Properties

Label 1080.2.q.c.721.1
Level $1080$
Weight $2$
Character 1080.721
Analytic conductor $8.624$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(361,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 721.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1080.721
Dual form 1080.2.q.c.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +2.00000 q^{17} -6.89898 q^{19} +(-3.72474 - 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.949490 + 1.64456i) q^{29} +(-0.550510 - 0.953512i) q^{31} +3.44949 q^{35} -6.00000 q^{37} +(-4.94949 - 8.57277i) q^{41} +(-5.89898 + 10.2173i) q^{43} +(-4.72474 + 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} -7.79796 q^{53} +(-0.550510 - 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(6.62372 + 11.4726i) q^{67} -9.79796 q^{71} +13.7980 q^{73} +(3.44949 - 5.97469i) q^{79} +(-2.72474 + 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} -2.79796 q^{89} +(3.44949 + 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 2 q^{7} + 8 q^{17} - 8 q^{19} - 10 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 4 q^{35} - 24 q^{37} - 10 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{53} - 12 q^{59} + 6 q^{61} + 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) −1.72474 + 2.98735i −0.651892 + 1.12911i 0.330771 + 0.943711i \(0.392691\pi\)
−0.982663 + 0.185399i \(0.940642\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −6.89898 −1.58273 −0.791367 0.611341i \(-0.790630\pi\)
−0.791367 + 0.611341i \(0.790630\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.72474 6.45145i −0.776663 1.34522i −0.933855 0.357652i \(-0.883578\pi\)
0.157192 0.987568i \(-0.449756\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.949490 + 1.64456i −0.176316 + 0.305388i −0.940616 0.339473i \(-0.889751\pi\)
0.764300 + 0.644861i \(0.223085\pi\)
\(30\) 0 0
\(31\) −0.550510 0.953512i −0.0988746 0.171256i 0.812345 0.583178i \(-0.198191\pi\)
−0.911219 + 0.411922i \(0.864858\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.44949 0.583070
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.94949 8.57277i −0.772980 1.33884i −0.935923 0.352206i \(-0.885432\pi\)
0.162942 0.986636i \(-0.447902\pi\)
\(42\) 0 0
\(43\) −5.89898 + 10.2173i −0.899586 + 1.55813i −0.0715617 + 0.997436i \(0.522798\pi\)
−0.828024 + 0.560692i \(0.810535\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.72474 + 8.18350i −0.689175 + 1.19369i 0.282930 + 0.959140i \(0.408693\pi\)
−0.972105 + 0.234545i \(0.924640\pi\)
\(48\) 0 0
\(49\) −2.44949 4.24264i −0.349927 0.606092i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.79796 −1.07113 −0.535566 0.844493i \(-0.679902\pi\)
−0.535566 + 0.844493i \(0.679902\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.550510 0.953512i −0.0716703 0.124137i 0.827963 0.560783i \(-0.189500\pi\)
−0.899633 + 0.436646i \(0.856166\pi\)
\(60\) 0 0
\(61\) 1.50000 2.59808i 0.192055 0.332650i −0.753876 0.657017i \(-0.771818\pi\)
0.945931 + 0.324367i \(0.105151\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.62372 + 11.4726i 0.809217 + 1.40160i 0.913407 + 0.407047i \(0.133442\pi\)
−0.104190 + 0.994557i \(0.533225\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.79796 −1.16280 −0.581402 0.813617i \(-0.697496\pi\)
−0.581402 + 0.813617i \(0.697496\pi\)
\(72\) 0 0
\(73\) 13.7980 1.61493 0.807464 0.589916i \(-0.200839\pi\)
0.807464 + 0.589916i \(0.200839\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.44949 5.97469i 0.388098 0.672205i −0.604096 0.796912i \(-0.706466\pi\)
0.992194 + 0.124706i \(0.0397989\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.72474 + 4.71940i −0.299080 + 0.518021i −0.975926 0.218104i \(-0.930013\pi\)
0.676846 + 0.736125i \(0.263346\pi\)
\(84\) 0 0
\(85\) −1.00000 1.73205i −0.108465 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.79796 −0.296583 −0.148292 0.988944i \(-0.547377\pi\)
−0.148292 + 0.988944i \(0.547377\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.44949 + 5.97469i 0.353910 + 0.612990i
\(96\) 0 0
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.00000 + 1.73205i −0.0995037 + 0.172345i −0.911479 0.411346i \(-0.865059\pi\)
0.811976 + 0.583691i \(0.198392\pi\)
\(102\) 0 0
\(103\) −5.00000 8.66025i −0.492665 0.853320i 0.507300 0.861770i \(-0.330644\pi\)
−0.999964 + 0.00844953i \(0.997310\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.3485 1.19377 0.596886 0.802326i \(-0.296405\pi\)
0.596886 + 0.802326i \(0.296405\pi\)
\(108\) 0 0
\(109\) −10.7980 −1.03426 −0.517128 0.855908i \(-0.672999\pi\)
−0.517128 + 0.855908i \(0.672999\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.89898 8.48528i −0.460857 0.798228i 0.538147 0.842851i \(-0.319125\pi\)
−0.999004 + 0.0446231i \(0.985791\pi\)
\(114\) 0 0
\(115\) −3.72474 + 6.45145i −0.347334 + 0.601601i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.44949 + 5.97469i −0.316214 + 0.547699i
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.348469 0.0309216 0.0154608 0.999880i \(-0.495078\pi\)
0.0154608 + 0.999880i \(0.495078\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.44949 + 2.51059i 0.126643 + 0.219351i 0.922374 0.386299i \(-0.126247\pi\)
−0.795731 + 0.605650i \(0.792913\pi\)
\(132\) 0 0
\(133\) 11.8990 20.6096i 1.03177 1.78708i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.79796 16.9706i 0.837096 1.44989i −0.0552162 0.998474i \(-0.517585\pi\)
0.892312 0.451419i \(-0.149082\pi\)
\(138\) 0 0
\(139\) −9.79796 16.9706i −0.831052 1.43942i −0.897205 0.441615i \(-0.854406\pi\)
0.0661527 0.997810i \(-0.478928\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.89898 0.157702
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.5000 18.1865i −0.860194 1.48990i −0.871742 0.489966i \(-0.837009\pi\)
0.0115483 0.999933i \(-0.496324\pi\)
\(150\) 0 0
\(151\) −6.00000 + 10.3923i −0.488273 + 0.845714i −0.999909 0.0134886i \(-0.995706\pi\)
0.511636 + 0.859202i \(0.329040\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.550510 + 0.953512i −0.0442180 + 0.0765879i
\(156\) 0 0
\(157\) 11.8990 + 20.6096i 0.949642 + 1.64483i 0.746179 + 0.665745i \(0.231886\pi\)
0.203463 + 0.979083i \(0.434780\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 25.6969 2.02520
\(162\) 0 0
\(163\) −7.79796 −0.610783 −0.305392 0.952227i \(-0.598787\pi\)
−0.305392 + 0.952227i \(0.598787\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.17423 + 7.22999i 0.323012 + 0.559473i 0.981108 0.193461i \(-0.0619713\pi\)
−0.658096 + 0.752934i \(0.728638\pi\)
\(168\) 0 0
\(169\) 6.50000 11.2583i 0.500000 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.00000 + 10.3923i −0.456172 + 0.790112i −0.998755 0.0498898i \(-0.984113\pi\)
0.542583 + 0.840002i \(0.317446\pi\)
\(174\) 0 0
\(175\) −1.72474 2.98735i −0.130378 0.225822i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.7980 1.03131 0.515654 0.856797i \(-0.327549\pi\)
0.515654 + 0.856797i \(0.327549\pi\)
\(180\) 0 0
\(181\) −9.69694 −0.720768 −0.360384 0.932804i \(-0.617354\pi\)
−0.360384 + 0.932804i \(0.617354\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 + 5.19615i 0.220564 + 0.382029i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.44949 + 12.9029i −0.539026 + 0.933621i 0.459931 + 0.887955i \(0.347874\pi\)
−0.998957 + 0.0456658i \(0.985459\pi\)
\(192\) 0 0
\(193\) 1.10102 + 1.90702i 0.0792532 + 0.137271i 0.902928 0.429792i \(-0.141413\pi\)
−0.823675 + 0.567063i \(0.808080\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.5959 1.11116 0.555582 0.831462i \(-0.312496\pi\)
0.555582 + 0.831462i \(0.312496\pi\)
\(198\) 0 0
\(199\) 22.8990 1.62327 0.811633 0.584168i \(-0.198579\pi\)
0.811633 + 0.584168i \(0.198579\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.27526 5.67291i −0.229878 0.398160i
\(204\) 0 0
\(205\) −4.94949 + 8.57277i −0.345687 + 0.598748i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.00000 10.3923i −0.413057 0.715436i 0.582165 0.813070i \(-0.302206\pi\)
−0.995222 + 0.0976347i \(0.968872\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.7980 0.804614
\(216\) 0 0
\(217\) 3.79796 0.257822
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.72474 + 4.71940i −0.182462 + 0.316034i −0.942718 0.333589i \(-0.891740\pi\)
0.760256 + 0.649623i \(0.225073\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.10102 + 7.10318i −0.272194 + 0.471454i −0.969423 0.245394i \(-0.921083\pi\)
0.697229 + 0.716848i \(0.254416\pi\)
\(228\) 0 0
\(229\) 2.05051 + 3.55159i 0.135502 + 0.234696i 0.925789 0.378041i \(-0.123402\pi\)
−0.790287 + 0.612736i \(0.790069\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.7980 1.29701 0.648504 0.761211i \(-0.275395\pi\)
0.648504 + 0.761211i \(0.275395\pi\)
\(234\) 0 0
\(235\) 9.44949 0.616417
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.10102 + 1.90702i 0.0712191 + 0.123355i 0.899436 0.437053i \(-0.143978\pi\)
−0.828217 + 0.560408i \(0.810644\pi\)
\(240\) 0 0
\(241\) 1.84847 3.20164i 0.119070 0.206236i −0.800329 0.599561i \(-0.795342\pi\)
0.919400 + 0.393325i \(0.128675\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.44949 + 4.24264i −0.156492 + 0.271052i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.89898 −0.182982 −0.0914910 0.995806i \(-0.529163\pi\)
−0.0914910 + 0.995806i \(0.529163\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.8990 + 24.0737i 0.866995 + 1.50168i 0.865053 + 0.501680i \(0.167285\pi\)
0.00194150 + 0.999998i \(0.499382\pi\)
\(258\) 0 0
\(259\) 10.3485 17.9241i 0.643023 1.11375i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000 15.5885i 0.554964 0.961225i −0.442943 0.896550i \(-0.646065\pi\)
0.997906 0.0646755i \(-0.0206012\pi\)
\(264\) 0 0
\(265\) 3.89898 + 6.75323i 0.239512 + 0.414848i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −22.5959 −1.37770 −0.688849 0.724905i \(-0.741884\pi\)
−0.688849 + 0.724905i \(0.741884\pi\)
\(270\) 0 0
\(271\) 30.8990 1.87698 0.938490 0.345307i \(-0.112225\pi\)
0.938490 + 0.345307i \(0.112225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.00000 12.1244i 0.420589 0.728482i −0.575408 0.817867i \(-0.695157\pi\)
0.995997 + 0.0893846i \(0.0284900\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.05051 + 7.01569i −0.241633 + 0.418521i −0.961180 0.275923i \(-0.911016\pi\)
0.719546 + 0.694444i \(0.244350\pi\)
\(282\) 0 0
\(283\) 1.82577 + 3.16232i 0.108530 + 0.187980i 0.915175 0.403056i \(-0.132052\pi\)
−0.806645 + 0.591037i \(0.798719\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 34.1464 2.01560
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.79796 + 15.2385i 0.513982 + 0.890243i 0.999868 + 0.0162213i \(0.00516362\pi\)
−0.485886 + 0.874022i \(0.661503\pi\)
\(294\) 0 0
\(295\) −0.550510 + 0.953512i −0.0320519 + 0.0555156i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −20.3485 35.2446i −1.17287 2.03146i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) −4.75255 −0.271242 −0.135621 0.990761i \(-0.543303\pi\)
−0.135621 + 0.990761i \(0.543303\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.44949 5.97469i −0.195603 0.338794i 0.751495 0.659738i \(-0.229333\pi\)
−0.947098 + 0.320945i \(0.896000\pi\)
\(312\) 0 0
\(313\) 7.79796 13.5065i 0.440767 0.763430i −0.556980 0.830526i \(-0.688040\pi\)
0.997747 + 0.0670957i \(0.0213733\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.8990 + 22.3417i −0.724479 + 1.25483i 0.234709 + 0.972066i \(0.424586\pi\)
−0.959188 + 0.282769i \(0.908747\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.7980 −0.767739
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.2980 28.2289i −0.898536 1.55631i
\(330\) 0 0
\(331\) 3.44949 5.97469i 0.189601 0.328399i −0.755516 0.655130i \(-0.772614\pi\)
0.945117 + 0.326731i \(0.105947\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.62372 11.4726i 0.361893 0.626817i
\(336\) 0 0
\(337\) 14.8990 + 25.8058i 0.811599 + 1.40573i 0.911744 + 0.410758i \(0.134736\pi\)
−0.100145 + 0.994973i \(0.531931\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.24745 −0.391325
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.7980 22.1667i −0.687030 1.18997i −0.972794 0.231671i \(-0.925581\pi\)
0.285764 0.958300i \(-0.407753\pi\)
\(348\) 0 0
\(349\) 1.15153 1.99451i 0.0616400 0.106764i −0.833559 0.552431i \(-0.813700\pi\)
0.895199 + 0.445667i \(0.147034\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 4.89898 + 8.48528i 0.260011 + 0.450352i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −28.6969 −1.51457 −0.757283 0.653087i \(-0.773474\pi\)
−0.757283 + 0.653087i \(0.773474\pi\)
\(360\) 0 0
\(361\) 28.5959 1.50505
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.89898 11.9494i −0.361109 0.625459i
\(366\) 0 0
\(367\) −11.0000 + 19.0526i −0.574195 + 0.994535i 0.421933 + 0.906627i \(0.361352\pi\)
−0.996129 + 0.0879086i \(0.971982\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13.4495 23.2952i 0.698263 1.20943i
\(372\) 0 0
\(373\) −10.7980 18.7026i −0.559097 0.968385i −0.997572 0.0696412i \(-0.977815\pi\)
0.438475 0.898743i \(-0.355519\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 26.8990 1.38171 0.690854 0.722994i \(-0.257235\pi\)
0.690854 + 0.722994i \(0.257235\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.00000 + 1.73205i 0.0510976 + 0.0885037i 0.890443 0.455095i \(-0.150395\pi\)
−0.839345 + 0.543599i \(0.817061\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.29796 + 14.3725i −0.420723 + 0.728714i −0.996010 0.0892375i \(-0.971557\pi\)
0.575287 + 0.817952i \(0.304890\pi\)
\(390\) 0 0
\(391\) −7.44949 12.9029i −0.376737 0.652527i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.89898 −0.347125
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.7980 + 22.1667i 0.639100 + 1.10695i 0.985631 + 0.168914i \(0.0540262\pi\)
−0.346531 + 0.938038i \(0.612641\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.79796 + 8.31031i 0.237244 + 0.410918i 0.959922 0.280266i \(-0.0904226\pi\)
−0.722679 + 0.691184i \(0.757089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.79796 0.186885
\(414\) 0 0
\(415\) 5.44949 0.267505
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.0000 + 17.3205i 0.488532 + 0.846162i 0.999913 0.0131919i \(-0.00419923\pi\)
−0.511381 + 0.859354i \(0.670866\pi\)
\(420\) 0 0
\(421\) −12.7980 + 22.1667i −0.623734 + 1.08034i 0.365050 + 0.930988i \(0.381052\pi\)
−0.988784 + 0.149352i \(0.952281\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 + 1.73205i −0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) 5.17423 + 8.96204i 0.250399 + 0.433703i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.8990 −0.717659 −0.358829 0.933403i \(-0.616824\pi\)
−0.358829 + 0.933403i \(0.616824\pi\)
\(432\) 0 0
\(433\) −11.7980 −0.566974 −0.283487 0.958976i \(-0.591491\pi\)
−0.283487 + 0.958976i \(0.591491\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.6969 + 44.5084i 1.22925 + 2.12913i
\(438\) 0 0
\(439\) 8.89898 15.4135i 0.424725 0.735645i −0.571670 0.820484i \(-0.693704\pi\)
0.996395 + 0.0848384i \(0.0270374\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.07321 + 7.05501i −0.193524 + 0.335194i −0.946416 0.322951i \(-0.895325\pi\)
0.752892 + 0.658145i \(0.228658\pi\)
\(444\) 0 0
\(445\) 1.39898 + 2.42310i 0.0663180 + 0.114866i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −21.5959 −1.01917 −0.509587 0.860419i \(-0.670202\pi\)
−0.509587 + 0.860419i \(0.670202\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.79796 + 15.2385i −0.411551 + 0.712828i −0.995060 0.0992796i \(-0.968346\pi\)
0.583508 + 0.812107i \(0.301680\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.05051 10.4798i 0.281800 0.488093i −0.690028 0.723783i \(-0.742402\pi\)
0.971828 + 0.235690i \(0.0757351\pi\)
\(462\) 0 0
\(463\) −6.10102 10.5673i −0.283538 0.491103i 0.688715 0.725032i \(-0.258175\pi\)
−0.972254 + 0.233929i \(0.924842\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.79796 −0.360847 −0.180423 0.983589i \(-0.557747\pi\)
−0.180423 + 0.983589i \(0.557747\pi\)
\(468\) 0 0
\(469\) −45.6969 −2.11009
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.44949 5.97469i 0.158273 0.274138i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.2474 26.4094i 0.696674 1.20667i −0.272939 0.962031i \(-0.587996\pi\)
0.969613 0.244643i \(-0.0786708\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −13.5959 −0.616090 −0.308045 0.951372i \(-0.599675\pi\)
−0.308045 + 0.951372i \(0.599675\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.89898 11.9494i −0.311347 0.539268i 0.667308 0.744782i \(-0.267447\pi\)
−0.978654 + 0.205514i \(0.934113\pi\)
\(492\) 0 0
\(493\) −1.89898 + 3.28913i −0.0855257 + 0.148135i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.8990 29.2699i 0.758023 1.31293i
\(498\) 0 0
\(499\) 14.0000 + 24.2487i 0.626726 + 1.08552i 0.988204 + 0.153141i \(0.0489388\pi\)
−0.361478 + 0.932381i \(0.617728\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.0454 −1.11672 −0.558360 0.829599i \(-0.688569\pi\)
−0.558360 + 0.829599i \(0.688569\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.05051 3.55159i −0.0908873 0.157421i 0.816997 0.576641i \(-0.195637\pi\)
−0.907885 + 0.419220i \(0.862304\pi\)
\(510\) 0 0
\(511\) −23.7980 + 41.2193i −1.05276 + 1.82343i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.00000 + 8.66025i −0.220326 + 0.381616i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.20204 −0.403149 −0.201574 0.979473i \(-0.564606\pi\)
−0.201574 + 0.979473i \(0.564606\pi\)
\(522\) 0 0
\(523\) −34.3485 −1.50195 −0.750977 0.660329i \(-0.770417\pi\)
−0.750977 + 0.660329i \(0.770417\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.10102 1.90702i −0.0479612 0.0830712i
\(528\) 0 0
\(529\) −16.2474 + 28.1414i −0.706411 + 1.22354i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.17423 10.6941i −0.266935 0.462346i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.1010 0.520264 0.260132 0.965573i \(-0.416234\pi\)
0.260132 + 0.965573i \(0.416234\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.39898 + 9.35131i 0.231267 + 0.400566i
\(546\) 0 0
\(547\) 8.37628 14.5081i 0.358144 0.620323i −0.629507 0.776995i \(-0.716743\pi\)
0.987651 + 0.156672i \(0.0500765\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.55051 11.3458i 0.279061 0.483348i
\(552\) 0 0
\(553\) 11.8990 + 20.6096i 0.505996 + 0.876411i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −43.7980 −1.85578 −0.927890 0.372855i \(-0.878379\pi\)
−0.927890 + 0.372855i \(0.878379\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.17423 8.96204i −0.218068 0.377705i 0.736149 0.676819i \(-0.236642\pi\)
−0.954217 + 0.299114i \(0.903309\pi\)
\(564\) 0 0
\(565\) −4.89898 + 8.48528i −0.206102 + 0.356978i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.00000 8.66025i 0.209611 0.363057i −0.741981 0.670421i \(-0.766114\pi\)
0.951592 + 0.307364i \(0.0994469\pi\)
\(570\) 0 0
\(571\) −12.8990 22.3417i −0.539805 0.934971i −0.998914 0.0465904i \(-0.985164\pi\)
0.459109 0.888380i \(-0.348169\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 7.44949 0.310665
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.39898 16.2795i −0.389935 0.675388i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.82577 6.62642i 0.157906 0.273502i −0.776207 0.630478i \(-0.782859\pi\)
0.934113 + 0.356976i \(0.116192\pi\)
\(588\) 0 0
\(589\) 3.79796 + 6.57826i 0.156492 + 0.271052i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 31.3939 1.28919 0.644596 0.764523i \(-0.277026\pi\)
0.644596 + 0.764523i \(0.277026\pi\)
\(594\) 0 0
\(595\) 6.89898 0.282831
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.3485 21.3882i −0.504545 0.873897i −0.999986 0.00525583i \(-0.998327\pi\)
0.495441 0.868641i \(-0.335006\pi\)
\(600\) 0 0
\(601\) −7.00000 + 12.1244i −0.285536 + 0.494563i −0.972739 0.231903i \(-0.925505\pi\)
0.687203 + 0.726465i \(0.258838\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.50000 9.52628i 0.223607 0.387298i
\(606\) 0 0
\(607\) −1.07321 1.85886i −0.0435604 0.0754489i 0.843423 0.537250i \(-0.180537\pi\)
−0.886984 + 0.461801i \(0.847203\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −29.5959 −1.19537 −0.597684 0.801732i \(-0.703912\pi\)
−0.597684 + 0.801732i \(0.703912\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.5959 30.4770i −0.708385 1.22696i −0.965456 0.260566i \(-0.916091\pi\)
0.257071 0.966393i \(-0.417243\pi\)
\(618\) 0 0
\(619\) 12.3485 21.3882i 0.496327 0.859663i −0.503664 0.863900i \(-0.668015\pi\)
0.999991 + 0.00423617i \(0.00134842\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.82577 8.35847i 0.193340 0.334875i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −15.5959 −0.620864 −0.310432 0.950596i \(-0.600474\pi\)
−0.310432 + 0.950596i \(0.600474\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.174235 0.301783i −0.00691429 0.0119759i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.94949 + 12.0369i −0.274488 + 0.475428i −0.970006 0.243081i \(-0.921842\pi\)
0.695518 + 0.718509i \(0.255175\pi\)
\(642\) 0 0
\(643\) 10.0732 + 17.4473i 0.397249 + 0.688055i 0.993385 0.114828i \(-0.0366317\pi\)
−0.596137 + 0.802883i \(0.703298\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.55051 −0.257527 −0.128764 0.991675i \(-0.541101\pi\)
−0.128764 + 0.991675i \(0.541101\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.8990 + 24.0737i 0.543909 + 0.942078i 0.998675 + 0.0514670i \(0.0163897\pi\)
−0.454766 + 0.890611i \(0.650277\pi\)
\(654\) 0 0
\(655\) 1.44949 2.51059i 0.0566363 0.0980969i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.10102 8.83523i 0.198708 0.344172i −0.749402 0.662115i \(-0.769659\pi\)
0.948110 + 0.317944i \(0.102992\pi\)
\(660\) 0 0
\(661\) −5.00000 8.66025i −0.194477 0.336845i 0.752252 0.658876i \(-0.228968\pi\)
−0.946729 + 0.322031i \(0.895634\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −23.7980 −0.922845
\(666\) 0 0
\(667\) 14.1464 0.547752
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000 13.8564i 0.308377 0.534125i −0.669630 0.742695i \(-0.733547\pi\)
0.978008 + 0.208569i \(0.0668807\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.10102 12.2993i 0.272914 0.472702i −0.696692 0.717370i \(-0.745346\pi\)
0.969607 + 0.244668i \(0.0786791\pi\)
\(678\) 0 0
\(679\) −3.44949 5.97469i −0.132379 0.229288i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23.3939 −0.895142 −0.447571 0.894248i \(-0.647711\pi\)
−0.447571 + 0.894248i \(0.647711\pi\)
\(684\) 0 0
\(685\) −19.5959 −0.748722
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −19.4495 + 33.6875i −0.739893 + 1.28153i 0.212649 + 0.977129i \(0.431791\pi\)
−0.952543 + 0.304404i \(0.901543\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.79796 + 16.9706i −0.371658 + 0.643730i
\(696\) 0 0
\(697\) −9.89898 17.1455i −0.374951 0.649433i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.3939 −1.14796 −0.573980 0.818869i \(-0.694601\pi\)
−0.573980 + 0.818869i \(0.694601\pi\)
\(702\) 0 0
\(703\) 41.3939 1.56120
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.44949 5.97469i −0.129731 0.224701i
\(708\) 0 0
\(709\) −4.84847 + 8.39780i −0.182088 + 0.315386i −0.942591 0.333948i \(-0.891619\pi\)
0.760503 + 0.649334i \(0.224952\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.10102 + 7.10318i −0.153584 + 0.266016i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 28.2929 1.05515 0.527573 0.849510i \(-0.323102\pi\)
0.527573 + 0.849510i \(0.323102\pi\)
\(720\) 0 0
\(721\) 34.4949 1.28466
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.949490 1.64456i −0.0352632 0.0610776i
\(726\) 0 0
\(727\) 15.6237 27.0611i 0.579452 1.00364i −0.416090 0.909323i \(-0.636600\pi\)
0.995542 0.0943168i \(-0.0300667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.7980 + 20.4347i −0.436363 + 0.755803i
\(732\) 0 0
\(733\) 14.7980 + 25.6308i 0.546575 + 0.946696i 0.998506 + 0.0546429i \(0.0174020\pi\)
−0.451931 + 0.892053i \(0.649265\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −33.1010 −1.21764 −0.608820 0.793308i \(-0.708357\pi\)
−0.608820 + 0.793308i \(0.708357\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.52270 6.10150i −0.129235 0.223842i 0.794145 0.607728i \(-0.207919\pi\)
−0.923381 + 0.383886i \(0.874586\pi\)
\(744\) 0 0
\(745\) −10.5000 + 18.1865i −0.384690 + 0.666303i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −21.2980 + 36.8891i −0.778210 + 1.34790i
\(750\) 0 0
\(751\) −13.1010 22.6916i −0.478063 0.828029i 0.521621 0.853177i \(-0.325328\pi\)
−0.999684 + 0.0251480i \(0.991994\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 31.5959 1.14837 0.574187 0.818724i \(-0.305318\pi\)
0.574187 + 0.818724i \(0.305318\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.5000 + 30.3109i 0.634375 + 1.09877i 0.986647 + 0.162872i \(0.0520756\pi\)
−0.352273 + 0.935897i \(0.614591\pi\)
\(762\) 0 0
\(763\) 18.6237 32.2572i 0.674224 1.16779i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.60102 + 4.50510i 0.0937952 + 0.162458i 0.909105 0.416567i \(-0.136767\pi\)
−0.815310 + 0.579025i \(0.803433\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.20204 −0.223072 −0.111536 0.993760i \(-0.535577\pi\)
−0.111536 + 0.993760i \(0.535577\pi\)
\(774\) 0 0
\(775\) 1.10102 0.0395498
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.1464 + 59.1433i 1.22342 + 2.11903i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.8990 20.6096i 0.424693 0.735589i
\(786\) 0 0
\(787\) 24.7980 + 42.9513i 0.883952 + 1.53105i 0.846910 + 0.531736i \(0.178460\pi\)
0.0370414 + 0.999314i \(0.488207\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 33.7980 1.20172
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.00000 6.92820i −0.141687 0.245410i 0.786445 0.617661i \(-0.211919\pi\)
−0.928132 + 0.372251i \(0.878586\pi\)
\(798\) 0 0
\(799\) −9.44949 + 16.3670i −0.334299 + 0.579023i
\(800\) 0