Properties

Label 10.24.a.a.1.1
Level $10$
Weight $24$
Character 10.1
Self dual yes
Analytic conductor $33.520$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,24,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, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 24 \)
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: \(33.5204037345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{219241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 54810 \) 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 1.1
Root \(234.616\) 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)
 
\(f(q)\) \(=\) \(q-2048.00 q^{2} -248056. q^{3} +4.19430e6 q^{4} +4.88281e7 q^{5} +5.08019e8 q^{6} -5.53518e9 q^{7} -8.58993e9 q^{8} -3.26113e10 q^{9} -1.00000e11 q^{10} +1.14906e12 q^{11} -1.04042e12 q^{12} +5.00457e12 q^{13} +1.13361e13 q^{14} -1.21121e13 q^{15} +1.75922e13 q^{16} +2.66925e14 q^{17} +6.67879e13 q^{18} -8.29918e14 q^{19} +2.04800e14 q^{20} +1.37304e15 q^{21} -2.35328e15 q^{22} +4.09173e15 q^{23} +2.13079e15 q^{24} +2.38419e15 q^{25} -1.02494e16 q^{26} +3.14422e16 q^{27} -2.32162e16 q^{28} -7.61938e16 q^{29} +2.48056e16 q^{30} -9.68453e16 q^{31} -3.60288e16 q^{32} -2.85032e17 q^{33} -5.46662e17 q^{34} -2.70273e17 q^{35} -1.36782e17 q^{36} +4.73319e16 q^{37} +1.69967e18 q^{38} -1.24142e18 q^{39} -4.19430e17 q^{40} -2.15162e18 q^{41} -2.81198e18 q^{42} -7.08076e18 q^{43} +4.81952e18 q^{44} -1.59235e18 q^{45} -8.37987e18 q^{46} -1.89843e19 q^{47} -4.36385e18 q^{48} +3.26951e18 q^{49} -4.88281e18 q^{50} -6.62123e19 q^{51} +2.09907e19 q^{52} +3.73319e19 q^{53} -6.43937e19 q^{54} +5.61066e19 q^{55} +4.75469e19 q^{56} +2.05866e20 q^{57} +1.56045e20 q^{58} +2.48317e20 q^{59} -5.08019e19 q^{60} -2.46263e20 q^{61} +1.98339e20 q^{62} +1.80510e20 q^{63} +7.37870e19 q^{64} +2.44364e20 q^{65} +5.83747e20 q^{66} -8.02335e20 q^{67} +1.11956e21 q^{68} -1.01498e21 q^{69} +5.53518e20 q^{70} -9.67342e20 q^{71} +2.80129e20 q^{72} +1.76339e21 q^{73} -9.69358e19 q^{74} -5.91412e20 q^{75} -3.48093e21 q^{76} -6.36028e21 q^{77} +2.54242e21 q^{78} -6.00291e21 q^{79} +8.58993e20 q^{80} -4.72931e21 q^{81} +4.40652e21 q^{82} -1.90063e22 q^{83} +5.75893e21 q^{84} +1.30334e22 q^{85} +1.45014e22 q^{86} +1.89003e22 q^{87} -9.87039e21 q^{88} -4.51150e22 q^{89} +3.26113e21 q^{90} -2.77012e22 q^{91} +1.71620e22 q^{92} +2.40231e22 q^{93} +3.88798e22 q^{94} -4.05233e22 q^{95} +8.93717e21 q^{96} +1.75770e22 q^{97} -6.69595e21 q^{98} -3.74725e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4096 q^{2} - 18516 q^{3} + 8388608 q^{4} + 97656250 q^{5} + 37920768 q^{6} - 4415735108 q^{7} - 17179869184 q^{8} - 74065768326 q^{9} - 200000000000 q^{10} + 195017239824 q^{11} - 77661732864 q^{12}+ \cdots + 20\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2048.00 −0.707107
\(3\) −248056. −0.808455 −0.404227 0.914659i \(-0.632460\pi\)
−0.404227 + 0.914659i \(0.632460\pi\)
\(4\) 4.19430e6 0.500000
\(5\) 4.88281e7 0.447214
\(6\) 5.08019e8 0.571664
\(7\) −5.53518e9 −1.05805 −0.529023 0.848607i \(-0.677441\pi\)
−0.529023 + 0.848607i \(0.677441\pi\)
\(8\) −8.58993e9 −0.353553
\(9\) −3.26113e10 −0.346401
\(10\) −1.00000e11 −0.316228
\(11\) 1.14906e12 1.21431 0.607153 0.794585i \(-0.292311\pi\)
0.607153 + 0.794585i \(0.292311\pi\)
\(12\) −1.04042e12 −0.404227
\(13\) 5.00457e12 0.774495 0.387247 0.921976i \(-0.373426\pi\)
0.387247 + 0.921976i \(0.373426\pi\)
\(14\) 1.13361e13 0.748152
\(15\) −1.21121e13 −0.361552
\(16\) 1.75922e13 0.250000
\(17\) 2.66925e14 1.88897 0.944487 0.328549i \(-0.106560\pi\)
0.944487 + 0.328549i \(0.106560\pi\)
\(18\) 6.67879e13 0.244943
\(19\) −8.29918e14 −1.63444 −0.817219 0.576327i \(-0.804486\pi\)
−0.817219 + 0.576327i \(0.804486\pi\)
\(20\) 2.04800e14 0.223607
\(21\) 1.37304e15 0.855382
\(22\) −2.35328e15 −0.858644
\(23\) 4.09173e15 0.895441 0.447721 0.894173i \(-0.352236\pi\)
0.447721 + 0.894173i \(0.352236\pi\)
\(24\) 2.13079e15 0.285832
\(25\) 2.38419e15 0.200000
\(26\) −1.02494e16 −0.547650
\(27\) 3.14422e16 1.08850
\(28\) −2.32162e16 −0.529023
\(29\) −7.61938e16 −1.15969 −0.579846 0.814726i \(-0.696887\pi\)
−0.579846 + 0.814726i \(0.696887\pi\)
\(30\) 2.48056e16 0.255656
\(31\) −9.68453e16 −0.684572 −0.342286 0.939596i \(-0.611201\pi\)
−0.342286 + 0.939596i \(0.611201\pi\)
\(32\) −3.60288e16 −0.176777
\(33\) −2.85032e17 −0.981712
\(34\) −5.46662e17 −1.33571
\(35\) −2.70273e17 −0.473173
\(36\) −1.36782e17 −0.173201
\(37\) 4.73319e16 0.0437355 0.0218678 0.999761i \(-0.493039\pi\)
0.0218678 + 0.999761i \(0.493039\pi\)
\(38\) 1.69967e18 1.15572
\(39\) −1.24142e18 −0.626144
\(40\) −4.19430e17 −0.158114
\(41\) −2.15162e18 −0.610592 −0.305296 0.952257i \(-0.598755\pi\)
−0.305296 + 0.952257i \(0.598755\pi\)
\(42\) −2.81198e18 −0.604847
\(43\) −7.08076e18 −1.16196 −0.580982 0.813916i \(-0.697331\pi\)
−0.580982 + 0.813916i \(0.697331\pi\)
\(44\) 4.81952e18 0.607153
\(45\) −1.59235e18 −0.154915
\(46\) −8.37987e18 −0.633173
\(47\) −1.89843e19 −1.12013 −0.560065 0.828448i \(-0.689224\pi\)
−0.560065 + 0.828448i \(0.689224\pi\)
\(48\) −4.36385e18 −0.202114
\(49\) 3.26951e18 0.119461
\(50\) −4.88281e18 −0.141421
\(51\) −6.62123e19 −1.52715
\(52\) 2.09907e19 0.387247
\(53\) 3.73319e19 0.553233 0.276616 0.960980i \(-0.410787\pi\)
0.276616 + 0.960980i \(0.410787\pi\)
\(54\) −6.43937e19 −0.769689
\(55\) 5.61066e19 0.543054
\(56\) 4.75469e19 0.374076
\(57\) 2.05866e20 1.32137
\(58\) 1.56045e20 0.820026
\(59\) 2.48317e20 1.07203 0.536016 0.844208i \(-0.319929\pi\)
0.536016 + 0.844208i \(0.319929\pi\)
\(60\) −5.08019e19 −0.180776
\(61\) −2.46263e20 −0.724613 −0.362306 0.932059i \(-0.618011\pi\)
−0.362306 + 0.932059i \(0.618011\pi\)
\(62\) 1.98339e20 0.484065
\(63\) 1.80510e20 0.366508
\(64\) 7.37870e19 0.125000
\(65\) 2.44364e20 0.346365
\(66\) 5.83747e20 0.694175
\(67\) −8.02335e20 −0.802594 −0.401297 0.915948i \(-0.631440\pi\)
−0.401297 + 0.915948i \(0.631440\pi\)
\(68\) 1.11956e21 0.944487
\(69\) −1.01498e21 −0.723924
\(70\) 5.53518e20 0.334584
\(71\) −9.67342e20 −0.496717 −0.248358 0.968668i \(-0.579891\pi\)
−0.248358 + 0.968668i \(0.579891\pi\)
\(72\) 2.80129e20 0.122471
\(73\) 1.76339e21 0.657864 0.328932 0.944354i \(-0.393311\pi\)
0.328932 + 0.944354i \(0.393311\pi\)
\(74\) −9.69358e19 −0.0309257
\(75\) −5.91412e20 −0.161691
\(76\) −3.48093e21 −0.817219
\(77\) −6.36028e21 −1.28479
\(78\) 2.54242e21 0.442751
\(79\) −6.00291e21 −0.902921 −0.451461 0.892291i \(-0.649097\pi\)
−0.451461 + 0.892291i \(0.649097\pi\)
\(80\) 8.58993e20 0.111803
\(81\) −4.72931e21 −0.533605
\(82\) 4.40652e21 0.431754
\(83\) −1.90063e22 −1.61994 −0.809969 0.586472i \(-0.800516\pi\)
−0.809969 + 0.586472i \(0.800516\pi\)
\(84\) 5.75893e21 0.427691
\(85\) 1.30334e22 0.844775
\(86\) 1.45014e22 0.821633
\(87\) 1.89003e22 0.937558
\(88\) −9.87039e21 −0.429322
\(89\) −4.51150e22 −1.72320 −0.861601 0.507587i \(-0.830538\pi\)
−0.861601 + 0.507587i \(0.830538\pi\)
\(90\) 3.26113e21 0.109542
\(91\) −2.77012e22 −0.819451
\(92\) 1.71620e22 0.447721
\(93\) 2.40231e22 0.553445
\(94\) 3.88798e22 0.792052
\(95\) −4.05233e22 −0.730943
\(96\) 8.93717e21 0.142916
\(97\) 1.75770e22 0.249499 0.124750 0.992188i \(-0.460187\pi\)
0.124750 + 0.992188i \(0.460187\pi\)
\(98\) −6.69595e21 −0.0844720
\(99\) −3.74725e22 −0.420637
\(100\) 1.00000e22 0.100000
\(101\) 1.72636e23 1.53970 0.769850 0.638225i \(-0.220331\pi\)
0.769850 + 0.638225i \(0.220331\pi\)
\(102\) 1.35603e23 1.07986
\(103\) −7.39196e22 −0.526176 −0.263088 0.964772i \(-0.584741\pi\)
−0.263088 + 0.964772i \(0.584741\pi\)
\(104\) −4.29890e22 −0.273825
\(105\) 6.70428e22 0.382539
\(106\) −7.64558e22 −0.391195
\(107\) 3.64152e23 1.67251 0.836255 0.548340i \(-0.184740\pi\)
0.836255 + 0.548340i \(0.184740\pi\)
\(108\) 1.31878e23 0.544252
\(109\) 2.32344e23 0.862434 0.431217 0.902248i \(-0.358084\pi\)
0.431217 + 0.902248i \(0.358084\pi\)
\(110\) −1.14906e23 −0.383997
\(111\) −1.17410e22 −0.0353582
\(112\) −9.73760e22 −0.264512
\(113\) −6.52015e23 −1.59902 −0.799512 0.600650i \(-0.794909\pi\)
−0.799512 + 0.600650i \(0.794909\pi\)
\(114\) −4.21614e23 −0.934349
\(115\) 1.99792e23 0.400454
\(116\) −3.19580e23 −0.579846
\(117\) −1.63206e23 −0.268286
\(118\) −5.08552e23 −0.758041
\(119\) −1.47748e24 −1.99862
\(120\) 1.04042e23 0.127828
\(121\) 4.24918e23 0.474541
\(122\) 5.04347e23 0.512379
\(123\) 5.33723e23 0.493636
\(124\) −4.06199e23 −0.342286
\(125\) 1.16415e23 0.0894427
\(126\) −3.69683e23 −0.259160
\(127\) −8.69127e23 −0.556340 −0.278170 0.960532i \(-0.589728\pi\)
−0.278170 + 0.960532i \(0.589728\pi\)
\(128\) −1.51116e23 −0.0883883
\(129\) 1.75643e24 0.939396
\(130\) −5.00457e23 −0.244917
\(131\) 1.54721e23 0.0693311 0.0346656 0.999399i \(-0.488963\pi\)
0.0346656 + 0.999399i \(0.488963\pi\)
\(132\) −1.19551e24 −0.490856
\(133\) 4.59375e24 1.72931
\(134\) 1.64318e24 0.567519
\(135\) 1.53527e24 0.486794
\(136\) −2.29286e24 −0.667853
\(137\) −5.27775e23 −0.141307 −0.0706533 0.997501i \(-0.522508\pi\)
−0.0706533 + 0.997501i \(0.522508\pi\)
\(138\) 2.07868e24 0.511891
\(139\) −3.13469e24 −0.710435 −0.355217 0.934784i \(-0.615593\pi\)
−0.355217 + 0.934784i \(0.615593\pi\)
\(140\) −1.13361e24 −0.236586
\(141\) 4.70917e24 0.905575
\(142\) 1.98112e24 0.351232
\(143\) 5.75057e24 0.940474
\(144\) −5.73704e23 −0.0866003
\(145\) −3.72040e24 −0.518630
\(146\) −3.61143e24 −0.465180
\(147\) −8.11022e23 −0.0965791
\(148\) 1.98525e23 0.0218678
\(149\) −2.68240e24 −0.273453 −0.136726 0.990609i \(-0.543658\pi\)
−0.136726 + 0.990609i \(0.543658\pi\)
\(150\) 1.21121e24 0.114333
\(151\) 9.87063e24 0.863197 0.431599 0.902066i \(-0.357950\pi\)
0.431599 + 0.902066i \(0.357950\pi\)
\(152\) 7.12894e24 0.577861
\(153\) −8.70476e24 −0.654343
\(154\) 1.30259e25 0.908485
\(155\) −4.72877e24 −0.306150
\(156\) −5.20687e24 −0.313072
\(157\) −2.58352e25 −1.44333 −0.721666 0.692242i \(-0.756623\pi\)
−0.721666 + 0.692242i \(0.756623\pi\)
\(158\) 1.22940e25 0.638462
\(159\) −9.26042e24 −0.447264
\(160\) −1.75922e24 −0.0790569
\(161\) −2.26485e25 −0.947418
\(162\) 9.68563e24 0.377316
\(163\) −3.31455e24 −0.120301 −0.0601503 0.998189i \(-0.519158\pi\)
−0.0601503 + 0.998189i \(0.519158\pi\)
\(164\) −9.02455e24 −0.305296
\(165\) −1.39176e25 −0.439035
\(166\) 3.89248e25 1.14547
\(167\) −9.23000e24 −0.253491 −0.126745 0.991935i \(-0.540453\pi\)
−0.126745 + 0.991935i \(0.540453\pi\)
\(168\) −1.17943e25 −0.302423
\(169\) −1.67082e25 −0.400158
\(170\) −2.66925e25 −0.597346
\(171\) 2.70647e25 0.566171
\(172\) −2.96989e25 −0.580982
\(173\) 2.50481e25 0.458400 0.229200 0.973379i \(-0.426389\pi\)
0.229200 + 0.973379i \(0.426389\pi\)
\(174\) −3.87079e25 −0.662954
\(175\) −1.31969e25 −0.211609
\(176\) 2.02145e25 0.303577
\(177\) −6.15965e25 −0.866689
\(178\) 9.23955e25 1.21849
\(179\) −7.76500e25 −0.960133 −0.480067 0.877232i \(-0.659388\pi\)
−0.480067 + 0.877232i \(0.659388\pi\)
\(180\) −6.67879e24 −0.0774576
\(181\) 1.79026e26 1.94810 0.974050 0.226331i \(-0.0726731\pi\)
0.974050 + 0.226331i \(0.0726731\pi\)
\(182\) 5.67321e25 0.579439
\(183\) 6.10871e25 0.585817
\(184\) −3.51477e25 −0.316586
\(185\) 2.31113e24 0.0195591
\(186\) −4.91993e25 −0.391345
\(187\) 3.06713e26 2.29379
\(188\) −7.96258e25 −0.560065
\(189\) −1.74039e26 −1.15169
\(190\) 8.29918e25 0.516855
\(191\) −1.87629e26 −1.10006 −0.550031 0.835144i \(-0.685384\pi\)
−0.550031 + 0.835144i \(0.685384\pi\)
\(192\) −1.83033e25 −0.101057
\(193\) 3.19948e26 1.66407 0.832033 0.554726i \(-0.187177\pi\)
0.832033 + 0.554726i \(0.187177\pi\)
\(194\) −3.59977e25 −0.176423
\(195\) −6.06160e25 −0.280020
\(196\) 1.37133e25 0.0597307
\(197\) 2.32413e26 0.954770 0.477385 0.878694i \(-0.341585\pi\)
0.477385 + 0.878694i \(0.341585\pi\)
\(198\) 7.67436e25 0.297435
\(199\) −1.44151e26 −0.527237 −0.263619 0.964627i \(-0.584916\pi\)
−0.263619 + 0.964627i \(0.584916\pi\)
\(200\) −2.04800e25 −0.0707107
\(201\) 1.99024e26 0.648861
\(202\) −3.53559e26 −1.08873
\(203\) 4.21747e26 1.22701
\(204\) −2.77714e26 −0.763575
\(205\) −1.05060e26 −0.273065
\(206\) 1.51387e26 0.372063
\(207\) −1.33437e26 −0.310182
\(208\) 8.80414e25 0.193624
\(209\) −9.53629e26 −1.98471
\(210\) −1.37304e26 −0.270496
\(211\) −9.00904e26 −1.68047 −0.840234 0.542223i \(-0.817583\pi\)
−0.840234 + 0.542223i \(0.817583\pi\)
\(212\) 1.56581e26 0.276616
\(213\) 2.39955e26 0.401573
\(214\) −7.45783e26 −1.18264
\(215\) −3.45740e26 −0.519646
\(216\) −2.70087e26 −0.384844
\(217\) 5.36056e26 0.724308
\(218\) −4.75840e26 −0.609833
\(219\) −4.37420e26 −0.531853
\(220\) 2.35328e26 0.271527
\(221\) 1.33584e27 1.46300
\(222\) 2.40455e25 0.0250020
\(223\) 1.12234e27 1.10820 0.554100 0.832450i \(-0.313063\pi\)
0.554100 + 0.832450i \(0.313063\pi\)
\(224\) 1.99426e26 0.187038
\(225\) −7.77514e25 −0.0692802
\(226\) 1.33533e27 1.13068
\(227\) −1.38912e26 −0.111800 −0.0558999 0.998436i \(-0.517803\pi\)
−0.0558999 + 0.998436i \(0.517803\pi\)
\(228\) 8.63466e26 0.660685
\(229\) −1.03506e27 −0.753105 −0.376553 0.926395i \(-0.622891\pi\)
−0.376553 + 0.926395i \(0.622891\pi\)
\(230\) −4.09173e26 −0.283163
\(231\) 1.57771e27 1.03870
\(232\) 6.54500e26 0.410013
\(233\) −1.03552e27 −0.617398 −0.308699 0.951160i \(-0.599894\pi\)
−0.308699 + 0.951160i \(0.599894\pi\)
\(234\) 3.34245e26 0.189707
\(235\) −9.26967e26 −0.500938
\(236\) 1.04151e27 0.536016
\(237\) 1.48906e27 0.729971
\(238\) 3.02587e27 1.41324
\(239\) −1.71220e27 −0.762041 −0.381021 0.924567i \(-0.624427\pi\)
−0.381021 + 0.924567i \(0.624427\pi\)
\(240\) −2.13079e26 −0.0903880
\(241\) −8.11502e26 −0.328166 −0.164083 0.986447i \(-0.552466\pi\)
−0.164083 + 0.986447i \(0.552466\pi\)
\(242\) −8.70232e26 −0.335551
\(243\) −1.78694e27 −0.657109
\(244\) −1.03290e27 −0.362306
\(245\) 1.59644e26 0.0534248
\(246\) −1.09306e27 −0.349053
\(247\) −4.15338e27 −1.26586
\(248\) 8.31895e26 0.242033
\(249\) 4.71462e27 1.30965
\(250\) −2.38419e26 −0.0632456
\(251\) 2.20197e27 0.557910 0.278955 0.960304i \(-0.410012\pi\)
0.278955 + 0.960304i \(0.410012\pi\)
\(252\) 7.57112e26 0.183254
\(253\) 4.70166e27 1.08734
\(254\) 1.77997e27 0.393392
\(255\) −3.23302e27 −0.682962
\(256\) 3.09485e26 0.0625000
\(257\) −8.24281e27 −1.59164 −0.795819 0.605534i \(-0.792959\pi\)
−0.795819 + 0.605534i \(0.792959\pi\)
\(258\) −3.59716e27 −0.664253
\(259\) −2.61991e26 −0.0462742
\(260\) 1.02494e27 0.173182
\(261\) 2.48478e27 0.401718
\(262\) −3.16868e26 −0.0490245
\(263\) 1.16638e27 0.172722 0.0863610 0.996264i \(-0.472476\pi\)
0.0863610 + 0.996264i \(0.472476\pi\)
\(264\) 2.44841e27 0.347088
\(265\) 1.82285e27 0.247413
\(266\) −9.40799e27 −1.22281
\(267\) 1.11911e28 1.39313
\(268\) −3.36524e27 −0.401297
\(269\) −1.48702e28 −1.69890 −0.849448 0.527673i \(-0.823065\pi\)
−0.849448 + 0.527673i \(0.823065\pi\)
\(270\) −3.14422e27 −0.344215
\(271\) −1.44238e28 −1.51333 −0.756665 0.653803i \(-0.773172\pi\)
−0.756665 + 0.653803i \(0.773172\pi\)
\(272\) 4.69579e27 0.472243
\(273\) 6.87146e27 0.662489
\(274\) 1.08088e27 0.0999188
\(275\) 2.73958e27 0.242861
\(276\) −4.25713e27 −0.361962
\(277\) −1.19544e28 −0.975013 −0.487507 0.873119i \(-0.662094\pi\)
−0.487507 + 0.873119i \(0.662094\pi\)
\(278\) 6.41984e27 0.502353
\(279\) 3.15825e27 0.237136
\(280\) 2.32162e27 0.167292
\(281\) 2.06996e28 1.43166 0.715831 0.698274i \(-0.246048\pi\)
0.715831 + 0.698274i \(0.246048\pi\)
\(282\) −9.64438e27 −0.640338
\(283\) −1.14948e28 −0.732753 −0.366376 0.930467i \(-0.619402\pi\)
−0.366376 + 0.930467i \(0.619402\pi\)
\(284\) −4.05733e27 −0.248358
\(285\) 1.00521e28 0.590934
\(286\) −1.17772e28 −0.665016
\(287\) 1.19096e28 0.646035
\(288\) 1.17495e27 0.0612356
\(289\) 5.12812e28 2.56822
\(290\) 7.61938e27 0.366727
\(291\) −4.36008e27 −0.201709
\(292\) 7.39620e27 0.328932
\(293\) −3.69791e28 −1.58117 −0.790586 0.612352i \(-0.790224\pi\)
−0.790586 + 0.612352i \(0.790224\pi\)
\(294\) 1.66097e27 0.0682917
\(295\) 1.21248e28 0.479427
\(296\) −4.06578e26 −0.0154628
\(297\) 3.61291e28 1.32178
\(298\) 5.49356e27 0.193360
\(299\) 2.04774e28 0.693515
\(300\) −2.48056e27 −0.0808455
\(301\) 3.91933e28 1.22941
\(302\) −2.02150e28 −0.610372
\(303\) −4.28235e28 −1.24478
\(304\) −1.46001e28 −0.408610
\(305\) −1.20246e28 −0.324057
\(306\) 1.78273e28 0.462690
\(307\) −3.17969e28 −0.794866 −0.397433 0.917631i \(-0.630099\pi\)
−0.397433 + 0.917631i \(0.630099\pi\)
\(308\) −2.66770e28 −0.642396
\(309\) 1.83362e28 0.425390
\(310\) 9.68453e27 0.216481
\(311\) −4.33022e28 −0.932750 −0.466375 0.884587i \(-0.654440\pi\)
−0.466375 + 0.884587i \(0.654440\pi\)
\(312\) 1.06637e28 0.221375
\(313\) 4.55810e26 0.00912061 0.00456030 0.999990i \(-0.498548\pi\)
0.00456030 + 0.999990i \(0.498548\pi\)
\(314\) 5.29105e28 1.02059
\(315\) 8.81394e27 0.163907
\(316\) −2.51780e28 −0.451461
\(317\) 5.36147e28 0.927049 0.463524 0.886084i \(-0.346585\pi\)
0.463524 + 0.886084i \(0.346585\pi\)
\(318\) 1.89653e28 0.316263
\(319\) −8.75515e28 −1.40822
\(320\) 3.60288e27 0.0559017
\(321\) −9.03301e28 −1.35215
\(322\) 4.63841e28 0.669926
\(323\) −2.21525e29 −3.08741
\(324\) −1.98362e28 −0.266803
\(325\) 1.19318e28 0.154899
\(326\) 6.78820e27 0.0850653
\(327\) −5.76343e28 −0.697239
\(328\) 1.84823e28 0.215877
\(329\) 1.05081e29 1.18515
\(330\) 2.85032e28 0.310445
\(331\) 3.04232e28 0.320025 0.160012 0.987115i \(-0.448847\pi\)
0.160012 + 0.987115i \(0.448847\pi\)
\(332\) −7.97180e28 −0.809969
\(333\) −1.54356e27 −0.0151500
\(334\) 1.89030e28 0.179245
\(335\) −3.91765e28 −0.358931
\(336\) 2.41547e28 0.213846
\(337\) 5.57310e28 0.476819 0.238409 0.971165i \(-0.423374\pi\)
0.238409 + 0.971165i \(0.423374\pi\)
\(338\) 3.42183e28 0.282954
\(339\) 1.61736e29 1.29274
\(340\) 5.46662e28 0.422387
\(341\) −1.11281e29 −0.831280
\(342\) −5.54285e28 −0.400344
\(343\) 1.33394e29 0.931650
\(344\) 6.08233e28 0.410816
\(345\) −4.95596e28 −0.323749
\(346\) −5.12985e28 −0.324138
\(347\) −2.05152e29 −1.25397 −0.626985 0.779031i \(-0.715711\pi\)
−0.626985 + 0.779031i \(0.715711\pi\)
\(348\) 7.92738e28 0.468779
\(349\) 2.36329e29 1.35215 0.676074 0.736834i \(-0.263680\pi\)
0.676074 + 0.736834i \(0.263680\pi\)
\(350\) 2.70273e28 0.149630
\(351\) 1.57355e29 0.843041
\(352\) −4.13994e28 −0.214661
\(353\) 1.09725e29 0.550677 0.275339 0.961347i \(-0.411210\pi\)
0.275339 + 0.961347i \(0.411210\pi\)
\(354\) 1.26150e29 0.612841
\(355\) −4.72335e28 −0.222139
\(356\) −1.89226e29 −0.861601
\(357\) 3.66497e29 1.61579
\(358\) 1.59027e29 0.678917
\(359\) −3.26602e29 −1.35031 −0.675153 0.737677i \(-0.735923\pi\)
−0.675153 + 0.737677i \(0.735923\pi\)
\(360\) 1.36782e28 0.0547708
\(361\) 4.30934e29 1.67139
\(362\) −3.66644e29 −1.37752
\(363\) −1.05404e29 −0.383645
\(364\) −1.16187e29 −0.409726
\(365\) 8.61031e28 0.294206
\(366\) −1.25106e29 −0.414235
\(367\) −8.85081e28 −0.284003 −0.142002 0.989866i \(-0.545354\pi\)
−0.142002 + 0.989866i \(0.545354\pi\)
\(368\) 7.19825e28 0.223860
\(369\) 7.01671e28 0.211510
\(370\) −4.73319e27 −0.0138304
\(371\) −2.06639e29 −0.585346
\(372\) 1.00760e29 0.276723
\(373\) −3.93901e29 −1.04890 −0.524452 0.851440i \(-0.675730\pi\)
−0.524452 + 0.851440i \(0.675730\pi\)
\(374\) −6.28149e29 −1.62196
\(375\) −2.88775e28 −0.0723104
\(376\) 1.63074e29 0.396026
\(377\) −3.81317e29 −0.898175
\(378\) 3.56431e29 0.814366
\(379\) 1.29839e29 0.287777 0.143888 0.989594i \(-0.454039\pi\)
0.143888 + 0.989594i \(0.454039\pi\)
\(380\) −1.69967e29 −0.365472
\(381\) 2.15592e29 0.449776
\(382\) 3.84265e29 0.777861
\(383\) 2.17964e29 0.428153 0.214076 0.976817i \(-0.431326\pi\)
0.214076 + 0.976817i \(0.431326\pi\)
\(384\) 3.74852e28 0.0714580
\(385\) −3.10561e29 −0.574577
\(386\) −6.55254e29 −1.17667
\(387\) 2.30913e29 0.402506
\(388\) 7.37232e28 0.124750
\(389\) −9.04905e29 −1.48656 −0.743280 0.668980i \(-0.766731\pi\)
−0.743280 + 0.668980i \(0.766731\pi\)
\(390\) 1.24142e29 0.198004
\(391\) 1.09218e30 1.69147
\(392\) −2.80849e28 −0.0422360
\(393\) −3.83794e28 −0.0560511
\(394\) −4.75982e29 −0.675125
\(395\) −2.93111e29 −0.403799
\(396\) −1.57171e29 −0.210319
\(397\) 1.18206e30 1.53656 0.768279 0.640116i \(-0.221114\pi\)
0.768279 + 0.640116i \(0.221114\pi\)
\(398\) 2.95221e29 0.372813
\(399\) −1.13951e30 −1.39807
\(400\) 4.19430e28 0.0500000
\(401\) 9.75353e29 1.12980 0.564900 0.825159i \(-0.308915\pi\)
0.564900 + 0.825159i \(0.308915\pi\)
\(402\) −4.07602e29 −0.458814
\(403\) −4.84669e29 −0.530197
\(404\) 7.24089e29 0.769850
\(405\) −2.30923e29 −0.238636
\(406\) −8.63737e29 −0.867625
\(407\) 5.43874e28 0.0531084
\(408\) 5.68759e29 0.539929
\(409\) 3.52564e29 0.325402 0.162701 0.986675i \(-0.447979\pi\)
0.162701 + 0.986675i \(0.447979\pi\)
\(410\) 2.15162e29 0.193086
\(411\) 1.30918e29 0.114240
\(412\) −3.10041e29 −0.263088
\(413\) −1.37448e30 −1.13426
\(414\) 2.73278e29 0.219332
\(415\) −9.28040e29 −0.724459
\(416\) −1.80309e29 −0.136913
\(417\) 7.77579e29 0.574354
\(418\) 1.95303e30 1.40340
\(419\) −1.40788e30 −0.984246 −0.492123 0.870526i \(-0.663779\pi\)
−0.492123 + 0.870526i \(0.663779\pi\)
\(420\) 2.81198e29 0.191269
\(421\) 7.36867e29 0.487691 0.243845 0.969814i \(-0.421591\pi\)
0.243845 + 0.969814i \(0.421591\pi\)
\(422\) 1.84505e30 1.18827
\(423\) 6.19102e29 0.388014
\(424\) −3.20679e29 −0.195597
\(425\) 6.36398e29 0.377795
\(426\) −4.91428e29 −0.283955
\(427\) 1.36311e30 0.766674
\(428\) 1.52736e30 0.836255
\(429\) −1.42647e30 −0.760331
\(430\) 7.08076e29 0.367445
\(431\) 1.71259e30 0.865298 0.432649 0.901562i \(-0.357579\pi\)
0.432649 + 0.901562i \(0.357579\pi\)
\(432\) 5.53138e29 0.272126
\(433\) 3.34373e30 1.60184 0.800921 0.598770i \(-0.204343\pi\)
0.800921 + 0.598770i \(0.204343\pi\)
\(434\) −1.09784e30 −0.512163
\(435\) 9.22868e29 0.419289
\(436\) 9.74520e29 0.431217
\(437\) −3.39580e30 −1.46354
\(438\) 8.95837e29 0.376077
\(439\) −3.95516e30 −1.61742 −0.808710 0.588208i \(-0.799834\pi\)
−0.808710 + 0.588208i \(0.799834\pi\)
\(440\) −4.81952e29 −0.191999
\(441\) −1.06623e29 −0.0413816
\(442\) −2.73581e30 −1.03450
\(443\) −3.40082e30 −1.25297 −0.626486 0.779433i \(-0.715507\pi\)
−0.626486 + 0.779433i \(0.715507\pi\)
\(444\) −4.92452e28 −0.0176791
\(445\) −2.20288e30 −0.770639
\(446\) −2.29855e30 −0.783616
\(447\) 6.65387e29 0.221074
\(448\) −4.08424e29 −0.132256
\(449\) 3.24298e30 1.02355 0.511777 0.859118i \(-0.328987\pi\)
0.511777 + 0.859118i \(0.328987\pi\)
\(450\) 1.59235e29 0.0489885
\(451\) −2.47235e30 −0.741446
\(452\) −2.73475e30 −0.799512
\(453\) −2.44847e30 −0.697856
\(454\) 2.84491e29 0.0790545
\(455\) −1.35260e30 −0.366470
\(456\) −1.76838e30 −0.467175
\(457\) −2.32028e30 −0.597729 −0.298864 0.954296i \(-0.596608\pi\)
−0.298864 + 0.954296i \(0.596608\pi\)
\(458\) 2.11979e30 0.532526
\(459\) 8.39270e30 2.05616
\(460\) 8.37987e29 0.200227
\(461\) −1.77997e30 −0.414813 −0.207406 0.978255i \(-0.566502\pi\)
−0.207406 + 0.978255i \(0.566502\pi\)
\(462\) −3.23114e30 −0.734469
\(463\) −1.16490e30 −0.258291 −0.129145 0.991626i \(-0.541223\pi\)
−0.129145 + 0.991626i \(0.541223\pi\)
\(464\) −1.34042e30 −0.289923
\(465\) 1.17300e30 0.247508
\(466\) 2.12074e30 0.436566
\(467\) 4.42414e30 0.888555 0.444278 0.895889i \(-0.353460\pi\)
0.444278 + 0.895889i \(0.353460\pi\)
\(468\) −6.84534e29 −0.134143
\(469\) 4.44107e30 0.849181
\(470\) 1.89843e30 0.354216
\(471\) 6.40858e30 1.16687
\(472\) −2.13302e30 −0.379020
\(473\) −8.13625e30 −1.41098
\(474\) −3.04959e30 −0.516167
\(475\) −1.97868e30 −0.326888
\(476\) −6.19699e30 −0.999311
\(477\) −1.21744e30 −0.191640
\(478\) 3.50659e30 0.538845
\(479\) 1.17747e31 1.76641 0.883207 0.468983i \(-0.155379\pi\)
0.883207 + 0.468983i \(0.155379\pi\)
\(480\) 4.36385e29 0.0639140
\(481\) 2.36876e29 0.0338729
\(482\) 1.66196e30 0.232048
\(483\) 5.61810e30 0.765945
\(484\) 1.78224e30 0.237270
\(485\) 8.58252e29 0.111580
\(486\) 3.65965e30 0.464646
\(487\) 8.71950e30 1.08121 0.540603 0.841278i \(-0.318196\pi\)
0.540603 + 0.841278i \(0.318196\pi\)
\(488\) 2.11538e30 0.256189
\(489\) 8.22195e29 0.0972575
\(490\) −3.26951e29 −0.0377770
\(491\) 3.31430e30 0.374071 0.187036 0.982353i \(-0.440112\pi\)
0.187036 + 0.982353i \(0.440112\pi\)
\(492\) 2.23860e30 0.246818
\(493\) −2.03380e31 −2.19063
\(494\) 8.50613e30 0.895101
\(495\) −1.82971e30 −0.188115
\(496\) −1.70372e30 −0.171143
\(497\) 5.35441e30 0.525549
\(498\) −9.65554e30 −0.926060
\(499\) −4.41221e30 −0.413523 −0.206762 0.978391i \(-0.566292\pi\)
−0.206762 + 0.978391i \(0.566292\pi\)
\(500\) 4.88281e29 0.0447214
\(501\) 2.28956e30 0.204936
\(502\) −4.50964e30 −0.394502
\(503\) 8.86520e30 0.757978 0.378989 0.925401i \(-0.376272\pi\)
0.378989 + 0.925401i \(0.376272\pi\)
\(504\) −1.55056e30 −0.129580
\(505\) 8.42950e30 0.688575
\(506\) −9.62901e30 −0.768866
\(507\) 4.14456e30 0.323510
\(508\) −3.64538e30 −0.278170
\(509\) 4.77364e30 0.356119 0.178059 0.984020i \(-0.443018\pi\)
0.178059 + 0.984020i \(0.443018\pi\)
\(510\) 6.62123e30 0.482927
\(511\) −9.76070e30 −0.696050
\(512\) −6.33825e29 −0.0441942
\(513\) −2.60945e31 −1.77909
\(514\) 1.68813e31 1.12546
\(515\) −3.60935e30 −0.235313
\(516\) 7.36699e30 0.469698
\(517\) −2.18142e31 −1.36018
\(518\) 5.36558e29 0.0327208
\(519\) −6.21334e30 −0.370596
\(520\) −2.09907e30 −0.122458
\(521\) −8.76488e30 −0.500164 −0.250082 0.968225i \(-0.580458\pi\)
−0.250082 + 0.968225i \(0.580458\pi\)
\(522\) −5.08882e30 −0.284058
\(523\) 1.88160e31 1.02744 0.513721 0.857957i \(-0.328266\pi\)
0.513721 + 0.857957i \(0.328266\pi\)
\(524\) 6.48945e29 0.0346656
\(525\) 3.27357e30 0.171076
\(526\) −2.38874e30 −0.122133
\(527\) −2.58504e31 −1.29314
\(528\) −5.01434e30 −0.245428
\(529\) −4.13819e30 −0.198185
\(530\) −3.73319e30 −0.174948
\(531\) −8.09792e30 −0.371353
\(532\) 1.92676e31 0.864656
\(533\) −1.07679e31 −0.472900
\(534\) −2.29193e31 −0.985092
\(535\) 1.77808e31 0.747970
\(536\) 6.89201e30 0.283760
\(537\) 1.92616e31 0.776224
\(538\) 3.04542e31 1.20130
\(539\) 3.75687e30 0.145063
\(540\) 6.43937e30 0.243397
\(541\) −2.61136e31 −0.966269 −0.483135 0.875546i \(-0.660502\pi\)
−0.483135 + 0.875546i \(0.660502\pi\)
\(542\) 2.95400e31 1.07009
\(543\) −4.44084e31 −1.57495
\(544\) −9.61697e30 −0.333927
\(545\) 1.13449e31 0.385692
\(546\) −1.40728e31 −0.468450
\(547\) 3.35569e29 0.0109377 0.00546887 0.999985i \(-0.498259\pi\)
0.00546887 + 0.999985i \(0.498259\pi\)
\(548\) −2.21365e30 −0.0706533
\(549\) 8.03095e30 0.251007
\(550\) −5.61066e30 −0.171729
\(551\) 6.32346e31 1.89544
\(552\) 8.71861e30 0.255946
\(553\) 3.32272e31 0.955332
\(554\) 2.44826e31 0.689438
\(555\) −5.73290e29 −0.0158127
\(556\) −1.31478e31 −0.355217
\(557\) 1.24154e31 0.328567 0.164284 0.986413i \(-0.447469\pi\)
0.164284 + 0.986413i \(0.447469\pi\)
\(558\) −6.46810e30 −0.167681
\(559\) −3.54362e31 −0.899935
\(560\) −4.75469e30 −0.118293
\(561\) −7.60822e31 −1.85443
\(562\) −4.23929e31 −1.01234
\(563\) 6.53842e31 1.52977 0.764885 0.644166i \(-0.222796\pi\)
0.764885 + 0.644166i \(0.222796\pi\)
\(564\) 1.97517e31 0.452787
\(565\) −3.18366e31 −0.715106
\(566\) 2.35413e31 0.518134
\(567\) 2.61776e31 0.564579
\(568\) 8.30940e30 0.175616
\(569\) 2.71677e31 0.562680 0.281340 0.959608i \(-0.409221\pi\)
0.281340 + 0.959608i \(0.409221\pi\)
\(570\) −2.05866e31 −0.417854
\(571\) 5.92712e31 1.17904 0.589520 0.807754i \(-0.299317\pi\)
0.589520 + 0.807754i \(0.299317\pi\)
\(572\) 2.41197e31 0.470237
\(573\) 4.65426e31 0.889350
\(574\) −2.43909e31 −0.456815
\(575\) 9.75545e30 0.179088
\(576\) −2.40629e30 −0.0433001
\(577\) −2.23935e31 −0.395003 −0.197502 0.980303i \(-0.563283\pi\)
−0.197502 + 0.980303i \(0.563283\pi\)
\(578\) −1.05024e32 −1.81601
\(579\) −7.93651e31 −1.34532
\(580\) −1.56045e31 −0.259315
\(581\) 1.05203e32 1.71397
\(582\) 8.92945e30 0.142630
\(583\) 4.28968e31 0.671794
\(584\) −1.51474e31 −0.232590
\(585\) −7.96902e30 −0.119981
\(586\) 7.57332e31 1.11806
\(587\) −6.49190e31 −0.939796 −0.469898 0.882721i \(-0.655709\pi\)
−0.469898 + 0.882721i \(0.655709\pi\)
\(588\) −3.40167e30 −0.0482896
\(589\) 8.03736e31 1.11889
\(590\) −2.48317e31 −0.339006
\(591\) −5.76515e31 −0.771889
\(592\) 8.32672e29 0.0109339
\(593\) 1.27637e32 1.64379 0.821895 0.569638i \(-0.192917\pi\)
0.821895 + 0.569638i \(0.192917\pi\)
\(594\) −7.39925e31 −0.934638
\(595\) −7.21424e31 −0.893811
\(596\) −1.12508e31 −0.136726
\(597\) 3.57575e31 0.426248
\(598\) −4.19377e31 −0.490389
\(599\) −4.35980e31 −0.500101 −0.250051 0.968233i \(-0.580447\pi\)
−0.250051 + 0.968233i \(0.580447\pi\)
\(600\) 5.08019e30 0.0571664
\(601\) −8.94982e31 −0.988002 −0.494001 0.869461i \(-0.664466\pi\)
−0.494001 + 0.869461i \(0.664466\pi\)
\(602\) −8.02679e31 −0.869326
\(603\) 2.61652e31 0.278019
\(604\) 4.14004e31 0.431599
\(605\) 2.07480e31 0.212221
\(606\) 8.77025e31 0.880191
\(607\) −8.25995e31 −0.813406 −0.406703 0.913560i \(-0.633322\pi\)
−0.406703 + 0.913560i \(0.633322\pi\)
\(608\) 2.99009e31 0.288931
\(609\) −1.04617e32 −0.991980
\(610\) 2.46263e31 0.229143
\(611\) −9.50082e31 −0.867535
\(612\) −3.65104e31 −0.327171
\(613\) −1.75363e32 −1.54221 −0.771104 0.636710i \(-0.780295\pi\)
−0.771104 + 0.636710i \(0.780295\pi\)
\(614\) 6.51201e31 0.562055
\(615\) 2.60607e31 0.220761
\(616\) 5.46344e31 0.454243
\(617\) −9.32726e31 −0.761158 −0.380579 0.924749i \(-0.624275\pi\)
−0.380579 + 0.924749i \(0.624275\pi\)
\(618\) −3.75526e31 −0.300796
\(619\) −2.11173e32 −1.66034 −0.830169 0.557512i \(-0.811756\pi\)
−0.830169 + 0.557512i \(0.811756\pi\)
\(620\) −1.98339e31 −0.153075
\(621\) 1.28653e32 0.974692
\(622\) 8.86828e31 0.659554
\(623\) 2.49720e32 1.82323
\(624\) −2.18392e31 −0.156536
\(625\) 5.68434e30 0.0400000
\(626\) −9.33498e29 −0.00644924
\(627\) 2.36553e32 1.60455
\(628\) −1.08361e32 −0.721666
\(629\) 1.26341e31 0.0826153
\(630\) −1.80510e31 −0.115900
\(631\) −1.01929e32 −0.642628 −0.321314 0.946973i \(-0.604125\pi\)
−0.321314 + 0.946973i \(0.604125\pi\)
\(632\) 5.15646e31 0.319231
\(633\) 2.23475e32 1.35858
\(634\) −1.09803e32 −0.655522
\(635\) −4.24379e31 −0.248803
\(636\) −3.88410e31 −0.223632
\(637\) 1.63625e31 0.0925222
\(638\) 1.79306e32 0.995763
\(639\) 3.15463e31 0.172063
\(640\) −7.37870e30 −0.0395285
\(641\) −8.95507e31 −0.471196 −0.235598 0.971851i \(-0.575705\pi\)
−0.235598 + 0.971851i \(0.575705\pi\)
\(642\) 1.84996e32 0.956114
\(643\) 1.55072e32 0.787242 0.393621 0.919273i \(-0.371222\pi\)
0.393621 + 0.919273i \(0.371222\pi\)
\(644\) −9.49947e31 −0.473709
\(645\) 8.57631e31 0.420110
\(646\) 4.53684e32 2.18313
\(647\) −7.18599e31 −0.339693 −0.169847 0.985470i \(-0.554327\pi\)
−0.169847 + 0.985470i \(0.554327\pi\)
\(648\) 4.06245e31 0.188658
\(649\) 2.85332e32 1.30177
\(650\) −2.44364e31 −0.109530
\(651\) −1.32972e32 −0.585571
\(652\) −1.39022e31 −0.0601503
\(653\) 1.03893e32 0.441658 0.220829 0.975313i \(-0.429124\pi\)
0.220829 + 0.975313i \(0.429124\pi\)
\(654\) 1.18035e32 0.493022
\(655\) 7.55472e30 0.0310058
\(656\) −3.78517e31 −0.152648
\(657\) −5.75065e31 −0.227885
\(658\) −2.15207e32 −0.838028
\(659\) −2.67825e31 −0.102487 −0.0512435 0.998686i \(-0.516318\pi\)
−0.0512435 + 0.998686i \(0.516318\pi\)
\(660\) −5.83747e31 −0.219517
\(661\) −3.87116e32 −1.43062 −0.715310 0.698807i \(-0.753715\pi\)
−0.715310 + 0.698807i \(0.753715\pi\)
\(662\) −6.23068e31 −0.226292
\(663\) −3.31364e32 −1.18277
\(664\) 1.63263e32 0.572735
\(665\) 2.24304e32 0.773371
\(666\) 3.16120e30 0.0107127
\(667\) −3.11765e32 −1.03844
\(668\) −3.87134e31 −0.126745
\(669\) −2.78403e32 −0.895929
\(670\) 8.02335e31 0.253802
\(671\) −2.82972e32 −0.879902
\(672\) −4.94689e31 −0.151212
\(673\) −2.33531e32 −0.701733 −0.350867 0.936425i \(-0.614113\pi\)
−0.350867 + 0.936425i \(0.614113\pi\)
\(674\) −1.14137e32 −0.337162
\(675\) 7.49641e31 0.217701
\(676\) −7.00791e31 −0.200079
\(677\) 6.22149e32 1.74632 0.873161 0.487432i \(-0.162066\pi\)
0.873161 + 0.487432i \(0.162066\pi\)
\(678\) −3.31236e32 −0.914104
\(679\) −9.72919e31 −0.263982
\(680\) −1.11956e32 −0.298673
\(681\) 3.44579e31 0.0903851
\(682\) 2.27904e32 0.587804
\(683\) −5.18176e32 −1.31413 −0.657067 0.753833i \(-0.728203\pi\)
−0.657067 + 0.753833i \(0.728203\pi\)
\(684\) 1.13518e32 0.283086
\(685\) −2.57703e31 −0.0631942
\(686\) −2.73190e32 −0.658776
\(687\) 2.56752e32 0.608851
\(688\) −1.24566e32 −0.290491
\(689\) 1.86830e32 0.428476
\(690\) 1.01498e32 0.228925
\(691\) 5.78541e32 1.28333 0.641663 0.766987i \(-0.278245\pi\)
0.641663 + 0.766987i \(0.278245\pi\)
\(692\) 1.05059e32 0.229200
\(693\) 2.07417e32 0.445053
\(694\) 4.20151e32 0.886690
\(695\) −1.53061e32 −0.317716
\(696\) −1.62353e32 −0.331477
\(697\) −5.74320e32 −1.15339
\(698\) −4.84001e32 −0.956113
\(699\) 2.56867e32 0.499138
\(700\) −5.53518e31 −0.105805
\(701\) 7.25665e32 1.36452 0.682259 0.731111i \(-0.260998\pi\)
0.682259 + 0.731111i \(0.260998\pi\)
\(702\) −3.22263e32 −0.596120
\(703\) −3.92816e31 −0.0714831
\(704\) 8.47860e31 0.151788
\(705\) 2.29940e32 0.404985
\(706\) −2.24717e32 −0.389388
\(707\) −9.55573e32 −1.62907
\(708\) −2.58354e32 −0.433344
\(709\) 6.96702e32 1.14978 0.574891 0.818230i \(-0.305044\pi\)
0.574891 + 0.818230i \(0.305044\pi\)
\(710\) 9.67342e31 0.157076
\(711\) 1.95763e32 0.312773
\(712\) 3.87535e32 0.609244
\(713\) −3.96265e32 −0.612994
\(714\) −7.50586e32 −1.14254
\(715\) 2.80790e32 0.420593
\(716\) −3.25688e32 −0.480067
\(717\) 4.24722e32 0.616076
\(718\) 6.68880e32 0.954811
\(719\) 7.58207e32 1.06514 0.532569 0.846387i \(-0.321227\pi\)
0.532569 + 0.846387i \(0.321227\pi\)
\(720\) −2.80129e31 −0.0387288
\(721\) 4.09158e32 0.556719
\(722\) −8.82552e32 −1.18185
\(723\) 2.01298e32 0.265307
\(724\) 7.50888e32 0.974050
\(725\) −1.81660e32 −0.231938
\(726\) 2.15866e32 0.271278
\(727\) 2.69772e32 0.333696 0.166848 0.985983i \(-0.446641\pi\)
0.166848 + 0.985983i \(0.446641\pi\)
\(728\) 2.37952e32 0.289720
\(729\) 8.88493e32 1.06485
\(730\) −1.76339e32 −0.208035
\(731\) −1.89003e33 −2.19492
\(732\) 2.56218e32 0.292908
\(733\) −3.67420e32 −0.413492 −0.206746 0.978395i \(-0.566287\pi\)
−0.206746 + 0.978395i \(0.566287\pi\)
\(734\) 1.81265e32 0.200821
\(735\) −3.96007e31 −0.0431915
\(736\) −1.47420e32 −0.158293
\(737\) −9.21935e32 −0.974595
\(738\) −1.43702e32 −0.149560
\(739\) 1.50918e33 1.54643 0.773216 0.634143i \(-0.218647\pi\)
0.773216 + 0.634143i \(0.218647\pi\)
\(740\) 9.69358e30 0.00977956
\(741\) 1.03027e33 1.02339
\(742\) 4.23197e32 0.413902
\(743\) 8.64193e32 0.832222 0.416111 0.909314i \(-0.363393\pi\)
0.416111 + 0.909314i \(0.363393\pi\)
\(744\) −2.06357e32 −0.195672
\(745\) −1.30977e32 −0.122292
\(746\) 8.06709e32 0.741687
\(747\) 6.19819e32 0.561148
\(748\) 1.28645e33 1.14690
\(749\) −2.01565e33 −1.76959
\(750\) 5.91412e31 0.0511312
\(751\) −8.02515e32 −0.683272 −0.341636 0.939832i \(-0.610981\pi\)
−0.341636 + 0.939832i \(0.610981\pi\)
\(752\) −3.33975e32 −0.280033
\(753\) −5.46213e32 −0.451045
\(754\) 7.80938e32 0.635106
\(755\) 4.81964e32 0.386033
\(756\) −7.29971e32 −0.575844
\(757\) −7.46047e32 −0.579647 −0.289823 0.957080i \(-0.593597\pi\)
−0.289823 + 0.957080i \(0.593597\pi\)
\(758\) −2.65911e32 −0.203489
\(759\) −1.16628e33 −0.879066
\(760\) 3.48093e32 0.258427
\(761\) −2.23694e33 −1.63580 −0.817901 0.575359i \(-0.804863\pi\)
−0.817901 + 0.575359i \(0.804863\pi\)
\(762\) −4.41533e32 −0.318040
\(763\) −1.28606e33 −0.912495
\(764\) −7.86974e32 −0.550031
\(765\) −4.25037e32 −0.292631
\(766\) −4.46390e32 −0.302750
\(767\) 1.24272e33 0.830283
\(768\) −7.67697e31 −0.0505284
\(769\) −8.84798e32 −0.573708 −0.286854 0.957974i \(-0.592609\pi\)
−0.286854 + 0.957974i \(0.592609\pi\)
\(770\) 6.36028e32 0.406287
\(771\) 2.04468e33 1.28677
\(772\) 1.34196e33 0.832033
\(773\) 4.12211e32 0.251800 0.125900 0.992043i \(-0.459818\pi\)
0.125900 + 0.992043i \(0.459818\pi\)
\(774\) −4.72910e32 −0.284615
\(775\) −2.30897e32 −0.136914
\(776\) −1.50985e32 −0.0882114
\(777\) 6.49885e31 0.0374106
\(778\) 1.85325e33 1.05116
\(779\) 1.78567e33 0.997975
\(780\) −2.54242e32 −0.140010
\(781\) −1.11154e33 −0.603167
\(782\) −2.23679e33 −1.19605
\(783\) −2.39570e33 −1.26233
\(784\) 5.75178e31 0.0298653
\(785\) −1.26148e33 −0.645477
\(786\) 7.86010e31 0.0396341
\(787\) −9.29443e31 −0.0461863 −0.0230931 0.999733i \(-0.507351\pi\)
−0.0230931 + 0.999733i \(0.507351\pi\)
\(788\) 9.74811e32 0.477385
\(789\) −2.89327e32 −0.139638
\(790\) 6.00291e32 0.285529
\(791\) 3.60902e33 1.69184
\(792\) 3.21886e32 0.148718
\(793\) −1.23244e33 −0.561209
\(794\) −2.42086e33 −1.08651
\(795\) −4.52169e32 −0.200022
\(796\) −6.04612e32 −0.263619
\(797\) −1.01244e32 −0.0435108 −0.0217554 0.999763i \(-0.506926\pi\)
−0.0217554 + 0.999763i \(0.506926\pi\)
\(798\) 2.33371e33 0.988585
\(799\) −5.06737e33 −2.11590
\(800\) −8.58993e31 −0.0353553
\(801\) 1.47126e33 0.596919
\(802\) −1.99752e33 −0.798890
\(803\) 2.02625e33 0.798848
\(804\) 8.34768e32 0.324430
\(805\) −1.10588e33 −0.423698
\(806\) 9.92603e32 0.374906
\(807\) 3.68865e33 1.37348
\(808\) −1.48293e33 −0.544366
\(809\) −3.30371e33 −1.19562 −0.597811 0.801637i \(-0.703963\pi\)
−0.597811 + 0.801637i \(0.703963\pi\)
\(810\) 4.72931e32 0.168741
\(811\) 3.27520e33 1.15212 0.576061 0.817407i \(-0.304589\pi\)
0.576061 + 0.817407i \(0.304589\pi\)
\(812\) 1.76893e33 0.613504
\(813\) 3.57792e33 1.22346
\(814\) −1.11385e32 −0.0375533
\(815\) −1.61843e32 −0.0538000
\(816\) −1.16482e33 −0.381787
\(817\) 5.87645e33 1.89916
\(818\) −7.22051e32 −0.230094
\(819\) 9.03373e32 0.283859
\(820\) −4.40652e32 −0.136533
\(821\) −2.62789e32 −0.0802900 −0.0401450 0.999194i \(-0.512782\pi\)
−0.0401450 + 0.999194i \(0.512782\pi\)
\(822\) −2.68120e32 −0.0807799
\(823\) 1.32429e33 0.393447 0.196724 0.980459i \(-0.436970\pi\)
0.196724 + 0.980459i \(0.436970\pi\)
\(824\) 6.34964e32 0.186031
\(825\) −6.79570e32 −0.196342
\(826\) 2.81493e33 0.802042
\(827\) 6.86820e32 0.192988 0.0964938 0.995334i \(-0.469237\pi\)
0.0964938 + 0.995334i \(0.469237\pi\)
\(828\) −5.59674e32 −0.155091
\(829\) −3.52526e33 −0.963416 −0.481708 0.876332i \(-0.659983\pi\)
−0.481708 + 0.876332i \(0.659983\pi\)
\(830\) 1.90063e33 0.512270
\(831\) 2.96536e33 0.788254
\(832\) 3.69272e32 0.0968118
\(833\) 8.72712e32 0.225659
\(834\) −1.59248e33 −0.406130
\(835\) −4.50684e32 −0.113365
\(836\) −3.99981e33 −0.992355
\(837\) −3.04503e33 −0.745159
\(838\) 2.88333e33 0.695967
\(839\) 7.77027e33 1.85001 0.925004 0.379958i \(-0.124061\pi\)
0.925004 + 0.379958i \(0.124061\pi\)
\(840\) −5.75893e32 −0.135248
\(841\) 1.48877e33 0.344885
\(842\) −1.50910e33 −0.344850
\(843\) −5.13468e33 −1.15743
\(844\) −3.77867e33 −0.840234
\(845\) −8.15828e32 −0.178956
\(846\) −1.26792e33 −0.274368
\(847\) −2.35200e33 −0.502086
\(848\) 6.56750e32 0.138308
\(849\) 2.85136e33 0.592397
\(850\) −1.30334e33 −0.267141
\(851\) 1.93670e32 0.0391626
\(852\) 1.00644e33 0.200787
\(853\) −5.19512e33 −1.02254 −0.511272 0.859419i \(-0.670825\pi\)
−0.511272 + 0.859419i \(0.670825\pi\)
\(854\) −2.79165e33 −0.542120
\(855\) 1.32152e33 0.253199
\(856\) −3.12804e33 −0.591322
\(857\) 9.62211e33 1.79469 0.897347 0.441325i \(-0.145491\pi\)
0.897347 + 0.441325i \(0.145491\pi\)
\(858\) 2.92140e33 0.537635
\(859\) −6.92148e33 −1.25683 −0.628417 0.777877i \(-0.716297\pi\)
−0.628417 + 0.777877i \(0.716297\pi\)
\(860\) −1.45014e33 −0.259823
\(861\) −2.95425e33 −0.522290
\(862\) −3.50739e33 −0.611858
\(863\) 4.39935e33 0.757294 0.378647 0.925541i \(-0.376389\pi\)
0.378647 + 0.925541i \(0.376389\pi\)
\(864\) −1.13283e33 −0.192422
\(865\) 1.22305e33 0.205003
\(866\) −6.84795e33 −1.13267
\(867\) −1.27206e34 −2.07629
\(868\) 2.24838e33 0.362154
\(869\) −6.89772e33 −1.09642
\(870\) −1.89003e33 −0.296482
\(871\) −4.01535e33 −0.621605
\(872\) −1.99582e33 −0.304917
\(873\) −5.73208e32 −0.0864269
\(874\) 6.95460e33 1.03488
\(875\) −6.44380e32 −0.0946345
\(876\) −1.83467e33 −0.265927
\(877\) −8.80187e33 −1.25916 −0.629578 0.776937i \(-0.716772\pi\)
−0.629578 + 0.776937i \(0.716772\pi\)
\(878\) 8.10018e33 1.14369
\(879\) 9.17290e33 1.27831
\(880\) 9.87039e32 0.135764
\(881\) 6.20312e33 0.842146 0.421073 0.907027i \(-0.361654\pi\)
0.421073 + 0.907027i \(0.361654\pi\)
\(882\) 2.18364e32 0.0292612
\(883\) −7.33525e32 −0.0970213 −0.0485106 0.998823i \(-0.515447\pi\)
−0.0485106 + 0.998823i \(0.515447\pi\)
\(884\) 5.60293e33 0.731500
\(885\) −3.00764e33 −0.387595
\(886\) 6.96489e33 0.885985
\(887\) −5.08588e33 −0.638623 −0.319311 0.947650i \(-0.603452\pi\)
−0.319311 + 0.947650i \(0.603452\pi\)
\(888\) 1.00854e32 0.0125010
\(889\) 4.81078e33 0.588634
\(890\) 4.51150e33 0.544924
\(891\) −5.43428e33 −0.647960
\(892\) 4.70743e33 0.554100
\(893\) 1.57554e34 1.83079
\(894\) −1.36271e33 −0.156323
\(895\) −3.79150e33 −0.429385
\(896\) 8.36453e32 0.0935189
\(897\) −5.07954e33 −0.560675
\(898\) −6.64162e33 −0.723763
\(899\) 7.37901e33 0.793892
\(900\) −3.26113e32 −0.0346401
\(901\) 9.96481e33 1.04504
\(902\) 5.06337e33 0.524282
\(903\) −9.72215e33 −0.993924
\(904\) 5.60076e33 0.565341
\(905\) 8.74148e33 0.871217
\(906\) 5.01447e33 0.493458
\(907\) 1.39867e34 1.35904 0.679521 0.733656i \(-0.262188\pi\)
0.679521 + 0.733656i \(0.262188\pi\)
\(908\) −5.82638e32 −0.0558999
\(909\) −5.62989e33 −0.533354
\(910\) 2.77012e33 0.259133
\(911\) −1.11628e34 −1.03112 −0.515561 0.856853i \(-0.672417\pi\)
−0.515561 + 0.856853i \(0.672417\pi\)
\(912\) 3.62164e33 0.330342
\(913\) −2.18394e34 −1.96710
\(914\) 4.75194e33 0.422658
\(915\) 2.98277e33 0.261985
\(916\) −4.34134e33 −0.376553
\(917\) −8.56407e32 −0.0733555
\(918\) −1.71883e34 −1.45392
\(919\) 8.24058e33 0.688382 0.344191 0.938900i \(-0.388153\pi\)
0.344191 + 0.938900i \(0.388153\pi\)
\(920\) −1.71620e33 −0.141582
\(921\) 7.88743e33 0.642613
\(922\) 3.64538e33 0.293317
\(923\) −4.84113e33 −0.384705
\(924\) 6.61738e33 0.519348
\(925\) 1.12848e32 0.00874711
\(926\) 2.38572e33 0.182639
\(927\) 2.41061e33 0.182268
\(928\) 2.74517e33 0.205006
\(929\) 1.03061e33 0.0760178 0.0380089 0.999277i \(-0.487898\pi\)
0.0380089 + 0.999277i \(0.487898\pi\)
\(930\) −2.40231e33 −0.175015
\(931\) −2.71342e33 −0.195252
\(932\) −4.34328e33 −0.308699
\(933\) 1.07414e34 0.754086
\(934\) −9.06063e33 −0.628303
\(935\) 1.49762e34 1.02582
\(936\) 1.40193e33 0.0948533
\(937\) −1.43845e34 −0.961368 −0.480684 0.876894i \(-0.659612\pi\)
−0.480684 + 0.876894i \(0.659612\pi\)
\(938\) −9.09532e33 −0.600462
\(939\) −1.13066e32 −0.00737360
\(940\) −3.88798e33 −0.250469
\(941\) −9.09450e33 −0.578760 −0.289380 0.957214i \(-0.593449\pi\)
−0.289380 + 0.957214i \(0.593449\pi\)
\(942\) −1.31248e34 −0.825100
\(943\) −8.80386e33 −0.546750
\(944\) 4.36843e33 0.268008
\(945\) −8.49798e33 −0.515050
\(946\) 1.66630e34 0.997714
\(947\) 9.97145e33 0.589839 0.294920 0.955522i \(-0.404707\pi\)
0.294920 + 0.955522i \(0.404707\pi\)
\(948\) 6.24556e33 0.364986
\(949\) 8.82502e33 0.509512
\(950\) 4.05233e33 0.231145
\(951\) −1.32995e34 −0.749477
\(952\) 1.26914e34 0.706619
\(953\) −7.81126e33 −0.429688 −0.214844 0.976648i \(-0.568924\pi\)
−0.214844 + 0.976648i \(0.568924\pi\)
\(954\) 2.49332e33 0.135510
\(955\) −9.16159e33 −0.491962
\(956\) −7.18149e33 −0.381021
\(957\) 2.17177e34 1.13848
\(958\) −2.41146e34 −1.24904
\(959\) 2.92133e33 0.149509
\(960\) −8.93717e32 −0.0451940
\(961\) −1.06343e34 −0.531361
\(962\) −4.85122e32 −0.0239518
\(963\) −1.18755e34 −0.579360
\(964\) −3.40368e33 −0.164083
\(965\) 1.56225e34 0.744193
\(966\) −1.15059e34 −0.541605
\(967\) 9.65442e33 0.449078 0.224539 0.974465i \(-0.427912\pi\)
0.224539 + 0.974465i \(0.427912\pi\)
\(968\) −3.65002e33 −0.167775
\(969\) 5.49508e34 2.49603
\(970\) −1.75770e33 −0.0788987
\(971\) −2.59685e34 −1.15193 −0.575965 0.817474i \(-0.695374\pi\)
−0.575965 + 0.817474i \(0.695374\pi\)
\(972\) −7.49496e33 −0.328554
\(973\) 1.73511e34 0.751673
\(974\) −1.78575e34 −0.764529
\(975\) −2.95976e33 −0.125229
\(976\) −4.33230e33 −0.181153
\(977\) −7.90048e33 −0.326487 −0.163244 0.986586i \(-0.552196\pi\)
−0.163244 + 0.986586i \(0.552196\pi\)
\(978\) −1.68386e33 −0.0687714
\(979\) −5.18400e34 −2.09249
\(980\) 6.69595e32 0.0267124
\(981\) −7.57703e33 −0.298748
\(982\) −6.78769e33 −0.264508
\(983\) 9.20116e33 0.354387 0.177193 0.984176i \(-0.443298\pi\)
0.177193 + 0.984176i \(0.443298\pi\)
\(984\) −4.58464e33 −0.174527
\(985\) 1.13483e34 0.426986
\(986\) 4.16522e34 1.54901
\(987\) −2.60661e34 −0.958140
\(988\) −1.74206e34 −0.632932
\(989\) −2.89726e34 −1.04047
\(990\) 3.74725e33 0.133017
\(991\) 2.65253e34 0.930708 0.465354 0.885125i \(-0.345927\pi\)
0.465354 + 0.885125i \(0.345927\pi\)
\(992\) 3.48922e33 0.121016
\(993\) −7.54667e33 −0.258725
\(994\) −1.09658e34 −0.371619
\(995\) −7.03861e33 −0.235788
\(996\) 1.97746e34 0.654824
\(997\) 4.79679e34 1.57021 0.785103 0.619365i \(-0.212610\pi\)
0.785103 + 0.619365i \(0.212610\pi\)
\(998\) 9.03621e33 0.292405
\(999\) 1.48822e33 0.0476063
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 10.24.a.a.1.1 2
5.2 odd 4 50.24.b.d.49.2 4
5.3 odd 4 50.24.b.d.49.3 4
5.4 even 2 50.24.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.24.a.a.1.1 2 1.1 even 1 trivial
50.24.a.d.1.2 2 5.4 even 2
50.24.b.d.49.2 4 5.2 odd 4
50.24.b.d.49.3 4 5.3 odd 4