Properties

Label 1176.2.k.a.881.13
Level $1176$
Weight $2$
Character 1176.881
Analytic conductor $9.390$
Analytic rank $0$
Dimension $16$
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] = [1176,2,Mod(881,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.k (of order \(2\), degree \(1\), 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: \(9.39040727770\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.13
Root \(-1.70742 - 0.291063i\) of defining polynomial
Character \(\chi\) \(=\) 1176.881
Dual form 1176.2.k.a.881.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.33314 - 1.10578i) q^{3} +0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} +O(q^{10})\) \(q+(1.33314 - 1.10578i) q^{3} +0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} -2.46977i q^{11} -2.04143i q^{13} +(0.193756 - 0.160712i) q^{15} +1.75684 q^{17} -4.25299i q^{19} +8.61812i q^{23} -4.97888 q^{25} +(-2.52094 - 4.54366i) q^{27} -7.08790i q^{29} -3.60050i q^{31} +(-2.73102 - 3.29255i) q^{33} +5.86986 q^{37} +(-2.25737 - 2.72151i) q^{39} +5.33255 q^{41} -9.19692 q^{43} +(0.0805917 - 0.428503i) q^{45} +9.30380 q^{47} +(2.34211 - 1.94267i) q^{51} -5.19128i q^{53} -0.358953i q^{55} +(-4.70286 - 5.66982i) q^{57} +11.2060 q^{59} +5.38498i q^{61} -0.296699i q^{65} -5.14835 q^{67} +(9.52973 + 11.4891i) q^{69} +7.79323i q^{71} -13.0496i q^{73} +(-6.63753 + 5.50553i) q^{75} -5.72151 q^{79} +(-8.38504 - 3.26973i) q^{81} -15.9818 q^{83} +0.255336 q^{85} +(-7.83764 - 9.44914i) q^{87} +8.68504 q^{89} +(-3.98135 - 4.79996i) q^{93} -0.618123i q^{95} +6.65337i q^{97} +(-7.28165 - 1.36951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{9} + 8 q^{15} + 36 q^{25} + 4 q^{37} + 44 q^{39} + 20 q^{43} - 12 q^{51} - 8 q^{57} - 28 q^{67} - 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.33314 1.10578i 0.769687 0.638421i
\(4\) 0 0
\(5\) 0.145339 0.0649974 0.0324987 0.999472i \(-0.489654\pi\)
0.0324987 + 0.999472i \(0.489654\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.554510 2.94831i 0.184837 0.982769i
\(10\) 0 0
\(11\) 2.46977i 0.744665i −0.928099 0.372332i \(-0.878558\pi\)
0.928099 0.372332i \(-0.121442\pi\)
\(12\) 0 0
\(13\) 2.04143i 0.566191i −0.959092 0.283096i \(-0.908639\pi\)
0.959092 0.283096i \(-0.0913613\pi\)
\(14\) 0 0
\(15\) 0.193756 0.160712i 0.0500276 0.0414957i
\(16\) 0 0
\(17\) 1.75684 0.426096 0.213048 0.977042i \(-0.431661\pi\)
0.213048 + 0.977042i \(0.431661\pi\)
\(18\) 0 0
\(19\) 4.25299i 0.975702i −0.872927 0.487851i \(-0.837781\pi\)
0.872927 0.487851i \(-0.162219\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.61812i 1.79700i 0.438971 + 0.898501i \(0.355343\pi\)
−0.438971 + 0.898501i \(0.644657\pi\)
\(24\) 0 0
\(25\) −4.97888 −0.995775
\(26\) 0 0
\(27\) −2.52094 4.54366i −0.485154 0.874429i
\(28\) 0 0
\(29\) 7.08790i 1.31619i −0.752935 0.658095i \(-0.771363\pi\)
0.752935 0.658095i \(-0.228637\pi\)
\(30\) 0 0
\(31\) 3.60050i 0.646669i −0.946285 0.323334i \(-0.895196\pi\)
0.946285 0.323334i \(-0.104804\pi\)
\(32\) 0 0
\(33\) −2.73102 3.29255i −0.475410 0.573159i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.86986 0.964999 0.482499 0.875896i \(-0.339729\pi\)
0.482499 + 0.875896i \(0.339729\pi\)
\(38\) 0 0
\(39\) −2.25737 2.72151i −0.361468 0.435790i
\(40\) 0 0
\(41\) 5.33255 0.832804 0.416402 0.909181i \(-0.363291\pi\)
0.416402 + 0.909181i \(0.363291\pi\)
\(42\) 0 0
\(43\) −9.19692 −1.40252 −0.701258 0.712907i \(-0.747378\pi\)
−0.701258 + 0.712907i \(0.747378\pi\)
\(44\) 0 0
\(45\) 0.0805917 0.428503i 0.0120139 0.0638774i
\(46\) 0 0
\(47\) 9.30380 1.35710 0.678549 0.734555i \(-0.262609\pi\)
0.678549 + 0.734555i \(0.262609\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.34211 1.94267i 0.327960 0.272029i
\(52\) 0 0
\(53\) 5.19128i 0.713077i −0.934281 0.356539i \(-0.883957\pi\)
0.934281 0.356539i \(-0.116043\pi\)
\(54\) 0 0
\(55\) 0.358953i 0.0484013i
\(56\) 0 0
\(57\) −4.70286 5.66982i −0.622909 0.750985i
\(58\) 0 0
\(59\) 11.2060 1.45889 0.729447 0.684037i \(-0.239777\pi\)
0.729447 + 0.684037i \(0.239777\pi\)
\(60\) 0 0
\(61\) 5.38498i 0.689476i 0.938699 + 0.344738i \(0.112032\pi\)
−0.938699 + 0.344738i \(0.887968\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.296699i 0.0368009i
\(66\) 0 0
\(67\) −5.14835 −0.628971 −0.314485 0.949262i \(-0.601832\pi\)
−0.314485 + 0.949262i \(0.601832\pi\)
\(68\) 0 0
\(69\) 9.52973 + 11.4891i 1.14724 + 1.38313i
\(70\) 0 0
\(71\) 7.79323i 0.924886i 0.886649 + 0.462443i \(0.153027\pi\)
−0.886649 + 0.462443i \(0.846973\pi\)
\(72\) 0 0
\(73\) 13.0496i 1.52734i −0.645604 0.763672i \(-0.723394\pi\)
0.645604 0.763672i \(-0.276606\pi\)
\(74\) 0 0
\(75\) −6.63753 + 5.50553i −0.766435 + 0.635724i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.72151 −0.643720 −0.321860 0.946787i \(-0.604308\pi\)
−0.321860 + 0.946787i \(0.604308\pi\)
\(80\) 0 0
\(81\) −8.38504 3.26973i −0.931671 0.363304i
\(82\) 0 0
\(83\) −15.9818 −1.75423 −0.877115 0.480280i \(-0.840535\pi\)
−0.877115 + 0.480280i \(0.840535\pi\)
\(84\) 0 0
\(85\) 0.255336 0.0276951
\(86\) 0 0
\(87\) −7.83764 9.44914i −0.840283 1.01305i
\(88\) 0 0
\(89\) 8.68504 0.920612 0.460306 0.887760i \(-0.347740\pi\)
0.460306 + 0.887760i \(0.347740\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.98135 4.79996i −0.412847 0.497733i
\(94\) 0 0
\(95\) 0.618123i 0.0634181i
\(96\) 0 0
\(97\) 6.65337i 0.675547i 0.941227 + 0.337774i \(0.109674\pi\)
−0.941227 + 0.337774i \(0.890326\pi\)
\(98\) 0 0
\(99\) −7.28165 1.36951i −0.731834 0.137641i
\(100\) 0 0
\(101\) 16.1271 1.60471 0.802355 0.596847i \(-0.203580\pi\)
0.802355 + 0.596847i \(0.203580\pi\)
\(102\) 0 0
\(103\) 0.170125i 0.0167629i −0.999965 0.00838147i \(-0.997332\pi\)
0.999965 0.00838147i \(-0.00266794\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.97324i 0.674129i 0.941482 + 0.337064i \(0.109434\pi\)
−0.941482 + 0.337064i \(0.890566\pi\)
\(108\) 0 0
\(109\) 1.35512 0.129797 0.0648984 0.997892i \(-0.479328\pi\)
0.0648984 + 0.997892i \(0.479328\pi\)
\(110\) 0 0
\(111\) 7.82532 6.49076i 0.742747 0.616076i
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 1.25255i 0.116800i
\(116\) 0 0
\(117\) −6.01877 1.13199i −0.556435 0.104653i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.90022 0.445474
\(122\) 0 0
\(123\) 7.10902 5.89662i 0.640999 0.531680i
\(124\) 0 0
\(125\) −1.45032 −0.129720
\(126\) 0 0
\(127\) −7.33399 −0.650787 −0.325393 0.945579i \(-0.605497\pi\)
−0.325393 + 0.945579i \(0.605497\pi\)
\(128\) 0 0
\(129\) −12.2608 + 10.1697i −1.07950 + 0.895396i
\(130\) 0 0
\(131\) −6.09665 −0.532667 −0.266333 0.963881i \(-0.585812\pi\)
−0.266333 + 0.963881i \(0.585812\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.366389 0.660370i −0.0315338 0.0568356i
\(136\) 0 0
\(137\) 20.0724i 1.71490i 0.514566 + 0.857451i \(0.327953\pi\)
−0.514566 + 0.857451i \(0.672047\pi\)
\(138\) 0 0
\(139\) 0.117694i 0.00998266i −0.999988 0.00499133i \(-0.998411\pi\)
0.999988 0.00499133i \(-0.00158880\pi\)
\(140\) 0 0
\(141\) 12.4032 10.2879i 1.04454 0.866401i
\(142\) 0 0
\(143\) −5.04187 −0.421623
\(144\) 0 0
\(145\) 1.03014i 0.0855489i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.95707i 0.733792i 0.930262 + 0.366896i \(0.119579\pi\)
−0.930262 + 0.366896i \(0.880421\pi\)
\(150\) 0 0
\(151\) −2.74263 −0.223192 −0.111596 0.993754i \(-0.535596\pi\)
−0.111596 + 0.993754i \(0.535596\pi\)
\(152\) 0 0
\(153\) 0.974184 5.17970i 0.0787581 0.418754i
\(154\) 0 0
\(155\) 0.523291i 0.0420318i
\(156\) 0 0
\(157\) 13.5450i 1.08101i 0.841342 + 0.540504i \(0.181766\pi\)
−0.841342 + 0.540504i \(0.818234\pi\)
\(158\) 0 0
\(159\) −5.74040 6.92069i −0.455244 0.548846i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.04857 −0.317108 −0.158554 0.987350i \(-0.550683\pi\)
−0.158554 + 0.987350i \(0.550683\pi\)
\(164\) 0 0
\(165\) −0.396923 0.478534i −0.0309004 0.0372538i
\(166\) 0 0
\(167\) 3.70521 0.286717 0.143359 0.989671i \(-0.454210\pi\)
0.143359 + 0.989671i \(0.454210\pi\)
\(168\) 0 0
\(169\) 8.83256 0.679428
\(170\) 0 0
\(171\) −12.5391 2.35832i −0.958890 0.180346i
\(172\) 0 0
\(173\) 22.4740 1.70867 0.854333 0.519726i \(-0.173966\pi\)
0.854333 + 0.519726i \(0.173966\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.9391 12.3913i 1.12289 0.931389i
\(178\) 0 0
\(179\) 3.67858i 0.274950i −0.990505 0.137475i \(-0.956101\pi\)
0.990505 0.137475i \(-0.0438986\pi\)
\(180\) 0 0
\(181\) 8.01062i 0.595425i −0.954656 0.297712i \(-0.903776\pi\)
0.954656 0.297712i \(-0.0962237\pi\)
\(182\) 0 0
\(183\) 5.95459 + 7.17892i 0.440176 + 0.530681i
\(184\) 0 0
\(185\) 0.853117 0.0627224
\(186\) 0 0
\(187\) 4.33899i 0.317298i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.12159i 0.0811554i −0.999176 0.0405777i \(-0.987080\pi\)
0.999176 0.0405777i \(-0.0129198\pi\)
\(192\) 0 0
\(193\) 18.3684 1.32219 0.661094 0.750303i \(-0.270092\pi\)
0.661094 + 0.750303i \(0.270092\pi\)
\(194\) 0 0
\(195\) −0.328083 0.395540i −0.0234945 0.0283252i
\(196\) 0 0
\(197\) 0.296699i 0.0211389i −0.999944 0.0105695i \(-0.996636\pi\)
0.999944 0.0105695i \(-0.00336442\pi\)
\(198\) 0 0
\(199\) 27.3518i 1.93892i 0.245251 + 0.969460i \(0.421130\pi\)
−0.245251 + 0.969460i \(0.578870\pi\)
\(200\) 0 0
\(201\) −6.86346 + 5.69293i −0.484111 + 0.401548i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.775025 0.0541301
\(206\) 0 0
\(207\) 25.4089 + 4.77884i 1.76604 + 0.332152i
\(208\) 0 0
\(209\) −10.5039 −0.726571
\(210\) 0 0
\(211\) 21.0295 1.44773 0.723864 0.689942i \(-0.242364\pi\)
0.723864 + 0.689942i \(0.242364\pi\)
\(212\) 0 0
\(213\) 8.61758 + 10.3894i 0.590467 + 0.711873i
\(214\) 0 0
\(215\) −1.33667 −0.0911599
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −14.4300 17.3970i −0.975089 1.17558i
\(220\) 0 0
\(221\) 3.58646i 0.241252i
\(222\) 0 0
\(223\) 6.89447i 0.461688i −0.972991 0.230844i \(-0.925851\pi\)
0.972991 0.230844i \(-0.0741487\pi\)
\(224\) 0 0
\(225\) −2.76084 + 14.6793i −0.184056 + 0.978617i
\(226\) 0 0
\(227\) −13.4147 −0.890363 −0.445182 0.895440i \(-0.646861\pi\)
−0.445182 + 0.895440i \(0.646861\pi\)
\(228\) 0 0
\(229\) 6.36254i 0.420448i −0.977653 0.210224i \(-0.932581\pi\)
0.977653 0.210224i \(-0.0674194\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.95707i 0.324748i 0.986729 + 0.162374i \(0.0519152\pi\)
−0.986729 + 0.162374i \(0.948085\pi\)
\(234\) 0 0
\(235\) 1.35220 0.0882079
\(236\) 0 0
\(237\) −7.62755 + 6.32672i −0.495463 + 0.410964i
\(238\) 0 0
\(239\) 17.3756i 1.12394i 0.827159 + 0.561968i \(0.189956\pi\)
−0.827159 + 0.561968i \(0.810044\pi\)
\(240\) 0 0
\(241\) 14.5060i 0.934416i −0.884147 0.467208i \(-0.845260\pi\)
0.884147 0.467208i \(-0.154740\pi\)
\(242\) 0 0
\(243\) −14.7940 + 4.91299i −0.949036 + 0.315168i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.68218 −0.552434
\(248\) 0 0
\(249\) −21.3059 + 17.6723i −1.35021 + 1.11994i
\(250\) 0 0
\(251\) −3.49783 −0.220781 −0.110391 0.993888i \(-0.535210\pi\)
−0.110391 + 0.993888i \(0.535210\pi\)
\(252\) 0 0
\(253\) 21.2848 1.33816
\(254\) 0 0
\(255\) 0.340398 0.282345i 0.0213166 0.0176811i
\(256\) 0 0
\(257\) −15.9356 −0.994036 −0.497018 0.867740i \(-0.665572\pi\)
−0.497018 + 0.867740i \(0.665572\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −20.8973 3.93031i −1.29351 0.243280i
\(262\) 0 0
\(263\) 14.3578i 0.885342i 0.896684 + 0.442671i \(0.145969\pi\)
−0.896684 + 0.442671i \(0.854031\pi\)
\(264\) 0 0
\(265\) 0.754493i 0.0463482i
\(266\) 0 0
\(267\) 11.5784 9.60373i 0.708584 0.587739i
\(268\) 0 0
\(269\) −7.36422 −0.449004 −0.224502 0.974474i \(-0.572076\pi\)
−0.224502 + 0.974474i \(0.572076\pi\)
\(270\) 0 0
\(271\) 12.5327i 0.761309i 0.924717 + 0.380654i \(0.124301\pi\)
−0.924717 + 0.380654i \(0.875699\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.2967i 0.741519i
\(276\) 0 0
\(277\) 32.4817 1.95164 0.975819 0.218580i \(-0.0701425\pi\)
0.975819 + 0.218580i \(0.0701425\pi\)
\(278\) 0 0
\(279\) −10.6154 1.99651i −0.635526 0.119528i
\(280\) 0 0
\(281\) 10.1758i 0.607037i 0.952826 + 0.303518i \(0.0981614\pi\)
−0.952826 + 0.303518i \(0.901839\pi\)
\(282\) 0 0
\(283\) 1.37024i 0.0814523i 0.999170 + 0.0407261i \(0.0129671\pi\)
−0.999170 + 0.0407261i \(0.987033\pi\)
\(284\) 0 0
\(285\) −0.683507 0.824043i −0.0404874 0.0488121i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.9135 −0.818442
\(290\) 0 0
\(291\) 7.35715 + 8.86986i 0.431284 + 0.519960i
\(292\) 0 0
\(293\) 16.9961 0.992923 0.496461 0.868059i \(-0.334632\pi\)
0.496461 + 0.868059i \(0.334632\pi\)
\(294\) 0 0
\(295\) 1.62866 0.0948243
\(296\) 0 0
\(297\) −11.2218 + 6.22614i −0.651156 + 0.361277i
\(298\) 0 0
\(299\) 17.5933 1.01745
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 21.4997 17.8330i 1.23513 1.02448i
\(304\) 0 0
\(305\) 0.782645i 0.0448141i
\(306\) 0 0
\(307\) 20.9023i 1.19296i 0.802629 + 0.596479i \(0.203434\pi\)
−0.802629 + 0.596479i \(0.796566\pi\)
\(308\) 0 0
\(309\) −0.188121 0.226800i −0.0107018 0.0129022i
\(310\) 0 0
\(311\) 11.4808 0.651017 0.325508 0.945539i \(-0.394465\pi\)
0.325508 + 0.945539i \(0.394465\pi\)
\(312\) 0 0
\(313\) 9.89777i 0.559455i 0.960079 + 0.279728i \(0.0902442\pi\)
−0.960079 + 0.279728i \(0.909756\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.63430i 0.372619i 0.982491 + 0.186310i \(0.0596527\pi\)
−0.982491 + 0.186310i \(0.940347\pi\)
\(318\) 0 0
\(319\) −17.5055 −0.980120
\(320\) 0 0
\(321\) 7.71086 + 9.29629i 0.430378 + 0.518868i
\(322\) 0 0
\(323\) 7.47181i 0.415742i
\(324\) 0 0
\(325\) 10.1640i 0.563799i
\(326\) 0 0
\(327\) 1.80656 1.49846i 0.0999029 0.0828650i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.7307 0.809675 0.404837 0.914389i \(-0.367328\pi\)
0.404837 + 0.914389i \(0.367328\pi\)
\(332\) 0 0
\(333\) 3.25489 17.3061i 0.178367 0.948371i
\(334\) 0 0
\(335\) −0.748254 −0.0408815
\(336\) 0 0
\(337\) −30.7209 −1.67347 −0.836737 0.547605i \(-0.815540\pi\)
−0.836737 + 0.547605i \(0.815540\pi\)
\(338\) 0 0
\(339\) 4.42311 + 5.33255i 0.240230 + 0.289624i
\(340\) 0 0
\(341\) −8.89242 −0.481551
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.38504 + 1.66982i 0.0745679 + 0.0898998i
\(346\) 0 0
\(347\) 16.7575i 0.899590i 0.893132 + 0.449795i \(0.148503\pi\)
−0.893132 + 0.449795i \(0.851497\pi\)
\(348\) 0 0
\(349\) 3.12385i 0.167216i 0.996499 + 0.0836080i \(0.0266443\pi\)
−0.996499 + 0.0836080i \(0.973356\pi\)
\(350\) 0 0
\(351\) −9.27558 + 5.14632i −0.495094 + 0.274690i
\(352\) 0 0
\(353\) −35.4901 −1.88895 −0.944473 0.328590i \(-0.893426\pi\)
−0.944473 + 0.328590i \(0.893426\pi\)
\(354\) 0 0
\(355\) 1.13266i 0.0601152i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.26791i 0.330807i −0.986226 0.165404i \(-0.947107\pi\)
0.986226 0.165404i \(-0.0528927\pi\)
\(360\) 0 0
\(361\) 0.912103 0.0480054
\(362\) 0 0
\(363\) 6.53266 5.41855i 0.342876 0.284400i
\(364\) 0 0
\(365\) 1.89662i 0.0992734i
\(366\) 0 0
\(367\) 16.8381i 0.878944i 0.898256 + 0.439472i \(0.144834\pi\)
−0.898256 + 0.439472i \(0.855166\pi\)
\(368\) 0 0
\(369\) 2.95695 15.7220i 0.153933 0.818454i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.23421 −0.0639051 −0.0319526 0.999489i \(-0.510173\pi\)
−0.0319526 + 0.999489i \(0.510173\pi\)
\(374\) 0 0
\(375\) −1.93347 + 1.60373i −0.0998439 + 0.0828161i
\(376\) 0 0
\(377\) −14.4695 −0.745215
\(378\) 0 0
\(379\) −14.3895 −0.739141 −0.369571 0.929203i \(-0.620495\pi\)
−0.369571 + 0.929203i \(0.620495\pi\)
\(380\) 0 0
\(381\) −9.77722 + 8.10977i −0.500902 + 0.415476i
\(382\) 0 0
\(383\) 9.91684 0.506727 0.253364 0.967371i \(-0.418463\pi\)
0.253364 + 0.967371i \(0.418463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.09978 + 27.1153i −0.259236 + 1.37835i
\(388\) 0 0
\(389\) 13.2861i 0.673633i −0.941570 0.336816i \(-0.890650\pi\)
0.941570 0.336816i \(-0.109350\pi\)
\(390\) 0 0
\(391\) 15.1406i 0.765695i
\(392\) 0 0
\(393\) −8.12767 + 6.74154i −0.409987 + 0.340066i
\(394\) 0 0
\(395\) −0.831556 −0.0418401
\(396\) 0 0
\(397\) 24.2961i 1.21939i 0.792638 + 0.609693i \(0.208707\pi\)
−0.792638 + 0.609693i \(0.791293\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.3327i 0.715741i −0.933771 0.357870i \(-0.883503\pi\)
0.933771 0.357870i \(-0.116497\pi\)
\(402\) 0 0
\(403\) −7.35017 −0.366138
\(404\) 0 0
\(405\) −1.21867 0.475218i −0.0605562 0.0236138i
\(406\) 0 0
\(407\) 14.4972i 0.718600i
\(408\) 0 0
\(409\) 20.0059i 0.989226i −0.869113 0.494613i \(-0.835310\pi\)
0.869113 0.494613i \(-0.164690\pi\)
\(410\) 0 0
\(411\) 22.1956 + 26.7593i 1.09483 + 1.31994i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.32277 −0.114020
\(416\) 0 0
\(417\) −0.130143 0.156902i −0.00637314 0.00768353i
\(418\) 0 0
\(419\) 27.7445 1.35541 0.677704 0.735335i \(-0.262975\pi\)
0.677704 + 0.735335i \(0.262975\pi\)
\(420\) 0 0
\(421\) −1.53586 −0.0748533 −0.0374267 0.999299i \(-0.511916\pi\)
−0.0374267 + 0.999299i \(0.511916\pi\)
\(422\) 0 0
\(423\) 5.15905 27.4305i 0.250842 1.33371i
\(424\) 0 0
\(425\) −8.74708 −0.424296
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −6.72151 + 5.57519i −0.324517 + 0.269173i
\(430\) 0 0
\(431\) 17.1216i 0.824718i −0.911021 0.412359i \(-0.864705\pi\)
0.911021 0.412359i \(-0.135295\pi\)
\(432\) 0 0
\(433\) 27.5219i 1.32262i −0.750113 0.661310i \(-0.770001\pi\)
0.750113 0.661310i \(-0.229999\pi\)
\(434\) 0 0
\(435\) −1.13911 1.37332i −0.0546162 0.0658459i
\(436\) 0 0
\(437\) 36.6528 1.75334
\(438\) 0 0
\(439\) 21.9303i 1.04667i −0.852126 0.523337i \(-0.824687\pi\)
0.852126 0.523337i \(-0.175313\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.5062i 0.974278i −0.873324 0.487139i \(-0.838040\pi\)
0.873324 0.487139i \(-0.161960\pi\)
\(444\) 0 0
\(445\) 1.26227 0.0598374
\(446\) 0 0
\(447\) 9.90453 + 11.9410i 0.468468 + 0.564790i
\(448\) 0 0
\(449\) 18.7692i 0.885773i 0.896578 + 0.442886i \(0.146046\pi\)
−0.896578 + 0.442886i \(0.853954\pi\)
\(450\) 0 0
\(451\) 13.1702i 0.620160i
\(452\) 0 0
\(453\) −3.65630 + 3.03274i −0.171788 + 0.142491i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.59340 0.355204 0.177602 0.984102i \(-0.443166\pi\)
0.177602 + 0.984102i \(0.443166\pi\)
\(458\) 0 0
\(459\) −4.42887 7.98248i −0.206722 0.372590i
\(460\) 0 0
\(461\) −29.2727 −1.36337 −0.681683 0.731648i \(-0.738752\pi\)
−0.681683 + 0.731648i \(0.738752\pi\)
\(462\) 0 0
\(463\) 11.8326 0.549906 0.274953 0.961458i \(-0.411338\pi\)
0.274953 + 0.961458i \(0.411338\pi\)
\(464\) 0 0
\(465\) −0.578644 0.697619i −0.0268340 0.0323513i
\(466\) 0 0
\(467\) −5.16565 −0.239038 −0.119519 0.992832i \(-0.538135\pi\)
−0.119519 + 0.992832i \(0.538135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.9777 + 18.0573i 0.690138 + 0.832038i
\(472\) 0 0
\(473\) 22.7143i 1.04440i
\(474\) 0 0
\(475\) 21.1751i 0.971580i
\(476\) 0 0
\(477\) −15.3055 2.87862i −0.700790 0.131803i
\(478\) 0 0
\(479\) 19.7099 0.900569 0.450284 0.892885i \(-0.351323\pi\)
0.450284 + 0.892885i \(0.351323\pi\)
\(480\) 0 0
\(481\) 11.9829i 0.546373i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.966991i 0.0439088i
\(486\) 0 0
\(487\) −5.00720 −0.226898 −0.113449 0.993544i \(-0.536190\pi\)
−0.113449 + 0.993544i \(0.536190\pi\)
\(488\) 0 0
\(489\) −5.39730 + 4.47682i −0.244074 + 0.202449i
\(490\) 0 0
\(491\) 3.55902i 0.160616i 0.996770 + 0.0803081i \(0.0255904\pi\)
−0.996770 + 0.0803081i \(0.974410\pi\)
\(492\) 0 0
\(493\) 12.4523i 0.560823i
\(494\) 0 0
\(495\) −1.05830 0.199043i −0.0475673 0.00894633i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −0.809404 −0.0362339 −0.0181170 0.999836i \(-0.505767\pi\)
−0.0181170 + 0.999836i \(0.505767\pi\)
\(500\) 0 0
\(501\) 4.93955 4.09713i 0.220683 0.183046i
\(502\) 0 0
\(503\) −9.47070 −0.422278 −0.211139 0.977456i \(-0.567717\pi\)
−0.211139 + 0.977456i \(0.567717\pi\)
\(504\) 0 0
\(505\) 2.34390 0.104302
\(506\) 0 0
\(507\) 11.7750 9.76685i 0.522947 0.433761i
\(508\) 0 0
\(509\) −10.4881 −0.464876 −0.232438 0.972611i \(-0.574670\pi\)
−0.232438 + 0.972611i \(0.574670\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −19.3241 + 10.7215i −0.853182 + 0.473366i
\(514\) 0 0
\(515\) 0.0247258i 0.00108955i
\(516\) 0 0
\(517\) 22.9783i 1.01058i
\(518\) 0 0
\(519\) 29.9609 24.8512i 1.31514 1.09085i
\(520\) 0 0
\(521\) −9.55707 −0.418703 −0.209351 0.977840i \(-0.567135\pi\)
−0.209351 + 0.977840i \(0.567135\pi\)
\(522\) 0 0
\(523\) 27.7480i 1.21334i −0.794955 0.606668i \(-0.792506\pi\)
0.794955 0.606668i \(-0.207494\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.32549i 0.275543i
\(528\) 0 0
\(529\) −51.2720 −2.22922
\(530\) 0 0
\(531\) 6.21383 33.0387i 0.269657 1.43376i
\(532\) 0 0
\(533\) 10.8860i 0.471526i
\(534\) 0 0
\(535\) 1.01348i 0.0438166i
\(536\) 0 0
\(537\) −4.06769 4.90405i −0.175534 0.211625i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.15555 −0.0496812 −0.0248406 0.999691i \(-0.507908\pi\)
−0.0248406 + 0.999691i \(0.507908\pi\)
\(542\) 0 0
\(543\) −8.85797 10.6793i −0.380132 0.458291i
\(544\) 0 0
\(545\) 0.196951 0.00843645
\(546\) 0 0
\(547\) 16.1394 0.690070 0.345035 0.938590i \(-0.387867\pi\)
0.345035 + 0.938590i \(0.387867\pi\)
\(548\) 0 0
\(549\) 15.8766 + 2.98603i 0.677596 + 0.127440i
\(550\) 0 0
\(551\) −30.1447 −1.28421
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.13732 0.943358i 0.0482766 0.0400433i
\(556\) 0 0
\(557\) 37.0744i 1.57089i −0.618929 0.785447i \(-0.712433\pi\)
0.618929 0.785447i \(-0.287567\pi\)
\(558\) 0 0
\(559\) 18.7749i 0.794092i
\(560\) 0 0
\(561\) −4.79796 5.78447i −0.202570 0.244221i
\(562\) 0 0
\(563\) 15.5917 0.657111 0.328556 0.944485i \(-0.393438\pi\)
0.328556 + 0.944485i \(0.393438\pi\)
\(564\) 0 0
\(565\) 0.581354i 0.0244578i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0429i 0.630633i −0.948987 0.315316i \(-0.897889\pi\)
0.948987 0.315316i \(-0.102111\pi\)
\(570\) 0 0
\(571\) 5.62668 0.235469 0.117735 0.993045i \(-0.462437\pi\)
0.117735 + 0.993045i \(0.462437\pi\)
\(572\) 0 0
\(573\) −1.24023 1.49523i −0.0518114 0.0624643i
\(574\) 0 0
\(575\) 42.9086i 1.78941i
\(576\) 0 0
\(577\) 22.2024i 0.924297i −0.886802 0.462149i \(-0.847079\pi\)
0.886802 0.462149i \(-0.152921\pi\)
\(578\) 0 0
\(579\) 24.4876 20.3114i 1.01767 0.844113i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.8213 −0.531004
\(584\) 0 0
\(585\) −0.874759 0.164522i −0.0361668 0.00680216i
\(586\) 0 0
\(587\) 20.9245 0.863648 0.431824 0.901958i \(-0.357870\pi\)
0.431824 + 0.901958i \(0.357870\pi\)
\(588\) 0 0
\(589\) −15.3129 −0.630956
\(590\) 0 0
\(591\) −0.328083 0.395540i −0.0134955 0.0162703i
\(592\) 0 0
\(593\) 21.1690 0.869307 0.434654 0.900598i \(-0.356871\pi\)
0.434654 + 0.900598i \(0.356871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 30.2450 + 36.4637i 1.23785 + 1.49236i
\(598\) 0 0
\(599\) 5.29601i 0.216389i 0.994130 + 0.108195i \(0.0345070\pi\)
−0.994130 + 0.108195i \(0.965493\pi\)
\(600\) 0 0
\(601\) 37.5346i 1.53107i −0.643396 0.765533i \(-0.722475\pi\)
0.643396 0.765533i \(-0.277525\pi\)
\(602\) 0 0
\(603\) −2.85481 + 15.1789i −0.116257 + 0.618133i
\(604\) 0 0
\(605\) 0.712191 0.0289547
\(606\) 0 0
\(607\) 39.5050i 1.60346i 0.597687 + 0.801729i \(0.296086\pi\)
−0.597687 + 0.801729i \(0.703914\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.9931i 0.768377i
\(612\) 0 0
\(613\) 14.3947 0.581397 0.290699 0.956815i \(-0.406112\pi\)
0.290699 + 0.956815i \(0.406112\pi\)
\(614\) 0 0
\(615\) 1.03321 0.857006i 0.0416632 0.0345578i
\(616\) 0 0
\(617\) 12.1573i 0.489435i 0.969594 + 0.244718i \(0.0786952\pi\)
−0.969594 + 0.244718i \(0.921305\pi\)
\(618\) 0 0
\(619\) 19.4835i 0.783107i 0.920155 + 0.391553i \(0.128062\pi\)
−0.920155 + 0.391553i \(0.871938\pi\)
\(620\) 0 0
\(621\) 39.1579 21.7257i 1.57135 0.871824i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 24.6836 0.987344
\(626\) 0 0
\(627\) −14.0032 + 11.6150i −0.559232 + 0.463858i
\(628\) 0 0
\(629\) 10.3124 0.411182
\(630\) 0 0
\(631\) −31.3846 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(632\) 0 0
\(633\) 28.0352 23.2539i 1.11430 0.924261i
\(634\) 0 0
\(635\) −1.06591 −0.0422994
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 22.9768 + 4.32142i 0.908950 + 0.170953i
\(640\) 0 0
\(641\) 41.5828i 1.64242i −0.570624 0.821211i \(-0.693299\pi\)
0.570624 0.821211i \(-0.306701\pi\)
\(642\) 0 0
\(643\) 13.5290i 0.533531i −0.963761 0.266766i \(-0.914045\pi\)
0.963761 0.266766i \(-0.0859549\pi\)
\(644\) 0 0
\(645\) −1.78196 + 1.47806i −0.0701646 + 0.0581984i
\(646\) 0 0
\(647\) −30.0884 −1.18290 −0.591449 0.806343i \(-0.701444\pi\)
−0.591449 + 0.806343i \(0.701444\pi\)
\(648\) 0 0
\(649\) 27.6762i 1.08639i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.2138i 0.830161i 0.909785 + 0.415081i \(0.136247\pi\)
−0.909785 + 0.415081i \(0.863753\pi\)
\(654\) 0 0
\(655\) −0.886078 −0.0346219
\(656\) 0 0
\(657\) −38.4743 7.23615i −1.50103 0.282309i
\(658\) 0 0
\(659\) 2.67926i 0.104369i −0.998637 0.0521846i \(-0.983382\pi\)
0.998637 0.0521846i \(-0.0166184\pi\)
\(660\) 0 0
\(661\) 5.54008i 0.215484i −0.994179 0.107742i \(-0.965638\pi\)
0.994179 0.107742i \(-0.0343621\pi\)
\(662\) 0 0
\(663\) −3.96583 4.78125i −0.154020 0.185688i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 61.0844 2.36520
\(668\) 0 0
\(669\) −7.62376 9.19128i −0.294752 0.355355i
\(670\) 0 0
\(671\) 13.2997 0.513429
\(672\) 0 0
\(673\) −17.1946 −0.662804 −0.331402 0.943490i \(-0.607522\pi\)
−0.331402 + 0.943490i \(0.607522\pi\)
\(674\) 0 0
\(675\) 12.5514 + 22.6223i 0.483105 + 0.870734i
\(676\) 0 0
\(677\) −14.4664 −0.555988 −0.277994 0.960583i \(-0.589670\pi\)
−0.277994 + 0.960583i \(0.589670\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −17.8836 + 14.8337i −0.685301 + 0.568427i
\(682\) 0 0
\(683\) 32.4882i 1.24313i −0.783363 0.621564i \(-0.786498\pi\)
0.783363 0.621564i \(-0.213502\pi\)
\(684\) 0 0
\(685\) 2.91730i 0.111464i
\(686\) 0 0
\(687\) −7.03555 8.48214i −0.268423 0.323614i
\(688\) 0 0
\(689\) −10.5976 −0.403738
\(690\) 0 0
\(691\) 32.4426i 1.23417i −0.786895 0.617087i \(-0.788312\pi\)
0.786895 0.617087i \(-0.211688\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0171055i 0.000648847i
\(696\) 0 0
\(697\) 9.36842 0.354854
\(698\) 0 0
\(699\) 5.48142 + 6.60845i 0.207326 + 0.249955i
\(700\) 0 0
\(701\) 30.3777i 1.14735i −0.819084 0.573674i \(-0.805518\pi\)
0.819084 0.573674i \(-0.194482\pi\)
\(702\) 0 0
\(703\) 24.9644i 0.941551i
\(704\) 0 0
\(705\) 1.80267 1.49523i 0.0678925 0.0563138i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −32.5139 −1.22108 −0.610542 0.791984i \(-0.709048\pi\)
−0.610542 + 0.791984i \(0.709048\pi\)
\(710\) 0 0
\(711\) −3.17263 + 16.8688i −0.118983 + 0.632628i
\(712\) 0 0
\(713\) 31.0295 1.16207
\(714\) 0 0
\(715\) −0.732778 −0.0274044
\(716\) 0 0
\(717\) 19.2136 + 23.1641i 0.717545 + 0.865079i
\(718\) 0 0
\(719\) 10.5973 0.395214 0.197607 0.980281i \(-0.436683\pi\)
0.197607 + 0.980281i \(0.436683\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16.0405 19.3385i −0.596551 0.719208i
\(724\) 0 0
\(725\) 35.2898i 1.31063i
\(726\) 0 0
\(727\) 31.3600i 1.16308i −0.813518 0.581540i \(-0.802451\pi\)
0.813518 0.581540i \(-0.197549\pi\)
\(728\) 0 0
\(729\) −14.2898 + 22.9086i −0.529251 + 0.848466i
\(730\) 0 0
\(731\) −16.1575 −0.597606
\(732\) 0 0
\(733\) 3.61634i 0.133573i −0.997767 0.0667863i \(-0.978725\pi\)
0.997767 0.0667863i \(-0.0212746\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.7153i 0.468373i
\(738\) 0 0
\(739\) −38.6926 −1.42333 −0.711665 0.702519i \(-0.752058\pi\)
−0.711665 + 0.702519i \(0.752058\pi\)
\(740\) 0 0
\(741\) −11.5745 + 9.60056i −0.425201 + 0.352685i
\(742\) 0 0
\(743\) 45.1194i 1.65527i −0.561266 0.827635i \(-0.689686\pi\)
0.561266 0.827635i \(-0.310314\pi\)
\(744\) 0 0
\(745\) 1.30181i 0.0476945i
\(746\) 0 0
\(747\) −8.86207 + 47.1193i −0.324246 + 1.72400i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.5683 −0.860018 −0.430009 0.902825i \(-0.641490\pi\)
−0.430009 + 0.902825i \(0.641490\pi\)
\(752\) 0 0
\(753\) −4.66309 + 3.86782i −0.169932 + 0.140951i
\(754\) 0 0
\(755\) −0.398610 −0.0145069
\(756\) 0 0
\(757\) 26.2967 0.955770 0.477885 0.878422i \(-0.341404\pi\)
0.477885 + 0.878422i \(0.341404\pi\)
\(758\) 0 0
\(759\) 28.3756 23.5363i 1.02997 0.854313i
\(760\) 0 0
\(761\) −25.9560 −0.940903 −0.470452 0.882426i \(-0.655909\pi\)
−0.470452 + 0.882426i \(0.655909\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.141586 0.752810i 0.00511907 0.0272179i
\(766\) 0 0
\(767\) 22.8762i 0.826013i
\(768\) 0 0
\(769\) 36.9215i 1.33142i 0.746209 + 0.665712i \(0.231872\pi\)
−0.746209 + 0.665712i \(0.768128\pi\)
\(770\) 0 0
\(771\) −21.2444 + 17.6212i −0.765097 + 0.634614i
\(772\) 0 0
\(773\) −11.5857 −0.416708 −0.208354 0.978054i \(-0.566811\pi\)
−0.208354 + 0.978054i \(0.566811\pi\)
\(774\) 0 0
\(775\) 17.9264i 0.643937i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22.6793i 0.812569i
\(780\) 0 0
\(781\) 19.2475 0.688730
\(782\) 0 0
\(783\) −32.2050 + 17.8681i −1.15091 + 0.638555i
\(784\) 0 0
\(785\) 1.96861i 0.0702627i
\(786\) 0 0
\(787\) 29.9244i 1.06669i −0.845898 0.533344i \(-0.820935\pi\)
0.845898 0.533344i \(-0.179065\pi\)
\(788\) 0 0
\(789\) 15.8766 + 19.1410i 0.565221 + 0.681436i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 10.9931 0.390375
\(794\) 0 0
\(795\) −0.834302 1.00584i −0.0295896 0.0356736i
\(796\) 0 0
\(797\) 20.2866 0.718587 0.359293 0.933225i \(-0.383018\pi\)
0.359293 + 0.933225i \(0.383018\pi\)
\(798\) 0 0
\(799\) 16.3453 0.578254
\(800\) 0 0
\(801\) 4.81594 25.6062i 0.170163 0.904750i
\(802\) 0 0
\(803\) −32.2297 −1.13736
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.81751 + 8.14319i −0.345593 + 0.286654i
\(808\) 0 0
\(809\) 2.49001i 0.0875442i 0.999042 + 0.0437721i \(0.0139375\pi\)
−0.999042 + 0.0437721i \(0.986062\pi\)
\(810\) 0 0
\(811\) 15.9838i 0.561269i 0.959815 + 0.280634i \(0.0905448\pi\)
−0.959815 + 0.280634i \(0.909455\pi\)
\(812\) 0 0
\(813\) 13.8584 + 16.7078i 0.486036 + 0.585970i
\(814\) 0 0
\(815\) −0.588413 −0.0206112
\(816\) 0 0
\(817\) 39.1144i 1.36844i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.75927i 0.0962991i 0.998840 + 0.0481495i \(0.0153324\pi\)
−0.998840 + 0.0481495i \(0.984668\pi\)
\(822\) 0 0
\(823\) −28.4424 −0.991440 −0.495720 0.868482i \(-0.665096\pi\)
−0.495720 + 0.868482i \(0.665096\pi\)
\(824\) 0 0
\(825\) 13.5974 + 16.3932i 0.473401 + 0.570738i
\(826\) 0 0
\(827\) 30.6070i 1.06431i 0.846647 + 0.532154i \(0.178617\pi\)
−0.846647 + 0.532154i \(0.821383\pi\)
\(828\) 0 0
\(829\) 10.0620i 0.349469i −0.984616 0.174734i \(-0.944093\pi\)
0.984616 0.174734i \(-0.0559067\pi\)
\(830\) 0 0
\(831\) 43.3026 35.9176i 1.50215 1.24597i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.538509 0.0186359
\(836\) 0 0
\(837\) −16.3595 + 9.07663i −0.565465 + 0.313734i
\(838\) 0 0
\(839\) 13.3067 0.459400 0.229700 0.973262i \(-0.426226\pi\)
0.229700 + 0.973262i \(0.426226\pi\)
\(840\) 0 0
\(841\) −21.2383 −0.732354
\(842\) 0 0
\(843\) 11.2522 + 13.5657i 0.387545 + 0.467229i
\(844\) 0 0
\(845\) 1.28371 0.0441610
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.51518 + 1.82672i 0.0520009 + 0.0626928i
\(850\) 0 0
\(851\) 50.5871i 1.73411i
\(852\) 0 0
\(853\) 42.4736i 1.45427i 0.686495 + 0.727134i \(0.259148\pi\)
−0.686495 + 0.727134i \(0.740852\pi\)
\(854\) 0 0
\(855\) −1.82242 0.342755i −0.0623253 0.0117220i
\(856\) 0 0
\(857\) −9.55707 −0.326463 −0.163232 0.986588i \(-0.552192\pi\)
−0.163232 + 0.986588i \(0.552192\pi\)
\(858\) 0 0
\(859\) 2.11241i 0.0720745i −0.999350 0.0360372i \(-0.988527\pi\)
0.999350 0.0360372i \(-0.0114735\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 35.2974i 1.20154i −0.799423 0.600768i \(-0.794861\pi\)
0.799423 0.600768i \(-0.205139\pi\)
\(864\) 0 0
\(865\) 3.26634 0.111059
\(866\) 0 0
\(867\) −18.5486 + 15.3853i −0.629945 + 0.522511i
\(868\) 0 0
\(869\) 14.1308i 0.479356i
\(870\) 0 0
\(871\) 10.5100i 0.356118i
\(872\) 0 0
\(873\) 19.6162 + 3.68936i 0.663907 + 0.124866i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.7377 −0.565193 −0.282596 0.959239i \(-0.591196\pi\)
−0.282596 + 0.959239i \(0.591196\pi\)
\(878\) 0 0
\(879\) 22.6581 18.7939i 0.764240 0.633903i
\(880\) 0 0
\(881\) −42.6152 −1.43574 −0.717871 0.696176i \(-0.754883\pi\)
−0.717871 + 0.696176i \(0.754883\pi\)
\(882\) 0 0
\(883\) −15.2392 −0.512839 −0.256419 0.966566i \(-0.582543\pi\)
−0.256419 + 0.966566i \(0.582543\pi\)
\(884\) 0 0
\(885\) 2.17123 1.80094i 0.0729850 0.0605378i
\(886\) 0 0
\(887\) −56.1267 −1.88455 −0.942275 0.334841i \(-0.891317\pi\)
−0.942275 + 0.334841i \(0.891317\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.07550 + 20.7091i −0.270539 + 0.693782i
\(892\) 0 0
\(893\) 39.5689i 1.32412i
\(894\) 0 0
\(895\) 0.534639i 0.0178710i
\(896\) 0 0
\(897\) 23.4543 19.4543i 0.783116 0.649560i
\(898\) 0 0
\(899\) −25.5200 −0.851138
\(900\) 0 0
\(901\) 9.12024i 0.303839i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.16425i 0.0387011i
\(906\) 0 0
\(907\) 40.7249 1.35225 0.676124 0.736788i \(-0.263658\pi\)
0.676124 + 0.736788i \(0.263658\pi\)
\(908\) 0 0
\(909\) 8.94266 47.5478i 0.296609 1.57706i
\(910\) 0 0
\(911\) 31.0308i 1.02810i 0.857761 + 0.514049i \(0.171855\pi\)
−0.857761 + 0.514049i \(0.828145\pi\)
\(912\) 0 0
\(913\) 39.4714i 1.30631i
\(914\) 0 0
\(915\) 0.865432 + 1.04337i 0.0286103 + 0.0344929i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −51.2356 −1.69011 −0.845053 0.534683i \(-0.820431\pi\)
−0.845053 + 0.534683i \(0.820431\pi\)
\(920\) 0 0
\(921\) 23.1133 + 27.8656i 0.761609 + 0.918204i
\(922\) 0 0
\(923\) 15.9093 0.523662
\(924\) 0 0
\(925\) −29.2253 −0.960922
\(926\) 0 0
\(927\) −0.501582 0.0943362i −0.0164741 0.00309841i
\(928\) 0 0
\(929\) 3.74232 0.122781 0.0613907 0.998114i \(-0.480446\pi\)
0.0613907 + 0.998114i \(0.480446\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 15.3055 12.6952i 0.501079 0.415623i
\(934\) 0 0
\(935\) 0.630623i 0.0206236i
\(936\) 0 0
\(937\) 3.23951i 0.105830i 0.998599 + 0.0529150i \(0.0168512\pi\)
−0.998599 + 0.0529150i \(0.983149\pi\)
\(938\) 0 0
\(939\) 10.9447 + 13.1951i 0.357168 + 0.430606i
\(940\) 0 0
\(941\) 22.0683 0.719407 0.359703 0.933067i \(-0.382878\pi\)
0.359703 + 0.933067i \(0.382878\pi\)
\(942\) 0 0
\(943\) 45.9566i 1.49655i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.2595i 1.17827i 0.808033 + 0.589137i \(0.200532\pi\)
−0.808033 + 0.589137i \(0.799468\pi\)
\(948\) 0 0
\(949\) −26.6399 −0.864769
\(950\) 0 0
\(951\) 7.33606 + 8.84443i 0.237888 + 0.286800i
\(952\) 0 0
\(953\) 13.4656i 0.436192i −0.975927 0.218096i \(-0.930015\pi\)
0.975927 0.218096i \(-0.0699846\pi\)
\(954\) 0 0
\(955\) 0.163010i 0.00527489i
\(956\) 0 0
\(957\) −23.3372 + 19.3572i −0.754386 + 0.625729i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0364 0.581820
\(962\) 0 0
\(963\) 20.5593 + 3.86673i 0.662513 + 0.124604i
\(964\) 0 0
\(965\) 2.66964 0.0859388
\(966\) 0 0
\(967\) −9.25940 −0.297762 −0.148881 0.988855i \(-0.547567\pi\)
−0.148881 + 0.988855i \(0.547567\pi\)
\(968\) 0 0
\(969\) −8.26216 9.96094i −0.265419 0.319992i
\(970\) 0 0
\(971\) 39.9290 1.28138 0.640691 0.767798i \(-0.278648\pi\)
0.640691 + 0.767798i \(0.278648\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 11.2392 + 13.5501i 0.359941 + 0.433949i
\(976\) 0 0
\(977\) 60.5408i 1.93687i 0.249261 + 0.968436i \(0.419812\pi\)
−0.249261 + 0.968436i \(0.580188\pi\)
\(978\) 0 0
\(979\) 21.4501i 0.685548i
\(980\) 0 0
\(981\) 0.751427 3.99531i 0.0239912 0.127560i
\(982\) 0 0
\(983\) −39.4054 −1.25684 −0.628419 0.777875i \(-0.716298\pi\)
−0.628419 + 0.777875i \(0.716298\pi\)
\(984\) 0 0
\(985\) 0.0431217i 0.00137397i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 79.2602i 2.52033i
\(990\) 0 0
\(991\) 30.1015 0.956205 0.478102 0.878304i \(-0.341325\pi\)
0.478102 + 0.878304i \(0.341325\pi\)
\(992\) 0 0
\(993\) 19.6381 16.2889i 0.623196 0.516914i
\(994\) 0 0
\(995\) 3.97527i 0.126025i
\(996\) 0 0
\(997\) 7.04907i 0.223246i 0.993751 + 0.111623i \(0.0356049\pi\)
−0.993751 + 0.111623i \(0.964395\pi\)
\(998\) 0 0
\(999\) −14.7975 26.6707i −0.468173 0.843822i
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 1176.2.k.a.881.13 16
3.2 odd 2 inner 1176.2.k.a.881.3 16
4.3 odd 2 2352.2.k.i.881.4 16
7.2 even 3 168.2.u.a.17.5 16
7.3 odd 6 168.2.u.a.89.7 yes 16
7.4 even 3 1176.2.u.b.1097.2 16
7.5 odd 6 1176.2.u.b.521.4 16
7.6 odd 2 inner 1176.2.k.a.881.4 16
12.11 even 2 2352.2.k.i.881.14 16
21.2 odd 6 168.2.u.a.17.7 yes 16
21.5 even 6 1176.2.u.b.521.2 16
21.11 odd 6 1176.2.u.b.1097.4 16
21.17 even 6 168.2.u.a.89.5 yes 16
21.20 even 2 inner 1176.2.k.a.881.14 16
28.3 even 6 336.2.bc.f.257.2 16
28.23 odd 6 336.2.bc.f.17.4 16
28.27 even 2 2352.2.k.i.881.13 16
84.23 even 6 336.2.bc.f.17.2 16
84.59 odd 6 336.2.bc.f.257.4 16
84.83 odd 2 2352.2.k.i.881.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.u.a.17.5 16 7.2 even 3
168.2.u.a.17.7 yes 16 21.2 odd 6
168.2.u.a.89.5 yes 16 21.17 even 6
168.2.u.a.89.7 yes 16 7.3 odd 6
336.2.bc.f.17.2 16 84.23 even 6
336.2.bc.f.17.4 16 28.23 odd 6
336.2.bc.f.257.2 16 28.3 even 6
336.2.bc.f.257.4 16 84.59 odd 6
1176.2.k.a.881.3 16 3.2 odd 2 inner
1176.2.k.a.881.4 16 7.6 odd 2 inner
1176.2.k.a.881.13 16 1.1 even 1 trivial
1176.2.k.a.881.14 16 21.20 even 2 inner
1176.2.u.b.521.2 16 21.5 even 6
1176.2.u.b.521.4 16 7.5 odd 6
1176.2.u.b.1097.2 16 7.4 even 3
1176.2.u.b.1097.4 16 21.11 odd 6
2352.2.k.i.881.3 16 84.83 odd 2
2352.2.k.i.881.4 16 4.3 odd 2
2352.2.k.i.881.13 16 28.27 even 2
2352.2.k.i.881.14 16 12.11 even 2