Properties

Label 2240.2.bd.b.561.6
Level $2240$
Weight $2$
Character 2240.561
Analytic conductor $17.886$
Analytic rank $0$
Dimension $52$
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] = [2240,2,Mod(561,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.561");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.bd (of order \(4\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 560)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 561.6
Character \(\chi\) \(=\) 2240.561
Dual form 2240.2.bd.b.1681.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.56700 + 1.56700i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} -1.91098i q^{9} +O(q^{10})\) \(q+(-1.56700 + 1.56700i) q^{3} +(-0.707107 - 0.707107i) q^{5} +1.00000i q^{7} -1.91098i q^{9} +(-1.19408 - 1.19408i) q^{11} +(-1.08857 + 1.08857i) q^{13} +2.21607 q^{15} -3.78951 q^{17} +(5.95387 - 5.95387i) q^{19} +(-1.56700 - 1.56700i) q^{21} +2.53073i q^{23} +1.00000i q^{25} +(-1.70649 - 1.70649i) q^{27} +(5.38242 - 5.38242i) q^{29} -1.98299 q^{31} +3.74224 q^{33} +(0.707107 - 0.707107i) q^{35} +(-2.14607 - 2.14607i) q^{37} -3.41159i q^{39} +9.95744i q^{41} +(-0.342507 - 0.342507i) q^{43} +(-1.35127 + 1.35127i) q^{45} +4.32467 q^{47} -1.00000 q^{49} +(5.93817 - 5.93817i) q^{51} +(-0.0660626 - 0.0660626i) q^{53} +1.68868i q^{55} +18.6595i q^{57} +(2.96474 + 2.96474i) q^{59} +(6.13245 - 6.13245i) q^{61} +1.91098 q^{63} +1.53948 q^{65} +(2.21747 - 2.21747i) q^{67} +(-3.96566 - 3.96566i) q^{69} -11.8350i q^{71} -8.17814i q^{73} +(-1.56700 - 1.56700i) q^{75} +(1.19408 - 1.19408i) q^{77} +8.46315 q^{79} +11.0811 q^{81} +(-4.79431 + 4.79431i) q^{83} +(2.67959 + 2.67959i) q^{85} +16.8685i q^{87} +17.9660i q^{89} +(-1.08857 - 1.08857i) q^{91} +(3.10734 - 3.10734i) q^{93} -8.42005 q^{95} -8.71168 q^{97} +(-2.28186 + 2.28186i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q+O(q^{10}) \) Copy content Toggle raw display \( 52 q + 4 q^{11} - 8 q^{15} - 8 q^{19} + 24 q^{27} + 4 q^{29} - 12 q^{37} - 36 q^{43} - 52 q^{49} + 8 q^{51} - 4 q^{53} - 24 q^{59} - 16 q^{61} + 68 q^{63} + 40 q^{65} + 12 q^{67} - 72 q^{69} - 4 q^{77} + 16 q^{79} - 116 q^{81} + 16 q^{85} + 8 q^{93} + 32 q^{95} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) −1.56700 + 1.56700i −0.904709 + 0.904709i −0.995839 0.0911305i \(-0.970952\pi\)
0.0911305 + 0.995839i \(0.470952\pi\)
\(4\) 0 0
\(5\) −0.707107 0.707107i −0.316228 0.316228i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.91098i 0.636995i
\(10\) 0 0
\(11\) −1.19408 1.19408i −0.360027 0.360027i 0.503796 0.863823i \(-0.331936\pi\)
−0.863823 + 0.503796i \(0.831936\pi\)
\(12\) 0 0
\(13\) −1.08857 + 1.08857i −0.301916 + 0.301916i −0.841763 0.539847i \(-0.818482\pi\)
0.539847 + 0.841763i \(0.318482\pi\)
\(14\) 0 0
\(15\) 2.21607 0.572188
\(16\) 0 0
\(17\) −3.78951 −0.919092 −0.459546 0.888154i \(-0.651988\pi\)
−0.459546 + 0.888154i \(0.651988\pi\)
\(18\) 0 0
\(19\) 5.95387 5.95387i 1.36591 1.36591i 0.499733 0.866180i \(-0.333432\pi\)
0.866180 0.499733i \(-0.166568\pi\)
\(20\) 0 0
\(21\) −1.56700 1.56700i −0.341948 0.341948i
\(22\) 0 0
\(23\) 2.53073i 0.527694i 0.964565 + 0.263847i \(0.0849914\pi\)
−0.964565 + 0.263847i \(0.915009\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) −1.70649 1.70649i −0.328414 0.328414i
\(28\) 0 0
\(29\) 5.38242 5.38242i 0.999489 0.999489i −0.000510422 1.00000i \(-0.500162\pi\)
1.00000 0.000510422i \(0.000162472\pi\)
\(30\) 0 0
\(31\) −1.98299 −0.356155 −0.178077 0.984016i \(-0.556988\pi\)
−0.178077 + 0.984016i \(0.556988\pi\)
\(32\) 0 0
\(33\) 3.74224 0.651440
\(34\) 0 0
\(35\) 0.707107 0.707107i 0.119523 0.119523i
\(36\) 0 0
\(37\) −2.14607 2.14607i −0.352811 0.352811i 0.508343 0.861155i \(-0.330258\pi\)
−0.861155 + 0.508343i \(0.830258\pi\)
\(38\) 0 0
\(39\) 3.41159i 0.546292i
\(40\) 0 0
\(41\) 9.95744i 1.55509i 0.628827 + 0.777546i \(0.283536\pi\)
−0.628827 + 0.777546i \(0.716464\pi\)
\(42\) 0 0
\(43\) −0.342507 0.342507i −0.0522318 0.0522318i 0.680508 0.732740i \(-0.261759\pi\)
−0.732740 + 0.680508i \(0.761759\pi\)
\(44\) 0 0
\(45\) −1.35127 + 1.35127i −0.201435 + 0.201435i
\(46\) 0 0
\(47\) 4.32467 0.630818 0.315409 0.948956i \(-0.397858\pi\)
0.315409 + 0.948956i \(0.397858\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 5.93817 5.93817i 0.831510 0.831510i
\(52\) 0 0
\(53\) −0.0660626 0.0660626i −0.00907440 0.00907440i 0.702555 0.711629i \(-0.252042\pi\)
−0.711629 + 0.702555i \(0.752042\pi\)
\(54\) 0 0
\(55\) 1.68868i 0.227701i
\(56\) 0 0
\(57\) 18.6595i 2.47151i
\(58\) 0 0
\(59\) 2.96474 + 2.96474i 0.385976 + 0.385976i 0.873250 0.487273i \(-0.162008\pi\)
−0.487273 + 0.873250i \(0.662008\pi\)
\(60\) 0 0
\(61\) 6.13245 6.13245i 0.785180 0.785180i −0.195520 0.980700i \(-0.562639\pi\)
0.980700 + 0.195520i \(0.0626395\pi\)
\(62\) 0 0
\(63\) 1.91098 0.240761
\(64\) 0 0
\(65\) 1.53948 0.190949
\(66\) 0 0
\(67\) 2.21747 2.21747i 0.270907 0.270907i −0.558558 0.829465i \(-0.688645\pi\)
0.829465 + 0.558558i \(0.188645\pi\)
\(68\) 0 0
\(69\) −3.96566 3.96566i −0.477409 0.477409i
\(70\) 0 0
\(71\) 11.8350i 1.40456i −0.711902 0.702279i \(-0.752166\pi\)
0.711902 0.702279i \(-0.247834\pi\)
\(72\) 0 0
\(73\) 8.17814i 0.957179i −0.878039 0.478589i \(-0.841148\pi\)
0.878039 0.478589i \(-0.158852\pi\)
\(74\) 0 0
\(75\) −1.56700 1.56700i −0.180942 0.180942i
\(76\) 0 0
\(77\) 1.19408 1.19408i 0.136078 0.136078i
\(78\) 0 0
\(79\) 8.46315 0.952179 0.476090 0.879397i \(-0.342054\pi\)
0.476090 + 0.879397i \(0.342054\pi\)
\(80\) 0 0
\(81\) 11.0811 1.23123
\(82\) 0 0
\(83\) −4.79431 + 4.79431i −0.526244 + 0.526244i −0.919450 0.393206i \(-0.871366\pi\)
0.393206 + 0.919450i \(0.371366\pi\)
\(84\) 0 0
\(85\) 2.67959 + 2.67959i 0.290642 + 0.290642i
\(86\) 0 0
\(87\) 16.8685i 1.80849i
\(88\) 0 0
\(89\) 17.9660i 1.90439i 0.305483 + 0.952197i \(0.401182\pi\)
−0.305483 + 0.952197i \(0.598818\pi\)
\(90\) 0 0
\(91\) −1.08857 1.08857i −0.114114 0.114114i
\(92\) 0 0
\(93\) 3.10734 3.10734i 0.322216 0.322216i
\(94\) 0 0
\(95\) −8.42005 −0.863879
\(96\) 0 0
\(97\) −8.71168 −0.884537 −0.442269 0.896883i \(-0.645826\pi\)
−0.442269 + 0.896883i \(0.645826\pi\)
\(98\) 0 0
\(99\) −2.28186 + 2.28186i −0.229336 + 0.229336i
\(100\) 0 0
\(101\) 2.52435 + 2.52435i 0.251182 + 0.251182i 0.821455 0.570273i \(-0.193163\pi\)
−0.570273 + 0.821455i \(0.693163\pi\)
\(102\) 0 0
\(103\) 1.60516i 0.158161i −0.996868 0.0790807i \(-0.974802\pi\)
0.996868 0.0790807i \(-0.0251985\pi\)
\(104\) 0 0
\(105\) 2.21607i 0.216267i
\(106\) 0 0
\(107\) 12.4612 + 12.4612i 1.20467 + 1.20467i 0.972731 + 0.231936i \(0.0745060\pi\)
0.231936 + 0.972731i \(0.425494\pi\)
\(108\) 0 0
\(109\) 2.73955 2.73955i 0.262401 0.262401i −0.563628 0.826029i \(-0.690595\pi\)
0.826029 + 0.563628i \(0.190595\pi\)
\(110\) 0 0
\(111\) 6.72578 0.638383
\(112\) 0 0
\(113\) 15.1618 1.42630 0.713150 0.701012i \(-0.247268\pi\)
0.713150 + 0.701012i \(0.247268\pi\)
\(114\) 0 0
\(115\) 1.78950 1.78950i 0.166871 0.166871i
\(116\) 0 0
\(117\) 2.08025 + 2.08025i 0.192319 + 0.192319i
\(118\) 0 0
\(119\) 3.78951i 0.347384i
\(120\) 0 0
\(121\) 8.14837i 0.740761i
\(122\) 0 0
\(123\) −15.6033 15.6033i −1.40690 1.40690i
\(124\) 0 0
\(125\) 0.707107 0.707107i 0.0632456 0.0632456i
\(126\) 0 0
\(127\) 15.4711 1.37284 0.686418 0.727208i \(-0.259182\pi\)
0.686418 + 0.727208i \(0.259182\pi\)
\(128\) 0 0
\(129\) 1.07342 0.0945091
\(130\) 0 0
\(131\) −7.44063 + 7.44063i −0.650091 + 0.650091i −0.953015 0.302924i \(-0.902037\pi\)
0.302924 + 0.953015i \(0.402037\pi\)
\(132\) 0 0
\(133\) 5.95387 + 5.95387i 0.516266 + 0.516266i
\(134\) 0 0
\(135\) 2.41334i 0.207707i
\(136\) 0 0
\(137\) 14.2513i 1.21757i 0.793337 + 0.608783i \(0.208342\pi\)
−0.793337 + 0.608783i \(0.791658\pi\)
\(138\) 0 0
\(139\) 13.6749 + 13.6749i 1.15989 + 1.15989i 0.984499 + 0.175393i \(0.0561196\pi\)
0.175393 + 0.984499i \(0.443880\pi\)
\(140\) 0 0
\(141\) −6.77677 + 6.77677i −0.570707 + 0.570707i
\(142\) 0 0
\(143\) 2.59968 0.217396
\(144\) 0 0
\(145\) −7.61188 −0.632133
\(146\) 0 0
\(147\) 1.56700 1.56700i 0.129244 0.129244i
\(148\) 0 0
\(149\) 6.08027 + 6.08027i 0.498115 + 0.498115i 0.910851 0.412736i \(-0.135427\pi\)
−0.412736 + 0.910851i \(0.635427\pi\)
\(150\) 0 0
\(151\) 7.92748i 0.645129i 0.946547 + 0.322565i \(0.104545\pi\)
−0.946547 + 0.322565i \(0.895455\pi\)
\(152\) 0 0
\(153\) 7.24170i 0.585457i
\(154\) 0 0
\(155\) 1.40218 + 1.40218i 0.112626 + 0.112626i
\(156\) 0 0
\(157\) 0.779619 0.779619i 0.0622204 0.0622204i −0.675312 0.737532i \(-0.735991\pi\)
0.737532 + 0.675312i \(0.235991\pi\)
\(158\) 0 0
\(159\) 0.207040 0.0164194
\(160\) 0 0
\(161\) −2.53073 −0.199449
\(162\) 0 0
\(163\) 6.01759 6.01759i 0.471334 0.471334i −0.431012 0.902346i \(-0.641843\pi\)
0.902346 + 0.431012i \(0.141843\pi\)
\(164\) 0 0
\(165\) −2.64616 2.64616i −0.206003 0.206003i
\(166\) 0 0
\(167\) 17.9564i 1.38951i 0.719248 + 0.694753i \(0.244486\pi\)
−0.719248 + 0.694753i \(0.755514\pi\)
\(168\) 0 0
\(169\) 10.6300i 0.817693i
\(170\) 0 0
\(171\) −11.3778 11.3778i −0.870079 0.870079i
\(172\) 0 0
\(173\) 11.5456 11.5456i 0.877798 0.877798i −0.115509 0.993306i \(-0.536850\pi\)
0.993306 + 0.115509i \(0.0368498\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −9.29150 −0.698392
\(178\) 0 0
\(179\) 9.30062 9.30062i 0.695161 0.695161i −0.268202 0.963363i \(-0.586429\pi\)
0.963363 + 0.268202i \(0.0864295\pi\)
\(180\) 0 0
\(181\) −11.1313 11.1313i −0.827387 0.827387i 0.159768 0.987155i \(-0.448925\pi\)
−0.987155 + 0.159768i \(0.948925\pi\)
\(182\) 0 0
\(183\) 19.2191i 1.42072i
\(184\) 0 0
\(185\) 3.03500i 0.223137i
\(186\) 0 0
\(187\) 4.52496 + 4.52496i 0.330898 + 0.330898i
\(188\) 0 0
\(189\) 1.70649 1.70649i 0.124129 0.124129i
\(190\) 0 0
\(191\) 18.4022 1.33153 0.665767 0.746160i \(-0.268105\pi\)
0.665767 + 0.746160i \(0.268105\pi\)
\(192\) 0 0
\(193\) 7.72357 0.555954 0.277977 0.960588i \(-0.410336\pi\)
0.277977 + 0.960588i \(0.410336\pi\)
\(194\) 0 0
\(195\) −2.41236 + 2.41236i −0.172753 + 0.172753i
\(196\) 0 0
\(197\) 3.23103 + 3.23103i 0.230201 + 0.230201i 0.812777 0.582575i \(-0.197955\pi\)
−0.582575 + 0.812777i \(0.697955\pi\)
\(198\) 0 0
\(199\) 8.14152i 0.577137i −0.957459 0.288568i \(-0.906821\pi\)
0.957459 0.288568i \(-0.0931793\pi\)
\(200\) 0 0
\(201\) 6.94955i 0.490183i
\(202\) 0 0
\(203\) 5.38242 + 5.38242i 0.377772 + 0.377772i
\(204\) 0 0
\(205\) 7.04097 7.04097i 0.491763 0.491763i
\(206\) 0 0
\(207\) 4.83619 0.336138
\(208\) 0 0
\(209\) −14.2188 −0.983532
\(210\) 0 0
\(211\) −2.19071 + 2.19071i −0.150814 + 0.150814i −0.778482 0.627667i \(-0.784010\pi\)
0.627667 + 0.778482i \(0.284010\pi\)
\(212\) 0 0
\(213\) 18.5455 + 18.5455i 1.27072 + 1.27072i
\(214\) 0 0
\(215\) 0.484378i 0.0330343i
\(216\) 0 0
\(217\) 1.98299i 0.134614i
\(218\) 0 0
\(219\) 12.8152 + 12.8152i 0.865968 + 0.865968i
\(220\) 0 0
\(221\) 4.12517 4.12517i 0.277489 0.277489i
\(222\) 0 0
\(223\) −10.0943 −0.675966 −0.337983 0.941152i \(-0.609745\pi\)
−0.337983 + 0.941152i \(0.609745\pi\)
\(224\) 0 0
\(225\) 1.91098 0.127399
\(226\) 0 0
\(227\) −14.7082 + 14.7082i −0.976218 + 0.976218i −0.999724 0.0235056i \(-0.992517\pi\)
0.0235056 + 0.999724i \(0.492517\pi\)
\(228\) 0 0
\(229\) −18.8816 18.8816i −1.24773 1.24773i −0.956718 0.291016i \(-0.906007\pi\)
−0.291016 0.956718i \(-0.593993\pi\)
\(230\) 0 0
\(231\) 3.74224i 0.246221i
\(232\) 0 0
\(233\) 17.0786i 1.11886i −0.828879 0.559428i \(-0.811021\pi\)
0.828879 0.559428i \(-0.188979\pi\)
\(234\) 0 0
\(235\) −3.05801 3.05801i −0.199482 0.199482i
\(236\) 0 0
\(237\) −13.2618 + 13.2618i −0.861445 + 0.861445i
\(238\) 0 0
\(239\) 7.76620 0.502354 0.251177 0.967941i \(-0.419182\pi\)
0.251177 + 0.967941i \(0.419182\pi\)
\(240\) 0 0
\(241\) −5.89836 −0.379947 −0.189973 0.981789i \(-0.560840\pi\)
−0.189973 + 0.981789i \(0.560840\pi\)
\(242\) 0 0
\(243\) −12.2446 + 12.2446i −0.785493 + 0.785493i
\(244\) 0 0
\(245\) 0.707107 + 0.707107i 0.0451754 + 0.0451754i
\(246\) 0 0
\(247\) 12.9625i 0.824782i
\(248\) 0 0
\(249\) 15.0254i 0.952195i
\(250\) 0 0
\(251\) −16.0065 16.0065i −1.01032 1.01032i −0.999946 0.0103768i \(-0.996697\pi\)
−0.0103768 0.999946i \(-0.503303\pi\)
\(252\) 0 0
\(253\) 3.02188 3.02188i 0.189984 0.189984i
\(254\) 0 0
\(255\) −8.39784 −0.525893
\(256\) 0 0
\(257\) 16.5127 1.03003 0.515015 0.857181i \(-0.327786\pi\)
0.515015 + 0.857181i \(0.327786\pi\)
\(258\) 0 0
\(259\) 2.14607 2.14607i 0.133350 0.133350i
\(260\) 0 0
\(261\) −10.2857 10.2857i −0.636670 0.636670i
\(262\) 0 0
\(263\) 8.22335i 0.507073i −0.967326 0.253537i \(-0.918406\pi\)
0.967326 0.253537i \(-0.0815939\pi\)
\(264\) 0 0
\(265\) 0.0934267i 0.00573915i
\(266\) 0 0
\(267\) −28.1528 28.1528i −1.72292 1.72292i
\(268\) 0 0
\(269\) 9.43495 9.43495i 0.575259 0.575259i −0.358334 0.933593i \(-0.616655\pi\)
0.933593 + 0.358334i \(0.116655\pi\)
\(270\) 0 0
\(271\) 27.9236 1.69624 0.848118 0.529807i \(-0.177736\pi\)
0.848118 + 0.529807i \(0.177736\pi\)
\(272\) 0 0
\(273\) 3.41159 0.206479
\(274\) 0 0
\(275\) 1.19408 1.19408i 0.0720055 0.0720055i
\(276\) 0 0
\(277\) 20.1281 + 20.1281i 1.20938 + 1.20938i 0.971228 + 0.238150i \(0.0765410\pi\)
0.238150 + 0.971228i \(0.423459\pi\)
\(278\) 0 0
\(279\) 3.78946i 0.226869i
\(280\) 0 0
\(281\) 29.4141i 1.75470i −0.479853 0.877349i \(-0.659310\pi\)
0.479853 0.877349i \(-0.340690\pi\)
\(282\) 0 0
\(283\) −13.3571 13.3571i −0.793999 0.793999i 0.188143 0.982142i \(-0.439753\pi\)
−0.982142 + 0.188143i \(0.939753\pi\)
\(284\) 0 0
\(285\) 13.1942 13.1942i 0.781559 0.781559i
\(286\) 0 0
\(287\) −9.95744 −0.587769
\(288\) 0 0
\(289\) −2.63959 −0.155270
\(290\) 0 0
\(291\) 13.6512 13.6512i 0.800248 0.800248i
\(292\) 0 0
\(293\) 0.359523 + 0.359523i 0.0210036 + 0.0210036i 0.717531 0.696527i \(-0.245272\pi\)
−0.696527 + 0.717531i \(0.745272\pi\)
\(294\) 0 0
\(295\) 4.19278i 0.244113i
\(296\) 0 0
\(297\) 4.07535i 0.236476i
\(298\) 0 0
\(299\) −2.75489 2.75489i −0.159319 0.159319i
\(300\) 0 0
\(301\) 0.342507 0.342507i 0.0197418 0.0197418i
\(302\) 0 0
\(303\) −7.91131 −0.454493
\(304\) 0 0
\(305\) −8.67259 −0.496591
\(306\) 0 0
\(307\) −22.1588 + 22.1588i −1.26467 + 1.26467i −0.315863 + 0.948805i \(0.602294\pi\)
−0.948805 + 0.315863i \(0.897706\pi\)
\(308\) 0 0
\(309\) 2.51529 + 2.51529i 0.143090 + 0.143090i
\(310\) 0 0
\(311\) 1.82146i 0.103286i −0.998666 0.0516429i \(-0.983554\pi\)
0.998666 0.0516429i \(-0.0164458\pi\)
\(312\) 0 0
\(313\) 2.18259i 0.123367i 0.998096 + 0.0616837i \(0.0196470\pi\)
−0.998096 + 0.0616837i \(0.980353\pi\)
\(314\) 0 0
\(315\) −1.35127 1.35127i −0.0761355 0.0761355i
\(316\) 0 0
\(317\) 4.76038 4.76038i 0.267370 0.267370i −0.560670 0.828039i \(-0.689456\pi\)
0.828039 + 0.560670i \(0.189456\pi\)
\(318\) 0 0
\(319\) −12.8540 −0.719687
\(320\) 0 0
\(321\) −39.0533 −2.17974
\(322\) 0 0
\(323\) −22.5623 + 22.5623i −1.25540 + 1.25540i
\(324\) 0 0
\(325\) −1.08857 1.08857i −0.0603832 0.0603832i
\(326\) 0 0
\(327\) 8.58576i 0.474794i
\(328\) 0 0
\(329\) 4.32467i 0.238427i
\(330\) 0 0
\(331\) 1.31659 + 1.31659i 0.0723661 + 0.0723661i 0.742363 0.669997i \(-0.233705\pi\)
−0.669997 + 0.742363i \(0.733705\pi\)
\(332\) 0 0
\(333\) −4.10110 + 4.10110i −0.224739 + 0.224739i
\(334\) 0 0
\(335\) −3.13597 −0.171336
\(336\) 0 0
\(337\) −4.65940 −0.253813 −0.126907 0.991915i \(-0.540505\pi\)
−0.126907 + 0.991915i \(0.540505\pi\)
\(338\) 0 0
\(339\) −23.7585 + 23.7585i −1.29039 + 1.29039i
\(340\) 0 0
\(341\) 2.36784 + 2.36784i 0.128225 + 0.128225i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 5.60829i 0.301940i
\(346\) 0 0
\(347\) −6.67385 6.67385i −0.358271 0.358271i 0.504904 0.863175i \(-0.331528\pi\)
−0.863175 + 0.504904i \(0.831528\pi\)
\(348\) 0 0
\(349\) 4.12213 4.12213i 0.220653 0.220653i −0.588121 0.808773i \(-0.700132\pi\)
0.808773 + 0.588121i \(0.200132\pi\)
\(350\) 0 0
\(351\) 3.71528 0.198307
\(352\) 0 0
\(353\) 18.0645 0.961478 0.480739 0.876864i \(-0.340368\pi\)
0.480739 + 0.876864i \(0.340368\pi\)
\(354\) 0 0
\(355\) −8.36862 + 8.36862i −0.444160 + 0.444160i
\(356\) 0 0
\(357\) 5.93817 + 5.93817i 0.314281 + 0.314281i
\(358\) 0 0
\(359\) 23.7729i 1.25469i −0.778743 0.627343i \(-0.784142\pi\)
0.778743 0.627343i \(-0.215858\pi\)
\(360\) 0 0
\(361\) 51.8972i 2.73143i
\(362\) 0 0
\(363\) 12.7685 + 12.7685i 0.670172 + 0.670172i
\(364\) 0 0
\(365\) −5.78282 + 5.78282i −0.302686 + 0.302686i
\(366\) 0 0
\(367\) 23.6120 1.23254 0.616269 0.787536i \(-0.288643\pi\)
0.616269 + 0.787536i \(0.288643\pi\)
\(368\) 0 0
\(369\) 19.0285 0.990585
\(370\) 0 0
\(371\) 0.0660626 0.0660626i 0.00342980 0.00342980i
\(372\) 0 0
\(373\) −19.1118 19.1118i −0.989572 0.989572i 0.0103739 0.999946i \(-0.496698\pi\)
−0.999946 + 0.0103739i \(0.996698\pi\)
\(374\) 0 0
\(375\) 2.21607i 0.114438i
\(376\) 0 0
\(377\) 11.7183i 0.603524i
\(378\) 0 0
\(379\) −2.04493 2.04493i −0.105041 0.105041i 0.652633 0.757674i \(-0.273664\pi\)
−0.757674 + 0.652633i \(0.773664\pi\)
\(380\) 0 0
\(381\) −24.2432 + 24.2432i −1.24202 + 1.24202i
\(382\) 0 0
\(383\) 33.5359 1.71360 0.856801 0.515647i \(-0.172448\pi\)
0.856801 + 0.515647i \(0.172448\pi\)
\(384\) 0 0
\(385\) −1.68868 −0.0860630
\(386\) 0 0
\(387\) −0.654525 + 0.654525i −0.0332714 + 0.0332714i
\(388\) 0 0
\(389\) −4.06325 4.06325i −0.206015 0.206015i 0.596556 0.802571i \(-0.296535\pi\)
−0.802571 + 0.596556i \(0.796535\pi\)
\(390\) 0 0
\(391\) 9.59023i 0.484999i
\(392\) 0 0
\(393\) 23.3190i 1.17629i
\(394\) 0 0
\(395\) −5.98435 5.98435i −0.301106 0.301106i
\(396\) 0 0
\(397\) 24.3755 24.3755i 1.22337 1.22337i 0.256946 0.966426i \(-0.417284\pi\)
0.966426 0.256946i \(-0.0827163\pi\)
\(398\) 0 0
\(399\) −18.6595 −0.934141
\(400\) 0 0
\(401\) −14.6859 −0.733378 −0.366689 0.930344i \(-0.619509\pi\)
−0.366689 + 0.930344i \(0.619509\pi\)
\(402\) 0 0
\(403\) 2.15863 2.15863i 0.107529 0.107529i
\(404\) 0 0
\(405\) −7.83551 7.83551i −0.389350 0.389350i
\(406\) 0 0
\(407\) 5.12513i 0.254043i
\(408\) 0 0
\(409\) 7.67269i 0.379390i 0.981843 + 0.189695i \(0.0607499\pi\)
−0.981843 + 0.189695i \(0.939250\pi\)
\(410\) 0 0
\(411\) −22.3317 22.3317i −1.10154 1.10154i
\(412\) 0 0
\(413\) −2.96474 + 2.96474i −0.145885 + 0.145885i
\(414\) 0 0
\(415\) 6.78018 0.332826
\(416\) 0 0
\(417\) −42.8572 −2.09873
\(418\) 0 0
\(419\) −7.61593 + 7.61593i −0.372062 + 0.372062i −0.868228 0.496166i \(-0.834741\pi\)
0.496166 + 0.868228i \(0.334741\pi\)
\(420\) 0 0
\(421\) −6.83236 6.83236i −0.332989 0.332989i 0.520732 0.853720i \(-0.325659\pi\)
−0.853720 + 0.520732i \(0.825659\pi\)
\(422\) 0 0
\(423\) 8.26439i 0.401828i
\(424\) 0 0
\(425\) 3.78951i 0.183818i
\(426\) 0 0
\(427\) 6.13245 + 6.13245i 0.296770 + 0.296770i
\(428\) 0 0
\(429\) −4.07370 + 4.07370i −0.196680 + 0.196680i
\(430\) 0 0
\(431\) −27.1817 −1.30930 −0.654649 0.755933i \(-0.727183\pi\)
−0.654649 + 0.755933i \(0.727183\pi\)
\(432\) 0 0
\(433\) 7.58371 0.364450 0.182225 0.983257i \(-0.441670\pi\)
0.182225 + 0.983257i \(0.441670\pi\)
\(434\) 0 0
\(435\) 11.9278 11.9278i 0.571896 0.571896i
\(436\) 0 0
\(437\) 15.0676 + 15.0676i 0.720783 + 0.720783i
\(438\) 0 0
\(439\) 39.4963i 1.88506i −0.334126 0.942528i \(-0.608441\pi\)
0.334126 0.942528i \(-0.391559\pi\)
\(440\) 0 0
\(441\) 1.91098i 0.0909993i
\(442\) 0 0
\(443\) −20.7530 20.7530i −0.986005 0.986005i 0.0138986 0.999903i \(-0.495576\pi\)
−0.999903 + 0.0138986i \(0.995576\pi\)
\(444\) 0 0
\(445\) 12.7039 12.7039i 0.602223 0.602223i
\(446\) 0 0
\(447\) −19.0556 −0.901298
\(448\) 0 0
\(449\) 20.5264 0.968702 0.484351 0.874874i \(-0.339056\pi\)
0.484351 + 0.874874i \(0.339056\pi\)
\(450\) 0 0
\(451\) 11.8899 11.8899i 0.559875 0.559875i
\(452\) 0 0
\(453\) −12.4224 12.4224i −0.583654 0.583654i
\(454\) 0 0
\(455\) 1.53948i 0.0721718i
\(456\) 0 0
\(457\) 16.9359i 0.792227i −0.918201 0.396114i \(-0.870359\pi\)
0.918201 0.396114i \(-0.129641\pi\)
\(458\) 0 0
\(459\) 6.46676 + 6.46676i 0.301842 + 0.301842i
\(460\) 0 0
\(461\) 8.72100 8.72100i 0.406178 0.406178i −0.474226 0.880403i \(-0.657272\pi\)
0.880403 + 0.474226i \(0.157272\pi\)
\(462\) 0 0
\(463\) 11.9544 0.555567 0.277783 0.960644i \(-0.410400\pi\)
0.277783 + 0.960644i \(0.410400\pi\)
\(464\) 0 0
\(465\) −4.39444 −0.203787
\(466\) 0 0
\(467\) 13.9823 13.9823i 0.647024 0.647024i −0.305249 0.952273i \(-0.598740\pi\)
0.952273 + 0.305249i \(0.0987397\pi\)
\(468\) 0 0
\(469\) 2.21747 + 2.21747i 0.102393 + 0.102393i
\(470\) 0 0
\(471\) 2.44333i 0.112583i
\(472\) 0 0
\(473\) 0.817958i 0.0376097i
\(474\) 0 0
\(475\) 5.95387 + 5.95387i 0.273182 + 0.273182i
\(476\) 0 0
\(477\) −0.126245 + 0.126245i −0.00578035 + 0.00578035i
\(478\) 0 0
\(479\) −28.6487 −1.30899 −0.654497 0.756064i \(-0.727120\pi\)
−0.654497 + 0.756064i \(0.727120\pi\)
\(480\) 0 0
\(481\) 4.67231 0.213039
\(482\) 0 0
\(483\) 3.96566 3.96566i 0.180444 0.180444i
\(484\) 0 0
\(485\) 6.16009 + 6.16009i 0.279715 + 0.279715i
\(486\) 0 0
\(487\) 22.1238i 1.00253i −0.865295 0.501263i \(-0.832869\pi\)
0.865295 0.501263i \(-0.167131\pi\)
\(488\) 0 0
\(489\) 18.8591i 0.852840i
\(490\) 0 0
\(491\) 2.29198 + 2.29198i 0.103436 + 0.103436i 0.756931 0.653495i \(-0.226698\pi\)
−0.653495 + 0.756931i \(0.726698\pi\)
\(492\) 0 0
\(493\) −20.3967 + 20.3967i −0.918623 + 0.918623i
\(494\) 0 0
\(495\) 3.22704 0.145045
\(496\) 0 0
\(497\) 11.8350 0.530873
\(498\) 0 0
\(499\) 25.9222 25.9222i 1.16044 1.16044i 0.176055 0.984380i \(-0.443666\pi\)
0.984380 0.176055i \(-0.0563338\pi\)
\(500\) 0 0
\(501\) −28.1377 28.1377i −1.25710 1.25710i
\(502\) 0 0
\(503\) 12.3871i 0.552314i −0.961113 0.276157i \(-0.910939\pi\)
0.961113 0.276157i \(-0.0890609\pi\)
\(504\) 0 0
\(505\) 3.56997i 0.158861i
\(506\) 0 0
\(507\) −16.6572 16.6572i −0.739774 0.739774i
\(508\) 0 0
\(509\) −25.8833 + 25.8833i −1.14726 + 1.14726i −0.160167 + 0.987090i \(0.551203\pi\)
−0.987090 + 0.160167i \(0.948797\pi\)
\(510\) 0 0
\(511\) 8.17814 0.361780
\(512\) 0 0
\(513\) −20.3204 −0.897169
\(514\) 0 0
\(515\) −1.13502 + 1.13502i −0.0500151 + 0.0500151i
\(516\) 0 0
\(517\) −5.16399 5.16399i −0.227112 0.227112i
\(518\) 0 0
\(519\) 36.1840i 1.58830i
\(520\) 0 0
\(521\) 15.6954i 0.687626i 0.939038 + 0.343813i \(0.111719\pi\)
−0.939038 + 0.343813i \(0.888281\pi\)
\(522\) 0 0
\(523\) 1.54028 + 1.54028i 0.0673516 + 0.0673516i 0.739980 0.672629i \(-0.234835\pi\)
−0.672629 + 0.739980i \(0.734835\pi\)
\(524\) 0 0
\(525\) 1.56700 1.56700i 0.0683895 0.0683895i
\(526\) 0 0
\(527\) 7.51455 0.327339
\(528\) 0 0
\(529\) 16.5954 0.721539
\(530\) 0 0
\(531\) 5.66557 5.66557i 0.245865 0.245865i
\(532\) 0 0
\(533\) −10.8394 10.8394i −0.469507 0.469507i
\(534\) 0 0
\(535\) 17.6228i 0.761898i
\(536\) 0 0
\(537\) 29.1481i 1.25784i
\(538\) 0 0
\(539\) 1.19408 + 1.19408i 0.0514325 + 0.0514325i
\(540\) 0 0
\(541\) 8.33236 8.33236i 0.358236 0.358236i −0.504926 0.863162i \(-0.668480\pi\)
0.863162 + 0.504926i \(0.168480\pi\)
\(542\) 0 0
\(543\) 34.8857 1.49709
\(544\) 0 0
\(545\) −3.87431 −0.165957
\(546\) 0 0
\(547\) 19.4887 19.4887i 0.833277 0.833277i −0.154687 0.987964i \(-0.549437\pi\)
0.987964 + 0.154687i \(0.0494369\pi\)
\(548\) 0 0
\(549\) −11.7190 11.7190i −0.500155 0.500155i
\(550\) 0 0
\(551\) 64.0924i 2.73043i
\(552\) 0 0
\(553\) 8.46315i 0.359890i
\(554\) 0 0
\(555\) −4.75584 4.75584i −0.201874 0.201874i
\(556\) 0 0
\(557\) −4.10316 + 4.10316i −0.173856 + 0.173856i −0.788671 0.614815i \(-0.789231\pi\)
0.614815 + 0.788671i \(0.289231\pi\)
\(558\) 0 0
\(559\) 0.745688 0.0315392
\(560\) 0 0
\(561\) −14.1812 −0.598733
\(562\) 0 0
\(563\) 1.37360 1.37360i 0.0578902 0.0578902i −0.677569 0.735459i \(-0.736966\pi\)
0.735459 + 0.677569i \(0.236966\pi\)
\(564\) 0 0
\(565\) −10.7210 10.7210i −0.451036 0.451036i
\(566\) 0 0
\(567\) 11.0811i 0.465362i
\(568\) 0 0
\(569\) 34.6455i 1.45241i 0.687476 + 0.726207i \(0.258719\pi\)
−0.687476 + 0.726207i \(0.741281\pi\)
\(570\) 0 0
\(571\) 28.0843 + 28.0843i 1.17529 + 1.17529i 0.980930 + 0.194361i \(0.0622632\pi\)
0.194361 + 0.980930i \(0.437737\pi\)
\(572\) 0 0
\(573\) −28.8362 + 28.8362i −1.20465 + 1.20465i
\(574\) 0 0
\(575\) −2.53073 −0.105539
\(576\) 0 0
\(577\) −23.3949 −0.973941 −0.486971 0.873418i \(-0.661898\pi\)
−0.486971 + 0.873418i \(0.661898\pi\)
\(578\) 0 0
\(579\) −12.1028 + 12.1028i −0.502977 + 0.502977i
\(580\) 0 0
\(581\) −4.79431 4.79431i −0.198902 0.198902i
\(582\) 0 0
\(583\) 0.157768i 0.00653406i
\(584\) 0 0
\(585\) 2.94192i 0.121633i
\(586\) 0 0
\(587\) −14.4086 14.4086i −0.594705 0.594705i 0.344194 0.938899i \(-0.388152\pi\)
−0.938899 + 0.344194i \(0.888152\pi\)
\(588\) 0 0
\(589\) −11.8064 + 11.8064i −0.486476 + 0.486476i
\(590\) 0 0
\(591\) −10.1260 −0.416530
\(592\) 0 0
\(593\) −6.97073 −0.286253 −0.143127 0.989704i \(-0.545716\pi\)
−0.143127 + 0.989704i \(0.545716\pi\)
\(594\) 0 0
\(595\) −2.67959 + 2.67959i −0.109852 + 0.109852i
\(596\) 0 0
\(597\) 12.7578 + 12.7578i 0.522141 + 0.522141i
\(598\) 0 0
\(599\) 0.671165i 0.0274231i −0.999906 0.0137115i \(-0.995635\pi\)
0.999906 0.0137115i \(-0.00436465\pi\)
\(600\) 0 0
\(601\) 33.3989i 1.36237i 0.732111 + 0.681185i \(0.238535\pi\)
−0.732111 + 0.681185i \(0.761465\pi\)
\(602\) 0 0
\(603\) −4.23755 4.23755i −0.172566 0.172566i
\(604\) 0 0
\(605\) −5.76177 + 5.76177i −0.234249 + 0.234249i
\(606\) 0 0
\(607\) −18.5572 −0.753212 −0.376606 0.926374i \(-0.622909\pi\)
−0.376606 + 0.926374i \(0.622909\pi\)
\(608\) 0 0
\(609\) −16.8685 −0.683546
\(610\) 0 0
\(611\) −4.70773 + 4.70773i −0.190454 + 0.190454i
\(612\) 0 0
\(613\) 16.6712 + 16.6712i 0.673345 + 0.673345i 0.958486 0.285141i \(-0.0920403\pi\)
−0.285141 + 0.958486i \(0.592040\pi\)
\(614\) 0 0
\(615\) 22.0664i 0.889804i
\(616\) 0 0
\(617\) 35.1103i 1.41349i 0.707470 + 0.706744i \(0.249837\pi\)
−0.707470 + 0.706744i \(0.750163\pi\)
\(618\) 0 0
\(619\) 6.33010 + 6.33010i 0.254428 + 0.254428i 0.822783 0.568355i \(-0.192420\pi\)
−0.568355 + 0.822783i \(0.692420\pi\)
\(620\) 0 0
\(621\) 4.31866 4.31866i 0.173302 0.173302i
\(622\) 0 0
\(623\) −17.9660 −0.719794
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 22.2808 22.2808i 0.889809 0.889809i
\(628\) 0 0
\(629\) 8.13255 + 8.13255i 0.324266 + 0.324266i
\(630\) 0 0
\(631\) 2.80873i 0.111814i −0.998436 0.0559069i \(-0.982195\pi\)
0.998436 0.0559069i \(-0.0178050\pi\)
\(632\) 0 0
\(633\) 6.86568i 0.272886i
\(634\) 0 0
\(635\) −10.9397 10.9397i −0.434129 0.434129i
\(636\) 0 0
\(637\) 1.08857 1.08857i 0.0431309 0.0431309i
\(638\) 0 0
\(639\) −22.6165 −0.894696
\(640\) 0 0
\(641\) 11.3083 0.446650 0.223325 0.974744i \(-0.428309\pi\)
0.223325 + 0.974744i \(0.428309\pi\)
\(642\) 0 0
\(643\) 2.41909 2.41909i 0.0953997 0.0953997i −0.657796 0.753196i \(-0.728511\pi\)
0.753196 + 0.657796i \(0.228511\pi\)
\(644\) 0 0
\(645\) −0.759020 0.759020i −0.0298864 0.0298864i
\(646\) 0 0
\(647\) 46.4567i 1.82640i 0.407512 + 0.913200i \(0.366396\pi\)
−0.407512 + 0.913200i \(0.633604\pi\)
\(648\) 0 0
\(649\) 7.08025i 0.277924i
\(650\) 0 0
\(651\) 3.10734 + 3.10734i 0.121786 + 0.121786i
\(652\) 0 0
\(653\) −27.9851 + 27.9851i −1.09514 + 1.09514i −0.100170 + 0.994970i \(0.531939\pi\)
−0.994970 + 0.100170i \(0.968061\pi\)
\(654\) 0 0
\(655\) 10.5226 0.411154
\(656\) 0 0
\(657\) −15.6283 −0.609718
\(658\) 0 0
\(659\) −29.2972 + 29.2972i −1.14126 + 1.14126i −0.153039 + 0.988220i \(0.548906\pi\)
−0.988220 + 0.153039i \(0.951094\pi\)
\(660\) 0 0
\(661\) 27.4083 + 27.4083i 1.06606 + 1.06606i 0.997658 + 0.0684007i \(0.0217896\pi\)
0.0684007 + 0.997658i \(0.478210\pi\)
\(662\) 0 0
\(663\) 12.9283i 0.502093i
\(664\) 0 0
\(665\) 8.42005i 0.326516i
\(666\) 0 0
\(667\) 13.6214 + 13.6214i 0.527424 + 0.527424i
\(668\) 0 0
\(669\) 15.8178 15.8178i 0.611552 0.611552i
\(670\) 0 0
\(671\) −14.6452 −0.565372
\(672\) 0 0
\(673\) −31.8026 −1.22590 −0.612950 0.790122i \(-0.710017\pi\)
−0.612950 + 0.790122i \(0.710017\pi\)
\(674\) 0 0
\(675\) 1.70649 1.70649i 0.0656827 0.0656827i
\(676\) 0 0
\(677\) −13.3856 13.3856i −0.514451 0.514451i 0.401436 0.915887i \(-0.368511\pi\)
−0.915887 + 0.401436i \(0.868511\pi\)
\(678\) 0 0
\(679\) 8.71168i 0.334324i
\(680\) 0 0
\(681\) 46.0956i 1.76639i
\(682\) 0 0
\(683\) −3.17288 3.17288i −0.121407 0.121407i 0.643793 0.765200i \(-0.277360\pi\)
−0.765200 + 0.643793i \(0.777360\pi\)
\(684\) 0 0
\(685\) 10.0772 10.0772i 0.385028 0.385028i
\(686\) 0 0
\(687\) 59.1751 2.25767
\(688\) 0 0
\(689\) 0.143828 0.00547942
\(690\) 0 0
\(691\) 19.7745 19.7745i 0.752257 0.752257i −0.222643 0.974900i \(-0.571468\pi\)
0.974900 + 0.222643i \(0.0714684\pi\)
\(692\) 0 0
\(693\) −2.28186 2.28186i −0.0866807 0.0866807i
\(694\) 0 0
\(695\) 19.3393i 0.733580i
\(696\) 0 0
\(697\) 37.7339i 1.42927i
\(698\) 0 0
\(699\) 26.7622 + 26.7622i 1.01224 + 1.01224i
\(700\) 0 0
\(701\) −16.7909 + 16.7909i −0.634183 + 0.634183i −0.949114 0.314931i \(-0.898018\pi\)
0.314931 + 0.949114i \(0.398018\pi\)
\(702\) 0 0
\(703\) −25.5548 −0.963819
\(704\) 0 0
\(705\) 9.58380 0.360947
\(706\) 0 0
\(707\) −2.52435 + 2.52435i −0.0949379 + 0.0949379i
\(708\) 0 0
\(709\) 10.0620 + 10.0620i 0.377888 + 0.377888i 0.870340 0.492452i \(-0.163899\pi\)
−0.492452 + 0.870340i \(0.663899\pi\)
\(710\) 0 0
\(711\) 16.1730i 0.606533i
\(712\) 0 0
\(713\) 5.01840i 0.187941i
\(714\) 0 0
\(715\) −1.83825 1.83825i −0.0687467 0.0687467i
\(716\) 0 0
\(717\) −12.1696 + 12.1696i −0.454484 + 0.454484i
\(718\) 0 0
\(719\) −38.0270 −1.41817 −0.709083 0.705125i \(-0.750891\pi\)
−0.709083 + 0.705125i \(0.750891\pi\)
\(720\) 0 0
\(721\) 1.60516 0.0597794
\(722\) 0 0
\(723\) 9.24274 9.24274i 0.343741 0.343741i
\(724\) 0 0
\(725\) 5.38242 + 5.38242i 0.199898 + 0.199898i
\(726\) 0 0
\(727\) 16.5488i 0.613761i 0.951748 + 0.306881i \(0.0992852\pi\)
−0.951748 + 0.306881i \(0.900715\pi\)
\(728\) 0 0
\(729\) 5.13139i 0.190051i
\(730\) 0 0
\(731\) 1.29793 + 1.29793i 0.0480058 + 0.0480058i
\(732\) 0 0
\(733\) 23.9078 23.9078i 0.883055 0.883055i −0.110789 0.993844i \(-0.535338\pi\)
0.993844 + 0.110789i \(0.0353376\pi\)
\(734\) 0 0
\(735\) −2.21607 −0.0817411
\(736\) 0 0
\(737\) −5.29565 −0.195068
\(738\) 0 0
\(739\) 1.69669 1.69669i 0.0624138 0.0624138i −0.675211 0.737625i \(-0.735947\pi\)
0.737625 + 0.675211i \(0.235947\pi\)
\(740\) 0 0
\(741\) −20.3122 20.3122i −0.746187 0.746187i
\(742\) 0 0
\(743\) 1.32561i 0.0486318i 0.999704 + 0.0243159i \(0.00774075\pi\)
−0.999704 + 0.0243159i \(0.992259\pi\)
\(744\) 0 0
\(745\) 8.59880i 0.315036i
\(746\) 0 0
\(747\) 9.16186 + 9.16186i 0.335215 + 0.335215i
\(748\) 0 0
\(749\) −12.4612 + 12.4612i −0.455321 + 0.455321i
\(750\) 0 0
\(751\) −49.2984 −1.79893 −0.899463 0.436998i \(-0.856042\pi\)
−0.899463 + 0.436998i \(0.856042\pi\)
\(752\) 0 0
\(753\) 50.1645 1.82810
\(754\) 0 0
\(755\) 5.60558 5.60558i 0.204008 0.204008i
\(756\) 0 0
\(757\) 0.110528 + 0.110528i 0.00401720 + 0.00401720i 0.709113 0.705095i \(-0.249096\pi\)
−0.705095 + 0.709113i \(0.749096\pi\)
\(758\) 0 0
\(759\) 9.47059i 0.343761i
\(760\) 0 0
\(761\) 46.0380i 1.66888i 0.551101 + 0.834438i \(0.314208\pi\)
−0.551101 + 0.834438i \(0.685792\pi\)
\(762\) 0 0
\(763\) 2.73955 + 2.73955i 0.0991784 + 0.0991784i
\(764\) 0 0
\(765\) 5.12066 5.12066i 0.185138 0.185138i
\(766\) 0 0
\(767\) −6.45468 −0.233065
\(768\) 0 0
\(769\) 48.8023 1.75986 0.879929 0.475106i \(-0.157590\pi\)
0.879929 + 0.475106i \(0.157590\pi\)
\(770\) 0 0
\(771\) −25.8753 + 25.8753i −0.931878 + 0.931878i
\(772\) 0 0
\(773\) 10.1791 + 10.1791i 0.366116 + 0.366116i 0.866058 0.499943i \(-0.166646\pi\)
−0.499943 + 0.866058i \(0.666646\pi\)
\(774\) 0 0
\(775\) 1.98299i 0.0712310i
\(776\) 0 0
\(777\) 6.72578i 0.241286i
\(778\) 0 0
\(779\) 59.2854 + 59.2854i 2.12412 + 2.12412i
\(780\) 0 0
\(781\) −14.1319 + 14.1319i −0.505679 + 0.505679i
\(782\) 0 0
\(783\) −18.3701 −0.656492
\(784\) 0 0
\(785\) −1.10255 −0.0393516
\(786\) 0 0
\(787\) 25.0676 25.0676i 0.893562 0.893562i −0.101294 0.994857i \(-0.532298\pi\)
0.994857 + 0.101294i \(0.0322983\pi\)
\(788\) 0 0
\(789\) 12.8860 + 12.8860i 0.458754 + 0.458754i
\(790\) 0 0
\(791\) 15.1618i 0.539091i
\(792\) 0 0
\(793\) 13.3512i 0.474117i
\(794\) 0 0
\(795\) −0.146400 0.146400i −0.00519226 0.00519226i
\(796\) 0 0
\(797\) 26.7500 26.7500i 0.947533 0.947533i −0.0511580 0.998691i \(-0.516291\pi\)
0.998691 + 0.0511580i \(0.0162912\pi\)
\(798\) 0 0
\(799\) −16.3884 −0.579780
\(800\) 0 0
\(801\) 34.3328 1.21309
\(802\) 0 0
\(803\) −9.76532 + 9.76532i −0.344610 + 0.344610i
\(804\) 0 0
\(805\) 1.78950 + 1.78950i 0.0630715 + 0.0630715i
\(806\) 0 0
\(807\) 29.5692i 1.04088i
\(808\) 0 0
\(809\) 32.3562i 1.13758i −0.822481 0.568792i \(-0.807411\pi\)
0.822481 0.568792i \(-0.192589\pi\)
\(810\) 0 0
\(811\) 14.1442 + 14.1442i 0.496670 + 0.496670i 0.910400 0.413730i \(-0.135774\pi\)
−0.413730 + 0.910400i \(0.635774\pi\)
\(812\) 0 0
\(813\) −43.7563 + 43.7563i −1.53460 + 1.53460i
\(814\) 0 0
\(815\) −8.51016 −0.298098
\(816\) 0 0
\(817\) −4.07848 −0.142688
\(818\) 0 0
\(819\) −2.08025 + 2.08025i −0.0726898 + 0.0726898i
\(820\) 0 0
\(821\) 9.55314 + 9.55314i 0.333407 + 0.333407i 0.853879 0.520472i \(-0.174244\pi\)
−0.520472 + 0.853879i \(0.674244\pi\)
\(822\) 0 0
\(823\) 0.435281i 0.0151729i 0.999971 + 0.00758647i \(0.00241487\pi\)
−0.999971 + 0.00758647i \(0.997585\pi\)
\(824\) 0 0
\(825\) 3.74224i 0.130288i
\(826\) 0 0
\(827\) 4.50281 + 4.50281i 0.156578 + 0.156578i 0.781048 0.624470i \(-0.214685\pi\)
−0.624470 + 0.781048i \(0.714685\pi\)
\(828\) 0 0
\(829\) −28.1416 + 28.1416i −0.977399 + 0.977399i −0.999750 0.0223513i \(-0.992885\pi\)
0.0223513 + 0.999750i \(0.492885\pi\)
\(830\) 0 0
\(831\) −63.0814 −2.18827
\(832\) 0 0
\(833\) 3.78951 0.131299
\(834\) 0 0
\(835\) 12.6971 12.6971i 0.439400 0.439400i
\(836\) 0 0
\(837\) 3.38394 + 3.38394i 0.116966 + 0.116966i
\(838\) 0 0
\(839\) 10.4652i 0.361299i −0.983548 0.180649i \(-0.942180\pi\)
0.983548 0.180649i \(-0.0578200\pi\)
\(840\) 0 0
\(841\) 28.9408i 0.997958i
\(842\) 0 0
\(843\) 46.0919 + 46.0919i 1.58749 + 1.58749i
\(844\) 0 0
\(845\) 7.51655 7.51655i 0.258577 0.258577i
\(846\) 0 0
\(847\) 8.14837 0.279981
\(848\) 0 0
\(849\) 41.8613 1.43668
\(850\) 0 0
\(851\) 5.43112 5.43112i 0.186176 0.186176i
\(852\) 0 0
\(853\) −3.20752 3.20752i −0.109823 0.109823i 0.650060 0.759883i \(-0.274744\pi\)
−0.759883 + 0.650060i \(0.774744\pi\)
\(854\) 0 0
\(855\) 16.0906i 0.550286i
\(856\) 0 0
\(857\) 1.88694i 0.0644566i −0.999481 0.0322283i \(-0.989740\pi\)
0.999481 0.0322283i \(-0.0102604\pi\)
\(858\) 0 0
\(859\) −18.8963 18.8963i −0.644735 0.644735i 0.306981 0.951716i \(-0.400681\pi\)
−0.951716 + 0.306981i \(0.900681\pi\)
\(860\) 0 0
\(861\) 15.6033 15.6033i 0.531760 0.531760i
\(862\) 0 0
\(863\) −23.3749 −0.795689 −0.397845 0.917453i \(-0.630242\pi\)
−0.397845 + 0.917453i \(0.630242\pi\)
\(864\) 0 0
\(865\) −16.3280 −0.555168
\(866\) 0 0
\(867\) 4.13624 4.13624i 0.140474 0.140474i
\(868\) 0 0
\(869\) −10.1056 10.1056i −0.342811 0.342811i
\(870\) 0 0
\(871\) 4.82775i 0.163582i
\(872\) 0 0
\(873\) 16.6479i 0.563446i
\(874\) 0 0
\(875\) 0.707107 + 0.707107i 0.0239046 + 0.0239046i
\(876\) 0 0
\(877\) 35.2855 35.2855i 1.19151 1.19151i 0.214864 0.976644i \(-0.431069\pi\)
0.976644 0.214864i \(-0.0689307\pi\)
\(878\) 0 0
\(879\) −1.12675 −0.0380042
\(880\) 0 0
\(881\) 36.7860 1.23935 0.619675 0.784858i \(-0.287264\pi\)
0.619675 + 0.784858i \(0.287264\pi\)
\(882\) 0 0
\(883\) −17.2710 + 17.2710i −0.581216 + 0.581216i −0.935237 0.354022i \(-0.884814\pi\)
0.354022 + 0.935237i \(0.384814\pi\)
\(884\) 0 0
\(885\) 6.57008 + 6.57008i 0.220851 + 0.220851i
\(886\) 0 0
\(887\) 38.2383i 1.28391i −0.766740 0.641957i \(-0.778123\pi\)
0.766740 0.641957i \(-0.221877\pi\)
\(888\) 0 0
\(889\) 15.4711i 0.518883i
\(890\) 0 0
\(891\) −13.2317 13.2317i −0.443277 0.443277i
\(892\) 0 0
\(893\) 25.7486 25.7486i 0.861643 0.861643i
\(894\) 0 0
\(895\) −13.1531 −0.439658
\(896\) 0 0
\(897\) 8.63382 0.288275
\(898\) 0 0
\(899\) −10.6733 + 10.6733i −0.355973 + 0.355973i
\(900\) 0 0
\(901\) 0.250345 + 0.250345i 0.00834021 + 0.00834021i
\(902\) 0 0
\(903\) 1.07342i 0.0357211i
\(904\) 0 0
\(905\) 15.7421i 0.523285i
\(906\) 0 0
\(907\) −3.62813 3.62813i −0.120470 0.120470i 0.644302 0.764772i \(-0.277148\pi\)
−0.764772 + 0.644302i \(0.777148\pi\)
\(908\) 0 0
\(909\) 4.82399 4.82399i 0.160002 0.160002i
\(910\) 0 0
\(911\) 39.8481 1.32023 0.660114 0.751166i \(-0.270508\pi\)
0.660114 + 0.751166i \(0.270508\pi\)
\(912\) 0 0
\(913\) 11.4495 0.378924
\(914\) 0 0
\(915\) 13.5900 13.5900i 0.449270 0.449270i
\(916\) 0 0
\(917\) −7.44063 7.44063i −0.245711 0.245711i
\(918\) 0 0
\(919\) 47.4733i 1.56600i 0.622022 + 0.783000i \(0.286311\pi\)
−0.622022 + 0.783000i \(0.713689\pi\)
\(920\) 0 0
\(921\) 69.4456i 2.28831i
\(922\) 0 0
\(923\) 12.8833 + 12.8833i 0.424059 + 0.424059i
\(924\) 0 0
\(925\) 2.14607 2.14607i 0.0705623 0.0705623i
\(926\) 0 0
\(927\) −3.06744 −0.100748
\(928\) 0 0
\(929\) −20.9057 −0.685894 −0.342947 0.939355i \(-0.611425\pi\)
−0.342947 + 0.939355i \(0.611425\pi\)
\(930\) 0 0
\(931\) −5.95387 + 5.95387i −0.195130 + 0.195130i
\(932\) 0 0
\(933\) 2.85424 + 2.85424i 0.0934435 + 0.0934435i
\(934\) 0 0
\(935\) 6.39927i 0.209278i
\(936\) 0 0
\(937\) 10.7301i 0.350536i 0.984521 + 0.175268i \(0.0560793\pi\)
−0.984521 + 0.175268i \(0.943921\pi\)
\(938\) 0 0
\(939\) −3.42012 3.42012i −0.111611 0.111611i
\(940\) 0 0
\(941\) 10.3926 10.3926i 0.338790 0.338790i −0.517122 0.855912i \(-0.672997\pi\)
0.855912 + 0.517122i \(0.172997\pi\)
\(942\) 0 0
\(943\) −25.1996 −0.820612
\(944\) 0 0
\(945\) −2.41334 −0.0785059
\(946\) 0 0
\(947\) 28.2724 28.2724i 0.918731 0.918731i −0.0782060 0.996937i \(-0.524919\pi\)
0.996937 + 0.0782060i \(0.0249192\pi\)
\(948\) 0 0
\(949\) 8.90251 + 8.90251i 0.288988 + 0.288988i
\(950\) 0 0
\(951\) 14.9190i 0.483783i
\(952\) 0 0
\(953\) 11.9583i 0.387366i −0.981064 0.193683i \(-0.937957\pi\)
0.981064 0.193683i \(-0.0620433\pi\)
\(954\) 0 0
\(955\) −13.0123 13.0123i −0.421068 0.421068i
\(956\) 0 0
\(957\) 20.1423 20.1423i 0.651107 0.651107i
\(958\) 0 0
\(959\) −14.2513 −0.460197
\(960\) 0 0
\(961\) −27.0678 −0.873154
\(962\) 0 0
\(963\) 23.8131 23.8131i 0.767367 0.767367i
\(964\) 0 0
\(965\) −5.46139 5.46139i −0.175808 0.175808i
\(966\) 0 0
\(967\) 22.2546i 0.715660i 0.933787 + 0.357830i \(0.116483\pi\)
−0.933787 + 0.357830i \(0.883517\pi\)
\(968\) 0 0
\(969\) 70.7102i 2.27154i
\(970\) 0 0
\(971\) 21.7588 + 21.7588i 0.698274 + 0.698274i 0.964038 0.265764i \(-0.0856243\pi\)
−0.265764 + 0.964038i \(0.585624\pi\)
\(972\) 0 0
\(973\) −13.6749 + 13.6749i −0.438398 + 0.438398i
\(974\) 0 0
\(975\) 3.41159 0.109258
\(976\) 0 0
\(977\) 57.4976 1.83951 0.919756 0.392491i \(-0.128387\pi\)
0.919756 + 0.392491i \(0.128387\pi\)
\(978\) 0 0
\(979\) 21.4528 21.4528i 0.685634 0.685634i
\(980\) 0 0
\(981\) −5.23524 5.23524i −0.167148 0.167148i
\(982\) 0 0
\(983\) 33.6196i 1.07230i 0.844123 + 0.536149i \(0.180121\pi\)
−0.844123 + 0.536149i \(0.819879\pi\)
\(984\) 0 0
\(985\) 4.56936i 0.145592i
\(986\) 0 0
\(987\) −6.77677 6.77677i −0.215707 0.215707i
\(988\) 0 0
\(989\) 0.866792 0.866792i 0.0275624 0.0275624i
\(990\) 0 0
\(991\) 5.02283 0.159555 0.0797777 0.996813i \(-0.474579\pi\)
0.0797777 + 0.996813i \(0.474579\pi\)
\(992\) 0 0
\(993\) −4.12618 −0.130940
\(994\) 0 0
\(995\) −5.75692 + 5.75692i −0.182507 + 0.182507i
\(996\) 0 0
\(997\) −17.7235 17.7235i −0.561308 0.561308i 0.368371 0.929679i \(-0.379916\pi\)
−0.929679 + 0.368371i \(0.879916\pi\)
\(998\) 0 0
\(999\) 7.32448i 0.231736i
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 2240.2.bd.b.561.6 52
4.3 odd 2 560.2.bd.b.421.26 yes 52
16.3 odd 4 560.2.bd.b.141.26 52
16.13 even 4 inner 2240.2.bd.b.1681.6 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.bd.b.141.26 52 16.3 odd 4
560.2.bd.b.421.26 yes 52 4.3 odd 2
2240.2.bd.b.561.6 52 1.1 even 1 trivial
2240.2.bd.b.1681.6 52 16.13 even 4 inner