Properties

Label 75.22.a.j.1.2
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(370.009\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1464.04 q^{2} -59049.0 q^{3} +46252.4 q^{4} +8.64499e7 q^{6} -1.40408e8 q^{7} +3.00259e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-1464.04 q^{2} -59049.0 q^{3} +46252.4 q^{4} +8.64499e7 q^{6} -1.40408e8 q^{7} +3.00259e9 q^{8} +3.48678e9 q^{9} +4.97624e9 q^{11} -2.73116e9 q^{12} +4.81063e11 q^{13} +2.05562e11 q^{14} -4.49291e12 q^{16} -5.10071e12 q^{17} -5.10478e12 q^{18} +3.13371e13 q^{19} +8.29094e12 q^{21} -7.28540e12 q^{22} -2.01713e14 q^{23} -1.77300e14 q^{24} -7.04294e14 q^{26} -2.05891e14 q^{27} -6.49420e12 q^{28} +2.45996e14 q^{29} -1.57216e15 q^{31} +2.80886e14 q^{32} -2.93842e14 q^{33} +7.46763e15 q^{34} +1.61272e14 q^{36} +3.62973e15 q^{37} -4.58786e16 q^{38} -2.84063e16 q^{39} -1.14093e17 q^{41} -1.21382e16 q^{42} -3.06780e16 q^{43} +2.30163e14 q^{44} +2.95316e17 q^{46} +4.22493e17 q^{47} +2.65302e17 q^{48} -5.38832e17 q^{49} +3.01192e17 q^{51} +2.22503e16 q^{52} +6.40096e17 q^{53} +3.01432e17 q^{54} -4.21588e17 q^{56} -1.85042e18 q^{57} -3.60148e17 q^{58} -3.11716e18 q^{59} -1.70405e18 q^{61} +2.30170e18 q^{62} -4.89572e17 q^{63} +9.01108e18 q^{64} +4.30196e17 q^{66} -7.05682e18 q^{67} -2.35920e17 q^{68} +1.19110e19 q^{69} +1.57307e19 q^{71} +1.04694e19 q^{72} +2.49081e19 q^{73} -5.31405e18 q^{74} +1.44941e18 q^{76} -6.98703e17 q^{77} +4.15879e19 q^{78} -1.06959e20 q^{79} +1.21577e19 q^{81} +1.67036e20 q^{82} +8.98326e19 q^{83} +3.83476e17 q^{84} +4.49137e19 q^{86} -1.45258e19 q^{87} +1.49416e19 q^{88} +1.82958e20 q^{89} -6.75450e19 q^{91} -9.32972e18 q^{92} +9.28344e19 q^{93} -6.18546e20 q^{94} -1.65860e19 q^{96} +7.66609e20 q^{97} +7.88869e20 q^{98} +1.73511e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1464.04 −1.01097 −0.505484 0.862836i \(-0.668686\pi\)
−0.505484 + 0.862836i \(0.668686\pi\)
\(3\) −59049.0 −0.577350
\(4\) 46252.4 0.0220549
\(5\) 0 0
\(6\) 8.64499e7 0.583682
\(7\) −1.40408e8 −0.187872 −0.0939360 0.995578i \(-0.529945\pi\)
−0.0939360 + 0.995578i \(0.529945\pi\)
\(8\) 3.00259e9 0.988671
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 4.97624e9 0.0578466 0.0289233 0.999582i \(-0.490792\pi\)
0.0289233 + 0.999582i \(0.490792\pi\)
\(12\) −2.73116e9 −0.0127334
\(13\) 4.81063e11 0.967825 0.483913 0.875116i \(-0.339215\pi\)
0.483913 + 0.875116i \(0.339215\pi\)
\(14\) 2.05562e11 0.189932
\(15\) 0 0
\(16\) −4.49291e12 −1.02157
\(17\) −5.10071e12 −0.613644 −0.306822 0.951767i \(-0.599266\pi\)
−0.306822 + 0.951767i \(0.599266\pi\)
\(18\) −5.10478e12 −0.336989
\(19\) 3.13371e13 1.17259 0.586294 0.810098i \(-0.300586\pi\)
0.586294 + 0.810098i \(0.300586\pi\)
\(20\) 0 0
\(21\) 8.29094e12 0.108468
\(22\) −7.28540e12 −0.0584811
\(23\) −2.01713e14 −1.01530 −0.507648 0.861565i \(-0.669485\pi\)
−0.507648 + 0.861565i \(0.669485\pi\)
\(24\) −1.77300e14 −0.570809
\(25\) 0 0
\(26\) −7.04294e14 −0.978439
\(27\) −2.05891e14 −0.192450
\(28\) −6.49420e12 −0.00414349
\(29\) 2.45996e14 0.108580 0.0542900 0.998525i \(-0.482710\pi\)
0.0542900 + 0.998525i \(0.482710\pi\)
\(30\) 0 0
\(31\) −1.57216e15 −0.344508 −0.172254 0.985053i \(-0.555105\pi\)
−0.172254 + 0.985053i \(0.555105\pi\)
\(32\) 2.80886e14 0.0441017
\(33\) −2.93842e14 −0.0333978
\(34\) 7.46763e15 0.620374
\(35\) 0 0
\(36\) 1.61272e14 0.00735162
\(37\) 3.62973e15 0.124095 0.0620477 0.998073i \(-0.480237\pi\)
0.0620477 + 0.998073i \(0.480237\pi\)
\(38\) −4.58786e16 −1.18545
\(39\) −2.84063e16 −0.558774
\(40\) 0 0
\(41\) −1.14093e17 −1.32748 −0.663741 0.747963i \(-0.731032\pi\)
−0.663741 + 0.747963i \(0.731032\pi\)
\(42\) −1.21382e16 −0.109657
\(43\) −3.06780e16 −0.216475 −0.108238 0.994125i \(-0.534521\pi\)
−0.108238 + 0.994125i \(0.534521\pi\)
\(44\) 2.30163e14 0.00127580
\(45\) 0 0
\(46\) 2.95316e17 1.02643
\(47\) 4.22493e17 1.17163 0.585817 0.810443i \(-0.300774\pi\)
0.585817 + 0.810443i \(0.300774\pi\)
\(48\) 2.65302e17 0.589803
\(49\) −5.38832e17 −0.964704
\(50\) 0 0
\(51\) 3.01192e17 0.354288
\(52\) 2.22503e16 0.0213452
\(53\) 6.40096e17 0.502746 0.251373 0.967890i \(-0.419118\pi\)
0.251373 + 0.967890i \(0.419118\pi\)
\(54\) 3.01432e17 0.194561
\(55\) 0 0
\(56\) −4.21588e17 −0.185743
\(57\) −1.85042e18 −0.676994
\(58\) −3.60148e17 −0.109771
\(59\) −3.11716e18 −0.793985 −0.396993 0.917822i \(-0.629946\pi\)
−0.396993 + 0.917822i \(0.629946\pi\)
\(60\) 0 0
\(61\) −1.70405e18 −0.305858 −0.152929 0.988237i \(-0.548871\pi\)
−0.152929 + 0.988237i \(0.548871\pi\)
\(62\) 2.30170e18 0.348286
\(63\) −4.89572e17 −0.0626240
\(64\) 9.01108e18 0.976983
\(65\) 0 0
\(66\) 4.30196e17 0.0337641
\(67\) −7.05682e18 −0.472959 −0.236480 0.971636i \(-0.575994\pi\)
−0.236480 + 0.971636i \(0.575994\pi\)
\(68\) −2.35920e17 −0.0135338
\(69\) 1.19110e19 0.586181
\(70\) 0 0
\(71\) 1.57307e19 0.573501 0.286751 0.958005i \(-0.407425\pi\)
0.286751 + 0.958005i \(0.407425\pi\)
\(72\) 1.04694e19 0.329557
\(73\) 2.49081e19 0.678344 0.339172 0.940724i \(-0.389853\pi\)
0.339172 + 0.940724i \(0.389853\pi\)
\(74\) −5.31405e18 −0.125456
\(75\) 0 0
\(76\) 1.44941e18 0.0258612
\(77\) −6.98703e17 −0.0108678
\(78\) 4.15879e19 0.564902
\(79\) −1.06959e20 −1.27096 −0.635481 0.772117i \(-0.719198\pi\)
−0.635481 + 0.772117i \(0.719198\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 1.67036e20 1.34204
\(83\) 8.98326e19 0.635498 0.317749 0.948175i \(-0.397073\pi\)
0.317749 + 0.948175i \(0.397073\pi\)
\(84\) 3.83476e17 0.00239224
\(85\) 0 0
\(86\) 4.49137e19 0.218849
\(87\) −1.45258e19 −0.0626887
\(88\) 1.49416e19 0.0571913
\(89\) 1.82958e20 0.621953 0.310976 0.950418i \(-0.399344\pi\)
0.310976 + 0.950418i \(0.399344\pi\)
\(90\) 0 0
\(91\) −6.75450e19 −0.181827
\(92\) −9.32972e18 −0.0223922
\(93\) 9.28344e19 0.198902
\(94\) −6.18546e20 −1.18448
\(95\) 0 0
\(96\) −1.65860e19 −0.0254621
\(97\) 7.66609e20 1.05553 0.527765 0.849390i \(-0.323030\pi\)
0.527765 + 0.849390i \(0.323030\pi\)
\(98\) 7.88869e20 0.975284
\(99\) 1.73511e19 0.0192822
\(100\) 0 0
\(101\) 2.74837e19 0.0247572 0.0123786 0.999923i \(-0.496060\pi\)
0.0123786 + 0.999923i \(0.496060\pi\)
\(102\) −4.40956e20 −0.358173
\(103\) −6.67325e20 −0.489268 −0.244634 0.969616i \(-0.578668\pi\)
−0.244634 + 0.969616i \(0.578668\pi\)
\(104\) 1.44444e21 0.956860
\(105\) 0 0
\(106\) −9.37125e20 −0.508260
\(107\) 2.44453e21 1.20134 0.600670 0.799497i \(-0.294901\pi\)
0.600670 + 0.799497i \(0.294901\pi\)
\(108\) −9.52295e18 −0.00424446
\(109\) 1.38123e21 0.558841 0.279420 0.960169i \(-0.409858\pi\)
0.279420 + 0.960169i \(0.409858\pi\)
\(110\) 0 0
\(111\) −2.14332e20 −0.0716465
\(112\) 6.30839e20 0.191924
\(113\) 1.88655e21 0.522812 0.261406 0.965229i \(-0.415814\pi\)
0.261406 + 0.965229i \(0.415814\pi\)
\(114\) 2.70909e21 0.684419
\(115\) 0 0
\(116\) 1.13779e19 0.00239471
\(117\) 1.67736e21 0.322608
\(118\) 4.56363e21 0.802693
\(119\) 7.16179e20 0.115287
\(120\) 0 0
\(121\) −7.37549e21 −0.996654
\(122\) 2.49480e21 0.309212
\(123\) 6.73708e21 0.766422
\(124\) −7.27161e19 −0.00759806
\(125\) 0 0
\(126\) 7.16751e20 0.0633108
\(127\) 2.12459e22 1.72718 0.863589 0.504197i \(-0.168211\pi\)
0.863589 + 0.504197i \(0.168211\pi\)
\(128\) −1.37816e22 −1.03180
\(129\) 1.81150e21 0.124982
\(130\) 0 0
\(131\) −1.93626e22 −1.13662 −0.568310 0.822814i \(-0.692403\pi\)
−0.568310 + 0.822814i \(0.692403\pi\)
\(132\) −1.35909e19 −0.000736583 0
\(133\) −4.39997e21 −0.220296
\(134\) 1.03314e22 0.478146
\(135\) 0 0
\(136\) −1.53154e22 −0.606692
\(137\) 2.44276e22 0.896013 0.448006 0.894030i \(-0.352134\pi\)
0.448006 + 0.894030i \(0.352134\pi\)
\(138\) −1.74381e22 −0.592610
\(139\) 5.09147e22 1.60394 0.801969 0.597366i \(-0.203786\pi\)
0.801969 + 0.597366i \(0.203786\pi\)
\(140\) 0 0
\(141\) −2.49478e22 −0.676443
\(142\) −2.30303e22 −0.579791
\(143\) 2.39389e21 0.0559854
\(144\) −1.56658e22 −0.340523
\(145\) 0 0
\(146\) −3.64663e22 −0.685784
\(147\) 3.18175e22 0.556972
\(148\) 1.67883e20 0.00273691
\(149\) 2.65231e22 0.402873 0.201437 0.979502i \(-0.435439\pi\)
0.201437 + 0.979502i \(0.435439\pi\)
\(150\) 0 0
\(151\) 2.94760e22 0.389234 0.194617 0.980879i \(-0.437654\pi\)
0.194617 + 0.980879i \(0.437654\pi\)
\(152\) 9.40924e22 1.15930
\(153\) −1.77851e22 −0.204548
\(154\) 1.02293e21 0.0109869
\(155\) 0 0
\(156\) −1.31386e21 −0.0123237
\(157\) 8.78473e22 0.770517 0.385258 0.922809i \(-0.374112\pi\)
0.385258 + 0.922809i \(0.374112\pi\)
\(158\) 1.56592e23 1.28490
\(159\) −3.77970e22 −0.290261
\(160\) 0 0
\(161\) 2.83221e22 0.190745
\(162\) −1.77993e22 −0.112330
\(163\) 1.56579e23 0.926323 0.463161 0.886274i \(-0.346715\pi\)
0.463161 + 0.886274i \(0.346715\pi\)
\(164\) −5.27707e21 −0.0292774
\(165\) 0 0
\(166\) −1.31518e23 −0.642467
\(167\) −1.30318e23 −0.597696 −0.298848 0.954301i \(-0.596602\pi\)
−0.298848 + 0.954301i \(0.596602\pi\)
\(168\) 2.48943e22 0.107239
\(169\) −1.56428e22 −0.0633147
\(170\) 0 0
\(171\) 1.09266e23 0.390863
\(172\) −1.41893e21 −0.00477432
\(173\) −4.20009e23 −1.32977 −0.664883 0.746948i \(-0.731518\pi\)
−0.664883 + 0.746948i \(0.731518\pi\)
\(174\) 2.12664e22 0.0633762
\(175\) 0 0
\(176\) −2.23578e22 −0.0590943
\(177\) 1.84065e23 0.458408
\(178\) −2.67858e23 −0.628774
\(179\) −5.61101e23 −1.24189 −0.620947 0.783853i \(-0.713252\pi\)
−0.620947 + 0.783853i \(0.713252\pi\)
\(180\) 0 0
\(181\) 4.19434e23 0.826111 0.413056 0.910706i \(-0.364461\pi\)
0.413056 + 0.910706i \(0.364461\pi\)
\(182\) 9.88884e22 0.183821
\(183\) 1.00623e23 0.176587
\(184\) −6.05663e23 −1.00379
\(185\) 0 0
\(186\) −1.35913e23 −0.201083
\(187\) −2.53824e22 −0.0354973
\(188\) 1.95413e22 0.0258402
\(189\) 2.89087e22 0.0361560
\(190\) 0 0
\(191\) 6.27733e22 0.0702950 0.0351475 0.999382i \(-0.488810\pi\)
0.0351475 + 0.999382i \(0.488810\pi\)
\(192\) −5.32095e23 −0.564061
\(193\) −4.93834e23 −0.495711 −0.247856 0.968797i \(-0.579726\pi\)
−0.247856 + 0.968797i \(0.579726\pi\)
\(194\) −1.12234e24 −1.06711
\(195\) 0 0
\(196\) −2.49222e22 −0.0212764
\(197\) −6.97018e23 −0.564090 −0.282045 0.959401i \(-0.591013\pi\)
−0.282045 + 0.959401i \(0.591013\pi\)
\(198\) −2.54026e22 −0.0194937
\(199\) −1.57604e24 −1.14712 −0.573560 0.819163i \(-0.694438\pi\)
−0.573560 + 0.819163i \(0.694438\pi\)
\(200\) 0 0
\(201\) 4.16698e23 0.273063
\(202\) −4.02372e22 −0.0250287
\(203\) −3.45398e22 −0.0203991
\(204\) 1.39308e22 0.00781377
\(205\) 0 0
\(206\) 9.76989e23 0.494634
\(207\) −7.03331e23 −0.338432
\(208\) −2.16137e24 −0.988699
\(209\) 1.55941e23 0.0678303
\(210\) 0 0
\(211\) 1.67419e20 6.58928e−5 0 3.29464e−5 1.00000i \(-0.499990\pi\)
3.29464e−5 1.00000i \(0.499990\pi\)
\(212\) 2.96060e22 0.0110880
\(213\) −9.28880e23 −0.331111
\(214\) −3.57888e24 −1.21452
\(215\) 0 0
\(216\) −6.18207e23 −0.190270
\(217\) 2.20743e23 0.0647233
\(218\) −2.02217e24 −0.564970
\(219\) −1.47080e24 −0.391642
\(220\) 0 0
\(221\) −2.45376e24 −0.593900
\(222\) 3.13790e23 0.0724323
\(223\) −4.20079e24 −0.924974 −0.462487 0.886626i \(-0.653043\pi\)
−0.462487 + 0.886626i \(0.653043\pi\)
\(224\) −3.94386e22 −0.00828547
\(225\) 0 0
\(226\) −2.76198e24 −0.528546
\(227\) −1.55102e24 −0.283366 −0.141683 0.989912i \(-0.545251\pi\)
−0.141683 + 0.989912i \(0.545251\pi\)
\(228\) −8.55864e22 −0.0149310
\(229\) −2.11100e24 −0.351735 −0.175867 0.984414i \(-0.556273\pi\)
−0.175867 + 0.984414i \(0.556273\pi\)
\(230\) 0 0
\(231\) 4.12577e22 0.00627450
\(232\) 7.38627e23 0.107350
\(233\) −1.12738e25 −1.56615 −0.783074 0.621929i \(-0.786349\pi\)
−0.783074 + 0.621929i \(0.786349\pi\)
\(234\) −2.45572e24 −0.326146
\(235\) 0 0
\(236\) −1.44176e23 −0.0175112
\(237\) 6.31581e24 0.733790
\(238\) −1.04851e24 −0.116551
\(239\) −1.38717e25 −1.47554 −0.737770 0.675052i \(-0.764121\pi\)
−0.737770 + 0.675052i \(0.764121\pi\)
\(240\) 0 0
\(241\) −1.88297e24 −0.183512 −0.0917561 0.995782i \(-0.529248\pi\)
−0.0917561 + 0.995782i \(0.529248\pi\)
\(242\) 1.07980e25 1.00758
\(243\) −7.17898e23 −0.0641500
\(244\) −7.88165e22 −0.00674565
\(245\) 0 0
\(246\) −9.86333e24 −0.774828
\(247\) 1.50751e25 1.13486
\(248\) −4.72056e24 −0.340605
\(249\) −5.30453e24 −0.366905
\(250\) 0 0
\(251\) −2.15841e25 −1.37265 −0.686324 0.727296i \(-0.740777\pi\)
−0.686324 + 0.727296i \(0.740777\pi\)
\(252\) −2.26439e22 −0.00138116
\(253\) −1.00377e24 −0.0587314
\(254\) −3.11049e25 −1.74612
\(255\) 0 0
\(256\) 1.27919e24 0.0661326
\(257\) 3.67753e25 1.82498 0.912491 0.409097i \(-0.134156\pi\)
0.912491 + 0.409097i \(0.134156\pi\)
\(258\) −2.65211e24 −0.126353
\(259\) −5.09642e23 −0.0233140
\(260\) 0 0
\(261\) 8.57736e23 0.0361933
\(262\) 2.83476e25 1.14909
\(263\) 3.68310e25 1.43443 0.717213 0.696854i \(-0.245417\pi\)
0.717213 + 0.696854i \(0.245417\pi\)
\(264\) −8.82288e23 −0.0330194
\(265\) 0 0
\(266\) 6.44171e24 0.222712
\(267\) −1.08035e25 −0.359085
\(268\) −3.26395e23 −0.0104310
\(269\) 2.87878e24 0.0884726 0.0442363 0.999021i \(-0.485915\pi\)
0.0442363 + 0.999021i \(0.485915\pi\)
\(270\) 0 0
\(271\) −9.41667e24 −0.267744 −0.133872 0.990999i \(-0.542741\pi\)
−0.133872 + 0.990999i \(0.542741\pi\)
\(272\) 2.29170e25 0.626880
\(273\) 3.98847e24 0.104978
\(274\) −3.57629e25 −0.905840
\(275\) 0 0
\(276\) 5.50911e23 0.0129281
\(277\) 5.57010e25 1.25842 0.629210 0.777235i \(-0.283379\pi\)
0.629210 + 0.777235i \(0.283379\pi\)
\(278\) −7.45409e25 −1.62153
\(279\) −5.48178e24 −0.114836
\(280\) 0 0
\(281\) −8.96103e25 −1.74157 −0.870786 0.491662i \(-0.836389\pi\)
−0.870786 + 0.491662i \(0.836389\pi\)
\(282\) 3.65245e25 0.683862
\(283\) −6.19488e25 −1.11757 −0.558786 0.829312i \(-0.688733\pi\)
−0.558786 + 0.829312i \(0.688733\pi\)
\(284\) 7.27581e23 0.0126485
\(285\) 0 0
\(286\) −3.50474e24 −0.0565994
\(287\) 1.60196e25 0.249397
\(288\) 9.79389e23 0.0147006
\(289\) −4.30747e25 −0.623440
\(290\) 0 0
\(291\) −4.52675e25 −0.609410
\(292\) 1.15206e24 0.0149608
\(293\) 3.64058e25 0.456101 0.228050 0.973649i \(-0.426765\pi\)
0.228050 + 0.973649i \(0.426765\pi\)
\(294\) −4.65819e25 −0.563081
\(295\) 0 0
\(296\) 1.08986e25 0.122689
\(297\) −1.02456e24 −0.0111326
\(298\) −3.88308e25 −0.407292
\(299\) −9.70368e25 −0.982628
\(300\) 0 0
\(301\) 4.30743e24 0.0406696
\(302\) −4.31539e25 −0.393503
\(303\) −1.62289e24 −0.0142936
\(304\) −1.40794e26 −1.19788
\(305\) 0 0
\(306\) 2.60380e25 0.206791
\(307\) 2.25914e26 1.73376 0.866882 0.498513i \(-0.166120\pi\)
0.866882 + 0.498513i \(0.166120\pi\)
\(308\) −3.23167e22 −0.000239687 0
\(309\) 3.94049e25 0.282479
\(310\) 0 0
\(311\) 7.99596e25 0.535657 0.267828 0.963467i \(-0.413694\pi\)
0.267828 + 0.963467i \(0.413694\pi\)
\(312\) −8.52925e25 −0.552443
\(313\) −2.18444e26 −1.36812 −0.684061 0.729425i \(-0.739788\pi\)
−0.684061 + 0.729425i \(0.739788\pi\)
\(314\) −1.28612e26 −0.778967
\(315\) 0 0
\(316\) −4.94710e24 −0.0280309
\(317\) −1.44597e26 −0.792570 −0.396285 0.918127i \(-0.629701\pi\)
−0.396285 + 0.918127i \(0.629701\pi\)
\(318\) 5.53363e25 0.293444
\(319\) 1.22414e24 0.00628099
\(320\) 0 0
\(321\) −1.44347e26 −0.693594
\(322\) −4.14646e25 −0.192837
\(323\) −1.59841e26 −0.719552
\(324\) 5.62321e23 0.00245054
\(325\) 0 0
\(326\) −2.29237e26 −0.936482
\(327\) −8.15603e25 −0.322647
\(328\) −3.42575e26 −1.31244
\(329\) −5.93213e25 −0.220117
\(330\) 0 0
\(331\) 4.17109e26 1.45230 0.726149 0.687537i \(-0.241308\pi\)
0.726149 + 0.687537i \(0.241308\pi\)
\(332\) 4.15497e24 0.0140158
\(333\) 1.26561e25 0.0413651
\(334\) 1.90790e26 0.604251
\(335\) 0 0
\(336\) −3.72504e25 −0.110807
\(337\) −5.04094e26 −1.45344 −0.726720 0.686934i \(-0.758956\pi\)
−0.726720 + 0.686934i \(0.758956\pi\)
\(338\) 2.29017e25 0.0640091
\(339\) −1.11399e26 −0.301846
\(340\) 0 0
\(341\) −7.82344e24 −0.0199286
\(342\) −1.59969e26 −0.395149
\(343\) 1.54080e26 0.369113
\(344\) −9.21135e25 −0.214022
\(345\) 0 0
\(346\) 6.14909e26 1.34435
\(347\) −1.48760e26 −0.315520 −0.157760 0.987478i \(-0.550427\pi\)
−0.157760 + 0.987478i \(0.550427\pi\)
\(348\) −6.71855e23 −0.00138259
\(349\) 3.27282e26 0.653513 0.326757 0.945108i \(-0.394044\pi\)
0.326757 + 0.945108i \(0.394044\pi\)
\(350\) 0 0
\(351\) −9.90466e25 −0.186258
\(352\) 1.39776e24 0.00255114
\(353\) −4.72680e25 −0.0837400 −0.0418700 0.999123i \(-0.513332\pi\)
−0.0418700 + 0.999123i \(0.513332\pi\)
\(354\) −2.69478e26 −0.463435
\(355\) 0 0
\(356\) 8.46227e24 0.0137171
\(357\) −4.22897e25 −0.0665607
\(358\) 8.21473e26 1.25551
\(359\) −6.32988e25 −0.0939516 −0.0469758 0.998896i \(-0.514958\pi\)
−0.0469758 + 0.998896i \(0.514958\pi\)
\(360\) 0 0
\(361\) 2.67801e26 0.374962
\(362\) −6.14067e26 −0.835172
\(363\) 4.35515e26 0.575418
\(364\) −3.12412e24 −0.00401017
\(365\) 0 0
\(366\) −1.47315e26 −0.178524
\(367\) 1.13022e27 1.33097 0.665487 0.746409i \(-0.268224\pi\)
0.665487 + 0.746409i \(0.268224\pi\)
\(368\) 9.06279e26 1.03719
\(369\) −3.97818e26 −0.442494
\(370\) 0 0
\(371\) −8.98745e25 −0.0944519
\(372\) 4.29381e24 0.00438674
\(373\) −8.91277e26 −0.885258 −0.442629 0.896705i \(-0.645954\pi\)
−0.442629 + 0.896705i \(0.645954\pi\)
\(374\) 3.71607e25 0.0358866
\(375\) 0 0
\(376\) 1.26857e27 1.15836
\(377\) 1.18340e26 0.105086
\(378\) −4.23234e25 −0.0365525
\(379\) 1.96849e27 1.65356 0.826782 0.562523i \(-0.190169\pi\)
0.826782 + 0.562523i \(0.190169\pi\)
\(380\) 0 0
\(381\) −1.25455e27 −0.997186
\(382\) −9.19025e25 −0.0710660
\(383\) 1.33558e27 1.00481 0.502404 0.864633i \(-0.332449\pi\)
0.502404 + 0.864633i \(0.332449\pi\)
\(384\) 8.13790e26 0.595710
\(385\) 0 0
\(386\) 7.22991e26 0.501148
\(387\) −1.06967e26 −0.0721583
\(388\) 3.54575e25 0.0232796
\(389\) 2.43758e27 1.55771 0.778856 0.627202i \(-0.215800\pi\)
0.778856 + 0.627202i \(0.215800\pi\)
\(390\) 0 0
\(391\) 1.02888e27 0.623030
\(392\) −1.61789e27 −0.953775
\(393\) 1.14334e27 0.656228
\(394\) 1.02046e27 0.570277
\(395\) 0 0
\(396\) 8.02529e23 0.000425266 0
\(397\) 2.21977e27 1.14553 0.572766 0.819719i \(-0.305870\pi\)
0.572766 + 0.819719i \(0.305870\pi\)
\(398\) 2.30738e27 1.15970
\(399\) 2.59814e26 0.127188
\(400\) 0 0
\(401\) −3.42519e27 −1.59099 −0.795496 0.605958i \(-0.792790\pi\)
−0.795496 + 0.605958i \(0.792790\pi\)
\(402\) −6.10061e26 −0.276058
\(403\) −7.56308e26 −0.333423
\(404\) 1.27119e24 0.000546016 0
\(405\) 0 0
\(406\) 5.05676e25 0.0206228
\(407\) 1.80624e25 0.00717850
\(408\) 9.04356e26 0.350274
\(409\) 4.43074e27 1.67256 0.836279 0.548304i \(-0.184726\pi\)
0.836279 + 0.548304i \(0.184726\pi\)
\(410\) 0 0
\(411\) −1.44242e27 −0.517313
\(412\) −3.08654e25 −0.0107907
\(413\) 4.37673e26 0.149168
\(414\) 1.02970e27 0.342143
\(415\) 0 0
\(416\) 1.35124e26 0.0426827
\(417\) −3.00646e27 −0.926034
\(418\) −2.28303e26 −0.0685742
\(419\) 5.60118e27 1.64071 0.820355 0.571854i \(-0.193776\pi\)
0.820355 + 0.571854i \(0.193776\pi\)
\(420\) 0 0
\(421\) 6.08394e27 1.69521 0.847603 0.530630i \(-0.178045\pi\)
0.847603 + 0.530630i \(0.178045\pi\)
\(422\) −2.45107e23 −6.66154e−5 0
\(423\) 1.47314e27 0.390545
\(424\) 1.92195e27 0.497050
\(425\) 0 0
\(426\) 1.35992e27 0.334743
\(427\) 2.39262e26 0.0574621
\(428\) 1.13065e26 0.0264954
\(429\) −1.41357e26 −0.0323232
\(430\) 0 0
\(431\) −4.32480e27 −0.941792 −0.470896 0.882189i \(-0.656069\pi\)
−0.470896 + 0.882189i \(0.656069\pi\)
\(432\) 9.25049e26 0.196601
\(433\) 1.04991e27 0.217785 0.108892 0.994054i \(-0.465270\pi\)
0.108892 + 0.994054i \(0.465270\pi\)
\(434\) −3.23177e26 −0.0654331
\(435\) 0 0
\(436\) 6.38852e25 0.0123251
\(437\) −6.32110e27 −1.19052
\(438\) 2.15330e27 0.395938
\(439\) −3.05313e26 −0.0548110 −0.0274055 0.999624i \(-0.508725\pi\)
−0.0274055 + 0.999624i \(0.508725\pi\)
\(440\) 0 0
\(441\) −1.87879e27 −0.321568
\(442\) 3.59240e27 0.600414
\(443\) 6.04289e26 0.0986292 0.0493146 0.998783i \(-0.484296\pi\)
0.0493146 + 0.998783i \(0.484296\pi\)
\(444\) −9.91335e24 −0.00158015
\(445\) 0 0
\(446\) 6.15011e27 0.935119
\(447\) −1.56616e27 −0.232599
\(448\) −1.26523e27 −0.183548
\(449\) −5.68434e27 −0.805551 −0.402776 0.915299i \(-0.631954\pi\)
−0.402776 + 0.915299i \(0.631954\pi\)
\(450\) 0 0
\(451\) −5.67754e26 −0.0767904
\(452\) 8.72576e25 0.0115305
\(453\) −1.74053e27 −0.224724
\(454\) 2.27076e27 0.286473
\(455\) 0 0
\(456\) −5.55606e27 −0.669324
\(457\) 1.03692e27 0.122074 0.0610371 0.998136i \(-0.480559\pi\)
0.0610371 + 0.998136i \(0.480559\pi\)
\(458\) 3.09058e27 0.355592
\(459\) 1.05019e27 0.118096
\(460\) 0 0
\(461\) 1.90353e27 0.204503 0.102252 0.994759i \(-0.467395\pi\)
0.102252 + 0.994759i \(0.467395\pi\)
\(462\) −6.04028e25 −0.00634332
\(463\) −1.89124e28 −1.94153 −0.970767 0.240022i \(-0.922845\pi\)
−0.970767 + 0.240022i \(0.922845\pi\)
\(464\) −1.10524e27 −0.110922
\(465\) 0 0
\(466\) 1.65052e28 1.58332
\(467\) −1.49943e28 −1.40637 −0.703186 0.711006i \(-0.748240\pi\)
−0.703186 + 0.711006i \(0.748240\pi\)
\(468\) 7.75820e25 0.00711508
\(469\) 9.90833e26 0.0888557
\(470\) 0 0
\(471\) −5.18730e27 −0.444858
\(472\) −9.35955e27 −0.784990
\(473\) −1.52661e26 −0.0125224
\(474\) −9.24658e27 −0.741837
\(475\) 0 0
\(476\) 3.31250e25 0.00254263
\(477\) 2.23188e27 0.167582
\(478\) 2.03086e28 1.49172
\(479\) −9.94822e27 −0.714862 −0.357431 0.933939i \(-0.616347\pi\)
−0.357431 + 0.933939i \(0.616347\pi\)
\(480\) 0 0
\(481\) 1.74613e27 0.120103
\(482\) 2.75674e27 0.185525
\(483\) −1.67239e27 −0.110127
\(484\) −3.41134e26 −0.0219811
\(485\) 0 0
\(486\) 1.05103e27 0.0648536
\(487\) −8.50467e27 −0.513574 −0.256787 0.966468i \(-0.582664\pi\)
−0.256787 + 0.966468i \(0.582664\pi\)
\(488\) −5.11658e27 −0.302393
\(489\) −9.24581e27 −0.534813
\(490\) 0 0
\(491\) 5.06842e27 0.280878 0.140439 0.990089i \(-0.455149\pi\)
0.140439 + 0.990089i \(0.455149\pi\)
\(492\) 3.11606e26 0.0169033
\(493\) −1.25476e27 −0.0666295
\(494\) −2.20705e28 −1.14731
\(495\) 0 0
\(496\) 7.06356e27 0.351938
\(497\) −2.20871e27 −0.107745
\(498\) 7.76602e27 0.370929
\(499\) −3.44904e27 −0.161303 −0.0806516 0.996742i \(-0.525700\pi\)
−0.0806516 + 0.996742i \(0.525700\pi\)
\(500\) 0 0
\(501\) 7.69513e27 0.345080
\(502\) 3.15999e28 1.38770
\(503\) −1.87485e28 −0.806312 −0.403156 0.915131i \(-0.632087\pi\)
−0.403156 + 0.915131i \(0.632087\pi\)
\(504\) −1.46998e27 −0.0619145
\(505\) 0 0
\(506\) 1.46956e27 0.0593755
\(507\) 9.23693e26 0.0365548
\(508\) 9.82676e26 0.0380926
\(509\) −1.58257e28 −0.600933 −0.300466 0.953792i \(-0.597142\pi\)
−0.300466 + 0.953792i \(0.597142\pi\)
\(510\) 0 0
\(511\) −3.49729e27 −0.127442
\(512\) 2.70294e28 0.964942
\(513\) −6.45202e27 −0.225665
\(514\) −5.38404e28 −1.84500
\(515\) 0 0
\(516\) 8.37863e25 0.00275646
\(517\) 2.10243e27 0.0677751
\(518\) 7.46135e26 0.0235697
\(519\) 2.48011e28 0.767740
\(520\) 0 0
\(521\) −1.95573e28 −0.581450 −0.290725 0.956807i \(-0.593896\pi\)
−0.290725 + 0.956807i \(0.593896\pi\)
\(522\) −1.25576e27 −0.0365903
\(523\) −4.97824e28 −1.42170 −0.710851 0.703343i \(-0.751690\pi\)
−0.710851 + 0.703343i \(0.751690\pi\)
\(524\) −8.95567e26 −0.0250680
\(525\) 0 0
\(526\) −5.39220e28 −1.45016
\(527\) 8.01913e27 0.211405
\(528\) 1.32020e27 0.0341181
\(529\) 1.21668e27 0.0308243
\(530\) 0 0
\(531\) −1.08689e28 −0.264662
\(532\) −2.03509e26 −0.00485860
\(533\) −5.48859e28 −1.28477
\(534\) 1.58167e28 0.363023
\(535\) 0 0
\(536\) −2.11888e28 −0.467601
\(537\) 3.31325e28 0.717008
\(538\) −4.21463e27 −0.0894429
\(539\) −2.68136e27 −0.0558049
\(540\) 0 0
\(541\) −8.94883e28 −1.79141 −0.895705 0.444648i \(-0.853329\pi\)
−0.895705 + 0.444648i \(0.853329\pi\)
\(542\) 1.37864e28 0.270680
\(543\) −2.47672e28 −0.476956
\(544\) −1.43272e27 −0.0270628
\(545\) 0 0
\(546\) −5.83926e27 −0.106129
\(547\) 7.47423e28 1.33260 0.666300 0.745684i \(-0.267877\pi\)
0.666300 + 0.745684i \(0.267877\pi\)
\(548\) 1.12983e27 0.0197614
\(549\) −5.94167e27 −0.101953
\(550\) 0 0
\(551\) 7.70880e27 0.127320
\(552\) 3.57638e28 0.579540
\(553\) 1.50179e28 0.238778
\(554\) −8.15484e28 −1.27222
\(555\) 0 0
\(556\) 2.35492e27 0.0353746
\(557\) −2.65398e28 −0.391217 −0.195608 0.980682i \(-0.562668\pi\)
−0.195608 + 0.980682i \(0.562668\pi\)
\(558\) 8.02553e27 0.116095
\(559\) −1.47580e28 −0.209510
\(560\) 0 0
\(561\) 1.49880e27 0.0204944
\(562\) 1.31193e29 1.76067
\(563\) −3.35112e27 −0.0441420 −0.0220710 0.999756i \(-0.507026\pi\)
−0.0220710 + 0.999756i \(0.507026\pi\)
\(564\) −1.15389e27 −0.0149189
\(565\) 0 0
\(566\) 9.06953e28 1.12983
\(567\) −1.70703e27 −0.0208747
\(568\) 4.72328e28 0.567004
\(569\) 6.04337e28 0.712197 0.356099 0.934448i \(-0.384107\pi\)
0.356099 + 0.934448i \(0.384107\pi\)
\(570\) 0 0
\(571\) −9.58264e28 −1.08844 −0.544222 0.838941i \(-0.683175\pi\)
−0.544222 + 0.838941i \(0.683175\pi\)
\(572\) 1.10723e26 0.00123475
\(573\) −3.70670e27 −0.0405849
\(574\) −2.34532e28 −0.252132
\(575\) 0 0
\(576\) 3.14197e28 0.325661
\(577\) −1.45577e29 −1.48166 −0.740829 0.671693i \(-0.765567\pi\)
−0.740829 + 0.671693i \(0.765567\pi\)
\(578\) 6.30630e28 0.630278
\(579\) 2.91604e28 0.286199
\(580\) 0 0
\(581\) −1.26132e28 −0.119392
\(582\) 6.62733e28 0.616094
\(583\) 3.18527e27 0.0290822
\(584\) 7.47888e28 0.670659
\(585\) 0 0
\(586\) −5.32995e28 −0.461103
\(587\) 4.78100e28 0.406273 0.203137 0.979150i \(-0.434886\pi\)
0.203137 + 0.979150i \(0.434886\pi\)
\(588\) 1.47163e27 0.0122839
\(589\) −4.92668e28 −0.403965
\(590\) 0 0
\(591\) 4.11582e28 0.325678
\(592\) −1.63080e28 −0.126772
\(593\) −2.30871e29 −1.76317 −0.881586 0.472023i \(-0.843524\pi\)
−0.881586 + 0.472023i \(0.843524\pi\)
\(594\) 1.50000e27 0.0112547
\(595\) 0 0
\(596\) 1.22676e27 0.00888531
\(597\) 9.30634e28 0.662291
\(598\) 1.42066e29 0.993405
\(599\) −3.65569e28 −0.251182 −0.125591 0.992082i \(-0.540083\pi\)
−0.125591 + 0.992082i \(0.540083\pi\)
\(600\) 0 0
\(601\) −5.74345e28 −0.381058 −0.190529 0.981682i \(-0.561020\pi\)
−0.190529 + 0.981682i \(0.561020\pi\)
\(602\) −6.30623e27 −0.0411156
\(603\) −2.46056e28 −0.157653
\(604\) 1.36333e27 0.00858449
\(605\) 0 0
\(606\) 2.37597e27 0.0144503
\(607\) −1.40009e29 −0.836900 −0.418450 0.908240i \(-0.637426\pi\)
−0.418450 + 0.908240i \(0.637426\pi\)
\(608\) 8.80214e27 0.0517131
\(609\) 2.03954e27 0.0117774
\(610\) 0 0
\(611\) 2.03246e29 1.13394
\(612\) −8.22602e26 −0.00451128
\(613\) 2.25020e29 1.21307 0.606535 0.795056i \(-0.292559\pi\)
0.606535 + 0.795056i \(0.292559\pi\)
\(614\) −3.30746e29 −1.75278
\(615\) 0 0
\(616\) −2.09792e27 −0.0107446
\(617\) −1.06133e29 −0.534387 −0.267193 0.963643i \(-0.586096\pi\)
−0.267193 + 0.963643i \(0.586096\pi\)
\(618\) −5.76902e28 −0.285577
\(619\) 1.21930e29 0.593416 0.296708 0.954968i \(-0.404111\pi\)
0.296708 + 0.954968i \(0.404111\pi\)
\(620\) 0 0
\(621\) 4.15310e28 0.195394
\(622\) −1.17064e29 −0.541531
\(623\) −2.56888e28 −0.116847
\(624\) 1.27627e29 0.570826
\(625\) 0 0
\(626\) 3.19810e29 1.38313
\(627\) −9.20814e27 −0.0391618
\(628\) 4.06315e27 0.0169936
\(629\) −1.85142e28 −0.0761505
\(630\) 0 0
\(631\) −2.07221e29 −0.824378 −0.412189 0.911098i \(-0.635236\pi\)
−0.412189 + 0.911098i \(0.635236\pi\)
\(632\) −3.21154e29 −1.25656
\(633\) −9.88590e24 −3.80432e−5 0
\(634\) 2.11696e29 0.801263
\(635\) 0 0
\(636\) −1.74820e27 −0.00640165
\(637\) −2.59212e29 −0.933665
\(638\) −1.79218e27 −0.00634987
\(639\) 5.48494e28 0.191167
\(640\) 0 0
\(641\) 2.45398e29 0.827679 0.413840 0.910350i \(-0.364187\pi\)
0.413840 + 0.910350i \(0.364187\pi\)
\(642\) 2.11330e29 0.701201
\(643\) 3.27757e29 1.06988 0.534942 0.844889i \(-0.320333\pi\)
0.534942 + 0.844889i \(0.320333\pi\)
\(644\) 1.30997e27 0.00420686
\(645\) 0 0
\(646\) 2.34013e29 0.727443
\(647\) −3.78689e29 −1.15821 −0.579106 0.815252i \(-0.696598\pi\)
−0.579106 + 0.815252i \(0.696598\pi\)
\(648\) 3.65045e28 0.109852
\(649\) −1.55117e28 −0.0459294
\(650\) 0 0
\(651\) −1.30347e28 −0.0373680
\(652\) 7.24213e27 0.0204299
\(653\) −1.97562e29 −0.548423 −0.274211 0.961669i \(-0.588417\pi\)
−0.274211 + 0.961669i \(0.588417\pi\)
\(654\) 1.19407e29 0.326185
\(655\) 0 0
\(656\) 5.12609e29 1.35611
\(657\) 8.68491e28 0.226115
\(658\) 8.68486e28 0.222531
\(659\) 1.47449e29 0.371831 0.185915 0.982566i \(-0.440475\pi\)
0.185915 + 0.982566i \(0.440475\pi\)
\(660\) 0 0
\(661\) −2.14683e29 −0.524424 −0.262212 0.965010i \(-0.584452\pi\)
−0.262212 + 0.965010i \(0.584452\pi\)
\(662\) −6.10663e29 −1.46823
\(663\) 1.44892e29 0.342889
\(664\) 2.69731e29 0.628298
\(665\) 0 0
\(666\) −1.85290e28 −0.0418188
\(667\) −4.96207e28 −0.110241
\(668\) −6.02750e27 −0.0131821
\(669\) 2.48052e29 0.534034
\(670\) 0 0
\(671\) −8.47978e27 −0.0176929
\(672\) 2.32881e27 0.00478362
\(673\) −2.29543e29 −0.464200 −0.232100 0.972692i \(-0.574560\pi\)
−0.232100 + 0.972692i \(0.574560\pi\)
\(674\) 7.38012e29 1.46938
\(675\) 0 0
\(676\) −7.23518e26 −0.00139640
\(677\) −7.69517e29 −1.46230 −0.731151 0.682215i \(-0.761017\pi\)
−0.731151 + 0.682215i \(0.761017\pi\)
\(678\) 1.63092e29 0.305156
\(679\) −1.07638e29 −0.198304
\(680\) 0 0
\(681\) 9.15865e28 0.163601
\(682\) 1.14538e28 0.0201472
\(683\) 4.29417e29 0.743809 0.371905 0.928271i \(-0.378705\pi\)
0.371905 + 0.928271i \(0.378705\pi\)
\(684\) 5.05379e27 0.00862042
\(685\) 0 0
\(686\) −2.25579e29 −0.373161
\(687\) 1.24652e29 0.203074
\(688\) 1.37833e29 0.221144
\(689\) 3.07927e29 0.486570
\(690\) 0 0
\(691\) −9.56053e29 −1.46542 −0.732711 0.680540i \(-0.761745\pi\)
−0.732711 + 0.680540i \(0.761745\pi\)
\(692\) −1.94264e28 −0.0293278
\(693\) −2.43623e27 −0.00362259
\(694\) 2.17790e29 0.318980
\(695\) 0 0
\(696\) −4.36152e28 −0.0619784
\(697\) 5.81955e29 0.814602
\(698\) −4.79152e29 −0.660681
\(699\) 6.65706e29 0.904216
\(700\) 0 0
\(701\) 7.06900e29 0.931792 0.465896 0.884839i \(-0.345732\pi\)
0.465896 + 0.884839i \(0.345732\pi\)
\(702\) 1.45008e29 0.188301
\(703\) 1.13745e29 0.145513
\(704\) 4.48413e28 0.0565152
\(705\) 0 0
\(706\) 6.92021e28 0.0846584
\(707\) −3.85893e27 −0.00465118
\(708\) 8.51344e27 0.0101101
\(709\) 1.15492e30 1.35135 0.675674 0.737201i \(-0.263853\pi\)
0.675674 + 0.737201i \(0.263853\pi\)
\(710\) 0 0
\(711\) −3.72942e29 −0.423654
\(712\) 5.49350e29 0.614907
\(713\) 3.17126e29 0.349777
\(714\) 6.19136e28 0.0672907
\(715\) 0 0
\(716\) −2.59523e28 −0.0273898
\(717\) 8.19109e29 0.851904
\(718\) 9.26718e28 0.0949819
\(719\) −1.74936e30 −1.76695 −0.883477 0.468475i \(-0.844804\pi\)
−0.883477 + 0.468475i \(0.844804\pi\)
\(720\) 0 0
\(721\) 9.36976e28 0.0919197
\(722\) −3.92071e29 −0.379074
\(723\) 1.11188e29 0.105951
\(724\) 1.93998e28 0.0182198
\(725\) 0 0
\(726\) −6.37610e29 −0.581729
\(727\) 1.44631e30 1.30062 0.650308 0.759671i \(-0.274640\pi\)
0.650308 + 0.759671i \(0.274640\pi\)
\(728\) −2.02810e29 −0.179767
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 1.56479e29 0.132839
\(732\) 4.65404e27 0.00389460
\(733\) 1.41576e30 1.16788 0.583941 0.811796i \(-0.301510\pi\)
0.583941 + 0.811796i \(0.301510\pi\)
\(734\) −1.65469e30 −1.34557
\(735\) 0 0
\(736\) −5.66585e28 −0.0447763
\(737\) −3.51164e28 −0.0273591
\(738\) 5.82420e29 0.447347
\(739\) 1.82688e30 1.38339 0.691694 0.722190i \(-0.256865\pi\)
0.691694 + 0.722190i \(0.256865\pi\)
\(740\) 0 0
\(741\) −8.90169e29 −0.655212
\(742\) 1.31580e29 0.0954877
\(743\) 1.23210e30 0.881580 0.440790 0.897610i \(-0.354698\pi\)
0.440790 + 0.897610i \(0.354698\pi\)
\(744\) 2.78744e29 0.196648
\(745\) 0 0
\(746\) 1.30486e30 0.894967
\(747\) 3.13227e29 0.211833
\(748\) −1.17399e27 −0.000782887 0
\(749\) −3.43231e29 −0.225698
\(750\) 0 0
\(751\) 2.55368e30 1.63285 0.816426 0.577451i \(-0.195952\pi\)
0.816426 + 0.577451i \(0.195952\pi\)
\(752\) −1.89822e30 −1.19690
\(753\) 1.27452e30 0.792499
\(754\) −1.73254e29 −0.106239
\(755\) 0 0
\(756\) 1.33710e27 0.000797414 0
\(757\) 3.09532e30 1.82054 0.910269 0.414018i \(-0.135875\pi\)
0.910269 + 0.414018i \(0.135875\pi\)
\(758\) −2.88194e30 −1.67170
\(759\) 5.92719e28 0.0339086
\(760\) 0 0
\(761\) −2.43048e30 −1.35255 −0.676274 0.736650i \(-0.736407\pi\)
−0.676274 + 0.736650i \(0.736407\pi\)
\(762\) 1.83671e30 1.00812
\(763\) −1.93936e29 −0.104990
\(764\) 2.90342e27 0.00155035
\(765\) 0 0
\(766\) −1.95534e30 −1.01583
\(767\) −1.49955e30 −0.768439
\(768\) −7.55349e28 −0.0381817
\(769\) −1.63118e30 −0.813347 −0.406674 0.913573i \(-0.633311\pi\)
−0.406674 + 0.913573i \(0.633311\pi\)
\(770\) 0 0
\(771\) −2.17155e30 −1.05365
\(772\) −2.28410e28 −0.0109328
\(773\) −1.50755e30 −0.711849 −0.355925 0.934515i \(-0.615834\pi\)
−0.355925 + 0.934515i \(0.615834\pi\)
\(774\) 1.56604e29 0.0729497
\(775\) 0 0
\(776\) 2.30181e30 1.04357
\(777\) 3.00938e28 0.0134604
\(778\) −3.56870e30 −1.57480
\(779\) −3.57534e30 −1.55659
\(780\) 0 0
\(781\) 7.82796e28 0.0331751
\(782\) −1.50632e30 −0.629863
\(783\) −5.06485e28 −0.0208962
\(784\) 2.42092e30 0.985511
\(785\) 0 0
\(786\) −1.67390e30 −0.663425
\(787\) 4.34068e29 0.169755 0.0848776 0.996391i \(-0.472950\pi\)
0.0848776 + 0.996391i \(0.472950\pi\)
\(788\) −3.22387e28 −0.0124409
\(789\) −2.17483e30 −0.828166
\(790\) 0 0
\(791\) −2.64887e29 −0.0982217
\(792\) 5.20982e28 0.0190638
\(793\) −8.19757e29 −0.296017
\(794\) −3.24982e30 −1.15810
\(795\) 0 0
\(796\) −7.28955e28 −0.0252996
\(797\) −3.70223e30 −1.26809 −0.634047 0.773295i \(-0.718607\pi\)
−0.634047 + 0.773295i \(0.718607\pi\)
\(798\) −3.80377e29 −0.128583
\(799\) −2.15501e30 −0.718967
\(800\) 0 0
\(801\) 6.37937e29 0.207318
\(802\) 5.01460e30 1.60844
\(803\) 1.23949e29 0.0392399
\(804\) 1.92733e28 0.00602237
\(805\) 0 0
\(806\) 1.10726e30 0.337080
\(807\) −1.69989e29 −0.0510797
\(808\) 8.25225e28 0.0244767
\(809\) −3.66144e30 −1.07199 −0.535997 0.844220i \(-0.680064\pi\)
−0.535997 + 0.844220i \(0.680064\pi\)
\(810\) 0 0
\(811\) 3.17404e30 0.905509 0.452755 0.891635i \(-0.350441\pi\)
0.452755 + 0.891635i \(0.350441\pi\)
\(812\) −1.59755e27 −0.000449900 0
\(813\) 5.56045e29 0.154582
\(814\) −2.64440e28 −0.00725723
\(815\) 0 0
\(816\) −1.35323e30 −0.361929
\(817\) −9.61357e29 −0.253836
\(818\) −6.48676e30 −1.69090
\(819\) −2.35515e29 −0.0606090
\(820\) 0 0
\(821\) 3.00812e30 0.754557 0.377278 0.926100i \(-0.376860\pi\)
0.377278 + 0.926100i \(0.376860\pi\)
\(822\) 2.11176e30 0.522987
\(823\) −4.14963e30 −1.01464 −0.507319 0.861758i \(-0.669363\pi\)
−0.507319 + 0.861758i \(0.669363\pi\)
\(824\) −2.00371e30 −0.483725
\(825\) 0 0
\(826\) −6.40770e29 −0.150804
\(827\) −3.19969e30 −0.743532 −0.371766 0.928326i \(-0.621248\pi\)
−0.371766 + 0.928326i \(0.621248\pi\)
\(828\) −3.25307e28 −0.00746406
\(829\) −6.86522e30 −1.55536 −0.777681 0.628659i \(-0.783604\pi\)
−0.777681 + 0.628659i \(0.783604\pi\)
\(830\) 0 0
\(831\) −3.28909e30 −0.726549
\(832\) 4.33490e30 0.945549
\(833\) 2.74842e30 0.591985
\(834\) 4.40157e30 0.936190
\(835\) 0 0
\(836\) 7.21263e27 0.00149599
\(837\) 3.23694e29 0.0663005
\(838\) −8.20033e30 −1.65870
\(839\) −8.08560e30 −1.61515 −0.807573 0.589767i \(-0.799219\pi\)
−0.807573 + 0.589767i \(0.799219\pi\)
\(840\) 0 0
\(841\) −5.07233e30 −0.988210
\(842\) −8.90711e30 −1.71380
\(843\) 5.29140e30 1.00550
\(844\) 7.74351e24 1.45326e−6 0
\(845\) 0 0
\(846\) −2.15674e30 −0.394828
\(847\) 1.03558e30 0.187243
\(848\) −2.87589e30 −0.513590
\(849\) 3.65802e30 0.645230
\(850\) 0 0
\(851\) −7.32164e29 −0.125993
\(852\) −4.29629e28 −0.00730261
\(853\) 3.94387e30 0.662152 0.331076 0.943604i \(-0.392588\pi\)
0.331076 + 0.943604i \(0.392588\pi\)
\(854\) −3.50289e29 −0.0580923
\(855\) 0 0
\(856\) 7.33993e30 1.18773
\(857\) −1.02177e31 −1.63325 −0.816625 0.577169i \(-0.804157\pi\)
−0.816625 + 0.577169i \(0.804157\pi\)
\(858\) 2.06951e29 0.0326777
\(859\) 6.98558e30 1.08962 0.544809 0.838560i \(-0.316602\pi\)
0.544809 + 0.838560i \(0.316602\pi\)
\(860\) 0 0
\(861\) −9.45939e29 −0.143989
\(862\) 6.33167e30 0.952120
\(863\) −7.69021e30 −1.14242 −0.571208 0.820805i \(-0.693525\pi\)
−0.571208 + 0.820805i \(0.693525\pi\)
\(864\) −5.78320e28 −0.00848738
\(865\) 0 0
\(866\) −1.53710e30 −0.220173
\(867\) 2.54352e30 0.359944
\(868\) 1.02099e28 0.00142746
\(869\) −5.32253e29 −0.0735208
\(870\) 0 0
\(871\) −3.39478e30 −0.457742
\(872\) 4.14727e30 0.552509
\(873\) 2.67300e30 0.351843
\(874\) 9.25433e30 1.20358
\(875\) 0 0
\(876\) −6.80279e28 −0.00863761
\(877\) 8.20562e30 1.02947 0.514737 0.857348i \(-0.327889\pi\)
0.514737 + 0.857348i \(0.327889\pi\)
\(878\) 4.46989e29 0.0554121
\(879\) −2.14973e30 −0.263330
\(880\) 0 0
\(881\) 9.14275e30 1.09353 0.546764 0.837287i \(-0.315860\pi\)
0.546764 + 0.837287i \(0.315860\pi\)
\(882\) 2.75062e30 0.325095
\(883\) 1.29725e31 1.51509 0.757544 0.652785i \(-0.226399\pi\)
0.757544 + 0.652785i \(0.226399\pi\)
\(884\) −1.13492e29 −0.0130984
\(885\) 0 0
\(886\) −8.84702e29 −0.0997109
\(887\) −1.76931e30 −0.197064 −0.0985319 0.995134i \(-0.531415\pi\)
−0.0985319 + 0.995134i \(0.531415\pi\)
\(888\) −6.43551e29 −0.0708348
\(889\) −2.98310e30 −0.324488
\(890\) 0 0
\(891\) 6.04995e28 0.00642740
\(892\) −1.94296e29 −0.0204002
\(893\) 1.32397e31 1.37384
\(894\) 2.29292e30 0.235150
\(895\) 0 0
\(896\) 1.93505e30 0.193846
\(897\) 5.72993e30 0.567321
\(898\) 8.32208e30 0.814386
\(899\) −3.86745e29 −0.0374066
\(900\) 0 0
\(901\) −3.26494e30 −0.308507
\(902\) 8.31213e29 0.0776326
\(903\) −2.54349e29 −0.0234806
\(904\) 5.66455e30 0.516889
\(905\) 0 0
\(906\) 2.54820e30 0.227189
\(907\) −1.12882e30 −0.0994830 −0.0497415 0.998762i \(-0.515840\pi\)
−0.0497415 + 0.998762i \(0.515840\pi\)
\(908\) −7.17386e28 −0.00624959
\(909\) 9.58298e28 0.00825240
\(910\) 0 0
\(911\) −6.82669e30 −0.574470 −0.287235 0.957860i \(-0.592736\pi\)
−0.287235 + 0.957860i \(0.592736\pi\)
\(912\) 8.31377e30 0.691595
\(913\) 4.47029e29 0.0367614
\(914\) −1.51808e30 −0.123413
\(915\) 0 0
\(916\) −9.76387e28 −0.00775746
\(917\) 2.71866e30 0.213539
\(918\) −1.53752e30 −0.119391
\(919\) 3.00199e30 0.230460 0.115230 0.993339i \(-0.463239\pi\)
0.115230 + 0.993339i \(0.463239\pi\)
\(920\) 0 0
\(921\) −1.33400e31 −1.00099
\(922\) −2.78684e30 −0.206746
\(923\) 7.56744e30 0.555049
\(924\) 1.90827e27 0.000138383 0
\(925\) 0 0
\(926\) 2.76884e31 1.96283
\(927\) −2.32682e30 −0.163089
\(928\) 6.90970e28 0.00478856
\(929\) 1.16096e31 0.795522 0.397761 0.917489i \(-0.369787\pi\)
0.397761 + 0.917489i \(0.369787\pi\)
\(930\) 0 0
\(931\) −1.68854e31 −1.13120
\(932\) −5.21440e29 −0.0345412
\(933\) −4.72153e30 −0.309262
\(934\) 2.19523e31 1.42180
\(935\) 0 0
\(936\) 5.03644e30 0.318953
\(937\) −1.17349e31 −0.734874 −0.367437 0.930048i \(-0.619765\pi\)
−0.367437 + 0.930048i \(0.619765\pi\)
\(938\) −1.45062e30 −0.0898302
\(939\) 1.28989e31 0.789886
\(940\) 0 0
\(941\) −4.00069e30 −0.239576 −0.119788 0.992799i \(-0.538222\pi\)
−0.119788 + 0.992799i \(0.538222\pi\)
\(942\) 7.59439e30 0.449737
\(943\) 2.30141e31 1.34779
\(944\) 1.40051e31 0.811110
\(945\) 0 0
\(946\) 2.23501e29 0.0126597
\(947\) −2.40112e31 −1.34505 −0.672527 0.740073i \(-0.734791\pi\)
−0.672527 + 0.740073i \(0.734791\pi\)
\(948\) 2.92121e29 0.0161836
\(949\) 1.19824e31 0.656519
\(950\) 0 0
\(951\) 8.53832e30 0.457591
\(952\) 2.15039e30 0.113980
\(953\) −5.91230e30 −0.309943 −0.154971 0.987919i \(-0.549528\pi\)
−0.154971 + 0.987919i \(0.549528\pi\)
\(954\) −3.26755e30 −0.169420
\(955\) 0 0
\(956\) −6.41598e29 −0.0325428
\(957\) −7.22841e28 −0.00362633
\(958\) 1.45646e31 0.722703
\(959\) −3.42982e30 −0.168336
\(960\) 0 0
\(961\) −1.83538e31 −0.881315
\(962\) −2.55639e30 −0.121420
\(963\) 8.52355e30 0.400447
\(964\) −8.70920e28 −0.00404734
\(965\) 0 0
\(966\) 2.44845e30 0.111335
\(967\) −8.05905e30 −0.362498 −0.181249 0.983437i \(-0.558014\pi\)
−0.181249 + 0.983437i \(0.558014\pi\)
\(968\) −2.21456e31 −0.985362
\(969\) 9.43846e30 0.415433
\(970\) 0 0
\(971\) 2.88231e31 1.24148 0.620739 0.784017i \(-0.286833\pi\)
0.620739 + 0.784017i \(0.286833\pi\)
\(972\) −3.32045e28 −0.00141482
\(973\) −7.14882e30 −0.301335
\(974\) 1.24512e31 0.519207
\(975\) 0 0
\(976\) 7.65615e30 0.312455
\(977\) −3.70139e31 −1.49442 −0.747210 0.664588i \(-0.768607\pi\)
−0.747210 + 0.664588i \(0.768607\pi\)
\(978\) 1.35362e31 0.540678
\(979\) 9.10445e29 0.0359779
\(980\) 0 0
\(981\) 4.81605e30 0.186280
\(982\) −7.42036e30 −0.283958
\(983\) −1.58254e31 −0.599162 −0.299581 0.954071i \(-0.596847\pi\)
−0.299581 + 0.954071i \(0.596847\pi\)
\(984\) 2.02287e31 0.757739
\(985\) 0 0
\(986\) 1.83701e30 0.0673602
\(987\) 3.50287e30 0.127085
\(988\) 6.97259e29 0.0250292
\(989\) 6.18816e30 0.219786
\(990\) 0 0
\(991\) −1.29881e31 −0.451621 −0.225810 0.974171i \(-0.572503\pi\)
−0.225810 + 0.974171i \(0.572503\pi\)
\(992\) −4.41598e29 −0.0151934
\(993\) −2.46299e31 −0.838485
\(994\) 3.23363e30 0.108926
\(995\) 0 0
\(996\) −2.45347e29 −0.00809203
\(997\) −4.42410e31 −1.44386 −0.721932 0.691964i \(-0.756746\pi\)
−0.721932 + 0.691964i \(0.756746\pi\)
\(998\) 5.04953e30 0.163072
\(999\) −7.47328e29 −0.0238822
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 75.22.a.j.1.2 yes 6
5.2 odd 4 75.22.b.i.49.4 12
5.3 odd 4 75.22.b.i.49.9 12
5.4 even 2 75.22.a.i.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.5 6 5.4 even 2
75.22.a.j.1.2 yes 6 1.1 even 1 trivial
75.22.b.i.49.4 12 5.2 odd 4
75.22.b.i.49.9 12 5.3 odd 4