Properties

Label 550.8.a.d.1.1
Level $550$
Weight $8$
Character 550.1
Self dual yes
Analytic conductor $171.812$
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] = [550,8,Mod(1,550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("550.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 550.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: \(171.811764016\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14881}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(61.4939\) of defining polynomial
Character \(\chi\) \(=\) 550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} -49.4939 q^{3} +64.0000 q^{4} +395.951 q^{6} -43.0861 q^{7} -512.000 q^{8} +262.641 q^{9} +O(q^{10})\) \(q-8.00000 q^{2} -49.4939 q^{3} +64.0000 q^{4} +395.951 q^{6} -43.0861 q^{7} -512.000 q^{8} +262.641 q^{9} +1331.00 q^{11} -3167.61 q^{12} -4494.27 q^{13} +344.689 q^{14} +4096.00 q^{16} +6878.55 q^{17} -2101.13 q^{18} -21063.0 q^{19} +2132.50 q^{21} -10648.0 q^{22} +60259.0 q^{23} +25340.9 q^{24} +35954.2 q^{26} +95243.9 q^{27} -2757.51 q^{28} -59039.5 q^{29} -88522.5 q^{31} -32768.0 q^{32} -65876.3 q^{33} -55028.4 q^{34} +16809.0 q^{36} -382131. q^{37} +168504. q^{38} +222439. q^{39} +550122. q^{41} -17060.0 q^{42} -693613. q^{43} +85184.0 q^{44} -482072. q^{46} +126233. q^{47} -202727. q^{48} -821687. q^{49} -340446. q^{51} -287634. q^{52} +1.19815e6 q^{53} -761951. q^{54} +22060.1 q^{56} +1.04249e6 q^{57} +472316. q^{58} +1.31641e6 q^{59} +440165. q^{61} +708180. q^{62} -11316.2 q^{63} +262144. q^{64} +527011. q^{66} -3.56366e6 q^{67} +440227. q^{68} -2.98245e6 q^{69} +3.13943e6 q^{71} -134472. q^{72} +4.07747e6 q^{73} +3.05705e6 q^{74} -1.34803e6 q^{76} -57347.6 q^{77} -1.77951e6 q^{78} +1.89723e6 q^{79} -5.28839e6 q^{81} -4.40097e6 q^{82} -6.49257e6 q^{83} +136480. q^{84} +5.54890e6 q^{86} +2.92209e6 q^{87} -681472. q^{88} +9.23033e6 q^{89} +193641. q^{91} +3.85658e6 q^{92} +4.38132e6 q^{93} -1.00986e6 q^{94} +1.62181e6 q^{96} +1.38062e7 q^{97} +6.57349e6 q^{98} +349576. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 23 q^{3} + 128 q^{4} - 184 q^{6} - 1794 q^{7} - 1024 q^{8} + 3331 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{2} + 23 q^{3} + 128 q^{4} - 184 q^{6} - 1794 q^{7} - 1024 q^{8} + 3331 q^{9} + 2662 q^{11} + 1472 q^{12} + 5406 q^{13} + 14352 q^{14} + 8192 q^{16} - 15032 q^{17} - 26648 q^{18} + 16916 q^{19} - 124798 q^{21} - 21296 q^{22} + 51351 q^{23} - 11776 q^{24} - 43248 q^{26} + 159137 q^{27} - 114816 q^{28} - 207130 q^{29} - 19071 q^{31} - 65536 q^{32} + 30613 q^{33} + 120256 q^{34} + 213184 q^{36} - 351333 q^{37} - 135328 q^{38} + 940148 q^{39} + 123610 q^{41} + 998384 q^{42} + 159822 q^{43} + 170368 q^{44} - 410808 q^{46} - 451160 q^{47} + 94208 q^{48} + 1420470 q^{49} - 1928826 q^{51} + 345984 q^{52} + 1260832 q^{53} - 1273096 q^{54} + 918528 q^{56} + 3795736 q^{57} + 1657040 q^{58} + 887547 q^{59} - 597918 q^{61} + 152568 q^{62} - 5383748 q^{63} + 524288 q^{64} - 244904 q^{66} - 2864711 q^{67} - 962048 q^{68} - 3628227 q^{69} + 1306267 q^{71} - 1705472 q^{72} + 4577530 q^{73} + 2810664 q^{74} + 1082624 q^{76} - 2387814 q^{77} - 7521184 q^{78} - 2946342 q^{79} - 7367030 q^{81} - 988880 q^{82} - 9965450 q^{83} - 7987072 q^{84} - 1278576 q^{86} - 7813560 q^{87} - 1362944 q^{88} + 10185377 q^{89} - 17140888 q^{91} + 3286464 q^{92} + 9416131 q^{93} + 3609280 q^{94} - 753664 q^{96} + 27765477 q^{97} - 11363760 q^{98} + 4433561 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) −49.4939 −1.05834 −0.529172 0.848515i \(-0.677497\pi\)
−0.529172 + 0.848515i \(0.677497\pi\)
\(4\) 64.0000 0.500000
\(5\) 0 0
\(6\) 395.951 0.748362
\(7\) −43.0861 −0.0474781 −0.0237391 0.999718i \(-0.507557\pi\)
−0.0237391 + 0.999718i \(0.507557\pi\)
\(8\) −512.000 −0.353553
\(9\) 262.641 0.120092
\(10\) 0 0
\(11\) 1331.00 0.301511
\(12\) −3167.61 −0.529172
\(13\) −4494.27 −0.567359 −0.283679 0.958919i \(-0.591555\pi\)
−0.283679 + 0.958919i \(0.591555\pi\)
\(14\) 344.689 0.0335721
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 6878.55 0.339567 0.169784 0.985481i \(-0.445693\pi\)
0.169784 + 0.985481i \(0.445693\pi\)
\(18\) −2101.13 −0.0849179
\(19\) −21063.0 −0.704503 −0.352252 0.935905i \(-0.614584\pi\)
−0.352252 + 0.935905i \(0.614584\pi\)
\(20\) 0 0
\(21\) 2132.50 0.0502482
\(22\) −10648.0 −0.213201
\(23\) 60259.0 1.03270 0.516350 0.856377i \(-0.327290\pi\)
0.516350 + 0.856377i \(0.327290\pi\)
\(24\) 25340.9 0.374181
\(25\) 0 0
\(26\) 35954.2 0.401183
\(27\) 95243.9 0.931245
\(28\) −2757.51 −0.0237391
\(29\) −59039.5 −0.449521 −0.224760 0.974414i \(-0.572160\pi\)
−0.224760 + 0.974414i \(0.572160\pi\)
\(30\) 0 0
\(31\) −88522.5 −0.533689 −0.266844 0.963740i \(-0.585981\pi\)
−0.266844 + 0.963740i \(0.585981\pi\)
\(32\) −32768.0 −0.176777
\(33\) −65876.3 −0.319103
\(34\) −55028.4 −0.240110
\(35\) 0 0
\(36\) 16809.0 0.0600460
\(37\) −382131. −1.24024 −0.620120 0.784507i \(-0.712916\pi\)
−0.620120 + 0.784507i \(0.712916\pi\)
\(38\) 168504. 0.498159
\(39\) 222439. 0.600461
\(40\) 0 0
\(41\) 550122. 1.24657 0.623283 0.781996i \(-0.285798\pi\)
0.623283 + 0.781996i \(0.285798\pi\)
\(42\) −17060.0 −0.0355309
\(43\) −693613. −1.33039 −0.665193 0.746671i \(-0.731651\pi\)
−0.665193 + 0.746671i \(0.731651\pi\)
\(44\) 85184.0 0.150756
\(45\) 0 0
\(46\) −482072. −0.730230
\(47\) 126233. 0.177349 0.0886745 0.996061i \(-0.471737\pi\)
0.0886745 + 0.996061i \(0.471737\pi\)
\(48\) −202727. −0.264586
\(49\) −821687. −0.997746
\(50\) 0 0
\(51\) −340446. −0.359379
\(52\) −287634. −0.283679
\(53\) 1.19815e6 1.10546 0.552732 0.833359i \(-0.313585\pi\)
0.552732 + 0.833359i \(0.313585\pi\)
\(54\) −761951. −0.658490
\(55\) 0 0
\(56\) 22060.1 0.0167861
\(57\) 1.04249e6 0.745607
\(58\) 472316. 0.317859
\(59\) 1.31641e6 0.834469 0.417234 0.908799i \(-0.362999\pi\)
0.417234 + 0.908799i \(0.362999\pi\)
\(60\) 0 0
\(61\) 440165. 0.248291 0.124145 0.992264i \(-0.460381\pi\)
0.124145 + 0.992264i \(0.460381\pi\)
\(62\) 708180. 0.377375
\(63\) −11316.2 −0.00570175
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 527011. 0.225640
\(67\) −3.56366e6 −1.44755 −0.723777 0.690034i \(-0.757596\pi\)
−0.723777 + 0.690034i \(0.757596\pi\)
\(68\) 440227. 0.169784
\(69\) −2.98245e6 −1.09295
\(70\) 0 0
\(71\) 3.13943e6 1.04099 0.520494 0.853865i \(-0.325748\pi\)
0.520494 + 0.853865i \(0.325748\pi\)
\(72\) −134472. −0.0424590
\(73\) 4.07747e6 1.22676 0.613382 0.789787i \(-0.289809\pi\)
0.613382 + 0.789787i \(0.289809\pi\)
\(74\) 3.05705e6 0.876982
\(75\) 0 0
\(76\) −1.34803e6 −0.352252
\(77\) −57347.6 −0.0143152
\(78\) −1.77951e6 −0.424590
\(79\) 1.89723e6 0.432937 0.216468 0.976290i \(-0.430546\pi\)
0.216468 + 0.976290i \(0.430546\pi\)
\(80\) 0 0
\(81\) −5.28839e6 −1.10567
\(82\) −4.40097e6 −0.881455
\(83\) −6.49257e6 −1.24636 −0.623179 0.782079i \(-0.714159\pi\)
−0.623179 + 0.782079i \(0.714159\pi\)
\(84\) 136480. 0.0251241
\(85\) 0 0
\(86\) 5.54890e6 0.940725
\(87\) 2.92209e6 0.475747
\(88\) −681472. −0.106600
\(89\) 9.23033e6 1.38788 0.693940 0.720032i \(-0.255873\pi\)
0.693940 + 0.720032i \(0.255873\pi\)
\(90\) 0 0
\(91\) 193641. 0.0269371
\(92\) 3.85658e6 0.516350
\(93\) 4.38132e6 0.564826
\(94\) −1.00986e6 −0.125405
\(95\) 0 0
\(96\) 1.62181e6 0.187091
\(97\) 1.38062e7 1.53593 0.767967 0.640489i \(-0.221268\pi\)
0.767967 + 0.640489i \(0.221268\pi\)
\(98\) 6.57349e6 0.705513
\(99\) 349576. 0.0362091
\(100\) 0 0
\(101\) 6.59478e6 0.636906 0.318453 0.947939i \(-0.396837\pi\)
0.318453 + 0.947939i \(0.396837\pi\)
\(102\) 2.72357e6 0.254119
\(103\) 1.98787e7 1.79250 0.896249 0.443551i \(-0.146282\pi\)
0.896249 + 0.443551i \(0.146282\pi\)
\(104\) 2.30107e6 0.200592
\(105\) 0 0
\(106\) −9.58517e6 −0.781681
\(107\) −2.09533e7 −1.65352 −0.826758 0.562558i \(-0.809817\pi\)
−0.826758 + 0.562558i \(0.809817\pi\)
\(108\) 6.09561e6 0.465623
\(109\) −598244. −0.0442472 −0.0221236 0.999755i \(-0.507043\pi\)
−0.0221236 + 0.999755i \(0.507043\pi\)
\(110\) 0 0
\(111\) 1.89131e7 1.31260
\(112\) −176481. −0.0118695
\(113\) 1.41807e6 0.0924535 0.0462267 0.998931i \(-0.485280\pi\)
0.0462267 + 0.998931i \(0.485280\pi\)
\(114\) −8.33992e6 −0.527224
\(115\) 0 0
\(116\) −3.77853e6 −0.224760
\(117\) −1.18038e6 −0.0681353
\(118\) −1.05313e7 −0.590058
\(119\) −296370. −0.0161220
\(120\) 0 0
\(121\) 1.77156e6 0.0909091
\(122\) −3.52132e6 −0.175568
\(123\) −2.72276e7 −1.31930
\(124\) −5.66544e6 −0.266844
\(125\) 0 0
\(126\) 90529.5 0.00403175
\(127\) 1.92988e7 0.836023 0.418012 0.908442i \(-0.362727\pi\)
0.418012 + 0.908442i \(0.362727\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 3.43296e7 1.40801
\(130\) 0 0
\(131\) 4.22451e7 1.64182 0.820912 0.571055i \(-0.193466\pi\)
0.820912 + 0.571055i \(0.193466\pi\)
\(132\) −4.21608e6 −0.159551
\(133\) 907523. 0.0334485
\(134\) 2.85093e7 1.02358
\(135\) 0 0
\(136\) −3.52182e6 −0.120055
\(137\) 7.65888e6 0.254474 0.127237 0.991872i \(-0.459389\pi\)
0.127237 + 0.991872i \(0.459389\pi\)
\(138\) 2.38596e7 0.772834
\(139\) −1.46018e7 −0.461164 −0.230582 0.973053i \(-0.574063\pi\)
−0.230582 + 0.973053i \(0.574063\pi\)
\(140\) 0 0
\(141\) −6.24773e6 −0.187696
\(142\) −2.51154e7 −0.736090
\(143\) −5.98188e6 −0.171065
\(144\) 1.07578e6 0.0300230
\(145\) 0 0
\(146\) −3.26198e7 −0.867453
\(147\) 4.06684e7 1.05596
\(148\) −2.44564e7 −0.620120
\(149\) −3.54730e7 −0.878507 −0.439254 0.898363i \(-0.644757\pi\)
−0.439254 + 0.898363i \(0.644757\pi\)
\(150\) 0 0
\(151\) 4.10281e7 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(152\) 1.07843e7 0.249080
\(153\) 1.80659e6 0.0407793
\(154\) 458780. 0.0101224
\(155\) 0 0
\(156\) 1.42361e7 0.300230
\(157\) 6.66977e7 1.37550 0.687752 0.725946i \(-0.258598\pi\)
0.687752 + 0.725946i \(0.258598\pi\)
\(158\) −1.51778e7 −0.306133
\(159\) −5.93009e7 −1.16996
\(160\) 0 0
\(161\) −2.59632e6 −0.0490307
\(162\) 4.23071e7 0.781827
\(163\) −9.29744e6 −0.168154 −0.0840769 0.996459i \(-0.526794\pi\)
−0.0840769 + 0.996459i \(0.526794\pi\)
\(164\) 3.52078e7 0.623283
\(165\) 0 0
\(166\) 5.19405e7 0.881309
\(167\) 5.93268e7 0.985696 0.492848 0.870115i \(-0.335956\pi\)
0.492848 + 0.870115i \(0.335956\pi\)
\(168\) −1.09184e6 −0.0177654
\(169\) −4.25500e7 −0.678104
\(170\) 0 0
\(171\) −5.53202e6 −0.0846053
\(172\) −4.43912e7 −0.665193
\(173\) −1.19997e8 −1.76201 −0.881004 0.473108i \(-0.843132\pi\)
−0.881004 + 0.473108i \(0.843132\pi\)
\(174\) −2.33767e7 −0.336404
\(175\) 0 0
\(176\) 5.45178e6 0.0753778
\(177\) −6.51543e7 −0.883155
\(178\) −7.38426e7 −0.981380
\(179\) 3.18894e7 0.415586 0.207793 0.978173i \(-0.433372\pi\)
0.207793 + 0.978173i \(0.433372\pi\)
\(180\) 0 0
\(181\) 2.14661e7 0.269079 0.134539 0.990908i \(-0.457045\pi\)
0.134539 + 0.990908i \(0.457045\pi\)
\(182\) −1.54913e6 −0.0190474
\(183\) −2.17854e7 −0.262777
\(184\) −3.08526e7 −0.365115
\(185\) 0 0
\(186\) −3.50506e7 −0.399392
\(187\) 9.15535e6 0.102383
\(188\) 8.07888e6 0.0886745
\(189\) −4.10369e6 −0.0442138
\(190\) 0 0
\(191\) −5.37382e7 −0.558042 −0.279021 0.960285i \(-0.590010\pi\)
−0.279021 + 0.960285i \(0.590010\pi\)
\(192\) −1.29745e7 −0.132293
\(193\) −1.72101e8 −1.72319 −0.861593 0.507600i \(-0.830533\pi\)
−0.861593 + 0.507600i \(0.830533\pi\)
\(194\) −1.10450e8 −1.08607
\(195\) 0 0
\(196\) −5.25879e7 −0.498873
\(197\) 1.14526e8 1.06726 0.533631 0.845718i \(-0.320827\pi\)
0.533631 + 0.845718i \(0.320827\pi\)
\(198\) −2.79661e6 −0.0256037
\(199\) −3.40644e7 −0.306418 −0.153209 0.988194i \(-0.548961\pi\)
−0.153209 + 0.988194i \(0.548961\pi\)
\(200\) 0 0
\(201\) 1.76379e8 1.53201
\(202\) −5.27583e7 −0.450361
\(203\) 2.54378e6 0.0213424
\(204\) −2.17885e7 −0.179689
\(205\) 0 0
\(206\) −1.59030e8 −1.26749
\(207\) 1.58265e7 0.124019
\(208\) −1.84085e7 −0.141840
\(209\) −2.80349e7 −0.212416
\(210\) 0 0
\(211\) −1.74288e8 −1.27726 −0.638628 0.769516i \(-0.720498\pi\)
−0.638628 + 0.769516i \(0.720498\pi\)
\(212\) 7.66814e7 0.552732
\(213\) −1.55382e8 −1.10172
\(214\) 1.67626e8 1.16921
\(215\) 0 0
\(216\) −4.87649e7 −0.329245
\(217\) 3.81409e6 0.0253385
\(218\) 4.78596e6 0.0312875
\(219\) −2.01810e8 −1.29834
\(220\) 0 0
\(221\) −3.09141e7 −0.192656
\(222\) −1.51305e8 −0.928149
\(223\) −1.45730e8 −0.879998 −0.439999 0.897998i \(-0.645021\pi\)
−0.439999 + 0.897998i \(0.645021\pi\)
\(224\) 1.41184e6 0.00839303
\(225\) 0 0
\(226\) −1.13446e7 −0.0653745
\(227\) 1.27102e8 0.721211 0.360606 0.932718i \(-0.382570\pi\)
0.360606 + 0.932718i \(0.382570\pi\)
\(228\) 6.67194e7 0.372804
\(229\) −1.29750e8 −0.713975 −0.356987 0.934109i \(-0.616196\pi\)
−0.356987 + 0.934109i \(0.616196\pi\)
\(230\) 0 0
\(231\) 2.83835e6 0.0151504
\(232\) 3.02282e7 0.158930
\(233\) 1.52210e8 0.788310 0.394155 0.919044i \(-0.371037\pi\)
0.394155 + 0.919044i \(0.371037\pi\)
\(234\) 9.44306e6 0.0481789
\(235\) 0 0
\(236\) 8.42504e7 0.417234
\(237\) −9.39011e7 −0.458196
\(238\) 2.37096e6 0.0114000
\(239\) −2.43061e8 −1.15165 −0.575827 0.817571i \(-0.695320\pi\)
−0.575827 + 0.817571i \(0.695320\pi\)
\(240\) 0 0
\(241\) −6.84592e7 −0.315045 −0.157522 0.987515i \(-0.550351\pi\)
−0.157522 + 0.987515i \(0.550351\pi\)
\(242\) −1.41725e7 −0.0642824
\(243\) 5.34441e7 0.238934
\(244\) 2.81705e7 0.124145
\(245\) 0 0
\(246\) 2.17821e8 0.932883
\(247\) 9.46630e7 0.399706
\(248\) 4.53235e7 0.188687
\(249\) 3.21342e8 1.31908
\(250\) 0 0
\(251\) 1.67292e8 0.667755 0.333878 0.942616i \(-0.391643\pi\)
0.333878 + 0.942616i \(0.391643\pi\)
\(252\) −724236. −0.00285087
\(253\) 8.02047e7 0.311371
\(254\) −1.54391e8 −0.591158
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −3.84789e8 −1.41402 −0.707012 0.707202i \(-0.749957\pi\)
−0.707012 + 0.707202i \(0.749957\pi\)
\(258\) −2.74637e8 −0.995611
\(259\) 1.64645e7 0.0588843
\(260\) 0 0
\(261\) −1.55062e7 −0.0539839
\(262\) −3.37960e8 −1.16094
\(263\) −5.09459e8 −1.72689 −0.863444 0.504445i \(-0.831697\pi\)
−0.863444 + 0.504445i \(0.831697\pi\)
\(264\) 3.37287e7 0.112820
\(265\) 0 0
\(266\) −7.26018e6 −0.0236517
\(267\) −4.56845e8 −1.46886
\(268\) −2.28074e8 −0.723777
\(269\) −5.51297e8 −1.72684 −0.863421 0.504484i \(-0.831683\pi\)
−0.863421 + 0.504484i \(0.831683\pi\)
\(270\) 0 0
\(271\) 3.42271e8 1.04467 0.522333 0.852742i \(-0.325062\pi\)
0.522333 + 0.852742i \(0.325062\pi\)
\(272\) 2.81745e7 0.0848918
\(273\) −9.58402e6 −0.0285088
\(274\) −6.12710e7 −0.179940
\(275\) 0 0
\(276\) −1.90877e8 −0.546476
\(277\) 5.10974e8 1.44451 0.722253 0.691629i \(-0.243107\pi\)
0.722253 + 0.691629i \(0.243107\pi\)
\(278\) 1.16815e8 0.326092
\(279\) −2.32497e7 −0.0640918
\(280\) 0 0
\(281\) −1.51750e7 −0.0407998 −0.0203999 0.999792i \(-0.506494\pi\)
−0.0203999 + 0.999792i \(0.506494\pi\)
\(282\) 4.99819e7 0.132721
\(283\) 2.25341e8 0.590999 0.295500 0.955343i \(-0.404514\pi\)
0.295500 + 0.955343i \(0.404514\pi\)
\(284\) 2.00923e8 0.520494
\(285\) 0 0
\(286\) 4.78550e7 0.120961
\(287\) −2.37026e7 −0.0591846
\(288\) −8.60623e6 −0.0212295
\(289\) −3.63024e8 −0.884694
\(290\) 0 0
\(291\) −6.83322e8 −1.62555
\(292\) 2.60958e8 0.613382
\(293\) −3.84004e8 −0.891864 −0.445932 0.895067i \(-0.647128\pi\)
−0.445932 + 0.895067i \(0.647128\pi\)
\(294\) −3.25347e8 −0.746675
\(295\) 0 0
\(296\) 1.95651e8 0.438491
\(297\) 1.26770e8 0.280781
\(298\) 2.83784e8 0.621199
\(299\) −2.70821e8 −0.585912
\(300\) 0 0
\(301\) 2.98851e7 0.0631643
\(302\) −3.28225e8 −0.685720
\(303\) −3.26401e8 −0.674066
\(304\) −8.62741e7 −0.176126
\(305\) 0 0
\(306\) −1.44527e7 −0.0288353
\(307\) −7.46289e8 −1.47205 −0.736026 0.676954i \(-0.763300\pi\)
−0.736026 + 0.676954i \(0.763300\pi\)
\(308\) −3.67024e6 −0.00715760
\(309\) −9.83876e8 −1.89708
\(310\) 0 0
\(311\) 6.40000e8 1.20648 0.603238 0.797561i \(-0.293877\pi\)
0.603238 + 0.797561i \(0.293877\pi\)
\(312\) −1.13889e8 −0.212295
\(313\) −1.35452e8 −0.249677 −0.124839 0.992177i \(-0.539841\pi\)
−0.124839 + 0.992177i \(0.539841\pi\)
\(314\) −5.33581e8 −0.972628
\(315\) 0 0
\(316\) 1.21423e8 0.216468
\(317\) −9.75198e7 −0.171943 −0.0859717 0.996298i \(-0.527399\pi\)
−0.0859717 + 0.996298i \(0.527399\pi\)
\(318\) 4.74407e8 0.827287
\(319\) −7.85816e7 −0.135536
\(320\) 0 0
\(321\) 1.03706e9 1.74999
\(322\) 2.07706e7 0.0346700
\(323\) −1.44883e8 −0.239226
\(324\) −3.38457e8 −0.552835
\(325\) 0 0
\(326\) 7.43795e7 0.118903
\(327\) 2.96094e7 0.0468288
\(328\) −2.81662e8 −0.440728
\(329\) −5.43886e6 −0.00842020
\(330\) 0 0
\(331\) 3.49842e8 0.530242 0.265121 0.964215i \(-0.414588\pi\)
0.265121 + 0.964215i \(0.414588\pi\)
\(332\) −4.15524e8 −0.623179
\(333\) −1.00363e8 −0.148943
\(334\) −4.74614e8 −0.696992
\(335\) 0 0
\(336\) 8.73470e6 0.0125621
\(337\) −7.20870e8 −1.02601 −0.513006 0.858385i \(-0.671468\pi\)
−0.513006 + 0.858385i \(0.671468\pi\)
\(338\) 3.40400e8 0.479492
\(339\) −7.01858e7 −0.0978476
\(340\) 0 0
\(341\) −1.17823e8 −0.160913
\(342\) 4.42562e7 0.0598250
\(343\) 7.08865e7 0.0948493
\(344\) 3.55130e8 0.470362
\(345\) 0 0
\(346\) 9.59974e8 1.24593
\(347\) −1.28079e8 −0.164560 −0.0822798 0.996609i \(-0.526220\pi\)
−0.0822798 + 0.996609i \(0.526220\pi\)
\(348\) 1.87014e8 0.237874
\(349\) −1.00379e8 −0.126402 −0.0632012 0.998001i \(-0.520131\pi\)
−0.0632012 + 0.998001i \(0.520131\pi\)
\(350\) 0 0
\(351\) −4.28052e8 −0.528350
\(352\) −4.36142e7 −0.0533002
\(353\) −7.51413e8 −0.909216 −0.454608 0.890692i \(-0.650221\pi\)
−0.454608 + 0.890692i \(0.650221\pi\)
\(354\) 5.21235e8 0.624485
\(355\) 0 0
\(356\) 5.90741e8 0.693940
\(357\) 1.46685e7 0.0170626
\(358\) −2.55115e8 −0.293864
\(359\) −9.17064e8 −1.04609 −0.523045 0.852305i \(-0.675204\pi\)
−0.523045 + 0.852305i \(0.675204\pi\)
\(360\) 0 0
\(361\) −4.50221e8 −0.503675
\(362\) −1.71729e8 −0.190267
\(363\) −8.76814e7 −0.0962131
\(364\) 1.23930e7 0.0134686
\(365\) 0 0
\(366\) 1.74283e8 0.185811
\(367\) 9.16435e8 0.967766 0.483883 0.875133i \(-0.339226\pi\)
0.483883 + 0.875133i \(0.339226\pi\)
\(368\) 2.46821e8 0.258175
\(369\) 1.44485e8 0.149703
\(370\) 0 0
\(371\) −5.16234e7 −0.0524854
\(372\) 2.80405e8 0.282413
\(373\) 1.56099e9 1.55747 0.778736 0.627352i \(-0.215861\pi\)
0.778736 + 0.627352i \(0.215861\pi\)
\(374\) −7.32428e7 −0.0723960
\(375\) 0 0
\(376\) −6.46311e7 −0.0627024
\(377\) 2.65340e8 0.255040
\(378\) 3.28295e7 0.0312639
\(379\) 4.34986e7 0.0410429 0.0205214 0.999789i \(-0.493467\pi\)
0.0205214 + 0.999789i \(0.493467\pi\)
\(380\) 0 0
\(381\) −9.55174e8 −0.884800
\(382\) 4.29906e8 0.394595
\(383\) −1.95422e8 −0.177737 −0.0888685 0.996043i \(-0.528325\pi\)
−0.0888685 + 0.996043i \(0.528325\pi\)
\(384\) 1.03796e8 0.0935453
\(385\) 0 0
\(386\) 1.37681e9 1.21848
\(387\) −1.82171e8 −0.159769
\(388\) 8.83596e8 0.767967
\(389\) −1.60846e9 −1.38543 −0.692716 0.721210i \(-0.743586\pi\)
−0.692716 + 0.721210i \(0.743586\pi\)
\(390\) 0 0
\(391\) 4.14495e8 0.350671
\(392\) 4.20704e8 0.352756
\(393\) −2.09087e9 −1.73761
\(394\) −9.16205e8 −0.754668
\(395\) 0 0
\(396\) 2.23728e7 0.0181046
\(397\) −1.20591e9 −0.967268 −0.483634 0.875270i \(-0.660683\pi\)
−0.483634 + 0.875270i \(0.660683\pi\)
\(398\) 2.72515e8 0.216670
\(399\) −4.49168e7 −0.0354000
\(400\) 0 0
\(401\) 7.40689e8 0.573628 0.286814 0.957986i \(-0.407404\pi\)
0.286814 + 0.957986i \(0.407404\pi\)
\(402\) −1.41104e9 −1.08330
\(403\) 3.97845e8 0.302793
\(404\) 4.22066e8 0.318453
\(405\) 0 0
\(406\) −2.03502e7 −0.0150914
\(407\) −5.08616e8 −0.373947
\(408\) 1.74308e8 0.127060
\(409\) 1.37187e7 0.00991476 0.00495738 0.999988i \(-0.498422\pi\)
0.00495738 + 0.999988i \(0.498422\pi\)
\(410\) 0 0
\(411\) −3.79067e8 −0.269321
\(412\) 1.27224e9 0.896249
\(413\) −5.67190e7 −0.0396190
\(414\) −1.26612e8 −0.0876948
\(415\) 0 0
\(416\) 1.47268e8 0.100296
\(417\) 7.22700e8 0.488070
\(418\) 2.24279e8 0.150201
\(419\) 1.95596e9 1.29901 0.649503 0.760359i \(-0.274977\pi\)
0.649503 + 0.760359i \(0.274977\pi\)
\(420\) 0 0
\(421\) −2.31110e9 −1.50950 −0.754748 0.656015i \(-0.772241\pi\)
−0.754748 + 0.656015i \(0.772241\pi\)
\(422\) 1.39430e9 0.903156
\(423\) 3.31539e7 0.0212982
\(424\) −6.13451e8 −0.390840
\(425\) 0 0
\(426\) 1.24306e9 0.779037
\(427\) −1.89650e7 −0.0117884
\(428\) −1.34101e9 −0.826758
\(429\) 2.96066e8 0.181046
\(430\) 0 0
\(431\) −2.03521e9 −1.22444 −0.612222 0.790686i \(-0.709724\pi\)
−0.612222 + 0.790686i \(0.709724\pi\)
\(432\) 3.90119e8 0.232811
\(433\) −2.81028e9 −1.66357 −0.831786 0.555096i \(-0.812682\pi\)
−0.831786 + 0.555096i \(0.812682\pi\)
\(434\) −3.05127e7 −0.0179171
\(435\) 0 0
\(436\) −3.82876e7 −0.0221236
\(437\) −1.26924e9 −0.727541
\(438\) 1.61448e9 0.918063
\(439\) 1.58225e9 0.892587 0.446293 0.894887i \(-0.352744\pi\)
0.446293 + 0.894887i \(0.352744\pi\)
\(440\) 0 0
\(441\) −2.15809e8 −0.119821
\(442\) 2.47313e8 0.136229
\(443\) 4.88989e8 0.267231 0.133615 0.991033i \(-0.457341\pi\)
0.133615 + 0.991033i \(0.457341\pi\)
\(444\) 1.21044e9 0.656301
\(445\) 0 0
\(446\) 1.16584e9 0.622253
\(447\) 1.75569e9 0.929763
\(448\) −1.12948e7 −0.00593477
\(449\) 1.81072e9 0.944036 0.472018 0.881589i \(-0.343526\pi\)
0.472018 + 0.881589i \(0.343526\pi\)
\(450\) 0 0
\(451\) 7.32212e8 0.375854
\(452\) 9.07565e7 0.0462267
\(453\) −2.03064e9 −1.02633
\(454\) −1.01682e9 −0.509973
\(455\) 0 0
\(456\) −5.33755e8 −0.263612
\(457\) 8.64311e8 0.423607 0.211804 0.977312i \(-0.432066\pi\)
0.211804 + 0.977312i \(0.432066\pi\)
\(458\) 1.03800e9 0.504857
\(459\) 6.55140e8 0.316220
\(460\) 0 0
\(461\) 4.85430e8 0.230767 0.115383 0.993321i \(-0.463190\pi\)
0.115383 + 0.993321i \(0.463190\pi\)
\(462\) −2.27068e7 −0.0107130
\(463\) −1.85376e9 −0.867999 −0.434000 0.900913i \(-0.642898\pi\)
−0.434000 + 0.900913i \(0.642898\pi\)
\(464\) −2.41826e8 −0.112380
\(465\) 0 0
\(466\) −1.21768e9 −0.557419
\(467\) −3.16192e7 −0.0143662 −0.00718310 0.999974i \(-0.502286\pi\)
−0.00718310 + 0.999974i \(0.502286\pi\)
\(468\) −7.55445e7 −0.0340677
\(469\) 1.53544e8 0.0687272
\(470\) 0 0
\(471\) −3.30112e9 −1.45576
\(472\) −6.74003e8 −0.295029
\(473\) −9.23199e8 −0.401126
\(474\) 7.51209e8 0.323994
\(475\) 0 0
\(476\) −1.89677e7 −0.00806101
\(477\) 3.14683e8 0.132757
\(478\) 1.94449e9 0.814343
\(479\) −3.11838e9 −1.29645 −0.648223 0.761451i \(-0.724487\pi\)
−0.648223 + 0.761451i \(0.724487\pi\)
\(480\) 0 0
\(481\) 1.71740e9 0.703661
\(482\) 5.47674e8 0.222770
\(483\) 1.28502e8 0.0518914
\(484\) 1.13380e8 0.0454545
\(485\) 0 0
\(486\) −4.27553e8 −0.168952
\(487\) 3.37608e9 1.32453 0.662265 0.749270i \(-0.269596\pi\)
0.662265 + 0.749270i \(0.269596\pi\)
\(488\) −2.25364e8 −0.0877840
\(489\) 4.60166e8 0.177965
\(490\) 0 0
\(491\) −4.55043e9 −1.73487 −0.867436 0.497548i \(-0.834234\pi\)
−0.867436 + 0.497548i \(0.834234\pi\)
\(492\) −1.74257e9 −0.659648
\(493\) −4.06106e8 −0.152642
\(494\) −7.57304e8 −0.282635
\(495\) 0 0
\(496\) −3.62588e8 −0.133422
\(497\) −1.35266e8 −0.0494242
\(498\) −2.57074e9 −0.932728
\(499\) −6.25733e8 −0.225443 −0.112722 0.993627i \(-0.535957\pi\)
−0.112722 + 0.993627i \(0.535957\pi\)
\(500\) 0 0
\(501\) −2.93631e9 −1.04321
\(502\) −1.33834e9 −0.472174
\(503\) 3.43240e9 1.20257 0.601285 0.799035i \(-0.294656\pi\)
0.601285 + 0.799035i \(0.294656\pi\)
\(504\) 5.79389e6 0.00201587
\(505\) 0 0
\(506\) −6.41638e8 −0.220173
\(507\) 2.10596e9 0.717667
\(508\) 1.23513e9 0.418012
\(509\) 1.36035e9 0.457234 0.228617 0.973516i \(-0.426580\pi\)
0.228617 + 0.973516i \(0.426580\pi\)
\(510\) 0 0
\(511\) −1.75682e8 −0.0582445
\(512\) −1.34218e8 −0.0441942
\(513\) −2.00613e9 −0.656066
\(514\) 3.07831e9 0.999866
\(515\) 0 0
\(516\) 2.19709e9 0.704003
\(517\) 1.68016e8 0.0534727
\(518\) −1.31716e8 −0.0416375
\(519\) 5.93910e9 1.86481
\(520\) 0 0
\(521\) −2.51109e9 −0.777912 −0.388956 0.921256i \(-0.627164\pi\)
−0.388956 + 0.921256i \(0.627164\pi\)
\(522\) 1.24050e8 0.0381724
\(523\) −4.25045e9 −1.29921 −0.649605 0.760272i \(-0.725066\pi\)
−0.649605 + 0.760272i \(0.725066\pi\)
\(524\) 2.70368e9 0.820912
\(525\) 0 0
\(526\) 4.07567e9 1.22109
\(527\) −6.08907e8 −0.181223
\(528\) −2.69829e8 −0.0797757
\(529\) 2.26323e8 0.0664713
\(530\) 0 0
\(531\) 3.45744e8 0.100213
\(532\) 5.80815e7 0.0167243
\(533\) −2.47240e9 −0.707250
\(534\) 3.65476e9 1.03864
\(535\) 0 0
\(536\) 1.82460e9 0.511788
\(537\) −1.57833e9 −0.439833
\(538\) 4.41038e9 1.22106
\(539\) −1.09366e9 −0.300832
\(540\) 0 0
\(541\) 1.06279e9 0.288574 0.144287 0.989536i \(-0.453911\pi\)
0.144287 + 0.989536i \(0.453911\pi\)
\(542\) −2.73817e9 −0.738690
\(543\) −1.06244e9 −0.284778
\(544\) −2.25396e8 −0.0600276
\(545\) 0 0
\(546\) 7.66722e7 0.0201587
\(547\) 3.28564e9 0.858350 0.429175 0.903221i \(-0.358804\pi\)
0.429175 + 0.903221i \(0.358804\pi\)
\(548\) 4.90168e8 0.127237
\(549\) 1.15605e8 0.0298178
\(550\) 0 0
\(551\) 1.24355e9 0.316689
\(552\) 1.52701e9 0.386417
\(553\) −8.17441e7 −0.0205550
\(554\) −4.08779e9 −1.02142
\(555\) 0 0
\(556\) −9.34516e8 −0.230582
\(557\) 4.04436e9 0.991645 0.495823 0.868424i \(-0.334867\pi\)
0.495823 + 0.868424i \(0.334867\pi\)
\(558\) 1.85997e8 0.0453197
\(559\) 3.11729e9 0.754806
\(560\) 0 0
\(561\) −4.53133e8 −0.108357
\(562\) 1.21400e8 0.0288498
\(563\) 1.60197e9 0.378333 0.189167 0.981945i \(-0.439421\pi\)
0.189167 + 0.981945i \(0.439421\pi\)
\(564\) −3.99855e8 −0.0938481
\(565\) 0 0
\(566\) −1.80272e9 −0.417899
\(567\) 2.27856e8 0.0524952
\(568\) −1.60739e9 −0.368045
\(569\) −5.94461e9 −1.35279 −0.676396 0.736539i \(-0.736459\pi\)
−0.676396 + 0.736539i \(0.736459\pi\)
\(570\) 0 0
\(571\) −1.01942e9 −0.229154 −0.114577 0.993414i \(-0.536551\pi\)
−0.114577 + 0.993414i \(0.536551\pi\)
\(572\) −3.82840e8 −0.0855326
\(573\) 2.65971e9 0.590600
\(574\) 1.89621e8 0.0418499
\(575\) 0 0
\(576\) 6.88499e7 0.0150115
\(577\) 5.05194e8 0.109482 0.0547410 0.998501i \(-0.482567\pi\)
0.0547410 + 0.998501i \(0.482567\pi\)
\(578\) 2.90419e9 0.625573
\(579\) 8.51792e9 1.82372
\(580\) 0 0
\(581\) 2.79739e8 0.0591748
\(582\) 5.46657e9 1.14944
\(583\) 1.59473e9 0.333310
\(584\) −2.08767e9 −0.433726
\(585\) 0 0
\(586\) 3.07203e9 0.630643
\(587\) −6.05441e8 −0.123549 −0.0617744 0.998090i \(-0.519676\pi\)
−0.0617744 + 0.998090i \(0.519676\pi\)
\(588\) 2.60278e9 0.527979
\(589\) 1.86455e9 0.375985
\(590\) 0 0
\(591\) −5.66831e9 −1.12953
\(592\) −1.56521e9 −0.310060
\(593\) 2.44359e8 0.0481212 0.0240606 0.999711i \(-0.492341\pi\)
0.0240606 + 0.999711i \(0.492341\pi\)
\(594\) −1.01416e9 −0.198542
\(595\) 0 0
\(596\) −2.27027e9 −0.439254
\(597\) 1.68598e9 0.324296
\(598\) 2.16656e9 0.414302
\(599\) 4.76209e9 0.905324 0.452662 0.891682i \(-0.350474\pi\)
0.452662 + 0.891682i \(0.350474\pi\)
\(600\) 0 0
\(601\) −4.27387e9 −0.803083 −0.401542 0.915841i \(-0.631525\pi\)
−0.401542 + 0.915841i \(0.631525\pi\)
\(602\) −2.39080e8 −0.0446639
\(603\) −9.35966e8 −0.173840
\(604\) 2.62580e9 0.484877
\(605\) 0 0
\(606\) 2.61121e9 0.476637
\(607\) −6.40384e9 −1.16220 −0.581099 0.813833i \(-0.697377\pi\)
−0.581099 + 0.813833i \(0.697377\pi\)
\(608\) 6.90193e8 0.124540
\(609\) −1.25901e8 −0.0225876
\(610\) 0 0
\(611\) −5.67324e8 −0.100621
\(612\) 1.15622e8 0.0203897
\(613\) 8.98899e9 1.57616 0.788078 0.615575i \(-0.211076\pi\)
0.788078 + 0.615575i \(0.211076\pi\)
\(614\) 5.97032e9 1.04090
\(615\) 0 0
\(616\) 2.93620e7 0.00506119
\(617\) 2.11391e9 0.362316 0.181158 0.983454i \(-0.442015\pi\)
0.181158 + 0.983454i \(0.442015\pi\)
\(618\) 7.87100e9 1.34144
\(619\) −4.25875e9 −0.721713 −0.360856 0.932621i \(-0.617516\pi\)
−0.360856 + 0.932621i \(0.617516\pi\)
\(620\) 0 0
\(621\) 5.73930e9 0.961698
\(622\) −5.12000e9 −0.853107
\(623\) −3.97699e8 −0.0658940
\(624\) 9.11110e8 0.150115
\(625\) 0 0
\(626\) 1.08361e9 0.176548
\(627\) 1.38755e9 0.224809
\(628\) 4.26865e9 0.687752
\(629\) −2.62850e9 −0.421145
\(630\) 0 0
\(631\) 9.55885e9 1.51462 0.757309 0.653057i \(-0.226514\pi\)
0.757309 + 0.653057i \(0.226514\pi\)
\(632\) −9.71380e8 −0.153066
\(633\) 8.62617e9 1.35178
\(634\) 7.80159e8 0.121582
\(635\) 0 0
\(636\) −3.79526e9 −0.584980
\(637\) 3.69289e9 0.566080
\(638\) 6.28652e8 0.0958381
\(639\) 8.24543e8 0.125015
\(640\) 0 0
\(641\) 7.93190e9 1.18953 0.594763 0.803901i \(-0.297246\pi\)
0.594763 + 0.803901i \(0.297246\pi\)
\(642\) −8.29646e9 −1.23743
\(643\) 3.67790e9 0.545583 0.272792 0.962073i \(-0.412053\pi\)
0.272792 + 0.962073i \(0.412053\pi\)
\(644\) −1.66165e8 −0.0245154
\(645\) 0 0
\(646\) 1.15906e9 0.169158
\(647\) 3.34066e9 0.484917 0.242459 0.970162i \(-0.422046\pi\)
0.242459 + 0.970162i \(0.422046\pi\)
\(648\) 2.70765e9 0.390913
\(649\) 1.75215e9 0.251602
\(650\) 0 0
\(651\) −1.88774e8 −0.0268169
\(652\) −5.95036e8 −0.0840769
\(653\) −1.23649e10 −1.73778 −0.868891 0.495004i \(-0.835167\pi\)
−0.868891 + 0.495004i \(0.835167\pi\)
\(654\) −2.36875e8 −0.0331129
\(655\) 0 0
\(656\) 2.25330e9 0.311642
\(657\) 1.07091e9 0.147325
\(658\) 4.35109e7 0.00595398
\(659\) 8.58087e9 1.16797 0.583986 0.811764i \(-0.301492\pi\)
0.583986 + 0.811764i \(0.301492\pi\)
\(660\) 0 0
\(661\) −8.09007e9 −1.08955 −0.544775 0.838582i \(-0.683385\pi\)
−0.544775 + 0.838582i \(0.683385\pi\)
\(662\) −2.79873e9 −0.374937
\(663\) 1.53006e9 0.203897
\(664\) 3.32419e9 0.440654
\(665\) 0 0
\(666\) 8.02907e8 0.105319
\(667\) −3.55766e9 −0.464220
\(668\) 3.79691e9 0.492848
\(669\) 7.21274e9 0.931341
\(670\) 0 0
\(671\) 5.85859e8 0.0748625
\(672\) −6.98776e7 −0.00888271
\(673\) 1.38929e10 1.75687 0.878436 0.477860i \(-0.158587\pi\)
0.878436 + 0.477860i \(0.158587\pi\)
\(674\) 5.76696e9 0.725500
\(675\) 0 0
\(676\) −2.72320e9 −0.339052
\(677\) −6.36018e9 −0.787788 −0.393894 0.919156i \(-0.628872\pi\)
−0.393894 + 0.919156i \(0.628872\pi\)
\(678\) 5.61486e8 0.0691887
\(679\) −5.94855e8 −0.0729233
\(680\) 0 0
\(681\) −6.29077e9 −0.763289
\(682\) 9.42588e8 0.113783
\(683\) 7.46187e9 0.896139 0.448069 0.893999i \(-0.352112\pi\)
0.448069 + 0.893999i \(0.352112\pi\)
\(684\) −3.54049e8 −0.0423026
\(685\) 0 0
\(686\) −5.67092e8 −0.0670686
\(687\) 6.42182e9 0.755631
\(688\) −2.84104e9 −0.332596
\(689\) −5.38480e9 −0.627195
\(690\) 0 0
\(691\) −1.01338e10 −1.16842 −0.584212 0.811601i \(-0.698596\pi\)
−0.584212 + 0.811601i \(0.698596\pi\)
\(692\) −7.67979e9 −0.881004
\(693\) −1.50618e7 −0.00171914
\(694\) 1.02463e9 0.116361
\(695\) 0 0
\(696\) −1.49611e9 −0.168202
\(697\) 3.78404e9 0.423293
\(698\) 8.03034e8 0.0893800
\(699\) −7.53345e9 −0.834303
\(700\) 0 0
\(701\) 1.31006e10 1.43641 0.718206 0.695831i \(-0.244964\pi\)
0.718206 + 0.695831i \(0.244964\pi\)
\(702\) 3.42442e9 0.373600
\(703\) 8.04883e9 0.873754
\(704\) 3.48914e8 0.0376889
\(705\) 0 0
\(706\) 6.01130e9 0.642913
\(707\) −2.84143e8 −0.0302391
\(708\) −4.16988e9 −0.441577
\(709\) −1.04601e10 −1.10223 −0.551116 0.834429i \(-0.685798\pi\)
−0.551116 + 0.834429i \(0.685798\pi\)
\(710\) 0 0
\(711\) 4.98290e8 0.0519923
\(712\) −4.72593e9 −0.490690
\(713\) −5.33428e9 −0.551141
\(714\) −1.17348e8 −0.0120651
\(715\) 0 0
\(716\) 2.04092e9 0.207793
\(717\) 1.20300e10 1.21885
\(718\) 7.33651e9 0.739697
\(719\) 6.14452e9 0.616505 0.308253 0.951305i \(-0.400256\pi\)
0.308253 + 0.951305i \(0.400256\pi\)
\(720\) 0 0
\(721\) −8.56497e8 −0.0851045
\(722\) 3.60177e9 0.356152
\(723\) 3.38831e9 0.333426
\(724\) 1.37383e9 0.134539
\(725\) 0 0
\(726\) 7.01451e8 0.0680329
\(727\) 1.47022e9 0.141910 0.0709550 0.997480i \(-0.477395\pi\)
0.0709550 + 0.997480i \(0.477395\pi\)
\(728\) −9.91440e7 −0.00952372
\(729\) 8.92054e9 0.852796
\(730\) 0 0
\(731\) −4.77105e9 −0.451755
\(732\) −1.39427e9 −0.131389
\(733\) −6.30989e8 −0.0591777 −0.0295888 0.999562i \(-0.509420\pi\)
−0.0295888 + 0.999562i \(0.509420\pi\)
\(734\) −7.33148e9 −0.684314
\(735\) 0 0
\(736\) −1.97457e9 −0.182557
\(737\) −4.74324e9 −0.436454
\(738\) −1.15588e9 −0.105856
\(739\) 7.22368e9 0.658420 0.329210 0.944257i \(-0.393218\pi\)
0.329210 + 0.944257i \(0.393218\pi\)
\(740\) 0 0
\(741\) −4.68524e9 −0.423027
\(742\) 4.12987e8 0.0371128
\(743\) −7.72789e9 −0.691194 −0.345597 0.938383i \(-0.612324\pi\)
−0.345597 + 0.938383i \(0.612324\pi\)
\(744\) −2.24324e9 −0.199696
\(745\) 0 0
\(746\) −1.24880e10 −1.10130
\(747\) −1.70522e9 −0.149678
\(748\) 5.85942e8 0.0511917
\(749\) 9.02793e8 0.0785059
\(750\) 0 0
\(751\) −1.54417e10 −1.33032 −0.665158 0.746702i \(-0.731636\pi\)
−0.665158 + 0.746702i \(0.731636\pi\)
\(752\) 5.17048e8 0.0443373
\(753\) −8.27993e9 −0.706715
\(754\) −2.12272e9 −0.180340
\(755\) 0 0
\(756\) −2.62636e8 −0.0221069
\(757\) −1.94159e10 −1.62676 −0.813378 0.581735i \(-0.802374\pi\)
−0.813378 + 0.581735i \(0.802374\pi\)
\(758\) −3.47989e8 −0.0290217
\(759\) −3.96964e9 −0.329538
\(760\) 0 0
\(761\) 5.03230e8 0.0413924 0.0206962 0.999786i \(-0.493412\pi\)
0.0206962 + 0.999786i \(0.493412\pi\)
\(762\) 7.64139e9 0.625648
\(763\) 2.57760e7 0.00210078
\(764\) −3.43925e9 −0.279021
\(765\) 0 0
\(766\) 1.56338e9 0.125679
\(767\) −5.91632e9 −0.473443
\(768\) −8.30369e8 −0.0661465
\(769\) −1.87701e10 −1.48842 −0.744208 0.667948i \(-0.767173\pi\)
−0.744208 + 0.667948i \(0.767173\pi\)
\(770\) 0 0
\(771\) 1.90447e10 1.49652
\(772\) −1.10144e10 −0.861593
\(773\) −2.39166e10 −1.86239 −0.931196 0.364518i \(-0.881234\pi\)
−0.931196 + 0.364518i \(0.881234\pi\)
\(774\) 1.45737e9 0.112974
\(775\) 0 0
\(776\) −7.06877e9 −0.543035
\(777\) −8.14892e8 −0.0623199
\(778\) 1.28676e10 0.979649
\(779\) −1.15872e10 −0.878210
\(780\) 0 0
\(781\) 4.17858e9 0.313870
\(782\) −3.31596e9 −0.247962
\(783\) −5.62315e9 −0.418614
\(784\) −3.36563e9 −0.249436
\(785\) 0 0
\(786\) 1.67270e10 1.22868
\(787\) −2.09777e10 −1.53408 −0.767038 0.641602i \(-0.778270\pi\)
−0.767038 + 0.641602i \(0.778270\pi\)
\(788\) 7.32964e9 0.533631
\(789\) 2.52151e10 1.82764
\(790\) 0 0
\(791\) −6.10991e7 −0.00438952
\(792\) −1.78983e8 −0.0128019
\(793\) −1.97822e9 −0.140870
\(794\) 9.64725e9 0.683962
\(795\) 0 0
\(796\) −2.18012e9 −0.153209
\(797\) −1.55513e10 −1.08809 −0.544043 0.839057i \(-0.683107\pi\)
−0.544043 + 0.839057i \(0.683107\pi\)
\(798\) 3.59334e8 0.0250316
\(799\) 8.68297e8 0.0602219
\(800\) 0 0
\(801\) 2.42427e9 0.166673
\(802\) −5.92551e9 −0.405617
\(803\) 5.42711e9 0.369883
\(804\) 1.12883e10 0.766005
\(805\) 0 0
\(806\) −3.18276e9 −0.214107
\(807\) 2.72858e10 1.82759
\(808\) −3.37653e9 −0.225180
\(809\) −1.19959e10 −0.796548 −0.398274 0.917266i \(-0.630391\pi\)
−0.398274 + 0.917266i \(0.630391\pi\)
\(810\) 0 0
\(811\) 1.18093e10 0.777412 0.388706 0.921362i \(-0.372922\pi\)
0.388706 + 0.921362i \(0.372922\pi\)
\(812\) 1.62802e8 0.0106712
\(813\) −1.69403e10 −1.10562
\(814\) 4.06893e9 0.264420
\(815\) 0 0
\(816\) −1.39447e9 −0.0898447
\(817\) 1.46096e10 0.937262
\(818\) −1.09750e8 −0.00701079
\(819\) 5.08580e7 0.00323494
\(820\) 0 0
\(821\) −1.96308e10 −1.23805 −0.619024 0.785372i \(-0.712472\pi\)
−0.619024 + 0.785372i \(0.712472\pi\)
\(822\) 3.03254e9 0.190439
\(823\) 2.67601e9 0.167336 0.0836679 0.996494i \(-0.473337\pi\)
0.0836679 + 0.996494i \(0.473337\pi\)
\(824\) −1.01779e10 −0.633744
\(825\) 0 0
\(826\) 4.53752e8 0.0280149
\(827\) −9.26722e8 −0.0569745 −0.0284872 0.999594i \(-0.509069\pi\)
−0.0284872 + 0.999594i \(0.509069\pi\)
\(828\) 1.01290e9 0.0620096
\(829\) −2.94467e10 −1.79513 −0.897565 0.440882i \(-0.854666\pi\)
−0.897565 + 0.440882i \(0.854666\pi\)
\(830\) 0 0
\(831\) −2.52900e10 −1.52878
\(832\) −1.17815e9 −0.0709199
\(833\) −5.65201e9 −0.338802
\(834\) −5.78160e9 −0.345117
\(835\) 0 0
\(836\) −1.79423e9 −0.106208
\(837\) −8.43123e9 −0.496995
\(838\) −1.56477e10 −0.918536
\(839\) −8.93988e9 −0.522595 −0.261297 0.965258i \(-0.584150\pi\)
−0.261297 + 0.965258i \(0.584150\pi\)
\(840\) 0 0
\(841\) −1.37642e10 −0.797931
\(842\) 1.84888e10 1.06737
\(843\) 7.51071e8 0.0431802
\(844\) −1.11544e10 −0.638628
\(845\) 0 0
\(846\) −2.65231e8 −0.0150601
\(847\) −7.63296e7 −0.00431619
\(848\) 4.90761e9 0.276366
\(849\) −1.11530e10 −0.625480
\(850\) 0 0
\(851\) −2.30268e10 −1.28080
\(852\) −9.94447e9 −0.550862
\(853\) −1.71708e10 −0.947257 −0.473628 0.880725i \(-0.657056\pi\)
−0.473628 + 0.880725i \(0.657056\pi\)
\(854\) 1.51720e8 0.00833565
\(855\) 0 0
\(856\) 1.07281e10 0.584606
\(857\) 7.71458e9 0.418677 0.209339 0.977843i \(-0.432869\pi\)
0.209339 + 0.977843i \(0.432869\pi\)
\(858\) −2.36853e9 −0.128019
\(859\) 1.68606e10 0.907606 0.453803 0.891102i \(-0.350067\pi\)
0.453803 + 0.891102i \(0.350067\pi\)
\(860\) 0 0
\(861\) 1.17313e9 0.0626377
\(862\) 1.62817e10 0.865813
\(863\) 2.42922e10 1.28656 0.643278 0.765633i \(-0.277574\pi\)
0.643278 + 0.765633i \(0.277574\pi\)
\(864\) −3.12095e9 −0.164622
\(865\) 0 0
\(866\) 2.24822e10 1.17632
\(867\) 1.79675e10 0.936311
\(868\) 2.44102e8 0.0126693
\(869\) 2.52521e9 0.130535
\(870\) 0 0
\(871\) 1.60161e10 0.821283
\(872\) 3.06301e8 0.0156438
\(873\) 3.62608e9 0.184454
\(874\) 1.01539e10 0.514449
\(875\) 0 0
\(876\) −1.29158e10 −0.649169
\(877\) −2.67740e10 −1.34034 −0.670169 0.742209i \(-0.733778\pi\)
−0.670169 + 0.742209i \(0.733778\pi\)
\(878\) −1.26580e10 −0.631154
\(879\) 1.90058e10 0.943899
\(880\) 0 0
\(881\) 1.10654e10 0.545197 0.272598 0.962128i \(-0.412117\pi\)
0.272598 + 0.962128i \(0.412117\pi\)
\(882\) 1.72647e9 0.0847265
\(883\) −2.06264e10 −1.00824 −0.504118 0.863635i \(-0.668182\pi\)
−0.504118 + 0.863635i \(0.668182\pi\)
\(884\) −1.97850e9 −0.0963282
\(885\) 0 0
\(886\) −3.91191e9 −0.188961
\(887\) 7.62436e9 0.366835 0.183417 0.983035i \(-0.441284\pi\)
0.183417 + 0.983035i \(0.441284\pi\)
\(888\) −9.68352e9 −0.464075
\(889\) −8.31512e8 −0.0396928
\(890\) 0 0
\(891\) −7.03884e9 −0.333372
\(892\) −9.32672e9 −0.439999
\(893\) −2.65884e9 −0.124943
\(894\) −1.40455e10 −0.657442
\(895\) 0 0
\(896\) 9.03580e7 0.00419651
\(897\) 1.34040e10 0.620097
\(898\) −1.44857e10 −0.667534
\(899\) 5.22633e9 0.239904
\(900\) 0 0
\(901\) 8.24151e9 0.375379
\(902\) −5.85770e9 −0.265769
\(903\) −1.47913e9 −0.0668495
\(904\) −7.26052e8 −0.0326872
\(905\) 0 0
\(906\) 1.62451e10 0.725728
\(907\) 3.36091e10 1.49566 0.747828 0.663893i \(-0.231097\pi\)
0.747828 + 0.663893i \(0.231097\pi\)
\(908\) 8.13454e9 0.360606
\(909\) 1.73206e9 0.0764874
\(910\) 0 0
\(911\) −3.89170e10 −1.70540 −0.852698 0.522404i \(-0.825035\pi\)
−0.852698 + 0.522404i \(0.825035\pi\)
\(912\) 4.27004e9 0.186402
\(913\) −8.64161e9 −0.375791
\(914\) −6.91449e9 −0.299536
\(915\) 0 0
\(916\) −8.30400e9 −0.356987
\(917\) −1.82017e9 −0.0779507
\(918\) −5.24112e9 −0.223601
\(919\) 4.35993e10 1.85300 0.926499 0.376296i \(-0.122803\pi\)
0.926499 + 0.376296i \(0.122803\pi\)
\(920\) 0 0
\(921\) 3.69367e10 1.55794
\(922\) −3.88344e9 −0.163177
\(923\) −1.41094e10 −0.590614
\(924\) 1.81655e8 0.00757520
\(925\) 0 0
\(926\) 1.48301e10 0.613768
\(927\) 5.22098e9 0.215265
\(928\) 1.93461e9 0.0794648
\(929\) −3.06911e10 −1.25591 −0.627954 0.778250i \(-0.716108\pi\)
−0.627954 + 0.778250i \(0.716108\pi\)
\(930\) 0 0
\(931\) 1.73072e10 0.702915
\(932\) 9.74143e9 0.394155
\(933\) −3.16761e10 −1.27687
\(934\) 2.52954e8 0.0101584
\(935\) 0 0
\(936\) 6.04356e8 0.0240895
\(937\) −3.51153e9 −0.139446 −0.0697232 0.997566i \(-0.522212\pi\)
−0.0697232 + 0.997566i \(0.522212\pi\)
\(938\) −1.22835e9 −0.0485975
\(939\) 6.70402e9 0.264244
\(940\) 0 0
\(941\) −1.84266e10 −0.720911 −0.360455 0.932776i \(-0.617379\pi\)
−0.360455 + 0.932776i \(0.617379\pi\)
\(942\) 2.64090e10 1.02938
\(943\) 3.31498e10 1.28733
\(944\) 5.39203e9 0.208617
\(945\) 0 0
\(946\) 7.38559e9 0.283639
\(947\) −8.78798e9 −0.336251 −0.168126 0.985766i \(-0.553771\pi\)
−0.168126 + 0.985766i \(0.553771\pi\)
\(948\) −6.00967e9 −0.229098
\(949\) −1.83253e10 −0.696015
\(950\) 0 0
\(951\) 4.82663e9 0.181975
\(952\) 1.51741e8 0.00569999
\(953\) −1.85304e10 −0.693521 −0.346761 0.937954i \(-0.612718\pi\)
−0.346761 + 0.937954i \(0.612718\pi\)
\(954\) −2.51746e9 −0.0938737
\(955\) 0 0
\(956\) −1.55559e10 −0.575827
\(957\) 3.88930e9 0.143443
\(958\) 2.49470e10 0.916725
\(959\) −3.29991e8 −0.0120819
\(960\) 0 0
\(961\) −1.96764e10 −0.715176
\(962\) −1.37392e10 −0.497564
\(963\) −5.50319e9 −0.198574
\(964\) −4.38139e9 −0.157522
\(965\) 0 0
\(966\) −1.02802e9 −0.0366927
\(967\) −4.71498e10 −1.67682 −0.838412 0.545037i \(-0.816516\pi\)
−0.838412 + 0.545037i \(0.816516\pi\)
\(968\) −9.07039e8 −0.0321412
\(969\) 7.17082e9 0.253184
\(970\) 0 0
\(971\) −8.76215e9 −0.307145 −0.153572 0.988137i \(-0.549078\pi\)
−0.153572 + 0.988137i \(0.549078\pi\)
\(972\) 3.42042e9 0.119467
\(973\) 6.29135e8 0.0218952
\(974\) −2.70086e10 −0.936583
\(975\) 0 0
\(976\) 1.80291e9 0.0620727
\(977\) 6.33661e9 0.217383 0.108692 0.994076i \(-0.465334\pi\)
0.108692 + 0.994076i \(0.465334\pi\)
\(978\) −3.68133e9 −0.125840
\(979\) 1.22856e10 0.418462
\(980\) 0 0
\(981\) −1.57124e8 −0.00531374
\(982\) 3.64035e10 1.22674
\(983\) −4.47312e10 −1.50201 −0.751006 0.660295i \(-0.770431\pi\)
−0.751006 + 0.660295i \(0.770431\pi\)
\(984\) 1.39406e10 0.466442
\(985\) 0 0
\(986\) 3.24885e9 0.107934
\(987\) 2.69190e8 0.00891147
\(988\) 6.05843e9 0.199853
\(989\) −4.17964e10 −1.37389
\(990\) 0 0
\(991\) 4.62109e10 1.50830 0.754148 0.656705i \(-0.228050\pi\)
0.754148 + 0.656705i \(0.228050\pi\)
\(992\) 2.90071e9 0.0943437
\(993\) −1.73150e10 −0.561178
\(994\) 1.08212e9 0.0349482
\(995\) 0 0
\(996\) 2.05659e10 0.659538
\(997\) 1.52392e10 0.487001 0.243501 0.969901i \(-0.421704\pi\)
0.243501 + 0.969901i \(0.421704\pi\)
\(998\) 5.00587e9 0.159413
\(999\) −3.63956e10 −1.15497
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 550.8.a.d.1.1 2
5.4 even 2 22.8.a.d.1.2 2
15.14 odd 2 198.8.a.f.1.2 2
20.19 odd 2 176.8.a.e.1.1 2
55.54 odd 2 242.8.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.8.a.d.1.2 2 5.4 even 2
176.8.a.e.1.1 2 20.19 odd 2
198.8.a.f.1.2 2 15.14 odd 2
242.8.a.h.1.2 2 55.54 odd 2
550.8.a.d.1.1 2 1.1 even 1 trivial