Properties

Label 315.10.a.a.1.1
Level $315$
Weight $10$
Character 315.1
Self dual yes
Analytic conductor $162.236$
Analytic rank $1$
Dimension $1$
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] = [315,10,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(162.236288392\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-28.0000 q^{2} +272.000 q^{4} -625.000 q^{5} +2401.00 q^{7} +6720.00 q^{8} +O(q^{10})\) \(q-28.0000 q^{2} +272.000 q^{4} -625.000 q^{5} +2401.00 q^{7} +6720.00 q^{8} +17500.0 q^{10} +25548.0 q^{11} -42306.0 q^{13} -67228.0 q^{14} -327424. q^{16} +526342. q^{17} -350060. q^{19} -170000. q^{20} -715344. q^{22} +621976. q^{23} +390625. q^{25} +1.18457e6 q^{26} +653072. q^{28} -6.72043e6 q^{29} -6.41221e6 q^{31} +5.72723e6 q^{32} -1.47376e7 q^{34} -1.50062e6 q^{35} -2.31768e6 q^{37} +9.80168e6 q^{38} -4.20000e6 q^{40} +1.02247e7 q^{41} +3.01140e7 q^{43} +6.94906e6 q^{44} -1.74153e7 q^{46} +2.36449e7 q^{47} +5.76480e6 q^{49} -1.09375e7 q^{50} -1.15072e7 q^{52} -5.72927e7 q^{53} -1.59675e7 q^{55} +1.61347e7 q^{56} +1.88172e8 q^{58} -8.49348e7 q^{59} +1.46778e7 q^{61} +1.79542e8 q^{62} +7.27859e6 q^{64} +2.64412e7 q^{65} -2.44558e8 q^{67} +1.43165e8 q^{68} +4.20175e7 q^{70} -6.19020e7 q^{71} -2.83764e8 q^{73} +6.48951e7 q^{74} -9.52163e7 q^{76} +6.13407e7 q^{77} +2.76107e8 q^{79} +2.04640e8 q^{80} -2.86291e8 q^{82} +7.29960e7 q^{83} -3.28964e8 q^{85} -8.43192e8 q^{86} +1.71683e8 q^{88} +8.96368e8 q^{89} -1.01577e8 q^{91} +1.69177e8 q^{92} -6.62058e8 q^{94} +2.18788e8 q^{95} +1.20581e9 q^{97} -1.61414e8 q^{98} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −28.0000 −1.23744 −0.618718 0.785613i \(-0.712348\pi\)
−0.618718 + 0.785613i \(0.712348\pi\)
\(3\) 0 0
\(4\) 272.000 0.531250
\(5\) −625.000 −0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 6720.00 0.580049
\(9\) 0 0
\(10\) 17500.0 0.553399
\(11\) 25548.0 0.526126 0.263063 0.964779i \(-0.415267\pi\)
0.263063 + 0.964779i \(0.415267\pi\)
\(12\) 0 0
\(13\) −42306.0 −0.410825 −0.205413 0.978675i \(-0.565854\pi\)
−0.205413 + 0.978675i \(0.565854\pi\)
\(14\) −67228.0 −0.467707
\(15\) 0 0
\(16\) −327424. −1.24902
\(17\) 526342. 1.52844 0.764219 0.644957i \(-0.223125\pi\)
0.764219 + 0.644957i \(0.223125\pi\)
\(18\) 0 0
\(19\) −350060. −0.616242 −0.308121 0.951347i \(-0.599700\pi\)
−0.308121 + 0.951347i \(0.599700\pi\)
\(20\) −170000. −0.237582
\(21\) 0 0
\(22\) −715344. −0.651048
\(23\) 621976. 0.463445 0.231723 0.972782i \(-0.425564\pi\)
0.231723 + 0.972782i \(0.425564\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 1.18457e6 0.508370
\(27\) 0 0
\(28\) 653072. 0.200794
\(29\) −6.72043e6 −1.76444 −0.882218 0.470841i \(-0.843951\pi\)
−0.882218 + 0.470841i \(0.843951\pi\)
\(30\) 0 0
\(31\) −6.41221e6 −1.24704 −0.623519 0.781808i \(-0.714298\pi\)
−0.623519 + 0.781808i \(0.714298\pi\)
\(32\) 5.72723e6 0.965539
\(33\) 0 0
\(34\) −1.47376e7 −1.89135
\(35\) −1.50062e6 −0.169031
\(36\) 0 0
\(37\) −2.31768e6 −0.203304 −0.101652 0.994820i \(-0.532413\pi\)
−0.101652 + 0.994820i \(0.532413\pi\)
\(38\) 9.80168e6 0.762561
\(39\) 0 0
\(40\) −4.20000e6 −0.259406
\(41\) 1.02247e7 0.565096 0.282548 0.959253i \(-0.408820\pi\)
0.282548 + 0.959253i \(0.408820\pi\)
\(42\) 0 0
\(43\) 3.01140e7 1.34326 0.671631 0.740886i \(-0.265594\pi\)
0.671631 + 0.740886i \(0.265594\pi\)
\(44\) 6.94906e6 0.279504
\(45\) 0 0
\(46\) −1.74153e7 −0.573484
\(47\) 2.36449e7 0.706801 0.353401 0.935472i \(-0.385025\pi\)
0.353401 + 0.935472i \(0.385025\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −1.09375e7 −0.247487
\(51\) 0 0
\(52\) −1.15072e7 −0.218251
\(53\) −5.72927e7 −0.997373 −0.498686 0.866782i \(-0.666184\pi\)
−0.498686 + 0.866782i \(0.666184\pi\)
\(54\) 0 0
\(55\) −1.59675e7 −0.235291
\(56\) 1.61347e7 0.219238
\(57\) 0 0
\(58\) 1.88172e8 2.18338
\(59\) −8.49348e7 −0.912539 −0.456270 0.889842i \(-0.650815\pi\)
−0.456270 + 0.889842i \(0.650815\pi\)
\(60\) 0 0
\(61\) 1.46778e7 0.135730 0.0678652 0.997694i \(-0.478381\pi\)
0.0678652 + 0.997694i \(0.478381\pi\)
\(62\) 1.79542e8 1.54313
\(63\) 0 0
\(64\) 7.27859e6 0.0542297
\(65\) 2.64412e7 0.183727
\(66\) 0 0
\(67\) −2.44558e8 −1.48267 −0.741336 0.671134i \(-0.765807\pi\)
−0.741336 + 0.671134i \(0.765807\pi\)
\(68\) 1.43165e8 0.811983
\(69\) 0 0
\(70\) 4.20175e7 0.209165
\(71\) −6.19020e7 −0.289096 −0.144548 0.989498i \(-0.546173\pi\)
−0.144548 + 0.989498i \(0.546173\pi\)
\(72\) 0 0
\(73\) −2.83764e8 −1.16951 −0.584755 0.811210i \(-0.698809\pi\)
−0.584755 + 0.811210i \(0.698809\pi\)
\(74\) 6.48951e7 0.251576
\(75\) 0 0
\(76\) −9.52163e7 −0.327379
\(77\) 6.13407e7 0.198857
\(78\) 0 0
\(79\) 2.76107e8 0.797547 0.398773 0.917049i \(-0.369436\pi\)
0.398773 + 0.917049i \(0.369436\pi\)
\(80\) 2.04640e8 0.558580
\(81\) 0 0
\(82\) −2.86291e8 −0.699271
\(83\) 7.29960e7 0.168829 0.0844146 0.996431i \(-0.473098\pi\)
0.0844146 + 0.996431i \(0.473098\pi\)
\(84\) 0 0
\(85\) −3.28964e8 −0.683538
\(86\) −8.43192e8 −1.66220
\(87\) 0 0
\(88\) 1.71683e8 0.305179
\(89\) 8.96368e8 1.51437 0.757184 0.653201i \(-0.226575\pi\)
0.757184 + 0.653201i \(0.226575\pi\)
\(90\) 0 0
\(91\) −1.01577e8 −0.155277
\(92\) 1.69177e8 0.246205
\(93\) 0 0
\(94\) −6.62058e8 −0.874622
\(95\) 2.18788e8 0.275592
\(96\) 0 0
\(97\) 1.20581e9 1.38295 0.691474 0.722401i \(-0.256962\pi\)
0.691474 + 0.722401i \(0.256962\pi\)
\(98\) −1.61414e8 −0.176777
\(99\) 0 0
\(100\) 1.06250e8 0.106250
\(101\) 1.46021e9 1.39627 0.698136 0.715965i \(-0.254013\pi\)
0.698136 + 0.715965i \(0.254013\pi\)
\(102\) 0 0
\(103\) 1.08009e9 0.945563 0.472782 0.881180i \(-0.343250\pi\)
0.472782 + 0.881180i \(0.343250\pi\)
\(104\) −2.84296e8 −0.238298
\(105\) 0 0
\(106\) 1.60419e9 1.23419
\(107\) −3.35949e8 −0.247769 −0.123884 0.992297i \(-0.539535\pi\)
−0.123884 + 0.992297i \(0.539535\pi\)
\(108\) 0 0
\(109\) −1.42521e9 −0.967072 −0.483536 0.875324i \(-0.660648\pi\)
−0.483536 + 0.875324i \(0.660648\pi\)
\(110\) 4.47090e8 0.291157
\(111\) 0 0
\(112\) −7.86145e8 −0.472086
\(113\) 2.84178e9 1.63960 0.819799 0.572651i \(-0.194085\pi\)
0.819799 + 0.572651i \(0.194085\pi\)
\(114\) 0 0
\(115\) −3.88735e8 −0.207259
\(116\) −1.82796e9 −0.937357
\(117\) 0 0
\(118\) 2.37817e9 1.12921
\(119\) 1.26375e9 0.577695
\(120\) 0 0
\(121\) −1.70525e9 −0.723191
\(122\) −4.10979e8 −0.167958
\(123\) 0 0
\(124\) −1.74412e9 −0.662489
\(125\) −2.44141e8 −0.0894427
\(126\) 0 0
\(127\) 3.49339e9 1.19160 0.595800 0.803133i \(-0.296835\pi\)
0.595800 + 0.803133i \(0.296835\pi\)
\(128\) −3.13614e9 −1.03264
\(129\) 0 0
\(130\) −7.40355e8 −0.227350
\(131\) 1.84697e9 0.547946 0.273973 0.961737i \(-0.411662\pi\)
0.273973 + 0.961737i \(0.411662\pi\)
\(132\) 0 0
\(133\) −8.40494e8 −0.232918
\(134\) 6.84762e9 1.83471
\(135\) 0 0
\(136\) 3.53702e9 0.886568
\(137\) −1.17238e9 −0.284331 −0.142166 0.989843i \(-0.545407\pi\)
−0.142166 + 0.989843i \(0.545407\pi\)
\(138\) 0 0
\(139\) −4.89001e9 −1.11108 −0.555538 0.831491i \(-0.687488\pi\)
−0.555538 + 0.831491i \(0.687488\pi\)
\(140\) −4.08170e8 −0.0897976
\(141\) 0 0
\(142\) 1.73325e9 0.357738
\(143\) −1.08083e9 −0.216146
\(144\) 0 0
\(145\) 4.20027e9 0.789080
\(146\) 7.94538e9 1.44720
\(147\) 0 0
\(148\) −6.30410e8 −0.108005
\(149\) 8.61488e9 1.43189 0.715947 0.698155i \(-0.245995\pi\)
0.715947 + 0.698155i \(0.245995\pi\)
\(150\) 0 0
\(151\) 5.48905e9 0.859213 0.429607 0.903016i \(-0.358652\pi\)
0.429607 + 0.903016i \(0.358652\pi\)
\(152\) −2.35240e9 −0.357450
\(153\) 0 0
\(154\) −1.71754e9 −0.246073
\(155\) 4.00763e9 0.557693
\(156\) 0 0
\(157\) −1.33110e9 −0.174849 −0.0874246 0.996171i \(-0.527864\pi\)
−0.0874246 + 0.996171i \(0.527864\pi\)
\(158\) −7.73101e9 −0.986914
\(159\) 0 0
\(160\) −3.57952e9 −0.431802
\(161\) 1.49336e9 0.175166
\(162\) 0 0
\(163\) 1.41097e9 0.156557 0.0782786 0.996932i \(-0.475058\pi\)
0.0782786 + 0.996932i \(0.475058\pi\)
\(164\) 2.78111e9 0.300207
\(165\) 0 0
\(166\) −2.04389e9 −0.208915
\(167\) −2.48555e8 −0.0247285 −0.0123642 0.999924i \(-0.503936\pi\)
−0.0123642 + 0.999924i \(0.503936\pi\)
\(168\) 0 0
\(169\) −8.81470e9 −0.831223
\(170\) 9.21098e9 0.845836
\(171\) 0 0
\(172\) 8.19101e9 0.713607
\(173\) −1.66522e10 −1.41340 −0.706699 0.707515i \(-0.749816\pi\)
−0.706699 + 0.707515i \(0.749816\pi\)
\(174\) 0 0
\(175\) 9.37891e8 0.0755929
\(176\) −8.36503e9 −0.657144
\(177\) 0 0
\(178\) −2.50983e10 −1.87394
\(179\) −2.02956e10 −1.47762 −0.738811 0.673913i \(-0.764612\pi\)
−0.738811 + 0.673913i \(0.764612\pi\)
\(180\) 0 0
\(181\) −1.36159e10 −0.942960 −0.471480 0.881877i \(-0.656280\pi\)
−0.471480 + 0.881877i \(0.656280\pi\)
\(182\) 2.84415e9 0.192146
\(183\) 0 0
\(184\) 4.17968e9 0.268821
\(185\) 1.44855e9 0.0909203
\(186\) 0 0
\(187\) 1.34470e10 0.804151
\(188\) 6.43142e9 0.375488
\(189\) 0 0
\(190\) −6.12605e9 −0.341027
\(191\) −1.82357e9 −0.0991453 −0.0495726 0.998771i \(-0.515786\pi\)
−0.0495726 + 0.998771i \(0.515786\pi\)
\(192\) 0 0
\(193\) 1.23747e10 0.641989 0.320995 0.947081i \(-0.395983\pi\)
0.320995 + 0.947081i \(0.395983\pi\)
\(194\) −3.37627e10 −1.71131
\(195\) 0 0
\(196\) 1.56803e9 0.0758929
\(197\) −1.93137e10 −0.913626 −0.456813 0.889563i \(-0.651009\pi\)
−0.456813 + 0.889563i \(0.651009\pi\)
\(198\) 0 0
\(199\) 1.52145e10 0.687733 0.343867 0.939019i \(-0.388263\pi\)
0.343867 + 0.939019i \(0.388263\pi\)
\(200\) 2.62500e9 0.116010
\(201\) 0 0
\(202\) −4.08860e10 −1.72780
\(203\) −1.61358e10 −0.666894
\(204\) 0 0
\(205\) −6.39042e9 −0.252719
\(206\) −3.02424e10 −1.17007
\(207\) 0 0
\(208\) 1.38520e10 0.513130
\(209\) −8.94333e9 −0.324221
\(210\) 0 0
\(211\) −3.89626e10 −1.35325 −0.676624 0.736329i \(-0.736558\pi\)
−0.676624 + 0.736329i \(0.736558\pi\)
\(212\) −1.55836e10 −0.529854
\(213\) 0 0
\(214\) 9.40658e9 0.306598
\(215\) −1.88213e10 −0.600725
\(216\) 0 0
\(217\) −1.53957e10 −0.471336
\(218\) 3.99058e10 1.19669
\(219\) 0 0
\(220\) −4.34316e9 −0.124998
\(221\) −2.22674e10 −0.627921
\(222\) 0 0
\(223\) 1.08324e9 0.0293328 0.0146664 0.999892i \(-0.495331\pi\)
0.0146664 + 0.999892i \(0.495331\pi\)
\(224\) 1.37511e10 0.364939
\(225\) 0 0
\(226\) −7.95698e10 −2.02890
\(227\) 4.94618e10 1.23639 0.618193 0.786027i \(-0.287865\pi\)
0.618193 + 0.786027i \(0.287865\pi\)
\(228\) 0 0
\(229\) −4.32776e10 −1.03993 −0.519965 0.854188i \(-0.674055\pi\)
−0.519965 + 0.854188i \(0.674055\pi\)
\(230\) 1.08846e10 0.256470
\(231\) 0 0
\(232\) −4.51613e10 −1.02346
\(233\) 7.55367e9 0.167902 0.0839511 0.996470i \(-0.473246\pi\)
0.0839511 + 0.996470i \(0.473246\pi\)
\(234\) 0 0
\(235\) −1.47781e10 −0.316091
\(236\) −2.31023e10 −0.484786
\(237\) 0 0
\(238\) −3.53849e10 −0.714862
\(239\) 2.76516e10 0.548188 0.274094 0.961703i \(-0.411622\pi\)
0.274094 + 0.961703i \(0.411622\pi\)
\(240\) 0 0
\(241\) −8.26006e10 −1.57727 −0.788635 0.614861i \(-0.789212\pi\)
−0.788635 + 0.614861i \(0.789212\pi\)
\(242\) 4.77469e10 0.894904
\(243\) 0 0
\(244\) 3.99237e9 0.0721068
\(245\) −3.60300e9 −0.0638877
\(246\) 0 0
\(247\) 1.48096e10 0.253168
\(248\) −4.30900e10 −0.723343
\(249\) 0 0
\(250\) 6.83594e9 0.110680
\(251\) −2.01817e10 −0.320942 −0.160471 0.987041i \(-0.551301\pi\)
−0.160471 + 0.987041i \(0.551301\pi\)
\(252\) 0 0
\(253\) 1.58902e10 0.243831
\(254\) −9.78150e10 −1.47453
\(255\) 0 0
\(256\) 8.40854e10 1.22360
\(257\) −2.82781e10 −0.404344 −0.202172 0.979350i \(-0.564800\pi\)
−0.202172 + 0.979350i \(0.564800\pi\)
\(258\) 0 0
\(259\) −5.56475e9 −0.0768417
\(260\) 7.19202e9 0.0976047
\(261\) 0 0
\(262\) −5.17150e10 −0.678049
\(263\) −1.39139e11 −1.79328 −0.896642 0.442756i \(-0.854001\pi\)
−0.896642 + 0.442756i \(0.854001\pi\)
\(264\) 0 0
\(265\) 3.58079e10 0.446039
\(266\) 2.35338e10 0.288221
\(267\) 0 0
\(268\) −6.65197e10 −0.787669
\(269\) −5.78883e9 −0.0674071 −0.0337035 0.999432i \(-0.510730\pi\)
−0.0337035 + 0.999432i \(0.510730\pi\)
\(270\) 0 0
\(271\) −7.46910e10 −0.841214 −0.420607 0.907243i \(-0.638183\pi\)
−0.420607 + 0.907243i \(0.638183\pi\)
\(272\) −1.72337e11 −1.90906
\(273\) 0 0
\(274\) 3.28265e10 0.351842
\(275\) 9.97969e9 0.105225
\(276\) 0 0
\(277\) 2.22355e10 0.226928 0.113464 0.993542i \(-0.463805\pi\)
0.113464 + 0.993542i \(0.463805\pi\)
\(278\) 1.36920e11 1.37489
\(279\) 0 0
\(280\) −1.00842e10 −0.0980461
\(281\) 1.36058e11 1.30180 0.650901 0.759162i \(-0.274391\pi\)
0.650901 + 0.759162i \(0.274391\pi\)
\(282\) 0 0
\(283\) −1.71084e11 −1.58551 −0.792757 0.609538i \(-0.791355\pi\)
−0.792757 + 0.609538i \(0.791355\pi\)
\(284\) −1.68373e10 −0.153582
\(285\) 0 0
\(286\) 3.02633e10 0.267467
\(287\) 2.45495e10 0.213586
\(288\) 0 0
\(289\) 1.58448e11 1.33612
\(290\) −1.17608e11 −0.976437
\(291\) 0 0
\(292\) −7.71837e10 −0.621302
\(293\) −1.05732e11 −0.838115 −0.419058 0.907960i \(-0.637639\pi\)
−0.419058 + 0.907960i \(0.637639\pi\)
\(294\) 0 0
\(295\) 5.30842e10 0.408100
\(296\) −1.55748e10 −0.117926
\(297\) 0 0
\(298\) −2.41217e11 −1.77188
\(299\) −2.63133e10 −0.190395
\(300\) 0 0
\(301\) 7.23037e10 0.507705
\(302\) −1.53693e11 −1.06322
\(303\) 0 0
\(304\) 1.14618e11 0.769701
\(305\) −9.17364e9 −0.0607005
\(306\) 0 0
\(307\) 1.74144e11 1.11889 0.559443 0.828869i \(-0.311015\pi\)
0.559443 + 0.828869i \(0.311015\pi\)
\(308\) 1.66847e10 0.105643
\(309\) 0 0
\(310\) −1.12214e11 −0.690110
\(311\) 1.05907e11 0.641954 0.320977 0.947087i \(-0.395989\pi\)
0.320977 + 0.947087i \(0.395989\pi\)
\(312\) 0 0
\(313\) −2.43558e11 −1.43434 −0.717171 0.696897i \(-0.754563\pi\)
−0.717171 + 0.696897i \(0.754563\pi\)
\(314\) 3.72709e10 0.216365
\(315\) 0 0
\(316\) 7.51012e10 0.423697
\(317\) −1.83776e11 −1.02217 −0.511083 0.859531i \(-0.670755\pi\)
−0.511083 + 0.859531i \(0.670755\pi\)
\(318\) 0 0
\(319\) −1.71694e11 −0.928316
\(320\) −4.54912e9 −0.0242523
\(321\) 0 0
\(322\) −4.18142e10 −0.216757
\(323\) −1.84251e11 −0.941888
\(324\) 0 0
\(325\) −1.65258e10 −0.0821650
\(326\) −3.95071e10 −0.193730
\(327\) 0 0
\(328\) 6.87098e10 0.327783
\(329\) 5.67714e10 0.267146
\(330\) 0 0
\(331\) −5.81760e10 −0.266390 −0.133195 0.991090i \(-0.542524\pi\)
−0.133195 + 0.991090i \(0.542524\pi\)
\(332\) 1.98549e10 0.0896905
\(333\) 0 0
\(334\) 6.95953e9 0.0305999
\(335\) 1.52849e11 0.663071
\(336\) 0 0
\(337\) −3.40267e11 −1.43709 −0.718547 0.695478i \(-0.755193\pi\)
−0.718547 + 0.695478i \(0.755193\pi\)
\(338\) 2.46812e11 1.02859
\(339\) 0 0
\(340\) −8.94781e10 −0.363130
\(341\) −1.63819e11 −0.656100
\(342\) 0 0
\(343\) 1.38413e10 0.0539949
\(344\) 2.02366e11 0.779157
\(345\) 0 0
\(346\) 4.66262e11 1.74899
\(347\) 5.02625e11 1.86107 0.930533 0.366208i \(-0.119344\pi\)
0.930533 + 0.366208i \(0.119344\pi\)
\(348\) 0 0
\(349\) 7.14710e10 0.257879 0.128939 0.991652i \(-0.458843\pi\)
0.128939 + 0.991652i \(0.458843\pi\)
\(350\) −2.62609e10 −0.0935414
\(351\) 0 0
\(352\) 1.46319e11 0.507995
\(353\) 2.55096e10 0.0874414 0.0437207 0.999044i \(-0.486079\pi\)
0.0437207 + 0.999044i \(0.486079\pi\)
\(354\) 0 0
\(355\) 3.86887e10 0.129288
\(356\) 2.43812e11 0.804508
\(357\) 0 0
\(358\) 5.68277e11 1.82846
\(359\) −1.49816e11 −0.476029 −0.238014 0.971262i \(-0.576497\pi\)
−0.238014 + 0.971262i \(0.576497\pi\)
\(360\) 0 0
\(361\) −2.00146e11 −0.620246
\(362\) 3.81246e11 1.16685
\(363\) 0 0
\(364\) −2.76289e10 −0.0824910
\(365\) 1.77352e11 0.523021
\(366\) 0 0
\(367\) −4.59514e11 −1.32221 −0.661107 0.750291i \(-0.729913\pi\)
−0.661107 + 0.750291i \(0.729913\pi\)
\(368\) −2.03650e11 −0.578854
\(369\) 0 0
\(370\) −4.05594e10 −0.112508
\(371\) −1.37560e11 −0.376971
\(372\) 0 0
\(373\) −5.04230e11 −1.34877 −0.674386 0.738379i \(-0.735592\pi\)
−0.674386 + 0.738379i \(0.735592\pi\)
\(374\) −3.76516e11 −0.995086
\(375\) 0 0
\(376\) 1.58894e11 0.409979
\(377\) 2.84315e11 0.724875
\(378\) 0 0
\(379\) −9.63136e10 −0.239779 −0.119889 0.992787i \(-0.538254\pi\)
−0.119889 + 0.992787i \(0.538254\pi\)
\(380\) 5.95102e10 0.146408
\(381\) 0 0
\(382\) 5.10599e10 0.122686
\(383\) −6.10835e11 −1.45054 −0.725269 0.688465i \(-0.758285\pi\)
−0.725269 + 0.688465i \(0.758285\pi\)
\(384\) 0 0
\(385\) −3.83380e10 −0.0889315
\(386\) −3.46492e11 −0.794421
\(387\) 0 0
\(388\) 3.27980e11 0.734691
\(389\) 6.49908e11 1.43906 0.719530 0.694461i \(-0.244357\pi\)
0.719530 + 0.694461i \(0.244357\pi\)
\(390\) 0 0
\(391\) 3.27372e11 0.708347
\(392\) 3.87395e10 0.0828641
\(393\) 0 0
\(394\) 5.40785e11 1.13055
\(395\) −1.72567e11 −0.356674
\(396\) 0 0
\(397\) −3.67168e11 −0.741836 −0.370918 0.928666i \(-0.620957\pi\)
−0.370918 + 0.928666i \(0.620957\pi\)
\(398\) −4.26007e11 −0.851026
\(399\) 0 0
\(400\) −1.27900e11 −0.249805
\(401\) 9.73985e11 1.88106 0.940530 0.339712i \(-0.110330\pi\)
0.940530 + 0.339712i \(0.110330\pi\)
\(402\) 0 0
\(403\) 2.71275e11 0.512315
\(404\) 3.97178e11 0.741770
\(405\) 0 0
\(406\) 4.51801e11 0.825240
\(407\) −5.92121e10 −0.106964
\(408\) 0 0
\(409\) 3.58196e11 0.632944 0.316472 0.948602i \(-0.397502\pi\)
0.316472 + 0.948602i \(0.397502\pi\)
\(410\) 1.78932e11 0.312723
\(411\) 0 0
\(412\) 2.93783e11 0.502330
\(413\) −2.03928e11 −0.344907
\(414\) 0 0
\(415\) −4.56225e10 −0.0755027
\(416\) −2.42296e11 −0.396668
\(417\) 0 0
\(418\) 2.50413e11 0.401203
\(419\) −4.84712e11 −0.768283 −0.384141 0.923274i \(-0.625502\pi\)
−0.384141 + 0.923274i \(0.625502\pi\)
\(420\) 0 0
\(421\) 7.18298e11 1.11438 0.557192 0.830384i \(-0.311879\pi\)
0.557192 + 0.830384i \(0.311879\pi\)
\(422\) 1.09095e12 1.67456
\(423\) 0 0
\(424\) −3.85007e11 −0.578525
\(425\) 2.05602e11 0.305688
\(426\) 0 0
\(427\) 3.52415e10 0.0513013
\(428\) −9.13782e10 −0.131627
\(429\) 0 0
\(430\) 5.26995e11 0.743359
\(431\) −8.27297e11 −1.15482 −0.577409 0.816455i \(-0.695936\pi\)
−0.577409 + 0.816455i \(0.695936\pi\)
\(432\) 0 0
\(433\) 8.73032e11 1.19353 0.596767 0.802415i \(-0.296452\pi\)
0.596767 + 0.802415i \(0.296452\pi\)
\(434\) 4.31080e11 0.583249
\(435\) 0 0
\(436\) −3.87657e11 −0.513757
\(437\) −2.17729e11 −0.285594
\(438\) 0 0
\(439\) 7.15061e11 0.918867 0.459433 0.888212i \(-0.348053\pi\)
0.459433 + 0.888212i \(0.348053\pi\)
\(440\) −1.07302e11 −0.136480
\(441\) 0 0
\(442\) 6.23488e11 0.777012
\(443\) −5.89691e11 −0.727457 −0.363729 0.931505i \(-0.618496\pi\)
−0.363729 + 0.931505i \(0.618496\pi\)
\(444\) 0 0
\(445\) −5.60230e11 −0.677246
\(446\) −3.03308e10 −0.0362975
\(447\) 0 0
\(448\) 1.74759e10 0.0204969
\(449\) 1.06477e12 1.23636 0.618181 0.786036i \(-0.287870\pi\)
0.618181 + 0.786036i \(0.287870\pi\)
\(450\) 0 0
\(451\) 2.61220e11 0.297312
\(452\) 7.72964e11 0.871037
\(453\) 0 0
\(454\) −1.38493e12 −1.52995
\(455\) 6.34854e10 0.0694421
\(456\) 0 0
\(457\) 1.54296e12 1.65475 0.827374 0.561651i \(-0.189834\pi\)
0.827374 + 0.561651i \(0.189834\pi\)
\(458\) 1.21177e12 1.28685
\(459\) 0 0
\(460\) −1.05736e11 −0.110106
\(461\) 8.38680e11 0.864852 0.432426 0.901669i \(-0.357658\pi\)
0.432426 + 0.901669i \(0.357658\pi\)
\(462\) 0 0
\(463\) −8.61819e11 −0.871569 −0.435784 0.900051i \(-0.643529\pi\)
−0.435784 + 0.900051i \(0.643529\pi\)
\(464\) 2.20043e12 2.20382
\(465\) 0 0
\(466\) −2.11503e11 −0.207768
\(467\) −1.46986e12 −1.43005 −0.715023 0.699101i \(-0.753584\pi\)
−0.715023 + 0.699101i \(0.753584\pi\)
\(468\) 0 0
\(469\) −5.87183e11 −0.560397
\(470\) 4.13786e11 0.391143
\(471\) 0 0
\(472\) −5.70762e11 −0.529317
\(473\) 7.69353e11 0.706725
\(474\) 0 0
\(475\) −1.36742e11 −0.123248
\(476\) 3.43739e11 0.306901
\(477\) 0 0
\(478\) −7.74245e11 −0.678348
\(479\) −2.20227e12 −1.91144 −0.955721 0.294275i \(-0.904922\pi\)
−0.955721 + 0.294275i \(0.904922\pi\)
\(480\) 0 0
\(481\) 9.80519e10 0.0835224
\(482\) 2.31282e12 1.95177
\(483\) 0 0
\(484\) −4.63827e11 −0.384195
\(485\) −7.53631e11 −0.618473
\(486\) 0 0
\(487\) 1.30073e12 1.04786 0.523932 0.851760i \(-0.324464\pi\)
0.523932 + 0.851760i \(0.324464\pi\)
\(488\) 9.86350e10 0.0787303
\(489\) 0 0
\(490\) 1.00884e11 0.0790569
\(491\) −6.88947e11 −0.534957 −0.267479 0.963564i \(-0.586190\pi\)
−0.267479 + 0.963564i \(0.586190\pi\)
\(492\) 0 0
\(493\) −3.53724e12 −2.69683
\(494\) −4.14670e11 −0.313279
\(495\) 0 0
\(496\) 2.09951e12 1.55758
\(497\) −1.48627e11 −0.109268
\(498\) 0 0
\(499\) 1.76710e12 1.27588 0.637939 0.770087i \(-0.279787\pi\)
0.637939 + 0.770087i \(0.279787\pi\)
\(500\) −6.64063e10 −0.0475164
\(501\) 0 0
\(502\) 5.65088e11 0.397145
\(503\) −2.63527e12 −1.83556 −0.917780 0.397089i \(-0.870020\pi\)
−0.917780 + 0.397089i \(0.870020\pi\)
\(504\) 0 0
\(505\) −9.12633e11 −0.624432
\(506\) −4.44927e11 −0.301725
\(507\) 0 0
\(508\) 9.50203e11 0.633038
\(509\) −2.22151e12 −1.46696 −0.733479 0.679712i \(-0.762105\pi\)
−0.733479 + 0.679712i \(0.762105\pi\)
\(510\) 0 0
\(511\) −6.81317e11 −0.442033
\(512\) −7.48685e11 −0.481487
\(513\) 0 0
\(514\) 7.91786e11 0.500350
\(515\) −6.75053e11 −0.422869
\(516\) 0 0
\(517\) 6.04080e11 0.371867
\(518\) 1.55813e11 0.0950868
\(519\) 0 0
\(520\) 1.77685e11 0.106570
\(521\) 1.87441e12 1.11454 0.557269 0.830332i \(-0.311849\pi\)
0.557269 + 0.830332i \(0.311849\pi\)
\(522\) 0 0
\(523\) −1.58412e12 −0.925828 −0.462914 0.886403i \(-0.653196\pi\)
−0.462914 + 0.886403i \(0.653196\pi\)
\(524\) 5.02375e11 0.291096
\(525\) 0 0
\(526\) 3.89590e12 2.21908
\(527\) −3.37501e12 −1.90602
\(528\) 0 0
\(529\) −1.41430e12 −0.785219
\(530\) −1.00262e12 −0.551945
\(531\) 0 0
\(532\) −2.28614e11 −0.123737
\(533\) −4.32565e11 −0.232156
\(534\) 0 0
\(535\) 2.09968e11 0.110806
\(536\) −1.64343e12 −0.860021
\(537\) 0 0
\(538\) 1.62087e11 0.0834120
\(539\) 1.47279e11 0.0751609
\(540\) 0 0
\(541\) 2.79888e12 1.40474 0.702370 0.711812i \(-0.252125\pi\)
0.702370 + 0.711812i \(0.252125\pi\)
\(542\) 2.09135e12 1.04095
\(543\) 0 0
\(544\) 3.01448e12 1.47577
\(545\) 8.90755e11 0.432488
\(546\) 0 0
\(547\) 1.16606e12 0.556901 0.278451 0.960451i \(-0.410179\pi\)
0.278451 + 0.960451i \(0.410179\pi\)
\(548\) −3.18886e11 −0.151051
\(549\) 0 0
\(550\) −2.79431e11 −0.130210
\(551\) 2.35255e12 1.08732
\(552\) 0 0
\(553\) 6.62934e11 0.301444
\(554\) −6.22594e11 −0.280809
\(555\) 0 0
\(556\) −1.33008e12 −0.590259
\(557\) −3.29033e12 −1.44841 −0.724204 0.689586i \(-0.757793\pi\)
−0.724204 + 0.689586i \(0.757793\pi\)
\(558\) 0 0
\(559\) −1.27400e12 −0.551845
\(560\) 4.91341e11 0.211123
\(561\) 0 0
\(562\) −3.80962e12 −1.61090
\(563\) 3.63871e12 1.52637 0.763186 0.646179i \(-0.223634\pi\)
0.763186 + 0.646179i \(0.223634\pi\)
\(564\) 0 0
\(565\) −1.77611e12 −0.733251
\(566\) 4.79035e12 1.96197
\(567\) 0 0
\(568\) −4.15981e11 −0.167690
\(569\) −4.35009e12 −1.73978 −0.869888 0.493250i \(-0.835809\pi\)
−0.869888 + 0.493250i \(0.835809\pi\)
\(570\) 0 0
\(571\) −4.91136e12 −1.93348 −0.966739 0.255767i \(-0.917672\pi\)
−0.966739 + 0.255767i \(0.917672\pi\)
\(572\) −2.93987e11 −0.114827
\(573\) 0 0
\(574\) −6.87385e11 −0.264299
\(575\) 2.42959e11 0.0926890
\(576\) 0 0
\(577\) −1.68758e12 −0.633830 −0.316915 0.948454i \(-0.602647\pi\)
−0.316915 + 0.948454i \(0.602647\pi\)
\(578\) −4.43654e12 −1.65337
\(579\) 0 0
\(580\) 1.14247e12 0.419199
\(581\) 1.75263e11 0.0638114
\(582\) 0 0
\(583\) −1.46371e12 −0.524744
\(584\) −1.90689e12 −0.678373
\(585\) 0 0
\(586\) 2.96051e12 1.03711
\(587\) 4.82739e12 1.67819 0.839094 0.543987i \(-0.183086\pi\)
0.839094 + 0.543987i \(0.183086\pi\)
\(588\) 0 0
\(589\) 2.24466e12 0.768478
\(590\) −1.48636e12 −0.504998
\(591\) 0 0
\(592\) 7.58865e11 0.253932
\(593\) 1.12076e12 0.372193 0.186096 0.982532i \(-0.440416\pi\)
0.186096 + 0.982532i \(0.440416\pi\)
\(594\) 0 0
\(595\) −7.89842e11 −0.258353
\(596\) 2.34325e12 0.760694
\(597\) 0 0
\(598\) 7.36773e11 0.235602
\(599\) −1.35944e12 −0.431460 −0.215730 0.976453i \(-0.569213\pi\)
−0.215730 + 0.976453i \(0.569213\pi\)
\(600\) 0 0
\(601\) 9.50421e11 0.297153 0.148577 0.988901i \(-0.452531\pi\)
0.148577 + 0.988901i \(0.452531\pi\)
\(602\) −2.02450e12 −0.628253
\(603\) 0 0
\(604\) 1.49302e12 0.456457
\(605\) 1.06578e12 0.323421
\(606\) 0 0
\(607\) 3.35939e12 1.00441 0.502205 0.864749i \(-0.332522\pi\)
0.502205 + 0.864749i \(0.332522\pi\)
\(608\) −2.00487e12 −0.595006
\(609\) 0 0
\(610\) 2.56862e11 0.0751131
\(611\) −1.00032e12 −0.290372
\(612\) 0 0
\(613\) −2.62257e12 −0.750162 −0.375081 0.926992i \(-0.622385\pi\)
−0.375081 + 0.926992i \(0.622385\pi\)
\(614\) −4.87603e12 −1.38455
\(615\) 0 0
\(616\) 4.12210e11 0.115347
\(617\) 3.35285e10 0.00931388 0.00465694 0.999989i \(-0.498518\pi\)
0.00465694 + 0.999989i \(0.498518\pi\)
\(618\) 0 0
\(619\) 5.51587e12 1.51010 0.755051 0.655666i \(-0.227612\pi\)
0.755051 + 0.655666i \(0.227612\pi\)
\(620\) 1.09008e12 0.296274
\(621\) 0 0
\(622\) −2.96540e12 −0.794377
\(623\) 2.15218e12 0.572377
\(624\) 0 0
\(625\) 1.52588e11 0.0400000
\(626\) 6.81962e12 1.77491
\(627\) 0 0
\(628\) −3.62060e11 −0.0928886
\(629\) −1.21989e12 −0.310738
\(630\) 0 0
\(631\) 2.72456e12 0.684170 0.342085 0.939669i \(-0.388867\pi\)
0.342085 + 0.939669i \(0.388867\pi\)
\(632\) 1.85544e12 0.462616
\(633\) 0 0
\(634\) 5.14572e12 1.26487
\(635\) −2.18337e12 −0.532900
\(636\) 0 0
\(637\) −2.43886e11 −0.0586893
\(638\) 4.80742e12 1.14873
\(639\) 0 0
\(640\) 1.96009e12 0.461813
\(641\) −3.79136e11 −0.0887022 −0.0443511 0.999016i \(-0.514122\pi\)
−0.0443511 + 0.999016i \(0.514122\pi\)
\(642\) 0 0
\(643\) −5.15446e12 −1.18914 −0.594571 0.804043i \(-0.702678\pi\)
−0.594571 + 0.804043i \(0.702678\pi\)
\(644\) 4.06195e11 0.0930568
\(645\) 0 0
\(646\) 5.15904e12 1.16553
\(647\) 2.68059e12 0.601397 0.300699 0.953719i \(-0.402780\pi\)
0.300699 + 0.953719i \(0.402780\pi\)
\(648\) 0 0
\(649\) −2.16991e12 −0.480111
\(650\) 4.62722e11 0.101674
\(651\) 0 0
\(652\) 3.83783e11 0.0831710
\(653\) 6.44986e12 1.38816 0.694082 0.719896i \(-0.255810\pi\)
0.694082 + 0.719896i \(0.255810\pi\)
\(654\) 0 0
\(655\) −1.15435e12 −0.245049
\(656\) −3.34780e12 −0.705818
\(657\) 0 0
\(658\) −1.58960e12 −0.330576
\(659\) 2.79549e12 0.577396 0.288698 0.957420i \(-0.406778\pi\)
0.288698 + 0.957420i \(0.406778\pi\)
\(660\) 0 0
\(661\) −4.93691e11 −0.100589 −0.0502943 0.998734i \(-0.516016\pi\)
−0.0502943 + 0.998734i \(0.516016\pi\)
\(662\) 1.62893e12 0.329641
\(663\) 0 0
\(664\) 4.90533e11 0.0979291
\(665\) 5.25309e11 0.104164
\(666\) 0 0
\(667\) −4.17995e12 −0.817720
\(668\) −6.76068e10 −0.0131370
\(669\) 0 0
\(670\) −4.27976e12 −0.820508
\(671\) 3.74989e11 0.0714113
\(672\) 0 0
\(673\) 2.83805e12 0.533277 0.266638 0.963797i \(-0.414087\pi\)
0.266638 + 0.963797i \(0.414087\pi\)
\(674\) 9.52747e12 1.77831
\(675\) 0 0
\(676\) −2.39760e12 −0.441587
\(677\) −7.71236e12 −1.41104 −0.705519 0.708691i \(-0.749286\pi\)
−0.705519 + 0.708691i \(0.749286\pi\)
\(678\) 0 0
\(679\) 2.89515e12 0.522705
\(680\) −2.21064e12 −0.396485
\(681\) 0 0
\(682\) 4.58693e12 0.811882
\(683\) −1.07677e13 −1.89334 −0.946670 0.322204i \(-0.895576\pi\)
−0.946670 + 0.322204i \(0.895576\pi\)
\(684\) 0 0
\(685\) 7.32735e11 0.127157
\(686\) −3.87556e11 −0.0668153
\(687\) 0 0
\(688\) −9.86005e12 −1.67776
\(689\) 2.42382e12 0.409746
\(690\) 0 0
\(691\) 2.02298e12 0.337552 0.168776 0.985654i \(-0.446019\pi\)
0.168776 + 0.985654i \(0.446019\pi\)
\(692\) −4.52940e12 −0.750867
\(693\) 0 0
\(694\) −1.40735e13 −2.30295
\(695\) 3.05626e12 0.496888
\(696\) 0 0
\(697\) 5.38168e12 0.863714
\(698\) −2.00119e12 −0.319109
\(699\) 0 0
\(700\) 2.55106e11 0.0401587
\(701\) −8.66031e11 −0.135457 −0.0677286 0.997704i \(-0.521575\pi\)
−0.0677286 + 0.997704i \(0.521575\pi\)
\(702\) 0 0
\(703\) 8.11328e11 0.125285
\(704\) 1.85953e11 0.0285317
\(705\) 0 0
\(706\) −7.14268e11 −0.108203
\(707\) 3.50597e12 0.527741
\(708\) 0 0
\(709\) −3.78834e12 −0.563042 −0.281521 0.959555i \(-0.590839\pi\)
−0.281521 + 0.959555i \(0.590839\pi\)
\(710\) −1.08328e12 −0.159985
\(711\) 0 0
\(712\) 6.02360e12 0.878407
\(713\) −3.98824e12 −0.577934
\(714\) 0 0
\(715\) 6.75521e11 0.0966633
\(716\) −5.52040e12 −0.784986
\(717\) 0 0
\(718\) 4.19485e12 0.589056
\(719\) −8.16972e11 −0.114006 −0.0570029 0.998374i \(-0.518154\pi\)
−0.0570029 + 0.998374i \(0.518154\pi\)
\(720\) 0 0
\(721\) 2.59328e12 0.357389
\(722\) 5.60408e12 0.767515
\(723\) 0 0
\(724\) −3.70353e12 −0.500947
\(725\) −2.62517e12 −0.352887
\(726\) 0 0
\(727\) 3.13227e12 0.415867 0.207933 0.978143i \(-0.433326\pi\)
0.207933 + 0.978143i \(0.433326\pi\)
\(728\) −6.82595e11 −0.0900683
\(729\) 0 0
\(730\) −4.96587e12 −0.647205
\(731\) 1.58503e13 2.05309
\(732\) 0 0
\(733\) −1.33197e13 −1.70422 −0.852112 0.523360i \(-0.824678\pi\)
−0.852112 + 0.523360i \(0.824678\pi\)
\(734\) 1.28664e13 1.63616
\(735\) 0 0
\(736\) 3.56220e12 0.447474
\(737\) −6.24796e12 −0.780072
\(738\) 0 0
\(739\) −1.56702e13 −1.93274 −0.966371 0.257154i \(-0.917215\pi\)
−0.966371 + 0.257154i \(0.917215\pi\)
\(740\) 3.94006e11 0.0483014
\(741\) 0 0
\(742\) 3.85167e12 0.466478
\(743\) −7.91108e12 −0.952327 −0.476164 0.879357i \(-0.657973\pi\)
−0.476164 + 0.879357i \(0.657973\pi\)
\(744\) 0 0
\(745\) −5.38430e12 −0.640362
\(746\) 1.41184e13 1.66902
\(747\) 0 0
\(748\) 3.65758e12 0.427205
\(749\) −8.06614e11 −0.0936478
\(750\) 0 0
\(751\) −4.01514e12 −0.460597 −0.230299 0.973120i \(-0.573970\pi\)
−0.230299 + 0.973120i \(0.573970\pi\)
\(752\) −7.74191e12 −0.882811
\(753\) 0 0
\(754\) −7.96081e12 −0.896987
\(755\) −3.43066e12 −0.384252
\(756\) 0 0
\(757\) 2.84075e12 0.314414 0.157207 0.987566i \(-0.449751\pi\)
0.157207 + 0.987566i \(0.449751\pi\)
\(758\) 2.69678e12 0.296711
\(759\) 0 0
\(760\) 1.47025e12 0.159857
\(761\) 7.10439e12 0.767884 0.383942 0.923357i \(-0.374566\pi\)
0.383942 + 0.923357i \(0.374566\pi\)
\(762\) 0 0
\(763\) −3.42192e12 −0.365519
\(764\) −4.96011e11 −0.0526709
\(765\) 0 0
\(766\) 1.71034e13 1.79495
\(767\) 3.59325e12 0.374894
\(768\) 0 0
\(769\) −4.97052e12 −0.512546 −0.256273 0.966604i \(-0.582495\pi\)
−0.256273 + 0.966604i \(0.582495\pi\)
\(770\) 1.07346e12 0.110047
\(771\) 0 0
\(772\) 3.36593e12 0.341057
\(773\) −1.11565e13 −1.12388 −0.561939 0.827179i \(-0.689944\pi\)
−0.561939 + 0.827179i \(0.689944\pi\)
\(774\) 0 0
\(775\) −2.50477e12 −0.249408
\(776\) 8.10304e12 0.802177
\(777\) 0 0
\(778\) −1.81974e13 −1.78075
\(779\) −3.57925e12 −0.348236
\(780\) 0 0
\(781\) −1.58147e12 −0.152101
\(782\) −9.16642e12 −0.876535
\(783\) 0 0
\(784\) −1.88753e12 −0.178432
\(785\) 8.31940e11 0.0781949
\(786\) 0 0
\(787\) −6.23763e12 −0.579607 −0.289803 0.957086i \(-0.593590\pi\)
−0.289803 + 0.957086i \(0.593590\pi\)
\(788\) −5.25334e12 −0.485364
\(789\) 0 0
\(790\) 4.83188e12 0.441361
\(791\) 6.82311e12 0.619710
\(792\) 0 0
\(793\) −6.20960e11 −0.0557615
\(794\) 1.02807e13 0.917975
\(795\) 0 0
\(796\) 4.13835e12 0.365358
\(797\) 1.83799e12 0.161355 0.0806773 0.996740i \(-0.474292\pi\)
0.0806773 + 0.996740i \(0.474292\pi\)
\(798\) 0 0
\(799\) 1.24453e13 1.08030
\(800\) 2.23720e12 0.193108
\(801\) 0 0
\(802\) −2.72716e13 −2.32769
\(803\) −7.24960e12 −0.615310
\(804\) 0 0
\(805\) −9.33353e11 −0.0783365
\(806\) −7.59570e12 −0.633957
\(807\) 0 0
\(808\) 9.81263e12 0.809906
\(809\) −1.92205e13 −1.57760 −0.788801 0.614649i \(-0.789298\pi\)
−0.788801 + 0.614649i \(0.789298\pi\)
\(810\) 0 0
\(811\) −1.70316e13 −1.38249 −0.691246 0.722620i \(-0.742938\pi\)
−0.691246 + 0.722620i \(0.742938\pi\)
\(812\) −4.38892e12 −0.354288
\(813\) 0 0
\(814\) 1.65794e12 0.132361
\(815\) −8.81855e11 −0.0700145
\(816\) 0 0
\(817\) −1.05417e13 −0.827774
\(818\) −1.00295e13 −0.783228
\(819\) 0 0
\(820\) −1.73820e12 −0.134257
\(821\) −1.33485e13 −1.02539 −0.512694 0.858572i \(-0.671352\pi\)
−0.512694 + 0.858572i \(0.671352\pi\)
\(822\) 0 0
\(823\) −1.72858e13 −1.31338 −0.656689 0.754161i \(-0.728044\pi\)
−0.656689 + 0.754161i \(0.728044\pi\)
\(824\) 7.25817e12 0.548473
\(825\) 0 0
\(826\) 5.71000e12 0.426801
\(827\) −1.64651e13 −1.22402 −0.612012 0.790849i \(-0.709639\pi\)
−0.612012 + 0.790849i \(0.709639\pi\)
\(828\) 0 0
\(829\) −5.91786e12 −0.435181 −0.217590 0.976040i \(-0.569820\pi\)
−0.217590 + 0.976040i \(0.569820\pi\)
\(830\) 1.27743e12 0.0934298
\(831\) 0 0
\(832\) −3.07928e11 −0.0222789
\(833\) 3.03426e12 0.218348
\(834\) 0 0
\(835\) 1.55347e11 0.0110589
\(836\) −2.43259e12 −0.172242
\(837\) 0 0
\(838\) 1.35719e13 0.950701
\(839\) −2.40380e13 −1.67483 −0.837414 0.546569i \(-0.815934\pi\)
−0.837414 + 0.546569i \(0.815934\pi\)
\(840\) 0 0
\(841\) 3.06570e13 2.11324
\(842\) −2.01123e13 −1.37898
\(843\) 0 0
\(844\) −1.05978e13 −0.718913
\(845\) 5.50919e12 0.371734
\(846\) 0 0
\(847\) −4.09430e12 −0.273341
\(848\) 1.87590e13 1.24574
\(849\) 0 0
\(850\) −5.75687e12 −0.378269
\(851\) −1.44154e12 −0.0942203
\(852\) 0 0
\(853\) 2.26446e13 1.46452 0.732258 0.681027i \(-0.238466\pi\)
0.732258 + 0.681027i \(0.238466\pi\)
\(854\) −9.86761e11 −0.0634821
\(855\) 0 0
\(856\) −2.25758e12 −0.143718
\(857\) 1.20223e13 0.761334 0.380667 0.924712i \(-0.375694\pi\)
0.380667 + 0.924712i \(0.375694\pi\)
\(858\) 0 0
\(859\) −9.44258e12 −0.591727 −0.295864 0.955230i \(-0.595607\pi\)
−0.295864 + 0.955230i \(0.595607\pi\)
\(860\) −5.11938e12 −0.319135
\(861\) 0 0
\(862\) 2.31643e13 1.42901
\(863\) −8.80380e12 −0.540283 −0.270142 0.962821i \(-0.587071\pi\)
−0.270142 + 0.962821i \(0.587071\pi\)
\(864\) 0 0
\(865\) 1.04076e13 0.632090
\(866\) −2.44449e13 −1.47692
\(867\) 0 0
\(868\) −4.18763e12 −0.250397
\(869\) 7.05399e12 0.419610
\(870\) 0 0
\(871\) 1.03463e13 0.609119
\(872\) −9.57740e12 −0.560949
\(873\) 0 0
\(874\) 6.09641e12 0.353405
\(875\) −5.86182e11 −0.0338062
\(876\) 0 0
\(877\) 6.73513e12 0.384457 0.192229 0.981350i \(-0.438429\pi\)
0.192229 + 0.981350i \(0.438429\pi\)
\(878\) −2.00217e13 −1.13704
\(879\) 0 0
\(880\) 5.22814e12 0.293884
\(881\) 7.58474e12 0.424179 0.212089 0.977250i \(-0.431973\pi\)
0.212089 + 0.977250i \(0.431973\pi\)
\(882\) 0 0
\(883\) −1.78234e13 −0.986662 −0.493331 0.869842i \(-0.664221\pi\)
−0.493331 + 0.869842i \(0.664221\pi\)
\(884\) −6.05674e12 −0.333583
\(885\) 0 0
\(886\) 1.65113e13 0.900182
\(887\) 2.45177e13 1.32991 0.664956 0.746882i \(-0.268450\pi\)
0.664956 + 0.746882i \(0.268450\pi\)
\(888\) 0 0
\(889\) 8.38764e12 0.450383
\(890\) 1.56864e13 0.838049
\(891\) 0 0
\(892\) 2.94642e11 0.0155830
\(893\) −8.27714e12 −0.435561
\(894\) 0 0
\(895\) 1.26847e13 0.660812
\(896\) −7.52988e12 −0.390303
\(897\) 0 0
\(898\) −2.98134e13 −1.52992
\(899\) 4.30928e13 2.20032
\(900\) 0 0
\(901\) −3.01555e13 −1.52442
\(902\) −7.31416e12 −0.367905
\(903\) 0 0
\(904\) 1.90968e13 0.951047
\(905\) 8.50995e12 0.421704
\(906\) 0 0
\(907\) −8.14231e12 −0.399498 −0.199749 0.979847i \(-0.564013\pi\)
−0.199749 + 0.979847i \(0.564013\pi\)
\(908\) 1.34536e13 0.656830
\(909\) 0 0
\(910\) −1.77759e12 −0.0859302
\(911\) 1.26965e13 0.610733 0.305367 0.952235i \(-0.401221\pi\)
0.305367 + 0.952235i \(0.401221\pi\)
\(912\) 0 0
\(913\) 1.86490e12 0.0888254
\(914\) −4.32029e13 −2.04765
\(915\) 0 0
\(916\) −1.17715e13 −0.552463
\(917\) 4.43456e12 0.207104
\(918\) 0 0
\(919\) −2.53575e13 −1.17270 −0.586350 0.810058i \(-0.699436\pi\)
−0.586350 + 0.810058i \(0.699436\pi\)
\(920\) −2.61230e12 −0.120220
\(921\) 0 0
\(922\) −2.34830e13 −1.07020
\(923\) 2.61882e12 0.118768
\(924\) 0 0
\(925\) −9.05345e11 −0.0406608
\(926\) 2.41309e13 1.07851
\(927\) 0 0
\(928\) −3.84895e13 −1.70363
\(929\) −1.89611e13 −0.835203 −0.417602 0.908630i \(-0.637129\pi\)
−0.417602 + 0.908630i \(0.637129\pi\)
\(930\) 0 0
\(931\) −2.01803e12 −0.0880346
\(932\) 2.05460e12 0.0891980
\(933\) 0 0
\(934\) 4.11561e13 1.76959
\(935\) −8.40437e12 −0.359627
\(936\) 0 0
\(937\) 8.65559e12 0.366833 0.183416 0.983035i \(-0.441284\pi\)
0.183416 + 0.983035i \(0.441284\pi\)
\(938\) 1.64411e13 0.693456
\(939\) 0 0
\(940\) −4.01964e12 −0.167923
\(941\) −3.42336e13 −1.42331 −0.711654 0.702530i \(-0.752054\pi\)
−0.711654 + 0.702530i \(0.752054\pi\)
\(942\) 0 0
\(943\) 6.35950e12 0.261891
\(944\) 2.78097e13 1.13978
\(945\) 0 0
\(946\) −2.15419e13 −0.874527
\(947\) −2.38597e13 −0.964030 −0.482015 0.876163i \(-0.660095\pi\)
−0.482015 + 0.876163i \(0.660095\pi\)
\(948\) 0 0
\(949\) 1.20049e13 0.480464
\(950\) 3.82878e12 0.152512
\(951\) 0 0
\(952\) 8.49238e12 0.335091
\(953\) 4.83481e13 1.89872 0.949360 0.314189i \(-0.101733\pi\)
0.949360 + 0.314189i \(0.101733\pi\)
\(954\) 0 0
\(955\) 1.13973e12 0.0443391
\(956\) 7.52123e12 0.291225
\(957\) 0 0
\(958\) 6.16636e13 2.36529
\(959\) −2.81488e12 −0.107467
\(960\) 0 0
\(961\) 1.46768e13 0.555106
\(962\) −2.74545e12 −0.103354
\(963\) 0 0
\(964\) −2.24674e13 −0.837925
\(965\) −7.73421e12 −0.287106
\(966\) 0 0
\(967\) −2.28025e13 −0.838617 −0.419308 0.907844i \(-0.637727\pi\)
−0.419308 + 0.907844i \(0.637727\pi\)
\(968\) −1.14593e13 −0.419486
\(969\) 0 0
\(970\) 2.11017e13 0.765322
\(971\) −7.62385e12 −0.275225 −0.137612 0.990486i \(-0.543943\pi\)
−0.137612 + 0.990486i \(0.543943\pi\)
\(972\) 0 0
\(973\) −1.17409e13 −0.419947
\(974\) −3.64203e13 −1.29667
\(975\) 0 0
\(976\) −4.80587e12 −0.169531
\(977\) −2.97018e13 −1.04293 −0.521467 0.853272i \(-0.674615\pi\)
−0.521467 + 0.853272i \(0.674615\pi\)
\(978\) 0 0
\(979\) 2.29004e13 0.796749
\(980\) −9.80016e11 −0.0339403
\(981\) 0 0
\(982\) 1.92905e13 0.661976
\(983\) 1.08292e13 0.369917 0.184958 0.982746i \(-0.440785\pi\)
0.184958 + 0.982746i \(0.440785\pi\)
\(984\) 0 0
\(985\) 1.20711e13 0.408586
\(986\) 9.90428e13 3.33716
\(987\) 0 0
\(988\) 4.02822e12 0.134495
\(989\) 1.87302e13 0.622528
\(990\) 0 0
\(991\) 5.29514e12 0.174400 0.0871999 0.996191i \(-0.472208\pi\)
0.0871999 + 0.996191i \(0.472208\pi\)
\(992\) −3.67242e13 −1.20406
\(993\) 0 0
\(994\) 4.16154e12 0.135212
\(995\) −9.50909e12 −0.307564
\(996\) 0 0
\(997\) 1.64214e13 0.526360 0.263180 0.964747i \(-0.415229\pi\)
0.263180 + 0.964747i \(0.415229\pi\)
\(998\) −4.94788e13 −1.57882
\(999\) 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 315.10.a.a.1.1 1
3.2 odd 2 35.10.a.a.1.1 1
15.2 even 4 175.10.b.a.99.2 2
15.8 even 4 175.10.b.a.99.1 2
15.14 odd 2 175.10.a.a.1.1 1
21.20 even 2 245.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.a.1.1 1 3.2 odd 2
175.10.a.a.1.1 1 15.14 odd 2
175.10.b.a.99.1 2 15.8 even 4
175.10.b.a.99.2 2 15.2 even 4
245.10.a.b.1.1 1 21.20 even 2
315.10.a.a.1.1 1 1.1 even 1 trivial