Properties

Label 1024.2.g.g.897.4
Level $1024$
Weight $2$
Character 1024.897
Analytic conductor $8.177$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1024,2,Mod(129,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.g (of order \(8\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 897.4
Root \(0.991445 + 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 1024.897
Dual form 1024.2.g.g.129.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.580775 + 1.40211i) q^{3} +(3.29788 + 1.36603i) q^{5} +(1.02642 + 1.02642i) q^{7} +(0.492694 - 0.492694i) q^{9} +O(q^{10})\) \(q+(0.580775 + 1.40211i) q^{3} +(3.29788 + 1.36603i) q^{5} +(1.02642 + 1.02642i) q^{7} +(0.492694 - 0.492694i) q^{9} +(-2.05935 + 4.97171i) q^{11} +(-3.78024 + 1.56583i) q^{13} +5.41736i q^{15} -2.35311i q^{17} +(2.79772 - 1.15885i) q^{19} +(-0.843039 + 2.03528i) q^{21} +(-1.06460 + 1.06460i) q^{23} +(5.47443 + 5.47443i) q^{25} +(5.18330 + 2.14699i) q^{27} +(-1.60110 - 3.86540i) q^{29} -10.5829 q^{31} -8.16693 q^{33} +(1.98289 + 4.78712i) q^{35} +(4.22973 + 1.75201i) q^{37} +(-4.39094 - 4.39094i) q^{39} +(6.27792 - 6.27792i) q^{41} +(3.56308 - 8.60205i) q^{43} +(2.29788 - 0.951812i) q^{45} -3.06910i q^{47} -4.89293i q^{49} +(3.29934 - 1.36663i) q^{51} +(0.159176 - 0.384286i) q^{53} +(-13.5830 + 13.5830i) q^{55} +(3.24969 + 3.24969i) q^{57} +(7.78071 + 3.22287i) q^{59} +(-1.06885 - 2.58044i) q^{61} +1.01142 q^{63} -14.6057 q^{65} +(1.98559 + 4.79363i) q^{67} +(-2.11099 - 0.874400i) q^{69} +(2.84718 + 2.84718i) q^{71} +(-8.43123 + 8.43123i) q^{73} +(-4.49637 + 10.8552i) q^{75} +(-7.21682 + 2.98930i) q^{77} +4.59983i q^{79} +6.42418i q^{81} +(-7.60263 + 3.14911i) q^{83} +(3.21441 - 7.76028i) q^{85} +(4.48986 - 4.48986i) q^{87} +(0.967128 + 0.967128i) q^{89} +(-5.48730 - 2.27292i) q^{91} +(-6.14626 - 14.8384i) q^{93} +10.8096 q^{95} +11.2672 q^{97} +(1.43490 + 3.46416i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} + 16 q^{9} - 24 q^{13} + 48 q^{21} + 32 q^{25} + 8 q^{29} - 80 q^{33} - 8 q^{37} + 16 q^{41} - 8 q^{45} - 40 q^{53} + 16 q^{57} - 8 q^{61} - 32 q^{65} - 32 q^{73} - 32 q^{77} + 32 q^{85} - 32 q^{89} - 48 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\)

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
\(3\) 0.580775 + 1.40211i 0.335311 + 0.809511i 0.998153 + 0.0607523i \(0.0193500\pi\)
−0.662842 + 0.748759i \(0.730650\pi\)
\(4\) 0 0
\(5\) 3.29788 + 1.36603i 1.47486 + 0.610905i 0.967960 0.251103i \(-0.0807932\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(6\) 0 0
\(7\) 1.02642 + 1.02642i 0.387950 + 0.387950i 0.873956 0.486006i \(-0.161547\pi\)
−0.486006 + 0.873956i \(0.661547\pi\)
\(8\) 0 0
\(9\) 0.492694 0.492694i 0.164231 0.164231i
\(10\) 0 0
\(11\) −2.05935 + 4.97171i −0.620917 + 1.49903i 0.229710 + 0.973259i \(0.426222\pi\)
−0.850628 + 0.525768i \(0.823778\pi\)
\(12\) 0 0
\(13\) −3.78024 + 1.56583i −1.04845 + 0.434282i −0.839338 0.543610i \(-0.817057\pi\)
−0.209112 + 0.977892i \(0.567057\pi\)
\(14\) 0 0
\(15\) 5.41736i 1.39876i
\(16\) 0 0
\(17\) 2.35311i 0.570714i −0.958421 0.285357i \(-0.907888\pi\)
0.958421 0.285357i \(-0.0921121\pi\)
\(18\) 0 0
\(19\) 2.79772 1.15885i 0.641840 0.265859i −0.0379340 0.999280i \(-0.512078\pi\)
0.679774 + 0.733421i \(0.262078\pi\)
\(20\) 0 0
\(21\) −0.843039 + 2.03528i −0.183966 + 0.444134i
\(22\) 0 0
\(23\) −1.06460 + 1.06460i −0.221985 + 0.221985i −0.809334 0.587349i \(-0.800172\pi\)
0.587349 + 0.809334i \(0.300172\pi\)
\(24\) 0 0
\(25\) 5.47443 + 5.47443i 1.09489 + 1.09489i
\(26\) 0 0
\(27\) 5.18330 + 2.14699i 0.997527 + 0.413189i
\(28\) 0 0
\(29\) −1.60110 3.86540i −0.297317 0.717787i −0.999980 0.00626466i \(-0.998006\pi\)
0.702663 0.711523i \(-0.251994\pi\)
\(30\) 0 0
\(31\) −10.5829 −1.90074 −0.950370 0.311123i \(-0.899295\pi\)
−0.950370 + 0.311123i \(0.899295\pi\)
\(32\) 0 0
\(33\) −8.16693 −1.42168
\(34\) 0 0
\(35\) 1.98289 + 4.78712i 0.335170 + 0.809171i
\(36\) 0 0
\(37\) 4.22973 + 1.75201i 0.695363 + 0.288029i 0.702233 0.711948i \(-0.252187\pi\)
−0.00686951 + 0.999976i \(0.502187\pi\)
\(38\) 0 0
\(39\) −4.39094 4.39094i −0.703113 0.703113i
\(40\) 0 0
\(41\) 6.27792 6.27792i 0.980446 0.980446i −0.0193666 0.999812i \(-0.506165\pi\)
0.999812 + 0.0193666i \(0.00616495\pi\)
\(42\) 0 0
\(43\) 3.56308 8.60205i 0.543365 1.31180i −0.378970 0.925409i \(-0.623722\pi\)
0.922335 0.386391i \(-0.126278\pi\)
\(44\) 0 0
\(45\) 2.29788 0.951812i 0.342547 0.141888i
\(46\) 0 0
\(47\) 3.06910i 0.447674i −0.974627 0.223837i \(-0.928142\pi\)
0.974627 0.223837i \(-0.0718583\pi\)
\(48\) 0 0
\(49\) 4.89293i 0.698990i
\(50\) 0 0
\(51\) 3.29934 1.36663i 0.461999 0.191366i
\(52\) 0 0
\(53\) 0.159176 0.384286i 0.0218646 0.0527857i −0.912570 0.408920i \(-0.865905\pi\)
0.934435 + 0.356134i \(0.115905\pi\)
\(54\) 0 0
\(55\) −13.5830 + 13.5830i −1.83153 + 1.83153i
\(56\) 0 0
\(57\) 3.24969 + 3.24969i 0.430432 + 0.430432i
\(58\) 0 0
\(59\) 7.78071 + 3.22287i 1.01296 + 0.419583i 0.826534 0.562886i \(-0.190309\pi\)
0.186427 + 0.982469i \(0.440309\pi\)
\(60\) 0 0
\(61\) −1.06885 2.58044i −0.136853 0.330391i 0.840564 0.541712i \(-0.182224\pi\)
−0.977417 + 0.211321i \(0.932224\pi\)
\(62\) 0 0
\(63\) 1.01142 0.127427
\(64\) 0 0
\(65\) −14.6057 −1.81162
\(66\) 0 0
\(67\) 1.98559 + 4.79363i 0.242578 + 0.585635i 0.997537 0.0701364i \(-0.0223434\pi\)
−0.754959 + 0.655772i \(0.772343\pi\)
\(68\) 0 0
\(69\) −2.11099 0.874400i −0.254133 0.105265i
\(70\) 0 0
\(71\) 2.84718 + 2.84718i 0.337898 + 0.337898i 0.855576 0.517678i \(-0.173203\pi\)
−0.517678 + 0.855576i \(0.673203\pi\)
\(72\) 0 0
\(73\) −8.43123 + 8.43123i −0.986801 + 0.986801i −0.999914 0.0131133i \(-0.995826\pi\)
0.0131133 + 0.999914i \(0.495826\pi\)
\(74\) 0 0
\(75\) −4.49637 + 10.8552i −0.519196 + 1.25345i
\(76\) 0 0
\(77\) −7.21682 + 2.98930i −0.822433 + 0.340663i
\(78\) 0 0
\(79\) 4.59983i 0.517521i 0.965941 + 0.258761i \(0.0833141\pi\)
−0.965941 + 0.258761i \(0.916686\pi\)
\(80\) 0 0
\(81\) 6.42418i 0.713798i
\(82\) 0 0
\(83\) −7.60263 + 3.14911i −0.834497 + 0.345660i −0.758681 0.651462i \(-0.774156\pi\)
−0.0758155 + 0.997122i \(0.524156\pi\)
\(84\) 0 0
\(85\) 3.21441 7.76028i 0.348652 0.841720i
\(86\) 0 0
\(87\) 4.48986 4.48986i 0.481363 0.481363i
\(88\) 0 0
\(89\) 0.967128 + 0.967128i 0.102515 + 0.102515i 0.756504 0.653989i \(-0.226906\pi\)
−0.653989 + 0.756504i \(0.726906\pi\)
\(90\) 0 0
\(91\) −5.48730 2.27292i −0.575226 0.238266i
\(92\) 0 0
\(93\) −6.14626 14.8384i −0.637338 1.53867i
\(94\) 0 0
\(95\) 10.8096 1.10904
\(96\) 0 0
\(97\) 11.2672 1.14401 0.572006 0.820249i \(-0.306165\pi\)
0.572006 + 0.820249i \(0.306165\pi\)
\(98\) 0 0
\(99\) 1.43490 + 3.46416i 0.144213 + 0.348161i
\(100\) 0 0
\(101\) 8.65855 + 3.58649i 0.861558 + 0.356869i 0.769317 0.638868i \(-0.220597\pi\)
0.0922416 + 0.995737i \(0.470597\pi\)
\(102\) 0 0
\(103\) −1.20450 1.20450i −0.118683 0.118683i 0.645271 0.763954i \(-0.276744\pi\)
−0.763954 + 0.645271i \(0.776744\pi\)
\(104\) 0 0
\(105\) −5.56048 + 5.56048i −0.542647 + 0.542647i
\(106\) 0 0
\(107\) 1.11406 2.68959i 0.107701 0.260012i −0.860836 0.508882i \(-0.830059\pi\)
0.968537 + 0.248870i \(0.0800591\pi\)
\(108\) 0 0
\(109\) −3.53055 + 1.46240i −0.338165 + 0.140073i −0.545303 0.838239i \(-0.683585\pi\)
0.207137 + 0.978312i \(0.433585\pi\)
\(110\) 0 0
\(111\) 6.94809i 0.659483i
\(112\) 0 0
\(113\) 2.13630i 0.200966i 0.994939 + 0.100483i \(0.0320388\pi\)
−0.994939 + 0.100483i \(0.967961\pi\)
\(114\) 0 0
\(115\) −4.96520 + 2.05665i −0.463007 + 0.191784i
\(116\) 0 0
\(117\) −1.09103 + 2.63397i −0.100866 + 0.243511i
\(118\) 0 0
\(119\) 2.41528 2.41528i 0.221408 0.221408i
\(120\) 0 0
\(121\) −12.6988 12.6988i −1.15444 1.15444i
\(122\) 0 0
\(123\) 12.4484 + 5.15630i 1.12244 + 0.464928i
\(124\) 0 0
\(125\) 3.74567 + 9.04284i 0.335023 + 0.808816i
\(126\) 0 0
\(127\) −1.09821 −0.0974502 −0.0487251 0.998812i \(-0.515516\pi\)
−0.0487251 + 0.998812i \(0.515516\pi\)
\(128\) 0 0
\(129\) 14.1304 1.24411
\(130\) 0 0
\(131\) −0.271379 0.655166i −0.0237105 0.0572421i 0.911581 0.411121i \(-0.134863\pi\)
−0.935291 + 0.353879i \(0.884863\pi\)
\(132\) 0 0
\(133\) 4.06110 + 1.68216i 0.352142 + 0.145862i
\(134\) 0 0
\(135\) 14.1610 + 14.1610i 1.21879 + 1.21879i
\(136\) 0 0
\(137\) 11.4887 11.4887i 0.981544 0.981544i −0.0182885 0.999833i \(-0.505822\pi\)
0.999833 + 0.0182885i \(0.00582173\pi\)
\(138\) 0 0
\(139\) 0.323537 0.781087i 0.0274421 0.0662510i −0.909564 0.415563i \(-0.863585\pi\)
0.937007 + 0.349312i \(0.113585\pi\)
\(140\) 0 0
\(141\) 4.30323 1.78245i 0.362397 0.150110i
\(142\) 0 0
\(143\) 22.0188i 1.84131i
\(144\) 0 0
\(145\) 14.9348i 1.24027i
\(146\) 0 0
\(147\) 6.86045 2.84169i 0.565840 0.234379i
\(148\) 0 0
\(149\) −2.70212 + 6.52350i −0.221367 + 0.534426i −0.995076 0.0991144i \(-0.968399\pi\)
0.773710 + 0.633541i \(0.218399\pi\)
\(150\) 0 0
\(151\) 9.87443 9.87443i 0.803569 0.803569i −0.180082 0.983652i \(-0.557636\pi\)
0.983652 + 0.180082i \(0.0576364\pi\)
\(152\) 0 0
\(153\) −1.15937 1.15937i −0.0937291 0.0937291i
\(154\) 0 0
\(155\) −34.9010 14.4565i −2.80332 1.16117i
\(156\) 0 0
\(157\) 9.26500 + 22.3677i 0.739428 + 1.78514i 0.608199 + 0.793785i \(0.291892\pi\)
0.131229 + 0.991352i \(0.458108\pi\)
\(158\) 0 0
\(159\) 0.631258 0.0500620
\(160\) 0 0
\(161\) −2.18546 −0.172238
\(162\) 0 0
\(163\) −6.70092 16.1774i −0.524856 1.26712i −0.934855 0.355029i \(-0.884471\pi\)
0.409999 0.912086i \(-0.365529\pi\)
\(164\) 0 0
\(165\) −26.9335 11.1562i −2.09677 0.868511i
\(166\) 0 0
\(167\) −15.9204 15.9204i −1.23196 1.23196i −0.963210 0.268751i \(-0.913389\pi\)
−0.268751 0.963210i \(-0.586611\pi\)
\(168\) 0 0
\(169\) 2.64601 2.64601i 0.203539 0.203539i
\(170\) 0 0
\(171\) 0.807459 1.94938i 0.0617479 0.149073i
\(172\) 0 0
\(173\) 8.36603 3.46532i 0.636057 0.263463i −0.0412671 0.999148i \(-0.513139\pi\)
0.677324 + 0.735685i \(0.263139\pi\)
\(174\) 0 0
\(175\) 11.2381i 0.849523i
\(176\) 0 0
\(177\) 12.7812i 0.960695i
\(178\) 0 0
\(179\) 0.132433 0.0548557i 0.00989854 0.00410011i −0.377729 0.925916i \(-0.623295\pi\)
0.387627 + 0.921816i \(0.373295\pi\)
\(180\) 0 0
\(181\) −0.984684 + 2.37724i −0.0731910 + 0.176699i −0.956241 0.292580i \(-0.905486\pi\)
0.883050 + 0.469279i \(0.155486\pi\)
\(182\) 0 0
\(183\) 2.99731 2.99731i 0.221567 0.221567i
\(184\) 0 0
\(185\) 11.5558 + 11.5558i 0.849602 + 0.849602i
\(186\) 0 0
\(187\) 11.6990 + 4.84588i 0.855516 + 0.354366i
\(188\) 0 0
\(189\) 3.11653 + 7.52396i 0.226694 + 0.547287i
\(190\) 0 0
\(191\) −16.4603 −1.19103 −0.595514 0.803345i \(-0.703052\pi\)
−0.595514 + 0.803345i \(0.703052\pi\)
\(192\) 0 0
\(193\) 9.84445 0.708619 0.354309 0.935128i \(-0.384716\pi\)
0.354309 + 0.935128i \(0.384716\pi\)
\(194\) 0 0
\(195\) −8.48264 20.4789i −0.607454 1.46652i
\(196\) 0 0
\(197\) −9.44074 3.91048i −0.672625 0.278610i 0.0201152 0.999798i \(-0.493597\pi\)
−0.692740 + 0.721187i \(0.743597\pi\)
\(198\) 0 0
\(199\) −2.41726 2.41726i −0.171355 0.171355i 0.616220 0.787574i \(-0.288663\pi\)
−0.787574 + 0.616220i \(0.788663\pi\)
\(200\) 0 0
\(201\) −5.56804 + 5.56804i −0.392739 + 0.392739i
\(202\) 0 0
\(203\) 2.32412 5.61093i 0.163121 0.393810i
\(204\) 0 0
\(205\) 29.2796 12.1280i 2.04498 0.847057i
\(206\) 0 0
\(207\) 1.04905i 0.0729137i
\(208\) 0 0
\(209\) 16.2959i 1.12721i
\(210\) 0 0
\(211\) 14.6429 6.06529i 1.00806 0.417552i 0.183312 0.983055i \(-0.441318\pi\)
0.824746 + 0.565503i \(0.191318\pi\)
\(212\) 0 0
\(213\) −2.33850 + 5.64564i −0.160231 + 0.386833i
\(214\) 0 0
\(215\) 23.5012 23.5012i 1.60277 1.60277i
\(216\) 0 0
\(217\) −10.8625 10.8625i −0.737392 0.737392i
\(218\) 0 0
\(219\) −16.7182 6.92490i −1.12971 0.467942i
\(220\) 0 0
\(221\) 3.68457 + 8.89533i 0.247851 + 0.598365i
\(222\) 0 0
\(223\) 20.9031 1.39978 0.699888 0.714253i \(-0.253233\pi\)
0.699888 + 0.714253i \(0.253233\pi\)
\(224\) 0 0
\(225\) 5.39444 0.359629
\(226\) 0 0
\(227\) −5.18600 12.5201i −0.344207 0.830989i −0.997281 0.0736948i \(-0.976521\pi\)
0.653074 0.757294i \(-0.273479\pi\)
\(228\) 0 0
\(229\) −5.35773 2.21925i −0.354049 0.146652i 0.198568 0.980087i \(-0.436371\pi\)
−0.552617 + 0.833435i \(0.686371\pi\)
\(230\) 0 0
\(231\) −8.38269 8.38269i −0.551541 0.551541i
\(232\) 0 0
\(233\) 0.0268197 0.0268197i 0.00175702 0.00175702i −0.706228 0.707985i \(-0.749605\pi\)
0.707985 + 0.706228i \(0.249605\pi\)
\(234\) 0 0
\(235\) 4.19246 10.1215i 0.273486 0.660254i
\(236\) 0 0
\(237\) −6.44949 + 2.67147i −0.418939 + 0.173530i
\(238\) 0 0
\(239\) 25.6128i 1.65676i −0.560169 0.828378i \(-0.689264\pi\)
0.560169 0.828378i \(-0.310736\pi\)
\(240\) 0 0
\(241\) 1.28132i 0.0825369i 0.999148 + 0.0412684i \(0.0131399\pi\)
−0.999148 + 0.0412684i \(0.986860\pi\)
\(242\) 0 0
\(243\) 6.54247 2.70998i 0.419699 0.173845i
\(244\) 0 0
\(245\) 6.68386 16.1363i 0.427016 1.03091i
\(246\) 0 0
\(247\) −8.76148 + 8.76148i −0.557480 + 0.557480i
\(248\) 0 0
\(249\) −8.83083 8.83083i −0.559631 0.559631i
\(250\) 0 0
\(251\) −9.68691 4.01245i −0.611432 0.253263i 0.0554086 0.998464i \(-0.482354\pi\)
−0.666841 + 0.745200i \(0.732354\pi\)
\(252\) 0 0
\(253\) −3.10051 7.48528i −0.194927 0.470596i
\(254\) 0 0
\(255\) 12.7477 0.798289
\(256\) 0 0
\(257\) 8.01513 0.499970 0.249985 0.968250i \(-0.419574\pi\)
0.249985 + 0.968250i \(0.419574\pi\)
\(258\) 0 0
\(259\) 2.54318 + 6.13977i 0.158025 + 0.381507i
\(260\) 0 0
\(261\) −2.69331 1.11561i −0.166712 0.0690544i
\(262\) 0 0
\(263\) 13.8663 + 13.8663i 0.855030 + 0.855030i 0.990748 0.135718i \(-0.0433340\pi\)
−0.135718 + 0.990748i \(0.543334\pi\)
\(264\) 0 0
\(265\) 1.04989 1.04989i 0.0644941 0.0644941i
\(266\) 0 0
\(267\) −0.794341 + 1.91771i −0.0486129 + 0.117362i
\(268\) 0 0
\(269\) −15.8038 + 6.54616i −0.963576 + 0.399126i −0.808317 0.588748i \(-0.799621\pi\)
−0.155259 + 0.987874i \(0.549621\pi\)
\(270\) 0 0
\(271\) 15.6152i 0.948557i −0.880375 0.474279i \(-0.842709\pi\)
0.880375 0.474279i \(-0.157291\pi\)
\(272\) 0 0
\(273\) 9.01388i 0.545545i
\(274\) 0 0
\(275\) −38.4911 + 15.9435i −2.32110 + 0.961431i
\(276\) 0 0
\(277\) −5.32219 + 12.8489i −0.319779 + 0.772016i 0.679486 + 0.733689i \(0.262203\pi\)
−0.999265 + 0.0383271i \(0.987797\pi\)
\(278\) 0 0
\(279\) −5.21412 + 5.21412i −0.312161 + 0.312161i
\(280\) 0 0
\(281\) −18.7289 18.7289i −1.11727 1.11727i −0.992140 0.125134i \(-0.960064\pi\)
−0.125134 0.992140i \(-0.539936\pi\)
\(282\) 0 0
\(283\) 0.989981 + 0.410064i 0.0588483 + 0.0243758i 0.411913 0.911223i \(-0.364861\pi\)
−0.353065 + 0.935599i \(0.614861\pi\)
\(284\) 0 0
\(285\) 6.27792 + 15.1562i 0.371872 + 0.897778i
\(286\) 0 0
\(287\) 12.8875 0.760728
\(288\) 0 0
\(289\) 11.4629 0.674286
\(290\) 0 0
\(291\) 6.54372 + 15.7979i 0.383600 + 0.926091i
\(292\) 0 0
\(293\) 16.4429 + 6.81087i 0.960604 + 0.397895i 0.807206 0.590269i \(-0.200978\pi\)
0.153398 + 0.988165i \(0.450978\pi\)
\(294\) 0 0
\(295\) 21.2573 + 21.2573i 1.23765 + 1.23765i
\(296\) 0 0
\(297\) −21.3485 + 21.3485i −1.23876 + 1.23876i
\(298\) 0 0
\(299\) 2.35747 5.69143i 0.136336 0.329144i
\(300\) 0 0
\(301\) 12.4865 5.17209i 0.719711 0.298114i
\(302\) 0 0
\(303\) 14.2232i 0.817103i
\(304\) 0 0
\(305\) 9.97005i 0.570883i
\(306\) 0 0
\(307\) −19.0498 + 7.89068i −1.08723 + 0.450345i −0.853040 0.521846i \(-0.825243\pi\)
−0.234189 + 0.972191i \(0.575243\pi\)
\(308\) 0 0
\(309\) 0.989303 2.38839i 0.0562795 0.135871i
\(310\) 0 0
\(311\) 1.42704 1.42704i 0.0809203 0.0809203i −0.665488 0.746408i \(-0.731777\pi\)
0.746408 + 0.665488i \(0.231777\pi\)
\(312\) 0 0
\(313\) 1.27186 + 1.27186i 0.0718900 + 0.0718900i 0.742138 0.670248i \(-0.233812\pi\)
−0.670248 + 0.742138i \(0.733812\pi\)
\(314\) 0 0
\(315\) 3.33554 + 1.38163i 0.187937 + 0.0778459i
\(316\) 0 0
\(317\) −9.48990 22.9106i −0.533006 1.28679i −0.929523 0.368763i \(-0.879781\pi\)
0.396518 0.918027i \(-0.370219\pi\)
\(318\) 0 0
\(319\) 22.5149 1.26059
\(320\) 0 0
\(321\) 4.41813 0.246596
\(322\) 0 0
\(323\) −2.72691 6.58335i −0.151729 0.366307i
\(324\) 0 0
\(325\) −29.2667 12.1227i −1.62342 0.672444i
\(326\) 0 0
\(327\) −4.10091 4.10091i −0.226781 0.226781i
\(328\) 0 0
\(329\) 3.15018 3.15018i 0.173675 0.173675i
\(330\) 0 0
\(331\) −8.77257 + 21.1789i −0.482184 + 1.16410i 0.476386 + 0.879236i \(0.341947\pi\)
−0.958570 + 0.284859i \(0.908053\pi\)
\(332\) 0 0
\(333\) 2.94717 1.22076i 0.161504 0.0668971i
\(334\) 0 0
\(335\) 18.5212i 1.01192i
\(336\) 0 0
\(337\) 27.4961i 1.49781i 0.662677 + 0.748905i \(0.269420\pi\)
−0.662677 + 0.748905i \(0.730580\pi\)
\(338\) 0 0
\(339\) −2.99533 + 1.24071i −0.162684 + 0.0673860i
\(340\) 0 0
\(341\) 21.7938 52.6150i 1.18020 2.84926i
\(342\) 0 0
\(343\) 12.2071 12.2071i 0.659123 0.659123i
\(344\) 0 0
\(345\) −5.76733 5.76733i −0.310502 0.310502i
\(346\) 0 0
\(347\) 4.53353 + 1.87785i 0.243373 + 0.100808i 0.501036 0.865427i \(-0.332952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(348\) 0 0
\(349\) −8.28327 19.9976i −0.443393 1.07045i −0.974750 0.223298i \(-0.928318\pi\)
0.531357 0.847148i \(-0.321682\pi\)
\(350\) 0 0
\(351\) −22.9559 −1.22530
\(352\) 0 0
\(353\) −11.5498 −0.614733 −0.307366 0.951591i \(-0.599448\pi\)
−0.307366 + 0.951591i \(0.599448\pi\)
\(354\) 0 0
\(355\) 5.50033 + 13.2790i 0.291927 + 0.704774i
\(356\) 0 0
\(357\) 4.78924 + 1.98377i 0.253473 + 0.104992i
\(358\) 0 0
\(359\) −19.0741 19.0741i −1.00669 1.00669i −0.999977 0.00671431i \(-0.997863\pi\)
−0.00671431 0.999977i \(-0.502137\pi\)
\(360\) 0 0
\(361\) −6.95074 + 6.95074i −0.365829 + 0.365829i
\(362\) 0 0
\(363\) 10.4300 25.1803i 0.547435 1.32163i
\(364\) 0 0
\(365\) −39.3224 + 16.2879i −2.05823 + 0.852547i
\(366\) 0 0
\(367\) 12.4379i 0.649251i −0.945843 0.324626i \(-0.894762\pi\)
0.945843 0.324626i \(-0.105238\pi\)
\(368\) 0 0
\(369\) 6.18618i 0.322040i
\(370\) 0 0
\(371\) 0.557820 0.231057i 0.0289606 0.0119959i
\(372\) 0 0
\(373\) −12.8919 + 31.1238i −0.667518 + 1.61153i 0.118232 + 0.992986i \(0.462277\pi\)
−0.785750 + 0.618544i \(0.787723\pi\)
\(374\) 0 0
\(375\) −10.5037 + 10.5037i −0.542409 + 0.542409i
\(376\) 0 0
\(377\) 12.1051 + 12.1051i 0.623444 + 0.623444i
\(378\) 0 0
\(379\) 13.0991 + 5.42584i 0.672858 + 0.278707i 0.692838 0.721094i \(-0.256360\pi\)
−0.0199800 + 0.999800i \(0.506360\pi\)
\(380\) 0 0
\(381\) −0.637812 1.53981i −0.0326761 0.0788870i
\(382\) 0 0
\(383\) 7.14287 0.364984 0.182492 0.983207i \(-0.441584\pi\)
0.182492 + 0.983207i \(0.441584\pi\)
\(384\) 0 0
\(385\) −27.8836 −1.42108
\(386\) 0 0
\(387\) −2.48267 5.99369i −0.126201 0.304676i
\(388\) 0 0
\(389\) 26.2131 + 10.8578i 1.32906 + 0.550514i 0.930387 0.366579i \(-0.119471\pi\)
0.398672 + 0.917093i \(0.369471\pi\)
\(390\) 0 0
\(391\) 2.50513 + 2.50513i 0.126690 + 0.126690i
\(392\) 0 0
\(393\) 0.761008 0.761008i 0.0383878 0.0383878i
\(394\) 0 0
\(395\) −6.28348 + 15.1697i −0.316156 + 0.763269i
\(396\) 0 0
\(397\) 8.57919 3.55362i 0.430577 0.178351i −0.156860 0.987621i \(-0.550137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(398\) 0 0
\(399\) 6.67109i 0.333972i
\(400\) 0 0
\(401\) 9.43274i 0.471049i 0.971868 + 0.235524i \(0.0756807\pi\)
−0.971868 + 0.235524i \(0.924319\pi\)
\(402\) 0 0
\(403\) 40.0058 16.5709i 1.99283 0.825457i
\(404\) 0 0
\(405\) −8.77559 + 21.1862i −0.436063 + 1.05275i
\(406\) 0 0
\(407\) −17.4210 + 17.4210i −0.863526 + 0.863526i
\(408\) 0 0
\(409\) 15.4495 + 15.4495i 0.763928 + 0.763928i 0.977030 0.213102i \(-0.0683566\pi\)
−0.213102 + 0.977030i \(0.568357\pi\)
\(410\) 0 0
\(411\) 22.7808 + 9.43611i 1.12369 + 0.465449i
\(412\) 0 0
\(413\) 4.67825 + 11.2943i 0.230202 + 0.555756i
\(414\) 0 0
\(415\) −29.3743 −1.44193
\(416\) 0 0
\(417\) 1.28308 0.0628325
\(418\) 0 0
\(419\) −8.18452 19.7592i −0.399840 0.965299i −0.987703 0.156339i \(-0.950031\pi\)
0.587864 0.808960i \(-0.299969\pi\)
\(420\) 0 0
\(421\) −24.5052 10.1504i −1.19431 0.494700i −0.305155 0.952303i \(-0.598708\pi\)
−0.889157 + 0.457602i \(0.848708\pi\)
\(422\) 0 0
\(423\) −1.51213 1.51213i −0.0735221 0.0735221i
\(424\) 0 0
\(425\) 12.8820 12.8820i 0.624867 0.624867i
\(426\) 0 0
\(427\) 1.55152 3.74570i 0.0750833 0.181267i
\(428\) 0 0
\(429\) 30.8729 12.7880i 1.49056 0.617410i
\(430\) 0 0
\(431\) 24.7162i 1.19054i −0.803528 0.595268i \(-0.797046\pi\)
0.803528 0.595268i \(-0.202954\pi\)
\(432\) 0 0
\(433\) 9.69501i 0.465913i −0.972487 0.232956i \(-0.925160\pi\)
0.972487 0.232956i \(-0.0748399\pi\)
\(434\) 0 0
\(435\) 20.9403 8.67374i 1.00401 0.415874i
\(436\) 0 0
\(437\) −1.74474 + 4.21217i −0.0834622 + 0.201496i
\(438\) 0 0
\(439\) −8.45429 + 8.45429i −0.403501 + 0.403501i −0.879465 0.475964i \(-0.842099\pi\)
0.475964 + 0.879465i \(0.342099\pi\)
\(440\) 0 0
\(441\) −2.41072 2.41072i −0.114796 0.114796i
\(442\) 0 0
\(443\) −35.0500 14.5182i −1.66528 0.689780i −0.666814 0.745224i \(-0.732343\pi\)
−0.998462 + 0.0554440i \(0.982343\pi\)
\(444\) 0 0
\(445\) 1.86835 + 4.51059i 0.0885682 + 0.213822i
\(446\) 0 0
\(447\) −10.7160 −0.506851
\(448\) 0 0
\(449\) −22.3365 −1.05413 −0.527063 0.849826i \(-0.676707\pi\)
−0.527063 + 0.849826i \(0.676707\pi\)
\(450\) 0 0
\(451\) 18.2836 + 44.1404i 0.860939 + 2.07849i
\(452\) 0 0
\(453\) 19.5799 + 8.11026i 0.919944 + 0.381053i
\(454\) 0 0
\(455\) −14.9916 14.9916i −0.702817 0.702817i
\(456\) 0 0
\(457\) 22.4049 22.4049i 1.04806 1.04806i 0.0492728 0.998785i \(-0.484310\pi\)
0.998785 0.0492728i \(-0.0156904\pi\)
\(458\) 0 0
\(459\) 5.05212 12.1969i 0.235813 0.569302i
\(460\) 0 0
\(461\) 2.92070 1.20979i 0.136031 0.0563457i −0.313630 0.949545i \(-0.601545\pi\)
0.449660 + 0.893200i \(0.351545\pi\)
\(462\) 0 0
\(463\) 26.3825i 1.22610i 0.790045 + 0.613048i \(0.210057\pi\)
−0.790045 + 0.613048i \(0.789943\pi\)
\(464\) 0 0
\(465\) 57.3312i 2.65867i
\(466\) 0 0
\(467\) −9.48183 + 3.92750i −0.438767 + 0.181743i −0.591121 0.806583i \(-0.701314\pi\)
0.152354 + 0.988326i \(0.451314\pi\)
\(468\) 0 0
\(469\) −2.88223 + 6.95832i −0.133089 + 0.321305i
\(470\) 0 0
\(471\) −25.9812 + 25.9812i −1.19715 + 1.19715i
\(472\) 0 0
\(473\) 35.4293 + 35.4293i 1.62904 + 1.62904i
\(474\) 0 0
\(475\) 21.6600 + 8.97186i 0.993828 + 0.411657i
\(476\) 0 0
\(477\) −0.110910 0.267761i −0.00507822 0.0122599i
\(478\) 0 0
\(479\) −26.4855 −1.21015 −0.605077 0.796167i \(-0.706858\pi\)
−0.605077 + 0.796167i \(0.706858\pi\)
\(480\) 0 0
\(481\) −18.7327 −0.854139
\(482\) 0 0
\(483\) −1.26926 3.06426i −0.0577532 0.139429i
\(484\) 0 0
\(485\) 37.1579 + 15.3913i 1.68725 + 0.698883i
\(486\) 0 0
\(487\) 5.38394 + 5.38394i 0.243970 + 0.243970i 0.818490 0.574521i \(-0.194811\pi\)
−0.574521 + 0.818490i \(0.694811\pi\)
\(488\) 0 0
\(489\) 18.7909 18.7909i 0.849754 0.849754i
\(490\) 0 0
\(491\) −6.62284 + 15.9889i −0.298884 + 0.721571i 0.701080 + 0.713083i \(0.252702\pi\)
−0.999964 + 0.00848784i \(0.997298\pi\)
\(492\) 0 0
\(493\) −9.09573 + 3.76758i −0.409651 + 0.169683i
\(494\) 0 0
\(495\) 13.3845i 0.601588i
\(496\) 0 0
\(497\) 5.84480i 0.262175i
\(498\) 0 0
\(499\) 4.07217 1.68675i 0.182295 0.0755091i −0.289669 0.957127i \(-0.593545\pi\)
0.471964 + 0.881618i \(0.343545\pi\)
\(500\) 0 0
\(501\) 13.0761 31.5685i 0.584197 1.41038i
\(502\) 0 0
\(503\) 22.1132 22.1132i 0.985978 0.985978i −0.0139248 0.999903i \(-0.504433\pi\)
0.999903 + 0.0139248i \(0.00443254\pi\)
\(504\) 0 0
\(505\) 23.6556 + 23.6556i 1.05266 + 1.05266i
\(506\) 0 0
\(507\) 5.24674 + 2.17327i 0.233016 + 0.0965183i
\(508\) 0 0
\(509\) 5.24094 + 12.6528i 0.232301 + 0.560823i 0.996447 0.0842193i \(-0.0268396\pi\)
−0.764147 + 0.645043i \(0.776840\pi\)
\(510\) 0 0
\(511\) −17.3080 −0.765659
\(512\) 0 0
\(513\) 16.9895 0.750103
\(514\) 0 0
\(515\) −2.32691 5.61767i −0.102536 0.247544i
\(516\) 0 0
\(517\) 15.2587 + 6.32034i 0.671075 + 0.277968i
\(518\) 0 0
\(519\) 9.71756 + 9.71756i 0.426553 + 0.426553i
\(520\) 0 0
\(521\) −7.89757 + 7.89757i −0.345999 + 0.345999i −0.858617 0.512618i \(-0.828676\pi\)
0.512618 + 0.858617i \(0.328676\pi\)
\(522\) 0 0
\(523\) −4.66429 + 11.2606i −0.203955 + 0.492391i −0.992450 0.122650i \(-0.960861\pi\)
0.788495 + 0.615041i \(0.210861\pi\)
\(524\) 0 0
\(525\) −15.7571 + 6.52682i −0.687698 + 0.284854i
\(526\) 0 0
\(527\) 24.9027i 1.08478i
\(528\) 0 0
\(529\) 20.7332i 0.901445i
\(530\) 0 0
\(531\) 5.42140 2.24562i 0.235269 0.0974515i
\(532\) 0 0
\(533\) −13.9019 + 33.5622i −0.602158 + 1.45374i
\(534\) 0 0
\(535\) 7.34809 7.34809i 0.317685 0.317685i
\(536\) 0 0
\(537\) 0.153828 + 0.153828i 0.00663817 + 0.00663817i
\(538\) 0 0
\(539\) 24.3262 + 10.0763i 1.04780 + 0.434015i
\(540\) 0 0
\(541\) −4.88194 11.7860i −0.209891 0.506722i 0.783515 0.621373i \(-0.213425\pi\)
−0.993406 + 0.114652i \(0.963425\pi\)
\(542\) 0 0
\(543\) −3.90504 −0.167581
\(544\) 0 0
\(545\) −13.6410 −0.584316
\(546\) 0 0
\(547\) 10.8254 + 26.1348i 0.462861 + 1.11745i 0.967217 + 0.253950i \(0.0817298\pi\)
−0.504356 + 0.863496i \(0.668270\pi\)
\(548\) 0 0
\(549\) −1.79798 0.744749i −0.0767361 0.0317851i
\(550\) 0 0
\(551\) −8.95887 8.95887i −0.381661 0.381661i
\(552\) 0 0
\(553\) −4.72135 + 4.72135i −0.200772 + 0.200772i
\(554\) 0 0
\(555\) −9.49127 + 22.9139i −0.402882 + 0.972643i
\(556\) 0 0
\(557\) −32.8089 + 13.5899i −1.39016 + 0.575822i −0.947178 0.320709i \(-0.896079\pi\)
−0.442980 + 0.896532i \(0.646079\pi\)
\(558\) 0 0
\(559\) 38.0970i 1.61133i
\(560\) 0 0
\(561\) 19.2177i 0.811372i
\(562\) 0 0
\(563\) 10.8336 4.48742i 0.456582 0.189122i −0.142526 0.989791i \(-0.545522\pi\)
0.599107 + 0.800669i \(0.295522\pi\)
\(564\) 0 0
\(565\) −2.91824 + 7.04524i −0.122771 + 0.296396i
\(566\) 0 0
\(567\) −6.59390 + 6.59390i −0.276918 + 0.276918i
\(568\) 0 0
\(569\) −18.7673 18.7673i −0.786767 0.786767i 0.194196 0.980963i \(-0.437790\pi\)
−0.980963 + 0.194196i \(0.937790\pi\)
\(570\) 0 0
\(571\) 12.1516 + 5.03334i 0.508527 + 0.210639i 0.622169 0.782883i \(-0.286252\pi\)
−0.113642 + 0.993522i \(0.536252\pi\)
\(572\) 0 0
\(573\) −9.55975 23.0793i −0.399364 0.964151i
\(574\) 0 0
\(575\) −11.6562 −0.486097
\(576\) 0 0
\(577\) 25.2275 1.05024 0.525118 0.851029i \(-0.324021\pi\)
0.525118 + 0.851029i \(0.324021\pi\)
\(578\) 0 0
\(579\) 5.71741 + 13.8030i 0.237607 + 0.573635i
\(580\) 0 0
\(581\) −11.0358 4.57117i −0.457842 0.189644i
\(582\) 0 0
\(583\) 1.58276 + 1.58276i 0.0655511 + 0.0655511i
\(584\) 0 0
\(585\) −7.19615 + 7.19615i −0.297524 + 0.297524i
\(586\) 0 0
\(587\) −6.83331 + 16.4971i −0.282041 + 0.680907i −0.999883 0.0152989i \(-0.995130\pi\)
0.717842 + 0.696206i \(0.245130\pi\)
\(588\) 0 0
\(589\) −29.6079 + 12.2640i −1.21997 + 0.505329i
\(590\) 0 0
\(591\) 15.5081i 0.637919i
\(592\) 0 0
\(593\) 41.3009i 1.69602i −0.529976 0.848012i \(-0.677799\pi\)
0.529976 0.848012i \(-0.322201\pi\)
\(594\) 0 0
\(595\) 11.2646 4.66596i 0.461805 0.191286i
\(596\) 0 0
\(597\) 1.98539 4.79315i 0.0812565 0.196171i
\(598\) 0 0
\(599\) −6.73920 + 6.73920i −0.275356 + 0.275356i −0.831252 0.555896i \(-0.812375\pi\)
0.555896 + 0.831252i \(0.312375\pi\)
\(600\) 0 0
\(601\) −23.2456 23.2456i −0.948206 0.948206i 0.0505170 0.998723i \(-0.483913\pi\)
−0.998723 + 0.0505170i \(0.983913\pi\)
\(602\) 0 0
\(603\) 3.34008 + 1.38351i 0.136019 + 0.0563407i
\(604\) 0 0
\(605\) −24.5322 59.2260i −0.997377 2.40788i
\(606\) 0 0
\(607\) 18.7402 0.760642 0.380321 0.924855i \(-0.375813\pi\)
0.380321 + 0.924855i \(0.375813\pi\)
\(608\) 0 0
\(609\) 9.21695 0.373490
\(610\) 0 0
\(611\) 4.80567 + 11.6019i 0.194417 + 0.469363i
\(612\) 0 0
\(613\) 34.4288 + 14.2609i 1.39057 + 0.575991i 0.947285 0.320391i \(-0.103814\pi\)
0.443281 + 0.896383i \(0.353814\pi\)
\(614\) 0 0
\(615\) 34.0097 + 34.0097i 1.37140 + 1.37140i
\(616\) 0 0
\(617\) 11.5470 11.5470i 0.464866 0.464866i −0.435381 0.900246i \(-0.643386\pi\)
0.900246 + 0.435381i \(0.143386\pi\)
\(618\) 0 0
\(619\) −16.5076 + 39.8529i −0.663498 + 1.60182i 0.128786 + 0.991672i \(0.458892\pi\)
−0.792284 + 0.610153i \(0.791108\pi\)
\(620\) 0 0
\(621\) −7.80385 + 3.23246i −0.313158 + 0.129714i
\(622\) 0 0
\(623\) 1.98536i 0.0795417i
\(624\) 0 0
\(625\) 3.77124i 0.150850i
\(626\) 0 0
\(627\) −22.8488 + 9.46427i −0.912492 + 0.377966i
\(628\) 0 0
\(629\) 4.12268 9.95303i 0.164382 0.396853i
\(630\) 0 0
\(631\) −4.48591 + 4.48591i −0.178581 + 0.178581i −0.790737 0.612156i \(-0.790302\pi\)
0.612156 + 0.790737i \(0.290302\pi\)
\(632\) 0 0
\(633\) 17.0085 + 17.0085i 0.676025 + 0.676025i
\(634\) 0 0
\(635\) −3.62175 1.50018i −0.143725 0.0595328i
\(636\) 0 0
\(637\) 7.66147 + 18.4964i 0.303559 + 0.732855i
\(638\) 0 0
\(639\) 2.80558 0.110987
\(640\) 0 0
\(641\) −39.5996 −1.56409 −0.782045 0.623221i \(-0.785824\pi\)
−0.782045 + 0.623221i \(0.785824\pi\)
\(642\) 0 0
\(643\) −9.36360 22.6057i −0.369264 0.891483i −0.993871 0.110544i \(-0.964741\pi\)
0.624607 0.780939i \(-0.285259\pi\)
\(644\) 0 0
\(645\) 46.6003 + 19.3025i 1.83489 + 0.760035i
\(646\) 0 0
\(647\) −25.4317 25.4317i −0.999823 0.999823i 0.000176614 1.00000i \(-0.499944\pi\)
−1.00000 0.000176614i \(0.999944\pi\)
\(648\) 0 0
\(649\) −32.0464 + 32.0464i −1.25793 + 1.25793i
\(650\) 0 0
\(651\) 8.92177 21.5391i 0.349672 0.844182i
\(652\) 0 0
\(653\) −22.6322 + 9.37458i −0.885668 + 0.366856i −0.778692 0.627406i \(-0.784117\pi\)
−0.106976 + 0.994262i \(0.534117\pi\)
\(654\) 0 0
\(655\) 2.53137i 0.0989087i
\(656\) 0 0
\(657\) 8.30803i 0.324127i
\(658\) 0 0
\(659\) −25.0100 + 10.3595i −0.974250 + 0.403547i −0.812292 0.583250i \(-0.801781\pi\)
−0.161957 + 0.986798i \(0.551781\pi\)
\(660\) 0 0
\(661\) 4.83467 11.6719i 0.188047 0.453985i −0.801537 0.597946i \(-0.795984\pi\)
0.989583 + 0.143961i \(0.0459838\pi\)
\(662\) 0 0
\(663\) −10.3324 + 10.3324i −0.401276 + 0.401276i
\(664\) 0 0
\(665\) 11.0951 + 11.0951i 0.430251 + 0.430251i
\(666\) 0 0
\(667\) 5.81965 + 2.41058i 0.225338 + 0.0933380i
\(668\) 0 0
\(669\) 12.1400 + 29.3085i 0.469359 + 1.13313i
\(670\) 0 0
\(671\) 15.0303 0.580240
\(672\) 0 0
\(673\) 9.29926 0.358460 0.179230 0.983807i \(-0.442639\pi\)
0.179230 + 0.983807i \(0.442639\pi\)
\(674\) 0 0
\(675\) 16.6221 + 40.1292i 0.639784 + 1.54457i
\(676\) 0 0
\(677\) −20.7549 8.59697i −0.797677 0.330409i −0.0536513 0.998560i \(-0.517086\pi\)
−0.744026 + 0.668151i \(0.767086\pi\)
\(678\) 0 0
\(679\) 11.5649 + 11.5649i 0.443820 + 0.443820i
\(680\) 0 0
\(681\) 14.5427 14.5427i 0.557279 0.557279i
\(682\) 0 0
\(683\) 2.13532 5.15512i 0.0817058 0.197255i −0.877747 0.479124i \(-0.840954\pi\)
0.959453 + 0.281869i \(0.0909543\pi\)
\(684\) 0 0
\(685\) 53.5821 22.1944i 2.04727 0.848005i
\(686\) 0 0
\(687\) 8.80104i 0.335781i
\(688\) 0 0
\(689\) 1.70193i 0.0648385i
\(690\) 0 0
\(691\) −22.9436 + 9.50357i −0.872817 + 0.361533i −0.773707 0.633544i \(-0.781600\pi\)
−0.0991103 + 0.995076i \(0.531600\pi\)
\(692\) 0 0
\(693\) −2.08287 + 5.02849i −0.0791217 + 0.191017i
\(694\) 0 0
\(695\) 2.13397 2.13397i 0.0809461 0.0809461i
\(696\) 0 0
\(697\) −14.7727 14.7727i −0.559554 0.559554i
\(698\) 0 0
\(699\) 0.0531806 + 0.0220281i 0.00201147 + 0.000833180i
\(700\) 0 0
\(701\) 7.60601 + 18.3625i 0.287275 + 0.693544i 0.999969 0.00793210i \(-0.00252489\pi\)
−0.712693 + 0.701476i \(0.752525\pi\)
\(702\) 0 0
\(703\) 13.8639 0.522887
\(704\) 0 0
\(705\) 16.6264 0.626186
\(706\) 0 0
\(707\) 5.20606 + 12.5685i 0.195794 + 0.472689i
\(708\) 0 0
\(709\) −24.3512 10.0866i −0.914529 0.378810i −0.124740 0.992189i \(-0.539810\pi\)
−0.789788 + 0.613379i \(0.789810\pi\)
\(710\) 0 0
\(711\) 2.26631 + 2.26631i 0.0849932 + 0.0849932i
\(712\) 0 0
\(713\) 11.2665 11.2665i 0.421935 0.421935i
\(714\) 0 0
\(715\) 30.0783 72.6154i 1.12486 2.71566i
\(716\) 0 0
\(717\) 35.9121 14.8753i 1.34116 0.555528i
\(718\) 0 0
\(719\) 34.6091i 1.29070i −0.763887 0.645350i \(-0.776711\pi\)
0.763887 0.645350i \(-0.223289\pi\)
\(720\) 0 0
\(721\) 2.47264i 0.0920860i
\(722\) 0 0
\(723\) −1.79655 + 0.744157i −0.0668145 + 0.0276755i
\(724\) 0 0
\(725\) 12.3958 29.9260i 0.460367 1.11142i
\(726\) 0 0
\(727\) −30.7749 + 30.7749i −1.14138 + 1.14138i −0.153181 + 0.988198i \(0.548952\pi\)
−0.988198 + 0.153181i \(0.951048\pi\)
\(728\) 0 0
\(729\) 21.2271 + 21.2271i 0.786191 + 0.786191i
\(730\) 0 0
\(731\) −20.2416 8.38434i −0.748662 0.310106i
\(732\) 0 0
\(733\) −17.4213 42.0588i −0.643472 1.55348i −0.821965 0.569537i \(-0.807122\pi\)
0.178494 0.983941i \(-0.442878\pi\)
\(734\) 0 0
\(735\) 26.5067 0.977715
\(736\) 0 0
\(737\) −27.9216 −1.02850
\(738\) 0 0
\(739\) −10.7099 25.8560i −0.393970 0.951128i −0.989066 0.147474i \(-0.952886\pi\)
0.595096 0.803655i \(-0.297114\pi\)
\(740\) 0 0
\(741\) −17.3730 7.19615i −0.638215 0.264357i
\(742\) 0 0
\(743\) 26.2978 + 26.2978i 0.964774 + 0.964774i 0.999400 0.0346265i \(-0.0110242\pi\)
−0.0346265 + 0.999400i \(0.511024\pi\)
\(744\) 0 0
\(745\) −17.8225 + 17.8225i −0.652967 + 0.652967i
\(746\) 0 0
\(747\) −2.19422 + 5.29732i −0.0802823 + 0.193819i
\(748\) 0 0
\(749\) 3.90414 1.61715i 0.142654 0.0590893i
\(750\) 0 0
\(751\) 37.4098i 1.36510i 0.730837 + 0.682552i \(0.239130\pi\)
−0.730837 + 0.682552i \(0.760870\pi\)
\(752\) 0 0
\(753\) 15.9125i 0.579883i
\(754\) 0 0
\(755\) 46.0534 19.0759i 1.67605 0.694244i
\(756\) 0 0
\(757\) 3.12025 7.53295i 0.113407 0.273790i −0.856977 0.515354i \(-0.827660\pi\)
0.970385 + 0.241564i \(0.0776604\pi\)
\(758\) 0 0
\(759\) 8.69453 8.69453i 0.315591 0.315591i
\(760\) 0 0
\(761\) −8.08516 8.08516i −0.293087 0.293087i 0.545212 0.838299i \(-0.316449\pi\)
−0.838299 + 0.545212i \(0.816449\pi\)
\(762\) 0 0
\(763\) −5.12486 2.12279i −0.185532 0.0768501i
\(764\) 0 0
\(765\) −2.23972 5.40717i −0.0809773 0.195496i
\(766\) 0 0
\(767\) −34.4594 −1.24426
\(768\) 0 0
\(769\) −30.6572 −1.10553 −0.552764 0.833338i \(-0.686427\pi\)
−0.552764 + 0.833338i \(0.686427\pi\)
\(770\) 0 0
\(771\) 4.65499 + 11.2381i 0.167645 + 0.404731i
\(772\) 0 0
\(773\) −29.4394 12.1942i −1.05886 0.438595i −0.215815 0.976434i \(-0.569241\pi\)
−0.843047 + 0.537839i \(0.819241\pi\)
\(774\) 0 0
\(775\) −57.9352 57.9352i −2.08109 2.08109i
\(776\) 0 0
\(777\) −7.13165 + 7.13165i −0.255847 + 0.255847i
\(778\) 0 0
\(779\) 10.2887 24.8390i 0.368629 0.889950i
\(780\) 0 0
\(781\) −20.0187 + 8.29201i −0.716325 + 0.296712i
\(782\) 0 0
\(783\) 23.4731i 0.838861i
\(784\) 0 0
\(785\) 86.4222i 3.08454i
\(786\) 0 0
\(787\) 0.505757 0.209491i 0.0180283 0.00746756i −0.373651 0.927569i \(-0.621894\pi\)
0.391679 + 0.920102i \(0.371894\pi\)
\(788\) 0 0
\(789\) −11.3889 + 27.4952i −0.405456 + 0.978857i
\(790\) 0 0
\(791\) −2.19274 + 2.19274i −0.0779647 + 0.0779647i
\(792\) 0 0
\(793\) 8.08104 + 8.08104i 0.286966 + 0.286966i
\(794\) 0 0
\(795\) 2.08181 + 0.862315i 0.0738343 + 0.0305832i
\(796\) 0 0
\(797\) 11.9978 + 28.9652i 0.424983 + 1.02600i 0.980856 + 0.194733i \(0.0623841\pi\)
−0.555873 + 0.831267i \(0.687616\pi\)
\(798\) 0 0
\(799\) −7.22193 −0.255494
\(800\) 0 0
\(801\) 0.952996 0.0336725
\(802\) 0 0
\(803\) −24.5548 59.2805i −0.866519 2.09196i
\(804\) 0 0
\(805\) −7.20736 2.98539i −0.254026 0.105221i
\(806\) 0 0
\(807\) −18.3569 18.3569i −0.646194 0.646194i
\(808\) 0 0
\(809\) −5.60821 + 5.60821i −0.197174 + 0.197174i −0.798787 0.601613i \(-0.794525\pi\)
0.601613 + 0.798787i \(0.294525\pi\)
\(810\) 0 0
\(811\) −15.7458 + 38.0138i −0.552910 + 1.33484i 0.362373 + 0.932033i \(0.381967\pi\)
−0.915283 + 0.402811i \(0.868033\pi\)
\(812\) 0 0
\(813\) 21.8943 9.06893i 0.767868 0.318061i
\(814\) 0 0
\(815\) 62.5049i 2.18945i
\(816\) 0 0
\(817\) 28.1952i 0.986425i
\(818\) 0 0
\(819\) −3.82341 + 1.58371i −0.133601 + 0.0553393i
\(820\) 0 0
\(821\) −10.2032 + 24.6328i −0.356095 + 0.859689i 0.639746 + 0.768586i \(0.279039\pi\)
−0.995841 + 0.0911034i \(0.970961\pi\)
\(822\) 0 0
\(823\) 26.4269 26.4269i 0.921185 0.921185i −0.0759281 0.997113i \(-0.524192\pi\)
0.997113 + 0.0759281i \(0.0241919\pi\)
\(824\) 0 0
\(825\) −44.7093 44.7093i −1.55658 1.55658i
\(826\) 0 0
\(827\) 4.59197 + 1.90206i 0.159678 + 0.0661410i 0.461091 0.887353i \(-0.347458\pi\)
−0.301413 + 0.953494i \(0.597458\pi\)
\(828\) 0 0
\(829\) −0.0153511 0.0370609i −0.000533166 0.00128718i 0.923613 0.383327i \(-0.125222\pi\)
−0.924146 + 0.382040i \(0.875222\pi\)
\(830\) 0 0
\(831\) −21.1066 −0.732181
\(832\) 0 0
\(833\) −11.5136 −0.398923
\(834\) 0 0
\(835\) −30.7559 74.2514i −1.06435 2.56957i
\(836\) 0 0
\(837\) −54.8542 22.7214i −1.89604 0.785365i
\(838\) 0 0
\(839\) 3.78014 + 3.78014i 0.130505 + 0.130505i 0.769342 0.638837i \(-0.220584\pi\)
−0.638837 + 0.769342i \(0.720584\pi\)
\(840\) 0 0
\(841\) 8.12828 8.12828i 0.280286 0.280286i
\(842\) 0 0
\(843\) 15.3828 37.1374i 0.529812 1.27908i
\(844\) 0 0
\(845\) 12.3407 5.11169i 0.424534 0.175848i
\(846\) 0 0
\(847\) 26.0686i 0.895728i
\(848\) 0 0
\(849\) 1.62622i 0.0558118i
\(850\) 0 0
\(851\) −6.36817 + 2.63778i −0.218298 + 0.0904220i
\(852\) 0 0
\(853\) 17.7231 42.7872i 0.606826 1.46501i −0.259607 0.965714i \(-0.583593\pi\)
0.866433 0.499293i \(-0.166407\pi\)
\(854\) 0 0
\(855\) 5.32580 5.32580i 0.182139 0.182139i
\(856\) 0 0
\(857\) 1.25991 + 1.25991i 0.0430375 + 0.0430375i 0.728298 0.685261i \(-0.240312\pi\)
−0.685261 + 0.728298i \(0.740312\pi\)
\(858\) 0 0
\(859\) 11.2626 + 4.66511i 0.384274 + 0.159171i 0.566453 0.824094i \(-0.308315\pi\)
−0.182179 + 0.983265i \(0.558315\pi\)
\(860\) 0 0
\(861\) 7.48477 + 18.0698i 0.255080 + 0.615818i
\(862\) 0 0
\(863\) 55.7303 1.89708 0.948541 0.316653i \(-0.102559\pi\)
0.948541 + 0.316653i \(0.102559\pi\)
\(864\) 0 0
\(865\) 32.3238 1.09904
\(866\) 0 0
\(867\) 6.65734 + 16.0722i 0.226095 + 0.545842i
\(868\) 0 0
\(869\) −22.8690 9.47266i −0.775779 0.321338i
\(870\) 0 0
\(871\) −15.0120 15.0120i −0.508662 0.508662i
\(872\) 0 0
\(873\) 5.55129 5.55129i 0.187883 0.187883i
\(874\) 0 0
\(875\) −5.43712 + 13.1264i −0.183808 + 0.443752i
\(876\) 0 0
\(877\) −10.3225 + 4.27570i −0.348565 + 0.144380i −0.550095 0.835102i \(-0.685409\pi\)
0.201530 + 0.979482i \(0.435409\pi\)
\(878\) 0 0
\(879\) 27.0104i 0.911039i
\(880\) 0 0
\(881\) 13.1185i 0.441972i 0.975277 + 0.220986i \(0.0709276\pi\)
−0.975277 + 0.220986i \(0.929072\pi\)
\(882\) 0 0
\(883\) −10.5395 + 4.36561i −0.354683 + 0.146915i −0.552909 0.833242i \(-0.686482\pi\)
0.198226 + 0.980156i \(0.436482\pi\)
\(884\) 0 0
\(885\) −17.4595 + 42.1509i −0.586893 + 1.41689i
\(886\) 0 0
\(887\) 32.1505 32.1505i 1.07951 1.07951i 0.0829550 0.996553i \(-0.473564\pi\)
0.996553 0.0829550i \(-0.0264358\pi\)
\(888\) 0 0
\(889\) −1.12722 1.12722i −0.0378058 0.0378058i
\(890\) 0 0
\(891\) −31.9392 13.2296i −1.07000 0.443210i
\(892\) 0 0
\(893\) −3.55663 8.58647i −0.119018 0.287335i
\(894\) 0 0
\(895\) 0.511683 0.0171037
\(896\) 0 0
\(897\) 9.34920 0.312161
\(898\) 0 0
\(899\) 16.9443 + 40.9070i 0.565123 + 1.36433i
\(900\) 0 0
\(901\) −0.904268 0.374560i −0.0301255 0.0124784i
\(902\) 0 0
\(903\) 14.5037 + 14.5037i 0.482654 + 0.482654i
\(904\) 0 0
\(905\) −6.49473 + 6.49473i −0.215892 + 0.215892i
\(906\) 0 0
\(907\) 12.1129 29.2431i 0.402202 0.971000i −0.584929 0.811084i \(-0.698878\pi\)
0.987131 0.159916i \(-0.0511223\pi\)
\(908\) 0 0
\(909\) 6.03306 2.49898i 0.200104 0.0828858i
\(910\) 0 0
\(911\) 41.1501i 1.36336i 0.731649 + 0.681681i \(0.238751\pi\)
−0.731649 + 0.681681i \(0.761249\pi\)
\(912\) 0 0
\(913\) 44.2832i 1.46556i
\(914\) 0 0
\(915\) 13.9792 5.79035i 0.462136 0.191423i
\(916\) 0 0
\(917\) 0.393927 0.951023i 0.0130086 0.0314055i
\(918\) 0 0
\(919\) −1.00540 + 1.00540i −0.0331651 + 0.0331651i −0.723495 0.690330i \(-0.757465\pi\)
0.690330 + 0.723495i \(0.257465\pi\)
\(920\) 0 0
\(921\) −22.1273 22.1273i −0.729119 0.729119i
\(922\) 0 0
\(923\) −15.2212 6.30483i −0.501012 0.207526i
\(924\) 0 0
\(925\) 13.5641 + 32.7466i 0.445985 + 1.07670i
\(926\) 0 0
\(927\) −1.18690 −0.0389829
\(928\) 0 0
\(929\) 52.8710 1.73464 0.867320 0.497751i \(-0.165841\pi\)
0.867320 + 0.497751i \(0.165841\pi\)
\(930\) 0 0
\(931\) −5.67018 13.6890i −0.185833 0.448640i
\(932\) 0 0
\(933\) 2.82967 + 1.17209i 0.0926393 + 0.0383724i
\(934\) 0 0
\(935\) 31.9623 + 31.9623i 1.04528 + 1.04528i
\(936\) 0 0
\(937\) −3.85201 + 3.85201i −0.125840 + 0.125840i −0.767222 0.641382i \(-0.778361\pi\)
0.641382 + 0.767222i \(0.278361\pi\)
\(938\) 0 0
\(939\) −1.04463 + 2.52197i −0.0340903 + 0.0823013i
\(940\) 0 0
\(941\) 39.2107 16.2416i 1.27823 0.529462i 0.362776 0.931876i \(-0.381829\pi\)
0.915458 + 0.402415i \(0.131829\pi\)
\(942\) 0 0
\(943\) 13.3670i 0.435288i
\(944\) 0 0
\(945\) 29.0703i 0.945658i
\(946\) 0 0
\(947\) −32.3640 + 13.4056i −1.05169 + 0.435624i −0.840494 0.541820i \(-0.817735\pi\)
−0.211194 + 0.977444i \(0.567735\pi\)
\(948\) 0 0
\(949\) 18.6702 45.0739i 0.606061 1.46316i
\(950\) 0 0
\(951\) 26.6119 26.6119i 0.862949 0.862949i
\(952\) 0 0
\(953\) 13.8976 + 13.8976i 0.450186 + 0.450186i 0.895416 0.445230i \(-0.146878\pi\)
−0.445230 + 0.895416i \(0.646878\pi\)
\(954\) 0 0
\(955\) −54.2842 22.4852i −1.75659 0.727605i
\(956\) 0 0
\(957\) 13.0761 + 31.5685i 0.422690 + 1.02046i
\(958\) 0 0
\(959\) 23.5844 0.761580
\(960\) 0 0
\(961\) 80.9971 2.61281
\(962\) 0 0
\(963\) −0.776251 1.87404i −0.0250143 0.0603900i
\(964\) 0 0
\(965\) 32.4658 + 13.4478i 1.04511 + 0.432899i
\(966\) 0 0
\(967\) 36.8527 + 36.8527i 1.18510 + 1.18510i 0.978404 + 0.206700i \(0.0662724\pi\)
0.206700 + 0.978404i \(0.433728\pi\)
\(968\) 0 0
\(969\) 7.64689 7.64689i 0.245653 0.245653i
\(970\) 0 0
\(971\) −15.5625 + 37.5712i −0.499425 + 1.20572i 0.450370 + 0.892842i \(0.351292\pi\)
−0.949794 + 0.312875i \(0.898708\pi\)
\(972\) 0 0
\(973\) 1.13381 0.469639i 0.0363482 0.0150559i
\(974\) 0 0
\(975\) 48.0758i 1.53966i
\(976\) 0 0
\(977\) 2.89004i 0.0924607i −0.998931 0.0462303i \(-0.985279\pi\)
0.998931 0.0462303i \(-0.0147208\pi\)
\(978\) 0 0
\(979\) −6.79994 + 2.81663i −0.217327 + 0.0900197i
\(980\) 0 0
\(981\) −1.01896 + 2.46000i −0.0325330 + 0.0785417i
\(982\) 0 0
\(983\) 10.1625 10.1625i 0.324135 0.324135i −0.526216 0.850351i \(-0.676390\pi\)
0.850351 + 0.526216i \(0.176390\pi\)
\(984\) 0 0
\(985\) −25.7926 25.7926i −0.821820 0.821820i
\(986\) 0 0
\(987\) 6.24646 + 2.58737i 0.198827 + 0.0823568i
\(988\) 0 0
\(989\) 5.36449 + 12.9510i 0.170581 + 0.411819i
\(990\) 0 0
\(991\) −51.7294 −1.64324 −0.821619 0.570038i \(-0.806929\pi\)
−0.821619 + 0.570038i \(0.806929\pi\)
\(992\) 0 0
\(993\) −34.7901 −1.10403
\(994\) 0 0
\(995\) −4.66978 11.2738i −0.148042 0.357405i
\(996\) 0 0
\(997\) 18.0280 + 7.46746i 0.570954 + 0.236497i 0.649433 0.760419i \(-0.275006\pi\)
−0.0784791 + 0.996916i \(0.525006\pi\)
\(998\) 0 0
\(999\) 18.1624 + 18.1624i 0.574633 + 0.574633i
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 1024.2.g.g.897.4 yes 16
4.3 odd 2 inner 1024.2.g.g.897.1 yes 16
8.3 odd 2 1024.2.g.d.897.4 yes 16
8.5 even 2 1024.2.g.d.897.1 yes 16
16.3 odd 4 1024.2.g.f.385.4 yes 16
16.5 even 4 1024.2.g.a.385.4 yes 16
16.11 odd 4 1024.2.g.a.385.1 16
16.13 even 4 1024.2.g.f.385.1 yes 16
32.3 odd 8 1024.2.g.d.129.4 yes 16
32.5 even 8 1024.2.g.f.641.1 yes 16
32.11 odd 8 1024.2.g.a.641.1 yes 16
32.13 even 8 inner 1024.2.g.g.129.4 yes 16
32.19 odd 8 inner 1024.2.g.g.129.1 yes 16
32.21 even 8 1024.2.g.a.641.4 yes 16
32.27 odd 8 1024.2.g.f.641.4 yes 16
32.29 even 8 1024.2.g.d.129.1 yes 16
64.13 even 16 4096.2.a.s.1.6 8
64.19 odd 16 4096.2.a.s.1.5 8
64.45 even 16 4096.2.a.i.1.3 8
64.51 odd 16 4096.2.a.i.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.a.385.1 16 16.11 odd 4
1024.2.g.a.385.4 yes 16 16.5 even 4
1024.2.g.a.641.1 yes 16 32.11 odd 8
1024.2.g.a.641.4 yes 16 32.21 even 8
1024.2.g.d.129.1 yes 16 32.29 even 8
1024.2.g.d.129.4 yes 16 32.3 odd 8
1024.2.g.d.897.1 yes 16 8.5 even 2
1024.2.g.d.897.4 yes 16 8.3 odd 2
1024.2.g.f.385.1 yes 16 16.13 even 4
1024.2.g.f.385.4 yes 16 16.3 odd 4
1024.2.g.f.641.1 yes 16 32.5 even 8
1024.2.g.f.641.4 yes 16 32.27 odd 8
1024.2.g.g.129.1 yes 16 32.19 odd 8 inner
1024.2.g.g.129.4 yes 16 32.13 even 8 inner
1024.2.g.g.897.1 yes 16 4.3 odd 2 inner
1024.2.g.g.897.4 yes 16 1.1 even 1 trivial
4096.2.a.i.1.3 8 64.45 even 16
4096.2.a.i.1.4 8 64.51 odd 16
4096.2.a.s.1.5 8 64.19 odd 16
4096.2.a.s.1.6 8 64.13 even 16