Properties

Label 175.10.b.a.99.1
Level $175$
Weight $10$
Character 175.99
Analytic conductor $90.131$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,10,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.10.b.a.99.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-28.0000i q^{2} -116.000i q^{3} -272.000 q^{4} -3248.00 q^{6} -2401.00i q^{7} -6720.00i q^{8} +6227.00 q^{9} +O(q^{10})\) \(q-28.0000i q^{2} -116.000i q^{3} -272.000 q^{4} -3248.00 q^{6} -2401.00i q^{7} -6720.00i q^{8} +6227.00 q^{9} -25548.0 q^{11} +31552.0i q^{12} -42306.0i q^{13} -67228.0 q^{14} -327424. q^{16} +526342. i q^{17} -174356. i q^{18} +350060. q^{19} -278516. q^{21} +715344. i q^{22} -621976. i q^{23} -779520. q^{24} -1.18457e6 q^{26} -3.00556e6i q^{27} +653072. i q^{28} -6.72043e6 q^{29} -6.41221e6 q^{31} +5.72723e6i q^{32} +2.96357e6i q^{33} +1.47376e7 q^{34} -1.69374e6 q^{36} +2.31768e6i q^{37} -9.80168e6i q^{38} -4.90750e6 q^{39} -1.02247e7 q^{41} +7.79845e6i q^{42} +3.01140e7i q^{43} +6.94906e6 q^{44} -1.74153e7 q^{46} +2.36449e7i q^{47} +3.79812e7i q^{48} -5.76480e6 q^{49} +6.10557e7 q^{51} +1.15072e7i q^{52} +5.72927e7i q^{53} -8.41557e7 q^{54} -1.61347e7 q^{56} -4.06070e7i q^{57} +1.88172e8i q^{58} -8.49348e7 q^{59} +1.46778e7 q^{61} +1.79542e8i q^{62} -1.49510e7i q^{63} -7.27859e6 q^{64} +8.29799e7 q^{66} +2.44558e8i q^{67} -1.43165e8i q^{68} -7.21492e7 q^{69} +6.19020e7 q^{71} -4.18454e7i q^{72} -2.83764e8i q^{73} +6.48951e7 q^{74} -9.52163e7 q^{76} +6.13407e7i q^{77} +1.37410e8i q^{78} -2.76107e8 q^{79} -2.26079e8 q^{81} +2.86291e8i q^{82} -7.29960e7i q^{83} +7.57564e7 q^{84} +8.43192e8 q^{86} +7.79570e8i q^{87} +1.71683e8i q^{88} +8.96368e8 q^{89} -1.01577e8 q^{91} +1.69177e8i q^{92} +7.43816e8i q^{93} +6.62058e8 q^{94} +6.64359e8 q^{96} -1.20581e9i q^{97} +1.61414e8i q^{98} -1.59087e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 544 q^{4} - 6496 q^{6} + 12454 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 544 q^{4} - 6496 q^{6} + 12454 q^{9} - 51096 q^{11} - 134456 q^{14} - 654848 q^{16} + 700120 q^{19} - 557032 q^{21} - 1559040 q^{24} - 2369136 q^{26} - 13440860 q^{29} - 12824416 q^{31} + 29475152 q^{34} - 3387488 q^{36} - 9814992 q^{39} - 20449356 q^{41} + 13898112 q^{44} - 34830656 q^{46} - 11529602 q^{49} + 122111344 q^{51} - 168311360 q^{54} - 32269440 q^{56} - 169869560 q^{59} + 29355644 q^{61} - 14557184 q^{64} + 165959808 q^{66} - 144298432 q^{69} + 123803904 q^{71} + 129790192 q^{74} - 190432640 q^{76} - 552214960 q^{79} - 452157838 q^{81} + 151512704 q^{84} + 1686384224 q^{86} + 1792736940 q^{89} - 203153412 q^{91} + 1324115072 q^{94} + 1328717824 q^{96} - 318174792 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 28.0000i − 1.23744i −0.785613 0.618718i \(-0.787652\pi\)
0.785613 0.618718i \(-0.212348\pi\)
\(3\) − 116.000i − 0.826823i −0.910544 0.413411i \(-0.864337\pi\)
0.910544 0.413411i \(-0.135663\pi\)
\(4\) −272.000 −0.531250
\(5\) 0 0
\(6\) −3248.00 −1.02314
\(7\) − 2401.00i − 0.377964i
\(8\) − 6720.00i − 0.580049i
\(9\) 6227.00 0.316364
\(10\) 0 0
\(11\) −25548.0 −0.526126 −0.263063 0.964779i \(-0.584733\pi\)
−0.263063 + 0.964779i \(0.584733\pi\)
\(12\) 31552.0i 0.439250i
\(13\) − 42306.0i − 0.410825i −0.978675 0.205413i \(-0.934146\pi\)
0.978675 0.205413i \(-0.0658536\pi\)
\(14\) −67228.0 −0.467707
\(15\) 0 0
\(16\) −327424. −1.24902
\(17\) 526342.i 1.52844i 0.644957 + 0.764219i \(0.276875\pi\)
−0.644957 + 0.764219i \(0.723125\pi\)
\(18\) − 174356.i − 0.391481i
\(19\) 350060. 0.616242 0.308121 0.951347i \(-0.400300\pi\)
0.308121 + 0.951347i \(0.400300\pi\)
\(20\) 0 0
\(21\) −278516. −0.312510
\(22\) 715344.i 0.651048i
\(23\) − 621976.i − 0.463445i −0.972782 0.231723i \(-0.925564\pi\)
0.972782 0.231723i \(-0.0744362\pi\)
\(24\) −779520. −0.479597
\(25\) 0 0
\(26\) −1.18457e6 −0.508370
\(27\) − 3.00556e6i − 1.08840i
\(28\) 653072.i 0.200794i
\(29\) −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 \(0.714298\pi\)
\(32\) 5.72723e6i 0.965539i
\(33\) 2.96357e6i 0.435013i
\(34\) 1.47376e7 1.89135
\(35\) 0 0
\(36\) −1.69374e6 −0.168069
\(37\) 2.31768e6i 0.203304i 0.994820 + 0.101652i \(0.0324128\pi\)
−0.994820 + 0.101652i \(0.967587\pi\)
\(38\) − 9.80168e6i − 0.762561i
\(39\) −4.90750e6 −0.339679
\(40\) 0 0
\(41\) −1.02247e7 −0.565096 −0.282548 0.959253i \(-0.591180\pi\)
−0.282548 + 0.959253i \(0.591180\pi\)
\(42\) 7.79845e6i 0.386711i
\(43\) 3.01140e7i 1.34326i 0.740886 + 0.671631i \(0.234406\pi\)
−0.740886 + 0.671631i \(0.765594\pi\)
\(44\) 6.94906e6 0.279504
\(45\) 0 0
\(46\) −1.74153e7 −0.573484
\(47\) 2.36449e7i 0.706801i 0.935472 + 0.353401i \(0.114975\pi\)
−0.935472 + 0.353401i \(0.885025\pi\)
\(48\) 3.79812e7i 1.03272i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) 6.10557e7 1.26375
\(52\) 1.15072e7i 0.218251i
\(53\) 5.72927e7i 0.997373i 0.866782 + 0.498686i \(0.166184\pi\)
−0.866782 + 0.498686i \(0.833816\pi\)
\(54\) −8.41557e7 −1.34683
\(55\) 0 0
\(56\) −1.61347e7 −0.219238
\(57\) − 4.06070e7i − 0.509523i
\(58\) 1.88172e8i 2.18338i
\(59\) −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 \(0.478381\pi\)
\(62\) 1.79542e8i 1.54313i
\(63\) − 1.49510e7i − 0.119574i
\(64\) −7.27859e6 −0.0542297
\(65\) 0 0
\(66\) 8.29799e7 0.538301
\(67\) 2.44558e8i 1.48267i 0.671134 + 0.741336i \(0.265807\pi\)
−0.671134 + 0.741336i \(0.734193\pi\)
\(68\) − 1.43165e8i − 0.811983i
\(69\) −7.21492e7 −0.383187
\(70\) 0 0
\(71\) 6.19020e7 0.289096 0.144548 0.989498i \(-0.453827\pi\)
0.144548 + 0.989498i \(0.453827\pi\)
\(72\) − 4.18454e7i − 0.183507i
\(73\) − 2.83764e8i − 1.16951i −0.811210 0.584755i \(-0.801191\pi\)
0.811210 0.584755i \(-0.198809\pi\)
\(74\) 6.48951e7 0.251576
\(75\) 0 0
\(76\) −9.52163e7 −0.327379
\(77\) 6.13407e7i 0.198857i
\(78\) 1.37410e8i 0.420332i
\(79\) −2.76107e8 −0.797547 −0.398773 0.917049i \(-0.630564\pi\)
−0.398773 + 0.917049i \(0.630564\pi\)
\(80\) 0 0
\(81\) −2.26079e8 −0.583549
\(82\) 2.86291e8i 0.699271i
\(83\) − 7.29960e7i − 0.168829i −0.996431 0.0844146i \(-0.973098\pi\)
0.996431 0.0844146i \(-0.0269020\pi\)
\(84\) 7.57564e7 0.166021
\(85\) 0 0
\(86\) 8.43192e8 1.66220
\(87\) 7.79570e8i 1.45888i
\(88\) 1.71683e8i 0.305179i
\(89\) 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
\(92\) 1.69177e8i 0.246205i
\(93\) 7.43816e8i 1.03108i
\(94\) 6.62058e8 0.874622
\(95\) 0 0
\(96\) 6.64359e8 0.798330
\(97\) − 1.20581e9i − 1.38295i −0.722401 0.691474i \(-0.756962\pi\)
0.722401 0.691474i \(-0.243038\pi\)
\(98\) 1.61414e8i 0.176777i
\(99\) −1.59087e8 −0.166448
\(100\) 0 0
\(101\) −1.46021e9 −1.39627 −0.698136 0.715965i \(-0.745987\pi\)
−0.698136 + 0.715965i \(0.745987\pi\)
\(102\) − 1.70956e9i − 1.56381i
\(103\) 1.08009e9i 0.945563i 0.881180 + 0.472782i \(0.156750\pi\)
−0.881180 + 0.472782i \(0.843250\pi\)
\(104\) −2.84296e8 −0.238298
\(105\) 0 0
\(106\) 1.60419e9 1.23419
\(107\) − 3.35949e8i − 0.247769i −0.992297 0.123884i \(-0.960465\pi\)
0.992297 0.123884i \(-0.0395352\pi\)
\(108\) 8.17512e8i 0.578212i
\(109\) 1.42521e9 0.967072 0.483536 0.875324i \(-0.339352\pi\)
0.483536 + 0.875324i \(0.339352\pi\)
\(110\) 0 0
\(111\) 2.68851e8 0.168096
\(112\) 7.86145e8i 0.472086i
\(113\) − 2.84178e9i − 1.63960i −0.572651 0.819799i \(-0.694085\pi\)
0.572651 0.819799i \(-0.305915\pi\)
\(114\) −1.13699e9 −0.630502
\(115\) 0 0
\(116\) 1.82796e9 0.937357
\(117\) − 2.63439e8i − 0.129970i
\(118\) 2.37817e9i 1.12921i
\(119\) 1.26375e9 0.577695
\(120\) 0 0
\(121\) −1.70525e9 −0.723191
\(122\) − 4.10979e8i − 0.167958i
\(123\) 1.18606e9i 0.467234i
\(124\) 1.74412e9 0.662489
\(125\) 0 0
\(126\) −4.18629e8 −0.147966
\(127\) − 3.49339e9i − 1.19160i −0.803133 0.595800i \(-0.796835\pi\)
0.803133 0.595800i \(-0.203165\pi\)
\(128\) 3.13614e9i 1.03264i
\(129\) 3.49322e9 1.11064
\(130\) 0 0
\(131\) −1.84697e9 −0.547946 −0.273973 0.961737i \(-0.588338\pi\)
−0.273973 + 0.961737i \(0.588338\pi\)
\(132\) − 8.06090e8i − 0.231101i
\(133\) − 8.40494e8i − 0.232918i
\(134\) 6.84762e9 1.83471
\(135\) 0 0
\(136\) 3.53702e9 0.886568
\(137\) − 1.17238e9i − 0.284331i −0.989843 0.142166i \(-0.954593\pi\)
0.989843 0.142166i \(-0.0454066\pi\)
\(138\) 2.02018e9i 0.474170i
\(139\) 4.89001e9 1.11108 0.555538 0.831491i \(-0.312512\pi\)
0.555538 + 0.831491i \(0.312512\pi\)
\(140\) 0 0
\(141\) 2.74281e9 0.584399
\(142\) − 1.73325e9i − 0.357738i
\(143\) 1.08083e9i 0.216146i
\(144\) −2.03887e9 −0.395147
\(145\) 0 0
\(146\) −7.94538e9 −1.44720
\(147\) 6.68717e8i 0.118118i
\(148\) − 6.30410e8i − 0.108005i
\(149\) 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 \(0.358652\pi\)
\(152\) − 2.35240e9i − 0.357450i
\(153\) 3.27753e9i 0.483543i
\(154\) 1.71754e9 0.246073
\(155\) 0 0
\(156\) 1.33484e9 0.180455
\(157\) 1.33110e9i 0.174849i 0.996171 + 0.0874246i \(0.0278637\pi\)
−0.996171 + 0.0874246i \(0.972136\pi\)
\(158\) 7.73101e9i 0.986914i
\(159\) 6.64595e9 0.824650
\(160\) 0 0
\(161\) −1.49336e9 −0.175166
\(162\) 6.33021e9i 0.722105i
\(163\) 1.41097e9i 0.156557i 0.996932 + 0.0782786i \(0.0249424\pi\)
−0.996932 + 0.0782786i \(0.975058\pi\)
\(164\) 2.78111e9 0.300207
\(165\) 0 0
\(166\) −2.04389e9 −0.208915
\(167\) − 2.48555e8i − 0.0247285i −0.999924 0.0123642i \(-0.996064\pi\)
0.999924 0.0123642i \(-0.00393576\pi\)
\(168\) 1.87163e9i 0.181271i
\(169\) 8.81470e9 0.831223
\(170\) 0 0
\(171\) 2.17982e9 0.194957
\(172\) − 8.19101e9i − 0.713607i
\(173\) 1.66522e10i 1.41340i 0.707515 + 0.706699i \(0.249816\pi\)
−0.707515 + 0.706699i \(0.750184\pi\)
\(174\) 2.18280e10 1.80527
\(175\) 0 0
\(176\) 8.36503e9 0.657144
\(177\) 9.85243e9i 0.754508i
\(178\) − 2.50983e10i − 1.87394i
\(179\) −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 \(0.656280\pi\)
\(182\) 2.84415e9i 0.192146i
\(183\) − 1.70263e9i − 0.112225i
\(184\) −4.17968e9 −0.268821
\(185\) 0 0
\(186\) 2.08269e10 1.27590
\(187\) − 1.34470e10i − 0.804151i
\(188\) − 6.43142e9i − 0.375488i
\(189\) −7.21635e9 −0.411376
\(190\) 0 0
\(191\) 1.82357e9 0.0991453 0.0495726 0.998771i \(-0.484214\pi\)
0.0495726 + 0.998771i \(0.484214\pi\)
\(192\) 8.44317e8i 0.0448384i
\(193\) 1.23747e10i 0.641989i 0.947081 + 0.320995i \(0.104017\pi\)
−0.947081 + 0.320995i \(0.895983\pi\)
\(194\) −3.37627e10 −1.71131
\(195\) 0 0
\(196\) 1.56803e9 0.0758929
\(197\) − 1.93137e10i − 0.913626i −0.889563 0.456813i \(-0.848991\pi\)
0.889563 0.456813i \(-0.151009\pi\)
\(198\) 4.45445e9i 0.205968i
\(199\) −1.52145e10 −0.687733 −0.343867 0.939019i \(-0.611737\pi\)
−0.343867 + 0.939019i \(0.611737\pi\)
\(200\) 0 0
\(201\) 2.83687e10 1.22591
\(202\) 4.08860e10i 1.72780i
\(203\) 1.61358e10i 0.666894i
\(204\) −1.66071e10 −0.671366
\(205\) 0 0
\(206\) 3.02424e10 1.17007
\(207\) − 3.87304e9i − 0.146618i
\(208\) 1.38520e10i 0.513130i
\(209\) −8.94333e9 −0.324221
\(210\) 0 0
\(211\) −3.89626e10 −1.35325 −0.676624 0.736329i \(-0.736558\pi\)
−0.676624 + 0.736329i \(0.736558\pi\)
\(212\) − 1.55836e10i − 0.529854i
\(213\) − 7.18063e9i − 0.239031i
\(214\) −9.40658e9 −0.306598
\(215\) 0 0
\(216\) −2.01974e10 −0.631325
\(217\) 1.53957e10i 0.471336i
\(218\) − 3.99058e10i − 1.19669i
\(219\) −3.29166e10 −0.966977
\(220\) 0 0
\(221\) 2.22674e10 0.627921
\(222\) − 7.52783e9i − 0.208009i
\(223\) 1.08324e9i 0.0293328i 0.999892 + 0.0146664i \(0.00466863\pi\)
−0.999892 + 0.0146664i \(0.995331\pi\)
\(224\) 1.37511e10 0.364939
\(225\) 0 0
\(226\) −7.95698e10 −2.02890
\(227\) 4.94618e10i 1.23639i 0.786027 + 0.618193i \(0.212135\pi\)
−0.786027 + 0.618193i \(0.787865\pi\)
\(228\) 1.10451e10i 0.270684i
\(229\) 4.32776e10 1.03993 0.519965 0.854188i \(-0.325945\pi\)
0.519965 + 0.854188i \(0.325945\pi\)
\(230\) 0 0
\(231\) 7.11553e9 0.164419
\(232\) 4.51613e10i 1.02346i
\(233\) − 7.55367e9i − 0.167902i −0.996470 0.0839511i \(-0.973246\pi\)
0.996470 0.0839511i \(-0.0267539\pi\)
\(234\) −7.37630e9 −0.160830
\(235\) 0 0
\(236\) 2.31023e10 0.484786
\(237\) 3.20285e10i 0.659430i
\(238\) − 3.53849e10i − 0.714862i
\(239\) 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 \(0.789212\pi\)
\(242\) 4.77469e10i 0.894904i
\(243\) − 3.29333e10i − 0.605908i
\(244\) −3.99237e9 −0.0721068
\(245\) 0 0
\(246\) 3.32098e10 0.578173
\(247\) − 1.48096e10i − 0.253168i
\(248\) 4.30900e10i 0.723343i
\(249\) −8.46753e9 −0.139592
\(250\) 0 0
\(251\) 2.01817e10 0.320942 0.160471 0.987041i \(-0.448699\pi\)
0.160471 + 0.987041i \(0.448699\pi\)
\(252\) 4.06668e9i 0.0635240i
\(253\) 1.58902e10i 0.243831i
\(254\) −9.78150e10 −1.47453
\(255\) 0 0
\(256\) 8.40854e10 1.22360
\(257\) − 2.82781e10i − 0.404344i −0.979350 0.202172i \(-0.935200\pi\)
0.979350 0.202172i \(-0.0648000\pi\)
\(258\) − 9.78103e10i − 1.37435i
\(259\) 5.56475e9 0.0768417
\(260\) 0 0
\(261\) −4.18481e10 −0.558205
\(262\) 5.17150e10i 0.678049i
\(263\) 1.39139e11i 1.79328i 0.442756 + 0.896642i \(0.354001\pi\)
−0.442756 + 0.896642i \(0.645999\pi\)
\(264\) 1.99152e10 0.252329
\(265\) 0 0
\(266\) −2.35338e10 −0.288221
\(267\) − 1.03979e11i − 1.25211i
\(268\) − 6.65197e10i − 0.787669i
\(269\) −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 \(0.638183\pi\)
\(272\) − 1.72337e11i − 1.90906i
\(273\) 1.17829e10i 0.128387i
\(274\) −3.28265e10 −0.351842
\(275\) 0 0
\(276\) 1.96246e10 0.203568
\(277\) − 2.22355e10i − 0.226928i −0.993542 0.113464i \(-0.963805\pi\)
0.993542 0.113464i \(-0.0361947\pi\)
\(278\) − 1.36920e11i − 1.37489i
\(279\) −3.99288e10 −0.394519
\(280\) 0 0
\(281\) −1.36058e11 −1.30180 −0.650901 0.759162i \(-0.725609\pi\)
−0.650901 + 0.759162i \(0.725609\pi\)
\(282\) − 7.67987e10i − 0.723157i
\(283\) − 1.71084e11i − 1.58551i −0.609538 0.792757i \(-0.708645\pi\)
0.609538 0.792757i \(-0.291355\pi\)
\(284\) −1.68373e10 −0.153582
\(285\) 0 0
\(286\) 3.02633e10 0.267467
\(287\) 2.45495e10i 0.213586i
\(288\) 3.56635e10i 0.305462i
\(289\) −1.58448e11 −1.33612
\(290\) 0 0
\(291\) −1.39874e11 −1.14345
\(292\) 7.71837e10i 0.621302i
\(293\) 1.05732e11i 0.838115i 0.907960 + 0.419058i \(0.137639\pi\)
−0.907960 + 0.419058i \(0.862361\pi\)
\(294\) 1.87241e10 0.146163
\(295\) 0 0
\(296\) 1.55748e10 0.117926
\(297\) 7.67860e10i 0.572636i
\(298\) − 2.41217e11i − 1.77188i
\(299\) −2.63133e10 −0.190395
\(300\) 0 0
\(301\) 7.23037e10 0.507705
\(302\) − 1.53693e11i − 1.06322i
\(303\) 1.69385e11i 1.15447i
\(304\) −1.14618e11 −0.769701
\(305\) 0 0
\(306\) 9.17709e10 0.598354
\(307\) − 1.74144e11i − 1.11889i −0.828869 0.559443i \(-0.811015\pi\)
0.828869 0.559443i \(-0.188985\pi\)
\(308\) − 1.66847e10i − 0.105643i
\(309\) 1.25290e11 0.781813
\(310\) 0 0
\(311\) −1.05907e11 −0.641954 −0.320977 0.947087i \(-0.604011\pi\)
−0.320977 + 0.947087i \(0.604011\pi\)
\(312\) 3.29784e10i 0.197031i
\(313\) − 2.43558e11i − 1.43434i −0.696897 0.717171i \(-0.745437\pi\)
0.696897 0.717171i \(-0.254563\pi\)
\(314\) 3.72709e10 0.216365
\(315\) 0 0
\(316\) 7.51012e10 0.423697
\(317\) − 1.83776e11i − 1.02217i −0.859531 0.511083i \(-0.829245\pi\)
0.859531 0.511083i \(-0.170755\pi\)
\(318\) − 1.86087e11i − 1.02045i
\(319\) 1.71694e11 0.928316
\(320\) 0 0
\(321\) −3.89701e10 −0.204861
\(322\) 4.18142e10i 0.216757i
\(323\) 1.84251e11i 0.941888i
\(324\) 6.14935e10 0.310011
\(325\) 0 0
\(326\) 3.95071e10 0.193730
\(327\) − 1.65324e11i − 0.799597i
\(328\) 6.87098e10i 0.327783i
\(329\) 5.67714e10 0.267146
\(330\) 0 0
\(331\) −5.81760e10 −0.266390 −0.133195 0.991090i \(-0.542524\pi\)
−0.133195 + 0.991090i \(0.542524\pi\)
\(332\) 1.98549e10i 0.0896905i
\(333\) 1.44322e10i 0.0643182i
\(334\) −6.95953e9 −0.0305999
\(335\) 0 0
\(336\) 9.11928e10 0.390332
\(337\) 3.40267e11i 1.43709i 0.695478 + 0.718547i \(0.255193\pi\)
−0.695478 + 0.718547i \(0.744807\pi\)
\(338\) − 2.46812e11i − 1.02859i
\(339\) −3.29646e11 −1.35566
\(340\) 0 0
\(341\) 1.63819e11 0.656100
\(342\) − 6.10351e10i − 0.241247i
\(343\) 1.38413e10i 0.0539949i
\(344\) 2.02366e11 0.779157
\(345\) 0 0
\(346\) 4.66262e11 1.74899
\(347\) 5.02625e11i 1.86107i 0.366208 + 0.930533i \(0.380656\pi\)
−0.366208 + 0.930533i \(0.619344\pi\)
\(348\) − 2.12043e11i − 0.775028i
\(349\) −7.14710e10 −0.257879 −0.128939 0.991652i \(-0.541157\pi\)
−0.128939 + 0.991652i \(0.541157\pi\)
\(350\) 0 0
\(351\) −1.27153e11 −0.447142
\(352\) − 1.46319e11i − 0.507995i
\(353\) − 2.55096e10i − 0.0874414i −0.999044 0.0437207i \(-0.986079\pi\)
0.999044 0.0437207i \(-0.0139212\pi\)
\(354\) 2.75868e11 0.933656
\(355\) 0 0
\(356\) −2.43812e11 −0.804508
\(357\) − 1.46595e11i − 0.477652i
\(358\) 5.68277e11i 1.82846i
\(359\) −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
\(362\) 3.81246e11i 1.16685i
\(363\) 1.97809e11i 0.597951i
\(364\) 2.76289e10 0.0824910
\(365\) 0 0
\(366\) −4.76736e10 −0.138871
\(367\) 4.59514e11i 1.32221i 0.750291 + 0.661107i \(0.229913\pi\)
−0.750291 + 0.661107i \(0.770087\pi\)
\(368\) 2.03650e11i 0.578854i
\(369\) −6.36691e10 −0.178776
\(370\) 0 0
\(371\) 1.37560e11 0.376971
\(372\) − 2.02318e11i − 0.547761i
\(373\) − 5.04230e11i − 1.34877i −0.738379 0.674386i \(-0.764408\pi\)
0.738379 0.674386i \(-0.235592\pi\)
\(374\) −3.76516e11 −0.995086
\(375\) 0 0
\(376\) 1.58894e11 0.409979
\(377\) 2.84315e11i 0.724875i
\(378\) 2.02058e11i 0.509052i
\(379\) 9.63136e10 0.239779 0.119889 0.992787i \(-0.461746\pi\)
0.119889 + 0.992787i \(0.461746\pi\)
\(380\) 0 0
\(381\) −4.05234e11 −0.985242
\(382\) − 5.10599e10i − 0.122686i
\(383\) 6.10835e11i 1.45054i 0.688465 + 0.725269i \(0.258285\pi\)
−0.688465 + 0.725269i \(0.741715\pi\)
\(384\) 3.63793e11 0.853814
\(385\) 0 0
\(386\) 3.46492e11 0.794421
\(387\) 1.87520e11i 0.424960i
\(388\) 3.27980e11i 0.734691i
\(389\) 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
\(392\) 3.87395e10i 0.0828641i
\(393\) 2.14248e11i 0.453054i
\(394\) −5.40785e11 −1.13055
\(395\) 0 0
\(396\) 4.32718e10 0.0884253
\(397\) 3.67168e11i 0.741836i 0.928666 + 0.370918i \(0.120957\pi\)
−0.928666 + 0.370918i \(0.879043\pi\)
\(398\) 4.26007e11i 0.851026i
\(399\) −9.74973e10 −0.192582
\(400\) 0 0
\(401\) −9.73985e11 −1.88106 −0.940530 0.339712i \(-0.889670\pi\)
−0.940530 + 0.339712i \(0.889670\pi\)
\(402\) − 7.94324e11i − 1.51698i
\(403\) 2.71275e11i 0.512315i
\(404\) 3.97178e11 0.741770
\(405\) 0 0
\(406\) 4.51801e11 0.825240
\(407\) − 5.92121e10i − 0.106964i
\(408\) − 4.10294e11i − 0.733035i
\(409\) −3.58196e11 −0.632944 −0.316472 0.948602i \(-0.602498\pi\)
−0.316472 + 0.948602i \(0.602498\pi\)
\(410\) 0 0
\(411\) −1.35996e11 −0.235091
\(412\) − 2.93783e11i − 0.502330i
\(413\) 2.03928e11i 0.344907i
\(414\) −1.08445e11 −0.181430
\(415\) 0 0
\(416\) 2.42296e11 0.396668
\(417\) − 5.67242e11i − 0.918662i
\(418\) 2.50413e11i 0.401203i
\(419\) −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 \(0.311879\pi\)
\(422\) 1.09095e12i 1.67456i
\(423\) 1.47237e11i 0.223607i
\(424\) 3.85007e11 0.578525
\(425\) 0 0
\(426\) −2.01058e11 −0.295786
\(427\) − 3.52415e10i − 0.0513013i
\(428\) 9.13782e10i 0.131627i
\(429\) 1.25377e11 0.178714
\(430\) 0 0
\(431\) 8.27297e11 1.15482 0.577409 0.816455i \(-0.304064\pi\)
0.577409 + 0.816455i \(0.304064\pi\)
\(432\) 9.84092e11i 1.35944i
\(433\) 8.73032e11i 1.19353i 0.802415 + 0.596767i \(0.203548\pi\)
−0.802415 + 0.596767i \(0.796452\pi\)
\(434\) 4.31080e11 0.583249
\(435\) 0 0
\(436\) −3.87657e11 −0.513757
\(437\) − 2.17729e11i − 0.285594i
\(438\) 9.21665e11i 1.19657i
\(439\) −7.15061e11 −0.918867 −0.459433 0.888212i \(-0.651947\pi\)
−0.459433 + 0.888212i \(0.651947\pi\)
\(440\) 0 0
\(441\) −3.58974e10 −0.0451949
\(442\) − 6.23488e11i − 0.777012i
\(443\) 5.89691e11i 0.727457i 0.931505 + 0.363729i \(0.118496\pi\)
−0.931505 + 0.363729i \(0.881504\pi\)
\(444\) −7.31275e10 −0.0893012
\(445\) 0 0
\(446\) 3.03308e10 0.0362975
\(447\) − 9.99326e11i − 1.18392i
\(448\) 1.74759e10i 0.0204969i
\(449\) 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
\(452\) 7.72964e11i 0.871037i
\(453\) − 6.36730e11i − 0.710417i
\(454\) 1.38493e12 1.52995
\(455\) 0 0
\(456\) −2.72879e11 −0.295548
\(457\) − 1.54296e12i − 1.65475i −0.561651 0.827374i \(-0.689834\pi\)
0.561651 0.827374i \(-0.310166\pi\)
\(458\) − 1.21177e12i − 1.28685i
\(459\) 1.58195e12 1.66355
\(460\) 0 0
\(461\) −8.38680e11 −0.864852 −0.432426 0.901669i \(-0.642342\pi\)
−0.432426 + 0.901669i \(0.642342\pi\)
\(462\) − 1.99235e11i − 0.203459i
\(463\) − 8.61819e11i − 0.871569i −0.900051 0.435784i \(-0.856471\pi\)
0.900051 0.435784i \(-0.143529\pi\)
\(464\) 2.20043e12 2.20382
\(465\) 0 0
\(466\) −2.11503e11 −0.207768
\(467\) − 1.46986e12i − 1.43005i −0.699101 0.715023i \(-0.746416\pi\)
0.699101 0.715023i \(-0.253584\pi\)
\(468\) 7.16555e10i 0.0690468i
\(469\) 5.87183e11 0.560397
\(470\) 0 0
\(471\) 1.54408e11 0.144569
\(472\) 5.70762e11i 0.529317i
\(473\) − 7.69353e11i − 0.706725i
\(474\) 8.96797e11 0.816003
\(475\) 0 0
\(476\) −3.43739e11 −0.306901
\(477\) 3.56761e11i 0.315533i
\(478\) − 7.74245e11i − 0.678348i
\(479\) −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
\(482\) 2.31282e12i 1.95177i
\(483\) 1.73230e11i 0.144831i
\(484\) 4.63827e11 0.384195
\(485\) 0 0
\(486\) −9.22132e11 −0.749773
\(487\) − 1.30073e12i − 1.04786i −0.851760 0.523932i \(-0.824464\pi\)
0.851760 0.523932i \(-0.175536\pi\)
\(488\) − 9.86350e10i − 0.0787303i
\(489\) 1.63672e11 0.129445
\(490\) 0 0
\(491\) 6.88947e11 0.534957 0.267479 0.963564i \(-0.413810\pi\)
0.267479 + 0.963564i \(0.413810\pi\)
\(492\) − 3.22609e11i − 0.248218i
\(493\) − 3.53724e12i − 2.69683i
\(494\) −4.14670e11 −0.313279
\(495\) 0 0
\(496\) 2.09951e12 1.55758
\(497\) − 1.48627e11i − 0.109268i
\(498\) 2.37091e11i 0.172736i
\(499\) −1.76710e12 −1.27588 −0.637939 0.770087i \(-0.720213\pi\)
−0.637939 + 0.770087i \(0.720213\pi\)
\(500\) 0 0
\(501\) −2.88323e10 −0.0204461
\(502\) − 5.65088e11i − 0.397145i
\(503\) 2.63527e12i 1.83556i 0.397089 + 0.917780i \(0.370020\pi\)
−0.397089 + 0.917780i \(0.629980\pi\)
\(504\) −1.00471e11 −0.0693590
\(505\) 0 0
\(506\) 4.44927e11 0.301725
\(507\) − 1.02251e12i − 0.687274i
\(508\) 9.50203e11i 0.633038i
\(509\) −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
\(512\) − 7.48685e11i − 0.481487i
\(513\) − 1.05213e12i − 0.670718i
\(514\) −7.91786e11 −0.500350
\(515\) 0 0
\(516\) −9.50157e11 −0.590027
\(517\) − 6.04080e11i − 0.371867i
\(518\) − 1.55813e11i − 0.0950868i
\(519\) 1.93166e12 1.16863
\(520\) 0 0
\(521\) −1.87441e12 −1.11454 −0.557269 0.830332i \(-0.688151\pi\)
−0.557269 + 0.830332i \(0.688151\pi\)
\(522\) 1.17175e12i 0.690743i
\(523\) − 1.58412e12i − 0.925828i −0.886403 0.462914i \(-0.846804\pi\)
0.886403 0.462914i \(-0.153196\pi\)
\(524\) 5.02375e11 0.291096
\(525\) 0 0
\(526\) 3.89590e12 2.21908
\(527\) − 3.37501e12i − 1.90602i
\(528\) − 9.70343e11i − 0.543341i
\(529\) 1.41430e12 0.785219
\(530\) 0 0
\(531\) −5.28889e11 −0.288695
\(532\) 2.28614e11i 0.123737i
\(533\) 4.32565e11i 0.232156i
\(534\) −2.91140e12 −1.54941
\(535\) 0 0
\(536\) 1.64343e12 0.860021
\(537\) 2.35429e12i 1.22173i
\(538\) 1.62087e11i 0.0834120i
\(539\) 1.47279e11 0.0751609
\(540\) 0 0
\(541\) 2.79888e12 1.40474 0.702370 0.711812i \(-0.252125\pi\)
0.702370 + 0.711812i \(0.252125\pi\)
\(542\) 2.09135e12i 1.04095i
\(543\) 1.57945e12i 0.779660i
\(544\) −3.01448e12 −1.47577
\(545\) 0 0
\(546\) 3.29921e11 0.158870
\(547\) − 1.16606e12i − 0.556901i −0.960451 0.278451i \(-0.910179\pi\)
0.960451 0.278451i \(-0.0898209\pi\)
\(548\) 3.18886e11i 0.151051i
\(549\) 9.13988e10 0.0429403
\(550\) 0 0
\(551\) −2.35255e12 −1.08732
\(552\) 4.84843e11i 0.222267i
\(553\) 6.62934e11i 0.301444i
\(554\) −6.22594e11 −0.280809
\(555\) 0 0
\(556\) −1.33008e12 −0.590259
\(557\) − 3.29033e12i − 1.44841i −0.689586 0.724204i \(-0.742207\pi\)
0.689586 0.724204i \(-0.257793\pi\)
\(558\) 1.11801e12i 0.488192i
\(559\) 1.27400e12 0.551845
\(560\) 0 0
\(561\) −1.55985e12 −0.664890
\(562\) 3.80962e12i 1.61090i
\(563\) − 3.63871e12i − 1.52637i −0.646179 0.763186i \(-0.723634\pi\)
0.646179 0.763186i \(-0.276366\pi\)
\(564\) −7.46044e11 −0.310462
\(565\) 0 0
\(566\) −4.79035e12 −1.96197
\(567\) 5.42815e11i 0.220561i
\(568\) − 4.15981e11i − 0.167690i
\(569\) −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 \(0.917672\pi\)
\(572\) − 2.93987e11i − 0.114827i
\(573\) − 2.11534e11i − 0.0819756i
\(574\) 6.87385e11 0.264299
\(575\) 0 0
\(576\) −4.53238e10 −0.0171564
\(577\) 1.68758e12i 0.633830i 0.948454 + 0.316915i \(0.102647\pi\)
−0.948454 + 0.316915i \(0.897353\pi\)
\(578\) 4.43654e12i 1.65337i
\(579\) 1.43547e12 0.530811
\(580\) 0 0
\(581\) −1.75263e11 −0.0638114
\(582\) 3.91647e12i 1.41495i
\(583\) − 1.46371e12i − 0.524744i
\(584\) −1.90689e12 −0.678373
\(585\) 0 0
\(586\) 2.96051e12 1.03711
\(587\) 4.82739e12i 1.67819i 0.543987 + 0.839094i \(0.316914\pi\)
−0.543987 + 0.839094i \(0.683086\pi\)
\(588\) − 1.81891e11i − 0.0627499i
\(589\) −2.24466e12 −0.768478
\(590\) 0 0
\(591\) −2.24039e12 −0.755407
\(592\) − 7.58865e11i − 0.253932i
\(593\) − 1.12076e12i − 0.372193i −0.982532 0.186096i \(-0.940416\pi\)
0.982532 0.186096i \(-0.0595837\pi\)
\(594\) 2.15001e12 0.708600
\(595\) 0 0
\(596\) −2.34325e12 −0.760694
\(597\) 1.76489e12i 0.568633i
\(598\) 7.36773e11i 0.235602i
\(599\) −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 \(0.452531\pi\)
\(602\) − 2.02450e12i − 0.628253i
\(603\) 1.52286e12i 0.469064i
\(604\) −1.49302e12 −0.456457
\(605\) 0 0
\(606\) 4.74277e12 1.42858
\(607\) − 3.35939e12i − 1.00441i −0.864749 0.502205i \(-0.832522\pi\)
0.864749 0.502205i \(-0.167478\pi\)
\(608\) 2.00487e12i 0.595006i
\(609\) 1.87175e12 0.551403
\(610\) 0 0
\(611\) 1.00032e12 0.290372
\(612\) − 8.91489e11i − 0.256882i
\(613\) − 2.62257e12i − 0.750162i −0.926992 0.375081i \(-0.877615\pi\)
0.926992 0.375081i \(-0.122385\pi\)
\(614\) −4.87603e12 −1.38455
\(615\) 0 0
\(616\) 4.12210e11 0.115347
\(617\) 3.35285e10i 0.00931388i 0.999989 + 0.00465694i \(0.00148236\pi\)
−0.999989 + 0.00465694i \(0.998518\pi\)
\(618\) − 3.50812e12i − 0.967444i
\(619\) −5.51587e12 −1.51010 −0.755051 0.655666i \(-0.772388\pi\)
−0.755051 + 0.655666i \(0.772388\pi\)
\(620\) 0 0
\(621\) −1.86939e12 −0.504414
\(622\) 2.96540e12i 0.794377i
\(623\) − 2.15218e12i − 0.572377i
\(624\) 1.60683e12 0.424268
\(625\) 0 0
\(626\) −6.81962e12 −1.77491
\(627\) 1.03743e12i 0.268073i
\(628\) − 3.62060e11i − 0.0928886i
\(629\) −1.21989e12 −0.310738
\(630\) 0 0
\(631\) 2.72456e12 0.684170 0.342085 0.939669i \(-0.388867\pi\)
0.342085 + 0.939669i \(0.388867\pi\)
\(632\) 1.85544e12i 0.462616i
\(633\) 4.51966e12i 1.11890i
\(634\) −5.14572e12 −1.26487
\(635\) 0 0
\(636\) −1.80770e12 −0.438096
\(637\) 2.43886e11i 0.0586893i
\(638\) − 4.80742e12i − 1.14873i
\(639\) 3.85463e11 0.0914596
\(640\) 0 0
\(641\) 3.79136e11 0.0887022 0.0443511 0.999016i \(-0.485878\pi\)
0.0443511 + 0.999016i \(0.485878\pi\)
\(642\) 1.09116e12i 0.253502i
\(643\) − 5.15446e12i − 1.18914i −0.804043 0.594571i \(-0.797322\pi\)
0.804043 0.594571i \(-0.202678\pi\)
\(644\) 4.06195e11 0.0930568
\(645\) 0 0
\(646\) 5.15904e12 1.16553
\(647\) 2.68059e12i 0.601397i 0.953719 + 0.300699i \(0.0972198\pi\)
−0.953719 + 0.300699i \(0.902780\pi\)
\(648\) 1.51925e12i 0.338487i
\(649\) 2.16991e12 0.480111
\(650\) 0 0
\(651\) 1.78590e12 0.389712
\(652\) − 3.83783e11i − 0.0831710i
\(653\) − 6.44986e12i − 1.38816i −0.719896 0.694082i \(-0.755810\pi\)
0.719896 0.694082i \(-0.244190\pi\)
\(654\) −4.62908e12 −0.989451
\(655\) 0 0
\(656\) 3.34780e12 0.705818
\(657\) − 1.76700e12i − 0.369991i
\(658\) − 1.58960e12i − 0.330576i
\(659\) 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 \(0.516016\pi\)
\(662\) 1.62893e12i 0.329641i
\(663\) − 2.58302e12i − 0.519179i
\(664\) −4.90533e11 −0.0979291
\(665\) 0 0
\(666\) 4.04102e11 0.0795897
\(667\) 4.17995e12i 0.817720i
\(668\) 6.76068e10i 0.0131370i
\(669\) 1.25656e11 0.0242530
\(670\) 0 0
\(671\) −3.74989e11 −0.0714113
\(672\) − 1.59513e12i − 0.301740i
\(673\) 2.83805e12i 0.533277i 0.963797 + 0.266638i \(0.0859130\pi\)
−0.963797 + 0.266638i \(0.914087\pi\)
\(674\) 9.52747e12 1.77831
\(675\) 0 0
\(676\) −2.39760e12 −0.441587
\(677\) − 7.71236e12i − 1.41104i −0.708691 0.705519i \(-0.750714\pi\)
0.708691 0.705519i \(-0.249286\pi\)
\(678\) 9.23010e12i 1.67754i
\(679\) −2.89515e12 −0.522705
\(680\) 0 0
\(681\) 5.73757e12 1.02227
\(682\) − 4.58693e12i − 0.811882i
\(683\) 1.07677e13i 1.89334i 0.322204 + 0.946670i \(0.395576\pi\)
−0.322204 + 0.946670i \(0.604424\pi\)
\(684\) −5.92912e11 −0.103571
\(685\) 0 0
\(686\) 3.87556e11 0.0668153
\(687\) − 5.02021e12i − 0.859837i
\(688\) − 9.86005e12i − 1.67776i
\(689\) 2.42382e12 0.409746
\(690\) 0 0
\(691\) 2.02298e12 0.337552 0.168776 0.985654i \(-0.446019\pi\)
0.168776 + 0.985654i \(0.446019\pi\)
\(692\) − 4.52940e12i − 0.750867i
\(693\) 3.81969e11i 0.0629113i
\(694\) 1.40735e13 2.30295
\(695\) 0 0
\(696\) 5.23871e12 0.846219
\(697\) − 5.38168e12i − 0.863714i
\(698\) 2.00119e12i 0.319109i
\(699\) −8.76226e11 −0.138825
\(700\) 0 0
\(701\) 8.66031e11 0.135457 0.0677286 0.997704i \(-0.478425\pi\)
0.0677286 + 0.997704i \(0.478425\pi\)
\(702\) 3.56029e12i 0.553310i
\(703\) 8.11328e11i 0.125285i
\(704\) 1.85953e11 0.0285317
\(705\) 0 0
\(706\) −7.14268e11 −0.108203
\(707\) 3.50597e12i 0.527741i
\(708\) − 2.67986e12i − 0.400832i
\(709\) 3.78834e12 0.563042 0.281521 0.959555i \(-0.409161\pi\)
0.281521 + 0.959555i \(0.409161\pi\)
\(710\) 0 0
\(711\) −1.71932e12 −0.252315
\(712\) − 6.02360e12i − 0.878407i
\(713\) 3.98824e12i 0.577934i
\(714\) −4.10465e12 −0.591064
\(715\) 0 0
\(716\) 5.52040e12 0.784986
\(717\) − 3.20758e12i − 0.453254i
\(718\) 4.19485e12i 0.589056i
\(719\) −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
\(722\) 5.60408e12i 0.767515i
\(723\) 9.58166e12i 1.30412i
\(724\) 3.70353e12 0.500947
\(725\) 0 0
\(726\) 5.53864e12 0.739927
\(727\) − 3.13227e12i − 0.415867i −0.978143 0.207933i \(-0.933326\pi\)
0.978143 0.207933i \(-0.0666737\pi\)
\(728\) 6.82595e11i 0.0900683i
\(729\) −8.27017e12 −1.08453
\(730\) 0 0
\(731\) −1.58503e13 −2.05309
\(732\) 4.63115e11i 0.0596195i
\(733\) − 1.33197e13i − 1.70422i −0.523360 0.852112i \(-0.675322\pi\)
0.523360 0.852112i \(-0.324678\pi\)
\(734\) 1.28664e13 1.63616
\(735\) 0 0
\(736\) 3.56220e12 0.447474
\(737\) − 6.24796e12i − 0.780072i
\(738\) 1.78273e12i 0.221224i
\(739\) 1.56702e13 1.93274 0.966371 0.257154i \(-0.0827847\pi\)
0.966371 + 0.257154i \(0.0827847\pi\)
\(740\) 0 0
\(741\) −1.71792e12 −0.209325
\(742\) − 3.85167e12i − 0.466478i
\(743\) 7.91108e12i 0.952327i 0.879357 + 0.476164i \(0.157973\pi\)
−0.879357 + 0.476164i \(0.842027\pi\)
\(744\) 4.99844e12 0.598076
\(745\) 0 0
\(746\) −1.41184e13 −1.66902
\(747\) − 4.54546e11i − 0.0534115i
\(748\) 3.65758e12i 0.427205i
\(749\) −8.06614e11 −0.0936478
\(750\) 0 0
\(751\) −4.01514e12 −0.460597 −0.230299 0.973120i \(-0.573970\pi\)
−0.230299 + 0.973120i \(0.573970\pi\)
\(752\) − 7.74191e12i − 0.882811i
\(753\) − 2.34108e12i − 0.265362i
\(754\) 7.96081e12 0.896987
\(755\) 0 0
\(756\) 1.96285e12 0.218544
\(757\) − 2.84075e12i − 0.314414i −0.987566 0.157207i \(-0.949751\pi\)
0.987566 0.157207i \(-0.0502489\pi\)
\(758\) − 2.69678e12i − 0.296711i
\(759\) 1.84327e12 0.201605
\(760\) 0 0
\(761\) −7.10439e12 −0.767884 −0.383942 0.923357i \(-0.625434\pi\)
−0.383942 + 0.923357i \(0.625434\pi\)
\(762\) 1.13465e13i 1.21918i
\(763\) − 3.42192e12i − 0.365519i
\(764\) −4.96011e11 −0.0526709
\(765\) 0 0
\(766\) 1.71034e13 1.79495
\(767\) 3.59325e12i 0.374894i
\(768\) − 9.75390e12i − 1.01170i
\(769\) 4.97052e12 0.512546 0.256273 0.966604i \(-0.417505\pi\)
0.256273 + 0.966604i \(0.417505\pi\)
\(770\) 0 0
\(771\) −3.28026e12 −0.334321
\(772\) − 3.36593e12i − 0.341057i
\(773\) 1.11565e13i 1.12388i 0.827179 + 0.561939i \(0.189944\pi\)
−0.827179 + 0.561939i \(0.810056\pi\)
\(774\) 5.25056e12 0.525861
\(775\) 0 0
\(776\) −8.10304e12 −0.802177
\(777\) − 6.45512e11i − 0.0635345i
\(778\) − 1.81974e13i − 1.78075i
\(779\) −3.57925e12 −0.348236
\(780\) 0 0
\(781\) −1.58147e12 −0.152101
\(782\) − 9.16642e12i − 0.876535i
\(783\) 2.01987e13i 1.92041i
\(784\) 1.88753e12 0.178432
\(785\) 0 0
\(786\) 5.99894e12 0.560626
\(787\) 6.23763e12i 0.579607i 0.957086 + 0.289803i \(0.0935899\pi\)
−0.957086 + 0.289803i \(0.906410\pi\)
\(788\) 5.25334e12i 0.485364i
\(789\) 1.61402e13 1.48273
\(790\) 0 0
\(791\) −6.82311e12 −0.619710
\(792\) 1.06907e12i 0.0965477i
\(793\) − 6.20960e11i − 0.0557615i
\(794\) 1.02807e13 0.917975
\(795\) 0 0
\(796\) 4.13835e12 0.365358
\(797\) 1.83799e12i 0.161355i 0.996740 + 0.0806773i \(0.0257083\pi\)
−0.996740 + 0.0806773i \(0.974292\pi\)
\(798\) 2.72992e12i 0.238307i
\(799\) −1.24453e13 −1.08030
\(800\) 0 0
\(801\) 5.58169e12 0.479092
\(802\) 2.72716e13i 2.32769i
\(803\) 7.24960e12i 0.615310i
\(804\) −7.71629e12 −0.651263
\(805\) 0 0
\(806\) 7.59570e12 0.633957
\(807\) 6.71504e11i 0.0557337i
\(808\) 9.81263e12i 0.809906i
\(809\) −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 \(0.742938\pi\)
\(812\) − 4.38892e12i − 0.354288i
\(813\) 8.66416e12i 0.695535i
\(814\) −1.65794e12 −0.132361
\(815\) 0 0
\(816\) −1.99911e13 −1.57845
\(817\) 1.05417e13i 0.827774i
\(818\) 1.00295e13i 0.783228i
\(819\) −6.32518e11 −0.0491242
\(820\) 0 0
\(821\) 1.33485e13 1.02539 0.512694 0.858572i \(-0.328648\pi\)
0.512694 + 0.858572i \(0.328648\pi\)
\(822\) 3.80788e12i 0.290911i
\(823\) − 1.72858e13i − 1.31338i −0.754161 0.656689i \(-0.771956\pi\)
0.754161 0.656689i \(-0.228044\pi\)
\(824\) 7.25817e12 0.548473
\(825\) 0 0
\(826\) 5.71000e12 0.426801
\(827\) − 1.64651e13i − 1.22402i −0.790849 0.612012i \(-0.790361\pi\)
0.790849 0.612012i \(-0.209639\pi\)
\(828\) 1.05347e12i 0.0778906i
\(829\) 5.91786e12 0.435181 0.217590 0.976040i \(-0.430180\pi\)
0.217590 + 0.976040i \(0.430180\pi\)
\(830\) 0 0
\(831\) −2.57932e12 −0.187629
\(832\) 3.07928e11i 0.0222789i
\(833\) − 3.03426e12i − 0.218348i
\(834\) −1.58828e13 −1.13679
\(835\) 0 0
\(836\) 2.43259e12 0.172242
\(837\) 1.92723e13i 1.35728i
\(838\) 1.35719e13i 0.950701i
\(839\) −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
\(842\) − 2.01123e13i − 1.37898i
\(843\) 1.57827e13i 1.07636i
\(844\) 1.05978e13 0.718913
\(845\) 0 0
\(846\) 4.12263e12 0.276699
\(847\) 4.09430e12i 0.273341i
\(848\) − 1.87590e13i − 1.24574i
\(849\) −1.98457e13 −1.31094
\(850\) 0 0
\(851\) 1.44154e12 0.0942203
\(852\) 1.95313e12i 0.126985i
\(853\) 2.26446e13i 1.46452i 0.681027 + 0.732258i \(0.261534\pi\)
−0.681027 + 0.732258i \(0.738466\pi\)
\(854\) −9.86761e11 −0.0634821
\(855\) 0 0
\(856\) −2.25758e12 −0.143718
\(857\) 1.20223e13i 0.761334i 0.924712 + 0.380667i \(0.124306\pi\)
−0.924712 + 0.380667i \(0.875694\pi\)
\(858\) − 3.51055e12i − 0.221148i
\(859\) 9.44258e12 0.591727 0.295864 0.955230i \(-0.404393\pi\)
0.295864 + 0.955230i \(0.404393\pi\)
\(860\) 0 0
\(861\) 2.84774e12 0.176598
\(862\) − 2.31643e13i − 1.42901i
\(863\) 8.80380e12i 0.540283i 0.962821 + 0.270142i \(0.0870705\pi\)
−0.962821 + 0.270142i \(0.912929\pi\)
\(864\) 1.72135e13 1.05089
\(865\) 0 0
\(866\) 2.44449e13 1.47692
\(867\) 1.83800e13i 1.10474i
\(868\) − 4.18763e12i − 0.250397i
\(869\) 7.05399e12 0.419610
\(870\) 0 0
\(871\) 1.03463e13 0.609119
\(872\) − 9.57740e12i − 0.560949i
\(873\) − 7.50858e12i − 0.437516i
\(874\) −6.09641e12 −0.353405
\(875\) 0 0
\(876\) 8.95331e12 0.513707
\(877\) − 6.73513e12i − 0.384457i −0.981350 0.192229i \(-0.938429\pi\)
0.981350 0.192229i \(-0.0615715\pi\)
\(878\) 2.00217e13i 1.13704i
\(879\) 1.22650e13 0.692973
\(880\) 0 0
\(881\) −7.58474e12 −0.424179 −0.212089 0.977250i \(-0.568027\pi\)
−0.212089 + 0.977250i \(0.568027\pi\)
\(882\) 1.00513e12i 0.0559258i
\(883\) − 1.78234e13i − 0.986662i −0.869842 0.493331i \(-0.835779\pi\)
0.869842 0.493331i \(-0.164221\pi\)
\(884\) −6.05674e12 −0.333583
\(885\) 0 0
\(886\) 1.65113e13 0.900182
\(887\) 2.45177e13i 1.32991i 0.746882 + 0.664956i \(0.231550\pi\)
−0.746882 + 0.664956i \(0.768450\pi\)
\(888\) − 1.80668e12i − 0.0975041i
\(889\) −8.38764e12 −0.450383
\(890\) 0 0
\(891\) 5.77586e12 0.307020
\(892\) − 2.94642e11i − 0.0155830i
\(893\) 8.27714e12i 0.435561i
\(894\) −2.79811e13 −1.46503
\(895\) 0 0
\(896\) 7.52988e12 0.390303
\(897\) 3.05234e12i 0.157423i
\(898\) − 2.98134e13i − 1.52992i
\(899\) 4.30928e13 2.20032
\(900\) 0 0
\(901\) −3.01555e13 −1.52442
\(902\) − 7.31416e12i − 0.367905i
\(903\) − 8.38723e12i − 0.419782i
\(904\) −1.90968e13 −0.951047
\(905\) 0 0
\(906\) −1.78284e13 −0.879096
\(907\) 8.14231e12i 0.399498i 0.979847 + 0.199749i \(0.0640127\pi\)
−0.979847 + 0.199749i \(0.935987\pi\)
\(908\) − 1.34536e13i − 0.656830i
\(909\) −9.09275e12 −0.441731
\(910\) 0 0
\(911\) −1.26965e13 −0.610733 −0.305367 0.952235i \(-0.598779\pi\)
−0.305367 + 0.952235i \(0.598779\pi\)
\(912\) 1.32957e13i 0.636406i
\(913\) 1.86490e12i 0.0888254i
\(914\) −4.32029e13 −2.04765
\(915\) 0 0
\(916\) −1.17715e13 −0.552463
\(917\) 4.43456e12i 0.207104i
\(918\) − 4.42947e13i − 2.05854i
\(919\) 2.53575e13 1.17270 0.586350 0.810058i \(-0.300564\pi\)
0.586350 + 0.810058i \(0.300564\pi\)
\(920\) 0 0
\(921\) −2.02007e13 −0.925120
\(922\) 2.34830e13i 1.07020i
\(923\) − 2.61882e12i − 0.118768i
\(924\) −1.93542e12 −0.0873478
\(925\) 0 0
\(926\) −2.41309e13 −1.07851
\(927\) 6.72569e12i 0.299143i
\(928\) − 3.84895e13i − 1.70363i
\(929\) −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
\(932\) 2.05460e12i 0.0891980i
\(933\) 1.22852e13i 0.530782i
\(934\) −4.11561e13 −1.76959
\(935\) 0 0
\(936\) −1.77031e12 −0.0753891
\(937\) − 8.65559e12i − 0.366833i −0.983035 0.183416i \(-0.941284\pi\)
0.983035 0.183416i \(-0.0587157\pi\)
\(938\) − 1.64411e13i − 0.693456i
\(939\) −2.82527e13 −1.18595
\(940\) 0 0
\(941\) 3.42336e13 1.42331 0.711654 0.702530i \(-0.247946\pi\)
0.711654 + 0.702530i \(0.247946\pi\)
\(942\) − 4.32343e12i − 0.178895i
\(943\) 6.35950e12i 0.261891i
\(944\) 2.78097e13 1.13978
\(945\) 0 0
\(946\) −2.15419e13 −0.874527
\(947\) − 2.38597e13i − 0.964030i −0.876163 0.482015i \(-0.839905\pi\)
0.876163 0.482015i \(-0.160095\pi\)
\(948\) − 8.71174e12i − 0.350322i
\(949\) −1.20049e13 −0.480464
\(950\) 0 0
\(951\) −2.13180e13 −0.845150
\(952\) − 8.49238e12i − 0.335091i
\(953\) − 4.83481e13i − 1.89872i −0.314189 0.949360i \(-0.601733\pi\)
0.314189 0.949360i \(-0.398267\pi\)
\(954\) 9.98932e12 0.390452
\(955\) 0 0
\(956\) −7.52123e12 −0.291225
\(957\) − 1.99165e13i − 0.767553i
\(958\) 6.16636e13i 2.36529i
\(959\) −2.81488e12 −0.107467
\(960\) 0 0
\(961\) 1.46768e13 0.555106
\(962\) − 2.74545e12i − 0.103354i
\(963\) − 2.09196e12i − 0.0783852i
\(964\) 2.24674e13 0.837925
\(965\) 0 0
\(966\) 4.85045e12 0.179219
\(967\) 2.28025e13i 0.838617i 0.907844 + 0.419308i \(0.137727\pi\)
−0.907844 + 0.419308i \(0.862273\pi\)
\(968\) 1.14593e13i 0.419486i
\(969\) 2.13731e13 0.778774
\(970\) 0 0
\(971\) 7.62385e12 0.275225 0.137612 0.990486i \(-0.456057\pi\)
0.137612 + 0.990486i \(0.456057\pi\)
\(972\) 8.95785e12i 0.321889i
\(973\) − 1.17409e13i − 0.419947i
\(974\) −3.64203e13 −1.29667
\(975\) 0 0
\(976\) −4.80587e12 −0.169531
\(977\) − 2.97018e13i − 1.04293i −0.853272 0.521467i \(-0.825385\pi\)
0.853272 0.521467i \(-0.174615\pi\)
\(978\) − 4.58282e12i − 0.160180i
\(979\) −2.29004e13 −0.796749
\(980\) 0 0
\(981\) 8.87477e12 0.305947
\(982\) − 1.92905e13i − 0.661976i
\(983\) − 1.08292e13i − 0.369917i −0.982746 0.184958i \(-0.940785\pi\)
0.982746 0.184958i \(-0.0592150\pi\)
\(984\) 7.97034e12 0.271019
\(985\) 0 0
\(986\) −9.90428e13 −3.33716
\(987\) − 6.58549e12i − 0.220882i
\(988\) 4.02822e12i 0.134495i
\(989\) 1.87302e13 0.622528
\(990\) 0 0
\(991\) 5.29514e12 0.174400 0.0871999 0.996191i \(-0.472208\pi\)
0.0871999 + 0.996191i \(0.472208\pi\)
\(992\) − 3.67242e13i − 1.20406i
\(993\) 6.74842e12i 0.220257i
\(994\) −4.16154e12 −0.135212
\(995\) 0 0
\(996\) 2.30317e12 0.0741581
\(997\) − 1.64214e13i − 0.526360i −0.964747 0.263180i \(-0.915229\pi\)
0.964747 0.263180i \(-0.0847712\pi\)
\(998\) 4.94788e13i 1.57882i
\(999\) 6.96593e12 0.221276
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 175.10.b.a.99.1 2
5.2 odd 4 35.10.a.a.1.1 1
5.3 odd 4 175.10.a.a.1.1 1
5.4 even 2 inner 175.10.b.a.99.2 2
15.2 even 4 315.10.a.a.1.1 1
35.27 even 4 245.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.a.1.1 1 5.2 odd 4
175.10.a.a.1.1 1 5.3 odd 4
175.10.b.a.99.1 2 1.1 even 1 trivial
175.10.b.a.99.2 2 5.4 even 2 inner
245.10.a.b.1.1 1 35.27 even 4
315.10.a.a.1.1 1 15.2 even 4