Properties

Label 175.8.b.c.99.1
Level $175$
Weight $8$
Character 175.99
Analytic conductor $54.667$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [175,8,Mod(99,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.99");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(54.6673794597\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.8.b.c.99.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.6332i q^{2} -54.7995i q^{3} -86.1320 q^{4} -801.895 q^{6} +343.000i q^{7} -612.665i q^{8} -815.985 q^{9} +O(q^{10})\) \(q-14.6332i q^{2} -54.7995i q^{3} -86.1320 q^{4} -801.895 q^{6} +343.000i q^{7} -612.665i q^{8} -815.985 q^{9} -6473.63 q^{11} +4719.99i q^{12} -11681.7i q^{13} +5019.20 q^{14} -19990.2 q^{16} -13460.5i q^{17} +11940.5i q^{18} -34955.5 q^{19} +18796.2 q^{21} +94730.3i q^{22} +77831.4i q^{23} -33573.7 q^{24} -170941. q^{26} -75130.9i q^{27} -29543.3i q^{28} +221135. q^{29} -23222.3 q^{31} +214100. i q^{32} +354752. i q^{33} -196971. q^{34} +70282.4 q^{36} +422392. i q^{37} +511512. i q^{38} -640151. q^{39} +191818. q^{41} -275050. i q^{42} +310754. i q^{43} +557587. q^{44} +1.13893e6 q^{46} +240747. i q^{47} +1.09545e6i q^{48} -117649. q^{49} -737628. q^{51} +1.00617e6i q^{52} -1.06654e6i q^{53} -1.09941e6 q^{54} +210144. q^{56} +1.91554e6i q^{57} -3.23592e6i q^{58} -451838. q^{59} -831659. q^{61} +339818. i q^{62} -279883. i q^{63} +574238. q^{64} +5.19117e6 q^{66} -2.26405e6i q^{67} +1.15938e6i q^{68} +4.26512e6 q^{69} -2.22036e6 q^{71} +499925. i q^{72} +4.99377e6i q^{73} +6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6i q^{77} +9.36749e6i q^{78} +2.72773e6 q^{79} -5.90170e6 q^{81} -2.80693e6i q^{82} -6.38392e6i q^{83} -1.61896e6 q^{84} +4.54734e6 q^{86} -1.21181e7i q^{87} +3.96617e6i q^{88} +7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6i q^{92} +1.27257e6i q^{93} +3.52291e6 q^{94} +1.17326e7 q^{96} +2.38676e6i q^{97} +1.72159e6i q^{98} +5.28239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 80 q^{4} - 1536 q^{6} + 1512 q^{9} - 15812 q^{11} + 10976 q^{14} - 8640 q^{16} + 7224 q^{19} + 20580 q^{21} - 49920 q^{24} - 358656 q^{26} + 253796 q^{29} + 505536 q^{31} - 350592 q^{34} + 537120 q^{36} - 975948 q^{39} - 223840 q^{41} + 753840 q^{44} + 2102944 q^{46} - 470596 q^{49} - 2261820 q^{51} - 1944000 q^{54} + 658560 q^{56} + 2720240 q^{59} - 3627360 q^{61} + 4979968 q^{64} + 10285248 q^{66} + 11703960 q^{69} - 2989856 q^{71} + 11934048 q^{74} + 15750752 q^{76} + 15885948 q^{79} - 9551196 q^{81} - 5383728 q^{84} + 10835424 q^{86} + 35887056 q^{89} + 12223148 q^{91} + 5646208 q^{94} + 26732544 q^{96} + 6061464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 14.6332i − 1.29341i −0.762741 0.646704i \(-0.776147\pi\)
0.762741 0.646704i \(-0.223853\pi\)
\(3\) − 54.7995i − 1.17180i −0.810385 0.585898i \(-0.800742\pi\)
0.810385 0.585898i \(-0.199258\pi\)
\(4\) −86.1320 −0.672906
\(5\) 0 0
\(6\) −801.895 −1.51561
\(7\) 343.000i 0.377964i
\(8\) − 612.665i − 0.423066i
\(9\) −815.985 −0.373107
\(10\) 0 0
\(11\) −6473.63 −1.46647 −0.733236 0.679974i \(-0.761991\pi\)
−0.733236 + 0.679974i \(0.761991\pi\)
\(12\) 4719.99i 0.788509i
\(13\) − 11681.7i − 1.47470i −0.675510 0.737351i \(-0.736076\pi\)
0.675510 0.737351i \(-0.263924\pi\)
\(14\) 5019.20 0.488863
\(15\) 0 0
\(16\) −19990.2 −1.22010
\(17\) − 13460.5i − 0.664491i −0.943193 0.332246i \(-0.892194\pi\)
0.943193 0.332246i \(-0.107806\pi\)
\(18\) 11940.5i 0.482580i
\(19\) −34955.5 −1.16917 −0.584585 0.811333i \(-0.698743\pi\)
−0.584585 + 0.811333i \(0.698743\pi\)
\(20\) 0 0
\(21\) 18796.2 0.442897
\(22\) 94730.3i 1.89675i
\(23\) 77831.4i 1.33385i 0.745124 + 0.666926i \(0.232390\pi\)
−0.745124 + 0.666926i \(0.767610\pi\)
\(24\) −33573.7 −0.495747
\(25\) 0 0
\(26\) −170941. −1.90739
\(27\) − 75130.9i − 0.734591i
\(28\) − 29543.3i − 0.254335i
\(29\) 221135. 1.68370 0.841848 0.539715i \(-0.181468\pi\)
0.841848 + 0.539715i \(0.181468\pi\)
\(30\) 0 0
\(31\) −23222.3 −0.140004 −0.0700018 0.997547i \(-0.522301\pi\)
−0.0700018 + 0.997547i \(0.522301\pi\)
\(32\) 214100.i 1.15503i
\(33\) 354752.i 1.71841i
\(34\) −196971. −0.859459
\(35\) 0 0
\(36\) 70282.4 0.251066
\(37\) 422392.i 1.37091i 0.728114 + 0.685456i \(0.240397\pi\)
−0.728114 + 0.685456i \(0.759603\pi\)
\(38\) 511512.i 1.51221i
\(39\) −640151. −1.72805
\(40\) 0 0
\(41\) 191818. 0.434657 0.217329 0.976099i \(-0.430266\pi\)
0.217329 + 0.976099i \(0.430266\pi\)
\(42\) − 275050.i − 0.572847i
\(43\) 310754.i 0.596042i 0.954559 + 0.298021i \(0.0963266\pi\)
−0.954559 + 0.298021i \(0.903673\pi\)
\(44\) 557587. 0.986798
\(45\) 0 0
\(46\) 1.13893e6 1.72522
\(47\) 240747.i 0.338235i 0.985596 + 0.169117i \(0.0540917\pi\)
−0.985596 + 0.169117i \(0.945908\pi\)
\(48\) 1.09545e6i 1.42971i
\(49\) −117649. −0.142857
\(50\) 0 0
\(51\) −737628. −0.778649
\(52\) 1.00617e6i 0.992336i
\(53\) − 1.06654e6i − 0.984040i −0.870584 0.492020i \(-0.836259\pi\)
0.870584 0.492020i \(-0.163741\pi\)
\(54\) −1.09941e6 −0.950126
\(55\) 0 0
\(56\) 210144. 0.159904
\(57\) 1.91554e6i 1.37003i
\(58\) − 3.23592e6i − 2.17771i
\(59\) −451838. −0.286418 −0.143209 0.989692i \(-0.545742\pi\)
−0.143209 + 0.989692i \(0.545742\pi\)
\(60\) 0 0
\(61\) −831659. −0.469127 −0.234564 0.972101i \(-0.575366\pi\)
−0.234564 + 0.972101i \(0.575366\pi\)
\(62\) 339818.i 0.181082i
\(63\) − 279883.i − 0.141021i
\(64\) 574238. 0.273818
\(65\) 0 0
\(66\) 5.19117e6 2.22260
\(67\) − 2.26405e6i − 0.919654i −0.888009 0.459827i \(-0.847912\pi\)
0.888009 0.459827i \(-0.152088\pi\)
\(68\) 1.15938e6i 0.447140i
\(69\) 4.26512e6 1.56300
\(70\) 0 0
\(71\) −2.22036e6 −0.736241 −0.368120 0.929778i \(-0.619999\pi\)
−0.368120 + 0.929778i \(0.619999\pi\)
\(72\) 499925.i 0.157849i
\(73\) 4.99377e6i 1.50244i 0.660049 + 0.751222i \(0.270535\pi\)
−0.660049 + 0.751222i \(0.729465\pi\)
\(74\) 6.18096e6 1.77315
\(75\) 0 0
\(76\) 3.01078e6 0.786742
\(77\) − 2.22046e6i − 0.554274i
\(78\) 9.36749e6i 2.23508i
\(79\) 2.72773e6 0.622452 0.311226 0.950336i \(-0.399260\pi\)
0.311226 + 0.950336i \(0.399260\pi\)
\(80\) 0 0
\(81\) −5.90170e6 −1.23390
\(82\) − 2.80693e6i − 0.562189i
\(83\) − 6.38392e6i − 1.22550i −0.790276 0.612751i \(-0.790063\pi\)
0.790276 0.612751i \(-0.209937\pi\)
\(84\) −1.61896e6 −0.298028
\(85\) 0 0
\(86\) 4.54734e6 0.770926
\(87\) − 1.21181e7i − 1.97295i
\(88\) 3.96617e6i 0.620414i
\(89\) 7.32978e6 1.10211 0.551056 0.834468i \(-0.314225\pi\)
0.551056 + 0.834468i \(0.314225\pi\)
\(90\) 0 0
\(91\) 4.00682e6 0.557385
\(92\) − 6.70378e6i − 0.897557i
\(93\) 1.27257e6i 0.164056i
\(94\) 3.52291e6 0.437476
\(95\) 0 0
\(96\) 1.17326e7 1.35346
\(97\) 2.38676e6i 0.265526i 0.991148 + 0.132763i \(0.0423850\pi\)
−0.991148 + 0.132763i \(0.957615\pi\)
\(98\) 1.72159e6i 0.184773i
\(99\) 5.28239e6 0.547151
\(100\) 0 0
\(101\) −1.92113e7 −1.85538 −0.927688 0.373356i \(-0.878207\pi\)
−0.927688 + 0.373356i \(0.878207\pi\)
\(102\) 1.07939e7i 1.00711i
\(103\) − 4.45359e6i − 0.401587i −0.979634 0.200793i \(-0.935648\pi\)
0.979634 0.200793i \(-0.0643520\pi\)
\(104\) −7.15697e6 −0.623896
\(105\) 0 0
\(106\) −1.56070e7 −1.27277
\(107\) − 7.61385e6i − 0.600843i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971273\pi\)
\(108\) 6.47118e6i 0.494311i
\(109\) 1.99698e7 1.47700 0.738502 0.674251i \(-0.235533\pi\)
0.738502 + 0.674251i \(0.235533\pi\)
\(110\) 0 0
\(111\) 2.31469e7 1.60643
\(112\) − 6.85663e6i − 0.461156i
\(113\) 2.57944e7i 1.68171i 0.541261 + 0.840855i \(0.317947\pi\)
−0.541261 + 0.840855i \(0.682053\pi\)
\(114\) 2.80306e7 1.77201
\(115\) 0 0
\(116\) −1.90468e7 −1.13297
\(117\) 9.53209e6i 0.550222i
\(118\) 6.61185e6i 0.370456i
\(119\) 4.61695e6 0.251154
\(120\) 0 0
\(121\) 2.24208e7 1.15054
\(122\) 1.21699e7i 0.606773i
\(123\) − 1.05116e7i − 0.509330i
\(124\) 2.00018e6 0.0942094
\(125\) 0 0
\(126\) −4.09560e6 −0.182398
\(127\) 6.75687e6i 0.292707i 0.989232 + 0.146353i \(0.0467536\pi\)
−0.989232 + 0.146353i \(0.953246\pi\)
\(128\) 1.90018e7i 0.800868i
\(129\) 1.70292e7 0.698440
\(130\) 0 0
\(131\) −2.21063e6 −0.0859144 −0.0429572 0.999077i \(-0.513678\pi\)
−0.0429572 + 0.999077i \(0.513678\pi\)
\(132\) − 3.05555e7i − 1.15633i
\(133\) − 1.19897e7i − 0.441905i
\(134\) −3.31304e7 −1.18949
\(135\) 0 0
\(136\) −8.24677e6 −0.281124
\(137\) 7.10581e6i 0.236097i 0.993008 + 0.118049i \(0.0376639\pi\)
−0.993008 + 0.118049i \(0.962336\pi\)
\(138\) − 6.24126e7i − 2.02160i
\(139\) −9.21848e6 −0.291144 −0.145572 0.989348i \(-0.546502\pi\)
−0.145572 + 0.989348i \(0.546502\pi\)
\(140\) 0 0
\(141\) 1.31928e7 0.396342
\(142\) 3.24911e7i 0.952260i
\(143\) 7.56230e7i 2.16261i
\(144\) 1.63117e7 0.455229
\(145\) 0 0
\(146\) 7.30751e7 1.94328
\(147\) 6.44711e6i 0.167399i
\(148\) − 3.63814e7i − 0.922495i
\(149\) −1.33298e7 −0.330119 −0.165059 0.986284i \(-0.552782\pi\)
−0.165059 + 0.986284i \(0.552782\pi\)
\(150\) 0 0
\(151\) −6.41939e7 −1.51731 −0.758656 0.651492i \(-0.774143\pi\)
−0.758656 + 0.651492i \(0.774143\pi\)
\(152\) 2.14160e7i 0.494636i
\(153\) 1.09836e7i 0.247926i
\(154\) −3.24925e7 −0.716903
\(155\) 0 0
\(156\) 5.51375e7 1.16282
\(157\) − 7.36596e7i − 1.51908i −0.650461 0.759540i \(-0.725424\pi\)
0.650461 0.759540i \(-0.274576\pi\)
\(158\) − 3.99155e7i − 0.805085i
\(159\) −5.84460e7 −1.15309
\(160\) 0 0
\(161\) −2.66962e7 −0.504149
\(162\) 8.63610e7i 1.59593i
\(163\) − 3.50642e7i − 0.634172i −0.948397 0.317086i \(-0.897296\pi\)
0.948397 0.317086i \(-0.102704\pi\)
\(164\) −1.65217e7 −0.292483
\(165\) 0 0
\(166\) −9.34175e7 −1.58508
\(167\) − 2.56950e6i − 0.0426915i −0.999772 0.0213458i \(-0.993205\pi\)
0.999772 0.0213458i \(-0.00679508\pi\)
\(168\) − 1.15158e7i − 0.187375i
\(169\) −7.37136e7 −1.17475
\(170\) 0 0
\(171\) 2.85231e7 0.436225
\(172\) − 2.67659e7i − 0.401081i
\(173\) 8.03463e7i 1.17979i 0.807480 + 0.589895i \(0.200831\pi\)
−0.807480 + 0.589895i \(0.799169\pi\)
\(174\) −1.77327e8 −2.55183
\(175\) 0 0
\(176\) 1.29409e8 1.78925
\(177\) 2.47605e7i 0.335624i
\(178\) − 1.07259e8i − 1.42548i
\(179\) −9.99074e7 −1.30200 −0.651001 0.759077i \(-0.725651\pi\)
−0.651001 + 0.759077i \(0.725651\pi\)
\(180\) 0 0
\(181\) −1.07414e8 −1.34644 −0.673221 0.739442i \(-0.735090\pi\)
−0.673221 + 0.739442i \(0.735090\pi\)
\(182\) − 5.86328e7i − 0.720927i
\(183\) 4.55745e7i 0.549722i
\(184\) 4.76846e7 0.564307
\(185\) 0 0
\(186\) 1.86218e7 0.212191
\(187\) 8.71382e7i 0.974458i
\(188\) − 2.07360e7i − 0.227600i
\(189\) 2.57699e7 0.277649
\(190\) 0 0
\(191\) −2.21085e7 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(192\) − 3.14679e7i − 0.320859i
\(193\) − 1.49793e8i − 1.49983i −0.661537 0.749913i \(-0.730095\pi\)
0.661537 0.749913i \(-0.269905\pi\)
\(194\) 3.49261e7 0.343434
\(195\) 0 0
\(196\) 1.01333e7 0.0961295
\(197\) 5.70107e7i 0.531281i 0.964072 + 0.265641i \(0.0855835\pi\)
−0.964072 + 0.265641i \(0.914417\pi\)
\(198\) − 7.72985e7i − 0.707690i
\(199\) 6.84161e7 0.615421 0.307711 0.951480i \(-0.400437\pi\)
0.307711 + 0.951480i \(0.400437\pi\)
\(200\) 0 0
\(201\) −1.24069e8 −1.07765
\(202\) 2.81124e8i 2.39976i
\(203\) 7.58492e7i 0.636377i
\(204\) 6.35333e7 0.523958
\(205\) 0 0
\(206\) −6.51704e7 −0.519416
\(207\) − 6.35093e7i − 0.497669i
\(208\) 2.33519e8i 1.79929i
\(209\) 2.26289e8 1.71455
\(210\) 0 0
\(211\) −1.36201e8 −0.998141 −0.499071 0.866561i \(-0.666325\pi\)
−0.499071 + 0.866561i \(0.666325\pi\)
\(212\) 9.18635e7i 0.662167i
\(213\) 1.21675e8i 0.862724i
\(214\) −1.11415e8 −0.777135
\(215\) 0 0
\(216\) −4.60301e7 −0.310780
\(217\) − 7.96525e6i − 0.0529164i
\(218\) − 2.92224e8i − 1.91037i
\(219\) 2.73656e8 1.76056
\(220\) 0 0
\(221\) −1.57241e8 −0.979927
\(222\) − 3.38714e8i − 2.07777i
\(223\) − 1.47728e8i − 0.892066i −0.895017 0.446033i \(-0.852836\pi\)
0.895017 0.446033i \(-0.147164\pi\)
\(224\) −7.34363e7 −0.436559
\(225\) 0 0
\(226\) 3.77456e8 2.17514
\(227\) − 3.22427e8i − 1.82954i −0.403980 0.914768i \(-0.632373\pi\)
0.403980 0.914768i \(-0.367627\pi\)
\(228\) − 1.64989e8i − 0.921901i
\(229\) −3.10033e8 −1.70602 −0.853010 0.521895i \(-0.825225\pi\)
−0.853010 + 0.521895i \(0.825225\pi\)
\(230\) 0 0
\(231\) −1.21680e8 −0.649497
\(232\) − 1.35481e8i − 0.712315i
\(233\) 1.80410e8i 0.934361i 0.884162 + 0.467181i \(0.154730\pi\)
−0.884162 + 0.467181i \(0.845270\pi\)
\(234\) 1.39485e8 0.711661
\(235\) 0 0
\(236\) 3.89177e7 0.192733
\(237\) − 1.49478e8i − 0.729387i
\(238\) − 6.75609e7i − 0.324845i
\(239\) −3.66489e8 −1.73647 −0.868237 0.496150i \(-0.834747\pi\)
−0.868237 + 0.496150i \(0.834747\pi\)
\(240\) 0 0
\(241\) 2.19729e8 1.01118 0.505589 0.862775i \(-0.331275\pi\)
0.505589 + 0.862775i \(0.331275\pi\)
\(242\) − 3.28089e8i − 1.48812i
\(243\) 1.59099e8i 0.711286i
\(244\) 7.16324e7 0.315679
\(245\) 0 0
\(246\) −1.53818e8 −0.658771
\(247\) 4.08339e8i 1.72418i
\(248\) 1.42275e7i 0.0592308i
\(249\) −3.49836e8 −1.43604
\(250\) 0 0
\(251\) −1.29875e8 −0.518404 −0.259202 0.965823i \(-0.583460\pi\)
−0.259202 + 0.965823i \(0.583460\pi\)
\(252\) 2.41069e7i 0.0948940i
\(253\) − 5.03852e8i − 1.95606i
\(254\) 9.88750e7 0.378589
\(255\) 0 0
\(256\) 3.51561e8 1.30967
\(257\) 2.94635e8i 1.08273i 0.840789 + 0.541363i \(0.182091\pi\)
−0.840789 + 0.541363i \(0.817909\pi\)
\(258\) − 2.49192e8i − 0.903369i
\(259\) −1.44880e8 −0.518156
\(260\) 0 0
\(261\) −1.80443e8 −0.628199
\(262\) 3.23486e7i 0.111122i
\(263\) 3.13955e8i 1.06420i 0.846683 + 0.532098i \(0.178596\pi\)
−0.846683 + 0.532098i \(0.821404\pi\)
\(264\) 2.17344e8 0.726999
\(265\) 0 0
\(266\) −1.75449e8 −0.571563
\(267\) − 4.01668e8i − 1.29145i
\(268\) 1.95007e8i 0.618841i
\(269\) 2.94745e8 0.923238 0.461619 0.887078i \(-0.347269\pi\)
0.461619 + 0.887078i \(0.347269\pi\)
\(270\) 0 0
\(271\) −8.47290e7 −0.258607 −0.129303 0.991605i \(-0.541274\pi\)
−0.129303 + 0.991605i \(0.541274\pi\)
\(272\) 2.69077e8i 0.810748i
\(273\) − 2.19572e8i − 0.653142i
\(274\) 1.03981e8 0.305371
\(275\) 0 0
\(276\) −3.67364e8 −1.05175
\(277\) − 1.96909e8i − 0.556655i −0.960486 0.278327i \(-0.910220\pi\)
0.960486 0.278327i \(-0.0897800\pi\)
\(278\) 1.34896e8i 0.376568i
\(279\) 1.89491e7 0.0522364
\(280\) 0 0
\(281\) 3.32330e8 0.893505 0.446753 0.894658i \(-0.352580\pi\)
0.446753 + 0.894658i \(0.352580\pi\)
\(282\) − 1.93054e8i − 0.512632i
\(283\) 4.40437e8i 1.15513i 0.816344 + 0.577566i \(0.195997\pi\)
−0.816344 + 0.577566i \(0.804003\pi\)
\(284\) 1.91244e8 0.495421
\(285\) 0 0
\(286\) 1.10661e9 2.79714
\(287\) 6.57937e7i 0.164285i
\(288\) − 1.74702e8i − 0.430948i
\(289\) 2.29154e8 0.558451
\(290\) 0 0
\(291\) 1.30793e8 0.311143
\(292\) − 4.30123e8i − 1.01100i
\(293\) − 3.05058e8i − 0.708510i −0.935149 0.354255i \(-0.884735\pi\)
0.935149 0.354255i \(-0.115265\pi\)
\(294\) 9.43421e7 0.216516
\(295\) 0 0
\(296\) 2.58785e8 0.579986
\(297\) 4.86370e8i 1.07726i
\(298\) 1.95058e8i 0.426979i
\(299\) 9.09203e8 1.96703
\(300\) 0 0
\(301\) −1.06589e8 −0.225283
\(302\) 9.39366e8i 1.96250i
\(303\) 1.05277e9i 2.17412i
\(304\) 6.98766e8 1.42651
\(305\) 0 0
\(306\) 1.60725e8 0.320670
\(307\) 2.41616e8i 0.476587i 0.971193 + 0.238293i \(0.0765880\pi\)
−0.971193 + 0.238293i \(0.923412\pi\)
\(308\) 1.91252e8i 0.372975i
\(309\) −2.44054e8 −0.470578
\(310\) 0 0
\(311\) −6.71768e7 −0.126636 −0.0633181 0.997993i \(-0.520168\pi\)
−0.0633181 + 0.997993i \(0.520168\pi\)
\(312\) 3.92198e8i 0.731079i
\(313\) − 6.75084e8i − 1.24438i −0.782867 0.622190i \(-0.786243\pi\)
0.782867 0.622190i \(-0.213757\pi\)
\(314\) −1.07788e9 −1.96479
\(315\) 0 0
\(316\) −2.34944e8 −0.418852
\(317\) 4.65149e8i 0.820135i 0.912055 + 0.410067i \(0.134495\pi\)
−0.912055 + 0.410067i \(0.865505\pi\)
\(318\) 8.55255e8i 1.49142i
\(319\) −1.43154e9 −2.46909
\(320\) 0 0
\(321\) −4.17235e8 −0.704066
\(322\) 3.90652e8i 0.652070i
\(323\) 4.70517e8i 0.776903i
\(324\) 5.08325e8 0.830298
\(325\) 0 0
\(326\) −5.13103e8 −0.820244
\(327\) − 1.09434e9i − 1.73075i
\(328\) − 1.17520e8i − 0.183889i
\(329\) −8.25762e7 −0.127841
\(330\) 0 0
\(331\) 2.69779e8 0.408893 0.204447 0.978878i \(-0.434460\pi\)
0.204447 + 0.978878i \(0.434460\pi\)
\(332\) 5.49860e8i 0.824648i
\(333\) − 3.44665e8i − 0.511497i
\(334\) −3.76002e7 −0.0552176
\(335\) 0 0
\(336\) −3.75740e8 −0.540381
\(337\) 6.29093e8i 0.895385i 0.894187 + 0.447693i \(0.147754\pi\)
−0.894187 + 0.447693i \(0.852246\pi\)
\(338\) 1.07867e9i 1.51943i
\(339\) 1.41352e9 1.97062
\(340\) 0 0
\(341\) 1.50333e8 0.205312
\(342\) − 4.17386e8i − 0.564218i
\(343\) − 4.03536e7i − 0.0539949i
\(344\) 1.90388e8 0.252165
\(345\) 0 0
\(346\) 1.17573e9 1.52595
\(347\) 7.61715e8i 0.978676i 0.872094 + 0.489338i \(0.162762\pi\)
−0.872094 + 0.489338i \(0.837238\pi\)
\(348\) 1.04375e9i 1.32761i
\(349\) 3.31639e8 0.417616 0.208808 0.977957i \(-0.433042\pi\)
0.208808 + 0.977957i \(0.433042\pi\)
\(350\) 0 0
\(351\) −8.77657e8 −1.08330
\(352\) − 1.38601e9i − 1.69381i
\(353\) 7.57419e8i 0.916484i 0.888828 + 0.458242i \(0.151521\pi\)
−0.888828 + 0.458242i \(0.848479\pi\)
\(354\) 3.62326e8 0.434099
\(355\) 0 0
\(356\) −6.31329e8 −0.741619
\(357\) − 2.53006e8i − 0.294302i
\(358\) 1.46197e9i 1.68402i
\(359\) −1.46796e9 −1.67449 −0.837246 0.546827i \(-0.815836\pi\)
−0.837246 + 0.546827i \(0.815836\pi\)
\(360\) 0 0
\(361\) 3.28013e8 0.366958
\(362\) 1.57182e9i 1.74150i
\(363\) − 1.22865e9i − 1.34820i
\(364\) −3.45116e8 −0.375068
\(365\) 0 0
\(366\) 6.66903e8 0.711015
\(367\) − 1.64615e9i − 1.73835i −0.494502 0.869177i \(-0.664649\pi\)
0.494502 0.869177i \(-0.335351\pi\)
\(368\) − 1.55586e9i − 1.62744i
\(369\) −1.56521e8 −0.162174
\(370\) 0 0
\(371\) 3.65824e8 0.371932
\(372\) − 1.09609e8i − 0.110394i
\(373\) − 1.16387e9i − 1.16124i −0.814173 0.580622i \(-0.802809\pi\)
0.814173 0.580622i \(-0.197191\pi\)
\(374\) 1.27512e9 1.26037
\(375\) 0 0
\(376\) 1.47497e8 0.143096
\(377\) − 2.58323e9i − 2.48295i
\(378\) − 3.77098e8i − 0.359114i
\(379\) −4.07762e8 −0.384742 −0.192371 0.981322i \(-0.561618\pi\)
−0.192371 + 0.981322i \(0.561618\pi\)
\(380\) 0 0
\(381\) 3.70273e8 0.342993
\(382\) 3.23519e8i 0.296946i
\(383\) 7.10345e8i 0.646061i 0.946389 + 0.323031i \(0.104702\pi\)
−0.946389 + 0.323031i \(0.895298\pi\)
\(384\) 1.04129e9 0.938454
\(385\) 0 0
\(386\) −2.19196e9 −1.93989
\(387\) − 2.53571e8i − 0.222388i
\(388\) − 2.05576e8i − 0.178674i
\(389\) −1.95091e8 −0.168040 −0.0840202 0.996464i \(-0.526776\pi\)
−0.0840202 + 0.996464i \(0.526776\pi\)
\(390\) 0 0
\(391\) 1.04765e9 0.886333
\(392\) 7.20794e7i 0.0604380i
\(393\) 1.21141e8i 0.100674i
\(394\) 8.34252e8 0.687164
\(395\) 0 0
\(396\) −4.54983e8 −0.368181
\(397\) − 1.58231e9i − 1.26919i −0.772846 0.634593i \(-0.781168\pi\)
0.772846 0.634593i \(-0.218832\pi\)
\(398\) − 1.00115e9i − 0.795991i
\(399\) −6.57031e8 −0.517822
\(400\) 0 0
\(401\) −2.13647e9 −1.65460 −0.827298 0.561763i \(-0.810123\pi\)
−0.827298 + 0.561763i \(0.810123\pi\)
\(402\) 1.81553e9i 1.39384i
\(403\) 2.71276e8i 0.206464i
\(404\) 1.65471e9 1.24849
\(405\) 0 0
\(406\) 1.10992e9 0.823096
\(407\) − 2.73441e9i − 2.01040i
\(408\) 4.51919e8i 0.329420i
\(409\) 4.05635e8 0.293159 0.146580 0.989199i \(-0.453174\pi\)
0.146580 + 0.989199i \(0.453174\pi\)
\(410\) 0 0
\(411\) 3.89395e8 0.276658
\(412\) 3.83596e8i 0.270230i
\(413\) − 1.54980e8i − 0.108256i
\(414\) −9.29347e8 −0.643690
\(415\) 0 0
\(416\) 2.50105e9 1.70332
\(417\) 5.05168e8i 0.341161i
\(418\) − 3.31134e9i − 2.21762i
\(419\) −1.07475e9 −0.713771 −0.356886 0.934148i \(-0.616161\pi\)
−0.356886 + 0.934148i \(0.616161\pi\)
\(420\) 0 0
\(421\) −8.32900e8 −0.544009 −0.272004 0.962296i \(-0.587686\pi\)
−0.272004 + 0.962296i \(0.587686\pi\)
\(422\) 1.99306e9i 1.29100i
\(423\) − 1.96446e8i − 0.126198i
\(424\) −6.53434e8 −0.416314
\(425\) 0 0
\(426\) 1.78050e9 1.11586
\(427\) − 2.85259e8i − 0.177313i
\(428\) 6.55796e8i 0.404311i
\(429\) 4.14411e9 2.53414
\(430\) 0 0
\(431\) 8.26292e6 0.00497122 0.00248561 0.999997i \(-0.499209\pi\)
0.00248561 + 0.999997i \(0.499209\pi\)
\(432\) 1.50188e9i 0.896277i
\(433\) − 3.10619e9i − 1.83874i −0.393396 0.919369i \(-0.628700\pi\)
0.393396 0.919369i \(-0.371300\pi\)
\(434\) −1.16558e8 −0.0684426
\(435\) 0 0
\(436\) −1.72004e9 −0.993885
\(437\) − 2.72063e9i − 1.55950i
\(438\) − 4.00448e9i − 2.27712i
\(439\) 1.28295e9 0.723742 0.361871 0.932228i \(-0.382138\pi\)
0.361871 + 0.932228i \(0.382138\pi\)
\(440\) 0 0
\(441\) 9.59998e7 0.0533010
\(442\) 2.30095e9i 1.26745i
\(443\) − 1.73263e9i − 0.946878i −0.880827 0.473439i \(-0.843012\pi\)
0.880827 0.473439i \(-0.156988\pi\)
\(444\) −1.99368e9 −1.08098
\(445\) 0 0
\(446\) −2.16175e9 −1.15381
\(447\) 7.30464e8i 0.386832i
\(448\) 1.96964e8i 0.103493i
\(449\) −1.86903e9 −0.974439 −0.487219 0.873280i \(-0.661989\pi\)
−0.487219 + 0.873280i \(0.661989\pi\)
\(450\) 0 0
\(451\) −1.24176e9 −0.637413
\(452\) − 2.22172e9i − 1.13163i
\(453\) 3.51780e9i 1.77798i
\(454\) −4.71816e9 −2.36634
\(455\) 0 0
\(456\) 1.17359e9 0.579613
\(457\) − 1.76868e9i − 0.866849i −0.901190 0.433425i \(-0.857305\pi\)
0.901190 0.433425i \(-0.142695\pi\)
\(458\) 4.53679e9i 2.20658i
\(459\) −1.01130e9 −0.488129
\(460\) 0 0
\(461\) −2.55825e8 −0.121616 −0.0608078 0.998149i \(-0.519368\pi\)
−0.0608078 + 0.998149i \(0.519368\pi\)
\(462\) 1.78057e9i 0.840065i
\(463\) − 4.19121e9i − 1.96249i −0.192777 0.981243i \(-0.561750\pi\)
0.192777 0.981243i \(-0.438250\pi\)
\(464\) −4.42052e9 −2.05428
\(465\) 0 0
\(466\) 2.63998e9 1.20851
\(467\) 2.94239e9i 1.33688i 0.743768 + 0.668438i \(0.233037\pi\)
−0.743768 + 0.668438i \(0.766963\pi\)
\(468\) − 8.21018e8i − 0.370247i
\(469\) 7.76569e8 0.347596
\(470\) 0 0
\(471\) −4.03651e9 −1.78005
\(472\) 2.76825e8i 0.121174i
\(473\) − 2.01171e9i − 0.874079i
\(474\) −2.18735e9 −0.943396
\(475\) 0 0
\(476\) −3.97667e8 −0.169003
\(477\) 8.70283e8i 0.367152i
\(478\) 5.36292e9i 2.24597i
\(479\) −4.57933e9 −1.90383 −0.951914 0.306366i \(-0.900887\pi\)
−0.951914 + 0.306366i \(0.900887\pi\)
\(480\) 0 0
\(481\) 4.93425e9 2.02169
\(482\) − 3.21535e9i − 1.30787i
\(483\) 1.46294e9i 0.590760i
\(484\) −1.93115e9 −0.774206
\(485\) 0 0
\(486\) 2.32813e9 0.919984
\(487\) 1.40131e9i 0.549771i 0.961477 + 0.274886i \(0.0886400\pi\)
−0.961477 + 0.274886i \(0.911360\pi\)
\(488\) 5.09528e8i 0.198472i
\(489\) −1.92150e9 −0.743121
\(490\) 0 0
\(491\) −4.79712e9 −1.82892 −0.914462 0.404672i \(-0.867386\pi\)
−0.914462 + 0.404672i \(0.867386\pi\)
\(492\) 9.05381e8i 0.342731i
\(493\) − 2.97658e9i − 1.11880i
\(494\) 5.97533e9 2.23007
\(495\) 0 0
\(496\) 4.64218e8 0.170819
\(497\) − 7.61585e8i − 0.278273i
\(498\) 5.11923e9i 1.85739i
\(499\) 2.01049e9 0.724351 0.362176 0.932110i \(-0.382034\pi\)
0.362176 + 0.932110i \(0.382034\pi\)
\(500\) 0 0
\(501\) −1.40807e8 −0.0500258
\(502\) 1.90050e9i 0.670509i
\(503\) 1.68618e9i 0.590766i 0.955379 + 0.295383i \(0.0954472\pi\)
−0.955379 + 0.295383i \(0.904553\pi\)
\(504\) −1.71474e8 −0.0596613
\(505\) 0 0
\(506\) −7.37300e9 −2.52998
\(507\) 4.03947e9i 1.37656i
\(508\) − 5.81983e8i − 0.196964i
\(509\) −1.63477e9 −0.549470 −0.274735 0.961520i \(-0.588590\pi\)
−0.274735 + 0.961520i \(0.588590\pi\)
\(510\) 0 0
\(511\) −1.71286e9 −0.567871
\(512\) − 2.71225e9i − 0.893068i
\(513\) 2.62624e9i 0.858862i
\(514\) 4.31147e9 1.40041
\(515\) 0 0
\(516\) −1.46676e9 −0.469985
\(517\) − 1.55851e9i − 0.496012i
\(518\) 2.12007e9i 0.670187i
\(519\) 4.40294e9 1.38247
\(520\) 0 0
\(521\) 3.28595e9 1.01796 0.508978 0.860779i \(-0.330023\pi\)
0.508978 + 0.860779i \(0.330023\pi\)
\(522\) 2.64046e9i 0.812518i
\(523\) 1.29734e9i 0.396549i 0.980147 + 0.198274i \(0.0635337\pi\)
−0.980147 + 0.198274i \(0.936466\pi\)
\(524\) 1.90406e8 0.0578123
\(525\) 0 0
\(526\) 4.59418e9 1.37644
\(527\) 3.12583e8i 0.0930313i
\(528\) − 7.09155e9i − 2.09663i
\(529\) −2.65291e9 −0.779161
\(530\) 0 0
\(531\) 3.68693e8 0.106865
\(532\) 1.03270e9i 0.297360i
\(533\) − 2.24076e9i − 0.640990i
\(534\) −5.87771e9 −1.67038
\(535\) 0 0
\(536\) −1.38710e9 −0.389074
\(537\) 5.47487e9i 1.52568i
\(538\) − 4.31308e9i − 1.19412i
\(539\) 7.61617e8 0.209496
\(540\) 0 0
\(541\) 1.23726e9 0.335948 0.167974 0.985791i \(-0.446278\pi\)
0.167974 + 0.985791i \(0.446278\pi\)
\(542\) 1.23986e9i 0.334484i
\(543\) 5.88625e9i 1.57775i
\(544\) 2.88189e9 0.767505
\(545\) 0 0
\(546\) −3.21305e9 −0.844779
\(547\) 4.27288e9i 1.11626i 0.829754 + 0.558130i \(0.188481\pi\)
−0.829754 + 0.558130i \(0.811519\pi\)
\(548\) − 6.12037e8i − 0.158871i
\(549\) 6.78621e8 0.175035
\(550\) 0 0
\(551\) −7.72986e9 −1.96853
\(552\) − 2.61309e9i − 0.661253i
\(553\) 9.35610e8i 0.235265i
\(554\) −2.88141e9 −0.719982
\(555\) 0 0
\(556\) 7.94006e8 0.195912
\(557\) 2.41921e9i 0.593171i 0.955006 + 0.296586i \(0.0958480\pi\)
−0.955006 + 0.296586i \(0.904152\pi\)
\(558\) − 2.77286e8i − 0.0675630i
\(559\) 3.63013e9 0.878985
\(560\) 0 0
\(561\) 4.77513e9 1.14187
\(562\) − 4.86306e9i − 1.15567i
\(563\) − 3.03839e9i − 0.717570i −0.933420 0.358785i \(-0.883191\pi\)
0.933420 0.358785i \(-0.116809\pi\)
\(564\) −1.13632e9 −0.266701
\(565\) 0 0
\(566\) 6.44503e9 1.49406
\(567\) − 2.02428e9i − 0.466370i
\(568\) 1.36034e9i 0.311478i
\(569\) −2.85539e9 −0.649788 −0.324894 0.945750i \(-0.605329\pi\)
−0.324894 + 0.945750i \(0.605329\pi\)
\(570\) 0 0
\(571\) 1.87867e9 0.422304 0.211152 0.977453i \(-0.432279\pi\)
0.211152 + 0.977453i \(0.432279\pi\)
\(572\) − 6.51356e9i − 1.45523i
\(573\) 1.21153e9i 0.269026i
\(574\) 9.62776e8 0.212488
\(575\) 0 0
\(576\) −4.68569e8 −0.102163
\(577\) − 2.19182e9i − 0.474996i −0.971388 0.237498i \(-0.923673\pi\)
0.971388 0.237498i \(-0.0763273\pi\)
\(578\) − 3.35327e9i − 0.722306i
\(579\) −8.20858e9 −1.75749
\(580\) 0 0
\(581\) 2.18969e9 0.463196
\(582\) − 1.91393e9i − 0.402435i
\(583\) 6.90441e9i 1.44307i
\(584\) 3.05951e9 0.635633
\(585\) 0 0
\(586\) −4.46399e9 −0.916392
\(587\) 4.70415e9i 0.959949i 0.877283 + 0.479974i \(0.159354\pi\)
−0.877283 + 0.479974i \(0.840646\pi\)
\(588\) − 5.55302e8i − 0.112644i
\(589\) 8.11747e8 0.163688
\(590\) 0 0
\(591\) 3.12416e9 0.622554
\(592\) − 8.44368e9i − 1.67265i
\(593\) 3.66996e9i 0.722719i 0.932427 + 0.361359i \(0.117687\pi\)
−0.932427 + 0.361359i \(0.882313\pi\)
\(594\) 7.11718e9 1.39333
\(595\) 0 0
\(596\) 1.14812e9 0.222139
\(597\) − 3.74917e9i − 0.721149i
\(598\) − 1.33046e10i − 2.54418i
\(599\) 5.46935e9 1.03978 0.519890 0.854233i \(-0.325973\pi\)
0.519890 + 0.854233i \(0.325973\pi\)
\(600\) 0 0
\(601\) 2.74417e9 0.515645 0.257822 0.966192i \(-0.416995\pi\)
0.257822 + 0.966192i \(0.416995\pi\)
\(602\) 1.55974e9i 0.291383i
\(603\) 1.84743e9i 0.343129i
\(604\) 5.52915e9 1.02101
\(605\) 0 0
\(606\) 1.54054e10 2.81203
\(607\) 2.26388e9i 0.410859i 0.978672 + 0.205430i \(0.0658591\pi\)
−0.978672 + 0.205430i \(0.934141\pi\)
\(608\) − 7.48397e9i − 1.35042i
\(609\) 4.15650e9 0.745705
\(610\) 0 0
\(611\) 2.81233e9 0.498795
\(612\) − 9.46035e8i − 0.166831i
\(613\) − 9.92731e9i − 1.74068i −0.492448 0.870342i \(-0.663898\pi\)
0.492448 0.870342i \(-0.336102\pi\)
\(614\) 3.53563e9 0.616422
\(615\) 0 0
\(616\) −1.36040e9 −0.234495
\(617\) − 2.27156e9i − 0.389338i −0.980869 0.194669i \(-0.937637\pi\)
0.980869 0.194669i \(-0.0623632\pi\)
\(618\) 3.57131e9i 0.608650i
\(619\) −6.90552e8 −0.117025 −0.0585126 0.998287i \(-0.518636\pi\)
−0.0585126 + 0.998287i \(0.518636\pi\)
\(620\) 0 0
\(621\) 5.84755e9 0.979836
\(622\) 9.83015e8i 0.163792i
\(623\) 2.51412e9i 0.416560i
\(624\) 1.27967e10 2.10840
\(625\) 0 0
\(626\) −9.87867e9 −1.60949
\(627\) − 1.24005e10i − 2.00911i
\(628\) 6.34445e9i 1.02220i
\(629\) 5.68560e9 0.910959
\(630\) 0 0
\(631\) −1.00992e10 −1.60023 −0.800116 0.599846i \(-0.795229\pi\)
−0.800116 + 0.599846i \(0.795229\pi\)
\(632\) − 1.67118e9i − 0.263338i
\(633\) 7.46375e9i 1.16962i
\(634\) 6.80665e9 1.06077
\(635\) 0 0
\(636\) 5.03407e9 0.775925
\(637\) 1.37434e9i 0.210672i
\(638\) 2.09482e10i 3.19355i
\(639\) 1.81178e9 0.274697
\(640\) 0 0
\(641\) 8.50418e8 0.127535 0.0637675 0.997965i \(-0.479688\pi\)
0.0637675 + 0.997965i \(0.479688\pi\)
\(642\) 6.10550e9i 0.910644i
\(643\) 1.56624e9i 0.232339i 0.993229 + 0.116169i \(0.0370615\pi\)
−0.993229 + 0.116169i \(0.962938\pi\)
\(644\) 2.29940e9 0.339245
\(645\) 0 0
\(646\) 6.88520e9 1.00485
\(647\) − 1.06053e10i − 1.53942i −0.638391 0.769712i \(-0.720400\pi\)
0.638391 0.769712i \(-0.279600\pi\)
\(648\) 3.61576e9i 0.522020i
\(649\) 2.92503e9 0.420024
\(650\) 0 0
\(651\) −4.36492e8 −0.0620073
\(652\) 3.02015e9i 0.426739i
\(653\) − 1.83644e9i − 0.258096i −0.991638 0.129048i \(-0.958808\pi\)
0.991638 0.129048i \(-0.0411921\pi\)
\(654\) −1.60137e10 −2.23856
\(655\) 0 0
\(656\) −3.83448e9 −0.530327
\(657\) − 4.07484e9i − 0.560573i
\(658\) 1.20836e9i 0.165350i
\(659\) 5.03583e9 0.685444 0.342722 0.939437i \(-0.388651\pi\)
0.342722 + 0.939437i \(0.388651\pi\)
\(660\) 0 0
\(661\) −1.27453e10 −1.71651 −0.858254 0.513225i \(-0.828451\pi\)
−0.858254 + 0.513225i \(0.828451\pi\)
\(662\) − 3.94774e9i − 0.528866i
\(663\) 8.61674e9i 1.14827i
\(664\) −3.91121e9 −0.518469
\(665\) 0 0
\(666\) −5.04357e9 −0.661574
\(667\) 1.72112e10i 2.24580i
\(668\) 2.21316e8i 0.0287274i
\(669\) −8.09544e9 −1.04532
\(670\) 0 0
\(671\) 5.38386e9 0.687962
\(672\) 4.02427e9i 0.511558i
\(673\) 1.02193e10i 1.29231i 0.763206 + 0.646155i \(0.223624\pi\)
−0.763206 + 0.646155i \(0.776376\pi\)
\(674\) 9.20567e9 1.15810
\(675\) 0 0
\(676\) 6.34910e9 0.790494
\(677\) − 1.04119e10i − 1.28965i −0.764332 0.644823i \(-0.776931\pi\)
0.764332 0.644823i \(-0.223069\pi\)
\(678\) − 2.06844e10i − 2.54882i
\(679\) −8.18659e8 −0.100360
\(680\) 0 0
\(681\) −1.76688e10 −2.14384
\(682\) − 2.19986e9i − 0.265552i
\(683\) 6.56705e9i 0.788675i 0.918966 + 0.394338i \(0.129026\pi\)
−0.918966 + 0.394338i \(0.870974\pi\)
\(684\) −2.45675e9 −0.293539
\(685\) 0 0
\(686\) −5.90504e8 −0.0698375
\(687\) 1.69897e10i 1.99911i
\(688\) − 6.21203e9i − 0.727233i
\(689\) −1.24590e10 −1.45117
\(690\) 0 0
\(691\) 4.44242e9 0.512208 0.256104 0.966649i \(-0.417561\pi\)
0.256104 + 0.966649i \(0.417561\pi\)
\(692\) − 6.92039e9i − 0.793888i
\(693\) 1.81186e9i 0.206804i
\(694\) 1.11464e10 1.26583
\(695\) 0 0
\(696\) −7.42431e9 −0.834688
\(697\) − 2.58197e9i − 0.288826i
\(698\) − 4.85296e9i − 0.540148i
\(699\) 9.88638e9 1.09488
\(700\) 0 0
\(701\) −7.92343e9 −0.868761 −0.434380 0.900729i \(-0.643033\pi\)
−0.434380 + 0.900729i \(0.643033\pi\)
\(702\) 1.28430e10i 1.40115i
\(703\) − 1.47649e10i − 1.60283i
\(704\) −3.71741e9 −0.401546
\(705\) 0 0
\(706\) 1.10835e10 1.18539
\(707\) − 6.58948e9i − 0.701266i
\(708\) − 2.13267e9i − 0.225843i
\(709\) −9.27216e9 −0.977055 −0.488528 0.872548i \(-0.662466\pi\)
−0.488528 + 0.872548i \(0.662466\pi\)
\(710\) 0 0
\(711\) −2.22578e9 −0.232241
\(712\) − 4.49070e9i − 0.466266i
\(713\) − 1.80743e9i − 0.186744i
\(714\) −3.70230e9 −0.380652
\(715\) 0 0
\(716\) 8.60522e9 0.876126
\(717\) 2.00834e10i 2.03479i
\(718\) 2.14810e10i 2.16580i
\(719\) 5.48244e9 0.550076 0.275038 0.961433i \(-0.411310\pi\)
0.275038 + 0.961433i \(0.411310\pi\)
\(720\) 0 0
\(721\) 1.52758e9 0.151786
\(722\) − 4.79990e9i − 0.474626i
\(723\) − 1.20410e10i − 1.18489i
\(724\) 9.25181e9 0.906029
\(725\) 0 0
\(726\) −1.79791e10 −1.74377
\(727\) 9.22691e9i 0.890606i 0.895380 + 0.445303i \(0.146904\pi\)
−0.895380 + 0.445303i \(0.853096\pi\)
\(728\) − 2.45484e9i − 0.235811i
\(729\) −4.18848e9 −0.400415
\(730\) 0 0
\(731\) 4.18290e9 0.396065
\(732\) − 3.92542e9i − 0.369911i
\(733\) 4.96865e9i 0.465988i 0.972478 + 0.232994i \(0.0748523\pi\)
−0.972478 + 0.232994i \(0.925148\pi\)
\(734\) −2.40885e10 −2.24840
\(735\) 0 0
\(736\) −1.66637e10 −1.54063
\(737\) 1.46566e10i 1.34865i
\(738\) 2.29041e9i 0.209757i
\(739\) 1.96084e9 0.178726 0.0893628 0.995999i \(-0.471517\pi\)
0.0893628 + 0.995999i \(0.471517\pi\)
\(740\) 0 0
\(741\) 2.23768e10 2.02038
\(742\) − 5.35320e9i − 0.481060i
\(743\) 9.56947e9i 0.855908i 0.903801 + 0.427954i \(0.140765\pi\)
−0.903801 + 0.427954i \(0.859235\pi\)
\(744\) 7.79660e8 0.0694064
\(745\) 0 0
\(746\) −1.70312e10 −1.50196
\(747\) 5.20918e9i 0.457244i
\(748\) − 7.50539e9i − 0.655719i
\(749\) 2.61155e9 0.227097
\(750\) 0 0
\(751\) 8.11719e9 0.699304 0.349652 0.936880i \(-0.386300\pi\)
0.349652 + 0.936880i \(0.386300\pi\)
\(752\) − 4.81257e9i − 0.412681i
\(753\) 7.11710e9i 0.607464i
\(754\) −3.78010e10 −3.21147
\(755\) 0 0
\(756\) −2.21961e9 −0.186832
\(757\) 9.11117e9i 0.763376i 0.924291 + 0.381688i \(0.124657\pi\)
−0.924291 + 0.381688i \(0.875343\pi\)
\(758\) 5.96689e9i 0.497629i
\(759\) −2.76109e10 −2.29210
\(760\) 0 0
\(761\) 1.71359e10 1.40948 0.704742 0.709464i \(-0.251063\pi\)
0.704742 + 0.709464i \(0.251063\pi\)
\(762\) − 5.41830e9i − 0.443630i
\(763\) 6.84965e9i 0.558255i
\(764\) 1.90425e9 0.154489
\(765\) 0 0
\(766\) 1.03947e10 0.835621
\(767\) 5.27823e9i 0.422381i
\(768\) − 1.92654e10i − 1.53466i
\(769\) 1.82316e10 1.44572 0.722858 0.690997i \(-0.242828\pi\)
0.722858 + 0.690997i \(0.242828\pi\)
\(770\) 0 0
\(771\) 1.61458e10 1.26873
\(772\) 1.29020e10i 1.00924i
\(773\) − 1.45965e10i − 1.13663i −0.822810 0.568316i \(-0.807595\pi\)
0.822810 0.568316i \(-0.192405\pi\)
\(774\) −3.71056e9 −0.287638
\(775\) 0 0
\(776\) 1.46228e9 0.112335
\(777\) 7.93937e9i 0.607173i
\(778\) 2.85481e9i 0.217345i
\(779\) −6.70510e9 −0.508188
\(780\) 0 0
\(781\) 1.43738e10 1.07968
\(782\) − 1.53305e10i − 1.14639i
\(783\) − 1.66141e10i − 1.23683i
\(784\) 2.35182e9 0.174300
\(785\) 0 0
\(786\) 1.77269e9 0.130213
\(787\) − 1.42358e10i − 1.04104i −0.853848 0.520522i \(-0.825737\pi\)
0.853848 0.520522i \(-0.174263\pi\)
\(788\) − 4.91045e9i − 0.357503i
\(789\) 1.72046e10 1.24702
\(790\) 0 0
\(791\) −8.84748e9 −0.635627
\(792\) − 3.23633e9i − 0.231481i
\(793\) 9.71519e9i 0.691823i
\(794\) −2.31544e10 −1.64158
\(795\) 0 0
\(796\) −5.89281e9 −0.414121
\(797\) − 1.03325e10i − 0.722935i −0.932385 0.361467i \(-0.882276\pi\)
0.932385 0.361467i \(-0.117724\pi\)
\(798\) 9.61450e9i 0.669756i
\(799\) 3.24057e9 0.224754
\(800\) 0 0
\(801\) −5.98099e9 −0.411206
\(802\) 3.12635e10i 2.14007i
\(803\) − 3.23278e10i − 2.20329i
\(804\) 1.06863e10 0.725155
\(805\) 0 0
\(806\) 3.96965e9 0.267042
\(807\) − 1.61519e10i − 1.08185i
\(808\) 1.17701e10i 0.784947i
\(809\) 1.58064e10 1.04957 0.524787 0.851233i \(-0.324145\pi\)
0.524787 + 0.851233i \(0.324145\pi\)
\(810\) 0 0
\(811\) 2.87547e9 0.189294 0.0946469 0.995511i \(-0.469828\pi\)
0.0946469 + 0.995511i \(0.469828\pi\)
\(812\) − 6.53304e9i − 0.428222i
\(813\) 4.64311e9i 0.303034i
\(814\) −4.00133e10 −2.60027
\(815\) 0 0
\(816\) 1.47453e10 0.950032
\(817\) − 1.08626e10i − 0.696875i
\(818\) − 5.93575e9i − 0.379175i
\(819\) −3.26951e9 −0.207964
\(820\) 0 0
\(821\) −1.42014e10 −0.895633 −0.447817 0.894125i \(-0.647798\pi\)
−0.447817 + 0.894125i \(0.647798\pi\)
\(822\) − 5.69811e9i − 0.357832i
\(823\) − 2.79354e10i − 1.74685i −0.486961 0.873424i \(-0.661895\pi\)
0.486961 0.873424i \(-0.338105\pi\)
\(824\) −2.72856e9 −0.169898
\(825\) 0 0
\(826\) −2.26787e9 −0.140019
\(827\) 1.48560e8i 0.00913341i 0.999990 + 0.00456670i \(0.00145363\pi\)
−0.999990 + 0.00456670i \(0.998546\pi\)
\(828\) 5.47018e9i 0.334885i
\(829\) −3.98861e9 −0.243154 −0.121577 0.992582i \(-0.538795\pi\)
−0.121577 + 0.992582i \(0.538795\pi\)
\(830\) 0 0
\(831\) −1.07905e10 −0.652286
\(832\) − 6.70807e9i − 0.403800i
\(833\) 1.58361e9i 0.0949273i
\(834\) 7.39225e9 0.441261
\(835\) 0 0
\(836\) −1.94907e10 −1.15373
\(837\) 1.74471e9i 0.102845i
\(838\) 1.57271e10i 0.923198i
\(839\) −2.43693e9 −0.142455 −0.0712273 0.997460i \(-0.522692\pi\)
−0.0712273 + 0.997460i \(0.522692\pi\)
\(840\) 0 0
\(841\) 3.16506e10 1.83483
\(842\) 1.21880e10i 0.703625i
\(843\) − 1.82115e10i − 1.04701i
\(844\) 1.17313e10 0.671655
\(845\) 0 0
\(846\) −2.87464e9 −0.163225
\(847\) 7.69033e9i 0.434863i
\(848\) 2.13204e10i 1.20063i
\(849\) 2.41357e10 1.35358
\(850\) 0 0
\(851\) −3.28754e10 −1.82859
\(852\) − 1.04801e10i − 0.580533i
\(853\) 1.54310e9i 0.0851282i 0.999094 + 0.0425641i \(0.0135527\pi\)
−0.999094 + 0.0425641i \(0.986447\pi\)
\(854\) −4.17427e9 −0.229339
\(855\) 0 0
\(856\) −4.66474e9 −0.254196
\(857\) 1.29972e10i 0.705369i 0.935742 + 0.352684i \(0.114731\pi\)
−0.935742 + 0.352684i \(0.885269\pi\)
\(858\) − 6.06417e10i − 3.27768i
\(859\) 1.98316e10 1.06754 0.533768 0.845631i \(-0.320776\pi\)
0.533768 + 0.845631i \(0.320776\pi\)
\(860\) 0 0
\(861\) 3.60546e9 0.192509
\(862\) − 1.20913e8i − 0.00642982i
\(863\) 4.94264e9i 0.261771i 0.991397 + 0.130886i \(0.0417820\pi\)
−0.991397 + 0.130886i \(0.958218\pi\)
\(864\) 1.60855e10 0.848472
\(865\) 0 0
\(866\) −4.54536e10 −2.37824
\(867\) − 1.25575e10i − 0.654391i
\(868\) 6.86063e8i 0.0356078i
\(869\) −1.76583e10 −0.912809
\(870\) 0 0
\(871\) −2.64480e10 −1.35621
\(872\) − 1.22348e10i − 0.624870i
\(873\) − 1.94756e9i − 0.0990697i
\(874\) −3.98117e10 −2.01707
\(875\) 0 0
\(876\) −2.35705e10 −1.18469
\(877\) 7.37011e9i 0.368957i 0.982837 + 0.184478i \(0.0590596\pi\)
−0.982837 + 0.184478i \(0.940940\pi\)
\(878\) − 1.87737e10i − 0.936094i
\(879\) −1.67170e10 −0.830229
\(880\) 0 0
\(881\) 9.74385e9 0.480081 0.240041 0.970763i \(-0.422839\pi\)
0.240041 + 0.970763i \(0.422839\pi\)
\(882\) − 1.40479e9i − 0.0689400i
\(883\) − 2.79540e10i − 1.36641i −0.730226 0.683206i \(-0.760585\pi\)
0.730226 0.683206i \(-0.239415\pi\)
\(884\) 1.35435e10 0.659399
\(885\) 0 0
\(886\) −2.53541e10 −1.22470
\(887\) 2.08176e10i 1.00161i 0.865561 + 0.500804i \(0.166962\pi\)
−0.865561 + 0.500804i \(0.833038\pi\)
\(888\) − 1.41813e10i − 0.679626i
\(889\) −2.31761e9 −0.110633
\(890\) 0 0
\(891\) 3.82054e10 1.80948
\(892\) 1.27241e10i 0.600276i
\(893\) − 8.41542e9i − 0.395454i
\(894\) 1.06891e10 0.500332
\(895\) 0 0
\(896\) −6.51763e9 −0.302700
\(897\) − 4.98239e10i − 2.30496i
\(898\) 2.73500e10i 1.26035i
\(899\) −5.13526e9 −0.235724
\(900\) 0 0
\(901\) −1.43562e10 −0.653886
\(902\) 1.81710e10i 0.824435i
\(903\) 5.84100e9i 0.263986i
\(904\) 1.58033e10 0.711474
\(905\) 0 0
\(906\) 5.14768e10 2.29965
\(907\) − 3.61602e10i − 1.60918i −0.593830 0.804591i \(-0.702385\pi\)
0.593830 0.804591i \(-0.297615\pi\)
\(908\) 2.77713e10i 1.23111i
\(909\) 1.56761e10 0.692254
\(910\) 0 0
\(911\) 1.79811e10 0.787954 0.393977 0.919120i \(-0.371099\pi\)
0.393977 + 0.919120i \(0.371099\pi\)
\(912\) − 3.82920e10i − 1.67158i
\(913\) 4.13272e10i 1.79717i
\(914\) −2.58816e10 −1.12119
\(915\) 0 0
\(916\) 2.67038e10 1.14799
\(917\) − 7.58245e8i − 0.0324726i
\(918\) 1.47986e10i 0.631351i
\(919\) 1.01373e10 0.430841 0.215421 0.976521i \(-0.430888\pi\)
0.215421 + 0.976521i \(0.430888\pi\)
\(920\) 0 0
\(921\) 1.32405e10 0.558463
\(922\) 3.74355e9i 0.157299i
\(923\) 2.59376e10i 1.08574i
\(924\) 1.04805e10 0.437050
\(925\) 0 0
\(926\) −6.13311e10 −2.53830
\(927\) 3.63406e9i 0.149835i
\(928\) 4.73449e10i 1.94471i
\(929\) 2.39605e10 0.980485 0.490242 0.871586i \(-0.336908\pi\)
0.490242 + 0.871586i \(0.336908\pi\)
\(930\) 0 0
\(931\) 4.11248e9 0.167024
\(932\) − 1.55391e10i − 0.628738i
\(933\) 3.68125e9i 0.148392i
\(934\) 4.30567e10 1.72913
\(935\) 0 0
\(936\) 5.83998e9 0.232780
\(937\) − 1.16627e10i − 0.463138i −0.972818 0.231569i \(-0.925614\pi\)
0.972818 0.231569i \(-0.0743859\pi\)
\(938\) − 1.13637e10i − 0.449584i
\(939\) −3.69943e10 −1.45816
\(940\) 0 0
\(941\) 3.03134e10 1.18596 0.592982 0.805216i \(-0.297951\pi\)
0.592982 + 0.805216i \(0.297951\pi\)
\(942\) 5.90672e10i 2.30233i
\(943\) 1.49295e10i 0.579768i
\(944\) 9.03231e9 0.349460
\(945\) 0 0
\(946\) −2.94378e10 −1.13054
\(947\) − 2.84339e10i − 1.08796i −0.839099 0.543979i \(-0.816917\pi\)
0.839099 0.543979i \(-0.183083\pi\)
\(948\) 1.28748e10i 0.490809i
\(949\) 5.83357e10 2.21566
\(950\) 0 0
\(951\) 2.54900e10 0.961031
\(952\) − 2.82864e9i − 0.106255i
\(953\) − 4.09127e10i − 1.53120i −0.643315 0.765602i \(-0.722441\pi\)
0.643315 0.765602i \(-0.277559\pi\)
\(954\) 1.27351e10 0.474878
\(955\) 0 0
\(956\) 3.15664e10 1.16848
\(957\) 7.84479e10i 2.89327i
\(958\) 6.70105e10i 2.46243i
\(959\) −2.43729e9 −0.0892365
\(960\) 0 0
\(961\) −2.69733e10 −0.980399
\(962\) − 7.22042e10i − 2.61487i
\(963\) 6.21278e9i 0.224179i
\(964\) −1.89257e10 −0.680428
\(965\) 0 0
\(966\) 2.14075e10 0.764094
\(967\) 4.11324e10i 1.46282i 0.681937 + 0.731411i \(0.261138\pi\)
−0.681937 + 0.731411i \(0.738862\pi\)
\(968\) − 1.37364e10i − 0.486754i
\(969\) 2.57841e10 0.910372
\(970\) 0 0
\(971\) 2.85539e10 1.00092 0.500459 0.865760i \(-0.333165\pi\)
0.500459 + 0.865760i \(0.333165\pi\)
\(972\) − 1.37035e10i − 0.478629i
\(973\) − 3.16194e9i − 0.110042i
\(974\) 2.05057e10 0.711079
\(975\) 0 0
\(976\) 1.66250e10 0.572384
\(977\) − 3.64395e10i − 1.25009i −0.780589 0.625045i \(-0.785081\pi\)
0.780589 0.625045i \(-0.214919\pi\)
\(978\) 2.81178e10i 0.961159i
\(979\) −4.74503e10 −1.61622
\(980\) 0 0
\(981\) −1.62951e10 −0.551081
\(982\) 7.01975e10i 2.36555i
\(983\) − 3.71636e10i − 1.24790i −0.781463 0.623951i \(-0.785526\pi\)
0.781463 0.623951i \(-0.214474\pi\)
\(984\) −6.44006e9 −0.215480
\(985\) 0 0
\(986\) −4.35570e10 −1.44707
\(987\) 4.52513e9i 0.149803i
\(988\) − 3.51711e10i − 1.16021i
\(989\) −2.41864e10 −0.795032
\(990\) 0 0
\(991\) 1.80657e10 0.589655 0.294827 0.955551i \(-0.404738\pi\)
0.294827 + 0.955551i \(0.404738\pi\)
\(992\) − 4.97190e9i − 0.161708i
\(993\) − 1.47838e10i − 0.479140i
\(994\) −1.11445e10 −0.359921
\(995\) 0 0
\(996\) 3.01321e10 0.966320
\(997\) 2.62408e10i 0.838580i 0.907852 + 0.419290i \(0.137721\pi\)
−0.907852 + 0.419290i \(0.862279\pi\)
\(998\) − 2.94199e10i − 0.936882i
\(999\) 3.17347e10 1.00706
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 175.8.b.c.99.1 4
5.2 odd 4 35.8.a.a.1.2 2
5.3 odd 4 175.8.a.b.1.1 2
5.4 even 2 inner 175.8.b.c.99.4 4
15.2 even 4 315.8.a.c.1.1 2
20.7 even 4 560.8.a.i.1.2 2
35.27 even 4 245.8.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 2 5.2 odd 4
175.8.a.b.1.1 2 5.3 odd 4
175.8.b.c.99.1 4 1.1 even 1 trivial
175.8.b.c.99.4 4 5.4 even 2 inner
245.8.a.b.1.2 2 35.27 even 4
315.8.a.c.1.1 2 15.2 even 4
560.8.a.i.1.2 2 20.7 even 4