Properties

Label 2.46.a.b.1.1
Level $2$
Weight $46$
Character 2.1
Self dual yes
Analytic conductor $25.651$
Analytic rank $0$
Dimension $2$
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] = [2,46,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 46, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 46);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 46 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(25.6511452149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 200169772272162 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.41481e7\) of defining polynomial
Character \(\chi\) \(=\) 2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.19430e6 q^{2} -2.43982e10 q^{3} +1.75922e13 q^{4} -4.75445e15 q^{5} -1.02334e17 q^{6} +5.49745e18 q^{7} +7.37870e19 q^{8} -2.35904e21 q^{9} +O(q^{10})\) \(q+4.19430e6 q^{2} -2.43982e10 q^{3} +1.75922e13 q^{4} -4.75445e15 q^{5} -1.02334e17 q^{6} +5.49745e18 q^{7} +7.37870e19 q^{8} -2.35904e21 q^{9} -1.99416e22 q^{10} +2.16058e23 q^{11} -4.29218e23 q^{12} +1.18584e25 q^{13} +2.30580e25 q^{14} +1.16000e26 q^{15} +3.09485e26 q^{16} +1.71231e27 q^{17} -9.89453e27 q^{18} -7.20975e26 q^{19} -8.36411e28 q^{20} -1.34128e29 q^{21} +9.06215e29 q^{22} +6.93777e30 q^{23} -1.80027e30 q^{24} -5.81693e30 q^{25} +4.97377e31 q^{26} +1.29636e32 q^{27} +9.67121e31 q^{28} +1.23261e33 q^{29} +4.86540e32 q^{30} +6.35535e33 q^{31} +1.29807e33 q^{32} -5.27144e33 q^{33} +7.18193e33 q^{34} -2.61373e34 q^{35} -4.15006e34 q^{36} -1.13513e35 q^{37} -3.02399e33 q^{38} -2.89324e35 q^{39} -3.50816e35 q^{40} -2.36419e36 q^{41} -5.62573e35 q^{42} +4.96460e36 q^{43} +3.80094e36 q^{44} +1.12159e37 q^{45} +2.90991e37 q^{46} +1.34643e37 q^{47} -7.55089e36 q^{48} -7.67850e37 q^{49} -2.43980e37 q^{50} -4.17773e37 q^{51} +2.08615e38 q^{52} -1.16156e39 q^{53} +5.43734e38 q^{54} -1.02724e39 q^{55} +4.05640e38 q^{56} +1.75905e37 q^{57} +5.16995e39 q^{58} -6.70116e39 q^{59} +2.04070e39 q^{60} +1.77152e40 q^{61} +2.66563e40 q^{62} -1.29687e40 q^{63} +5.44452e39 q^{64} -5.63802e40 q^{65} -2.21100e40 q^{66} -1.34297e41 q^{67} +3.01232e40 q^{68} -1.69269e41 q^{69} -1.09628e41 q^{70} +4.73439e41 q^{71} -1.74066e41 q^{72} +1.25402e42 q^{73} -4.76108e41 q^{74} +1.41923e41 q^{75} -1.26835e40 q^{76} +1.18777e42 q^{77} -1.21351e42 q^{78} +2.71951e42 q^{79} -1.47143e42 q^{80} +3.80644e42 q^{81} -9.91615e42 q^{82} -1.70726e43 q^{83} -2.35960e42 q^{84} -8.14107e42 q^{85} +2.08230e43 q^{86} -3.00736e43 q^{87} +1.59423e43 q^{88} +5.81345e43 q^{89} +4.70430e43 q^{90} +6.51909e43 q^{91} +1.22051e44 q^{92} -1.55059e44 q^{93} +5.64732e43 q^{94} +3.42784e42 q^{95} -3.16707e43 q^{96} -4.76787e44 q^{97} -3.22060e44 q^{98} -5.09690e44 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8388608 q^{2} + 59861217192 q^{3} + 35184372088832 q^{4} + 43\!\cdots\!00 q^{5}+ \cdots + 17\!\cdots\!46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8388608 q^{2} + 59861217192 q^{3} + 35184372088832 q^{4} + 43\!\cdots\!00 q^{5}+ \cdots - 60\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.19430e6 0.707107
\(3\) −2.43982e10 −0.448880 −0.224440 0.974488i \(-0.572055\pi\)
−0.224440 + 0.974488i \(0.572055\pi\)
\(4\) 1.75922e13 0.500000
\(5\) −4.75445e15 −0.891816 −0.445908 0.895079i \(-0.647119\pi\)
−0.445908 + 0.895079i \(0.647119\pi\)
\(6\) −1.02334e17 −0.317406
\(7\) 5.49745e18 0.531441 0.265720 0.964050i \(-0.414390\pi\)
0.265720 + 0.964050i \(0.414390\pi\)
\(8\) 7.37870e19 0.353553
\(9\) −2.35904e21 −0.798507
\(10\) −1.99416e22 −0.630609
\(11\) 2.16058e23 0.800268 0.400134 0.916457i \(-0.368964\pi\)
0.400134 + 0.916457i \(0.368964\pi\)
\(12\) −4.29218e23 −0.224440
\(13\) 1.18584e25 1.02400 0.512001 0.858985i \(-0.328904\pi\)
0.512001 + 0.858985i \(0.328904\pi\)
\(14\) 2.30580e25 0.375786
\(15\) 1.16000e26 0.400318
\(16\) 3.09485e26 0.250000
\(17\) 1.71231e27 0.353574 0.176787 0.984249i \(-0.443430\pi\)
0.176787 + 0.984249i \(0.443430\pi\)
\(18\) −9.89453e27 −0.564630
\(19\) −7.20975e26 −0.0121889 −0.00609444 0.999981i \(-0.501940\pi\)
−0.00609444 + 0.999981i \(0.501940\pi\)
\(20\) −8.36411e28 −0.445908
\(21\) −1.34128e29 −0.238553
\(22\) 9.06215e29 0.565875
\(23\) 6.93777e30 1.59347 0.796735 0.604329i \(-0.206559\pi\)
0.796735 + 0.604329i \(0.206559\pi\)
\(24\) −1.80027e30 −0.158703
\(25\) −5.81693e30 −0.204665
\(26\) 4.97377e31 0.724079
\(27\) 1.29636e32 0.807314
\(28\) 9.67121e31 0.265720
\(29\) 1.23261e33 1.53770 0.768850 0.639429i \(-0.220829\pi\)
0.768850 + 0.639429i \(0.220829\pi\)
\(30\) 4.86540e32 0.283068
\(31\) 6.35535e33 1.76808 0.884039 0.467413i \(-0.154814\pi\)
0.884039 + 0.467413i \(0.154814\pi\)
\(32\) 1.29807e33 0.176777
\(33\) −5.27144e33 −0.359224
\(34\) 7.18193e33 0.250015
\(35\) −2.61373e34 −0.473947
\(36\) −4.15006e34 −0.399253
\(37\) −1.13513e35 −0.589531 −0.294765 0.955570i \(-0.595242\pi\)
−0.294765 + 0.955570i \(0.595242\pi\)
\(38\) −3.02399e33 −0.00861884
\(39\) −2.89324e35 −0.459654
\(40\) −3.50816e35 −0.315304
\(41\) −2.36419e36 −1.21912 −0.609560 0.792740i \(-0.708654\pi\)
−0.609560 + 0.792740i \(0.708654\pi\)
\(42\) −5.62573e35 −0.168683
\(43\) 4.96460e36 0.876686 0.438343 0.898808i \(-0.355566\pi\)
0.438343 + 0.898808i \(0.355566\pi\)
\(44\) 3.80094e36 0.400134
\(45\) 1.12159e37 0.712121
\(46\) 2.90991e37 1.12675
\(47\) 1.34643e37 0.321352 0.160676 0.987007i \(-0.448633\pi\)
0.160676 + 0.987007i \(0.448633\pi\)
\(48\) −7.55089e36 −0.112220
\(49\) −7.67850e37 −0.717571
\(50\) −2.43980e37 −0.144720
\(51\) −4.17773e37 −0.158712
\(52\) 2.08615e38 0.512001
\(53\) −1.16156e39 −1.85710 −0.928551 0.371205i \(-0.878945\pi\)
−0.928551 + 0.371205i \(0.878945\pi\)
\(54\) 5.43734e38 0.570857
\(55\) −1.02724e39 −0.713692
\(56\) 4.05640e38 0.187893
\(57\) 1.75905e37 0.00547135
\(58\) 5.16995e39 1.08732
\(59\) −6.70116e39 −0.959356 −0.479678 0.877444i \(-0.659247\pi\)
−0.479678 + 0.877444i \(0.659247\pi\)
\(60\) 2.04070e39 0.200159
\(61\) 1.77152e40 1.19791 0.598954 0.800783i \(-0.295583\pi\)
0.598954 + 0.800783i \(0.295583\pi\)
\(62\) 2.66563e40 1.25022
\(63\) −1.29687e40 −0.424359
\(64\) 5.44452e39 0.125000
\(65\) −5.63802e40 −0.913221
\(66\) −2.21100e40 −0.254010
\(67\) −1.34297e41 −1.09997 −0.549987 0.835174i \(-0.685367\pi\)
−0.549987 + 0.835174i \(0.685367\pi\)
\(68\) 3.01232e40 0.176787
\(69\) −1.69269e41 −0.715277
\(70\) −1.09628e41 −0.335131
\(71\) 4.73439e41 1.05184 0.525922 0.850533i \(-0.323720\pi\)
0.525922 + 0.850533i \(0.323720\pi\)
\(72\) −1.74066e41 −0.282315
\(73\) 1.25402e42 1.49122 0.745609 0.666384i \(-0.232159\pi\)
0.745609 + 0.666384i \(0.232159\pi\)
\(74\) −4.76108e41 −0.416861
\(75\) 1.41923e41 0.0918700
\(76\) −1.26835e40 −0.00609444
\(77\) 1.18777e42 0.425295
\(78\) −1.21351e42 −0.325024
\(79\) 2.71951e42 0.546866 0.273433 0.961891i \(-0.411841\pi\)
0.273433 + 0.961891i \(0.411841\pi\)
\(80\) −1.47143e42 −0.222954
\(81\) 3.80644e42 0.436120
\(82\) −9.91615e42 −0.862048
\(83\) −1.70726e43 −1.12991 −0.564955 0.825122i \(-0.691106\pi\)
−0.564955 + 0.825122i \(0.691106\pi\)
\(84\) −2.35960e42 −0.119277
\(85\) −8.14107e42 −0.315323
\(86\) 2.08230e43 0.619911
\(87\) −3.00736e43 −0.690243
\(88\) 1.59423e43 0.282938
\(89\) 5.81345e43 0.800127 0.400063 0.916487i \(-0.368988\pi\)
0.400063 + 0.916487i \(0.368988\pi\)
\(90\) 4.70430e43 0.503545
\(91\) 6.51909e43 0.544197
\(92\) 1.22051e44 0.796735
\(93\) −1.55059e44 −0.793655
\(94\) 5.64732e43 0.227230
\(95\) 3.42784e42 0.0108702
\(96\) −3.16707e43 −0.0793515
\(97\) −4.76787e44 −0.946150 −0.473075 0.881022i \(-0.656856\pi\)
−0.473075 + 0.881022i \(0.656856\pi\)
\(98\) −3.22060e44 −0.507399
\(99\) −5.09690e44 −0.639020
\(100\) −1.02332e44 −0.102332
\(101\) 1.26653e44 0.101248 0.0506238 0.998718i \(-0.483879\pi\)
0.0506238 + 0.998718i \(0.483879\pi\)
\(102\) −1.75227e44 −0.112227
\(103\) 2.48579e45 1.27828 0.639141 0.769089i \(-0.279290\pi\)
0.639141 + 0.769089i \(0.279290\pi\)
\(104\) 8.74996e44 0.362039
\(105\) 6.37705e44 0.212745
\(106\) −4.87195e45 −1.31317
\(107\) 1.38278e45 0.301729 0.150864 0.988554i \(-0.451794\pi\)
0.150864 + 0.988554i \(0.451794\pi\)
\(108\) 2.28059e45 0.403657
\(109\) 8.73749e45 1.25687 0.628435 0.777862i \(-0.283696\pi\)
0.628435 + 0.777862i \(0.283696\pi\)
\(110\) −4.30855e45 −0.504656
\(111\) 2.76952e45 0.264629
\(112\) 1.70138e45 0.132860
\(113\) −7.03342e45 −0.449678 −0.224839 0.974396i \(-0.572186\pi\)
−0.224839 + 0.974396i \(0.572186\pi\)
\(114\) 7.37799e43 0.00386883
\(115\) −3.29853e46 −1.42108
\(116\) 2.16843e46 0.768850
\(117\) −2.79744e46 −0.817672
\(118\) −2.81067e46 −0.678367
\(119\) 9.41331e45 0.187904
\(120\) 8.55930e45 0.141534
\(121\) −2.62093e46 −0.359570
\(122\) 7.43027e46 0.847049
\(123\) 5.76822e46 0.547238
\(124\) 1.11805e47 0.884039
\(125\) 1.62786e47 1.07434
\(126\) −5.43946e46 −0.300067
\(127\) −1.52817e47 −0.705649 −0.352824 0.935690i \(-0.614779\pi\)
−0.352824 + 0.935690i \(0.614779\pi\)
\(128\) 2.28360e46 0.0883883
\(129\) −1.21127e47 −0.393527
\(130\) −2.36476e47 −0.645745
\(131\) 1.24762e47 0.286734 0.143367 0.989670i \(-0.454207\pi\)
0.143367 + 0.989670i \(0.454207\pi\)
\(132\) −9.27362e46 −0.179612
\(133\) −3.96352e45 −0.00647767
\(134\) −5.63281e47 −0.777798
\(135\) −6.16350e47 −0.719975
\(136\) 1.26346e47 0.125007
\(137\) 1.15687e48 0.970670 0.485335 0.874328i \(-0.338698\pi\)
0.485335 + 0.874328i \(0.338698\pi\)
\(138\) −7.09968e47 −0.505777
\(139\) 1.44859e48 0.877226 0.438613 0.898676i \(-0.355470\pi\)
0.438613 + 0.898676i \(0.355470\pi\)
\(140\) −4.59813e47 −0.236974
\(141\) −3.28504e47 −0.144248
\(142\) 1.98575e48 0.743766
\(143\) 2.56211e48 0.819476
\(144\) −7.30087e47 −0.199627
\(145\) −5.86039e48 −1.37134
\(146\) 5.25975e48 1.05445
\(147\) 1.87342e48 0.322103
\(148\) −1.99694e48 −0.294765
\(149\) 1.96992e48 0.249895 0.124947 0.992163i \(-0.460124\pi\)
0.124947 + 0.992163i \(0.460124\pi\)
\(150\) 5.95267e47 0.0649619
\(151\) −1.22959e49 −1.15553 −0.577764 0.816204i \(-0.696075\pi\)
−0.577764 + 0.816204i \(0.696075\pi\)
\(152\) −5.31985e46 −0.00430942
\(153\) −4.03940e48 −0.282332
\(154\) 4.98186e48 0.300729
\(155\) −3.02162e49 −1.57680
\(156\) −5.08984e48 −0.229827
\(157\) 1.07839e49 0.421729 0.210865 0.977515i \(-0.432372\pi\)
0.210865 + 0.977515i \(0.432372\pi\)
\(158\) 1.14065e49 0.386693
\(159\) 2.83401e49 0.833616
\(160\) −6.17163e48 −0.157652
\(161\) 3.81400e49 0.846835
\(162\) 1.59654e49 0.308383
\(163\) 9.21779e49 1.55026 0.775132 0.631799i \(-0.217683\pi\)
0.775132 + 0.631799i \(0.217683\pi\)
\(164\) −4.15913e49 −0.609560
\(165\) 2.50628e49 0.320362
\(166\) −7.16078e49 −0.798966
\(167\) −1.28707e50 −1.25453 −0.627266 0.778805i \(-0.715826\pi\)
−0.627266 + 0.778805i \(0.715826\pi\)
\(168\) −9.89690e48 −0.0843413
\(169\) 6.51488e48 0.0485798
\(170\) −3.41461e49 −0.222967
\(171\) 1.70081e48 0.00973291
\(172\) 8.73382e49 0.438343
\(173\) −1.85770e50 −0.818350 −0.409175 0.912456i \(-0.634183\pi\)
−0.409175 + 0.912456i \(0.634183\pi\)
\(174\) −1.26138e50 −0.488075
\(175\) −3.19782e49 −0.108767
\(176\) 6.68668e49 0.200067
\(177\) 1.63497e50 0.430636
\(178\) 2.43834e50 0.565775
\(179\) 1.41352e50 0.289141 0.144571 0.989494i \(-0.453820\pi\)
0.144571 + 0.989494i \(0.453820\pi\)
\(180\) 1.97313e50 0.356060
\(181\) −2.62197e50 −0.417695 −0.208848 0.977948i \(-0.566971\pi\)
−0.208848 + 0.977948i \(0.566971\pi\)
\(182\) 2.73431e50 0.384805
\(183\) −4.32219e50 −0.537717
\(184\) 5.11917e50 0.563377
\(185\) 5.39691e50 0.525753
\(186\) −6.50366e50 −0.561199
\(187\) 3.69958e50 0.282954
\(188\) 2.36866e50 0.160676
\(189\) 7.12669e50 0.429040
\(190\) 1.43774e49 0.00768642
\(191\) −3.56974e51 −1.69584 −0.847922 0.530121i \(-0.822146\pi\)
−0.847922 + 0.530121i \(0.822146\pi\)
\(192\) −1.32837e50 −0.0561100
\(193\) −9.65667e50 −0.362901 −0.181450 0.983400i \(-0.558079\pi\)
−0.181450 + 0.983400i \(0.558079\pi\)
\(194\) −1.99979e51 −0.669029
\(195\) 1.37558e51 0.409927
\(196\) −1.35082e51 −0.358785
\(197\) 4.05849e51 0.961335 0.480668 0.876903i \(-0.340394\pi\)
0.480668 + 0.876903i \(0.340394\pi\)
\(198\) −2.13780e51 −0.451855
\(199\) 3.22222e50 0.0608081 0.0304040 0.999538i \(-0.490321\pi\)
0.0304040 + 0.999538i \(0.490321\pi\)
\(200\) −4.29214e50 −0.0723600
\(201\) 3.27660e51 0.493756
\(202\) 5.31221e50 0.0715928
\(203\) 6.77622e51 0.817197
\(204\) −7.34953e50 −0.0793562
\(205\) 1.12404e52 1.08723
\(206\) 1.04262e52 0.903882
\(207\) −1.63665e52 −1.27240
\(208\) 3.67000e51 0.256000
\(209\) −1.55773e50 −0.00975438
\(210\) 2.67473e51 0.150434
\(211\) −2.89921e52 −1.46529 −0.732647 0.680608i \(-0.761716\pi\)
−0.732647 + 0.680608i \(0.761716\pi\)
\(212\) −2.04344e52 −0.928551
\(213\) −1.15511e52 −0.472151
\(214\) 5.79978e51 0.213355
\(215\) −2.36039e52 −0.781842
\(216\) 9.56548e51 0.285428
\(217\) 3.49382e52 0.939629
\(218\) 3.66477e52 0.888741
\(219\) −3.05959e52 −0.669378
\(220\) −1.80714e52 −0.356846
\(221\) 2.03052e52 0.362061
\(222\) 1.16162e52 0.187121
\(223\) 5.82693e52 0.848360 0.424180 0.905578i \(-0.360562\pi\)
0.424180 + 0.905578i \(0.360562\pi\)
\(224\) 7.13609e51 0.0939464
\(225\) 1.37224e52 0.163426
\(226\) −2.95003e52 −0.317970
\(227\) −6.91309e52 −0.674668 −0.337334 0.941385i \(-0.609525\pi\)
−0.337334 + 0.941385i \(0.609525\pi\)
\(228\) 3.09456e50 0.00273567
\(229\) 2.12040e53 1.69871 0.849355 0.527822i \(-0.176991\pi\)
0.849355 + 0.527822i \(0.176991\pi\)
\(230\) −1.38350e53 −1.00486
\(231\) −2.89795e52 −0.190907
\(232\) 9.09507e52 0.543659
\(233\) 5.96910e52 0.323892 0.161946 0.986800i \(-0.448223\pi\)
0.161946 + 0.986800i \(0.448223\pi\)
\(234\) −1.17333e53 −0.578182
\(235\) −6.40152e52 −0.286587
\(236\) −1.17888e53 −0.479678
\(237\) −6.63513e52 −0.245477
\(238\) 3.94823e52 0.132868
\(239\) −5.65103e53 −1.73051 −0.865256 0.501330i \(-0.832844\pi\)
−0.865256 + 0.501330i \(0.832844\pi\)
\(240\) 3.59003e52 0.100080
\(241\) 5.60656e53 1.42335 0.711677 0.702507i \(-0.247936\pi\)
0.711677 + 0.702507i \(0.247936\pi\)
\(242\) −1.09930e53 −0.254255
\(243\) −4.75857e53 −1.00308
\(244\) 3.11648e53 0.598954
\(245\) 3.65070e53 0.639941
\(246\) 2.41937e53 0.386956
\(247\) −8.54961e51 −0.0124814
\(248\) 4.68942e53 0.625110
\(249\) 4.16542e53 0.507194
\(250\) 6.82773e53 0.759672
\(251\) −1.07353e54 −1.09183 −0.545915 0.837840i \(-0.683818\pi\)
−0.545915 + 0.837840i \(0.683818\pi\)
\(252\) −2.28148e53 −0.212180
\(253\) 1.49896e54 1.27520
\(254\) −6.40962e53 −0.498969
\(255\) 1.98628e53 0.141542
\(256\) 9.57810e52 0.0625000
\(257\) −2.16725e54 −1.29543 −0.647715 0.761883i \(-0.724275\pi\)
−0.647715 + 0.761883i \(0.724275\pi\)
\(258\) −5.08045e53 −0.278265
\(259\) −6.24031e53 −0.313301
\(260\) −9.91850e53 −0.456610
\(261\) −2.90778e54 −1.22786
\(262\) 5.23292e53 0.202752
\(263\) 8.56704e53 0.304668 0.152334 0.988329i \(-0.451321\pi\)
0.152334 + 0.988329i \(0.451321\pi\)
\(264\) −3.88964e53 −0.127005
\(265\) 5.52259e54 1.65619
\(266\) −1.66242e52 −0.00458041
\(267\) −1.41838e54 −0.359161
\(268\) −2.36257e54 −0.549987
\(269\) 5.69316e54 1.21879 0.609393 0.792868i \(-0.291413\pi\)
0.609393 + 0.792868i \(0.291413\pi\)
\(270\) −2.58516e54 −0.509099
\(271\) 5.25688e54 0.952618 0.476309 0.879278i \(-0.341974\pi\)
0.476309 + 0.879278i \(0.341974\pi\)
\(272\) 5.29933e53 0.0883936
\(273\) −1.59054e54 −0.244279
\(274\) 4.85226e54 0.686367
\(275\) −1.25680e54 −0.163787
\(276\) −2.97782e54 −0.357638
\(277\) 5.03893e54 0.557883 0.278942 0.960308i \(-0.410016\pi\)
0.278942 + 0.960308i \(0.410016\pi\)
\(278\) 6.07581e54 0.620292
\(279\) −1.49925e55 −1.41182
\(280\) −1.92859e54 −0.167566
\(281\) −1.97183e55 −1.58117 −0.790584 0.612354i \(-0.790223\pi\)
−0.790584 + 0.612354i \(0.790223\pi\)
\(282\) −1.37785e54 −0.101999
\(283\) 1.50647e55 1.02983 0.514916 0.857241i \(-0.327823\pi\)
0.514916 + 0.857241i \(0.327823\pi\)
\(284\) 8.32882e54 0.525922
\(285\) −8.36332e52 −0.00487943
\(286\) 1.07463e55 0.579457
\(287\) −1.29970e55 −0.647890
\(288\) −3.06221e54 −0.141157
\(289\) −2.05212e55 −0.874985
\(290\) −2.45803e55 −0.969687
\(291\) 1.16328e55 0.424708
\(292\) 2.20610e55 0.745609
\(293\) −2.45911e55 −0.769584 −0.384792 0.923003i \(-0.625727\pi\)
−0.384792 + 0.923003i \(0.625727\pi\)
\(294\) 7.85769e54 0.227761
\(295\) 3.18603e55 0.855569
\(296\) −8.37578e54 −0.208431
\(297\) 2.80090e55 0.646068
\(298\) 8.26243e54 0.176702
\(299\) 8.22709e55 1.63172
\(300\) 2.49673e54 0.0459350
\(301\) 2.72926e55 0.465907
\(302\) −5.15729e55 −0.817081
\(303\) −3.09011e54 −0.0454480
\(304\) −2.23131e53 −0.00304722
\(305\) −8.42258e55 −1.06831
\(306\) −1.69425e55 −0.199639
\(307\) −7.26437e54 −0.0795398 −0.0397699 0.999209i \(-0.512662\pi\)
−0.0397699 + 0.999209i \(0.512662\pi\)
\(308\) 2.08955e55 0.212648
\(309\) −6.06489e55 −0.573795
\(310\) −1.26736e56 −1.11497
\(311\) 5.05100e55 0.413303 0.206652 0.978415i \(-0.433743\pi\)
0.206652 + 0.978415i \(0.433743\pi\)
\(312\) −2.13484e55 −0.162512
\(313\) −1.13054e56 −0.800827 −0.400413 0.916335i \(-0.631133\pi\)
−0.400413 + 0.916335i \(0.631133\pi\)
\(314\) 4.52310e55 0.298208
\(315\) 6.16590e55 0.378450
\(316\) 4.78421e55 0.273433
\(317\) −1.02352e56 −0.544834 −0.272417 0.962179i \(-0.587823\pi\)
−0.272417 + 0.962179i \(0.587823\pi\)
\(318\) 1.18867e56 0.589455
\(319\) 2.66316e56 1.23057
\(320\) −2.58857e55 −0.111477
\(321\) −3.37373e55 −0.135440
\(322\) 1.59971e56 0.598803
\(323\) −1.23453e54 −0.00430968
\(324\) 6.69636e55 0.218060
\(325\) −6.89795e55 −0.209577
\(326\) 3.86622e56 1.09620
\(327\) −2.13179e56 −0.564184
\(328\) −1.74447e56 −0.431024
\(329\) 7.40191e55 0.170780
\(330\) 1.05121e56 0.226530
\(331\) −4.11537e56 −0.828474 −0.414237 0.910169i \(-0.635952\pi\)
−0.414237 + 0.910169i \(0.635952\pi\)
\(332\) −3.00345e56 −0.564955
\(333\) 2.67781e56 0.470744
\(334\) −5.39836e56 −0.887088
\(335\) 6.38506e56 0.980973
\(336\) −4.15106e55 −0.0596383
\(337\) −1.30888e57 −1.75884 −0.879422 0.476044i \(-0.842070\pi\)
−0.879422 + 0.476044i \(0.842070\pi\)
\(338\) 2.73254e55 0.0343511
\(339\) 1.71603e56 0.201851
\(340\) −1.43219e56 −0.157662
\(341\) 1.37313e57 1.41494
\(342\) 7.13370e54 0.00688220
\(343\) −1.01039e57 −0.912787
\(344\) 3.66323e56 0.309955
\(345\) 8.04783e56 0.637895
\(346\) −7.79177e56 −0.578661
\(347\) −2.43581e56 −0.169524 −0.0847620 0.996401i \(-0.527013\pi\)
−0.0847620 + 0.996401i \(0.527013\pi\)
\(348\) −5.29060e56 −0.345121
\(349\) 9.47821e56 0.579634 0.289817 0.957082i \(-0.406405\pi\)
0.289817 + 0.957082i \(0.406405\pi\)
\(350\) −1.34126e56 −0.0769101
\(351\) 1.53728e57 0.826691
\(352\) 2.80460e56 0.141469
\(353\) 2.19394e57 1.03823 0.519115 0.854704i \(-0.326262\pi\)
0.519115 + 0.854704i \(0.326262\pi\)
\(354\) 6.85754e56 0.304506
\(355\) −2.25094e57 −0.938050
\(356\) 1.02271e57 0.400063
\(357\) −2.29668e56 −0.0843463
\(358\) 5.92875e56 0.204454
\(359\) −1.03080e57 −0.333848 −0.166924 0.985970i \(-0.553384\pi\)
−0.166924 + 0.985970i \(0.553384\pi\)
\(360\) 8.27589e56 0.251773
\(361\) −3.49822e57 −0.999851
\(362\) −1.09973e57 −0.295355
\(363\) 6.39460e56 0.161404
\(364\) 1.14685e57 0.272098
\(365\) −5.96218e57 −1.32989
\(366\) −1.81286e57 −0.380223
\(367\) −6.34847e57 −1.25223 −0.626113 0.779733i \(-0.715355\pi\)
−0.626113 + 0.779733i \(0.715355\pi\)
\(368\) 2.14714e57 0.398367
\(369\) 5.57723e57 0.973475
\(370\) 2.26363e57 0.371763
\(371\) −6.38563e57 −0.986940
\(372\) −2.72783e57 −0.396828
\(373\) 1.18275e58 1.61974 0.809869 0.586610i \(-0.199538\pi\)
0.809869 + 0.586610i \(0.199538\pi\)
\(374\) 1.55172e57 0.200079
\(375\) −3.97169e57 −0.482249
\(376\) 9.93488e56 0.113615
\(377\) 1.46168e58 1.57461
\(378\) 2.98915e57 0.303377
\(379\) 1.27405e58 1.21844 0.609219 0.793002i \(-0.291483\pi\)
0.609219 + 0.793002i \(0.291483\pi\)
\(380\) 6.03031e55 0.00543512
\(381\) 3.72847e57 0.316752
\(382\) −1.49726e58 −1.19914
\(383\) −7.37288e57 −0.556756 −0.278378 0.960472i \(-0.589797\pi\)
−0.278378 + 0.960472i \(0.589797\pi\)
\(384\) −5.57157e56 −0.0396758
\(385\) −5.64719e57 −0.379285
\(386\) −4.05030e57 −0.256610
\(387\) −1.17117e58 −0.700040
\(388\) −8.38772e57 −0.473075
\(389\) 2.64413e57 0.140740 0.0703699 0.997521i \(-0.477582\pi\)
0.0703699 + 0.997521i \(0.477582\pi\)
\(390\) 5.76959e57 0.289862
\(391\) 1.18796e58 0.563410
\(392\) −5.66573e57 −0.253699
\(393\) −3.04398e57 −0.128709
\(394\) 1.70226e58 0.679767
\(395\) −1.29298e58 −0.487704
\(396\) −8.96656e57 −0.319510
\(397\) −3.15430e58 −1.06198 −0.530991 0.847378i \(-0.678180\pi\)
−0.530991 + 0.847378i \(0.678180\pi\)
\(398\) 1.35150e57 0.0429978
\(399\) 9.67029e55 0.00290770
\(400\) −1.80025e57 −0.0511662
\(401\) −4.27820e57 −0.114951 −0.0574755 0.998347i \(-0.518305\pi\)
−0.0574755 + 0.998347i \(0.518305\pi\)
\(402\) 1.37431e58 0.349138
\(403\) 7.53643e58 1.81052
\(404\) 2.22810e57 0.0506238
\(405\) −1.80975e58 −0.388939
\(406\) 2.84215e58 0.577845
\(407\) −2.45254e58 −0.471783
\(408\) −3.08262e57 −0.0561133
\(409\) 6.30167e57 0.108563 0.0542814 0.998526i \(-0.482713\pi\)
0.0542814 + 0.998526i \(0.482713\pi\)
\(410\) 4.71458e58 0.768788
\(411\) −2.82256e58 −0.435714
\(412\) 4.37305e58 0.639141
\(413\) −3.68393e58 −0.509841
\(414\) −6.86460e58 −0.899720
\(415\) 8.11710e58 1.00767
\(416\) 1.53931e58 0.181020
\(417\) −3.53429e58 −0.393769
\(418\) −6.53358e56 −0.00689739
\(419\) 5.08887e56 0.00509104 0.00254552 0.999997i \(-0.499190\pi\)
0.00254552 + 0.999997i \(0.499190\pi\)
\(420\) 1.12186e58 0.106373
\(421\) −2.18083e58 −0.196009 −0.0980044 0.995186i \(-0.531246\pi\)
−0.0980044 + 0.995186i \(0.531246\pi\)
\(422\) −1.21602e59 −1.03612
\(423\) −3.17627e58 −0.256602
\(424\) −8.57082e58 −0.656585
\(425\) −9.96036e57 −0.0723643
\(426\) −4.84487e58 −0.333861
\(427\) 9.73881e58 0.636618
\(428\) 2.43261e58 0.150864
\(429\) −6.25109e58 −0.367847
\(430\) −9.90021e58 −0.552846
\(431\) 9.32922e58 0.494432 0.247216 0.968960i \(-0.420484\pi\)
0.247216 + 0.968960i \(0.420484\pi\)
\(432\) 4.01205e58 0.201828
\(433\) −3.45886e58 −0.165179 −0.0825896 0.996584i \(-0.526319\pi\)
−0.0825896 + 0.996584i \(0.526319\pi\)
\(434\) 1.46541e59 0.664418
\(435\) 1.42983e59 0.615569
\(436\) 1.53711e59 0.628435
\(437\) −5.00196e57 −0.0194226
\(438\) −1.28329e59 −0.473321
\(439\) 5.43939e59 1.90589 0.952946 0.303139i \(-0.0980348\pi\)
0.952946 + 0.303139i \(0.0980348\pi\)
\(440\) −7.57968e58 −0.252328
\(441\) 1.81139e59 0.572985
\(442\) 8.51663e58 0.256016
\(443\) −4.79483e59 −1.36990 −0.684951 0.728590i \(-0.740176\pi\)
−0.684951 + 0.728590i \(0.740176\pi\)
\(444\) 4.87218e58 0.132314
\(445\) −2.76398e59 −0.713566
\(446\) 2.44399e59 0.599881
\(447\) −4.80625e58 −0.112173
\(448\) 2.99309e58 0.0664301
\(449\) −8.20387e59 −1.73171 −0.865857 0.500292i \(-0.833226\pi\)
−0.865857 + 0.500292i \(0.833226\pi\)
\(450\) 5.75557e58 0.115560
\(451\) −5.10804e59 −0.975623
\(452\) −1.23733e59 −0.224839
\(453\) 2.99999e59 0.518693
\(454\) −2.89956e59 −0.477062
\(455\) −3.09947e59 −0.485323
\(456\) 1.29795e57 0.00193441
\(457\) 6.40430e59 0.908567 0.454284 0.890857i \(-0.349895\pi\)
0.454284 + 0.890857i \(0.349895\pi\)
\(458\) 8.89360e59 1.20117
\(459\) 2.21977e59 0.285445
\(460\) −5.80283e59 −0.710541
\(461\) 5.31409e58 0.0619667 0.0309834 0.999520i \(-0.490136\pi\)
0.0309834 + 0.999520i \(0.490136\pi\)
\(462\) −1.21549e59 −0.134991
\(463\) −6.75536e59 −0.714622 −0.357311 0.933985i \(-0.616306\pi\)
−0.357311 + 0.933985i \(0.616306\pi\)
\(464\) 3.81475e59 0.384425
\(465\) 7.37222e59 0.707794
\(466\) 2.50362e59 0.229027
\(467\) 1.81391e60 1.58119 0.790597 0.612337i \(-0.209770\pi\)
0.790597 + 0.612337i \(0.209770\pi\)
\(468\) −4.92131e59 −0.408836
\(469\) −7.38288e59 −0.584571
\(470\) −2.68499e59 −0.202647
\(471\) −2.63108e59 −0.189306
\(472\) −4.94459e59 −0.339184
\(473\) 1.07264e60 0.701584
\(474\) −2.78297e59 −0.173579
\(475\) 4.19386e57 0.00249464
\(476\) 1.65601e59 0.0939519
\(477\) 2.74017e60 1.48291
\(478\) −2.37021e60 −1.22366
\(479\) −1.52375e60 −0.750522 −0.375261 0.926919i \(-0.622447\pi\)
−0.375261 + 0.926919i \(0.622447\pi\)
\(480\) 1.50577e59 0.0707669
\(481\) −1.34608e60 −0.603681
\(482\) 2.35156e60 1.00646
\(483\) −9.30550e59 −0.380127
\(484\) −4.61078e59 −0.179785
\(485\) 2.26686e60 0.843791
\(486\) −1.99589e60 −0.709284
\(487\) −4.56850e60 −1.55014 −0.775072 0.631873i \(-0.782287\pi\)
−0.775072 + 0.631873i \(0.782287\pi\)
\(488\) 1.30715e60 0.423525
\(489\) −2.24898e60 −0.695882
\(490\) 1.53122e60 0.452506
\(491\) −4.69070e60 −1.32405 −0.662024 0.749483i \(-0.730302\pi\)
−0.662024 + 0.749483i \(0.730302\pi\)
\(492\) 1.01476e60 0.273619
\(493\) 2.11061e60 0.543691
\(494\) −3.58597e58 −0.00882571
\(495\) 2.42330e60 0.569888
\(496\) 1.96689e60 0.442020
\(497\) 2.60270e60 0.558993
\(498\) 1.74710e60 0.358640
\(499\) 2.78439e60 0.546347 0.273173 0.961965i \(-0.411927\pi\)
0.273173 + 0.961965i \(0.411927\pi\)
\(500\) 2.86376e60 0.537169
\(501\) 3.14022e60 0.563134
\(502\) −4.50271e60 −0.772041
\(503\) −3.32687e59 −0.0545451 −0.0272726 0.999628i \(-0.508682\pi\)
−0.0272726 + 0.999628i \(0.508682\pi\)
\(504\) −9.56920e59 −0.150034
\(505\) −6.02165e59 −0.0902941
\(506\) 6.28711e60 0.901705
\(507\) −1.58952e59 −0.0218065
\(508\) −2.68839e60 −0.352824
\(509\) −8.27552e60 −1.03907 −0.519535 0.854449i \(-0.673895\pi\)
−0.519535 + 0.854449i \(0.673895\pi\)
\(510\) 8.33105e59 0.100085
\(511\) 6.89392e60 0.792494
\(512\) 4.01735e59 0.0441942
\(513\) −9.34646e58 −0.00984025
\(514\) −9.09011e60 −0.916007
\(515\) −1.18186e61 −1.13999
\(516\) −2.13090e60 −0.196763
\(517\) 2.90907e60 0.257168
\(518\) −2.61738e60 −0.221537
\(519\) 4.53247e60 0.367341
\(520\) −4.16012e60 −0.322872
\(521\) −1.78155e61 −1.32419 −0.662093 0.749422i \(-0.730332\pi\)
−0.662093 + 0.749422i \(0.730332\pi\)
\(522\) −1.21961e61 −0.868231
\(523\) 1.77356e61 1.20937 0.604683 0.796466i \(-0.293300\pi\)
0.604683 + 0.796466i \(0.293300\pi\)
\(524\) 2.19484e60 0.143367
\(525\) 7.80213e59 0.0488235
\(526\) 3.59328e60 0.215433
\(527\) 1.08823e61 0.625147
\(528\) −1.63143e60 −0.0898061
\(529\) 2.91765e61 1.53915
\(530\) 2.31634e61 1.17111
\(531\) 1.58083e61 0.766053
\(532\) −6.97270e58 −0.00323884
\(533\) −2.80356e61 −1.24838
\(534\) −5.94912e60 −0.253965
\(535\) −6.57434e60 −0.269087
\(536\) −9.90934e60 −0.388899
\(537\) −3.44875e60 −0.129790
\(538\) 2.38789e61 0.861812
\(539\) −1.65900e61 −0.574249
\(540\) −1.08429e61 −0.359987
\(541\) 5.18437e60 0.165104 0.0825520 0.996587i \(-0.473693\pi\)
0.0825520 + 0.996587i \(0.473693\pi\)
\(542\) 2.20489e61 0.673603
\(543\) 6.39714e60 0.187495
\(544\) 2.22270e60 0.0625037
\(545\) −4.15419e61 −1.12090
\(546\) −6.67122e60 −0.172731
\(547\) −4.57431e60 −0.113661 −0.0568304 0.998384i \(-0.518099\pi\)
−0.0568304 + 0.998384i \(0.518099\pi\)
\(548\) 2.03519e61 0.485335
\(549\) −4.17907e61 −0.956538
\(550\) −5.27138e60 −0.115815
\(551\) −8.88682e59 −0.0187428
\(552\) −1.24899e61 −0.252889
\(553\) 1.49504e61 0.290627
\(554\) 2.11348e61 0.394483
\(555\) −1.31675e61 −0.236000
\(556\) 2.54838e61 0.438613
\(557\) −1.47434e61 −0.243700 −0.121850 0.992549i \(-0.538883\pi\)
−0.121850 + 0.992549i \(0.538883\pi\)
\(558\) −6.28832e61 −0.998309
\(559\) 5.88722e61 0.897728
\(560\) −8.08911e60 −0.118487
\(561\) −9.02633e60 −0.127013
\(562\) −8.27047e61 −1.11805
\(563\) 6.02307e61 0.782310 0.391155 0.920325i \(-0.372076\pi\)
0.391155 + 0.920325i \(0.372076\pi\)
\(564\) −5.77911e60 −0.0721242
\(565\) 3.34400e61 0.401030
\(566\) 6.31860e61 0.728201
\(567\) 2.09257e61 0.231772
\(568\) 3.49336e61 0.371883
\(569\) −1.83240e61 −0.187497 −0.0937487 0.995596i \(-0.529885\pi\)
−0.0937487 + 0.995596i \(0.529885\pi\)
\(570\) −3.50783e59 −0.00345028
\(571\) 1.05375e62 0.996379 0.498189 0.867068i \(-0.333998\pi\)
0.498189 + 0.867068i \(0.333998\pi\)
\(572\) 4.50731e61 0.409738
\(573\) 8.70954e61 0.761230
\(574\) −5.45135e61 −0.458128
\(575\) −4.03565e61 −0.326127
\(576\) −1.28438e61 −0.0998133
\(577\) 3.46971e60 0.0259321 0.0129660 0.999916i \(-0.495873\pi\)
0.0129660 + 0.999916i \(0.495873\pi\)
\(578\) −8.60720e61 −0.618708
\(579\) 2.35606e61 0.162899
\(580\) −1.03097e62 −0.685672
\(581\) −9.38559e61 −0.600480
\(582\) 4.87913e61 0.300314
\(583\) −2.50965e62 −1.48618
\(584\) 9.25305e61 0.527225
\(585\) 1.33003e62 0.729213
\(586\) −1.03142e62 −0.544178
\(587\) −3.47497e62 −1.76439 −0.882194 0.470887i \(-0.843934\pi\)
−0.882194 + 0.470887i \(0.843934\pi\)
\(588\) 3.29575e61 0.161052
\(589\) −4.58205e60 −0.0215509
\(590\) 1.33632e62 0.604979
\(591\) −9.90201e61 −0.431524
\(592\) −3.51305e61 −0.147383
\(593\) 2.38315e62 0.962546 0.481273 0.876571i \(-0.340175\pi\)
0.481273 + 0.876571i \(0.340175\pi\)
\(594\) 1.17478e62 0.456839
\(595\) −4.47551e61 −0.167576
\(596\) 3.46552e61 0.124947
\(597\) −7.86166e60 −0.0272955
\(598\) 3.45069e62 1.15380
\(599\) −1.44253e61 −0.0464539 −0.0232270 0.999730i \(-0.507394\pi\)
−0.0232270 + 0.999730i \(0.507394\pi\)
\(600\) 1.04721e61 0.0324809
\(601\) 1.16486e62 0.348016 0.174008 0.984744i \(-0.444328\pi\)
0.174008 + 0.984744i \(0.444328\pi\)
\(602\) 1.14474e62 0.329446
\(603\) 3.16811e62 0.878336
\(604\) −2.16312e62 −0.577764
\(605\) 1.24611e62 0.320670
\(606\) −1.29609e61 −0.0321366
\(607\) −9.22722e61 −0.220458 −0.110229 0.993906i \(-0.535158\pi\)
−0.110229 + 0.993906i \(0.535158\pi\)
\(608\) −9.35879e59 −0.00215471
\(609\) −1.65328e62 −0.366823
\(610\) −3.53269e62 −0.755412
\(611\) 1.59665e62 0.329065
\(612\) −7.10618e61 −0.141166
\(613\) 6.43322e61 0.123188 0.0615939 0.998101i \(-0.480382\pi\)
0.0615939 + 0.998101i \(0.480382\pi\)
\(614\) −3.04690e61 −0.0562431
\(615\) −2.74247e62 −0.488036
\(616\) 8.76419e61 0.150365
\(617\) 3.21910e62 0.532498 0.266249 0.963904i \(-0.414216\pi\)
0.266249 + 0.963904i \(0.414216\pi\)
\(618\) −2.54380e62 −0.405735
\(619\) −1.15460e63 −1.77579 −0.887895 0.460046i \(-0.847833\pi\)
−0.887895 + 0.460046i \(0.847833\pi\)
\(620\) −5.31569e62 −0.788400
\(621\) 8.99388e62 1.28643
\(622\) 2.11854e62 0.292249
\(623\) 3.19591e62 0.425220
\(624\) −8.95415e61 −0.114913
\(625\) −6.08630e62 −0.753447
\(626\) −4.74183e62 −0.566270
\(627\) 3.80058e60 0.00437855
\(628\) 1.89712e62 0.210865
\(629\) −1.94369e62 −0.208443
\(630\) 2.58616e62 0.267605
\(631\) 1.06339e63 1.06178 0.530889 0.847441i \(-0.321858\pi\)
0.530889 + 0.847441i \(0.321858\pi\)
\(632\) 2.00664e62 0.193346
\(633\) 7.07357e62 0.657741
\(634\) −4.29296e62 −0.385256
\(635\) 7.26562e62 0.629309
\(636\) 4.98564e62 0.416808
\(637\) −9.10547e62 −0.734794
\(638\) 1.11701e63 0.870146
\(639\) −1.11686e63 −0.839904
\(640\) −1.08572e62 −0.0788261
\(641\) −1.23129e63 −0.863085 −0.431543 0.902093i \(-0.642031\pi\)
−0.431543 + 0.902093i \(0.642031\pi\)
\(642\) −1.41505e62 −0.0957706
\(643\) 6.42176e62 0.419669 0.209834 0.977737i \(-0.432707\pi\)
0.209834 + 0.977737i \(0.432707\pi\)
\(644\) 6.70967e62 0.423418
\(645\) 5.75894e62 0.350953
\(646\) −5.17799e60 −0.00304740
\(647\) 2.16731e63 1.23190 0.615950 0.787785i \(-0.288772\pi\)
0.615950 + 0.787785i \(0.288772\pi\)
\(648\) 2.80866e62 0.154192
\(649\) −1.44784e63 −0.767743
\(650\) −2.89321e62 −0.148194
\(651\) −8.52431e62 −0.421781
\(652\) 1.62161e63 0.775132
\(653\) 5.79373e61 0.0267554 0.0133777 0.999911i \(-0.495742\pi\)
0.0133777 + 0.999911i \(0.495742\pi\)
\(654\) −8.94139e62 −0.398938
\(655\) −5.93177e62 −0.255714
\(656\) −7.31683e62 −0.304780
\(657\) −2.95829e63 −1.19075
\(658\) 3.10459e62 0.120759
\(659\) −2.42846e63 −0.912870 −0.456435 0.889757i \(-0.650874\pi\)
−0.456435 + 0.889757i \(0.650874\pi\)
\(660\) 4.40910e62 0.160181
\(661\) −1.52653e63 −0.536008 −0.268004 0.963418i \(-0.586364\pi\)
−0.268004 + 0.963418i \(0.586364\pi\)
\(662\) −1.72611e63 −0.585820
\(663\) −4.95411e62 −0.162522
\(664\) −1.25974e63 −0.399483
\(665\) 1.88443e61 0.00577689
\(666\) 1.12316e63 0.332867
\(667\) 8.55159e63 2.45028
\(668\) −2.26424e63 −0.627266
\(669\) −1.42167e63 −0.380812
\(670\) 2.67809e63 0.693653
\(671\) 3.82751e63 0.958648
\(672\) −1.74108e62 −0.0421706
\(673\) 6.53256e63 1.53019 0.765094 0.643919i \(-0.222693\pi\)
0.765094 + 0.643919i \(0.222693\pi\)
\(674\) −5.48984e63 −1.24369
\(675\) −7.54086e62 −0.165229
\(676\) 1.14611e62 0.0242899
\(677\) 4.70923e63 0.965394 0.482697 0.875787i \(-0.339657\pi\)
0.482697 + 0.875787i \(0.339657\pi\)
\(678\) 7.19755e62 0.142730
\(679\) −2.62111e63 −0.502823
\(680\) −6.00705e62 −0.111484
\(681\) 1.68667e63 0.302845
\(682\) 5.75931e63 1.00051
\(683\) 4.97563e62 0.0836339 0.0418169 0.999125i \(-0.486685\pi\)
0.0418169 + 0.999125i \(0.486685\pi\)
\(684\) 2.99209e61 0.00486645
\(685\) −5.50027e63 −0.865658
\(686\) −4.23787e63 −0.645438
\(687\) −5.17340e63 −0.762517
\(688\) 1.53647e63 0.219171
\(689\) −1.37743e64 −1.90168
\(690\) 3.37550e63 0.451060
\(691\) 9.66259e63 1.24979 0.624896 0.780708i \(-0.285142\pi\)
0.624896 + 0.780708i \(0.285142\pi\)
\(692\) −3.26811e63 −0.409175
\(693\) −2.80199e63 −0.339601
\(694\) −1.02165e63 −0.119872
\(695\) −6.88723e63 −0.782324
\(696\) −2.21904e63 −0.244038
\(697\) −4.04822e63 −0.431049
\(698\) 3.97545e63 0.409863
\(699\) −1.45636e63 −0.145389
\(700\) −5.62567e62 −0.0543837
\(701\) 1.15174e64 1.07819 0.539097 0.842243i \(-0.318765\pi\)
0.539097 + 0.842243i \(0.318765\pi\)
\(702\) 6.44782e63 0.584559
\(703\) 8.18399e61 0.00718572
\(704\) 1.17633e63 0.100034
\(705\) 1.56186e63 0.128643
\(706\) 9.20204e63 0.734139
\(707\) 6.96268e62 0.0538071
\(708\) 2.87626e63 0.215318
\(709\) −2.22559e64 −1.61401 −0.807003 0.590548i \(-0.798912\pi\)
−0.807003 + 0.590548i \(0.798912\pi\)
\(710\) −9.44113e63 −0.663302
\(711\) −6.41543e63 −0.436677
\(712\) 4.28957e63 0.282888
\(713\) 4.40920e64 2.81738
\(714\) −9.63298e62 −0.0596418
\(715\) −1.21814e64 −0.730822
\(716\) 2.48670e63 0.144571
\(717\) 1.37875e64 0.776793
\(718\) −4.32349e63 −0.236066
\(719\) 4.32783e62 0.0229018 0.0114509 0.999934i \(-0.496355\pi\)
0.0114509 + 0.999934i \(0.496355\pi\)
\(720\) 3.47116e63 0.178030
\(721\) 1.36655e64 0.679332
\(722\) −1.46726e64 −0.707002
\(723\) −1.36790e64 −0.638915
\(724\) −4.61261e63 −0.208848
\(725\) −7.17002e63 −0.314713
\(726\) 2.68209e63 0.114130
\(727\) −7.31606e63 −0.301824 −0.150912 0.988547i \(-0.548221\pi\)
−0.150912 + 0.988547i \(0.548221\pi\)
\(728\) 4.81024e63 0.192403
\(729\) 3.64659e62 0.0141422
\(730\) −2.50072e64 −0.940375
\(731\) 8.50092e63 0.309974
\(732\) −7.60367e63 −0.268859
\(733\) 4.17807e64 1.43264 0.716318 0.697774i \(-0.245826\pi\)
0.716318 + 0.697774i \(0.245826\pi\)
\(734\) −2.66274e64 −0.885457
\(735\) −8.90707e63 −0.287256
\(736\) 9.00575e63 0.281688
\(737\) −2.90159e64 −0.880274
\(738\) 2.33926e64 0.688351
\(739\) −6.90768e64 −1.97166 −0.985831 0.167739i \(-0.946353\pi\)
−0.985831 + 0.167739i \(0.946353\pi\)
\(740\) 9.49435e63 0.262876
\(741\) 2.08595e62 0.00560267
\(742\) −2.67833e64 −0.697872
\(743\) 5.54554e64 1.40183 0.700916 0.713244i \(-0.252775\pi\)
0.700916 + 0.713244i \(0.252775\pi\)
\(744\) −1.14414e64 −0.280599
\(745\) −9.36587e63 −0.222860
\(746\) 4.96082e64 1.14533
\(747\) 4.02750e64 0.902240
\(748\) 6.50837e63 0.141477
\(749\) 7.60174e63 0.160351
\(750\) −1.66585e64 −0.341002
\(751\) −5.42200e64 −1.07711 −0.538556 0.842589i \(-0.681030\pi\)
−0.538556 + 0.842589i \(0.681030\pi\)
\(752\) 4.16699e63 0.0803380
\(753\) 2.61922e64 0.490101
\(754\) 6.13074e64 1.11342
\(755\) 5.84604e64 1.03052
\(756\) 1.25374e64 0.214520
\(757\) −4.84159e64 −0.804138 −0.402069 0.915609i \(-0.631709\pi\)
−0.402069 + 0.915609i \(0.631709\pi\)
\(758\) 5.34374e64 0.861566
\(759\) −3.65721e64 −0.572413
\(760\) 2.52930e62 0.00384321
\(761\) −6.15512e63 −0.0907991 −0.0453995 0.998969i \(-0.514456\pi\)
−0.0453995 + 0.998969i \(0.514456\pi\)
\(762\) 1.56383e64 0.223977
\(763\) 4.80339e64 0.667952
\(764\) −6.27995e64 −0.847922
\(765\) 1.92051e64 0.251788
\(766\) −3.09241e64 −0.393686
\(767\) −7.94651e64 −0.982383
\(768\) −2.33689e63 −0.0280550
\(769\) −8.09959e64 −0.944322 −0.472161 0.881512i \(-0.656526\pi\)
−0.472161 + 0.881512i \(0.656526\pi\)
\(770\) −2.36860e64 −0.268195
\(771\) 5.28771e64 0.581493
\(772\) −1.69882e64 −0.181450
\(773\) −1.09275e65 −1.13366 −0.566829 0.823835i \(-0.691830\pi\)
−0.566829 + 0.823835i \(0.691830\pi\)
\(774\) −4.91224e64 −0.495003
\(775\) −3.69686e64 −0.361864
\(776\) −3.51806e64 −0.334515
\(777\) 1.52253e64 0.140634
\(778\) 1.10903e64 0.0995181
\(779\) 1.70452e63 0.0148597
\(780\) 2.41994e64 0.204963
\(781\) 1.02290e65 0.841757
\(782\) 4.98266e64 0.398391
\(783\) 1.59791e65 1.24141
\(784\) −2.37638e64 −0.179393
\(785\) −5.12715e64 −0.376105
\(786\) −1.27674e64 −0.0910112
\(787\) 2.41478e65 1.67281 0.836403 0.548115i \(-0.184654\pi\)
0.836403 + 0.548115i \(0.184654\pi\)
\(788\) 7.13978e64 0.480668
\(789\) −2.09021e64 −0.136759
\(790\) −5.42314e64 −0.344859
\(791\) −3.86658e64 −0.238977
\(792\) −3.76085e64 −0.225928
\(793\) 2.10073e65 1.22666
\(794\) −1.32301e65 −0.750934
\(795\) −1.34742e65 −0.743432
\(796\) 5.66860e63 0.0304040
\(797\) −2.14766e64 −0.111984 −0.0559918 0.998431i \(-0.517832\pi\)
−0.0559918 + 0.998431i \(0.517832\pi\)
\(798\) 4.05601e62 0.00205605
\(799\) 2.30550e64 0.113622
\(800\) −7.55080e63 −0.0361800
\(801\) −1.37142e65 −0.638907
\(802\) −1.79441e64 −0.0812826
\(803\) 2.70942e65 1.19337
\(804\) 5.76426e64 0.246878
\(805\) −1.81335e65 −0.755221
\(806\) 3.16101e65 1.28023
\(807\) −1.38903e65 −0.547089
\(808\) 9.34534e63 0.0357964
\(809\) −4.54729e65 −1.69399 −0.846996 0.531600i \(-0.821591\pi\)
−0.846996 + 0.531600i \(0.821591\pi\)
\(810\) −7.59065e64 −0.275021
\(811\) 2.51365e65 0.885801 0.442901 0.896571i \(-0.353949\pi\)
0.442901 + 0.896571i \(0.353949\pi\)
\(812\) 1.19209e65 0.408598
\(813\) −1.28259e65 −0.427611
\(814\) −1.02867e65 −0.333601
\(815\) −4.38255e65 −1.38255
\(816\) −1.29294e64 −0.0396781
\(817\) −3.57935e63 −0.0106858
\(818\) 2.64311e64 0.0767656
\(819\) −1.53788e65 −0.434545
\(820\) 1.97744e65 0.543615
\(821\) 3.27797e65 0.876765 0.438383 0.898788i \(-0.355551\pi\)
0.438383 + 0.898788i \(0.355551\pi\)
\(822\) −1.18387e65 −0.308096
\(823\) −8.90049e63 −0.0225381 −0.0112691 0.999937i \(-0.503587\pi\)
−0.0112691 + 0.999937i \(0.503587\pi\)
\(824\) 1.83419e65 0.451941
\(825\) 3.06636e64 0.0735207
\(826\) −1.54515e65 −0.360512
\(827\) 1.63577e65 0.371405 0.185702 0.982606i \(-0.440544\pi\)
0.185702 + 0.982606i \(0.440544\pi\)
\(828\) −2.87922e65 −0.636198
\(829\) 1.84764e65 0.397320 0.198660 0.980068i \(-0.436341\pi\)
0.198660 + 0.980068i \(0.436341\pi\)
\(830\) 3.40456e65 0.712531
\(831\) −1.22941e65 −0.250423
\(832\) 6.45633e64 0.128000
\(833\) −1.31479e65 −0.253715
\(834\) −1.48239e65 −0.278437
\(835\) 6.11931e65 1.11881
\(836\) −2.74038e63 −0.00487719
\(837\) 8.23885e65 1.42739
\(838\) 2.13443e63 0.00359991
\(839\) −7.98160e65 −1.31053 −0.655263 0.755401i \(-0.727442\pi\)
−0.655263 + 0.755401i \(0.727442\pi\)
\(840\) 4.70543e64 0.0752169
\(841\) 8.76779e65 1.36452
\(842\) −9.14708e64 −0.138599
\(843\) 4.81092e65 0.709754
\(844\) −5.10035e65 −0.732647
\(845\) −3.09747e64 −0.0433242
\(846\) −1.33223e65 −0.181445
\(847\) −1.44084e65 −0.191090
\(848\) −3.59486e65 −0.464276
\(849\) −3.67553e65 −0.462271
\(850\) −4.17768e64 −0.0511693
\(851\) −7.87527e65 −0.939400
\(852\) −2.03209e65 −0.236076
\(853\) −1.07480e66 −1.21612 −0.608058 0.793892i \(-0.708051\pi\)
−0.608058 + 0.793892i \(0.708051\pi\)
\(854\) 4.08475e65 0.450157
\(855\) −8.08640e63 −0.00867996
\(856\) 1.02031e65 0.106677
\(857\) 3.80613e65 0.387628 0.193814 0.981038i \(-0.437914\pi\)
0.193814 + 0.981038i \(0.437914\pi\)
\(858\) −2.62190e65 −0.260107
\(859\) −5.67906e65 −0.548821 −0.274410 0.961613i \(-0.588483\pi\)
−0.274410 + 0.961613i \(0.588483\pi\)
\(860\) −4.15245e65 −0.390921
\(861\) 3.17105e65 0.290825
\(862\) 3.91296e65 0.349616
\(863\) −6.96770e65 −0.606522 −0.303261 0.952908i \(-0.598075\pi\)
−0.303261 + 0.952908i \(0.598075\pi\)
\(864\) 1.68278e65 0.142714
\(865\) 8.83236e65 0.729817
\(866\) −1.45075e65 −0.116799
\(867\) 5.00680e65 0.392763
\(868\) 6.14639e65 0.469815
\(869\) 5.87573e65 0.437640
\(870\) 5.99715e65 0.435273
\(871\) −1.59254e66 −1.12637
\(872\) 6.44713e65 0.444371
\(873\) 1.12476e66 0.755507
\(874\) −2.09797e64 −0.0137339
\(875\) 8.94906e65 0.570948
\(876\) −5.38249e65 −0.334689
\(877\) −3.00025e65 −0.181830 −0.0909151 0.995859i \(-0.528979\pi\)
−0.0909151 + 0.995859i \(0.528979\pi\)
\(878\) 2.28144e66 1.34767
\(879\) 5.99979e65 0.345451
\(880\) −3.17915e65 −0.178423
\(881\) 7.45578e65 0.407882 0.203941 0.978983i \(-0.434625\pi\)
0.203941 + 0.978983i \(0.434625\pi\)
\(882\) 7.59751e65 0.405162
\(883\) −6.77149e65 −0.352021 −0.176010 0.984388i \(-0.556319\pi\)
−0.176010 + 0.984388i \(0.556319\pi\)
\(884\) 3.57213e65 0.181030
\(885\) −7.77336e65 −0.384048
\(886\) −2.01110e66 −0.968666
\(887\) −3.77816e66 −1.77419 −0.887093 0.461592i \(-0.847279\pi\)
−0.887093 + 0.461592i \(0.847279\pi\)
\(888\) 2.04354e65 0.0935603
\(889\) −8.40104e65 −0.375011
\(890\) −1.15930e66 −0.504567
\(891\) 8.22413e65 0.349013
\(892\) 1.02508e66 0.424180
\(893\) −9.70740e63 −0.00391692
\(894\) −2.01589e65 −0.0793181
\(895\) −6.72052e65 −0.257861
\(896\) 1.25539e65 0.0469732
\(897\) −2.00727e66 −0.732445
\(898\) −3.44095e66 −1.22451
\(899\) 7.83369e66 2.71877
\(900\) 2.41406e65 0.0817132
\(901\) −1.98895e66 −0.656624
\(902\) −2.14247e66 −0.689870
\(903\) −6.65892e65 −0.209136
\(904\) −5.18975e65 −0.158985
\(905\) 1.24660e66 0.372507
\(906\) 1.25829e66 0.366771
\(907\) 5.56448e66 1.58220 0.791098 0.611690i \(-0.209510\pi\)
0.791098 + 0.611690i \(0.209510\pi\)
\(908\) −1.21616e66 −0.337334
\(909\) −2.98779e65 −0.0808468
\(910\) −1.30001e66 −0.343175
\(911\) −2.38244e66 −0.613562 −0.306781 0.951780i \(-0.599252\pi\)
−0.306781 + 0.951780i \(0.599252\pi\)
\(912\) 5.44400e63 0.00136784
\(913\) −3.68869e66 −0.904231
\(914\) 2.68616e66 0.642454
\(915\) 2.05496e66 0.479544
\(916\) 3.73025e66 0.849355
\(917\) 6.85875e65 0.152382
\(918\) 9.31040e65 0.201840
\(919\) −3.52107e65 −0.0744862 −0.0372431 0.999306i \(-0.511858\pi\)
−0.0372431 + 0.999306i \(0.511858\pi\)
\(920\) −2.43389e66 −0.502428
\(921\) 1.77238e65 0.0357038
\(922\) 2.22889e65 0.0438171
\(923\) 5.61423e66 1.07709
\(924\) −5.09812e65 −0.0954533
\(925\) 6.60297e65 0.120656
\(926\) −2.83340e66 −0.505314
\(927\) −5.86407e66 −1.02072
\(928\) 1.60002e66 0.271829
\(929\) 5.51562e66 0.914619 0.457310 0.889308i \(-0.348813\pi\)
0.457310 + 0.889308i \(0.348813\pi\)
\(930\) 3.09213e66 0.500486
\(931\) 5.53600e64 0.00874638
\(932\) 1.05010e66 0.161946
\(933\) −1.23236e66 −0.185524
\(934\) 7.60808e66 1.11807
\(935\) −1.75895e66 −0.252343
\(936\) −2.06415e66 −0.289091
\(937\) −1.16406e67 −1.59160 −0.795798 0.605562i \(-0.792948\pi\)
−0.795798 + 0.605562i \(0.792948\pi\)
\(938\) −3.09661e66 −0.413354
\(939\) 2.75832e66 0.359475
\(940\) −1.12617e66 −0.143293
\(941\) −1.01573e67 −1.26186 −0.630931 0.775839i \(-0.717327\pi\)
−0.630931 + 0.775839i \(0.717327\pi\)
\(942\) −1.10356e66 −0.133859
\(943\) −1.64022e67 −1.94263
\(944\) −2.07391e66 −0.239839
\(945\) −3.38835e66 −0.382624
\(946\) 4.49899e66 0.496095
\(947\) 6.79284e66 0.731437 0.365718 0.930726i \(-0.380823\pi\)
0.365718 + 0.930726i \(0.380823\pi\)
\(948\) −1.16726e66 −0.122739
\(949\) 1.48707e67 1.52701
\(950\) 1.75903e64 0.00176398
\(951\) 2.49721e66 0.244565
\(952\) 6.94580e65 0.0664341
\(953\) −1.02194e67 −0.954626 −0.477313 0.878733i \(-0.658389\pi\)
−0.477313 + 0.878733i \(0.658389\pi\)
\(954\) 1.14931e67 1.04857
\(955\) 1.69721e67 1.51238
\(956\) −9.94140e66 −0.865256
\(957\) −6.49765e66 −0.552379
\(958\) −6.39105e66 −0.530699
\(959\) 6.35982e66 0.515854
\(960\) 6.31565e65 0.0500398
\(961\) 2.74701e67 2.12610
\(962\) −5.64588e66 −0.426867
\(963\) −3.26202e66 −0.240933
\(964\) 9.86316e66 0.711677
\(965\) 4.59122e66 0.323641
\(966\) −3.90301e66 −0.268791
\(967\) −2.40504e66 −0.161818 −0.0809090 0.996721i \(-0.525782\pi\)
−0.0809090 + 0.996721i \(0.525782\pi\)
\(968\) −1.93390e66 −0.127127
\(969\) 3.01203e64 0.00193453
\(970\) 9.50789e66 0.596651
\(971\) −3.51609e66 −0.215589 −0.107795 0.994173i \(-0.534379\pi\)
−0.107795 + 0.994173i \(0.534379\pi\)
\(972\) −8.37136e66 −0.501540
\(973\) 7.96352e66 0.466194
\(974\) −1.91617e67 −1.09612
\(975\) 1.68298e66 0.0940750
\(976\) 5.48257e66 0.299477
\(977\) −2.59994e66 −0.138783 −0.0693914 0.997590i \(-0.522106\pi\)
−0.0693914 + 0.997590i \(0.522106\pi\)
\(978\) −9.43290e66 −0.492063
\(979\) 1.25604e67 0.640316
\(980\) 6.42239e66 0.319970
\(981\) −2.06121e67 −1.00362
\(982\) −1.96742e67 −0.936243
\(983\) −3.00612e67 −1.39815 −0.699073 0.715050i \(-0.746404\pi\)
−0.699073 + 0.715050i \(0.746404\pi\)
\(984\) 4.25619e66 0.193478
\(985\) −1.92959e67 −0.857334
\(986\) 8.85254e66 0.384448
\(987\) −1.80593e66 −0.0766595
\(988\) −1.50406e65 −0.00624072
\(989\) 3.44433e67 1.39697
\(990\) 1.01640e67 0.402972
\(991\) −4.87826e67 −1.89064 −0.945318 0.326149i \(-0.894249\pi\)
−0.945318 + 0.326149i \(0.894249\pi\)
\(992\) 8.24972e66 0.312555
\(993\) 1.00408e67 0.371885
\(994\) 1.09165e67 0.395267
\(995\) −1.53199e66 −0.0542296
\(996\) 7.32789e66 0.253597
\(997\) 3.67161e67 1.24227 0.621134 0.783704i \(-0.286672\pi\)
0.621134 + 0.783704i \(0.286672\pi\)
\(998\) 1.16786e67 0.386325
\(999\) −1.47154e67 −0.475936
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 2.46.a.b.1.1 2
3.2 odd 2 18.46.a.b.1.2 2
4.3 odd 2 16.46.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.46.a.b.1.1 2 1.1 even 1 trivial
16.46.a.a.1.2 2 4.3 odd 2
18.46.a.b.1.2 2 3.2 odd 2