Properties

Label 560.3.f.a.321.2
Level $560$
Weight $3$
Character 560.321
Analytic conductor $15.259$
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] = [560,3,Mod(321,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.f (of order \(2\), degree \(1\), 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.2
Root \(2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 560.321
Dual form 560.3.f.a.321.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.47214i q^{3} -2.23607i q^{5} -7.00000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q+4.47214i q^{3} -2.23607i q^{5} -7.00000 q^{7} -11.0000 q^{9} -2.00000 q^{11} +13.4164i q^{13} +10.0000 q^{15} -26.8328i q^{17} -13.4164i q^{19} -31.3050i q^{21} -26.0000 q^{23} -5.00000 q^{25} -8.94427i q^{27} -22.0000 q^{29} -53.6656i q^{31} -8.94427i q^{33} +15.6525i q^{35} +14.0000 q^{37} -60.0000 q^{39} +26.8328i q^{41} +34.0000 q^{43} +24.5967i q^{45} -26.8328i q^{47} +49.0000 q^{49} +120.000 q^{51} -34.0000 q^{53} +4.47214i q^{55} +60.0000 q^{57} -40.2492i q^{59} -93.9149i q^{61} +77.0000 q^{63} +30.0000 q^{65} -14.0000 q^{67} -116.276i q^{69} -62.0000 q^{71} +53.6656i q^{73} -22.3607i q^{75} +14.0000 q^{77} -38.0000 q^{79} -59.0000 q^{81} +40.2492i q^{83} -60.0000 q^{85} -98.3870i q^{87} +26.8328i q^{89} -93.9149i q^{91} +240.000 q^{93} -30.0000 q^{95} +26.8328i q^{97} +22.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{7} - 22 q^{9} - 4 q^{11} + 20 q^{15} - 52 q^{23} - 10 q^{25} - 44 q^{29} + 28 q^{37} - 120 q^{39} + 68 q^{43} + 98 q^{49} + 240 q^{51} - 68 q^{53} + 120 q^{57} + 154 q^{63} + 60 q^{65} - 28 q^{67} - 124 q^{71} + 28 q^{77} - 76 q^{79} - 118 q^{81} - 120 q^{85} + 480 q^{93} - 60 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.47214i 1.49071i 0.666667 + 0.745356i \(0.267720\pi\)
−0.666667 + 0.745356i \(0.732280\pi\)
\(4\) 0 0
\(5\) − 2.23607i − 0.447214i
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) −11.0000 −1.22222
\(10\) 0 0
\(11\) −2.00000 −0.181818 −0.0909091 0.995859i \(-0.528977\pi\)
−0.0909091 + 0.995859i \(0.528977\pi\)
\(12\) 0 0
\(13\) 13.4164i 1.03203i 0.856579 + 0.516016i \(0.172585\pi\)
−0.856579 + 0.516016i \(0.827415\pi\)
\(14\) 0 0
\(15\) 10.0000 0.666667
\(16\) 0 0
\(17\) − 26.8328i − 1.57840i −0.614136 0.789200i \(-0.710495\pi\)
0.614136 0.789200i \(-0.289505\pi\)
\(18\) 0 0
\(19\) − 13.4164i − 0.706127i −0.935599 0.353063i \(-0.885140\pi\)
0.935599 0.353063i \(-0.114860\pi\)
\(20\) 0 0
\(21\) − 31.3050i − 1.49071i
\(22\) 0 0
\(23\) −26.0000 −1.13043 −0.565217 0.824942i \(-0.691208\pi\)
−0.565217 + 0.824942i \(0.691208\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) − 8.94427i − 0.331269i
\(28\) 0 0
\(29\) −22.0000 −0.758621 −0.379310 0.925270i \(-0.623839\pi\)
−0.379310 + 0.925270i \(0.623839\pi\)
\(30\) 0 0
\(31\) − 53.6656i − 1.73115i −0.500780 0.865575i \(-0.666953\pi\)
0.500780 0.865575i \(-0.333047\pi\)
\(32\) 0 0
\(33\) − 8.94427i − 0.271039i
\(34\) 0 0
\(35\) 15.6525i 0.447214i
\(36\) 0 0
\(37\) 14.0000 0.378378 0.189189 0.981941i \(-0.439414\pi\)
0.189189 + 0.981941i \(0.439414\pi\)
\(38\) 0 0
\(39\) −60.0000 −1.53846
\(40\) 0 0
\(41\) 26.8328i 0.654459i 0.944945 + 0.327229i \(0.106115\pi\)
−0.944945 + 0.327229i \(0.893885\pi\)
\(42\) 0 0
\(43\) 34.0000 0.790698 0.395349 0.918531i \(-0.370624\pi\)
0.395349 + 0.918531i \(0.370624\pi\)
\(44\) 0 0
\(45\) 24.5967i 0.546594i
\(46\) 0 0
\(47\) − 26.8328i − 0.570911i −0.958392 0.285455i \(-0.907855\pi\)
0.958392 0.285455i \(-0.0921449\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 120.000 2.35294
\(52\) 0 0
\(53\) −34.0000 −0.641509 −0.320755 0.947162i \(-0.603937\pi\)
−0.320755 + 0.947162i \(0.603937\pi\)
\(54\) 0 0
\(55\) 4.47214i 0.0813116i
\(56\) 0 0
\(57\) 60.0000 1.05263
\(58\) 0 0
\(59\) − 40.2492i − 0.682190i −0.940029 0.341095i \(-0.889202\pi\)
0.940029 0.341095i \(-0.110798\pi\)
\(60\) 0 0
\(61\) − 93.9149i − 1.53959i −0.638293 0.769794i \(-0.720359\pi\)
0.638293 0.769794i \(-0.279641\pi\)
\(62\) 0 0
\(63\) 77.0000 1.22222
\(64\) 0 0
\(65\) 30.0000 0.461538
\(66\) 0 0
\(67\) −14.0000 −0.208955 −0.104478 0.994527i \(-0.533317\pi\)
−0.104478 + 0.994527i \(0.533317\pi\)
\(68\) 0 0
\(69\) − 116.276i − 1.68515i
\(70\) 0 0
\(71\) −62.0000 −0.873239 −0.436620 0.899646i \(-0.643824\pi\)
−0.436620 + 0.899646i \(0.643824\pi\)
\(72\) 0 0
\(73\) 53.6656i 0.735146i 0.929995 + 0.367573i \(0.119811\pi\)
−0.929995 + 0.367573i \(0.880189\pi\)
\(74\) 0 0
\(75\) − 22.3607i − 0.298142i
\(76\) 0 0
\(77\) 14.0000 0.181818
\(78\) 0 0
\(79\) −38.0000 −0.481013 −0.240506 0.970648i \(-0.577313\pi\)
−0.240506 + 0.970648i \(0.577313\pi\)
\(80\) 0 0
\(81\) −59.0000 −0.728395
\(82\) 0 0
\(83\) 40.2492i 0.484930i 0.970160 + 0.242465i \(0.0779560\pi\)
−0.970160 + 0.242465i \(0.922044\pi\)
\(84\) 0 0
\(85\) −60.0000 −0.705882
\(86\) 0 0
\(87\) − 98.3870i − 1.13088i
\(88\) 0 0
\(89\) 26.8328i 0.301492i 0.988573 + 0.150746i \(0.0481676\pi\)
−0.988573 + 0.150746i \(0.951832\pi\)
\(90\) 0 0
\(91\) − 93.9149i − 1.03203i
\(92\) 0 0
\(93\) 240.000 2.58065
\(94\) 0 0
\(95\) −30.0000 −0.315789
\(96\) 0 0
\(97\) 26.8328i 0.276627i 0.990388 + 0.138313i \(0.0441681\pi\)
−0.990388 + 0.138313i \(0.955832\pi\)
\(98\) 0 0
\(99\) 22.0000 0.222222
\(100\) 0 0
\(101\) − 67.0820i − 0.664179i −0.943248 0.332089i \(-0.892246\pi\)
0.943248 0.332089i \(-0.107754\pi\)
\(102\) 0 0
\(103\) 160.997i 1.56308i 0.623857 + 0.781538i \(0.285565\pi\)
−0.623857 + 0.781538i \(0.714435\pi\)
\(104\) 0 0
\(105\) −70.0000 −0.666667
\(106\) 0 0
\(107\) 106.000 0.990654 0.495327 0.868707i \(-0.335048\pi\)
0.495327 + 0.868707i \(0.335048\pi\)
\(108\) 0 0
\(109\) −142.000 −1.30275 −0.651376 0.758755i \(-0.725808\pi\)
−0.651376 + 0.758755i \(0.725808\pi\)
\(110\) 0 0
\(111\) 62.6099i 0.564053i
\(112\) 0 0
\(113\) −34.0000 −0.300885 −0.150442 0.988619i \(-0.548070\pi\)
−0.150442 + 0.988619i \(0.548070\pi\)
\(114\) 0 0
\(115\) 58.1378i 0.505546i
\(116\) 0 0
\(117\) − 147.580i − 1.26137i
\(118\) 0 0
\(119\) 187.830i 1.57840i
\(120\) 0 0
\(121\) −117.000 −0.966942
\(122\) 0 0
\(123\) −120.000 −0.975610
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −194.000 −1.52756 −0.763780 0.645477i \(-0.776659\pi\)
−0.763780 + 0.645477i \(0.776659\pi\)
\(128\) 0 0
\(129\) 152.053i 1.17870i
\(130\) 0 0
\(131\) − 120.748i − 0.921738i −0.887468 0.460869i \(-0.847538\pi\)
0.887468 0.460869i \(-0.152462\pi\)
\(132\) 0 0
\(133\) 93.9149i 0.706127i
\(134\) 0 0
\(135\) −20.0000 −0.148148
\(136\) 0 0
\(137\) −166.000 −1.21168 −0.605839 0.795587i \(-0.707163\pi\)
−0.605839 + 0.795587i \(0.707163\pi\)
\(138\) 0 0
\(139\) − 93.9149i − 0.675646i −0.941210 0.337823i \(-0.890309\pi\)
0.941210 0.337823i \(-0.109691\pi\)
\(140\) 0 0
\(141\) 120.000 0.851064
\(142\) 0 0
\(143\) − 26.8328i − 0.187642i
\(144\) 0 0
\(145\) 49.1935i 0.339265i
\(146\) 0 0
\(147\) 219.135i 1.49071i
\(148\) 0 0
\(149\) −142.000 −0.953020 −0.476510 0.879169i \(-0.658098\pi\)
−0.476510 + 0.879169i \(0.658098\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.0132450 −0.00662252 0.999978i \(-0.502108\pi\)
−0.00662252 + 0.999978i \(0.502108\pi\)
\(152\) 0 0
\(153\) 295.161i 1.92916i
\(154\) 0 0
\(155\) −120.000 −0.774194
\(156\) 0 0
\(157\) 67.0820i 0.427274i 0.976913 + 0.213637i \(0.0685310\pi\)
−0.976913 + 0.213637i \(0.931469\pi\)
\(158\) 0 0
\(159\) − 152.053i − 0.956306i
\(160\) 0 0
\(161\) 182.000 1.13043
\(162\) 0 0
\(163\) 34.0000 0.208589 0.104294 0.994546i \(-0.466742\pi\)
0.104294 + 0.994546i \(0.466742\pi\)
\(164\) 0 0
\(165\) −20.0000 −0.121212
\(166\) 0 0
\(167\) − 107.331i − 0.642702i −0.946960 0.321351i \(-0.895863\pi\)
0.946960 0.321351i \(-0.104137\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.0650888
\(170\) 0 0
\(171\) 147.580i 0.863044i
\(172\) 0 0
\(173\) − 147.580i − 0.853066i −0.904472 0.426533i \(-0.859735\pi\)
0.904472 0.426533i \(-0.140265\pi\)
\(174\) 0 0
\(175\) 35.0000 0.200000
\(176\) 0 0
\(177\) 180.000 1.01695
\(178\) 0 0
\(179\) −218.000 −1.21788 −0.608939 0.793217i \(-0.708404\pi\)
−0.608939 + 0.793217i \(0.708404\pi\)
\(180\) 0 0
\(181\) 254.912i 1.40835i 0.710025 + 0.704176i \(0.248683\pi\)
−0.710025 + 0.704176i \(0.751317\pi\)
\(182\) 0 0
\(183\) 420.000 2.29508
\(184\) 0 0
\(185\) − 31.3050i − 0.169216i
\(186\) 0 0
\(187\) 53.6656i 0.286982i
\(188\) 0 0
\(189\) 62.6099i 0.331269i
\(190\) 0 0
\(191\) 58.0000 0.303665 0.151832 0.988406i \(-0.451483\pi\)
0.151832 + 0.988406i \(0.451483\pi\)
\(192\) 0 0
\(193\) 206.000 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(194\) 0 0
\(195\) 134.164i 0.688021i
\(196\) 0 0
\(197\) −226.000 −1.14721 −0.573604 0.819133i \(-0.694455\pi\)
−0.573604 + 0.819133i \(0.694455\pi\)
\(198\) 0 0
\(199\) 134.164i 0.674191i 0.941470 + 0.337096i \(0.109445\pi\)
−0.941470 + 0.337096i \(0.890555\pi\)
\(200\) 0 0
\(201\) − 62.6099i − 0.311492i
\(202\) 0 0
\(203\) 154.000 0.758621
\(204\) 0 0
\(205\) 60.0000 0.292683
\(206\) 0 0
\(207\) 286.000 1.38164
\(208\) 0 0
\(209\) 26.8328i 0.128387i
\(210\) 0 0
\(211\) 118.000 0.559242 0.279621 0.960111i \(-0.409791\pi\)
0.279621 + 0.960111i \(0.409791\pi\)
\(212\) 0 0
\(213\) − 277.272i − 1.30175i
\(214\) 0 0
\(215\) − 76.0263i − 0.353611i
\(216\) 0 0
\(217\) 375.659i 1.73115i
\(218\) 0 0
\(219\) −240.000 −1.09589
\(220\) 0 0
\(221\) 360.000 1.62896
\(222\) 0 0
\(223\) 80.4984i 0.360980i 0.983577 + 0.180490i \(0.0577683\pi\)
−0.983577 + 0.180490i \(0.942232\pi\)
\(224\) 0 0
\(225\) 55.0000 0.244444
\(226\) 0 0
\(227\) 254.912i 1.12296i 0.827491 + 0.561480i \(0.189768\pi\)
−0.827491 + 0.561480i \(0.810232\pi\)
\(228\) 0 0
\(229\) − 13.4164i − 0.0585869i −0.999571 0.0292935i \(-0.990674\pi\)
0.999571 0.0292935i \(-0.00932573\pi\)
\(230\) 0 0
\(231\) 62.6099i 0.271039i
\(232\) 0 0
\(233\) −214.000 −0.918455 −0.459227 0.888319i \(-0.651874\pi\)
−0.459227 + 0.888319i \(0.651874\pi\)
\(234\) 0 0
\(235\) −60.0000 −0.255319
\(236\) 0 0
\(237\) − 169.941i − 0.717051i
\(238\) 0 0
\(239\) −98.0000 −0.410042 −0.205021 0.978758i \(-0.565726\pi\)
−0.205021 + 0.978758i \(0.565726\pi\)
\(240\) 0 0
\(241\) 160.997i 0.668037i 0.942567 + 0.334018i \(0.108405\pi\)
−0.942567 + 0.334018i \(0.891595\pi\)
\(242\) 0 0
\(243\) − 344.354i − 1.41710i
\(244\) 0 0
\(245\) − 109.567i − 0.447214i
\(246\) 0 0
\(247\) 180.000 0.728745
\(248\) 0 0
\(249\) −180.000 −0.722892
\(250\) 0 0
\(251\) 335.410i 1.33630i 0.744029 + 0.668148i \(0.232913\pi\)
−0.744029 + 0.668148i \(0.767087\pi\)
\(252\) 0 0
\(253\) 52.0000 0.205534
\(254\) 0 0
\(255\) − 268.328i − 1.05227i
\(256\) 0 0
\(257\) 134.164i 0.522039i 0.965333 + 0.261020i \(0.0840587\pi\)
−0.965333 + 0.261020i \(0.915941\pi\)
\(258\) 0 0
\(259\) −98.0000 −0.378378
\(260\) 0 0
\(261\) 242.000 0.927203
\(262\) 0 0
\(263\) 34.0000 0.129278 0.0646388 0.997909i \(-0.479410\pi\)
0.0646388 + 0.997909i \(0.479410\pi\)
\(264\) 0 0
\(265\) 76.0263i 0.286892i
\(266\) 0 0
\(267\) −120.000 −0.449438
\(268\) 0 0
\(269\) − 254.912i − 0.947627i −0.880625 0.473814i \(-0.842877\pi\)
0.880625 0.473814i \(-0.157123\pi\)
\(270\) 0 0
\(271\) 321.994i 1.18817i 0.804403 + 0.594084i \(0.202486\pi\)
−0.804403 + 0.594084i \(0.797514\pi\)
\(272\) 0 0
\(273\) 420.000 1.53846
\(274\) 0 0
\(275\) 10.0000 0.0363636
\(276\) 0 0
\(277\) 14.0000 0.0505415 0.0252708 0.999681i \(-0.491955\pi\)
0.0252708 + 0.999681i \(0.491955\pi\)
\(278\) 0 0
\(279\) 590.322i 2.11585i
\(280\) 0 0
\(281\) 2.00000 0.00711744 0.00355872 0.999994i \(-0.498867\pi\)
0.00355872 + 0.999994i \(0.498867\pi\)
\(282\) 0 0
\(283\) − 93.9149i − 0.331855i −0.986138 0.165927i \(-0.946938\pi\)
0.986138 0.165927i \(-0.0530617\pi\)
\(284\) 0 0
\(285\) − 134.164i − 0.470751i
\(286\) 0 0
\(287\) − 187.830i − 0.654459i
\(288\) 0 0
\(289\) −431.000 −1.49135
\(290\) 0 0
\(291\) −120.000 −0.412371
\(292\) 0 0
\(293\) − 335.410i − 1.14474i −0.819994 0.572372i \(-0.806023\pi\)
0.819994 0.572372i \(-0.193977\pi\)
\(294\) 0 0
\(295\) −90.0000 −0.305085
\(296\) 0 0
\(297\) 17.8885i 0.0602308i
\(298\) 0 0
\(299\) − 348.827i − 1.16664i
\(300\) 0 0
\(301\) −238.000 −0.790698
\(302\) 0 0
\(303\) 300.000 0.990099
\(304\) 0 0
\(305\) −210.000 −0.688525
\(306\) 0 0
\(307\) 201.246i 0.655525i 0.944760 + 0.327762i \(0.106295\pi\)
−0.944760 + 0.327762i \(0.893705\pi\)
\(308\) 0 0
\(309\) −720.000 −2.33010
\(310\) 0 0
\(311\) − 509.823i − 1.63930i −0.572862 0.819652i \(-0.694167\pi\)
0.572862 0.819652i \(-0.305833\pi\)
\(312\) 0 0
\(313\) − 321.994i − 1.02873i −0.857570 0.514367i \(-0.828027\pi\)
0.857570 0.514367i \(-0.171973\pi\)
\(314\) 0 0
\(315\) − 172.177i − 0.546594i
\(316\) 0 0
\(317\) 374.000 1.17981 0.589905 0.807472i \(-0.299165\pi\)
0.589905 + 0.807472i \(0.299165\pi\)
\(318\) 0 0
\(319\) 44.0000 0.137931
\(320\) 0 0
\(321\) 474.046i 1.47678i
\(322\) 0 0
\(323\) −360.000 −1.11455
\(324\) 0 0
\(325\) − 67.0820i − 0.206406i
\(326\) 0 0
\(327\) − 635.043i − 1.94203i
\(328\) 0 0
\(329\) 187.830i 0.570911i
\(330\) 0 0
\(331\) −482.000 −1.45619 −0.728097 0.685474i \(-0.759595\pi\)
−0.728097 + 0.685474i \(0.759595\pi\)
\(332\) 0 0
\(333\) −154.000 −0.462462
\(334\) 0 0
\(335\) 31.3050i 0.0934476i
\(336\) 0 0
\(337\) 494.000 1.46588 0.732938 0.680296i \(-0.238149\pi\)
0.732938 + 0.680296i \(0.238149\pi\)
\(338\) 0 0
\(339\) − 152.053i − 0.448533i
\(340\) 0 0
\(341\) 107.331i 0.314754i
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) −260.000 −0.753623
\(346\) 0 0
\(347\) 346.000 0.997118 0.498559 0.866856i \(-0.333863\pi\)
0.498559 + 0.866856i \(0.333863\pi\)
\(348\) 0 0
\(349\) 335.410i 0.961061i 0.876978 + 0.480530i \(0.159556\pi\)
−0.876978 + 0.480530i \(0.840444\pi\)
\(350\) 0 0
\(351\) 120.000 0.341880
\(352\) 0 0
\(353\) − 26.8328i − 0.0760136i −0.999277 0.0380068i \(-0.987899\pi\)
0.999277 0.0380068i \(-0.0121009\pi\)
\(354\) 0 0
\(355\) 138.636i 0.390525i
\(356\) 0 0
\(357\) −840.000 −2.35294
\(358\) 0 0
\(359\) −338.000 −0.941504 −0.470752 0.882266i \(-0.656017\pi\)
−0.470752 + 0.882266i \(0.656017\pi\)
\(360\) 0 0
\(361\) 181.000 0.501385
\(362\) 0 0
\(363\) − 523.240i − 1.44143i
\(364\) 0 0
\(365\) 120.000 0.328767
\(366\) 0 0
\(367\) 295.161i 0.804253i 0.915584 + 0.402127i \(0.131729\pi\)
−0.915584 + 0.402127i \(0.868271\pi\)
\(368\) 0 0
\(369\) − 295.161i − 0.799894i
\(370\) 0 0
\(371\) 238.000 0.641509
\(372\) 0 0
\(373\) 86.0000 0.230563 0.115282 0.993333i \(-0.463223\pi\)
0.115282 + 0.993333i \(0.463223\pi\)
\(374\) 0 0
\(375\) −50.0000 −0.133333
\(376\) 0 0
\(377\) − 295.161i − 0.782920i
\(378\) 0 0
\(379\) 262.000 0.691293 0.345646 0.938365i \(-0.387660\pi\)
0.345646 + 0.938365i \(0.387660\pi\)
\(380\) 0 0
\(381\) − 867.594i − 2.27715i
\(382\) 0 0
\(383\) − 563.489i − 1.47125i −0.677388 0.735625i \(-0.736888\pi\)
0.677388 0.735625i \(-0.263112\pi\)
\(384\) 0 0
\(385\) − 31.3050i − 0.0813116i
\(386\) 0 0
\(387\) −374.000 −0.966408
\(388\) 0 0
\(389\) 698.000 1.79434 0.897172 0.441681i \(-0.145618\pi\)
0.897172 + 0.441681i \(0.145618\pi\)
\(390\) 0 0
\(391\) 697.653i 1.78428i
\(392\) 0 0
\(393\) 540.000 1.37405
\(394\) 0 0
\(395\) 84.9706i 0.215115i
\(396\) 0 0
\(397\) − 308.577i − 0.777273i −0.921391 0.388636i \(-0.872946\pi\)
0.921391 0.388636i \(-0.127054\pi\)
\(398\) 0 0
\(399\) −420.000 −1.05263
\(400\) 0 0
\(401\) −538.000 −1.34165 −0.670823 0.741618i \(-0.734059\pi\)
−0.670823 + 0.741618i \(0.734059\pi\)
\(402\) 0 0
\(403\) 720.000 1.78660
\(404\) 0 0
\(405\) 131.928i 0.325748i
\(406\) 0 0
\(407\) −28.0000 −0.0687961
\(408\) 0 0
\(409\) − 295.161i − 0.721665i −0.932631 0.360832i \(-0.882493\pi\)
0.932631 0.360832i \(-0.117507\pi\)
\(410\) 0 0
\(411\) − 742.375i − 1.80626i
\(412\) 0 0
\(413\) 281.745i 0.682190i
\(414\) 0 0
\(415\) 90.0000 0.216867
\(416\) 0 0
\(417\) 420.000 1.00719
\(418\) 0 0
\(419\) − 818.401i − 1.95322i −0.215009 0.976612i \(-0.568978\pi\)
0.215009 0.976612i \(-0.431022\pi\)
\(420\) 0 0
\(421\) −118.000 −0.280285 −0.140143 0.990131i \(-0.544756\pi\)
−0.140143 + 0.990131i \(0.544756\pi\)
\(422\) 0 0
\(423\) 295.161i 0.697780i
\(424\) 0 0
\(425\) 134.164i 0.315680i
\(426\) 0 0
\(427\) 657.404i 1.53959i
\(428\) 0 0
\(429\) 120.000 0.279720
\(430\) 0 0
\(431\) 718.000 1.66589 0.832947 0.553353i \(-0.186652\pi\)
0.832947 + 0.553353i \(0.186652\pi\)
\(432\) 0 0
\(433\) − 509.823i − 1.17742i −0.808344 0.588711i \(-0.799636\pi\)
0.808344 0.588711i \(-0.200364\pi\)
\(434\) 0 0
\(435\) −220.000 −0.505747
\(436\) 0 0
\(437\) 348.827i 0.798230i
\(438\) 0 0
\(439\) − 26.8328i − 0.0611226i −0.999533 0.0305613i \(-0.990271\pi\)
0.999533 0.0305613i \(-0.00972948\pi\)
\(440\) 0 0
\(441\) −539.000 −1.22222
\(442\) 0 0
\(443\) 634.000 1.43115 0.715576 0.698535i \(-0.246164\pi\)
0.715576 + 0.698535i \(0.246164\pi\)
\(444\) 0 0
\(445\) 60.0000 0.134831
\(446\) 0 0
\(447\) − 635.043i − 1.42068i
\(448\) 0 0
\(449\) 338.000 0.752784 0.376392 0.926461i \(-0.377165\pi\)
0.376392 + 0.926461i \(0.377165\pi\)
\(450\) 0 0
\(451\) − 53.6656i − 0.118993i
\(452\) 0 0
\(453\) − 8.94427i − 0.0197445i
\(454\) 0 0
\(455\) −210.000 −0.461538
\(456\) 0 0
\(457\) −466.000 −1.01969 −0.509847 0.860265i \(-0.670298\pi\)
−0.509847 + 0.860265i \(0.670298\pi\)
\(458\) 0 0
\(459\) −240.000 −0.522876
\(460\) 0 0
\(461\) 442.741i 0.960394i 0.877161 + 0.480197i \(0.159435\pi\)
−0.877161 + 0.480197i \(0.840565\pi\)
\(462\) 0 0
\(463\) −206.000 −0.444924 −0.222462 0.974941i \(-0.571409\pi\)
−0.222462 + 0.974941i \(0.571409\pi\)
\(464\) 0 0
\(465\) − 536.656i − 1.15410i
\(466\) 0 0
\(467\) 362.243i 0.775681i 0.921727 + 0.387840i \(0.126779\pi\)
−0.921727 + 0.387840i \(0.873221\pi\)
\(468\) 0 0
\(469\) 98.0000 0.208955
\(470\) 0 0
\(471\) −300.000 −0.636943
\(472\) 0 0
\(473\) −68.0000 −0.143763
\(474\) 0 0
\(475\) 67.0820i 0.141225i
\(476\) 0 0
\(477\) 374.000 0.784067
\(478\) 0 0
\(479\) 214.663i 0.448147i 0.974572 + 0.224074i \(0.0719356\pi\)
−0.974572 + 0.224074i \(0.928064\pi\)
\(480\) 0 0
\(481\) 187.830i 0.390498i
\(482\) 0 0
\(483\) 813.929i 1.68515i
\(484\) 0 0
\(485\) 60.0000 0.123711
\(486\) 0 0
\(487\) 166.000 0.340862 0.170431 0.985370i \(-0.445484\pi\)
0.170431 + 0.985370i \(0.445484\pi\)
\(488\) 0 0
\(489\) 152.053i 0.310946i
\(490\) 0 0
\(491\) 838.000 1.70672 0.853360 0.521321i \(-0.174561\pi\)
0.853360 + 0.521321i \(0.174561\pi\)
\(492\) 0 0
\(493\) 590.322i 1.19741i
\(494\) 0 0
\(495\) − 49.1935i − 0.0993808i
\(496\) 0 0
\(497\) 434.000 0.873239
\(498\) 0 0
\(499\) 262.000 0.525050 0.262525 0.964925i \(-0.415445\pi\)
0.262525 + 0.964925i \(0.415445\pi\)
\(500\) 0 0
\(501\) 480.000 0.958084
\(502\) 0 0
\(503\) 429.325i 0.853529i 0.904363 + 0.426764i \(0.140347\pi\)
−0.904363 + 0.426764i \(0.859653\pi\)
\(504\) 0 0
\(505\) −150.000 −0.297030
\(506\) 0 0
\(507\) − 49.1935i − 0.0970286i
\(508\) 0 0
\(509\) − 898.899i − 1.76601i −0.469363 0.883005i \(-0.655516\pi\)
0.469363 0.883005i \(-0.344484\pi\)
\(510\) 0 0
\(511\) − 375.659i − 0.735146i
\(512\) 0 0
\(513\) −120.000 −0.233918
\(514\) 0 0
\(515\) 360.000 0.699029
\(516\) 0 0
\(517\) 53.6656i 0.103802i
\(518\) 0 0
\(519\) 660.000 1.27168
\(520\) 0 0
\(521\) − 724.486i − 1.39057i −0.718735 0.695284i \(-0.755279\pi\)
0.718735 0.695284i \(-0.244721\pi\)
\(522\) 0 0
\(523\) − 523.240i − 1.00046i −0.865893 0.500229i \(-0.833249\pi\)
0.865893 0.500229i \(-0.166751\pi\)
\(524\) 0 0
\(525\) 156.525i 0.298142i
\(526\) 0 0
\(527\) −1440.00 −2.73245
\(528\) 0 0
\(529\) 147.000 0.277883
\(530\) 0 0
\(531\) 442.741i 0.833788i
\(532\) 0 0
\(533\) −360.000 −0.675422
\(534\) 0 0
\(535\) − 237.023i − 0.443034i
\(536\) 0 0
\(537\) − 974.926i − 1.81550i
\(538\) 0 0
\(539\) −98.0000 −0.181818
\(540\) 0 0
\(541\) 842.000 1.55638 0.778189 0.628031i \(-0.216139\pi\)
0.778189 + 0.628031i \(0.216139\pi\)
\(542\) 0 0
\(543\) −1140.00 −2.09945
\(544\) 0 0
\(545\) 317.522i 0.582609i
\(546\) 0 0
\(547\) −134.000 −0.244973 −0.122486 0.992470i \(-0.539087\pi\)
−0.122486 + 0.992470i \(0.539087\pi\)
\(548\) 0 0
\(549\) 1033.06i 1.88172i
\(550\) 0 0
\(551\) 295.161i 0.535682i
\(552\) 0 0
\(553\) 266.000 0.481013
\(554\) 0 0
\(555\) 140.000 0.252252
\(556\) 0 0
\(557\) −706.000 −1.26750 −0.633752 0.773536i \(-0.718486\pi\)
−0.633752 + 0.773536i \(0.718486\pi\)
\(558\) 0 0
\(559\) 456.158i 0.816025i
\(560\) 0 0
\(561\) −240.000 −0.427807
\(562\) 0 0
\(563\) − 13.4164i − 0.0238302i −0.999929 0.0119151i \(-0.996207\pi\)
0.999929 0.0119151i \(-0.00379279\pi\)
\(564\) 0 0
\(565\) 76.0263i 0.134560i
\(566\) 0 0
\(567\) 413.000 0.728395
\(568\) 0 0
\(569\) −82.0000 −0.144112 −0.0720562 0.997401i \(-0.522956\pi\)
−0.0720562 + 0.997401i \(0.522956\pi\)
\(570\) 0 0
\(571\) 118.000 0.206655 0.103327 0.994647i \(-0.467051\pi\)
0.103327 + 0.994647i \(0.467051\pi\)
\(572\) 0 0
\(573\) 259.384i 0.452677i
\(574\) 0 0
\(575\) 130.000 0.226087
\(576\) 0 0
\(577\) 885.483i 1.53463i 0.641269 + 0.767316i \(0.278408\pi\)
−0.641269 + 0.767316i \(0.721592\pi\)
\(578\) 0 0
\(579\) 921.260i 1.59112i
\(580\) 0 0
\(581\) − 281.745i − 0.484930i
\(582\) 0 0
\(583\) 68.0000 0.116638
\(584\) 0 0
\(585\) −330.000 −0.564103
\(586\) 0 0
\(587\) − 791.568i − 1.34850i −0.738504 0.674249i \(-0.764468\pi\)
0.738504 0.674249i \(-0.235532\pi\)
\(588\) 0 0
\(589\) −720.000 −1.22241
\(590\) 0 0
\(591\) − 1010.70i − 1.71016i
\(592\) 0 0
\(593\) 134.164i 0.226246i 0.993581 + 0.113123i \(0.0360855\pi\)
−0.993581 + 0.113123i \(0.963915\pi\)
\(594\) 0 0
\(595\) 420.000 0.705882
\(596\) 0 0
\(597\) −600.000 −1.00503
\(598\) 0 0
\(599\) −398.000 −0.664441 −0.332220 0.943202i \(-0.607798\pi\)
−0.332220 + 0.943202i \(0.607798\pi\)
\(600\) 0 0
\(601\) − 134.164i − 0.223235i −0.993751 0.111617i \(-0.964397\pi\)
0.993751 0.111617i \(-0.0356031\pi\)
\(602\) 0 0
\(603\) 154.000 0.255390
\(604\) 0 0
\(605\) 261.620i 0.432430i
\(606\) 0 0
\(607\) − 939.149i − 1.54720i −0.633676 0.773598i \(-0.718455\pi\)
0.633676 0.773598i \(-0.281545\pi\)
\(608\) 0 0
\(609\) 688.709i 1.13088i
\(610\) 0 0
\(611\) 360.000 0.589198
\(612\) 0 0
\(613\) 206.000 0.336052 0.168026 0.985783i \(-0.446261\pi\)
0.168026 + 0.985783i \(0.446261\pi\)
\(614\) 0 0
\(615\) 268.328i 0.436306i
\(616\) 0 0
\(617\) 494.000 0.800648 0.400324 0.916374i \(-0.368898\pi\)
0.400324 + 0.916374i \(0.368898\pi\)
\(618\) 0 0
\(619\) 120.748i 0.195069i 0.995232 + 0.0975345i \(0.0310956\pi\)
−0.995232 + 0.0975345i \(0.968904\pi\)
\(620\) 0 0
\(621\) 232.551i 0.374478i
\(622\) 0 0
\(623\) − 187.830i − 0.301492i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) −120.000 −0.191388
\(628\) 0 0
\(629\) − 375.659i − 0.597233i
\(630\) 0 0
\(631\) −542.000 −0.858954 −0.429477 0.903078i \(-0.641302\pi\)
−0.429477 + 0.903078i \(0.641302\pi\)
\(632\) 0 0
\(633\) 527.712i 0.833668i
\(634\) 0 0
\(635\) 433.797i 0.683145i
\(636\) 0 0
\(637\) 657.404i 1.03203i
\(638\) 0 0
\(639\) 682.000 1.06729
\(640\) 0 0
\(641\) −298.000 −0.464899 −0.232449 0.972609i \(-0.574674\pi\)
−0.232449 + 0.972609i \(0.574674\pi\)
\(642\) 0 0
\(643\) 1006.23i 1.56490i 0.622714 + 0.782450i \(0.286030\pi\)
−0.622714 + 0.782450i \(0.713970\pi\)
\(644\) 0 0
\(645\) 340.000 0.527132
\(646\) 0 0
\(647\) 643.988i 0.995344i 0.867365 + 0.497672i \(0.165812\pi\)
−0.867365 + 0.497672i \(0.834188\pi\)
\(648\) 0 0
\(649\) 80.4984i 0.124035i
\(650\) 0 0
\(651\) −1680.00 −2.58065
\(652\) 0 0
\(653\) −154.000 −0.235835 −0.117917 0.993023i \(-0.537622\pi\)
−0.117917 + 0.993023i \(0.537622\pi\)
\(654\) 0 0
\(655\) −270.000 −0.412214
\(656\) 0 0
\(657\) − 590.322i − 0.898511i
\(658\) 0 0
\(659\) −338.000 −0.512898 −0.256449 0.966558i \(-0.582553\pi\)
−0.256449 + 0.966558i \(0.582553\pi\)
\(660\) 0 0
\(661\) 576.906i 0.872777i 0.899758 + 0.436388i \(0.143743\pi\)
−0.899758 + 0.436388i \(0.856257\pi\)
\(662\) 0 0
\(663\) 1609.97i 2.42831i
\(664\) 0 0
\(665\) 210.000 0.315789
\(666\) 0 0
\(667\) 572.000 0.857571
\(668\) 0 0
\(669\) −360.000 −0.538117
\(670\) 0 0
\(671\) 187.830i 0.279925i
\(672\) 0 0
\(673\) −814.000 −1.20951 −0.604755 0.796412i \(-0.706729\pi\)
−0.604755 + 0.796412i \(0.706729\pi\)
\(674\) 0 0
\(675\) 44.7214i 0.0662539i
\(676\) 0 0
\(677\) 684.237i 1.01069i 0.862918 + 0.505345i \(0.168635\pi\)
−0.862918 + 0.505345i \(0.831365\pi\)
\(678\) 0 0
\(679\) − 187.830i − 0.276627i
\(680\) 0 0
\(681\) −1140.00 −1.67401
\(682\) 0 0
\(683\) −926.000 −1.35578 −0.677892 0.735162i \(-0.737106\pi\)
−0.677892 + 0.735162i \(0.737106\pi\)
\(684\) 0 0
\(685\) 371.187i 0.541879i
\(686\) 0 0
\(687\) 60.0000 0.0873362
\(688\) 0 0
\(689\) − 456.158i − 0.662058i
\(690\) 0 0
\(691\) 576.906i 0.834885i 0.908703 + 0.417443i \(0.137073\pi\)
−0.908703 + 0.417443i \(0.862927\pi\)
\(692\) 0 0
\(693\) −154.000 −0.222222
\(694\) 0 0
\(695\) −210.000 −0.302158
\(696\) 0 0
\(697\) 720.000 1.03300
\(698\) 0 0
\(699\) − 957.037i − 1.36915i
\(700\) 0 0
\(701\) 362.000 0.516405 0.258203 0.966091i \(-0.416870\pi\)
0.258203 + 0.966091i \(0.416870\pi\)
\(702\) 0 0
\(703\) − 187.830i − 0.267183i
\(704\) 0 0
\(705\) − 268.328i − 0.380607i
\(706\) 0 0
\(707\) 469.574i 0.664179i
\(708\) 0 0
\(709\) 1058.00 1.49224 0.746121 0.665810i \(-0.231914\pi\)
0.746121 + 0.665810i \(0.231914\pi\)
\(710\) 0 0
\(711\) 418.000 0.587904
\(712\) 0 0
\(713\) 1395.31i 1.95695i
\(714\) 0 0
\(715\) −60.0000 −0.0839161
\(716\) 0 0
\(717\) − 438.269i − 0.611254i
\(718\) 0 0
\(719\) − 482.991i − 0.671753i −0.941906 0.335877i \(-0.890967\pi\)
0.941906 0.335877i \(-0.109033\pi\)
\(720\) 0 0
\(721\) − 1126.98i − 1.56308i
\(722\) 0 0
\(723\) −720.000 −0.995851
\(724\) 0 0
\(725\) 110.000 0.151724
\(726\) 0 0
\(727\) 1126.98i 1.55018i 0.631853 + 0.775088i \(0.282295\pi\)
−0.631853 + 0.775088i \(0.717705\pi\)
\(728\) 0 0
\(729\) 1009.00 1.38409
\(730\) 0 0
\(731\) − 912.316i − 1.24804i
\(732\) 0 0
\(733\) 1301.39i 1.77543i 0.460392 + 0.887716i \(0.347709\pi\)
−0.460392 + 0.887716i \(0.652291\pi\)
\(734\) 0 0
\(735\) 490.000 0.666667
\(736\) 0 0
\(737\) 28.0000 0.0379919
\(738\) 0 0
\(739\) 982.000 1.32882 0.664411 0.747367i \(-0.268682\pi\)
0.664411 + 0.747367i \(0.268682\pi\)
\(740\) 0 0
\(741\) 804.984i 1.08635i
\(742\) 0 0
\(743\) 694.000 0.934051 0.467026 0.884244i \(-0.345326\pi\)
0.467026 + 0.884244i \(0.345326\pi\)
\(744\) 0 0
\(745\) 317.522i 0.426204i
\(746\) 0 0
\(747\) − 442.741i − 0.592693i
\(748\) 0 0
\(749\) −742.000 −0.990654
\(750\) 0 0
\(751\) −242.000 −0.322237 −0.161119 0.986935i \(-0.551510\pi\)
−0.161119 + 0.986935i \(0.551510\pi\)
\(752\) 0 0
\(753\) −1500.00 −1.99203
\(754\) 0 0
\(755\) 4.47214i 0.00592336i
\(756\) 0 0
\(757\) −106.000 −0.140026 −0.0700132 0.997546i \(-0.522304\pi\)
−0.0700132 + 0.997546i \(0.522304\pi\)
\(758\) 0 0
\(759\) 232.551i 0.306391i
\(760\) 0 0
\(761\) − 1100.15i − 1.44566i −0.691027 0.722829i \(-0.742842\pi\)
0.691027 0.722829i \(-0.257158\pi\)
\(762\) 0 0
\(763\) 994.000 1.30275
\(764\) 0 0
\(765\) 660.000 0.862745
\(766\) 0 0
\(767\) 540.000 0.704042
\(768\) 0 0
\(769\) − 1126.98i − 1.46551i −0.680492 0.732756i \(-0.738234\pi\)
0.680492 0.732756i \(-0.261766\pi\)
\(770\) 0 0
\(771\) −600.000 −0.778210
\(772\) 0 0
\(773\) − 818.401i − 1.05873i −0.848393 0.529367i \(-0.822430\pi\)
0.848393 0.529367i \(-0.177570\pi\)
\(774\) 0 0
\(775\) 268.328i 0.346230i
\(776\) 0 0
\(777\) − 438.269i − 0.564053i
\(778\) 0 0
\(779\) 360.000 0.462131
\(780\) 0 0
\(781\) 124.000 0.158771
\(782\) 0 0
\(783\) 196.774i 0.251308i
\(784\) 0 0
\(785\) 150.000 0.191083
\(786\) 0 0
\(787\) 684.237i 0.869424i 0.900569 + 0.434712i \(0.143150\pi\)
−0.900569 + 0.434712i \(0.856850\pi\)
\(788\) 0 0
\(789\) 152.053i 0.192716i
\(790\) 0 0
\(791\) 238.000 0.300885
\(792\) 0 0
\(793\) 1260.00 1.58890
\(794\) 0 0
\(795\) −340.000 −0.427673
\(796\) 0 0
\(797\) − 308.577i − 0.387174i −0.981083 0.193587i \(-0.937988\pi\)
0.981083 0.193587i \(-0.0620121\pi\)
\(798\) 0 0
\(799\) −720.000 −0.901126
\(800\) 0 0
\(801\) − 295.161i − 0.368491i
\(802\) 0 0
\(803\) − 107.331i − 0.133663i
\(804\) 0 0
\(805\) − 406.964i − 0.505546i
\(806\) 0 0
\(807\) 1140.00 1.41264
\(808\) 0 0
\(809\) 1358.00 1.67862 0.839308 0.543657i \(-0.182961\pi\)
0.839308 + 0.543657i \(0.182961\pi\)
\(810\) 0 0
\(811\) − 308.577i − 0.380490i −0.981737 0.190245i \(-0.939072\pi\)
0.981737 0.190245i \(-0.0609282\pi\)
\(812\) 0 0
\(813\) −1440.00 −1.77122
\(814\) 0 0
\(815\) − 76.0263i − 0.0932838i
\(816\) 0 0
\(817\) − 456.158i − 0.558333i
\(818\) 0 0
\(819\) 1033.06i 1.26137i
\(820\) 0 0
\(821\) 482.000 0.587089 0.293544 0.955945i \(-0.405165\pi\)
0.293544 + 0.955945i \(0.405165\pi\)
\(822\) 0 0
\(823\) −926.000 −1.12515 −0.562576 0.826746i \(-0.690190\pi\)
−0.562576 + 0.826746i \(0.690190\pi\)
\(824\) 0 0
\(825\) 44.7214i 0.0542077i
\(826\) 0 0
\(827\) 226.000 0.273277 0.136638 0.990621i \(-0.456370\pi\)
0.136638 + 0.990621i \(0.456370\pi\)
\(828\) 0 0
\(829\) 1462.39i 1.76404i 0.471213 + 0.882020i \(0.343816\pi\)
−0.471213 + 0.882020i \(0.656184\pi\)
\(830\) 0 0
\(831\) 62.6099i 0.0753428i
\(832\) 0 0
\(833\) − 1314.81i − 1.57840i
\(834\) 0 0
\(835\) −240.000 −0.287425
\(836\) 0 0
\(837\) −480.000 −0.573477
\(838\) 0 0
\(839\) − 831.817i − 0.991439i −0.868483 0.495719i \(-0.834904\pi\)
0.868483 0.495719i \(-0.165096\pi\)
\(840\) 0 0
\(841\) −357.000 −0.424495
\(842\) 0 0
\(843\) 8.94427i 0.0106100i
\(844\) 0 0
\(845\) 24.5967i 0.0291086i
\(846\) 0 0
\(847\) 819.000 0.966942
\(848\) 0 0
\(849\) 420.000 0.494700
\(850\) 0 0
\(851\) −364.000 −0.427732
\(852\) 0 0
\(853\) 40.2492i 0.0471855i 0.999722 + 0.0235927i \(0.00751050\pi\)
−0.999722 + 0.0235927i \(0.992489\pi\)
\(854\) 0 0
\(855\) 330.000 0.385965
\(856\) 0 0
\(857\) − 268.328i − 0.313102i −0.987670 0.156551i \(-0.949962\pi\)
0.987670 0.156551i \(-0.0500375\pi\)
\(858\) 0 0
\(859\) − 308.577i − 0.359229i −0.983737 0.179614i \(-0.942515\pi\)
0.983737 0.179614i \(-0.0574850\pi\)
\(860\) 0 0
\(861\) 840.000 0.975610
\(862\) 0 0
\(863\) 514.000 0.595597 0.297798 0.954629i \(-0.403748\pi\)
0.297798 + 0.954629i \(0.403748\pi\)
\(864\) 0 0
\(865\) −330.000 −0.381503
\(866\) 0 0
\(867\) − 1927.49i − 2.22317i
\(868\) 0 0
\(869\) 76.0000 0.0874568
\(870\) 0 0
\(871\) − 187.830i − 0.215648i
\(872\) 0 0
\(873\) − 295.161i − 0.338100i
\(874\) 0 0
\(875\) − 78.2624i − 0.0894427i
\(876\) 0 0
\(877\) −1306.00 −1.48917 −0.744584 0.667529i \(-0.767352\pi\)
−0.744584 + 0.667529i \(0.767352\pi\)
\(878\) 0 0
\(879\) 1500.00 1.70648
\(880\) 0 0
\(881\) 1126.98i 1.27920i 0.768706 + 0.639602i \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(882\) 0 0
\(883\) −1526.00 −1.72820 −0.864100 0.503321i \(-0.832111\pi\)
−0.864100 + 0.503321i \(0.832111\pi\)
\(884\) 0 0
\(885\) − 402.492i − 0.454793i
\(886\) 0 0
\(887\) 1556.30i 1.75457i 0.479970 + 0.877285i \(0.340648\pi\)
−0.479970 + 0.877285i \(0.659352\pi\)
\(888\) 0 0
\(889\) 1358.00 1.52756
\(890\) 0 0
\(891\) 118.000 0.132435
\(892\) 0 0
\(893\) −360.000 −0.403135
\(894\) 0 0
\(895\) 487.463i 0.544651i
\(896\) 0 0
\(897\) 1560.00 1.73913
\(898\) 0 0
\(899\) 1180.64i 1.31329i
\(900\) 0 0
\(901\) 912.316i 1.01256i
\(902\) 0 0
\(903\) − 1064.37i − 1.17870i
\(904\) 0 0
\(905\) 570.000 0.629834
\(906\) 0 0
\(907\) −734.000 −0.809261 −0.404631 0.914480i \(-0.632600\pi\)
−0.404631 + 0.914480i \(0.632600\pi\)
\(908\) 0 0
\(909\) 737.902i 0.811774i
\(910\) 0 0
\(911\) −1202.00 −1.31943 −0.659715 0.751516i \(-0.729323\pi\)
−0.659715 + 0.751516i \(0.729323\pi\)
\(912\) 0 0
\(913\) − 80.4984i − 0.0881692i
\(914\) 0 0
\(915\) − 939.149i − 1.02639i
\(916\) 0 0
\(917\) 845.234i 0.921738i
\(918\) 0 0
\(919\) 1282.00 1.39499 0.697497 0.716587i \(-0.254297\pi\)
0.697497 + 0.716587i \(0.254297\pi\)
\(920\) 0 0
\(921\) −900.000 −0.977199
\(922\) 0 0
\(923\) − 831.817i − 0.901210i
\(924\) 0 0
\(925\) −70.0000 −0.0756757
\(926\) 0 0
\(927\) − 1770.97i − 1.91043i
\(928\) 0 0
\(929\) 1126.98i 1.21311i 0.795042 + 0.606554i \(0.207449\pi\)
−0.795042 + 0.606554i \(0.792551\pi\)
\(930\) 0 0
\(931\) − 657.404i − 0.706127i
\(932\) 0 0
\(933\) 2280.00 2.44373
\(934\) 0 0
\(935\) 120.000 0.128342
\(936\) 0 0
\(937\) − 214.663i − 0.229096i −0.993418 0.114548i \(-0.963458\pi\)
0.993418 0.114548i \(-0.0365419\pi\)
\(938\) 0 0
\(939\) 1440.00 1.53355
\(940\) 0 0
\(941\) − 845.234i − 0.898229i −0.893474 0.449115i \(-0.851739\pi\)
0.893474 0.449115i \(-0.148261\pi\)
\(942\) 0 0
\(943\) − 697.653i − 0.739823i
\(944\) 0 0
\(945\) 140.000 0.148148
\(946\) 0 0
\(947\) −734.000 −0.775079 −0.387540 0.921853i \(-0.626675\pi\)
−0.387540 + 0.921853i \(0.626675\pi\)
\(948\) 0 0
\(949\) −720.000 −0.758693
\(950\) 0 0
\(951\) 1672.58i 1.75876i
\(952\) 0 0
\(953\) −934.000 −0.980063 −0.490031 0.871705i \(-0.663015\pi\)
−0.490031 + 0.871705i \(0.663015\pi\)
\(954\) 0 0
\(955\) − 129.692i − 0.135803i
\(956\) 0 0
\(957\) 196.774i 0.205615i
\(958\) 0 0
\(959\) 1162.00 1.21168
\(960\) 0 0
\(961\) −1919.00 −1.99688
\(962\) 0 0
\(963\) −1166.00 −1.21080
\(964\) 0 0
\(965\) − 460.630i − 0.477337i
\(966\) 0 0
\(967\) −314.000 −0.324716 −0.162358 0.986732i \(-0.551910\pi\)
−0.162358 + 0.986732i \(0.551910\pi\)
\(968\) 0 0
\(969\) − 1609.97i − 1.66147i
\(970\) 0 0
\(971\) − 147.580i − 0.151988i −0.997108 0.0759941i \(-0.975787\pi\)
0.997108 0.0759941i \(-0.0242130\pi\)
\(972\) 0 0
\(973\) 657.404i 0.675646i
\(974\) 0 0
\(975\) 300.000 0.307692
\(976\) 0 0
\(977\) −1486.00 −1.52098 −0.760491 0.649348i \(-0.775042\pi\)
−0.760491 + 0.649348i \(0.775042\pi\)
\(978\) 0 0
\(979\) − 53.6656i − 0.0548168i
\(980\) 0 0
\(981\) 1562.00 1.59225
\(982\) 0 0
\(983\) − 965.981i − 0.982687i −0.870966 0.491344i \(-0.836506\pi\)
0.870966 0.491344i \(-0.163494\pi\)
\(984\) 0 0
\(985\) 505.351i 0.513047i
\(986\) 0 0
\(987\) −840.000 −0.851064
\(988\) 0 0
\(989\) −884.000 −0.893832
\(990\) 0 0
\(991\) 58.0000 0.0585267 0.0292634 0.999572i \(-0.490684\pi\)
0.0292634 + 0.999572i \(0.490684\pi\)
\(992\) 0 0
\(993\) − 2155.57i − 2.17076i
\(994\) 0 0
\(995\) 300.000 0.301508
\(996\) 0 0
\(997\) 630.571i 0.632469i 0.948681 + 0.316234i \(0.102419\pi\)
−0.948681 + 0.316234i \(0.897581\pi\)
\(998\) 0 0
\(999\) − 125.220i − 0.125345i
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 560.3.f.a.321.2 2
4.3 odd 2 35.3.d.a.6.1 2
7.6 odd 2 inner 560.3.f.a.321.1 2
12.11 even 2 315.3.h.b.181.2 2
20.3 even 4 175.3.c.d.174.4 4
20.7 even 4 175.3.c.d.174.1 4
20.19 odd 2 175.3.d.f.76.2 2
28.3 even 6 245.3.h.b.166.2 4
28.11 odd 6 245.3.h.b.166.1 4
28.19 even 6 245.3.h.b.31.1 4
28.23 odd 6 245.3.h.b.31.2 4
28.27 even 2 35.3.d.a.6.2 yes 2
84.83 odd 2 315.3.h.b.181.1 2
140.27 odd 4 175.3.c.d.174.2 4
140.83 odd 4 175.3.c.d.174.3 4
140.139 even 2 175.3.d.f.76.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.d.a.6.1 2 4.3 odd 2
35.3.d.a.6.2 yes 2 28.27 even 2
175.3.c.d.174.1 4 20.7 even 4
175.3.c.d.174.2 4 140.27 odd 4
175.3.c.d.174.3 4 140.83 odd 4
175.3.c.d.174.4 4 20.3 even 4
175.3.d.f.76.1 2 140.139 even 2
175.3.d.f.76.2 2 20.19 odd 2
245.3.h.b.31.1 4 28.19 even 6
245.3.h.b.31.2 4 28.23 odd 6
245.3.h.b.166.1 4 28.11 odd 6
245.3.h.b.166.2 4 28.3 even 6
315.3.h.b.181.1 2 84.83 odd 2
315.3.h.b.181.2 2 12.11 even 2
560.3.f.a.321.1 2 7.6 odd 2 inner
560.3.f.a.321.2 2 1.1 even 1 trivial