Properties

Label 50.26.a.i.1.3
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 1.3
Root \(-18822.3\) 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)
 
\(f(q)\) \(=\) \(q-4096.00 q^{2} +43423.8 q^{3} +1.67772e7 q^{4} -1.77864e8 q^{6} +1.17021e10 q^{7} -6.87195e10 q^{8} -8.45403e11 q^{9} +1.64780e13 q^{11} +7.28531e11 q^{12} -1.00150e14 q^{13} -4.79317e13 q^{14} +2.81475e14 q^{16} -4.09920e15 q^{17} +3.46277e15 q^{18} +6.47386e15 q^{19} +5.08149e14 q^{21} -6.74937e16 q^{22} +1.11414e17 q^{23} -2.98406e15 q^{24} +4.10213e17 q^{26} -7.35031e16 q^{27} +1.96328e17 q^{28} -2.49627e18 q^{29} -7.13708e18 q^{31} -1.15292e18 q^{32} +7.15536e17 q^{33} +1.67903e19 q^{34} -1.41835e19 q^{36} +1.51372e19 q^{37} -2.65169e19 q^{38} -4.34889e18 q^{39} +2.55381e20 q^{41} -2.08138e18 q^{42} +1.31517e20 q^{43} +2.76454e20 q^{44} -4.56351e20 q^{46} +5.62025e20 q^{47} +1.22227e19 q^{48} -1.20413e21 q^{49} -1.78003e20 q^{51} -1.68023e21 q^{52} -5.11971e21 q^{53} +3.01069e20 q^{54} -8.04160e20 q^{56} +2.81120e20 q^{57} +1.02247e22 q^{58} +1.95274e22 q^{59} +2.40784e22 q^{61} +2.92335e22 q^{62} -9.89297e21 q^{63} +4.72237e21 q^{64} -2.93083e21 q^{66} -3.73512e22 q^{67} -6.87732e22 q^{68} +4.83801e21 q^{69} -1.22452e23 q^{71} +5.80957e22 q^{72} +1.02009e23 q^{73} -6.20018e22 q^{74} +1.08613e23 q^{76} +1.92826e23 q^{77} +1.78130e22 q^{78} -3.06262e23 q^{79} +7.13109e23 q^{81} -1.04604e24 q^{82} -1.24177e23 q^{83} +8.52532e21 q^{84} -5.38694e23 q^{86} -1.08398e23 q^{87} -1.13236e24 q^{88} +6.70415e23 q^{89} -1.17196e24 q^{91} +1.86921e24 q^{92} -3.09919e23 q^{93} -2.30205e24 q^{94} -5.00642e22 q^{96} -2.48231e24 q^{97} +4.93212e24 q^{98} -1.39305e25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20480 q^{2} - 723995 q^{3} + 83886080 q^{4} + 2965483520 q^{6} - 49218886190 q^{7} - 343597383680 q^{8} + 975375361390 q^{9} - 8837033983815 q^{11} - 12146620497920 q^{12} - 67609989586220 q^{13}+ \cdots - 23\!\cdots\!70 q^{99}+O(q^{100}) \) 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
\(3\) 43423.8 0.0471750 0.0235875 0.999722i \(-0.492491\pi\)
0.0235875 + 0.999722i \(0.492491\pi\)
\(4\) 1.67772e7 0.500000
\(5\) 0 0
\(6\) −1.77864e8 −0.0333578
\(7\) 1.17021e10 0.319549 0.159774 0.987154i \(-0.448923\pi\)
0.159774 + 0.987154i \(0.448923\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) −8.45403e11 −0.997775
\(10\) 0 0
\(11\) 1.64780e13 1.58305 0.791525 0.611137i \(-0.209287\pi\)
0.791525 + 0.611137i \(0.209287\pi\)
\(12\) 7.28531e11 0.0235875
\(13\) −1.00150e14 −1.19222 −0.596112 0.802901i \(-0.703289\pi\)
−0.596112 + 0.802901i \(0.703289\pi\)
\(14\) −4.79317e13 −0.225955
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) −4.09920e15 −1.70643 −0.853214 0.521561i \(-0.825350\pi\)
−0.853214 + 0.521561i \(0.825350\pi\)
\(18\) 3.46277e15 0.705533
\(19\) 6.47386e15 0.671032 0.335516 0.942035i \(-0.391089\pi\)
0.335516 + 0.942035i \(0.391089\pi\)
\(20\) 0 0
\(21\) 5.08149e14 0.0150747
\(22\) −6.74937e16 −1.11939
\(23\) 1.11414e17 1.06009 0.530043 0.847971i \(-0.322176\pi\)
0.530043 + 0.847971i \(0.322176\pi\)
\(24\) −2.98406e15 −0.0166789
\(25\) 0 0
\(26\) 4.10213e17 0.843030
\(27\) −7.35031e16 −0.0942451
\(28\) 1.96328e17 0.159774
\(29\) −2.49627e18 −1.31014 −0.655070 0.755569i \(-0.727361\pi\)
−0.655070 + 0.755569i \(0.727361\pi\)
\(30\) 0 0
\(31\) −7.13708e18 −1.62742 −0.813710 0.581271i \(-0.802556\pi\)
−0.813710 + 0.581271i \(0.802556\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) 7.15536e17 0.0746805
\(34\) 1.67903e19 1.20663
\(35\) 0 0
\(36\) −1.41835e19 −0.498887
\(37\) 1.51372e19 0.378027 0.189013 0.981974i \(-0.439471\pi\)
0.189013 + 0.981974i \(0.439471\pi\)
\(38\) −2.65169e19 −0.474491
\(39\) −4.34889e18 −0.0562432
\(40\) 0 0
\(41\) 2.55381e20 1.76762 0.883811 0.467843i \(-0.154969\pi\)
0.883811 + 0.467843i \(0.154969\pi\)
\(42\) −2.08138e18 −0.0106594
\(43\) 1.31517e20 0.501911 0.250955 0.967999i \(-0.419255\pi\)
0.250955 + 0.967999i \(0.419255\pi\)
\(44\) 2.76454e20 0.791525
\(45\) 0 0
\(46\) −4.56351e20 −0.749594
\(47\) 5.62025e20 0.705557 0.352778 0.935707i \(-0.385237\pi\)
0.352778 + 0.935707i \(0.385237\pi\)
\(48\) 1.22227e19 0.0117938
\(49\) −1.20413e21 −0.897889
\(50\) 0 0
\(51\) −1.78003e20 −0.0805008
\(52\) −1.68023e21 −0.596112
\(53\) −5.11971e21 −1.43152 −0.715759 0.698348i \(-0.753919\pi\)
−0.715759 + 0.698348i \(0.753919\pi\)
\(54\) 3.01069e20 0.0666413
\(55\) 0 0
\(56\) −8.04160e20 −0.112978
\(57\) 2.81120e20 0.0316560
\(58\) 1.02247e22 0.926408
\(59\) 1.95274e22 1.42888 0.714438 0.699699i \(-0.246682\pi\)
0.714438 + 0.699699i \(0.246682\pi\)
\(60\) 0 0
\(61\) 2.40784e22 1.16146 0.580731 0.814096i \(-0.302767\pi\)
0.580731 + 0.814096i \(0.302767\pi\)
\(62\) 2.92335e22 1.15076
\(63\) −9.89297e21 −0.318838
\(64\) 4.72237e21 0.125000
\(65\) 0 0
\(66\) −2.93083e21 −0.0528071
\(67\) −3.73512e22 −0.557660 −0.278830 0.960340i \(-0.589947\pi\)
−0.278830 + 0.960340i \(0.589947\pi\)
\(68\) −6.87732e22 −0.853214
\(69\) 4.83801e21 0.0500096
\(70\) 0 0
\(71\) −1.22452e23 −0.885601 −0.442800 0.896620i \(-0.646015\pi\)
−0.442800 + 0.896620i \(0.646015\pi\)
\(72\) 5.80957e22 0.352767
\(73\) 1.02009e23 0.521316 0.260658 0.965431i \(-0.416060\pi\)
0.260658 + 0.965431i \(0.416060\pi\)
\(74\) −6.20018e22 −0.267305
\(75\) 0 0
\(76\) 1.08613e23 0.335516
\(77\) 1.92826e23 0.505862
\(78\) 1.78130e22 0.0397700
\(79\) −3.06262e23 −0.583115 −0.291558 0.956553i \(-0.594173\pi\)
−0.291558 + 0.956553i \(0.594173\pi\)
\(80\) 0 0
\(81\) 7.13109e23 0.993328
\(82\) −1.04604e24 −1.24990
\(83\) −1.24177e23 −0.127516 −0.0637581 0.997965i \(-0.520309\pi\)
−0.0637581 + 0.997965i \(0.520309\pi\)
\(84\) 8.52532e21 0.00753736
\(85\) 0 0
\(86\) −5.38694e23 −0.354905
\(87\) −1.08398e23 −0.0618059
\(88\) −1.13236e24 −0.559693
\(89\) 6.70415e23 0.287719 0.143859 0.989598i \(-0.454049\pi\)
0.143859 + 0.989598i \(0.454049\pi\)
\(90\) 0 0
\(91\) −1.17196e24 −0.380974
\(92\) 1.86921e24 0.530043
\(93\) −3.09919e23 −0.0767736
\(94\) −2.30205e24 −0.498904
\(95\) 0 0
\(96\) −5.00642e22 −0.00833945
\(97\) −2.48231e24 −0.363253 −0.181626 0.983368i \(-0.558136\pi\)
−0.181626 + 0.983368i \(0.558136\pi\)
\(98\) 4.93212e24 0.634903
\(99\) −1.39305e25 −1.57953
\(100\) 0 0
\(101\) 5.99639e24 0.529508 0.264754 0.964316i \(-0.414709\pi\)
0.264754 + 0.964316i \(0.414709\pi\)
\(102\) 7.29100e23 0.0569227
\(103\) −1.43755e25 −0.993474 −0.496737 0.867901i \(-0.665469\pi\)
−0.496737 + 0.867901i \(0.665469\pi\)
\(104\) 6.88224e24 0.421515
\(105\) 0 0
\(106\) 2.09703e25 1.01224
\(107\) −2.52288e25 −1.08293 −0.541463 0.840725i \(-0.682129\pi\)
−0.541463 + 0.840725i \(0.682129\pi\)
\(108\) −1.23318e24 −0.0471225
\(109\) 1.90602e25 0.649078 0.324539 0.945872i \(-0.394791\pi\)
0.324539 + 0.945872i \(0.394791\pi\)
\(110\) 0 0
\(111\) 6.57313e23 0.0178334
\(112\) 3.29384e24 0.0798872
\(113\) −7.00404e25 −1.52009 −0.760043 0.649873i \(-0.774822\pi\)
−0.760043 + 0.649873i \(0.774822\pi\)
\(114\) −1.15147e24 −0.0223841
\(115\) 0 0
\(116\) −4.18805e25 −0.655070
\(117\) 8.46669e25 1.18957
\(118\) −7.99843e25 −1.01037
\(119\) −4.79692e25 −0.545287
\(120\) 0 0
\(121\) 1.63176e26 1.50605
\(122\) −9.86252e25 −0.821277
\(123\) 1.10896e25 0.0833877
\(124\) −1.19740e26 −0.813710
\(125\) 0 0
\(126\) 4.05216e25 0.225452
\(127\) −1.37993e26 −0.695519 −0.347759 0.937584i \(-0.613057\pi\)
−0.347759 + 0.937584i \(0.613057\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 5.71097e24 0.0236777
\(130\) 0 0
\(131\) −5.92280e25 −0.202598 −0.101299 0.994856i \(-0.532300\pi\)
−0.101299 + 0.994856i \(0.532300\pi\)
\(132\) 1.20047e25 0.0373402
\(133\) 7.57576e25 0.214427
\(134\) 1.52991e26 0.394325
\(135\) 0 0
\(136\) 2.81695e26 0.603313
\(137\) −4.40818e26 −0.861494 −0.430747 0.902473i \(-0.641750\pi\)
−0.430747 + 0.902473i \(0.641750\pi\)
\(138\) −1.98165e25 −0.0353621
\(139\) −8.03278e26 −1.30973 −0.654863 0.755747i \(-0.727274\pi\)
−0.654863 + 0.755747i \(0.727274\pi\)
\(140\) 0 0
\(141\) 2.44053e25 0.0332847
\(142\) 5.01565e26 0.626214
\(143\) −1.65026e27 −1.88735
\(144\) −2.37960e26 −0.249444
\(145\) 0 0
\(146\) −4.17828e26 −0.368626
\(147\) −5.22879e25 −0.0423579
\(148\) 2.53959e26 0.189013
\(149\) 1.20493e27 0.824393 0.412197 0.911095i \(-0.364762\pi\)
0.412197 + 0.911095i \(0.364762\pi\)
\(150\) 0 0
\(151\) 1.12194e27 0.649767 0.324884 0.945754i \(-0.394675\pi\)
0.324884 + 0.945754i \(0.394675\pi\)
\(152\) −4.44880e26 −0.237246
\(153\) 3.46548e27 1.70263
\(154\) −7.89816e26 −0.357698
\(155\) 0 0
\(156\) −7.29622e25 −0.0281216
\(157\) −7.34635e26 −0.261412 −0.130706 0.991421i \(-0.541724\pi\)
−0.130706 + 0.991421i \(0.541724\pi\)
\(158\) 1.25445e27 0.412325
\(159\) −2.22317e26 −0.0675319
\(160\) 0 0
\(161\) 1.30377e27 0.338749
\(162\) −2.92089e27 −0.702389
\(163\) −3.30397e27 −0.735683 −0.367841 0.929889i \(-0.619903\pi\)
−0.367841 + 0.929889i \(0.619903\pi\)
\(164\) 4.28457e27 0.883811
\(165\) 0 0
\(166\) 5.08629e26 0.0901676
\(167\) 8.04476e27 1.32299 0.661496 0.749949i \(-0.269922\pi\)
0.661496 + 0.749949i \(0.269922\pi\)
\(168\) −3.49197e25 −0.00532972
\(169\) 2.97357e27 0.421399
\(170\) 0 0
\(171\) −5.47302e27 −0.669539
\(172\) 2.20649e27 0.250955
\(173\) 1.78239e28 1.88550 0.942749 0.333504i \(-0.108231\pi\)
0.942749 + 0.333504i \(0.108231\pi\)
\(174\) 4.43997e26 0.0437033
\(175\) 0 0
\(176\) 4.63813e27 0.395763
\(177\) 8.47955e26 0.0674073
\(178\) −2.74602e27 −0.203448
\(179\) 2.18421e28 1.50880 0.754399 0.656416i \(-0.227928\pi\)
0.754399 + 0.656416i \(0.227928\pi\)
\(180\) 0 0
\(181\) 3.17113e27 0.190648 0.0953238 0.995446i \(-0.469611\pi\)
0.0953238 + 0.995446i \(0.469611\pi\)
\(182\) 4.80035e27 0.269389
\(183\) 1.04558e27 0.0547920
\(184\) −7.65630e27 −0.374797
\(185\) 0 0
\(186\) 1.26943e27 0.0542871
\(187\) −6.75465e28 −2.70136
\(188\) 9.42921e27 0.352778
\(189\) −8.60139e26 −0.0301159
\(190\) 0 0
\(191\) 3.72320e28 1.14288 0.571439 0.820645i \(-0.306385\pi\)
0.571439 + 0.820645i \(0.306385\pi\)
\(192\) 2.05063e26 0.00589688
\(193\) 5.76046e28 1.55235 0.776176 0.630516i \(-0.217157\pi\)
0.776176 + 0.630516i \(0.217157\pi\)
\(194\) 1.01675e28 0.256859
\(195\) 0 0
\(196\) −2.02020e28 −0.448944
\(197\) 7.70781e28 1.60732 0.803662 0.595086i \(-0.202882\pi\)
0.803662 + 0.595086i \(0.202882\pi\)
\(198\) 5.70594e28 1.11689
\(199\) −3.56814e28 −0.655811 −0.327906 0.944711i \(-0.606343\pi\)
−0.327906 + 0.944711i \(0.606343\pi\)
\(200\) 0 0
\(201\) −1.62193e27 −0.0263076
\(202\) −2.45612e28 −0.374419
\(203\) −2.92116e28 −0.418653
\(204\) −2.98640e27 −0.0402504
\(205\) 0 0
\(206\) 5.88819e28 0.702492
\(207\) −9.41896e28 −1.05773
\(208\) −2.81897e28 −0.298056
\(209\) 1.06676e29 1.06228
\(210\) 0 0
\(211\) 1.75441e29 1.55096 0.775480 0.631373i \(-0.217508\pi\)
0.775480 + 0.631373i \(0.217508\pi\)
\(212\) −8.58944e28 −0.715759
\(213\) −5.31735e27 −0.0417783
\(214\) 1.03337e29 0.765744
\(215\) 0 0
\(216\) 5.05110e27 0.0333207
\(217\) −8.35187e28 −0.520040
\(218\) −7.80707e28 −0.458968
\(219\) 4.42961e27 0.0245931
\(220\) 0 0
\(221\) 4.10534e29 2.03445
\(222\) −2.69235e27 −0.0126101
\(223\) 3.77614e29 1.67200 0.836001 0.548728i \(-0.184888\pi\)
0.836001 + 0.548728i \(0.184888\pi\)
\(224\) −1.34916e28 −0.0564888
\(225\) 0 0
\(226\) 2.86886e29 1.07486
\(227\) −4.12672e29 −1.46312 −0.731562 0.681774i \(-0.761209\pi\)
−0.731562 + 0.681774i \(0.761209\pi\)
\(228\) 4.71641e27 0.0158280
\(229\) −1.29184e29 −0.410454 −0.205227 0.978714i \(-0.565793\pi\)
−0.205227 + 0.978714i \(0.565793\pi\)
\(230\) 0 0
\(231\) 8.37325e27 0.0238641
\(232\) 1.71543e29 0.463204
\(233\) −4.29301e29 −1.09853 −0.549266 0.835648i \(-0.685093\pi\)
−0.549266 + 0.835648i \(0.685093\pi\)
\(234\) −3.46796e29 −0.841154
\(235\) 0 0
\(236\) 3.27616e29 0.714438
\(237\) −1.32991e28 −0.0275085
\(238\) 1.96482e29 0.385576
\(239\) 6.70528e29 1.24866 0.624329 0.781162i \(-0.285373\pi\)
0.624329 + 0.781162i \(0.285373\pi\)
\(240\) 0 0
\(241\) 8.94387e29 1.50076 0.750382 0.661004i \(-0.229870\pi\)
0.750382 + 0.661004i \(0.229870\pi\)
\(242\) −6.68368e29 −1.06494
\(243\) 9.32443e28 0.141105
\(244\) 4.03969e29 0.580731
\(245\) 0 0
\(246\) −4.54230e28 −0.0589640
\(247\) −6.48356e29 −0.800021
\(248\) 4.90457e29 0.575380
\(249\) −5.39224e27 −0.00601558
\(250\) 0 0
\(251\) 1.46535e30 1.47917 0.739587 0.673061i \(-0.235021\pi\)
0.739587 + 0.673061i \(0.235021\pi\)
\(252\) −1.65976e29 −0.159419
\(253\) 1.83587e30 1.67817
\(254\) 5.65217e29 0.491806
\(255\) 0 0
\(256\) 7.92282e28 0.0625000
\(257\) −1.69807e30 −1.27582 −0.637912 0.770110i \(-0.720201\pi\)
−0.637912 + 0.770110i \(0.720201\pi\)
\(258\) −2.33922e28 −0.0167426
\(259\) 1.77136e29 0.120798
\(260\) 0 0
\(261\) 2.11036e30 1.30722
\(262\) 2.42598e29 0.143259
\(263\) 2.11180e30 1.18907 0.594534 0.804070i \(-0.297336\pi\)
0.594534 + 0.804070i \(0.297336\pi\)
\(264\) −4.91712e28 −0.0264035
\(265\) 0 0
\(266\) −3.10303e29 −0.151623
\(267\) 2.91120e28 0.0135731
\(268\) −6.26649e29 −0.278830
\(269\) −1.68743e30 −0.716676 −0.358338 0.933592i \(-0.616656\pi\)
−0.358338 + 0.933592i \(0.616656\pi\)
\(270\) 0 0
\(271\) −1.00044e30 −0.387324 −0.193662 0.981068i \(-0.562036\pi\)
−0.193662 + 0.981068i \(0.562036\pi\)
\(272\) −1.15382e30 −0.426607
\(273\) −5.08910e28 −0.0179725
\(274\) 1.80559e30 0.609168
\(275\) 0 0
\(276\) 8.11684e28 0.0250048
\(277\) −7.83715e29 −0.230760 −0.115380 0.993321i \(-0.536809\pi\)
−0.115380 + 0.993321i \(0.536809\pi\)
\(278\) 3.29023e30 0.926117
\(279\) 6.03371e30 1.62380
\(280\) 0 0
\(281\) 1.80627e30 0.444584 0.222292 0.974980i \(-0.428646\pi\)
0.222292 + 0.974980i \(0.428646\pi\)
\(282\) −9.99639e28 −0.0235358
\(283\) 4.85439e30 1.09346 0.546732 0.837308i \(-0.315872\pi\)
0.546732 + 0.837308i \(0.315872\pi\)
\(284\) −2.05441e30 −0.442800
\(285\) 0 0
\(286\) 6.75948e30 1.33456
\(287\) 2.98848e30 0.564842
\(288\) 9.74683e29 0.176383
\(289\) 1.10328e31 1.91190
\(290\) 0 0
\(291\) −1.07791e29 −0.0171365
\(292\) 1.71142e30 0.260658
\(293\) 1.46069e30 0.213163 0.106582 0.994304i \(-0.466009\pi\)
0.106582 + 0.994304i \(0.466009\pi\)
\(294\) 2.14171e29 0.0299516
\(295\) 0 0
\(296\) −1.04022e30 −0.133653
\(297\) −1.21118e30 −0.149195
\(298\) −4.93541e30 −0.582934
\(299\) −1.11581e31 −1.26386
\(300\) 0 0
\(301\) 1.53902e30 0.160385
\(302\) −4.59547e30 −0.459455
\(303\) 2.60386e29 0.0249796
\(304\) 1.82223e30 0.167758
\(305\) 0 0
\(306\) −1.41946e31 −1.20394
\(307\) 1.83807e30 0.149669 0.0748345 0.997196i \(-0.476157\pi\)
0.0748345 + 0.997196i \(0.476157\pi\)
\(308\) 3.23509e30 0.252931
\(309\) −6.24238e29 −0.0468672
\(310\) 0 0
\(311\) −3.33152e30 −0.230748 −0.115374 0.993322i \(-0.536807\pi\)
−0.115374 + 0.993322i \(0.536807\pi\)
\(312\) 2.98853e29 0.0198850
\(313\) −3.48183e30 −0.222589 −0.111294 0.993787i \(-0.535500\pi\)
−0.111294 + 0.993787i \(0.535500\pi\)
\(314\) 3.00907e30 0.184846
\(315\) 0 0
\(316\) −5.13822e30 −0.291558
\(317\) 3.11024e30 0.169650 0.0848249 0.996396i \(-0.472967\pi\)
0.0848249 + 0.996396i \(0.472967\pi\)
\(318\) 9.10611e29 0.0477523
\(319\) −4.11335e31 −2.07402
\(320\) 0 0
\(321\) −1.09553e30 −0.0510871
\(322\) −5.34025e30 −0.239532
\(323\) −2.65377e31 −1.14507
\(324\) 1.19640e31 0.496664
\(325\) 0 0
\(326\) 1.35330e31 0.520206
\(327\) 8.27668e29 0.0306203
\(328\) −1.75496e31 −0.624949
\(329\) 6.57685e30 0.225460
\(330\) 0 0
\(331\) 2.69668e31 0.856998 0.428499 0.903542i \(-0.359043\pi\)
0.428499 + 0.903542i \(0.359043\pi\)
\(332\) −2.08335e30 −0.0637581
\(333\) −1.27970e31 −0.377186
\(334\) −3.29513e31 −0.935497
\(335\) 0 0
\(336\) 1.43031e29 0.00376868
\(337\) 4.07016e31 1.03333 0.516664 0.856188i \(-0.327174\pi\)
0.516664 + 0.856188i \(0.327174\pi\)
\(338\) −1.21797e31 −0.297974
\(339\) −3.04142e30 −0.0717101
\(340\) 0 0
\(341\) −1.17605e32 −2.57629
\(342\) 2.24175e31 0.473435
\(343\) −2.97841e31 −0.606468
\(344\) −9.03779e30 −0.177452
\(345\) 0 0
\(346\) −7.30066e31 −1.33325
\(347\) 1.58471e31 0.279145 0.139573 0.990212i \(-0.455427\pi\)
0.139573 + 0.990212i \(0.455427\pi\)
\(348\) −1.81861e30 −0.0309029
\(349\) −2.28191e30 −0.0374094 −0.0187047 0.999825i \(-0.505954\pi\)
−0.0187047 + 0.999825i \(0.505954\pi\)
\(350\) 0 0
\(351\) 7.36132e30 0.112361
\(352\) −1.89978e31 −0.279846
\(353\) −8.46470e31 −1.20345 −0.601725 0.798704i \(-0.705520\pi\)
−0.601725 + 0.798704i \(0.705520\pi\)
\(354\) −3.47322e30 −0.0476642
\(355\) 0 0
\(356\) 1.12477e31 0.143859
\(357\) −2.08300e30 −0.0257239
\(358\) −8.94652e31 −1.06688
\(359\) −2.47468e31 −0.284996 −0.142498 0.989795i \(-0.545513\pi\)
−0.142498 + 0.989795i \(0.545513\pi\)
\(360\) 0 0
\(361\) −5.11656e31 −0.549716
\(362\) −1.29889e31 −0.134808
\(363\) 7.08572e30 0.0710479
\(364\) −1.96622e31 −0.190487
\(365\) 0 0
\(366\) −4.28268e30 −0.0387438
\(367\) 2.15791e32 1.88672 0.943359 0.331773i \(-0.107647\pi\)
0.943359 + 0.331773i \(0.107647\pi\)
\(368\) 3.13602e31 0.265022
\(369\) −2.15899e32 −1.76369
\(370\) 0 0
\(371\) −5.99112e31 −0.457440
\(372\) −5.19958e30 −0.0383868
\(373\) 1.43569e32 1.02494 0.512471 0.858705i \(-0.328730\pi\)
0.512471 + 0.858705i \(0.328730\pi\)
\(374\) 2.76670e32 1.91015
\(375\) 0 0
\(376\) −3.86220e31 −0.249452
\(377\) 2.50001e32 1.56198
\(378\) 3.52313e30 0.0212952
\(379\) 1.53691e32 0.898791 0.449396 0.893333i \(-0.351639\pi\)
0.449396 + 0.893333i \(0.351639\pi\)
\(380\) 0 0
\(381\) −5.99216e30 −0.0328111
\(382\) −1.52502e32 −0.808137
\(383\) −2.05714e32 −1.05506 −0.527532 0.849535i \(-0.676882\pi\)
−0.527532 + 0.849535i \(0.676882\pi\)
\(384\) −8.39939e29 −0.00416972
\(385\) 0 0
\(386\) −2.35948e32 −1.09768
\(387\) −1.11185e32 −0.500794
\(388\) −4.16462e31 −0.181626
\(389\) −1.97058e32 −0.832192 −0.416096 0.909321i \(-0.636602\pi\)
−0.416096 + 0.909321i \(0.636602\pi\)
\(390\) 0 0
\(391\) −4.56708e32 −1.80896
\(392\) 8.27472e31 0.317452
\(393\) −2.57191e30 −0.00955759
\(394\) −3.15712e32 −1.13655
\(395\) 0 0
\(396\) −2.33715e32 −0.789764
\(397\) 2.04618e31 0.0669980 0.0334990 0.999439i \(-0.489335\pi\)
0.0334990 + 0.999439i \(0.489335\pi\)
\(398\) 1.46151e32 0.463728
\(399\) 3.28968e30 0.0101156
\(400\) 0 0
\(401\) 4.79432e32 1.38491 0.692456 0.721460i \(-0.256529\pi\)
0.692456 + 0.721460i \(0.256529\pi\)
\(402\) 6.64343e30 0.0186023
\(403\) 7.14777e32 1.94025
\(404\) 1.00603e32 0.264754
\(405\) 0 0
\(406\) 1.19651e32 0.296033
\(407\) 2.49429e32 0.598436
\(408\) 1.22323e31 0.0284613
\(409\) 2.61432e32 0.589952 0.294976 0.955505i \(-0.404688\pi\)
0.294976 + 0.955505i \(0.404688\pi\)
\(410\) 0 0
\(411\) −1.91420e31 −0.0406410
\(412\) −2.41180e32 −0.496737
\(413\) 2.28511e32 0.456596
\(414\) 3.85801e32 0.747926
\(415\) 0 0
\(416\) 1.15465e32 0.210758
\(417\) −3.48814e31 −0.0617864
\(418\) −4.36945e32 −0.751144
\(419\) −2.37687e32 −0.396579 −0.198290 0.980143i \(-0.563539\pi\)
−0.198290 + 0.980143i \(0.563539\pi\)
\(420\) 0 0
\(421\) 7.17539e32 1.12803 0.564013 0.825766i \(-0.309257\pi\)
0.564013 + 0.825766i \(0.309257\pi\)
\(422\) −7.18606e32 −1.09669
\(423\) −4.75137e32 −0.703987
\(424\) 3.51824e32 0.506118
\(425\) 0 0
\(426\) 2.17799e31 0.0295417
\(427\) 2.81767e32 0.371144
\(428\) −4.23269e32 −0.541463
\(429\) −7.16607e31 −0.0890359
\(430\) 0 0
\(431\) 4.94839e32 0.580094 0.290047 0.957012i \(-0.406329\pi\)
0.290047 + 0.957012i \(0.406329\pi\)
\(432\) −2.06893e31 −0.0235613
\(433\) −1.87459e32 −0.207399 −0.103700 0.994609i \(-0.533068\pi\)
−0.103700 + 0.994609i \(0.533068\pi\)
\(434\) 3.42092e32 0.367724
\(435\) 0 0
\(436\) 3.19778e32 0.324539
\(437\) 7.21278e32 0.711352
\(438\) −1.81437e31 −0.0173900
\(439\) −2.54653e32 −0.237215 −0.118607 0.992941i \(-0.537843\pi\)
−0.118607 + 0.992941i \(0.537843\pi\)
\(440\) 0 0
\(441\) 1.01798e33 0.895890
\(442\) −1.68155e33 −1.43857
\(443\) 1.63920e33 1.36328 0.681642 0.731685i \(-0.261266\pi\)
0.681642 + 0.731685i \(0.261266\pi\)
\(444\) 1.10279e31 0.00891672
\(445\) 0 0
\(446\) −1.54671e33 −1.18228
\(447\) 5.23228e31 0.0388908
\(448\) 5.52615e31 0.0399436
\(449\) −1.07131e33 −0.753068 −0.376534 0.926403i \(-0.622884\pi\)
−0.376534 + 0.926403i \(0.622884\pi\)
\(450\) 0 0
\(451\) 4.20815e33 2.79824
\(452\) −1.17508e33 −0.760043
\(453\) 4.87189e31 0.0306528
\(454\) 1.69030e33 1.03459
\(455\) 0 0
\(456\) −1.93184e31 −0.0111921
\(457\) −1.92628e33 −1.08584 −0.542921 0.839784i \(-0.682682\pi\)
−0.542921 + 0.839784i \(0.682682\pi\)
\(458\) 5.29137e32 0.290235
\(459\) 3.01304e32 0.160822
\(460\) 0 0
\(461\) 1.76482e33 0.892152 0.446076 0.894995i \(-0.352821\pi\)
0.446076 + 0.894995i \(0.352821\pi\)
\(462\) −3.42968e31 −0.0168744
\(463\) −6.39982e32 −0.306482 −0.153241 0.988189i \(-0.548971\pi\)
−0.153241 + 0.988189i \(0.548971\pi\)
\(464\) −7.02639e32 −0.327535
\(465\) 0 0
\(466\) 1.75842e33 0.776779
\(467\) −1.87926e33 −0.808213 −0.404106 0.914712i \(-0.632417\pi\)
−0.404106 + 0.914712i \(0.632417\pi\)
\(468\) 1.42048e33 0.594786
\(469\) −4.37087e32 −0.178200
\(470\) 0 0
\(471\) −3.19007e31 −0.0123321
\(472\) −1.34191e33 −0.505184
\(473\) 2.16713e33 0.794550
\(474\) 5.44729e31 0.0194514
\(475\) 0 0
\(476\) −8.04789e32 −0.272643
\(477\) 4.32821e33 1.42833
\(478\) −2.74648e33 −0.882934
\(479\) −4.49267e32 −0.140705 −0.0703527 0.997522i \(-0.522412\pi\)
−0.0703527 + 0.997522i \(0.522412\pi\)
\(480\) 0 0
\(481\) −1.51598e33 −0.450693
\(482\) −3.66341e33 −1.06120
\(483\) 5.66148e31 0.0159805
\(484\) 2.73764e33 0.753024
\(485\) 0 0
\(486\) −3.81928e32 −0.0997766
\(487\) 4.38487e33 1.11646 0.558232 0.829685i \(-0.311480\pi\)
0.558232 + 0.829685i \(0.311480\pi\)
\(488\) −1.65466e33 −0.410639
\(489\) −1.43471e32 −0.0347059
\(490\) 0 0
\(491\) 8.31613e33 1.91162 0.955812 0.293978i \(-0.0949794\pi\)
0.955812 + 0.293978i \(0.0949794\pi\)
\(492\) 1.86053e32 0.0416938
\(493\) 1.02327e34 2.23566
\(494\) 2.65566e33 0.565700
\(495\) 0 0
\(496\) −2.00891e33 −0.406855
\(497\) −1.43295e33 −0.282993
\(498\) 2.20866e31 0.00425366
\(499\) −5.09164e33 −0.956315 −0.478158 0.878274i \(-0.658695\pi\)
−0.478158 + 0.878274i \(0.658695\pi\)
\(500\) 0 0
\(501\) 3.49334e32 0.0624122
\(502\) −6.00207e33 −1.04593
\(503\) 1.81699e33 0.308853 0.154427 0.988004i \(-0.450647\pi\)
0.154427 + 0.988004i \(0.450647\pi\)
\(504\) 6.79839e32 0.112726
\(505\) 0 0
\(506\) −7.51973e33 −1.18665
\(507\) 1.29124e32 0.0198795
\(508\) −2.31513e33 −0.347759
\(509\) 5.80999e33 0.851537 0.425768 0.904832i \(-0.360004\pi\)
0.425768 + 0.904832i \(0.360004\pi\)
\(510\) 0 0
\(511\) 1.19371e33 0.166586
\(512\) −3.24519e32 −0.0441942
\(513\) −4.75849e32 −0.0632415
\(514\) 6.95528e33 0.902143
\(515\) 0 0
\(516\) 9.58143e31 0.0118388
\(517\) 9.26101e33 1.11693
\(518\) −7.25549e32 −0.0854171
\(519\) 7.73981e32 0.0889484
\(520\) 0 0
\(521\) −5.31585e33 −0.582238 −0.291119 0.956687i \(-0.594028\pi\)
−0.291119 + 0.956687i \(0.594028\pi\)
\(522\) −8.64403e33 −0.924347
\(523\) 6.52606e33 0.681365 0.340682 0.940178i \(-0.389342\pi\)
0.340682 + 0.940178i \(0.389342\pi\)
\(524\) −9.93681e32 −0.101299
\(525\) 0 0
\(526\) −8.64995e33 −0.840798
\(527\) 2.92564e34 2.77707
\(528\) 2.01405e32 0.0186701
\(529\) 1.36728e33 0.123783
\(530\) 0 0
\(531\) −1.65085e34 −1.42570
\(532\) 1.27100e33 0.107214
\(533\) −2.55763e34 −2.10740
\(534\) −1.19243e32 −0.00959767
\(535\) 0 0
\(536\) 2.56676e33 0.197163
\(537\) 9.48467e32 0.0711776
\(538\) 6.91172e33 0.506766
\(539\) −1.98416e34 −1.42140
\(540\) 0 0
\(541\) 1.24335e34 0.850410 0.425205 0.905097i \(-0.360202\pi\)
0.425205 + 0.905097i \(0.360202\pi\)
\(542\) 4.09779e33 0.273879
\(543\) 1.37702e32 0.00899381
\(544\) 4.72606e33 0.301657
\(545\) 0 0
\(546\) 2.08449e32 0.0127084
\(547\) −1.77086e34 −1.05522 −0.527610 0.849487i \(-0.676912\pi\)
−0.527610 + 0.849487i \(0.676912\pi\)
\(548\) −7.39570e33 −0.430747
\(549\) −2.03560e34 −1.15888
\(550\) 0 0
\(551\) −1.61605e34 −0.879145
\(552\) −3.32466e32 −0.0176811
\(553\) −3.58390e33 −0.186334
\(554\) 3.21010e33 0.163172
\(555\) 0 0
\(556\) −1.34768e34 −0.654863
\(557\) 8.12072e33 0.385838 0.192919 0.981215i \(-0.438205\pi\)
0.192919 + 0.981215i \(0.438205\pi\)
\(558\) −2.47141e34 −1.14820
\(559\) −1.31714e34 −0.598390
\(560\) 0 0
\(561\) −2.93313e33 −0.127437
\(562\) −7.39849e33 −0.314369
\(563\) 1.21025e34 0.502946 0.251473 0.967864i \(-0.419085\pi\)
0.251473 + 0.967864i \(0.419085\pi\)
\(564\) 4.09452e32 0.0166423
\(565\) 0 0
\(566\) −1.98836e34 −0.773195
\(567\) 8.34485e33 0.317417
\(568\) 8.41487e33 0.313107
\(569\) −9.61211e33 −0.349877 −0.174938 0.984579i \(-0.555973\pi\)
−0.174938 + 0.984579i \(0.555973\pi\)
\(570\) 0 0
\(571\) −1.48318e34 −0.516704 −0.258352 0.966051i \(-0.583179\pi\)
−0.258352 + 0.966051i \(0.583179\pi\)
\(572\) −2.76868e34 −0.943676
\(573\) 1.61676e33 0.0539153
\(574\) −1.22408e34 −0.399403
\(575\) 0 0
\(576\) −3.99230e33 −0.124722
\(577\) 4.94604e34 1.51203 0.756014 0.654556i \(-0.227144\pi\)
0.756014 + 0.654556i \(0.227144\pi\)
\(578\) −4.51905e34 −1.35191
\(579\) 2.50141e33 0.0732323
\(580\) 0 0
\(581\) −1.45313e33 −0.0407477
\(582\) 4.41513e32 0.0121173
\(583\) −8.43623e34 −2.26616
\(584\) −7.00998e33 −0.184313
\(585\) 0 0
\(586\) −5.98297e33 −0.150729
\(587\) 5.81688e34 1.43454 0.717272 0.696793i \(-0.245390\pi\)
0.717272 + 0.696793i \(0.245390\pi\)
\(588\) −8.77246e32 −0.0211790
\(589\) −4.62045e34 −1.09205
\(590\) 0 0
\(591\) 3.34703e33 0.0758256
\(592\) 4.26073e33 0.0945067
\(593\) −1.10616e34 −0.240234 −0.120117 0.992760i \(-0.538327\pi\)
−0.120117 + 0.992760i \(0.538327\pi\)
\(594\) 4.96100e33 0.105497
\(595\) 0 0
\(596\) 2.02154e34 0.412197
\(597\) −1.54942e33 −0.0309379
\(598\) 4.57035e34 0.893685
\(599\) 7.59828e34 1.45506 0.727529 0.686077i \(-0.240669\pi\)
0.727529 + 0.686077i \(0.240669\pi\)
\(600\) 0 0
\(601\) 3.55188e34 0.652421 0.326210 0.945297i \(-0.394228\pi\)
0.326210 + 0.945297i \(0.394228\pi\)
\(602\) −6.30384e33 −0.113409
\(603\) 3.15768e34 0.556419
\(604\) 1.88230e34 0.324884
\(605\) 0 0
\(606\) −1.06654e33 −0.0176632
\(607\) −7.28541e34 −1.18194 −0.590970 0.806694i \(-0.701255\pi\)
−0.590970 + 0.806694i \(0.701255\pi\)
\(608\) −7.46385e33 −0.118623
\(609\) −1.26848e33 −0.0197500
\(610\) 0 0
\(611\) −5.62866e34 −0.841182
\(612\) 5.81411e34 0.851315
\(613\) 1.55275e34 0.222765 0.111382 0.993778i \(-0.464472\pi\)
0.111382 + 0.993778i \(0.464472\pi\)
\(614\) −7.52872e33 −0.105832
\(615\) 0 0
\(616\) −1.32509e34 −0.178849
\(617\) −1.18825e35 −1.57160 −0.785800 0.618481i \(-0.787748\pi\)
−0.785800 + 0.618481i \(0.787748\pi\)
\(618\) 2.55688e33 0.0331401
\(619\) −1.31383e34 −0.166881 −0.0834406 0.996513i \(-0.526591\pi\)
−0.0834406 + 0.996513i \(0.526591\pi\)
\(620\) 0 0
\(621\) −8.18927e33 −0.0999079
\(622\) 1.36459e34 0.163163
\(623\) 7.84524e33 0.0919402
\(624\) −1.22410e33 −0.0140608
\(625\) 0 0
\(626\) 1.42616e34 0.157394
\(627\) 4.63228e33 0.0501130
\(628\) −1.23251e34 −0.130706
\(629\) −6.20503e34 −0.645076
\(630\) 0 0
\(631\) 1.62969e35 1.62831 0.814154 0.580649i \(-0.197201\pi\)
0.814154 + 0.580649i \(0.197201\pi\)
\(632\) 2.10462e34 0.206162
\(633\) 7.61832e33 0.0731666
\(634\) −1.27395e34 −0.119961
\(635\) 0 0
\(636\) −3.72986e33 −0.0337659
\(637\) 1.20593e35 1.07048
\(638\) 1.68483e35 1.46655
\(639\) 1.03522e35 0.883630
\(640\) 0 0
\(641\) −2.15302e35 −1.76735 −0.883675 0.468100i \(-0.844939\pi\)
−0.883675 + 0.468100i \(0.844939\pi\)
\(642\) 4.48729e33 0.0361240
\(643\) 9.81888e34 0.775220 0.387610 0.921823i \(-0.373301\pi\)
0.387610 + 0.921823i \(0.373301\pi\)
\(644\) 2.18737e34 0.169375
\(645\) 0 0
\(646\) 1.08698e35 0.809685
\(647\) −9.66650e34 −0.706262 −0.353131 0.935574i \(-0.614883\pi\)
−0.353131 + 0.935574i \(0.614883\pi\)
\(648\) −4.90044e34 −0.351195
\(649\) 3.21772e35 2.26198
\(650\) 0 0
\(651\) −3.62670e33 −0.0245329
\(652\) −5.54314e34 −0.367841
\(653\) 2.65692e35 1.72967 0.864835 0.502056i \(-0.167423\pi\)
0.864835 + 0.502056i \(0.167423\pi\)
\(654\) −3.39013e33 −0.0216518
\(655\) 0 0
\(656\) 7.18832e34 0.441906
\(657\) −8.62384e34 −0.520156
\(658\) −2.69388e34 −0.159424
\(659\) −1.12403e35 −0.652697 −0.326348 0.945250i \(-0.605818\pi\)
−0.326348 + 0.945250i \(0.605818\pi\)
\(660\) 0 0
\(661\) 1.74137e35 0.973582 0.486791 0.873518i \(-0.338167\pi\)
0.486791 + 0.873518i \(0.338167\pi\)
\(662\) −1.10456e35 −0.605989
\(663\) 1.78270e34 0.0959750
\(664\) 8.53339e33 0.0450838
\(665\) 0 0
\(666\) 5.24165e34 0.266711
\(667\) −2.78120e35 −1.38886
\(668\) 1.34969e35 0.661496
\(669\) 1.63974e34 0.0788768
\(670\) 0 0
\(671\) 3.96763e35 1.83865
\(672\) −5.85855e32 −0.00266486
\(673\) 7.64859e34 0.341502 0.170751 0.985314i \(-0.445381\pi\)
0.170751 + 0.985314i \(0.445381\pi\)
\(674\) −1.66714e35 −0.730673
\(675\) 0 0
\(676\) 4.98882e34 0.210700
\(677\) −3.11236e34 −0.129042 −0.0645211 0.997916i \(-0.520552\pi\)
−0.0645211 + 0.997916i \(0.520552\pi\)
\(678\) 1.24577e34 0.0507067
\(679\) −2.90481e34 −0.116077
\(680\) 0 0
\(681\) −1.79198e34 −0.0690230
\(682\) 4.81708e35 1.82171
\(683\) −1.85231e35 −0.687790 −0.343895 0.939008i \(-0.611746\pi\)
−0.343895 + 0.939008i \(0.611746\pi\)
\(684\) −9.18220e34 −0.334769
\(685\) 0 0
\(686\) 1.21996e35 0.428838
\(687\) −5.60966e33 −0.0193632
\(688\) 3.70188e34 0.125478
\(689\) 5.12737e35 1.70669
\(690\) 0 0
\(691\) −4.90286e35 −1.57389 −0.786944 0.617024i \(-0.788338\pi\)
−0.786944 + 0.617024i \(0.788338\pi\)
\(692\) 2.99035e35 0.942749
\(693\) −1.63016e35 −0.504736
\(694\) −6.49096e34 −0.197386
\(695\) 0 0
\(696\) 7.44904e33 0.0218517
\(697\) −1.04686e36 −3.01632
\(698\) 9.34669e33 0.0264524
\(699\) −1.86419e34 −0.0518233
\(700\) 0 0
\(701\) 4.65445e35 1.24851 0.624257 0.781219i \(-0.285402\pi\)
0.624257 + 0.781219i \(0.285402\pi\)
\(702\) −3.01520e34 −0.0794514
\(703\) 9.79958e34 0.253668
\(704\) 7.78149e34 0.197881
\(705\) 0 0
\(706\) 3.46714e35 0.850967
\(707\) 7.01702e34 0.169204
\(708\) 1.42263e34 0.0337036
\(709\) −1.61968e35 −0.377009 −0.188504 0.982072i \(-0.560364\pi\)
−0.188504 + 0.982072i \(0.560364\pi\)
\(710\) 0 0
\(711\) 2.58915e35 0.581817
\(712\) −4.60705e34 −0.101724
\(713\) −7.95170e35 −1.72521
\(714\) 8.53198e33 0.0181896
\(715\) 0 0
\(716\) 3.66449e35 0.754399
\(717\) 2.91169e34 0.0589055
\(718\) 1.01363e35 0.201523
\(719\) 3.86289e35 0.754747 0.377374 0.926061i \(-0.376827\pi\)
0.377374 + 0.926061i \(0.376827\pi\)
\(720\) 0 0
\(721\) −1.68223e35 −0.317464
\(722\) 2.09574e35 0.388708
\(723\) 3.88377e34 0.0707986
\(724\) 5.32027e34 0.0953238
\(725\) 0 0
\(726\) −2.90231e34 −0.0502384
\(727\) 3.11776e35 0.530471 0.265236 0.964184i \(-0.414550\pi\)
0.265236 + 0.964184i \(0.414550\pi\)
\(728\) 8.05365e34 0.134695
\(729\) −6.00160e35 −0.986672
\(730\) 0 0
\(731\) −5.39115e35 −0.856474
\(732\) 1.75419e34 0.0273960
\(733\) 2.23983e35 0.343886 0.171943 0.985107i \(-0.444995\pi\)
0.171943 + 0.985107i \(0.444995\pi\)
\(734\) −8.83879e35 −1.33411
\(735\) 0 0
\(736\) −1.28451e35 −0.187399
\(737\) −6.15471e35 −0.882804
\(738\) 8.84324e35 1.24712
\(739\) 5.23044e34 0.0725241 0.0362621 0.999342i \(-0.488455\pi\)
0.0362621 + 0.999342i \(0.488455\pi\)
\(740\) 0 0
\(741\) −2.81541e34 −0.0377410
\(742\) 2.45396e35 0.323459
\(743\) −7.47724e35 −0.969127 −0.484564 0.874756i \(-0.661022\pi\)
−0.484564 + 0.874756i \(0.661022\pi\)
\(744\) 2.12975e34 0.0271436
\(745\) 0 0
\(746\) −5.88057e35 −0.724743
\(747\) 1.04980e35 0.127232
\(748\) −1.13324e36 −1.35068
\(749\) −2.95229e35 −0.346048
\(750\) 0 0
\(751\) −1.40867e36 −1.59702 −0.798510 0.601982i \(-0.794378\pi\)
−0.798510 + 0.601982i \(0.794378\pi\)
\(752\) 1.58196e35 0.176389
\(753\) 6.36310e34 0.0697801
\(754\) −1.02401e36 −1.10449
\(755\) 0 0
\(756\) −1.44307e34 −0.0150580
\(757\) 1.39455e36 1.43132 0.715658 0.698451i \(-0.246127\pi\)
0.715658 + 0.698451i \(0.246127\pi\)
\(758\) −6.29519e35 −0.635541
\(759\) 7.97206e34 0.0791677
\(760\) 0 0
\(761\) −1.19222e36 −1.14564 −0.572818 0.819682i \(-0.694150\pi\)
−0.572818 + 0.819682i \(0.694150\pi\)
\(762\) 2.45439e34 0.0232010
\(763\) 2.23044e35 0.207412
\(764\) 6.24650e35 0.571439
\(765\) 0 0
\(766\) 8.42603e35 0.746042
\(767\) −1.95567e36 −1.70354
\(768\) 3.44039e33 0.00294844
\(769\) 4.70446e35 0.396671 0.198336 0.980134i \(-0.436446\pi\)
0.198336 + 0.980134i \(0.436446\pi\)
\(770\) 0 0
\(771\) −7.37365e34 −0.0601870
\(772\) 9.66444e35 0.776176
\(773\) −4.82216e35 −0.381064 −0.190532 0.981681i \(-0.561021\pi\)
−0.190532 + 0.981681i \(0.561021\pi\)
\(774\) 4.55414e35 0.354115
\(775\) 0 0
\(776\) 1.70583e35 0.128429
\(777\) 7.69192e33 0.00569865
\(778\) 8.07148e35 0.588448
\(779\) 1.65330e36 1.18613
\(780\) 0 0
\(781\) −2.01777e36 −1.40195
\(782\) 1.87068e36 1.27913
\(783\) 1.83484e35 0.123474
\(784\) −3.38933e35 −0.224472
\(785\) 0 0
\(786\) 1.05345e34 0.00675824
\(787\) 1.69806e36 1.07219 0.536093 0.844159i \(-0.319900\pi\)
0.536093 + 0.844159i \(0.319900\pi\)
\(788\) 1.29316e36 0.803662
\(789\) 9.17026e34 0.0560944
\(790\) 0 0
\(791\) −8.19618e35 −0.485742
\(792\) 9.57297e35 0.558447
\(793\) −2.41145e36 −1.38472
\(794\) −8.38114e34 −0.0473747
\(795\) 0 0
\(796\) −5.98635e35 −0.327906
\(797\) 2.41463e36 1.30203 0.651017 0.759063i \(-0.274343\pi\)
0.651017 + 0.759063i \(0.274343\pi\)
\(798\) −1.34745e34 −0.00715283
\(799\) −2.30385e36 −1.20398
\(800\) 0 0
\(801\) −5.66770e35 −0.287079
\(802\) −1.96375e36 −0.979281
\(803\) 1.68089e36 0.825270
\(804\) −2.72115e34 −0.0131538
\(805\) 0 0
\(806\) −2.92773e36 −1.37196
\(807\) −7.32747e34 −0.0338092
\(808\) −4.12069e35 −0.187209
\(809\) −4.34243e35 −0.194257 −0.0971283 0.995272i \(-0.530966\pi\)
−0.0971283 + 0.995272i \(0.530966\pi\)
\(810\) 0 0
\(811\) −1.52344e35 −0.0660792 −0.0330396 0.999454i \(-0.510519\pi\)
−0.0330396 + 0.999454i \(0.510519\pi\)
\(812\) −4.90089e35 −0.209327
\(813\) −4.34428e34 −0.0182720
\(814\) −1.02166e36 −0.423158
\(815\) 0 0
\(816\) −5.01034e34 −0.0201252
\(817\) 8.51423e35 0.336798
\(818\) −1.07082e36 −0.417159
\(819\) 9.90778e35 0.380126
\(820\) 0 0
\(821\) 1.12855e36 0.419983 0.209992 0.977703i \(-0.432656\pi\)
0.209992 + 0.977703i \(0.432656\pi\)
\(822\) 7.84056e34 0.0287375
\(823\) −3.43120e36 −1.23865 −0.619325 0.785135i \(-0.712594\pi\)
−0.619325 + 0.785135i \(0.712594\pi\)
\(824\) 9.87875e35 0.351246
\(825\) 0 0
\(826\) −9.35982e35 −0.322862
\(827\) −2.18743e36 −0.743216 −0.371608 0.928390i \(-0.621193\pi\)
−0.371608 + 0.928390i \(0.621193\pi\)
\(828\) −1.58024e36 −0.528864
\(829\) 2.16830e36 0.714804 0.357402 0.933951i \(-0.383663\pi\)
0.357402 + 0.933951i \(0.383663\pi\)
\(830\) 0 0
\(831\) −3.40319e34 −0.0108861
\(832\) −4.72944e35 −0.149028
\(833\) 4.93597e36 1.53218
\(834\) 1.42874e35 0.0436896
\(835\) 0 0
\(836\) 1.78973e36 0.531139
\(837\) 5.24598e35 0.153376
\(838\) 9.73566e35 0.280424
\(839\) −6.23659e36 −1.76980 −0.884898 0.465786i \(-0.845772\pi\)
−0.884898 + 0.465786i \(0.845772\pi\)
\(840\) 0 0
\(841\) 2.60103e36 0.716465
\(842\) −2.93904e36 −0.797635
\(843\) 7.84352e34 0.0209733
\(844\) 2.94341e36 0.775480
\(845\) 0 0
\(846\) 1.94616e36 0.497794
\(847\) 1.90950e36 0.481256
\(848\) −1.44107e36 −0.357879
\(849\) 2.10796e35 0.0515842
\(850\) 0 0
\(851\) 1.68649e36 0.400741
\(852\) −8.92103e34 −0.0208891
\(853\) 1.57022e36 0.362325 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(854\) −1.15412e36 −0.262438
\(855\) 0 0
\(856\) 1.73371e36 0.382872
\(857\) −1.90353e36 −0.414284 −0.207142 0.978311i \(-0.566416\pi\)
−0.207142 + 0.978311i \(0.566416\pi\)
\(858\) 2.93522e35 0.0629579
\(859\) 1.49712e36 0.316476 0.158238 0.987401i \(-0.449419\pi\)
0.158238 + 0.987401i \(0.449419\pi\)
\(860\) 0 0
\(861\) 1.29771e35 0.0266464
\(862\) −2.02686e36 −0.410188
\(863\) −6.49700e36 −1.29592 −0.647960 0.761675i \(-0.724378\pi\)
−0.647960 + 0.761675i \(0.724378\pi\)
\(864\) 8.47433e34 0.0166603
\(865\) 0 0
\(866\) 7.67830e35 0.146653
\(867\) 4.79088e35 0.0901937
\(868\) −1.40121e36 −0.260020
\(869\) −5.04657e36 −0.923101
\(870\) 0 0
\(871\) 3.74072e36 0.664856
\(872\) −1.30981e36 −0.229484
\(873\) 2.09855e36 0.362444
\(874\) −2.95435e36 −0.503002
\(875\) 0 0
\(876\) 7.43165e34 0.0122966
\(877\) −4.25958e36 −0.694819 −0.347410 0.937713i \(-0.612939\pi\)
−0.347410 + 0.937713i \(0.612939\pi\)
\(878\) 1.04306e36 0.167736
\(879\) 6.34286e34 0.0100560
\(880\) 0 0
\(881\) 1.17230e36 0.180652 0.0903258 0.995912i \(-0.471209\pi\)
0.0903258 + 0.995912i \(0.471209\pi\)
\(882\) −4.16963e36 −0.633490
\(883\) 8.13959e36 1.21925 0.609626 0.792689i \(-0.291320\pi\)
0.609626 + 0.792689i \(0.291320\pi\)
\(884\) 6.88762e36 1.01722
\(885\) 0 0
\(886\) −6.71418e36 −0.963988
\(887\) −3.20938e36 −0.454335 −0.227168 0.973856i \(-0.572947\pi\)
−0.227168 + 0.973856i \(0.572947\pi\)
\(888\) −4.51702e34 −0.00630507
\(889\) −1.61480e36 −0.222252
\(890\) 0 0
\(891\) 1.17506e37 1.57249
\(892\) 6.33530e36 0.836001
\(893\) 3.63847e36 0.473451
\(894\) −2.14314e35 −0.0274999
\(895\) 0 0
\(896\) −2.26351e35 −0.0282444
\(897\) −4.84526e35 −0.0596227
\(898\) 4.38807e36 0.532499
\(899\) 1.78161e37 2.13215
\(900\) 0 0
\(901\) 2.09867e37 2.44278
\(902\) −1.72366e37 −1.97865
\(903\) 6.68302e34 0.00756617
\(904\) 4.81314e36 0.537432
\(905\) 0 0
\(906\) −1.99553e35 −0.0216748
\(907\) 1.37794e37 1.47618 0.738091 0.674702i \(-0.235728\pi\)
0.738091 + 0.674702i \(0.235728\pi\)
\(908\) −6.92348e36 −0.731562
\(909\) −5.06937e36 −0.528330
\(910\) 0 0
\(911\) 1.00864e37 1.02272 0.511359 0.859367i \(-0.329142\pi\)
0.511359 + 0.859367i \(0.329142\pi\)
\(912\) 7.91282e34 0.00791399
\(913\) −2.04618e36 −0.201865
\(914\) 7.89003e36 0.767806
\(915\) 0 0
\(916\) −2.16735e36 −0.205227
\(917\) −6.93090e35 −0.0647401
\(918\) −1.23414e36 −0.113719
\(919\) −1.70873e37 −1.55321 −0.776603 0.629991i \(-0.783059\pi\)
−0.776603 + 0.629991i \(0.783059\pi\)
\(920\) 0 0
\(921\) 7.98159e34 0.00706064
\(922\) −7.22870e36 −0.630846
\(923\) 1.22636e37 1.05584
\(924\) 1.40480e35 0.0119320
\(925\) 0 0
\(926\) 2.62137e36 0.216716
\(927\) 1.21531e37 0.991263
\(928\) 2.87801e36 0.231602
\(929\) 1.39851e37 1.11037 0.555185 0.831727i \(-0.312647\pi\)
0.555185 + 0.831727i \(0.312647\pi\)
\(930\) 0 0
\(931\) −7.79537e36 −0.602512
\(932\) −7.20247e36 −0.549266
\(933\) −1.44667e35 −0.0108855
\(934\) 7.69744e36 0.571493
\(935\) 0 0
\(936\) −5.81827e36 −0.420577
\(937\) −5.96148e36 −0.425216 −0.212608 0.977138i \(-0.568196\pi\)
−0.212608 + 0.977138i \(0.568196\pi\)
\(938\) 1.79031e36 0.126006
\(939\) −1.51194e35 −0.0105006
\(940\) 0 0
\(941\) −2.77897e37 −1.87937 −0.939686 0.342038i \(-0.888883\pi\)
−0.939686 + 0.342038i \(0.888883\pi\)
\(942\) 1.30665e35 0.00872014
\(943\) 2.84529e37 1.87383
\(944\) 5.49648e36 0.357219
\(945\) 0 0
\(946\) −8.87658e36 −0.561832
\(947\) 3.29072e36 0.205549 0.102775 0.994705i \(-0.467228\pi\)
0.102775 + 0.994705i \(0.467228\pi\)
\(948\) −2.23121e35 −0.0137542
\(949\) −1.02161e37 −0.621526
\(950\) 0 0
\(951\) 1.35059e35 0.00800324
\(952\) 3.29642e36 0.192788
\(953\) −3.30322e36 −0.190667 −0.0953337 0.995445i \(-0.530392\pi\)
−0.0953337 + 0.995445i \(0.530392\pi\)
\(954\) −1.77284e37 −1.00998
\(955\) 0 0
\(956\) 1.12496e37 0.624329
\(957\) −1.78617e36 −0.0978418
\(958\) 1.84020e36 0.0994938
\(959\) −5.15848e36 −0.275289
\(960\) 0 0
\(961\) 3.17052e37 1.64850
\(962\) 6.20947e36 0.318688
\(963\) 2.13285e37 1.08052
\(964\) 1.50053e37 0.750382
\(965\) 0 0
\(966\) −2.31894e35 −0.0112999
\(967\) −1.07284e37 −0.516064 −0.258032 0.966136i \(-0.583074\pi\)
−0.258032 + 0.966136i \(0.583074\pi\)
\(968\) −1.12134e37 −0.532468
\(969\) −1.15237e36 −0.0540186
\(970\) 0 0
\(971\) −3.81170e36 −0.174132 −0.0870659 0.996203i \(-0.527749\pi\)
−0.0870659 + 0.996203i \(0.527749\pi\)
\(972\) 1.56438e36 0.0705527
\(973\) −9.40001e36 −0.418522
\(974\) −1.79604e37 −0.789459
\(975\) 0 0
\(976\) 6.77747e36 0.290365
\(977\) 4.24334e37 1.79484 0.897420 0.441178i \(-0.145439\pi\)
0.897420 + 0.441178i \(0.145439\pi\)
\(978\) 5.87657e35 0.0245408
\(979\) 1.10471e37 0.455473
\(980\) 0 0
\(981\) −1.61136e37 −0.647634
\(982\) −3.40629e37 −1.35172
\(983\) −2.83202e37 −1.10963 −0.554814 0.831974i \(-0.687211\pi\)
−0.554814 + 0.831974i \(0.687211\pi\)
\(984\) −7.62071e35 −0.0294820
\(985\) 0 0
\(986\) −4.19133e37 −1.58085
\(987\) 2.85592e35 0.0106361
\(988\) −1.08776e37 −0.400010
\(989\) 1.46528e37 0.532069
\(990\) 0 0
\(991\) −2.64697e37 −0.937193 −0.468596 0.883412i \(-0.655240\pi\)
−0.468596 + 0.883412i \(0.655240\pi\)
\(992\) 8.22850e36 0.287690
\(993\) 1.17100e36 0.0404289
\(994\) 5.86935e36 0.200106
\(995\) 0 0
\(996\) −9.04668e34 −0.00300779
\(997\) 2.72983e37 0.896287 0.448143 0.893962i \(-0.352085\pi\)
0.448143 + 0.893962i \(0.352085\pi\)
\(998\) 2.08554e37 0.676217
\(999\) −1.11263e36 −0.0356272
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.1.3 5
5.2 odd 4 50.26.b.h.49.3 10
5.3 odd 4 50.26.b.h.49.8 10
5.4 even 2 50.26.a.j.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.26.a.i.1.3 5 1.1 even 1 trivial
50.26.a.j.1.3 yes 5 5.4 even 2
50.26.b.h.49.3 10 5.2 odd 4
50.26.b.h.49.8 10 5.3 odd 4