Properties

Label 1520.2.g.g.1519.11
Level $1520$
Weight $2$
Character 1520.1519
Analytic conductor $12.137$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(1519,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.1519");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.g (of order \(2\), degree \(1\), 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1519.11
Root \(1.34352 + 0.775681i\) of defining polynomial
Character \(\chi\) \(=\) 1520.1519
Dual form 1520.2.g.g.1519.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.55136i q^{3} +(-2.03407 - 0.928731i) q^{5} -1.46240 q^{7} +0.593276 q^{9} +O(q^{10})\) \(q+1.55136i q^{3} +(-2.03407 - 0.928731i) q^{5} -1.46240 q^{7} +0.593276 q^{9} +2.19396i q^{11} +3.35861 q^{13} +(1.44080 - 3.15559i) q^{15} +3.77822i q^{17} +(-4.27492 - 0.851518i) q^{19} -2.26871i q^{21} -7.11926 q^{23} +(3.27492 + 3.77822i) q^{25} +5.57447i q^{27} -6.65663i q^{29} +1.93185 q^{31} -3.40362 q^{33} +(2.97463 + 1.35818i) q^{35} -4.72463 q^{37} +5.21042i q^{39} -7.88349i q^{41} -3.50000 q^{43} +(-1.20677 - 0.550994i) q^{45} +5.37408 q^{47} -4.86138 q^{49} -5.86138 q^{51} -10.9763 q^{53} +(2.03760 - 4.46267i) q^{55} +(1.32101 - 6.63194i) q^{57} -13.4615 q^{59} -3.80004 q^{61} -0.867609 q^{63} +(-6.83166 - 3.11924i) q^{65} +0.493890i q^{67} -11.0445i q^{69} -1.93185 q^{71} +11.0857i q^{73} +(-5.86138 + 5.08058i) q^{75} -3.20845i q^{77} -0.949745 q^{79} -6.86819 q^{81} -3.79241 q^{83} +(3.50895 - 7.68517i) q^{85} +10.3268 q^{87} -18.3043i q^{89} -4.91164 q^{91} +2.99700i q^{93} +(7.90467 + 5.70230i) q^{95} -0.649442 q^{97} +1.30162i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.55136i 0.895679i 0.894114 + 0.447840i \(0.147806\pi\)
−0.894114 + 0.447840i \(0.852194\pi\)
\(4\) 0 0
\(5\) −2.03407 0.928731i −0.909666 0.415341i
\(6\) 0 0
\(7\) −1.46240 −0.552736 −0.276368 0.961052i \(-0.589131\pi\)
−0.276368 + 0.961052i \(0.589131\pi\)
\(8\) 0 0
\(9\) 0.593276 0.197759
\(10\) 0 0
\(11\) 2.19396i 0.661503i 0.943718 + 0.330751i \(0.107302\pi\)
−0.943718 + 0.330751i \(0.892698\pi\)
\(12\) 0 0
\(13\) 3.35861 0.931510 0.465755 0.884914i \(-0.345783\pi\)
0.465755 + 0.884914i \(0.345783\pi\)
\(14\) 0 0
\(15\) 1.44080 3.15559i 0.372012 0.814769i
\(16\) 0 0
\(17\) 3.77822i 0.916352i 0.888861 + 0.458176i \(0.151497\pi\)
−0.888861 + 0.458176i \(0.848503\pi\)
\(18\) 0 0
\(19\) −4.27492 0.851518i −0.980733 0.195352i
\(20\) 0 0
\(21\) 2.26871i 0.495074i
\(22\) 0 0
\(23\) −7.11926 −1.48447 −0.742234 0.670141i \(-0.766234\pi\)
−0.742234 + 0.670141i \(0.766234\pi\)
\(24\) 0 0
\(25\) 3.27492 + 3.77822i 0.654983 + 0.755643i
\(26\) 0 0
\(27\) 5.57447i 1.07281i
\(28\) 0 0
\(29\) 6.65663i 1.23610i −0.786137 0.618052i \(-0.787922\pi\)
0.786137 0.618052i \(-0.212078\pi\)
\(30\) 0 0
\(31\) 1.93185 0.346971 0.173485 0.984836i \(-0.444497\pi\)
0.173485 + 0.984836i \(0.444497\pi\)
\(32\) 0 0
\(33\) −3.40362 −0.592494
\(34\) 0 0
\(35\) 2.97463 + 1.35818i 0.502805 + 0.229574i
\(36\) 0 0
\(37\) −4.72463 −0.776725 −0.388362 0.921507i \(-0.626959\pi\)
−0.388362 + 0.921507i \(0.626959\pi\)
\(38\) 0 0
\(39\) 5.21042i 0.834334i
\(40\) 0 0
\(41\) 7.88349i 1.23119i −0.788061 0.615597i \(-0.788915\pi\)
0.788061 0.615597i \(-0.211085\pi\)
\(42\) 0 0
\(43\) −3.50000 −0.533745 −0.266872 0.963732i \(-0.585990\pi\)
−0.266872 + 0.963732i \(0.585990\pi\)
\(44\) 0 0
\(45\) −1.20677 0.550994i −0.179894 0.0821374i
\(46\) 0 0
\(47\) 5.37408 0.783889 0.391945 0.919989i \(-0.371802\pi\)
0.391945 + 0.919989i \(0.371802\pi\)
\(48\) 0 0
\(49\) −4.86138 −0.694483
\(50\) 0 0
\(51\) −5.86138 −0.820757
\(52\) 0 0
\(53\) −10.9763 −1.50771 −0.753854 0.657042i \(-0.771807\pi\)
−0.753854 + 0.657042i \(0.771807\pi\)
\(54\) 0 0
\(55\) 2.03760 4.46267i 0.274749 0.601747i
\(56\) 0 0
\(57\) 1.32101 6.63194i 0.174972 0.878422i
\(58\) 0 0
\(59\) −13.4615 −1.75253 −0.876267 0.481826i \(-0.839974\pi\)
−0.876267 + 0.481826i \(0.839974\pi\)
\(60\) 0 0
\(61\) −3.80004 −0.486546 −0.243273 0.969958i \(-0.578221\pi\)
−0.243273 + 0.969958i \(0.578221\pi\)
\(62\) 0 0
\(63\) −0.867609 −0.109308
\(64\) 0 0
\(65\) −6.83166 3.11924i −0.847363 0.386895i
\(66\) 0 0
\(67\) 0.493890i 0.0603383i 0.999545 + 0.0301692i \(0.00960460\pi\)
−0.999545 + 0.0301692i \(0.990395\pi\)
\(68\) 0 0
\(69\) 11.0445i 1.32961i
\(70\) 0 0
\(71\) −1.93185 −0.229269 −0.114634 0.993408i \(-0.536570\pi\)
−0.114634 + 0.993408i \(0.536570\pi\)
\(72\) 0 0
\(73\) 11.0857i 1.29748i 0.761011 + 0.648739i \(0.224703\pi\)
−0.761011 + 0.648739i \(0.775297\pi\)
\(74\) 0 0
\(75\) −5.86138 + 5.08058i −0.676814 + 0.586655i
\(76\) 0 0
\(77\) 3.20845i 0.365636i
\(78\) 0 0
\(79\) −0.949745 −0.106855 −0.0534273 0.998572i \(-0.517015\pi\)
−0.0534273 + 0.998572i \(0.517015\pi\)
\(80\) 0 0
\(81\) −6.86819 −0.763133
\(82\) 0 0
\(83\) −3.79241 −0.416271 −0.208136 0.978100i \(-0.566740\pi\)
−0.208136 + 0.978100i \(0.566740\pi\)
\(84\) 0 0
\(85\) 3.50895 7.68517i 0.380599 0.833574i
\(86\) 0 0
\(87\) 10.3268 1.10715
\(88\) 0 0
\(89\) 18.3043i 1.94025i −0.242596 0.970127i \(-0.577999\pi\)
0.242596 0.970127i \(-0.422001\pi\)
\(90\) 0 0
\(91\) −4.91164 −0.514879
\(92\) 0 0
\(93\) 2.99700i 0.310774i
\(94\) 0 0
\(95\) 7.90467 + 5.70230i 0.811002 + 0.585044i
\(96\) 0 0
\(97\) −0.649442 −0.0659408 −0.0329704 0.999456i \(-0.510497\pi\)
−0.0329704 + 0.999456i \(0.510497\pi\)
\(98\) 0 0
\(99\) 1.30162i 0.130818i
\(100\) 0 0
\(101\) 0.749790 0.0746068 0.0373034 0.999304i \(-0.488123\pi\)
0.0373034 + 0.999304i \(0.488123\pi\)
\(102\) 0 0
\(103\) 2.28806i 0.225449i 0.993626 + 0.112725i \(0.0359578\pi\)
−0.993626 + 0.112725i \(0.964042\pi\)
\(104\) 0 0
\(105\) −2.10703 + 4.61473i −0.205625 + 0.450352i
\(106\) 0 0
\(107\) 13.7578i 1.33001i 0.746837 + 0.665007i \(0.231572\pi\)
−0.746837 + 0.665007i \(0.768428\pi\)
\(108\) 0 0
\(109\) 3.76421i 0.360545i −0.983617 0.180273i \(-0.942302\pi\)
0.983617 0.180273i \(-0.0576980\pi\)
\(110\) 0 0
\(111\) 7.32962i 0.695696i
\(112\) 0 0
\(113\) 1.32101 0.124270 0.0621352 0.998068i \(-0.480209\pi\)
0.0621352 + 0.998068i \(0.480209\pi\)
\(114\) 0 0
\(115\) 14.4811 + 6.61187i 1.35037 + 0.616560i
\(116\) 0 0
\(117\) 1.99258 0.184214
\(118\) 0 0
\(119\) 5.52527i 0.506501i
\(120\) 0 0
\(121\) 6.18655 0.562414
\(122\) 0 0
\(123\) 12.2302 1.10276
\(124\) 0 0
\(125\) −3.15248 10.7267i −0.281966 0.959424i
\(126\) 0 0
\(127\) 20.8056i 1.84620i −0.384557 0.923101i \(-0.625646\pi\)
0.384557 0.923101i \(-0.374354\pi\)
\(128\) 0 0
\(129\) 5.42976i 0.478064i
\(130\) 0 0
\(131\) 8.00153i 0.699097i −0.936918 0.349549i \(-0.886335\pi\)
0.936918 0.349549i \(-0.113665\pi\)
\(132\) 0 0
\(133\) 6.25165 + 1.24526i 0.542087 + 0.107978i
\(134\) 0 0
\(135\) 5.17719 11.3389i 0.445581 0.975896i
\(136\) 0 0
\(137\) 7.61972i 0.650997i 0.945543 + 0.325498i \(0.105532\pi\)
−0.945543 + 0.325498i \(0.894468\pi\)
\(138\) 0 0
\(139\) 9.81188i 0.832232i 0.909311 + 0.416116i \(0.136609\pi\)
−0.909311 + 0.416116i \(0.863391\pi\)
\(140\) 0 0
\(141\) 8.33713i 0.702113i
\(142\) 0 0
\(143\) 7.36864i 0.616197i
\(144\) 0 0
\(145\) −6.18222 + 13.5401i −0.513405 + 1.12444i
\(146\) 0 0
\(147\) 7.54176i 0.622034i
\(148\) 0 0
\(149\) −15.3678 −1.25898 −0.629488 0.777010i \(-0.716736\pi\)
−0.629488 + 0.777010i \(0.716736\pi\)
\(150\) 0 0
\(151\) 11.7228 0.953985 0.476992 0.878907i \(-0.341727\pi\)
0.476992 + 0.878907i \(0.341727\pi\)
\(152\) 0 0
\(153\) 2.24153i 0.181217i
\(154\) 0 0
\(155\) −3.92953 1.79417i −0.315627 0.144111i
\(156\) 0 0
\(157\) 10.0470i 0.801834i −0.916114 0.400917i \(-0.868692\pi\)
0.916114 0.400917i \(-0.131308\pi\)
\(158\) 0 0
\(159\) 17.0282i 1.35042i
\(160\) 0 0
\(161\) 10.4112 0.820519
\(162\) 0 0
\(163\) −17.1730 −1.34509 −0.672545 0.740056i \(-0.734799\pi\)
−0.672545 + 0.740056i \(0.734799\pi\)
\(164\) 0 0
\(165\) 6.92322 + 3.16105i 0.538972 + 0.246087i
\(166\) 0 0
\(167\) 9.76243i 0.755439i 0.925920 + 0.377720i \(0.123292\pi\)
−0.925920 + 0.377720i \(0.876708\pi\)
\(168\) 0 0
\(169\) −1.71975 −0.132288
\(170\) 0 0
\(171\) −2.53621 0.505186i −0.193949 0.0386325i
\(172\) 0 0
\(173\) −6.96823 −0.529785 −0.264892 0.964278i \(-0.585336\pi\)
−0.264892 + 0.964278i \(0.585336\pi\)
\(174\) 0 0
\(175\) −4.78925 5.52527i −0.362033 0.417671i
\(176\) 0 0
\(177\) 20.8836i 1.56971i
\(178\) 0 0
\(179\) −3.73639 −0.279271 −0.139635 0.990203i \(-0.544593\pi\)
−0.139635 + 0.990203i \(0.544593\pi\)
\(180\) 0 0
\(181\) 10.5703i 0.785688i 0.919605 + 0.392844i \(0.128509\pi\)
−0.919605 + 0.392844i \(0.871491\pi\)
\(182\) 0 0
\(183\) 5.89524i 0.435789i
\(184\) 0 0
\(185\) 9.61026 + 4.38791i 0.706560 + 0.322606i
\(186\) 0 0
\(187\) −8.28924 −0.606170
\(188\) 0 0
\(189\) 8.15212i 0.592979i
\(190\) 0 0
\(191\) 23.1684i 1.67641i 0.545357 + 0.838204i \(0.316394\pi\)
−0.545357 + 0.838204i \(0.683606\pi\)
\(192\) 0 0
\(193\) 26.6772 1.92027 0.960133 0.279543i \(-0.0901828\pi\)
0.960133 + 0.279543i \(0.0901828\pi\)
\(194\) 0 0
\(195\) 4.83908 10.5984i 0.346533 0.758965i
\(196\) 0 0
\(197\) 22.9268i 1.63347i 0.577016 + 0.816733i \(0.304217\pi\)
−0.577016 + 0.816733i \(0.695783\pi\)
\(198\) 0 0
\(199\) 14.7447i 1.04523i 0.852570 + 0.522613i \(0.175043\pi\)
−0.852570 + 0.522613i \(0.824957\pi\)
\(200\) 0 0
\(201\) −0.766203 −0.0540438
\(202\) 0 0
\(203\) 9.73467i 0.683240i
\(204\) 0 0
\(205\) −7.32165 + 16.0356i −0.511366 + 1.11998i
\(206\) 0 0
\(207\) −4.22369 −0.293567
\(208\) 0 0
\(209\) 1.86819 9.37898i 0.129226 0.648758i
\(210\) 0 0
\(211\) 18.2749 1.25810 0.629049 0.777366i \(-0.283445\pi\)
0.629049 + 0.777366i \(0.283445\pi\)
\(212\) 0 0
\(213\) 2.99700i 0.205351i
\(214\) 0 0
\(215\) 7.11926 + 3.25056i 0.485529 + 0.221686i
\(216\) 0 0
\(217\) −2.82514 −0.191783
\(218\) 0 0
\(219\) −17.1979 −1.16212
\(220\) 0 0
\(221\) 12.6895i 0.853591i
\(222\) 0 0
\(223\) 8.18331i 0.547995i −0.961730 0.273997i \(-0.911654\pi\)
0.961730 0.273997i \(-0.0883459\pi\)
\(224\) 0 0
\(225\) 1.94293 + 2.24153i 0.129529 + 0.149435i
\(226\) 0 0
\(227\) 19.1869i 1.27348i −0.771079 0.636739i \(-0.780283\pi\)
0.771079 0.636739i \(-0.219717\pi\)
\(228\) 0 0
\(229\) 18.4180 1.21710 0.608549 0.793516i \(-0.291752\pi\)
0.608549 + 0.793516i \(0.291752\pi\)
\(230\) 0 0
\(231\) 4.97746 0.327493
\(232\) 0 0
\(233\) 23.6447i 1.54902i 0.632564 + 0.774508i \(0.282002\pi\)
−0.632564 + 0.774508i \(0.717998\pi\)
\(234\) 0 0
\(235\) −10.9313 4.99107i −0.713077 0.325582i
\(236\) 0 0
\(237\) 1.47340i 0.0957075i
\(238\) 0 0
\(239\) 1.28195i 0.0829222i 0.999140 + 0.0414611i \(0.0132013\pi\)
−0.999140 + 0.0414611i \(0.986799\pi\)
\(240\) 0 0
\(241\) 16.2057i 1.04390i 0.852976 + 0.521950i \(0.174795\pi\)
−0.852976 + 0.521950i \(0.825205\pi\)
\(242\) 0 0
\(243\) 6.06836i 0.389286i
\(244\) 0 0
\(245\) 9.88841 + 4.51491i 0.631747 + 0.288447i
\(246\) 0 0
\(247\) −14.3578 2.85992i −0.913563 0.181972i
\(248\) 0 0
\(249\) 5.88340i 0.372846i
\(250\) 0 0
\(251\) 9.07288i 0.572675i 0.958129 + 0.286338i \(0.0924379\pi\)
−0.958129 + 0.286338i \(0.907562\pi\)
\(252\) 0 0
\(253\) 15.6193i 0.981980i
\(254\) 0 0
\(255\) 11.9225 + 5.44365i 0.746615 + 0.340894i
\(256\) 0 0
\(257\) −22.3731 −1.39560 −0.697798 0.716294i \(-0.745837\pi\)
−0.697798 + 0.716294i \(0.745837\pi\)
\(258\) 0 0
\(259\) 6.90931 0.429324
\(260\) 0 0
\(261\) 3.94922i 0.244451i
\(262\) 0 0
\(263\) −23.5342 −1.45118 −0.725591 0.688126i \(-0.758434\pi\)
−0.725591 + 0.688126i \(0.758434\pi\)
\(264\) 0 0
\(265\) 22.3266 + 10.1940i 1.37151 + 0.626213i
\(266\) 0 0
\(267\) 28.3966 1.73785
\(268\) 0 0
\(269\) 11.9163i 0.726551i −0.931682 0.363276i \(-0.881658\pi\)
0.931682 0.363276i \(-0.118342\pi\)
\(270\) 0 0
\(271\) 6.28374i 0.381710i −0.981618 0.190855i \(-0.938874\pi\)
0.981618 0.190855i \(-0.0611261\pi\)
\(272\) 0 0
\(273\) 7.61972i 0.461167i
\(274\) 0 0
\(275\) −8.28924 + 7.18503i −0.499860 + 0.433273i
\(276\) 0 0
\(277\) 19.6808i 1.18250i 0.806487 + 0.591252i \(0.201366\pi\)
−0.806487 + 0.591252i \(0.798634\pi\)
\(278\) 0 0
\(279\) 1.14612 0.0686165
\(280\) 0 0
\(281\) 15.4119i 0.919397i 0.888075 + 0.459699i \(0.152043\pi\)
−0.888075 + 0.459699i \(0.847957\pi\)
\(282\) 0 0
\(283\) −8.49163 −0.504775 −0.252388 0.967626i \(-0.581216\pi\)
−0.252388 + 0.967626i \(0.581216\pi\)
\(284\) 0 0
\(285\) −8.84633 + 12.2630i −0.524011 + 0.726397i
\(286\) 0 0
\(287\) 11.5288i 0.680526i
\(288\) 0 0
\(289\) 2.72508 0.160299
\(290\) 0 0
\(291\) 1.00752i 0.0590618i
\(292\) 0 0
\(293\) −16.2603 −0.949939 −0.474969 0.880002i \(-0.657541\pi\)
−0.474969 + 0.880002i \(0.657541\pi\)
\(294\) 0 0
\(295\) 27.3816 + 12.5021i 1.59422 + 0.727900i
\(296\) 0 0
\(297\) −12.2302 −0.709665
\(298\) 0 0
\(299\) −23.9108 −1.38280
\(300\) 0 0
\(301\) 5.11840 0.295020
\(302\) 0 0
\(303\) 1.16319i 0.0668238i
\(304\) 0 0
\(305\) 7.72957 + 3.52922i 0.442594 + 0.202083i
\(306\) 0 0
\(307\) 7.61145i 0.434408i 0.976126 + 0.217204i \(0.0696938\pi\)
−0.976126 + 0.217204i \(0.930306\pi\)
\(308\) 0 0
\(309\) −3.54961 −0.201930
\(310\) 0 0
\(311\) 25.3938i 1.43995i −0.693998 0.719976i \(-0.744153\pi\)
0.693998 0.719976i \(-0.255847\pi\)
\(312\) 0 0
\(313\) 8.75926i 0.495103i −0.968875 0.247551i \(-0.920374\pi\)
0.968875 0.247551i \(-0.0796259\pi\)
\(314\) 0 0
\(315\) 1.76478 + 0.805775i 0.0994341 + 0.0454003i
\(316\) 0 0
\(317\) 22.5349 1.26568 0.632842 0.774281i \(-0.281888\pi\)
0.632842 + 0.774281i \(0.281888\pi\)
\(318\) 0 0
\(319\) 14.6044 0.817687
\(320\) 0 0
\(321\) −21.3433 −1.19127
\(322\) 0 0
\(323\) 3.21722 16.1516i 0.179011 0.898697i
\(324\) 0 0
\(325\) 10.9992 + 12.6895i 0.610124 + 0.703890i
\(326\) 0 0
\(327\) 5.83964 0.322933
\(328\) 0 0
\(329\) −7.85906 −0.433284
\(330\) 0 0
\(331\) 16.3107 0.896517 0.448259 0.893904i \(-0.352044\pi\)
0.448259 + 0.893904i \(0.352044\pi\)
\(332\) 0 0
\(333\) −2.80301 −0.153604
\(334\) 0 0
\(335\) 0.458691 1.00461i 0.0250610 0.0548877i
\(336\) 0 0
\(337\) 32.7228 1.78253 0.891263 0.453486i \(-0.149820\pi\)
0.891263 + 0.453486i \(0.149820\pi\)
\(338\) 0 0
\(339\) 2.04937i 0.111306i
\(340\) 0 0
\(341\) 4.23840i 0.229522i
\(342\) 0 0
\(343\) 17.3461 0.936602
\(344\) 0 0
\(345\) −10.2574 + 22.4654i −0.552240 + 1.20950i
\(346\) 0 0
\(347\) −21.7333 −1.16671 −0.583353 0.812219i \(-0.698259\pi\)
−0.583353 + 0.812219i \(0.698259\pi\)
\(348\) 0 0
\(349\) −22.0090 −1.17811 −0.589057 0.808092i \(-0.700501\pi\)
−0.589057 + 0.808092i \(0.700501\pi\)
\(350\) 0 0
\(351\) 18.7225i 0.999331i
\(352\) 0 0
\(353\) 19.7441i 1.05087i −0.850833 0.525435i \(-0.823902\pi\)
0.850833 0.525435i \(-0.176098\pi\)
\(354\) 0 0
\(355\) 3.92953 + 1.79417i 0.208558 + 0.0952247i
\(356\) 0 0
\(357\) 8.57169 0.453662
\(358\) 0 0
\(359\) 4.91631i 0.259473i −0.991549 0.129737i \(-0.958587\pi\)
0.991549 0.129737i \(-0.0414132\pi\)
\(360\) 0 0
\(361\) 17.5498 + 7.28034i 0.923675 + 0.383176i
\(362\) 0 0
\(363\) 9.59758i 0.503742i
\(364\) 0 0
\(365\) 10.2956 22.5490i 0.538896 1.18027i
\(366\) 0 0
\(367\) −12.1813 −0.635860 −0.317930 0.948114i \(-0.602988\pi\)
−0.317930 + 0.948114i \(0.602988\pi\)
\(368\) 0 0
\(369\) 4.67709i 0.243480i
\(370\) 0 0
\(371\) 16.0517 0.833365
\(372\) 0 0
\(373\) 6.99111 0.361986 0.180993 0.983484i \(-0.442069\pi\)
0.180993 + 0.983484i \(0.442069\pi\)
\(374\) 0 0
\(375\) 16.6410 4.89063i 0.859336 0.252551i
\(376\) 0 0
\(377\) 22.3570i 1.15144i
\(378\) 0 0
\(379\) −20.1744 −1.03629 −0.518145 0.855293i \(-0.673377\pi\)
−0.518145 + 0.855293i \(0.673377\pi\)
\(380\) 0 0
\(381\) 32.2771 1.65361
\(382\) 0 0
\(383\) 29.1621i 1.49011i −0.667002 0.745056i \(-0.732423\pi\)
0.667002 0.745056i \(-0.267577\pi\)
\(384\) 0 0
\(385\) −2.97978 + 6.52622i −0.151864 + 0.332607i
\(386\) 0 0
\(387\) −2.07647 −0.105553
\(388\) 0 0
\(389\) 13.6278 0.690957 0.345479 0.938427i \(-0.387717\pi\)
0.345479 + 0.938427i \(0.387717\pi\)
\(390\) 0 0
\(391\) 26.8981i 1.36029i
\(392\) 0 0
\(393\) 12.4133 0.626167
\(394\) 0 0
\(395\) 1.93185 + 0.882057i 0.0972020 + 0.0443811i
\(396\) 0 0
\(397\) 13.3863i 0.671839i 0.941891 + 0.335920i \(0.109047\pi\)
−0.941891 + 0.335920i \(0.890953\pi\)
\(398\) 0 0
\(399\) −1.93185 + 9.69857i −0.0967135 + 0.485536i
\(400\) 0 0
\(401\) 18.3043i 0.914074i 0.889448 + 0.457037i \(0.151089\pi\)
−0.889448 + 0.457037i \(0.848911\pi\)
\(402\) 0 0
\(403\) 6.48833 0.323207
\(404\) 0 0
\(405\) 13.9704 + 6.37870i 0.694196 + 0.316960i
\(406\) 0 0
\(407\) 10.3656i 0.513806i
\(408\) 0 0
\(409\) 23.6505i 1.16944i 0.811235 + 0.584721i \(0.198796\pi\)
−0.811235 + 0.584721i \(0.801204\pi\)
\(410\) 0 0
\(411\) −11.8209 −0.583084
\(412\) 0 0
\(413\) 19.6861 0.968689
\(414\) 0 0
\(415\) 7.71405 + 3.52213i 0.378668 + 0.172895i
\(416\) 0 0
\(417\) −15.2218 −0.745413
\(418\) 0 0
\(419\) 24.2792i 1.18612i 0.805159 + 0.593059i \(0.202080\pi\)
−0.805159 + 0.593059i \(0.797920\pi\)
\(420\) 0 0
\(421\) 18.3249i 0.893100i 0.894759 + 0.446550i \(0.147347\pi\)
−0.894759 + 0.446550i \(0.852653\pi\)
\(422\) 0 0
\(423\) 3.18831 0.155021
\(424\) 0 0
\(425\) −14.2749 + 12.3733i −0.692435 + 0.600195i
\(426\) 0 0
\(427\) 5.55719 0.268931
\(428\) 0 0
\(429\) −11.4314 −0.551915
\(430\) 0 0
\(431\) −4.78109 −0.230297 −0.115148 0.993348i \(-0.536734\pi\)
−0.115148 + 0.993348i \(0.536734\pi\)
\(432\) 0 0
\(433\) −29.9008 −1.43694 −0.718470 0.695558i \(-0.755157\pi\)
−0.718470 + 0.695558i \(0.755157\pi\)
\(434\) 0 0
\(435\) −21.0056 9.59086i −1.00714 0.459846i
\(436\) 0 0
\(437\) 30.4342 + 6.06217i 1.45587 + 0.289993i
\(438\) 0 0
\(439\) 11.7551 0.561041 0.280521 0.959848i \(-0.409493\pi\)
0.280521 + 0.959848i \(0.409493\pi\)
\(440\) 0 0
\(441\) −2.88414 −0.137340
\(442\) 0 0
\(443\) 39.8004 1.89097 0.945486 0.325662i \(-0.105587\pi\)
0.945486 + 0.325662i \(0.105587\pi\)
\(444\) 0 0
\(445\) −16.9998 + 37.2324i −0.805868 + 1.76498i
\(446\) 0 0
\(447\) 23.8410i 1.12764i
\(448\) 0 0
\(449\) 3.72870i 0.175968i −0.996122 0.0879842i \(-0.971957\pi\)
0.996122 0.0879842i \(-0.0280425\pi\)
\(450\) 0 0
\(451\) 17.2960 0.814439
\(452\) 0 0
\(453\) 18.1862i 0.854464i
\(454\) 0 0
\(455\) 9.99063 + 4.56159i 0.468368 + 0.213851i
\(456\) 0 0
\(457\) 22.1456i 1.03593i −0.855403 0.517963i \(-0.826691\pi\)
0.855403 0.517963i \(-0.173309\pi\)
\(458\) 0 0
\(459\) −21.0616 −0.983069
\(460\) 0 0
\(461\) −10.0413 −0.467672 −0.233836 0.972276i \(-0.575128\pi\)
−0.233836 + 0.972276i \(0.575128\pi\)
\(462\) 0 0
\(463\) 21.7598 1.01126 0.505632 0.862749i \(-0.331259\pi\)
0.505632 + 0.862749i \(0.331259\pi\)
\(464\) 0 0
\(465\) 2.78341 6.09612i 0.129077 0.282701i
\(466\) 0 0
\(467\) −3.84631 −0.177986 −0.0889929 0.996032i \(-0.528365\pi\)
−0.0889929 + 0.996032i \(0.528365\pi\)
\(468\) 0 0
\(469\) 0.722266i 0.0333512i
\(470\) 0 0
\(471\) 15.5865 0.718186
\(472\) 0 0
\(473\) 7.67884i 0.353074i
\(474\) 0 0
\(475\) −10.7828 18.9402i −0.494748 0.869037i
\(476\) 0 0
\(477\) −6.51197 −0.298163
\(478\) 0 0
\(479\) 2.72216i 0.124379i −0.998064 0.0621893i \(-0.980192\pi\)
0.998064 0.0621893i \(-0.0198082\pi\)
\(480\) 0 0
\(481\) −15.8682 −0.723527
\(482\) 0 0
\(483\) 16.1516i 0.734921i
\(484\) 0 0
\(485\) 1.32101 + 0.603157i 0.0599841 + 0.0273879i
\(486\) 0 0
\(487\) 0.553011i 0.0250593i 0.999922 + 0.0125297i \(0.00398842\pi\)
−0.999922 + 0.0125297i \(0.996012\pi\)
\(488\) 0 0
\(489\) 26.6415i 1.20477i
\(490\) 0 0
\(491\) 23.5295i 1.06187i −0.847412 0.530936i \(-0.821841\pi\)
0.847412 0.530936i \(-0.178159\pi\)
\(492\) 0 0
\(493\) 25.1502 1.13271
\(494\) 0 0
\(495\) 1.20886 2.64760i 0.0543341 0.119001i
\(496\) 0 0
\(497\) 2.82514 0.126725
\(498\) 0 0
\(499\) 37.2206i 1.66622i −0.553106 0.833111i \(-0.686558\pi\)
0.553106 0.833111i \(-0.313442\pi\)
\(500\) 0 0
\(501\) −15.1451 −0.676631
\(502\) 0 0
\(503\) 33.4033 1.48938 0.744690 0.667411i \(-0.232598\pi\)
0.744690 + 0.667411i \(0.232598\pi\)
\(504\) 0 0
\(505\) −1.52513 0.696353i −0.0678673 0.0309873i
\(506\) 0 0
\(507\) 2.66795i 0.118488i
\(508\) 0 0
\(509\) 16.0562i 0.711677i −0.934547 0.355839i \(-0.884195\pi\)
0.934547 0.355839i \(-0.115805\pi\)
\(510\) 0 0
\(511\) 16.2117i 0.717162i
\(512\) 0 0
\(513\) 4.74676 23.8304i 0.209575 1.05214i
\(514\) 0 0
\(515\) 2.12499 4.65409i 0.0936384 0.205084i
\(516\) 0 0
\(517\) 11.7905i 0.518545i
\(518\) 0 0
\(519\) 10.8102i 0.474517i
\(520\) 0 0
\(521\) 16.3042i 0.714302i 0.934047 + 0.357151i \(0.116252\pi\)
−0.934047 + 0.357151i \(0.883748\pi\)
\(522\) 0 0
\(523\) 7.96826i 0.348428i 0.984708 + 0.174214i \(0.0557384\pi\)
−0.984708 + 0.174214i \(0.944262\pi\)
\(524\) 0 0
\(525\) 8.57169 7.42985i 0.374099 0.324265i
\(526\) 0 0
\(527\) 7.29895i 0.317947i
\(528\) 0 0
\(529\) 27.6838 1.20364
\(530\) 0 0
\(531\) −7.98637 −0.346579
\(532\) 0 0
\(533\) 26.4776i 1.14687i
\(534\) 0 0
\(535\) 12.7773 27.9843i 0.552410 1.20987i
\(536\) 0 0
\(537\) 5.79649i 0.250137i
\(538\) 0 0
\(539\) 10.6657i 0.459402i
\(540\) 0 0
\(541\) 2.79022 0.119961 0.0599805 0.998200i \(-0.480896\pi\)
0.0599805 + 0.998200i \(0.480896\pi\)
\(542\) 0 0
\(543\) −16.3984 −0.703724
\(544\) 0 0
\(545\) −3.49593 + 7.65667i −0.149749 + 0.327976i
\(546\) 0 0
\(547\) 15.0858i 0.645023i 0.946566 + 0.322511i \(0.104527\pi\)
−0.946566 + 0.322511i \(0.895473\pi\)
\(548\) 0 0
\(549\) −2.25448 −0.0962187
\(550\) 0 0
\(551\) −5.66824 + 28.4565i −0.241475 + 1.21229i
\(552\) 0 0
\(553\) 1.38891 0.0590624
\(554\) 0 0
\(555\) −6.80724 + 14.9090i −0.288951 + 0.632851i
\(556\) 0 0
\(557\) 8.40942i 0.356319i 0.984002 + 0.178159i \(0.0570143\pi\)
−0.984002 + 0.178159i \(0.942986\pi\)
\(558\) 0 0
\(559\) −11.7551 −0.497189
\(560\) 0 0
\(561\) 12.8596i 0.542933i
\(562\) 0 0
\(563\) 13.7472i 0.579376i −0.957121 0.289688i \(-0.906449\pi\)
0.957121 0.289688i \(-0.0935515\pi\)
\(564\) 0 0
\(565\) −2.68704 1.22687i −0.113045 0.0516146i
\(566\) 0 0
\(567\) 10.0441 0.421811
\(568\) 0 0
\(569\) 34.9636i 1.46575i −0.680362 0.732876i \(-0.738177\pi\)
0.680362 0.732876i \(-0.261823\pi\)
\(570\) 0 0
\(571\) 39.3338i 1.64607i 0.567993 + 0.823033i \(0.307720\pi\)
−0.567993 + 0.823033i \(0.692280\pi\)
\(572\) 0 0
\(573\) −35.9426 −1.50152
\(574\) 0 0
\(575\) −23.3150 26.8981i −0.972302 1.12173i
\(576\) 0 0
\(577\) 21.3816i 0.890128i −0.895499 0.445064i \(-0.853181\pi\)
0.895499 0.445064i \(-0.146819\pi\)
\(578\) 0 0
\(579\) 41.3860i 1.71994i
\(580\) 0 0
\(581\) 5.54603 0.230088
\(582\) 0 0
\(583\) 24.0815i 0.997353i
\(584\) 0 0
\(585\) −4.05306 1.85057i −0.167574 0.0765118i
\(586\) 0 0
\(587\) −0.143920 −0.00594020 −0.00297010 0.999996i \(-0.500945\pi\)
−0.00297010 + 0.999996i \(0.500945\pi\)
\(588\) 0 0
\(589\) −8.25850 1.64501i −0.340286 0.0677813i
\(590\) 0 0
\(591\) −35.5677 −1.46306
\(592\) 0 0
\(593\) 6.83002i 0.280475i 0.990118 + 0.140238i \(0.0447867\pi\)
−0.990118 + 0.140238i \(0.955213\pi\)
\(594\) 0 0
\(595\) −5.13149 + 11.2388i −0.210371 + 0.460746i
\(596\) 0 0
\(597\) −22.8744 −0.936187
\(598\) 0 0
\(599\) 9.79091 0.400046 0.200023 0.979791i \(-0.435898\pi\)
0.200023 + 0.979791i \(0.435898\pi\)
\(600\) 0 0
\(601\) 0.892334i 0.0363991i −0.999834 0.0181995i \(-0.994207\pi\)
0.999834 0.0181995i \(-0.00579341\pi\)
\(602\) 0 0
\(603\) 0.293014i 0.0119324i
\(604\) 0 0
\(605\) −12.5839 5.74564i −0.511609 0.233594i
\(606\) 0 0
\(607\) 19.0511i 0.773261i 0.922235 + 0.386630i \(0.126361\pi\)
−0.922235 + 0.386630i \(0.873639\pi\)
\(608\) 0 0
\(609\) −15.1020 −0.611964
\(610\) 0 0
\(611\) 18.0494 0.730201
\(612\) 0 0
\(613\) 1.40119i 0.0565935i −0.999600 0.0282968i \(-0.990992\pi\)
0.999600 0.0282968i \(-0.00900834\pi\)
\(614\) 0 0
\(615\) −24.8770 11.3585i −1.00314 0.458020i
\(616\) 0 0
\(617\) 17.6034i 0.708686i 0.935116 + 0.354343i \(0.115295\pi\)
−0.935116 + 0.354343i \(0.884705\pi\)
\(618\) 0 0
\(619\) 24.7967i 0.996663i −0.866987 0.498332i \(-0.833946\pi\)
0.866987 0.498332i \(-0.166054\pi\)
\(620\) 0 0
\(621\) 39.6861i 1.59255i
\(622\) 0 0
\(623\) 26.7683i 1.07245i
\(624\) 0 0
\(625\) −3.54983 + 24.7467i −0.141993 + 0.989868i
\(626\) 0 0
\(627\) 14.5502 + 2.89824i 0.581079 + 0.115745i
\(628\) 0 0
\(629\) 17.8507i 0.711753i
\(630\) 0 0
\(631\) 42.4713i 1.69076i 0.534168 + 0.845378i \(0.320625\pi\)
−0.534168 + 0.845378i \(0.679375\pi\)
\(632\) 0 0
\(633\) 28.3510i 1.12685i
\(634\) 0 0
\(635\) −19.3229 + 42.3202i −0.766804 + 1.67943i
\(636\) 0 0
\(637\) −16.3275 −0.646918
\(638\) 0 0
\(639\) −1.14612 −0.0453399
\(640\) 0 0
\(641\) 12.4209i 0.490597i −0.969448 0.245298i \(-0.921114\pi\)
0.969448 0.245298i \(-0.0788860\pi\)
\(642\) 0 0
\(643\) 36.5374 1.44089 0.720447 0.693510i \(-0.243937\pi\)
0.720447 + 0.693510i \(0.243937\pi\)
\(644\) 0 0
\(645\) −5.04279 + 11.0445i −0.198560 + 0.434878i
\(646\) 0 0
\(647\) −4.56726 −0.179557 −0.0897787 0.995962i \(-0.528616\pi\)
−0.0897787 + 0.995962i \(0.528616\pi\)
\(648\) 0 0
\(649\) 29.5339i 1.15931i
\(650\) 0 0
\(651\) 4.38282i 0.171776i
\(652\) 0 0
\(653\) 11.3938i 0.445873i 0.974833 + 0.222936i \(0.0715642\pi\)
−0.974833 + 0.222936i \(0.928436\pi\)
\(654\) 0 0
\(655\) −7.43127 + 16.2757i −0.290364 + 0.635945i
\(656\) 0 0
\(657\) 6.57686i 0.256588i
\(658\) 0 0
\(659\) −5.73407 −0.223367 −0.111684 0.993744i \(-0.535624\pi\)
−0.111684 + 0.993744i \(0.535624\pi\)
\(660\) 0 0
\(661\) 6.21794i 0.241850i 0.992662 + 0.120925i \(0.0385860\pi\)
−0.992662 + 0.120925i \(0.961414\pi\)
\(662\) 0 0
\(663\) −19.6861 −0.764544
\(664\) 0 0
\(665\) −11.5598 8.33905i −0.448270 0.323375i
\(666\) 0 0
\(667\) 47.3902i 1.83496i
\(668\) 0 0
\(669\) 12.6953 0.490827
\(670\) 0 0
\(671\) 8.33713i 0.321852i
\(672\) 0 0
\(673\) −13.6412 −0.525829 −0.262915 0.964819i \(-0.584684\pi\)
−0.262915 + 0.964819i \(0.584684\pi\)
\(674\) 0 0
\(675\) −21.0616 + 18.2559i −0.810660 + 0.702671i
\(676\) 0 0
\(677\) −27.1885 −1.04494 −0.522470 0.852658i \(-0.674989\pi\)
−0.522470 + 0.852658i \(0.674989\pi\)
\(678\) 0 0
\(679\) 0.949745 0.0364479
\(680\) 0 0
\(681\) 29.7658 1.14063
\(682\) 0 0
\(683\) 7.54176i 0.288577i −0.989536 0.144289i \(-0.953911\pi\)
0.989536 0.144289i \(-0.0460894\pi\)
\(684\) 0 0
\(685\) 7.07667 15.4991i 0.270386 0.592190i
\(686\) 0 0
\(687\) 28.5730i 1.09013i
\(688\) 0 0
\(689\) −36.8650 −1.40445
\(690\) 0 0
\(691\) 8.05081i 0.306267i −0.988205 0.153134i \(-0.951064\pi\)
0.988205 0.153134i \(-0.0489365\pi\)
\(692\) 0 0
\(693\) 1.90350i 0.0723078i
\(694\) 0 0
\(695\) 9.11259 19.9581i 0.345660 0.757053i
\(696\) 0 0
\(697\) 29.7855 1.12821
\(698\) 0 0
\(699\) −36.6815 −1.38742
\(700\) 0 0
\(701\) −24.0914 −0.909919 −0.454959 0.890512i \(-0.650346\pi\)
−0.454959 + 0.890512i \(0.650346\pi\)
\(702\) 0 0
\(703\) 20.1974 + 4.02311i 0.761760 + 0.151734i
\(704\) 0 0
\(705\) 7.74296 16.9584i 0.291617 0.638689i
\(706\) 0 0
\(707\) −1.09649 −0.0412379
\(708\) 0 0
\(709\) 20.8492 0.783009 0.391505 0.920176i \(-0.371955\pi\)
0.391505 + 0.920176i \(0.371955\pi\)
\(710\) 0 0
\(711\) −0.563461 −0.0211314
\(712\) 0 0
\(713\) −13.7533 −0.515067
\(714\) 0 0
\(715\) 6.84349 14.9884i 0.255932 0.560533i
\(716\) 0 0
\(717\) −1.98876 −0.0742717
\(718\) 0 0
\(719\) 31.5723i 1.17745i 0.808334 + 0.588725i \(0.200370\pi\)
−0.808334 + 0.588725i \(0.799630\pi\)
\(720\) 0 0
\(721\) 3.34606i 0.124614i
\(722\) 0 0
\(723\) −25.1409 −0.934999
\(724\) 0 0
\(725\) 25.1502 21.7999i 0.934054 0.809628i
\(726\) 0 0
\(727\) −23.4978 −0.871484 −0.435742 0.900072i \(-0.643514\pi\)
−0.435742 + 0.900072i \(0.643514\pi\)
\(728\) 0 0
\(729\) −30.0188 −1.11181
\(730\) 0 0
\(731\) 13.2237i 0.489098i
\(732\) 0 0
\(733\) 19.5959i 0.723792i 0.932219 + 0.361896i \(0.117870\pi\)
−0.932219 + 0.361896i \(0.882130\pi\)
\(734\) 0 0
\(735\) −7.00427 + 15.3405i −0.258356 + 0.565843i
\(736\) 0 0
\(737\) −1.08357 −0.0399140
\(738\) 0 0
\(739\) 24.9089i 0.916290i −0.888878 0.458145i \(-0.848514\pi\)
0.888878 0.458145i \(-0.151486\pi\)
\(740\) 0 0
\(741\) 4.43676 22.2741i 0.162989 0.818260i
\(742\) 0 0
\(743\) 32.3345i 1.18624i −0.805115 0.593119i \(-0.797897\pi\)
0.805115 0.593119i \(-0.202103\pi\)
\(744\) 0 0
\(745\) 31.2592 + 14.2725i 1.14525 + 0.522905i
\(746\) 0 0
\(747\) −2.24995 −0.0823213
\(748\) 0 0
\(749\) 20.1194i 0.735147i
\(750\) 0 0
\(751\) −43.0405 −1.57057 −0.785285 0.619134i \(-0.787484\pi\)
−0.785285 + 0.619134i \(0.787484\pi\)
\(752\) 0 0
\(753\) −14.0753 −0.512933
\(754\) 0 0
\(755\) −23.8450 10.8873i −0.867807 0.396229i
\(756\) 0 0
\(757\) 18.6163i 0.676622i 0.941034 + 0.338311i \(0.109856\pi\)
−0.941034 + 0.338311i \(0.890144\pi\)
\(758\) 0 0
\(759\) 24.2313 0.879539
\(760\) 0 0
\(761\) 9.35647 0.339172 0.169586 0.985515i \(-0.445757\pi\)
0.169586 + 0.985515i \(0.445757\pi\)
\(762\) 0 0
\(763\) 5.50478i 0.199286i
\(764\) 0 0
\(765\) 2.08178 4.55943i 0.0752668 0.164847i
\(766\) 0 0
\(767\) −45.2118 −1.63250
\(768\) 0 0
\(769\) −3.57965 −0.129085 −0.0645427 0.997915i \(-0.520559\pi\)
−0.0645427 + 0.997915i \(0.520559\pi\)
\(770\) 0 0
\(771\) 34.7088i 1.25001i
\(772\) 0 0
\(773\) −3.31435 −0.119209 −0.0596044 0.998222i \(-0.518984\pi\)
−0.0596044 + 0.998222i \(0.518984\pi\)
\(774\) 0 0
\(775\) 6.32665 + 7.29895i 0.227260 + 0.262186i
\(776\) 0 0
\(777\) 10.7188i 0.384536i
\(778\) 0 0
\(779\) −6.71294 + 33.7013i −0.240516 + 1.20747i
\(780\) 0 0
\(781\) 4.23840i 0.151662i
\(782\) 0 0
\(783\) 37.1072 1.32610
\(784\) 0 0
\(785\) −9.33092 + 20.4363i −0.333035 + 0.729401i
\(786\) 0 0
\(787\) 46.4213i 1.65474i −0.561657 0.827370i \(-0.689836\pi\)
0.561657 0.827370i \(-0.310164\pi\)
\(788\) 0 0
\(789\) 36.5101i 1.29979i
\(790\) 0 0
\(791\) −1.93185 −0.0686887
\(792\) 0 0
\(793\) −12.7629 −0.453223
\(794\) 0 0
\(795\) −15.8146 + 34.6366i −0.560886 + 1.22843i
\(796\) 0 0
\(797\) −32.4726 −1.15024 −0.575119 0.818070i \(-0.695044\pi\)
−0.575119 + 0.818070i \(0.695044\pi\)
\(798\) 0 0
\(799\) 20.3044i 0.718319i
\(800\) 0 0
\(801\) 10.8595i 0.383703i
\(802\) 0 0
\(803\) −24.3214 −0.858285
\(804\) 0 0
\(805\) −21.1772 9.66922i −0.746398 0.340795i
\(806\) 0 0
\(807\) 18.4865 0.650757
\(808\) 0 0
\(809\) 9.69828 0.340973 0.170487 0.985360i \(-0.445466\pi\)
0.170487 + 0.985360i \(0.445466\pi\)
\(810\) 0 0
\(811\) 21.1242 0.741769 0.370885 0.928679i \(-0.379055\pi\)
0.370885 + 0.928679i \(0.379055\pi\)
\(812\) 0 0
\(813\) 9.74836 0.341890
\(814\) 0 0
\(815\) 34.9311 + 15.9491i 1.22358 + 0.558671i
\(816\) 0 0
\(817\) 14.9622 + 2.98031i 0.523461 + 0.104268i
\(818\) 0 0
\(819\) −2.91396 −0.101822
\(820\) 0 0
\(821\) 16.1721 0.564410 0.282205 0.959354i \(-0.408934\pi\)
0.282205 + 0.959354i \(0.408934\pi\)
\(822\) 0 0
\(823\) −38.5211 −1.34276 −0.671380 0.741113i \(-0.734298\pi\)
−0.671380 + 0.741113i \(0.734298\pi\)
\(824\) 0 0
\(825\) −11.1466 12.8596i −0.388074 0.447714i
\(826\) 0 0
\(827\) 17.4324i 0.606182i 0.952962 + 0.303091i \(0.0980187\pi\)
−0.952962 + 0.303091i \(0.901981\pi\)
\(828\) 0 0
\(829\) 14.9228i 0.518289i 0.965839 + 0.259144i \(0.0834406\pi\)
−0.965839 + 0.259144i \(0.916559\pi\)
\(830\) 0 0
\(831\) −30.5320 −1.05914
\(832\) 0 0
\(833\) 18.3673i 0.636391i
\(834\) 0 0
\(835\) 9.06667 19.8575i 0.313765 0.687197i
\(836\) 0 0
\(837\) 10.7691i 0.372233i
\(838\) 0 0
\(839\) −18.4357 −0.636471 −0.318235 0.948012i \(-0.603090\pi\)
−0.318235 + 0.948012i \(0.603090\pi\)
\(840\) 0 0
\(841\) −15.3107 −0.527955
\(842\) 0 0
\(843\) −23.9094 −0.823485
\(844\) 0 0
\(845\) 3.49810 + 1.59718i 0.120338 + 0.0549448i
\(846\) 0 0
\(847\) −9.04723 −0.310866
\(848\) 0 0
\(849\) 13.1736i 0.452117i
\(850\) 0 0
\(851\) 33.6359 1.15302
\(852\) 0 0
\(853\) 52.1480i 1.78551i −0.450538 0.892757i \(-0.648768\pi\)
0.450538 0.892757i \(-0.351232\pi\)
\(854\) 0 0
\(855\) 4.68965 + 3.38304i 0.160383 + 0.115698i
\(856\) 0 0
\(857\) −40.1131 −1.37024 −0.685119 0.728431i \(-0.740250\pi\)
−0.685119 + 0.728431i \(0.740250\pi\)
\(858\) 0 0
\(859\) 48.6430i 1.65968i 0.558003 + 0.829839i \(0.311568\pi\)
−0.558003 + 0.829839i \(0.688432\pi\)
\(860\) 0 0
\(861\) −17.8854 −0.609533
\(862\) 0 0
\(863\) 18.3735i 0.625442i 0.949845 + 0.312721i \(0.101241\pi\)
−0.949845 + 0.312721i \(0.898759\pi\)
\(864\) 0 0
\(865\) 14.1739 + 6.47161i 0.481927 + 0.220041i
\(866\) 0 0
\(867\) 4.22759i 0.143576i
\(868\) 0 0
\(869\) 2.08370i 0.0706846i
\(870\) 0 0
\(871\) 1.65878i 0.0562058i
\(872\) 0 0
\(873\) −0.385298 −0.0130404
\(874\) 0 0
\(875\) 4.61019 + 15.6867i 0.155853 + 0.530308i
\(876\) 0 0
\(877\) 27.1870 0.918041 0.459020 0.888426i \(-0.348201\pi\)
0.459020 + 0.888426i \(0.348201\pi\)
\(878\) 0 0
\(879\) 25.2257i 0.850840i
\(880\) 0 0
\(881\) 19.3814 0.652976 0.326488 0.945201i \(-0.394135\pi\)
0.326488 + 0.945201i \(0.394135\pi\)
\(882\) 0 0
\(883\) 4.74348 0.159631 0.0798154 0.996810i \(-0.474567\pi\)
0.0798154 + 0.996810i \(0.474567\pi\)
\(884\) 0 0
\(885\) −19.3953 + 42.4788i −0.651964 + 1.42791i
\(886\) 0 0
\(887\) 29.1621i 0.979166i −0.871957 0.489583i \(-0.837149\pi\)
0.871957 0.489583i \(-0.162851\pi\)
\(888\) 0 0
\(889\) 30.4262i 1.02046i
\(890\) 0 0
\(891\) 15.0685i 0.504814i
\(892\) 0 0
\(893\) −22.9737 4.57612i −0.768786 0.153134i
\(894\) 0 0
\(895\) 7.60009 + 3.47010i 0.254043 + 0.115993i
\(896\) 0 0
\(897\) 37.0943i 1.23854i
\(898\) 0 0
\(899\) 12.8596i 0.428892i
\(900\) 0 0
\(901\) 41.4708i 1.38159i
\(902\) 0 0
\(903\) 7.94050i 0.264243i
\(904\) 0 0
\(905\) 9.81701 21.5009i 0.326329 0.714713i
\(906\) 0 0
\(907\) 34.2149i 1.13609i −0.822999 0.568043i \(-0.807701\pi\)
0.822999 0.568043i \(-0.192299\pi\)
\(908\) 0 0
\(909\) 0.444832 0.0147542
\(910\) 0 0
\(911\) −38.6457 −1.28039 −0.640195 0.768213i \(-0.721146\pi\)
−0.640195 + 0.768213i \(0.721146\pi\)
\(912\) 0 0
\(913\) 8.32039i 0.275365i
\(914\) 0 0
\(915\) −5.47510 + 11.9914i −0.181001 + 0.396422i
\(916\) 0 0
\(917\) 11.7015i 0.386416i
\(918\) 0 0
\(919\) 34.4725i 1.13714i −0.822634 0.568572i \(-0.807496\pi\)
0.822634 0.568572i \(-0.192504\pi\)
\(920\) 0 0
\(921\) −11.8081 −0.389091
\(922\) 0 0
\(923\) −6.48833 −0.213566
\(924\) 0 0
\(925\) −15.4728 17.8507i −0.508742 0.586927i
\(926\) 0 0
\(927\) 1.35745i 0.0445846i
\(928\) 0 0
\(929\) 3.59777 0.118039 0.0590195 0.998257i \(-0.481203\pi\)
0.0590195 + 0.998257i \(0.481203\pi\)
\(930\) 0 0
\(931\) 20.7820 + 4.13955i 0.681102 + 0.135668i
\(932\) 0 0
\(933\) 39.3950 1.28974
\(934\) 0 0
\(935\) 16.8609 + 7.69848i 0.551412 + 0.251767i
\(936\) 0 0
\(937\) 37.2584i 1.21718i 0.793485 + 0.608590i \(0.208265\pi\)
−0.793485 + 0.608590i \(0.791735\pi\)
\(938\) 0 0
\(939\) 13.5888 0.443453
\(940\) 0 0
\(941\) 35.4378i 1.15524i 0.816306 + 0.577620i \(0.196019\pi\)
−0.816306 + 0.577620i \(0.803981\pi\)
\(942\) 0 0
\(943\) 56.1246i 1.82767i
\(944\) 0 0
\(945\) −7.57113 + 16.5820i −0.246289 + 0.539413i
\(946\) 0 0
\(947\) 3.40034 0.110496 0.0552481 0.998473i \(-0.482405\pi\)
0.0552481 + 0.998473i \(0.482405\pi\)
\(948\) 0 0
\(949\) 37.2324i 1.20861i
\(950\) 0 0
\(951\) 34.9597i 1.13365i
\(952\) 0 0
\(953\) 28.6783 0.928982 0.464491 0.885578i \(-0.346237\pi\)
0.464491 + 0.885578i \(0.346237\pi\)
\(954\) 0 0
\(955\) 21.5172 47.1263i 0.696281 1.52497i
\(956\) 0 0
\(957\) 22.6566i 0.732385i
\(958\) 0 0
\(959\) 11.1431i 0.359829i
\(960\) 0 0
\(961\) −27.2680 −0.879611
\(962\) 0 0
\(963\) 8.16217i 0.263022i
\(964\) 0 0
\(965\) −54.2634 24.7759i −1.74680 0.797566i
\(966\) 0 0
\(967\) −14.4948 −0.466121 −0.233061 0.972462i \(-0.574874\pi\)
−0.233061 + 0.972462i \(0.574874\pi\)
\(968\) 0 0
\(969\) 25.0569 + 4.99107i 0.804944 + 0.160336i
\(970\) 0 0
\(971\) 16.7760 0.538367 0.269183 0.963089i \(-0.413246\pi\)
0.269183 + 0.963089i \(0.413246\pi\)
\(972\) 0 0
\(973\) 14.3489i 0.460005i
\(974\) 0 0
\(975\) −19.6861 + 17.0637i −0.630459 + 0.546475i
\(976\) 0 0
\(977\) 50.1439 1.60425 0.802124 0.597158i \(-0.203704\pi\)
0.802124 + 0.597158i \(0.203704\pi\)
\(978\) 0 0
\(979\) 40.1589 1.28348
\(980\) 0 0
\(981\) 2.23321i 0.0713010i
\(982\) 0 0
\(983\) 27.2922i 0.870487i 0.900313 + 0.435243i \(0.143338\pi\)
−0.900313 + 0.435243i \(0.856662\pi\)
\(984\) 0 0
\(985\) 21.2928 46.6348i 0.678445 1.48591i
\(986\) 0 0
\(987\) 12.1922i 0.388083i
\(988\) 0 0
\(989\) 24.9174 0.792327
\(990\) 0 0
\(991\) −58.5622 −1.86029 −0.930145 0.367191i \(-0.880319\pi\)
−0.930145 + 0.367191i \(0.880319\pi\)
\(992\) 0 0
\(993\) 25.3038i 0.802992i
\(994\) 0 0
\(995\) 13.6939 29.9919i 0.434125 0.950806i
\(996\) 0 0
\(997\) 48.1715i 1.52561i −0.646631 0.762803i \(-0.723823\pi\)
0.646631 0.762803i \(-0.276177\pi\)
\(998\) 0 0
\(999\) 26.3373i 0.833276i
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 1520.2.g.g.1519.11 yes 16
4.3 odd 2 1520.2.g.e.1519.6 yes 16
5.4 even 2 inner 1520.2.g.g.1519.6 yes 16
19.18 odd 2 1520.2.g.e.1519.5 16
20.19 odd 2 1520.2.g.e.1519.11 yes 16
76.75 even 2 inner 1520.2.g.g.1519.12 yes 16
95.94 odd 2 1520.2.g.e.1519.12 yes 16
380.379 even 2 inner 1520.2.g.g.1519.5 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1520.2.g.e.1519.5 16 19.18 odd 2
1520.2.g.e.1519.6 yes 16 4.3 odd 2
1520.2.g.e.1519.11 yes 16 20.19 odd 2
1520.2.g.e.1519.12 yes 16 95.94 odd 2
1520.2.g.g.1519.5 yes 16 380.379 even 2 inner
1520.2.g.g.1519.6 yes 16 5.4 even 2 inner
1520.2.g.g.1519.11 yes 16 1.1 even 1 trivial
1520.2.g.g.1519.12 yes 16 76.75 even 2 inner