Properties

Label 560.2.q.j.401.2
Level $560$
Weight $2$
Character 560.401
Analytic conductor $4.472$
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] = [560,2,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.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: \(4.47162251319\)
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 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 401.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 560.401
Dual form 560.2.q.j.81.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.207107 + 0.358719i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(1.41421 - 2.44949i) q^{9} +O(q^{10})\) \(q+(0.207107 + 0.358719i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-1.62132 + 2.09077i) q^{7} +(1.41421 - 2.44949i) q^{9} +(0.414214 + 0.717439i) q^{11} +2.00000 q^{13} -0.414214 q^{15} +(3.82843 + 6.63103i) q^{17} +(-2.82843 + 4.89898i) q^{19} +(-1.08579 - 0.148586i) q^{21} +(-2.79289 + 4.83743i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.41421 q^{27} -7.82843 q^{29} +(0.414214 + 0.717439i) q^{31} +(-0.171573 + 0.297173i) q^{33} +(-1.00000 - 2.44949i) q^{35} +(-2.82843 + 4.89898i) q^{37} +(0.414214 + 0.717439i) q^{39} +5.82843 q^{41} +6.89949 q^{43} +(1.41421 + 2.44949i) q^{45} +(5.82843 - 10.0951i) q^{47} +(-1.74264 - 6.77962i) q^{49} +(-1.58579 + 2.74666i) q^{51} +(2.82843 + 4.89898i) q^{53} -0.828427 q^{55} -2.34315 q^{57} +(-2.00000 - 3.46410i) q^{59} +(-3.32843 + 5.76500i) q^{61} +(2.82843 + 6.92820i) q^{63} +(-1.00000 + 1.73205i) q^{65} +(-6.44975 - 11.1713i) q^{67} -2.31371 q^{69} +12.0000 q^{71} +(-1.82843 - 3.16693i) q^{73} +(0.207107 - 0.358719i) q^{75} +(-2.17157 - 0.297173i) q^{77} +(-2.00000 + 3.46410i) q^{79} +(-3.74264 - 6.48244i) q^{81} +4.75736 q^{83} -7.65685 q^{85} +(-1.62132 - 2.80821i) q^{87} +(2.67157 - 4.62730i) q^{89} +(-3.24264 + 4.18154i) q^{91} +(-0.171573 + 0.297173i) q^{93} +(-2.82843 - 4.89898i) q^{95} +6.00000 q^{97} +2.34315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{11} + 8 q^{13} + 4 q^{15} + 4 q^{17} - 10 q^{21} - 14 q^{23} - 2 q^{25} + 4 q^{27} - 20 q^{29} - 4 q^{31} - 12 q^{33} - 4 q^{35} - 4 q^{39} + 12 q^{41} - 12 q^{43} + 12 q^{47} + 10 q^{49} - 12 q^{51} + 8 q^{55} - 32 q^{57} - 8 q^{59} - 2 q^{61} - 4 q^{65} - 6 q^{67} + 36 q^{69} + 48 q^{71} + 4 q^{73} - 2 q^{75} - 20 q^{77} - 8 q^{79} + 2 q^{81} + 36 q^{83} - 8 q^{85} + 2 q^{87} + 22 q^{89} + 4 q^{91} - 12 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 \(-0.128514\pi\)
−0.800025 + 0.599966i \(0.795181\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −1.62132 + 2.09077i −0.612801 + 0.790237i
\(8\) 0 0
\(9\) 1.41421 2.44949i 0.471405 0.816497i
\(10\) 0 0
\(11\) 0.414214 + 0.717439i 0.124890 + 0.216316i 0.921690 0.387927i \(-0.126809\pi\)
−0.796800 + 0.604243i \(0.793476\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −0.414214 −0.106949
\(16\) 0 0
\(17\) 3.82843 + 6.63103i 0.928530 + 1.60826i 0.785783 + 0.618502i \(0.212260\pi\)
0.142747 + 0.989759i \(0.454407\pi\)
\(18\) 0 0
\(19\) −2.82843 + 4.89898i −0.648886 + 1.12390i 0.334504 + 0.942394i \(0.391431\pi\)
−0.983389 + 0.181509i \(0.941902\pi\)
\(20\) 0 0
\(21\) −1.08579 0.148586i −0.236938 0.0324242i
\(22\) 0 0
\(23\) −2.79289 + 4.83743i −0.582358 + 1.00867i 0.412841 + 0.910803i \(0.364537\pi\)
−0.995199 + 0.0978712i \(0.968797\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −7.82843 −1.45370 −0.726851 0.686795i \(-0.759017\pi\)
−0.726851 + 0.686795i \(0.759017\pi\)
\(30\) 0 0
\(31\) 0.414214 + 0.717439i 0.0743950 + 0.128856i 0.900823 0.434187i \(-0.142964\pi\)
−0.826428 + 0.563042i \(0.809631\pi\)
\(32\) 0 0
\(33\) −0.171573 + 0.297173i −0.0298670 + 0.0517312i
\(34\) 0 0
\(35\) −1.00000 2.44949i −0.169031 0.414039i
\(36\) 0 0
\(37\) −2.82843 + 4.89898i −0.464991 + 0.805387i −0.999201 0.0399642i \(-0.987276\pi\)
0.534211 + 0.845351i \(0.320609\pi\)
\(38\) 0 0
\(39\) 0.414214 + 0.717439i 0.0663273 + 0.114882i
\(40\) 0 0
\(41\) 5.82843 0.910247 0.455124 0.890428i \(-0.349595\pi\)
0.455124 + 0.890428i \(0.349595\pi\)
\(42\) 0 0
\(43\) 6.89949 1.05216 0.526082 0.850434i \(-0.323661\pi\)
0.526082 + 0.850434i \(0.323661\pi\)
\(44\) 0 0
\(45\) 1.41421 + 2.44949i 0.210819 + 0.365148i
\(46\) 0 0
\(47\) 5.82843 10.0951i 0.850163 1.47253i −0.0308969 0.999523i \(-0.509836\pi\)
0.881060 0.473004i \(-0.156830\pi\)
\(48\) 0 0
\(49\) −1.74264 6.77962i −0.248949 0.968517i
\(50\) 0 0
\(51\) −1.58579 + 2.74666i −0.222055 + 0.384610i
\(52\) 0 0
\(53\) 2.82843 + 4.89898i 0.388514 + 0.672927i 0.992250 0.124258i \(-0.0396551\pi\)
−0.603736 + 0.797185i \(0.706322\pi\)
\(54\) 0 0
\(55\) −0.828427 −0.111705
\(56\) 0 0
\(57\) −2.34315 −0.310357
\(58\) 0 0
\(59\) −2.00000 3.46410i −0.260378 0.450988i 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) −3.32843 + 5.76500i −0.426161 + 0.738133i −0.996528 0.0832569i \(-0.973468\pi\)
0.570367 + 0.821390i \(0.306801\pi\)
\(62\) 0 0
\(63\) 2.82843 + 6.92820i 0.356348 + 0.872872i
\(64\) 0 0
\(65\) −1.00000 + 1.73205i −0.124035 + 0.214834i
\(66\) 0 0
\(67\) −6.44975 11.1713i −0.787962 1.36479i −0.927213 0.374533i \(-0.877803\pi\)
0.139251 0.990257i \(-0.455530\pi\)
\(68\) 0 0
\(69\) −2.31371 −0.278538
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.82843 3.16693i −0.214001 0.370661i 0.738962 0.673747i \(-0.235316\pi\)
−0.952963 + 0.303086i \(0.901983\pi\)
\(74\) 0 0
\(75\) 0.207107 0.358719i 0.0239146 0.0414214i
\(76\) 0 0
\(77\) −2.17157 0.297173i −0.247474 0.0338660i
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −3.74264 6.48244i −0.415849 0.720272i
\(82\) 0 0
\(83\) 4.75736 0.522188 0.261094 0.965313i \(-0.415917\pi\)
0.261094 + 0.965313i \(0.415917\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) −1.62132 2.80821i −0.173824 0.301072i
\(88\) 0 0
\(89\) 2.67157 4.62730i 0.283186 0.490493i −0.688982 0.724779i \(-0.741942\pi\)
0.972168 + 0.234286i \(0.0752752\pi\)
\(90\) 0 0
\(91\) −3.24264 + 4.18154i −0.339921 + 0.438345i
\(92\) 0 0
\(93\) −0.171573 + 0.297173i −0.0177913 + 0.0308154i
\(94\) 0 0
\(95\) −2.82843 4.89898i −0.290191 0.502625i
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 2.34315 0.235495
\(100\) 0 0
\(101\) 5.74264 + 9.94655i 0.571414 + 0.989718i 0.996421 + 0.0845282i \(0.0269383\pi\)
−0.425007 + 0.905190i \(0.639728\pi\)
\(102\) 0 0
\(103\) 3.79289 6.56948i 0.373725 0.647310i −0.616410 0.787425i \(-0.711414\pi\)
0.990135 + 0.140115i \(0.0447471\pi\)
\(104\) 0 0
\(105\) 0.671573 0.866025i 0.0655388 0.0845154i
\(106\) 0 0
\(107\) −2.79289 + 4.83743i −0.269999 + 0.467652i −0.968861 0.247604i \(-0.920357\pi\)
0.698862 + 0.715256i \(0.253690\pi\)
\(108\) 0 0
\(109\) −9.15685 15.8601i −0.877068 1.51913i −0.854544 0.519379i \(-0.826163\pi\)
−0.0225237 0.999746i \(-0.507170\pi\)
\(110\) 0 0
\(111\) −2.34315 −0.222402
\(112\) 0 0
\(113\) 11.3137 1.06430 0.532152 0.846649i \(-0.321383\pi\)
0.532152 + 0.846649i \(0.321383\pi\)
\(114\) 0 0
\(115\) −2.79289 4.83743i −0.260439 0.451093i
\(116\) 0 0
\(117\) 2.82843 4.89898i 0.261488 0.452911i
\(118\) 0 0
\(119\) −20.0711 2.74666i −1.83991 0.251786i
\(120\) 0 0
\(121\) 5.15685 8.93193i 0.468805 0.811994i
\(122\) 0 0
\(123\) 1.20711 + 2.09077i 0.108841 + 0.188518i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) 0 0
\(129\) 1.42893 + 2.47498i 0.125810 + 0.217910i
\(130\) 0 0
\(131\) 6.82843 11.8272i 0.596602 1.03335i −0.396716 0.917941i \(-0.629850\pi\)
0.993319 0.115404i \(-0.0368164\pi\)
\(132\) 0 0
\(133\) −5.65685 13.8564i −0.490511 1.20150i
\(134\) 0 0
\(135\) −1.20711 + 2.09077i −0.103891 + 0.179945i
\(136\) 0 0
\(137\) 2.00000 + 3.46410i 0.170872 + 0.295958i 0.938725 0.344668i \(-0.112008\pi\)
−0.767853 + 0.640626i \(0.778675\pi\)
\(138\) 0 0
\(139\) −2.48528 −0.210799 −0.105399 0.994430i \(-0.533612\pi\)
−0.105399 + 0.994430i \(0.533612\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 0 0
\(143\) 0.828427 + 1.43488i 0.0692766 + 0.119991i
\(144\) 0 0
\(145\) 3.91421 6.77962i 0.325058 0.563017i
\(146\) 0 0
\(147\) 2.07107 2.02922i 0.170819 0.167368i
\(148\) 0 0
\(149\) −2.32843 + 4.03295i −0.190752 + 0.330392i −0.945500 0.325623i \(-0.894426\pi\)
0.754748 + 0.656015i \(0.227759\pi\)
\(150\) 0 0
\(151\) −5.58579 9.67487i −0.454565 0.787329i 0.544098 0.839022i \(-0.316872\pi\)
−0.998663 + 0.0516921i \(0.983539\pi\)
\(152\) 0 0
\(153\) 21.6569 1.75085
\(154\) 0 0
\(155\) −0.828427 −0.0665409
\(156\) 0 0
\(157\) −0.656854 1.13770i −0.0524227 0.0907987i 0.838623 0.544712i \(-0.183361\pi\)
−0.891046 + 0.453913i \(0.850028\pi\)
\(158\) 0 0
\(159\) −1.17157 + 2.02922i −0.0929118 + 0.160928i
\(160\) 0 0
\(161\) −5.58579 13.6823i −0.440222 1.07832i
\(162\) 0 0
\(163\) −7.82843 + 13.5592i −0.613170 + 1.06204i 0.377533 + 0.925996i \(0.376773\pi\)
−0.990703 + 0.136045i \(0.956561\pi\)
\(164\) 0 0
\(165\) −0.171573 0.297173i −0.0133569 0.0231349i
\(166\) 0 0
\(167\) −2.07107 −0.160264 −0.0801320 0.996784i \(-0.525534\pi\)
−0.0801320 + 0.996784i \(0.525534\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 8.00000 + 13.8564i 0.611775 + 1.05963i
\(172\) 0 0
\(173\) −5.17157 + 8.95743i −0.393187 + 0.681021i −0.992868 0.119219i \(-0.961961\pi\)
0.599681 + 0.800239i \(0.295294\pi\)
\(174\) 0 0
\(175\) 2.62132 + 0.358719i 0.198153 + 0.0271166i
\(176\) 0 0
\(177\) 0.828427 1.43488i 0.0622684 0.107852i
\(178\) 0 0
\(179\) −3.24264 5.61642i −0.242366 0.419791i 0.719022 0.694988i \(-0.244590\pi\)
−0.961388 + 0.275197i \(0.911257\pi\)
\(180\) 0 0
\(181\) −4.17157 −0.310071 −0.155035 0.987909i \(-0.549549\pi\)
−0.155035 + 0.987909i \(0.549549\pi\)
\(182\) 0 0
\(183\) −2.75736 −0.203830
\(184\) 0 0
\(185\) −2.82843 4.89898i −0.207950 0.360180i
\(186\) 0 0
\(187\) −3.17157 + 5.49333i −0.231928 + 0.401712i
\(188\) 0 0
\(189\) −3.91421 + 5.04757i −0.284717 + 0.367156i
\(190\) 0 0
\(191\) −2.75736 + 4.77589i −0.199516 + 0.345571i −0.948371 0.317162i \(-0.897270\pi\)
0.748856 + 0.662733i \(0.230603\pi\)
\(192\) 0 0
\(193\) 2.65685 + 4.60181i 0.191245 + 0.331245i 0.945663 0.325149i \(-0.105414\pi\)
−0.754418 + 0.656394i \(0.772081\pi\)
\(194\) 0 0
\(195\) −0.828427 −0.0593249
\(196\) 0 0
\(197\) 0.343146 0.0244481 0.0122241 0.999925i \(-0.496109\pi\)
0.0122241 + 0.999925i \(0.496109\pi\)
\(198\) 0 0
\(199\) −11.6569 20.1903i −0.826332 1.43125i −0.900897 0.434034i \(-0.857090\pi\)
0.0745642 0.997216i \(-0.476243\pi\)
\(200\) 0 0
\(201\) 2.67157 4.62730i 0.188438 0.326385i
\(202\) 0 0
\(203\) 12.6924 16.3674i 0.890831 1.14877i
\(204\) 0 0
\(205\) −2.91421 + 5.04757i −0.203538 + 0.352537i
\(206\) 0 0
\(207\) 7.89949 + 13.6823i 0.549053 + 0.950987i
\(208\) 0 0
\(209\) −4.68629 −0.324158
\(210\) 0 0
\(211\) −26.6274 −1.83311 −0.916553 0.399912i \(-0.869041\pi\)
−0.916553 + 0.399912i \(0.869041\pi\)
\(212\) 0 0
\(213\) 2.48528 + 4.30463i 0.170289 + 0.294949i
\(214\) 0 0
\(215\) −3.44975 + 5.97514i −0.235271 + 0.407501i
\(216\) 0 0
\(217\) −2.17157 0.297173i −0.147416 0.0201734i
\(218\) 0 0
\(219\) 0.757359 1.31178i 0.0511776 0.0886422i
\(220\) 0 0
\(221\) 7.65685 + 13.2621i 0.515056 + 0.892103i
\(222\) 0 0
\(223\) 14.9706 1.00250 0.501252 0.865302i \(-0.332873\pi\)
0.501252 + 0.865302i \(0.332873\pi\)
\(224\) 0 0
\(225\) −2.82843 −0.188562
\(226\) 0 0
\(227\) 7.00000 + 12.1244i 0.464606 + 0.804722i 0.999184 0.0403978i \(-0.0128625\pi\)
−0.534577 + 0.845120i \(0.679529\pi\)
\(228\) 0 0
\(229\) 7.00000 12.1244i 0.462573 0.801200i −0.536515 0.843891i \(-0.680260\pi\)
0.999088 + 0.0426906i \(0.0135930\pi\)
\(230\) 0 0
\(231\) −0.343146 0.840532i −0.0225773 0.0553029i
\(232\) 0 0
\(233\) −5.82843 + 10.0951i −0.381833 + 0.661354i −0.991324 0.131439i \(-0.958040\pi\)
0.609491 + 0.792793i \(0.291374\pi\)
\(234\) 0 0
\(235\) 5.82843 + 10.0951i 0.380205 + 0.658534i
\(236\) 0 0
\(237\) −1.65685 −0.107624
\(238\) 0 0
\(239\) 30.4853 1.97193 0.985964 0.166955i \(-0.0533936\pi\)
0.985964 + 0.166955i \(0.0533936\pi\)
\(240\) 0 0
\(241\) 5.00000 + 8.66025i 0.322078 + 0.557856i 0.980917 0.194429i \(-0.0622852\pi\)
−0.658838 + 0.752285i \(0.728952\pi\)
\(242\) 0 0
\(243\) 5.17157 8.95743i 0.331757 0.574619i
\(244\) 0 0
\(245\) 6.74264 + 1.88064i 0.430772 + 0.120150i
\(246\) 0 0
\(247\) −5.65685 + 9.79796i −0.359937 + 0.623429i
\(248\) 0 0
\(249\) 0.985281 + 1.70656i 0.0624397 + 0.108149i
\(250\) 0 0
\(251\) 27.4558 1.73300 0.866499 0.499179i \(-0.166365\pi\)
0.866499 + 0.499179i \(0.166365\pi\)
\(252\) 0 0
\(253\) −4.62742 −0.290923
\(254\) 0 0
\(255\) −1.58579 2.74666i −0.0993058 0.172003i
\(256\) 0 0
\(257\) −7.65685 + 13.2621i −0.477621 + 0.827265i −0.999671 0.0256506i \(-0.991834\pi\)
0.522050 + 0.852915i \(0.325168\pi\)
\(258\) 0 0
\(259\) −5.65685 13.8564i −0.351500 0.860995i
\(260\) 0 0
\(261\) −11.0711 + 19.1757i −0.685282 + 1.18694i
\(262\) 0 0
\(263\) −7.86396 13.6208i −0.484913 0.839893i 0.514937 0.857228i \(-0.327815\pi\)
−0.999850 + 0.0173347i \(0.994482\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) 0 0
\(267\) 2.21320 0.135446
\(268\) 0 0
\(269\) 3.32843 + 5.76500i 0.202938 + 0.351499i 0.949474 0.313847i \(-0.101618\pi\)
−0.746536 + 0.665345i \(0.768284\pi\)
\(270\) 0 0
\(271\) −1.65685 + 2.86976i −0.100647 + 0.174325i −0.911951 0.410298i \(-0.865425\pi\)
0.811305 + 0.584624i \(0.198758\pi\)
\(272\) 0 0
\(273\) −2.17157 0.297173i −0.131430 0.0179857i
\(274\) 0 0
\(275\) 0.414214 0.717439i 0.0249780 0.0432632i
\(276\) 0 0
\(277\) −14.3137 24.7921i −0.860027 1.48961i −0.871901 0.489682i \(-0.837113\pi\)
0.0118739 0.999930i \(-0.496220\pi\)
\(278\) 0 0
\(279\) 2.34315 0.140280
\(280\) 0 0
\(281\) 2.68629 0.160251 0.0801254 0.996785i \(-0.474468\pi\)
0.0801254 + 0.996785i \(0.474468\pi\)
\(282\) 0 0
\(283\) 9.00000 + 15.5885i 0.534994 + 0.926638i 0.999164 + 0.0408910i \(0.0130196\pi\)
−0.464169 + 0.885747i \(0.653647\pi\)
\(284\) 0 0
\(285\) 1.17157 2.02922i 0.0693980 0.120201i
\(286\) 0 0
\(287\) −9.44975 + 12.1859i −0.557801 + 0.719311i
\(288\) 0 0
\(289\) −20.8137 + 36.0504i −1.22434 + 2.12061i
\(290\) 0 0
\(291\) 1.24264 + 2.15232i 0.0728449 + 0.126171i
\(292\) 0 0
\(293\) 16.9706 0.991431 0.495715 0.868485i \(-0.334906\pi\)
0.495715 + 0.868485i \(0.334906\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 1.00000 + 1.73205i 0.0580259 + 0.100504i
\(298\) 0 0
\(299\) −5.58579 + 9.67487i −0.323034 + 0.559512i
\(300\) 0 0
\(301\) −11.1863 + 14.4253i −0.644767 + 0.831458i
\(302\) 0 0
\(303\) −2.37868 + 4.11999i −0.136652 + 0.236687i
\(304\) 0 0
\(305\) −3.32843 5.76500i −0.190585 0.330103i
\(306\) 0 0
\(307\) −4.75736 −0.271517 −0.135758 0.990742i \(-0.543347\pi\)
−0.135758 + 0.990742i \(0.543347\pi\)
\(308\) 0 0
\(309\) 3.14214 0.178750
\(310\) 0 0
\(311\) 10.8284 + 18.7554i 0.614024 + 1.06352i 0.990555 + 0.137116i \(0.0437834\pi\)
−0.376531 + 0.926404i \(0.622883\pi\)
\(312\) 0 0
\(313\) −10.4853 + 18.1610i −0.592663 + 1.02652i 0.401209 + 0.915987i \(0.368590\pi\)
−0.993872 + 0.110536i \(0.964743\pi\)
\(314\) 0 0
\(315\) −7.41421 1.01461i −0.417744 0.0571669i
\(316\) 0 0
\(317\) 11.0000 19.0526i 0.617822 1.07010i −0.372061 0.928208i \(-0.621349\pi\)
0.989882 0.141890i \(-0.0453179\pi\)
\(318\) 0 0
\(319\) −3.24264 5.61642i −0.181553 0.314459i
\(320\) 0 0
\(321\) −2.31371 −0.129139
\(322\) 0 0
\(323\) −43.3137 −2.41004
\(324\) 0 0
\(325\) −1.00000 1.73205i −0.0554700 0.0960769i
\(326\) 0 0
\(327\) 3.79289 6.56948i 0.209747 0.363293i
\(328\) 0 0
\(329\) 11.6569 + 28.5533i 0.642663 + 1.57420i
\(330\) 0 0
\(331\) 13.2426 22.9369i 0.727881 1.26073i −0.229896 0.973215i \(-0.573838\pi\)
0.957777 0.287512i \(-0.0928282\pi\)
\(332\) 0 0
\(333\) 8.00000 + 13.8564i 0.438397 + 0.759326i
\(334\) 0 0
\(335\) 12.8995 0.704775
\(336\) 0 0
\(337\) 24.9706 1.36023 0.680117 0.733104i \(-0.261929\pi\)
0.680117 + 0.733104i \(0.261929\pi\)
\(338\) 0 0
\(339\) 2.34315 + 4.05845i 0.127262 + 0.220425i
\(340\) 0 0
\(341\) −0.343146 + 0.594346i −0.0185824 + 0.0321856i
\(342\) 0 0
\(343\) 17.0000 + 7.34847i 0.917914 + 0.396780i
\(344\) 0 0
\(345\) 1.15685 2.00373i 0.0622829 0.107877i
\(346\) 0 0
\(347\) 5.69239 + 9.85951i 0.305583 + 0.529286i 0.977391 0.211440i \(-0.0678152\pi\)
−0.671808 + 0.740726i \(0.734482\pi\)
\(348\) 0 0
\(349\) 9.82843 0.526104 0.263052 0.964782i \(-0.415271\pi\)
0.263052 + 0.964782i \(0.415271\pi\)
\(350\) 0 0
\(351\) 4.82843 0.257722
\(352\) 0 0
\(353\) −16.8284 29.1477i −0.895687 1.55138i −0.832952 0.553345i \(-0.813351\pi\)
−0.0627345 0.998030i \(-0.519982\pi\)
\(354\) 0 0
\(355\) −6.00000 + 10.3923i −0.318447 + 0.551566i
\(356\) 0 0
\(357\) −3.17157 7.76874i −0.167857 0.411165i
\(358\) 0 0
\(359\) −3.24264 + 5.61642i −0.171140 + 0.296423i −0.938819 0.344412i \(-0.888078\pi\)
0.767679 + 0.640835i \(0.221412\pi\)
\(360\) 0 0
\(361\) −6.50000 11.2583i −0.342105 0.592544i
\(362\) 0 0
\(363\) 4.27208 0.224226
\(364\) 0 0
\(365\) 3.65685 0.191408
\(366\) 0 0
\(367\) 10.7929 + 18.6938i 0.563384 + 0.975810i 0.997198 + 0.0748078i \(0.0238343\pi\)
−0.433814 + 0.901003i \(0.642832\pi\)
\(368\) 0 0
\(369\) 8.24264 14.2767i 0.429095 0.743214i
\(370\) 0 0
\(371\) −14.8284 2.02922i −0.769854 0.105352i
\(372\) 0 0
\(373\) −6.00000 + 10.3923i −0.310668 + 0.538093i −0.978507 0.206213i \(-0.933886\pi\)
0.667839 + 0.744306i \(0.267219\pi\)
\(374\) 0 0
\(375\) 0.207107 + 0.358719i 0.0106949 + 0.0185242i
\(376\) 0 0
\(377\) −15.6569 −0.806369
\(378\) 0 0
\(379\) −4.68629 −0.240719 −0.120359 0.992730i \(-0.538405\pi\)
−0.120359 + 0.992730i \(0.538405\pi\)
\(380\) 0 0
\(381\) 0.899495 + 1.55797i 0.0460825 + 0.0798173i
\(382\) 0 0
\(383\) −0.449747 + 0.778985i −0.0229810 + 0.0398043i −0.877287 0.479966i \(-0.840649\pi\)
0.854306 + 0.519770i \(0.173982\pi\)
\(384\) 0 0
\(385\) 1.34315 1.73205i 0.0684530 0.0882735i
\(386\) 0 0
\(387\) 9.75736 16.9002i 0.495994 0.859088i
\(388\) 0 0
\(389\) −2.65685 4.60181i −0.134708 0.233321i 0.790778 0.612103i \(-0.209676\pi\)
−0.925486 + 0.378782i \(0.876343\pi\)
\(390\) 0 0
\(391\) −42.7696 −2.16295
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) −2.00000 3.46410i −0.100631 0.174298i
\(396\) 0 0
\(397\) 12.3137 21.3280i 0.618007 1.07042i −0.371842 0.928296i \(-0.621274\pi\)
0.989849 0.142124i \(-0.0453931\pi\)
\(398\) 0 0
\(399\) 3.79899 4.89898i 0.190187 0.245256i
\(400\) 0 0
\(401\) 16.1569 27.9845i 0.806835 1.39748i −0.108211 0.994128i \(-0.534512\pi\)
0.915045 0.403351i \(-0.132155\pi\)
\(402\) 0 0
\(403\) 0.828427 + 1.43488i 0.0412669 + 0.0714764i
\(404\) 0 0
\(405\) 7.48528 0.371947
\(406\) 0 0
\(407\) −4.68629 −0.232291
\(408\) 0 0
\(409\) 12.5711 + 21.7737i 0.621599 + 1.07664i 0.989188 + 0.146653i \(0.0468501\pi\)
−0.367589 + 0.929988i \(0.619817\pi\)
\(410\) 0 0
\(411\) −0.828427 + 1.43488i −0.0408633 + 0.0707773i
\(412\) 0 0
\(413\) 10.4853 + 1.43488i 0.515947 + 0.0706057i
\(414\) 0 0
\(415\) −2.37868 + 4.11999i −0.116765 + 0.202243i
\(416\) 0 0
\(417\) −0.514719 0.891519i −0.0252059 0.0436579i
\(418\) 0 0
\(419\) 15.3137 0.748124 0.374062 0.927404i \(-0.377965\pi\)
0.374062 + 0.927404i \(0.377965\pi\)
\(420\) 0 0
\(421\) 27.3431 1.33262 0.666312 0.745673i \(-0.267872\pi\)
0.666312 + 0.745673i \(0.267872\pi\)
\(422\) 0 0
\(423\) −16.4853 28.5533i −0.801542 1.38831i
\(424\) 0 0
\(425\) 3.82843 6.63103i 0.185706 0.321652i
\(426\) 0 0
\(427\) −6.65685 16.3059i −0.322148 0.789098i
\(428\) 0 0
\(429\) −0.343146 + 0.594346i −0.0165672 + 0.0286953i
\(430\) 0 0
\(431\) −0.414214 0.717439i −0.0199520 0.0345578i 0.855877 0.517179i \(-0.173018\pi\)
−0.875829 + 0.482622i \(0.839685\pi\)
\(432\) 0 0
\(433\) −19.3137 −0.928158 −0.464079 0.885794i \(-0.653615\pi\)
−0.464079 + 0.885794i \(0.653615\pi\)
\(434\) 0 0
\(435\) 3.24264 0.155473
\(436\) 0 0
\(437\) −15.7990 27.3647i −0.755768 1.30903i
\(438\) 0 0
\(439\) −9.17157 + 15.8856i −0.437735 + 0.758180i −0.997514 0.0704621i \(-0.977553\pi\)
0.559779 + 0.828642i \(0.310886\pi\)
\(440\) 0 0
\(441\) −19.0711 5.31925i −0.908146 0.253297i
\(442\) 0 0
\(443\) −7.79289 + 13.4977i −0.370252 + 0.641294i −0.989604 0.143819i \(-0.954062\pi\)
0.619353 + 0.785113i \(0.287395\pi\)
\(444\) 0 0
\(445\) 2.67157 + 4.62730i 0.126645 + 0.219355i
\(446\) 0 0
\(447\) −1.92893 −0.0912354
\(448\) 0 0
\(449\) −7.48528 −0.353252 −0.176626 0.984278i \(-0.556518\pi\)
−0.176626 + 0.984278i \(0.556518\pi\)
\(450\) 0 0
\(451\) 2.41421 + 4.18154i 0.113681 + 0.196901i
\(452\) 0 0
\(453\) 2.31371 4.00746i 0.108708 0.188287i
\(454\) 0 0
\(455\) −2.00000 4.89898i −0.0937614 0.229668i
\(456\) 0 0
\(457\) −14.4853 + 25.0892i −0.677593 + 1.17363i 0.298111 + 0.954531i \(0.403643\pi\)
−0.975704 + 0.219094i \(0.929690\pi\)
\(458\) 0 0
\(459\) 9.24264 + 16.0087i 0.431410 + 0.747223i
\(460\) 0 0
\(461\) 1.31371 0.0611855 0.0305928 0.999532i \(-0.490261\pi\)
0.0305928 + 0.999532i \(0.490261\pi\)
\(462\) 0 0
\(463\) −14.8995 −0.692438 −0.346219 0.938154i \(-0.612535\pi\)
−0.346219 + 0.938154i \(0.612535\pi\)
\(464\) 0 0
\(465\) −0.171573 0.297173i −0.00795650 0.0137811i
\(466\) 0 0
\(467\) −18.9350 + 32.7964i −0.876209 + 1.51764i −0.0207390 + 0.999785i \(0.506602\pi\)
−0.855470 + 0.517853i \(0.826731\pi\)
\(468\) 0 0
\(469\) 33.8137 + 4.62730i 1.56137 + 0.213669i
\(470\) 0 0
\(471\) 0.272078 0.471253i 0.0125367 0.0217142i
\(472\) 0 0
\(473\) 2.85786 + 4.94997i 0.131405 + 0.227600i
\(474\) 0 0
\(475\) 5.65685 0.259554
\(476\) 0 0
\(477\) 16.0000 0.732590
\(478\) 0 0
\(479\) −0.757359 1.31178i −0.0346046 0.0599370i 0.848204 0.529669i \(-0.177684\pi\)
−0.882809 + 0.469732i \(0.844351\pi\)
\(480\) 0 0
\(481\) −5.65685 + 9.79796i −0.257930 + 0.446748i
\(482\) 0 0
\(483\) 3.75126 4.83743i 0.170688 0.220111i
\(484\) 0 0
\(485\) −3.00000 + 5.19615i −0.136223 + 0.235945i
\(486\) 0 0
\(487\) −19.1421 33.1552i −0.867413 1.50240i −0.864631 0.502407i \(-0.832448\pi\)
−0.00278182 0.999996i \(-0.500885\pi\)
\(488\) 0 0
\(489\) −6.48528 −0.293275
\(490\) 0 0
\(491\) −1.51472 −0.0683583 −0.0341791 0.999416i \(-0.510882\pi\)
−0.0341791 + 0.999416i \(0.510882\pi\)
\(492\) 0 0
\(493\) −29.9706 51.9105i −1.34981 2.33793i
\(494\) 0 0
\(495\) −1.17157 + 2.02922i −0.0526583 + 0.0912068i
\(496\) 0 0
\(497\) −19.4558 + 25.0892i −0.872714 + 1.12541i
\(498\) 0 0
\(499\) 22.0711 38.2282i 0.988037 1.71133i 0.360461 0.932774i \(-0.382619\pi\)
0.627576 0.778555i \(-0.284047\pi\)
\(500\) 0 0
\(501\) −0.428932 0.742932i −0.0191633 0.0331918i
\(502\) 0 0
\(503\) 3.92893 0.175182 0.0875912 0.996157i \(-0.472083\pi\)
0.0875912 + 0.996157i \(0.472083\pi\)
\(504\) 0 0
\(505\) −11.4853 −0.511088
\(506\) 0 0
\(507\) −1.86396 3.22848i −0.0827814 0.143382i
\(508\) 0 0
\(509\) 16.7426 28.9991i 0.742105 1.28536i −0.209431 0.977823i \(-0.567161\pi\)
0.951535 0.307539i \(-0.0995055\pi\)
\(510\) 0 0
\(511\) 9.58579 + 1.31178i 0.424050 + 0.0580299i
\(512\) 0 0
\(513\) −6.82843 + 11.8272i −0.301482 + 0.522183i
\(514\) 0 0
\(515\) 3.79289 + 6.56948i 0.167135 + 0.289486i
\(516\) 0 0
\(517\) 9.65685 0.424708
\(518\) 0 0
\(519\) −4.28427 −0.188059
\(520\) 0 0
\(521\) −18.3137 31.7203i −0.802338 1.38969i −0.918073 0.396410i \(-0.870256\pi\)
0.115735 0.993280i \(-0.463078\pi\)
\(522\) 0 0
\(523\) 13.9706 24.1977i 0.610890 1.05809i −0.380201 0.924904i \(-0.624145\pi\)
0.991091 0.133189i \(-0.0425216\pi\)
\(524\) 0 0
\(525\) 0.414214 + 1.01461i 0.0180778 + 0.0442813i
\(526\) 0 0
\(527\) −3.17157 + 5.49333i −0.138156 + 0.239293i
\(528\) 0 0
\(529\) −4.10051 7.10228i −0.178283 0.308795i
\(530\) 0 0
\(531\) −11.3137 −0.490973
\(532\) 0 0
\(533\) 11.6569 0.504914
\(534\) 0 0
\(535\) −2.79289 4.83743i −0.120747 0.209140i
\(536\) 0 0
\(537\) 1.34315 2.32640i 0.0579610 0.100391i
\(538\) 0 0
\(539\) 4.14214 4.05845i 0.178414 0.174810i
\(540\) 0 0
\(541\) 11.7426 20.3389i 0.504856 0.874435i −0.495129 0.868820i \(-0.664879\pi\)
0.999984 0.00561582i \(-0.00178758\pi\)
\(542\) 0 0
\(543\) −0.863961 1.49642i −0.0370761 0.0642177i
\(544\) 0 0
\(545\) 18.3137 0.784473
\(546\) 0 0
\(547\) 2.27208 0.0971470 0.0485735 0.998820i \(-0.484532\pi\)
0.0485735 + 0.998820i \(0.484532\pi\)
\(548\) 0 0
\(549\) 9.41421 + 16.3059i 0.401789 + 0.695919i
\(550\) 0 0
\(551\) 22.1421 38.3513i 0.943287 1.63382i
\(552\) 0 0
\(553\) −4.00000 9.79796i −0.170097 0.416652i
\(554\) 0 0
\(555\) 1.17157 2.02922i 0.0497305 0.0861358i
\(556\) 0 0
\(557\) 8.65685 + 14.9941i 0.366803 + 0.635321i 0.989064 0.147489i \(-0.0471191\pi\)
−0.622261 + 0.782810i \(0.713786\pi\)
\(558\) 0 0
\(559\) 13.7990 0.583635
\(560\) 0 0
\(561\) −2.62742 −0.110930
\(562\) 0 0
\(563\) −2.03553 3.52565i −0.0857875 0.148588i 0.819939 0.572451i \(-0.194007\pi\)
−0.905727 + 0.423863i \(0.860674\pi\)
\(564\) 0 0
\(565\) −5.65685 + 9.79796i −0.237986 + 0.412203i
\(566\) 0 0
\(567\) 19.6213 + 2.68512i 0.824018 + 0.112764i
\(568\) 0 0
\(569\) 18.3137 31.7203i 0.767751 1.32978i −0.171029 0.985266i \(-0.554709\pi\)
0.938780 0.344517i \(-0.111957\pi\)
\(570\) 0 0
\(571\) −10.4853 18.1610i −0.438795 0.760016i 0.558801 0.829301i \(-0.311261\pi\)
−0.997597 + 0.0692856i \(0.977928\pi\)
\(572\) 0 0
\(573\) −2.28427 −0.0954268
\(574\) 0 0
\(575\) 5.58579 0.232943
\(576\) 0 0
\(577\) 11.1421 + 19.2987i 0.463853 + 0.803417i 0.999149 0.0412474i \(-0.0131332\pi\)
−0.535296 + 0.844665i \(0.679800\pi\)
\(578\) 0 0
\(579\) −1.10051 + 1.90613i −0.0457354 + 0.0792161i
\(580\) 0 0
\(581\) −7.71320 + 9.94655i −0.319998 + 0.412652i
\(582\) 0 0
\(583\) −2.34315 + 4.05845i −0.0970432 + 0.168084i
\(584\) 0 0
\(585\) 2.82843 + 4.89898i 0.116941 + 0.202548i
\(586\) 0 0
\(587\) 22.6863 0.936363 0.468182 0.883632i \(-0.344909\pi\)
0.468182 + 0.883632i \(0.344909\pi\)
\(588\) 0 0
\(589\) −4.68629 −0.193095
\(590\) 0 0
\(591\) 0.0710678 + 0.123093i 0.00292334 + 0.00506337i
\(592\) 0 0
\(593\) −14.9706 + 25.9298i −0.614767 + 1.06481i 0.375658 + 0.926758i \(0.377417\pi\)
−0.990425 + 0.138050i \(0.955917\pi\)
\(594\) 0 0
\(595\) 12.4142 16.0087i 0.508933 0.656294i
\(596\) 0 0
\(597\) 4.82843 8.36308i 0.197614 0.342278i
\(598\) 0 0
\(599\) 21.3137 + 36.9164i 0.870855 + 1.50836i 0.861114 + 0.508412i \(0.169767\pi\)
0.00974040 + 0.999953i \(0.496899\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) −36.4853 −1.48580
\(604\) 0 0
\(605\) 5.15685 + 8.93193i 0.209656 + 0.363135i
\(606\) 0 0
\(607\) 13.6213 23.5928i 0.552872 0.957603i −0.445193 0.895434i \(-0.646865\pi\)
0.998066 0.0621685i \(-0.0198016\pi\)
\(608\) 0 0
\(609\) 8.50000 + 1.16320i 0.344437 + 0.0471352i
\(610\) 0 0
\(611\) 11.6569 20.1903i 0.471586 0.816811i
\(612\) 0 0
\(613\) −19.4853 33.7495i −0.787003 1.36313i −0.927795 0.373090i \(-0.878298\pi\)
0.140792 0.990039i \(-0.455035\pi\)
\(614\) 0 0
\(615\) −2.41421 −0.0973505
\(616\) 0 0
\(617\) −12.6863 −0.510731 −0.255365 0.966845i \(-0.582196\pi\)
−0.255365 + 0.966845i \(0.582196\pi\)
\(618\) 0 0
\(619\) 19.7279 + 34.1698i 0.792932 + 1.37340i 0.924144 + 0.382044i \(0.124780\pi\)
−0.131212 + 0.991354i \(0.541887\pi\)
\(620\) 0 0
\(621\) −6.74264 + 11.6786i −0.270573 + 0.468646i
\(622\) 0 0
\(623\) 5.34315 + 13.0880i 0.214069 + 0.524359i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −0.970563 1.68106i −0.0387605 0.0671352i
\(628\) 0 0
\(629\) −43.3137 −1.72703
\(630\) 0 0
\(631\) 1.51472 0.0603000 0.0301500 0.999545i \(-0.490402\pi\)
0.0301500 + 0.999545i \(0.490402\pi\)
\(632\) 0 0
\(633\) −5.51472 9.55177i −0.219190 0.379649i
\(634\) 0 0
\(635\) −2.17157 + 3.76127i −0.0861762 + 0.149262i
\(636\) 0 0
\(637\) −3.48528 13.5592i −0.138092 0.537236i
\(638\) 0 0
\(639\) 16.9706 29.3939i 0.671345 1.16280i
\(640\) 0 0
\(641\) −7.05635 12.2220i −0.278709 0.482738i 0.692355 0.721557i \(-0.256573\pi\)
−0.971064 + 0.238819i \(0.923240\pi\)
\(642\) 0 0
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) −2.85786 −0.112528
\(646\) 0 0
\(647\) −10.3787 17.9764i −0.408028 0.706725i 0.586641 0.809847i \(-0.300450\pi\)
−0.994669 + 0.103122i \(0.967117\pi\)
\(648\) 0 0
\(649\) 1.65685 2.86976i 0.0650372 0.112648i
\(650\) 0 0
\(651\) −0.343146 0.840532i −0.0134489 0.0329430i
\(652\) 0 0
\(653\) 7.82843 13.5592i 0.306350 0.530614i −0.671211 0.741266i \(-0.734226\pi\)
0.977561 + 0.210653i \(0.0675589\pi\)
\(654\) 0 0
\(655\) 6.82843 + 11.8272i 0.266809 + 0.462126i
\(656\) 0 0
\(657\) −10.3431 −0.403525
\(658\) 0 0
\(659\) −12.6863 −0.494188 −0.247094 0.968992i \(-0.579476\pi\)
−0.247094 + 0.968992i \(0.579476\pi\)
\(660\) 0 0
\(661\) 25.1569 + 43.5729i 0.978488 + 1.69479i 0.667907 + 0.744244i \(0.267190\pi\)
0.310581 + 0.950547i \(0.399476\pi\)
\(662\) 0 0
\(663\) −3.17157 + 5.49333i −0.123174 + 0.213343i
\(664\) 0 0
\(665\) 14.8284 + 2.02922i 0.575022 + 0.0786899i
\(666\) 0 0
\(667\) 21.8640 37.8695i 0.846576 1.46631i
\(668\) 0 0
\(669\) 3.10051 + 5.37023i 0.119872 + 0.207625i
\(670\) 0 0
\(671\) −5.51472 −0.212893
\(672\) 0 0
\(673\) −5.65685 −0.218056 −0.109028 0.994039i \(-0.534774\pi\)
−0.109028 + 0.994039i \(0.534774\pi\)
\(674\) 0 0
\(675\) −1.20711 2.09077i −0.0464616 0.0804738i
\(676\) 0 0
\(677\) −11.4853 + 19.8931i −0.441415 + 0.764554i −0.997795 0.0663747i \(-0.978857\pi\)
0.556380 + 0.830928i \(0.312190\pi\)
\(678\) 0 0
\(679\) −9.72792 + 12.5446i −0.373323 + 0.481418i
\(680\) 0 0
\(681\) −2.89949 + 5.02207i −0.111109 + 0.192446i
\(682\) 0 0
\(683\) −0.964466 1.67050i −0.0369043 0.0639201i 0.846983 0.531619i \(-0.178416\pi\)
−0.883888 + 0.467699i \(0.845083\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 5.79899 0.221245
\(688\) 0 0
\(689\) 5.65685 + 9.79796i 0.215509 + 0.373273i
\(690\) 0 0
\(691\) −21.3848 + 37.0395i −0.813515 + 1.40905i 0.0968739 + 0.995297i \(0.469116\pi\)
−0.910389 + 0.413753i \(0.864218\pi\)
\(692\) 0 0
\(693\) −3.79899 + 4.89898i −0.144312 + 0.186097i
\(694\) 0 0
\(695\) 1.24264 2.15232i 0.0471360 0.0816420i
\(696\) 0 0
\(697\) 22.3137 + 38.6485i 0.845192 + 1.46392i
\(698\) 0 0
\(699\) −4.82843 −0.182628
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 0 0
\(703\) −16.0000 27.7128i −0.603451 1.04521i
\(704\) 0 0
\(705\) −2.41421 + 4.18154i −0.0909245 + 0.157486i
\(706\) 0 0
\(707\) −30.1066 4.11999i −1.13228 0.154948i
\(708\) 0 0
\(709\) 17.7132 30.6802i 0.665233 1.15222i −0.313989 0.949427i \(-0.601665\pi\)
0.979222 0.202791i \(-0.0650013\pi\)
\(710\) 0 0
\(711\) 5.65685 + 9.79796i 0.212149 + 0.367452i
\(712\) 0 0
\(713\) −4.62742 −0.173298
\(714\) 0 0
\(715\) −1.65685 −0.0619628
\(716\) 0 0
\(717\) 6.31371 + 10.9357i 0.235790 + 0.408400i
\(718\) 0 0
\(719\) −12.8995 + 22.3426i −0.481070 + 0.833238i −0.999764 0.0217223i \(-0.993085\pi\)
0.518694 + 0.854960i \(0.326418\pi\)
\(720\) 0 0
\(721\) 7.58579 + 18.5813i 0.282509 + 0.692004i
\(722\) 0 0
\(723\) −2.07107 + 3.58719i −0.0770238 + 0.133409i
\(724\) 0 0
\(725\) 3.91421 + 6.77962i 0.145370 + 0.251789i
\(726\) 0 0
\(727\) −38.0711 −1.41198 −0.705989 0.708223i \(-0.749497\pi\)
−0.705989 + 0.708223i \(0.749497\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 26.4142 + 45.7508i 0.976965 + 1.69215i
\(732\) 0 0
\(733\) −3.82843 + 6.63103i −0.141406 + 0.244923i −0.928026 0.372514i \(-0.878496\pi\)
0.786620 + 0.617437i \(0.211829\pi\)
\(734\) 0 0
\(735\) 0.721825 + 2.80821i 0.0266249 + 0.103582i
\(736\) 0 0
\(737\) 5.34315 9.25460i 0.196817 0.340898i
\(738\) 0 0
\(739\) 8.41421 + 14.5738i 0.309522 + 0.536108i 0.978258 0.207392i \(-0.0664977\pi\)
−0.668736 + 0.743500i \(0.733164\pi\)
\(740\) 0 0
\(741\) −4.68629 −0.172155
\(742\) 0 0
\(743\) 10.7574 0.394649 0.197325 0.980338i \(-0.436775\pi\)
0.197325 + 0.980338i \(0.436775\pi\)
\(744\) 0 0
\(745\) −2.32843 4.03295i −0.0853070 0.147756i
\(746\) 0 0
\(747\) 6.72792 11.6531i 0.246162 0.426365i
\(748\) 0 0
\(749\) −5.58579 13.6823i −0.204100 0.499941i
\(750\) 0 0
\(751\) −6.00000 + 10.3923i −0.218943 + 0.379221i −0.954485 0.298259i \(-0.903594\pi\)
0.735542 + 0.677479i \(0.236928\pi\)
\(752\) 0 0
\(753\) 5.68629 + 9.84895i 0.207220 + 0.358916i
\(754\) 0 0
\(755\) 11.1716 0.406575
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 0 0
\(759\) −0.958369 1.65994i −0.0347866 0.0602522i
\(760\) 0 0
\(761\) 21.9706 38.0541i 0.796432 1.37946i −0.125493 0.992094i \(-0.540051\pi\)
0.921926 0.387367i \(-0.126615\pi\)
\(762\) 0 0
\(763\) 48.0061 + 6.56948i 1.73794 + 0.237831i
\(764\) 0 0
\(765\) −10.8284 + 18.7554i −0.391503 + 0.678102i
\(766\) 0 0
\(767\) −4.00000 6.92820i −0.144432 0.250163i
\(768\) 0 0
\(769\) 43.2548 1.55981 0.779905 0.625898i \(-0.215268\pi\)
0.779905 + 0.625898i \(0.215268\pi\)
\(770\) 0 0
\(771\) −6.34315 −0.228443
\(772\) 0 0
\(773\) −12.1716 21.0818i −0.437781 0.758259i 0.559737 0.828670i \(-0.310902\pi\)
−0.997518 + 0.0704113i \(0.977569\pi\)
\(774\) 0 0
\(775\) 0.414214 0.717439i 0.0148790 0.0257712i
\(776\) 0 0
\(777\) 3.79899 4.89898i 0.136288 0.175750i
\(778\) 0 0
\(779\) −16.4853 + 28.5533i −0.590647 + 1.02303i
\(780\) 0 0
\(781\) 4.97056 + 8.60927i 0.177861 + 0.308064i
\(782\) 0 0
\(783\) −18.8995 −0.675413
\(784\) 0 0
\(785\) 1.31371 0.0468883
\(786\) 0 0
\(787\) −0.792893 1.37333i −0.0282636 0.0489540i 0.851548 0.524277i \(-0.175664\pi\)
−0.879811 + 0.475323i \(0.842331\pi\)
\(788\) 0 0
\(789\) 3.25736 5.64191i 0.115965 0.200857i
\(790\) 0 0
\(791\) −18.3431 + 23.6544i −0.652207 + 0.841052i
\(792\) 0 0
\(793\) −6.65685 + 11.5300i −0.236392 + 0.409443i
\(794\) 0 0
\(795\) −1.17157 2.02922i −0.0415514 0.0719691i
\(796\) 0 0
\(797\) −35.3137 −1.25088 −0.625438 0.780274i \(-0.715080\pi\)
−0.625438 + 0.780274i \(0.715080\pi\)
\(798\) 0 0
\(799\) 89.2548 3.15761
\(800\) 0 0
\(801\) −7.55635 13.0880i −0.266990 0.462441i
\(802\) 0 0
\(803\) 1.51472 2.62357i 0.0534533 0.0925838i
\(804\) 0 0
\(805\) 14.6421 + 2.00373i 0.516067 + 0.0706222i
\(806\) 0 0
\(807\) −1.37868 + 2.38794i −0.0485318 + 0.0840596i
\(808\) 0 0
\(809\) 17.9853 + 31.1514i 0.632329 + 1.09523i 0.987074 + 0.160263i \(0.0512342\pi\)
−0.354746 + 0.934963i \(0.615433\pi\)
\(810\) 0 0
\(811\) −52.1421 −1.83096 −0.915479 0.402366i \(-0.868188\pi\)
−0.915479 + 0.402366i \(0.868188\pi\)
\(812\) 0 0
\(813\) −1.37258 −0.0481386
\(814\) 0 0
\(815\) −7.82843 13.5592i −0.274218 0.474959i
\(816\) 0 0
\(817\) −19.5147 + 33.8005i −0.682734 + 1.18253i
\(818\) 0 0
\(819\) 5.65685 + 13.8564i 0.197666 + 0.484182i
\(820\) 0 0
\(821\) 16.3137 28.2562i 0.569352 0.986147i −0.427278 0.904120i \(-0.640527\pi\)
0.996630 0.0820268i \(-0.0261393\pi\)
\(822\) 0 0
\(823\) 15.0061 + 25.9913i 0.523080 + 0.906001i 0.999639 + 0.0268584i \(0.00855031\pi\)
−0.476560 + 0.879142i \(0.658116\pi\)
\(824\) 0 0
\(825\) 0.343146 0.0119468
\(826\) 0 0
\(827\) −13.5269 −0.470377 −0.235188 0.971950i \(-0.575571\pi\)
−0.235188 + 0.971950i \(0.575571\pi\)
\(828\) 0 0
\(829\) −2.65685 4.60181i −0.0922764 0.159827i 0.816192 0.577780i \(-0.196081\pi\)
−0.908469 + 0.417953i \(0.862748\pi\)
\(830\) 0 0
\(831\) 5.92893 10.2692i 0.205672 0.356235i
\(832\) 0 0
\(833\) 38.2843 37.5108i 1.32647 1.29967i
\(834\) 0 0
\(835\) 1.03553 1.79360i 0.0358361 0.0620700i
\(836\) 0 0
\(837\) 1.00000 + 1.73205i 0.0345651 + 0.0598684i
\(838\) 0 0
\(839\) −20.1421 −0.695384 −0.347692 0.937609i \(-0.613034\pi\)
−0.347692 + 0.937609i \(0.613034\pi\)
\(840\) 0 0
\(841\) 32.2843 1.11325
\(842\) 0 0
\(843\) 0.556349 + 0.963625i 0.0191617 + 0.0331890i
\(844\) 0 0
\(845\) 4.50000 7.79423i 0.154805 0.268130i
\(846\) 0 0
\(847\) 10.3137 + 25.2633i 0.354383 + 0.868058i
\(848\) 0 0
\(849\) −3.72792 + 6.45695i −0.127942 + 0.221602i
\(850\) 0 0
\(851\) −15.7990 27.3647i −0.541582 0.938048i
\(852\) 0 0
\(853\) 12.6863 0.434370 0.217185 0.976130i \(-0.430312\pi\)
0.217185 + 0.976130i \(0.430312\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) 9.48528 + 16.4290i 0.324011 + 0.561204i 0.981312 0.192425i \(-0.0616352\pi\)
−0.657301 + 0.753628i \(0.728302\pi\)
\(858\) 0 0
\(859\) −15.3137 + 26.5241i −0.522497 + 0.904991i 0.477160 + 0.878816i \(0.341666\pi\)
−0.999657 + 0.0261751i \(0.991667\pi\)
\(860\) 0 0
\(861\) −6.32843 0.866025i −0.215672 0.0295141i
\(862\) 0 0
\(863\) −28.0061 + 48.5080i −0.953339 + 1.65123i −0.215214 + 0.976567i \(0.569045\pi\)
−0.738125 + 0.674664i \(0.764289\pi\)
\(864\) 0 0
\(865\) −5.17157 8.95743i −0.175839 0.304562i
\(866\) 0 0
\(867\) −17.2426 −0.585591
\(868\) 0 0
\(869\) −3.31371 −0.112410
\(870\) 0 0
\(871\) −12.8995 22.3426i −0.437083 0.757049i
\(872\) 0 0
\(873\) 8.48528 14.6969i 0.287183 0.497416i
\(874\) 0 0
\(875\) −1.62132 + 2.09077i −0.0548106 + 0.0706809i
\(876\) 0 0
\(877\) 14.1421 24.4949i 0.477546 0.827134i −0.522123 0.852870i \(-0.674860\pi\)
0.999669 + 0.0257364i \(0.00819306\pi\)
\(878\) 0 0
\(879\) 3.51472 + 6.08767i 0.118549 + 0.205332i
\(880\) 0 0
\(881\) −26.4558 −0.891320 −0.445660 0.895202i \(-0.647031\pi\)
−0.445660 + 0.895202i \(0.647031\pi\)
\(882\) 0 0
\(883\) −6.68629 −0.225012 −0.112506 0.993651i \(-0.535888\pi\)
−0.112506 + 0.993651i \(0.535888\pi\)
\(884\) 0 0
\(885\) 0.828427 + 1.43488i 0.0278473 + 0.0482329i
\(886\) 0 0
\(887\) 8.79289 15.2297i 0.295236 0.511365i −0.679803 0.733394i \(-0.737935\pi\)
0.975040 + 0.222030i \(0.0712682\pi\)
\(888\) 0 0
\(889\) −7.04163 + 9.08052i −0.236169 + 0.304551i
\(890\) 0 0
\(891\) 3.10051 5.37023i 0.103871 0.179910i
\(892\) 0 0
\(893\) 32.9706 + 57.1067i 1.10332 + 1.91100i
\(894\) 0 0
\(895\) 6.48528 0.216779
\(896\) 0 0
\(897\) −4.62742 −0.154505
\(898\) 0 0
\(899\) −3.24264 5.61642i −0.108148 0.187318i
\(900\) 0 0
\(901\) −21.6569 + 37.5108i −0.721494 + 1.24966i
\(902\) 0 0
\(903\) −7.49138 1.02517i −0.249297 0.0341156i
\(904\) 0 0
\(905\) 2.08579 3.61269i 0.0693339 0.120090i
\(906\) 0 0
\(907\) 23.1066 + 40.0218i 0.767242 + 1.32890i 0.939053 + 0.343772i \(0.111705\pi\)
−0.171811 + 0.985130i \(0.554962\pi\)
\(908\) 0 0
\(909\) 32.4853 1.07747
\(910\) 0 0
\(911\) 18.4853 0.612445 0.306222 0.951960i \(-0.400935\pi\)
0.306222 + 0.951960i \(0.400935\pi\)
\(912\) 0 0
\(913\) 1.97056 + 3.41311i 0.0652161 + 0.112958i
\(914\) 0 0
\(915\) 1.37868 2.38794i 0.0455777 0.0789430i
\(916\) 0 0
\(917\) 13.6569 + 33.4523i 0.450989 + 1.10469i
\(918\) 0 0
\(919\) −16.6274 + 28.7995i −0.548488 + 0.950009i 0.449891 + 0.893084i \(0.351463\pi\)
−0.998378 + 0.0569252i \(0.981870\pi\)
\(920\) 0 0
\(921\) −0.985281 1.70656i −0.0324661 0.0562330i
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 5.65685 0.185996
\(926\) 0 0
\(927\) −10.7279 18.5813i −0.352351 0.610290i
\(928\) 0 0
\(929\) −4.60051 + 7.96831i −0.150938 + 0.261432i −0.931572 0.363556i \(-0.881563\pi\)
0.780635 + 0.624988i \(0.214896\pi\)
\(930\) 0 0
\(931\) 38.1421 + 10.6385i 1.25006 + 0.348662i
\(932\) 0 0
\(933\) −4.48528 + 7.76874i −0.146842 + 0.254337i
\(934\) 0 0
\(935\) −3.17157 5.49333i −0.103722 0.179651i
\(936\) 0 0
\(937\) 18.6274 0.608531 0.304266 0.952587i \(-0.401589\pi\)
0.304266 + 0.952587i \(0.401589\pi\)
\(938\) 0 0
\(939\) −8.68629 −0.283466
\(940\) 0 0
\(941\) 5.00000 + 8.66025i 0.162995 + 0.282316i 0.935942 0.352155i \(-0.114551\pi\)
−0.772946 + 0.634472i \(0.781218\pi\)
\(942\) 0 0
\(943\) −16.2782 + 28.1946i −0.530090 + 0.918143i
\(944\) 0 0
\(945\) −2.41421 5.91359i −0.0785344 0.192369i
\(946\) 0 0
\(947\) 5.55025 9.61332i 0.180359 0.312391i −0.761644 0.647996i \(-0.775607\pi\)
0.942003 + 0.335605i \(0.108941\pi\)
\(948\) 0 0
\(949\) −3.65685 6.33386i −0.118707 0.205606i
\(950\) 0 0
\(951\) 9.11270 0.295499
\(952\) 0 0
\(953\) −54.6274 −1.76956 −0.884778 0.466013i \(-0.845690\pi\)
−0.884778 + 0.466013i \(0.845690\pi\)
\(954\) 0 0
\(955\) −2.75736 4.77589i −0.0892261 0.154544i
\(956\) 0 0
\(957\) 1.34315 2.32640i 0.0434177 0.0752017i
\(958\) 0 0
\(959\) −10.4853 1.43488i −0.338587 0.0463346i
\(960\) 0 0
\(961\) 15.1569 26.2524i 0.488931 0.846853i
\(962\) 0 0
\(963\) 7.89949 + 13.6823i 0.254558 + 0.440907i
\(964\) 0 0
\(965\) −5.31371 −0.171054
\(966\) 0 0
\(967\) 24.7574 0.796143 0.398072 0.917354i \(-0.369680\pi\)
0.398072 + 0.917354i \(0.369680\pi\)
\(968\) 0 0
\(969\) −8.97056 15.5375i −0.288176 0.499135i
\(970\) 0 0
\(971\) 4.00000 6.92820i 0.128366 0.222337i −0.794678 0.607032i \(-0.792360\pi\)
0.923044 + 0.384695i \(0.125693\pi\)
\(972\) 0 0
\(973\) 4.02944 5.19615i 0.129178 0.166581i
\(974\) 0 0
\(975\) 0.414214 0.717439i 0.0132655 0.0229764i
\(976\) 0 0
\(977\) −15.1421 26.2269i −0.484440 0.839074i 0.515400 0.856949i \(-0.327643\pi\)
−0.999840 + 0.0178751i \(0.994310\pi\)
\(978\) 0 0
\(979\) 4.42641 0.141469
\(980\) 0 0
\(981\) −51.7990 −1.65381
\(982\) 0 0
\(983\) −20.6924 35.8403i −0.659985 1.14313i −0.980619 0.195924i \(-0.937229\pi\)
0.320634 0.947203i \(-0.396104\pi\)
\(984\) 0 0
\(985\) −0.171573 + 0.297173i −0.00546677 + 0.00946872i
\(986\) 0 0
\(987\) −7.82843 + 10.0951i −0.249182 + 0.321332i
\(988\) 0 0
\(989\) −19.2696 + 33.3758i −0.612736 + 1.06129i
\(990\) 0 0
\(991\) 2.41421 + 4.18154i 0.0766900 + 0.132831i 0.901820 0.432112i \(-0.142231\pi\)
−0.825130 + 0.564943i \(0.808898\pi\)
\(992\) 0 0
\(993\) 10.9706 0.348140
\(994\) 0 0
\(995\) 23.3137 0.739094
\(996\) 0 0
\(997\) 7.17157 + 12.4215i 0.227126 + 0.393394i 0.956955 0.290236i \(-0.0937337\pi\)
−0.729829 + 0.683630i \(0.760400\pi\)
\(998\) 0 0
\(999\) −6.82843 + 11.8272i −0.216042 + 0.374196i
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 560.2.q.j.401.2 4
4.3 odd 2 280.2.q.d.121.1 yes 4
7.2 even 3 3920.2.a.bz.1.1 2
7.4 even 3 inner 560.2.q.j.81.2 4
7.5 odd 6 3920.2.a.bp.1.2 2
12.11 even 2 2520.2.bi.k.1801.2 4
20.3 even 4 1400.2.bh.g.849.3 8
20.7 even 4 1400.2.bh.g.849.2 8
20.19 odd 2 1400.2.q.h.401.2 4
28.3 even 6 1960.2.q.q.361.2 4
28.11 odd 6 280.2.q.d.81.1 4
28.19 even 6 1960.2.a.t.1.1 2
28.23 odd 6 1960.2.a.p.1.2 2
28.27 even 2 1960.2.q.q.961.2 4
84.11 even 6 2520.2.bi.k.361.2 4
140.19 even 6 9800.2.a.br.1.2 2
140.39 odd 6 1400.2.q.h.1201.2 4
140.67 even 12 1400.2.bh.g.249.3 8
140.79 odd 6 9800.2.a.bz.1.1 2
140.123 even 12 1400.2.bh.g.249.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.d.81.1 4 28.11 odd 6
280.2.q.d.121.1 yes 4 4.3 odd 2
560.2.q.j.81.2 4 7.4 even 3 inner
560.2.q.j.401.2 4 1.1 even 1 trivial
1400.2.q.h.401.2 4 20.19 odd 2
1400.2.q.h.1201.2 4 140.39 odd 6
1400.2.bh.g.249.2 8 140.123 even 12
1400.2.bh.g.249.3 8 140.67 even 12
1400.2.bh.g.849.2 8 20.7 even 4
1400.2.bh.g.849.3 8 20.3 even 4
1960.2.a.p.1.2 2 28.23 odd 6
1960.2.a.t.1.1 2 28.19 even 6
1960.2.q.q.361.2 4 28.3 even 6
1960.2.q.q.961.2 4 28.27 even 2
2520.2.bi.k.361.2 4 84.11 even 6
2520.2.bi.k.1801.2 4 12.11 even 2
3920.2.a.bp.1.2 2 7.5 odd 6
3920.2.a.bz.1.1 2 7.2 even 3
9800.2.a.br.1.2 2 140.19 even 6
9800.2.a.bz.1.1 2 140.79 odd 6