Properties

Label 1425.2.c.j.799.4
Level $1425$
Weight $2$
Character 1425.799
Analytic conductor $11.379$
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] = [1425,2,Mod(799,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.799");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.c (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: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.j.799.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} +3.41421i q^{7} -4.41421i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} +3.41421i q^{7} -4.41421i q^{8} -1.00000 q^{9} -1.41421 q^{11} -3.82843i q^{12} -2.58579i q^{13} -8.24264 q^{14} +3.00000 q^{16} -6.82843i q^{17} -2.41421i q^{18} -1.00000 q^{19} -3.41421 q^{21} -3.41421i q^{22} +3.65685i q^{23} +4.41421 q^{24} +6.24264 q^{26} -1.00000i q^{27} -13.0711i q^{28} -5.07107 q^{29} -10.4853 q^{31} -1.58579i q^{32} -1.41421i q^{33} +16.4853 q^{34} +3.82843 q^{36} -3.07107i q^{37} -2.41421i q^{38} +2.58579 q^{39} -4.58579 q^{41} -8.24264i q^{42} -3.41421i q^{43} +5.41421 q^{44} -8.82843 q^{46} +11.6569i q^{47} +3.00000i q^{48} -4.65685 q^{49} +6.82843 q^{51} +9.89949i q^{52} -4.00000i q^{53} +2.41421 q^{54} +15.0711 q^{56} -1.00000i q^{57} -12.2426i q^{58} +8.48528 q^{59} -5.65685 q^{61} -25.3137i q^{62} -3.41421i q^{63} +9.82843 q^{64} +3.41421 q^{66} +12.0000i q^{67} +26.1421i q^{68} -3.65685 q^{69} +12.4853 q^{71} +4.41421i q^{72} +2.00000i q^{73} +7.41421 q^{74} +3.82843 q^{76} -4.82843i q^{77} +6.24264i q^{78} -11.3137 q^{79} +1.00000 q^{81} -11.0711i q^{82} -6.48528i q^{83} +13.0711 q^{84} +8.24264 q^{86} -5.07107i q^{87} +6.24264i q^{88} +14.7279 q^{89} +8.82843 q^{91} -14.0000i q^{92} -10.4853i q^{93} -28.1421 q^{94} +1.58579 q^{96} +4.24264i q^{97} -11.2426i q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 16 q^{14} + 12 q^{16} - 4 q^{19} - 8 q^{21} + 12 q^{24} + 8 q^{26} + 8 q^{29} - 8 q^{31} + 32 q^{34} + 4 q^{36} + 16 q^{39} - 24 q^{41} + 16 q^{44} - 24 q^{46} + 4 q^{49} + 16 q^{51} + 4 q^{54} + 32 q^{56} + 28 q^{64} + 8 q^{66} + 8 q^{69} + 16 q^{71} + 24 q^{74} + 4 q^{76} + 4 q^{81} + 24 q^{84} + 16 q^{86} + 8 q^{89} + 24 q^{91} - 56 q^{94} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421i 1.70711i 0.521005 + 0.853553i \(0.325557\pi\)
−0.521005 + 0.853553i \(0.674443\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) 3.41421i 1.29045i 0.763992 + 0.645226i \(0.223237\pi\)
−0.763992 + 0.645226i \(0.776763\pi\)
\(8\) − 4.41421i − 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) − 3.82843i − 1.10517i
\(13\) − 2.58579i − 0.717168i −0.933497 0.358584i \(-0.883260\pi\)
0.933497 0.358584i \(-0.116740\pi\)
\(14\) −8.24264 −2.20294
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 6.82843i − 1.65614i −0.560627 0.828068i \(-0.689440\pi\)
0.560627 0.828068i \(-0.310560\pi\)
\(18\) − 2.41421i − 0.569036i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.41421 −0.745042
\(22\) − 3.41421i − 0.727913i
\(23\) 3.65685i 0.762507i 0.924471 + 0.381253i \(0.124507\pi\)
−0.924471 + 0.381253i \(0.875493\pi\)
\(24\) 4.41421 0.901048
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) − 1.00000i − 0.192450i
\(28\) − 13.0711i − 2.47020i
\(29\) −5.07107 −0.941674 −0.470837 0.882220i \(-0.656048\pi\)
−0.470837 + 0.882220i \(0.656048\pi\)
\(30\) 0 0
\(31\) −10.4853 −1.88321 −0.941606 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) − 1.58579i − 0.280330i
\(33\) − 1.41421i − 0.246183i
\(34\) 16.4853 2.82720
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) − 3.07107i − 0.504880i −0.967612 0.252440i \(-0.918767\pi\)
0.967612 0.252440i \(-0.0812331\pi\)
\(38\) − 2.41421i − 0.391637i
\(39\) 2.58579 0.414057
\(40\) 0 0
\(41\) −4.58579 −0.716180 −0.358090 0.933687i \(-0.616572\pi\)
−0.358090 + 0.933687i \(0.616572\pi\)
\(42\) − 8.24264i − 1.27187i
\(43\) − 3.41421i − 0.520663i −0.965519 0.260331i \(-0.916168\pi\)
0.965519 0.260331i \(-0.0838318\pi\)
\(44\) 5.41421 0.816223
\(45\) 0 0
\(46\) −8.82843 −1.30168
\(47\) 11.6569i 1.70033i 0.526519 + 0.850163i \(0.323497\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −4.65685 −0.665265
\(50\) 0 0
\(51\) 6.82843 0.956171
\(52\) 9.89949i 1.37281i
\(53\) − 4.00000i − 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) 2.41421 0.328533
\(55\) 0 0
\(56\) 15.0711 2.01396
\(57\) − 1.00000i − 0.132453i
\(58\) − 12.2426i − 1.60754i
\(59\) 8.48528 1.10469 0.552345 0.833616i \(-0.313733\pi\)
0.552345 + 0.833616i \(0.313733\pi\)
\(60\) 0 0
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) − 25.3137i − 3.21484i
\(63\) − 3.41421i − 0.430150i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) 3.41421 0.420261
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 26.1421i 3.17020i
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 4.41421i 0.520220i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 7.41421 0.861885
\(75\) 0 0
\(76\) 3.82843 0.439151
\(77\) − 4.82843i − 0.550250i
\(78\) 6.24264i 0.706840i
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 11.0711i − 1.22259i
\(83\) − 6.48528i − 0.711852i −0.934514 0.355926i \(-0.884165\pi\)
0.934514 0.355926i \(-0.115835\pi\)
\(84\) 13.0711 1.42617
\(85\) 0 0
\(86\) 8.24264 0.888827
\(87\) − 5.07107i − 0.543676i
\(88\) 6.24264i 0.665468i
\(89\) 14.7279 1.56116 0.780578 0.625058i \(-0.214925\pi\)
0.780578 + 0.625058i \(0.214925\pi\)
\(90\) 0 0
\(91\) 8.82843 0.925471
\(92\) − 14.0000i − 1.45960i
\(93\) − 10.4853i − 1.08727i
\(94\) −28.1421 −2.90264
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) 4.24264i 0.430775i 0.976529 + 0.215387i \(0.0691014\pi\)
−0.976529 + 0.215387i \(0.930899\pi\)
\(98\) − 11.2426i − 1.13568i
\(99\) 1.41421 0.142134
\(100\) 0 0
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 16.4853i 1.63229i
\(103\) − 9.65685i − 0.951518i −0.879576 0.475759i \(-0.842173\pi\)
0.879576 0.475759i \(-0.157827\pi\)
\(104\) −11.4142 −1.11926
\(105\) 0 0
\(106\) 9.65685 0.937957
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) 3.82843i 0.368391i
\(109\) −8.82843 −0.845610 −0.422805 0.906221i \(-0.638954\pi\)
−0.422805 + 0.906221i \(0.638954\pi\)
\(110\) 0 0
\(111\) 3.07107 0.291493
\(112\) 10.2426i 0.967839i
\(113\) 4.48528i 0.421940i 0.977493 + 0.210970i \(0.0676622\pi\)
−0.977493 + 0.210970i \(0.932338\pi\)
\(114\) 2.41421 0.226112
\(115\) 0 0
\(116\) 19.4142 1.80256
\(117\) 2.58579i 0.239056i
\(118\) 20.4853i 1.88582i
\(119\) 23.3137 2.13716
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) − 13.6569i − 1.23643i
\(123\) − 4.58579i − 0.413486i
\(124\) 40.1421 3.60487
\(125\) 0 0
\(126\) 8.24264 0.734313
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 20.5563i 1.81694i
\(129\) 3.41421 0.300605
\(130\) 0 0
\(131\) −8.72792 −0.762562 −0.381281 0.924459i \(-0.624517\pi\)
−0.381281 + 0.924459i \(0.624517\pi\)
\(132\) 5.41421i 0.471247i
\(133\) − 3.41421i − 0.296050i
\(134\) −28.9706 −2.50268
\(135\) 0 0
\(136\) −30.1421 −2.58467
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) − 8.82843i − 0.751526i
\(139\) −6.82843 −0.579180 −0.289590 0.957151i \(-0.593519\pi\)
−0.289590 + 0.957151i \(0.593519\pi\)
\(140\) 0 0
\(141\) −11.6569 −0.981684
\(142\) 30.1421i 2.52947i
\(143\) 3.65685i 0.305802i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −4.82843 −0.399603
\(147\) − 4.65685i − 0.384091i
\(148\) 11.7574i 0.966449i
\(149\) −7.65685 −0.627274 −0.313637 0.949543i \(-0.601547\pi\)
−0.313637 + 0.949543i \(0.601547\pi\)
\(150\) 0 0
\(151\) −21.7990 −1.77398 −0.886988 0.461792i \(-0.847207\pi\)
−0.886988 + 0.461792i \(0.847207\pi\)
\(152\) 4.41421i 0.358040i
\(153\) 6.82843i 0.552046i
\(154\) 11.6569 0.939336
\(155\) 0 0
\(156\) −9.89949 −0.792594
\(157\) 17.7990i 1.42051i 0.703942 + 0.710257i \(0.251421\pi\)
−0.703942 + 0.710257i \(0.748579\pi\)
\(158\) − 27.3137i − 2.17296i
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) −12.4853 −0.983978
\(162\) 2.41421i 0.189679i
\(163\) 11.8995i 0.932040i 0.884774 + 0.466020i \(0.154313\pi\)
−0.884774 + 0.466020i \(0.845687\pi\)
\(164\) 17.5563 1.37092
\(165\) 0 0
\(166\) 15.6569 1.21521
\(167\) 10.0000i 0.773823i 0.922117 + 0.386912i \(0.126458\pi\)
−0.922117 + 0.386912i \(0.873542\pi\)
\(168\) 15.0711i 1.16276i
\(169\) 6.31371 0.485670
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 13.0711i 0.996660i
\(173\) − 22.1421i − 1.68344i −0.539918 0.841718i \(-0.681545\pi\)
0.539918 0.841718i \(-0.318455\pi\)
\(174\) 12.2426 0.928112
\(175\) 0 0
\(176\) −4.24264 −0.319801
\(177\) 8.48528i 0.637793i
\(178\) 35.5563i 2.66506i
\(179\) −22.8284 −1.70628 −0.853138 0.521685i \(-0.825304\pi\)
−0.853138 + 0.521685i \(0.825304\pi\)
\(180\) 0 0
\(181\) −24.8284 −1.84548 −0.922741 0.385420i \(-0.874057\pi\)
−0.922741 + 0.385420i \(0.874057\pi\)
\(182\) 21.3137i 1.57988i
\(183\) − 5.65685i − 0.418167i
\(184\) 16.1421 1.19001
\(185\) 0 0
\(186\) 25.3137 1.85609
\(187\) 9.65685i 0.706179i
\(188\) − 44.6274i − 3.25479i
\(189\) 3.41421 0.248347
\(190\) 0 0
\(191\) −17.8995 −1.29516 −0.647581 0.761997i \(-0.724219\pi\)
−0.647581 + 0.761997i \(0.724219\pi\)
\(192\) 9.82843i 0.709306i
\(193\) 0.928932i 0.0668660i 0.999441 + 0.0334330i \(0.0106440\pi\)
−0.999441 + 0.0334330i \(0.989356\pi\)
\(194\) −10.2426 −0.735379
\(195\) 0 0
\(196\) 17.8284 1.27346
\(197\) − 9.17157i − 0.653448i −0.945120 0.326724i \(-0.894055\pi\)
0.945120 0.326724i \(-0.105945\pi\)
\(198\) 3.41421i 0.242638i
\(199\) −0.485281 −0.0344007 −0.0172003 0.999852i \(-0.505475\pi\)
−0.0172003 + 0.999852i \(0.505475\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 2.00000i 0.140720i
\(203\) − 17.3137i − 1.21518i
\(204\) −26.1421 −1.83032
\(205\) 0 0
\(206\) 23.3137 1.62434
\(207\) − 3.65685i − 0.254169i
\(208\) − 7.75736i − 0.537876i
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 7.31371 0.503496 0.251748 0.967793i \(-0.418995\pi\)
0.251748 + 0.967793i \(0.418995\pi\)
\(212\) 15.3137i 1.05175i
\(213\) 12.4853i 0.855477i
\(214\) −19.3137 −1.32026
\(215\) 0 0
\(216\) −4.41421 −0.300349
\(217\) − 35.7990i − 2.43019i
\(218\) − 21.3137i − 1.44355i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −17.6569 −1.18773
\(222\) 7.41421i 0.497609i
\(223\) − 17.6569i − 1.18239i −0.806529 0.591195i \(-0.798656\pi\)
0.806529 0.591195i \(-0.201344\pi\)
\(224\) 5.41421 0.361752
\(225\) 0 0
\(226\) −10.8284 −0.720296
\(227\) 14.9706i 0.993631i 0.867856 + 0.496816i \(0.165497\pi\)
−0.867856 + 0.496816i \(0.834503\pi\)
\(228\) 3.82843i 0.253544i
\(229\) −9.65685 −0.638143 −0.319071 0.947731i \(-0.603371\pi\)
−0.319071 + 0.947731i \(0.603371\pi\)
\(230\) 0 0
\(231\) 4.82843 0.317687
\(232\) 22.3848i 1.46963i
\(233\) 19.6569i 1.28776i 0.765125 + 0.643882i \(0.222677\pi\)
−0.765125 + 0.643882i \(0.777323\pi\)
\(234\) −6.24264 −0.408094
\(235\) 0 0
\(236\) −32.4853 −2.11461
\(237\) − 11.3137i − 0.734904i
\(238\) 56.2843i 3.64837i
\(239\) −5.41421 −0.350216 −0.175108 0.984549i \(-0.556028\pi\)
−0.175108 + 0.984549i \(0.556028\pi\)
\(240\) 0 0
\(241\) 18.9706 1.22200 0.611001 0.791630i \(-0.290767\pi\)
0.611001 + 0.791630i \(0.290767\pi\)
\(242\) − 21.7279i − 1.39672i
\(243\) 1.00000i 0.0641500i
\(244\) 21.6569 1.38644
\(245\) 0 0
\(246\) 11.0711 0.705866
\(247\) 2.58579i 0.164530i
\(248\) 46.2843i 2.93905i
\(249\) 6.48528 0.410988
\(250\) 0 0
\(251\) 27.0711 1.70871 0.854355 0.519689i \(-0.173952\pi\)
0.854355 + 0.519689i \(0.173952\pi\)
\(252\) 13.0711i 0.823400i
\(253\) − 5.17157i − 0.325134i
\(254\) −19.3137 −1.21185
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 6.82843i 0.425946i 0.977058 + 0.212973i \(0.0683146\pi\)
−0.977058 + 0.212973i \(0.931685\pi\)
\(258\) 8.24264i 0.513164i
\(259\) 10.4853 0.651524
\(260\) 0 0
\(261\) 5.07107 0.313891
\(262\) − 21.0711i − 1.30177i
\(263\) 3.85786i 0.237886i 0.992901 + 0.118943i \(0.0379506\pi\)
−0.992901 + 0.118943i \(0.962049\pi\)
\(264\) −6.24264 −0.384208
\(265\) 0 0
\(266\) 8.24264 0.505389
\(267\) 14.7279i 0.901334i
\(268\) − 45.9411i − 2.80630i
\(269\) 28.3848 1.73065 0.865325 0.501211i \(-0.167112\pi\)
0.865325 + 0.501211i \(0.167112\pi\)
\(270\) 0 0
\(271\) −25.1716 −1.52906 −0.764532 0.644586i \(-0.777030\pi\)
−0.764532 + 0.644586i \(0.777030\pi\)
\(272\) − 20.4853i − 1.24210i
\(273\) 8.82843i 0.534321i
\(274\) 33.7990 2.04187
\(275\) 0 0
\(276\) 14.0000 0.842701
\(277\) − 22.9706i − 1.38017i −0.723730 0.690084i \(-0.757574\pi\)
0.723730 0.690084i \(-0.242426\pi\)
\(278\) − 16.4853i − 0.988721i
\(279\) 10.4853 0.627737
\(280\) 0 0
\(281\) 10.7279 0.639974 0.319987 0.947422i \(-0.396321\pi\)
0.319987 + 0.947422i \(0.396321\pi\)
\(282\) − 28.1421i − 1.67584i
\(283\) 2.24264i 0.133311i 0.997776 + 0.0666556i \(0.0212329\pi\)
−0.997776 + 0.0666556i \(0.978767\pi\)
\(284\) −47.7990 −2.83635
\(285\) 0 0
\(286\) −8.82843 −0.522036
\(287\) − 15.6569i − 0.924195i
\(288\) 1.58579i 0.0934434i
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) −4.24264 −0.248708
\(292\) − 7.65685i − 0.448084i
\(293\) − 7.79899i − 0.455622i −0.973705 0.227811i \(-0.926843\pi\)
0.973705 0.227811i \(-0.0731568\pi\)
\(294\) 11.2426 0.655684
\(295\) 0 0
\(296\) −13.5563 −0.787947
\(297\) 1.41421i 0.0820610i
\(298\) − 18.4853i − 1.07082i
\(299\) 9.45584 0.546846
\(300\) 0 0
\(301\) 11.6569 0.671890
\(302\) − 52.6274i − 3.02837i
\(303\) 0.828427i 0.0475919i
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −16.4853 −0.942401
\(307\) − 31.7990i − 1.81486i −0.420199 0.907432i \(-0.638040\pi\)
0.420199 0.907432i \(-0.361960\pi\)
\(308\) 18.4853i 1.05330i
\(309\) 9.65685 0.549359
\(310\) 0 0
\(311\) −23.7574 −1.34716 −0.673578 0.739116i \(-0.735244\pi\)
−0.673578 + 0.739116i \(0.735244\pi\)
\(312\) − 11.4142i − 0.646203i
\(313\) 26.4853i 1.49704i 0.663114 + 0.748518i \(0.269234\pi\)
−0.663114 + 0.748518i \(0.730766\pi\)
\(314\) −42.9706 −2.42497
\(315\) 0 0
\(316\) 43.3137 2.43659
\(317\) − 11.3137i − 0.635441i −0.948184 0.317721i \(-0.897083\pi\)
0.948184 0.317721i \(-0.102917\pi\)
\(318\) 9.65685i 0.541529i
\(319\) 7.17157 0.401531
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) − 30.1421i − 1.67976i
\(323\) 6.82843i 0.379944i
\(324\) −3.82843 −0.212690
\(325\) 0 0
\(326\) −28.7279 −1.59109
\(327\) − 8.82843i − 0.488213i
\(328\) 20.2426i 1.11771i
\(329\) −39.7990 −2.19419
\(330\) 0 0
\(331\) 12.8284 0.705114 0.352557 0.935790i \(-0.385312\pi\)
0.352557 + 0.935790i \(0.385312\pi\)
\(332\) 24.8284i 1.36264i
\(333\) 3.07107i 0.168293i
\(334\) −24.1421 −1.32100
\(335\) 0 0
\(336\) −10.2426 −0.558782
\(337\) 19.7574i 1.07625i 0.842864 + 0.538126i \(0.180868\pi\)
−0.842864 + 0.538126i \(0.819132\pi\)
\(338\) 15.2426i 0.829090i
\(339\) −4.48528 −0.243607
\(340\) 0 0
\(341\) 14.8284 0.803004
\(342\) 2.41421i 0.130546i
\(343\) 8.00000i 0.431959i
\(344\) −15.0711 −0.812578
\(345\) 0 0
\(346\) 53.4558 2.87380
\(347\) − 6.48528i − 0.348148i −0.984733 0.174074i \(-0.944307\pi\)
0.984733 0.174074i \(-0.0556932\pi\)
\(348\) 19.4142i 1.04071i
\(349\) −6.68629 −0.357909 −0.178954 0.983857i \(-0.557271\pi\)
−0.178954 + 0.983857i \(0.557271\pi\)
\(350\) 0 0
\(351\) −2.58579 −0.138019
\(352\) 2.24264i 0.119533i
\(353\) 7.65685i 0.407533i 0.979019 + 0.203767i \(0.0653184\pi\)
−0.979019 + 0.203767i \(0.934682\pi\)
\(354\) −20.4853 −1.08878
\(355\) 0 0
\(356\) −56.3848 −2.98839
\(357\) 23.3137i 1.23389i
\(358\) − 55.1127i − 2.91280i
\(359\) 9.89949 0.522475 0.261238 0.965275i \(-0.415869\pi\)
0.261238 + 0.965275i \(0.415869\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 59.9411i − 3.15044i
\(363\) − 9.00000i − 0.472377i
\(364\) −33.7990 −1.77155
\(365\) 0 0
\(366\) 13.6569 0.713855
\(367\) − 16.5858i − 0.865771i −0.901449 0.432886i \(-0.857495\pi\)
0.901449 0.432886i \(-0.142505\pi\)
\(368\) 10.9706i 0.571880i
\(369\) 4.58579 0.238727
\(370\) 0 0
\(371\) 13.6569 0.709029
\(372\) 40.1421i 2.08127i
\(373\) − 9.89949i − 0.512576i −0.966600 0.256288i \(-0.917500\pi\)
0.966600 0.256288i \(-0.0824996\pi\)
\(374\) −23.3137 −1.20552
\(375\) 0 0
\(376\) 51.4558 2.65363
\(377\) 13.1127i 0.675338i
\(378\) 8.24264i 0.423956i
\(379\) 4.14214 0.212767 0.106384 0.994325i \(-0.466073\pi\)
0.106384 + 0.994325i \(0.466073\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 43.2132i − 2.21098i
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) −20.5563 −1.04901
\(385\) 0 0
\(386\) −2.24264 −0.114147
\(387\) 3.41421i 0.173554i
\(388\) − 16.2426i − 0.824595i
\(389\) 18.9706 0.961846 0.480923 0.876763i \(-0.340302\pi\)
0.480923 + 0.876763i \(0.340302\pi\)
\(390\) 0 0
\(391\) 24.9706 1.26282
\(392\) 20.5563i 1.03825i
\(393\) − 8.72792i − 0.440265i
\(394\) 22.1421 1.11550
\(395\) 0 0
\(396\) −5.41421 −0.272074
\(397\) 24.3431i 1.22175i 0.791728 + 0.610874i \(0.209182\pi\)
−0.791728 + 0.610874i \(0.790818\pi\)
\(398\) − 1.17157i − 0.0587256i
\(399\) 3.41421 0.170924
\(400\) 0 0
\(401\) −18.0416 −0.900956 −0.450478 0.892788i \(-0.648746\pi\)
−0.450478 + 0.892788i \(0.648746\pi\)
\(402\) − 28.9706i − 1.44492i
\(403\) 27.1127i 1.35058i
\(404\) −3.17157 −0.157792
\(405\) 0 0
\(406\) 41.7990 2.07445
\(407\) 4.34315i 0.215282i
\(408\) − 30.1421i − 1.49226i
\(409\) −26.4853 −1.30961 −0.654806 0.755797i \(-0.727250\pi\)
−0.654806 + 0.755797i \(0.727250\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 36.9706i 1.82141i
\(413\) 28.9706i 1.42555i
\(414\) 8.82843 0.433894
\(415\) 0 0
\(416\) −4.10051 −0.201044
\(417\) − 6.82843i − 0.334390i
\(418\) 3.41421i 0.166995i
\(419\) −18.8701 −0.921863 −0.460931 0.887436i \(-0.652485\pi\)
−0.460931 + 0.887436i \(0.652485\pi\)
\(420\) 0 0
\(421\) −37.3137 −1.81856 −0.909279 0.416186i \(-0.863366\pi\)
−0.909279 + 0.416186i \(0.863366\pi\)
\(422\) 17.6569i 0.859522i
\(423\) − 11.6569i − 0.566776i
\(424\) −17.6569 −0.857493
\(425\) 0 0
\(426\) −30.1421 −1.46039
\(427\) − 19.3137i − 0.934656i
\(428\) − 30.6274i − 1.48043i
\(429\) −3.65685 −0.176555
\(430\) 0 0
\(431\) 20.4853 0.986741 0.493371 0.869819i \(-0.335765\pi\)
0.493371 + 0.869819i \(0.335765\pi\)
\(432\) − 3.00000i − 0.144338i
\(433\) − 15.0711i − 0.724269i −0.932126 0.362135i \(-0.882048\pi\)
0.932126 0.362135i \(-0.117952\pi\)
\(434\) 86.4264 4.14860
\(435\) 0 0
\(436\) 33.7990 1.61868
\(437\) − 3.65685i − 0.174931i
\(438\) − 4.82843i − 0.230711i
\(439\) 32.9706 1.57360 0.786800 0.617209i \(-0.211737\pi\)
0.786800 + 0.617209i \(0.211737\pi\)
\(440\) 0 0
\(441\) 4.65685 0.221755
\(442\) − 42.6274i − 2.02758i
\(443\) − 21.3137i − 1.01264i −0.862344 0.506322i \(-0.831005\pi\)
0.862344 0.506322i \(-0.168995\pi\)
\(444\) −11.7574 −0.557980
\(445\) 0 0
\(446\) 42.6274 2.01847
\(447\) − 7.65685i − 0.362157i
\(448\) 33.5563i 1.58539i
\(449\) 11.8995 0.561572 0.280786 0.959770i \(-0.409405\pi\)
0.280786 + 0.959770i \(0.409405\pi\)
\(450\) 0 0
\(451\) 6.48528 0.305380
\(452\) − 17.1716i − 0.807683i
\(453\) − 21.7990i − 1.02421i
\(454\) −36.1421 −1.69623
\(455\) 0 0
\(456\) −4.41421 −0.206714
\(457\) 23.1716i 1.08392i 0.840404 + 0.541960i \(0.182318\pi\)
−0.840404 + 0.541960i \(0.817682\pi\)
\(458\) − 23.3137i − 1.08938i
\(459\) −6.82843 −0.318724
\(460\) 0 0
\(461\) 33.3137 1.55157 0.775787 0.630995i \(-0.217353\pi\)
0.775787 + 0.630995i \(0.217353\pi\)
\(462\) 11.6569i 0.542326i
\(463\) 27.8995i 1.29660i 0.761385 + 0.648300i \(0.224520\pi\)
−0.761385 + 0.648300i \(0.775480\pi\)
\(464\) −15.2132 −0.706255
\(465\) 0 0
\(466\) −47.4558 −2.19835
\(467\) 27.6569i 1.27981i 0.768455 + 0.639903i \(0.221026\pi\)
−0.768455 + 0.639903i \(0.778974\pi\)
\(468\) − 9.89949i − 0.457604i
\(469\) −40.9706 −1.89184
\(470\) 0 0
\(471\) −17.7990 −0.820134
\(472\) − 37.4558i − 1.72404i
\(473\) 4.82843i 0.222011i
\(474\) 27.3137 1.25456
\(475\) 0 0
\(476\) −89.2548 −4.09099
\(477\) 4.00000i 0.183147i
\(478\) − 13.0711i − 0.597857i
\(479\) −38.1838 −1.74466 −0.872330 0.488917i \(-0.837392\pi\)
−0.872330 + 0.488917i \(0.837392\pi\)
\(480\) 0 0
\(481\) −7.94113 −0.362084
\(482\) 45.7990i 2.08609i
\(483\) − 12.4853i − 0.568100i
\(484\) 34.4558 1.56617
\(485\) 0 0
\(486\) −2.41421 −0.109511
\(487\) 2.82843i 0.128168i 0.997944 + 0.0640841i \(0.0204126\pi\)
−0.997944 + 0.0640841i \(0.979587\pi\)
\(488\) 24.9706i 1.13036i
\(489\) −11.8995 −0.538114
\(490\) 0 0
\(491\) 2.10051 0.0947945 0.0473972 0.998876i \(-0.484907\pi\)
0.0473972 + 0.998876i \(0.484907\pi\)
\(492\) 17.5563i 0.791501i
\(493\) 34.6274i 1.55954i
\(494\) −6.24264 −0.280870
\(495\) 0 0
\(496\) −31.4558 −1.41241
\(497\) 42.6274i 1.91210i
\(498\) 15.6569i 0.701600i
\(499\) −15.7990 −0.707260 −0.353630 0.935385i \(-0.615053\pi\)
−0.353630 + 0.935385i \(0.615053\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) 65.3553i 2.91695i
\(503\) 4.82843i 0.215289i 0.994189 + 0.107644i \(0.0343308\pi\)
−0.994189 + 0.107644i \(0.965669\pi\)
\(504\) −15.0711 −0.671319
\(505\) 0 0
\(506\) 12.4853 0.555038
\(507\) 6.31371i 0.280402i
\(508\) − 30.6274i − 1.35887i
\(509\) 4.38478 0.194352 0.0971759 0.995267i \(-0.469019\pi\)
0.0971759 + 0.995267i \(0.469019\pi\)
\(510\) 0 0
\(511\) −6.82843 −0.302072
\(512\) − 31.2426i − 1.38074i
\(513\) 1.00000i 0.0441511i
\(514\) −16.4853 −0.727135
\(515\) 0 0
\(516\) −13.0711 −0.575422
\(517\) − 16.4853i − 0.725022i
\(518\) 25.3137i 1.11222i
\(519\) 22.1421 0.971932
\(520\) 0 0
\(521\) 12.3848 0.542587 0.271293 0.962497i \(-0.412549\pi\)
0.271293 + 0.962497i \(0.412549\pi\)
\(522\) 12.2426i 0.535846i
\(523\) − 23.7990i − 1.04066i −0.853966 0.520329i \(-0.825809\pi\)
0.853966 0.520329i \(-0.174191\pi\)
\(524\) 33.4142 1.45971
\(525\) 0 0
\(526\) −9.31371 −0.406097
\(527\) 71.5980i 3.11886i
\(528\) − 4.24264i − 0.184637i
\(529\) 9.62742 0.418583
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 13.0711i 0.566703i
\(533\) 11.8579i 0.513621i
\(534\) −35.5563 −1.53867
\(535\) 0 0
\(536\) 52.9706 2.28798
\(537\) − 22.8284i − 0.985119i
\(538\) 68.5269i 2.95440i
\(539\) 6.58579 0.283670
\(540\) 0 0
\(541\) 12.6274 0.542895 0.271448 0.962453i \(-0.412498\pi\)
0.271448 + 0.962453i \(0.412498\pi\)
\(542\) − 60.7696i − 2.61028i
\(543\) − 24.8284i − 1.06549i
\(544\) −10.8284 −0.464265
\(545\) 0 0
\(546\) −21.3137 −0.912143
\(547\) − 5.85786i − 0.250464i −0.992127 0.125232i \(-0.960032\pi\)
0.992127 0.125232i \(-0.0399676\pi\)
\(548\) 53.5980i 2.28959i
\(549\) 5.65685 0.241429
\(550\) 0 0
\(551\) 5.07107 0.216035
\(552\) 16.1421i 0.687055i
\(553\) − 38.6274i − 1.64260i
\(554\) 55.4558 2.35609
\(555\) 0 0
\(556\) 26.1421 1.10867
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) 25.3137i 1.07161i
\(559\) −8.82843 −0.373403
\(560\) 0 0
\(561\) −9.65685 −0.407713
\(562\) 25.8995i 1.09250i
\(563\) 14.2843i 0.602010i 0.953623 + 0.301005i \(0.0973221\pi\)
−0.953623 + 0.301005i \(0.902678\pi\)
\(564\) 44.6274 1.87915
\(565\) 0 0
\(566\) −5.41421 −0.227576
\(567\) 3.41421i 0.143383i
\(568\) − 55.1127i − 2.31248i
\(569\) −18.7279 −0.785115 −0.392558 0.919727i \(-0.628410\pi\)
−0.392558 + 0.919727i \(0.628410\pi\)
\(570\) 0 0
\(571\) −19.7990 −0.828562 −0.414281 0.910149i \(-0.635967\pi\)
−0.414281 + 0.910149i \(0.635967\pi\)
\(572\) − 14.0000i − 0.585369i
\(573\) − 17.8995i − 0.747762i
\(574\) 37.7990 1.57770
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) 1.79899i 0.0748929i 0.999299 + 0.0374465i \(0.0119224\pi\)
−0.999299 + 0.0374465i \(0.988078\pi\)
\(578\) − 71.5269i − 2.97513i
\(579\) −0.928932 −0.0386051
\(580\) 0 0
\(581\) 22.1421 0.918611
\(582\) − 10.2426i − 0.424571i
\(583\) 5.65685i 0.234283i
\(584\) 8.82843 0.365323
\(585\) 0 0
\(586\) 18.8284 0.777795
\(587\) 0.343146i 0.0141631i 0.999975 + 0.00708157i \(0.00225415\pi\)
−0.999975 + 0.00708157i \(0.997746\pi\)
\(588\) 17.8284i 0.735232i
\(589\) 10.4853 0.432038
\(590\) 0 0
\(591\) 9.17157 0.377268
\(592\) − 9.21320i − 0.378660i
\(593\) 6.68629i 0.274573i 0.990531 + 0.137287i \(0.0438381\pi\)
−0.990531 + 0.137287i \(0.956162\pi\)
\(594\) −3.41421 −0.140087
\(595\) 0 0
\(596\) 29.3137 1.20074
\(597\) − 0.485281i − 0.0198612i
\(598\) 22.8284i 0.933524i
\(599\) −45.9411 −1.87710 −0.938552 0.345139i \(-0.887832\pi\)
−0.938552 + 0.345139i \(0.887832\pi\)
\(600\) 0 0
\(601\) −23.1716 −0.945188 −0.472594 0.881280i \(-0.656682\pi\)
−0.472594 + 0.881280i \(0.656682\pi\)
\(602\) 28.1421i 1.14699i
\(603\) − 12.0000i − 0.488678i
\(604\) 83.4558 3.39577
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) 26.1421i 1.06108i 0.847661 + 0.530538i \(0.178010\pi\)
−0.847661 + 0.530538i \(0.821990\pi\)
\(608\) 1.58579i 0.0643121i
\(609\) 17.3137 0.701587
\(610\) 0 0
\(611\) 30.1421 1.21942
\(612\) − 26.1421i − 1.05673i
\(613\) 37.1127i 1.49897i 0.662023 + 0.749484i \(0.269698\pi\)
−0.662023 + 0.749484i \(0.730302\pi\)
\(614\) 76.7696 3.09817
\(615\) 0 0
\(616\) −21.3137 −0.858754
\(617\) − 18.1421i − 0.730375i −0.930934 0.365187i \(-0.881005\pi\)
0.930934 0.365187i \(-0.118995\pi\)
\(618\) 23.3137i 0.937815i
\(619\) 4.20101 0.168853 0.0844264 0.996430i \(-0.473094\pi\)
0.0844264 + 0.996430i \(0.473094\pi\)
\(620\) 0 0
\(621\) 3.65685 0.146745
\(622\) − 57.3553i − 2.29974i
\(623\) 50.2843i 2.01460i
\(624\) 7.75736 0.310543
\(625\) 0 0
\(626\) −63.9411 −2.55560
\(627\) 1.41421i 0.0564782i
\(628\) − 68.1421i − 2.71917i
\(629\) −20.9706 −0.836151
\(630\) 0 0
\(631\) −22.6274 −0.900783 −0.450392 0.892831i \(-0.648716\pi\)
−0.450392 + 0.892831i \(0.648716\pi\)
\(632\) 49.9411i 1.98655i
\(633\) 7.31371i 0.290694i
\(634\) 27.3137 1.08477
\(635\) 0 0
\(636\) −15.3137 −0.607228
\(637\) 12.0416i 0.477107i
\(638\) 17.3137i 0.685456i
\(639\) −12.4853 −0.493910
\(640\) 0 0
\(641\) −11.4142 −0.450834 −0.225417 0.974262i \(-0.572375\pi\)
−0.225417 + 0.974262i \(0.572375\pi\)
\(642\) − 19.3137i − 0.762251i
\(643\) 42.0416i 1.65796i 0.559278 + 0.828980i \(0.311078\pi\)
−0.559278 + 0.828980i \(0.688922\pi\)
\(644\) 47.7990 1.88354
\(645\) 0 0
\(646\) −16.4853 −0.648605
\(647\) − 17.1127i − 0.672770i −0.941725 0.336385i \(-0.890796\pi\)
0.941725 0.336385i \(-0.109204\pi\)
\(648\) − 4.41421i − 0.173407i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 35.7990 1.40307
\(652\) − 45.5563i − 1.78412i
\(653\) 42.4264i 1.66027i 0.557560 + 0.830137i \(0.311738\pi\)
−0.557560 + 0.830137i \(0.688262\pi\)
\(654\) 21.3137 0.833432
\(655\) 0 0
\(656\) −13.7574 −0.537135
\(657\) − 2.00000i − 0.0780274i
\(658\) − 96.0833i − 3.74572i
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 1.51472 0.0589157 0.0294579 0.999566i \(-0.490622\pi\)
0.0294579 + 0.999566i \(0.490622\pi\)
\(662\) 30.9706i 1.20371i
\(663\) − 17.6569i − 0.685735i
\(664\) −28.6274 −1.11096
\(665\) 0 0
\(666\) −7.41421 −0.287295
\(667\) − 18.5442i − 0.718033i
\(668\) − 38.2843i − 1.48126i
\(669\) 17.6569 0.682653
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 5.41421i 0.208858i
\(673\) − 2.10051i − 0.0809685i −0.999180 0.0404843i \(-0.987110\pi\)
0.999180 0.0404843i \(-0.0128901\pi\)
\(674\) −47.6985 −1.83728
\(675\) 0 0
\(676\) −24.1716 −0.929676
\(677\) − 11.0294i − 0.423896i −0.977281 0.211948i \(-0.932019\pi\)
0.977281 0.211948i \(-0.0679807\pi\)
\(678\) − 10.8284i − 0.415863i
\(679\) −14.4853 −0.555894
\(680\) 0 0
\(681\) −14.9706 −0.573673
\(682\) 35.7990i 1.37081i
\(683\) 5.65685i 0.216454i 0.994126 + 0.108227i \(0.0345173\pi\)
−0.994126 + 0.108227i \(0.965483\pi\)
\(684\) −3.82843 −0.146384
\(685\) 0 0
\(686\) −19.3137 −0.737401
\(687\) − 9.65685i − 0.368432i
\(688\) − 10.2426i − 0.390497i
\(689\) −10.3431 −0.394042
\(690\) 0 0
\(691\) 39.1127 1.48792 0.743959 0.668226i \(-0.232946\pi\)
0.743959 + 0.668226i \(0.232946\pi\)
\(692\) 84.7696i 3.22245i
\(693\) 4.82843i 0.183417i
\(694\) 15.6569 0.594326
\(695\) 0 0
\(696\) −22.3848 −0.848493
\(697\) 31.3137i 1.18609i
\(698\) − 16.1421i − 0.610989i
\(699\) −19.6569 −0.743491
\(700\) 0 0
\(701\) −11.6569 −0.440273 −0.220137 0.975469i \(-0.570650\pi\)
−0.220137 + 0.975469i \(0.570650\pi\)
\(702\) − 6.24264i − 0.235613i
\(703\) 3.07107i 0.115828i
\(704\) −13.8995 −0.523857
\(705\) 0 0
\(706\) −18.4853 −0.695703
\(707\) 2.82843i 0.106374i
\(708\) − 32.4853i − 1.22087i
\(709\) 12.6863 0.476444 0.238222 0.971211i \(-0.423435\pi\)
0.238222 + 0.971211i \(0.423435\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) − 65.0122i − 2.43643i
\(713\) − 38.3431i − 1.43596i
\(714\) −56.2843 −2.10639
\(715\) 0 0
\(716\) 87.3970 3.26618
\(717\) − 5.41421i − 0.202198i
\(718\) 23.8995i 0.891921i
\(719\) 47.5563 1.77355 0.886776 0.462199i \(-0.152939\pi\)
0.886776 + 0.462199i \(0.152939\pi\)
\(720\) 0 0
\(721\) 32.9706 1.22789
\(722\) 2.41421i 0.0898477i
\(723\) 18.9706i 0.705523i
\(724\) 95.0538 3.53265
\(725\) 0 0
\(726\) 21.7279 0.806399
\(727\) − 7.41421i − 0.274978i −0.990503 0.137489i \(-0.956097\pi\)
0.990503 0.137489i \(-0.0439032\pi\)
\(728\) − 38.9706i − 1.44435i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −23.3137 −0.862289
\(732\) 21.6569i 0.800460i
\(733\) − 21.3137i − 0.787240i −0.919273 0.393620i \(-0.871223\pi\)
0.919273 0.393620i \(-0.128777\pi\)
\(734\) 40.0416 1.47796
\(735\) 0 0
\(736\) 5.79899 0.213754
\(737\) − 16.9706i − 0.625119i
\(738\) 11.0711i 0.407532i
\(739\) −1.65685 −0.0609484 −0.0304742 0.999536i \(-0.509702\pi\)
−0.0304742 + 0.999536i \(0.509702\pi\)
\(740\) 0 0
\(741\) −2.58579 −0.0949912
\(742\) 32.9706i 1.21039i
\(743\) − 7.31371i − 0.268314i −0.990960 0.134157i \(-0.957167\pi\)
0.990960 0.134157i \(-0.0428326\pi\)
\(744\) −46.2843 −1.69686
\(745\) 0 0
\(746\) 23.8995 0.875023
\(747\) 6.48528i 0.237284i
\(748\) − 36.9706i − 1.35178i
\(749\) −27.3137 −0.998021
\(750\) 0 0
\(751\) 25.1127 0.916375 0.458188 0.888855i \(-0.348499\pi\)
0.458188 + 0.888855i \(0.348499\pi\)
\(752\) 34.9706i 1.27525i
\(753\) 27.0711i 0.986525i
\(754\) −31.6569 −1.15287
\(755\) 0 0
\(756\) −13.0711 −0.475390
\(757\) 11.1716i 0.406038i 0.979175 + 0.203019i \(0.0650753\pi\)
−0.979175 + 0.203019i \(0.934925\pi\)
\(758\) 10.0000i 0.363216i
\(759\) 5.17157 0.187716
\(760\) 0 0
\(761\) 46.2843 1.67780 0.838902 0.544283i \(-0.183198\pi\)
0.838902 + 0.544283i \(0.183198\pi\)
\(762\) − 19.3137i − 0.699662i
\(763\) − 30.1421i − 1.09122i
\(764\) 68.5269 2.47922
\(765\) 0 0
\(766\) 28.9706 1.04675
\(767\) − 21.9411i − 0.792248i
\(768\) − 29.9706i − 1.08147i
\(769\) 24.3431 0.877836 0.438918 0.898527i \(-0.355362\pi\)
0.438918 + 0.898527i \(0.355362\pi\)
\(770\) 0 0
\(771\) −6.82843 −0.245920
\(772\) − 3.55635i − 0.127996i
\(773\) − 0.970563i − 0.0349087i −0.999848 0.0174544i \(-0.994444\pi\)
0.999848 0.0174544i \(-0.00555618\pi\)
\(774\) −8.24264 −0.296276
\(775\) 0 0
\(776\) 18.7279 0.672293
\(777\) 10.4853i 0.376157i
\(778\) 45.7990i 1.64197i
\(779\) 4.58579 0.164303
\(780\) 0 0
\(781\) −17.6569 −0.631812
\(782\) 60.2843i 2.15576i
\(783\) 5.07107i 0.181225i
\(784\) −13.9706 −0.498949
\(785\) 0 0
\(786\) 21.0711 0.751580
\(787\) − 37.4558i − 1.33516i −0.744540 0.667578i \(-0.767331\pi\)
0.744540 0.667578i \(-0.232669\pi\)
\(788\) 35.1127i 1.25084i
\(789\) −3.85786 −0.137344
\(790\) 0 0
\(791\) −15.3137 −0.544493
\(792\) − 6.24264i − 0.221823i
\(793\) 14.6274i 0.519435i
\(794\) −58.7696 −2.08565
\(795\) 0 0
\(796\) 1.85786 0.0658503
\(797\) 5.17157i 0.183187i 0.995797 + 0.0915933i \(0.0291960\pi\)
−0.995797 + 0.0915933i \(0.970804\pi\)
\(798\) 8.24264i 0.291786i
\(799\) 79.5980 2.81597
\(800\) 0 0
\(801\) −14.7279 −0.520386
\(802\) − 43.5563i − 1.53803i
\(803\) − 2.82843i − 0.0998130i
\(804\) 45.9411 1.62022
\(805\) 0 0
\(806\) −65.4558 −2.30558
\(807\) 28.3848i 0.999191i
\(808\) − 3.65685i − 0.128648i
\(809\) −26.6863 −0.938240 −0.469120 0.883134i \(-0.655429\pi\)
−0.469120 + 0.883134i \(0.655429\pi\)
\(810\) 0 0
\(811\) 7.31371 0.256819 0.128410 0.991721i \(-0.459013\pi\)
0.128410 + 0.991721i \(0.459013\pi\)
\(812\) 66.2843i 2.32612i
\(813\) − 25.1716i − 0.882806i
\(814\) −10.4853 −0.367509
\(815\) 0 0
\(816\) 20.4853 0.717128
\(817\) 3.41421i 0.119448i
\(818\) − 63.9411i − 2.23565i
\(819\) −8.82843 −0.308490
\(820\) 0 0
\(821\) 0.544156 0.0189912 0.00949559 0.999955i \(-0.496977\pi\)
0.00949559 + 0.999955i \(0.496977\pi\)
\(822\) 33.7990i 1.17888i
\(823\) − 22.7279i − 0.792246i −0.918198 0.396123i \(-0.870355\pi\)
0.918198 0.396123i \(-0.129645\pi\)
\(824\) −42.6274 −1.48500
\(825\) 0 0
\(826\) −69.9411 −2.43356
\(827\) 3.37258i 0.117276i 0.998279 + 0.0586381i \(0.0186758\pi\)
−0.998279 + 0.0586381i \(0.981324\pi\)
\(828\) 14.0000i 0.486534i
\(829\) 21.5147 0.747237 0.373619 0.927582i \(-0.378117\pi\)
0.373619 + 0.927582i \(0.378117\pi\)
\(830\) 0 0
\(831\) 22.9706 0.796840
\(832\) − 25.4142i − 0.881079i
\(833\) 31.7990i 1.10177i
\(834\) 16.4853 0.570839
\(835\) 0 0
\(836\) −5.41421 −0.187254
\(837\) 10.4853i 0.362424i
\(838\) − 45.5563i − 1.57372i
\(839\) 35.1127 1.21222 0.606112 0.795379i \(-0.292728\pi\)
0.606112 + 0.795379i \(0.292728\pi\)
\(840\) 0 0
\(841\) −3.28427 −0.113251
\(842\) − 90.0833i − 3.10447i
\(843\) 10.7279i 0.369489i
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) 28.1421 0.967547
\(847\) − 30.7279i − 1.05582i
\(848\) − 12.0000i − 0.412082i
\(849\) −2.24264 −0.0769672
\(850\) 0 0
\(851\) 11.2304 0.384975
\(852\) − 47.7990i − 1.63757i
\(853\) 26.4853i 0.906839i 0.891297 + 0.453419i \(0.149796\pi\)
−0.891297 + 0.453419i \(0.850204\pi\)
\(854\) 46.6274 1.59556
\(855\) 0 0
\(856\) 35.3137 1.20700
\(857\) − 29.9411i − 1.02277i −0.859352 0.511385i \(-0.829133\pi\)
0.859352 0.511385i \(-0.170867\pi\)
\(858\) − 8.82843i − 0.301398i
\(859\) 41.9411 1.43101 0.715506 0.698606i \(-0.246196\pi\)
0.715506 + 0.698606i \(0.246196\pi\)
\(860\) 0 0
\(861\) 15.6569 0.533584
\(862\) 49.4558i 1.68447i
\(863\) 8.68629i 0.295685i 0.989011 + 0.147842i \(0.0472328\pi\)
−0.989011 + 0.147842i \(0.952767\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 0 0
\(866\) 36.3848 1.23641
\(867\) − 29.6274i − 1.00620i
\(868\) 137.054i 4.65191i
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 31.0294 1.05139
\(872\) 38.9706i 1.31971i
\(873\) − 4.24264i − 0.143592i
\(874\) 8.82843 0.298626
\(875\) 0 0
\(876\) 7.65685 0.258701
\(877\) − 1.89949i − 0.0641414i −0.999486 0.0320707i \(-0.989790\pi\)
0.999486 0.0320707i \(-0.0102102\pi\)
\(878\) 79.5980i 2.68630i
\(879\) 7.79899 0.263053
\(880\) 0 0
\(881\) 3.45584 0.116430 0.0582152 0.998304i \(-0.481459\pi\)
0.0582152 + 0.998304i \(0.481459\pi\)
\(882\) 11.2426i 0.378559i
\(883\) 18.5269i 0.623480i 0.950167 + 0.311740i \(0.100912\pi\)
−0.950167 + 0.311740i \(0.899088\pi\)
\(884\) 67.5980 2.27357
\(885\) 0 0
\(886\) 51.4558 1.72869
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) − 13.5563i − 0.454921i
\(889\) −27.3137 −0.916072
\(890\) 0 0
\(891\) −1.41421 −0.0473779
\(892\) 67.5980i 2.26335i
\(893\) − 11.6569i − 0.390082i
\(894\) 18.4853 0.618240
\(895\) 0 0
\(896\) −70.1838 −2.34468
\(897\) 9.45584i 0.315721i
\(898\) 28.7279i 0.958663i
\(899\) 53.1716 1.77337
\(900\) 0 0
\(901\) −27.3137 −0.909952
\(902\) 15.6569i 0.521316i
\(903\) 11.6569i 0.387916i
\(904\) 19.7990 0.658505
\(905\) 0 0
\(906\) 52.6274 1.74843
\(907\) 9.85786i 0.327325i 0.986516 + 0.163663i \(0.0523308\pi\)
−0.986516 + 0.163663i \(0.947669\pi\)
\(908\) − 57.3137i − 1.90202i
\(909\) −0.828427 −0.0274772
\(910\) 0 0
\(911\) −0.686292 −0.0227379 −0.0113689 0.999935i \(-0.503619\pi\)
−0.0113689 + 0.999935i \(0.503619\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) 9.17157i 0.303535i
\(914\) −55.9411 −1.85037
\(915\) 0 0
\(916\) 36.9706 1.22154
\(917\) − 29.7990i − 0.984049i
\(918\) − 16.4853i − 0.544095i
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 31.7990 1.04781
\(922\) 80.4264i 2.64870i
\(923\) − 32.2843i − 1.06265i
\(924\) −18.4853 −0.608121
\(925\) 0 0
\(926\) −67.3553 −2.21343
\(927\) 9.65685i 0.317173i
\(928\) 8.04163i 0.263979i
\(929\) 0.544156 0.0178532 0.00892659 0.999960i \(-0.497159\pi\)
0.00892659 + 0.999960i \(0.497159\pi\)
\(930\) 0 0
\(931\) 4.65685 0.152622
\(932\) − 75.2548i − 2.46505i
\(933\) − 23.7574i − 0.777781i
\(934\) −66.7696 −2.18477
\(935\) 0 0
\(936\) 11.4142 0.373085
\(937\) 54.7696i 1.78924i 0.446824 + 0.894622i \(0.352555\pi\)
−0.446824 + 0.894622i \(0.647445\pi\)
\(938\) − 98.9117i − 3.22958i
\(939\) −26.4853 −0.864314
\(940\) 0 0
\(941\) 13.5563 0.441924 0.220962 0.975282i \(-0.429080\pi\)
0.220962 + 0.975282i \(0.429080\pi\)
\(942\) − 42.9706i − 1.40006i
\(943\) − 16.7696i − 0.546092i
\(944\) 25.4558 0.828517
\(945\) 0 0
\(946\) −11.6569 −0.378997
\(947\) 7.17157i 0.233045i 0.993188 + 0.116522i \(0.0371747\pi\)
−0.993188 + 0.116522i \(0.962825\pi\)
\(948\) 43.3137i 1.40676i
\(949\) 5.17157 0.167876
\(950\) 0 0
\(951\) 11.3137 0.366872
\(952\) − 102.912i − 3.33539i
\(953\) 34.1421i 1.10597i 0.833190 + 0.552986i \(0.186512\pi\)
−0.833190 + 0.552986i \(0.813488\pi\)
\(954\) −9.65685 −0.312652
\(955\) 0 0
\(956\) 20.7279 0.670389
\(957\) 7.17157i 0.231824i
\(958\) − 92.1838i − 2.97832i
\(959\) 47.7990 1.54351
\(960\) 0 0
\(961\) 78.9411 2.54649
\(962\) − 19.1716i − 0.618116i
\(963\) − 8.00000i − 0.257796i
\(964\) −72.6274 −2.33917
\(965\) 0 0
\(966\) 30.1421 0.969807
\(967\) − 8.10051i − 0.260495i −0.991482 0.130247i \(-0.958423\pi\)
0.991482 0.130247i \(-0.0415771\pi\)
\(968\) 39.7279i 1.27690i
\(969\) −6.82843 −0.219361
\(970\) 0 0
\(971\) 6.34315 0.203561 0.101781 0.994807i \(-0.467546\pi\)
0.101781 + 0.994807i \(0.467546\pi\)
\(972\) − 3.82843i − 0.122797i
\(973\) − 23.3137i − 0.747403i
\(974\) −6.82843 −0.218797
\(975\) 0 0
\(976\) −16.9706 −0.543214
\(977\) − 39.5980i − 1.26685i −0.773803 0.633426i \(-0.781648\pi\)
0.773803 0.633426i \(-0.218352\pi\)
\(978\) − 28.7279i − 0.918618i
\(979\) −20.8284 −0.665679
\(980\) 0 0
\(981\) 8.82843 0.281870
\(982\) 5.07107i 0.161824i
\(983\) 35.9411i 1.14634i 0.819435 + 0.573172i \(0.194287\pi\)
−0.819435 + 0.573172i \(0.805713\pi\)
\(984\) −20.2426 −0.645312
\(985\) 0 0
\(986\) −83.5980 −2.66230
\(987\) − 39.7990i − 1.26682i
\(988\) − 9.89949i − 0.314945i
\(989\) 12.4853 0.397009
\(990\) 0 0
\(991\) −50.3431 −1.59920 −0.799601 0.600531i \(-0.794956\pi\)
−0.799601 + 0.600531i \(0.794956\pi\)
\(992\) 16.6274i 0.527921i
\(993\) 12.8284i 0.407098i
\(994\) −102.912 −3.26416
\(995\) 0 0
\(996\) −24.8284 −0.786719
\(997\) − 33.5147i − 1.06142i −0.847553 0.530711i \(-0.821925\pi\)
0.847553 0.530711i \(-0.178075\pi\)
\(998\) − 38.1421i − 1.20737i
\(999\) −3.07107 −0.0971643
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 1425.2.c.j.799.4 4
5.2 odd 4 1425.2.a.l.1.1 2
5.3 odd 4 285.2.a.f.1.2 2
5.4 even 2 inner 1425.2.c.j.799.1 4
15.2 even 4 4275.2.a.x.1.2 2
15.8 even 4 855.2.a.e.1.1 2
20.3 even 4 4560.2.a.bj.1.1 2
95.18 even 4 5415.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 2 5.3 odd 4
855.2.a.e.1.1 2 15.8 even 4
1425.2.a.l.1.1 2 5.2 odd 4
1425.2.c.j.799.1 4 5.4 even 2 inner
1425.2.c.j.799.4 4 1.1 even 1 trivial
4275.2.a.x.1.2 2 15.2 even 4
4560.2.a.bj.1.1 2 20.3 even 4
5415.2.a.p.1.1 2 95.18 even 4