Properties

Label 2268.2.i.l.2053.4
Level $2268$
Weight $2$
Character 2268.2053
Analytic conductor $18.110$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), 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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.310217769.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2053.4
Root \(0.882007 + 1.52768i\) of defining polynomial
Character \(\chi\) \(=\) 2268.2053
Dual form 2268.2.i.l.865.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.25300 + 2.17026i) q^{5} +(1.71031 + 2.01862i) q^{7} +O(q^{10})\) \(q+(1.25300 + 2.17026i) q^{5} +(1.71031 + 2.01862i) q^{7} +(0.542689 - 0.939965i) q^{11} +(-0.457311 + 0.792086i) q^{13} +(1.39302 + 2.41278i) q^{17} +(0.667623 - 1.15636i) q^{19} +(3.06064 + 5.30119i) q^{23} +(-0.640021 + 1.10855i) q^{25} +(-0.414622 - 0.718147i) q^{29} +2.61529 q^{31} +(-2.23791 + 6.24116i) q^{35} +(3.14602 - 5.44907i) q^{37} +(-0.817290 + 1.41559i) q^{41} +(0.795689 + 1.37817i) q^{43} -6.59138 q^{47} +(-1.14967 + 6.90495i) q^{49} +(3.95967 + 6.85835i) q^{53} +2.71996 q^{55} +5.07809 q^{59} -11.6393 q^{61} -2.29204 q^{65} +15.8954 q^{67} -9.64658 q^{71} +(-5.03904 - 8.72788i) q^{73} +(2.82560 - 0.512149i) q^{77} -9.34725 q^{79} +(4.09733 + 7.09679i) q^{83} +(-3.49091 + 6.04644i) q^{85} +(-6.36829 + 11.0302i) q^{89} +(-2.38107 + 0.431576i) q^{91} +3.34613 q^{95} +(-8.46931 - 14.6693i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59} + 24 q^{61} + 24 q^{65} + 14 q^{67} - 22 q^{71} - 10 q^{73} - 11 q^{77} + 35 q^{83} + 13 q^{85} - 18 q^{89} - 9 q^{91} + 20 q^{95} - 19 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.25300 + 2.17026i 0.560359 + 0.970570i 0.997465 + 0.0711601i \(0.0226701\pi\)
−0.437106 + 0.899410i \(0.643997\pi\)
\(6\) 0 0
\(7\) 1.71031 + 2.01862i 0.646437 + 0.762967i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.542689 0.939965i 0.163627 0.283410i −0.772540 0.634966i \(-0.781014\pi\)
0.936167 + 0.351556i \(0.114347\pi\)
\(12\) 0 0
\(13\) −0.457311 + 0.792086i −0.126835 + 0.219685i −0.922449 0.386119i \(-0.873815\pi\)
0.795614 + 0.605804i \(0.207149\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.39302 + 2.41278i 0.337857 + 0.585186i 0.984029 0.178006i \(-0.0569645\pi\)
−0.646172 + 0.763192i \(0.723631\pi\)
\(18\) 0 0
\(19\) 0.667623 1.15636i 0.153163 0.265286i −0.779225 0.626744i \(-0.784387\pi\)
0.932389 + 0.361457i \(0.117721\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.06064 + 5.30119i 0.638189 + 1.10537i 0.985830 + 0.167747i \(0.0536493\pi\)
−0.347641 + 0.937628i \(0.613017\pi\)
\(24\) 0 0
\(25\) −0.640021 + 1.10855i −0.128004 + 0.221710i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.414622 0.718147i −0.0769934 0.133357i 0.824958 0.565194i \(-0.191199\pi\)
−0.901951 + 0.431838i \(0.857865\pi\)
\(30\) 0 0
\(31\) 2.61529 0.469720 0.234860 0.972029i \(-0.424537\pi\)
0.234860 + 0.972029i \(0.424537\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.23791 + 6.24116i −0.378276 + 1.05495i
\(36\) 0 0
\(37\) 3.14602 5.44907i 0.517203 0.895822i −0.482597 0.875842i \(-0.660307\pi\)
0.999800 0.0199794i \(-0.00636008\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.817290 + 1.41559i −0.127639 + 0.221078i −0.922762 0.385371i \(-0.874073\pi\)
0.795122 + 0.606449i \(0.207407\pi\)
\(42\) 0 0
\(43\) 0.795689 + 1.37817i 0.121342 + 0.210170i 0.920297 0.391221i \(-0.127947\pi\)
−0.798955 + 0.601390i \(0.794614\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.59138 −0.961451 −0.480726 0.876871i \(-0.659627\pi\)
−0.480726 + 0.876871i \(0.659627\pi\)
\(48\) 0 0
\(49\) −1.14967 + 6.90495i −0.164238 + 0.986421i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.95967 + 6.85835i 0.543902 + 0.942066i 0.998675 + 0.0514588i \(0.0163871\pi\)
−0.454773 + 0.890607i \(0.650280\pi\)
\(54\) 0 0
\(55\) 2.71996 0.366759
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.07809 0.661111 0.330555 0.943787i \(-0.392764\pi\)
0.330555 + 0.943787i \(0.392764\pi\)
\(60\) 0 0
\(61\) −11.6393 −1.49026 −0.745129 0.666920i \(-0.767612\pi\)
−0.745129 + 0.666920i \(0.767612\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.29204 −0.284293
\(66\) 0 0
\(67\) 15.8954 1.94194 0.970968 0.239211i \(-0.0768887\pi\)
0.970968 + 0.239211i \(0.0768887\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.64658 −1.14484 −0.572419 0.819961i \(-0.693995\pi\)
−0.572419 + 0.819961i \(0.693995\pi\)
\(72\) 0 0
\(73\) −5.03904 8.72788i −0.589776 1.02152i −0.994262 0.106977i \(-0.965883\pi\)
0.404486 0.914544i \(-0.367450\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.82560 0.512149i 0.322007 0.0583648i
\(78\) 0 0
\(79\) −9.34725 −1.05165 −0.525824 0.850594i \(-0.676243\pi\)
−0.525824 + 0.850594i \(0.676243\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.09733 + 7.09679i 0.449741 + 0.778974i 0.998369 0.0570928i \(-0.0181831\pi\)
−0.548628 + 0.836066i \(0.684850\pi\)
\(84\) 0 0
\(85\) −3.49091 + 6.04644i −0.378643 + 0.655829i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.36829 + 11.0302i −0.675037 + 1.16920i 0.301421 + 0.953491i \(0.402539\pi\)
−0.976458 + 0.215708i \(0.930794\pi\)
\(90\) 0 0
\(91\) −2.38107 + 0.431576i −0.249604 + 0.0452415i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.34613 0.343305
\(96\) 0 0
\(97\) −8.46931 14.6693i −0.859929 1.48944i −0.871996 0.489513i \(-0.837175\pi\)
0.0120677 0.999927i \(-0.496159\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.56064 + 2.70312i −0.155290 + 0.268970i −0.933165 0.359449i \(-0.882965\pi\)
0.777875 + 0.628420i \(0.216298\pi\)
\(102\) 0 0
\(103\) −2.40867 4.17194i −0.237333 0.411073i 0.722615 0.691251i \(-0.242940\pi\)
−0.959948 + 0.280178i \(0.909607\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.12729 + 12.3448i −0.689021 + 1.19342i 0.283134 + 0.959080i \(0.408626\pi\)
−0.972155 + 0.234339i \(0.924707\pi\)
\(108\) 0 0
\(109\) −2.94767 5.10551i −0.282335 0.489019i 0.689624 0.724167i \(-0.257776\pi\)
−0.971959 + 0.235149i \(0.924442\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.58173 7.93579i 0.431013 0.746537i −0.565948 0.824441i \(-0.691490\pi\)
0.996961 + 0.0779045i \(0.0248229\pi\)
\(114\) 0 0
\(115\) −7.66998 + 13.2848i −0.715229 + 1.23881i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.48800 + 6.93860i −0.228074 + 0.636060i
\(120\) 0 0
\(121\) 4.91098 + 8.50606i 0.446453 + 0.773278i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.32222 0.833805
\(126\) 0 0
\(127\) −5.48671 −0.486867 −0.243433 0.969918i \(-0.578274\pi\)
−0.243433 + 0.969918i \(0.578274\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.8587 + 18.8078i 0.948728 + 1.64325i 0.748109 + 0.663576i \(0.230962\pi\)
0.200620 + 0.979669i \(0.435704\pi\)
\(132\) 0 0
\(133\) 3.47609 0.630053i 0.301415 0.0546325i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.2012 17.6690i 0.871550 1.50957i 0.0111561 0.999938i \(-0.496449\pi\)
0.860393 0.509630i \(-0.170218\pi\)
\(138\) 0 0
\(139\) 2.41698 4.18633i 0.205005 0.355080i −0.745129 0.666920i \(-0.767612\pi\)
0.950134 + 0.311840i \(0.100945\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.496355 + 0.859713i 0.0415073 + 0.0718928i
\(144\) 0 0
\(145\) 1.03904 1.79968i 0.0862879 0.149455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.90867 + 5.03796i 0.238287 + 0.412726i 0.960223 0.279234i \(-0.0900806\pi\)
−0.721936 + 0.691960i \(0.756747\pi\)
\(150\) 0 0
\(151\) 4.30769 7.46114i 0.350555 0.607179i −0.635792 0.771861i \(-0.719326\pi\)
0.986347 + 0.164682i \(0.0526596\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.27696 + 5.67586i 0.263212 + 0.455896i
\(156\) 0 0
\(157\) −8.26187 −0.659369 −0.329685 0.944091i \(-0.606942\pi\)
−0.329685 + 0.944091i \(0.606942\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.46644 + 15.2450i −0.430816 + 1.20147i
\(162\) 0 0
\(163\) 1.02160 1.76946i 0.0800179 0.138595i −0.823239 0.567694i \(-0.807836\pi\)
0.903257 + 0.429099i \(0.141169\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.31965 + 14.4101i −0.643794 + 1.11508i 0.340785 + 0.940141i \(0.389307\pi\)
−0.984579 + 0.174942i \(0.944026\pi\)
\(168\) 0 0
\(169\) 6.08173 + 10.5339i 0.467826 + 0.810298i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −20.4685 −1.55619 −0.778097 0.628144i \(-0.783815\pi\)
−0.778097 + 0.628144i \(0.783815\pi\)
\(174\) 0 0
\(175\) −3.33238 + 0.604004i −0.251904 + 0.0456584i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.624422 + 1.08153i 0.0466715 + 0.0808374i 0.888417 0.459036i \(-0.151805\pi\)
−0.841746 + 0.539874i \(0.818472\pi\)
\(180\) 0 0
\(181\) 14.6153 1.08635 0.543173 0.839621i \(-0.317223\pi\)
0.543173 + 0.839621i \(0.317223\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.7679 1.15928
\(186\) 0 0
\(187\) 3.02391 0.221130
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.9495 1.29878 0.649390 0.760455i \(-0.275024\pi\)
0.649390 + 0.760455i \(0.275024\pi\)
\(192\) 0 0
\(193\) 23.1812 1.66862 0.834310 0.551296i \(-0.185866\pi\)
0.834310 + 0.551296i \(0.185866\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.31134 −0.449664 −0.224832 0.974398i \(-0.572183\pi\)
−0.224832 + 0.974398i \(0.572183\pi\)
\(198\) 0 0
\(199\) 3.82873 + 6.63156i 0.271412 + 0.470099i 0.969224 0.246182i \(-0.0791762\pi\)
−0.697812 + 0.716281i \(0.745843\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.740534 2.06522i 0.0519753 0.144950i
\(204\) 0 0
\(205\) −4.09626 −0.286095
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −0.724623 1.25508i −0.0501232 0.0868160i
\(210\) 0 0
\(211\) −2.73740 + 4.74132i −0.188450 + 0.326406i −0.944734 0.327838i \(-0.893680\pi\)
0.756283 + 0.654244i \(0.227013\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.99400 + 3.45371i −0.135990 + 0.235541i
\(216\) 0 0
\(217\) 4.47296 + 5.27928i 0.303644 + 0.358381i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.54818 −0.171409
\(222\) 0 0
\(223\) 3.93807 + 6.82093i 0.263712 + 0.456763i 0.967225 0.253919i \(-0.0817197\pi\)
−0.703513 + 0.710682i \(0.748386\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.0913311 0.158190i 0.00606186 0.0104994i −0.862979 0.505241i \(-0.831404\pi\)
0.869040 + 0.494741i \(0.164737\pi\)
\(228\) 0 0
\(229\) −7.32329 12.6843i −0.483937 0.838203i 0.515893 0.856653i \(-0.327460\pi\)
−0.999830 + 0.0184500i \(0.994127\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.4693 18.1334i 0.685868 1.18796i −0.287296 0.957842i \(-0.592756\pi\)
0.973163 0.230116i \(-0.0739104\pi\)
\(234\) 0 0
\(235\) −8.25900 14.3050i −0.538758 0.933156i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.16398 + 14.1404i −0.528084 + 0.914668i 0.471380 + 0.881930i \(0.343756\pi\)
−0.999464 + 0.0327379i \(0.989577\pi\)
\(240\) 0 0
\(241\) 0.152536 0.264200i 0.00982571 0.0170186i −0.861071 0.508485i \(-0.830206\pi\)
0.870896 + 0.491467i \(0.163539\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −16.4261 + 6.15682i −1.04942 + 0.393345i
\(246\) 0 0
\(247\) 0.610623 + 1.05763i 0.0388530 + 0.0672954i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 28.2619 1.78387 0.891937 0.452160i \(-0.149346\pi\)
0.891937 + 0.452160i \(0.149346\pi\)
\(252\) 0 0
\(253\) 6.64391 0.417699
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.1208 17.5297i −0.631317 1.09347i −0.987283 0.158974i \(-0.949181\pi\)
0.355966 0.934499i \(-0.384152\pi\)
\(258\) 0 0
\(259\) 16.3803 2.96898i 1.01782 0.184483i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.81078 11.7966i 0.419970 0.727410i −0.575966 0.817474i \(-0.695374\pi\)
0.995936 + 0.0900638i \(0.0287071\pi\)
\(264\) 0 0
\(265\) −9.92293 + 17.1870i −0.609561 + 1.05579i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.30169 2.25459i −0.0793655 0.137465i 0.823611 0.567155i \(-0.191956\pi\)
−0.902976 + 0.429690i \(0.858623\pi\)
\(270\) 0 0
\(271\) −0.362288 + 0.627501i −0.0220074 + 0.0381179i −0.876819 0.480820i \(-0.840339\pi\)
0.854812 + 0.518938i \(0.173672\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.694665 + 1.20319i 0.0418899 + 0.0725554i
\(276\) 0 0
\(277\) −7.46331 + 12.9268i −0.448427 + 0.776698i −0.998284 0.0585606i \(-0.981349\pi\)
0.549857 + 0.835259i \(0.314682\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.67311 16.7543i −0.577049 0.999479i −0.995816 0.0913849i \(-0.970871\pi\)
0.418766 0.908094i \(-0.362463\pi\)
\(282\) 0 0
\(283\) 1.38471 0.0823126 0.0411563 0.999153i \(-0.486896\pi\)
0.0411563 + 0.999153i \(0.486896\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.25536 + 0.771297i −0.251186 + 0.0455282i
\(288\) 0 0
\(289\) 4.61898 8.00031i 0.271705 0.470606i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.61220 11.4527i 0.386289 0.669072i −0.605658 0.795725i \(-0.707090\pi\)
0.991947 + 0.126653i \(0.0404234\pi\)
\(294\) 0 0
\(295\) 6.36285 + 11.0208i 0.370459 + 0.641654i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.59867 −0.323779
\(300\) 0 0
\(301\) −1.42114 + 3.96330i −0.0819129 + 0.228441i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.5840 25.2603i −0.835080 1.44640i
\(306\) 0 0
\(307\) 13.3342 0.761024 0.380512 0.924776i \(-0.375748\pi\)
0.380512 + 0.924776i \(0.375748\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −25.2427 −1.43138 −0.715690 0.698418i \(-0.753888\pi\)
−0.715690 + 0.698418i \(0.753888\pi\)
\(312\) 0 0
\(313\) 12.7933 0.723122 0.361561 0.932348i \(-0.382244\pi\)
0.361561 + 0.932348i \(0.382244\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.8028 −0.775240 −0.387620 0.921819i \(-0.626703\pi\)
−0.387620 + 0.921819i \(0.626703\pi\)
\(318\) 0 0
\(319\) −0.900044 −0.0503928
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.72005 0.206989
\(324\) 0 0
\(325\) −0.585378 1.01390i −0.0324709 0.0562413i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.2733 13.3055i −0.621518 0.733556i
\(330\) 0 0
\(331\) −12.4951 −0.686794 −0.343397 0.939190i \(-0.611578\pi\)
−0.343397 + 0.939190i \(0.611578\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 19.9170 + 34.4972i 1.08818 + 1.88478i
\(336\) 0 0
\(337\) 3.56716 6.17850i 0.194315 0.336564i −0.752360 0.658752i \(-0.771085\pi\)
0.946676 + 0.322187i \(0.104418\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.41929 2.45828i 0.0768587 0.133123i
\(342\) 0 0
\(343\) −15.9048 + 9.48887i −0.858776 + 0.512351i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.10200 −0.273890 −0.136945 0.990579i \(-0.543728\pi\)
−0.136945 + 0.990579i \(0.543728\pi\)
\(348\) 0 0
\(349\) −15.7764 27.3256i −0.844494 1.46271i −0.886060 0.463570i \(-0.846568\pi\)
0.0415664 0.999136i \(-0.486765\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.52082 13.0265i 0.400293 0.693328i −0.593468 0.804858i \(-0.702242\pi\)
0.993761 + 0.111530i \(0.0355750\pi\)
\(354\) 0 0
\(355\) −12.0872 20.9356i −0.641521 1.11115i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.49322 12.9786i 0.395477 0.684987i −0.597685 0.801731i \(-0.703913\pi\)
0.993162 + 0.116745i \(0.0372459\pi\)
\(360\) 0 0
\(361\) 8.60856 + 14.9105i 0.453082 + 0.784761i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.6279 21.8721i 0.660972 1.14484i
\(366\) 0 0
\(367\) 8.73006 15.1209i 0.455706 0.789305i −0.543023 0.839718i \(-0.682720\pi\)
0.998728 + 0.0504126i \(0.0160536\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.07214 + 19.7230i −0.367167 + 1.02397i
\(372\) 0 0
\(373\) −5.56716 9.64260i −0.288257 0.499275i 0.685137 0.728414i \(-0.259742\pi\)
−0.973394 + 0.229139i \(0.926409\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.758446 0.0390619
\(378\) 0 0
\(379\) −30.0958 −1.54592 −0.772959 0.634455i \(-0.781224\pi\)
−0.772959 + 0.634455i \(0.781224\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.07029 8.78200i −0.259080 0.448739i 0.706916 0.707298i \(-0.250086\pi\)
−0.965996 + 0.258558i \(0.916753\pi\)
\(384\) 0 0
\(385\) 4.65198 + 5.49057i 0.237087 + 0.279825i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.64367 14.9713i 0.438251 0.759073i −0.559303 0.828963i \(-0.688931\pi\)
0.997555 + 0.0698895i \(0.0222647\pi\)
\(390\) 0 0
\(391\) −8.52709 + 14.7694i −0.431233 + 0.746918i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.7121 20.2860i −0.589300 1.02070i
\(396\) 0 0
\(397\) −12.5666 + 21.7661i −0.630702 + 1.09241i 0.356707 + 0.934216i \(0.383900\pi\)
−0.987408 + 0.158191i \(0.949434\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −11.9956 20.7770i −0.599031 1.03755i −0.992964 0.118413i \(-0.962219\pi\)
0.393934 0.919139i \(-0.371114\pi\)
\(402\) 0 0
\(403\) −1.19600 + 2.07153i −0.0595770 + 0.103190i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.41462 5.91430i −0.169257 0.293161i
\(408\) 0 0
\(409\) 31.5898 1.56202 0.781008 0.624521i \(-0.214706\pi\)
0.781008 + 0.624521i \(0.214706\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.68511 + 10.2507i 0.427367 + 0.504406i
\(414\) 0 0
\(415\) −10.2679 + 17.7846i −0.504032 + 0.873010i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.12729 + 7.14868i −0.201631 + 0.349236i −0.949054 0.315113i \(-0.897958\pi\)
0.747423 + 0.664349i \(0.231291\pi\)
\(420\) 0 0
\(421\) −15.5557 26.9433i −0.758139 1.31314i −0.943799 0.330521i \(-0.892775\pi\)
0.185660 0.982614i \(-0.440558\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.56625 −0.172989
\(426\) 0 0
\(427\) −19.9068 23.4953i −0.963359 1.13702i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.0078 29.4585i −0.819239 1.41896i −0.906244 0.422756i \(-0.861063\pi\)
0.0870045 0.996208i \(-0.472271\pi\)
\(432\) 0 0
\(433\) −5.46751 −0.262752 −0.131376 0.991333i \(-0.541940\pi\)
−0.131376 + 0.991333i \(0.541940\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.17343 0.390988
\(438\) 0 0
\(439\) 22.9303 1.09440 0.547202 0.837001i \(-0.315693\pi\)
0.547202 + 0.837001i \(0.315693\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.5045 1.68687 0.843436 0.537230i \(-0.180529\pi\)
0.843436 + 0.537230i \(0.180529\pi\)
\(444\) 0 0
\(445\) −31.9179 −1.51305
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.82093 −0.463478 −0.231739 0.972778i \(-0.574442\pi\)
−0.231739 + 0.972778i \(0.574442\pi\)
\(450\) 0 0
\(451\) 0.887068 + 1.53645i 0.0417704 + 0.0723485i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.92011 4.62677i −0.183778 0.216906i
\(456\) 0 0
\(457\) 22.0350 1.03075 0.515377 0.856964i \(-0.327652\pi\)
0.515377 + 0.856964i \(0.327652\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.1026 + 36.5507i 0.982844 + 1.70234i 0.651150 + 0.758949i \(0.274287\pi\)
0.331694 + 0.943387i \(0.392380\pi\)
\(462\) 0 0
\(463\) 17.7637 30.7677i 0.825550 1.42989i −0.0759485 0.997112i \(-0.524198\pi\)
0.901498 0.432783i \(-0.142468\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.39015 + 9.33602i −0.249427 + 0.432019i −0.963367 0.268187i \(-0.913575\pi\)
0.713940 + 0.700207i \(0.246909\pi\)
\(468\) 0 0
\(469\) 27.1861 + 32.0869i 1.25534 + 1.48163i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.72725 0.0794189
\(474\) 0 0
\(475\) 0.854586 + 1.48019i 0.0392111 + 0.0679156i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.4844 19.8916i 0.524738 0.908873i −0.474847 0.880068i \(-0.657497\pi\)
0.999585 0.0288044i \(-0.00916998\pi\)
\(480\) 0 0
\(481\) 2.87742 + 4.98384i 0.131199 + 0.227244i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.2241 36.7612i 0.963737 1.66924i
\(486\) 0 0
\(487\) −14.5432 25.1896i −0.659018 1.14145i −0.980870 0.194662i \(-0.937639\pi\)
0.321853 0.946790i \(-0.395694\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.81005 + 16.9915i −0.442721 + 0.766816i −0.997890 0.0649218i \(-0.979320\pi\)
0.555169 + 0.831738i \(0.312654\pi\)
\(492\) 0 0
\(493\) 1.15516 2.00079i 0.0520256 0.0901110i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.4987 19.4728i −0.740066 0.873474i
\(498\) 0 0
\(499\) −1.29805 2.24828i −0.0581085 0.100647i 0.835508 0.549479i \(-0.185174\pi\)
−0.893616 + 0.448832i \(0.851840\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 36.0865 1.60902 0.804509 0.593941i \(-0.202429\pi\)
0.804509 + 0.593941i \(0.202429\pi\)
\(504\) 0 0
\(505\) −7.82196 −0.348072
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.3826 30.1076i −0.770472 1.33450i −0.937304 0.348512i \(-0.886687\pi\)
0.166832 0.985985i \(-0.446646\pi\)
\(510\) 0 0
\(511\) 8.99995 25.0993i 0.398134 1.11033i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.03613 10.4549i 0.265984 0.460697i
\(516\) 0 0
\(517\) −3.57707 + 6.19566i −0.157319 + 0.272485i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.37609 11.0437i −0.279341 0.483834i 0.691880 0.722013i \(-0.256783\pi\)
−0.971221 + 0.238179i \(0.923449\pi\)
\(522\) 0 0
\(523\) −19.0906 + 33.0659i −0.834774 + 1.44587i 0.0594407 + 0.998232i \(0.481068\pi\)
−0.894214 + 0.447639i \(0.852265\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.64315 + 6.31013i 0.158698 + 0.274873i
\(528\) 0 0
\(529\) −7.23509 + 12.5315i −0.314569 + 0.544850i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −0.747512 1.29473i −0.0323783 0.0560809i
\(534\) 0 0
\(535\) −35.7220 −1.54440
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.86649 + 4.82788i 0.252688 + 0.207952i
\(540\) 0 0
\(541\) 7.26136 12.5770i 0.312190 0.540729i −0.666646 0.745374i \(-0.732271\pi\)
0.978836 + 0.204645i \(0.0656041\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.38685 12.7944i 0.316418 0.548052i
\(546\) 0 0
\(547\) 13.2907 + 23.0202i 0.568270 + 0.984272i 0.996737 + 0.0807148i \(0.0257203\pi\)
−0.428468 + 0.903557i \(0.640946\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.10725 −0.0471703
\(552\) 0 0
\(553\) −15.9867 18.8686i −0.679824 0.802373i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.3017 23.0392i −0.563611 0.976202i −0.997177 0.0750809i \(-0.976078\pi\)
0.433567 0.901121i \(-0.357255\pi\)
\(558\) 0 0
\(559\) −1.45551 −0.0615615
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.8220 −1.08827 −0.544133 0.838999i \(-0.683141\pi\)
−0.544133 + 0.838999i \(0.683141\pi\)
\(564\) 0 0
\(565\) 22.9637 0.966088
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.7152 0.491129 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(570\) 0 0
\(571\) 10.0973 0.422558 0.211279 0.977426i \(-0.432237\pi\)
0.211279 + 0.977426i \(0.432237\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.83551 −0.326763
\(576\) 0 0
\(577\) 4.92842 + 8.53628i 0.205173 + 0.355370i 0.950188 0.311678i \(-0.100891\pi\)
−0.745015 + 0.667048i \(0.767558\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.31801 + 20.4087i −0.303602 + 0.846695i
\(582\) 0 0
\(583\) 8.59547 0.355988
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.2506 + 17.7546i 0.423089 + 0.732812i 0.996240 0.0866378i \(-0.0276123\pi\)
−0.573151 + 0.819450i \(0.694279\pi\)
\(588\) 0 0
\(589\) 1.74603 3.02421i 0.0719438 0.124610i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.94458 8.56427i 0.203050 0.351692i −0.746460 0.665430i \(-0.768248\pi\)
0.949510 + 0.313738i \(0.101581\pi\)
\(594\) 0 0
\(595\) −18.1760 + 3.29446i −0.745145 + 0.135060i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.34263 0.259153 0.129576 0.991569i \(-0.458638\pi\)
0.129576 + 0.991569i \(0.458638\pi\)
\(600\) 0 0
\(601\) −7.74469 13.4142i −0.315913 0.547177i 0.663719 0.747982i \(-0.268977\pi\)
−0.979631 + 0.200806i \(0.935644\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.3069 + 21.3162i −0.500347 + 0.866627i
\(606\) 0 0
\(607\) 1.07942 + 1.86962i 0.0438125 + 0.0758854i 0.887100 0.461577i \(-0.152716\pi\)
−0.843288 + 0.537463i \(0.819383\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.01431 5.22094i 0.121946 0.211217i
\(612\) 0 0
\(613\) 0.944844 + 1.63652i 0.0381619 + 0.0660983i 0.884476 0.466587i \(-0.154516\pi\)
−0.846314 + 0.532685i \(0.821183\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.91093 13.7021i 0.318482 0.551627i −0.661689 0.749778i \(-0.730160\pi\)
0.980172 + 0.198151i \(0.0634936\pi\)
\(618\) 0 0
\(619\) 3.87742 6.71589i 0.155847 0.269935i −0.777520 0.628858i \(-0.783523\pi\)
0.933367 + 0.358923i \(0.116856\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −33.1576 + 6.00992i −1.32843 + 0.240782i
\(624\) 0 0
\(625\) 14.8809 + 25.7744i 0.595234 + 1.03098i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.5299 0.698963
\(630\) 0 0
\(631\) 39.7148 1.58102 0.790511 0.612448i \(-0.209815\pi\)
0.790511 + 0.612448i \(0.209815\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.87485 11.9076i −0.272820 0.472538i
\(636\) 0 0
\(637\) −4.94356 4.06834i −0.195871 0.161194i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.4289 26.7237i 0.609406 1.05552i −0.381932 0.924190i \(-0.624741\pi\)
0.991338 0.131332i \(-0.0419255\pi\)
\(642\) 0 0
\(643\) −8.93807 + 15.4812i −0.352483 + 0.610518i −0.986684 0.162650i \(-0.947996\pi\)
0.634201 + 0.773168i \(0.281329\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.3928 23.1969i −0.526524 0.911966i −0.999522 0.0309026i \(-0.990162\pi\)
0.472999 0.881063i \(-0.343172\pi\)
\(648\) 0 0
\(649\) 2.75582 4.77322i 0.108175 0.187365i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.2093 22.8792i −0.516919 0.895331i −0.999807 0.0196482i \(-0.993745\pi\)
0.482888 0.875682i \(-0.339588\pi\)
\(654\) 0 0
\(655\) −27.2119 + 47.1324i −1.06326 + 1.84161i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19.3473 33.5105i −0.753664 1.30538i −0.946036 0.324062i \(-0.894951\pi\)
0.192372 0.981322i \(-0.438382\pi\)
\(660\) 0 0
\(661\) −7.90004 −0.307276 −0.153638 0.988127i \(-0.549099\pi\)
−0.153638 + 0.988127i \(0.549099\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.72292 + 6.75457i 0.221925 + 0.261931i
\(666\) 0 0
\(667\) 2.53802 4.39599i 0.0982727 0.170213i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.31651 + 10.9405i −0.243846 + 0.422354i
\(672\) 0 0
\(673\) 0.777686 + 1.34699i 0.0299776 + 0.0519227i 0.880625 0.473814i \(-0.157123\pi\)
−0.850647 + 0.525737i \(0.823790\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −29.1811 −1.12152 −0.560761 0.827978i \(-0.689491\pi\)
−0.560761 + 0.827978i \(0.689491\pi\)
\(678\) 0 0
\(679\) 15.1266 42.1854i 0.580504 1.61893i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −17.9993 31.1757i −0.688723 1.19290i −0.972251 0.233939i \(-0.924838\pi\)
0.283528 0.958964i \(-0.408495\pi\)
\(684\) 0 0
\(685\) 51.1286 1.95352
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.24320 −0.275944
\(690\) 0 0
\(691\) 43.8664 1.66876 0.834378 0.551193i \(-0.185827\pi\)
0.834378 + 0.551193i \(0.185827\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.1139 0.459507
\(696\) 0 0
\(697\) −4.55401 −0.172495
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.5873 1.75958 0.879790 0.475363i \(-0.157683\pi\)
0.879790 + 0.475363i \(0.157683\pi\)
\(702\) 0 0
\(703\) −4.20071 7.27585i −0.158433 0.274414i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.12576 + 1.47282i −0.305601 + 0.0553911i
\(708\) 0 0
\(709\) 18.4805 0.694051 0.347026 0.937856i \(-0.387192\pi\)
0.347026 + 0.937856i \(0.387192\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00447 + 13.8641i 0.299770 + 0.519216i
\(714\) 0 0
\(715\) −1.24387 + 2.15444i −0.0465180 + 0.0805715i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.7978 + 34.2908i −0.738333 + 1.27883i 0.214913 + 0.976633i \(0.431053\pi\)
−0.953246 + 0.302197i \(0.902280\pi\)
\(720\) 0 0
\(721\) 4.30199 11.9975i 0.160214 0.446810i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.06147 0.0394219
\(726\) 0 0
\(727\) 11.9170 + 20.6409i 0.441978 + 0.765528i 0.997836 0.0657487i \(-0.0209436\pi\)
−0.555858 + 0.831277i \(0.687610\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.21683 + 3.83965i −0.0819923 + 0.142015i
\(732\) 0 0
\(733\) 8.02036 + 13.8917i 0.296239 + 0.513101i 0.975272 0.221007i \(-0.0709343\pi\)
−0.679034 + 0.734107i \(0.737601\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.62627 14.9411i 0.317753 0.550364i
\(738\) 0 0
\(739\) 2.74335 + 4.75163i 0.100916 + 0.174792i 0.912062 0.410052i \(-0.134489\pi\)
−0.811146 + 0.584843i \(0.801156\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.182662 + 0.316380i −0.00670122 + 0.0116069i −0.869357 0.494185i \(-0.835466\pi\)
0.862655 + 0.505792i \(0.168800\pi\)
\(744\) 0 0
\(745\) −7.28913 + 12.6251i −0.267053 + 0.462549i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −37.1094 + 6.72620i −1.35595 + 0.245770i
\(750\) 0 0
\(751\) 5.67855 + 9.83554i 0.207213 + 0.358904i 0.950836 0.309696i \(-0.100227\pi\)
−0.743622 + 0.668600i \(0.766894\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.5902 0.785746
\(756\) 0 0
\(757\) −27.4794 −0.998757 −0.499378 0.866384i \(-0.666438\pi\)
−0.499378 + 0.866384i \(0.666438\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.0210882 + 0.0365259i 0.000764448 + 0.00132406i 0.866407 0.499338i \(-0.166423\pi\)
−0.865643 + 0.500662i \(0.833090\pi\)
\(762\) 0 0
\(763\) 5.26466 14.6822i 0.190593 0.531532i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.32227 + 4.02228i −0.0838522 + 0.145236i
\(768\) 0 0
\(769\) −13.3233 + 23.0766i −0.480450 + 0.832164i −0.999748 0.0224290i \(-0.992860\pi\)
0.519298 + 0.854593i \(0.326193\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.6206 + 39.1800i 0.813605 + 1.40921i 0.910325 + 0.413894i \(0.135831\pi\)
−0.0967202 + 0.995312i \(0.530835\pi\)
\(774\) 0 0
\(775\) −1.67384 + 2.89918i −0.0601261 + 0.104141i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.09128 + 1.89016i 0.0390993 + 0.0677219i
\(780\) 0 0
\(781\) −5.23509 + 9.06745i −0.187326 + 0.324459i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.3521 17.9304i −0.369483 0.639964i
\(786\) 0 0
\(787\) 36.2843 1.29340 0.646698 0.762746i \(-0.276149\pi\)
0.646698 + 0.762746i \(0.276149\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 23.8556 4.32390i 0.848206 0.153740i
\(792\) 0 0
\(793\) 5.32278 9.21932i 0.189017 0.327388i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.35398 + 11.0054i −0.225069 + 0.389832i −0.956340 0.292256i \(-0.905594\pi\)
0.731271 + 0.682087i \(0.238928\pi\)
\(798\) 0 0
\(799\) −9.18193 15.9036i −0.324833 0.562628i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −10.9385 −0.386012
\(804\) 0 0
\(805\) −39.9350 + 7.23835i −1.40752 + 0.255118i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.6398 37.4812i −0.760815 1.31777i −0.942431 0.334401i \(-0.891466\pi\)
0.181616 0.983370i \(-0.441867\pi\)
\(810\) 0 0
\(811\) 15.4941 0.544071 0.272036 0.962287i \(-0.412303\pi\)
0.272036 + 0.962287i \(0.412303\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.12026 0.179355
\(816\) 0 0
\(817\) 2.12488 0.0743402
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 36.7704 1.28330 0.641649 0.766999i \(-0.278251\pi\)
0.641649 + 0.766999i \(0.278251\pi\)
\(822\) 0 0
\(823\) 7.13422 0.248683 0.124342 0.992239i \(-0.460318\pi\)
0.124342 + 0.992239i \(0.460318\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −40.7236 −1.41610 −0.708050 0.706162i \(-0.750425\pi\)
−0.708050 + 0.706162i \(0.750425\pi\)
\(828\) 0 0
\(829\) −15.0794 26.1183i −0.523730 0.907127i −0.999618 0.0276214i \(-0.991207\pi\)
0.475888 0.879506i \(-0.342127\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.2617 + 6.84484i −0.632729 + 0.237160i
\(834\) 0 0
\(835\) −41.6981 −1.44302
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.37871 + 7.58415i 0.151170 + 0.261834i 0.931658 0.363337i \(-0.118363\pi\)
−0.780488 + 0.625171i \(0.785029\pi\)
\(840\) 0 0
\(841\) 14.1562 24.5192i 0.488144 0.845490i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −15.2408 + 26.3979i −0.524301 + 0.908115i
\(846\) 0 0
\(847\) −8.77122 + 24.4614i −0.301383 + 0.840505i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 38.5154 1.32029
\(852\) 0 0
\(853\) −22.8129 39.5130i −0.781098 1.35290i −0.931303 0.364246i \(-0.881327\pi\)
0.150205 0.988655i \(-0.452007\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.3497 49.1030i 0.968406 1.67733i 0.268233 0.963354i \(-0.413560\pi\)
0.700172 0.713974i \(-0.253107\pi\)
\(858\) 0 0
\(859\) 5.29287 + 9.16752i 0.180590 + 0.312792i 0.942082 0.335383i \(-0.108866\pi\)
−0.761491 + 0.648175i \(0.775533\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.3124 + 35.1821i −0.691442 + 1.19761i 0.279923 + 0.960022i \(0.409691\pi\)
−0.971365 + 0.237591i \(0.923642\pi\)
\(864\) 0 0
\(865\) −25.6471 44.4221i −0.872027 1.51040i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −5.07265 + 8.78608i −0.172078 + 0.298047i
\(870\) 0 0
\(871\) −7.26916 + 12.5905i −0.246306 + 0.426614i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 15.9439 + 18.8180i 0.539002 + 0.636166i
\(876\) 0 0
\(877\) −6.01744 10.4225i −0.203195 0.351943i 0.746361 0.665541i \(-0.231799\pi\)
−0.949556 + 0.313597i \(0.898466\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.07911 0.204811 0.102405 0.994743i \(-0.467346\pi\)
0.102405 + 0.994743i \(0.467346\pi\)
\(882\) 0 0
\(883\) 11.6345 0.391532 0.195766 0.980651i \(-0.437281\pi\)
0.195766 + 0.980651i \(0.437281\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.243915 + 0.422473i 0.00818986 + 0.0141853i 0.870091 0.492891i \(-0.164060\pi\)
−0.861901 + 0.507076i \(0.830726\pi\)
\(888\) 0 0
\(889\) −9.38398 11.0756i −0.314729 0.371463i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.40056 + 7.62199i −0.147259 + 0.255060i
\(894\) 0 0
\(895\) −1.56480 + 2.71032i −0.0523056 + 0.0905959i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.08436 1.87816i −0.0361653 0.0626402i
\(900\) 0 0
\(901\) −11.0318 + 19.1076i −0.367523 + 0.636568i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.3130 + 31.7190i 0.608744 + 1.05437i
\(906\) 0 0
\(907\) 8.02889 13.9064i 0.266595 0.461756i −0.701385 0.712782i \(-0.747435\pi\)
0.967980 + 0.251027i \(0.0807681\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 19.9439 + 34.5438i 0.660769 + 1.14449i 0.980414 + 0.196949i \(0.0631032\pi\)
−0.319645 + 0.947538i \(0.603563\pi\)
\(912\) 0 0
\(913\) 8.89431 0.294359
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −19.3941 + 54.0868i −0.640449 + 1.78610i
\(918\) 0 0
\(919\) −8.47711 + 14.6828i −0.279634 + 0.484340i −0.971294 0.237883i \(-0.923546\pi\)
0.691660 + 0.722224i \(0.256880\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.41149 7.64092i 0.145206 0.251504i
\(924\) 0 0
\(925\) 4.02704 + 6.97504i 0.132408 + 0.229338i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.7179 −0.515688 −0.257844 0.966187i \(-0.583012\pi\)
−0.257844 + 0.966187i \(0.583012\pi\)
\(930\) 0 0
\(931\) 7.21704 + 5.93933i 0.236529 + 0.194654i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.78896 + 6.56267i 0.123912 + 0.214622i
\(936\) 0 0
\(937\) −21.0501 −0.687675 −0.343838 0.939029i \(-0.611727\pi\)
−0.343838 + 0.939029i \(0.611727\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.3571 1.02221 0.511106 0.859518i \(-0.329236\pi\)
0.511106 + 0.859518i \(0.329236\pi\)
\(942\) 0 0
\(943\) −10.0057 −0.325832
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.6365 −0.508119 −0.254059 0.967189i \(-0.581766\pi\)
−0.254059 + 0.967189i \(0.581766\pi\)
\(948\) 0 0
\(949\) 9.21764 0.299217
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −27.7543 −0.899051 −0.449525 0.893268i \(-0.648407\pi\)
−0.449525 + 0.893268i \(0.648407\pi\)
\(954\) 0 0
\(955\) 22.4907 + 38.9551i 0.727783 + 1.26056i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 53.1144 9.62715i 1.71515 0.310877i
\(960\) 0 0
\(961\) −24.1603 −0.779363
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 29.0461 + 50.3093i 0.935026 + 1.61951i
\(966\) 0 0
\(967\) 13.6893 23.7105i 0.440217 0.762479i −0.557488 0.830185i \(-0.688235\pi\)
0.997705 + 0.0677063i \(0.0215681\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.88209 + 6.72398i −0.124582 + 0.215783i −0.921570 0.388213i \(-0.873092\pi\)
0.796987 + 0.603996i \(0.206426\pi\)
\(972\) 0 0
\(973\) 12.5844 2.28096i 0.403437 0.0731244i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.3134 0.585897 0.292948 0.956128i \(-0.405364\pi\)
0.292948 + 0.956128i \(0.405364\pi\)
\(978\) 0 0
\(979\) 6.91200 + 11.9719i 0.220908 + 0.382625i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 8.09420 14.0196i 0.258165 0.447155i −0.707585 0.706628i \(-0.750216\pi\)
0.965750 + 0.259473i \(0.0835489\pi\)
\(984\) 0 0
\(985\) −7.90811 13.6972i −0.251973 0.436431i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.87065 + 8.43621i −0.154878 + 0.268256i
\(990\) 0 0
\(991\) −19.1897 33.2376i −0.609582 1.05583i −0.991309 0.131552i \(-0.958004\pi\)
0.381727 0.924275i \(-0.375329\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.59481 + 16.6187i −0.304176 + 0.526848i
\(996\) 0 0
\(997\) 11.5661 20.0331i 0.366303 0.634456i −0.622681 0.782476i \(-0.713957\pi\)
0.988984 + 0.148020i \(0.0472900\pi\)
\(998\) 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 2268.2.i.l.2053.4 8
3.2 odd 2 2268.2.i.m.2053.1 8
7.4 even 3 2268.2.l.m.109.1 8
9.2 odd 6 2268.2.l.l.541.4 8
9.4 even 3 2268.2.k.c.1297.4 8
9.5 odd 6 2268.2.k.d.1297.1 yes 8
9.7 even 3 2268.2.l.m.541.1 8
21.11 odd 6 2268.2.l.l.109.4 8
63.4 even 3 2268.2.k.c.1621.4 yes 8
63.11 odd 6 2268.2.i.m.865.1 8
63.25 even 3 inner 2268.2.i.l.865.4 8
63.32 odd 6 2268.2.k.d.1621.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2268.2.i.l.865.4 8 63.25 even 3 inner
2268.2.i.l.2053.4 8 1.1 even 1 trivial
2268.2.i.m.865.1 8 63.11 odd 6
2268.2.i.m.2053.1 8 3.2 odd 2
2268.2.k.c.1297.4 8 9.4 even 3
2268.2.k.c.1621.4 yes 8 63.4 even 3
2268.2.k.d.1297.1 yes 8 9.5 odd 6
2268.2.k.d.1621.1 yes 8 63.32 odd 6
2268.2.l.l.109.4 8 21.11 odd 6
2268.2.l.l.541.4 8 9.2 odd 6
2268.2.l.m.109.1 8 7.4 even 3
2268.2.l.m.541.1 8 9.7 even 3