Properties

Label 160.10.a.c.1.3
Level $160$
Weight $10$
Character 160.1
Self dual yes
Analytic conductor $82.406$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [160,10,Mod(1,160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("160.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-2500,0,0,0,23364] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(82.4057337862\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 353x^{2} + 354x + 2313 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.12853\) of defining polynomial
Character \(\chi\) \(=\) 160.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+31.5423 q^{3} -625.000 q^{5} -5763.01 q^{7} -18688.1 q^{9} +40083.3 q^{11} -66880.5 q^{13} -19714.0 q^{15} -91431.8 q^{17} -901894. q^{19} -181779. q^{21} +260886. q^{23} +390625. q^{25} -1.21031e6 q^{27} -822128. q^{29} +8.24915e6 q^{31} +1.26432e6 q^{33} +3.60188e6 q^{35} -3.26345e6 q^{37} -2.10957e6 q^{39} -721750. q^{41} +4.71283e6 q^{43} +1.16801e7 q^{45} +1.54235e6 q^{47} -7.14133e6 q^{49} -2.88397e6 q^{51} +4.95803e7 q^{53} -2.50521e7 q^{55} -2.84479e7 q^{57} -1.47617e8 q^{59} +1.31460e8 q^{61} +1.07700e8 q^{63} +4.18003e7 q^{65} +3.13535e8 q^{67} +8.22897e6 q^{69} -1.43087e8 q^{71} -7.49397e6 q^{73} +1.23212e7 q^{75} -2.31000e8 q^{77} -4.72822e7 q^{79} +3.29661e8 q^{81} -5.06399e8 q^{83} +5.71449e7 q^{85} -2.59319e7 q^{87} +2.62827e8 q^{89} +3.85433e8 q^{91} +2.60198e8 q^{93} +5.63684e8 q^{95} +1.33714e9 q^{97} -7.49080e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2500 q^{5} + 23364 q^{9} + 201256 q^{13} - 442040 q^{17} - 3496176 q^{21} + 1562500 q^{25} + 6719352 q^{29} - 36991008 q^{33} + 29092616 q^{37} + 49561624 q^{41} - 14602500 q^{45} + 3039028 q^{49}+ \cdots + 464919272 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 31.5423 0.224827 0.112413 0.993662i \(-0.464142\pi\)
0.112413 + 0.993662i \(0.464142\pi\)
\(4\) 0 0
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) −5763.01 −0.907211 −0.453605 0.891203i \(-0.649862\pi\)
−0.453605 + 0.891203i \(0.649862\pi\)
\(8\) 0 0
\(9\) −18688.1 −0.949453
\(10\) 0 0
\(11\) 40083.3 0.825461 0.412730 0.910853i \(-0.364575\pi\)
0.412730 + 0.910853i \(0.364575\pi\)
\(12\) 0 0
\(13\) −66880.5 −0.649463 −0.324731 0.945806i \(-0.605274\pi\)
−0.324731 + 0.945806i \(0.605274\pi\)
\(14\) 0 0
\(15\) −19714.0 −0.100546
\(16\) 0 0
\(17\) −91431.8 −0.265508 −0.132754 0.991149i \(-0.542382\pi\)
−0.132754 + 0.991149i \(0.542382\pi\)
\(18\) 0 0
\(19\) −901894. −1.58768 −0.793842 0.608124i \(-0.791923\pi\)
−0.793842 + 0.608124i \(0.791923\pi\)
\(20\) 0 0
\(21\) −181779. −0.203965
\(22\) 0 0
\(23\) 260886. 0.194391 0.0971955 0.995265i \(-0.469013\pi\)
0.0971955 + 0.995265i \(0.469013\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 0 0
\(27\) −1.21031e6 −0.438290
\(28\) 0 0
\(29\) −822128. −0.215848 −0.107924 0.994159i \(-0.534420\pi\)
−0.107924 + 0.994159i \(0.534420\pi\)
\(30\) 0 0
\(31\) 8.24915e6 1.60429 0.802143 0.597132i \(-0.203693\pi\)
0.802143 + 0.597132i \(0.203693\pi\)
\(32\) 0 0
\(33\) 1.26432e6 0.185586
\(34\) 0 0
\(35\) 3.60188e6 0.405717
\(36\) 0 0
\(37\) −3.26345e6 −0.286266 −0.143133 0.989703i \(-0.545718\pi\)
−0.143133 + 0.989703i \(0.545718\pi\)
\(38\) 0 0
\(39\) −2.10957e6 −0.146017
\(40\) 0 0
\(41\) −721750. −0.0398896 −0.0199448 0.999801i \(-0.506349\pi\)
−0.0199448 + 0.999801i \(0.506349\pi\)
\(42\) 0 0
\(43\) 4.71283e6 0.210220 0.105110 0.994461i \(-0.466481\pi\)
0.105110 + 0.994461i \(0.466481\pi\)
\(44\) 0 0
\(45\) 1.16801e7 0.424608
\(46\) 0 0
\(47\) 1.54235e6 0.0461044 0.0230522 0.999734i \(-0.492662\pi\)
0.0230522 + 0.999734i \(0.492662\pi\)
\(48\) 0 0
\(49\) −7.14133e6 −0.176969
\(50\) 0 0
\(51\) −2.88397e6 −0.0596933
\(52\) 0 0
\(53\) 4.95803e7 0.863113 0.431557 0.902086i \(-0.357965\pi\)
0.431557 + 0.902086i \(0.357965\pi\)
\(54\) 0 0
\(55\) −2.50521e7 −0.369157
\(56\) 0 0
\(57\) −2.84479e7 −0.356954
\(58\) 0 0
\(59\) −1.47617e8 −1.58600 −0.793000 0.609222i \(-0.791482\pi\)
−0.793000 + 0.609222i \(0.791482\pi\)
\(60\) 0 0
\(61\) 1.31460e8 1.21565 0.607826 0.794070i \(-0.292042\pi\)
0.607826 + 0.794070i \(0.292042\pi\)
\(62\) 0 0
\(63\) 1.07700e8 0.861354
\(64\) 0 0
\(65\) 4.18003e7 0.290449
\(66\) 0 0
\(67\) 3.13535e8 1.90086 0.950429 0.310942i \(-0.100645\pi\)
0.950429 + 0.310942i \(0.100645\pi\)
\(68\) 0 0
\(69\) 8.22897e6 0.0437043
\(70\) 0 0
\(71\) −1.43087e8 −0.668247 −0.334123 0.942529i \(-0.608440\pi\)
−0.334123 + 0.942529i \(0.608440\pi\)
\(72\) 0 0
\(73\) −7.49397e6 −0.0308858 −0.0154429 0.999881i \(-0.504916\pi\)
−0.0154429 + 0.999881i \(0.504916\pi\)
\(74\) 0 0
\(75\) 1.23212e7 0.0449654
\(76\) 0 0
\(77\) −2.31000e8 −0.748867
\(78\) 0 0
\(79\) −4.72822e7 −0.136577 −0.0682883 0.997666i \(-0.521754\pi\)
−0.0682883 + 0.997666i \(0.521754\pi\)
\(80\) 0 0
\(81\) 3.29661e8 0.850914
\(82\) 0 0
\(83\) −5.06399e8 −1.17123 −0.585614 0.810590i \(-0.699147\pi\)
−0.585614 + 0.810590i \(0.699147\pi\)
\(84\) 0 0
\(85\) 5.71449e7 0.118739
\(86\) 0 0
\(87\) −2.59319e7 −0.0485285
\(88\) 0 0
\(89\) 2.62827e8 0.444032 0.222016 0.975043i \(-0.428736\pi\)
0.222016 + 0.975043i \(0.428736\pi\)
\(90\) 0 0
\(91\) 3.85433e8 0.589200
\(92\) 0 0
\(93\) 2.60198e8 0.360687
\(94\) 0 0
\(95\) 5.63684e8 0.710034
\(96\) 0 0
\(97\) 1.33714e9 1.53357 0.766784 0.641905i \(-0.221856\pi\)
0.766784 + 0.641905i \(0.221856\pi\)
\(98\) 0 0
\(99\) −7.49080e8 −0.783736
\(100\) 0 0
\(101\) 1.59609e9 1.52620 0.763099 0.646282i \(-0.223677\pi\)
0.763099 + 0.646282i \(0.223677\pi\)
\(102\) 0 0
\(103\) −4.80627e8 −0.420766 −0.210383 0.977619i \(-0.567471\pi\)
−0.210383 + 0.977619i \(0.567471\pi\)
\(104\) 0 0
\(105\) 1.13612e8 0.0912161
\(106\) 0 0
\(107\) 3.78945e7 0.0279479 0.0139739 0.999902i \(-0.495552\pi\)
0.0139739 + 0.999902i \(0.495552\pi\)
\(108\) 0 0
\(109\) 1.75532e9 1.19107 0.595535 0.803330i \(-0.296940\pi\)
0.595535 + 0.803330i \(0.296940\pi\)
\(110\) 0 0
\(111\) −1.02937e8 −0.0643602
\(112\) 0 0
\(113\) 8.94453e8 0.516065 0.258033 0.966136i \(-0.416926\pi\)
0.258033 + 0.966136i \(0.416926\pi\)
\(114\) 0 0
\(115\) −1.63054e8 −0.0869343
\(116\) 0 0
\(117\) 1.24987e9 0.616634
\(118\) 0 0
\(119\) 5.26922e8 0.240871
\(120\) 0 0
\(121\) −7.51276e8 −0.318615
\(122\) 0 0
\(123\) −2.27657e7 −0.00896825
\(124\) 0 0
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) 3.04135e9 1.03741 0.518705 0.854954i \(-0.326414\pi\)
0.518705 + 0.854954i \(0.326414\pi\)
\(128\) 0 0
\(129\) 1.48654e8 0.0472631
\(130\) 0 0
\(131\) −8.80696e7 −0.0261280 −0.0130640 0.999915i \(-0.504159\pi\)
−0.0130640 + 0.999915i \(0.504159\pi\)
\(132\) 0 0
\(133\) 5.19762e9 1.44036
\(134\) 0 0
\(135\) 7.56446e8 0.196009
\(136\) 0 0
\(137\) 5.03962e9 1.22224 0.611118 0.791540i \(-0.290720\pi\)
0.611118 + 0.791540i \(0.290720\pi\)
\(138\) 0 0
\(139\) 3.67230e9 0.834395 0.417197 0.908816i \(-0.363012\pi\)
0.417197 + 0.908816i \(0.363012\pi\)
\(140\) 0 0
\(141\) 4.86493e7 0.0103655
\(142\) 0 0
\(143\) −2.68079e9 −0.536106
\(144\) 0 0
\(145\) 5.13830e8 0.0965303
\(146\) 0 0
\(147\) −2.25254e8 −0.0397874
\(148\) 0 0
\(149\) 9.50765e9 1.58028 0.790142 0.612924i \(-0.210007\pi\)
0.790142 + 0.612924i \(0.210007\pi\)
\(150\) 0 0
\(151\) −5.53508e9 −0.866418 −0.433209 0.901294i \(-0.642619\pi\)
−0.433209 + 0.901294i \(0.642619\pi\)
\(152\) 0 0
\(153\) 1.70869e9 0.252087
\(154\) 0 0
\(155\) −5.15572e9 −0.717459
\(156\) 0 0
\(157\) 3.26124e9 0.428385 0.214193 0.976791i \(-0.431288\pi\)
0.214193 + 0.976791i \(0.431288\pi\)
\(158\) 0 0
\(159\) 1.56388e9 0.194051
\(160\) 0 0
\(161\) −1.50349e9 −0.176354
\(162\) 0 0
\(163\) 5.90330e9 0.655014 0.327507 0.944849i \(-0.393792\pi\)
0.327507 + 0.944849i \(0.393792\pi\)
\(164\) 0 0
\(165\) −7.90201e8 −0.0829965
\(166\) 0 0
\(167\) −1.41098e10 −1.40377 −0.701884 0.712291i \(-0.747658\pi\)
−0.701884 + 0.712291i \(0.747658\pi\)
\(168\) 0 0
\(169\) −6.13150e9 −0.578198
\(170\) 0 0
\(171\) 1.68547e10 1.50743
\(172\) 0 0
\(173\) 8.67737e9 0.736514 0.368257 0.929724i \(-0.379955\pi\)
0.368257 + 0.929724i \(0.379955\pi\)
\(174\) 0 0
\(175\) −2.25118e9 −0.181442
\(176\) 0 0
\(177\) −4.65620e9 −0.356576
\(178\) 0 0
\(179\) −1.99826e10 −1.45483 −0.727417 0.686196i \(-0.759279\pi\)
−0.727417 + 0.686196i \(0.759279\pi\)
\(180\) 0 0
\(181\) 1.09751e10 0.760072 0.380036 0.924972i \(-0.375912\pi\)
0.380036 + 0.924972i \(0.375912\pi\)
\(182\) 0 0
\(183\) 4.14656e9 0.273311
\(184\) 0 0
\(185\) 2.03966e9 0.128022
\(186\) 0 0
\(187\) −3.66489e9 −0.219166
\(188\) 0 0
\(189\) 6.97505e9 0.397621
\(190\) 0 0
\(191\) 1.88576e10 1.02527 0.512633 0.858608i \(-0.328670\pi\)
0.512633 + 0.858608i \(0.328670\pi\)
\(192\) 0 0
\(193\) −2.12010e10 −1.09989 −0.549944 0.835201i \(-0.685351\pi\)
−0.549944 + 0.835201i \(0.685351\pi\)
\(194\) 0 0
\(195\) 1.31848e9 0.0653007
\(196\) 0 0
\(197\) 4.20226e9 0.198785 0.0993927 0.995048i \(-0.468310\pi\)
0.0993927 + 0.995048i \(0.468310\pi\)
\(198\) 0 0
\(199\) 3.42438e10 1.54790 0.773950 0.633247i \(-0.218278\pi\)
0.773950 + 0.633247i \(0.218278\pi\)
\(200\) 0 0
\(201\) 9.88963e9 0.427364
\(202\) 0 0
\(203\) 4.73793e9 0.195820
\(204\) 0 0
\(205\) 4.51094e8 0.0178392
\(206\) 0 0
\(207\) −4.87547e9 −0.184565
\(208\) 0 0
\(209\) −3.61509e10 −1.31057
\(210\) 0 0
\(211\) −3.15040e10 −1.09419 −0.547097 0.837069i \(-0.684267\pi\)
−0.547097 + 0.837069i \(0.684267\pi\)
\(212\) 0 0
\(213\) −4.51329e9 −0.150240
\(214\) 0 0
\(215\) −2.94552e9 −0.0940132
\(216\) 0 0
\(217\) −4.75400e10 −1.45543
\(218\) 0 0
\(219\) −2.36378e8 −0.00694397
\(220\) 0 0
\(221\) 6.11501e9 0.172437
\(222\) 0 0
\(223\) −6.76841e10 −1.83280 −0.916399 0.400267i \(-0.868918\pi\)
−0.916399 + 0.400267i \(0.868918\pi\)
\(224\) 0 0
\(225\) −7.30003e9 −0.189891
\(226\) 0 0
\(227\) 2.32392e10 0.580904 0.290452 0.956890i \(-0.406194\pi\)
0.290452 + 0.956890i \(0.406194\pi\)
\(228\) 0 0
\(229\) −3.87727e10 −0.931678 −0.465839 0.884869i \(-0.654247\pi\)
−0.465839 + 0.884869i \(0.654247\pi\)
\(230\) 0 0
\(231\) −7.28630e9 −0.168365
\(232\) 0 0
\(233\) −3.74668e10 −0.832809 −0.416404 0.909180i \(-0.636710\pi\)
−0.416404 + 0.909180i \(0.636710\pi\)
\(234\) 0 0
\(235\) −9.63968e8 −0.0206185
\(236\) 0 0
\(237\) −1.49139e9 −0.0307061
\(238\) 0 0
\(239\) 1.01374e10 0.200973 0.100487 0.994938i \(-0.467960\pi\)
0.100487 + 0.994938i \(0.467960\pi\)
\(240\) 0 0
\(241\) −6.77474e10 −1.29365 −0.646824 0.762639i \(-0.723903\pi\)
−0.646824 + 0.762639i \(0.723903\pi\)
\(242\) 0 0
\(243\) 3.42209e10 0.629598
\(244\) 0 0
\(245\) 4.46333e9 0.0791429
\(246\) 0 0
\(247\) 6.03191e10 1.03114
\(248\) 0 0
\(249\) −1.59730e10 −0.263324
\(250\) 0 0
\(251\) 1.49908e9 0.0238393 0.0119197 0.999929i \(-0.496206\pi\)
0.0119197 + 0.999929i \(0.496206\pi\)
\(252\) 0 0
\(253\) 1.04572e10 0.160462
\(254\) 0 0
\(255\) 1.80248e9 0.0266957
\(256\) 0 0
\(257\) −8.73728e10 −1.24933 −0.624665 0.780893i \(-0.714764\pi\)
−0.624665 + 0.780893i \(0.714764\pi\)
\(258\) 0 0
\(259\) 1.88073e10 0.259703
\(260\) 0 0
\(261\) 1.53640e10 0.204938
\(262\) 0 0
\(263\) 8.15504e9 0.105105 0.0525527 0.998618i \(-0.483264\pi\)
0.0525527 + 0.998618i \(0.483264\pi\)
\(264\) 0 0
\(265\) −3.09877e10 −0.385996
\(266\) 0 0
\(267\) 8.29018e9 0.0998305
\(268\) 0 0
\(269\) 4.12445e9 0.0480265 0.0240133 0.999712i \(-0.492356\pi\)
0.0240133 + 0.999712i \(0.492356\pi\)
\(270\) 0 0
\(271\) 4.44948e10 0.501127 0.250563 0.968100i \(-0.419384\pi\)
0.250563 + 0.968100i \(0.419384\pi\)
\(272\) 0 0
\(273\) 1.21575e10 0.132468
\(274\) 0 0
\(275\) 1.56575e10 0.165092
\(276\) 0 0
\(277\) 9.73881e10 0.993909 0.496955 0.867777i \(-0.334452\pi\)
0.496955 + 0.867777i \(0.334452\pi\)
\(278\) 0 0
\(279\) −1.54161e11 −1.52319
\(280\) 0 0
\(281\) −1.38133e11 −1.32165 −0.660827 0.750539i \(-0.729794\pi\)
−0.660827 + 0.750539i \(0.729794\pi\)
\(282\) 0 0
\(283\) −1.44541e11 −1.33952 −0.669762 0.742575i \(-0.733604\pi\)
−0.669762 + 0.742575i \(0.733604\pi\)
\(284\) 0 0
\(285\) 1.77799e10 0.159635
\(286\) 0 0
\(287\) 4.15945e9 0.0361883
\(288\) 0 0
\(289\) −1.10228e11 −0.929506
\(290\) 0 0
\(291\) 4.21765e10 0.344788
\(292\) 0 0
\(293\) 9.30787e10 0.737813 0.368906 0.929467i \(-0.379732\pi\)
0.368906 + 0.929467i \(0.379732\pi\)
\(294\) 0 0
\(295\) 9.22608e10 0.709281
\(296\) 0 0
\(297\) −4.85134e10 −0.361791
\(298\) 0 0
\(299\) −1.74482e10 −0.126250
\(300\) 0 0
\(301\) −2.71601e10 −0.190714
\(302\) 0 0
\(303\) 5.03444e10 0.343130
\(304\) 0 0
\(305\) −8.21625e10 −0.543656
\(306\) 0 0
\(307\) −8.83221e10 −0.567475 −0.283738 0.958902i \(-0.591574\pi\)
−0.283738 + 0.958902i \(0.591574\pi\)
\(308\) 0 0
\(309\) −1.51601e10 −0.0945996
\(310\) 0 0
\(311\) −4.39960e10 −0.266681 −0.133340 0.991070i \(-0.542570\pi\)
−0.133340 + 0.991070i \(0.542570\pi\)
\(312\) 0 0
\(313\) −9.54876e10 −0.562338 −0.281169 0.959658i \(-0.590722\pi\)
−0.281169 + 0.959658i \(0.590722\pi\)
\(314\) 0 0
\(315\) −6.73122e10 −0.385209
\(316\) 0 0
\(317\) 1.05730e11 0.588073 0.294037 0.955794i \(-0.405001\pi\)
0.294037 + 0.955794i \(0.405001\pi\)
\(318\) 0 0
\(319\) −3.29536e10 −0.178174
\(320\) 0 0
\(321\) 1.19528e9 0.00628344
\(322\) 0 0
\(323\) 8.24618e10 0.421543
\(324\) 0 0
\(325\) −2.61252e10 −0.129893
\(326\) 0 0
\(327\) 5.53669e10 0.267784
\(328\) 0 0
\(329\) −8.88857e9 −0.0418264
\(330\) 0 0
\(331\) −3.74481e10 −0.171476 −0.0857380 0.996318i \(-0.527325\pi\)
−0.0857380 + 0.996318i \(0.527325\pi\)
\(332\) 0 0
\(333\) 6.09876e10 0.271796
\(334\) 0 0
\(335\) −1.95959e11 −0.850089
\(336\) 0 0
\(337\) 2.00187e11 0.845474 0.422737 0.906252i \(-0.361069\pi\)
0.422737 + 0.906252i \(0.361069\pi\)
\(338\) 0 0
\(339\) 2.82131e10 0.116025
\(340\) 0 0
\(341\) 3.30653e11 1.32428
\(342\) 0 0
\(343\) 2.73714e11 1.06776
\(344\) 0 0
\(345\) −5.14311e9 −0.0195452
\(346\) 0 0
\(347\) 3.12228e11 1.15609 0.578043 0.816007i \(-0.303817\pi\)
0.578043 + 0.816007i \(0.303817\pi\)
\(348\) 0 0
\(349\) −1.24537e10 −0.0449349 −0.0224675 0.999748i \(-0.507152\pi\)
−0.0224675 + 0.999748i \(0.507152\pi\)
\(350\) 0 0
\(351\) 8.09464e10 0.284653
\(352\) 0 0
\(353\) 4.00381e11 1.37242 0.686210 0.727404i \(-0.259273\pi\)
0.686210 + 0.727404i \(0.259273\pi\)
\(354\) 0 0
\(355\) 8.94292e10 0.298849
\(356\) 0 0
\(357\) 1.66204e10 0.0541544
\(358\) 0 0
\(359\) 5.64723e10 0.179437 0.0897183 0.995967i \(-0.471403\pi\)
0.0897183 + 0.995967i \(0.471403\pi\)
\(360\) 0 0
\(361\) 4.90725e11 1.52074
\(362\) 0 0
\(363\) −2.36970e10 −0.0716331
\(364\) 0 0
\(365\) 4.68373e9 0.0138126
\(366\) 0 0
\(367\) −6.09656e11 −1.75424 −0.877118 0.480276i \(-0.840537\pi\)
−0.877118 + 0.480276i \(0.840537\pi\)
\(368\) 0 0
\(369\) 1.34881e10 0.0378733
\(370\) 0 0
\(371\) −2.85732e11 −0.783025
\(372\) 0 0
\(373\) 3.30775e11 0.884796 0.442398 0.896819i \(-0.354128\pi\)
0.442398 + 0.896819i \(0.354128\pi\)
\(374\) 0 0
\(375\) −7.70077e9 −0.0201091
\(376\) 0 0
\(377\) 5.49843e10 0.140185
\(378\) 0 0
\(379\) 5.91235e11 1.47192 0.735959 0.677026i \(-0.236732\pi\)
0.735959 + 0.677026i \(0.236732\pi\)
\(380\) 0 0
\(381\) 9.59314e10 0.233238
\(382\) 0 0
\(383\) 1.13650e11 0.269882 0.134941 0.990854i \(-0.456915\pi\)
0.134941 + 0.990854i \(0.456915\pi\)
\(384\) 0 0
\(385\) 1.44375e11 0.334903
\(386\) 0 0
\(387\) −8.80737e10 −0.199594
\(388\) 0 0
\(389\) −1.04287e11 −0.230917 −0.115458 0.993312i \(-0.536834\pi\)
−0.115458 + 0.993312i \(0.536834\pi\)
\(390\) 0 0
\(391\) −2.38533e10 −0.0516123
\(392\) 0 0
\(393\) −2.77792e9 −0.00587427
\(394\) 0 0
\(395\) 2.95514e10 0.0610789
\(396\) 0 0
\(397\) −6.43948e10 −0.130105 −0.0650524 0.997882i \(-0.520721\pi\)
−0.0650524 + 0.997882i \(0.520721\pi\)
\(398\) 0 0
\(399\) 1.63945e11 0.323833
\(400\) 0 0
\(401\) 5.25190e11 1.01430 0.507151 0.861857i \(-0.330699\pi\)
0.507151 + 0.861857i \(0.330699\pi\)
\(402\) 0 0
\(403\) −5.51708e11 −1.04192
\(404\) 0 0
\(405\) −2.06038e11 −0.380540
\(406\) 0 0
\(407\) −1.30810e11 −0.236301
\(408\) 0 0
\(409\) −4.93589e11 −0.872189 −0.436094 0.899901i \(-0.643639\pi\)
−0.436094 + 0.899901i \(0.643639\pi\)
\(410\) 0 0
\(411\) 1.58961e11 0.274792
\(412\) 0 0
\(413\) 8.50720e11 1.43884
\(414\) 0 0
\(415\) 3.16500e11 0.523789
\(416\) 0 0
\(417\) 1.15833e11 0.187594
\(418\) 0 0
\(419\) −1.44705e11 −0.229362 −0.114681 0.993402i \(-0.536585\pi\)
−0.114681 + 0.993402i \(0.536585\pi\)
\(420\) 0 0
\(421\) 2.14833e11 0.333297 0.166648 0.986016i \(-0.446706\pi\)
0.166648 + 0.986016i \(0.446706\pi\)
\(422\) 0 0
\(423\) −2.88235e10 −0.0437740
\(424\) 0 0
\(425\) −3.57156e10 −0.0531016
\(426\) 0 0
\(427\) −7.57605e11 −1.10285
\(428\) 0 0
\(429\) −8.45585e10 −0.120531
\(430\) 0 0
\(431\) 1.60116e11 0.223504 0.111752 0.993736i \(-0.464354\pi\)
0.111752 + 0.993736i \(0.464354\pi\)
\(432\) 0 0
\(433\) 1.08023e12 1.47679 0.738397 0.674366i \(-0.235583\pi\)
0.738397 + 0.674366i \(0.235583\pi\)
\(434\) 0 0
\(435\) 1.62074e10 0.0217026
\(436\) 0 0
\(437\) −2.35292e11 −0.308632
\(438\) 0 0
\(439\) −5.70666e11 −0.733317 −0.366659 0.930356i \(-0.619498\pi\)
−0.366659 + 0.930356i \(0.619498\pi\)
\(440\) 0 0
\(441\) 1.33458e11 0.168024
\(442\) 0 0
\(443\) 8.64148e11 1.06604 0.533018 0.846104i \(-0.321058\pi\)
0.533018 + 0.846104i \(0.321058\pi\)
\(444\) 0 0
\(445\) −1.64267e11 −0.198577
\(446\) 0 0
\(447\) 2.99894e11 0.355290
\(448\) 0 0
\(449\) −8.23785e11 −0.956545 −0.478273 0.878211i \(-0.658737\pi\)
−0.478273 + 0.878211i \(0.658737\pi\)
\(450\) 0 0
\(451\) −2.89301e10 −0.0329273
\(452\) 0 0
\(453\) −1.74589e11 −0.194794
\(454\) 0 0
\(455\) −2.40896e11 −0.263498
\(456\) 0 0
\(457\) 1.18240e12 1.26807 0.634033 0.773306i \(-0.281398\pi\)
0.634033 + 0.773306i \(0.281398\pi\)
\(458\) 0 0
\(459\) 1.10661e11 0.116369
\(460\) 0 0
\(461\) −8.13722e11 −0.839116 −0.419558 0.907728i \(-0.637815\pi\)
−0.419558 + 0.907728i \(0.637815\pi\)
\(462\) 0 0
\(463\) 3.50433e11 0.354398 0.177199 0.984175i \(-0.443296\pi\)
0.177199 + 0.984175i \(0.443296\pi\)
\(464\) 0 0
\(465\) −1.62624e11 −0.161304
\(466\) 0 0
\(467\) 8.86558e11 0.862543 0.431272 0.902222i \(-0.358065\pi\)
0.431272 + 0.902222i \(0.358065\pi\)
\(468\) 0 0
\(469\) −1.80691e12 −1.72448
\(470\) 0 0
\(471\) 1.02867e11 0.0963126
\(472\) 0 0
\(473\) 1.88906e11 0.173528
\(474\) 0 0
\(475\) −3.52302e11 −0.317537
\(476\) 0 0
\(477\) −9.26561e11 −0.819485
\(478\) 0 0
\(479\) −1.88439e10 −0.0163554 −0.00817770 0.999967i \(-0.502603\pi\)
−0.00817770 + 0.999967i \(0.502603\pi\)
\(480\) 0 0
\(481\) 2.18261e11 0.185919
\(482\) 0 0
\(483\) −4.74236e10 −0.0396490
\(484\) 0 0
\(485\) −8.35711e11 −0.685833
\(486\) 0 0
\(487\) 1.88234e12 1.51642 0.758208 0.652013i \(-0.226075\pi\)
0.758208 + 0.652013i \(0.226075\pi\)
\(488\) 0 0
\(489\) 1.86204e11 0.147265
\(490\) 0 0
\(491\) −2.33032e12 −1.80946 −0.904729 0.425988i \(-0.859927\pi\)
−0.904729 + 0.425988i \(0.859927\pi\)
\(492\) 0 0
\(493\) 7.51687e10 0.0573094
\(494\) 0 0
\(495\) 4.68175e11 0.350497
\(496\) 0 0
\(497\) 8.24611e11 0.606241
\(498\) 0 0
\(499\) −1.76574e12 −1.27489 −0.637446 0.770495i \(-0.720009\pi\)
−0.637446 + 0.770495i \(0.720009\pi\)
\(500\) 0 0
\(501\) −4.45055e11 −0.315605
\(502\) 0 0
\(503\) −2.75964e12 −1.92219 −0.961095 0.276219i \(-0.910918\pi\)
−0.961095 + 0.276219i \(0.910918\pi\)
\(504\) 0 0
\(505\) −9.97555e11 −0.682536
\(506\) 0 0
\(507\) −1.93402e11 −0.129994
\(508\) 0 0
\(509\) −2.34774e12 −1.55031 −0.775156 0.631770i \(-0.782329\pi\)
−0.775156 + 0.631770i \(0.782329\pi\)
\(510\) 0 0
\(511\) 4.31878e10 0.0280200
\(512\) 0 0
\(513\) 1.09157e12 0.695866
\(514\) 0 0
\(515\) 3.00392e11 0.188172
\(516\) 0 0
\(517\) 6.18225e10 0.0380574
\(518\) 0 0
\(519\) 2.73705e11 0.165588
\(520\) 0 0
\(521\) 1.00091e12 0.595151 0.297576 0.954698i \(-0.403822\pi\)
0.297576 + 0.954698i \(0.403822\pi\)
\(522\) 0 0
\(523\) 3.01151e12 1.76006 0.880030 0.474919i \(-0.157523\pi\)
0.880030 + 0.474919i \(0.157523\pi\)
\(524\) 0 0
\(525\) −7.10074e10 −0.0407931
\(526\) 0 0
\(527\) −7.54235e11 −0.425950
\(528\) 0 0
\(529\) −1.73309e12 −0.962212
\(530\) 0 0
\(531\) 2.75868e12 1.50583
\(532\) 0 0
\(533\) 4.82710e10 0.0259068
\(534\) 0 0
\(535\) −2.36840e10 −0.0124987
\(536\) 0 0
\(537\) −6.30298e11 −0.327086
\(538\) 0 0
\(539\) −2.86248e11 −0.146081
\(540\) 0 0
\(541\) −1.79083e12 −0.898807 −0.449404 0.893329i \(-0.648363\pi\)
−0.449404 + 0.893329i \(0.648363\pi\)
\(542\) 0 0
\(543\) 3.46180e11 0.170885
\(544\) 0 0
\(545\) −1.09707e12 −0.532662
\(546\) 0 0
\(547\) 1.42084e12 0.678580 0.339290 0.940682i \(-0.389813\pi\)
0.339290 + 0.940682i \(0.389813\pi\)
\(548\) 0 0
\(549\) −2.45673e12 −1.15420
\(550\) 0 0
\(551\) 7.41472e11 0.342699
\(552\) 0 0
\(553\) 2.72488e11 0.123904
\(554\) 0 0
\(555\) 6.43355e10 0.0287828
\(556\) 0 0
\(557\) −2.06883e12 −0.910704 −0.455352 0.890311i \(-0.650487\pi\)
−0.455352 + 0.890311i \(0.650487\pi\)
\(558\) 0 0
\(559\) −3.15196e11 −0.136530
\(560\) 0 0
\(561\) −1.15599e11 −0.0492745
\(562\) 0 0
\(563\) −1.67871e12 −0.704188 −0.352094 0.935965i \(-0.614530\pi\)
−0.352094 + 0.935965i \(0.614530\pi\)
\(564\) 0 0
\(565\) −5.59033e11 −0.230791
\(566\) 0 0
\(567\) −1.89984e12 −0.771958
\(568\) 0 0
\(569\) −2.89714e12 −1.15868 −0.579342 0.815085i \(-0.696690\pi\)
−0.579342 + 0.815085i \(0.696690\pi\)
\(570\) 0 0
\(571\) 5.19889e11 0.204667 0.102334 0.994750i \(-0.467369\pi\)
0.102334 + 0.994750i \(0.467369\pi\)
\(572\) 0 0
\(573\) 5.94813e11 0.230507
\(574\) 0 0
\(575\) 1.01909e11 0.0388782
\(576\) 0 0
\(577\) −1.35855e12 −0.510252 −0.255126 0.966908i \(-0.582117\pi\)
−0.255126 + 0.966908i \(0.582117\pi\)
\(578\) 0 0
\(579\) −6.68730e11 −0.247285
\(580\) 0 0
\(581\) 2.91838e12 1.06255
\(582\) 0 0
\(583\) 1.98734e12 0.712466
\(584\) 0 0
\(585\) −7.81168e11 −0.275767
\(586\) 0 0
\(587\) 5.99152e11 0.208289 0.104144 0.994562i \(-0.466790\pi\)
0.104144 + 0.994562i \(0.466790\pi\)
\(588\) 0 0
\(589\) −7.43986e12 −2.54710
\(590\) 0 0
\(591\) 1.32549e11 0.0446923
\(592\) 0 0
\(593\) 5.46699e12 1.81553 0.907763 0.419483i \(-0.137789\pi\)
0.907763 + 0.419483i \(0.137789\pi\)
\(594\) 0 0
\(595\) −3.29327e11 −0.107721
\(596\) 0 0
\(597\) 1.08013e12 0.348010
\(598\) 0 0
\(599\) 4.52056e12 1.43473 0.717367 0.696695i \(-0.245347\pi\)
0.717367 + 0.696695i \(0.245347\pi\)
\(600\) 0 0
\(601\) 3.06316e12 0.957710 0.478855 0.877894i \(-0.341052\pi\)
0.478855 + 0.877894i \(0.341052\pi\)
\(602\) 0 0
\(603\) −5.85937e12 −1.80477
\(604\) 0 0
\(605\) 4.69548e11 0.142489
\(606\) 0 0
\(607\) −5.44730e12 −1.62867 −0.814333 0.580398i \(-0.802897\pi\)
−0.814333 + 0.580398i \(0.802897\pi\)
\(608\) 0 0
\(609\) 1.49446e11 0.0440256
\(610\) 0 0
\(611\) −1.03153e11 −0.0299431
\(612\) 0 0
\(613\) 4.89823e12 1.40109 0.700547 0.713607i \(-0.252940\pi\)
0.700547 + 0.713607i \(0.252940\pi\)
\(614\) 0 0
\(615\) 1.42286e10 0.00401072
\(616\) 0 0
\(617\) 2.36830e11 0.0657890 0.0328945 0.999459i \(-0.489527\pi\)
0.0328945 + 0.999459i \(0.489527\pi\)
\(618\) 0 0
\(619\) 1.58066e12 0.432743 0.216372 0.976311i \(-0.430578\pi\)
0.216372 + 0.976311i \(0.430578\pi\)
\(620\) 0 0
\(621\) −3.15754e11 −0.0851996
\(622\) 0 0
\(623\) −1.51467e12 −0.402831
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 0 0
\(627\) −1.14028e12 −0.294652
\(628\) 0 0
\(629\) 2.98383e11 0.0760057
\(630\) 0 0
\(631\) −5.75117e12 −1.44419 −0.722094 0.691795i \(-0.756820\pi\)
−0.722094 + 0.691795i \(0.756820\pi\)
\(632\) 0 0
\(633\) −9.93709e11 −0.246004
\(634\) 0 0
\(635\) −1.90085e12 −0.463943
\(636\) 0 0
\(637\) 4.77616e11 0.114935
\(638\) 0 0
\(639\) 2.67402e12 0.634469
\(640\) 0 0
\(641\) 5.42860e12 1.27007 0.635033 0.772485i \(-0.280986\pi\)
0.635033 + 0.772485i \(0.280986\pi\)
\(642\) 0 0
\(643\) 3.08107e12 0.710808 0.355404 0.934713i \(-0.384343\pi\)
0.355404 + 0.934713i \(0.384343\pi\)
\(644\) 0 0
\(645\) −9.29086e10 −0.0211367
\(646\) 0 0
\(647\) 7.88099e12 1.76812 0.884059 0.467375i \(-0.154800\pi\)
0.884059 + 0.467375i \(0.154800\pi\)
\(648\) 0 0
\(649\) −5.91699e12 −1.30918
\(650\) 0 0
\(651\) −1.49952e12 −0.327219
\(652\) 0 0
\(653\) 9.00773e12 1.93868 0.969339 0.245725i \(-0.0790261\pi\)
0.969339 + 0.245725i \(0.0790261\pi\)
\(654\) 0 0
\(655\) 5.50435e10 0.0116848
\(656\) 0 0
\(657\) 1.40048e11 0.0293246
\(658\) 0 0
\(659\) 4.22272e12 0.872183 0.436091 0.899902i \(-0.356362\pi\)
0.436091 + 0.899902i \(0.356362\pi\)
\(660\) 0 0
\(661\) 4.48023e12 0.912837 0.456419 0.889765i \(-0.349132\pi\)
0.456419 + 0.889765i \(0.349132\pi\)
\(662\) 0 0
\(663\) 1.92882e11 0.0387686
\(664\) 0 0
\(665\) −3.24851e12 −0.644151
\(666\) 0 0
\(667\) −2.14482e11 −0.0419590
\(668\) 0 0
\(669\) −2.13491e12 −0.412062
\(670\) 0 0
\(671\) 5.26935e12 1.00347
\(672\) 0 0
\(673\) 1.92638e12 0.361971 0.180986 0.983486i \(-0.442071\pi\)
0.180986 + 0.983486i \(0.442071\pi\)
\(674\) 0 0
\(675\) −4.72779e11 −0.0876579
\(676\) 0 0
\(677\) 9.32257e12 1.70564 0.852819 0.522207i \(-0.174891\pi\)
0.852819 + 0.522207i \(0.174891\pi\)
\(678\) 0 0
\(679\) −7.70594e12 −1.39127
\(680\) 0 0
\(681\) 7.33018e11 0.130603
\(682\) 0 0
\(683\) 4.38750e12 0.771478 0.385739 0.922608i \(-0.373946\pi\)
0.385739 + 0.922608i \(0.373946\pi\)
\(684\) 0 0
\(685\) −3.14976e12 −0.546600
\(686\) 0 0
\(687\) −1.22298e12 −0.209466
\(688\) 0 0
\(689\) −3.31596e12 −0.560560
\(690\) 0 0
\(691\) 6.05838e12 1.01089 0.505447 0.862858i \(-0.331328\pi\)
0.505447 + 0.862858i \(0.331328\pi\)
\(692\) 0 0
\(693\) 4.31696e12 0.711014
\(694\) 0 0
\(695\) −2.29519e12 −0.373153
\(696\) 0 0
\(697\) 6.59909e10 0.0105910
\(698\) 0 0
\(699\) −1.18179e12 −0.187238
\(700\) 0 0
\(701\) 2.15758e12 0.337471 0.168736 0.985661i \(-0.446032\pi\)
0.168736 + 0.985661i \(0.446032\pi\)
\(702\) 0 0
\(703\) 2.94328e12 0.454499
\(704\) 0 0
\(705\) −3.04058e10 −0.00463560
\(706\) 0 0
\(707\) −9.19827e12 −1.38458
\(708\) 0 0
\(709\) 2.60621e12 0.387348 0.193674 0.981066i \(-0.437960\pi\)
0.193674 + 0.981066i \(0.437960\pi\)
\(710\) 0 0
\(711\) 8.83614e11 0.129673
\(712\) 0 0
\(713\) 2.15209e12 0.311859
\(714\) 0 0
\(715\) 1.67549e12 0.239754
\(716\) 0 0
\(717\) 3.19759e11 0.0451842
\(718\) 0 0
\(719\) −6.17622e12 −0.861872 −0.430936 0.902382i \(-0.641817\pi\)
−0.430936 + 0.902382i \(0.641817\pi\)
\(720\) 0 0
\(721\) 2.76986e12 0.381723
\(722\) 0 0
\(723\) −2.13691e12 −0.290847
\(724\) 0 0
\(725\) −3.21144e11 −0.0431697
\(726\) 0 0
\(727\) −5.75173e11 −0.0763649 −0.0381825 0.999271i \(-0.512157\pi\)
−0.0381825 + 0.999271i \(0.512157\pi\)
\(728\) 0 0
\(729\) −5.40932e12 −0.709363
\(730\) 0 0
\(731\) −4.30903e11 −0.0558150
\(732\) 0 0
\(733\) 1.41134e13 1.80577 0.902886 0.429880i \(-0.141444\pi\)
0.902886 + 0.429880i \(0.141444\pi\)
\(734\) 0 0
\(735\) 1.40784e11 0.0177935
\(736\) 0 0
\(737\) 1.25675e13 1.56908
\(738\) 0 0
\(739\) −7.30570e11 −0.0901077 −0.0450538 0.998985i \(-0.514346\pi\)
−0.0450538 + 0.998985i \(0.514346\pi\)
\(740\) 0 0
\(741\) 1.90261e12 0.231829
\(742\) 0 0
\(743\) −1.38765e12 −0.167044 −0.0835222 0.996506i \(-0.526617\pi\)
−0.0835222 + 0.996506i \(0.526617\pi\)
\(744\) 0 0
\(745\) −5.94228e12 −0.706724
\(746\) 0 0
\(747\) 9.46363e12 1.11203
\(748\) 0 0
\(749\) −2.18386e11 −0.0253546
\(750\) 0 0
\(751\) 3.54749e12 0.406951 0.203475 0.979080i \(-0.434776\pi\)
0.203475 + 0.979080i \(0.434776\pi\)
\(752\) 0 0
\(753\) 4.72846e10 0.00535972
\(754\) 0 0
\(755\) 3.45942e12 0.387474
\(756\) 0 0
\(757\) 3.77008e11 0.0417272 0.0208636 0.999782i \(-0.493358\pi\)
0.0208636 + 0.999782i \(0.493358\pi\)
\(758\) 0 0
\(759\) 3.29844e11 0.0360762
\(760\) 0 0
\(761\) 7.57515e12 0.818767 0.409383 0.912362i \(-0.365744\pi\)
0.409383 + 0.912362i \(0.365744\pi\)
\(762\) 0 0
\(763\) −1.01159e13 −1.08055
\(764\) 0 0
\(765\) −1.06793e12 −0.112737
\(766\) 0 0
\(767\) 9.87272e12 1.03005
\(768\) 0 0
\(769\) −6.31842e12 −0.651539 −0.325769 0.945449i \(-0.605623\pi\)
−0.325769 + 0.945449i \(0.605623\pi\)
\(770\) 0 0
\(771\) −2.75594e12 −0.280883
\(772\) 0 0
\(773\) 1.62451e13 1.63650 0.818249 0.574864i \(-0.194945\pi\)
0.818249 + 0.574864i \(0.194945\pi\)
\(774\) 0 0
\(775\) 3.22233e12 0.320857
\(776\) 0 0
\(777\) 5.93226e11 0.0583883
\(778\) 0 0
\(779\) 6.50942e11 0.0633321
\(780\) 0 0
\(781\) −5.73539e12 −0.551612
\(782\) 0 0
\(783\) 9.95033e11 0.0946040
\(784\) 0 0
\(785\) −2.03828e12 −0.191580
\(786\) 0 0
\(787\) −6.25892e12 −0.581585 −0.290793 0.956786i \(-0.593919\pi\)
−0.290793 + 0.956786i \(0.593919\pi\)
\(788\) 0 0
\(789\) 2.57229e11 0.0236305
\(790\) 0 0
\(791\) −5.15474e12 −0.468180
\(792\) 0 0
\(793\) −8.79211e12 −0.789521
\(794\) 0 0
\(795\) −9.77425e11 −0.0867823
\(796\) 0 0
\(797\) −1.35039e12 −0.118549 −0.0592744 0.998242i \(-0.518879\pi\)
−0.0592744 + 0.998242i \(0.518879\pi\)
\(798\) 0 0
\(799\) −1.41020e11 −0.0122411
\(800\) 0 0
\(801\) −4.91173e12 −0.421588
\(802\) 0 0
\(803\) −3.00383e11 −0.0254950
\(804\) 0 0
\(805\) 9.39682e11 0.0788677
\(806\) 0 0
\(807\) 1.30095e11 0.0107977
\(808\) 0 0
\(809\) −1.96020e13 −1.60891 −0.804455 0.594014i \(-0.797543\pi\)
−0.804455 + 0.594014i \(0.797543\pi\)
\(810\) 0 0
\(811\) −1.40454e13 −1.14009 −0.570046 0.821612i \(-0.693075\pi\)
−0.570046 + 0.821612i \(0.693075\pi\)
\(812\) 0 0
\(813\) 1.40347e12 0.112667
\(814\) 0 0
\(815\) −3.68956e12 −0.292931
\(816\) 0 0
\(817\) −4.25047e12 −0.333763
\(818\) 0 0
\(819\) −7.20300e12 −0.559417
\(820\) 0 0
\(821\) 6.06800e12 0.466124 0.233062 0.972462i \(-0.425125\pi\)
0.233062 + 0.972462i \(0.425125\pi\)
\(822\) 0 0
\(823\) 1.46415e13 1.11246 0.556232 0.831027i \(-0.312247\pi\)
0.556232 + 0.831027i \(0.312247\pi\)
\(824\) 0 0
\(825\) 4.93876e11 0.0371172
\(826\) 0 0
\(827\) 1.85325e13 1.37772 0.688859 0.724895i \(-0.258112\pi\)
0.688859 + 0.724895i \(0.258112\pi\)
\(828\) 0 0
\(829\) −1.09144e13 −0.802612 −0.401306 0.915944i \(-0.631444\pi\)
−0.401306 + 0.915944i \(0.631444\pi\)
\(830\) 0 0
\(831\) 3.07185e12 0.223458
\(832\) 0 0
\(833\) 6.52945e11 0.0469866
\(834\) 0 0
\(835\) 8.81860e12 0.627784
\(836\) 0 0
\(837\) −9.98407e12 −0.703142
\(838\) 0 0
\(839\) 2.71319e12 0.189039 0.0945196 0.995523i \(-0.469868\pi\)
0.0945196 + 0.995523i \(0.469868\pi\)
\(840\) 0 0
\(841\) −1.38313e13 −0.953410
\(842\) 0 0
\(843\) −4.35702e12 −0.297143
\(844\) 0 0
\(845\) 3.83219e12 0.258578
\(846\) 0 0
\(847\) 4.32961e12 0.289050
\(848\) 0 0
\(849\) −4.55915e12 −0.301161
\(850\) 0 0
\(851\) −8.51389e11 −0.0556474
\(852\) 0 0
\(853\) 1.54689e13 1.00043 0.500217 0.865900i \(-0.333253\pi\)
0.500217 + 0.865900i \(0.333253\pi\)
\(854\) 0 0
\(855\) −1.05342e13 −0.674144
\(856\) 0 0
\(857\) −6.70023e12 −0.424303 −0.212151 0.977237i \(-0.568047\pi\)
−0.212151 + 0.977237i \(0.568047\pi\)
\(858\) 0 0
\(859\) 1.12634e13 0.705832 0.352916 0.935655i \(-0.385190\pi\)
0.352916 + 0.935655i \(0.385190\pi\)
\(860\) 0 0
\(861\) 1.31199e11 0.00813610
\(862\) 0 0
\(863\) −2.44794e13 −1.50229 −0.751144 0.660139i \(-0.770497\pi\)
−0.751144 + 0.660139i \(0.770497\pi\)
\(864\) 0 0
\(865\) −5.42336e12 −0.329379
\(866\) 0 0
\(867\) −3.47685e12 −0.208978
\(868\) 0 0
\(869\) −1.89523e12 −0.112739
\(870\) 0 0
\(871\) −2.09694e13 −1.23454
\(872\) 0 0
\(873\) −2.49885e13 −1.45605
\(874\) 0 0
\(875\) 1.40698e12 0.0811434
\(876\) 0 0
\(877\) −2.25259e13 −1.28583 −0.642914 0.765938i \(-0.722275\pi\)
−0.642914 + 0.765938i \(0.722275\pi\)
\(878\) 0 0
\(879\) 2.93592e12 0.165880
\(880\) 0 0
\(881\) 2.29048e13 1.28096 0.640480 0.767975i \(-0.278736\pi\)
0.640480 + 0.767975i \(0.278736\pi\)
\(882\) 0 0
\(883\) 2.33032e13 1.29001 0.645005 0.764178i \(-0.276855\pi\)
0.645005 + 0.764178i \(0.276855\pi\)
\(884\) 0 0
\(885\) 2.91012e12 0.159465
\(886\) 0 0
\(887\) −3.30975e13 −1.79531 −0.897655 0.440699i \(-0.854731\pi\)
−0.897655 + 0.440699i \(0.854731\pi\)
\(888\) 0 0
\(889\) −1.75273e13 −0.941149
\(890\) 0 0
\(891\) 1.32139e13 0.702396
\(892\) 0 0
\(893\) −1.39104e12 −0.0731993
\(894\) 0 0
\(895\) 1.24891e13 0.650621
\(896\) 0 0
\(897\) −5.50358e11 −0.0283844
\(898\) 0 0
\(899\) −6.78186e12 −0.346282
\(900\) 0 0
\(901\) −4.53322e12 −0.229163
\(902\) 0 0
\(903\) −8.56693e11 −0.0428776
\(904\) 0 0
\(905\) −6.85943e12 −0.339914
\(906\) 0 0
\(907\) −2.75737e13 −1.35289 −0.676445 0.736493i \(-0.736480\pi\)
−0.676445 + 0.736493i \(0.736480\pi\)
\(908\) 0 0
\(909\) −2.98278e13 −1.44905
\(910\) 0 0
\(911\) 2.32541e13 1.11858 0.559291 0.828972i \(-0.311074\pi\)
0.559291 + 0.828972i \(0.311074\pi\)
\(912\) 0 0
\(913\) −2.02982e13 −0.966803
\(914\) 0 0
\(915\) −2.59160e12 −0.122229
\(916\) 0 0
\(917\) 5.07546e11 0.0237036
\(918\) 0 0
\(919\) 2.51992e13 1.16538 0.582689 0.812695i \(-0.302001\pi\)
0.582689 + 0.812695i \(0.302001\pi\)
\(920\) 0 0
\(921\) −2.78589e12 −0.127584
\(922\) 0 0
\(923\) 9.56972e12 0.434002
\(924\) 0 0
\(925\) −1.27478e12 −0.0572531
\(926\) 0 0
\(927\) 8.98200e12 0.399498
\(928\) 0 0
\(929\) 3.57780e13 1.57596 0.787980 0.615701i \(-0.211127\pi\)
0.787980 + 0.615701i \(0.211127\pi\)
\(930\) 0 0
\(931\) 6.44072e12 0.280971
\(932\) 0 0
\(933\) −1.38774e12 −0.0599570
\(934\) 0 0
\(935\) 2.29056e12 0.0980141
\(936\) 0 0
\(937\) −2.23802e13 −0.948498 −0.474249 0.880391i \(-0.657280\pi\)
−0.474249 + 0.880391i \(0.657280\pi\)
\(938\) 0 0
\(939\) −3.01190e12 −0.126429
\(940\) 0 0
\(941\) −3.36203e13 −1.39781 −0.698906 0.715214i \(-0.746329\pi\)
−0.698906 + 0.715214i \(0.746329\pi\)
\(942\) 0 0
\(943\) −1.88295e11 −0.00775418
\(944\) 0 0
\(945\) −4.35941e12 −0.177821
\(946\) 0 0
\(947\) −1.09703e13 −0.443245 −0.221622 0.975133i \(-0.571135\pi\)
−0.221622 + 0.975133i \(0.571135\pi\)
\(948\) 0 0
\(949\) 5.01201e11 0.0200592
\(950\) 0 0
\(951\) 3.33497e12 0.132215
\(952\) 0 0
\(953\) 2.40941e12 0.0946220 0.0473110 0.998880i \(-0.484935\pi\)
0.0473110 + 0.998880i \(0.484935\pi\)
\(954\) 0 0
\(955\) −1.17860e13 −0.458512
\(956\) 0 0
\(957\) −1.03943e12 −0.0400584
\(958\) 0 0
\(959\) −2.90434e13 −1.10883
\(960\) 0 0
\(961\) 4.16089e13 1.57373
\(962\) 0 0
\(963\) −7.08175e11 −0.0265352
\(964\) 0 0
\(965\) 1.32506e13 0.491885
\(966\) 0 0
\(967\) 4.16349e13 1.53122 0.765611 0.643304i \(-0.222437\pi\)
0.765611 + 0.643304i \(0.222437\pi\)
\(968\) 0 0
\(969\) 2.60104e12 0.0947742
\(970\) 0 0
\(971\) 2.19480e13 0.792334 0.396167 0.918178i \(-0.370340\pi\)
0.396167 + 0.918178i \(0.370340\pi\)
\(972\) 0 0
\(973\) −2.11635e13 −0.756972
\(974\) 0 0
\(975\) −8.24050e11 −0.0292034
\(976\) 0 0
\(977\) 5.29108e13 1.85788 0.928942 0.370225i \(-0.120719\pi\)
0.928942 + 0.370225i \(0.120719\pi\)
\(978\) 0 0
\(979\) 1.05350e13 0.366531
\(980\) 0 0
\(981\) −3.28036e13 −1.13086
\(982\) 0 0
\(983\) 4.37280e13 1.49372 0.746860 0.664981i \(-0.231560\pi\)
0.746860 + 0.664981i \(0.231560\pi\)
\(984\) 0 0
\(985\) −2.62641e12 −0.0888995
\(986\) 0 0
\(987\) −2.80366e11 −0.00940370
\(988\) 0 0
\(989\) 1.22951e12 0.0408649
\(990\) 0 0
\(991\) −5.51408e13 −1.81611 −0.908055 0.418852i \(-0.862433\pi\)
−0.908055 + 0.418852i \(0.862433\pi\)
\(992\) 0 0
\(993\) −1.18120e12 −0.0385524
\(994\) 0 0
\(995\) −2.14024e13 −0.692242
\(996\) 0 0
\(997\) 1.52889e13 0.490060 0.245030 0.969516i \(-0.421202\pi\)
0.245030 + 0.969516i \(0.421202\pi\)
\(998\) 0 0
\(999\) 3.94980e12 0.125467
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 160.10.a.c.1.3 yes 4
4.3 odd 2 inner 160.10.a.c.1.2 4
8.3 odd 2 320.10.a.z.1.3 4
8.5 even 2 320.10.a.z.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.10.a.c.1.2 4 4.3 odd 2 inner
160.10.a.c.1.3 yes 4 1.1 even 1 trivial
320.10.a.z.1.2 4 8.5 even 2
320.10.a.z.1.3 4 8.3 odd 2