Properties

Label 2052.3.m.a.881.7
Level $2052$
Weight $3$
Character 2052.881
Analytic conductor $55.913$
Analytic rank $0$
Dimension $80$
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] = [2052,3,Mod(881,2052)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2052, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2052.881");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2052.m (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: \(55.9129502467\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 881.7
Character \(\chi\) \(=\) 2052.881
Dual form 2052.3.m.a.1493.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.32167i q^{5} +(2.00114 + 3.46607i) q^{7} +O(q^{10})\) \(q-6.32167i q^{5} +(2.00114 + 3.46607i) q^{7} +(6.04816 - 3.49191i) q^{11} +(4.14076 + 7.17201i) q^{13} +(-23.9657 + 13.8366i) q^{17} +(-17.9139 - 6.33195i) q^{19} +(27.0361 - 15.6093i) q^{23} -14.9636 q^{25} -18.4925i q^{29} +(22.7611 - 39.4233i) q^{31} +(21.9114 - 12.6505i) q^{35} -40.3326 q^{37} +9.03661i q^{41} +(-8.29062 + 14.3598i) q^{43} -54.4672i q^{47} +(16.4909 - 28.5631i) q^{49} +(26.2519 + 15.1566i) q^{53} +(-22.0747 - 38.2345i) q^{55} -40.5469i q^{59} +37.9214 q^{61} +(45.3391 - 26.1766i) q^{65} +(13.6512 + 23.6445i) q^{67} +(-16.8327 + 9.71834i) q^{71} +(-32.2512 - 55.8608i) q^{73} +(24.2064 + 13.9756i) q^{77} +(-43.0462 + 74.5583i) q^{79} +(-122.748 + 70.8687i) q^{83} +(87.4705 + 151.503i) q^{85} +(-72.8836 - 42.0794i) q^{89} +(-16.5725 + 28.7044i) q^{91} +(-40.0285 + 113.246i) q^{95} +(73.5940 - 127.469i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1027\) \(1217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 6.32167i 1.26433i −0.774832 0.632167i \(-0.782165\pi\)
0.774832 0.632167i \(-0.217835\pi\)
\(6\) 0 0
\(7\) 2.00114 + 3.46607i 0.285877 + 0.495153i 0.972821 0.231557i \(-0.0743819\pi\)
−0.686945 + 0.726710i \(0.741049\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.04816 3.49191i 0.549833 0.317446i −0.199222 0.979954i \(-0.563841\pi\)
0.749055 + 0.662508i \(0.230508\pi\)
\(12\) 0 0
\(13\) 4.14076 + 7.17201i 0.318520 + 0.551693i 0.980180 0.198111i \(-0.0634807\pi\)
−0.661659 + 0.749805i \(0.730147\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −23.9657 + 13.8366i −1.40975 + 0.813918i −0.995364 0.0961842i \(-0.969336\pi\)
−0.414384 + 0.910102i \(0.636003\pi\)
\(18\) 0 0
\(19\) −17.9139 6.33195i −0.942835 0.333261i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 27.0361 15.6093i 1.17548 0.678665i 0.220517 0.975383i \(-0.429225\pi\)
0.954965 + 0.296718i \(0.0958921\pi\)
\(24\) 0 0
\(25\) −14.9636 −0.598543
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 18.4925i 0.637671i −0.947810 0.318835i \(-0.896708\pi\)
0.947810 0.318835i \(-0.103292\pi\)
\(30\) 0 0
\(31\) 22.7611 39.4233i 0.734227 1.27172i −0.220834 0.975311i \(-0.570878\pi\)
0.955061 0.296408i \(-0.0957888\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.9114 12.6505i 0.626039 0.361444i
\(36\) 0 0
\(37\) −40.3326 −1.09007 −0.545035 0.838413i \(-0.683484\pi\)
−0.545035 + 0.838413i \(0.683484\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.03661i 0.220405i 0.993909 + 0.110203i \(0.0351499\pi\)
−0.993909 + 0.110203i \(0.964850\pi\)
\(42\) 0 0
\(43\) −8.29062 + 14.3598i −0.192805 + 0.333948i −0.946179 0.323644i \(-0.895092\pi\)
0.753374 + 0.657593i \(0.228425\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 54.4672i 1.15888i −0.815016 0.579438i \(-0.803272\pi\)
0.815016 0.579438i \(-0.196728\pi\)
\(48\) 0 0
\(49\) 16.4909 28.5631i 0.336549 0.582920i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 26.2519 + 15.1566i 0.495320 + 0.285973i 0.726779 0.686872i \(-0.241017\pi\)
−0.231459 + 0.972845i \(0.574350\pi\)
\(54\) 0 0
\(55\) −22.0747 38.2345i −0.401358 0.695173i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 40.5469i 0.687236i −0.939110 0.343618i \(-0.888347\pi\)
0.939110 0.343618i \(-0.111653\pi\)
\(60\) 0 0
\(61\) 37.9214 0.621661 0.310831 0.950465i \(-0.399393\pi\)
0.310831 + 0.950465i \(0.399393\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 45.3391 26.1766i 0.697525 0.402716i
\(66\) 0 0
\(67\) 13.6512 + 23.6445i 0.203749 + 0.352904i 0.949733 0.313060i \(-0.101354\pi\)
−0.745984 + 0.665963i \(0.768021\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −16.8327 + 9.71834i −0.237080 + 0.136878i −0.613834 0.789435i \(-0.710374\pi\)
0.376754 + 0.926313i \(0.377040\pi\)
\(72\) 0 0
\(73\) −32.2512 55.8608i −0.441798 0.765216i 0.556025 0.831165i \(-0.312326\pi\)
−0.997823 + 0.0659493i \(0.978992\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 24.2064 + 13.9756i 0.314369 + 0.181501i
\(78\) 0 0
\(79\) −43.0462 + 74.5583i −0.544889 + 0.943776i 0.453725 + 0.891142i \(0.350095\pi\)
−0.998614 + 0.0526337i \(0.983238\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −122.748 + 70.8687i −1.47889 + 0.853840i −0.999715 0.0238776i \(-0.992399\pi\)
−0.479179 + 0.877717i \(0.659065\pi\)
\(84\) 0 0
\(85\) 87.4705 + 151.503i 1.02906 + 1.78239i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −72.8836 42.0794i −0.818917 0.472802i 0.0311261 0.999515i \(-0.490091\pi\)
−0.850043 + 0.526714i \(0.823424\pi\)
\(90\) 0 0
\(91\) −16.5725 + 28.7044i −0.182115 + 0.315433i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −40.0285 + 113.246i −0.421353 + 1.19206i
\(96\) 0 0
\(97\) 73.5940 127.469i 0.758701 1.31411i −0.184813 0.982774i \(-0.559168\pi\)
0.943513 0.331335i \(-0.107499\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.4881i 0.133545i −0.997768 0.0667726i \(-0.978730\pi\)
0.997768 0.0667726i \(-0.0212702\pi\)
\(102\) 0 0
\(103\) 19.7030 34.1266i 0.191292 0.331327i −0.754387 0.656430i \(-0.772066\pi\)
0.945679 + 0.325103i \(0.105399\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 38.1059i 0.356130i −0.984019 0.178065i \(-0.943016\pi\)
0.984019 0.178065i \(-0.0569837\pi\)
\(108\) 0 0
\(109\) −13.3506 23.1239i −0.122483 0.212146i 0.798263 0.602308i \(-0.205752\pi\)
−0.920746 + 0.390162i \(0.872419\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −161.778 93.4027i −1.43167 0.826573i −0.434418 0.900712i \(-0.643046\pi\)
−0.997248 + 0.0741390i \(0.976379\pi\)
\(114\) 0 0
\(115\) −98.6769 170.913i −0.858060 1.48620i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −95.9173 55.3779i −0.806028 0.465360i
\(120\) 0 0
\(121\) −36.1131 + 62.5498i −0.298456 + 0.516941i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 63.4471i 0.507577i
\(126\) 0 0
\(127\) −59.6131 + 103.253i −0.469394 + 0.813015i −0.999388 0.0349868i \(-0.988861\pi\)
0.529993 + 0.848002i \(0.322194\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 148.423i 1.13300i −0.824063 0.566498i \(-0.808298\pi\)
0.824063 0.566498i \(-0.191702\pi\)
\(132\) 0 0
\(133\) −13.9011 74.7618i −0.104520 0.562119i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 211.149i 1.54124i 0.637298 + 0.770618i \(0.280052\pi\)
−0.637298 + 0.770618i \(0.719948\pi\)
\(138\) 0 0
\(139\) −108.298 187.578i −0.779124 1.34948i −0.932447 0.361307i \(-0.882331\pi\)
0.153322 0.988176i \(-0.451003\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 50.0880 + 28.9183i 0.350266 + 0.202226i
\(144\) 0 0
\(145\) −116.903 −0.806230
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 228.955i 1.53661i −0.640084 0.768305i \(-0.721100\pi\)
0.640084 0.768305i \(-0.278900\pi\)
\(150\) 0 0
\(151\) 51.3731 + 88.9809i 0.340219 + 0.589277i 0.984473 0.175534i \(-0.0561653\pi\)
−0.644254 + 0.764812i \(0.722832\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −249.221 143.888i −1.60788 0.928309i
\(156\) 0 0
\(157\) 182.678 1.16355 0.581777 0.813348i \(-0.302357\pi\)
0.581777 + 0.813348i \(0.302357\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 108.206 + 62.4727i 0.672086 + 0.388029i
\(162\) 0 0
\(163\) −279.869 −1.71699 −0.858495 0.512822i \(-0.828600\pi\)
−0.858495 + 0.512822i \(0.828600\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −152.723 + 88.1747i −0.914510 + 0.527992i −0.881879 0.471475i \(-0.843722\pi\)
−0.0326302 + 0.999467i \(0.510388\pi\)
\(168\) 0 0
\(169\) 50.2081 86.9630i 0.297090 0.514574i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −66.4404 38.3594i −0.384049 0.221731i 0.295530 0.955334i \(-0.404504\pi\)
−0.679578 + 0.733603i \(0.737837\pi\)
\(174\) 0 0
\(175\) −29.9441 51.8648i −0.171109 0.296370i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 42.0832i 0.235102i −0.993067 0.117551i \(-0.962496\pi\)
0.993067 0.117551i \(-0.0375043\pi\)
\(180\) 0 0
\(181\) 70.5353 122.171i 0.389698 0.674977i −0.602711 0.797960i \(-0.705913\pi\)
0.992409 + 0.122983i \(0.0392461\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 254.970i 1.37821i
\(186\) 0 0
\(187\) −96.6323 + 167.372i −0.516750 + 0.895038i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −96.4688 + 55.6963i −0.505072 + 0.291604i −0.730806 0.682585i \(-0.760856\pi\)
0.225733 + 0.974189i \(0.427522\pi\)
\(192\) 0 0
\(193\) −35.8077 −0.185532 −0.0927661 0.995688i \(-0.529571\pi\)
−0.0927661 + 0.995688i \(0.529571\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 201.997i 1.02537i 0.858578 + 0.512683i \(0.171348\pi\)
−0.858578 + 0.512683i \(0.828652\pi\)
\(198\) 0 0
\(199\) 68.6408 118.889i 0.344929 0.597434i −0.640412 0.768032i \(-0.721236\pi\)
0.985341 + 0.170597i \(0.0545697\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 64.0962 37.0059i 0.315745 0.182295i
\(204\) 0 0
\(205\) 57.1265 0.278666
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −130.457 + 24.2569i −0.624194 + 0.116062i
\(210\) 0 0
\(211\) 148.420 0.703413 0.351706 0.936110i \(-0.385602\pi\)
0.351706 + 0.936110i \(0.385602\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 90.7778 + 52.4106i 0.422222 + 0.243770i
\(216\) 0 0
\(217\) 182.192 0.839594
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −198.473 114.588i −0.898066 0.518499i
\(222\) 0 0
\(223\) 183.175 317.269i 0.821414 1.42273i −0.0832152 0.996532i \(-0.526519\pi\)
0.904629 0.426199i \(-0.140148\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −229.354 + 132.417i −1.01037 + 0.583336i −0.911299 0.411744i \(-0.864920\pi\)
−0.0990686 + 0.995081i \(0.531586\pi\)
\(228\) 0 0
\(229\) 176.804 306.233i 0.772068 1.33726i −0.164360 0.986400i \(-0.552556\pi\)
0.936428 0.350860i \(-0.114111\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −65.2399 + 37.6663i −0.280000 + 0.161658i −0.633423 0.773805i \(-0.718351\pi\)
0.353424 + 0.935463i \(0.385017\pi\)
\(234\) 0 0
\(235\) −344.324 −1.46521
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.42144 2.55272i −0.0184998 0.0106808i 0.490722 0.871316i \(-0.336733\pi\)
−0.509221 + 0.860636i \(0.670067\pi\)
\(240\) 0 0
\(241\) 175.057 0.726378 0.363189 0.931716i \(-0.381688\pi\)
0.363189 + 0.931716i \(0.381688\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −180.567 104.250i −0.737006 0.425511i
\(246\) 0 0
\(247\) −28.7642 154.698i −0.116454 0.626306i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 416.120 + 240.247i 1.65785 + 0.957160i 0.973704 + 0.227816i \(0.0731584\pi\)
0.684146 + 0.729345i \(0.260175\pi\)
\(252\) 0 0
\(253\) 109.012 188.815i 0.430879 0.746305i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −269.498 + 155.595i −1.04863 + 0.605426i −0.922265 0.386558i \(-0.873664\pi\)
−0.126364 + 0.991984i \(0.540331\pi\)
\(258\) 0 0
\(259\) −80.7111 139.796i −0.311626 0.539752i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −48.0406 27.7362i −0.182664 0.105461i 0.405880 0.913926i \(-0.366965\pi\)
−0.588544 + 0.808465i \(0.700298\pi\)
\(264\) 0 0
\(265\) 95.8148 165.956i 0.361565 0.626250i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −179.719 + 103.761i −0.668102 + 0.385729i −0.795357 0.606141i \(-0.792717\pi\)
0.127255 + 0.991870i \(0.459383\pi\)
\(270\) 0 0
\(271\) 108.517 + 187.957i 0.400433 + 0.693570i 0.993778 0.111378i \(-0.0355265\pi\)
−0.593345 + 0.804948i \(0.702193\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −90.5021 + 52.2514i −0.329098 + 0.190005i
\(276\) 0 0
\(277\) −66.9535 115.967i −0.241709 0.418653i 0.719492 0.694501i \(-0.244375\pi\)
−0.961201 + 0.275848i \(0.911041\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 325.724i 1.15916i −0.814915 0.579580i \(-0.803216\pi\)
0.814915 0.579580i \(-0.196784\pi\)
\(282\) 0 0
\(283\) −406.710 −1.43714 −0.718570 0.695455i \(-0.755203\pi\)
−0.718570 + 0.695455i \(0.755203\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.3215 + 18.0835i −0.109134 + 0.0630087i
\(288\) 0 0
\(289\) 238.403 412.927i 0.824925 1.42881i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 218.194 + 125.974i 0.744688 + 0.429946i 0.823771 0.566922i \(-0.191866\pi\)
−0.0790834 + 0.996868i \(0.525199\pi\)
\(294\) 0 0
\(295\) −256.324 −0.868896
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 223.900 + 129.269i 0.748830 + 0.432337i
\(300\) 0 0
\(301\) −66.3626 −0.220474
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 239.726i 0.785988i
\(306\) 0 0
\(307\) −58.6501 101.585i −0.191043 0.330896i 0.754553 0.656239i \(-0.227854\pi\)
−0.945596 + 0.325343i \(0.894520\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 75.4767 + 43.5765i 0.242690 + 0.140117i 0.616413 0.787423i \(-0.288585\pi\)
−0.373722 + 0.927541i \(0.621919\pi\)
\(312\) 0 0
\(313\) −321.081 −1.02582 −0.512909 0.858443i \(-0.671432\pi\)
−0.512909 + 0.858443i \(0.671432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 461.392i 1.45550i −0.685845 0.727748i \(-0.740567\pi\)
0.685845 0.727748i \(-0.259433\pi\)
\(318\) 0 0
\(319\) −64.5740 111.845i −0.202426 0.350613i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 516.931 96.1174i 1.60041 0.297577i
\(324\) 0 0
\(325\) −61.9606 107.319i −0.190648 0.330212i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 188.787 108.996i 0.573821 0.331296i
\(330\) 0 0
\(331\) −106.818 185.014i −0.322712 0.558954i 0.658334 0.752726i \(-0.271261\pi\)
−0.981047 + 0.193771i \(0.937928\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 149.473 86.2983i 0.446188 0.257607i
\(336\) 0 0
\(337\) 182.989 0.542995 0.271497 0.962439i \(-0.412481\pi\)
0.271497 + 0.962439i \(0.412481\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 317.918i 0.932311i
\(342\) 0 0
\(343\) 328.114 0.956599
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 110.589i 0.318700i 0.987222 + 0.159350i \(0.0509399\pi\)
−0.987222 + 0.159350i \(0.949060\pi\)
\(348\) 0 0
\(349\) −110.819 191.944i −0.317532 0.549982i 0.662440 0.749115i \(-0.269521\pi\)
−0.979972 + 0.199133i \(0.936187\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −322.547 + 186.223i −0.913732 + 0.527543i −0.881630 0.471941i \(-0.843553\pi\)
−0.0321020 + 0.999485i \(0.510220\pi\)
\(354\) 0 0
\(355\) 61.4362 + 106.411i 0.173060 + 0.299748i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 64.0955 37.0056i 0.178539 0.103080i −0.408067 0.912952i \(-0.633797\pi\)
0.586606 + 0.809872i \(0.300464\pi\)
\(360\) 0 0
\(361\) 280.813 + 226.859i 0.777875 + 0.628419i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −353.134 + 203.882i −0.967489 + 0.558580i
\(366\) 0 0
\(367\) 377.052 1.02739 0.513695 0.857973i \(-0.328276\pi\)
0.513695 + 0.857973i \(0.328276\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 121.321i 0.327012i
\(372\) 0 0
\(373\) −3.67579 + 6.36666i −0.00985467 + 0.0170688i −0.870911 0.491441i \(-0.836470\pi\)
0.861056 + 0.508510i \(0.169804\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 132.628 76.5729i 0.351799 0.203111i
\(378\) 0 0
\(379\) −366.651 −0.967417 −0.483709 0.875229i \(-0.660711\pi\)
−0.483709 + 0.875229i \(0.660711\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 274.610i 0.716997i −0.933530 0.358498i \(-0.883289\pi\)
0.933530 0.358498i \(-0.116711\pi\)
\(384\) 0 0
\(385\) 88.3490 153.025i 0.229478 0.397467i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 664.121i 1.70725i −0.520887 0.853626i \(-0.674399\pi\)
0.520887 0.853626i \(-0.325601\pi\)
\(390\) 0 0
\(391\) −431.959 + 748.176i −1.10476 + 1.91349i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 471.333 + 272.124i 1.19325 + 0.688922i
\(396\) 0 0
\(397\) 156.466 + 271.007i 0.394120 + 0.682636i 0.992988 0.118211i \(-0.0377160\pi\)
−0.598868 + 0.800848i \(0.704383\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 473.726i 1.18136i 0.806906 + 0.590680i \(0.201141\pi\)
−0.806906 + 0.590680i \(0.798859\pi\)
\(402\) 0 0
\(403\) 376.993 0.935466
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −243.938 + 140.838i −0.599357 + 0.346039i
\(408\) 0 0
\(409\) 276.393 + 478.727i 0.675777 + 1.17048i 0.976241 + 0.216688i \(0.0695253\pi\)
−0.300464 + 0.953793i \(0.597141\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 140.538 81.1399i 0.340287 0.196465i
\(414\) 0 0
\(415\) 448.009 + 775.974i 1.07954 + 1.86982i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.32796 + 0.766697i 0.00316935 + 0.00182983i 0.501584 0.865109i \(-0.332751\pi\)
−0.498414 + 0.866939i \(0.666084\pi\)
\(420\) 0 0
\(421\) −97.1313 + 168.236i −0.230716 + 0.399611i −0.958019 0.286705i \(-0.907440\pi\)
0.727303 + 0.686316i \(0.240773\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 358.612 207.045i 0.843794 0.487165i
\(426\) 0 0
\(427\) 75.8858 + 131.438i 0.177719 + 0.307817i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 60.2968 + 34.8124i 0.139900 + 0.0807711i 0.568316 0.822810i \(-0.307595\pi\)
−0.428416 + 0.903581i \(0.640928\pi\)
\(432\) 0 0
\(433\) −125.295 + 217.018i −0.289365 + 0.501196i −0.973658 0.228012i \(-0.926778\pi\)
0.684293 + 0.729207i \(0.260111\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −583.158 + 108.432i −1.33446 + 0.248127i
\(438\) 0 0
\(439\) −359.264 + 622.264i −0.818369 + 1.41746i 0.0885135 + 0.996075i \(0.471788\pi\)
−0.906883 + 0.421383i \(0.861545\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 595.462i 1.34416i −0.740479 0.672079i \(-0.765402\pi\)
0.740479 0.672079i \(-0.234598\pi\)
\(444\) 0 0
\(445\) −266.012 + 460.746i −0.597780 + 1.03538i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 649.900i 1.44744i 0.690094 + 0.723720i \(0.257569\pi\)
−0.690094 + 0.723720i \(0.742431\pi\)
\(450\) 0 0
\(451\) 31.5550 + 54.6549i 0.0699668 + 0.121186i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 181.460 + 104.766i 0.398812 + 0.230254i
\(456\) 0 0
\(457\) −386.945 670.209i −0.846707 1.46654i −0.884130 0.467241i \(-0.845248\pi\)
0.0374227 0.999300i \(-0.488085\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −108.581 62.6890i −0.235533 0.135985i 0.377589 0.925973i \(-0.376753\pi\)
−0.613122 + 0.789988i \(0.710087\pi\)
\(462\) 0 0
\(463\) −295.738 + 512.234i −0.638744 + 1.10634i 0.346965 + 0.937878i \(0.387212\pi\)
−0.985709 + 0.168459i \(0.946121\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 411.110i 0.880320i −0.897919 0.440160i \(-0.854922\pi\)
0.897919 0.440160i \(-0.145078\pi\)
\(468\) 0 0
\(469\) −54.6357 + 94.6319i −0.116494 + 0.201774i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 115.800i 0.244821i
\(474\) 0 0
\(475\) 268.055 + 94.7485i 0.564327 + 0.199471i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 123.171i 0.257141i −0.991700 0.128571i \(-0.958961\pi\)
0.991700 0.128571i \(-0.0410389\pi\)
\(480\) 0 0
\(481\) −167.008 289.266i −0.347210 0.601385i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −805.814 465.237i −1.66147 0.959252i
\(486\) 0 0
\(487\) −682.651 −1.40175 −0.700874 0.713285i \(-0.747207\pi\)
−0.700874 + 0.713285i \(0.747207\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 198.231i 0.403729i 0.979413 + 0.201865i \(0.0647001\pi\)
−0.979413 + 0.201865i \(0.935300\pi\)
\(492\) 0 0
\(493\) 255.873 + 443.185i 0.519012 + 0.898955i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −67.3689 38.8954i −0.135551 0.0782605i
\(498\) 0 0
\(499\) 543.154 1.08849 0.544243 0.838928i \(-0.316817\pi\)
0.544243 + 0.838928i \(0.316817\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 344.739 + 199.035i 0.685365 + 0.395696i 0.801873 0.597494i \(-0.203837\pi\)
−0.116508 + 0.993190i \(0.537170\pi\)
\(504\) 0 0
\(505\) −85.2672 −0.168846
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 296.983 171.463i 0.583463 0.336863i −0.179045 0.983841i \(-0.557301\pi\)
0.762509 + 0.646978i \(0.223968\pi\)
\(510\) 0 0
\(511\) 129.078 223.570i 0.252599 0.437515i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −215.738 124.556i −0.418908 0.241857i
\(516\) 0 0
\(517\) −190.194 329.427i −0.367881 0.637189i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 664.298i 1.27504i −0.770432 0.637522i \(-0.779960\pi\)
0.770432 0.637522i \(-0.220040\pi\)
\(522\) 0 0
\(523\) −259.118 + 448.806i −0.495446 + 0.858138i −0.999986 0.00525011i \(-0.998329\pi\)
0.504540 + 0.863388i \(0.331662\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1259.74i 2.39040i
\(528\) 0 0
\(529\) 222.800 385.902i 0.421173 0.729493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −64.8107 + 37.4185i −0.121596 + 0.0702035i
\(534\) 0 0
\(535\) −240.893 −0.450267
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 230.339i 0.427345i
\(540\) 0 0
\(541\) 430.351 745.389i 0.795472 1.37780i −0.127066 0.991894i \(-0.540556\pi\)
0.922539 0.385905i \(-0.126111\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −146.182 + 84.3982i −0.268224 + 0.154859i
\(546\) 0 0
\(547\) 117.128 0.214127 0.107064 0.994252i \(-0.465855\pi\)
0.107064 + 0.994252i \(0.465855\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −117.093 + 331.271i −0.212511 + 0.601218i
\(552\) 0 0
\(553\) −344.566 −0.623084
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 200.752 + 115.904i 0.360416 + 0.208086i 0.669263 0.743025i \(-0.266610\pi\)
−0.308847 + 0.951112i \(0.599943\pi\)
\(558\) 0 0
\(559\) −137.318 −0.245649
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 465.717 + 268.882i 0.827205 + 0.477587i 0.852895 0.522083i \(-0.174845\pi\)
−0.0256895 + 0.999670i \(0.508178\pi\)
\(564\) 0 0
\(565\) −590.461 + 1022.71i −1.04506 + 1.81010i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 880.392 508.295i 1.54726 0.893312i 0.548912 0.835880i \(-0.315042\pi\)
0.998349 0.0574321i \(-0.0182913\pi\)
\(570\) 0 0
\(571\) −58.4285 + 101.201i −0.102327 + 0.177235i −0.912643 0.408758i \(-0.865962\pi\)
0.810316 + 0.585993i \(0.199295\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −404.556 + 233.571i −0.703576 + 0.406210i
\(576\) 0 0
\(577\) 115.022 0.199345 0.0996724 0.995020i \(-0.468221\pi\)
0.0996724 + 0.995020i \(0.468221\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −491.272 283.636i −0.845562 0.488186i
\(582\) 0 0
\(583\) 211.701 0.363124
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 862.770 + 498.120i 1.46980 + 0.848587i 0.999426 0.0338825i \(-0.0107872\pi\)
0.470370 + 0.882469i \(0.344121\pi\)
\(588\) 0 0
\(589\) −657.365 + 562.102i −1.11607 + 0.954332i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −109.232 63.0651i −0.184202 0.106349i 0.405063 0.914289i \(-0.367249\pi\)
−0.589266 + 0.807939i \(0.700583\pi\)
\(594\) 0 0
\(595\) −350.081 + 606.358i −0.588371 + 1.01909i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −740.979 + 427.804i −1.23703 + 0.714197i −0.968485 0.249071i \(-0.919875\pi\)
−0.268541 + 0.963268i \(0.586541\pi\)
\(600\) 0 0
\(601\) 527.598 + 913.826i 0.877866 + 1.52051i 0.853678 + 0.520801i \(0.174367\pi\)
0.0241883 + 0.999707i \(0.492300\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 395.419 + 228.296i 0.653586 + 0.377348i
\(606\) 0 0
\(607\) 330.385 572.243i 0.544291 0.942740i −0.454360 0.890818i \(-0.650132\pi\)
0.998651 0.0519218i \(-0.0165347\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 390.640 225.536i 0.639345 0.369126i
\(612\) 0 0
\(613\) 303.879 + 526.335i 0.495725 + 0.858621i 0.999988 0.00492941i \(-0.00156909\pi\)
−0.504263 + 0.863550i \(0.668236\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 801.853 462.950i 1.29960 0.750324i 0.319265 0.947666i \(-0.396564\pi\)
0.980335 + 0.197342i \(0.0632308\pi\)
\(618\) 0 0
\(619\) 62.0433 + 107.462i 0.100231 + 0.173606i 0.911780 0.410679i \(-0.134708\pi\)
−0.811549 + 0.584285i \(0.801375\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 336.826i 0.540652i
\(624\) 0 0
\(625\) −775.181 −1.24029
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 966.600 558.067i 1.53672 0.887228i
\(630\) 0 0
\(631\) 464.866 805.172i 0.736713 1.27603i −0.217254 0.976115i \(-0.569710\pi\)
0.953967 0.299910i \(-0.0969567\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 652.731 + 376.855i 1.02792 + 0.593472i
\(636\) 0 0
\(637\) 273.140 0.428791
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 830.818 + 479.673i 1.29613 + 0.748320i 0.979733 0.200307i \(-0.0641941\pi\)
0.316395 + 0.948627i \(0.397527\pi\)
\(642\) 0 0
\(643\) 1034.89 1.60947 0.804736 0.593633i \(-0.202307\pi\)
0.804736 + 0.593633i \(0.202307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1029.68i 1.59147i 0.605647 + 0.795733i \(0.292914\pi\)
−0.605647 + 0.795733i \(0.707086\pi\)
\(648\) 0 0
\(649\) −141.586 245.234i −0.218160 0.377865i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 193.285 + 111.593i 0.295996 + 0.170893i 0.640643 0.767839i \(-0.278668\pi\)
−0.344647 + 0.938732i \(0.612001\pi\)
\(654\) 0 0
\(655\) −938.279 −1.43249
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 470.828i 0.714458i 0.934017 + 0.357229i \(0.116278\pi\)
−0.934017 + 0.357229i \(0.883722\pi\)
\(660\) 0 0
\(661\) −184.554 319.657i −0.279204 0.483596i 0.691983 0.721914i \(-0.256737\pi\)
−0.971187 + 0.238318i \(0.923404\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −472.620 + 87.8782i −0.710706 + 0.132148i
\(666\) 0 0
\(667\) −288.654 499.964i −0.432765 0.749571i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 229.355 132.418i 0.341810 0.197344i
\(672\) 0 0
\(673\) 556.859 + 964.509i 0.827429 + 1.43315i 0.900049 + 0.435789i \(0.143531\pi\)
−0.0726204 + 0.997360i \(0.523136\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −83.3953 + 48.1483i −0.123184 + 0.0711201i −0.560326 0.828272i \(-0.689324\pi\)
0.437142 + 0.899392i \(0.355991\pi\)
\(678\) 0 0
\(679\) 589.086 0.867579
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 56.7307i 0.0830611i 0.999137 + 0.0415305i \(0.0132234\pi\)
−0.999137 + 0.0415305i \(0.986777\pi\)
\(684\) 0 0
\(685\) 1334.82 1.94864
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 251.039i 0.364353i
\(690\) 0 0
\(691\) −199.549 345.629i −0.288783 0.500187i 0.684736 0.728791i \(-0.259917\pi\)
−0.973520 + 0.228604i \(0.926584\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1185.81 + 684.627i −1.70620 + 0.985074i
\(696\) 0 0
\(697\) −125.036 216.569i −0.179392 0.310715i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 401.210 231.639i 0.572339 0.330440i −0.185744 0.982598i \(-0.559469\pi\)
0.758083 + 0.652158i \(0.226136\pi\)
\(702\) 0 0
\(703\) 722.513 + 255.384i 1.02776 + 0.363278i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 46.7506 26.9915i 0.0661253 0.0381775i
\(708\) 0 0
\(709\) 254.458 0.358898 0.179449 0.983767i \(-0.442569\pi\)
0.179449 + 0.983767i \(0.442569\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1421.14i 1.99318i
\(714\) 0 0
\(715\) 182.812 316.640i 0.255682 0.442854i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.1832 18.5810i 0.0447610 0.0258428i −0.477453 0.878658i \(-0.658440\pi\)
0.522214 + 0.852815i \(0.325106\pi\)
\(720\) 0 0
\(721\) 157.714 0.218743
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 276.713i 0.381673i
\(726\) 0 0
\(727\) −125.786 + 217.868i −0.173021 + 0.299681i −0.939475 0.342619i \(-0.888686\pi\)
0.766454 + 0.642299i \(0.222019\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 458.856i 0.627710i
\(732\) 0 0
\(733\) 424.518 735.287i 0.579151 1.00312i −0.416426 0.909170i \(-0.636717\pi\)
0.995577 0.0939498i \(-0.0299493\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 165.129 + 95.3373i 0.224056 + 0.129359i
\(738\) 0 0
\(739\) −222.140 384.757i −0.300595 0.520646i 0.675676 0.737199i \(-0.263852\pi\)
−0.976271 + 0.216553i \(0.930519\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 294.169i 0.395921i −0.980210 0.197960i \(-0.936568\pi\)
0.980210 0.197960i \(-0.0634317\pi\)
\(744\) 0 0
\(745\) −1447.38 −1.94279
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 132.078 76.2551i 0.176339 0.101809i
\(750\) 0 0
\(751\) 748.371 + 1296.22i 0.996499 + 1.72599i 0.570654 + 0.821191i \(0.306690\pi\)
0.425845 + 0.904796i \(0.359977\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 562.508 324.764i 0.745044 0.430151i
\(756\) 0 0
\(757\) 11.2134 + 19.4222i 0.0148130 + 0.0256568i 0.873337 0.487117i \(-0.161951\pi\)
−0.858524 + 0.512774i \(0.828618\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 410.105 + 236.774i 0.538903 + 0.311136i 0.744634 0.667473i \(-0.232624\pi\)
−0.205731 + 0.978608i \(0.565957\pi\)
\(762\) 0 0
\(763\) 53.4328 92.5483i 0.0700299 0.121295i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 290.803 167.895i 0.379144 0.218899i
\(768\) 0 0
\(769\) −476.791 825.826i −0.620014 1.07390i −0.989483 0.144652i \(-0.953794\pi\)
0.369469 0.929243i \(-0.379540\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 776.662 + 448.406i 1.00474 + 0.580085i 0.909647 0.415383i \(-0.136352\pi\)
0.0950909 + 0.995469i \(0.469686\pi\)
\(774\) 0 0
\(775\) −340.586 + 589.913i −0.439466 + 0.761178i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 57.2193 161.881i 0.0734523 0.207806i
\(780\) 0 0
\(781\) −67.8711 + 117.556i −0.0869028 + 0.150520i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1154.83i 1.47112i
\(786\) 0 0
\(787\) 200.246 346.836i 0.254442 0.440706i −0.710302 0.703897i \(-0.751442\pi\)
0.964744 + 0.263191i \(0.0847750\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 747.646i 0.945191i
\(792\) 0 0
\(793\) 157.023 + 271.972i 0.198012 + 0.342967i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −537.021 310.049i −0.673803 0.389020i 0.123713 0.992318i \(-0.460520\pi\)
−0.797516 + 0.603298i \(0.793853\pi\)
\(798\) 0 0
\(799\) 753.641 + 1305.34i 0.943230 + 1.63372i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −390.121 225.237i −0.485830 0.280494i
\(804\) 0 0
\(805\) 394.932 684.042i 0.490599 0.849742i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 365.081i 0.451275i 0.974211 + 0.225637i \(0.0724465\pi\)
−0.974211 + 0.225637i \(0.927554\pi\)
\(810\) 0 0
\(811\) 452.295 783.397i 0.557700 0.965965i −0.439988 0.898004i \(-0.645017\pi\)
0.997688 0.0679610i \(-0.0216493\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1769.24i 2.17085i
\(816\) 0 0
\(817\) 239.442 204.743i 0.293075 0.250604i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1249.88i 1.52239i −0.648522 0.761196i \(-0.724613\pi\)
0.648522 0.761196i \(-0.275387\pi\)
\(822\) 0 0
\(823\) −720.887 1248.61i −0.875926 1.51715i −0.855773 0.517352i \(-0.826918\pi\)
−0.0201535 0.999797i \(-0.506415\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1076.73 + 621.651i 1.30197 + 0.751694i 0.980742 0.195307i \(-0.0625704\pi\)
0.321230 + 0.947001i \(0.395904\pi\)
\(828\) 0 0
\(829\) 1617.54 1.95119 0.975594 0.219581i \(-0.0704691\pi\)
0.975594 + 0.219581i \(0.0704691\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 912.713i 1.09569i
\(834\) 0 0
\(835\) 557.412 + 965.466i 0.667559 + 1.15625i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 441.625 + 254.973i 0.526371 + 0.303901i 0.739538 0.673115i \(-0.235044\pi\)
−0.213166 + 0.977016i \(0.568378\pi\)
\(840\) 0 0
\(841\) 499.029 0.593376
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −549.752 317.399i −0.650594 0.375621i
\(846\) 0 0
\(847\) −289.069 −0.341286
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1090.44 + 629.564i −1.28136 + 0.739793i
\(852\) 0 0
\(853\) −545.610 + 945.024i −0.639636 + 1.10788i 0.345876 + 0.938280i \(0.387582\pi\)
−0.985513 + 0.169602i \(0.945752\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 896.786 + 517.760i 1.04642 + 0.604154i 0.921646 0.388031i \(-0.126845\pi\)
0.124779 + 0.992185i \(0.460178\pi\)
\(858\) 0 0
\(859\) −263.394 456.211i −0.306628 0.531096i 0.670994 0.741462i \(-0.265867\pi\)
−0.977623 + 0.210367i \(0.932534\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 717.519i 0.831424i 0.909496 + 0.415712i \(0.136468\pi\)
−0.909496 + 0.415712i \(0.863532\pi\)
\(864\) 0 0
\(865\) −242.496 + 420.015i −0.280342 + 0.485566i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 601.254i 0.691892i
\(870\) 0 0
\(871\) −113.053 + 195.813i −0.129796 + 0.224814i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 219.912 126.966i 0.251328 0.145104i
\(876\) 0 0
\(877\) −88.2260 −0.100600 −0.0502999 0.998734i \(-0.516018\pi\)
−0.0502999 + 0.998734i \(0.516018\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 439.288i 0.498625i 0.968423 + 0.249312i \(0.0802046\pi\)
−0.968423 + 0.249312i \(0.919795\pi\)
\(882\) 0 0
\(883\) −482.731 + 836.115i −0.546695 + 0.946903i 0.451804 + 0.892117i \(0.350781\pi\)
−0.998498 + 0.0547853i \(0.982553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1154.40 666.494i 1.30147 0.751403i 0.320812 0.947143i \(-0.396044\pi\)
0.980656 + 0.195740i \(0.0627108\pi\)
\(888\) 0 0
\(889\) −477.176 −0.536756
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −344.884 + 975.718i −0.386208 + 1.09263i
\(894\) 0 0
\(895\) −266.036 −0.297247
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −729.034 420.908i −0.810938 0.468196i
\(900\) 0 0
\(901\) −838.861 −0.931034
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −772.324 445.901i −0.853397 0.492709i
\(906\) 0 0
\(907\) 431.199 746.858i 0.475412 0.823438i −0.524191 0.851601i \(-0.675632\pi\)
0.999603 + 0.0281627i \(0.00896566\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1310.94 + 756.870i −1.43901 + 0.830813i −0.997781 0.0665817i \(-0.978791\pi\)
−0.441229 + 0.897395i \(0.645457\pi\)
\(912\) 0 0
\(913\) −494.934 + 857.251i −0.542096 + 0.938939i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 514.443 297.014i 0.561006 0.323897i
\(918\) 0 0
\(919\) −863.054 −0.939123 −0.469561 0.882900i \(-0.655588\pi\)
−0.469561 + 0.882900i \(0.655588\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −139.400 80.4827i −0.151029 0.0871969i
\(924\) 0 0
\(925\) 603.520 0.652454
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1065.48 + 615.154i 1.14691 + 0.662167i 0.948132 0.317878i \(-0.102970\pi\)
0.198776 + 0.980045i \(0.436303\pi\)
\(930\) 0 0
\(931\) −476.276 + 407.256i −0.511574 + 0.437439i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1058.07 + 610.878i 1.13163 + 0.653346i
\(936\) 0 0
\(937\) 562.399 974.104i 0.600212 1.03960i −0.392576 0.919719i \(-0.628416\pi\)
0.992789 0.119879i \(-0.0382506\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −670.472 + 387.097i −0.712511 + 0.411368i −0.811990 0.583671i \(-0.801616\pi\)
0.0994794 + 0.995040i \(0.468282\pi\)
\(942\) 0 0
\(943\) 141.055 + 244.315i 0.149581 + 0.259082i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1025.04 591.807i −1.08241 0.624929i −0.150863 0.988555i \(-0.548205\pi\)
−0.931545 + 0.363626i \(0.881539\pi\)
\(948\) 0 0
\(949\) 267.090 462.613i 0.281443 0.487474i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 563.432 325.298i 0.591219 0.341341i −0.174360 0.984682i \(-0.555786\pi\)
0.765579 + 0.643341i \(0.222452\pi\)
\(954\) 0 0
\(955\) 352.094 + 609.845i 0.368685 + 0.638581i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −731.858 + 422.538i −0.763147 + 0.440603i
\(960\) 0 0
\(961\) −555.631 962.381i −0.578180 1.00144i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 226.365i 0.234575i
\(966\) 0 0
\(967\) −1473.44 −1.52373 −0.761863 0.647738i \(-0.775715\pi\)
−0.761863 + 0.647738i \(0.775715\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −168.684 + 97.3900i −0.173722 + 0.100299i −0.584340 0.811509i \(-0.698646\pi\)
0.410618 + 0.911808i \(0.365313\pi\)
\(972\) 0 0
\(973\) 433.439 750.739i 0.445467 0.771571i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −203.255 117.349i −0.208040 0.120112i 0.392360 0.919812i \(-0.371659\pi\)
−0.600400 + 0.799700i \(0.704992\pi\)
\(978\) 0 0
\(979\) −587.749 −0.600357
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 270.658 + 156.265i 0.275339 + 0.158967i 0.631312 0.775529i \(-0.282517\pi\)
−0.355972 + 0.934496i \(0.615850\pi\)
\(984\) 0 0
\(985\) 1276.96 1.29641
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 517.643i 0.523400i
\(990\) 0 0
\(991\) −327.808 567.781i −0.330786 0.572937i 0.651881 0.758322i \(-0.273980\pi\)
−0.982666 + 0.185384i \(0.940647\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −751.580 433.925i −0.755357 0.436106i
\(996\) 0 0
\(997\) 1365.95 1.37006 0.685029 0.728516i \(-0.259790\pi\)
0.685029 + 0.728516i \(0.259790\pi\)
\(998\) 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 2052.3.m.a.881.7 80
3.2 odd 2 684.3.m.a.653.36 yes 80
9.2 odd 6 2052.3.be.a.197.7 80
9.7 even 3 684.3.be.a.425.18 yes 80
19.11 even 3 2052.3.be.a.125.7 80
57.11 odd 6 684.3.be.a.581.18 yes 80
171.11 odd 6 inner 2052.3.m.a.1493.34 80
171.106 even 3 684.3.m.a.353.36 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.m.a.353.36 80 171.106 even 3
684.3.m.a.653.36 yes 80 3.2 odd 2
684.3.be.a.425.18 yes 80 9.7 even 3
684.3.be.a.581.18 yes 80 57.11 odd 6
2052.3.m.a.881.7 80 1.1 even 1 trivial
2052.3.m.a.1493.34 80 171.11 odd 6 inner
2052.3.be.a.125.7 80 19.11 even 3
2052.3.be.a.197.7 80 9.2 odd 6