Properties

Label 1350.2.f.a.593.1
Level $1350$
Weight $2$
Character 1350.593
Analytic conductor $10.780$
Analytic rank $0$
Dimension $8$
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] = [1350,2,Mod(107,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([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("1350.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.1
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1350.593
Dual form 1350.2.f.a.107.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{2} +1.00000i q^{4} +(-1.36603 + 1.36603i) q^{7} +(0.707107 - 0.707107i) q^{8} +O(q^{10})\) \(q+(-0.707107 - 0.707107i) q^{2} +1.00000i q^{4} +(-1.36603 + 1.36603i) q^{7} +(0.707107 - 0.707107i) q^{8} +2.31079i q^{11} +(3.46410 + 3.46410i) q^{13} +1.93185 q^{14} -1.00000 q^{16} +(-3.86370 - 3.86370i) q^{17} -0.535898i q^{19} +(1.63397 - 1.63397i) q^{22} +(1.41421 - 1.41421i) q^{23} -4.89898i q^{26} +(-1.36603 - 1.36603i) q^{28} -3.48477 q^{29} -10.6603 q^{31} +(0.707107 + 0.707107i) q^{32} +5.46410i q^{34} +(-8.19615 + 8.19615i) q^{37} +(-0.378937 + 0.378937i) q^{38} -9.52056i q^{41} +(4.73205 + 4.73205i) q^{43} -2.31079 q^{44} -2.00000 q^{46} +(4.24264 + 4.24264i) q^{47} +3.26795i q^{49} +(-3.46410 + 3.46410i) q^{52} +(2.26002 - 2.26002i) q^{53} +1.93185i q^{56} +(2.46410 + 2.46410i) q^{58} -10.1769 q^{59} +2.53590 q^{61} +(7.53794 + 7.53794i) q^{62} -1.00000i q^{64} +(-4.19615 + 4.19615i) q^{67} +(3.86370 - 3.86370i) q^{68} +4.14110i q^{71} +(-7.63397 - 7.63397i) q^{73} +11.5911 q^{74} +0.535898 q^{76} +(-3.15660 - 3.15660i) q^{77} +3.46410i q^{79} +(-6.73205 + 6.73205i) q^{82} +(-11.2629 + 11.2629i) q^{83} -6.69213i q^{86} +(1.63397 + 1.63397i) q^{88} -8.76268 q^{89} -9.46410 q^{91} +(1.41421 + 1.41421i) q^{92} -6.00000i q^{94} +(1.43782 - 1.43782i) q^{97} +(2.31079 - 2.31079i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 8 q^{16} + 20 q^{22} - 4 q^{28} - 16 q^{31} - 24 q^{37} + 24 q^{43} - 16 q^{46} - 8 q^{58} + 48 q^{61} + 8 q^{67} - 68 q^{73} + 32 q^{76} - 40 q^{82} + 20 q^{88} - 48 q^{91} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 0.707107i −0.500000 0.500000i
\(3\) 0 0
\(4\) 1.00000i 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) −1.36603 + 1.36603i −0.516309 + 0.516309i −0.916453 0.400143i \(-0.868960\pi\)
0.400143 + 0.916453i \(0.368960\pi\)
\(8\) 0.707107 0.707107i 0.250000 0.250000i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.31079i 0.696729i 0.937359 + 0.348365i \(0.113263\pi\)
−0.937359 + 0.348365i \(0.886737\pi\)
\(12\) 0 0
\(13\) 3.46410 + 3.46410i 0.960769 + 0.960769i 0.999259 0.0384901i \(-0.0122548\pi\)
−0.0384901 + 0.999259i \(0.512255\pi\)
\(14\) 1.93185 0.516309
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −3.86370 3.86370i −0.937086 0.937086i 0.0610491 0.998135i \(-0.480555\pi\)
−0.998135 + 0.0610491i \(0.980555\pi\)
\(18\) 0 0
\(19\) 0.535898i 0.122944i −0.998109 0.0614718i \(-0.980421\pi\)
0.998109 0.0614718i \(-0.0195794\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.63397 1.63397i 0.348365 0.348365i
\(23\) 1.41421 1.41421i 0.294884 0.294884i −0.544122 0.839006i \(-0.683137\pi\)
0.839006 + 0.544122i \(0.183137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.89898i 0.960769i
\(27\) 0 0
\(28\) −1.36603 1.36603i −0.258155 0.258155i
\(29\) −3.48477 −0.647105 −0.323552 0.946210i \(-0.604877\pi\)
−0.323552 + 0.946210i \(0.604877\pi\)
\(30\) 0 0
\(31\) −10.6603 −1.91464 −0.957319 0.289033i \(-0.906666\pi\)
−0.957319 + 0.289033i \(0.906666\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) 0 0
\(34\) 5.46410i 0.937086i
\(35\) 0 0
\(36\) 0 0
\(37\) −8.19615 + 8.19615i −1.34744 + 1.34744i −0.459006 + 0.888433i \(0.651794\pi\)
−0.888433 + 0.459006i \(0.848206\pi\)
\(38\) −0.378937 + 0.378937i −0.0614718 + 0.0614718i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.52056i 1.48686i −0.668813 0.743431i \(-0.733197\pi\)
0.668813 0.743431i \(-0.266803\pi\)
\(42\) 0 0
\(43\) 4.73205 + 4.73205i 0.721631 + 0.721631i 0.968937 0.247306i \(-0.0795454\pi\)
−0.247306 + 0.968937i \(0.579545\pi\)
\(44\) −2.31079 −0.348365
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 4.24264 + 4.24264i 0.618853 + 0.618853i 0.945237 0.326384i \(-0.105830\pi\)
−0.326384 + 0.945237i \(0.605830\pi\)
\(48\) 0 0
\(49\) 3.26795i 0.466850i
\(50\) 0 0
\(51\) 0 0
\(52\) −3.46410 + 3.46410i −0.480384 + 0.480384i
\(53\) 2.26002 2.26002i 0.310438 0.310438i −0.534641 0.845079i \(-0.679553\pi\)
0.845079 + 0.534641i \(0.179553\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.93185i 0.258155i
\(57\) 0 0
\(58\) 2.46410 + 2.46410i 0.323552 + 0.323552i
\(59\) −10.1769 −1.32492 −0.662460 0.749098i \(-0.730487\pi\)
−0.662460 + 0.749098i \(0.730487\pi\)
\(60\) 0 0
\(61\) 2.53590 0.324689 0.162344 0.986734i \(-0.448094\pi\)
0.162344 + 0.986734i \(0.448094\pi\)
\(62\) 7.53794 + 7.53794i 0.957319 + 0.957319i
\(63\) 0 0
\(64\) 1.00000i 0.125000i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.19615 + 4.19615i −0.512642 + 0.512642i −0.915335 0.402693i \(-0.868074\pi\)
0.402693 + 0.915335i \(0.368074\pi\)
\(68\) 3.86370 3.86370i 0.468543 0.468543i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.14110i 0.491459i 0.969338 + 0.245729i \(0.0790274\pi\)
−0.969338 + 0.245729i \(0.920973\pi\)
\(72\) 0 0
\(73\) −7.63397 7.63397i −0.893489 0.893489i 0.101361 0.994850i \(-0.467680\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 11.5911 1.34744
\(75\) 0 0
\(76\) 0.535898 0.0614718
\(77\) −3.15660 3.15660i −0.359728 0.359728i
\(78\) 0 0
\(79\) 3.46410i 0.389742i 0.980829 + 0.194871i \(0.0624288\pi\)
−0.980829 + 0.194871i \(0.937571\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.73205 + 6.73205i −0.743431 + 0.743431i
\(83\) −11.2629 + 11.2629i −1.23627 + 1.23627i −0.274754 + 0.961515i \(0.588596\pi\)
−0.961515 + 0.274754i \(0.911404\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.69213i 0.721631i
\(87\) 0 0
\(88\) 1.63397 + 1.63397i 0.174182 + 0.174182i
\(89\) −8.76268 −0.928843 −0.464421 0.885614i \(-0.653738\pi\)
−0.464421 + 0.885614i \(0.653738\pi\)
\(90\) 0 0
\(91\) −9.46410 −0.992107
\(92\) 1.41421 + 1.41421i 0.147442 + 0.147442i
\(93\) 0 0
\(94\) 6.00000i 0.618853i
\(95\) 0 0
\(96\) 0 0
\(97\) 1.43782 1.43782i 0.145989 0.145989i −0.630335 0.776323i \(-0.717082\pi\)
0.776323 + 0.630335i \(0.217082\pi\)
\(98\) 2.31079 2.31079i 0.233425 0.233425i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.2103i 1.21497i 0.794332 + 0.607484i \(0.207821\pi\)
−0.794332 + 0.607484i \(0.792179\pi\)
\(102\) 0 0
\(103\) 6.26795 + 6.26795i 0.617599 + 0.617599i 0.944915 0.327316i \(-0.106144\pi\)
−0.327316 + 0.944915i \(0.606144\pi\)
\(104\) 4.89898 0.480384
\(105\) 0 0
\(106\) −3.19615 −0.310438
\(107\) −8.43451 8.43451i −0.815395 0.815395i 0.170042 0.985437i \(-0.445610\pi\)
−0.985437 + 0.170042i \(0.945610\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.36603 1.36603i 0.129077 0.129077i
\(113\) 2.17209 2.17209i 0.204333 0.204333i −0.597521 0.801854i \(-0.703847\pi\)
0.801854 + 0.597521i \(0.203847\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.48477i 0.323552i
\(117\) 0 0
\(118\) 7.19615 + 7.19615i 0.662460 + 0.662460i
\(119\) 10.5558 0.967652
\(120\) 0 0
\(121\) 5.66025 0.514569
\(122\) −1.79315 1.79315i −0.162344 0.162344i
\(123\) 0 0
\(124\) 10.6603i 0.957319i
\(125\) 0 0
\(126\) 0 0
\(127\) 9.83013 9.83013i 0.872283 0.872283i −0.120438 0.992721i \(-0.538430\pi\)
0.992721 + 0.120438i \(0.0384299\pi\)
\(128\) −0.707107 + 0.707107i −0.0625000 + 0.0625000i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.7651i 1.11529i 0.830079 + 0.557645i \(0.188295\pi\)
−0.830079 + 0.557645i \(0.811705\pi\)
\(132\) 0 0
\(133\) 0.732051 + 0.732051i 0.0634769 + 0.0634769i
\(134\) 5.93426 0.512642
\(135\) 0 0
\(136\) −5.46410 −0.468543
\(137\) −2.44949 2.44949i −0.209274 0.209274i 0.594685 0.803959i \(-0.297277\pi\)
−0.803959 + 0.594685i \(0.797277\pi\)
\(138\) 0 0
\(139\) 17.8564i 1.51456i 0.653090 + 0.757280i \(0.273472\pi\)
−0.653090 + 0.757280i \(0.726528\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.92820 2.92820i 0.245729 0.245729i
\(143\) −8.00481 + 8.00481i −0.669396 + 0.669396i
\(144\) 0 0
\(145\) 0 0
\(146\) 10.7961i 0.893489i
\(147\) 0 0
\(148\) −8.19615 8.19615i −0.673720 0.673720i
\(149\) −0.240237 −0.0196810 −0.00984048 0.999952i \(-0.503132\pi\)
−0.00984048 + 0.999952i \(0.503132\pi\)
\(150\) 0 0
\(151\) −1.53590 −0.124990 −0.0624948 0.998045i \(-0.519906\pi\)
−0.0624948 + 0.998045i \(0.519906\pi\)
\(152\) −0.378937 0.378937i −0.0307359 0.0307359i
\(153\) 0 0
\(154\) 4.46410i 0.359728i
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0000 + 12.0000i −0.957704 + 0.957704i −0.999141 0.0414369i \(-0.986806\pi\)
0.0414369 + 0.999141i \(0.486806\pi\)
\(158\) 2.44949 2.44949i 0.194871 0.194871i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.86370i 0.304502i
\(162\) 0 0
\(163\) 1.46410 + 1.46410i 0.114677 + 0.114677i 0.762117 0.647440i \(-0.224160\pi\)
−0.647440 + 0.762117i \(0.724160\pi\)
\(164\) 9.52056 0.743431
\(165\) 0 0
\(166\) 15.9282 1.23627
\(167\) −2.17209 2.17209i −0.168081 0.168081i 0.618054 0.786135i \(-0.287921\pi\)
−0.786135 + 0.618054i \(0.787921\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) −4.73205 + 4.73205i −0.360815 + 0.360815i
\(173\) −2.50026 + 2.50026i −0.190091 + 0.190091i −0.795735 0.605644i \(-0.792915\pi\)
0.605644 + 0.795735i \(0.292915\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.31079i 0.174182i
\(177\) 0 0
\(178\) 6.19615 + 6.19615i 0.464421 + 0.464421i
\(179\) 6.83083 0.510560 0.255280 0.966867i \(-0.417832\pi\)
0.255280 + 0.966867i \(0.417832\pi\)
\(180\) 0 0
\(181\) 1.07180 0.0796660 0.0398330 0.999206i \(-0.487317\pi\)
0.0398330 + 0.999206i \(0.487317\pi\)
\(182\) 6.69213 + 6.69213i 0.496054 + 0.496054i
\(183\) 0 0
\(184\) 2.00000i 0.147442i
\(185\) 0 0
\(186\) 0 0
\(187\) 8.92820 8.92820i 0.652895 0.652895i
\(188\) −4.24264 + 4.24264i −0.309426 + 0.309426i
\(189\) 0 0
\(190\) 0 0
\(191\) 26.0106i 1.88206i 0.338317 + 0.941032i \(0.390142\pi\)
−0.338317 + 0.941032i \(0.609858\pi\)
\(192\) 0 0
\(193\) −11.0263 11.0263i −0.793689 0.793689i 0.188403 0.982092i \(-0.439669\pi\)
−0.982092 + 0.188403i \(0.939669\pi\)
\(194\) −2.03339 −0.145989
\(195\) 0 0
\(196\) −3.26795 −0.233425
\(197\) 5.46739 + 5.46739i 0.389535 + 0.389535i 0.874522 0.484987i \(-0.161175\pi\)
−0.484987 + 0.874522i \(0.661175\pi\)
\(198\) 0 0
\(199\) 20.3205i 1.44048i −0.693724 0.720241i \(-0.744031\pi\)
0.693724 0.720241i \(-0.255969\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.63397 8.63397i 0.607484 0.607484i
\(203\) 4.76028 4.76028i 0.334106 0.334106i
\(204\) 0 0
\(205\) 0 0
\(206\) 8.86422i 0.617599i
\(207\) 0 0
\(208\) −3.46410 3.46410i −0.240192 0.240192i
\(209\) 1.23835 0.0856583
\(210\) 0 0
\(211\) 13.3205 0.917022 0.458511 0.888689i \(-0.348383\pi\)
0.458511 + 0.888689i \(0.348383\pi\)
\(212\) 2.26002 + 2.26002i 0.155219 + 0.155219i
\(213\) 0 0
\(214\) 11.9282i 0.815395i
\(215\) 0 0
\(216\) 0 0
\(217\) 14.5622 14.5622i 0.988545 0.988545i
\(218\) −1.41421 + 1.41421i −0.0957826 + 0.0957826i
\(219\) 0 0
\(220\) 0 0
\(221\) 26.7685i 1.80065i
\(222\) 0 0
\(223\) −7.73205 7.73205i −0.517776 0.517776i 0.399122 0.916898i \(-0.369315\pi\)
−0.916898 + 0.399122i \(0.869315\pi\)
\(224\) −1.93185 −0.129077
\(225\) 0 0
\(226\) −3.07180 −0.204333
\(227\) 8.10634 + 8.10634i 0.538037 + 0.538037i 0.922952 0.384915i \(-0.125769\pi\)
−0.384915 + 0.922952i \(0.625769\pi\)
\(228\) 0 0
\(229\) 22.3923i 1.47973i −0.672758 0.739863i \(-0.734890\pi\)
0.672758 0.739863i \(-0.265110\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.46410 + 2.46410i −0.161776 + 0.161776i
\(233\) 14.0406 14.0406i 0.919830 0.919830i −0.0771864 0.997017i \(-0.524594\pi\)
0.997017 + 0.0771864i \(0.0245937\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.1769i 0.662460i
\(237\) 0 0
\(238\) −7.46410 7.46410i −0.483826 0.483826i
\(239\) −20.0764 −1.29863 −0.649317 0.760518i \(-0.724945\pi\)
−0.649317 + 0.760518i \(0.724945\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −4.00240 4.00240i −0.257284 0.257284i
\(243\) 0 0
\(244\) 2.53590i 0.162344i
\(245\) 0 0
\(246\) 0 0
\(247\) 1.85641 1.85641i 0.118120 0.118120i
\(248\) −7.53794 + 7.53794i −0.478660 + 0.478660i
\(249\) 0 0
\(250\) 0 0
\(251\) 19.4201i 1.22578i −0.790167 0.612891i \(-0.790006\pi\)
0.790167 0.612891i \(-0.209994\pi\)
\(252\) 0 0
\(253\) 3.26795 + 3.26795i 0.205454 + 0.205454i
\(254\) −13.9019 −0.872283
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.04008 9.04008i −0.563905 0.563905i 0.366509 0.930414i \(-0.380553\pi\)
−0.930414 + 0.366509i \(0.880553\pi\)
\(258\) 0 0
\(259\) 22.3923i 1.39139i
\(260\) 0 0
\(261\) 0 0
\(262\) 9.02628 9.02628i 0.557645 0.557645i
\(263\) −9.89949 + 9.89949i −0.610429 + 0.610429i −0.943058 0.332629i \(-0.892064\pi\)
0.332629 + 0.943058i \(0.392064\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.03528i 0.0634769i
\(267\) 0 0
\(268\) −4.19615 4.19615i −0.256321 0.256321i
\(269\) −21.7680 −1.32722 −0.663609 0.748079i \(-0.730976\pi\)
−0.663609 + 0.748079i \(0.730976\pi\)
\(270\) 0 0
\(271\) 25.3923 1.54247 0.771236 0.636549i \(-0.219639\pi\)
0.771236 + 0.636549i \(0.219639\pi\)
\(272\) 3.86370 + 3.86370i 0.234271 + 0.234271i
\(273\) 0 0
\(274\) 3.46410i 0.209274i
\(275\) 0 0
\(276\) 0 0
\(277\) 7.85641 7.85641i 0.472046 0.472046i −0.430530 0.902576i \(-0.641673\pi\)
0.902576 + 0.430530i \(0.141673\pi\)
\(278\) 12.6264 12.6264i 0.757280 0.757280i
\(279\) 0 0
\(280\) 0 0
\(281\) 3.58630i 0.213941i 0.994262 + 0.106970i \(0.0341150\pi\)
−0.994262 + 0.106970i \(0.965885\pi\)
\(282\) 0 0
\(283\) 5.26795 + 5.26795i 0.313147 + 0.313147i 0.846128 0.532980i \(-0.178928\pi\)
−0.532980 + 0.846128i \(0.678928\pi\)
\(284\) −4.14110 −0.245729
\(285\) 0 0
\(286\) 11.3205 0.669396
\(287\) 13.0053 + 13.0053i 0.767680 + 0.767680i
\(288\) 0 0
\(289\) 12.8564i 0.756259i
\(290\) 0 0
\(291\) 0 0
\(292\) 7.63397 7.63397i 0.446745 0.446745i
\(293\) −8.48528 + 8.48528i −0.495715 + 0.495715i −0.910101 0.414386i \(-0.863996\pi\)
0.414386 + 0.910101i \(0.363996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 11.5911i 0.673720i
\(297\) 0 0
\(298\) 0.169873 + 0.169873i 0.00984048 + 0.00984048i
\(299\) 9.79796 0.566631
\(300\) 0 0
\(301\) −12.9282 −0.745169
\(302\) 1.08604 + 1.08604i 0.0624948 + 0.0624948i
\(303\) 0 0
\(304\) 0.535898i 0.0307359i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 3.15660 3.15660i 0.179864 0.179864i
\(309\) 0 0
\(310\) 0 0
\(311\) 1.03528i 0.0587051i 0.999569 + 0.0293526i \(0.00934455\pi\)
−0.999569 + 0.0293526i \(0.990655\pi\)
\(312\) 0 0
\(313\) 0.973721 + 0.973721i 0.0550379 + 0.0550379i 0.734090 0.679052i \(-0.237609\pi\)
−0.679052 + 0.734090i \(0.737609\pi\)
\(314\) 16.9706 0.957704
\(315\) 0 0
\(316\) −3.46410 −0.194871
\(317\) 3.91447 + 3.91447i 0.219859 + 0.219859i 0.808439 0.588580i \(-0.200313\pi\)
−0.588580 + 0.808439i \(0.700313\pi\)
\(318\) 0 0
\(319\) 8.05256i 0.450857i
\(320\) 0 0
\(321\) 0 0
\(322\) 2.73205 2.73205i 0.152251 0.152251i
\(323\) −2.07055 + 2.07055i −0.115209 + 0.115209i
\(324\) 0 0
\(325\) 0 0
\(326\) 2.07055i 0.114677i
\(327\) 0 0
\(328\) −6.73205 6.73205i −0.371715 0.371715i
\(329\) −11.5911 −0.639039
\(330\) 0 0
\(331\) 0.535898 0.0294556 0.0147278 0.999892i \(-0.495312\pi\)
0.0147278 + 0.999892i \(0.495312\pi\)
\(332\) −11.2629 11.2629i −0.618134 0.618134i
\(333\) 0 0
\(334\) 3.07180i 0.168081i
\(335\) 0 0
\(336\) 0 0
\(337\) 6.46410 6.46410i 0.352122 0.352122i −0.508777 0.860899i \(-0.669902\pi\)
0.860899 + 0.508777i \(0.169902\pi\)
\(338\) 7.77817 7.77817i 0.423077 0.423077i
\(339\) 0 0
\(340\) 0 0
\(341\) 24.6336i 1.33398i
\(342\) 0 0
\(343\) −14.0263 14.0263i −0.757348 0.757348i
\(344\) 6.69213 0.360815
\(345\) 0 0
\(346\) 3.53590 0.190091
\(347\) −8.01841 8.01841i −0.430451 0.430451i 0.458331 0.888782i \(-0.348448\pi\)
−0.888782 + 0.458331i \(0.848448\pi\)
\(348\) 0 0
\(349\) 26.3923i 1.41275i −0.707839 0.706374i \(-0.750330\pi\)
0.707839 0.706374i \(-0.249670\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.63397 + 1.63397i −0.0870911 + 0.0870911i
\(353\) 5.55532 5.55532i 0.295680 0.295680i −0.543639 0.839319i \(-0.682954\pi\)
0.839319 + 0.543639i \(0.182954\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8.76268i 0.464421i
\(357\) 0 0
\(358\) −4.83013 4.83013i −0.255280 0.255280i
\(359\) 20.6312 1.08887 0.544436 0.838802i \(-0.316744\pi\)
0.544436 + 0.838802i \(0.316744\pi\)
\(360\) 0 0
\(361\) 18.7128 0.984885
\(362\) −0.757875 0.757875i −0.0398330 0.0398330i
\(363\) 0 0
\(364\) 9.46410i 0.496054i
\(365\) 0 0
\(366\) 0 0
\(367\) −16.0263 + 16.0263i −0.836565 + 0.836565i −0.988405 0.151840i \(-0.951480\pi\)
0.151840 + 0.988405i \(0.451480\pi\)
\(368\) −1.41421 + 1.41421i −0.0737210 + 0.0737210i
\(369\) 0 0
\(370\) 0 0
\(371\) 6.17449i 0.320564i
\(372\) 0 0
\(373\) 12.3923 + 12.3923i 0.641649 + 0.641649i 0.950961 0.309312i \(-0.100099\pi\)
−0.309312 + 0.950961i \(0.600099\pi\)
\(374\) −12.6264 −0.652895
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −12.0716 12.0716i −0.621718 0.621718i
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 18.3923 18.3923i 0.941032 0.941032i
\(383\) 24.9754 24.9754i 1.27618 1.27618i 0.333394 0.942787i \(-0.391806\pi\)
0.942787 0.333394i \(-0.108194\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.5935i 0.793689i
\(387\) 0 0
\(388\) 1.43782 + 1.43782i 0.0729944 + 0.0729944i
\(389\) 8.90138 0.451318 0.225659 0.974206i \(-0.427546\pi\)
0.225659 + 0.974206i \(0.427546\pi\)
\(390\) 0 0
\(391\) −10.9282 −0.552663
\(392\) 2.31079 + 2.31079i 0.116712 + 0.116712i
\(393\) 0 0
\(394\) 7.73205i 0.389535i
\(395\) 0 0
\(396\) 0 0
\(397\) 8.92820 8.92820i 0.448094 0.448094i −0.446627 0.894720i \(-0.647375\pi\)
0.894720 + 0.446627i \(0.147375\pi\)
\(398\) −14.3688 + 14.3688i −0.720241 + 0.720241i
\(399\) 0 0
\(400\) 0 0
\(401\) 1.31268i 0.0655520i −0.999463 0.0327760i \(-0.989565\pi\)
0.999463 0.0327760i \(-0.0104348\pi\)
\(402\) 0 0
\(403\) −36.9282 36.9282i −1.83952 1.83952i
\(404\) −12.2103 −0.607484
\(405\) 0 0
\(406\) −6.73205 −0.334106
\(407\) −18.9396 18.9396i −0.938800 0.938800i
\(408\) 0 0
\(409\) 20.6603i 1.02158i 0.859704 + 0.510792i \(0.170648\pi\)
−0.859704 + 0.510792i \(0.829352\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.26795 + 6.26795i −0.308800 + 0.308800i
\(413\) 13.9019 13.9019i 0.684068 0.684068i
\(414\) 0 0
\(415\) 0 0
\(416\) 4.89898i 0.240192i
\(417\) 0 0
\(418\) −0.875644 0.875644i −0.0428292 0.0428292i
\(419\) 37.7033 1.84193 0.920963 0.389650i \(-0.127404\pi\)
0.920963 + 0.389650i \(0.127404\pi\)
\(420\) 0 0
\(421\) 9.85641 0.480372 0.240186 0.970727i \(-0.422792\pi\)
0.240186 + 0.970727i \(0.422792\pi\)
\(422\) −9.41902 9.41902i −0.458511 0.458511i
\(423\) 0 0
\(424\) 3.19615i 0.155219i
\(425\) 0 0
\(426\) 0 0
\(427\) −3.46410 + 3.46410i −0.167640 + 0.167640i
\(428\) 8.43451 8.43451i 0.407698 0.407698i
\(429\) 0 0
\(430\) 0 0
\(431\) 11.0363i 0.531600i 0.964028 + 0.265800i \(0.0856361\pi\)
−0.964028 + 0.265800i \(0.914364\pi\)
\(432\) 0 0
\(433\) 1.16987 + 1.16987i 0.0562205 + 0.0562205i 0.734658 0.678438i \(-0.237343\pi\)
−0.678438 + 0.734658i \(0.737343\pi\)
\(434\) −20.5940 −0.988545
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −0.757875 0.757875i −0.0362541 0.0362541i
\(438\) 0 0
\(439\) 19.2487i 0.918691i 0.888258 + 0.459345i \(0.151916\pi\)
−0.888258 + 0.459345i \(0.848084\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18.9282 + 18.9282i −0.900323 + 0.900323i
\(443\) −8.66115 + 8.66115i −0.411504 + 0.411504i −0.882262 0.470759i \(-0.843980\pi\)
0.470759 + 0.882262i \(0.343980\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 10.9348i 0.517776i
\(447\) 0 0
\(448\) 1.36603 + 1.36603i 0.0645386 + 0.0645386i
\(449\) −29.1165 −1.37409 −0.687046 0.726614i \(-0.741093\pi\)
−0.687046 + 0.726614i \(0.741093\pi\)
\(450\) 0 0
\(451\) 22.0000 1.03594
\(452\) 2.17209 + 2.17209i 0.102166 + 0.102166i
\(453\) 0 0
\(454\) 11.4641i 0.538037i
\(455\) 0 0
\(456\) 0 0
\(457\) 24.2942 24.2942i 1.13644 1.13644i 0.147352 0.989084i \(-0.452925\pi\)
0.989084 0.147352i \(-0.0470750\pi\)
\(458\) −15.8338 + 15.8338i −0.739863 + 0.739863i
\(459\) 0 0
\(460\) 0 0
\(461\) 30.6694i 1.42842i −0.699934 0.714208i \(-0.746787\pi\)
0.699934 0.714208i \(-0.253213\pi\)
\(462\) 0 0
\(463\) 7.29423 + 7.29423i 0.338992 + 0.338992i 0.855988 0.516996i \(-0.172950\pi\)
−0.516996 + 0.855988i \(0.672950\pi\)
\(464\) 3.48477 0.161776
\(465\) 0 0
\(466\) −19.8564 −0.919830
\(467\) 9.19239 + 9.19239i 0.425373 + 0.425373i 0.887049 0.461676i \(-0.152752\pi\)
−0.461676 + 0.887049i \(0.652752\pi\)
\(468\) 0 0
\(469\) 11.4641i 0.529363i
\(470\) 0 0
\(471\) 0 0
\(472\) −7.19615 + 7.19615i −0.331230 + 0.331230i
\(473\) −10.9348 + 10.9348i −0.502781 + 0.502781i
\(474\) 0 0
\(475\) 0 0
\(476\) 10.5558i 0.483826i
\(477\) 0 0
\(478\) 14.1962 + 14.1962i 0.649317 + 0.649317i
\(479\) 25.7332 1.17578 0.587891 0.808940i \(-0.299958\pi\)
0.587891 + 0.808940i \(0.299958\pi\)
\(480\) 0 0
\(481\) −56.7846 −2.58916
\(482\) −2.82843 2.82843i −0.128831 0.128831i
\(483\) 0 0
\(484\) 5.66025i 0.257284i
\(485\) 0 0
\(486\) 0 0
\(487\) 12.8038 12.8038i 0.580198 0.580198i −0.354760 0.934957i \(-0.615437\pi\)
0.934957 + 0.354760i \(0.115437\pi\)
\(488\) 1.79315 1.79315i 0.0811721 0.0811721i
\(489\) 0 0
\(490\) 0 0
\(491\) 23.9773i 1.08208i 0.840997 + 0.541039i \(0.181969\pi\)
−0.840997 + 0.541039i \(0.818031\pi\)
\(492\) 0 0
\(493\) 13.4641 + 13.4641i 0.606393 + 0.606393i
\(494\) −2.62536 −0.118120
\(495\) 0 0
\(496\) 10.6603 0.478660
\(497\) −5.65685 5.65685i −0.253745 0.253745i
\(498\) 0 0
\(499\) 8.14359i 0.364557i 0.983247 + 0.182279i \(0.0583473\pi\)
−0.983247 + 0.182279i \(0.941653\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −13.7321 + 13.7321i −0.612891 + 0.612891i
\(503\) −15.8338 + 15.8338i −0.705992 + 0.705992i −0.965690 0.259698i \(-0.916377\pi\)
0.259698 + 0.965690i \(0.416377\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.62158i 0.205454i
\(507\) 0 0
\(508\) 9.83013 + 9.83013i 0.436141 + 0.436141i
\(509\) 20.7699 0.920609 0.460305 0.887761i \(-0.347740\pi\)
0.460305 + 0.887761i \(0.347740\pi\)
\(510\) 0 0
\(511\) 20.8564 0.922633
\(512\) −0.707107 0.707107i −0.0312500 0.0312500i
\(513\) 0 0
\(514\) 12.7846i 0.563905i
\(515\) 0 0
\(516\) 0 0
\(517\) −9.80385 + 9.80385i −0.431173 + 0.431173i
\(518\) −15.8338 + 15.8338i −0.695695 + 0.695695i
\(519\) 0 0
\(520\) 0 0
\(521\) 13.3843i 0.586375i −0.956055 0.293188i \(-0.905284\pi\)
0.956055 0.293188i \(-0.0947160\pi\)
\(522\) 0 0
\(523\) 14.5359 + 14.5359i 0.635610 + 0.635610i 0.949470 0.313859i \(-0.101622\pi\)
−0.313859 + 0.949470i \(0.601622\pi\)
\(524\) −12.7651 −0.557645
\(525\) 0 0
\(526\) 14.0000 0.610429
\(527\) 41.1881 + 41.1881i 1.79418 + 1.79418i
\(528\) 0 0
\(529\) 19.0000i 0.826087i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.732051 + 0.732051i −0.0317384 + 0.0317384i
\(533\) 32.9802 32.9802i 1.42853 1.42853i
\(534\) 0 0
\(535\) 0 0
\(536\) 5.93426i 0.256321i
\(537\) 0 0
\(538\) 15.3923 + 15.3923i 0.663609 + 0.663609i
\(539\) −7.55154 −0.325268
\(540\) 0 0
\(541\) −7.32051 −0.314733 −0.157367 0.987540i \(-0.550300\pi\)
−0.157367 + 0.987540i \(0.550300\pi\)
\(542\) −17.9551 17.9551i −0.771236 0.771236i
\(543\) 0 0
\(544\) 5.46410i 0.234271i
\(545\) 0 0
\(546\) 0 0
\(547\) −12.9282 + 12.9282i −0.552770 + 0.552770i −0.927239 0.374469i \(-0.877825\pi\)
0.374469 + 0.927239i \(0.377825\pi\)
\(548\) 2.44949 2.44949i 0.104637 0.104637i
\(549\) 0 0
\(550\) 0 0
\(551\) 1.86748i 0.0795573i
\(552\) 0 0
\(553\) −4.73205 4.73205i −0.201227 0.201227i
\(554\) −11.1106 −0.472046
\(555\) 0 0
\(556\) −17.8564 −0.757280
\(557\) 10.3664 + 10.3664i 0.439237 + 0.439237i 0.891755 0.452518i \(-0.149474\pi\)
−0.452518 + 0.891755i \(0.649474\pi\)
\(558\) 0 0
\(559\) 32.7846i 1.38664i
\(560\) 0 0
\(561\) 0 0
\(562\) 2.53590 2.53590i 0.106970 0.106970i
\(563\) −12.1595 + 12.1595i −0.512462 + 0.512462i −0.915280 0.402818i \(-0.868031\pi\)
0.402818 + 0.915280i \(0.368031\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.45001i 0.313147i
\(567\) 0 0
\(568\) 2.92820 + 2.92820i 0.122865 + 0.122865i
\(569\) −36.0117 −1.50969 −0.754844 0.655904i \(-0.772287\pi\)
−0.754844 + 0.655904i \(0.772287\pi\)
\(570\) 0 0
\(571\) 5.46410 0.228666 0.114333 0.993443i \(-0.463527\pi\)
0.114333 + 0.993443i \(0.463527\pi\)
\(572\) −8.00481 8.00481i −0.334698 0.334698i
\(573\) 0 0
\(574\) 18.3923i 0.767680i
\(575\) 0 0
\(576\) 0 0
\(577\) −5.39230 + 5.39230i −0.224485 + 0.224485i −0.810384 0.585899i \(-0.800741\pi\)
0.585899 + 0.810384i \(0.300741\pi\)
\(578\) 9.09085 9.09085i 0.378130 0.378130i
\(579\) 0 0
\(580\) 0 0
\(581\) 30.7709i 1.27659i
\(582\) 0 0
\(583\) 5.22243 + 5.22243i 0.216291 + 0.216291i
\(584\) −10.7961 −0.446745
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 27.2726 + 27.2726i 1.12566 + 1.12566i 0.990875 + 0.134784i \(0.0430340\pi\)
0.134784 + 0.990875i \(0.456966\pi\)
\(588\) 0 0
\(589\) 5.71281i 0.235392i
\(590\) 0 0
\(591\) 0 0
\(592\) 8.19615 8.19615i 0.336860 0.336860i
\(593\) −13.7632 + 13.7632i −0.565187 + 0.565187i −0.930776 0.365589i \(-0.880867\pi\)
0.365589 + 0.930776i \(0.380867\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.240237i 0.00984048i
\(597\) 0 0
\(598\) −6.92820 6.92820i −0.283315 0.283315i
\(599\) 12.3490 0.504566 0.252283 0.967654i \(-0.418819\pi\)
0.252283 + 0.967654i \(0.418819\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 9.14162 + 9.14162i 0.372585 + 0.372585i
\(603\) 0 0
\(604\) 1.53590i 0.0624948i
\(605\) 0 0
\(606\) 0 0
\(607\) −9.58846 + 9.58846i −0.389183 + 0.389183i −0.874396 0.485213i \(-0.838742\pi\)
0.485213 + 0.874396i \(0.338742\pi\)
\(608\) 0.378937 0.378937i 0.0153679 0.0153679i
\(609\) 0 0
\(610\) 0 0
\(611\) 29.3939i 1.18915i
\(612\) 0 0
\(613\) −0.339746 0.339746i −0.0137222 0.0137222i 0.700212 0.713935i \(-0.253089\pi\)
−0.713935 + 0.700212i \(0.753089\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −4.46410 −0.179864
\(617\) 4.72311 + 4.72311i 0.190145 + 0.190145i 0.795759 0.605614i \(-0.207072\pi\)
−0.605614 + 0.795759i \(0.707072\pi\)
\(618\) 0 0
\(619\) 28.3923i 1.14118i −0.821234 0.570592i \(-0.806714\pi\)
0.821234 0.570592i \(-0.193286\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.732051 0.732051i 0.0293526 0.0293526i
\(623\) 11.9700 11.9700i 0.479570 0.479570i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.37705i 0.0550379i
\(627\) 0 0
\(628\) −12.0000 12.0000i −0.478852 0.478852i
\(629\) 63.3350 2.52533
\(630\) 0 0
\(631\) −34.9090 −1.38970 −0.694852 0.719153i \(-0.744530\pi\)
−0.694852 + 0.719153i \(0.744530\pi\)
\(632\) 2.44949 + 2.44949i 0.0974355 + 0.0974355i
\(633\) 0 0
\(634\) 5.53590i 0.219859i
\(635\) 0 0
\(636\) 0 0
\(637\) −11.3205 + 11.3205i −0.448535 + 0.448535i
\(638\) −5.69402 + 5.69402i −0.225428 + 0.225428i
\(639\) 0 0
\(640\) 0 0
\(641\) 11.0363i 0.435908i −0.975959 0.217954i \(-0.930062\pi\)
0.975959 0.217954i \(-0.0699383\pi\)
\(642\) 0 0
\(643\) 3.46410 + 3.46410i 0.136611 + 0.136611i 0.772105 0.635495i \(-0.219204\pi\)
−0.635495 + 0.772105i \(0.719204\pi\)
\(644\) −3.86370 −0.152251
\(645\) 0 0
\(646\) 2.92820 0.115209
\(647\) 18.5606 + 18.5606i 0.729694 + 0.729694i 0.970559 0.240865i \(-0.0774310\pi\)
−0.240865 + 0.970559i \(0.577431\pi\)
\(648\) 0 0
\(649\) 23.5167i 0.923110i
\(650\) 0 0
\(651\) 0 0
\(652\) −1.46410 + 1.46410i −0.0573386 + 0.0573386i
\(653\) 14.3688 14.3688i 0.562293 0.562293i −0.367665 0.929958i \(-0.619843\pi\)
0.929958 + 0.367665i \(0.119843\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.52056i 0.371715i
\(657\) 0 0
\(658\) 8.19615 + 8.19615i 0.319519 + 0.319519i
\(659\) −26.4540 −1.03050 −0.515250 0.857040i \(-0.672301\pi\)
−0.515250 + 0.857040i \(0.672301\pi\)
\(660\) 0 0
\(661\) 18.3923 0.715378 0.357689 0.933841i \(-0.383565\pi\)
0.357689 + 0.933841i \(0.383565\pi\)
\(662\) −0.378937 0.378937i −0.0147278 0.0147278i
\(663\) 0 0
\(664\) 15.9282i 0.618134i
\(665\) 0 0
\(666\) 0 0
\(667\) −4.92820 + 4.92820i −0.190821 + 0.190821i
\(668\) 2.17209 2.17209i 0.0840406 0.0840406i
\(669\) 0 0
\(670\) 0 0
\(671\) 5.85993i 0.226220i
\(672\) 0 0
\(673\) 16.0263 + 16.0263i 0.617768 + 0.617768i 0.944958 0.327191i \(-0.106102\pi\)
−0.327191 + 0.944958i \(0.606102\pi\)
\(674\) −9.14162 −0.352122
\(675\) 0 0
\(676\) −11.0000 −0.423077
\(677\) 9.69642 + 9.69642i 0.372664 + 0.372664i 0.868447 0.495783i \(-0.165119\pi\)
−0.495783 + 0.868447i \(0.665119\pi\)
\(678\) 0 0
\(679\) 3.92820i 0.150751i
\(680\) 0 0
\(681\) 0 0
\(682\) −17.4186 + 17.4186i −0.666992 + 0.666992i
\(683\) −12.0716 + 12.0716i −0.461906 + 0.461906i −0.899280 0.437374i \(-0.855909\pi\)
0.437374 + 0.899280i \(0.355909\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 19.8362i 0.757348i
\(687\) 0 0
\(688\) −4.73205 4.73205i −0.180408 0.180408i
\(689\) 15.6579 0.596518
\(690\) 0 0
\(691\) −34.6410 −1.31781 −0.658903 0.752228i \(-0.728979\pi\)
−0.658903 + 0.752228i \(0.728979\pi\)
\(692\) −2.50026 2.50026i −0.0950455 0.0950455i
\(693\) 0 0
\(694\) 11.3397i 0.430451i
\(695\) 0 0
\(696\) 0 0
\(697\) −36.7846 + 36.7846i −1.39332 + 1.39332i
\(698\) −18.6622 + 18.6622i −0.706374 + 0.706374i
\(699\) 0 0
\(700\) 0 0
\(701\) 35.1151i 1.32628i 0.748496 + 0.663140i \(0.230776\pi\)
−0.748496 + 0.663140i \(0.769224\pi\)
\(702\) 0 0
\(703\) 4.39230 + 4.39230i 0.165659 + 0.165659i
\(704\) 2.31079 0.0870911
\(705\) 0 0
\(706\) −7.85641 −0.295680
\(707\) −16.6796 16.6796i −0.627299 0.627299i
\(708\) 0 0
\(709\) 28.7846i 1.08103i 0.841335 + 0.540514i \(0.181770\pi\)
−0.841335 + 0.540514i \(0.818230\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −6.19615 + 6.19615i −0.232211 + 0.232211i
\(713\) −15.0759 + 15.0759i −0.564596 + 0.564596i
\(714\) 0 0
\(715\) 0 0
\(716\) 6.83083i 0.255280i
\(717\) 0 0
\(718\) −14.5885 14.5885i −0.544436 0.544436i
\(719\) 20.3538 0.759068 0.379534 0.925178i \(-0.376084\pi\)
0.379534 + 0.925178i \(0.376084\pi\)
\(720\) 0 0
\(721\) −17.1244 −0.637744
\(722\) −13.2320 13.2320i −0.492442 0.492442i
\(723\) 0 0
\(724\) 1.07180i 0.0398330i
\(725\) 0 0
\(726\) 0 0
\(727\) −20.0263 + 20.0263i −0.742734 + 0.742734i −0.973103 0.230370i \(-0.926006\pi\)
0.230370 + 0.973103i \(0.426006\pi\)
\(728\) −6.69213 + 6.69213i −0.248027 + 0.248027i
\(729\) 0 0
\(730\) 0 0
\(731\) 36.5665i 1.35246i
\(732\) 0 0
\(733\) −13.1244 13.1244i −0.484759 0.484759i 0.421889 0.906648i \(-0.361367\pi\)
−0.906648 + 0.421889i \(0.861367\pi\)
\(734\) 22.6646 0.836565
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) −9.69642 9.69642i −0.357172 0.357172i
\(738\) 0 0
\(739\) 22.9282i 0.843428i 0.906729 + 0.421714i \(0.138571\pi\)
−0.906729 + 0.421714i \(0.861429\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.36603 4.36603i 0.160282 0.160282i
\(743\) −33.2576 + 33.2576i −1.22010 + 1.22010i −0.252507 + 0.967595i \(0.581255\pi\)
−0.967595 + 0.252507i \(0.918745\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.5254i 0.641649i
\(747\) 0 0
\(748\) 8.92820 + 8.92820i 0.326447 + 0.326447i
\(749\) 23.0435 0.841992
\(750\) 0 0
\(751\) 33.5885 1.22566 0.612830 0.790215i \(-0.290031\pi\)
0.612830 + 0.790215i \(0.290031\pi\)
\(752\) −4.24264 4.24264i −0.154713 0.154713i
\(753\) 0 0
\(754\) 17.0718i 0.621718i
\(755\) 0 0
\(756\) 0 0
\(757\) −8.19615 + 8.19615i −0.297894 + 0.297894i −0.840189 0.542294i \(-0.817556\pi\)
0.542294 + 0.840189i \(0.317556\pi\)
\(758\) 5.65685 5.65685i 0.205466 0.205466i
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6990i 1.25784i 0.777471 + 0.628919i \(0.216502\pi\)
−0.777471 + 0.628919i \(0.783498\pi\)
\(762\) 0 0
\(763\) 2.73205 + 2.73205i 0.0989069 + 0.0989069i
\(764\) −26.0106 −0.941032
\(765\) 0 0
\(766\) −35.3205 −1.27618
\(767\) −35.2538 35.2538i −1.27294 1.27294i
\(768\) 0 0
\(769\) 9.78461i 0.352842i 0.984315 + 0.176421i \(0.0564520\pi\)
−0.984315 + 0.176421i \(0.943548\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 11.0263 11.0263i 0.396844 0.396844i
\(773\) 31.0112 31.0112i 1.11539 1.11539i 0.122986 0.992408i \(-0.460753\pi\)
0.992408 0.122986i \(-0.0392469\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.03339i 0.0729944i
\(777\) 0 0
\(778\) −6.29423 6.29423i −0.225659 0.225659i
\(779\) −5.10205 −0.182800
\(780\) 0 0
\(781\) −9.56922 −0.342414
\(782\) 7.72741 + 7.72741i 0.276331 + 0.276331i
\(783\) 0 0
\(784\) 3.26795i 0.116712i
\(785\) 0 0
\(786\) 0 0
\(787\) −16.9282 + 16.9282i −0.603425 + 0.603425i −0.941220 0.337795i \(-0.890319\pi\)
0.337795 + 0.941220i \(0.390319\pi\)
\(788\) −5.46739 + 5.46739i −0.194768 + 0.194768i
\(789\) 0 0
\(790\) 0 0
\(791\) 5.93426i 0.210998i
\(792\) 0 0
\(793\) 8.78461 + 8.78461i 0.311951 + 0.311951i
\(794\) −12.6264 −0.448094
\(795\) 0 0
\(796\) 20.3205 0.720241
\(797\) 10.7825 + 10.7825i 0.381935 + 0.381935i 0.871799 0.489864i \(-0.162954\pi\)
−0.489864 + 0.871799i \(0.662954\pi\)
\(798\) 0 0
\(799\) 32.7846i 1.15984i
\(800\) 0 0
\(801\) 0 0
\(802\) −0.928203 + 0.928203i −0.0327760 + 0.0327760i
\(803\) 17.6405 17.6405i 0.622520 0.622520i
\(804\) 0 0
\(805\) 0 0
\(806\) 52.2244i 1.83952i
\(807\) 0 0
\(808\) 8.63397 + 8.63397i 0.303742 + 0.303742i
\(809\) −5.17638 −0.181992 −0.0909959 0.995851i \(-0.529005\pi\)
−0.0909959 + 0.995851i \(0.529005\pi\)
\(810\) 0 0
\(811\) −15.0718 −0.529242 −0.264621 0.964352i \(-0.585247\pi\)
−0.264621 + 0.964352i \(0.585247\pi\)
\(812\) 4.76028 + 4.76028i 0.167053 + 0.167053i
\(813\) 0 0
\(814\) 26.7846i 0.938800i
\(815\) 0 0
\(816\) 0 0
\(817\) 2.53590 2.53590i 0.0887199 0.0887199i
\(818\) 14.6090 14.6090i 0.510792 0.510792i
\(819\) 0 0
\(820\) 0 0
\(821\) 40.0512i 1.39780i −0.715220 0.698899i \(-0.753674\pi\)
0.715220 0.698899i \(-0.246326\pi\)
\(822\) 0 0
\(823\) 15.7058 + 15.7058i 0.547469 + 0.547469i 0.925708 0.378239i \(-0.123470\pi\)
−0.378239 + 0.925708i \(0.623470\pi\)
\(824\) 8.86422 0.308800
\(825\) 0 0
\(826\) −19.6603 −0.684068
\(827\) −29.2180 29.2180i −1.01601 1.01601i −0.999870 0.0161401i \(-0.994862\pi\)
−0.0161401 0.999870i \(-0.505138\pi\)
\(828\) 0 0
\(829\) 37.3205i 1.29619i −0.761557 0.648097i \(-0.775565\pi\)
0.761557 0.648097i \(-0.224435\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.46410 3.46410i 0.120096 0.120096i
\(833\) 12.6264 12.6264i 0.437478 0.437478i
\(834\) 0 0
\(835\) 0 0
\(836\) 1.23835i 0.0428292i
\(837\) 0 0
\(838\) −26.6603 26.6603i −0.920963 0.920963i
\(839\) 0.554803 0.0191539 0.00957696 0.999954i \(-0.496952\pi\)
0.00957696 + 0.999954i \(0.496952\pi\)
\(840\) 0 0
\(841\) −16.8564 −0.581255
\(842\) −6.96953 6.96953i −0.240186 0.240186i
\(843\) 0 0
\(844\) 13.3205i 0.458511i
\(845\) 0 0
\(846\) 0 0
\(847\) −7.73205 + 7.73205i −0.265676 + 0.265676i
\(848\) −2.26002 + 2.26002i −0.0776094 + 0.0776094i
\(849\) 0 0
\(850\) 0 0
\(851\) 23.1822i 0.794676i
\(852\) 0 0
\(853\) 35.1244 + 35.1244i 1.20264 + 1.20264i 0.973362 + 0.229273i \(0.0736348\pi\)
0.229273 + 0.973362i \(0.426365\pi\)
\(854\) 4.89898 0.167640
\(855\) 0 0
\(856\) −11.9282 −0.407698
\(857\) −16.0096 16.0096i −0.546878 0.546878i 0.378658 0.925536i \(-0.376386\pi\)
−0.925536 + 0.378658i \(0.876386\pi\)
\(858\) 0 0
\(859\) 3.32051i 0.113294i 0.998394 + 0.0566471i \(0.0180410\pi\)
−0.998394 + 0.0566471i \(0.981959\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 7.80385 7.80385i 0.265800 0.265800i
\(863\) 5.17638 5.17638i 0.176206 0.176206i −0.613494 0.789700i \(-0.710236\pi\)
0.789700 + 0.613494i \(0.210236\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.65445i 0.0562205i
\(867\) 0 0
\(868\) 14.5622 + 14.5622i 0.494273 + 0.494273i
\(869\) −8.00481 −0.271545
\(870\) 0 0
\(871\) −29.0718 −0.985060
\(872\) −1.41421 1.41421i −0.0478913 0.0478913i
\(873\) 0 0
\(874\) 1.07180i 0.0362541i
\(875\) 0 0
\(876\) 0 0
\(877\) 4.73205 4.73205i 0.159790 0.159790i −0.622684 0.782474i \(-0.713958\pi\)
0.782474 + 0.622684i \(0.213958\pi\)
\(878\) 13.6109 13.6109i 0.459345 0.459345i
\(879\) 0 0
\(880\) 0 0
\(881\) 16.2127i 0.546219i −0.961983 0.273110i \(-0.911948\pi\)
0.961983 0.273110i \(-0.0880522\pi\)
\(882\) 0 0
\(883\) 5.12436 + 5.12436i 0.172448 + 0.172448i 0.788054 0.615606i \(-0.211089\pi\)
−0.615606 + 0.788054i \(0.711089\pi\)
\(884\) 26.7685 0.900323
\(885\) 0 0
\(886\) 12.2487 0.411504
\(887\) 4.41851 + 4.41851i 0.148359 + 0.148359i 0.777385 0.629026i \(-0.216546\pi\)
−0.629026 + 0.777385i \(0.716546\pi\)
\(888\) 0 0
\(889\) 26.8564i 0.900735i
\(890\) 0 0
\(891\) 0 0
\(892\) 7.73205 7.73205i 0.258888 0.258888i
\(893\) 2.27362 2.27362i 0.0760839 0.0760839i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.93185i 0.0645386i
\(897\) 0 0
\(898\) 20.5885 + 20.5885i 0.687046 + 0.687046i
\(899\) 37.1485 1.23897
\(900\) 0 0
\(901\) −17.4641 −0.581814
\(902\) −15.5563 15.5563i −0.517970 0.517970i
\(903\) 0 0
\(904\) 3.07180i 0.102166i
\(905\) 0 0
\(906\) 0 0
\(907\) 24.9808 24.9808i 0.829473 0.829473i −0.157971 0.987444i \(-0.550495\pi\)
0.987444 + 0.157971i \(0.0504953\pi\)
\(908\) −8.10634 + 8.10634i −0.269018 + 0.269018i
\(909\) 0 0
\(910\) 0 0
\(911\) 36.4922i 1.20904i −0.796590 0.604519i \(-0.793365\pi\)
0.796590 0.604519i \(-0.206635\pi\)
\(912\) 0 0
\(913\) −26.0263 26.0263i −0.861344 0.861344i
\(914\) −34.3572 −1.13644
\(915\) 0 0
\(916\) 22.3923 0.739863
\(917\) −17.4374 17.4374i −0.575835 0.575835i
\(918\) 0 0
\(919\) 10.4115i 0.343445i −0.985145 0.171723i \(-0.945067\pi\)
0.985145 0.171723i \(-0.0549333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −21.6865 + 21.6865i −0.714208 + 0.714208i
\(923\) −14.3452 + 14.3452i −0.472178 + 0.472178i
\(924\) 0 0
\(925\) 0 0
\(926\) 10.3156i 0.338992i
\(927\) 0 0
\(928\) −2.46410 2.46410i −0.0808881 0.0808881i
\(929\) −56.3655 −1.84929 −0.924646 0.380829i \(-0.875639\pi\)
−0.924646 + 0.380829i \(0.875639\pi\)
\(930\) 0 0
\(931\) 1.75129 0.0573962
\(932\) 14.0406 + 14.0406i 0.459915 + 0.459915i
\(933\) 0 0
\(934\) 13.0000i 0.425373i
\(935\) 0 0
\(936\) 0 0
\(937\) −10.1699 + 10.1699i −0.332235 + 0.332235i −0.853435 0.521200i \(-0.825485\pi\)
0.521200 + 0.853435i \(0.325485\pi\)
\(938\) −8.10634 + 8.10634i −0.264682 + 0.264682i
\(939\) 0 0
\(940\) 0 0
\(941\) 11.6555i 0.379958i 0.981788 + 0.189979i \(0.0608420\pi\)
−0.981788 + 0.189979i \(0.939158\pi\)
\(942\) 0 0
\(943\) −13.4641 13.4641i −0.438451 0.438451i
\(944\) 10.1769 0.331230
\(945\) 0 0
\(946\) 15.4641 0.502781
\(947\) 39.8618 + 39.8618i 1.29533 + 1.29533i 0.931441 + 0.363893i \(0.118553\pi\)
0.363893 + 0.931441i \(0.381447\pi\)
\(948\) 0 0
\(949\) 52.8897i 1.71687i
\(950\) 0 0
\(951\) 0 0
\(952\) 7.46410 7.46410i 0.241913 0.241913i
\(953\) −15.1774 + 15.1774i −0.491645 + 0.491645i −0.908824 0.417180i \(-0.863019\pi\)
0.417180 + 0.908824i \(0.363019\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.0764i 0.649317i
\(957\) 0 0
\(958\) −18.1962 18.1962i −0.587891 0.587891i
\(959\) 6.69213 0.216100
\(960\) 0 0
\(961\) 82.6410 2.66584
\(962\) 40.1528 + 40.1528i 1.29458 + 1.29458i
\(963\) 0 0
\(964\) 4.00000i 0.128831i
\(965\) 0 0
\(966\) 0 0
\(967\) 28.4904 28.4904i 0.916189 0.916189i −0.0805608 0.996750i \(-0.525671\pi\)
0.996750 + 0.0805608i \(0.0256711\pi\)
\(968\) 4.00240 4.00240i 0.128642 0.128642i
\(969\) 0 0
\(970\) 0 0
\(971\) 29.1808i 0.936458i 0.883607 + 0.468229i \(0.155108\pi\)
−0.883607 + 0.468229i \(0.844892\pi\)
\(972\) 0 0
\(973\) −24.3923 24.3923i −0.781981 0.781981i
\(974\) −18.1074 −0.580198
\(975\) 0 0
\(976\) −2.53590 −0.0811721
\(977\) 22.5259 + 22.5259i 0.720667 + 0.720667i 0.968741 0.248074i \(-0.0797977\pi\)
−0.248074 + 0.968741i \(0.579798\pi\)
\(978\) 0 0
\(979\) 20.2487i 0.647152i
\(980\) 0 0
\(981\) 0 0
\(982\) 16.9545 16.9545i 0.541039 0.541039i
\(983\) 3.00429 3.00429i 0.0958221 0.0958221i −0.657571 0.753393i \(-0.728416\pi\)
0.753393 + 0.657571i \(0.228416\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19.0411i 0.606393i
\(987\) 0 0
\(988\) 1.85641 + 1.85641i 0.0590602 + 0.0590602i
\(989\) 13.3843 0.425595
\(990\) 0 0
\(991\) 23.3923 0.743081 0.371541 0.928417i \(-0.378830\pi\)
0.371541 + 0.928417i \(0.378830\pi\)
\(992\) −7.53794 7.53794i −0.239330 0.239330i
\(993\) 0 0
\(994\) 8.00000i 0.253745i
\(995\) 0 0
\(996\) 0 0
\(997\) 6.14359 6.14359i 0.194570 0.194570i −0.603098 0.797667i \(-0.706067\pi\)
0.797667 + 0.603098i \(0.206067\pi\)
\(998\) 5.75839 5.75839i 0.182279 0.182279i
\(999\) 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 1350.2.f.a.593.1 8
3.2 odd 2 inner 1350.2.f.a.593.3 8
5.2 odd 4 inner 1350.2.f.a.107.3 8
5.3 odd 4 270.2.f.b.107.1 yes 8
5.4 even 2 270.2.f.b.53.4 yes 8
15.2 even 4 inner 1350.2.f.a.107.1 8
15.8 even 4 270.2.f.b.107.4 yes 8
15.14 odd 2 270.2.f.b.53.1 8
20.3 even 4 2160.2.w.b.1457.1 8
20.19 odd 2 2160.2.w.b.593.4 8
45.4 even 6 810.2.m.e.593.1 8
45.13 odd 12 810.2.m.d.107.1 8
45.14 odd 6 810.2.m.e.593.2 8
45.23 even 12 810.2.m.d.107.2 8
45.29 odd 6 810.2.m.d.53.1 8
45.34 even 6 810.2.m.d.53.2 8
45.38 even 12 810.2.m.e.377.1 8
45.43 odd 12 810.2.m.e.377.2 8
60.23 odd 4 2160.2.w.b.1457.4 8
60.59 even 2 2160.2.w.b.593.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.f.b.53.1 8 15.14 odd 2
270.2.f.b.53.4 yes 8 5.4 even 2
270.2.f.b.107.1 yes 8 5.3 odd 4
270.2.f.b.107.4 yes 8 15.8 even 4
810.2.m.d.53.1 8 45.29 odd 6
810.2.m.d.53.2 8 45.34 even 6
810.2.m.d.107.1 8 45.13 odd 12
810.2.m.d.107.2 8 45.23 even 12
810.2.m.e.377.1 8 45.38 even 12
810.2.m.e.377.2 8 45.43 odd 12
810.2.m.e.593.1 8 45.4 even 6
810.2.m.e.593.2 8 45.14 odd 6
1350.2.f.a.107.1 8 15.2 even 4 inner
1350.2.f.a.107.3 8 5.2 odd 4 inner
1350.2.f.a.593.1 8 1.1 even 1 trivial
1350.2.f.a.593.3 8 3.2 odd 2 inner
2160.2.w.b.593.1 8 60.59 even 2
2160.2.w.b.593.4 8 20.19 odd 2
2160.2.w.b.1457.1 8 20.3 even 4
2160.2.w.b.1457.4 8 60.23 odd 4