Properties

Label 1080.2.q.e.721.3
Level $1080$
Weight $2$
Character 1080.721
Analytic conductor $8.624$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 721.3
Root \(0.385731i\) of defining polynomial
Character \(\chi\) \(=\) 1080.721
Dual form 1080.2.q.e.361.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(0.165947 - 0.287429i) q^{7} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(0.165947 - 0.287429i) q^{7} +(2.20380 - 3.81710i) q^{11} +(-3.49031 - 6.04539i) q^{13} +3.07139 q^{17} -7.55364 q^{19} +(0.834053 + 1.44462i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.78651 - 3.09432i) q^{29} +(-2.82220 - 4.88820i) q^{31} +0.331895 q^{35} +7.64441 q^{37} +(2.74919 + 4.76174i) q^{41} +(3.53570 - 6.12401i) q^{43} +(3.98815 - 6.90768i) q^{47} +(3.44492 + 5.96678i) q^{49} +5.47900 q^{53} +4.40761 q^{55} +(-3.69411 - 6.39839i) q^{59} +(-0.296197 + 0.513029i) q^{61} +(3.49031 - 6.04539i) q^{65} +(-2.61087 - 4.52216i) q^{67} -8.98062 q^{71} -8.05202 q^{73} +(-0.731430 - 1.26687i) q^{77} +(5.49838 - 9.52347i) q^{79} +(-3.14657 + 5.45002i) q^{83} +(1.53570 + 2.65991i) q^{85} -11.9060 q^{89} -2.31683 q^{91} +(-3.77682 - 6.54164i) q^{95} +(0.622718 - 1.07858i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + q^{7} + q^{11} - 4 q^{13} - 10 q^{17} + 2 q^{19} + 7 q^{23} - 4 q^{25} + 7 q^{29} + 2 q^{31} + 2 q^{35} + 12 q^{37} + 12 q^{41} + 11 q^{43} + 7 q^{47} - 3 q^{49} - 24 q^{53} + 2 q^{55} + 11 q^{59} - 19 q^{61} + 4 q^{65} + 10 q^{67} - 24 q^{71} + 18 q^{73} + 32 q^{77} + 24 q^{79} + 23 q^{83} - 5 q^{85} - 42 q^{89} + 28 q^{91} + q^{95} - q^{97}+O(q^{100}) \) 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
\(7\) 0.165947 0.287429i 0.0627222 0.108638i −0.832959 0.553335i \(-0.813355\pi\)
0.895681 + 0.444696i \(0.146688\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.20380 3.81710i 0.664472 1.15090i −0.314957 0.949106i \(-0.601990\pi\)
0.979428 0.201792i \(-0.0646766\pi\)
\(12\) 0 0
\(13\) −3.49031 6.04539i −0.968038 1.67669i −0.701222 0.712943i \(-0.747362\pi\)
−0.266816 0.963747i \(-0.585972\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.07139 0.744923 0.372461 0.928048i \(-0.378514\pi\)
0.372461 + 0.928048i \(0.378514\pi\)
\(18\) 0 0
\(19\) −7.55364 −1.73292 −0.866461 0.499244i \(-0.833611\pi\)
−0.866461 + 0.499244i \(0.833611\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.834053 + 1.44462i 0.173912 + 0.301224i 0.939784 0.341768i \(-0.111026\pi\)
−0.765872 + 0.642993i \(0.777693\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.78651 3.09432i 0.331746 0.574601i −0.651108 0.758985i \(-0.725696\pi\)
0.982854 + 0.184384i \(0.0590289\pi\)
\(30\) 0 0
\(31\) −2.82220 4.88820i −0.506883 0.877947i −0.999968 0.00796608i \(-0.997464\pi\)
0.493085 0.869981i \(-0.335869\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.331895 0.0561004
\(36\) 0 0
\(37\) 7.64441 1.25673 0.628367 0.777917i \(-0.283724\pi\)
0.628367 + 0.777917i \(0.283724\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.74919 + 4.76174i 0.429351 + 0.743658i 0.996816 0.0797396i \(-0.0254089\pi\)
−0.567464 + 0.823398i \(0.692076\pi\)
\(42\) 0 0
\(43\) 3.53570 6.12401i 0.539189 0.933902i −0.459759 0.888044i \(-0.652064\pi\)
0.998948 0.0458587i \(-0.0146024\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.98815 6.90768i 0.581732 1.00759i −0.413542 0.910485i \(-0.635709\pi\)
0.995274 0.0971042i \(-0.0309580\pi\)
\(48\) 0 0
\(49\) 3.44492 + 5.96678i 0.492132 + 0.852397i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.47900 0.752599 0.376299 0.926498i \(-0.377196\pi\)
0.376299 + 0.926498i \(0.377196\pi\)
\(54\) 0 0
\(55\) 4.40761 0.594321
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.69411 6.39839i −0.480933 0.833000i 0.518828 0.854879i \(-0.326368\pi\)
−0.999761 + 0.0218790i \(0.993035\pi\)
\(60\) 0 0
\(61\) −0.296197 + 0.513029i −0.0379242 + 0.0656866i −0.884364 0.466797i \(-0.845408\pi\)
0.846440 + 0.532484i \(0.178741\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.49031 6.04539i 0.432920 0.749839i
\(66\) 0 0
\(67\) −2.61087 4.52216i −0.318969 0.552470i 0.661305 0.750118i \(-0.270003\pi\)
−0.980273 + 0.197648i \(0.936670\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.98062 −1.06580 −0.532902 0.846177i \(-0.678898\pi\)
−0.532902 + 0.846177i \(0.678898\pi\)
\(72\) 0 0
\(73\) −8.05202 −0.942417 −0.471209 0.882022i \(-0.656182\pi\)
−0.471209 + 0.882022i \(0.656182\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.731430 1.26687i −0.0833542 0.144374i
\(78\) 0 0
\(79\) 5.49838 9.52347i 0.618616 1.07147i −0.371122 0.928584i \(-0.621027\pi\)
0.989739 0.142891i \(-0.0456397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.14657 + 5.45002i −0.345381 + 0.598217i −0.985423 0.170123i \(-0.945584\pi\)
0.640042 + 0.768340i \(0.278917\pi\)
\(84\) 0 0
\(85\) 1.53570 + 2.65991i 0.166570 + 0.288507i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.9060 −1.26203 −0.631016 0.775770i \(-0.717362\pi\)
−0.631016 + 0.775770i \(0.717362\pi\)
\(90\) 0 0
\(91\) −2.31683 −0.242870
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.77682 6.54164i −0.387493 0.671158i
\(96\) 0 0
\(97\) 0.622718 1.07858i 0.0632274 0.109513i −0.832679 0.553756i \(-0.813194\pi\)
0.895906 + 0.444243i \(0.146527\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.33189 4.03896i 0.232032 0.401892i −0.726374 0.687300i \(-0.758796\pi\)
0.958406 + 0.285408i \(0.0921292\pi\)
\(102\) 0 0
\(103\) 2.08270 + 3.60735i 0.205215 + 0.355443i 0.950201 0.311637i \(-0.100877\pi\)
−0.744986 + 0.667080i \(0.767544\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.57409 0.635541 0.317771 0.948168i \(-0.397066\pi\)
0.317771 + 0.948168i \(0.397066\pi\)
\(108\) 0 0
\(109\) 0.743816 0.0712446 0.0356223 0.999365i \(-0.488659\pi\)
0.0356223 + 0.999365i \(0.488659\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.31252 + 14.3977i 0.781976 + 1.35442i 0.930789 + 0.365556i \(0.119121\pi\)
−0.148814 + 0.988865i \(0.547545\pi\)
\(114\) 0 0
\(115\) −0.834053 + 1.44462i −0.0777758 + 0.134712i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.509690 0.882809i 0.0467232 0.0809269i
\(120\) 0 0
\(121\) −4.21349 7.29798i −0.383045 0.663453i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.79152 0.425178 0.212589 0.977142i \(-0.431810\pi\)
0.212589 + 0.977142i \(0.431810\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.88985 + 17.1297i 0.864080 + 1.49663i 0.867958 + 0.496638i \(0.165432\pi\)
−0.00387809 + 0.999992i \(0.501234\pi\)
\(132\) 0 0
\(133\) −1.25351 + 2.17114i −0.108693 + 0.188261i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.26713 16.0511i 0.791744 1.37134i −0.133142 0.991097i \(-0.542507\pi\)
0.924886 0.380244i \(-0.124160\pi\)
\(138\) 0 0
\(139\) −2.68980 4.65887i −0.228146 0.395160i 0.729113 0.684393i \(-0.239933\pi\)
−0.957259 + 0.289234i \(0.906600\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −30.7678 −2.57293
\(144\) 0 0
\(145\) 3.57301 0.296723
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.51794 + 9.55735i 0.452047 + 0.782969i 0.998513 0.0545128i \(-0.0173606\pi\)
−0.546466 + 0.837481i \(0.684027\pi\)
\(150\) 0 0
\(151\) 0.750810 1.30044i 0.0611001 0.105828i −0.833857 0.551980i \(-0.813872\pi\)
0.894957 + 0.446152i \(0.147206\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.82220 4.88820i 0.226685 0.392630i
\(156\) 0 0
\(157\) 11.8109 + 20.4571i 0.942612 + 1.63265i 0.760463 + 0.649381i \(0.224972\pi\)
0.182149 + 0.983271i \(0.441695\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.553635 0.0436326
\(162\) 0 0
\(163\) −9.16217 −0.717636 −0.358818 0.933407i \(-0.616820\pi\)
−0.358818 + 0.933407i \(0.616820\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.324363 0.561813i −0.0250999 0.0434744i 0.853203 0.521580i \(-0.174657\pi\)
−0.878303 + 0.478105i \(0.841324\pi\)
\(168\) 0 0
\(169\) −17.8645 + 30.9423i −1.37419 + 2.38017i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.40761 7.63420i 0.335104 0.580417i −0.648401 0.761299i \(-0.724562\pi\)
0.983505 + 0.180882i \(0.0578952\pi\)
\(174\) 0 0
\(175\) 0.165947 + 0.287429i 0.0125444 + 0.0217276i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.16217 −0.684813 −0.342406 0.939552i \(-0.611242\pi\)
−0.342406 + 0.939552i \(0.611242\pi\)
\(180\) 0 0
\(181\) 13.5730 1.00887 0.504437 0.863448i \(-0.331700\pi\)
0.504437 + 0.863448i \(0.331700\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.82220 + 6.62025i 0.281014 + 0.486731i
\(186\) 0 0
\(187\) 6.76875 11.7238i 0.494980 0.857330i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.13472 + 8.89360i −0.371535 + 0.643518i −0.989802 0.142450i \(-0.954502\pi\)
0.618267 + 0.785968i \(0.287835\pi\)
\(192\) 0 0
\(193\) −4.53570 7.85606i −0.326487 0.565491i 0.655325 0.755347i \(-0.272531\pi\)
−0.981812 + 0.189855i \(0.939198\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.80658 0.342455 0.171227 0.985232i \(-0.445227\pi\)
0.171227 + 0.985232i \(0.445227\pi\)
\(198\) 0 0
\(199\) −18.4596 −1.30857 −0.654284 0.756249i \(-0.727030\pi\)
−0.654284 + 0.756249i \(0.727030\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.592932 1.02699i −0.0416157 0.0720805i
\(204\) 0 0
\(205\) −2.74919 + 4.76174i −0.192012 + 0.332574i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.6467 + 28.8330i −1.15148 + 1.99442i
\(210\) 0 0
\(211\) −0.177795 0.307950i −0.0122399 0.0212002i 0.859841 0.510563i \(-0.170563\pi\)
−0.872080 + 0.489362i \(0.837230\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.07139 0.482265
\(216\) 0 0
\(217\) −1.87335 −0.127171
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −10.7201 18.5678i −0.721113 1.24900i
\(222\) 0 0
\(223\) −1.65194 + 2.86125i −0.110622 + 0.191603i −0.916021 0.401130i \(-0.868618\pi\)
0.805399 + 0.592733i \(0.201951\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.36921 + 4.10360i −0.157250 + 0.272365i −0.933876 0.357597i \(-0.883596\pi\)
0.776626 + 0.629962i \(0.216930\pi\)
\(228\) 0 0
\(229\) −9.53032 16.5070i −0.629781 1.09081i −0.987595 0.157021i \(-0.949811\pi\)
0.357814 0.933793i \(-0.383522\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.3721 0.876035 0.438017 0.898967i \(-0.355681\pi\)
0.438017 + 0.898967i \(0.355681\pi\)
\(234\) 0 0
\(235\) 7.97630 0.520317
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.64873 + 16.7121i 0.624124 + 1.08101i 0.988710 + 0.149845i \(0.0478774\pi\)
−0.364585 + 0.931170i \(0.618789\pi\)
\(240\) 0 0
\(241\) −9.99031 + 17.3037i −0.643532 + 1.11463i 0.341106 + 0.940025i \(0.389199\pi\)
−0.984638 + 0.174606i \(0.944135\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.44492 + 5.96678i −0.220088 + 0.381204i
\(246\) 0 0
\(247\) 26.3645 + 45.6647i 1.67753 + 2.90558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.889846 0.0561666 0.0280833 0.999606i \(-0.491060\pi\)
0.0280833 + 0.999606i \(0.491060\pi\)
\(252\) 0 0
\(253\) 7.35235 0.462238
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.36222 + 14.4838i 0.521621 + 0.903474i 0.999684 + 0.0251482i \(0.00800576\pi\)
−0.478063 + 0.878326i \(0.658661\pi\)
\(258\) 0 0
\(259\) 1.26857 2.19723i 0.0788251 0.136529i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.56170 + 6.16905i −0.219624 + 0.380400i −0.954693 0.297592i \(-0.903816\pi\)
0.735069 + 0.677992i \(0.237150\pi\)
\(264\) 0 0
\(265\) 2.73950 + 4.74495i 0.168286 + 0.291480i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −23.5030 −1.43300 −0.716502 0.697585i \(-0.754258\pi\)
−0.716502 + 0.697585i \(0.754258\pi\)
\(270\) 0 0
\(271\) −8.49838 −0.516240 −0.258120 0.966113i \(-0.583103\pi\)
−0.258120 + 0.966113i \(0.583103\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.20380 + 3.81710i 0.132894 + 0.230180i
\(276\) 0 0
\(277\) 11.1153 19.2523i 0.667856 1.15676i −0.310646 0.950526i \(-0.600545\pi\)
0.978502 0.206235i \(-0.0661212\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.87029 + 10.1676i −0.350192 + 0.606550i −0.986283 0.165064i \(-0.947217\pi\)
0.636091 + 0.771614i \(0.280550\pi\)
\(282\) 0 0
\(283\) 11.8667 + 20.5537i 0.705401 + 1.22179i 0.966546 + 0.256491i \(0.0825666\pi\)
−0.261145 + 0.965300i \(0.584100\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.82488 0.107719
\(288\) 0 0
\(289\) −7.56653 −0.445090
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.1347 17.5538i −0.592077 1.02551i −0.993952 0.109813i \(-0.964975\pi\)
0.401876 0.915694i \(-0.368358\pi\)
\(294\) 0 0
\(295\) 3.69411 6.39839i 0.215080 0.372529i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.82220 10.0844i 0.336707 0.583193i
\(300\) 0 0
\(301\) −1.17348 2.03253i −0.0676382 0.117153i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.592395 −0.0339204
\(306\) 0 0
\(307\) 2.05094 0.117053 0.0585267 0.998286i \(-0.481360\pi\)
0.0585267 + 0.998286i \(0.481360\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.89792 17.1437i −0.561259 0.972130i −0.997387 0.0722448i \(-0.976984\pi\)
0.436128 0.899885i \(-0.356350\pi\)
\(312\) 0 0
\(313\) 0.0577727 0.100065i 0.00326551 0.00565603i −0.864388 0.502825i \(-0.832294\pi\)
0.867654 + 0.497169i \(0.165627\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.8936 20.6003i 0.668011 1.15703i −0.310448 0.950590i \(-0.600479\pi\)
0.978459 0.206439i \(-0.0661875\pi\)
\(318\) 0 0
\(319\) −7.87422 13.6385i −0.440872 0.763612i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −23.2002 −1.29089
\(324\) 0 0
\(325\) 6.98062 0.387215
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.32365 2.29262i −0.0729750 0.126396i
\(330\) 0 0
\(331\) −17.3125 + 29.9862i −0.951582 + 1.64819i −0.209579 + 0.977792i \(0.567209\pi\)
−0.742003 + 0.670396i \(0.766124\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.61087 4.52216i 0.142647 0.247072i
\(336\) 0 0
\(337\) 5.26713 + 9.12293i 0.286919 + 0.496958i 0.973073 0.230499i \(-0.0740357\pi\)
−0.686154 + 0.727456i \(0.740702\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −24.8783 −1.34724
\(342\) 0 0
\(343\) 4.60997 0.248915
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.94654 5.10356i −0.158179 0.273974i 0.776033 0.630692i \(-0.217229\pi\)
−0.934212 + 0.356718i \(0.883896\pi\)
\(348\) 0 0
\(349\) 11.5066 19.9301i 0.615936 1.06683i −0.374284 0.927314i \(-0.622111\pi\)
0.990220 0.139518i \(-0.0445552\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.36222 + 12.7517i −0.391851 + 0.678706i −0.992694 0.120661i \(-0.961499\pi\)
0.600842 + 0.799367i \(0.294832\pi\)
\(354\) 0 0
\(355\) −4.49031 7.77745i −0.238321 0.412784i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.02262 −0.212306 −0.106153 0.994350i \(-0.533853\pi\)
−0.106153 + 0.994350i \(0.533853\pi\)
\(360\) 0 0
\(361\) 38.0574 2.00302
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.02601 6.97325i −0.210731 0.364997i
\(366\) 0 0
\(367\) 4.43830 7.68735i 0.231677 0.401277i −0.726625 0.687035i \(-0.758912\pi\)
0.958302 + 0.285758i \(0.0922454\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.909226 1.57482i 0.0472046 0.0817608i
\(372\) 0 0
\(373\) −15.3645 26.6121i −0.795545 1.37792i −0.922492 0.386015i \(-0.873851\pi\)
0.126947 0.991909i \(-0.459482\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.9419 −1.28457
\(378\) 0 0
\(379\) −4.24004 −0.217796 −0.108898 0.994053i \(-0.534732\pi\)
−0.108898 + 0.994053i \(0.534732\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0439 + 27.7889i 0.819807 + 1.41995i 0.905824 + 0.423654i \(0.139253\pi\)
−0.0860167 + 0.996294i \(0.527414\pi\)
\(384\) 0 0
\(385\) 0.731430 1.26687i 0.0372771 0.0645659i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14.1667 + 24.5374i −0.718278 + 1.24409i 0.243403 + 0.969925i \(0.421736\pi\)
−0.961681 + 0.274169i \(0.911597\pi\)
\(390\) 0 0
\(391\) 2.56170 + 4.43700i 0.129551 + 0.224389i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.9968 0.553307
\(396\) 0 0
\(397\) 13.1395 0.659455 0.329728 0.944076i \(-0.393043\pi\)
0.329728 + 0.944076i \(0.393043\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.59471 + 13.1544i 0.379262 + 0.656900i 0.990955 0.134195i \(-0.0428447\pi\)
−0.611693 + 0.791095i \(0.709511\pi\)
\(402\) 0 0
\(403\) −19.7007 + 34.1227i −0.981364 + 1.69977i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.8468 29.1795i 0.835063 1.44637i
\(408\) 0 0
\(409\) 5.10064 + 8.83457i 0.252211 + 0.436841i 0.964134 0.265415i \(-0.0855091\pi\)
−0.711924 + 0.702257i \(0.752176\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.45211 −0.120661
\(414\) 0 0
\(415\) −6.29314 −0.308918
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.32058 + 10.9476i 0.308781 + 0.534824i 0.978096 0.208155i \(-0.0667457\pi\)
−0.669315 + 0.742979i \(0.733412\pi\)
\(420\) 0 0
\(421\) −0.501620 + 0.868832i −0.0244475 + 0.0423443i −0.877990 0.478678i \(-0.841116\pi\)
0.853543 + 0.521023i \(0.174449\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.53570 + 2.65991i −0.0744923 + 0.129024i
\(426\) 0 0
\(427\) 0.0983063 + 0.170272i 0.00475738 + 0.00824002i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 39.0685 1.88186 0.940932 0.338596i \(-0.109952\pi\)
0.940932 + 0.338596i \(0.109952\pi\)
\(432\) 0 0
\(433\) −29.6663 −1.42567 −0.712836 0.701331i \(-0.752589\pi\)
−0.712836 + 0.701331i \(0.752589\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.30013 10.9121i −0.301376 0.521999i
\(438\) 0 0
\(439\) 16.1223 27.9247i 0.769477 1.33277i −0.168370 0.985724i \(-0.553850\pi\)
0.937847 0.347049i \(-0.112816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.22372 + 3.85160i −0.105652 + 0.182995i −0.914004 0.405704i \(-0.867026\pi\)
0.808352 + 0.588699i \(0.200360\pi\)
\(444\) 0 0
\(445\) −5.95299 10.3109i −0.282199 0.488783i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.0639 0.616523 0.308261 0.951302i \(-0.400253\pi\)
0.308261 + 0.951302i \(0.400253\pi\)
\(450\) 0 0
\(451\) 24.2347 1.14117
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.15842 2.00643i −0.0543074 0.0940631i
\(456\) 0 0
\(457\) −8.16504 + 14.1423i −0.381945 + 0.661547i −0.991340 0.131319i \(-0.958079\pi\)
0.609396 + 0.792866i \(0.291412\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.7639 18.6436i 0.501324 0.868319i −0.498675 0.866789i \(-0.666180\pi\)
0.999999 0.00152975i \(-0.000486935\pi\)
\(462\) 0 0
\(463\) 11.1584 + 19.3269i 0.518576 + 0.898199i 0.999767 + 0.0215837i \(0.00687083\pi\)
−0.481192 + 0.876616i \(0.659796\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.5203 1.68996 0.844978 0.534801i \(-0.179613\pi\)
0.844978 + 0.534801i \(0.179613\pi\)
\(468\) 0 0
\(469\) −1.73307 −0.0800256
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −15.5840 26.9922i −0.716551 1.24110i
\(474\) 0 0
\(475\) 3.77682 6.54164i 0.173292 0.300151i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.47900 7.75786i 0.204651 0.354465i −0.745371 0.666650i \(-0.767727\pi\)
0.950021 + 0.312185i \(0.101061\pi\)
\(480\) 0 0
\(481\) −26.6814 46.2135i −1.21657 2.10715i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.24544 0.0565523
\(486\) 0 0
\(487\) −29.5016 −1.33685 −0.668423 0.743781i \(-0.733030\pi\)
−0.668423 + 0.743781i \(0.733030\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.17635 8.96571i −0.233606 0.404617i 0.725261 0.688474i \(-0.241719\pi\)
−0.958867 + 0.283857i \(0.908386\pi\)
\(492\) 0 0
\(493\) 5.48707 9.50388i 0.247125 0.428033i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.49031 + 2.58129i −0.0668495 + 0.115787i
\(498\) 0 0
\(499\) −11.1925 19.3860i −0.501045 0.867835i −0.999999 0.00120683i \(-0.999616\pi\)
0.498954 0.866628i \(-0.333717\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 42.8654 1.91127 0.955637 0.294546i \(-0.0951685\pi\)
0.955637 + 0.294546i \(0.0951685\pi\)
\(504\) 0 0
\(505\) 4.66379 0.207536
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.54539 6.14079i −0.157147 0.272186i 0.776692 0.629881i \(-0.216896\pi\)
−0.933839 + 0.357695i \(0.883563\pi\)
\(510\) 0 0
\(511\) −1.33621 + 2.31438i −0.0591105 + 0.102382i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.08270 + 3.60735i −0.0917749 + 0.158959i
\(516\) 0 0
\(517\) −17.5782 30.4463i −0.773088 1.33903i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 39.5863 1.73431 0.867153 0.498042i \(-0.165947\pi\)
0.867153 + 0.498042i \(0.165947\pi\)
\(522\) 0 0
\(523\) 25.9375 1.13417 0.567085 0.823659i \(-0.308071\pi\)
0.567085 + 0.823659i \(0.308071\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.66811 15.0136i −0.377589 0.654003i
\(528\) 0 0
\(529\) 10.1087 17.5088i 0.439509 0.761252i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 19.1911 33.2399i 0.831257 1.43978i
\(534\) 0 0
\(535\) 3.28705 + 5.69333i 0.142111 + 0.246144i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 30.3677 1.30803
\(540\) 0 0
\(541\) −38.7929 −1.66784 −0.833920 0.551886i \(-0.813908\pi\)
−0.833920 + 0.551886i \(0.813908\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.371908 + 0.644164i 0.0159308 + 0.0275929i
\(546\) 0 0
\(547\) 5.02655 8.70623i 0.214920 0.372252i −0.738328 0.674442i \(-0.764384\pi\)
0.953248 + 0.302190i \(0.0977177\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13.4946 + 23.3734i −0.574890 + 0.995739i
\(552\) 0 0
\(553\) −1.82488 3.16079i −0.0776019 0.134410i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.4316 0.992829 0.496415 0.868086i \(-0.334650\pi\)
0.496415 + 0.868086i \(0.334650\pi\)
\(558\) 0 0
\(559\) −49.3627 −2.08782
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −20.7580 35.9539i −0.874844 1.51527i −0.856928 0.515435i \(-0.827630\pi\)
−0.0179159 0.999839i \(-0.505703\pi\)
\(564\) 0 0
\(565\) −8.31252 + 14.3977i −0.349710 + 0.605716i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.5990 + 30.4824i −0.737789 + 1.27789i 0.215699 + 0.976460i \(0.430797\pi\)
−0.953489 + 0.301429i \(0.902536\pi\)
\(570\) 0 0
\(571\) 0.689797 + 1.19476i 0.0288671 + 0.0499993i 0.880098 0.474792i \(-0.157477\pi\)
−0.851231 + 0.524791i \(0.824143\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.66811 −0.0695648
\(576\) 0 0
\(577\) −17.3322 −0.721549 −0.360775 0.932653i \(-0.617488\pi\)
−0.360775 + 0.932653i \(0.617488\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.04433 + 1.80883i 0.0433261 + 0.0750430i
\(582\) 0 0
\(583\) 12.0746 20.9139i 0.500080 0.866164i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22.8450 + 39.5687i −0.942914 + 1.63317i −0.183039 + 0.983106i \(0.558593\pi\)
−0.759875 + 0.650069i \(0.774740\pi\)
\(588\) 0 0
\(589\) 21.3179 + 36.9237i 0.878389 + 1.52141i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −10.4661 −0.429791 −0.214896 0.976637i \(-0.568941\pi\)
−0.214896 + 0.976637i \(0.568941\pi\)
\(594\) 0 0
\(595\) 1.01938 0.0417905
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.5698 + 18.3074i 0.431869 + 0.748020i 0.997034 0.0769583i \(-0.0245208\pi\)
−0.565165 + 0.824978i \(0.691188\pi\)
\(600\) 0 0
\(601\) 5.33421 9.23912i 0.217587 0.376871i −0.736483 0.676456i \(-0.763515\pi\)
0.954070 + 0.299585i \(0.0968481\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.21349 7.29798i 0.171303 0.296705i
\(606\) 0 0
\(607\) −0.651942 1.12920i −0.0264615 0.0458327i 0.852491 0.522741i \(-0.175091\pi\)
−0.878953 + 0.476909i \(0.841757\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −55.6796 −2.25255
\(612\) 0 0
\(613\) 30.9741 1.25103 0.625517 0.780211i \(-0.284888\pi\)
0.625517 + 0.780211i \(0.284888\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.29190 2.23763i −0.0520099 0.0900838i 0.838848 0.544365i \(-0.183229\pi\)
−0.890858 + 0.454281i \(0.849896\pi\)
\(618\) 0 0
\(619\) 0.465378 0.806058i 0.0187051 0.0323982i −0.856521 0.516112i \(-0.827379\pi\)
0.875226 + 0.483713i \(0.160712\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.97577 + 3.42213i −0.0791574 + 0.137105i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.4790 0.936169
\(630\) 0 0
\(631\) 20.6250 0.821069 0.410535 0.911845i \(-0.365342\pi\)
0.410535 + 0.911845i \(0.365342\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.39576 + 4.14957i 0.0950727 + 0.164671i
\(636\) 0 0
\(637\) 24.0477 41.6518i 0.952805 1.65031i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.9516 20.7007i 0.472058 0.817628i −0.527431 0.849598i \(-0.676845\pi\)
0.999489 + 0.0319697i \(0.0101780\pi\)
\(642\) 0 0
\(643\) −1.12863 1.95484i −0.0445088 0.0770915i 0.842913 0.538050i \(-0.180839\pi\)
−0.887422 + 0.460959i \(0.847506\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.7262 0.500320 0.250160 0.968204i \(-0.419517\pi\)
0.250160 + 0.968204i \(0.419517\pi\)
\(648\) 0 0
\(649\) −32.5644 −1.27826
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.585400 + 1.01394i 0.0229085 + 0.0396787i 0.877252 0.480029i \(-0.159374\pi\)
−0.854344 + 0.519708i \(0.826041\pi\)
\(654\) 0 0
\(655\) −9.88985 + 17.1297i −0.386428 + 0.669313i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.3678 30.0819i 0.676552 1.17182i −0.299460 0.954109i \(-0.596807\pi\)
0.976013 0.217714i \(-0.0698601\pi\)
\(660\) 0 0
\(661\) −17.5730 30.4374i −0.683511 1.18388i −0.973902 0.226968i \(-0.927119\pi\)
0.290391 0.956908i \(-0.406215\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.50701 −0.0972177
\(666\) 0 0
\(667\) 5.96017 0.230779
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.30552 + 2.26123i 0.0503991 + 0.0872938i
\(672\) 0 0
\(673\) 13.6374 23.6207i 0.525684 0.910511i −0.473869 0.880596i \(-0.657143\pi\)
0.999552 0.0299154i \(-0.00952380\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.56602 7.90858i 0.175486 0.303951i −0.764843 0.644217i \(-0.777184\pi\)
0.940330 + 0.340265i \(0.110517\pi\)
\(678\) 0 0
\(679\) −0.206677 0.357975i −0.00793153 0.0137378i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.0326 1.11090 0.555451 0.831549i \(-0.312545\pi\)
0.555451 + 0.831549i \(0.312545\pi\)
\(684\) 0 0
\(685\) 18.5343 0.708158
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −19.1234 33.1227i −0.728544 1.26188i
\(690\) 0 0
\(691\) 8.38823 14.5288i 0.319103 0.552703i −0.661198 0.750211i \(-0.729952\pi\)
0.980301 + 0.197509i \(0.0632850\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.68980 4.65887i 0.102030 0.176721i
\(696\) 0 0
\(697\) 8.44385 + 14.6252i 0.319834 + 0.553968i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.93400 −0.110816 −0.0554078 0.998464i \(-0.517646\pi\)
−0.0554078 + 0.998464i \(0.517646\pi\)
\(702\) 0 0
\(703\) −57.7431 −2.17782
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.773944 1.34051i −0.0291071 0.0504150i
\(708\) 0 0
\(709\) −13.9293 + 24.1263i −0.523126 + 0.906080i 0.476512 + 0.879168i \(0.341901\pi\)
−0.999638 + 0.0269124i \(0.991432\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.70773 8.15404i 0.176306 0.305371i
\(714\) 0 0
\(715\) −15.3839 26.6457i −0.575326 0.996493i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.95585 −0.259409 −0.129705 0.991553i \(-0.541403\pi\)
−0.129705 + 0.991553i \(0.541403\pi\)
\(720\) 0 0
\(721\) 1.38248 0.0514861
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.78651 + 3.09432i 0.0663492 + 0.114920i
\(726\) 0 0
\(727\) 2.51023 4.34784i 0.0930992 0.161253i −0.815715 0.578455i \(-0.803656\pi\)
0.908814 + 0.417202i \(0.136989\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.8595 18.8092i 0.401654 0.695685i
\(732\) 0 0
\(733\) 10.2368 + 17.7307i 0.378105 + 0.654897i 0.990787 0.135433i \(-0.0432425\pi\)
−0.612682 + 0.790330i \(0.709909\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.0154 −0.847782
\(738\) 0 0
\(739\) 51.1614 1.88200 0.941002 0.338401i \(-0.109886\pi\)
0.941002 + 0.338401i \(0.109886\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.07086 5.31888i −0.112659 0.195131i 0.804183 0.594382i \(-0.202603\pi\)
−0.916841 + 0.399251i \(0.869270\pi\)
\(744\) 0 0
\(745\) −5.51794 + 9.55735i −0.202162 + 0.350154i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.09095 1.88959i 0.0398626 0.0690440i
\(750\) 0 0
\(751\) 7.54557 + 13.0693i 0.275342 + 0.476906i 0.970221 0.242220i \(-0.0778757\pi\)
−0.694880 + 0.719126i \(0.744542\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.50162 0.0546496
\(756\) 0 0
\(757\) −2.87659 −0.104551 −0.0522757 0.998633i \(-0.516647\pi\)
−0.0522757 + 0.998633i \(0.516647\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.119478 + 0.206941i 0.00433106 + 0.00750162i 0.868183 0.496245i \(-0.165288\pi\)
−0.863852 + 0.503746i \(0.831955\pi\)
\(762\) 0 0
\(763\) 0.123434 0.213794i 0.00446862 0.00773988i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.7872 + 44.6647i −0.931122 + 1.61275i
\(768\) 0 0
\(769\) 1.92930 + 3.34164i 0.0695722 + 0.120503i 0.898713 0.438537i \(-0.144503\pi\)
−0.829141 + 0.559040i \(0.811170\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.22355 0.187878 0.0939390 0.995578i \(-0.470054\pi\)
0.0939390 + 0.995578i \(0.470054\pi\)
\(774\) 0 0
\(775\) 5.64441 0.202753
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.7664 35.9684i −0.744033 1.28870i
\(780\) 0 0
\(781\) −19.7915 + 34.2799i −0.708196 + 1.22663i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −11.8109 + 20.4571i −0.421549 + 0.730144i
\(786\) 0 0
\(787\) 10.8818 + 18.8478i 0.387893 + 0.671851i 0.992166 0.124927i \(-0.0398695\pi\)
−0.604273 + 0.796778i \(0.706536\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.51776 0.196189
\(792\) 0 0
\(793\) 4.13528 0.146848
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.7228 + 22.0365i 0.450665 + 0.780574i 0.998427 0.0560596i \(-0.0178537\pi\)
−0.547763 + 0.836634i \(0.684520\pi\)
\(798\) 0 0
\(799\) 12.2492 21.2162i 0.433345 0.750576i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −17.7451 + 30.7353i −0.626209 + 1.08463i
\(804\) 0 0
\(805\) 0.276818 + 0.479462i 0.00975654 + 0.0168988i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 11.0854 0.389741 0.194871 0.980829i \(-0.437571\pi\)
0.194871 + 0.980829i \(0.437571\pi\)
\(810\) 0 0
\(811\) 42.4869 1.49192 0.745958 0.665993i \(-0.231992\pi\)
0.745958 + 0.665993i \(0.231992\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.58108 7.93467i −0.160468 0.277939i
\(816\) 0 0
\(817\) −26.7074 + 46.2585i −0.934373 + 1.61838i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −15.0673 + 26.0973i −0.525851 + 0.910800i 0.473696 + 0.880689i \(0.342920\pi\)
−0.999547 + 0.0301119i \(0.990414\pi\)
\(822\) 0 0
\(823\) −10.8109 18.7251i −0.376845 0.652715i 0.613756 0.789495i \(-0.289658\pi\)
−0.990601 + 0.136781i \(0.956324\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.82488 0.0634574 0.0317287 0.999497i \(-0.489899\pi\)
0.0317287 + 0.999497i \(0.489899\pi\)
\(828\) 0 0
\(829\) 30.4374 1.05713 0.528567 0.848892i \(-0.322730\pi\)
0.528567 + 0.848892i \(0.322730\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 10.5807 + 18.3263i 0.366600 + 0.634970i
\(834\) 0 0
\(835\) 0.324363 0.561813i 0.0112250 0.0194423i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 16.7834 29.0698i 0.579429 1.00360i −0.416116 0.909312i \(-0.636609\pi\)
0.995545 0.0942887i \(-0.0300577\pi\)
\(840\) 0 0
\(841\) 8.11678 + 14.0587i 0.279889 + 0.484782i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −35.7291 −1.22912
\(846\) 0 0
\(847\) −2.79687 −0.0961016
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.37584 + 11.0433i 0.218561 + 0.378559i
\(852\) 0 0
\(853\) 11.8775 20.5724i 0.406676 0.704384i −0.587839 0.808978i \(-0.700021\pi\)
0.994515 + 0.104594i \(0.0333543\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.0052 + 22.5256i −0.444249 + 0.769461i −0.998000 0.0632211i \(-0.979863\pi\)
0.553751 + 0.832682i \(0.313196\pi\)
\(858\) 0 0
\(859\) 25.3835 + 43.9656i 0.866075 + 1.50009i 0.865975 + 0.500086i \(0.166698\pi\)
9.98605e−5 1.00000i \(0.499968\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.9343 −1.12110 −0.560548 0.828122i \(-0.689410\pi\)
−0.560548 + 0.828122i \(0.689410\pi\)
\(864\) 0 0
\(865\) 8.81521 0.299726
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −24.2347 41.9757i −0.822105 1.42393i
\(870\) 0 0
\(871\) −18.2255 + 31.5675i −0.617547 + 1.06962i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.165947 + 0.287429i −0.00561004 + 0.00971688i
\(876\) 0 0
\(877\) −8.56170 14.8293i −0.289108 0.500750i 0.684489 0.729023i \(-0.260025\pi\)
−0.973597 + 0.228273i \(0.926692\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 46.7617 1.57544 0.787721 0.616032i \(-0.211261\pi\)
0.787721 + 0.616032i \(0.211261\pi\)
\(882\) 0 0
\(883\) −21.1356 −0.711269 −0.355635 0.934625i \(-0.615735\pi\)
−0.355635 + 0.934625i \(0.615735\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.9887 39.8176i −0.771885 1.33694i −0.936529 0.350590i \(-0.885981\pi\)
0.164644 0.986353i \(-0.447352\pi\)
\(888\) 0 0
\(889\) 0.795139 1.37722i 0.0266681 0.0461905i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.1250 + 52.1781i −1.00810 + 1.74607i
\(894\) 0 0
\(895\) −4.58108 7.93467i −0.153129 0.265227i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −20.1676 −0.672626
\(900\) 0 0
\(901\) 16.8282 0.560628
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.78651 + 11.7546i 0.225591 + 0.390735i
\(906\) 0 0
\(907\) 3.29404 5.70544i 0.109377 0.189446i −0.806141 0.591723i \(-0.798448\pi\)
0.915518 + 0.402277i \(0.131781\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −11.8855 + 20.5863i −0.393785 + 0.682056i −0.992945 0.118573i \(-0.962168\pi\)
0.599160 + 0.800629i \(0.295501\pi\)
\(912\) 0 0
\(913\) 13.8688 + 24.0215i 0.458991 + 0.794996i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.56477 0.216788
\(918\) 0 0
\(919\) 46.1305 1.52171 0.760853 0.648924i \(-0.224781\pi\)
0.760853 + 0.648924i \(0.224781\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 31.3452 + 54.2914i 1.03174 + 1.78702i
\(924\) 0 0
\(925\) −3.82220 + 6.62025i −0.125673 + 0.217673i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 11.3082 19.5864i 0.371010 0.642608i −0.618711 0.785619i \(-0.712345\pi\)
0.989721 + 0.143010i \(0.0456782\pi\)
\(930\) 0 0
\(931\) −26.0217 45.0709i −0.852826 1.47714i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 13.5375 0.442723
\(936\) 0 0
\(937\) −47.5830 −1.55447 −0.777235 0.629211i \(-0.783378\pi\)
−0.777235 + 0.629211i \(0.783378\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21.4991 37.2375i −0.700850 1.21391i −0.968169 0.250299i \(-0.919471\pi\)
0.267319 0.963608i \(-0.413862\pi\)
\(942\) 0 0
\(943\) −4.58594 + 7.94308i −0.149339 + 0.258662i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −24.0274 + 41.6167i −0.780786 + 1.35236i 0.150698 + 0.988580i \(0.451848\pi\)
−0.931484 + 0.363781i \(0.881486\pi\)
\(948\) 0 0
\(949\) 28.1040 + 48.6776i 0.912295 + 1.58014i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.0413 1.26467 0.632335 0.774695i \(-0.282096\pi\)
0.632335 + 0.774695i \(0.282096\pi\)
\(954\) 0 0
\(955\) −10.2694 −0.332311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.07571 5.32729i −0.0993199 0.172027i
\(960\) 0 0
\(961\) −0.429681 + 0.744229i −0.0138607 + 0.0240074i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.53570 7.85606i 0.146009 0.252895i
\(966\) 0 0
\(967\) 1.02423 + 1.77402i 0.0329371 + 0.0570488i 0.882024 0.471204i \(-0.156181\pi\)
−0.849087 + 0.528253i \(0.822847\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −27.4068 −0.879527 −0.439764 0.898114i \(-0.644938\pi\)
−0.439764 + 0.898114i \(0.644938\pi\)
\(972\) 0 0
\(973\) −1.78546 −0.0572392
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.04914 3.54922i −0.0655578 0.113549i 0.831383 0.555699i \(-0.187549\pi\)
−0.896941 + 0.442150i \(0.854216\pi\)
\(978\) 0 0
\(979\) −26.2384 + 45.4463i −0.838584 + 1.45247i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.05903 + 5.29840i −0.0975680 + 0.168993i −0.910677 0.413118i \(-0.864440\pi\)
0.813109 + 0.582111i \(0.197773\pi\)
\(984\) 0 0
\(985\) 2.40329 + 4.16262i 0.0765752 + 0.132632i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.7958 0.375086
\(990\) 0 0
\(991\) 14.9881 0.476114 0.238057 0.971251i \(-0.423490\pi\)
0.238057 + 0.971251i \(0.423490\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.22981 15.9865i −0.292605 0.506806i
\(996\) 0 0
\(997\) 6.74813 11.6881i 0.213715 0.370166i −0.739159 0.673531i \(-0.764777\pi\)
0.952874 + 0.303365i \(0.0981101\pi\)
\(998\) 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 1080.2.q.e.721.3 8
3.2 odd 2 360.2.q.e.241.1 yes 8
4.3 odd 2 2160.2.q.l.721.2 8
9.2 odd 6 3240.2.a.u.1.2 4
9.4 even 3 inner 1080.2.q.e.361.3 8
9.5 odd 6 360.2.q.e.121.1 8
9.7 even 3 3240.2.a.s.1.2 4
12.11 even 2 720.2.q.l.241.4 8
36.7 odd 6 6480.2.a.bz.1.3 4
36.11 even 6 6480.2.a.cb.1.3 4
36.23 even 6 720.2.q.l.481.4 8
36.31 odd 6 2160.2.q.l.1441.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.e.121.1 8 9.5 odd 6
360.2.q.e.241.1 yes 8 3.2 odd 2
720.2.q.l.241.4 8 12.11 even 2
720.2.q.l.481.4 8 36.23 even 6
1080.2.q.e.361.3 8 9.4 even 3 inner
1080.2.q.e.721.3 8 1.1 even 1 trivial
2160.2.q.l.721.2 8 4.3 odd 2
2160.2.q.l.1441.2 8 36.31 odd 6
3240.2.a.s.1.2 4 9.7 even 3
3240.2.a.u.1.2 4 9.2 odd 6
6480.2.a.bz.1.3 4 36.7 odd 6
6480.2.a.cb.1.3 4 36.11 even 6