Properties

Label 1008.4.bt.a.17.4
Level $1008$
Weight $4$
Character 1008.17
Analytic conductor $59.474$
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] = [1008,4,Mod(17,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.bt (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: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.4
Root \(-3.91663 - 2.26127i\) of defining polynomial
Character \(\chi\) \(=\) 1008.17
Dual form 1008.4.bt.a.593.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.632851 - 1.09613i) q^{5} +(-13.4032 + 12.7810i) q^{7} +O(q^{10})\) \(q+(-0.632851 - 1.09613i) q^{5} +(-13.4032 + 12.7810i) q^{7} +(-36.0248 - 20.7989i) q^{11} +85.7355i q^{13} +(38.8929 - 67.3645i) q^{17} +(-42.1638 + 24.3433i) q^{19} +(-78.7639 + 45.4743i) q^{23} +(61.6990 - 106.866i) q^{25} +151.196i q^{29} +(-76.3661 - 44.0900i) q^{31} +(22.4919 + 6.60319i) q^{35} +(-45.2914 - 78.4470i) q^{37} +383.530 q^{41} +227.894 q^{43} +(-69.5529 - 120.469i) q^{47} +(16.2918 - 342.613i) q^{49} +(-289.749 - 167.287i) q^{53} +52.6505i q^{55} +(440.050 - 762.189i) q^{59} +(11.3944 - 6.57854i) q^{61} +(93.9774 - 54.2579i) q^{65} +(-221.212 + 383.151i) q^{67} +341.552i q^{71} +(-798.218 - 460.851i) q^{73} +(748.678 - 181.661i) q^{77} +(-206.564 - 357.780i) q^{79} +954.307 q^{83} -98.4538 q^{85} +(-14.8490 - 25.7193i) q^{89} +(-1095.79 - 1149.13i) q^{91} +(53.3668 + 30.8113i) q^{95} +1199.63i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 56 q^{7} + 612 q^{19} - 20 q^{25} - 1128 q^{31} - 1196 q^{37} - 328 q^{43} + 784 q^{49} - 1632 q^{61} - 308 q^{67} + 4068 q^{73} + 2176 q^{79} - 4608 q^{85} - 924 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(1\) \(-1\)

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
\(5\) −0.632851 1.09613i −0.0566040 0.0980409i 0.836335 0.548219i \(-0.184694\pi\)
−0.892939 + 0.450178i \(0.851361\pi\)
\(6\) 0 0
\(7\) −13.4032 + 12.7810i −0.723705 + 0.690109i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −36.0248 20.7989i −0.987443 0.570101i −0.0829344 0.996555i \(-0.526429\pi\)
−0.904509 + 0.426454i \(0.859763\pi\)
\(12\) 0 0
\(13\) 85.7355i 1.82914i 0.404433 + 0.914568i \(0.367469\pi\)
−0.404433 + 0.914568i \(0.632531\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 38.8929 67.3645i 0.554878 0.961076i −0.443035 0.896504i \(-0.646098\pi\)
0.997913 0.0645722i \(-0.0205683\pi\)
\(18\) 0 0
\(19\) −42.1638 + 24.3433i −0.509107 + 0.293933i −0.732466 0.680803i \(-0.761631\pi\)
0.223360 + 0.974736i \(0.428298\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −78.7639 + 45.4743i −0.714061 + 0.412263i −0.812563 0.582874i \(-0.801928\pi\)
0.0985019 + 0.995137i \(0.468595\pi\)
\(24\) 0 0
\(25\) 61.6990 106.866i 0.493592 0.854926i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 151.196i 0.968151i 0.875026 + 0.484075i \(0.160844\pi\)
−0.875026 + 0.484075i \(0.839156\pi\)
\(30\) 0 0
\(31\) −76.3661 44.0900i −0.442444 0.255445i 0.262190 0.965016i \(-0.415555\pi\)
−0.704634 + 0.709571i \(0.748889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 22.4919 + 6.60319i 0.108624 + 0.0318898i
\(36\) 0 0
\(37\) −45.2914 78.4470i −0.201239 0.348557i 0.747689 0.664050i \(-0.231164\pi\)
−0.948928 + 0.315493i \(0.897830\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 383.530 1.46091 0.730455 0.682961i \(-0.239308\pi\)
0.730455 + 0.682961i \(0.239308\pi\)
\(42\) 0 0
\(43\) 227.894 0.808222 0.404111 0.914710i \(-0.367581\pi\)
0.404111 + 0.914710i \(0.367581\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −69.5529 120.469i −0.215858 0.373877i 0.737680 0.675151i \(-0.235922\pi\)
−0.953538 + 0.301274i \(0.902588\pi\)
\(48\) 0 0
\(49\) 16.2918 342.613i 0.0474981 0.998871i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −289.749 167.287i −0.750945 0.433558i 0.0750904 0.997177i \(-0.476075\pi\)
−0.826035 + 0.563619i \(0.809409\pi\)
\(54\) 0 0
\(55\) 52.6505i 0.129080i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 440.050 762.189i 0.971010 1.68184i 0.278493 0.960438i \(-0.410165\pi\)
0.692518 0.721401i \(-0.256501\pi\)
\(60\) 0 0
\(61\) 11.3944 6.57854i 0.0239164 0.0138081i −0.487994 0.872847i \(-0.662271\pi\)
0.511911 + 0.859039i \(0.328938\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 93.9774 54.2579i 0.179330 0.103536i
\(66\) 0 0
\(67\) −221.212 + 383.151i −0.403364 + 0.698647i −0.994130 0.108197i \(-0.965492\pi\)
0.590766 + 0.806843i \(0.298826\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 341.552i 0.570912i 0.958392 + 0.285456i \(0.0921450\pi\)
−0.958392 + 0.285456i \(0.907855\pi\)
\(72\) 0 0
\(73\) −798.218 460.851i −1.27979 0.738885i −0.302977 0.952998i \(-0.597981\pi\)
−0.976809 + 0.214113i \(0.931314\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 748.678 181.661i 1.10805 0.268859i
\(78\) 0 0
\(79\) −206.564 357.780i −0.294181 0.509537i 0.680613 0.732643i \(-0.261714\pi\)
−0.974794 + 0.223107i \(0.928380\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 954.307 1.26203 0.631017 0.775769i \(-0.282638\pi\)
0.631017 + 0.775769i \(0.282638\pi\)
\(84\) 0 0
\(85\) −98.4538 −0.125633
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.8490 25.7193i −0.0176853 0.0306319i 0.857047 0.515238i \(-0.172296\pi\)
−0.874733 + 0.484606i \(0.838963\pi\)
\(90\) 0 0
\(91\) −1095.79 1149.13i −1.26230 1.32375i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 53.3668 + 30.8113i 0.0576349 + 0.0332755i
\(96\) 0 0
\(97\) 1199.63i 1.25572i 0.778328 + 0.627858i \(0.216068\pi\)
−0.778328 + 0.627858i \(0.783932\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 327.422 567.111i 0.322571 0.558710i −0.658447 0.752628i \(-0.728786\pi\)
0.981018 + 0.193918i \(0.0621195\pi\)
\(102\) 0 0
\(103\) 1186.01 684.744i 1.13457 0.655047i 0.189493 0.981882i \(-0.439316\pi\)
0.945081 + 0.326836i \(0.105982\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 371.311 214.377i 0.335477 0.193688i −0.322793 0.946470i \(-0.604622\pi\)
0.658270 + 0.752782i \(0.271289\pi\)
\(108\) 0 0
\(109\) 334.261 578.957i 0.293728 0.508752i −0.680960 0.732321i \(-0.738437\pi\)
0.974688 + 0.223568i \(0.0717706\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 914.837i 0.761598i −0.924658 0.380799i \(-0.875649\pi\)
0.924658 0.380799i \(-0.124351\pi\)
\(114\) 0 0
\(115\) 99.6917 + 57.5570i 0.0808374 + 0.0466715i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 339.696 + 1399.99i 0.261680 + 1.07846i
\(120\) 0 0
\(121\) 199.690 + 345.873i 0.150030 + 0.259859i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −314.398 −0.224965
\(126\) 0 0
\(127\) −1260.95 −0.881034 −0.440517 0.897744i \(-0.645205\pi\)
−0.440517 + 0.897744i \(0.645205\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −683.600 1184.03i −0.455926 0.789688i 0.542814 0.839853i \(-0.317359\pi\)
−0.998741 + 0.0501648i \(0.984025\pi\)
\(132\) 0 0
\(133\) 253.998 865.173i 0.165597 0.564060i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −953.631 550.579i −0.594702 0.343351i 0.172252 0.985053i \(-0.444896\pi\)
−0.766955 + 0.641701i \(0.778229\pi\)
\(138\) 0 0
\(139\) 2306.56i 1.40748i −0.710458 0.703739i \(-0.751512\pi\)
0.710458 0.703739i \(-0.248488\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1783.21 3088.60i 1.04279 1.80617i
\(144\) 0 0
\(145\) 165.730 95.6845i 0.0949184 0.0548012i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1520.57 877.901i 0.836040 0.482688i −0.0198764 0.999802i \(-0.506327\pi\)
0.855916 + 0.517115i \(0.172994\pi\)
\(150\) 0 0
\(151\) −262.491 + 454.647i −0.141465 + 0.245024i −0.928048 0.372460i \(-0.878515\pi\)
0.786584 + 0.617484i \(0.211848\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 111.610i 0.0578368i
\(156\) 0 0
\(157\) −1141.44 659.009i −0.580233 0.334998i 0.180993 0.983484i \(-0.442069\pi\)
−0.761226 + 0.648487i \(0.775402\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 474.481 1616.18i 0.232263 0.791137i
\(162\) 0 0
\(163\) −223.916 387.834i −0.107598 0.186365i 0.807199 0.590280i \(-0.200983\pi\)
−0.914797 + 0.403915i \(0.867649\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −811.124 −0.375848 −0.187924 0.982184i \(-0.560176\pi\)
−0.187924 + 0.982184i \(0.560176\pi\)
\(168\) 0 0
\(169\) −5153.58 −2.34574
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1121.24 + 1942.04i 0.492751 + 0.853470i 0.999965 0.00834994i \(-0.00265790\pi\)
−0.507214 + 0.861820i \(0.669325\pi\)
\(174\) 0 0
\(175\) 538.888 + 2220.92i 0.232778 + 0.959347i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2531.77 + 1461.72i 1.05717 + 0.610357i 0.924648 0.380824i \(-0.124359\pi\)
0.132521 + 0.991180i \(0.457693\pi\)
\(180\) 0 0
\(181\) 282.859i 0.116159i −0.998312 0.0580794i \(-0.981502\pi\)
0.998312 0.0580794i \(-0.0184977\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −57.3255 + 99.2906i −0.0227819 + 0.0394594i
\(186\) 0 0
\(187\) −2802.22 + 1617.86i −1.09582 + 0.632672i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3998.63 2308.61i 1.51482 0.874582i 0.514971 0.857208i \(-0.327803\pi\)
0.999849 0.0173741i \(-0.00553064\pi\)
\(192\) 0 0
\(193\) 2077.73 3598.73i 0.774912 1.34219i −0.159933 0.987128i \(-0.551128\pi\)
0.934844 0.355058i \(-0.115539\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1626.36i 0.588190i −0.955776 0.294095i \(-0.904982\pi\)
0.955776 0.294095i \(-0.0950183\pi\)
\(198\) 0 0
\(199\) −150.861 87.0995i −0.0537399 0.0310267i 0.472889 0.881122i \(-0.343211\pi\)
−0.526629 + 0.850095i \(0.676544\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1932.44 2026.51i −0.668130 0.700656i
\(204\) 0 0
\(205\) −242.718 420.399i −0.0826933 0.143229i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2025.25 0.670286
\(210\) 0 0
\(211\) 2942.35 0.959999 0.479999 0.877269i \(-0.340637\pi\)
0.479999 + 0.877269i \(0.340637\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −144.223 249.802i −0.0457486 0.0792389i
\(216\) 0 0
\(217\) 1587.06 385.088i 0.496484 0.120468i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5775.53 + 3334.51i 1.75794 + 1.01495i
\(222\) 0 0
\(223\) 3374.75i 1.01341i −0.862120 0.506704i \(-0.830864\pi\)
0.862120 0.506704i \(-0.169136\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1515.43 + 2624.79i −0.443094 + 0.767461i −0.997917 0.0645069i \(-0.979453\pi\)
0.554823 + 0.831968i \(0.312786\pi\)
\(228\) 0 0
\(229\) 960.030 554.274i 0.277033 0.159945i −0.355046 0.934849i \(-0.615535\pi\)
0.632080 + 0.774904i \(0.282202\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2684.57 1549.94i 0.754815 0.435793i −0.0726160 0.997360i \(-0.523135\pi\)
0.827431 + 0.561567i \(0.189801\pi\)
\(234\) 0 0
\(235\) −88.0333 + 152.478i −0.0244368 + 0.0423259i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1735.25i 0.469640i −0.972039 0.234820i \(-0.924550\pi\)
0.972039 0.234820i \(-0.0754501\pi\)
\(240\) 0 0
\(241\) −1039.26 600.019i −0.277779 0.160376i 0.354638 0.935004i \(-0.384604\pi\)
−0.632418 + 0.774628i \(0.717937\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −385.859 + 198.965i −0.100619 + 0.0518833i
\(246\) 0 0
\(247\) −2087.08 3614.93i −0.537643 0.931225i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3712.56 0.933603 0.466802 0.884362i \(-0.345406\pi\)
0.466802 + 0.884362i \(0.345406\pi\)
\(252\) 0 0
\(253\) 3783.27 0.940126
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −389.574 674.762i −0.0945563 0.163776i 0.814867 0.579648i \(-0.196810\pi\)
−0.909423 + 0.415872i \(0.863477\pi\)
\(258\) 0 0
\(259\) 1609.68 + 472.572i 0.386180 + 0.113375i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1702.07 982.690i −0.399065 0.230400i 0.287015 0.957926i \(-0.407337\pi\)
−0.686080 + 0.727526i \(0.740670\pi\)
\(264\) 0 0
\(265\) 423.470i 0.0981644i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1236.41 + 2141.53i −0.280243 + 0.485395i −0.971444 0.237267i \(-0.923748\pi\)
0.691201 + 0.722662i \(0.257082\pi\)
\(270\) 0 0
\(271\) −4095.79 + 2364.71i −0.918088 + 0.530058i −0.883025 0.469327i \(-0.844497\pi\)
−0.0350633 + 0.999385i \(0.511163\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4445.39 + 2566.54i −0.974788 + 0.562794i
\(276\) 0 0
\(277\) −586.579 + 1015.98i −0.127235 + 0.220378i −0.922604 0.385747i \(-0.873944\pi\)
0.795369 + 0.606125i \(0.207277\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8195.18i 1.73980i −0.493229 0.869899i \(-0.664184\pi\)
0.493229 0.869899i \(-0.335816\pi\)
\(282\) 0 0
\(283\) −2242.44 1294.67i −0.471021 0.271944i 0.245646 0.969360i \(-0.421000\pi\)
−0.716667 + 0.697415i \(0.754333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5140.53 + 4901.90i −1.05727 + 1.00819i
\(288\) 0 0
\(289\) −568.820 985.225i −0.115779 0.200534i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8871.16 −1.76880 −0.884400 0.466729i \(-0.845432\pi\)
−0.884400 + 0.466729i \(0.845432\pi\)
\(294\) 0 0
\(295\) −1113.94 −0.219852
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3898.77 6752.86i −0.754085 1.30611i
\(300\) 0 0
\(301\) −3054.51 + 2912.72i −0.584915 + 0.557762i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.4219 8.32647i −0.00270752 0.00156319i
\(306\) 0 0
\(307\) 2707.52i 0.503344i 0.967813 + 0.251672i \(0.0809804\pi\)
−0.967813 + 0.251672i \(0.919020\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1080.55 1871.56i 0.197017 0.341243i −0.750543 0.660822i \(-0.770208\pi\)
0.947560 + 0.319579i \(0.103541\pi\)
\(312\) 0 0
\(313\) −7300.25 + 4214.80i −1.31832 + 0.761133i −0.983459 0.181133i \(-0.942023\pi\)
−0.334863 + 0.942267i \(0.608690\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8310.07 + 4797.82i −1.47237 + 0.850071i −0.999517 0.0310734i \(-0.990107\pi\)
−0.472848 + 0.881144i \(0.656774\pi\)
\(318\) 0 0
\(319\) 3144.71 5446.80i 0.551943 0.955994i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3787.12i 0.652387i
\(324\) 0 0
\(325\) 9162.20 + 5289.80i 1.56378 + 0.902847i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2471.95 + 725.717i 0.414234 + 0.121611i
\(330\) 0 0
\(331\) 4271.96 + 7399.25i 0.709390 + 1.22870i 0.965084 + 0.261941i \(0.0843626\pi\)
−0.255694 + 0.966758i \(0.582304\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 559.978 0.0913280
\(336\) 0 0
\(337\) 598.875 0.0968036 0.0484018 0.998828i \(-0.484587\pi\)
0.0484018 + 0.998828i \(0.484587\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1834.05 + 3176.66i 0.291259 + 0.504475i
\(342\) 0 0
\(343\) 4160.57 + 4800.34i 0.654956 + 0.755667i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6149.62 + 3550.49i 0.951381 + 0.549280i 0.893510 0.449044i \(-0.148235\pi\)
0.0578712 + 0.998324i \(0.481569\pi\)
\(348\) 0 0
\(349\) 3620.71i 0.555336i −0.960677 0.277668i \(-0.910438\pi\)
0.960677 0.277668i \(-0.0895616\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1164.89 2017.64i 0.175639 0.304216i −0.764743 0.644335i \(-0.777134\pi\)
0.940382 + 0.340119i \(0.110467\pi\)
\(354\) 0 0
\(355\) 374.386 216.152i 0.0559727 0.0323159i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1522.43 878.975i 0.223818 0.129222i −0.383899 0.923375i \(-0.625419\pi\)
0.607717 + 0.794154i \(0.292086\pi\)
\(360\) 0 0
\(361\) −2244.31 + 3887.26i −0.327207 + 0.566739i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1166.60i 0.167295i
\(366\) 0 0
\(367\) −1458.89 842.290i −0.207503 0.119802i 0.392648 0.919689i \(-0.371559\pi\)
−0.600150 + 0.799887i \(0.704893\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6021.65 1461.11i 0.842665 0.204466i
\(372\) 0 0
\(373\) 148.646 + 257.462i 0.0206343 + 0.0357397i 0.876158 0.482024i \(-0.160098\pi\)
−0.855524 + 0.517763i \(0.826765\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12962.9 −1.77088
\(378\) 0 0
\(379\) 7402.78 1.00331 0.501656 0.865067i \(-0.332724\pi\)
0.501656 + 0.865067i \(0.332724\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6263.77 + 10849.2i 0.835676 + 1.44743i 0.893479 + 0.449105i \(0.148257\pi\)
−0.0578031 + 0.998328i \(0.518410\pi\)
\(384\) 0 0
\(385\) −672.926 705.685i −0.0890792 0.0934157i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6904.14 + 3986.11i 0.899881 + 0.519547i 0.877162 0.480195i \(-0.159434\pi\)
0.0227196 + 0.999742i \(0.492767\pi\)
\(390\) 0 0
\(391\) 7074.52i 0.915023i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −261.449 + 452.843i −0.0333036 + 0.0576836i
\(396\) 0 0
\(397\) 10832.5 6254.16i 1.36944 0.790648i 0.378586 0.925566i \(-0.376410\pi\)
0.990857 + 0.134918i \(0.0430770\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6943.55 + 4008.86i −0.864699 + 0.499234i −0.865583 0.500765i \(-0.833052\pi\)
0.000883860 1.00000i \(0.499719\pi\)
\(402\) 0 0
\(403\) 3780.08 6547.29i 0.467243 0.809289i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3768.05i 0.458907i
\(408\) 0 0
\(409\) −7566.04 4368.26i −0.914711 0.528109i −0.0327670 0.999463i \(-0.510432\pi\)
−0.881944 + 0.471354i \(0.843765\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3843.46 + 15840.1i 0.457928 + 1.88726i
\(414\) 0 0
\(415\) −603.935 1046.05i −0.0714361 0.123731i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3926.67 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(420\) 0 0
\(421\) 1443.44 0.167100 0.0835499 0.996504i \(-0.473374\pi\)
0.0835499 + 0.996504i \(0.473374\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4799.31 8312.65i −0.547766 0.948759i
\(426\) 0 0
\(427\) −68.6406 + 233.805i −0.00777928 + 0.0264979i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12820.5 7401.92i −1.43281 0.827234i −0.435477 0.900200i \(-0.643420\pi\)
−0.997334 + 0.0729655i \(0.976754\pi\)
\(432\) 0 0
\(433\) 15872.1i 1.76158i 0.473508 + 0.880790i \(0.342988\pi\)
−0.473508 + 0.880790i \(0.657012\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2213.99 3834.74i 0.242356 0.419772i
\(438\) 0 0
\(439\) 2626.58 1516.45i 0.285557 0.164867i −0.350379 0.936608i \(-0.613947\pi\)
0.635937 + 0.771741i \(0.280614\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11126.8 + 6424.08i −1.19334 + 0.688978i −0.959064 0.283191i \(-0.908607\pi\)
−0.234281 + 0.972169i \(0.575274\pi\)
\(444\) 0 0
\(445\) −18.7945 + 32.5530i −0.00200212 + 0.00346777i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 107.668i 0.0113166i −0.999984 0.00565831i \(-0.998199\pi\)
0.999984 0.00565831i \(-0.00180111\pi\)
\(450\) 0 0
\(451\) −13816.6 7977.01i −1.44257 0.832866i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −566.128 + 1928.35i −0.0583308 + 0.198687i
\(456\) 0 0
\(457\) 4888.53 + 8467.18i 0.500385 + 0.866691i 1.00000 0.000444115i \(0.000141366\pi\)
−0.499615 + 0.866247i \(0.666525\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −638.874 −0.0645452 −0.0322726 0.999479i \(-0.510274\pi\)
−0.0322726 + 0.999479i \(0.510274\pi\)
\(462\) 0 0
\(463\) 5602.26 0.562331 0.281165 0.959659i \(-0.409279\pi\)
0.281165 + 0.959659i \(0.409279\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2759.44 4779.49i −0.273430 0.473594i 0.696308 0.717743i \(-0.254825\pi\)
−0.969738 + 0.244149i \(0.921491\pi\)
\(468\) 0 0
\(469\) −1932.10 7962.76i −0.190226 0.783979i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8209.84 4739.95i −0.798074 0.460768i
\(474\) 0 0
\(475\) 6007.82i 0.580332i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2734.40 + 4736.11i −0.260830 + 0.451771i −0.966463 0.256807i \(-0.917330\pi\)
0.705632 + 0.708578i \(0.250663\pi\)
\(480\) 0 0
\(481\) 6725.70 3883.08i 0.637558 0.368094i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1314.96 759.190i 0.123112 0.0710785i
\(486\) 0 0
\(487\) −5866.72 + 10161.5i −0.545886 + 0.945502i 0.452665 + 0.891681i \(0.350473\pi\)
−0.998551 + 0.0538213i \(0.982860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3514.92i 0.323068i 0.986867 + 0.161534i \(0.0516441\pi\)
−0.986867 + 0.161534i \(0.948356\pi\)
\(492\) 0 0
\(493\) 10185.2 + 5880.45i 0.930467 + 0.537205i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4365.38 4577.89i −0.393992 0.413172i
\(498\) 0 0
\(499\) 4944.49 + 8564.11i 0.443579 + 0.768301i 0.997952 0.0639672i \(-0.0203753\pi\)
−0.554373 + 0.832268i \(0.687042\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10172.2 −0.901698 −0.450849 0.892600i \(-0.648879\pi\)
−0.450849 + 0.892600i \(0.648879\pi\)
\(504\) 0 0
\(505\) −828.838 −0.0730353
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2149.56 + 3723.15i 0.187186 + 0.324216i 0.944311 0.329054i \(-0.106730\pi\)
−0.757125 + 0.653270i \(0.773397\pi\)
\(510\) 0 0
\(511\) 16588.8 4025.14i 1.43610 0.348458i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1501.14 866.682i −0.128443 0.0741565i
\(516\) 0 0
\(517\) 5786.50i 0.492243i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5496.48 9520.18i 0.462198 0.800550i −0.536872 0.843664i \(-0.680394\pi\)
0.999070 + 0.0431133i \(0.0137276\pi\)
\(522\) 0 0
\(523\) −7386.80 + 4264.77i −0.617595 + 0.356569i −0.775932 0.630817i \(-0.782720\pi\)
0.158337 + 0.987385i \(0.449387\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5940.20 + 3429.58i −0.491004 + 0.283481i
\(528\) 0 0
\(529\) −1947.67 + 3373.46i −0.160078 + 0.277263i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 32882.2i 2.67220i
\(534\) 0 0
\(535\) −469.970 271.337i −0.0379786 0.0219270i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7712.89 + 12003.7i −0.616359 + 0.959250i
\(540\) 0 0
\(541\) 4352.93 + 7539.49i 0.345928 + 0.599165i 0.985522 0.169548i \(-0.0542308\pi\)
−0.639594 + 0.768713i \(0.720897\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −846.150 −0.0665047
\(546\) 0 0
\(547\) −17183.8 −1.34319 −0.671596 0.740917i \(-0.734391\pi\)
−0.671596 + 0.740917i \(0.734391\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3680.60 6374.99i −0.284571 0.492892i
\(552\) 0 0
\(553\) 7341.41 + 2155.30i 0.564536 + 0.165737i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8989.79 + 5190.26i 0.683859 + 0.394826i 0.801307 0.598253i \(-0.204138\pi\)
−0.117448 + 0.993079i \(0.537471\pi\)
\(558\) 0 0
\(559\) 19538.6i 1.47835i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9248.22 16018.4i 0.692302 1.19910i −0.278780 0.960355i \(-0.589930\pi\)
0.971082 0.238747i \(-0.0767366\pi\)
\(564\) 0 0
\(565\) −1002.78 + 578.956i −0.0746678 + 0.0431095i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3493.45 2016.94i 0.257386 0.148602i −0.365755 0.930711i \(-0.619189\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(570\) 0 0
\(571\) 6430.01 11137.1i 0.471257 0.816241i −0.528203 0.849118i \(-0.677134\pi\)
0.999459 + 0.0328777i \(0.0104672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11222.9i 0.813959i
\(576\) 0 0
\(577\) −17669.2 10201.3i −1.27483 0.736026i −0.298940 0.954272i \(-0.596633\pi\)
−0.975894 + 0.218246i \(0.929967\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12790.8 + 12197.0i −0.913340 + 0.870941i
\(582\) 0 0
\(583\) 6958.76 + 12052.9i 0.494344 + 0.856228i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16279.7 −1.14470 −0.572348 0.820011i \(-0.693967\pi\)
−0.572348 + 0.820011i \(0.693967\pi\)
\(588\) 0 0
\(589\) 4293.17 0.300335
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1342.16 2324.68i −0.0929440 0.160984i 0.815805 0.578328i \(-0.196294\pi\)
−0.908749 + 0.417344i \(0.862961\pi\)
\(594\) 0 0
\(595\) 1319.60 1258.34i 0.0909213 0.0867006i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12224.6 + 7057.90i 0.833865 + 0.481432i 0.855174 0.518341i \(-0.173450\pi\)
−0.0213091 + 0.999773i \(0.506783\pi\)
\(600\) 0 0
\(601\) 11096.1i 0.753109i −0.926394 0.376555i \(-0.877109\pi\)
0.926394 0.376555i \(-0.122891\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 252.748 437.772i 0.0169846 0.0294181i
\(606\) 0 0
\(607\) 9592.70 5538.35i 0.641442 0.370337i −0.143728 0.989617i \(-0.545909\pi\)
0.785170 + 0.619280i \(0.212576\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 10328.5 5963.15i 0.683872 0.394834i
\(612\) 0 0
\(613\) 3801.34 6584.11i 0.250464 0.433817i −0.713189 0.700971i \(-0.752750\pi\)
0.963654 + 0.267154i \(0.0860834\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11325.9i 0.738998i −0.929231 0.369499i \(-0.879529\pi\)
0.929231 0.369499i \(-0.120471\pi\)
\(618\) 0 0
\(619\) 16595.2 + 9581.22i 1.07757 + 0.622136i 0.930240 0.366952i \(-0.119599\pi\)
0.147330 + 0.989087i \(0.452932\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 527.743 + 154.935i 0.0339383 + 0.00996365i
\(624\) 0 0
\(625\) −7513.41 13013.6i −0.480858 0.832871i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7046.06 −0.446653
\(630\) 0 0
\(631\) −10140.7 −0.639768 −0.319884 0.947457i \(-0.603644\pi\)
−0.319884 + 0.947457i \(0.603644\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 797.995 + 1382.17i 0.0498700 + 0.0863774i
\(636\) 0 0
\(637\) 29374.1 + 1396.79i 1.82707 + 0.0868804i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2950.66 + 1703.56i 0.181816 + 0.104972i 0.588146 0.808755i \(-0.299858\pi\)
−0.406330 + 0.913727i \(0.633192\pi\)
\(642\) 0 0
\(643\) 659.110i 0.0404242i 0.999796 + 0.0202121i \(0.00643415\pi\)
−0.999796 + 0.0202121i \(0.993566\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3303.47 5721.78i 0.200731 0.347676i −0.748033 0.663661i \(-0.769002\pi\)
0.948764 + 0.315985i \(0.102335\pi\)
\(648\) 0 0
\(649\) −31705.4 + 18305.1i −1.91764 + 1.10715i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22417.1 + 12942.5i −1.34342 + 0.775622i −0.987307 0.158823i \(-0.949230\pi\)
−0.356109 + 0.934444i \(0.615897\pi\)
\(654\) 0 0
\(655\) −865.234 + 1498.63i −0.0516145 + 0.0893989i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7468.86i 0.441495i −0.975331 0.220748i \(-0.929150\pi\)
0.975331 0.220748i \(-0.0708497\pi\)
\(660\) 0 0
\(661\) −5501.96 3176.56i −0.323754 0.186919i 0.329311 0.944222i \(-0.393184\pi\)
−0.653065 + 0.757302i \(0.726517\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1109.09 + 269.111i −0.0646744 + 0.0156927i
\(666\) 0 0
\(667\) −6875.53 11908.8i −0.399133 0.691319i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −547.306 −0.0314881
\(672\) 0 0
\(673\) −20238.2 −1.15918 −0.579589 0.814909i \(-0.696787\pi\)
−0.579589 + 0.814909i \(0.696787\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5658.37 9800.59i −0.321224 0.556377i 0.659516 0.751690i \(-0.270761\pi\)
−0.980741 + 0.195313i \(0.937428\pi\)
\(678\) 0 0
\(679\) −15332.5 16078.9i −0.866581 0.908768i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14869.4 + 8584.87i 0.833035 + 0.480953i 0.854891 0.518808i \(-0.173624\pi\)
−0.0218557 + 0.999761i \(0.506957\pi\)
\(684\) 0 0
\(685\) 1393.74i 0.0777402i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14342.4 24841.8i 0.793037 1.37358i
\(690\) 0 0
\(691\) 7238.59 4179.20i 0.398508 0.230079i −0.287332 0.957831i \(-0.592768\pi\)
0.685840 + 0.727752i \(0.259435\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2528.29 + 1459.71i −0.137990 + 0.0796688i
\(696\) 0 0
\(697\) 14916.6 25836.3i 0.810627 1.40405i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19235.8i 1.03641i −0.855256 0.518206i \(-0.826600\pi\)
0.855256 0.518206i \(-0.173400\pi\)
\(702\) 0 0
\(703\) 3819.31 + 2205.08i 0.204905 + 0.118302i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2859.75 + 11785.9i 0.152124 + 0.626951i
\(708\) 0 0
\(709\) 5160.17 + 8937.67i 0.273334 + 0.473429i 0.969714 0.244245i \(-0.0785401\pi\)
−0.696379 + 0.717674i \(0.745207\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8019.85 0.421242
\(714\) 0 0
\(715\) −4514.02 −0.236105
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −11679.8 20229.9i −0.605815 1.04930i −0.991922 0.126849i \(-0.959514\pi\)
0.386107 0.922454i \(-0.373820\pi\)
\(720\) 0 0
\(721\) −7144.63 + 24336.2i −0.369043 + 1.25704i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16157.7 + 9328.63i 0.827698 + 0.477871i
\(726\) 0 0
\(727\) 22260.4i 1.13561i −0.823162 0.567807i \(-0.807792\pi\)
0.823162 0.567807i \(-0.192208\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8863.48 15352.0i 0.448464 0.776763i
\(732\) 0 0
\(733\) 9047.84 5223.77i 0.455920 0.263226i −0.254407 0.967097i \(-0.581880\pi\)
0.710327 + 0.703872i \(0.248547\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15938.2 9201.95i 0.796598 0.459916i
\(738\) 0 0
\(739\) 6595.44 11423.6i 0.328304 0.568640i −0.653871 0.756606i \(-0.726856\pi\)
0.982176 + 0.187966i \(0.0601895\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 35382.2i 1.74703i −0.486795 0.873517i \(-0.661834\pi\)
0.486795 0.873517i \(-0.338166\pi\)
\(744\) 0 0
\(745\) −1924.59 1111.16i −0.0946463 0.0546441i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2236.81 + 7619.07i −0.109121 + 0.371688i
\(750\) 0 0
\(751\) −14692.3 25447.8i −0.713888 1.23649i −0.963387 0.268116i \(-0.913599\pi\)
0.249498 0.968375i \(-0.419734\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 664.470 0.0320299
\(756\) 0 0
\(757\) 11329.1 0.543939 0.271969 0.962306i \(-0.412325\pi\)
0.271969 + 0.962306i \(0.412325\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12696.5 21990.9i −0.604792 1.04753i −0.992084 0.125574i \(-0.959923\pi\)
0.387292 0.921957i \(-0.373411\pi\)
\(762\) 0 0
\(763\) 2919.48 + 12032.1i 0.138522 + 0.570891i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 65346.7 + 37727.9i 3.07631 + 1.77611i
\(768\) 0 0
\(769\) 18120.8i 0.849744i 0.905253 + 0.424872i \(0.139681\pi\)
−0.905253 + 0.424872i \(0.860319\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8673.05 + 15022.2i −0.403555 + 0.698978i −0.994152 0.107989i \(-0.965559\pi\)
0.590597 + 0.806967i \(0.298892\pi\)
\(774\) 0 0
\(775\) −9423.42 + 5440.61i −0.436773 + 0.252171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16171.1 + 9336.37i −0.743759 + 0.429410i
\(780\) 0 0
\(781\) 7103.91 12304.3i 0.325477 0.563743i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1668.22i 0.0758488i
\(786\) 0 0
\(787\) −1801.52 1040.11i −0.0815977 0.0471104i 0.458646 0.888619i \(-0.348335\pi\)
−0.540244 + 0.841509i \(0.681668\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11692.5 + 12261.7i 0.525586 + 0.551172i
\(792\) 0 0
\(793\) 564.014 + 976.901i 0.0252569 + 0.0437463i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39737.3 1.76608 0.883041 0.469297i \(-0.155493\pi\)
0.883041 + 0.469297i \(0.155493\pi\)
\(798\) 0 0
\(799\) −10820.5 −0.479099
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19170.4 + 33204.1i 0.842477 + 1.45921i
\(804\) 0 0
\(805\) −2071.82 + 502.711i −0.0907108 + 0.0220102i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9713.37 + 5608.02i 0.422131 + 0.243717i 0.695989 0.718053i \(-0.254966\pi\)
−0.273858 + 0.961770i \(0.588300\pi\)
\(810\) 0 0
\(811\) 14792.0i 0.640465i 0.947339 + 0.320232i \(0.103761\pi\)
−0.947339 + 0.320232i \(0.896239\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −283.411 + 490.882i −0.0121809 + 0.0210980i
\(816\) 0 0
\(817\) −9608.88 + 5547.69i −0.411471 + 0.237563i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 28632.7 16531.1i 1.21716 0.702726i 0.252849 0.967506i \(-0.418632\pi\)
0.964309 + 0.264779i \(0.0852992\pi\)
\(822\) 0 0
\(823\) 10045.4 17399.1i 0.425467 0.736930i −0.570997 0.820952i \(-0.693443\pi\)
0.996464 + 0.0840220i \(0.0267766\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36882.5i 1.55082i −0.631458 0.775410i \(-0.717543\pi\)
0.631458 0.775410i \(-0.282457\pi\)
\(828\) 0 0
\(829\) −32697.0 18877.6i −1.36986 0.790888i −0.378949 0.925418i \(-0.623714\pi\)
−0.990910 + 0.134530i \(0.957048\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −22446.3 14422.7i −0.933636 0.599901i
\(834\) 0 0
\(835\) 513.321 + 889.098i 0.0212745 + 0.0368485i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −44250.4 −1.82085 −0.910426 0.413672i \(-0.864246\pi\)
−0.910426 + 0.413672i \(0.864246\pi\)
\(840\) 0 0
\(841\) 1528.81 0.0626844
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3261.45 + 5649.00i 0.132778 + 0.229978i
\(846\) 0 0
\(847\) −7097.08 2083.57i −0.287909 0.0845245i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7134.65 + 4119.19i 0.287394 + 0.165927i
\(852\) 0 0
\(853\) 14952.2i 0.600179i 0.953911 + 0.300089i \(0.0970165\pi\)
−0.953911 + 0.300089i \(0.902983\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1224.76 2121.35i 0.0488181 0.0845555i −0.840584 0.541682i \(-0.817788\pi\)
0.889402 + 0.457126i \(0.151121\pi\)
\(858\) 0 0
\(859\) 5751.97 3320.90i 0.228469 0.131906i −0.381397 0.924411i \(-0.624557\pi\)
0.609865 + 0.792505i \(0.291223\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16339.2 9433.46i 0.644489 0.372096i −0.141853 0.989888i \(-0.545306\pi\)
0.786342 + 0.617792i \(0.211973\pi\)
\(864\) 0 0
\(865\) 1419.15 2458.04i 0.0557833 0.0966196i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 17185.3i 0.670851i
\(870\) 0 0
\(871\) −32849.6 18965.8i −1.27792 0.737807i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4213.94 4018.32i 0.162808 0.155250i
\(876\) 0 0
\(877\) −5989.84 10374.7i −0.230630 0.399463i 0.727364 0.686252i \(-0.240745\pi\)
−0.957994 + 0.286789i \(0.907412\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −34504.0 −1.31949 −0.659743 0.751491i \(-0.729335\pi\)
−0.659743 + 0.751491i \(0.729335\pi\)
\(882\) 0 0
\(883\) 8148.85 0.310567 0.155283 0.987870i \(-0.450371\pi\)
0.155283 + 0.987870i \(0.450371\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17477.6 30272.1i −0.661602 1.14593i −0.980195 0.198036i \(-0.936544\pi\)
0.318593 0.947892i \(-0.396790\pi\)
\(888\) 0 0
\(889\) 16900.8 16116.2i 0.637609 0.608010i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5865.22 + 3386.29i 0.219790 + 0.126896i
\(894\) 0 0
\(895\) 3700.20i 0.138194i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6666.22 11546.2i 0.247309 0.428352i
\(900\) 0 0
\(901\) −22538.4 + 13012.5i −0.833365 + 0.481144i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −310.051 + 179.008i −0.0113883 + 0.00657505i
\(906\) 0 0
\(907\) −12494.5 + 21641.1i −0.457412 + 0.792261i −0.998823 0.0484970i \(-0.984557\pi\)
0.541411 + 0.840758i \(0.317890\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 41609.1i 1.51325i −0.653849 0.756625i \(-0.726847\pi\)
0.653849 0.756625i \(-0.273153\pi\)
\(912\) 0 0
\(913\) −34378.7 19848.5i −1.24619 0.719486i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24295.5 + 7132.70i 0.874927 + 0.256862i
\(918\) 0 0
\(919\) 25496.0 + 44160.4i 0.915164 + 1.58511i 0.806661 + 0.591014i \(0.201272\pi\)
0.108502 + 0.994096i \(0.465395\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −29283.1 −1.04428
\(924\) 0 0
\(925\) −11177.7 −0.397321
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −23746.5 41130.2i −0.838642 1.45257i −0.891031 0.453943i \(-0.850017\pi\)
0.0523888 0.998627i \(-0.483316\pi\)
\(930\) 0 0
\(931\) 7653.39 + 14842.4i 0.269420 + 0.522493i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3546.78 + 2047.73i 0.124056 + 0.0716235i
\(936\) 0 0
\(937\) 6811.98i 0.237500i −0.992924 0.118750i \(-0.962111\pi\)
0.992924 0.118750i \(-0.0378887\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 16271.0 28182.2i 0.563677 0.976318i −0.433494 0.901156i \(-0.642720\pi\)
0.997171 0.0751612i \(-0.0239471\pi\)
\(942\) 0 0
\(943\) −30208.3 + 17440.8i −1.04318 + 0.602280i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10119.9 5842.70i 0.347256 0.200488i −0.316220 0.948686i \(-0.602414\pi\)
0.663476 + 0.748198i \(0.269080\pi\)
\(948\) 0 0
\(949\) 39511.3 68435.7i 1.35152 2.34090i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 46457.5i 1.57912i −0.613671 0.789562i \(-0.710308\pi\)
0.613671 0.789562i \(-0.289692\pi\)
\(954\) 0 0
\(955\) −5061.08 2922.01i −0.171490 0.0990096i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19818.7 4808.84i 0.667339 0.161924i
\(960\) 0 0
\(961\) −11007.6 19065.8i −0.369496 0.639986i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −5259.57 −0.175452
\(966\) 0 0
\(967\) 27949.1 0.929455 0.464728 0.885454i \(-0.346152\pi\)
0.464728 + 0.885454i \(0.346152\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1609.85 + 2788.33i 0.0532053 + 0.0921544i 0.891401 0.453215i \(-0.149723\pi\)
−0.838196 + 0.545369i \(0.816390\pi\)
\(972\) 0 0
\(973\) 29480.1 + 30915.2i 0.971314 + 1.01860i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33258.6 + 19201.9i 1.08909 + 0.628785i 0.933333 0.359011i \(-0.116886\pi\)
0.155754 + 0.987796i \(0.450219\pi\)
\(978\) 0 0
\(979\) 1235.38i 0.0403297i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13709.1 23744.8i 0.444814 0.770440i −0.553226 0.833031i \(-0.686603\pi\)
0.998039 + 0.0625917i \(0.0199366\pi\)
\(984\) 0 0
\(985\) −1782.71 + 1029.25i −0.0576667 + 0.0332939i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17949.8 + 10363.3i −0.577120 + 0.333200i
\(990\) 0 0
\(991\) 21857.9 37859.1i 0.700646 1.21355i −0.267594 0.963532i \(-0.586229\pi\)
0.968240 0.250023i \(-0.0804381\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 220.484i 0.00702495i
\(996\) 0 0
\(997\) −21930.8 12661.7i −0.696644 0.402208i 0.109452 0.993992i \(-0.465090\pi\)
−0.806096 + 0.591784i \(0.798424\pi\)
\(998\) 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 1008.4.bt.a.17.4 16
3.2 odd 2 inner 1008.4.bt.a.17.5 16
4.3 odd 2 63.4.p.a.17.2 16
7.5 odd 6 inner 1008.4.bt.a.593.5 16
12.11 even 2 63.4.p.a.17.7 yes 16
21.5 even 6 inner 1008.4.bt.a.593.4 16
28.3 even 6 441.4.c.a.440.3 16
28.11 odd 6 441.4.c.a.440.4 16
28.19 even 6 63.4.p.a.26.7 yes 16
28.23 odd 6 441.4.p.c.215.7 16
28.27 even 2 441.4.p.c.80.2 16
84.11 even 6 441.4.c.a.440.13 16
84.23 even 6 441.4.p.c.215.2 16
84.47 odd 6 63.4.p.a.26.2 yes 16
84.59 odd 6 441.4.c.a.440.14 16
84.83 odd 2 441.4.p.c.80.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.p.a.17.2 16 4.3 odd 2
63.4.p.a.17.7 yes 16 12.11 even 2
63.4.p.a.26.2 yes 16 84.47 odd 6
63.4.p.a.26.7 yes 16 28.19 even 6
441.4.c.a.440.3 16 28.3 even 6
441.4.c.a.440.4 16 28.11 odd 6
441.4.c.a.440.13 16 84.11 even 6
441.4.c.a.440.14 16 84.59 odd 6
441.4.p.c.80.2 16 28.27 even 2
441.4.p.c.80.7 16 84.83 odd 2
441.4.p.c.215.2 16 84.23 even 6
441.4.p.c.215.7 16 28.23 odd 6
1008.4.bt.a.17.4 16 1.1 even 1 trivial
1008.4.bt.a.17.5 16 3.2 odd 2 inner
1008.4.bt.a.593.4 16 21.5 even 6 inner
1008.4.bt.a.593.5 16 7.5 odd 6 inner