Properties

Label 672.2.bk.a.529.7
Level $672$
Weight $2$
Character 672.529
Analytic conductor $5.366$
Analytic rank $0$
Dimension $32$
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] = [672,2,Mod(529,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bk (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 529.7
Character \(\chi\) \(=\) 672.529
Dual form 672.2.bk.a.625.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(1.98722 - 1.14732i) q^{5} +(1.05630 - 2.42574i) q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(1.98722 - 1.14732i) q^{5} +(1.05630 - 2.42574i) q^{7} +(0.500000 + 0.866025i) q^{9} +(-3.36596 - 1.94334i) q^{11} -3.33413i q^{13} -2.29465 q^{15} +(-0.143560 + 0.248654i) q^{17} +(-2.41210 + 1.39263i) q^{19} +(-2.12766 + 1.57260i) q^{21} +(3.26713 + 5.65883i) q^{23} +(0.132707 - 0.229856i) q^{25} -1.00000i q^{27} -5.53374i q^{29} +(3.72154 - 6.44589i) q^{31} +(1.94334 + 3.36596i) q^{33} +(-0.684001 - 6.03242i) q^{35} +(-5.15752 + 2.97769i) q^{37} +(-1.66706 + 2.88744i) q^{39} -3.51617 q^{41} -11.2465i q^{43} +(1.98722 + 1.14732i) q^{45} +(0.0435037 + 0.0753507i) q^{47} +(-4.76845 - 5.12464i) q^{49} +(0.248654 - 0.143560i) q^{51} +(-6.11258 - 3.52910i) q^{53} -8.91855 q^{55} +2.78525 q^{57} +(3.76909 + 2.17609i) q^{59} +(6.20693 - 3.58357i) q^{61} +(2.62891 - 0.298085i) q^{63} +(-3.82533 - 6.62566i) q^{65} +(11.2804 + 6.51276i) q^{67} -6.53425i q^{69} +6.18835 q^{71} +(6.93978 - 12.0201i) q^{73} +(-0.229856 + 0.132707i) q^{75} +(-8.26950 + 6.11219i) q^{77} +(-4.49926 - 7.79295i) q^{79} +(-0.500000 + 0.866025i) q^{81} +17.6313i q^{83} +0.658842i q^{85} +(-2.76687 + 4.79236i) q^{87} +(8.59194 + 14.8817i) q^{89} +(-8.08774 - 3.52185i) q^{91} +(-6.44589 + 3.72154i) q^{93} +(-3.19559 + 5.53493i) q^{95} +6.46528 q^{97} -3.88667i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{9} + 8 q^{23} + 16 q^{25} + 24 q^{31} + 24 q^{47} + 8 q^{49} + 64 q^{55} - 16 q^{57} + 80 q^{71} + 8 q^{73} - 8 q^{79} - 16 q^{81} - 24 q^{87} - 24 q^{95} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.866025 0.500000i −0.500000 0.288675i
\(4\) 0 0
\(5\) 1.98722 1.14732i 0.888714 0.513099i 0.0151922 0.999885i \(-0.495164\pi\)
0.873522 + 0.486785i \(0.161831\pi\)
\(6\) 0 0
\(7\) 1.05630 2.42574i 0.399245 0.916844i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −3.36596 1.94334i −1.01487 0.585938i −0.102259 0.994758i \(-0.532607\pi\)
−0.912615 + 0.408820i \(0.865940\pi\)
\(12\) 0 0
\(13\) 3.33413i 0.924721i −0.886692 0.462361i \(-0.847003\pi\)
0.886692 0.462361i \(-0.152997\pi\)
\(14\) 0 0
\(15\) −2.29465 −0.592476
\(16\) 0 0
\(17\) −0.143560 + 0.248654i −0.0348185 + 0.0603074i −0.882910 0.469543i \(-0.844419\pi\)
0.848091 + 0.529851i \(0.177752\pi\)
\(18\) 0 0
\(19\) −2.41210 + 1.39263i −0.553374 + 0.319491i −0.750482 0.660891i \(-0.770178\pi\)
0.197108 + 0.980382i \(0.436845\pi\)
\(20\) 0 0
\(21\) −2.12766 + 1.57260i −0.464293 + 0.343170i
\(22\) 0 0
\(23\) 3.26713 + 5.65883i 0.681243 + 1.17995i 0.974602 + 0.223945i \(0.0718935\pi\)
−0.293359 + 0.956002i \(0.594773\pi\)
\(24\) 0 0
\(25\) 0.132707 0.229856i 0.0265415 0.0459712i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 5.53374i 1.02759i −0.857913 0.513795i \(-0.828239\pi\)
0.857913 0.513795i \(-0.171761\pi\)
\(30\) 0 0
\(31\) 3.72154 6.44589i 0.668408 1.15772i −0.309941 0.950756i \(-0.600309\pi\)
0.978349 0.206961i \(-0.0663573\pi\)
\(32\) 0 0
\(33\) 1.94334 + 3.36596i 0.338291 + 0.585938i
\(34\) 0 0
\(35\) −0.684001 6.03242i −0.115617 1.01966i
\(36\) 0 0
\(37\) −5.15752 + 2.97769i −0.847890 + 0.489530i −0.859939 0.510398i \(-0.829498\pi\)
0.0120481 + 0.999927i \(0.496165\pi\)
\(38\) 0 0
\(39\) −1.66706 + 2.88744i −0.266944 + 0.462361i
\(40\) 0 0
\(41\) −3.51617 −0.549134 −0.274567 0.961568i \(-0.588534\pi\)
−0.274567 + 0.961568i \(0.588534\pi\)
\(42\) 0 0
\(43\) 11.2465i 1.71508i −0.514416 0.857541i \(-0.671991\pi\)
0.514416 0.857541i \(-0.328009\pi\)
\(44\) 0 0
\(45\) 1.98722 + 1.14732i 0.296238 + 0.171033i
\(46\) 0 0
\(47\) 0.0435037 + 0.0753507i 0.00634567 + 0.0109910i 0.869181 0.494494i \(-0.164647\pi\)
−0.862835 + 0.505485i \(0.831313\pi\)
\(48\) 0 0
\(49\) −4.76845 5.12464i −0.681207 0.732091i
\(50\) 0 0
\(51\) 0.248654 0.143560i 0.0348185 0.0201025i
\(52\) 0 0
\(53\) −6.11258 3.52910i −0.839627 0.484759i 0.0175103 0.999847i \(-0.494426\pi\)
−0.857137 + 0.515088i \(0.827759\pi\)
\(54\) 0 0
\(55\) −8.91855 −1.20258
\(56\) 0 0
\(57\) 2.78525 0.368916
\(58\) 0 0
\(59\) 3.76909 + 2.17609i 0.490694 + 0.283302i 0.724862 0.688894i \(-0.241903\pi\)
−0.234168 + 0.972196i \(0.575237\pi\)
\(60\) 0 0
\(61\) 6.20693 3.58357i 0.794716 0.458829i −0.0469043 0.998899i \(-0.514936\pi\)
0.841620 + 0.540070i \(0.181602\pi\)
\(62\) 0 0
\(63\) 2.62891 0.298085i 0.331211 0.0375552i
\(64\) 0 0
\(65\) −3.82533 6.62566i −0.474474 0.821812i
\(66\) 0 0
\(67\) 11.2804 + 6.51276i 1.37812 + 0.795661i 0.991934 0.126758i \(-0.0404573\pi\)
0.386191 + 0.922419i \(0.373791\pi\)
\(68\) 0 0
\(69\) 6.53425i 0.786631i
\(70\) 0 0
\(71\) 6.18835 0.734421 0.367211 0.930138i \(-0.380313\pi\)
0.367211 + 0.930138i \(0.380313\pi\)
\(72\) 0 0
\(73\) 6.93978 12.0201i 0.812240 1.40684i −0.0990526 0.995082i \(-0.531581\pi\)
0.911293 0.411759i \(-0.135085\pi\)
\(74\) 0 0
\(75\) −0.229856 + 0.132707i −0.0265415 + 0.0153237i
\(76\) 0 0
\(77\) −8.26950 + 6.11219i −0.942397 + 0.696549i
\(78\) 0 0
\(79\) −4.49926 7.79295i −0.506207 0.876776i −0.999974 0.00718179i \(-0.997714\pi\)
0.493767 0.869594i \(-0.335619\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 17.6313i 1.93529i 0.252312 + 0.967646i \(0.418809\pi\)
−0.252312 + 0.967646i \(0.581191\pi\)
\(84\) 0 0
\(85\) 0.658842i 0.0714614i
\(86\) 0 0
\(87\) −2.76687 + 4.79236i −0.296640 + 0.513795i
\(88\) 0 0
\(89\) 8.59194 + 14.8817i 0.910744 + 1.57745i 0.813016 + 0.582242i \(0.197824\pi\)
0.0977280 + 0.995213i \(0.468842\pi\)
\(90\) 0 0
\(91\) −8.08774 3.52185i −0.847825 0.369190i
\(92\) 0 0
\(93\) −6.44589 + 3.72154i −0.668408 + 0.385905i
\(94\) 0 0
\(95\) −3.19559 + 5.53493i −0.327861 + 0.567871i
\(96\) 0 0
\(97\) 6.46528 0.656450 0.328225 0.944600i \(-0.393550\pi\)
0.328225 + 0.944600i \(0.393550\pi\)
\(98\) 0 0
\(99\) 3.88667i 0.390625i
\(100\) 0 0
\(101\) 10.2590 + 5.92305i 1.02081 + 0.589365i 0.914339 0.404951i \(-0.132711\pi\)
0.106472 + 0.994316i \(0.466045\pi\)
\(102\) 0 0
\(103\) 0.810948 + 1.40460i 0.0799051 + 0.138400i 0.903209 0.429201i \(-0.141205\pi\)
−0.823304 + 0.567601i \(0.807872\pi\)
\(104\) 0 0
\(105\) −2.42385 + 5.56623i −0.236543 + 0.543208i
\(106\) 0 0
\(107\) −4.35546 + 2.51463i −0.421059 + 0.243098i −0.695530 0.718497i \(-0.744830\pi\)
0.274471 + 0.961595i \(0.411497\pi\)
\(108\) 0 0
\(109\) −0.177033 0.102210i −0.0169567 0.00978994i 0.491498 0.870879i \(-0.336450\pi\)
−0.508454 + 0.861089i \(0.669783\pi\)
\(110\) 0 0
\(111\) 5.95539 0.565260
\(112\) 0 0
\(113\) −14.9203 −1.40358 −0.701792 0.712382i \(-0.747616\pi\)
−0.701792 + 0.712382i \(0.747616\pi\)
\(114\) 0 0
\(115\) 12.9850 + 7.49691i 1.21086 + 0.699090i
\(116\) 0 0
\(117\) 2.88744 1.66706i 0.266944 0.154120i
\(118\) 0 0
\(119\) 0.451527 + 0.610895i 0.0413914 + 0.0560006i
\(120\) 0 0
\(121\) 2.05311 + 3.55609i 0.186646 + 0.323281i
\(122\) 0 0
\(123\) 3.04509 + 1.75809i 0.274567 + 0.158521i
\(124\) 0 0
\(125\) 10.8642i 0.971725i
\(126\) 0 0
\(127\) 6.29267 0.558384 0.279192 0.960235i \(-0.409933\pi\)
0.279192 + 0.960235i \(0.409933\pi\)
\(128\) 0 0
\(129\) −5.62327 + 9.73979i −0.495101 + 0.857541i
\(130\) 0 0
\(131\) −1.07470 + 0.620480i −0.0938973 + 0.0542116i −0.546213 0.837646i \(-0.683931\pi\)
0.452316 + 0.891858i \(0.350598\pi\)
\(132\) 0 0
\(133\) 0.830243 + 7.32217i 0.0719912 + 0.634913i
\(134\) 0 0
\(135\) −1.14732 1.98722i −0.0987460 0.171033i
\(136\) 0 0
\(137\) −0.795096 + 1.37715i −0.0679296 + 0.117658i −0.897990 0.440016i \(-0.854973\pi\)
0.830060 + 0.557674i \(0.188306\pi\)
\(138\) 0 0
\(139\) 1.02981i 0.0873470i 0.999046 + 0.0436735i \(0.0139061\pi\)
−0.999046 + 0.0436735i \(0.986094\pi\)
\(140\) 0 0
\(141\) 0.0870075i 0.00732735i
\(142\) 0 0
\(143\) −6.47933 + 11.2225i −0.541829 + 0.938476i
\(144\) 0 0
\(145\) −6.34900 10.9968i −0.527256 0.913234i
\(146\) 0 0
\(147\) 1.56728 + 6.82229i 0.129267 + 0.562693i
\(148\) 0 0
\(149\) 8.30672 4.79589i 0.680513 0.392895i −0.119535 0.992830i \(-0.538140\pi\)
0.800048 + 0.599935i \(0.204807\pi\)
\(150\) 0 0
\(151\) 3.49611 6.05544i 0.284510 0.492785i −0.687981 0.725729i \(-0.741503\pi\)
0.972490 + 0.232944i \(0.0748359\pi\)
\(152\) 0 0
\(153\) −0.287121 −0.0232123
\(154\) 0 0
\(155\) 17.0792i 1.37184i
\(156\) 0 0
\(157\) 15.0324 + 8.67895i 1.19971 + 0.692655i 0.960492 0.278309i \(-0.0897740\pi\)
0.239223 + 0.970965i \(0.423107\pi\)
\(158\) 0 0
\(159\) 3.52910 + 6.11258i 0.279876 + 0.484759i
\(160\) 0 0
\(161\) 17.1779 1.94776i 1.35381 0.153505i
\(162\) 0 0
\(163\) −13.6156 + 7.86099i −1.06646 + 0.615720i −0.927212 0.374537i \(-0.877802\pi\)
−0.139247 + 0.990258i \(0.544468\pi\)
\(164\) 0 0
\(165\) 7.72369 + 4.45927i 0.601288 + 0.347154i
\(166\) 0 0
\(167\) 0.102064 0.00789795 0.00394897 0.999992i \(-0.498743\pi\)
0.00394897 + 0.999992i \(0.498743\pi\)
\(168\) 0 0
\(169\) 1.88358 0.144891
\(170\) 0 0
\(171\) −2.41210 1.39263i −0.184458 0.106497i
\(172\) 0 0
\(173\) 1.98126 1.14388i 0.150633 0.0869678i −0.422789 0.906228i \(-0.638949\pi\)
0.573422 + 0.819260i \(0.305616\pi\)
\(174\) 0 0
\(175\) −0.417392 0.564711i −0.0315518 0.0426881i
\(176\) 0 0
\(177\) −2.17609 3.76909i −0.163565 0.283302i
\(178\) 0 0
\(179\) 16.1281 + 9.31154i 1.20547 + 0.695978i 0.961766 0.273873i \(-0.0883046\pi\)
0.243702 + 0.969850i \(0.421638\pi\)
\(180\) 0 0
\(181\) 13.6786i 1.01672i 0.861144 + 0.508361i \(0.169748\pi\)
−0.861144 + 0.508361i \(0.830252\pi\)
\(182\) 0 0
\(183\) −7.16714 −0.529810
\(184\) 0 0
\(185\) −6.83276 + 11.8347i −0.502355 + 0.870104i
\(186\) 0 0
\(187\) 0.966437 0.557972i 0.0706728 0.0408030i
\(188\) 0 0
\(189\) −2.42574 1.05630i −0.176447 0.0768348i
\(190\) 0 0
\(191\) −6.31729 10.9419i −0.457103 0.791726i 0.541703 0.840570i \(-0.317780\pi\)
−0.998806 + 0.0488436i \(0.984446\pi\)
\(192\) 0 0
\(193\) −1.30291 + 2.25671i −0.0937856 + 0.162441i −0.909101 0.416576i \(-0.863230\pi\)
0.815316 + 0.579017i \(0.196563\pi\)
\(194\) 0 0
\(195\) 7.65066i 0.547875i
\(196\) 0 0
\(197\) 0.794777i 0.0566255i −0.999599 0.0283127i \(-0.990987\pi\)
0.999599 0.0283127i \(-0.00901343\pi\)
\(198\) 0 0
\(199\) 5.36714 9.29616i 0.380466 0.658987i −0.610663 0.791891i \(-0.709097\pi\)
0.991129 + 0.132904i \(0.0424301\pi\)
\(200\) 0 0
\(201\) −6.51276 11.2804i −0.459375 0.795661i
\(202\) 0 0
\(203\) −13.4234 5.84531i −0.942140 0.410260i
\(204\) 0 0
\(205\) −6.98742 + 4.03419i −0.488023 + 0.281760i
\(206\) 0 0
\(207\) −3.26713 + 5.65883i −0.227081 + 0.393316i
\(208\) 0 0
\(209\) 10.8254 0.748807
\(210\) 0 0
\(211\) 0.556345i 0.0383004i 0.999817 + 0.0191502i \(0.00609607\pi\)
−0.999817 + 0.0191502i \(0.993904\pi\)
\(212\) 0 0
\(213\) −5.35926 3.09417i −0.367211 0.212009i
\(214\) 0 0
\(215\) −12.9034 22.3494i −0.880007 1.52422i
\(216\) 0 0
\(217\) −11.7050 15.8363i −0.794587 1.07504i
\(218\) 0 0
\(219\) −12.0201 + 6.93978i −0.812240 + 0.468947i
\(220\) 0 0
\(221\) 0.829044 + 0.478649i 0.0557676 + 0.0321974i
\(222\) 0 0
\(223\) −11.1076 −0.743820 −0.371910 0.928269i \(-0.621297\pi\)
−0.371910 + 0.928269i \(0.621297\pi\)
\(224\) 0 0
\(225\) 0.265415 0.0176943
\(226\) 0 0
\(227\) 7.59321 + 4.38394i 0.503979 + 0.290972i 0.730355 0.683068i \(-0.239355\pi\)
−0.226376 + 0.974040i \(0.572688\pi\)
\(228\) 0 0
\(229\) 17.9235 10.3481i 1.18442 0.683822i 0.227383 0.973805i \(-0.426983\pi\)
0.957032 + 0.289983i \(0.0936496\pi\)
\(230\) 0 0
\(231\) 10.2177 1.15856i 0.672275 0.0762276i
\(232\) 0 0
\(233\) 2.86427 + 4.96105i 0.187644 + 0.325009i 0.944464 0.328614i \(-0.106581\pi\)
−0.756820 + 0.653623i \(0.773248\pi\)
\(234\) 0 0
\(235\) 0.172903 + 0.0998258i 0.0112790 + 0.00651192i
\(236\) 0 0
\(237\) 8.99853i 0.584517i
\(238\) 0 0
\(239\) −2.27651 −0.147255 −0.0736276 0.997286i \(-0.523458\pi\)
−0.0736276 + 0.997286i \(0.523458\pi\)
\(240\) 0 0
\(241\) −11.7925 + 20.4252i −0.759623 + 1.31570i 0.183421 + 0.983035i \(0.441283\pi\)
−0.943043 + 0.332670i \(0.892050\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) −15.3556 4.71285i −0.981033 0.301093i
\(246\) 0 0
\(247\) 4.64320 + 8.04226i 0.295440 + 0.511717i
\(248\) 0 0
\(249\) 8.81567 15.2692i 0.558671 0.967646i
\(250\) 0 0
\(251\) 12.4322i 0.784716i −0.919813 0.392358i \(-0.871659\pi\)
0.919813 0.392358i \(-0.128341\pi\)
\(252\) 0 0
\(253\) 25.3965i 1.59666i
\(254\) 0 0
\(255\) 0.329421 0.570574i 0.0206291 0.0357307i
\(256\) 0 0
\(257\) −0.141492 0.245072i −0.00882604 0.0152871i 0.861579 0.507624i \(-0.169476\pi\)
−0.870405 + 0.492337i \(0.836143\pi\)
\(258\) 0 0
\(259\) 1.77521 + 15.6562i 0.110306 + 0.972826i
\(260\) 0 0
\(261\) 4.79236 2.76687i 0.296640 0.171265i
\(262\) 0 0
\(263\) 7.15962 12.4008i 0.441481 0.764667i −0.556319 0.830969i \(-0.687787\pi\)
0.997800 + 0.0663018i \(0.0211200\pi\)
\(264\) 0 0
\(265\) −16.1961 −0.994918
\(266\) 0 0
\(267\) 17.1839i 1.05164i
\(268\) 0 0
\(269\) 14.5614 + 8.40704i 0.887825 + 0.512586i 0.873230 0.487307i \(-0.162021\pi\)
0.0145946 + 0.999893i \(0.495354\pi\)
\(270\) 0 0
\(271\) 13.4228 + 23.2490i 0.815378 + 1.41228i 0.909056 + 0.416674i \(0.136804\pi\)
−0.0936779 + 0.995603i \(0.529862\pi\)
\(272\) 0 0
\(273\) 5.24326 + 7.09388i 0.317336 + 0.429341i
\(274\) 0 0
\(275\) −0.893374 + 0.515790i −0.0538725 + 0.0311033i
\(276\) 0 0
\(277\) −19.4881 11.2515i −1.17093 0.676036i −0.217030 0.976165i \(-0.569637\pi\)
−0.953899 + 0.300129i \(0.902970\pi\)
\(278\) 0 0
\(279\) 7.44308 0.445605
\(280\) 0 0
\(281\) 12.8375 0.765824 0.382912 0.923785i \(-0.374921\pi\)
0.382912 + 0.923785i \(0.374921\pi\)
\(282\) 0 0
\(283\) −6.32017 3.64895i −0.375695 0.216908i 0.300249 0.953861i \(-0.402930\pi\)
−0.675944 + 0.736953i \(0.736264\pi\)
\(284\) 0 0
\(285\) 5.53493 3.19559i 0.327861 0.189290i
\(286\) 0 0
\(287\) −3.71414 + 8.52932i −0.219239 + 0.503470i
\(288\) 0 0
\(289\) 8.45878 + 14.6510i 0.497575 + 0.861826i
\(290\) 0 0
\(291\) −5.59910 3.23264i −0.328225 0.189501i
\(292\) 0 0
\(293\) 9.95674i 0.581679i −0.956772 0.290840i \(-0.906065\pi\)
0.956772 0.290840i \(-0.0939346\pi\)
\(294\) 0 0
\(295\) 9.98671 0.581449
\(296\) 0 0
\(297\) −1.94334 + 3.36596i −0.112764 + 0.195313i
\(298\) 0 0
\(299\) 18.8673 10.8930i 1.09112 0.629959i
\(300\) 0 0
\(301\) −27.2812 11.8798i −1.57246 0.684738i
\(302\) 0 0
\(303\) −5.92305 10.2590i −0.340270 0.589365i
\(304\) 0 0
\(305\) 8.22304 14.2427i 0.470850 0.815536i
\(306\) 0 0
\(307\) 14.9479i 0.853122i −0.904459 0.426561i \(-0.859725\pi\)
0.904459 0.426561i \(-0.140275\pi\)
\(308\) 0 0
\(309\) 1.62190i 0.0922664i
\(310\) 0 0
\(311\) −15.6295 + 27.0712i −0.886270 + 1.53506i −0.0420192 + 0.999117i \(0.513379\pi\)
−0.844251 + 0.535948i \(0.819954\pi\)
\(312\) 0 0
\(313\) 1.53234 + 2.65409i 0.0866130 + 0.150018i 0.906077 0.423112i \(-0.139062\pi\)
−0.819464 + 0.573130i \(0.805729\pi\)
\(314\) 0 0
\(315\) 4.88223 3.60857i 0.275082 0.203320i
\(316\) 0 0
\(317\) −14.1145 + 8.14904i −0.792752 + 0.457695i −0.840930 0.541143i \(-0.817992\pi\)
0.0481787 + 0.998839i \(0.484658\pi\)
\(318\) 0 0
\(319\) −10.7539 + 18.6263i −0.602104 + 1.04287i
\(320\) 0 0
\(321\) 5.02926 0.280706
\(322\) 0 0
\(323\) 0.799705i 0.0444968i
\(324\) 0 0
\(325\) −0.766369 0.442463i −0.0425105 0.0245434i
\(326\) 0 0
\(327\) 0.102210 + 0.177033i 0.00565222 + 0.00978994i
\(328\) 0 0
\(329\) 0.228734 0.0259356i 0.0126105 0.00142988i
\(330\) 0 0
\(331\) 2.39289 1.38153i 0.131525 0.0759359i −0.432794 0.901493i \(-0.642472\pi\)
0.564319 + 0.825557i \(0.309139\pi\)
\(332\) 0 0
\(333\) −5.15752 2.97769i −0.282630 0.163177i
\(334\) 0 0
\(335\) 29.8890 1.63301
\(336\) 0 0
\(337\) 25.8259 1.40682 0.703412 0.710782i \(-0.251659\pi\)
0.703412 + 0.710782i \(0.251659\pi\)
\(338\) 0 0
\(339\) 12.9214 + 7.46015i 0.701792 + 0.405180i
\(340\) 0 0
\(341\) −25.0531 + 14.4644i −1.35670 + 0.783291i
\(342\) 0 0
\(343\) −17.4680 + 6.15385i −0.943182 + 0.332276i
\(344\) 0 0
\(345\) −7.49691 12.9850i −0.403620 0.699090i
\(346\) 0 0
\(347\) −10.6309 6.13775i −0.570696 0.329492i 0.186731 0.982411i \(-0.440211\pi\)
−0.757427 + 0.652919i \(0.773544\pi\)
\(348\) 0 0
\(349\) 0.209753i 0.0112278i 0.999984 + 0.00561390i \(0.00178697\pi\)
−0.999984 + 0.00561390i \(0.998213\pi\)
\(350\) 0 0
\(351\) −3.33413 −0.177963
\(352\) 0 0
\(353\) 9.79924 16.9728i 0.521561 0.903370i −0.478124 0.878292i \(-0.658683\pi\)
0.999685 0.0250782i \(-0.00798347\pi\)
\(354\) 0 0
\(355\) 12.2976 7.10004i 0.652690 0.376831i
\(356\) 0 0
\(357\) −0.0855865 0.754814i −0.00452972 0.0399490i
\(358\) 0 0
\(359\) −2.85556 4.94598i −0.150711 0.261039i 0.780778 0.624808i \(-0.214823\pi\)
−0.931489 + 0.363770i \(0.881490\pi\)
\(360\) 0 0
\(361\) −5.62118 + 9.73617i −0.295852 + 0.512430i
\(362\) 0 0
\(363\) 4.10622i 0.215521i
\(364\) 0 0
\(365\) 31.8487i 1.66704i
\(366\) 0 0
\(367\) 14.0638 24.3591i 0.734122 1.27154i −0.220985 0.975277i \(-0.570927\pi\)
0.955107 0.296260i \(-0.0957394\pi\)
\(368\) 0 0
\(369\) −1.75809 3.04509i −0.0915223 0.158521i
\(370\) 0 0
\(371\) −15.0174 + 11.0997i −0.779666 + 0.576270i
\(372\) 0 0
\(373\) −14.8987 + 8.60176i −0.771425 + 0.445382i −0.833383 0.552697i \(-0.813599\pi\)
0.0619580 + 0.998079i \(0.480266\pi\)
\(374\) 0 0
\(375\) 5.43211 9.40868i 0.280513 0.485862i
\(376\) 0 0
\(377\) −18.4502 −0.950234
\(378\) 0 0
\(379\) 10.1872i 0.523281i −0.965165 0.261640i \(-0.915737\pi\)
0.965165 0.261640i \(-0.0842635\pi\)
\(380\) 0 0
\(381\) −5.44961 3.14633i −0.279192 0.161192i
\(382\) 0 0
\(383\) −9.92164 17.1848i −0.506972 0.878102i −0.999967 0.00806963i \(-0.997431\pi\)
0.492995 0.870032i \(-0.335902\pi\)
\(384\) 0 0
\(385\) −9.42069 + 21.6341i −0.480123 + 1.10258i
\(386\) 0 0
\(387\) 9.73979 5.62327i 0.495101 0.285847i
\(388\) 0 0
\(389\) 16.0503 + 9.26666i 0.813784 + 0.469838i 0.848268 0.529567i \(-0.177646\pi\)
−0.0344843 + 0.999405i \(0.510979\pi\)
\(390\) 0 0
\(391\) −1.87612 −0.0948794
\(392\) 0 0
\(393\) 1.24096 0.0625982
\(394\) 0 0
\(395\) −17.8821 10.3242i −0.899746 0.519468i
\(396\) 0 0
\(397\) −6.91207 + 3.99068i −0.346907 + 0.200287i −0.663322 0.748334i \(-0.730854\pi\)
0.316415 + 0.948621i \(0.397521\pi\)
\(398\) 0 0
\(399\) 2.94207 6.75631i 0.147288 0.338238i
\(400\) 0 0
\(401\) −18.0210 31.2132i −0.899925 1.55872i −0.827589 0.561335i \(-0.810288\pi\)
−0.0723360 0.997380i \(-0.523045\pi\)
\(402\) 0 0
\(403\) −21.4914 12.4081i −1.07056 0.618091i
\(404\) 0 0
\(405\) 2.29465i 0.114022i
\(406\) 0 0
\(407\) 23.1466 1.14734
\(408\) 0 0
\(409\) 14.7833 25.6054i 0.730987 1.26611i −0.225474 0.974249i \(-0.572393\pi\)
0.956462 0.291858i \(-0.0942735\pi\)
\(410\) 0 0
\(411\) 1.37715 0.795096i 0.0679296 0.0392192i
\(412\) 0 0
\(413\) 9.25993 6.84424i 0.455652 0.336783i
\(414\) 0 0
\(415\) 20.2289 + 35.0374i 0.992996 + 1.71992i
\(416\) 0 0
\(417\) 0.514903 0.891838i 0.0252149 0.0436735i
\(418\) 0 0
\(419\) 8.73304i 0.426637i 0.976983 + 0.213318i \(0.0684271\pi\)
−0.976983 + 0.213318i \(0.931573\pi\)
\(420\) 0 0
\(421\) 1.77349i 0.0864344i 0.999066 + 0.0432172i \(0.0137607\pi\)
−0.999066 + 0.0432172i \(0.986239\pi\)
\(422\) 0 0
\(423\) −0.0435037 + 0.0753507i −0.00211522 + 0.00366368i
\(424\) 0 0
\(425\) 0.0381030 + 0.0659964i 0.00184827 + 0.00320130i
\(426\) 0 0
\(427\) −2.13642 18.8417i −0.103389 0.911816i
\(428\) 0 0
\(429\) 11.2225 6.47933i 0.541829 0.312825i
\(430\) 0 0
\(431\) 14.1836 24.5667i 0.683200 1.18334i −0.290799 0.956784i \(-0.593921\pi\)
0.973999 0.226553i \(-0.0727456\pi\)
\(432\) 0 0
\(433\) −6.53217 −0.313916 −0.156958 0.987605i \(-0.550169\pi\)
−0.156958 + 0.987605i \(0.550169\pi\)
\(434\) 0 0
\(435\) 12.6980i 0.608822i
\(436\) 0 0
\(437\) −15.7613 9.09977i −0.753964 0.435301i
\(438\) 0 0
\(439\) 0.266161 + 0.461004i 0.0127032 + 0.0220025i 0.872307 0.488958i \(-0.162623\pi\)
−0.859604 + 0.510961i \(0.829290\pi\)
\(440\) 0 0
\(441\) 2.05384 6.69191i 0.0978021 0.318663i
\(442\) 0 0
\(443\) −7.37827 + 4.25985i −0.350552 + 0.202392i −0.664929 0.746907i \(-0.731538\pi\)
0.314376 + 0.949299i \(0.398205\pi\)
\(444\) 0 0
\(445\) 34.1482 + 19.7155i 1.61878 + 0.934604i
\(446\) 0 0
\(447\) −9.59178 −0.453676
\(448\) 0 0
\(449\) −15.0180 −0.708745 −0.354373 0.935104i \(-0.615306\pi\)
−0.354373 + 0.935104i \(0.615306\pi\)
\(450\) 0 0
\(451\) 11.8353 + 6.83310i 0.557302 + 0.321758i
\(452\) 0 0
\(453\) −6.05544 + 3.49611i −0.284510 + 0.164262i
\(454\) 0 0
\(455\) −20.1129 + 2.28055i −0.942905 + 0.106914i
\(456\) 0 0
\(457\) 15.7276 + 27.2410i 0.735705 + 1.27428i 0.954413 + 0.298488i \(0.0964823\pi\)
−0.218708 + 0.975790i \(0.570184\pi\)
\(458\) 0 0
\(459\) 0.248654 + 0.143560i 0.0116062 + 0.00670083i
\(460\) 0 0
\(461\) 4.36281i 0.203196i −0.994826 0.101598i \(-0.967604\pi\)
0.994826 0.101598i \(-0.0323956\pi\)
\(462\) 0 0
\(463\) −13.9787 −0.649646 −0.324823 0.945775i \(-0.605305\pi\)
−0.324823 + 0.945775i \(0.605305\pi\)
\(464\) 0 0
\(465\) −8.53962 + 14.7911i −0.396016 + 0.685919i
\(466\) 0 0
\(467\) 14.3226 8.26916i 0.662771 0.382651i −0.130561 0.991440i \(-0.541678\pi\)
0.793332 + 0.608789i \(0.208345\pi\)
\(468\) 0 0
\(469\) 27.7138 20.4840i 1.27971 0.945862i
\(470\) 0 0
\(471\) −8.67895 15.0324i −0.399905 0.692655i
\(472\) 0 0
\(473\) −21.8558 + 37.8554i −1.00493 + 1.74059i
\(474\) 0 0
\(475\) 0.739247i 0.0339190i
\(476\) 0 0
\(477\) 7.05820i 0.323173i
\(478\) 0 0
\(479\) −0.592007 + 1.02539i −0.0270495 + 0.0468511i −0.879233 0.476391i \(-0.841945\pi\)
0.852184 + 0.523243i \(0.175278\pi\)
\(480\) 0 0
\(481\) 9.92801 + 17.1958i 0.452679 + 0.784062i
\(482\) 0 0
\(483\) −15.8504 6.90215i −0.721218 0.314059i
\(484\) 0 0
\(485\) 12.8480 7.41778i 0.583396 0.336824i
\(486\) 0 0
\(487\) −12.0448 + 20.8622i −0.545801 + 0.945356i 0.452755 + 0.891635i \(0.350441\pi\)
−0.998556 + 0.0537205i \(0.982892\pi\)
\(488\) 0 0
\(489\) 15.7220 0.710973
\(490\) 0 0
\(491\) 18.4971i 0.834761i 0.908732 + 0.417381i \(0.137052\pi\)
−0.908732 + 0.417381i \(0.862948\pi\)
\(492\) 0 0
\(493\) 1.37599 + 0.794427i 0.0619714 + 0.0357792i
\(494\) 0 0
\(495\) −4.45927 7.72369i −0.200429 0.347154i
\(496\) 0 0
\(497\) 6.53677 15.0113i 0.293214 0.673350i
\(498\) 0 0
\(499\) −10.5034 + 6.06414i −0.470197 + 0.271468i −0.716322 0.697770i \(-0.754176\pi\)
0.246125 + 0.969238i \(0.420843\pi\)
\(500\) 0 0
\(501\) −0.0883900 0.0510320i −0.00394897 0.00227994i
\(502\) 0 0
\(503\) 7.61078 0.339348 0.169674 0.985500i \(-0.445729\pi\)
0.169674 + 0.985500i \(0.445729\pi\)
\(504\) 0 0
\(505\) 27.1826 1.20961
\(506\) 0 0
\(507\) −1.63123 0.941791i −0.0724455 0.0418264i
\(508\) 0 0
\(509\) 13.9392 8.04779i 0.617844 0.356712i −0.158185 0.987409i \(-0.550564\pi\)
0.776029 + 0.630697i \(0.217231\pi\)
\(510\) 0 0
\(511\) −21.8270 29.5309i −0.965571 1.30637i
\(512\) 0 0
\(513\) 1.39263 + 2.41210i 0.0614860 + 0.106497i
\(514\) 0 0
\(515\) 3.22307 + 1.86084i 0.142025 + 0.0819985i
\(516\) 0 0
\(517\) 0.338169i 0.0148727i
\(518\) 0 0
\(519\) −2.28777 −0.100422
\(520\) 0 0
\(521\) 9.64668 16.7085i 0.422629 0.732015i −0.573567 0.819159i \(-0.694441\pi\)
0.996196 + 0.0871440i \(0.0277740\pi\)
\(522\) 0 0
\(523\) −27.9337 + 16.1275i −1.22146 + 0.705208i −0.965228 0.261409i \(-0.915813\pi\)
−0.256227 + 0.966617i \(0.582480\pi\)
\(524\) 0 0
\(525\) 0.0791162 + 0.697750i 0.00345291 + 0.0304523i
\(526\) 0 0
\(527\) 1.06853 + 1.85075i 0.0465459 + 0.0806199i
\(528\) 0 0
\(529\) −9.84821 + 17.0576i −0.428183 + 0.741635i
\(530\) 0 0
\(531\) 4.35217i 0.188868i
\(532\) 0 0
\(533\) 11.7234i 0.507796i
\(534\) 0 0
\(535\) −5.77019 + 9.99426i −0.249467 + 0.432090i
\(536\) 0 0
\(537\) −9.31154 16.1281i −0.401823 0.695978i
\(538\) 0 0
\(539\) 6.09149 + 26.5160i 0.262379 + 1.14213i
\(540\) 0 0
\(541\) −3.45438 + 1.99439i −0.148515 + 0.0857454i −0.572416 0.819963i \(-0.693994\pi\)
0.423901 + 0.905709i \(0.360660\pi\)
\(542\) 0 0
\(543\) 6.83930 11.8460i 0.293502 0.508361i
\(544\) 0 0
\(545\) −0.469072 −0.0200928
\(546\) 0 0
\(547\) 12.1824i 0.520882i 0.965490 + 0.260441i \(0.0838679\pi\)
−0.965490 + 0.260441i \(0.916132\pi\)
\(548\) 0 0
\(549\) 6.20693 + 3.58357i 0.264905 + 0.152943i
\(550\) 0 0
\(551\) 7.70644 + 13.3479i 0.328305 + 0.568642i
\(552\) 0 0
\(553\) −23.6563 + 2.68233i −1.00597 + 0.114064i
\(554\) 0 0
\(555\) 11.8347 6.83276i 0.502355 0.290035i
\(556\) 0 0
\(557\) 2.58791 + 1.49413i 0.109653 + 0.0633084i 0.553824 0.832634i \(-0.313168\pi\)
−0.444170 + 0.895942i \(0.646502\pi\)
\(558\) 0 0
\(559\) −37.4974 −1.58597
\(560\) 0 0
\(561\) −1.11594 −0.0471152
\(562\) 0 0
\(563\) 3.82170 + 2.20646i 0.161065 + 0.0929911i 0.578366 0.815777i \(-0.303691\pi\)
−0.417301 + 0.908768i \(0.637024\pi\)
\(564\) 0 0
\(565\) −29.6500 + 17.1184i −1.24738 + 0.720177i
\(566\) 0 0
\(567\) 1.57260 + 2.12766i 0.0660431 + 0.0893532i
\(568\) 0 0
\(569\) 6.15432 + 10.6596i 0.258002 + 0.446873i 0.965707 0.259635i \(-0.0836024\pi\)
−0.707704 + 0.706509i \(0.750269\pi\)
\(570\) 0 0
\(571\) −26.2766 15.1708i −1.09964 0.634877i −0.163513 0.986541i \(-0.552283\pi\)
−0.936126 + 0.351664i \(0.885616\pi\)
\(572\) 0 0
\(573\) 12.6346i 0.527818i
\(574\) 0 0
\(575\) 1.73429 0.0723247
\(576\) 0 0
\(577\) 6.02559 10.4366i 0.250848 0.434482i −0.712911 0.701254i \(-0.752624\pi\)
0.963760 + 0.266772i \(0.0859571\pi\)
\(578\) 0 0
\(579\) 2.25671 1.30291i 0.0937856 0.0541471i
\(580\) 0 0
\(581\) 42.7691 + 18.6241i 1.77436 + 0.772656i
\(582\) 0 0
\(583\) 13.7164 + 23.7576i 0.568077 + 0.983939i
\(584\) 0 0
\(585\) 3.82533 6.62566i 0.158158 0.273937i
\(586\) 0 0
\(587\) 5.34331i 0.220542i −0.993902 0.110271i \(-0.964828\pi\)
0.993902 0.110271i \(-0.0351719\pi\)
\(588\) 0 0
\(589\) 20.7309i 0.854200i
\(590\) 0 0
\(591\) −0.397388 + 0.688297i −0.0163464 + 0.0283127i
\(592\) 0 0
\(593\) −4.74530 8.21910i −0.194866 0.337518i 0.751991 0.659174i \(-0.229094\pi\)
−0.946857 + 0.321656i \(0.895761\pi\)
\(594\) 0 0
\(595\) 1.59818 + 0.695937i 0.0655190 + 0.0285306i
\(596\) 0 0
\(597\) −9.29616 + 5.36714i −0.380466 + 0.219662i
\(598\) 0 0
\(599\) −3.86976 + 6.70262i −0.158114 + 0.273861i −0.934189 0.356780i \(-0.883875\pi\)
0.776075 + 0.630641i \(0.217208\pi\)
\(600\) 0 0
\(601\) −19.0198 −0.775835 −0.387917 0.921694i \(-0.626805\pi\)
−0.387917 + 0.921694i \(0.626805\pi\)
\(602\) 0 0
\(603\) 13.0255i 0.530440i
\(604\) 0 0
\(605\) 8.15998 + 4.71117i 0.331751 + 0.191536i
\(606\) 0 0
\(607\) 4.59791 + 7.96381i 0.186623 + 0.323241i 0.944122 0.329595i \(-0.106912\pi\)
−0.757499 + 0.652836i \(0.773579\pi\)
\(608\) 0 0
\(609\) 8.70238 + 11.7739i 0.352638 + 0.477103i
\(610\) 0 0
\(611\) 0.251229 0.145047i 0.0101636 0.00586798i
\(612\) 0 0
\(613\) 22.8688 + 13.2033i 0.923663 + 0.533277i 0.884802 0.465967i \(-0.154294\pi\)
0.0388615 + 0.999245i \(0.487627\pi\)
\(614\) 0 0
\(615\) 8.06838 0.325348
\(616\) 0 0
\(617\) −45.5837 −1.83513 −0.917565 0.397585i \(-0.869848\pi\)
−0.917565 + 0.397585i \(0.869848\pi\)
\(618\) 0 0
\(619\) 42.4644 + 24.5168i 1.70679 + 0.985415i 0.938479 + 0.345337i \(0.112235\pi\)
0.768310 + 0.640078i \(0.221098\pi\)
\(620\) 0 0
\(621\) 5.65883 3.26713i 0.227081 0.131105i
\(622\) 0 0
\(623\) 45.1748 5.12226i 1.80989 0.205219i
\(624\) 0 0
\(625\) 13.1283 + 22.7389i 0.525133 + 0.909556i
\(626\) 0 0
\(627\) −9.37505 5.41269i −0.374403 0.216162i
\(628\) 0 0
\(629\) 1.70992i 0.0681788i
\(630\) 0 0
\(631\) 12.5801 0.500804 0.250402 0.968142i \(-0.419437\pi\)
0.250402 + 0.968142i \(0.419437\pi\)
\(632\) 0 0
\(633\) 0.278173 0.481809i 0.0110564 0.0191502i
\(634\) 0 0
\(635\) 12.5049 7.21973i 0.496244 0.286506i
\(636\) 0 0
\(637\) −17.0862 + 15.8986i −0.676980 + 0.629926i
\(638\) 0 0
\(639\) 3.09417 + 5.35926i 0.122404 + 0.212009i
\(640\) 0 0
\(641\) 17.1129 29.6404i 0.675920 1.17073i −0.300280 0.953851i \(-0.597080\pi\)
0.976199 0.216876i \(-0.0695867\pi\)
\(642\) 0 0
\(643\) 23.0442i 0.908776i 0.890804 + 0.454388i \(0.150142\pi\)
−0.890804 + 0.454388i \(0.849858\pi\)
\(644\) 0 0
\(645\) 25.8069i 1.01614i
\(646\) 0 0
\(647\) 7.79098 13.4944i 0.306295 0.530519i −0.671254 0.741228i \(-0.734244\pi\)
0.977549 + 0.210709i \(0.0675773\pi\)
\(648\) 0 0
\(649\) −8.45774 14.6492i −0.331995 0.575033i
\(650\) 0 0
\(651\) 2.21867 + 19.5671i 0.0869566 + 0.766897i
\(652\) 0 0
\(653\) −42.6318 + 24.6135i −1.66831 + 0.963199i −0.699761 + 0.714377i \(0.746710\pi\)
−0.968549 + 0.248822i \(0.919957\pi\)
\(654\) 0 0
\(655\) −1.42378 + 2.46607i −0.0556319 + 0.0963572i
\(656\) 0 0
\(657\) 13.8796 0.541493
\(658\) 0 0
\(659\) 5.33204i 0.207707i 0.994593 + 0.103853i \(0.0331173\pi\)
−0.994593 + 0.103853i \(0.966883\pi\)
\(660\) 0 0
\(661\) −33.5057 19.3445i −1.30322 0.752416i −0.322267 0.946649i \(-0.604445\pi\)
−0.980955 + 0.194233i \(0.937778\pi\)
\(662\) 0 0
\(663\) −0.478649 0.829044i −0.0185892 0.0321974i
\(664\) 0 0
\(665\) 10.0508 + 13.5982i 0.389753 + 0.527317i
\(666\) 0 0
\(667\) 31.3145 18.0794i 1.21250 0.700038i
\(668\) 0 0
\(669\) 9.61947 + 5.55380i 0.371910 + 0.214722i
\(670\) 0 0
\(671\) −27.8563 −1.07538
\(672\) 0 0
\(673\) −9.56678 −0.368772 −0.184386 0.982854i \(-0.559030\pi\)
−0.184386 + 0.982854i \(0.559030\pi\)
\(674\) 0 0
\(675\) −0.229856 0.132707i −0.00884715 0.00510791i
\(676\) 0 0
\(677\) 15.3536 8.86439i 0.590086 0.340686i −0.175046 0.984560i \(-0.556007\pi\)
0.765131 + 0.643874i \(0.222674\pi\)
\(678\) 0 0
\(679\) 6.82930 15.6831i 0.262084 0.601862i
\(680\) 0 0
\(681\) −4.38394 7.59321i −0.167993 0.290972i
\(682\) 0 0
\(683\) 7.59766 + 4.38651i 0.290716 + 0.167845i 0.638265 0.769817i \(-0.279652\pi\)
−0.347549 + 0.937662i \(0.612986\pi\)
\(684\) 0 0
\(685\) 3.64893i 0.139418i
\(686\) 0 0
\(687\) −20.6962 −0.789610
\(688\) 0 0
\(689\) −11.7665 + 20.3801i −0.448267 + 0.776421i
\(690\) 0 0
\(691\) 18.5176 10.6911i 0.704443 0.406710i −0.104557 0.994519i \(-0.533343\pi\)
0.809000 + 0.587809i \(0.200009\pi\)
\(692\) 0 0
\(693\) −9.42806 4.10551i −0.358143 0.155955i
\(694\) 0 0
\(695\) 1.18152 + 2.04646i 0.0448177 + 0.0776265i
\(696\) 0 0
\(697\) 0.504783 0.874310i 0.0191200 0.0331169i
\(698\) 0 0
\(699\) 5.72853i 0.216673i
\(700\) 0 0
\(701\) 13.4233i 0.506990i −0.967337 0.253495i \(-0.918420\pi\)
0.967337 0.253495i \(-0.0815802\pi\)
\(702\) 0 0
\(703\) 8.29363 14.3650i 0.312800 0.541786i
\(704\) 0 0
\(705\) −0.0998258 0.172903i −0.00375966 0.00651192i
\(706\) 0 0
\(707\) 25.2044 18.6292i 0.947910 0.700623i
\(708\) 0 0
\(709\) 43.8660 25.3260i 1.64742 0.951139i 0.669330 0.742965i \(-0.266581\pi\)
0.978092 0.208174i \(-0.0667520\pi\)
\(710\) 0 0
\(711\) 4.49926 7.79295i 0.168736 0.292259i
\(712\) 0 0
\(713\) 48.6349 1.82139
\(714\) 0 0
\(715\) 29.7356i 1.11205i
\(716\) 0 0
\(717\) 1.97152 + 1.13826i 0.0736276 + 0.0425089i
\(718\) 0 0
\(719\) 14.4800 + 25.0801i 0.540013 + 0.935330i 0.998903 + 0.0468366i \(0.0149140\pi\)
−0.458890 + 0.888493i \(0.651753\pi\)
\(720\) 0 0
\(721\) 4.26381 0.483463i 0.158793 0.0180051i
\(722\) 0 0
\(723\) 20.4252 11.7925i 0.759623 0.438568i
\(724\) 0 0
\(725\) −1.27196 0.734368i −0.0472395 0.0272738i
\(726\) 0 0
\(727\) 10.5092 0.389766 0.194883 0.980827i \(-0.437567\pi\)
0.194883 + 0.980827i \(0.437567\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 2.79650 + 1.61456i 0.103432 + 0.0597166i
\(732\) 0 0
\(733\) −24.3360 + 14.0504i −0.898871 + 0.518963i −0.876834 0.480794i \(-0.840349\pi\)
−0.0220370 + 0.999757i \(0.507015\pi\)
\(734\) 0 0
\(735\) 10.9419 + 11.7592i 0.403598 + 0.433746i
\(736\) 0 0
\(737\) −25.3130 43.8434i −0.932415 1.61499i
\(738\) 0 0
\(739\) 30.1856 + 17.4276i 1.11039 + 0.641086i 0.938931 0.344104i \(-0.111817\pi\)
0.171463 + 0.985191i \(0.445151\pi\)
\(740\) 0 0
\(741\) 9.28640i 0.341144i
\(742\) 0 0
\(743\) −38.4805 −1.41171 −0.705857 0.708354i \(-0.749438\pi\)
−0.705857 + 0.708354i \(0.749438\pi\)
\(744\) 0 0
\(745\) 11.0049 19.0610i 0.403188 0.698342i
\(746\) 0 0
\(747\) −15.2692 + 8.81567i −0.558671 + 0.322549i
\(748\) 0 0
\(749\) 1.49915 + 13.2214i 0.0547777 + 0.483101i
\(750\) 0 0
\(751\) 0.0112661 + 0.0195135i 0.000411106 + 0.000712057i 0.866231 0.499644i \(-0.166536\pi\)
−0.865820 + 0.500356i \(0.833202\pi\)
\(752\) 0 0
\(753\) −6.21612 + 10.7666i −0.226528 + 0.392358i
\(754\) 0 0
\(755\) 16.0447i 0.583926i
\(756\) 0 0
\(757\) 30.0479i 1.09211i 0.837749 + 0.546055i \(0.183871\pi\)
−0.837749 + 0.546055i \(0.816129\pi\)
\(758\) 0 0
\(759\) −12.6982 + 21.9940i −0.460917 + 0.798332i
\(760\) 0 0
\(761\) 8.58063 + 14.8621i 0.311048 + 0.538751i 0.978589 0.205822i \(-0.0659867\pi\)
−0.667542 + 0.744572i \(0.732653\pi\)
\(762\) 0 0
\(763\) −0.434935 + 0.321471i −0.0157457 + 0.0116380i
\(764\) 0 0
\(765\) −0.570574 + 0.329421i −0.0206291 + 0.0119102i
\(766\) 0 0
\(767\) 7.25536 12.5666i 0.261976 0.453755i
\(768\) 0 0
\(769\) −47.2945 −1.70548 −0.852742 0.522332i \(-0.825062\pi\)
−0.852742 + 0.522332i \(0.825062\pi\)
\(770\) 0 0
\(771\) 0.282984i 0.0101914i
\(772\) 0 0
\(773\) −37.0445 21.3877i −1.33240 0.769260i −0.346731 0.937964i \(-0.612708\pi\)
−0.985667 + 0.168704i \(0.946042\pi\)
\(774\) 0 0
\(775\) −0.987751 1.71083i −0.0354810 0.0614550i
\(776\) 0 0
\(777\) 6.29070 14.4462i 0.225677 0.518256i
\(778\) 0 0
\(779\) 8.48136 4.89672i 0.303876 0.175443i
\(780\) 0 0
\(781\) −20.8297 12.0260i −0.745345 0.430325i
\(782\) 0 0
\(783\) −5.53374 −0.197760
\(784\) 0 0
\(785\) 39.8303 1.42160
\(786\) 0 0
\(787\) −42.8358 24.7312i −1.52693 0.881574i −0.999488 0.0319826i \(-0.989818\pi\)
−0.527442 0.849591i \(-0.676849\pi\)
\(788\) 0 0
\(789\) −12.4008 + 7.15962i −0.441481 + 0.254889i
\(790\) 0 0
\(791\) −15.7604 + 36.1928i −0.560374 + 1.28687i
\(792\) 0 0
\(793\) −11.9481 20.6947i −0.424289 0.734890i
\(794\) 0 0
\(795\) 14.0262 + 8.09804i 0.497459 + 0.287208i
\(796\) 0 0
\(797\) 13.3755i 0.473783i 0.971536 + 0.236892i \(0.0761286\pi\)
−0.971536 + 0.236892i \(0.923871\pi\)
\(798\) 0 0
\(799\) −0.0249817 −0.000883788
\(800\) 0 0
\(801\) −8.59194 + 14.8817i −0.303581 + 0.525818i
\(802\) 0 0
\(803\) −46.7180 + 26.9727i −1.64864 + 0.951845i
\(804\) 0 0
\(805\) 31.9017 23.5793i 1.12439 0.831061i
\(806\) 0 0
\(807\) −8.40704 14.5614i −0.295942 0.512586i
\(808\) 0 0
\(809\) −7.11266 + 12.3195i −0.250068 + 0.433130i −0.963544 0.267549i \(-0.913786\pi\)
0.713476 + 0.700679i \(0.247120\pi\)
\(810\) 0 0
\(811\) 20.7849i 0.729855i 0.931036 + 0.364928i \(0.118906\pi\)
−0.931036 + 0.364928i \(0.881094\pi\)
\(812\) 0 0
\(813\) 26.8456i 0.941518i
\(814\) 0 0
\(815\) −18.0382 + 31.2431i −0.631851 + 1.09440i
\(816\) 0 0
\(817\) 15.6622 + 27.1278i 0.547952 + 0.949081i
\(818\) 0 0
\(819\) −0.993855 8.76511i −0.0347281 0.306278i
\(820\) 0 0
\(821\) −4.29903 + 2.48205i −0.150037 + 0.0866241i −0.573139 0.819458i \(-0.694275\pi\)
0.423102 + 0.906082i \(0.360941\pi\)
\(822\) 0 0
\(823\) 4.91749 8.51734i 0.171413 0.296896i −0.767501 0.641048i \(-0.778500\pi\)
0.938914 + 0.344152i \(0.111833\pi\)
\(824\) 0 0
\(825\) 1.03158 0.0359150
\(826\) 0 0
\(827\) 41.4331i 1.44077i 0.693574 + 0.720385i \(0.256035\pi\)
−0.693574 + 0.720385i \(0.743965\pi\)
\(828\) 0 0
\(829\) 21.6531 + 12.5014i 0.752044 + 0.434193i 0.826432 0.563036i \(-0.190367\pi\)
−0.0743878 + 0.997229i \(0.523700\pi\)
\(830\) 0 0
\(831\) 11.2515 + 19.4881i 0.390310 + 0.676036i
\(832\) 0 0
\(833\) 1.95882 0.449998i 0.0678692 0.0155915i
\(834\) 0 0
\(835\) 0.202824 0.117100i 0.00701901 0.00405243i
\(836\) 0 0
\(837\) −6.44589 3.72154i −0.222803 0.128635i
\(838\) 0 0
\(839\) −2.43551 −0.0840833 −0.0420416 0.999116i \(-0.513386\pi\)
−0.0420416 + 0.999116i \(0.513386\pi\)
\(840\) 0 0
\(841\) −1.62232 −0.0559420
\(842\) 0 0
\(843\) −11.1176 6.41877i −0.382912 0.221074i
\(844\) 0 0
\(845\) 3.74310 2.16108i 0.128767 0.0743434i
\(846\) 0 0
\(847\) 10.7949 1.22400i 0.370916 0.0420573i
\(848\) 0 0
\(849\) 3.64895 + 6.32017i 0.125232 + 0.216908i
\(850\) 0 0
\(851\) −33.7005 19.4570i −1.15524 0.666977i
\(852\) 0 0
\(853\) 51.1404i 1.75101i −0.483206 0.875507i \(-0.660528\pi\)
0.483206 0.875507i \(-0.339472\pi\)
\(854\) 0 0
\(855\) −6.39118 −0.218574
\(856\) 0 0
\(857\) −26.7314 + 46.3001i −0.913127 + 1.58158i −0.103506 + 0.994629i \(0.533006\pi\)
−0.809621 + 0.586954i \(0.800327\pi\)
\(858\) 0 0
\(859\) 14.3724 8.29790i 0.490380 0.283121i −0.234352 0.972152i \(-0.575297\pi\)
0.724732 + 0.689031i \(0.241964\pi\)
\(860\) 0 0
\(861\) 7.48121 5.52954i 0.254959 0.188446i
\(862\) 0 0
\(863\) −20.4256 35.3782i −0.695296 1.20429i −0.970081 0.242782i \(-0.921940\pi\)
0.274785 0.961506i \(-0.411393\pi\)
\(864\) 0 0
\(865\) 2.62481 4.54631i 0.0892463 0.154579i
\(866\) 0 0
\(867\) 16.9176i 0.574551i
\(868\) 0 0
\(869\) 34.9743i 1.18642i
\(870\) 0 0
\(871\) 21.7144 37.6104i 0.735764 1.27438i
\(872\) 0 0
\(873\) 3.23264 + 5.59910i 0.109408 + 0.189501i
\(874\) 0 0
\(875\) 26.3538 + 11.4759i 0.890920 + 0.387956i
\(876\) 0 0
\(877\) −1.31595 + 0.759763i −0.0444364 + 0.0256554i −0.522054 0.852913i \(-0.674834\pi\)
0.477617 + 0.878568i \(0.341501\pi\)
\(878\) 0 0
\(879\) −4.97837 + 8.62279i −0.167916 + 0.290840i
\(880\) 0 0
\(881\) 12.3299 0.415406 0.207703 0.978192i \(-0.433401\pi\)
0.207703 + 0.978192i \(0.433401\pi\)
\(882\) 0 0
\(883\) 12.0145i 0.404319i −0.979353 0.202159i \(-0.935204\pi\)
0.979353 0.202159i \(-0.0647959\pi\)
\(884\) 0 0
\(885\) −8.64875 4.99336i −0.290724 0.167850i
\(886\) 0 0
\(887\) −11.5240 19.9601i −0.386938 0.670196i 0.605098 0.796151i \(-0.293134\pi\)
−0.992036 + 0.125955i \(0.959800\pi\)
\(888\) 0 0
\(889\) 6.64697 15.2644i 0.222932 0.511951i
\(890\) 0 0
\(891\) 3.36596 1.94334i 0.112764 0.0651042i
\(892\) 0 0
\(893\) −0.209871 0.121169i −0.00702306 0.00405476i
\(894\) 0 0
\(895\) 42.7335 1.42842
\(896\) 0 0
\(897\) −21.7860 −0.727415
\(898\) 0 0
\(899\) −35.6699 20.5940i −1.18966 0.686850i
\(900\) 0 0
\(901\) 1.75505 1.01328i 0.0584692 0.0337572i
\(902\) 0 0
\(903\) 17.6863 + 23.9288i 0.588564 + 0.796300i
\(904\) 0 0
\(905\) 15.6938 + 27.1824i 0.521679 + 0.903575i
\(906\) 0 0
\(907\) −28.3744 16.3820i −0.942156 0.543954i −0.0515204 0.998672i \(-0.516407\pi\)
−0.890635 + 0.454718i \(0.849740\pi\)
\(908\) 0 0
\(909\) 11.8461i 0.392910i
\(910\) 0 0
\(911\) −35.5607 −1.17818 −0.589090 0.808067i \(-0.700514\pi\)
−0.589090 + 0.808067i \(0.700514\pi\)
\(912\) 0 0
\(913\) 34.2636 59.3464i 1.13396 1.96408i
\(914\) 0 0
\(915\) −14.2427 + 8.22304i −0.470850 + 0.271845i
\(916\) 0 0
\(917\) 0.369912 + 3.26237i 0.0122156 + 0.107733i
\(918\) 0 0
\(919\) −13.7621 23.8366i −0.453969 0.786297i 0.544660 0.838657i \(-0.316659\pi\)
−0.998628 + 0.0523606i \(0.983325\pi\)
\(920\) 0 0
\(921\) −7.47395 + 12.9453i −0.246275 + 0.426561i
\(922\) 0 0
\(923\) 20.6327i 0.679135i
\(924\) 0 0
\(925\) 1.58065i 0.0519713i
\(926\) 0 0
\(927\) −0.810948 + 1.40460i −0.0266350 + 0.0461332i
\(928\) 0 0
\(929\) −17.4740 30.2659i −0.573303 0.992991i −0.996224 0.0868237i \(-0.972328\pi\)
0.422920 0.906167i \(-0.361005\pi\)
\(930\) 0 0
\(931\) 18.6387 + 5.72048i 0.610858 + 0.187481i
\(932\) 0 0
\(933\) 27.0712 15.6295i 0.886270 0.511688i
\(934\) 0 0
\(935\) 1.28035 2.21763i 0.0418719 0.0725243i
\(936\) 0 0
\(937\) 17.7423 0.579615 0.289807 0.957085i \(-0.406409\pi\)
0.289807 + 0.957085i \(0.406409\pi\)
\(938\) 0 0
\(939\) 3.06468i 0.100012i
\(940\) 0 0
\(941\) −19.5486 11.2864i −0.637267 0.367926i 0.146294 0.989241i \(-0.453265\pi\)
−0.783561 + 0.621315i \(0.786599\pi\)
\(942\) 0 0
\(943\) −11.4878 19.8974i −0.374093 0.647949i
\(944\) 0 0
\(945\) −6.03242 + 0.684001i −0.196235 + 0.0222506i
\(946\) 0 0
\(947\) 23.6178 13.6357i 0.767474 0.443102i −0.0644985 0.997918i \(-0.520545\pi\)
0.831973 + 0.554816i \(0.187211\pi\)
\(948\) 0 0
\(949\) −40.0764 23.1381i −1.30094 0.751096i
\(950\) 0 0
\(951\) 16.2981 0.528501
\(952\) 0 0
\(953\) −5.90986 −0.191439 −0.0957196 0.995408i \(-0.530515\pi\)
−0.0957196 + 0.995408i \(0.530515\pi\)
\(954\) 0 0
\(955\) −25.1078 14.4960i −0.812468 0.469079i
\(956\) 0 0
\(957\) 18.6263 10.7539i 0.602104 0.347625i
\(958\) 0 0
\(959\) 2.50074 + 3.38338i 0.0807531 + 0.109255i
\(960\) 0 0
\(961\) −12.1997 21.1305i −0.393538 0.681628i
\(962\) 0 0
\(963\) −4.35546 2.51463i −0.140353 0.0810328i
\(964\) 0 0
\(965\) 5.97945i 0.192485i
\(966\) 0 0
\(967\) −14.0032 −0.450312 −0.225156 0.974323i \(-0.572289\pi\)
−0.225156 + 0.974323i \(0.572289\pi\)
\(968\) 0 0
\(969\) −0.399852 + 0.692565i −0.0128451 + 0.0222484i
\(970\) 0 0
\(971\) −7.91244 + 4.56825i −0.253922 + 0.146602i −0.621559 0.783367i \(-0.713500\pi\)
0.367637 + 0.929970i \(0.380167\pi\)
\(972\) 0 0
\(973\) 2.49804 + 1.08779i 0.0800836 + 0.0348729i
\(974\) 0 0
\(975\) 0.442463 + 0.766369i 0.0141702 + 0.0245434i
\(976\) 0 0
\(977\) −10.5767 + 18.3194i −0.338379 + 0.586090i −0.984128 0.177460i \(-0.943212\pi\)
0.645749 + 0.763550i \(0.276545\pi\)
\(978\) 0 0
\(979\) 66.7881i 2.13456i
\(980\) 0 0
\(981\) 0.204420i 0.00652662i
\(982\) 0 0
\(983\) 23.2004 40.1843i 0.739979 1.28168i −0.212526 0.977156i \(-0.568169\pi\)
0.952504 0.304525i \(-0.0984978\pi\)
\(984\) 0 0
\(985\) −0.911867 1.57940i −0.0290545 0.0503239i
\(986\) 0 0
\(987\) −0.211058 0.0919063i −0.00671804 0.00292541i
\(988\) 0 0
\(989\) 63.6422 36.7439i 2.02370 1.16839i
\(990\) 0 0
\(991\) 1.29280 2.23920i 0.0410672 0.0711305i −0.844761 0.535143i \(-0.820258\pi\)
0.885828 + 0.464013i \(0.153591\pi\)
\(992\) 0 0
\(993\) −2.76307 −0.0876833
\(994\) 0 0
\(995\) 24.6314i 0.780868i
\(996\) 0 0
\(997\) 17.9557 + 10.3667i 0.568662 + 0.328317i 0.756615 0.653861i \(-0.226852\pi\)
−0.187953 + 0.982178i \(0.560185\pi\)
\(998\) 0 0
\(999\) 2.97769 + 5.15752i 0.0942101 + 0.163177i
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 672.2.bk.a.529.7 32
3.2 odd 2 2016.2.cr.e.1873.4 32
4.3 odd 2 168.2.bc.a.109.5 yes 32
7.2 even 3 inner 672.2.bk.a.625.10 32
7.3 odd 6 4704.2.c.f.2353.2 16
7.4 even 3 4704.2.c.e.2353.15 16
8.3 odd 2 168.2.bc.a.109.6 yes 32
8.5 even 2 inner 672.2.bk.a.529.10 32
12.11 even 2 504.2.cj.e.109.12 32
21.2 odd 6 2016.2.cr.e.1297.13 32
24.5 odd 2 2016.2.cr.e.1873.13 32
24.11 even 2 504.2.cj.e.109.11 32
28.3 even 6 1176.2.c.f.589.15 16
28.11 odd 6 1176.2.c.e.589.15 16
28.23 odd 6 168.2.bc.a.37.6 yes 32
56.3 even 6 1176.2.c.f.589.16 16
56.11 odd 6 1176.2.c.e.589.16 16
56.37 even 6 inner 672.2.bk.a.625.7 32
56.45 odd 6 4704.2.c.f.2353.15 16
56.51 odd 6 168.2.bc.a.37.5 32
56.53 even 6 4704.2.c.e.2353.2 16
84.23 even 6 504.2.cj.e.37.11 32
168.107 even 6 504.2.cj.e.37.12 32
168.149 odd 6 2016.2.cr.e.1297.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.bc.a.37.5 32 56.51 odd 6
168.2.bc.a.37.6 yes 32 28.23 odd 6
168.2.bc.a.109.5 yes 32 4.3 odd 2
168.2.bc.a.109.6 yes 32 8.3 odd 2
504.2.cj.e.37.11 32 84.23 even 6
504.2.cj.e.37.12 32 168.107 even 6
504.2.cj.e.109.11 32 24.11 even 2
504.2.cj.e.109.12 32 12.11 even 2
672.2.bk.a.529.7 32 1.1 even 1 trivial
672.2.bk.a.529.10 32 8.5 even 2 inner
672.2.bk.a.625.7 32 56.37 even 6 inner
672.2.bk.a.625.10 32 7.2 even 3 inner
1176.2.c.e.589.15 16 28.11 odd 6
1176.2.c.e.589.16 16 56.11 odd 6
1176.2.c.f.589.15 16 28.3 even 6
1176.2.c.f.589.16 16 56.3 even 6
2016.2.cr.e.1297.4 32 168.149 odd 6
2016.2.cr.e.1297.13 32 21.2 odd 6
2016.2.cr.e.1873.4 32 3.2 odd 2
2016.2.cr.e.1873.13 32 24.5 odd 2
4704.2.c.e.2353.2 16 56.53 even 6
4704.2.c.e.2353.15 16 7.4 even 3
4704.2.c.f.2353.2 16 7.3 odd 6
4704.2.c.f.2353.15 16 56.45 odd 6