Properties

Label 1600.2.l.h.401.4
Level $1600$
Weight $2$
Character 1600.401
Analytic conductor $12.776$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(401,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.534694406811304329216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 401.4
Root \(1.32661 - 0.490008i\) of defining polynomial
Character \(\chi\) \(=\) 1600.401
Dual form 1600.2.l.h.1201.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.183790 - 0.183790i) q^{3} -3.84853i q^{7} -2.93244i q^{9} +O(q^{10})\) \(q+(-0.183790 - 0.183790i) q^{3} -3.84853i q^{7} -2.93244i q^{9} +(-1.60020 + 1.60020i) q^{11} +(1.80775 + 1.80775i) q^{13} +4.93886 q^{17} +(-4.77162 - 4.77162i) q^{19} +(-0.707323 + 0.707323i) q^{21} +0.134544i q^{23} +(-1.09033 + 1.09033i) q^{27} +(-2.17142 - 2.17142i) q^{29} -2.26371 q^{31} +0.588201 q^{33} +(-4.35066 + 4.35066i) q^{37} -0.664493i q^{39} -3.34709i q^{41} +(2.70896 - 2.70896i) q^{43} -7.03343 q^{47} -7.81119 q^{49} +(-0.907714 - 0.907714i) q^{51} +(-3.40020 + 3.40020i) q^{53} +1.75396i q^{57} +(-0.107127 + 0.107127i) q^{59} +(-3.46410 - 3.46410i) q^{61} -11.2856 q^{63} +(1.91078 + 1.91078i) q^{67} +(0.0247279 - 0.0247279i) q^{69} -9.32899i q^{71} -9.82769i q^{73} +(6.15840 + 6.15840i) q^{77} -11.0073 q^{79} -8.39654 q^{81} +(8.80967 + 8.80967i) q^{83} +0.798174i q^{87} +1.12125i q^{89} +(6.95717 - 6.95717i) q^{91} +(0.416048 + 0.416048i) q^{93} -6.10461 q^{97} +(4.69248 + 4.69248i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{11} - 8 q^{19} - 16 q^{21} + 16 q^{29} - 16 q^{31} - 16 q^{49} + 16 q^{51} - 24 q^{59} - 32 q^{69} + 16 q^{79} - 16 q^{81} + 16 q^{91} + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −0.183790 0.183790i −0.106111 0.106111i 0.652058 0.758169i \(-0.273906\pi\)
−0.758169 + 0.652058i \(0.773906\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.84853i 1.45461i −0.686315 0.727304i \(-0.740773\pi\)
0.686315 0.727304i \(-0.259227\pi\)
\(8\) 0 0
\(9\) 2.93244i 0.977481i
\(10\) 0 0
\(11\) −1.60020 + 1.60020i −0.482477 + 0.482477i −0.905922 0.423445i \(-0.860821\pi\)
0.423445 + 0.905922i \(0.360821\pi\)
\(12\) 0 0
\(13\) 1.80775 + 1.80775i 0.501379 + 0.501379i 0.911866 0.410487i \(-0.134641\pi\)
−0.410487 + 0.911866i \(0.634641\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.93886 1.19785 0.598924 0.800806i \(-0.295595\pi\)
0.598924 + 0.800806i \(0.295595\pi\)
\(18\) 0 0
\(19\) −4.77162 4.77162i −1.09468 1.09468i −0.995021 0.0996636i \(-0.968223\pi\)
−0.0996636 0.995021i \(-0.531777\pi\)
\(20\) 0 0
\(21\) −0.707323 + 0.707323i −0.154351 + 0.154351i
\(22\) 0 0
\(23\) 0.134544i 0.0280543i 0.999902 + 0.0140272i \(0.00446513\pi\)
−0.999902 + 0.0140272i \(0.995535\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.09033 + 1.09033i −0.209833 + 0.209833i
\(28\) 0 0
\(29\) −2.17142 2.17142i −0.403223 0.403223i 0.476144 0.879367i \(-0.342034\pi\)
−0.879367 + 0.476144i \(0.842034\pi\)
\(30\) 0 0
\(31\) −2.26371 −0.406574 −0.203287 0.979119i \(-0.565163\pi\)
−0.203287 + 0.979119i \(0.565163\pi\)
\(32\) 0 0
\(33\) 0.588201 0.102393
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.35066 + 4.35066i −0.715243 + 0.715243i −0.967627 0.252384i \(-0.918785\pi\)
0.252384 + 0.967627i \(0.418785\pi\)
\(38\) 0 0
\(39\) 0.664493i 0.106404i
\(40\) 0 0
\(41\) 3.34709i 0.522727i −0.965240 0.261364i \(-0.915828\pi\)
0.965240 0.261364i \(-0.0841722\pi\)
\(42\) 0 0
\(43\) 2.70896 2.70896i 0.413112 0.413112i −0.469709 0.882821i \(-0.655641\pi\)
0.882821 + 0.469709i \(0.155641\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.03343 −1.02593 −0.512966 0.858409i \(-0.671453\pi\)
−0.512966 + 0.858409i \(0.671453\pi\)
\(48\) 0 0
\(49\) −7.81119 −1.11588
\(50\) 0 0
\(51\) −0.907714 0.907714i −0.127105 0.127105i
\(52\) 0 0
\(53\) −3.40020 + 3.40020i −0.467053 + 0.467053i −0.900958 0.433905i \(-0.857135\pi\)
0.433905 + 0.900958i \(0.357135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.75396i 0.232317i
\(58\) 0 0
\(59\) −0.107127 + 0.107127i −0.0139468 + 0.0139468i −0.714046 0.700099i \(-0.753139\pi\)
0.700099 + 0.714046i \(0.253139\pi\)
\(60\) 0 0
\(61\) −3.46410 3.46410i −0.443533 0.443533i 0.449665 0.893197i \(-0.351543\pi\)
−0.893197 + 0.449665i \(0.851543\pi\)
\(62\) 0 0
\(63\) −11.2856 −1.42185
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.91078 + 1.91078i 0.233440 + 0.233440i 0.814127 0.580687i \(-0.197216\pi\)
−0.580687 + 0.814127i \(0.697216\pi\)
\(68\) 0 0
\(69\) 0.0247279 0.0247279i 0.00297689 0.00297689i
\(70\) 0 0
\(71\) 9.32899i 1.10715i −0.832800 0.553573i \(-0.813264\pi\)
0.832800 0.553573i \(-0.186736\pi\)
\(72\) 0 0
\(73\) 9.82769i 1.15024i −0.818068 0.575122i \(-0.804955\pi\)
0.818068 0.575122i \(-0.195045\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.15840 + 6.15840i 0.701815 + 0.701815i
\(78\) 0 0
\(79\) −11.0073 −1.23842 −0.619211 0.785224i \(-0.712548\pi\)
−0.619211 + 0.785224i \(0.712548\pi\)
\(80\) 0 0
\(81\) −8.39654 −0.932949
\(82\) 0 0
\(83\) 8.80967 + 8.80967i 0.966987 + 0.966987i 0.999472 0.0324850i \(-0.0103421\pi\)
−0.0324850 + 0.999472i \(0.510342\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.798174i 0.0855732i
\(88\) 0 0
\(89\) 1.12125i 0.118853i 0.998233 + 0.0594263i \(0.0189271\pi\)
−0.998233 + 0.0594263i \(0.981073\pi\)
\(90\) 0 0
\(91\) 6.95717 6.95717i 0.729310 0.729310i
\(92\) 0 0
\(93\) 0.416048 + 0.416048i 0.0431422 + 0.0431422i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.10461 −0.619829 −0.309915 0.950764i \(-0.600300\pi\)
−0.309915 + 0.950764i \(0.600300\pi\)
\(98\) 0 0
\(99\) 4.69248 + 4.69248i 0.471612 + 0.471612i
\(100\) 0 0
\(101\) 2.17142 2.17142i 0.216065 0.216065i −0.590773 0.806838i \(-0.701177\pi\)
0.806838 + 0.590773i \(0.201177\pi\)
\(102\) 0 0
\(103\) 15.3778i 1.51522i −0.652707 0.757610i \(-0.726367\pi\)
0.652707 0.757610i \(-0.273633\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.98419 6.98419i 0.675187 0.675187i −0.283720 0.958907i \(-0.591569\pi\)
0.958907 + 0.283720i \(0.0915688\pi\)
\(108\) 0 0
\(109\) −8.59694 8.59694i −0.823437 0.823437i 0.163162 0.986599i \(-0.447831\pi\)
−0.986599 + 0.163162i \(0.947831\pi\)
\(110\) 0 0
\(111\) 1.59922 0.151791
\(112\) 0 0
\(113\) 14.5329 1.36714 0.683572 0.729883i \(-0.260426\pi\)
0.683572 + 0.729883i \(0.260426\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.30111 5.30111i 0.490088 0.490088i
\(118\) 0 0
\(119\) 19.0073i 1.74240i
\(120\) 0 0
\(121\) 5.87875i 0.534432i
\(122\) 0 0
\(123\) −0.615163 + 0.615163i −0.0554673 + 0.0554673i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.96617 0.529412 0.264706 0.964329i \(-0.414725\pi\)
0.264706 + 0.964329i \(0.414725\pi\)
\(128\) 0 0
\(129\) −0.995761 −0.0876719
\(130\) 0 0
\(131\) −2.37084 2.37084i −0.207141 0.207141i 0.595910 0.803051i \(-0.296791\pi\)
−0.803051 + 0.595910i \(0.796791\pi\)
\(132\) 0 0
\(133\) −18.3637 + 18.3637i −1.59234 + 1.59234i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.2745i 1.56129i −0.624973 0.780646i \(-0.714890\pi\)
0.624973 0.780646i \(-0.285110\pi\)
\(138\) 0 0
\(139\) −0.136094 + 0.136094i −0.0115433 + 0.0115433i −0.712855 0.701312i \(-0.752598\pi\)
0.701312 + 0.712855i \(0.252598\pi\)
\(140\) 0 0
\(141\) 1.29268 + 1.29268i 0.108863 + 0.108863i
\(142\) 0 0
\(143\) −5.78550 −0.483808
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.43562 + 1.43562i 0.118408 + 0.118408i
\(148\) 0 0
\(149\) 2.40078 2.40078i 0.196680 0.196680i −0.601895 0.798575i \(-0.705588\pi\)
0.798575 + 0.601895i \(0.205588\pi\)
\(150\) 0 0
\(151\) 17.9935i 1.46429i 0.681150 + 0.732144i \(0.261480\pi\)
−0.681150 + 0.732144i \(0.738520\pi\)
\(152\) 0 0
\(153\) 14.4829i 1.17087i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.6359 12.6359i −1.00846 1.00846i −0.999964 0.00849213i \(-0.997297\pi\)
−0.00849213 0.999964i \(-0.502703\pi\)
\(158\) 0 0
\(159\) 1.24985 0.0991193
\(160\) 0 0
\(161\) 0.517796 0.0408081
\(162\) 0 0
\(163\) 15.8470 + 15.8470i 1.24123 + 1.24123i 0.959490 + 0.281743i \(0.0909126\pi\)
0.281743 + 0.959490i \(0.409087\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.7559i 0.987083i 0.869722 + 0.493541i \(0.164298\pi\)
−0.869722 + 0.493541i \(0.835702\pi\)
\(168\) 0 0
\(169\) 6.46410i 0.497239i
\(170\) 0 0
\(171\) −13.9925 + 13.9925i −1.07003 + 1.07003i
\(172\) 0 0
\(173\) −2.64673 2.64673i −0.201227 0.201227i 0.599299 0.800525i \(-0.295446\pi\)
−0.800525 + 0.599299i \(0.795446\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.0393779 0.00295983
\(178\) 0 0
\(179\) −11.6497 11.6497i −0.870736 0.870736i 0.121817 0.992553i \(-0.461128\pi\)
−0.992553 + 0.121817i \(0.961128\pi\)
\(180\) 0 0
\(181\) 1.24322 1.24322i 0.0924079 0.0924079i −0.659392 0.751800i \(-0.729186\pi\)
0.751800 + 0.659392i \(0.229186\pi\)
\(182\) 0 0
\(183\) 1.27334i 0.0941278i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.90314 + 7.90314i −0.577935 + 0.577935i
\(188\) 0 0
\(189\) 4.19615 + 4.19615i 0.305225 + 0.305225i
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) 0 0
\(193\) 2.30278 0.165758 0.0828788 0.996560i \(-0.473589\pi\)
0.0828788 + 0.996560i \(0.473589\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.06997 + 8.06997i −0.574961 + 0.574961i −0.933511 0.358549i \(-0.883271\pi\)
0.358549 + 0.933511i \(0.383271\pi\)
\(198\) 0 0
\(199\) 21.8564i 1.54936i 0.632354 + 0.774680i \(0.282089\pi\)
−0.632354 + 0.774680i \(0.717911\pi\)
\(200\) 0 0
\(201\) 0.702368i 0.0495412i
\(202\) 0 0
\(203\) −8.35679 + 8.35679i −0.586532 + 0.586532i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.394542 0.0274226
\(208\) 0 0
\(209\) 15.2711 1.05632
\(210\) 0 0
\(211\) −0.478227 0.478227i −0.0329225 0.0329225i 0.690454 0.723376i \(-0.257411\pi\)
−0.723376 + 0.690454i \(0.757411\pi\)
\(212\) 0 0
\(213\) −1.71458 + 1.71458i −0.117481 + 0.117481i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.71196i 0.591406i
\(218\) 0 0
\(219\) −1.80623 + 1.80623i −0.122054 + 0.122054i
\(220\) 0 0
\(221\) 8.92820 + 8.92820i 0.600576 + 0.600576i
\(222\) 0 0
\(223\) −7.50859 −0.502813 −0.251406 0.967882i \(-0.580893\pi\)
−0.251406 + 0.967882i \(0.580893\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.8788 + 12.8788i 0.854798 + 0.854798i 0.990720 0.135922i \(-0.0433997\pi\)
−0.135922 + 0.990720i \(0.543400\pi\)
\(228\) 0 0
\(229\) −8.62166 + 8.62166i −0.569736 + 0.569736i −0.932054 0.362319i \(-0.881985\pi\)
0.362319 + 0.932054i \(0.381985\pi\)
\(230\) 0 0
\(231\) 2.26371i 0.148941i
\(232\) 0 0
\(233\) 4.63429i 0.303602i −0.988411 0.151801i \(-0.951493\pi\)
0.988411 0.151801i \(-0.0485073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.02304 + 2.02304i 0.131411 + 0.131411i
\(238\) 0 0
\(239\) 18.4220 1.19162 0.595810 0.803126i \(-0.296831\pi\)
0.595810 + 0.803126i \(0.296831\pi\)
\(240\) 0 0
\(241\) 18.3247 1.18040 0.590200 0.807257i \(-0.299049\pi\)
0.590200 + 0.807257i \(0.299049\pi\)
\(242\) 0 0
\(243\) 4.81418 + 4.81418i 0.308830 + 0.308830i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.2518i 1.09770i
\(248\) 0 0
\(249\) 3.23827i 0.205217i
\(250\) 0 0
\(251\) 16.0222 16.0222i 1.01131 1.01131i 0.0113760 0.999935i \(-0.496379\pi\)
0.999935 0.0113760i \(-0.00362116\pi\)
\(252\) 0 0
\(253\) −0.215297 0.215297i −0.0135356 0.0135356i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.82098 −0.363103 −0.181551 0.983381i \(-0.558112\pi\)
−0.181551 + 0.983381i \(0.558112\pi\)
\(258\) 0 0
\(259\) 16.7436 + 16.7436i 1.04040 + 1.04040i
\(260\) 0 0
\(261\) −6.36758 + 6.36758i −0.394143 + 0.394143i
\(262\) 0 0
\(263\) 0.806693i 0.0497428i −0.999691 0.0248714i \(-0.992082\pi\)
0.999691 0.0248714i \(-0.00791763\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.206075 0.206075i 0.0126116 0.0126116i
\(268\) 0 0
\(269\) 15.1939 + 15.1939i 0.926387 + 0.926387i 0.997470 0.0710837i \(-0.0226458\pi\)
−0.0710837 + 0.997470i \(0.522646\pi\)
\(270\) 0 0
\(271\) 10.8491 0.659034 0.329517 0.944150i \(-0.393114\pi\)
0.329517 + 0.944150i \(0.393114\pi\)
\(272\) 0 0
\(273\) −2.55732 −0.154776
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.27334 + 1.27334i −0.0765074 + 0.0765074i −0.744325 0.667818i \(-0.767229\pi\)
0.667818 + 0.744325i \(0.267229\pi\)
\(278\) 0 0
\(279\) 6.63820i 0.397419i
\(280\) 0 0
\(281\) 20.2174i 1.20607i −0.797716 0.603033i \(-0.793959\pi\)
0.797716 0.603033i \(-0.206041\pi\)
\(282\) 0 0
\(283\) 21.3741 21.3741i 1.27056 1.27056i 0.324759 0.945797i \(-0.394717\pi\)
0.945797 0.324759i \(-0.105283\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.8814 −0.760363
\(288\) 0 0
\(289\) 7.39230 0.434841
\(290\) 0 0
\(291\) 1.12197 + 1.12197i 0.0657710 + 0.0657710i
\(292\) 0 0
\(293\) 11.1656 11.1656i 0.652301 0.652301i −0.301246 0.953547i \(-0.597402\pi\)
0.953547 + 0.301246i \(0.0974024\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.48947i 0.202480i
\(298\) 0 0
\(299\) −0.243221 + 0.243221i −0.0140659 + 0.0140659i
\(300\) 0 0
\(301\) −10.4255 10.4255i −0.600916 0.600916i
\(302\) 0 0
\(303\) −0.798174 −0.0458539
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.18143 2.18143i −0.124501 0.124501i 0.642111 0.766612i \(-0.278059\pi\)
−0.766612 + 0.642111i \(0.778059\pi\)
\(308\) 0 0
\(309\) −2.82629 + 2.82629i −0.160782 + 0.160782i
\(310\) 0 0
\(311\) 11.5517i 0.655038i −0.944845 0.327519i \(-0.893787\pi\)
0.944845 0.327519i \(-0.106213\pi\)
\(312\) 0 0
\(313\) 10.4265i 0.589343i 0.955599 + 0.294671i \(0.0952102\pi\)
−0.955599 + 0.294671i \(0.904790\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0785 + 10.0785i 0.566063 + 0.566063i 0.931023 0.364960i \(-0.118917\pi\)
−0.364960 + 0.931023i \(0.618917\pi\)
\(318\) 0 0
\(319\) 6.94941 0.389092
\(320\) 0 0
\(321\) −2.56725 −0.143290
\(322\) 0 0
\(323\) −23.5663 23.5663i −1.31127 1.31127i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.16007i 0.174752i
\(328\) 0 0
\(329\) 27.0684i 1.49233i
\(330\) 0 0
\(331\) −8.77162 + 8.77162i −0.482132 + 0.482132i −0.905812 0.423680i \(-0.860738\pi\)
0.423680 + 0.905812i \(0.360738\pi\)
\(332\) 0 0
\(333\) 12.7580 + 12.7580i 0.699137 + 0.699137i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.0226 1.79885 0.899427 0.437072i \(-0.143984\pi\)
0.899427 + 0.437072i \(0.143984\pi\)
\(338\) 0 0
\(339\) −2.67101 2.67101i −0.145070 0.145070i
\(340\) 0 0
\(341\) 3.62238 3.62238i 0.196163 0.196163i
\(342\) 0 0
\(343\) 3.12189i 0.168566i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.1412 11.1412i 0.598090 0.598090i −0.341714 0.939804i \(-0.611007\pi\)
0.939804 + 0.341714i \(0.111007\pi\)
\(348\) 0 0
\(349\) 10.8656 + 10.8656i 0.581622 + 0.581622i 0.935349 0.353727i \(-0.115086\pi\)
−0.353727 + 0.935349i \(0.615086\pi\)
\(350\) 0 0
\(351\) −3.94207 −0.210412
\(352\) 0 0
\(353\) −11.3480 −0.603995 −0.301998 0.953309i \(-0.597653\pi\)
−0.301998 + 0.953309i \(0.597653\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.49337 + 3.49337i −0.184889 + 0.184889i
\(358\) 0 0
\(359\) 26.5788i 1.40278i 0.712779 + 0.701389i \(0.247436\pi\)
−0.712779 + 0.701389i \(0.752564\pi\)
\(360\) 0 0
\(361\) 26.5367i 1.39667i
\(362\) 0 0
\(363\) 1.08046 1.08046i 0.0567093 0.0567093i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.90729 0.151760 0.0758798 0.997117i \(-0.475823\pi\)
0.0758798 + 0.997117i \(0.475823\pi\)
\(368\) 0 0
\(369\) −9.81514 −0.510956
\(370\) 0 0
\(371\) 13.0858 + 13.0858i 0.679379 + 0.679379i
\(372\) 0 0
\(373\) 4.65522 4.65522i 0.241038 0.241038i −0.576241 0.817280i \(-0.695481\pi\)
0.817280 + 0.576241i \(0.195481\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.85077i 0.404335i
\(378\) 0 0
\(379\) 9.52106 9.52106i 0.489064 0.489064i −0.418947 0.908011i \(-0.637601\pi\)
0.908011 + 0.418947i \(0.137601\pi\)
\(380\) 0 0
\(381\) −1.09652 1.09652i −0.0561767 0.0561767i
\(382\) 0 0
\(383\) 6.92429 0.353815 0.176907 0.984228i \(-0.443391\pi\)
0.176907 + 0.984228i \(0.443391\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.94386 7.94386i −0.403809 0.403809i
\(388\) 0 0
\(389\) 20.0232 20.0232i 1.01521 1.01521i 0.0153322 0.999882i \(-0.495119\pi\)
0.999882 0.0153322i \(-0.00488057\pi\)
\(390\) 0 0
\(391\) 0.664493i 0.0336049i
\(392\) 0 0
\(393\) 0.871474i 0.0439601i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −3.81625 3.81625i −0.191532 0.191532i 0.604826 0.796358i \(-0.293243\pi\)
−0.796358 + 0.604826i \(0.793243\pi\)
\(398\) 0 0
\(399\) 6.75015 0.337930
\(400\) 0 0
\(401\) 1.68031 0.0839108 0.0419554 0.999119i \(-0.486641\pi\)
0.0419554 + 0.999119i \(0.486641\pi\)
\(402\) 0 0
\(403\) −4.09222 4.09222i −0.203848 0.203848i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.9238i 0.690177i
\(408\) 0 0
\(409\) 20.1317i 0.995448i 0.867335 + 0.497724i \(0.165831\pi\)
−0.867335 + 0.497724i \(0.834169\pi\)
\(410\) 0 0
\(411\) −3.35867 + 3.35867i −0.165671 + 0.165671i
\(412\) 0 0
\(413\) 0.412282 + 0.412282i 0.0202871 + 0.0202871i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.0500256 0.00244976
\(418\) 0 0
\(419\) 22.6570 + 22.6570i 1.10687 + 1.10687i 0.993560 + 0.113306i \(0.0361442\pi\)
0.113306 + 0.993560i \(0.463856\pi\)
\(420\) 0 0
\(421\) −10.1583 + 10.1583i −0.495084 + 0.495084i −0.909904 0.414820i \(-0.863845\pi\)
0.414820 + 0.909904i \(0.363845\pi\)
\(422\) 0 0
\(423\) 20.6251i 1.00283i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −13.3317 + 13.3317i −0.645166 + 0.645166i
\(428\) 0 0
\(429\) 1.06332 + 1.06332i 0.0513375 + 0.0513375i
\(430\) 0 0
\(431\) 26.1518 1.25969 0.629843 0.776723i \(-0.283119\pi\)
0.629843 + 0.776723i \(0.283119\pi\)
\(432\) 0 0
\(433\) −9.30795 −0.447312 −0.223656 0.974668i \(-0.571799\pi\)
−0.223656 + 0.974668i \(0.571799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.641992 0.641992i 0.0307107 0.0307107i
\(438\) 0 0
\(439\) 30.4799i 1.45473i −0.686252 0.727364i \(-0.740745\pi\)
0.686252 0.727364i \(-0.259255\pi\)
\(440\) 0 0
\(441\) 22.9059i 1.09076i
\(442\) 0 0
\(443\) −16.7437 + 16.7437i −0.795516 + 0.795516i −0.982385 0.186869i \(-0.940166\pi\)
0.186869 + 0.982385i \(0.440166\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.882482 −0.0417399
\(448\) 0 0
\(449\) −5.40502 −0.255079 −0.127539 0.991834i \(-0.540708\pi\)
−0.127539 + 0.991834i \(0.540708\pi\)
\(450\) 0 0
\(451\) 5.35600 + 5.35600i 0.252204 + 0.252204i
\(452\) 0 0
\(453\) 3.30703 3.30703i 0.155378 0.155378i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 34.5929i 1.61819i 0.587678 + 0.809095i \(0.300042\pi\)
−0.587678 + 0.809095i \(0.699958\pi\)
\(458\) 0 0
\(459\) −5.38496 + 5.38496i −0.251349 + 0.251349i
\(460\) 0 0
\(461\) 14.3876 + 14.3876i 0.670099 + 0.670099i 0.957739 0.287640i \(-0.0928706\pi\)
−0.287640 + 0.957739i \(0.592871\pi\)
\(462\) 0 0
\(463\) 20.6591 0.960108 0.480054 0.877239i \(-0.340617\pi\)
0.480054 + 0.877239i \(0.340617\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.3222 16.3222i −0.755300 0.755300i 0.220163 0.975463i \(-0.429341\pi\)
−0.975463 + 0.220163i \(0.929341\pi\)
\(468\) 0 0
\(469\) 7.35371 7.35371i 0.339563 0.339563i
\(470\) 0 0
\(471\) 4.64472i 0.214017i
\(472\) 0 0
\(473\) 8.66973i 0.398635i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 9.97088 + 9.97088i 0.456535 + 0.456535i
\(478\) 0 0
\(479\) −19.5136 −0.891597 −0.445799 0.895133i \(-0.647080\pi\)
−0.445799 + 0.895133i \(0.647080\pi\)
\(480\) 0 0
\(481\) −15.7298 −0.717216
\(482\) 0 0
\(483\) −0.0951660 0.0951660i −0.00433020 0.00433020i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15.6638i 0.709794i −0.934905 0.354897i \(-0.884516\pi\)
0.934905 0.354897i \(-0.115484\pi\)
\(488\) 0 0
\(489\) 5.82505i 0.263418i
\(490\) 0 0
\(491\) 17.9076 17.9076i 0.808157 0.808157i −0.176198 0.984355i \(-0.556380\pi\)
0.984355 + 0.176198i \(0.0563799\pi\)
\(492\) 0 0
\(493\) −10.7244 10.7244i −0.483001 0.483001i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −35.9029 −1.61046
\(498\) 0 0
\(499\) −2.32067 2.32067i −0.103887 0.103887i 0.653253 0.757140i \(-0.273404\pi\)
−0.757140 + 0.653253i \(0.773404\pi\)
\(500\) 0 0
\(501\) 2.34441 2.34441i 0.104741 0.104741i
\(502\) 0 0
\(503\) 6.18913i 0.275960i −0.990435 0.137980i \(-0.955939\pi\)
0.990435 0.137980i \(-0.0440609\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.18804 + 1.18804i −0.0527627 + 0.0527627i
\(508\) 0 0
\(509\) −18.6217 18.6217i −0.825391 0.825391i 0.161485 0.986875i \(-0.448372\pi\)
−0.986875 + 0.161485i \(0.948372\pi\)
\(510\) 0 0
\(511\) −37.8222 −1.67315
\(512\) 0 0
\(513\) 10.4052 0.459403
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 11.2549 11.2549i 0.494988 0.494988i
\(518\) 0 0
\(519\) 0.972885i 0.0427049i
\(520\) 0 0
\(521\) 24.0232i 1.05247i −0.850338 0.526237i \(-0.823603\pi\)
0.850338 0.526237i \(-0.176397\pi\)
\(522\) 0 0
\(523\) −16.1791 + 16.1791i −0.707463 + 0.707463i −0.966001 0.258538i \(-0.916759\pi\)
0.258538 + 0.966001i \(0.416759\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.1801 −0.487015
\(528\) 0 0
\(529\) 22.9819 0.999213
\(530\) 0 0
\(531\) 0.314144 + 0.314144i 0.0136327 + 0.0136327i
\(532\) 0 0
\(533\) 6.05069 6.05069i 0.262084 0.262084i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.28219i 0.184790i
\(538\) 0 0
\(539\) 12.4994 12.4994i 0.538389 0.538389i
\(540\) 0 0
\(541\) 6.76526 + 6.76526i 0.290861 + 0.290861i 0.837420 0.546559i \(-0.184063\pi\)
−0.546559 + 0.837420i \(0.684063\pi\)
\(542\) 0 0
\(543\) −0.456984 −0.0196111
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.38359 4.38359i −0.187429 0.187429i 0.607155 0.794584i \(-0.292311\pi\)
−0.794584 + 0.607155i \(0.792311\pi\)
\(548\) 0 0
\(549\) −10.1583 + 10.1583i −0.433545 + 0.433545i
\(550\) 0 0
\(551\) 20.7224i 0.882805i
\(552\) 0 0
\(553\) 42.3621i 1.80142i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.92396 3.92396i −0.166264 0.166264i 0.619071 0.785335i \(-0.287509\pi\)
−0.785335 + 0.619071i \(0.787509\pi\)
\(558\) 0 0
\(559\) 9.79422 0.414252
\(560\) 0 0
\(561\) 2.90504 0.122651
\(562\) 0 0
\(563\) −6.61660 6.61660i −0.278857 0.278857i 0.553796 0.832652i \(-0.313179\pi\)
−0.832652 + 0.553796i \(0.813179\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 32.3144i 1.35708i
\(568\) 0 0
\(569\) 40.2900i 1.68904i −0.535521 0.844522i \(-0.679885\pi\)
0.535521 0.844522i \(-0.320115\pi\)
\(570\) 0 0
\(571\) −22.6010 + 22.6010i −0.945823 + 0.945823i −0.998606 0.0527829i \(-0.983191\pi\)
0.0527829 + 0.998606i \(0.483191\pi\)
\(572\) 0 0
\(573\) 0.932147 + 0.932147i 0.0389410 + 0.0389410i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.8020 0.782737 0.391368 0.920234i \(-0.372002\pi\)
0.391368 + 0.920234i \(0.372002\pi\)
\(578\) 0 0
\(579\) −0.423229 0.423229i −0.0175888 0.0175888i
\(580\) 0 0
\(581\) 33.9043 33.9043i 1.40659 1.40659i
\(582\) 0 0
\(583\) 10.8820i 0.450685i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.71961 2.71961i 0.112250 0.112250i −0.648751 0.761001i \(-0.724708\pi\)
0.761001 + 0.648751i \(0.224708\pi\)
\(588\) 0 0
\(589\) 10.8016 + 10.8016i 0.445071 + 0.445071i
\(590\) 0 0
\(591\) 2.96636 0.122020
\(592\) 0 0
\(593\) −4.04894 −0.166270 −0.0831350 0.996538i \(-0.526493\pi\)
−0.0831350 + 0.996538i \(0.526493\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.01700 4.01700i 0.164405 0.164405i
\(598\) 0 0
\(599\) 19.0455i 0.778178i −0.921200 0.389089i \(-0.872790\pi\)
0.921200 0.389089i \(-0.127210\pi\)
\(600\) 0 0
\(601\) 14.4406i 0.589045i −0.955645 0.294522i \(-0.904839\pi\)
0.955645 0.294522i \(-0.0951606\pi\)
\(602\) 0 0
\(603\) 5.60327 5.60327i 0.228183 0.228183i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 46.3473 1.88118 0.940589 0.339546i \(-0.110273\pi\)
0.940589 + 0.339546i \(0.110273\pi\)
\(608\) 0 0
\(609\) 3.07180 0.124475
\(610\) 0 0
\(611\) −12.7147 12.7147i −0.514380 0.514380i
\(612\) 0 0
\(613\) 0.961106 0.961106i 0.0388187 0.0388187i −0.687431 0.726250i \(-0.741262\pi\)
0.726250 + 0.687431i \(0.241262\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.44724i 0.138781i 0.997590 + 0.0693903i \(0.0221054\pi\)
−0.997590 + 0.0693903i \(0.977895\pi\)
\(618\) 0 0
\(619\) −24.5574 + 24.5574i −0.987044 + 0.987044i −0.999917 0.0128733i \(-0.995902\pi\)
0.0128733 + 0.999917i \(0.495902\pi\)
\(620\) 0 0
\(621\) −0.146697 0.146697i −0.00588673 0.00588673i
\(622\) 0 0
\(623\) 4.31517 0.172884
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.80667 2.80667i −0.112088 0.112088i
\(628\) 0 0
\(629\) −21.4873 + 21.4873i −0.856753 + 0.856753i
\(630\) 0 0
\(631\) 22.7950i 0.907456i −0.891140 0.453728i \(-0.850094\pi\)
0.891140 0.453728i \(-0.149906\pi\)
\(632\) 0 0
\(633\) 0.175787i 0.00698691i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14.1207 14.1207i −0.559481 0.559481i
\(638\) 0 0
\(639\) −27.3567 −1.08221
\(640\) 0 0
\(641\) 30.4468 1.20258 0.601289 0.799032i \(-0.294654\pi\)
0.601289 + 0.799032i \(0.294654\pi\)
\(642\) 0 0
\(643\) −20.4452 20.4452i −0.806282 0.806282i 0.177787 0.984069i \(-0.443106\pi\)
−0.984069 + 0.177787i \(0.943106\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.5876i 1.16321i 0.813472 + 0.581604i \(0.197575\pi\)
−0.813472 + 0.581604i \(0.802425\pi\)
\(648\) 0 0
\(649\) 0.342849i 0.0134580i
\(650\) 0 0
\(651\) 1.60117 1.60117i 0.0627550 0.0627550i
\(652\) 0 0
\(653\) −21.2334 21.2334i −0.830928 0.830928i 0.156716 0.987644i \(-0.449909\pi\)
−0.987644 + 0.156716i \(0.949909\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −28.8191 −1.12434
\(658\) 0 0
\(659\) −20.0222 20.0222i −0.779954 0.779954i 0.199869 0.979823i \(-0.435948\pi\)
−0.979823 + 0.199869i \(0.935948\pi\)
\(660\) 0 0
\(661\) −19.9536 + 19.9536i −0.776107 + 0.776107i −0.979166 0.203059i \(-0.934912\pi\)
0.203059 + 0.979166i \(0.434912\pi\)
\(662\) 0 0
\(663\) 3.28184i 0.127456i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.292152 0.292152i 0.0113122 0.0113122i
\(668\) 0 0
\(669\) 1.38001 + 1.38001i 0.0533542 + 0.0533542i
\(670\) 0 0
\(671\) 11.0865 0.427989
\(672\) 0 0
\(673\) −2.91192 −0.112246 −0.0561231 0.998424i \(-0.517874\pi\)
−0.0561231 + 0.998424i \(0.517874\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.6045 34.6045i 1.32996 1.32996i 0.424558 0.905401i \(-0.360430\pi\)
0.905401 0.424558i \(-0.139570\pi\)
\(678\) 0 0
\(679\) 23.4938i 0.901609i
\(680\) 0 0
\(681\) 4.73401i 0.181408i
\(682\) 0 0
\(683\) 24.7435 24.7435i 0.946785 0.946785i −0.0518690 0.998654i \(-0.516518\pi\)
0.998654 + 0.0518690i \(0.0165178\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.16916 0.120911
\(688\) 0 0
\(689\) −12.2934 −0.468341
\(690\) 0 0
\(691\) −25.3782 25.3782i −0.965431 0.965431i 0.0339907 0.999422i \(-0.489178\pi\)
−0.999422 + 0.0339907i \(0.989178\pi\)
\(692\) 0 0
\(693\) 18.0592 18.0592i 0.686011 0.686011i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 16.5308i 0.626148i
\(698\) 0 0
\(699\) −0.851738 + 0.851738i −0.0322157 + 0.0322157i
\(700\) 0 0
\(701\) −32.3544 32.3544i −1.22201 1.22201i −0.966916 0.255094i \(-0.917894\pi\)
−0.255094 0.966916i \(-0.582106\pi\)
\(702\) 0 0
\(703\) 41.5194 1.56593
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.35679 8.35679i −0.314290 0.314290i
\(708\) 0 0
\(709\) −6.64939 + 6.64939i −0.249723 + 0.249723i −0.820857 0.571134i \(-0.806504\pi\)
0.571134 + 0.820857i \(0.306504\pi\)
\(710\) 0 0
\(711\) 32.2784i 1.21053i
\(712\) 0 0
\(713\) 0.304568i 0.0114062i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.38578 3.38578i −0.126444 0.126444i
\(718\) 0 0
\(719\) 45.0785 1.68115 0.840573 0.541699i \(-0.182219\pi\)
0.840573 + 0.541699i \(0.182219\pi\)
\(720\) 0 0
\(721\) −59.1820 −2.20405
\(722\) 0 0
\(723\) −3.36791 3.36791i −0.125254 0.125254i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.6075i 0.690116i 0.938581 + 0.345058i \(0.112141\pi\)
−0.938581 + 0.345058i \(0.887859\pi\)
\(728\) 0 0
\(729\) 23.4200i 0.867409i
\(730\) 0 0
\(731\) 13.3792 13.3792i 0.494846 0.494846i
\(732\) 0 0
\(733\) −7.95550 7.95550i −0.293843 0.293843i 0.544753 0.838596i \(-0.316623\pi\)
−0.838596 + 0.544753i \(0.816623\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.11526 −0.225258
\(738\) 0 0
\(739\) −15.2636 15.2636i −0.561479 0.561479i 0.368249 0.929727i \(-0.379958\pi\)
−0.929727 + 0.368249i \(0.879958\pi\)
\(740\) 0 0
\(741\) −3.17071 + 3.17071i −0.116479 + 0.116479i
\(742\) 0 0
\(743\) 33.3017i 1.22172i 0.791738 + 0.610861i \(0.209177\pi\)
−0.791738 + 0.610861i \(0.790823\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 25.8339 25.8339i 0.945211 0.945211i
\(748\) 0 0
\(749\) −26.8789 26.8789i −0.982132 0.982132i
\(750\) 0 0
\(751\) −1.17214 −0.0427720 −0.0213860 0.999771i \(-0.506808\pi\)
−0.0213860 + 0.999771i \(0.506808\pi\)
\(752\) 0 0
\(753\) −5.88945 −0.214623
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −17.9408 + 17.9408i −0.652069 + 0.652069i −0.953491 0.301422i \(-0.902539\pi\)
0.301422 + 0.953491i \(0.402539\pi\)
\(758\) 0 0
\(759\) 0.0791389i 0.00287256i
\(760\) 0 0
\(761\) 15.4641i 0.560573i 0.959916 + 0.280287i \(0.0904295\pi\)
−0.959916 + 0.280287i \(0.909570\pi\)
\(762\) 0 0
\(763\) −33.0856 + 33.0856i −1.19778 + 1.19778i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.387318 −0.0139852
\(768\) 0 0
\(769\) −14.9777 −0.540108 −0.270054 0.962845i \(-0.587042\pi\)
−0.270054 + 0.962845i \(0.587042\pi\)
\(770\) 0 0
\(771\) 1.06984 + 1.06984i 0.0385293 + 0.0385293i
\(772\) 0 0
\(773\) 33.0120 33.0120i 1.18736 1.18736i 0.209566 0.977794i \(-0.432795\pi\)
0.977794 0.209566i \(-0.0672052\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.15464i 0.220796i
\(778\) 0 0
\(779\) −15.9710 + 15.9710i −0.572222 + 0.572222i
\(780\) 0 0
\(781\) 14.9282 + 14.9282i 0.534173 + 0.534173i
\(782\) 0 0
\(783\) 4.73512 0.169219
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.8326 + 28.8326i 1.02777 + 1.02777i 0.999603 + 0.0281690i \(0.00896767\pi\)
0.0281690 + 0.999603i \(0.491032\pi\)
\(788\) 0 0
\(789\) −0.148262 + 0.148262i −0.00527828 + 0.00527828i
\(790\) 0 0
\(791\) 55.9305i 1.98866i
\(792\) 0 0
\(793\) 12.5244i 0.444756i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 23.1556 + 23.1556i 0.820214 + 0.820214i 0.986139 0.165924i \(-0.0530607\pi\)
−0.165924 + 0.986139i \(0.553061\pi\)
\(798\) 0 0
\(799\) −34.7371 −1.22891
\(800\) 0 0
\(801\) 3.28801 0.116176
\(802\) 0 0
\(803\) 15.7262 + 15.7262i 0.554966 + 0.554966i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.58497i 0.196600i
\(808\) 0 0
\(809\) 36.2210i 1.27346i −0.771086 0.636731i \(-0.780286\pi\)
0.771086 0.636731i \(-0.219714\pi\)
\(810\) 0 0
\(811\) −9.17312 + 9.17312i −0.322112 + 0.322112i −0.849577 0.527465i \(-0.823143\pi\)
0.527465 + 0.849577i \(0.323143\pi\)
\(812\) 0 0
\(813\) −1.99395 1.99395i −0.0699310 0.0699310i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −25.8522 −0.904456
\(818\) 0 0
\(819\) −20.4015 20.4015i −0.712886 0.712886i
\(820\) 0 0
\(821\) 7.26795 7.26795i 0.253653 0.253653i −0.568813 0.822467i \(-0.692597\pi\)
0.822467 + 0.568813i \(0.192597\pi\)
\(822\) 0 0
\(823\) 28.2974i 0.986384i 0.869920 + 0.493192i \(0.164170\pi\)
−0.869920 + 0.493192i \(0.835830\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.1661 + 18.1661i −0.631697 + 0.631697i −0.948494 0.316797i \(-0.897393\pi\)
0.316797 + 0.948494i \(0.397393\pi\)
\(828\) 0 0
\(829\) −11.0865 11.0865i −0.385049 0.385049i 0.487868 0.872917i \(-0.337775\pi\)
−0.872917 + 0.487868i \(0.837775\pi\)
\(830\) 0 0
\(831\) 0.468054 0.0162366
\(832\) 0 0
\(833\) −38.5783 −1.33666
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.46818 2.46818i 0.0853128 0.0853128i
\(838\) 0 0
\(839\) 11.9093i 0.411153i −0.978641 0.205577i \(-0.934093\pi\)
0.978641 0.205577i \(-0.0659069\pi\)
\(840\) 0 0
\(841\) 19.5698i 0.674822i
\(842\) 0 0
\(843\) −3.71576 + 3.71576i −0.127977 + 0.127977i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.6245 0.777388
\(848\) 0 0
\(849\) −7.85669 −0.269641
\(850\) 0 0
\(851\) −0.585354 0.585354i −0.0200657 0.0200657i
\(852\) 0 0
\(853\) 2.44597 2.44597i 0.0837485 0.0837485i −0.663992 0.747740i \(-0.731139\pi\)
0.747740 + 0.663992i \(0.231139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.57862i 0.0880839i 0.999030 + 0.0440419i \(0.0140235\pi\)
−0.999030 + 0.0440419i \(0.985976\pi\)
\(858\) 0 0
\(859\) 33.0076 33.0076i 1.12620 1.12620i 0.135416 0.990789i \(-0.456763\pi\)
0.990789 0.135416i \(-0.0432371\pi\)
\(860\) 0 0
\(861\) 2.36747 + 2.36747i 0.0806832 + 0.0806832i
\(862\) 0 0
\(863\) 23.5500 0.801652 0.400826 0.916154i \(-0.368723\pi\)
0.400826 + 0.916154i \(0.368723\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.35863 1.35863i −0.0461416 0.0461416i
\(868\) 0 0
\(869\) 17.6139 17.6139i 0.597511 0.597511i
\(870\) 0 0
\(871\) 6.90843i 0.234083i
\(872\) 0 0
\(873\) 17.9014i 0.605871i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −34.2135 34.2135i −1.15531 1.15531i −0.985472 0.169836i \(-0.945676\pi\)
−0.169836 0.985472i \(-0.554324\pi\)
\(878\) 0 0
\(879\) −4.10426 −0.138433
\(880\) 0 0
\(881\) 40.6823 1.37062 0.685310 0.728251i \(-0.259667\pi\)
0.685310 + 0.728251i \(0.259667\pi\)
\(882\) 0 0
\(883\) −35.8531 35.8531i −1.20655 1.20655i −0.972137 0.234415i \(-0.924682\pi\)
−0.234415 0.972137i \(-0.575318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.33231i 0.313348i 0.987650 + 0.156674i \(0.0500773\pi\)
−0.987650 + 0.156674i \(0.949923\pi\)
\(888\) 0 0
\(889\) 22.9610i 0.770087i
\(890\) 0 0
\(891\) 13.4361 13.4361i 0.450127 0.450127i
\(892\) 0 0
\(893\) 33.5609 + 33.5609i 1.12307 + 1.12307i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.0894035 0.00298510
\(898\) 0 0
\(899\) 4.91548 + 4.91548i 0.163940 + 0.163940i
\(900\) 0 0
\(901\) −16.7931 + 16.7931i −0.559459 + 0.559459i
\(902\) 0 0
\(903\) 3.83222i 0.127528i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −33.2170 + 33.2170i −1.10295 + 1.10295i −0.108899 + 0.994053i \(0.534733\pi\)
−0.994053 + 0.108899i \(0.965267\pi\)
\(908\) 0 0
\(909\) −6.36758 6.36758i −0.211199 0.211199i
\(910\) 0 0
\(911\) −5.77870 −0.191457 −0.0957284 0.995407i \(-0.530518\pi\)
−0.0957284 + 0.995407i \(0.530518\pi\)
\(912\) 0 0
\(913\) −28.1944 −0.933098
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.12424 + 9.12424i −0.301309 + 0.301309i
\(918\) 0 0
\(919\) 50.8572i 1.67763i −0.544420 0.838813i \(-0.683250\pi\)
0.544420 0.838813i \(-0.316750\pi\)
\(920\) 0 0
\(921\) 0.801852i 0.0264219i
\(922\) 0 0
\(923\) 16.8644 16.8644i 0.555100 0.555100i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −45.0945 −1.48110
\(928\) 0 0
\(929\) −21.6815 −0.711346 −0.355673 0.934611i \(-0.615748\pi\)
−0.355673 + 0.934611i \(0.615748\pi\)
\(930\) 0 0
\(931\) 37.2720 + 37.2720i 1.22154 + 1.22154i
\(932\) 0 0
\(933\) −2.12309 + 2.12309i −0.0695070 + 0.0695070i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.7948i 0.646668i 0.946285 + 0.323334i \(0.104804\pi\)
−0.946285 + 0.323334i \(0.895196\pi\)
\(938\) 0 0
\(939\) 1.91630 1.91630i 0.0625360 0.0625360i
\(940\) 0 0
\(941\) −29.4510 29.4510i −0.960074 0.960074i 0.0391593 0.999233i \(-0.487532\pi\)
−0.999233 + 0.0391593i \(0.987532\pi\)
\(942\) 0 0
\(943\) 0.450330 0.0146648
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.11783 + 4.11783i 0.133811 + 0.133811i 0.770840 0.637029i \(-0.219837\pi\)
−0.637029 + 0.770840i \(0.719837\pi\)
\(948\) 0 0
\(949\) 17.7660 17.7660i 0.576708 0.576708i
\(950\) 0 0
\(951\) 3.70465i 0.120131i
\(952\) 0 0
\(953\) 40.3245i 1.30624i 0.757255 + 0.653119i \(0.226540\pi\)
−0.757255 + 0.653119i \(0.773460\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.27723 1.27723i −0.0412871 0.0412871i
\(958\) 0 0
\(959\) −70.3298 −2.27107
\(960\) 0 0
\(961\) −25.8756 −0.834697
\(962\) 0 0
\(963\) −20.4807 20.4807i −0.659982 0.659982i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 58.1740i 1.87075i −0.353656 0.935375i \(-0.615062\pi\)
0.353656 0.935375i \(-0.384938\pi\)
\(968\) 0 0
\(969\) 8.66254i 0.278281i
\(970\) 0 0
\(971\) 1.70830 1.70830i 0.0548220 0.0548220i −0.679164 0.733986i \(-0.737658\pi\)
0.733986 + 0.679164i \(0.237658\pi\)
\(972\) 0 0
\(973\) 0.523762 + 0.523762i 0.0167910 + 0.0167910i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.1811 1.12554 0.562772 0.826612i \(-0.309735\pi\)
0.562772 + 0.826612i \(0.309735\pi\)
\(978\) 0 0
\(979\) −1.79422 1.79422i −0.0573436 0.0573436i
\(980\) 0 0
\(981\) −25.2100 + 25.2100i −0.804894 + 0.804894i
\(982\) 0 0
\(983\) 27.7257i 0.884312i 0.896938 + 0.442156i \(0.145786\pi\)
−0.896938 + 0.442156i \(0.854214\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.97491 4.97491i 0.158353 0.158353i
\(988\) 0 0
\(989\) 0.364474 + 0.364474i 0.0115896 + 0.0115896i
\(990\) 0 0
\(991\) 7.02711 0.223224 0.111612 0.993752i \(-0.464399\pi\)
0.111612 + 0.993752i \(0.464399\pi\)
\(992\) 0 0
\(993\) 3.22428 0.102319
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.2467 + 14.2467i −0.451197 + 0.451197i −0.895752 0.444555i \(-0.853362\pi\)
0.444555 + 0.895752i \(0.353362\pi\)
\(998\) 0 0
\(999\) 9.48726i 0.300164i
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 1600.2.l.h.401.4 16
4.3 odd 2 400.2.l.i.301.4 16
5.2 odd 4 320.2.q.c.209.4 16
5.3 odd 4 320.2.q.c.209.5 16
5.4 even 2 inner 1600.2.l.h.401.5 16
16.5 even 4 inner 1600.2.l.h.1201.4 16
16.11 odd 4 400.2.l.i.101.4 16
20.3 even 4 80.2.q.c.29.1 16
20.7 even 4 80.2.q.c.29.8 yes 16
20.19 odd 2 400.2.l.i.301.5 16
40.3 even 4 640.2.q.e.289.5 16
40.13 odd 4 640.2.q.f.289.4 16
40.27 even 4 640.2.q.e.289.4 16
40.37 odd 4 640.2.q.f.289.5 16
60.23 odd 4 720.2.bm.f.109.8 16
60.47 odd 4 720.2.bm.f.109.1 16
80.3 even 4 640.2.q.e.609.4 16
80.13 odd 4 640.2.q.f.609.5 16
80.27 even 4 80.2.q.c.69.1 yes 16
80.37 odd 4 320.2.q.c.49.5 16
80.43 even 4 80.2.q.c.69.8 yes 16
80.53 odd 4 320.2.q.c.49.4 16
80.59 odd 4 400.2.l.i.101.5 16
80.67 even 4 640.2.q.e.609.5 16
80.69 even 4 inner 1600.2.l.h.1201.5 16
80.77 odd 4 640.2.q.f.609.4 16
240.107 odd 4 720.2.bm.f.469.8 16
240.203 odd 4 720.2.bm.f.469.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.q.c.29.1 16 20.3 even 4
80.2.q.c.29.8 yes 16 20.7 even 4
80.2.q.c.69.1 yes 16 80.27 even 4
80.2.q.c.69.8 yes 16 80.43 even 4
320.2.q.c.49.4 16 80.53 odd 4
320.2.q.c.49.5 16 80.37 odd 4
320.2.q.c.209.4 16 5.2 odd 4
320.2.q.c.209.5 16 5.3 odd 4
400.2.l.i.101.4 16 16.11 odd 4
400.2.l.i.101.5 16 80.59 odd 4
400.2.l.i.301.4 16 4.3 odd 2
400.2.l.i.301.5 16 20.19 odd 2
640.2.q.e.289.4 16 40.27 even 4
640.2.q.e.289.5 16 40.3 even 4
640.2.q.e.609.4 16 80.3 even 4
640.2.q.e.609.5 16 80.67 even 4
640.2.q.f.289.4 16 40.13 odd 4
640.2.q.f.289.5 16 40.37 odd 4
640.2.q.f.609.4 16 80.77 odd 4
640.2.q.f.609.5 16 80.13 odd 4
720.2.bm.f.109.1 16 60.47 odd 4
720.2.bm.f.109.8 16 60.23 odd 4
720.2.bm.f.469.1 16 240.203 odd 4
720.2.bm.f.469.8 16 240.107 odd 4
1600.2.l.h.401.4 16 1.1 even 1 trivial
1600.2.l.h.401.5 16 5.4 even 2 inner
1600.2.l.h.1201.4 16 16.5 even 4 inner
1600.2.l.h.1201.5 16 80.69 even 4 inner