Properties

Label 640.2.s.c.287.9
Level $640$
Weight $2$
Character 640.287
Analytic conductor $5.110$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,2,Mod(223,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.s (of order \(4\), 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: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 287.9
Root \(-0.480367 - 1.33013i\) of defining polynomial
Character \(\chi\) \(=\) 640.287
Dual form 640.2.s.c.223.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.85601 q^{3} +(1.71489 - 1.43498i) q^{5} +(0.458895 + 0.458895i) q^{7} +5.15678 q^{9} +O(q^{10})\) \(q+2.85601 q^{3} +(1.71489 - 1.43498i) q^{5} +(0.458895 + 0.458895i) q^{7} +5.15678 q^{9} +(-0.492763 + 0.492763i) q^{11} +4.52109i q^{13} +(4.89773 - 4.09831i) q^{15} +(-3.12823 - 3.12823i) q^{17} +(-4.04508 + 4.04508i) q^{19} +(1.31061 + 1.31061i) q^{21} +(1.80660 - 1.80660i) q^{23} +(0.881683 - 4.92165i) q^{25} +6.15978 q^{27} +(-3.83926 - 3.83926i) q^{29} +0.139949i q^{31} +(-1.40733 + 1.40733i) q^{33} +(1.44546 + 0.128450i) q^{35} -5.84330i q^{37} +12.9123i q^{39} -4.55648i q^{41} +7.49928i q^{43} +(8.84330 - 7.39986i) q^{45} +(-4.14073 + 4.14073i) q^{47} -6.57883i q^{49} +(-8.93426 - 8.93426i) q^{51} -2.75773 q^{53} +(-0.137930 + 1.55214i) q^{55} +(-11.5528 + 11.5528i) q^{57} +(3.62521 + 3.62521i) q^{59} +(-3.72781 + 3.72781i) q^{61} +(2.36642 + 2.36642i) q^{63} +(6.48766 + 7.75317i) q^{65} +3.32677i q^{67} +(5.15965 - 5.15965i) q^{69} -1.37056 q^{71} +(2.55028 + 2.55028i) q^{73} +(2.51809 - 14.0563i) q^{75} -0.452252 q^{77} -3.86426 q^{79} +2.12204 q^{81} +14.4698 q^{83} +(-9.85351 - 0.875628i) q^{85} +(-10.9650 - 10.9650i) q^{87} +3.35011 q^{89} +(-2.07470 + 2.07470i) q^{91} +0.399696i q^{93} +(-1.13226 + 12.7415i) q^{95} +(-4.95582 - 4.95582i) q^{97} +(-2.54107 + 2.54107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5} - 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{5} - 2 q^{7} + 10 q^{9} - 2 q^{11} + 20 q^{15} - 6 q^{17} - 2 q^{19} + 16 q^{21} + 2 q^{23} - 6 q^{25} - 24 q^{27} - 14 q^{29} - 8 q^{33} + 2 q^{35} + 14 q^{45} - 38 q^{47} + 8 q^{51} - 12 q^{53} + 6 q^{55} - 24 q^{57} + 10 q^{59} - 14 q^{61} + 6 q^{63} + 32 q^{69} - 24 q^{71} - 14 q^{73} + 16 q^{75} + 44 q^{77} + 16 q^{79} + 2 q^{81} + 40 q^{83} - 14 q^{85} - 24 q^{87} + 12 q^{89} - 34 q^{95} + 18 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.85601 1.64892 0.824458 0.565923i \(-0.191480\pi\)
0.824458 + 0.565923i \(0.191480\pi\)
\(4\) 0 0
\(5\) 1.71489 1.43498i 0.766921 0.641741i
\(6\) 0 0
\(7\) 0.458895 + 0.458895i 0.173446 + 0.173446i 0.788491 0.615046i \(-0.210862\pi\)
−0.615046 + 0.788491i \(0.710862\pi\)
\(8\) 0 0
\(9\) 5.15678 1.71893
\(10\) 0 0
\(11\) −0.492763 + 0.492763i −0.148574 + 0.148574i −0.777481 0.628907i \(-0.783503\pi\)
0.628907 + 0.777481i \(0.283503\pi\)
\(12\) 0 0
\(13\) 4.52109i 1.25393i 0.779049 + 0.626963i \(0.215702\pi\)
−0.779049 + 0.626963i \(0.784298\pi\)
\(14\) 0 0
\(15\) 4.89773 4.09831i 1.26459 1.05818i
\(16\) 0 0
\(17\) −3.12823 3.12823i −0.758708 0.758708i 0.217379 0.976087i \(-0.430249\pi\)
−0.976087 + 0.217379i \(0.930249\pi\)
\(18\) 0 0
\(19\) −4.04508 + 4.04508i −0.928005 + 0.928005i −0.997577 0.0695721i \(-0.977837\pi\)
0.0695721 + 0.997577i \(0.477837\pi\)
\(20\) 0 0
\(21\) 1.31061 + 1.31061i 0.285998 + 0.285998i
\(22\) 0 0
\(23\) 1.80660 1.80660i 0.376701 0.376701i −0.493209 0.869911i \(-0.664176\pi\)
0.869911 + 0.493209i \(0.164176\pi\)
\(24\) 0 0
\(25\) 0.881683 4.92165i 0.176337 0.984330i
\(26\) 0 0
\(27\) 6.15978 1.18545
\(28\) 0 0
\(29\) −3.83926 3.83926i −0.712932 0.712932i 0.254215 0.967148i \(-0.418183\pi\)
−0.967148 + 0.254215i \(0.918183\pi\)
\(30\) 0 0
\(31\) 0.139949i 0.0251356i 0.999921 + 0.0125678i \(0.00400057\pi\)
−0.999921 + 0.0125678i \(0.995999\pi\)
\(32\) 0 0
\(33\) −1.40733 + 1.40733i −0.244985 + 0.244985i
\(34\) 0 0
\(35\) 1.44546 + 0.128450i 0.244327 + 0.0217120i
\(36\) 0 0
\(37\) 5.84330i 0.960633i −0.877095 0.480317i \(-0.840522\pi\)
0.877095 0.480317i \(-0.159478\pi\)
\(38\) 0 0
\(39\) 12.9123i 2.06762i
\(40\) 0 0
\(41\) 4.55648i 0.711602i −0.934562 0.355801i \(-0.884208\pi\)
0.934562 0.355801i \(-0.115792\pi\)
\(42\) 0 0
\(43\) 7.49928i 1.14363i 0.820383 + 0.571815i \(0.193760\pi\)
−0.820383 + 0.571815i \(0.806240\pi\)
\(44\) 0 0
\(45\) 8.84330 7.39986i 1.31828 1.10311i
\(46\) 0 0
\(47\) −4.14073 + 4.14073i −0.603987 + 0.603987i −0.941368 0.337381i \(-0.890459\pi\)
0.337381 + 0.941368i \(0.390459\pi\)
\(48\) 0 0
\(49\) 6.57883i 0.939833i
\(50\) 0 0
\(51\) −8.93426 8.93426i −1.25105 1.25105i
\(52\) 0 0
\(53\) −2.75773 −0.378803 −0.189402 0.981900i \(-0.560655\pi\)
−0.189402 + 0.981900i \(0.560655\pi\)
\(54\) 0 0
\(55\) −0.137930 + 1.55214i −0.0185985 + 0.209290i
\(56\) 0 0
\(57\) −11.5528 + 11.5528i −1.53020 + 1.53020i
\(58\) 0 0
\(59\) 3.62521 + 3.62521i 0.471962 + 0.471962i 0.902549 0.430587i \(-0.141694\pi\)
−0.430587 + 0.902549i \(0.641694\pi\)
\(60\) 0 0
\(61\) −3.72781 + 3.72781i −0.477298 + 0.477298i −0.904266 0.426969i \(-0.859581\pi\)
0.426969 + 0.904266i \(0.359581\pi\)
\(62\) 0 0
\(63\) 2.36642 + 2.36642i 0.298141 + 0.298141i
\(64\) 0 0
\(65\) 6.48766 + 7.75317i 0.804696 + 0.961662i
\(66\) 0 0
\(67\) 3.32677i 0.406430i 0.979134 + 0.203215i \(0.0651390\pi\)
−0.979134 + 0.203215i \(0.934861\pi\)
\(68\) 0 0
\(69\) 5.15965 5.15965i 0.621149 0.621149i
\(70\) 0 0
\(71\) −1.37056 −0.162655 −0.0813275 0.996687i \(-0.525916\pi\)
−0.0813275 + 0.996687i \(0.525916\pi\)
\(72\) 0 0
\(73\) 2.55028 + 2.55028i 0.298488 + 0.298488i 0.840422 0.541933i \(-0.182307\pi\)
−0.541933 + 0.840422i \(0.682307\pi\)
\(74\) 0 0
\(75\) 2.51809 14.0563i 0.290764 1.62308i
\(76\) 0 0
\(77\) −0.452252 −0.0515389
\(78\) 0 0
\(79\) −3.86426 −0.434763 −0.217382 0.976087i \(-0.569752\pi\)
−0.217382 + 0.976087i \(0.569752\pi\)
\(80\) 0 0
\(81\) 2.12204 0.235782
\(82\) 0 0
\(83\) 14.4698 1.58827 0.794133 0.607744i \(-0.207925\pi\)
0.794133 + 0.607744i \(0.207925\pi\)
\(84\) 0 0
\(85\) −9.85351 0.875628i −1.06876 0.0949752i
\(86\) 0 0
\(87\) −10.9650 10.9650i −1.17557 1.17557i
\(88\) 0 0
\(89\) 3.35011 0.355111 0.177556 0.984111i \(-0.443181\pi\)
0.177556 + 0.984111i \(0.443181\pi\)
\(90\) 0 0
\(91\) −2.07470 + 2.07470i −0.217488 + 0.217488i
\(92\) 0 0
\(93\) 0.399696i 0.0414466i
\(94\) 0 0
\(95\) −1.13226 + 12.7415i −0.116168 + 1.30725i
\(96\) 0 0
\(97\) −4.95582 4.95582i −0.503187 0.503187i 0.409240 0.912427i \(-0.365794\pi\)
−0.912427 + 0.409240i \(0.865794\pi\)
\(98\) 0 0
\(99\) −2.54107 + 2.54107i −0.255387 + 0.255387i
\(100\) 0 0
\(101\) 1.84536 + 1.84536i 0.183621 + 0.183621i 0.792931 0.609311i \(-0.208554\pi\)
−0.609311 + 0.792931i \(0.708554\pi\)
\(102\) 0 0
\(103\) 11.6655 11.6655i 1.14944 1.14944i 0.162773 0.986664i \(-0.447956\pi\)
0.986664 0.162773i \(-0.0520437\pi\)
\(104\) 0 0
\(105\) 4.12823 + 0.366853i 0.402874 + 0.0358012i
\(106\) 0 0
\(107\) −15.3106 −1.48013 −0.740067 0.672534i \(-0.765206\pi\)
−0.740067 + 0.672534i \(0.765206\pi\)
\(108\) 0 0
\(109\) 12.4798 + 12.4798i 1.19535 + 1.19535i 0.975544 + 0.219803i \(0.0705416\pi\)
0.219803 + 0.975544i \(0.429458\pi\)
\(110\) 0 0
\(111\) 16.6885i 1.58400i
\(112\) 0 0
\(113\) 2.53557 2.53557i 0.238526 0.238526i −0.577713 0.816240i \(-0.696055\pi\)
0.816240 + 0.577713i \(0.196055\pi\)
\(114\) 0 0
\(115\) 0.505686 5.69053i 0.0471555 0.530645i
\(116\) 0 0
\(117\) 23.3143i 2.15541i
\(118\) 0 0
\(119\) 2.87106i 0.263189i
\(120\) 0 0
\(121\) 10.5144i 0.955852i
\(122\) 0 0
\(123\) 13.0133i 1.17337i
\(124\) 0 0
\(125\) −5.55047 9.70527i −0.496449 0.868066i
\(126\) 0 0
\(127\) −0.615790 + 0.615790i −0.0546426 + 0.0546426i −0.733900 0.679257i \(-0.762302\pi\)
0.679257 + 0.733900i \(0.262302\pi\)
\(128\) 0 0
\(129\) 21.4180i 1.88575i
\(130\) 0 0
\(131\) 9.55413 + 9.55413i 0.834748 + 0.834748i 0.988162 0.153414i \(-0.0490268\pi\)
−0.153414 + 0.988162i \(0.549027\pi\)
\(132\) 0 0
\(133\) −3.71253 −0.321917
\(134\) 0 0
\(135\) 10.5633 8.83914i 0.909147 0.760752i
\(136\) 0 0
\(137\) 3.70277 3.70277i 0.316349 0.316349i −0.531014 0.847363i \(-0.678189\pi\)
0.847363 + 0.531014i \(0.178189\pi\)
\(138\) 0 0
\(139\) −5.46761 5.46761i −0.463756 0.463756i 0.436128 0.899885i \(-0.356349\pi\)
−0.899885 + 0.436128i \(0.856349\pi\)
\(140\) 0 0
\(141\) −11.8260 + 11.8260i −0.995925 + 0.995925i
\(142\) 0 0
\(143\) −2.22783 2.22783i −0.186300 0.186300i
\(144\) 0 0
\(145\) −12.0931 1.07465i −1.00428 0.0892450i
\(146\) 0 0
\(147\) 18.7892i 1.54971i
\(148\) 0 0
\(149\) 4.21561 4.21561i 0.345356 0.345356i −0.513021 0.858376i \(-0.671474\pi\)
0.858376 + 0.513021i \(0.171474\pi\)
\(150\) 0 0
\(151\) −12.4417 −1.01249 −0.506244 0.862390i \(-0.668966\pi\)
−0.506244 + 0.862390i \(0.668966\pi\)
\(152\) 0 0
\(153\) −16.1316 16.1316i −1.30416 1.30416i
\(154\) 0 0
\(155\) 0.200824 + 0.239997i 0.0161306 + 0.0192771i
\(156\) 0 0
\(157\) −7.50500 −0.598964 −0.299482 0.954102i \(-0.596814\pi\)
−0.299482 + 0.954102i \(0.596814\pi\)
\(158\) 0 0
\(159\) −7.87609 −0.624615
\(160\) 0 0
\(161\) 1.65807 0.130675
\(162\) 0 0
\(163\) −23.7284 −1.85855 −0.929277 0.369383i \(-0.879569\pi\)
−0.929277 + 0.369383i \(0.879569\pi\)
\(164\) 0 0
\(165\) −0.393929 + 4.43291i −0.0306673 + 0.345102i
\(166\) 0 0
\(167\) 0.402976 + 0.402976i 0.0311832 + 0.0311832i 0.722526 0.691343i \(-0.242981\pi\)
−0.691343 + 0.722526i \(0.742981\pi\)
\(168\) 0 0
\(169\) −7.44028 −0.572330
\(170\) 0 0
\(171\) −20.8596 + 20.8596i −1.59517 + 1.59517i
\(172\) 0 0
\(173\) 15.4500i 1.17464i −0.809355 0.587320i \(-0.800183\pi\)
0.809355 0.587320i \(-0.199817\pi\)
\(174\) 0 0
\(175\) 2.66312 1.85392i 0.201313 0.140143i
\(176\) 0 0
\(177\) 10.3536 + 10.3536i 0.778225 + 0.778225i
\(178\) 0 0
\(179\) −5.20444 + 5.20444i −0.388998 + 0.388998i −0.874330 0.485332i \(-0.838699\pi\)
0.485332 + 0.874330i \(0.338699\pi\)
\(180\) 0 0
\(181\) 9.08925 + 9.08925i 0.675599 + 0.675599i 0.959001 0.283402i \(-0.0914632\pi\)
−0.283402 + 0.959001i \(0.591463\pi\)
\(182\) 0 0
\(183\) −10.6467 + 10.6467i −0.787024 + 0.787024i
\(184\) 0 0
\(185\) −8.38500 10.0206i −0.616478 0.736730i
\(186\) 0 0
\(187\) 3.08295 0.225448
\(188\) 0 0
\(189\) 2.82669 + 2.82669i 0.205611 + 0.205611i
\(190\) 0 0
\(191\) 15.1075i 1.09314i 0.837413 + 0.546571i \(0.184067\pi\)
−0.837413 + 0.546571i \(0.815933\pi\)
\(192\) 0 0
\(193\) 4.19166 4.19166i 0.301722 0.301722i −0.539965 0.841687i \(-0.681563\pi\)
0.841687 + 0.539965i \(0.181563\pi\)
\(194\) 0 0
\(195\) 18.5288 + 22.1431i 1.32688 + 1.58570i
\(196\) 0 0
\(197\) 4.03184i 0.287256i 0.989632 + 0.143628i \(0.0458769\pi\)
−0.989632 + 0.143628i \(0.954123\pi\)
\(198\) 0 0
\(199\) 5.43055i 0.384961i 0.981301 + 0.192481i \(0.0616533\pi\)
−0.981301 + 0.192481i \(0.938347\pi\)
\(200\) 0 0
\(201\) 9.50129i 0.670169i
\(202\) 0 0
\(203\) 3.52363i 0.247310i
\(204\) 0 0
\(205\) −6.53844 7.81385i −0.456664 0.545743i
\(206\) 0 0
\(207\) 9.31622 9.31622i 0.647522 0.647522i
\(208\) 0 0
\(209\) 3.98653i 0.275754i
\(210\) 0 0
\(211\) 3.23020 + 3.23020i 0.222376 + 0.222376i 0.809498 0.587122i \(-0.199739\pi\)
−0.587122 + 0.809498i \(0.699739\pi\)
\(212\) 0 0
\(213\) −3.91432 −0.268205
\(214\) 0 0
\(215\) 10.7613 + 12.8604i 0.733914 + 0.877074i
\(216\) 0 0
\(217\) −0.0642220 + 0.0642220i −0.00435967 + 0.00435967i
\(218\) 0 0
\(219\) 7.28363 + 7.28363i 0.492182 + 0.492182i
\(220\) 0 0
\(221\) 14.1430 14.1430i 0.951363 0.951363i
\(222\) 0 0
\(223\) −8.17319 8.17319i −0.547317 0.547317i 0.378347 0.925664i \(-0.376493\pi\)
−0.925664 + 0.378347i \(0.876493\pi\)
\(224\) 0 0
\(225\) 4.54664 25.3799i 0.303110 1.69199i
\(226\) 0 0
\(227\) 1.54068i 0.102258i −0.998692 0.0511292i \(-0.983718\pi\)
0.998692 0.0511292i \(-0.0162820\pi\)
\(228\) 0 0
\(229\) −17.5646 + 17.5646i −1.16070 + 1.16070i −0.176378 + 0.984322i \(0.556438\pi\)
−0.984322 + 0.176378i \(0.943562\pi\)
\(230\) 0 0
\(231\) −1.29164 −0.0849834
\(232\) 0 0
\(233\) −9.99018 9.99018i −0.654479 0.654479i 0.299590 0.954068i \(-0.403150\pi\)
−0.954068 + 0.299590i \(0.903150\pi\)
\(234\) 0 0
\(235\) −1.15904 + 13.0427i −0.0756072 + 0.850814i
\(236\) 0 0
\(237\) −11.0364 −0.716889
\(238\) 0 0
\(239\) 26.2762 1.69967 0.849833 0.527052i \(-0.176703\pi\)
0.849833 + 0.527052i \(0.176703\pi\)
\(240\) 0 0
\(241\) −0.113242 −0.00729456 −0.00364728 0.999993i \(-0.501161\pi\)
−0.00364728 + 0.999993i \(0.501161\pi\)
\(242\) 0 0
\(243\) −12.4188 −0.796665
\(244\) 0 0
\(245\) −9.44047 11.2820i −0.603130 0.720778i
\(246\) 0 0
\(247\) −18.2882 18.2882i −1.16365 1.16365i
\(248\) 0 0
\(249\) 41.3258 2.61892
\(250\) 0 0
\(251\) 19.2220 19.2220i 1.21328 1.21328i 0.243339 0.969941i \(-0.421757\pi\)
0.969941 0.243339i \(-0.0782427\pi\)
\(252\) 0 0
\(253\) 1.78045i 0.111936i
\(254\) 0 0
\(255\) −28.1417 2.50080i −1.76230 0.156606i
\(256\) 0 0
\(257\) −0.757800 0.757800i −0.0472703 0.0472703i 0.683077 0.730347i \(-0.260642\pi\)
−0.730347 + 0.683077i \(0.760642\pi\)
\(258\) 0 0
\(259\) 2.68146 2.68146i 0.166618 0.166618i
\(260\) 0 0
\(261\) −19.7982 19.7982i −1.22548 1.22548i
\(262\) 0 0
\(263\) −5.73017 + 5.73017i −0.353338 + 0.353338i −0.861350 0.508012i \(-0.830380\pi\)
0.508012 + 0.861350i \(0.330380\pi\)
\(264\) 0 0
\(265\) −4.72919 + 3.95728i −0.290512 + 0.243094i
\(266\) 0 0
\(267\) 9.56795 0.585549
\(268\) 0 0
\(269\) 9.78879 + 9.78879i 0.596833 + 0.596833i 0.939468 0.342635i \(-0.111320\pi\)
−0.342635 + 0.939468i \(0.611320\pi\)
\(270\) 0 0
\(271\) 4.10159i 0.249154i 0.992210 + 0.124577i \(0.0397574\pi\)
−0.992210 + 0.124577i \(0.960243\pi\)
\(272\) 0 0
\(273\) −5.92537 + 5.92537i −0.358620 + 0.358620i
\(274\) 0 0
\(275\) 1.99075 + 2.85967i 0.120046 + 0.172444i
\(276\) 0 0
\(277\) 24.6755i 1.48261i −0.671169 0.741305i \(-0.734207\pi\)
0.671169 0.741305i \(-0.265793\pi\)
\(278\) 0 0
\(279\) 0.721688i 0.0432063i
\(280\) 0 0
\(281\) 23.6688i 1.41196i 0.708230 + 0.705981i \(0.249494\pi\)
−0.708230 + 0.705981i \(0.750506\pi\)
\(282\) 0 0
\(283\) 13.0492i 0.775694i −0.921724 0.387847i \(-0.873219\pi\)
0.921724 0.387847i \(-0.126781\pi\)
\(284\) 0 0
\(285\) −3.23375 + 36.3897i −0.191551 + 2.15554i
\(286\) 0 0
\(287\) 2.09094 2.09094i 0.123424 0.123424i
\(288\) 0 0
\(289\) 2.57168i 0.151275i
\(290\) 0 0
\(291\) −14.1539 14.1539i −0.829714 0.829714i
\(292\) 0 0
\(293\) 31.6731 1.85036 0.925181 0.379526i \(-0.123913\pi\)
0.925181 + 0.379526i \(0.123913\pi\)
\(294\) 0 0
\(295\) 11.4189 + 1.01474i 0.664835 + 0.0590802i
\(296\) 0 0
\(297\) −3.03531 + 3.03531i −0.176127 + 0.176127i
\(298\) 0 0
\(299\) 8.16779 + 8.16779i 0.472355 + 0.472355i
\(300\) 0 0
\(301\) −3.44138 + 3.44138i −0.198358 + 0.198358i
\(302\) 0 0
\(303\) 5.27037 + 5.27037i 0.302775 + 0.302775i
\(304\) 0 0
\(305\) −1.04346 + 11.7421i −0.0597482 + 0.672351i
\(306\) 0 0
\(307\) 27.3597i 1.56150i −0.624843 0.780751i \(-0.714837\pi\)
0.624843 0.780751i \(-0.285163\pi\)
\(308\) 0 0
\(309\) 33.3168 33.3168i 1.89532 1.89532i
\(310\) 0 0
\(311\) 15.8076 0.896368 0.448184 0.893941i \(-0.352071\pi\)
0.448184 + 0.893941i \(0.352071\pi\)
\(312\) 0 0
\(313\) 13.8388 + 13.8388i 0.782217 + 0.782217i 0.980205 0.197988i \(-0.0634406\pi\)
−0.197988 + 0.980205i \(0.563441\pi\)
\(314\) 0 0
\(315\) 7.45390 + 0.662387i 0.419980 + 0.0373213i
\(316\) 0 0
\(317\) 35.0092 1.96631 0.983156 0.182766i \(-0.0585051\pi\)
0.983156 + 0.182766i \(0.0585051\pi\)
\(318\) 0 0
\(319\) 3.78369 0.211846
\(320\) 0 0
\(321\) −43.7272 −2.44062
\(322\) 0 0
\(323\) 25.3079 1.40817
\(324\) 0 0
\(325\) 22.2512 + 3.98617i 1.23428 + 0.221113i
\(326\) 0 0
\(327\) 35.6424 + 35.6424i 1.97103 + 1.97103i
\(328\) 0 0
\(329\) −3.80032 −0.209518
\(330\) 0 0
\(331\) 16.8212 16.8212i 0.924578 0.924578i −0.0727709 0.997349i \(-0.523184\pi\)
0.997349 + 0.0727709i \(0.0231842\pi\)
\(332\) 0 0
\(333\) 30.1326i 1.65126i
\(334\) 0 0
\(335\) 4.77384 + 5.70504i 0.260823 + 0.311700i
\(336\) 0 0
\(337\) 14.4984 + 14.4984i 0.789777 + 0.789777i 0.981457 0.191680i \(-0.0613937\pi\)
−0.191680 + 0.981457i \(0.561394\pi\)
\(338\) 0 0
\(339\) 7.24160 7.24160i 0.393310 0.393310i
\(340\) 0 0
\(341\) −0.0689618 0.0689618i −0.00373449 0.00373449i
\(342\) 0 0
\(343\) 6.23125 6.23125i 0.336456 0.336456i
\(344\) 0 0
\(345\) 1.44424 16.2522i 0.0777555 0.874989i
\(346\) 0 0
\(347\) −16.7705 −0.900286 −0.450143 0.892956i \(-0.648627\pi\)
−0.450143 + 0.892956i \(0.648627\pi\)
\(348\) 0 0
\(349\) 1.86337 + 1.86337i 0.0997439 + 0.0997439i 0.755218 0.655474i \(-0.227531\pi\)
−0.655474 + 0.755218i \(0.727531\pi\)
\(350\) 0 0
\(351\) 27.8489i 1.48647i
\(352\) 0 0
\(353\) 24.1362 24.1362i 1.28464 1.28464i 0.346642 0.937998i \(-0.387322\pi\)
0.937998 0.346642i \(-0.112678\pi\)
\(354\) 0 0
\(355\) −2.35035 + 1.96672i −0.124744 + 0.104382i
\(356\) 0 0
\(357\) 8.19976i 0.433978i
\(358\) 0 0
\(359\) 12.2500i 0.646532i −0.946308 0.323266i \(-0.895219\pi\)
0.946308 0.323266i \(-0.104781\pi\)
\(360\) 0 0
\(361\) 13.7253i 0.722386i
\(362\) 0 0
\(363\) 30.0291i 1.57612i
\(364\) 0 0
\(365\) 8.03305 + 0.713853i 0.420469 + 0.0373648i
\(366\) 0 0
\(367\) 2.71307 2.71307i 0.141621 0.141621i −0.632742 0.774363i \(-0.718071\pi\)
0.774363 + 0.632742i \(0.218071\pi\)
\(368\) 0 0
\(369\) 23.4967i 1.22319i
\(370\) 0 0
\(371\) −1.26551 1.26551i −0.0657018 0.0657018i
\(372\) 0 0
\(373\) −16.4846 −0.853541 −0.426771 0.904360i \(-0.640349\pi\)
−0.426771 + 0.904360i \(0.640349\pi\)
\(374\) 0 0
\(375\) −15.8522 27.7183i −0.818603 1.43137i
\(376\) 0 0
\(377\) 17.3576 17.3576i 0.893964 0.893964i
\(378\) 0 0
\(379\) 13.7716 + 13.7716i 0.707401 + 0.707401i 0.965988 0.258587i \(-0.0832568\pi\)
−0.258587 + 0.965988i \(0.583257\pi\)
\(380\) 0 0
\(381\) −1.75870 + 1.75870i −0.0901011 + 0.0901011i
\(382\) 0 0
\(383\) 11.5530 + 11.5530i 0.590332 + 0.590332i 0.937721 0.347389i \(-0.112932\pi\)
−0.347389 + 0.937721i \(0.612932\pi\)
\(384\) 0 0
\(385\) −0.775562 + 0.648972i −0.0395263 + 0.0330747i
\(386\) 0 0
\(387\) 38.6722i 1.96582i
\(388\) 0 0
\(389\) −15.7728 + 15.7728i −0.799712 + 0.799712i −0.983050 0.183338i \(-0.941310\pi\)
0.183338 + 0.983050i \(0.441310\pi\)
\(390\) 0 0
\(391\) −11.3029 −0.571612
\(392\) 0 0
\(393\) 27.2867 + 27.2867i 1.37643 + 1.37643i
\(394\) 0 0
\(395\) −6.62677 + 5.54512i −0.333429 + 0.279006i
\(396\) 0 0
\(397\) −29.9558 −1.50344 −0.751720 0.659483i \(-0.770775\pi\)
−0.751720 + 0.659483i \(0.770775\pi\)
\(398\) 0 0
\(399\) −10.6030 −0.530815
\(400\) 0 0
\(401\) 19.9241 0.994963 0.497481 0.867475i \(-0.334258\pi\)
0.497481 + 0.867475i \(0.334258\pi\)
\(402\) 0 0
\(403\) −0.632724 −0.0315182
\(404\) 0 0
\(405\) 3.63906 3.04508i 0.180826 0.151311i
\(406\) 0 0
\(407\) 2.87936 + 2.87936i 0.142725 + 0.142725i
\(408\) 0 0
\(409\) −5.89856 −0.291665 −0.145832 0.989309i \(-0.546586\pi\)
−0.145832 + 0.989309i \(0.546586\pi\)
\(410\) 0 0
\(411\) 10.5751 10.5751i 0.521634 0.521634i
\(412\) 0 0
\(413\) 3.32717i 0.163720i
\(414\) 0 0
\(415\) 24.8141 20.7638i 1.21808 1.01926i
\(416\) 0 0
\(417\) −15.6155 15.6155i −0.764696 0.764696i
\(418\) 0 0
\(419\) −8.24430 + 8.24430i −0.402760 + 0.402760i −0.879205 0.476444i \(-0.841925\pi\)
0.476444 + 0.879205i \(0.341925\pi\)
\(420\) 0 0
\(421\) 17.1776 + 17.1776i 0.837184 + 0.837184i 0.988487 0.151304i \(-0.0483471\pi\)
−0.151304 + 0.988487i \(0.548347\pi\)
\(422\) 0 0
\(423\) −21.3528 + 21.3528i −1.03821 + 1.03821i
\(424\) 0 0
\(425\) −18.1542 + 12.6380i −0.880607 + 0.613031i
\(426\) 0 0
\(427\) −3.42135 −0.165571
\(428\) 0 0
\(429\) −6.36269 6.36269i −0.307194 0.307194i
\(430\) 0 0
\(431\) 32.1769i 1.54990i −0.632020 0.774952i \(-0.717774\pi\)
0.632020 0.774952i \(-0.282226\pi\)
\(432\) 0 0
\(433\) −20.3383 + 20.3383i −0.977396 + 0.977396i −0.999750 0.0223540i \(-0.992884\pi\)
0.0223540 + 0.999750i \(0.492884\pi\)
\(434\) 0 0
\(435\) −34.5381 3.06921i −1.65598 0.147158i
\(436\) 0 0
\(437\) 14.6156i 0.699161i
\(438\) 0 0
\(439\) 35.4180i 1.69041i 0.534444 + 0.845204i \(0.320521\pi\)
−0.534444 + 0.845204i \(0.679479\pi\)
\(440\) 0 0
\(441\) 33.9256i 1.61550i
\(442\) 0 0
\(443\) 3.03787i 0.144333i 0.997393 + 0.0721667i \(0.0229913\pi\)
−0.997393 + 0.0721667i \(0.977009\pi\)
\(444\) 0 0
\(445\) 5.74507 4.80733i 0.272342 0.227890i
\(446\) 0 0
\(447\) 12.0398 12.0398i 0.569463 0.569463i
\(448\) 0 0
\(449\) 8.65559i 0.408483i −0.978921 0.204241i \(-0.934527\pi\)
0.978921 0.204241i \(-0.0654727\pi\)
\(450\) 0 0
\(451\) 2.24526 + 2.24526i 0.105725 + 0.105725i
\(452\) 0 0
\(453\) −35.5335 −1.66951
\(454\) 0 0
\(455\) −0.580733 + 6.53504i −0.0272252 + 0.306367i
\(456\) 0 0
\(457\) 13.5575 13.5575i 0.634193 0.634193i −0.314924 0.949117i \(-0.601979\pi\)
0.949117 + 0.314924i \(0.101979\pi\)
\(458\) 0 0
\(459\) −19.2692 19.2692i −0.899411 0.899411i
\(460\) 0 0
\(461\) 1.19682 1.19682i 0.0557416 0.0557416i −0.678687 0.734428i \(-0.737450\pi\)
0.734428 + 0.678687i \(0.237450\pi\)
\(462\) 0 0
\(463\) −21.1815 21.1815i −0.984390 0.984390i 0.0154904 0.999880i \(-0.495069\pi\)
−0.999880 + 0.0154904i \(0.995069\pi\)
\(464\) 0 0
\(465\) 0.573555 + 0.685435i 0.0265980 + 0.0317863i
\(466\) 0 0
\(467\) 24.8448i 1.14968i 0.818266 + 0.574840i \(0.194936\pi\)
−0.818266 + 0.574840i \(0.805064\pi\)
\(468\) 0 0
\(469\) −1.52664 + 1.52664i −0.0704936 + 0.0704936i
\(470\) 0 0
\(471\) −21.4343 −0.987642
\(472\) 0 0
\(473\) −3.69537 3.69537i −0.169913 0.169913i
\(474\) 0 0
\(475\) 16.3420 + 23.4749i 0.749822 + 1.07710i
\(476\) 0 0
\(477\) −14.2210 −0.651135
\(478\) 0 0
\(479\) −23.5766 −1.07724 −0.538621 0.842548i \(-0.681054\pi\)
−0.538621 + 0.842548i \(0.681054\pi\)
\(480\) 0 0
\(481\) 26.4181 1.20456
\(482\) 0 0
\(483\) 4.73547 0.215471
\(484\) 0 0
\(485\) −15.6102 1.38719i −0.708821 0.0629891i
\(486\) 0 0
\(487\) 2.63011 + 2.63011i 0.119182 + 0.119182i 0.764182 0.645001i \(-0.223143\pi\)
−0.645001 + 0.764182i \(0.723143\pi\)
\(488\) 0 0
\(489\) −67.7686 −3.06460
\(490\) 0 0
\(491\) −18.6899 + 18.6899i −0.843465 + 0.843465i −0.989308 0.145843i \(-0.953411\pi\)
0.145843 + 0.989308i \(0.453411\pi\)
\(492\) 0 0
\(493\) 24.0202i 1.08182i
\(494\) 0 0
\(495\) −0.711274 + 8.00403i −0.0319694 + 0.359754i
\(496\) 0 0
\(497\) −0.628940 0.628940i −0.0282118 0.0282118i
\(498\) 0 0
\(499\) 9.69342 9.69342i 0.433937 0.433937i −0.456028 0.889965i \(-0.650728\pi\)
0.889965 + 0.456028i \(0.150728\pi\)
\(500\) 0 0
\(501\) 1.15090 + 1.15090i 0.0514185 + 0.0514185i
\(502\) 0 0
\(503\) 13.0434 13.0434i 0.581577 0.581577i −0.353759 0.935336i \(-0.615097\pi\)
0.935336 + 0.353759i \(0.115097\pi\)
\(504\) 0 0
\(505\) 5.81265 + 0.516538i 0.258659 + 0.0229856i
\(506\) 0 0
\(507\) −21.2495 −0.943724
\(508\) 0 0
\(509\) −25.8539 25.8539i −1.14595 1.14595i −0.987341 0.158611i \(-0.949298\pi\)
−0.158611 0.987341i \(-0.550702\pi\)
\(510\) 0 0
\(511\) 2.34062i 0.103543i
\(512\) 0 0
\(513\) −24.9168 + 24.9168i −1.10010 + 1.10010i
\(514\) 0 0
\(515\) 3.26531 36.7448i 0.143887 1.61917i
\(516\) 0 0
\(517\) 4.08080i 0.179473i
\(518\) 0 0
\(519\) 44.1252i 1.93688i
\(520\) 0 0
\(521\) 25.0528i 1.09758i −0.835959 0.548792i \(-0.815088\pi\)
0.835959 0.548792i \(-0.184912\pi\)
\(522\) 0 0
\(523\) 40.3434i 1.76410i −0.471160 0.882048i \(-0.656165\pi\)
0.471160 0.882048i \(-0.343835\pi\)
\(524\) 0 0
\(525\) 7.60588 5.29481i 0.331948 0.231084i
\(526\) 0 0
\(527\) 0.437794 0.437794i 0.0190706 0.0190706i
\(528\) 0 0
\(529\) 16.4724i 0.716192i
\(530\) 0 0
\(531\) 18.6944 + 18.6944i 0.811267 + 0.811267i
\(532\) 0 0
\(533\) 20.6003 0.892296
\(534\) 0 0
\(535\) −26.2560 + 21.9704i −1.13515 + 0.949862i
\(536\) 0 0
\(537\) −14.8639 + 14.8639i −0.641425 + 0.641425i
\(538\) 0 0
\(539\) 3.24180 + 3.24180i 0.139634 + 0.139634i
\(540\) 0 0
\(541\) 24.7446 24.7446i 1.06385 1.06385i 0.0660360 0.997817i \(-0.478965\pi\)
0.997817 0.0660360i \(-0.0210352\pi\)
\(542\) 0 0
\(543\) 25.9590 + 25.9590i 1.11401 + 1.11401i
\(544\) 0 0
\(545\) 39.3097 + 3.49324i 1.68384 + 0.149634i
\(546\) 0 0
\(547\) 19.0254i 0.813465i 0.913547 + 0.406733i \(0.133332\pi\)
−0.913547 + 0.406733i \(0.866668\pi\)
\(548\) 0 0
\(549\) −19.2235 + 19.2235i −0.820440 + 0.820440i
\(550\) 0 0
\(551\) 31.0602 1.32321
\(552\) 0 0
\(553\) −1.77329 1.77329i −0.0754079 0.0754079i
\(554\) 0 0
\(555\) −23.9476 28.6189i −1.01652 1.21481i
\(556\) 0 0
\(557\) −30.9517 −1.31146 −0.655732 0.754993i \(-0.727640\pi\)
−0.655732 + 0.754993i \(0.727640\pi\)
\(558\) 0 0
\(559\) −33.9050 −1.43403
\(560\) 0 0
\(561\) 8.80494 0.371745
\(562\) 0 0
\(563\) 3.50238 0.147608 0.0738039 0.997273i \(-0.476486\pi\)
0.0738039 + 0.997273i \(0.476486\pi\)
\(564\) 0 0
\(565\) 0.709734 7.98670i 0.0298587 0.336003i
\(566\) 0 0
\(567\) 0.973793 + 0.973793i 0.0408955 + 0.0408955i
\(568\) 0 0
\(569\) 0.525780 0.0220418 0.0110209 0.999939i \(-0.496492\pi\)
0.0110209 + 0.999939i \(0.496492\pi\)
\(570\) 0 0
\(571\) −11.2487 + 11.2487i −0.470743 + 0.470743i −0.902155 0.431412i \(-0.858016\pi\)
0.431412 + 0.902155i \(0.358016\pi\)
\(572\) 0 0
\(573\) 43.1472i 1.80250i
\(574\) 0 0
\(575\) −7.29859 10.4843i −0.304372 0.437224i
\(576\) 0 0
\(577\) −2.92884 2.92884i −0.121929 0.121929i 0.643509 0.765438i \(-0.277478\pi\)
−0.765438 + 0.643509i \(0.777478\pi\)
\(578\) 0 0
\(579\) 11.9714 11.9714i 0.497515 0.497515i
\(580\) 0 0
\(581\) 6.64011 + 6.64011i 0.275478 + 0.275478i
\(582\) 0 0
\(583\) 1.35891 1.35891i 0.0562801 0.0562801i
\(584\) 0 0
\(585\) 33.4555 + 39.9814i 1.38321 + 1.65303i
\(586\) 0 0
\(587\) 23.1574 0.955809 0.477905 0.878412i \(-0.341396\pi\)
0.477905 + 0.878412i \(0.341396\pi\)
\(588\) 0 0
\(589\) −0.566106 0.566106i −0.0233260 0.0233260i
\(590\) 0 0
\(591\) 11.5150i 0.473662i
\(592\) 0 0
\(593\) −13.9325 + 13.9325i −0.572141 + 0.572141i −0.932726 0.360585i \(-0.882577\pi\)
0.360585 + 0.932726i \(0.382577\pi\)
\(594\) 0 0
\(595\) −4.11990 4.92354i −0.168900 0.201846i
\(596\) 0 0
\(597\) 15.5097i 0.634769i
\(598\) 0 0
\(599\) 33.5311i 1.37004i −0.728523 0.685021i \(-0.759793\pi\)
0.728523 0.685021i \(-0.240207\pi\)
\(600\) 0 0
\(601\) 19.4164i 0.792011i 0.918248 + 0.396005i \(0.129604\pi\)
−0.918248 + 0.396005i \(0.870396\pi\)
\(602\) 0 0
\(603\) 17.1554i 0.698623i
\(604\) 0 0
\(605\) 15.0879 + 18.0310i 0.613409 + 0.733063i
\(606\) 0 0
\(607\) −9.51495 + 9.51495i −0.386200 + 0.386200i −0.873330 0.487130i \(-0.838044\pi\)
0.487130 + 0.873330i \(0.338044\pi\)
\(608\) 0 0
\(609\) 10.0635i 0.407794i
\(610\) 0 0
\(611\) −18.7206 18.7206i −0.757355 0.757355i
\(612\) 0 0
\(613\) 9.37947 0.378833 0.189417 0.981897i \(-0.439340\pi\)
0.189417 + 0.981897i \(0.439340\pi\)
\(614\) 0 0
\(615\) −18.6738 22.3164i −0.753002 0.899884i
\(616\) 0 0
\(617\) 3.54768 3.54768i 0.142824 0.142824i −0.632079 0.774904i \(-0.717798\pi\)
0.774904 + 0.632079i \(0.217798\pi\)
\(618\) 0 0
\(619\) −24.6158 24.6158i −0.989392 0.989392i 0.0105527 0.999944i \(-0.496641\pi\)
−0.999944 + 0.0105527i \(0.996641\pi\)
\(620\) 0 0
\(621\) 11.1282 11.1282i 0.446561 0.446561i
\(622\) 0 0
\(623\) 1.53735 + 1.53735i 0.0615926 + 0.0615926i
\(624\) 0 0
\(625\) −23.4453 8.67867i −0.937811 0.347147i
\(626\) 0 0
\(627\) 11.3856i 0.454695i
\(628\) 0 0
\(629\) −18.2792 + 18.2792i −0.728840 + 0.728840i
\(630\) 0 0
\(631\) 28.8921 1.15018 0.575088 0.818092i \(-0.304968\pi\)
0.575088 + 0.818092i \(0.304968\pi\)
\(632\) 0 0
\(633\) 9.22547 + 9.22547i 0.366679 + 0.366679i
\(634\) 0 0
\(635\) −0.172367 + 1.93966i −0.00684016 + 0.0769729i
\(636\) 0 0
\(637\) 29.7435 1.17848
\(638\) 0 0
\(639\) −7.06765 −0.279592
\(640\) 0 0
\(641\) −16.6914 −0.659271 −0.329636 0.944108i \(-0.606926\pi\)
−0.329636 + 0.944108i \(0.606926\pi\)
\(642\) 0 0
\(643\) −5.22468 −0.206041 −0.103021 0.994679i \(-0.532851\pi\)
−0.103021 + 0.994679i \(0.532851\pi\)
\(644\) 0 0
\(645\) 30.7343 + 36.7295i 1.21016 + 1.44622i
\(646\) 0 0
\(647\) −21.6797 21.6797i −0.852318 0.852318i 0.138100 0.990418i \(-0.455900\pi\)
−0.990418 + 0.138100i \(0.955900\pi\)
\(648\) 0 0
\(649\) −3.57273 −0.140242
\(650\) 0 0
\(651\) −0.183418 + 0.183418i −0.00718874 + 0.00718874i
\(652\) 0 0
\(653\) 22.7642i 0.890833i 0.895323 + 0.445417i \(0.146944\pi\)
−0.895323 + 0.445417i \(0.853056\pi\)
\(654\) 0 0
\(655\) 30.0942 + 2.67431i 1.17588 + 0.104494i
\(656\) 0 0
\(657\) 13.1513 + 13.1513i 0.513079 + 0.513079i
\(658\) 0 0
\(659\) 1.66201 1.66201i 0.0647427 0.0647427i −0.673994 0.738737i \(-0.735423\pi\)
0.738737 + 0.673994i \(0.235423\pi\)
\(660\) 0 0
\(661\) 5.62818 + 5.62818i 0.218911 + 0.218911i 0.808039 0.589129i \(-0.200529\pi\)
−0.589129 + 0.808039i \(0.700529\pi\)
\(662\) 0 0
\(663\) 40.3926 40.3926i 1.56872 1.56872i
\(664\) 0 0
\(665\) −6.36657 + 5.32739i −0.246885 + 0.206587i
\(666\) 0 0
\(667\) −13.8720 −0.537125
\(668\) 0 0
\(669\) −23.3427 23.3427i −0.902481 0.902481i
\(670\) 0 0
\(671\) 3.67386i 0.141828i
\(672\) 0 0
\(673\) 0.278251 0.278251i 0.0107258 0.0107258i −0.701724 0.712449i \(-0.747586\pi\)
0.712449 + 0.701724i \(0.247586\pi\)
\(674\) 0 0
\(675\) 5.43097 30.3163i 0.209038 1.16687i
\(676\) 0 0
\(677\) 26.3591i 1.01306i −0.862222 0.506531i \(-0.830928\pi\)
0.862222 0.506531i \(-0.169072\pi\)
\(678\) 0 0
\(679\) 4.54840i 0.174551i
\(680\) 0 0
\(681\) 4.40019i 0.168616i
\(682\) 0 0
\(683\) 2.83023i 0.108296i 0.998533 + 0.0541479i \(0.0172442\pi\)
−0.998533 + 0.0541479i \(0.982756\pi\)
\(684\) 0 0
\(685\) 1.03645 11.6632i 0.0396006 0.445629i
\(686\) 0 0
\(687\) −50.1646 + 50.1646i −1.91390 + 1.91390i
\(688\) 0 0
\(689\) 12.4679i 0.474991i
\(690\) 0 0
\(691\) 22.1815 + 22.1815i 0.843825 + 0.843825i 0.989354 0.145529i \(-0.0464884\pi\)
−0.145529 + 0.989354i \(0.546488\pi\)
\(692\) 0 0
\(693\) −2.33217 −0.0885917
\(694\) 0 0
\(695\) −17.2222 1.53044i −0.653276 0.0580531i
\(696\) 0 0
\(697\) −14.2537 + 14.2537i −0.539898 + 0.539898i
\(698\) 0 0
\(699\) −28.5320 28.5320i −1.07918 1.07918i
\(700\) 0 0
\(701\) −16.2264 + 16.2264i −0.612864 + 0.612864i −0.943691 0.330828i \(-0.892672\pi\)
0.330828 + 0.943691i \(0.392672\pi\)
\(702\) 0 0
\(703\) 23.6366 + 23.6366i 0.891472 + 0.891472i
\(704\) 0 0
\(705\) −3.31022 + 37.2502i −0.124670 + 1.40292i
\(706\) 0 0
\(707\) 1.69365i 0.0636965i
\(708\) 0 0
\(709\) 25.3577 25.3577i 0.952329 0.952329i −0.0465856 0.998914i \(-0.514834\pi\)
0.998914 + 0.0465856i \(0.0148340\pi\)
\(710\) 0 0
\(711\) −19.9271 −0.747326
\(712\) 0 0
\(713\) 0.252832 + 0.252832i 0.00946863 + 0.00946863i
\(714\) 0 0
\(715\) −7.01735 0.623594i −0.262434 0.0233211i
\(716\) 0 0
\(717\) 75.0450 2.80261
\(718\) 0 0
\(719\) 41.3374 1.54163 0.770813 0.637061i \(-0.219850\pi\)
0.770813 + 0.637061i \(0.219850\pi\)
\(720\) 0 0
\(721\) 10.7065 0.398730
\(722\) 0 0
\(723\) −0.323420 −0.0120281
\(724\) 0 0
\(725\) −22.2805 + 15.5105i −0.827477 + 0.576045i
\(726\) 0 0
\(727\) 23.4630 + 23.4630i 0.870193 + 0.870193i 0.992493 0.122300i \(-0.0390271\pi\)
−0.122300 + 0.992493i \(0.539027\pi\)
\(728\) 0 0
\(729\) −41.8342 −1.54942
\(730\) 0 0
\(731\) 23.4595 23.4595i 0.867681 0.867681i
\(732\) 0 0
\(733\) 15.1628i 0.560051i −0.959993 0.280025i \(-0.909657\pi\)
0.959993 0.280025i \(-0.0903429\pi\)
\(734\) 0 0
\(735\) −26.9621 32.2214i −0.994511 1.18850i
\(736\) 0 0
\(737\) −1.63931 1.63931i −0.0603848 0.0603848i
\(738\) 0 0
\(739\) 0.974343 0.974343i 0.0358418 0.0358418i −0.688959 0.724801i \(-0.741932\pi\)
0.724801 + 0.688959i \(0.241932\pi\)
\(740\) 0 0
\(741\) −52.2312 52.2312i −1.91876 1.91876i
\(742\) 0 0
\(743\) −29.0897 + 29.0897i −1.06720 + 1.06720i −0.0696259 + 0.997573i \(0.522181\pi\)
−0.997573 + 0.0696259i \(0.977819\pi\)
\(744\) 0 0
\(745\) 1.18000 13.2786i 0.0432317 0.486490i
\(746\) 0 0
\(747\) 74.6176 2.73011
\(748\) 0 0
\(749\) −7.02596 7.02596i −0.256723 0.256723i
\(750\) 0 0
\(751\) 7.77705i 0.283789i −0.989882 0.141894i \(-0.954681\pi\)
0.989882 0.141894i \(-0.0453193\pi\)
\(752\) 0 0
\(753\) 54.8981 54.8981i 2.00060 2.00060i
\(754\) 0 0
\(755\) −21.3361 + 17.8535i −0.776498 + 0.649755i
\(756\) 0 0
\(757\) 1.42073i 0.0516372i −0.999667 0.0258186i \(-0.991781\pi\)
0.999667 0.0258186i \(-0.00821923\pi\)
\(758\) 0 0
\(759\) 5.08497i 0.184573i
\(760\) 0 0
\(761\) 26.6737i 0.966921i 0.875366 + 0.483460i \(0.160620\pi\)
−0.875366 + 0.483460i \(0.839380\pi\)
\(762\) 0 0
\(763\) 11.4538i 0.414656i
\(764\) 0 0
\(765\) −50.8124 4.51542i −1.83713 0.163255i
\(766\) 0 0
\(767\) −16.3899 + 16.3899i −0.591805 + 0.591805i
\(768\) 0 0
\(769\) 45.8210i 1.65235i 0.563415 + 0.826174i \(0.309487\pi\)
−0.563415 + 0.826174i \(0.690513\pi\)
\(770\) 0 0
\(771\) −2.16428 2.16428i −0.0779447 0.0779447i
\(772\) 0 0
\(773\) 18.5473 0.667101 0.333550 0.942732i \(-0.391753\pi\)
0.333550 + 0.942732i \(0.391753\pi\)
\(774\) 0 0
\(775\) 0.688782 + 0.123391i 0.0247418 + 0.00443233i
\(776\) 0 0
\(777\) 7.65827 7.65827i 0.274739 0.274739i
\(778\) 0 0
\(779\) 18.4313 + 18.4313i 0.660370 + 0.660370i
\(780\) 0 0
\(781\) 0.675359 0.675359i 0.0241662 0.0241662i
\(782\) 0 0
\(783\) −23.6490 23.6490i −0.845146 0.845146i
\(784\) 0 0
\(785\) −12.8702 + 10.7695i −0.459358 + 0.384380i
\(786\) 0 0
\(787\) 21.3016i 0.759319i −0.925126 0.379659i \(-0.876041\pi\)
0.925126 0.379659i \(-0.123959\pi\)
\(788\) 0 0
\(789\) −16.3654 + 16.3654i −0.582624 + 0.582624i
\(790\) 0 0
\(791\) 2.32712 0.0827427
\(792\) 0 0
\(793\) −16.8538 16.8538i −0.598496 0.598496i
\(794\) 0 0
\(795\) −13.5066 + 11.3020i −0.479030 + 0.400841i
\(796\) 0 0
\(797\) 2.35457 0.0834033 0.0417016 0.999130i \(-0.486722\pi\)
0.0417016 + 0.999130i \(0.486722\pi\)
\(798\) 0 0
\(799\) 25.9063 0.916500
\(800\) 0 0
\(801\) 17.2758 0.610410
\(802\) 0 0
\(803\) −2.51337 −0.0886949
\(804\) 0 0
\(805\) 2.84341 2.37930i 0.100217 0.0838592i
\(806\) 0 0
\(807\) 27.9569 + 27.9569i 0.984128 + 0.984128i
\(808\) 0 0
\(809\) 23.9476 0.841952 0.420976 0.907072i \(-0.361688\pi\)
0.420976 + 0.907072i \(0.361688\pi\)
\(810\) 0 0
\(811\) −1.33006 + 1.33006i −0.0467048 + 0.0467048i −0.730073 0.683369i \(-0.760514\pi\)
0.683369 + 0.730073i \(0.260514\pi\)
\(812\) 0 0
\(813\) 11.7142i 0.410834i
\(814\) 0 0
\(815\) −40.6916 + 34.0498i −1.42537 + 1.19271i
\(816\) 0 0
\(817\) −30.3352 30.3352i −1.06129 1.06129i
\(818\) 0 0
\(819\) −10.6988 + 10.6988i −0.373846 + 0.373846i
\(820\) 0 0
\(821\) −36.4676 36.4676i −1.27273 1.27273i −0.944651 0.328076i \(-0.893600\pi\)
−0.328076 0.944651i \(-0.606400\pi\)
\(822\) 0 0
\(823\) −26.3978 + 26.3978i −0.920170 + 0.920170i −0.997041 0.0768712i \(-0.975507\pi\)
0.0768712 + 0.997041i \(0.475507\pi\)
\(824\) 0 0
\(825\) 5.68559 + 8.16723i 0.197947 + 0.284346i
\(826\) 0 0
\(827\) −1.99830 −0.0694878 −0.0347439 0.999396i \(-0.511062\pi\)
−0.0347439 + 0.999396i \(0.511062\pi\)
\(828\) 0 0
\(829\) −13.0376 13.0376i −0.452813 0.452813i 0.443474 0.896287i \(-0.353746\pi\)
−0.896287 + 0.443474i \(0.853746\pi\)
\(830\) 0 0
\(831\) 70.4735i 2.44470i
\(832\) 0 0
\(833\) −20.5801 + 20.5801i −0.713059 + 0.713059i
\(834\) 0 0
\(835\) 1.26932 + 0.112797i 0.0439266 + 0.00390352i
\(836\) 0 0
\(837\) 0.862057i 0.0297971i
\(838\) 0 0
\(839\) 15.4102i 0.532018i 0.963971 + 0.266009i \(0.0857050\pi\)
−0.963971 + 0.266009i \(0.914295\pi\)
\(840\) 0 0
\(841\) 0.479815i 0.0165453i
\(842\) 0 0
\(843\) 67.5983i 2.32821i
\(844\) 0 0
\(845\) −12.7593 + 10.6766i −0.438932 + 0.367287i
\(846\) 0 0
\(847\) −4.82499 + 4.82499i −0.165789 + 0.165789i
\(848\) 0 0
\(849\) 37.2686i 1.27906i
\(850\) 0 0
\(851\) −10.5565 10.5565i −0.361872 0.361872i
\(852\) 0 0
\(853\) −7.96419 −0.272689 −0.136344 0.990662i \(-0.543535\pi\)
−0.136344 + 0.990662i \(0.543535\pi\)
\(854\) 0 0
\(855\) −5.83883 + 65.7049i −0.199684 + 2.24706i
\(856\) 0 0
\(857\) −5.35407 + 5.35407i −0.182891 + 0.182891i −0.792615 0.609723i \(-0.791281\pi\)
0.609723 + 0.792615i \(0.291281\pi\)
\(858\) 0 0
\(859\) −35.0058 35.0058i −1.19438 1.19438i −0.975824 0.218559i \(-0.929864\pi\)
−0.218559 0.975824i \(-0.570136\pi\)
\(860\) 0 0
\(861\) 5.97175 5.97175i 0.203517 0.203517i
\(862\) 0 0
\(863\) −36.7138 36.7138i −1.24975 1.24975i −0.955829 0.293923i \(-0.905039\pi\)
−0.293923 0.955829i \(-0.594961\pi\)
\(864\) 0 0
\(865\) −22.1703 26.4950i −0.753814 0.900856i
\(866\) 0 0
\(867\) 7.34475i 0.249441i
\(868\) 0 0
\(869\) 1.90416 1.90416i 0.0645943 0.0645943i
\(870\) 0 0
\(871\) −15.0406 −0.509633
\(872\) 0 0
\(873\) −25.5561 25.5561i −0.864942 0.864942i
\(874\) 0 0
\(875\) 1.90662 7.00078i 0.0644555 0.236669i
\(876\) 0 0
\(877\) 14.3410 0.484262 0.242131 0.970244i \(-0.422154\pi\)
0.242131 + 0.970244i \(0.422154\pi\)
\(878\) 0 0
\(879\) 90.4586 3.05109
\(880\) 0 0
\(881\) −13.6397 −0.459533 −0.229767 0.973246i \(-0.573796\pi\)
−0.229767 + 0.973246i \(0.573796\pi\)
\(882\) 0 0
\(883\) 6.12563 0.206144 0.103072 0.994674i \(-0.467133\pi\)
0.103072 + 0.994674i \(0.467133\pi\)
\(884\) 0 0
\(885\) 32.6125 + 2.89809i 1.09626 + 0.0974184i
\(886\) 0 0
\(887\) −25.5187 25.5187i −0.856834 0.856834i 0.134130 0.990964i \(-0.457176\pi\)
−0.990964 + 0.134130i \(0.957176\pi\)
\(888\) 0 0
\(889\) −0.565166 −0.0189551
\(890\) 0 0
\(891\) −1.04566 + 1.04566i −0.0350310 + 0.0350310i
\(892\) 0 0
\(893\) 33.4992i 1.12101i
\(894\) 0 0
\(895\) −1.45678 + 16.3933i −0.0486948 + 0.547967i
\(896\) 0 0
\(897\) 23.3273 + 23.3273i 0.778875 + 0.778875i
\(898\) 0 0
\(899\) 0.537302 0.537302i 0.0179200 0.0179200i
\(900\) 0 0
\(901\) 8.62682 + 8.62682i 0.287401 + 0.287401i
\(902\) 0 0
\(903\) −9.82861 + 9.82861i −0.327076 + 0.327076i
\(904\) 0 0
\(905\) 28.6299 + 2.54418i 0.951691 + 0.0845715i
\(906\) 0 0
\(907\) 32.1815 1.06857 0.534284 0.845305i \(-0.320581\pi\)
0.534284 + 0.845305i \(0.320581\pi\)
\(908\) 0 0
\(909\) 9.51614 + 9.51614i 0.315630 + 0.315630i
\(910\) 0 0
\(911\) 38.6282i 1.27981i 0.768455 + 0.639904i \(0.221026\pi\)
−0.768455 + 0.639904i \(0.778974\pi\)
\(912\) 0 0
\(913\) −7.13018 + 7.13018i −0.235974 + 0.235974i
\(914\) 0 0
\(915\) −2.98012 + 33.5356i −0.0985198 + 1.10865i
\(916\) 0 0
\(917\) 8.76867i 0.289567i
\(918\) 0 0
\(919\) 19.1924i 0.633099i −0.948576 0.316550i \(-0.897476\pi\)
0.948576 0.316550i \(-0.102524\pi\)
\(920\) 0 0
\(921\) 78.1395i 2.57479i
\(922\) 0 0
\(923\) 6.19641i 0.203957i
\(924\) 0 0
\(925\) −28.7587 5.15194i −0.945580 0.169395i
\(926\) 0 0
\(927\) 60.1564 60.1564i 1.97580 1.97580i
\(928\) 0 0
\(929\) 16.8576i 0.553081i −0.961002 0.276541i \(-0.910812\pi\)
0.961002 0.276541i \(-0.0891880\pi\)
\(930\) 0 0
\(931\) 26.6119 + 26.6119i 0.872170 + 0.872170i
\(932\) 0 0
\(933\) 45.1467 1.47804
\(934\) 0 0
\(935\) 5.28692 4.42397i 0.172901 0.144679i
\(936\) 0 0
\(937\) 23.9511 23.9511i 0.782449 0.782449i −0.197795 0.980243i \(-0.563378\pi\)
0.980243 + 0.197795i \(0.0633779\pi\)
\(938\) 0 0
\(939\) 39.5238 + 39.5238i 1.28981 + 1.28981i
\(940\) 0 0
\(941\) −14.2496 + 14.2496i −0.464525 + 0.464525i −0.900135 0.435610i \(-0.856533\pi\)
0.435610 + 0.900135i \(0.356533\pi\)
\(942\) 0 0
\(943\) −8.23171 8.23171i −0.268061 0.268061i
\(944\) 0 0
\(945\) 8.90369 + 0.791222i 0.289637 + 0.0257385i
\(946\) 0 0
\(947\) 20.2943i 0.659477i 0.944072 + 0.329738i \(0.106961\pi\)
−0.944072 + 0.329738i \(0.893039\pi\)
\(948\) 0 0
\(949\) −11.5301 + 11.5301i −0.374282 + 0.374282i
\(950\) 0 0
\(951\) 99.9866 3.24229
\(952\) 0 0
\(953\) 10.9257 + 10.9257i 0.353919 + 0.353919i 0.861565 0.507647i \(-0.169484\pi\)
−0.507647 + 0.861565i \(0.669484\pi\)
\(954\) 0 0
\(955\) 21.6789 + 25.9077i 0.701514 + 0.838353i
\(956\) 0 0
\(957\) 10.8062 0.349316
\(958\) 0 0
\(959\) 3.39836 0.109739
\(960\) 0 0
\(961\) 30.9804 0.999368
\(962\) 0 0
\(963\) −78.9535 −2.54424
\(964\) 0 0
\(965\) 1.17329 13.2032i 0.0377696 0.425025i
\(966\) 0 0
\(967\) −10.7569 10.7569i −0.345918 0.345918i 0.512669 0.858586i \(-0.328657\pi\)
−0.858586 + 0.512669i \(0.828657\pi\)
\(968\) 0 0
\(969\) 72.2796 2.32195
\(970\) 0 0
\(971\) −18.7456 + 18.7456i −0.601574 + 0.601574i −0.940730 0.339156i \(-0.889858\pi\)
0.339156 + 0.940730i \(0.389858\pi\)
\(972\) 0 0
\(973\) 5.01811i 0.160873i
\(974\) 0 0
\(975\) 63.5497 + 11.3845i 2.03522 + 0.364597i
\(976\) 0 0
\(977\) −26.3906 26.3906i −0.844309 0.844309i 0.145107 0.989416i \(-0.453647\pi\)
−0.989416 + 0.145107i \(0.953647\pi\)
\(978\) 0 0
\(979\) −1.65081 + 1.65081i −0.0527602 + 0.0527602i
\(980\) 0 0
\(981\) 64.3556 + 64.3556i 2.05472 + 2.05472i
\(982\) 0 0
\(983\) 4.87875 4.87875i 0.155608 0.155608i −0.625009 0.780617i \(-0.714905\pi\)
0.780617 + 0.625009i \(0.214905\pi\)
\(984\) 0 0
\(985\) 5.78559 + 6.91415i 0.184344 + 0.220303i
\(986\) 0 0
\(987\) −10.8537 −0.345478
\(988\) 0 0
\(989\) 13.5482 + 13.5482i 0.430807 + 0.430807i
\(990\) 0 0
\(991\) 61.2103i 1.94441i 0.234130 + 0.972205i \(0.424776\pi\)
−0.234130 + 0.972205i \(0.575224\pi\)
\(992\) 0 0
\(993\) 48.0415 48.0415i 1.52455 1.52455i
\(994\) 0 0
\(995\) 7.79271 + 9.31279i 0.247046 + 0.295235i
\(996\) 0 0
\(997\) 39.1082i 1.23857i −0.785167 0.619284i \(-0.787423\pi\)
0.785167 0.619284i \(-0.212577\pi\)
\(998\) 0 0
\(999\) 35.9935i 1.13878i
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 640.2.s.c.287.9 18
4.3 odd 2 640.2.s.d.287.1 18
5.3 odd 4 640.2.j.c.543.9 18
8.3 odd 2 80.2.s.b.27.1 yes 18
8.5 even 2 320.2.s.b.207.1 18
16.3 odd 4 640.2.j.c.607.1 18
16.5 even 4 80.2.j.b.67.5 yes 18
16.11 odd 4 320.2.j.b.47.9 18
16.13 even 4 640.2.j.d.607.9 18
20.3 even 4 640.2.j.d.543.1 18
24.11 even 2 720.2.z.g.667.9 18
40.3 even 4 80.2.j.b.43.5 18
40.13 odd 4 320.2.j.b.143.1 18
40.19 odd 2 400.2.s.d.107.9 18
40.27 even 4 400.2.j.d.43.5 18
40.29 even 2 1600.2.s.d.207.9 18
40.37 odd 4 1600.2.j.d.143.9 18
48.5 odd 4 720.2.bd.g.307.5 18
80.3 even 4 inner 640.2.s.c.223.9 18
80.13 odd 4 640.2.s.d.223.1 18
80.27 even 4 1600.2.s.d.943.9 18
80.37 odd 4 400.2.s.d.243.9 18
80.43 even 4 320.2.s.b.303.1 18
80.53 odd 4 80.2.s.b.3.1 yes 18
80.59 odd 4 1600.2.j.d.1007.1 18
80.69 even 4 400.2.j.d.307.5 18
120.83 odd 4 720.2.bd.g.523.5 18
240.53 even 4 720.2.z.g.163.9 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.5 18 40.3 even 4
80.2.j.b.67.5 yes 18 16.5 even 4
80.2.s.b.3.1 yes 18 80.53 odd 4
80.2.s.b.27.1 yes 18 8.3 odd 2
320.2.j.b.47.9 18 16.11 odd 4
320.2.j.b.143.1 18 40.13 odd 4
320.2.s.b.207.1 18 8.5 even 2
320.2.s.b.303.1 18 80.43 even 4
400.2.j.d.43.5 18 40.27 even 4
400.2.j.d.307.5 18 80.69 even 4
400.2.s.d.107.9 18 40.19 odd 2
400.2.s.d.243.9 18 80.37 odd 4
640.2.j.c.543.9 18 5.3 odd 4
640.2.j.c.607.1 18 16.3 odd 4
640.2.j.d.543.1 18 20.3 even 4
640.2.j.d.607.9 18 16.13 even 4
640.2.s.c.223.9 18 80.3 even 4 inner
640.2.s.c.287.9 18 1.1 even 1 trivial
640.2.s.d.223.1 18 80.13 odd 4
640.2.s.d.287.1 18 4.3 odd 2
720.2.z.g.163.9 18 240.53 even 4
720.2.z.g.667.9 18 24.11 even 2
720.2.bd.g.307.5 18 48.5 odd 4
720.2.bd.g.523.5 18 120.83 odd 4
1600.2.j.d.143.9 18 40.37 odd 4
1600.2.j.d.1007.1 18 80.59 odd 4
1600.2.s.d.207.9 18 40.29 even 2
1600.2.s.d.943.9 18 80.27 even 4