Properties

Label 1620.4.i.w.541.6
Level $1620$
Weight $4$
Character 1620.541
Analytic conductor $95.583$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,4,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (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: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 306x^{10} + 30777x^{8} + 1381040x^{6} + 28918584x^{4} + 243888288x^{2} + 366645904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.6
Root \(7.39017i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.w.1081.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{5} +(11.6578 - 20.1920i) q^{7} +O(q^{10})\) \(q+(-2.50000 - 4.33013i) q^{5} +(11.6578 - 20.1920i) q^{7} +(-16.8745 + 29.2275i) q^{11} +(44.4469 + 76.9843i) q^{13} +108.661 q^{17} +21.4799 q^{19} +(-42.9612 - 74.4110i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(9.48907 - 16.4355i) q^{29} +(83.1724 + 144.059i) q^{31} -116.578 q^{35} -167.664 q^{37} +(-189.143 - 327.605i) q^{41} +(-190.222 + 329.474i) q^{43} +(-123.414 + 213.760i) q^{47} +(-100.311 - 173.744i) q^{49} -284.344 q^{53} +168.745 q^{55} +(363.699 + 629.945i) q^{59} +(-235.434 + 407.783i) q^{61} +(222.235 - 384.922i) q^{65} +(375.978 + 651.213i) q^{67} -864.963 q^{71} -1003.53 q^{73} +(393.440 + 681.459i) q^{77} +(419.251 - 726.163i) q^{79} +(-238.057 + 412.327i) q^{83} +(-271.653 - 470.517i) q^{85} +554.055 q^{89} +2072.62 q^{91} +(-53.6996 - 93.0105i) q^{95} +(523.994 - 907.584i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 30 q^{5} - 12 q^{7} + 84 q^{13} - 24 q^{17} - 228 q^{19} + 30 q^{23} - 150 q^{25} + 168 q^{29} + 324 q^{31} + 120 q^{35} - 984 q^{37} + 312 q^{41} + 156 q^{43} + 462 q^{47} + 588 q^{49} - 2028 q^{53} + 1008 q^{59} - 36 q^{61} + 420 q^{65} - 144 q^{67} - 2424 q^{71} - 1800 q^{73} + 672 q^{77} + 936 q^{79} + 288 q^{83} + 60 q^{85} + 240 q^{89} + 4572 q^{91} + 570 q^{95} + 1188 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.50000 4.33013i −0.223607 0.387298i
\(6\) 0 0
\(7\) 11.6578 20.1920i 0.629465 1.09026i −0.358195 0.933647i \(-0.616608\pi\)
0.987659 0.156618i \(-0.0500591\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.8745 + 29.2275i −0.462532 + 0.801128i −0.999086 0.0427371i \(-0.986392\pi\)
0.536555 + 0.843866i \(0.319726\pi\)
\(12\) 0 0
\(13\) 44.4469 + 76.9843i 0.948258 + 1.64243i 0.749092 + 0.662466i \(0.230490\pi\)
0.199166 + 0.979966i \(0.436177\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 108.661 1.55025 0.775124 0.631809i \(-0.217687\pi\)
0.775124 + 0.631809i \(0.217687\pi\)
\(18\) 0 0
\(19\) 21.4799 0.259359 0.129679 0.991556i \(-0.458605\pi\)
0.129679 + 0.991556i \(0.458605\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −42.9612 74.4110i −0.389480 0.674599i 0.602900 0.797817i \(-0.294012\pi\)
−0.992380 + 0.123218i \(0.960679\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.48907 16.4355i 0.0607612 0.105242i −0.834045 0.551697i \(-0.813981\pi\)
0.894806 + 0.446455i \(0.147314\pi\)
\(30\) 0 0
\(31\) 83.1724 + 144.059i 0.481877 + 0.834636i 0.999784 0.0208014i \(-0.00662176\pi\)
−0.517906 + 0.855437i \(0.673288\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −116.578 −0.563010
\(36\) 0 0
\(37\) −167.664 −0.744966 −0.372483 0.928039i \(-0.621493\pi\)
−0.372483 + 0.928039i \(0.621493\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −189.143 327.605i −0.720466 1.24788i −0.960813 0.277197i \(-0.910595\pi\)
0.240347 0.970687i \(-0.422739\pi\)
\(42\) 0 0
\(43\) −190.222 + 329.474i −0.674618 + 1.16847i 0.301962 + 0.953320i \(0.402358\pi\)
−0.976580 + 0.215153i \(0.930975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −123.414 + 213.760i −0.383018 + 0.663406i −0.991492 0.130167i \(-0.958449\pi\)
0.608474 + 0.793574i \(0.291782\pi\)
\(48\) 0 0
\(49\) −100.311 173.744i −0.292452 0.506541i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −284.344 −0.736937 −0.368469 0.929640i \(-0.620118\pi\)
−0.368469 + 0.929640i \(0.620118\pi\)
\(54\) 0 0
\(55\) 168.745 0.413701
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 363.699 + 629.945i 0.802534 + 1.39003i 0.917943 + 0.396712i \(0.129849\pi\)
−0.115409 + 0.993318i \(0.536818\pi\)
\(60\) 0 0
\(61\) −235.434 + 407.783i −0.494168 + 0.855923i −0.999977 0.00672169i \(-0.997860\pi\)
0.505810 + 0.862645i \(0.331194\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 222.235 384.922i 0.424074 0.734518i
\(66\) 0 0
\(67\) 375.978 + 651.213i 0.685567 + 1.18744i 0.973258 + 0.229714i \(0.0737791\pi\)
−0.287691 + 0.957723i \(0.592888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −864.963 −1.44581 −0.722903 0.690949i \(-0.757193\pi\)
−0.722903 + 0.690949i \(0.757193\pi\)
\(72\) 0 0
\(73\) −1003.53 −1.60896 −0.804479 0.593982i \(-0.797555\pi\)
−0.804479 + 0.593982i \(0.797555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 393.440 + 681.459i 0.582295 + 1.00856i
\(78\) 0 0
\(79\) 419.251 726.163i 0.597081 1.03417i −0.396169 0.918178i \(-0.629661\pi\)
0.993250 0.115996i \(-0.0370061\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −238.057 + 412.327i −0.314821 + 0.545287i −0.979399 0.201932i \(-0.935278\pi\)
0.664578 + 0.747219i \(0.268611\pi\)
\(84\) 0 0
\(85\) −271.653 470.517i −0.346646 0.600408i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 554.055 0.659884 0.329942 0.944001i \(-0.392971\pi\)
0.329942 + 0.944001i \(0.392971\pi\)
\(90\) 0 0
\(91\) 2072.62 2.38758
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −53.6996 93.0105i −0.0579944 0.100449i
\(96\) 0 0
\(97\) 523.994 907.584i 0.548490 0.950012i −0.449888 0.893085i \(-0.648536\pi\)
0.998378 0.0569275i \(-0.0181304\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 209.199 362.343i 0.206100 0.356975i −0.744383 0.667753i \(-0.767256\pi\)
0.950483 + 0.310778i \(0.100590\pi\)
\(102\) 0 0
\(103\) 640.394 + 1109.19i 0.612620 + 1.06109i 0.990797 + 0.135356i \(0.0432178\pi\)
−0.378177 + 0.925733i \(0.623449\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 351.116 0.317231 0.158615 0.987340i \(-0.449297\pi\)
0.158615 + 0.987340i \(0.449297\pi\)
\(108\) 0 0
\(109\) 1762.43 1.54872 0.774360 0.632746i \(-0.218072\pi\)
0.774360 + 0.632746i \(0.218072\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 500.931 + 867.638i 0.417023 + 0.722306i 0.995639 0.0932949i \(-0.0297399\pi\)
−0.578615 + 0.815601i \(0.696407\pi\)
\(114\) 0 0
\(115\) −214.806 + 372.055i −0.174181 + 0.301690i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1266.76 2194.09i 0.975826 1.69018i
\(120\) 0 0
\(121\) 96.0034 + 166.283i 0.0721288 + 0.124931i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 852.753 0.595824 0.297912 0.954593i \(-0.403710\pi\)
0.297912 + 0.954593i \(0.403710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1339.12 2319.42i −0.893123 1.54693i −0.836110 0.548562i \(-0.815176\pi\)
−0.0570131 0.998373i \(-0.518158\pi\)
\(132\) 0 0
\(133\) 250.409 433.721i 0.163257 0.282770i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 211.997 367.189i 0.132205 0.228986i −0.792321 0.610104i \(-0.791128\pi\)
0.924526 + 0.381118i \(0.124461\pi\)
\(138\) 0 0
\(139\) 335.757 + 581.549i 0.204882 + 0.354866i 0.950095 0.311960i \(-0.100986\pi\)
−0.745213 + 0.666826i \(0.767652\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3000.08 −1.75440
\(144\) 0 0
\(145\) −94.8907 −0.0543465
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1198.72 + 2076.24i 0.659080 + 1.14156i 0.980854 + 0.194743i \(0.0623874\pi\)
−0.321774 + 0.946816i \(0.604279\pi\)
\(150\) 0 0
\(151\) 151.005 261.547i 0.0813813 0.140957i −0.822462 0.568820i \(-0.807400\pi\)
0.903843 + 0.427863i \(0.140734\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 415.862 720.294i 0.215502 0.373261i
\(156\) 0 0
\(157\) 1117.50 + 1935.56i 0.568063 + 0.983914i 0.996757 + 0.0804642i \(0.0256403\pi\)
−0.428695 + 0.903449i \(0.641026\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2003.34 −0.980655
\(162\) 0 0
\(163\) −45.7276 −0.0219734 −0.0109867 0.999940i \(-0.503497\pi\)
−0.0109867 + 0.999940i \(0.503497\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1238.07 + 2144.40i 0.573681 + 0.993645i 0.996184 + 0.0872833i \(0.0278185\pi\)
−0.422502 + 0.906362i \(0.638848\pi\)
\(168\) 0 0
\(169\) −2852.56 + 4940.77i −1.29839 + 2.24887i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1574.72 2727.49i 0.692044 1.19866i −0.279123 0.960255i \(-0.590044\pi\)
0.971167 0.238400i \(-0.0766228\pi\)
\(174\) 0 0
\(175\) 291.446 + 504.800i 0.125893 + 0.218053i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4111.57 1.71683 0.858417 0.512953i \(-0.171448\pi\)
0.858417 + 0.512953i \(0.171448\pi\)
\(180\) 0 0
\(181\) −1344.52 −0.552138 −0.276069 0.961138i \(-0.589032\pi\)
−0.276069 + 0.961138i \(0.589032\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 419.159 + 726.005i 0.166579 + 0.288524i
\(186\) 0 0
\(187\) −1833.60 + 3175.89i −0.717039 + 1.24195i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1264.78 + 2190.67i −0.479144 + 0.829901i −0.999714 0.0239175i \(-0.992386\pi\)
0.520570 + 0.853819i \(0.325719\pi\)
\(192\) 0 0
\(193\) 2004.82 + 3472.45i 0.747722 + 1.29509i 0.948912 + 0.315540i \(0.102186\pi\)
−0.201191 + 0.979552i \(0.564481\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2489.65 −0.900408 −0.450204 0.892926i \(-0.648649\pi\)
−0.450204 + 0.892926i \(0.648649\pi\)
\(198\) 0 0
\(199\) 1358.91 0.484073 0.242037 0.970267i \(-0.422185\pi\)
0.242037 + 0.970267i \(0.422185\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −221.244 383.206i −0.0764941 0.132492i
\(204\) 0 0
\(205\) −945.713 + 1638.02i −0.322202 + 0.558071i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −362.462 + 627.802i −0.119962 + 0.207780i
\(210\) 0 0
\(211\) 1897.31 + 3286.24i 0.619034 + 1.07220i 0.989662 + 0.143417i \(0.0458090\pi\)
−0.370628 + 0.928781i \(0.620858\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1902.22 0.603397
\(216\) 0 0
\(217\) 3878.44 1.21330
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4829.66 + 8365.21i 1.47004 + 2.54618i
\(222\) 0 0
\(223\) 664.359 1150.70i 0.199501 0.345546i −0.748866 0.662722i \(-0.769401\pi\)
0.948367 + 0.317176i \(0.102734\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 495.789 858.731i 0.144963 0.251084i −0.784396 0.620260i \(-0.787027\pi\)
0.929359 + 0.369177i \(0.120360\pi\)
\(228\) 0 0
\(229\) 1987.89 + 3443.13i 0.573640 + 0.993574i 0.996188 + 0.0872336i \(0.0278026\pi\)
−0.422547 + 0.906341i \(0.638864\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4850.18 −1.36372 −0.681858 0.731484i \(-0.738828\pi\)
−0.681858 + 0.731484i \(0.738828\pi\)
\(234\) 0 0
\(235\) 1234.14 0.342582
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1835.50 3179.18i −0.496773 0.860437i 0.503220 0.864159i \(-0.332149\pi\)
−0.999993 + 0.00372178i \(0.998815\pi\)
\(240\) 0 0
\(241\) 1137.89 1970.88i 0.304140 0.526786i −0.672929 0.739707i \(-0.734964\pi\)
0.977069 + 0.212921i \(0.0682976\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −501.555 + 868.718i −0.130788 + 0.226532i
\(246\) 0 0
\(247\) 954.713 + 1653.61i 0.245939 + 0.425979i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 816.764 0.205393 0.102697 0.994713i \(-0.467253\pi\)
0.102697 + 0.994713i \(0.467253\pi\)
\(252\) 0 0
\(253\) 2899.80 0.720587
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3147.07 5450.88i −0.763847 1.32302i −0.940854 0.338812i \(-0.889975\pi\)
0.177008 0.984209i \(-0.443358\pi\)
\(258\) 0 0
\(259\) −1954.60 + 3385.46i −0.468930 + 0.812210i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −927.493 + 1606.47i −0.217459 + 0.376650i −0.954030 0.299710i \(-0.903110\pi\)
0.736572 + 0.676360i \(0.236443\pi\)
\(264\) 0 0
\(265\) 710.861 + 1231.25i 0.164784 + 0.285415i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3602.16 −0.816460 −0.408230 0.912879i \(-0.633854\pi\)
−0.408230 + 0.912879i \(0.633854\pi\)
\(270\) 0 0
\(271\) 2920.88 0.654727 0.327363 0.944898i \(-0.393840\pi\)
0.327363 + 0.944898i \(0.393840\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −421.862 730.687i −0.0925063 0.160226i
\(276\) 0 0
\(277\) 268.940 465.817i 0.0583358 0.101041i −0.835383 0.549669i \(-0.814754\pi\)
0.893718 + 0.448628i \(0.148087\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2035.26 3525.18i 0.432077 0.748379i −0.564975 0.825108i \(-0.691114\pi\)
0.997052 + 0.0767289i \(0.0244476\pi\)
\(282\) 0 0
\(283\) 805.835 + 1395.75i 0.169265 + 0.293175i 0.938162 0.346198i \(-0.112527\pi\)
−0.768897 + 0.639373i \(0.779194\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8819.98 −1.81403
\(288\) 0 0
\(289\) 6894.26 1.40327
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2228.29 3859.51i −0.444293 0.769539i 0.553709 0.832710i \(-0.313212\pi\)
−0.998003 + 0.0631712i \(0.979879\pi\)
\(294\) 0 0
\(295\) 1818.49 3149.72i 0.358904 0.621640i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3818.99 6614.68i 0.738655 1.27939i
\(300\) 0 0
\(301\) 4435.16 + 7681.92i 0.849297 + 1.47102i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2354.34 0.441997
\(306\) 0 0
\(307\) −11.9533 −0.00222219 −0.00111109 0.999999i \(-0.500354\pi\)
−0.00111109 + 0.999999i \(0.500354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4672.56 + 8093.12i 0.851951 + 1.47562i 0.879445 + 0.476000i \(0.157914\pi\)
−0.0274941 + 0.999622i \(0.508753\pi\)
\(312\) 0 0
\(313\) 4384.12 7593.51i 0.791709 1.37128i −0.133199 0.991089i \(-0.542525\pi\)
0.924908 0.380191i \(-0.124142\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 988.087 1711.42i 0.175068 0.303226i −0.765117 0.643891i \(-0.777319\pi\)
0.940185 + 0.340665i \(0.110652\pi\)
\(318\) 0 0
\(319\) 320.246 + 554.683i 0.0562080 + 0.0973551i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2334.03 0.402070
\(324\) 0 0
\(325\) −2222.35 −0.379303
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2877.49 + 4983.96i 0.482192 + 0.835182i
\(330\) 0 0
\(331\) −3644.28 + 6312.08i −0.605159 + 1.04817i 0.386867 + 0.922135i \(0.373557\pi\)
−0.992026 + 0.126031i \(0.959776\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1879.89 3256.06i 0.306595 0.531038i
\(336\) 0 0
\(337\) −3649.89 6321.80i −0.589977 1.02187i −0.994235 0.107225i \(-0.965803\pi\)
0.404258 0.914645i \(-0.367530\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5613.96 −0.891534
\(342\) 0 0
\(343\) 3319.65 0.522577
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3381.18 + 5856.37i 0.523087 + 0.906013i 0.999639 + 0.0268668i \(0.00855299\pi\)
−0.476552 + 0.879146i \(0.658114\pi\)
\(348\) 0 0
\(349\) 3436.48 5952.16i 0.527079 0.912928i −0.472423 0.881372i \(-0.656621\pi\)
0.999502 0.0315556i \(-0.0100461\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4805.59 8323.53i 0.724578 1.25501i −0.234570 0.972099i \(-0.575368\pi\)
0.959148 0.282906i \(-0.0912984\pi\)
\(354\) 0 0
\(355\) 2162.41 + 3745.40i 0.323292 + 0.559958i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7951.69 −1.16901 −0.584504 0.811391i \(-0.698711\pi\)
−0.584504 + 0.811391i \(0.698711\pi\)
\(360\) 0 0
\(361\) −6397.62 −0.932733
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2508.82 + 4345.40i 0.359774 + 0.623146i
\(366\) 0 0
\(367\) −3194.49 + 5533.02i −0.454362 + 0.786978i −0.998651 0.0519193i \(-0.983466\pi\)
0.544289 + 0.838898i \(0.316799\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3314.84 + 5741.48i −0.463876 + 0.803457i
\(372\) 0 0
\(373\) 4581.58 + 7935.52i 0.635992 + 1.10157i 0.986304 + 0.164938i \(0.0527423\pi\)
−0.350312 + 0.936633i \(0.613924\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1687.04 0.230469
\(378\) 0 0
\(379\) −10303.5 −1.39646 −0.698228 0.715875i \(-0.746028\pi\)
−0.698228 + 0.715875i \(0.746028\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1859.95 + 3221.53i 0.248144 + 0.429798i 0.963011 0.269463i \(-0.0868461\pi\)
−0.714867 + 0.699260i \(0.753513\pi\)
\(384\) 0 0
\(385\) 1967.20 3407.29i 0.260410 0.451044i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2235.23 + 3871.54i −0.291339 + 0.504614i −0.974127 0.226003i \(-0.927434\pi\)
0.682788 + 0.730617i \(0.260767\pi\)
\(390\) 0 0
\(391\) −4668.22 8085.59i −0.603790 1.04580i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4192.51 −0.534045
\(396\) 0 0
\(397\) 4848.35 0.612927 0.306463 0.951882i \(-0.400854\pi\)
0.306463 + 0.951882i \(0.400854\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −74.7667 129.500i −0.00931090 0.0161269i 0.861332 0.508042i \(-0.169630\pi\)
−0.870643 + 0.491915i \(0.836297\pi\)
\(402\) 0 0
\(403\) −7393.51 + 12805.9i −0.913888 + 1.58290i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2829.24 4900.38i 0.344570 0.596813i
\(408\) 0 0
\(409\) −5324.84 9222.89i −0.643756 1.11502i −0.984587 0.174894i \(-0.944042\pi\)
0.340831 0.940125i \(-0.389292\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16959.8 2.02067
\(414\) 0 0
\(415\) 2380.57 0.281585
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −169.897 294.269i −0.0198090 0.0343103i 0.855951 0.517057i \(-0.172972\pi\)
−0.875760 + 0.482747i \(0.839639\pi\)
\(420\) 0 0
\(421\) −1066.02 + 1846.40i −0.123408 + 0.213748i −0.921109 0.389304i \(-0.872716\pi\)
0.797702 + 0.603052i \(0.206049\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1358.27 + 2352.58i −0.155025 + 0.268511i
\(426\) 0 0
\(427\) 5489.31 + 9507.76i 0.622122 + 1.07755i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2509.67 −0.280480 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(432\) 0 0
\(433\) 2792.17 0.309891 0.154946 0.987923i \(-0.450480\pi\)
0.154946 + 0.987923i \(0.450480\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −922.801 1598.34i −0.101015 0.174963i
\(438\) 0 0
\(439\) 3278.50 5678.53i 0.356433 0.617361i −0.630929 0.775841i \(-0.717326\pi\)
0.987362 + 0.158480i \(0.0506594\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −949.888 + 1645.25i −0.101875 + 0.176452i −0.912457 0.409172i \(-0.865817\pi\)
0.810582 + 0.585625i \(0.199151\pi\)
\(444\) 0 0
\(445\) −1385.14 2399.13i −0.147555 0.255572i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9468.83 −0.995238 −0.497619 0.867396i \(-0.665792\pi\)
−0.497619 + 0.867396i \(0.665792\pi\)
\(450\) 0 0
\(451\) 12766.7 1.33295
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5181.55 8974.72i −0.533879 0.924706i
\(456\) 0 0
\(457\) −1449.85 + 2511.21i −0.148405 + 0.257045i −0.930638 0.365941i \(-0.880747\pi\)
0.782233 + 0.622986i \(0.214081\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2758.81 + 4778.39i −0.278721 + 0.482759i −0.971067 0.238806i \(-0.923244\pi\)
0.692346 + 0.721566i \(0.256577\pi\)
\(462\) 0 0
\(463\) 4485.15 + 7768.50i 0.450200 + 0.779769i 0.998398 0.0565795i \(-0.0180194\pi\)
−0.548198 + 0.836348i \(0.684686\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12314.8 1.22026 0.610130 0.792302i \(-0.291117\pi\)
0.610130 + 0.792302i \(0.291117\pi\)
\(468\) 0 0
\(469\) 17532.4 1.72616
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6419.80 11119.4i −0.624065 1.08091i
\(474\) 0 0
\(475\) −268.498 + 465.053i −0.0259359 + 0.0449223i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6364.47 + 11023.6i −0.607098 + 1.05153i 0.384618 + 0.923076i \(0.374333\pi\)
−0.991716 + 0.128449i \(0.959000\pi\)
\(480\) 0 0
\(481\) −7452.13 12907.5i −0.706420 1.22356i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5239.94 −0.490584
\(486\) 0 0
\(487\) −7230.67 −0.672798 −0.336399 0.941719i \(-0.609209\pi\)
−0.336399 + 0.941719i \(0.609209\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8803.45 15248.0i −0.809153 1.40149i −0.913451 0.406949i \(-0.866593\pi\)
0.104298 0.994546i \(-0.466741\pi\)
\(492\) 0 0
\(493\) 1031.09 1785.91i 0.0941950 0.163150i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10083.6 + 17465.3i −0.910084 + 1.57631i
\(498\) 0 0
\(499\) 10175.8 + 17625.0i 0.912886 + 1.58116i 0.809967 + 0.586475i \(0.199485\pi\)
0.102918 + 0.994690i \(0.467182\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14477.1 1.28331 0.641654 0.766995i \(-0.278249\pi\)
0.641654 + 0.766995i \(0.278249\pi\)
\(504\) 0 0
\(505\) −2091.99 −0.184341
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1841.81 + 3190.11i 0.160387 + 0.277798i 0.935007 0.354628i \(-0.115393\pi\)
−0.774621 + 0.632426i \(0.782059\pi\)
\(510\) 0 0
\(511\) −11699.0 + 20263.2i −1.01278 + 1.75419i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3201.97 5545.97i 0.273972 0.474534i
\(516\) 0 0
\(517\) −4165.11 7214.18i −0.354316 0.613693i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20684.0 1.73931 0.869655 0.493660i \(-0.164341\pi\)
0.869655 + 0.493660i \(0.164341\pi\)
\(522\) 0 0
\(523\) −18781.8 −1.57031 −0.785153 0.619302i \(-0.787416\pi\)
−0.785153 + 0.619302i \(0.787416\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9037.61 + 15653.6i 0.747029 + 1.29389i
\(528\) 0 0
\(529\) 2392.16 4143.35i 0.196611 0.340540i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16813.6 29122.0i 1.36638 2.36663i
\(534\) 0 0
\(535\) −877.791 1520.38i −0.0709349 0.122863i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6770.78 0.541073
\(540\) 0 0
\(541\) 9322.54 0.740864 0.370432 0.928860i \(-0.379210\pi\)
0.370432 + 0.928860i \(0.379210\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4406.08 7631.56i −0.346304 0.599816i
\(546\) 0 0
\(547\) 4072.48 7053.74i 0.318330 0.551364i −0.661809 0.749672i \(-0.730211\pi\)
0.980140 + 0.198308i \(0.0635446\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 203.824 353.033i 0.0157590 0.0272953i
\(552\) 0 0
\(553\) −9775.12 16931.0i −0.751682 1.30195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1025.83 −0.0780355 −0.0390177 0.999239i \(-0.512423\pi\)
−0.0390177 + 0.999239i \(0.512423\pi\)
\(558\) 0 0
\(559\) −33819.1 −2.55885
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4406.31 7631.95i −0.329847 0.571311i 0.652634 0.757673i \(-0.273664\pi\)
−0.982481 + 0.186362i \(0.940330\pi\)
\(564\) 0 0
\(565\) 2504.66 4338.19i 0.186499 0.323025i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3186.43 + 5519.07i −0.234767 + 0.406628i −0.959205 0.282712i \(-0.908766\pi\)
0.724438 + 0.689340i \(0.242099\pi\)
\(570\) 0 0
\(571\) −5420.99 9389.44i −0.397306 0.688154i 0.596087 0.802920i \(-0.296721\pi\)
−0.993393 + 0.114766i \(0.963388\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2148.06 0.155792
\(576\) 0 0
\(577\) −1494.00 −0.107792 −0.0538959 0.998547i \(-0.517164\pi\)
−0.0538959 + 0.998547i \(0.517164\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5550.47 + 9613.70i 0.396338 + 0.686477i
\(582\) 0 0
\(583\) 4798.16 8310.66i 0.340857 0.590382i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7829.53 + 13561.1i −0.550527 + 0.953541i 0.447710 + 0.894179i \(0.352240\pi\)
−0.998237 + 0.0593617i \(0.981093\pi\)
\(588\) 0 0
\(589\) 1786.53 + 3094.36i 0.124979 + 0.216470i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1722.59 −0.119289 −0.0596444 0.998220i \(-0.518997\pi\)
−0.0596444 + 0.998220i \(0.518997\pi\)
\(594\) 0 0
\(595\) −12667.6 −0.872806
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3301.86 5718.98i −0.225226 0.390102i 0.731161 0.682205i \(-0.238979\pi\)
−0.956387 + 0.292102i \(0.905645\pi\)
\(600\) 0 0
\(601\) 3237.98 5608.35i 0.219767 0.380648i −0.734970 0.678100i \(-0.762804\pi\)
0.954737 + 0.297452i \(0.0961369\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 480.017 831.414i 0.0322570 0.0558707i
\(606\) 0 0
\(607\) 3965.93 + 6869.20i 0.265193 + 0.459328i 0.967614 0.252434i \(-0.0812309\pi\)
−0.702421 + 0.711762i \(0.747898\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21941.6 −1.45280
\(612\) 0 0
\(613\) −16332.6 −1.07613 −0.538065 0.842903i \(-0.680845\pi\)
−0.538065 + 0.842903i \(0.680845\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5835.76 + 10107.8i 0.380776 + 0.659524i 0.991173 0.132572i \(-0.0423235\pi\)
−0.610397 + 0.792095i \(0.708990\pi\)
\(618\) 0 0
\(619\) 5361.24 9285.93i 0.348120 0.602961i −0.637796 0.770206i \(-0.720154\pi\)
0.985915 + 0.167244i \(0.0534869\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6459.09 11187.5i 0.415374 0.719448i
\(624\) 0 0
\(625\) −312.500 541.266i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18218.5 −1.15488
\(630\) 0 0
\(631\) 9754.15 0.615383 0.307691 0.951486i \(-0.400444\pi\)
0.307691 + 0.951486i \(0.400444\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2131.88 3692.53i −0.133230 0.230761i
\(636\) 0 0
\(637\) 8917.02 15444.7i 0.554639 0.960664i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9682.73 + 16771.0i −0.596638 + 1.03341i 0.396676 + 0.917959i \(0.370164\pi\)
−0.993313 + 0.115448i \(0.963170\pi\)
\(642\) 0 0
\(643\) −14537.7 25180.0i −0.891616 1.54432i −0.837938 0.545766i \(-0.816239\pi\)
−0.0536786 0.998558i \(-0.517095\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17400.3 1.05730 0.528651 0.848839i \(-0.322698\pi\)
0.528651 + 0.848839i \(0.322698\pi\)
\(648\) 0 0
\(649\) −24548.9 −1.48479
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4233.84 + 7333.23i 0.253726 + 0.439466i 0.964549 0.263905i \(-0.0850105\pi\)
−0.710823 + 0.703371i \(0.751677\pi\)
\(654\) 0 0
\(655\) −6695.58 + 11597.1i −0.399417 + 0.691810i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11991.3 + 20769.5i −0.708822 + 1.22772i 0.256472 + 0.966552i \(0.417440\pi\)
−0.965294 + 0.261164i \(0.915894\pi\)
\(660\) 0 0
\(661\) −13361.5 23142.9i −0.786238 1.36180i −0.928256 0.371941i \(-0.878692\pi\)
0.142018 0.989864i \(-0.454641\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2504.09 −0.146022
\(666\) 0 0
\(667\) −1630.65 −0.0946611
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7945.65 13762.3i −0.457136 0.791783i
\(672\) 0 0
\(673\) 7560.71 13095.5i 0.433052 0.750068i −0.564082 0.825719i \(-0.690770\pi\)
0.997134 + 0.0756503i \(0.0241033\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7468.82 12936.4i 0.424003 0.734395i −0.572324 0.820028i \(-0.693958\pi\)
0.996327 + 0.0856328i \(0.0272912\pi\)
\(678\) 0 0
\(679\) −12217.3 21161.0i −0.690510 1.19600i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11141.0 0.624158 0.312079 0.950056i \(-0.398975\pi\)
0.312079 + 0.950056i \(0.398975\pi\)
\(684\) 0 0
\(685\) −2119.97 −0.118248
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12638.2 21890.0i −0.698807 1.21037i
\(690\) 0 0
\(691\) 5981.42 10360.1i 0.329297 0.570359i −0.653076 0.757293i \(-0.726522\pi\)
0.982373 + 0.186934i \(0.0598550\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1678.79 2907.74i 0.0916259 0.158701i
\(696\) 0 0
\(697\) −20552.5 35597.9i −1.11690 1.93453i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17436.0 −0.939442 −0.469721 0.882815i \(-0.655646\pi\)
−0.469721 + 0.882815i \(0.655646\pi\)
\(702\) 0 0
\(703\) −3601.39 −0.193213
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4877.62 8448.28i −0.259465 0.449406i
\(708\) 0 0
\(709\) −5664.21 + 9810.70i −0.300034 + 0.519673i −0.976143 0.217128i \(-0.930331\pi\)
0.676110 + 0.736801i \(0.263665\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7146.37 12377.9i 0.375363 0.650148i
\(714\) 0 0
\(715\) 7500.19 + 12990.7i 0.392295 + 0.679475i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36889.9 1.91344 0.956718 0.291015i \(-0.0939931\pi\)
0.956718 + 0.291015i \(0.0939931\pi\)
\(720\) 0 0
\(721\) 29862.5 1.54249
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 237.227 + 410.889i 0.0121522 + 0.0210483i
\(726\) 0 0
\(727\) −5477.89 + 9487.99i −0.279455 + 0.484030i −0.971249 0.238064i \(-0.923487\pi\)
0.691794 + 0.722095i \(0.256820\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −20669.7 + 35801.1i −1.04583 + 1.81142i
\(732\) 0 0
\(733\) 11191.0 + 19383.4i 0.563915 + 0.976730i 0.997150 + 0.0754488i \(0.0240390\pi\)
−0.433234 + 0.901281i \(0.642628\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25377.7 −1.26839
\(738\) 0 0
\(739\) 27800.0 1.38382 0.691908 0.721985i \(-0.256770\pi\)
0.691908 + 0.721985i \(0.256770\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8963.13 15524.6i −0.442564 0.766544i 0.555315 0.831640i \(-0.312598\pi\)
−0.997879 + 0.0650965i \(0.979264\pi\)
\(744\) 0 0
\(745\) 5993.59 10381.2i 0.294749 0.510521i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4093.26 7089.74i 0.199686 0.345866i
\(750\) 0 0
\(751\) −12059.9 20888.4i −0.585982 1.01495i −0.994752 0.102313i \(-0.967376\pi\)
0.408770 0.912637i \(-0.365958\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1510.05 −0.0727896
\(756\) 0 0
\(757\) 10582.4 0.508088 0.254044 0.967193i \(-0.418239\pi\)
0.254044 + 0.967193i \(0.418239\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18291.5 + 31681.9i 0.871312 + 1.50916i 0.860641 + 0.509213i \(0.170063\pi\)
0.0106710 + 0.999943i \(0.496603\pi\)
\(762\) 0 0
\(763\) 20546.2 35587.0i 0.974864 1.68851i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −32330.6 + 55998.2i −1.52202 + 2.63622i
\(768\) 0 0
\(769\) −7388.24 12796.8i −0.346459 0.600084i 0.639159 0.769075i \(-0.279283\pi\)
−0.985618 + 0.168991i \(0.945949\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3195.12 0.148668 0.0743340 0.997233i \(-0.476317\pi\)
0.0743340 + 0.997233i \(0.476317\pi\)
\(774\) 0 0
\(775\) −4158.62 −0.192751
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4062.76 7036.90i −0.186859 0.323650i
\(780\) 0 0
\(781\) 14595.8 25280.7i 0.668731 1.15828i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5587.48 9677.80i 0.254045 0.440020i
\(786\) 0 0
\(787\) −15493.6 26835.6i −0.701761 1.21549i −0.967848 0.251537i \(-0.919064\pi\)
0.266087 0.963949i \(-0.414269\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 23359.1 1.05001
\(792\) 0 0
\(793\) −41857.2 −1.87439
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18541.5 32114.9i −0.824059 1.42731i −0.902636 0.430405i \(-0.858371\pi\)
0.0785767 0.996908i \(-0.474962\pi\)
\(798\) 0 0
\(799\) −13410.4 + 23227.4i −0.593772 + 1.02844i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16934.0 29330.5i 0.744194 1.28898i
\(804\) 0 0
\(805\) 5008.36 + 8674.73i 0.219281 + 0.379806i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −648.034 −0.0281627 −0.0140814 0.999901i \(-0.504482\pi\)
−0.0140814 + 0.999901i \(0.504482\pi\)
\(810\) 0 0
\(811\) −36017.4 −1.55948 −0.779742 0.626101i \(-0.784650\pi\)
−0.779742 + 0.626101i \(0.784650\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 114.319 + 198.006i 0.00491340 + 0.00851025i
\(816\) 0 0
\(817\) −4085.94 + 7077.06i −0.174968 + 0.303054i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14404.5 + 24949.3i −0.612326 + 1.06058i 0.378522 + 0.925592i \(0.376433\pi\)
−0.990847 + 0.134987i \(0.956901\pi\)
\(822\) 0 0
\(823\) 13777.3 + 23863.0i 0.583533 + 1.01071i 0.995057 + 0.0993094i \(0.0316634\pi\)
−0.411524 + 0.911399i \(0.635003\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34914.4 1.46807 0.734035 0.679111i \(-0.237635\pi\)
0.734035 + 0.679111i \(0.237635\pi\)
\(828\) 0 0
\(829\) 749.124 0.0313850 0.0156925 0.999877i \(-0.495005\pi\)
0.0156925 + 0.999877i \(0.495005\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −10899.9 18879.2i −0.453373 0.785264i
\(834\) 0 0
\(835\) 6190.35 10722.0i 0.256558 0.444372i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18727.9 + 32437.6i −0.770629 + 1.33477i 0.166590 + 0.986026i \(0.446724\pi\)
−0.937219 + 0.348742i \(0.886609\pi\)
\(840\) 0 0
\(841\) 12014.4 + 20809.6i 0.492616 + 0.853236i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 28525.6 1.16131
\(846\) 0 0
\(847\) 4476.77 0.181610
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7203.04 + 12476.0i 0.290149 + 0.502553i
\(852\) 0 0
\(853\) 5903.02 10224.3i 0.236947 0.410404i −0.722890 0.690963i \(-0.757187\pi\)
0.959837 + 0.280559i \(0.0905199\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12034.6 20844.5i 0.479688 0.830844i −0.520041 0.854141i \(-0.674083\pi\)
0.999729 + 0.0232978i \(0.00741659\pi\)
\(858\) 0 0
\(859\) 5683.98 + 9844.94i 0.225768 + 0.391042i 0.956550 0.291570i \(-0.0941775\pi\)
−0.730782 + 0.682611i \(0.760844\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 10733.7 0.423382 0.211691 0.977337i \(-0.432103\pi\)
0.211691 + 0.977337i \(0.432103\pi\)
\(864\) 0 0
\(865\) −15747.2 −0.618983
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14149.3 + 24507.3i 0.552338 + 0.956677i
\(870\) 0 0
\(871\) −33422.1 + 57888.8i −1.30019 + 2.25199i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1457.23 2524.00i 0.0563010 0.0975163i
\(876\) 0 0
\(877\) −9775.48 16931.6i −0.376391 0.651928i 0.614143 0.789194i \(-0.289502\pi\)
−0.990534 + 0.137267i \(0.956168\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35047.7 1.34028 0.670140 0.742235i \(-0.266234\pi\)
0.670140 + 0.742235i \(0.266234\pi\)
\(882\) 0 0
\(883\) −35769.3 −1.36323 −0.681615 0.731711i \(-0.738722\pi\)
−0.681615 + 0.731711i \(0.738722\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9019.69 15622.6i −0.341434 0.591380i 0.643266 0.765643i \(-0.277579\pi\)
−0.984699 + 0.174263i \(0.944246\pi\)
\(888\) 0 0
\(889\) 9941.27 17218.8i 0.375050 0.649606i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2650.92 + 4591.53i −0.0993390 + 0.172060i
\(894\) 0 0
\(895\) −10278.9 17803.6i −0.383896 0.664927i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3156.91 0.117118
\(900\) 0 0
\(901\) −30897.2 −1.14244
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3361.29 + 5821.92i 0.123462 + 0.213842i
\(906\) 0 0
\(907\) −14525.1 + 25158.2i −0.531750 + 0.921019i 0.467563 + 0.883960i \(0.345132\pi\)
−0.999313 + 0.0370589i \(0.988201\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5273.31 + 9133.64i −0.191781 + 0.332174i −0.945841 0.324632i \(-0.894760\pi\)
0.754060 + 0.656806i \(0.228093\pi\)
\(912\) 0 0
\(913\) −8034.19 13915.6i −0.291230 0.504425i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −62444.9 −2.24876
\(918\) 0 0
\(919\) 2975.12 0.106790 0.0533951 0.998573i \(-0.482996\pi\)
0.0533951 + 0.998573i \(0.482996\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −38444.9 66588.6i −1.37100 2.37464i
\(924\) 0 0
\(925\) 2095.80 3630.02i 0.0744966 0.129032i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 25152.3 43565.1i 0.888290 1.53856i 0.0463940 0.998923i \(-0.485227\pi\)
0.841896 0.539640i \(-0.181440\pi\)
\(930\) 0 0
\(931\) −2154.66 3731.99i −0.0758499 0.131376i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18336.0 0.641339
\(936\) 0 0
\(937\) −44253.7 −1.54291 −0.771454 0.636285i \(-0.780470\pi\)
−0.771454 + 0.636285i \(0.780470\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23676.1 + 41008.1i 0.820210 + 1.42064i 0.905526 + 0.424291i \(0.139477\pi\)
−0.0853163 + 0.996354i \(0.527190\pi\)
\(942\) 0 0
\(943\) −16251.6 + 28148.6i −0.561214 + 0.972051i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11982.1 20753.5i 0.411156 0.712143i −0.583861 0.811854i \(-0.698459\pi\)
0.995016 + 0.0997113i \(0.0317919\pi\)
\(948\) 0 0
\(949\) −44603.7 77255.8i −1.52571 2.64260i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6697.41 −0.227650 −0.113825 0.993501i \(-0.536310\pi\)
−0.113825 + 0.993501i \(0.536310\pi\)
\(954\) 0 0
\(955\) 12647.8 0.428559
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4942.85 8561.26i −0.166437 0.288277i
\(960\) 0 0
\(961\) 1060.22 1836.35i 0.0355885 0.0616411i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 10024.1 17362.3i 0.334391 0.579183i
\(966\) 0 0
\(967\) 18497.8 + 32039.2i 0.615150 + 1.06547i 0.990358 + 0.138530i \(0.0442379\pi\)
−0.375208 + 0.926941i \(0.622429\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −32987.1 −1.09022 −0.545111 0.838364i \(-0.683513\pi\)
−0.545111 + 0.838364i \(0.683513\pi\)
\(972\) 0 0
\(973\) 15656.8 0.515864
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9924.79 + 17190.2i 0.324997 + 0.562911i 0.981512 0.191402i \(-0.0613034\pi\)
−0.656515 + 0.754313i \(0.727970\pi\)
\(978\) 0 0
\(979\) −9349.39 + 16193.6i −0.305217 + 0.528652i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6503.35 11264.1i 0.211012 0.365483i −0.741020 0.671483i \(-0.765657\pi\)
0.952031 + 0.306000i \(0.0989908\pi\)
\(984\) 0 0
\(985\) 6224.13 + 10780.5i 0.201337 + 0.348727i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32688.7 1.05100
\(990\) 0 0
\(991\) 14119.3 0.452588 0.226294 0.974059i \(-0.427339\pi\)
0.226294 + 0.974059i \(0.427339\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3397.28 5884.25i −0.108242 0.187481i
\(996\) 0 0
\(997\) −9958.43 + 17248.5i −0.316336 + 0.547910i −0.979721 0.200369i \(-0.935786\pi\)
0.663385 + 0.748278i \(0.269119\pi\)
\(998\) 0 0
\(999\) 0 0
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 1620.4.i.w.541.6 12
3.2 odd 2 1620.4.i.x.541.6 12
9.2 odd 6 1620.4.a.i.1.1 6
9.4 even 3 inner 1620.4.i.w.1081.6 12
9.5 odd 6 1620.4.i.x.1081.6 12
9.7 even 3 1620.4.a.j.1.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.4.a.i.1.1 6 9.2 odd 6
1620.4.a.j.1.1 yes 6 9.7 even 3
1620.4.i.w.541.6 12 1.1 even 1 trivial
1620.4.i.w.1081.6 12 9.4 even 3 inner
1620.4.i.x.541.6 12 3.2 odd 2
1620.4.i.x.1081.6 12 9.5 odd 6