Properties

Label 9.54.a.b.1.4
Level $9$
Weight $54$
Character 9.1
Self dual yes
Analytic conductor $160.113$
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] = [9,54,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 54, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 54);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 54 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(160.112796847\)
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: \( x^{4} - x^{3} - 2315873743412x^{2} - 421178019174503472x + 612167648493870378955584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{11}\cdot 5^{3}\cdot 7\cdot 13 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.26509e6\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.38568e8 q^{2} +1.01938e16 q^{4} +5.35900e18 q^{5} +2.55363e22 q^{7} +1.64427e23 q^{8} +O(q^{10})\) \(q+1.38568e8 q^{2} +1.01938e16 q^{4} +5.35900e18 q^{5} +2.55363e22 q^{7} +1.64427e23 q^{8} +7.42585e26 q^{10} -2.25708e27 q^{11} +1.44397e29 q^{13} +3.53851e30 q^{14} -6.90335e31 q^{16} +6.08256e32 q^{17} +5.84282e33 q^{19} +5.46287e34 q^{20} -3.12758e35 q^{22} -9.54096e35 q^{23} +1.76167e37 q^{25} +2.00088e37 q^{26} +2.60313e38 q^{28} +4.38838e37 q^{29} +8.43957e38 q^{31} -1.10468e40 q^{32} +8.42846e40 q^{34} +1.36849e41 q^{35} +1.19964e41 q^{37} +8.09626e41 q^{38} +8.81165e41 q^{40} +8.08650e42 q^{41} +1.66354e43 q^{43} -2.30083e43 q^{44} -1.32207e44 q^{46} +7.16487e42 q^{47} +3.52310e43 q^{49} +2.44110e45 q^{50} +1.47196e45 q^{52} -3.19144e45 q^{53} -1.20957e46 q^{55} +4.19887e45 q^{56} +6.08088e45 q^{58} -3.07882e46 q^{59} -2.46965e47 q^{61} +1.16945e47 q^{62} -9.08937e47 q^{64} +7.73825e47 q^{65} +1.37146e48 q^{67} +6.20045e48 q^{68} +1.89629e49 q^{70} -1.40028e49 q^{71} -1.59560e49 q^{73} +1.66231e49 q^{74} +5.95606e49 q^{76} -5.76375e49 q^{77} -1.73480e50 q^{79} -3.69950e50 q^{80} +1.12053e51 q^{82} +1.32890e51 q^{83} +3.25964e51 q^{85} +2.30513e51 q^{86} -3.71125e50 q^{88} +6.74530e51 q^{89} +3.68737e51 q^{91} -9.72588e51 q^{92} +9.92819e50 q^{94} +3.13117e52 q^{95} +4.44346e52 q^{97} +4.88188e51 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 68476320 q^{2} + 78\!\cdots\!28 q^{4}+ \cdots - 13\!\cdots\!40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 68476320 q^{2} + 78\!\cdots\!28 q^{4}+ \cdots + 88\!\cdots\!40 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.38568e8 1.46005 0.730024 0.683421i \(-0.239509\pi\)
0.730024 + 0.683421i \(0.239509\pi\)
\(3\) 0 0
\(4\) 1.01938e16 1.13174
\(5\) 5.35900e18 1.60834 0.804172 0.594397i \(-0.202609\pi\)
0.804172 + 0.594397i \(0.202609\pi\)
\(6\) 0 0
\(7\) 2.55363e22 1.02816 0.514080 0.857742i \(-0.328134\pi\)
0.514080 + 0.857742i \(0.328134\pi\)
\(8\) 1.64427e23 0.192348
\(9\) 0 0
\(10\) 7.42585e26 2.34826
\(11\) −2.25708e27 −0.571006 −0.285503 0.958378i \(-0.592161\pi\)
−0.285503 + 0.958378i \(0.592161\pi\)
\(12\) 0 0
\(13\) 1.44397e29 0.436576 0.218288 0.975884i \(-0.429953\pi\)
0.218288 + 0.975884i \(0.429953\pi\)
\(14\) 3.53851e30 1.50116
\(15\) 0 0
\(16\) −6.90335e31 −0.850903
\(17\) 6.08256e32 1.50380 0.751899 0.659278i \(-0.229138\pi\)
0.751899 + 0.659278i \(0.229138\pi\)
\(18\) 0 0
\(19\) 5.84282e33 0.757970 0.378985 0.925403i \(-0.376273\pi\)
0.378985 + 0.925403i \(0.376273\pi\)
\(20\) 5.46287e34 1.82023
\(21\) 0 0
\(22\) −3.12758e35 −0.833696
\(23\) −9.54096e35 −0.783077 −0.391538 0.920162i \(-0.628057\pi\)
−0.391538 + 0.920162i \(0.628057\pi\)
\(24\) 0 0
\(25\) 1.76167e37 1.58677
\(26\) 2.00088e37 0.637422
\(27\) 0 0
\(28\) 2.60313e38 1.16361
\(29\) 4.38838e37 0.0774029 0.0387015 0.999251i \(-0.487678\pi\)
0.0387015 + 0.999251i \(0.487678\pi\)
\(30\) 0 0
\(31\) 8.43957e38 0.254235 0.127118 0.991888i \(-0.459427\pi\)
0.127118 + 0.991888i \(0.459427\pi\)
\(32\) −1.10468e40 −1.43471
\(33\) 0 0
\(34\) 8.42846e40 2.19562
\(35\) 1.36849e41 1.65363
\(36\) 0 0
\(37\) 1.19964e41 0.332433 0.166216 0.986089i \(-0.446845\pi\)
0.166216 + 0.986089i \(0.446845\pi\)
\(38\) 8.09626e41 1.10667
\(39\) 0 0
\(40\) 8.81165e41 0.309362
\(41\) 8.08650e42 1.47567 0.737834 0.674983i \(-0.235849\pi\)
0.737834 + 0.674983i \(0.235849\pi\)
\(42\) 0 0
\(43\) 1.66354e43 0.859249 0.429624 0.903008i \(-0.358646\pi\)
0.429624 + 0.903008i \(0.358646\pi\)
\(44\) −2.30083e43 −0.646231
\(45\) 0 0
\(46\) −1.32207e44 −1.14333
\(47\) 7.16487e42 0.0350441 0.0175220 0.999846i \(-0.494422\pi\)
0.0175220 + 0.999846i \(0.494422\pi\)
\(48\) 0 0
\(49\) 3.52310e43 0.0571122
\(50\) 2.44110e45 2.31676
\(51\) 0 0
\(52\) 1.47196e45 0.494091
\(53\) −3.19144e45 −0.646660 −0.323330 0.946286i \(-0.604802\pi\)
−0.323330 + 0.946286i \(0.604802\pi\)
\(54\) 0 0
\(55\) −1.20957e46 −0.918374
\(56\) 4.19887e45 0.197765
\(57\) 0 0
\(58\) 6.08088e45 0.113012
\(59\) −3.07882e46 −0.363752 −0.181876 0.983321i \(-0.558217\pi\)
−0.181876 + 0.983321i \(0.558217\pi\)
\(60\) 0 0
\(61\) −2.46965e47 −1.20613 −0.603066 0.797691i \(-0.706054\pi\)
−0.603066 + 0.797691i \(0.706054\pi\)
\(62\) 1.16945e47 0.371195
\(63\) 0 0
\(64\) −9.08937e47 −1.24384
\(65\) 7.73825e47 0.702164
\(66\) 0 0
\(67\) 1.37146e48 0.557445 0.278723 0.960372i \(-0.410089\pi\)
0.278723 + 0.960372i \(0.410089\pi\)
\(68\) 6.20045e48 1.70191
\(69\) 0 0
\(70\) 1.89629e49 2.41439
\(71\) −1.40028e49 −1.22424 −0.612122 0.790763i \(-0.709684\pi\)
−0.612122 + 0.790763i \(0.709684\pi\)
\(72\) 0 0
\(73\) −1.59560e49 −0.668140 −0.334070 0.942548i \(-0.608422\pi\)
−0.334070 + 0.942548i \(0.608422\pi\)
\(74\) 1.66231e49 0.485368
\(75\) 0 0
\(76\) 5.95606e49 0.857825
\(77\) −5.76375e49 −0.587085
\(78\) 0 0
\(79\) −1.73480e50 −0.895635 −0.447818 0.894125i \(-0.647799\pi\)
−0.447818 + 0.894125i \(0.647799\pi\)
\(80\) −3.69950e50 −1.36854
\(81\) 0 0
\(82\) 1.12053e51 2.15455
\(83\) 1.32890e51 1.85320 0.926600 0.376048i \(-0.122717\pi\)
0.926600 + 0.376048i \(0.122717\pi\)
\(84\) 0 0
\(85\) 3.25964e51 2.41862
\(86\) 2.30513e51 1.25454
\(87\) 0 0
\(88\) −3.71125e50 −0.109832
\(89\) 6.74530e51 1.47967 0.739837 0.672786i \(-0.234902\pi\)
0.739837 + 0.672786i \(0.234902\pi\)
\(90\) 0 0
\(91\) 3.68737e51 0.448870
\(92\) −9.72588e51 −0.886240
\(93\) 0 0
\(94\) 9.92819e50 0.0511661
\(95\) 3.13117e52 1.21908
\(96\) 0 0
\(97\) 4.44346e52 0.996025 0.498012 0.867170i \(-0.334063\pi\)
0.498012 + 0.867170i \(0.334063\pi\)
\(98\) 4.88188e51 0.0833866
\(99\) 0 0
\(100\) 1.79581e53 1.79581
\(101\) −1.33654e53 −1.02675 −0.513377 0.858163i \(-0.671606\pi\)
−0.513377 + 0.858163i \(0.671606\pi\)
\(102\) 0 0
\(103\) 7.98514e52 0.364835 0.182417 0.983221i \(-0.441608\pi\)
0.182417 + 0.983221i \(0.441608\pi\)
\(104\) 2.37428e52 0.0839747
\(105\) 0 0
\(106\) −4.42230e53 −0.944155
\(107\) 1.04658e54 1.74222 0.871108 0.491092i \(-0.163402\pi\)
0.871108 + 0.491092i \(0.163402\pi\)
\(108\) 0 0
\(109\) −5.02767e53 −0.512348 −0.256174 0.966631i \(-0.582462\pi\)
−0.256174 + 0.966631i \(0.582462\pi\)
\(110\) −1.67607e54 −1.34087
\(111\) 0 0
\(112\) −1.76286e54 −0.874864
\(113\) −3.92048e54 −1.53731 −0.768654 0.639665i \(-0.779073\pi\)
−0.768654 + 0.639665i \(0.779073\pi\)
\(114\) 0 0
\(115\) −5.11300e54 −1.25946
\(116\) 4.47344e53 0.0876001
\(117\) 0 0
\(118\) −4.26625e54 −0.531096
\(119\) 1.55326e55 1.54615
\(120\) 0 0
\(121\) −1.05303e55 −0.673952
\(122\) −3.42214e55 −1.76101
\(123\) 0 0
\(124\) 8.60314e54 0.287728
\(125\) 3.49109e55 0.943726
\(126\) 0 0
\(127\) 2.95271e55 0.524110 0.262055 0.965053i \(-0.415600\pi\)
0.262055 + 0.965053i \(0.415600\pi\)
\(128\) −2.64483e55 −0.381359
\(129\) 0 0
\(130\) 1.07227e56 1.02519
\(131\) 1.59227e55 0.124259 0.0621296 0.998068i \(-0.480211\pi\)
0.0621296 + 0.998068i \(0.480211\pi\)
\(132\) 0 0
\(133\) 1.49204e56 0.779314
\(134\) 1.90041e56 0.813897
\(135\) 0 0
\(136\) 1.00014e56 0.289253
\(137\) −1.20350e56 −0.286650 −0.143325 0.989676i \(-0.545779\pi\)
−0.143325 + 0.989676i \(0.545779\pi\)
\(138\) 0 0
\(139\) 1.03961e57 1.68647 0.843235 0.537546i \(-0.180648\pi\)
0.843235 + 0.537546i \(0.180648\pi\)
\(140\) 1.39502e57 1.87149
\(141\) 0 0
\(142\) −1.94033e57 −1.78745
\(143\) −3.25916e56 −0.249287
\(144\) 0 0
\(145\) 2.35173e56 0.124490
\(146\) −2.21098e57 −0.975516
\(147\) 0 0
\(148\) 1.22289e57 0.376228
\(149\) 5.27498e57 1.35764 0.678820 0.734305i \(-0.262492\pi\)
0.678820 + 0.734305i \(0.262492\pi\)
\(150\) 0 0
\(151\) −1.14345e57 −0.206695 −0.103347 0.994645i \(-0.532955\pi\)
−0.103347 + 0.994645i \(0.532955\pi\)
\(152\) 9.60717e56 0.145794
\(153\) 0 0
\(154\) −7.98670e57 −0.857173
\(155\) 4.52277e57 0.408897
\(156\) 0 0
\(157\) 2.71579e58 1.74806 0.874030 0.485873i \(-0.161498\pi\)
0.874030 + 0.485873i \(0.161498\pi\)
\(158\) −2.40387e58 −1.30767
\(159\) 0 0
\(160\) −5.92000e58 −2.30750
\(161\) −2.43641e58 −0.805128
\(162\) 0 0
\(163\) −1.18765e58 −0.282955 −0.141478 0.989941i \(-0.545185\pi\)
−0.141478 + 0.989941i \(0.545185\pi\)
\(164\) 8.24323e58 1.67007
\(165\) 0 0
\(166\) 1.84142e59 2.70576
\(167\) 1.03522e59 1.29732 0.648661 0.761078i \(-0.275329\pi\)
0.648661 + 0.761078i \(0.275329\pi\)
\(168\) 0 0
\(169\) −8.85445e58 −0.809402
\(170\) 4.51681e59 3.53131
\(171\) 0 0
\(172\) 1.69578e59 0.972447
\(173\) −1.05509e59 −0.518881 −0.259441 0.965759i \(-0.583538\pi\)
−0.259441 + 0.965759i \(0.583538\pi\)
\(174\) 0 0
\(175\) 4.49865e59 1.63145
\(176\) 1.55814e59 0.485871
\(177\) 0 0
\(178\) 9.34681e59 2.16040
\(179\) 8.35545e58 0.166481 0.0832405 0.996529i \(-0.473473\pi\)
0.0832405 + 0.996529i \(0.473473\pi\)
\(180\) 0 0
\(181\) −7.95749e59 −1.18112 −0.590559 0.806994i \(-0.701093\pi\)
−0.590559 + 0.806994i \(0.701093\pi\)
\(182\) 5.10951e59 0.655371
\(183\) 0 0
\(184\) −1.56879e59 −0.150624
\(185\) 6.42885e59 0.534666
\(186\) 0 0
\(187\) −1.37288e60 −0.858678
\(188\) 7.30373e58 0.0396608
\(189\) 0 0
\(190\) 4.33879e60 1.77991
\(191\) 3.99509e60 1.42607 0.713037 0.701126i \(-0.247319\pi\)
0.713037 + 0.701126i \(0.247319\pi\)
\(192\) 0 0
\(193\) −2.31728e60 −0.627638 −0.313819 0.949483i \(-0.601609\pi\)
−0.313819 + 0.949483i \(0.601609\pi\)
\(194\) 6.15720e60 1.45424
\(195\) 0 0
\(196\) 3.59139e59 0.0646362
\(197\) −6.97303e60 −1.09665 −0.548323 0.836267i \(-0.684733\pi\)
−0.548323 + 0.836267i \(0.684733\pi\)
\(198\) 0 0
\(199\) −7.01618e60 −0.844295 −0.422148 0.906527i \(-0.638724\pi\)
−0.422148 + 0.906527i \(0.638724\pi\)
\(200\) 2.89666e60 0.305213
\(201\) 0 0
\(202\) −1.85202e61 −1.49911
\(203\) 1.12063e60 0.0795826
\(204\) 0 0
\(205\) 4.33356e61 2.37338
\(206\) 1.10648e61 0.532676
\(207\) 0 0
\(208\) −9.96823e60 −0.371484
\(209\) −1.31877e61 −0.432805
\(210\) 0 0
\(211\) 7.11365e60 0.181388 0.0906938 0.995879i \(-0.471092\pi\)
0.0906938 + 0.995879i \(0.471092\pi\)
\(212\) −3.25329e61 −0.731852
\(213\) 0 0
\(214\) 1.45022e62 2.54372
\(215\) 8.91491e61 1.38197
\(216\) 0 0
\(217\) 2.15516e61 0.261394
\(218\) −6.96673e61 −0.748052
\(219\) 0 0
\(220\) −1.23301e62 −1.03936
\(221\) 8.78304e61 0.656522
\(222\) 0 0
\(223\) −1.07039e62 −0.630176 −0.315088 0.949062i \(-0.602034\pi\)
−0.315088 + 0.949062i \(0.602034\pi\)
\(224\) −2.82096e62 −1.47511
\(225\) 0 0
\(226\) −5.43252e62 −2.24454
\(227\) −2.87663e62 −1.05730 −0.528651 0.848840i \(-0.677302\pi\)
−0.528651 + 0.848840i \(0.677302\pi\)
\(228\) 0 0
\(229\) −5.16513e62 −1.50467 −0.752334 0.658782i \(-0.771072\pi\)
−0.752334 + 0.658782i \(0.771072\pi\)
\(230\) −7.08497e62 −1.83887
\(231\) 0 0
\(232\) 7.21569e60 0.0148883
\(233\) 1.10244e62 0.202965 0.101483 0.994837i \(-0.467641\pi\)
0.101483 + 0.994837i \(0.467641\pi\)
\(234\) 0 0
\(235\) 3.83965e61 0.0563629
\(236\) −3.13849e62 −0.411673
\(237\) 0 0
\(238\) 2.15232e63 2.25745
\(239\) −4.32058e61 −0.0405507 −0.0202753 0.999794i \(-0.506454\pi\)
−0.0202753 + 0.999794i \(0.506454\pi\)
\(240\) 0 0
\(241\) −2.12916e63 −1.60235 −0.801175 0.598431i \(-0.795791\pi\)
−0.801175 + 0.598431i \(0.795791\pi\)
\(242\) −1.45916e63 −0.984003
\(243\) 0 0
\(244\) −2.51752e63 −1.36503
\(245\) 1.88803e62 0.0918561
\(246\) 0 0
\(247\) 8.43686e62 0.330911
\(248\) 1.38769e62 0.0489017
\(249\) 0 0
\(250\) 4.83753e63 1.37789
\(251\) 2.49739e63 0.639930 0.319965 0.947429i \(-0.396329\pi\)
0.319965 + 0.947429i \(0.396329\pi\)
\(252\) 0 0
\(253\) 2.15347e63 0.447141
\(254\) 4.09150e63 0.765226
\(255\) 0 0
\(256\) 4.52210e63 0.687038
\(257\) −3.80580e63 −0.521457 −0.260728 0.965412i \(-0.583963\pi\)
−0.260728 + 0.965412i \(0.583963\pi\)
\(258\) 0 0
\(259\) 3.06343e63 0.341794
\(260\) 7.88823e63 0.794668
\(261\) 0 0
\(262\) 2.20638e63 0.181425
\(263\) 1.69696e64 1.26137 0.630687 0.776037i \(-0.282773\pi\)
0.630687 + 0.776037i \(0.282773\pi\)
\(264\) 0 0
\(265\) −1.71029e64 −1.04005
\(266\) 2.06749e64 1.13784
\(267\) 0 0
\(268\) 1.39805e64 0.630884
\(269\) −6.10982e63 −0.249800 −0.124900 0.992169i \(-0.539861\pi\)
−0.124900 + 0.992169i \(0.539861\pi\)
\(270\) 0 0
\(271\) −9.96186e63 −0.334699 −0.167350 0.985898i \(-0.553521\pi\)
−0.167350 + 0.985898i \(0.553521\pi\)
\(272\) −4.19900e64 −1.27959
\(273\) 0 0
\(274\) −1.66767e64 −0.418523
\(275\) −3.97622e64 −0.906055
\(276\) 0 0
\(277\) 2.47062e64 0.464615 0.232307 0.972642i \(-0.425372\pi\)
0.232307 + 0.972642i \(0.425372\pi\)
\(278\) 1.44056e65 2.46233
\(279\) 0 0
\(280\) 2.25017e64 0.318074
\(281\) 1.25270e64 0.161113 0.0805564 0.996750i \(-0.474330\pi\)
0.0805564 + 0.996750i \(0.474330\pi\)
\(282\) 0 0
\(283\) −2.70393e64 −0.288174 −0.144087 0.989565i \(-0.546025\pi\)
−0.144087 + 0.989565i \(0.546025\pi\)
\(284\) −1.42742e65 −1.38553
\(285\) 0 0
\(286\) −4.51614e64 −0.363972
\(287\) 2.06500e65 1.51722
\(288\) 0 0
\(289\) 2.06371e65 1.26141
\(290\) 3.25874e64 0.181762
\(291\) 0 0
\(292\) −1.62652e65 −0.756161
\(293\) 6.85669e64 0.291154 0.145577 0.989347i \(-0.453496\pi\)
0.145577 + 0.989347i \(0.453496\pi\)
\(294\) 0 0
\(295\) −1.64994e65 −0.585039
\(296\) 1.97253e64 0.0639429
\(297\) 0 0
\(298\) 7.30942e65 1.98222
\(299\) −1.37769e65 −0.341872
\(300\) 0 0
\(301\) 4.24807e65 0.883445
\(302\) −1.58446e65 −0.301784
\(303\) 0 0
\(304\) −4.03350e65 −0.644959
\(305\) −1.32349e66 −1.93987
\(306\) 0 0
\(307\) −1.46437e66 −1.80502 −0.902512 0.430664i \(-0.858279\pi\)
−0.902512 + 0.430664i \(0.858279\pi\)
\(308\) −5.87547e65 −0.664429
\(309\) 0 0
\(310\) 6.26710e65 0.597010
\(311\) −2.08500e66 −1.82371 −0.911856 0.410511i \(-0.865350\pi\)
−0.911856 + 0.410511i \(0.865350\pi\)
\(312\) 0 0
\(313\) −1.88530e66 −1.39141 −0.695706 0.718327i \(-0.744908\pi\)
−0.695706 + 0.718327i \(0.744908\pi\)
\(314\) 3.76322e66 2.55225
\(315\) 0 0
\(316\) −1.76842e66 −1.01363
\(317\) −1.84865e66 −0.974503 −0.487252 0.873262i \(-0.662001\pi\)
−0.487252 + 0.873262i \(0.662001\pi\)
\(318\) 0 0
\(319\) −9.90492e64 −0.0441975
\(320\) −4.87100e66 −2.00052
\(321\) 0 0
\(322\) −3.37608e66 −1.17553
\(323\) 3.55393e66 1.13983
\(324\) 0 0
\(325\) 2.54380e66 0.692745
\(326\) −1.64570e66 −0.413129
\(327\) 0 0
\(328\) 1.32964e66 0.283842
\(329\) 1.82964e65 0.0360309
\(330\) 0 0
\(331\) 1.90131e66 0.318867 0.159434 0.987209i \(-0.449033\pi\)
0.159434 + 0.987209i \(0.449033\pi\)
\(332\) 1.35466e67 2.09734
\(333\) 0 0
\(334\) 1.43449e67 1.89415
\(335\) 7.34968e66 0.896564
\(336\) 0 0
\(337\) −9.17803e66 −0.956218 −0.478109 0.878300i \(-0.658678\pi\)
−0.478109 + 0.878300i \(0.658678\pi\)
\(338\) −1.22694e67 −1.18177
\(339\) 0 0
\(340\) 3.32282e67 2.73726
\(341\) −1.90488e66 −0.145170
\(342\) 0 0
\(343\) −1.48530e67 −0.969439
\(344\) 2.73531e66 0.165275
\(345\) 0 0
\(346\) −1.46202e67 −0.757592
\(347\) −1.14278e67 −0.548569 −0.274285 0.961649i \(-0.588441\pi\)
−0.274285 + 0.961649i \(0.588441\pi\)
\(348\) 0 0
\(349\) 2.72467e67 1.12316 0.561578 0.827424i \(-0.310195\pi\)
0.561578 + 0.827424i \(0.310195\pi\)
\(350\) 6.23368e67 2.38200
\(351\) 0 0
\(352\) 2.49336e67 0.819227
\(353\) 3.17343e67 0.967163 0.483582 0.875299i \(-0.339336\pi\)
0.483582 + 0.875299i \(0.339336\pi\)
\(354\) 0 0
\(355\) −7.50409e67 −1.96900
\(356\) 6.87604e67 1.67461
\(357\) 0 0
\(358\) 1.15780e67 0.243070
\(359\) −3.93676e67 −0.767601 −0.383801 0.923416i \(-0.625385\pi\)
−0.383801 + 0.923416i \(0.625385\pi\)
\(360\) 0 0
\(361\) −2.52826e67 −0.425482
\(362\) −1.10265e68 −1.72449
\(363\) 0 0
\(364\) 3.75884e67 0.508004
\(365\) −8.55081e67 −1.07460
\(366\) 0 0
\(367\) −1.53609e68 −1.67020 −0.835098 0.550101i \(-0.814589\pi\)
−0.835098 + 0.550101i \(0.814589\pi\)
\(368\) 6.58645e67 0.666322
\(369\) 0 0
\(370\) 8.90831e67 0.780638
\(371\) −8.14977e67 −0.664870
\(372\) 0 0
\(373\) −1.29701e68 −0.917611 −0.458805 0.888537i \(-0.651722\pi\)
−0.458805 + 0.888537i \(0.651722\pi\)
\(374\) −1.90237e68 −1.25371
\(375\) 0 0
\(376\) 1.17810e66 0.00674068
\(377\) 6.33670e66 0.0337922
\(378\) 0 0
\(379\) 1.95972e68 0.908354 0.454177 0.890911i \(-0.349933\pi\)
0.454177 + 0.890911i \(0.349933\pi\)
\(380\) 3.19185e68 1.37968
\(381\) 0 0
\(382\) 5.53590e68 2.08214
\(383\) 7.90039e67 0.277256 0.138628 0.990345i \(-0.455731\pi\)
0.138628 + 0.990345i \(0.455731\pi\)
\(384\) 0 0
\(385\) −3.08880e68 −0.944235
\(386\) −3.21100e68 −0.916382
\(387\) 0 0
\(388\) 4.52958e68 1.12724
\(389\) 2.30262e68 0.535252 0.267626 0.963523i \(-0.413761\pi\)
0.267626 + 0.963523i \(0.413761\pi\)
\(390\) 0 0
\(391\) −5.80334e68 −1.17759
\(392\) 5.79293e66 0.0109854
\(393\) 0 0
\(394\) −9.66236e68 −1.60116
\(395\) −9.29679e68 −1.44049
\(396\) 0 0
\(397\) 4.76387e68 0.645671 0.322835 0.946455i \(-0.395364\pi\)
0.322835 + 0.946455i \(0.395364\pi\)
\(398\) −9.72217e68 −1.23271
\(399\) 0 0
\(400\) −1.21614e69 −1.35019
\(401\) 5.31851e68 0.552668 0.276334 0.961062i \(-0.410880\pi\)
0.276334 + 0.961062i \(0.410880\pi\)
\(402\) 0 0
\(403\) 1.21865e68 0.110993
\(404\) −1.36245e69 −1.16202
\(405\) 0 0
\(406\) 1.55283e68 0.116194
\(407\) −2.70767e68 −0.189821
\(408\) 0 0
\(409\) −2.07340e69 −1.27649 −0.638243 0.769835i \(-0.720338\pi\)
−0.638243 + 0.769835i \(0.720338\pi\)
\(410\) 6.00491e69 3.46525
\(411\) 0 0
\(412\) 8.13991e68 0.412898
\(413\) −7.86217e68 −0.373995
\(414\) 0 0
\(415\) 7.12157e69 2.98058
\(416\) −1.59513e69 −0.626359
\(417\) 0 0
\(418\) −1.82739e69 −0.631916
\(419\) −4.81874e69 −1.56409 −0.782045 0.623221i \(-0.785824\pi\)
−0.782045 + 0.623221i \(0.785824\pi\)
\(420\) 0 0
\(421\) −4.17794e68 −0.119533 −0.0597664 0.998212i \(-0.519036\pi\)
−0.0597664 + 0.998212i \(0.519036\pi\)
\(422\) 9.85723e68 0.264835
\(423\) 0 0
\(424\) −5.24759e68 −0.124384
\(425\) 1.07154e70 2.38618
\(426\) 0 0
\(427\) −6.30658e69 −1.24010
\(428\) 1.06686e70 1.97174
\(429\) 0 0
\(430\) 1.23532e70 2.01774
\(431\) −2.97872e69 −0.457490 −0.228745 0.973486i \(-0.573462\pi\)
−0.228745 + 0.973486i \(0.573462\pi\)
\(432\) 0 0
\(433\) 2.69910e69 0.366682 0.183341 0.983049i \(-0.441309\pi\)
0.183341 + 0.983049i \(0.441309\pi\)
\(434\) 2.98635e69 0.381648
\(435\) 0 0
\(436\) −5.12512e69 −0.579845
\(437\) −5.57460e69 −0.593548
\(438\) 0 0
\(439\) −3.57805e68 −0.0337549 −0.0168774 0.999858i \(-0.505373\pi\)
−0.0168774 + 0.999858i \(0.505373\pi\)
\(440\) −1.98886e69 −0.176648
\(441\) 0 0
\(442\) 1.21705e70 0.958554
\(443\) 9.79799e69 0.726839 0.363419 0.931626i \(-0.381609\pi\)
0.363419 + 0.931626i \(0.381609\pi\)
\(444\) 0 0
\(445\) 3.61481e70 2.37983
\(446\) −1.48321e70 −0.920088
\(447\) 0 0
\(448\) −2.32109e70 −1.27887
\(449\) 2.01028e70 1.04407 0.522033 0.852925i \(-0.325174\pi\)
0.522033 + 0.852925i \(0.325174\pi\)
\(450\) 0 0
\(451\) −1.82519e70 −0.842615
\(452\) −3.99647e70 −1.73983
\(453\) 0 0
\(454\) −3.98608e70 −1.54371
\(455\) 1.97606e70 0.721937
\(456\) 0 0
\(457\) −3.35756e70 −1.09205 −0.546027 0.837768i \(-0.683860\pi\)
−0.546027 + 0.837768i \(0.683860\pi\)
\(458\) −7.15720e70 −2.19689
\(459\) 0 0
\(460\) −5.21210e70 −1.42538
\(461\) −4.09546e70 −1.05737 −0.528686 0.848817i \(-0.677315\pi\)
−0.528686 + 0.848817i \(0.677315\pi\)
\(462\) 0 0
\(463\) 1.86380e69 0.0429045 0.0214523 0.999770i \(-0.493171\pi\)
0.0214523 + 0.999770i \(0.493171\pi\)
\(464\) −3.02945e69 −0.0658624
\(465\) 0 0
\(466\) 1.52762e70 0.296339
\(467\) −4.04901e69 −0.0742081 −0.0371040 0.999311i \(-0.511813\pi\)
−0.0371040 + 0.999311i \(0.511813\pi\)
\(468\) 0 0
\(469\) 3.50222e70 0.573143
\(470\) 5.32052e69 0.0822926
\(471\) 0 0
\(472\) −5.06241e69 −0.0699672
\(473\) −3.75474e70 −0.490636
\(474\) 0 0
\(475\) 1.02931e71 1.20272
\(476\) 1.58337e71 1.74984
\(477\) 0 0
\(478\) −5.98694e69 −0.0592059
\(479\) 1.14957e71 1.07558 0.537790 0.843079i \(-0.319259\pi\)
0.537790 + 0.843079i \(0.319259\pi\)
\(480\) 0 0
\(481\) 1.73224e70 0.145132
\(482\) −2.95033e71 −2.33951
\(483\) 0 0
\(484\) −1.07344e71 −0.762739
\(485\) 2.38125e71 1.60195
\(486\) 0 0
\(487\) 2.40380e71 1.45004 0.725022 0.688726i \(-0.241830\pi\)
0.725022 + 0.688726i \(0.241830\pi\)
\(488\) −4.06077e70 −0.231998
\(489\) 0 0
\(490\) 2.61620e70 0.134114
\(491\) −4.81194e70 −0.233701 −0.116850 0.993150i \(-0.537280\pi\)
−0.116850 + 0.993150i \(0.537280\pi\)
\(492\) 0 0
\(493\) 2.66926e70 0.116398
\(494\) 1.16908e71 0.483146
\(495\) 0 0
\(496\) −5.82613e70 −0.216329
\(497\) −3.57580e71 −1.25872
\(498\) 0 0
\(499\) 4.07670e69 0.0129016 0.00645082 0.999979i \(-0.497947\pi\)
0.00645082 + 0.999979i \(0.497947\pi\)
\(500\) 3.55875e71 1.06805
\(501\) 0 0
\(502\) 3.46057e71 0.934329
\(503\) 2.27736e71 0.583284 0.291642 0.956527i \(-0.405798\pi\)
0.291642 + 0.956527i \(0.405798\pi\)
\(504\) 0 0
\(505\) −7.16254e71 −1.65137
\(506\) 2.98401e71 0.652848
\(507\) 0 0
\(508\) 3.00994e71 0.593157
\(509\) −3.77057e71 −0.705319 −0.352659 0.935752i \(-0.614723\pi\)
−0.352659 + 0.935752i \(0.614723\pi\)
\(510\) 0 0
\(511\) −4.07457e71 −0.686954
\(512\) 8.64842e71 1.38447
\(513\) 0 0
\(514\) −5.27361e71 −0.761352
\(515\) 4.27924e71 0.586779
\(516\) 0 0
\(517\) −1.61717e70 −0.0200104
\(518\) 4.24492e71 0.499035
\(519\) 0 0
\(520\) 1.27238e71 0.135060
\(521\) −1.53660e72 −1.55011 −0.775055 0.631894i \(-0.782278\pi\)
−0.775055 + 0.631894i \(0.782278\pi\)
\(522\) 0 0
\(523\) −5.24951e71 −0.478436 −0.239218 0.970966i \(-0.576891\pi\)
−0.239218 + 0.970966i \(0.576891\pi\)
\(524\) 1.62314e71 0.140629
\(525\) 0 0
\(526\) 2.35144e72 1.84167
\(527\) 5.13342e71 0.382318
\(528\) 0 0
\(529\) −5.74185e71 −0.386791
\(530\) −2.36991e72 −1.51853
\(531\) 0 0
\(532\) 1.52096e72 0.881981
\(533\) 1.16767e72 0.644241
\(534\) 0 0
\(535\) 5.60862e72 2.80208
\(536\) 2.25506e71 0.107224
\(537\) 0 0
\(538\) −8.46624e71 −0.364721
\(539\) −7.95192e70 −0.0326114
\(540\) 0 0
\(541\) −4.62761e72 −1.72040 −0.860200 0.509956i \(-0.829662\pi\)
−0.860200 + 0.509956i \(0.829662\pi\)
\(542\) −1.38039e72 −0.488677
\(543\) 0 0
\(544\) −6.71930e72 −2.15751
\(545\) −2.69433e72 −0.824031
\(546\) 0 0
\(547\) 1.78991e72 0.496782 0.248391 0.968660i \(-0.420098\pi\)
0.248391 + 0.968660i \(0.420098\pi\)
\(548\) −1.22683e72 −0.324413
\(549\) 0 0
\(550\) −5.50976e72 −1.32288
\(551\) 2.56405e71 0.0586691
\(552\) 0 0
\(553\) −4.43004e72 −0.920856
\(554\) 3.42349e72 0.678360
\(555\) 0 0
\(556\) 1.05976e73 1.90865
\(557\) −2.84794e72 −0.489067 −0.244533 0.969641i \(-0.578635\pi\)
−0.244533 + 0.969641i \(0.578635\pi\)
\(558\) 0 0
\(559\) 2.40210e72 0.375127
\(560\) −9.44718e72 −1.40708
\(561\) 0 0
\(562\) 1.73584e72 0.235232
\(563\) 8.39092e72 1.08477 0.542385 0.840130i \(-0.317522\pi\)
0.542385 + 0.840130i \(0.317522\pi\)
\(564\) 0 0
\(565\) −2.10099e73 −2.47252
\(566\) −3.74678e72 −0.420748
\(567\) 0 0
\(568\) −2.30244e72 −0.235481
\(569\) −6.11471e72 −0.596899 −0.298450 0.954425i \(-0.596469\pi\)
−0.298450 + 0.954425i \(0.596469\pi\)
\(570\) 0 0
\(571\) −1.26705e73 −1.12703 −0.563515 0.826106i \(-0.690551\pi\)
−0.563515 + 0.826106i \(0.690551\pi\)
\(572\) −3.32233e72 −0.282129
\(573\) 0 0
\(574\) 2.86142e73 2.21522
\(575\) −1.68080e73 −1.24256
\(576\) 0 0
\(577\) 1.54639e73 1.04270 0.521350 0.853343i \(-0.325429\pi\)
0.521350 + 0.853343i \(0.325429\pi\)
\(578\) 2.85964e73 1.84172
\(579\) 0 0
\(580\) 2.39732e72 0.140891
\(581\) 3.39352e73 1.90539
\(582\) 0 0
\(583\) 7.20333e72 0.369247
\(584\) −2.62359e72 −0.128516
\(585\) 0 0
\(586\) 9.50116e72 0.425099
\(587\) 2.89131e73 1.23647 0.618237 0.785992i \(-0.287847\pi\)
0.618237 + 0.785992i \(0.287847\pi\)
\(588\) 0 0
\(589\) 4.93109e72 0.192702
\(590\) −2.28628e73 −0.854185
\(591\) 0 0
\(592\) −8.28150e72 −0.282868
\(593\) 1.35508e73 0.442606 0.221303 0.975205i \(-0.428969\pi\)
0.221303 + 0.975205i \(0.428969\pi\)
\(594\) 0 0
\(595\) 8.32393e73 2.48673
\(596\) 5.37722e73 1.53650
\(597\) 0 0
\(598\) −1.90903e73 −0.499150
\(599\) 1.38679e73 0.346897 0.173448 0.984843i \(-0.444509\pi\)
0.173448 + 0.984843i \(0.444509\pi\)
\(600\) 0 0
\(601\) −4.98700e73 −1.14200 −0.571000 0.820950i \(-0.693444\pi\)
−0.571000 + 0.820950i \(0.693444\pi\)
\(602\) 5.88645e73 1.28987
\(603\) 0 0
\(604\) −1.16561e73 −0.233925
\(605\) −5.64320e73 −1.08395
\(606\) 0 0
\(607\) 4.33744e73 0.763367 0.381683 0.924293i \(-0.375344\pi\)
0.381683 + 0.924293i \(0.375344\pi\)
\(608\) −6.45446e73 −1.08747
\(609\) 0 0
\(610\) −1.83392e74 −2.83231
\(611\) 1.03459e72 0.0152994
\(612\) 0 0
\(613\) 1.11056e74 1.50605 0.753027 0.657990i \(-0.228593\pi\)
0.753027 + 0.657990i \(0.228593\pi\)
\(614\) −2.02914e74 −2.63542
\(615\) 0 0
\(616\) −9.47717e72 −0.112925
\(617\) −6.32555e73 −0.722007 −0.361004 0.932564i \(-0.617566\pi\)
−0.361004 + 0.932564i \(0.617566\pi\)
\(618\) 0 0
\(619\) −2.98152e73 −0.312345 −0.156173 0.987730i \(-0.549916\pi\)
−0.156173 + 0.987730i \(0.549916\pi\)
\(620\) 4.61043e73 0.462766
\(621\) 0 0
\(622\) −2.88914e74 −2.66271
\(623\) 1.72250e74 1.52134
\(624\) 0 0
\(625\) −8.49672e72 −0.0689336
\(626\) −2.61242e74 −2.03153
\(627\) 0 0
\(628\) 2.76843e74 1.97835
\(629\) 7.29685e73 0.499912
\(630\) 0 0
\(631\) 2.66754e74 1.68009 0.840046 0.542515i \(-0.182528\pi\)
0.840046 + 0.542515i \(0.182528\pi\)
\(632\) −2.85248e73 −0.172274
\(633\) 0 0
\(634\) −2.56163e74 −1.42282
\(635\) 1.58236e74 0.842949
\(636\) 0 0
\(637\) 5.08726e72 0.0249338
\(638\) −1.37250e73 −0.0645305
\(639\) 0 0
\(640\) −1.41736e74 −0.613356
\(641\) 7.34082e72 0.0304794 0.0152397 0.999884i \(-0.495149\pi\)
0.0152397 + 0.999884i \(0.495149\pi\)
\(642\) 0 0
\(643\) −2.35192e74 −0.899150 −0.449575 0.893243i \(-0.648425\pi\)
−0.449575 + 0.893243i \(0.648425\pi\)
\(644\) −2.48363e74 −0.911196
\(645\) 0 0
\(646\) 4.92459e74 1.66421
\(647\) −1.32555e74 −0.429963 −0.214981 0.976618i \(-0.568969\pi\)
−0.214981 + 0.976618i \(0.568969\pi\)
\(648\) 0 0
\(649\) 6.94914e73 0.207705
\(650\) 3.52488e74 1.01144
\(651\) 0 0
\(652\) −1.21067e74 −0.320232
\(653\) 8.70965e73 0.221208 0.110604 0.993865i \(-0.464721\pi\)
0.110604 + 0.993865i \(0.464721\pi\)
\(654\) 0 0
\(655\) 8.53300e73 0.199852
\(656\) −5.58239e74 −1.25565
\(657\) 0 0
\(658\) 2.53530e73 0.0526069
\(659\) −2.94646e74 −0.587270 −0.293635 0.955918i \(-0.594865\pi\)
−0.293635 + 0.955918i \(0.594865\pi\)
\(660\) 0 0
\(661\) −1.59089e73 −0.0292618 −0.0146309 0.999893i \(-0.504657\pi\)
−0.0146309 + 0.999893i \(0.504657\pi\)
\(662\) 2.63460e74 0.465562
\(663\) 0 0
\(664\) 2.18507e74 0.356460
\(665\) 7.99585e74 1.25340
\(666\) 0 0
\(667\) −4.18693e73 −0.0606124
\(668\) 1.05529e75 1.46823
\(669\) 0 0
\(670\) 1.01843e75 1.30903
\(671\) 5.57420e74 0.688708
\(672\) 0 0
\(673\) −1.05609e75 −1.20587 −0.602935 0.797790i \(-0.706002\pi\)
−0.602935 + 0.797790i \(0.706002\pi\)
\(674\) −1.27178e75 −1.39612
\(675\) 0 0
\(676\) −9.02607e74 −0.916033
\(677\) 1.20175e75 1.17277 0.586386 0.810032i \(-0.300550\pi\)
0.586386 + 0.810032i \(0.300550\pi\)
\(678\) 0 0
\(679\) 1.13470e75 1.02407
\(680\) 5.35974e74 0.465219
\(681\) 0 0
\(682\) −2.63955e74 −0.211955
\(683\) −1.42292e75 −1.09909 −0.549543 0.835466i \(-0.685198\pi\)
−0.549543 + 0.835466i \(0.685198\pi\)
\(684\) 0 0
\(685\) −6.44957e74 −0.461031
\(686\) −2.05815e75 −1.41543
\(687\) 0 0
\(688\) −1.14840e75 −0.731138
\(689\) −4.60835e74 −0.282316
\(690\) 0 0
\(691\) −1.83314e75 −1.03998 −0.519992 0.854171i \(-0.674065\pi\)
−0.519992 + 0.854171i \(0.674065\pi\)
\(692\) −1.07554e75 −0.587239
\(693\) 0 0
\(694\) −1.58352e75 −0.800938
\(695\) 5.57127e75 2.71242
\(696\) 0 0
\(697\) 4.91866e75 2.21911
\(698\) 3.77551e75 1.63986
\(699\) 0 0
\(700\) 4.58584e75 1.84638
\(701\) 1.33614e75 0.517992 0.258996 0.965878i \(-0.416608\pi\)
0.258996 + 0.965878i \(0.416608\pi\)
\(702\) 0 0
\(703\) 7.00925e74 0.251974
\(704\) 2.05154e75 0.710240
\(705\) 0 0
\(706\) 4.39736e75 1.41211
\(707\) −3.41304e75 −1.05567
\(708\) 0 0
\(709\) −2.59964e75 −0.746084 −0.373042 0.927815i \(-0.621685\pi\)
−0.373042 + 0.927815i \(0.621685\pi\)
\(710\) −1.03982e76 −2.87484
\(711\) 0 0
\(712\) 1.10911e75 0.284613
\(713\) −8.05216e74 −0.199086
\(714\) 0 0
\(715\) −1.74658e75 −0.400940
\(716\) 8.51739e74 0.188413
\(717\) 0 0
\(718\) −5.45508e75 −1.12074
\(719\) −5.36453e75 −1.06222 −0.531112 0.847302i \(-0.678226\pi\)
−0.531112 + 0.847302i \(0.678226\pi\)
\(720\) 0 0
\(721\) 2.03911e75 0.375108
\(722\) −3.50336e75 −0.621225
\(723\) 0 0
\(724\) −8.11172e75 −1.33672
\(725\) 7.73087e74 0.122821
\(726\) 0 0
\(727\) −4.93191e75 −0.728373 −0.364186 0.931326i \(-0.618653\pi\)
−0.364186 + 0.931326i \(0.618653\pi\)
\(728\) 6.06304e74 0.0863394
\(729\) 0 0
\(730\) −1.18487e76 −1.56897
\(731\) 1.01186e76 1.29214
\(732\) 0 0
\(733\) −9.00902e75 −1.07010 −0.535048 0.844822i \(-0.679706\pi\)
−0.535048 + 0.844822i \(0.679706\pi\)
\(734\) −2.12853e76 −2.43857
\(735\) 0 0
\(736\) 1.05397e76 1.12349
\(737\) −3.09550e75 −0.318305
\(738\) 0 0
\(739\) 4.93120e75 0.471927 0.235964 0.971762i \(-0.424175\pi\)
0.235964 + 0.971762i \(0.424175\pi\)
\(740\) 6.55345e75 0.605103
\(741\) 0 0
\(742\) −1.12929e76 −0.970742
\(743\) 1.49779e76 1.24236 0.621181 0.783667i \(-0.286653\pi\)
0.621181 + 0.783667i \(0.286653\pi\)
\(744\) 0 0
\(745\) 2.82686e76 2.18355
\(746\) −1.79723e76 −1.33976
\(747\) 0 0
\(748\) −1.39949e76 −0.971801
\(749\) 2.67258e76 1.79128
\(750\) 0 0
\(751\) 4.11227e75 0.256817 0.128408 0.991721i \(-0.459013\pi\)
0.128408 + 0.991721i \(0.459013\pi\)
\(752\) −4.94616e74 −0.0298191
\(753\) 0 0
\(754\) 8.78062e74 0.0493383
\(755\) −6.12777e75 −0.332436
\(756\) 0 0
\(757\) 3.57142e76 1.80634 0.903172 0.429278i \(-0.141232\pi\)
0.903172 + 0.429278i \(0.141232\pi\)
\(758\) 2.71554e76 1.32624
\(759\) 0 0
\(760\) 5.14848e75 0.234487
\(761\) −2.38006e76 −1.04687 −0.523437 0.852064i \(-0.675351\pi\)
−0.523437 + 0.852064i \(0.675351\pi\)
\(762\) 0 0
\(763\) −1.28388e76 −0.526775
\(764\) 4.07252e76 1.61395
\(765\) 0 0
\(766\) 1.09474e76 0.404807
\(767\) −4.44572e75 −0.158805
\(768\) 0 0
\(769\) 2.37124e76 0.790546 0.395273 0.918564i \(-0.370650\pi\)
0.395273 + 0.918564i \(0.370650\pi\)
\(770\) −4.28008e76 −1.37863
\(771\) 0 0
\(772\) −2.36219e76 −0.710324
\(773\) 6.25079e76 1.81626 0.908131 0.418686i \(-0.137509\pi\)
0.908131 + 0.418686i \(0.137509\pi\)
\(774\) 0 0
\(775\) 1.48677e76 0.403412
\(776\) 7.30625e75 0.191584
\(777\) 0 0
\(778\) 3.19069e76 0.781494
\(779\) 4.72479e76 1.11851
\(780\) 0 0
\(781\) 3.16054e76 0.699051
\(782\) −8.04156e76 −1.71934
\(783\) 0 0
\(784\) −2.43212e75 −0.0485970
\(785\) 1.45540e77 2.81148
\(786\) 0 0
\(787\) −1.33252e76 −0.240627 −0.120314 0.992736i \(-0.538390\pi\)
−0.120314 + 0.992736i \(0.538390\pi\)
\(788\) −7.10818e76 −1.24112
\(789\) 0 0
\(790\) −1.28824e77 −2.10318
\(791\) −1.00115e77 −1.58060
\(792\) 0 0
\(793\) −3.56610e76 −0.526568
\(794\) 6.60118e76 0.942710
\(795\) 0 0
\(796\) −7.15217e76 −0.955524
\(797\) 2.10248e76 0.271698 0.135849 0.990730i \(-0.456624\pi\)
0.135849 + 0.990730i \(0.456624\pi\)
\(798\) 0 0
\(799\) 4.35807e75 0.0526993
\(800\) −1.94609e77 −2.27655
\(801\) 0 0
\(802\) 7.36975e76 0.806922
\(803\) 3.60139e76 0.381512
\(804\) 0 0
\(805\) −1.30567e77 −1.29492
\(806\) 1.68866e76 0.162055
\(807\) 0 0
\(808\) −2.19764e76 −0.197495
\(809\) −1.17401e77 −1.02103 −0.510514 0.859870i \(-0.670545\pi\)
−0.510514 + 0.859870i \(0.670545\pi\)
\(810\) 0 0
\(811\) 9.03233e75 0.0735779 0.0367889 0.999323i \(-0.488287\pi\)
0.0367889 + 0.999323i \(0.488287\pi\)
\(812\) 1.14235e76 0.0900669
\(813\) 0 0
\(814\) −3.75196e76 −0.277148
\(815\) −6.36464e76 −0.455090
\(816\) 0 0
\(817\) 9.71975e76 0.651285
\(818\) −2.87307e77 −1.86373
\(819\) 0 0
\(820\) 4.41755e77 2.68605
\(821\) −8.19852e76 −0.482660 −0.241330 0.970443i \(-0.577584\pi\)
−0.241330 + 0.970443i \(0.577584\pi\)
\(822\) 0 0
\(823\) 1.15061e76 0.0635088 0.0317544 0.999496i \(-0.489891\pi\)
0.0317544 + 0.999496i \(0.489891\pi\)
\(824\) 1.31297e76 0.0701754
\(825\) 0 0
\(826\) −1.08944e77 −0.546051
\(827\) −4.29383e76 −0.208425 −0.104212 0.994555i \(-0.533232\pi\)
−0.104212 + 0.994555i \(0.533232\pi\)
\(828\) 0 0
\(829\) −1.25368e77 −0.570808 −0.285404 0.958407i \(-0.592128\pi\)
−0.285404 + 0.958407i \(0.592128\pi\)
\(830\) 9.86820e77 4.35180
\(831\) 0 0
\(832\) −1.31248e77 −0.543031
\(833\) 2.14295e76 0.0858853
\(834\) 0 0
\(835\) 5.54776e77 2.08654
\(836\) −1.34433e77 −0.489823
\(837\) 0 0
\(838\) −6.67722e77 −2.28365
\(839\) 1.07457e77 0.356075 0.178038 0.984024i \(-0.443025\pi\)
0.178038 + 0.984024i \(0.443025\pi\)
\(840\) 0 0
\(841\) −3.19510e77 −0.994009
\(842\) −5.78928e76 −0.174524
\(843\) 0 0
\(844\) 7.25153e76 0.205284
\(845\) −4.74510e77 −1.30180
\(846\) 0 0
\(847\) −2.68906e77 −0.692930
\(848\) 2.20316e77 0.550245
\(849\) 0 0
\(850\) 1.48481e78 3.48394
\(851\) −1.14457e77 −0.260320
\(852\) 0 0
\(853\) −7.22678e77 −1.54453 −0.772263 0.635303i \(-0.780875\pi\)
−0.772263 + 0.635303i \(0.780875\pi\)
\(854\) −8.73889e77 −1.81060
\(855\) 0 0
\(856\) 1.72086e77 0.335113
\(857\) 6.80686e77 1.28515 0.642577 0.766221i \(-0.277865\pi\)
0.642577 + 0.766221i \(0.277865\pi\)
\(858\) 0 0
\(859\) 4.28760e77 0.761019 0.380510 0.924777i \(-0.375749\pi\)
0.380510 + 0.924777i \(0.375749\pi\)
\(860\) 9.08770e77 1.56403
\(861\) 0 0
\(862\) −4.12755e77 −0.667958
\(863\) −6.31202e77 −0.990564 −0.495282 0.868732i \(-0.664935\pi\)
−0.495282 + 0.868732i \(0.664935\pi\)
\(864\) 0 0
\(865\) −5.65423e77 −0.834539
\(866\) 3.74009e77 0.535374
\(867\) 0 0
\(868\) 2.19693e77 0.295831
\(869\) 3.91558e77 0.511413
\(870\) 0 0
\(871\) 1.98036e77 0.243367
\(872\) −8.26685e76 −0.0985493
\(873\) 0 0
\(874\) −7.72460e77 −0.866609
\(875\) 8.91497e77 0.970301
\(876\) 0 0
\(877\) −8.46824e77 −0.867569 −0.433785 0.901017i \(-0.642822\pi\)
−0.433785 + 0.901017i \(0.642822\pi\)
\(878\) −4.95802e76 −0.0492837
\(879\) 0 0
\(880\) 8.35008e77 0.781447
\(881\) −1.78575e78 −1.62166 −0.810830 0.585281i \(-0.800984\pi\)
−0.810830 + 0.585281i \(0.800984\pi\)
\(882\) 0 0
\(883\) 8.63389e77 0.738325 0.369163 0.929365i \(-0.379645\pi\)
0.369163 + 0.929365i \(0.379645\pi\)
\(884\) 8.95327e77 0.743013
\(885\) 0 0
\(886\) 1.35768e78 1.06122
\(887\) −1.09908e78 −0.833785 −0.416893 0.908956i \(-0.636881\pi\)
−0.416893 + 0.908956i \(0.636881\pi\)
\(888\) 0 0
\(889\) 7.54014e77 0.538869
\(890\) 5.00896e78 3.47466
\(891\) 0 0
\(892\) −1.09113e78 −0.713197
\(893\) 4.18630e76 0.0265624
\(894\) 0 0
\(895\) 4.47769e77 0.267759
\(896\) −6.75393e77 −0.392097
\(897\) 0 0
\(898\) 2.78560e78 1.52439
\(899\) 3.70360e76 0.0196785
\(900\) 0 0
\(901\) −1.94121e78 −0.972446
\(902\) −2.52912e78 −1.23026
\(903\) 0 0
\(904\) −6.44633e77 −0.295699
\(905\) −4.26442e78 −1.89964
\(906\) 0 0
\(907\) −1.67275e78 −0.702812 −0.351406 0.936223i \(-0.614296\pi\)
−0.351406 + 0.936223i \(0.614296\pi\)
\(908\) −2.93238e78 −1.19659
\(909\) 0 0
\(910\) 2.73819e78 1.05406
\(911\) 3.79500e78 1.41898 0.709488 0.704718i \(-0.248926\pi\)
0.709488 + 0.704718i \(0.248926\pi\)
\(912\) 0 0
\(913\) −2.99943e78 −1.05819
\(914\) −4.65249e78 −1.59445
\(915\) 0 0
\(916\) −5.26524e78 −1.70289
\(917\) 4.06609e77 0.127758
\(918\) 0 0
\(919\) 2.27887e78 0.675863 0.337932 0.941171i \(-0.390273\pi\)
0.337932 + 0.941171i \(0.390273\pi\)
\(920\) −8.40716e77 −0.242254
\(921\) 0 0
\(922\) −5.67499e78 −1.54382
\(923\) −2.02196e78 −0.534475
\(924\) 0 0
\(925\) 2.11336e78 0.527494
\(926\) 2.58263e77 0.0626427
\(927\) 0 0
\(928\) −4.84777e77 −0.111051
\(929\) 1.94979e78 0.434082 0.217041 0.976162i \(-0.430359\pi\)
0.217041 + 0.976162i \(0.430359\pi\)
\(930\) 0 0
\(931\) 2.05848e77 0.0432893
\(932\) 1.12380e78 0.229704
\(933\) 0 0
\(934\) −5.61062e77 −0.108347
\(935\) −7.35727e78 −1.38105
\(936\) 0 0
\(937\) 7.29807e78 1.29452 0.647260 0.762269i \(-0.275915\pi\)
0.647260 + 0.762269i \(0.275915\pi\)
\(938\) 4.85295e78 0.836816
\(939\) 0 0
\(940\) 3.91407e77 0.0637883
\(941\) 8.24697e78 1.30668 0.653341 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(942\) 0 0
\(943\) −7.71529e78 −1.15556
\(944\) 2.12541e78 0.309518
\(945\) 0 0
\(946\) −5.20286e78 −0.716353
\(947\) −1.41218e79 −1.89067 −0.945335 0.326100i \(-0.894266\pi\)
−0.945335 + 0.326100i \(0.894266\pi\)
\(948\) 0 0
\(949\) −2.30400e78 −0.291694
\(950\) 1.42629e79 1.75603
\(951\) 0 0
\(952\) 2.55398e78 0.297399
\(953\) 1.39912e79 1.58451 0.792253 0.610192i \(-0.208908\pi\)
0.792253 + 0.610192i \(0.208908\pi\)
\(954\) 0 0
\(955\) 2.14097e79 2.29362
\(956\) −4.40432e77 −0.0458929
\(957\) 0 0
\(958\) 1.59293e79 1.57040
\(959\) −3.07330e78 −0.294722
\(960\) 0 0
\(961\) −1.03074e79 −0.935365
\(962\) 2.40032e78 0.211900
\(963\) 0 0
\(964\) −2.17043e79 −1.81344
\(965\) −1.24183e79 −1.00946
\(966\) 0 0
\(967\) 4.89301e78 0.376508 0.188254 0.982120i \(-0.439717\pi\)
0.188254 + 0.982120i \(0.439717\pi\)
\(968\) −1.73147e78 −0.129634
\(969\) 0 0
\(970\) 3.29964e79 2.33892
\(971\) 5.58018e78 0.384892 0.192446 0.981308i \(-0.438358\pi\)
0.192446 + 0.981308i \(0.438358\pi\)
\(972\) 0 0
\(973\) 2.65478e79 1.73396
\(974\) 3.33089e79 2.11713
\(975\) 0 0
\(976\) 1.70488e79 1.02630
\(977\) 2.57199e79 1.50683 0.753414 0.657546i \(-0.228406\pi\)
0.753414 + 0.657546i \(0.228406\pi\)
\(978\) 0 0
\(979\) −1.52247e79 −0.844903
\(980\) 1.92462e78 0.103957
\(981\) 0 0
\(982\) −6.66779e78 −0.341215
\(983\) −3.41218e79 −1.69967 −0.849833 0.527051i \(-0.823298\pi\)
−0.849833 + 0.527051i \(0.823298\pi\)
\(984\) 0 0
\(985\) −3.73685e79 −1.76378
\(986\) 3.69873e78 0.169947
\(987\) 0 0
\(988\) 8.60038e78 0.374506
\(989\) −1.58718e79 −0.672858
\(990\) 0 0
\(991\) −2.88433e79 −1.15902 −0.579512 0.814964i \(-0.696757\pi\)
−0.579512 + 0.814964i \(0.696757\pi\)
\(992\) −9.32306e78 −0.364753
\(993\) 0 0
\(994\) −4.95490e79 −1.83779
\(995\) −3.75997e79 −1.35792
\(996\) 0 0
\(997\) −3.11454e79 −1.06653 −0.533264 0.845949i \(-0.679035\pi\)
−0.533264 + 0.845949i \(0.679035\pi\)
\(998\) 5.64900e77 0.0188370
\(999\) 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 9.54.a.b.1.4 4
3.2 odd 2 1.54.a.a.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.54.a.a.1.1 4 3.2 odd 2
9.54.a.b.1.4 4 1.1 even 1 trivial