Properties

Label 1300.2.bs.d.193.5
Level $1300$
Weight $2$
Character 1300.193
Analytic conductor $10.381$
Analytic rank $0$
Dimension $20$
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] = [1300,2,Mod(193,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bs (of order \(12\), degree \(4\), 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 193.5
Root \(0.125665i\) of defining polynomial
Character \(\chi\) \(=\) 1300.193
Dual form 1300.2.bs.d.357.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.690650 - 2.57754i) q^{3} +(-1.87342 - 1.08162i) q^{7} +(-3.56864 - 2.06036i) q^{9} +O(q^{10})\) \(q+(0.690650 - 2.57754i) q^{3} +(-1.87342 - 1.08162i) q^{7} +(-3.56864 - 2.06036i) q^{9} +(-5.13447 - 1.37578i) q^{11} +(-2.35570 + 2.72959i) q^{13} +(-1.07583 + 0.288267i) q^{17} +(1.69845 + 6.33871i) q^{19} +(-4.08179 + 4.08179i) q^{21} +(0.718837 + 0.192612i) q^{23} +(-2.11466 + 2.11466i) q^{27} +(-0.0866681 + 0.0500378i) q^{29} +(3.90035 + 3.90035i) q^{31} +(-7.09224 + 12.2841i) q^{33} +(-5.91709 + 3.41623i) q^{37} +(5.40866 + 7.95711i) q^{39} +(1.36667 - 5.10047i) q^{41} +(-0.959150 - 3.57960i) q^{43} +2.04263i q^{47} +(-1.16021 - 2.00954i) q^{49} +2.97208i q^{51} +(-8.28330 - 8.28330i) q^{53} +17.5113 q^{57} +(12.1618 - 3.25875i) q^{59} +(-4.66503 + 8.08007i) q^{61} +(4.45703 + 7.71981i) q^{63} +(-5.51132 - 9.54588i) q^{67} +(0.992930 - 1.71981i) q^{69} +(-9.19754 + 2.46447i) q^{71} -10.7436 q^{73} +(8.13093 + 8.13093i) q^{77} -15.3154i q^{79} +(-2.19093 - 3.79481i) q^{81} -0.473480i q^{83} +(0.0691173 + 0.257949i) q^{87} +(-0.560208 + 2.09072i) q^{89} +(7.36558 - 2.56569i) q^{91} +(12.7471 - 7.35953i) q^{93} +(-6.34981 + 10.9982i) q^{97} +(15.4885 + 15.4885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 6 q^{7} - 12 q^{9} - 8 q^{13} + 20 q^{19} - 12 q^{21} - 6 q^{23} + 20 q^{27} + 24 q^{29} + 8 q^{31} + 10 q^{33} + 4 q^{39} + 6 q^{41} - 38 q^{43} + 14 q^{49} - 30 q^{53} + 76 q^{57} - 24 q^{59} - 32 q^{61} + 24 q^{63} - 22 q^{67} - 16 q^{69} + 44 q^{73} + 12 q^{77} + 2 q^{81} - 38 q^{87} - 30 q^{89} - 72 q^{91} + 48 q^{93} - 46 q^{97} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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 0
\(3\) 0.690650 2.57754i 0.398747 1.48814i −0.416557 0.909110i \(-0.636763\pi\)
0.815303 0.579034i \(-0.196570\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.87342 1.08162i −0.708085 0.408813i 0.102267 0.994757i \(-0.467390\pi\)
−0.810351 + 0.585944i \(0.800724\pi\)
\(8\) 0 0
\(9\) −3.56864 2.06036i −1.18955 0.686785i
\(10\) 0 0
\(11\) −5.13447 1.37578i −1.54810 0.414812i −0.619228 0.785211i \(-0.712554\pi\)
−0.928873 + 0.370399i \(0.879221\pi\)
\(12\) 0 0
\(13\) −2.35570 + 2.72959i −0.653355 + 0.757052i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.07583 + 0.288267i −0.260927 + 0.0699151i −0.386911 0.922117i \(-0.626458\pi\)
0.125984 + 0.992032i \(0.459791\pi\)
\(18\) 0 0
\(19\) 1.69845 + 6.33871i 0.389652 + 1.45420i 0.830702 + 0.556717i \(0.187939\pi\)
−0.441050 + 0.897482i \(0.645394\pi\)
\(20\) 0 0
\(21\) −4.08179 + 4.08179i −0.890719 + 0.890719i
\(22\) 0 0
\(23\) 0.718837 + 0.192612i 0.149888 + 0.0401624i 0.332983 0.942933i \(-0.391945\pi\)
−0.183095 + 0.983095i \(0.558612\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.11466 + 2.11466i −0.406967 + 0.406967i
\(28\) 0 0
\(29\) −0.0866681 + 0.0500378i −0.0160939 + 0.00929179i −0.508025 0.861342i \(-0.669624\pi\)
0.491931 + 0.870634i \(0.336291\pi\)
\(30\) 0 0
\(31\) 3.90035 + 3.90035i 0.700523 + 0.700523i 0.964523 0.264000i \(-0.0850418\pi\)
−0.264000 + 0.964523i \(0.585042\pi\)
\(32\) 0 0
\(33\) −7.09224 + 12.2841i −1.23460 + 2.13839i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.91709 + 3.41623i −0.972764 + 0.561625i −0.900078 0.435729i \(-0.856490\pi\)
−0.0726861 + 0.997355i \(0.523157\pi\)
\(38\) 0 0
\(39\) 5.40866 + 7.95711i 0.866079 + 1.27416i
\(40\) 0 0
\(41\) 1.36667 5.10047i 0.213437 0.796559i −0.773273 0.634073i \(-0.781382\pi\)
0.986711 0.162487i \(-0.0519514\pi\)
\(42\) 0 0
\(43\) −0.959150 3.57960i −0.146269 0.545883i −0.999696 0.0246704i \(-0.992146\pi\)
0.853427 0.521213i \(-0.174520\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.04263i 0.297948i 0.988841 + 0.148974i \(0.0475971\pi\)
−0.988841 + 0.148974i \(0.952403\pi\)
\(48\) 0 0
\(49\) −1.16021 2.00954i −0.165744 0.287077i
\(50\) 0 0
\(51\) 2.97208i 0.416175i
\(52\) 0 0
\(53\) −8.28330 8.28330i −1.13780 1.13780i −0.988844 0.148955i \(-0.952409\pi\)
−0.148955 0.988844i \(-0.547591\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 17.5113 2.31943
\(58\) 0 0
\(59\) 12.1618 3.25875i 1.58333 0.424253i 0.643378 0.765549i \(-0.277533\pi\)
0.939956 + 0.341296i \(0.110866\pi\)
\(60\) 0 0
\(61\) −4.66503 + 8.08007i −0.597296 + 1.03455i 0.395922 + 0.918284i \(0.370425\pi\)
−0.993218 + 0.116263i \(0.962908\pi\)
\(62\) 0 0
\(63\) 4.45703 + 7.71981i 0.561533 + 0.972604i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.51132 9.54588i −0.673315 1.16622i −0.976958 0.213429i \(-0.931537\pi\)
0.303644 0.952786i \(-0.401797\pi\)
\(68\) 0 0
\(69\) 0.992930 1.71981i 0.119535 0.207040i
\(70\) 0 0
\(71\) −9.19754 + 2.46447i −1.09155 + 0.292479i −0.759318 0.650720i \(-0.774467\pi\)
−0.332229 + 0.943199i \(0.607801\pi\)
\(72\) 0 0
\(73\) −10.7436 −1.25744 −0.628720 0.777632i \(-0.716421\pi\)
−0.628720 + 0.777632i \(0.716421\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.13093 + 8.13093i 0.926606 + 0.926606i
\(78\) 0 0
\(79\) 15.3154i 1.72312i −0.507655 0.861560i \(-0.669488\pi\)
0.507655 0.861560i \(-0.330512\pi\)
\(80\) 0 0
\(81\) −2.19093 3.79481i −0.243437 0.421645i
\(82\) 0 0
\(83\) 0.473480i 0.0519712i −0.999662 0.0259856i \(-0.991728\pi\)
0.999662 0.0259856i \(-0.00827240\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.0691173 + 0.257949i 0.00741015 + 0.0276551i
\(88\) 0 0
\(89\) −0.560208 + 2.09072i −0.0593819 + 0.221616i −0.989240 0.146302i \(-0.953263\pi\)
0.929858 + 0.367918i \(0.119929\pi\)
\(90\) 0 0
\(91\) 7.36558 2.56569i 0.772123 0.268957i
\(92\) 0 0
\(93\) 12.7471 7.35953i 1.32181 0.763147i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.34981 + 10.9982i −0.644725 + 1.11670i 0.339640 + 0.940556i \(0.389695\pi\)
−0.984365 + 0.176141i \(0.943638\pi\)
\(98\) 0 0
\(99\) 15.4885 + 15.4885i 1.55665 + 1.55665i
\(100\) 0 0
\(101\) −15.3907 + 8.88581i −1.53143 + 0.884172i −0.532134 + 0.846660i \(0.678610\pi\)
−0.999296 + 0.0375113i \(0.988057\pi\)
\(102\) 0 0
\(103\) 5.01350 5.01350i 0.493995 0.493995i −0.415567 0.909562i \(-0.636417\pi\)
0.909562 + 0.415567i \(0.136417\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.3783 + 2.78085i 1.00331 + 0.268835i 0.722830 0.691026i \(-0.242841\pi\)
0.280476 + 0.959861i \(0.409508\pi\)
\(108\) 0 0
\(109\) −2.36072 + 2.36072i −0.226116 + 0.226116i −0.811068 0.584952i \(-0.801113\pi\)
0.584952 + 0.811068i \(0.301113\pi\)
\(110\) 0 0
\(111\) 4.71884 + 17.6110i 0.447893 + 1.67156i
\(112\) 0 0
\(113\) −6.65737 + 1.78384i −0.626273 + 0.167809i −0.557978 0.829856i \(-0.688423\pi\)
−0.0682952 + 0.997665i \(0.521756\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 14.0306 4.88734i 1.29713 0.451834i
\(118\) 0 0
\(119\) 2.32727 + 0.623590i 0.213340 + 0.0571644i
\(120\) 0 0
\(121\) 14.9437 + 8.62776i 1.35852 + 0.784342i
\(122\) 0 0
\(123\) −12.2028 7.04528i −1.10029 0.635251i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.58896 + 13.3942i −0.318469 + 1.18854i 0.602248 + 0.798309i \(0.294272\pi\)
−0.920717 + 0.390232i \(0.872395\pi\)
\(128\) 0 0
\(129\) −9.88899 −0.870677
\(130\) 0 0
\(131\) 5.00046 0.436892 0.218446 0.975849i \(-0.429901\pi\)
0.218446 + 0.975849i \(0.429901\pi\)
\(132\) 0 0
\(133\) 3.67415 13.7121i 0.318589 1.18899i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.74706 5.62747i −0.832748 0.480787i 0.0220448 0.999757i \(-0.492982\pi\)
−0.854793 + 0.518970i \(0.826316\pi\)
\(138\) 0 0
\(139\) −7.54341 4.35519i −0.639823 0.369402i 0.144723 0.989472i \(-0.453771\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(140\) 0 0
\(141\) 5.26497 + 1.41074i 0.443390 + 0.118806i
\(142\) 0 0
\(143\) 15.8506 10.7741i 1.32549 0.900973i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.98097 + 1.60260i −0.493302 + 0.132180i
\(148\) 0 0
\(149\) 1.07157 + 3.99916i 0.0877865 + 0.327624i 0.995827 0.0912581i \(-0.0290888\pi\)
−0.908041 + 0.418882i \(0.862422\pi\)
\(150\) 0 0
\(151\) −4.01320 + 4.01320i −0.326590 + 0.326590i −0.851288 0.524698i \(-0.824178\pi\)
0.524698 + 0.851288i \(0.324178\pi\)
\(152\) 0 0
\(153\) 4.43318 + 1.18787i 0.358401 + 0.0960333i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.09083 + 6.09083i −0.486101 + 0.486101i −0.907073 0.420972i \(-0.861689\pi\)
0.420972 + 0.907073i \(0.361689\pi\)
\(158\) 0 0
\(159\) −27.0714 + 15.6297i −2.14690 + 1.23951i
\(160\) 0 0
\(161\) −1.13835 1.13835i −0.0897145 0.0897145i
\(162\) 0 0
\(163\) −5.59090 + 9.68372i −0.437913 + 0.758488i −0.997528 0.0702649i \(-0.977616\pi\)
0.559615 + 0.828752i \(0.310949\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.72554 3.88299i 0.520438 0.300475i −0.216676 0.976244i \(-0.569521\pi\)
0.737114 + 0.675769i \(0.236188\pi\)
\(168\) 0 0
\(169\) −1.90131 12.8602i −0.146255 0.989247i
\(170\) 0 0
\(171\) 6.99883 26.1200i 0.535214 1.99745i
\(172\) 0 0
\(173\) 3.86973 + 14.4420i 0.294210 + 1.09801i 0.941843 + 0.336054i \(0.109092\pi\)
−0.647633 + 0.761952i \(0.724241\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 33.5982i 2.52540i
\(178\) 0 0
\(179\) −0.627120 1.08620i −0.0468731 0.0811867i 0.841637 0.540044i \(-0.181592\pi\)
−0.888510 + 0.458857i \(0.848259\pi\)
\(180\) 0 0
\(181\) 7.23242i 0.537582i −0.963199 0.268791i \(-0.913376\pi\)
0.963199 0.268791i \(-0.0866240\pi\)
\(182\) 0 0
\(183\) 17.6048 + 17.6048i 1.30139 + 1.30139i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.92040 0.432942
\(188\) 0 0
\(189\) 6.24889 1.67439i 0.454540 0.121794i
\(190\) 0 0
\(191\) 10.5227 18.2259i 0.761397 1.31878i −0.180734 0.983532i \(-0.557847\pi\)
0.942131 0.335246i \(-0.108819\pi\)
\(192\) 0 0
\(193\) −4.97251 8.61264i −0.357929 0.619952i 0.629686 0.776850i \(-0.283184\pi\)
−0.987615 + 0.156899i \(0.949850\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.68592 + 4.65216i 0.191364 + 0.331453i 0.945703 0.325033i \(-0.105376\pi\)
−0.754338 + 0.656486i \(0.772042\pi\)
\(198\) 0 0
\(199\) 3.24212 5.61552i 0.229828 0.398073i −0.727929 0.685652i \(-0.759517\pi\)
0.957757 + 0.287579i \(0.0928504\pi\)
\(200\) 0 0
\(201\) −28.4113 + 7.61278i −2.00398 + 0.536964i
\(202\) 0 0
\(203\) 0.216487 0.0151944
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.16842 2.16842i −0.150716 0.150716i
\(208\) 0 0
\(209\) 34.8826i 2.41288i
\(210\) 0 0
\(211\) −13.6792 23.6930i −0.941713 1.63110i −0.762202 0.647340i \(-0.775882\pi\)
−0.179512 0.983756i \(-0.557452\pi\)
\(212\) 0 0
\(213\) 25.4091i 1.74100i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.08829 11.5257i −0.209647 0.782412i
\(218\) 0 0
\(219\) −7.42004 + 27.6920i −0.501400 + 1.87125i
\(220\) 0 0
\(221\) 1.74748 3.61564i 0.117548 0.243214i
\(222\) 0 0
\(223\) −1.84198 + 1.06347i −0.123348 + 0.0712151i −0.560405 0.828219i \(-0.689354\pi\)
0.437056 + 0.899434i \(0.356021\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.2288 + 19.4489i −0.745282 + 1.29087i 0.204781 + 0.978808i \(0.434352\pi\)
−0.950063 + 0.312058i \(0.898982\pi\)
\(228\) 0 0
\(229\) 0.742654 + 0.742654i 0.0490760 + 0.0490760i 0.731219 0.682143i \(-0.238952\pi\)
−0.682143 + 0.731219i \(0.738952\pi\)
\(230\) 0 0
\(231\) 26.5734 15.3422i 1.74840 1.00944i
\(232\) 0 0
\(233\) −6.72311 + 6.72311i −0.440446 + 0.440446i −0.892162 0.451716i \(-0.850812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −39.4761 10.5776i −2.56425 0.687089i
\(238\) 0 0
\(239\) 1.41635 1.41635i 0.0916161 0.0916161i −0.659813 0.751430i \(-0.729365\pi\)
0.751430 + 0.659813i \(0.229365\pi\)
\(240\) 0 0
\(241\) 0.0381404 + 0.142342i 0.00245684 + 0.00916904i 0.967143 0.254232i \(-0.0818226\pi\)
−0.964687 + 0.263401i \(0.915156\pi\)
\(242\) 0 0
\(243\) −19.9605 + 5.34840i −1.28047 + 0.343100i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −21.3031 10.2960i −1.35549 0.655122i
\(248\) 0 0
\(249\) −1.22041 0.327009i −0.0773406 0.0207233i
\(250\) 0 0
\(251\) −12.1871 7.03620i −0.769240 0.444121i 0.0633631 0.997991i \(-0.479817\pi\)
−0.832604 + 0.553869i \(0.813151\pi\)
\(252\) 0 0
\(253\) −3.42586 1.97792i −0.215382 0.124351i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.98961 18.6215i 0.311243 1.16158i −0.616193 0.787595i \(-0.711326\pi\)
0.927436 0.373981i \(-0.122007\pi\)
\(258\) 0 0
\(259\) 14.7802 0.918399
\(260\) 0 0
\(261\) 0.412383 0.0255259
\(262\) 0 0
\(263\) 5.68609 21.2208i 0.350619 1.30853i −0.535289 0.844669i \(-0.679797\pi\)
0.885908 0.463860i \(-0.153536\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.00202 + 2.88792i 0.306118 + 0.176738i
\(268\) 0 0
\(269\) 9.22286 + 5.32482i 0.562328 + 0.324660i 0.754079 0.656783i \(-0.228083\pi\)
−0.191751 + 0.981444i \(0.561417\pi\)
\(270\) 0 0
\(271\) 3.71082 + 0.994312i 0.225416 + 0.0604001i 0.369759 0.929128i \(-0.379440\pi\)
−0.144343 + 0.989528i \(0.546107\pi\)
\(272\) 0 0
\(273\) −1.52612 20.7571i −0.0923649 1.25628i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 13.6128 3.64753i 0.817912 0.219159i 0.174479 0.984661i \(-0.444176\pi\)
0.643434 + 0.765502i \(0.277509\pi\)
\(278\) 0 0
\(279\) −5.88284 21.9550i −0.352196 1.31441i
\(280\) 0 0
\(281\) 12.6788 12.6788i 0.756354 0.756354i −0.219303 0.975657i \(-0.570378\pi\)
0.975657 + 0.219303i \(0.0703782\pi\)
\(282\) 0 0
\(283\) 8.40291 + 2.25155i 0.499501 + 0.133841i 0.499768 0.866159i \(-0.333418\pi\)
−0.000267339 1.00000i \(0.500085\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.07709 + 8.07709i −0.476776 + 0.476776i
\(288\) 0 0
\(289\) −13.6481 + 7.87975i −0.802831 + 0.463515i
\(290\) 0 0
\(291\) 23.9628 + 23.9628i 1.40472 + 1.40472i
\(292\) 0 0
\(293\) 5.98879 10.3729i 0.349869 0.605991i −0.636357 0.771395i \(-0.719559\pi\)
0.986226 + 0.165404i \(0.0528928\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.7670 7.94836i 0.798840 0.461211i
\(298\) 0 0
\(299\) −2.21912 + 1.50839i −0.128335 + 0.0872327i
\(300\) 0 0
\(301\) −2.07487 + 7.74351i −0.119593 + 0.446328i
\(302\) 0 0
\(303\) 12.2740 + 45.8071i 0.705121 + 2.63155i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0009i 1.14151i 0.821120 + 0.570755i \(0.193350\pi\)
−0.821120 + 0.570755i \(0.806650\pi\)
\(308\) 0 0
\(309\) −9.45993 16.3851i −0.538157 0.932115i
\(310\) 0 0
\(311\) 14.8121i 0.839916i −0.907543 0.419958i \(-0.862045\pi\)
0.907543 0.419958i \(-0.137955\pi\)
\(312\) 0 0
\(313\) −2.35138 2.35138i −0.132908 0.132908i 0.637523 0.770431i \(-0.279959\pi\)
−0.770431 + 0.637523i \(0.779959\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.2905 0.858799 0.429399 0.903115i \(-0.358725\pi\)
0.429399 + 0.903115i \(0.358725\pi\)
\(318\) 0 0
\(319\) 0.513835 0.137682i 0.0287693 0.00770870i
\(320\) 0 0
\(321\) 14.3355 24.8298i 0.800130 1.38587i
\(322\) 0 0
\(323\) −3.65449 6.32975i −0.203341 0.352197i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.45443 + 7.71529i 0.246330 + 0.426657i
\(328\) 0 0
\(329\) 2.20935 3.82670i 0.121805 0.210973i
\(330\) 0 0
\(331\) 0.900776 0.241362i 0.0495111 0.0132665i −0.233978 0.972242i \(-0.575175\pi\)
0.283490 + 0.958975i \(0.408508\pi\)
\(332\) 0 0
\(333\) 28.1546 1.54286
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.9698 15.9698i −0.869930 0.869930i 0.122534 0.992464i \(-0.460898\pi\)
−0.992464 + 0.122534i \(0.960898\pi\)
\(338\) 0 0
\(339\) 18.3917i 0.998898i
\(340\) 0 0
\(341\) −14.6602 25.3922i −0.793894 1.37507i
\(342\) 0 0
\(343\) 20.1622i 1.08866i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.63370 9.82910i −0.141384 0.527654i −0.999890 0.0148492i \(-0.995273\pi\)
0.858505 0.512805i \(-0.171393\pi\)
\(348\) 0 0
\(349\) −0.766967 + 2.86236i −0.0410548 + 0.153219i −0.983410 0.181394i \(-0.941939\pi\)
0.942356 + 0.334613i \(0.108606\pi\)
\(350\) 0 0
\(351\) −0.790640 10.7537i −0.0422012 0.573989i
\(352\) 0 0
\(353\) 27.9018 16.1091i 1.48506 0.857402i 0.485209 0.874398i \(-0.338744\pi\)
0.999856 + 0.0169962i \(0.00541032\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.21466 5.56795i 0.170138 0.294687i
\(358\) 0 0
\(359\) 1.11254 + 1.11254i 0.0587178 + 0.0587178i 0.735856 0.677138i \(-0.236780\pi\)
−0.677138 + 0.735856i \(0.736780\pi\)
\(360\) 0 0
\(361\) −20.8400 + 12.0320i −1.09684 + 0.633262i
\(362\) 0 0
\(363\) 32.5593 32.5593i 1.70892 1.70892i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 20.3821 + 5.46138i 1.06394 + 0.285082i 0.748001 0.663698i \(-0.231014\pi\)
0.315938 + 0.948780i \(0.397681\pi\)
\(368\) 0 0
\(369\) −15.3859 + 15.3859i −0.800959 + 0.800959i
\(370\) 0 0
\(371\) 6.55871 + 24.4774i 0.340511 + 1.27080i
\(372\) 0 0
\(373\) 24.3817 6.53307i 1.26244 0.338269i 0.435309 0.900281i \(-0.356639\pi\)
0.827129 + 0.562012i \(0.189972\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.0675816 0.354443i 0.00348063 0.0182547i
\(378\) 0 0
\(379\) −25.4336 6.81491i −1.30644 0.350058i −0.462556 0.886590i \(-0.653067\pi\)
−0.843880 + 0.536532i \(0.819734\pi\)
\(380\) 0 0
\(381\) 32.0453 + 18.5014i 1.64173 + 0.947854i
\(382\) 0 0
\(383\) −31.3842 18.1197i −1.60366 0.925873i −0.990747 0.135722i \(-0.956664\pi\)
−0.612913 0.790151i \(-0.710002\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.95238 + 14.7505i −0.200911 + 0.749810i
\(388\) 0 0
\(389\) 16.9315 0.858462 0.429231 0.903195i \(-0.358785\pi\)
0.429231 + 0.903195i \(0.358785\pi\)
\(390\) 0 0
\(391\) −0.828869 −0.0419177
\(392\) 0 0
\(393\) 3.45356 12.8889i 0.174209 0.650158i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 18.5248 + 10.6953i 0.929732 + 0.536781i 0.886727 0.462294i \(-0.152973\pi\)
0.0430054 + 0.999075i \(0.486307\pi\)
\(398\) 0 0
\(399\) −32.8060 18.9405i −1.64235 0.948213i
\(400\) 0 0
\(401\) 34.4734 + 9.23713i 1.72152 + 0.461280i 0.978202 0.207657i \(-0.0665839\pi\)
0.743319 + 0.668937i \(0.233251\pi\)
\(402\) 0 0
\(403\) −19.8344 + 1.45828i −0.988022 + 0.0726421i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 35.0811 9.39995i 1.73891 0.465938i
\(408\) 0 0
\(409\) 1.43399 + 5.35172i 0.0709062 + 0.264625i 0.992274 0.124067i \(-0.0395937\pi\)
−0.921368 + 0.388692i \(0.872927\pi\)
\(410\) 0 0
\(411\) −21.2368 + 21.2368i −1.04754 + 1.04754i
\(412\) 0 0
\(413\) −26.3089 7.04944i −1.29457 0.346880i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.4355 + 16.4355i −0.804851 + 0.804851i
\(418\) 0 0
\(419\) 10.8355 6.25590i 0.529351 0.305621i −0.211401 0.977399i \(-0.567803\pi\)
0.740752 + 0.671779i \(0.234469\pi\)
\(420\) 0 0
\(421\) −0.936634 0.936634i −0.0456488 0.0456488i 0.683914 0.729563i \(-0.260276\pi\)
−0.729563 + 0.683914i \(0.760276\pi\)
\(422\) 0 0
\(423\) 4.20855 7.28942i 0.204627 0.354424i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 17.4791 10.0916i 0.845873 0.488365i
\(428\) 0 0
\(429\) −16.8234 48.2967i −0.812240 2.33178i
\(430\) 0 0
\(431\) −3.95753 + 14.7697i −0.190628 + 0.711432i 0.802728 + 0.596346i \(0.203381\pi\)
−0.993355 + 0.115087i \(0.963285\pi\)
\(432\) 0 0
\(433\) −2.95228 11.0180i −0.141877 0.529493i −0.999875 0.0158391i \(-0.994958\pi\)
0.857997 0.513654i \(-0.171709\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.88364i 0.233616i
\(438\) 0 0
\(439\) 17.3246 + 30.0071i 0.826858 + 1.43216i 0.900491 + 0.434875i \(0.143208\pi\)
−0.0736324 + 0.997285i \(0.523459\pi\)
\(440\) 0 0
\(441\) 9.56177i 0.455322i
\(442\) 0 0
\(443\) −4.85964 4.85964i −0.230888 0.230888i 0.582175 0.813063i \(-0.302202\pi\)
−0.813063 + 0.582175i \(0.802202\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 11.0481 0.522556
\(448\) 0 0
\(449\) 28.2416 7.56732i 1.33280 0.357124i 0.479044 0.877791i \(-0.340983\pi\)
0.853760 + 0.520667i \(0.174317\pi\)
\(450\) 0 0
\(451\) −14.0342 + 24.3080i −0.660845 + 1.14462i
\(452\) 0 0
\(453\) 7.57248 + 13.1159i 0.355786 + 0.616240i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.2322 21.1868i −0.572198 0.991076i −0.996340 0.0854803i \(-0.972758\pi\)
0.424142 0.905596i \(-0.360576\pi\)
\(458\) 0 0
\(459\) 1.66542 2.88460i 0.0777354 0.134642i
\(460\) 0 0
\(461\) −13.4819 + 3.61245i −0.627912 + 0.168249i −0.558722 0.829355i \(-0.688708\pi\)
−0.0691902 + 0.997603i \(0.522042\pi\)
\(462\) 0 0
\(463\) 7.08508 0.329272 0.164636 0.986354i \(-0.447355\pi\)
0.164636 + 0.986354i \(0.447355\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.40762 + 4.40762i 0.203960 + 0.203960i 0.801694 0.597734i \(-0.203932\pi\)
−0.597734 + 0.801694i \(0.703932\pi\)
\(468\) 0 0
\(469\) 23.8445i 1.10104i
\(470\) 0 0
\(471\) 11.4927 + 19.9060i 0.529557 + 0.917219i
\(472\) 0 0
\(473\) 19.6989i 0.905757i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.4936 + 46.6267i 0.572042 + 2.13489i
\(478\) 0 0
\(479\) 8.30384 30.9903i 0.379412 1.41598i −0.467378 0.884058i \(-0.654801\pi\)
0.846790 0.531927i \(-0.178532\pi\)
\(480\) 0 0
\(481\) 4.61400 24.1989i 0.210380 1.10337i
\(482\) 0 0
\(483\) −3.72034 + 2.14794i −0.169281 + 0.0977347i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.99150 10.3776i 0.271501 0.470253i −0.697746 0.716346i \(-0.745813\pi\)
0.969246 + 0.246093i \(0.0791467\pi\)
\(488\) 0 0
\(489\) 21.0988 + 21.0988i 0.954122 + 0.954122i
\(490\) 0 0
\(491\) 20.9648 12.1041i 0.946130 0.546248i 0.0542534 0.998527i \(-0.482722\pi\)
0.891877 + 0.452279i \(0.149389\pi\)
\(492\) 0 0
\(493\) 0.0788157 0.0788157i 0.00354968 0.00354968i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.8964 + 5.33123i 0.892477 + 0.239138i
\(498\) 0 0
\(499\) −23.7196 + 23.7196i −1.06183 + 1.06183i −0.0638768 + 0.997958i \(0.520346\pi\)
−0.997958 + 0.0638768i \(0.979654\pi\)
\(500\) 0 0
\(501\) −5.36358 20.0171i −0.239627 0.894300i
\(502\) 0 0
\(503\) 0.473787 0.126951i 0.0211251 0.00566046i −0.248241 0.968698i \(-0.579852\pi\)
0.269366 + 0.963038i \(0.413186\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −34.4609 3.98119i −1.53046 0.176811i
\(508\) 0 0
\(509\) 14.5334 + 3.89421i 0.644181 + 0.172608i 0.566097 0.824339i \(-0.308453\pi\)
0.0780847 + 0.996947i \(0.475120\pi\)
\(510\) 0 0
\(511\) 20.1272 + 11.6204i 0.890373 + 0.514057i
\(512\) 0 0
\(513\) −16.9959 9.81257i −0.750386 0.433236i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.81020 10.4878i 0.123593 0.461254i
\(518\) 0 0
\(519\) 39.8975 1.75131
\(520\) 0 0
\(521\) −38.1804 −1.67271 −0.836357 0.548185i \(-0.815319\pi\)
−0.836357 + 0.548185i \(0.815319\pi\)
\(522\) 0 0
\(523\) −5.48576 + 20.4731i −0.239875 + 0.895227i 0.736015 + 0.676965i \(0.236705\pi\)
−0.975890 + 0.218262i \(0.929961\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.32045 3.07176i −0.231762 0.133808i
\(528\) 0 0
\(529\) −19.4390 11.2231i −0.845172 0.487960i
\(530\) 0 0
\(531\) −50.1153 13.4284i −2.17482 0.582741i
\(532\) 0 0
\(533\) 10.7027 + 15.7456i 0.463586 + 0.682019i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.23285 + 0.866241i −0.139508 + 0.0373810i
\(538\) 0 0
\(539\) 3.19238 + 11.9141i 0.137505 + 0.513177i
\(540\) 0 0
\(541\) −0.700130 + 0.700130i −0.0301009 + 0.0301009i −0.721997 0.691896i \(-0.756776\pi\)
0.691896 + 0.721997i \(0.256776\pi\)
\(542\) 0 0
\(543\) −18.6419 4.99507i −0.799999 0.214359i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.28589 7.28589i 0.311522 0.311522i −0.533977 0.845499i \(-0.679303\pi\)
0.845499 + 0.533977i \(0.179303\pi\)
\(548\) 0 0
\(549\) 33.2957 19.2233i 1.42102 0.820429i
\(550\) 0 0
\(551\) −0.464377 0.464377i −0.0197831 0.0197831i
\(552\) 0 0
\(553\) −16.5654 + 28.6922i −0.704434 + 1.22012i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.5259 + 11.8506i −0.869710 + 0.502127i −0.867252 0.497870i \(-0.834116\pi\)
−0.00245824 + 0.999997i \(0.500782\pi\)
\(558\) 0 0
\(559\) 12.0303 + 5.81439i 0.508828 + 0.245922i
\(560\) 0 0
\(561\) 4.08892 15.2601i 0.172634 0.644280i
\(562\) 0 0
\(563\) −5.47068 20.4169i −0.230562 0.860468i −0.980100 0.198507i \(-0.936391\pi\)
0.749538 0.661962i \(-0.230276\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.47901i 0.398081i
\(568\) 0 0
\(569\) 5.92902 + 10.2694i 0.248557 + 0.430514i 0.963126 0.269052i \(-0.0867102\pi\)
−0.714568 + 0.699566i \(0.753377\pi\)
\(570\) 0 0
\(571\) 6.26224i 0.262067i 0.991378 + 0.131033i \(0.0418295\pi\)
−0.991378 + 0.131033i \(0.958171\pi\)
\(572\) 0 0
\(573\) −39.7104 39.7104i −1.65893 1.65893i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −45.6328 −1.89972 −0.949858 0.312681i \(-0.898773\pi\)
−0.949858 + 0.312681i \(0.898773\pi\)
\(578\) 0 0
\(579\) −25.6337 + 6.86853i −1.06530 + 0.285446i
\(580\) 0 0
\(581\) −0.512124 + 0.887025i −0.0212465 + 0.0368000i
\(582\) 0 0
\(583\) 31.1344 + 53.9263i 1.28945 + 2.23340i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.44141 2.49660i −0.0594935 0.103046i 0.834745 0.550637i \(-0.185615\pi\)
−0.894238 + 0.447592i \(0.852282\pi\)
\(588\) 0 0
\(589\) −18.0986 + 31.3477i −0.745740 + 1.29166i
\(590\) 0 0
\(591\) 13.8462 3.71007i 0.569555 0.152612i
\(592\) 0 0
\(593\) 17.0136 0.698663 0.349332 0.936999i \(-0.386409\pi\)
0.349332 + 0.936999i \(0.386409\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.2351 12.2351i −0.500747 0.500747i
\(598\) 0 0
\(599\) 15.8501i 0.647618i −0.946123 0.323809i \(-0.895037\pi\)
0.946123 0.323809i \(-0.104963\pi\)
\(600\) 0 0
\(601\) −7.73635 13.3998i −0.315572 0.546587i 0.663987 0.747744i \(-0.268863\pi\)
−0.979559 + 0.201157i \(0.935530\pi\)
\(602\) 0 0
\(603\) 45.4211i 1.84969i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.254323 + 0.949146i 0.0103227 + 0.0385247i 0.970895 0.239505i \(-0.0769852\pi\)
−0.960572 + 0.278030i \(0.910319\pi\)
\(608\) 0 0
\(609\) 0.149517 0.558004i 0.00605873 0.0226115i
\(610\) 0 0
\(611\) −5.57554 4.81184i −0.225562 0.194666i
\(612\) 0 0
\(613\) 7.48436 4.32110i 0.302291 0.174528i −0.341181 0.939998i \(-0.610827\pi\)
0.643471 + 0.765470i \(0.277494\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.38476 + 7.59463i −0.176524 + 0.305748i −0.940688 0.339274i \(-0.889819\pi\)
0.764164 + 0.645022i \(0.223152\pi\)
\(618\) 0 0
\(619\) 8.20840 + 8.20840i 0.329924 + 0.329924i 0.852557 0.522634i \(-0.175050\pi\)
−0.522634 + 0.852557i \(0.675050\pi\)
\(620\) 0 0
\(621\) −1.92741 + 1.11279i −0.0773442 + 0.0446547i
\(622\) 0 0
\(623\) 3.31086 3.31086i 0.132647 0.132647i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −89.9113 24.0917i −3.59071 0.962128i
\(628\) 0 0
\(629\) 5.38099 5.38099i 0.214554 0.214554i
\(630\) 0 0
\(631\) 1.58993 + 5.93370i 0.0632941 + 0.236217i 0.990325 0.138769i \(-0.0443144\pi\)
−0.927031 + 0.374985i \(0.877648\pi\)
\(632\) 0 0
\(633\) −70.5173 + 18.8950i −2.80281 + 0.751011i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.21833 + 1.56699i 0.325622 + 0.0620864i
\(638\) 0 0
\(639\) 37.9004 + 10.1554i 1.49932 + 0.401741i
\(640\) 0 0
\(641\) −24.8739 14.3609i −0.982459 0.567223i −0.0794473 0.996839i \(-0.525316\pi\)
−0.903012 + 0.429616i \(0.858649\pi\)
\(642\) 0 0
\(643\) −27.4892 15.8709i −1.08407 0.625887i −0.152078 0.988369i \(-0.548596\pi\)
−0.931991 + 0.362481i \(0.881930\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.01862 3.80155i 0.0400462 0.149454i −0.943008 0.332771i \(-0.892017\pi\)
0.983054 + 0.183316i \(0.0586832\pi\)
\(648\) 0 0
\(649\) −66.9278 −2.62714
\(650\) 0 0
\(651\) −31.8408 −1.24794
\(652\) 0 0
\(653\) −7.77659 + 29.0226i −0.304321 + 1.13574i 0.629207 + 0.777238i \(0.283380\pi\)
−0.933528 + 0.358505i \(0.883287\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 38.3399 + 22.1356i 1.49578 + 0.863591i
\(658\) 0 0
\(659\) −17.0616 9.85051i −0.664625 0.383721i 0.129412 0.991591i \(-0.458691\pi\)
−0.794037 + 0.607870i \(0.792024\pi\)
\(660\) 0 0
\(661\) −12.2450 3.28105i −0.476277 0.127618i 0.0126917 0.999919i \(-0.495960\pi\)
−0.488968 + 0.872301i \(0.662627\pi\)
\(662\) 0 0
\(663\) −8.11256 7.00135i −0.315066 0.271910i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −0.0719382 + 0.0192758i −0.00278546 + 0.000746361i
\(668\) 0 0
\(669\) 1.46897 + 5.48227i 0.0567936 + 0.211957i
\(670\) 0 0
\(671\) 35.0688 35.0688i 1.35382 1.35382i
\(672\) 0 0
\(673\) 43.9051 + 11.7643i 1.69242 + 0.453482i 0.971012 0.239031i \(-0.0768296\pi\)
0.721406 + 0.692513i \(0.243496\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.14523 7.14523i 0.274614 0.274614i −0.556341 0.830954i \(-0.687795\pi\)
0.830954 + 0.556341i \(0.187795\pi\)
\(678\) 0 0
\(679\) 23.7917 13.7361i 0.913040 0.527144i
\(680\) 0 0
\(681\) 42.3750 + 42.3750i 1.62381 + 1.62381i
\(682\) 0 0
\(683\) 4.65327 8.05969i 0.178052 0.308396i −0.763161 0.646208i \(-0.776354\pi\)
0.941213 + 0.337813i \(0.109687\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 2.42714 1.40131i 0.0926010 0.0534632i
\(688\) 0 0
\(689\) 42.1230 3.09700i 1.60476 0.117986i
\(690\) 0 0
\(691\) −0.156512 + 0.584112i −0.00595401 + 0.0222207i −0.968839 0.247692i \(-0.920328\pi\)
0.962885 + 0.269912i \(0.0869947\pi\)
\(692\) 0 0
\(693\) −12.2638 45.7690i −0.465862 1.73862i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.88119i 0.222766i
\(698\) 0 0
\(699\) 12.6858 + 21.9724i 0.479820 + 0.831072i
\(700\) 0 0
\(701\) 42.5349i 1.60652i 0.595627 + 0.803261i \(0.296904\pi\)
−0.595627 + 0.803261i \(0.703096\pi\)
\(702\) 0 0
\(703\) −31.7044 31.7044i −1.19575 1.19575i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 38.4442 1.44584
\(708\) 0 0
\(709\) −47.9316 + 12.8432i −1.80011 + 0.482337i −0.993995 0.109425i \(-0.965099\pi\)
−0.806112 + 0.591762i \(0.798432\pi\)
\(710\) 0 0
\(711\) −31.5552 + 54.6553i −1.18341 + 2.04973i
\(712\) 0 0
\(713\) 2.05246 + 3.55497i 0.0768653 + 0.133135i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2.67250 4.62890i −0.0998063 0.172870i
\(718\) 0 0
\(719\) −3.49009 + 6.04502i −0.130159 + 0.225441i −0.923738 0.383026i \(-0.874882\pi\)
0.793579 + 0.608467i \(0.208215\pi\)
\(720\) 0 0
\(721\) −14.8151 + 3.96969i −0.551742 + 0.147839i
\(722\) 0 0
\(723\) 0.393233 0.0146245
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −18.2655 18.2655i −0.677429 0.677429i 0.281989 0.959418i \(-0.409006\pi\)
−0.959418 + 0.281989i \(0.909006\pi\)
\(728\) 0 0
\(729\) 41.9972i 1.55545i
\(730\) 0 0
\(731\) 2.06376 + 3.57454i 0.0763310 + 0.132209i
\(732\) 0 0
\(733\) 30.7161i 1.13452i 0.823537 + 0.567262i \(0.191997\pi\)
−0.823537 + 0.567262i \(0.808003\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.1647 + 56.5954i 0.558598 + 2.08472i
\(738\) 0 0
\(739\) 7.17064 26.7612i 0.263776 0.984427i −0.699219 0.714908i \(-0.746469\pi\)
0.962995 0.269519i \(-0.0868647\pi\)
\(740\) 0 0
\(741\) −41.2515 + 47.7987i −1.51541 + 1.75593i
\(742\) 0 0
\(743\) −6.90167 + 3.98468i −0.253198 + 0.146184i −0.621228 0.783630i \(-0.713366\pi\)
0.368030 + 0.929814i \(0.380032\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.975537 + 1.68968i −0.0356930 + 0.0618222i
\(748\) 0 0
\(749\) −16.4350 16.4350i −0.600522 0.600522i
\(750\) 0 0
\(751\) −27.1108 + 15.6524i −0.989287 + 0.571165i −0.905061 0.425282i \(-0.860175\pi\)
−0.0842259 + 0.996447i \(0.526842\pi\)
\(752\) 0 0
\(753\) −26.5531 + 26.5531i −0.967648 + 0.967648i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37.8020 + 10.1290i 1.37394 + 0.368145i 0.868915 0.494962i \(-0.164818\pi\)
0.505021 + 0.863107i \(0.331485\pi\)
\(758\) 0 0
\(759\) −7.46424 + 7.46424i −0.270935 + 0.270935i
\(760\) 0 0
\(761\) −7.61145 28.4063i −0.275915 1.02973i −0.955226 0.295877i \(-0.904388\pi\)
0.679311 0.733850i \(-0.262279\pi\)
\(762\) 0 0
\(763\) 6.97602 1.86922i 0.252549 0.0676702i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −19.7546 + 40.8734i −0.713297 + 1.47585i
\(768\) 0 0
\(769\) 32.5217 + 8.71417i 1.17276 + 0.314241i 0.792052 0.610453i \(-0.209013\pi\)
0.380711 + 0.924694i \(0.375679\pi\)
\(770\) 0 0
\(771\) −44.5516 25.7219i −1.60449 0.926350i
\(772\) 0 0
\(773\) 22.7618 + 13.1416i 0.818687 + 0.472669i 0.849963 0.526842i \(-0.176624\pi\)
−0.0312766 + 0.999511i \(0.509957\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.2080 38.0966i 0.366209 1.36671i
\(778\) 0 0
\(779\) 34.6516 1.24152
\(780\) 0 0
\(781\) 50.6150 1.81115
\(782\) 0 0
\(783\) 0.0774606 0.289087i 0.00276821 0.0103311i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 17.8020 + 10.2780i 0.634573 + 0.366371i 0.782521 0.622624i \(-0.213933\pi\)
−0.147948 + 0.988995i \(0.547267\pi\)
\(788\) 0 0
\(789\) −50.7703 29.3123i −1.80747 1.04354i
\(790\) 0 0
\(791\) 14.4015 + 3.85886i 0.512057 + 0.137205i
\(792\) 0 0
\(793\) −11.0658 31.7679i −0.392960 1.12811i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.6043 + 6.59270i −0.871529 + 0.233526i −0.666749 0.745282i \(-0.732315\pi\)
−0.204780 + 0.978808i \(0.565648\pi\)
\(798\) 0 0
\(799\) −0.588824 2.19752i −0.0208311 0.0777427i
\(800\) 0 0
\(801\) 6.30681 6.30681i 0.222840 0.222840i
\(802\) 0 0
\(803\) 55.1625 + 14.7807i 1.94664 + 0.521601i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20.0947 20.0947i 0.707368 0.707368i
\(808\) 0 0
\(809\) 14.4124 8.32102i 0.506714 0.292551i −0.224768 0.974412i \(-0.572162\pi\)
0.731482 + 0.681861i \(0.238829\pi\)
\(810\) 0 0
\(811\) −12.1054 12.1054i −0.425079 0.425079i 0.461869 0.886948i \(-0.347179\pi\)
−0.886948 + 0.461869i \(0.847179\pi\)
\(812\) 0 0
\(813\) 5.12576 8.87807i 0.179768 0.311368i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21.0610 12.1595i 0.736830 0.425409i
\(818\) 0 0
\(819\) −31.5714 6.01971i −1.10319 0.210346i
\(820\) 0 0
\(821\) −8.36891 + 31.2332i −0.292077 + 1.09005i 0.651434 + 0.758705i \(0.274168\pi\)
−0.943511 + 0.331341i \(0.892499\pi\)
\(822\) 0 0
\(823\) −1.80480 6.73560i −0.0629114 0.234788i 0.927310 0.374295i \(-0.122115\pi\)
−0.990221 + 0.139506i \(0.955448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.2158i 1.18980i −0.803800 0.594900i \(-0.797192\pi\)
0.803800 0.594900i \(-0.202808\pi\)
\(828\) 0 0
\(829\) 13.9995 + 24.2478i 0.486222 + 0.842161i 0.999875 0.0158373i \(-0.00504139\pi\)
−0.513653 + 0.857998i \(0.671708\pi\)
\(830\) 0 0
\(831\) 37.6066i 1.30456i
\(832\) 0 0
\(833\) 1.82747 + 1.82747i 0.0633181 + 0.0633181i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −16.4958 −0.570179
\(838\) 0 0
\(839\) −44.5978 + 11.9500i −1.53969 + 0.412558i −0.926165 0.377119i \(-0.876915\pi\)
−0.613523 + 0.789677i \(0.710248\pi\)
\(840\) 0 0
\(841\) −14.4950 + 25.1061i −0.499827 + 0.865726i
\(842\) 0 0
\(843\) −23.9235 41.4368i −0.823970 1.42716i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −18.6639 32.3268i −0.641298 1.11076i
\(848\) 0 0
\(849\) 11.6069 20.1038i 0.398349 0.689961i
\(850\) 0 0
\(851\) −4.91143 + 1.31601i −0.168362 + 0.0451124i
\(852\) 0 0
\(853\) 0.328048 0.0112321 0.00561607 0.999984i \(-0.498212\pi\)
0.00561607 + 0.999984i \(0.498212\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.20279 + 7.20279i 0.246043 + 0.246043i 0.819344 0.573302i \(-0.194338\pi\)
−0.573302 + 0.819344i \(0.694338\pi\)
\(858\) 0 0
\(859\) 19.2793i 0.657800i −0.944365 0.328900i \(-0.893322\pi\)
0.944365 0.328900i \(-0.106678\pi\)
\(860\) 0 0
\(861\) 15.2406 + 26.3975i 0.519398 + 0.899623i
\(862\) 0 0
\(863\) 16.7097i 0.568806i −0.958705 0.284403i \(-0.908205\pi\)
0.958705 0.284403i \(-0.0917954\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 10.8843 + 40.6207i 0.369650 + 1.37955i
\(868\) 0 0
\(869\) −21.0706 + 78.6366i −0.714772 + 2.66756i
\(870\) 0 0
\(871\) 39.0394 + 7.44364i 1.32280 + 0.252218i
\(872\) 0 0
\(873\) 45.3204 26.1657i 1.53386 0.885576i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −12.5974 + 21.8193i −0.425383 + 0.736784i −0.996456 0.0841148i \(-0.973194\pi\)
0.571074 + 0.820899i \(0.306527\pi\)
\(878\) 0 0
\(879\) −22.6004 22.6004i −0.762292 0.762292i
\(880\) 0 0
\(881\) 46.8238 27.0337i 1.57753 0.910790i 0.582333 0.812951i \(-0.302140\pi\)
0.995202 0.0978396i \(-0.0311932\pi\)
\(882\) 0 0
\(883\) 5.21525 5.21525i 0.175507 0.175507i −0.613887 0.789394i \(-0.710395\pi\)
0.789394 + 0.613887i \(0.210395\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.8074 + 7.45098i 0.933682 + 0.250179i 0.693424 0.720530i \(-0.256101\pi\)
0.240258 + 0.970709i \(0.422768\pi\)
\(888\) 0 0
\(889\) 21.2110 21.2110i 0.711394 0.711394i
\(890\) 0 0
\(891\) 6.02847 + 22.4986i 0.201961 + 0.753730i
\(892\) 0 0
\(893\) −12.9476 + 3.46931i −0.433276 + 0.116096i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.35531 + 6.76164i 0.0786416 + 0.225765i
\(898\) 0 0
\(899\) −0.533201 0.142871i −0.0177832 0.00476500i
\(900\) 0 0
\(901\) 11.2992 + 6.52360i 0.376431 + 0.217333i
\(902\) 0 0
\(903\) 18.5262 + 10.6961i 0.616513 + 0.355944i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.40058 + 35.0834i −0.312141 + 1.16493i 0.614481 + 0.788932i \(0.289365\pi\)
−0.926622 + 0.375994i \(0.877301\pi\)
\(908\) 0 0
\(909\) 73.2318 2.42894
\(910\) 0 0
\(911\) 1.89993 0.0629473 0.0314737 0.999505i \(-0.489980\pi\)
0.0314737 + 0.999505i \(0.489980\pi\)
\(912\) 0 0
\(913\) −0.651403 + 2.43107i −0.0215583 + 0.0804566i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.36793 5.40858i −0.309356 0.178607i
\(918\) 0 0
\(919\) 33.0949 + 19.1073i 1.09170 + 0.630293i 0.934028 0.357199i \(-0.116268\pi\)
0.157671 + 0.987492i \(0.449601\pi\)
\(920\) 0 0
\(921\) 51.5531 + 13.8136i 1.69873 + 0.455174i
\(922\) 0 0
\(923\) 14.9397 30.9111i 0.491745 1.01745i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −28.2210 + 7.56180i −0.926899 + 0.248362i
\(928\) 0 0
\(929\) −0.860601 3.21181i −0.0282354 0.105376i 0.950370 0.311122i \(-0.100705\pi\)
−0.978605 + 0.205746i \(0.934038\pi\)
\(930\) 0 0
\(931\) 10.7673 10.7673i 0.352885 0.352885i
\(932\) 0 0
\(933\) −38.1788 10.2300i −1.24992 0.334914i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −8.40073 + 8.40073i −0.274440 + 0.274440i −0.830885 0.556445i \(-0.812165\pi\)
0.556445 + 0.830885i \(0.312165\pi\)
\(938\) 0 0
\(939\) −7.68475 + 4.43679i −0.250782 + 0.144789i
\(940\) 0 0
\(941\) −22.2587 22.2587i −0.725614 0.725614i 0.244129 0.969743i \(-0.421498\pi\)
−0.969743 + 0.244129i \(0.921498\pi\)
\(942\) 0 0
\(943\) 1.96482 3.40317i 0.0639834 0.110823i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.3428 + 8.28083i −0.466079 + 0.269091i −0.714597 0.699536i \(-0.753390\pi\)
0.248518 + 0.968627i \(0.420057\pi\)
\(948\) 0 0
\(949\) 25.3087 29.3255i 0.821554 0.951947i
\(950\) 0 0
\(951\) 10.5604 39.4118i 0.342443 1.27802i
\(952\) 0 0
\(953\) 6.48521 + 24.2031i 0.210077 + 0.784016i 0.987842 + 0.155462i \(0.0496865\pi\)
−0.777765 + 0.628555i \(0.783647\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.41952i 0.0458866i
\(958\) 0 0
\(959\) 12.1735 + 21.0852i 0.393104 + 0.680876i
\(960\) 0 0
\(961\) 0.574591i 0.0185352i
\(962\) 0 0
\(963\) −31.3068 31.3068i −1.00885 1.00885i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 31.4641 1.01182 0.505908 0.862587i \(-0.331157\pi\)
0.505908 + 0.862587i \(0.331157\pi\)
\(968\) 0 0
\(969\) −18.8392 + 5.04794i −0.605201 + 0.162163i
\(970\) 0 0
\(971\) −0.0733855 + 0.127107i −0.00235505 + 0.00407907i −0.867201 0.497959i \(-0.834083\pi\)
0.864845 + 0.502038i \(0.167416\pi\)
\(972\) 0 0
\(973\) 9.42129 + 16.3182i 0.302033 + 0.523136i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.07986 15.7268i −0.290490 0.503144i 0.683435 0.730011i \(-0.260485\pi\)
−0.973926 + 0.226867i \(0.927152\pi\)
\(978\) 0 0
\(979\) 5.75274 9.96403i 0.183858 0.318452i
\(980\) 0 0
\(981\) 13.2885 3.56064i 0.424269 0.113683i
\(982\) 0 0
\(983\) 42.3986 1.35230 0.676152 0.736762i \(-0.263646\pi\)
0.676152 + 0.736762i \(0.263646\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.33759 8.33759i −0.265388 0.265388i
\(988\) 0 0
\(989\) 2.75789i 0.0876959i
\(990\) 0 0
\(991\) 16.8511 + 29.1869i 0.535292 + 0.927152i 0.999149 + 0.0412426i \(0.0131317\pi\)
−0.463857 + 0.885910i \(0.653535\pi\)
\(992\) 0 0
\(993\) 2.48848i 0.0789697i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.27444 + 12.2204i 0.103703 + 0.387023i 0.998195 0.0600602i \(-0.0191293\pi\)
−0.894492 + 0.447084i \(0.852463\pi\)
\(998\) 0 0
\(999\) 5.28846 19.7368i 0.167320 0.624445i
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 1300.2.bs.d.193.5 20
5.2 odd 4 1300.2.bn.d.557.1 20
5.3 odd 4 260.2.bf.c.37.5 20
5.4 even 2 260.2.bk.c.193.1 yes 20
13.6 odd 12 1300.2.bn.d.1293.1 20
65.19 odd 12 260.2.bf.c.253.5 yes 20
65.32 even 12 inner 1300.2.bs.d.357.5 20
65.58 even 12 260.2.bk.c.97.1 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.bf.c.37.5 20 5.3 odd 4
260.2.bf.c.253.5 yes 20 65.19 odd 12
260.2.bk.c.97.1 yes 20 65.58 even 12
260.2.bk.c.193.1 yes 20 5.4 even 2
1300.2.bn.d.557.1 20 5.2 odd 4
1300.2.bn.d.1293.1 20 13.6 odd 12
1300.2.bs.d.193.5 20 1.1 even 1 trivial
1300.2.bs.d.357.5 20 65.32 even 12 inner