44 0.940528
\(266\) 5.05078e44 0.696703
\(267\) 0 0
\(268\) −9.88352e44 −1.18691
\(269\) −1.21815e45 −1.36548 −0.682738 0.730663i \(-0.739211\pi\)
−0.682738 + 0.730663i \(0.739211\pi\)
\(270\) 0 0
\(271\) 7.74428e44 0.756918 0.378459 0.925618i \(-0.376454\pi\)
0.378459 + 0.925618i \(0.376454\pi\)
\(272\) −2.97738e44 −0.271838
\(273\) 0 0
\(274\) 1.34056e45 1.06881
\(275\) −4.36512e44 −0.325343
\(276\) 0 0
\(277\) −6.27119e44 −0.408766 −0.204383 0.978891i \(-0.565519\pi\)
−0.204383 + 0.978891i \(0.565519\pi\)
\(278\) 2.62122e45 1.59836
\(279\) 0 0
\(280\) −2.88855e45 −1.54261
\(281\) 1.10245e45 0.551180 0.275590 0.961275i \(-0.411127\pi\)
0.275590 + 0.961275i \(0.411127\pi\)
\(282\) 0 0
\(283\) 3.71020e45 1.62685 0.813427 0.581667i \(-0.197599\pi\)
0.813427 + 0.581667i \(0.197599\pi\)
\(284\) −5.23572e45 −2.15073
\(285\) 0 0
\(286\) 6.32668e45 2.28246
\(287\) 1.99340e44 0.0674186
\(288\) 0 0
\(289\) −3.34474e45 −0.994838
\(290\) −2.57739e45 −0.719145
\(291\) 0 0
\(292\) −4.06529e45 −0.998869
\(293\) 1.22173e45 0.281788 0.140894 0.990025i \(-0.455002\pi\)
0.140894 + 0.990025i \(0.455002\pi\)
\(294\) 0 0
\(295\) 8.19925e45 1.66750
\(296\) −5.45786e45 −1.04262
\(297\) 0 0
\(298\) −1.69701e46 −2.86210
\(299\) 1.25164e46 1.98410
\(300\) 0 0
\(301\) 1.25176e45 0.175404
\(302\) −2.17967e46 −2.87251
\(303\) 0 0
\(304\) −2.88604e46 −3.36621
\(305\) 5.83964e45 0.640972
\(306\) 0 0
\(307\) −2.41884e45 −0.235260 −0.117630 0.993057i \(-0.537530\pi\)
−0.117630 + 0.993057i \(0.537530\pi\)
\(308\) −1.42622e46 −1.30617
\(309\) 0 0
\(310\) 1.17115e46 0.951531
\(311\) 2.34597e46 1.79579 0.897893 0.440214i \(-0.145098\pi\)
0.897893 + 0.440214i \(0.145098\pi\)
\(312\) 0 0
\(313\) 1.37353e46 0.933832 0.466916 0.884302i \(-0.345365\pi\)
0.466916 + 0.884302i \(0.345365\pi\)
\(314\) 9.97062e45 0.639033
\(315\) 0 0
\(316\) 9.19392e46 5.23950
\(317\) −1.81588e46 −0.976091 −0.488046 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(318\) 0 0
\(319\) −8.08738e45 −0.386973
\(320\) 9.64082e46 4.35351
\(321\) 0 0
\(322\) −3.85001e46 −1.54927
\(323\) 1.68233e45 0.0639241
\(324\) 0 0
\(325\) 8.18255e45 0.277362
\(326\) −5.97957e46 −1.91489
\(327\) 0 0
\(328\) −1.96500e46 −0.561948
\(329\) −5.29205e44 −0.0143054
\(330\) 0 0
\(331\) 6.35602e46 1.53592 0.767958 0.640500i \(-0.221273\pi\)
0.767958 + 0.640500i \(0.221273\pi\)
\(332\) 1.34802e46 0.308065
\(333\) 0 0
\(334\) −3.77852e46 −0.772701
\(335\) −2.52614e46 −0.488797
\(336\) 0 0
\(337\) −8.32564e46 −1.44299 −0.721497 0.692417i \(-0.756546\pi\)
−0.721497 + 0.692417i \(0.756546\pi\)
\(338\) −6.72034e44 −0.0110264
\(339\) 0 0
\(340\) −1.51395e46 −0.222716
\(341\) 3.67488e46 0.512020
\(342\) 0 0
\(343\) −5.94318e46 −0.743152
\(344\) −1.23392e47 −1.46203
\(345\) 0 0
\(346\) −2.72707e47 −2.90261
\(347\) 2.08182e46 0.210062 0.105031 0.994469i \(-0.466506\pi\)
0.105031 + 0.994469i \(0.466506\pi\)
\(348\) 0 0
\(349\) 1.34295e47 1.21840 0.609201 0.793016i \(-0.291490\pi\)
0.609201 + 0.793016i \(0.291490\pi\)
\(350\) −2.51693e46 −0.216577
\(351\) 0 0
\(352\) 5.99555e47 4.64291
\(353\) 2.08412e47 1.53141 0.765705 0.643192i \(-0.222390\pi\)
0.765705 + 0.643192i \(0.222390\pi\)
\(354\) 0 0
\(355\) −1.33820e47 −0.885723
\(356\) −4.83625e47 −3.03867
\(357\) 0 0
\(358\) −4.11808e47 −2.33270
\(359\) −6.98095e46 −0.375550 −0.187775 0.982212i \(-0.560128\pi\)
−0.187775 + 0.982212i \(0.560128\pi\)
\(360\) 0 0
\(361\) −4.29354e46 −0.208416
\(362\) −6.64740e47 −3.06580
\(363\) 0 0
\(364\) 2.67350e47 1.11354
\(365\) −1.03905e47 −0.411357
\(366\) 0 0
\(367\) 4.35331e46 0.155775 0.0778875 0.996962i \(-0.475183\pi\)
0.0778875 + 0.996962i \(0.475183\pi\)
\(368\) 2.19992e48 7.48553
\(369\) 0 0
\(370\) −2.19506e47 −0.675640
\(371\) 1.15060e47 0.336904
\(372\) 0 0
\(373\) 4.60312e47 1.22022 0.610111 0.792316i \(-0.291125\pi\)
0.610111 + 0.792316i \(0.291125\pi\)
\(374\) −6.48203e46 −0.163526
\(375\) 0 0
\(376\) 5.21666e46 0.119239
\(377\) 1.51601e47 0.329903
\(378\) 0 0
\(379\) −1.24559e47 −0.245783 −0.122891 0.992420i \(-0.539217\pi\)
−0.122891 + 0.992420i \(0.539217\pi\)
\(380\) −1.46751e48 −2.75793
\(381\) 0 0
\(382\) 9.66823e47 1.64882
\(383\) 5.34361e47 0.868275 0.434137 0.900847i \(-0.357053\pi\)
0.434137 + 0.900847i \(0.357053\pi\)
\(384\) 0 0
\(385\) −3.64530e47 −0.537910
\(386\) −1.85365e48 −2.60713
\(387\) 0 0
\(388\) −3.28191e48 −4.19507
\(389\) −5.89339e47 −0.718286 −0.359143 0.933283i \(-0.616931\pi\)
−0.359143 + 0.933283i \(0.616931\pi\)
\(390\) 0 0
\(391\) −1.28238e47 −0.142149
\(392\) 2.66795e48 2.82088
\(393\) 0 0
\(394\) 2.32717e48 2.23948
\(395\) 2.34989e48 2.15775
\(396\) 0 0
\(397\) 6.73632e47 0.563376 0.281688 0.959506i \(-0.409106\pi\)
0.281688 + 0.959506i \(0.409106\pi\)
\(398\) −3.58169e48 −2.85924
\(399\) 0 0
\(400\) 1.43819e48 1.04642
\(401\) 1.51806e48 1.05468 0.527338 0.849656i \(-0.323190\pi\)
0.527338 + 0.849656i \(0.323190\pi\)
\(402\) 0 0
\(403\) −6.88867e47 −0.436508
\(404\) 4.16408e48 2.52037
\(405\) 0 0
\(406\) −4.66319e47 −0.257603
\(407\) −6.88772e47 −0.363563
\(408\) 0 0
\(409\) 3.44449e48 1.66052 0.830259 0.557377i \(-0.188192\pi\)
0.830259 + 0.557377i \(0.188192\pi\)
\(410\) −7.90292e47 −0.364155
\(411\) 0 0
\(412\) −1.12354e48 −0.473141
\(413\) 1.48347e48 0.597313
\(414\) 0 0
\(415\) 3.44543e47 0.126868
\(416\) −1.12389e49 −3.95818
\(417\) 0 0
\(418\) −6.28318e48 −2.02497
\(419\) 7.49755e47 0.231185 0.115593 0.993297i \(-0.463123\pi\)
0.115593 + 0.993297i \(0.463123\pi\)
\(420\) 0 0
\(421\) 5.66375e48 1.59913 0.799567 0.600576i \(-0.205062\pi\)
0.799567 + 0.600576i \(0.205062\pi\)
\(422\) −3.43207e48 −0.927418
\(423\) 0 0
\(424\) −1.13421e49 −2.80817
\(425\) −8.38347e46 −0.0198714
\(426\) 0 0
\(427\) 1.05655e48 0.229601
\(428\) −2.95245e48 −0.614431
\(429\) 0 0
\(430\) −4.96265e48 −0.947430
\(431\) 1.73243e47 0.0316829 0.0158415 0.999875i \(-0.494957\pi\)
0.0158415 + 0.999875i \(0.494957\pi\)
\(432\) 0 0
\(433\) 1.13654e49 1.90792 0.953960 0.299934i \(-0.0969646\pi\)
0.953960 + 0.299934i \(0.0969646\pi\)
\(434\) 2.11893e48 0.340845
\(435\) 0 0
\(436\) 4.43981e48 0.655941
\(437\) −1.24304e49 −1.76026
\(438\) 0 0
\(439\) −1.20746e49 −1.57137 −0.785684 0.618629i \(-0.787689\pi\)
−0.785684 + 0.618629i \(0.787689\pi\)
\(440\) 3.59336e49 4.48360
\(441\) 0 0
\(442\) 1.21508e48 0.139409
\(443\) 4.29789e48 0.472919 0.236459 0.971641i \(-0.424013\pi\)
0.236459 + 0.971641i \(0.424013\pi\)
\(444\) 0 0
\(445\) −1.23610e49 −1.25140
\(446\) 1.68915e49 1.64049
\(447\) 0 0
\(448\) 1.74428e49 1.55946
\(449\) −9.38321e48 −0.804997 −0.402498 0.915421i \(-0.631858\pi\)
−0.402498 + 0.915421i \(0.631858\pi\)
\(450\) 0 0
\(451\) −2.47980e48 −0.195952
\(452\) −4.96677e49 −3.76715
\(453\) 0 0
\(454\) 2.15959e49 1.50952
\(455\) 6.83323e48 0.458580
\(456\) 0 0
\(457\) −2.71199e49 −1.67818 −0.839091 0.543992i \(-0.816912\pi\)
−0.839091 + 0.543992i \(0.816912\pi\)
\(458\) 3.65794e49 2.17383
\(459\) 0 0
\(460\) 1.11862e50 6.13288
\(461\) −3.50105e48 −0.184387 −0.0921937 0.995741i \(-0.529388\pi\)
−0.0921937 + 0.995741i \(0.529388\pi\)
\(462\) 0 0
\(463\) −2.14736e49 −1.04389 −0.521946 0.852979i \(-0.674794\pi\)
−0.521946 + 0.852979i \(0.674794\pi\)
\(464\) 2.66457e49 1.24464
\(465\) 0 0
\(466\) 8.28179e48 0.357260
\(467\) 1.07777e49 0.446853 0.223427 0.974721i \(-0.428276\pi\)
0.223427 + 0.974721i \(0.428276\pi\)
\(468\) 0 0
\(469\) −4.57047e48 −0.175091
\(470\) 2.09806e48 0.0772693
\(471\) 0 0
\(472\) −1.46233e50 −4.97873
\(473\) −1.55719e49 −0.509813
\(474\) 0 0
\(475\) −8.12629e48 −0.246071
\(476\) −2.73914e48 −0.0797786
\(477\) 0 0
\(478\) −6.91711e49 −1.86427
\(479\) −4.42983e49 −1.14863 −0.574314 0.818635i \(-0.694731\pi\)
−0.574314 + 0.818635i \(0.694731\pi\)
\(480\) 0 0
\(481\) 1.29113e49 0.309945
\(482\) −2.07205e49 −0.478664
\(483\) 0 0
\(484\) 4.92037e49 1.05283
\(485\) −8.38827e49 −1.72763
\(486\) 0 0
\(487\) 8.17887e48 0.156102 0.0780508 0.996949i \(-0.475130\pi\)
0.0780508 + 0.996949i \(0.475130\pi\)
\(488\) −1.04150e50 −1.91378
\(489\) 0 0
\(490\) 1.07301e50 1.82799
\(491\) 8.00334e49 1.31300 0.656498 0.754328i \(-0.272037\pi\)
0.656498 + 0.754328i \(0.272037\pi\)
\(492\) 0 0
\(493\) −1.55323e48 −0.0236357
\(494\) 1.17780e50 1.72633
\(495\) 0 0
\(496\) −1.21077e50 −1.64684
\(497\) −2.42117e49 −0.317273
\(498\) 0 0
\(499\) −2.89490e49 −0.352187 −0.176094 0.984373i \(-0.556346\pi\)
−0.176094 + 0.984373i \(0.556346\pi\)
\(500\) −1.91282e50 −2.24248
\(501\) 0 0
\(502\) 2.31544e49 0.252124
\(503\) 6.85440e49 0.719385 0.359693 0.933071i \(-0.382882\pi\)
0.359693 + 0.933071i \(0.382882\pi\)
\(504\) 0 0
\(505\) 1.06430e50 1.03795
\(506\) 4.78943e50 4.50297
\(507\) 0 0
\(508\) 4.65844e50 4.07157
\(509\) −5.38706e49 −0.454018 −0.227009 0.973893i \(-0.572895\pi\)
−0.227009 + 0.973893i \(0.572895\pi\)
\(510\) 0 0
\(511\) −1.87993e49 −0.147351
\(512\) −2.87022e50 −2.16981
\(513\) 0 0
\(514\) −2.00136e50 −1.40769
\(515\) −2.87167e49 −0.194850
\(516\) 0 0
\(517\) 6.58333e48 0.0415787
\(518\) −3.97146e49 −0.242019
\(519\) 0 0
\(520\) −6.73587e50 −3.82236
\(521\) −3.05490e50 −1.67301 −0.836506 0.547958i \(-0.815405\pi\)
−0.836506 + 0.547958i \(0.815405\pi\)
\(522\) 0 0
\(523\) −2.66823e50 −1.36126 −0.680631 0.732626i \(-0.738294\pi\)
−0.680631 + 0.732626i \(0.738294\pi\)
\(524\) 8.36574e50 4.11980
\(525\) 0 0
\(526\) 1.07888e49 0.0495152
\(527\) 7.05782e48 0.0312733
\(528\) 0 0
\(529\) 7.05455e50 2.91433
\(530\) −4.56159e50 −1.81976
\(531\) 0 0
\(532\) −2.65512e50 −0.987912
\(533\) 4.64846e49 0.167054
\(534\) 0 0
\(535\) −7.54621e49 −0.253037
\(536\) 4.50536e50 1.45942
\(537\) 0 0
\(538\) 8.73771e50 2.64196
\(539\) 3.36690e50 0.983645
\(540\) 0 0
\(541\) −3.15729e50 −0.861322 −0.430661 0.902514i \(-0.641720\pi\)
−0.430661 + 0.902514i \(0.641720\pi\)
\(542\) −5.55491e50 −1.46450
\(543\) 0 0
\(544\) 1.15148e50 0.283581
\(545\) 1.13478e50 0.270132
\(546\) 0 0
\(547\) 1.90478e50 0.423720 0.211860 0.977300i \(-0.432048\pi\)
0.211860 + 0.977300i \(0.432048\pi\)
\(548\) −7.04713e50 −1.51556
\(549\) 0 0
\(550\) 3.13107e50 0.629481
\(551\) −1.50558e50 −0.292685
\(552\) 0 0
\(553\) 4.25158e50 0.772922
\(554\) 4.49828e50 0.790890
\(555\) 0 0
\(556\) −1.37794e51 −2.26645
\(557\) −1.91249e50 −0.304283 −0.152142 0.988359i \(-0.548617\pi\)
−0.152142 + 0.988359i \(0.548617\pi\)
\(558\) 0 0
\(559\) 2.91901e50 0.434627
\(560\) 1.20103e51 1.73011
\(561\) 0 0
\(562\) −7.90782e50 −1.06644
\(563\) 4.86098e50 0.634335 0.317167 0.948370i \(-0.397268\pi\)
0.317167 + 0.948370i \(0.397268\pi\)
\(564\) 0 0
\(565\) −1.26946e51 −1.55140
\(566\) −2.66130e51 −3.14768
\(567\) 0 0
\(568\) 2.38668e51 2.64454
\(569\) 9.00352e50 0.965685 0.482842 0.875707i \(-0.339604\pi\)
0.482842 + 0.875707i \(0.339604\pi\)
\(570\) 0 0
\(571\) −1.09945e51 −1.10512 −0.552560 0.833473i \(-0.686349\pi\)
−0.552560 + 0.833473i \(0.686349\pi\)
\(572\) −3.32584e51 −3.23649
\(573\) 0 0
\(574\) −1.42985e50 −0.130443
\(575\) 6.19436e50 0.547194
\(576\) 0 0
\(577\) −8.24476e50 −0.683007 −0.341504 0.939880i \(-0.610936\pi\)
−0.341504 + 0.939880i \(0.610936\pi\)
\(578\) 2.39916e51 1.92484
\(579\) 0 0
\(580\) 1.35489e51 1.01973
\(581\) 6.23371e49 0.0454452
\(582\) 0 0
\(583\) −1.43135e51 −0.979213
\(584\) 1.85314e51 1.22821
\(585\) 0 0
\(586\) −8.76336e50 −0.545211
\(587\) −2.47843e51 −1.49407 −0.747035 0.664784i \(-0.768523\pi\)
−0.747035 + 0.664784i \(0.768523\pi\)
\(588\) 0 0
\(589\) 6.84131e50 0.387263
\(590\) −5.88127e51 −3.22633
\(591\) 0 0
\(592\) 2.26932e51 1.16935
\(593\) 2.66024e50 0.132864 0.0664322 0.997791i \(-0.478838\pi\)
0.0664322 + 0.997791i \(0.478838\pi\)
\(594\) 0 0
\(595\) −7.00102e49 −0.0328547
\(596\) 8.92094e51 4.05840
\(597\) 0 0
\(598\) −8.97793e51 −3.83888
\(599\) −7.18446e50 −0.297850 −0.148925 0.988848i \(-0.547581\pi\)
−0.148925 + 0.988848i \(0.547581\pi\)
\(600\) 0 0
\(601\) 4.14069e51 1.61397 0.806984 0.590574i \(-0.201099\pi\)
0.806984 + 0.590574i \(0.201099\pi\)
\(602\) −8.97877e50 −0.339377
\(603\) 0 0
\(604\) 1.14582e52 4.07317
\(605\) 1.25760e51 0.433580
\(606\) 0 0
\(607\) 2.73180e51 0.886049 0.443025 0.896509i \(-0.353905\pi\)
0.443025 + 0.896509i \(0.353905\pi\)
\(608\) 1.11616e52 3.51164
\(609\) 0 0
\(610\) −4.18873e51 −1.24017
\(611\) −1.23407e50 −0.0354467
\(612\) 0 0
\(613\) 4.36236e51 1.17951 0.589756 0.807581i \(-0.299224\pi\)
0.589756 + 0.807581i \(0.299224\pi\)
\(614\) 1.73502e51 0.455187
\(615\) 0 0
\(616\) 6.50136e51 1.60606
\(617\) 3.01014e51 0.721624 0.360812 0.932639i \(-0.382500\pi\)
0.360812 + 0.932639i \(0.382500\pi\)
\(618\) 0 0
\(619\) −1.23535e51 −0.278942 −0.139471 0.990226i \(-0.544540\pi\)
−0.139471 + 0.990226i \(0.544540\pi\)
\(620\) −6.15658e51 −1.34925
\(621\) 0 0
\(622\) −1.68274e52 −3.47453
\(623\) −2.23644e51 −0.448259
\(624\) 0 0
\(625\) −6.35318e51 −1.20008
\(626\) −9.85225e51 −1.80680
\(627\) 0 0
\(628\) −5.24140e51 −0.906138
\(629\) −1.32283e50 −0.0222058
\(630\) 0 0
\(631\) 1.05432e52 1.66890 0.834450 0.551083i \(-0.185785\pi\)
0.834450 + 0.551083i \(0.185785\pi\)
\(632\) −4.19101e52 −6.44247
\(633\) 0 0
\(634\) 1.30252e52 1.88857
\(635\) 1.19066e52 1.67677
\(636\) 0 0
\(637\) −6.31137e51 −0.838579
\(638\) 5.80102e51 0.748724
\(639\) 0 0
\(640\) −3.25417e52 −3.96379
\(641\) 8.08669e51 0.956968 0.478484 0.878096i \(-0.341187\pi\)
0.478484 + 0.878096i \(0.341187\pi\)
\(642\) 0 0
\(643\) 5.36035e51 0.598812 0.299406 0.954126i \(-0.403211\pi\)
0.299406 + 0.954126i \(0.403211\pi\)
\(644\) 2.02389e52 2.19684
\(645\) 0 0
\(646\) −1.20672e51 −0.123682
\(647\) 6.77154e51 0.674463 0.337232 0.941422i \(-0.390509\pi\)
0.337232 + 0.941422i \(0.390509\pi\)
\(648\) 0 0
\(649\) −1.84544e52 −1.73609
\(650\) −5.86928e51 −0.536646
\(651\) 0 0
\(652\) 3.14337e52 2.71528
\(653\) 9.50332e51 0.797961 0.398981 0.916959i \(-0.369364\pi\)
0.398981 + 0.916959i \(0.369364\pi\)
\(654\) 0 0
\(655\) 2.13821e52 1.69663
\(656\) 8.17026e51 0.630253
\(657\) 0 0
\(658\) 3.79595e50 0.0276784
\(659\) −1.31463e52 −0.932018 −0.466009 0.884780i \(-0.654309\pi\)
−0.466009 + 0.884780i \(0.654309\pi\)
\(660\) 0 0
\(661\) 1.11659e52 0.748459 0.374229 0.927336i \(-0.377907\pi\)
0.374229 + 0.927336i \(0.377907\pi\)
\(662\) −4.55913e52 −2.97173
\(663\) 0 0
\(664\) −6.14490e51 −0.378796
\(665\) −6.78625e51 −0.406845
\(666\) 0 0
\(667\) 1.14765e52 0.650850
\(668\) 1.98631e52 1.09568
\(669\) 0 0
\(670\) 1.81198e52 0.945737
\(671\) −1.31435e52 −0.667336
\(672\) 0 0
\(673\) −2.40983e52 −1.15800 −0.578998 0.815329i \(-0.696556\pi\)
−0.578998 + 0.815329i \(0.696556\pi\)
\(674\) 5.97192e52 2.79194
\(675\) 0 0
\(676\) 3.53278e50 0.0156352
\(677\) −2.62781e52 −1.13163 −0.565814 0.824533i \(-0.691438\pi\)
−0.565814 + 0.824533i \(0.691438\pi\)
\(678\) 0 0
\(679\) −1.51767e52 −0.618850
\(680\) 6.90127e51 0.273851
\(681\) 0 0
\(682\) −2.63596e52 −0.990668
\(683\) −2.18094e52 −0.797741 −0.398870 0.917007i \(-0.630598\pi\)
−0.398870 + 0.917007i \(0.630598\pi\)
\(684\) 0 0
\(685\) −1.80119e52 −0.624142
\(686\) 4.26300e52 1.43787
\(687\) 0 0
\(688\) 5.13053e52 1.63974
\(689\) 2.68311e52 0.834800
\(690\) 0 0
\(691\) 2.89991e51 0.0855148 0.0427574 0.999085i \(-0.486386\pi\)
0.0427574 + 0.999085i \(0.486386\pi\)
\(692\) 1.43358e53 4.11585
\(693\) 0 0
\(694\) −1.49327e52 −0.406433
\(695\) −3.52188e52 −0.933375
\(696\) 0 0
\(697\) −4.76260e50 −0.0119684
\(698\) −9.63291e52 −2.35740
\(699\) 0 0
\(700\) 1.32311e52 0.307102
\(701\) 7.75630e51 0.175336 0.0876681 0.996150i \(-0.472059\pi\)
0.0876681 + 0.996150i \(0.472059\pi\)
\(702\) 0 0
\(703\) −1.28225e52 −0.274979
\(704\) −2.16990e53 −4.53257
\(705\) 0 0
\(706\) −1.49493e53 −2.96301
\(707\) 1.92561e52 0.371800
\(708\) 0 0
\(709\) −2.37217e52 −0.434700 −0.217350 0.976094i \(-0.569741\pi\)
−0.217350 + 0.976094i \(0.569741\pi\)
\(710\) 9.59884e52 1.71372
\(711\) 0 0
\(712\) 2.20458e53 3.73634
\(713\) −5.21487e52 −0.861166
\(714\) 0 0
\(715\) −8.50056e52 −1.33286
\(716\) 2.16481e53 3.30773
\(717\) 0 0
\(718\) 5.00739e52 0.726623
\(719\) −1.39711e53 −1.97581 −0.987904 0.155068i \(-0.950440\pi\)
−0.987904 + 0.155068i \(0.950440\pi\)
\(720\) 0 0
\(721\) −5.19563e51 −0.0697969
\(722\) 3.07972e52 0.403249
\(723\) 0 0
\(724\) 3.49444e53 4.34724
\(725\) 7.50269e51 0.0909838
\(726\) 0 0
\(727\) 1.82625e52 0.210462 0.105231 0.994448i \(-0.466442\pi\)
0.105231 + 0.994448i \(0.466442\pi\)
\(728\) −1.21870e53 −1.36920
\(729\) 0 0
\(730\) 7.45305e52 0.795904
\(731\) −2.99068e51 −0.0311386
\(732\) 0 0
\(733\) −2.47287e52 −0.244781 −0.122390 0.992482i \(-0.539056\pi\)
−0.122390 + 0.992482i \(0.539056\pi\)
\(734\) −3.12260e52 −0.301397
\(735\) 0 0
\(736\) −8.50805e53 −7.80891
\(737\) 5.68569e52 0.508902
\(738\) 0 0
\(739\) −3.06278e52 −0.260732 −0.130366 0.991466i \(-0.541615\pi\)
−0.130366 + 0.991466i \(0.541615\pi\)
\(740\) 1.15391e53 0.958045
\(741\) 0 0
\(742\) −8.25316e52 −0.651850
\(743\) −2.58567e52 −0.199196 −0.0995978 0.995028i \(-0.531756\pi\)
−0.0995978 + 0.995028i \(0.531756\pi\)
\(744\) 0 0
\(745\) 2.28011e53 1.67134
\(746\) −3.30179e53 −2.36092
\(747\) 0 0
\(748\) 3.40751e52 0.231877
\(749\) −1.36531e52 −0.0906398
\(750\) 0 0
\(751\) −1.09214e53 −0.690141 −0.345071 0.938577i \(-0.612145\pi\)
−0.345071 + 0.938577i \(0.612145\pi\)
\(752\) −2.16903e52 −0.133732
\(753\) 0 0
\(754\) −1.08742e53 −0.638304
\(755\) 2.92862e53 1.67743
\(756\) 0 0
\(757\) 5.53806e52 0.302054 0.151027 0.988530i \(-0.451742\pi\)
0.151027 + 0.988530i \(0.451742\pi\)
\(758\) 8.93456e52 0.475546
\(759\) 0 0
\(760\) 6.68956e53 3.39114
\(761\) −1.91021e53 −0.945072 −0.472536 0.881311i \(-0.656661\pi\)
−0.472536 + 0.881311i \(0.656661\pi\)
\(762\) 0 0
\(763\) 2.05312e52 0.0967632
\(764\) −5.08244e53 −2.33800
\(765\) 0 0
\(766\) −3.83293e53 −1.67996
\(767\) 3.45933e53 1.48006
\(768\) 0 0
\(769\) −3.71712e52 −0.151555 −0.0757773 0.997125i \(-0.524144\pi\)
−0.0757773 + 0.997125i \(0.524144\pi\)
\(770\) 2.61475e53 1.04076
\(771\) 0 0
\(772\) 9.74436e53 3.69686
\(773\) 1.53846e53 0.569858 0.284929 0.958549i \(-0.408030\pi\)
0.284929 + 0.958549i \(0.408030\pi\)
\(774\) 0 0
\(775\) −3.40920e52 −0.120384
\(776\) 1.49604e54 5.15825
\(777\) 0 0
\(778\) 4.22728e53 1.38976
\(779\) −4.61650e52 −0.148207
\(780\) 0 0
\(781\) 3.01195e53 0.922153
\(782\) 9.19839e52 0.275034
\(783\) 0 0
\(784\) −1.10930e54 −3.16376
\(785\) −1.33966e53 −0.373169
\(786\) 0 0
\(787\) 1.23159e53 0.327292 0.163646 0.986519i \(-0.447675\pi\)
0.163646 + 0.986519i \(0.447675\pi\)
\(788\) −1.22336e54 −3.17554
\(789\) 0 0
\(790\) −1.68556e54 −4.17486
\(791\) −2.29680e53 −0.555723
\(792\) 0 0
\(793\) 2.46379e53 0.568918
\(794\) −4.83192e53 −1.09003
\(795\) 0 0
\(796\) 1.88284e54 4.05436
\(797\) 1.87818e53 0.395147 0.197574 0.980288i \(-0.436694\pi\)
0.197574 + 0.980288i \(0.436694\pi\)
\(798\) 0 0
\(799\) 1.26437e51 0.00253956
\(800\) −5.56209e53 −1.09163
\(801\) 0 0
\(802\) −1.08890e54 −2.04061
\(803\) 2.33863e53 0.428277
\(804\) 0 0
\(805\) 5.17289e53 0.904711
\(806\) 4.94119e53 0.844566
\(807\) 0 0
\(808\) −1.89818e54 −3.09904
\(809\) −1.37996e53 −0.220201 −0.110100 0.993920i \(-0.535117\pi\)
−0.110100 + 0.993920i \(0.535117\pi\)
\(810\) 0 0
\(811\) 2.88787e53 0.440243 0.220121 0.975472i \(-0.429355\pi\)
0.220121 + 0.975472i \(0.429355\pi\)
\(812\) 2.45137e53 0.365276
\(813\) 0 0
\(814\) 4.94051e53 0.703430
\(815\) 8.03418e53 1.11822
\(816\) 0 0
\(817\) −2.89894e53 −0.385595
\(818\) −2.47071e54 −3.21281
\(819\) 0 0
\(820\) 4.15445e53 0.516365
\(821\) −9.46819e53 −1.15058 −0.575292 0.817948i \(-0.695112\pi\)
−0.575292 + 0.817948i \(0.695112\pi\)
\(822\) 0 0
\(823\) −9.45005e53 −1.09783 −0.548917 0.835877i \(-0.684960\pi\)
−0.548917 + 0.835877i \(0.684960\pi\)
\(824\) 5.12161e53 0.581772
\(825\) 0 0
\(826\) −1.06408e54 −1.15570
\(827\) −6.22799e53 −0.661449 −0.330724 0.943727i \(-0.607293\pi\)
−0.330724 + 0.943727i \(0.607293\pi\)
\(828\) 0 0
\(829\) −3.74356e53 −0.380212 −0.190106 0.981764i \(-0.560883\pi\)
−0.190106 + 0.981764i \(0.560883\pi\)
\(830\) −2.47138e53 −0.245468
\(831\) 0 0
\(832\) 4.06754e54 3.86412
\(833\) 6.46634e52 0.0600795
\(834\) 0 0
\(835\) 5.07683e53 0.451225
\(836\) 3.30297e54 2.87137
\(837\) 0 0
\(838\) −5.37794e53 −0.447303
\(839\) 1.42400e54 1.15855 0.579274 0.815133i \(-0.303336\pi\)
0.579274 + 0.815133i \(0.303336\pi\)
\(840\) 0 0
\(841\) −1.14547e54 −0.891781
\(842\) −4.06257e54 −3.09405
\(843\) 0 0
\(844\) 1.80419e54 1.31506
\(845\) 9.02949e51 0.00643893
\(846\) 0 0
\(847\) 2.27535e53 0.155312
\(848\) 4.71590e54 3.14950
\(849\) 0 0
\(850\) 6.01340e52 0.0384477
\(851\) 9.77409e53 0.611476
\(852\) 0 0
\(853\) −1.93589e54 −1.15964 −0.579821 0.814744i \(-0.696877\pi\)
−0.579821 + 0.814744i \(0.696877\pi\)
\(854\) −7.57855e53 −0.444238
\(855\) 0 0
\(856\) 1.34586e54 0.755502
\(857\) 2.55656e54 1.40447 0.702233 0.711947i \(-0.252187\pi\)
0.702233 + 0.711947i \(0.252187\pi\)
\(858\) 0 0
\(859\) −1.64382e54 −0.864927 −0.432464 0.901651i \(-0.642356\pi\)
−0.432464 + 0.901651i \(0.642356\pi\)
\(860\) 2.60879e54 1.34344
\(861\) 0 0
\(862\) −1.24266e53 −0.0613009
\(863\) 1.78555e54 0.862128 0.431064 0.902321i \(-0.358138\pi\)
0.431064 + 0.902321i \(0.358138\pi\)
\(864\) 0 0
\(865\) 3.66411e54 1.69500
\(866\) −8.15235e54 −3.69149
\(867\) 0 0
\(868\) −1.11389e54 −0.483313
\(869\) −5.28898e54 −2.24650
\(870\) 0 0
\(871\) −1.06580e54 −0.433850
\(872\) −2.02387e54 −0.806543
\(873\) 0 0
\(874\) 8.91621e54 3.40580
\(875\) −8.84553e53 −0.330807
\(876\) 0 0
\(877\) −1.18991e54 −0.426602 −0.213301 0.976987i \(-0.568421\pi\)
−0.213301 + 0.976987i \(0.568421\pi\)
\(878\) 8.66099e54 3.04032
\(879\) 0 0
\(880\) −1.49408e55 −5.02858
\(881\) −6.27215e53 −0.206710 −0.103355 0.994645i \(-0.532958\pi\)
−0.103355 + 0.994645i \(0.532958\pi\)
\(882\) 0 0
\(883\) −7.94118e53 −0.250964 −0.125482 0.992096i \(-0.540048\pi\)
−0.125482 + 0.992096i \(0.540048\pi\)
\(884\) −6.38748e53 −0.197680
\(885\) 0 0
\(886\) −3.08285e54 −0.915014
\(887\) −4.43676e54 −1.28967 −0.644836 0.764321i \(-0.723074\pi\)
−0.644836 + 0.764321i \(0.723074\pi\)
\(888\) 0 0
\(889\) 2.15422e54 0.600630
\(890\) 8.86648e54 2.42123
\(891\) 0 0
\(892\) −8.87961e54 −2.32618
\(893\) 1.22558e53 0.0314478
\(894\) 0 0
\(895\) 5.53308e54 1.36220
\(896\) −5.88767e54 −1.41986
\(897\) 0 0
\(898\) 6.73051e54 1.55753
\(899\) −6.31632e53 −0.143189
\(900\) 0 0
\(901\) −2.74899e53 −0.0598087
\(902\) 1.77874e54 0.379133
\(903\) 0 0
\(904\) 2.26408e55 4.63207
\(905\) 8.93148e54 1.79030
\(906\) 0 0
\(907\) 8.87474e54 1.70774 0.853868 0.520490i \(-0.174251\pi\)
0.853868 + 0.520490i \(0.174251\pi\)
\(908\) −1.13526e55 −2.14047
\(909\) 0 0
\(910\) −4.90142e54 −0.887271
\(911\) −2.41119e54 −0.427702 −0.213851 0.976866i \(-0.568601\pi\)
−0.213851 + 0.976866i \(0.568601\pi\)
\(912\) 0 0
\(913\) −7.75476e53 −0.132087
\(914\) 1.94529e55 3.24699
\(915\) 0 0
\(916\) −1.92293e55 −3.08245
\(917\) 3.86860e54 0.607745
\(918\) 0 0
\(919\) −9.87356e54 −1.48983 −0.744915 0.667159i \(-0.767510\pi\)
−0.744915 + 0.667159i \(0.767510\pi\)
\(920\) −5.09920e55 −7.54096
\(921\) 0 0
\(922\) 2.51128e54 0.356757
\(923\) −5.64599e54 −0.786156
\(924\) 0 0
\(925\) 6.38977e53 0.0854797
\(926\) 1.54028e55 2.01975
\(927\) 0 0
\(928\) −1.03051e55 −1.29841
\(929\) 6.84483e54 0.845418 0.422709 0.906265i \(-0.361079\pi\)
0.422709 + 0.906265i \(0.361079\pi\)
\(930\) 0 0
\(931\) 6.26798e54 0.743975
\(932\) −4.35361e54 −0.506589
\(933\) 0 0
\(934\) −7.73078e54 −0.864582
\(935\) 8.70929e53 0.0954922
\(936\) 0 0
\(937\) −9.21588e53 −0.0971302 −0.0485651 0.998820i \(-0.515465\pi\)
−0.0485651 + 0.998820i \(0.515465\pi\)
\(938\) 3.27837e54 0.338770
\(939\) 0 0
\(940\) −1.10292e54 −0.109566
\(941\) −1.10225e55 −1.07367 −0.536835 0.843687i \(-0.680380\pi\)
−0.536835 + 0.843687i \(0.680380\pi\)
\(942\) 0 0
\(943\) 3.51898e54 0.329572
\(944\) 6.08022e55 5.58390
\(945\) 0 0
\(946\) 1.11696e55 0.986399
\(947\) −2.12495e55 −1.84024 −0.920120 0.391636i \(-0.871909\pi\)
−0.920120 + 0.391636i \(0.871909\pi\)
\(948\) 0 0
\(949\) −4.38384e54 −0.365115
\(950\) 5.82893e54 0.476104
\(951\) 0 0
\(952\) 1.24863e54 0.0980955
\(953\) −1.66851e55 −1.28562 −0.642808 0.766028i \(-0.722231\pi\)
−0.642808 + 0.766028i \(0.722231\pi\)
\(954\) 0 0
\(955\) −1.29903e55 −0.962844
\(956\) 3.63622e55 2.64350
\(957\) 0 0
\(958\) 3.17748e55 2.22240
\(959\) −3.25883e54 −0.223572
\(960\) 0 0
\(961\) −1.22788e55 −0.810541
\(962\) −9.26115e54 −0.599689
\(963\) 0 0
\(964\) 1.08925e55 0.678737
\(965\) 2.49057e55 1.52245
\(966\) 0 0
\(967\) −1.94586e55 −1.14478 −0.572389 0.819982i \(-0.693983\pi\)
−0.572389 + 0.819982i \(0.693983\pi\)
\(968\) −2.24293e55 −1.29456
\(969\) 0 0
\(970\) 6.01685e55 3.34266
\(971\) 1.46805e55 0.800173 0.400087 0.916477i \(-0.368980\pi\)
0.400087 + 0.916477i \(0.368980\pi\)
\(972\) 0 0
\(973\) −6.37204e54 −0.334342
\(974\) −5.86664e54 −0.302029
\(975\) 0 0
\(976\) 4.33043e55 2.14639
\(977\) −2.32343e55 −1.13001 −0.565003 0.825089i \(-0.691125\pi\)
−0.565003 + 0.825089i \(0.691125\pi\)
\(978\) 0 0
\(979\) 2.78215e55 1.30287
\(980\) −5.64063e55 −2.59206
\(981\) 0 0
\(982\) −5.74073e55 −2.54042
\(983\) −1.19994e55 −0.521097 −0.260549 0.965461i \(-0.583903\pi\)
−0.260549 + 0.965461i \(0.583903\pi\)
\(984\) 0 0
\(985\) −3.12679e55 −1.30776
\(986\) 1.11412e54 0.0457309
\(987\) 0 0
\(988\) −6.19153e55 −2.44790
\(989\) 2.20975e55 0.857455
\(990\) 0 0
\(991\) −3.02946e54 −0.113241 −0.0566203 0.998396i \(-0.518032\pi\)
−0.0566203 + 0.998396i \(0.518032\pi\)
\(992\) 4.68258e55 1.71799
\(993\) 0 0
\(994\) 1.73669e55 0.613867
\(995\) 4.81238e55 1.66968
\(996\) 0 0
\(997\) −3.34593e55 −1.11855 −0.559277 0.828981i \(-0.688921\pi\)
−0.559277 + 0.828981i \(0.688921\pi\)
\(998\) 2.07649e55 0.681421
\(999\) 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 9.38.a.c.1.1 4
3.2 odd 2 3.38.a.b.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.38.a.b.1.4 4 3.2 odd 2
9.38.a.c.1.1 4 1.1 even 1 trivial