Properties

Label 1280.2.n.m.767.1
Level $1280$
Weight $2$
Character 1280.767
Analytic conductor $10.221$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(767,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.767");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.n (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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 767.1
Root \(-0.951057 + 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 1280.767
Dual form 1280.2.n.m.1023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.61803 - 1.61803i) q^{3} +(-1.17557 + 1.90211i) q^{5} +(1.17557 - 1.17557i) q^{7} +2.23607i q^{9} +O(q^{10})\) \(q+(-1.61803 - 1.61803i) q^{3} +(-1.17557 + 1.90211i) q^{5} +(1.17557 - 1.17557i) q^{7} +2.23607i q^{9} -1.23607i q^{11} +(-3.07768 + 3.07768i) q^{13} +(4.97980 - 1.17557i) q^{15} +(-1.00000 - 1.00000i) q^{17} +2.00000 q^{19} -3.80423 q^{21} +(2.62866 + 2.62866i) q^{23} +(-2.23607 - 4.47214i) q^{25} +(-1.23607 + 1.23607i) q^{27} +1.45309i q^{29} +5.25731i q^{31} +(-2.00000 + 2.00000i) q^{33} +(0.854102 + 3.61803i) q^{35} +(3.07768 + 3.07768i) q^{37} +9.95959 q^{39} +7.70820 q^{41} +(-2.38197 - 2.38197i) q^{43} +(-4.25325 - 2.62866i) q^{45} +(7.33094 - 7.33094i) q^{47} +4.23607i q^{49} +3.23607i q^{51} +(0.726543 - 0.726543i) q^{53} +(2.35114 + 1.45309i) q^{55} +(-3.23607 - 3.23607i) q^{57} +8.47214 q^{59} +9.95959 q^{61} +(2.62866 + 2.62866i) q^{63} +(-2.23607 - 9.47214i) q^{65} +(2.38197 - 2.38197i) q^{67} -8.50651i q^{69} -7.05342i q^{71} +(-8.70820 + 8.70820i) q^{73} +(-3.61803 + 10.8541i) q^{75} +(-1.45309 - 1.45309i) q^{77} +12.3107 q^{79} +10.7082 q^{81} +(-4.38197 - 4.38197i) q^{83} +(3.07768 - 0.726543i) q^{85} +(2.35114 - 2.35114i) q^{87} +6.47214i q^{89} +7.23607i q^{91} +(8.50651 - 8.50651i) q^{93} +(-2.35114 + 3.80423i) q^{95} +(-0.236068 - 0.236068i) q^{97} +2.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 8 q^{17} + 16 q^{19} + 8 q^{27} - 16 q^{33} - 20 q^{35} + 8 q^{41} - 28 q^{43} - 8 q^{57} + 32 q^{59} + 28 q^{67} - 16 q^{73} - 20 q^{75} + 32 q^{81} - 44 q^{83} + 16 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(3\) −1.61803 1.61803i −0.934172 0.934172i 0.0637909 0.997963i \(-0.479681\pi\)
−0.997963 + 0.0637909i \(0.979681\pi\)
\(4\) 0 0
\(5\) −1.17557 + 1.90211i −0.525731 + 0.850651i
\(6\) 0 0
\(7\) 1.17557 1.17557i 0.444324 0.444324i −0.449138 0.893462i \(-0.648269\pi\)
0.893462 + 0.449138i \(0.148269\pi\)
\(8\) 0 0
\(9\) 2.23607i 0.745356i
\(10\) 0 0
\(11\) 1.23607i 0.372689i −0.982485 0.186344i \(-0.940336\pi\)
0.982485 0.186344i \(-0.0596640\pi\)
\(12\) 0 0
\(13\) −3.07768 + 3.07768i −0.853596 + 0.853596i −0.990574 0.136978i \(-0.956261\pi\)
0.136978 + 0.990574i \(0.456261\pi\)
\(14\) 0 0
\(15\) 4.97980 1.17557i 1.28578 0.303531i
\(16\) 0 0
\(17\) −1.00000 1.00000i −0.242536 0.242536i 0.575363 0.817898i \(-0.304861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −3.80423 −0.830150
\(22\) 0 0
\(23\) 2.62866 + 2.62866i 0.548113 + 0.548113i 0.925895 0.377782i \(-0.123313\pi\)
−0.377782 + 0.925895i \(0.623313\pi\)
\(24\) 0 0
\(25\) −2.23607 4.47214i −0.447214 0.894427i
\(26\) 0 0
\(27\) −1.23607 + 1.23607i −0.237881 + 0.237881i
\(28\) 0 0
\(29\) 1.45309i 0.269831i 0.990857 + 0.134916i \(0.0430763\pi\)
−0.990857 + 0.134916i \(0.956924\pi\)
\(30\) 0 0
\(31\) 5.25731i 0.944241i 0.881534 + 0.472120i \(0.156511\pi\)
−0.881534 + 0.472120i \(0.843489\pi\)
\(32\) 0 0
\(33\) −2.00000 + 2.00000i −0.348155 + 0.348155i
\(34\) 0 0
\(35\) 0.854102 + 3.61803i 0.144370 + 0.611559i
\(36\) 0 0
\(37\) 3.07768 + 3.07768i 0.505968 + 0.505968i 0.913286 0.407318i \(-0.133536\pi\)
−0.407318 + 0.913286i \(0.633536\pi\)
\(38\) 0 0
\(39\) 9.95959 1.59481
\(40\) 0 0
\(41\) 7.70820 1.20382 0.601910 0.798564i \(-0.294407\pi\)
0.601910 + 0.798564i \(0.294407\pi\)
\(42\) 0 0
\(43\) −2.38197 2.38197i −0.363246 0.363246i 0.501760 0.865007i \(-0.332686\pi\)
−0.865007 + 0.501760i \(0.832686\pi\)
\(44\) 0 0
\(45\) −4.25325 2.62866i −0.634038 0.391857i
\(46\) 0 0
\(47\) 7.33094 7.33094i 1.06933 1.06933i 0.0719165 0.997411i \(-0.477088\pi\)
0.997411 0.0719165i \(-0.0229115\pi\)
\(48\) 0 0
\(49\) 4.23607i 0.605153i
\(50\) 0 0
\(51\) 3.23607i 0.453140i
\(52\) 0 0
\(53\) 0.726543 0.726543i 0.0997983 0.0997983i −0.655445 0.755243i \(-0.727519\pi\)
0.755243 + 0.655445i \(0.227519\pi\)
\(54\) 0 0
\(55\) 2.35114 + 1.45309i 0.317028 + 0.195934i
\(56\) 0 0
\(57\) −3.23607 3.23607i −0.428628 0.428628i
\(58\) 0 0
\(59\) 8.47214 1.10298 0.551489 0.834182i \(-0.314060\pi\)
0.551489 + 0.834182i \(0.314060\pi\)
\(60\) 0 0
\(61\) 9.95959 1.27520 0.637598 0.770370i \(-0.279928\pi\)
0.637598 + 0.770370i \(0.279928\pi\)
\(62\) 0 0
\(63\) 2.62866 + 2.62866i 0.331179 + 0.331179i
\(64\) 0 0
\(65\) −2.23607 9.47214i −0.277350 1.17487i
\(66\) 0 0
\(67\) 2.38197 2.38197i 0.291003 0.291003i −0.546473 0.837477i \(-0.684030\pi\)
0.837477 + 0.546473i \(0.184030\pi\)
\(68\) 0 0
\(69\) 8.50651i 1.02406i
\(70\) 0 0
\(71\) 7.05342i 0.837087i −0.908197 0.418544i \(-0.862541\pi\)
0.908197 0.418544i \(-0.137459\pi\)
\(72\) 0 0
\(73\) −8.70820 + 8.70820i −1.01922 + 1.01922i −0.0194065 + 0.999812i \(0.506178\pi\)
−0.999812 + 0.0194065i \(0.993822\pi\)
\(74\) 0 0
\(75\) −3.61803 + 10.8541i −0.417775 + 1.25332i
\(76\) 0 0
\(77\) −1.45309 1.45309i −0.165594 0.165594i
\(78\) 0 0
\(79\) 12.3107 1.38507 0.692533 0.721386i \(-0.256495\pi\)
0.692533 + 0.721386i \(0.256495\pi\)
\(80\) 0 0
\(81\) 10.7082 1.18980
\(82\) 0 0
\(83\) −4.38197 4.38197i −0.480983 0.480983i 0.424462 0.905446i \(-0.360463\pi\)
−0.905446 + 0.424462i \(0.860463\pi\)
\(84\) 0 0
\(85\) 3.07768 0.726543i 0.333822 0.0788046i
\(86\) 0 0
\(87\) 2.35114 2.35114i 0.252069 0.252069i
\(88\) 0 0
\(89\) 6.47214i 0.686045i 0.939327 + 0.343023i \(0.111451\pi\)
−0.939327 + 0.343023i \(0.888549\pi\)
\(90\) 0 0
\(91\) 7.23607i 0.758546i
\(92\) 0 0
\(93\) 8.50651 8.50651i 0.882084 0.882084i
\(94\) 0 0
\(95\) −2.35114 + 3.80423i −0.241222 + 0.390305i
\(96\) 0 0
\(97\) −0.236068 0.236068i −0.0239691 0.0239691i 0.695021 0.718990i \(-0.255395\pi\)
−0.718990 + 0.695021i \(0.755395\pi\)
\(98\) 0 0
\(99\) 2.76393 0.277786
\(100\) 0 0
\(101\) 12.3107 1.22496 0.612482 0.790485i \(-0.290171\pi\)
0.612482 + 0.790485i \(0.290171\pi\)
\(102\) 0 0
\(103\) 7.33094 + 7.33094i 0.722339 + 0.722339i 0.969081 0.246742i \(-0.0793601\pi\)
−0.246742 + 0.969081i \(0.579360\pi\)
\(104\) 0 0
\(105\) 4.47214 7.23607i 0.436436 0.706168i
\(106\) 0 0
\(107\) 12.0902 12.0902i 1.16880 1.16880i 0.186310 0.982491i \(-0.440347\pi\)
0.982491 0.186310i \(-0.0596528\pi\)
\(108\) 0 0
\(109\) 6.71040i 0.642739i −0.946954 0.321370i \(-0.895857\pi\)
0.946954 0.321370i \(-0.104143\pi\)
\(110\) 0 0
\(111\) 9.95959i 0.945323i
\(112\) 0 0
\(113\) −4.70820 + 4.70820i −0.442911 + 0.442911i −0.892989 0.450078i \(-0.851396\pi\)
0.450078 + 0.892989i \(0.351396\pi\)
\(114\) 0 0
\(115\) −8.09017 + 1.90983i −0.754412 + 0.178093i
\(116\) 0 0
\(117\) −6.88191 6.88191i −0.636233 0.636233i
\(118\) 0 0
\(119\) −2.35114 −0.215529
\(120\) 0 0
\(121\) 9.47214 0.861103
\(122\) 0 0
\(123\) −12.4721 12.4721i −1.12457 1.12457i
\(124\) 0 0
\(125\) 11.1352 + 1.00406i 0.995959 + 0.0898056i
\(126\) 0 0
\(127\) −8.78402 + 8.78402i −0.779456 + 0.779456i −0.979738 0.200282i \(-0.935814\pi\)
0.200282 + 0.979738i \(0.435814\pi\)
\(128\) 0 0
\(129\) 7.70820i 0.678670i
\(130\) 0 0
\(131\) 0.291796i 0.0254943i 0.999919 + 0.0127472i \(0.00405766\pi\)
−0.999919 + 0.0127472i \(0.995942\pi\)
\(132\) 0 0
\(133\) 2.35114 2.35114i 0.203870 0.203870i
\(134\) 0 0
\(135\) −0.898056 3.80423i −0.0772924 0.327416i
\(136\) 0 0
\(137\) −3.47214 3.47214i −0.296645 0.296645i 0.543054 0.839698i \(-0.317268\pi\)
−0.839698 + 0.543054i \(0.817268\pi\)
\(138\) 0 0
\(139\) −5.41641 −0.459414 −0.229707 0.973260i \(-0.573777\pi\)
−0.229707 + 0.973260i \(0.573777\pi\)
\(140\) 0 0
\(141\) −23.7234 −1.99787
\(142\) 0 0
\(143\) 3.80423 + 3.80423i 0.318125 + 0.318125i
\(144\) 0 0
\(145\) −2.76393 1.70820i −0.229532 0.141859i
\(146\) 0 0
\(147\) 6.85410 6.85410i 0.565317 0.565317i
\(148\) 0 0
\(149\) 13.2088i 1.08211i 0.840988 + 0.541053i \(0.181974\pi\)
−0.840988 + 0.541053i \(0.818026\pi\)
\(150\) 0 0
\(151\) 14.6619i 1.19317i 0.802551 + 0.596583i \(0.203475\pi\)
−0.802551 + 0.596583i \(0.796525\pi\)
\(152\) 0 0
\(153\) 2.23607 2.23607i 0.180775 0.180775i
\(154\) 0 0
\(155\) −10.0000 6.18034i −0.803219 0.496417i
\(156\) 0 0
\(157\) 9.78808 + 9.78808i 0.781174 + 0.781174i 0.980029 0.198855i \(-0.0637223\pi\)
−0.198855 + 0.980029i \(0.563722\pi\)
\(158\) 0 0
\(159\) −2.35114 −0.186458
\(160\) 0 0
\(161\) 6.18034 0.487079
\(162\) 0 0
\(163\) −7.14590 7.14590i −0.559710 0.559710i 0.369515 0.929225i \(-0.379524\pi\)
−0.929225 + 0.369515i \(0.879524\pi\)
\(164\) 0 0
\(165\) −1.45309 6.15537i −0.113123 0.479195i
\(166\) 0 0
\(167\) 0.277515 0.277515i 0.0214747 0.0214747i −0.696288 0.717763i \(-0.745166\pi\)
0.717763 + 0.696288i \(0.245166\pi\)
\(168\) 0 0
\(169\) 5.94427i 0.457252i
\(170\) 0 0
\(171\) 4.47214i 0.341993i
\(172\) 0 0
\(173\) −6.32688 + 6.32688i −0.481024 + 0.481024i −0.905459 0.424435i \(-0.860473\pi\)
0.424435 + 0.905459i \(0.360473\pi\)
\(174\) 0 0
\(175\) −7.88597 2.62866i −0.596123 0.198708i
\(176\) 0 0
\(177\) −13.7082 13.7082i −1.03037 1.03037i
\(178\) 0 0
\(179\) 16.4721 1.23119 0.615593 0.788065i \(-0.288917\pi\)
0.615593 + 0.788065i \(0.288917\pi\)
\(180\) 0 0
\(181\) −9.40456 −0.699036 −0.349518 0.936930i \(-0.613655\pi\)
−0.349518 + 0.936930i \(0.613655\pi\)
\(182\) 0 0
\(183\) −16.1150 16.1150i −1.19125 1.19125i
\(184\) 0 0
\(185\) −9.47214 + 2.23607i −0.696405 + 0.164399i
\(186\) 0 0
\(187\) −1.23607 + 1.23607i −0.0903902 + 0.0903902i
\(188\) 0 0
\(189\) 2.90617i 0.211393i
\(190\) 0 0
\(191\) 12.8658i 0.930934i −0.885065 0.465467i \(-0.845886\pi\)
0.885065 0.465467i \(-0.154114\pi\)
\(192\) 0 0
\(193\) −7.47214 + 7.47214i −0.537856 + 0.537856i −0.922899 0.385043i \(-0.874187\pi\)
0.385043 + 0.922899i \(0.374187\pi\)
\(194\) 0 0
\(195\) −11.7082 + 18.9443i −0.838442 + 1.35663i
\(196\) 0 0
\(197\) 2.17963 + 2.17963i 0.155292 + 0.155292i 0.780477 0.625185i \(-0.214976\pi\)
−0.625185 + 0.780477i \(0.714976\pi\)
\(198\) 0 0
\(199\) −18.1231 −1.28471 −0.642355 0.766407i \(-0.722043\pi\)
−0.642355 + 0.766407i \(0.722043\pi\)
\(200\) 0 0
\(201\) −7.70820 −0.543695
\(202\) 0 0
\(203\) 1.70820 + 1.70820i 0.119892 + 0.119892i
\(204\) 0 0
\(205\) −9.06154 + 14.6619i −0.632885 + 1.02403i
\(206\) 0 0
\(207\) −5.87785 + 5.87785i −0.408539 + 0.408539i
\(208\) 0 0
\(209\) 2.47214i 0.171001i
\(210\) 0 0
\(211\) 15.7082i 1.08140i 0.841216 + 0.540699i \(0.181840\pi\)
−0.841216 + 0.540699i \(0.818160\pi\)
\(212\) 0 0
\(213\) −11.4127 + 11.4127i −0.781984 + 0.781984i
\(214\) 0 0
\(215\) 7.33094 1.73060i 0.499966 0.118026i
\(216\) 0 0
\(217\) 6.18034 + 6.18034i 0.419549 + 0.419549i
\(218\) 0 0
\(219\) 28.1803 1.90425
\(220\) 0 0
\(221\) 6.15537 0.414055
\(222\) 0 0
\(223\) −1.73060 1.73060i −0.115890 0.115890i 0.646784 0.762673i \(-0.276114\pi\)
−0.762673 + 0.646784i \(0.776114\pi\)
\(224\) 0 0
\(225\) 10.0000 5.00000i 0.666667 0.333333i
\(226\) 0 0
\(227\) 11.6180 11.6180i 0.771116 0.771116i −0.207186 0.978302i \(-0.566430\pi\)
0.978302 + 0.207186i \(0.0664304\pi\)
\(228\) 0 0
\(229\) 21.3723i 1.41232i 0.708053 + 0.706160i \(0.249574\pi\)
−0.708053 + 0.706160i \(0.750426\pi\)
\(230\) 0 0
\(231\) 4.70228i 0.309387i
\(232\) 0 0
\(233\) 3.47214 3.47214i 0.227467 0.227467i −0.584167 0.811634i \(-0.698579\pi\)
0.811634 + 0.584167i \(0.198579\pi\)
\(234\) 0 0
\(235\) 5.32624 + 22.5623i 0.347445 + 1.47180i
\(236\) 0 0
\(237\) −19.9192 19.9192i −1.29389 1.29389i
\(238\) 0 0
\(239\) −29.3238 −1.89680 −0.948398 0.317083i \(-0.897297\pi\)
−0.948398 + 0.317083i \(0.897297\pi\)
\(240\) 0 0
\(241\) 6.76393 0.435703 0.217852 0.975982i \(-0.430095\pi\)
0.217852 + 0.975982i \(0.430095\pi\)
\(242\) 0 0
\(243\) −13.6180 13.6180i −0.873597 0.873597i
\(244\) 0 0
\(245\) −8.05748 4.97980i −0.514774 0.318148i
\(246\) 0 0
\(247\) −6.15537 + 6.15537i −0.391657 + 0.391657i
\(248\) 0 0
\(249\) 14.1803i 0.898643i
\(250\) 0 0
\(251\) 22.1803i 1.40001i 0.714138 + 0.700005i \(0.246819\pi\)
−0.714138 + 0.700005i \(0.753181\pi\)
\(252\) 0 0
\(253\) 3.24920 3.24920i 0.204275 0.204275i
\(254\) 0 0
\(255\) −6.15537 3.80423i −0.385464 0.238230i
\(256\) 0 0
\(257\) 18.7082 + 18.7082i 1.16699 + 1.16699i 0.982913 + 0.184073i \(0.0589283\pi\)
0.184073 + 0.982913i \(0.441072\pi\)
\(258\) 0 0
\(259\) 7.23607 0.449627
\(260\) 0 0
\(261\) −3.24920 −0.201120
\(262\) 0 0
\(263\) −16.3925 16.3925i −1.01080 1.01080i −0.999941 0.0108623i \(-0.996542\pi\)
−0.0108623 0.999941i \(-0.503458\pi\)
\(264\) 0 0
\(265\) 0.527864 + 2.23607i 0.0324264 + 0.137361i
\(266\) 0 0
\(267\) 10.4721 10.4721i 0.640884 0.640884i
\(268\) 0 0
\(269\) 17.9111i 1.09206i 0.837766 + 0.546029i \(0.183861\pi\)
−0.837766 + 0.546029i \(0.816139\pi\)
\(270\) 0 0
\(271\) 31.6749i 1.92411i 0.272851 + 0.962056i \(0.412033\pi\)
−0.272851 + 0.962056i \(0.587967\pi\)
\(272\) 0 0
\(273\) 11.7082 11.7082i 0.708613 0.708613i
\(274\) 0 0
\(275\) −5.52786 + 2.76393i −0.333343 + 0.166671i
\(276\) 0 0
\(277\) −2.17963 2.17963i −0.130961 0.130961i 0.638588 0.769549i \(-0.279519\pi\)
−0.769549 + 0.638588i \(0.779519\pi\)
\(278\) 0 0
\(279\) −11.7557 −0.703796
\(280\) 0 0
\(281\) −3.70820 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(282\) 0 0
\(283\) −15.6180 15.6180i −0.928396 0.928396i 0.0692066 0.997602i \(-0.477953\pi\)
−0.997602 + 0.0692066i \(0.977953\pi\)
\(284\) 0 0
\(285\) 9.95959 2.35114i 0.589955 0.139270i
\(286\) 0 0
\(287\) 9.06154 9.06154i 0.534886 0.534886i
\(288\) 0 0
\(289\) 15.0000i 0.882353i
\(290\) 0 0
\(291\) 0.763932i 0.0447825i
\(292\) 0 0
\(293\) −0.726543 + 0.726543i −0.0424451 + 0.0424451i −0.728011 0.685566i \(-0.759555\pi\)
0.685566 + 0.728011i \(0.259555\pi\)
\(294\) 0 0
\(295\) −9.95959 + 16.1150i −0.579870 + 0.938249i
\(296\) 0 0
\(297\) 1.52786 + 1.52786i 0.0886557 + 0.0886557i
\(298\) 0 0
\(299\) −16.1803 −0.935733
\(300\) 0 0
\(301\) −5.60034 −0.322798
\(302\) 0 0
\(303\) −19.9192 19.9192i −1.14433 1.14433i
\(304\) 0 0
\(305\) −11.7082 + 18.9443i −0.670410 + 1.08475i
\(306\) 0 0
\(307\) −6.56231 + 6.56231i −0.374531 + 0.374531i −0.869124 0.494594i \(-0.835317\pi\)
0.494594 + 0.869124i \(0.335317\pi\)
\(308\) 0 0
\(309\) 23.7234i 1.34958i
\(310\) 0 0
\(311\) 8.16348i 0.462909i −0.972846 0.231454i \(-0.925652\pi\)
0.972846 0.231454i \(-0.0743484\pi\)
\(312\) 0 0
\(313\) 6.23607 6.23607i 0.352483 0.352483i −0.508549 0.861033i \(-0.669818\pi\)
0.861033 + 0.508549i \(0.169818\pi\)
\(314\) 0 0
\(315\) −8.09017 + 1.90983i −0.455829 + 0.107607i
\(316\) 0 0
\(317\) 10.6861 + 10.6861i 0.600193 + 0.600193i 0.940364 0.340171i \(-0.110485\pi\)
−0.340171 + 0.940364i \(0.610485\pi\)
\(318\) 0 0
\(319\) 1.79611 0.100563
\(320\) 0 0
\(321\) −39.1246 −2.18372
\(322\) 0 0
\(323\) −2.00000 2.00000i −0.111283 0.111283i
\(324\) 0 0
\(325\) 20.6457 + 6.88191i 1.14522 + 0.381740i
\(326\) 0 0
\(327\) −10.8576 + 10.8576i −0.600429 + 0.600429i
\(328\) 0 0
\(329\) 17.2361i 0.950255i
\(330\) 0 0
\(331\) 28.0689i 1.54281i −0.636347 0.771403i \(-0.719555\pi\)
0.636347 0.771403i \(-0.280445\pi\)
\(332\) 0 0
\(333\) −6.88191 + 6.88191i −0.377126 + 0.377126i
\(334\) 0 0
\(335\) 1.73060 + 7.33094i 0.0945528 + 0.400532i
\(336\) 0 0
\(337\) −2.05573 2.05573i −0.111983 0.111983i 0.648895 0.760878i \(-0.275231\pi\)
−0.760878 + 0.648895i \(0.775231\pi\)
\(338\) 0 0
\(339\) 15.2361 0.827510
\(340\) 0 0
\(341\) 6.49839 0.351908
\(342\) 0 0
\(343\) 13.2088 + 13.2088i 0.713208 + 0.713208i
\(344\) 0 0
\(345\) 16.1803 + 10.0000i 0.871120 + 0.538382i
\(346\) 0 0
\(347\) −3.03444 + 3.03444i −0.162897 + 0.162897i −0.783849 0.620952i \(-0.786746\pi\)
0.620952 + 0.783849i \(0.286746\pi\)
\(348\) 0 0
\(349\) 15.5599i 0.832904i −0.909158 0.416452i \(-0.863273\pi\)
0.909158 0.416452i \(-0.136727\pi\)
\(350\) 0 0
\(351\) 7.60845i 0.406109i
\(352\) 0 0
\(353\) 4.41641 4.41641i 0.235062 0.235062i −0.579740 0.814802i \(-0.696846\pi\)
0.814802 + 0.579740i \(0.196846\pi\)
\(354\) 0 0
\(355\) 13.4164 + 8.29180i 0.712069 + 0.440083i
\(356\) 0 0
\(357\) 3.80423 + 3.80423i 0.201341 + 0.201341i
\(358\) 0 0
\(359\) −1.79611 −0.0947952 −0.0473976 0.998876i \(-0.515093\pi\)
−0.0473976 + 0.998876i \(0.515093\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −15.3262 15.3262i −0.804419 0.804419i
\(364\) 0 0
\(365\) −6.32688 26.8011i −0.331164 1.40283i
\(366\) 0 0
\(367\) −24.0009 + 24.0009i −1.25284 + 1.25284i −0.298396 + 0.954442i \(0.596452\pi\)
−0.954442 + 0.298396i \(0.903548\pi\)
\(368\) 0 0
\(369\) 17.2361i 0.897274i
\(370\) 0 0
\(371\) 1.70820i 0.0886855i
\(372\) 0 0
\(373\) 15.0454 15.0454i 0.779021 0.779021i −0.200644 0.979664i \(-0.564303\pi\)
0.979664 + 0.200644i \(0.0643033\pi\)
\(374\) 0 0
\(375\) −16.3925 19.6417i −0.846504 1.01429i
\(376\) 0 0
\(377\) −4.47214 4.47214i −0.230327 0.230327i
\(378\) 0 0
\(379\) 35.8885 1.84347 0.921735 0.387820i \(-0.126772\pi\)
0.921735 + 0.387820i \(0.126772\pi\)
\(380\) 0 0
\(381\) 28.4257 1.45629
\(382\) 0 0
\(383\) −11.1352 11.1352i −0.568980 0.568980i 0.362862 0.931843i \(-0.381799\pi\)
−0.931843 + 0.362862i \(0.881799\pi\)
\(384\) 0 0
\(385\) 4.47214 1.05573i 0.227921 0.0538049i
\(386\) 0 0
\(387\) 5.32624 5.32624i 0.270748 0.270748i
\(388\) 0 0
\(389\) 23.7234i 1.20282i −0.798939 0.601412i \(-0.794605\pi\)
0.798939 0.601412i \(-0.205395\pi\)
\(390\) 0 0
\(391\) 5.25731i 0.265874i
\(392\) 0 0
\(393\) 0.472136 0.472136i 0.0238161 0.0238161i
\(394\) 0 0
\(395\) −14.4721 + 23.4164i −0.728172 + 1.17821i
\(396\) 0 0
\(397\) 7.22494 + 7.22494i 0.362609 + 0.362609i 0.864773 0.502164i \(-0.167462\pi\)
−0.502164 + 0.864773i \(0.667462\pi\)
\(398\) 0 0
\(399\) −7.60845 −0.380899
\(400\) 0 0
\(401\) −3.88854 −0.194185 −0.0970923 0.995275i \(-0.530954\pi\)
−0.0970923 + 0.995275i \(0.530954\pi\)
\(402\) 0 0
\(403\) −16.1803 16.1803i −0.806000 0.806000i
\(404\) 0 0
\(405\) −12.5882 + 20.3682i −0.625515 + 1.01210i
\(406\) 0 0
\(407\) 3.80423 3.80423i 0.188568 0.188568i
\(408\) 0 0
\(409\) 27.5967i 1.36457i −0.731086 0.682286i \(-0.760986\pi\)
0.731086 0.682286i \(-0.239014\pi\)
\(410\) 0 0
\(411\) 11.2361i 0.554234i
\(412\) 0 0
\(413\) 9.95959 9.95959i 0.490080 0.490080i
\(414\) 0 0
\(415\) 13.4863 3.18368i 0.662017 0.156281i
\(416\) 0 0
\(417\) 8.76393 + 8.76393i 0.429172 + 0.429172i
\(418\) 0 0
\(419\) −24.8328 −1.21316 −0.606581 0.795022i \(-0.707460\pi\)
−0.606581 + 0.795022i \(0.707460\pi\)
\(420\) 0 0
\(421\) −3.46120 −0.168689 −0.0843443 0.996437i \(-0.526880\pi\)
−0.0843443 + 0.996437i \(0.526880\pi\)
\(422\) 0 0
\(423\) 16.3925 + 16.3925i 0.797029 + 0.797029i
\(424\) 0 0
\(425\) −2.23607 + 6.70820i −0.108465 + 0.325396i
\(426\) 0 0
\(427\) 11.7082 11.7082i 0.566600 0.566600i
\(428\) 0 0
\(429\) 12.3107i 0.594368i
\(430\) 0 0
\(431\) 11.7557i 0.566252i −0.959083 0.283126i \(-0.908628\pi\)
0.959083 0.283126i \(-0.0913715\pi\)
\(432\) 0 0
\(433\) 23.1803 23.1803i 1.11398 1.11398i 0.121369 0.992608i \(-0.461272\pi\)
0.992608 0.121369i \(-0.0387283\pi\)
\(434\) 0 0
\(435\) 1.70820 + 7.23607i 0.0819021 + 0.346943i
\(436\) 0 0
\(437\) 5.25731 + 5.25731i 0.251491 + 0.251491i
\(438\) 0 0
\(439\) 11.2007 0.534579 0.267290 0.963616i \(-0.413872\pi\)
0.267290 + 0.963616i \(0.413872\pi\)
\(440\) 0 0
\(441\) −9.47214 −0.451054
\(442\) 0 0
\(443\) −10.0902 10.0902i −0.479398 0.479398i 0.425541 0.904939i \(-0.360084\pi\)
−0.904939 + 0.425541i \(0.860084\pi\)
\(444\) 0 0
\(445\) −12.3107 7.60845i −0.583585 0.360675i
\(446\) 0 0
\(447\) 21.3723 21.3723i 1.01087 1.01087i
\(448\) 0 0
\(449\) 31.5967i 1.49114i −0.666426 0.745571i \(-0.732177\pi\)
0.666426 0.745571i \(-0.267823\pi\)
\(450\) 0 0
\(451\) 9.52786i 0.448650i
\(452\) 0 0
\(453\) 23.7234 23.7234i 1.11462 1.11462i
\(454\) 0 0
\(455\) −13.7638 8.50651i −0.645258 0.398791i
\(456\) 0 0
\(457\) 21.6525 + 21.6525i 1.01286 + 1.01286i 0.999916 + 0.0129439i \(0.00412028\pi\)
0.0129439 + 0.999916i \(0.495880\pi\)
\(458\) 0 0
\(459\) 2.47214 0.115389
\(460\) 0 0
\(461\) 6.49839 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(462\) 0 0
\(463\) 17.2905 + 17.2905i 0.803559 + 0.803559i 0.983650 0.180091i \(-0.0576392\pi\)
−0.180091 + 0.983650i \(0.557639\pi\)
\(464\) 0 0
\(465\) 6.18034 + 26.1803i 0.286606 + 1.21408i
\(466\) 0 0
\(467\) 2.67376 2.67376i 0.123727 0.123727i −0.642532 0.766259i \(-0.722116\pi\)
0.766259 + 0.642532i \(0.222116\pi\)
\(468\) 0 0
\(469\) 5.60034i 0.258600i
\(470\) 0 0
\(471\) 31.6749i 1.45950i
\(472\) 0 0
\(473\) −2.94427 + 2.94427i −0.135378 + 0.135378i
\(474\) 0 0
\(475\) −4.47214 8.94427i −0.205196 0.410391i
\(476\) 0 0
\(477\) 1.62460 + 1.62460i 0.0743853 + 0.0743853i
\(478\) 0 0
\(479\) 7.60845 0.347639 0.173820 0.984778i \(-0.444389\pi\)
0.173820 + 0.984778i \(0.444389\pi\)
\(480\) 0 0
\(481\) −18.9443 −0.863784
\(482\) 0 0
\(483\) −10.0000 10.0000i −0.455016 0.455016i
\(484\) 0 0
\(485\) 0.726543 0.171513i 0.0329906 0.00778802i
\(486\) 0 0
\(487\) −9.33905 + 9.33905i −0.423193 + 0.423193i −0.886302 0.463109i \(-0.846734\pi\)
0.463109 + 0.886302i \(0.346734\pi\)
\(488\) 0 0
\(489\) 23.1246i 1.04573i
\(490\) 0 0
\(491\) 10.7639i 0.485769i 0.970055 + 0.242885i \(0.0780937\pi\)
−0.970055 + 0.242885i \(0.921906\pi\)
\(492\) 0 0
\(493\) 1.45309 1.45309i 0.0654437 0.0654437i
\(494\) 0 0
\(495\) −3.24920 + 5.25731i −0.146041 + 0.236299i
\(496\) 0 0
\(497\) −8.29180 8.29180i −0.371938 0.371938i
\(498\) 0 0
\(499\) 23.8885 1.06940 0.534699 0.845043i \(-0.320425\pi\)
0.534699 + 0.845043i \(0.320425\pi\)
\(500\) 0 0
\(501\) −0.898056 −0.0401222
\(502\) 0 0
\(503\) 5.53483 + 5.53483i 0.246786 + 0.246786i 0.819650 0.572864i \(-0.194168\pi\)
−0.572864 + 0.819650i \(0.694168\pi\)
\(504\) 0 0
\(505\) −14.4721 + 23.4164i −0.644002 + 1.04202i
\(506\) 0 0
\(507\) −9.61803 + 9.61803i −0.427152 + 0.427152i
\(508\) 0 0
\(509\) 9.06154i 0.401646i 0.979628 + 0.200823i \(0.0643615\pi\)
−0.979628 + 0.200823i \(0.935638\pi\)
\(510\) 0 0
\(511\) 20.4742i 0.905726i
\(512\) 0 0
\(513\) −2.47214 + 2.47214i −0.109147 + 0.109147i
\(514\) 0 0
\(515\) −22.5623 + 5.32624i −0.994214 + 0.234702i
\(516\) 0 0
\(517\) −9.06154 9.06154i −0.398526 0.398526i
\(518\) 0 0
\(519\) 20.4742 0.898718
\(520\) 0 0
\(521\) −8.47214 −0.371171 −0.185586 0.982628i \(-0.559418\pi\)
−0.185586 + 0.982628i \(0.559418\pi\)
\(522\) 0 0
\(523\) 16.7426 + 16.7426i 0.732105 + 0.732105i 0.971036 0.238932i \(-0.0767972\pi\)
−0.238932 + 0.971036i \(0.576797\pi\)
\(524\) 0 0
\(525\) 8.50651 + 17.0130i 0.371254 + 0.742509i
\(526\) 0 0
\(527\) 5.25731 5.25731i 0.229012 0.229012i
\(528\) 0 0
\(529\) 9.18034i 0.399145i
\(530\) 0 0
\(531\) 18.9443i 0.822111i
\(532\) 0 0
\(533\) −23.7234 + 23.7234i −1.02758 + 1.02758i
\(534\) 0 0
\(535\) 8.78402 + 37.2097i 0.379766 + 1.60872i
\(536\) 0 0
\(537\) −26.6525 26.6525i −1.15014 1.15014i
\(538\) 0 0
\(539\) 5.23607 0.225533
\(540\) 0 0
\(541\) 2.90617 0.124946 0.0624730 0.998047i \(-0.480101\pi\)
0.0624730 + 0.998047i \(0.480101\pi\)
\(542\) 0 0
\(543\) 15.2169 + 15.2169i 0.653020 + 0.653020i
\(544\) 0 0
\(545\) 12.7639 + 7.88854i 0.546747 + 0.337908i
\(546\) 0 0
\(547\) 8.56231 8.56231i 0.366098 0.366098i −0.499954 0.866052i \(-0.666650\pi\)
0.866052 + 0.499954i \(0.166650\pi\)
\(548\) 0 0
\(549\) 22.2703i 0.950474i
\(550\) 0 0
\(551\) 2.90617i 0.123807i
\(552\) 0 0
\(553\) 14.4721 14.4721i 0.615418 0.615418i
\(554\) 0 0
\(555\) 18.9443 + 11.7082i 0.804140 + 0.496986i
\(556\) 0 0
\(557\) −17.1845 17.1845i −0.728132 0.728132i 0.242116 0.970247i \(-0.422159\pi\)
−0.970247 + 0.242116i \(0.922159\pi\)
\(558\) 0 0
\(559\) 14.6619 0.620131
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 0 0
\(563\) 11.3262 + 11.3262i 0.477344 + 0.477344i 0.904281 0.426937i \(-0.140407\pi\)
−0.426937 + 0.904281i \(0.640407\pi\)
\(564\) 0 0
\(565\) −3.42071 14.4904i −0.143910 0.609614i
\(566\) 0 0
\(567\) 12.5882 12.5882i 0.528657 0.528657i
\(568\) 0 0
\(569\) 13.1246i 0.550212i −0.961414 0.275106i \(-0.911287\pi\)
0.961414 0.275106i \(-0.0887130\pi\)
\(570\) 0 0
\(571\) 8.65248i 0.362095i 0.983474 + 0.181047i \(0.0579488\pi\)
−0.983474 + 0.181047i \(0.942051\pi\)
\(572\) 0 0
\(573\) −20.8172 + 20.8172i −0.869653 + 0.869653i
\(574\) 0 0
\(575\) 5.87785 17.6336i 0.245123 0.735370i
\(576\) 0 0
\(577\) 21.7639 + 21.7639i 0.906044 + 0.906044i 0.995950 0.0899059i \(-0.0286566\pi\)
−0.0899059 + 0.995950i \(0.528657\pi\)
\(578\) 0 0
\(579\) 24.1803 1.00490
\(580\) 0 0
\(581\) −10.3026 −0.427425
\(582\) 0 0
\(583\) −0.898056 0.898056i −0.0371937 0.0371937i
\(584\) 0 0
\(585\) 21.1803 5.00000i 0.875699 0.206725i
\(586\) 0 0
\(587\) 5.90983 5.90983i 0.243925 0.243925i −0.574547 0.818472i \(-0.694822\pi\)
0.818472 + 0.574547i \(0.194822\pi\)
\(588\) 0 0
\(589\) 10.5146i 0.433247i
\(590\) 0 0
\(591\) 7.05342i 0.290139i
\(592\) 0 0
\(593\) −6.41641 + 6.41641i −0.263490 + 0.263490i −0.826470 0.562980i \(-0.809655\pi\)
0.562980 + 0.826470i \(0.309655\pi\)
\(594\) 0 0
\(595\) 2.76393 4.47214i 0.113310 0.183340i
\(596\) 0 0
\(597\) 29.3238 + 29.3238i 1.20014 + 1.20014i
\(598\) 0 0
\(599\) 27.5276 1.12475 0.562374 0.826883i \(-0.309888\pi\)
0.562374 + 0.826883i \(0.309888\pi\)
\(600\) 0 0
\(601\) 4.29180 0.175066 0.0875330 0.996162i \(-0.472102\pi\)
0.0875330 + 0.996162i \(0.472102\pi\)
\(602\) 0 0
\(603\) 5.32624 + 5.32624i 0.216901 + 0.216901i
\(604\) 0 0
\(605\) −11.1352 + 18.0171i −0.452709 + 0.732498i
\(606\) 0 0
\(607\) −6.08985 + 6.08985i −0.247180 + 0.247180i −0.819812 0.572633i \(-0.805922\pi\)
0.572633 + 0.819812i \(0.305922\pi\)
\(608\) 0 0
\(609\) 5.52786i 0.224000i
\(610\) 0 0
\(611\) 45.1246i 1.82555i
\(612\) 0 0
\(613\) −27.3561 + 27.3561i −1.10490 + 1.10490i −0.111094 + 0.993810i \(0.535435\pi\)
−0.993810 + 0.111094i \(0.964565\pi\)
\(614\) 0 0
\(615\) 38.3853 9.06154i 1.54784 0.365396i
\(616\) 0 0
\(617\) 30.8885 + 30.8885i 1.24353 + 1.24353i 0.958528 + 0.284998i \(0.0919929\pi\)
0.284998 + 0.958528i \(0.408007\pi\)
\(618\) 0 0
\(619\) −27.3050 −1.09748 −0.548739 0.835994i \(-0.684892\pi\)
−0.548739 + 0.835994i \(0.684892\pi\)
\(620\) 0 0
\(621\) −6.49839 −0.260772
\(622\) 0 0
\(623\) 7.60845 + 7.60845i 0.304826 + 0.304826i
\(624\) 0 0
\(625\) −15.0000 + 20.0000i −0.600000 + 0.800000i
\(626\) 0 0
\(627\) −4.00000 + 4.00000i −0.159745 + 0.159745i
\(628\) 0 0
\(629\) 6.15537i 0.245431i
\(630\) 0 0
\(631\) 28.0827i 1.11795i 0.829183 + 0.558977i \(0.188806\pi\)
−0.829183 + 0.558977i \(0.811194\pi\)
\(632\) 0 0
\(633\) 25.4164 25.4164i 1.01021 1.01021i
\(634\) 0 0
\(635\) −6.38197 27.0344i −0.253261 1.07283i
\(636\) 0 0
\(637\) −13.0373 13.0373i −0.516556 0.516556i
\(638\) 0 0
\(639\) 15.7719 0.623928
\(640\) 0 0
\(641\) −7.34752 −0.290210 −0.145105 0.989416i \(-0.546352\pi\)
−0.145105 + 0.989416i \(0.546352\pi\)
\(642\) 0 0
\(643\) 8.56231 + 8.56231i 0.337664 + 0.337664i 0.855488 0.517823i \(-0.173258\pi\)
−0.517823 + 0.855488i \(0.673258\pi\)
\(644\) 0 0
\(645\) −14.6619 9.06154i −0.577311 0.356798i
\(646\) 0 0
\(647\) 22.8909 22.8909i 0.899933 0.899933i −0.0954968 0.995430i \(-0.530444\pi\)
0.995430 + 0.0954968i \(0.0304440\pi\)
\(648\) 0 0
\(649\) 10.4721i 0.411067i
\(650\) 0 0
\(651\) 20.0000i 0.783862i
\(652\) 0 0
\(653\) −11.2412 + 11.2412i −0.439901 + 0.439901i −0.891979 0.452078i \(-0.850683\pi\)
0.452078 + 0.891979i \(0.350683\pi\)
\(654\) 0 0
\(655\) −0.555029 0.343027i −0.0216868 0.0134032i
\(656\) 0 0
\(657\) −19.4721 19.4721i −0.759680 0.759680i
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) −2.35114 −0.0914488 −0.0457244 0.998954i \(-0.514560\pi\)
−0.0457244 + 0.998954i \(0.514560\pi\)
\(662\) 0 0
\(663\) −9.95959 9.95959i −0.386799 0.386799i
\(664\) 0 0
\(665\) 1.70820 + 7.23607i 0.0662413 + 0.280603i
\(666\) 0 0
\(667\) −3.81966 + 3.81966i −0.147898 + 0.147898i
\(668\) 0 0
\(669\) 5.60034i 0.216522i
\(670\) 0 0
\(671\) 12.3107i 0.475251i
\(672\) 0 0
\(673\) −30.7082 + 30.7082i −1.18371 + 1.18371i −0.204940 + 0.978775i \(0.565700\pi\)
−0.978775 + 0.204940i \(0.934300\pi\)
\(674\) 0 0
\(675\) 8.29180 + 2.76393i 0.319151 + 0.106384i
\(676\) 0 0
\(677\) −8.33499 8.33499i −0.320340 0.320340i 0.528558 0.848897i \(-0.322733\pi\)
−0.848897 + 0.528558i \(0.822733\pi\)
\(678\) 0 0
\(679\) −0.555029 −0.0213001
\(680\) 0 0
\(681\) −37.5967 −1.44071
\(682\) 0 0
\(683\) −1.79837 1.79837i −0.0688129 0.0688129i 0.671863 0.740676i \(-0.265494\pi\)
−0.740676 + 0.671863i \(0.765494\pi\)
\(684\) 0 0
\(685\) 10.6861 2.52265i 0.408296 0.0963857i
\(686\) 0 0
\(687\) 34.5811 34.5811i 1.31935 1.31935i
\(688\) 0 0
\(689\) 4.47214i 0.170375i
\(690\) 0 0
\(691\) 31.1246i 1.18404i −0.805925 0.592018i \(-0.798331\pi\)
0.805925 0.592018i \(-0.201669\pi\)
\(692\) 0 0
\(693\) 3.24920 3.24920i 0.123427 0.123427i
\(694\) 0 0
\(695\) 6.36737 10.3026i 0.241528 0.390801i
\(696\) 0 0
\(697\) −7.70820 7.70820i −0.291969 0.291969i
\(698\) 0 0
\(699\) −11.2361 −0.424987
\(700\) 0 0
\(701\) −40.3934 −1.52564 −0.762819 0.646612i \(-0.776185\pi\)
−0.762819 + 0.646612i \(0.776185\pi\)
\(702\) 0 0
\(703\) 6.15537 + 6.15537i 0.232154 + 0.232154i
\(704\) 0 0
\(705\) 27.8885 45.1246i 1.05034 1.69949i
\(706\) 0 0
\(707\) 14.4721 14.4721i 0.544281 0.544281i
\(708\) 0 0
\(709\) 13.7638i 0.516911i 0.966023 + 0.258456i \(0.0832136\pi\)
−0.966023 + 0.258456i \(0.916786\pi\)
\(710\) 0 0
\(711\) 27.5276i 1.03237i
\(712\) 0 0
\(713\) −13.8197 + 13.8197i −0.517550 + 0.517550i
\(714\) 0 0
\(715\) −11.7082 + 2.76393i −0.437862 + 0.103365i
\(716\) 0 0
\(717\) 47.4468 + 47.4468i 1.77193 + 1.77193i
\(718\) 0 0
\(719\) 44.5407 1.66109 0.830543 0.556954i \(-0.188030\pi\)
0.830543 + 0.556954i \(0.188030\pi\)
\(720\) 0 0
\(721\) 17.2361 0.641905
\(722\) 0 0
\(723\) −10.9443 10.9443i −0.407022 0.407022i
\(724\) 0 0
\(725\) 6.49839 3.24920i 0.241344 0.120672i
\(726\) 0 0
\(727\) 5.87785 5.87785i 0.217997 0.217997i −0.589657 0.807654i \(-0.700737\pi\)
0.807654 + 0.589657i \(0.200737\pi\)
\(728\) 0 0
\(729\) 11.9443i 0.442380i
\(730\) 0 0
\(731\) 4.76393i 0.176200i
\(732\) 0 0
\(733\) 1.28157 1.28157i 0.0473359 0.0473359i −0.683043 0.730379i \(-0.739344\pi\)
0.730379 + 0.683043i \(0.239344\pi\)
\(734\) 0 0
\(735\) 4.97980 + 21.0948i 0.183683 + 0.778092i
\(736\) 0 0
\(737\) −2.94427 2.94427i −0.108454 0.108454i
\(738\) 0 0
\(739\) −17.4164 −0.640673 −0.320336 0.947304i \(-0.603796\pi\)
−0.320336 + 0.947304i \(0.603796\pi\)
\(740\) 0 0
\(741\) 19.9192 0.731750
\(742\) 0 0
\(743\) −13.4863 13.4863i −0.494765 0.494765i 0.415039 0.909804i \(-0.363768\pi\)
−0.909804 + 0.415039i \(0.863768\pi\)
\(744\) 0 0
\(745\) −25.1246 15.5279i −0.920495 0.568897i
\(746\) 0 0
\(747\) 9.79837 9.79837i 0.358504 0.358504i
\(748\) 0 0
\(749\) 28.4257i 1.03865i
\(750\) 0 0
\(751\) 7.05342i 0.257383i −0.991685 0.128692i \(-0.958922\pi\)
0.991685 0.128692i \(-0.0410777\pi\)
\(752\) 0 0
\(753\) 35.8885 35.8885i 1.30785 1.30785i
\(754\) 0 0
\(755\) −27.8885 17.2361i −1.01497 0.627285i
\(756\) 0 0
\(757\) −38.0018 38.0018i −1.38120 1.38120i −0.842497 0.538701i \(-0.818915\pi\)
−0.538701 0.842497i \(-0.681085\pi\)
\(758\) 0 0
\(759\) −10.5146 −0.381657
\(760\) 0 0
\(761\) 14.9443 0.541729 0.270865 0.962617i \(-0.412690\pi\)
0.270865 + 0.962617i \(0.412690\pi\)
\(762\) 0 0
\(763\) −7.88854 7.88854i −0.285584 0.285584i
\(764\) 0 0
\(765\) 1.62460 + 6.88191i 0.0587375 + 0.248816i
\(766\) 0 0
\(767\) −26.0746 + 26.0746i −0.941498 + 0.941498i
\(768\) 0 0
\(769\) 2.47214i 0.0891475i 0.999006 + 0.0445738i \(0.0141930\pi\)
−0.999006 + 0.0445738i \(0.985807\pi\)
\(770\) 0 0
\(771\) 60.5410i 2.18033i
\(772\) 0 0
\(773\) 4.18774 4.18774i 0.150623 0.150623i −0.627773 0.778396i \(-0.716034\pi\)
0.778396 + 0.627773i \(0.216034\pi\)
\(774\) 0 0
\(775\) 23.5114 11.7557i 0.844555 0.422277i
\(776\) 0 0
\(777\) −11.7082 11.7082i −0.420029 0.420029i
\(778\) 0 0
\(779\) 15.4164 0.552350
\(780\) 0 0
\(781\) −8.71851 −0.311973
\(782\) 0 0
\(783\) −1.79611 1.79611i −0.0641878 0.0641878i
\(784\) 0 0
\(785\) −30.1246 + 7.11146i −1.07519 + 0.253819i
\(786\) 0 0
\(787\) −11.1459 + 11.1459i −0.397308 + 0.397308i −0.877283 0.479974i \(-0.840646\pi\)
0.479974 + 0.877283i \(0.340646\pi\)
\(788\) 0 0
\(789\) 53.0472i 1.88853i
\(790\) 0 0
\(791\) 11.0697i 0.393591i
\(792\) 0 0
\(793\) −30.6525 + 30.6525i −1.08850 + 1.08850i
\(794\) 0 0
\(795\) 2.76393 4.47214i 0.0980266 0.158610i
\(796\) 0 0
\(797\) −19.7477 19.7477i −0.699498 0.699498i 0.264804 0.964302i \(-0.414693\pi\)
−0.964302 + 0.264804i \(0.914693\pi\)
\(798\) 0 0
\(799\) −14.6619 −0.518700
\(800\) 0 0
\(801\) −14.4721 −0.511348
\(802\) 0 0
\(803\) 10.7639 + 10.7639i 0.379851 + 0.379851i
\(804\) 0 0
\(805\) −7.26543 + 11.7557i −0.256073 + 0.414334i
\(806\) 0 0
\(807\) 28.9807 28.9807i 1.02017 1.02017i
\(808\) 0 0
\(809\) 12.9443i 0.455096i −0.973767 0.227548i \(-0.926929\pi\)
0.973767 0.227548i \(-0.0730709\pi\)
\(810\) 0 0
\(811\) 32.0689i 1.12609i 0.826426 + 0.563045i \(0.190370\pi\)
−0.826426 + 0.563045i \(0.809630\pi\)
\(812\) 0 0
\(813\) 51.2511 51.2511i 1.79745 1.79745i
\(814\) 0 0
\(815\) 21.9928 5.19180i 0.770375 0.181861i
\(816\) 0 0
\(817\) −4.76393 4.76393i −0.166669 0.166669i
\(818\) 0 0
\(819\) −16.1803 −0.565387
\(820\) 0 0
\(821\) 20.4742 0.714555 0.357278 0.933998i \(-0.383705\pi\)
0.357278 + 0.933998i \(0.383705\pi\)
\(822\) 0 0
\(823\) 24.3440 + 24.3440i 0.848577 + 0.848577i 0.989956 0.141379i \(-0.0451535\pi\)
−0.141379 + 0.989956i \(0.545154\pi\)
\(824\) 0 0
\(825\) 13.4164 + 4.47214i 0.467099 + 0.155700i
\(826\) 0 0
\(827\) 14.8541 14.8541i 0.516528 0.516528i −0.399991 0.916519i \(-0.630987\pi\)
0.916519 + 0.399991i \(0.130987\pi\)
\(828\) 0 0
\(829\) 3.11817i 0.108299i −0.998533 0.0541493i \(-0.982755\pi\)
0.998533 0.0541493i \(-0.0172447\pi\)
\(830\) 0 0
\(831\) 7.05342i 0.244681i
\(832\) 0 0
\(833\) 4.23607 4.23607i 0.146771 0.146771i
\(834\) 0 0
\(835\) 0.201626 + 0.854102i 0.00697756 + 0.0295574i
\(836\) 0 0
\(837\) −6.49839 6.49839i −0.224617 0.224617i
\(838\) 0 0
\(839\) −9.40456 −0.324682 −0.162341 0.986735i \(-0.551904\pi\)
−0.162341 + 0.986735i \(0.551904\pi\)
\(840\) 0 0
\(841\) 26.8885 0.927191
\(842\) 0 0
\(843\) 6.00000 + 6.00000i 0.206651 + 0.206651i
\(844\) 0 0
\(845\) 11.3067 + 6.98791i 0.388962 + 0.240391i
\(846\) 0 0
\(847\) 11.1352 11.1352i 0.382609 0.382609i
\(848\) 0 0
\(849\) 50.5410i 1.73456i
\(850\) 0 0
\(851\) 16.1803i 0.554655i
\(852\) 0 0
\(853\) 24.7930 24.7930i 0.848896 0.848896i −0.141100 0.989995i \(-0.545064\pi\)
0.989995 + 0.141100i \(0.0450639\pi\)
\(854\) 0 0
\(855\) −8.50651 5.25731i −0.290916 0.179796i
\(856\) 0 0
\(857\) 17.8328 + 17.8328i 0.609157 + 0.609157i 0.942726 0.333568i \(-0.108253\pi\)
−0.333568 + 0.942726i \(0.608253\pi\)
\(858\) 0 0
\(859\) −7.52786 −0.256847 −0.128424 0.991719i \(-0.540992\pi\)
−0.128424 + 0.991719i \(0.540992\pi\)
\(860\) 0 0
\(861\) −29.3238 −0.999351
\(862\) 0 0
\(863\) −6.22088 6.22088i −0.211761 0.211761i 0.593254 0.805015i \(-0.297843\pi\)
−0.805015 + 0.593254i \(0.797843\pi\)
\(864\) 0 0
\(865\) −4.59675 19.4721i −0.156294 0.662072i
\(866\) 0 0
\(867\) −24.2705 + 24.2705i −0.824270 + 0.824270i
\(868\) 0 0
\(869\) 15.2169i 0.516198i
\(870\) 0 0
\(871\) 14.6619i 0.496799i
\(872\) 0 0
\(873\) 0.527864 0.527864i 0.0178655 0.0178655i
\(874\) 0 0
\(875\) 14.2705 11.9098i 0.482431 0.402626i
\(876\) 0 0
\(877\) 5.08580 + 5.08580i 0.171735 + 0.171735i 0.787741 0.616006i \(-0.211250\pi\)
−0.616006 + 0.787741i \(0.711250\pi\)
\(878\) 0 0
\(879\) 2.35114 0.0793020
\(880\) 0 0
\(881\) 47.1246 1.58767 0.793834 0.608134i \(-0.208082\pi\)
0.793834 + 0.608134i \(0.208082\pi\)
\(882\) 0 0
\(883\) 21.7984 + 21.7984i 0.733574 + 0.733574i 0.971326 0.237752i \(-0.0764106\pi\)
−0.237752 + 0.971326i \(0.576411\pi\)
\(884\) 0 0
\(885\) 42.1895 9.95959i 1.41818 0.334788i
\(886\) 0 0
\(887\) −28.1482 + 28.1482i −0.945123 + 0.945123i −0.998571 0.0534473i \(-0.982979\pi\)
0.0534473 + 0.998571i \(0.482979\pi\)
\(888\) 0 0
\(889\) 20.6525i 0.692662i
\(890\) 0 0
\(891\) 13.2361i 0.443425i
\(892\) 0 0
\(893\) 14.6619 14.6619i 0.490641 0.490641i
\(894\) 0 0
\(895\) −19.3642 + 31.3319i −0.647272 + 1.04731i
\(896\) 0 0
\(897\) 26.1803 + 26.1803i 0.874136 + 0.874136i
\(898\) 0 0
\(899\) −7.63932 −0.254786
\(900\) 0 0
\(901\) −1.45309 −0.0484093
\(902\) 0 0
\(903\) 9.06154 + 9.06154i 0.301549 + 0.301549i
\(904\) 0 0
\(905\) 11.0557 17.8885i 0.367505 0.594635i
\(906\) 0 0
\(907\) −23.3262 + 23.3262i −0.774535 + 0.774535i −0.978896 0.204361i \(-0.934488\pi\)
0.204361 + 0.978896i \(0.434488\pi\)
\(908\) 0 0
\(909\) 27.5276i 0.913034i
\(910\) 0 0
\(911\) 25.1765i 0.834135i −0.908876 0.417067i \(-0.863058\pi\)
0.908876 0.417067i \(-0.136942\pi\)
\(912\) 0 0
\(913\) −5.41641 + 5.41641i −0.179257 + 0.179257i
\(914\) 0 0
\(915\) 49.5967 11.7082i 1.63962 0.387061i
\(916\) 0 0
\(917\) 0.343027 + 0.343027i 0.0113277 + 0.0113277i
\(918\) 0 0
\(919\) −21.7153 −0.716322 −0.358161 0.933660i \(-0.616596\pi\)
−0.358161 + 0.933660i \(0.616596\pi\)
\(920\) 0 0
\(921\) 21.2361 0.699752
\(922\) 0 0
\(923\) 21.7082 + 21.7082i 0.714534 + 0.714534i
\(924\) 0 0
\(925\) 6.88191 20.6457i 0.226276 0.678827i
\(926\) 0 0
\(927\) −16.3925 + 16.3925i −0.538400 + 0.538400i
\(928\) 0 0
\(929\) 5.34752i 0.175447i −0.996145 0.0877233i \(-0.972041\pi\)
0.996145 0.0877233i \(-0.0279591\pi\)
\(930\) 0 0
\(931\) 8.47214i 0.277663i
\(932\) 0 0
\(933\) −13.2088 + 13.2088i −0.432436 + 0.432436i
\(934\) 0 0
\(935\) −0.898056 3.80423i −0.0293696 0.124411i
\(936\) 0 0
\(937\) −20.3050 20.3050i −0.663334 0.663334i 0.292831 0.956164i \(-0.405403\pi\)
−0.956164 + 0.292831i \(0.905403\pi\)
\(938\) 0 0
\(939\) −20.1803 −0.658561
\(940\) 0 0
\(941\) 60.8676 1.98423 0.992114 0.125340i \(-0.0400023\pi\)
0.992114 + 0.125340i \(0.0400023\pi\)
\(942\) 0 0
\(943\) 20.2622 + 20.2622i 0.659828 + 0.659828i
\(944\) 0 0
\(945\) −5.52786 3.41641i −0.179821 0.111136i
\(946\) 0 0
\(947\) 8.85410 8.85410i 0.287720 0.287720i −0.548458 0.836178i \(-0.684785\pi\)
0.836178 + 0.548458i \(0.184785\pi\)
\(948\) 0 0
\(949\) 53.6022i 1.74000i
\(950\) 0 0
\(951\) 34.5811i 1.12137i
\(952\) 0 0
\(953\) 6.81966 6.81966i 0.220910 0.220910i −0.587971 0.808882i \(-0.700073\pi\)
0.808882 + 0.587971i \(0.200073\pi\)
\(954\) 0 0
\(955\) 24.4721 + 15.1246i 0.791900 + 0.489421i
\(956\) 0 0
\(957\) −2.90617 2.90617i −0.0939431 0.0939431i
\(958\) 0 0
\(959\) −8.16348 −0.263613
\(960\) 0 0
\(961\) 3.36068 0.108409
\(962\) 0 0
\(963\) 27.0344 + 27.0344i 0.871173 + 0.871173i
\(964\) 0 0
\(965\) −5.42882 22.9969i −0.174760 0.740295i
\(966\) 0 0
\(967\) −12.0332 + 12.0332i −0.386962 + 0.386962i −0.873602 0.486640i \(-0.838222\pi\)
0.486640 + 0.873602i \(0.338222\pi\)
\(968\) 0 0
\(969\) 6.47214i 0.207915i
\(970\) 0 0
\(971\) 33.5967i 1.07817i −0.842251 0.539085i \(-0.818770\pi\)
0.842251 0.539085i \(-0.181230\pi\)
\(972\) 0 0
\(973\) −6.36737 + 6.36737i −0.204128 + 0.204128i
\(974\) 0 0
\(975\) −22.2703 44.5407i −0.713221 1.42644i
\(976\) 0 0
\(977\) −28.2361 28.2361i −0.903352 0.903352i 0.0923727 0.995725i \(-0.470555\pi\)
−0.995725 + 0.0923727i \(0.970555\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 15.0049 0.479070
\(982\) 0 0
\(983\) 0.620541 + 0.620541i 0.0197922 + 0.0197922i 0.716934 0.697141i \(-0.245545\pi\)
−0.697141 + 0.716934i \(0.745545\pi\)
\(984\) 0 0
\(985\) −6.70820 + 1.58359i −0.213741 + 0.0504574i
\(986\) 0 0
\(987\) −27.8885 + 27.8885i −0.887702 + 0.887702i
\(988\) 0 0
\(989\) 12.5227i 0.398200i
\(990\) 0 0
\(991\) 19.3642i 0.615123i 0.951528 + 0.307561i \(0.0995129\pi\)
−0.951528 + 0.307561i \(0.900487\pi\)
\(992\) 0 0
\(993\) −45.4164 + 45.4164i −1.44125 + 1.44125i
\(994\) 0 0
\(995\) 21.3050 34.4721i 0.675412 1.09284i
\(996\) 0 0
\(997\) 35.6506 + 35.6506i 1.12907 + 1.12907i 0.990329 + 0.138738i \(0.0443045\pi\)
0.138738 + 0.990329i \(0.455696\pi\)
\(998\) 0 0
\(999\) −7.60845 −0.240721
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 1280.2.n.m.767.1 8
4.3 odd 2 1280.2.n.q.767.3 8
5.3 odd 4 1280.2.n.q.1023.3 8
8.3 odd 2 inner 1280.2.n.m.767.2 8
8.5 even 2 1280.2.n.q.767.4 8
16.3 odd 4 40.2.k.a.27.2 yes 8
16.5 even 4 40.2.k.a.27.4 yes 8
16.11 odd 4 160.2.o.a.47.3 8
16.13 even 4 160.2.o.a.47.4 8
20.3 even 4 inner 1280.2.n.m.1023.1 8
40.3 even 4 1280.2.n.q.1023.4 8
40.13 odd 4 inner 1280.2.n.m.1023.2 8
48.5 odd 4 360.2.w.c.307.1 8
48.11 even 4 1440.2.bi.c.847.4 8
48.29 odd 4 1440.2.bi.c.847.1 8
48.35 even 4 360.2.w.c.307.3 8
80.3 even 4 40.2.k.a.3.4 yes 8
80.13 odd 4 160.2.o.a.143.3 8
80.19 odd 4 200.2.k.h.107.3 8
80.27 even 4 800.2.o.g.143.2 8
80.29 even 4 800.2.o.g.207.2 8
80.37 odd 4 200.2.k.h.43.3 8
80.43 even 4 160.2.o.a.143.4 8
80.53 odd 4 40.2.k.a.3.2 8
80.59 odd 4 800.2.o.g.207.1 8
80.67 even 4 200.2.k.h.43.1 8
80.69 even 4 200.2.k.h.107.1 8
80.77 odd 4 800.2.o.g.143.1 8
240.53 even 4 360.2.w.c.163.3 8
240.83 odd 4 360.2.w.c.163.1 8
240.173 even 4 1440.2.bi.c.1423.4 8
240.203 odd 4 1440.2.bi.c.1423.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.k.a.3.2 8 80.53 odd 4
40.2.k.a.3.4 yes 8 80.3 even 4
40.2.k.a.27.2 yes 8 16.3 odd 4
40.2.k.a.27.4 yes 8 16.5 even 4
160.2.o.a.47.3 8 16.11 odd 4
160.2.o.a.47.4 8 16.13 even 4
160.2.o.a.143.3 8 80.13 odd 4
160.2.o.a.143.4 8 80.43 even 4
200.2.k.h.43.1 8 80.67 even 4
200.2.k.h.43.3 8 80.37 odd 4
200.2.k.h.107.1 8 80.69 even 4
200.2.k.h.107.3 8 80.19 odd 4
360.2.w.c.163.1 8 240.83 odd 4
360.2.w.c.163.3 8 240.53 even 4
360.2.w.c.307.1 8 48.5 odd 4
360.2.w.c.307.3 8 48.35 even 4
800.2.o.g.143.1 8 80.77 odd 4
800.2.o.g.143.2 8 80.27 even 4
800.2.o.g.207.1 8 80.59 odd 4
800.2.o.g.207.2 8 80.29 even 4
1280.2.n.m.767.1 8 1.1 even 1 trivial
1280.2.n.m.767.2 8 8.3 odd 2 inner
1280.2.n.m.1023.1 8 20.3 even 4 inner
1280.2.n.m.1023.2 8 40.13 odd 4 inner
1280.2.n.q.767.3 8 4.3 odd 2
1280.2.n.q.767.4 8 8.5 even 2
1280.2.n.q.1023.3 8 5.3 odd 4
1280.2.n.q.1023.4 8 40.3 even 4
1440.2.bi.c.847.1 8 48.29 odd 4
1440.2.bi.c.847.4 8 48.11 even 4
1440.2.bi.c.1423.1 8 240.203 odd 4
1440.2.bi.c.1423.4 8 240.173 even 4