Properties

Label 588.4.i.d.361.1
Level $588$
Weight $4$
Character 588.361
Analytic conductor $34.693$
Analytic rank $0$
Dimension $2$
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] = [588,4,Mod(361,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.4.i.d.373.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{3} +(9.00000 + 15.5885i) q^{5} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{3} +(9.00000 + 15.5885i) q^{5} +(-4.50000 - 7.79423i) q^{9} +(-18.0000 + 31.1769i) q^{11} -10.0000 q^{13} -54.0000 q^{15} +(-9.00000 + 15.5885i) q^{17} +(50.0000 + 86.6025i) q^{19} +(-36.0000 - 62.3538i) q^{23} +(-99.5000 + 172.339i) q^{25} +27.0000 q^{27} -234.000 q^{29} +(8.00000 - 13.8564i) q^{31} +(-54.0000 - 93.5307i) q^{33} +(113.000 + 195.722i) q^{37} +(15.0000 - 25.9808i) q^{39} +90.0000 q^{41} +452.000 q^{43} +(81.0000 - 140.296i) q^{45} +(-216.000 - 374.123i) q^{47} +(-27.0000 - 46.7654i) q^{51} +(-207.000 + 358.535i) q^{53} -648.000 q^{55} -300.000 q^{57} +(342.000 - 592.361i) q^{59} +(-211.000 - 365.463i) q^{61} +(-90.0000 - 155.885i) q^{65} +(-166.000 + 287.520i) q^{67} +216.000 q^{69} -360.000 q^{71} +(-13.0000 + 22.5167i) q^{73} +(-298.500 - 517.017i) q^{75} +(-256.000 - 443.405i) q^{79} +(-40.5000 + 70.1481i) q^{81} -1188.00 q^{83} -324.000 q^{85} +(351.000 - 607.950i) q^{87} +(315.000 + 545.596i) q^{89} +(24.0000 + 41.5692i) q^{93} +(-900.000 + 1558.85i) q^{95} -1054.00 q^{97} +324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 18 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 18 q^{5} - 9 q^{9} - 36 q^{11} - 20 q^{13} - 108 q^{15} - 18 q^{17} + 100 q^{19} - 72 q^{23} - 199 q^{25} + 54 q^{27} - 468 q^{29} + 16 q^{31} - 108 q^{33} + 226 q^{37} + 30 q^{39} + 180 q^{41} + 904 q^{43} + 162 q^{45} - 432 q^{47} - 54 q^{51} - 414 q^{53} - 1296 q^{55} - 600 q^{57} + 684 q^{59} - 422 q^{61} - 180 q^{65} - 332 q^{67} + 432 q^{69} - 720 q^{71} - 26 q^{73} - 597 q^{75} - 512 q^{79} - 81 q^{81} - 2376 q^{83} - 648 q^{85} + 702 q^{87} + 630 q^{89} + 48 q^{93} - 1800 q^{95} - 2108 q^{97} + 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 9.00000 + 15.5885i 0.804984 + 1.39427i 0.916302 + 0.400489i \(0.131160\pi\)
−0.111317 + 0.993785i \(0.535507\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −4.50000 7.79423i −0.166667 0.288675i
\(10\) 0 0
\(11\) −18.0000 + 31.1769i −0.493382 + 0.854563i −0.999971 0.00762479i \(-0.997573\pi\)
0.506589 + 0.862188i \(0.330906\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.213346 −0.106673 0.994294i \(-0.534020\pi\)
−0.106673 + 0.994294i \(0.534020\pi\)
\(14\) 0 0
\(15\) −54.0000 −0.929516
\(16\) 0 0
\(17\) −9.00000 + 15.5885i −0.128401 + 0.222397i −0.923057 0.384662i \(-0.874318\pi\)
0.794656 + 0.607060i \(0.207651\pi\)
\(18\) 0 0
\(19\) 50.0000 + 86.6025i 0.603726 + 1.04568i 0.992251 + 0.124246i \(0.0396511\pi\)
−0.388526 + 0.921438i \(0.627016\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −36.0000 62.3538i −0.326370 0.565290i 0.655418 0.755266i \(-0.272492\pi\)
−0.981789 + 0.189976i \(0.939159\pi\)
\(24\) 0 0
\(25\) −99.5000 + 172.339i −0.796000 + 1.37871i
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −234.000 −1.49837 −0.749185 0.662361i \(-0.769554\pi\)
−0.749185 + 0.662361i \(0.769554\pi\)
\(30\) 0 0
\(31\) 8.00000 13.8564i 0.0463498 0.0802801i −0.841920 0.539603i \(-0.818574\pi\)
0.888270 + 0.459323i \(0.151908\pi\)
\(32\) 0 0
\(33\) −54.0000 93.5307i −0.284854 0.493382i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 113.000 + 195.722i 0.502083 + 0.869634i 0.999997 + 0.00240737i \(0.000766290\pi\)
−0.497914 + 0.867227i \(0.665900\pi\)
\(38\) 0 0
\(39\) 15.0000 25.9808i 0.0615878 0.106673i
\(40\) 0 0
\(41\) 90.0000 0.342820 0.171410 0.985200i \(-0.445168\pi\)
0.171410 + 0.985200i \(0.445168\pi\)
\(42\) 0 0
\(43\) 452.000 1.60301 0.801504 0.597989i \(-0.204033\pi\)
0.801504 + 0.597989i \(0.204033\pi\)
\(44\) 0 0
\(45\) 81.0000 140.296i 0.268328 0.464758i
\(46\) 0 0
\(47\) −216.000 374.123i −0.670358 1.16109i −0.977803 0.209528i \(-0.932807\pi\)
0.307444 0.951566i \(-0.400526\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −27.0000 46.7654i −0.0741325 0.128401i
\(52\) 0 0
\(53\) −207.000 + 358.535i −0.536484 + 0.929217i 0.462606 + 0.886564i \(0.346914\pi\)
−0.999090 + 0.0426532i \(0.986419\pi\)
\(54\) 0 0
\(55\) −648.000 −1.58866
\(56\) 0 0
\(57\) −300.000 −0.697122
\(58\) 0 0
\(59\) 342.000 592.361i 0.754654 1.30710i −0.190892 0.981611i \(-0.561138\pi\)
0.945546 0.325489i \(-0.105529\pi\)
\(60\) 0 0
\(61\) −211.000 365.463i −0.442882 0.767094i 0.555020 0.831837i \(-0.312710\pi\)
−0.997902 + 0.0647433i \(0.979377\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −90.0000 155.885i −0.171740 0.297463i
\(66\) 0 0
\(67\) −166.000 + 287.520i −0.302688 + 0.524272i −0.976744 0.214409i \(-0.931218\pi\)
0.674056 + 0.738681i \(0.264551\pi\)
\(68\) 0 0
\(69\) 216.000 0.376860
\(70\) 0 0
\(71\) −360.000 −0.601748 −0.300874 0.953664i \(-0.597278\pi\)
−0.300874 + 0.953664i \(0.597278\pi\)
\(72\) 0 0
\(73\) −13.0000 + 22.5167i −0.0208429 + 0.0361010i −0.876259 0.481841i \(-0.839968\pi\)
0.855416 + 0.517942i \(0.173302\pi\)
\(74\) 0 0
\(75\) −298.500 517.017i −0.459571 0.796000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −256.000 443.405i −0.364585 0.631481i 0.624124 0.781325i \(-0.285456\pi\)
−0.988710 + 0.149845i \(0.952123\pi\)
\(80\) 0 0
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −1188.00 −1.57108 −0.785542 0.618809i \(-0.787616\pi\)
−0.785542 + 0.618809i \(0.787616\pi\)
\(84\) 0 0
\(85\) −324.000 −0.413444
\(86\) 0 0
\(87\) 351.000 607.950i 0.432542 0.749185i
\(88\) 0 0
\(89\) 315.000 + 545.596i 0.375168 + 0.649810i 0.990352 0.138574i \(-0.0442518\pi\)
−0.615184 + 0.788383i \(0.710918\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 24.0000 + 41.5692i 0.0267600 + 0.0463498i
\(94\) 0 0
\(95\) −900.000 + 1558.85i −0.971979 + 1.68352i
\(96\) 0 0
\(97\) −1054.00 −1.10327 −0.551637 0.834085i \(-0.685996\pi\)
−0.551637 + 0.834085i \(0.685996\pi\)
\(98\) 0 0
\(99\) 324.000 0.328921
\(100\) 0 0
\(101\) −279.000 + 483.242i −0.274867 + 0.476083i −0.970101 0.242700i \(-0.921967\pi\)
0.695235 + 0.718783i \(0.255300\pi\)
\(102\) 0 0
\(103\) −4.00000 6.92820i −0.00382652 0.00662773i 0.864106 0.503310i \(-0.167885\pi\)
−0.867932 + 0.496682i \(0.834551\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −882.000 1527.67i −0.796880 1.38024i −0.921639 0.388049i \(-0.873149\pi\)
0.124759 0.992187i \(-0.460184\pi\)
\(108\) 0 0
\(109\) −811.000 + 1404.69i −0.712658 + 1.23436i 0.251198 + 0.967936i \(0.419175\pi\)
−0.963856 + 0.266424i \(0.914158\pi\)
\(110\) 0 0
\(111\) −678.000 −0.579756
\(112\) 0 0
\(113\) −1134.00 −0.944051 −0.472025 0.881585i \(-0.656477\pi\)
−0.472025 + 0.881585i \(0.656477\pi\)
\(114\) 0 0
\(115\) 648.000 1122.37i 0.525446 0.910099i
\(116\) 0 0
\(117\) 45.0000 + 77.9423i 0.0355577 + 0.0615878i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 17.5000 + 30.3109i 0.0131480 + 0.0227730i
\(122\) 0 0
\(123\) −135.000 + 233.827i −0.0989637 + 0.171410i
\(124\) 0 0
\(125\) −1332.00 −0.953102
\(126\) 0 0
\(127\) −592.000 −0.413634 −0.206817 0.978380i \(-0.566310\pi\)
−0.206817 + 0.978380i \(0.566310\pi\)
\(128\) 0 0
\(129\) −678.000 + 1174.33i −0.462749 + 0.801504i
\(130\) 0 0
\(131\) 954.000 + 1652.38i 0.636270 + 1.10205i 0.986245 + 0.165293i \(0.0528569\pi\)
−0.349975 + 0.936759i \(0.613810\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 243.000 + 420.888i 0.154919 + 0.268328i
\(136\) 0 0
\(137\) −477.000 + 826.188i −0.297466 + 0.515226i −0.975556 0.219753i \(-0.929475\pi\)
0.678089 + 0.734979i \(0.262808\pi\)
\(138\) 0 0
\(139\) 2564.00 1.56457 0.782286 0.622919i \(-0.214053\pi\)
0.782286 + 0.622919i \(0.214053\pi\)
\(140\) 0 0
\(141\) 1296.00 0.774063
\(142\) 0 0
\(143\) 180.000 311.769i 0.105261 0.182318i
\(144\) 0 0
\(145\) −2106.00 3647.70i −1.20616 2.08914i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 369.000 + 639.127i 0.202884 + 0.351405i 0.949456 0.313899i \(-0.101635\pi\)
−0.746573 + 0.665304i \(0.768302\pi\)
\(150\) 0 0
\(151\) 1220.00 2113.10i 0.657498 1.13882i −0.323763 0.946138i \(-0.604948\pi\)
0.981261 0.192682i \(-0.0617185\pi\)
\(152\) 0 0
\(153\) 162.000 0.0856008
\(154\) 0 0
\(155\) 288.000 0.149243
\(156\) 0 0
\(157\) 1277.00 2211.83i 0.649145 1.12435i −0.334183 0.942508i \(-0.608460\pi\)
0.983328 0.181843i \(-0.0582063\pi\)
\(158\) 0 0
\(159\) −621.000 1075.60i −0.309739 0.536484i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 410.000 + 710.141i 0.197016 + 0.341242i 0.947560 0.319579i \(-0.103541\pi\)
−0.750543 + 0.660821i \(0.770208\pi\)
\(164\) 0 0
\(165\) 972.000 1683.55i 0.458607 0.794330i
\(166\) 0 0
\(167\) 1944.00 0.900786 0.450393 0.892830i \(-0.351284\pi\)
0.450393 + 0.892830i \(0.351284\pi\)
\(168\) 0 0
\(169\) −2097.00 −0.954483
\(170\) 0 0
\(171\) 450.000 779.423i 0.201242 0.348561i
\(172\) 0 0
\(173\) 621.000 + 1075.60i 0.272912 + 0.472697i 0.969606 0.244671i \(-0.0786799\pi\)
−0.696694 + 0.717368i \(0.745347\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1026.00 + 1777.08i 0.435700 + 0.754654i
\(178\) 0 0
\(179\) −558.000 + 966.484i −0.232999 + 0.403567i −0.958689 0.284455i \(-0.908187\pi\)
0.725690 + 0.688022i \(0.241521\pi\)
\(180\) 0 0
\(181\) 1070.00 0.439406 0.219703 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(182\) 0 0
\(183\) 1266.00 0.511396
\(184\) 0 0
\(185\) −2034.00 + 3522.99i −0.808339 + 1.40008i
\(186\) 0 0
\(187\) −324.000 561.184i −0.126702 0.219454i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 288.000 + 498.831i 0.109104 + 0.188974i 0.915408 0.402528i \(-0.131868\pi\)
−0.806303 + 0.591502i \(0.798535\pi\)
\(192\) 0 0
\(193\) 671.000 1162.21i 0.250257 0.433458i −0.713339 0.700819i \(-0.752818\pi\)
0.963597 + 0.267361i \(0.0861515\pi\)
\(194\) 0 0
\(195\) 540.000 0.198309
\(196\) 0 0
\(197\) 1422.00 0.514281 0.257140 0.966374i \(-0.417220\pi\)
0.257140 + 0.966374i \(0.417220\pi\)
\(198\) 0 0
\(199\) −436.000 + 755.174i −0.155313 + 0.269009i −0.933173 0.359428i \(-0.882972\pi\)
0.777860 + 0.628437i \(0.216305\pi\)
\(200\) 0 0
\(201\) −498.000 862.561i −0.174757 0.302688i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 810.000 + 1402.96i 0.275965 + 0.477986i
\(206\) 0 0
\(207\) −324.000 + 561.184i −0.108790 + 0.188430i
\(208\) 0 0
\(209\) −3600.00 −1.19147
\(210\) 0 0
\(211\) 1340.00 0.437201 0.218600 0.975814i \(-0.429851\pi\)
0.218600 + 0.975814i \(0.429851\pi\)
\(212\) 0 0
\(213\) 540.000 935.307i 0.173710 0.300874i
\(214\) 0 0
\(215\) 4068.00 + 7045.98i 1.29040 + 2.23503i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −39.0000 67.5500i −0.0120337 0.0208429i
\(220\) 0 0
\(221\) 90.0000 155.885i 0.0273939 0.0474477i
\(222\) 0 0
\(223\) 4880.00 1.46542 0.732711 0.680540i \(-0.238255\pi\)
0.732711 + 0.680540i \(0.238255\pi\)
\(224\) 0 0
\(225\) 1791.00 0.530667
\(226\) 0 0
\(227\) −1350.00 + 2338.27i −0.394725 + 0.683684i −0.993066 0.117557i \(-0.962494\pi\)
0.598341 + 0.801242i \(0.295827\pi\)
\(228\) 0 0
\(229\) −127.000 219.970i −0.0366480 0.0634762i 0.847120 0.531402i \(-0.178335\pi\)
−0.883768 + 0.467926i \(0.845001\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2205.00 3819.17i −0.619976 1.07383i −0.989490 0.144604i \(-0.953809\pi\)
0.369514 0.929225i \(-0.379524\pi\)
\(234\) 0 0
\(235\) 3888.00 6734.21i 1.07926 1.86933i
\(236\) 0 0
\(237\) 1536.00 0.420987
\(238\) 0 0
\(239\) −3888.00 −1.05228 −0.526138 0.850399i \(-0.676360\pi\)
−0.526138 + 0.850399i \(0.676360\pi\)
\(240\) 0 0
\(241\) −2569.00 + 4449.64i −0.686655 + 1.18932i 0.286259 + 0.958152i \(0.407588\pi\)
−0.972914 + 0.231169i \(0.925745\pi\)
\(242\) 0 0
\(243\) −121.500 210.444i −0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −500.000 866.025i −0.128803 0.223093i
\(248\) 0 0
\(249\) 1782.00 3086.51i 0.453533 0.785542i
\(250\) 0 0
\(251\) 4788.00 1.20405 0.602024 0.798478i \(-0.294361\pi\)
0.602024 + 0.798478i \(0.294361\pi\)
\(252\) 0 0
\(253\) 2592.00 0.644101
\(254\) 0 0
\(255\) 486.000 841.777i 0.119351 0.206722i
\(256\) 0 0
\(257\) 2943.00 + 5097.43i 0.714316 + 1.23723i 0.963223 + 0.268705i \(0.0865955\pi\)
−0.248906 + 0.968528i \(0.580071\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1053.00 + 1823.85i 0.249728 + 0.432542i
\(262\) 0 0
\(263\) −1116.00 + 1932.97i −0.261656 + 0.453201i −0.966682 0.255980i \(-0.917602\pi\)
0.705026 + 0.709181i \(0.250935\pi\)
\(264\) 0 0
\(265\) −7452.00 −1.72744
\(266\) 0 0
\(267\) −1890.00 −0.433206
\(268\) 0 0
\(269\) 333.000 576.773i 0.0754772 0.130730i −0.825816 0.563939i \(-0.809285\pi\)
0.901294 + 0.433209i \(0.142619\pi\)
\(270\) 0 0
\(271\) 2768.00 + 4794.32i 0.620458 + 1.07466i 0.989401 + 0.145212i \(0.0463865\pi\)
−0.368943 + 0.929452i \(0.620280\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3582.00 6204.21i −0.785464 1.36046i
\(276\) 0 0
\(277\) −1063.00 + 1841.17i −0.230576 + 0.399369i −0.957978 0.286843i \(-0.907394\pi\)
0.727402 + 0.686212i \(0.240728\pi\)
\(278\) 0 0
\(279\) −144.000 −0.0308998
\(280\) 0 0
\(281\) −2934.00 −0.622875 −0.311437 0.950267i \(-0.600810\pi\)
−0.311437 + 0.950267i \(0.600810\pi\)
\(282\) 0 0
\(283\) −1018.00 + 1763.23i −0.213830 + 0.370364i −0.952910 0.303253i \(-0.901927\pi\)
0.739080 + 0.673617i \(0.235260\pi\)
\(284\) 0 0
\(285\) −2700.00 4676.54i −0.561173 0.971979i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2294.50 + 3974.19i 0.467026 + 0.808913i
\(290\) 0 0
\(291\) 1581.00 2738.37i 0.318487 0.551637i
\(292\) 0 0
\(293\) 2286.00 0.455800 0.227900 0.973684i \(-0.426814\pi\)
0.227900 + 0.973684i \(0.426814\pi\)
\(294\) 0 0
\(295\) 12312.0 2.42994
\(296\) 0 0
\(297\) −486.000 + 841.777i −0.0949514 + 0.164461i
\(298\) 0 0
\(299\) 360.000 + 623.538i 0.0696299 + 0.120603i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −837.000 1449.73i −0.158694 0.274867i
\(304\) 0 0
\(305\) 3798.00 6578.33i 0.713026 1.23500i
\(306\) 0 0
\(307\) 1244.00 0.231267 0.115633 0.993292i \(-0.463110\pi\)
0.115633 + 0.993292i \(0.463110\pi\)
\(308\) 0 0
\(309\) 24.0000 0.00441849
\(310\) 0 0
\(311\) −612.000 + 1060.02i −0.111586 + 0.193273i −0.916410 0.400241i \(-0.868926\pi\)
0.804824 + 0.593514i \(0.202260\pi\)
\(312\) 0 0
\(313\) −949.000 1643.72i −0.171376 0.296832i 0.767525 0.641019i \(-0.221488\pi\)
−0.938901 + 0.344187i \(0.888155\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4581.00 + 7934.52i 0.811655 + 1.40583i 0.911705 + 0.410845i \(0.134766\pi\)
−0.100050 + 0.994982i \(0.531900\pi\)
\(318\) 0 0
\(319\) 4212.00 7295.40i 0.739269 1.28045i
\(320\) 0 0
\(321\) 5292.00 0.920158
\(322\) 0 0
\(323\) −1800.00 −0.310076
\(324\) 0 0
\(325\) 995.000 1723.39i 0.169824 0.294143i
\(326\) 0 0
\(327\) −2433.00 4214.08i −0.411453 0.712658i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2174.00 + 3765.48i 0.361009 + 0.625285i 0.988127 0.153639i \(-0.0490992\pi\)
−0.627119 + 0.778924i \(0.715766\pi\)
\(332\) 0 0
\(333\) 1017.00 1761.50i 0.167361 0.289878i
\(334\) 0 0
\(335\) −5976.00 −0.974638
\(336\) 0 0
\(337\) 7154.00 1.15639 0.578195 0.815899i \(-0.303757\pi\)
0.578195 + 0.815899i \(0.303757\pi\)
\(338\) 0 0
\(339\) 1701.00 2946.22i 0.272524 0.472025i
\(340\) 0 0
\(341\) 288.000 + 498.831i 0.0457363 + 0.0792176i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1944.00 + 3367.11i 0.303366 + 0.525446i
\(346\) 0 0
\(347\) 918.000 1590.02i 0.142020 0.245985i −0.786237 0.617925i \(-0.787974\pi\)
0.928257 + 0.371939i \(0.121307\pi\)
\(348\) 0 0
\(349\) 5894.00 0.904007 0.452004 0.892016i \(-0.350709\pi\)
0.452004 + 0.892016i \(0.350709\pi\)
\(350\) 0 0
\(351\) −270.000 −0.0410585
\(352\) 0 0
\(353\) −5553.00 + 9618.08i −0.837270 + 1.45019i 0.0548984 + 0.998492i \(0.482516\pi\)
−0.892169 + 0.451703i \(0.850817\pi\)
\(354\) 0 0
\(355\) −3240.00 5611.84i −0.484398 0.839002i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6588.00 11410.8i −0.968527 1.67754i −0.699823 0.714316i \(-0.746738\pi\)
−0.268704 0.963223i \(-0.586595\pi\)
\(360\) 0 0
\(361\) −1570.50 + 2720.19i −0.228969 + 0.396586i
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) −468.000 −0.0671130
\(366\) 0 0
\(367\) 3056.00 5293.15i 0.434665 0.752861i −0.562604 0.826727i \(-0.690200\pi\)
0.997268 + 0.0738656i \(0.0235336\pi\)
\(368\) 0 0
\(369\) −405.000 701.481i −0.0571367 0.0989637i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6809.00 + 11793.5i 0.945192 + 1.63712i 0.755366 + 0.655303i \(0.227459\pi\)
0.189826 + 0.981818i \(0.439208\pi\)
\(374\) 0 0
\(375\) 1998.00 3460.64i 0.275137 0.476551i
\(376\) 0 0
\(377\) 2340.00 0.319671
\(378\) 0 0
\(379\) 692.000 0.0937880 0.0468940 0.998900i \(-0.485068\pi\)
0.0468940 + 0.998900i \(0.485068\pi\)
\(380\) 0 0
\(381\) 888.000 1538.06i 0.119406 0.206817i
\(382\) 0 0
\(383\) 4032.00 + 6983.63i 0.537926 + 0.931715i 0.999016 + 0.0443613i \(0.0141253\pi\)
−0.461090 + 0.887353i \(0.652541\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2034.00 3522.99i −0.267168 0.462749i
\(388\) 0 0
\(389\) −6327.00 + 10958.7i −0.824657 + 1.42835i 0.0775239 + 0.996990i \(0.475299\pi\)
−0.902181 + 0.431358i \(0.858035\pi\)
\(390\) 0 0
\(391\) 1296.00 0.167625
\(392\) 0 0
\(393\) −5724.00 −0.734701
\(394\) 0 0
\(395\) 4608.00 7981.29i 0.586971 1.01666i
\(396\) 0 0
\(397\) 53.0000 + 91.7987i 0.00670024 + 0.0116051i 0.869356 0.494186i \(-0.164534\pi\)
−0.862656 + 0.505791i \(0.831201\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2007.00 + 3476.23i 0.249937 + 0.432904i 0.963508 0.267679i \(-0.0862567\pi\)
−0.713571 + 0.700583i \(0.752923\pi\)
\(402\) 0 0
\(403\) −80.0000 + 138.564i −0.00988855 + 0.0171275i
\(404\) 0 0
\(405\) −1458.00 −0.178885
\(406\) 0 0
\(407\) −8136.00 −0.990876
\(408\) 0 0
\(409\) −1957.00 + 3389.62i −0.236595 + 0.409795i −0.959735 0.280907i \(-0.909365\pi\)
0.723140 + 0.690702i \(0.242698\pi\)
\(410\) 0 0
\(411\) −1431.00 2478.56i −0.171742 0.297466i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −10692.0 18519.1i −1.26470 2.19052i
\(416\) 0 0
\(417\) −3846.00 + 6661.47i −0.451653 + 0.782286i
\(418\) 0 0
\(419\) 4428.00 0.516282 0.258141 0.966107i \(-0.416890\pi\)
0.258141 + 0.966107i \(0.416890\pi\)
\(420\) 0 0
\(421\) −15490.0 −1.79320 −0.896599 0.442843i \(-0.853970\pi\)
−0.896599 + 0.442843i \(0.853970\pi\)
\(422\) 0 0
\(423\) −1944.00 + 3367.11i −0.223453 + 0.387032i
\(424\) 0 0
\(425\) −1791.00 3102.10i −0.204415 0.354057i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 540.000 + 935.307i 0.0607726 + 0.105261i
\(430\) 0 0
\(431\) −3384.00 + 5861.26i −0.378194 + 0.655051i −0.990800 0.135338i \(-0.956788\pi\)
0.612606 + 0.790389i \(0.290121\pi\)
\(432\) 0 0
\(433\) 1298.00 0.144060 0.0720299 0.997402i \(-0.477052\pi\)
0.0720299 + 0.997402i \(0.477052\pi\)
\(434\) 0 0
\(435\) 12636.0 1.39276
\(436\) 0 0
\(437\) 3600.00 6235.38i 0.394076 0.682560i
\(438\) 0 0
\(439\) 1124.00 + 1946.83i 0.122200 + 0.211656i 0.920635 0.390425i \(-0.127672\pi\)
−0.798435 + 0.602081i \(0.794339\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4806.00 + 8324.24i 0.515440 + 0.892768i 0.999839 + 0.0179214i \(0.00570486\pi\)
−0.484399 + 0.874847i \(0.660962\pi\)
\(444\) 0 0
\(445\) −5670.00 + 9820.73i −0.604008 + 1.04617i
\(446\) 0 0
\(447\) −2214.00 −0.234270
\(448\) 0 0
\(449\) 162.000 0.0170273 0.00851364 0.999964i \(-0.497290\pi\)
0.00851364 + 0.999964i \(0.497290\pi\)
\(450\) 0 0
\(451\) −1620.00 + 2805.92i −0.169142 + 0.292962i
\(452\) 0 0
\(453\) 3660.00 + 6339.31i 0.379607 + 0.657498i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −685.000 1186.45i −0.0701159 0.121444i 0.828836 0.559492i \(-0.189004\pi\)
−0.898952 + 0.438047i \(0.855670\pi\)
\(458\) 0 0
\(459\) −243.000 + 420.888i −0.0247108 + 0.0428004i
\(460\) 0 0
\(461\) −15354.0 −1.55121 −0.775604 0.631220i \(-0.782555\pi\)
−0.775604 + 0.631220i \(0.782555\pi\)
\(462\) 0 0
\(463\) −13024.0 −1.30729 −0.653646 0.756800i \(-0.726762\pi\)
−0.653646 + 0.756800i \(0.726762\pi\)
\(464\) 0 0
\(465\) −432.000 + 748.246i −0.0430828 + 0.0746217i
\(466\) 0 0
\(467\) 7218.00 + 12501.9i 0.715223 + 1.23880i 0.962873 + 0.269953i \(0.0870083\pi\)
−0.247650 + 0.968849i \(0.579658\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3831.00 + 6635.49i 0.374784 + 0.649145i
\(472\) 0 0
\(473\) −8136.00 + 14092.0i −0.790896 + 1.36987i
\(474\) 0 0
\(475\) −19900.0 −1.92226
\(476\) 0 0
\(477\) 3726.00 0.357656
\(478\) 0 0
\(479\) −6048.00 + 10475.4i −0.576911 + 0.999238i 0.418921 + 0.908023i \(0.362409\pi\)
−0.995831 + 0.0912156i \(0.970925\pi\)
\(480\) 0 0
\(481\) −1130.00 1957.22i −0.107118 0.185533i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9486.00 16430.2i −0.888118 1.53826i
\(486\) 0 0
\(487\) −3028.00 + 5244.65i −0.281749 + 0.488004i −0.971816 0.235742i \(-0.924248\pi\)
0.690067 + 0.723746i \(0.257581\pi\)
\(488\) 0 0
\(489\) −2460.00 −0.227495
\(490\) 0 0
\(491\) 7524.00 0.691555 0.345777 0.938317i \(-0.387615\pi\)
0.345777 + 0.938317i \(0.387615\pi\)
\(492\) 0 0
\(493\) 2106.00 3647.70i 0.192392 0.333233i
\(494\) 0 0
\(495\) 2916.00 + 5050.66i 0.264777 + 0.458607i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2638.00 4569.15i −0.236660 0.409906i 0.723094 0.690749i \(-0.242719\pi\)
−0.959754 + 0.280843i \(0.909386\pi\)
\(500\) 0 0
\(501\) −2916.00 + 5050.66i −0.260034 + 0.450393i
\(502\) 0 0
\(503\) 4968.00 0.440382 0.220191 0.975457i \(-0.429332\pi\)
0.220191 + 0.975457i \(0.429332\pi\)
\(504\) 0 0
\(505\) −10044.0 −0.885054
\(506\) 0 0
\(507\) 3145.50 5448.17i 0.275536 0.477242i
\(508\) 0 0
\(509\) −5499.00 9524.55i −0.478858 0.829407i 0.520848 0.853650i \(-0.325616\pi\)
−0.999706 + 0.0242426i \(0.992283\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1350.00 + 2338.27i 0.116187 + 0.201242i
\(514\) 0 0
\(515\) 72.0000 124.708i 0.00616058 0.0106704i
\(516\) 0 0
\(517\) 15552.0 1.32297
\(518\) 0 0
\(519\) −3726.00 −0.315131
\(520\) 0 0
\(521\) 4419.00 7653.93i 0.371593 0.643618i −0.618218 0.786007i \(-0.712145\pi\)
0.989811 + 0.142389i \(0.0454784\pi\)
\(522\) 0 0
\(523\) −11218.0 19430.1i −0.937914 1.62451i −0.769354 0.638823i \(-0.779422\pi\)
−0.168560 0.985691i \(-0.553912\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 144.000 + 249.415i 0.0119027 + 0.0206161i
\(528\) 0 0
\(529\) 3491.50 6047.46i 0.286965 0.497038i
\(530\) 0 0
\(531\) −6156.00 −0.503103
\(532\) 0 0
\(533\) −900.000 −0.0731395
\(534\) 0 0
\(535\) 15876.0 27498.0i 1.28295 2.22214i
\(536\) 0 0
\(537\) −1674.00 2899.45i −0.134522 0.232999i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2381.00 + 4124.01i 0.189218 + 0.327736i 0.944990 0.327100i \(-0.106071\pi\)
−0.755771 + 0.654835i \(0.772738\pi\)
\(542\) 0 0
\(543\) −1605.00 + 2779.94i −0.126846 + 0.219703i
\(544\) 0 0
\(545\) −29196.0 −2.29471
\(546\) 0 0
\(547\) −6004.00 −0.469310 −0.234655 0.972079i \(-0.575396\pi\)
−0.234655 + 0.972079i \(0.575396\pi\)
\(548\) 0 0
\(549\) −1899.00 + 3289.16i −0.147627 + 0.255698i
\(550\) 0 0
\(551\) −11700.0 20265.0i −0.904604 1.56682i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −6102.00 10569.0i −0.466695 0.808339i
\(556\) 0 0
\(557\) 2637.00 4567.42i 0.200598 0.347447i −0.748123 0.663560i \(-0.769045\pi\)
0.948721 + 0.316114i \(0.102378\pi\)
\(558\) 0 0
\(559\) −4520.00 −0.341996
\(560\) 0 0
\(561\) 1944.00 0.146303
\(562\) 0 0
\(563\) 6210.00 10756.0i 0.464867 0.805174i −0.534328 0.845277i \(-0.679435\pi\)
0.999196 + 0.0401033i \(0.0127687\pi\)
\(564\) 0 0
\(565\) −10206.0 17677.3i −0.759946 1.31627i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10683.0 + 18503.5i 0.787091 + 1.36328i 0.927742 + 0.373222i \(0.121747\pi\)
−0.140651 + 0.990059i \(0.544920\pi\)
\(570\) 0 0
\(571\) −10570.0 + 18307.8i −0.774677 + 1.34178i 0.160298 + 0.987069i \(0.448754\pi\)
−0.934976 + 0.354712i \(0.884579\pi\)
\(572\) 0 0
\(573\) −1728.00 −0.125983
\(574\) 0 0
\(575\) 14328.0 1.03916
\(576\) 0 0
\(577\) −1633.00 + 2828.44i −0.117821 + 0.204072i −0.918904 0.394482i \(-0.870924\pi\)
0.801083 + 0.598553i \(0.204258\pi\)
\(578\) 0 0
\(579\) 2013.00 + 3486.62i 0.144486 + 0.250257i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −7452.00 12907.2i −0.529383 0.916918i
\(584\) 0 0
\(585\) −810.000 + 1402.96i −0.0572468 + 0.0991544i
\(586\) 0 0
\(587\) 17028.0 1.19731 0.598655 0.801007i \(-0.295702\pi\)
0.598655 + 0.801007i \(0.295702\pi\)
\(588\) 0 0
\(589\) 1600.00 0.111930
\(590\) 0 0
\(591\) −2133.00 + 3694.46i −0.148460 + 0.257140i
\(592\) 0 0
\(593\) −4761.00 8246.29i −0.329698 0.571053i 0.652754 0.757570i \(-0.273614\pi\)
−0.982452 + 0.186517i \(0.940280\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1308.00 2265.52i −0.0896698 0.155313i
\(598\) 0 0
\(599\) 5148.00 8916.60i 0.351155 0.608218i −0.635297 0.772268i \(-0.719123\pi\)
0.986452 + 0.164050i \(0.0524558\pi\)
\(600\) 0 0
\(601\) −3382.00 −0.229542 −0.114771 0.993392i \(-0.536613\pi\)
−0.114771 + 0.993392i \(0.536613\pi\)
\(602\) 0 0
\(603\) 2988.00 0.201792
\(604\) 0 0
\(605\) −315.000 + 545.596i −0.0211679 + 0.0366639i
\(606\) 0 0
\(607\) 10328.0 + 17888.6i 0.690611 + 1.19617i 0.971638 + 0.236473i \(0.0759914\pi\)
−0.281028 + 0.959700i \(0.590675\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2160.00 + 3741.23i 0.143018 + 0.247715i
\(612\) 0 0
\(613\) 11057.0 19151.3i 0.728529 1.26185i −0.228976 0.973432i \(-0.573538\pi\)
0.957505 0.288417i \(-0.0931288\pi\)
\(614\) 0 0
\(615\) −4860.00 −0.318657
\(616\) 0 0
\(617\) 19962.0 1.30250 0.651248 0.758865i \(-0.274246\pi\)
0.651248 + 0.758865i \(0.274246\pi\)
\(618\) 0 0
\(619\) 302.000 523.079i 0.0196097 0.0339650i −0.856054 0.516886i \(-0.827091\pi\)
0.875664 + 0.482921i \(0.160424\pi\)
\(620\) 0 0
\(621\) −972.000 1683.55i −0.0628100 0.108790i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 449.500 + 778.557i 0.0287680 + 0.0498276i
\(626\) 0 0
\(627\) 5400.00 9353.07i 0.343948 0.595735i
\(628\) 0 0
\(629\) −4068.00 −0.257872
\(630\) 0 0
\(631\) 152.000 0.00958958 0.00479479 0.999989i \(-0.498474\pi\)
0.00479479 + 0.999989i \(0.498474\pi\)
\(632\) 0 0
\(633\) −2010.00 + 3481.42i −0.126209 + 0.218600i
\(634\) 0 0
\(635\) −5328.00 9228.37i −0.332969 0.576719i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 1620.00 + 2805.92i 0.100291 + 0.173710i
\(640\) 0 0
\(641\) −2097.00 + 3632.11i −0.129215 + 0.223806i −0.923372 0.383905i \(-0.874579\pi\)
0.794158 + 0.607711i \(0.207912\pi\)
\(642\) 0 0
\(643\) −7252.00 −0.444776 −0.222388 0.974958i \(-0.571385\pi\)
−0.222388 + 0.974958i \(0.571385\pi\)
\(644\) 0 0
\(645\) −24408.0 −1.49002
\(646\) 0 0
\(647\) 3348.00 5798.91i 0.203437 0.352362i −0.746197 0.665725i \(-0.768122\pi\)
0.949633 + 0.313363i \(0.101456\pi\)
\(648\) 0 0
\(649\) 12312.0 + 21325.0i 0.744666 + 1.28980i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14211.0 24614.2i −0.851638 1.47508i −0.879730 0.475474i \(-0.842276\pi\)
0.0280919 0.999605i \(-0.491057\pi\)
\(654\) 0 0
\(655\) −17172.0 + 29742.8i −1.02437 + 1.77427i
\(656\) 0 0
\(657\) 234.000 0.0138953
\(658\) 0 0
\(659\) −19908.0 −1.17679 −0.588396 0.808573i \(-0.700240\pi\)
−0.588396 + 0.808573i \(0.700240\pi\)
\(660\) 0 0
\(661\) −7159.00 + 12399.8i −0.421260 + 0.729644i −0.996063 0.0886482i \(-0.971745\pi\)
0.574803 + 0.818292i \(0.305079\pi\)
\(662\) 0 0
\(663\) 270.000 + 467.654i 0.0158159 + 0.0273939i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8424.00 + 14590.8i 0.489023 + 0.847013i
\(668\) 0 0
\(669\) −7320.00 + 12678.6i −0.423031 + 0.732711i
\(670\) 0 0
\(671\) 15192.0 0.874040
\(672\) 0 0
\(673\) 30050.0 1.72116 0.860581 0.509313i \(-0.170101\pi\)
0.860581 + 0.509313i \(0.170101\pi\)
\(674\) 0 0
\(675\) −2686.50 + 4653.15i −0.153190 + 0.265333i
\(676\) 0 0
\(677\) −11079.0 19189.4i −0.628952 1.08938i −0.987762 0.155966i \(-0.950151\pi\)
0.358810 0.933411i \(-0.383183\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −4050.00 7014.81i −0.227895 0.394725i
\(682\) 0 0
\(683\) 1566.00 2712.39i 0.0877325 0.151957i −0.818820 0.574051i \(-0.805371\pi\)
0.906552 + 0.422093i \(0.138705\pi\)
\(684\) 0 0
\(685\) −17172.0 −0.957822
\(686\) 0 0
\(687\) 762.000 0.0423175
\(688\) 0 0
\(689\) 2070.00 3585.35i 0.114457 0.198245i
\(690\) 0 0
\(691\) 10466.0 + 18127.6i 0.576187 + 0.997986i 0.995912 + 0.0903340i \(0.0287934\pi\)
−0.419724 + 0.907652i \(0.637873\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23076.0 + 39968.8i 1.25946 + 2.18144i
\(696\) 0 0
\(697\) −810.000 + 1402.96i −0.0440186 + 0.0762424i
\(698\) 0 0
\(699\) 13230.0 0.715886
\(700\) 0 0
\(701\) −21834.0 −1.17640 −0.588202 0.808714i \(-0.700164\pi\)
−0.588202 + 0.808714i \(0.700164\pi\)
\(702\) 0 0
\(703\) −11300.0 + 19572.2i −0.606241 + 1.05004i
\(704\) 0 0
\(705\) 11664.0 + 20202.6i 0.623109 + 1.07926i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6223.00 10778.6i −0.329633 0.570941i 0.652806 0.757525i \(-0.273592\pi\)
−0.982439 + 0.186584i \(0.940258\pi\)
\(710\) 0 0
\(711\) −2304.00 + 3990.65i −0.121528 + 0.210494i
\(712\) 0 0
\(713\) −1152.00 −0.0605088
\(714\) 0 0
\(715\) 6480.00 0.338935
\(716\) 0 0
\(717\) 5832.00 10101.3i 0.303766 0.526138i
\(718\) 0 0
\(719\) 6264.00 + 10849.6i 0.324907 + 0.562755i 0.981493 0.191496i \(-0.0613338\pi\)
−0.656587 + 0.754250i \(0.728000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −7707.00 13348.9i −0.396440 0.686655i
\(724\) 0 0
\(725\) 23283.0 40327.3i 1.19270 2.06582i
\(726\) 0 0
\(727\) 11576.0 0.590550 0.295275 0.955412i \(-0.404589\pi\)
0.295275 + 0.955412i \(0.404589\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −4068.00 + 7045.98i −0.205828 + 0.356505i
\(732\) 0 0
\(733\) 14669.0 + 25407.5i 0.739170 + 1.28028i 0.952869 + 0.303381i \(0.0981155\pi\)
−0.213699 + 0.976900i \(0.568551\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5976.00 10350.7i −0.298682 0.517333i
\(738\) 0 0
\(739\) −1270.00 + 2199.70i −0.0632175 + 0.109496i −0.895902 0.444252i \(-0.853470\pi\)
0.832684 + 0.553748i \(0.186803\pi\)
\(740\) 0 0
\(741\) 3000.00 0.148728
\(742\) 0 0
\(743\) −18792.0 −0.927876 −0.463938 0.885868i \(-0.653564\pi\)
−0.463938 + 0.885868i \(0.653564\pi\)
\(744\) 0 0
\(745\) −6642.00 + 11504.3i −0.326636 + 0.565751i
\(746\) 0 0
\(747\) 5346.00 + 9259.54i 0.261847 + 0.453533i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2416.00 4184.63i −0.117392 0.203328i 0.801342 0.598207i \(-0.204120\pi\)
−0.918733 + 0.394879i \(0.870787\pi\)
\(752\) 0 0
\(753\) −7182.00 + 12439.6i −0.347579 + 0.602024i
\(754\) 0 0
\(755\) 43920.0 2.11710
\(756\) 0 0
\(757\) −20818.0 −0.999529 −0.499764 0.866161i \(-0.666580\pi\)
−0.499764 + 0.866161i \(0.666580\pi\)
\(758\) 0 0
\(759\) −3888.00 + 6734.21i −0.185936 + 0.322051i
\(760\) 0 0
\(761\) −6021.00 10428.7i −0.286808 0.496766i 0.686238 0.727377i \(-0.259261\pi\)
−0.973046 + 0.230611i \(0.925928\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1458.00 + 2525.33i 0.0689073 + 0.119351i
\(766\) 0 0
\(767\) −3420.00 + 5923.61i −0.161003 + 0.278865i
\(768\) 0 0
\(769\) 13058.0 0.612332 0.306166 0.951978i \(-0.400954\pi\)
0.306166 + 0.951978i \(0.400954\pi\)
\(770\) 0 0
\(771\) −17658.0 −0.824821
\(772\) 0 0
\(773\) 5913.00 10241.6i 0.275130 0.476540i −0.695038 0.718973i \(-0.744612\pi\)
0.970168 + 0.242434i \(0.0779456\pi\)
\(774\) 0 0
\(775\) 1592.00 + 2757.42i 0.0737888 + 0.127806i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4500.00 + 7794.23i 0.206969 + 0.358482i
\(780\) 0 0
\(781\) 6480.00 11223.7i 0.296892 0.514232i
\(782\) 0 0
\(783\) −6318.00 −0.288361
\(784\) 0 0
\(785\) 45972.0 2.09021
\(786\) 0 0
\(787\) −5998.00 + 10388.8i −0.271672 + 0.470549i −0.969290 0.245920i \(-0.920910\pi\)
0.697618 + 0.716470i \(0.254243\pi\)
\(788\) 0 0
\(789\) −3348.00 5798.91i −0.151067 0.261656i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2110.00 + 3654.63i 0.0944871 + 0.163657i
\(794\) 0 0
\(795\) 11178.0 19360.9i 0.498670 0.863722i
\(796\) 0 0
\(797\) 6966.00 0.309596 0.154798 0.987946i \(-0.450527\pi\)
0.154798 + 0.987946i \(0.450527\pi\)
\(798\) 0 0
\(799\) 7776.00 0.344299
\(800\) 0 0
\(801\) 2835.00 4910.36i 0.125056 0.216603i
\(802\) 0 0
\(803\) −468.000 810.600i −0.0205671 0.0356232i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 999.000 + 1730.32i 0.0435768 + 0.0754772i
\(808\) 0 0
\(809\) 20403.0 35339.0i 0.886689 1.53579i 0.0429232 0.999078i \(-0.486333\pi\)
0.843766 0.536712i \(-0.180334\pi\)
\(810\) 0 0
\(811\) −17980.0 −0.778500 −0.389250 0.921132i \(-0.627266\pi\)
−0.389250 + 0.921132i \(0.627266\pi\)
\(812\) 0 0
\(813\) −16608.0 −0.716443
\(814\) 0 0
\(815\) −7380.00 + 12782.5i −0.317190 + 0.549390i
\(816\) 0 0
\(817\) 22600.0 + 39144.3i 0.967777 + 1.67624i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6417.00 + 11114.6i 0.272783 + 0.472474i 0.969573 0.244801i \(-0.0787226\pi\)
−0.696790 + 0.717275i \(0.745389\pi\)
\(822\) 0 0
\(823\) 18932.0 32791.2i 0.801857 1.38886i −0.116536 0.993186i \(-0.537179\pi\)
0.918393 0.395670i \(-0.129488\pi\)
\(824\) 0 0
\(825\) 21492.0 0.906976
\(826\) 0 0
\(827\) 42516.0 1.78770 0.893849 0.448368i \(-0.147995\pi\)
0.893849 + 0.448368i \(0.147995\pi\)
\(828\) 0 0
\(829\) −22819.0 + 39523.7i −0.956015 + 1.65587i −0.223988 + 0.974592i \(0.571908\pi\)
−0.732028 + 0.681275i \(0.761426\pi\)
\(830\) 0 0
\(831\) −3189.00 5523.51i −0.133123 0.230576i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 17496.0 + 30304.0i 0.725119 + 1.25594i
\(836\) 0 0
\(837\) 216.000 374.123i 0.00892001 0.0154499i
\(838\) 0 0
\(839\) 17496.0 0.719939 0.359970 0.932964i \(-0.382787\pi\)
0.359970 + 0.932964i \(0.382787\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) 4401.00 7622.76i 0.179808 0.311437i
\(844\) 0 0
\(845\) −18873.0 32689.0i −0.768344 1.33081i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3054.00 5289.68i −0.123455 0.213830i
\(850\) 0 0
\(851\) 8136.00 14092.0i 0.327730 0.567646i
\(852\) 0 0
\(853\) 32174.0 1.29146 0.645731 0.763565i \(-0.276553\pi\)
0.645731 + 0.763565i \(0.276553\pi\)
\(854\) 0 0
\(855\) 16200.0 0.647986
\(856\) 0 0
\(857\) 19467.0 33717.8i 0.775939 1.34397i −0.158326 0.987387i \(-0.550610\pi\)
0.934265 0.356579i \(-0.116057\pi\)
\(858\) 0 0
\(859\) −14890.0 25790.2i −0.591432 1.02439i −0.994040 0.109018i \(-0.965229\pi\)
0.402608 0.915373i \(-0.368104\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24048.0 + 41652.4i 0.948556 + 1.64295i 0.748471 + 0.663168i \(0.230788\pi\)
0.200085 + 0.979779i \(0.435878\pi\)
\(864\) 0 0
\(865\) −11178.0 + 19360.9i −0.439380 + 0.761028i
\(866\) 0 0
\(867\) −13767.0 −0.539275
\(868\) 0 0
\(869\) 18432.0 0.719520
\(870\) 0 0
\(871\) 1660.00 2875.20i 0.0645774 0.111851i
\(872\) 0 0
\(873\) 4743.00 + 8215.12i 0.183879 + 0.318487i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −10651.0 18448.1i −0.410101 0.710316i 0.584799 0.811178i \(-0.301173\pi\)
−0.994900 + 0.100862i \(0.967840\pi\)
\(878\) 0 0
\(879\) −3429.00 + 5939.20i −0.131578 + 0.227900i
\(880\) 0 0
\(881\) −7470.00 −0.285665 −0.142832 0.989747i \(-0.545621\pi\)
−0.142832 + 0.989747i \(0.545621\pi\)
\(882\) 0 0
\(883\) 764.000 0.0291174 0.0145587 0.999894i \(-0.495366\pi\)
0.0145587 + 0.999894i \(0.495366\pi\)
\(884\) 0 0
\(885\) −18468.0 + 31987.5i −0.701463 + 1.21497i
\(886\) 0 0
\(887\) −16164.0 27996.9i −0.611876 1.05980i −0.990924 0.134423i \(-0.957082\pi\)
0.379048 0.925377i \(-0.376252\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1458.00 2525.33i −0.0548202 0.0949514i
\(892\) 0 0
\(893\) 21600.0 37412.3i 0.809425 1.40197i
\(894\) 0 0
\(895\) −20088.0 −0.750243
\(896\) 0 0
\(897\) −2160.00 −0.0804017
\(898\) 0 0
\(899\) −1872.00 + 3242.40i −0.0694490 + 0.120289i
\(900\) 0 0
\(901\) −3726.00 6453.62i −0.137770 0.238625i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9630.00 + 16679.6i 0.353715 + 0.612652i
\(906\) 0 0
\(907\) 18158.0 31450.6i 0.664748 1.15138i −0.314605 0.949223i \(-0.601872\pi\)
0.979354 0.202155i \(-0.0647944\pi\)
\(908\) 0 0
\(909\) 5022.00 0.183244
\(910\) 0 0
\(911\) −13392.0 −0.487044 −0.243522 0.969895i \(-0.578303\pi\)
−0.243522 + 0.969895i \(0.578303\pi\)
\(912\) 0 0
\(913\) 21384.0 37038.2i 0.775145 1.34259i
\(914\) 0 0
\(915\) 11394.0 + 19735.0i 0.411666 + 0.713026i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −19036.0 32971.3i −0.683286 1.18349i −0.973972 0.226667i \(-0.927217\pi\)
0.290686 0.956818i \(-0.406116\pi\)
\(920\) 0 0
\(921\) −1866.00 + 3232.01i −0.0667609 + 0.115633i
\(922\) 0 0
\(923\) 3600.00 0.128381
\(924\) 0 0
\(925\) −44974.0 −1.59863
\(926\) 0 0
\(927\) −36.0000 + 62.3538i −0.00127551 + 0.00220924i
\(928\) 0 0
\(929\) 6399.00 + 11083.4i 0.225990 + 0.391426i 0.956616 0.291352i \(-0.0941051\pi\)
−0.730626 + 0.682778i \(0.760772\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1836.00 3180.05i −0.0644244 0.111586i
\(934\) 0 0
\(935\) 5832.00 10101.3i 0.203986 0.353314i
\(936\) 0 0
\(937\) 34874.0 1.21588 0.607942 0.793981i \(-0.291995\pi\)
0.607942 + 0.793981i \(0.291995\pi\)
\(938\) 0 0
\(939\) 5694.00 0.197888
\(940\) 0 0
\(941\) −8595.00 + 14887.0i −0.297757 + 0.515730i −0.975622 0.219456i \(-0.929572\pi\)
0.677866 + 0.735186i \(0.262905\pi\)
\(942\) 0 0
\(943\) −3240.00 5611.84i −0.111886 0.193793i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20142.0 34887.0i −0.691158 1.19712i −0.971459 0.237209i \(-0.923767\pi\)
0.280300 0.959912i \(-0.409566\pi\)
\(948\) 0 0
\(949\) 130.000 225.167i 0.00444676 0.00770202i
\(950\) 0 0
\(951\) −27486.0 −0.937218
\(952\) 0 0
\(953\) 15498.0 0.526789 0.263394 0.964688i \(-0.415158\pi\)
0.263394 + 0.964688i \(0.415158\pi\)
\(954\) 0 0
\(955\) −5184.00 + 8978.95i −0.175655 + 0.304243i
\(956\) 0 0
\(957\) 12636.0 + 21886.2i 0.426817 + 0.739269i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 14767.5 + 25578.1i 0.495703 + 0.858583i
\(962\) 0 0
\(963\) −7938.00 + 13749.0i −0.265627 + 0.460079i
\(964\) 0 0
\(965\) 24156.0 0.805813
\(966\) 0 0
\(967\) 37160.0 1.23577 0.617883 0.786270i \(-0.287991\pi\)
0.617883 + 0.786270i \(0.287991\pi\)
\(968\) 0 0
\(969\) 2700.00 4676.54i 0.0895113 0.155038i
\(970\) 0 0
\(971\) −9234.00 15993.8i −0.305183 0.528593i 0.672119 0.740443i \(-0.265384\pi\)
−0.977302 + 0.211850i \(0.932051\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2985.00 + 5170.17i 0.0980477 + 0.169824i
\(976\) 0 0
\(977\) −5193.00 + 8994.54i −0.170050 + 0.294535i −0.938437 0.345450i \(-0.887726\pi\)
0.768387 + 0.639985i \(0.221060\pi\)
\(978\) 0 0
\(979\) −22680.0 −0.740404
\(980\) 0 0
\(981\) 14598.0 0.475105
\(982\) 0 0
\(983\) −22068.0 + 38222.9i −0.716032 + 1.24020i 0.246528 + 0.969136i \(0.420710\pi\)
−0.962560 + 0.271069i \(0.912623\pi\)
\(984\) 0 0
\(985\) 12798.0 + 22166.8i 0.413988 + 0.717048i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16272.0 28183.9i −0.523174 0.906165i
\(990\) 0 0
\(991\) 14216.0 24622.8i 0.455687 0.789274i −0.543040 0.839707i \(-0.682727\pi\)
0.998727 + 0.0504332i \(0.0160602\pi\)
\(992\) 0 0
\(993\) −13044.0 −0.416857
\(994\) 0 0
\(995\) −15696.0 −0.500097
\(996\) 0 0
\(997\) 19889.0 34448.8i 0.631786 1.09429i −0.355400 0.934714i \(-0.615655\pi\)
0.987186 0.159572i \(-0.0510113\pi\)
\(998\) 0 0
\(999\) 3051.00 + 5284.49i 0.0966260 + 0.167361i
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 588.4.i.d.361.1 2
3.2 odd 2 1764.4.k.b.361.1 2
7.2 even 3 inner 588.4.i.d.373.1 2
7.3 odd 6 588.4.a.c.1.1 1
7.4 even 3 12.4.a.a.1.1 1
7.5 odd 6 588.4.i.e.373.1 2
7.6 odd 2 588.4.i.e.361.1 2
21.2 odd 6 1764.4.k.b.1549.1 2
21.5 even 6 1764.4.k.o.1549.1 2
21.11 odd 6 36.4.a.a.1.1 1
21.17 even 6 1764.4.a.b.1.1 1
21.20 even 2 1764.4.k.o.361.1 2
28.3 even 6 2352.4.a.bk.1.1 1
28.11 odd 6 48.4.a.a.1.1 1
35.4 even 6 300.4.a.b.1.1 1
35.18 odd 12 300.4.d.e.49.2 2
35.32 odd 12 300.4.d.e.49.1 2
56.11 odd 6 192.4.a.l.1.1 1
56.53 even 6 192.4.a.f.1.1 1
63.4 even 3 324.4.e.h.217.1 2
63.11 odd 6 324.4.e.a.109.1 2
63.25 even 3 324.4.e.h.109.1 2
63.32 odd 6 324.4.e.a.217.1 2
77.32 odd 6 1452.4.a.d.1.1 1
84.11 even 6 144.4.a.g.1.1 1
91.18 odd 12 2028.4.b.c.337.2 2
91.25 even 6 2028.4.a.c.1.1 1
91.60 odd 12 2028.4.b.c.337.1 2
105.32 even 12 900.4.d.c.649.2 2
105.53 even 12 900.4.d.c.649.1 2
105.74 odd 6 900.4.a.g.1.1 1
112.11 odd 12 768.4.d.j.385.2 2
112.53 even 12 768.4.d.g.385.1 2
112.67 odd 12 768.4.d.j.385.1 2
112.109 even 12 768.4.d.g.385.2 2
140.39 odd 6 1200.4.a.be.1.1 1
140.67 even 12 1200.4.f.d.49.2 2
140.123 even 12 1200.4.f.d.49.1 2
168.11 even 6 576.4.a.a.1.1 1
168.53 odd 6 576.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.4.a.a.1.1 1 7.4 even 3
36.4.a.a.1.1 1 21.11 odd 6
48.4.a.a.1.1 1 28.11 odd 6
144.4.a.g.1.1 1 84.11 even 6
192.4.a.f.1.1 1 56.53 even 6
192.4.a.l.1.1 1 56.11 odd 6
300.4.a.b.1.1 1 35.4 even 6
300.4.d.e.49.1 2 35.32 odd 12
300.4.d.e.49.2 2 35.18 odd 12
324.4.e.a.109.1 2 63.11 odd 6
324.4.e.a.217.1 2 63.32 odd 6
324.4.e.h.109.1 2 63.25 even 3
324.4.e.h.217.1 2 63.4 even 3
576.4.a.a.1.1 1 168.11 even 6
576.4.a.b.1.1 1 168.53 odd 6
588.4.a.c.1.1 1 7.3 odd 6
588.4.i.d.361.1 2 1.1 even 1 trivial
588.4.i.d.373.1 2 7.2 even 3 inner
588.4.i.e.361.1 2 7.6 odd 2
588.4.i.e.373.1 2 7.5 odd 6
768.4.d.g.385.1 2 112.53 even 12
768.4.d.g.385.2 2 112.109 even 12
768.4.d.j.385.1 2 112.67 odd 12
768.4.d.j.385.2 2 112.11 odd 12
900.4.a.g.1.1 1 105.74 odd 6
900.4.d.c.649.1 2 105.53 even 12
900.4.d.c.649.2 2 105.32 even 12
1200.4.a.be.1.1 1 140.39 odd 6
1200.4.f.d.49.1 2 140.123 even 12
1200.4.f.d.49.2 2 140.67 even 12
1452.4.a.d.1.1 1 77.32 odd 6
1764.4.a.b.1.1 1 21.17 even 6
1764.4.k.b.361.1 2 3.2 odd 2
1764.4.k.b.1549.1 2 21.2 odd 6
1764.4.k.o.361.1 2 21.20 even 2
1764.4.k.o.1549.1 2 21.5 even 6
2028.4.a.c.1.1 1 91.25 even 6
2028.4.b.c.337.1 2 91.60 odd 12
2028.4.b.c.337.2 2 91.18 odd 12
2352.4.a.bk.1.1 1 28.3 even 6