Properties

Label 2100.2.d.i.1301.2
Level $2100$
Weight $2$
Character 2100.1301
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1301,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.d (of order \(2\), degree \(1\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1301.2
Root \(-0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1301
Dual form 2100.2.d.i.1301.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.41421i) q^{3} +(2.23607 + 1.41421i) q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\) \(q+(1.00000 - 1.41421i) q^{3} +(2.23607 + 1.41421i) q^{7} +(-1.00000 - 2.82843i) q^{9} +2.82843i q^{11} +2.82843i q^{13} +4.00000 q^{17} +6.32456i q^{19} +(4.23607 - 1.74806i) q^{21} +6.32456i q^{23} +(-5.00000 - 1.41421i) q^{27} -5.65685i q^{29} +6.32456i q^{31} +(4.00000 + 2.82843i) q^{33} -8.94427 q^{37} +(4.00000 + 2.82843i) q^{39} +4.47214 q^{41} -4.47214 q^{43} +6.00000 q^{47} +(3.00000 + 6.32456i) q^{49} +(4.00000 - 5.65685i) q^{51} +6.32456i q^{53} +(8.94427 + 6.32456i) q^{57} +8.94427 q^{59} +(1.76393 - 7.73877i) q^{63} -4.47214 q^{67} +(8.94427 + 6.32456i) q^{69} -14.1421i q^{71} -8.48528i q^{73} +(-4.00000 + 6.32456i) q^{77} +12.0000 q^{79} +(-7.00000 + 5.65685i) q^{81} -10.0000 q^{83} +(-8.00000 - 5.65685i) q^{87} +4.47214 q^{89} +(-4.00000 + 6.32456i) q^{91} +(8.94427 + 6.32456i) q^{93} +8.48528i q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{9} + 16 q^{17} + 8 q^{21} - 20 q^{27} + 16 q^{33} + 16 q^{39} + 24 q^{47} + 12 q^{49} + 16 q^{51} + 16 q^{63} - 16 q^{77} + 48 q^{79} - 28 q^{81} - 40 q^{83} - 32 q^{87} - 16 q^{91} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 0
\(3\) 1.00000 1.41421i 0.577350 0.816497i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.23607 + 1.41421i 0.845154 + 0.534522i
\(8\) 0 0
\(9\) −1.00000 2.82843i −0.333333 0.942809i
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.32456i 1.45095i 0.688247 + 0.725476i \(0.258380\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 4.23607 1.74806i 0.924386 0.381459i
\(22\) 0 0
\(23\) 6.32456i 1.31876i 0.751809 + 0.659380i \(0.229181\pi\)
−0.751809 + 0.659380i \(0.770819\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 1.41421i −0.962250 0.272166i
\(28\) 0 0
\(29\) 5.65685i 1.05045i −0.850963 0.525226i \(-0.823981\pi\)
0.850963 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i 0.823055 + 0.567962i \(0.192268\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 0 0
\(33\) 4.00000 + 2.82843i 0.696311 + 0.492366i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.94427 −1.47043 −0.735215 0.677834i \(-0.762919\pi\)
−0.735215 + 0.677834i \(0.762919\pi\)
\(38\) 0 0
\(39\) 4.00000 + 2.82843i 0.640513 + 0.452911i
\(40\) 0 0
\(41\) 4.47214 0.698430 0.349215 0.937043i \(-0.386448\pi\)
0.349215 + 0.937043i \(0.386448\pi\)
\(42\) 0 0
\(43\) −4.47214 −0.681994 −0.340997 0.940064i \(-0.610765\pi\)
−0.340997 + 0.940064i \(0.610765\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 3.00000 + 6.32456i 0.428571 + 0.903508i
\(50\) 0 0
\(51\) 4.00000 5.65685i 0.560112 0.792118i
\(52\) 0 0
\(53\) 6.32456i 0.868744i 0.900733 + 0.434372i \(0.143030\pi\)
−0.900733 + 0.434372i \(0.856970\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.94427 + 6.32456i 1.18470 + 0.837708i
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.76393 7.73877i 0.222235 0.974993i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.47214 −0.546358 −0.273179 0.961963i \(-0.588075\pi\)
−0.273179 + 0.961963i \(0.588075\pi\)
\(68\) 0 0
\(69\) 8.94427 + 6.32456i 1.07676 + 0.761387i
\(70\) 0 0
\(71\) 14.1421i 1.67836i −0.543852 0.839181i \(-0.683035\pi\)
0.543852 0.839181i \(-0.316965\pi\)
\(72\) 0 0
\(73\) 8.48528i 0.993127i −0.868000 0.496564i \(-0.834595\pi\)
0.868000 0.496564i \(-0.165405\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 + 6.32456i −0.455842 + 0.720750i
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 0 0
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 5.65685i −0.857690 0.606478i
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) 0 0
\(91\) −4.00000 + 6.32456i −0.419314 + 0.662994i
\(92\) 0 0
\(93\) 8.94427 + 6.32456i 0.927478 + 0.655826i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.48528i 0.861550i 0.902459 + 0.430775i \(0.141760\pi\)
−0.902459 + 0.430775i \(0.858240\pi\)
\(98\) 0 0
\(99\) 8.00000 2.82843i 0.804030 0.284268i
\(100\) 0 0
\(101\) −4.47214 −0.444994 −0.222497 0.974933i \(-0.571421\pi\)
−0.222497 + 0.974933i \(0.571421\pi\)
\(102\) 0 0
\(103\) 2.82843i 0.278693i 0.990244 + 0.139347i \(0.0445002\pi\)
−0.990244 + 0.139347i \(0.955500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.9737i 1.83425i −0.398596 0.917127i \(-0.630502\pi\)
0.398596 0.917127i \(-0.369498\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −8.94427 + 12.6491i −0.848953 + 1.20060i
\(112\) 0 0
\(113\) 18.9737i 1.78489i −0.451154 0.892446i \(-0.648987\pi\)
0.451154 0.892446i \(-0.351013\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000 2.82843i 0.739600 0.261488i
\(118\) 0 0
\(119\) 8.94427 + 5.65685i 0.819920 + 0.518563i
\(120\) 0 0
\(121\) 3.00000 0.272727
\(122\) 0 0
\(123\) 4.47214 6.32456i 0.403239 0.570266i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −4.47214 −0.396838 −0.198419 0.980117i \(-0.563581\pi\)
−0.198419 + 0.980117i \(0.563581\pi\)
\(128\) 0 0
\(129\) −4.47214 + 6.32456i −0.393750 + 0.556846i
\(130\) 0 0
\(131\) 8.94427 0.781465 0.390732 0.920504i \(-0.372222\pi\)
0.390732 + 0.920504i \(0.372222\pi\)
\(132\) 0 0
\(133\) −8.94427 + 14.1421i −0.775567 + 1.22628i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.32456i 0.540343i −0.962812 0.270172i \(-0.912920\pi\)
0.962812 0.270172i \(-0.0870804\pi\)
\(138\) 0 0
\(139\) 6.32456i 0.536442i 0.963357 + 0.268221i \(0.0864357\pi\)
−0.963357 + 0.268221i \(0.913564\pi\)
\(140\) 0 0
\(141\) 6.00000 8.48528i 0.505291 0.714590i
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.9443 + 2.08191i 0.985147 + 0.171713i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) −4.00000 11.3137i −0.323381 0.914659i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.82843i 0.225733i −0.993610 0.112867i \(-0.963997\pi\)
0.993610 0.112867i \(-0.0360032\pi\)
\(158\) 0 0
\(159\) 8.94427 + 6.32456i 0.709327 + 0.501570i
\(160\) 0 0
\(161\) −8.94427 + 14.1421i −0.704907 + 1.11456i
\(162\) 0 0
\(163\) 13.4164 1.05085 0.525427 0.850839i \(-0.323906\pi\)
0.525427 + 0.850839i \(0.323906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 17.8885 6.32456i 1.36797 0.483651i
\(172\) 0 0
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.94427 12.6491i 0.672293 0.950765i
\(178\) 0 0
\(179\) 14.1421i 1.05703i −0.848923 0.528516i \(-0.822748\pi\)
0.848923 0.528516i \(-0.177252\pi\)
\(180\) 0 0
\(181\) 12.6491i 0.940201i −0.882613 0.470100i \(-0.844218\pi\)
0.882613 0.470100i \(-0.155782\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.3137i 0.827340i
\(188\) 0 0
\(189\) −9.18034 10.2333i −0.667771 0.744366i
\(190\) 0 0
\(191\) 2.82843i 0.204658i −0.994751 0.102329i \(-0.967371\pi\)
0.994751 0.102329i \(-0.0326294\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.9737i 1.35182i 0.736985 + 0.675909i \(0.236249\pi\)
−0.736985 + 0.675909i \(0.763751\pi\)
\(198\) 0 0
\(199\) 18.9737i 1.34501i 0.740094 + 0.672504i \(0.234781\pi\)
−0.740094 + 0.672504i \(0.765219\pi\)
\(200\) 0 0
\(201\) −4.47214 + 6.32456i −0.315440 + 0.446100i
\(202\) 0 0
\(203\) 8.00000 12.6491i 0.561490 0.887794i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 17.8885 6.32456i 1.24334 0.439587i
\(208\) 0 0
\(209\) −17.8885 −1.23738
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −20.0000 14.1421i −1.37038 0.969003i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.94427 + 14.1421i −0.607177 + 0.960031i
\(218\) 0 0
\(219\) −12.0000 8.48528i −0.810885 0.573382i
\(220\) 0 0
\(221\) 11.3137i 0.761042i
\(222\) 0 0
\(223\) 2.82843i 0.189405i −0.995506 0.0947027i \(-0.969810\pi\)
0.995506 0.0947027i \(-0.0301901\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) 12.6491i 0.835877i −0.908475 0.417938i \(-0.862753\pi\)
0.908475 0.417938i \(-0.137247\pi\)
\(230\) 0 0
\(231\) 4.94427 + 11.9814i 0.325309 + 0.788319i
\(232\) 0 0
\(233\) 6.32456i 0.414335i 0.978305 + 0.207168i \(0.0664246\pi\)
−0.978305 + 0.207168i \(0.933575\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 16.9706i 0.779484 1.10236i
\(238\) 0 0
\(239\) 8.48528i 0.548867i −0.961606 0.274434i \(-0.911510\pi\)
0.961606 0.274434i \(-0.0884904\pi\)
\(240\) 0 0
\(241\) 12.6491i 0.814801i −0.913250 0.407400i \(-0.866435\pi\)
0.913250 0.407400i \(-0.133565\pi\)
\(242\) 0 0
\(243\) 1.00000 + 15.5563i 0.0641500 + 0.997940i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −17.8885 −1.13822
\(248\) 0 0
\(249\) −10.0000 + 14.1421i −0.633724 + 0.896221i
\(250\) 0 0
\(251\) 26.8328 1.69367 0.846836 0.531854i \(-0.178504\pi\)
0.846836 + 0.531854i \(0.178504\pi\)
\(252\) 0 0
\(253\) −17.8885 −1.12464
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) 0 0
\(259\) −20.0000 12.6491i −1.24274 0.785977i
\(260\) 0 0
\(261\) −16.0000 + 5.65685i −0.990375 + 0.350150i
\(262\) 0 0
\(263\) 6.32456i 0.389989i −0.980804 0.194994i \(-0.937531\pi\)
0.980804 0.194994i \(-0.0624689\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.47214 6.32456i 0.273690 0.387056i
\(268\) 0 0
\(269\) 4.47214 0.272671 0.136335 0.990663i \(-0.456467\pi\)
0.136335 + 0.990663i \(0.456467\pi\)
\(270\) 0 0
\(271\) 6.32456i 0.384189i 0.981376 + 0.192095i \(0.0615281\pi\)
−0.981376 + 0.192095i \(0.938472\pi\)
\(272\) 0 0
\(273\) 4.94427 + 11.9814i 0.299241 + 0.725148i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −26.8328 −1.61223 −0.806114 0.591761i \(-0.798433\pi\)
−0.806114 + 0.591761i \(0.798433\pi\)
\(278\) 0 0
\(279\) 17.8885 6.32456i 1.07096 0.378641i
\(280\) 0 0
\(281\) 16.9706i 1.01238i −0.862422 0.506189i \(-0.831054\pi\)
0.862422 0.506189i \(-0.168946\pi\)
\(282\) 0 0
\(283\) 25.4558i 1.51319i −0.653882 0.756596i \(-0.726861\pi\)
0.653882 0.756596i \(-0.273139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 + 6.32456i 0.590281 + 0.373327i
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 12.0000 + 8.48528i 0.703452 + 0.497416i
\(292\) 0 0
\(293\) 32.0000 1.86946 0.934730 0.355359i \(-0.115641\pi\)
0.934730 + 0.355359i \(0.115641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 14.1421i 0.232104 0.820610i
\(298\) 0 0
\(299\) −17.8885 −1.03452
\(300\) 0 0
\(301\) −10.0000 6.32456i −0.576390 0.364541i
\(302\) 0 0
\(303\) −4.47214 + 6.32456i −0.256917 + 0.363336i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.48528i 0.484281i −0.970241 0.242140i \(-0.922151\pi\)
0.970241 0.242140i \(-0.0778494\pi\)
\(308\) 0 0
\(309\) 4.00000 + 2.82843i 0.227552 + 0.160904i
\(310\) 0 0
\(311\) −17.8885 −1.01437 −0.507183 0.861838i \(-0.669313\pi\)
−0.507183 + 0.861838i \(0.669313\pi\)
\(312\) 0 0
\(313\) 25.4558i 1.43885i 0.694570 + 0.719425i \(0.255594\pi\)
−0.694570 + 0.719425i \(0.744406\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.9737i 1.06567i 0.846220 + 0.532834i \(0.178873\pi\)
−0.846220 + 0.532834i \(0.821127\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −26.8328 18.9737i −1.49766 1.05901i
\(322\) 0 0
\(323\) 25.2982i 1.40763i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.00000 + 2.82843i −0.110600 + 0.156412i
\(328\) 0 0
\(329\) 13.4164 + 8.48528i 0.739671 + 0.467809i
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 8.94427 + 25.2982i 0.490143 + 1.38633i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) −26.8328 18.9737i −1.45736 1.03051i
\(340\) 0 0
\(341\) −17.8885 −0.968719
\(342\) 0 0
\(343\) −2.23607 + 18.3848i −0.120736 + 0.992685i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.9737i 1.01856i 0.860601 + 0.509280i \(0.170088\pi\)
−0.860601 + 0.509280i \(0.829912\pi\)
\(348\) 0 0
\(349\) 12.6491i 0.677091i −0.940950 0.338546i \(-0.890065\pi\)
0.940950 0.338546i \(-0.109935\pi\)
\(350\) 0 0
\(351\) 4.00000 14.1421i 0.213504 0.754851i
\(352\) 0 0
\(353\) −28.0000 −1.49029 −0.745145 0.666903i \(-0.767620\pi\)
−0.745145 + 0.666903i \(0.767620\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 16.9443 6.99226i 0.896786 0.370069i
\(358\) 0 0
\(359\) 14.1421i 0.746393i 0.927752 + 0.373197i \(0.121738\pi\)
−0.927752 + 0.373197i \(0.878262\pi\)
\(360\) 0 0
\(361\) −21.0000 −1.10526
\(362\) 0 0
\(363\) 3.00000 4.24264i 0.157459 0.222681i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) −4.47214 12.6491i −0.232810 0.658486i
\(370\) 0 0
\(371\) −8.94427 + 14.1421i −0.464363 + 0.734223i
\(372\) 0 0
\(373\) −26.8328 −1.38935 −0.694675 0.719323i \(-0.744452\pi\)
−0.694675 + 0.719323i \(0.744452\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) −4.47214 + 6.32456i −0.229114 + 0.324017i
\(382\) 0 0
\(383\) 30.0000 1.53293 0.766464 0.642287i \(-0.222014\pi\)
0.766464 + 0.642287i \(0.222014\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.47214 + 12.6491i 0.227331 + 0.642990i
\(388\) 0 0
\(389\) 33.9411i 1.72088i 0.509549 + 0.860442i \(0.329812\pi\)
−0.509549 + 0.860442i \(0.670188\pi\)
\(390\) 0 0
\(391\) 25.2982i 1.27939i
\(392\) 0 0
\(393\) 8.94427 12.6491i 0.451179 0.638063i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.7990i 0.993683i 0.867841 + 0.496841i \(0.165507\pi\)
−0.867841 + 0.496841i \(0.834493\pi\)
\(398\) 0 0
\(399\) 11.0557 + 26.7912i 0.553479 + 1.34124i
\(400\) 0 0
\(401\) 11.3137i 0.564980i −0.959270 0.282490i \(-0.908840\pi\)
0.959270 0.282490i \(-0.0911603\pi\)
\(402\) 0 0
\(403\) −17.8885 −0.891092
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.2982i 1.25399i
\(408\) 0 0
\(409\) 37.9473i 1.87637i −0.346128 0.938187i \(-0.612504\pi\)
0.346128 0.938187i \(-0.387496\pi\)
\(410\) 0 0
\(411\) −8.94427 6.32456i −0.441188 0.311967i
\(412\) 0 0
\(413\) 20.0000 + 12.6491i 0.984136 + 0.622422i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.94427 + 6.32456i 0.438003 + 0.309715i
\(418\) 0 0
\(419\) −26.8328 −1.31087 −0.655434 0.755252i \(-0.727514\pi\)
−0.655434 + 0.755252i \(0.727514\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) −6.00000 16.9706i −0.291730 0.825137i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 + 11.3137i −0.386244 + 0.546231i
\(430\) 0 0
\(431\) 2.82843i 0.136241i −0.997677 0.0681203i \(-0.978300\pi\)
0.997677 0.0681203i \(-0.0217002\pi\)
\(432\) 0 0
\(433\) 8.48528i 0.407777i −0.978994 0.203888i \(-0.934642\pi\)
0.978994 0.203888i \(-0.0653579\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −40.0000 −1.91346
\(438\) 0 0
\(439\) 6.32456i 0.301855i −0.988545 0.150927i \(-0.951774\pi\)
0.988545 0.150927i \(-0.0482259\pi\)
\(440\) 0 0
\(441\) 14.8885 14.8098i 0.708978 0.705230i
\(442\) 0 0
\(443\) 6.32456i 0.300489i −0.988649 0.150244i \(-0.951994\pi\)
0.988649 0.150244i \(-0.0480060\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.65685i 0.266963i 0.991051 + 0.133482i \(0.0426157\pi\)
−0.991051 + 0.133482i \(0.957384\pi\)
\(450\) 0 0
\(451\) 12.6491i 0.595623i
\(452\) 0 0
\(453\) 4.00000 5.65685i 0.187936 0.265782i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) −20.0000 5.65685i −0.933520 0.264039i
\(460\) 0 0
\(461\) 4.47214 0.208288 0.104144 0.994562i \(-0.466790\pi\)
0.104144 + 0.994562i \(0.466790\pi\)
\(462\) 0 0
\(463\) 40.2492 1.87054 0.935270 0.353935i \(-0.115157\pi\)
0.935270 + 0.353935i \(0.115157\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.0000 1.20314 0.601568 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) −10.0000 6.32456i −0.461757 0.292041i
\(470\) 0 0
\(471\) −4.00000 2.82843i −0.184310 0.130327i
\(472\) 0 0
\(473\) 12.6491i 0.581607i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 17.8885 6.32456i 0.819060 0.289581i
\(478\) 0 0
\(479\) 17.8885 0.817348 0.408674 0.912680i \(-0.365991\pi\)
0.408674 + 0.912680i \(0.365991\pi\)
\(480\) 0 0
\(481\) 25.2982i 1.15350i
\(482\) 0 0
\(483\) 11.0557 + 26.7912i 0.503053 + 1.21904i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.4164 0.607955 0.303978 0.952679i \(-0.401685\pi\)
0.303978 + 0.952679i \(0.401685\pi\)
\(488\) 0 0
\(489\) 13.4164 18.9737i 0.606711 0.858019i
\(490\) 0 0
\(491\) 2.82843i 0.127645i 0.997961 + 0.0638226i \(0.0203292\pi\)
−0.997961 + 0.0638226i \(0.979671\pi\)
\(492\) 0 0
\(493\) 22.6274i 1.01909i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.0000 31.6228i 0.897123 1.41848i
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 0 0
\(501\) −6.00000 + 8.48528i −0.268060 + 0.379094i
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 5.00000 7.07107i 0.222058 0.314037i
\(508\) 0 0
\(509\) −13.4164 −0.594672 −0.297336 0.954773i \(-0.596098\pi\)
−0.297336 + 0.954773i \(0.596098\pi\)
\(510\) 0 0
\(511\) 12.0000 18.9737i 0.530849 0.839346i
\(512\) 0 0
\(513\) 8.94427 31.6228i 0.394899 1.39618i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.9706i 0.746364i
\(518\) 0 0
\(519\) 8.00000 11.3137i 0.351161 0.496617i
\(520\) 0 0
\(521\) −13.4164 −0.587784 −0.293892 0.955839i \(-0.594951\pi\)
−0.293892 + 0.955839i \(0.594951\pi\)
\(522\) 0 0
\(523\) 36.7696i 1.60782i 0.594751 + 0.803910i \(0.297251\pi\)
−0.594751 + 0.803910i \(0.702749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.2982i 1.10201i
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) −8.94427 25.2982i −0.388148 1.09785i
\(532\) 0 0
\(533\) 12.6491i 0.547894i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −20.0000 14.1421i −0.863064 0.610278i
\(538\) 0 0
\(539\) −17.8885 + 8.48528i −0.770514 + 0.365487i
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) −17.8885 12.6491i −0.767671 0.542825i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −40.2492 −1.72093 −0.860466 0.509507i \(-0.829828\pi\)
−0.860466 + 0.509507i \(0.829828\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 35.7771 1.52416
\(552\) 0 0
\(553\) 26.8328 + 16.9706i 1.14105 + 0.721662i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.6228i 1.33990i −0.742406 0.669950i \(-0.766316\pi\)
0.742406 0.669950i \(-0.233684\pi\)
\(558\) 0 0
\(559\) 12.6491i 0.535000i
\(560\) 0 0
\(561\) 16.0000 + 11.3137i 0.675521 + 0.477665i
\(562\) 0 0
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −23.6525 + 2.74962i −0.993311 + 0.115473i
\(568\) 0 0
\(569\) 28.2843i 1.18574i −0.805299 0.592869i \(-0.797995\pi\)
0.805299 0.592869i \(-0.202005\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) −4.00000 2.82843i −0.167102 0.118159i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 36.7696i 1.53074i −0.643593 0.765368i \(-0.722557\pi\)
0.643593 0.765368i \(-0.277443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.3607 14.1421i −0.927677 0.586715i
\(582\) 0 0
\(583\) −17.8885 −0.740868
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.0000 −0.412744 −0.206372 0.978474i \(-0.566166\pi\)
−0.206372 + 0.978474i \(0.566166\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) 26.8328 + 18.9737i 1.10375 + 0.780472i
\(592\) 0 0
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 26.8328 + 18.9737i 1.09819 + 0.776540i
\(598\) 0 0
\(599\) 19.7990i 0.808965i −0.914546 0.404482i \(-0.867452\pi\)
0.914546 0.404482i \(-0.132548\pi\)
\(600\) 0 0
\(601\) 12.6491i 0.515968i 0.966149 + 0.257984i \(0.0830582\pi\)
−0.966149 + 0.257984i \(0.916942\pi\)
\(602\) 0 0
\(603\) 4.47214 + 12.6491i 0.182119 + 0.515112i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.7990i 0.803616i 0.915724 + 0.401808i \(0.131618\pi\)
−0.915724 + 0.401808i \(0.868382\pi\)
\(608\) 0 0
\(609\) −9.88854 23.9628i −0.400704 0.971022i
\(610\) 0 0
\(611\) 16.9706i 0.686555i
\(612\) 0 0
\(613\) 8.94427 0.361256 0.180628 0.983552i \(-0.442187\pi\)
0.180628 + 0.983552i \(0.442187\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.32456i 0.254617i −0.991863 0.127309i \(-0.959366\pi\)
0.991863 0.127309i \(-0.0406338\pi\)
\(618\) 0 0
\(619\) 18.9737i 0.762616i −0.924448 0.381308i \(-0.875474\pi\)
0.924448 0.381308i \(-0.124526\pi\)
\(620\) 0 0
\(621\) 8.94427 31.6228i 0.358921 1.26898i
\(622\) 0 0
\(623\) 10.0000 + 6.32456i 0.400642 + 0.253388i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −17.8885 + 25.2982i −0.714400 + 1.01031i
\(628\) 0 0
\(629\) −35.7771 −1.42653
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 16.0000 22.6274i 0.635943 0.899359i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −17.8885 + 8.48528i −0.708770 + 0.336199i
\(638\) 0 0
\(639\) −40.0000 + 14.1421i −1.58238 + 0.559454i
\(640\) 0 0
\(641\) 28.2843i 1.11716i 0.829450 + 0.558581i \(0.188654\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 31.1127i 1.22697i −0.789708 0.613483i \(-0.789768\pi\)
0.789708 0.613483i \(-0.210232\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.0000 0.393141 0.196570 0.980490i \(-0.437020\pi\)
0.196570 + 0.980490i \(0.437020\pi\)
\(648\) 0 0
\(649\) 25.2982i 0.993042i
\(650\) 0 0
\(651\) 11.0557 + 26.7912i 0.433308 + 1.05003i
\(652\) 0 0
\(653\) 44.2719i 1.73249i −0.499617 0.866246i \(-0.666526\pi\)
0.499617 0.866246i \(-0.333474\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −24.0000 + 8.48528i −0.936329 + 0.331042i
\(658\) 0 0
\(659\) 8.48528i 0.330540i 0.986248 + 0.165270i \(0.0528495\pi\)
−0.986248 + 0.165270i \(0.947151\pi\)
\(660\) 0 0
\(661\) 25.2982i 0.983987i −0.870599 0.491993i \(-0.836268\pi\)
0.870599 0.491993i \(-0.163732\pi\)
\(662\) 0 0
\(663\) 16.0000 + 11.3137i 0.621389 + 0.439388i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7771 1.38529
\(668\) 0 0
\(669\) −4.00000 2.82843i −0.154649 0.109353i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −35.7771 −1.37911 −0.689553 0.724236i \(-0.742193\pi\)
−0.689553 + 0.724236i \(0.742193\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.0000 0.922395 0.461197 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) 0 0
\(679\) −12.0000 + 18.9737i −0.460518 + 0.728142i
\(680\) 0 0
\(681\) −6.00000 + 8.48528i −0.229920 + 0.325157i
\(682\) 0 0
\(683\) 6.32456i 0.242002i −0.992652 0.121001i \(-0.961390\pi\)
0.992652 0.121001i \(-0.0386105\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −17.8885 12.6491i −0.682491 0.482594i
\(688\) 0 0
\(689\) −17.8885 −0.681499
\(690\) 0 0
\(691\) 18.9737i 0.721792i 0.932606 + 0.360896i \(0.117529\pi\)
−0.932606 + 0.360896i \(0.882471\pi\)
\(692\) 0 0
\(693\) 21.8885 + 4.98915i 0.831477 + 0.189522i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 17.8885 0.677577
\(698\) 0 0
\(699\) 8.94427 + 6.32456i 0.338303 + 0.239217i
\(700\) 0 0
\(701\) 45.2548i 1.70925i 0.519244 + 0.854626i \(0.326213\pi\)
−0.519244 + 0.854626i \(0.673787\pi\)
\(702\) 0 0
\(703\) 56.5685i 2.13352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.0000 6.32456i −0.376089 0.237859i
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) −12.0000 33.9411i −0.450035 1.27289i
\(712\) 0 0
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −12.0000 8.48528i −0.448148 0.316889i
\(718\) 0 0
\(719\) 17.8885 0.667130 0.333565 0.942727i \(-0.391748\pi\)
0.333565 + 0.942727i \(0.391748\pi\)
\(720\) 0 0
\(721\) −4.00000 + 6.32456i −0.148968 + 0.235539i
\(722\) 0 0
\(723\) −17.8885 12.6491i −0.665282 0.470425i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 25.4558i 0.944105i −0.881570 0.472052i \(-0.843513\pi\)
0.881570 0.472052i \(-0.156487\pi\)
\(728\) 0 0
\(729\) 23.0000 + 14.1421i 0.851852 + 0.523783i
\(730\) 0 0
\(731\) −17.8885 −0.661632
\(732\) 0 0
\(733\) 25.4558i 0.940233i 0.882604 + 0.470117i \(0.155788\pi\)
−0.882604 + 0.470117i \(0.844212\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.6491i 0.465936i
\(738\) 0 0
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) −17.8885 + 25.2982i −0.657152 + 0.929353i
\(742\) 0 0
\(743\) 18.9737i 0.696076i 0.937480 + 0.348038i \(0.113152\pi\)
−0.937480 + 0.348038i \(0.886848\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10.0000 + 28.2843i 0.365881 + 1.03487i
\(748\) 0 0
\(749\) 26.8328 42.4264i 0.980450 1.55023i
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 26.8328 37.9473i 0.977842 1.38288i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.94427 −0.325085 −0.162543 0.986702i \(-0.551969\pi\)
−0.162543 + 0.986702i \(0.551969\pi\)
\(758\) 0 0
\(759\) −17.8885 + 25.2982i −0.649313 + 0.918267i
\(760\) 0 0
\(761\) −4.47214 −0.162115 −0.0810574 0.996709i \(-0.525830\pi\)
−0.0810574 + 0.996709i \(0.525830\pi\)
\(762\) 0 0
\(763\) −4.47214 2.82843i −0.161902 0.102396i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.2982i 0.913466i
\(768\) 0 0
\(769\) 25.2982i 0.912277i 0.889909 + 0.456139i \(0.150768\pi\)
−0.889909 + 0.456139i \(0.849232\pi\)
\(770\) 0 0
\(771\) −20.0000 + 28.2843i −0.720282 + 1.01863i
\(772\) 0 0
\(773\) −32.0000 −1.15096 −0.575480 0.817816i \(-0.695185\pi\)
−0.575480 + 0.817816i \(0.695185\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −37.8885 + 15.6352i −1.35924 + 0.560908i
\(778\) 0 0
\(779\) 28.2843i 1.01339i
\(780\) 0 0
\(781\) 40.0000 1.43131
\(782\) 0 0
\(783\) −8.00000 + 28.2843i −0.285897 + 1.01080i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 8.48528i 0.302468i −0.988498 0.151234i \(-0.951675\pi\)
0.988498 0.151234i \(-0.0483246\pi\)
\(788\) 0 0
\(789\) −8.94427 6.32456i −0.318425 0.225160i
\(790\) 0 0
\(791\) 26.8328 42.4264i 0.954065 1.50851i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.0000 −0.566749 −0.283375 0.959009i \(-0.591454\pi\)
−0.283375 + 0.959009i \(0.591454\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −4.47214 12.6491i −0.158015 0.446934i
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.47214 6.32456i 0.157427 0.222635i
\(808\) 0 0
\(809\) 22.6274i 0.795538i 0.917486 + 0.397769i \(0.130215\pi\)
−0.917486 + 0.397769i \(0.869785\pi\)
\(810\) 0 0
\(811\) 31.6228i 1.11043i −0.831708 0.555213i \(-0.812637\pi\)
0.831708 0.555213i \(-0.187363\pi\)
\(812\) 0 0
\(813\) 8.94427 + 6.32456i 0.313689 + 0.221812i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 28.2843i 0.989541i
\(818\) 0 0
\(819\) 21.8885 + 4.98915i 0.764848 + 0.174335i
\(820\) 0 0
\(821\) 11.3137i 0.394851i −0.980318 0.197426i \(-0.936742\pi\)
0.980318 0.197426i \(-0.0632581\pi\)
\(822\) 0 0
\(823\) 22.3607 0.779444 0.389722 0.920932i \(-0.372571\pi\)
0.389722 + 0.920932i \(0.372571\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.9737i 0.659779i −0.944020 0.329890i \(-0.892989\pi\)
0.944020 0.329890i \(-0.107011\pi\)
\(828\) 0 0
\(829\) 25.2982i 0.878644i 0.898330 + 0.439322i \(0.144781\pi\)
−0.898330 + 0.439322i \(0.855219\pi\)
\(830\) 0 0
\(831\) −26.8328 + 37.9473i −0.930820 + 1.31638i
\(832\) 0 0
\(833\) 12.0000 + 25.2982i 0.415775 + 0.876531i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.94427 31.6228i 0.309159 1.09304i
\(838\) 0 0
\(839\) 35.7771 1.23516 0.617581 0.786507i \(-0.288113\pi\)
0.617581 + 0.786507i \(0.288113\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 0 0
\(843\) −24.0000 16.9706i −0.826604 0.584497i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.70820 + 4.24264i 0.230497 + 0.145779i
\(848\) 0 0
\(849\) −36.0000 25.4558i −1.23552 0.873642i
\(850\) 0 0
\(851\) 56.5685i 1.93914i
\(852\) 0 0
\(853\) 36.7696i 1.25897i 0.777014 + 0.629483i \(0.216733\pi\)
−0.777014 + 0.629483i \(0.783267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) 0 0
\(859\) 44.2719i 1.51054i −0.655415 0.755269i \(-0.727506\pi\)
0.655415 0.755269i \(-0.272494\pi\)
\(860\) 0 0
\(861\) 18.9443 7.81758i 0.645619 0.266422i
\(862\) 0 0
\(863\) 18.9737i 0.645871i −0.946421 0.322936i \(-0.895330\pi\)
0.946421 0.322936i \(-0.104670\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.00000 + 1.41421i −0.0339618 + 0.0480292i
\(868\) 0 0
\(869\) 33.9411i 1.15137i
\(870\) 0 0
\(871\) 12.6491i 0.428599i
\(872\) 0 0
\(873\) 24.0000 8.48528i 0.812277 0.287183i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.8328 0.906080 0.453040 0.891490i \(-0.350340\pi\)
0.453040 + 0.891490i \(0.350340\pi\)
\(878\) 0 0
\(879\) 32.0000 45.2548i 1.07933 1.52641i
\(880\) 0 0
\(881\) 31.3050 1.05469 0.527345 0.849651i \(-0.323187\pi\)
0.527345 + 0.849651i \(0.323187\pi\)
\(882\) 0 0
\(883\) 40.2492 1.35449 0.677247 0.735756i \(-0.263173\pi\)
0.677247 + 0.735756i \(0.263173\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 30.0000 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) 0 0
\(889\) −10.0000 6.32456i −0.335389 0.212119i
\(890\) 0 0
\(891\) −16.0000 19.7990i −0.536020 0.663291i
\(892\) 0 0
\(893\) 37.9473i 1.26986i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −17.8885 + 25.2982i −0.597281 + 0.844683i
\(898\) 0 0
\(899\) 35.7771 1.19323
\(900\) 0 0
\(901\) 25.2982i 0.842806i
\(902\) 0 0
\(903\) −18.9443 + 7.81758i −0.630426 + 0.260153i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 13.4164 0.445485 0.222742 0.974877i \(-0.428499\pi\)
0.222742 + 0.974877i \(0.428499\pi\)
\(908\) 0 0
\(909\) 4.47214 + 12.6491i 0.148331 + 0.419545i
\(910\) 0 0
\(911\) 14.1421i 0.468550i −0.972170 0.234275i \(-0.924728\pi\)
0.972170 0.234275i \(-0.0752716\pi\)
\(912\) 0 0
\(913\) 28.2843i 0.936073i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 20.0000 + 12.6491i 0.660458 + 0.417710i
\(918\) 0 0
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −12.0000 8.48528i −0.395413 0.279600i
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000 2.82843i 0.262754 0.0928977i
\(928\) 0 0
\(929\) 4.47214 0.146726 0.0733630 0.997305i \(-0.476627\pi\)
0.0733630 + 0.997305i \(0.476627\pi\)
\(930\) 0 0
\(931\) −40.0000 + 18.9737i −1.31095 + 0.621837i
\(932\) 0 0
\(933\) −17.8885 + 25.2982i −0.585645 + 0.828227i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8.48528i 0.277202i 0.990348 + 0.138601i \(0.0442606\pi\)
−0.990348 + 0.138601i \(0.955739\pi\)
\(938\) 0 0
\(939\) 36.0000 + 25.4558i 1.17482 + 0.830720i
\(940\) 0 0
\(941\) 40.2492 1.31209 0.656044 0.754723i \(-0.272229\pi\)
0.656044 + 0.754723i \(0.272229\pi\)
\(942\) 0 0
\(943\) 28.2843i 0.921063i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 18.9737i 0.616561i 0.951295 + 0.308281i \(0.0997536\pi\)
−0.951295 + 0.308281i \(0.900246\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 26.8328 + 18.9737i 0.870114 + 0.615263i
\(952\) 0 0
\(953\) 18.9737i 0.614617i −0.951610 0.307309i \(-0.900572\pi\)
0.951610 0.307309i \(-0.0994284\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 16.0000 22.6274i 0.517207 0.731441i
\(958\) 0 0
\(959\) 8.94427 14.1421i 0.288826 0.456673i
\(960\) 0 0
\(961\) −9.00000 −0.290323
\(962\) 0 0
\(963\) −53.6656 + 18.9737i −1.72935 + 0.611418i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 40.2492 1.29433 0.647164 0.762351i \(-0.275955\pi\)
0.647164 + 0.762351i \(0.275955\pi\)
\(968\) 0 0
\(969\) 35.7771 + 25.2982i 1.14933 + 0.812696i
\(970\) 0 0
\(971\) −44.7214 −1.43518 −0.717588 0.696467i \(-0.754754\pi\)
−0.717588 + 0.696467i \(0.754754\pi\)
\(972\) 0 0
\(973\) −8.94427 + 14.1421i −0.286740 + 0.453376i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.9737i 0.607021i 0.952828 + 0.303511i \(0.0981588\pi\)
−0.952828 + 0.303511i \(0.901841\pi\)
\(978\) 0 0
\(979\) 12.6491i 0.404267i
\(980\) 0 0
\(981\) 2.00000 + 5.65685i 0.0638551 + 0.180609i
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 25.4164 10.4884i 0.809013 0.333849i
\(988\) 0 0
\(989\) 28.2843i 0.899388i
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 0 0
\(993\) 8.00000 11.3137i 0.253872 0.359030i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.48528i 0.268732i 0.990932 + 0.134366i \(0.0428997\pi\)
−0.990932 + 0.134366i \(0.957100\pi\)
\(998\) 0 0
\(999\) 44.7214 + 12.6491i 1.41492 + 0.400200i
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 2100.2.d.i.1301.2 4
3.2 odd 2 2100.2.d.f.1301.2 4
5.2 odd 4 420.2.f.b.209.2 yes 8
5.3 odd 4 420.2.f.b.209.8 yes 8
5.4 even 2 2100.2.d.f.1301.3 4
7.6 odd 2 2100.2.d.f.1301.4 4
15.2 even 4 420.2.f.b.209.3 yes 8
15.8 even 4 420.2.f.b.209.5 yes 8
15.14 odd 2 inner 2100.2.d.i.1301.3 4
20.3 even 4 1680.2.k.g.209.2 8
20.7 even 4 1680.2.k.g.209.8 8
21.20 even 2 inner 2100.2.d.i.1301.4 4
35.13 even 4 420.2.f.b.209.1 8
35.27 even 4 420.2.f.b.209.7 yes 8
35.34 odd 2 inner 2100.2.d.i.1301.1 4
60.23 odd 4 1680.2.k.g.209.3 8
60.47 odd 4 1680.2.k.g.209.5 8
105.62 odd 4 420.2.f.b.209.6 yes 8
105.83 odd 4 420.2.f.b.209.4 yes 8
105.104 even 2 2100.2.d.f.1301.1 4
140.27 odd 4 1680.2.k.g.209.1 8
140.83 odd 4 1680.2.k.g.209.7 8
420.83 even 4 1680.2.k.g.209.6 8
420.167 even 4 1680.2.k.g.209.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.f.b.209.1 8 35.13 even 4
420.2.f.b.209.2 yes 8 5.2 odd 4
420.2.f.b.209.3 yes 8 15.2 even 4
420.2.f.b.209.4 yes 8 105.83 odd 4
420.2.f.b.209.5 yes 8 15.8 even 4
420.2.f.b.209.6 yes 8 105.62 odd 4
420.2.f.b.209.7 yes 8 35.27 even 4
420.2.f.b.209.8 yes 8 5.3 odd 4
1680.2.k.g.209.1 8 140.27 odd 4
1680.2.k.g.209.2 8 20.3 even 4
1680.2.k.g.209.3 8 60.23 odd 4
1680.2.k.g.209.4 8 420.167 even 4
1680.2.k.g.209.5 8 60.47 odd 4
1680.2.k.g.209.6 8 420.83 even 4
1680.2.k.g.209.7 8 140.83 odd 4
1680.2.k.g.209.8 8 20.7 even 4
2100.2.d.f.1301.1 4 105.104 even 2
2100.2.d.f.1301.2 4 3.2 odd 2
2100.2.d.f.1301.3 4 5.4 even 2
2100.2.d.f.1301.4 4 7.6 odd 2
2100.2.d.i.1301.1 4 35.34 odd 2 inner
2100.2.d.i.1301.2 4 1.1 even 1 trivial
2100.2.d.i.1301.3 4 15.14 odd 2 inner
2100.2.d.i.1301.4 4 21.20 even 2 inner