Properties

Label 50.26.a.i.1.2
Level $50$
Weight $26$
Character 50.1
Self dual yes
Analytic conductor $197.998$
Analytic rank $0$
Dimension $5$
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] = [50,26,Mod(1,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(197.998389976\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 25534923283x^{3} - 31863478542482x^{2} + 141941149085067124800x + 2515032055818200956928000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{5}\cdot 5^{12} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(107996.\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} -1.22475e6 q^{3} +1.67772e7 q^{4} +5.01659e9 q^{6} +5.40343e10 q^{7} -6.87195e10 q^{8} +6.52735e11 q^{9} -1.79977e13 q^{11} -2.05480e13 q^{12} -5.05810e13 q^{13} -2.21324e14 q^{14} +2.81475e14 q^{16} -1.91806e15 q^{17} -2.67360e15 q^{18} -3.52431e15 q^{19} -6.61787e16 q^{21} +7.37184e16 q^{22} -1.31301e17 q^{23} +8.41645e16 q^{24} +2.07180e17 q^{26} +2.38280e17 q^{27} +9.06545e17 q^{28} +3.38023e18 q^{29} -1.09603e18 q^{31} -1.15292e18 q^{32} +2.20427e19 q^{33} +7.85638e18 q^{34} +1.09511e19 q^{36} -4.59139e19 q^{37} +1.44356e19 q^{38} +6.19494e19 q^{39} -4.75209e19 q^{41} +2.71068e20 q^{42} +1.40138e20 q^{43} -3.01950e20 q^{44} +5.37807e20 q^{46} -9.40719e20 q^{47} -3.44738e20 q^{48} +1.57863e21 q^{49} +2.34916e21 q^{51} -8.48609e20 q^{52} -4.94559e21 q^{53} -9.75996e20 q^{54} -3.71321e21 q^{56} +4.31641e21 q^{57} -1.38454e22 q^{58} +8.56554e21 q^{59} -2.83544e22 q^{61} +4.48934e21 q^{62} +3.52701e22 q^{63} +4.72237e21 q^{64} -9.02869e22 q^{66} -5.48848e22 q^{67} -3.21797e22 q^{68} +1.60811e23 q^{69} +2.54981e23 q^{71} -4.48556e22 q^{72} -7.34354e22 q^{73} +1.88063e23 q^{74} -5.91281e22 q^{76} -9.72490e23 q^{77} -2.53745e23 q^{78} +3.27839e22 q^{79} -8.44890e23 q^{81} +1.94646e23 q^{82} -1.62267e24 q^{83} -1.11029e24 q^{84} -5.74004e23 q^{86} -4.13995e24 q^{87} +1.23679e24 q^{88} +1.92566e24 q^{89} -2.73311e24 q^{91} -2.20286e24 q^{92} +1.34237e24 q^{93} +3.85318e24 q^{94} +1.41205e24 q^{96} +9.42112e24 q^{97} -6.46609e24 q^{98} -1.17477e25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20480 q^{2} - 723995 q^{3} + 83886080 q^{4} + 2965483520 q^{6} - 49218886190 q^{7} - 343597383680 q^{8} + 975375361390 q^{9} - 8837033983815 q^{11} - 12146620497920 q^{12} - 67609989586220 q^{13}+ \cdots - 23\!\cdots\!70 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) −1.22475e6 −1.33056 −0.665278 0.746595i \(-0.731687\pi\)
−0.665278 + 0.746595i \(0.731687\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) 5.01659e9 0.940846
\(7\) 5.40343e10 1.47552 0.737758 0.675065i \(-0.235885\pi\)
0.737758 + 0.675065i \(0.235885\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 6.52735e11 0.770381
\(10\) 0 0
\(11\) −1.79977e13 −1.72905 −0.864524 0.502591i \(-0.832380\pi\)
−0.864524 + 0.502591i \(0.832380\pi\)
\(12\) −2.05480e13 −0.665278
\(13\) −5.05810e13 −0.602138 −0.301069 0.953602i \(-0.597343\pi\)
−0.301069 + 0.953602i \(0.597343\pi\)
\(14\) −2.21324e14 −1.04335
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −1.91806e15 −0.798456 −0.399228 0.916852i \(-0.630722\pi\)
−0.399228 + 0.916852i \(0.630722\pi\)
\(18\) −2.67360e15 −0.544742
\(19\) −3.52431e15 −0.365303 −0.182652 0.983178i \(-0.558468\pi\)
−0.182652 + 0.983178i \(0.558468\pi\)
\(20\) 0 0
\(21\) −6.61787e16 −1.96326
\(22\) 7.37184e16 1.22262
\(23\) −1.31301e17 −1.24931 −0.624653 0.780902i \(-0.714760\pi\)
−0.624653 + 0.780902i \(0.714760\pi\)
\(24\) 8.41645e16 0.470423
\(25\) 0 0
\(26\) 2.07180e17 0.425776
\(27\) 2.38280e17 0.305521
\(28\) 9.06545e17 0.737758
\(29\) 3.38023e18 1.77407 0.887037 0.461699i \(-0.152760\pi\)
0.887037 + 0.461699i \(0.152760\pi\)
\(30\) 0 0
\(31\) −1.09603e18 −0.249920 −0.124960 0.992162i \(-0.539880\pi\)
−0.124960 + 0.992162i \(0.539880\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 2.20427e19 2.30060
\(34\) 7.85638e18 0.564594
\(35\) 0 0
\(36\) 1.09511e19 0.385190
\(37\) −4.59139e19 −1.14663 −0.573314 0.819336i \(-0.694343\pi\)
−0.573314 + 0.819336i \(0.694343\pi\)
\(38\) 1.44356e19 0.258309
\(39\) 6.19494e19 0.801178
\(40\) 0 0
\(41\) −4.75209e19 −0.328917 −0.164458 0.986384i \(-0.552588\pi\)
−0.164458 + 0.986384i \(0.552588\pi\)
\(42\) 2.71068e20 1.38823
\(43\) 1.40138e20 0.534810 0.267405 0.963584i \(-0.413834\pi\)
0.267405 + 0.963584i \(0.413834\pi\)
\(44\) −3.01950e20 −0.864524
\(45\) 0 0
\(46\) 5.37807e20 0.883393
\(47\) −9.40719e20 −1.18096 −0.590482 0.807051i \(-0.701062\pi\)
−0.590482 + 0.807051i \(0.701062\pi\)
\(48\) −3.44738e20 −0.332639
\(49\) 1.57863e21 1.17715
\(50\) 0 0
\(51\) 2.34916e21 1.06239
\(52\) −8.48609e20 −0.301069
\(53\) −4.94559e21 −1.38283 −0.691416 0.722457i \(-0.743013\pi\)
−0.691416 + 0.722457i \(0.743013\pi\)
\(54\) −9.75996e20 −0.216036
\(55\) 0 0
\(56\) −3.71321e21 −0.521674
\(57\) 4.31641e21 0.486057
\(58\) −1.38454e22 −1.25446
\(59\) 8.56554e21 0.626765 0.313382 0.949627i \(-0.398538\pi\)
0.313382 + 0.949627i \(0.398538\pi\)
\(60\) 0 0
\(61\) −2.83544e22 −1.36772 −0.683861 0.729612i \(-0.739701\pi\)
−0.683861 + 0.729612i \(0.739701\pi\)
\(62\) 4.48934e21 0.176720
\(63\) 3.52701e22 1.13671
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −9.02869e22 −1.62677
\(67\) −5.48848e22 −0.819440 −0.409720 0.912211i \(-0.634374\pi\)
−0.409720 + 0.912211i \(0.634374\pi\)
\(68\) −3.21797e22 −0.399228
\(69\) 1.60811e23 1.66227
\(70\) 0 0
\(71\) 2.54981e23 1.84408 0.922038 0.387100i \(-0.126523\pi\)
0.922038 + 0.387100i \(0.126523\pi\)
\(72\) −4.48556e22 −0.272371
\(73\) −7.34354e22 −0.375292 −0.187646 0.982237i \(-0.560086\pi\)
−0.187646 + 0.982237i \(0.560086\pi\)
\(74\) 1.88063e23 0.810789
\(75\) 0 0
\(76\) −5.91281e22 −0.182652
\(77\) −9.72490e23 −2.55124
\(78\) −2.53745e23 −0.566519
\(79\) 3.27839e22 0.0624197 0.0312099 0.999513i \(-0.490064\pi\)
0.0312099 + 0.999513i \(0.490064\pi\)
\(80\) 0 0
\(81\) −8.44890e23 −1.17689
\(82\) 1.94646e23 0.232579
\(83\) −1.62267e24 −1.66630 −0.833149 0.553048i \(-0.813465\pi\)
−0.833149 + 0.553048i \(0.813465\pi\)
\(84\) −1.11029e24 −0.981629
\(85\) 0 0
\(86\) −5.74004e23 −0.378167
\(87\) −4.13995e24 −2.36051
\(88\) 1.23679e24 0.611311
\(89\) 1.92566e24 0.826429 0.413214 0.910634i \(-0.364406\pi\)
0.413214 + 0.910634i \(0.364406\pi\)
\(90\) 0 0
\(91\) −2.73311e24 −0.888464
\(92\) −2.20286e24 −0.624653
\(93\) 1.34237e24 0.332533
\(94\) 3.85318e24 0.835067
\(95\) 0 0
\(96\) 1.41205e24 0.235211
\(97\) 9.42112e24 1.37866 0.689328 0.724450i \(-0.257906\pi\)
0.689328 + 0.724450i \(0.257906\pi\)
\(98\) −6.46609e24 −0.832368
\(99\) −1.17477e25 −1.33203
\(100\) 0 0
\(101\) −6.26202e24 −0.552964 −0.276482 0.961019i \(-0.589169\pi\)
−0.276482 + 0.961019i \(0.589169\pi\)
\(102\) −9.62214e24 −0.751224
\(103\) −9.51652e24 −0.657677 −0.328839 0.944386i \(-0.606657\pi\)
−0.328839 + 0.944386i \(0.606657\pi\)
\(104\) 3.47590e24 0.212888
\(105\) 0 0
\(106\) 2.02571e25 0.977810
\(107\) 5.03847e24 0.216273 0.108136 0.994136i \(-0.465512\pi\)
0.108136 + 0.994136i \(0.465512\pi\)
\(108\) 3.99768e24 0.152761
\(109\) −1.74857e25 −0.595461 −0.297730 0.954650i \(-0.596230\pi\)
−0.297730 + 0.954650i \(0.596230\pi\)
\(110\) 0 0
\(111\) 5.62333e25 1.52565
\(112\) 1.52093e25 0.368879
\(113\) −4.83834e25 −1.05006 −0.525032 0.851082i \(-0.675947\pi\)
−0.525032 + 0.851082i \(0.675947\pi\)
\(114\) −1.76800e25 −0.343694
\(115\) 0 0
\(116\) 5.67109e25 0.887037
\(117\) −3.30160e25 −0.463875
\(118\) −3.50844e25 −0.443189
\(119\) −1.03641e26 −1.17813
\(120\) 0 0
\(121\) 2.15568e26 1.98961
\(122\) 1.16140e26 0.967126
\(123\) 5.82014e25 0.437643
\(124\) −1.83883e25 −0.124960
\(125\) 0 0
\(126\) −1.44466e26 −0.803775
\(127\) 2.16513e26 1.09128 0.545640 0.838019i \(-0.316286\pi\)
0.545640 + 0.838019i \(0.316286\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) −1.71634e26 −0.711594
\(130\) 0 0
\(131\) −3.82649e26 −1.30891 −0.654454 0.756102i \(-0.727102\pi\)
−0.654454 + 0.756102i \(0.727102\pi\)
\(132\) 3.69815e26 1.15030
\(133\) −1.90433e26 −0.539011
\(134\) 2.24808e26 0.579431
\(135\) 0 0
\(136\) 1.31808e26 0.282297
\(137\) −8.94431e26 −1.74799 −0.873996 0.485933i \(-0.838480\pi\)
−0.873996 + 0.485933i \(0.838480\pi\)
\(138\) −6.58682e26 −1.17540
\(139\) −6.14787e26 −1.00240 −0.501199 0.865332i \(-0.667107\pi\)
−0.501199 + 0.865332i \(0.667107\pi\)
\(140\) 0 0
\(141\) 1.15215e27 1.57134
\(142\) −1.04440e27 −1.30396
\(143\) 9.10340e26 1.04113
\(144\) 1.83729e26 0.192595
\(145\) 0 0
\(146\) 3.00791e26 0.265372
\(147\) −1.93344e27 −1.56626
\(148\) −7.70307e26 −0.573314
\(149\) −1.52978e27 −1.04665 −0.523325 0.852133i \(-0.675309\pi\)
−0.523325 + 0.852133i \(0.675309\pi\)
\(150\) 0 0
\(151\) −1.66948e27 −0.966873 −0.483436 0.875379i \(-0.660612\pi\)
−0.483436 + 0.875379i \(0.660612\pi\)
\(152\) 2.42189e26 0.129154
\(153\) −1.25199e27 −0.615116
\(154\) 3.98332e27 1.80400
\(155\) 0 0
\(156\) 1.03934e27 0.400589
\(157\) −1.97558e27 −0.702988 −0.351494 0.936190i \(-0.614326\pi\)
−0.351494 + 0.936190i \(0.614326\pi\)
\(158\) −1.34283e26 −0.0441374
\(159\) 6.05713e27 1.83994
\(160\) 0 0
\(161\) −7.09473e27 −1.84337
\(162\) 3.46067e27 0.832190
\(163\) 4.18722e27 0.932353 0.466176 0.884692i \(-0.345631\pi\)
0.466176 + 0.884692i \(0.345631\pi\)
\(164\) −7.97268e26 −0.164458
\(165\) 0 0
\(166\) 6.64644e27 1.17825
\(167\) 1.83560e27 0.301871 0.150936 0.988544i \(-0.451771\pi\)
0.150936 + 0.988544i \(0.451771\pi\)
\(168\) 4.54777e27 0.694116
\(169\) −4.49797e27 −0.637430
\(170\) 0 0
\(171\) −2.30044e27 −0.281423
\(172\) 2.35112e27 0.267405
\(173\) 7.54567e27 0.798218 0.399109 0.916903i \(-0.369319\pi\)
0.399109 + 0.916903i \(0.369319\pi\)
\(174\) 1.69573e28 1.66913
\(175\) 0 0
\(176\) −5.06589e27 −0.432262
\(177\) −1.04907e28 −0.833946
\(178\) −7.88752e27 −0.584374
\(179\) −2.40551e28 −1.66167 −0.830834 0.556521i \(-0.812136\pi\)
−0.830834 + 0.556521i \(0.812136\pi\)
\(180\) 0 0
\(181\) 3.05290e28 1.83540 0.917699 0.397277i \(-0.130045\pi\)
0.917699 + 0.397277i \(0.130045\pi\)
\(182\) 1.11948e28 0.628239
\(183\) 3.47272e28 1.81983
\(184\) 9.02291e27 0.441696
\(185\) 0 0
\(186\) −5.49834e27 −0.235136
\(187\) 3.45206e28 1.38057
\(188\) −1.57826e28 −0.590482
\(189\) 1.28753e28 0.450801
\(190\) 0 0
\(191\) 3.33829e28 1.02472 0.512362 0.858769i \(-0.328770\pi\)
0.512362 + 0.858769i \(0.328770\pi\)
\(192\) −5.78374e27 −0.166320
\(193\) 2.89791e28 0.780942 0.390471 0.920615i \(-0.372312\pi\)
0.390471 + 0.920615i \(0.372312\pi\)
\(194\) −3.85889e28 −0.974857
\(195\) 0 0
\(196\) 2.64851e28 0.588573
\(197\) −5.37586e28 −1.12104 −0.560519 0.828142i \(-0.689398\pi\)
−0.560519 + 0.828142i \(0.689398\pi\)
\(198\) 4.81186e28 0.941885
\(199\) 2.53738e28 0.466361 0.233180 0.972434i \(-0.425087\pi\)
0.233180 + 0.972434i \(0.425087\pi\)
\(200\) 0 0
\(201\) 6.72204e28 1.09031
\(202\) 2.56492e28 0.391005
\(203\) 1.82648e29 2.61767
\(204\) 3.94123e28 0.531196
\(205\) 0 0
\(206\) 3.89797e28 0.465048
\(207\) −8.57045e28 −0.962442
\(208\) −1.42373e28 −0.150534
\(209\) 6.34293e28 0.631627
\(210\) 0 0
\(211\) −7.53161e28 −0.665821 −0.332910 0.942958i \(-0.608031\pi\)
−0.332910 + 0.942958i \(0.608031\pi\)
\(212\) −8.29731e28 −0.691416
\(213\) −3.12289e29 −2.45365
\(214\) −2.06376e28 −0.152928
\(215\) 0 0
\(216\) −1.63745e28 −0.108018
\(217\) −5.92232e28 −0.368761
\(218\) 7.16216e28 0.421054
\(219\) 8.99404e28 0.499348
\(220\) 0 0
\(221\) 9.70176e28 0.480781
\(222\) −2.30331e29 −1.07880
\(223\) −2.07743e29 −0.919848 −0.459924 0.887958i \(-0.652123\pi\)
−0.459924 + 0.887958i \(0.652123\pi\)
\(224\) −6.22973e28 −0.260837
\(225\) 0 0
\(226\) 1.98179e29 0.742508
\(227\) −4.74953e29 −1.68394 −0.841971 0.539523i \(-0.818605\pi\)
−0.841971 + 0.539523i \(0.818605\pi\)
\(228\) 7.24174e28 0.243028
\(229\) 4.47901e29 1.42311 0.711555 0.702630i \(-0.247991\pi\)
0.711555 + 0.702630i \(0.247991\pi\)
\(230\) 0 0
\(231\) 1.19106e30 3.39457
\(232\) −2.32288e29 −0.627230
\(233\) −1.19571e29 −0.305970 −0.152985 0.988229i \(-0.548889\pi\)
−0.152985 + 0.988229i \(0.548889\pi\)
\(234\) 1.35234e29 0.328009
\(235\) 0 0
\(236\) 1.43706e29 0.313382
\(237\) −4.01522e28 −0.0830530
\(238\) 4.24514e29 0.833067
\(239\) −1.73647e29 −0.323365 −0.161682 0.986843i \(-0.551692\pi\)
−0.161682 + 0.986843i \(0.551692\pi\)
\(240\) 0 0
\(241\) 6.54337e29 1.09796 0.548982 0.835834i \(-0.315016\pi\)
0.548982 + 0.835834i \(0.315016\pi\)
\(242\) −8.82968e29 −1.40687
\(243\) 8.32891e29 1.26040
\(244\) −4.75708e29 −0.683861
\(245\) 0 0
\(246\) −2.38393e29 −0.309460
\(247\) 1.78263e29 0.219963
\(248\) 7.53186e28 0.0883601
\(249\) 1.98737e30 2.21711
\(250\) 0 0
\(251\) 6.53799e29 0.659968 0.329984 0.943987i \(-0.392957\pi\)
0.329984 + 0.943987i \(0.392957\pi\)
\(252\) 5.91734e29 0.568355
\(253\) 2.36310e30 2.16011
\(254\) −8.86835e29 −0.771652
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −1.59934e29 −0.120165 −0.0600823 0.998193i \(-0.519136\pi\)
−0.0600823 + 0.998193i \(0.519136\pi\)
\(258\) 7.03014e29 0.503173
\(259\) −2.48092e30 −1.69187
\(260\) 0 0
\(261\) 2.20640e30 1.36671
\(262\) 1.56733e30 0.925538
\(263\) −1.44294e30 −0.812461 −0.406231 0.913771i \(-0.633157\pi\)
−0.406231 + 0.913771i \(0.633157\pi\)
\(264\) −1.51476e30 −0.813384
\(265\) 0 0
\(266\) 7.80015e29 0.381138
\(267\) −2.35847e30 −1.09961
\(268\) −9.20814e29 −0.409720
\(269\) 3.68677e30 1.56582 0.782911 0.622134i \(-0.213734\pi\)
0.782911 + 0.622134i \(0.213734\pi\)
\(270\) 0 0
\(271\) 3.98811e30 1.54401 0.772007 0.635613i \(-0.219253\pi\)
0.772007 + 0.635613i \(0.219253\pi\)
\(272\) −5.39887e29 −0.199614
\(273\) 3.34739e30 1.18215
\(274\) 3.66359e30 1.23602
\(275\) 0 0
\(276\) 2.69796e30 0.831136
\(277\) 3.61317e30 1.06388 0.531939 0.846783i \(-0.321464\pi\)
0.531939 + 0.846783i \(0.321464\pi\)
\(278\) 2.51817e30 0.708802
\(279\) −7.15417e29 −0.192534
\(280\) 0 0
\(281\) 1.28210e29 0.0315568 0.0157784 0.999876i \(-0.494977\pi\)
0.0157784 + 0.999876i \(0.494977\pi\)
\(282\) −4.71920e30 −1.11110
\(283\) 7.90843e30 1.78139 0.890697 0.454598i \(-0.150217\pi\)
0.890697 + 0.454598i \(0.150217\pi\)
\(284\) 4.27787e30 0.922038
\(285\) 0 0
\(286\) −3.72875e30 −0.736187
\(287\) −2.56776e30 −0.485322
\(288\) −7.52552e29 −0.136185
\(289\) −2.09166e30 −0.362467
\(290\) 0 0
\(291\) −1.15386e31 −1.83438
\(292\) −1.23204e30 −0.187646
\(293\) −6.25018e30 −0.912110 −0.456055 0.889952i \(-0.650738\pi\)
−0.456055 + 0.889952i \(0.650738\pi\)
\(294\) 7.91937e30 1.10751
\(295\) 0 0
\(296\) 3.15518e30 0.405394
\(297\) −4.28849e30 −0.528261
\(298\) 6.26600e30 0.740094
\(299\) 6.64132e30 0.752254
\(300\) 0 0
\(301\) 7.57224e30 0.789120
\(302\) 6.83819e30 0.683682
\(303\) 7.66943e30 0.735750
\(304\) −9.92004e29 −0.0913259
\(305\) 0 0
\(306\) 5.12814e30 0.434952
\(307\) −2.00082e31 −1.62922 −0.814609 0.580010i \(-0.803049\pi\)
−0.814609 + 0.580010i \(0.803049\pi\)
\(308\) −1.63157e31 −1.27562
\(309\) 1.16554e31 0.875077
\(310\) 0 0
\(311\) −2.26965e31 −1.57201 −0.786003 0.618223i \(-0.787853\pi\)
−0.786003 + 0.618223i \(0.787853\pi\)
\(312\) −4.25713e30 −0.283259
\(313\) −1.47728e31 −0.944405 −0.472203 0.881490i \(-0.656541\pi\)
−0.472203 + 0.881490i \(0.656541\pi\)
\(314\) 8.09196e30 0.497088
\(315\) 0 0
\(316\) 5.50022e29 0.0312099
\(317\) 2.36061e30 0.128761 0.0643805 0.997925i \(-0.479493\pi\)
0.0643805 + 0.997925i \(0.479493\pi\)
\(318\) −2.48100e31 −1.30103
\(319\) −6.08362e31 −3.06746
\(320\) 0 0
\(321\) −6.17089e30 −0.287763
\(322\) 2.90600e31 1.30346
\(323\) 6.75984e30 0.291679
\(324\) −1.41749e31 −0.588447
\(325\) 0 0
\(326\) −1.71508e31 −0.659273
\(327\) 2.14157e31 0.792294
\(328\) 3.26561e30 0.116290
\(329\) −5.08311e31 −1.74253
\(330\) 0 0
\(331\) 2.93789e31 0.933653 0.466826 0.884349i \(-0.345397\pi\)
0.466826 + 0.884349i \(0.345397\pi\)
\(332\) −2.72238e31 −0.833149
\(333\) −2.99696e31 −0.883341
\(334\) −7.51861e30 −0.213455
\(335\) 0 0
\(336\) −1.86277e31 −0.490814
\(337\) 6.82670e31 1.73315 0.866577 0.499044i \(-0.166315\pi\)
0.866577 + 0.499044i \(0.166315\pi\)
\(338\) 1.84237e31 0.450731
\(339\) 5.92578e31 1.39717
\(340\) 0 0
\(341\) 1.97260e31 0.432124
\(342\) 9.42260e30 0.198996
\(343\) 1.28367e31 0.261383
\(344\) −9.63019e30 −0.189084
\(345\) 0 0
\(346\) −3.09071e31 −0.564425
\(347\) 1.59692e31 0.281296 0.140648 0.990060i \(-0.455081\pi\)
0.140648 + 0.990060i \(0.455081\pi\)
\(348\) −6.94569e31 −1.18025
\(349\) −4.91783e31 −0.806225 −0.403112 0.915150i \(-0.632072\pi\)
−0.403112 + 0.915150i \(0.632072\pi\)
\(350\) 0 0
\(351\) −1.20525e31 −0.183966
\(352\) 2.07499e31 0.305656
\(353\) 3.19773e30 0.0454630 0.0227315 0.999742i \(-0.492764\pi\)
0.0227315 + 0.999742i \(0.492764\pi\)
\(354\) 4.29698e31 0.589689
\(355\) 0 0
\(356\) 3.23073e31 0.413214
\(357\) 1.26935e32 1.56758
\(358\) 9.85296e31 1.17498
\(359\) −3.57527e31 −0.411745 −0.205872 0.978579i \(-0.566003\pi\)
−0.205872 + 0.978579i \(0.566003\pi\)
\(360\) 0 0
\(361\) −8.06558e31 −0.866553
\(362\) −1.25047e32 −1.29782
\(363\) −2.64018e32 −2.64729
\(364\) −4.58540e31 −0.444232
\(365\) 0 0
\(366\) −1.42243e32 −1.28682
\(367\) 2.22256e32 1.94325 0.971623 0.236536i \(-0.0760122\pi\)
0.971623 + 0.236536i \(0.0760122\pi\)
\(368\) −3.69578e31 −0.312327
\(369\) −3.10185e31 −0.253391
\(370\) 0 0
\(371\) −2.67231e32 −2.04039
\(372\) 2.25212e31 0.166266
\(373\) −2.11651e32 −1.51099 −0.755493 0.655157i \(-0.772603\pi\)
−0.755493 + 0.655157i \(0.772603\pi\)
\(374\) −1.41396e32 −0.976210
\(375\) 0 0
\(376\) 6.46457e31 0.417534
\(377\) −1.70976e32 −1.06824
\(378\) −5.27373e31 −0.318765
\(379\) 1.57098e32 0.918713 0.459357 0.888252i \(-0.348080\pi\)
0.459357 + 0.888252i \(0.348080\pi\)
\(380\) 0 0
\(381\) −2.65175e32 −1.45201
\(382\) −1.36736e32 −0.724590
\(383\) 2.04365e32 1.04815 0.524073 0.851673i \(-0.324412\pi\)
0.524073 + 0.851673i \(0.324412\pi\)
\(384\) 2.36902e31 0.117606
\(385\) 0 0
\(386\) −1.18699e32 −0.552210
\(387\) 9.14728e31 0.412007
\(388\) 1.58060e32 0.689328
\(389\) 1.78110e32 0.752174 0.376087 0.926584i \(-0.377269\pi\)
0.376087 + 0.926584i \(0.377269\pi\)
\(390\) 0 0
\(391\) 2.51843e32 0.997516
\(392\) −1.08483e32 −0.416184
\(393\) 4.68651e32 1.74158
\(394\) 2.20195e32 0.792694
\(395\) 0 0
\(396\) −1.97094e32 −0.666013
\(397\) −1.81768e32 −0.595163 −0.297581 0.954696i \(-0.596180\pi\)
−0.297581 + 0.954696i \(0.596180\pi\)
\(398\) −1.03931e32 −0.329767
\(399\) 2.33234e32 0.717185
\(400\) 0 0
\(401\) −1.87592e32 −0.541889 −0.270945 0.962595i \(-0.587336\pi\)
−0.270945 + 0.962595i \(0.587336\pi\)
\(402\) −2.75335e32 −0.770966
\(403\) 5.54383e31 0.150486
\(404\) −1.05059e32 −0.276482
\(405\) 0 0
\(406\) −7.48128e32 −1.85097
\(407\) 8.26342e32 1.98258
\(408\) −1.61433e32 −0.375612
\(409\) −3.48178e32 −0.785706 −0.392853 0.919601i \(-0.628512\pi\)
−0.392853 + 0.919601i \(0.628512\pi\)
\(410\) 0 0
\(411\) 1.09546e33 2.32580
\(412\) −1.59661e32 −0.328839
\(413\) 4.62833e32 0.924801
\(414\) 3.51046e32 0.680549
\(415\) 0 0
\(416\) 5.83160e31 0.106444
\(417\) 7.52964e32 1.33375
\(418\) −2.59806e32 −0.446628
\(419\) −8.02659e32 −1.33923 −0.669616 0.742708i \(-0.733541\pi\)
−0.669616 + 0.742708i \(0.733541\pi\)
\(420\) 0 0
\(421\) 4.26783e32 0.670936 0.335468 0.942052i \(-0.391106\pi\)
0.335468 + 0.942052i \(0.391106\pi\)
\(422\) 3.08495e32 0.470806
\(423\) −6.14040e32 −0.909792
\(424\) 3.39858e32 0.488905
\(425\) 0 0
\(426\) 1.27914e33 1.73499
\(427\) −1.53211e33 −2.01810
\(428\) 8.45315e31 0.108136
\(429\) −1.11494e33 −1.38528
\(430\) 0 0
\(431\) 6.47631e32 0.759210 0.379605 0.925149i \(-0.376060\pi\)
0.379605 + 0.925149i \(0.376060\pi\)
\(432\) 6.70700e31 0.0763803
\(433\) 5.41225e32 0.598796 0.299398 0.954128i \(-0.403214\pi\)
0.299398 + 0.954128i \(0.403214\pi\)
\(434\) 2.42578e32 0.260753
\(435\) 0 0
\(436\) −2.93362e32 −0.297730
\(437\) 4.62744e32 0.456376
\(438\) −3.68396e32 −0.353092
\(439\) −4.19028e32 −0.390334 −0.195167 0.980770i \(-0.562525\pi\)
−0.195167 + 0.980770i \(0.562525\pi\)
\(440\) 0 0
\(441\) 1.03043e33 0.906851
\(442\) −3.97384e32 −0.339963
\(443\) 5.75377e32 0.478527 0.239263 0.970955i \(-0.423094\pi\)
0.239263 + 0.970955i \(0.423094\pi\)
\(444\) 9.43437e32 0.762827
\(445\) 0 0
\(446\) 8.50916e32 0.650431
\(447\) 1.87361e33 1.39263
\(448\) 2.55170e32 0.184439
\(449\) 8.31503e32 0.584500 0.292250 0.956342i \(-0.405596\pi\)
0.292250 + 0.956342i \(0.405596\pi\)
\(450\) 0 0
\(451\) 8.55264e32 0.568714
\(452\) −8.11739e32 −0.525032
\(453\) 2.04470e33 1.28648
\(454\) 1.94541e33 1.19073
\(455\) 0 0
\(456\) −2.96622e32 −0.171847
\(457\) −1.74441e33 −0.983322 −0.491661 0.870787i \(-0.663610\pi\)
−0.491661 + 0.870787i \(0.663610\pi\)
\(458\) −1.83460e33 −1.00629
\(459\) −4.57037e32 −0.243945
\(460\) 0 0
\(461\) 1.31968e33 0.667126 0.333563 0.942728i \(-0.391749\pi\)
0.333563 + 0.942728i \(0.391749\pi\)
\(462\) −4.87859e33 −2.40032
\(463\) 1.40349e33 0.672121 0.336061 0.941840i \(-0.390905\pi\)
0.336061 + 0.941840i \(0.390905\pi\)
\(464\) 9.51451e32 0.443518
\(465\) 0 0
\(466\) 4.89765e32 0.216353
\(467\) −7.68890e32 −0.330676 −0.165338 0.986237i \(-0.552872\pi\)
−0.165338 + 0.986237i \(0.552872\pi\)
\(468\) −5.53917e32 −0.231938
\(469\) −2.96566e33 −1.20910
\(470\) 0 0
\(471\) 2.41960e33 0.935366
\(472\) −5.88619e32 −0.221595
\(473\) −2.52215e33 −0.924712
\(474\) 1.64463e32 0.0587273
\(475\) 0 0
\(476\) −1.73881e33 −0.589067
\(477\) −3.22816e33 −1.06531
\(478\) 7.11256e32 0.228653
\(479\) 2.68490e33 0.840879 0.420440 0.907321i \(-0.361876\pi\)
0.420440 + 0.907321i \(0.361876\pi\)
\(480\) 0 0
\(481\) 2.32237e33 0.690428
\(482\) −2.68016e33 −0.776378
\(483\) 8.68931e33 2.45271
\(484\) 3.61664e33 0.994805
\(485\) 0 0
\(486\) −3.41152e33 −0.891240
\(487\) −4.59053e33 −1.16883 −0.584415 0.811455i \(-0.698676\pi\)
−0.584415 + 0.811455i \(0.698676\pi\)
\(488\) 1.94850e33 0.483563
\(489\) −5.12831e33 −1.24055
\(490\) 0 0
\(491\) 5.15180e33 1.18424 0.592120 0.805849i \(-0.298291\pi\)
0.592120 + 0.805849i \(0.298291\pi\)
\(492\) 9.76458e32 0.218821
\(493\) −6.48350e33 −1.41652
\(494\) −7.30166e32 −0.155537
\(495\) 0 0
\(496\) −3.08505e32 −0.0624800
\(497\) 1.37777e34 2.72096
\(498\) −8.14026e33 −1.56773
\(499\) 7.46002e33 1.40115 0.700573 0.713581i \(-0.252928\pi\)
0.700573 + 0.713581i \(0.252928\pi\)
\(500\) 0 0
\(501\) −2.24816e33 −0.401657
\(502\) −2.67796e33 −0.466668
\(503\) 2.55159e33 0.433722 0.216861 0.976203i \(-0.430418\pi\)
0.216861 + 0.976203i \(0.430418\pi\)
\(504\) −2.42374e33 −0.401887
\(505\) 0 0
\(506\) −9.67927e33 −1.52743
\(507\) 5.50891e33 0.848137
\(508\) 3.63248e33 0.545640
\(509\) 8.95543e33 1.31254 0.656272 0.754524i \(-0.272132\pi\)
0.656272 + 0.754524i \(0.272132\pi\)
\(510\) 0 0
\(511\) −3.96803e33 −0.553750
\(512\) −3.24519e32 −0.0441942
\(513\) −8.39773e32 −0.111608
\(514\) 6.55089e32 0.0849692
\(515\) 0 0
\(516\) −2.87954e33 −0.355797
\(517\) 1.69307e34 2.04194
\(518\) 1.01619e34 1.19633
\(519\) −9.24159e33 −1.06207
\(520\) 0 0
\(521\) 8.35519e33 0.915134 0.457567 0.889175i \(-0.348721\pi\)
0.457567 + 0.889175i \(0.348721\pi\)
\(522\) −9.03740e33 −0.966412
\(523\) 1.81744e34 1.89753 0.948765 0.315981i \(-0.102334\pi\)
0.948765 + 0.315981i \(0.102334\pi\)
\(524\) −6.41978e33 −0.654454
\(525\) 0 0
\(526\) 5.91030e33 0.574497
\(527\) 2.10225e33 0.199550
\(528\) 6.20447e33 0.575149
\(529\) 6.19409e33 0.560766
\(530\) 0 0
\(531\) 5.59103e33 0.482848
\(532\) −3.19494e33 −0.269505
\(533\) 2.40366e33 0.198053
\(534\) 9.66028e33 0.777542
\(535\) 0 0
\(536\) 3.77165e33 0.289716
\(537\) 2.94616e34 2.21094
\(538\) −1.51010e34 −1.10720
\(539\) −2.84117e34 −2.03534
\(540\) 0 0
\(541\) −6.80008e33 −0.465102 −0.232551 0.972584i \(-0.574707\pi\)
−0.232551 + 0.972584i \(0.574707\pi\)
\(542\) −1.63353e34 −1.09178
\(543\) −3.73905e34 −2.44210
\(544\) 2.21138e33 0.141148
\(545\) 0 0
\(546\) −1.37109e34 −0.835907
\(547\) 2.41615e34 1.43973 0.719866 0.694113i \(-0.244203\pi\)
0.719866 + 0.694113i \(0.244203\pi\)
\(548\) −1.50061e34 −0.873996
\(549\) −1.85079e34 −1.05367
\(550\) 0 0
\(551\) −1.19130e34 −0.648075
\(552\) −1.10509e34 −0.587702
\(553\) 1.77145e33 0.0921013
\(554\) −1.47996e34 −0.752275
\(555\) 0 0
\(556\) −1.03144e34 −0.501199
\(557\) 1.04132e33 0.0494762 0.0247381 0.999694i \(-0.492125\pi\)
0.0247381 + 0.999694i \(0.492125\pi\)
\(558\) 2.93035e33 0.136142
\(559\) −7.08831e33 −0.322029
\(560\) 0 0
\(561\) −4.22793e34 −1.83693
\(562\) −5.25148e32 −0.0223140
\(563\) −3.05082e34 −1.26783 −0.633915 0.773403i \(-0.718553\pi\)
−0.633915 + 0.773403i \(0.718553\pi\)
\(564\) 1.93299e34 0.785670
\(565\) 0 0
\(566\) −3.23929e34 −1.25964
\(567\) −4.56530e34 −1.73653
\(568\) −1.75222e34 −0.651979
\(569\) 1.30060e34 0.473413 0.236707 0.971581i \(-0.423932\pi\)
0.236707 + 0.971581i \(0.423932\pi\)
\(570\) 0 0
\(571\) 2.50202e34 0.871646 0.435823 0.900032i \(-0.356457\pi\)
0.435823 + 0.900032i \(0.356457\pi\)
\(572\) 1.52730e34 0.520563
\(573\) −4.08859e34 −1.36345
\(574\) 1.05175e34 0.343175
\(575\) 0 0
\(576\) 3.08245e33 0.0962976
\(577\) −6.21101e33 −0.189874 −0.0949368 0.995483i \(-0.530265\pi\)
−0.0949368 + 0.995483i \(0.530265\pi\)
\(578\) 8.56746e33 0.256303
\(579\) −3.54923e34 −1.03909
\(580\) 0 0
\(581\) −8.76796e34 −2.45865
\(582\) 4.72619e34 1.29710
\(583\) 8.90089e34 2.39098
\(584\) 5.04644e33 0.132686
\(585\) 0 0
\(586\) 2.56007e34 0.644959
\(587\) −2.21731e34 −0.546827 −0.273413 0.961897i \(-0.588153\pi\)
−0.273413 + 0.961897i \(0.588153\pi\)
\(588\) −3.24377e34 −0.783130
\(589\) 3.86275e33 0.0912967
\(590\) 0 0
\(591\) 6.58411e34 1.49160
\(592\) −1.29236e34 −0.286657
\(593\) −4.72404e34 −1.02596 −0.512980 0.858401i \(-0.671458\pi\)
−0.512980 + 0.858401i \(0.671458\pi\)
\(594\) 1.75656e34 0.373537
\(595\) 0 0
\(596\) −2.56655e34 −0.523325
\(597\) −3.10767e34 −0.620519
\(598\) −2.72029e34 −0.531924
\(599\) 9.34743e34 1.79002 0.895008 0.446050i \(-0.147170\pi\)
0.895008 + 0.446050i \(0.147170\pi\)
\(600\) 0 0
\(601\) 1.28681e34 0.236365 0.118183 0.992992i \(-0.462293\pi\)
0.118183 + 0.992992i \(0.462293\pi\)
\(602\) −3.10159e34 −0.557992
\(603\) −3.58252e34 −0.631281
\(604\) −2.80092e34 −0.483436
\(605\) 0 0
\(606\) −3.14140e34 −0.520254
\(607\) 6.95915e34 1.12901 0.564505 0.825430i \(-0.309067\pi\)
0.564505 + 0.825430i \(0.309067\pi\)
\(608\) 4.06325e33 0.0645771
\(609\) −2.23699e35 −3.48296
\(610\) 0 0
\(611\) 4.75825e34 0.711103
\(612\) −2.10048e34 −0.307558
\(613\) 1.91426e34 0.274628 0.137314 0.990528i \(-0.456153\pi\)
0.137314 + 0.990528i \(0.456153\pi\)
\(614\) 8.19538e34 1.15203
\(615\) 0 0
\(616\) 6.68290e34 0.901999
\(617\) 1.16629e35 1.54255 0.771277 0.636499i \(-0.219618\pi\)
0.771277 + 0.636499i \(0.219618\pi\)
\(618\) −4.77405e34 −0.618773
\(619\) −7.21348e33 −0.0916246 −0.0458123 0.998950i \(-0.514588\pi\)
−0.0458123 + 0.998950i \(0.514588\pi\)
\(620\) 0 0
\(621\) −3.12864e34 −0.381689
\(622\) 9.29649e34 1.11158
\(623\) 1.04052e35 1.21941
\(624\) 1.74372e34 0.200295
\(625\) 0 0
\(626\) 6.05093e34 0.667795
\(627\) −7.76853e34 −0.840416
\(628\) −3.31447e34 −0.351494
\(629\) 8.80657e34 0.915533
\(630\) 0 0
\(631\) 3.50645e34 0.350349 0.175174 0.984537i \(-0.443951\pi\)
0.175174 + 0.984537i \(0.443951\pi\)
\(632\) −2.25289e33 −0.0220687
\(633\) 9.22438e34 0.885912
\(634\) −9.66908e33 −0.0910478
\(635\) 0 0
\(636\) 1.01622e35 0.919968
\(637\) −7.98490e34 −0.708804
\(638\) 2.49185e35 2.16902
\(639\) 1.66435e35 1.42064
\(640\) 0 0
\(641\) −1.17260e35 −0.962551 −0.481276 0.876569i \(-0.659826\pi\)
−0.481276 + 0.876569i \(0.659826\pi\)
\(642\) 2.52760e34 0.203479
\(643\) 1.79181e35 1.41467 0.707336 0.706877i \(-0.249897\pi\)
0.707336 + 0.706877i \(0.249897\pi\)
\(644\) −1.19030e35 −0.921685
\(645\) 0 0
\(646\) −2.76883e34 −0.206248
\(647\) 1.07946e35 0.788686 0.394343 0.918963i \(-0.370972\pi\)
0.394343 + 0.918963i \(0.370972\pi\)
\(648\) 5.80604e34 0.416095
\(649\) −1.54160e35 −1.08371
\(650\) 0 0
\(651\) 7.25338e34 0.490657
\(652\) 7.02498e34 0.466176
\(653\) −1.14134e35 −0.743017 −0.371508 0.928430i \(-0.621159\pi\)
−0.371508 + 0.928430i \(0.621159\pi\)
\(654\) −8.77189e34 −0.560236
\(655\) 0 0
\(656\) −1.33759e34 −0.0822292
\(657\) −4.79339e34 −0.289118
\(658\) 2.08204e35 1.23216
\(659\) 2.00321e35 1.16321 0.581605 0.813471i \(-0.302425\pi\)
0.581605 + 0.813471i \(0.302425\pi\)
\(660\) 0 0
\(661\) −2.41113e35 −1.34804 −0.674019 0.738714i \(-0.735433\pi\)
−0.674019 + 0.738714i \(0.735433\pi\)
\(662\) −1.20336e35 −0.660192
\(663\) −1.18823e35 −0.639706
\(664\) 1.11509e35 0.589126
\(665\) 0 0
\(666\) 1.22756e35 0.624616
\(667\) −4.43827e35 −2.21636
\(668\) 3.07962e34 0.150936
\(669\) 2.54434e35 1.22391
\(670\) 0 0
\(671\) 5.10313e35 2.36486
\(672\) 7.62989e34 0.347058
\(673\) 4.10824e34 0.183429 0.0917145 0.995785i \(-0.470765\pi\)
0.0917145 + 0.995785i \(0.470765\pi\)
\(674\) −2.79622e35 −1.22552
\(675\) 0 0
\(676\) −7.54634e34 −0.318715
\(677\) −2.21198e35 −0.917111 −0.458556 0.888666i \(-0.651633\pi\)
−0.458556 + 0.888666i \(0.651633\pi\)
\(678\) −2.42720e35 −0.987949
\(679\) 5.09063e35 2.03423
\(680\) 0 0
\(681\) 5.81700e35 2.24058
\(682\) −8.07975e34 −0.305558
\(683\) 1.93443e35 0.718280 0.359140 0.933284i \(-0.383070\pi\)
0.359140 + 0.933284i \(0.383070\pi\)
\(684\) −3.85950e34 −0.140711
\(685\) 0 0
\(686\) −5.25791e34 −0.184825
\(687\) −5.48569e35 −1.89353
\(688\) 3.94452e34 0.133702
\(689\) 2.50153e35 0.832655
\(690\) 0 0
\(691\) −1.57151e33 −0.00504479 −0.00252239 0.999997i \(-0.500803\pi\)
−0.00252239 + 0.999997i \(0.500803\pi\)
\(692\) 1.26595e35 0.399109
\(693\) −6.34778e35 −1.96543
\(694\) −6.54097e34 −0.198907
\(695\) 0 0
\(696\) 2.84496e35 0.834565
\(697\) 9.11480e34 0.262626
\(698\) 2.01435e35 0.570087
\(699\) 1.46446e35 0.407110
\(700\) 0 0
\(701\) −4.76280e35 −1.27758 −0.638789 0.769382i \(-0.720564\pi\)
−0.638789 + 0.769382i \(0.720564\pi\)
\(702\) 4.93669e34 0.130083
\(703\) 1.61815e35 0.418867
\(704\) −8.49915e34 −0.216131
\(705\) 0 0
\(706\) −1.30979e34 −0.0321472
\(707\) −3.38363e35 −0.815907
\(708\) −1.76004e35 −0.416973
\(709\) −2.73484e35 −0.636580 −0.318290 0.947993i \(-0.603109\pi\)
−0.318290 + 0.947993i \(0.603109\pi\)
\(710\) 0 0
\(711\) 2.13992e34 0.0480870
\(712\) −1.32331e35 −0.292187
\(713\) 1.43909e35 0.312227
\(714\) −5.19925e35 −1.10844
\(715\) 0 0
\(716\) −4.03577e35 −0.830834
\(717\) 2.12674e35 0.430255
\(718\) 1.46443e35 0.291148
\(719\) −1.61873e35 −0.316274 −0.158137 0.987417i \(-0.550549\pi\)
−0.158137 + 0.987417i \(0.550549\pi\)
\(720\) 0 0
\(721\) −5.14218e35 −0.970413
\(722\) 3.30366e35 0.612746
\(723\) −8.01402e35 −1.46090
\(724\) 5.12191e35 0.917699
\(725\) 0 0
\(726\) 1.08142e36 1.87192
\(727\) 3.12165e35 0.531134 0.265567 0.964092i \(-0.414441\pi\)
0.265567 + 0.964092i \(0.414441\pi\)
\(728\) 1.87818e35 0.314119
\(729\) −3.04221e35 −0.500144
\(730\) 0 0
\(731\) −2.68793e35 −0.427022
\(732\) 5.82626e35 0.909916
\(733\) −2.02938e35 −0.311576 −0.155788 0.987791i \(-0.549792\pi\)
−0.155788 + 0.987791i \(0.549792\pi\)
\(734\) −9.10360e35 −1.37408
\(735\) 0 0
\(736\) 1.51379e35 0.220848
\(737\) 9.87798e35 1.41685
\(738\) 1.27052e35 0.179175
\(739\) 6.40360e34 0.0887910 0.0443955 0.999014i \(-0.485864\pi\)
0.0443955 + 0.999014i \(0.485864\pi\)
\(740\) 0 0
\(741\) −2.18329e35 −0.292673
\(742\) 1.09458e36 1.44277
\(743\) 9.29648e35 1.20492 0.602460 0.798149i \(-0.294187\pi\)
0.602460 + 0.798149i \(0.294187\pi\)
\(744\) −9.22468e34 −0.117568
\(745\) 0 0
\(746\) 8.66924e35 1.06843
\(747\) −1.05917e36 −1.28369
\(748\) 5.79160e35 0.690285
\(749\) 2.72250e35 0.319113
\(750\) 0 0
\(751\) 3.56197e35 0.403823 0.201912 0.979404i \(-0.435285\pi\)
0.201912 + 0.979404i \(0.435285\pi\)
\(752\) −2.64789e35 −0.295241
\(753\) −8.00743e35 −0.878124
\(754\) 7.00316e35 0.755357
\(755\) 0 0
\(756\) 2.16012e35 0.225401
\(757\) −1.75236e36 −1.79856 −0.899278 0.437378i \(-0.855907\pi\)
−0.899278 + 0.437378i \(0.855907\pi\)
\(758\) −6.43473e35 −0.649629
\(759\) −2.89422e36 −2.87415
\(760\) 0 0
\(761\) −1.03372e36 −0.993332 −0.496666 0.867942i \(-0.665443\pi\)
−0.496666 + 0.867942i \(0.665443\pi\)
\(762\) 1.08616e36 1.02673
\(763\) −9.44830e35 −0.878611
\(764\) 5.60072e35 0.512362
\(765\) 0 0
\(766\) −8.37079e35 −0.741151
\(767\) −4.33254e35 −0.377399
\(768\) −9.70351e34 −0.0831598
\(769\) 1.96647e35 0.165809 0.0829044 0.996558i \(-0.473580\pi\)
0.0829044 + 0.996558i \(0.473580\pi\)
\(770\) 0 0
\(771\) 1.95880e35 0.159886
\(772\) 4.86189e35 0.390471
\(773\) 3.07830e35 0.243258 0.121629 0.992576i \(-0.461188\pi\)
0.121629 + 0.992576i \(0.461188\pi\)
\(774\) −3.74672e35 −0.291333
\(775\) 0 0
\(776\) −6.47414e35 −0.487428
\(777\) 3.03852e36 2.25113
\(778\) −7.29538e35 −0.531867
\(779\) 1.67478e35 0.120155
\(780\) 0 0
\(781\) −4.58906e36 −3.18850
\(782\) −1.03155e36 −0.705351
\(783\) 8.05443e35 0.542017
\(784\) 4.44346e35 0.294287
\(785\) 0 0
\(786\) −1.91959e36 −1.23148
\(787\) 2.22687e36 1.40608 0.703042 0.711148i \(-0.251825\pi\)
0.703042 + 0.711148i \(0.251825\pi\)
\(788\) −9.01920e35 −0.560519
\(789\) 1.76725e36 1.08103
\(790\) 0 0
\(791\) −2.61436e36 −1.54939
\(792\) 8.07296e35 0.470942
\(793\) 1.43420e36 0.823557
\(794\) 7.44521e35 0.420844
\(795\) 0 0
\(796\) 4.25702e35 0.233180
\(797\) −9.40522e35 −0.507154 −0.253577 0.967315i \(-0.581607\pi\)
−0.253577 + 0.967315i \(0.581607\pi\)
\(798\) −9.55327e35 −0.507126
\(799\) 1.80436e36 0.942948
\(800\) 0 0
\(801\) 1.25695e36 0.636665
\(802\) 7.68378e35 0.383174
\(803\) 1.32167e36 0.648899
\(804\) 1.12777e36 0.545156
\(805\) 0 0
\(806\) −2.27075e35 −0.106410
\(807\) −4.51539e36 −2.08341
\(808\) 4.30322e35 0.195502
\(809\) 2.23971e36 1.00193 0.500963 0.865468i \(-0.332979\pi\)
0.500963 + 0.865468i \(0.332979\pi\)
\(810\) 0 0
\(811\) −2.70491e35 −0.117325 −0.0586627 0.998278i \(-0.518684\pi\)
−0.0586627 + 0.998278i \(0.518684\pi\)
\(812\) 3.06433e36 1.30884
\(813\) −4.88446e36 −2.05440
\(814\) −3.38470e36 −1.40189
\(815\) 0 0
\(816\) 6.61228e35 0.265598
\(817\) −4.93888e35 −0.195368
\(818\) 1.42614e36 0.555578
\(819\) −1.78400e36 −0.684455
\(820\) 0 0
\(821\) 8.72543e35 0.324711 0.162356 0.986732i \(-0.448091\pi\)
0.162356 + 0.986732i \(0.448091\pi\)
\(822\) −4.48700e36 −1.64459
\(823\) −5.40468e35 −0.195107 −0.0975534 0.995230i \(-0.531102\pi\)
−0.0975534 + 0.995230i \(0.531102\pi\)
\(824\) 6.53970e35 0.232524
\(825\) 0 0
\(826\) −1.89576e36 −0.653933
\(827\) 9.59407e35 0.325975 0.162987 0.986628i \(-0.447887\pi\)
0.162987 + 0.986628i \(0.447887\pi\)
\(828\) −1.43788e36 −0.481221
\(829\) −2.44895e36 −0.807326 −0.403663 0.914908i \(-0.632263\pi\)
−0.403663 + 0.914908i \(0.632263\pi\)
\(830\) 0 0
\(831\) −4.42525e36 −1.41555
\(832\) −2.38862e35 −0.0752672
\(833\) −3.02792e36 −0.939900
\(834\) −3.08414e36 −0.943101
\(835\) 0 0
\(836\) 1.06417e36 0.315814
\(837\) −2.61162e35 −0.0763559
\(838\) 3.28769e36 0.946980
\(839\) 5.26267e36 1.49342 0.746709 0.665151i \(-0.231633\pi\)
0.746709 + 0.665151i \(0.231633\pi\)
\(840\) 0 0
\(841\) 7.79561e36 2.14734
\(842\) −1.74810e36 −0.474423
\(843\) −1.57026e35 −0.0419881
\(844\) −1.26359e36 −0.332910
\(845\) 0 0
\(846\) 2.51511e36 0.643320
\(847\) 1.16481e37 2.93570
\(848\) −1.39206e36 −0.345708
\(849\) −9.68588e36 −2.37024
\(850\) 0 0
\(851\) 6.02852e36 1.43249
\(852\) −5.23934e36 −1.22682
\(853\) 2.08360e36 0.480786 0.240393 0.970676i \(-0.422724\pi\)
0.240393 + 0.970676i \(0.422724\pi\)
\(854\) 6.27553e36 1.42701
\(855\) 0 0
\(856\) −3.46241e35 −0.0764639
\(857\) −4.78596e36 −1.04162 −0.520810 0.853673i \(-0.674370\pi\)
−0.520810 + 0.853673i \(0.674370\pi\)
\(858\) 4.56681e36 0.979538
\(859\) −4.96565e36 −1.04969 −0.524846 0.851197i \(-0.675877\pi\)
−0.524846 + 0.851197i \(0.675877\pi\)
\(860\) 0 0
\(861\) 3.14487e36 0.645749
\(862\) −2.65270e36 −0.536843
\(863\) 3.87006e36 0.771939 0.385969 0.922512i \(-0.373867\pi\)
0.385969 + 0.922512i \(0.373867\pi\)
\(864\) −2.74719e35 −0.0540090
\(865\) 0 0
\(866\) −2.21686e36 −0.423413
\(867\) 2.56178e36 0.482284
\(868\) −9.93600e35 −0.184381
\(869\) −5.90033e35 −0.107927
\(870\) 0 0
\(871\) 2.77613e36 0.493416
\(872\) 1.20161e36 0.210527
\(873\) 6.14949e36 1.06209
\(874\) −1.89540e36 −0.322706
\(875\) 0 0
\(876\) 1.50895e36 0.249674
\(877\) 9.31488e36 1.51944 0.759718 0.650253i \(-0.225337\pi\)
0.759718 + 0.650253i \(0.225337\pi\)
\(878\) 1.71634e36 0.276008
\(879\) 7.65493e36 1.21361
\(880\) 0 0
\(881\) 3.30000e36 0.508528 0.254264 0.967135i \(-0.418167\pi\)
0.254264 + 0.967135i \(0.418167\pi\)
\(882\) −4.22064e36 −0.641241
\(883\) −6.02750e36 −0.902876 −0.451438 0.892302i \(-0.649089\pi\)
−0.451438 + 0.892302i \(0.649089\pi\)
\(884\) 1.62768e36 0.240390
\(885\) 0 0
\(886\) −2.35675e36 −0.338369
\(887\) 3.09975e36 0.438814 0.219407 0.975633i \(-0.429588\pi\)
0.219407 + 0.975633i \(0.429588\pi\)
\(888\) −3.86432e36 −0.539400
\(889\) 1.16991e37 1.61020
\(890\) 0 0
\(891\) 1.52060e37 2.03491
\(892\) −3.48535e36 −0.459924
\(893\) 3.31538e36 0.431410
\(894\) −7.67431e36 −0.984737
\(895\) 0 0
\(896\) −1.04517e36 −0.130418
\(897\) −8.13399e36 −1.00092
\(898\) −3.40583e36 −0.413304
\(899\) −3.70483e36 −0.443377
\(900\) 0 0
\(901\) 9.48594e36 1.10413
\(902\) −3.50316e36 −0.402141
\(903\) −9.27413e36 −1.04997
\(904\) 3.32488e36 0.371254
\(905\) 0 0
\(906\) −8.37510e36 −0.909678
\(907\) −1.94270e36 −0.208120 −0.104060 0.994571i \(-0.533183\pi\)
−0.104060 + 0.994571i \(0.533183\pi\)
\(908\) −7.96838e36 −0.841971
\(909\) −4.08744e36 −0.425993
\(910\) 0 0
\(911\) −1.71377e37 −1.73769 −0.868845 0.495085i \(-0.835137\pi\)
−0.868845 + 0.495085i \(0.835137\pi\)
\(912\) 1.21496e36 0.121514
\(913\) 2.92042e37 2.88111
\(914\) 7.14510e36 0.695314
\(915\) 0 0
\(916\) 7.51453e36 0.711555
\(917\) −2.06762e37 −1.93132
\(918\) 1.87202e36 0.172495
\(919\) −8.92079e35 −0.0810886 −0.0405443 0.999178i \(-0.512909\pi\)
−0.0405443 + 0.999178i \(0.512909\pi\)
\(920\) 0 0
\(921\) 2.45052e37 2.16777
\(922\) −5.40542e36 −0.471729
\(923\) −1.28972e37 −1.11039
\(924\) 1.99827e37 1.69728
\(925\) 0 0
\(926\) −5.74870e36 −0.475261
\(927\) −6.21177e36 −0.506662
\(928\) −3.89714e36 −0.313615
\(929\) −1.38002e36 −0.109569 −0.0547846 0.998498i \(-0.517447\pi\)
−0.0547846 + 0.998498i \(0.517447\pi\)
\(930\) 0 0
\(931\) −5.56359e36 −0.430016
\(932\) −2.00608e36 −0.152985
\(933\) 2.77976e37 2.09164
\(934\) 3.14937e36 0.233823
\(935\) 0 0
\(936\) 2.26884e36 0.164005
\(937\) −1.08251e37 −0.772126 −0.386063 0.922472i \(-0.626165\pi\)
−0.386063 + 0.922472i \(0.626165\pi\)
\(938\) 1.21473e37 0.854960
\(939\) 1.80930e37 1.25658
\(940\) 0 0
\(941\) −7.75055e36 −0.524158 −0.262079 0.965046i \(-0.584408\pi\)
−0.262079 + 0.965046i \(0.584408\pi\)
\(942\) −9.91066e36 −0.661403
\(943\) 6.23952e36 0.410918
\(944\) 2.41098e36 0.156691
\(945\) 0 0
\(946\) 1.03307e37 0.653870
\(947\) −5.61736e36 −0.350879 −0.175440 0.984490i \(-0.556135\pi\)
−0.175440 + 0.984490i \(0.556135\pi\)
\(948\) −6.73642e35 −0.0415265
\(949\) 3.71444e36 0.225978
\(950\) 0 0
\(951\) −2.89117e36 −0.171324
\(952\) 7.12216e36 0.416534
\(953\) 8.09802e36 0.467431 0.233715 0.972305i \(-0.424912\pi\)
0.233715 + 0.972305i \(0.424912\pi\)
\(954\) 1.32225e37 0.753286
\(955\) 0 0
\(956\) −2.91331e36 −0.161682
\(957\) 7.45095e37 4.08143
\(958\) −1.09973e37 −0.594591
\(959\) −4.83299e37 −2.57919
\(960\) 0 0
\(961\) −1.80315e37 −0.937540
\(962\) −9.51244e36 −0.488206
\(963\) 3.28878e36 0.166612
\(964\) 1.09779e37 0.548982
\(965\) 0 0
\(966\) −3.55914e37 −1.73433
\(967\) 8.79417e36 0.423023 0.211511 0.977376i \(-0.432161\pi\)
0.211511 + 0.977376i \(0.432161\pi\)
\(968\) −1.48137e37 −0.703433
\(969\) −8.27915e36 −0.388095
\(970\) 0 0
\(971\) −1.66027e37 −0.758472 −0.379236 0.925300i \(-0.623813\pi\)
−0.379236 + 0.925300i \(0.623813\pi\)
\(972\) 1.39736e37 0.630202
\(973\) −3.32196e37 −1.47905
\(974\) 1.88028e37 0.826487
\(975\) 0 0
\(976\) −7.98106e36 −0.341931
\(977\) −2.34433e37 −0.991601 −0.495801 0.868436i \(-0.665125\pi\)
−0.495801 + 0.868436i \(0.665125\pi\)
\(978\) 2.10056e37 0.877200
\(979\) −3.46574e37 −1.42894
\(980\) 0 0
\(981\) −1.14136e37 −0.458732
\(982\) −2.11018e37 −0.837385
\(983\) 3.99283e37 1.56445 0.782225 0.622996i \(-0.214085\pi\)
0.782225 + 0.622996i \(0.214085\pi\)
\(984\) −3.99957e36 −0.154730
\(985\) 0 0
\(986\) 2.65564e37 1.00163
\(987\) 6.22556e37 2.31854
\(988\) 2.99076e36 0.109981
\(989\) −1.84002e37 −0.668141
\(990\) 0 0
\(991\) 2.11236e37 0.747907 0.373954 0.927447i \(-0.378002\pi\)
0.373954 + 0.927447i \(0.378002\pi\)
\(992\) 1.26364e36 0.0441800
\(993\) −3.59819e37 −1.24228
\(994\) −5.64335e37 −1.92401
\(995\) 0 0
\(996\) 3.33425e37 1.10855
\(997\) −2.11846e37 −0.695555 −0.347777 0.937577i \(-0.613064\pi\)
−0.347777 + 0.937577i \(0.613064\pi\)
\(998\) −3.05562e37 −0.990760
\(999\) −1.09404e37 −0.350319
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 50.26.a.i.1.2 5
5.2 odd 4 50.26.b.h.49.4 10
5.3 odd 4 50.26.b.h.49.7 10
5.4 even 2 50.26.a.j.1.4 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.26.a.i.1.2 5 1.1 even 1 trivial
50.26.a.j.1.4 yes 5 5.4 even 2
50.26.b.h.49.4 10 5.2 odd 4
50.26.b.h.49.7 10 5.3 odd 4