Properties

Label 9.54.a.b.1.2
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 1.2
Root \(447512.\) 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)
 
\(f(q)\) \(=\) \(q-2.58420e7 q^{2} -8.33939e15 q^{4} -5.38136e18 q^{5} +2.33436e22 q^{7} +4.48271e23 q^{8} +O(q^{10})\) \(q-2.58420e7 q^{2} -8.33939e15 q^{4} -5.38136e18 q^{5} +2.33436e22 q^{7} +4.48271e23 q^{8} +1.39065e26 q^{10} +3.50807e27 q^{11} -3.76599e28 q^{13} -6.03245e29 q^{14} +6.35303e31 q^{16} +5.80067e32 q^{17} -9.40144e33 q^{19} +4.48772e34 q^{20} -9.06556e34 q^{22} +5.69353e35 q^{23} +1.78568e37 q^{25} +9.73207e35 q^{26} -1.94671e38 q^{28} +2.46957e38 q^{29} -4.25695e39 q^{31} -5.67942e39 q^{32} -1.49901e40 q^{34} -1.25620e41 q^{35} -5.85991e41 q^{37} +2.42952e41 q^{38} -2.41231e42 q^{40} -5.37725e42 q^{41} +1.54488e43 q^{43} -2.92552e43 q^{44} -1.47132e43 q^{46} +1.14085e44 q^{47} -7.19501e43 q^{49} -4.61455e44 q^{50} +3.14060e44 q^{52} -8.51129e44 q^{53} -1.88782e46 q^{55} +1.04643e46 q^{56} -6.38187e45 q^{58} +9.48774e46 q^{59} +3.37366e47 q^{61} +1.10008e47 q^{62} -4.25463e47 q^{64} +2.02661e47 q^{65} -3.70994e48 q^{67} -4.83740e48 q^{68} +3.24628e48 q^{70} -2.16582e48 q^{71} -1.03616e49 q^{73} +1.51432e49 q^{74} +7.84022e49 q^{76} +8.18910e49 q^{77} +5.02034e49 q^{79} -3.41879e50 q^{80} +1.38959e50 q^{82} +3.18305e50 q^{83} -3.12155e51 q^{85} -3.99229e50 q^{86} +1.57257e51 q^{88} -3.04162e51 q^{89} -8.79117e50 q^{91} -4.74806e51 q^{92} -2.94817e51 q^{94} +5.05925e52 q^{95} +4.29374e52 q^{97} +1.85933e51 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 68476320 q^{2} + 78\!\cdots\!28 q^{4}+ \cdots - 13\!\cdots\!40 q^{8}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58420e7 −0.272290 −0.136145 0.990689i \(-0.543471\pi\)
−0.136145 + 0.990689i \(0.543471\pi\)
\(3\) 0 0
\(4\) −8.33939e15 −0.925858
\(5\) −5.38136e18 −1.61505 −0.807527 0.589831i \(-0.799194\pi\)
−0.807527 + 0.589831i \(0.799194\pi\)
\(6\) 0 0
\(7\) 2.33436e22 0.939874 0.469937 0.882700i \(-0.344277\pi\)
0.469937 + 0.882700i \(0.344277\pi\)
\(8\) 4.48271e23 0.524392
\(9\) 0 0
\(10\) 1.39065e26 0.439763
\(11\) 3.50807e27 0.887487 0.443744 0.896154i \(-0.353650\pi\)
0.443744 + 0.896154i \(0.353650\pi\)
\(12\) 0 0
\(13\) −3.76599e28 −0.113862 −0.0569312 0.998378i \(-0.518132\pi\)
−0.0569312 + 0.998378i \(0.518132\pi\)
\(14\) −6.03245e29 −0.255918
\(15\) 0 0
\(16\) 6.35303e31 0.783072
\(17\) 5.80067e32 1.43411 0.717053 0.697018i \(-0.245490\pi\)
0.717053 + 0.697018i \(0.245490\pi\)
\(18\) 0 0
\(19\) −9.40144e33 −1.21962 −0.609809 0.792548i \(-0.708754\pi\)
−0.609809 + 0.792548i \(0.708754\pi\)
\(20\) 4.48772e34 1.49531
\(21\) 0 0
\(22\) −9.06556e34 −0.241654
\(23\) 5.69353e35 0.467298 0.233649 0.972321i \(-0.424933\pi\)
0.233649 + 0.972321i \(0.424933\pi\)
\(24\) 0 0
\(25\) 1.78568e37 1.60840
\(26\) 9.73207e35 0.0310035
\(27\) 0 0
\(28\) −1.94671e38 −0.870190
\(29\) 2.46957e38 0.435587 0.217794 0.975995i \(-0.430114\pi\)
0.217794 + 0.975995i \(0.430114\pi\)
\(30\) 0 0
\(31\) −4.25695e39 −1.28237 −0.641185 0.767386i \(-0.721557\pi\)
−0.641185 + 0.767386i \(0.721557\pi\)
\(32\) −5.67942e39 −0.737614
\(33\) 0 0
\(34\) −1.49901e40 −0.390493
\(35\) −1.25620e41 −1.51795
\(36\) 0 0
\(37\) −5.85991e41 −1.62385 −0.811924 0.583764i \(-0.801580\pi\)
−0.811924 + 0.583764i \(0.801580\pi\)
\(38\) 2.42952e41 0.332090
\(39\) 0 0
\(40\) −2.41231e42 −0.846920
\(41\) −5.37725e42 −0.981269 −0.490634 0.871366i \(-0.663235\pi\)
−0.490634 + 0.871366i \(0.663235\pi\)
\(42\) 0 0
\(43\) 1.54488e43 0.797961 0.398981 0.916959i \(-0.369364\pi\)
0.398981 + 0.916959i \(0.369364\pi\)
\(44\) −2.92552e43 −0.821687
\(45\) 0 0
\(46\) −1.47132e43 −0.127241
\(47\) 1.14085e44 0.557999 0.279000 0.960291i \(-0.409997\pi\)
0.279000 + 0.960291i \(0.409997\pi\)
\(48\) 0 0
\(49\) −7.19501e43 −0.116637
\(50\) −4.61455e44 −0.437950
\(51\) 0 0
\(52\) 3.14060e44 0.105420
\(53\) −8.51129e44 −0.172459 −0.0862293 0.996275i \(-0.527482\pi\)
−0.0862293 + 0.996275i \(0.527482\pi\)
\(54\) 0 0
\(55\) −1.88782e46 −1.43334
\(56\) 1.04643e46 0.492862
\(57\) 0 0
\(58\) −6.38187e45 −0.118606
\(59\) 9.48774e46 1.12095 0.560473 0.828173i \(-0.310620\pi\)
0.560473 + 0.828173i \(0.310620\pi\)
\(60\) 0 0
\(61\) 3.37366e47 1.64764 0.823818 0.566854i \(-0.191840\pi\)
0.823818 + 0.566854i \(0.191840\pi\)
\(62\) 1.10008e47 0.349176
\(63\) 0 0
\(64\) −4.25463e47 −0.582227
\(65\) 2.02661e47 0.183894
\(66\) 0 0
\(67\) −3.70994e48 −1.50794 −0.753971 0.656908i \(-0.771864\pi\)
−0.753971 + 0.656908i \(0.771864\pi\)
\(68\) −4.83740e48 −1.32778
\(69\) 0 0
\(70\) 3.24628e48 0.413321
\(71\) −2.16582e48 −0.189355 −0.0946774 0.995508i \(-0.530182\pi\)
−0.0946774 + 0.995508i \(0.530182\pi\)
\(72\) 0 0
\(73\) −1.03616e49 −0.433882 −0.216941 0.976185i \(-0.569608\pi\)
−0.216941 + 0.976185i \(0.569608\pi\)
\(74\) 1.51432e49 0.442157
\(75\) 0 0
\(76\) 7.84022e49 1.12919
\(77\) 8.18910e49 0.834126
\(78\) 0 0
\(79\) 5.02034e49 0.259188 0.129594 0.991567i \(-0.458633\pi\)
0.129594 + 0.991567i \(0.458633\pi\)
\(80\) −3.41879e50 −1.26470
\(81\) 0 0
\(82\) 1.38959e50 0.267189
\(83\) 3.18305e50 0.443889 0.221944 0.975059i \(-0.428760\pi\)
0.221944 + 0.975059i \(0.428760\pi\)
\(84\) 0 0
\(85\) −3.12155e51 −2.31616
\(86\) −3.99229e50 −0.217277
\(87\) 0 0
\(88\) 1.57257e51 0.465391
\(89\) −3.04162e51 −0.667222 −0.333611 0.942711i \(-0.608267\pi\)
−0.333611 + 0.942711i \(0.608267\pi\)
\(90\) 0 0
\(91\) −8.79117e50 −0.107016
\(92\) −4.74806e51 −0.432652
\(93\) 0 0
\(94\) −2.94817e51 −0.151938
\(95\) 5.05925e52 1.96975
\(96\) 0 0
\(97\) 4.29374e52 0.962465 0.481232 0.876593i \(-0.340189\pi\)
0.481232 + 0.876593i \(0.340189\pi\)
\(98\) 1.85933e51 0.0317590
\(99\) 0 0
\(100\) −1.48915e53 −1.48915
\(101\) −2.40879e53 −1.85047 −0.925236 0.379392i \(-0.876133\pi\)
−0.925236 + 0.379392i \(0.876133\pi\)
\(102\) 0 0
\(103\) 5.77425e52 0.263821 0.131910 0.991262i \(-0.457889\pi\)
0.131910 + 0.991262i \(0.457889\pi\)
\(104\) −1.68818e52 −0.0597084
\(105\) 0 0
\(106\) 2.19949e52 0.0469587
\(107\) 2.30554e53 0.383797 0.191899 0.981415i \(-0.438535\pi\)
0.191899 + 0.981415i \(0.438535\pi\)
\(108\) 0 0
\(109\) 5.84013e53 0.595142 0.297571 0.954700i \(-0.403824\pi\)
0.297571 + 0.954700i \(0.403824\pi\)
\(110\) 4.87850e53 0.390284
\(111\) 0 0
\(112\) 1.48303e54 0.735989
\(113\) 4.64679e53 0.182211 0.0911055 0.995841i \(-0.470960\pi\)
0.0911055 + 0.995841i \(0.470960\pi\)
\(114\) 0 0
\(115\) −3.06389e54 −0.754711
\(116\) −2.05947e54 −0.403292
\(117\) 0 0
\(118\) −2.45182e54 −0.305222
\(119\) 1.35408e55 1.34788
\(120\) 0 0
\(121\) −3.31816e54 −0.212366
\(122\) −8.71823e54 −0.448634
\(123\) 0 0
\(124\) 3.55003e55 1.18729
\(125\) −3.63487e55 −0.982593
\(126\) 0 0
\(127\) 1.05751e54 0.0187709 0.00938545 0.999956i \(-0.497012\pi\)
0.00938545 + 0.999956i \(0.497012\pi\)
\(128\) 6.21504e55 0.896149
\(129\) 0 0
\(130\) −5.23717e54 −0.0500724
\(131\) −9.48285e55 −0.740031 −0.370015 0.929026i \(-0.620648\pi\)
−0.370015 + 0.929026i \(0.620648\pi\)
\(132\) 0 0
\(133\) −2.19463e56 −1.14629
\(134\) 9.58722e55 0.410597
\(135\) 0 0
\(136\) 2.60027e56 0.752034
\(137\) −1.67624e56 −0.399248 −0.199624 0.979873i \(-0.563972\pi\)
−0.199624 + 0.979873i \(0.563972\pi\)
\(138\) 0 0
\(139\) −6.36794e55 −0.103301 −0.0516507 0.998665i \(-0.516448\pi\)
−0.0516507 + 0.998665i \(0.516448\pi\)
\(140\) 1.04760e57 1.40540
\(141\) 0 0
\(142\) 5.59692e55 0.0515594
\(143\) −1.32113e56 −0.101051
\(144\) 0 0
\(145\) −1.32897e57 −0.703496
\(146\) 2.67765e56 0.118142
\(147\) 0 0
\(148\) 4.88681e57 1.50345
\(149\) −5.30162e55 −0.0136450 −0.00682249 0.999977i \(-0.502172\pi\)
−0.00682249 + 0.999977i \(0.502172\pi\)
\(150\) 0 0
\(151\) −3.81438e56 −0.0689500 −0.0344750 0.999406i \(-0.510976\pi\)
−0.0344750 + 0.999406i \(0.510976\pi\)
\(152\) −4.21439e57 −0.639557
\(153\) 0 0
\(154\) −2.11623e57 −0.227124
\(155\) 2.29082e58 2.07110
\(156\) 0 0
\(157\) −2.25055e57 −0.144860 −0.0724298 0.997374i \(-0.523075\pi\)
−0.0724298 + 0.997374i \(0.523075\pi\)
\(158\) −1.29736e57 −0.0705744
\(159\) 0 0
\(160\) 3.05630e58 1.19129
\(161\) 1.32907e58 0.439201
\(162\) 0 0
\(163\) 5.68137e58 1.35357 0.676786 0.736180i \(-0.263372\pi\)
0.676786 + 0.736180i \(0.263372\pi\)
\(164\) 4.48430e58 0.908516
\(165\) 0 0
\(166\) −8.22564e57 −0.120866
\(167\) 5.90727e58 0.740288 0.370144 0.928974i \(-0.379308\pi\)
0.370144 + 0.928974i \(0.379308\pi\)
\(168\) 0 0
\(169\) −1.07977e59 −0.987035
\(170\) 8.06670e58 0.630666
\(171\) 0 0
\(172\) −1.28834e59 −0.738799
\(173\) −5.53658e58 −0.272282 −0.136141 0.990689i \(-0.543470\pi\)
−0.136141 + 0.990689i \(0.543470\pi\)
\(174\) 0 0
\(175\) 4.16842e59 1.51169
\(176\) 2.22869e59 0.694966
\(177\) 0 0
\(178\) 7.86016e58 0.181678
\(179\) 7.91775e59 1.57760 0.788799 0.614651i \(-0.210703\pi\)
0.788799 + 0.614651i \(0.210703\pi\)
\(180\) 0 0
\(181\) 6.76480e59 1.00409 0.502045 0.864841i \(-0.332581\pi\)
0.502045 + 0.864841i \(0.332581\pi\)
\(182\) 2.27181e58 0.0291394
\(183\) 0 0
\(184\) 2.55224e59 0.245047
\(185\) 3.15343e60 2.62260
\(186\) 0 0
\(187\) 2.03491e60 1.27275
\(188\) −9.51396e59 −0.516628
\(189\) 0 0
\(190\) −1.30741e60 −0.536342
\(191\) −5.29979e59 −0.189180 −0.0945899 0.995516i \(-0.530154\pi\)
−0.0945899 + 0.995516i \(0.530154\pi\)
\(192\) 0 0
\(193\) −9.36608e59 −0.253682 −0.126841 0.991923i \(-0.540484\pi\)
−0.126841 + 0.991923i \(0.540484\pi\)
\(194\) −1.10959e60 −0.262069
\(195\) 0 0
\(196\) 6.00020e59 0.107989
\(197\) 1.14277e61 1.79723 0.898615 0.438737i \(-0.144574\pi\)
0.898615 + 0.438737i \(0.144574\pi\)
\(198\) 0 0
\(199\) 7.32212e59 0.0881110 0.0440555 0.999029i \(-0.485972\pi\)
0.0440555 + 0.999029i \(0.485972\pi\)
\(200\) 8.00468e60 0.843430
\(201\) 0 0
\(202\) 6.22480e60 0.503865
\(203\) 5.76487e60 0.409397
\(204\) 0 0
\(205\) 2.89369e61 1.58480
\(206\) −1.49218e60 −0.0718357
\(207\) 0 0
\(208\) −2.39254e60 −0.0891624
\(209\) −3.29809e61 −1.08240
\(210\) 0 0
\(211\) −7.50377e61 −1.91335 −0.956676 0.291156i \(-0.905960\pi\)
−0.956676 + 0.291156i \(0.905960\pi\)
\(212\) 7.09790e60 0.159672
\(213\) 0 0
\(214\) −5.95797e60 −0.104504
\(215\) −8.31358e61 −1.28875
\(216\) 0 0
\(217\) −9.93724e61 −1.20527
\(218\) −1.50921e61 −0.162051
\(219\) 0 0
\(220\) 1.57433e62 1.32707
\(221\) −2.18452e61 −0.163291
\(222\) 0 0
\(223\) −1.22197e61 −0.0719419 −0.0359710 0.999353i \(-0.511452\pi\)
−0.0359710 + 0.999353i \(0.511452\pi\)
\(224\) −1.32578e62 −0.693264
\(225\) 0 0
\(226\) −1.20082e61 −0.0496142
\(227\) 5.02981e61 0.184870 0.0924350 0.995719i \(-0.470535\pi\)
0.0924350 + 0.995719i \(0.470535\pi\)
\(228\) 0 0
\(229\) −3.41310e62 −0.994281 −0.497140 0.867670i \(-0.665617\pi\)
−0.497140 + 0.867670i \(0.665617\pi\)
\(230\) 7.91771e61 0.205500
\(231\) 0 0
\(232\) 1.10704e62 0.228418
\(233\) −5.82407e62 −1.07225 −0.536123 0.844140i \(-0.680112\pi\)
−0.536123 + 0.844140i \(0.680112\pi\)
\(234\) 0 0
\(235\) −6.13930e62 −0.901198
\(236\) −7.91219e62 −1.03784
\(237\) 0 0
\(238\) −3.49923e62 −0.367014
\(239\) −1.01634e63 −0.953879 −0.476939 0.878936i \(-0.658254\pi\)
−0.476939 + 0.878936i \(0.658254\pi\)
\(240\) 0 0
\(241\) 2.10470e63 1.58394 0.791970 0.610560i \(-0.209056\pi\)
0.791970 + 0.610560i \(0.209056\pi\)
\(242\) 8.57480e61 0.0578252
\(243\) 0 0
\(244\) −2.81343e63 −1.52548
\(245\) 3.87189e62 0.188374
\(246\) 0 0
\(247\) 3.54057e62 0.138868
\(248\) −1.90826e63 −0.672464
\(249\) 0 0
\(250\) 9.39324e62 0.267550
\(251\) 3.46671e63 0.888309 0.444154 0.895950i \(-0.353504\pi\)
0.444154 + 0.895950i \(0.353504\pi\)
\(252\) 0 0
\(253\) 1.99733e63 0.414721
\(254\) −2.73281e61 −0.00511113
\(255\) 0 0
\(256\) 2.22614e63 0.338215
\(257\) 7.98120e63 1.09355 0.546777 0.837278i \(-0.315854\pi\)
0.546777 + 0.837278i \(0.315854\pi\)
\(258\) 0 0
\(259\) −1.36791e64 −1.52621
\(260\) −1.69007e63 −0.170259
\(261\) 0 0
\(262\) 2.45056e63 0.201503
\(263\) 6.73466e63 0.500597 0.250299 0.968169i \(-0.419471\pi\)
0.250299 + 0.968169i \(0.419471\pi\)
\(264\) 0 0
\(265\) 4.58023e63 0.278530
\(266\) 5.67137e63 0.312122
\(267\) 0 0
\(268\) 3.09386e64 1.39614
\(269\) −1.44464e64 −0.590640 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(270\) 0 0
\(271\) 2.71685e64 0.912810 0.456405 0.889772i \(-0.349137\pi\)
0.456405 + 0.889772i \(0.349137\pi\)
\(272\) 3.68518e64 1.12301
\(273\) 0 0
\(274\) 4.33175e63 0.108711
\(275\) 6.26429e64 1.42743
\(276\) 0 0
\(277\) 2.80661e64 0.527798 0.263899 0.964550i \(-0.414991\pi\)
0.263899 + 0.964550i \(0.414991\pi\)
\(278\) 1.64560e63 0.0281279
\(279\) 0 0
\(280\) −5.63119e64 −0.795998
\(281\) 2.78031e64 0.357582 0.178791 0.983887i \(-0.442781\pi\)
0.178791 + 0.983887i \(0.442781\pi\)
\(282\) 0 0
\(283\) −6.35233e64 −0.677006 −0.338503 0.940965i \(-0.609920\pi\)
−0.338503 + 0.940965i \(0.609920\pi\)
\(284\) 1.80616e64 0.175316
\(285\) 0 0
\(286\) 3.41408e63 0.0275153
\(287\) −1.25524e65 −0.922269
\(288\) 0 0
\(289\) 1.72874e65 1.05666
\(290\) 3.43432e64 0.191555
\(291\) 0 0
\(292\) 8.64095e64 0.401713
\(293\) 1.18763e65 0.504300 0.252150 0.967688i \(-0.418862\pi\)
0.252150 + 0.967688i \(0.418862\pi\)
\(294\) 0 0
\(295\) −5.10569e65 −1.81039
\(296\) −2.62683e65 −0.851532
\(297\) 0 0
\(298\) 1.37005e63 0.00371539
\(299\) −2.14418e64 −0.0532076
\(300\) 0 0
\(301\) 3.60632e65 0.749983
\(302\) 9.85711e63 0.0187744
\(303\) 0 0
\(304\) −5.97276e65 −0.955048
\(305\) −1.81549e66 −2.66102
\(306\) 0 0
\(307\) −1.02441e65 −0.126271 −0.0631357 0.998005i \(-0.520110\pi\)
−0.0631357 + 0.998005i \(0.520110\pi\)
\(308\) −6.82921e65 −0.772283
\(309\) 0 0
\(310\) −5.91993e65 −0.563938
\(311\) 2.08869e66 1.82694 0.913469 0.406909i \(-0.133393\pi\)
0.913469 + 0.406909i \(0.133393\pi\)
\(312\) 0 0
\(313\) 8.41675e65 0.621183 0.310591 0.950543i \(-0.399473\pi\)
0.310591 + 0.950543i \(0.399473\pi\)
\(314\) 5.81587e64 0.0394438
\(315\) 0 0
\(316\) −4.18666e65 −0.239972
\(317\) −1.89204e66 −0.997378 −0.498689 0.866781i \(-0.666185\pi\)
−0.498689 + 0.866781i \(0.666185\pi\)
\(318\) 0 0
\(319\) 8.66344e65 0.386578
\(320\) 2.28957e66 0.940328
\(321\) 0 0
\(322\) −3.43460e65 −0.119590
\(323\) −5.45346e66 −1.74906
\(324\) 0 0
\(325\) −6.72484e65 −0.183136
\(326\) −1.46818e66 −0.368564
\(327\) 0 0
\(328\) −2.41046e66 −0.514569
\(329\) 2.66314e66 0.524449
\(330\) 0 0
\(331\) −3.72579e66 −0.624850 −0.312425 0.949942i \(-0.601141\pi\)
−0.312425 + 0.949942i \(0.601141\pi\)
\(332\) −2.65447e66 −0.410978
\(333\) 0 0
\(334\) −1.52656e66 −0.201573
\(335\) 1.99645e67 2.43540
\(336\) 0 0
\(337\) 9.36606e66 0.975808 0.487904 0.872897i \(-0.337762\pi\)
0.487904 + 0.872897i \(0.337762\pi\)
\(338\) 2.79034e66 0.268760
\(339\) 0 0
\(340\) 2.60318e67 2.14443
\(341\) −1.49337e67 −1.13809
\(342\) 0 0
\(343\) −1.60796e67 −1.04950
\(344\) 6.92526e66 0.418444
\(345\) 0 0
\(346\) 1.43076e66 0.0741397
\(347\) 2.74862e67 1.31942 0.659711 0.751520i \(-0.270679\pi\)
0.659711 + 0.751520i \(0.270679\pi\)
\(348\) 0 0
\(349\) 1.18632e67 0.489024 0.244512 0.969646i \(-0.421372\pi\)
0.244512 + 0.969646i \(0.421372\pi\)
\(350\) −1.07720e67 −0.411618
\(351\) 0 0
\(352\) −1.99238e67 −0.654623
\(353\) 2.74837e67 0.837618 0.418809 0.908074i \(-0.362448\pi\)
0.418809 + 0.908074i \(0.362448\pi\)
\(354\) 0 0
\(355\) 1.16551e67 0.305818
\(356\) 2.53653e67 0.617753
\(357\) 0 0
\(358\) −2.04610e67 −0.429564
\(359\) −3.20694e67 −0.625299 −0.312649 0.949869i \(-0.601216\pi\)
−0.312649 + 0.949869i \(0.601216\pi\)
\(360\) 0 0
\(361\) 2.89659e67 0.487468
\(362\) −1.74816e67 −0.273404
\(363\) 0 0
\(364\) 7.33130e66 0.0990819
\(365\) 5.57595e67 0.700742
\(366\) 0 0
\(367\) 6.14979e67 0.668666 0.334333 0.942455i \(-0.391489\pi\)
0.334333 + 0.942455i \(0.391489\pi\)
\(368\) 3.61712e67 0.365928
\(369\) 0 0
\(370\) −8.14909e67 −0.714107
\(371\) −1.98684e67 −0.162089
\(372\) 0 0
\(373\) 1.08656e68 0.768723 0.384362 0.923183i \(-0.374422\pi\)
0.384362 + 0.923183i \(0.374422\pi\)
\(374\) −5.25863e67 −0.346557
\(375\) 0 0
\(376\) 5.11408e67 0.292610
\(377\) −9.30038e66 −0.0495970
\(378\) 0 0
\(379\) −2.80807e68 −1.30158 −0.650788 0.759259i \(-0.725561\pi\)
−0.650788 + 0.759259i \(0.725561\pi\)
\(380\) −4.21911e68 −1.82371
\(381\) 0 0
\(382\) 1.36957e67 0.0515117
\(383\) −4.61965e68 −1.62122 −0.810610 0.585586i \(-0.800864\pi\)
−0.810610 + 0.585586i \(0.800864\pi\)
\(384\) 0 0
\(385\) −4.40685e68 −1.34716
\(386\) 2.42038e67 0.0690749
\(387\) 0 0
\(388\) −3.58072e68 −0.891106
\(389\) 3.34377e68 0.777271 0.388636 0.921392i \(-0.372947\pi\)
0.388636 + 0.921392i \(0.372947\pi\)
\(390\) 0 0
\(391\) 3.30263e68 0.670155
\(392\) −3.22531e67 −0.0611633
\(393\) 0 0
\(394\) −2.95315e68 −0.489368
\(395\) −2.70163e68 −0.418603
\(396\) 0 0
\(397\) 2.89097e68 0.391827 0.195914 0.980621i \(-0.437233\pi\)
0.195914 + 0.980621i \(0.437233\pi\)
\(398\) −1.89218e67 −0.0239917
\(399\) 0 0
\(400\) 1.13445e69 1.25949
\(401\) −4.66806e67 −0.0485077 −0.0242538 0.999706i \(-0.507721\pi\)
−0.0242538 + 0.999706i \(0.507721\pi\)
\(402\) 0 0
\(403\) 1.60316e68 0.146014
\(404\) 2.00878e69 1.71327
\(405\) 0 0
\(406\) −1.48976e68 −0.111475
\(407\) −2.05570e69 −1.44114
\(408\) 0 0
\(409\) −5.42643e68 −0.334077 −0.167039 0.985950i \(-0.553420\pi\)
−0.167039 + 0.985950i \(0.553420\pi\)
\(410\) −7.47788e68 −0.431525
\(411\) 0 0
\(412\) −4.81537e68 −0.244261
\(413\) 2.21478e69 1.05355
\(414\) 0 0
\(415\) −1.71291e69 −0.716904
\(416\) 2.13886e68 0.0839864
\(417\) 0 0
\(418\) 8.52293e68 0.294725
\(419\) −2.80482e69 −0.910403 −0.455202 0.890388i \(-0.650433\pi\)
−0.455202 + 0.890388i \(0.650433\pi\)
\(420\) 0 0
\(421\) 4.47945e69 1.28159 0.640795 0.767712i \(-0.278605\pi\)
0.640795 + 0.767712i \(0.278605\pi\)
\(422\) 1.93913e69 0.520986
\(423\) 0 0
\(424\) −3.81536e68 −0.0904359
\(425\) 1.03581e70 2.30661
\(426\) 0 0
\(427\) 7.87535e69 1.54857
\(428\) −1.92268e69 −0.355342
\(429\) 0 0
\(430\) 2.14840e69 0.350914
\(431\) 1.07297e70 1.64793 0.823967 0.566638i \(-0.191756\pi\)
0.823967 + 0.566638i \(0.191756\pi\)
\(432\) 0 0
\(433\) −1.28857e70 −1.75057 −0.875286 0.483606i \(-0.839327\pi\)
−0.875286 + 0.483606i \(0.839327\pi\)
\(434\) 2.56798e69 0.328182
\(435\) 0 0
\(436\) −4.87031e69 −0.551017
\(437\) −5.35274e69 −0.569925
\(438\) 0 0
\(439\) 2.10965e69 0.199022 0.0995111 0.995036i \(-0.468272\pi\)
0.0995111 + 0.995036i \(0.468272\pi\)
\(440\) −8.46254e69 −0.751631
\(441\) 0 0
\(442\) 5.64525e68 0.0444624
\(443\) −1.77505e70 −1.31678 −0.658389 0.752678i \(-0.728762\pi\)
−0.658389 + 0.752678i \(0.728762\pi\)
\(444\) 0 0
\(445\) 1.63681e70 1.07760
\(446\) 3.15782e68 0.0195891
\(447\) 0 0
\(448\) −9.93183e69 −0.547220
\(449\) 7.87557e69 0.409029 0.204515 0.978864i \(-0.434438\pi\)
0.204515 + 0.978864i \(0.434438\pi\)
\(450\) 0 0
\(451\) −1.88638e70 −0.870863
\(452\) −3.87514e69 −0.168702
\(453\) 0 0
\(454\) −1.29980e69 −0.0503382
\(455\) 4.73084e69 0.172837
\(456\) 0 0
\(457\) 3.48981e70 1.13507 0.567536 0.823349i \(-0.307897\pi\)
0.567536 + 0.823349i \(0.307897\pi\)
\(458\) 8.82014e69 0.270733
\(459\) 0 0
\(460\) 2.55510e70 0.698756
\(461\) −4.79728e70 −1.23857 −0.619286 0.785166i \(-0.712578\pi\)
−0.619286 + 0.785166i \(0.712578\pi\)
\(462\) 0 0
\(463\) −5.09571e70 −1.17303 −0.586514 0.809939i \(-0.699500\pi\)
−0.586514 + 0.809939i \(0.699500\pi\)
\(464\) 1.56893e70 0.341096
\(465\) 0 0
\(466\) 1.50506e70 0.291962
\(467\) −8.74085e70 −1.60198 −0.800989 0.598680i \(-0.795692\pi\)
−0.800989 + 0.598680i \(0.795692\pi\)
\(468\) 0 0
\(469\) −8.66033e70 −1.41727
\(470\) 1.58652e70 0.245387
\(471\) 0 0
\(472\) 4.25307e70 0.587814
\(473\) 5.41956e70 0.708181
\(474\) 0 0
\(475\) −1.67879e71 −1.96163
\(476\) −1.12922e71 −1.24795
\(477\) 0 0
\(478\) 2.62642e70 0.259732
\(479\) 9.38882e70 0.878455 0.439228 0.898376i \(-0.355252\pi\)
0.439228 + 0.898376i \(0.355252\pi\)
\(480\) 0 0
\(481\) 2.20683e70 0.184895
\(482\) −5.43897e70 −0.431291
\(483\) 0 0
\(484\) 2.76715e70 0.196621
\(485\) −2.31062e71 −1.55443
\(486\) 0 0
\(487\) 3.27176e71 1.97362 0.986810 0.161885i \(-0.0517573\pi\)
0.986810 + 0.161885i \(0.0517573\pi\)
\(488\) 1.51232e71 0.864006
\(489\) 0 0
\(490\) −1.00057e70 −0.0512924
\(491\) 2.76660e71 1.34365 0.671826 0.740709i \(-0.265510\pi\)
0.671826 + 0.740709i \(0.265510\pi\)
\(492\) 0 0
\(493\) 1.43252e71 0.624679
\(494\) −9.14954e69 −0.0378125
\(495\) 0 0
\(496\) −2.70445e71 −1.00419
\(497\) −5.05580e70 −0.177970
\(498\) 0 0
\(499\) −4.81364e71 −1.52338 −0.761692 0.647939i \(-0.775631\pi\)
−0.761692 + 0.647939i \(0.775631\pi\)
\(500\) 3.03126e71 0.909742
\(501\) 0 0
\(502\) −8.95868e70 −0.241877
\(503\) 4.97279e71 1.27364 0.636822 0.771011i \(-0.280249\pi\)
0.636822 + 0.771011i \(0.280249\pi\)
\(504\) 0 0
\(505\) 1.29626e72 2.98861
\(506\) −5.16150e70 −0.112924
\(507\) 0 0
\(508\) −8.81898e69 −0.0173792
\(509\) 4.86664e71 0.910349 0.455174 0.890402i \(-0.349577\pi\)
0.455174 + 0.890402i \(0.349577\pi\)
\(510\) 0 0
\(511\) −2.41877e71 −0.407794
\(512\) −6.17329e71 −0.988241
\(513\) 0 0
\(514\) −2.06250e71 −0.297764
\(515\) −3.10733e71 −0.426084
\(516\) 0 0
\(517\) 4.00217e71 0.495217
\(518\) 3.53496e71 0.415572
\(519\) 0 0
\(520\) 9.08471e70 0.0964323
\(521\) 1.04553e72 1.05472 0.527358 0.849643i \(-0.323183\pi\)
0.527358 + 0.849643i \(0.323183\pi\)
\(522\) 0 0
\(523\) −2.01240e71 −0.183408 −0.0917042 0.995786i \(-0.529231\pi\)
−0.0917042 + 0.995786i \(0.529231\pi\)
\(524\) 7.90812e71 0.685163
\(525\) 0 0
\(526\) −1.74037e71 −0.136308
\(527\) −2.46931e72 −1.83906
\(528\) 0 0
\(529\) −1.16032e72 −0.781633
\(530\) −1.18362e71 −0.0758409
\(531\) 0 0
\(532\) 1.83019e72 1.06130
\(533\) 2.02506e71 0.111730
\(534\) 0 0
\(535\) −1.24069e72 −0.619853
\(536\) −1.66306e72 −0.790752
\(537\) 0 0
\(538\) 3.73323e71 0.160825
\(539\) −2.52406e71 −0.103514
\(540\) 0 0
\(541\) −2.67923e72 −0.996053 −0.498026 0.867162i \(-0.665942\pi\)
−0.498026 + 0.867162i \(0.665942\pi\)
\(542\) −7.02090e71 −0.248549
\(543\) 0 0
\(544\) −3.29444e72 −1.05782
\(545\) −3.14278e72 −0.961186
\(546\) 0 0
\(547\) 3.99196e72 1.10795 0.553977 0.832532i \(-0.313110\pi\)
0.553977 + 0.832532i \(0.313110\pi\)
\(548\) 1.39789e72 0.369647
\(549\) 0 0
\(550\) −1.61882e72 −0.388675
\(551\) −2.32175e72 −0.531250
\(552\) 0 0
\(553\) 1.17193e72 0.243604
\(554\) −7.25284e71 −0.143714
\(555\) 0 0
\(556\) 5.31047e71 0.0956425
\(557\) 4.75050e72 0.815786 0.407893 0.913030i \(-0.366264\pi\)
0.407893 + 0.913030i \(0.366264\pi\)
\(558\) 0 0
\(559\) −5.81801e71 −0.0908577
\(560\) −7.98070e72 −1.18866
\(561\) 0 0
\(562\) −7.18488e71 −0.0973660
\(563\) −2.08122e72 −0.269058 −0.134529 0.990910i \(-0.542952\pi\)
−0.134529 + 0.990910i \(0.542952\pi\)
\(564\) 0 0
\(565\) −2.50061e72 −0.294280
\(566\) 1.64157e72 0.184342
\(567\) 0 0
\(568\) −9.70874e71 −0.0992961
\(569\) 8.16033e72 0.796586 0.398293 0.917258i \(-0.369603\pi\)
0.398293 + 0.917258i \(0.369603\pi\)
\(570\) 0 0
\(571\) −5.30282e72 −0.471683 −0.235841 0.971792i \(-0.575785\pi\)
−0.235841 + 0.971792i \(0.575785\pi\)
\(572\) 1.10175e72 0.0935592
\(573\) 0 0
\(574\) 3.24380e72 0.251124
\(575\) 1.01668e73 0.751601
\(576\) 0 0
\(577\) 2.14077e73 1.44348 0.721740 0.692164i \(-0.243343\pi\)
0.721740 + 0.692164i \(0.243343\pi\)
\(578\) −4.46741e72 −0.287718
\(579\) 0 0
\(580\) 1.10828e73 0.651338
\(581\) 7.43039e72 0.417200
\(582\) 0 0
\(583\) −2.98582e72 −0.153055
\(584\) −4.64481e72 −0.227524
\(585\) 0 0
\(586\) −3.06907e72 −0.137316
\(587\) 1.26033e73 0.538981 0.269490 0.963003i \(-0.413145\pi\)
0.269490 + 0.963003i \(0.413145\pi\)
\(588\) 0 0
\(589\) 4.00214e73 1.56400
\(590\) 1.31941e73 0.492950
\(591\) 0 0
\(592\) −3.72282e73 −1.27159
\(593\) 3.38540e73 1.10576 0.552880 0.833261i \(-0.313529\pi\)
0.552880 + 0.833261i \(0.313529\pi\)
\(594\) 0 0
\(595\) −7.28681e73 −2.17690
\(596\) 4.42123e71 0.0126333
\(597\) 0 0
\(598\) 5.54098e71 0.0144879
\(599\) −4.50976e73 −1.12809 −0.564043 0.825745i \(-0.690755\pi\)
−0.564043 + 0.825745i \(0.690755\pi\)
\(600\) 0 0
\(601\) 5.21159e73 1.19343 0.596714 0.802454i \(-0.296473\pi\)
0.596714 + 0.802454i \(0.296473\pi\)
\(602\) −9.31944e72 −0.204213
\(603\) 0 0
\(604\) 3.18096e72 0.0638379
\(605\) 1.78562e73 0.342983
\(606\) 0 0
\(607\) 1.40353e72 0.0247015 0.0123507 0.999924i \(-0.496069\pi\)
0.0123507 + 0.999924i \(0.496069\pi\)
\(608\) 5.33947e73 0.899607
\(609\) 0 0
\(610\) 4.69159e73 0.724569
\(611\) −4.29641e72 −0.0635351
\(612\) 0 0
\(613\) 9.09089e73 1.23283 0.616416 0.787421i \(-0.288584\pi\)
0.616416 + 0.787421i \(0.288584\pi\)
\(614\) 2.64727e72 0.0343824
\(615\) 0 0
\(616\) 3.67093e73 0.437409
\(617\) 1.15906e74 1.32297 0.661486 0.749957i \(-0.269926\pi\)
0.661486 + 0.749957i \(0.269926\pi\)
\(618\) 0 0
\(619\) −1.53802e74 −1.61124 −0.805621 0.592432i \(-0.798168\pi\)
−0.805621 + 0.592432i \(0.798168\pi\)
\(620\) −1.91040e74 −1.91754
\(621\) 0 0
\(622\) −5.39760e73 −0.497457
\(623\) −7.10024e73 −0.627104
\(624\) 0 0
\(625\) −2.64476e72 −0.0214568
\(626\) −2.17506e73 −0.169142
\(627\) 0 0
\(628\) 1.87682e73 0.134119
\(629\) −3.39914e74 −2.32877
\(630\) 0 0
\(631\) −1.90940e74 −1.20259 −0.601297 0.799025i \(-0.705349\pi\)
−0.601297 + 0.799025i \(0.705349\pi\)
\(632\) 2.25047e73 0.135916
\(633\) 0 0
\(634\) 4.88941e73 0.271576
\(635\) −5.69083e72 −0.0303160
\(636\) 0 0
\(637\) 2.70963e72 0.0132805
\(638\) −2.23881e73 −0.105261
\(639\) 0 0
\(640\) −3.34454e74 −1.44733
\(641\) 2.13596e74 0.886860 0.443430 0.896309i \(-0.353761\pi\)
0.443430 + 0.896309i \(0.353761\pi\)
\(642\) 0 0
\(643\) 1.06521e74 0.407235 0.203617 0.979051i \(-0.434730\pi\)
0.203617 + 0.979051i \(0.434730\pi\)
\(644\) −1.10837e74 −0.406638
\(645\) 0 0
\(646\) 1.40928e74 0.476252
\(647\) 3.74264e74 1.21399 0.606995 0.794706i \(-0.292375\pi\)
0.606995 + 0.794706i \(0.292375\pi\)
\(648\) 0 0
\(649\) 3.32836e74 0.994824
\(650\) 1.73783e73 0.0498660
\(651\) 0 0
\(652\) −4.73791e74 −1.25322
\(653\) 5.40144e73 0.137186 0.0685930 0.997645i \(-0.478149\pi\)
0.0685930 + 0.997645i \(0.478149\pi\)
\(654\) 0 0
\(655\) 5.10306e74 1.19519
\(656\) −3.41618e74 −0.768404
\(657\) 0 0
\(658\) −6.88210e73 −0.142802
\(659\) −7.60313e74 −1.51541 −0.757703 0.652599i \(-0.773678\pi\)
−0.757703 + 0.652599i \(0.773678\pi\)
\(660\) 0 0
\(661\) −2.63286e72 −0.00484272 −0.00242136 0.999997i \(-0.500771\pi\)
−0.00242136 + 0.999997i \(0.500771\pi\)
\(662\) 9.62818e73 0.170140
\(663\) 0 0
\(664\) 1.42687e74 0.232772
\(665\) 1.18101e75 1.85131
\(666\) 0 0
\(667\) 1.40606e74 0.203549
\(668\) −4.92630e74 −0.685401
\(669\) 0 0
\(670\) −5.15923e74 −0.663136
\(671\) 1.18351e75 1.46226
\(672\) 0 0
\(673\) −6.59799e74 −0.753378 −0.376689 0.926340i \(-0.622937\pi\)
−0.376689 + 0.926340i \(0.622937\pi\)
\(674\) −2.42038e74 −0.265703
\(675\) 0 0
\(676\) 9.00460e74 0.913855
\(677\) 9.68261e74 0.944915 0.472457 0.881354i \(-0.343367\pi\)
0.472457 + 0.881354i \(0.343367\pi\)
\(678\) 0 0
\(679\) 1.00231e75 0.904596
\(680\) −1.39930e75 −1.21457
\(681\) 0 0
\(682\) 3.85916e74 0.309890
\(683\) 2.38917e75 1.84543 0.922715 0.385483i \(-0.125965\pi\)
0.922715 + 0.385483i \(0.125965\pi\)
\(684\) 0 0
\(685\) 9.02047e74 0.644806
\(686\) 4.15530e74 0.285768
\(687\) 0 0
\(688\) 9.81470e74 0.624861
\(689\) 3.20534e73 0.0196365
\(690\) 0 0
\(691\) −2.74982e74 −0.156004 −0.0780020 0.996953i \(-0.524854\pi\)
−0.0780020 + 0.996953i \(0.524854\pi\)
\(692\) 4.61717e74 0.252095
\(693\) 0 0
\(694\) −7.10298e74 −0.359265
\(695\) 3.42681e74 0.166837
\(696\) 0 0
\(697\) −3.11916e75 −1.40724
\(698\) −3.06570e74 −0.133156
\(699\) 0 0
\(700\) −3.47621e75 −1.39961
\(701\) −5.35713e74 −0.207685 −0.103842 0.994594i \(-0.533114\pi\)
−0.103842 + 0.994594i \(0.533114\pi\)
\(702\) 0 0
\(703\) 5.50916e75 1.98047
\(704\) −1.49255e75 −0.516719
\(705\) 0 0
\(706\) −7.10235e74 −0.228075
\(707\) −5.62298e75 −1.73921
\(708\) 0 0
\(709\) −3.32161e75 −0.953287 −0.476644 0.879097i \(-0.658147\pi\)
−0.476644 + 0.879097i \(0.658147\pi\)
\(710\) −3.01190e74 −0.0832711
\(711\) 0 0
\(712\) −1.36347e75 −0.349885
\(713\) −2.42371e75 −0.599249
\(714\) 0 0
\(715\) 7.10950e74 0.163203
\(716\) −6.60292e75 −1.46063
\(717\) 0 0
\(718\) 8.28737e74 0.170262
\(719\) 6.15825e75 1.21939 0.609694 0.792637i \(-0.291292\pi\)
0.609694 + 0.792637i \(0.291292\pi\)
\(720\) 0 0
\(721\) 1.34792e75 0.247958
\(722\) −7.48536e74 −0.132732
\(723\) 0 0
\(724\) −5.64143e75 −0.929645
\(725\) 4.40987e75 0.700597
\(726\) 0 0
\(727\) 1.07750e76 1.59131 0.795657 0.605747i \(-0.207126\pi\)
0.795657 + 0.605747i \(0.207126\pi\)
\(728\) −3.94082e74 −0.0561184
\(729\) 0 0
\(730\) −1.44094e75 −0.190805
\(731\) 8.96136e75 1.14436
\(732\) 0 0
\(733\) −1.46870e75 −0.174453 −0.0872263 0.996189i \(-0.527800\pi\)
−0.0872263 + 0.996189i \(0.527800\pi\)
\(734\) −1.58923e75 −0.182071
\(735\) 0 0
\(736\) −3.23359e75 −0.344686
\(737\) −1.30147e76 −1.33828
\(738\) 0 0
\(739\) −7.45777e75 −0.713726 −0.356863 0.934157i \(-0.616154\pi\)
−0.356863 + 0.934157i \(0.616154\pi\)
\(740\) −2.62977e76 −2.42816
\(741\) 0 0
\(742\) 5.13440e74 0.0441353
\(743\) 2.17472e76 1.80385 0.901923 0.431897i \(-0.142155\pi\)
0.901923 + 0.431897i \(0.142155\pi\)
\(744\) 0 0
\(745\) 2.85299e74 0.0220374
\(746\) −2.80789e75 −0.209316
\(747\) 0 0
\(748\) −1.69699e76 −1.17839
\(749\) 5.38195e75 0.360721
\(750\) 0 0
\(751\) 8.74960e75 0.546424 0.273212 0.961954i \(-0.411914\pi\)
0.273212 + 0.961954i \(0.411914\pi\)
\(752\) 7.24783e75 0.436953
\(753\) 0 0
\(754\) 2.40341e74 0.0135047
\(755\) 2.05265e75 0.111358
\(756\) 0 0
\(757\) 2.13811e76 1.08141 0.540705 0.841212i \(-0.318158\pi\)
0.540705 + 0.841212i \(0.318158\pi\)
\(758\) 7.25663e75 0.354406
\(759\) 0 0
\(760\) 2.26791e76 1.03292
\(761\) 2.22724e76 0.979658 0.489829 0.871819i \(-0.337059\pi\)
0.489829 + 0.871819i \(0.337059\pi\)
\(762\) 0 0
\(763\) 1.36330e76 0.559358
\(764\) 4.41970e75 0.175154
\(765\) 0 0
\(766\) 1.19381e76 0.441442
\(767\) −3.57307e75 −0.127633
\(768\) 0 0
\(769\) −1.21344e75 −0.0404548 −0.0202274 0.999795i \(-0.506439\pi\)
−0.0202274 + 0.999795i \(0.506439\pi\)
\(770\) 1.13882e76 0.366817
\(771\) 0 0
\(772\) 7.81074e75 0.234873
\(773\) −8.70709e75 −0.252998 −0.126499 0.991967i \(-0.540374\pi\)
−0.126499 + 0.991967i \(0.540374\pi\)
\(774\) 0 0
\(775\) −7.60154e76 −2.06256
\(776\) 1.92476e76 0.504709
\(777\) 0 0
\(778\) −8.64098e75 −0.211643
\(779\) 5.05539e76 1.19677
\(780\) 0 0
\(781\) −7.59785e75 −0.168050
\(782\) −8.53465e75 −0.182476
\(783\) 0 0
\(784\) −4.57101e75 −0.0913349
\(785\) 1.21110e76 0.233956
\(786\) 0 0
\(787\) 7.21858e76 1.30353 0.651766 0.758420i \(-0.274029\pi\)
0.651766 + 0.758420i \(0.274029\pi\)
\(788\) −9.53000e76 −1.66398
\(789\) 0 0
\(790\) 6.98155e75 0.113981
\(791\) 1.08473e76 0.171255
\(792\) 0 0
\(793\) −1.27052e76 −0.187604
\(794\) −7.47084e75 −0.106691
\(795\) 0 0
\(796\) −6.10620e75 −0.0815783
\(797\) 7.22104e76 0.933155 0.466578 0.884480i \(-0.345487\pi\)
0.466578 + 0.884480i \(0.345487\pi\)
\(798\) 0 0
\(799\) 6.61767e76 0.800231
\(800\) −1.01416e77 −1.18638
\(801\) 0 0
\(802\) 1.20632e75 0.0132082
\(803\) −3.63493e76 −0.385065
\(804\) 0 0
\(805\) −7.15223e76 −0.709334
\(806\) −4.14289e75 −0.0397580
\(807\) 0 0
\(808\) −1.07979e77 −0.970372
\(809\) 1.58042e77 1.37447 0.687235 0.726435i \(-0.258824\pi\)
0.687235 + 0.726435i \(0.258824\pi\)
\(810\) 0 0
\(811\) 1.16585e77 0.949707 0.474853 0.880065i \(-0.342501\pi\)
0.474853 + 0.880065i \(0.342501\pi\)
\(812\) −4.80755e76 −0.379044
\(813\) 0 0
\(814\) 5.31234e76 0.392409
\(815\) −3.05735e77 −2.18609
\(816\) 0 0
\(817\) −1.45241e77 −0.973208
\(818\) 1.40230e76 0.0909658
\(819\) 0 0
\(820\) −2.41316e77 −1.46730
\(821\) −2.60575e77 −1.53405 −0.767023 0.641619i \(-0.778263\pi\)
−0.767023 + 0.641619i \(0.778263\pi\)
\(822\) 0 0
\(823\) 3.34341e76 0.184541 0.0922707 0.995734i \(-0.470588\pi\)
0.0922707 + 0.995734i \(0.470588\pi\)
\(824\) 2.58843e76 0.138345
\(825\) 0 0
\(826\) −5.72343e76 −0.286870
\(827\) 1.56452e77 0.759426 0.379713 0.925104i \(-0.376023\pi\)
0.379713 + 0.925104i \(0.376023\pi\)
\(828\) 0 0
\(829\) 2.97591e77 1.35495 0.677477 0.735544i \(-0.263073\pi\)
0.677477 + 0.735544i \(0.263073\pi\)
\(830\) 4.42651e76 0.195206
\(831\) 0 0
\(832\) 1.60229e76 0.0662937
\(833\) −4.17359e76 −0.167269
\(834\) 0 0
\(835\) −3.17891e77 −1.19560
\(836\) 2.75041e77 1.00214
\(837\) 0 0
\(838\) 7.24822e76 0.247893
\(839\) −2.75386e77 −0.912535 −0.456267 0.889843i \(-0.650814\pi\)
−0.456267 + 0.889843i \(0.650814\pi\)
\(840\) 0 0
\(841\) −2.60448e77 −0.810264
\(842\) −1.15758e77 −0.348964
\(843\) 0 0
\(844\) 6.25769e77 1.77149
\(845\) 5.81062e77 1.59411
\(846\) 0 0
\(847\) −7.74579e76 −0.199598
\(848\) −5.40725e76 −0.135048
\(849\) 0 0
\(850\) −2.67675e77 −0.628067
\(851\) −3.33636e77 −0.758821
\(852\) 0 0
\(853\) 6.83575e77 1.46096 0.730478 0.682936i \(-0.239297\pi\)
0.730478 + 0.682936i \(0.239297\pi\)
\(854\) −2.03515e77 −0.421660
\(855\) 0 0
\(856\) 1.03350e77 0.201260
\(857\) 6.22171e77 1.17467 0.587337 0.809342i \(-0.300176\pi\)
0.587337 + 0.809342i \(0.300176\pi\)
\(858\) 0 0
\(859\) −4.11261e77 −0.729960 −0.364980 0.931015i \(-0.618924\pi\)
−0.364980 + 0.931015i \(0.618924\pi\)
\(860\) 6.93302e77 1.19320
\(861\) 0 0
\(862\) −2.77277e77 −0.448716
\(863\) 3.72761e77 0.584984 0.292492 0.956268i \(-0.405516\pi\)
0.292492 + 0.956268i \(0.405516\pi\)
\(864\) 0 0
\(865\) 2.97943e77 0.439751
\(866\) 3.32993e77 0.476663
\(867\) 0 0
\(868\) 8.28706e77 1.11591
\(869\) 1.76117e77 0.230026
\(870\) 0 0
\(871\) 1.39716e77 0.171698
\(872\) 2.61796e77 0.312087
\(873\) 0 0
\(874\) 1.38325e77 0.155185
\(875\) −8.48510e77 −0.923514
\(876\) 0 0
\(877\) 4.87970e77 0.499924 0.249962 0.968256i \(-0.419582\pi\)
0.249962 + 0.968256i \(0.419582\pi\)
\(878\) −5.45177e76 −0.0541917
\(879\) 0 0
\(880\) −1.19934e78 −1.12241
\(881\) −1.30154e78 −1.18194 −0.590970 0.806693i \(-0.701255\pi\)
−0.590970 + 0.806693i \(0.701255\pi\)
\(882\) 0 0
\(883\) 3.47958e77 0.297556 0.148778 0.988871i \(-0.452466\pi\)
0.148778 + 0.988871i \(0.452466\pi\)
\(884\) 1.82176e77 0.151184
\(885\) 0 0
\(886\) 4.58710e77 0.358545
\(887\) 1.85101e78 1.40421 0.702107 0.712071i \(-0.252243\pi\)
0.702107 + 0.712071i \(0.252243\pi\)
\(888\) 0 0
\(889\) 2.46861e76 0.0176423
\(890\) −4.22984e77 −0.293419
\(891\) 0 0
\(892\) 1.01905e77 0.0666080
\(893\) −1.07256e78 −0.680546
\(894\) 0 0
\(895\) −4.26082e78 −2.54790
\(896\) 1.45081e78 0.842267
\(897\) 0 0
\(898\) −2.03521e77 −0.111374
\(899\) −1.05128e78 −0.558584
\(900\) 0 0
\(901\) −4.93712e77 −0.247324
\(902\) 4.87478e77 0.237127
\(903\) 0 0
\(904\) 2.08302e77 0.0955499
\(905\) −3.64038e78 −1.62166
\(906\) 0 0
\(907\) 1.94140e78 0.815686 0.407843 0.913052i \(-0.366281\pi\)
0.407843 + 0.913052i \(0.366281\pi\)
\(908\) −4.19456e77 −0.171163
\(909\) 0 0
\(910\) −1.22254e77 −0.0470617
\(911\) −6.52188e75 −0.00243857 −0.00121929 0.999999i \(-0.500388\pi\)
−0.00121929 + 0.999999i \(0.500388\pi\)
\(912\) 0 0
\(913\) 1.11664e78 0.393946
\(914\) −9.01838e77 −0.309068
\(915\) 0 0
\(916\) 2.84632e78 0.920563
\(917\) −2.21364e78 −0.695536
\(918\) 0 0
\(919\) 1.20106e78 0.356208 0.178104 0.984012i \(-0.443004\pi\)
0.178104 + 0.984012i \(0.443004\pi\)
\(920\) −1.37345e78 −0.395764
\(921\) 0 0
\(922\) 1.23971e78 0.337250
\(923\) 8.15645e76 0.0215604
\(924\) 0 0
\(925\) −1.04639e79 −2.61179
\(926\) 1.31683e78 0.319404
\(927\) 0 0
\(928\) −1.40257e78 −0.321295
\(929\) 1.35743e78 0.302205 0.151103 0.988518i \(-0.451718\pi\)
0.151103 + 0.988518i \(0.451718\pi\)
\(930\) 0 0
\(931\) 6.76434e77 0.142252
\(932\) 4.85692e78 0.992748
\(933\) 0 0
\(934\) 2.25881e78 0.436202
\(935\) −1.09506e79 −2.05556
\(936\) 0 0
\(937\) −7.78543e77 −0.138097 −0.0690483 0.997613i \(-0.521996\pi\)
−0.0690483 + 0.997613i \(0.521996\pi\)
\(938\) 2.23800e78 0.385909
\(939\) 0 0
\(940\) 5.11980e78 0.834382
\(941\) 8.40474e77 0.133168 0.0665839 0.997781i \(-0.478790\pi\)
0.0665839 + 0.997781i \(0.478790\pi\)
\(942\) 0 0
\(943\) −3.06155e78 −0.458545
\(944\) 6.02759e78 0.877780
\(945\) 0 0
\(946\) −1.40052e78 −0.192830
\(947\) −6.73558e78 −0.901780 −0.450890 0.892579i \(-0.648893\pi\)
−0.450890 + 0.892579i \(0.648893\pi\)
\(948\) 0 0
\(949\) 3.90217e77 0.0494028
\(950\) 4.33834e78 0.534132
\(951\) 0 0
\(952\) 6.06996e78 0.706817
\(953\) 1.66845e79 1.88952 0.944758 0.327768i \(-0.106296\pi\)
0.944758 + 0.327768i \(0.106296\pi\)
\(954\) 0 0
\(955\) 2.85201e78 0.305535
\(956\) 8.47563e78 0.883157
\(957\) 0 0
\(958\) −2.42626e78 −0.239194
\(959\) −3.91296e78 −0.375242
\(960\) 0 0
\(961\) 7.10190e78 0.644473
\(962\) −5.70290e77 −0.0503450
\(963\) 0 0
\(964\) −1.75519e79 −1.46650
\(965\) 5.04022e78 0.409709
\(966\) 0 0
\(967\) 2.07490e79 1.59660 0.798301 0.602259i \(-0.205733\pi\)
0.798301 + 0.602259i \(0.205733\pi\)
\(968\) −1.48744e78 −0.111363
\(969\) 0 0
\(970\) 5.97110e78 0.423256
\(971\) 7.04775e78 0.486117 0.243059 0.970012i \(-0.421849\pi\)
0.243059 + 0.970012i \(0.421849\pi\)
\(972\) 0 0
\(973\) −1.48651e78 −0.0970904
\(974\) −8.45487e78 −0.537396
\(975\) 0 0
\(976\) 2.14330e79 1.29022
\(977\) −1.67318e79 −0.980247 −0.490123 0.871653i \(-0.663048\pi\)
−0.490123 + 0.871653i \(0.663048\pi\)
\(978\) 0 0
\(979\) −1.06702e79 −0.592151
\(980\) −3.22892e78 −0.174408
\(981\) 0 0
\(982\) −7.14945e78 −0.365863
\(983\) 5.74735e78 0.286286 0.143143 0.989702i \(-0.454279\pi\)
0.143143 + 0.989702i \(0.454279\pi\)
\(984\) 0 0
\(985\) −6.14966e79 −2.90262
\(986\) −3.70191e78 −0.170094
\(987\) 0 0
\(988\) −2.95262e78 −0.128573
\(989\) 8.79585e78 0.372886
\(990\) 0 0
\(991\) 2.54858e79 1.02411 0.512055 0.858953i \(-0.328885\pi\)
0.512055 + 0.858953i \(0.328885\pi\)
\(992\) 2.41770e79 0.945894
\(993\) 0 0
\(994\) 1.30652e78 0.0484593
\(995\) −3.94029e78 −0.142304
\(996\) 0 0
\(997\) 4.70743e79 1.61199 0.805994 0.591924i \(-0.201631\pi\)
0.805994 + 0.591924i \(0.201631\pi\)
\(998\) 1.24394e79 0.414802
\(999\) 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.1.2 4
3.2 odd 2 1.54.a.a.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.54.a.a.1.3 4 3.2 odd 2
9.54.a.b.1.2 4 1.1 even 1 trivial