Properties

Label 1764.2.x.a.293.3
Level $1764$
Weight $2$
Character 1764.293
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
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] = [1764,2,Mod(293,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.x (of order \(6\), degree \(2\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} - 156 x^{7} + 558 x^{6} - 837 x^{5} + 1782 x^{4} - 4131 x^{3} + 3645 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 293.3
Root \(-0.544978 + 1.64408i\) of defining polynomial
Character \(\chi\) \(=\) 1764.293
Dual form 1764.2.x.a.1469.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.09400 + 1.34282i) q^{3} +(-1.95741 + 3.39033i) q^{5} +(-0.606348 - 2.93808i) q^{9} +O(q^{10})\) \(q+(-1.09400 + 1.34282i) q^{3} +(-1.95741 + 3.39033i) q^{5} +(-0.606348 - 2.93808i) q^{9} +(3.19958 - 1.84728i) q^{11} +(0.480242 + 0.277268i) q^{13} +(-2.41122 - 6.33747i) q^{15} -5.83832 q^{17} -5.33973i q^{19} +(1.96965 + 1.13718i) q^{23} +(-5.16291 - 8.94242i) q^{25} +(4.60867 + 2.40003i) q^{27} +(3.53638 - 2.04173i) q^{29} +(-7.00132 - 4.04222i) q^{31} +(-1.01976 + 6.31739i) q^{33} -7.79699 q^{37} +(-0.897704 + 0.341550i) q^{39} +(3.59234 - 6.22212i) q^{41} +(-0.754009 - 1.30598i) q^{43} +(11.1480 + 3.69531i) q^{45} +(1.41416 + 2.44940i) q^{47} +(6.38709 - 7.83983i) q^{51} -0.0479960i q^{53} +14.4635i q^{55} +(7.17031 + 5.84164i) q^{57} +(4.45656 - 7.71900i) q^{59} +(6.03343 - 3.48340i) q^{61} +(-1.88006 + 1.08545i) q^{65} +(-0.587402 + 1.01741i) q^{67} +(-3.68181 + 1.40082i) q^{69} +6.71061i q^{71} -4.07253i q^{73} +(17.6563 + 2.85009i) q^{75} +(1.97374 + 3.41861i) q^{79} +(-8.26468 + 3.56300i) q^{81} +(3.84674 + 6.66275i) q^{83} +(11.4280 - 19.7938i) q^{85} +(-1.12710 + 6.98238i) q^{87} +5.42600 q^{89} +(13.0874 - 4.97937i) q^{93} +(18.1035 + 10.4520i) q^{95} +(13.9874 - 8.07563i) q^{97} +(-7.36753 - 8.28055i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{9} + 6 q^{11} - 3 q^{13} - 3 q^{15} - 18 q^{17} - 21 q^{23} - 8 q^{25} + 9 q^{27} + 6 q^{29} + 6 q^{31} - 27 q^{33} - 2 q^{37} + 6 q^{39} - 6 q^{41} - 2 q^{43} + 15 q^{45} + 18 q^{47} + 18 q^{51} + 15 q^{57} + 15 q^{59} + 3 q^{61} + 39 q^{65} - 7 q^{67} + 21 q^{69} + 42 q^{75} - q^{79} - 18 q^{81} + 6 q^{85} - 51 q^{87} - 42 q^{89} + 48 q^{93} - 6 q^{95} - 3 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\)

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
\(3\) −1.09400 + 1.34282i −0.631619 + 0.775279i
\(4\) 0 0
\(5\) −1.95741 + 3.39033i −0.875381 + 1.51620i −0.0190238 + 0.999819i \(0.506056\pi\)
−0.856357 + 0.516385i \(0.827278\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.606348 2.93808i −0.202116 0.979362i
\(10\) 0 0
\(11\) 3.19958 1.84728i 0.964710 0.556976i 0.0670908 0.997747i \(-0.478628\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(12\) 0 0
\(13\) 0.480242 + 0.277268i 0.133195 + 0.0769002i 0.565117 0.825011i \(-0.308831\pi\)
−0.431922 + 0.901911i \(0.642164\pi\)
\(14\) 0 0
\(15\) −2.41122 6.33747i −0.622575 1.63633i
\(16\) 0 0
\(17\) −5.83832 −1.41600 −0.708000 0.706213i \(-0.750402\pi\)
−0.708000 + 0.706213i \(0.750402\pi\)
\(18\) 0 0
\(19\) 5.33973i 1.22502i −0.790464 0.612509i \(-0.790160\pi\)
0.790464 0.612509i \(-0.209840\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.96965 + 1.13718i 0.410700 + 0.237118i 0.691090 0.722768i \(-0.257131\pi\)
−0.280390 + 0.959886i \(0.590464\pi\)
\(24\) 0 0
\(25\) −5.16291 8.94242i −1.03258 1.78848i
\(26\) 0 0
\(27\) 4.60867 + 2.40003i 0.886939 + 0.461887i
\(28\) 0 0
\(29\) 3.53638 2.04173i 0.656690 0.379140i −0.134325 0.990937i \(-0.542887\pi\)
0.791014 + 0.611797i \(0.209553\pi\)
\(30\) 0 0
\(31\) −7.00132 4.04222i −1.25748 0.726004i −0.284892 0.958560i \(-0.591958\pi\)
−0.972583 + 0.232556i \(0.925291\pi\)
\(32\) 0 0
\(33\) −1.01976 + 6.31739i −0.177517 + 1.09972i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.79699 −1.28182 −0.640909 0.767617i \(-0.721442\pi\)
−0.640909 + 0.767617i \(0.721442\pi\)
\(38\) 0 0
\(39\) −0.897704 + 0.341550i −0.143748 + 0.0546918i
\(40\) 0 0
\(41\) 3.59234 6.22212i 0.561030 0.971732i −0.436377 0.899764i \(-0.643739\pi\)
0.997407 0.0719684i \(-0.0229281\pi\)
\(42\) 0 0
\(43\) −0.754009 1.30598i −0.114985 0.199160i 0.802789 0.596264i \(-0.203349\pi\)
−0.917774 + 0.397103i \(0.870015\pi\)
\(44\) 0 0
\(45\) 11.1480 + 3.69531i 1.66184 + 0.550865i
\(46\) 0 0
\(47\) 1.41416 + 2.44940i 0.206277 + 0.357282i 0.950539 0.310606i \(-0.100532\pi\)
−0.744262 + 0.667888i \(0.767199\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 6.38709 7.83983i 0.894372 1.09780i
\(52\) 0 0
\(53\) 0.0479960i 0.00659276i −0.999995 0.00329638i \(-0.998951\pi\)
0.999995 0.00329638i \(-0.00104927\pi\)
\(54\) 0 0
\(55\) 14.4635i 1.95026i
\(56\) 0 0
\(57\) 7.17031 + 5.84164i 0.949731 + 0.773744i
\(58\) 0 0
\(59\) 4.45656 7.71900i 0.580195 1.00493i −0.415261 0.909703i \(-0.636310\pi\)
0.995456 0.0952251i \(-0.0303571\pi\)
\(60\) 0 0
\(61\) 6.03343 3.48340i 0.772501 0.446004i −0.0612648 0.998122i \(-0.519513\pi\)
0.833766 + 0.552118i \(0.186180\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.88006 + 1.08545i −0.233193 + 0.134634i
\(66\) 0 0
\(67\) −0.587402 + 1.01741i −0.0717626 + 0.124296i −0.899674 0.436563i \(-0.856196\pi\)
0.827911 + 0.560859i \(0.189529\pi\)
\(68\) 0 0
\(69\) −3.68181 + 1.40082i −0.443238 + 0.168639i
\(70\) 0 0
\(71\) 6.71061i 0.796403i 0.917298 + 0.398202i \(0.130366\pi\)
−0.917298 + 0.398202i \(0.869634\pi\)
\(72\) 0 0
\(73\) 4.07253i 0.476654i −0.971185 0.238327i \(-0.923401\pi\)
0.971185 0.238327i \(-0.0765990\pi\)
\(74\) 0 0
\(75\) 17.6563 + 2.85009i 2.03877 + 0.329101i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.97374 + 3.41861i 0.222063 + 0.384624i 0.955434 0.295204i \(-0.0953877\pi\)
−0.733371 + 0.679828i \(0.762054\pi\)
\(80\) 0 0
\(81\) −8.26468 + 3.56300i −0.918298 + 0.395889i
\(82\) 0 0
\(83\) 3.84674 + 6.66275i 0.422235 + 0.731332i 0.996158 0.0875774i \(-0.0279125\pi\)
−0.573923 + 0.818909i \(0.694579\pi\)
\(84\) 0 0
\(85\) 11.4280 19.7938i 1.23954 2.14694i
\(86\) 0 0
\(87\) −1.12710 + 6.98238i −0.120838 + 0.748590i
\(88\) 0 0
\(89\) 5.42600 0.575155 0.287577 0.957757i \(-0.407150\pi\)
0.287577 + 0.957757i \(0.407150\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 13.0874 4.97937i 1.35710 0.516337i
\(94\) 0 0
\(95\) 18.1035 + 10.4520i 1.85738 + 1.07236i
\(96\) 0 0
\(97\) 13.9874 8.07563i 1.42021 0.819956i 0.423890 0.905714i \(-0.360664\pi\)
0.996316 + 0.0857571i \(0.0273309\pi\)
\(98\) 0 0
\(99\) −7.36753 8.28055i −0.740464 0.832227i
\(100\) 0 0
\(101\) 0.811750 + 1.40599i 0.0807722 + 0.139901i 0.903582 0.428416i \(-0.140928\pi\)
−0.822810 + 0.568317i \(0.807595\pi\)
\(102\) 0 0
\(103\) 0.342653 + 0.197831i 0.0337626 + 0.0194929i 0.516786 0.856114i \(-0.327128\pi\)
−0.483024 + 0.875607i \(0.660461\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.66700i 0.547850i 0.961751 + 0.273925i \(0.0883219\pi\)
−0.961751 + 0.273925i \(0.911678\pi\)
\(108\) 0 0
\(109\) 13.5133 1.29434 0.647171 0.762345i \(-0.275952\pi\)
0.647171 + 0.762345i \(0.275952\pi\)
\(110\) 0 0
\(111\) 8.52987 10.4700i 0.809619 0.993766i
\(112\) 0 0
\(113\) 1.13651 + 0.656162i 0.106913 + 0.0617265i 0.552503 0.833511i \(-0.313673\pi\)
−0.445590 + 0.895237i \(0.647006\pi\)
\(114\) 0 0
\(115\) −7.71082 + 4.45184i −0.719038 + 0.415137i
\(116\) 0 0
\(117\) 0.523442 1.57911i 0.0483922 0.145989i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.32489 2.29477i 0.120444 0.208615i
\(122\) 0 0
\(123\) 4.42520 + 11.6309i 0.399007 + 1.04872i
\(124\) 0 0
\(125\) 20.8496 1.86485
\(126\) 0 0
\(127\) −17.3935 −1.54342 −0.771710 0.635975i \(-0.780598\pi\)
−0.771710 + 0.635975i \(0.780598\pi\)
\(128\) 0 0
\(129\) 2.57858 + 0.416237i 0.227032 + 0.0366476i
\(130\) 0 0
\(131\) 5.45361 9.44593i 0.476484 0.825295i −0.523153 0.852239i \(-0.675244\pi\)
0.999637 + 0.0269442i \(0.00857764\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −17.1580 + 10.9271i −1.47672 + 0.940454i
\(136\) 0 0
\(137\) −7.62547 + 4.40257i −0.651488 + 0.376137i −0.789026 0.614360i \(-0.789414\pi\)
0.137538 + 0.990496i \(0.456081\pi\)
\(138\) 0 0
\(139\) −14.2352 8.21869i −1.20741 0.697100i −0.245220 0.969468i \(-0.578860\pi\)
−0.962193 + 0.272367i \(0.912193\pi\)
\(140\) 0 0
\(141\) −4.83621 0.780664i −0.407282 0.0657438i
\(142\) 0 0
\(143\) 2.04876 0.171326
\(144\) 0 0
\(145\) 15.9860i 1.32757i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.5814 7.26390i −1.03071 0.595082i −0.113523 0.993535i \(-0.536214\pi\)
−0.917188 + 0.398454i \(0.869547\pi\)
\(150\) 0 0
\(151\) −2.80307 4.85505i −0.228110 0.395099i 0.729138 0.684367i \(-0.239921\pi\)
−0.957248 + 0.289268i \(0.906588\pi\)
\(152\) 0 0
\(153\) 3.54005 + 17.1535i 0.286196 + 1.38678i
\(154\) 0 0
\(155\) 27.4089 15.8246i 2.20154 1.27106i
\(156\) 0 0
\(157\) −15.4411 8.91493i −1.23233 0.711489i −0.264819 0.964298i \(-0.585312\pi\)
−0.967516 + 0.252809i \(0.918645\pi\)
\(158\) 0 0
\(159\) 0.0644502 + 0.0525074i 0.00511123 + 0.00416411i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.15399 0.0903874 0.0451937 0.998978i \(-0.485610\pi\)
0.0451937 + 0.998978i \(0.485610\pi\)
\(164\) 0 0
\(165\) −19.4220 15.8230i −1.51200 1.23182i
\(166\) 0 0
\(167\) −8.95550 + 15.5114i −0.692997 + 1.20031i 0.277854 + 0.960623i \(0.410377\pi\)
−0.970851 + 0.239683i \(0.922957\pi\)
\(168\) 0 0
\(169\) −6.34625 10.9920i −0.488173 0.845540i
\(170\) 0 0
\(171\) −15.6886 + 3.23773i −1.19974 + 0.247596i
\(172\) 0 0
\(173\) −3.74814 6.49197i −0.284966 0.493576i 0.687635 0.726057i \(-0.258649\pi\)
−0.972601 + 0.232481i \(0.925316\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.48979 + 14.4289i 0.412638 + 1.08454i
\(178\) 0 0
\(179\) 0.720974i 0.0538881i 0.999637 + 0.0269441i \(0.00857760\pi\)
−0.999637 + 0.0269441i \(0.991422\pi\)
\(180\) 0 0
\(181\) 5.07121i 0.376940i −0.982079 0.188470i \(-0.939647\pi\)
0.982079 0.188470i \(-0.0603529\pi\)
\(182\) 0 0
\(183\) −1.92295 + 11.9127i −0.142149 + 0.880609i
\(184\) 0 0
\(185\) 15.2619 26.4344i 1.12208 1.94350i
\(186\) 0 0
\(187\) −18.6802 + 10.7850i −1.36603 + 0.788678i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.0005 6.35111i 0.795965 0.459551i −0.0460934 0.998937i \(-0.514677\pi\)
0.842058 + 0.539387i \(0.181344\pi\)
\(192\) 0 0
\(193\) 11.4076 19.7586i 0.821140 1.42226i −0.0836931 0.996492i \(-0.526672\pi\)
0.904834 0.425765i \(-0.139995\pi\)
\(194\) 0 0
\(195\) 0.599205 3.71207i 0.0429100 0.265827i
\(196\) 0 0
\(197\) 0.0311360i 0.00221835i 0.999999 + 0.00110918i \(0.000353062\pi\)
−0.999999 + 0.00110918i \(0.999647\pi\)
\(198\) 0 0
\(199\) 22.9952i 1.63008i 0.579402 + 0.815042i \(0.303286\pi\)
−0.579402 + 0.815042i \(0.696714\pi\)
\(200\) 0 0
\(201\) −0.723587 1.90182i −0.0510379 0.134144i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0634 + 24.3585i 0.982229 + 1.70127i
\(206\) 0 0
\(207\) 2.14683 6.47652i 0.149215 0.450149i
\(208\) 0 0
\(209\) −9.86397 17.0849i −0.682305 1.18179i
\(210\) 0 0
\(211\) 8.55841 14.8236i 0.589185 1.02050i −0.405154 0.914248i \(-0.632782\pi\)
0.994339 0.106250i \(-0.0338845\pi\)
\(212\) 0 0
\(213\) −9.01117 7.34138i −0.617435 0.503023i
\(214\) 0 0
\(215\) 5.90362 0.402623
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.46869 + 4.45533i 0.369540 + 0.301064i
\(220\) 0 0
\(221\) −2.80380 1.61878i −0.188604 0.108891i
\(222\) 0 0
\(223\) 1.25230 0.723016i 0.0838602 0.0484167i −0.457484 0.889218i \(-0.651249\pi\)
0.541344 + 0.840801i \(0.317916\pi\)
\(224\) 0 0
\(225\) −23.1431 + 20.5913i −1.54287 + 1.37275i
\(226\) 0 0
\(227\) −2.23596 3.87280i −0.148406 0.257047i 0.782232 0.622987i \(-0.214081\pi\)
−0.930639 + 0.365940i \(0.880748\pi\)
\(228\) 0 0
\(229\) −2.24072 1.29368i −0.148071 0.0854888i 0.424134 0.905599i \(-0.360579\pi\)
−0.572205 + 0.820111i \(0.693912\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.4078i 1.14042i −0.821498 0.570211i \(-0.806862\pi\)
0.821498 0.570211i \(-0.193138\pi\)
\(234\) 0 0
\(235\) −11.0724 −0.722284
\(236\) 0 0
\(237\) −6.74985 1.08957i −0.438450 0.0707750i
\(238\) 0 0
\(239\) 4.23642 + 2.44590i 0.274031 + 0.158212i 0.630718 0.776012i \(-0.282760\pi\)
−0.356687 + 0.934224i \(0.616094\pi\)
\(240\) 0 0
\(241\) −7.04282 + 4.06618i −0.453668 + 0.261925i −0.709378 0.704828i \(-0.751024\pi\)
0.255710 + 0.966754i \(0.417691\pi\)
\(242\) 0 0
\(243\) 4.25704 14.9959i 0.273089 0.961989i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.48053 2.56436i 0.0942041 0.163166i
\(248\) 0 0
\(249\) −13.1552 2.12353i −0.833678 0.134573i
\(250\) 0 0
\(251\) −25.9341 −1.63694 −0.818472 0.574546i \(-0.805179\pi\)
−0.818472 + 0.574546i \(0.805179\pi\)
\(252\) 0 0
\(253\) 8.40274 0.528276
\(254\) 0 0
\(255\) 14.0775 + 37.0001i 0.881566 + 2.31704i
\(256\) 0 0
\(257\) 15.4115 26.6935i 0.961344 1.66510i 0.242213 0.970223i \(-0.422127\pi\)
0.719131 0.694874i \(-0.244540\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.14306 9.15219i −0.504043 0.566506i
\(262\) 0 0
\(263\) 15.6625 9.04276i 0.965792 0.557600i 0.0678413 0.997696i \(-0.478389\pi\)
0.897951 + 0.440096i \(0.145056\pi\)
\(264\) 0 0
\(265\) 0.162723 + 0.0939479i 0.00999597 + 0.00577117i
\(266\) 0 0
\(267\) −5.93602 + 7.28616i −0.363278 + 0.445906i
\(268\) 0 0
\(269\) 21.6406 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(270\) 0 0
\(271\) 14.2551i 0.865937i 0.901409 + 0.432968i \(0.142534\pi\)
−0.901409 + 0.432968i \(0.857466\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −33.0383 19.0747i −1.99229 1.15025i
\(276\) 0 0
\(277\) −4.40164 7.62386i −0.264469 0.458073i 0.702956 0.711234i \(-0.251863\pi\)
−0.967424 + 0.253160i \(0.918530\pi\)
\(278\) 0 0
\(279\) −7.63113 + 23.0215i −0.456864 + 1.37826i
\(280\) 0 0
\(281\) −16.6889 + 9.63537i −0.995579 + 0.574798i −0.906937 0.421266i \(-0.861586\pi\)
−0.0886417 + 0.996064i \(0.528253\pi\)
\(282\) 0 0
\(283\) −8.32822 4.80830i −0.495061 0.285824i 0.231611 0.972809i \(-0.425601\pi\)
−0.726672 + 0.686985i \(0.758934\pi\)
\(284\) 0 0
\(285\) −33.8403 + 12.8753i −2.00453 + 0.762665i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0859 1.00506
\(290\) 0 0
\(291\) −4.45801 + 27.6173i −0.261333 + 1.61896i
\(292\) 0 0
\(293\) 1.22598 2.12346i 0.0716225 0.124054i −0.827990 0.560743i \(-0.810516\pi\)
0.899613 + 0.436689i \(0.143849\pi\)
\(294\) 0 0
\(295\) 17.4467 + 30.2185i 1.01578 + 1.75939i
\(296\) 0 0
\(297\) 19.1794 0.834401i 1.11290 0.0484168i
\(298\) 0 0
\(299\) 0.630605 + 1.09224i 0.0364688 + 0.0631658i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −2.77605 0.448112i −0.159480 0.0257434i
\(304\) 0 0
\(305\) 27.2738i 1.56169i
\(306\) 0 0
\(307\) 10.6839i 0.609760i −0.952391 0.304880i \(-0.901384\pi\)
0.952391 0.304880i \(-0.0986163\pi\)
\(308\) 0 0
\(309\) −0.640513 + 0.243697i −0.0364375 + 0.0138634i
\(310\) 0 0
\(311\) 10.3833 17.9843i 0.588780 1.01980i −0.405612 0.914045i \(-0.632942\pi\)
0.994393 0.105752i \(-0.0337250\pi\)
\(312\) 0 0
\(313\) 3.40449 1.96558i 0.192433 0.111101i −0.400688 0.916215i \(-0.631229\pi\)
0.593121 + 0.805113i \(0.297896\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.98369 + 1.14528i −0.111415 + 0.0643256i −0.554672 0.832069i \(-0.687156\pi\)
0.443257 + 0.896395i \(0.353823\pi\)
\(318\) 0 0
\(319\) 7.54330 13.0654i 0.422344 0.731521i
\(320\) 0 0
\(321\) −7.60978 6.19967i −0.424736 0.346032i
\(322\) 0 0
\(323\) 31.1750i 1.73462i
\(324\) 0 0
\(325\) 5.72603i 0.317623i
\(326\) 0 0
\(327\) −14.7835 + 18.1460i −0.817531 + 1.00348i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.46788 6.00655i −0.190612 0.330150i 0.754841 0.655908i \(-0.227714\pi\)
−0.945453 + 0.325758i \(0.894381\pi\)
\(332\) 0 0
\(333\) 4.72769 + 22.9082i 0.259076 + 1.25536i
\(334\) 0 0
\(335\) −2.29957 3.98298i −0.125639 0.217613i
\(336\) 0 0
\(337\) −9.59771 + 16.6237i −0.522821 + 0.905552i 0.476827 + 0.878997i \(0.341787\pi\)
−0.999647 + 0.0265545i \(0.991546\pi\)
\(338\) 0 0
\(339\) −2.12444 + 0.808288i −0.115384 + 0.0439002i
\(340\) 0 0
\(341\) −29.8684 −1.61747
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.45756 15.2246i 0.132311 0.819663i
\(346\) 0 0
\(347\) 7.35287 + 4.24518i 0.394723 + 0.227893i 0.684204 0.729290i \(-0.260150\pi\)
−0.289482 + 0.957184i \(0.593483\pi\)
\(348\) 0 0
\(349\) −16.5478 + 9.55386i −0.885782 + 0.511407i −0.872560 0.488506i \(-0.837542\pi\)
−0.0132216 + 0.999913i \(0.504209\pi\)
\(350\) 0 0
\(351\) 1.54782 + 2.43043i 0.0826167 + 0.129727i
\(352\) 0 0
\(353\) 6.82951 + 11.8291i 0.363498 + 0.629597i 0.988534 0.150999i \(-0.0482490\pi\)
−0.625036 + 0.780596i \(0.714916\pi\)
\(354\) 0 0
\(355\) −22.7512 13.1354i −1.20751 0.697156i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.1945i 0.907490i 0.891132 + 0.453745i \(0.149912\pi\)
−0.891132 + 0.453745i \(0.850088\pi\)
\(360\) 0 0
\(361\) −9.51270 −0.500668
\(362\) 0 0
\(363\) 1.63205 + 4.28955i 0.0856604 + 0.225143i
\(364\) 0 0
\(365\) 13.8073 + 7.97162i 0.722705 + 0.417254i
\(366\) 0 0
\(367\) −14.6001 + 8.42936i −0.762118 + 0.440009i −0.830056 0.557680i \(-0.811691\pi\)
0.0679376 + 0.997690i \(0.478358\pi\)
\(368\) 0 0
\(369\) −20.4593 6.78184i −1.06507 0.353048i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.704288 1.21986i 0.0364667 0.0631621i −0.847216 0.531248i \(-0.821723\pi\)
0.883683 + 0.468086i \(0.155056\pi\)
\(374\) 0 0
\(375\) −22.8094 + 27.9974i −1.17787 + 1.44578i
\(376\) 0 0
\(377\) 2.26442 0.116624
\(378\) 0 0
\(379\) −0.598572 −0.0307466 −0.0153733 0.999882i \(-0.504894\pi\)
−0.0153733 + 0.999882i \(0.504894\pi\)
\(380\) 0 0
\(381\) 19.0284 23.3563i 0.974853 1.19658i
\(382\) 0 0
\(383\) −4.26039 + 7.37921i −0.217696 + 0.377060i −0.954103 0.299478i \(-0.903187\pi\)
0.736407 + 0.676538i \(0.236521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.37989 + 3.00722i −0.171810 + 0.152866i
\(388\) 0 0
\(389\) 29.9624 17.2988i 1.51915 0.877084i 0.519409 0.854526i \(-0.326152\pi\)
0.999746 0.0225587i \(-0.00718126\pi\)
\(390\) 0 0
\(391\) −11.4994 6.63920i −0.581551 0.335759i
\(392\) 0 0
\(393\) 6.71799 + 17.6570i 0.338878 + 0.890680i
\(394\) 0 0
\(395\) −15.4537 −0.777558
\(396\) 0 0
\(397\) 32.2821i 1.62019i −0.586296 0.810097i \(-0.699414\pi\)
0.586296 0.810097i \(-0.300586\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.3473 6.55139i −0.566659 0.327161i 0.189155 0.981947i \(-0.439425\pi\)
−0.755814 + 0.654787i \(0.772758\pi\)
\(402\) 0 0
\(403\) −2.24155 3.88248i −0.111660 0.193400i
\(404\) 0 0
\(405\) 4.09760 34.9943i 0.203611 1.73888i
\(406\) 0 0
\(407\) −24.9471 + 14.4032i −1.23658 + 0.713941i
\(408\) 0 0
\(409\) 32.3493 + 18.6769i 1.59957 + 0.923513i 0.991569 + 0.129577i \(0.0413620\pi\)
0.608002 + 0.793936i \(0.291971\pi\)
\(410\) 0 0
\(411\) 2.43036 15.0561i 0.119881 0.742660i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −30.1186 −1.47846
\(416\) 0 0
\(417\) 26.6095 10.1241i 1.30307 0.495781i
\(418\) 0 0
\(419\) 14.1954 24.5871i 0.693490 1.20116i −0.277198 0.960813i \(-0.589406\pi\)
0.970687 0.240346i \(-0.0772610\pi\)
\(420\) 0 0
\(421\) −17.3359 30.0267i −0.844901 1.46341i −0.885707 0.464245i \(-0.846326\pi\)
0.0408054 0.999167i \(-0.487008\pi\)
\(422\) 0 0
\(423\) 6.33908 5.64013i 0.308217 0.274232i
\(424\) 0 0
\(425\) 30.1427 + 52.2087i 1.46214 + 2.53249i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −2.24134 + 2.75113i −0.108213 + 0.132826i
\(430\) 0 0
\(431\) 15.2240i 0.733315i 0.930356 + 0.366657i \(0.119498\pi\)
−0.930356 + 0.366657i \(0.880502\pi\)
\(432\) 0 0
\(433\) 3.97041i 0.190806i 0.995439 + 0.0954028i \(0.0304139\pi\)
−0.995439 + 0.0954028i \(0.969586\pi\)
\(434\) 0 0
\(435\) −21.4664 17.4886i −1.02924 0.838516i
\(436\) 0 0
\(437\) 6.07222 10.5174i 0.290474 0.503115i
\(438\) 0 0
\(439\) 8.21910 4.74530i 0.392276 0.226481i −0.290870 0.956763i \(-0.593945\pi\)
0.683146 + 0.730282i \(0.260611\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.3955 16.3942i 1.34911 0.778910i 0.360989 0.932570i \(-0.382439\pi\)
0.988124 + 0.153660i \(0.0491060\pi\)
\(444\) 0 0
\(445\) −10.6209 + 18.3960i −0.503479 + 0.872052i
\(446\) 0 0
\(447\) 23.5182 8.94798i 1.11237 0.423225i
\(448\) 0 0
\(449\) 0.658896i 0.0310952i −0.999879 0.0155476i \(-0.995051\pi\)
0.999879 0.0155476i \(-0.00494916\pi\)
\(450\) 0 0
\(451\) 26.5443i 1.24992i
\(452\) 0 0
\(453\) 9.58602 + 1.54738i 0.450390 + 0.0727024i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.94514 13.7614i −0.371658 0.643730i 0.618163 0.786050i \(-0.287877\pi\)
−0.989821 + 0.142320i \(0.954544\pi\)
\(458\) 0 0
\(459\) −26.9069 14.0122i −1.25591 0.654031i
\(460\) 0 0
\(461\) −9.81626 17.0023i −0.457189 0.791874i 0.541622 0.840622i \(-0.317810\pi\)
−0.998811 + 0.0487477i \(0.984477\pi\)
\(462\) 0 0
\(463\) 0.600159 1.03951i 0.0278918 0.0483099i −0.851743 0.523960i \(-0.824454\pi\)
0.879634 + 0.475651i \(0.157787\pi\)
\(464\) 0 0
\(465\) −8.73567 + 54.1173i −0.405107 + 2.50963i
\(466\) 0 0
\(467\) −38.5618 −1.78443 −0.892213 0.451614i \(-0.850848\pi\)
−0.892213 + 0.451614i \(0.850848\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 28.8637 10.9818i 1.32997 0.506014i
\(472\) 0 0
\(473\) −4.82503 2.78573i −0.221855 0.128088i
\(474\) 0 0
\(475\) −47.7501 + 27.5685i −2.19093 + 1.26493i
\(476\) 0 0
\(477\) −0.141016 + 0.0291023i −0.00645670 + 0.00133250i
\(478\) 0 0
\(479\) −3.61289 6.25771i −0.165077 0.285922i 0.771606 0.636101i \(-0.219454\pi\)
−0.936683 + 0.350179i \(0.886121\pi\)
\(480\) 0 0
\(481\) −3.74444 2.16185i −0.170732 0.0985720i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 63.2293i 2.87110i
\(486\) 0 0
\(487\) −9.71539 −0.440247 −0.220123 0.975472i \(-0.570646\pi\)
−0.220123 + 0.975472i \(0.570646\pi\)
\(488\) 0 0
\(489\) −1.26246 + 1.54960i −0.0570903 + 0.0700754i
\(490\) 0 0
\(491\) 17.2480 + 9.95814i 0.778392 + 0.449405i 0.835860 0.548943i \(-0.184969\pi\)
−0.0574682 + 0.998347i \(0.518303\pi\)
\(492\) 0 0
\(493\) −20.6465 + 11.9203i −0.929872 + 0.536862i
\(494\) 0 0
\(495\) 42.4951 8.76994i 1.91001 0.394179i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17.1920 + 29.7774i −0.769619 + 1.33302i 0.168150 + 0.985761i \(0.446221\pi\)
−0.937770 + 0.347258i \(0.887113\pi\)
\(500\) 0 0
\(501\) −11.0318 28.9950i −0.492863 1.29540i
\(502\) 0 0
\(503\) 1.22542 0.0546388 0.0273194 0.999627i \(-0.491303\pi\)
0.0273194 + 0.999627i \(0.491303\pi\)
\(504\) 0 0
\(505\) −6.35571 −0.282826
\(506\) 0 0
\(507\) 21.7031 + 3.50333i 0.963869 + 0.155588i
\(508\) 0 0
\(509\) 5.05078 8.74820i 0.223872 0.387757i −0.732109 0.681188i \(-0.761464\pi\)
0.955980 + 0.293431i \(0.0947970\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 12.8155 24.6090i 0.565819 1.08652i
\(514\) 0 0
\(515\) −1.34143 + 0.774473i −0.0591103 + 0.0341274i
\(516\) 0 0
\(517\) 9.04947 + 5.22471i 0.397995 + 0.229783i
\(518\) 0 0
\(519\) 12.8180 + 2.06910i 0.562649 + 0.0908233i
\(520\) 0 0
\(521\) −21.0781 −0.923446 −0.461723 0.887024i \(-0.652769\pi\)
−0.461723 + 0.887024i \(0.652769\pi\)
\(522\) 0 0
\(523\) 19.7145i 0.862057i −0.902338 0.431028i \(-0.858151\pi\)
0.902338 0.431028i \(-0.141849\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.8760 + 23.5997i 1.78058 + 1.02802i
\(528\) 0 0
\(529\) −8.91366 15.4389i −0.387550 0.671257i
\(530\) 0 0
\(531\) −25.3813 8.41336i −1.10145 0.365109i
\(532\) 0 0
\(533\) 3.45039 1.99208i 0.149453 0.0862866i
\(534\) 0 0
\(535\) −19.2130 11.0926i −0.830652 0.479577i
\(536\) 0 0
\(537\) −0.968140 0.788742i −0.0417783 0.0340367i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.44950 0.363272 0.181636 0.983366i \(-0.441861\pi\)
0.181636 + 0.983366i \(0.441861\pi\)
\(542\) 0 0
\(543\) 6.80974 + 5.54788i 0.292234 + 0.238082i
\(544\) 0 0
\(545\) −26.4511 + 45.8147i −1.13304 + 1.96249i
\(546\) 0 0
\(547\) −4.02889 6.97824i −0.172263 0.298368i 0.766948 0.641709i \(-0.221774\pi\)
−0.939211 + 0.343342i \(0.888441\pi\)
\(548\) 0 0
\(549\) −13.8929 15.6146i −0.592934 0.666414i
\(550\) 0 0
\(551\) −10.9023 18.8833i −0.464453 0.804456i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 18.8003 + 49.4132i 0.798027 + 2.09747i
\(556\) 0 0
\(557\) 21.0495i 0.891895i −0.895059 0.445947i \(-0.852867\pi\)
0.895059 0.445947i \(-0.147133\pi\)
\(558\) 0 0
\(559\) 0.836249i 0.0353696i
\(560\) 0 0
\(561\) 5.95367 36.8829i 0.251364 1.55720i
\(562\) 0 0
\(563\) −20.6410 + 35.7513i −0.869916 + 1.50674i −0.00783378 + 0.999969i \(0.502494\pi\)
−0.862082 + 0.506769i \(0.830840\pi\)
\(564\) 0 0
\(565\) −4.44922 + 2.56876i −0.187180 + 0.108068i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.2691 + 18.0532i −1.31087 + 0.756829i −0.982240 0.187630i \(-0.939919\pi\)
−0.328627 + 0.944460i \(0.606586\pi\)
\(570\) 0 0
\(571\) 9.62111 16.6642i 0.402631 0.697377i −0.591412 0.806370i \(-0.701429\pi\)
0.994043 + 0.108993i \(0.0347625\pi\)
\(572\) 0 0
\(573\) −3.50602 + 21.7198i −0.146466 + 0.907356i
\(574\) 0 0
\(575\) 23.4846i 0.979375i
\(576\) 0 0
\(577\) 29.8031i 1.24072i −0.784318 0.620359i \(-0.786987\pi\)
0.784318 0.620359i \(-0.213013\pi\)
\(578\) 0 0
\(579\) 14.0524 + 36.9343i 0.583999 + 1.53494i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.0886621 0.153567i −0.00367201 0.00636010i
\(584\) 0 0
\(585\) 4.32912 + 4.86561i 0.178987 + 0.201168i
\(586\) 0 0
\(587\) 4.72218 + 8.17905i 0.194905 + 0.337586i 0.946869 0.321618i \(-0.104227\pi\)
−0.751964 + 0.659204i \(0.770893\pi\)
\(588\) 0 0
\(589\) −21.5843 + 37.3852i −0.889367 + 1.54043i
\(590\) 0 0
\(591\) −0.0418102 0.0340627i −0.00171984 0.00140115i
\(592\) 0 0
\(593\) −24.8352 −1.01986 −0.509929 0.860216i \(-0.670328\pi\)
−0.509929 + 0.860216i \(0.670328\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −30.8784 25.1566i −1.26377 1.02959i
\(598\) 0 0
\(599\) 10.3052 + 5.94974i 0.421061 + 0.243100i 0.695531 0.718496i \(-0.255169\pi\)
−0.274470 + 0.961596i \(0.588502\pi\)
\(600\) 0 0
\(601\) 22.1276 12.7754i 0.902604 0.521118i 0.0245596 0.999698i \(-0.492182\pi\)
0.878044 + 0.478580i \(0.158848\pi\)
\(602\) 0 0
\(603\) 3.34541 + 1.10893i 0.136236 + 0.0451592i
\(604\) 0 0
\(605\) 5.18669 + 8.98361i 0.210869 + 0.365236i
\(606\) 0 0
\(607\) 19.5544 + 11.2897i 0.793687 + 0.458235i 0.841259 0.540632i \(-0.181815\pi\)
−0.0475718 + 0.998868i \(0.515148\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.56841i 0.0634510i
\(612\) 0 0
\(613\) 22.8588 0.923257 0.461628 0.887073i \(-0.347265\pi\)
0.461628 + 0.887073i \(0.347265\pi\)
\(614\) 0 0
\(615\) −48.0944 7.76344i −1.93935 0.313052i
\(616\) 0 0
\(617\) 1.78792 + 1.03226i 0.0719791 + 0.0415572i 0.535558 0.844499i \(-0.320101\pi\)
−0.463578 + 0.886056i \(0.653435\pi\)
\(618\) 0 0
\(619\) 28.2233 16.2947i 1.13439 0.654940i 0.189354 0.981909i \(-0.439361\pi\)
0.945035 + 0.326969i \(0.106027\pi\)
\(620\) 0 0
\(621\) 6.34820 + 9.96809i 0.254744 + 0.400006i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14.9967 + 25.9751i −0.599870 + 1.03901i
\(626\) 0 0
\(627\) 33.7331 + 5.44523i 1.34717 + 0.217462i
\(628\) 0 0
\(629\) 45.5213 1.81505
\(630\) 0 0
\(631\) 38.4706 1.53149 0.765744 0.643145i \(-0.222371\pi\)
0.765744 + 0.643145i \(0.222371\pi\)
\(632\) 0 0
\(633\) 10.5426 + 27.7094i 0.419031 + 1.10135i
\(634\) 0 0
\(635\) 34.0461 58.9696i 1.35108 2.34014i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 19.7163 4.06897i 0.779967 0.160966i
\(640\) 0 0
\(641\) −41.3645 + 23.8818i −1.63380 + 0.943274i −0.650892 + 0.759170i \(0.725605\pi\)
−0.982907 + 0.184104i \(0.941062\pi\)
\(642\) 0 0
\(643\) 29.2346 + 16.8786i 1.15290 + 0.665626i 0.949592 0.313489i \(-0.101498\pi\)
0.203306 + 0.979115i \(0.434831\pi\)
\(644\) 0 0
\(645\) −6.45853 + 7.92752i −0.254304 + 0.312146i
\(646\) 0 0
\(647\) −1.07202 −0.0421453 −0.0210727 0.999778i \(-0.506708\pi\)
−0.0210727 + 0.999778i \(0.506708\pi\)
\(648\) 0 0
\(649\) 32.9301i 1.29262i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −28.8503 16.6567i −1.12900 0.651828i −0.185317 0.982679i \(-0.559331\pi\)
−0.943683 + 0.330851i \(0.892664\pi\)
\(654\) 0 0
\(655\) 21.3499 + 36.9791i 0.834210 + 1.44489i
\(656\) 0 0
\(657\) −11.9654 + 2.46937i −0.466817 + 0.0963394i
\(658\) 0 0
\(659\) −8.41890 + 4.86065i −0.327954 + 0.189344i −0.654932 0.755688i \(-0.727303\pi\)
0.326979 + 0.945032i \(0.393969\pi\)
\(660\) 0 0
\(661\) −14.7856 8.53647i −0.575093 0.332030i 0.184088 0.982910i \(-0.441067\pi\)
−0.759181 + 0.650880i \(0.774400\pi\)
\(662\) 0 0
\(663\) 5.24108 1.99408i 0.203547 0.0774436i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.28724 0.359603
\(668\) 0 0
\(669\) −0.399128 + 2.47259i −0.0154312 + 0.0955960i
\(670\) 0 0
\(671\) 12.8696 22.2909i 0.496827 0.860529i
\(672\) 0 0
\(673\) −18.3359 31.7588i −0.706798 1.22421i −0.966039 0.258398i \(-0.916805\pi\)
0.259240 0.965813i \(-0.416528\pi\)
\(674\) 0 0
\(675\) −2.33204 53.6038i −0.0897604 2.06321i
\(676\) 0 0
\(677\) 20.1769 + 34.9474i 0.775461 + 1.34314i 0.934535 + 0.355872i \(0.115816\pi\)
−0.159073 + 0.987267i \(0.550851\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.64663 + 1.23432i 0.293019 + 0.0472994i
\(682\) 0 0
\(683\) 9.51083i 0.363922i 0.983306 + 0.181961i \(0.0582444\pi\)
−0.983306 + 0.181961i \(0.941756\pi\)
\(684\) 0 0
\(685\) 34.4705i 1.31705i
\(686\) 0 0
\(687\) 4.18852 1.59361i 0.159802 0.0608000i
\(688\) 0 0
\(689\) 0.0133077 0.0230497i 0.000506985 0.000878123i
\(690\) 0 0
\(691\) −6.67519 + 3.85392i −0.253936 + 0.146610i −0.621565 0.783362i \(-0.713503\pi\)
0.367629 + 0.929972i \(0.380170\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.7282 32.1747i 2.11389 1.22046i
\(696\) 0 0
\(697\) −20.9732 + 36.3267i −0.794418 + 1.37597i
\(698\) 0 0
\(699\) 23.3756 + 19.0440i 0.884145 + 0.720311i
\(700\) 0 0
\(701\) 15.6388i 0.590671i −0.955394 0.295336i \(-0.904569\pi\)
0.955394 0.295336i \(-0.0954314\pi\)
\(702\) 0 0
\(703\) 41.6338i 1.57025i
\(704\) 0 0
\(705\) 12.1132 14.8683i 0.456208 0.559972i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.72025 11.6398i −0.252384 0.437142i 0.711797 0.702385i \(-0.247881\pi\)
−0.964182 + 0.265242i \(0.914548\pi\)
\(710\) 0 0
\(711\) 8.84740 7.87187i 0.331803 0.295218i
\(712\) 0 0
\(713\) −9.19343 15.9235i −0.344297 0.596339i
\(714\) 0 0
\(715\) −4.01027 + 6.94599i −0.149976 + 0.259765i
\(716\) 0 0
\(717\) −7.91903 + 3.01296i −0.295742 + 0.112521i
\(718\) 0 0
\(719\) −40.0619 −1.49405 −0.747027 0.664793i \(-0.768520\pi\)
−0.747027 + 0.664793i \(0.768520\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 2.24466 13.9056i 0.0834798 0.517156i
\(724\) 0 0
\(725\) −36.5161 21.0826i −1.35617 0.782986i
\(726\) 0 0
\(727\) 43.2091 24.9468i 1.60254 0.925225i 0.611560 0.791198i \(-0.290542\pi\)
0.990978 0.134027i \(-0.0427910\pi\)
\(728\) 0 0
\(729\) 15.4797 + 22.1219i 0.573322 + 0.819330i
\(730\) 0 0
\(731\) 4.40214 + 7.62473i 0.162819 + 0.282011i
\(732\) 0 0
\(733\) 9.91430 + 5.72402i 0.366193 + 0.211422i 0.671794 0.740738i \(-0.265524\pi\)
−0.305601 + 0.952160i \(0.598857\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.34038i 0.159880i
\(738\) 0 0
\(739\) −8.92607 −0.328351 −0.164175 0.986431i \(-0.552496\pi\)
−0.164175 + 0.986431i \(0.552496\pi\)
\(740\) 0 0
\(741\) 1.82378 + 4.79349i 0.0669984 + 0.176093i
\(742\) 0 0
\(743\) −45.8621 26.4785i −1.68252 0.971403i −0.959979 0.280074i \(-0.909641\pi\)
−0.722540 0.691329i \(-0.757026\pi\)
\(744\) 0 0
\(745\) 49.2541 28.4369i 1.80453 1.04185i
\(746\) 0 0
\(747\) 17.2433 15.3420i 0.630898 0.561334i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.2326 22.9195i 0.482865 0.836346i −0.516942 0.856021i \(-0.672930\pi\)
0.999806 + 0.0196744i \(0.00626295\pi\)
\(752\) 0 0
\(753\) 28.3718 34.8249i 1.03392 1.26909i
\(754\) 0 0
\(755\) 21.9470 0.798733
\(756\) 0 0
\(757\) 8.46749 0.307756 0.153878 0.988090i \(-0.450824\pi\)
0.153878 + 0.988090i \(0.450824\pi\)
\(758\) 0 0
\(759\) −9.19256 + 11.2834i −0.333669 + 0.409561i
\(760\) 0 0
\(761\) −26.9968 + 46.7599i −0.978635 + 1.69505i −0.311258 + 0.950325i \(0.600750\pi\)
−0.667377 + 0.744720i \(0.732583\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −65.0853 21.5744i −2.35317 0.780025i
\(766\) 0 0
\(767\) 4.28046 2.47132i 0.154558 0.0892343i
\(768\) 0 0
\(769\) 30.1912 + 17.4309i 1.08872 + 0.628575i 0.933236 0.359263i \(-0.116972\pi\)
0.155487 + 0.987838i \(0.450305\pi\)
\(770\) 0 0
\(771\) 18.9846 + 49.8976i 0.683713 + 1.79702i
\(772\) 0 0
\(773\) 2.12749 0.0765207 0.0382603 0.999268i \(-0.487818\pi\)
0.0382603 + 0.999268i \(0.487818\pi\)
\(774\) 0 0
\(775\) 83.4784i 2.99863i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33.2244 19.1821i −1.19039 0.687272i
\(780\) 0 0
\(781\) 12.3964 + 21.4712i 0.443577 + 0.768298i
\(782\) 0 0
\(783\) 21.1982 0.922233i 0.757563 0.0329579i
\(784\) 0 0
\(785\) 60.4492 34.9004i 2.15752 1.24565i
\(786\) 0 0
\(787\) 24.5457 + 14.1715i 0.874959 + 0.505158i 0.868993 0.494824i \(-0.164768\pi\)
0.00596615 + 0.999982i \(0.498101\pi\)
\(788\) 0 0
\(789\) −4.99190 + 30.9247i −0.177716 + 1.10095i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.86334 0.137191
\(794\) 0 0
\(795\) −0.304173 + 0.115729i −0.0107879 + 0.00410449i
\(796\) 0 0
\(797\) −18.9123 + 32.7570i −0.669907 + 1.16031i 0.308022 + 0.951379i \(0.400333\pi\)
−0.977930 + 0.208935i \(0.933000\pi\)
\(798\) 0 0
\(799\) −8.25634 14.3004i −0.292088 0.505912i
\(800\) 0 0
\(801\) −3.29005 15.9420i −0.116248 0.563285i
\(802\) 0 0
\(803\) −7.52311 13.0304i −0.265485 0.459833i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −23.6747 + 29.0595i −0.833389 + 1.02294i
\(808\) 0 0
\(809\) 45.3191i 1.59333i 0.604419 + 0.796667i \(0.293405\pi\)
−0.604419 + 0.796667i \(0.706595\pi\)
\(810\) 0 0
\(811\) 5.45145i 0.191426i −0.995409 0.0957132i \(-0.969487\pi\)
0.995409 0.0957132i \(-0.0305132\pi\)
\(812\) 0 0
\(813\) −19.1421 15.5950i −0.671343 0.546942i
\(814\) 0 0
\(815\) −2.25883 + 3.91241i −0.0791233 + 0.137046i
\(816\) 0 0
\(817\) −6.97359 + 4.02620i −0.243975 + 0.140859i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.7121 24.6598i 1.49066 0.860634i 0.490718 0.871318i \(-0.336734\pi\)
0.999943 + 0.0106847i \(0.00340111\pi\)
\(822\) 0 0
\(823\) 11.8496 20.5241i 0.413050 0.715424i −0.582171 0.813066i \(-0.697797\pi\)
0.995222 + 0.0976419i \(0.0311300\pi\)
\(824\) 0 0
\(825\) 61.7577 23.4970i 2.15013 0.818061i
\(826\) 0 0
\(827\) 19.9706i 0.694445i 0.937783 + 0.347222i \(0.112875\pi\)
−0.937783 + 0.347222i \(0.887125\pi\)
\(828\) 0 0
\(829\) 15.4431i 0.536361i −0.963369 0.268181i \(-0.913578\pi\)
0.963369 0.268181i \(-0.0864224\pi\)
\(830\) 0 0
\(831\) 15.0529 + 2.42985i 0.522178 + 0.0842904i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −35.0592 60.7242i −1.21327 2.10145i
\(836\) 0 0
\(837\) −22.5653 35.4327i −0.779972 1.22473i
\(838\) 0 0
\(839\) −5.53910 9.59401i −0.191231 0.331222i 0.754427 0.656383i \(-0.227915\pi\)
−0.945658 + 0.325162i \(0.894581\pi\)
\(840\) 0 0
\(841\) −6.16267 + 10.6741i −0.212506 + 0.368071i
\(842\) 0 0
\(843\) 5.31903 32.9513i 0.183197 1.13490i
\(844\) 0 0
\(845\) 49.6888 1.70935
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 15.5677 5.92307i 0.534283 0.203279i
\(850\) 0 0
\(851\) −15.3573 8.86656i −0.526442 0.303942i
\(852\) 0 0
\(853\) −42.1706 + 24.3472i −1.44389 + 0.833633i −0.998107 0.0615058i \(-0.980410\pi\)
−0.445788 + 0.895139i \(0.647076\pi\)
\(854\) 0 0
\(855\) 19.7320 59.5271i 0.674819 2.03578i
\(856\) 0 0
\(857\) 8.39130 + 14.5342i 0.286641 + 0.496477i 0.973006 0.230780i \(-0.0741279\pi\)
−0.686365 + 0.727258i \(0.740795\pi\)
\(858\) 0 0
\(859\) 21.7682 + 12.5679i 0.742722 + 0.428811i 0.823058 0.567957i \(-0.192266\pi\)
−0.0803361 + 0.996768i \(0.525599\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.78244i 0.230877i −0.993315 0.115438i \(-0.963173\pi\)
0.993315 0.115438i \(-0.0368273\pi\)
\(864\) 0 0
\(865\) 29.3466 0.997815
\(866\) 0 0
\(867\) −18.6919 + 22.9434i −0.634812 + 0.779199i
\(868\) 0 0
\(869\) 12.6303 + 7.29209i 0.428452 + 0.247367i
\(870\) 0 0
\(871\) −0.564190 + 0.325735i −0.0191168 + 0.0110371i
\(872\) 0 0
\(873\) −32.2081 36.1995i −1.09008 1.22517i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21.8630 37.8678i 0.738260 1.27870i −0.215019 0.976610i \(-0.568981\pi\)
0.953278 0.302093i \(-0.0976854\pi\)
\(878\) 0 0
\(879\) 1.51021 + 3.96933i 0.0509382 + 0.133882i
\(880\) 0 0
\(881\) −27.5307 −0.927531 −0.463766 0.885958i \(-0.653502\pi\)
−0.463766 + 0.885958i \(0.653502\pi\)
\(882\) 0 0
\(883\) 5.56040 0.187122 0.0935612 0.995614i \(-0.470175\pi\)
0.0935612 + 0.995614i \(0.470175\pi\)
\(884\) 0 0
\(885\) −59.6646 9.63112i −2.00560 0.323746i
\(886\) 0 0
\(887\) 12.3092 21.3202i 0.413303 0.715862i −0.581945 0.813228i \(-0.697708\pi\)
0.995249 + 0.0973655i \(0.0310416\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −19.8617 + 26.6673i −0.665391 + 0.893388i
\(892\) 0 0
\(893\) 13.0792 7.55125i 0.437677 0.252693i
\(894\) 0 0
\(895\) −2.44434 1.41124i −0.0817054 0.0471726i
\(896\) 0 0
\(897\) −2.15656 0.348114i −0.0720056 0.0116232i
\(898\) 0 0
\(899\) −33.0125 −1.10103
\(900\) 0 0
\(901\) 0.280216i 0.00933535i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.1931 + 9.92644i 0.571518 + 0.329966i
\(906\) 0 0
\(907\) 5.04337 + 8.73537i 0.167462 + 0.290053i 0.937527 0.347913i \(-0.113109\pi\)
−0.770065 + 0.637966i \(0.779776\pi\)
\(908\) 0 0
\(909\) 3.63872 3.23751i 0.120689 0.107381i
\(910\) 0 0
\(911\) 23.5808 13.6144i 0.781267 0.451065i −0.0556121 0.998452i \(-0.517711\pi\)
0.836879 + 0.547388i \(0.184378\pi\)
\(912\) 0 0
\(913\) 24.6159 + 14.2120i 0.814668 + 0.470349i
\(914\) 0 0
\(915\) −36.6239 29.8374i −1.21075 0.986394i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 39.6986 1.30954 0.654769 0.755829i \(-0.272766\pi\)
0.654769 + 0.755829i \(0.272766\pi\)
\(920\) 0 0
\(921\) 14.3465 + 11.6881i 0.472734 + 0.385136i
\(922\) 0 0
\(923\) −1.86064 + 3.22272i −0.0612436 + 0.106077i
\(924\) 0 0
\(925\) 40.2552 + 69.7240i 1.32358 + 2.29251i
\(926\) 0 0
\(927\) 0.373477 1.12670i 0.0122666 0.0370057i
\(928\) 0 0
\(929\) 0.142283 + 0.246442i 0.00466816 + 0.00808550i 0.868350 0.495952i \(-0.165181\pi\)
−0.863682 + 0.504037i \(0.831847\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.7905 + 33.6176i 0.418743 + 1.10059i
\(934\) 0 0
\(935\) 84.4427i 2.76157i
\(936\) 0 0
\(937\) 21.7298i 0.709881i 0.934889 + 0.354940i \(0.115499\pi\)
−0.934889 + 0.354940i \(0.884501\pi\)
\(938\) 0 0
\(939\) −1.08507 + 6.72197i −0.0354098 + 0.219363i
\(940\) 0 0
\(941\) 5.64242 9.77295i 0.183938 0.318589i −0.759280 0.650764i \(-0.774449\pi\)
0.943218 + 0.332174i \(0.107782\pi\)
\(942\) 0 0
\(943\) 14.1513 8.17026i 0.460830 0.266060i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19.6701 + 11.3566i −0.639194 + 0.369039i −0.784304 0.620377i \(-0.786980\pi\)
0.145110 + 0.989416i \(0.453646\pi\)
\(948\) 0 0
\(949\) 1.12918 1.95580i 0.0366548 0.0634880i
\(950\) 0 0
\(951\) 0.632234 3.91668i 0.0205016 0.127007i
\(952\) 0 0
\(953\) 16.5638i 0.536554i −0.963342 0.268277i \(-0.913546\pi\)
0.963342 0.268277i \(-0.0864543\pi\)
\(954\) 0 0
\(955\) 49.7270i 1.60913i
\(956\) 0 0
\(957\) 9.29215 + 24.4228i 0.300373 + 0.789476i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 17.1790 + 29.7550i 0.554162 + 0.959837i
\(962\) 0 0
\(963\) 16.6501 3.43618i 0.536543 0.110729i
\(964\) 0 0
\(965\) 44.6589 + 77.3515i 1.43762 + 2.49003i
\(966\) 0 0
\(967\) 8.38867 14.5296i 0.269762 0.467241i −0.699039 0.715084i \(-0.746388\pi\)
0.968800 + 0.247843i \(0.0797218\pi\)
\(968\) 0 0
\(969\) −41.8625 34.1053i −1.34482 1.09562i
\(970\) 0 0
\(971\) 31.3640 1.00652 0.503259 0.864136i \(-0.332134\pi\)
0.503259 + 0.864136i \(0.332134\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 7.68905 + 6.26425i 0.246247 + 0.200617i
\(976\) 0 0
\(977\) −49.0953 28.3452i −1.57070 0.906843i −0.996083 0.0884183i \(-0.971819\pi\)
−0.574614 0.818424i \(-0.694848\pi\)
\(978\) 0 0
\(979\) 17.3609 10.0233i 0.554858 0.320347i
\(980\) 0 0
\(981\) −8.19378 39.7033i −0.261607 1.26763i
\(982\) 0 0
\(983\) 19.9204 + 34.5032i 0.635362 + 1.10048i 0.986438 + 0.164133i \(0.0524825\pi\)
−0.351076 + 0.936347i \(0.614184\pi\)
\(984\) 0 0
\(985\) −0.105562 0.0609460i −0.00336347 0.00194190i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.42977i 0.109060i
\(990\) 0 0
\(991\) −62.5951 −1.98840 −0.994199 0.107558i \(-0.965697\pi\)
−0.994199 + 0.107558i \(0.965697\pi\)
\(992\) 0 0
\(993\) 11.8596 + 1.91438i 0.376352 + 0.0607511i
\(994\) 0 0
\(995\) −77.9613 45.0110i −2.47154 1.42694i
\(996\) 0 0
\(997\) −39.0613 + 22.5520i −1.23708 + 0.714230i −0.968497 0.249025i \(-0.919890\pi\)
−0.268586 + 0.963256i \(0.586556\pi\)
\(998\) 0 0
\(999\) −35.9338 18.7130i −1.13689 0.592054i
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 1764.2.x.a.293.3 16
3.2 odd 2 5292.2.x.a.881.8 16
7.2 even 3 1764.2.bm.a.1697.3 16
7.3 odd 6 1764.2.w.b.509.1 16
7.4 even 3 252.2.w.a.5.8 16
7.5 odd 6 252.2.bm.a.185.6 yes 16
7.6 odd 2 1764.2.x.b.293.6 16
9.2 odd 6 1764.2.x.b.1469.6 16
9.7 even 3 5292.2.x.b.4409.1 16
21.2 odd 6 5292.2.bm.a.2285.1 16
21.5 even 6 756.2.bm.a.17.8 16
21.11 odd 6 756.2.w.a.341.8 16
21.17 even 6 5292.2.w.b.1097.1 16
21.20 even 2 5292.2.x.b.881.1 16
28.11 odd 6 1008.2.ca.d.257.1 16
28.19 even 6 1008.2.df.d.689.3 16
63.2 odd 6 1764.2.w.b.1109.1 16
63.4 even 3 2268.2.t.b.2105.1 16
63.5 even 6 2268.2.t.b.1781.1 16
63.11 odd 6 252.2.bm.a.173.6 yes 16
63.16 even 3 5292.2.w.b.521.1 16
63.20 even 6 inner 1764.2.x.a.1469.3 16
63.25 even 3 756.2.bm.a.89.8 16
63.32 odd 6 2268.2.t.a.2105.8 16
63.34 odd 6 5292.2.x.a.4409.8 16
63.38 even 6 1764.2.bm.a.1685.3 16
63.40 odd 6 2268.2.t.a.1781.8 16
63.47 even 6 252.2.w.a.101.8 yes 16
63.52 odd 6 5292.2.bm.a.4625.1 16
63.61 odd 6 756.2.w.a.521.8 16
84.11 even 6 3024.2.ca.d.2609.8 16
84.47 odd 6 3024.2.df.d.17.8 16
252.11 even 6 1008.2.df.d.929.3 16
252.47 odd 6 1008.2.ca.d.353.1 16
252.151 odd 6 3024.2.df.d.1601.8 16
252.187 even 6 3024.2.ca.d.2033.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.w.a.5.8 16 7.4 even 3
252.2.w.a.101.8 yes 16 63.47 even 6
252.2.bm.a.173.6 yes 16 63.11 odd 6
252.2.bm.a.185.6 yes 16 7.5 odd 6
756.2.w.a.341.8 16 21.11 odd 6
756.2.w.a.521.8 16 63.61 odd 6
756.2.bm.a.17.8 16 21.5 even 6
756.2.bm.a.89.8 16 63.25 even 3
1008.2.ca.d.257.1 16 28.11 odd 6
1008.2.ca.d.353.1 16 252.47 odd 6
1008.2.df.d.689.3 16 28.19 even 6
1008.2.df.d.929.3 16 252.11 even 6
1764.2.w.b.509.1 16 7.3 odd 6
1764.2.w.b.1109.1 16 63.2 odd 6
1764.2.x.a.293.3 16 1.1 even 1 trivial
1764.2.x.a.1469.3 16 63.20 even 6 inner
1764.2.x.b.293.6 16 7.6 odd 2
1764.2.x.b.1469.6 16 9.2 odd 6
1764.2.bm.a.1685.3 16 63.38 even 6
1764.2.bm.a.1697.3 16 7.2 even 3
2268.2.t.a.1781.8 16 63.40 odd 6
2268.2.t.a.2105.8 16 63.32 odd 6
2268.2.t.b.1781.1 16 63.5 even 6
2268.2.t.b.2105.1 16 63.4 even 3
3024.2.ca.d.2033.8 16 252.187 even 6
3024.2.ca.d.2609.8 16 84.11 even 6
3024.2.df.d.17.8 16 84.47 odd 6
3024.2.df.d.1601.8 16 252.151 odd 6
5292.2.w.b.521.1 16 63.16 even 3
5292.2.w.b.1097.1 16 21.17 even 6
5292.2.x.a.881.8 16 3.2 odd 2
5292.2.x.a.4409.8 16 63.34 odd 6
5292.2.x.b.881.1 16 21.20 even 2
5292.2.x.b.4409.1 16 9.7 even 3
5292.2.bm.a.2285.1 16 21.2 odd 6
5292.2.bm.a.4625.1 16 63.52 odd 6