Properties

Label 160.2.o.a.143.3
Level $160$
Weight $2$
Character 160.143
Analytic conductor $1.278$
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] = [160,2,Mod(47,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.o (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: \(1.27760643234\)
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 143.3
Root \(-0.951057 - 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 160.143
Dual form 160.2.o.a.47.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.61803 + 1.61803i) q^{3} +(-1.90211 + 1.17557i) q^{5} +(1.17557 + 1.17557i) q^{7} +2.23607i q^{9} +O(q^{10})\) \(q+(1.61803 + 1.61803i) q^{3} +(-1.90211 + 1.17557i) q^{5} +(1.17557 + 1.17557i) q^{7} +2.23607i q^{9} -1.23607 q^{11} +(3.07768 - 3.07768i) q^{13} +(-4.97980 - 1.17557i) q^{15} +(-1.00000 + 1.00000i) q^{17} -2.00000i q^{19} +3.80423i q^{21} +(2.62866 - 2.62866i) q^{23} +(2.23607 - 4.47214i) q^{25} +(1.23607 - 1.23607i) q^{27} +1.45309 q^{29} +5.25731i q^{31} +(-2.00000 - 2.00000i) q^{33} +(-3.61803 - 0.854102i) q^{35} +(-3.07768 - 3.07768i) q^{37} +9.95959 q^{39} -7.70820 q^{41} +(-2.38197 - 2.38197i) q^{43} +(-2.62866 - 4.25325i) q^{45} +(-7.33094 - 7.33094i) q^{47} -4.23607i q^{49} -3.23607 q^{51} +(0.726543 - 0.726543i) q^{53} +(2.35114 - 1.45309i) q^{55} +(3.23607 - 3.23607i) q^{57} +8.47214i q^{59} +9.95959i q^{61} +(-2.62866 + 2.62866i) q^{63} +(-2.23607 + 9.47214i) q^{65} +(2.38197 - 2.38197i) q^{67} +8.50651 q^{69} +7.05342i q^{71} +(8.70820 + 8.70820i) q^{73} +(10.8541 - 3.61803i) q^{75} +(-1.45309 - 1.45309i) q^{77} -12.3107 q^{79} +10.7082 q^{81} +(4.38197 + 4.38197i) q^{83} +(0.726543 - 3.07768i) q^{85} +(2.35114 + 2.35114i) q^{87} +6.47214i q^{89} +7.23607 q^{91} +(-8.50651 + 8.50651i) q^{93} +(2.35114 + 3.80423i) q^{95} +(-0.236068 + 0.236068i) q^{97} -2.76393i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 8 q^{11} - 8 q^{17} - 8 q^{27} - 16 q^{33} - 20 q^{35} - 8 q^{41} - 28 q^{43} - 8 q^{51} + 8 q^{57} + 28 q^{67} + 16 q^{73} + 60 q^{75} + 32 q^{81} + 44 q^{83} + 40 q^{91} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) 1.61803 + 1.61803i 0.934172 + 0.934172i 0.997963 0.0637909i \(-0.0203191\pi\)
−0.0637909 + 0.997963i \(0.520319\pi\)
\(4\) 0 0
\(5\) −1.90211 + 1.17557i −0.850651 + 0.525731i
\(6\) 0 0
\(7\) 1.17557 + 1.17557i 0.444324 + 0.444324i 0.893462 0.449138i \(-0.148269\pi\)
−0.449138 + 0.893462i \(0.648269\pi\)
\(8\) 0 0
\(9\) 2.23607i 0.745356i
\(10\) 0 0
\(11\) −1.23607 −0.372689 −0.186344 0.982485i \(-0.559664\pi\)
−0.186344 + 0.982485i \(0.559664\pi\)
\(12\) 0 0
\(13\) 3.07768 3.07768i 0.853596 0.853596i −0.136978 0.990574i \(-0.543739\pi\)
0.990574 + 0.136978i \(0.0437390\pi\)
\(14\) 0 0
\(15\) −4.97980 1.17557i −1.28578 0.303531i
\(16\) 0 0
\(17\) −1.00000 + 1.00000i −0.242536 + 0.242536i −0.817898 0.575363i \(-0.804861\pi\)
0.575363 + 0.817898i \(0.304861\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 3.80423i 0.830150i
\(22\) 0 0
\(23\) 2.62866 2.62866i 0.548113 0.548113i −0.377782 0.925895i \(-0.623313\pi\)
0.925895 + 0.377782i \(0.123313\pi\)
\(24\) 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
\(29\) 1.45309 0.269831 0.134916 0.990857i \(-0.456924\pi\)
0.134916 + 0.990857i \(0.456924\pi\)
\(30\) 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
\(35\) −3.61803 0.854102i −0.611559 0.144370i
\(36\) 0 0
\(37\) −3.07768 3.07768i −0.505968 0.505968i 0.407318 0.913286i \(-0.366464\pi\)
−0.913286 + 0.407318i \(0.866464\pi\)
\(38\) 0 0
\(39\) 9.95959 1.59481
\(40\) 0 0
\(41\) −7.70820 −1.20382 −0.601910 0.798564i \(-0.705593\pi\)
−0.601910 + 0.798564i \(0.705593\pi\)
\(42\) 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
\(45\) −2.62866 4.25325i −0.391857 0.634038i
\(46\) 0 0
\(47\) −7.33094 7.33094i −1.06933 1.06933i −0.997411 0.0719165i \(-0.977088\pi\)
−0.0719165 0.997411i \(-0.522912\pi\)
\(48\) 0 0
\(49\) 4.23607i 0.605153i
\(50\) 0 0
\(51\) −3.23607 −0.453140
\(52\) 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
\(59\) 8.47214i 1.10298i 0.834182 + 0.551489i \(0.185940\pi\)
−0.834182 + 0.551489i \(0.814060\pi\)
\(60\) 0 0
\(61\) 9.95959i 1.27520i 0.770370 + 0.637598i \(0.220072\pi\)
−0.770370 + 0.637598i \(0.779928\pi\)
\(62\) 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
\(69\) 8.50651 1.02406
\(70\) 0 0
\(71\) 7.05342i 0.837087i 0.908197 + 0.418544i \(0.137459\pi\)
−0.908197 + 0.418544i \(0.862541\pi\)
\(72\) 0 0
\(73\) 8.70820 + 8.70820i 1.01922 + 1.01922i 0.999812 + 0.0194065i \(0.00617767\pi\)
0.0194065 + 0.999812i \(0.493822\pi\)
\(74\) 0 0
\(75\) 10.8541 3.61803i 1.25332 0.417775i
\(76\) 0 0
\(77\) −1.45309 1.45309i −0.165594 0.165594i
\(78\) 0 0
\(79\) −12.3107 −1.38507 −0.692533 0.721386i \(-0.743505\pi\)
−0.692533 + 0.721386i \(0.743505\pi\)
\(80\) 0 0
\(81\) 10.7082 1.18980
\(82\) 0 0
\(83\) 4.38197 + 4.38197i 0.480983 + 0.480983i 0.905446 0.424462i \(-0.139537\pi\)
−0.424462 + 0.905446i \(0.639537\pi\)
\(84\) 0 0
\(85\) 0.726543 3.07768i 0.0788046 0.333822i
\(86\) 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
\(91\) 7.23607 0.758546
\(92\) 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 −0.718990 0.695021i \(-0.755395\pi\)
0.695021 + 0.718990i \(0.255395\pi\)
\(98\) 0 0
\(99\) 2.76393i 0.277786i
\(100\) 0 0
\(101\) 12.3107i 1.22496i −0.790485 0.612482i \(-0.790171\pi\)
0.790485 0.612482i \(-0.209829\pi\)
\(102\) 0 0
\(103\) 7.33094 7.33094i 0.722339 0.722339i −0.246742 0.969081i \(-0.579360\pi\)
0.969081 + 0.246742i \(0.0793601\pi\)
\(104\) 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 \(0.559653\pi\)
−0.982491 + 0.186310i \(0.940347\pi\)
\(108\) 0 0
\(109\) −6.71040 −0.642739 −0.321370 0.946954i \(-0.604143\pi\)
−0.321370 + 0.946954i \(0.604143\pi\)
\(110\) 0 0
\(111\) 9.95959i 0.945323i
\(112\) 0 0
\(113\) −4.70820 4.70820i −0.442911 0.442911i 0.450078 0.892989i \(-0.351396\pi\)
−0.892989 + 0.450078i \(0.851396\pi\)
\(114\) 0 0
\(115\) −1.90983 + 8.09017i −0.178093 + 0.754412i
\(116\) 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
\(125\) 1.00406 + 11.1352i 0.0898056 + 0.995959i
\(126\) 0 0
\(127\) 8.78402 + 8.78402i 0.779456 + 0.779456i 0.979738 0.200282i \(-0.0641859\pi\)
−0.200282 + 0.979738i \(0.564186\pi\)
\(128\) 0 0
\(129\) 7.70820i 0.678670i
\(130\) 0 0
\(131\) −0.291796 −0.0254943 −0.0127472 0.999919i \(-0.504058\pi\)
−0.0127472 + 0.999919i \(0.504058\pi\)
\(132\) 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 \(-0.682732\pi\)
0.839698 + 0.543054i \(0.182732\pi\)
\(138\) 0 0
\(139\) 5.41641i 0.459414i −0.973260 0.229707i \(-0.926223\pi\)
0.973260 0.229707i \(-0.0737767\pi\)
\(140\) 0 0
\(141\) 23.7234i 1.99787i
\(142\) 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
\(149\) −13.2088 −1.08211 −0.541053 0.840988i \(-0.681974\pi\)
−0.541053 + 0.840988i \(0.681974\pi\)
\(150\) 0 0
\(151\) 14.6619i 1.19317i −0.802551 0.596583i \(-0.796525\pi\)
0.802551 0.596583i \(-0.203475\pi\)
\(152\) 0 0
\(153\) −2.23607 2.23607i −0.180775 0.180775i
\(154\) 0 0
\(155\) −6.18034 10.0000i −0.496417 0.803219i
\(156\) 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 0.929225 0.369515i \(-0.120476\pi\)
−0.369515 + 0.929225i \(0.620476\pi\)
\(164\) 0 0
\(165\) 6.15537 + 1.45309i 0.479195 + 0.113123i
\(166\) 0 0
\(167\) 0.277515 + 0.277515i 0.0214747 + 0.0214747i 0.717763 0.696288i \(-0.245166\pi\)
−0.696288 + 0.717763i \(0.745166\pi\)
\(168\) 0 0
\(169\) 5.94427i 0.457252i
\(170\) 0 0
\(171\) 4.47214 0.341993
\(172\) 0 0
\(173\) 6.32688 6.32688i 0.481024 0.481024i −0.424435 0.905459i \(-0.639527\pi\)
0.905459 + 0.424435i \(0.139527\pi\)
\(174\) 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
\(179\) 16.4721i 1.23119i −0.788065 0.615593i \(-0.788917\pi\)
0.788065 0.615593i \(-0.211083\pi\)
\(180\) 0 0
\(181\) 9.40456i 0.699036i 0.936930 + 0.349518i \(0.113655\pi\)
−0.936930 + 0.349518i \(0.886345\pi\)
\(182\) 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
\(189\) 2.90617 0.211393
\(190\) 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 0.385043 0.922899i \(-0.374187\pi\)
−0.922899 + 0.385043i \(0.874187\pi\)
\(194\) 0 0
\(195\) −18.9443 + 11.7082i −1.35663 + 0.838442i
\(196\) 0 0
\(197\) −2.17963 2.17963i −0.155292 0.155292i 0.625185 0.780477i \(-0.285024\pi\)
−0.780477 + 0.625185i \(0.785024\pi\)
\(198\) 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
\(205\) 14.6619 9.06154i 1.02403 0.632885i
\(206\) 0 0
\(207\) 5.87785 + 5.87785i 0.408539 + 0.408539i
\(208\) 0 0
\(209\) 2.47214i 0.171001i
\(210\) 0 0
\(211\) −15.7082 −1.08140 −0.540699 0.841216i \(-0.681840\pi\)
−0.540699 + 0.841216i \(0.681840\pi\)
\(212\) 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
\(219\) 28.1803i 1.90425i
\(220\) 0 0
\(221\) 6.15537i 0.414055i
\(222\) 0 0
\(223\) 1.73060 1.73060i 0.115890 0.115890i −0.646784 0.762673i \(-0.723886\pi\)
0.762673 + 0.646784i \(0.223886\pi\)
\(224\) 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
\(229\) −21.3723 −1.41232 −0.706160 0.708053i \(-0.749574\pi\)
−0.706160 + 0.708053i \(0.749574\pi\)
\(230\) 0 0
\(231\) 4.70228i 0.309387i
\(232\) 0 0
\(233\) −3.47214 3.47214i −0.227467 0.227467i 0.584167 0.811634i \(-0.301421\pi\)
−0.811634 + 0.584167i \(0.801421\pi\)
\(234\) 0 0
\(235\) 22.5623 + 5.32624i 1.47180 + 0.347445i
\(236\) 0 0
\(237\) −19.9192 19.9192i −1.29389 1.29389i
\(238\) 0 0
\(239\) 29.3238 1.89680 0.948398 0.317083i \(-0.102703\pi\)
0.948398 + 0.317083i \(0.102703\pi\)
\(240\) 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
\(245\) 4.97980 + 8.05748i 0.318148 + 0.514774i
\(246\) 0 0
\(247\) −6.15537 6.15537i −0.391657 0.391657i
\(248\) 0 0
\(249\) 14.1803i 0.898643i
\(250\) 0 0
\(251\) 22.1803 1.40001 0.700005 0.714138i \(-0.253181\pi\)
0.700005 + 0.714138i \(0.253181\pi\)
\(252\) 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 0.184073 0.982913i \(-0.441072\pi\)
0.982913 0.184073i \(-0.0589283\pi\)
\(258\) 0 0
\(259\) 7.23607i 0.449627i
\(260\) 0 0
\(261\) 3.24920i 0.201120i
\(262\) 0 0
\(263\) −16.3925 + 16.3925i −1.01080 + 1.01080i −0.0108623 + 0.999941i \(0.503458\pi\)
−0.999941 + 0.0108623i \(0.996542\pi\)
\(264\) 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
\(269\) 17.9111 1.09206 0.546029 0.837766i \(-0.316139\pi\)
0.546029 + 0.837766i \(0.316139\pi\)
\(270\) 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
\(275\) −2.76393 + 5.52786i −0.166671 + 0.333343i
\(276\) 0 0
\(277\) 2.17963 + 2.17963i 0.130961 + 0.130961i 0.769549 0.638588i \(-0.220481\pi\)
−0.638588 + 0.769549i \(0.720481\pi\)
\(278\) 0 0
\(279\) −11.7557 −0.703796
\(280\) 0 0
\(281\) 3.70820 0.221213 0.110606 0.993864i \(-0.464721\pi\)
0.110606 + 0.993864i \(0.464721\pi\)
\(282\) 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
\(285\) −2.35114 + 9.95959i −0.139270 + 0.589955i
\(286\) 0 0
\(287\) −9.06154 9.06154i −0.534886 0.534886i
\(288\) 0 0
\(289\) 15.0000i 0.882353i
\(290\) 0 0
\(291\) −0.763932 −0.0447825
\(292\) 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
\(299\) 16.1803i 0.935733i
\(300\) 0 0
\(301\) 5.60034i 0.322798i
\(302\) 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
\(309\) 23.7234 1.34958
\(310\) 0 0
\(311\) 8.16348i 0.462909i 0.972846 + 0.231454i \(0.0743484\pi\)
−0.972846 + 0.231454i \(0.925652\pi\)
\(312\) 0 0
\(313\) −6.23607 6.23607i −0.352483 0.352483i 0.508549 0.861033i \(-0.330182\pi\)
−0.861033 + 0.508549i \(0.830182\pi\)
\(314\) 0 0
\(315\) 1.90983 8.09017i 0.107607 0.455829i
\(316\) 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
\(325\) −6.88191 20.6457i −0.381740 1.14522i
\(326\) 0 0
\(327\) −10.8576 10.8576i −0.600429 0.600429i
\(328\) 0 0
\(329\) 17.2361i 0.950255i
\(330\) 0 0
\(331\) −28.0689 −1.54281 −0.771403 0.636347i \(-0.780445\pi\)
−0.771403 + 0.636347i \(0.780445\pi\)
\(332\) 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 −0.760878 0.648895i \(-0.775231\pi\)
0.648895 + 0.760878i \(0.275231\pi\)
\(338\) 0 0
\(339\) 15.2361i 0.827510i
\(340\) 0 0
\(341\) 6.49839i 0.351908i
\(342\) 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 −0.620952 0.783849i \(-0.713254\pi\)
0.783849 + 0.620952i \(0.213254\pi\)
\(348\) 0 0
\(349\) −15.5599 −0.832904 −0.416452 0.909158i \(-0.636727\pi\)
−0.416452 + 0.909158i \(0.636727\pi\)
\(350\) 0 0
\(351\) 7.60845i 0.406109i
\(352\) 0 0
\(353\) 4.41641 + 4.41641i 0.235062 + 0.235062i 0.814802 0.579740i \(-0.196846\pi\)
−0.579740 + 0.814802i \(0.696846\pi\)
\(354\) 0 0
\(355\) −8.29180 13.4164i −0.440083 0.712069i
\(356\) 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
\(365\) −26.8011 6.32688i −1.40283 0.331164i
\(366\) 0 0
\(367\) 24.0009 + 24.0009i 1.25284 + 1.25284i 0.954442 + 0.298396i \(0.0964516\pi\)
0.298396 + 0.954442i \(0.403548\pi\)
\(368\) 0 0
\(369\) 17.2361i 0.897274i
\(370\) 0 0
\(371\) 1.70820 0.0886855
\(372\) 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
\(379\) 35.8885i 1.84347i 0.387820 + 0.921735i \(0.373228\pi\)
−0.387820 + 0.921735i \(0.626772\pi\)
\(380\) 0 0
\(381\) 28.4257i 1.45629i
\(382\) 0 0
\(383\) 11.1352 11.1352i 0.568980 0.568980i −0.362862 0.931843i \(-0.618201\pi\)
0.931843 + 0.362862i \(0.118201\pi\)
\(384\) 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
\(389\) 23.7234 1.20282 0.601412 0.798939i \(-0.294605\pi\)
0.601412 + 0.798939i \(0.294605\pi\)
\(390\) 0 0
\(391\) 5.25731i 0.265874i
\(392\) 0 0
\(393\) −0.472136 0.472136i −0.0238161 0.0238161i
\(394\) 0 0
\(395\) 23.4164 14.4721i 1.17821 0.728172i
\(396\) 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
\(405\) −20.3682 + 12.5882i −1.01210 + 0.625515i
\(406\) 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
\(411\) 11.2361 0.554234
\(412\) 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
\(419\) 24.8328i 1.21316i 0.795022 + 0.606581i \(0.207460\pi\)
−0.795022 + 0.606581i \(0.792540\pi\)
\(420\) 0 0
\(421\) 3.46120i 0.168689i 0.996437 + 0.0843443i \(0.0268795\pi\)
−0.996437 + 0.0843443i \(0.973120\pi\)
\(422\) 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
\(429\) −12.3107 −0.594368
\(430\) 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 0.992608 + 0.121369i \(0.0387283\pi\)
0.121369 + 0.992608i \(0.461272\pi\)
\(434\) 0 0
\(435\) −7.23607 1.70820i −0.346943 0.0819021i
\(436\) 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
\(445\) −7.60845 12.3107i −0.360675 0.583585i
\(446\) 0 0
\(447\) −21.3723 21.3723i −1.01087 1.01087i
\(448\) 0 0
\(449\) 31.5967i 1.49114i 0.666426 + 0.745571i \(0.267823\pi\)
−0.666426 + 0.745571i \(0.732177\pi\)
\(450\) 0 0
\(451\) 9.52786 0.448650
\(452\) 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 −0.0129439 + 0.999916i \(0.504120\pi\)
−0.999916 + 0.0129439i \(0.995880\pi\)
\(458\) 0 0
\(459\) 2.47214i 0.115389i
\(460\) 0 0
\(461\) 6.49839i 0.302660i 0.988483 + 0.151330i \(0.0483557\pi\)
−0.988483 + 0.151330i \(0.951644\pi\)
\(462\) 0 0
\(463\) −17.2905 + 17.2905i −0.803559 + 0.803559i −0.983650 0.180091i \(-0.942361\pi\)
0.180091 + 0.983650i \(0.442361\pi\)
\(464\) 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
\(469\) 5.60034 0.258600
\(470\) 0 0
\(471\) 31.6749i 1.45950i
\(472\) 0 0
\(473\) 2.94427 + 2.94427i 0.135378 + 0.135378i
\(474\) 0 0
\(475\) −8.94427 4.47214i −0.410391 0.205196i
\(476\) 0 0
\(477\) 1.62460 + 1.62460i 0.0743853 + 0.0743853i
\(478\) 0 0
\(479\) −7.60845 −0.347639 −0.173820 0.984778i \(-0.555611\pi\)
−0.173820 + 0.984778i \(0.555611\pi\)
\(480\) 0 0
\(481\) −18.9443 −0.863784
\(482\) 0 0
\(483\) 10.0000 + 10.0000i 0.455016 + 0.455016i
\(484\) 0 0
\(485\) 0.171513 0.726543i 0.00778802 0.0329906i
\(486\) 0 0
\(487\) −9.33905 9.33905i −0.423193 0.423193i 0.463109 0.886302i \(-0.346734\pi\)
−0.886302 + 0.463109i \(0.846734\pi\)
\(488\) 0 0
\(489\) 23.1246i 1.04573i
\(490\) 0 0
\(491\) 10.7639 0.485769 0.242885 0.970055i \(-0.421906\pi\)
0.242885 + 0.970055i \(0.421906\pi\)
\(492\) 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
\(499\) 23.8885i 1.06940i −0.845043 0.534699i \(-0.820425\pi\)
0.845043 0.534699i \(-0.179575\pi\)
\(500\) 0 0
\(501\) 0.898056i 0.0401222i
\(502\) 0 0
\(503\) 5.53483 5.53483i 0.246786 0.246786i −0.572864 0.819650i \(-0.694168\pi\)
0.819650 + 0.572864i \(0.194168\pi\)
\(504\) 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
\(509\) 9.06154 0.401646 0.200823 0.979628i \(-0.435638\pi\)
0.200823 + 0.979628i \(0.435638\pi\)
\(510\) 0 0
\(511\) 20.4742i 0.905726i
\(512\) 0 0
\(513\) −2.47214 2.47214i −0.109147 0.109147i
\(514\) 0 0
\(515\) −5.32624 + 22.5623i −0.234702 + 0.994214i
\(516\) 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 \(-0.440582\pi\)
0.185586 + 0.982628i \(0.440582\pi\)
\(522\) 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
\(525\) 17.0130 + 8.50651i 0.742509 + 0.371254i
\(526\) 0 0
\(527\) −5.25731 5.25731i −0.229012 0.229012i
\(528\) 0 0
\(529\) 9.18034i 0.399145i
\(530\) 0 0
\(531\) −18.9443 −0.822111
\(532\) 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
\(539\) 5.23607i 0.225533i
\(540\) 0 0
\(541\) 2.90617i 0.124946i 0.998047 + 0.0624730i \(0.0198987\pi\)
−0.998047 + 0.0624730i \(0.980101\pi\)
\(542\) 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
\(549\) −22.2703 −0.950474
\(550\) 0 0
\(551\) 2.90617i 0.123807i
\(552\) 0 0
\(553\) −14.4721 14.4721i −0.615418 0.615418i
\(554\) 0 0
\(555\) 11.7082 + 18.9443i 0.496986 + 0.804140i
\(556\) 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 0.426937 0.904281i \(-0.359593\pi\)
−0.904281 + 0.426937i \(0.859593\pi\)
\(564\) 0 0
\(565\) 14.4904 + 3.42071i 0.609614 + 0.143910i
\(566\) 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
\(571\) 8.65248 0.362095 0.181047 0.983474i \(-0.442051\pi\)
0.181047 + 0.983474i \(0.442051\pi\)
\(572\) 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 −0.0899059 0.995950i \(-0.528657\pi\)
0.995950 + 0.0899059i \(0.0286566\pi\)
\(578\) 0 0
\(579\) 24.1803i 1.00490i
\(580\) 0 0
\(581\) 10.3026i 0.427425i
\(582\) 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 −0.818472 0.574547i \(-0.805178\pi\)
0.574547 + 0.818472i \(0.305178\pi\)
\(588\) 0 0
\(589\) 10.5146 0.433247
\(590\) 0 0
\(591\) 7.05342i 0.290139i
\(592\) 0 0
\(593\) −6.41641 6.41641i −0.263490 0.263490i 0.562980 0.826470i \(-0.309655\pi\)
−0.826470 + 0.562980i \(0.809655\pi\)
\(594\) 0 0
\(595\) 4.47214 2.76393i 0.183340 0.113310i
\(596\) 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 \(-0.527898\pi\)
−0.0875330 + 0.996162i \(0.527898\pi\)
\(602\) 0 0
\(603\) 5.32624 + 5.32624i 0.216901 + 0.216901i
\(604\) 0 0
\(605\) 18.0171 11.1352i 0.732498 0.452709i
\(606\) 0 0
\(607\) 6.08985 + 6.08985i 0.247180 + 0.247180i 0.819812 0.572633i \(-0.194078\pi\)
−0.572633 + 0.819812i \(0.694078\pi\)
\(608\) 0 0
\(609\) 5.52786i 0.224000i
\(610\) 0 0
\(611\) −45.1246 −1.82555
\(612\) 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 −0.284998 + 0.958528i \(0.591993\pi\)
−0.958528 + 0.284998i \(0.908007\pi\)
\(618\) 0 0
\(619\) 27.3050i 1.09748i −0.835994 0.548739i \(-0.815108\pi\)
0.835994 0.548739i \(-0.184892\pi\)
\(620\) 0 0
\(621\) 6.49839i 0.260772i
\(622\) 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
\(629\) 6.15537 0.245431
\(630\) 0 0
\(631\) 28.0827i 1.11795i −0.829183 0.558977i \(-0.811194\pi\)
0.829183 0.558977i \(-0.188806\pi\)
\(632\) 0 0
\(633\) −25.4164 25.4164i −1.01021 1.01021i
\(634\) 0 0
\(635\) −27.0344 6.38197i −1.07283 0.253261i
\(636\) 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 0.517823 0.855488i \(-0.326742\pi\)
−0.855488 + 0.517823i \(0.826742\pi\)
\(644\) 0 0
\(645\) 9.06154 + 14.6619i 0.356798 + 0.577311i
\(646\) 0 0
\(647\) 22.8909 + 22.8909i 0.899933 + 0.899933i 0.995430 0.0954968i \(-0.0304440\pi\)
−0.0954968 + 0.995430i \(0.530444\pi\)
\(648\) 0 0
\(649\) 10.4721i 0.411067i
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 11.2412 11.2412i 0.439901 0.439901i −0.452078 0.891979i \(-0.649317\pi\)
0.891979 + 0.452078i \(0.149317\pi\)
\(654\) 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
\(659\) 18.0000i 0.701180i 0.936529 + 0.350590i \(0.114019\pi\)
−0.936529 + 0.350590i \(0.885981\pi\)
\(660\) 0 0
\(661\) 2.35114i 0.0914488i 0.998954 + 0.0457244i \(0.0145596\pi\)
−0.998954 + 0.0457244i \(0.985440\pi\)
\(662\) 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
\(669\) 5.60034 0.216522
\(670\) 0 0
\(671\) 12.3107i 0.475251i
\(672\) 0 0
\(673\) −30.7082 30.7082i −1.18371 1.18371i −0.978775 0.204940i \(-0.934300\pi\)
−0.204940 0.978775i \(-0.565700\pi\)
\(674\) 0 0
\(675\) −2.76393 8.29180i −0.106384 0.319151i
\(676\) 0 0
\(677\) 8.33499 + 8.33499i 0.320340 + 0.320340i 0.848897 0.528558i \(-0.177267\pi\)
−0.528558 + 0.848897i \(0.677267\pi\)
\(678\) 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
\(685\) −2.52265 + 10.6861i −0.0963857 + 0.408296i
\(686\) 0 0
\(687\) −34.5811 34.5811i −1.31935 1.31935i
\(688\) 0 0
\(689\) 4.47214i 0.170375i
\(690\) 0 0
\(691\) 31.1246 1.18404 0.592018 0.805925i \(-0.298331\pi\)
0.592018 + 0.805925i \(0.298331\pi\)
\(692\) 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
\(699\) 11.2361i 0.424987i
\(700\) 0 0
\(701\) 40.3934i 1.52564i −0.646612 0.762819i \(-0.723815\pi\)
0.646612 0.762819i \(-0.276185\pi\)
\(702\) 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
\(709\) −13.7638 −0.516911 −0.258456 0.966023i \(-0.583214\pi\)
−0.258456 + 0.966023i \(0.583214\pi\)
\(710\) 0 0
\(711\) 27.5276i 1.03237i
\(712\) 0 0
\(713\) 13.8197 + 13.8197i 0.517550 + 0.517550i
\(714\) 0 0
\(715\) 2.76393 11.7082i 0.103365 0.437862i
\(716\) 0 0
\(717\) 47.4468 + 47.4468i 1.77193 + 1.77193i
\(718\) 0 0
\(719\) −44.5407 −1.66109 −0.830543 0.556954i \(-0.811970\pi\)
−0.830543 + 0.556954i \(0.811970\pi\)
\(720\) 0 0
\(721\) 17.2361 0.641905
\(722\) 0 0
\(723\) 10.9443 + 10.9443i 0.407022 + 0.407022i
\(724\) 0 0
\(725\) 3.24920 6.49839i 0.120672 0.241344i
\(726\) 0 0
\(727\) 5.87785 + 5.87785i 0.217997 + 0.217997i 0.807654 0.589657i \(-0.200737\pi\)
−0.589657 + 0.807654i \(0.700737\pi\)
\(728\) 0 0
\(729\) 11.9443i 0.442380i
\(730\) 0 0
\(731\) 4.76393 0.176200
\(732\) 0 0
\(733\) −1.28157 + 1.28157i −0.0473359 + 0.0473359i −0.730379 0.683043i \(-0.760656\pi\)
0.683043 + 0.730379i \(0.260656\pi\)
\(734\) 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
\(739\) 17.4164i 0.640673i 0.947304 + 0.320336i \(0.103796\pi\)
−0.947304 + 0.320336i \(0.896204\pi\)
\(740\) 0 0
\(741\) 19.9192i 0.731750i
\(742\) 0 0
\(743\) −13.4863 + 13.4863i −0.494765 + 0.494765i −0.909804 0.415039i \(-0.863768\pi\)
0.415039 + 0.909804i \(0.363768\pi\)
\(744\) 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
\(749\) −28.4257 −1.03865
\(750\) 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
\(755\) 17.2361 + 27.8885i 0.627285 + 1.01497i
\(756\) 0 0
\(757\) 38.0018 + 38.0018i 1.38120 + 1.38120i 0.842497 + 0.538701i \(0.181085\pi\)
0.538701 + 0.842497i \(0.318915\pi\)
\(758\) 0 0
\(759\) −10.5146 −0.381657
\(760\) 0 0
\(761\) −14.9443 −0.541729 −0.270865 0.962617i \(-0.587310\pi\)
−0.270865 + 0.962617i \(0.587310\pi\)
\(762\) 0 0
\(763\) −7.88854 7.88854i −0.285584 0.285584i
\(764\) 0 0
\(765\) 6.88191 + 1.62460i 0.248816 + 0.0587375i
\(766\) 0 0
\(767\) 26.0746 + 26.0746i 0.941498 + 0.941498i
\(768\) 0 0
\(769\) 2.47214i 0.0891475i −0.999006 0.0445738i \(-0.985807\pi\)
0.999006 0.0445738i \(-0.0141930\pi\)
\(770\) 0 0
\(771\) 60.5410 2.18033
\(772\) 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
\(779\) 15.4164i 0.552350i
\(780\) 0 0
\(781\) 8.71851i 0.311973i
\(782\) 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
\(789\) −53.0472 −1.88853
\(790\) 0 0
\(791\) 11.0697i 0.393591i
\(792\) 0 0
\(793\) 30.6525 + 30.6525i 1.08850 + 1.08850i
\(794\) 0 0
\(795\) −4.47214 + 2.76393i −0.158610 + 0.0980266i
\(796\) 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
\(805\) −11.7557 + 7.26543i −0.414334 + 0.256073i
\(806\) 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
\(811\) 32.0689 1.12609 0.563045 0.826426i \(-0.309630\pi\)
0.563045 + 0.826426i \(0.309630\pi\)
\(812\) 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
\(819\) 16.1803i 0.565387i
\(820\) 0 0
\(821\) 20.4742i 0.714555i −0.933998 0.357278i \(-0.883705\pi\)
0.933998 0.357278i \(-0.116295\pi\)
\(822\) 0 0
\(823\) 24.3440 24.3440i 0.848577 0.848577i −0.141379 0.989956i \(-0.545154\pi\)
0.989956 + 0.141379i \(0.0451535\pi\)
\(824\) 0 0
\(825\) −13.4164 + 4.47214i −0.467099 + 0.155700i
\(826\) 0 0
\(827\) −14.8541 + 14.8541i −0.516528 + 0.516528i −0.916519 0.399991i \(-0.869013\pi\)
0.399991 + 0.916519i \(0.369013\pi\)
\(828\) 0 0
\(829\) −3.11817 −0.108299 −0.0541493 0.998533i \(-0.517245\pi\)
−0.0541493 + 0.998533i \(0.517245\pi\)
\(830\) 0 0
\(831\) 7.05342i 0.244681i
\(832\) 0 0
\(833\) 4.23607 + 4.23607i 0.146771 + 0.146771i
\(834\) 0 0
\(835\) −0.854102 0.201626i −0.0295574 0.00697756i
\(836\) 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
\(845\) 6.98791 + 11.3067i 0.240391 + 0.388962i
\(846\) 0 0
\(847\) −11.1352 11.1352i −0.382609 0.382609i
\(848\) 0 0
\(849\) 50.5410i 1.73456i
\(850\) 0 0
\(851\) −16.1803 −0.554655
\(852\) 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 \(-0.891747\pi\)
0.333568 + 0.942726i \(0.391747\pi\)
\(858\) 0 0
\(859\) 7.52786i 0.256847i −0.991719 0.128424i \(-0.959008\pi\)
0.991719 0.128424i \(-0.0409917\pi\)
\(860\) 0 0
\(861\) 29.3238i 0.999351i
\(862\) 0 0
\(863\) 6.22088 6.22088i 0.211761 0.211761i −0.593254 0.805015i \(-0.702157\pi\)
0.805015 + 0.593254i \(0.202157\pi\)
\(864\) 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
\(869\) 15.2169 0.516198
\(870\) 0 0
\(871\) 14.6619i 0.496799i
\(872\) 0 0
\(873\) −0.527864 0.527864i −0.0178655 0.0178655i
\(874\) 0 0
\(875\) −11.9098 + 14.2705i −0.402626 + 0.482431i
\(876\) 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 0.237752 0.971326i \(-0.423589\pi\)
−0.971326 + 0.237752i \(0.923589\pi\)
\(884\) 0 0
\(885\) 9.95959 42.1895i 0.334788 1.41818i
\(886\) 0 0
\(887\) −28.1482 28.1482i −0.945123 0.945123i 0.0534473 0.998571i \(-0.482979\pi\)
−0.998571 + 0.0534473i \(0.982979\pi\)
\(888\) 0 0
\(889\) 20.6525i 0.692662i
\(890\) 0 0
\(891\) −13.2361 −0.443425
\(892\) 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
\(899\) 7.63932i 0.254786i
\(900\) 0 0
\(901\) 1.45309i 0.0484093i
\(902\) 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 −0.204361 0.978896i \(-0.565512\pi\)
0.978896 + 0.204361i \(0.0655115\pi\)
\(908\) 0 0
\(909\) 27.5276 0.913034
\(910\) 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
\(915\) 11.7082 49.5967i 0.387061 1.63962i
\(916\) 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
\(925\) −20.6457 + 6.88191i −0.678827 + 0.226276i
\(926\) 0 0
\(927\) 16.3925 + 16.3925i 0.538400 + 0.538400i
\(928\) 0 0
\(929\) 5.34752i 0.175447i 0.996145 + 0.0877233i \(0.0279591\pi\)
−0.996145 + 0.0877233i \(0.972041\pi\)
\(930\) 0 0
\(931\) −8.47214 −0.277663
\(932\) 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 \(-0.594597\pi\)
0.956164 + 0.292831i \(0.0945972\pi\)
\(938\) 0 0
\(939\) 20.1803i 0.658561i
\(940\) 0 0
\(941\) 60.8676i 1.98423i 0.125340 + 0.992114i \(0.459998\pi\)
−0.125340 + 0.992114i \(0.540002\pi\)
\(942\) 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
\(949\) 53.6022 1.74000
\(950\) 0 0
\(951\) 34.5811i 1.12137i
\(952\) 0 0
\(953\) −6.81966 6.81966i −0.220910 0.220910i 0.587971 0.808882i \(-0.299927\pi\)
−0.808882 + 0.587971i \(0.799927\pi\)
\(954\) 0 0
\(955\) 15.1246 + 24.4721i 0.489421 + 0.791900i
\(956\) 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
\(965\) 22.9969 + 5.42882i 0.740295 + 0.174760i
\(966\) 0 0
\(967\) −12.0332 12.0332i −0.386962 0.386962i 0.486640 0.873602i \(-0.338222\pi\)
−0.873602 + 0.486640i \(0.838222\pi\)
\(968\) 0 0
\(969\) 6.47214i 0.207915i
\(970\) 0 0
\(971\) −33.5967 −1.07817 −0.539085 0.842251i \(-0.681230\pi\)
−0.539085 + 0.842251i \(0.681230\pi\)
\(972\) 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 −0.995725 0.0923727i \(-0.970555\pi\)
0.0923727 + 0.995725i \(0.470555\pi\)
\(978\) 0 0
\(979\) 8.00000i 0.255681i
\(980\) 0 0
\(981\) 15.0049i 0.479070i
\(982\) 0 0
\(983\) 0.620541 0.620541i 0.0197922 0.0197922i −0.697141 0.716934i \(-0.745545\pi\)
0.716934 + 0.697141i \(0.245545\pi\)
\(984\) 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
\(989\) −12.5227 −0.398200
\(990\) 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
\(995\) 34.4721 21.3050i 1.09284 0.675412i
\(996\) 0 0
\(997\) −35.6506 35.6506i −1.12907 1.12907i −0.990329 0.138738i \(-0.955696\pi\)
−0.138738 0.990329i \(-0.544304\pi\)
\(998\) 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 160.2.o.a.143.3 8
3.2 odd 2 1440.2.bi.c.1423.4 8
4.3 odd 2 40.2.k.a.3.4 yes 8
5.2 odd 4 inner 160.2.o.a.47.4 8
5.3 odd 4 800.2.o.g.207.2 8
5.4 even 2 800.2.o.g.143.1 8
8.3 odd 2 inner 160.2.o.a.143.4 8
8.5 even 2 40.2.k.a.3.2 8
12.11 even 2 360.2.w.c.163.1 8
15.2 even 4 1440.2.bi.c.847.1 8
16.3 odd 4 1280.2.n.q.1023.4 8
16.5 even 4 1280.2.n.q.1023.3 8
16.11 odd 4 1280.2.n.m.1023.1 8
16.13 even 4 1280.2.n.m.1023.2 8
20.3 even 4 200.2.k.h.107.3 8
20.7 even 4 40.2.k.a.27.2 yes 8
20.19 odd 2 200.2.k.h.43.1 8
24.5 odd 2 360.2.w.c.163.3 8
24.11 even 2 1440.2.bi.c.1423.1 8
40.3 even 4 800.2.o.g.207.1 8
40.13 odd 4 200.2.k.h.107.1 8
40.19 odd 2 800.2.o.g.143.2 8
40.27 even 4 inner 160.2.o.a.47.3 8
40.29 even 2 200.2.k.h.43.3 8
40.37 odd 4 40.2.k.a.27.4 yes 8
60.47 odd 4 360.2.w.c.307.3 8
80.27 even 4 1280.2.n.q.767.3 8
80.37 odd 4 1280.2.n.m.767.1 8
80.67 even 4 1280.2.n.m.767.2 8
80.77 odd 4 1280.2.n.q.767.4 8
120.77 even 4 360.2.w.c.307.1 8
120.107 odd 4 1440.2.bi.c.847.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.k.a.3.2 8 8.5 even 2
40.2.k.a.3.4 yes 8 4.3 odd 2
40.2.k.a.27.2 yes 8 20.7 even 4
40.2.k.a.27.4 yes 8 40.37 odd 4
160.2.o.a.47.3 8 40.27 even 4 inner
160.2.o.a.47.4 8 5.2 odd 4 inner
160.2.o.a.143.3 8 1.1 even 1 trivial
160.2.o.a.143.4 8 8.3 odd 2 inner
200.2.k.h.43.1 8 20.19 odd 2
200.2.k.h.43.3 8 40.29 even 2
200.2.k.h.107.1 8 40.13 odd 4
200.2.k.h.107.3 8 20.3 even 4
360.2.w.c.163.1 8 12.11 even 2
360.2.w.c.163.3 8 24.5 odd 2
360.2.w.c.307.1 8 120.77 even 4
360.2.w.c.307.3 8 60.47 odd 4
800.2.o.g.143.1 8 5.4 even 2
800.2.o.g.143.2 8 40.19 odd 2
800.2.o.g.207.1 8 40.3 even 4
800.2.o.g.207.2 8 5.3 odd 4
1280.2.n.m.767.1 8 80.37 odd 4
1280.2.n.m.767.2 8 80.67 even 4
1280.2.n.m.1023.1 8 16.11 odd 4
1280.2.n.m.1023.2 8 16.13 even 4
1280.2.n.q.767.3 8 80.27 even 4
1280.2.n.q.767.4 8 80.77 odd 4
1280.2.n.q.1023.3 8 16.5 even 4
1280.2.n.q.1023.4 8 16.3 odd 4
1440.2.bi.c.847.1 8 15.2 even 4
1440.2.bi.c.847.4 8 120.107 odd 4
1440.2.bi.c.1423.1 8 24.11 even 2
1440.2.bi.c.1423.4 8 3.2 odd 2