Properties

Label 1008.2.bt.c.17.4
Level $1008$
Weight $2$
Character 1008.17
Analytic conductor $8.049$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1008,2,Mod(17,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (of order \(6\), 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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.4
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1008.17
Dual form 1008.2.bt.c.593.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.09077 + 3.62132i) q^{5} +(1.62132 + 2.09077i) q^{7} +O(q^{10})\) \(q+(2.09077 + 3.62132i) q^{5} +(1.62132 + 2.09077i) q^{7} +(2.59808 + 1.50000i) q^{11} -2.44949i q^{13} +(-0.507306 + 0.878680i) q^{17} +(0.878680 - 0.507306i) q^{19} +(3.67423 - 2.12132i) q^{23} +(-6.24264 + 10.8126i) q^{25} -1.24264i q^{29} +(-4.86396 - 2.80821i) q^{31} +(-4.18154 + 10.2426i) q^{35} +(-4.12132 - 7.13834i) q^{37} +2.02922 q^{41} -8.24264 q^{43} +(0.507306 + 0.878680i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(1.07616 + 0.621320i) q^{53} +12.5446i q^{55} +(5.76500 - 9.98528i) q^{59} +(5.12132 - 2.95680i) q^{61} +(8.87039 - 5.12132i) q^{65} +(-5.00000 + 8.66025i) q^{67} +10.2426i q^{71} +(7.24264 + 4.18154i) q^{73} +(1.07616 + 7.86396i) q^{77} +(-5.62132 - 9.73641i) q^{79} +3.16693 q^{83} -4.24264 q^{85} +(5.19615 + 9.00000i) q^{89} +(5.12132 - 3.97141i) q^{91} +(3.67423 + 2.12132i) q^{95} -3.76127i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 24 q^{19} - 16 q^{25} + 12 q^{31} - 16 q^{37} - 32 q^{43} + 20 q^{49} + 24 q^{61} - 40 q^{67} + 24 q^{73} - 28 q^{79} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.09077 + 3.62132i 0.935021 + 1.61950i 0.774597 + 0.632456i \(0.217953\pi\)
0.160424 + 0.987048i \(0.448714\pi\)
\(6\) 0 0
\(7\) 1.62132 + 2.09077i 0.612801 + 0.790237i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.59808 + 1.50000i 0.783349 + 0.452267i 0.837616 0.546259i \(-0.183949\pi\)
−0.0542666 + 0.998526i \(0.517282\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i −0.940540 0.339683i \(-0.889680\pi\)
0.940540 0.339683i \(-0.110320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.507306 + 0.878680i −0.123040 + 0.213111i −0.920965 0.389645i \(-0.872598\pi\)
0.797925 + 0.602756i \(0.205931\pi\)
\(18\) 0 0
\(19\) 0.878680 0.507306i 0.201583 0.116384i −0.395811 0.918332i \(-0.629536\pi\)
0.597394 + 0.801948i \(0.296203\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.67423 2.12132i 0.766131 0.442326i −0.0653618 0.997862i \(-0.520820\pi\)
0.831493 + 0.555536i \(0.187487\pi\)
\(24\) 0 0
\(25\) −6.24264 + 10.8126i −1.24853 + 2.16251i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.24264i 0.230753i −0.993322 0.115376i \(-0.963193\pi\)
0.993322 0.115376i \(-0.0368074\pi\)
\(30\) 0 0
\(31\) −4.86396 2.80821i −0.873593 0.504369i −0.00505256 0.999987i \(-0.501608\pi\)
−0.868541 + 0.495618i \(0.834942\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.18154 + 10.2426i −0.706809 + 1.73132i
\(36\) 0 0
\(37\) −4.12132 7.13834i −0.677541 1.17354i −0.975719 0.219025i \(-0.929712\pi\)
0.298178 0.954510i \(-0.403621\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.02922 0.316912 0.158456 0.987366i \(-0.449348\pi\)
0.158456 + 0.987366i \(0.449348\pi\)
\(42\) 0 0
\(43\) −8.24264 −1.25699 −0.628495 0.777813i \(-0.716329\pi\)
−0.628495 + 0.777813i \(0.716329\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.507306 + 0.878680i 0.0739982 + 0.128169i 0.900650 0.434545i \(-0.143091\pi\)
−0.826652 + 0.562713i \(0.809757\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.07616 + 0.621320i 0.147822 + 0.0853449i 0.572087 0.820193i \(-0.306134\pi\)
−0.424265 + 0.905538i \(0.639467\pi\)
\(54\) 0 0
\(55\) 12.5446i 1.69152i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.76500 9.98528i 0.750540 1.29997i −0.197022 0.980399i \(-0.563127\pi\)
0.947561 0.319574i \(-0.103540\pi\)
\(60\) 0 0
\(61\) 5.12132 2.95680i 0.655718 0.378579i −0.134926 0.990856i \(-0.543080\pi\)
0.790643 + 0.612277i \(0.209746\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.87039 5.12132i 1.10024 0.635222i
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2426i 1.21558i 0.794099 + 0.607789i \(0.207943\pi\)
−0.794099 + 0.607789i \(0.792057\pi\)
\(72\) 0 0
\(73\) 7.24264 + 4.18154i 0.847687 + 0.489412i 0.859870 0.510513i \(-0.170545\pi\)
−0.0121828 + 0.999926i \(0.503878\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.07616 + 7.86396i 0.122640 + 0.896182i
\(78\) 0 0
\(79\) −5.62132 9.73641i −0.632448 1.09543i −0.987050 0.160415i \(-0.948717\pi\)
0.354602 0.935017i \(-0.384616\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.16693 0.347616 0.173808 0.984780i \(-0.444393\pi\)
0.173808 + 0.984780i \(0.444393\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 + 9.00000i 0.550791 + 0.953998i 0.998218 + 0.0596775i \(0.0190072\pi\)
−0.447427 + 0.894321i \(0.647659\pi\)
\(90\) 0 0
\(91\) 5.12132 3.97141i 0.536860 0.416317i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.67423 + 2.12132i 0.376969 + 0.217643i
\(96\) 0 0
\(97\) 3.76127i 0.381900i −0.981600 0.190950i \(-0.938843\pi\)
0.981600 0.190950i \(-0.0611568\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 13.2426 7.64564i 1.30484 0.753348i 0.323607 0.946192i \(-0.395105\pi\)
0.981229 + 0.192844i \(0.0617712\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.75039 + 2.74264i −0.459238 + 0.265141i −0.711724 0.702459i \(-0.752085\pi\)
0.252486 + 0.967601i \(0.418752\pi\)
\(108\) 0 0
\(109\) −0.757359 + 1.31178i −0.0725419 + 0.125646i −0.900015 0.435860i \(-0.856444\pi\)
0.827473 + 0.561506i \(0.189778\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.48528i 0.798228i −0.916901 0.399114i \(-0.869318\pi\)
0.916901 0.399114i \(-0.130682\pi\)
\(114\) 0 0
\(115\) 15.3640 + 8.87039i 1.43270 + 0.827168i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.65962 + 0.363961i −0.243807 + 0.0333643i
\(120\) 0 0
\(121\) −1.00000 1.73205i −0.0909091 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −31.3000 −2.79956
\(126\) 0 0
\(127\) 5.24264 0.465209 0.232605 0.972571i \(-0.425275\pi\)
0.232605 + 0.972571i \(0.425275\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.59808 + 4.50000i 0.226995 + 0.393167i 0.956916 0.290365i \(-0.0937766\pi\)
−0.729921 + 0.683531i \(0.760443\pi\)
\(132\) 0 0
\(133\) 2.48528 + 1.01461i 0.215501 + 0.0879780i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.5446 + 7.24264i 1.07176 + 0.618781i 0.928662 0.370928i \(-0.120960\pi\)
0.143098 + 0.989709i \(0.454294\pi\)
\(138\) 0 0
\(139\) 20.1903i 1.71252i −0.516549 0.856258i \(-0.672783\pi\)
0.516549 0.856258i \(-0.327217\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.67423 6.36396i 0.307255 0.532181i
\(144\) 0 0
\(145\) 4.50000 2.59808i 0.373705 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.7408 + 10.2426i −1.45338 + 0.839110i −0.998671 0.0515300i \(-0.983590\pi\)
−0.454709 + 0.890640i \(0.650257\pi\)
\(150\) 0 0
\(151\) −1.62132 + 2.80821i −0.131941 + 0.228529i −0.924425 0.381364i \(-0.875454\pi\)
0.792484 + 0.609893i \(0.208788\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 23.4853i 1.88638i
\(156\) 0 0
\(157\) −12.7279 7.34847i −1.01580 0.586472i −0.102915 0.994690i \(-0.532817\pi\)
−0.912884 + 0.408219i \(0.866150\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.3923 + 4.24264i 0.819028 + 0.334367i
\(162\) 0 0
\(163\) 3.12132 + 5.40629i 0.244481 + 0.423453i 0.961985 0.273101i \(-0.0880492\pi\)
−0.717505 + 0.696554i \(0.754716\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.0600 −1.78444 −0.892219 0.451603i \(-0.850852\pi\)
−0.892219 + 0.451603i \(0.850852\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.3923 18.0000i −0.790112 1.36851i −0.925897 0.377776i \(-0.876689\pi\)
0.135785 0.990738i \(-0.456644\pi\)
\(174\) 0 0
\(175\) −32.7279 + 4.47871i −2.47400 + 0.338559i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.23999 4.75736i −0.615886 0.355582i 0.159380 0.987217i \(-0.449051\pi\)
−0.775265 + 0.631636i \(0.782384\pi\)
\(180\) 0 0
\(181\) 2.02922i 0.150831i −0.997152 0.0754155i \(-0.975972\pi\)
0.997152 0.0754155i \(-0.0240283\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 17.2335 29.8492i 1.26703 2.19456i
\(186\) 0 0
\(187\) −2.63604 + 1.52192i −0.192766 + 0.111294i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.34847 4.24264i 0.531717 0.306987i −0.209999 0.977702i \(-0.567346\pi\)
0.741715 + 0.670715i \(0.234013\pi\)
\(192\) 0 0
\(193\) 3.74264 6.48244i 0.269401 0.466617i −0.699306 0.714822i \(-0.746508\pi\)
0.968707 + 0.248206i \(0.0798409\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.51472i 0.677896i −0.940805 0.338948i \(-0.889929\pi\)
0.940805 0.338948i \(-0.110071\pi\)
\(198\) 0 0
\(199\) 13.9706 + 8.06591i 0.990347 + 0.571777i 0.905378 0.424607i \(-0.139588\pi\)
0.0849690 + 0.996384i \(0.472921\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.59808 2.01472i 0.182349 0.141406i
\(204\) 0 0
\(205\) 4.24264 + 7.34847i 0.296319 + 0.513239i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.04384 0.210547
\(210\) 0 0
\(211\) −8.24264 −0.567447 −0.283723 0.958906i \(-0.591570\pi\)
−0.283723 + 0.958906i \(0.591570\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −17.2335 29.8492i −1.17531 2.03570i
\(216\) 0 0
\(217\) −2.01472 14.7224i −0.136768 0.999424i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.15232 + 1.24264i 0.144780 + 0.0835891i
\(222\) 0 0
\(223\) 12.5446i 0.840050i −0.907513 0.420025i \(-0.862021\pi\)
0.907513 0.420025i \(-0.137979\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.79423 + 13.5000i −0.517321 + 0.896026i 0.482476 + 0.875909i \(0.339737\pi\)
−0.999798 + 0.0201176i \(0.993596\pi\)
\(228\) 0 0
\(229\) −12.0000 + 6.92820i −0.792982 + 0.457829i −0.841011 0.541017i \(-0.818039\pi\)
0.0480291 + 0.998846i \(0.484706\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.82655 + 3.36396i −0.381710 + 0.220380i −0.678562 0.734543i \(-0.737397\pi\)
0.296852 + 0.954924i \(0.404063\pi\)
\(234\) 0 0
\(235\) −2.12132 + 3.67423i −0.138380 + 0.239681i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.7279i 0.823301i 0.911342 + 0.411650i \(0.135048\pi\)
−0.911342 + 0.411650i \(0.864952\pi\)
\(240\) 0 0
\(241\) 14.7426 + 8.51167i 0.949657 + 0.548285i 0.892974 0.450108i \(-0.148614\pi\)
0.0566826 + 0.998392i \(0.481948\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −28.1946 + 7.86396i −1.80129 + 0.502410i
\(246\) 0 0
\(247\) −1.24264 2.15232i −0.0790673 0.136949i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.6177 1.11202 0.556009 0.831176i \(-0.312332\pi\)
0.556009 + 0.831176i \(0.312332\pi\)
\(252\) 0 0
\(253\) 12.7279 0.800198
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.5446 21.7279i −0.782512 1.35535i −0.930474 0.366358i \(-0.880605\pi\)
0.147962 0.988993i \(-0.452729\pi\)
\(258\) 0 0
\(259\) 8.24264 20.1903i 0.512173 1.25456i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 23.5673 + 13.6066i 1.45322 + 0.839019i 0.998663 0.0516967i \(-0.0164629\pi\)
0.454561 + 0.890716i \(0.349796\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.25770 9.10660i 0.320568 0.555239i −0.660038 0.751232i \(-0.729460\pi\)
0.980605 + 0.195993i \(0.0627930\pi\)
\(270\) 0 0
\(271\) 9.62132 5.55487i 0.584454 0.337434i −0.178448 0.983949i \(-0.557108\pi\)
0.762901 + 0.646515i \(0.223774\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −32.4377 + 18.7279i −1.95607 + 1.12934i
\(276\) 0 0
\(277\) 10.4853 18.1610i 0.630000 1.09119i −0.357552 0.933893i \(-0.616388\pi\)
0.987551 0.157298i \(-0.0502783\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000i 0.357930i 0.983855 + 0.178965i \(0.0572749\pi\)
−0.983855 + 0.178965i \(0.942725\pi\)
\(282\) 0 0
\(283\) 5.63604 + 3.25397i 0.335028 + 0.193428i 0.658071 0.752956i \(-0.271373\pi\)
−0.323043 + 0.946384i \(0.604706\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.29002 + 4.24264i 0.194204 + 0.250435i
\(288\) 0 0
\(289\) 7.98528 + 13.8309i 0.469722 + 0.813583i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.18154 0.244288 0.122144 0.992512i \(-0.461023\pi\)
0.122144 + 0.992512i \(0.461023\pi\)
\(294\) 0 0
\(295\) 48.2132 2.80708
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.19615 9.00000i −0.300501 0.520483i
\(300\) 0 0
\(301\) −13.3640 17.2335i −0.770286 0.993321i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 21.4150 + 12.3640i 1.22622 + 0.707958i
\(306\) 0 0
\(307\) 24.6690i 1.40793i −0.710233 0.703966i \(-0.751411\pi\)
0.710233 0.703966i \(-0.248589\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.37769 16.2426i 0.531760 0.921036i −0.467552 0.883965i \(-0.654864\pi\)
0.999313 0.0370703i \(-0.0118026\pi\)
\(312\) 0 0
\(313\) −0.985281 + 0.568852i −0.0556914 + 0.0321534i −0.527587 0.849501i \(-0.676903\pi\)
0.471896 + 0.881654i \(0.343570\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.27231 + 3.62132i −0.352288 + 0.203394i −0.665693 0.746226i \(-0.731864\pi\)
0.313404 + 0.949620i \(0.398530\pi\)
\(318\) 0 0
\(319\) 1.86396 3.22848i 0.104362 0.180760i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.02944i 0.0572794i
\(324\) 0 0
\(325\) 26.4853 + 15.2913i 1.46914 + 0.848208i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.01461 + 2.48528i −0.0559374 + 0.137018i
\(330\) 0 0
\(331\) 8.72792 + 15.1172i 0.479730 + 0.830917i 0.999730 0.0232497i \(-0.00740129\pi\)
−0.520000 + 0.854166i \(0.674068\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −41.8154 −2.28462
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.42463 14.5919i −0.456219 0.790195i
\(342\) 0 0
\(343\) −17.0000 + 7.34847i −0.917914 + 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.5446 + 7.24264i 0.673431 + 0.388805i 0.797375 0.603484i \(-0.206221\pi\)
−0.123945 + 0.992289i \(0.539555\pi\)
\(348\) 0 0
\(349\) 36.9164i 1.97609i −0.154163 0.988045i \(-0.549268\pi\)
0.154163 0.988045i \(-0.450732\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.37769 + 16.2426i −0.499124 + 0.864509i −0.999999 0.00101095i \(-0.999678\pi\)
0.500875 + 0.865519i \(0.333012\pi\)
\(354\) 0 0
\(355\) −37.0919 + 21.4150i −1.96863 + 1.13659i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.5885 9.00000i 0.822727 0.475002i −0.0286287 0.999590i \(-0.509114\pi\)
0.851356 + 0.524588i \(0.175781\pi\)
\(360\) 0 0
\(361\) −8.98528 + 15.5630i −0.472910 + 0.819103i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 34.9706i 1.83044i
\(366\) 0 0
\(367\) −16.3492 9.43924i −0.853424 0.492724i 0.00838099 0.999965i \(-0.497332\pi\)
−0.861804 + 0.507241i \(0.830666\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.445759 + 3.25736i 0.0231427 + 0.169114i
\(372\) 0 0
\(373\) −10.7279 18.5813i −0.555471 0.962104i −0.997867 0.0652837i \(-0.979205\pi\)
0.442396 0.896820i \(-0.354129\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.04384 −0.156766
\(378\) 0 0
\(379\) 4.48528 0.230393 0.115197 0.993343i \(-0.463250\pi\)
0.115197 + 0.993343i \(0.463250\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.21076 + 10.7574i 0.317355 + 0.549675i 0.979935 0.199316i \(-0.0638719\pi\)
−0.662580 + 0.748991i \(0.730539\pi\)
\(384\) 0 0
\(385\) −26.2279 + 20.3389i −1.33670 + 1.03656i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 16.8493 + 9.72792i 0.854291 + 0.493225i 0.862096 0.506744i \(-0.169151\pi\)
−0.00780525 + 0.999970i \(0.502485\pi\)
\(390\) 0 0
\(391\) 4.30463i 0.217695i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 23.5058 40.7132i 1.18270 2.04850i
\(396\) 0 0
\(397\) 12.0000 6.92820i 0.602263 0.347717i −0.167668 0.985843i \(-0.553624\pi\)
0.769931 + 0.638127i \(0.220290\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(402\) 0 0
\(403\) −6.87868 + 11.9142i −0.342651 + 0.593490i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.7279i 1.22572i
\(408\) 0 0
\(409\) −3.98528 2.30090i −0.197059 0.113772i 0.398224 0.917288i \(-0.369627\pi\)
−0.595283 + 0.803516i \(0.702960\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 30.2238 4.13604i 1.48722 0.203521i
\(414\) 0 0
\(415\) 6.62132 + 11.4685i 0.325028 + 0.562965i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.05845 −0.198268 −0.0991341 0.995074i \(-0.531607\pi\)
−0.0991341 + 0.995074i \(0.531607\pi\)
\(420\) 0 0
\(421\) −5.75736 −0.280597 −0.140298 0.990109i \(-0.544806\pi\)
−0.140298 + 0.990109i \(0.544806\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.33386 10.9706i −0.307237 0.532150i
\(426\) 0 0
\(427\) 14.4853 + 5.91359i 0.700992 + 0.286179i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.7408 + 10.2426i 0.854543 + 0.493371i 0.862181 0.506600i \(-0.169098\pi\)
−0.00763808 + 0.999971i \(0.502431\pi\)
\(432\) 0 0
\(433\) 3.46410i 0.166474i 0.996530 + 0.0832370i \(0.0265259\pi\)
−0.996530 + 0.0832370i \(0.973474\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.15232 3.72792i 0.102959 0.178331i
\(438\) 0 0
\(439\) −23.5919 + 13.6208i −1.12598 + 0.650084i −0.942921 0.333018i \(-0.891933\pi\)
−0.183059 + 0.983102i \(0.558600\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −29.8396 + 17.2279i −1.41772 + 0.818523i −0.996099 0.0882469i \(-0.971874\pi\)
−0.421625 + 0.906770i \(0.638540\pi\)
\(444\) 0 0
\(445\) −21.7279 + 37.6339i −1.03000 + 1.78402i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.2426i 0.483380i −0.970354 0.241690i \(-0.922298\pi\)
0.970354 0.241690i \(-0.0777017\pi\)
\(450\) 0 0
\(451\) 5.27208 + 3.04384i 0.248252 + 0.143329i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 25.0892 + 10.2426i 1.17620 + 0.480182i
\(456\) 0 0
\(457\) 11.5000 + 19.9186i 0.537947 + 0.931752i 0.999014 + 0.0443868i \(0.0141334\pi\)
−0.461067 + 0.887365i \(0.652533\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 22.8138 1.06255 0.531273 0.847201i \(-0.321714\pi\)
0.531273 + 0.847201i \(0.321714\pi\)
\(462\) 0 0
\(463\) −21.4558 −0.997138 −0.498569 0.866850i \(-0.666141\pi\)
−0.498569 + 0.866850i \(0.666141\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.50079 + 16.4558i 0.439644 + 0.761486i 0.997662 0.0683432i \(-0.0217713\pi\)
−0.558018 + 0.829829i \(0.688438\pi\)
\(468\) 0 0
\(469\) −26.2132 + 3.58719i −1.21041 + 0.165641i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −21.4150 12.3640i −0.984663 0.568496i
\(474\) 0 0
\(475\) 12.6677i 0.581235i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −18.2481 + 31.6066i −0.833776 + 1.44414i 0.0612470 + 0.998123i \(0.480492\pi\)
−0.895023 + 0.446020i \(0.852841\pi\)
\(480\) 0 0
\(481\) −17.4853 + 10.0951i −0.797260 + 0.460298i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.6208 7.86396i 0.618488 0.357084i
\(486\) 0 0
\(487\) 14.1066 24.4334i 0.639231 1.10718i −0.346371 0.938098i \(-0.612586\pi\)
0.985602 0.169083i \(-0.0540806\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.9706i 0.901259i −0.892711 0.450629i \(-0.851200\pi\)
0.892711 0.450629i \(-0.148800\pi\)
\(492\) 0 0
\(493\) 1.09188 + 0.630399i 0.0491759 + 0.0283917i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −21.4150 + 16.6066i −0.960594 + 0.744908i
\(498\) 0 0
\(499\) −17.9706 31.1259i −0.804473 1.39339i −0.916646 0.399699i \(-0.869115\pi\)
0.112173 0.993689i \(-0.464219\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.29002 −0.146695 −0.0733474 0.997306i \(-0.523368\pi\)
−0.0733474 + 0.997306i \(0.523368\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.8462 + 36.1066i 0.923990 + 1.60040i 0.793178 + 0.608990i \(0.208425\pi\)
0.130812 + 0.991407i \(0.458242\pi\)
\(510\) 0 0
\(511\) 3.00000 + 21.9223i 0.132712 + 0.969786i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 55.3746 + 31.9706i 2.44010 + 1.40879i
\(516\) 0 0
\(517\) 3.04384i 0.133868i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 10.0081 17.3345i 0.438462 0.759439i −0.559109 0.829094i \(-0.688857\pi\)
0.997571 + 0.0696551i \(0.0221899\pi\)
\(522\) 0 0
\(523\) 23.8492 13.7694i 1.04285 0.602092i 0.122214 0.992504i \(-0.461001\pi\)
0.920641 + 0.390411i \(0.127667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.93503 2.84924i 0.214973 0.124115i
\(528\) 0 0
\(529\) −2.50000 + 4.33013i −0.108696 + 0.188266i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.97056i 0.215299i
\(534\) 0 0
\(535\) −19.8640 11.4685i −0.858794 0.495825i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −14.6969 + 15.0000i −0.633042 + 0.646096i
\(540\) 0 0
\(541\) 5.36396 + 9.29065i 0.230615 + 0.399436i 0.957989 0.286804i \(-0.0925930\pi\)
−0.727374 + 0.686241i \(0.759260\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.33386 −0.271313
\(546\) 0 0
\(547\) −19.6985 −0.842246 −0.421123 0.907003i \(-0.638364\pi\)
−0.421123 + 0.907003i \(0.638364\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.630399 1.09188i −0.0268559 0.0465158i
\(552\) 0 0
\(553\) 11.2426 27.5387i 0.478086 1.17107i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.6208 7.86396i −0.577131 0.333207i 0.182861 0.983139i \(-0.441464\pi\)
−0.759992 + 0.649932i \(0.774797\pi\)
\(558\) 0 0
\(559\) 20.1903i 0.853957i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0989 20.9558i 0.509906 0.883184i −0.490028 0.871707i \(-0.663013\pi\)
0.999934 0.0114768i \(-0.00365325\pi\)
\(564\) 0 0
\(565\) 30.7279 17.7408i 1.29273 0.746360i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.52192 + 0.878680i −0.0638021 + 0.0368362i −0.531562 0.847020i \(-0.678395\pi\)
0.467760 + 0.883856i \(0.345061\pi\)
\(570\) 0 0
\(571\) −8.36396 + 14.4868i −0.350021 + 0.606254i −0.986253 0.165244i \(-0.947159\pi\)
0.636232 + 0.771498i \(0.280492\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 52.9706i 2.20903i
\(576\) 0 0
\(577\) 17.7426 + 10.2437i 0.738636 + 0.426452i 0.821573 0.570103i \(-0.193097\pi\)
−0.0829373 + 0.996555i \(0.526430\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.13461 + 6.62132i 0.213019 + 0.274699i
\(582\) 0 0
\(583\) 1.86396 + 3.22848i 0.0771974 + 0.133710i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.19615 −0.214468 −0.107234 0.994234i \(-0.534199\pi\)
−0.107234 + 0.994234i \(0.534199\pi\)
\(588\) 0 0
\(589\) −5.69848 −0.234802
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.2042 + 26.3345i 0.624363 + 1.08143i 0.988664 + 0.150148i \(0.0479749\pi\)
−0.364300 + 0.931282i \(0.618692\pi\)
\(594\) 0 0
\(595\) −6.87868 8.87039i −0.281998 0.363650i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −37.6339 21.7279i −1.53768 0.887779i −0.998974 0.0452836i \(-0.985581\pi\)
−0.538704 0.842495i \(-0.681086\pi\)
\(600\) 0 0
\(601\) 6.03668i 0.246241i 0.992392 + 0.123121i \(0.0392902\pi\)
−0.992392 + 0.123121i \(0.960710\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.18154 7.24264i 0.170004 0.294455i
\(606\) 0 0
\(607\) 21.6213 12.4831i 0.877582 0.506672i 0.00772182 0.999970i \(-0.497542\pi\)
0.869860 + 0.493298i \(0.164209\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.15232 1.24264i 0.0870734 0.0502719i
\(612\) 0 0
\(613\) −2.60660 + 4.51477i −0.105280 + 0.182350i −0.913852 0.406046i \(-0.866907\pi\)
0.808573 + 0.588396i \(0.200240\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.6985i 1.67872i 0.543578 + 0.839359i \(0.317069\pi\)
−0.543578 + 0.839359i \(0.682931\pi\)
\(618\) 0 0
\(619\) 41.3345 + 23.8645i 1.66137 + 0.959195i 0.972060 + 0.234733i \(0.0754217\pi\)
0.689315 + 0.724462i \(0.257912\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 + 25.4558i −0.416359 + 1.01987i
\(624\) 0 0
\(625\) −34.2279 59.2845i −1.36912 2.37138i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 8.36308 0.333458
\(630\) 0 0
\(631\) −33.2426 −1.32337 −0.661684 0.749783i \(-0.730158\pi\)
−0.661684 + 0.749783i \(0.730158\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.9612 + 18.9853i 0.434980 + 0.753408i
\(636\) 0 0
\(637\) 16.6066 + 4.26858i 0.657978 + 0.169127i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 36.1119 + 20.8492i 1.42634 + 0.823496i 0.996829 0.0795681i \(-0.0253541\pi\)
0.429507 + 0.903064i \(0.358687\pi\)
\(642\) 0 0
\(643\) 2.62357i 0.103463i 0.998661 + 0.0517317i \(0.0164741\pi\)
−0.998661 + 0.0517317i \(0.983526\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.82655 + 10.0919i −0.229065 + 0.396753i −0.957531 0.288329i \(-0.906900\pi\)
0.728466 + 0.685082i \(0.240234\pi\)
\(648\) 0 0
\(649\) 29.9558 17.2950i 1.17587 0.678889i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.31615 5.37868i 0.364569 0.210484i −0.306514 0.951866i \(-0.599163\pi\)
0.671083 + 0.741382i \(0.265829\pi\)
\(654\) 0 0
\(655\) −10.8640 + 18.8169i −0.424490 + 0.735238i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000i 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) 35.1213 + 20.2773i 1.36606 + 0.788696i 0.990422 0.138071i \(-0.0440901\pi\)
0.375639 + 0.926766i \(0.377423\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.52192 + 11.1213i 0.0590174 + 0.431266i
\(666\) 0 0
\(667\) −2.63604 4.56575i −0.102068 0.176787i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 17.7408 0.684875
\(672\) 0 0
\(673\) −15.9706 −0.615620 −0.307810 0.951448i \(-0.599596\pi\)
−0.307810 + 0.951448i \(0.599596\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.27231 10.8640i −0.241064 0.417536i 0.719953 0.694023i \(-0.244163\pi\)
−0.961018 + 0.276487i \(0.910830\pi\)
\(678\) 0 0
\(679\) 7.86396 6.09823i 0.301791 0.234029i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.4912 12.9853i −0.860601 0.496868i 0.00361277 0.999993i \(-0.498850\pi\)
−0.864213 + 0.503125i \(0.832183\pi\)
\(684\) 0 0
\(685\) 60.5708i 2.31429i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.52192 2.63604i 0.0579805 0.100425i
\(690\) 0 0
\(691\) 0.727922 0.420266i 0.0276915 0.0159877i −0.486090 0.873909i \(-0.661577\pi\)
0.513782 + 0.857921i \(0.328244\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 73.1154 42.2132i 2.77343 1.60124i
\(696\) 0 0
\(697\) −1.02944 + 1.78304i −0.0389927 + 0.0675374i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 38.6985i 1.46162i −0.682580 0.730811i \(-0.739142\pi\)
0.682580 0.730811i \(-0.260858\pi\)
\(702\) 0 0
\(703\) −7.24264 4.18154i −0.273161 0.157710i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.48528 + 6.03668i 0.130892 + 0.226712i 0.924021 0.382342i \(-0.124882\pi\)
−0.793128 + 0.609055i \(0.791549\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −23.8284 −0.892382
\(714\) 0 0
\(715\) 30.7279 1.14916
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11.5300 19.9706i −0.429997 0.744776i 0.566876 0.823803i \(-0.308152\pi\)
−0.996872 + 0.0790270i \(0.974819\pi\)
\(720\) 0 0
\(721\) 37.4558 + 15.2913i 1.39493 + 0.569477i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.4361 + 7.75736i 0.499006 + 0.288101i
\(726\) 0 0
\(727\) 26.4010i 0.979160i −0.871958 0.489580i \(-0.837150\pi\)
0.871958 0.489580i \(-0.162850\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.18154 7.24264i 0.154660 0.267879i
\(732\) 0 0
\(733\) −34.0919 + 19.6830i −1.25921 + 0.727007i −0.972921 0.231136i \(-0.925756\pi\)
−0.286291 + 0.958143i \(0.592422\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.9808 + 15.0000i −0.957014 + 0.552532i
\(738\) 0 0
\(739\) −17.7279 + 30.7057i −0.652132 + 1.12953i 0.330472 + 0.943816i \(0.392792\pi\)
−0.982605 + 0.185710i \(0.940541\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.5147i 0.789298i −0.918832 0.394649i \(-0.870866\pi\)
0.918832 0.394649i \(-0.129134\pi\)
\(744\) 0 0
\(745\) −74.1838 42.8300i −2.71788 1.56917i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −13.4361 5.48528i −0.490946 0.200428i
\(750\) 0 0
\(751\) 13.3787 + 23.1726i 0.488195 + 0.845578i 0.999908 0.0135781i \(-0.00432217\pi\)
−0.511713 + 0.859157i \(0.670989\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13.5592 −0.493471
\(756\) 0 0
\(757\) −42.2426 −1.53533 −0.767667 0.640848i \(-0.778583\pi\)
−0.767667 + 0.640848i \(0.778583\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.53653 4.39340i −0.0919491 0.159261i 0.816382 0.577512i \(-0.195976\pi\)
−0.908331 + 0.418252i \(0.862643\pi\)
\(762\) 0 0
\(763\) −3.97056 + 0.543359i −0.143744 + 0.0196709i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.4588 14.1213i −0.883158 0.509891i
\(768\) 0 0
\(769\) 49.0408i 1.76846i −0.467056 0.884228i \(-0.654685\pi\)
0.467056 0.884228i \(-0.345315\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.5300 + 19.9706i −0.414706 + 0.718291i −0.995398 0.0958322i \(-0.969449\pi\)
0.580692 + 0.814123i \(0.302782\pi\)
\(774\) 0 0
\(775\) 60.7279 35.0613i 2.18141 1.25944i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.78304 1.02944i 0.0638840 0.0368834i
\(780\) 0 0
\(781\) −15.3640 + 26.6112i −0.549766 + 0.952222i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 61.4558i 2.19345i
\(786\) 0 0
\(787\) −32.1213 18.5453i −1.14500 0.661067i −0.197337 0.980336i \(-0.563229\pi\)
−0.947664 + 0.319269i \(0.896563\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 17.7408 13.7574i 0.630789 0.489155i
\(792\) 0 0
\(793\) −7.24264 12.5446i −0.257194 0.445473i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −37.6339 −1.33306 −0.666530 0.745478i \(-0.732221\pi\)
−0.666530 + 0.745478i \(0.732221\pi\)
\(798\) 0 0
\(799\) −1.02944 −0.0364189
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.5446 + 21.7279i 0.442690 + 0.766762i
\(804\) 0 0
\(805\) 6.36396 + 46.5043i 0.224300 + 1.63906i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −35.4815 20.4853i −1.24746 0.720224i −0.276862 0.960910i \(-0.589294\pi\)
−0.970603 + 0.240686i \(0.922628\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i 0.836881 + 0.547385i \(0.184377\pi\)
−0.836881 + 0.547385i \(0.815623\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13.0519 + 22.6066i −0.457189 + 0.791875i
\(816\) 0 0
\(817\) −7.24264 + 4.18154i −0.253388 + 0.146294i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.6454 24.6213i 1.48833 0.859290i 0.488423 0.872607i \(-0.337572\pi\)
0.999911 + 0.0133172i \(0.00423912\pi\)
\(822\) 0 0
\(823\) −18.9706 + 32.8580i −0.661272 + 1.14536i 0.319009 + 0.947752i \(0.396650\pi\)
−0.980282 + 0.197606i \(0.936683\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.02944i 0.140117i −0.997543 0.0700586i \(-0.977681\pi\)
0.997543 0.0700586i \(-0.0223186\pi\)
\(828\) 0 0
\(829\) −35.3345 20.4004i −1.22722 0.708535i −0.260772 0.965401i \(-0.583977\pi\)
−0.966447 + 0.256865i \(0.917310\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.07306 4.97056i −0.175771 0.172220i
\(834\) 0 0
\(835\) −48.2132 83.5077i −1.66849 2.88990i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0746 0.831149 0.415574 0.909559i \(-0.363581\pi\)
0.415574 + 0.909559i \(0.363581\pi\)
\(840\) 0 0
\(841\) 27.4558 0.946753
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.6354 + 25.3492i 0.503473 + 0.872040i
\(846\) 0 0
\(847\) 2.00000 4.89898i 0.0687208 0.168331i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −30.2854 17.4853i −1.03817 0.599388i
\(852\) 0 0
\(853\) 2.27541i 0.0779085i 0.999241 + 0.0389543i \(0.0124027\pi\)
−0.999241 + 0.0389543i \(0.987597\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.0081 + 17.3345i −0.341870 + 0.592136i −0.984780 0.173806i \(-0.944394\pi\)
0.642910 + 0.765942i \(0.277727\pi\)
\(858\) 0 0
\(859\) −3.87868 + 2.23936i −0.132339 + 0.0764059i −0.564708 0.825291i \(-0.691011\pi\)
0.432369 + 0.901697i \(0.357678\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −43.7215 + 25.2426i −1.48830 + 0.859269i −0.999911 0.0133573i \(-0.995748\pi\)
−0.488388 + 0.872627i \(0.662415\pi\)
\(864\) 0 0
\(865\) 43.4558 75.2677i 1.47754 2.55918i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 33.7279i 1.14414i
\(870\) 0 0
\(871\) 21.2132 + 12.2474i 0.718782 + 0.414989i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −50.7473 65.4411i −1.71557 2.21231i
\(876\) 0 0
\(877\) 6.24264 + 10.8126i 0.210799 + 0.365115i 0.951965 0.306207i \(-0.0990601\pi\)
−0.741166 + 0.671322i \(0.765727\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −39.7862 −1.34043 −0.670215 0.742167i \(-0.733798\pi\)
−0.670215 + 0.742167i \(0.733798\pi\)
\(882\) 0 0
\(883\) 9.45584 0.318214 0.159107 0.987261i \(-0.449138\pi\)
0.159107 + 0.987261i \(0.449138\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.4296 38.8492i −0.753113 1.30443i −0.946307 0.323269i \(-0.895218\pi\)
0.193194 0.981161i \(-0.438115\pi\)
\(888\) 0 0
\(889\) 8.50000 + 10.9612i 0.285081 + 0.367625i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.891519 + 0.514719i 0.0298335 + 0.0172244i
\(894\) 0 0
\(895\) 39.7862i 1.32991i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.48960 + 6.04416i −0.116385 + 0.201584i
\(900\) 0 0
\(901\) −1.09188 + 0.630399i −0.0363759 + 0.0210016i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.34847 4.24264i 0.244271 0.141030i
\(906\) 0 0
\(907\) 13.8492 23.9876i 0.459857 0.796495i −0.539096 0.842244i \(-0.681234\pi\)
0.998953 + 0.0457492i \(0.0145675\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 18.7279i 0.620484i −0.950658 0.310242i \(-0.899590\pi\)
0.950658 0.310242i \(-0.100410\pi\)
\(912\) 0 0
\(913\) 8.22792 + 4.75039i 0.272304 + 0.157215i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.19615 + 12.7279i −0.171592 + 0.420313i
\(918\) 0 0
\(919\) 9.75736 + 16.9002i 0.321866 + 0.557488i 0.980873 0.194649i \(-0.0623567\pi\)
−0.659007 + 0.752136i \(0.729023\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 25.0892 0.825823
\(924\) 0 0
\(925\) 102.912 3.38372
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.64501 + 2.84924i 0.0539711 + 0.0934806i 0.891749 0.452531i \(-0.149479\pi\)
−0.837778 + 0.546012i \(0.816145\pi\)
\(930\) 0 0
\(931\) 1.90812 + 6.84116i 0.0625360 + 0.224210i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −11.0227 6.36396i −0.360481 0.208124i
\(936\) 0 0
\(937\) 4.00746i 0.130918i −0.997855 0.0654590i \(-0.979149\pi\)
0.997855 0.0654590i \(-0.0208512\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 26.0423 45.1066i 0.848955 1.47043i −0.0331867 0.999449i \(-0.510566\pi\)
0.882142 0.470984i \(-0.156101\pi\)
\(942\) 0 0
\(943\) 7.45584 4.30463i 0.242796 0.140178i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.8931 + 11.4853i −0.646439 + 0.373221i −0.787090 0.616838i \(-0.788414\pi\)
0.140652 + 0.990059i \(0.455080\pi\)
\(948\) 0 0
\(949\) 10.2426 17.7408i 0.332490 0.575890i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 41.6985i 1.35075i 0.737476 + 0.675373i \(0.236017\pi\)
−0.737476 + 0.675373i \(0.763983\pi\)
\(954\) 0 0
\(955\) 30.7279 + 17.7408i 0.994332 + 0.574078i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.19615 + 37.9706i 0.167793 + 1.22613i
\(960\) 0 0
\(961\) 0.272078 + 0.471253i 0.00877671 + 0.0152017i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 31.3000 1.00758
\(966\) 0 0
\(967\) −22.2721 −0.716222 −0.358111 0.933679i \(-0.616579\pi\)
−0.358111 + 0.933679i \(0.616579\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25.6581 + 44.4411i 0.823407 + 1.42618i 0.903130 + 0.429367i \(0.141263\pi\)
−0.0797229 + 0.996817i \(0.525404\pi\)
\(972\) 0 0
\(973\) 42.2132 32.7349i 1.35329 1.04943i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.5027 15.8787i −0.879889 0.508004i −0.00926698 0.999957i \(-0.502950\pi\)
−0.870622 + 0.491953i \(0.836283\pi\)
\(978\) 0 0
\(979\) 31.1769i 0.996419i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.05845 7.02944i 0.129444 0.224204i −0.794017 0.607895i \(-0.792014\pi\)
0.923461 + 0.383691i \(0.125347\pi\)
\(984\) 0 0
\(985\) 34.4558 19.8931i 1.09785 0.633847i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.2854 + 17.4853i −0.963020 + 0.556000i
\(990\) 0 0
\(991\) 4.89340 8.47561i 0.155444 0.269237i −0.777777 0.628541i \(-0.783653\pi\)
0.933221 + 0.359304i \(0.116986\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 67.4558i 2.13849i
\(996\) 0 0
\(997\) −8.27208 4.77589i −0.261979 0.151254i 0.363258 0.931689i \(-0.381664\pi\)
−0.625237 + 0.780435i \(0.714998\pi\)
\(998\) 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 1008.2.bt.c.17.4 8
3.2 odd 2 inner 1008.2.bt.c.17.1 8
4.3 odd 2 126.2.k.a.17.4 yes 8
7.3 odd 6 7056.2.k.f.881.7 8
7.4 even 3 7056.2.k.f.881.1 8
7.5 odd 6 inner 1008.2.bt.c.593.1 8
12.11 even 2 126.2.k.a.17.1 8
20.3 even 4 3150.2.bp.b.899.1 8
20.7 even 4 3150.2.bp.e.899.4 8
20.19 odd 2 3150.2.bf.a.1151.2 8
21.5 even 6 inner 1008.2.bt.c.593.4 8
21.11 odd 6 7056.2.k.f.881.8 8
21.17 even 6 7056.2.k.f.881.2 8
28.3 even 6 882.2.d.a.881.8 8
28.11 odd 6 882.2.d.a.881.5 8
28.19 even 6 126.2.k.a.89.1 yes 8
28.23 odd 6 882.2.k.a.215.2 8
28.27 even 2 882.2.k.a.521.3 8
36.7 odd 6 1134.2.t.e.1025.1 8
36.11 even 6 1134.2.t.e.1025.4 8
36.23 even 6 1134.2.l.f.269.1 8
36.31 odd 6 1134.2.l.f.269.4 8
60.23 odd 4 3150.2.bp.e.899.1 8
60.47 odd 4 3150.2.bp.b.899.4 8
60.59 even 2 3150.2.bf.a.1151.4 8
84.11 even 6 882.2.d.a.881.4 8
84.23 even 6 882.2.k.a.215.3 8
84.47 odd 6 126.2.k.a.89.4 yes 8
84.59 odd 6 882.2.d.a.881.1 8
84.83 odd 2 882.2.k.a.521.2 8
140.19 even 6 3150.2.bf.a.1601.4 8
140.47 odd 12 3150.2.bp.e.1349.1 8
140.103 odd 12 3150.2.bp.b.1349.4 8
252.47 odd 6 1134.2.l.f.215.2 8
252.103 even 6 1134.2.t.e.593.4 8
252.131 odd 6 1134.2.t.e.593.1 8
252.187 even 6 1134.2.l.f.215.3 8
420.47 even 12 3150.2.bp.b.1349.1 8
420.299 odd 6 3150.2.bf.a.1601.2 8
420.383 even 12 3150.2.bp.e.1349.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 8 12.11 even 2
126.2.k.a.17.4 yes 8 4.3 odd 2
126.2.k.a.89.1 yes 8 28.19 even 6
126.2.k.a.89.4 yes 8 84.47 odd 6
882.2.d.a.881.1 8 84.59 odd 6
882.2.d.a.881.4 8 84.11 even 6
882.2.d.a.881.5 8 28.11 odd 6
882.2.d.a.881.8 8 28.3 even 6
882.2.k.a.215.2 8 28.23 odd 6
882.2.k.a.215.3 8 84.23 even 6
882.2.k.a.521.2 8 84.83 odd 2
882.2.k.a.521.3 8 28.27 even 2
1008.2.bt.c.17.1 8 3.2 odd 2 inner
1008.2.bt.c.17.4 8 1.1 even 1 trivial
1008.2.bt.c.593.1 8 7.5 odd 6 inner
1008.2.bt.c.593.4 8 21.5 even 6 inner
1134.2.l.f.215.2 8 252.47 odd 6
1134.2.l.f.215.3 8 252.187 even 6
1134.2.l.f.269.1 8 36.23 even 6
1134.2.l.f.269.4 8 36.31 odd 6
1134.2.t.e.593.1 8 252.131 odd 6
1134.2.t.e.593.4 8 252.103 even 6
1134.2.t.e.1025.1 8 36.7 odd 6
1134.2.t.e.1025.4 8 36.11 even 6
3150.2.bf.a.1151.2 8 20.19 odd 2
3150.2.bf.a.1151.4 8 60.59 even 2
3150.2.bf.a.1601.2 8 420.299 odd 6
3150.2.bf.a.1601.4 8 140.19 even 6
3150.2.bp.b.899.1 8 20.3 even 4
3150.2.bp.b.899.4 8 60.47 odd 4
3150.2.bp.b.1349.1 8 420.47 even 12
3150.2.bp.b.1349.4 8 140.103 odd 12
3150.2.bp.e.899.1 8 60.23 odd 4
3150.2.bp.e.899.4 8 20.7 even 4
3150.2.bp.e.1349.1 8 140.47 odd 12
3150.2.bp.e.1349.4 8 420.383 even 12
7056.2.k.f.881.1 8 7.4 even 3
7056.2.k.f.881.2 8 21.17 even 6
7056.2.k.f.881.7 8 7.3 odd 6
7056.2.k.f.881.8 8 21.11 odd 6