Properties

Label 1425.2.c.j.799.2
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 799.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1425.799
Dual form 1425.2.c.j.799.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} +0.585786i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} +0.585786i q^{7} -1.58579i q^{8} -1.00000 q^{9} +1.41421 q^{11} +1.82843i q^{12} -5.41421i q^{13} +0.242641 q^{14} +3.00000 q^{16} -1.17157i q^{17} +0.414214i q^{18} -1.00000 q^{19} -0.585786 q^{21} -0.585786i q^{22} -7.65685i q^{23} +1.58579 q^{24} -2.24264 q^{26} -1.00000i q^{27} +1.07107i q^{28} +9.07107 q^{29} +6.48528 q^{31} -4.41421i q^{32} +1.41421i q^{33} -0.485281 q^{34} -1.82843 q^{36} +11.0711i q^{37} +0.414214i q^{38} +5.41421 q^{39} -7.41421 q^{41} +0.242641i q^{42} -0.585786i q^{43} +2.58579 q^{44} -3.17157 q^{46} +0.343146i q^{47} +3.00000i q^{48} +6.65685 q^{49} +1.17157 q^{51} -9.89949i q^{52} -4.00000i q^{53} -0.414214 q^{54} +0.928932 q^{56} -1.00000i q^{57} -3.75736i q^{58} -8.48528 q^{59} +5.65685 q^{61} -2.68629i q^{62} -0.585786i q^{63} +4.17157 q^{64} +0.585786 q^{66} +12.0000i q^{67} -2.14214i q^{68} +7.65685 q^{69} -4.48528 q^{71} +1.58579i q^{72} +2.00000i q^{73} +4.58579 q^{74} -1.82843 q^{76} +0.828427i q^{77} -2.24264i q^{78} +11.3137 q^{79} +1.00000 q^{81} +3.07107i q^{82} +10.4853i q^{83} -1.07107 q^{84} -0.242641 q^{86} +9.07107i q^{87} -2.24264i q^{88} -10.7279 q^{89} +3.17157 q^{91} -14.0000i q^{92} +6.48528i q^{93} +0.142136 q^{94} +4.41421 q^{96} -4.24264i q^{97} -2.75736i q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) 0.414214 0.169102
\(7\) 0.585786i 0.221406i 0.993854 + 0.110703i \(0.0353103\pi\)
−0.993854 + 0.110703i \(0.964690\pi\)
\(8\) − 1.58579i − 0.560660i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 1.82843i 0.527821i
\(13\) − 5.41421i − 1.50163i −0.660511 0.750816i \(-0.729660\pi\)
0.660511 0.750816i \(-0.270340\pi\)
\(14\) 0.242641 0.0648485
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 1.17157i − 0.284148i −0.989856 0.142074i \(-0.954623\pi\)
0.989856 0.142074i \(-0.0453771\pi\)
\(18\) 0.414214i 0.0976311i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) − 0.585786i − 0.124890i
\(23\) − 7.65685i − 1.59656i −0.602284 0.798282i \(-0.705742\pi\)
0.602284 0.798282i \(-0.294258\pi\)
\(24\) 1.58579 0.323697
\(25\) 0 0
\(26\) −2.24264 −0.439818
\(27\) − 1.00000i − 0.192450i
\(28\) 1.07107i 0.202413i
\(29\) 9.07107 1.68446 0.842228 0.539122i \(-0.181244\pi\)
0.842228 + 0.539122i \(0.181244\pi\)
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) 1.41421i 0.246183i
\(34\) −0.485281 −0.0832251
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 11.0711i 1.82007i 0.414529 + 0.910036i \(0.363946\pi\)
−0.414529 + 0.910036i \(0.636054\pi\)
\(38\) 0.414214i 0.0671943i
\(39\) 5.41421 0.866968
\(40\) 0 0
\(41\) −7.41421 −1.15791 −0.578953 0.815361i \(-0.696538\pi\)
−0.578953 + 0.815361i \(0.696538\pi\)
\(42\) 0.242641i 0.0374403i
\(43\) − 0.585786i − 0.0893316i −0.999002 0.0446658i \(-0.985778\pi\)
0.999002 0.0446658i \(-0.0142223\pi\)
\(44\) 2.58579 0.389822
\(45\) 0 0
\(46\) −3.17157 −0.467623
\(47\) 0.343146i 0.0500530i 0.999687 + 0.0250265i \(0.00796701\pi\)
−0.999687 + 0.0250265i \(0.992033\pi\)
\(48\) 3.00000i 0.433013i
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) 1.17157 0.164053
\(52\) − 9.89949i − 1.37281i
\(53\) − 4.00000i − 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) 0.928932 0.124134
\(57\) − 1.00000i − 0.132453i
\(58\) − 3.75736i − 0.493365i
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) − 2.68629i − 0.341159i
\(63\) − 0.585786i − 0.0738022i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0.585786 0.0721053
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) − 2.14214i − 0.259772i
\(69\) 7.65685 0.921777
\(70\) 0 0
\(71\) −4.48528 −0.532305 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(72\) 1.58579i 0.186887i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 4.58579 0.533087
\(75\) 0 0
\(76\) −1.82843 −0.209735
\(77\) 0.828427i 0.0944080i
\(78\) − 2.24264i − 0.253929i
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.07107i 0.339143i
\(83\) 10.4853i 1.15091i 0.817834 + 0.575455i \(0.195175\pi\)
−0.817834 + 0.575455i \(0.804825\pi\)
\(84\) −1.07107 −0.116863
\(85\) 0 0
\(86\) −0.242641 −0.0261646
\(87\) 9.07107i 0.972521i
\(88\) − 2.24264i − 0.239066i
\(89\) −10.7279 −1.13716 −0.568579 0.822629i \(-0.692507\pi\)
−0.568579 + 0.822629i \(0.692507\pi\)
\(90\) 0 0
\(91\) 3.17157 0.332471
\(92\) − 14.0000i − 1.45960i
\(93\) 6.48528i 0.672492i
\(94\) 0.142136 0.0146602
\(95\) 0 0
\(96\) 4.41421 0.450524
\(97\) − 4.24264i − 0.430775i −0.976529 0.215387i \(-0.930899\pi\)
0.976529 0.215387i \(-0.0691014\pi\)
\(98\) − 2.75736i − 0.278535i
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) − 0.485281i − 0.0480500i
\(103\) 1.65685i 0.163255i 0.996663 + 0.0816274i \(0.0260117\pi\)
−0.996663 + 0.0816274i \(0.973988\pi\)
\(104\) −8.58579 −0.841906
\(105\) 0 0
\(106\) −1.65685 −0.160928
\(107\) 8.00000i 0.773389i 0.922208 + 0.386695i \(0.126383\pi\)
−0.922208 + 0.386695i \(0.873617\pi\)
\(108\) − 1.82843i − 0.175940i
\(109\) −3.17157 −0.303782 −0.151891 0.988397i \(-0.548536\pi\)
−0.151891 + 0.988397i \(0.548536\pi\)
\(110\) 0 0
\(111\) −11.0711 −1.05082
\(112\) 1.75736i 0.166055i
\(113\) − 12.4853i − 1.17452i −0.809400 0.587258i \(-0.800207\pi\)
0.809400 0.587258i \(-0.199793\pi\)
\(114\) −0.414214 −0.0387947
\(115\) 0 0
\(116\) 16.5858 1.53995
\(117\) 5.41421i 0.500544i
\(118\) 3.51472i 0.323556i
\(119\) 0.686292 0.0629122
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) − 2.34315i − 0.212138i
\(123\) − 7.41421i − 0.668517i
\(124\) 11.8579 1.06487
\(125\) 0 0
\(126\) −0.242641 −0.0216162
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) 0.585786 0.0515756
\(130\) 0 0
\(131\) 16.7279 1.46153 0.730763 0.682632i \(-0.239165\pi\)
0.730763 + 0.682632i \(0.239165\pi\)
\(132\) 2.58579i 0.225064i
\(133\) − 0.585786i − 0.0507941i
\(134\) 4.97056 0.429391
\(135\) 0 0
\(136\) −1.85786 −0.159311
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) − 3.17157i − 0.269982i
\(139\) −1.17157 −0.0993715 −0.0496858 0.998765i \(-0.515822\pi\)
−0.0496858 + 0.998765i \(0.515822\pi\)
\(140\) 0 0
\(141\) −0.343146 −0.0288981
\(142\) 1.85786i 0.155909i
\(143\) − 7.65685i − 0.640298i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 0.828427 0.0685611
\(147\) 6.65685i 0.549048i
\(148\) 20.2426i 1.66393i
\(149\) 3.65685 0.299581 0.149791 0.988718i \(-0.452140\pi\)
0.149791 + 0.988718i \(0.452140\pi\)
\(150\) 0 0
\(151\) 17.7990 1.44846 0.724231 0.689558i \(-0.242195\pi\)
0.724231 + 0.689558i \(0.242195\pi\)
\(152\) 1.58579i 0.128624i
\(153\) 1.17157i 0.0947161i
\(154\) 0.343146 0.0276515
\(155\) 0 0
\(156\) 9.89949 0.792594
\(157\) − 21.7990i − 1.73975i −0.493273 0.869874i \(-0.664200\pi\)
0.493273 0.869874i \(-0.335800\pi\)
\(158\) − 4.68629i − 0.372821i
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 4.48528 0.353490
\(162\) − 0.414214i − 0.0325437i
\(163\) − 7.89949i − 0.618736i −0.950942 0.309368i \(-0.899882\pi\)
0.950942 0.309368i \(-0.100118\pi\)
\(164\) −13.5563 −1.05857
\(165\) 0 0
\(166\) 4.34315 0.337093
\(167\) 10.0000i 0.773823i 0.922117 + 0.386912i \(0.126458\pi\)
−0.922117 + 0.386912i \(0.873542\pi\)
\(168\) 0.928932i 0.0716687i
\(169\) −16.3137 −1.25490
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 1.07107i − 0.0816682i
\(173\) 6.14214i 0.466978i 0.972359 + 0.233489i \(0.0750143\pi\)
−0.972359 + 0.233489i \(0.924986\pi\)
\(174\) 3.75736 0.284845
\(175\) 0 0
\(176\) 4.24264 0.319801
\(177\) − 8.48528i − 0.637793i
\(178\) 4.44365i 0.333066i
\(179\) −17.1716 −1.28346 −0.641732 0.766929i \(-0.721784\pi\)
−0.641732 + 0.766929i \(0.721784\pi\)
\(180\) 0 0
\(181\) −19.1716 −1.42501 −0.712506 0.701666i \(-0.752440\pi\)
−0.712506 + 0.701666i \(0.752440\pi\)
\(182\) − 1.31371i − 0.0973786i
\(183\) 5.65685i 0.418167i
\(184\) −12.1421 −0.895130
\(185\) 0 0
\(186\) 2.68629 0.196968
\(187\) − 1.65685i − 0.121161i
\(188\) 0.627417i 0.0457591i
\(189\) 0.585786 0.0426097
\(190\) 0 0
\(191\) 1.89949 0.137443 0.0687213 0.997636i \(-0.478108\pi\)
0.0687213 + 0.997636i \(0.478108\pi\)
\(192\) 4.17157i 0.301057i
\(193\) 15.0711i 1.08484i 0.840108 + 0.542420i \(0.182492\pi\)
−0.840108 + 0.542420i \(0.817508\pi\)
\(194\) −1.75736 −0.126171
\(195\) 0 0
\(196\) 12.1716 0.869398
\(197\) − 14.8284i − 1.05648i −0.849095 0.528241i \(-0.822852\pi\)
0.849095 0.528241i \(-0.177148\pi\)
\(198\) 0.585786i 0.0416300i
\(199\) 16.4853 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 2.00000i 0.140720i
\(203\) 5.31371i 0.372949i
\(204\) 2.14214 0.149979
\(205\) 0 0
\(206\) 0.686292 0.0478162
\(207\) 7.65685i 0.532188i
\(208\) − 16.2426i − 1.12622i
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) −15.3137 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(212\) − 7.31371i − 0.502308i
\(213\) − 4.48528i − 0.307326i
\(214\) 3.31371 0.226520
\(215\) 0 0
\(216\) −1.58579 −0.107899
\(217\) 3.79899i 0.257892i
\(218\) 1.31371i 0.0889756i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −6.34315 −0.426686
\(222\) 4.58579i 0.307778i
\(223\) − 6.34315i − 0.424768i −0.977186 0.212384i \(-0.931877\pi\)
0.977186 0.212384i \(-0.0681228\pi\)
\(224\) 2.58579 0.172770
\(225\) 0 0
\(226\) −5.17157 −0.344008
\(227\) − 18.9706i − 1.25912i −0.776952 0.629560i \(-0.783235\pi\)
0.776952 0.629560i \(-0.216765\pi\)
\(228\) − 1.82843i − 0.121091i
\(229\) 1.65685 0.109488 0.0547440 0.998500i \(-0.482566\pi\)
0.0547440 + 0.998500i \(0.482566\pi\)
\(230\) 0 0
\(231\) −0.828427 −0.0545065
\(232\) − 14.3848i − 0.944407i
\(233\) 8.34315i 0.546578i 0.961932 + 0.273289i \(0.0881115\pi\)
−0.961932 + 0.273289i \(0.911889\pi\)
\(234\) 2.24264 0.146606
\(235\) 0 0
\(236\) −15.5147 −1.00992
\(237\) 11.3137i 0.734904i
\(238\) − 0.284271i − 0.0184266i
\(239\) −2.58579 −0.167261 −0.0836303 0.996497i \(-0.526651\pi\)
−0.0836303 + 0.996497i \(0.526651\pi\)
\(240\) 0 0
\(241\) −14.9706 −0.964339 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(242\) 3.72792i 0.239640i
\(243\) 1.00000i 0.0641500i
\(244\) 10.3431 0.662152
\(245\) 0 0
\(246\) −3.07107 −0.195804
\(247\) 5.41421i 0.344498i
\(248\) − 10.2843i − 0.653052i
\(249\) −10.4853 −0.664478
\(250\) 0 0
\(251\) 12.9289 0.816067 0.408033 0.912967i \(-0.366215\pi\)
0.408033 + 0.912967i \(0.366215\pi\)
\(252\) − 1.07107i − 0.0674709i
\(253\) − 10.8284i − 0.680777i
\(254\) 3.31371 0.207921
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 1.17157i 0.0730807i 0.999332 + 0.0365404i \(0.0116337\pi\)
−0.999332 + 0.0365404i \(0.988366\pi\)
\(258\) − 0.242641i − 0.0151061i
\(259\) −6.48528 −0.402976
\(260\) 0 0
\(261\) −9.07107 −0.561485
\(262\) − 6.92893i − 0.428071i
\(263\) 32.1421i 1.98197i 0.133977 + 0.990984i \(0.457225\pi\)
−0.133977 + 0.990984i \(0.542775\pi\)
\(264\) 2.24264 0.138025
\(265\) 0 0
\(266\) −0.242641 −0.0148773
\(267\) − 10.7279i − 0.656538i
\(268\) 21.9411i 1.34027i
\(269\) −8.38478 −0.511229 −0.255614 0.966779i \(-0.582278\pi\)
−0.255614 + 0.966779i \(0.582278\pi\)
\(270\) 0 0
\(271\) −30.8284 −1.87269 −0.936347 0.351076i \(-0.885816\pi\)
−0.936347 + 0.351076i \(0.885816\pi\)
\(272\) − 3.51472i − 0.213111i
\(273\) 3.17157i 0.191952i
\(274\) −5.79899 −0.350330
\(275\) 0 0
\(276\) 14.0000 0.842701
\(277\) 10.9706i 0.659157i 0.944128 + 0.329579i \(0.106907\pi\)
−0.944128 + 0.329579i \(0.893093\pi\)
\(278\) 0.485281i 0.0291052i
\(279\) −6.48528 −0.388264
\(280\) 0 0
\(281\) −14.7279 −0.878594 −0.439297 0.898342i \(-0.644772\pi\)
−0.439297 + 0.898342i \(0.644772\pi\)
\(282\) 0.142136i 0.00846405i
\(283\) − 6.24264i − 0.371086i −0.982636 0.185543i \(-0.940596\pi\)
0.982636 0.185543i \(-0.0594045\pi\)
\(284\) −8.20101 −0.486640
\(285\) 0 0
\(286\) −3.17157 −0.187539
\(287\) − 4.34315i − 0.256368i
\(288\) 4.41421i 0.260110i
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) 4.24264 0.248708
\(292\) 3.65685i 0.214001i
\(293\) 31.7990i 1.85772i 0.370435 + 0.928858i \(0.379209\pi\)
−0.370435 + 0.928858i \(0.620791\pi\)
\(294\) 2.75736 0.160812
\(295\) 0 0
\(296\) 17.5563 1.02044
\(297\) − 1.41421i − 0.0820610i
\(298\) − 1.51472i − 0.0877453i
\(299\) −41.4558 −2.39745
\(300\) 0 0
\(301\) 0.343146 0.0197786
\(302\) − 7.37258i − 0.424244i
\(303\) − 4.82843i − 0.277386i
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) 0.485281 0.0277417
\(307\) 7.79899i 0.445112i 0.974920 + 0.222556i \(0.0714400\pi\)
−0.974920 + 0.222556i \(0.928560\pi\)
\(308\) 1.51472i 0.0863091i
\(309\) −1.65685 −0.0942551
\(310\) 0 0
\(311\) −32.2426 −1.82831 −0.914156 0.405362i \(-0.867145\pi\)
−0.914156 + 0.405362i \(0.867145\pi\)
\(312\) − 8.58579i − 0.486074i
\(313\) 9.51472i 0.537804i 0.963168 + 0.268902i \(0.0866607\pi\)
−0.963168 + 0.268902i \(0.913339\pi\)
\(314\) −9.02944 −0.509561
\(315\) 0 0
\(316\) 20.6863 1.16369
\(317\) 11.3137i 0.635441i 0.948184 + 0.317721i \(0.102917\pi\)
−0.948184 + 0.317721i \(0.897083\pi\)
\(318\) − 1.65685i − 0.0929118i
\(319\) 12.8284 0.718254
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) − 1.85786i − 0.103535i
\(323\) 1.17157i 0.0651881i
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) −3.27208 −0.181224
\(327\) − 3.17157i − 0.175388i
\(328\) 11.7574i 0.649192i
\(329\) −0.201010 −0.0110820
\(330\) 0 0
\(331\) 7.17157 0.394185 0.197093 0.980385i \(-0.436850\pi\)
0.197093 + 0.980385i \(0.436850\pi\)
\(332\) 19.1716i 1.05218i
\(333\) − 11.0711i − 0.606691i
\(334\) 4.14214 0.226648
\(335\) 0 0
\(336\) −1.75736 −0.0958718
\(337\) 28.2426i 1.53847i 0.638963 + 0.769237i \(0.279364\pi\)
−0.638963 + 0.769237i \(0.720636\pi\)
\(338\) 6.75736i 0.367552i
\(339\) 12.4853 0.678107
\(340\) 0 0
\(341\) 9.17157 0.496669
\(342\) − 0.414214i − 0.0223981i
\(343\) 8.00000i 0.431959i
\(344\) −0.928932 −0.0500847
\(345\) 0 0
\(346\) 2.54416 0.136775
\(347\) 10.4853i 0.562879i 0.959579 + 0.281440i \(0.0908119\pi\)
−0.959579 + 0.281440i \(0.909188\pi\)
\(348\) 16.5858i 0.889091i
\(349\) −29.3137 −1.56913 −0.784563 0.620049i \(-0.787113\pi\)
−0.784563 + 0.620049i \(0.787113\pi\)
\(350\) 0 0
\(351\) −5.41421 −0.288989
\(352\) − 6.24264i − 0.332734i
\(353\) − 3.65685i − 0.194635i −0.995253 0.0973174i \(-0.968974\pi\)
0.995253 0.0973174i \(-0.0310262\pi\)
\(354\) −3.51472 −0.186805
\(355\) 0 0
\(356\) −19.6152 −1.03960
\(357\) 0.686292i 0.0363224i
\(358\) 7.11270i 0.375918i
\(359\) −9.89949 −0.522475 −0.261238 0.965275i \(-0.584131\pi\)
−0.261238 + 0.965275i \(0.584131\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 7.94113i 0.417376i
\(363\) − 9.00000i − 0.472377i
\(364\) 5.79899 0.303950
\(365\) 0 0
\(366\) 2.34315 0.122478
\(367\) − 19.4142i − 1.01341i −0.862118 0.506707i \(-0.830863\pi\)
0.862118 0.506707i \(-0.169137\pi\)
\(368\) − 22.9706i − 1.19742i
\(369\) 7.41421 0.385969
\(370\) 0 0
\(371\) 2.34315 0.121650
\(372\) 11.8579i 0.614802i
\(373\) 9.89949i 0.512576i 0.966600 + 0.256288i \(0.0824996\pi\)
−0.966600 + 0.256288i \(0.917500\pi\)
\(374\) −0.686292 −0.0354873
\(375\) 0 0
\(376\) 0.544156 0.0280627
\(377\) − 49.1127i − 2.52943i
\(378\) − 0.242641i − 0.0124801i
\(379\) −24.1421 −1.24010 −0.620049 0.784563i \(-0.712887\pi\)
−0.620049 + 0.784563i \(0.712887\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 0.786797i − 0.0402560i
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 6.24264 0.317742
\(387\) 0.585786i 0.0297772i
\(388\) − 7.75736i − 0.393820i
\(389\) −14.9706 −0.759038 −0.379519 0.925184i \(-0.623910\pi\)
−0.379519 + 0.925184i \(0.623910\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) − 10.5563i − 0.533176i
\(393\) 16.7279i 0.843812i
\(394\) −6.14214 −0.309436
\(395\) 0 0
\(396\) −2.58579 −0.129941
\(397\) 35.6569i 1.78957i 0.446501 + 0.894783i \(0.352670\pi\)
−0.446501 + 0.894783i \(0.647330\pi\)
\(398\) − 6.82843i − 0.342278i
\(399\) 0.585786 0.0293260
\(400\) 0 0
\(401\) 30.0416 1.50021 0.750104 0.661320i \(-0.230004\pi\)
0.750104 + 0.661320i \(0.230004\pi\)
\(402\) 4.97056i 0.247909i
\(403\) − 35.1127i − 1.74909i
\(404\) −8.82843 −0.439231
\(405\) 0 0
\(406\) 2.20101 0.109234
\(407\) 15.6569i 0.776081i
\(408\) − 1.85786i − 0.0919780i
\(409\) −9.51472 −0.470473 −0.235236 0.971938i \(-0.575586\pi\)
−0.235236 + 0.971938i \(0.575586\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) 3.02944i 0.149250i
\(413\) − 4.97056i − 0.244585i
\(414\) 3.17157 0.155874
\(415\) 0 0
\(416\) −23.8995 −1.17177
\(417\) − 1.17157i − 0.0573722i
\(418\) 0.585786i 0.0286518i
\(419\) 34.8701 1.70351 0.851757 0.523937i \(-0.175537\pi\)
0.851757 + 0.523937i \(0.175537\pi\)
\(420\) 0 0
\(421\) −14.6863 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(422\) 6.34315i 0.308780i
\(423\) − 0.343146i − 0.0166843i
\(424\) −6.34315 −0.308050
\(425\) 0 0
\(426\) −1.85786 −0.0900138
\(427\) 3.31371i 0.160362i
\(428\) 14.6274i 0.707043i
\(429\) 7.65685 0.369676
\(430\) 0 0
\(431\) 3.51472 0.169298 0.0846490 0.996411i \(-0.473023\pi\)
0.0846490 + 0.996411i \(0.473023\pi\)
\(432\) − 3.00000i − 0.144338i
\(433\) − 0.928932i − 0.0446416i −0.999751 0.0223208i \(-0.992894\pi\)
0.999751 0.0223208i \(-0.00710553\pi\)
\(434\) 1.57359 0.0755349
\(435\) 0 0
\(436\) −5.79899 −0.277721
\(437\) 7.65685i 0.366277i
\(438\) 0.828427i 0.0395838i
\(439\) −0.970563 −0.0463224 −0.0231612 0.999732i \(-0.507373\pi\)
−0.0231612 + 0.999732i \(0.507373\pi\)
\(440\) 0 0
\(441\) −6.65685 −0.316993
\(442\) 2.62742i 0.124973i
\(443\) 1.31371i 0.0624162i 0.999513 + 0.0312081i \(0.00993546\pi\)
−0.999513 + 0.0312081i \(0.990065\pi\)
\(444\) −20.2426 −0.960673
\(445\) 0 0
\(446\) −2.62742 −0.124412
\(447\) 3.65685i 0.172963i
\(448\) 2.44365i 0.115452i
\(449\) −7.89949 −0.372800 −0.186400 0.982474i \(-0.559682\pi\)
−0.186400 + 0.982474i \(0.559682\pi\)
\(450\) 0 0
\(451\) −10.4853 −0.493733
\(452\) − 22.8284i − 1.07376i
\(453\) 17.7990i 0.836269i
\(454\) −7.85786 −0.368788
\(455\) 0 0
\(456\) −1.58579 −0.0742613
\(457\) 28.8284i 1.34854i 0.738486 + 0.674268i \(0.235541\pi\)
−0.738486 + 0.674268i \(0.764459\pi\)
\(458\) − 0.686292i − 0.0320683i
\(459\) −1.17157 −0.0546843
\(460\) 0 0
\(461\) 10.6863 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(462\) 0.343146i 0.0159646i
\(463\) 8.10051i 0.376462i 0.982125 + 0.188231i \(0.0602754\pi\)
−0.982125 + 0.188231i \(0.939725\pi\)
\(464\) 27.2132 1.26334
\(465\) 0 0
\(466\) 3.45584 0.160089
\(467\) 16.3431i 0.756271i 0.925750 + 0.378135i \(0.123435\pi\)
−0.925750 + 0.378135i \(0.876565\pi\)
\(468\) 9.89949i 0.457604i
\(469\) −7.02944 −0.324589
\(470\) 0 0
\(471\) 21.7990 1.00444
\(472\) 13.4558i 0.619355i
\(473\) − 0.828427i − 0.0380911i
\(474\) 4.68629 0.215248
\(475\) 0 0
\(476\) 1.25483 0.0575152
\(477\) 4.00000i 0.183147i
\(478\) 1.07107i 0.0489895i
\(479\) 38.1838 1.74466 0.872330 0.488917i \(-0.162608\pi\)
0.872330 + 0.488917i \(0.162608\pi\)
\(480\) 0 0
\(481\) 59.9411 2.73308
\(482\) 6.20101i 0.282448i
\(483\) 4.48528i 0.204087i
\(484\) −16.4558 −0.747993
\(485\) 0 0
\(486\) 0.414214 0.0187891
\(487\) − 2.82843i − 0.128168i −0.997944 0.0640841i \(-0.979587\pi\)
0.997944 0.0640841i \(-0.0204126\pi\)
\(488\) − 8.97056i − 0.406078i
\(489\) 7.89949 0.357228
\(490\) 0 0
\(491\) 21.8995 0.988310 0.494155 0.869374i \(-0.335477\pi\)
0.494155 + 0.869374i \(0.335477\pi\)
\(492\) − 13.5563i − 0.611167i
\(493\) − 10.6274i − 0.478635i
\(494\) 2.24264 0.100901
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) − 2.62742i − 0.117856i
\(498\) 4.34315i 0.194621i
\(499\) 23.7990 1.06539 0.532695 0.846308i \(-0.321179\pi\)
0.532695 + 0.846308i \(0.321179\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) − 5.35534i − 0.239020i
\(503\) − 0.828427i − 0.0369377i −0.999829 0.0184689i \(-0.994121\pi\)
0.999829 0.0184689i \(-0.00587916\pi\)
\(504\) −0.928932 −0.0413779
\(505\) 0 0
\(506\) −4.48528 −0.199395
\(507\) − 16.3137i − 0.724517i
\(508\) 14.6274i 0.648987i
\(509\) −32.3848 −1.43543 −0.717715 0.696337i \(-0.754812\pi\)
−0.717715 + 0.696337i \(0.754812\pi\)
\(510\) 0 0
\(511\) −1.17157 −0.0518273
\(512\) − 22.7574i − 1.00574i
\(513\) 1.00000i 0.0441511i
\(514\) 0.485281 0.0214048
\(515\) 0 0
\(516\) 1.07107 0.0471511
\(517\) 0.485281i 0.0213427i
\(518\) 2.68629i 0.118029i
\(519\) −6.14214 −0.269610
\(520\) 0 0
\(521\) −24.3848 −1.06832 −0.534158 0.845385i \(-0.679371\pi\)
−0.534158 + 0.845385i \(0.679371\pi\)
\(522\) 3.75736i 0.164455i
\(523\) 15.7990i 0.690842i 0.938448 + 0.345421i \(0.112264\pi\)
−0.938448 + 0.345421i \(0.887736\pi\)
\(524\) 30.5858 1.33615
\(525\) 0 0
\(526\) 13.3137 0.580505
\(527\) − 7.59798i − 0.330973i
\(528\) 4.24264i 0.184637i
\(529\) −35.6274 −1.54902
\(530\) 0 0
\(531\) 8.48528 0.368230
\(532\) − 1.07107i − 0.0464367i
\(533\) 40.1421i 1.73875i
\(534\) −4.44365 −0.192296
\(535\) 0 0
\(536\) 19.0294 0.821946
\(537\) − 17.1716i − 0.741008i
\(538\) 3.47309i 0.149735i
\(539\) 9.41421 0.405499
\(540\) 0 0
\(541\) −32.6274 −1.40276 −0.701381 0.712786i \(-0.747433\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(542\) 12.7696i 0.548499i
\(543\) − 19.1716i − 0.822731i
\(544\) −5.17157 −0.221729
\(545\) 0 0
\(546\) 1.31371 0.0562215
\(547\) − 34.1421i − 1.45981i −0.683547 0.729906i \(-0.739564\pi\)
0.683547 0.729906i \(-0.260436\pi\)
\(548\) − 25.5980i − 1.09349i
\(549\) −5.65685 −0.241429
\(550\) 0 0
\(551\) −9.07107 −0.386440
\(552\) − 12.1421i − 0.516804i
\(553\) 6.62742i 0.281826i
\(554\) 4.54416 0.193063
\(555\) 0 0
\(556\) −2.14214 −0.0908468
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) 2.68629i 0.113720i
\(559\) −3.17157 −0.134143
\(560\) 0 0
\(561\) 1.65685 0.0699524
\(562\) 6.10051i 0.257334i
\(563\) − 42.2843i − 1.78207i −0.453935 0.891035i \(-0.649980\pi\)
0.453935 0.891035i \(-0.350020\pi\)
\(564\) −0.627417 −0.0264190
\(565\) 0 0
\(566\) −2.58579 −0.108689
\(567\) 0.585786i 0.0246007i
\(568\) 7.11270i 0.298442i
\(569\) 6.72792 0.282049 0.141025 0.990006i \(-0.454960\pi\)
0.141025 + 0.990006i \(0.454960\pi\)
\(570\) 0 0
\(571\) 19.7990 0.828562 0.414281 0.910149i \(-0.364033\pi\)
0.414281 + 0.910149i \(0.364033\pi\)
\(572\) − 14.0000i − 0.585369i
\(573\) 1.89949i 0.0793525i
\(574\) −1.79899 −0.0750884
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) − 37.7990i − 1.57359i −0.617213 0.786796i \(-0.711738\pi\)
0.617213 0.786796i \(-0.288262\pi\)
\(578\) − 6.47309i − 0.269245i
\(579\) −15.0711 −0.626332
\(580\) 0 0
\(581\) −6.14214 −0.254819
\(582\) − 1.75736i − 0.0728449i
\(583\) − 5.65685i − 0.234283i
\(584\) 3.17157 0.131241
\(585\) 0 0
\(586\) 13.1716 0.544113
\(587\) 11.6569i 0.481130i 0.970633 + 0.240565i \(0.0773327\pi\)
−0.970633 + 0.240565i \(0.922667\pi\)
\(588\) 12.1716i 0.501947i
\(589\) −6.48528 −0.267221
\(590\) 0 0
\(591\) 14.8284 0.609960
\(592\) 33.2132i 1.36505i
\(593\) 29.3137i 1.20377i 0.798583 + 0.601885i \(0.205583\pi\)
−0.798583 + 0.601885i \(0.794417\pi\)
\(594\) −0.585786 −0.0240351
\(595\) 0 0
\(596\) 6.68629 0.273881
\(597\) 16.4853i 0.674698i
\(598\) 17.1716i 0.702198i
\(599\) 21.9411 0.896490 0.448245 0.893911i \(-0.352049\pi\)
0.448245 + 0.893911i \(0.352049\pi\)
\(600\) 0 0
\(601\) −28.8284 −1.17594 −0.587968 0.808884i \(-0.700072\pi\)
−0.587968 + 0.808884i \(0.700072\pi\)
\(602\) − 0.142136i − 0.00579302i
\(603\) − 12.0000i − 0.488678i
\(604\) 32.5442 1.32420
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) − 2.14214i − 0.0869466i −0.999055 0.0434733i \(-0.986158\pi\)
0.999055 0.0434733i \(-0.0138423\pi\)
\(608\) 4.41421i 0.179020i
\(609\) −5.31371 −0.215322
\(610\) 0 0
\(611\) 1.85786 0.0751611
\(612\) 2.14214i 0.0865907i
\(613\) − 25.1127i − 1.01429i −0.861860 0.507146i \(-0.830700\pi\)
0.861860 0.507146i \(-0.169300\pi\)
\(614\) 3.23045 0.130370
\(615\) 0 0
\(616\) 1.31371 0.0529308
\(617\) 10.1421i 0.408307i 0.978939 + 0.204154i \(0.0654442\pi\)
−0.978939 + 0.204154i \(0.934556\pi\)
\(618\) 0.686292i 0.0276067i
\(619\) 43.7990 1.76043 0.880215 0.474575i \(-0.157398\pi\)
0.880215 + 0.474575i \(0.157398\pi\)
\(620\) 0 0
\(621\) −7.65685 −0.307259
\(622\) 13.3553i 0.535500i
\(623\) − 6.28427i − 0.251774i
\(624\) 16.2426 0.650226
\(625\) 0 0
\(626\) 3.94113 0.157519
\(627\) − 1.41421i − 0.0564782i
\(628\) − 39.8579i − 1.59050i
\(629\) 12.9706 0.517170
\(630\) 0 0
\(631\) 22.6274 0.900783 0.450392 0.892831i \(-0.351284\pi\)
0.450392 + 0.892831i \(0.351284\pi\)
\(632\) − 17.9411i − 0.713660i
\(633\) − 15.3137i − 0.608665i
\(634\) 4.68629 0.186116
\(635\) 0 0
\(636\) 7.31371 0.290007
\(637\) − 36.0416i − 1.42802i
\(638\) − 5.31371i − 0.210372i
\(639\) 4.48528 0.177435
\(640\) 0 0
\(641\) −8.58579 −0.339118 −0.169559 0.985520i \(-0.554234\pi\)
−0.169559 + 0.985520i \(0.554234\pi\)
\(642\) 3.31371i 0.130782i
\(643\) − 6.04163i − 0.238259i −0.992879 0.119129i \(-0.961990\pi\)
0.992879 0.119129i \(-0.0380103\pi\)
\(644\) 8.20101 0.323165
\(645\) 0 0
\(646\) 0.485281 0.0190931
\(647\) 45.1127i 1.77356i 0.462189 + 0.886782i \(0.347064\pi\)
−0.462189 + 0.886782i \(0.652936\pi\)
\(648\) − 1.58579i − 0.0622956i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −3.79899 −0.148894
\(652\) − 14.4437i − 0.565657i
\(653\) − 42.4264i − 1.66027i −0.557560 0.830137i \(-0.688262\pi\)
0.557560 0.830137i \(-0.311738\pi\)
\(654\) −1.31371 −0.0513701
\(655\) 0 0
\(656\) −22.2426 −0.868429
\(657\) − 2.00000i − 0.0780274i
\(658\) 0.0832611i 0.00324586i
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 18.4853 0.718994 0.359497 0.933146i \(-0.382948\pi\)
0.359497 + 0.933146i \(0.382948\pi\)
\(662\) − 2.97056i − 0.115454i
\(663\) − 6.34315i − 0.246347i
\(664\) 16.6274 0.645269
\(665\) 0 0
\(666\) −4.58579 −0.177696
\(667\) − 69.4558i − 2.68934i
\(668\) 18.2843i 0.707440i
\(669\) 6.34315 0.245240
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 2.58579i 0.0997489i
\(673\) − 21.8995i − 0.844163i −0.906558 0.422082i \(-0.861300\pi\)
0.906558 0.422082i \(-0.138700\pi\)
\(674\) 11.6985 0.450609
\(675\) 0 0
\(676\) −29.8284 −1.14725
\(677\) − 44.9706i − 1.72836i −0.503184 0.864180i \(-0.667838\pi\)
0.503184 0.864180i \(-0.332162\pi\)
\(678\) − 5.17157i − 0.198613i
\(679\) 2.48528 0.0953763
\(680\) 0 0
\(681\) 18.9706 0.726954
\(682\) − 3.79899i − 0.145471i
\(683\) − 5.65685i − 0.216454i −0.994126 0.108227i \(-0.965483\pi\)
0.994126 0.108227i \(-0.0345173\pi\)
\(684\) 1.82843 0.0699117
\(685\) 0 0
\(686\) 3.31371 0.126518
\(687\) 1.65685i 0.0632129i
\(688\) − 1.75736i − 0.0669987i
\(689\) −21.6569 −0.825060
\(690\) 0 0
\(691\) −23.1127 −0.879248 −0.439624 0.898182i \(-0.644888\pi\)
−0.439624 + 0.898182i \(0.644888\pi\)
\(692\) 11.2304i 0.426918i
\(693\) − 0.828427i − 0.0314693i
\(694\) 4.34315 0.164864
\(695\) 0 0
\(696\) 14.3848 0.545254
\(697\) 8.68629i 0.329017i
\(698\) 12.1421i 0.459587i
\(699\) −8.34315 −0.315567
\(700\) 0 0
\(701\) −0.343146 −0.0129604 −0.00648022 0.999979i \(-0.502063\pi\)
−0.00648022 + 0.999979i \(0.502063\pi\)
\(702\) 2.24264i 0.0846430i
\(703\) − 11.0711i − 0.417553i
\(704\) 5.89949 0.222346
\(705\) 0 0
\(706\) −1.51472 −0.0570072
\(707\) − 2.82843i − 0.106374i
\(708\) − 15.5147i − 0.583079i
\(709\) 35.3137 1.32623 0.663117 0.748516i \(-0.269233\pi\)
0.663117 + 0.748516i \(0.269233\pi\)
\(710\) 0 0
\(711\) −11.3137 −0.424297
\(712\) 17.0122i 0.637559i
\(713\) − 49.6569i − 1.85966i
\(714\) 0.284271 0.0106386
\(715\) 0 0
\(716\) −31.3970 −1.17336
\(717\) − 2.58579i − 0.0965680i
\(718\) 4.10051i 0.153029i
\(719\) 16.4437 0.613245 0.306622 0.951831i \(-0.400801\pi\)
0.306622 + 0.951831i \(0.400801\pi\)
\(720\) 0 0
\(721\) −0.970563 −0.0361456
\(722\) − 0.414214i − 0.0154154i
\(723\) − 14.9706i − 0.556761i
\(724\) −35.0538 −1.30277
\(725\) 0 0
\(726\) −3.72792 −0.138356
\(727\) − 4.58579i − 0.170077i −0.996378 0.0850387i \(-0.972899\pi\)
0.996378 0.0850387i \(-0.0271014\pi\)
\(728\) − 5.02944i − 0.186403i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −0.686292 −0.0253834
\(732\) 10.3431i 0.382294i
\(733\) 1.31371i 0.0485229i 0.999706 + 0.0242615i \(0.00772342\pi\)
−0.999706 + 0.0242615i \(0.992277\pi\)
\(734\) −8.04163 −0.296822
\(735\) 0 0
\(736\) −33.7990 −1.24585
\(737\) 16.9706i 0.625119i
\(738\) − 3.07107i − 0.113048i
\(739\) 9.65685 0.355233 0.177617 0.984100i \(-0.443161\pi\)
0.177617 + 0.984100i \(0.443161\pi\)
\(740\) 0 0
\(741\) −5.41421 −0.198896
\(742\) − 0.970563i − 0.0356305i
\(743\) 15.3137i 0.561805i 0.959736 + 0.280903i \(0.0906338\pi\)
−0.959736 + 0.280903i \(0.909366\pi\)
\(744\) 10.2843 0.377040
\(745\) 0 0
\(746\) 4.10051 0.150130
\(747\) − 10.4853i − 0.383636i
\(748\) − 3.02944i − 0.110767i
\(749\) −4.68629 −0.171233
\(750\) 0 0
\(751\) −37.1127 −1.35426 −0.677131 0.735863i \(-0.736777\pi\)
−0.677131 + 0.735863i \(0.736777\pi\)
\(752\) 1.02944i 0.0375397i
\(753\) 12.9289i 0.471156i
\(754\) −20.3431 −0.740854
\(755\) 0 0
\(756\) 1.07107 0.0389544
\(757\) 16.8284i 0.611640i 0.952089 + 0.305820i \(0.0989305\pi\)
−0.952089 + 0.305820i \(0.901069\pi\)
\(758\) 10.0000i 0.363216i
\(759\) 10.8284 0.393047
\(760\) 0 0
\(761\) −10.2843 −0.372805 −0.186402 0.982474i \(-0.559683\pi\)
−0.186402 + 0.982474i \(0.559683\pi\)
\(762\) 3.31371i 0.120043i
\(763\) − 1.85786i − 0.0672592i
\(764\) 3.47309 0.125652
\(765\) 0 0
\(766\) −4.97056 −0.179594
\(767\) 45.9411i 1.65884i
\(768\) 3.97056i 0.143275i
\(769\) 35.6569 1.28582 0.642910 0.765942i \(-0.277727\pi\)
0.642910 + 0.765942i \(0.277727\pi\)
\(770\) 0 0
\(771\) −1.17157 −0.0421932
\(772\) 27.5563i 0.991775i
\(773\) 32.9706i 1.18587i 0.805251 + 0.592934i \(0.202031\pi\)
−0.805251 + 0.592934i \(0.797969\pi\)
\(774\) 0.242641 0.00872154
\(775\) 0 0
\(776\) −6.72792 −0.241518
\(777\) − 6.48528i − 0.232658i
\(778\) 6.20101i 0.222317i
\(779\) 7.41421 0.265642
\(780\) 0 0
\(781\) −6.34315 −0.226976
\(782\) 3.71573i 0.132874i
\(783\) − 9.07107i − 0.324174i
\(784\) 19.9706 0.713234
\(785\) 0 0
\(786\) 6.92893 0.247147
\(787\) 13.4558i 0.479649i 0.970816 + 0.239825i \(0.0770899\pi\)
−0.970816 + 0.239825i \(0.922910\pi\)
\(788\) − 27.1127i − 0.965850i
\(789\) −32.1421 −1.14429
\(790\) 0 0
\(791\) 7.31371 0.260046
\(792\) 2.24264i 0.0796888i
\(793\) − 30.6274i − 1.08761i
\(794\) 14.7696 0.524152
\(795\) 0 0
\(796\) 30.1421 1.06836
\(797\) 10.8284i 0.383563i 0.981438 + 0.191781i \(0.0614264\pi\)
−0.981438 + 0.191781i \(0.938574\pi\)
\(798\) − 0.242641i − 0.00858939i
\(799\) 0.402020 0.0142225
\(800\) 0 0
\(801\) 10.7279 0.379052
\(802\) − 12.4437i − 0.439401i
\(803\) 2.82843i 0.0998130i
\(804\) −21.9411 −0.773804
\(805\) 0 0
\(806\) −14.5442 −0.512296
\(807\) − 8.38478i − 0.295158i
\(808\) 7.65685i 0.269367i
\(809\) −49.3137 −1.73378 −0.866889 0.498502i \(-0.833884\pi\)
−0.866889 + 0.498502i \(0.833884\pi\)
\(810\) 0 0
\(811\) −15.3137 −0.537737 −0.268869 0.963177i \(-0.586650\pi\)
−0.268869 + 0.963177i \(0.586650\pi\)
\(812\) 9.71573i 0.340955i
\(813\) − 30.8284i − 1.08120i
\(814\) 6.48528 0.227309
\(815\) 0 0
\(816\) 3.51472 0.123040
\(817\) 0.585786i 0.0204941i
\(818\) 3.94113i 0.137798i
\(819\) −3.17157 −0.110824
\(820\) 0 0
\(821\) 51.4558 1.79582 0.897911 0.440178i \(-0.145085\pi\)
0.897911 + 0.440178i \(0.145085\pi\)
\(822\) − 5.79899i − 0.202263i
\(823\) 2.72792i 0.0950894i 0.998869 + 0.0475447i \(0.0151397\pi\)
−0.998869 + 0.0475447i \(0.984860\pi\)
\(824\) 2.62742 0.0915304
\(825\) 0 0
\(826\) −2.05887 −0.0716374
\(827\) 48.6274i 1.69094i 0.534022 + 0.845470i \(0.320680\pi\)
−0.534022 + 0.845470i \(0.679320\pi\)
\(828\) 14.0000i 0.486534i
\(829\) 38.4853 1.33665 0.668325 0.743870i \(-0.267012\pi\)
0.668325 + 0.743870i \(0.267012\pi\)
\(830\) 0 0
\(831\) −10.9706 −0.380565
\(832\) − 22.5858i − 0.783021i
\(833\) − 7.79899i − 0.270219i
\(834\) −0.485281 −0.0168039
\(835\) 0 0
\(836\) −2.58579 −0.0894313
\(837\) − 6.48528i − 0.224164i
\(838\) − 14.4437i − 0.498948i
\(839\) −27.1127 −0.936034 −0.468017 0.883719i \(-0.655031\pi\)
−0.468017 + 0.883719i \(0.655031\pi\)
\(840\) 0 0
\(841\) 53.2843 1.83739
\(842\) 6.08326i 0.209643i
\(843\) − 14.7279i − 0.507257i
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) −0.142136 −0.00488672
\(847\) − 5.27208i − 0.181151i
\(848\) − 12.0000i − 0.412082i
\(849\) 6.24264 0.214247
\(850\) 0 0
\(851\) 84.7696 2.90586
\(852\) − 8.20101i − 0.280962i
\(853\) 9.51472i 0.325778i 0.986644 + 0.162889i \(0.0520812\pi\)
−0.986644 + 0.162889i \(0.947919\pi\)
\(854\) 1.37258 0.0469688
\(855\) 0 0
\(856\) 12.6863 0.433609
\(857\) 37.9411i 1.29604i 0.761622 + 0.648022i \(0.224404\pi\)
−0.761622 + 0.648022i \(0.775596\pi\)
\(858\) − 3.17157i − 0.108276i
\(859\) −25.9411 −0.885100 −0.442550 0.896744i \(-0.645926\pi\)
−0.442550 + 0.896744i \(0.645926\pi\)
\(860\) 0 0
\(861\) 4.34315 0.148014
\(862\) − 1.45584i − 0.0495862i
\(863\) 31.3137i 1.06593i 0.846137 + 0.532966i \(0.178922\pi\)
−0.846137 + 0.532966i \(0.821078\pi\)
\(864\) −4.41421 −0.150175
\(865\) 0 0
\(866\) −0.384776 −0.0130752
\(867\) 15.6274i 0.530735i
\(868\) 6.94618i 0.235769i
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 64.9706 2.20144
\(872\) 5.02944i 0.170318i
\(873\) 4.24264i 0.143592i
\(874\) 3.17157 0.107280
\(875\) 0 0
\(876\) −3.65685 −0.123554
\(877\) 17.8995i 0.604423i 0.953241 + 0.302211i \(0.0977249\pi\)
−0.953241 + 0.302211i \(0.902275\pi\)
\(878\) 0.402020i 0.0135675i
\(879\) −31.7990 −1.07255
\(880\) 0 0
\(881\) −47.4558 −1.59883 −0.799414 0.600781i \(-0.794857\pi\)
−0.799414 + 0.600781i \(0.794857\pi\)
\(882\) 2.75736i 0.0928451i
\(883\) − 46.5269i − 1.56576i −0.622176 0.782878i \(-0.713751\pi\)
0.622176 0.782878i \(-0.286249\pi\)
\(884\) −11.5980 −0.390082
\(885\) 0 0
\(886\) 0.544156 0.0182813
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 17.5563i 0.589153i
\(889\) −4.68629 −0.157173
\(890\) 0 0
\(891\) 1.41421 0.0473779
\(892\) − 11.5980i − 0.388329i
\(893\) − 0.343146i − 0.0114829i
\(894\) 1.51472 0.0506598
\(895\) 0 0
\(896\) 6.18377 0.206585
\(897\) − 41.4558i − 1.38417i
\(898\) 3.27208i 0.109191i
\(899\) 58.8284 1.96204
\(900\) 0 0
\(901\) −4.68629 −0.156123
\(902\) 4.34315i 0.144611i
\(903\) 0.343146i 0.0114192i
\(904\) −19.7990 −0.658505
\(905\) 0 0
\(906\) 7.37258 0.244938
\(907\) 38.1421i 1.26649i 0.773952 + 0.633244i \(0.218277\pi\)
−0.773952 + 0.633244i \(0.781723\pi\)
\(908\) − 34.6863i − 1.15111i
\(909\) 4.82843 0.160149
\(910\) 0 0
\(911\) −23.3137 −0.772418 −0.386209 0.922411i \(-0.626216\pi\)
−0.386209 + 0.922411i \(0.626216\pi\)
\(912\) − 3.00000i − 0.0993399i
\(913\) 14.8284i 0.490749i
\(914\) 11.9411 0.394977
\(915\) 0 0
\(916\) 3.02944 0.100095
\(917\) 9.79899i 0.323591i
\(918\) 0.485281i 0.0160167i
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) −7.79899 −0.256985
\(922\) − 4.42641i − 0.145776i
\(923\) 24.2843i 0.799327i
\(924\) −1.51472 −0.0498306
\(925\) 0 0
\(926\) 3.35534 0.110263
\(927\) − 1.65685i − 0.0544182i
\(928\) − 40.0416i − 1.31443i
\(929\) 51.4558 1.68821 0.844106 0.536177i \(-0.180132\pi\)
0.844106 + 0.536177i \(0.180132\pi\)
\(930\) 0 0
\(931\) −6.65685 −0.218170
\(932\) 15.2548i 0.499689i
\(933\) − 32.2426i − 1.05558i
\(934\) 6.76955 0.221507
\(935\) 0 0
\(936\) 8.58579 0.280635
\(937\) − 18.7696i − 0.613175i −0.951843 0.306587i \(-0.900813\pi\)
0.951843 0.306587i \(-0.0991871\pi\)
\(938\) 2.91169i 0.0950700i
\(939\) −9.51472 −0.310501
\(940\) 0 0
\(941\) −17.5563 −0.572321 −0.286160 0.958182i \(-0.592379\pi\)
−0.286160 + 0.958182i \(0.592379\pi\)
\(942\) − 9.02944i − 0.294195i
\(943\) 56.7696i 1.84867i
\(944\) −25.4558 −0.828517
\(945\) 0 0
\(946\) −0.343146 −0.0111566
\(947\) 12.8284i 0.416868i 0.978036 + 0.208434i \(0.0668366\pi\)
−0.978036 + 0.208434i \(0.933163\pi\)
\(948\) 20.6863i 0.671860i
\(949\) 10.8284 0.351506
\(950\) 0 0
\(951\) −11.3137 −0.366872
\(952\) − 1.08831i − 0.0352724i
\(953\) 5.85786i 0.189755i 0.995489 + 0.0948774i \(0.0302459\pi\)
−0.995489 + 0.0948774i \(0.969754\pi\)
\(954\) 1.65685 0.0536426
\(955\) 0 0
\(956\) −4.72792 −0.152912
\(957\) 12.8284i 0.414684i
\(958\) − 15.8162i − 0.510999i
\(959\) 8.20101 0.264824
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) − 24.8284i − 0.800501i
\(963\) − 8.00000i − 0.257796i
\(964\) −27.3726 −0.881612
\(965\) 0 0
\(966\) 1.85786 0.0597758
\(967\) − 27.8995i − 0.897187i −0.893736 0.448594i \(-0.851925\pi\)
0.893736 0.448594i \(-0.148075\pi\)
\(968\) 14.2721i 0.458722i
\(969\) −1.17157 −0.0376363
\(970\) 0 0
\(971\) 17.6569 0.566635 0.283318 0.959026i \(-0.408565\pi\)
0.283318 + 0.959026i \(0.408565\pi\)
\(972\) 1.82843i 0.0586468i
\(973\) − 0.686292i − 0.0220015i
\(974\) −1.17157 −0.0375396
\(975\) 0 0
\(976\) 16.9706 0.543214
\(977\) 39.5980i 1.26685i 0.773803 + 0.633426i \(0.218352\pi\)
−0.773803 + 0.633426i \(0.781648\pi\)
\(978\) − 3.27208i − 0.104630i
\(979\) −15.1716 −0.484886
\(980\) 0 0
\(981\) 3.17157 0.101261
\(982\) − 9.07107i − 0.289469i
\(983\) − 31.9411i − 1.01876i −0.860541 0.509382i \(-0.829874\pi\)
0.860541 0.509382i \(-0.170126\pi\)
\(984\) −11.7574 −0.374811
\(985\) 0 0
\(986\) −4.40202 −0.140189
\(987\) − 0.201010i − 0.00639822i
\(988\) 9.89949i 0.314945i
\(989\) −4.48528 −0.142624
\(990\) 0 0
\(991\) −61.6569 −1.95859 −0.979297 0.202427i \(-0.935117\pi\)
−0.979297 + 0.202427i \(0.935117\pi\)
\(992\) − 28.6274i − 0.908921i
\(993\) 7.17157i 0.227583i
\(994\) −1.08831 −0.0345192
\(995\) 0 0
\(996\) −19.1716 −0.607475
\(997\) − 50.4853i − 1.59888i −0.600743 0.799442i \(-0.705128\pi\)
0.600743 0.799442i \(-0.294872\pi\)
\(998\) − 9.85786i − 0.312045i
\(999\) 11.0711 0.350273
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.799.2 4
5.2 odd 4 1425.2.a.l.1.2 2
5.3 odd 4 285.2.a.f.1.1 2
5.4 even 2 inner 1425.2.c.j.799.3 4
15.2 even 4 4275.2.a.x.1.1 2
15.8 even 4 855.2.a.e.1.2 2
20.3 even 4 4560.2.a.bj.1.2 2
95.18 even 4 5415.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 5.3 odd 4
855.2.a.e.1.2 2 15.8 even 4
1425.2.a.l.1.2 2 5.2 odd 4
1425.2.c.j.799.2 4 1.1 even 1 trivial
1425.2.c.j.799.3 4 5.4 even 2 inner
4275.2.a.x.1.1 2 15.2 even 4
4560.2.a.bj.1.2 2 20.3 even 4
5415.2.a.p.1.2 2 95.18 even 4