Properties

Label 1008.2.r.n.673.1
Level $1008$
Weight $2$
Character 1008.673
Analytic conductor $8.049$
Analytic rank $0$
Dimension $10$
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] = [1008,2,Mod(337,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (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: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.6095158642368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 6x^{8} - 7x^{7} + 25x^{6} - 66x^{5} + 75x^{4} - 63x^{3} + 162x^{2} - 324x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 673.1
Root \(1.11541 - 1.32509i\) of defining polynomial
Character \(\chi\) \(=\) 1008.673
Dual form 1008.2.r.n.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.62713 - 0.593666i) q^{3} +(1.50470 + 2.60622i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(2.29512 + 1.93195i) q^{9} +O(q^{10})\) \(q+(-1.62713 - 0.593666i) q^{3} +(1.50470 + 2.60622i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(2.29512 + 1.93195i) q^{9} +(-2.74254 + 4.75022i) q^{11} +(-0.421867 - 0.730695i) q^{13} +(-0.901126 - 5.13396i) q^{15} -4.59024 q^{17} +3.80203 q^{19} +(1.32770 - 1.11231i) q^{21} +(-4.48385 - 7.76626i) q^{23} +(-2.02826 + 3.51305i) q^{25} +(-2.58754 - 4.50607i) q^{27} +(-0.974447 + 1.68779i) q^{29} +(1.68873 + 2.92497i) q^{31} +(7.28252 - 6.10109i) q^{33} -3.00941 q^{35} -7.49050 q^{37} +(0.252645 + 1.43939i) q^{39} +(-3.69144 - 6.39375i) q^{41} +(3.11330 - 5.39240i) q^{43} +(-1.58160 + 8.88860i) q^{45} +(-6.57013 + 11.3798i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(7.46894 + 2.72507i) q^{51} +8.05652 q^{53} -16.5068 q^{55} +(-6.18641 - 2.25714i) q^{57} +(5.04037 + 8.73018i) q^{59} +(-2.84423 + 4.92635i) q^{61} +(-2.82068 + 1.02166i) q^{63} +(1.26957 - 2.19896i) q^{65} +(-2.71969 - 4.71065i) q^{67} +(2.68526 + 15.2986i) q^{69} -5.81144 q^{71} -10.7167 q^{73} +(5.38582 - 4.51209i) q^{75} +(-2.74254 - 4.75022i) q^{77} +(1.37055 - 2.37386i) q^{79} +(1.53517 + 8.86810i) q^{81} +(-6.35164 + 11.0014i) q^{83} +(-6.90695 - 11.9632i) q^{85} +(2.58754 - 2.16776i) q^{87} -8.74898 q^{89} +0.843734 q^{91} +(-1.01134 - 5.76185i) q^{93} +(5.72093 + 9.90894i) q^{95} +(3.83699 - 6.64586i) q^{97} +(-15.4716 + 5.60390i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{5} - 5 q^{7} - 4 q^{11} - 3 q^{13} - 15 q^{15} - 2 q^{19} - 8 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} + 3 q^{31} + 30 q^{33} + 6 q^{35} - 6 q^{37} + 18 q^{39} - 12 q^{41} + 5 q^{43} - 9 q^{45} - 3 q^{47} - 5 q^{49} - 9 q^{51} + 60 q^{53} - 44 q^{55} - 21 q^{57} - 7 q^{59} - 14 q^{61} - 6 q^{63} - 11 q^{65} + 8 q^{67} + 21 q^{69} + 18 q^{71} + 30 q^{73} + 51 q^{75} - 4 q^{77} + 3 q^{79} - 12 q^{81} - 20 q^{83} - 21 q^{85} - 9 q^{87} - 24 q^{89} + 6 q^{91} - 39 q^{93} + 12 q^{95} - 37 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −1.62713 0.593666i −0.939426 0.342753i
\(4\) 0 0
\(5\) 1.50470 + 2.60622i 0.672923 + 1.16554i 0.977071 + 0.212913i \(0.0682950\pi\)
−0.304148 + 0.952625i \(0.598372\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 2.29512 + 1.93195i 0.765041 + 0.643982i
\(10\) 0 0
\(11\) −2.74254 + 4.75022i −0.826908 + 1.43225i 0.0735450 + 0.997292i \(0.476569\pi\)
−0.900453 + 0.434954i \(0.856765\pi\)
\(12\) 0 0
\(13\) −0.421867 0.730695i −0.117005 0.202658i 0.801575 0.597895i \(-0.203996\pi\)
−0.918579 + 0.395236i \(0.870663\pi\)
\(14\) 0 0
\(15\) −0.901126 5.13396i −0.232670 1.32558i
\(16\) 0 0
\(17\) −4.59024 −1.11330 −0.556649 0.830748i \(-0.687913\pi\)
−0.556649 + 0.830748i \(0.687913\pi\)
\(18\) 0 0
\(19\) 3.80203 0.872246 0.436123 0.899887i \(-0.356351\pi\)
0.436123 + 0.899887i \(0.356351\pi\)
\(20\) 0 0
\(21\) 1.32770 1.11231i 0.289727 0.242725i
\(22\) 0 0
\(23\) −4.48385 7.76626i −0.934948 1.61938i −0.774727 0.632296i \(-0.782113\pi\)
−0.160221 0.987081i \(-0.551221\pi\)
\(24\) 0 0
\(25\) −2.02826 + 3.51305i −0.405652 + 0.702610i
\(26\) 0 0
\(27\) −2.58754 4.50607i −0.497972 0.867193i
\(28\) 0 0
\(29\) −0.974447 + 1.68779i −0.180950 + 0.313415i −0.942204 0.335039i \(-0.891251\pi\)
0.761254 + 0.648454i \(0.224584\pi\)
\(30\) 0 0
\(31\) 1.68873 + 2.92497i 0.303305 + 0.525339i 0.976882 0.213777i \(-0.0685766\pi\)
−0.673578 + 0.739116i \(0.735243\pi\)
\(32\) 0 0
\(33\) 7.28252 6.10109i 1.26772 1.06206i
\(34\) 0 0
\(35\) −3.00941 −0.508682
\(36\) 0 0
\(37\) −7.49050 −1.23143 −0.615715 0.787969i \(-0.711133\pi\)
−0.615715 + 0.787969i \(0.711133\pi\)
\(38\) 0 0
\(39\) 0.252645 + 1.43939i 0.0404556 + 0.230486i
\(40\) 0 0
\(41\) −3.69144 6.39375i −0.576505 0.998537i −0.995876 0.0907217i \(-0.971083\pi\)
0.419371 0.907815i \(-0.362251\pi\)
\(42\) 0 0
\(43\) 3.11330 5.39240i 0.474774 0.822333i −0.524808 0.851220i \(-0.675863\pi\)
0.999583 + 0.0288873i \(0.00919639\pi\)
\(44\) 0 0
\(45\) −1.58160 + 8.88860i −0.235771 + 1.32503i
\(46\) 0 0
\(47\) −6.57013 + 11.3798i −0.958352 + 1.65991i −0.231847 + 0.972752i \(0.574477\pi\)
−0.726505 + 0.687162i \(0.758856\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 7.46894 + 2.72507i 1.04586 + 0.381586i
\(52\) 0 0
\(53\) 8.05652 1.10665 0.553324 0.832966i \(-0.313359\pi\)
0.553324 + 0.832966i \(0.313359\pi\)
\(54\) 0 0
\(55\) −16.5068 −2.22578
\(56\) 0 0
\(57\) −6.18641 2.25714i −0.819410 0.298965i
\(58\) 0 0
\(59\) 5.04037 + 8.73018i 0.656200 + 1.13657i 0.981591 + 0.190993i \(0.0611707\pi\)
−0.325391 + 0.945580i \(0.605496\pi\)
\(60\) 0 0
\(61\) −2.84423 + 4.92635i −0.364167 + 0.630755i −0.988642 0.150289i \(-0.951979\pi\)
0.624475 + 0.781044i \(0.285313\pi\)
\(62\) 0 0
\(63\) −2.82068 + 1.02166i −0.355372 + 0.128717i
\(64\) 0 0
\(65\) 1.26957 2.19896i 0.157471 0.272747i
\(66\) 0 0
\(67\) −2.71969 4.71065i −0.332264 0.575497i 0.650692 0.759342i \(-0.274479\pi\)
−0.982955 + 0.183845i \(0.941146\pi\)
\(68\) 0 0
\(69\) 2.68526 + 15.2986i 0.323267 + 1.84174i
\(70\) 0 0
\(71\) −5.81144 −0.689691 −0.344845 0.938660i \(-0.612069\pi\)
−0.344845 + 0.938660i \(0.612069\pi\)
\(72\) 0 0
\(73\) −10.7167 −1.25429 −0.627146 0.778902i \(-0.715777\pi\)
−0.627146 + 0.778902i \(0.715777\pi\)
\(74\) 0 0
\(75\) 5.38582 4.51209i 0.621901 0.521011i
\(76\) 0 0
\(77\) −2.74254 4.75022i −0.312542 0.541338i
\(78\) 0 0
\(79\) 1.37055 2.37386i 0.154199 0.267080i −0.778568 0.627560i \(-0.784054\pi\)
0.932767 + 0.360480i \(0.117387\pi\)
\(80\) 0 0
\(81\) 1.53517 + 8.86810i 0.170574 + 0.985345i
\(82\) 0 0
\(83\) −6.35164 + 11.0014i −0.697183 + 1.20756i 0.272256 + 0.962225i \(0.412230\pi\)
−0.969439 + 0.245332i \(0.921103\pi\)
\(84\) 0 0
\(85\) −6.90695 11.9632i −0.749164 1.29759i
\(86\) 0 0
\(87\) 2.58754 2.16776i 0.277413 0.232409i
\(88\) 0 0
\(89\) −8.74898 −0.927390 −0.463695 0.885995i \(-0.653477\pi\)
−0.463695 + 0.885995i \(0.653477\pi\)
\(90\) 0 0
\(91\) 0.843734 0.0884474
\(92\) 0 0
\(93\) −1.01134 5.76185i −0.104871 0.597476i
\(94\) 0 0
\(95\) 5.72093 + 9.90894i 0.586955 + 1.01664i
\(96\) 0 0
\(97\) 3.83699 6.64586i 0.389587 0.674785i −0.602807 0.797887i \(-0.705951\pi\)
0.992394 + 0.123102i \(0.0392843\pi\)
\(98\) 0 0
\(99\) −15.4716 + 5.60390i −1.55496 + 0.563213i
\(100\) 0 0
\(101\) −3.74033 + 6.47845i −0.372177 + 0.644630i −0.989900 0.141766i \(-0.954722\pi\)
0.617723 + 0.786396i \(0.288055\pi\)
\(102\) 0 0
\(103\) 2.55258 + 4.42120i 0.251513 + 0.435634i 0.963943 0.266110i \(-0.0857385\pi\)
−0.712429 + 0.701744i \(0.752405\pi\)
\(104\) 0 0
\(105\) 4.89670 + 1.78658i 0.477869 + 0.174352i
\(106\) 0 0
\(107\) −1.48256 −0.143325 −0.0716623 0.997429i \(-0.522830\pi\)
−0.0716623 + 0.997429i \(0.522830\pi\)
\(108\) 0 0
\(109\) 0.878554 0.0841502 0.0420751 0.999114i \(-0.486603\pi\)
0.0420751 + 0.999114i \(0.486603\pi\)
\(110\) 0 0
\(111\) 12.1880 + 4.44685i 1.15684 + 0.422076i
\(112\) 0 0
\(113\) 5.30474 + 9.18808i 0.499028 + 0.864342i 0.999999 0.00112222i \(-0.000357215\pi\)
−0.500972 + 0.865464i \(0.667024\pi\)
\(114\) 0 0
\(115\) 13.4937 23.3718i 1.25830 2.17943i
\(116\) 0 0
\(117\) 0.443427 2.49206i 0.0409949 0.230391i
\(118\) 0 0
\(119\) 2.29512 3.97527i 0.210393 0.364412i
\(120\) 0 0
\(121\) −9.54308 16.5291i −0.867552 1.50264i
\(122\) 0 0
\(123\) 2.21070 + 12.5950i 0.199332 + 1.13565i
\(124\) 0 0
\(125\) 2.83932 0.253956
\(126\) 0 0
\(127\) 9.55795 0.848131 0.424065 0.905632i \(-0.360603\pi\)
0.424065 + 0.905632i \(0.360603\pi\)
\(128\) 0 0
\(129\) −8.26704 + 6.92589i −0.727872 + 0.609790i
\(130\) 0 0
\(131\) −0.785008 1.35967i −0.0685864 0.118795i 0.829693 0.558220i \(-0.188516\pi\)
−0.898279 + 0.439425i \(0.855182\pi\)
\(132\) 0 0
\(133\) −1.90102 + 3.29266i −0.164839 + 0.285510i
\(134\) 0 0
\(135\) 7.85033 13.5240i 0.675649 1.16396i
\(136\) 0 0
\(137\) −1.15500 + 2.00053i −0.0986787 + 0.170916i −0.911138 0.412101i \(-0.864795\pi\)
0.812459 + 0.583018i \(0.198128\pi\)
\(138\) 0 0
\(139\) 9.11779 + 15.7925i 0.773361 + 1.33950i 0.935711 + 0.352768i \(0.114759\pi\)
−0.162350 + 0.986733i \(0.551907\pi\)
\(140\) 0 0
\(141\) 17.4463 14.6160i 1.46924 1.23089i
\(142\) 0 0
\(143\) 4.62795 0.387009
\(144\) 0 0
\(145\) −5.86501 −0.487063
\(146\) 0 0
\(147\) 0.299437 + 1.70597i 0.0246971 + 0.140706i
\(148\) 0 0
\(149\) 1.64279 + 2.84540i 0.134583 + 0.233105i 0.925438 0.378899i \(-0.123697\pi\)
−0.790855 + 0.612003i \(0.790364\pi\)
\(150\) 0 0
\(151\) −6.12791 + 10.6138i −0.498682 + 0.863743i −0.999999 0.00152112i \(-0.999516\pi\)
0.501317 + 0.865264i \(0.332849\pi\)
\(152\) 0 0
\(153\) −10.5352 8.86810i −0.851718 0.716944i
\(154\) 0 0
\(155\) −5.08207 + 8.80241i −0.408202 + 0.707026i
\(156\) 0 0
\(157\) 1.58407 + 2.74369i 0.126422 + 0.218970i 0.922288 0.386503i \(-0.126317\pi\)
−0.795866 + 0.605473i \(0.792984\pi\)
\(158\) 0 0
\(159\) −13.1090 4.78288i −1.03961 0.379307i
\(160\) 0 0
\(161\) 8.96770 0.706754
\(162\) 0 0
\(163\) −8.06739 −0.631887 −0.315943 0.948778i \(-0.602321\pi\)
−0.315943 + 0.948778i \(0.602321\pi\)
\(164\) 0 0
\(165\) 26.8588 + 9.79955i 2.09096 + 0.762894i
\(166\) 0 0
\(167\) 0.162965 + 0.282264i 0.0126106 + 0.0218422i 0.872262 0.489039i \(-0.162652\pi\)
−0.859651 + 0.510881i \(0.829319\pi\)
\(168\) 0 0
\(169\) 6.14406 10.6418i 0.472620 0.818601i
\(170\) 0 0
\(171\) 8.72613 + 7.34532i 0.667304 + 0.561711i
\(172\) 0 0
\(173\) −4.06220 + 7.03593i −0.308843 + 0.534932i −0.978110 0.208091i \(-0.933275\pi\)
0.669267 + 0.743022i \(0.266608\pi\)
\(174\) 0 0
\(175\) −2.02826 3.51305i −0.153322 0.265561i
\(176\) 0 0
\(177\) −3.01854 17.1975i −0.226888 1.29264i
\(178\) 0 0
\(179\) 17.5470 1.31153 0.655763 0.754967i \(-0.272347\pi\)
0.655763 + 0.754967i \(0.272347\pi\)
\(180\) 0 0
\(181\) 14.7991 1.10001 0.550004 0.835162i \(-0.314626\pi\)
0.550004 + 0.835162i \(0.314626\pi\)
\(182\) 0 0
\(183\) 7.55255 6.32731i 0.558301 0.467728i
\(184\) 0 0
\(185\) −11.2710 19.5219i −0.828658 1.43528i
\(186\) 0 0
\(187\) 12.5889 21.8047i 0.920594 1.59452i
\(188\) 0 0
\(189\) 5.19614 + 0.0121603i 0.377963 + 0.000884530i
\(190\) 0 0
\(191\) 11.6859 20.2406i 0.845564 1.46456i −0.0395667 0.999217i \(-0.512598\pi\)
0.885131 0.465343i \(-0.154069\pi\)
\(192\) 0 0
\(193\) −8.80056 15.2430i −0.633478 1.09722i −0.986835 0.161728i \(-0.948293\pi\)
0.353357 0.935488i \(-0.385040\pi\)
\(194\) 0 0
\(195\) −3.37120 + 2.82430i −0.241417 + 0.202252i
\(196\) 0 0
\(197\) 9.54413 0.679991 0.339995 0.940427i \(-0.389574\pi\)
0.339995 + 0.940427i \(0.389574\pi\)
\(198\) 0 0
\(199\) −8.05652 −0.571111 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(200\) 0 0
\(201\) 1.62875 + 9.27944i 0.114883 + 0.654521i
\(202\) 0 0
\(203\) −0.974447 1.68779i −0.0683928 0.118460i
\(204\) 0 0
\(205\) 11.1090 19.2414i 0.775888 1.34388i
\(206\) 0 0
\(207\) 4.71301 26.4871i 0.327577 1.84098i
\(208\) 0 0
\(209\) −10.4272 + 18.0605i −0.721267 + 1.24927i
\(210\) 0 0
\(211\) −9.08203 15.7305i −0.625233 1.08293i −0.988496 0.151248i \(-0.951671\pi\)
0.363263 0.931687i \(-0.381663\pi\)
\(212\) 0 0
\(213\) 9.45598 + 3.45005i 0.647913 + 0.236394i
\(214\) 0 0
\(215\) 18.7384 1.27795
\(216\) 0 0
\(217\) −3.37746 −0.229277
\(218\) 0 0
\(219\) 17.4375 + 6.36213i 1.17831 + 0.429913i
\(220\) 0 0
\(221\) 1.93647 + 3.35407i 0.130261 + 0.225619i
\(222\) 0 0
\(223\) 3.47051 6.01110i 0.232402 0.402533i −0.726112 0.687576i \(-0.758675\pi\)
0.958515 + 0.285043i \(0.0920080\pi\)
\(224\) 0 0
\(225\) −11.4421 + 4.14439i −0.762808 + 0.276292i
\(226\) 0 0
\(227\) 5.01660 8.68901i 0.332964 0.576710i −0.650128 0.759825i \(-0.725285\pi\)
0.983092 + 0.183115i \(0.0586180\pi\)
\(228\) 0 0
\(229\) 3.24325 + 5.61748i 0.214320 + 0.371213i 0.953062 0.302775i \(-0.0979131\pi\)
−0.738742 + 0.673988i \(0.764580\pi\)
\(230\) 0 0
\(231\) 1.64244 + 9.35740i 0.108064 + 0.615671i
\(232\) 0 0
\(233\) 14.9215 0.977538 0.488769 0.872413i \(-0.337446\pi\)
0.488769 + 0.872413i \(0.337446\pi\)
\(234\) 0 0
\(235\) −39.5444 −2.57959
\(236\) 0 0
\(237\) −3.63934 + 3.04894i −0.236401 + 0.198050i
\(238\) 0 0
\(239\) −4.76389 8.25130i −0.308150 0.533732i 0.669807 0.742535i \(-0.266377\pi\)
−0.977958 + 0.208803i \(0.933043\pi\)
\(240\) 0 0
\(241\) −13.0471 + 22.5982i −0.840436 + 1.45568i 0.0490912 + 0.998794i \(0.484367\pi\)
−0.889527 + 0.456883i \(0.848966\pi\)
\(242\) 0 0
\(243\) 2.76676 15.3410i 0.177488 0.984123i
\(244\) 0 0
\(245\) 1.50470 2.60622i 0.0961319 0.166505i
\(246\) 0 0
\(247\) −1.60395 2.77813i −0.102057 0.176768i
\(248\) 0 0
\(249\) 16.8661 14.1299i 1.06885 0.895448i
\(250\) 0 0
\(251\) −26.0711 −1.64560 −0.822798 0.568334i \(-0.807588\pi\)
−0.822798 + 0.568334i \(0.807588\pi\)
\(252\) 0 0
\(253\) 49.1886 3.09246
\(254\) 0 0
\(255\) 4.13639 + 23.5661i 0.259031 + 1.47577i
\(256\) 0 0
\(257\) 2.43398 + 4.21577i 0.151827 + 0.262973i 0.931899 0.362717i \(-0.118151\pi\)
−0.780072 + 0.625690i \(0.784818\pi\)
\(258\) 0 0
\(259\) 3.74525 6.48696i 0.232718 0.403080i
\(260\) 0 0
\(261\) −5.49720 + 1.99111i −0.340268 + 0.123247i
\(262\) 0 0
\(263\) −15.6106 + 27.0383i −0.962588 + 1.66725i −0.246627 + 0.969110i \(0.579322\pi\)
−0.715961 + 0.698141i \(0.754011\pi\)
\(264\) 0 0
\(265\) 12.1227 + 20.9971i 0.744689 + 1.28984i
\(266\) 0 0
\(267\) 14.2357 + 5.19397i 0.871213 + 0.317866i
\(268\) 0 0
\(269\) 11.3736 0.693459 0.346730 0.937965i \(-0.387292\pi\)
0.346730 + 0.937965i \(0.387292\pi\)
\(270\) 0 0
\(271\) 2.74404 0.166689 0.0833444 0.996521i \(-0.473440\pi\)
0.0833444 + 0.996521i \(0.473440\pi\)
\(272\) 0 0
\(273\) −1.37287 0.500896i −0.0830897 0.0303156i
\(274\) 0 0
\(275\) −11.1252 19.2694i −0.670873 1.16199i
\(276\) 0 0
\(277\) −16.1725 + 28.0116i −0.971710 + 1.68305i −0.281321 + 0.959614i \(0.590773\pi\)
−0.690389 + 0.723438i \(0.742561\pi\)
\(278\) 0 0
\(279\) −1.77503 + 9.97569i −0.106269 + 0.597229i
\(280\) 0 0
\(281\) 3.74131 6.48014i 0.223188 0.386573i −0.732586 0.680674i \(-0.761687\pi\)
0.955774 + 0.294101i \(0.0950203\pi\)
\(282\) 0 0
\(283\) 5.02902 + 8.71052i 0.298944 + 0.517787i 0.975895 0.218242i \(-0.0700322\pi\)
−0.676950 + 0.736029i \(0.736699\pi\)
\(284\) 0 0
\(285\) −3.42611 19.5195i −0.202945 1.15623i
\(286\) 0 0
\(287\) 7.38287 0.435797
\(288\) 0 0
\(289\) 4.07034 0.239432
\(290\) 0 0
\(291\) −10.1887 + 8.53581i −0.597273 + 0.500378i
\(292\) 0 0
\(293\) 12.7055 + 22.0066i 0.742263 + 1.28564i 0.951463 + 0.307764i \(0.0995806\pi\)
−0.209200 + 0.977873i \(0.567086\pi\)
\(294\) 0 0
\(295\) −15.1685 + 26.2726i −0.883145 + 1.52965i
\(296\) 0 0
\(297\) 28.5013 + 0.0667001i 1.65381 + 0.00387033i
\(298\) 0 0
\(299\) −3.78318 + 6.55266i −0.218787 + 0.378950i
\(300\) 0 0
\(301\) 3.11330 + 5.39240i 0.179448 + 0.310813i
\(302\) 0 0
\(303\) 9.93205 8.32079i 0.570582 0.478017i
\(304\) 0 0
\(305\) −17.1189 −0.980225
\(306\) 0 0
\(307\) 9.82625 0.560814 0.280407 0.959881i \(-0.409531\pi\)
0.280407 + 0.959881i \(0.409531\pi\)
\(308\) 0 0
\(309\) −1.52867 8.70925i −0.0869631 0.495452i
\(310\) 0 0
\(311\) 1.69543 + 2.93657i 0.0961389 + 0.166518i 0.910083 0.414425i \(-0.136017\pi\)
−0.813944 + 0.580943i \(0.802684\pi\)
\(312\) 0 0
\(313\) 11.6346 20.1517i 0.657627 1.13904i −0.323601 0.946193i \(-0.604894\pi\)
0.981228 0.192850i \(-0.0617730\pi\)
\(314\) 0 0
\(315\) −6.90695 5.81401i −0.389163 0.327582i
\(316\) 0 0
\(317\) −17.2509 + 29.8794i −0.968904 + 1.67819i −0.270167 + 0.962813i \(0.587079\pi\)
−0.698737 + 0.715378i \(0.746254\pi\)
\(318\) 0 0
\(319\) −5.34492 9.25768i −0.299258 0.518330i
\(320\) 0 0
\(321\) 2.41232 + 0.880146i 0.134643 + 0.0491249i
\(322\) 0 0
\(323\) −17.4523 −0.971070
\(324\) 0 0
\(325\) 3.42262 0.189853
\(326\) 0 0
\(327\) −1.42952 0.521567i −0.0790529 0.0288427i
\(328\) 0 0
\(329\) −6.57013 11.3798i −0.362223 0.627388i
\(330\) 0 0
\(331\) −7.99741 + 13.8519i −0.439577 + 0.761370i −0.997657 0.0684173i \(-0.978205\pi\)
0.558080 + 0.829787i \(0.311538\pi\)
\(332\) 0 0
\(333\) −17.1916 14.4712i −0.942094 0.793019i
\(334\) 0 0
\(335\) 8.18466 14.1763i 0.447176 0.774531i
\(336\) 0 0
\(337\) −2.83257 4.90616i −0.154300 0.267256i 0.778504 0.627640i \(-0.215979\pi\)
−0.932804 + 0.360384i \(0.882646\pi\)
\(338\) 0 0
\(339\) −3.17687 18.0995i −0.172544 0.983028i
\(340\) 0 0
\(341\) −18.5257 −1.00322
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −35.8311 + 30.0183i −1.92908 + 1.61613i
\(346\) 0 0
\(347\) −14.2090 24.6107i −0.762780 1.32117i −0.941412 0.337257i \(-0.890501\pi\)
0.178633 0.983916i \(-0.442833\pi\)
\(348\) 0 0
\(349\) −8.48661 + 14.6992i −0.454278 + 0.786832i −0.998646 0.0520138i \(-0.983436\pi\)
0.544368 + 0.838846i \(0.316769\pi\)
\(350\) 0 0
\(351\) −2.20096 + 3.79166i −0.117479 + 0.202384i
\(352\) 0 0
\(353\) −8.83029 + 15.2945i −0.469989 + 0.814045i −0.999411 0.0343137i \(-0.989075\pi\)
0.529422 + 0.848359i \(0.322409\pi\)
\(354\) 0 0
\(355\) −8.74448 15.1459i −0.464109 0.803860i
\(356\) 0 0
\(357\) −6.09445 + 5.10575i −0.322552 + 0.270225i
\(358\) 0 0
\(359\) 13.7147 0.723834 0.361917 0.932210i \(-0.382122\pi\)
0.361917 + 0.932210i \(0.382122\pi\)
\(360\) 0 0
\(361\) −4.54455 −0.239187
\(362\) 0 0
\(363\) 5.71510 + 32.5604i 0.299965 + 1.70898i
\(364\) 0 0
\(365\) −16.1254 27.9300i −0.844043 1.46192i
\(366\) 0 0
\(367\) −7.33008 + 12.6961i −0.382627 + 0.662730i −0.991437 0.130587i \(-0.958314\pi\)
0.608810 + 0.793316i \(0.291647\pi\)
\(368\) 0 0
\(369\) 3.88009 21.8061i 0.201990 1.13518i
\(370\) 0 0
\(371\) −4.02826 + 6.97715i −0.209137 + 0.362236i
\(372\) 0 0
\(373\) 2.10543 + 3.64670i 0.109015 + 0.188819i 0.915371 0.402610i \(-0.131897\pi\)
−0.806357 + 0.591430i \(0.798564\pi\)
\(374\) 0 0
\(375\) −4.61995 1.68561i −0.238573 0.0870443i
\(376\) 0 0
\(377\) 1.64435 0.0846882
\(378\) 0 0
\(379\) 17.4637 0.897048 0.448524 0.893771i \(-0.351950\pi\)
0.448524 + 0.893771i \(0.351950\pi\)
\(380\) 0 0
\(381\) −15.5520 5.67423i −0.796756 0.290699i
\(382\) 0 0
\(383\) 12.8196 + 22.2042i 0.655050 + 1.13458i 0.981881 + 0.189497i \(0.0606858\pi\)
−0.326831 + 0.945083i \(0.605981\pi\)
\(384\) 0 0
\(385\) 8.25342 14.2953i 0.420633 0.728558i
\(386\) 0 0
\(387\) 17.5632 6.36148i 0.892789 0.323372i
\(388\) 0 0
\(389\) 3.73580 6.47059i 0.189413 0.328072i −0.755642 0.654985i \(-0.772675\pi\)
0.945055 + 0.326913i \(0.106008\pi\)
\(390\) 0 0
\(391\) 20.5820 + 35.6490i 1.04088 + 1.80285i
\(392\) 0 0
\(393\) 0.470120 + 2.67840i 0.0237144 + 0.135107i
\(394\) 0 0
\(395\) 8.24907 0.415056
\(396\) 0 0
\(397\) 2.65217 0.133109 0.0665544 0.997783i \(-0.478799\pi\)
0.0665544 + 0.997783i \(0.478799\pi\)
\(398\) 0 0
\(399\) 5.04794 4.22902i 0.252713 0.211716i
\(400\) 0 0
\(401\) 8.78586 + 15.2175i 0.438745 + 0.759928i 0.997593 0.0693419i \(-0.0220899\pi\)
−0.558848 + 0.829270i \(0.688757\pi\)
\(402\) 0 0
\(403\) 1.42484 2.46789i 0.0709763 0.122935i
\(404\) 0 0
\(405\) −20.8023 + 17.3448i −1.03367 + 0.861872i
\(406\) 0 0
\(407\) 20.5430 35.5815i 1.01828 1.76371i
\(408\) 0 0
\(409\) 5.22907 + 9.05702i 0.258561 + 0.447841i 0.965857 0.259077i \(-0.0834184\pi\)
−0.707296 + 0.706918i \(0.750085\pi\)
\(410\) 0 0
\(411\) 3.06699 2.56944i 0.151283 0.126741i
\(412\) 0 0
\(413\) −10.0807 −0.496041
\(414\) 0 0
\(415\) −38.2293 −1.87660
\(416\) 0 0
\(417\) −5.46041 31.1094i −0.267397 1.52343i
\(418\) 0 0
\(419\) −4.44830 7.70468i −0.217314 0.376398i 0.736672 0.676250i \(-0.236396\pi\)
−0.953986 + 0.299852i \(0.903063\pi\)
\(420\) 0 0
\(421\) −10.5280 + 18.2350i −0.513103 + 0.888721i 0.486781 + 0.873524i \(0.338171\pi\)
−0.999885 + 0.0151973i \(0.995162\pi\)
\(422\) 0 0
\(423\) −37.0644 + 13.4249i −1.80213 + 0.652740i
\(424\) 0 0
\(425\) 9.31020 16.1257i 0.451611 0.782214i
\(426\) 0 0
\(427\) −2.84423 4.92635i −0.137642 0.238403i
\(428\) 0 0
\(429\) −7.53029 2.74746i −0.363566 0.132648i
\(430\) 0 0
\(431\) 26.1233 1.25831 0.629157 0.777278i \(-0.283400\pi\)
0.629157 + 0.777278i \(0.283400\pi\)
\(432\) 0 0
\(433\) 30.3505 1.45855 0.729276 0.684219i \(-0.239857\pi\)
0.729276 + 0.684219i \(0.239857\pi\)
\(434\) 0 0
\(435\) 9.54315 + 3.48186i 0.457559 + 0.166942i
\(436\) 0 0
\(437\) −17.0477 29.5276i −0.815504 1.41250i
\(438\) 0 0
\(439\) 4.82759 8.36162i 0.230408 0.399079i −0.727520 0.686086i \(-0.759327\pi\)
0.957928 + 0.287008i \(0.0926606\pi\)
\(440\) 0 0
\(441\) 0.525553 2.95361i 0.0250263 0.140648i
\(442\) 0 0
\(443\) −5.35965 + 9.28318i −0.254644 + 0.441057i −0.964799 0.262989i \(-0.915292\pi\)
0.710154 + 0.704046i \(0.248625\pi\)
\(444\) 0 0
\(445\) −13.1646 22.8018i −0.624062 1.08091i
\(446\) 0 0
\(447\) −0.983826 5.60512i −0.0465334 0.265113i
\(448\) 0 0
\(449\) −14.1126 −0.666015 −0.333008 0.942924i \(-0.608064\pi\)
−0.333008 + 0.942924i \(0.608064\pi\)
\(450\) 0 0
\(451\) 40.4957 1.90687
\(452\) 0 0
\(453\) 16.2720 13.6322i 0.764525 0.640497i
\(454\) 0 0
\(455\) 1.26957 + 2.19896i 0.0595183 + 0.103089i
\(456\) 0 0
\(457\) 14.7102 25.4788i 0.688114 1.19185i −0.284334 0.958725i \(-0.591772\pi\)
0.972447 0.233123i \(-0.0748943\pi\)
\(458\) 0 0
\(459\) 11.8774 + 20.6840i 0.554391 + 0.965444i
\(460\) 0 0
\(461\) −15.5313 + 26.9009i −0.723363 + 1.25290i 0.236282 + 0.971685i \(0.424071\pi\)
−0.959644 + 0.281216i \(0.909262\pi\)
\(462\) 0 0
\(463\) −6.00276 10.3971i −0.278972 0.483194i 0.692158 0.721746i \(-0.256660\pi\)
−0.971130 + 0.238553i \(0.923327\pi\)
\(464\) 0 0
\(465\) 13.4949 11.3056i 0.625811 0.524286i
\(466\) 0 0
\(467\) 0.115560 0.00534750 0.00267375 0.999996i \(-0.499149\pi\)
0.00267375 + 0.999996i \(0.499149\pi\)
\(468\) 0 0
\(469\) 5.43939 0.251168
\(470\) 0 0
\(471\) −0.948657 5.40475i −0.0437118 0.249038i
\(472\) 0 0
\(473\) 17.0767 + 29.5778i 0.785189 + 1.35999i
\(474\) 0 0
\(475\) −7.71151 + 13.3567i −0.353828 + 0.612848i
\(476\) 0 0
\(477\) 18.4907 + 15.5648i 0.846631 + 0.712661i
\(478\) 0 0
\(479\) −7.92835 + 13.7323i −0.362256 + 0.627445i −0.988332 0.152317i \(-0.951327\pi\)
0.626076 + 0.779762i \(0.284660\pi\)
\(480\) 0 0
\(481\) 3.15999 + 5.47327i 0.144083 + 0.249560i
\(482\) 0 0
\(483\) −14.5916 5.32382i −0.663943 0.242242i
\(484\) 0 0
\(485\) 23.0941 1.04865
\(486\) 0 0
\(487\) 34.9185 1.58231 0.791154 0.611617i \(-0.209481\pi\)
0.791154 + 0.611617i \(0.209481\pi\)
\(488\) 0 0
\(489\) 13.1267 + 4.78933i 0.593611 + 0.216581i
\(490\) 0 0
\(491\) −11.7104 20.2831i −0.528484 0.915362i −0.999448 0.0332092i \(-0.989427\pi\)
0.470964 0.882152i \(-0.343906\pi\)
\(492\) 0 0
\(493\) 4.47295 7.74737i 0.201451 0.348924i
\(494\) 0 0
\(495\) −37.8852 31.8903i −1.70281 1.43336i
\(496\) 0 0
\(497\) 2.90572 5.03285i 0.130339 0.225754i
\(498\) 0 0
\(499\) 2.22661 + 3.85659i 0.0996765 + 0.172645i 0.911551 0.411188i \(-0.134886\pi\)
−0.811874 + 0.583832i \(0.801553\pi\)
\(500\) 0 0
\(501\) −0.0975954 0.556027i −0.00436024 0.0248415i
\(502\) 0 0
\(503\) 13.1429 0.586014 0.293007 0.956110i \(-0.405344\pi\)
0.293007 + 0.956110i \(0.405344\pi\)
\(504\) 0 0
\(505\) −22.5124 −1.00179
\(506\) 0 0
\(507\) −16.3149 + 13.6681i −0.724569 + 0.607023i
\(508\) 0 0
\(509\) −5.41360 9.37662i −0.239953 0.415612i 0.720747 0.693198i \(-0.243799\pi\)
−0.960701 + 0.277586i \(0.910466\pi\)
\(510\) 0 0
\(511\) 5.35834 9.28092i 0.237039 0.410564i
\(512\) 0 0
\(513\) −9.83790 17.1322i −0.434354 0.756406i
\(514\) 0 0
\(515\) −7.68175 + 13.3052i −0.338498 + 0.586296i
\(516\) 0 0
\(517\) −36.0377 62.4191i −1.58494 2.74519i
\(518\) 0 0
\(519\) 10.7867 9.03681i 0.473485 0.396672i
\(520\) 0 0
\(521\) −19.9378 −0.873489 −0.436745 0.899586i \(-0.643869\pi\)
−0.436745 + 0.899586i \(0.643869\pi\)
\(522\) 0 0
\(523\) −23.6481 −1.03406 −0.517029 0.855968i \(-0.672962\pi\)
−0.517029 + 0.855968i \(0.672962\pi\)
\(524\) 0 0
\(525\) 1.21467 + 6.92030i 0.0530126 + 0.302027i
\(526\) 0 0
\(527\) −7.75168 13.4263i −0.337669 0.584859i
\(528\) 0 0
\(529\) −28.7099 + 49.7269i −1.24825 + 2.16204i
\(530\) 0 0
\(531\) −5.29797 + 29.7745i −0.229912 + 1.29211i
\(532\) 0 0
\(533\) −3.11459 + 5.39463i −0.134908 + 0.233667i
\(534\) 0 0
\(535\) −2.23081 3.86388i −0.0964465 0.167050i
\(536\) 0 0
\(537\) −28.5513 10.4171i −1.23208 0.449529i
\(538\) 0 0
\(539\) 5.48508 0.236259
\(540\) 0 0
\(541\) 1.93883 0.0833567 0.0416783 0.999131i \(-0.486730\pi\)
0.0416783 + 0.999131i \(0.486730\pi\)
\(542\) 0 0
\(543\) −24.0801 8.78571i −1.03338 0.377031i
\(544\) 0 0
\(545\) 1.32196 + 2.28971i 0.0566266 + 0.0980802i
\(546\) 0 0
\(547\) −19.8625 + 34.4029i −0.849259 + 1.47096i 0.0326112 + 0.999468i \(0.489618\pi\)
−0.881870 + 0.471492i \(0.843716\pi\)
\(548\) 0 0
\(549\) −16.0453 + 5.81168i −0.684797 + 0.248037i
\(550\) 0 0
\(551\) −3.70488 + 6.41704i −0.157833 + 0.273375i
\(552\) 0 0
\(553\) 1.37055 + 2.37386i 0.0582817 + 0.100947i
\(554\) 0 0
\(555\) 6.74988 + 38.4559i 0.286517 + 1.63236i
\(556\) 0 0
\(557\) 24.8445 1.05269 0.526347 0.850270i \(-0.323561\pi\)
0.526347 + 0.850270i \(0.323561\pi\)
\(558\) 0 0
\(559\) −5.25360 −0.222204
\(560\) 0 0
\(561\) −33.4286 + 28.0055i −1.41136 + 1.18239i
\(562\) 0 0
\(563\) −19.7945 34.2851i −0.834240 1.44495i −0.894648 0.446772i \(-0.852574\pi\)
0.0604081 0.998174i \(-0.480760\pi\)
\(564\) 0 0
\(565\) −15.9641 + 27.6506i −0.671615 + 1.16327i
\(566\) 0 0
\(567\) −8.44759 3.10456i −0.354765 0.130379i
\(568\) 0 0
\(569\) 18.6846 32.3626i 0.783298 1.35671i −0.146713 0.989179i \(-0.546869\pi\)
0.930011 0.367532i \(-0.119797\pi\)
\(570\) 0 0
\(571\) −2.24275 3.88456i −0.0938563 0.162564i 0.815274 0.579075i \(-0.196586\pi\)
−0.909131 + 0.416511i \(0.863253\pi\)
\(572\) 0 0
\(573\) −31.0307 + 25.9966i −1.29633 + 1.08603i
\(574\) 0 0
\(575\) 36.3777 1.51705
\(576\) 0 0
\(577\) 29.6060 1.23251 0.616256 0.787546i \(-0.288649\pi\)
0.616256 + 0.787546i \(0.288649\pi\)
\(578\) 0 0
\(579\) 5.27042 + 30.0270i 0.219031 + 1.24788i
\(580\) 0 0
\(581\) −6.35164 11.0014i −0.263510 0.456413i
\(582\) 0 0
\(583\) −22.0953 + 38.2703i −0.915096 + 1.58499i
\(584\) 0 0
\(585\) 7.16208 2.59414i 0.296116 0.107254i
\(586\) 0 0
\(587\) 6.99150 12.1096i 0.288570 0.499818i −0.684899 0.728638i \(-0.740154\pi\)
0.973469 + 0.228820i \(0.0734869\pi\)
\(588\) 0 0
\(589\) 6.42061 + 11.1208i 0.264556 + 0.458225i
\(590\) 0 0
\(591\) −15.5296 5.66602i −0.638801 0.233069i
\(592\) 0 0
\(593\) 27.1799 1.11614 0.558072 0.829793i \(-0.311541\pi\)
0.558072 + 0.829793i \(0.311541\pi\)
\(594\) 0 0
\(595\) 13.8139 0.566315
\(596\) 0 0
\(597\) 13.1090 + 4.78288i 0.536517 + 0.195750i
\(598\) 0 0
\(599\) 15.4745 + 26.8026i 0.632270 + 1.09512i 0.987086 + 0.160188i \(0.0512101\pi\)
−0.354816 + 0.934936i \(0.615457\pi\)
\(600\) 0 0
\(601\) −7.69296 + 13.3246i −0.313803 + 0.543522i −0.979182 0.202983i \(-0.934936\pi\)
0.665380 + 0.746505i \(0.268270\pi\)
\(602\) 0 0
\(603\) 2.85869 16.0658i 0.116415 0.654251i
\(604\) 0 0
\(605\) 28.7190 49.7427i 1.16759 2.02233i
\(606\) 0 0
\(607\) −7.06075 12.2296i −0.286587 0.496383i 0.686406 0.727219i \(-0.259187\pi\)
−0.972993 + 0.230835i \(0.925854\pi\)
\(608\) 0 0
\(609\) 0.583570 + 3.32476i 0.0236475 + 0.134726i
\(610\) 0 0
\(611\) 11.0869 0.448527
\(612\) 0 0
\(613\) −28.9248 −1.16826 −0.584130 0.811660i \(-0.698564\pi\)
−0.584130 + 0.811660i \(0.698564\pi\)
\(614\) 0 0
\(615\) −29.4988 + 24.7133i −1.18951 + 0.996535i
\(616\) 0 0
\(617\) −12.2196 21.1649i −0.491941 0.852066i 0.508016 0.861347i \(-0.330379\pi\)
−0.999957 + 0.00928127i \(0.997046\pi\)
\(618\) 0 0
\(619\) 14.6690 25.4074i 0.589595 1.02121i −0.404690 0.914454i \(-0.632621\pi\)
0.994285 0.106755i \(-0.0340461\pi\)
\(620\) 0 0
\(621\) −23.3932 + 40.3000i −0.938735 + 1.61718i
\(622\) 0 0
\(623\) 4.37449 7.57684i 0.175260 0.303559i
\(624\) 0 0
\(625\) 14.4136 + 24.9651i 0.576545 + 0.998605i
\(626\) 0 0
\(627\) 27.6884 23.1965i 1.10577 0.926381i
\(628\) 0 0
\(629\) 34.3832 1.37095
\(630\) 0 0
\(631\) 12.9995 0.517501 0.258750 0.965944i \(-0.416689\pi\)
0.258750 + 0.965944i \(0.416689\pi\)
\(632\) 0 0
\(633\) 5.43899 + 30.9874i 0.216180 + 1.23164i
\(634\) 0 0
\(635\) 14.3819 + 24.9101i 0.570727 + 0.988528i
\(636\) 0 0
\(637\) −0.421867 + 0.730695i −0.0167150 + 0.0289512i
\(638\) 0 0
\(639\) −13.3380 11.2274i −0.527641 0.444148i
\(640\) 0 0
\(641\) −9.02698 + 15.6352i −0.356544 + 0.617553i −0.987381 0.158363i \(-0.949378\pi\)
0.630837 + 0.775916i \(0.282712\pi\)
\(642\) 0 0
\(643\) −1.88663 3.26773i −0.0744013 0.128867i 0.826424 0.563048i \(-0.190371\pi\)
−0.900826 + 0.434181i \(0.857038\pi\)
\(644\) 0 0
\(645\) −30.4898 11.1243i −1.20054 0.438020i
\(646\) 0 0
\(647\) 0.383345 0.0150709 0.00753543 0.999972i \(-0.497601\pi\)
0.00753543 + 0.999972i \(0.497601\pi\)
\(648\) 0 0
\(649\) −55.2937 −2.17047
\(650\) 0 0
\(651\) 5.49557 + 2.00508i 0.215389 + 0.0785854i
\(652\) 0 0
\(653\) −0.0983475 0.170343i −0.00384864 0.00666603i 0.864095 0.503329i \(-0.167892\pi\)
−0.867943 + 0.496663i \(0.834558\pi\)
\(654\) 0 0
\(655\) 2.36241 4.09181i 0.0923068 0.159880i
\(656\) 0 0
\(657\) −24.5961 20.7040i −0.959585 0.807742i
\(658\) 0 0
\(659\) −0.919372 + 1.59240i −0.0358137 + 0.0620311i −0.883377 0.468664i \(-0.844736\pi\)
0.847563 + 0.530695i \(0.178069\pi\)
\(660\) 0 0
\(661\) 13.5492 + 23.4679i 0.527004 + 0.912797i 0.999505 + 0.0314671i \(0.0100179\pi\)
−0.472501 + 0.881330i \(0.656649\pi\)
\(662\) 0 0
\(663\) −1.15970 6.60713i −0.0450391 0.256600i
\(664\) 0 0
\(665\) −11.4419 −0.443696
\(666\) 0 0
\(667\) 17.4771 0.676716
\(668\) 0 0
\(669\) −9.21556 + 7.72053i −0.356294 + 0.298493i
\(670\) 0 0
\(671\) −15.6009 27.0215i −0.602264 1.04315i
\(672\) 0 0
\(673\) 4.70611 8.15122i 0.181407 0.314207i −0.760953 0.648807i \(-0.775268\pi\)
0.942360 + 0.334601i \(0.108601\pi\)
\(674\) 0 0
\(675\) 21.0782 + 0.0493284i 0.811301 + 0.00189865i
\(676\) 0 0
\(677\) 8.05519 13.9520i 0.309586 0.536219i −0.668686 0.743545i \(-0.733143\pi\)
0.978272 + 0.207326i \(0.0664762\pi\)
\(678\) 0 0
\(679\) 3.83699 + 6.64586i 0.147250 + 0.255045i
\(680\) 0 0
\(681\) −13.3210 + 11.1600i −0.510464 + 0.427652i
\(682\) 0 0
\(683\) 7.35812 0.281551 0.140775 0.990042i \(-0.455040\pi\)
0.140775 + 0.990042i \(0.455040\pi\)
\(684\) 0 0
\(685\) −6.95175 −0.265613
\(686\) 0 0
\(687\) −1.94230 11.0658i −0.0741033 0.422186i
\(688\) 0 0
\(689\) −3.39878 5.88686i −0.129483 0.224271i
\(690\) 0 0
\(691\) −2.60616 + 4.51400i −0.0991430 + 0.171721i −0.911330 0.411676i \(-0.864943\pi\)
0.812187 + 0.583397i \(0.198277\pi\)
\(692\) 0 0
\(693\) 2.88270 16.2008i 0.109505 0.615417i
\(694\) 0 0
\(695\) −27.4391 + 47.5260i −1.04083 + 1.80276i
\(696\) 0 0
\(697\) 16.9446 + 29.3489i 0.641822 + 1.11167i
\(698\) 0 0
\(699\) −24.2792 8.85837i −0.918325 0.335054i
\(700\) 0 0
\(701\) −1.50384 −0.0567994 −0.0283997 0.999597i \(-0.509041\pi\)
−0.0283997 + 0.999597i \(0.509041\pi\)
\(702\) 0 0
\(703\) −28.4791 −1.07411
\(704\) 0 0
\(705\) 64.3439 + 23.4761i 2.42333 + 0.884162i
\(706\) 0 0
\(707\) −3.74033 6.47845i −0.140670 0.243647i
\(708\) 0 0
\(709\) 22.8248 39.5338i 0.857205 1.48472i −0.0173798 0.999849i \(-0.505532\pi\)
0.874585 0.484873i \(-0.161134\pi\)
\(710\) 0 0
\(711\) 7.73175 2.80047i 0.289963 0.105026i
\(712\) 0 0
\(713\) 15.1440 26.2302i 0.567148 0.982330i
\(714\) 0 0
\(715\) 6.96369 + 12.0615i 0.260427 + 0.451073i
\(716\) 0 0
\(717\) 2.85297 + 16.2541i 0.106546 + 0.607021i
\(718\) 0 0
\(719\) −18.1940 −0.678522 −0.339261 0.940692i \(-0.610177\pi\)
−0.339261 + 0.940692i \(0.610177\pi\)
\(720\) 0 0
\(721\) −5.10516 −0.190126
\(722\) 0 0
\(723\) 34.6451 29.0247i 1.28846 1.07944i
\(724\) 0 0
\(725\) −3.95286 6.84656i −0.146806 0.254275i
\(726\) 0 0
\(727\) 8.38218 14.5184i 0.310878 0.538456i −0.667675 0.744453i \(-0.732710\pi\)
0.978553 + 0.205997i \(0.0660436\pi\)
\(728\) 0 0
\(729\) −13.6093 + 23.3192i −0.504048 + 0.863676i
\(730\) 0 0
\(731\) −14.2908 + 24.7524i −0.528565 + 0.915501i
\(732\) 0 0
\(733\) 7.73528 + 13.3979i 0.285709 + 0.494862i 0.972781 0.231727i \(-0.0744376\pi\)
−0.687072 + 0.726589i \(0.741104\pi\)
\(734\) 0 0
\(735\) −3.99557 + 3.34738i −0.147379 + 0.123470i
\(736\) 0 0
\(737\) 29.8355 1.09901
\(738\) 0 0
\(739\) −7.01244 −0.257957 −0.128978 0.991647i \(-0.541170\pi\)
−0.128978 + 0.991647i \(0.541170\pi\)
\(740\) 0 0
\(741\) 0.960564 + 5.47259i 0.0352872 + 0.201041i
\(742\) 0 0
\(743\) 9.29512 + 16.0996i 0.341005 + 0.590638i 0.984620 0.174712i \(-0.0558994\pi\)
−0.643615 + 0.765350i \(0.722566\pi\)
\(744\) 0 0
\(745\) −4.94383 + 8.56297i −0.181128 + 0.313723i
\(746\) 0 0
\(747\) −35.8318 + 12.9784i −1.31102 + 0.474857i
\(748\) 0 0
\(749\) 0.741281 1.28394i 0.0270858 0.0469140i
\(750\) 0 0
\(751\) 27.1237 + 46.9797i 0.989759 + 1.71431i 0.618501 + 0.785784i \(0.287740\pi\)
0.371259 + 0.928529i \(0.378926\pi\)
\(752\) 0 0
\(753\) 42.4212 + 15.4775i 1.54591 + 0.564033i
\(754\) 0 0
\(755\) −36.8827 −1.34230
\(756\) 0 0
\(757\) −16.7679 −0.609439 −0.304719 0.952442i \(-0.598563\pi\)
−0.304719 + 0.952442i \(0.598563\pi\)
\(758\) 0 0
\(759\) −80.0364 29.2016i −2.90514 1.05995i
\(760\) 0 0
\(761\) −0.762796 1.32120i −0.0276514 0.0478935i 0.851869 0.523756i \(-0.175469\pi\)
−0.879520 + 0.475862i \(0.842136\pi\)
\(762\) 0 0
\(763\) −0.439277 + 0.760850i −0.0159029 + 0.0275446i
\(764\) 0 0
\(765\) 7.25994 40.8008i 0.262484 1.47516i
\(766\) 0 0
\(767\) 4.25273 7.36595i 0.153557 0.265969i
\(768\) 0 0
\(769\) 22.9904 + 39.8205i 0.829053 + 1.43596i 0.898782 + 0.438397i \(0.144453\pi\)
−0.0697283 + 0.997566i \(0.522213\pi\)
\(770\) 0 0
\(771\) −1.45764 8.30459i −0.0524958 0.299083i
\(772\) 0 0
\(773\) 16.2008 0.582701 0.291351 0.956616i \(-0.405895\pi\)
0.291351 + 0.956616i \(0.405895\pi\)
\(774\) 0 0
\(775\) −13.7007 −0.492145
\(776\) 0 0
\(777\) −9.94510 + 8.33172i −0.356779 + 0.298899i
\(778\) 0 0
\(779\) −14.0350 24.3093i −0.502855 0.870970i
\(780\) 0 0
\(781\) 15.9381 27.6056i 0.570311 0.987807i
\(782\) 0 0
\(783\) 10.1267 + 0.0236991i 0.361899 + 0.000846936i
\(784\) 0 0
\(785\) −4.76710 + 8.25686i −0.170145 + 0.294700i
\(786\) 0 0
\(787\) −10.3187 17.8726i −0.367823 0.637088i 0.621402 0.783492i \(-0.286563\pi\)
−0.989225 + 0.146404i \(0.953230\pi\)
\(788\) 0 0
\(789\) 41.4521 34.7274i 1.47574 1.23633i
\(790\) 0 0
\(791\) −10.6095 −0.377230
\(792\) 0 0
\(793\) 4.79955 0.170437
\(794\) 0 0
\(795\) −7.25994 41.3618i −0.257484 1.46695i
\(796\) 0 0
\(797\) −24.3590 42.1910i −0.862839 1.49448i −0.869177 0.494501i \(-0.835351\pi\)
0.00633764 0.999980i \(-0.497983\pi\)
\(798\) 0 0
\(799\) 30.1585 52.2360i 1.06693 1.84798i
\(800\) 0 0
\(801\) −20.0800 16.9025i −0.709491 0.597222i
\(802\) 0 0
\(803\) 29.3909 50.9066i 1.03718 1.79646i
\(804\) 0 0
\(805\) 13.4937 + 23.3718i 0.475591 + 0.823748i
\(806\) 0 0
\(807\) −18.5063 6.75210i −0.651453 0.237685i
\(808\) 0 0
\(809\) −0.903622 −0.0317696 −0.0158848 0.999874i \(-0.505057\pi\)
−0.0158848 + 0.999874i \(0.505057\pi\)
\(810\) 0 0
\(811\) 22.7487 0.798816 0.399408 0.916773i \(-0.369216\pi\)
0.399408 + 0.916773i \(0.369216\pi\)
\(812\) 0 0
\(813\) −4.46492 1.62905i −0.156592 0.0571331i
\(814\) 0 0
\(815\) −12.1390 21.0254i −0.425212 0.736488i
\(816\) 0 0
\(817\) 11.8369 20.5021i 0.414120 0.717277i
\(818\) 0 0
\(819\) 1.93647 + 1.63005i 0.0676658 + 0.0569585i
\(820\) 0 0
\(821\) −27.0367 + 46.8290i −0.943589 + 1.63434i −0.185036 + 0.982732i \(0.559240\pi\)
−0.758553 + 0.651612i \(0.774093\pi\)
\(822\) 0 0
\(823\) 9.27471 + 16.0643i 0.323296 + 0.559966i 0.981166 0.193166i \(-0.0618756\pi\)
−0.657870 + 0.753132i \(0.728542\pi\)
\(824\) 0 0
\(825\) 6.66257 + 37.9584i 0.231961 + 1.32154i
\(826\) 0 0
\(827\) −40.1856 −1.39739 −0.698695 0.715419i \(-0.746236\pi\)
−0.698695 + 0.715419i \(0.746236\pi\)
\(828\) 0 0
\(829\) −2.26946 −0.0788217 −0.0394108 0.999223i \(-0.512548\pi\)
−0.0394108 + 0.999223i \(0.512548\pi\)
\(830\) 0 0
\(831\) 42.9443 35.9775i 1.48972 1.24804i
\(832\) 0 0
\(833\) 2.29512 + 3.97527i 0.0795213 + 0.137735i
\(834\) 0 0
\(835\) −0.490428 + 0.849446i −0.0169720 + 0.0293963i
\(836\) 0 0
\(837\) 8.81044 15.1780i 0.304533 0.524628i
\(838\) 0 0
\(839\) 7.96058 13.7881i 0.274830 0.476019i −0.695262 0.718756i \(-0.744712\pi\)
0.970092 + 0.242737i \(0.0780452\pi\)
\(840\) 0 0
\(841\) 12.6009 + 21.8254i 0.434514 + 0.752600i
\(842\) 0 0
\(843\) −9.93464 + 8.32296i −0.342167 + 0.286658i
\(844\) 0 0
\(845\) 36.9799 1.27215
\(846\) 0 0
\(847\) 19.0862 0.655808
\(848\) 0 0
\(849\) −3.01175 17.1587i −0.103363 0.588886i
\(850\) 0 0
\(851\) 33.5863 + 58.1731i 1.15132 + 1.99415i
\(852\) 0 0
\(853\) 3.87812 6.71711i 0.132784 0.229989i −0.791964 0.610567i \(-0.790942\pi\)
0.924749 + 0.380578i \(0.124275\pi\)
\(854\) 0 0
\(855\) −6.01330 + 33.7947i −0.205651 + 1.15576i
\(856\) 0 0
\(857\) 3.17118 5.49264i 0.108325 0.187625i −0.806767 0.590870i \(-0.798785\pi\)
0.915092 + 0.403245i \(0.132118\pi\)
\(858\) 0 0
\(859\) 14.6252 + 25.3315i 0.499004 + 0.864300i 0.999999 0.00114976i \(-0.000365979\pi\)
−0.500995 + 0.865450i \(0.667033\pi\)
\(860\) 0 0
\(861\) −12.0129 4.38296i −0.409399 0.149371i
\(862\) 0 0
\(863\) −41.2373 −1.40374 −0.701868 0.712307i \(-0.747650\pi\)
−0.701868 + 0.712307i \(0.747650\pi\)
\(864\) 0 0
\(865\) −24.4496 −0.831311
\(866\) 0 0
\(867\) −6.62298 2.41642i −0.224928 0.0820660i
\(868\) 0 0
\(869\) 7.51758 + 13.0208i 0.255016 + 0.441701i
\(870\) 0 0
\(871\) −2.29470 + 3.97454i −0.0777529 + 0.134672i
\(872\) 0 0
\(873\) 21.6458 7.84021i 0.732600 0.265351i
\(874\) 0 0
\(875\) −1.41966 + 2.45892i −0.0479932 + 0.0831267i
\(876\) 0 0
\(877\) 18.7306 + 32.4424i 0.632488 + 1.09550i 0.987041 + 0.160466i \(0.0512997\pi\)
−0.354553 + 0.935036i \(0.615367\pi\)
\(878\) 0 0
\(879\) −7.60898 43.3504i −0.256645 1.46217i
\(880\) 0 0
\(881\) −42.2889 −1.42475 −0.712375 0.701799i \(-0.752380\pi\)
−0.712375 + 0.701799i \(0.752380\pi\)
\(882\) 0 0
\(883\) 51.1718 1.72207 0.861034 0.508547i \(-0.169817\pi\)
0.861034 + 0.508547i \(0.169817\pi\)
\(884\) 0 0
\(885\) 40.2784 33.7440i 1.35394 1.13429i
\(886\) 0 0
\(887\) 4.40377 + 7.62755i 0.147864 + 0.256108i 0.930438 0.366450i \(-0.119427\pi\)
−0.782574 + 0.622558i \(0.786094\pi\)
\(888\) 0 0
\(889\) −4.77897 + 8.27743i −0.160282 + 0.277616i
\(890\) 0 0
\(891\) −46.3357 17.0287i −1.55231 0.570485i
\(892\) 0 0
\(893\) −24.9798 + 43.2664i −0.835918 + 1.44785i
\(894\) 0 0
\(895\) 26.4030 + 45.7314i 0.882556 + 1.52863i
\(896\) 0 0
\(897\) 10.0458 8.41610i 0.335420 0.281005i
\(898\) 0 0
\(899\) −6.58231 −0.219532
\(900\) 0 0
\(901\) −36.9814 −1.23203
\(902\) 0 0
\(903\) −1.86447 10.6224i −0.0620458 0.353492i
\(904\) 0 0
\(905\) 22.2682 + 38.5697i 0.740221 + 1.28210i
\(906\) 0 0
\(907\) 8.25169 14.2923i 0.273993 0.474570i −0.695888 0.718151i \(-0.744989\pi\)
0.969881 + 0.243581i \(0.0783222\pi\)
\(908\) 0 0
\(909\) −21.1005 + 7.64271i −0.699861 + 0.253493i
\(910\) 0 0
\(911\) 11.4836 19.8902i 0.380469 0.658992i −0.610660 0.791893i \(-0.709096\pi\)
0.991129 + 0.132901i \(0.0424292\pi\)
\(912\) 0 0
\(913\) −34.8393 60.3434i −1.15301 1.99708i
\(914\) 0 0
\(915\) 27.8547 + 10.1629i 0.920848 + 0.335975i
\(916\) 0 0
\(917\) 1.57002 0.0518465
\(918\) 0 0
\(919\) 23.4537 0.773668 0.386834 0.922149i \(-0.373569\pi\)
0.386834 + 0.922149i \(0.373569\pi\)
\(920\) 0 0
\(921\) −15.9886 5.83351i −0.526843 0.192221i
\(922\) 0 0
\(923\) 2.45165 + 4.24639i 0.0806972 + 0.139772i
\(924\) 0 0
\(925\) 15.1927 26.3145i 0.499532 0.865215i
\(926\) 0 0
\(927\) −2.68303 + 15.0786i −0.0881224 + 0.495247i
\(928\) 0 0
\(929\) 18.8182 32.5940i 0.617404 1.06938i −0.372553 0.928011i \(-0.621518\pi\)
0.989958 0.141365i \(-0.0451491\pi\)
\(930\) 0 0
\(931\) −1.90102 3.29266i −0.0623033 0.107912i
\(932\) 0 0
\(933\) −1.01535 5.78471i −0.0332410 0.189383i
\(934\) 0 0
\(935\) 75.7704 2.47796
\(936\) 0 0
\(937\) 41.5719 1.35809 0.679047 0.734094i \(-0.262393\pi\)
0.679047 + 0.734094i \(0.262393\pi\)
\(938\) 0 0
\(939\) −30.8944 + 25.8825i −1.00820 + 0.844643i
\(940\) 0 0
\(941\) −13.5785 23.5187i −0.442647 0.766686i 0.555238 0.831691i \(-0.312627\pi\)
−0.997885 + 0.0650049i \(0.979294\pi\)
\(942\) 0 0
\(943\) −33.1037 + 57.3373i −1.07800 + 1.86716i
\(944\) 0 0
\(945\) 7.78695 + 13.5606i 0.253309 + 0.441126i
\(946\) 0 0
\(947\) 15.9444 27.6166i 0.518124 0.897418i −0.481654 0.876362i \(-0.659964\pi\)
0.999778 0.0210563i \(-0.00670293\pi\)
\(948\) 0 0
\(949\) 4.52101 + 7.83062i 0.146758 + 0.254193i
\(950\) 0 0
\(951\) 45.8078 38.3764i 1.48542 1.24444i
\(952\) 0 0
\(953\) 53.5094 1.73334 0.866670 0.498881i \(-0.166256\pi\)
0.866670 + 0.498881i \(0.166256\pi\)
\(954\) 0 0
\(955\) 70.3354 2.27600
\(956\) 0 0
\(957\) 3.20093 + 18.2366i 0.103471 + 0.589505i
\(958\) 0 0
\(959\) −1.15500 2.00053i −0.0372970 0.0646004i
\(960\) 0 0
\(961\) 9.79638 16.9678i 0.316012 0.547349i
\(962\) 0 0
\(963\) −3.40266 2.86423i −0.109649 0.0922985i
\(964\) 0 0
\(965\) 26.4844 45.8724i 0.852565 1.47669i
\(966\) 0 0
\(967\) −2.46065 4.26197i −0.0791292 0.137056i 0.823745 0.566960i \(-0.191881\pi\)
−0.902874 + 0.429904i \(0.858547\pi\)
\(968\) 0 0
\(969\) 28.3971 + 10.3608i 0.912248 + 0.332837i
\(970\) 0 0
\(971\) 2.43251 0.0780629 0.0390315 0.999238i \(-0.487573\pi\)
0.0390315 + 0.999238i \(0.487573\pi\)
\(972\) 0 0
\(973\) −18.2356 −0.584606
\(974\) 0 0
\(975\) −5.56906 2.03189i −0.178353 0.0650727i
\(976\) 0 0
\(977\) −3.15608 5.46650i −0.100972 0.174889i 0.811113 0.584889i \(-0.198862\pi\)
−0.912085 + 0.410000i \(0.865529\pi\)
\(978\) 0 0
\(979\) 23.9944 41.5596i 0.766865 1.32825i
\(980\) 0 0
\(981\) 2.01639 + 1.69732i 0.0643783 + 0.0541912i
\(982\) 0 0
\(983\) −12.9727 + 22.4694i −0.413766 + 0.716664i −0.995298 0.0968597i \(-0.969120\pi\)
0.581532 + 0.813524i \(0.302454\pi\)
\(984\) 0 0
\(985\) 14.3611 + 24.8741i 0.457582 + 0.792555i
\(986\) 0 0
\(987\) 3.93468 + 22.4169i 0.125242 + 0.713538i
\(988\) 0 0
\(989\) −55.8384 −1.77556
\(990\) 0 0
\(991\) −40.7413 −1.29419 −0.647094 0.762410i \(-0.724016\pi\)
−0.647094 + 0.762410i \(0.724016\pi\)
\(992\) 0 0
\(993\) 21.2363 17.7911i 0.673912 0.564584i
\(994\) 0 0
\(995\) −12.1227 20.9971i −0.384314 0.665652i
\(996\) 0 0
\(997\) −1.92260 + 3.33004i −0.0608894 + 0.105464i −0.894863 0.446340i \(-0.852727\pi\)
0.833974 + 0.551804i \(0.186060\pi\)
\(998\) 0 0
\(999\) 19.3819 + 33.7527i 0.613218 + 1.06789i
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 1008.2.r.n.673.1 10
3.2 odd 2 3024.2.r.n.2017.1 10
4.3 odd 2 504.2.r.f.169.5 10
9.2 odd 6 9072.2.a.cm.1.5 5
9.4 even 3 inner 1008.2.r.n.337.1 10
9.5 odd 6 3024.2.r.n.1009.1 10
9.7 even 3 9072.2.a.cn.1.1 5
12.11 even 2 1512.2.r.f.505.1 10
36.7 odd 6 4536.2.a.bd.1.1 5
36.11 even 6 4536.2.a.bc.1.5 5
36.23 even 6 1512.2.r.f.1009.1 10
36.31 odd 6 504.2.r.f.337.5 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.f.169.5 10 4.3 odd 2
504.2.r.f.337.5 yes 10 36.31 odd 6
1008.2.r.n.337.1 10 9.4 even 3 inner
1008.2.r.n.673.1 10 1.1 even 1 trivial
1512.2.r.f.505.1 10 12.11 even 2
1512.2.r.f.1009.1 10 36.23 even 6
3024.2.r.n.1009.1 10 9.5 odd 6
3024.2.r.n.2017.1 10 3.2 odd 2
4536.2.a.bc.1.5 5 36.11 even 6
4536.2.a.bd.1.1 5 36.7 odd 6
9072.2.a.cm.1.5 5 9.2 odd 6
9072.2.a.cn.1.1 5 9.7 even 3