Properties

Label 3997.1.cz.a.97.1
Level $3997$
Weight $1$
Character 3997.97
Analytic conductor $1.995$
Analytic rank $0$
Dimension $144$
Projective image $D_{285}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3997,1,Mod(13,3997)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3997, base_ring=CyclotomicField(570))
 
chi = DirichletCharacter(H, H._module([285, 352]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3997.13");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3997 = 7 \cdot 571 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3997.cz (of order \(570\), degree \(144\), 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: \(1.99476285549\)
Analytic rank: \(0\)
Dimension: \(144\)
Coefficient field: \(\Q(\zeta_{570})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{144} - x^{143} + x^{142} + x^{139} - x^{138} + x^{137} - x^{129} + x^{128} - x^{127} + x^{125} - 2 x^{124} + 2 x^{123} - x^{122} + x^{120} - x^{119} + x^{118} + x^{114} - x^{113} + x^{112} - x^{110} + 2 x^{109} - 2 x^{108} + x^{107} - x^{105} + x^{104} - x^{103} - x^{99} + x^{98} - x^{97} + x^{95} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{285}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{285} - \cdots)\)

Embedding invariants

Embedding label 97.1
Root \(0.868768 + 0.495219i\) of defining polynomial
Character \(\chi\) \(=\) 3997.97
Dual form 3997.1.cz.a.2596.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.62671 + 1.12794i) q^{2} +(1.02331 + 2.73313i) q^{4} +(0.490424 + 0.871484i) q^{7} +(-0.932234 + 3.68131i) q^{8} +(-0.997024 - 0.0770854i) q^{9} +O(q^{10})\) \(q+(1.62671 + 1.12794i) q^{2} +(1.02331 + 2.73313i) q^{4} +(0.490424 + 0.871484i) q^{7} +(-0.932234 + 3.68131i) q^{8} +(-0.997024 - 0.0770854i) q^{9} +(-0.978802 - 1.27197i) q^{11} +(-0.185201 + 1.97082i) q^{14} +(-3.46795 + 3.02024i) q^{16} +(-1.53492 - 1.24997i) q^{18} +(-0.157530 - 3.17315i) q^{22} +(-0.252301 - 0.0167112i) q^{23} +(0.411766 + 0.911290i) q^{25} +(-1.88003 + 2.23219i) q^{28} +(0.248433 + 1.79157i) q^{29} +(-5.27352 + 0.583674i) q^{32} +(-0.809578 - 2.80388i) q^{36} +(1.87393 + 0.463568i) q^{37} +(0.373713 - 1.54723i) q^{43} +(2.47484 - 3.97681i) q^{44} +(-0.391571 - 0.311763i) q^{46} +(-0.518970 + 0.854793i) q^{49} +(-0.358053 + 1.94685i) q^{50} +(0.372974 - 1.06725i) q^{53} +(-3.66539 + 0.992973i) q^{56} +(-1.61665 + 3.19458i) q^{58} +(-0.421786 - 0.906696i) q^{63} +(-5.19235 - 2.80996i) q^{64} +(1.61478 - 0.306192i) q^{67} +(-0.353856 + 0.0752144i) q^{71} +(1.21324 - 3.59849i) q^{72} +(2.52547 + 2.86776i) q^{74} +(0.628472 - 1.47681i) q^{77} +(1.64652 - 0.647707i) q^{79} +(0.988116 + 0.153712i) q^{81} +(2.35310 - 2.09537i) q^{86} +(5.59498 - 2.41750i) q^{88} +(-0.212507 - 0.706672i) q^{92} +(-1.80836 + 0.805135i) q^{98} +(0.877839 + 1.34363i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q + q^{2} + 5 q^{4} + 2 q^{7} + 21 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 144 q + q^{2} + 5 q^{4} + 2 q^{7} + 21 q^{8} - q^{9} + 6 q^{11} + q^{14} + 6 q^{16} - 9 q^{18} + 21 q^{22} + 3 q^{23} - q^{25} - 10 q^{28} - 4 q^{29} - 5 q^{32} - 2 q^{37} - 9 q^{43} - 20 q^{44} - 34 q^{46} + 2 q^{49} - 2 q^{50} + 6 q^{53} - 8 q^{56} - q^{58} - q^{63} + 11 q^{64} + 20 q^{67} + 3 q^{71} + 23 q^{72} - 31 q^{74} + q^{77} + 6 q^{79} - q^{81} + 7 q^{86} - 9 q^{88} + 9 q^{92} + 6 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3997\mathbb{Z}\right)^\times\).

\(n\) \(1716\) \(2285\)
\(\chi(n)\) \(e\left(\frac{146}{285}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.62671 + 1.12794i 1.62671 + 1.12794i 0.894729 + 0.446609i \(0.147368\pi\)
0.731980 + 0.681326i \(0.238596\pi\)
\(3\) 0 0 0.0385714 0.999256i \(-0.487719\pi\)
−0.0385714 + 0.999256i \(0.512281\pi\)
\(4\) 1.02331 + 2.73313i 1.02331 + 2.73313i
\(5\) 0 0 −0.840168 0.542326i \(-0.817544\pi\)
0.840168 + 0.542326i \(0.182456\pi\)
\(6\) 0 0
\(7\) 0.490424 + 0.871484i 0.490424 + 0.871484i
\(8\) −0.932234 + 3.68131i −0.932234 + 3.68131i
\(9\) −0.997024 0.0770854i −0.997024 0.0770854i
\(10\) 0 0
\(11\) −0.978802 1.27197i −0.978802 1.27197i −0.962268 0.272103i \(-0.912281\pi\)
−0.0165339 0.999863i \(-0.505263\pi\)
\(12\) 0 0
\(13\) 0 0 −0.528360 0.849020i \(-0.677193\pi\)
0.528360 + 0.849020i \(0.322807\pi\)
\(14\) −0.185201 + 1.97082i −0.185201 + 1.97082i
\(15\) 0 0
\(16\) −3.46795 + 3.02024i −3.46795 + 3.02024i
\(17\) 0 0 0.224056 0.974576i \(-0.428070\pi\)
−0.224056 + 0.974576i \(0.571930\pi\)
\(18\) −1.53492 1.24997i −1.53492 1.24997i
\(19\) 0 0 −0.938430 0.345471i \(-0.887719\pi\)
0.938430 + 0.345471i \(0.112281\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.157530 3.17315i −0.157530 3.17315i
\(23\) −0.252301 0.0167112i −0.252301 0.0167112i −0.0605901 0.998163i \(-0.519298\pi\)
−0.191711 + 0.981451i \(0.561404\pi\)
\(24\) 0 0
\(25\) 0.411766 + 0.911290i 0.411766 + 0.911290i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.88003 + 2.23219i −1.88003 + 2.23219i
\(29\) 0.248433 + 1.79157i 0.248433 + 1.79157i 0.546948 + 0.837166i \(0.315789\pi\)
−0.298515 + 0.954405i \(0.596491\pi\)
\(30\) 0 0
\(31\) 0 0 −0.945817 0.324699i \(-0.894737\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(32\) −5.27352 + 0.583674i −5.27352 + 0.583674i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.809578 2.80388i −0.809578 2.80388i
\(37\) 1.87393 + 0.463568i 1.87393 + 0.463568i 0.999757 0.0220445i \(-0.00701754\pi\)
0.874174 + 0.485613i \(0.161404\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.998482 0.0550878i \(-0.0175439\pi\)
−0.998482 + 0.0550878i \(0.982456\pi\)
\(42\) 0 0
\(43\) 0.373713 1.54723i 0.373713 1.54723i −0.401695 0.915773i \(-0.631579\pi\)
0.775409 0.631460i \(-0.217544\pi\)
\(44\) 2.47484 3.97681i 2.47484 3.97681i
\(45\) 0 0
\(46\) −0.391571 0.311763i −0.391571 0.311763i
\(47\) 0 0 −0.191711 0.981451i \(-0.561404\pi\)
0.191711 + 0.981451i \(0.438596\pi\)
\(48\) 0 0
\(49\) −0.518970 + 0.854793i −0.518970 + 0.854793i
\(50\) −0.358053 + 1.94685i −0.358053 + 1.94685i
\(51\) 0 0
\(52\) 0 0
\(53\) 0.372974 1.06725i 0.372974 1.06725i −0.592235 0.805765i \(-0.701754\pi\)
0.965209 0.261480i \(-0.0842105\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.66539 + 0.992973i −3.66539 + 0.992973i
\(57\) 0 0
\(58\) −1.61665 + 3.19458i −1.61665 + 3.19458i
\(59\) 0 0 0.677282 0.735724i \(-0.263158\pi\)
−0.677282 + 0.735724i \(0.736842\pi\)
\(60\) 0 0
\(61\) 0 0 −0.126427 0.991976i \(-0.540351\pi\)
0.126427 + 0.991976i \(0.459649\pi\)
\(62\) 0 0
\(63\) −0.421786 0.906696i −0.421786 0.906696i
\(64\) −5.19235 2.80996i −5.19235 2.80996i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.61478 0.306192i 1.61478 0.306192i 0.701237 0.712928i \(-0.252632\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.353856 + 0.0752144i −0.353856 + 0.0752144i −0.381410 0.924406i \(-0.624561\pi\)
0.0275543 + 0.999620i \(0.491228\pi\)
\(72\) 1.21324 3.59849i 1.21324 3.59849i
\(73\) 0 0 0.693336 0.720615i \(-0.256140\pi\)
−0.693336 + 0.720615i \(0.743860\pi\)
\(74\) 2.52547 + 2.86776i 2.52547 + 2.86776i
\(75\) 0 0
\(76\) 0 0
\(77\) 0.628472 1.47681i 0.628472 1.47681i
\(78\) 0 0
\(79\) 1.64652 0.647707i 1.64652 0.647707i 0.652586 0.757715i \(-0.273684\pi\)
0.993931 + 0.110008i \(0.0350877\pi\)
\(80\) 0 0
\(81\) 0.988116 + 0.153712i 0.988116 + 0.153712i
\(82\) 0 0
\(83\) 0 0 −0.992658 0.120958i \(-0.961404\pi\)
0.992658 + 0.120958i \(0.0385965\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.35310 2.09537i 2.35310 2.09537i
\(87\) 0 0
\(88\) 5.59498 2.41750i 5.59498 2.41750i
\(89\) 0 0 −0.391577 0.920146i \(-0.628070\pi\)
0.391577 + 0.920146i \(0.371930\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.212507 0.706672i −0.212507 0.706672i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.266796 0.963753i \(-0.414035\pi\)
−0.266796 + 0.963753i \(0.585965\pi\)
\(98\) −1.80836 + 0.805135i −1.80836 + 0.805135i
\(99\) 0.877839 + 1.34363i 0.877839 + 1.34363i
\(100\) −2.06931 + 2.05794i −2.06931 + 2.05794i
\(101\) 0 0 0.834139 0.551554i \(-0.185965\pi\)
−0.834139 + 0.551554i \(0.814035\pi\)
\(102\) 0 0
\(103\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.81050 1.31541i 1.81050 1.31541i
\(107\) −0.174175 0.151690i −0.174175 0.151690i 0.565270 0.824906i \(-0.308772\pi\)
−0.739446 + 0.673216i \(0.764912\pi\)
\(108\) 0 0
\(109\) −0.922290 1.59745i −0.922290 1.59745i −0.795863 0.605477i \(-0.792982\pi\)
−0.126427 0.991976i \(-0.540351\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.33286 1.54107i −4.33286 1.54107i
\(113\) 0.0320799 0.446962i 0.0320799 0.446962i −0.956036 0.293250i \(-0.905263\pi\)
0.988116 0.153712i \(-0.0491228\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.64238 + 2.51233i −4.64238 + 2.51233i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.403690 + 1.52337i −0.403690 + 1.52337i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.336571 1.95068i 0.336571 1.95068i
\(127\) 1.98424 0.153413i 1.98424 0.153413i 0.984487 0.175457i \(-0.0561404\pi\)
0.999757 + 0.0220445i \(0.00701754\pi\)
\(128\) −2.88128 5.69356i −2.88128 5.69356i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.986361 0.164595i \(-0.947368\pi\)
0.986361 + 0.164595i \(0.0526316\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.97215 + 1.32328i 2.97215 + 1.32328i
\(135\) 0 0
\(136\) 0 0
\(137\) 0.0770492 0.170520i 0.0770492 0.170520i −0.868768 0.495219i \(-0.835088\pi\)
0.945817 + 0.324699i \(0.105263\pi\)
\(138\) 0 0
\(139\) 0 0 −0.565270 0.824906i \(-0.691228\pi\)
0.565270 + 0.824906i \(0.308772\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −0.660457 0.276774i −0.660457 0.276774i
\(143\) 0 0
\(144\) 3.69045 2.74393i 3.69045 2.74393i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0.650614 + 5.59608i 0.650614 + 5.59608i
\(149\) −0.382583 + 0.0253405i −0.382583 + 0.0253405i −0.256156 0.966635i \(-0.582456\pi\)
−0.126427 + 0.991976i \(0.540351\pi\)
\(150\) 0 0
\(151\) −0.468021 1.76613i −0.468021 1.76613i −0.627176 0.778877i \(-0.715789\pi\)
0.159156 0.987253i \(-0.449123\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 2.68809 1.69347i 2.68809 1.69347i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.731980 0.681326i \(-0.761404\pi\)
0.731980 + 0.681326i \(0.238596\pi\)
\(158\) 3.40897 + 0.803533i 3.40897 + 0.803533i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.109171 0.228072i −0.109171 0.228072i
\(162\) 1.43400 + 1.36458i 1.43400 + 1.36458i
\(163\) −0.710550 + 1.26265i −0.710550 + 1.26265i 0.245485 + 0.969400i \(0.421053\pi\)
−0.956036 + 0.293250i \(0.905263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(168\) 0 0
\(169\) −0.441671 + 0.897177i −0.441671 + 0.897177i
\(170\) 0 0
\(171\) 0 0
\(172\) 4.61122 0.561887i 4.61122 0.561887i
\(173\) 0 0 −0.815447 0.578832i \(-0.803509\pi\)
0.815447 + 0.578832i \(0.196491\pi\)
\(174\) 0 0
\(175\) −0.592235 + 0.805765i −0.592235 + 0.805765i
\(176\) 7.23609 + 1.45490i 7.23609 + 1.45490i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.26166 1.28270i −1.26166 1.28270i −0.942181 0.335105i \(-0.891228\pi\)
−0.319482 0.947592i \(-0.603509\pi\)
\(180\) 0 0
\(181\) 0 0 −0.789141 0.614213i \(-0.789474\pi\)
0.789141 + 0.614213i \(0.210526\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0.296723 0.913218i 0.296723 0.913218i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.627284 + 0.983642i −0.627284 + 0.983642i 0.371197 + 0.928554i \(0.378947\pi\)
−0.998482 + 0.0550878i \(0.982456\pi\)
\(192\) 0 0
\(193\) −0.766039 0.162827i −0.766039 0.162827i −0.191711 0.981451i \(-0.561404\pi\)
−0.574329 + 0.818625i \(0.694737\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.86733 0.543697i −2.86733 0.543697i
\(197\) −1.93450 0.344771i −1.93450 0.344771i −0.934564 0.355794i \(-0.884211\pi\)
−0.999939 + 0.0110229i \(0.996491\pi\)
\(198\) −0.0875416 + 3.17585i −0.0875416 + 3.17585i
\(199\) 0 0 −0.999939 0.0110229i \(-0.996491\pi\)
0.999939 + 0.0110229i \(0.00350877\pi\)
\(200\) −3.73860 + 0.666301i −3.73860 + 0.666301i
\(201\) 0 0
\(202\) 0 0
\(203\) −1.43949 + 1.09513i −1.43949 + 1.09513i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.250262 + 0.0361102i 0.250262 + 0.0361102i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.0948019 0.0754800i 0.0948019 0.0754800i −0.574329 0.818625i \(-0.694737\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(212\) 3.29859 0.0727334i 3.29859 0.0727334i
\(213\) 0 0
\(214\) −0.112237 0.443213i −0.112237 0.443213i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.301526 3.63887i 0.301526 3.63887i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.441671 0.897177i \(-0.645614\pi\)
0.441671 + 0.897177i \(0.354386\pi\)
\(224\) −3.09492 4.30955i −3.09492 4.30955i
\(225\) −0.340293 0.940319i −0.340293 0.940319i
\(226\) 0.556329 0.690893i 0.556329 0.690893i
\(227\) 0 0 −0.884667 0.466224i \(-0.845614\pi\)
0.884667 + 0.466224i \(0.154386\pi\)
\(228\) 0 0
\(229\) 0 0 −0.509516 0.860461i \(-0.670175\pi\)
0.509516 + 0.860461i \(0.329825\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.82693 0.755604i −6.82693 0.755604i
\(233\) −0.372922 + 0.207162i −0.372922 + 0.207162i −0.660898 0.750475i \(-0.729825\pi\)
0.287976 + 0.957638i \(0.407018\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.435054 + 1.80119i 0.435054 + 1.80119i 0.583317 + 0.812244i \(0.301754\pi\)
−0.148264 + 0.988948i \(0.547368\pi\)
\(240\) 0 0
\(241\) 0 0 −0.846095 0.533032i \(-0.821053\pi\)
0.846095 + 0.533032i \(0.178947\pi\)
\(242\) −2.37495 + 2.02275i −2.37495 + 2.02275i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.480787 0.876837i \(-0.659649\pi\)
0.480787 + 0.876837i \(0.340351\pi\)
\(252\) 2.04650 2.08062i 2.04650 2.08062i
\(253\) 0.225696 + 0.337275i 0.225696 + 0.337275i
\(254\) 3.40083 + 1.98854i 3.40083 + 1.98854i
\(255\) 0 0
\(256\) 0.924040 6.66370i 0.924040 6.66370i
\(257\) 0 0 −0.213300 0.976987i \(-0.568421\pi\)
0.213300 + 0.976987i \(0.431579\pi\)
\(258\) 0 0
\(259\) 0.515027 + 1.86045i 0.515027 + 1.86045i
\(260\) 0 0
\(261\) −0.109590 1.80539i −0.109590 1.80539i
\(262\) 0 0
\(263\) 1.65441 + 0.0729943i 1.65441 + 0.0729943i 0.851919 0.523673i \(-0.175439\pi\)
0.802489 + 0.596667i \(0.203509\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.48928 + 4.10009i 2.48928 + 4.10009i
\(269\) 0 0 0.340293 0.940319i \(-0.389474\pi\)
−0.340293 + 0.940319i \(0.610526\pi\)
\(270\) 0 0
\(271\) 0 0 −0.677282 0.735724i \(-0.736842\pi\)
0.677282 + 0.735724i \(0.263158\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.317672 0.190480i 0.317672 0.190480i
\(275\) 0.756094 1.41572i 0.756094 1.41572i
\(276\) 0 0
\(277\) −0.628136 1.57129i −0.628136 1.57129i −0.809017 0.587785i \(-0.800000\pi\)
0.180881 0.983505i \(-0.442105\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.35069 + 0.789779i −1.35069 + 0.789779i −0.989750 0.142811i \(-0.954386\pi\)
−0.360939 + 0.932589i \(0.617544\pi\)
\(282\) 0 0
\(283\) 0 0 0.999453 0.0330634i \(-0.0105263\pi\)
−0.999453 + 0.0330634i \(0.989474\pi\)
\(284\) −0.567674 0.890167i −0.567674 0.890167i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 5.30283 0.175425i 5.30283 0.175425i
\(289\) −0.899598 0.436719i −0.899598 0.436719i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.00551154 0.999985i \(-0.498246\pi\)
−0.00551154 + 0.999985i \(0.501754\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.45348 + 6.46637i −3.45348 + 6.46637i
\(297\) 0 0
\(298\) −0.650934 0.390307i −0.650934 0.390307i
\(299\) 0 0
\(300\) 0 0
\(301\) 1.53167 0.433114i 1.53167 0.433114i
\(302\) 1.23075 3.40088i 1.23075 3.40088i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.360939 0.932589i \(-0.617544\pi\)
0.360939 + 0.932589i \(0.382456\pi\)
\(308\) 4.67945 + 0.206462i 4.67945 + 0.206462i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.942181 0.335105i \(-0.108772\pi\)
−0.942181 + 0.335105i \(0.891228\pi\)
\(312\) 0 0
\(313\) 0 0 −0.635724 0.771917i \(-0.719298\pi\)
0.635724 + 0.771917i \(0.280702\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.45516 + 3.83735i 3.45516 + 3.83735i
\(317\) 1.18327 + 0.691882i 1.18327 + 0.691882i 0.959210 0.282694i \(-0.0912281\pi\)
0.224056 + 0.974576i \(0.428070\pi\)
\(318\) 0 0
\(319\) 2.03565 2.06959i 2.03565 2.06959i
\(320\) 0 0
\(321\) 0 0
\(322\) 0.0796612 0.494144i 0.0796612 0.494144i
\(323\) 0 0
\(324\) 0.591031 + 2.85795i 0.591031 + 2.85795i
\(325\) 0 0
\(326\) −2.58005 + 1.25251i −2.58005 + 1.25251i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.160640 + 1.52838i 0.160640 + 1.52838i 0.716783 + 0.697297i \(0.245614\pi\)
−0.556143 + 0.831087i \(0.687719\pi\)
\(332\) 0 0
\(333\) −1.83262 0.606642i −1.83262 0.606642i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.106650 0.836802i 0.106650 0.836802i −0.846095 0.533032i \(-0.821053\pi\)
0.952745 0.303771i \(-0.0982456\pi\)
\(338\) −1.73043 + 0.961271i −1.73043 + 0.961271i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.999453 0.0330634i −0.999453 0.0330634i
\(344\) 5.34746 + 2.81814i 5.34746 + 2.81814i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.286392 + 0.398788i 0.286392 + 0.398788i 0.930586 0.366074i \(-0.119298\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 0 0 0.775409 0.631460i \(-0.217544\pi\)
−0.775409 + 0.631460i \(0.782456\pi\)
\(350\) −1.87225 + 0.642743i −1.87225 + 0.642743i
\(351\) 0 0
\(352\) 5.90415 + 6.13645i 5.90415 + 6.13645i
\(353\) 0 0 0.0825793 0.996584i \(-0.473684\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.605558 3.50965i −0.605558 3.50965i
\(359\) −1.94618 + 0.0429130i −1.94618 + 0.0429130i −0.978148 0.207912i \(-0.933333\pi\)
−0.968033 + 0.250825i \(0.919298\pi\)
\(360\) 0 0
\(361\) 0.761300 + 0.648400i 0.761300 + 0.648400i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.0935596 0.995614i \(-0.529825\pi\)
0.0935596 + 0.995614i \(0.470175\pi\)
\(368\) 0.925439 0.704056i 0.925439 0.704056i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.11300 0.198361i 1.11300 0.198361i
\(372\) 0 0
\(373\) 0.0204562 0.742113i 0.0204562 0.742113i −0.917973 0.396642i \(-0.870175\pi\)
0.938430 0.345471i \(-0.112281\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.830966 1.40332i 0.830966 1.40332i −0.0825793 0.996584i \(-0.526316\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.12989 + 0.892563i −2.12989 + 0.892563i
\(383\) 0 0 0.851919 0.523673i \(-0.175439\pi\)
−0.851919 + 0.523673i \(0.824561\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.06247 1.12891i −1.06247 1.12891i
\(387\) −0.491870 + 1.51382i −0.491870 + 1.51382i
\(388\) 0 0
\(389\) 0.00986267 0.00492300i 0.00986267 0.00492300i −0.441671 0.897177i \(-0.645614\pi\)
0.451533 + 0.892254i \(0.350877\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.66296 2.70736i −2.66296 2.70736i
\(393\) 0 0
\(394\) −2.75800 2.74284i −2.75800 2.74284i
\(395\) 0 0
\(396\) −2.77403 + 3.77420i −2.77403 + 3.77420i
\(397\) 0 0 −0.874174 0.485613i \(-0.838596\pi\)
0.874174 + 0.485613i \(0.161404\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −4.18030 1.91668i −4.18030 1.91668i
\(401\) −1.25411 + 0.209273i −1.25411 + 0.209273i −0.754107 0.656752i \(-0.771930\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) −3.57687 + 0.157816i −3.57687 + 0.157816i
\(407\) −1.24456 2.83732i −1.24456 2.83732i
\(408\) 0 0
\(409\) 0 0 −0.724425 0.689353i \(-0.757895\pi\)
0.724425 + 0.689353i \(0.242105\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0.366373 + 0.341020i 0.366373 + 0.341020i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.565270 0.824906i \(-0.308772\pi\)
−0.565270 + 0.824906i \(0.691228\pi\)
\(420\) 0 0
\(421\) 1.92132 + 0.430577i 1.92132 + 0.430577i 0.999028 + 0.0440782i \(0.0140351\pi\)
0.922290 + 0.386499i \(0.126316\pi\)
\(422\) 0.239352 0.0158536i 0.239352 0.0158536i
\(423\) 0 0
\(424\) 3.58116 + 2.36795i 3.58116 + 2.36795i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0.236353 0.631270i 0.236353 0.631270i
\(429\) 0 0
\(430\) 0 0
\(431\) 1.06339 0.326178i 1.06339 0.326178i 0.287976 0.957638i \(-0.407018\pi\)
0.775409 + 0.631460i \(0.217544\pi\)
\(432\) 0 0
\(433\) 0 0 0.652586 0.757715i \(-0.273684\pi\)
−0.652586 + 0.757715i \(0.726316\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.42227 4.15543i 3.42227 4.15543i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.828009 0.560715i \(-0.810526\pi\)
0.828009 + 0.560715i \(0.189474\pi\)
\(440\) 0 0
\(441\) 0.583317 0.812244i 0.583317 0.812244i
\(442\) 0 0
\(443\) −0.269579 0.532702i −0.269579 0.532702i 0.716783 0.697297i \(-0.245614\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.0976152 5.90312i −0.0976152 5.90312i
\(449\) −0.919097 + 1.76755i −0.919097 + 1.76755i −0.381410 + 0.924406i \(0.624561\pi\)
−0.537687 + 0.843145i \(0.680702\pi\)
\(450\) 0.507061 1.91345i 0.507061 1.91345i
\(451\) 0 0
\(452\) 1.25443 0.369701i 1.25443 0.369701i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.05387 + 0.680268i −1.05387 + 0.680268i −0.949339 0.314254i \(-0.898246\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 0.876642 0.216861i 0.876642 0.216861i 0.224056 0.974576i \(-0.428070\pi\)
0.652586 + 0.757715i \(0.273684\pi\)
\(464\) −6.27254 5.46276i −6.27254 5.46276i
\(465\) 0 0
\(466\) −0.840301 0.0836393i −0.840301 0.0836393i
\(467\) 0 0 0.952745 0.303771i \(-0.0982456\pi\)
−0.952745 + 0.303771i \(0.901754\pi\)
\(468\) 0 0
\(469\) 1.05877 + 1.25709i 1.05877 + 1.25709i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.33382 + 1.03908i −2.33382 + 1.03908i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.454133 + 1.03532i −0.454133 + 1.03532i
\(478\) −1.32392 + 3.42073i −1.32392 + 3.42073i
\(479\) 0 0 −0.287976 0.957638i \(-0.592982\pi\)
0.287976 + 0.957638i \(0.407018\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −4.57668 + 0.455539i −4.57668 + 0.455539i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.47566 1.34349i −1.47566 1.34349i −0.782322 0.622874i \(-0.785965\pi\)
−0.693336 0.720615i \(-0.743860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.84997 0.266932i 1.84997 0.266932i 0.874174 0.485613i \(-0.161404\pi\)
0.975796 + 0.218681i \(0.0701754\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.239087 0.271493i −0.239087 0.271493i
\(498\) 0 0
\(499\) 0.422290 1.25252i 0.422290 1.25252i −0.500000 0.866025i \(-0.666667\pi\)
0.922290 0.386499i \(-0.126316\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.991264 0.131892i \(-0.0421053\pi\)
−0.991264 + 0.131892i \(0.957895\pi\)
\(504\) 3.73103 0.707471i 3.73103 0.707471i
\(505\) 0 0
\(506\) −0.0132822 + 0.803219i −0.0132822 + 0.803219i
\(507\) 0 0
\(508\) 2.44979 + 5.26622i 2.44979 + 5.26622i
\(509\) 0 0 −0.592235 0.805765i \(-0.701754\pi\)
0.592235 + 0.805765i \(0.298246\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 4.69756 5.10291i 4.69756 5.10291i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −1.26066 + 3.60732i −1.26066 + 3.60732i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.180881 0.983505i \(-0.442105\pi\)
−0.180881 + 0.983505i \(0.557895\pi\)
\(522\) 1.85809 3.06046i 1.85809 3.06046i
\(523\) 0 0 −0.627176 0.778877i \(-0.715789\pi\)
0.627176 + 0.778877i \(0.284211\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.60891 + 1.98481i 2.60891 + 1.98481i
\(527\) 0 0
\(528\) 0 0
\(529\) −0.927888 0.123460i −0.927888 0.123460i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.378168 + 6.22996i −0.378168 + 6.22996i
\(537\) 0 0
\(538\) 0 0
\(539\) 1.59524 0.176561i 1.59524 0.176561i
\(540\) 0 0
\(541\) 1.86547 0.686748i 1.86547 0.686748i 0.894729 0.446609i \(-0.147368\pi\)
0.970739 0.240139i \(-0.0771930\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.55203 0.457408i −1.55203 0.457408i −0.609854 0.792514i \(-0.708772\pi\)
−0.942181 + 0.335105i \(0.891228\pi\)
\(548\) 0.544898 + 0.0360916i 0.544898 + 0.0360916i
\(549\) 0 0
\(550\) 2.82679 1.45015i 2.82679 1.45015i
\(551\) 0 0
\(552\) 0 0
\(553\) 1.37196 + 1.11726i 1.37196 + 1.11726i
\(554\) 0.750518 3.26453i 0.750518 3.26453i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.187074 1.99074i 0.187074 1.99074i 0.0715891 0.997434i \(-0.477193\pi\)
0.115485 0.993309i \(-0.463158\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −3.08800 0.238750i −3.08800 0.238750i
\(563\) 0 0 0.245485 0.969400i \(-0.421053\pi\)
−0.245485 + 0.969400i \(0.578947\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.350638 + 0.936511i 0.350638 + 0.936511i
\(568\) 0.0529891 1.37277i 0.0529891 1.37277i
\(569\) 1.42787 + 0.990062i 1.42787 + 0.990062i 0.996114 + 0.0880708i \(0.0280702\pi\)
0.431754 + 0.901991i \(0.357895\pi\)
\(570\) 0 0
\(571\) −0.754107 + 0.656752i −0.754107 + 0.656752i
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.0886600 0.236800i −0.0886600 0.236800i
\(576\) 4.96029 + 3.20185i 4.96029 + 3.20185i
\(577\) 0 0 0.601081 0.799188i \(-0.294737\pi\)
−0.601081 + 0.799188i \(0.705263\pi\)
\(578\) −0.970793 1.72510i −0.970793 1.72510i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.72257 + 0.570211i −1.72257 + 0.570211i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.754107 0.656752i \(-0.228070\pi\)
−0.754107 + 0.656752i \(0.771930\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −7.89879 + 4.05209i −7.89879 + 4.05209i
\(593\) 0 0 −0.0495838 0.998770i \(-0.515789\pi\)
0.0495838 + 0.998770i \(0.484211\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.460759 1.01972i −0.460759 1.01972i
\(597\) 0 0
\(598\) 0 0
\(599\) 0.491404 0.583452i 0.491404 0.583452i −0.461341 0.887223i \(-0.652632\pi\)
0.952745 + 0.303771i \(0.0982456\pi\)
\(600\) 0 0
\(601\) 0 0 0.938430 0.345471i \(-0.112281\pi\)
−0.938430 + 0.345471i \(0.887719\pi\)
\(602\) 2.98010 + 1.02307i 2.98010 + 1.02307i
\(603\) −1.63358 + 0.180805i −1.63358 + 0.180805i
\(604\) 4.34814 3.08646i 4.34814 3.08646i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.277403 0.960754i \(-0.589474\pi\)
0.277403 + 0.960754i \(0.410526\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.77383 + 0.236016i 1.77383 + 0.236016i 0.945817 0.324699i \(-0.105263\pi\)
0.828009 + 0.560715i \(0.189474\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 4.85072 + 3.69033i 4.85072 + 3.69033i
\(617\) −0.644267 0.512956i −0.644267 0.512956i 0.245485 0.969400i \(-0.421053\pi\)
−0.889752 + 0.456444i \(0.849123\pi\)
\(618\) 0 0
\(619\) 0 0 −0.627176 0.778877i \(-0.715789\pi\)
0.627176 + 0.778877i \(0.284211\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.660898 + 0.750475i −0.660898 + 0.750475i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.986456 0.255598i −0.986456 0.255598i −0.277403 0.960754i \(-0.589474\pi\)
−0.709053 + 0.705155i \(0.750877\pi\)
\(632\) 0.849470 + 6.66515i 0.849470 + 6.66515i
\(633\) 0 0
\(634\) 1.14443 + 2.46014i 1.14443 + 2.46014i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 5.64578 1.07054i 5.64578 1.07054i
\(639\) 0.358601 0.0477135i 0.358601 0.0477135i
\(640\) 0 0
\(641\) 0.886736 1.12641i 0.886736 1.12641i −0.104528 0.994522i \(-0.533333\pi\)
0.991264 0.131892i \(-0.0421053\pi\)
\(642\) 0 0
\(643\) 0 0 0.319482 0.947592i \(-0.396491\pi\)
−0.319482 + 0.947592i \(0.603509\pi\)
\(644\) 0.511635 0.531765i 0.511635 0.531765i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.00551154 0.999985i \(-0.501754\pi\)
0.00551154 + 0.999985i \(0.498246\pi\)
\(648\) −1.48702 + 3.49426i −1.48702 + 3.49426i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.17810 0.649949i −4.17810 0.649949i
\(653\) 0.345057 + 0.212106i 0.345057 + 0.212106i 0.685350 0.728214i \(-0.259649\pi\)
−0.340293 + 0.940319i \(0.610526\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.21337 0.524280i 1.21337 0.524280i 0.309017 0.951057i \(-0.400000\pi\)
0.904357 + 0.426776i \(0.140351\pi\)
\(660\) 0 0
\(661\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(662\) −1.46260 + 2.66743i −1.46260 + 2.66743i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −2.29689 3.05391i −2.29689 3.05391i
\(667\) −0.0327405 0.456166i −0.0327405 0.456166i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.0355007 0.0421506i −0.0355007 0.0421506i 0.746821 0.665025i \(-0.231579\pi\)
−0.782322 + 0.622874i \(0.785965\pi\)
\(674\) 1.11735 1.24094i 1.11735 1.24094i
\(675\) 0 0
\(676\) −2.90407 0.289056i −2.90407 0.289056i
\(677\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.81326 + 0.644922i 1.81326 + 0.644922i 0.997814 + 0.0660906i \(0.0210526\pi\)
0.815447 + 0.578832i \(0.196491\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.58853 1.18110i −1.58853 1.18110i
\(687\) 0 0
\(688\) 3.37700 + 6.49444i 3.37700 + 6.49444i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.652586 0.757715i \(-0.726316\pi\)
0.652586 + 0.757715i \(0.273684\pi\)
\(692\) 0 0
\(693\) −0.740442 + 1.42397i −0.740442 + 1.42397i
\(694\) 0.0160690 + 0.971744i 0.0160690 + 0.971744i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −2.80830 0.794112i −2.80830 0.794112i
\(701\) −1.01353 + 1.41130i −1.01353 + 1.41130i −0.104528 + 0.994522i \(0.533333\pi\)
−0.909007 + 0.416782i \(0.863158\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.50811 + 9.35489i 1.50811 + 9.35489i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.23905 + 1.43866i −1.23905 + 1.43866i −0.381410 + 0.924406i \(0.624561\pi\)
−0.857640 + 0.514250i \(0.828070\pi\)
\(710\) 0 0
\(711\) −1.69155 + 0.518857i −1.69155 + 0.518857i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 2.21472 4.76089i 2.21472 4.76089i
\(717\) 0 0
\(718\) −3.21427 2.12536i −3.21427 2.12536i
\(719\) 0 0 −0.115485 0.993309i \(-0.536842\pi\)
0.115485 + 0.993309i \(0.463158\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.507061 + 1.91345i 0.507061 + 1.91345i
\(723\) 0 0
\(724\) 0 0
\(725\) −1.53034 + 0.964102i −1.53034 + 0.964102i
\(726\) 0 0
\(727\) 0 0 −0.768401 0.639969i \(-0.778947\pi\)
0.768401 + 0.639969i \(0.221053\pi\)
\(728\) 0 0
\(729\) −0.973327 0.229424i −0.973327 0.229424i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.724425 0.689353i \(-0.757895\pi\)
0.724425 + 0.689353i \(0.242105\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.34027 0.0591341i 1.34027 0.0591341i
\(737\) −1.97002 1.75425i −1.97002 1.75425i
\(738\) 0 0
\(739\) 0.0615498 + 0.267723i 0.0615498 + 0.267723i 0.996114 0.0880708i \(-0.0280702\pi\)
−0.934564 + 0.355794i \(0.884211\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.03427 + 0.932718i 2.03427 + 0.932718i
\(743\) 0.294350 0.0358672i 0.294350 0.0358672i 0.0275543 0.999620i \(-0.491228\pi\)
0.266796 + 0.963753i \(0.414035\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.870331 1.18413i 0.870331 1.18413i
\(747\) 0 0
\(748\) 0 0
\(749\) 0.0467753 0.226183i 0.0467753 0.226183i
\(750\) 0 0
\(751\) 0.454508 1.51142i 0.454508 1.51142i −0.360939 0.932589i \(-0.617544\pi\)
0.815447 0.578832i \(-0.196491\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.65511 + 1.12082i −1.65511 + 1.12082i −0.809017 + 0.587785i \(0.800000\pi\)
−0.846095 + 0.533032i \(0.821053\pi\)
\(758\) 2.93460 1.34552i 2.93460 1.34552i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.0385714 0.999256i \(-0.512281\pi\)
0.0385714 + 0.999256i \(0.487719\pi\)
\(762\) 0 0
\(763\) 0.939842 1.58719i 0.939842 1.58719i
\(764\) −3.33033 0.707883i −3.33033 0.707883i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 0.0275543 0.999620i \(-0.491228\pi\)
−0.0275543 + 0.999620i \(0.508772\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.338867 2.26031i −0.338867 2.26031i
\(773\) 0 0 −0.999757 0.0220445i \(-0.992982\pi\)
0.999757 + 0.0220445i \(0.00701754\pi\)
\(774\) −2.50762 + 1.90775i −2.50762 + 1.90775i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0.0215965 + 0.00311616i 0.0215965 + 0.00311616i
\(779\) 0 0
\(780\) 0 0
\(781\) 0.442025 + 0.376473i 0.442025 + 0.376473i
\(782\) 0 0
\(783\) 0 0
\(784\) −0.781920 4.53180i −0.781920 4.53180i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.934564 0.355794i \(-0.884211\pi\)
0.934564 + 0.355794i \(0.115789\pi\)
\(788\) −1.03729 5.64006i −1.03729 5.64006i
\(789\) 0 0
\(790\) 0 0
\(791\) 0.405253 0.191244i 0.405253 0.191244i
\(792\) −5.76468 + 1.97902i −5.76468 + 1.97902i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.627176 0.778877i \(-0.284211\pi\)
−0.627176 + 0.778877i \(0.715789\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.70335 4.56537i −2.70335 4.56537i
\(801\) 0 0
\(802\) −2.27611 1.07412i −2.27611 1.07412i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.445781 + 0.147564i 0.445781 + 0.147564i 0.528360 0.849020i \(-0.322807\pi\)
−0.0825793 + 0.996584i \(0.526316\pi\)
\(810\) 0 0
\(811\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(812\) −4.46619 2.81366i −4.46619 2.81366i
\(813\) 0 0
\(814\) 1.17577 6.01928i 1.17577 6.01928i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.36537 1.45077i 1.36537 1.45077i 0.618553 0.785743i \(-0.287719\pi\)
0.746821 0.665025i \(-0.231579\pi\)
\(822\) 0 0
\(823\) −0.779976 + 0.792980i −0.779976 + 0.792980i −0.982493 0.186298i \(-0.940351\pi\)
0.202517 + 0.979279i \(0.435088\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.0906279 0.653561i 0.0906279 0.653561i −0.889752 0.456444i \(-0.849123\pi\)
0.980380 0.197117i \(-0.0631579\pi\)
\(828\) 0.157401 + 0.720951i 0.157401 + 0.720951i
\(829\) 0 0 −0.635724 0.771917i \(-0.719298\pi\)
0.635724 + 0.771917i \(0.280702\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.518970 0.854793i \(-0.673684\pi\)
0.518970 + 0.854793i \(0.326316\pi\)
\(840\) 0 0
\(841\) −2.18574 + 0.618069i −2.18574 + 0.618069i
\(842\) 2.63976 + 2.86755i 2.63976 + 2.86755i
\(843\) 0 0
\(844\) 0.303308 + 0.181867i 0.303308 + 0.181867i
\(845\) 0 0
\(846\) 0 0
\(847\) −1.52557 + 0.395288i −1.52557 + 0.395288i
\(848\) 1.92988 + 4.82763i 1.92988 + 4.82763i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.465047 0.148274i −0.465047 0.148274i
\(852\) 0 0
\(853\) 0 0 −0.899598 0.436719i \(-0.856140\pi\)
0.899598 + 0.436719i \(0.143860\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0.720789 0.499784i 0.720789 0.499784i
\(857\) 0 0 0.821778 0.569808i \(-0.192982\pi\)
−0.821778 + 0.569808i \(0.807018\pi\)
\(858\) 0 0
\(859\) 0 0 0.999453 0.0330634i \(-0.0105263\pi\)
−0.999453 + 0.0330634i \(0.989474\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.09773 + 0.668833i 2.09773 + 0.668833i
\(863\) −1.70851 0.738219i −1.70851 0.738219i −0.999453 0.0330634i \(-0.989474\pi\)
−0.709053 0.705155i \(-0.750877\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.43547 1.46034i −2.43547 1.46034i
\(870\) 0 0
\(871\) 0 0
\(872\) 6.74051 1.90603i 6.74051 1.90603i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.55676 + 1.21167i −1.55676 + 1.21167i −0.677282 + 0.735724i \(0.736842\pi\)
−0.879474 + 0.475947i \(0.842105\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.0605901 0.998163i \(-0.519298\pi\)
0.0605901 + 0.998163i \(0.480702\pi\)
\(882\) 1.86505 0.663341i 1.86505 0.663341i
\(883\) −0.0557755 0.201479i −0.0557755 0.201479i 0.930586 0.366074i \(-0.119298\pi\)
−0.986361 + 0.164595i \(0.947368\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.162327 1.17062i 0.162327 1.17062i
\(887\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(888\) 0 0
\(889\) 1.10682 + 1.65400i 1.10682 + 1.65400i
\(890\) 0 0
\(891\) −0.771653 1.40730i −0.771653 1.40730i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 3.54880 5.30325i 3.54880 5.30325i
\(897\) 0 0
\(898\) −3.48879 + 1.83861i −3.48879 + 1.83861i
\(899\) 0 0
\(900\) 2.22179 1.89230i 2.22179 1.89230i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 1.61550 + 0.534769i 1.61550 + 0.534769i
\(905\) 0 0
\(906\) 0 0
\(907\) −0.741267 + 1.85429i −0.741267 + 1.85429i −0.319482 + 0.947592i \(0.603509\pi\)
−0.421786 + 0.906696i \(0.638596\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.76047 0.830786i −1.76047 0.830786i −0.978148 0.207912i \(-0.933333\pi\)
−0.782322 0.622874i \(-0.785965\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.48163 0.0820961i −2.48163 0.0820961i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.300595 + 0.610607i 0.300595 + 0.610607i 0.993931 0.110008i \(-0.0350877\pi\)
−0.693336 + 0.720615i \(0.743860\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.349175 + 1.89858i 0.349175 + 1.89858i
\(926\) 1.67065 + 0.636024i 1.67065 + 0.636024i
\(927\) 0 0
\(928\) −2.35581 9.30289i −2.35581 9.30289i
\(929\) 0 0 −0.170028 0.985439i \(-0.554386\pi\)
0.170028 + 0.985439i \(0.445614\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.947816 0.807255i −0.947816 0.807255i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.868768 0.495219i \(-0.835088\pi\)
0.868768 + 0.495219i \(0.164912\pi\)
\(938\) 0.304388 + 3.23915i 0.304388 + 3.23915i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.148264 0.988948i \(-0.547368\pi\)
0.148264 + 0.988948i \(0.452632\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −4.96847 0.942111i −4.96847 0.942111i
\(947\) 1.77680 + 0.0980289i 1.77680 + 0.0980289i 0.913545 0.406737i \(-0.133333\pi\)
0.863256 + 0.504766i \(0.168421\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.09243 + 0.457797i −1.09243 + 0.457797i −0.857640 0.514250i \(-0.828070\pi\)
−0.234785 + 0.972047i \(0.575439\pi\)
\(954\) −1.90651 + 1.17193i −1.90651 + 1.17193i
\(955\) 0 0
\(956\) −4.47770 + 3.03223i −4.47770 + 3.03223i
\(957\) 0 0
\(958\) 0 0
\(959\) 0.186392 0.0164797i 0.186392 0.0164797i
\(960\) 0 0
\(961\) 0.789141 + 0.614213i 0.789141 + 0.614213i
\(962\) 0 0
\(963\) 0.161964 + 0.164665i 0.161964 + 0.164665i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.792565 + 1.07832i −0.792565 + 1.07832i 0.202517 + 0.979279i \(0.435088\pi\)
−0.995083 + 0.0990455i \(0.968421\pi\)
\(968\) −5.23167 2.90625i −5.23167 2.90625i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.909007 0.416782i \(-0.863158\pi\)
0.909007 + 0.416782i \(0.136842\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −0.885098 3.84991i −0.885098 3.84991i
\(975\) 0 0
\(976\) 0 0
\(977\) −1.03693 + 0.0457505i −1.03693 + 0.0457505i −0.556143 0.831087i \(-0.687719\pi\)
−0.480787 + 0.876837i \(0.659649\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.796405 + 1.66379i 0.796405 + 1.66379i
\(982\) 3.31045 + 1.65243i 3.31045 + 1.65243i
\(983\) 0 0 −0.930586 0.366074i \(-0.880702\pi\)
0.930586 + 0.366074i \(0.119298\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.120144 + 0.384123i −0.120144 + 0.384123i
\(990\) 0 0
\(991\) −0.499913 1.88648i −0.499913 1.88648i −0.461341 0.887223i \(-0.652632\pi\)
−0.0385714 0.999256i \(-0.512281\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.0826992 0.711315i −0.0826992 0.711315i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.421786 0.906696i \(-0.361404\pi\)
−0.421786 + 0.906696i \(0.638596\pi\)
\(998\) 2.09971 1.56118i 2.09971 1.56118i
\(999\) 0 0
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 3997.1.cz.a.97.1 144
7.6 odd 2 CM 3997.1.cz.a.97.1 144
571.312 even 285 inner 3997.1.cz.a.2596.1 yes 144
3997.2596 odd 570 inner 3997.1.cz.a.2596.1 yes 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3997.1.cz.a.97.1 144 1.1 even 1 trivial
3997.1.cz.a.97.1 144 7.6 odd 2 CM
3997.1.cz.a.2596.1 yes 144 571.312 even 285 inner
3997.1.cz.a.2596.1 yes 144 3997.2596 odd 570 inner