Properties

Label 117.3.bd.a.46.1
Level $117$
Weight $3$
Character 117.46
Analytic conductor $3.188$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(19,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.bd (of order \(12\), degree \(4\), 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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 46.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 117.46
Dual form 117.3.bd.a.28.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.46410 + 2.00000i) q^{4} +(-12.4282 - 3.33013i) q^{7} +(-12.9904 - 0.500000i) q^{13} +(8.00000 - 13.8564i) q^{16} +(-9.83013 + 36.6865i) q^{19} -25.0000i q^{25} +(49.7128 - 13.3205i) q^{28} +(23.8109 + 23.8109i) q^{31} +(-7.92116 - 29.5622i) q^{37} +(-19.5000 + 11.2583i) q^{43} +(100.935 + 58.2750i) q^{49} +(46.0000 - 24.2487i) q^{52} +(-56.2917 - 97.5000i) q^{61} +64.0000i q^{64} +(-99.9711 + 26.7872i) q^{67} +(-86.2224 + 86.2224i) q^{73} +(-39.3205 - 146.746i) q^{76} -157.617 q^{79} +(159.782 + 49.4737i) q^{91} +(-13.4948 + 50.3634i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{7} + 32 q^{16} - 22 q^{19} + 88 q^{28} - 26 q^{31} - 146 q^{37} - 78 q^{43} + 234 q^{49} + 184 q^{52} - 244 q^{67} - 286 q^{73} - 88 q^{76} + 362 q^{91} + 334 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{11}{12}\right)\) \(1\)

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 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0 0
\(4\) −3.46410 + 2.00000i −0.866025 + 0.500000i
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) −12.4282 3.33013i −1.77546 0.475732i −0.785714 0.618590i \(-0.787704\pi\)
−0.989743 + 0.142857i \(0.954371\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) 0 0
\(13\) −12.9904 0.500000i −0.999260 0.0384615i
\(14\) 0 0
\(15\) 0 0
\(16\) 8.00000 13.8564i 0.500000 0.866025i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −9.83013 + 36.6865i −0.517375 + 1.93087i −0.227901 + 0.973684i \(0.573186\pi\)
−0.289474 + 0.957186i \(0.593480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 25.0000i 1.00000i
\(26\) 0 0
\(27\) 0 0
\(28\) 49.7128 13.3205i 1.77546 0.475732i
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 23.8109 + 23.8109i 0.768093 + 0.768093i 0.977771 0.209677i \(-0.0672414\pi\)
−0.209677 + 0.977771i \(0.567241\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.92116 29.5622i −0.214085 0.798978i −0.986486 0.163843i \(-0.947611\pi\)
0.772401 0.635135i \(-0.219056\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(42\) 0 0
\(43\) −19.5000 + 11.2583i −0.453488 + 0.261822i −0.709302 0.704904i \(-0.750990\pi\)
0.255814 + 0.966726i \(0.417657\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) 0 0
\(49\) 100.935 + 58.2750i 2.05990 + 1.18929i
\(50\) 0 0
\(51\) 0 0
\(52\) 46.0000 24.2487i 0.884615 0.466321i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0 0
\(61\) −56.2917 97.5000i −0.922814 1.59836i −0.795040 0.606557i \(-0.792550\pi\)
−0.127774 0.991803i \(-0.540783\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −99.9711 + 26.7872i −1.49211 + 0.399809i −0.910448 0.413624i \(-0.864263\pi\)
−0.581659 + 0.813433i \(0.697596\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(72\) 0 0
\(73\) −86.2224 + 86.2224i −1.18113 + 1.18113i −0.201677 + 0.979452i \(0.564639\pi\)
−0.979452 + 0.201677i \(0.935361\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −39.3205 146.746i −0.517375 1.93087i
\(77\) 0 0
\(78\) 0 0
\(79\) −157.617 −1.99515 −0.997574 0.0696203i \(-0.977821\pi\)
−0.997574 + 0.0696203i \(0.977821\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(90\) 0 0
\(91\) 159.782 + 49.4737i 1.75585 + 0.543667i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.4948 + 50.3634i −0.139122 + 0.519211i 0.860825 + 0.508902i \(0.169948\pi\)
−0.999947 + 0.0103093i \(0.996718\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 50.0000 + 86.6025i 0.500000 + 0.866025i
\(101\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(102\) 0 0
\(103\) 202.650i 1.96748i −0.179612 0.983738i \(-0.557484\pi\)
0.179612 0.983738i \(-0.442516\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(108\) 0 0
\(109\) 10.7724 + 10.7724i 0.0988295 + 0.0988295i 0.754793 0.655963i \(-0.227737\pi\)
−0.655963 + 0.754793i \(0.727737\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −145.569 + 145.569i −1.29973 + 1.29973i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 104.789 60.5000i 0.866025 0.500000i
\(122\) 0 0
\(123\) 0 0
\(124\) −130.105 34.8616i −1.04923 0.281142i
\(125\) 0 0
\(126\) 0 0
\(127\) 219.104 + 126.500i 1.72523 + 0.996063i 0.906940 + 0.421260i \(0.138412\pi\)
0.818292 + 0.574803i \(0.194921\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 244.342 423.212i 1.83716 3.18205i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(138\) 0 0
\(139\) 114.500 + 198.320i 0.823741 + 1.42676i 0.902878 + 0.429898i \(0.141450\pi\)
−0.0791367 + 0.996864i \(0.525216\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 86.5641 + 86.5641i 0.584892 + 0.584892i
\(149\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(150\) 0 0
\(151\) −94.5026 + 94.5026i −0.625845 + 0.625845i −0.947020 0.321175i \(-0.895922\pi\)
0.321175 + 0.947020i \(0.395922\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 193.000 1.22930 0.614650 0.788800i \(-0.289297\pi\)
0.614650 + 0.788800i \(0.289297\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −292.947 78.4948i −1.79722 0.481563i −0.803681 0.595060i \(-0.797128\pi\)
−0.993538 + 0.113497i \(0.963795\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(168\) 0 0
\(169\) 168.500 + 12.9904i 0.997041 + 0.0768662i
\(170\) 0 0
\(171\) 0 0
\(172\) 45.0333 78.0000i 0.261822 0.453488i
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) −83.2532 + 310.705i −0.475732 + 1.77546i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 180.133i 0.995212i −0.867403 0.497606i \(-0.834213\pi\)
0.867403 0.497606i \(-0.165787\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) −91.7872 342.554i −0.475581 1.77489i −0.619171 0.785256i \(-0.712531\pi\)
0.143590 0.989637i \(-0.454135\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −466.200 −2.37857
\(197\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(198\) 0 0
\(199\) −94.3968 + 54.5000i −0.474356 + 0.273869i −0.718061 0.695980i \(-0.754970\pi\)
0.243706 + 0.969849i \(0.421637\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −110.851 + 176.000i −0.532939 + 0.846154i
\(209\) 0 0
\(210\) 0 0
\(211\) −168.875 + 292.500i −0.800355 + 1.38626i 0.119027 + 0.992891i \(0.462022\pi\)
−0.919382 + 0.393365i \(0.871311\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −216.633 375.220i −0.998310 1.72912i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 32.1122 8.60443i 0.144001 0.0385849i −0.186099 0.982531i \(-0.559584\pi\)
0.330099 + 0.943946i \(0.392918\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(228\) 0 0
\(229\) −241.631 + 241.631i −1.05516 + 1.05516i −0.0567686 + 0.998387i \(0.518080\pi\)
−0.998387 + 0.0567686i \(0.981920\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) 460.336 + 123.347i 1.91011 + 0.511812i 0.993776 + 0.111397i \(0.0355327\pi\)
0.916334 + 0.400415i \(0.131134\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 390.000 + 225.167i 1.59836 + 0.922814i
\(245\) 0 0
\(246\) 0 0
\(247\) 146.040 471.657i 0.591257 1.90954i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −128.000 221.703i −0.500000 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 393.783i 1.52040i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 292.736 292.736i 1.09230 1.09230i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −28.0129 104.546i −0.103369 0.385777i 0.894786 0.446494i \(-0.147328\pi\)
−0.998155 + 0.0607176i \(0.980661\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −468.000 + 270.200i −1.68953 + 0.975451i −0.734657 + 0.678439i \(0.762657\pi\)
−0.954874 + 0.297012i \(0.904010\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 51.0955 + 29.5000i 0.180549 + 0.104240i 0.587551 0.809187i \(-0.300092\pi\)
−0.407001 + 0.913428i \(0.633426\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −144.500 + 250.281i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 126.238 471.128i 0.432323 1.61345i
\(293\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 279.842 74.9833i 0.929706 0.249114i
\(302\) 0 0
\(303\) 0 0
\(304\) 429.703 + 429.703i 1.41350 + 1.41350i
\(305\) 0 0
\(306\) 0 0
\(307\) 275.189 275.189i 0.896381 0.896381i −0.0987325 0.995114i \(-0.531479\pi\)
0.995114 + 0.0987325i \(0.0314788\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 427.817 1.36683 0.683413 0.730032i \(-0.260495\pi\)
0.683413 + 0.730032i \(0.260495\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 546.000 315.233i 1.72785 0.997574i
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.5000 + 324.760i −0.0384615 + 0.999260i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −162.813 + 607.628i −0.491884 + 1.83573i 0.0549442 + 0.998489i \(0.482502\pi\)
−0.546828 + 0.837245i \(0.684165\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 167.000i 0.495549i −0.968818 0.247774i \(-0.920301\pi\)
0.968818 0.247774i \(-0.0796992\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −614.575 614.575i −1.79176 1.79176i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 154.871 + 577.987i 0.443757 + 1.65612i 0.719198 + 0.694805i \(0.244510\pi\)
−0.275441 + 0.961318i \(0.588824\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) −936.635 540.767i −2.59456 1.49797i
\(362\) 0 0
\(363\) 0 0
\(364\) −652.449 + 148.182i −1.79244 + 0.407094i
\(365\) 0 0
\(366\) 0 0
\(367\) 113.500 196.588i 0.309264 0.535661i −0.668937 0.743319i \(-0.733251\pi\)
0.978202 + 0.207657i \(0.0665839\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 236.425 + 409.500i 0.633847 + 1.09786i 0.986758 + 0.162198i \(0.0518585\pi\)
−0.352911 + 0.935657i \(0.614808\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −110.920 + 29.7211i −0.292666 + 0.0784197i −0.402165 0.915567i \(-0.631742\pi\)
0.109499 + 0.993987i \(0.465075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −53.9794 201.454i −0.139122 0.519211i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −514.420 137.838i −1.29577 0.347200i −0.455919 0.890021i \(-0.650689\pi\)
−0.839848 + 0.542821i \(0.817356\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −346.410 200.000i −0.866025 0.500000i
\(401\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(402\) 0 0
\(403\) −297.407 321.218i −0.737983 0.797067i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 121.228 452.430i 0.296402 1.10619i −0.643696 0.765281i \(-0.722600\pi\)
0.940098 0.340905i \(-0.110733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 405.300 + 702.000i 0.983738 + 1.70388i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −105.044 105.044i −0.249511 0.249511i 0.571259 0.820770i \(-0.306455\pi\)
−0.820770 + 0.571259i \(0.806455\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 374.917 + 1399.21i 0.878025 + 3.27683i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(432\) 0 0
\(433\) −310.903 + 179.500i −0.718021 + 0.414550i −0.814024 0.580831i \(-0.802728\pi\)
0.0960028 + 0.995381i \(0.469394\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −58.8616 15.7719i −0.135004 0.0361741i
\(437\) 0 0
\(438\) 0 0
\(439\) −448.500 258.942i −1.02164 0.589844i −0.107062 0.994252i \(-0.534144\pi\)
−0.914579 + 0.404408i \(0.867478\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 213.128 795.405i 0.475732 1.77546i
\(449\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −863.395 + 231.346i −1.88927 + 0.506228i −0.890591 + 0.454805i \(0.849709\pi\)
−0.998677 + 0.0514223i \(0.983625\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(462\) 0 0
\(463\) 498.739 498.739i 1.07719 1.07719i 0.0804300 0.996760i \(-0.474371\pi\)
0.996760 0.0804300i \(-0.0256293\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 1331.67 2.83937
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 917.163 + 245.753i 1.93087 + 0.517375i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(480\) 0 0
\(481\) 88.1178 + 387.985i 0.183197 + 0.806621i
\(482\) 0 0
\(483\) 0 0
\(484\) −242.000 + 419.156i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 148.163 552.953i 0.304237 1.13543i −0.629363 0.777111i \(-0.716684\pi\)
0.933600 0.358316i \(-0.116649\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 520.420 139.446i 1.04923 0.281142i
\(497\) 0 0
\(498\) 0 0
\(499\) −485.831 485.831i −0.973608 0.973608i 0.0260521 0.999661i \(-0.491706\pi\)
−0.999661 + 0.0260521i \(0.991706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −1012.00 −1.99213
\(509\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(510\) 0 0
\(511\) 1358.72 784.458i 2.65895 1.53514i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −491.000 + 850.437i −0.938815 + 1.62607i −0.171128 + 0.985249i \(0.554741\pi\)
−0.767686 + 0.640826i \(0.778592\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −264.500 458.127i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 1954.73i 3.67431i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 764.561 764.561i 1.41324 1.41324i 0.680334 0.732902i \(-0.261835\pi\)
0.732902 0.680334i \(-0.238165\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 587.000 1.07313 0.536563 0.843860i \(-0.319722\pi\)
0.536563 + 0.843860i \(0.319722\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1958.89 + 524.883i 3.54230 + 0.949156i
\(554\) 0 0
\(555\) 0 0
\(556\) −793.279 458.000i −1.42676 0.823741i
\(557\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(558\) 0 0
\(559\) 258.942 136.500i 0.463223 0.244186i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 886.000i 1.55166i −0.630940 0.775832i \(-0.717330\pi\)
0.630940 0.775832i \(-0.282670\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 162.303 + 162.303i 0.281287 + 0.281287i 0.833622 0.552335i \(-0.186263\pi\)
−0.552335 + 0.833622i \(0.686263\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(588\) 0 0
\(589\) −1107.60 + 639.475i −1.88048 + 1.08570i
\(590\) 0 0
\(591\) 0 0
\(592\) −472.995 126.739i −0.798978 0.214085i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −540.400 + 936.000i −0.899168 + 1.55740i −0.0706077 + 0.997504i \(0.522494\pi\)
−0.828560 + 0.559900i \(0.810839\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 138.361 516.372i 0.229075 0.854920i
\(605\) 0 0
\(606\) 0 0
\(607\) −407.000 704.945i −0.670511 1.16136i −0.977759 0.209729i \(-0.932742\pi\)
0.307249 0.951629i \(-0.400592\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −929.329 + 249.013i −1.51603 + 0.406220i −0.918434 0.395574i \(-0.870546\pi\)
−0.597600 + 0.801794i \(0.703879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(618\) 0 0
\(619\) 76.9943 76.9943i 0.124385 0.124385i −0.642174 0.766559i \(-0.721967\pi\)
0.766559 + 0.642174i \(0.221967\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −625.000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −668.572 + 386.000i −1.06460 + 0.614650i
\(629\) 0 0
\(630\) 0 0
\(631\) −895.586 239.972i −1.41931 0.380304i −0.534073 0.845438i \(-0.679339\pi\)
−0.885240 + 0.465135i \(0.846006\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1282.05 807.482i −2.01264 1.26763i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) −5.03829 + 18.8032i −0.00783560 + 0.0292429i −0.969733 0.244168i \(-0.921485\pi\)
0.961897 + 0.273411i \(0.0881518\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1171.79 313.979i 1.79722 0.481563i
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(660\) 0 0
\(661\) 337.262 + 1258.68i 0.510230 + 1.90420i 0.417946 + 0.908472i \(0.362750\pi\)
0.0922844 + 0.995733i \(0.470583\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 565.500 + 326.492i 0.840267 + 0.485129i 0.857355 0.514725i \(-0.172106\pi\)
−0.0170877 + 0.999854i \(0.505439\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −609.682 + 292.000i −0.901896 + 0.431953i
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 335.433 580.988i 0.494011 0.855652i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 360.267i 0.523643i
\(689\) 0 0
\(690\) 0 0
\(691\) 717.346 192.212i 1.03813 0.278165i 0.300790 0.953690i \(-0.402750\pi\)
0.737337 + 0.675525i \(0.236083\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −333.013 1242.82i −0.475732 1.77546i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1162.40 1.65348
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −761.972 204.170i −1.07471 0.287969i −0.322285 0.946643i \(-0.604451\pi\)
−0.752428 + 0.658674i \(0.771118\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) −674.850 + 2518.57i −0.935992 + 3.49317i
\(722\) 0 0
\(723\) 0 0
\(724\) 360.267 + 624.000i 0.497606 + 0.861878i
\(725\) 0 0
\(726\) 0 0
\(727\) 947.000i 1.30261i −0.758815 0.651307i \(-0.774221\pi\)
0.758815 0.651307i \(-0.225779\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −520.572 520.572i −0.710194 0.710194i 0.256381 0.966576i \(-0.417470\pi\)
−0.966576 + 0.256381i \(0.917470\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 71.4876 + 266.795i 0.0967356 + 0.361022i 0.997277 0.0737483i \(-0.0234961\pi\)
−0.900541 + 0.434771i \(0.856829\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −780.000 450.333i −1.03862 0.599645i −0.119174 0.992873i \(-0.538025\pi\)
−0.919441 + 0.393229i \(0.871358\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 630.466 1092.00i 0.832849 1.44254i −0.0629213 0.998018i \(-0.520042\pi\)
0.895770 0.444518i \(-0.146625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(762\) 0 0
\(763\) −98.0082 169.755i −0.128451 0.222484i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1123.45 301.029i 1.46093 0.391455i 0.561118 0.827736i \(-0.310371\pi\)
0.899811 + 0.436281i \(0.143705\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1003.07 + 1003.07i 1.29931 + 1.29931i
\(773\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(774\) 0 0
\(775\) 595.272 595.272i 0.768093 0.768093i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1614.96 932.400i 2.05990 1.18929i
\(785\) 0 0
\(786\) 0 0
\(787\) −209.505 56.1367i −0.266207 0.0713300i 0.123246 0.992376i \(-0.460669\pi\)
−0.389454 + 0.921046i \(0.627336\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 682.500 + 1294.71i 0.860656 + 1.63267i
\(794\) 0 0
\(795\) 0 0
\(796\) 218.000 377.587i 0.273869 0.474356i
\(797\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0