Properties

Label 3024.2.cc.a.2897.3
Level $3024$
Weight $2$
Character 3024.2897
Analytic conductor $24.147$
Analytic rank $0$
Dimension $12$
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] = [3024,2,Mod(881,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("3024.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cc (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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 2897.3
Root \(1.29589 - 0.748185i\) of defining polynomial
Character \(\chi\) \(=\) 3024.2897
Dual form 3024.2.cc.a.881.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.717144 - 1.24213i) q^{5} +(-2.16235 - 1.52455i) q^{7} +O(q^{10})\) \(q+(-0.717144 - 1.24213i) q^{5} +(-2.16235 - 1.52455i) q^{7} +(2.80150 + 1.61745i) q^{11} +(-4.43334 + 2.55959i) q^{13} +1.09132 q^{17} +4.48911i q^{19} +(3.47141 - 2.00422i) q^{23} +(1.47141 - 2.54856i) q^{25} +(-1.02859 - 0.593857i) q^{29} +(-3.24275 + 1.87220i) q^{31} +(-0.342971 + 3.77924i) q^{35} -0.239123 q^{37} +(3.71620 + 6.43664i) q^{41} +(3.82326 - 6.62208i) q^{43} +(2.11042 - 3.65536i) q^{47} +(2.35150 + 6.59321i) q^{49} +7.01414i q^{53} -4.63977i q^{55} +(4.73531 + 8.20179i) q^{59} +(2.82757 + 1.63250i) q^{61} +(6.35868 + 3.67119i) q^{65} +(0.330095 + 0.571741i) q^{67} -3.82347i q^{71} -7.31073i q^{73} +(-3.59195 - 7.76852i) q^{77} +(1.83009 - 3.16982i) q^{79} +(5.45245 - 9.44392i) q^{83} +(-0.782630 - 1.35556i) q^{85} +13.6915 q^{89} +(13.4887 + 1.22412i) q^{91} +(5.57605 - 3.21934i) q^{95} +(-2.69709 - 1.55716i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{7} + 24 q^{23} - 30 q^{29} - 4 q^{37} + 10 q^{43} + 6 q^{49} + 78 q^{65} - 12 q^{67} + 24 q^{77} + 6 q^{79} - 6 q^{85} + 24 q^{91} + 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.717144 1.24213i −0.320716 0.555497i 0.659920 0.751336i \(-0.270590\pi\)
−0.980636 + 0.195839i \(0.937257\pi\)
\(6\) 0 0
\(7\) −2.16235 1.52455i −0.817291 0.576225i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.80150 + 1.61745i 0.844686 + 0.487679i 0.858854 0.512220i \(-0.171177\pi\)
−0.0141686 + 0.999900i \(0.504510\pi\)
\(12\) 0 0
\(13\) −4.43334 + 2.55959i −1.22959 + 0.709903i −0.966944 0.254990i \(-0.917928\pi\)
−0.262644 + 0.964893i \(0.584594\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.09132 0.264683 0.132341 0.991204i \(-0.457750\pi\)
0.132341 + 0.991204i \(0.457750\pi\)
\(18\) 0 0
\(19\) 4.48911i 1.02987i 0.857228 + 0.514936i \(0.172184\pi\)
−0.857228 + 0.514936i \(0.827816\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.47141 2.00422i 0.723839 0.417909i −0.0923250 0.995729i \(-0.529430\pi\)
0.816164 + 0.577820i \(0.196097\pi\)
\(24\) 0 0
\(25\) 1.47141 2.54856i 0.294282 0.509711i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.02859 0.593857i −0.191004 0.110276i 0.401448 0.915882i \(-0.368507\pi\)
−0.592453 + 0.805605i \(0.701840\pi\)
\(30\) 0 0
\(31\) −3.24275 + 1.87220i −0.582414 + 0.336257i −0.762092 0.647468i \(-0.775828\pi\)
0.179678 + 0.983726i \(0.442494\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.342971 + 3.77924i −0.0579728 + 0.638808i
\(36\) 0 0
\(37\) −0.239123 −0.0393116 −0.0196558 0.999807i \(-0.506257\pi\)
−0.0196558 + 0.999807i \(0.506257\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.71620 + 6.43664i 0.580373 + 1.00523i 0.995435 + 0.0954418i \(0.0304264\pi\)
−0.415062 + 0.909793i \(0.636240\pi\)
\(42\) 0 0
\(43\) 3.82326 6.62208i 0.583041 1.00986i −0.412075 0.911150i \(-0.635196\pi\)
0.995116 0.0987075i \(-0.0314708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.11042 3.65536i 0.307837 0.533189i −0.670052 0.742314i \(-0.733728\pi\)
0.977889 + 0.209125i \(0.0670615\pi\)
\(48\) 0 0
\(49\) 2.35150 + 6.59321i 0.335929 + 0.941887i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.01414i 0.963466i 0.876318 + 0.481733i \(0.159992\pi\)
−0.876318 + 0.481733i \(0.840008\pi\)
\(54\) 0 0
\(55\) 4.63977i 0.625627i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.73531 + 8.20179i 0.616484 + 1.06778i 0.990122 + 0.140208i \(0.0447770\pi\)
−0.373638 + 0.927575i \(0.621890\pi\)
\(60\) 0 0
\(61\) 2.82757 + 1.63250i 0.362033 + 0.209020i 0.669972 0.742386i \(-0.266306\pi\)
−0.307939 + 0.951406i \(0.599639\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.35868 + 3.67119i 0.788698 + 0.455355i
\(66\) 0 0
\(67\) 0.330095 + 0.571741i 0.0403275 + 0.0698493i 0.885485 0.464669i \(-0.153827\pi\)
−0.845157 + 0.534518i \(0.820493\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.82347i 0.453762i −0.973922 0.226881i \(-0.927147\pi\)
0.973922 0.226881i \(-0.0728529\pi\)
\(72\) 0 0
\(73\) 7.31073i 0.855656i −0.903860 0.427828i \(-0.859279\pi\)
0.903860 0.427828i \(-0.140721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.59195 7.76852i −0.409341 0.885305i
\(78\) 0 0
\(79\) 1.83009 3.16982i 0.205902 0.356632i −0.744518 0.667602i \(-0.767321\pi\)
0.950420 + 0.310970i \(0.100654\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.45245 9.44392i 0.598484 1.03660i −0.394561 0.918870i \(-0.629103\pi\)
0.993045 0.117735i \(-0.0375634\pi\)
\(84\) 0 0
\(85\) −0.782630 1.35556i −0.0848882 0.147031i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.6915 1.45129 0.725646 0.688068i \(-0.241541\pi\)
0.725646 + 0.688068i \(0.241541\pi\)
\(90\) 0 0
\(91\) 13.4887 + 1.22412i 1.41399 + 0.128322i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.57605 3.21934i 0.572091 0.330297i
\(96\) 0 0
\(97\) −2.69709 1.55716i −0.273848 0.158106i 0.356787 0.934186i \(-0.383872\pi\)
−0.630635 + 0.776080i \(0.717205\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.54471 6.13962i 0.352712 0.610915i −0.634012 0.773324i \(-0.718593\pi\)
0.986724 + 0.162408i \(0.0519262\pi\)
\(102\) 0 0
\(103\) 1.47529 0.851761i 0.145365 0.0839265i −0.425553 0.904933i \(-0.639921\pi\)
0.570918 + 0.821007i \(0.306587\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.93582i 0.477164i 0.971122 + 0.238582i \(0.0766826\pi\)
−0.971122 + 0.238582i \(0.923317\pi\)
\(108\) 0 0
\(109\) 8.13844 0.779521 0.389760 0.920916i \(-0.372558\pi\)
0.389760 + 0.920916i \(0.372558\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.39699 1.96125i 0.319562 0.184499i −0.331635 0.943408i \(-0.607600\pi\)
0.651197 + 0.758908i \(0.274267\pi\)
\(114\) 0 0
\(115\) −4.97900 2.87463i −0.464294 0.268060i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.35981 1.66376i −0.216323 0.152517i
\(120\) 0 0
\(121\) −0.267713 0.463693i −0.0243376 0.0421539i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.3923 −1.01896
\(126\) 0 0
\(127\) −6.16827 −0.547345 −0.273673 0.961823i \(-0.588239\pi\)
−0.273673 + 0.961823i \(0.588239\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.13138 7.15575i −0.360960 0.625201i 0.627159 0.778891i \(-0.284218\pi\)
−0.988119 + 0.153690i \(0.950884\pi\)
\(132\) 0 0
\(133\) 6.84387 9.70702i 0.593438 0.841706i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.96169 + 5.17404i 0.765649 + 0.442048i 0.831320 0.555794i \(-0.187585\pi\)
−0.0656711 + 0.997841i \(0.520919\pi\)
\(138\) 0 0
\(139\) 15.4589 8.92521i 1.31121 0.757026i 0.328912 0.944361i \(-0.393318\pi\)
0.982296 + 0.187334i \(0.0599848\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −16.5600 −1.38482
\(144\) 0 0
\(145\) 1.70352i 0.141470i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.1758 8.76175i 1.24325 0.717790i 0.273495 0.961873i \(-0.411820\pi\)
0.969754 + 0.244083i \(0.0784869\pi\)
\(150\) 0 0
\(151\) 0.550343 0.953223i 0.0447863 0.0775722i −0.842763 0.538284i \(-0.819073\pi\)
0.887550 + 0.460712i \(0.152406\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.65103 + 2.68527i 0.373580 + 0.215686i
\(156\) 0 0
\(157\) 8.45150 4.87948i 0.674503 0.389425i −0.123277 0.992372i \(-0.539340\pi\)
0.797781 + 0.602947i \(0.206007\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.5619 0.958511i −0.832397 0.0755413i
\(162\) 0 0
\(163\) 7.22545 0.565941 0.282970 0.959129i \(-0.408680\pi\)
0.282970 + 0.959129i \(0.408680\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.65419 + 14.9895i 0.669681 + 1.15992i 0.977993 + 0.208637i \(0.0669027\pi\)
−0.308312 + 0.951285i \(0.599764\pi\)
\(168\) 0 0
\(169\) 6.60301 11.4367i 0.507924 0.879750i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.978103 1.69412i 0.0743638 0.128802i −0.826446 0.563017i \(-0.809641\pi\)
0.900809 + 0.434215i \(0.142974\pi\)
\(174\) 0 0
\(175\) −7.06710 + 3.26763i −0.534223 + 0.247010i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 23.2017i 1.73418i 0.498152 + 0.867090i \(0.334012\pi\)
−0.498152 + 0.867090i \(0.665988\pi\)
\(180\) 0 0
\(181\) 10.2744i 0.763689i 0.924226 + 0.381845i \(0.124711\pi\)
−0.924226 + 0.381845i \(0.875289\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.171486 + 0.297022i 0.0126079 + 0.0218375i
\(186\) 0 0
\(187\) 3.05733 + 1.76515i 0.223574 + 0.129080i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.6758 + 11.3598i 1.42369 + 0.821968i 0.996612 0.0822464i \(-0.0262094\pi\)
0.427079 + 0.904215i \(0.359543\pi\)
\(192\) 0 0
\(193\) −8.43598 14.6116i −0.607235 1.05176i −0.991694 0.128620i \(-0.958945\pi\)
0.384459 0.923142i \(-0.374388\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.94426i 0.637252i 0.947880 + 0.318626i \(0.103221\pi\)
−0.947880 + 0.318626i \(0.896779\pi\)
\(198\) 0 0
\(199\) 5.78528i 0.410108i −0.978751 0.205054i \(-0.934263\pi\)
0.978751 0.205054i \(-0.0657369\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.31881 + 2.85226i 0.0925621 + 0.200189i
\(204\) 0 0
\(205\) 5.33009 9.23200i 0.372270 0.644791i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.26091 + 12.5763i −0.502248 + 0.869918i
\(210\) 0 0
\(211\) 12.9451 + 22.4216i 0.891180 + 1.54357i 0.838462 + 0.544960i \(0.183455\pi\)
0.0527186 + 0.998609i \(0.483211\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.9673 −0.747964
\(216\) 0 0
\(217\) 9.86621 + 0.895374i 0.669762 + 0.0607819i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.83818 + 2.79332i −0.325451 + 0.187899i
\(222\) 0 0
\(223\) 15.4827 + 8.93892i 1.03680 + 0.598594i 0.918924 0.394435i \(-0.129060\pi\)
0.117871 + 0.993029i \(0.462393\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.48365 9.49796i 0.363963 0.630402i −0.624646 0.780908i \(-0.714757\pi\)
0.988609 + 0.150506i \(0.0480902\pi\)
\(228\) 0 0
\(229\) 16.8349 9.71965i 1.11248 0.642293i 0.173012 0.984920i \(-0.444650\pi\)
0.939471 + 0.342627i \(0.111317\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.94031i 0.192626i −0.995351 0.0963131i \(-0.969295\pi\)
0.995351 0.0963131i \(-0.0307050\pi\)
\(234\) 0 0
\(235\) −6.05391 −0.394914
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.7255 + 6.19234i −0.693772 + 0.400549i −0.805023 0.593243i \(-0.797847\pi\)
0.111252 + 0.993792i \(0.464514\pi\)
\(240\) 0 0
\(241\) −11.6943 6.75168i −0.753293 0.434914i 0.0735896 0.997289i \(-0.476555\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.50325 7.64915i 0.415478 0.488686i
\(246\) 0 0
\(247\) −11.4903 19.9018i −0.731109 1.26632i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.51441 −0.474305 −0.237153 0.971472i \(-0.576214\pi\)
−0.237153 + 0.971472i \(0.576214\pi\)
\(252\) 0 0
\(253\) 12.9669 0.815222
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.87788 + 6.71668i 0.241895 + 0.418975i 0.961254 0.275664i \(-0.0888976\pi\)
−0.719359 + 0.694639i \(0.755564\pi\)
\(258\) 0 0
\(259\) 0.517068 + 0.364555i 0.0321290 + 0.0226523i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.1127 6.99329i −0.746903 0.431224i 0.0776710 0.996979i \(-0.475252\pi\)
−0.824574 + 0.565755i \(0.808585\pi\)
\(264\) 0 0
\(265\) 8.71246 5.03014i 0.535202 0.308999i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.8321 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(270\) 0 0
\(271\) 16.6537i 1.01164i 0.862639 + 0.505821i \(0.168810\pi\)
−0.862639 + 0.505821i \(0.831190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.24433 4.75986i 0.497152 0.287031i
\(276\) 0 0
\(277\) −15.7044 + 27.2008i −0.943585 + 1.63434i −0.185025 + 0.982734i \(0.559237\pi\)
−0.758560 + 0.651603i \(0.774097\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.10464 4.67922i −0.483483 0.279139i 0.238384 0.971171i \(-0.423382\pi\)
−0.721867 + 0.692032i \(0.756716\pi\)
\(282\) 0 0
\(283\) −13.6603 + 7.88676i −0.812018 + 0.468819i −0.847656 0.530546i \(-0.821987\pi\)
0.0356380 + 0.999365i \(0.488654\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.77726 19.5838i 0.104908 1.15599i
\(288\) 0 0
\(289\) −15.8090 −0.929943
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.4287 + 21.5271i 0.726090 + 1.25762i 0.958524 + 0.285013i \(0.0919978\pi\)
−0.232434 + 0.972612i \(0.574669\pi\)
\(294\) 0 0
\(295\) 6.79179 11.7637i 0.395433 0.684911i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.2600 + 17.7708i −0.593349 + 1.02771i
\(300\) 0 0
\(301\) −18.3629 + 8.49050i −1.05842 + 0.489384i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.68294i 0.268144i
\(306\) 0 0
\(307\) 18.8878i 1.07799i −0.842310 0.538993i \(-0.818805\pi\)
0.842310 0.538993i \(-0.181195\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.97716 6.88864i −0.225524 0.390619i 0.730953 0.682428i \(-0.239076\pi\)
−0.956476 + 0.291809i \(0.905743\pi\)
\(312\) 0 0
\(313\) −9.64210 5.56687i −0.545004 0.314658i 0.202101 0.979365i \(-0.435223\pi\)
−0.747104 + 0.664707i \(0.768556\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.1380 + 11.6267i 1.13107 + 0.653021i 0.944203 0.329365i \(-0.106835\pi\)
0.186863 + 0.982386i \(0.440168\pi\)
\(318\) 0 0
\(319\) −1.92107 3.32738i −0.107559 0.186298i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.89904i 0.272590i
\(324\) 0 0
\(325\) 15.0648i 0.835646i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.1363 + 4.68672i −0.558830 + 0.258387i
\(330\) 0 0
\(331\) −9.57962 + 16.5924i −0.526544 + 0.912000i 0.472978 + 0.881074i \(0.343179\pi\)
−0.999522 + 0.0309261i \(0.990154\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.473451 0.820041i 0.0258674 0.0448036i
\(336\) 0 0
\(337\) 14.2781 + 24.7304i 0.777779 + 1.34715i 0.933219 + 0.359307i \(0.116987\pi\)
−0.155441 + 0.987845i \(0.549680\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.1128 −0.655943
\(342\) 0 0
\(343\) 4.96690 17.8418i 0.268187 0.963367i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.56690 1.48200i 0.137798 0.0795578i −0.429516 0.903059i \(-0.641316\pi\)
0.567314 + 0.823501i \(0.307983\pi\)
\(348\) 0 0
\(349\) 23.3885 + 13.5034i 1.25196 + 0.722818i 0.971498 0.237048i \(-0.0761797\pi\)
0.280460 + 0.959866i \(0.409513\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.8238 25.6755i 0.788990 1.36657i −0.137596 0.990488i \(-0.543937\pi\)
0.926586 0.376083i \(-0.122729\pi\)
\(354\) 0 0
\(355\) −4.74924 + 2.74198i −0.252064 + 0.145529i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.6261i 1.29972i −0.760056 0.649858i \(-0.774828\pi\)
0.760056 0.649858i \(-0.225172\pi\)
\(360\) 0 0
\(361\) −1.15211 −0.0606373
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.08087 + 5.24284i −0.475314 + 0.274423i
\(366\) 0 0
\(367\) −4.85598 2.80360i −0.253480 0.146347i 0.367877 0.929875i \(-0.380085\pi\)
−0.621357 + 0.783528i \(0.713418\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.6934 15.1670i 0.555173 0.787432i
\(372\) 0 0
\(373\) 1.86677 + 3.23333i 0.0966574 + 0.167416i 0.910299 0.413951i \(-0.135852\pi\)
−0.813642 + 0.581367i \(0.802518\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.08012 0.313142
\(378\) 0 0
\(379\) 30.4419 1.56369 0.781847 0.623470i \(-0.214278\pi\)
0.781847 + 0.623470i \(0.214278\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.49251 + 14.7095i 0.433947 + 0.751618i 0.997209 0.0746601i \(-0.0237872\pi\)
−0.563262 + 0.826278i \(0.690454\pi\)
\(384\) 0 0
\(385\) −7.07356 + 10.0328i −0.360502 + 0.511319i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.43310 5.44621i −0.478277 0.276134i 0.241421 0.970420i \(-0.422387\pi\)
−0.719698 + 0.694287i \(0.755720\pi\)
\(390\) 0 0
\(391\) 3.78840 2.18724i 0.191588 0.110613i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.24976 −0.264144
\(396\) 0 0
\(397\) 22.3035i 1.11938i −0.828702 0.559690i \(-0.810920\pi\)
0.828702 0.559690i \(-0.189080\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −20.8554 + 12.0409i −1.04147 + 0.601293i −0.920249 0.391333i \(-0.872014\pi\)
−0.121221 + 0.992626i \(0.538681\pi\)
\(402\) 0 0
\(403\) 9.58414 16.6002i 0.477420 0.826915i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.669905 0.386770i −0.0332060 0.0191715i
\(408\) 0 0
\(409\) −22.8191 + 13.1746i −1.12833 + 0.651443i −0.943515 0.331330i \(-0.892503\pi\)
−0.184817 + 0.982773i \(0.559169\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.26464 24.9543i 0.111436 1.22792i
\(414\) 0 0
\(415\) −15.6408 −0.767775
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16.1761 + 28.0178i 0.790252 + 1.36876i 0.925811 + 0.377988i \(0.123384\pi\)
−0.135558 + 0.990769i \(0.543283\pi\)
\(420\) 0 0
\(421\) −5.54746 + 9.60849i −0.270367 + 0.468289i −0.968956 0.247234i \(-0.920478\pi\)
0.698589 + 0.715523i \(0.253812\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.60577 2.78128i 0.0778914 0.134912i
\(426\) 0 0
\(427\) −3.62537 7.84079i −0.175444 0.379443i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.3047i 0.785368i −0.919673 0.392684i \(-0.871547\pi\)
0.919673 0.392684i \(-0.128453\pi\)
\(432\) 0 0
\(433\) 12.5359i 0.602438i −0.953555 0.301219i \(-0.902606\pi\)
0.953555 0.301219i \(-0.0973936\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.99716 + 15.5835i 0.430393 + 0.745462i
\(438\) 0 0
\(439\) −16.1276 9.31127i −0.769728 0.444403i 0.0630496 0.998010i \(-0.479917\pi\)
−0.832778 + 0.553608i \(0.813251\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.11436 + 2.37543i 0.195479 + 0.112860i 0.594545 0.804062i \(-0.297332\pi\)
−0.399066 + 0.916922i \(0.630666\pi\)
\(444\) 0 0
\(445\) −9.81875 17.0066i −0.465453 0.806189i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.2393i 0.766379i 0.923670 + 0.383189i \(0.125174\pi\)
−0.923670 + 0.383189i \(0.874826\pi\)
\(450\) 0 0
\(451\) 24.0431i 1.13214i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.15279 17.6325i −0.382209 0.826625i
\(456\) 0 0
\(457\) 2.87360 4.97722i 0.134421 0.232825i −0.790955 0.611874i \(-0.790416\pi\)
0.925376 + 0.379050i \(0.123749\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.1346 31.4101i 0.844613 1.46291i −0.0413440 0.999145i \(-0.513164\pi\)
0.885957 0.463768i \(-0.153503\pi\)
\(462\) 0 0
\(463\) −14.6202 25.3230i −0.679461 1.17686i −0.975144 0.221574i \(-0.928881\pi\)
0.295683 0.955286i \(-0.404453\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.64215 −0.122264 −0.0611320 0.998130i \(-0.519471\pi\)
−0.0611320 + 0.998130i \(0.519471\pi\)
\(468\) 0 0
\(469\) 0.157867 1.73955i 0.00728961 0.0803249i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.4218 12.3679i 0.984973 0.568675i
\(474\) 0 0
\(475\) 11.4408 + 6.60532i 0.524938 + 0.303073i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.5409 26.9177i 0.710083 1.22990i −0.254742 0.967009i \(-0.581991\pi\)
0.964826 0.262891i \(-0.0846760\pi\)
\(480\) 0 0
\(481\) 1.06012 0.612058i 0.0483371 0.0279074i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.46684i 0.202829i
\(486\) 0 0
\(487\) −34.8720 −1.58020 −0.790100 0.612978i \(-0.789971\pi\)
−0.790100 + 0.612978i \(0.789971\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −22.6758 + 13.0919i −1.02334 + 0.590828i −0.915071 0.403293i \(-0.867866\pi\)
−0.108273 + 0.994121i \(0.534532\pi\)
\(492\) 0 0
\(493\) −1.12252 0.648085i −0.0505556 0.0291883i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.82906 + 8.26767i −0.261469 + 0.370856i
\(498\) 0 0
\(499\) 6.23912 + 10.8065i 0.279302 + 0.483764i 0.971211 0.238220i \(-0.0765638\pi\)
−0.691910 + 0.721984i \(0.743230\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.8479 1.68756 0.843778 0.536693i \(-0.180327\pi\)
0.843778 + 0.536693i \(0.180327\pi\)
\(504\) 0 0
\(505\) −10.1683 −0.452482
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.6924 + 30.6441i 0.784200 + 1.35827i 0.929476 + 0.368883i \(0.120260\pi\)
−0.145276 + 0.989391i \(0.546407\pi\)
\(510\) 0 0
\(511\) −11.1456 + 15.8083i −0.493050 + 0.699320i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.11599 1.22167i −0.0932419 0.0538332i
\(516\) 0 0
\(517\) 11.8247 6.82701i 0.520051 0.300252i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.31879 −0.101588 −0.0507940 0.998709i \(-0.516175\pi\)
−0.0507940 + 0.998709i \(0.516175\pi\)
\(522\) 0 0
\(523\) 20.1840i 0.882585i −0.897363 0.441293i \(-0.854520\pi\)
0.897363 0.441293i \(-0.145480\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.53886 + 2.04316i −0.154155 + 0.0890015i
\(528\) 0 0
\(529\) −3.46621 + 6.00365i −0.150705 + 0.261028i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −32.9503 19.0239i −1.42724 0.824016i
\(534\) 0 0
\(535\) 6.13093 3.53970i 0.265063 0.153034i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.07644 + 22.2744i −0.175585 + 0.959424i
\(540\) 0 0
\(541\) −22.7713 −0.979014 −0.489507 0.871999i \(-0.662823\pi\)
−0.489507 + 0.871999i \(0.662823\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.83643 10.1090i −0.250005 0.433022i
\(546\) 0 0
\(547\) −14.7918 + 25.6201i −0.632451 + 1.09544i 0.354598 + 0.935019i \(0.384618\pi\)
−0.987049 + 0.160419i \(0.948716\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.66589 4.61745i 0.113571 0.196710i
\(552\) 0 0
\(553\) −8.78984 + 4.06418i −0.373782 + 0.172827i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.71407i 0.199741i −0.995000 0.0998707i \(-0.968157\pi\)
0.995000 0.0998707i \(-0.0318429\pi\)
\(558\) 0 0
\(559\) 39.1439i 1.65561i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.6742 23.6844i −0.576299 0.998179i −0.995899 0.0904697i \(-0.971163\pi\)
0.419601 0.907709i \(-0.362170\pi\)
\(564\) 0 0
\(565\) −4.87226 2.81300i −0.204977 0.118344i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20.4018 11.7790i −0.855288 0.493801i 0.00714355 0.999974i \(-0.497726\pi\)
−0.862432 + 0.506174i \(0.831059\pi\)
\(570\) 0 0
\(571\) 9.59385 + 16.6170i 0.401490 + 0.695401i 0.993906 0.110231i \(-0.0351591\pi\)
−0.592416 + 0.805632i \(0.701826\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.7961i 0.491932i
\(576\) 0 0
\(577\) 2.23413i 0.0930079i 0.998918 + 0.0465039i \(0.0148080\pi\)
−0.998918 + 0.0465039i \(0.985192\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −26.1878 + 12.1085i −1.08645 + 0.502346i
\(582\) 0 0
\(583\) −11.3450 + 19.6501i −0.469862 + 0.813826i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.9883 + 22.4963i −0.536083 + 0.928522i 0.463028 + 0.886344i \(0.346763\pi\)
−0.999110 + 0.0421784i \(0.986570\pi\)
\(588\) 0 0
\(589\) −8.40451 14.5570i −0.346302 0.599813i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −5.71754 −0.234791 −0.117396 0.993085i \(-0.537455\pi\)
−0.117396 + 0.993085i \(0.537455\pi\)
\(594\) 0 0
\(595\) −0.374290 + 4.12434i −0.0153444 + 0.169081i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.8662 12.6245i 0.893429 0.515822i 0.0183665 0.999831i \(-0.494153\pi\)
0.875063 + 0.484010i \(0.160820\pi\)
\(600\) 0 0
\(601\) 40.2546 + 23.2410i 1.64202 + 0.948021i 0.980114 + 0.198435i \(0.0635858\pi\)
0.661907 + 0.749586i \(0.269748\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.383978 + 0.665069i −0.0156109 + 0.0270389i
\(606\) 0 0
\(607\) 6.09405 3.51840i 0.247350 0.142808i −0.371200 0.928553i \(-0.621054\pi\)
0.618550 + 0.785745i \(0.287720\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.6073i 0.874138i
\(612\) 0 0
\(613\) 6.54256 0.264252 0.132126 0.991233i \(-0.457820\pi\)
0.132126 + 0.991233i \(0.457820\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0043 17.3230i 1.20793 0.697396i 0.245620 0.969366i \(-0.421008\pi\)
0.962306 + 0.271970i \(0.0876751\pi\)
\(618\) 0 0
\(619\) −14.7072 8.49123i −0.591134 0.341291i 0.174412 0.984673i \(-0.444198\pi\)
−0.765546 + 0.643381i \(0.777531\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −29.6057 20.8733i −1.18613 0.836271i
\(624\) 0 0
\(625\) 0.812855 + 1.40791i 0.0325142 + 0.0563162i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.260959 −0.0104051
\(630\) 0 0
\(631\) −26.2438 −1.04475 −0.522374 0.852716i \(-0.674953\pi\)
−0.522374 + 0.852716i \(0.674953\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.42354 + 7.66179i 0.175543 + 0.304049i
\(636\) 0 0
\(637\) −27.3009 23.2111i −1.08170 0.919656i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −16.5092 9.53157i −0.652073 0.376474i 0.137177 0.990547i \(-0.456197\pi\)
−0.789250 + 0.614072i \(0.789530\pi\)
\(642\) 0 0
\(643\) −15.3447 + 8.85928i −0.605136 + 0.349376i −0.771060 0.636763i \(-0.780273\pi\)
0.165923 + 0.986139i \(0.446940\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.7902 −0.856661 −0.428330 0.903622i \(-0.640898\pi\)
−0.428330 + 0.903622i \(0.640898\pi\)
\(648\) 0 0
\(649\) 30.6365i 1.20259i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.0852 7.55475i 0.512064 0.295640i −0.221618 0.975134i \(-0.571134\pi\)
0.733682 + 0.679493i \(0.237800\pi\)
\(654\) 0 0
\(655\) −5.92558 + 10.2634i −0.231532 + 0.401024i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −27.1850 15.6952i −1.05898 0.611400i −0.133827 0.991005i \(-0.542727\pi\)
−0.925149 + 0.379605i \(0.876060\pi\)
\(660\) 0 0
\(661\) −37.8554 + 21.8558i −1.47240 + 0.850093i −0.999518 0.0310314i \(-0.990121\pi\)
−0.472885 + 0.881124i \(0.656787\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −16.9654 1.53964i −0.657890 0.0597046i
\(666\) 0 0
\(667\) −4.76088 −0.184342
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.28096 + 9.14690i 0.203869 + 0.353112i
\(672\) 0 0
\(673\) −4.60589 + 7.97763i −0.177544 + 0.307515i −0.941039 0.338299i \(-0.890149\pi\)
0.763495 + 0.645814i \(0.223482\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.4194 19.7789i 0.438882 0.760165i −0.558722 0.829355i \(-0.688708\pi\)
0.997604 + 0.0691899i \(0.0220414\pi\)
\(678\) 0 0
\(679\) 3.45807 + 7.47898i 0.132709 + 0.287017i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.1826i 1.30796i 0.756511 + 0.653981i \(0.226902\pi\)
−0.756511 + 0.653981i \(0.773098\pi\)
\(684\) 0 0
\(685\) 14.8421i 0.567088i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.9533 31.0961i −0.683967 1.18467i
\(690\) 0 0
\(691\) 0.224082 + 0.129374i 0.00852446 + 0.00492160i 0.504256 0.863554i \(-0.331767\pi\)
−0.495732 + 0.868476i \(0.665100\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.1725 12.8013i −0.841052 0.485582i
\(696\) 0 0
\(697\) 4.05555 + 7.02441i 0.153615 + 0.266069i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.16189i 0.194962i −0.995237 0.0974810i \(-0.968921\pi\)
0.995237 0.0974810i \(-0.0310785\pi\)
\(702\) 0 0
\(703\) 1.07345i 0.0404860i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.0251 + 7.87192i −0.640293 + 0.296054i
\(708\) 0 0
\(709\) −11.7472 + 20.3468i −0.441175 + 0.764138i −0.997777 0.0666412i \(-0.978772\pi\)
0.556602 + 0.830780i \(0.312105\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.50460 + 12.9984i −0.281050 + 0.486792i
\(714\) 0 0
\(715\) 11.8759 + 20.5697i 0.444134 + 0.769263i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.1566 −0.378776 −0.189388 0.981902i \(-0.560650\pi\)
−0.189388 + 0.981902i \(0.560650\pi\)
\(720\) 0 0
\(721\) −4.48865 0.407352i −0.167166 0.0151706i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.02696 + 1.74761i −0.112418 + 0.0649047i
\(726\) 0 0
\(727\) 5.74874 + 3.31904i 0.213209 + 0.123096i 0.602802 0.797891i \(-0.294051\pi\)
−0.389593 + 0.920987i \(0.627384\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.17238 7.22678i 0.154321 0.267292i
\(732\) 0 0
\(733\) 5.20130 3.00297i 0.192114 0.110917i −0.400858 0.916140i \(-0.631288\pi\)
0.592972 + 0.805223i \(0.297954\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.13565i 0.0786676i
\(738\) 0 0
\(739\) 15.6386 0.575275 0.287638 0.957739i \(-0.407130\pi\)
0.287638 + 0.957739i \(0.407130\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.3807 15.8083i 1.00450 0.579949i 0.0949246 0.995484i \(-0.469739\pi\)
0.909577 + 0.415535i \(0.136406\pi\)
\(744\) 0 0
\(745\) −21.7664 12.5669i −0.797461 0.460414i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.52490 10.6730i 0.274954 0.389982i
\(750\) 0 0
\(751\) −7.13680 12.3613i −0.260426 0.451070i 0.705929 0.708282i \(-0.250530\pi\)
−0.966355 + 0.257212i \(0.917196\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.57870 −0.0574548
\(756\) 0 0
\(757\) −10.8227 −0.393358 −0.196679 0.980468i \(-0.563016\pi\)
−0.196679 + 0.980468i \(0.563016\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.93098 5.07660i −0.106248 0.184027i 0.808000 0.589183i \(-0.200550\pi\)
−0.914247 + 0.405157i \(0.867217\pi\)
\(762\) 0 0
\(763\) −17.5981 12.4074i −0.637095 0.449180i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.9865 24.2409i −1.51604 0.875288i
\(768\) 0 0
\(769\) 27.5683 15.9166i 0.994140 0.573967i 0.0876307 0.996153i \(-0.472070\pi\)
0.906509 + 0.422186i \(0.138737\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.0382 −0.684755 −0.342378 0.939562i \(-0.611232\pi\)
−0.342378 + 0.939562i \(0.611232\pi\)
\(774\) 0 0
\(775\) 11.0191i 0.395818i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.8948 + 16.6824i −1.03526 + 0.597710i
\(780\) 0 0
\(781\) 6.18427 10.7115i 0.221290 0.383286i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.1219 6.99857i −0.432649 0.249790i
\(786\) 0 0
\(787\) 16.4123 9.47564i 0.585035 0.337770i −0.178097 0.984013i \(-0.556994\pi\)
0.763132 + 0.646243i \(0.223661\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −10.3355 0.937963i −0.367488 0.0333501i
\(792\) 0 0
\(793\) −16.7141 −0.593535
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.7207 + 46.2816i 0.946497 + 1.63938i 0.752727 + 0.658333i \(0.228738\pi\)
0.193770 + 0.981047i \(0.437929\pi\)
\(798\) 0 0
\(799\) 2.30314 3.98916i 0.0814792 0.141126i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.8247 20.4810i 0.417286 0.722760i
\(804\) 0 0
\(805\) 6.38383 + 13.8067i 0.225000 + 0.486621i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.58095i 0.0907413i 0.998970 + 0.0453706i \(0.0144469\pi\)
−0.998970 + 0.0453706i \(0.985553\pi\)
\(810\) 0 0
\(811\) 6.06938i 0.213125i −0.994306 0.106562i \(-0.966016\pi\)
0.994306 0.106562i \(-0.0339844\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.18169 8.97494i −0.181507 0.314379i
\(816\) 0 0
\(817\) 29.7272 + 17.1630i 1.04002 + 0.600458i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.03938 4.64154i −0.280576 0.161991i 0.353108 0.935583i \(-0.385125\pi\)
−0.633684 + 0.773592i \(0.718458\pi\)
\(822\) 0 0
\(823\) 9.03448 + 15.6482i 0.314922 + 0.545461i 0.979421 0.201828i \(-0.0646882\pi\)
−0.664499 + 0.747289i \(0.731355\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.5440i 1.68804i −0.536310 0.844021i \(-0.680182\pi\)
0.536310 0.844021i \(-0.319818\pi\)
\(828\) 0 0
\(829\) 5.44792i 0.189214i −0.995515 0.0946071i \(-0.969841\pi\)
0.995515 0.0946071i \(-0.0301595\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.56623 + 7.19527i 0.0889147 + 0.249302i
\(834\) 0 0
\(835\) 12.4126 21.4992i 0.429556 0.744012i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24.2673 + 42.0322i −0.837801 + 1.45111i 0.0539281 + 0.998545i \(0.482826\pi\)
−0.891729 + 0.452569i \(0.850508\pi\)
\(840\) 0 0
\(841\) −13.7947 23.8931i −0.475678 0.823899i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −18.9412 −0.651598
\(846\) 0 0
\(847\) −0.128033 + 1.41081i −0.00439926 + 0.0484759i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −0.830095 + 0.479256i −0.0284553 + 0.0164287i
\(852\) 0 0
\(853\) −10.7703 6.21823i −0.368768 0.212908i 0.304152 0.952623i \(-0.401627\pi\)
−0.672920 + 0.739715i \(0.734960\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.29077 + 9.16388i −0.180729 + 0.313032i −0.942129 0.335250i \(-0.891179\pi\)
0.761400 + 0.648283i \(0.224512\pi\)
\(858\) 0 0
\(859\) 28.1452 16.2496i 0.960302 0.554431i 0.0640360 0.997948i \(-0.479603\pi\)
0.896266 + 0.443517i \(0.146269\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.2203i 0.858510i −0.903183 0.429255i \(-0.858776\pi\)
0.903183 0.429255i \(-0.141224\pi\)
\(864\) 0 0
\(865\) −2.80576 −0.0953987
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.2540 5.92017i 0.347844 0.200828i
\(870\) 0 0
\(871\) −2.92685 1.68982i −0.0991724 0.0572572i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 24.6341 + 17.3681i 0.832784 + 0.587149i
\(876\) 0 0
\(877\) 7.47893 + 12.9539i 0.252546 + 0.437422i 0.964226 0.265082i \(-0.0853989\pi\)
−0.711680 + 0.702503i \(0.752066\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.4482 1.22797 0.613985 0.789318i \(-0.289566\pi\)
0.613985 + 0.789318i \(0.289566\pi\)
\(882\) 0 0
\(883\) −15.9831 −0.537873 −0.268936 0.963158i \(-0.586672\pi\)
−0.268936 + 0.963158i \(0.586672\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.5208 + 42.4713i 0.823329 + 1.42605i 0.903190 + 0.429241i \(0.141219\pi\)
−0.0798613 + 0.996806i \(0.525448\pi\)
\(888\) 0 0
\(889\) 13.3380 + 9.40383i 0.447341 + 0.315394i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 16.4093 + 9.47393i 0.549117 + 0.317033i
\(894\) 0 0
\(895\) 28.8196 16.6390i 0.963332 0.556180i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.44728 0.148325
\(900\) 0 0
\(901\) 7.65464i 0.255013i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.7621 7.36821i 0.424227 0.244928i
\(906\) 0 0
\(907\) −2.42915 + 4.20741i −0.0806585 + 0.139705i −0.903533 0.428519i \(-0.859036\pi\)
0.822874 + 0.568223i \(0.192369\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 14.4945 + 8.36843i 0.480226 + 0.277258i 0.720510 0.693444i \(-0.243908\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(912\) 0 0
\(913\) 30.5501 17.6381i 1.01106 0.583737i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.97582 + 21.7717i −0.0652472 + 0.718965i
\(918\) 0 0
\(919\) −30.6400 −1.01072 −0.505360 0.862909i \(-0.668640\pi\)
−0.505360 + 0.862909i \(0.668640\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.78651 + 16.9507i 0.322127 + 0.557940i
\(924\) 0 0
\(925\) −0.351848 + 0.609419i −0.0115687 + 0.0200376i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.8723 + 25.7595i −0.487943 + 0.845142i −0.999904 0.0138670i \(-0.995586\pi\)
0.511961 + 0.859009i \(0.328919\pi\)
\(930\) 0 0
\(931\) −29.5976 + 10.5562i −0.970024 + 0.345964i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.06346i 0.165593i
\(936\) 0 0
\(937\) 4.03712i 0.131887i 0.997823 + 0.0659434i \(0.0210057\pi\)
−0.997823 + 0.0659434i \(0.978994\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.20264 12.4753i −0.234799 0.406684i 0.724415 0.689364i \(-0.242110\pi\)
−0.959214 + 0.282680i \(0.908777\pi\)
\(942\) 0 0
\(943\) 25.8009 + 14.8962i 0.840193 + 0.485085i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 27.0334 + 15.6077i 0.878467 + 0.507183i 0.870153 0.492782i \(-0.164020\pi\)
0.00831468 + 0.999965i \(0.497353\pi\)
\(948\) 0 0
\(949\) 18.7125 + 32.4109i 0.607432 + 1.05210i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.55869i 0.277243i −0.990345 0.138622i \(-0.955733\pi\)
0.990345 0.138622i \(-0.0442672\pi\)
\(954\) 0 0
\(955\) 32.5865i 1.05447i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.4902 24.8506i −0.371039 0.802468i
\(960\) 0 0
\(961\) −8.48973 + 14.7046i −0.273862 + 0.474343i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.0996 + 20.9572i −0.389501 + 0.674635i
\(966\) 0 0
\(967\) −16.0280 27.7614i −0.515427 0.892745i −0.999840 0.0179059i \(-0.994300\pi\)
0.484413 0.874840i \(-0.339033\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.2366 1.06661 0.533307 0.845922i \(-0.320949\pi\)
0.533307 + 0.845922i \(0.320949\pi\)
\(972\) 0 0
\(973\) −47.0345 4.26845i −1.50786 0.136840i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −45.1558 + 26.0707i −1.44466 + 0.834076i −0.998156 0.0607042i \(-0.980665\pi\)
−0.446507 + 0.894780i \(0.647332\pi\)
\(978\) 0 0
\(979\) 38.3567 + 22.1453i 1.22589 + 0.707765i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.1192 + 20.9911i −0.386544 + 0.669513i −0.991982 0.126379i \(-0.959664\pi\)
0.605438 + 0.795892i \(0.292998\pi\)
\(984\) 0 0
\(985\) 11.1099 6.41432i 0.353992 0.204377i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30.6506i 0.974632i
\(990\) 0 0
\(991\) 24.1981 0.768678 0.384339 0.923192i \(-0.374429\pi\)
0.384339 + 0.923192i \(0.374429\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −7.18607 + 4.14888i −0.227814 + 0.131528i
\(996\) 0 0
\(997\) 8.81920 + 5.09177i 0.279307 + 0.161258i 0.633110 0.774062i \(-0.281778\pi\)
−0.353803 + 0.935320i \(0.615112\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3024.2.cc.a.2897.3 12
3.2 odd 2 1008.2.cc.a.209.5 12
4.3 odd 2 189.2.o.a.62.5 12
7.6 odd 2 inner 3024.2.cc.a.2897.4 12
9.4 even 3 1008.2.cc.a.545.2 12
9.5 odd 6 inner 3024.2.cc.a.881.4 12
12.11 even 2 63.2.o.a.20.1 12
21.20 even 2 1008.2.cc.a.209.2 12
28.3 even 6 1323.2.s.c.656.1 12
28.11 odd 6 1323.2.s.c.656.2 12
28.19 even 6 1323.2.i.c.521.2 12
28.23 odd 6 1323.2.i.c.521.1 12
28.27 even 2 189.2.o.a.62.6 12
36.7 odd 6 567.2.c.c.566.2 12
36.11 even 6 567.2.c.c.566.11 12
36.23 even 6 189.2.o.a.125.6 12
36.31 odd 6 63.2.o.a.41.2 yes 12
63.13 odd 6 1008.2.cc.a.545.5 12
63.41 even 6 inner 3024.2.cc.a.881.3 12
84.11 even 6 441.2.s.c.362.6 12
84.23 even 6 441.2.i.c.227.5 12
84.47 odd 6 441.2.i.c.227.6 12
84.59 odd 6 441.2.s.c.362.5 12
84.83 odd 2 63.2.o.a.20.2 yes 12
252.23 even 6 1323.2.s.c.962.1 12
252.31 even 6 441.2.i.c.68.1 12
252.59 odd 6 1323.2.i.c.1097.5 12
252.67 odd 6 441.2.i.c.68.2 12
252.83 odd 6 567.2.c.c.566.12 12
252.95 even 6 1323.2.i.c.1097.6 12
252.103 even 6 441.2.s.c.374.6 12
252.131 odd 6 1323.2.s.c.962.2 12
252.139 even 6 63.2.o.a.41.1 yes 12
252.167 odd 6 189.2.o.a.125.5 12
252.223 even 6 567.2.c.c.566.1 12
252.247 odd 6 441.2.s.c.374.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.o.a.20.1 12 12.11 even 2
63.2.o.a.20.2 yes 12 84.83 odd 2
63.2.o.a.41.1 yes 12 252.139 even 6
63.2.o.a.41.2 yes 12 36.31 odd 6
189.2.o.a.62.5 12 4.3 odd 2
189.2.o.a.62.6 12 28.27 even 2
189.2.o.a.125.5 12 252.167 odd 6
189.2.o.a.125.6 12 36.23 even 6
441.2.i.c.68.1 12 252.31 even 6
441.2.i.c.68.2 12 252.67 odd 6
441.2.i.c.227.5 12 84.23 even 6
441.2.i.c.227.6 12 84.47 odd 6
441.2.s.c.362.5 12 84.59 odd 6
441.2.s.c.362.6 12 84.11 even 6
441.2.s.c.374.5 12 252.247 odd 6
441.2.s.c.374.6 12 252.103 even 6
567.2.c.c.566.1 12 252.223 even 6
567.2.c.c.566.2 12 36.7 odd 6
567.2.c.c.566.11 12 36.11 even 6
567.2.c.c.566.12 12 252.83 odd 6
1008.2.cc.a.209.2 12 21.20 even 2
1008.2.cc.a.209.5 12 3.2 odd 2
1008.2.cc.a.545.2 12 9.4 even 3
1008.2.cc.a.545.5 12 63.13 odd 6
1323.2.i.c.521.1 12 28.23 odd 6
1323.2.i.c.521.2 12 28.19 even 6
1323.2.i.c.1097.5 12 252.59 odd 6
1323.2.i.c.1097.6 12 252.95 even 6
1323.2.s.c.656.1 12 28.3 even 6
1323.2.s.c.656.2 12 28.11 odd 6
1323.2.s.c.962.1 12 252.23 even 6
1323.2.s.c.962.2 12 252.131 odd 6
3024.2.cc.a.881.3 12 63.41 even 6 inner
3024.2.cc.a.881.4 12 9.5 odd 6 inner
3024.2.cc.a.2897.3 12 1.1 even 1 trivial
3024.2.cc.a.2897.4 12 7.6 odd 2 inner