Properties

Label 9.38.a.c.1.4
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 1.4
Root \(105009.\)
Character \(\chi\) = 9.1

$q$-expansion

\(f(q)\) \(=\) \(q+520663. q^{2} +1.33651e11 q^{4} +2.63441e12 q^{5} +8.34393e15 q^{7} -1.97212e15 q^{8} +O(q^{10})\) \(q+520663. q^{2} +1.33651e11 q^{4} +2.63441e12 q^{5} +8.34393e15 q^{7} -1.97212e15 q^{8} +1.37164e18 q^{10} +2.00696e19 q^{11} -1.05373e20 q^{13} +4.34438e21 q^{14} -1.93957e22 q^{16} -1.77033e22 q^{17} +4.00776e23 q^{19} +3.52092e23 q^{20} +1.04495e25 q^{22} +1.07722e25 q^{23} -6.58195e25 q^{25} -5.48640e25 q^{26} +1.11518e27 q^{28} -1.59166e27 q^{29} +4.20936e27 q^{31} -9.82758e27 q^{32} -9.21743e27 q^{34} +2.19813e28 q^{35} +1.71917e29 q^{37} +2.08669e29 q^{38} -5.19537e27 q^{40} +3.95960e29 q^{41} +3.48956e29 q^{43} +2.68233e30 q^{44} +5.60870e30 q^{46} +2.42346e30 q^{47} +5.10591e31 q^{49} -3.42698e31 q^{50} -1.40833e31 q^{52} +1.34925e32 q^{53} +5.28715e31 q^{55} -1.64552e31 q^{56} -8.28718e32 q^{58} -1.06678e33 q^{59} -7.04574e32 q^{61} +2.19166e33 q^{62} -2.45114e33 q^{64} -2.77596e32 q^{65} +7.19450e33 q^{67} -2.36606e33 q^{68} +1.14449e34 q^{70} -2.35209e34 q^{71} +2.69218e33 q^{73} +8.95108e34 q^{74} +5.35642e34 q^{76} +1.67459e35 q^{77} +2.20044e34 q^{79} -5.10962e34 q^{80} +2.06162e35 q^{82} +2.35934e35 q^{83} -4.66376e34 q^{85} +1.81689e35 q^{86} -3.95797e34 q^{88} -1.13980e35 q^{89} -8.79227e35 q^{91} +1.43972e36 q^{92} +1.26181e36 q^{94} +1.05581e36 q^{95} +8.44145e36 q^{97} +2.65846e37 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} + O(q^{10}) \) \( 4q - 437562q^{2} + 346098955492q^{4} + 4099829756904q^{5} + 6605809948153184q^{7} - 140484113342159976q^{8} - 21511023001649316q^{10} - 20953708852195292976q^{11} + 51830892788989874168q^{13} + \)\(18\!\cdots\!12\)\(q^{14} + \)\(55\!\cdots\!40\)\(q^{16} - \)\(81\!\cdots\!28\)\(q^{17} - \)\(54\!\cdots\!32\)\(q^{19} + \)\(35\!\cdots\!76\)\(q^{20} + \)\(23\!\cdots\!76\)\(q^{22} + \)\(61\!\cdots\!88\)\(q^{23} + \)\(95\!\cdots\!16\)\(q^{25} - \)\(42\!\cdots\!88\)\(q^{26} + \)\(24\!\cdots\!48\)\(q^{28} - \)\(41\!\cdots\!36\)\(q^{29} + \)\(89\!\cdots\!64\)\(q^{31} - \)\(36\!\cdots\!84\)\(q^{32} + \)\(31\!\cdots\!24\)\(q^{34} - \)\(42\!\cdots\!40\)\(q^{35} + \)\(55\!\cdots\!24\)\(q^{37} + \)\(73\!\cdots\!68\)\(q^{38} - \)\(26\!\cdots\!52\)\(q^{40} + \)\(86\!\cdots\!76\)\(q^{41} - \)\(50\!\cdots\!80\)\(q^{43} - \)\(28\!\cdots\!36\)\(q^{44} - \)\(14\!\cdots\!96\)\(q^{46} - \)\(42\!\cdots\!20\)\(q^{47} + \)\(40\!\cdots\!20\)\(q^{49} - \)\(10\!\cdots\!14\)\(q^{50} + \)\(18\!\cdots\!68\)\(q^{52} + \)\(12\!\cdots\!88\)\(q^{53} - \)\(32\!\cdots\!24\)\(q^{55} - \)\(49\!\cdots\!80\)\(q^{56} - \)\(75\!\cdots\!56\)\(q^{58} + \)\(13\!\cdots\!88\)\(q^{59} - \)\(12\!\cdots\!60\)\(q^{61} + \)\(37\!\cdots\!92\)\(q^{62} + \)\(85\!\cdots\!28\)\(q^{64} - \)\(15\!\cdots\!68\)\(q^{65} + \)\(16\!\cdots\!48\)\(q^{67} - \)\(63\!\cdots\!84\)\(q^{68} + \)\(82\!\cdots\!60\)\(q^{70} - \)\(10\!\cdots\!88\)\(q^{71} - \)\(19\!\cdots\!48\)\(q^{73} + \)\(89\!\cdots\!12\)\(q^{74} - \)\(95\!\cdots\!68\)\(q^{76} + \)\(25\!\cdots\!92\)\(q^{77} + \)\(42\!\cdots\!20\)\(q^{79} + \)\(95\!\cdots\!36\)\(q^{80} + \)\(33\!\cdots\!48\)\(q^{82} + \)\(46\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!12\)\(q^{85} + \)\(36\!\cdots\!04\)\(q^{86} + \)\(42\!\cdots\!56\)\(q^{88} + \)\(31\!\cdots\!52\)\(q^{89} + \)\(26\!\cdots\!24\)\(q^{91} + \)\(16\!\cdots\!16\)\(q^{92} - \)\(57\!\cdots\!48\)\(q^{94} + \)\(89\!\cdots\!56\)\(q^{95} + \)\(44\!\cdots\!48\)\(q^{97} + \)\(43\!\cdots\!78\)\(q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 520663. 1.40444 0.702218 0.711962i \(-0.252193\pi\)
0.702218 + 0.711962i \(0.252193\pi\)
\(3\) 0 0
\(4\) 1.33651e11 0.972441
\(5\) 2.63441e12 0.308843 0.154422 0.988005i \(-0.450649\pi\)
0.154422 + 0.988005i \(0.450649\pi\)
\(6\) 0 0
\(7\) 8.34393e15 1.93668 0.968338 0.249642i \(-0.0803128\pi\)
0.968338 + 0.249642i \(0.0803128\pi\)
\(8\) −1.97212e15 −0.0387051
\(9\) 0 0
\(10\) 1.37164e18 0.433750
\(11\) 2.00696e19 1.08836 0.544182 0.838967i \(-0.316840\pi\)
0.544182 + 0.838967i \(0.316840\pi\)
\(12\) 0 0
\(13\) −1.05373e20 −0.259883 −0.129942 0.991522i \(-0.541479\pi\)
−0.129942 + 0.991522i \(0.541479\pi\)
\(14\) 4.34438e21 2.71994
\(15\) 0 0
\(16\) −1.93957e22 −1.02680
\(17\) −1.77033e22 −0.305315 −0.152657 0.988279i \(-0.548783\pi\)
−0.152657 + 0.988279i \(0.548783\pi\)
\(18\) 0 0
\(19\) 4.00776e23 0.882999 0.441499 0.897262i \(-0.354447\pi\)
0.441499 + 0.897262i \(0.354447\pi\)
\(20\) 3.52092e23 0.300332
\(21\) 0 0
\(22\) 1.04495e25 1.52854
\(23\) 1.07722e25 0.692374 0.346187 0.938166i \(-0.387476\pi\)
0.346187 + 0.938166i \(0.387476\pi\)
\(24\) 0 0
\(25\) −6.58195e25 −0.904616
\(26\) −5.48640e25 −0.364989
\(27\) 0 0
\(28\) 1.11518e27 1.88330
\(29\) −1.59166e27 −1.40439 −0.702193 0.711986i \(-0.747796\pi\)
−0.702193 + 0.711986i \(0.747796\pi\)
\(30\) 0 0
\(31\) 4.20936e27 1.08149 0.540747 0.841185i \(-0.318142\pi\)
0.540747 + 0.841185i \(0.318142\pi\)
\(32\) −9.82758e27 −1.40337
\(33\) 0 0
\(34\) −9.21743e27 −0.428795
\(35\) 2.19813e28 0.598129
\(36\) 0 0
\(37\) 1.71917e29 1.67335 0.836675 0.547700i \(-0.184496\pi\)
0.836675 + 0.547700i \(0.184496\pi\)
\(38\) 2.08669e29 1.24012
\(39\) 0 0
\(40\) −5.19537e27 −0.0119538
\(41\) 3.95960e29 0.576965 0.288483 0.957485i \(-0.406849\pi\)
0.288483 + 0.957485i \(0.406849\pi\)
\(42\) 0 0
\(43\) 3.48956e29 0.210671 0.105335 0.994437i \(-0.466408\pi\)
0.105335 + 0.994437i \(0.466408\pi\)
\(44\) 2.68233e30 1.05837
\(45\) 0 0
\(46\) 5.60870e30 0.972395
\(47\) 2.42346e30 0.282244 0.141122 0.989992i \(-0.454929\pi\)
0.141122 + 0.989992i \(0.454929\pi\)
\(48\) 0 0
\(49\) 5.10591e31 2.75072
\(50\) −3.42698e31 −1.27048
\(51\) 0 0
\(52\) −1.40833e31 −0.252721
\(53\) 1.34925e32 1.70212 0.851060 0.525068i \(-0.175960\pi\)
0.851060 + 0.525068i \(0.175960\pi\)
\(54\) 0 0
\(55\) 5.28715e31 0.336134
\(56\) −1.64552e31 −0.0749593
\(57\) 0 0
\(58\) −8.28718e32 −1.97237
\(59\) −1.06678e33 −1.85061 −0.925304 0.379226i \(-0.876190\pi\)
−0.925304 + 0.379226i \(0.876190\pi\)
\(60\) 0 0
\(61\) −7.04574e32 −0.659667 −0.329833 0.944039i \(-0.606993\pi\)
−0.329833 + 0.944039i \(0.606993\pi\)
\(62\) 2.19166e33 1.51889
\(63\) 0 0
\(64\) −2.45114e33 −0.944143
\(65\) −2.77596e32 −0.0802632
\(66\) 0 0
\(67\) 7.19450e33 1.18745 0.593727 0.804667i \(-0.297656\pi\)
0.593727 + 0.804667i \(0.297656\pi\)
\(68\) −2.36606e33 −0.296901
\(69\) 0 0
\(70\) 1.14449e34 0.840034
\(71\) −2.35209e34 −1.32793 −0.663964 0.747765i \(-0.731127\pi\)
−0.663964 + 0.747765i \(0.731127\pi\)
\(72\) 0 0
\(73\) 2.69218e33 0.0909140 0.0454570 0.998966i \(-0.485526\pi\)
0.0454570 + 0.998966i \(0.485526\pi\)
\(74\) 8.95108e34 2.35011
\(75\) 0 0
\(76\) 5.35642e34 0.858664
\(77\) 1.67459e35 2.10781
\(78\) 0 0
\(79\) 2.20044e34 0.172349 0.0861746 0.996280i \(-0.472536\pi\)
0.0861746 + 0.996280i \(0.472536\pi\)
\(80\) −5.10962e34 −0.317120
\(81\) 0 0
\(82\) 2.06162e35 0.810311
\(83\) 2.35934e35 0.741046 0.370523 0.928823i \(-0.379178\pi\)
0.370523 + 0.928823i \(0.379178\pi\)
\(84\) 0 0
\(85\) −4.66376e34 −0.0942944
\(86\) 1.81689e35 0.295874
\(87\) 0 0
\(88\) −3.95797e34 −0.0421253
\(89\) −1.13980e35 −0.0984271 −0.0492135 0.998788i \(-0.515671\pi\)
−0.0492135 + 0.998788i \(0.515671\pi\)
\(90\) 0 0
\(91\) −8.79227e35 −0.503310
\(92\) 1.43972e36 0.673293
\(93\) 0 0
\(94\) 1.26181e36 0.396394
\(95\) 1.05581e36 0.272708
\(96\) 0 0
\(97\) 8.44145e36 1.48300 0.741498 0.670955i \(-0.234116\pi\)
0.741498 + 0.670955i \(0.234116\pi\)
\(98\) 2.65846e37 3.86321
\(99\) 0 0
\(100\) −8.79685e36 −0.879685
\(101\) −6.86236e36 −0.570858 −0.285429 0.958400i \(-0.592136\pi\)
−0.285429 + 0.958400i \(0.592136\pi\)
\(102\) 0 0
\(103\) 1.30579e37 0.755760 0.377880 0.925855i \(-0.376653\pi\)
0.377880 + 0.925855i \(0.376653\pi\)
\(104\) 2.07809e35 0.0100588
\(105\) 0 0
\(106\) 7.02507e37 2.39052
\(107\) −4.38206e37 −1.25337 −0.626683 0.779274i \(-0.715588\pi\)
−0.626683 + 0.779274i \(0.715588\pi\)
\(108\) 0 0
\(109\) −1.05626e37 −0.214476 −0.107238 0.994233i \(-0.534201\pi\)
−0.107238 + 0.994233i \(0.534201\pi\)
\(110\) 2.75282e37 0.472078
\(111\) 0 0
\(112\) −1.61836e38 −1.98858
\(113\) 6.38435e37 0.665527 0.332763 0.943010i \(-0.392019\pi\)
0.332763 + 0.943010i \(0.392019\pi\)
\(114\) 0 0
\(115\) 2.83784e37 0.213835
\(116\) −2.12727e38 −1.36568
\(117\) 0 0
\(118\) −5.55435e38 −2.59906
\(119\) −1.47715e38 −0.591296
\(120\) 0 0
\(121\) 6.27494e37 0.184536
\(122\) −3.66846e38 −0.926460
\(123\) 0 0
\(124\) 5.62586e38 1.05169
\(125\) −3.65074e38 −0.588227
\(126\) 0 0
\(127\) −2.81304e38 −0.337915 −0.168957 0.985623i \(-0.554040\pi\)
−0.168957 + 0.985623i \(0.554040\pi\)
\(128\) 7.44763e37 0.0773809
\(129\) 0 0
\(130\) −1.44534e38 −0.112724
\(131\) −1.95973e39 −1.32641 −0.663206 0.748437i \(-0.730805\pi\)
−0.663206 + 0.748437i \(0.730805\pi\)
\(132\) 0 0
\(133\) 3.34405e39 1.71008
\(134\) 3.74591e39 1.66770
\(135\) 0 0
\(136\) 3.49129e37 0.0118173
\(137\) −1.41955e39 −0.419586 −0.209793 0.977746i \(-0.567279\pi\)
−0.209793 + 0.977746i \(0.567279\pi\)
\(138\) 0 0
\(139\) −2.88414e39 −0.651992 −0.325996 0.945371i \(-0.605700\pi\)
−0.325996 + 0.945371i \(0.605700\pi\)
\(140\) 2.93783e39 0.581645
\(141\) 0 0
\(142\) −1.22465e40 −1.86499
\(143\) −2.11480e39 −0.282848
\(144\) 0 0
\(145\) −4.19308e39 −0.433735
\(146\) 1.40172e39 0.127683
\(147\) 0 0
\(148\) 2.29769e40 1.62723
\(149\) 1.54836e40 0.968111 0.484055 0.875037i \(-0.339163\pi\)
0.484055 + 0.875037i \(0.339163\pi\)
\(150\) 0 0
\(151\) −5.65320e39 −0.276198 −0.138099 0.990418i \(-0.544099\pi\)
−0.138099 + 0.990418i \(0.544099\pi\)
\(152\) −7.90378e38 −0.0341766
\(153\) 0 0
\(154\) 8.71900e40 2.96028
\(155\) 1.10892e40 0.334012
\(156\) 0 0
\(157\) −6.59332e40 −1.56661 −0.783304 0.621638i \(-0.786467\pi\)
−0.783304 + 0.621638i \(0.786467\pi\)
\(158\) 1.14569e40 0.242053
\(159\) 0 0
\(160\) −2.58899e40 −0.433421
\(161\) 8.98828e40 1.34090
\(162\) 0 0
\(163\) 3.78789e40 0.449704 0.224852 0.974393i \(-0.427810\pi\)
0.224852 + 0.974393i \(0.427810\pi\)
\(164\) 5.29205e40 0.561065
\(165\) 0 0
\(166\) 1.22842e41 1.04075
\(167\) −1.83545e41 −1.39151 −0.695757 0.718277i \(-0.744931\pi\)
−0.695757 + 0.718277i \(0.744931\pi\)
\(168\) 0 0
\(169\) −1.53297e41 −0.932461
\(170\) −2.42825e40 −0.132430
\(171\) 0 0
\(172\) 4.66384e40 0.204865
\(173\) −1.65118e41 −0.651540 −0.325770 0.945449i \(-0.605624\pi\)
−0.325770 + 0.945449i \(0.605624\pi\)
\(174\) 0 0
\(175\) −5.49193e41 −1.75195
\(176\) −3.89264e41 −1.11753
\(177\) 0 0
\(178\) −5.93454e40 −0.138235
\(179\) −4.97609e41 −1.04498 −0.522489 0.852646i \(-0.674996\pi\)
−0.522489 + 0.852646i \(0.674996\pi\)
\(180\) 0 0
\(181\) −7.52929e40 −0.128736 −0.0643680 0.997926i \(-0.520503\pi\)
−0.0643680 + 0.997926i \(0.520503\pi\)
\(182\) −4.57781e41 −0.706867
\(183\) 0 0
\(184\) −2.12441e40 −0.0267984
\(185\) 4.52899e41 0.516803
\(186\) 0 0
\(187\) −3.55297e41 −0.332294
\(188\) 3.23898e41 0.274466
\(189\) 0 0
\(190\) 5.49720e41 0.383001
\(191\) 3.49406e41 0.220909 0.110454 0.993881i \(-0.464769\pi\)
0.110454 + 0.993881i \(0.464769\pi\)
\(192\) 0 0
\(193\) −8.39189e41 −0.437572 −0.218786 0.975773i \(-0.570210\pi\)
−0.218786 + 0.975773i \(0.570210\pi\)
\(194\) 4.39515e42 2.08277
\(195\) 0 0
\(196\) 6.82411e42 2.67491
\(197\) 7.30118e41 0.260476 0.130238 0.991483i \(-0.458426\pi\)
0.130238 + 0.991483i \(0.458426\pi\)
\(198\) 0 0
\(199\) 5.82003e41 0.172244 0.0861218 0.996285i \(-0.472553\pi\)
0.0861218 + 0.996285i \(0.472553\pi\)
\(200\) 1.29804e41 0.0350133
\(201\) 0 0
\(202\) −3.57298e42 −0.801733
\(203\) −1.32807e43 −2.71984
\(204\) 0 0
\(205\) 1.04312e42 0.178192
\(206\) 6.79875e42 1.06142
\(207\) 0 0
\(208\) 2.04379e42 0.266848
\(209\) 8.04341e42 0.961024
\(210\) 0 0
\(211\) −1.25054e43 −1.25277 −0.626384 0.779515i \(-0.715466\pi\)
−0.626384 + 0.779515i \(0.715466\pi\)
\(212\) 1.80330e43 1.65521
\(213\) 0 0
\(214\) −2.28158e43 −1.76027
\(215\) 9.19292e41 0.0650642
\(216\) 0 0
\(217\) 3.51226e43 2.09450
\(218\) −5.49955e42 −0.301219
\(219\) 0 0
\(220\) 7.06634e42 0.326870
\(221\) 1.86545e42 0.0793462
\(222\) 0 0
\(223\) −1.91165e43 −0.688287 −0.344143 0.938917i \(-0.611831\pi\)
−0.344143 + 0.938917i \(0.611831\pi\)
\(224\) −8.20007e43 −2.71787
\(225\) 0 0
\(226\) 3.32410e43 0.934690
\(227\) 2.06794e43 0.535871 0.267936 0.963437i \(-0.413659\pi\)
0.267936 + 0.963437i \(0.413659\pi\)
\(228\) 0 0
\(229\) 6.73090e42 0.148292 0.0741458 0.997247i \(-0.476377\pi\)
0.0741458 + 0.997247i \(0.476377\pi\)
\(230\) 1.47756e43 0.300317
\(231\) 0 0
\(232\) 3.13894e42 0.0543570
\(233\) 5.29076e43 0.846123 0.423062 0.906101i \(-0.360955\pi\)
0.423062 + 0.906101i \(0.360955\pi\)
\(234\) 0 0
\(235\) 6.38437e42 0.0871692
\(236\) −1.42577e44 −1.79961
\(237\) 0 0
\(238\) −7.69096e43 −0.830438
\(239\) −8.90649e43 −0.889908 −0.444954 0.895553i \(-0.646780\pi\)
−0.444954 + 0.895553i \(0.646780\pi\)
\(240\) 0 0
\(241\) −1.14469e44 −0.980331 −0.490166 0.871629i \(-0.663064\pi\)
−0.490166 + 0.871629i \(0.663064\pi\)
\(242\) 3.26713e43 0.259168
\(243\) 0 0
\(244\) −9.41672e43 −0.641487
\(245\) 1.34510e44 0.849540
\(246\) 0 0
\(247\) −4.22311e43 −0.229477
\(248\) −8.30135e42 −0.0418594
\(249\) 0 0
\(250\) −1.90080e44 −0.826128
\(251\) 1.57202e44 0.634590 0.317295 0.948327i \(-0.397226\pi\)
0.317295 + 0.948327i \(0.397226\pi\)
\(252\) 0 0
\(253\) 2.16194e44 0.753555
\(254\) −1.46465e44 −0.474579
\(255\) 0 0
\(256\) 3.75659e44 1.05282
\(257\) −2.35997e44 −0.615380 −0.307690 0.951487i \(-0.599556\pi\)
−0.307690 + 0.951487i \(0.599556\pi\)
\(258\) 0 0
\(259\) 1.43446e45 3.24074
\(260\) −3.71011e43 −0.0780512
\(261\) 0 0
\(262\) −1.02036e45 −1.86286
\(263\) 1.02621e45 1.74604 0.873022 0.487680i \(-0.162157\pi\)
0.873022 + 0.487680i \(0.162157\pi\)
\(264\) 0 0
\(265\) 3.55449e44 0.525688
\(266\) 1.74112e45 2.40170
\(267\) 0 0
\(268\) 9.61554e44 1.15473
\(269\) −2.39378e44 −0.268328 −0.134164 0.990959i \(-0.542835\pi\)
−0.134164 + 0.990959i \(0.542835\pi\)
\(270\) 0 0
\(271\) −1.10518e45 −1.08019 −0.540094 0.841605i \(-0.681611\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(272\) 3.43367e44 0.313497
\(273\) 0 0
\(274\) −7.39109e44 −0.589282
\(275\) −1.32097e45 −0.984551
\(276\) 0 0
\(277\) −2.08812e45 −1.36107 −0.680534 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(278\) −1.50166e45 −0.915681
\(279\) 0 0
\(280\) −4.33498e43 −0.0231507
\(281\) −2.01962e45 −1.00973 −0.504863 0.863200i \(-0.668457\pi\)
−0.504863 + 0.863200i \(0.668457\pi\)
\(282\) 0 0
\(283\) 2.43661e45 1.06841 0.534204 0.845356i \(-0.320612\pi\)
0.534204 + 0.845356i \(0.320612\pi\)
\(284\) −3.14360e45 −1.29133
\(285\) 0 0
\(286\) −1.10110e45 −0.397241
\(287\) 3.30386e45 1.11740
\(288\) 0 0
\(289\) −3.04869e45 −0.906783
\(290\) −2.18318e45 −0.609153
\(291\) 0 0
\(292\) 3.59813e44 0.0884084
\(293\) −5.41508e44 −0.124898 −0.0624488 0.998048i \(-0.519891\pi\)
−0.0624488 + 0.998048i \(0.519891\pi\)
\(294\) 0 0
\(295\) −2.81034e45 −0.571547
\(296\) −3.39041e44 −0.0647672
\(297\) 0 0
\(298\) 8.06172e45 1.35965
\(299\) −1.13511e45 −0.179936
\(300\) 0 0
\(301\) 2.91167e45 0.408001
\(302\) −2.94341e45 −0.387903
\(303\) 0 0
\(304\) −7.77333e45 −0.906663
\(305\) −1.85614e45 −0.203734
\(306\) 0 0
\(307\) 3.09102e45 0.300637 0.150318 0.988638i \(-0.451970\pi\)
0.150318 + 0.988638i \(0.451970\pi\)
\(308\) 2.23812e46 2.04972
\(309\) 0 0
\(310\) 5.77372e45 0.469099
\(311\) 2.15029e46 1.64600 0.823001 0.568040i \(-0.192298\pi\)
0.823001 + 0.568040i \(0.192298\pi\)
\(312\) 0 0
\(313\) −1.50052e42 −0.000102017 0 −5.10083e−5 1.00000i \(-0.500016\pi\)
−5.10083e−5 1.00000i \(0.500016\pi\)
\(314\) −3.43290e46 −2.20020
\(315\) 0 0
\(316\) 2.94092e45 0.167599
\(317\) 7.41293e45 0.398468 0.199234 0.979952i \(-0.436155\pi\)
0.199234 + 0.979952i \(0.436155\pi\)
\(318\) 0 0
\(319\) −3.19439e46 −1.52848
\(320\) −6.45729e45 −0.291592
\(321\) 0 0
\(322\) 4.67987e46 1.88321
\(323\) −7.09504e45 −0.269593
\(324\) 0 0
\(325\) 6.93561e45 0.235095
\(326\) 1.97221e46 0.631580
\(327\) 0 0
\(328\) −7.80880e44 −0.0223315
\(329\) 2.02212e46 0.546616
\(330\) 0 0
\(331\) −2.02313e46 −0.488883 −0.244441 0.969664i \(-0.578605\pi\)
−0.244441 + 0.969664i \(0.578605\pi\)
\(332\) 3.15328e46 0.720623
\(333\) 0 0
\(334\) −9.55651e46 −1.95429
\(335\) 1.89532e46 0.366737
\(336\) 0 0
\(337\) 1.40951e46 0.244295 0.122148 0.992512i \(-0.461022\pi\)
0.122148 + 0.992512i \(0.461022\pi\)
\(338\) −7.98163e46 −1.30958
\(339\) 0 0
\(340\) −6.23317e45 −0.0916957
\(341\) 8.44801e46 1.17706
\(342\) 0 0
\(343\) 2.71153e47 3.39057
\(344\) −6.88183e44 −0.00815404
\(345\) 0 0
\(346\) −8.59708e46 −0.915046
\(347\) −1.91978e46 −0.193711 −0.0968557 0.995298i \(-0.530879\pi\)
−0.0968557 + 0.995298i \(0.530879\pi\)
\(348\) 0 0
\(349\) 1.61862e47 1.46851 0.734253 0.678876i \(-0.237533\pi\)
0.734253 + 0.678876i \(0.237533\pi\)
\(350\) −2.85945e47 −2.46050
\(351\) 0 0
\(352\) −1.97236e47 −1.52738
\(353\) 9.35860e46 0.687668 0.343834 0.939030i \(-0.388274\pi\)
0.343834 + 0.939030i \(0.388274\pi\)
\(354\) 0 0
\(355\) −6.19636e46 −0.410121
\(356\) −1.52336e46 −0.0957145
\(357\) 0 0
\(358\) −2.59086e47 −1.46760
\(359\) −3.05449e47 −1.64321 −0.821603 0.570060i \(-0.806920\pi\)
−0.821603 + 0.570060i \(0.806920\pi\)
\(360\) 0 0
\(361\) −4.53862e46 −0.220313
\(362\) −3.92022e46 −0.180802
\(363\) 0 0
\(364\) −1.17510e47 −0.489439
\(365\) 7.09229e45 0.0280781
\(366\) 0 0
\(367\) −3.45206e47 −1.23526 −0.617628 0.786471i \(-0.711906\pi\)
−0.617628 + 0.786471i \(0.711906\pi\)
\(368\) −2.08935e47 −0.710929
\(369\) 0 0
\(370\) 2.35808e47 0.725816
\(371\) 1.12581e48 3.29646
\(372\) 0 0
\(373\) 5.25222e47 1.39229 0.696145 0.717901i \(-0.254897\pi\)
0.696145 + 0.717901i \(0.254897\pi\)
\(374\) −1.84990e47 −0.466685
\(375\) 0 0
\(376\) −4.77935e45 −0.0109243
\(377\) 1.67718e47 0.364977
\(378\) 0 0
\(379\) 1.07624e47 0.212365 0.106183 0.994347i \(-0.466137\pi\)
0.106183 + 0.994347i \(0.466137\pi\)
\(380\) 1.41110e47 0.265192
\(381\) 0 0
\(382\) 1.81923e47 0.310252
\(383\) −3.92097e44 −0.000637112 0 −0.000318556 1.00000i \(-0.500101\pi\)
−0.000318556 1.00000i \(0.500101\pi\)
\(384\) 0 0
\(385\) 4.41156e47 0.650982
\(386\) −4.36935e47 −0.614542
\(387\) 0 0
\(388\) 1.12821e48 1.44213
\(389\) 1.07652e48 1.31206 0.656029 0.754736i \(-0.272235\pi\)
0.656029 + 0.754736i \(0.272235\pi\)
\(390\) 0 0
\(391\) −1.90704e47 −0.211392
\(392\) −1.00695e47 −0.106467
\(393\) 0 0
\(394\) 3.80145e47 0.365822
\(395\) 5.79687e46 0.0532288
\(396\) 0 0
\(397\) −9.91329e47 −0.829074 −0.414537 0.910032i \(-0.636056\pi\)
−0.414537 + 0.910032i \(0.636056\pi\)
\(398\) 3.03027e47 0.241905
\(399\) 0 0
\(400\) 1.27661e48 0.928859
\(401\) −5.74839e47 −0.399370 −0.199685 0.979860i \(-0.563992\pi\)
−0.199685 + 0.979860i \(0.563992\pi\)
\(402\) 0 0
\(403\) −4.43553e47 −0.281062
\(404\) −9.17163e47 −0.555125
\(405\) 0 0
\(406\) −6.91477e48 −3.81985
\(407\) 3.45030e48 1.82121
\(408\) 0 0
\(409\) −1.42222e48 −0.685623 −0.342812 0.939404i \(-0.611379\pi\)
−0.342812 + 0.939404i \(0.611379\pi\)
\(410\) 5.43114e47 0.250259
\(411\) 0 0
\(412\) 1.74520e48 0.734932
\(413\) −8.90118e48 −3.58403
\(414\) 0 0
\(415\) 6.21545e47 0.228867
\(416\) 1.03556e48 0.364712
\(417\) 0 0
\(418\) 4.18791e48 1.34970
\(419\) 2.58159e47 0.0796026 0.0398013 0.999208i \(-0.487328\pi\)
0.0398013 + 0.999208i \(0.487328\pi\)
\(420\) 0 0
\(421\) 2.31697e48 0.654186 0.327093 0.944992i \(-0.393931\pi\)
0.327093 + 0.944992i \(0.393931\pi\)
\(422\) −6.51108e48 −1.75943
\(423\) 0 0
\(424\) −2.66089e47 −0.0658808
\(425\) 1.16522e48 0.276193
\(426\) 0 0
\(427\) −5.87892e48 −1.27756
\(428\) −5.85668e48 −1.21883
\(429\) 0 0
\(430\) 4.78642e47 0.0913785
\(431\) 3.89804e48 0.712881 0.356441 0.934318i \(-0.383990\pi\)
0.356441 + 0.934318i \(0.383990\pi\)
\(432\) 0 0
\(433\) 9.77220e48 1.64046 0.820231 0.572032i \(-0.193845\pi\)
0.820231 + 0.572032i \(0.193845\pi\)
\(434\) 1.82870e49 2.94160
\(435\) 0 0
\(436\) −1.41170e48 −0.208566
\(437\) 4.31725e48 0.611365
\(438\) 0 0
\(439\) −1.19379e48 −0.155359 −0.0776793 0.996978i \(-0.524751\pi\)
−0.0776793 + 0.996978i \(0.524751\pi\)
\(440\) −1.04269e47 −0.0130101
\(441\) 0 0
\(442\) 9.71271e47 0.111437
\(443\) 1.49848e48 0.164886 0.0824429 0.996596i \(-0.473728\pi\)
0.0824429 + 0.996596i \(0.473728\pi\)
\(444\) 0 0
\(445\) −3.00271e47 −0.0303985
\(446\) −9.95328e48 −0.966655
\(447\) 0 0
\(448\) −2.04521e49 −1.82850
\(449\) −9.14003e48 −0.784134 −0.392067 0.919937i \(-0.628240\pi\)
−0.392067 + 0.919937i \(0.628240\pi\)
\(450\) 0 0
\(451\) 7.94675e48 0.627948
\(452\) 8.53277e48 0.647185
\(453\) 0 0
\(454\) 1.07670e49 0.752597
\(455\) −2.31624e48 −0.155444
\(456\) 0 0
\(457\) −7.21363e48 −0.446379 −0.223190 0.974775i \(-0.571647\pi\)
−0.223190 + 0.974775i \(0.571647\pi\)
\(458\) 3.50453e48 0.208266
\(459\) 0 0
\(460\) 3.79281e48 0.207942
\(461\) 3.62582e48 0.190959 0.0954793 0.995431i \(-0.469562\pi\)
0.0954793 + 0.995431i \(0.469562\pi\)
\(462\) 0 0
\(463\) −2.42342e49 −1.17810 −0.589048 0.808098i \(-0.700497\pi\)
−0.589048 + 0.808098i \(0.700497\pi\)
\(464\) 3.08713e49 1.44202
\(465\) 0 0
\(466\) 2.75471e49 1.18833
\(467\) −2.11890e49 −0.878514 −0.439257 0.898362i \(-0.644758\pi\)
−0.439257 + 0.898362i \(0.644758\pi\)
\(468\) 0 0
\(469\) 6.00304e49 2.29971
\(470\) 3.32411e48 0.122424
\(471\) 0 0
\(472\) 2.10383e48 0.0716280
\(473\) 7.00341e48 0.229286
\(474\) 0 0
\(475\) −2.63789e49 −0.798775
\(476\) −1.97423e49 −0.575001
\(477\) 0 0
\(478\) −4.63728e49 −1.24982
\(479\) 3.96549e49 1.02823 0.514115 0.857721i \(-0.328121\pi\)
0.514115 + 0.857721i \(0.328121\pi\)
\(480\) 0 0
\(481\) −1.81154e49 −0.434876
\(482\) −5.95998e49 −1.37681
\(483\) 0 0
\(484\) 8.38653e48 0.179450
\(485\) 2.22382e49 0.458013
\(486\) 0 0
\(487\) −3.49809e49 −0.667644 −0.333822 0.942636i \(-0.608338\pi\)
−0.333822 + 0.942636i \(0.608338\pi\)
\(488\) 1.38950e48 0.0255325
\(489\) 0 0
\(490\) 7.00347e49 1.19312
\(491\) −1.54468e49 −0.253414 −0.126707 0.991940i \(-0.540441\pi\)
−0.126707 + 0.991940i \(0.540441\pi\)
\(492\) 0 0
\(493\) 2.81775e49 0.428780
\(494\) −2.19882e49 −0.322285
\(495\) 0 0
\(496\) −8.16434e49 −1.11048
\(497\) −1.96257e50 −2.57177
\(498\) 0 0
\(499\) −8.88148e49 −1.08050 −0.540250 0.841504i \(-0.681670\pi\)
−0.540250 + 0.841504i \(0.681670\pi\)
\(500\) −4.87926e49 −0.572016
\(501\) 0 0
\(502\) 8.18491e49 0.891240
\(503\) 4.16645e49 0.437279 0.218639 0.975806i \(-0.429838\pi\)
0.218639 + 0.975806i \(0.429838\pi\)
\(504\) 0 0
\(505\) −1.80782e49 −0.176305
\(506\) 1.12564e50 1.05832
\(507\) 0 0
\(508\) −3.75966e49 −0.328602
\(509\) −1.88211e50 −1.58623 −0.793114 0.609074i \(-0.791541\pi\)
−0.793114 + 0.609074i \(0.791541\pi\)
\(510\) 0 0
\(511\) 2.24633e49 0.176071
\(512\) 1.85356e50 1.40124
\(513\) 0 0
\(514\) −1.22875e50 −0.864262
\(515\) 3.43998e49 0.233411
\(516\) 0 0
\(517\) 4.86378e49 0.307184
\(518\) 7.46872e50 4.55141
\(519\) 0 0
\(520\) 5.47453e47 0.00310660
\(521\) −2.50866e50 −1.37386 −0.686931 0.726723i \(-0.741043\pi\)
−0.686931 + 0.726723i \(0.741043\pi\)
\(522\) 0 0
\(523\) 1.68115e49 0.0857682 0.0428841 0.999080i \(-0.486345\pi\)
0.0428841 + 0.999080i \(0.486345\pi\)
\(524\) −2.61921e50 −1.28986
\(525\) 0 0
\(526\) 5.34310e50 2.45221
\(527\) −7.45193e49 −0.330196
\(528\) 0 0
\(529\) −1.26023e50 −0.520618
\(530\) 1.85069e50 0.738295
\(531\) 0 0
\(532\) 4.46936e50 1.66295
\(533\) −4.17236e49 −0.149944
\(534\) 0 0
\(535\) −1.15441e50 −0.387094
\(536\) −1.41884e49 −0.0459606
\(537\) 0 0
\(538\) −1.24635e50 −0.376850
\(539\) 1.02474e51 2.99378
\(540\) 0 0
\(541\) 9.81522e49 0.267763 0.133882 0.990997i \(-0.457256\pi\)
0.133882 + 0.990997i \(0.457256\pi\)
\(542\) −5.75425e50 −1.51706
\(543\) 0 0
\(544\) 1.73980e50 0.428470
\(545\) −2.78261e49 −0.0662396
\(546\) 0 0
\(547\) −6.05828e50 −1.34767 −0.673836 0.738881i \(-0.735354\pi\)
−0.673836 + 0.738881i \(0.735354\pi\)
\(548\) −1.89725e50 −0.408023
\(549\) 0 0
\(550\) −6.87781e50 −1.38274
\(551\) −6.37898e50 −1.24007
\(552\) 0 0
\(553\) 1.83604e50 0.333785
\(554\) −1.08721e51 −1.91153
\(555\) 0 0
\(556\) −3.85469e50 −0.634023
\(557\) −1.04564e50 −0.166365 −0.0831826 0.996534i \(-0.526508\pi\)
−0.0831826 + 0.996534i \(0.526508\pi\)
\(558\) 0 0
\(559\) −3.67706e49 −0.0547498
\(560\) −4.26343e50 −0.614159
\(561\) 0 0
\(562\) −1.05154e51 −1.41809
\(563\) 6.57533e50 0.858050 0.429025 0.903293i \(-0.358857\pi\)
0.429025 + 0.903293i \(0.358857\pi\)
\(564\) 0 0
\(565\) 1.68190e50 0.205543
\(566\) 1.26865e51 1.50051
\(567\) 0 0
\(568\) 4.63860e49 0.0513976
\(569\) 2.74695e50 0.294628 0.147314 0.989090i \(-0.452937\pi\)
0.147314 + 0.989090i \(0.452937\pi\)
\(570\) 0 0
\(571\) −2.57528e50 −0.258855 −0.129428 0.991589i \(-0.541314\pi\)
−0.129428 + 0.991589i \(0.541314\pi\)
\(572\) −2.82646e50 −0.275053
\(573\) 0 0
\(574\) 1.72020e51 1.56931
\(575\) −7.09023e50 −0.626332
\(576\) 0 0
\(577\) 1.01888e51 0.844058 0.422029 0.906582i \(-0.361318\pi\)
0.422029 + 0.906582i \(0.361318\pi\)
\(578\) −1.58734e51 −1.27352
\(579\) 0 0
\(580\) −5.60410e50 −0.421782
\(581\) 1.96861e51 1.43517
\(582\) 0 0
\(583\) 2.70790e51 1.85253
\(584\) −5.30929e48 −0.00351884
\(585\) 0 0
\(586\) −2.81943e50 −0.175411
\(587\) −2.83210e51 −1.70728 −0.853638 0.520867i \(-0.825609\pi\)
−0.853638 + 0.520867i \(0.825609\pi\)
\(588\) 0 0
\(589\) 1.68701e51 0.954958
\(590\) −1.46324e51 −0.802702
\(591\) 0 0
\(592\) −3.33445e51 −1.71820
\(593\) 3.45126e51 1.72372 0.861858 0.507150i \(-0.169301\pi\)
0.861858 + 0.507150i \(0.169301\pi\)
\(594\) 0 0
\(595\) −3.89141e50 −0.182618
\(596\) 2.06940e51 0.941430
\(597\) 0 0
\(598\) −5.91007e50 −0.252709
\(599\) 4.18318e51 1.73425 0.867123 0.498094i \(-0.165966\pi\)
0.867123 + 0.498094i \(0.165966\pi\)
\(600\) 0 0
\(601\) −2.49077e51 −0.970860 −0.485430 0.874276i \(-0.661337\pi\)
−0.485430 + 0.874276i \(0.661337\pi\)
\(602\) 1.51600e51 0.573011
\(603\) 0 0
\(604\) −7.55557e50 −0.268586
\(605\) 1.65307e50 0.0569925
\(606\) 0 0
\(607\) 1.98679e50 0.0644409 0.0322204 0.999481i \(-0.489742\pi\)
0.0322204 + 0.999481i \(0.489742\pi\)
\(608\) −3.93866e51 −1.23917
\(609\) 0 0
\(610\) −9.66421e50 −0.286131
\(611\) −2.55368e50 −0.0733505
\(612\) 0 0
\(613\) 3.22821e50 0.0872856 0.0436428 0.999047i \(-0.486104\pi\)
0.0436428 + 0.999047i \(0.486104\pi\)
\(614\) 1.60938e51 0.422225
\(615\) 0 0
\(616\) −3.30250e50 −0.0815830
\(617\) −2.17195e51 −0.520683 −0.260342 0.965517i \(-0.583835\pi\)
−0.260342 + 0.965517i \(0.583835\pi\)
\(618\) 0 0
\(619\) 3.90977e51 0.882824 0.441412 0.897305i \(-0.354478\pi\)
0.441412 + 0.897305i \(0.354478\pi\)
\(620\) 1.48208e51 0.324807
\(621\) 0 0
\(622\) 1.11958e52 2.31171
\(623\) −9.51044e50 −0.190621
\(624\) 0 0
\(625\) 3.82724e51 0.722946
\(626\) −7.81264e47 −0.000143276 0
\(627\) 0 0
\(628\) −8.81205e51 −1.52343
\(629\) −3.04349e51 −0.510899
\(630\) 0 0
\(631\) −7.04074e51 −1.11449 −0.557244 0.830349i \(-0.688141\pi\)
−0.557244 + 0.830349i \(0.688141\pi\)
\(632\) −4.33954e49 −0.00667080
\(633\) 0 0
\(634\) 3.85964e51 0.559623
\(635\) −7.41069e50 −0.104363
\(636\) 0 0
\(637\) −5.38026e51 −0.714865
\(638\) −1.66320e52 −2.14666
\(639\) 0 0
\(640\) 1.96201e50 0.0238985
\(641\) 1.31488e52 1.55601 0.778003 0.628260i \(-0.216233\pi\)
0.778003 + 0.628260i \(0.216233\pi\)
\(642\) 0 0
\(643\) 1.39124e52 1.55417 0.777084 0.629397i \(-0.216698\pi\)
0.777084 + 0.629397i \(0.216698\pi\)
\(644\) 1.20129e52 1.30395
\(645\) 0 0
\(646\) −3.69413e51 −0.378626
\(647\) −1.43133e52 −1.42564 −0.712821 0.701346i \(-0.752583\pi\)
−0.712821 + 0.701346i \(0.752583\pi\)
\(648\) 0 0
\(649\) −2.14099e52 −2.01413
\(650\) 3.61112e51 0.330175
\(651\) 0 0
\(652\) 5.06256e51 0.437310
\(653\) −5.93896e51 −0.498674 −0.249337 0.968417i \(-0.580213\pi\)
−0.249337 + 0.968417i \(0.580213\pi\)
\(654\) 0 0
\(655\) −5.16274e51 −0.409653
\(656\) −7.67992e51 −0.592428
\(657\) 0 0
\(658\) 1.05284e52 0.767687
\(659\) 1.23688e52 0.876898 0.438449 0.898756i \(-0.355528\pi\)
0.438449 + 0.898756i \(0.355528\pi\)
\(660\) 0 0
\(661\) −2.33294e52 −1.56378 −0.781891 0.623415i \(-0.785745\pi\)
−0.781891 + 0.623415i \(0.785745\pi\)
\(662\) −1.05337e52 −0.686605
\(663\) 0 0
\(664\) −4.65289e50 −0.0286823
\(665\) 8.80959e51 0.528147
\(666\) 0 0
\(667\) −1.71457e52 −0.972361
\(668\) −2.45310e52 −1.35316
\(669\) 0 0
\(670\) 9.86826e51 0.515058
\(671\) −1.41405e52 −0.717958
\(672\) 0 0
\(673\) 1.92388e52 0.924484 0.462242 0.886754i \(-0.347045\pi\)
0.462242 + 0.886754i \(0.347045\pi\)
\(674\) 7.33880e51 0.343097
\(675\) 0 0
\(676\) −2.04884e52 −0.906763
\(677\) 6.78443e51 0.292162 0.146081 0.989273i \(-0.453334\pi\)
0.146081 + 0.989273i \(0.453334\pi\)
\(678\) 0 0
\(679\) 7.04349e52 2.87208
\(680\) 9.19749e49 0.00364968
\(681\) 0 0
\(682\) 4.39857e52 1.65310
\(683\) 1.97532e52 0.722528 0.361264 0.932464i \(-0.382345\pi\)
0.361264 + 0.932464i \(0.382345\pi\)
\(684\) 0 0
\(685\) −3.73968e51 −0.129586
\(686\) 1.41179e53 4.76184
\(687\) 0 0
\(688\) −6.76824e51 −0.216317
\(689\) −1.42175e52 −0.442353
\(690\) 0 0
\(691\) −2.82676e52 −0.833575 −0.416788 0.909004i \(-0.636844\pi\)
−0.416788 + 0.909004i \(0.636844\pi\)
\(692\) −2.20682e52 −0.633584
\(693\) 0 0
\(694\) −9.99557e51 −0.272055
\(695\) −7.59799e51 −0.201363
\(696\) 0 0
\(697\) −7.00977e51 −0.176156
\(698\) 8.42758e52 2.06242
\(699\) 0 0
\(700\) −7.34004e52 −1.70367
\(701\) −6.97961e52 −1.57778 −0.788892 0.614532i \(-0.789345\pi\)
−0.788892 + 0.614532i \(0.789345\pi\)
\(702\) 0 0
\(703\) 6.89002e52 1.47757
\(704\) −4.91933e52 −1.02757
\(705\) 0 0
\(706\) 4.87268e52 0.965785
\(707\) −5.72591e52 −1.10557
\(708\) 0 0
\(709\) −2.71632e52 −0.497766 −0.248883 0.968534i \(-0.580063\pi\)
−0.248883 + 0.968534i \(0.580063\pi\)
\(710\) −3.22622e52 −0.575989
\(711\) 0 0
\(712\) 2.24783e50 0.00380963
\(713\) 4.53441e52 0.748798
\(714\) 0 0
\(715\) −5.57124e51 −0.0873555
\(716\) −6.65060e52 −1.01618
\(717\) 0 0
\(718\) −1.59036e53 −2.30778
\(719\) −1.18694e53 −1.67860 −0.839298 0.543672i \(-0.817033\pi\)
−0.839298 + 0.543672i \(0.817033\pi\)
\(720\) 0 0
\(721\) 1.08954e53 1.46366
\(722\) −2.36309e52 −0.309416
\(723\) 0 0
\(724\) −1.00630e52 −0.125188
\(725\) 1.04762e53 1.27043
\(726\) 0 0
\(727\) −5.48820e52 −0.632475 −0.316237 0.948680i \(-0.602420\pi\)
−0.316237 + 0.948680i \(0.602420\pi\)
\(728\) 1.73394e51 0.0194807
\(729\) 0 0
\(730\) 3.69269e51 0.0394340
\(731\) −6.17765e51 −0.0643209
\(732\) 0 0
\(733\) −1.43675e53 −1.42219 −0.711097 0.703094i \(-0.751801\pi\)
−0.711097 + 0.703094i \(0.751801\pi\)
\(734\) −1.79736e53 −1.73484
\(735\) 0 0
\(736\) −1.05865e53 −0.971656
\(737\) 1.44391e53 1.29238
\(738\) 0 0
\(739\) −1.14941e53 −0.978484 −0.489242 0.872148i \(-0.662727\pi\)
−0.489242 + 0.872148i \(0.662727\pi\)
\(740\) 6.05306e52 0.502560
\(741\) 0 0
\(742\) 5.86167e53 4.62966
\(743\) 7.23269e52 0.557195 0.278597 0.960408i \(-0.410130\pi\)
0.278597 + 0.960408i \(0.410130\pi\)
\(744\) 0 0
\(745\) 4.07900e52 0.298994
\(746\) 2.73464e53 1.95538
\(747\) 0 0
\(748\) −4.74859e52 −0.323136
\(749\) −3.65636e53 −2.42737
\(750\) 0 0
\(751\) −1.89145e53 −1.19524 −0.597621 0.801779i \(-0.703887\pi\)
−0.597621 + 0.801779i \(0.703887\pi\)
\(752\) −4.70046e52 −0.289808
\(753\) 0 0
\(754\) 8.73247e52 0.512586
\(755\) −1.48928e52 −0.0853018
\(756\) 0 0
\(757\) 1.52507e53 0.831796 0.415898 0.909411i \(-0.363467\pi\)
0.415898 + 0.909411i \(0.363467\pi\)
\(758\) 5.60358e52 0.298253
\(759\) 0 0
\(760\) −2.08218e51 −0.0105552
\(761\) 1.93715e53 0.958399 0.479199 0.877706i \(-0.340927\pi\)
0.479199 + 0.877706i \(0.340927\pi\)
\(762\) 0 0
\(763\) −8.81334e52 −0.415372
\(764\) 4.66986e52 0.214821
\(765\) 0 0
\(766\) −2.04150e50 −0.000894783 0
\(767\) 1.12411e53 0.480942
\(768\) 0 0
\(769\) 3.76633e53 1.53561 0.767805 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(770\) 2.29694e53 0.914263
\(771\) 0 0
\(772\) −1.12159e53 −0.425513
\(773\) −2.46250e53 −0.912127 −0.456064 0.889947i \(-0.650741\pi\)
−0.456064 + 0.889947i \(0.650741\pi\)
\(774\) 0 0
\(775\) −2.77058e53 −0.978337
\(776\) −1.66475e52 −0.0573995
\(777\) 0 0
\(778\) 5.60503e53 1.84270
\(779\) 1.58691e53 0.509460
\(780\) 0 0
\(781\) −4.72055e53 −1.44527
\(782\) −9.92923e52 −0.296887
\(783\) 0 0
\(784\) −9.90327e53 −2.82443
\(785\) −1.73695e53 −0.483836
\(786\) 0 0
\(787\) 3.69252e53 0.981275 0.490638 0.871364i \(-0.336764\pi\)
0.490638 + 0.871364i \(0.336764\pi\)
\(788\) 9.75812e52 0.253298
\(789\) 0 0
\(790\) 3.01822e52 0.0747565
\(791\) 5.32706e53 1.28891
\(792\) 0 0
\(793\) 7.42433e52 0.171436
\(794\) −5.16148e53 −1.16438
\(795\) 0 0
\(796\) 7.77854e52 0.167497
\(797\) −5.45878e53 −1.14846 −0.574231 0.818694i \(-0.694699\pi\)
−0.574231 + 0.818694i \(0.694699\pi\)
\(798\) 0 0
\(799\) −4.29031e52 −0.0861733
\(800\) 6.46846e53 1.26951
\(801\) 0 0
\(802\) −2.99297e53 −0.560889
\(803\) 5.40309e52 0.0989475
\(804\) 0 0
\(805\) 2.36788e53 0.414129
\(806\) −2.30942e53 −0.394734
\(807\) 0 0
\(808\) 1.35334e52 0.0220951
\(809\) 1.10283e54 1.75979 0.879893 0.475171i \(-0.157614\pi\)
0.879893 + 0.475171i \(0.157614\pi\)
\(810\) 0 0
\(811\) 7.85452e53 1.19739 0.598693 0.800979i \(-0.295687\pi\)
0.598693 + 0.800979i \(0.295687\pi\)
\(812\) −1.77498e54 −2.64489
\(813\) 0 0
\(814\) 1.79645e54 2.55778
\(815\) 9.97884e52 0.138888
\(816\) 0 0
\(817\) 1.39853e53 0.186022
\(818\) −7.40497e53 −0.962914
\(819\) 0 0
\(820\) 1.39414e53 0.173281
\(821\) −1.01998e54 −1.23949 −0.619744 0.784804i \(-0.712764\pi\)
−0.619744 + 0.784804i \(0.712764\pi\)
\(822\) 0 0
\(823\) −1.92578e52 −0.0223722 −0.0111861 0.999937i \(-0.503561\pi\)
−0.0111861 + 0.999937i \(0.503561\pi\)
\(824\) −2.57517e52 −0.0292518
\(825\) 0 0
\(826\) −4.63452e54 −5.03354
\(827\) 5.47526e53 0.581504 0.290752 0.956798i \(-0.406095\pi\)
0.290752 + 0.956798i \(0.406095\pi\)
\(828\) 0 0
\(829\) −1.39168e53 −0.141346 −0.0706728 0.997500i \(-0.522515\pi\)
−0.0706728 + 0.997500i \(0.522515\pi\)
\(830\) 3.23616e53 0.321429
\(831\) 0 0
\(832\) 2.58284e53 0.245367
\(833\) −9.03912e53 −0.839835
\(834\) 0 0
\(835\) −4.83532e53 −0.429759
\(836\) 1.07501e54 0.934539
\(837\) 0 0
\(838\) 1.34414e53 0.111797
\(839\) 1.43760e54 1.16961 0.584804 0.811174i \(-0.301171\pi\)
0.584804 + 0.811174i \(0.301171\pi\)
\(840\) 0 0
\(841\) 1.24890e54 0.972302
\(842\) 1.20636e54 0.918762
\(843\) 0 0
\(844\) −1.67136e54 −1.21824
\(845\) −4.03848e53 −0.287984
\(846\) 0 0
\(847\) 5.23577e53 0.357386
\(848\) −2.61697e54 −1.74774
\(849\) 0 0
\(850\) 6.06687e53 0.387895
\(851\) 1.85193e54 1.15858
\(852\) 0 0
\(853\) 1.36560e54 0.818024 0.409012 0.912529i \(-0.365873\pi\)
0.409012 + 0.912529i \(0.365873\pi\)
\(854\) −3.06094e54 −1.79425
\(855\) 0 0
\(856\) 8.64194e52 0.0485117
\(857\) 3.06618e54 1.68443 0.842213 0.539145i \(-0.181253\pi\)
0.842213 + 0.539145i \(0.181253\pi\)
\(858\) 0 0
\(859\) −2.99741e53 −0.157715 −0.0788575 0.996886i \(-0.525127\pi\)
−0.0788575 + 0.996886i \(0.525127\pi\)
\(860\) 1.22865e53 0.0632711
\(861\) 0 0
\(862\) 2.02957e54 1.00120
\(863\) 4.97031e53 0.239985 0.119992 0.992775i \(-0.461713\pi\)
0.119992 + 0.992775i \(0.461713\pi\)
\(864\) 0 0
\(865\) −4.34988e53 −0.201224
\(866\) 5.08802e54 2.30392
\(867\) 0 0
\(868\) 4.69418e54 2.03678
\(869\) 4.41620e53 0.187579
\(870\) 0 0
\(871\) −7.58108e53 −0.308599
\(872\) 2.08307e52 0.00830134
\(873\) 0 0
\(874\) 2.24783e54 0.858624
\(875\) −3.04615e54 −1.13921
\(876\) 0 0
\(877\) 5.81431e53 0.208452 0.104226 0.994554i \(-0.466764\pi\)
0.104226 + 0.994554i \(0.466764\pi\)
\(878\) −6.21564e53 −0.218191
\(879\) 0 0
\(880\) −1.02548e54 −0.345142
\(881\) 2.08913e54 0.688510 0.344255 0.938876i \(-0.388131\pi\)
0.344255 + 0.938876i \(0.388131\pi\)
\(882\) 0 0
\(883\) −3.47147e54 −1.09709 −0.548543 0.836122i \(-0.684817\pi\)
−0.548543 + 0.836122i \(0.684817\pi\)
\(884\) 2.49320e53 0.0771595
\(885\) 0 0
\(886\) 7.80205e53 0.231572
\(887\) −5.12164e54 −1.48875 −0.744376 0.667761i \(-0.767253\pi\)
−0.744376 + 0.667761i \(0.767253\pi\)
\(888\) 0 0
\(889\) −2.34718e54 −0.654431
\(890\) −1.56340e53 −0.0426928
\(891\) 0 0
\(892\) −2.55495e54 −0.669318
\(893\) 9.71264e53 0.249221
\(894\) 0 0
\(895\) −1.31090e54 −0.322734
\(896\) 6.21425e53 0.149862
\(897\) 0 0
\(898\) −4.75888e54 −1.10127
\(899\) −6.69985e54 −1.51884
\(900\) 0 0
\(901\) −2.38862e54 −0.519683
\(902\) 4.13758e54 0.881913
\(903\) 0 0
\(904\) −1.25907e53 −0.0257593
\(905\) −1.98352e53 −0.0397592
\(906\) 0 0
\(907\) −6.23293e53 −0.119938 −0.0599691 0.998200i \(-0.519100\pi\)
−0.0599691 + 0.998200i \(0.519100\pi\)
\(908\) 2.76383e54 0.521103
\(909\) 0 0
\(910\) −1.20598e54 −0.218311
\(911\) −2.11014e54 −0.374302 −0.187151 0.982331i \(-0.559925\pi\)
−0.187151 + 0.982331i \(0.559925\pi\)
\(912\) 0 0
\(913\) 4.73509e54 0.806527
\(914\) −3.75587e54 −0.626911
\(915\) 0 0
\(916\) 8.99593e53 0.144205
\(917\) −1.63519e55 −2.56883
\(918\) 0 0
\(919\) 1.59651e52 0.00240899 0.00120450 0.999999i \(-0.499617\pi\)
0.00120450 + 0.999999i \(0.499617\pi\)
\(920\) −5.59657e52 −0.00827651
\(921\) 0 0
\(922\) 1.88783e54 0.268189
\(923\) 2.47847e54 0.345106
\(924\) 0 0
\(925\) −1.13155e55 −1.51374
\(926\) −1.26179e55 −1.65456
\(927\) 0 0
\(928\) 1.56421e55 1.97087
\(929\) 4.23691e54 0.523309 0.261654 0.965162i \(-0.415732\pi\)
0.261654 + 0.965162i \(0.415732\pi\)
\(930\) 0 0
\(931\) 2.04633e55 2.42888
\(932\) 7.07117e54 0.822805
\(933\) 0 0
\(934\) −1.10323e55 −1.23382
\(935\) −9.35998e53 −0.102627
\(936\) 0 0
\(937\) −1.28910e55 −1.35863 −0.679317 0.733845i \(-0.737724\pi\)
−0.679317 + 0.733845i \(0.737724\pi\)
\(938\) 3.12556e55 3.22980
\(939\) 0 0
\(940\) 8.53279e53 0.0847668
\(941\) 4.90525e54 0.477807 0.238904 0.971043i \(-0.423212\pi\)
0.238904 + 0.971043i \(0.423212\pi\)
\(942\) 0 0
\(943\) 4.26537e54 0.399476
\(944\) 2.06910e55 1.90020
\(945\) 0 0
\(946\) 3.64642e54 0.322018
\(947\) 7.97016e54 0.690228 0.345114 0.938561i \(-0.387840\pi\)
0.345114 + 0.938561i \(0.387840\pi\)
\(948\) 0 0
\(949\) −2.83683e53 −0.0236270
\(950\) −1.37345e55 −1.12183
\(951\) 0 0
\(952\) 2.91311e53 0.0228862
\(953\) 1.23666e55 0.952864 0.476432 0.879211i \(-0.341930\pi\)
0.476432 + 0.879211i \(0.341930\pi\)
\(954\) 0 0
\(955\) 9.20478e53 0.0682262
\(956\) −1.19036e55 −0.865383
\(957\) 0 0
\(958\) 2.06469e55 1.44408
\(959\) −1.18447e55 −0.812603
\(960\) 0 0
\(961\) 2.56972e54 0.169630
\(962\) −9.43205e54 −0.610755
\(963\) 0 0
\(964\) −1.52989e55 −0.953314
\(965\) −2.21077e54 −0.135141
\(966\) 0 0
\(967\) −1.79944e55 −1.05864 −0.529318 0.848424i \(-0.677552\pi\)
−0.529318 + 0.848424i \(0.677552\pi\)
\(968\) −1.23749e53 −0.00714247
\(969\) 0 0
\(970\) 1.15786e55 0.643250
\(971\) −1.81554e55 −0.989577 −0.494789 0.869013i \(-0.664755\pi\)
−0.494789 + 0.869013i \(0.664755\pi\)
\(972\) 0 0
\(973\) −2.40651e55 −1.26270
\(974\) −1.82133e55 −0.937664
\(975\) 0 0
\(976\) 1.36657e55 0.677346
\(977\) 1.52699e54 0.0742656 0.0371328 0.999310i \(-0.488178\pi\)
0.0371328 + 0.999310i \(0.488178\pi\)
\(978\) 0 0
\(979\) −2.28754e54 −0.107124
\(980\) 1.79775e55 0.826127
\(981\) 0 0
\(982\) −8.04259e54 −0.355904
\(983\) 3.38697e55 1.47086 0.735431 0.677600i \(-0.236980\pi\)
0.735431 + 0.677600i \(0.236980\pi\)
\(984\) 0 0
\(985\) 1.92343e54 0.0804462
\(986\) 1.46710e55 0.602194
\(987\) 0 0
\(988\) −5.64424e54 −0.223152
\(989\) 3.75903e54 0.145863
\(990\) 0 0
\(991\) 4.67890e55 1.74897 0.874483 0.485057i \(-0.161201\pi\)
0.874483 + 0.485057i \(0.161201\pi\)
\(992\) −4.13678e55 −1.51774
\(993\) 0 0
\(994\) −1.02184e56 −3.61188
\(995\) 1.53323e54 0.0531962
\(996\) 0 0
\(997\) 4.89388e55 1.63603 0.818017 0.575193i \(-0.195073\pi\)
0.818017 + 0.575193i \(0.195073\pi\)
\(998\) −4.62426e55 −1.51749
\(999\) 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.1.4 4
3.2 odd 2 3.38.a.b.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.38.a.b.1.1 4 3.2 odd 2
9.38.a.c.1.4 4 1.1 even 1 trivial