Properties

Label 2.52.a.b.1.1
Level $2$
Weight $52$
Character 2.1
Self dual yes
Analytic conductor $32.946$
Analytic rank $1$
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] = [2,52,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, 52, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 52);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 52 \)
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: \(32.9462706828\)
Analytic rank: \(1\)
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: \( x^{2} - x - 313153613520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{5}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(559602.\) 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)
 
\(f(q)\) \(=\) \(q+3.35544e7 q^{2} +2.70751e11 q^{3} +1.12590e15 q^{4} +5.90004e17 q^{5} +9.08491e18 q^{6} -5.92711e21 q^{7} +3.77789e22 q^{8} -2.08039e24 q^{9} +O(q^{10})\) \(q+3.35544e7 q^{2} +2.70751e11 q^{3} +1.12590e15 q^{4} +5.90004e17 q^{5} +9.08491e18 q^{6} -5.92711e21 q^{7} +3.77789e22 q^{8} -2.08039e24 q^{9} +1.97972e25 q^{10} -2.67424e26 q^{11} +3.04839e26 q^{12} +1.41683e28 q^{13} -1.98881e29 q^{14} +1.59744e29 q^{15} +1.26765e30 q^{16} -1.60476e31 q^{17} -6.98062e31 q^{18} -4.18466e32 q^{19} +6.64285e32 q^{20} -1.60477e33 q^{21} -8.97327e33 q^{22} +6.82511e34 q^{23} +1.02287e34 q^{24} -9.59850e34 q^{25} +4.75411e35 q^{26} -1.14638e36 q^{27} -6.67333e36 q^{28} -6.03711e36 q^{29} +5.36013e36 q^{30} -7.93779e37 q^{31} +4.25353e37 q^{32} -7.24055e37 q^{33} -5.38469e38 q^{34} -3.49702e39 q^{35} -2.34231e39 q^{36} -1.71766e40 q^{37} -1.40414e40 q^{38} +3.83610e39 q^{39} +2.22897e40 q^{40} -2.28639e41 q^{41} -5.38473e40 q^{42} +6.51175e41 q^{43} -3.01093e41 q^{44} -1.22744e42 q^{45} +2.29013e42 q^{46} +5.04013e42 q^{47} +3.43218e41 q^{48} +2.25414e43 q^{49} -3.22072e42 q^{50} -4.34492e42 q^{51} +1.59521e43 q^{52} -9.15542e43 q^{53} -3.84663e43 q^{54} -1.57781e44 q^{55} -2.23920e44 q^{56} -1.13300e44 q^{57} -2.02572e44 q^{58} +6.44776e44 q^{59} +1.79856e44 q^{60} -2.22244e45 q^{61} -2.66348e45 q^{62} +1.23307e46 q^{63} +1.42725e45 q^{64} +8.35937e45 q^{65} -2.42953e45 q^{66} +5.61205e46 q^{67} -1.80680e46 q^{68} +1.84791e46 q^{69} -1.17340e47 q^{70} -1.91073e47 q^{71} -7.85948e46 q^{72} +2.40551e47 q^{73} -5.76350e47 q^{74} -2.59881e46 q^{75} -4.71150e47 q^{76} +1.58505e48 q^{77} +1.28718e47 q^{78} +1.22693e48 q^{79} +7.47918e47 q^{80} +4.17013e48 q^{81} -7.67184e48 q^{82} +8.82136e48 q^{83} -1.80681e48 q^{84} -9.46815e48 q^{85} +2.18498e49 q^{86} -1.63456e48 q^{87} -1.01030e49 q^{88} -5.56843e49 q^{89} -4.11859e49 q^{90} -8.39773e49 q^{91} +7.68439e49 q^{92} -2.14917e49 q^{93} +1.69119e50 q^{94} -2.46896e50 q^{95} +1.15165e49 q^{96} +5.94385e50 q^{97} +7.56363e50 q^{98} +5.56346e50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 67108864 q^{2} + 889619774904 q^{3} + 22\!\cdots\!48 q^{4}+ \cdots - 38\!\cdots\!86 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 67108864 q^{2} + 889619774904 q^{3} + 22\!\cdots\!48 q^{4}+ \cdots + 67\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.35544e7 0.707107
\(3\) 2.70751e11 0.184493 0.0922463 0.995736i \(-0.470595\pi\)
0.0922463 + 0.995736i \(0.470595\pi\)
\(4\) 1.12590e15 0.500000
\(5\) 5.90004e17 0.885359 0.442680 0.896680i \(-0.354028\pi\)
0.442680 + 0.896680i \(0.354028\pi\)
\(6\) 9.08491e18 0.130456
\(7\) −5.92711e21 −1.67049 −0.835243 0.549881i \(-0.814673\pi\)
−0.835243 + 0.549881i \(0.814673\pi\)
\(8\) 3.77789e22 0.353553
\(9\) −2.08039e24 −0.965962
\(10\) 1.97972e25 0.626043
\(11\) −2.67424e26 −0.744196 −0.372098 0.928194i \(-0.621361\pi\)
−0.372098 + 0.928194i \(0.621361\pi\)
\(12\) 3.04839e26 0.0922463
\(13\) 1.41683e28 0.556882 0.278441 0.960453i \(-0.410182\pi\)
0.278441 + 0.960453i \(0.410182\pi\)
\(14\) −1.98881e29 −1.18121
\(15\) 1.59744e29 0.163342
\(16\) 1.26765e30 0.250000
\(17\) −1.60476e31 −0.674471 −0.337235 0.941420i \(-0.609492\pi\)
−0.337235 + 0.941420i \(0.609492\pi\)
\(18\) −6.98062e31 −0.683039
\(19\) −4.18466e32 −1.03144 −0.515719 0.856758i \(-0.672475\pi\)
−0.515719 + 0.856758i \(0.672475\pi\)
\(20\) 6.64285e32 0.442680
\(21\) −1.60477e33 −0.308192
\(22\) −8.97327e33 −0.526226
\(23\) 6.82511e34 1.28840 0.644198 0.764859i \(-0.277191\pi\)
0.644198 + 0.764859i \(0.277191\pi\)
\(24\) 1.02287e34 0.0652280
\(25\) −9.59850e34 −0.216139
\(26\) 4.75411e35 0.393775
\(27\) −1.14638e36 −0.362705
\(28\) −6.67333e36 −0.835243
\(29\) −6.03711e36 −0.308802 −0.154401 0.988008i \(-0.549345\pi\)
−0.154401 + 0.988008i \(0.549345\pi\)
\(30\) 5.36013e36 0.115500
\(31\) −7.93779e37 −0.741270 −0.370635 0.928779i \(-0.620860\pi\)
−0.370635 + 0.928779i \(0.620860\pi\)
\(32\) 4.25353e37 0.176777
\(33\) −7.24055e37 −0.137299
\(34\) −5.38469e38 −0.476923
\(35\) −3.49702e39 −1.47898
\(36\) −2.34231e39 −0.482981
\(37\) −1.71766e40 −1.76113 −0.880567 0.473921i \(-0.842838\pi\)
−0.880567 + 0.473921i \(0.842838\pi\)
\(38\) −1.40414e40 −0.729337
\(39\) 3.83610e39 0.102741
\(40\) 2.22897e40 0.313022
\(41\) −2.28639e41 −1.71065 −0.855326 0.518090i \(-0.826643\pi\)
−0.855326 + 0.518090i \(0.826643\pi\)
\(42\) −5.38473e40 −0.217925
\(43\) 6.51175e41 1.44628 0.723140 0.690702i \(-0.242698\pi\)
0.723140 + 0.690702i \(0.242698\pi\)
\(44\) −3.01093e41 −0.372098
\(45\) −1.22744e42 −0.855224
\(46\) 2.29013e42 0.911034
\(47\) 5.04013e42 1.15863 0.579317 0.815103i \(-0.303319\pi\)
0.579317 + 0.815103i \(0.303319\pi\)
\(48\) 3.43218e41 0.0461231
\(49\) 2.25414e43 1.79052
\(50\) −3.22072e42 −0.152833
\(51\) −4.34492e42 −0.124435
\(52\) 1.59521e43 0.278441
\(53\) −9.15542e43 −0.983204 −0.491602 0.870820i \(-0.663589\pi\)
−0.491602 + 0.870820i \(0.663589\pi\)
\(54\) −3.84663e43 −0.256471
\(55\) −1.57781e44 −0.658881
\(56\) −2.23920e44 −0.590606
\(57\) −1.13300e44 −0.190293
\(58\) −2.02572e44 −0.218356
\(59\) 6.44776e44 0.449452 0.224726 0.974422i \(-0.427851\pi\)
0.224726 + 0.974422i \(0.427851\pi\)
\(60\) 1.79856e44 0.0816711
\(61\) −2.22244e45 −0.662093 −0.331047 0.943614i \(-0.607402\pi\)
−0.331047 + 0.943614i \(0.607402\pi\)
\(62\) −2.66348e45 −0.524157
\(63\) 1.23307e46 1.61363
\(64\) 1.42725e45 0.125000
\(65\) 8.35937e45 0.493041
\(66\) −2.42953e45 −0.0970848
\(67\) 5.61205e46 1.52832 0.764159 0.645028i \(-0.223154\pi\)
0.764159 + 0.645028i \(0.223154\pi\)
\(68\) −1.80680e46 −0.337235
\(69\) 1.84791e46 0.237700
\(70\) −1.17340e47 −1.04580
\(71\) −1.91073e47 −1.18608 −0.593038 0.805175i \(-0.702072\pi\)
−0.593038 + 0.805175i \(0.702072\pi\)
\(72\) −7.85948e46 −0.341519
\(73\) 2.40551e47 0.735316 0.367658 0.929961i \(-0.380160\pi\)
0.367658 + 0.929961i \(0.380160\pi\)
\(74\) −5.76350e47 −1.24531
\(75\) −2.59881e46 −0.0398761
\(76\) −4.71150e47 −0.515719
\(77\) 1.58505e48 1.24317
\(78\) 1.28718e47 0.0726486
\(79\) 1.22693e48 0.500413 0.250206 0.968193i \(-0.419502\pi\)
0.250206 + 0.968193i \(0.419502\pi\)
\(80\) 7.47918e47 0.221340
\(81\) 4.17013e48 0.899046
\(82\) −7.67184e48 −1.20961
\(83\) 8.82136e48 1.02104 0.510522 0.859865i \(-0.329452\pi\)
0.510522 + 0.859865i \(0.329452\pi\)
\(84\) −1.80681e48 −0.154096
\(85\) −9.46815e48 −0.597149
\(86\) 2.18498e49 1.02267
\(87\) −1.63456e48 −0.0569716
\(88\) −1.01030e49 −0.263113
\(89\) −5.56843e49 −1.08715 −0.543573 0.839362i \(-0.682929\pi\)
−0.543573 + 0.839362i \(0.682929\pi\)
\(90\) −4.11859e49 −0.604735
\(91\) −8.39773e49 −0.930264
\(92\) 7.68439e49 0.644198
\(93\) −2.14917e49 −0.136759
\(94\) 1.69119e50 0.819278
\(95\) −2.46896e50 −0.913193
\(96\) 1.15165e49 0.0326140
\(97\) 5.94385e50 1.29238 0.646188 0.763178i \(-0.276362\pi\)
0.646188 + 0.763178i \(0.276362\pi\)
\(98\) 7.56363e50 1.26609
\(99\) 5.56346e50 0.718865
\(100\) −1.08070e50 −0.108070
\(101\) −1.51456e51 −1.17514 −0.587571 0.809172i \(-0.699916\pi\)
−0.587571 + 0.809172i \(0.699916\pi\)
\(102\) −1.45791e50 −0.0879887
\(103\) −8.50653e50 −0.400316 −0.200158 0.979764i \(-0.564146\pi\)
−0.200158 + 0.979764i \(0.564146\pi\)
\(104\) 5.35265e50 0.196888
\(105\) −9.46822e50 −0.272861
\(106\) −3.07205e51 −0.695230
\(107\) −2.42869e51 −0.432599 −0.216299 0.976327i \(-0.569399\pi\)
−0.216299 + 0.976327i \(0.569399\pi\)
\(108\) −1.29071e51 −0.181353
\(109\) 9.70083e51 1.07754 0.538770 0.842453i \(-0.318889\pi\)
0.538770 + 0.842453i \(0.318889\pi\)
\(110\) −5.29426e51 −0.465899
\(111\) −4.65058e51 −0.324916
\(112\) −7.51350e51 −0.417622
\(113\) 2.32574e52 1.03053 0.515265 0.857031i \(-0.327693\pi\)
0.515265 + 0.857031i \(0.327693\pi\)
\(114\) −3.80172e51 −0.134557
\(115\) 4.02684e52 1.14069
\(116\) −6.79718e51 −0.154401
\(117\) −2.94756e52 −0.537927
\(118\) 2.16351e52 0.317810
\(119\) 9.51160e52 1.12669
\(120\) 6.03497e51 0.0577502
\(121\) −5.76143e52 −0.446173
\(122\) −7.45727e52 −0.468171
\(123\) −6.19043e52 −0.315603
\(124\) −8.93716e52 −0.370635
\(125\) −3.18646e53 −1.07672
\(126\) 4.13749e53 1.14101
\(127\) −8.04886e53 −1.81443 −0.907213 0.420671i \(-0.861795\pi\)
−0.907213 + 0.420671i \(0.861795\pi\)
\(128\) 4.78905e52 0.0883883
\(129\) 1.76307e53 0.266828
\(130\) 2.80494e53 0.348633
\(131\) 3.68828e53 0.377056 0.188528 0.982068i \(-0.439628\pi\)
0.188528 + 0.982068i \(0.439628\pi\)
\(132\) −8.15213e52 −0.0686493
\(133\) 2.48029e54 1.72300
\(134\) 1.88309e54 1.08068
\(135\) −6.76371e53 −0.321125
\(136\) −6.06262e53 −0.238461
\(137\) −5.14641e54 −1.67931 −0.839654 0.543122i \(-0.817242\pi\)
−0.839654 + 0.543122i \(0.817242\pi\)
\(138\) 6.20055e53 0.168079
\(139\) 5.59390e54 1.26136 0.630678 0.776045i \(-0.282777\pi\)
0.630678 + 0.776045i \(0.282777\pi\)
\(140\) −3.93729e54 −0.739490
\(141\) 1.36462e54 0.213759
\(142\) −6.41136e54 −0.838682
\(143\) −3.78896e54 −0.414429
\(144\) −2.63720e54 −0.241491
\(145\) −3.56191e54 −0.273401
\(146\) 8.07155e54 0.519947
\(147\) 6.10311e54 0.330338
\(148\) −1.93391e55 −0.880567
\(149\) 1.41490e55 0.542596 0.271298 0.962495i \(-0.412547\pi\)
0.271298 + 0.962495i \(0.412547\pi\)
\(150\) −8.72016e53 −0.0281966
\(151\) 1.21063e55 0.330444 0.165222 0.986256i \(-0.447166\pi\)
0.165222 + 0.986256i \(0.447166\pi\)
\(152\) −1.58092e55 −0.364668
\(153\) 3.33853e55 0.651513
\(154\) 5.31855e55 0.879053
\(155\) −4.68332e55 −0.656290
\(156\) 4.31906e54 0.0513703
\(157\) 1.15196e56 1.16412 0.582060 0.813146i \(-0.302247\pi\)
0.582060 + 0.813146i \(0.302247\pi\)
\(158\) 4.11689e55 0.353845
\(159\) −2.47884e55 −0.181394
\(160\) 2.50960e55 0.156511
\(161\) −4.04531e56 −2.15225
\(162\) 1.39926e56 0.635722
\(163\) −2.58863e56 −1.00528 −0.502640 0.864496i \(-0.667638\pi\)
−0.502640 + 0.864496i \(0.667638\pi\)
\(164\) −2.57424e56 −0.855326
\(165\) −4.27195e55 −0.121559
\(166\) 2.95996e56 0.721987
\(167\) 2.51396e56 0.526125 0.263062 0.964779i \(-0.415267\pi\)
0.263062 + 0.964779i \(0.415267\pi\)
\(168\) −6.06266e55 −0.108962
\(169\) −4.46566e56 −0.689882
\(170\) −3.17698e56 −0.422248
\(171\) 8.70571e56 0.996330
\(172\) 7.33158e56 0.723140
\(173\) 1.45880e57 1.24114 0.620568 0.784153i \(-0.286902\pi\)
0.620568 + 0.784153i \(0.286902\pi\)
\(174\) −5.48466e55 −0.0402850
\(175\) 5.68914e56 0.361057
\(176\) −3.39000e56 −0.186049
\(177\) 1.74574e56 0.0829205
\(178\) −1.86845e57 −0.768728
\(179\) −1.87738e57 −0.669576 −0.334788 0.942293i \(-0.608665\pi\)
−0.334788 + 0.942293i \(0.608665\pi\)
\(180\) −1.38197e57 −0.427612
\(181\) 3.52839e57 0.947923 0.473962 0.880546i \(-0.342824\pi\)
0.473962 + 0.880546i \(0.342824\pi\)
\(182\) −2.81781e57 −0.657796
\(183\) −6.01729e56 −0.122151
\(184\) 2.57845e57 0.455517
\(185\) −1.01342e58 −1.55924
\(186\) −7.21141e56 −0.0967031
\(187\) 4.29152e57 0.501938
\(188\) 5.67468e57 0.579317
\(189\) 6.79474e57 0.605894
\(190\) −8.28446e57 −0.645725
\(191\) −7.54559e57 −0.514449 −0.257224 0.966352i \(-0.582808\pi\)
−0.257224 + 0.966352i \(0.582808\pi\)
\(192\) 3.86429e56 0.0230616
\(193\) −3.90265e57 −0.204009 −0.102004 0.994784i \(-0.532526\pi\)
−0.102004 + 0.994784i \(0.532526\pi\)
\(194\) 1.99443e58 0.913848
\(195\) 2.26331e57 0.0909624
\(196\) 2.53793e58 0.895262
\(197\) 5.55147e57 0.171996 0.0859981 0.996295i \(-0.472592\pi\)
0.0859981 + 0.996295i \(0.472592\pi\)
\(198\) 1.86679e58 0.508314
\(199\) −1.39534e58 −0.334138 −0.167069 0.985945i \(-0.553430\pi\)
−0.167069 + 0.985945i \(0.553430\pi\)
\(200\) −3.62621e57 −0.0764167
\(201\) 1.51947e58 0.281963
\(202\) −5.08201e58 −0.830951
\(203\) 3.57826e58 0.515849
\(204\) −4.89194e57 −0.0622174
\(205\) −1.34898e59 −1.51454
\(206\) −2.85432e58 −0.283066
\(207\) −1.41989e59 −1.24454
\(208\) 1.79605e58 0.139221
\(209\) 1.11908e59 0.767592
\(210\) −3.17701e58 −0.192942
\(211\) 1.98069e59 1.06565 0.532825 0.846226i \(-0.321130\pi\)
0.532825 + 0.846226i \(0.321130\pi\)
\(212\) −1.03081e59 −0.491602
\(213\) −5.17334e58 −0.218822
\(214\) −8.14933e58 −0.305894
\(215\) 3.84196e59 1.28048
\(216\) −4.33092e58 −0.128236
\(217\) 4.70481e59 1.23828
\(218\) 3.25506e59 0.761935
\(219\) 6.51295e58 0.135660
\(220\) −1.77646e59 −0.329440
\(221\) −2.27368e59 −0.375601
\(222\) −1.56048e59 −0.229750
\(223\) −1.05167e60 −1.38072 −0.690360 0.723466i \(-0.742548\pi\)
−0.690360 + 0.723466i \(0.742548\pi\)
\(224\) −2.52111e59 −0.295303
\(225\) 1.99686e59 0.208782
\(226\) 7.80389e59 0.728695
\(227\) −2.32315e59 −0.193829 −0.0969145 0.995293i \(-0.530897\pi\)
−0.0969145 + 0.995293i \(0.530897\pi\)
\(228\) −1.27565e59 −0.0951463
\(229\) −2.08303e60 −1.38960 −0.694800 0.719203i \(-0.744507\pi\)
−0.694800 + 0.719203i \(0.744507\pi\)
\(230\) 1.35118e60 0.806592
\(231\) 4.29155e59 0.229355
\(232\) −2.28075e59 −0.109178
\(233\) −2.17828e60 −0.934410 −0.467205 0.884149i \(-0.654739\pi\)
−0.467205 + 0.884149i \(0.654739\pi\)
\(234\) −9.89038e59 −0.380372
\(235\) 2.97369e60 1.02581
\(236\) 7.25954e59 0.224726
\(237\) 3.32193e59 0.0923225
\(238\) 3.19156e60 0.796693
\(239\) −1.01088e59 −0.0226754 −0.0113377 0.999936i \(-0.503609\pi\)
−0.0113377 + 0.999936i \(0.503609\pi\)
\(240\) 2.02500e59 0.0408355
\(241\) −5.46605e59 −0.0991378 −0.0495689 0.998771i \(-0.515785\pi\)
−0.0495689 + 0.998771i \(0.515785\pi\)
\(242\) −1.93321e60 −0.315492
\(243\) 3.59803e60 0.528573
\(244\) −2.50224e60 −0.331047
\(245\) 1.32995e61 1.58526
\(246\) −2.07716e60 −0.223165
\(247\) −5.92896e60 −0.574389
\(248\) −2.99881e60 −0.262079
\(249\) 2.38840e60 0.188375
\(250\) −1.06920e61 −0.761356
\(251\) −3.50691e60 −0.225551 −0.112776 0.993620i \(-0.535974\pi\)
−0.112776 + 0.993620i \(0.535974\pi\)
\(252\) 1.38831e61 0.806813
\(253\) −1.82520e61 −0.958819
\(254\) −2.70075e61 −1.28299
\(255\) −2.56352e60 −0.110170
\(256\) 1.60694e60 0.0625000
\(257\) −5.05435e61 −1.77980 −0.889900 0.456157i \(-0.849226\pi\)
−0.889900 + 0.456157i \(0.849226\pi\)
\(258\) 5.91587e60 0.188676
\(259\) 1.01807e62 2.94195
\(260\) 9.41181e60 0.246520
\(261\) 1.25595e61 0.298291
\(262\) 1.23758e61 0.266619
\(263\) 1.44972e61 0.283408 0.141704 0.989909i \(-0.454742\pi\)
0.141704 + 0.989909i \(0.454742\pi\)
\(264\) −2.73540e60 −0.0485424
\(265\) −5.40173e61 −0.870489
\(266\) 8.32248e61 1.21835
\(267\) −1.50766e61 −0.200570
\(268\) 6.31860e61 0.764159
\(269\) −1.34234e61 −0.147632 −0.0738158 0.997272i \(-0.523518\pi\)
−0.0738158 + 0.997272i \(0.523518\pi\)
\(270\) −2.26952e61 −0.227069
\(271\) −8.45578e61 −0.769905 −0.384952 0.922936i \(-0.625782\pi\)
−0.384952 + 0.922936i \(0.625782\pi\)
\(272\) −2.03428e61 −0.168618
\(273\) −2.27370e61 −0.171627
\(274\) −1.72685e62 −1.18745
\(275\) 2.56687e61 0.160850
\(276\) 2.08056e61 0.118850
\(277\) −1.96004e62 −1.02101 −0.510506 0.859874i \(-0.670542\pi\)
−0.510506 + 0.859874i \(0.670542\pi\)
\(278\) 1.87700e62 0.891913
\(279\) 1.65137e62 0.716039
\(280\) −1.32113e62 −0.522898
\(281\) −2.29277e61 −0.0828608 −0.0414304 0.999141i \(-0.513191\pi\)
−0.0414304 + 0.999141i \(0.513191\pi\)
\(282\) 4.57891e61 0.151151
\(283\) −2.92760e61 −0.0882992 −0.0441496 0.999025i \(-0.514058\pi\)
−0.0441496 + 0.999025i \(0.514058\pi\)
\(284\) −2.15130e62 −0.593038
\(285\) −6.68475e61 −0.168477
\(286\) −1.27136e62 −0.293046
\(287\) 1.35517e63 2.85762
\(288\) −8.84899e61 −0.170760
\(289\) −3.08576e62 −0.545089
\(290\) −1.19518e62 −0.193323
\(291\) 1.60931e62 0.238434
\(292\) 2.70836e62 0.367658
\(293\) −4.39216e62 −0.546454 −0.273227 0.961950i \(-0.588091\pi\)
−0.273227 + 0.961950i \(0.588091\pi\)
\(294\) 2.04786e62 0.233584
\(295\) 3.80420e62 0.397926
\(296\) −6.48913e62 −0.622655
\(297\) 3.06571e62 0.269924
\(298\) 4.74762e62 0.383673
\(299\) 9.67004e62 0.717485
\(300\) −2.92600e61 −0.0199380
\(301\) −3.85959e63 −2.41599
\(302\) 4.06219e62 0.233659
\(303\) −4.10069e62 −0.216805
\(304\) −5.30468e62 −0.257859
\(305\) −1.31125e63 −0.586190
\(306\) 1.12022e63 0.460690
\(307\) −2.38716e63 −0.903342 −0.451671 0.892185i \(-0.649172\pi\)
−0.451671 + 0.892185i \(0.649172\pi\)
\(308\) 1.78461e63 0.621584
\(309\) −2.30316e62 −0.0738554
\(310\) −1.57146e63 −0.464067
\(311\) 4.89938e63 1.33276 0.666379 0.745614i \(-0.267844\pi\)
0.666379 + 0.745614i \(0.267844\pi\)
\(312\) 1.44924e62 0.0363243
\(313\) −1.11505e63 −0.257580 −0.128790 0.991672i \(-0.541109\pi\)
−0.128790 + 0.991672i \(0.541109\pi\)
\(314\) 3.86535e63 0.823158
\(315\) 7.27515e63 1.42864
\(316\) 1.38140e63 0.250206
\(317\) −2.45255e63 −0.409833 −0.204917 0.978779i \(-0.565692\pi\)
−0.204917 + 0.978779i \(0.565692\pi\)
\(318\) −8.31762e62 −0.128265
\(319\) 1.61447e63 0.229809
\(320\) 8.42081e62 0.110670
\(321\) −6.57571e62 −0.0798113
\(322\) −1.35738e64 −1.52187
\(323\) 6.71538e63 0.695675
\(324\) 4.69515e63 0.449523
\(325\) −1.35995e63 −0.120364
\(326\) −8.68602e63 −0.710840
\(327\) 2.62651e63 0.198798
\(328\) −8.63773e63 −0.604807
\(329\) −2.98734e64 −1.93548
\(330\) −1.43343e63 −0.0859549
\(331\) −1.52520e64 −0.846666 −0.423333 0.905974i \(-0.639140\pi\)
−0.423333 + 0.905974i \(0.639140\pi\)
\(332\) 9.93197e63 0.510522
\(333\) 3.57339e64 1.70119
\(334\) 8.43546e63 0.372026
\(335\) 3.31113e64 1.35311
\(336\) −2.03429e63 −0.0770481
\(337\) −2.16648e64 −0.760664 −0.380332 0.924850i \(-0.624190\pi\)
−0.380332 + 0.924850i \(0.624190\pi\)
\(338\) −1.49843e64 −0.487820
\(339\) 6.29698e63 0.190125
\(340\) −1.06602e64 −0.298574
\(341\) 2.12276e64 0.551650
\(342\) 2.92115e64 0.704512
\(343\) −5.89872e64 −1.32056
\(344\) 2.46007e64 0.511337
\(345\) 1.09027e64 0.210450
\(346\) 4.89491e64 0.877615
\(347\) 5.29970e64 0.882774 0.441387 0.897317i \(-0.354487\pi\)
0.441387 + 0.897317i \(0.354487\pi\)
\(348\) −1.84035e63 −0.0284858
\(349\) −1.00163e65 −1.44099 −0.720494 0.693461i \(-0.756085\pi\)
−0.720494 + 0.693461i \(0.756085\pi\)
\(350\) 1.90896e64 0.255306
\(351\) −1.62424e64 −0.201984
\(352\) −1.13750e64 −0.131556
\(353\) 6.03904e64 0.649700 0.324850 0.945766i \(-0.394686\pi\)
0.324850 + 0.945766i \(0.394686\pi\)
\(354\) 5.85774e63 0.0586336
\(355\) −1.12734e65 −1.05010
\(356\) −6.26949e64 −0.543573
\(357\) 2.57528e64 0.207867
\(358\) −6.29944e64 −0.473462
\(359\) −7.01548e64 −0.491076 −0.245538 0.969387i \(-0.578965\pi\)
−0.245538 + 0.969387i \(0.578965\pi\)
\(360\) −4.63712e64 −0.302367
\(361\) 1.05121e64 0.0638637
\(362\) 1.18393e65 0.670283
\(363\) −1.55991e64 −0.0823156
\(364\) −9.45500e64 −0.465132
\(365\) 1.41926e65 0.651019
\(366\) −2.01907e64 −0.0863740
\(367\) −1.18243e65 −0.471836 −0.235918 0.971773i \(-0.575810\pi\)
−0.235918 + 0.971773i \(0.575810\pi\)
\(368\) 8.65185e64 0.322099
\(369\) 4.75657e65 1.65243
\(370\) −3.40049e65 −1.10255
\(371\) 5.42652e65 1.64243
\(372\) −2.41975e64 −0.0683794
\(373\) 7.20138e64 0.190038 0.0950190 0.995475i \(-0.469709\pi\)
0.0950190 + 0.995475i \(0.469709\pi\)
\(374\) 1.44000e65 0.354924
\(375\) −8.62738e64 −0.198647
\(376\) 1.90411e65 0.409639
\(377\) −8.55357e64 −0.171966
\(378\) 2.27994e65 0.428432
\(379\) −6.43875e65 −1.13110 −0.565551 0.824714i \(-0.691336\pi\)
−0.565551 + 0.824714i \(0.691336\pi\)
\(380\) −2.77980e65 −0.456596
\(381\) −2.17924e65 −0.334748
\(382\) −2.53188e65 −0.363770
\(383\) −1.01890e66 −1.36950 −0.684751 0.728777i \(-0.740089\pi\)
−0.684751 + 0.728777i \(0.740089\pi\)
\(384\) 1.29664e64 0.0163070
\(385\) 9.35186e65 1.10065
\(386\) −1.30951e65 −0.144256
\(387\) −1.35470e66 −1.39705
\(388\) 6.69218e65 0.646188
\(389\) 3.06541e65 0.277188 0.138594 0.990349i \(-0.455742\pi\)
0.138594 + 0.990349i \(0.455742\pi\)
\(390\) 7.59441e64 0.0643201
\(391\) −1.09527e66 −0.868986
\(392\) 8.51589e65 0.633046
\(393\) 9.98608e64 0.0695641
\(394\) 1.86276e65 0.121620
\(395\) 7.23892e65 0.443045
\(396\) 6.26390e65 0.359433
\(397\) 3.60543e66 1.93999 0.969993 0.243131i \(-0.0781745\pi\)
0.969993 + 0.243131i \(0.0781745\pi\)
\(398\) −4.68198e65 −0.236271
\(399\) 6.71543e65 0.317881
\(400\) −1.21675e65 −0.0540348
\(401\) −4.04832e66 −1.68691 −0.843457 0.537196i \(-0.819483\pi\)
−0.843457 + 0.537196i \(0.819483\pi\)
\(402\) 5.09849e65 0.199378
\(403\) −1.12465e66 −0.412800
\(404\) −1.70524e66 −0.587571
\(405\) 2.46039e66 0.795979
\(406\) 1.20066e66 0.364760
\(407\) 4.59343e66 1.31063
\(408\) −1.64146e65 −0.0439944
\(409\) −2.76141e65 −0.0695324 −0.0347662 0.999395i \(-0.511069\pi\)
−0.0347662 + 0.999395i \(0.511069\pi\)
\(410\) −4.52642e66 −1.07094
\(411\) −1.39340e66 −0.309820
\(412\) −9.57750e65 −0.200158
\(413\) −3.82166e66 −0.750803
\(414\) −4.76435e66 −0.880025
\(415\) 5.20463e66 0.903990
\(416\) 6.02654e65 0.0984438
\(417\) 1.51456e66 0.232711
\(418\) 3.75500e66 0.542769
\(419\) −1.14203e67 −1.55317 −0.776585 0.630013i \(-0.783050\pi\)
−0.776585 + 0.630013i \(0.783050\pi\)
\(420\) −1.06603e66 −0.136430
\(421\) 6.12212e66 0.737409 0.368705 0.929547i \(-0.379801\pi\)
0.368705 + 0.929547i \(0.379801\pi\)
\(422\) 6.64610e66 0.753528
\(423\) −1.04854e67 −1.11920
\(424\) −3.45882e66 −0.347615
\(425\) 1.54033e66 0.145779
\(426\) −1.73589e66 −0.154731
\(427\) 1.31726e67 1.10602
\(428\) −2.73446e66 −0.216299
\(429\) −1.02587e66 −0.0764591
\(430\) 1.28915e67 0.905434
\(431\) 1.51854e67 1.00521 0.502604 0.864517i \(-0.332375\pi\)
0.502604 + 0.864517i \(0.332375\pi\)
\(432\) −1.45321e66 −0.0906764
\(433\) 1.75181e66 0.103050 0.0515248 0.998672i \(-0.483592\pi\)
0.0515248 + 0.998672i \(0.483592\pi\)
\(434\) 1.57867e67 0.875597
\(435\) −9.64393e65 −0.0504404
\(436\) 1.09222e67 0.538770
\(437\) −2.85607e67 −1.32890
\(438\) 2.18538e66 0.0959264
\(439\) 1.63861e66 0.0678626 0.0339313 0.999424i \(-0.489197\pi\)
0.0339313 + 0.999424i \(0.489197\pi\)
\(440\) −5.96081e66 −0.232949
\(441\) −4.68948e67 −1.72958
\(442\) −7.62921e66 −0.265590
\(443\) 2.54855e67 0.837526 0.418763 0.908095i \(-0.362464\pi\)
0.418763 + 0.908095i \(0.362464\pi\)
\(444\) −5.23609e66 −0.162458
\(445\) −3.28539e67 −0.962514
\(446\) −3.52882e67 −0.976316
\(447\) 3.83087e66 0.100105
\(448\) −8.45945e66 −0.208811
\(449\) 3.94691e66 0.0920398 0.0460199 0.998941i \(-0.485346\pi\)
0.0460199 + 0.998941i \(0.485346\pi\)
\(450\) 6.70035e66 0.147631
\(451\) 6.11435e67 1.27306
\(452\) 2.61855e67 0.515265
\(453\) 3.27779e66 0.0609645
\(454\) −7.79521e66 −0.137058
\(455\) −4.95469e67 −0.823618
\(456\) −4.28036e66 −0.0672786
\(457\) 9.44235e67 1.40351 0.701757 0.712416i \(-0.252399\pi\)
0.701757 + 0.712416i \(0.252399\pi\)
\(458\) −6.98947e67 −0.982595
\(459\) 1.83967e67 0.244634
\(460\) 4.53381e67 0.570347
\(461\) 6.24242e67 0.742984 0.371492 0.928436i \(-0.378846\pi\)
0.371492 + 0.928436i \(0.378846\pi\)
\(462\) 1.44001e67 0.162179
\(463\) 1.16253e68 1.23905 0.619525 0.784977i \(-0.287325\pi\)
0.619525 + 0.784977i \(0.287325\pi\)
\(464\) −7.65294e66 −0.0772004
\(465\) −1.26802e67 −0.121081
\(466\) −7.30909e67 −0.660727
\(467\) 3.38070e67 0.289352 0.144676 0.989479i \(-0.453786\pi\)
0.144676 + 0.989479i \(0.453786\pi\)
\(468\) −3.31866e67 −0.268964
\(469\) −3.32632e68 −2.55303
\(470\) 9.97806e67 0.725355
\(471\) 3.11896e67 0.214772
\(472\) 2.43590e67 0.158905
\(473\) −1.74140e68 −1.07631
\(474\) 1.11465e67 0.0652818
\(475\) 4.01664e67 0.222934
\(476\) 1.07091e68 0.563347
\(477\) 1.90468e68 0.949738
\(478\) −3.39196e66 −0.0160339
\(479\) −2.32823e68 −1.04345 −0.521724 0.853115i \(-0.674711\pi\)
−0.521724 + 0.853115i \(0.674711\pi\)
\(480\) 6.79477e66 0.0288751
\(481\) −2.43363e68 −0.980745
\(482\) −1.83410e67 −0.0701010
\(483\) −1.09527e68 −0.397074
\(484\) −6.48679e67 −0.223086
\(485\) 3.50689e68 1.14422
\(486\) 1.20730e68 0.373757
\(487\) 5.88848e67 0.172988 0.0864938 0.996252i \(-0.472434\pi\)
0.0864938 + 0.996252i \(0.472434\pi\)
\(488\) −8.39614e67 −0.234085
\(489\) −7.00877e67 −0.185467
\(490\) 4.46257e68 1.12095
\(491\) −2.11706e68 −0.504841 −0.252420 0.967618i \(-0.581227\pi\)
−0.252420 + 0.967618i \(0.581227\pi\)
\(492\) −6.96980e67 −0.157801
\(493\) 9.68812e67 0.208278
\(494\) −1.98943e68 −0.406155
\(495\) 3.28246e68 0.636454
\(496\) −1.00623e68 −0.185317
\(497\) 1.13251e69 1.98132
\(498\) 8.01413e67 0.133201
\(499\) −4.72866e67 −0.0746748 −0.0373374 0.999303i \(-0.511888\pi\)
−0.0373374 + 0.999303i \(0.511888\pi\)
\(500\) −3.58763e68 −0.538360
\(501\) 6.80659e67 0.0970661
\(502\) −1.17673e68 −0.159489
\(503\) −3.67158e68 −0.473009 −0.236505 0.971630i \(-0.576002\pi\)
−0.236505 + 0.971630i \(0.576002\pi\)
\(504\) 4.65840e68 0.570503
\(505\) −8.93594e68 −1.04042
\(506\) −6.12435e68 −0.677988
\(507\) −1.20908e68 −0.127278
\(508\) −9.06221e68 −0.907213
\(509\) 1.22746e69 1.16870 0.584351 0.811501i \(-0.301349\pi\)
0.584351 + 0.811501i \(0.301349\pi\)
\(510\) −8.60173e67 −0.0779016
\(511\) −1.42577e69 −1.22834
\(512\) 5.39199e67 0.0441942
\(513\) 4.79722e68 0.374108
\(514\) −1.69596e69 −1.25851
\(515\) −5.01888e68 −0.354424
\(516\) 1.98504e68 0.133414
\(517\) −1.34785e69 −0.862250
\(518\) 3.41609e69 2.08027
\(519\) 3.94971e68 0.228980
\(520\) 3.15808e68 0.174316
\(521\) 2.83740e69 1.49128 0.745640 0.666349i \(-0.232144\pi\)
0.745640 + 0.666349i \(0.232144\pi\)
\(522\) 4.21428e68 0.210924
\(523\) −1.04293e69 −0.497122 −0.248561 0.968616i \(-0.579958\pi\)
−0.248561 + 0.968616i \(0.579958\pi\)
\(524\) 4.15264e68 0.188528
\(525\) 1.54034e68 0.0666124
\(526\) 4.86445e68 0.200400
\(527\) 1.27383e69 0.499965
\(528\) −9.17849e67 −0.0343246
\(529\) 1.85200e69 0.659966
\(530\) −1.81252e69 −0.615529
\(531\) −1.34138e69 −0.434153
\(532\) 2.79256e69 0.861501
\(533\) −3.23943e69 −0.952632
\(534\) −5.05887e68 −0.141825
\(535\) −1.43293e69 −0.383005
\(536\) 2.12017e69 0.540342
\(537\) −5.08304e68 −0.123532
\(538\) −4.50414e68 −0.104391
\(539\) −6.02810e69 −1.33250
\(540\) −7.61526e68 −0.160562
\(541\) −1.50880e69 −0.303459 −0.151730 0.988422i \(-0.548484\pi\)
−0.151730 + 0.988422i \(0.548484\pi\)
\(542\) −2.83729e69 −0.544405
\(543\) 9.55317e68 0.174885
\(544\) −6.82590e68 −0.119231
\(545\) 5.72352e69 0.954009
\(546\) −7.62926e68 −0.121358
\(547\) 4.40510e69 0.668773 0.334387 0.942436i \(-0.391471\pi\)
0.334387 + 0.942436i \(0.391471\pi\)
\(548\) −5.79435e69 −0.839654
\(549\) 4.62354e69 0.639557
\(550\) 8.61299e68 0.113738
\(551\) 2.52632e69 0.318510
\(552\) 6.98120e68 0.0840395
\(553\) −7.27214e69 −0.835933
\(554\) −6.57681e69 −0.721965
\(555\) −2.74386e69 −0.287668
\(556\) 6.29817e69 0.630678
\(557\) 2.55341e69 0.244238 0.122119 0.992515i \(-0.461031\pi\)
0.122119 + 0.992515i \(0.461031\pi\)
\(558\) 5.54107e69 0.506316
\(559\) 9.22607e69 0.805407
\(560\) −4.43299e69 −0.369745
\(561\) 1.16194e69 0.0926039
\(562\) −7.69325e68 −0.0585914
\(563\) 2.20159e70 1.60240 0.801201 0.598395i \(-0.204195\pi\)
0.801201 + 0.598395i \(0.204195\pi\)
\(564\) 1.53643e69 0.106880
\(565\) 1.37220e70 0.912390
\(566\) −9.82338e68 −0.0624370
\(567\) −2.47168e70 −1.50184
\(568\) −7.21855e69 −0.419341
\(569\) −2.89847e70 −1.60993 −0.804965 0.593323i \(-0.797816\pi\)
−0.804965 + 0.593323i \(0.797816\pi\)
\(570\) −2.24303e69 −0.119131
\(571\) 1.91490e70 0.972581 0.486291 0.873797i \(-0.338350\pi\)
0.486291 + 0.873797i \(0.338350\pi\)
\(572\) −4.26598e69 −0.207215
\(573\) −2.04298e69 −0.0949120
\(574\) 4.54719e70 2.02064
\(575\) −6.55108e69 −0.278473
\(576\) −2.96923e69 −0.120745
\(577\) −3.95868e70 −1.54017 −0.770083 0.637944i \(-0.779785\pi\)
−0.770083 + 0.637944i \(0.779785\pi\)
\(578\) −1.03541e70 −0.385436
\(579\) −1.05665e69 −0.0376381
\(580\) −4.01036e69 −0.136700
\(581\) −5.22851e70 −1.70564
\(582\) 5.39994e69 0.168598
\(583\) 2.44838e70 0.731696
\(584\) 9.08776e69 0.259973
\(585\) −1.73907e70 −0.476259
\(586\) −1.47376e70 −0.386401
\(587\) 1.26116e70 0.316592 0.158296 0.987392i \(-0.449400\pi\)
0.158296 + 0.987392i \(0.449400\pi\)
\(588\) 6.87149e69 0.165169
\(589\) 3.32169e70 0.764574
\(590\) 1.27648e70 0.281376
\(591\) 1.50307e69 0.0317320
\(592\) −2.17739e70 −0.440284
\(593\) 5.89249e70 1.14131 0.570656 0.821189i \(-0.306689\pi\)
0.570656 + 0.821189i \(0.306689\pi\)
\(594\) 1.02868e70 0.190865
\(595\) 5.61188e70 0.997529
\(596\) 1.59304e70 0.271298
\(597\) −3.77790e69 −0.0616460
\(598\) 3.24473e70 0.507339
\(599\) 5.88782e70 0.882208 0.441104 0.897456i \(-0.354587\pi\)
0.441104 + 0.897456i \(0.354587\pi\)
\(600\) −9.81802e68 −0.0140983
\(601\) 3.71667e70 0.511511 0.255755 0.966742i \(-0.417676\pi\)
0.255755 + 0.966742i \(0.417676\pi\)
\(602\) −1.29506e71 −1.70836
\(603\) −1.16752e71 −1.47630
\(604\) 1.36304e70 0.165222
\(605\) −3.39926e70 −0.395023
\(606\) −1.37596e70 −0.153304
\(607\) −1.80800e70 −0.193146 −0.0965731 0.995326i \(-0.530788\pi\)
−0.0965731 + 0.995326i \(0.530788\pi\)
\(608\) −1.77996e70 −0.182334
\(609\) 9.68819e69 0.0951703
\(610\) −4.39982e70 −0.414499
\(611\) 7.14102e70 0.645222
\(612\) 3.75885e70 0.325757
\(613\) −1.17454e71 −0.976397 −0.488198 0.872733i \(-0.662346\pi\)
−0.488198 + 0.872733i \(0.662346\pi\)
\(614\) −8.00997e70 −0.638759
\(615\) −3.65238e70 −0.279422
\(616\) 5.98816e70 0.439526
\(617\) 6.20543e70 0.437019 0.218509 0.975835i \(-0.429881\pi\)
0.218509 + 0.975835i \(0.429881\pi\)
\(618\) −7.72811e69 −0.0522236
\(619\) −8.66432e70 −0.561853 −0.280927 0.959729i \(-0.590642\pi\)
−0.280927 + 0.959729i \(0.590642\pi\)
\(620\) −5.27295e70 −0.328145
\(621\) −7.82419e70 −0.467308
\(622\) 1.64396e71 0.942402
\(623\) 3.30047e71 1.81606
\(624\) 4.86283e69 0.0256852
\(625\) −1.45376e71 −0.737145
\(626\) −3.74148e70 −0.182137
\(627\) 3.02992e70 0.141615
\(628\) 1.29700e71 0.582060
\(629\) 2.75643e71 1.18783
\(630\) 2.44113e71 1.01020
\(631\) −4.75034e71 −1.88789 −0.943944 0.330107i \(-0.892915\pi\)
−0.943944 + 0.330107i \(0.892915\pi\)
\(632\) 4.63521e70 0.176923
\(633\) 5.36276e70 0.196604
\(634\) −8.22940e70 −0.289796
\(635\) −4.74885e71 −1.60642
\(636\) −2.79093e70 −0.0906969
\(637\) 3.19374e71 0.997111
\(638\) 5.41726e70 0.162499
\(639\) 3.97507e71 1.14570
\(640\) 2.82556e70 0.0782554
\(641\) 3.18961e71 0.848904 0.424452 0.905450i \(-0.360467\pi\)
0.424452 + 0.905450i \(0.360467\pi\)
\(642\) −2.20644e70 −0.0564351
\(643\) −3.61532e71 −0.888724 −0.444362 0.895847i \(-0.646570\pi\)
−0.444362 + 0.895847i \(0.646570\pi\)
\(644\) −4.55462e71 −1.07612
\(645\) 1.04022e71 0.236238
\(646\) 2.25331e71 0.491916
\(647\) −8.45895e71 −1.77524 −0.887621 0.460575i \(-0.847643\pi\)
−0.887621 + 0.460575i \(0.847643\pi\)
\(648\) 1.57543e71 0.317861
\(649\) −1.72429e71 −0.334480
\(650\) −4.56323e70 −0.0851102
\(651\) 1.27384e71 0.228454
\(652\) −2.91454e71 −0.502640
\(653\) 2.78198e71 0.461389 0.230695 0.973026i \(-0.425900\pi\)
0.230695 + 0.973026i \(0.425900\pi\)
\(654\) 8.81312e70 0.140571
\(655\) 2.17610e71 0.333830
\(656\) −2.89834e71 −0.427663
\(657\) −5.00439e71 −0.710288
\(658\) −1.00238e72 −1.36859
\(659\) 1.17155e72 1.53881 0.769404 0.638762i \(-0.220553\pi\)
0.769404 + 0.638762i \(0.220553\pi\)
\(660\) −4.80979e70 −0.0607793
\(661\) 3.69963e70 0.0449802 0.0224901 0.999747i \(-0.492841\pi\)
0.0224901 + 0.999747i \(0.492841\pi\)
\(662\) −5.11771e71 −0.598683
\(663\) −6.15602e70 −0.0692956
\(664\) 3.33261e71 0.360993
\(665\) 1.46338e72 1.52548
\(666\) 1.19903e72 1.20292
\(667\) −4.12039e71 −0.397859
\(668\) 2.83047e71 0.263062
\(669\) −2.84741e71 −0.254732
\(670\) 1.11103e72 0.956794
\(671\) 5.94334e71 0.492727
\(672\) −6.82595e70 −0.0544812
\(673\) −8.02511e71 −0.616690 −0.308345 0.951275i \(-0.599775\pi\)
−0.308345 + 0.951275i \(0.599775\pi\)
\(674\) −7.26951e71 −0.537870
\(675\) 1.10036e71 0.0783948
\(676\) −5.02789e71 −0.344941
\(677\) 2.75308e72 1.81890 0.909448 0.415817i \(-0.136504\pi\)
0.909448 + 0.415817i \(0.136504\pi\)
\(678\) 2.11291e71 0.134439
\(679\) −3.52299e72 −2.15890
\(680\) −3.57697e71 −0.211124
\(681\) −6.28997e70 −0.0357600
\(682\) 7.12279e71 0.390075
\(683\) −7.59014e71 −0.400426 −0.200213 0.979752i \(-0.564163\pi\)
−0.200213 + 0.979752i \(0.564163\pi\)
\(684\) 9.80176e71 0.498165
\(685\) −3.03640e72 −1.48679
\(686\) −1.97928e72 −0.933776
\(687\) −5.63982e71 −0.256371
\(688\) 8.25462e71 0.361570
\(689\) −1.29717e72 −0.547529
\(690\) 3.65835e71 0.148810
\(691\) 2.56440e71 0.100530 0.0502651 0.998736i \(-0.483993\pi\)
0.0502651 + 0.998736i \(0.483993\pi\)
\(692\) 1.64246e72 0.620568
\(693\) −3.29752e72 −1.20085
\(694\) 1.77828e72 0.624216
\(695\) 3.30042e72 1.11675
\(696\) −6.17518e70 −0.0201425
\(697\) 3.66911e72 1.15378
\(698\) −3.36092e72 −1.01893
\(699\) −5.89772e71 −0.172392
\(700\) 6.40540e71 0.180529
\(701\) −2.26409e72 −0.615295 −0.307647 0.951500i \(-0.599542\pi\)
−0.307647 + 0.951500i \(0.599542\pi\)
\(702\) −5.45003e71 −0.142824
\(703\) 7.18781e72 1.81650
\(704\) −3.81681e71 −0.0930245
\(705\) 8.05132e71 0.189254
\(706\) 2.02636e72 0.459407
\(707\) 8.97694e72 1.96306
\(708\) 1.96553e71 0.0414602
\(709\) −1.99310e72 −0.405555 −0.202778 0.979225i \(-0.564997\pi\)
−0.202778 + 0.979225i \(0.564997\pi\)
\(710\) −3.78273e72 −0.742535
\(711\) −2.55249e72 −0.483380
\(712\) −2.10369e72 −0.384364
\(713\) −5.41763e72 −0.955050
\(714\) 8.64120e71 0.146984
\(715\) −2.23550e72 −0.366919
\(716\) −2.11374e72 −0.334788
\(717\) −2.73698e70 −0.00418344
\(718\) −2.35401e72 −0.347244
\(719\) 3.49023e72 0.496899 0.248449 0.968645i \(-0.420079\pi\)
0.248449 + 0.968645i \(0.420079\pi\)
\(720\) −1.55596e72 −0.213806
\(721\) 5.04191e72 0.668723
\(722\) 3.52726e71 0.0451584
\(723\) −1.47994e71 −0.0182902
\(724\) 3.97262e72 0.473962
\(725\) 5.79472e71 0.0667441
\(726\) −5.23420e71 −0.0582059
\(727\) −4.54890e72 −0.488403 −0.244202 0.969724i \(-0.578526\pi\)
−0.244202 + 0.969724i \(0.578526\pi\)
\(728\) −3.17257e72 −0.328898
\(729\) −8.00702e72 −0.801528
\(730\) 4.76224e72 0.460340
\(731\) −1.04498e73 −0.975473
\(732\) −6.77486e71 −0.0610756
\(733\) −1.23171e72 −0.107240 −0.0536202 0.998561i \(-0.517076\pi\)
−0.0536202 + 0.998561i \(0.517076\pi\)
\(734\) −3.96758e72 −0.333639
\(735\) 3.60086e72 0.292468
\(736\) 2.90308e72 0.227758
\(737\) −1.50080e73 −1.13737
\(738\) 1.59604e73 1.16844
\(739\) 2.26780e73 1.60388 0.801940 0.597404i \(-0.203801\pi\)
0.801940 + 0.597404i \(0.203801\pi\)
\(740\) −1.14101e73 −0.779618
\(741\) −1.60528e72 −0.105971
\(742\) 1.82084e73 1.16137
\(743\) 2.66153e72 0.164028 0.0820138 0.996631i \(-0.473865\pi\)
0.0820138 + 0.996631i \(0.473865\pi\)
\(744\) −8.11933e71 −0.0483515
\(745\) 8.34797e72 0.480392
\(746\) 2.41638e72 0.134377
\(747\) −1.83518e73 −0.986289
\(748\) 4.83182e72 0.250969
\(749\) 1.43951e73 0.722651
\(750\) −2.89487e72 −0.140465
\(751\) 2.46245e72 0.115491 0.0577457 0.998331i \(-0.481609\pi\)
0.0577457 + 0.998331i \(0.481609\pi\)
\(752\) 6.38912e72 0.289658
\(753\) −9.49502e71 −0.0416125
\(754\) −2.87010e72 −0.121598
\(755\) 7.14274e72 0.292562
\(756\) 7.65020e72 0.302947
\(757\) 1.10950e73 0.424796 0.212398 0.977183i \(-0.431873\pi\)
0.212398 + 0.977183i \(0.431873\pi\)
\(758\) −2.16048e73 −0.799809
\(759\) −4.94175e72 −0.176895
\(760\) −9.32748e72 −0.322862
\(761\) −9.14926e72 −0.306251 −0.153125 0.988207i \(-0.548934\pi\)
−0.153125 + 0.988207i \(0.548934\pi\)
\(762\) −7.31232e72 −0.236703
\(763\) −5.74979e73 −1.80001
\(764\) −8.49557e72 −0.257224
\(765\) 1.96974e73 0.576823
\(766\) −3.41886e73 −0.968384
\(767\) 9.13541e72 0.250292
\(768\) 4.35081e71 0.0115308
\(769\) −3.24419e72 −0.0831735 −0.0415867 0.999135i \(-0.513241\pi\)
−0.0415867 + 0.999135i \(0.513241\pi\)
\(770\) 3.13796e73 0.778278
\(771\) −1.36847e73 −0.328360
\(772\) −4.39399e72 −0.102004
\(773\) −9.02670e72 −0.202746 −0.101373 0.994849i \(-0.532324\pi\)
−0.101373 + 0.994849i \(0.532324\pi\)
\(774\) −4.54561e73 −0.987865
\(775\) 7.61909e72 0.160217
\(776\) 2.24552e73 0.456924
\(777\) 2.75645e73 0.542768
\(778\) 1.02858e73 0.196001
\(779\) 9.56775e73 1.76443
\(780\) 2.54826e72 0.0454812
\(781\) 5.10976e73 0.882672
\(782\) −3.67511e73 −0.614466
\(783\) 6.92084e72 0.112004
\(784\) 2.85746e73 0.447631
\(785\) 6.79663e73 1.03066
\(786\) 3.35077e72 0.0491892
\(787\) −8.00870e73 −1.13817 −0.569084 0.822279i \(-0.692702\pi\)
−0.569084 + 0.822279i \(0.692702\pi\)
\(788\) 6.25040e72 0.0859981
\(789\) 3.92513e72 0.0522867
\(790\) 2.42898e73 0.313280
\(791\) −1.37849e74 −1.72149
\(792\) 2.10182e73 0.254157
\(793\) −3.14883e73 −0.368708
\(794\) 1.20978e74 1.37178
\(795\) −1.46253e73 −0.160599
\(796\) −1.57101e73 −0.167069
\(797\) −9.52170e72 −0.0980680 −0.0490340 0.998797i \(-0.515614\pi\)
−0.0490340 + 0.998797i \(0.515614\pi\)
\(798\) 2.25332e73 0.224776
\(799\) −8.08820e73 −0.781464
\(800\) −4.08275e72 −0.0382084
\(801\) 1.15845e74 1.05014
\(802\) −1.35839e74 −1.19283
\(803\) −6.43291e73 −0.547219
\(804\) 1.71077e73 0.140982
\(805\) −2.38675e74 −1.90551
\(806\) −3.77371e73 −0.291894
\(807\) −3.63440e72 −0.0272369
\(808\) −5.72183e73 −0.415476
\(809\) 2.01261e74 1.41603 0.708016 0.706197i \(-0.249591\pi\)
0.708016 + 0.706197i \(0.249591\pi\)
\(810\) 8.25571e73 0.562842
\(811\) 1.25620e74 0.829900 0.414950 0.909844i \(-0.363799\pi\)
0.414950 + 0.909844i \(0.363799\pi\)
\(812\) 4.02876e73 0.257925
\(813\) −2.28942e73 −0.142042
\(814\) 1.54130e74 0.926755
\(815\) −1.52730e74 −0.890034
\(816\) −5.50784e72 −0.0311087
\(817\) −2.72494e74 −1.49175
\(818\) −9.26577e72 −0.0491668
\(819\) 1.74705e74 0.898600
\(820\) −1.51881e74 −0.757271
\(821\) 1.17682e74 0.568799 0.284400 0.958706i \(-0.408206\pi\)
0.284400 + 0.958706i \(0.408206\pi\)
\(822\) −4.67547e73 −0.219076
\(823\) 4.09028e74 1.85805 0.929025 0.370018i \(-0.120649\pi\)
0.929025 + 0.370018i \(0.120649\pi\)
\(824\) −3.21368e73 −0.141533
\(825\) 6.94984e72 0.0296756
\(826\) −1.28234e74 −0.530898
\(827\) −2.32690e73 −0.0934086 −0.0467043 0.998909i \(-0.514872\pi\)
−0.0467043 + 0.998909i \(0.514872\pi\)
\(828\) −1.59865e74 −0.622271
\(829\) 4.64969e74 1.75503 0.877513 0.479553i \(-0.159201\pi\)
0.877513 + 0.479553i \(0.159201\pi\)
\(830\) 1.74638e74 0.639217
\(831\) −5.30684e73 −0.188369
\(832\) 2.02217e73 0.0696103
\(833\) −3.61735e74 −1.20766
\(834\) 5.08201e73 0.164551
\(835\) 1.48325e74 0.465810
\(836\) 1.25997e74 0.383796
\(837\) 9.09975e73 0.268863
\(838\) −3.83201e74 −1.09826
\(839\) 1.05353e74 0.292898 0.146449 0.989218i \(-0.453216\pi\)
0.146449 + 0.989218i \(0.453216\pi\)
\(840\) −3.57699e73 −0.0964709
\(841\) −3.45760e74 −0.904641
\(842\) 2.05424e74 0.521427
\(843\) −6.20770e72 −0.0152872
\(844\) 2.23006e74 0.532825
\(845\) −2.63476e74 −0.610793
\(846\) −3.51832e74 −0.791391
\(847\) 3.41486e74 0.745325
\(848\) −1.16059e74 −0.245801
\(849\) −7.92651e72 −0.0162906
\(850\) 5.16849e73 0.103082
\(851\) −1.17232e75 −2.26904
\(852\) −5.82466e73 −0.109411
\(853\) −8.76449e74 −1.59781 −0.798907 0.601454i \(-0.794588\pi\)
−0.798907 + 0.601454i \(0.794588\pi\)
\(854\) 4.42000e74 0.782073
\(855\) 5.13640e74 0.882110
\(856\) −9.17533e73 −0.152947
\(857\) −1.75132e74 −0.283370 −0.141685 0.989912i \(-0.545252\pi\)
−0.141685 + 0.989912i \(0.545252\pi\)
\(858\) −3.44223e73 −0.0540648
\(859\) 4.14489e74 0.631956 0.315978 0.948766i \(-0.397667\pi\)
0.315978 + 0.948766i \(0.397667\pi\)
\(860\) 4.32566e74 0.640238
\(861\) 3.66913e74 0.527210
\(862\) 5.09537e74 0.710789
\(863\) 1.46481e74 0.198383 0.0991917 0.995068i \(-0.468374\pi\)
0.0991917 + 0.995068i \(0.468374\pi\)
\(864\) −4.87618e73 −0.0641179
\(865\) 8.60695e74 1.09885
\(866\) 5.87810e73 0.0728670
\(867\) −8.35475e73 −0.100565
\(868\) 5.29715e74 0.619141
\(869\) −3.28110e74 −0.372405
\(870\) −3.23597e73 −0.0356667
\(871\) 7.95134e74 0.851093
\(872\) 3.66487e74 0.380968
\(873\) −1.23655e75 −1.24839
\(874\) −9.58339e74 −0.939675
\(875\) 1.88865e75 1.79865
\(876\) 7.33293e73 0.0678302
\(877\) −5.01962e74 −0.451005 −0.225503 0.974243i \(-0.572402\pi\)
−0.225503 + 0.974243i \(0.572402\pi\)
\(878\) 5.49826e73 0.0479861
\(879\) −1.18918e74 −0.100817
\(880\) −2.00011e74 −0.164720
\(881\) −1.02163e75 −0.817349 −0.408675 0.912680i \(-0.634009\pi\)
−0.408675 + 0.912680i \(0.634009\pi\)
\(882\) −1.57353e75 −1.22300
\(883\) −1.42175e75 −1.07356 −0.536780 0.843722i \(-0.680359\pi\)
−0.536780 + 0.843722i \(0.680359\pi\)
\(884\) −2.55994e74 −0.187800
\(885\) 1.02999e74 0.0734144
\(886\) 8.55152e74 0.592221
\(887\) −3.69071e74 −0.248346 −0.124173 0.992261i \(-0.539628\pi\)
−0.124173 + 0.992261i \(0.539628\pi\)
\(888\) −1.75694e74 −0.114875
\(889\) 4.77064e75 3.03097
\(890\) −1.10239e75 −0.680600
\(891\) −1.11519e75 −0.669066
\(892\) −1.18408e75 −0.690360
\(893\) −2.10912e75 −1.19506
\(894\) 1.28543e74 0.0707849
\(895\) −1.10766e75 −0.592815
\(896\) −2.83852e74 −0.147652
\(897\) 2.61818e74 0.132371
\(898\) 1.32436e74 0.0650820
\(899\) 4.79213e74 0.228905
\(900\) 2.24827e74 0.104391
\(901\) 1.46923e75 0.663143
\(902\) 2.05164e75 0.900189
\(903\) −1.04499e75 −0.445732
\(904\) 8.78640e74 0.364348
\(905\) 2.08176e75 0.839253
\(906\) 1.09984e74 0.0431084
\(907\) 2.44006e75 0.929853 0.464927 0.885349i \(-0.346081\pi\)
0.464927 + 0.885349i \(0.346081\pi\)
\(908\) −2.61564e74 −0.0969145
\(909\) 3.15087e75 1.13514
\(910\) −1.66252e75 −0.582386
\(911\) −4.25501e75 −1.44938 −0.724690 0.689076i \(-0.758017\pi\)
−0.724690 + 0.689076i \(0.758017\pi\)
\(912\) −1.43625e74 −0.0475731
\(913\) −2.35904e75 −0.759856
\(914\) 3.16833e75 0.992435
\(915\) −3.55022e74 −0.108148
\(916\) −2.34528e75 −0.694800
\(917\) −2.18608e75 −0.629867
\(918\) 6.17292e74 0.172983
\(919\) 4.21824e75 1.14970 0.574851 0.818258i \(-0.305060\pi\)
0.574851 + 0.818258i \(0.305060\pi\)
\(920\) 1.52130e75 0.403296
\(921\) −6.46326e74 −0.166660
\(922\) 2.09461e75 0.525369
\(923\) −2.70719e75 −0.660504
\(924\) 4.83186e74 0.114678
\(925\) 1.64869e75 0.380650
\(926\) 3.90080e75 0.876141
\(927\) 1.76969e75 0.386691
\(928\) −2.56790e74 −0.0545890
\(929\) −8.38446e75 −1.73410 −0.867050 0.498221i \(-0.833987\pi\)
−0.867050 + 0.498221i \(0.833987\pi\)
\(930\) −4.25476e74 −0.0856170
\(931\) −9.43279e75 −1.84681
\(932\) −2.45252e75 −0.467205
\(933\) 1.32651e75 0.245884
\(934\) 1.13438e75 0.204603
\(935\) 2.53201e75 0.444396
\(936\) −1.11356e75 −0.190186
\(937\) 3.50330e75 0.582261 0.291131 0.956683i \(-0.405969\pi\)
0.291131 + 0.956683i \(0.405969\pi\)
\(938\) −1.11613e76 −1.80527
\(939\) −3.01901e74 −0.0475217
\(940\) 3.34808e75 0.512903
\(941\) 1.26513e76 1.88625 0.943127 0.332433i \(-0.107870\pi\)
0.943127 + 0.332433i \(0.107870\pi\)
\(942\) 1.04655e75 0.151866
\(943\) −1.56048e76 −2.20400
\(944\) 8.17351e74 0.112363
\(945\) 4.00892e75 0.536434
\(946\) −5.84317e75 −0.761069
\(947\) 5.93978e75 0.753088 0.376544 0.926399i \(-0.377112\pi\)
0.376544 + 0.926399i \(0.377112\pi\)
\(948\) 3.74016e74 0.0461612
\(949\) 3.40821e75 0.409484
\(950\) 1.34776e75 0.157638
\(951\) −6.64033e74 −0.0756112
\(952\) 3.59338e75 0.398347
\(953\) 7.28890e75 0.786671 0.393335 0.919395i \(-0.371321\pi\)
0.393335 + 0.919395i \(0.371321\pi\)
\(954\) 6.39106e75 0.671567
\(955\) −4.45192e75 −0.455472
\(956\) −1.13815e74 −0.0113377
\(957\) 4.37120e74 0.0423980
\(958\) −7.81225e75 −0.737828
\(959\) 3.05033e76 2.80526
\(960\) 2.27995e74 0.0204178
\(961\) −5.16606e75 −0.450519
\(962\) −8.16592e75 −0.693491
\(963\) 5.05261e75 0.417874
\(964\) −6.15423e74 −0.0495689
\(965\) −2.30258e75 −0.180621
\(966\) −3.67513e75 −0.280774
\(967\) −2.01514e74 −0.0149945 −0.00749723 0.999972i \(-0.502386\pi\)
−0.00749723 + 0.999972i \(0.502386\pi\)
\(968\) −2.17661e75 −0.157746
\(969\) 1.81820e75 0.128347
\(970\) 1.17672e76 0.809084
\(971\) −2.14332e76 −1.43548 −0.717740 0.696311i \(-0.754823\pi\)
−0.717740 + 0.696311i \(0.754823\pi\)
\(972\) 4.05102e75 0.264286
\(973\) −3.31557e76 −2.10708
\(974\) 1.97585e75 0.122321
\(975\) −3.68208e74 −0.0222063
\(976\) −2.81728e75 −0.165523
\(977\) −2.17676e76 −1.24595 −0.622973 0.782243i \(-0.714076\pi\)
−0.622973 + 0.782243i \(0.714076\pi\)
\(978\) −2.35175e75 −0.131145
\(979\) 1.48913e76 0.809049
\(980\) 1.49739e76 0.792628
\(981\) −2.01815e76 −1.04086
\(982\) −7.10366e75 −0.356976
\(983\) 2.22369e75 0.108883 0.0544414 0.998517i \(-0.482662\pi\)
0.0544414 + 0.998517i \(0.482662\pi\)
\(984\) −2.33868e75 −0.111582
\(985\) 3.27539e75 0.152279
\(986\) 3.25079e75 0.147275
\(987\) −8.08826e75 −0.357082
\(988\) −6.67542e75 −0.287195
\(989\) 4.44434e76 1.86338
\(990\) 1.10141e76 0.450041
\(991\) 2.17380e76 0.865646 0.432823 0.901479i \(-0.357517\pi\)
0.432823 + 0.901479i \(0.357517\pi\)
\(992\) −3.37636e75 −0.131039
\(993\) −4.12949e75 −0.156204
\(994\) 3.80008e76 1.40101
\(995\) −8.23255e75 −0.295832
\(996\) 2.68909e75 0.0941874
\(997\) −4.09289e76 −1.39734 −0.698672 0.715442i \(-0.746225\pi\)
−0.698672 + 0.715442i \(0.746225\pi\)
\(998\) −1.58668e75 −0.0528031
\(999\) 1.96909e76 0.638773
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 2.52.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.52.a.b.1.1 2 1.1 even 1 trivial