Properties

Label 3150.2.d.a.3149.5
Level $3150$
Weight $2$
Character 3150.3149
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(3149,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.3149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.d (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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.5
Root \(1.91681i\) of defining polynomial
Character \(\chi\) \(=\) 3150.3149
Dual form 3150.2.d.a.3149.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(1.35539 - 2.27220i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(1.35539 - 2.27220i) q^{7} -1.00000 q^{8} +2.71078i q^{11} -6.54441 q^{13} +(-1.35539 + 2.27220i) q^{14} +1.00000 q^{16} +1.53186i q^{17} -2.30177i q^{19} -2.71078i q^{22} +3.83363 q^{23} +6.54441 q^{26} +(1.35539 - 2.27220i) q^{28} +3.83363i q^{29} +3.25519i q^{31} -1.00000 q^{32} -1.53186i q^{34} -3.01255i q^{37} +2.30177i q^{38} +6.54441 q^{41} -0.468142i q^{43} +2.71078i q^{44} -3.83363 q^{46} -9.11980i q^{47} +(-3.32583 - 6.15945i) q^{49} -6.54441 q^{52} +9.25519 q^{53} +(-1.35539 + 2.27220i) q^{56} -3.83363i q^{58} +11.1961 q^{59} -4.78705i q^{61} -3.25519i q^{62} +1.00000 q^{64} -13.5570i q^{67} +1.53186i q^{68} +2.30177i q^{71} +11.4216 q^{73} +3.01255i q^{74} -2.30177i q^{76} +(6.15945 + 3.67417i) q^{77} +12.6768 q^{79} -6.54441 q^{82} -13.0888i q^{83} +0.468142i q^{86} -2.71078i q^{88} -9.60812 q^{89} +(-8.87024 + 14.8702i) q^{91} +3.83363 q^{92} +9.11980i q^{94} -16.9224 q^{97} +(3.32583 + 6.15945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} - 8 q^{13} + 8 q^{16} + 8 q^{23} + 8 q^{26} - 8 q^{32} + 8 q^{41} - 8 q^{46} - 4 q^{49} - 8 q^{52} + 8 q^{53} + 8 q^{64} + 48 q^{73} + 4 q^{77} - 8 q^{79} - 8 q^{82} - 8 q^{89} - 4 q^{91} + 8 q^{92} - 24 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.35539 2.27220i 0.512290 0.858813i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 2.71078i 0.817332i 0.912684 + 0.408666i \(0.134006\pi\)
−0.912684 + 0.408666i \(0.865994\pi\)
\(12\) 0 0
\(13\) −6.54441 −1.81509 −0.907546 0.419952i \(-0.862047\pi\)
−0.907546 + 0.419952i \(0.862047\pi\)
\(14\) −1.35539 + 2.27220i −0.362244 + 0.607272i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.53186i 0.371530i 0.982594 + 0.185765i \(0.0594763\pi\)
−0.982594 + 0.185765i \(0.940524\pi\)
\(18\) 0 0
\(19\) 2.30177i 0.528062i −0.964514 0.264031i \(-0.914948\pi\)
0.964514 0.264031i \(-0.0850521\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.71078i 0.577941i
\(23\) 3.83363 0.799366 0.399683 0.916653i \(-0.369120\pi\)
0.399683 + 0.916653i \(0.369120\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.54441 1.28346
\(27\) 0 0
\(28\) 1.35539 2.27220i 0.256145 0.429406i
\(29\) 3.83363i 0.711887i 0.934508 + 0.355943i \(0.115840\pi\)
−0.934508 + 0.355943i \(0.884160\pi\)
\(30\) 0 0
\(31\) 3.25519i 0.584650i 0.956319 + 0.292325i \(0.0944289\pi\)
−0.956319 + 0.292325i \(0.905571\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.53186i 0.262711i
\(35\) 0 0
\(36\) 0 0
\(37\) 3.01255i 0.495260i −0.968855 0.247630i \(-0.920348\pi\)
0.968855 0.247630i \(-0.0796518\pi\)
\(38\) 2.30177i 0.373396i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.54441 1.02207 0.511033 0.859561i \(-0.329263\pi\)
0.511033 + 0.859561i \(0.329263\pi\)
\(42\) 0 0
\(43\) 0.468142i 0.0713910i −0.999363 0.0356955i \(-0.988635\pi\)
0.999363 0.0356955i \(-0.0113647\pi\)
\(44\) 2.71078i 0.408666i
\(45\) 0 0
\(46\) −3.83363 −0.565237
\(47\) 9.11980i 1.33026i −0.746728 0.665130i \(-0.768376\pi\)
0.746728 0.665130i \(-0.231624\pi\)
\(48\) 0 0
\(49\) −3.32583 6.15945i −0.475118 0.879922i
\(50\) 0 0
\(51\) 0 0
\(52\) −6.54441 −0.907546
\(53\) 9.25519 1.27130 0.635649 0.771978i \(-0.280732\pi\)
0.635649 + 0.771978i \(0.280732\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.35539 + 2.27220i −0.181122 + 0.303636i
\(57\) 0 0
\(58\) 3.83363i 0.503380i
\(59\) 11.1961 1.45760 0.728802 0.684725i \(-0.240078\pi\)
0.728802 + 0.684725i \(0.240078\pi\)
\(60\) 0 0
\(61\) 4.78705i 0.612919i −0.951884 0.306459i \(-0.900856\pi\)
0.951884 0.306459i \(-0.0991444\pi\)
\(62\) 3.25519i 0.413410i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.5570i 1.65625i −0.560546 0.828123i \(-0.689409\pi\)
0.560546 0.828123i \(-0.310591\pi\)
\(68\) 1.53186i 0.185765i
\(69\) 0 0
\(70\) 0 0
\(71\) 2.30177i 0.273170i 0.990628 + 0.136585i \(0.0436126\pi\)
−0.990628 + 0.136585i \(0.956387\pi\)
\(72\) 0 0
\(73\) 11.4216 1.33679 0.668397 0.743805i \(-0.266981\pi\)
0.668397 + 0.743805i \(0.266981\pi\)
\(74\) 3.01255i 0.350202i
\(75\) 0 0
\(76\) 2.30177i 0.264031i
\(77\) 6.15945 + 3.67417i 0.701935 + 0.418711i
\(78\) 0 0
\(79\) 12.6768 1.42625 0.713123 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.54441 −0.722709
\(83\) 13.0888i 1.43668i −0.695690 0.718342i \(-0.744901\pi\)
0.695690 0.718342i \(-0.255099\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.468142i 0.0504811i
\(87\) 0 0
\(88\) 2.71078i 0.288970i
\(89\) −9.60812 −1.01846 −0.509230 0.860631i \(-0.670070\pi\)
−0.509230 + 0.860631i \(0.670070\pi\)
\(90\) 0 0
\(91\) −8.87024 + 14.8702i −0.929853 + 1.55882i
\(92\) 3.83363 0.399683
\(93\) 0 0
\(94\) 9.11980i 0.940635i
\(95\) 0 0
\(96\) 0 0
\(97\) −16.9224 −1.71821 −0.859107 0.511796i \(-0.828980\pi\)
−0.859107 + 0.511796i \(0.828980\pi\)
\(98\) 3.32583 + 6.15945i 0.335959 + 0.622199i
\(99\) 0 0
\(100\) 0 0
\(101\) −17.7405 −1.76524 −0.882622 0.470084i \(-0.844224\pi\)
−0.882622 + 0.470084i \(0.844224\pi\)
\(102\) 0 0
\(103\) −4.05913 −0.399958 −0.199979 0.979800i \(-0.564087\pi\)
−0.199979 + 0.979800i \(0.564087\pi\)
\(104\) 6.54441 0.641732
\(105\) 0 0
\(106\) −9.25519 −0.898944
\(107\) 6.31891 0.610872 0.305436 0.952213i \(-0.401198\pi\)
0.305436 + 0.952213i \(0.401198\pi\)
\(108\) 0 0
\(109\) 12.0251 1.15180 0.575898 0.817522i \(-0.304653\pi\)
0.575898 + 0.817522i \(0.304653\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.35539 2.27220i 0.128072 0.214703i
\(113\) −9.06372 −0.852643 −0.426321 0.904572i \(-0.640191\pi\)
−0.426321 + 0.904572i \(0.640191\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.83363i 0.355943i
\(117\) 0 0
\(118\) −11.1961 −1.03068
\(119\) 3.48069 + 2.07627i 0.319075 + 0.190331i
\(120\) 0 0
\(121\) 3.65166 0.331969
\(122\) 4.78705i 0.433399i
\(123\) 0 0
\(124\) 3.25519i 0.292325i
\(125\) 0 0
\(126\) 0 0
\(127\) 21.6332i 1.91964i −0.280619 0.959819i \(-0.590540\pi\)
0.280619 0.959819i \(-0.409460\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 2.51931 0.220113 0.110056 0.993925i \(-0.464897\pi\)
0.110056 + 0.993925i \(0.464897\pi\)
\(132\) 0 0
\(133\) −5.23009 3.11980i −0.453506 0.270521i
\(134\) 13.5570i 1.17114i
\(135\) 0 0
\(136\) 1.53186i 0.131356i
\(137\) −1.39646 −0.119308 −0.0596539 0.998219i \(-0.519000\pi\)
−0.0596539 + 0.998219i \(0.519000\pi\)
\(138\) 0 0
\(139\) 6.13539i 0.520397i −0.965555 0.260199i \(-0.916212\pi\)
0.965555 0.260199i \(-0.0837881\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.30177i 0.193160i
\(143\) 17.7405i 1.48353i
\(144\) 0 0
\(145\) 0 0
\(146\) −11.4216 −0.945256
\(147\) 0 0
\(148\) 3.01255i 0.247630i
\(149\) 4.65166i 0.381078i 0.981680 + 0.190539i \(0.0610236\pi\)
−0.981680 + 0.190539i \(0.938976\pi\)
\(150\) 0 0
\(151\) −3.58794 −0.291982 −0.145991 0.989286i \(-0.546637\pi\)
−0.145991 + 0.989286i \(0.546637\pi\)
\(152\) 2.30177i 0.186698i
\(153\) 0 0
\(154\) −6.15945 3.67417i −0.496343 0.296073i
\(155\) 0 0
\(156\) 0 0
\(157\) −13.1228 −1.04732 −0.523658 0.851928i \(-0.675433\pi\)
−0.523658 + 0.851928i \(0.675433\pi\)
\(158\) −12.6768 −1.00851
\(159\) 0 0
\(160\) 0 0
\(161\) 5.19606 8.71078i 0.409507 0.686506i
\(162\) 0 0
\(163\) 16.6207i 1.30183i 0.759150 + 0.650916i \(0.225615\pi\)
−0.759150 + 0.650916i \(0.774385\pi\)
\(164\) 6.54441 0.511033
\(165\) 0 0
\(166\) 13.0888i 1.01589i
\(167\) 8.62068i 0.667088i 0.942735 + 0.333544i \(0.108245\pi\)
−0.942735 + 0.333544i \(0.891755\pi\)
\(168\) 0 0
\(169\) 29.8293 2.29456
\(170\) 0 0
\(171\) 0 0
\(172\) 0.468142i 0.0356955i
\(173\) 10.6517i 0.809830i 0.914354 + 0.404915i \(0.132699\pi\)
−0.914354 + 0.404915i \(0.867301\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.71078i 0.204333i
\(177\) 0 0
\(178\) 9.60812 0.720159
\(179\) 23.1961i 1.73376i −0.498521 0.866878i \(-0.666123\pi\)
0.498521 0.866878i \(-0.333877\pi\)
\(180\) 0 0
\(181\) 9.11980i 0.677869i −0.940810 0.338935i \(-0.889933\pi\)
0.940810 0.338935i \(-0.110067\pi\)
\(182\) 8.87024 14.8702i 0.657506 1.10226i
\(183\) 0 0
\(184\) −3.83363 −0.282619
\(185\) 0 0
\(186\) 0 0
\(187\) −4.15253 −0.303663
\(188\) 9.11980i 0.665130i
\(189\) 0 0
\(190\) 0 0
\(191\) 9.69823i 0.701739i 0.936424 + 0.350870i \(0.114114\pi\)
−0.936424 + 0.350870i \(0.885886\pi\)
\(192\) 0 0
\(193\) 18.1525i 1.30665i −0.757078 0.653324i \(-0.773374\pi\)
0.757078 0.653324i \(-0.226626\pi\)
\(194\) 16.9224 1.21496
\(195\) 0 0
\(196\) −3.32583 6.15945i −0.237559 0.439961i
\(197\) 4.60354 0.327988 0.163994 0.986461i \(-0.447562\pi\)
0.163994 + 0.986461i \(0.447562\pi\)
\(198\) 0 0
\(199\) 11.7405i 0.832260i −0.909305 0.416130i \(-0.863386\pi\)
0.909305 0.416130i \(-0.136614\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 17.7405 1.24822
\(203\) 8.71078 + 5.19606i 0.611377 + 0.364692i
\(204\) 0 0
\(205\) 0 0
\(206\) 4.05913 0.282813
\(207\) 0 0
\(208\) −6.54441 −0.453773
\(209\) 6.23960 0.431602
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 9.25519 0.635649
\(213\) 0 0
\(214\) −6.31891 −0.431952
\(215\) 0 0
\(216\) 0 0
\(217\) 7.39646 + 4.41206i 0.502105 + 0.299510i
\(218\) −12.0251 −0.814443
\(219\) 0 0
\(220\) 0 0
\(221\) 10.0251i 0.674361i
\(222\) 0 0
\(223\) 26.4513 1.77131 0.885654 0.464347i \(-0.153711\pi\)
0.885654 + 0.464347i \(0.153711\pi\)
\(224\) −1.35539 + 2.27220i −0.0905609 + 0.151818i
\(225\) 0 0
\(226\) 9.06372 0.602909
\(227\) 15.0637i 0.999814i 0.866079 + 0.499907i \(0.166632\pi\)
−0.866079 + 0.499907i \(0.833368\pi\)
\(228\) 0 0
\(229\) 11.3655i 0.751052i −0.926812 0.375526i \(-0.877462\pi\)
0.926812 0.375526i \(-0.122538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.83363i 0.251690i
\(233\) 20.9957 1.37547 0.687736 0.725961i \(-0.258605\pi\)
0.687736 + 0.725961i \(0.258605\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 11.1961 0.728802
\(237\) 0 0
\(238\) −3.48069 2.07627i −0.225620 0.134584i
\(239\) 9.69823i 0.627326i 0.949534 + 0.313663i \(0.101556\pi\)
−0.949534 + 0.313663i \(0.898444\pi\)
\(240\) 0 0
\(241\) 24.7019i 1.59119i −0.605831 0.795593i \(-0.707159\pi\)
0.605831 0.795593i \(-0.292841\pi\)
\(242\) −3.65166 −0.234737
\(243\) 0 0
\(244\) 4.78705i 0.306459i
\(245\) 0 0
\(246\) 0 0
\(247\) 15.0637i 0.958481i
\(248\) 3.25519i 0.206705i
\(249\) 0 0
\(250\) 0 0
\(251\) −3.60812 −0.227743 −0.113871 0.993495i \(-0.536325\pi\)
−0.113871 + 0.993495i \(0.536325\pi\)
\(252\) 0 0
\(253\) 10.3921i 0.653348i
\(254\) 21.6332i 1.35739i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.7983i 1.79639i −0.439598 0.898195i \(-0.644879\pi\)
0.439598 0.898195i \(-0.355121\pi\)
\(258\) 0 0
\(259\) −6.84513 4.08319i −0.425336 0.253717i
\(260\) 0 0
\(261\) 0 0
\(262\) −2.51931 −0.155643
\(263\) 12.7699 0.787426 0.393713 0.919233i \(-0.371190\pi\)
0.393713 + 0.919233i \(0.371190\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.23009 + 3.11980i 0.320677 + 0.191287i
\(267\) 0 0
\(268\) 13.5570i 0.828123i
\(269\) −8.43716 −0.514423 −0.257211 0.966355i \(-0.582804\pi\)
−0.257211 + 0.966355i \(0.582804\pi\)
\(270\) 0 0
\(271\) 2.16637i 0.131598i −0.997833 0.0657989i \(-0.979040\pi\)
0.997833 0.0657989i \(-0.0209596\pi\)
\(272\) 1.53186i 0.0928825i
\(273\) 0 0
\(274\) 1.39646 0.0843634
\(275\) 0 0
\(276\) 0 0
\(277\) 5.92373i 0.355923i −0.984037 0.177961i \(-0.943050\pi\)
0.984037 0.177961i \(-0.0569502\pi\)
\(278\) 6.13539i 0.367977i
\(279\) 0 0
\(280\) 0 0
\(281\) 27.3566i 1.63196i −0.578083 0.815978i \(-0.696199\pi\)
0.578083 0.815978i \(-0.303801\pi\)
\(282\) 0 0
\(283\) −18.0594 −1.07352 −0.536759 0.843735i \(-0.680352\pi\)
−0.536759 + 0.843735i \(0.680352\pi\)
\(284\) 2.30177i 0.136585i
\(285\) 0 0
\(286\) 17.7405i 1.04902i
\(287\) 8.87024 14.8702i 0.523594 0.877762i
\(288\) 0 0
\(289\) 14.6534 0.861965
\(290\) 0 0
\(291\) 0 0
\(292\) 11.4216 0.668397
\(293\) 29.1139i 1.70085i 0.526094 + 0.850427i \(0.323656\pi\)
−0.526094 + 0.850427i \(0.676344\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.01255i 0.175101i
\(297\) 0 0
\(298\) 4.65166i 0.269463i
\(299\) −25.0888 −1.45092
\(300\) 0 0
\(301\) −1.06372 0.634516i −0.0613115 0.0365729i
\(302\) 3.58794 0.206463
\(303\) 0 0
\(304\) 2.30177i 0.132015i
\(305\) 0 0
\(306\) 0 0
\(307\) −7.39646 −0.422138 −0.211069 0.977471i \(-0.567695\pi\)
−0.211069 + 0.977471i \(0.567695\pi\)
\(308\) 6.15945 + 3.67417i 0.350967 + 0.209355i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.97490 −0.111986 −0.0559931 0.998431i \(-0.517832\pi\)
−0.0559931 + 0.998431i \(0.517832\pi\)
\(312\) 0 0
\(313\) 0.961388 0.0543409 0.0271704 0.999631i \(-0.491350\pi\)
0.0271704 + 0.999631i \(0.491350\pi\)
\(314\) 13.1228 0.740565
\(315\) 0 0
\(316\) 12.6768 0.713123
\(317\) 17.1620 0.963916 0.481958 0.876194i \(-0.339926\pi\)
0.481958 + 0.876194i \(0.339926\pi\)
\(318\) 0 0
\(319\) −10.3921 −0.581848
\(320\) 0 0
\(321\) 0 0
\(322\) −5.19606 + 8.71078i −0.289565 + 0.485433i
\(323\) 3.52598 0.196191
\(324\) 0 0
\(325\) 0 0
\(326\) 16.6207i 0.920534i
\(327\) 0 0
\(328\) −6.54441 −0.361355
\(329\) −20.7220 12.3609i −1.14244 0.681478i
\(330\) 0 0
\(331\) 9.74047 0.535385 0.267692 0.963504i \(-0.413739\pi\)
0.267692 + 0.963504i \(0.413739\pi\)
\(332\) 13.0888i 0.718342i
\(333\) 0 0
\(334\) 8.62068i 0.471702i
\(335\) 0 0
\(336\) 0 0
\(337\) 3.15078i 0.171634i −0.996311 0.0858169i \(-0.972650\pi\)
0.996311 0.0858169i \(-0.0273500\pi\)
\(338\) −29.8293 −1.62250
\(339\) 0 0
\(340\) 0 0
\(341\) −8.82412 −0.477853
\(342\) 0 0
\(343\) −18.5033 0.791511i −0.999086 0.0427376i
\(344\) 0.468142i 0.0252405i
\(345\) 0 0
\(346\) 10.6517i 0.572637i
\(347\) −30.0594 −1.61367 −0.806836 0.590775i \(-0.798822\pi\)
−0.806836 + 0.590775i \(0.798822\pi\)
\(348\) 0 0
\(349\) 27.5180i 1.47301i 0.676434 + 0.736503i \(0.263524\pi\)
−0.676434 + 0.736503i \(0.736476\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.71078i 0.144485i
\(353\) 16.5956i 0.883293i −0.897189 0.441647i \(-0.854395\pi\)
0.897189 0.441647i \(-0.145605\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.60812 −0.509230
\(357\) 0 0
\(358\) 23.1961i 1.22595i
\(359\) 15.0076i 0.792073i 0.918235 + 0.396036i \(0.129615\pi\)
−0.918235 + 0.396036i \(0.870385\pi\)
\(360\) 0 0
\(361\) 13.7019 0.721151
\(362\) 9.11980i 0.479326i
\(363\) 0 0
\(364\) −8.87024 + 14.8702i −0.464927 + 0.779412i
\(365\) 0 0
\(366\) 0 0
\(367\) 14.2598 0.744354 0.372177 0.928162i \(-0.378611\pi\)
0.372177 + 0.928162i \(0.378611\pi\)
\(368\) 3.83363 0.199842
\(369\) 0 0
\(370\) 0 0
\(371\) 12.5444 21.0297i 0.651273 1.09181i
\(372\) 0 0
\(373\) 8.98745i 0.465352i 0.972554 + 0.232676i \(0.0747482\pi\)
−0.972554 + 0.232676i \(0.925252\pi\)
\(374\) 4.15253 0.214722
\(375\) 0 0
\(376\) 9.11980i 0.470318i
\(377\) 25.0888i 1.29214i
\(378\) 0 0
\(379\) 34.5447 1.77444 0.887220 0.461346i \(-0.152633\pi\)
0.887220 + 0.461346i \(0.152633\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.69823i 0.496205i
\(383\) 2.88020i 0.147171i −0.997289 0.0735857i \(-0.976556\pi\)
0.997289 0.0735857i \(-0.0234443\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.1525i 0.923940i
\(387\) 0 0
\(388\) −16.9224 −0.859107
\(389\) 35.1620i 1.78279i −0.453231 0.891393i \(-0.649729\pi\)
0.453231 0.891393i \(-0.350271\pi\)
\(390\) 0 0
\(391\) 5.87257i 0.296989i
\(392\) 3.32583 + 6.15945i 0.167980 + 0.311099i
\(393\) 0 0
\(394\) −4.60354 −0.231923
\(395\) 0 0
\(396\) 0 0
\(397\) 16.1185 0.808965 0.404482 0.914546i \(-0.367452\pi\)
0.404482 + 0.914546i \(0.367452\pi\)
\(398\) 11.7405i 0.588497i
\(399\) 0 0
\(400\) 0 0
\(401\) 16.6256i 0.830243i −0.909766 0.415121i \(-0.863739\pi\)
0.909766 0.415121i \(-0.136261\pi\)
\(402\) 0 0
\(403\) 21.3033i 1.06119i
\(404\) −17.7405 −0.882622
\(405\) 0 0
\(406\) −8.71078 5.19606i −0.432309 0.257876i
\(407\) 8.16637 0.404792
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.05913 −0.199979
\(413\) 15.1751 25.4397i 0.746715 1.25181i
\(414\) 0 0
\(415\) 0 0
\(416\) 6.54441 0.320866
\(417\) 0 0
\(418\) −6.23960 −0.305189
\(419\) −36.2849 −1.77263 −0.886316 0.463080i \(-0.846744\pi\)
−0.886316 + 0.463080i \(0.846744\pi\)
\(420\) 0 0
\(421\) 19.2414 0.937766 0.468883 0.883260i \(-0.344657\pi\)
0.468883 + 0.883260i \(0.344657\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −9.25519 −0.449472
\(425\) 0 0
\(426\) 0 0
\(427\) −10.8772 6.48833i −0.526383 0.313992i
\(428\) 6.31891 0.305436
\(429\) 0 0
\(430\) 0 0
\(431\) 17.2974i 0.833188i 0.909093 + 0.416594i \(0.136776\pi\)
−0.909093 + 0.416594i \(0.863224\pi\)
\(432\) 0 0
\(433\) 31.6473 1.52087 0.760437 0.649412i \(-0.224985\pi\)
0.760437 + 0.649412i \(0.224985\pi\)
\(434\) −7.39646 4.41206i −0.355042 0.211786i
\(435\) 0 0
\(436\) 12.0251 0.575898
\(437\) 8.82412i 0.422115i
\(438\) 0 0
\(439\) 13.0095i 0.620910i −0.950588 0.310455i \(-0.899519\pi\)
0.950588 0.310455i \(-0.100481\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.0251i 0.476846i
\(443\) −3.78551 −0.179855 −0.0899275 0.995948i \(-0.528664\pi\)
−0.0899275 + 0.995948i \(0.528664\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −26.4513 −1.25250
\(447\) 0 0
\(448\) 1.35539 2.27220i 0.0640362 0.107352i
\(449\) 24.9307i 1.17655i −0.808661 0.588275i \(-0.799807\pi\)
0.808661 0.588275i \(-0.200193\pi\)
\(450\) 0 0
\(451\) 17.7405i 0.835366i
\(452\) −9.06372 −0.426321
\(453\) 0 0
\(454\) 15.0637i 0.706975i
\(455\) 0 0
\(456\) 0 0
\(457\) 31.3535i 1.46666i 0.679875 + 0.733328i \(0.262034\pi\)
−0.679875 + 0.733328i \(0.737966\pi\)
\(458\) 11.3655i 0.531074i
\(459\) 0 0
\(460\) 0 0
\(461\) −9.52598 −0.443669 −0.221835 0.975084i \(-0.571205\pi\)
−0.221835 + 0.975084i \(0.571205\pi\)
\(462\) 0 0
\(463\) 34.0003i 1.58013i 0.613026 + 0.790063i \(0.289952\pi\)
−0.613026 + 0.790063i \(0.710048\pi\)
\(464\) 3.83363i 0.177972i
\(465\) 0 0
\(466\) −20.9957 −0.972605
\(467\) 39.2665i 1.81703i 0.417847 + 0.908517i \(0.362785\pi\)
−0.417847 + 0.908517i \(0.637215\pi\)
\(468\) 0 0
\(469\) −30.8042 18.3750i −1.42241 0.848478i
\(470\) 0 0
\(471\) 0 0
\(472\) −11.1961 −0.515341
\(473\) 1.26903 0.0583502
\(474\) 0 0
\(475\) 0 0
\(476\) 3.48069 + 2.07627i 0.159537 + 0.0951655i
\(477\) 0 0
\(478\) 9.69823i 0.443587i
\(479\) 34.0251 1.55465 0.777323 0.629101i \(-0.216577\pi\)
0.777323 + 0.629101i \(0.216577\pi\)
\(480\) 0 0
\(481\) 19.7154i 0.898944i
\(482\) 24.7019i 1.12514i
\(483\) 0 0
\(484\) 3.65166 0.165984
\(485\) 0 0
\(486\) 0 0
\(487\) 12.8091i 0.580436i 0.956961 + 0.290218i \(0.0937278\pi\)
−0.956961 + 0.290218i \(0.906272\pi\)
\(488\) 4.78705i 0.216700i
\(489\) 0 0
\(490\) 0 0
\(491\) 11.6471i 0.525625i 0.964847 + 0.262812i \(0.0846500\pi\)
−0.964847 + 0.262812i \(0.915350\pi\)
\(492\) 0 0
\(493\) −5.87257 −0.264487
\(494\) 15.0637i 0.677749i
\(495\) 0 0
\(496\) 3.25519i 0.146162i
\(497\) 5.23009 + 3.11980i 0.234602 + 0.139942i
\(498\) 0 0
\(499\) −35.5511 −1.59149 −0.795743 0.605635i \(-0.792919\pi\)
−0.795743 + 0.605635i \(0.792919\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.60812 0.161038
\(503\) 8.23372i 0.367123i −0.983008 0.183562i \(-0.941237\pi\)
0.983008 0.183562i \(-0.0587627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.3921i 0.461986i
\(507\) 0 0
\(508\) 21.6332i 0.959819i
\(509\) 31.0888 1.37799 0.688994 0.724767i \(-0.258053\pi\)
0.688994 + 0.724767i \(0.258053\pi\)
\(510\) 0 0
\(511\) 15.4807 25.9521i 0.684826 1.14805i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 28.7983i 1.27024i
\(515\) 0 0
\(516\) 0 0
\(517\) 24.7218 1.08726
\(518\) 6.84513 + 4.08319i 0.300758 + 0.179405i
\(519\) 0 0
\(520\) 0 0
\(521\) −1.39363 −0.0610561 −0.0305281 0.999534i \(-0.509719\pi\)
−0.0305281 + 0.999534i \(0.509719\pi\)
\(522\) 0 0
\(523\) 20.4853 0.895759 0.447879 0.894094i \(-0.352179\pi\)
0.447879 + 0.894094i \(0.352179\pi\)
\(524\) 2.51931 0.110056
\(525\) 0 0
\(526\) −12.7699 −0.556795
\(527\) −4.98649 −0.217215
\(528\) 0 0
\(529\) −8.30331 −0.361013
\(530\) 0 0
\(531\) 0 0
\(532\) −5.23009 3.11980i −0.226753 0.135260i
\(533\) −42.8293 −1.85514
\(534\) 0 0
\(535\) 0 0
\(536\) 13.5570i 0.585572i
\(537\) 0 0
\(538\) 8.43716 0.363752
\(539\) 16.6969 9.01560i 0.719188 0.388329i
\(540\) 0 0
\(541\) −0.0870615 −0.00374306 −0.00187153 0.999998i \(-0.500596\pi\)
−0.00187153 + 0.999998i \(0.500596\pi\)
\(542\) 2.16637i 0.0930537i
\(543\) 0 0
\(544\) 1.53186i 0.0656779i
\(545\) 0 0
\(546\) 0 0
\(547\) 5.40443i 0.231077i 0.993303 + 0.115538i \(0.0368593\pi\)
−0.993303 + 0.115538i \(0.963141\pi\)
\(548\) −1.39646 −0.0596539
\(549\) 0 0
\(550\) 0 0
\(551\) 8.82412 0.375920
\(552\) 0 0
\(553\) 17.1820 28.8042i 0.730652 1.22488i
\(554\) 5.92373i 0.251675i
\(555\) 0 0
\(556\) 6.13539i 0.260199i
\(557\) −0.127431 −0.00539940 −0.00269970 0.999996i \(-0.500859\pi\)
−0.00269970 + 0.999996i \(0.500859\pi\)
\(558\) 0 0
\(559\) 3.06372i 0.129581i
\(560\) 0 0
\(561\) 0 0
\(562\) 27.3566i 1.15397i
\(563\) 6.23960i 0.262968i 0.991318 + 0.131484i \(0.0419741\pi\)
−0.991318 + 0.131484i \(0.958026\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 18.0594 0.759092
\(567\) 0 0
\(568\) 2.30177i 0.0965801i
\(569\) 11.6391i 0.487937i 0.969783 + 0.243968i \(0.0784493\pi\)
−0.969783 + 0.243968i \(0.921551\pi\)
\(570\) 0 0
\(571\) −35.9181 −1.50313 −0.751563 0.659661i \(-0.770700\pi\)
−0.751563 + 0.659661i \(0.770700\pi\)
\(572\) 17.7405i 0.741766i
\(573\) 0 0
\(574\) −8.87024 + 14.8702i −0.370237 + 0.620672i
\(575\) 0 0
\(576\) 0 0
\(577\) 7.15687 0.297944 0.148972 0.988841i \(-0.452404\pi\)
0.148972 + 0.988841i \(0.452404\pi\)
\(578\) −14.6534 −0.609502
\(579\) 0 0
\(580\) 0 0
\(581\) −29.7405 17.7405i −1.23384 0.735999i
\(582\) 0 0
\(583\) 25.0888i 1.03907i
\(584\) −11.4216 −0.472628
\(585\) 0 0
\(586\) 29.1139i 1.20269i
\(587\) 4.15253i 0.171393i −0.996321 0.0856967i \(-0.972688\pi\)
0.996321 0.0856967i \(-0.0273116\pi\)
\(588\) 0 0
\(589\) 7.49270 0.308731
\(590\) 0 0
\(591\) 0 0
\(592\) 3.01255i 0.123815i
\(593\) 29.7966i 1.22360i −0.791013 0.611799i \(-0.790446\pi\)
0.791013 0.611799i \(-0.209554\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.65166i 0.190539i
\(597\) 0 0
\(598\) 25.0888 1.02596
\(599\) 35.4249i 1.44742i −0.690104 0.723710i \(-0.742435\pi\)
0.690104 0.723710i \(-0.257565\pi\)
\(600\) 0 0
\(601\) 27.4948i 1.12154i −0.827973 0.560768i \(-0.810506\pi\)
0.827973 0.560768i \(-0.189494\pi\)
\(602\) 1.06372 + 0.634516i 0.0433538 + 0.0258609i
\(603\) 0 0
\(604\) −3.58794 −0.145991
\(605\) 0 0
\(606\) 0 0
\(607\) 4.76499 0.193405 0.0967025 0.995313i \(-0.469170\pi\)
0.0967025 + 0.995313i \(0.469170\pi\)
\(608\) 2.30177i 0.0933490i
\(609\) 0 0
\(610\) 0 0
\(611\) 59.6837i 2.41454i
\(612\) 0 0
\(613\) 10.7981i 0.436130i 0.975934 + 0.218065i \(0.0699744\pi\)
−0.975934 + 0.218065i \(0.930026\pi\)
\(614\) 7.39646 0.298497
\(615\) 0 0
\(616\) −6.15945 3.67417i −0.248171 0.148037i
\(617\) −34.9025 −1.40512 −0.702561 0.711623i \(-0.747960\pi\)
−0.702561 + 0.711623i \(0.747960\pi\)
\(618\) 0 0
\(619\) 1.26107i 0.0506866i −0.999679 0.0253433i \(-0.991932\pi\)
0.999679 0.0253433i \(-0.00806789\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.97490 0.0791861
\(623\) −13.0228 + 21.8316i −0.521746 + 0.874666i
\(624\) 0 0
\(625\) 0 0
\(626\) −0.961388 −0.0384248
\(627\) 0 0
\(628\) −13.1228 −0.523658
\(629\) 4.61480 0.184004
\(630\) 0 0
\(631\) 16.2145 0.645489 0.322744 0.946486i \(-0.395395\pi\)
0.322744 + 0.946486i \(0.395395\pi\)
\(632\) −12.6768 −0.504254
\(633\) 0 0
\(634\) −17.1620 −0.681592
\(635\) 0 0
\(636\) 0 0
\(637\) 21.7656 + 40.3100i 0.862384 + 1.59714i
\(638\) 10.3921 0.411428
\(639\) 0 0
\(640\) 0 0
\(641\) 19.6893i 0.777681i −0.921305 0.388840i \(-0.872876\pi\)
0.921305 0.388840i \(-0.127124\pi\)
\(642\) 0 0
\(643\) −17.1508 −0.676361 −0.338180 0.941081i \(-0.609811\pi\)
−0.338180 + 0.941081i \(0.609811\pi\)
\(644\) 5.19606 8.71078i 0.204754 0.343253i
\(645\) 0 0
\(646\) −3.52598 −0.138728
\(647\) 29.0578i 1.14238i 0.820817 + 0.571191i \(0.193518\pi\)
−0.820817 + 0.571191i \(0.806482\pi\)
\(648\) 0 0
\(649\) 30.3501i 1.19135i
\(650\) 0 0
\(651\) 0 0
\(652\) 16.6207i 0.650916i
\(653\) 9.11183 0.356574 0.178287 0.983979i \(-0.442945\pi\)
0.178287 + 0.983979i \(0.442945\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.54441 0.255516
\(657\) 0 0
\(658\) 20.7220 + 12.3609i 0.807829 + 0.481878i
\(659\) 37.5539i 1.46289i 0.681899 + 0.731446i \(0.261154\pi\)
−0.681899 + 0.731446i \(0.738846\pi\)
\(660\) 0 0
\(661\) 9.50275i 0.369614i 0.982775 + 0.184807i \(0.0591660\pi\)
−0.982775 + 0.184807i \(0.940834\pi\)
\(662\) −9.74047 −0.378574
\(663\) 0 0
\(664\) 13.0888i 0.507945i
\(665\) 0 0
\(666\) 0 0
\(667\) 14.6967i 0.569058i
\(668\) 8.62068i 0.333544i
\(669\) 0 0
\(670\) 0 0
\(671\) 12.9767 0.500958
\(672\) 0 0
\(673\) 49.6335i 1.91323i 0.291355 + 0.956615i \(0.405894\pi\)
−0.291355 + 0.956615i \(0.594106\pi\)
\(674\) 3.15078i 0.121363i
\(675\) 0 0
\(676\) 29.8293 1.14728
\(677\) 37.5312i 1.44244i 0.692706 + 0.721220i \(0.256418\pi\)
−0.692706 + 0.721220i \(0.743582\pi\)
\(678\) 0 0
\(679\) −22.9365 + 38.4513i −0.880224 + 1.47562i
\(680\) 0 0
\(681\) 0 0
\(682\) 8.82412 0.337893
\(683\) 9.01560 0.344972 0.172486 0.985012i \(-0.444820\pi\)
0.172486 + 0.985012i \(0.444820\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18.5033 + 0.791511i 0.706461 + 0.0302200i
\(687\) 0 0
\(688\) 0.468142i 0.0178478i
\(689\) −60.5698 −2.30752
\(690\) 0 0
\(691\) 8.15841i 0.310361i −0.987886 0.155180i \(-0.950404\pi\)
0.987886 0.155180i \(-0.0495958\pi\)
\(692\) 10.6517i 0.404915i
\(693\) 0 0
\(694\) 30.0594 1.14104
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0251i 0.379728i
\(698\) 27.5180i 1.04157i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.3440i 1.29716i 0.761148 + 0.648578i \(0.224636\pi\)
−0.761148 + 0.648578i \(0.775364\pi\)
\(702\) 0 0
\(703\) −6.93420 −0.261528
\(704\) 2.71078i 0.102166i
\(705\) 0 0
\(706\) 16.5956i 0.624583i
\(707\) −24.0453 + 40.3100i −0.904316 + 1.51601i
\(708\) 0 0
\(709\) 5.06372 0.190172 0.0950859 0.995469i \(-0.469687\pi\)
0.0950859 + 0.995469i \(0.469687\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.60812 0.360080
\(713\) 12.4792i 0.467349i
\(714\) 0 0
\(715\) 0 0
\(716\) 23.1961i 0.866878i
\(717\) 0 0
\(718\) 15.0076i 0.560080i
\(719\) 18.1274 0.676039 0.338020 0.941139i \(-0.390243\pi\)
0.338020 + 0.941139i \(0.390243\pi\)
\(720\) 0 0
\(721\) −5.50171 + 9.22317i −0.204894 + 0.343489i
\(722\) −13.7019 −0.509930
\(723\) 0 0
\(724\) 9.11980i 0.338935i
\(725\) 0 0
\(726\) 0 0
\(727\) −11.1168 −0.412298 −0.206149 0.978521i \(-0.566093\pi\)
−0.206149 + 0.978521i \(0.566093\pi\)
\(728\) 8.87024 14.8702i 0.328753 0.551128i
\(729\) 0 0
\(730\) 0 0
\(731\) 0.717127 0.0265239
\(732\) 0 0
\(733\) −20.8342 −0.769529 −0.384765 0.923015i \(-0.625717\pi\)
−0.384765 + 0.923015i \(0.625717\pi\)
\(734\) −14.2598 −0.526338
\(735\) 0 0
\(736\) −3.83363 −0.141309
\(737\) 36.7500 1.35370
\(738\) 0 0
\(739\) −30.9818 −1.13968 −0.569842 0.821754i \(-0.692996\pi\)
−0.569842 + 0.821754i \(0.692996\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.5444 + 21.0297i −0.460520 + 0.772024i
\(743\) −45.8449 −1.68189 −0.840943 0.541124i \(-0.817999\pi\)
−0.840943 + 0.541124i \(0.817999\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8.98745i 0.329054i
\(747\) 0 0
\(748\) −4.15253 −0.151832
\(749\) 8.56459 14.3579i 0.312943 0.524624i
\(750\) 0 0
\(751\) −35.2665 −1.28689 −0.643446 0.765492i \(-0.722496\pi\)
−0.643446 + 0.765492i \(0.722496\pi\)
\(752\) 9.11980i 0.332565i
\(753\) 0 0
\(754\) 25.0888i 0.913681i
\(755\) 0 0
\(756\) 0 0
\(757\) 12.3661i 0.449452i 0.974422 + 0.224726i \(0.0721488\pi\)
−0.974422 + 0.224726i \(0.927851\pi\)
\(758\) −34.5447 −1.25472
\(759\) 0 0
\(760\) 0 0
\(761\) 1.50580 0.0545851 0.0272926 0.999627i \(-0.491311\pi\)
0.0272926 + 0.999627i \(0.491311\pi\)
\(762\) 0 0
\(763\) 16.2987 27.3235i 0.590053 0.989177i
\(764\) 9.69823i 0.350870i
\(765\) 0 0
\(766\) 2.88020i 0.104066i
\(767\) −73.2716 −2.64569
\(768\) 0 0
\(769\) 3.76558i 0.135790i 0.997692 + 0.0678951i \(0.0216283\pi\)
−0.997692 + 0.0678951i \(0.978372\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.1525i 0.653324i
\(773\) 15.1911i 0.546388i 0.961959 + 0.273194i \(0.0880800\pi\)
−0.961959 + 0.273194i \(0.911920\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 16.9224 0.607480
\(777\) 0 0
\(778\) 35.1620i 1.26062i
\(779\) 15.0637i 0.539714i
\(780\) 0 0
\(781\) −6.23960 −0.223270
\(782\) 5.87257i 0.210003i
\(783\) 0 0
\(784\) −3.32583 6.15945i −0.118780 0.219980i
\(785\) 0 0
\(786\) 0 0
\(787\) 23.8198 0.849084 0.424542 0.905408i \(-0.360435\pi\)
0.424542 + 0.905408i \(0.360435\pi\)
\(788\) 4.60354 0.163994
\(789\) 0 0
\(790\) 0 0
\(791\) −12.2849 + 20.5946i −0.436800 + 0.732260i
\(792\) 0 0
\(793\) 31.3284i 1.11250i
\(794\) −16.1185 −0.572024
\(795\) 0 0
\(796\) 11.7405i 0.416130i
\(797\) 10.6517i 0.377301i 0.982044 + 0.188650i \(0.0604113\pi\)
−0.982044 + 0.188650i \(0.939589\pi\)
\(798\) 0 0
\(799\) 13.9702 0.494231
\(800\) 0 0
\(801\) 0 0
\(802\) 16.6256i 0.587070i
\(803\) 30.9614i 1.09260i
\(804\) 0 0
\(805\) 0 0
\(806\) 21.3033i 0.750377i
\(807\) 0 0
\(808\) 17.7405 0.624108
\(809\) 32.8462i 1.15481i −0.816458 0.577405i \(-0.804065\pi\)
0.816458 0.577405i \(-0.195935\pi\)
\(810\) 0 0
\(811\) 26.0734i 0.915562i −0.889065 0.457781i \(-0.848644\pi\)
0.889065 0.457781i \(-0.151356\pi\)
\(812\) 8.71078 + 5.19606i 0.305689 + 0.182346i
\(813\) 0 0
\(814\) −8.16637 −0.286231
\(815\) 0 0
\(816\) 0 0
\(817\) −1.07756 −0.0376989
\(818\) 12.0000i 0.419570i
\(819\) 0 0
\(820\) 0 0
\(821\) 14.1352i 0.493321i 0.969102 + 0.246661i \(0.0793333\pi\)
−0.969102 + 0.246661i \(0.920667\pi\)
\(822\) 0 0
\(823\) 16.9114i 0.589496i −0.955575 0.294748i \(-0.904764\pi\)
0.955575 0.294748i \(-0.0952356\pi\)
\(824\) 4.05913 0.141406
\(825\) 0 0
\(826\) −15.1751 + 25.4397i −0.528008 + 0.885162i
\(827\) −34.2959 −1.19259 −0.596293 0.802767i \(-0.703360\pi\)
−0.596293 + 0.802767i \(0.703360\pi\)
\(828\) 0 0
\(829\) 53.4408i 1.85608i −0.372486 0.928038i \(-0.621495\pi\)
0.372486 0.928038i \(-0.378505\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −6.54441 −0.226887
\(833\) 9.43541 5.09469i 0.326917 0.176521i
\(834\) 0 0
\(835\) 0 0
\(836\) 6.23960 0.215801
\(837\) 0 0
\(838\) 36.2849 1.25344
\(839\) −14.2898 −0.493339 −0.246669 0.969100i \(-0.579336\pi\)
−0.246669 + 0.969100i \(0.579336\pi\)
\(840\) 0 0
\(841\) 14.3033 0.493218
\(842\) −19.2414 −0.663101
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 4.94942 8.29731i 0.170064 0.285099i
\(848\) 9.25519 0.317825
\(849\) 0 0
\(850\) 0 0
\(851\) 11.5490i 0.395895i
\(852\) 0 0
\(853\) −47.0118 −1.60966 −0.804828 0.593509i \(-0.797742\pi\)
−0.804828 + 0.593509i \(0.797742\pi\)
\(854\) 10.8772 + 6.48833i 0.372209 + 0.222026i
\(855\) 0 0
\(856\) −6.31891 −0.215976
\(857\) 7.65929i 0.261636i −0.991406 0.130818i \(-0.958240\pi\)
0.991406 0.130818i \(-0.0417604\pi\)
\(858\) 0 0
\(859\) 47.8559i 1.63282i 0.577470 + 0.816412i \(0.304040\pi\)
−0.577470 + 0.816412i \(0.695960\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 17.2974i 0.589153i
\(863\) 41.4922 1.41241 0.706206 0.708007i \(-0.250405\pi\)
0.706206 + 0.708007i \(0.250405\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −31.6473 −1.07542
\(867\) 0 0
\(868\) 7.39646 + 4.41206i 0.251052 + 0.149755i
\(869\) 34.3639i 1.16572i
\(870\) 0 0
\(871\) 88.7223i 3.00624i
\(872\) −12.0251 −0.407221
\(873\) 0 0
\(874\) 8.82412i 0.298480i
\(875\) 0 0
\(876\) 0 0
\(877\) 17.2925i 0.583927i −0.956429 0.291963i \(-0.905691\pi\)
0.956429 0.291963i \(-0.0943085\pi\)
\(878\) 13.0095i 0.439050i
\(879\) 0 0
\(880\) 0 0
\(881\) 31.7454 1.06953 0.534765 0.845001i \(-0.320400\pi\)
0.534765 + 0.845001i \(0.320400\pi\)
\(882\) 0 0
\(883\) 2.07601i 0.0698634i −0.999390 0.0349317i \(-0.988879\pi\)
0.999390 0.0349317i \(-0.0111214\pi\)
\(884\) 10.0251i 0.337181i
\(885\) 0 0
\(886\) 3.78551 0.127177
\(887\) 5.18994i 0.174261i −0.996197 0.0871305i \(-0.972230\pi\)
0.996197 0.0871305i \(-0.0277697\pi\)
\(888\) 0 0
\(889\) −49.1551 29.3215i −1.64861 0.983411i
\(890\) 0 0
\(891\) 0 0
\(892\) 26.4513 0.885654
\(893\) −20.9917 −0.702459
\(894\) 0 0
\(895\) 0 0
\(896\) −1.35539 + 2.27220i −0.0452805 + 0.0759090i
\(897\) 0 0
\(898\) 24.9307i 0.831947i
\(899\) −12.4792 −0.416204
\(900\) 0 0
\(901\) 14.1776i 0.472326i
\(902\) 17.7405i 0.590693i
\(903\) 0 0
\(904\) 9.06372 0.301455
\(905\) 0 0
\(906\) 0 0
\(907\) 32.9473i 1.09400i −0.837133 0.546999i \(-0.815770\pi\)
0.837133 0.546999i \(-0.184230\pi\)
\(908\) 15.0637i 0.499907i
\(909\) 0 0
\(910\) 0 0
\(911\) 3.02356i 0.100175i 0.998745 + 0.0500875i \(0.0159500\pi\)
−0.998745 + 0.0500875i \(0.984050\pi\)
\(912\) 0 0
\(913\) 35.4809 1.17425
\(914\) 31.3535i 1.03708i
\(915\) 0 0
\(916\) 11.3655i 0.375526i
\(917\) 3.41465 5.72438i 0.112762 0.189036i
\(918\) 0 0
\(919\) −13.9129 −0.458945 −0.229473 0.973315i \(-0.573700\pi\)
−0.229473 + 0.973315i \(0.573700\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 9.52598 0.313721
\(923\) 15.0637i 0.495828i
\(924\) 0 0
\(925\) 0 0
\(926\) 34.0003i 1.11732i
\(927\) 0 0
\(928\) 3.83363i 0.125845i
\(929\) −8.15228 −0.267468 −0.133734 0.991017i \(-0.542697\pi\)
−0.133734 + 0.991017i \(0.542697\pi\)
\(930\) 0 0
\(931\) −14.1776 + 7.65529i −0.464653 + 0.250892i
\(932\) 20.9957 0.687736
\(933\) 0 0
\(934\) 39.2665i 1.28484i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.22576 0.0727122 0.0363561 0.999339i \(-0.488425\pi\)
0.0363561 + 0.999339i \(0.488425\pi\)
\(938\) 30.8042 + 18.3750i 1.00579 + 0.599965i
\(939\) 0 0
\(940\) 0 0
\(941\) 40.6517 1.32521 0.662603 0.748971i \(-0.269452\pi\)
0.662603 + 0.748971i \(0.269452\pi\)
\(942\) 0 0
\(943\) 25.0888 0.817004
\(944\) 11.1961 0.364401
\(945\) 0 0
\(946\) −1.26903 −0.0412598
\(947\) −34.4874 −1.12069 −0.560344 0.828260i \(-0.689331\pi\)
−0.560344 + 0.828260i \(0.689331\pi\)
\(948\) 0 0
\(949\) −74.7474 −2.42640
\(950\) 0 0
\(951\) 0 0
\(952\) −3.48069 2.07627i −0.112810 0.0672922i
\(953\) −23.9688 −0.776426 −0.388213 0.921570i \(-0.626907\pi\)
−0.388213 + 0.921570i \(0.626907\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9.69823i 0.313663i
\(957\) 0 0
\(958\) −34.0251 −1.09930
\(959\) −1.89275 + 3.17305i −0.0611202 + 0.102463i
\(960\) 0 0
\(961\) 20.4037 0.658185
\(962\) 19.7154i 0.635649i
\(963\) 0 0
\(964\) 24.7019i 0.795593i
\(965\) 0 0
\(966\) 0 0
\(967\) 6.45735i 0.207654i −0.994595 0.103827i \(-0.966891\pi\)
0.994595 0.103827i \(-0.0331089\pi\)
\(968\) −3.65166 −0.117369
\(969\) 0 0
\(970\) 0 0
\(971\) 48.7471 1.56437 0.782185 0.623046i \(-0.214105\pi\)
0.782185 + 0.623046i \(0.214105\pi\)
\(972\) 0 0
\(973\) −13.9409 8.31586i −0.446924 0.266594i
\(974\) 12.8091i 0.410430i
\(975\) 0 0
\(976\) 4.78705i 0.153230i
\(977\) 14.4853 0.463425 0.231713 0.972784i \(-0.425567\pi\)
0.231713 + 0.972784i \(0.425567\pi\)
\(978\) 0 0
\(979\) 26.0455i 0.832419i
\(980\) 0 0
\(981\) 0 0
\(982\) 11.6471i 0.371673i
\(983\) 46.7784i 1.49200i 0.665947 + 0.745999i \(0.268028\pi\)
−0.665947 + 0.745999i \(0.731972\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 5.87257 0.187021
\(987\) 0 0
\(988\) 15.0637i 0.479241i
\(989\) 1.79468i 0.0570676i
\(990\) 0 0
\(991\) −37.1292 −1.17945 −0.589724 0.807605i \(-0.700763\pi\)
−0.589724 + 0.807605i \(0.700763\pi\)
\(992\) 3.25519i 0.103352i
\(993\) 0 0
\(994\) −5.23009 3.11980i −0.165888 0.0989540i
\(995\) 0 0
\(996\) 0 0
\(997\) 35.5309 1.12527 0.562637 0.826704i \(-0.309787\pi\)
0.562637 + 0.826704i \(0.309787\pi\)
\(998\) 35.5511 1.12535
\(999\) 0 0
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 3150.2.d.a.3149.5 8
3.2 odd 2 3150.2.d.d.3149.5 8
5.2 odd 4 630.2.b.b.251.4 yes 8
5.3 odd 4 3150.2.b.f.251.5 8
5.4 even 2 3150.2.d.f.3149.4 8
7.6 odd 2 3150.2.d.c.3149.3 8
15.2 even 4 630.2.b.a.251.8 yes 8
15.8 even 4 3150.2.b.e.251.1 8
15.14 odd 2 3150.2.d.c.3149.4 8
20.7 even 4 5040.2.f.i.881.1 8
21.20 even 2 3150.2.d.f.3149.3 8
35.13 even 4 3150.2.b.e.251.5 8
35.27 even 4 630.2.b.a.251.4 8
35.34 odd 2 3150.2.d.d.3149.6 8
60.47 odd 4 5040.2.f.f.881.1 8
105.62 odd 4 630.2.b.b.251.8 yes 8
105.83 odd 4 3150.2.b.f.251.1 8
105.104 even 2 inner 3150.2.d.a.3149.6 8
140.27 odd 4 5040.2.f.f.881.2 8
420.167 even 4 5040.2.f.i.881.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.b.a.251.4 8 35.27 even 4
630.2.b.a.251.8 yes 8 15.2 even 4
630.2.b.b.251.4 yes 8 5.2 odd 4
630.2.b.b.251.8 yes 8 105.62 odd 4
3150.2.b.e.251.1 8 15.8 even 4
3150.2.b.e.251.5 8 35.13 even 4
3150.2.b.f.251.1 8 105.83 odd 4
3150.2.b.f.251.5 8 5.3 odd 4
3150.2.d.a.3149.5 8 1.1 even 1 trivial
3150.2.d.a.3149.6 8 105.104 even 2 inner
3150.2.d.c.3149.3 8 7.6 odd 2
3150.2.d.c.3149.4 8 15.14 odd 2
3150.2.d.d.3149.5 8 3.2 odd 2
3150.2.d.d.3149.6 8 35.34 odd 2
3150.2.d.f.3149.3 8 21.20 even 2
3150.2.d.f.3149.4 8 5.4 even 2
5040.2.f.f.881.1 8 60.47 odd 4
5040.2.f.f.881.2 8 140.27 odd 4
5040.2.f.i.881.1 8 20.7 even 4
5040.2.f.i.881.2 8 420.167 even 4