Properties

Label 1232.2.q.h.529.1
Level $1232$
Weight $2$
Character 1232.529
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,2,Mod(177,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1232.529
Dual form 1232.2.q.h.177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.207107 + 0.358719i) q^{3} +(-0.707107 - 1.22474i) q^{5} +(-2.62132 + 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +O(q^{10})\) \(q+(-0.207107 + 0.358719i) q^{3} +(-0.707107 - 1.22474i) q^{5} +(-2.62132 + 0.358719i) q^{7} +(1.41421 + 2.44949i) q^{9} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +0.585786 q^{15} +(1.00000 - 1.73205i) q^{17} +(-1.29289 - 2.23936i) q^{19} +(0.414214 - 1.01461i) q^{21} +(2.70711 + 4.68885i) q^{23} +(1.50000 - 2.59808i) q^{25} -2.41421 q^{27} +1.00000 q^{29} +(-4.82843 + 8.36308i) q^{31} +(0.207107 + 0.358719i) q^{33} +(2.29289 + 2.95680i) q^{35} +(4.53553 + 7.85578i) q^{37} +(-0.378680 + 0.655892i) q^{39} +6.24264 q^{41} +8.00000 q^{43} +(2.00000 - 3.46410i) q^{45} +(2.41421 + 4.18154i) q^{47} +(6.74264 - 1.88064i) q^{49} +(0.414214 + 0.717439i) q^{51} +(-1.29289 + 2.23936i) q^{53} -1.41421 q^{55} +1.07107 q^{57} +(2.20711 - 3.82282i) q^{59} +(4.08579 + 7.07679i) q^{61} +(-4.58579 - 5.91359i) q^{63} +(-1.29289 - 2.23936i) q^{65} +(4.03553 - 6.98975i) q^{67} -2.24264 q^{69} +5.75736 q^{71} +(1.70711 - 2.95680i) q^{73} +(0.621320 + 1.07616i) q^{75} +(-1.00000 + 2.44949i) q^{77} +(4.03553 + 6.98975i) q^{79} +(-3.74264 + 6.48244i) q^{81} +10.4853 q^{83} -2.82843 q^{85} +(-0.207107 + 0.358719i) q^{87} +(4.58579 + 7.94282i) q^{89} +(-4.79289 + 0.655892i) q^{91} +(-2.00000 - 3.46410i) q^{93} +(-1.82843 + 3.16693i) q^{95} -15.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{7} + 2 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{17} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{27} + 4 q^{29} - 8 q^{31} - 2 q^{33} + 12 q^{35} + 4 q^{37} - 10 q^{39} + 8 q^{41} + 32 q^{43} + 8 q^{45} + 4 q^{47} + 10 q^{49} - 4 q^{51} - 8 q^{53} - 24 q^{57} + 6 q^{59} + 22 q^{61} - 24 q^{63} - 8 q^{65} + 2 q^{67} + 8 q^{69} + 40 q^{71} + 4 q^{73} - 6 q^{75} - 4 q^{77} + 2 q^{79} + 2 q^{81} + 8 q^{83} + 2 q^{87} + 24 q^{89} - 22 q^{91} - 8 q^{93} + 4 q^{95} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.207107 + 0.358719i −0.119573 + 0.207107i −0.919599 0.392859i \(-0.871486\pi\)
0.800025 + 0.599966i \(0.204819\pi\)
\(4\) 0 0
\(5\) −0.707107 1.22474i −0.316228 0.547723i 0.663470 0.748203i \(-0.269083\pi\)
−0.979698 + 0.200480i \(0.935750\pi\)
\(6\) 0 0
\(7\) −2.62132 + 0.358719i −0.990766 + 0.135583i
\(8\) 0 0
\(9\) 1.41421 + 2.44949i 0.471405 + 0.816497i
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 0 0
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) −1.29289 2.23936i −0.296610 0.513744i 0.678748 0.734371i \(-0.262523\pi\)
−0.975358 + 0.220628i \(0.929189\pi\)
\(20\) 0 0
\(21\) 0.414214 1.01461i 0.0903888 0.221406i
\(22\) 0 0
\(23\) 2.70711 + 4.68885i 0.564471 + 0.977692i 0.997099 + 0.0761195i \(0.0242531\pi\)
−0.432628 + 0.901573i \(0.642414\pi\)
\(24\) 0 0
\(25\) 1.50000 2.59808i 0.300000 0.519615i
\(26\) 0 0
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −4.82843 + 8.36308i −0.867211 + 1.50205i −0.00237631 + 0.999997i \(0.500756\pi\)
−0.864835 + 0.502057i \(0.832577\pi\)
\(32\) 0 0
\(33\) 0.207107 + 0.358719i 0.0360527 + 0.0624450i
\(34\) 0 0
\(35\) 2.29289 + 2.95680i 0.387570 + 0.499790i
\(36\) 0 0
\(37\) 4.53553 + 7.85578i 0.745637 + 1.29148i 0.949896 + 0.312565i \(0.101188\pi\)
−0.204259 + 0.978917i \(0.565479\pi\)
\(38\) 0 0
\(39\) −0.378680 + 0.655892i −0.0606373 + 0.105027i
\(40\) 0 0
\(41\) 6.24264 0.974937 0.487468 0.873141i \(-0.337920\pi\)
0.487468 + 0.873141i \(0.337920\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 2.00000 3.46410i 0.298142 0.516398i
\(46\) 0 0
\(47\) 2.41421 + 4.18154i 0.352149 + 0.609940i 0.986626 0.163002i \(-0.0521176\pi\)
−0.634477 + 0.772942i \(0.718784\pi\)
\(48\) 0 0
\(49\) 6.74264 1.88064i 0.963234 0.268662i
\(50\) 0 0
\(51\) 0.414214 + 0.717439i 0.0580015 + 0.100462i
\(52\) 0 0
\(53\) −1.29289 + 2.23936i −0.177593 + 0.307599i −0.941055 0.338252i \(-0.890164\pi\)
0.763463 + 0.645852i \(0.223498\pi\)
\(54\) 0 0
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) 1.07107 0.141866
\(58\) 0 0
\(59\) 2.20711 3.82282i 0.287341 0.497689i −0.685833 0.727759i \(-0.740562\pi\)
0.973174 + 0.230070i \(0.0738954\pi\)
\(60\) 0 0
\(61\) 4.08579 + 7.07679i 0.523131 + 0.906090i 0.999638 + 0.0269190i \(0.00856962\pi\)
−0.476506 + 0.879171i \(0.658097\pi\)
\(62\) 0 0
\(63\) −4.58579 5.91359i −0.577755 0.745042i
\(64\) 0 0
\(65\) −1.29289 2.23936i −0.160364 0.277758i
\(66\) 0 0
\(67\) 4.03553 6.98975i 0.493019 0.853934i −0.506949 0.861976i \(-0.669227\pi\)
0.999968 + 0.00804237i \(0.00255999\pi\)
\(68\) 0 0
\(69\) −2.24264 −0.269982
\(70\) 0 0
\(71\) 5.75736 0.683273 0.341636 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(72\) 0 0
\(73\) 1.70711 2.95680i 0.199802 0.346067i −0.748662 0.662952i \(-0.769304\pi\)
0.948464 + 0.316885i \(0.102637\pi\)
\(74\) 0 0
\(75\) 0.621320 + 1.07616i 0.0717439 + 0.124264i
\(76\) 0 0
\(77\) −1.00000 + 2.44949i −0.113961 + 0.279145i
\(78\) 0 0
\(79\) 4.03553 + 6.98975i 0.454033 + 0.786408i 0.998632 0.0522883i \(-0.0166515\pi\)
−0.544599 + 0.838697i \(0.683318\pi\)
\(80\) 0 0
\(81\) −3.74264 + 6.48244i −0.415849 + 0.720272i
\(82\) 0 0
\(83\) 10.4853 1.15091 0.575455 0.817834i \(-0.304825\pi\)
0.575455 + 0.817834i \(0.304825\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) −0.207107 + 0.358719i −0.0222042 + 0.0384588i
\(88\) 0 0
\(89\) 4.58579 + 7.94282i 0.486092 + 0.841937i 0.999872 0.0159854i \(-0.00508851\pi\)
−0.513780 + 0.857922i \(0.671755\pi\)
\(90\) 0 0
\(91\) −4.79289 + 0.655892i −0.502432 + 0.0687562i
\(92\) 0 0
\(93\) −2.00000 3.46410i −0.207390 0.359211i
\(94\) 0 0
\(95\) −1.82843 + 3.16693i −0.187593 + 0.324920i
\(96\) 0 0
\(97\) −15.8284 −1.60713 −0.803567 0.595215i \(-0.797067\pi\)
−0.803567 + 0.595215i \(0.797067\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) 3.74264 6.48244i 0.372407 0.645027i −0.617529 0.786548i \(-0.711866\pi\)
0.989935 + 0.141521i \(0.0451993\pi\)
\(102\) 0 0
\(103\) −7.94975 13.7694i −0.783312 1.35674i −0.930002 0.367554i \(-0.880195\pi\)
0.146690 0.989182i \(-0.453138\pi\)
\(104\) 0 0
\(105\) −1.53553 + 0.210133i −0.149853 + 0.0205069i
\(106\) 0 0
\(107\) 6.77817 + 11.7401i 0.655271 + 1.13496i 0.981826 + 0.189784i \(0.0607788\pi\)
−0.326555 + 0.945178i \(0.605888\pi\)
\(108\) 0 0
\(109\) 4.24264 7.34847i 0.406371 0.703856i −0.588109 0.808782i \(-0.700127\pi\)
0.994480 + 0.104926i \(0.0334606\pi\)
\(110\) 0 0
\(111\) −3.75736 −0.356633
\(112\) 0 0
\(113\) −7.48528 −0.704156 −0.352078 0.935971i \(-0.614525\pi\)
−0.352078 + 0.935971i \(0.614525\pi\)
\(114\) 0 0
\(115\) 3.82843 6.63103i 0.357003 0.618347i
\(116\) 0 0
\(117\) 2.58579 + 4.47871i 0.239056 + 0.414057i
\(118\) 0 0
\(119\) −2.00000 + 4.89898i −0.183340 + 0.449089i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.0454545 0.0787296i
\(122\) 0 0
\(123\) −1.29289 + 2.23936i −0.116576 + 0.201916i
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −6.41421 −0.569169 −0.284585 0.958651i \(-0.591856\pi\)
−0.284585 + 0.958651i \(0.591856\pi\)
\(128\) 0 0
\(129\) −1.65685 + 2.86976i −0.145878 + 0.252668i
\(130\) 0 0
\(131\) −2.46447 4.26858i −0.215321 0.372948i 0.738051 0.674745i \(-0.235747\pi\)
−0.953372 + 0.301798i \(0.902413\pi\)
\(132\) 0 0
\(133\) 4.19239 + 5.40629i 0.363526 + 0.468784i
\(134\) 0 0
\(135\) 1.70711 + 2.95680i 0.146924 + 0.254480i
\(136\) 0 0
\(137\) −8.50000 + 14.7224i −0.726204 + 1.25782i 0.232273 + 0.972651i \(0.425384\pi\)
−0.958477 + 0.285171i \(0.907949\pi\)
\(138\) 0 0
\(139\) 3.31371 0.281065 0.140533 0.990076i \(-0.455119\pi\)
0.140533 + 0.990076i \(0.455119\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) 0.914214 1.58346i 0.0764504 0.132416i
\(144\) 0 0
\(145\) −0.707107 1.22474i −0.0587220 0.101710i
\(146\) 0 0
\(147\) −0.721825 + 2.80821i −0.0595352 + 0.231617i
\(148\) 0 0
\(149\) 8.00000 + 13.8564i 0.655386 + 1.13516i 0.981797 + 0.189933i \(0.0608272\pi\)
−0.326411 + 0.945228i \(0.605840\pi\)
\(150\) 0 0
\(151\) 2.79289 4.83743i 0.227282 0.393665i −0.729719 0.683747i \(-0.760349\pi\)
0.957002 + 0.290082i \(0.0936826\pi\)
\(152\) 0 0
\(153\) 5.65685 0.457330
\(154\) 0 0
\(155\) 13.6569 1.09694
\(156\) 0 0
\(157\) 3.65685 6.33386i 0.291849 0.505497i −0.682398 0.730981i \(-0.739063\pi\)
0.974247 + 0.225484i \(0.0723963\pi\)
\(158\) 0 0
\(159\) −0.535534 0.927572i −0.0424706 0.0735612i
\(160\) 0 0
\(161\) −8.77817 11.3199i −0.691817 0.892131i
\(162\) 0 0
\(163\) 2.55025 + 4.41717i 0.199751 + 0.345979i 0.948448 0.316934i \(-0.102653\pi\)
−0.748696 + 0.662913i \(0.769320\pi\)
\(164\) 0 0
\(165\) 0.292893 0.507306i 0.0228017 0.0394937i
\(166\) 0 0
\(167\) 0.272078 0.0210540 0.0105270 0.999945i \(-0.496649\pi\)
0.0105270 + 0.999945i \(0.496649\pi\)
\(168\) 0 0
\(169\) −9.65685 −0.742835
\(170\) 0 0
\(171\) 3.65685 6.33386i 0.279647 0.484362i
\(172\) 0 0
\(173\) −1.08579 1.88064i −0.0825508 0.142982i 0.821794 0.569785i \(-0.192973\pi\)
−0.904345 + 0.426802i \(0.859640\pi\)
\(174\) 0 0
\(175\) −3.00000 + 7.34847i −0.226779 + 0.555492i
\(176\) 0 0
\(177\) 0.914214 + 1.58346i 0.0687165 + 0.119020i
\(178\) 0 0
\(179\) 9.44975 16.3674i 0.706307 1.22336i −0.259910 0.965633i \(-0.583693\pi\)
0.966218 0.257727i \(-0.0829736\pi\)
\(180\) 0 0
\(181\) −13.3137 −0.989600 −0.494800 0.869007i \(-0.664759\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(182\) 0 0
\(183\) −3.38478 −0.250210
\(184\) 0 0
\(185\) 6.41421 11.1097i 0.471582 0.816805i
\(186\) 0 0
\(187\) −1.00000 1.73205i −0.0731272 0.126660i
\(188\) 0 0
\(189\) 6.32843 0.866025i 0.460325 0.0629941i
\(190\) 0 0
\(191\) −3.24264 5.61642i −0.234629 0.406390i 0.724536 0.689237i \(-0.242054\pi\)
−0.959165 + 0.282848i \(0.908721\pi\)
\(192\) 0 0
\(193\) −11.6066 + 20.1032i −0.835461 + 1.44706i 0.0581927 + 0.998305i \(0.481466\pi\)
−0.893654 + 0.448756i \(0.851867\pi\)
\(194\) 0 0
\(195\) 1.07107 0.0767008
\(196\) 0 0
\(197\) −11.1421 −0.793844 −0.396922 0.917852i \(-0.629922\pi\)
−0.396922 + 0.917852i \(0.629922\pi\)
\(198\) 0 0
\(199\) 12.0208 20.8207i 0.852133 1.47594i −0.0271465 0.999631i \(-0.508642\pi\)
0.879279 0.476306i \(-0.158025\pi\)
\(200\) 0 0
\(201\) 1.67157 + 2.89525i 0.117904 + 0.204215i
\(202\) 0 0
\(203\) −2.62132 + 0.358719i −0.183981 + 0.0251772i
\(204\) 0 0
\(205\) −4.41421 7.64564i −0.308302 0.533995i
\(206\) 0 0
\(207\) −7.65685 + 13.2621i −0.532188 + 0.921777i
\(208\) 0 0
\(209\) −2.58579 −0.178863
\(210\) 0 0
\(211\) 4.92893 0.339322 0.169661 0.985503i \(-0.445733\pi\)
0.169661 + 0.985503i \(0.445733\pi\)
\(212\) 0 0
\(213\) −1.19239 + 2.06528i −0.0817011 + 0.141510i
\(214\) 0 0
\(215\) −5.65685 9.79796i −0.385794 0.668215i
\(216\) 0 0
\(217\) 9.65685 23.6544i 0.655550 1.60576i
\(218\) 0 0
\(219\) 0.707107 + 1.22474i 0.0477818 + 0.0827606i
\(220\) 0 0
\(221\) 1.82843 3.16693i 0.122993 0.213031i
\(222\) 0 0
\(223\) −19.5563 −1.30959 −0.654795 0.755807i \(-0.727245\pi\)
−0.654795 + 0.755807i \(0.727245\pi\)
\(224\) 0 0
\(225\) 8.48528 0.565685
\(226\) 0 0
\(227\) −14.0711 + 24.3718i −0.933930 + 1.61761i −0.157398 + 0.987535i \(0.550311\pi\)
−0.776532 + 0.630078i \(0.783023\pi\)
\(228\) 0 0
\(229\) 2.00000 + 3.46410i 0.132164 + 0.228914i 0.924510 0.381157i \(-0.124474\pi\)
−0.792347 + 0.610071i \(0.791141\pi\)
\(230\) 0 0
\(231\) −0.671573 0.866025i −0.0441863 0.0569803i
\(232\) 0 0
\(233\) −10.1924 17.6537i −0.667726 1.15653i −0.978539 0.206064i \(-0.933935\pi\)
0.310813 0.950471i \(-0.399399\pi\)
\(234\) 0 0
\(235\) 3.41421 5.91359i 0.222719 0.385760i
\(236\) 0 0
\(237\) −3.34315 −0.217161
\(238\) 0 0
\(239\) 11.7279 0.758616 0.379308 0.925270i \(-0.376162\pi\)
0.379308 + 0.925270i \(0.376162\pi\)
\(240\) 0 0
\(241\) 11.3640 19.6830i 0.732017 1.26789i −0.224003 0.974589i \(-0.571912\pi\)
0.956020 0.293302i \(-0.0947542\pi\)
\(242\) 0 0
\(243\) −5.17157 8.95743i −0.331757 0.574619i
\(244\) 0 0
\(245\) −7.07107 6.92820i −0.451754 0.442627i
\(246\) 0 0
\(247\) −2.36396 4.09450i −0.150415 0.260527i
\(248\) 0 0
\(249\) −2.17157 + 3.76127i −0.137618 + 0.238361i
\(250\) 0 0
\(251\) −0.485281 −0.0306307 −0.0153153 0.999883i \(-0.504875\pi\)
−0.0153153 + 0.999883i \(0.504875\pi\)
\(252\) 0 0
\(253\) 5.41421 0.340389
\(254\) 0 0
\(255\) 0.585786 1.01461i 0.0366834 0.0635375i
\(256\) 0 0
\(257\) 9.74264 + 16.8747i 0.607729 + 1.05262i 0.991614 + 0.129237i \(0.0412527\pi\)
−0.383885 + 0.923381i \(0.625414\pi\)
\(258\) 0 0
\(259\) −14.7071 18.9655i −0.913855 1.17846i
\(260\) 0 0
\(261\) 1.41421 + 2.44949i 0.0875376 + 0.151620i
\(262\) 0 0
\(263\) −11.5208 + 19.9546i −0.710404 + 1.23046i 0.254302 + 0.967125i \(0.418154\pi\)
−0.964706 + 0.263331i \(0.915179\pi\)
\(264\) 0 0
\(265\) 3.65685 0.224639
\(266\) 0 0
\(267\) −3.79899 −0.232494
\(268\) 0 0
\(269\) 6.82843 11.8272i 0.416337 0.721116i −0.579231 0.815163i \(-0.696647\pi\)
0.995568 + 0.0940473i \(0.0299805\pi\)
\(270\) 0 0
\(271\) −0.0355339 0.0615465i −0.00215853 0.00373869i 0.864944 0.501868i \(-0.167354\pi\)
−0.867103 + 0.498129i \(0.834020\pi\)
\(272\) 0 0
\(273\) 0.757359 1.85514i 0.0458375 0.112278i
\(274\) 0 0
\(275\) −1.50000 2.59808i −0.0904534 0.156670i
\(276\) 0 0
\(277\) 12.3995 21.4766i 0.745013 1.29040i −0.205175 0.978725i \(-0.565776\pi\)
0.950188 0.311676i \(-0.100890\pi\)
\(278\) 0 0
\(279\) −27.3137 −1.63523
\(280\) 0 0
\(281\) −11.8995 −0.709864 −0.354932 0.934892i \(-0.615496\pi\)
−0.354932 + 0.934892i \(0.615496\pi\)
\(282\) 0 0
\(283\) 13.7782 23.8645i 0.819027 1.41860i −0.0873721 0.996176i \(-0.527847\pi\)
0.906400 0.422421i \(-0.138820\pi\)
\(284\) 0 0
\(285\) −0.757359 1.31178i −0.0448621 0.0777034i
\(286\) 0 0
\(287\) −16.3640 + 2.23936i −0.965934 + 0.132185i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 3.27817 5.67796i 0.192170 0.332848i
\(292\) 0 0
\(293\) −13.1716 −0.769492 −0.384746 0.923023i \(-0.625711\pi\)
−0.384746 + 0.923023i \(0.625711\pi\)
\(294\) 0 0
\(295\) −6.24264 −0.363461
\(296\) 0 0
\(297\) −1.20711 + 2.09077i −0.0700434 + 0.121319i
\(298\) 0 0
\(299\) 4.94975 + 8.57321i 0.286251 + 0.495802i
\(300\) 0 0
\(301\) −20.9706 + 2.86976i −1.20872 + 0.165410i
\(302\) 0 0
\(303\) 1.55025 + 2.68512i 0.0890597 + 0.154256i
\(304\) 0 0
\(305\) 5.77817 10.0081i 0.330857 0.573062i
\(306\) 0 0
\(307\) −8.58579 −0.490017 −0.245008 0.969521i \(-0.578791\pi\)
−0.245008 + 0.969521i \(0.578791\pi\)
\(308\) 0 0
\(309\) 6.58579 0.374652
\(310\) 0 0
\(311\) −13.6066 + 23.5673i −0.771560 + 1.33638i 0.165148 + 0.986269i \(0.447190\pi\)
−0.936708 + 0.350112i \(0.886144\pi\)
\(312\) 0 0
\(313\) 4.67157 + 8.09140i 0.264053 + 0.457353i 0.967315 0.253578i \(-0.0816073\pi\)
−0.703262 + 0.710931i \(0.748274\pi\)
\(314\) 0 0
\(315\) −4.00000 + 9.79796i −0.225374 + 0.552052i
\(316\) 0 0
\(317\) −10.4853 18.1610i −0.588912 1.02003i −0.994375 0.105914i \(-0.966223\pi\)
0.405463 0.914111i \(-0.367110\pi\)
\(318\) 0 0
\(319\) 0.500000 0.866025i 0.0279946 0.0484881i
\(320\) 0 0
\(321\) −5.61522 −0.313411
\(322\) 0 0
\(323\) −5.17157 −0.287754
\(324\) 0 0
\(325\) 2.74264 4.75039i 0.152134 0.263504i
\(326\) 0 0
\(327\) 1.75736 + 3.04384i 0.0971822 + 0.168324i
\(328\) 0 0
\(329\) −7.82843 10.0951i −0.431595 0.556563i
\(330\) 0 0
\(331\) −2.69239 4.66335i −0.147987 0.256321i 0.782496 0.622655i \(-0.213946\pi\)
−0.930483 + 0.366334i \(0.880613\pi\)
\(332\) 0 0
\(333\) −12.8284 + 22.2195i −0.702993 + 1.21762i
\(334\) 0 0
\(335\) −11.4142 −0.623625
\(336\) 0 0
\(337\) −2.24264 −0.122164 −0.0610822 0.998133i \(-0.519455\pi\)
−0.0610822 + 0.998133i \(0.519455\pi\)
\(338\) 0 0
\(339\) 1.55025 2.68512i 0.0841982 0.145835i
\(340\) 0 0
\(341\) 4.82843 + 8.36308i 0.261474 + 0.452886i
\(342\) 0 0
\(343\) −17.0000 + 7.34847i −0.917914 + 0.396780i
\(344\) 0 0
\(345\) 1.58579 + 2.74666i 0.0853759 + 0.147875i
\(346\) 0 0
\(347\) −4.29289 + 7.43551i −0.230455 + 0.399159i −0.957942 0.286962i \(-0.907355\pi\)
0.727487 + 0.686121i \(0.240688\pi\)
\(348\) 0 0
\(349\) 24.2843 1.29991 0.649954 0.759974i \(-0.274788\pi\)
0.649954 + 0.759974i \(0.274788\pi\)
\(350\) 0 0
\(351\) −4.41421 −0.235613
\(352\) 0 0
\(353\) 1.34315 2.32640i 0.0714884 0.123822i −0.828065 0.560632i \(-0.810558\pi\)
0.899554 + 0.436810i \(0.143892\pi\)
\(354\) 0 0
\(355\) −4.07107 7.05130i −0.216070 0.374244i
\(356\) 0 0
\(357\) −1.34315 1.73205i −0.0710868 0.0916698i
\(358\) 0 0
\(359\) 6.96447 + 12.0628i 0.367570 + 0.636651i 0.989185 0.146672i \(-0.0468563\pi\)
−0.621615 + 0.783323i \(0.713523\pi\)
\(360\) 0 0
\(361\) 6.15685 10.6640i 0.324045 0.561262i
\(362\) 0 0
\(363\) 0.414214 0.0217406
\(364\) 0 0
\(365\) −4.82843 −0.252731
\(366\) 0 0
\(367\) 4.12132 7.13834i 0.215131 0.372618i −0.738182 0.674602i \(-0.764315\pi\)
0.953313 + 0.301984i \(0.0976488\pi\)
\(368\) 0 0
\(369\) 8.82843 + 15.2913i 0.459590 + 0.796032i
\(370\) 0 0
\(371\) 2.58579 6.33386i 0.134247 0.328837i
\(372\) 0 0
\(373\) 6.81371 + 11.8017i 0.352800 + 0.611068i 0.986739 0.162315i \(-0.0518961\pi\)
−0.633939 + 0.773383i \(0.718563\pi\)
\(374\) 0 0
\(375\) 2.34315 4.05845i 0.121000 0.209577i
\(376\) 0 0
\(377\) 1.82843 0.0941688
\(378\) 0 0
\(379\) 7.38478 0.379330 0.189665 0.981849i \(-0.439260\pi\)
0.189665 + 0.981849i \(0.439260\pi\)
\(380\) 0 0
\(381\) 1.32843 2.30090i 0.0680574 0.117879i
\(382\) 0 0
\(383\) −4.05025 7.01524i −0.206958 0.358462i 0.743797 0.668406i \(-0.233023\pi\)
−0.950755 + 0.309944i \(0.899690\pi\)
\(384\) 0 0
\(385\) 3.70711 0.507306i 0.188932 0.0258547i
\(386\) 0 0
\(387\) 11.3137 + 19.5959i 0.575108 + 0.996116i
\(388\) 0 0
\(389\) −16.6066 + 28.7635i −0.841988 + 1.45837i 0.0462232 + 0.998931i \(0.485281\pi\)
−0.888211 + 0.459435i \(0.848052\pi\)
\(390\) 0 0
\(391\) 10.8284 0.547617
\(392\) 0 0
\(393\) 2.04163 0.102987
\(394\) 0 0
\(395\) 5.70711 9.88500i 0.287156 0.497368i
\(396\) 0 0
\(397\) −17.8284 30.8797i −0.894783 1.54981i −0.834073 0.551654i \(-0.813997\pi\)
−0.0607103 0.998155i \(-0.519337\pi\)
\(398\) 0 0
\(399\) −2.80761 + 0.384213i −0.140556 + 0.0192347i
\(400\) 0 0
\(401\) 12.5000 + 21.6506i 0.624220 + 1.08118i 0.988691 + 0.149966i \(0.0479165\pi\)
−0.364471 + 0.931215i \(0.618750\pi\)
\(402\) 0 0
\(403\) −8.82843 + 15.2913i −0.439775 + 0.761713i
\(404\) 0 0
\(405\) 10.5858 0.526012
\(406\) 0 0
\(407\) 9.07107 0.449636
\(408\) 0 0
\(409\) 16.9497 29.3578i 0.838111 1.45165i −0.0533610 0.998575i \(-0.516993\pi\)
0.891472 0.453076i \(-0.149673\pi\)
\(410\) 0 0
\(411\) −3.52082 6.09823i −0.173669 0.300804i
\(412\) 0 0
\(413\) −4.41421 + 10.8126i −0.217209 + 0.532052i
\(414\) 0 0
\(415\) −7.41421 12.8418i −0.363949 0.630379i
\(416\) 0 0
\(417\) −0.686292 + 1.18869i −0.0336078 + 0.0582105i
\(418\) 0 0
\(419\) 25.4558 1.24360 0.621800 0.783176i \(-0.286402\pi\)
0.621800 + 0.783176i \(0.286402\pi\)
\(420\) 0 0
\(421\) 9.65685 0.470646 0.235323 0.971917i \(-0.424385\pi\)
0.235323 + 0.971917i \(0.424385\pi\)
\(422\) 0 0
\(423\) −6.82843 + 11.8272i −0.332009 + 0.575057i
\(424\) 0 0
\(425\) −3.00000 5.19615i −0.145521 0.252050i
\(426\) 0 0
\(427\) −13.2487 17.0849i −0.641151 0.826795i
\(428\) 0 0
\(429\) 0.378680 + 0.655892i 0.0182828 + 0.0316668i
\(430\) 0 0
\(431\) 3.86396 6.69258i 0.186120 0.322370i −0.757833 0.652449i \(-0.773742\pi\)
0.943954 + 0.330078i \(0.107075\pi\)
\(432\) 0 0
\(433\) −13.4558 −0.646647 −0.323323 0.946289i \(-0.604800\pi\)
−0.323323 + 0.946289i \(0.604800\pi\)
\(434\) 0 0
\(435\) 0.585786 0.0280863
\(436\) 0 0
\(437\) 7.00000 12.1244i 0.334855 0.579987i
\(438\) 0 0
\(439\) 0.964466 + 1.67050i 0.0460315 + 0.0797288i 0.888123 0.459606i \(-0.152009\pi\)
−0.842092 + 0.539334i \(0.818676\pi\)
\(440\) 0 0
\(441\) 14.1421 + 13.8564i 0.673435 + 0.659829i
\(442\) 0 0
\(443\) 2.65685 + 4.60181i 0.126231 + 0.218638i 0.922213 0.386681i \(-0.126379\pi\)
−0.795982 + 0.605320i \(0.793045\pi\)
\(444\) 0 0
\(445\) 6.48528 11.2328i 0.307432 0.532488i
\(446\) 0 0
\(447\) −6.62742 −0.313466
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) 3.12132 5.40629i 0.146977 0.254572i
\(452\) 0 0
\(453\) 1.15685 + 2.00373i 0.0543538 + 0.0941435i
\(454\) 0 0
\(455\) 4.19239 + 5.40629i 0.196542 + 0.253451i
\(456\) 0 0
\(457\) −14.6569 25.3864i −0.685619 1.18753i −0.973242 0.229783i \(-0.926198\pi\)
0.287623 0.957744i \(-0.407135\pi\)
\(458\) 0 0
\(459\) −2.41421 + 4.18154i −0.112686 + 0.195178i
\(460\) 0 0
\(461\) −13.0000 −0.605470 −0.302735 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 0 0
\(463\) 13.7990 0.641293 0.320647 0.947199i \(-0.396100\pi\)
0.320647 + 0.947199i \(0.396100\pi\)
\(464\) 0 0
\(465\) −2.82843 + 4.89898i −0.131165 + 0.227185i
\(466\) 0 0
\(467\) 17.9706 + 31.1259i 0.831578 + 1.44034i 0.896786 + 0.442464i \(0.145896\pi\)
−0.0652076 + 0.997872i \(0.520771\pi\)
\(468\) 0 0
\(469\) −8.07107 + 19.7700i −0.372687 + 0.912894i
\(470\) 0 0
\(471\) 1.51472 + 2.62357i 0.0697946 + 0.120888i
\(472\) 0 0
\(473\) 4.00000 6.92820i 0.183920 0.318559i
\(474\) 0 0
\(475\) −7.75736 −0.355932
\(476\) 0 0
\(477\) −7.31371 −0.334872
\(478\) 0 0
\(479\) 7.03553 12.1859i 0.321462 0.556788i −0.659328 0.751855i \(-0.729159\pi\)
0.980790 + 0.195067i \(0.0624925\pi\)
\(480\) 0 0
\(481\) 8.29289 + 14.3637i 0.378123 + 0.654929i
\(482\) 0 0
\(483\) 5.87868 0.804479i 0.267489 0.0366051i
\(484\) 0 0
\(485\) 11.1924 + 19.3858i 0.508220 + 0.880263i
\(486\) 0 0
\(487\) −11.6569 + 20.1903i −0.528222 + 0.914908i 0.471236 + 0.882007i \(0.343808\pi\)
−0.999459 + 0.0329011i \(0.989525\pi\)
\(488\) 0 0
\(489\) −2.11270 −0.0955395
\(490\) 0 0
\(491\) 0.828427 0.0373864 0.0186932 0.999825i \(-0.494049\pi\)
0.0186932 + 0.999825i \(0.494049\pi\)
\(492\) 0 0
\(493\) 1.00000 1.73205i 0.0450377 0.0780076i
\(494\) 0 0
\(495\) −2.00000 3.46410i −0.0898933 0.155700i
\(496\) 0 0
\(497\) −15.0919 + 2.06528i −0.676963 + 0.0926403i
\(498\) 0 0
\(499\) −11.8995 20.6105i −0.532695 0.922654i −0.999271 0.0381732i \(-0.987846\pi\)
0.466577 0.884481i \(-0.345487\pi\)
\(500\) 0 0
\(501\) −0.0563492 + 0.0975997i −0.00251750 + 0.00436043i
\(502\) 0 0
\(503\) 19.7279 0.879625 0.439812 0.898090i \(-0.355045\pi\)
0.439812 + 0.898090i \(0.355045\pi\)
\(504\) 0 0
\(505\) −10.5858 −0.471061
\(506\) 0 0
\(507\) 2.00000 3.46410i 0.0888231 0.153846i
\(508\) 0 0
\(509\) −17.8284 30.8797i −0.790231 1.36872i −0.925824 0.377956i \(-0.876627\pi\)
0.135593 0.990765i \(-0.456706\pi\)
\(510\) 0 0
\(511\) −3.41421 + 8.36308i −0.151036 + 0.369961i
\(512\) 0 0
\(513\) 3.12132 + 5.40629i 0.137810 + 0.238693i
\(514\) 0 0
\(515\) −11.2426 + 19.4728i −0.495410 + 0.858075i
\(516\) 0 0
\(517\) 4.82843 0.212354
\(518\) 0 0
\(519\) 0.899495 0.0394834
\(520\) 0 0
\(521\) −6.00000 + 10.3923i −0.262865 + 0.455295i −0.967002 0.254769i \(-0.918001\pi\)
0.704137 + 0.710064i \(0.251334\pi\)
\(522\) 0 0
\(523\) −6.53553 11.3199i −0.285779 0.494984i 0.687019 0.726640i \(-0.258919\pi\)
−0.972798 + 0.231656i \(0.925586\pi\)
\(524\) 0 0
\(525\) −2.01472 2.59808i −0.0879295 0.113389i
\(526\) 0 0
\(527\) 9.65685 + 16.7262i 0.420659 + 0.728603i
\(528\) 0 0
\(529\) −3.15685 + 5.46783i −0.137255 + 0.237732i
\(530\) 0 0
\(531\) 12.4853 0.541815
\(532\) 0 0
\(533\) 11.4142 0.494404
\(534\) 0 0
\(535\) 9.58579 16.6031i 0.414430 0.717813i
\(536\) 0 0
\(537\) 3.91421 + 6.77962i 0.168911 + 0.292562i
\(538\) 0 0
\(539\) 1.74264 6.77962i 0.0750608 0.292019i
\(540\) 0 0
\(541\) 4.42893 + 7.67114i 0.190415 + 0.329808i 0.945388 0.325948i \(-0.105683\pi\)
−0.754973 + 0.655756i \(0.772350\pi\)
\(542\) 0 0
\(543\) 2.75736 4.77589i 0.118330 0.204953i
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 43.6985 1.86841 0.934206 0.356734i \(-0.116110\pi\)
0.934206 + 0.356734i \(0.116110\pi\)
\(548\) 0 0
\(549\) −11.5563 + 20.0162i −0.493213 + 0.854270i
\(550\) 0 0
\(551\) −1.29289 2.23936i −0.0550791 0.0953998i
\(552\) 0 0
\(553\) −13.0858 16.8747i −0.556464 0.717587i
\(554\) 0 0
\(555\) 2.65685 + 4.60181i 0.112777 + 0.195336i
\(556\) 0 0
\(557\) 9.41421 16.3059i 0.398893 0.690903i −0.594697 0.803950i \(-0.702728\pi\)
0.993590 + 0.113047i \(0.0360611\pi\)
\(558\) 0 0
\(559\) 14.6274 0.618674
\(560\) 0 0
\(561\) 0.828427 0.0349762
\(562\) 0 0
\(563\) 1.39340 2.41344i 0.0587247 0.101714i −0.835168 0.549994i \(-0.814630\pi\)
0.893893 + 0.448280i \(0.147963\pi\)
\(564\) 0 0
\(565\) 5.29289 + 9.16756i 0.222674 + 0.385682i
\(566\) 0 0
\(567\) 7.48528 18.3351i 0.314352 0.770003i
\(568\) 0 0
\(569\) 6.48528 + 11.2328i 0.271877 + 0.470905i 0.969343 0.245713i \(-0.0790222\pi\)
−0.697465 + 0.716618i \(0.745689\pi\)
\(570\) 0 0
\(571\) 3.70711 6.42090i 0.155138 0.268706i −0.777972 0.628300i \(-0.783751\pi\)
0.933109 + 0.359593i \(0.117085\pi\)
\(572\) 0 0
\(573\) 2.68629 0.112221
\(574\) 0 0
\(575\) 16.2426 0.677365
\(576\) 0 0
\(577\) −7.98528 + 13.8309i −0.332432 + 0.575788i −0.982988 0.183669i \(-0.941202\pi\)
0.650556 + 0.759458i \(0.274536\pi\)
\(578\) 0 0
\(579\) −4.80761 8.32703i −0.199798 0.346059i
\(580\) 0 0
\(581\) −27.4853 + 3.76127i −1.14028 + 0.156044i
\(582\) 0 0
\(583\) 1.29289 + 2.23936i 0.0535462 + 0.0927447i
\(584\) 0 0
\(585\) 3.65685 6.33386i 0.151192 0.261873i
\(586\) 0 0
\(587\) −16.2132 −0.669191 −0.334595 0.942362i \(-0.608600\pi\)
−0.334595 + 0.942362i \(0.608600\pi\)
\(588\) 0 0
\(589\) 24.9706 1.02889
\(590\) 0 0
\(591\) 2.30761 3.99690i 0.0949225 0.164411i
\(592\) 0 0
\(593\) 5.29289 + 9.16756i 0.217353 + 0.376467i 0.953998 0.299813i \(-0.0969244\pi\)
−0.736645 + 0.676280i \(0.763591\pi\)
\(594\) 0 0
\(595\) 7.41421 1.01461i 0.303953 0.0415950i
\(596\) 0 0
\(597\) 4.97918 + 8.62420i 0.203784 + 0.352965i
\(598\) 0 0
\(599\) 22.4853 38.9456i 0.918724 1.59128i 0.117368 0.993089i \(-0.462554\pi\)
0.801356 0.598188i \(-0.204112\pi\)
\(600\) 0 0
\(601\) −23.6569 −0.964983 −0.482492 0.875901i \(-0.660268\pi\)
−0.482492 + 0.875901i \(0.660268\pi\)
\(602\) 0 0
\(603\) 22.8284 0.929645
\(604\) 0 0
\(605\) −0.707107 + 1.22474i −0.0287480 + 0.0497930i
\(606\) 0 0
\(607\) 11.1421 + 19.2987i 0.452245 + 0.783312i 0.998525 0.0542908i \(-0.0172898\pi\)
−0.546280 + 0.837603i \(0.683956\pi\)
\(608\) 0 0
\(609\) 0.414214 1.01461i 0.0167848 0.0411141i
\(610\) 0 0
\(611\) 4.41421 + 7.64564i 0.178580 + 0.309310i
\(612\) 0 0
\(613\) −14.3137 + 24.7921i −0.578125 + 1.00134i 0.417569 + 0.908645i \(0.362882\pi\)
−0.995694 + 0.0926971i \(0.970451\pi\)
\(614\) 0 0
\(615\) 3.65685 0.147459
\(616\) 0 0
\(617\) 33.6274 1.35379 0.676894 0.736080i \(-0.263325\pi\)
0.676894 + 0.736080i \(0.263325\pi\)
\(618\) 0 0
\(619\) 0.343146 0.594346i 0.0137922 0.0238888i −0.859047 0.511897i \(-0.828943\pi\)
0.872839 + 0.488008i \(0.162276\pi\)
\(620\) 0 0
\(621\) −6.53553 11.3199i −0.262262 0.454251i
\(622\) 0 0
\(623\) −14.8701 19.1757i −0.595756 0.768256i
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0.535534 0.927572i 0.0213872 0.0370437i
\(628\) 0 0
\(629\) 18.1421 0.723374
\(630\) 0 0
\(631\) 4.38478 0.174555 0.0872776 0.996184i \(-0.472183\pi\)
0.0872776 + 0.996184i \(0.472183\pi\)
\(632\) 0 0
\(633\) −1.02082 + 1.76810i −0.0405738 + 0.0702758i
\(634\) 0 0
\(635\) 4.53553 + 7.85578i 0.179987 + 0.311747i
\(636\) 0 0
\(637\) 12.3284 3.43861i 0.488470 0.136243i
\(638\) 0 0
\(639\) 8.14214 + 14.1026i 0.322098 + 0.557890i
\(640\) 0 0
\(641\) −22.1274 + 38.3258i −0.873980 + 1.51378i −0.0161355 + 0.999870i \(0.505136\pi\)
−0.857845 + 0.513909i \(0.828197\pi\)
\(642\) 0 0
\(643\) −35.5858 −1.40337 −0.701683 0.712489i \(-0.747568\pi\)
−0.701683 + 0.712489i \(0.747568\pi\)
\(644\) 0 0
\(645\) 4.68629 0.184523
\(646\) 0 0
\(647\) −6.53553 + 11.3199i −0.256938 + 0.445030i −0.965420 0.260699i \(-0.916047\pi\)
0.708482 + 0.705729i \(0.249380\pi\)
\(648\) 0 0
\(649\) −2.20711 3.82282i −0.0866365 0.150059i
\(650\) 0 0
\(651\) 6.48528 + 8.36308i 0.254178 + 0.327775i
\(652\) 0 0
\(653\) 24.5355 + 42.4968i 0.960150 + 1.66303i 0.722117 + 0.691771i \(0.243169\pi\)
0.238032 + 0.971257i \(0.423498\pi\)
\(654\) 0 0
\(655\) −3.48528 + 6.03668i −0.136181 + 0.235873i
\(656\) 0 0
\(657\) 9.65685 0.376750
\(658\) 0 0
\(659\) −31.3137 −1.21981 −0.609904 0.792475i \(-0.708792\pi\)
−0.609904 + 0.792475i \(0.708792\pi\)
\(660\) 0 0
\(661\) 2.31371 4.00746i 0.0899928 0.155872i −0.817515 0.575907i \(-0.804649\pi\)
0.907508 + 0.420035i \(0.137982\pi\)
\(662\) 0 0
\(663\) 0.757359 + 1.31178i 0.0294134 + 0.0509455i
\(664\) 0 0
\(665\) 3.65685 8.95743i 0.141807 0.347354i
\(666\) 0 0
\(667\) 2.70711 + 4.68885i 0.104820 + 0.181553i
\(668\) 0 0
\(669\) 4.05025 7.01524i 0.156592 0.271225i
\(670\) 0 0
\(671\) 8.17157 0.315460
\(672\) 0 0
\(673\) 16.2426 0.626108 0.313054 0.949735i \(-0.398648\pi\)
0.313054 + 0.949735i \(0.398648\pi\)
\(674\) 0 0
\(675\) −3.62132 + 6.27231i −0.139385 + 0.241421i
\(676\) 0 0
\(677\) −4.14214 7.17439i −0.159195 0.275734i 0.775383 0.631491i \(-0.217557\pi\)
−0.934579 + 0.355756i \(0.884223\pi\)
\(678\) 0 0
\(679\) 41.4914 5.67796i 1.59229 0.217900i
\(680\) 0 0
\(681\) −5.82843 10.0951i −0.223346 0.386846i
\(682\) 0 0
\(683\) 7.44975 12.9033i 0.285057 0.493733i −0.687566 0.726122i \(-0.741321\pi\)
0.972623 + 0.232389i \(0.0746542\pi\)
\(684\) 0 0
\(685\) 24.0416 0.918583
\(686\) 0 0
\(687\) −1.65685 −0.0632129
\(688\) 0 0
\(689\) −2.36396 + 4.09450i −0.0900597 + 0.155988i
\(690\) 0 0
\(691\) 14.6924 + 25.4480i 0.558925 + 0.968086i 0.997587 + 0.0694339i \(0.0221193\pi\)
−0.438662 + 0.898652i \(0.644547\pi\)
\(692\) 0 0
\(693\) −7.41421 + 1.01461i −0.281643 + 0.0385419i
\(694\) 0 0
\(695\) −2.34315 4.05845i −0.0888806 0.153946i
\(696\) 0 0
\(697\) 6.24264 10.8126i 0.236457 0.409555i
\(698\) 0 0
\(699\) 8.44365 0.319368
\(700\) 0 0
\(701\) 42.4558 1.60354 0.801768 0.597636i \(-0.203893\pi\)
0.801768 + 0.597636i \(0.203893\pi\)
\(702\) 0 0
\(703\) 11.7279 20.3134i 0.442327 0.766133i
\(704\) 0 0
\(705\) 1.41421 + 2.44949i 0.0532624 + 0.0922531i
\(706\) 0 0
\(707\) −7.48528 + 18.3351i −0.281513 + 0.689563i
\(708\) 0 0
\(709\) −9.46447 16.3929i −0.355445 0.615650i 0.631749 0.775173i \(-0.282337\pi\)
−0.987194 + 0.159524i \(0.949004\pi\)
\(710\) 0 0
\(711\) −11.4142 + 19.7700i −0.428066 + 0.741433i
\(712\) 0 0
\(713\) −52.2843 −1.95806
\(714\) 0 0
\(715\) −2.58579 −0.0967029
\(716\) 0 0
\(717\) −2.42893 + 4.20703i −0.0907101 + 0.157115i
\(718\) 0 0
\(719\) 14.4142 + 24.9662i 0.537559 + 0.931080i 0.999035 + 0.0439272i \(0.0139870\pi\)
−0.461475 + 0.887153i \(0.652680\pi\)
\(720\) 0 0
\(721\) 25.7782 + 33.2422i 0.960029 + 1.23800i
\(722\) 0 0
\(723\) 4.70711 + 8.15295i 0.175059 + 0.303211i
\(724\) 0 0
\(725\) 1.50000 2.59808i 0.0557086 0.0964901i
\(726\) 0 0
\(727\) 13.7990 0.511776 0.255888 0.966706i \(-0.417632\pi\)
0.255888 + 0.966706i \(0.417632\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 8.00000 13.8564i 0.295891 0.512498i
\(732\) 0 0
\(733\) −14.3284 24.8176i −0.529233 0.916658i −0.999419 0.0340904i \(-0.989147\pi\)
0.470186 0.882567i \(-0.344187\pi\)
\(734\) 0 0
\(735\) 3.94975 1.10165i 0.145689 0.0406350i
\(736\) 0 0
\(737\) −4.03553 6.98975i −0.148651 0.257471i
\(738\) 0 0
\(739\) 18.2132 31.5462i 0.669984 1.16045i −0.307925 0.951411i \(-0.599634\pi\)
0.977908 0.209035i \(-0.0670322\pi\)
\(740\) 0 0
\(741\) 1.95837 0.0719425
\(742\) 0 0
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) 11.3137 19.5959i 0.414502 0.717939i
\(746\) 0 0
\(747\) 14.8284 + 25.6836i 0.542544 + 0.939713i
\(748\) 0 0
\(749\) −21.9792 28.3432i −0.803102 1.03564i
\(750\) 0 0
\(751\) 5.65685 + 9.79796i 0.206422 + 0.357533i 0.950585 0.310465i \(-0.100485\pi\)
−0.744163 + 0.667998i \(0.767152\pi\)
\(752\) 0 0
\(753\) 0.100505 0.174080i 0.00366261 0.00634382i
\(754\) 0 0
\(755\) −7.89949 −0.287492
\(756\) 0 0
\(757\) 17.3137 0.629277 0.314639 0.949212i \(-0.398117\pi\)
0.314639 + 0.949212i \(0.398117\pi\)
\(758\) 0 0
\(759\) −1.12132 + 1.94218i −0.0407013 + 0.0704968i
\(760\) 0 0
\(761\) 13.9706 + 24.1977i 0.506433 + 0.877167i 0.999972 + 0.00744361i \(0.00236940\pi\)
−0.493540 + 0.869723i \(0.664297\pi\)
\(762\) 0 0
\(763\) −8.48528 + 20.7846i −0.307188 + 0.752453i
\(764\) 0 0
\(765\) −4.00000 6.92820i −0.144620 0.250490i
\(766\) 0 0
\(767\) 4.03553 6.98975i 0.145715 0.252385i
\(768\) 0 0
\(769\) 30.9706 1.11683 0.558414 0.829563i \(-0.311410\pi\)
0.558414 + 0.829563i \(0.311410\pi\)
\(770\) 0 0
\(771\) −8.07107 −0.290672
\(772\) 0 0
\(773\) −17.3137 + 29.9882i −0.622731 + 1.07860i 0.366244 + 0.930519i \(0.380643\pi\)
−0.988975 + 0.148083i \(0.952690\pi\)
\(774\) 0 0
\(775\) 14.4853 + 25.0892i 0.520327 + 0.901232i
\(776\) 0 0
\(777\) 9.84924 1.34784i 0.353340 0.0483534i
\(778\) 0 0
\(779\) −8.07107 13.9795i −0.289176 0.500868i
\(780\) 0 0
\(781\) 2.87868 4.98602i 0.103007 0.178414i
\(782\) 0 0
\(783\) −2.41421 −0.0862770
\(784\) 0 0
\(785\) −10.3431 −0.369163
\(786\) 0 0
\(787\) 2.80761 4.86293i 0.100081 0.173345i −0.811637 0.584162i \(-0.801423\pi\)
0.911718 + 0.410817i \(0.134757\pi\)
\(788\) 0 0
\(789\) −4.77208 8.26548i −0.169890 0.294259i
\(790\) 0 0
\(791\) 19.6213 2.68512i 0.697654 0.0954717i
\(792\) 0 0
\(793\) 7.47056 + 12.9394i 0.265287 + 0.459491i
\(794\) 0 0
\(795\) −0.757359 + 1.31178i −0.0268608 + 0.0465242i
\(796\) 0 0
\(797\) 9.31371 0.329908 0.164954 0.986301i \(-0.447252\pi\)
0.164954 + 0.986301i \(0.447252\pi\)
\(798\) 0 0
\(799\) 9.65685 0.341635
\(800\) 0 0
\(801\) −12.9706 + 22.4657i −0.458292 + 0.793786i
\(802\) 0 0
\(803\) −1.70711 2.95680i −0.0602425 0.104343i
\(804\) 0 0
\(805\) −7.65685 + 18.7554i −0.269869 + 0.661040i
\(806\) 0 0
\(807\) 2.82843 + 4.89898i 0.0995654 + 0.172452i
\(808\) 0 0
\(809\) 4.00000 6.92820i 0.140633 0.243583i −0.787102 0.616822i \(-0.788420\pi\)
0.927735 + 0.373240i \(0.121753\pi\)
\(810\) 0 0
\(811\) −49.6569 −1.74369 −0.871844 0.489784i \(-0.837076\pi\)
−0.871844 + 0.489784i \(0.837076\pi\)
\(812\) 0 0
\(813\) 0.0294373 0.00103241
\(814\) 0 0
\(815\) 3.60660 6.24682i 0.126334 0.218816i
\(816\) 0 0
\(817\) −10.3431 17.9149i −0.361861 0.626761i
\(818\) 0 0
\(819\) −8.38478 10.8126i −0.292988 0.377822i
\(820\) 0 0
\(821\) 11.0858 + 19.2011i 0.386897 + 0.670125i 0.992030 0.125999i \(-0.0402137\pi\)
−0.605134 + 0.796124i \(0.706880\pi\)
\(822\) 0 0
\(823\) −13.5858 + 23.5313i −0.473571 + 0.820249i −0.999542 0.0302536i \(-0.990369\pi\)
0.525972 + 0.850502i \(0.323702\pi\)
\(824\) 0 0
\(825\) 1.24264 0.0432632
\(826\) 0 0
\(827\) −32.5269 −1.13107 −0.565536 0.824724i \(-0.691331\pi\)
−0.565536 + 0.824724i \(0.691331\pi\)
\(828\) 0 0
\(829\) −18.7071 + 32.4017i −0.649725 + 1.12536i 0.333464 + 0.942763i \(0.391782\pi\)
−0.983189 + 0.182593i \(0.941551\pi\)
\(830\) 0 0
\(831\) 5.13604 + 8.89588i 0.178167 + 0.308595i
\(832\) 0 0
\(833\) 3.48528 13.5592i 0.120758 0.469800i
\(834\) 0 0
\(835\) −0.192388 0.333226i −0.00665787 0.0115318i
\(836\) 0 0
\(837\) 11.6569 20.1903i 0.402920 0.697878i
\(838\) 0 0
\(839\) −26.4853 −0.914373 −0.457187 0.889371i \(-0.651143\pi\)
−0.457187 + 0.889371i \(0.651143\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 2.46447 4.26858i 0.0848807 0.147018i
\(844\) 0 0
\(845\) 6.82843 + 11.8272i 0.234905 + 0.406867i
\(846\) 0 0
\(847\) 1.62132 + 2.09077i 0.0557092 + 0.0718397i
\(848\) 0 0
\(849\) 5.70711 + 9.88500i 0.195867 + 0.339252i
\(850\) 0 0
\(851\) −24.5563 + 42.5328i −0.841781 + 1.45801i
\(852\) 0 0
\(853\) −50.0000 −1.71197 −0.855984 0.517003i \(-0.827048\pi\)
−0.855984 + 0.517003i \(0.827048\pi\)
\(854\) 0 0
\(855\) −10.3431 −0.353728
\(856\) 0 0
\(857\) 24.1421 41.8154i 0.824680 1.42839i −0.0774842 0.996994i \(-0.524689\pi\)
0.902164 0.431393i \(-0.141978\pi\)
\(858\) 0 0
\(859\) −6.20711 10.7510i −0.211784 0.366820i 0.740489 0.672068i \(-0.234594\pi\)
−0.952273 + 0.305248i \(0.901261\pi\)
\(860\) 0 0
\(861\) 2.58579 6.33386i 0.0881234 0.215857i
\(862\) 0 0
\(863\) −13.1924 22.8499i −0.449074 0.777819i 0.549252 0.835657i \(-0.314913\pi\)
−0.998326 + 0.0578377i \(0.981579\pi\)
\(864\) 0 0
\(865\) −1.53553 + 2.65962i −0.0522097 + 0.0904299i
\(866\) 0 0
\(867\) −5.38478 −0.182877
\(868\) 0 0
\(869\) 8.07107 0.273792
\(870\) 0 0
\(871\) 7.37868 12.7802i 0.250017 0.433042i
\(872\) 0 0
\(873\) −22.3848 38.7716i −0.757610 1.31222i
\(874\) 0 0
\(875\) 29.6569 4.05845i 1.00258 0.137201i
\(876\) 0 0
\(877\) −6.64214 11.5045i −0.224289 0.388480i 0.731817 0.681501i \(-0.238673\pi\)
−0.956106 + 0.293021i \(0.905339\pi\)
\(878\) 0 0
\(879\) 2.72792 4.72490i 0.0920105 0.159367i
\(880\) 0 0
\(881\) 19.8284 0.668037 0.334018 0.942567i \(-0.391595\pi\)
0.334018 + 0.942567i \(0.391595\pi\)
\(882\) 0 0
\(883\) −39.6690 −1.33497 −0.667485 0.744623i \(-0.732629\pi\)
−0.667485 + 0.744623i \(0.732629\pi\)
\(884\) 0 0
\(885\) 1.29289 2.23936i 0.0434601 0.0752752i
\(886\) 0 0
\(887\) −15.8640 27.4772i −0.532660 0.922594i −0.999273 0.0381321i \(-0.987859\pi\)
0.466613 0.884462i \(-0.345474\pi\)
\(888\) 0 0
\(889\) 16.8137 2.30090i 0.563914 0.0771698i
\(890\) 0 0
\(891\) 3.74264 + 6.48244i 0.125383 + 0.217170i
\(892\) 0 0
\(893\) 6.24264 10.8126i 0.208902 0.361829i
\(894\) 0 0
\(895\) −26.7279 −0.893416
\(896\) 0 0
\(897\) −4.10051 −0.136912
\(898\) 0 0
\(899\) −4.82843 + 8.36308i −0.161037 + 0.278924i
\(900\) 0 0
\(901\) 2.58579 + 4.47871i 0.0861450 + 0.149208i
\(902\) 0 0
\(903\) 3.31371 8.11689i 0.110273 0.270113i
\(904\) 0 0
\(905\) 9.41421 + 16.3059i 0.312939 + 0.542026i
\(906\) 0 0
\(907\) −6.31371 + 10.9357i −0.209643 + 0.363113i −0.951602 0.307333i \(-0.900564\pi\)
0.741959 + 0.670445i \(0.233897\pi\)
\(908\) 0 0
\(909\) 21.1716 0.702217
\(910\) 0 0
\(911\) −38.4853 −1.27507 −0.637537 0.770420i \(-0.720047\pi\)
−0.637537 + 0.770420i \(0.720047\pi\)
\(912\) 0 0
\(913\) 5.24264 9.08052i 0.173506 0.300521i
\(914\) 0 0
\(915\) 2.39340 + 4.14549i 0.0791233 + 0.137046i
\(916\) 0 0
\(917\) 7.99138 + 10.3053i 0.263899 + 0.340310i
\(918\) 0 0
\(919\) 2.72792 + 4.72490i 0.0899858 + 0.155860i 0.907505 0.420041i \(-0.137984\pi\)
−0.817519 + 0.575902i \(0.804651\pi\)
\(920\) 0 0
\(921\) 1.77817 3.07989i 0.0585928 0.101486i
\(922\) 0 0
\(923\) 10.5269 0.346498
\(924\) 0 0
\(925\) 27.2132 0.894765
\(926\) 0 0
\(927\) 22.4853 38.9456i 0.738514 1.27914i
\(928\) 0 0
\(929\) 20.7132 + 35.8763i 0.679578 + 1.17706i 0.975108 + 0.221730i \(0.0711705\pi\)
−0.295530 + 0.955334i \(0.595496\pi\)
\(930\) 0 0
\(931\) −12.9289 12.6677i −0.423729 0.415168i
\(932\) 0 0
\(933\) −5.63604 9.76191i −0.184516 0.319590i
\(934\) 0 0
\(935\) −1.41421 + 2.44949i −0.0462497 + 0.0801069i
\(936\) 0 0
\(937\) −21.4142 −0.699572 −0.349786 0.936830i \(-0.613746\pi\)
−0.349786 + 0.936830i \(0.613746\pi\)
\(938\) 0 0
\(939\) −3.87006 −0.126295
\(940\) 0 0
\(941\) 25.4706 44.1163i 0.830317 1.43815i −0.0674707 0.997721i \(-0.521493\pi\)
0.897787 0.440429i \(-0.145174\pi\)
\(942\) 0 0
\(943\) 16.8995 + 29.2708i 0.550323 + 0.953188i
\(944\) 0 0
\(945\) −5.53553 7.13834i −0.180071 0.232210i
\(946\) 0 0
\(947\) 0.100505 + 0.174080i 0.00326598 + 0.00565684i 0.867654 0.497169i \(-0.165627\pi\)
−0.864388 + 0.502826i \(0.832294\pi\)
\(948\) 0 0
\(949\) 3.12132 5.40629i 0.101322 0.175495i
\(950\) 0 0
\(951\) 8.68629 0.281672
\(952\) 0 0
\(953\) 17.6152 0.570613 0.285307 0.958436i \(-0.407905\pi\)
0.285307 + 0.958436i \(0.407905\pi\)
\(954\) 0 0
\(955\) −4.58579 + 7.94282i −0.148393 + 0.257023i
\(956\) 0 0
\(957\) 0.207107 + 0.358719i 0.00669481 + 0.0115958i
\(958\) 0 0
\(959\) 17.0000 41.6413i 0.548959 1.34467i
\(960\) 0 0
\(961\) −31.1274 53.9143i −1.00411 1.73917i
\(962\) 0 0
\(963\) −19.1716 + 33.2061i −0.617795 + 1.07005i
\(964\) 0 0
\(965\) 32.8284 1.05678
\(966\) 0 0
\(967\) 18.3431 0.589876 0.294938 0.955516i \(-0.404701\pi\)
0.294938 + 0.955516i \(0.404701\pi\)
\(968\) 0 0
\(969\) 1.07107 1.85514i 0.0344077 0.0595958i
\(970\) 0 0
\(971\) 15.1066 + 26.1654i 0.484794 + 0.839688i 0.999847 0.0174704i \(-0.00556127\pi\)
−0.515053 + 0.857158i \(0.672228\pi\)
\(972\) 0 0
\(973\) −8.68629 + 1.18869i −0.278470 + 0.0381077i
\(974\) 0 0
\(975\) 1.13604 + 1.96768i 0.0363824 + 0.0630161i
\(976\) 0 0
\(977\) −1.51472 + 2.62357i −0.0484601 + 0.0839354i −0.889238 0.457445i \(-0.848765\pi\)
0.840778 + 0.541380i \(0.182098\pi\)
\(978\) 0 0
\(979\) 9.17157 0.293125
\(980\) 0 0
\(981\) 24.0000 0.766261
\(982\) 0 0
\(983\) −14.7574 + 25.5605i −0.470687 + 0.815253i −0.999438 0.0335236i \(-0.989327\pi\)
0.528751 + 0.848777i \(0.322660\pi\)
\(984\) 0 0
\(985\) 7.87868 + 13.6463i 0.251036 + 0.434806i
\(986\) 0 0
\(987\) 5.24264 0.717439i 0.166875 0.0228363i
\(988\) 0 0
\(989\) 21.6569 + 37.5108i 0.688648 + 1.19277i
\(990\) 0 0
\(991\) 0.0502525 0.0870399i 0.00159632 0.00276491i −0.865226 0.501382i \(-0.832825\pi\)
0.866822 + 0.498617i \(0.166159\pi\)
\(992\) 0 0
\(993\) 2.23045 0.0707811
\(994\) 0 0
\(995\) −34.0000 −1.07787
\(996\) 0 0
\(997\) −20.7279 + 35.9018i −0.656460 + 1.13702i 0.325066 + 0.945691i \(0.394614\pi\)
−0.981526 + 0.191330i \(0.938720\pi\)
\(998\) 0 0
\(999\) −10.9497 18.9655i −0.346435 0.600042i
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 1232.2.q.h.529.1 4
4.3 odd 2 616.2.q.b.529.2 yes 4
7.2 even 3 inner 1232.2.q.h.177.1 4
7.3 odd 6 8624.2.a.cd.1.1 2
7.4 even 3 8624.2.a.bg.1.2 2
28.3 even 6 4312.2.a.m.1.2 2
28.11 odd 6 4312.2.a.u.1.1 2
28.23 odd 6 616.2.q.b.177.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.q.b.177.2 4 28.23 odd 6
616.2.q.b.529.2 yes 4 4.3 odd 2
1232.2.q.h.177.1 4 7.2 even 3 inner
1232.2.q.h.529.1 4 1.1 even 1 trivial
4312.2.a.m.1.2 2 28.3 even 6
4312.2.a.u.1.1 2 28.11 odd 6
8624.2.a.bg.1.2 2 7.4 even 3
8624.2.a.cd.1.1 2 7.3 odd 6