Properties

Label 180.2.k.b.163.1
Level $180$
Weight $2$
Character 180.163
Analytic conductor $1.437$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,2,Mod(127,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("180.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.k (of order \(4\), degree \(2\), 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: \(1.43730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 163.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 180.163
Dual form 180.2.k.b.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\) \(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-1.00000 - 2.00000i) q^{5} +(-2.00000 - 2.00000i) q^{8} +(-3.00000 - 1.00000i) q^{10} +(5.00000 + 5.00000i) q^{13} -4.00000 q^{16} +(3.00000 - 3.00000i) q^{17} +(-4.00000 + 2.00000i) q^{20} +(-3.00000 + 4.00000i) q^{25} +10.0000 q^{26} +10.0000i q^{29} +(-4.00000 + 4.00000i) q^{32} -6.00000i q^{34} +(5.00000 - 5.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} -10.0000 q^{41} -7.00000i q^{49} +(1.00000 + 7.00000i) q^{50} +(10.0000 - 10.0000i) q^{52} +(9.00000 + 9.00000i) q^{53} +(10.0000 + 10.0000i) q^{58} -12.0000 q^{61} +8.00000i q^{64} +(5.00000 - 15.0000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-5.00000 - 5.00000i) q^{73} -10.0000i q^{74} +(4.00000 + 8.00000i) q^{80} +(-10.0000 + 10.0000i) q^{82} +(-9.00000 - 3.00000i) q^{85} -10.0000i q^{89} +(-5.00000 + 5.00000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{5} - 4 q^{8} - 6 q^{10} + 10 q^{13} - 8 q^{16} + 6 q^{17} - 8 q^{20} - 6 q^{25} + 20 q^{26} - 8 q^{32} + 10 q^{37} - 4 q^{40} - 20 q^{41} + 2 q^{50} + 20 q^{52} + 18 q^{53} + 20 q^{58} - 24 q^{61} + 10 q^{65} - 12 q^{68} - 10 q^{73} + 8 q^{80} - 20 q^{82} - 18 q^{85} - 10 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 1.00000i 0.707107 0.707107i
\(3\) 0 0
\(4\) 2.00000i 1.00000i
\(5\) −1.00000 2.00000i −0.447214 0.894427i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −2.00000 2.00000i −0.707107 0.707107i
\(9\) 0 0
\(10\) −3.00000 1.00000i −0.948683 0.316228i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 5.00000 + 5.00000i 1.38675 + 1.38675i 0.832050 + 0.554700i \(0.187167\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 3.00000 3.00000i 0.727607 0.727607i −0.242536 0.970143i \(-0.577979\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −4.00000 + 2.00000i −0.894427 + 0.447214i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 10.0000 1.96116
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000i 1.85695i 0.371391 + 0.928477i \(0.378881\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.00000 + 4.00000i −0.707107 + 0.707107i
\(33\) 0 0
\(34\) 6.00000i 1.02899i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 5.00000i 0.821995 0.821995i −0.164399 0.986394i \(-0.552568\pi\)
0.986394 + 0.164399i \(0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.00000 + 6.00000i −0.316228 + 0.948683i
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 1.00000 + 7.00000i 0.141421 + 0.989949i
\(51\) 0 0
\(52\) 10.0000 10.0000i 1.38675 1.38675i
\(53\) 9.00000 + 9.00000i 1.23625 + 1.23625i 0.961524 + 0.274721i \(0.0885855\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 + 10.0000i 1.31306 + 1.31306i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 5.00000 15.0000i 0.620174 1.86052i
\(66\) 0 0
\(67\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) −6.00000 6.00000i −0.727607 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −5.00000 5.00000i −0.585206 0.585206i 0.351123 0.936329i \(-0.385800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 10.0000i 1.16248i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 + 8.00000i 0.447214 + 0.894427i
\(81\) 0 0
\(82\) −10.0000 + 10.0000i −1.10432 + 1.10432i
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) −9.00000 3.00000i −0.976187 0.325396i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00000 + 5.00000i −0.507673 + 0.507673i −0.913812 0.406138i \(-0.866875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −7.00000 7.00000i −0.707107 0.707107i
\(99\) 0 0
\(100\) 8.00000 + 6.00000i 0.800000 + 0.600000i
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(104\) 20.0000i 1.96116i
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.00000 + 1.00000i 0.0940721 + 0.0940721i 0.752577 0.658505i \(-0.228811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 20.0000 1.85695
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) −12.0000 + 12.0000i −1.08643 + 1.08643i
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0000 + 2.00000i 0.983870 + 0.178885i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 8.00000 + 8.00000i 0.707107 + 0.707107i
\(129\) 0 0
\(130\) −10.0000 20.0000i −0.877058 1.75412i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 7.00000 7.00000i 0.598050 0.598050i −0.341743 0.939793i \(-0.611017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 20.0000 10.0000i 1.66091 0.830455i
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −10.0000 10.0000i −0.821995 0.821995i
\(149\) 20.0000i 1.63846i 0.573462 + 0.819232i \(0.305600\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −5.00000 + 5.00000i −0.399043 + 0.399043i −0.877896 0.478852i \(-0.841053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 12.0000 + 4.00000i 0.948683 + 0.316228i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 20.0000i 1.56174i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 37.0000i 2.84615i
\(170\) −12.0000 + 6.00000i −0.920358 + 0.460179i
\(171\) 0 0
\(172\) 0 0
\(173\) 11.0000 + 11.0000i 0.836315 + 0.836315i 0.988372 0.152057i \(-0.0485898\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0000 10.0000i −0.749532 0.749532i
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −15.0000 5.00000i −1.10282 0.367607i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −5.00000 5.00000i −0.359908 0.359908i 0.503871 0.863779i \(-0.331909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 10.0000i 0.717958i
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) −13.0000 + 13.0000i −0.926212 + 0.926212i −0.997459 0.0712470i \(-0.977302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 14.0000 2.00000i 0.989949 0.141421i
\(201\) 0 0
\(202\) −20.0000 + 20.0000i −1.40720 + 1.40720i
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 + 20.0000i 0.698430 + 1.39686i
\(206\) 0 0
\(207\) 0 0
\(208\) −20.0000 20.0000i −1.38675 1.38675i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 18.0000 18.0000i 1.23625 1.23625i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −6.00000 6.00000i −0.406371 0.406371i
\(219\) 0 0
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 4.00000i 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 20.0000 20.0000i 1.31306 1.31306i
\(233\) −21.0000 21.0000i −1.37576 1.37576i −0.851658 0.524097i \(-0.824403\pi\)
−0.524097 0.851658i \(-0.675597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 11.0000 11.0000i 0.707107 0.707107i
\(243\) 0 0
\(244\) 24.0000i 1.53644i
\(245\) −14.0000 + 7.00000i −0.894427 + 0.447214i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 13.0000 9.00000i 0.822192 0.569210i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 17.0000 17.0000i 1.06043 1.06043i 0.0623783 0.998053i \(-0.480131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −30.0000 10.0000i −1.86052 0.620174i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 9.00000 27.0000i 0.552866 1.65860i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.0000i 1.21942i −0.792624 0.609711i \(-0.791286\pi\)
0.792624 0.609711i \(-0.208714\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −12.0000 + 12.0000i −0.727607 + 0.727607i
\(273\) 0 0
\(274\) 14.0000i 0.845771i
\(275\) 0 0
\(276\) 0 0
\(277\) 5.00000 5.00000i 0.300421 0.300421i −0.540758 0.841178i \(-0.681862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 10.0000 30.0000i 0.587220 1.76166i
\(291\) 0 0
\(292\) −10.0000 + 10.0000i −0.585206 + 0.585206i
\(293\) −19.0000 19.0000i −1.10999 1.10999i −0.993151 0.116841i \(-0.962723\pi\)
−0.116841 0.993151i \(-0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −20.0000 −1.16248
\(297\) 0 0
\(298\) 20.0000 + 20.0000i 1.15857 + 1.15857i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 12.0000 + 24.0000i 0.687118 + 1.37424i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −25.0000 25.0000i −1.41308 1.41308i −0.734803 0.678280i \(-0.762726\pi\)
−0.678280 0.734803i \(-0.737274\pi\)
\(314\) 10.0000i 0.564333i
\(315\) 0 0
\(316\) 0 0
\(317\) 3.00000 3.00000i 0.168497 0.168497i −0.617822 0.786318i \(-0.711985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 16.0000 8.00000i 0.894427 0.447214i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −35.0000 + 5.00000i −1.94145 + 0.277350i
\(326\) 0 0
\(327\) 0 0
\(328\) 20.0000 + 20.0000i 1.10432 + 1.10432i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −25.0000 + 25.0000i −1.36184 + 1.36184i −0.490261 + 0.871576i \(0.663099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 37.0000 + 37.0000i 2.01253 + 2.01253i
\(339\) 0 0
\(340\) −6.00000 + 18.0000i −0.325396 + 0.976187i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 36.0000i 1.92704i −0.267644 0.963518i \(-0.586245\pi\)
0.267644 0.963518i \(-0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.00000 + 9.00000i 0.479022 + 0.479022i 0.904819 0.425797i \(-0.140006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −20.0000 −1.06000
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 18.0000 18.0000i 0.946059 0.946059i
\(363\) 0 0
\(364\) 0 0
\(365\) −5.00000 + 15.0000i −0.261712 + 0.785136i
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −20.0000 + 10.0000i −1.03975 + 0.519875i
\(371\) 0 0
\(372\) 0 0
\(373\) 25.0000 + 25.0000i 1.29445 + 1.29445i 0.932005 + 0.362446i \(0.118058\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −50.0000 + 50.0000i −2.57513 + 2.57513i
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) 10.0000 + 10.0000i 0.507673 + 0.507673i
\(389\) 20.0000i 1.01404i −0.861934 0.507020i \(-0.830747\pi\)
0.861934 0.507020i \(-0.169253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −14.0000 + 14.0000i −0.707107 + 0.707107i
\(393\) 0 0
\(394\) 26.0000i 1.30986i
\(395\) 0 0
\(396\) 0 0
\(397\) 25.0000 25.0000i 1.25471 1.25471i 0.301131 0.953583i \(-0.402636\pi\)
0.953583 0.301131i \(-0.0973643\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 12.0000 16.0000i 0.600000 0.800000i
\(401\) 40.0000 1.99750 0.998752 0.0499376i \(-0.0159023\pi\)
0.998752 + 0.0499376i \(0.0159023\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 40.0000i 1.99007i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.00000i 0.296681i −0.988936 0.148340i \(-0.952607\pi\)
0.988936 0.148340i \(-0.0473931\pi\)
\(410\) 30.0000 + 10.0000i 1.48159 + 0.493865i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −40.0000 −1.96116
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 36.0000i 1.74831i
\(425\) 3.00000 + 21.0000i 0.145521 + 1.01865i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 5.00000 + 5.00000i 0.240285 + 0.240285i 0.816968 0.576683i \(-0.195653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 30.0000 30.0000i 1.42695 1.42695i
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −20.0000 + 10.0000i −0.948091 + 0.474045i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.0000i 1.88772i −0.330350 0.943858i \(-0.607167\pi\)
0.330350 0.943858i \(-0.392833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.00000 2.00000i 0.0940721 0.0940721i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 25.0000 25.0000i 1.16945 1.16945i 0.187112 0.982339i \(-0.440087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) −4.00000 4.00000i −0.186908 0.186908i
\(459\) 0 0
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 0 0
\(463\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(464\) 40.0000i 1.85695i
\(465\) 0 0
\(466\) −42.0000 −1.94561
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 50.0000 2.27980
\(482\) −8.00000 + 8.00000i −0.364390 + 0.364390i
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 15.0000 + 5.00000i 0.681115 + 0.227038i
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 24.0000 + 24.0000i 1.08643 + 1.08643i
\(489\) 0 0
\(490\) −7.00000 + 21.0000i −0.316228 + 0.948683i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 30.0000 + 30.0000i 1.35113 + 1.35113i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 4.00000 22.0000i 0.178885 0.983870i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 20.0000 + 40.0000i 0.889988 + 1.77998i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.0000i 0.443242i −0.975133 0.221621i \(-0.928865\pi\)
0.975133 0.221621i \(-0.0711348\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 16.0000 16.0000i 0.707107 0.707107i
\(513\) 0 0
\(514\) 34.0000i 1.49968i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −40.0000 + 20.0000i −1.75412 + 0.877058i
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) −18.0000 36.0000i −0.781870 1.56374i
\(531\) 0 0
\(532\) 0 0
\(533\) −50.0000 50.0000i −2.16574 2.16574i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −20.0000 20.0000i −0.862261 0.862261i
\(539\) 0 0
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 24.0000i 1.02899i
\(545\) −12.0000 + 6.00000i −0.514024 + 0.257012i
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) −14.0000 14.0000i −0.598050 0.598050i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000i 0.424859i
\(555\) 0 0
\(556\) 0 0
\(557\) 33.0000 33.0000i 1.39825 1.39825i 0.593199 0.805056i \(-0.297865\pi\)
0.805056 0.593199i \(-0.202135\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 10.0000i 0.421825 0.421825i
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 1.00000 3.00000i 0.0420703 0.126211i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 40.0000i 1.67689i 0.544988 + 0.838444i \(0.316534\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −25.0000 + 25.0000i −1.04076 + 1.04076i −0.0416305 + 0.999133i \(0.513255\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) −1.00000 1.00000i −0.0415945 0.0415945i
\(579\) 0 0
\(580\) −20.0000 40.0000i −0.830455 1.66091i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 20.0000i 0.827606i
\(585\) 0 0
\(586\) −38.0000 −1.56977
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −20.0000 + 20.0000i −0.821995 + 0.821995i
\(593\) 31.0000 + 31.0000i 1.27302 + 1.27302i 0.944497 + 0.328521i \(0.106550\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 40.0000 1.63846
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 48.0000 1.95796 0.978980 0.203954i \(-0.0653794\pi\)
0.978980 + 0.203954i \(0.0653794\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 22.0000i −0.447214 0.894427i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 36.0000 + 12.0000i 1.45760 + 0.485866i
\(611\) 0 0
\(612\) 0 0
\(613\) 35.0000 + 35.0000i 1.41364 + 1.41364i 0.727013 + 0.686624i \(0.240908\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 3.00000i 0.120775 0.120775i −0.644136 0.764911i \(-0.722783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) −50.0000 −1.99840
\(627\) 0 0
\(628\) 10.0000 + 10.0000i 0.399043 + 0.399043i
\(629\) 30.0000i 1.19618i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 6.00000i 0.238290i
\(635\) 0 0
\(636\) 0 0
\(637\) 35.0000 35.0000i 1.38675 1.38675i
\(638\) 0 0
\(639\) 0 0
\(640\) 8.00000 24.0000i 0.316228 0.948683i
\(641\) 50.0000 1.97488 0.987441 0.157991i \(-0.0505015\pi\)
0.987441 + 0.157991i \(0.0505015\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −30.0000 + 40.0000i −1.17670 + 1.56893i
\(651\) 0 0
\(652\) 0 0
\(653\) 9.00000 + 9.00000i 0.352197 + 0.352197i 0.860927 0.508729i \(-0.169885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 40.0000 1.56174
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 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
\(673\) −35.0000 35.0000i −1.34915 1.34915i −0.886585 0.462566i \(-0.846929\pi\)
−0.462566 0.886585i \(-0.653071\pi\)
\(674\) 50.0000i 1.92593i
\(675\) 0 0
\(676\) 74.0000 2.84615
\(677\) −27.0000 + 27.0000i −1.03769 + 1.03769i −0.0384331 + 0.999261i \(0.512237\pi\)
−0.999261 + 0.0384331i \(0.987763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 12.0000 + 24.0000i 0.460179 + 0.920358i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 0 0
\(685\) −21.0000 7.00000i −0.802369 0.267456i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 90.0000i 3.42873i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 22.0000 22.0000i 0.836315 0.836315i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −30.0000 + 30.0000i −1.13633 + 1.13633i
\(698\) −36.0000 36.0000i −1.36262 1.36262i
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000i 1.65245i −0.563337 0.826227i \(-0.690483\pi\)
0.563337 0.826227i \(-0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −20.0000 + 20.0000i −0.749532 + 0.749532i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −19.0000 + 19.0000i −0.707107 + 0.707107i
\(723\) 0 0
\(724\) 36.0000i 1.33793i
\(725\) −40.0000 30.0000i −1.48556 1.11417i
\(726\) 0 0
\(727\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 10.0000 + 20.0000i 0.370117 + 0.740233i
\(731\) 0 0
\(732\) 0 0
\(733\) −25.0000 25.0000i −0.923396 0.923396i 0.0738717 0.997268i \(-0.476464\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −10.0000 + 30.0000i −0.367607 + 1.10282i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 40.0000 20.0000i 1.46549 0.732743i
\(746\) 50.0000 1.83063
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 100.000i 3.64179i
\(755\) 0 0
\(756\) 0 0
\(757\) −35.0000 + 35.0000i −1.27210 + 1.27210i −0.327111 + 0.944986i \(0.606075\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 24.0000i 0.865462i 0.901523 + 0.432731i \(0.142450\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 + 10.0000i −0.359908 + 0.359908i
\(773\) 39.0000 + 39.0000i 1.40273 + 1.40273i 0.791285 + 0.611448i \(0.209412\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 20.0000 0.717958
\(777\) 0 0
\(778\) −20.0000 20.0000i −0.717035 0.717035i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) 15.0000 + 5.00000i 0.535373 + 0.178458i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 26.0000 + 26.0000i 0.926212 + 0.926212i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −60.0000 60.0000i −2.13066 2.13066i
\(794\) 50.0000i 1.77443i
\(795\) 0 0
\(796\) 0 0
\(797\) 37.0000 37.0000i 1.31061 1.31061i 0.389640 0.920967i \(-0.372599\pi\)
0.920967 0.389640i \(-0.127401\pi\)
\(798\) 0 0