Properties

Label 48.28.a.l.1.3
Level $48$
Weight $28$
Character 48.1
Self dual yes
Analytic conductor $221.691$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [48,28,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(221.690675922\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{42}\cdot 3^{8}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(637889.\) of defining polynomial
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.59432e6 q^{3} +3.92930e9 q^{5} -4.56279e11 q^{7} +2.54187e12 q^{9} +O(q^{10})\) \(q-1.59432e6 q^{3} +3.92930e9 q^{5} -4.56279e11 q^{7} +2.54187e12 q^{9} +2.13831e14 q^{11} -1.01747e14 q^{13} -6.26458e15 q^{15} +7.98275e16 q^{17} -2.06157e17 q^{19} +7.27455e17 q^{21} +4.83007e17 q^{23} +7.98885e18 q^{25} -4.05256e18 q^{27} +5.24815e19 q^{29} +1.20851e19 q^{31} -3.40916e20 q^{33} -1.79286e21 q^{35} -1.22666e21 q^{37} +1.62218e20 q^{39} -1.90592e21 q^{41} +1.48523e22 q^{43} +9.98777e21 q^{45} -3.30691e22 q^{47} +1.42478e23 q^{49} -1.27271e23 q^{51} +1.44510e23 q^{53} +8.40208e23 q^{55} +3.28680e23 q^{57} +1.21957e24 q^{59} -9.71114e23 q^{61} -1.15980e24 q^{63} -3.99795e23 q^{65} +7.48681e23 q^{67} -7.70070e23 q^{69} -1.68002e25 q^{71} +6.29934e24 q^{73} -1.27368e25 q^{75} -9.75666e25 q^{77} -5.57420e25 q^{79} +6.46108e24 q^{81} +1.13665e26 q^{83} +3.13666e26 q^{85} -8.36725e25 q^{87} -1.35435e26 q^{89} +4.64250e25 q^{91} -1.92675e25 q^{93} -8.10052e26 q^{95} -7.49440e26 q^{97} +5.43530e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6377292 q^{3} + 4949050424 q^{5} + 88249731552 q^{7} + 10167463313316 q^{9} + 111189633716848 q^{11} - 7339657642664 q^{13} - 78\!\cdots\!52 q^{15}+ \cdots + 28\!\cdots\!92 q^{99}+O(q^{100}) \) 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
\(3\) −1.59432e6 −0.577350
\(4\) 0 0
\(5\) 3.92930e9 1.43953 0.719765 0.694218i \(-0.244250\pi\)
0.719765 + 0.694218i \(0.244250\pi\)
\(6\) 0 0
\(7\) −4.56279e11 −1.77994 −0.889972 0.456014i \(-0.849277\pi\)
−0.889972 + 0.456014i \(0.849277\pi\)
\(8\) 0 0
\(9\) 2.54187e12 0.333333
\(10\) 0 0
\(11\) 2.13831e14 1.86754 0.933770 0.357873i \(-0.116498\pi\)
0.933770 + 0.357873i \(0.116498\pi\)
\(12\) 0 0
\(13\) −1.01747e14 −0.0931722 −0.0465861 0.998914i \(-0.514834\pi\)
−0.0465861 + 0.998914i \(0.514834\pi\)
\(14\) 0 0
\(15\) −6.26458e15 −0.831113
\(16\) 0 0
\(17\) 7.98275e16 1.95475 0.977377 0.211506i \(-0.0678368\pi\)
0.977377 + 0.211506i \(0.0678368\pi\)
\(18\) 0 0
\(19\) −2.06157e17 −1.12467 −0.562333 0.826911i \(-0.690096\pi\)
−0.562333 + 0.826911i \(0.690096\pi\)
\(20\) 0 0
\(21\) 7.27455e17 1.02765
\(22\) 0 0
\(23\) 4.83007e17 0.199815 0.0999074 0.994997i \(-0.468145\pi\)
0.0999074 + 0.994997i \(0.468145\pi\)
\(24\) 0 0
\(25\) 7.98885e18 1.07225
\(26\) 0 0
\(27\) −4.05256e18 −0.192450
\(28\) 0 0
\(29\) 5.24815e19 0.949802 0.474901 0.880039i \(-0.342484\pi\)
0.474901 + 0.880039i \(0.342484\pi\)
\(30\) 0 0
\(31\) 1.20851e19 0.0888927 0.0444464 0.999012i \(-0.485848\pi\)
0.0444464 + 0.999012i \(0.485848\pi\)
\(32\) 0 0
\(33\) −3.40916e20 −1.07822
\(34\) 0 0
\(35\) −1.79286e21 −2.56228
\(36\) 0 0
\(37\) −1.22666e21 −0.827944 −0.413972 0.910290i \(-0.635859\pi\)
−0.413972 + 0.910290i \(0.635859\pi\)
\(38\) 0 0
\(39\) 1.62218e20 0.0537930
\(40\) 0 0
\(41\) −1.90592e21 −0.321752 −0.160876 0.986975i \(-0.551432\pi\)
−0.160876 + 0.986975i \(0.551432\pi\)
\(42\) 0 0
\(43\) 1.48523e22 1.31816 0.659081 0.752072i \(-0.270946\pi\)
0.659081 + 0.752072i \(0.270946\pi\)
\(44\) 0 0
\(45\) 9.98777e21 0.479843
\(46\) 0 0
\(47\) −3.30691e22 −0.883286 −0.441643 0.897191i \(-0.645604\pi\)
−0.441643 + 0.897191i \(0.645604\pi\)
\(48\) 0 0
\(49\) 1.42478e23 2.16820
\(50\) 0 0
\(51\) −1.27271e23 −1.12858
\(52\) 0 0
\(53\) 1.44510e23 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(54\) 0 0
\(55\) 8.40208e23 2.68838
\(56\) 0 0
\(57\) 3.28680e23 0.649326
\(58\) 0 0
\(59\) 1.21957e24 1.51254 0.756268 0.654262i \(-0.227021\pi\)
0.756268 + 0.654262i \(0.227021\pi\)
\(60\) 0 0
\(61\) −9.71114e23 −0.767922 −0.383961 0.923349i \(-0.625440\pi\)
−0.383961 + 0.923349i \(0.625440\pi\)
\(62\) 0 0
\(63\) −1.15980e24 −0.593315
\(64\) 0 0
\(65\) −3.99795e23 −0.134124
\(66\) 0 0
\(67\) 7.48681e23 0.166835 0.0834174 0.996515i \(-0.473417\pi\)
0.0834174 + 0.996515i \(0.473417\pi\)
\(68\) 0 0
\(69\) −7.70070e23 −0.115363
\(70\) 0 0
\(71\) −1.68002e25 −1.71130 −0.855652 0.517551i \(-0.826844\pi\)
−0.855652 + 0.517551i \(0.826844\pi\)
\(72\) 0 0
\(73\) 6.29934e24 0.440998 0.220499 0.975387i \(-0.429231\pi\)
0.220499 + 0.975387i \(0.429231\pi\)
\(74\) 0 0
\(75\) −1.27368e25 −0.619061
\(76\) 0 0
\(77\) −9.75666e25 −3.32412
\(78\) 0 0
\(79\) −5.57420e25 −1.34343 −0.671717 0.740807i \(-0.734443\pi\)
−0.671717 + 0.740807i \(0.734443\pi\)
\(80\) 0 0
\(81\) 6.46108e24 0.111111
\(82\) 0 0
\(83\) 1.13665e26 1.40629 0.703144 0.711048i \(-0.251779\pi\)
0.703144 + 0.711048i \(0.251779\pi\)
\(84\) 0 0
\(85\) 3.13666e26 2.81393
\(86\) 0 0
\(87\) −8.36725e25 −0.548369
\(88\) 0 0
\(89\) −1.35435e26 −0.653080 −0.326540 0.945183i \(-0.605883\pi\)
−0.326540 + 0.945183i \(0.605883\pi\)
\(90\) 0 0
\(91\) 4.64250e25 0.165841
\(92\) 0 0
\(93\) −1.92675e25 −0.0513222
\(94\) 0 0
\(95\) −8.10052e26 −1.61899
\(96\) 0 0
\(97\) −7.49440e26 −1.13062 −0.565312 0.824877i \(-0.691244\pi\)
−0.565312 + 0.824877i \(0.691244\pi\)
\(98\) 0 0
\(99\) 5.43530e26 0.622513
\(100\) 0 0
\(101\) 2.43787e26 0.213143 0.106572 0.994305i \(-0.466013\pi\)
0.106572 + 0.994305i \(0.466013\pi\)
\(102\) 0 0
\(103\) −4.27217e26 −0.286646 −0.143323 0.989676i \(-0.545779\pi\)
−0.143323 + 0.989676i \(0.545779\pi\)
\(104\) 0 0
\(105\) 2.85839e27 1.47933
\(106\) 0 0
\(107\) −1.13880e27 −0.456844 −0.228422 0.973562i \(-0.573357\pi\)
−0.228422 + 0.973562i \(0.573357\pi\)
\(108\) 0 0
\(109\) −1.40070e27 −0.437609 −0.218805 0.975769i \(-0.570216\pi\)
−0.218805 + 0.975769i \(0.570216\pi\)
\(110\) 0 0
\(111\) 1.95569e27 0.478014
\(112\) 0 0
\(113\) 1.96674e27 0.377736 0.188868 0.982002i \(-0.439518\pi\)
0.188868 + 0.982002i \(0.439518\pi\)
\(114\) 0 0
\(115\) 1.89788e27 0.287639
\(116\) 0 0
\(117\) −2.58627e26 −0.0310574
\(118\) 0 0
\(119\) −3.64236e28 −3.47935
\(120\) 0 0
\(121\) 3.26138e28 2.48771
\(122\) 0 0
\(123\) 3.03865e27 0.185764
\(124\) 0 0
\(125\) 2.11504e27 0.104000
\(126\) 0 0
\(127\) −3.39337e26 −0.0134673 −0.00673365 0.999977i \(-0.502143\pi\)
−0.00673365 + 0.999977i \(0.502143\pi\)
\(128\) 0 0
\(129\) −2.36793e28 −0.761041
\(130\) 0 0
\(131\) −1.79815e28 −0.469529 −0.234765 0.972052i \(-0.575432\pi\)
−0.234765 + 0.972052i \(0.575432\pi\)
\(132\) 0 0
\(133\) 9.40648e28 2.00184
\(134\) 0 0
\(135\) −1.59237e28 −0.277038
\(136\) 0 0
\(137\) −4.06607e28 −0.580025 −0.290013 0.957023i \(-0.593660\pi\)
−0.290013 + 0.957023i \(0.593660\pi\)
\(138\) 0 0
\(139\) 7.26950e28 0.852716 0.426358 0.904554i \(-0.359796\pi\)
0.426358 + 0.904554i \(0.359796\pi\)
\(140\) 0 0
\(141\) 5.27228e28 0.509965
\(142\) 0 0
\(143\) −2.17567e28 −0.174003
\(144\) 0 0
\(145\) 2.06216e29 1.36727
\(146\) 0 0
\(147\) −2.27156e29 −1.25181
\(148\) 0 0
\(149\) 2.49136e29 1.14399 0.571994 0.820258i \(-0.306170\pi\)
0.571994 + 0.820258i \(0.306170\pi\)
\(150\) 0 0
\(151\) −3.04402e29 −1.16751 −0.583753 0.811931i \(-0.698416\pi\)
−0.583753 + 0.811931i \(0.698416\pi\)
\(152\) 0 0
\(153\) 2.02911e29 0.651584
\(154\) 0 0
\(155\) 4.74859e28 0.127964
\(156\) 0 0
\(157\) 1.02275e29 0.231806 0.115903 0.993261i \(-0.463024\pi\)
0.115903 + 0.993261i \(0.463024\pi\)
\(158\) 0 0
\(159\) −2.30396e29 −0.440163
\(160\) 0 0
\(161\) −2.20386e29 −0.355659
\(162\) 0 0
\(163\) 9.82757e29 1.34250 0.671249 0.741232i \(-0.265758\pi\)
0.671249 + 0.741232i \(0.265758\pi\)
\(164\) 0 0
\(165\) −1.33956e30 −1.55214
\(166\) 0 0
\(167\) −8.93485e29 −0.879863 −0.439931 0.898031i \(-0.644997\pi\)
−0.439931 + 0.898031i \(0.644997\pi\)
\(168\) 0 0
\(169\) −1.18218e30 −0.991319
\(170\) 0 0
\(171\) −5.24022e29 −0.374889
\(172\) 0 0
\(173\) 1.18482e30 0.724486 0.362243 0.932084i \(-0.382011\pi\)
0.362243 + 0.932084i \(0.382011\pi\)
\(174\) 0 0
\(175\) −3.64514e30 −1.90854
\(176\) 0 0
\(177\) −1.94439e30 −0.873263
\(178\) 0 0
\(179\) 2.65362e30 1.02406 0.512028 0.858969i \(-0.328894\pi\)
0.512028 + 0.858969i \(0.328894\pi\)
\(180\) 0 0
\(181\) 3.09227e30 1.02711 0.513555 0.858057i \(-0.328328\pi\)
0.513555 + 0.858057i \(0.328328\pi\)
\(182\) 0 0
\(183\) 1.54827e30 0.443360
\(184\) 0 0
\(185\) −4.81992e30 −1.19185
\(186\) 0 0
\(187\) 1.70696e31 3.65058
\(188\) 0 0
\(189\) 1.84909e30 0.342551
\(190\) 0 0
\(191\) 7.72343e30 1.24125 0.620625 0.784107i \(-0.286879\pi\)
0.620625 + 0.784107i \(0.286879\pi\)
\(192\) 0 0
\(193\) 2.36421e30 0.330113 0.165056 0.986284i \(-0.447219\pi\)
0.165056 + 0.986284i \(0.447219\pi\)
\(194\) 0 0
\(195\) 6.37402e29 0.0774366
\(196\) 0 0
\(197\) 1.16045e31 1.22838 0.614192 0.789157i \(-0.289482\pi\)
0.614192 + 0.789157i \(0.289482\pi\)
\(198\) 0 0
\(199\) 1.70480e31 1.57455 0.787274 0.616604i \(-0.211492\pi\)
0.787274 + 0.616604i \(0.211492\pi\)
\(200\) 0 0
\(201\) −1.19364e30 −0.0963221
\(202\) 0 0
\(203\) −2.39462e31 −1.69060
\(204\) 0 0
\(205\) −7.48893e30 −0.463172
\(206\) 0 0
\(207\) 1.22774e30 0.0666050
\(208\) 0 0
\(209\) −4.40827e31 −2.10036
\(210\) 0 0
\(211\) −2.89826e31 −1.21429 −0.607147 0.794590i \(-0.707686\pi\)
−0.607147 + 0.794590i \(0.707686\pi\)
\(212\) 0 0
\(213\) 2.67850e31 0.988022
\(214\) 0 0
\(215\) 5.83591e31 1.89753
\(216\) 0 0
\(217\) −5.51416e30 −0.158224
\(218\) 0 0
\(219\) −1.00432e31 −0.254610
\(220\) 0 0
\(221\) −8.12221e30 −0.182129
\(222\) 0 0
\(223\) 4.44673e30 0.0882928 0.0441464 0.999025i \(-0.485943\pi\)
0.0441464 + 0.999025i \(0.485943\pi\)
\(224\) 0 0
\(225\) 2.03066e31 0.357415
\(226\) 0 0
\(227\) 4.46381e31 0.697198 0.348599 0.937272i \(-0.386657\pi\)
0.348599 + 0.937272i \(0.386657\pi\)
\(228\) 0 0
\(229\) 6.31550e31 0.876251 0.438126 0.898914i \(-0.355643\pi\)
0.438126 + 0.898914i \(0.355643\pi\)
\(230\) 0 0
\(231\) 1.55553e32 1.91918
\(232\) 0 0
\(233\) 1.56723e32 1.72119 0.860594 0.509291i \(-0.170092\pi\)
0.860594 + 0.509291i \(0.170092\pi\)
\(234\) 0 0
\(235\) −1.29939e32 −1.27152
\(236\) 0 0
\(237\) 8.88707e31 0.775632
\(238\) 0 0
\(239\) −6.78134e31 −0.528377 −0.264188 0.964471i \(-0.585104\pi\)
−0.264188 + 0.964471i \(0.585104\pi\)
\(240\) 0 0
\(241\) −6.75257e31 −0.470153 −0.235076 0.971977i \(-0.575534\pi\)
−0.235076 + 0.971977i \(0.575534\pi\)
\(242\) 0 0
\(243\) −1.03011e31 −0.0641500
\(244\) 0 0
\(245\) 5.59838e32 3.12119
\(246\) 0 0
\(247\) 2.09758e31 0.104788
\(248\) 0 0
\(249\) −1.81219e32 −0.811921
\(250\) 0 0
\(251\) 5.61236e30 0.0225710 0.0112855 0.999936i \(-0.496408\pi\)
0.0112855 + 0.999936i \(0.496408\pi\)
\(252\) 0 0
\(253\) 1.03282e32 0.373162
\(254\) 0 0
\(255\) −5.00086e32 −1.62462
\(256\) 0 0
\(257\) 1.97027e32 0.576008 0.288004 0.957629i \(-0.407008\pi\)
0.288004 + 0.957629i \(0.407008\pi\)
\(258\) 0 0
\(259\) 5.59699e32 1.47369
\(260\) 0 0
\(261\) 1.33401e32 0.316601
\(262\) 0 0
\(263\) 3.79991e31 0.0813525 0.0406763 0.999172i \(-0.487049\pi\)
0.0406763 + 0.999172i \(0.487049\pi\)
\(264\) 0 0
\(265\) 5.67825e32 1.09748
\(266\) 0 0
\(267\) 2.15927e32 0.377056
\(268\) 0 0
\(269\) −7.38156e31 −0.116545 −0.0582723 0.998301i \(-0.518559\pi\)
−0.0582723 + 0.998301i \(0.518559\pi\)
\(270\) 0 0
\(271\) 5.37105e32 0.767316 0.383658 0.923475i \(-0.374664\pi\)
0.383658 + 0.923475i \(0.374664\pi\)
\(272\) 0 0
\(273\) −7.40164e31 −0.0957485
\(274\) 0 0
\(275\) 1.70827e33 2.00246
\(276\) 0 0
\(277\) −4.00236e32 −0.425441 −0.212721 0.977113i \(-0.568232\pi\)
−0.212721 + 0.977113i \(0.568232\pi\)
\(278\) 0 0
\(279\) 3.07186e31 0.0296309
\(280\) 0 0
\(281\) 2.28381e32 0.200044 0.100022 0.994985i \(-0.468109\pi\)
0.100022 + 0.994985i \(0.468109\pi\)
\(282\) 0 0
\(283\) −1.98592e32 −0.158068 −0.0790342 0.996872i \(-0.525184\pi\)
−0.0790342 + 0.996872i \(0.525184\pi\)
\(284\) 0 0
\(285\) 1.29148e33 0.934724
\(286\) 0 0
\(287\) 8.69629e32 0.572701
\(288\) 0 0
\(289\) 4.70472e33 2.82106
\(290\) 0 0
\(291\) 1.19485e33 0.652766
\(292\) 0 0
\(293\) 3.08347e32 0.153577 0.0767885 0.997047i \(-0.475533\pi\)
0.0767885 + 0.997047i \(0.475533\pi\)
\(294\) 0 0
\(295\) 4.79208e33 2.17734
\(296\) 0 0
\(297\) −8.66563e32 −0.359408
\(298\) 0 0
\(299\) −4.91445e31 −0.0186172
\(300\) 0 0
\(301\) −6.77677e33 −2.34625
\(302\) 0 0
\(303\) −3.88675e32 −0.123058
\(304\) 0 0
\(305\) −3.81580e33 −1.10545
\(306\) 0 0
\(307\) 4.04200e33 1.07208 0.536042 0.844192i \(-0.319919\pi\)
0.536042 + 0.844192i \(0.319919\pi\)
\(308\) 0 0
\(309\) 6.81121e32 0.165495
\(310\) 0 0
\(311\) −6.83389e32 −0.152196 −0.0760979 0.997100i \(-0.524246\pi\)
−0.0760979 + 0.997100i \(0.524246\pi\)
\(312\) 0 0
\(313\) 2.56142e33 0.523157 0.261579 0.965182i \(-0.415757\pi\)
0.261579 + 0.965182i \(0.415757\pi\)
\(314\) 0 0
\(315\) −4.55720e33 −0.854094
\(316\) 0 0
\(317\) −1.32333e33 −0.227703 −0.113852 0.993498i \(-0.536319\pi\)
−0.113852 + 0.993498i \(0.536319\pi\)
\(318\) 0 0
\(319\) 1.12222e34 1.77379
\(320\) 0 0
\(321\) 1.81562e33 0.263759
\(322\) 0 0
\(323\) −1.64570e34 −2.19844
\(324\) 0 0
\(325\) −8.12842e32 −0.0999035
\(326\) 0 0
\(327\) 2.23316e33 0.252654
\(328\) 0 0
\(329\) 1.50887e34 1.57220
\(330\) 0 0
\(331\) 1.16843e34 1.12182 0.560911 0.827876i \(-0.310451\pi\)
0.560911 + 0.827876i \(0.310451\pi\)
\(332\) 0 0
\(333\) −3.11801e33 −0.275981
\(334\) 0 0
\(335\) 2.94179e33 0.240164
\(336\) 0 0
\(337\) −1.02074e34 −0.768973 −0.384487 0.923131i \(-0.625622\pi\)
−0.384487 + 0.923131i \(0.625622\pi\)
\(338\) 0 0
\(339\) −3.13562e33 −0.218086
\(340\) 0 0
\(341\) 2.58416e33 0.166011
\(342\) 0 0
\(343\) −3.50264e34 −2.07934
\(344\) 0 0
\(345\) −3.02584e33 −0.166069
\(346\) 0 0
\(347\) 9.21786e33 0.467931 0.233966 0.972245i \(-0.424830\pi\)
0.233966 + 0.972245i \(0.424830\pi\)
\(348\) 0 0
\(349\) −2.76261e34 −1.29771 −0.648853 0.760914i \(-0.724751\pi\)
−0.648853 + 0.760914i \(0.724751\pi\)
\(350\) 0 0
\(351\) 4.12335e32 0.0179310
\(352\) 0 0
\(353\) 3.06885e34 1.23600 0.617998 0.786180i \(-0.287944\pi\)
0.617998 + 0.786180i \(0.287944\pi\)
\(354\) 0 0
\(355\) −6.60132e34 −2.46347
\(356\) 0 0
\(357\) 5.80709e34 2.00881
\(358\) 0 0
\(359\) 4.08395e34 1.31010 0.655051 0.755585i \(-0.272647\pi\)
0.655051 + 0.755585i \(0.272647\pi\)
\(360\) 0 0
\(361\) 8.89991e33 0.264873
\(362\) 0 0
\(363\) −5.19970e34 −1.43628
\(364\) 0 0
\(365\) 2.47520e34 0.634830
\(366\) 0 0
\(367\) 4.56480e34 1.08750 0.543751 0.839247i \(-0.317004\pi\)
0.543751 + 0.839247i \(0.317004\pi\)
\(368\) 0 0
\(369\) −4.84458e33 −0.107251
\(370\) 0 0
\(371\) −6.59369e34 −1.35700
\(372\) 0 0
\(373\) −1.98822e34 −0.380536 −0.190268 0.981732i \(-0.560936\pi\)
−0.190268 + 0.981732i \(0.560936\pi\)
\(374\) 0 0
\(375\) −3.37206e33 −0.0600445
\(376\) 0 0
\(377\) −5.33983e33 −0.0884952
\(378\) 0 0
\(379\) 7.87808e34 1.21560 0.607801 0.794090i \(-0.292052\pi\)
0.607801 + 0.794090i \(0.292052\pi\)
\(380\) 0 0
\(381\) 5.41013e32 0.00777535
\(382\) 0 0
\(383\) −9.13886e34 −1.22380 −0.611898 0.790937i \(-0.709594\pi\)
−0.611898 + 0.790937i \(0.709594\pi\)
\(384\) 0 0
\(385\) −3.83369e35 −4.78517
\(386\) 0 0
\(387\) 3.77525e34 0.439387
\(388\) 0 0
\(389\) −6.28351e34 −0.682154 −0.341077 0.940035i \(-0.610792\pi\)
−0.341077 + 0.940035i \(0.610792\pi\)
\(390\) 0 0
\(391\) 3.85573e34 0.390589
\(392\) 0 0
\(393\) 2.86683e34 0.271083
\(394\) 0 0
\(395\) −2.19027e35 −1.93391
\(396\) 0 0
\(397\) 7.07826e34 0.583787 0.291894 0.956451i \(-0.405715\pi\)
0.291894 + 0.956451i \(0.405715\pi\)
\(398\) 0 0
\(399\) −1.49970e35 −1.15576
\(400\) 0 0
\(401\) −3.30509e34 −0.238087 −0.119043 0.992889i \(-0.537983\pi\)
−0.119043 + 0.992889i \(0.537983\pi\)
\(402\) 0 0
\(403\) −1.22962e33 −0.00828233
\(404\) 0 0
\(405\) 2.53876e34 0.159948
\(406\) 0 0
\(407\) −2.62298e35 −1.54622
\(408\) 0 0
\(409\) −1.27557e35 −0.703784 −0.351892 0.936041i \(-0.614462\pi\)
−0.351892 + 0.936041i \(0.614462\pi\)
\(410\) 0 0
\(411\) 6.48262e34 0.334878
\(412\) 0 0
\(413\) −5.56465e35 −2.69223
\(414\) 0 0
\(415\) 4.46626e35 2.02439
\(416\) 0 0
\(417\) −1.15899e35 −0.492316
\(418\) 0 0
\(419\) 3.81380e35 1.51869 0.759343 0.650690i \(-0.225520\pi\)
0.759343 + 0.650690i \(0.225520\pi\)
\(420\) 0 0
\(421\) 3.74630e35 1.39892 0.699462 0.714670i \(-0.253423\pi\)
0.699462 + 0.714670i \(0.253423\pi\)
\(422\) 0 0
\(423\) −8.40572e34 −0.294429
\(424\) 0 0
\(425\) 6.37730e35 2.09598
\(426\) 0 0
\(427\) 4.43098e35 1.36686
\(428\) 0 0
\(429\) 3.46872e34 0.100461
\(430\) 0 0
\(431\) −8.21144e34 −0.223345 −0.111673 0.993745i \(-0.535621\pi\)
−0.111673 + 0.993745i \(0.535621\pi\)
\(432\) 0 0
\(433\) −7.41271e35 −1.89405 −0.947023 0.321166i \(-0.895925\pi\)
−0.947023 + 0.321166i \(0.895925\pi\)
\(434\) 0 0
\(435\) −3.28775e35 −0.789393
\(436\) 0 0
\(437\) −9.95751e34 −0.224725
\(438\) 0 0
\(439\) 2.99056e35 0.634574 0.317287 0.948330i \(-0.397228\pi\)
0.317287 + 0.948330i \(0.397228\pi\)
\(440\) 0 0
\(441\) 3.62159e35 0.722734
\(442\) 0 0
\(443\) −5.83039e34 −0.109458 −0.0547290 0.998501i \(-0.517429\pi\)
−0.0547290 + 0.998501i \(0.517429\pi\)
\(444\) 0 0
\(445\) −5.32166e35 −0.940128
\(446\) 0 0
\(447\) −3.97203e35 −0.660481
\(448\) 0 0
\(449\) 8.82420e34 0.138150 0.0690748 0.997611i \(-0.477995\pi\)
0.0690748 + 0.997611i \(0.477995\pi\)
\(450\) 0 0
\(451\) −4.07545e35 −0.600886
\(452\) 0 0
\(453\) 4.85316e35 0.674060
\(454\) 0 0
\(455\) 1.82418e35 0.238734
\(456\) 0 0
\(457\) 3.17142e35 0.391187 0.195594 0.980685i \(-0.437337\pi\)
0.195594 + 0.980685i \(0.437337\pi\)
\(458\) 0 0
\(459\) −3.23505e35 −0.376192
\(460\) 0 0
\(461\) 1.38230e36 1.51579 0.757894 0.652377i \(-0.226228\pi\)
0.757894 + 0.652377i \(0.226228\pi\)
\(462\) 0 0
\(463\) 1.00795e35 0.104255 0.0521275 0.998640i \(-0.483400\pi\)
0.0521275 + 0.998640i \(0.483400\pi\)
\(464\) 0 0
\(465\) −7.57079e34 −0.0738799
\(466\) 0 0
\(467\) −2.32864e35 −0.214449 −0.107225 0.994235i \(-0.534196\pi\)
−0.107225 + 0.994235i \(0.534196\pi\)
\(468\) 0 0
\(469\) −3.41607e35 −0.296957
\(470\) 0 0
\(471\) −1.63059e35 −0.133833
\(472\) 0 0
\(473\) 3.17588e36 2.46172
\(474\) 0 0
\(475\) −1.64695e36 −1.20592
\(476\) 0 0
\(477\) 3.67326e35 0.254128
\(478\) 0 0
\(479\) 2.90344e35 0.189838 0.0949189 0.995485i \(-0.469741\pi\)
0.0949189 + 0.995485i \(0.469741\pi\)
\(480\) 0 0
\(481\) 1.24809e35 0.0771414
\(482\) 0 0
\(483\) 3.51366e35 0.205340
\(484\) 0 0
\(485\) −2.94478e36 −1.62757
\(486\) 0 0
\(487\) 2.05345e36 1.07360 0.536802 0.843708i \(-0.319632\pi\)
0.536802 + 0.843708i \(0.319632\pi\)
\(488\) 0 0
\(489\) −1.56683e36 −0.775091
\(490\) 0 0
\(491\) 3.18417e35 0.149072 0.0745359 0.997218i \(-0.476252\pi\)
0.0745359 + 0.997218i \(0.476252\pi\)
\(492\) 0 0
\(493\) 4.18947e36 1.85663
\(494\) 0 0
\(495\) 2.13570e36 0.896127
\(496\) 0 0
\(497\) 7.66558e36 3.04603
\(498\) 0 0
\(499\) 4.61143e36 1.73571 0.867857 0.496814i \(-0.165497\pi\)
0.867857 + 0.496814i \(0.165497\pi\)
\(500\) 0 0
\(501\) 1.42450e36 0.507989
\(502\) 0 0
\(503\) 7.31010e35 0.247033 0.123517 0.992343i \(-0.460583\pi\)
0.123517 + 0.992343i \(0.460583\pi\)
\(504\) 0 0
\(505\) 9.57913e35 0.306826
\(506\) 0 0
\(507\) 1.88478e36 0.572338
\(508\) 0 0
\(509\) −5.01805e36 −1.44492 −0.722462 0.691411i \(-0.756989\pi\)
−0.722462 + 0.691411i \(0.756989\pi\)
\(510\) 0 0
\(511\) −2.87425e36 −0.784952
\(512\) 0 0
\(513\) 8.35461e35 0.216442
\(514\) 0 0
\(515\) −1.67866e36 −0.412635
\(516\) 0 0
\(517\) −7.07121e36 −1.64957
\(518\) 0 0
\(519\) −1.88899e36 −0.418282
\(520\) 0 0
\(521\) −8.47455e36 −1.78159 −0.890795 0.454406i \(-0.849852\pi\)
−0.890795 + 0.454406i \(0.849852\pi\)
\(522\) 0 0
\(523\) −3.12621e36 −0.624087 −0.312043 0.950068i \(-0.601013\pi\)
−0.312043 + 0.950068i \(0.601013\pi\)
\(524\) 0 0
\(525\) 5.81154e36 1.10190
\(526\) 0 0
\(527\) 9.64720e35 0.173763
\(528\) 0 0
\(529\) −5.60992e36 −0.960074
\(530\) 0 0
\(531\) 3.09999e36 0.504179
\(532\) 0 0
\(533\) 1.93921e35 0.0299784
\(534\) 0 0
\(535\) −4.47471e36 −0.657641
\(536\) 0 0
\(537\) −4.23074e36 −0.591239
\(538\) 0 0
\(539\) 3.04662e37 4.04921
\(540\) 0 0
\(541\) 5.64527e36 0.713710 0.356855 0.934160i \(-0.383849\pi\)
0.356855 + 0.934160i \(0.383849\pi\)
\(542\) 0 0
\(543\) −4.93008e36 −0.593002
\(544\) 0 0
\(545\) −5.50376e36 −0.629951
\(546\) 0 0
\(547\) 7.57035e36 0.824683 0.412342 0.911029i \(-0.364711\pi\)
0.412342 + 0.911029i \(0.364711\pi\)
\(548\) 0 0
\(549\) −2.46844e36 −0.255974
\(550\) 0 0
\(551\) −1.08194e37 −1.06821
\(552\) 0 0
\(553\) 2.54339e37 2.39124
\(554\) 0 0
\(555\) 7.68451e36 0.688115
\(556\) 0 0
\(557\) 2.24392e37 1.91409 0.957044 0.289944i \(-0.0936366\pi\)
0.957044 + 0.289944i \(0.0936366\pi\)
\(558\) 0 0
\(559\) −1.51117e36 −0.122816
\(560\) 0 0
\(561\) −2.72145e37 −2.10766
\(562\) 0 0
\(563\) −7.70935e36 −0.569055 −0.284528 0.958668i \(-0.591837\pi\)
−0.284528 + 0.958668i \(0.591837\pi\)
\(564\) 0 0
\(565\) 7.72793e36 0.543763
\(566\) 0 0
\(567\) −2.94805e36 −0.197772
\(568\) 0 0
\(569\) −6.03687e36 −0.386186 −0.193093 0.981181i \(-0.561852\pi\)
−0.193093 + 0.981181i \(0.561852\pi\)
\(570\) 0 0
\(571\) −2.07491e37 −1.26594 −0.632968 0.774178i \(-0.718164\pi\)
−0.632968 + 0.774178i \(0.718164\pi\)
\(572\) 0 0
\(573\) −1.23136e37 −0.716636
\(574\) 0 0
\(575\) 3.85867e36 0.214251
\(576\) 0 0
\(577\) 4.36983e36 0.231521 0.115761 0.993277i \(-0.463069\pi\)
0.115761 + 0.993277i \(0.463069\pi\)
\(578\) 0 0
\(579\) −3.76931e36 −0.190591
\(580\) 0 0
\(581\) −5.18631e37 −2.50311
\(582\) 0 0
\(583\) 3.09008e37 1.42379
\(584\) 0 0
\(585\) −1.01623e36 −0.0447080
\(586\) 0 0
\(587\) −8.18673e36 −0.343951 −0.171975 0.985101i \(-0.555015\pi\)
−0.171975 + 0.985101i \(0.555015\pi\)
\(588\) 0 0
\(589\) −2.49142e36 −0.0999746
\(590\) 0 0
\(591\) −1.85014e37 −0.709208
\(592\) 0 0
\(593\) −1.10108e37 −0.403254 −0.201627 0.979462i \(-0.564623\pi\)
−0.201627 + 0.979462i \(0.564623\pi\)
\(594\) 0 0
\(595\) −1.43119e38 −5.00863
\(596\) 0 0
\(597\) −2.71799e37 −0.909065
\(598\) 0 0
\(599\) −4.15783e37 −1.32924 −0.664622 0.747179i \(-0.731408\pi\)
−0.664622 + 0.747179i \(0.731408\pi\)
\(600\) 0 0
\(601\) −3.10857e37 −0.950070 −0.475035 0.879967i \(-0.657565\pi\)
−0.475035 + 0.879967i \(0.657565\pi\)
\(602\) 0 0
\(603\) 1.90305e36 0.0556116
\(604\) 0 0
\(605\) 1.28150e38 3.58113
\(606\) 0 0
\(607\) 1.57503e37 0.420960 0.210480 0.977598i \(-0.432497\pi\)
0.210480 + 0.977598i \(0.432497\pi\)
\(608\) 0 0
\(609\) 3.81779e37 0.976066
\(610\) 0 0
\(611\) 3.36468e36 0.0822977
\(612\) 0 0
\(613\) 2.28357e37 0.534438 0.267219 0.963636i \(-0.413895\pi\)
0.267219 + 0.963636i \(0.413895\pi\)
\(614\) 0 0
\(615\) 1.19398e37 0.267413
\(616\) 0 0
\(617\) −2.21751e37 −0.475353 −0.237676 0.971344i \(-0.576386\pi\)
−0.237676 + 0.971344i \(0.576386\pi\)
\(618\) 0 0
\(619\) −6.90120e37 −1.41612 −0.708061 0.706151i \(-0.750430\pi\)
−0.708061 + 0.706151i \(0.750430\pi\)
\(620\) 0 0
\(621\) −1.95741e36 −0.0384544
\(622\) 0 0
\(623\) 6.17962e37 1.16245
\(624\) 0 0
\(625\) −5.12110e37 −0.922535
\(626\) 0 0
\(627\) 7.02821e37 1.21264
\(628\) 0 0
\(629\) −9.79212e37 −1.61843
\(630\) 0 0
\(631\) −3.32303e37 −0.526184 −0.263092 0.964771i \(-0.584742\pi\)
−0.263092 + 0.964771i \(0.584742\pi\)
\(632\) 0 0
\(633\) 4.62076e37 0.701072
\(634\) 0 0
\(635\) −1.33336e36 −0.0193866
\(636\) 0 0
\(637\) −1.44967e37 −0.202016
\(638\) 0 0
\(639\) −4.27039e37 −0.570435
\(640\) 0 0
\(641\) −9.61681e37 −1.23154 −0.615769 0.787927i \(-0.711154\pi\)
−0.615769 + 0.787927i \(0.711154\pi\)
\(642\) 0 0
\(643\) 6.93703e37 0.851776 0.425888 0.904776i \(-0.359962\pi\)
0.425888 + 0.904776i \(0.359962\pi\)
\(644\) 0 0
\(645\) −9.30432e37 −1.09554
\(646\) 0 0
\(647\) 1.21489e38 1.37192 0.685958 0.727641i \(-0.259383\pi\)
0.685958 + 0.727641i \(0.259383\pi\)
\(648\) 0 0
\(649\) 2.60783e38 2.82472
\(650\) 0 0
\(651\) 8.79135e36 0.0913507
\(652\) 0 0
\(653\) −1.01371e38 −1.01062 −0.505308 0.862939i \(-0.668621\pi\)
−0.505308 + 0.862939i \(0.668621\pi\)
\(654\) 0 0
\(655\) −7.06547e37 −0.675902
\(656\) 0 0
\(657\) 1.60121e37 0.146999
\(658\) 0 0
\(659\) 1.87758e38 1.65442 0.827208 0.561896i \(-0.189928\pi\)
0.827208 + 0.561896i \(0.189928\pi\)
\(660\) 0 0
\(661\) −1.84502e38 −1.56056 −0.780281 0.625430i \(-0.784924\pi\)
−0.780281 + 0.625430i \(0.784924\pi\)
\(662\) 0 0
\(663\) 1.29494e37 0.105152
\(664\) 0 0
\(665\) 3.69609e38 2.88171
\(666\) 0 0
\(667\) 2.53489e37 0.189785
\(668\) 0 0
\(669\) −7.08953e36 −0.0509759
\(670\) 0 0
\(671\) −2.07654e38 −1.43413
\(672\) 0 0
\(673\) −2.94215e38 −1.95192 −0.975960 0.217950i \(-0.930063\pi\)
−0.975960 + 0.217950i \(0.930063\pi\)
\(674\) 0 0
\(675\) −3.23753e37 −0.206354
\(676\) 0 0
\(677\) 2.45383e38 1.50279 0.751395 0.659853i \(-0.229381\pi\)
0.751395 + 0.659853i \(0.229381\pi\)
\(678\) 0 0
\(679\) 3.41953e38 2.01245
\(680\) 0 0
\(681\) −7.11675e37 −0.402528
\(682\) 0 0
\(683\) 1.45172e38 0.789230 0.394615 0.918847i \(-0.370878\pi\)
0.394615 + 0.918847i \(0.370878\pi\)
\(684\) 0 0
\(685\) −1.59768e38 −0.834964
\(686\) 0 0
\(687\) −1.00689e38 −0.505904
\(688\) 0 0
\(689\) −1.47035e37 −0.0710331
\(690\) 0 0
\(691\) −5.22201e37 −0.242596 −0.121298 0.992616i \(-0.538706\pi\)
−0.121298 + 0.992616i \(0.538706\pi\)
\(692\) 0 0
\(693\) −2.48001e38 −1.10804
\(694\) 0 0
\(695\) 2.85641e38 1.22751
\(696\) 0 0
\(697\) −1.52144e38 −0.628947
\(698\) 0 0
\(699\) −2.49867e38 −0.993729
\(700\) 0 0
\(701\) 4.70202e38 1.79925 0.899624 0.436665i \(-0.143840\pi\)
0.899624 + 0.436665i \(0.143840\pi\)
\(702\) 0 0
\(703\) 2.52884e38 0.931161
\(704\) 0 0
\(705\) 2.07164e38 0.734110
\(706\) 0 0
\(707\) −1.11235e38 −0.379383
\(708\) 0 0
\(709\) 2.24631e38 0.737471 0.368735 0.929534i \(-0.379791\pi\)
0.368735 + 0.929534i \(0.379791\pi\)
\(710\) 0 0
\(711\) −1.41689e38 −0.447812
\(712\) 0 0
\(713\) 5.83717e36 0.0177621
\(714\) 0 0
\(715\) −8.54887e37 −0.250482
\(716\) 0 0
\(717\) 1.08116e38 0.305058
\(718\) 0 0
\(719\) −1.19163e38 −0.323819 −0.161909 0.986806i \(-0.551765\pi\)
−0.161909 + 0.986806i \(0.551765\pi\)
\(720\) 0 0
\(721\) 1.94930e38 0.510214
\(722\) 0 0
\(723\) 1.07658e38 0.271443
\(724\) 0 0
\(725\) 4.19267e38 1.01842
\(726\) 0 0
\(727\) 3.10009e38 0.725536 0.362768 0.931879i \(-0.381832\pi\)
0.362768 + 0.931879i \(0.381832\pi\)
\(728\) 0 0
\(729\) 1.64232e37 0.0370370
\(730\) 0 0
\(731\) 1.18562e39 2.57668
\(732\) 0 0
\(733\) −1.56749e38 −0.328323 −0.164162 0.986433i \(-0.552492\pi\)
−0.164162 + 0.986433i \(0.552492\pi\)
\(734\) 0 0
\(735\) −8.92563e38 −1.80202
\(736\) 0 0
\(737\) 1.60091e38 0.311571
\(738\) 0 0
\(739\) −8.84062e38 −1.65876 −0.829378 0.558688i \(-0.811305\pi\)
−0.829378 + 0.558688i \(0.811305\pi\)
\(740\) 0 0
\(741\) −3.34422e37 −0.0604991
\(742\) 0 0
\(743\) 5.30279e36 0.00925030 0.00462515 0.999989i \(-0.498528\pi\)
0.00462515 + 0.999989i \(0.498528\pi\)
\(744\) 0 0
\(745\) 9.78930e38 1.64680
\(746\) 0 0
\(747\) 2.88922e38 0.468763
\(748\) 0 0
\(749\) 5.19612e38 0.813158
\(750\) 0 0
\(751\) 9.49178e38 1.43288 0.716438 0.697651i \(-0.245771\pi\)
0.716438 + 0.697651i \(0.245771\pi\)
\(752\) 0 0
\(753\) −8.94791e36 −0.0130314
\(754\) 0 0
\(755\) −1.19609e39 −1.68066
\(756\) 0 0
\(757\) 4.83283e37 0.0655250 0.0327625 0.999463i \(-0.489570\pi\)
0.0327625 + 0.999463i \(0.489570\pi\)
\(758\) 0 0
\(759\) −1.64665e38 −0.215445
\(760\) 0 0
\(761\) 4.16452e38 0.525862 0.262931 0.964815i \(-0.415311\pi\)
0.262931 + 0.964815i \(0.415311\pi\)
\(762\) 0 0
\(763\) 6.39108e38 0.778920
\(764\) 0 0
\(765\) 7.97298e38 0.937975
\(766\) 0 0
\(767\) −1.24088e38 −0.140926
\(768\) 0 0
\(769\) −1.60444e39 −1.75921 −0.879606 0.475703i \(-0.842194\pi\)
−0.879606 + 0.475703i \(0.842194\pi\)
\(770\) 0 0
\(771\) −3.14125e38 −0.332559
\(772\) 0 0
\(773\) 1.24479e39 1.27254 0.636270 0.771466i \(-0.280476\pi\)
0.636270 + 0.771466i \(0.280476\pi\)
\(774\) 0 0
\(775\) 9.65458e37 0.0953148
\(776\) 0 0
\(777\) −8.92341e38 −0.850838
\(778\) 0 0
\(779\) 3.92917e38 0.361864
\(780\) 0 0
\(781\) −3.59241e39 −3.19593
\(782\) 0 0
\(783\) −2.12684e38 −0.182790
\(784\) 0 0
\(785\) 4.01870e38 0.333691
\(786\) 0 0
\(787\) 1.02898e38 0.0825557 0.0412779 0.999148i \(-0.486857\pi\)
0.0412779 + 0.999148i \(0.486857\pi\)
\(788\) 0 0
\(789\) −6.05829e37 −0.0469689
\(790\) 0 0
\(791\) −8.97383e38 −0.672350
\(792\) 0 0
\(793\) 9.88079e37 0.0715490
\(794\) 0 0
\(795\) −9.05296e38 −0.633628
\(796\) 0 0
\(797\) −1.10350e39 −0.746596 −0.373298 0.927711i \(-0.621773\pi\)
−0.373298 + 0.927711i \(0.621773\pi\)
\(798\) 0 0
\(799\) −2.63982e39 −1.72661
\(800\) 0 0
\(801\) −3.44258e38 −0.217693
\(802\) 0 0
\(803\) 1.34700e39 0.823582
\(804\) 0 0
\(805\) −8.65963e38 −0.511982
\(806\) 0 0
\(807\) 1.17686e38 0.0672871
\(808\) 0 0
\(809\) −3.55425e38 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(810\) 0 0
\(811\) −5.26124e38 −0.281389 −0.140694 0.990053i \(-0.544933\pi\)
−0.140694 + 0.990053i \(0.544933\pi\)
\(812\) 0 0
\(813\) −8.56319e38 −0.443010
\(814\) 0 0
\(815\) 3.86155e39 1.93257
\(816\) 0 0
\(817\) −3.06189e39 −1.48249
\(818\) 0 0
\(819\) 1.18006e38 0.0552804
\(820\) 0 0
\(821\) −1.61146e39 −0.730445 −0.365222 0.930920i \(-0.619007\pi\)
−0.365222 + 0.930920i \(0.619007\pi\)
\(822\) 0 0
\(823\) 7.83371e38 0.343613 0.171807 0.985131i \(-0.445040\pi\)
0.171807 + 0.985131i \(0.445040\pi\)
\(824\) 0 0
\(825\) −2.72353e39 −1.15612
\(826\) 0 0
\(827\) 3.84813e39 1.58098 0.790488 0.612477i \(-0.209827\pi\)
0.790488 + 0.612477i \(0.209827\pi\)
\(828\) 0 0
\(829\) −1.78227e39 −0.708743 −0.354371 0.935105i \(-0.615305\pi\)
−0.354371 + 0.935105i \(0.615305\pi\)
\(830\) 0 0
\(831\) 6.38106e38 0.245628
\(832\) 0 0
\(833\) 1.13736e40 4.23830
\(834\) 0 0
\(835\) −3.51078e39 −1.26659
\(836\) 0 0
\(837\) −4.89754e37 −0.0171074
\(838\) 0 0
\(839\) 3.30219e39 1.11690 0.558451 0.829538i \(-0.311396\pi\)
0.558451 + 0.829538i \(0.311396\pi\)
\(840\) 0 0
\(841\) −2.98827e38 −0.0978756
\(842\) 0 0
\(843\) −3.64113e38 −0.115495
\(844\) 0 0
\(845\) −4.64515e39 −1.42703
\(846\) 0 0
\(847\) −1.48810e40 −4.42798
\(848\) 0 0
\(849\) 3.16620e38 0.0912608
\(850\) 0 0
\(851\) −5.92486e38 −0.165436
\(852\) 0 0
\(853\) 4.91422e38 0.132936 0.0664680 0.997789i \(-0.478827\pi\)
0.0664680 + 0.997789i \(0.478827\pi\)
\(854\) 0 0
\(855\) −2.05904e39 −0.539663
\(856\) 0 0
\(857\) −1.11566e39 −0.283328 −0.141664 0.989915i \(-0.545245\pi\)
−0.141664 + 0.989915i \(0.545245\pi\)
\(858\) 0 0
\(859\) 6.42063e39 1.58005 0.790023 0.613077i \(-0.210069\pi\)
0.790023 + 0.613077i \(0.210069\pi\)
\(860\) 0 0
\(861\) −1.38647e39 −0.330649
\(862\) 0 0
\(863\) −3.97200e39 −0.918045 −0.459023 0.888425i \(-0.651800\pi\)
−0.459023 + 0.888425i \(0.651800\pi\)
\(864\) 0 0
\(865\) 4.65552e39 1.04292
\(866\) 0 0
\(867\) −7.50084e39 −1.62874
\(868\) 0 0
\(869\) −1.19194e40 −2.50892
\(870\) 0 0
\(871\) −7.61760e37 −0.0155444
\(872\) 0 0
\(873\) −1.90498e39 −0.376875
\(874\) 0 0
\(875\) −9.65048e38 −0.185114
\(876\) 0 0
\(877\) 6.93582e39 1.29004 0.645020 0.764165i \(-0.276849\pi\)
0.645020 + 0.764165i \(0.276849\pi\)
\(878\) 0 0
\(879\) −4.91604e38 −0.0886677
\(880\) 0 0
\(881\) 7.70788e39 1.34822 0.674109 0.738632i \(-0.264528\pi\)
0.674109 + 0.738632i \(0.264528\pi\)
\(882\) 0 0
\(883\) 4.09177e39 0.694131 0.347065 0.937841i \(-0.387178\pi\)
0.347065 + 0.937841i \(0.387178\pi\)
\(884\) 0 0
\(885\) −7.64012e39 −1.25709
\(886\) 0 0
\(887\) 4.75140e39 0.758320 0.379160 0.925331i \(-0.376213\pi\)
0.379160 + 0.925331i \(0.376213\pi\)
\(888\) 0 0
\(889\) 1.54832e38 0.0239711
\(890\) 0 0
\(891\) 1.38158e39 0.207504
\(892\) 0 0
\(893\) 6.81741e39 0.993402
\(894\) 0 0
\(895\) 1.04269e40 1.47416
\(896\) 0 0
\(897\) 7.83523e37 0.0107486
\(898\) 0 0
\(899\) 6.34242e38 0.0844305
\(900\) 0 0
\(901\) 1.15359e40 1.49028
\(902\) 0 0
\(903\) 1.08044e40 1.35461
\(904\) 0 0
\(905\) 1.21505e40 1.47856
\(906\) 0 0
\(907\) −3.50727e39 −0.414258 −0.207129 0.978314i \(-0.566412\pi\)
−0.207129 + 0.978314i \(0.566412\pi\)
\(908\) 0 0
\(909\) 6.19674e38 0.0710478
\(910\) 0 0
\(911\) −1.45346e40 −1.61773 −0.808865 0.587995i \(-0.799918\pi\)
−0.808865 + 0.587995i \(0.799918\pi\)
\(912\) 0 0
\(913\) 2.43052e40 2.62630
\(914\) 0 0
\(915\) 6.08362e39 0.638230
\(916\) 0 0
\(917\) 8.20456e39 0.835736
\(918\) 0 0
\(919\) −1.52616e40 −1.50953 −0.754765 0.655995i \(-0.772249\pi\)
−0.754765 + 0.655995i \(0.772249\pi\)
\(920\) 0 0
\(921\) −6.44426e39 −0.618968
\(922\) 0 0
\(923\) 1.70937e39 0.159446
\(924\) 0 0
\(925\) −9.79961e39 −0.887760
\(926\) 0 0
\(927\) −1.08593e39 −0.0955486
\(928\) 0 0
\(929\) −1.42822e38 −0.0122063 −0.00610314 0.999981i \(-0.501943\pi\)
−0.00610314 + 0.999981i \(0.501943\pi\)
\(930\) 0 0
\(931\) −2.93727e40 −2.43850
\(932\) 0 0
\(933\) 1.08954e39 0.0878703
\(934\) 0 0
\(935\) 6.70717e40 5.25512
\(936\) 0 0
\(937\) −1.74167e40 −1.32581 −0.662907 0.748702i \(-0.730677\pi\)
−0.662907 + 0.748702i \(0.730677\pi\)
\(938\) 0 0
\(939\) −4.08373e39 −0.302045
\(940\) 0 0
\(941\) −1.37643e40 −0.989220 −0.494610 0.869115i \(-0.664689\pi\)
−0.494610 + 0.869115i \(0.664689\pi\)
\(942\) 0 0
\(943\) −9.20571e38 −0.0642909
\(944\) 0 0
\(945\) 7.26565e39 0.493112
\(946\) 0 0
\(947\) −1.36420e40 −0.899817 −0.449909 0.893075i \(-0.648543\pi\)
−0.449909 + 0.893075i \(0.648543\pi\)
\(948\) 0 0
\(949\) −6.40939e38 −0.0410887
\(950\) 0 0
\(951\) 2.10982e39 0.131465
\(952\) 0 0
\(953\) −9.35719e39 −0.566750 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(954\) 0 0
\(955\) 3.03477e40 1.78682
\(956\) 0 0
\(957\) −1.78918e40 −1.02410
\(958\) 0 0
\(959\) 1.85526e40 1.03241
\(960\) 0 0
\(961\) −1.83367e40 −0.992098
\(962\) 0 0
\(963\) −2.89469e39 −0.152281
\(964\) 0 0
\(965\) 9.28969e39 0.475207
\(966\) 0 0
\(967\) −5.99631e39 −0.298282 −0.149141 0.988816i \(-0.547651\pi\)
−0.149141 + 0.988816i \(0.547651\pi\)
\(968\) 0 0
\(969\) 2.62377e40 1.26927
\(970\) 0 0
\(971\) 7.15564e39 0.336658 0.168329 0.985731i \(-0.446163\pi\)
0.168329 + 0.985731i \(0.446163\pi\)
\(972\) 0 0
\(973\) −3.31692e40 −1.51779
\(974\) 0 0
\(975\) 1.29593e39 0.0576793
\(976\) 0 0
\(977\) 1.38682e40 0.600406 0.300203 0.953875i \(-0.402946\pi\)
0.300203 + 0.953875i \(0.402946\pi\)
\(978\) 0 0
\(979\) −2.89603e40 −1.21965
\(980\) 0 0
\(981\) −3.56038e39 −0.145870
\(982\) 0 0
\(983\) −1.79367e40 −0.714941 −0.357470 0.933924i \(-0.616361\pi\)
−0.357470 + 0.933924i \(0.616361\pi\)
\(984\) 0 0
\(985\) 4.55978e40 1.76829
\(986\) 0 0
\(987\) −2.40563e40 −0.907710
\(988\) 0 0
\(989\) 7.17375e39 0.263388
\(990\) 0 0
\(991\) 5.16699e40 1.84605 0.923025 0.384739i \(-0.125708\pi\)
0.923025 + 0.384739i \(0.125708\pi\)
\(992\) 0 0
\(993\) −1.86285e40 −0.647684
\(994\) 0 0
\(995\) 6.69866e40 2.26661
\(996\) 0 0
\(997\) −1.37268e40 −0.452049 −0.226025 0.974122i \(-0.572573\pi\)
−0.226025 + 0.974122i \(0.572573\pi\)
\(998\) 0 0
\(999\) 4.97111e39 0.159338
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 48.28.a.l.1.3 4
4.3 odd 2 24.28.a.d.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.28.a.d.1.3 4 4.3 odd 2
48.28.a.l.1.3 4 1.1 even 1 trivial