Properties

Label 136.3.t.b.57.3
Level $136$
Weight $3$
Character 136.57
Analytic conductor $3.706$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [136,3,Mod(41,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.41");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 136.t (of order \(16\), degree \(8\), 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: \(3.70573159530\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label 57.3
Character \(\chi\) \(=\) 136.57
Dual form 136.3.t.b.105.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.63930 - 0.326077i) q^{3} +(-4.61793 + 6.91122i) q^{5} +(2.96901 + 4.44344i) q^{7} +(-5.73394 + 2.37508i) q^{9} +O(q^{10})\) \(q+(1.63930 - 0.326077i) q^{3} +(-4.61793 + 6.91122i) q^{5} +(2.96901 + 4.44344i) q^{7} +(-5.73394 + 2.37508i) q^{9} +(-2.84284 + 14.2919i) q^{11} +(15.6679 - 15.6679i) q^{13} +(-5.31657 + 12.8353i) q^{15} +(16.0654 + 5.55912i) q^{17} +(-21.1674 - 8.76784i) q^{19} +(6.31600 + 6.31600i) q^{21} +(34.3844 + 6.83948i) q^{23} +(-16.8726 - 40.7340i) q^{25} +(-21.1328 + 14.1205i) q^{27} +(-11.1864 - 7.47449i) q^{29} +(6.22691 + 31.3048i) q^{31} +24.3557i q^{33} -44.4202 q^{35} +(17.3336 - 3.44786i) q^{37} +(20.5754 - 30.7933i) q^{39} +(-26.6426 - 39.8734i) q^{41} +(15.6376 - 6.47729i) q^{43} +(10.0643 - 50.5964i) q^{45} +(49.2607 - 49.2607i) q^{47} +(7.82237 - 18.8849i) q^{49} +(28.1487 + 3.87451i) q^{51} +(-8.90219 - 3.68741i) q^{53} +(-85.6466 - 85.6466i) q^{55} +(-37.5587 - 7.47090i) q^{57} +(13.7735 + 33.2521i) q^{59} +(2.92989 - 1.95769i) q^{61} +(-27.5776 - 18.4268i) q^{63} +(35.9310 + 180.638i) q^{65} +44.1965i q^{67} +58.5965 q^{69} +(27.2022 - 5.41086i) q^{71} +(22.0828 - 33.0492i) q^{73} +(-40.9416 - 61.2734i) q^{75} +(-71.9457 + 29.8009i) q^{77} +(-14.9727 + 75.2727i) q^{79} +(9.45855 - 9.45855i) q^{81} +(-17.9264 + 43.2783i) q^{83} +(-112.609 + 85.3596i) q^{85} +(-20.7751 - 8.60531i) q^{87} +(82.2618 + 82.2618i) q^{89} +(116.138 + 23.1012i) q^{91} +(20.4155 + 49.2875i) q^{93} +(158.346 - 105.803i) q^{95} +(-39.3793 - 26.3124i) q^{97} +(-17.6437 - 88.7011i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 8 q^{3} - 8 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 8 q^{3} - 8 q^{7} - 16 q^{9} + 24 q^{11} - 48 q^{13} - 96 q^{15} - 40 q^{19} + 80 q^{21} + 48 q^{23} + 48 q^{25} + 224 q^{27} + 24 q^{29} + 88 q^{31} + 32 q^{35} - 176 q^{37} - 120 q^{39} - 352 q^{43} + 264 q^{45} - 48 q^{47} - 208 q^{49} + 400 q^{51} - 472 q^{53} - 208 q^{55} + 24 q^{57} - 576 q^{59} - 632 q^{63} - 32 q^{65} + 160 q^{69} - 160 q^{71} + 256 q^{73} + 1128 q^{75} - 208 q^{77} + 1000 q^{79} + 24 q^{81} + 312 q^{83} + 1240 q^{85} - 664 q^{87} + 720 q^{89} + 664 q^{91} - 432 q^{93} + 736 q^{95} - 288 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{15}{16}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.63930 0.326077i 0.546433 0.108692i 0.0858486 0.996308i \(-0.472640\pi\)
0.460584 + 0.887616i \(0.347640\pi\)
\(4\) 0 0
\(5\) −4.61793 + 6.91122i −0.923585 + 1.38224i 0.000476532 1.00000i \(0.499848\pi\)
−0.924062 + 0.382243i \(0.875152\pi\)
\(6\) 0 0
\(7\) 2.96901 + 4.44344i 0.424144 + 0.634777i 0.980581 0.196113i \(-0.0628320\pi\)
−0.556437 + 0.830890i \(0.687832\pi\)
\(8\) 0 0
\(9\) −5.73394 + 2.37508i −0.637104 + 0.263897i
\(10\) 0 0
\(11\) −2.84284 + 14.2919i −0.258440 + 1.29927i 0.605570 + 0.795792i \(0.292945\pi\)
−0.864010 + 0.503475i \(0.832055\pi\)
\(12\) 0 0
\(13\) 15.6679 15.6679i 1.20522 1.20522i 0.232667 0.972556i \(-0.425255\pi\)
0.972556 0.232667i \(-0.0747453\pi\)
\(14\) 0 0
\(15\) −5.31657 + 12.8353i −0.354438 + 0.855690i
\(16\) 0 0
\(17\) 16.0654 + 5.55912i 0.945022 + 0.327007i
\(18\) 0 0
\(19\) −21.1674 8.76784i −1.11408 0.461465i −0.251736 0.967796i \(-0.581001\pi\)
−0.862339 + 0.506331i \(0.831001\pi\)
\(20\) 0 0
\(21\) 6.31600 + 6.31600i 0.300762 + 0.300762i
\(22\) 0 0
\(23\) 34.3844 + 6.83948i 1.49497 + 0.297368i 0.873792 0.486299i \(-0.161654\pi\)
0.621180 + 0.783668i \(0.286654\pi\)
\(24\) 0 0
\(25\) −16.8726 40.7340i −0.674903 1.62936i
\(26\) 0 0
\(27\) −21.1328 + 14.1205i −0.782695 + 0.522980i
\(28\) 0 0
\(29\) −11.1864 7.47449i −0.385737 0.257741i 0.347549 0.937662i \(-0.387014\pi\)
−0.733285 + 0.679921i \(0.762014\pi\)
\(30\) 0 0
\(31\) 6.22691 + 31.3048i 0.200868 + 1.00983i 0.941267 + 0.337664i \(0.109637\pi\)
−0.740399 + 0.672168i \(0.765363\pi\)
\(32\) 0 0
\(33\) 24.3557i 0.738053i
\(34\) 0 0
\(35\) −44.4202 −1.26915
\(36\) 0 0
\(37\) 17.3336 3.44786i 0.468474 0.0931854i 0.0447924 0.998996i \(-0.485737\pi\)
0.423682 + 0.905811i \(0.360737\pi\)
\(38\) 0 0
\(39\) 20.5754 30.7933i 0.527576 0.789573i
\(40\) 0 0
\(41\) −26.6426 39.8734i −0.649819 0.972523i −0.999366 0.0356040i \(-0.988665\pi\)
0.349547 0.936919i \(-0.386335\pi\)
\(42\) 0 0
\(43\) 15.6376 6.47729i 0.363664 0.150635i −0.193366 0.981127i \(-0.561941\pi\)
0.557031 + 0.830492i \(0.311941\pi\)
\(44\) 0 0
\(45\) 10.0643 50.5964i 0.223650 1.12436i
\(46\) 0 0
\(47\) 49.2607 49.2607i 1.04810 1.04810i 0.0493173 0.998783i \(-0.484295\pi\)
0.998783 0.0493173i \(-0.0157046\pi\)
\(48\) 0 0
\(49\) 7.82237 18.8849i 0.159640 0.385406i
\(50\) 0 0
\(51\) 28.1487 + 3.87451i 0.551934 + 0.0759708i
\(52\) 0 0
\(53\) −8.90219 3.68741i −0.167966 0.0695737i 0.297116 0.954841i \(-0.403975\pi\)
−0.465082 + 0.885268i \(0.653975\pi\)
\(54\) 0 0
\(55\) −85.6466 85.6466i −1.55721 1.55721i
\(56\) 0 0
\(57\) −37.5587 7.47090i −0.658925 0.131068i
\(58\) 0 0
\(59\) 13.7735 + 33.2521i 0.233449 + 0.563596i 0.996579 0.0826498i \(-0.0263383\pi\)
−0.763130 + 0.646245i \(0.776338\pi\)
\(60\) 0 0
\(61\) 2.92989 1.95769i 0.0480310 0.0320933i −0.531323 0.847169i \(-0.678305\pi\)
0.579354 + 0.815076i \(0.303305\pi\)
\(62\) 0 0
\(63\) −27.5776 18.4268i −0.437740 0.292489i
\(64\) 0 0
\(65\) 35.9310 + 180.638i 0.552785 + 2.77904i
\(66\) 0 0
\(67\) 44.1965i 0.659649i 0.944042 + 0.329825i \(0.106990\pi\)
−0.944042 + 0.329825i \(0.893010\pi\)
\(68\) 0 0
\(69\) 58.5965 0.849224
\(70\) 0 0
\(71\) 27.2022 5.41086i 0.383130 0.0762093i 0.000232228 1.00000i \(-0.499926\pi\)
0.382898 + 0.923791i \(0.374926\pi\)
\(72\) 0 0
\(73\) 22.0828 33.0492i 0.302504 0.452729i −0.648811 0.760950i \(-0.724733\pi\)
0.951315 + 0.308220i \(0.0997334\pi\)
\(74\) 0 0
\(75\) −40.9416 61.2734i −0.545888 0.816979i
\(76\) 0 0
\(77\) −71.9457 + 29.8009i −0.934360 + 0.387025i
\(78\) 0 0
\(79\) −14.9727 + 75.2727i −0.189528 + 0.952819i 0.762542 + 0.646938i \(0.223951\pi\)
−0.952070 + 0.305881i \(0.901049\pi\)
\(80\) 0 0
\(81\) 9.45855 9.45855i 0.116772 0.116772i
\(82\) 0 0
\(83\) −17.9264 + 43.2783i −0.215981 + 0.521425i −0.994321 0.106418i \(-0.966062\pi\)
0.778340 + 0.627843i \(0.216062\pi\)
\(84\) 0 0
\(85\) −112.609 + 85.3596i −1.32481 + 1.00423i
\(86\) 0 0
\(87\) −20.7751 8.60531i −0.238794 0.0989116i
\(88\) 0 0
\(89\) 82.2618 + 82.2618i 0.924290 + 0.924290i 0.997329 0.0730393i \(-0.0232698\pi\)
−0.0730393 + 0.997329i \(0.523270\pi\)
\(90\) 0 0
\(91\) 116.138 + 23.1012i 1.27624 + 0.253859i
\(92\) 0 0
\(93\) 20.4155 + 49.2875i 0.219522 + 0.529973i
\(94\) 0 0
\(95\) 158.346 105.803i 1.66680 1.11372i
\(96\) 0 0
\(97\) −39.3793 26.3124i −0.405972 0.271262i 0.335774 0.941943i \(-0.391002\pi\)
−0.741746 + 0.670681i \(0.766002\pi\)
\(98\) 0 0
\(99\) −17.6437 88.7011i −0.178220 0.895971i
\(100\) 0 0
\(101\) 61.1665i 0.605609i 0.953053 + 0.302804i \(0.0979229\pi\)
−0.953053 + 0.302804i \(0.902077\pi\)
\(102\) 0 0
\(103\) 133.129 1.29251 0.646255 0.763122i \(-0.276334\pi\)
0.646255 + 0.763122i \(0.276334\pi\)
\(104\) 0 0
\(105\) −72.8180 + 14.4844i −0.693505 + 0.137947i
\(106\) 0 0
\(107\) −55.0149 + 82.3356i −0.514158 + 0.769492i −0.994176 0.107767i \(-0.965630\pi\)
0.480018 + 0.877258i \(0.340630\pi\)
\(108\) 0 0
\(109\) 4.98036 + 7.45364i 0.0456914 + 0.0683820i 0.853612 0.520909i \(-0.174407\pi\)
−0.807921 + 0.589291i \(0.799407\pi\)
\(110\) 0 0
\(111\) 27.2906 11.3041i 0.245861 0.101839i
\(112\) 0 0
\(113\) 24.3650 122.491i 0.215619 1.08399i −0.709614 0.704591i \(-0.751131\pi\)
0.925233 0.379400i \(-0.123869\pi\)
\(114\) 0 0
\(115\) −206.054 + 206.054i −1.79177 + 1.79177i
\(116\) 0 0
\(117\) −52.6264 + 127.051i −0.449798 + 1.08591i
\(118\) 0 0
\(119\) 22.9966 + 87.8905i 0.193249 + 0.738576i
\(120\) 0 0
\(121\) −84.3883 34.9548i −0.697424 0.288882i
\(122\) 0 0
\(123\) −56.6770 56.6770i −0.460788 0.460788i
\(124\) 0 0
\(125\) 155.629 + 30.9566i 1.24503 + 0.247653i
\(126\) 0 0
\(127\) −60.6875 146.513i −0.477854 1.15364i −0.960613 0.277889i \(-0.910365\pi\)
0.482759 0.875753i \(-0.339635\pi\)
\(128\) 0 0
\(129\) 23.5226 15.7173i 0.182345 0.121839i
\(130\) 0 0
\(131\) −125.928 84.1427i −0.961286 0.642311i −0.0273026 0.999627i \(-0.508692\pi\)
−0.933983 + 0.357316i \(0.883692\pi\)
\(132\) 0 0
\(133\) −23.8870 120.088i −0.179601 0.902917i
\(134\) 0 0
\(135\) 211.260i 1.56489i
\(136\) 0 0
\(137\) 107.514 0.784772 0.392386 0.919801i \(-0.371650\pi\)
0.392386 + 0.919801i \(0.371650\pi\)
\(138\) 0 0
\(139\) 26.2161 5.21470i 0.188605 0.0375158i −0.0998840 0.994999i \(-0.531847\pi\)
0.288489 + 0.957483i \(0.406847\pi\)
\(140\) 0 0
\(141\) 64.6903 96.8158i 0.458796 0.686637i
\(142\) 0 0
\(143\) 179.383 + 268.466i 1.25443 + 1.87739i
\(144\) 0 0
\(145\) 103.316 42.7947i 0.712521 0.295136i
\(146\) 0 0
\(147\) 6.66529 33.5087i 0.0453421 0.227950i
\(148\) 0 0
\(149\) 45.5277 45.5277i 0.305555 0.305555i −0.537628 0.843182i \(-0.680679\pi\)
0.843182 + 0.537628i \(0.180679\pi\)
\(150\) 0 0
\(151\) −27.1000 + 65.4252i −0.179470 + 0.433280i −0.987856 0.155374i \(-0.950342\pi\)
0.808385 + 0.588654i \(0.200342\pi\)
\(152\) 0 0
\(153\) −105.321 + 6.28082i −0.688374 + 0.0410511i
\(154\) 0 0
\(155\) −245.110 101.528i −1.58135 0.655017i
\(156\) 0 0
\(157\) −71.9058 71.9058i −0.457999 0.457999i 0.439999 0.897998i \(-0.354979\pi\)
−0.897998 + 0.439999i \(0.854979\pi\)
\(158\) 0 0
\(159\) −15.7957 3.14196i −0.0993442 0.0197608i
\(160\) 0 0
\(161\) 71.6967 + 173.091i 0.445321 + 1.07510i
\(162\) 0 0
\(163\) 202.124 135.055i 1.24002 0.828558i 0.249835 0.968288i \(-0.419624\pi\)
0.990190 + 0.139730i \(0.0446236\pi\)
\(164\) 0 0
\(165\) −168.328 112.473i −1.02017 0.681655i
\(166\) 0 0
\(167\) 29.3721 + 147.664i 0.175881 + 0.884214i 0.963430 + 0.267961i \(0.0863499\pi\)
−0.787549 + 0.616252i \(0.788650\pi\)
\(168\) 0 0
\(169\) 321.967i 1.90513i
\(170\) 0 0
\(171\) 142.197 0.831562
\(172\) 0 0
\(173\) −129.875 + 25.8337i −0.750720 + 0.149328i −0.555598 0.831451i \(-0.687510\pi\)
−0.195123 + 0.980779i \(0.562510\pi\)
\(174\) 0 0
\(175\) 130.904 195.912i 0.748023 1.11950i
\(176\) 0 0
\(177\) 33.4216 + 50.0190i 0.188823 + 0.282593i
\(178\) 0 0
\(179\) −143.287 + 59.3513i −0.800485 + 0.331572i −0.745150 0.666896i \(-0.767622\pi\)
−0.0553340 + 0.998468i \(0.517622\pi\)
\(180\) 0 0
\(181\) 38.2243 192.166i 0.211184 1.06169i −0.719115 0.694891i \(-0.755453\pi\)
0.930299 0.366802i \(-0.119547\pi\)
\(182\) 0 0
\(183\) 4.16461 4.16461i 0.0227574 0.0227574i
\(184\) 0 0
\(185\) −56.2162 + 135.718i −0.303871 + 0.733610i
\(186\) 0 0
\(187\) −125.122 + 213.802i −0.669101 + 1.14332i
\(188\) 0 0
\(189\) −125.487 51.9783i −0.663951 0.275018i
\(190\) 0 0
\(191\) −111.071 111.071i −0.581523 0.581523i 0.353799 0.935322i \(-0.384890\pi\)
−0.935322 + 0.353799i \(0.884890\pi\)
\(192\) 0 0
\(193\) −133.384 26.5317i −0.691109 0.137470i −0.162974 0.986630i \(-0.552109\pi\)
−0.528135 + 0.849160i \(0.677109\pi\)
\(194\) 0 0
\(195\) 117.803 + 284.403i 0.604120 + 1.45848i
\(196\) 0 0
\(197\) −114.722 + 76.6551i −0.582347 + 0.389112i −0.811564 0.584263i \(-0.801384\pi\)
0.229217 + 0.973375i \(0.426384\pi\)
\(198\) 0 0
\(199\) 72.6120 + 48.5178i 0.364884 + 0.243808i 0.724473 0.689303i \(-0.242083\pi\)
−0.359589 + 0.933111i \(0.617083\pi\)
\(200\) 0 0
\(201\) 14.4115 + 72.4513i 0.0716988 + 0.360454i
\(202\) 0 0
\(203\) 71.8977i 0.354176i
\(204\) 0 0
\(205\) 398.607 1.94443
\(206\) 0 0
\(207\) −213.402 + 42.4483i −1.03093 + 0.205064i
\(208\) 0 0
\(209\) 185.485 277.598i 0.887488 1.32822i
\(210\) 0 0
\(211\) −182.735 273.483i −0.866044 1.29613i −0.953940 0.299996i \(-0.903015\pi\)
0.0878969 0.996130i \(-0.471985\pi\)
\(212\) 0 0
\(213\) 42.8283 17.7400i 0.201072 0.0832866i
\(214\) 0 0
\(215\) −27.4472 + 137.986i −0.127661 + 0.641797i
\(216\) 0 0
\(217\) −120.613 + 120.613i −0.555821 + 0.555821i
\(218\) 0 0
\(219\) 25.4237 61.3783i 0.116090 0.280266i
\(220\) 0 0
\(221\) 338.811 164.611i 1.53308 0.744846i
\(222\) 0 0
\(223\) 330.436 + 136.871i 1.48177 + 0.613771i 0.969509 0.245056i \(-0.0788064\pi\)
0.512266 + 0.858827i \(0.328806\pi\)
\(224\) 0 0
\(225\) 193.493 + 193.493i 0.859967 + 0.859967i
\(226\) 0 0
\(227\) 197.677 + 39.3203i 0.870822 + 0.173217i 0.610223 0.792230i \(-0.291080\pi\)
0.260599 + 0.965447i \(0.416080\pi\)
\(228\) 0 0
\(229\) −57.0684 137.775i −0.249207 0.601639i 0.748930 0.662649i \(-0.230568\pi\)
−0.998137 + 0.0610101i \(0.980568\pi\)
\(230\) 0 0
\(231\) −108.223 + 72.3124i −0.468499 + 0.313041i
\(232\) 0 0
\(233\) −133.734 89.3582i −0.573965 0.383511i 0.234446 0.972129i \(-0.424672\pi\)
−0.808412 + 0.588618i \(0.799672\pi\)
\(234\) 0 0
\(235\) 112.969 + 567.934i 0.480719 + 2.41674i
\(236\) 0 0
\(237\) 128.277i 0.541252i
\(238\) 0 0
\(239\) −194.043 −0.811897 −0.405949 0.913896i \(-0.633059\pi\)
−0.405949 + 0.913896i \(0.633059\pi\)
\(240\) 0 0
\(241\) 4.42729 0.880642i 0.0183705 0.00365412i −0.185897 0.982569i \(-0.559519\pi\)
0.204267 + 0.978915i \(0.434519\pi\)
\(242\) 0 0
\(243\) 139.505 208.784i 0.574096 0.859195i
\(244\) 0 0
\(245\) 94.3943 + 141.271i 0.385283 + 0.576617i
\(246\) 0 0
\(247\) −469.023 + 194.276i −1.89888 + 0.786541i
\(248\) 0 0
\(249\) −15.2748 + 76.7914i −0.0613444 + 0.308399i
\(250\) 0 0
\(251\) −123.725 + 123.725i −0.492928 + 0.492928i −0.909228 0.416299i \(-0.863327\pi\)
0.416299 + 0.909228i \(0.363327\pi\)
\(252\) 0 0
\(253\) −195.499 + 471.976i −0.772722 + 1.86552i
\(254\) 0 0
\(255\) −156.766 + 176.649i −0.614769 + 0.692742i
\(256\) 0 0
\(257\) −374.071 154.945i −1.45553 0.602900i −0.492022 0.870583i \(-0.663742\pi\)
−0.963507 + 0.267682i \(0.913742\pi\)
\(258\) 0 0
\(259\) 66.7838 + 66.7838i 0.257853 + 0.257853i
\(260\) 0 0
\(261\) 81.8944 + 16.2898i 0.313772 + 0.0624131i
\(262\) 0 0
\(263\) −127.880 308.730i −0.486236 1.17388i −0.956600 0.291405i \(-0.905877\pi\)
0.470364 0.882472i \(-0.344123\pi\)
\(264\) 0 0
\(265\) 66.5941 44.4967i 0.251298 0.167912i
\(266\) 0 0
\(267\) 161.675 + 108.028i 0.605526 + 0.404599i
\(268\) 0 0
\(269\) 74.7720 + 375.904i 0.277963 + 1.39741i 0.827276 + 0.561796i \(0.189889\pi\)
−0.549313 + 0.835617i \(0.685111\pi\)
\(270\) 0 0
\(271\) 174.280i 0.643099i 0.946893 + 0.321550i \(0.104204\pi\)
−0.946893 + 0.321550i \(0.895796\pi\)
\(272\) 0 0
\(273\) 197.917 0.724970
\(274\) 0 0
\(275\) 630.133 125.341i 2.29139 0.455787i
\(276\) 0 0
\(277\) −68.8045 + 102.973i −0.248392 + 0.371744i −0.934624 0.355638i \(-0.884263\pi\)
0.686232 + 0.727383i \(0.259263\pi\)
\(278\) 0 0
\(279\) −110.056 164.710i −0.394466 0.590360i
\(280\) 0 0
\(281\) −495.657 + 205.308i −1.76390 + 0.730633i −0.767976 + 0.640478i \(0.778736\pi\)
−0.995928 + 0.0901546i \(0.971264\pi\)
\(282\) 0 0
\(283\) 40.4243 203.227i 0.142842 0.718116i −0.841277 0.540605i \(-0.818195\pi\)
0.984119 0.177511i \(-0.0568046\pi\)
\(284\) 0 0
\(285\) 225.076 225.076i 0.789742 0.789742i
\(286\) 0 0
\(287\) 98.0730 236.769i 0.341718 0.824980i
\(288\) 0 0
\(289\) 227.192 + 178.619i 0.786133 + 0.618058i
\(290\) 0 0
\(291\) −73.1344 30.2932i −0.251321 0.104100i
\(292\) 0 0
\(293\) −226.842 226.842i −0.774206 0.774206i 0.204633 0.978839i \(-0.434400\pi\)
−0.978839 + 0.204633i \(0.934400\pi\)
\(294\) 0 0
\(295\) −293.418 58.3644i −0.994636 0.197845i
\(296\) 0 0
\(297\) −141.732 342.170i −0.477211 1.15209i
\(298\) 0 0
\(299\) 645.891 431.571i 2.16017 1.44338i
\(300\) 0 0
\(301\) 75.2095 + 50.2534i 0.249866 + 0.166955i
\(302\) 0 0
\(303\) 19.9450 + 100.270i 0.0658250 + 0.330925i
\(304\) 0 0
\(305\) 29.2896i 0.0960314i
\(306\) 0 0
\(307\) −290.459 −0.946119 −0.473060 0.881030i \(-0.656850\pi\)
−0.473060 + 0.881030i \(0.656850\pi\)
\(308\) 0 0
\(309\) 218.237 43.4101i 0.706270 0.140486i
\(310\) 0 0
\(311\) 236.493 353.937i 0.760428 1.13806i −0.226042 0.974118i \(-0.572579\pi\)
0.986470 0.163943i \(-0.0524214\pi\)
\(312\) 0 0
\(313\) 154.057 + 230.562i 0.492195 + 0.736621i 0.991543 0.129780i \(-0.0414272\pi\)
−0.499348 + 0.866401i \(0.666427\pi\)
\(314\) 0 0
\(315\) 254.703 105.501i 0.808581 0.334925i
\(316\) 0 0
\(317\) −47.2492 + 237.538i −0.149051 + 0.749331i 0.831877 + 0.554960i \(0.187266\pi\)
−0.980928 + 0.194371i \(0.937734\pi\)
\(318\) 0 0
\(319\) 138.626 138.626i 0.434564 0.434564i
\(320\) 0 0
\(321\) −63.3381 + 152.912i −0.197315 + 0.476361i
\(322\) 0 0
\(323\) −291.321 258.531i −0.901923 0.800405i
\(324\) 0 0
\(325\) −902.574 373.858i −2.77715 1.15033i
\(326\) 0 0
\(327\) 10.5948 + 10.5948i 0.0323999 + 0.0323999i
\(328\) 0 0
\(329\) 365.142 + 72.6314i 1.10986 + 0.220764i
\(330\) 0 0
\(331\) −49.7048 119.998i −0.150166 0.362532i 0.830840 0.556511i \(-0.187860\pi\)
−0.981006 + 0.193980i \(0.937860\pi\)
\(332\) 0 0
\(333\) −91.2006 + 60.9383i −0.273876 + 0.182998i
\(334\) 0 0
\(335\) −305.451 204.096i −0.911795 0.609242i
\(336\) 0 0
\(337\) −37.2805 187.422i −0.110625 0.556148i −0.995852 0.0909890i \(-0.970997\pi\)
0.885227 0.465159i \(-0.154003\pi\)
\(338\) 0 0
\(339\) 208.744i 0.615764i
\(340\) 0 0
\(341\) −465.108 −1.36395
\(342\) 0 0
\(343\) 363.967 72.3975i 1.06113 0.211071i
\(344\) 0 0
\(345\) −270.594 + 404.973i −0.784331 + 1.17383i
\(346\) 0 0
\(347\) −281.533 421.344i −0.811334 1.21425i −0.973772 0.227528i \(-0.926936\pi\)
0.162438 0.986719i \(-0.448064\pi\)
\(348\) 0 0
\(349\) 594.610 246.296i 1.70375 0.705718i 0.703765 0.710433i \(-0.251501\pi\)
0.999989 + 0.00471518i \(0.00150090\pi\)
\(350\) 0 0
\(351\) −109.868 + 552.344i −0.313015 + 1.57363i
\(352\) 0 0
\(353\) 276.652 276.652i 0.783716 0.783716i −0.196740 0.980456i \(-0.563035\pi\)
0.980456 + 0.196740i \(0.0630354\pi\)
\(354\) 0 0
\(355\) −88.2223 + 212.988i −0.248514 + 0.599965i
\(356\) 0 0
\(357\) 66.3575 + 136.580i 0.185875 + 0.382578i
\(358\) 0 0
\(359\) 69.2901 + 28.7009i 0.193009 + 0.0799468i 0.477094 0.878852i \(-0.341690\pi\)
−0.284086 + 0.958799i \(0.591690\pi\)
\(360\) 0 0
\(361\) 115.920 + 115.920i 0.321107 + 0.321107i
\(362\) 0 0
\(363\) −149.736 29.7843i −0.412495 0.0820503i
\(364\) 0 0
\(365\) 126.434 + 305.238i 0.346394 + 0.836268i
\(366\) 0 0
\(367\) −279.201 + 186.556i −0.760766 + 0.508327i −0.874406 0.485195i \(-0.838749\pi\)
0.113641 + 0.993522i \(0.463749\pi\)
\(368\) 0 0
\(369\) 247.469 + 165.354i 0.670649 + 0.448113i
\(370\) 0 0
\(371\) −10.0459 50.5042i −0.0270779 0.136130i
\(372\) 0 0
\(373\) 246.692i 0.661372i −0.943741 0.330686i \(-0.892720\pi\)
0.943741 0.330686i \(-0.107280\pi\)
\(374\) 0 0
\(375\) 265.217 0.707245
\(376\) 0 0
\(377\) −292.377 + 58.1573i −0.775535 + 0.154263i
\(378\) 0 0
\(379\) −5.66749 + 8.48200i −0.0149538 + 0.0223799i −0.838871 0.544330i \(-0.816784\pi\)
0.823917 + 0.566710i \(0.191784\pi\)
\(380\) 0 0
\(381\) −147.259 220.389i −0.386507 0.578449i
\(382\) 0 0
\(383\) −459.825 + 190.466i −1.20059 + 0.497299i −0.891189 0.453633i \(-0.850128\pi\)
−0.309398 + 0.950932i \(0.600128\pi\)
\(384\) 0 0
\(385\) 126.280 634.851i 0.327999 1.64896i
\(386\) 0 0
\(387\) −74.2808 + 74.2808i −0.191940 + 0.191940i
\(388\) 0 0
\(389\) 176.119 425.188i 0.452747 1.09303i −0.518526 0.855062i \(-0.673519\pi\)
0.971273 0.237967i \(-0.0764810\pi\)
\(390\) 0 0
\(391\) 514.376 + 301.026i 1.31554 + 0.769886i
\(392\) 0 0
\(393\) −233.871 96.8727i −0.595093 0.246495i
\(394\) 0 0
\(395\) −451.083 451.083i −1.14198 1.14198i
\(396\) 0 0
\(397\) 614.279 + 122.188i 1.54730 + 0.307777i 0.893561 0.448941i \(-0.148199\pi\)
0.653741 + 0.756719i \(0.273199\pi\)
\(398\) 0 0
\(399\) −78.3158 189.071i −0.196280 0.473862i
\(400\) 0 0
\(401\) −151.745 + 101.393i −0.378418 + 0.252851i −0.730203 0.683230i \(-0.760575\pi\)
0.351786 + 0.936081i \(0.385575\pi\)
\(402\) 0 0
\(403\) 588.043 + 392.918i 1.45916 + 0.974982i
\(404\) 0 0
\(405\) 21.6912 + 109.049i 0.0535585 + 0.269257i
\(406\) 0 0
\(407\) 257.532i 0.632756i
\(408\) 0 0
\(409\) −3.15321 −0.00770956 −0.00385478 0.999993i \(-0.501227\pi\)
−0.00385478 + 0.999993i \(0.501227\pi\)
\(410\) 0 0
\(411\) 176.247 35.0577i 0.428825 0.0852986i
\(412\) 0 0
\(413\) −106.860 + 159.928i −0.258741 + 0.387234i
\(414\) 0 0
\(415\) −216.322 323.749i −0.521259 0.780119i
\(416\) 0 0
\(417\) 41.2756 17.0969i 0.0989822 0.0409998i
\(418\) 0 0
\(419\) 15.7759 79.3110i 0.0376514 0.189286i −0.957383 0.288823i \(-0.906736\pi\)
0.995034 + 0.0995367i \(0.0317361\pi\)
\(420\) 0 0
\(421\) −98.2149 + 98.2149i −0.233290 + 0.233290i −0.814064 0.580775i \(-0.802750\pi\)
0.580775 + 0.814064i \(0.302750\pi\)
\(422\) 0 0
\(423\) −165.460 + 399.456i −0.391159 + 0.944340i
\(424\) 0 0
\(425\) −44.6190 748.203i −0.104986 1.76048i
\(426\) 0 0
\(427\) 17.3977 + 7.20638i 0.0407441 + 0.0168768i
\(428\) 0 0
\(429\) 381.604 + 381.604i 0.889519 + 0.889519i
\(430\) 0 0
\(431\) 318.284 + 63.3106i 0.738478 + 0.146892i 0.549976 0.835180i \(-0.314637\pi\)
0.188502 + 0.982073i \(0.439637\pi\)
\(432\) 0 0
\(433\) 228.502 + 551.654i 0.527719 + 1.27403i 0.933014 + 0.359841i \(0.117169\pi\)
−0.405294 + 0.914186i \(0.632831\pi\)
\(434\) 0 0
\(435\) 155.411 103.842i 0.357266 0.238718i
\(436\) 0 0
\(437\) −667.861 446.251i −1.52829 1.02117i
\(438\) 0 0
\(439\) 20.5765 + 103.445i 0.0468713 + 0.235638i 0.997119 0.0758586i \(-0.0241697\pi\)
−0.950247 + 0.311497i \(0.899170\pi\)
\(440\) 0 0
\(441\) 126.864i 0.287672i
\(442\) 0 0
\(443\) 312.080 0.704469 0.352235 0.935912i \(-0.385422\pi\)
0.352235 + 0.935912i \(0.385422\pi\)
\(444\) 0 0
\(445\) −948.408 + 188.650i −2.13125 + 0.423933i
\(446\) 0 0
\(447\) 59.7879 89.4790i 0.133754 0.200177i
\(448\) 0 0
\(449\) 261.386 + 391.192i 0.582152 + 0.871252i 0.999293 0.0375900i \(-0.0119681\pi\)
−0.417142 + 0.908842i \(0.636968\pi\)
\(450\) 0 0
\(451\) 645.609 267.420i 1.43151 0.592949i
\(452\) 0 0
\(453\) −23.0914 + 116.088i −0.0509744 + 0.256265i
\(454\) 0 0
\(455\) −695.972 + 695.972i −1.52961 + 1.52961i
\(456\) 0 0
\(457\) −36.5035 + 88.1273i −0.0798764 + 0.192839i −0.958773 0.284174i \(-0.908281\pi\)
0.878896 + 0.477013i \(0.158281\pi\)
\(458\) 0 0
\(459\) −418.003 + 109.371i −0.910682 + 0.238281i
\(460\) 0 0
\(461\) 658.460 + 272.743i 1.42833 + 0.591634i 0.956938 0.290291i \(-0.0937520\pi\)
0.471391 + 0.881924i \(0.343752\pi\)
\(462\) 0 0
\(463\) −133.625 133.625i −0.288606 0.288606i 0.547923 0.836529i \(-0.315419\pi\)
−0.836529 + 0.547923i \(0.815419\pi\)
\(464\) 0 0
\(465\) −434.914 86.5097i −0.935298 0.186042i
\(466\) 0 0
\(467\) 245.812 + 593.443i 0.526364 + 1.27076i 0.933890 + 0.357562i \(0.116392\pi\)
−0.407525 + 0.913194i \(0.633608\pi\)
\(468\) 0 0
\(469\) −196.384 + 131.220i −0.418730 + 0.279786i
\(470\) 0 0
\(471\) −141.322 94.4283i −0.300046 0.200485i
\(472\) 0 0
\(473\) 48.1179 + 241.905i 0.101729 + 0.511427i
\(474\) 0 0
\(475\) 1010.17i 2.12667i
\(476\) 0 0
\(477\) 59.8025 0.125372
\(478\) 0 0
\(479\) −770.025 + 153.168i −1.60757 + 0.319765i −0.915578 0.402141i \(-0.868266\pi\)
−0.691991 + 0.721906i \(0.743266\pi\)
\(480\) 0 0
\(481\) 217.560 325.601i 0.452307 0.676926i
\(482\) 0 0
\(483\) 173.973 + 260.370i 0.360193 + 0.539068i
\(484\) 0 0
\(485\) 363.702 150.650i 0.749900 0.310619i
\(486\) 0 0
\(487\) −27.1911 + 136.699i −0.0558338 + 0.280695i −0.998609 0.0527349i \(-0.983206\pi\)
0.942775 + 0.333430i \(0.108206\pi\)
\(488\) 0 0
\(489\) 287.303 287.303i 0.587533 0.587533i
\(490\) 0 0
\(491\) 111.558 269.326i 0.227207 0.548525i −0.768629 0.639695i \(-0.779061\pi\)
0.995835 + 0.0911698i \(0.0290606\pi\)
\(492\) 0 0
\(493\) −138.162 182.267i −0.280246 0.369709i
\(494\) 0 0
\(495\) 694.510 + 287.675i 1.40305 + 0.581162i
\(496\) 0 0
\(497\) 104.807 + 104.807i 0.210878 + 0.210878i
\(498\) 0 0
\(499\) −474.259 94.3360i −0.950419 0.189050i −0.304548 0.952497i \(-0.598506\pi\)
−0.645871 + 0.763447i \(0.723506\pi\)
\(500\) 0 0
\(501\) 96.2994 + 232.487i 0.192214 + 0.464047i
\(502\) 0 0
\(503\) 328.366 219.407i 0.652814 0.436196i −0.184564 0.982820i \(-0.559087\pi\)
0.837378 + 0.546624i \(0.184087\pi\)
\(504\) 0 0
\(505\) −422.735 282.462i −0.837099 0.559331i
\(506\) 0 0
\(507\) −104.986 527.800i −0.207073 1.04103i
\(508\) 0 0
\(509\) 605.145i 1.18889i −0.804136 0.594445i \(-0.797372\pi\)
0.804136 0.594445i \(-0.202628\pi\)
\(510\) 0 0
\(511\) 212.416 0.415687
\(512\) 0 0
\(513\) 571.132 113.605i 1.11332 0.221453i
\(514\) 0 0
\(515\) −614.778 + 920.080i −1.19374 + 1.78656i
\(516\) 0 0
\(517\) 563.991 + 844.072i 1.09089 + 1.63263i
\(518\) 0 0
\(519\) −204.480 + 84.6982i −0.393988 + 0.163195i
\(520\) 0 0
\(521\) 151.096 759.610i 0.290011 1.45798i −0.511127 0.859505i \(-0.670772\pi\)
0.801138 0.598479i \(-0.204228\pi\)
\(522\) 0 0
\(523\) 390.295 390.295i 0.746261 0.746261i −0.227513 0.973775i \(-0.573060\pi\)
0.973775 + 0.227513i \(0.0730596\pi\)
\(524\) 0 0
\(525\) 150.709 363.843i 0.287064 0.693034i
\(526\) 0 0
\(527\) −73.9894 + 537.539i −0.140397 + 1.02000i
\(528\) 0 0
\(529\) 646.774 + 267.903i 1.22263 + 0.506432i
\(530\) 0 0
\(531\) −157.953 157.953i −0.297463 0.297463i
\(532\) 0 0
\(533\) −1042.17 207.300i −1.95528 0.388930i
\(534\) 0 0
\(535\) −314.984 760.440i −0.588756 1.42138i
\(536\) 0 0
\(537\) −215.537 + 144.017i −0.401372 + 0.268188i
\(538\) 0 0
\(539\) 247.664 + 165.484i 0.459488 + 0.307020i
\(540\) 0 0
\(541\) −165.940 834.234i −0.306727 1.54202i −0.759566 0.650430i \(-0.774589\pi\)
0.452838 0.891593i \(-0.350411\pi\)
\(542\) 0 0
\(543\) 327.482i 0.603098i
\(544\) 0 0
\(545\) −74.5127 −0.136721
\(546\) 0 0
\(547\) −34.1837 + 6.79956i −0.0624930 + 0.0124306i −0.226238 0.974072i \(-0.572643\pi\)
0.163745 + 0.986503i \(0.447643\pi\)
\(548\) 0 0
\(549\) −12.1502 + 18.1840i −0.0221314 + 0.0331220i
\(550\) 0 0
\(551\) 171.251 + 256.296i 0.310801 + 0.465147i
\(552\) 0 0
\(553\) −378.924 + 156.955i −0.685214 + 0.283825i
\(554\) 0 0
\(555\) −47.9007 + 240.813i −0.0863075 + 0.433897i
\(556\) 0 0
\(557\) 6.67112 6.67112i 0.0119769 0.0119769i −0.701093 0.713070i \(-0.747304\pi\)
0.713070 + 0.701093i \(0.247304\pi\)
\(558\) 0 0
\(559\) 143.522 346.494i 0.256748 0.619846i
\(560\) 0 0
\(561\) −135.396 + 391.284i −0.241348 + 0.697476i
\(562\) 0 0
\(563\) 644.559 + 266.985i 1.14486 + 0.474218i 0.872808 0.488063i \(-0.162296\pi\)
0.272056 + 0.962281i \(0.412296\pi\)
\(564\) 0 0
\(565\) 734.046 + 734.046i 1.29920 + 1.29920i
\(566\) 0 0
\(567\) 70.1110 + 13.9459i 0.123653 + 0.0245960i
\(568\) 0 0
\(569\) −35.2433 85.0848i −0.0619390 0.149534i 0.889880 0.456195i \(-0.150788\pi\)
−0.951819 + 0.306661i \(0.900788\pi\)
\(570\) 0 0
\(571\) −838.846 + 560.499i −1.46908 + 0.981609i −0.474216 + 0.880409i \(0.657268\pi\)
−0.994866 + 0.101201i \(0.967732\pi\)
\(572\) 0 0
\(573\) −218.296 145.861i −0.380970 0.254556i
\(574\) 0 0
\(575\) −301.553 1516.01i −0.524441 2.63654i
\(576\) 0 0
\(577\) 902.184i 1.56358i 0.623544 + 0.781788i \(0.285692\pi\)
−0.623544 + 0.781788i \(0.714308\pi\)
\(578\) 0 0
\(579\) −227.308 −0.392587
\(580\) 0 0
\(581\) −245.528 + 48.8385i −0.422595 + 0.0840595i
\(582\) 0 0
\(583\) 78.0077 116.747i 0.133804 0.200252i
\(584\) 0 0
\(585\) −635.054 950.426i −1.08556 1.62466i
\(586\) 0 0
\(587\) 382.237 158.328i 0.651171 0.269724i −0.0325470 0.999470i \(-0.510362\pi\)
0.683718 + 0.729746i \(0.260362\pi\)
\(588\) 0 0
\(589\) 142.668 717.238i 0.242220 1.21772i
\(590\) 0 0
\(591\) −163.069 + 163.069i −0.275920 + 0.275920i
\(592\) 0 0
\(593\) 306.441 739.813i 0.516763 1.24758i −0.423118 0.906075i \(-0.639064\pi\)
0.939881 0.341502i \(-0.110936\pi\)
\(594\) 0 0
\(595\) −713.627 246.937i −1.19937 0.415021i
\(596\) 0 0
\(597\) 134.853 + 55.8581i 0.225885 + 0.0935646i
\(598\) 0 0
\(599\) −74.2393 74.2393i −0.123939 0.123939i 0.642417 0.766356i \(-0.277932\pi\)
−0.766356 + 0.642417i \(0.777932\pi\)
\(600\) 0 0
\(601\) −155.429 30.9167i −0.258617 0.0514422i 0.0640787 0.997945i \(-0.479589\pi\)
−0.322696 + 0.946503i \(0.604589\pi\)
\(602\) 0 0
\(603\) −104.970 253.420i −0.174080 0.420265i
\(604\) 0 0
\(605\) 631.279 421.807i 1.04344 0.697202i
\(606\) 0 0
\(607\) 86.2065 + 57.6013i 0.142021 + 0.0948951i 0.624551 0.780984i \(-0.285282\pi\)
−0.482530 + 0.875879i \(0.660282\pi\)
\(608\) 0 0
\(609\) −23.4442 117.862i −0.0384962 0.193533i
\(610\) 0 0
\(611\) 1543.62i 2.52639i
\(612\) 0 0
\(613\) 106.904 0.174394 0.0871971 0.996191i \(-0.472209\pi\)
0.0871971 + 0.996191i \(0.472209\pi\)
\(614\) 0 0
\(615\) 653.437 129.977i 1.06250 0.211344i
\(616\) 0 0
\(617\) −167.765 + 251.078i −0.271904 + 0.406934i −0.942143 0.335212i \(-0.891192\pi\)
0.670238 + 0.742146i \(0.266192\pi\)
\(618\) 0 0
\(619\) 161.719 + 242.029i 0.261258 + 0.391001i 0.938788 0.344494i \(-0.111950\pi\)
−0.677530 + 0.735495i \(0.736950\pi\)
\(620\) 0 0
\(621\) −823.213 + 340.986i −1.32563 + 0.549092i
\(622\) 0 0
\(623\) −121.289 + 609.761i −0.194685 + 0.978750i
\(624\) 0 0
\(625\) −153.221 + 153.221i −0.245153 + 0.245153i
\(626\) 0 0
\(627\) 213.547 515.549i 0.340586 0.822246i
\(628\) 0 0
\(629\) 297.637 + 40.9682i 0.473191 + 0.0651322i
\(630\) 0 0
\(631\) 58.1522 + 24.0874i 0.0921588 + 0.0381734i 0.428287 0.903643i \(-0.359117\pi\)
−0.336128 + 0.941816i \(0.609117\pi\)
\(632\) 0 0
\(633\) −388.734 388.734i −0.614114 0.614114i
\(634\) 0 0
\(635\) 1292.83 + 257.160i 2.03595 + 0.404976i
\(636\) 0 0
\(637\) −173.326 418.447i −0.272098 0.656902i
\(638\) 0 0
\(639\) −143.125 + 95.6330i −0.223983 + 0.149660i
\(640\) 0 0
\(641\) −33.3553 22.2873i −0.0520364 0.0347696i 0.529280 0.848447i \(-0.322462\pi\)
−0.581316 + 0.813678i \(0.697462\pi\)
\(642\) 0 0
\(643\) −68.0029 341.873i −0.105759 0.531685i −0.996949 0.0780554i \(-0.975129\pi\)
0.891190 0.453630i \(-0.149871\pi\)
\(644\) 0 0
\(645\) 235.151i 0.364575i
\(646\) 0 0
\(647\) −383.840 −0.593261 −0.296631 0.954992i \(-0.595863\pi\)
−0.296631 + 0.954992i \(0.595863\pi\)
\(648\) 0 0
\(649\) −514.393 + 102.319i −0.792594 + 0.157657i
\(650\) 0 0
\(651\) −158.392 + 237.050i −0.243305 + 0.364132i
\(652\) 0 0
\(653\) 181.018 + 270.913i 0.277210 + 0.414874i 0.943783 0.330565i \(-0.107239\pi\)
−0.666573 + 0.745440i \(0.732239\pi\)
\(654\) 0 0
\(655\) 1163.06 481.754i 1.77566 0.735502i
\(656\) 0 0
\(657\) −48.1270 + 241.951i −0.0732526 + 0.368266i
\(658\) 0 0
\(659\) −505.846 + 505.846i −0.767596 + 0.767596i −0.977683 0.210086i \(-0.932625\pi\)
0.210086 + 0.977683i \(0.432625\pi\)
\(660\) 0 0
\(661\) −116.507 + 281.272i −0.176258 + 0.425525i −0.987176 0.159635i \(-0.948968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(662\) 0 0
\(663\) 501.736 380.325i 0.756766 0.573642i
\(664\) 0 0
\(665\) 940.262 + 389.469i 1.41393 + 0.585668i
\(666\) 0 0
\(667\) −333.514 333.514i −0.500022 0.500022i
\(668\) 0 0
\(669\) 586.313 + 116.625i 0.876403 + 0.174327i
\(670\) 0 0
\(671\) 19.6500 + 47.4392i 0.0292846 + 0.0706993i
\(672\) 0 0
\(673\) −665.108 + 444.411i −0.988273 + 0.660343i −0.940953 0.338537i \(-0.890068\pi\)
−0.0473200 + 0.998880i \(0.515068\pi\)
\(674\) 0 0
\(675\) 931.746 + 622.573i 1.38036 + 0.922330i
\(676\) 0 0
\(677\) −29.0546 146.067i −0.0429167 0.215757i 0.953376 0.301785i \(-0.0975825\pi\)
−0.996293 + 0.0860281i \(0.972583\pi\)
\(678\) 0 0
\(679\) 253.101i 0.372756i
\(680\) 0 0
\(681\) 336.872 0.494673
\(682\) 0 0
\(683\) −548.888 + 109.181i −0.803643 + 0.159855i −0.579788 0.814768i \(-0.696864\pi\)
−0.223856 + 0.974622i \(0.571864\pi\)
\(684\) 0 0
\(685\) −496.490 + 743.051i −0.724804 + 1.08475i
\(686\) 0 0
\(687\) −138.477 207.246i −0.201568 0.301668i
\(688\) 0 0
\(689\) −197.253 + 81.7047i −0.286288 + 0.118584i
\(690\) 0 0
\(691\) −139.685 + 702.242i −0.202148 + 1.01627i 0.737817 + 0.675000i \(0.235857\pi\)
−0.939966 + 0.341269i \(0.889143\pi\)
\(692\) 0 0
\(693\) 341.753 341.753i 0.493150 0.493150i
\(694\) 0 0
\(695\) −85.0240 + 205.266i −0.122337 + 0.295347i
\(696\) 0 0
\(697\) −206.362 788.691i −0.296071 1.13155i
\(698\) 0 0
\(699\) −248.368 102.877i −0.355318 0.147178i
\(700\) 0 0
\(701\) 83.2890 + 83.2890i 0.118815 + 0.118815i 0.764014 0.645200i \(-0.223226\pi\)
−0.645200 + 0.764014i \(0.723226\pi\)
\(702\) 0 0
\(703\) −397.137 78.9955i −0.564918 0.112369i
\(704\) 0 0
\(705\) 370.380 + 894.177i 0.525362 + 1.26834i
\(706\) 0 0
\(707\) −271.789 + 181.604i −0.384426 + 0.256866i
\(708\) 0 0
\(709\) 655.789 + 438.184i 0.924949 + 0.618031i 0.924175 0.381969i \(-0.124754\pi\)
0.000773505 1.00000i \(0.499754\pi\)
\(710\) 0 0
\(711\) −92.9260 467.170i −0.130698 0.657061i
\(712\) 0 0
\(713\) 1118.98i 1.56940i
\(714\) 0 0
\(715\) −2683.81 −3.75358
\(716\) 0 0
\(717\) −318.095 + 63.2731i −0.443647 + 0.0882470i
\(718\) 0 0
\(719\) −272.711 + 408.141i −0.379292 + 0.567651i −0.971173 0.238375i \(-0.923385\pi\)
0.591881 + 0.806025i \(0.298385\pi\)
\(720\) 0 0
\(721\) 395.260 + 591.548i 0.548211 + 0.820455i
\(722\) 0 0
\(723\) 6.97049 2.88727i 0.00964107 0.00399346i
\(724\) 0 0
\(725\) −115.723 + 581.779i −0.159618 + 0.802454i
\(726\) 0 0
\(727\) 836.092 836.092i 1.15006 1.15006i 0.163518 0.986540i \(-0.447716\pi\)
0.986540 0.163518i \(-0.0522841\pi\)
\(728\) 0 0
\(729\) 114.541 276.526i 0.157120 0.379322i
\(730\) 0 0
\(731\) 287.231 17.1290i 0.392929 0.0234323i
\(732\) 0 0
\(733\) 233.598 + 96.7596i 0.318688 + 0.132005i 0.536293 0.844032i \(-0.319824\pi\)
−0.217605 + 0.976037i \(0.569824\pi\)
\(734\) 0 0
\(735\) 200.806 + 200.806i 0.273205 + 0.273205i
\(736\) 0 0
\(737\) −631.653 125.644i −0.857060 0.170480i
\(738\) 0 0
\(739\) 9.19672 + 22.2029i 0.0124448 + 0.0300445i 0.929980 0.367611i \(-0.119824\pi\)
−0.917535 + 0.397655i \(0.869824\pi\)
\(740\) 0 0
\(741\) −705.520 + 471.414i −0.952119 + 0.636186i
\(742\) 0 0
\(743\) 414.988 + 277.286i 0.558530 + 0.373198i 0.802561 0.596569i \(-0.203470\pi\)
−0.244031 + 0.969767i \(0.578470\pi\)
\(744\) 0 0
\(745\) 104.408 + 524.895i 0.140145 + 0.704557i
\(746\) 0 0
\(747\) 290.732i 0.389199i
\(748\) 0 0
\(749\) −529.193 −0.706532
\(750\) 0 0
\(751\) 1176.01 233.924i 1.56593 0.311483i 0.665471 0.746423i \(-0.268231\pi\)
0.900459 + 0.434941i \(0.143231\pi\)
\(752\) 0 0
\(753\) −162.478 + 243.166i −0.215775 + 0.322930i
\(754\) 0 0
\(755\) −327.022 489.423i −0.433142 0.648242i
\(756\) 0 0
\(757\) 907.665 375.967i 1.19903 0.496654i 0.308344 0.951275i \(-0.400225\pi\)
0.890685 + 0.454621i \(0.150225\pi\)
\(758\) 0 0
\(759\) −166.581 + 837.457i −0.219474 + 1.10337i
\(760\) 0 0
\(761\) 96.4662 96.4662i 0.126762 0.126762i −0.640879 0.767642i \(-0.721430\pi\)
0.767642 + 0.640879i \(0.221430\pi\)
\(762\) 0 0
\(763\) −18.3330 + 44.2599i −0.0240276 + 0.0580077i
\(764\) 0 0
\(765\) 442.958 756.902i 0.579029 0.989414i
\(766\) 0 0
\(767\) 736.793 + 305.190i 0.960617 + 0.397901i
\(768\) 0 0
\(769\) 211.871 + 211.871i 0.275515 + 0.275515i 0.831315 0.555801i \(-0.187588\pi\)
−0.555801 + 0.831315i \(0.687588\pi\)
\(770\) 0 0
\(771\) −663.739 132.026i −0.860880 0.171240i
\(772\) 0 0
\(773\) −143.142 345.575i −0.185177 0.447057i 0.803842 0.594842i \(-0.202786\pi\)
−0.989019 + 0.147785i \(0.952786\pi\)
\(774\) 0 0
\(775\) 1170.10 781.839i 1.50981 1.00882i
\(776\) 0 0
\(777\) 131.255 + 87.7020i 0.168926 + 0.112873i
\(778\) 0 0
\(779\) 214.351 + 1077.62i 0.275162 + 1.38333i
\(780\) 0 0
\(781\) 404.155i 0.517484i
\(782\) 0 0
\(783\) 341.942 0.436707
\(784\) 0 0
\(785\) 829.012 164.901i 1.05607 0.210065i
\(786\) 0 0
\(787\) 488.026 730.382i 0.620109 0.928059i −0.379886 0.925033i \(-0.624037\pi\)
0.999995 0.00302581i \(-0.000963146\pi\)
\(788\) 0 0
\(789\) −310.303 464.402i −0.393287 0.588595i
\(790\) 0 0
\(791\) 616.621 255.413i 0.779546 0.322898i
\(792\) 0 0
\(793\) 15.2323 76.5782i 0.0192085 0.0965677i
\(794\) 0 0
\(795\) 94.6583 94.6583i 0.119067 0.119067i
\(796\) 0 0
\(797\) 33.8659 81.7594i 0.0424917 0.102584i −0.901209 0.433385i \(-0.857319\pi\)
0.943700 + 0.330801i \(0.107319\pi\)
\(798\) 0 0
\(799\) 1065.24 517.546i 1.33321 0.647742i
\(800\) 0 0
\(801\) −667.062 276.306i −0.832787 0.344952i
\(802\) 0 0
\(803\) 409.560 + 409.560i 0.510037 + 0.510037i
\(804\) 0 0
\(805\) −1527.36 303.811i −1.89734 0.377405i
\(806\) 0 0
\(807\) 245.147 + 591.838i 0.303776 + 0.733380i
\(808\) 0 0
\(809\) −1156.53 + 772.769i −1.42958 + 0.955216i −0.430979 + 0.902362i \(0.641832\pi\)
−0.998602 + 0.0528535i \(0.983168\pi\)
\(810\) 0 0
\(811\) −441.389 294.927i −0.544253 0.363658i 0.252846 0.967506i \(-0.418633\pi\)
−0.797099 + 0.603848i \(0.793633\pi\)
\(812\) 0 0
\(813\) 56.8286 + 285.697i 0.0698999 + 0.351411i
\(814\) 0 0
\(815\) 2020.60i 2.47926i
\(816\) 0 0
\(817\) −387.799 −0.474662
\(818\) 0 0
\(819\) −720.793 + 143.375i −0.880089 + 0.175061i
\(820\) 0 0
\(821\) −201.598 + 301.713i −0.245552 + 0.367494i −0.933688 0.358087i \(-0.883429\pi\)
0.688136 + 0.725581i \(0.258429\pi\)
\(822\) 0 0
\(823\) −501.260 750.189i −0.609065 0.911530i 0.390895 0.920435i \(-0.372165\pi\)
−0.999960 + 0.00890509i \(0.997165\pi\)
\(824\) 0 0
\(825\) 992.106 410.944i 1.20255 0.498114i
\(826\) 0 0
\(827\) 16.1095 80.9880i 0.0194795 0.0979299i −0.969823 0.243812i \(-0.921602\pi\)
0.989302 + 0.145882i \(0.0466020\pi\)
\(828\) 0 0
\(829\) −667.698 + 667.698i −0.805426 + 0.805426i −0.983938 0.178511i \(-0.942872\pi\)
0.178511 + 0.983938i \(0.442872\pi\)
\(830\) 0 0
\(831\) −79.2140 + 191.239i −0.0953236 + 0.230132i
\(832\) 0 0
\(833\) 230.653 259.907i 0.276894 0.312013i
\(834\) 0 0
\(835\) −1156.17 478.903i −1.38464 0.573536i
\(836\) 0 0
\(837\) −573.630 573.630i −0.685340 0.685340i
\(838\) 0 0
\(839\) −1374.83 273.470i −1.63865 0.325947i −0.712087 0.702092i \(-0.752250\pi\)
−0.926561 + 0.376144i \(0.877250\pi\)
\(840\) 0 0
\(841\) −252.570 609.758i −0.300321 0.725039i
\(842\) 0 0
\(843\) −745.584 + 498.183i −0.884441 + 0.590965i
\(844\) 0 0
\(845\) 2225.18 + 1486.82i 2.63335 + 1.75955i
\(846\) 0 0
\(847\) −95.2303 478.755i −0.112432 0.565236i
\(848\) 0 0
\(849\) 346.331i 0.407928i
\(850\) 0 0
\(851\) 619.585 0.728067
\(852\) 0 0
\(853\) 1671.12 332.407i 1.95911 0.389691i 0.969790 0.243942i \(-0.0784407\pi\)
0.989322 0.145749i \(-0.0465593\pi\)
\(854\) 0 0
\(855\) −656.656 + 982.754i −0.768018 + 1.14942i
\(856\) 0 0
\(857\) −164.307 245.903i −0.191723 0.286934i 0.723137 0.690705i \(-0.242700\pi\)
−0.914860 + 0.403770i \(0.867700\pi\)
\(858\) 0 0
\(859\) −978.055 + 405.124i −1.13860 + 0.471622i −0.870695 0.491824i \(-0.836330\pi\)
−0.267902 + 0.963446i \(0.586330\pi\)
\(860\) 0 0
\(861\) 83.5660 420.115i 0.0970570 0.487938i
\(862\) 0 0
\(863\) 333.846 333.846i 0.386844 0.386844i −0.486716 0.873560i \(-0.661805\pi\)
0.873560 + 0.486716i \(0.161805\pi\)
\(864\) 0 0
\(865\) 421.209 1016.89i 0.486947 1.17559i
\(866\) 0 0
\(867\) 430.680 + 218.727i 0.496747 + 0.252281i
\(868\) 0 0
\(869\) −1033.23 427.977i −1.18898 0.492494i
\(870\) 0 0
\(871\) 692.467 + 692.467i 0.795025 + 0.795025i
\(872\) 0 0
\(873\) 288.293 + 57.3450i 0.330232 + 0.0656873i
\(874\) 0 0
\(875\) 324.511 + 783.439i 0.370870 + 0.895359i
\(876\) 0 0
\(877\) 110.050 73.5328i 0.125484 0.0838459i −0.491244 0.871022i \(-0.663458\pi\)
0.616728 + 0.787176i \(0.288458\pi\)
\(878\) 0 0
\(879\) −445.830 297.894i −0.507202 0.338901i
\(880\) 0 0
\(881\) −290.402 1459.95i −0.329628 1.65715i −0.689602 0.724188i \(-0.742215\pi\)
0.359975 0.932962i \(-0.382785\pi\)
\(882\) 0 0
\(883\) 850.214i 0.962870i 0.876482 + 0.481435i \(0.159884\pi\)
−0.876482 + 0.481435i \(0.840116\pi\)
\(884\) 0 0
\(885\) −500.030 −0.565006
\(886\) 0 0
\(887\) −764.270 + 152.023i −0.861635 + 0.171390i −0.606079 0.795404i \(-0.707259\pi\)
−0.255556 + 0.966794i \(0.582259\pi\)
\(888\) 0 0
\(889\) 470.837 704.658i 0.529626 0.792641i
\(890\) 0 0
\(891\) 108.292 + 162.070i 0.121540 + 0.181897i
\(892\) 0 0
\(893\) −1474.63 + 610.813i −1.65132 + 0.684001i
\(894\) 0 0
\(895\) 251.498 1264.37i 0.281003 1.41270i
\(896\) 0 0
\(897\) 918.084 918.084i 1.02350 1.02350i
\(898\) 0 0
\(899\) 164.331 396.730i 0.182793 0.441301i
\(900\) 0 0
\(901\) −122.518 108.728i −0.135980 0.120675i
\(902\) 0 0
\(903\) 139.677 + 57.8563i 0.154682 + 0.0640712i
\(904\) 0 0
\(905\) 1151.59 + 1151.59i 1.27247 + 1.27247i
\(906\) 0 0
\(907\) 886.642 + 176.364i 0.977555 + 0.194448i 0.657916 0.753092i \(-0.271438\pi\)
0.319639 + 0.947539i \(0.396438\pi\)
\(908\) 0 0
\(909\) −145.275 350.725i −0.159819 0.385836i
\(910\) 0 0
\(911\) 641.980 428.957i 0.704698 0.470864i −0.150871 0.988553i \(-0.548208\pi\)
0.855569 + 0.517689i \(0.173208\pi\)
\(912\) 0 0
\(913\) −567.568 379.237i −0.621652 0.415374i
\(914\) 0 0
\(915\) 9.55065 + 48.0144i 0.0104379 + 0.0524747i
\(916\) 0 0
\(917\) 809.376i 0.882634i
\(918\) 0 0
\(919\) −1260.71 −1.37183 −0.685913 0.727684i \(-0.740597\pi\)
−0.685913 + 0.727684i \(0.740597\pi\)
\(920\) 0 0
\(921\) −476.148 + 94.7118i −0.516991 + 0.102836i
\(922\) 0 0
\(923\) 341.425 510.979i 0.369908 0.553607i
\(924\) 0 0
\(925\) −432.907 647.890i −0.468007 0.700422i
\(926\) 0 0
\(927\) −763.351 + 316.190i −0.823464 + 0.341090i
\(928\) 0 0
\(929\) −46.8279 + 235.420i −0.0504067 + 0.253412i −0.997770 0.0667496i \(-0.978737\pi\)
0.947363 + 0.320161i \(0.103737\pi\)
\(930\) 0 0
\(931\) −331.159 + 331.159i −0.355703 + 0.355703i
\(932\) 0 0
\(933\) 272.272 657.323i 0.291825 0.704527i
\(934\) 0 0
\(935\) −899.825 1852.06i −0.962380 1.98082i
\(936\) 0 0
\(937\) 1368.87 + 567.005i 1.46091 + 0.605128i 0.964765 0.263113i \(-0.0847492\pi\)
0.496143 + 0.868241i \(0.334749\pi\)
\(938\) 0 0
\(939\) 327.726 + 327.726i 0.349016 + 0.349016i
\(940\) 0 0
\(941\) 1508.49 + 300.056i 1.60307 + 0.318870i 0.913964 0.405794i \(-0.133005\pi\)
0.689102 + 0.724664i \(0.258005\pi\)
\(942\) 0 0
\(943\) −643.375 1553.24i −0.682264 1.64713i
\(944\) 0 0
\(945\) 938.722 627.234i 0.993356 0.663740i
\(946\) 0 0
\(947\) 195.098 + 130.360i 0.206017 + 0.137656i 0.654299 0.756236i \(-0.272964\pi\)
−0.448282 + 0.893892i \(0.647964\pi\)
\(948\) 0 0
\(949\) −171.821 863.804i −0.181055 0.910225i
\(950\) 0 0
\(951\) 404.802i 0.425660i
\(952\) 0 0
\(953\) −577.880 −0.606380 −0.303190 0.952930i \(-0.598052\pi\)
−0.303190 + 0.952930i \(0.598052\pi\)
\(954\) 0 0
\(955\) 1280.55 254.718i 1.34089 0.266720i
\(956\) 0 0
\(957\) 182.047 272.452i 0.190226 0.284694i
\(958\) 0 0
\(959\) 319.209 + 477.730i 0.332856 + 0.498155i
\(960\) 0 0
\(961\) −53.3672 + 22.1054i −0.0555330 + 0.0230025i
\(962\) 0 0
\(963\) 119.899 602.772i 0.124506 0.625931i
\(964\) 0 0
\(965\) 799.324 799.324i 0.828315 0.828315i
\(966\) 0 0
\(967\) 290.511 701.356i 0.300425 0.725290i −0.699518 0.714615i \(-0.746602\pi\)
0.999943 0.0106752i \(-0.00339810\pi\)
\(968\) 0 0
\(969\) −561.863 328.816i −0.579838 0.339336i
\(970\) 0 0
\(971\) 1394.29 + 577.535i 1.43593 + 0.594783i 0.958809 0.284052i \(-0.0916788\pi\)
0.477125 + 0.878835i \(0.341679\pi\)
\(972\) 0 0
\(973\) 101.007 + 101.007i 0.103810 + 0.103810i
\(974\) 0 0
\(975\) −1601.50 318.557i −1.64256 0.326725i
\(976\) 0 0
\(977\) −533.687 1288.43i −0.546251 1.31877i −0.920248 0.391336i \(-0.872013\pi\)
0.373997 0.927430i \(-0.377987\pi\)
\(978\) 0 0
\(979\) −1409.54 + 941.823i −1.43977 + 0.962026i
\(980\) 0 0
\(981\) −46.2601 30.9100i −0.0471560 0.0315087i
\(982\) 0 0
\(983\) 209.895 + 1055.21i 0.213525 + 1.07346i 0.927652 + 0.373446i \(0.121824\pi\)
−0.714127 + 0.700016i \(0.753176\pi\)
\(984\) 0 0
\(985\) 1146.86i 1.16432i
\(986\) 0 0
\(987\) 622.261 0.630457
\(988\) 0 0
\(989\) 581.989 115.765i 0.588462 0.117052i
\(990\) 0 0
\(991\) −939.515 + 1406.08i −0.948048 + 1.41885i −0.0403703 + 0.999185i \(0.512854\pi\)
−0.907677 + 0.419669i \(0.862146\pi\)
\(992\) 0 0
\(993\) −120.610 180.505i −0.121460 0.181778i
\(994\) 0 0
\(995\) −670.633 + 277.785i −0.674004 + 0.279181i
\(996\) 0 0
\(997\) −140.134 + 704.501i −0.140556 + 0.706620i 0.844660 + 0.535303i \(0.179802\pi\)
−0.985216 + 0.171318i \(0.945198\pi\)
\(998\) 0 0
\(999\) −317.621 + 317.621i −0.317938 + 0.317938i
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 136.3.t.b.57.3 40
4.3 odd 2 272.3.bh.g.193.3 40
17.3 odd 16 inner 136.3.t.b.105.3 yes 40
68.3 even 16 272.3.bh.g.241.3 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.3.t.b.57.3 40 1.1 even 1 trivial
136.3.t.b.105.3 yes 40 17.3 odd 16 inner
272.3.bh.g.193.3 40 4.3 odd 2
272.3.bh.g.241.3 40 68.3 even 16