Properties

Label 10.30.a.a.1.2
Level $10$
Weight $30$
Character 10.1
Self dual yes
Analytic conductor $53.278$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,30,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 30 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(53.2780423830\)
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: \( x^{2} - x - 2788867122 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-52809.2\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16384.0 q^{2} +7.68338e6 q^{3} +2.68435e8 q^{4} +6.10352e9 q^{5} +1.25884e11 q^{6} -1.48072e12 q^{7} +4.39805e12 q^{8} -9.59610e12 q^{9} +1.00000e14 q^{10} -1.49204e15 q^{11} +2.06249e15 q^{12} -1.87567e16 q^{13} -2.42601e16 q^{14} +4.68956e16 q^{15} +7.20576e16 q^{16} -5.52201e17 q^{17} -1.57223e17 q^{18} -3.63993e18 q^{19} +1.63840e18 q^{20} -1.13769e19 q^{21} -2.44455e19 q^{22} +1.19154e19 q^{23} +3.37918e19 q^{24} +3.72529e19 q^{25} -3.07309e20 q^{26} -6.01043e20 q^{27} -3.97477e20 q^{28} +1.49220e21 q^{29} +7.68338e20 q^{30} +2.12272e20 q^{31} +1.18059e21 q^{32} -1.14639e22 q^{33} -9.04727e21 q^{34} -9.03758e21 q^{35} -2.57593e21 q^{36} +4.36693e22 q^{37} -5.96365e22 q^{38} -1.44114e23 q^{39} +2.68435e22 q^{40} +1.24529e23 q^{41} -1.86399e23 q^{42} +4.89124e23 q^{43} -4.00515e23 q^{44} -5.85700e22 q^{45} +1.95221e23 q^{46} +3.20595e24 q^{47} +5.53646e23 q^{48} -1.02738e24 q^{49} +6.10352e23 q^{50} -4.24277e24 q^{51} -5.03495e24 q^{52} -1.61509e25 q^{53} -9.84750e24 q^{54} -9.10667e24 q^{55} -6.51226e24 q^{56} -2.79669e25 q^{57} +2.44481e25 q^{58} -6.74135e25 q^{59} +1.25884e25 q^{60} -4.59568e25 q^{61} +3.47787e24 q^{62} +1.42091e25 q^{63} +1.93428e25 q^{64} -1.14482e26 q^{65} -1.87824e26 q^{66} -3.04003e26 q^{67} -1.48230e26 q^{68} +9.15502e25 q^{69} -1.48072e26 q^{70} +8.77630e25 q^{71} -4.22041e25 q^{72} -1.54604e27 q^{73} +7.15478e26 q^{74} +2.86228e26 q^{75} -9.77085e26 q^{76} +2.20928e27 q^{77} -2.36117e27 q^{78} +8.64681e25 q^{79} +4.39805e26 q^{80} -3.95946e27 q^{81} +2.04028e27 q^{82} +8.92312e27 q^{83} -3.05396e27 q^{84} -3.37037e27 q^{85} +8.01381e27 q^{86} +1.14651e28 q^{87} -6.56204e27 q^{88} +4.85579e27 q^{89} -9.59610e26 q^{90} +2.77733e28 q^{91} +3.19851e27 q^{92} +1.63097e27 q^{93} +5.25263e28 q^{94} -2.22163e28 q^{95} +9.07093e27 q^{96} +6.50601e28 q^{97} -1.68327e28 q^{98} +1.43177e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32768 q^{2} - 3644748 q^{3} + 536870912 q^{4} + 12207031250 q^{5} - 59715551232 q^{6} - 2619992530316 q^{7} + 8796093022208 q^{8} + 50099928783786 q^{9} + 200000000000000 q^{10} - 609601386608016 q^{11}+ \cdots + 66\!\cdots\!12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16384.0 0.707107
\(3\) 7.68338e6 0.927457 0.463729 0.885977i \(-0.346511\pi\)
0.463729 + 0.885977i \(0.346511\pi\)
\(4\) 2.68435e8 0.500000
\(5\) 6.10352e9 0.447214
\(6\) 1.25884e11 0.655811
\(7\) −1.48072e12 −0.825183 −0.412592 0.910916i \(-0.635376\pi\)
−0.412592 + 0.910916i \(0.635376\pi\)
\(8\) 4.39805e12 0.353553
\(9\) −9.59610e12 −0.139823
\(10\) 1.00000e14 0.316228
\(11\) −1.49204e15 −1.18464 −0.592319 0.805704i \(-0.701787\pi\)
−0.592319 + 0.805704i \(0.701787\pi\)
\(12\) 2.06249e15 0.463729
\(13\) −1.87567e16 −1.32122 −0.660612 0.750727i \(-0.729703\pi\)
−0.660612 + 0.750727i \(0.729703\pi\)
\(14\) −2.42601e16 −0.583493
\(15\) 4.68956e16 0.414772
\(16\) 7.20576e16 0.250000
\(17\) −5.52201e17 −0.795405 −0.397702 0.917515i \(-0.630192\pi\)
−0.397702 + 0.917515i \(0.630192\pi\)
\(18\) −1.57223e17 −0.0988697
\(19\) −3.63993e18 −1.04512 −0.522559 0.852603i \(-0.675023\pi\)
−0.522559 + 0.852603i \(0.675023\pi\)
\(20\) 1.63840e18 0.223607
\(21\) −1.13769e19 −0.765322
\(22\) −2.44455e19 −0.837665
\(23\) 1.19154e19 0.214316 0.107158 0.994242i \(-0.465825\pi\)
0.107158 + 0.994242i \(0.465825\pi\)
\(24\) 3.37918e19 0.327906
\(25\) 3.72529e19 0.200000
\(26\) −3.07309e20 −0.934247
\(27\) −6.01043e20 −1.05714
\(28\) −3.97477e20 −0.412592
\(29\) 1.49220e21 0.931226 0.465613 0.884988i \(-0.345834\pi\)
0.465613 + 0.884988i \(0.345834\pi\)
\(30\) 7.68338e20 0.293288
\(31\) 2.12272e20 0.0503673 0.0251836 0.999683i \(-0.491983\pi\)
0.0251836 + 0.999683i \(0.491983\pi\)
\(32\) 1.18059e21 0.176777
\(33\) −1.14639e22 −1.09870
\(34\) −9.04727e21 −0.562436
\(35\) −9.03758e21 −0.369033
\(36\) −2.57593e21 −0.0699115
\(37\) 4.36693e22 0.796620 0.398310 0.917251i \(-0.369597\pi\)
0.398310 + 0.917251i \(0.369597\pi\)
\(38\) −5.96365e22 −0.739010
\(39\) −1.44114e23 −1.22538
\(40\) 2.68435e22 0.158114
\(41\) 1.24529e23 0.512749 0.256375 0.966578i \(-0.417472\pi\)
0.256375 + 0.966578i \(0.417472\pi\)
\(42\) −1.86399e23 −0.541164
\(43\) 4.89124e23 1.00955 0.504774 0.863252i \(-0.331576\pi\)
0.504774 + 0.863252i \(0.331576\pi\)
\(44\) −4.00515e23 −0.592319
\(45\) −5.85700e22 −0.0625307
\(46\) 1.95221e23 0.151544
\(47\) 3.20595e24 1.82195 0.910977 0.412456i \(-0.135329\pi\)
0.910977 + 0.412456i \(0.135329\pi\)
\(48\) 5.53646e23 0.231864
\(49\) −1.02738e24 −0.319073
\(50\) 6.10352e23 0.141421
\(51\) −4.24277e24 −0.737704
\(52\) −5.03495e24 −0.660612
\(53\) −1.61509e25 −1.60767 −0.803834 0.594854i \(-0.797210\pi\)
−0.803834 + 0.594854i \(0.797210\pi\)
\(54\) −9.84750e24 −0.747509
\(55\) −9.10667e24 −0.529786
\(56\) −6.51226e24 −0.291746
\(57\) −2.79669e25 −0.969302
\(58\) 2.44481e25 0.658476
\(59\) −6.74135e25 −1.41708 −0.708538 0.705673i \(-0.750645\pi\)
−0.708538 + 0.705673i \(0.750645\pi\)
\(60\) 1.25884e25 0.207386
\(61\) −4.59568e25 −0.595755 −0.297877 0.954604i \(-0.596279\pi\)
−0.297877 + 0.954604i \(0.596279\pi\)
\(62\) 3.47787e24 0.0356151
\(63\) 1.42091e25 0.115380
\(64\) 1.93428e25 0.125000
\(65\) −1.14482e26 −0.590870
\(66\) −1.87824e26 −0.776899
\(67\) −3.04003e26 −1.01110 −0.505548 0.862798i \(-0.668710\pi\)
−0.505548 + 0.862798i \(0.668710\pi\)
\(68\) −1.48230e26 −0.397702
\(69\) 9.15502e25 0.198769
\(70\) −1.48072e26 −0.260946
\(71\) 8.77630e25 0.125912 0.0629558 0.998016i \(-0.479947\pi\)
0.0629558 + 0.998016i \(0.479947\pi\)
\(72\) −4.22041e25 −0.0494349
\(73\) −1.54604e27 −1.48265 −0.741327 0.671144i \(-0.765803\pi\)
−0.741327 + 0.671144i \(0.765803\pi\)
\(74\) 7.15478e26 0.563295
\(75\) 2.86228e26 0.185491
\(76\) −9.77085e26 −0.522559
\(77\) 2.20928e27 0.977543
\(78\) −2.36117e27 −0.866474
\(79\) 8.64681e25 0.0263793 0.0131896 0.999913i \(-0.495801\pi\)
0.0131896 + 0.999913i \(0.495801\pi\)
\(80\) 4.39805e26 0.111803
\(81\) −3.95946e27 −0.840627
\(82\) 2.04028e27 0.362568
\(83\) 8.92312e27 1.33010 0.665051 0.746798i \(-0.268410\pi\)
0.665051 + 0.746798i \(0.268410\pi\)
\(84\) −3.05396e27 −0.382661
\(85\) −3.37037e27 −0.355716
\(86\) 8.01381e27 0.713858
\(87\) 1.14651e28 0.863672
\(88\) −6.56204e27 −0.418833
\(89\) 4.85579e27 0.263090 0.131545 0.991310i \(-0.458006\pi\)
0.131545 + 0.991310i \(0.458006\pi\)
\(90\) −9.59610e26 −0.0442159
\(91\) 2.77733e28 1.09025
\(92\) 3.19851e27 0.107158
\(93\) 1.63097e27 0.0467135
\(94\) 5.25263e28 1.28832
\(95\) −2.22163e28 −0.467391
\(96\) 9.07093e27 0.163953
\(97\) 6.50601e28 1.01187 0.505934 0.862572i \(-0.331148\pi\)
0.505934 + 0.862572i \(0.331148\pi\)
\(98\) −1.68327e28 −0.225619
\(99\) 1.43177e28 0.165639
\(100\) 1.00000e28 0.100000
\(101\) 2.13267e29 1.84614 0.923068 0.384638i \(-0.125674\pi\)
0.923068 + 0.384638i \(0.125674\pi\)
\(102\) −6.95136e28 −0.521635
\(103\) −8.52648e28 −0.555431 −0.277715 0.960663i \(-0.589577\pi\)
−0.277715 + 0.960663i \(0.589577\pi\)
\(104\) −8.24926e28 −0.467124
\(105\) −6.94391e28 −0.342262
\(106\) −2.64616e29 −1.13679
\(107\) −1.53318e29 −0.574814 −0.287407 0.957809i \(-0.592793\pi\)
−0.287407 + 0.957809i \(0.592793\pi\)
\(108\) −1.61341e29 −0.528569
\(109\) −4.55386e28 −0.130525 −0.0652627 0.997868i \(-0.520789\pi\)
−0.0652627 + 0.997868i \(0.520789\pi\)
\(110\) −1.49204e29 −0.374615
\(111\) 3.35528e29 0.738831
\(112\) −1.06697e29 −0.206296
\(113\) −9.49638e29 −1.61406 −0.807031 0.590508i \(-0.798927\pi\)
−0.807031 + 0.590508i \(0.798927\pi\)
\(114\) −4.58210e29 −0.685400
\(115\) 7.27256e28 0.0958448
\(116\) 4.00558e29 0.465613
\(117\) 1.79991e29 0.184738
\(118\) −1.10450e30 −1.00202
\(119\) 8.17654e29 0.656354
\(120\) 2.06249e29 0.146644
\(121\) 6.39862e29 0.403365
\(122\) −7.52957e29 −0.421262
\(123\) 9.56804e29 0.475553
\(124\) 5.69814e28 0.0251836
\(125\) 2.27374e29 0.0894427
\(126\) 2.32802e29 0.0815856
\(127\) −6.99268e28 −0.0218519 −0.0109260 0.999940i \(-0.503478\pi\)
−0.0109260 + 0.999940i \(0.503478\pi\)
\(128\) 3.16913e29 0.0883883
\(129\) 3.75813e30 0.936312
\(130\) −1.87567e30 −0.417808
\(131\) −1.54928e30 −0.308814 −0.154407 0.988007i \(-0.549347\pi\)
−0.154407 + 0.988007i \(0.549347\pi\)
\(132\) −3.07731e30 −0.549350
\(133\) 5.38970e30 0.862414
\(134\) −4.98078e30 −0.714953
\(135\) −3.66848e30 −0.472766
\(136\) −2.42861e30 −0.281218
\(137\) −5.90514e29 −0.0614868 −0.0307434 0.999527i \(-0.509787\pi\)
−0.0307434 + 0.999527i \(0.509787\pi\)
\(138\) 1.49996e30 0.140551
\(139\) 1.35271e31 1.14154 0.570769 0.821111i \(-0.306645\pi\)
0.570769 + 0.821111i \(0.306645\pi\)
\(140\) −2.42601e30 −0.184517
\(141\) 2.46325e31 1.68979
\(142\) 1.43791e30 0.0890329
\(143\) 2.79856e31 1.56517
\(144\) −6.91472e29 −0.0349557
\(145\) 9.10764e30 0.416457
\(146\) −2.53304e31 −1.04839
\(147\) −7.89378e30 −0.295927
\(148\) 1.17224e31 0.398310
\(149\) 3.78242e31 1.16565 0.582826 0.812597i \(-0.301947\pi\)
0.582826 + 0.812597i \(0.301947\pi\)
\(150\) 4.68956e30 0.131162
\(151\) −5.96443e31 −1.51497 −0.757483 0.652854i \(-0.773571\pi\)
−0.757483 + 0.652854i \(0.773571\pi\)
\(152\) −1.60086e31 −0.369505
\(153\) 5.29898e30 0.111216
\(154\) 3.61969e31 0.691227
\(155\) 1.29561e30 0.0225249
\(156\) −3.86854e31 −0.612690
\(157\) −6.30671e31 −0.910454 −0.455227 0.890375i \(-0.650442\pi\)
−0.455227 + 0.890375i \(0.650442\pi\)
\(158\) 1.41669e30 0.0186530
\(159\) −1.24093e32 −1.49104
\(160\) 7.20576e30 0.0790569
\(161\) −1.76433e31 −0.176850
\(162\) −6.48718e31 −0.594413
\(163\) −1.15204e29 −0.000965486 0 −0.000482743 1.00000i \(-0.500154\pi\)
−0.000482743 1.00000i \(0.500154\pi\)
\(164\) 3.34280e31 0.256375
\(165\) −6.99699e31 −0.491354
\(166\) 1.46196e32 0.940523
\(167\) −3.21378e32 −1.89508 −0.947539 0.319641i \(-0.896438\pi\)
−0.947539 + 0.319641i \(0.896438\pi\)
\(168\) −5.00361e31 −0.270582
\(169\) 1.50274e32 0.745635
\(170\) −5.52201e31 −0.251529
\(171\) 3.49291e31 0.146131
\(172\) 1.31298e32 0.504774
\(173\) 2.25727e32 0.797840 0.398920 0.916986i \(-0.369385\pi\)
0.398920 + 0.916986i \(0.369385\pi\)
\(174\) 1.87844e32 0.610708
\(175\) −5.51610e31 −0.165037
\(176\) −1.07513e32 −0.296159
\(177\) −5.17964e32 −1.31428
\(178\) 7.95572e31 0.186033
\(179\) −1.67639e32 −0.361415 −0.180707 0.983537i \(-0.557839\pi\)
−0.180707 + 0.983537i \(0.557839\pi\)
\(180\) −1.57223e31 −0.0312654
\(181\) 5.21905e32 0.957751 0.478875 0.877883i \(-0.341045\pi\)
0.478875 + 0.877883i \(0.341045\pi\)
\(182\) 4.55038e32 0.770925
\(183\) −3.53104e32 −0.552537
\(184\) 5.24043e31 0.0757720
\(185\) 2.66536e32 0.356259
\(186\) 2.67218e31 0.0330314
\(187\) 8.23904e32 0.942266
\(188\) 8.60590e32 0.910977
\(189\) 8.89975e32 0.872332
\(190\) −3.63993e32 −0.330495
\(191\) 2.11179e33 1.77692 0.888460 0.458955i \(-0.151776\pi\)
0.888460 + 0.458955i \(0.151776\pi\)
\(192\) 1.48618e32 0.115932
\(193\) −1.46578e33 −1.06045 −0.530224 0.847858i \(-0.677892\pi\)
−0.530224 + 0.847858i \(0.677892\pi\)
\(194\) 1.06594e33 0.715499
\(195\) −8.79605e32 −0.548006
\(196\) −2.75786e32 −0.159536
\(197\) −8.00038e32 −0.429883 −0.214942 0.976627i \(-0.568956\pi\)
−0.214942 + 0.976627i \(0.568956\pi\)
\(198\) 2.34582e32 0.117125
\(199\) −3.40186e33 −1.57887 −0.789436 0.613833i \(-0.789627\pi\)
−0.789436 + 0.613833i \(0.789627\pi\)
\(200\) 1.63840e32 0.0707107
\(201\) −2.33577e33 −0.937749
\(202\) 3.49416e33 1.30541
\(203\) −2.20952e33 −0.768432
\(204\) −1.13891e33 −0.368852
\(205\) 7.60065e32 0.229308
\(206\) −1.39698e33 −0.392749
\(207\) −1.14341e32 −0.0299662
\(208\) −1.35156e33 −0.330306
\(209\) 5.43090e33 1.23809
\(210\) −1.13769e33 −0.242016
\(211\) −1.67250e33 −0.332102 −0.166051 0.986117i \(-0.553102\pi\)
−0.166051 + 0.986117i \(0.553102\pi\)
\(212\) −4.33547e33 −0.803834
\(213\) 6.74316e32 0.116778
\(214\) −2.51196e33 −0.406455
\(215\) 2.98538e33 0.451483
\(216\) −2.64342e33 −0.373754
\(217\) −3.14315e32 −0.0415622
\(218\) −7.46104e32 −0.0922954
\(219\) −1.18788e34 −1.37510
\(220\) −2.44455e33 −0.264893
\(221\) 1.03575e34 1.05091
\(222\) 5.49729e33 0.522432
\(223\) 1.89519e34 1.68745 0.843726 0.536775i \(-0.180357\pi\)
0.843726 + 0.536775i \(0.180357\pi\)
\(224\) −1.74812e33 −0.145873
\(225\) −3.57483e32 −0.0279646
\(226\) −1.55589e34 −1.14131
\(227\) 2.37544e33 0.163444 0.0817222 0.996655i \(-0.473958\pi\)
0.0817222 + 0.996655i \(0.473958\pi\)
\(228\) −7.50731e33 −0.484651
\(229\) −3.16033e34 −1.91478 −0.957388 0.288804i \(-0.906742\pi\)
−0.957388 + 0.288804i \(0.906742\pi\)
\(230\) 1.19154e33 0.0677725
\(231\) 1.69747e34 0.906629
\(232\) 6.56275e33 0.329238
\(233\) 1.58340e34 0.746328 0.373164 0.927765i \(-0.378273\pi\)
0.373164 + 0.927765i \(0.378273\pi\)
\(234\) 2.94897e33 0.130629
\(235\) 1.95676e34 0.814803
\(236\) −1.80962e34 −0.708538
\(237\) 6.64367e32 0.0244657
\(238\) 1.33964e34 0.464113
\(239\) −1.69580e34 −0.552849 −0.276424 0.961036i \(-0.589150\pi\)
−0.276424 + 0.961036i \(0.589150\pi\)
\(240\) 3.37918e33 0.103693
\(241\) −6.52685e33 −0.188563 −0.0942815 0.995546i \(-0.530055\pi\)
−0.0942815 + 0.995546i \(0.530055\pi\)
\(242\) 1.04835e34 0.285222
\(243\) 1.08278e34 0.277492
\(244\) −1.23364e34 −0.297877
\(245\) −6.27066e33 −0.142694
\(246\) 1.56763e34 0.336267
\(247\) 6.82728e34 1.38084
\(248\) 9.33583e32 0.0178075
\(249\) 6.85597e34 1.23361
\(250\) 3.72529e33 0.0632456
\(251\) 8.19176e34 1.31253 0.656264 0.754532i \(-0.272136\pi\)
0.656264 + 0.754532i \(0.272136\pi\)
\(252\) 3.81423e33 0.0576898
\(253\) −1.77782e34 −0.253886
\(254\) −1.14568e33 −0.0154516
\(255\) −2.58958e34 −0.329911
\(256\) 5.19230e33 0.0625000
\(257\) −1.00126e35 −1.13899 −0.569493 0.821996i \(-0.692861\pi\)
−0.569493 + 0.821996i \(0.692861\pi\)
\(258\) 6.15731e34 0.662073
\(259\) −6.46618e34 −0.657357
\(260\) −3.07309e34 −0.295435
\(261\) −1.43193e34 −0.130207
\(262\) −2.53835e34 −0.218365
\(263\) −3.41235e34 −0.277776 −0.138888 0.990308i \(-0.544353\pi\)
−0.138888 + 0.990308i \(0.544353\pi\)
\(264\) −5.04187e34 −0.388449
\(265\) −9.85772e34 −0.718971
\(266\) 8.83048e34 0.609818
\(267\) 3.73088e34 0.244005
\(268\) −8.16051e34 −0.505548
\(269\) −3.29203e35 −1.93222 −0.966109 0.258133i \(-0.916893\pi\)
−0.966109 + 0.258133i \(0.916893\pi\)
\(270\) −6.01043e34 −0.334296
\(271\) −2.96956e35 −1.56544 −0.782721 0.622373i \(-0.786169\pi\)
−0.782721 + 0.622373i \(0.786169\pi\)
\(272\) −3.97903e34 −0.198851
\(273\) 2.13393e35 1.01116
\(274\) −9.67498e33 −0.0434778
\(275\) −5.55827e34 −0.236927
\(276\) 2.45753e34 0.0993843
\(277\) −1.65686e35 −0.635813 −0.317906 0.948122i \(-0.602980\pi\)
−0.317906 + 0.948122i \(0.602980\pi\)
\(278\) 2.21628e35 0.807189
\(279\) −2.03699e33 −0.00704250
\(280\) −3.97477e34 −0.130473
\(281\) 3.67502e35 1.14556 0.572781 0.819708i \(-0.305865\pi\)
0.572781 + 0.819708i \(0.305865\pi\)
\(282\) 4.03579e35 1.19486
\(283\) −2.61559e35 −0.735641 −0.367820 0.929897i \(-0.619896\pi\)
−0.367820 + 0.929897i \(0.619896\pi\)
\(284\) 2.35587e34 0.0629558
\(285\) −1.70697e35 −0.433485
\(286\) 4.58516e35 1.10674
\(287\) −1.84392e35 −0.423112
\(288\) −1.13291e34 −0.0247174
\(289\) −1.77042e35 −0.367332
\(290\) 1.49220e35 0.294479
\(291\) 4.99881e35 0.938465
\(292\) −4.15013e35 −0.741327
\(293\) 8.43142e35 1.43324 0.716622 0.697462i \(-0.245687\pi\)
0.716622 + 0.697462i \(0.245687\pi\)
\(294\) −1.29332e35 −0.209252
\(295\) −4.11460e35 −0.633735
\(296\) 1.92060e35 0.281648
\(297\) 8.96779e35 1.25232
\(298\) 6.19712e35 0.824240
\(299\) −2.23492e35 −0.283159
\(300\) 7.68338e34 0.0927457
\(301\) −7.24254e35 −0.833061
\(302\) −9.77212e35 −1.07124
\(303\) 1.63861e36 1.71221
\(304\) −2.62284e35 −0.261279
\(305\) −2.80498e35 −0.266430
\(306\) 8.68185e34 0.0786415
\(307\) −1.18090e36 −1.02025 −0.510124 0.860101i \(-0.670400\pi\)
−0.510124 + 0.860101i \(0.670400\pi\)
\(308\) 5.93050e35 0.488771
\(309\) −6.55121e35 −0.515138
\(310\) 2.12272e34 0.0159275
\(311\) 1.04336e36 0.747149 0.373574 0.927600i \(-0.378132\pi\)
0.373574 + 0.927600i \(0.378132\pi\)
\(312\) −6.33822e35 −0.433237
\(313\) 2.76507e36 1.80432 0.902161 0.431400i \(-0.141980\pi\)
0.902161 + 0.431400i \(0.141980\pi\)
\(314\) −1.03329e36 −0.643788
\(315\) 8.67255e34 0.0515993
\(316\) 2.32111e34 0.0131896
\(317\) 2.19277e36 1.19024 0.595119 0.803638i \(-0.297105\pi\)
0.595119 + 0.803638i \(0.297105\pi\)
\(318\) −2.03315e36 −1.05433
\(319\) −2.22641e36 −1.10316
\(320\) 1.18059e35 0.0559017
\(321\) −1.17800e36 −0.533116
\(322\) −2.89067e35 −0.125052
\(323\) 2.00997e36 0.831292
\(324\) −1.06286e36 −0.420313
\(325\) −6.98740e35 −0.264245
\(326\) −1.88750e33 −0.000682702 0
\(327\) −3.49890e35 −0.121057
\(328\) 5.47685e35 0.181284
\(329\) −4.74710e36 −1.50345
\(330\) −1.14639e36 −0.347440
\(331\) 3.32856e35 0.0965496 0.0482748 0.998834i \(-0.484628\pi\)
0.0482748 + 0.998834i \(0.484628\pi\)
\(332\) 2.39528e36 0.665051
\(333\) −4.19055e35 −0.111386
\(334\) −5.26545e36 −1.34002
\(335\) −1.85548e36 −0.452176
\(336\) −8.19792e35 −0.191331
\(337\) 4.57852e36 1.02351 0.511755 0.859132i \(-0.328996\pi\)
0.511755 + 0.859132i \(0.328996\pi\)
\(338\) 2.46209e36 0.527244
\(339\) −7.29642e36 −1.49697
\(340\) −9.04727e35 −0.177858
\(341\) −3.16718e35 −0.0596670
\(342\) 5.72278e35 0.103331
\(343\) 6.28903e36 1.08848
\(344\) 2.15119e36 0.356929
\(345\) 5.58778e35 0.0888920
\(346\) 3.69832e36 0.564158
\(347\) 3.69118e36 0.539993 0.269997 0.962861i \(-0.412977\pi\)
0.269997 + 0.962861i \(0.412977\pi\)
\(348\) 3.07764e36 0.431836
\(349\) −2.78937e36 −0.375438 −0.187719 0.982223i \(-0.560109\pi\)
−0.187719 + 0.982223i \(0.560109\pi\)
\(350\) −9.03758e35 −0.116699
\(351\) 1.12736e37 1.39672
\(352\) −1.76149e36 −0.209416
\(353\) −1.23430e37 −1.40827 −0.704135 0.710066i \(-0.748665\pi\)
−0.704135 + 0.710066i \(0.748665\pi\)
\(354\) −8.48631e36 −0.929334
\(355\) 5.35663e35 0.0563093
\(356\) 1.30347e36 0.131545
\(357\) 6.28234e36 0.608741
\(358\) −2.74660e36 −0.255559
\(359\) 1.60585e36 0.143494 0.0717472 0.997423i \(-0.477143\pi\)
0.0717472 + 0.997423i \(0.477143\pi\)
\(360\) −2.57593e35 −0.0221079
\(361\) 1.11924e36 0.0922713
\(362\) 8.55090e36 0.677232
\(363\) 4.91630e36 0.374104
\(364\) 7.45533e36 0.545126
\(365\) −9.43629e36 −0.663063
\(366\) −5.78525e36 −0.390703
\(367\) −3.15301e36 −0.204677 −0.102338 0.994750i \(-0.532632\pi\)
−0.102338 + 0.994750i \(0.532632\pi\)
\(368\) 8.58592e35 0.0535789
\(369\) −1.19499e36 −0.0716941
\(370\) 4.36693e36 0.251913
\(371\) 2.39149e37 1.32662
\(372\) 4.37810e35 0.0233568
\(373\) 1.47949e37 0.759162 0.379581 0.925158i \(-0.376068\pi\)
0.379581 + 0.925158i \(0.376068\pi\)
\(374\) 1.34988e37 0.666283
\(375\) 1.74700e36 0.0829543
\(376\) 1.40999e37 0.644158
\(377\) −2.79886e37 −1.23036
\(378\) 1.45813e37 0.616832
\(379\) −3.91070e37 −1.59216 −0.796079 0.605193i \(-0.793096\pi\)
−0.796079 + 0.605193i \(0.793096\pi\)
\(380\) −5.96365e36 −0.233695
\(381\) −5.37274e35 −0.0202667
\(382\) 3.45996e37 1.25647
\(383\) 1.25775e37 0.439757 0.219879 0.975527i \(-0.429434\pi\)
0.219879 + 0.975527i \(0.429434\pi\)
\(384\) 2.43496e36 0.0819764
\(385\) 1.34844e37 0.437170
\(386\) −2.40154e37 −0.749850
\(387\) −4.69369e36 −0.141158
\(388\) 1.74644e37 0.505934
\(389\) −5.78321e37 −1.61398 −0.806992 0.590562i \(-0.798906\pi\)
−0.806992 + 0.590562i \(0.798906\pi\)
\(390\) −1.44114e37 −0.387499
\(391\) −6.57968e36 −0.170468
\(392\) −4.51849e36 −0.112809
\(393\) −1.19037e37 −0.286412
\(394\) −1.31078e37 −0.303973
\(395\) 5.27759e35 0.0117972
\(396\) 3.84339e36 0.0828197
\(397\) −6.66269e37 −1.38416 −0.692081 0.721820i \(-0.743306\pi\)
−0.692081 + 0.721820i \(0.743306\pi\)
\(398\) −5.57361e37 −1.11643
\(399\) 4.14111e37 0.799852
\(400\) 2.68435e36 0.0500000
\(401\) −8.06663e37 −1.44910 −0.724551 0.689221i \(-0.757953\pi\)
−0.724551 + 0.689221i \(0.757953\pi\)
\(402\) −3.82692e37 −0.663088
\(403\) −3.98152e36 −0.0665465
\(404\) 5.72484e37 0.923068
\(405\) −2.41666e37 −0.375940
\(406\) −3.62008e37 −0.543363
\(407\) −6.51562e37 −0.943706
\(408\) −1.86599e37 −0.260818
\(409\) 1.28165e38 1.72895 0.864473 0.502679i \(-0.167652\pi\)
0.864473 + 0.502679i \(0.167652\pi\)
\(410\) 1.24529e37 0.162145
\(411\) −4.53714e36 −0.0570264
\(412\) −2.28881e37 −0.277715
\(413\) 9.98203e37 1.16935
\(414\) −1.87336e36 −0.0211893
\(415\) 5.44624e37 0.594839
\(416\) −2.21439e37 −0.233562
\(417\) 1.03934e38 1.05873
\(418\) 8.89799e37 0.875459
\(419\) 1.46843e38 1.39556 0.697780 0.716312i \(-0.254171\pi\)
0.697780 + 0.716312i \(0.254171\pi\)
\(420\) −1.86399e37 −0.171131
\(421\) 2.12573e36 0.0188546 0.00942732 0.999956i \(-0.496999\pi\)
0.00942732 + 0.999956i \(0.496999\pi\)
\(422\) −2.74023e37 −0.234832
\(423\) −3.07646e37 −0.254751
\(424\) −7.10324e37 −0.568396
\(425\) −2.05711e37 −0.159081
\(426\) 1.10480e37 0.0825742
\(427\) 6.80490e37 0.491607
\(428\) −4.11559e37 −0.287407
\(429\) 2.15024e38 1.45163
\(430\) 4.89124e37 0.319247
\(431\) 8.37130e37 0.528290 0.264145 0.964483i \(-0.414910\pi\)
0.264145 + 0.964483i \(0.414910\pi\)
\(432\) −4.33097e37 −0.264284
\(433\) −1.09694e38 −0.647305 −0.323652 0.946176i \(-0.604911\pi\)
−0.323652 + 0.946176i \(0.604911\pi\)
\(434\) −5.14974e36 −0.0293889
\(435\) 6.99775e37 0.386246
\(436\) −1.22242e37 −0.0652627
\(437\) −4.33710e37 −0.223985
\(438\) −1.94623e38 −0.972341
\(439\) 1.66935e38 0.806885 0.403442 0.915005i \(-0.367814\pi\)
0.403442 + 0.915005i \(0.367814\pi\)
\(440\) −4.00515e37 −0.187308
\(441\) 9.85889e36 0.0446137
\(442\) 1.69696e38 0.743104
\(443\) −3.71165e38 −1.57294 −0.786472 0.617626i \(-0.788094\pi\)
−0.786472 + 0.617626i \(0.788094\pi\)
\(444\) 9.00675e37 0.369416
\(445\) 2.96374e37 0.117657
\(446\) 3.10507e38 1.19321
\(447\) 2.90618e38 1.08109
\(448\) −2.86412e37 −0.103148
\(449\) −3.75572e38 −1.30955 −0.654774 0.755825i \(-0.727236\pi\)
−0.654774 + 0.755825i \(0.727236\pi\)
\(450\) −5.85700e36 −0.0197739
\(451\) −1.85802e38 −0.607422
\(452\) −2.54916e38 −0.807031
\(453\) −4.58270e38 −1.40507
\(454\) 3.89193e37 0.115573
\(455\) 1.69515e38 0.487576
\(456\) −1.23000e38 −0.342700
\(457\) −2.86202e38 −0.772482 −0.386241 0.922398i \(-0.626227\pi\)
−0.386241 + 0.922398i \(0.626227\pi\)
\(458\) −5.17789e38 −1.35395
\(459\) 3.31897e38 0.840852
\(460\) 1.95221e37 0.0479224
\(461\) −3.46285e38 −0.823703 −0.411851 0.911251i \(-0.635118\pi\)
−0.411851 + 0.911251i \(0.635118\pi\)
\(462\) 2.78114e38 0.641084
\(463\) 5.93445e38 1.32573 0.662867 0.748738i \(-0.269340\pi\)
0.662867 + 0.748738i \(0.269340\pi\)
\(464\) 1.07524e38 0.232806
\(465\) 9.95464e36 0.0208909
\(466\) 2.59424e38 0.527734
\(467\) 9.62200e37 0.189745 0.0948725 0.995489i \(-0.469756\pi\)
0.0948725 + 0.995489i \(0.469756\pi\)
\(468\) 4.83159e37 0.0923688
\(469\) 4.50141e38 0.834340
\(470\) 3.20595e38 0.576153
\(471\) −4.84569e38 −0.844408
\(472\) −2.96488e38 −0.501012
\(473\) −7.29791e38 −1.19595
\(474\) 1.08850e37 0.0172998
\(475\) −1.35598e38 −0.209024
\(476\) 2.19487e38 0.328177
\(477\) 1.54986e38 0.224789
\(478\) −2.77841e38 −0.390923
\(479\) −4.88061e38 −0.666207 −0.333104 0.942890i \(-0.608096\pi\)
−0.333104 + 0.942890i \(0.608096\pi\)
\(480\) 5.53646e37 0.0733219
\(481\) −8.19090e38 −1.05251
\(482\) −1.06936e38 −0.133334
\(483\) −1.35560e38 −0.164020
\(484\) 1.71762e38 0.201683
\(485\) 3.97095e38 0.452522
\(486\) 1.77403e38 0.196216
\(487\) 1.54915e38 0.166312 0.0831558 0.996537i \(-0.473500\pi\)
0.0831558 + 0.996537i \(0.473500\pi\)
\(488\) −2.02120e38 −0.210631
\(489\) −8.85153e35 −0.000895447 0
\(490\) −1.02738e38 −0.100900
\(491\) −8.20640e37 −0.0782476 −0.0391238 0.999234i \(-0.512457\pi\)
−0.0391238 + 0.999234i \(0.512457\pi\)
\(492\) 2.56840e38 0.237776
\(493\) −8.23993e38 −0.740701
\(494\) 1.11858e39 0.976398
\(495\) 8.73885e37 0.0740762
\(496\) 1.52958e37 0.0125918
\(497\) −1.29952e38 −0.103900
\(498\) 1.12328e39 0.872295
\(499\) 2.36296e39 1.78237 0.891187 0.453637i \(-0.149874\pi\)
0.891187 + 0.453637i \(0.149874\pi\)
\(500\) 6.10352e37 0.0447214
\(501\) −2.46927e39 −1.75760
\(502\) 1.34214e39 0.928097
\(503\) 1.38727e39 0.932023 0.466011 0.884779i \(-0.345691\pi\)
0.466011 + 0.884779i \(0.345691\pi\)
\(504\) 6.24923e37 0.0407928
\(505\) 1.30168e39 0.825617
\(506\) −2.91277e38 −0.179525
\(507\) 1.15461e39 0.691545
\(508\) −1.87708e37 −0.0109260
\(509\) −2.30259e39 −1.30259 −0.651297 0.758823i \(-0.725775\pi\)
−0.651297 + 0.758823i \(0.725775\pi\)
\(510\) −4.24277e38 −0.233282
\(511\) 2.28925e39 1.22346
\(512\) 8.50706e37 0.0441942
\(513\) 2.18775e39 1.10483
\(514\) −1.64047e39 −0.805385
\(515\) −5.20415e38 −0.248396
\(516\) 1.00881e39 0.468156
\(517\) −4.78339e39 −2.15836
\(518\) −1.05942e39 −0.464822
\(519\) 1.73435e39 0.739962
\(520\) −5.03495e38 −0.208904
\(521\) 1.16195e39 0.468857 0.234429 0.972133i \(-0.424678\pi\)
0.234429 + 0.972133i \(0.424678\pi\)
\(522\) −2.34607e38 −0.0920701
\(523\) 4.58242e39 1.74912 0.874561 0.484915i \(-0.161149\pi\)
0.874561 + 0.484915i \(0.161149\pi\)
\(524\) −4.15883e38 −0.154407
\(525\) −4.23823e38 −0.153064
\(526\) −5.59080e38 −0.196418
\(527\) −1.17217e38 −0.0400624
\(528\) −8.26059e38 −0.274675
\(529\) −2.94908e39 −0.954069
\(530\) −1.61509e39 −0.508389
\(531\) 6.46907e38 0.198140
\(532\) 1.44679e39 0.431207
\(533\) −2.33575e39 −0.677457
\(534\) 6.11268e38 0.172537
\(535\) −9.35777e38 −0.257065
\(536\) −1.33702e39 −0.357477
\(537\) −1.28803e39 −0.335197
\(538\) −5.39367e39 −1.36628
\(539\) 1.53290e39 0.377986
\(540\) −9.84750e38 −0.236383
\(541\) 3.15300e39 0.736824 0.368412 0.929663i \(-0.379902\pi\)
0.368412 + 0.929663i \(0.379902\pi\)
\(542\) −4.86532e39 −1.10693
\(543\) 4.01000e39 0.888273
\(544\) −6.51924e38 −0.140609
\(545\) −2.77945e38 −0.0583728
\(546\) 3.49622e39 0.715000
\(547\) −1.24403e39 −0.247750 −0.123875 0.992298i \(-0.539532\pi\)
−0.123875 + 0.992298i \(0.539532\pi\)
\(548\) −1.58515e38 −0.0307434
\(549\) 4.41006e38 0.0833002
\(550\) −9.10667e38 −0.167533
\(551\) −5.43148e39 −0.973241
\(552\) 4.02642e38 0.0702753
\(553\) −1.28035e38 −0.0217677
\(554\) −2.71460e39 −0.449588
\(555\) 2.04790e39 0.330415
\(556\) 3.63116e39 0.570769
\(557\) −7.59373e39 −1.16293 −0.581467 0.813570i \(-0.697521\pi\)
−0.581467 + 0.813570i \(0.697521\pi\)
\(558\) −3.33740e37 −0.00497980
\(559\) −9.17434e39 −1.33384
\(560\) −6.51226e38 −0.0922583
\(561\) 6.33037e39 0.873911
\(562\) 6.02116e39 0.810035
\(563\) −8.33986e38 −0.109342 −0.0546710 0.998504i \(-0.517411\pi\)
−0.0546710 + 0.998504i \(0.517411\pi\)
\(564\) 6.61224e39 0.844893
\(565\) −5.79613e39 −0.721831
\(566\) −4.28538e39 −0.520177
\(567\) 5.86284e39 0.693671
\(568\) 3.85986e38 0.0445164
\(569\) −1.07248e40 −1.20576 −0.602879 0.797833i \(-0.705980\pi\)
−0.602879 + 0.797833i \(0.705980\pi\)
\(570\) −2.79669e39 −0.306520
\(571\) 1.45529e40 1.55498 0.777490 0.628895i \(-0.216493\pi\)
0.777490 + 0.628895i \(0.216493\pi\)
\(572\) 7.51233e39 0.782586
\(573\) 1.62257e40 1.64802
\(574\) −3.02108e39 −0.299185
\(575\) 4.43882e38 0.0428631
\(576\) −1.85616e38 −0.0174779
\(577\) 1.89246e40 1.73771 0.868855 0.495067i \(-0.164856\pi\)
0.868855 + 0.495067i \(0.164856\pi\)
\(578\) −2.90066e39 −0.259743
\(579\) −1.12622e40 −0.983520
\(580\) 2.44481e39 0.208228
\(581\) −1.32126e40 −1.09758
\(582\) 8.19005e39 0.663595
\(583\) 2.40977e40 1.90450
\(584\) −6.79956e39 −0.524197
\(585\) 1.09858e39 0.0826171
\(586\) 1.38140e40 1.01346
\(587\) 1.86692e39 0.133621 0.0668104 0.997766i \(-0.478718\pi\)
0.0668104 + 0.997766i \(0.478718\pi\)
\(588\) −2.11897e39 −0.147963
\(589\) −7.72655e38 −0.0526398
\(590\) −6.74135e39 −0.448119
\(591\) −6.14700e39 −0.398698
\(592\) 3.14670e39 0.199155
\(593\) 2.24511e40 1.38658 0.693289 0.720659i \(-0.256161\pi\)
0.693289 + 0.720659i \(0.256161\pi\)
\(594\) 1.46928e40 0.885527
\(595\) 4.99056e39 0.293531
\(596\) 1.01534e40 0.582826
\(597\) −2.61378e40 −1.46434
\(598\) −3.66170e39 −0.200224
\(599\) −1.43934e40 −0.768198 −0.384099 0.923292i \(-0.625488\pi\)
−0.384099 + 0.923292i \(0.625488\pi\)
\(600\) 1.25884e39 0.0655811
\(601\) −3.10527e40 −1.57914 −0.789569 0.613662i \(-0.789696\pi\)
−0.789569 + 0.613662i \(0.789696\pi\)
\(602\) −1.18662e40 −0.589063
\(603\) 2.91724e39 0.141374
\(604\) −1.60106e40 −0.757483
\(605\) 3.90541e39 0.180391
\(606\) 2.68470e40 1.21072
\(607\) −3.10127e39 −0.136554 −0.0682769 0.997666i \(-0.521750\pi\)
−0.0682769 + 0.997666i \(0.521750\pi\)
\(608\) −4.29727e39 −0.184752
\(609\) −1.69766e40 −0.712688
\(610\) −4.59568e39 −0.188394
\(611\) −6.01329e40 −2.40721
\(612\) 1.42243e39 0.0556079
\(613\) −3.61509e38 −0.0138020 −0.00690101 0.999976i \(-0.502197\pi\)
−0.00690101 + 0.999976i \(0.502197\pi\)
\(614\) −1.93478e40 −0.721424
\(615\) 5.83987e39 0.212674
\(616\) 9.71653e39 0.345614
\(617\) −1.26407e40 −0.439173 −0.219586 0.975593i \(-0.570471\pi\)
−0.219586 + 0.975593i \(0.570471\pi\)
\(618\) −1.07335e40 −0.364258
\(619\) 8.23888e39 0.273120 0.136560 0.990632i \(-0.456395\pi\)
0.136560 + 0.990632i \(0.456395\pi\)
\(620\) 3.47787e38 0.0112625
\(621\) −7.16165e39 −0.226561
\(622\) 1.70944e40 0.528314
\(623\) −7.19004e39 −0.217097
\(624\) −1.03845e40 −0.306345
\(625\) 1.38778e39 0.0400000
\(626\) 4.53030e40 1.27585
\(627\) 4.17277e40 1.14827
\(628\) −1.69295e40 −0.455227
\(629\) −2.41142e40 −0.633635
\(630\) 1.42091e39 0.0364862
\(631\) 5.22318e40 1.31072 0.655359 0.755317i \(-0.272517\pi\)
0.655359 + 0.755317i \(0.272517\pi\)
\(632\) 3.80291e38 0.00932648
\(633\) −1.28505e40 −0.308011
\(634\) 3.59263e40 0.841625
\(635\) −4.26799e38 −0.00977247
\(636\) −3.33111e40 −0.745522
\(637\) 1.92703e40 0.421567
\(638\) −3.64775e40 −0.780055
\(639\) −8.42183e38 −0.0176053
\(640\) 1.93428e39 0.0395285
\(641\) −4.31938e40 −0.862940 −0.431470 0.902127i \(-0.642005\pi\)
−0.431470 + 0.902127i \(0.642005\pi\)
\(642\) −1.93003e40 −0.376970
\(643\) 8.02353e40 1.53217 0.766084 0.642740i \(-0.222202\pi\)
0.766084 + 0.642740i \(0.222202\pi\)
\(644\) −4.73608e39 −0.0884248
\(645\) 2.29378e40 0.418731
\(646\) 3.29314e40 0.587812
\(647\) −8.30129e40 −1.44888 −0.724442 0.689336i \(-0.757902\pi\)
−0.724442 + 0.689336i \(0.757902\pi\)
\(648\) −1.74139e40 −0.297206
\(649\) 1.00583e41 1.67872
\(650\) −1.14482e40 −0.186849
\(651\) −2.41500e39 −0.0385472
\(652\) −3.09247e37 −0.000482743 0
\(653\) −6.89266e40 −1.05232 −0.526158 0.850387i \(-0.676368\pi\)
−0.526158 + 0.850387i \(0.676368\pi\)
\(654\) −5.73260e39 −0.0856001
\(655\) −9.45608e39 −0.138106
\(656\) 8.97327e39 0.128187
\(657\) 1.48360e40 0.207309
\(658\) −7.77765e40 −1.06310
\(659\) 4.28089e40 0.572395 0.286198 0.958171i \(-0.407609\pi\)
0.286198 + 0.958171i \(0.407609\pi\)
\(660\) −1.87824e40 −0.245677
\(661\) −3.05389e40 −0.390780 −0.195390 0.980726i \(-0.562597\pi\)
−0.195390 + 0.980726i \(0.562597\pi\)
\(662\) 5.45351e39 0.0682709
\(663\) 7.95802e40 0.974673
\(664\) 3.92443e40 0.470262
\(665\) 3.28961e40 0.385683
\(666\) −6.86580e39 −0.0787616
\(667\) 1.77801e40 0.199576
\(668\) −8.62692e40 −0.947539
\(669\) 1.45614e41 1.56504
\(670\) −3.04003e40 −0.319737
\(671\) 6.85692e40 0.705753
\(672\) −1.34315e40 −0.135291
\(673\) 1.17559e41 1.15887 0.579437 0.815017i \(-0.303273\pi\)
0.579437 + 0.815017i \(0.303273\pi\)
\(674\) 7.50145e40 0.723730
\(675\) −2.23906e40 −0.211427
\(676\) 4.03389e40 0.372818
\(677\) 8.31378e40 0.752078 0.376039 0.926604i \(-0.377286\pi\)
0.376039 + 0.926604i \(0.377286\pi\)
\(678\) −1.19545e41 −1.05852
\(679\) −9.63355e40 −0.834977
\(680\) −1.48230e40 −0.125765
\(681\) 1.82514e40 0.151588
\(682\) −5.18911e39 −0.0421909
\(683\) −6.88330e40 −0.547894 −0.273947 0.961745i \(-0.588329\pi\)
−0.273947 + 0.961745i \(0.588329\pi\)
\(684\) 9.37621e39 0.0730657
\(685\) −3.60421e39 −0.0274977
\(686\) 1.03040e41 0.769669
\(687\) −2.42820e41 −1.77587
\(688\) 3.52451e40 0.252387
\(689\) 3.02937e41 2.12409
\(690\) 9.15502e39 0.0628561
\(691\) −2.86827e41 −1.92836 −0.964182 0.265243i \(-0.914548\pi\)
−0.964182 + 0.265243i \(0.914548\pi\)
\(692\) 6.05932e40 0.398920
\(693\) −2.12005e40 −0.136683
\(694\) 6.04764e40 0.381833
\(695\) 8.25629e40 0.510511
\(696\) 5.04241e40 0.305354
\(697\) −6.87651e40 −0.407843
\(698\) −4.57011e40 −0.265475
\(699\) 1.21659e41 0.692188
\(700\) −1.48072e40 −0.0825183
\(701\) −3.31493e41 −1.80952 −0.904759 0.425923i \(-0.859949\pi\)
−0.904759 + 0.425923i \(0.859949\pi\)
\(702\) 1.84706e41 0.987627
\(703\) −1.58953e41 −0.832562
\(704\) −2.88602e40 −0.148080
\(705\) 1.50345e41 0.755695
\(706\) −2.02227e41 −0.995797
\(707\) −3.15788e41 −1.52340
\(708\) −1.39040e41 −0.657138
\(709\) −3.21463e41 −1.48854 −0.744271 0.667878i \(-0.767203\pi\)
−0.744271 + 0.667878i \(0.767203\pi\)
\(710\) 8.77630e39 0.0398167
\(711\) −8.29756e38 −0.00368843
\(712\) 2.13560e40 0.0930164
\(713\) 2.52930e39 0.0107945
\(714\) 1.02930e41 0.430445
\(715\) 1.70811e41 0.699966
\(716\) −4.50002e40 −0.180707
\(717\) −1.30295e41 −0.512743
\(718\) 2.63102e40 0.101466
\(719\) −1.53997e41 −0.582029 −0.291015 0.956719i \(-0.593993\pi\)
−0.291015 + 0.956719i \(0.593993\pi\)
\(720\) −4.22041e39 −0.0156327
\(721\) 1.26253e41 0.458332
\(722\) 1.83375e40 0.0652457
\(723\) −5.01482e40 −0.174884
\(724\) 1.40098e41 0.478875
\(725\) 5.55886e40 0.186245
\(726\) 8.05487e40 0.264532
\(727\) −2.73405e40 −0.0880153 −0.0440076 0.999031i \(-0.514013\pi\)
−0.0440076 + 0.999031i \(0.514013\pi\)
\(728\) 1.22148e41 0.385462
\(729\) 3.54933e41 1.09799
\(730\) −1.54604e41 −0.468856
\(731\) −2.70095e41 −0.802999
\(732\) −9.47855e40 −0.276269
\(733\) −7.89799e40 −0.225688 −0.112844 0.993613i \(-0.535996\pi\)
−0.112844 + 0.993613i \(0.535996\pi\)
\(734\) −5.16590e40 −0.144728
\(735\) −4.81798e40 −0.132342
\(736\) 1.40672e40 0.0378860
\(737\) 4.53583e41 1.19778
\(738\) −1.95788e40 −0.0506954
\(739\) −4.82157e41 −1.22418 −0.612088 0.790789i \(-0.709670\pi\)
−0.612088 + 0.790789i \(0.709670\pi\)
\(740\) 7.15478e40 0.178130
\(741\) 5.24566e41 1.28067
\(742\) 3.91822e41 0.938062
\(743\) 2.55089e41 0.598900 0.299450 0.954112i \(-0.403197\pi\)
0.299450 + 0.954112i \(0.403197\pi\)
\(744\) 7.17307e39 0.0165157
\(745\) 2.30861e41 0.521295
\(746\) 2.42400e41 0.536809
\(747\) −8.56272e40 −0.185979
\(748\) 2.21165e41 0.471133
\(749\) 2.27020e41 0.474327
\(750\) 2.86228e40 0.0586576
\(751\) −4.47925e40 −0.0900380 −0.0450190 0.998986i \(-0.514335\pi\)
−0.0450190 + 0.998986i \(0.514335\pi\)
\(752\) 2.31013e41 0.455489
\(753\) 6.29404e41 1.21731
\(754\) −4.58565e41 −0.869995
\(755\) −3.64040e41 −0.677514
\(756\) 2.38901e41 0.436166
\(757\) −1.78881e41 −0.320387 −0.160194 0.987086i \(-0.551212\pi\)
−0.160194 + 0.987086i \(0.551212\pi\)
\(758\) −6.40729e41 −1.12583
\(759\) −1.36596e41 −0.235469
\(760\) −9.77085e40 −0.165248
\(761\) 4.40276e40 0.0730546 0.0365273 0.999333i \(-0.488370\pi\)
0.0365273 + 0.999333i \(0.488370\pi\)
\(762\) −8.80269e39 −0.0143307
\(763\) 6.74297e40 0.107707
\(764\) 5.66880e41 0.888460
\(765\) 3.23424e40 0.0497372
\(766\) 2.06070e41 0.310955
\(767\) 1.26445e42 1.87228
\(768\) 3.98944e40 0.0579661
\(769\) −1.76688e41 −0.251927 −0.125964 0.992035i \(-0.540202\pi\)
−0.125964 + 0.992035i \(0.540202\pi\)
\(770\) 2.20928e41 0.309126
\(771\) −7.69309e41 −1.05636
\(772\) −3.93469e41 −0.530224
\(773\) −1.91963e41 −0.253873 −0.126936 0.991911i \(-0.540514\pi\)
−0.126936 + 0.991911i \(0.540514\pi\)
\(774\) −7.69014e40 −0.0998137
\(775\) 7.90776e39 0.0100735
\(776\) 2.86137e41 0.357750
\(777\) −4.96821e41 −0.609671
\(778\) −9.47521e41 −1.14126
\(779\) −4.53277e41 −0.535883
\(780\) −2.36117e41 −0.274003
\(781\) −1.30946e41 −0.149159
\(782\) −1.07801e41 −0.120539
\(783\) −8.96875e41 −0.984433
\(784\) −7.40309e40 −0.0797682
\(785\) −3.84931e41 −0.407168
\(786\) −1.95031e41 −0.202524
\(787\) 1.48600e42 1.51491 0.757454 0.652888i \(-0.226443\pi\)
0.757454 + 0.652888i \(0.226443\pi\)
\(788\) −2.14759e41 −0.214942
\(789\) −2.62184e41 −0.257626
\(790\) 8.64681e39 0.00834186
\(791\) 1.40614e42 1.33190
\(792\) 6.29700e40 0.0585624
\(793\) 8.61996e41 0.787126
\(794\) −1.09161e42 −0.978751
\(795\) −7.57406e41 −0.666815
\(796\) −9.13180e41 −0.789436
\(797\) 6.29245e41 0.534163 0.267082 0.963674i \(-0.413941\pi\)
0.267082 + 0.963674i \(0.413941\pi\)
\(798\) 6.78479e41 0.565581
\(799\) −1.77033e42 −1.44919
\(800\) 4.39805e40 0.0353553
\(801\) −4.65966e40 −0.0367860
\(802\) −1.32164e42 −1.02467
\(803\) 2.30675e42 1.75641
\(804\) −6.27002e41 −0.468874
\(805\) −1.07686e41 −0.0790895
\(806\) −6.52332e40 −0.0470555
\(807\) −2.52939e42 −1.79205
\(808\) 9.37958e41 0.652707
\(809\) 8.35891e41 0.571342 0.285671 0.958328i \(-0.407784\pi\)
0.285671 + 0.958328i \(0.407784\pi\)
\(810\) −3.95946e41 −0.265829
\(811\) 8.80958e41 0.580969 0.290484 0.956880i \(-0.406184\pi\)
0.290484 + 0.956880i \(0.406184\pi\)
\(812\) −5.93113e41 −0.384216
\(813\) −2.28162e42 −1.45188
\(814\) −1.06752e42 −0.667301
\(815\) −7.03147e38 −0.000431779 0
\(816\) −3.05724e41 −0.184426
\(817\) −1.78038e42 −1.05510
\(818\) 2.09985e42 1.22255
\(819\) −2.66515e41 −0.152442
\(820\) 2.04028e41 0.114654
\(821\) 1.16383e42 0.642559 0.321280 0.946984i \(-0.395887\pi\)
0.321280 + 0.946984i \(0.395887\pi\)
\(822\) −7.43365e40 −0.0403238
\(823\) 3.33222e42 1.77597 0.887987 0.459869i \(-0.152103\pi\)
0.887987 + 0.459869i \(0.152103\pi\)
\(824\) −3.74998e41 −0.196374
\(825\) −4.27063e41 −0.219740
\(826\) 1.63546e42 0.826853
\(827\) 2.56265e42 1.27309 0.636545 0.771239i \(-0.280363\pi\)
0.636545 + 0.771239i \(0.280363\pi\)
\(828\) −3.06932e40 −0.0149831
\(829\) 3.15692e42 1.51434 0.757169 0.653219i \(-0.226582\pi\)
0.757169 + 0.653219i \(0.226582\pi\)
\(830\) 8.92312e41 0.420615
\(831\) −1.27303e42 −0.589689
\(832\) −3.62806e41 −0.165153
\(833\) 5.67323e41 0.253792
\(834\) 1.70285e42 0.748634
\(835\) −1.96153e42 −0.847504
\(836\) 1.45785e42 0.619043
\(837\) −1.27585e41 −0.0532451
\(838\) 2.40587e42 0.986810
\(839\) −2.03367e42 −0.819846 −0.409923 0.912120i \(-0.634444\pi\)
−0.409923 + 0.912120i \(0.634444\pi\)
\(840\) −3.05396e41 −0.121008
\(841\) −3.41036e41 −0.132819
\(842\) 3.48280e40 0.0133322
\(843\) 2.82366e42 1.06246
\(844\) −4.48959e41 −0.166051
\(845\) 9.17200e41 0.333458
\(846\) −5.04047e41 −0.180136
\(847\) −9.47455e41 −0.332850
\(848\) −1.16379e42 −0.401917
\(849\) −2.00965e42 −0.682275
\(850\) −3.37037e41 −0.112487
\(851\) 5.20336e41 0.170728
\(852\) 1.81010e41 0.0583888
\(853\) −4.30318e42 −1.36467 −0.682337 0.731038i \(-0.739036\pi\)
−0.682337 + 0.731038i \(0.739036\pi\)
\(854\) 1.11492e42 0.347618
\(855\) 2.13190e41 0.0653520
\(856\) −6.74298e41 −0.203228
\(857\) 1.85145e40 0.00548642 0.00274321 0.999996i \(-0.499127\pi\)
0.00274321 + 0.999996i \(0.499127\pi\)
\(858\) 3.52295e42 1.02646
\(859\) −4.76577e42 −1.36531 −0.682656 0.730740i \(-0.739175\pi\)
−0.682656 + 0.730740i \(0.739175\pi\)
\(860\) 8.01381e41 0.225742
\(861\) −1.41676e42 −0.392418
\(862\) 1.37155e42 0.373558
\(863\) 5.09150e40 0.0136361 0.00681804 0.999977i \(-0.497830\pi\)
0.00681804 + 0.999977i \(0.497830\pi\)
\(864\) −7.09587e41 −0.186877
\(865\) 1.37773e42 0.356805
\(866\) −1.79723e42 −0.457713
\(867\) −1.36028e42 −0.340684
\(868\) −8.43733e40 −0.0207811
\(869\) −1.29013e41 −0.0312499
\(870\) 1.14651e42 0.273117
\(871\) 5.70207e42 1.33589
\(872\) −2.00281e41 −0.0461477
\(873\) −6.24323e41 −0.141482
\(874\) −7.10591e41 −0.158381
\(875\) −3.36676e41 −0.0738066
\(876\) −3.18870e42 −0.687549
\(877\) 5.73938e42 1.21722 0.608612 0.793468i \(-0.291727\pi\)
0.608612 + 0.793468i \(0.291727\pi\)
\(878\) 2.73506e42 0.570554
\(879\) 6.47818e42 1.32927
\(880\) −6.56204e41 −0.132446
\(881\) −4.61884e42 −0.917028 −0.458514 0.888687i \(-0.651618\pi\)
−0.458514 + 0.888687i \(0.651618\pi\)
\(882\) 1.61528e41 0.0315467
\(883\) −6.74072e42 −1.29502 −0.647509 0.762058i \(-0.724189\pi\)
−0.647509 + 0.762058i \(0.724189\pi\)
\(884\) 2.78031e42 0.525454
\(885\) −3.16140e42 −0.587762
\(886\) −6.08117e42 −1.11224
\(887\) −3.54607e42 −0.638050 −0.319025 0.947746i \(-0.603355\pi\)
−0.319025 + 0.947746i \(0.603355\pi\)
\(888\) 1.47567e42 0.261216
\(889\) 1.03542e41 0.0180318
\(890\) 4.85579e41 0.0831964
\(891\) 5.90766e42 0.995838
\(892\) 5.08735e42 0.843726
\(893\) −1.16694e43 −1.90416
\(894\) 4.76148e42 0.764448
\(895\) −1.02319e42 −0.161630
\(896\) −4.69258e41 −0.0729366
\(897\) −1.71718e42 −0.262618
\(898\) −6.15337e42 −0.925990
\(899\) 3.16752e41 0.0469033
\(900\) −9.59610e40 −0.0139823
\(901\) 8.91854e42 1.27875
\(902\) −3.04418e42 −0.429512
\(903\) −5.56472e42 −0.772629
\(904\) −4.17655e42 −0.570657
\(905\) 3.18546e42 0.428319
\(906\) −7.50829e42 −0.993532
\(907\) 8.10190e42 1.05507 0.527535 0.849533i \(-0.323116\pi\)
0.527535 + 0.849533i \(0.323116\pi\)
\(908\) 6.37654e41 0.0817222
\(909\) −2.04653e42 −0.258132
\(910\) 2.77733e42 0.344768
\(911\) 7.96842e42 0.973544 0.486772 0.873529i \(-0.338174\pi\)
0.486772 + 0.873529i \(0.338174\pi\)
\(912\) −2.01523e42 −0.242326
\(913\) −1.33136e43 −1.57569
\(914\) −4.68914e42 −0.546227
\(915\) −2.15517e42 −0.247102
\(916\) −8.48345e42 −0.957388
\(917\) 2.29405e42 0.254828
\(918\) 5.43780e42 0.594572
\(919\) −2.53647e42 −0.272995 −0.136497 0.990640i \(-0.543585\pi\)
−0.136497 + 0.990640i \(0.543585\pi\)
\(920\) 3.19851e41 0.0338863
\(921\) −9.07328e42 −0.946236
\(922\) −5.67354e42 −0.582446
\(923\) −1.64614e42 −0.166357
\(924\) 4.55662e42 0.453315
\(925\) 1.62681e42 0.159324
\(926\) 9.72300e42 0.937435
\(927\) 8.18209e41 0.0776619
\(928\) 1.76167e42 0.164619
\(929\) 3.99948e42 0.367939 0.183970 0.982932i \(-0.441105\pi\)
0.183970 + 0.982932i \(0.441105\pi\)
\(930\) 1.63097e41 0.0147721
\(931\) 3.73960e42 0.333469
\(932\) 4.25041e42 0.373164
\(933\) 8.01650e42 0.692949
\(934\) 1.57647e42 0.134170
\(935\) 5.02871e42 0.421394
\(936\) 7.91608e41 0.0653146
\(937\) 1.52825e42 0.124157 0.0620784 0.998071i \(-0.480227\pi\)
0.0620784 + 0.998071i \(0.480227\pi\)
\(938\) 7.37512e42 0.589967
\(939\) 2.12451e43 1.67343
\(940\) 5.25263e42 0.407401
\(941\) 1.35671e42 0.103619 0.0518093 0.998657i \(-0.483501\pi\)
0.0518093 + 0.998657i \(0.483501\pi\)
\(942\) −7.93917e42 −0.597086
\(943\) 1.48381e42 0.109890
\(944\) −4.85766e42 −0.354269
\(945\) 5.43198e42 0.390119
\(946\) −1.19569e43 −0.845662
\(947\) 2.29402e43 1.59780 0.798901 0.601463i \(-0.205415\pi\)
0.798901 + 0.601463i \(0.205415\pi\)
\(948\) 1.78340e41 0.0122328
\(949\) 2.89986e43 1.95892
\(950\) −2.22163e42 −0.147802
\(951\) 1.68479e43 1.10389
\(952\) 3.59608e42 0.232056
\(953\) −1.86788e43 −1.18714 −0.593570 0.804782i \(-0.702282\pi\)
−0.593570 + 0.804782i \(0.702282\pi\)
\(954\) 2.53928e42 0.158950
\(955\) 1.28894e43 0.794663
\(956\) −4.55214e42 −0.276424
\(957\) −1.71064e43 −1.02314
\(958\) −7.99639e42 −0.471080
\(959\) 8.74384e41 0.0507379
\(960\) 9.07093e41 0.0518464
\(961\) −1.77168e43 −0.997463
\(962\) −1.34200e43 −0.744240
\(963\) 1.47125e42 0.0803722
\(964\) −1.75204e42 −0.0942815
\(965\) −8.94644e42 −0.474247
\(966\) −2.22101e42 −0.115980
\(967\) −4.98614e42 −0.256496 −0.128248 0.991742i \(-0.540935\pi\)
−0.128248 + 0.991742i \(0.540935\pi\)
\(968\) 2.81414e42 0.142611
\(969\) 1.54434e43 0.770987
\(970\) 6.50601e42 0.319981
\(971\) −1.80120e43 −0.872735 −0.436368 0.899768i \(-0.643735\pi\)
−0.436368 + 0.899768i \(0.643735\pi\)
\(972\) 2.90657e42 0.138746
\(973\) −2.00298e43 −0.941978
\(974\) 2.53812e42 0.117600
\(975\) −5.36868e42 −0.245076
\(976\) −3.31154e42 −0.148939
\(977\) −2.15005e43 −0.952747 −0.476373 0.879243i \(-0.658049\pi\)
−0.476373 + 0.879243i \(0.658049\pi\)
\(978\) −1.45023e40 −0.000633177 0
\(979\) −7.24501e42 −0.311666
\(980\) −1.68327e42 −0.0713469
\(981\) 4.36993e41 0.0182505
\(982\) −1.34454e42 −0.0553294
\(983\) 2.69873e42 0.109429 0.0547146 0.998502i \(-0.482575\pi\)
0.0547146 + 0.998502i \(0.482575\pi\)
\(984\) 4.20807e42 0.168133
\(985\) −4.88305e42 −0.192250
\(986\) −1.35003e43 −0.523755
\(987\) −3.64738e43 −1.39438
\(988\) 1.83268e43 0.690418
\(989\) 5.82809e42 0.216362
\(990\) 1.43177e42 0.0523798
\(991\) −1.68950e43 −0.609101 −0.304551 0.952496i \(-0.598506\pi\)
−0.304551 + 0.952496i \(0.598506\pi\)
\(992\) 2.50607e41 0.00890376
\(993\) 2.55746e42 0.0895456
\(994\) −2.12914e42 −0.0734684
\(995\) −2.07633e43 −0.706093
\(996\) 1.84039e43 0.616806
\(997\) 2.21106e41 0.00730333 0.00365167 0.999993i \(-0.498838\pi\)
0.00365167 + 0.999993i \(0.498838\pi\)
\(998\) 3.87148e43 1.26033
\(999\) −2.62471e43 −0.842137
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 10.30.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.30.a.a.1.2 2 1.1 even 1 trivial