Properties

Label 504.2.r.f.169.4
Level $504$
Weight $2$
Character 504.169
Analytic conductor $4.024$
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] = [504,2,Mod(169,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 169.4
Root \(1.34147 + 1.09565i\) of defining polynomial
Character \(\chi\) \(=\) 504.169
Dual form 504.2.r.f.337.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.742385 + 1.56488i) q^{3} +(-1.76217 - 3.05216i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.89773 + 2.32349i) q^{9} +O(q^{10})\) \(q+(0.742385 + 1.56488i) q^{3} +(-1.76217 - 3.05216i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.89773 + 2.32349i) q^{9} +(2.08386 - 3.60934i) q^{11} +(-1.52051 - 2.63361i) q^{13} +(3.46807 - 5.02347i) q^{15} +3.79546 q^{17} +5.40544 q^{19} +(1.72642 + 0.139518i) q^{21} +(-4.48540 - 7.76894i) q^{23} +(-3.71046 + 6.42670i) q^{25} +(-5.04485 - 1.24480i) q^{27} +(1.46107 - 2.53065i) q^{29} +(3.08767 + 5.34801i) q^{31} +(7.19523 + 0.581470i) q^{33} -3.52433 q^{35} -4.66649 q^{37} +(2.99249 - 4.33458i) q^{39} +(1.83828 + 3.18400i) q^{41} +(1.31777 - 2.28244i) q^{43} +(10.4358 + 1.69779i) q^{45} +(-0.860518 + 1.49046i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(2.81769 + 5.93945i) q^{51} +11.4209 q^{53} -14.6884 q^{55} +(4.01292 + 8.45890i) q^{57} +(0.564482 + 0.977712i) q^{59} +(-6.03597 + 10.4546i) q^{61} +(1.06334 + 2.80523i) q^{63} +(-5.35880 + 9.28171i) q^{65} +(-1.12782 - 1.95345i) q^{67} +(8.82761 - 12.7867i) q^{69} -9.92978 q^{71} +2.50106 q^{73} +(-12.8116 - 1.03535i) q^{75} +(-2.08386 - 3.60934i) q^{77} +(3.16763 - 5.48650i) q^{79} +(-1.79726 - 8.81872i) q^{81} +(5.52319 - 9.56645i) q^{83} +(-6.68823 - 11.5843i) q^{85} +(5.04485 + 0.407691i) q^{87} -13.4697 q^{89} -3.04103 q^{91} +(-6.07677 + 8.80214i) q^{93} +(-9.52529 - 16.4983i) q^{95} +(-9.46871 + 16.4003i) q^{97} +(4.43170 + 11.6914i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{5} + 5 q^{7}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0.742385 + 1.56488i 0.428616 + 0.903487i
\(4\) 0 0
\(5\) −1.76217 3.05216i −0.788065 1.36497i −0.927151 0.374688i \(-0.877750\pi\)
0.139086 0.990280i \(-0.455583\pi\)
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.188982 0.327327i
\(8\) 0 0
\(9\) −1.89773 + 2.32349i −0.632576 + 0.774498i
\(10\) 0 0
\(11\) 2.08386 3.60934i 0.628306 1.08826i −0.359585 0.933112i \(-0.617082\pi\)
0.987892 0.155146i \(-0.0495848\pi\)
\(12\) 0 0
\(13\) −1.52051 2.63361i −0.421715 0.730431i 0.574393 0.818580i \(-0.305238\pi\)
−0.996107 + 0.0881486i \(0.971905\pi\)
\(14\) 0 0
\(15\) 3.46807 5.02347i 0.895453 1.29705i
\(16\) 0 0
\(17\) 3.79546 0.920533 0.460267 0.887781i \(-0.347754\pi\)
0.460267 + 0.887781i \(0.347754\pi\)
\(18\) 0 0
\(19\) 5.40544 1.24009 0.620047 0.784565i \(-0.287114\pi\)
0.620047 + 0.784565i \(0.287114\pi\)
\(20\) 0 0
\(21\) 1.72642 + 0.139518i 0.376736 + 0.0304453i
\(22\) 0 0
\(23\) −4.48540 7.76894i −0.935271 1.61994i −0.774150 0.633002i \(-0.781822\pi\)
−0.161121 0.986935i \(-0.551511\pi\)
\(24\) 0 0
\(25\) −3.71046 + 6.42670i −0.742092 + 1.28534i
\(26\) 0 0
\(27\) −5.04485 1.24480i −0.970881 0.239561i
\(28\) 0 0
\(29\) 1.46107 2.53065i 0.271314 0.469929i −0.697885 0.716210i \(-0.745875\pi\)
0.969199 + 0.246281i \(0.0792086\pi\)
\(30\) 0 0
\(31\) 3.08767 + 5.34801i 0.554563 + 0.960531i 0.997937 + 0.0641943i \(0.0204477\pi\)
−0.443375 + 0.896336i \(0.646219\pi\)
\(32\) 0 0
\(33\) 7.19523 + 0.581470i 1.25253 + 0.101221i
\(34\) 0 0
\(35\) −3.52433 −0.595721
\(36\) 0 0
\(37\) −4.66649 −0.767166 −0.383583 0.923506i \(-0.625310\pi\)
−0.383583 + 0.923506i \(0.625310\pi\)
\(38\) 0 0
\(39\) 2.99249 4.33458i 0.479181 0.694088i
\(40\) 0 0
\(41\) 1.83828 + 3.18400i 0.287092 + 0.497257i 0.973114 0.230323i \(-0.0739782\pi\)
−0.686023 + 0.727580i \(0.740645\pi\)
\(42\) 0 0
\(43\) 1.31777 2.28244i 0.200958 0.348069i −0.747879 0.663835i \(-0.768928\pi\)
0.948837 + 0.315765i \(0.102261\pi\)
\(44\) 0 0
\(45\) 10.4358 + 1.69779i 1.55568 + 0.253091i
\(46\) 0 0
\(47\) −0.860518 + 1.49046i −0.125519 + 0.217406i −0.921936 0.387343i \(-0.873393\pi\)
0.796416 + 0.604749i \(0.206726\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 2.81769 + 5.93945i 0.394556 + 0.831690i
\(52\) 0 0
\(53\) 11.4209 1.56878 0.784392 0.620265i \(-0.212975\pi\)
0.784392 + 0.620265i \(0.212975\pi\)
\(54\) 0 0
\(55\) −14.6884 −1.98058
\(56\) 0 0
\(57\) 4.01292 + 8.45890i 0.531524 + 1.12041i
\(58\) 0 0
\(59\) 0.564482 + 0.977712i 0.0734894 + 0.127287i 0.900428 0.435004i \(-0.143253\pi\)
−0.826939 + 0.562292i \(0.809920\pi\)
\(60\) 0 0
\(61\) −6.03597 + 10.4546i −0.772826 + 1.33857i 0.163182 + 0.986596i \(0.447824\pi\)
−0.936008 + 0.351979i \(0.885509\pi\)
\(62\) 0 0
\(63\) 1.06334 + 2.80523i 0.133968 + 0.353426i
\(64\) 0 0
\(65\) −5.35880 + 9.28171i −0.664677 + 1.15125i
\(66\) 0 0
\(67\) −1.12782 1.95345i −0.137786 0.238652i 0.788873 0.614557i \(-0.210665\pi\)
−0.926658 + 0.375905i \(0.877332\pi\)
\(68\) 0 0
\(69\) 8.82761 12.7867i 1.06272 1.53934i
\(70\) 0 0
\(71\) −9.92978 −1.17845 −0.589224 0.807970i \(-0.700566\pi\)
−0.589224 + 0.807970i \(0.700566\pi\)
\(72\) 0 0
\(73\) 2.50106 0.292727 0.146364 0.989231i \(-0.453243\pi\)
0.146364 + 0.989231i \(0.453243\pi\)
\(74\) 0 0
\(75\) −12.8116 1.03535i −1.47936 0.119552i
\(76\) 0 0
\(77\) −2.08386 3.60934i −0.237477 0.411323i
\(78\) 0 0
\(79\) 3.16763 5.48650i 0.356387 0.617280i −0.630968 0.775809i \(-0.717342\pi\)
0.987354 + 0.158530i \(0.0506753\pi\)
\(80\) 0 0
\(81\) −1.79726 8.81872i −0.199695 0.979858i
\(82\) 0 0
\(83\) 5.52319 9.56645i 0.606249 1.05005i −0.385604 0.922664i \(-0.626007\pi\)
0.991853 0.127390i \(-0.0406599\pi\)
\(84\) 0 0
\(85\) −6.68823 11.5843i −0.725440 1.25650i
\(86\) 0 0
\(87\) 5.04485 + 0.407691i 0.540864 + 0.0437090i
\(88\) 0 0
\(89\) −13.4697 −1.42779 −0.713895 0.700253i \(-0.753071\pi\)
−0.713895 + 0.700253i \(0.753071\pi\)
\(90\) 0 0
\(91\) −3.04103 −0.318786
\(92\) 0 0
\(93\) −6.07677 + 8.80214i −0.630132 + 0.912739i
\(94\) 0 0
\(95\) −9.52529 16.4983i −0.977274 1.69269i
\(96\) 0 0
\(97\) −9.46871 + 16.4003i −0.961401 + 1.66520i −0.242414 + 0.970173i \(0.577939\pi\)
−0.718987 + 0.695023i \(0.755394\pi\)
\(98\) 0 0
\(99\) 4.43170 + 11.6914i 0.445403 + 1.17503i
\(100\) 0 0
\(101\) −4.82940 + 8.36477i −0.480544 + 0.832326i −0.999751 0.0223224i \(-0.992894\pi\)
0.519207 + 0.854648i \(0.326227\pi\)
\(102\) 0 0
\(103\) 0.981584 + 1.70015i 0.0967183 + 0.167521i 0.910324 0.413895i \(-0.135832\pi\)
−0.813606 + 0.581416i \(0.802499\pi\)
\(104\) 0 0
\(105\) −2.61641 5.51517i −0.255336 0.538226i
\(106\) 0 0
\(107\) 18.5072 1.78916 0.894578 0.446912i \(-0.147476\pi\)
0.894578 + 0.446912i \(0.147476\pi\)
\(108\) 0 0
\(109\) 2.67237 0.255967 0.127983 0.991776i \(-0.459150\pi\)
0.127983 + 0.991776i \(0.459150\pi\)
\(110\) 0 0
\(111\) −3.46433 7.30252i −0.328820 0.693125i
\(112\) 0 0
\(113\) −4.65605 8.06452i −0.438005 0.758646i 0.559531 0.828809i \(-0.310981\pi\)
−0.997536 + 0.0701632i \(0.977648\pi\)
\(114\) 0 0
\(115\) −15.8080 + 27.3803i −1.47411 + 2.55323i
\(116\) 0 0
\(117\) 9.00470 + 1.46497i 0.832485 + 0.135436i
\(118\) 0 0
\(119\) 1.89773 3.28696i 0.173964 0.301315i
\(120\) 0 0
\(121\) −3.18491 5.51642i −0.289537 0.501493i
\(122\) 0 0
\(123\) −3.61788 + 5.24046i −0.326213 + 0.472516i
\(124\) 0 0
\(125\) 8.53213 0.763137
\(126\) 0 0
\(127\) 16.7663 1.48777 0.743883 0.668310i \(-0.232982\pi\)
0.743883 + 0.668310i \(0.232982\pi\)
\(128\) 0 0
\(129\) 4.55005 + 0.367705i 0.400610 + 0.0323746i
\(130\) 0 0
\(131\) 1.36566 + 2.36539i 0.119318 + 0.206665i 0.919498 0.393096i \(-0.128596\pi\)
−0.800180 + 0.599761i \(0.795263\pi\)
\(132\) 0 0
\(133\) 2.70272 4.68125i 0.234356 0.405916i
\(134\) 0 0
\(135\) 5.09054 + 17.5912i 0.438124 + 1.51401i
\(136\) 0 0
\(137\) −8.12870 + 14.0793i −0.694482 + 1.20288i 0.275873 + 0.961194i \(0.411033\pi\)
−0.970355 + 0.241684i \(0.922300\pi\)
\(138\) 0 0
\(139\) −0.654057 1.13286i −0.0554764 0.0960879i 0.836954 0.547274i \(-0.184334\pi\)
−0.892430 + 0.451186i \(0.851001\pi\)
\(140\) 0 0
\(141\) −2.97124 0.240115i −0.250223 0.0202214i
\(142\) 0 0
\(143\) −12.6741 −1.05986
\(144\) 0 0
\(145\) −10.2986 −0.855251
\(146\) 0 0
\(147\) 0.984037 1.42537i 0.0811620 0.117562i
\(148\) 0 0
\(149\) 6.54580 + 11.3377i 0.536253 + 0.928818i 0.999102 + 0.0423802i \(0.0134941\pi\)
−0.462848 + 0.886437i \(0.653173\pi\)
\(150\) 0 0
\(151\) −7.10933 + 12.3137i −0.578549 + 1.00208i 0.417097 + 0.908862i \(0.363048\pi\)
−0.995646 + 0.0932143i \(0.970286\pi\)
\(152\) 0 0
\(153\) −7.20274 + 8.81872i −0.582307 + 0.712951i
\(154\) 0 0
\(155\) 10.8820 18.8482i 0.874062 1.51392i
\(156\) 0 0
\(157\) 4.87043 + 8.43584i 0.388703 + 0.673253i 0.992275 0.124055i \(-0.0395899\pi\)
−0.603572 + 0.797308i \(0.706257\pi\)
\(158\) 0 0
\(159\) 8.47872 + 17.8724i 0.672406 + 1.41738i
\(160\) 0 0
\(161\) −8.97080 −0.706998
\(162\) 0 0
\(163\) 18.2129 1.42655 0.713274 0.700886i \(-0.247212\pi\)
0.713274 + 0.700886i \(0.247212\pi\)
\(164\) 0 0
\(165\) −10.9045 22.9857i −0.848910 1.78943i
\(166\) 0 0
\(167\) 7.81475 + 13.5355i 0.604724 + 1.04741i 0.992095 + 0.125488i \(0.0400498\pi\)
−0.387371 + 0.921924i \(0.626617\pi\)
\(168\) 0 0
\(169\) 1.87607 3.24945i 0.144313 0.249958i
\(170\) 0 0
\(171\) −10.2581 + 12.5595i −0.784454 + 0.960450i
\(172\) 0 0
\(173\) 5.23991 9.07579i 0.398383 0.690019i −0.595144 0.803619i \(-0.702905\pi\)
0.993527 + 0.113600i \(0.0362382\pi\)
\(174\) 0 0
\(175\) 3.71046 + 6.42670i 0.280484 + 0.485813i
\(176\) 0 0
\(177\) −1.11094 + 1.60919i −0.0835036 + 0.120954i
\(178\) 0 0
\(179\) −18.0874 −1.35192 −0.675958 0.736940i \(-0.736270\pi\)
−0.675958 + 0.736940i \(0.736270\pi\)
\(180\) 0 0
\(181\) 22.4020 1.66513 0.832564 0.553929i \(-0.186872\pi\)
0.832564 + 0.553929i \(0.186872\pi\)
\(182\) 0 0
\(183\) −20.8413 1.68425i −1.54063 0.124503i
\(184\) 0 0
\(185\) 8.22313 + 14.2429i 0.604577 + 1.04716i
\(186\) 0 0
\(187\) 7.90918 13.6991i 0.578377 1.00178i
\(188\) 0 0
\(189\) −3.60045 + 3.74657i −0.261894 + 0.272523i
\(190\) 0 0
\(191\) 1.69490 2.93566i 0.122639 0.212417i −0.798169 0.602434i \(-0.794198\pi\)
0.920808 + 0.390017i \(0.127531\pi\)
\(192\) 0 0
\(193\) −7.99828 13.8534i −0.575729 0.997191i −0.995962 0.0897754i \(-0.971385\pi\)
0.420233 0.907416i \(-0.361948\pi\)
\(194\) 0 0
\(195\) −18.5031 1.49530i −1.32503 0.107080i
\(196\) 0 0
\(197\) −6.75083 −0.480977 −0.240488 0.970652i \(-0.577308\pi\)
−0.240488 + 0.970652i \(0.577308\pi\)
\(198\) 0 0
\(199\) 11.4209 0.809608 0.404804 0.914404i \(-0.367340\pi\)
0.404804 + 0.914404i \(0.367340\pi\)
\(200\) 0 0
\(201\) 2.21964 3.21513i 0.156561 0.226777i
\(202\) 0 0
\(203\) −1.46107 2.53065i −0.102547 0.177617i
\(204\) 0 0
\(205\) 6.47872 11.2215i 0.452494 0.783742i
\(206\) 0 0
\(207\) 26.5632 + 4.32154i 1.84627 + 0.300368i
\(208\) 0 0
\(209\) 11.2642 19.5101i 0.779158 1.34954i
\(210\) 0 0
\(211\) −6.40147 11.0877i −0.440695 0.763307i 0.557046 0.830482i \(-0.311935\pi\)
−0.997741 + 0.0671751i \(0.978601\pi\)
\(212\) 0 0
\(213\) −7.37172 15.5390i −0.505102 1.06471i
\(214\) 0 0
\(215\) −9.28851 −0.633471
\(216\) 0 0
\(217\) 6.17535 0.419210
\(218\) 0 0
\(219\) 1.85675 + 3.91387i 0.125468 + 0.264475i
\(220\) 0 0
\(221\) −5.77105 9.99574i −0.388203 0.672387i
\(222\) 0 0
\(223\) 5.86357 10.1560i 0.392654 0.680096i −0.600145 0.799891i \(-0.704890\pi\)
0.992799 + 0.119795i \(0.0382237\pi\)
\(224\) 0 0
\(225\) −7.89097 20.8174i −0.526065 1.38782i
\(226\) 0 0
\(227\) 6.30312 10.9173i 0.418353 0.724608i −0.577421 0.816446i \(-0.695941\pi\)
0.995774 + 0.0918384i \(0.0292743\pi\)
\(228\) 0 0
\(229\) 4.34480 + 7.52542i 0.287113 + 0.497294i 0.973119 0.230302i \(-0.0739713\pi\)
−0.686007 + 0.727595i \(0.740638\pi\)
\(230\) 0 0
\(231\) 4.10118 5.94052i 0.269838 0.390857i
\(232\) 0 0
\(233\) 12.9251 0.846748 0.423374 0.905955i \(-0.360846\pi\)
0.423374 + 0.905955i \(0.360846\pi\)
\(234\) 0 0
\(235\) 6.06550 0.395670
\(236\) 0 0
\(237\) 10.9373 + 0.883883i 0.710457 + 0.0574143i
\(238\) 0 0
\(239\) 10.8020 + 18.7097i 0.698725 + 1.21023i 0.968909 + 0.247419i \(0.0795823\pi\)
−0.270183 + 0.962809i \(0.587084\pi\)
\(240\) 0 0
\(241\) −1.66179 + 2.87831i −0.107046 + 0.185408i −0.914572 0.404423i \(-0.867472\pi\)
0.807527 + 0.589831i \(0.200806\pi\)
\(242\) 0 0
\(243\) 12.4660 9.35939i 0.799696 0.600405i
\(244\) 0 0
\(245\) −1.76217 + 3.05216i −0.112581 + 0.194995i
\(246\) 0 0
\(247\) −8.21905 14.2358i −0.522966 0.905803i
\(248\) 0 0
\(249\) 19.0707 + 1.54117i 1.20856 + 0.0976675i
\(250\) 0 0
\(251\) 2.67969 0.169140 0.0845701 0.996418i \(-0.473048\pi\)
0.0845701 + 0.996418i \(0.473048\pi\)
\(252\) 0 0
\(253\) −37.3877 −2.35055
\(254\) 0 0
\(255\) 13.1629 19.0663i 0.824294 1.19398i
\(256\) 0 0
\(257\) −3.75443 6.50286i −0.234195 0.405637i 0.724844 0.688913i \(-0.241912\pi\)
−0.959038 + 0.283276i \(0.908579\pi\)
\(258\) 0 0
\(259\) −2.33325 + 4.04130i −0.144981 + 0.251114i
\(260\) 0 0
\(261\) 3.10723 + 8.19727i 0.192333 + 0.507398i
\(262\) 0 0
\(263\) −9.35062 + 16.1957i −0.576584 + 0.998672i 0.419284 + 0.907855i \(0.362281\pi\)
−0.995868 + 0.0908171i \(0.971052\pi\)
\(264\) 0 0
\(265\) −20.1256 34.8585i −1.23630 2.14134i
\(266\) 0 0
\(267\) −9.99974 21.0786i −0.611974 1.28999i
\(268\) 0 0
\(269\) −17.0035 −1.03672 −0.518360 0.855163i \(-0.673457\pi\)
−0.518360 + 0.855163i \(0.673457\pi\)
\(270\) 0 0
\(271\) 23.1427 1.40582 0.702910 0.711279i \(-0.251884\pi\)
0.702910 + 0.711279i \(0.251884\pi\)
\(272\) 0 0
\(273\) −2.25761 4.75886i −0.136637 0.288019i
\(274\) 0 0
\(275\) 15.4641 + 26.7846i 0.932522 + 1.61518i
\(276\) 0 0
\(277\) 6.93092 12.0047i 0.416438 0.721293i −0.579140 0.815228i \(-0.696611\pi\)
0.995578 + 0.0939357i \(0.0299448\pi\)
\(278\) 0 0
\(279\) −18.2856 2.97487i −1.09473 0.178101i
\(280\) 0 0
\(281\) −4.56926 + 7.91419i −0.272579 + 0.472121i −0.969521 0.245006i \(-0.921210\pi\)
0.696942 + 0.717127i \(0.254543\pi\)
\(282\) 0 0
\(283\) 3.45421 + 5.98287i 0.205331 + 0.355645i 0.950238 0.311524i \(-0.100839\pi\)
−0.744907 + 0.667169i \(0.767506\pi\)
\(284\) 0 0
\(285\) 18.7465 27.1541i 1.11045 1.60847i
\(286\) 0 0
\(287\) 3.67657 0.217021
\(288\) 0 0
\(289\) −2.59451 −0.152618
\(290\) 0 0
\(291\) −32.6940 2.64211i −1.91655 0.154883i
\(292\) 0 0
\(293\) 9.30083 + 16.1095i 0.543361 + 0.941128i 0.998708 + 0.0508143i \(0.0161817\pi\)
−0.455348 + 0.890314i \(0.650485\pi\)
\(294\) 0 0
\(295\) 1.98942 3.44578i 0.115829 0.200621i
\(296\) 0 0
\(297\) −15.0056 + 15.6146i −0.870715 + 0.906051i
\(298\) 0 0
\(299\) −13.6402 + 23.6256i −0.788835 + 1.36630i
\(300\) 0 0
\(301\) −1.31777 2.28244i −0.0759550 0.131558i
\(302\) 0 0
\(303\) −16.6752 1.34758i −0.957964 0.0774162i
\(304\) 0 0
\(305\) 42.5455 2.43615
\(306\) 0 0
\(307\) 13.9553 0.796473 0.398236 0.917283i \(-0.369622\pi\)
0.398236 + 0.917283i \(0.369622\pi\)
\(308\) 0 0
\(309\) −1.93183 + 2.79823i −0.109898 + 0.159186i
\(310\) 0 0
\(311\) 8.86140 + 15.3484i 0.502484 + 0.870327i 0.999996 + 0.00287031i \(0.000913649\pi\)
−0.497512 + 0.867457i \(0.665753\pi\)
\(312\) 0 0
\(313\) −7.38305 + 12.7878i −0.417315 + 0.722810i −0.995668 0.0929758i \(-0.970362\pi\)
0.578354 + 0.815786i \(0.303695\pi\)
\(314\) 0 0
\(315\) 6.68823 8.18877i 0.376839 0.461385i
\(316\) 0 0
\(317\) 1.28820 2.23122i 0.0723523 0.125318i −0.827580 0.561348i \(-0.810283\pi\)
0.899932 + 0.436031i \(0.143616\pi\)
\(318\) 0 0
\(319\) −6.08932 10.5470i −0.340936 0.590519i
\(320\) 0 0
\(321\) 13.7395 + 28.9616i 0.766861 + 1.61648i
\(322\) 0 0
\(323\) 20.5161 1.14155
\(324\) 0 0
\(325\) 22.5672 1.25180
\(326\) 0 0
\(327\) 1.98393 + 4.18195i 0.109712 + 0.231263i
\(328\) 0 0
\(329\) 0.860518 + 1.49046i 0.0474419 + 0.0821718i
\(330\) 0 0
\(331\) 0.158777 0.275009i 0.00872716 0.0151159i −0.861629 0.507539i \(-0.830555\pi\)
0.870356 + 0.492423i \(0.163889\pi\)
\(332\) 0 0
\(333\) 8.85573 10.8426i 0.485291 0.594169i
\(334\) 0 0
\(335\) −3.97483 + 6.88460i −0.217168 + 0.376146i
\(336\) 0 0
\(337\) 6.97761 + 12.0856i 0.380095 + 0.658343i 0.991075 0.133302i \(-0.0425581\pi\)
−0.610981 + 0.791645i \(0.709225\pi\)
\(338\) 0 0
\(339\) 9.16346 13.2732i 0.497691 0.720899i
\(340\) 0 0
\(341\) 25.7371 1.39374
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −54.5827 4.41101i −2.93864 0.237481i
\(346\) 0 0
\(347\) −12.0401 20.8540i −0.646344 1.11950i −0.983989 0.178228i \(-0.942964\pi\)
0.337645 0.941274i \(-0.390370\pi\)
\(348\) 0 0
\(349\) 13.1616 22.7966i 0.704526 1.22027i −0.262337 0.964976i \(-0.584493\pi\)
0.966862 0.255298i \(-0.0821736\pi\)
\(350\) 0 0
\(351\) 4.39245 + 15.1789i 0.234452 + 0.810189i
\(352\) 0 0
\(353\) −1.30502 + 2.26036i −0.0694590 + 0.120307i −0.898663 0.438639i \(-0.855461\pi\)
0.829204 + 0.558946i \(0.188794\pi\)
\(354\) 0 0
\(355\) 17.4979 + 30.3073i 0.928693 + 1.60854i
\(356\) 0 0
\(357\) 6.55256 + 0.529534i 0.346798 + 0.0280259i
\(358\) 0 0
\(359\) 3.48082 0.183711 0.0918553 0.995772i \(-0.470720\pi\)
0.0918553 + 0.995772i \(0.470720\pi\)
\(360\) 0 0
\(361\) 10.2188 0.537832
\(362\) 0 0
\(363\) 6.26814 9.07932i 0.328992 0.476541i
\(364\) 0 0
\(365\) −4.40729 7.63365i −0.230688 0.399563i
\(366\) 0 0
\(367\) −0.960991 + 1.66449i −0.0501633 + 0.0868854i −0.890017 0.455928i \(-0.849308\pi\)
0.839853 + 0.542813i \(0.182641\pi\)
\(368\) 0 0
\(369\) −10.8866 1.77113i −0.566732 0.0922011i
\(370\) 0 0
\(371\) 5.71046 9.89081i 0.296472 0.513505i
\(372\) 0 0
\(373\) −16.1228 27.9255i −0.834806 1.44593i −0.894189 0.447691i \(-0.852247\pi\)
0.0593828 0.998235i \(-0.481087\pi\)
\(374\) 0 0
\(375\) 6.33412 + 13.3518i 0.327093 + 0.689484i
\(376\) 0 0
\(377\) −8.88631 −0.457668
\(378\) 0 0
\(379\) −4.98892 −0.256264 −0.128132 0.991757i \(-0.540898\pi\)
−0.128132 + 0.991757i \(0.540898\pi\)
\(380\) 0 0
\(381\) 12.4470 + 26.2373i 0.637680 + 1.34418i
\(382\) 0 0
\(383\) 0.284377 + 0.492556i 0.0145310 + 0.0251684i 0.873199 0.487363i \(-0.162041\pi\)
−0.858668 + 0.512531i \(0.828708\pi\)
\(384\) 0 0
\(385\) −7.34420 + 12.7205i −0.374295 + 0.648298i
\(386\) 0 0
\(387\) 2.80248 + 7.39329i 0.142458 + 0.375822i
\(388\) 0 0
\(389\) −12.4259 + 21.5223i −0.630017 + 1.09122i 0.357531 + 0.933901i \(0.383619\pi\)
−0.987548 + 0.157320i \(0.949715\pi\)
\(390\) 0 0
\(391\) −17.0241 29.4867i −0.860948 1.49121i
\(392\) 0 0
\(393\) −2.68772 + 3.89313i −0.135577 + 0.196382i
\(394\) 0 0
\(395\) −22.3276 −1.12342
\(396\) 0 0
\(397\) −3.87576 −0.194519 −0.0972593 0.995259i \(-0.531008\pi\)
−0.0972593 + 0.995259i \(0.531008\pi\)
\(398\) 0 0
\(399\) 9.33208 + 0.754156i 0.467188 + 0.0377550i
\(400\) 0 0
\(401\) 7.43200 + 12.8726i 0.371136 + 0.642827i 0.989741 0.142876i \(-0.0456349\pi\)
−0.618604 + 0.785703i \(0.712302\pi\)
\(402\) 0 0
\(403\) 9.38970 16.2634i 0.467734 0.810140i
\(404\) 0 0
\(405\) −23.7491 + 21.0256i −1.18010 + 1.04477i
\(406\) 0 0
\(407\) −9.72429 + 16.8430i −0.482015 + 0.834875i
\(408\) 0 0
\(409\) 11.6707 + 20.2142i 0.577078 + 0.999529i 0.995812 + 0.0914201i \(0.0291406\pi\)
−0.418734 + 0.908109i \(0.637526\pi\)
\(410\) 0 0
\(411\) −28.0671 2.26820i −1.38445 0.111882i
\(412\) 0 0
\(413\) 1.12896 0.0555527
\(414\) 0 0
\(415\) −38.9311 −1.91105
\(416\) 0 0
\(417\) 1.28723 1.86454i 0.0630360 0.0913070i
\(418\) 0 0
\(419\) −10.7749 18.6627i −0.526389 0.911733i −0.999527 0.0307446i \(-0.990212\pi\)
0.473138 0.880988i \(-0.343121\pi\)
\(420\) 0 0
\(421\) 9.83901 17.0417i 0.479524 0.830560i −0.520200 0.854044i \(-0.674143\pi\)
0.999724 + 0.0234847i \(0.00747609\pi\)
\(422\) 0 0
\(423\) −1.83005 4.82790i −0.0889800 0.234740i
\(424\) 0 0
\(425\) −14.0829 + 24.3923i −0.683120 + 1.18320i
\(426\) 0 0
\(427\) 6.03597 + 10.4546i 0.292101 + 0.505934i
\(428\) 0 0
\(429\) −9.40909 19.8335i −0.454275 0.957573i
\(430\) 0 0
\(431\) −19.3862 −0.933798 −0.466899 0.884311i \(-0.654629\pi\)
−0.466899 + 0.884311i \(0.654629\pi\)
\(432\) 0 0
\(433\) 34.9983 1.68191 0.840956 0.541104i \(-0.181993\pi\)
0.840956 + 0.541104i \(0.181993\pi\)
\(434\) 0 0
\(435\) −7.64552 16.1161i −0.366575 0.772708i
\(436\) 0 0
\(437\) −24.2456 41.9946i −1.15982 2.00887i
\(438\) 0 0
\(439\) 1.94437 3.36775i 0.0927999 0.160734i −0.815888 0.578209i \(-0.803752\pi\)
0.908688 + 0.417475i \(0.137085\pi\)
\(440\) 0 0
\(441\) 2.96107 + 0.481734i 0.141003 + 0.0229397i
\(442\) 0 0
\(443\) 10.8632 18.8156i 0.516126 0.893956i −0.483699 0.875235i \(-0.660707\pi\)
0.999825 0.0187219i \(-0.00595970\pi\)
\(444\) 0 0
\(445\) 23.7359 + 41.1118i 1.12519 + 1.94889i
\(446\) 0 0
\(447\) −12.8826 + 18.6603i −0.609327 + 0.882604i
\(448\) 0 0
\(449\) −19.3098 −0.911287 −0.455643 0.890162i \(-0.650591\pi\)
−0.455643 + 0.890162i \(0.650591\pi\)
\(450\) 0 0
\(451\) 15.3229 0.721526
\(452\) 0 0
\(453\) −24.5474 1.98376i −1.15334 0.0932050i
\(454\) 0 0
\(455\) 5.35880 + 9.28171i 0.251224 + 0.435133i
\(456\) 0 0
\(457\) 8.03867 13.9234i 0.376033 0.651308i −0.614448 0.788957i \(-0.710621\pi\)
0.990481 + 0.137649i \(0.0439546\pi\)
\(458\) 0 0
\(459\) −19.1475 4.72458i −0.893729 0.220524i
\(460\) 0 0
\(461\) −16.6888 + 28.9058i −0.777274 + 1.34628i 0.156233 + 0.987720i \(0.450065\pi\)
−0.933507 + 0.358558i \(0.883269\pi\)
\(462\) 0 0
\(463\) −6.67623 11.5636i −0.310271 0.537404i 0.668150 0.744026i \(-0.267086\pi\)
−0.978421 + 0.206622i \(0.933753\pi\)
\(464\) 0 0
\(465\) 37.5738 + 3.03646i 1.74244 + 0.140813i
\(466\) 0 0
\(467\) 11.4976 0.532047 0.266023 0.963967i \(-0.414290\pi\)
0.266023 + 0.963967i \(0.414290\pi\)
\(468\) 0 0
\(469\) −2.25565 −0.104156
\(470\) 0 0
\(471\) −9.58537 + 13.8843i −0.441671 + 0.639755i
\(472\) 0 0
\(473\) −5.49208 9.51256i −0.252526 0.437388i
\(474\) 0 0
\(475\) −20.0567 + 34.7392i −0.920264 + 1.59394i
\(476\) 0 0
\(477\) −21.6738 + 26.5364i −0.992375 + 1.21502i
\(478\) 0 0
\(479\) 2.48078 4.29684i 0.113350 0.196328i −0.803769 0.594941i \(-0.797175\pi\)
0.917119 + 0.398614i \(0.130509\pi\)
\(480\) 0 0
\(481\) 7.09547 + 12.2897i 0.323525 + 0.560363i
\(482\) 0 0
\(483\) −6.65979 14.0383i −0.303031 0.638764i
\(484\) 0 0
\(485\) 66.7417 3.03059
\(486\) 0 0
\(487\) 26.4801 1.19993 0.599964 0.800027i \(-0.295182\pi\)
0.599964 + 0.800027i \(0.295182\pi\)
\(488\) 0 0
\(489\) 13.5210 + 28.5011i 0.611441 + 1.28887i
\(490\) 0 0
\(491\) 10.7019 + 18.5362i 0.482970 + 0.836529i 0.999809 0.0195540i \(-0.00622463\pi\)
−0.516839 + 0.856083i \(0.672891\pi\)
\(492\) 0 0
\(493\) 5.54543 9.60496i 0.249753 0.432586i
\(494\) 0 0
\(495\) 27.8746 34.1284i 1.25287 1.53396i
\(496\) 0 0
\(497\) −4.96489 + 8.59944i −0.222706 + 0.385738i
\(498\) 0 0
\(499\) 6.63554 + 11.4931i 0.297048 + 0.514501i 0.975459 0.220181i \(-0.0706647\pi\)
−0.678412 + 0.734682i \(0.737331\pi\)
\(500\) 0 0
\(501\) −15.3800 + 22.2778i −0.687128 + 0.995298i
\(502\) 0 0
\(503\) −32.3009 −1.44022 −0.720112 0.693857i \(-0.755910\pi\)
−0.720112 + 0.693857i \(0.755910\pi\)
\(504\) 0 0
\(505\) 34.0409 1.51480
\(506\) 0 0
\(507\) 6.47779 + 0.523491i 0.287689 + 0.0232491i
\(508\) 0 0
\(509\) −11.3401 19.6417i −0.502642 0.870601i −0.999995 0.00305334i \(-0.999028\pi\)
0.497353 0.867548i \(-0.334305\pi\)
\(510\) 0 0
\(511\) 1.25053 2.16598i 0.0553203 0.0958175i
\(512\) 0 0
\(513\) −27.2696 6.72868i −1.20398 0.297079i
\(514\) 0 0
\(515\) 3.45943 5.99190i 0.152441 0.264035i
\(516\) 0 0
\(517\) 3.58639 + 6.21181i 0.157729 + 0.273195i
\(518\) 0 0
\(519\) 18.0926 + 1.46212i 0.794177 + 0.0641800i
\(520\) 0 0
\(521\) −23.2159 −1.01710 −0.508552 0.861031i \(-0.669819\pi\)
−0.508552 + 0.861031i \(0.669819\pi\)
\(522\) 0 0
\(523\) −20.4196 −0.892885 −0.446442 0.894812i \(-0.647309\pi\)
−0.446442 + 0.894812i \(0.647309\pi\)
\(524\) 0 0
\(525\) −7.30246 + 10.5775i −0.318706 + 0.461641i
\(526\) 0 0
\(527\) 11.7191 + 20.2981i 0.510493 + 0.884200i
\(528\) 0 0
\(529\) −28.7377 + 49.7751i −1.24946 + 2.16413i
\(530\) 0 0
\(531\) −3.34294 0.543860i −0.145071 0.0236015i
\(532\) 0 0
\(533\) 5.59027 9.68264i 0.242142 0.419402i
\(534\) 0 0
\(535\) −32.6127 56.4869i −1.40997 2.44214i
\(536\) 0 0
\(537\) −13.4278 28.3047i −0.579454 1.22144i
\(538\) 0 0
\(539\) −4.16771 −0.179516
\(540\) 0 0
\(541\) −39.4226 −1.69491 −0.847455 0.530868i \(-0.821866\pi\)
−0.847455 + 0.530868i \(0.821866\pi\)
\(542\) 0 0
\(543\) 16.6309 + 35.0566i 0.713701 + 1.50442i
\(544\) 0 0
\(545\) −4.70916 8.15651i −0.201718 0.349386i
\(546\) 0 0
\(547\) −12.2911 + 21.2888i −0.525529 + 0.910243i 0.474029 + 0.880509i \(0.342799\pi\)
−0.999558 + 0.0297339i \(0.990534\pi\)
\(548\) 0 0
\(549\) −12.8366 33.8645i −0.547852 1.44530i
\(550\) 0 0
\(551\) 7.89773 13.6793i 0.336455 0.582756i
\(552\) 0 0
\(553\) −3.16763 5.48650i −0.134701 0.233310i
\(554\) 0 0
\(555\) −16.1837 + 23.4420i −0.686962 + 0.995056i
\(556\) 0 0
\(557\) −12.4619 −0.528026 −0.264013 0.964519i \(-0.585046\pi\)
−0.264013 + 0.964519i \(0.585046\pi\)
\(558\) 0 0
\(559\) −8.01475 −0.338988
\(560\) 0 0
\(561\) 27.3092 + 2.20694i 1.15299 + 0.0931773i
\(562\) 0 0
\(563\) 5.24949 + 9.09239i 0.221240 + 0.383198i 0.955185 0.296011i \(-0.0956563\pi\)
−0.733945 + 0.679209i \(0.762323\pi\)
\(564\) 0 0
\(565\) −16.4095 + 28.4221i −0.690352 + 1.19572i
\(566\) 0 0
\(567\) −8.53587 2.85289i −0.358473 0.119810i
\(568\) 0 0
\(569\) 5.11698 8.86288i 0.214515 0.371551i −0.738607 0.674136i \(-0.764516\pi\)
0.953122 + 0.302585i \(0.0978495\pi\)
\(570\) 0 0
\(571\) 2.34986 + 4.07008i 0.0983387 + 0.170328i 0.910997 0.412413i \(-0.135314\pi\)
−0.812658 + 0.582740i \(0.801980\pi\)
\(572\) 0 0
\(573\) 5.85224 + 0.472939i 0.244481 + 0.0197573i
\(574\) 0 0
\(575\) 66.5716 2.77623
\(576\) 0 0
\(577\) −27.3980 −1.14060 −0.570298 0.821438i \(-0.693172\pi\)
−0.570298 + 0.821438i \(0.693172\pi\)
\(578\) 0 0
\(579\) 15.7412 22.8010i 0.654182 0.947576i
\(580\) 0 0
\(581\) −5.52319 9.56645i −0.229141 0.396883i
\(582\) 0 0
\(583\) 23.7995 41.2220i 0.985677 1.70724i
\(584\) 0 0
\(585\) −11.3965 30.0653i −0.471186 1.24305i
\(586\) 0 0
\(587\) 0.334943 0.580138i 0.0138246 0.0239448i −0.859030 0.511925i \(-0.828933\pi\)
0.872855 + 0.487980i \(0.162266\pi\)
\(588\) 0 0
\(589\) 16.6902 + 28.9084i 0.687710 + 1.19115i
\(590\) 0 0
\(591\) −5.01172 10.5643i −0.206155 0.434556i
\(592\) 0 0
\(593\) −9.83861 −0.404023 −0.202012 0.979383i \(-0.564748\pi\)
−0.202012 + 0.979383i \(0.564748\pi\)
\(594\) 0 0
\(595\) −13.3765 −0.548381
\(596\) 0 0
\(597\) 8.47872 + 17.8724i 0.347011 + 0.731470i
\(598\) 0 0
\(599\) 0.0135159 + 0.0234103i 0.000552245 + 0.000956517i 0.866301 0.499522i \(-0.166491\pi\)
−0.865749 + 0.500478i \(0.833158\pi\)
\(600\) 0 0
\(601\) 18.1676 31.4672i 0.741073 1.28358i −0.210934 0.977500i \(-0.567651\pi\)
0.952007 0.306076i \(-0.0990160\pi\)
\(602\) 0 0
\(603\) 6.67913 + 1.08662i 0.271995 + 0.0442506i
\(604\) 0 0
\(605\) −11.2247 + 19.4417i −0.456348 + 0.790418i
\(606\) 0 0
\(607\) −10.6590 18.4620i −0.432637 0.749349i 0.564463 0.825459i \(-0.309083\pi\)
−0.997099 + 0.0761098i \(0.975750\pi\)
\(608\) 0 0
\(609\) 2.87549 4.16512i 0.116521 0.168779i
\(610\) 0 0
\(611\) 5.23372 0.211734
\(612\) 0 0
\(613\) −12.5229 −0.505793 −0.252897 0.967493i \(-0.581383\pi\)
−0.252897 + 0.967493i \(0.581383\pi\)
\(614\) 0 0
\(615\) 22.3700 + 1.80779i 0.902046 + 0.0728973i
\(616\) 0 0
\(617\) 22.2691 + 38.5713i 0.896521 + 1.55282i 0.831911 + 0.554910i \(0.187247\pi\)
0.0646104 + 0.997911i \(0.479420\pi\)
\(618\) 0 0
\(619\) −14.4100 + 24.9588i −0.579186 + 1.00318i 0.416387 + 0.909187i \(0.363296\pi\)
−0.995573 + 0.0939920i \(0.970037\pi\)
\(620\) 0 0
\(621\) 12.9574 + 44.7766i 0.519963 + 1.79682i
\(622\) 0 0
\(623\) −6.73487 + 11.6651i −0.269827 + 0.467354i
\(624\) 0 0
\(625\) 3.51728 + 6.09210i 0.140691 + 0.243684i
\(626\) 0 0
\(627\) 38.8934 + 3.14310i 1.55325 + 0.125523i
\(628\) 0 0
\(629\) −17.7115 −0.706202
\(630\) 0 0
\(631\) −35.3192 −1.40604 −0.703018 0.711172i \(-0.748165\pi\)
−0.703018 + 0.711172i \(0.748165\pi\)
\(632\) 0 0
\(633\) 12.5986 18.2489i 0.500748 0.725328i
\(634\) 0 0
\(635\) −29.5449 51.1733i −1.17246 2.03075i
\(636\) 0 0
\(637\) −1.52051 + 2.63361i −0.0602450 + 0.104347i
\(638\) 0 0
\(639\) 18.8440 23.0718i 0.745458 0.912706i
\(640\) 0 0
\(641\) 18.2261 31.5685i 0.719888 1.24688i −0.241156 0.970486i \(-0.577526\pi\)
0.961044 0.276396i \(-0.0891402\pi\)
\(642\) 0 0
\(643\) −19.8629 34.4036i −0.783317 1.35674i −0.929999 0.367561i \(-0.880193\pi\)
0.146682 0.989184i \(-0.453140\pi\)
\(644\) 0 0
\(645\) −6.89566 14.5355i −0.271516 0.572333i
\(646\) 0 0
\(647\) −0.649904 −0.0255504 −0.0127752 0.999918i \(-0.504067\pi\)
−0.0127752 + 0.999918i \(0.504067\pi\)
\(648\) 0 0
\(649\) 4.70520 0.184695
\(650\) 0 0
\(651\) 4.58449 + 9.66371i 0.179680 + 0.378751i
\(652\) 0 0
\(653\) −7.40239 12.8213i −0.289678 0.501737i 0.684055 0.729430i \(-0.260215\pi\)
−0.973733 + 0.227694i \(0.926881\pi\)
\(654\) 0 0
\(655\) 4.81303 8.33641i 0.188061 0.325731i
\(656\) 0 0
\(657\) −4.74634 + 5.81121i −0.185172 + 0.226717i
\(658\) 0 0
\(659\) 3.53713 6.12649i 0.137787 0.238654i −0.788872 0.614558i \(-0.789334\pi\)
0.926659 + 0.375904i \(0.122668\pi\)
\(660\) 0 0
\(661\) 12.3419 + 21.3767i 0.480043 + 0.831459i 0.999738 0.0228933i \(-0.00728778\pi\)
−0.519695 + 0.854352i \(0.673954\pi\)
\(662\) 0 0
\(663\) 11.3578 16.4517i 0.441102 0.638932i
\(664\) 0 0
\(665\) −19.0506 −0.738750
\(666\) 0 0
\(667\) −26.2139 −1.01501
\(668\) 0 0
\(669\) 20.2460 + 1.63615i 0.782756 + 0.0632571i
\(670\) 0 0
\(671\) 25.1562 + 43.5718i 0.971143 + 1.68207i
\(672\) 0 0
\(673\) 16.5508 28.6669i 0.637988 1.10503i −0.347886 0.937537i \(-0.613100\pi\)
0.985874 0.167491i \(-0.0535664\pi\)
\(674\) 0 0
\(675\) 26.7186 27.8030i 1.02840 1.07014i
\(676\) 0 0
\(677\) −5.59003 + 9.68222i −0.214842 + 0.372118i −0.953224 0.302265i \(-0.902257\pi\)
0.738381 + 0.674383i \(0.235590\pi\)
\(678\) 0 0
\(679\) 9.46871 + 16.4003i 0.363376 + 0.629385i
\(680\) 0 0
\(681\) 21.7637 + 1.75879i 0.833986 + 0.0673972i
\(682\) 0 0
\(683\) −33.1925 −1.27008 −0.635039 0.772480i \(-0.719016\pi\)
−0.635039 + 0.772480i \(0.719016\pi\)
\(684\) 0 0
\(685\) 57.2965 2.18919
\(686\) 0 0
\(687\) −8.55089 + 12.3859i −0.326237 + 0.472551i
\(688\) 0 0
\(689\) −17.3657 30.0782i −0.661579 1.14589i
\(690\) 0 0
\(691\) −8.96460 + 15.5271i −0.341030 + 0.590680i −0.984624 0.174686i \(-0.944109\pi\)
0.643595 + 0.765367i \(0.277442\pi\)
\(692\) 0 0
\(693\) 12.3409 + 2.00773i 0.468791 + 0.0762672i
\(694\) 0 0
\(695\) −2.30511 + 3.99257i −0.0874379 + 0.151447i
\(696\) 0 0
\(697\) 6.97713 + 12.0847i 0.264277 + 0.457742i
\(698\) 0 0
\(699\) 9.59537 + 20.2262i 0.362930 + 0.765026i
\(700\) 0 0
\(701\) 22.6754 0.856438 0.428219 0.903675i \(-0.359141\pi\)
0.428219 + 0.903675i \(0.359141\pi\)
\(702\) 0 0
\(703\) −25.2245 −0.951358
\(704\) 0 0
\(705\) 4.50294 + 9.49181i 0.169591 + 0.357482i
\(706\) 0 0
\(707\) 4.82940 + 8.36477i 0.181628 + 0.314590i
\(708\) 0 0
\(709\) −4.47722 + 7.75477i −0.168146 + 0.291237i −0.937768 0.347263i \(-0.887111\pi\)
0.769622 + 0.638499i \(0.220445\pi\)
\(710\) 0 0
\(711\) 6.73655 + 17.7719i 0.252640 + 0.666497i
\(712\) 0 0
\(713\) 27.6989 47.9759i 1.03733 1.79671i
\(714\) 0 0
\(715\) 22.3339 + 38.6835i 0.835241 + 1.44668i
\(716\) 0 0
\(717\) −21.2592 + 30.7937i −0.793939 + 1.15001i
\(718\) 0 0
\(719\) −33.9896 −1.26760 −0.633799 0.773498i \(-0.718505\pi\)
−0.633799 + 0.773498i \(0.718505\pi\)
\(720\) 0 0
\(721\) 1.96317 0.0731122
\(722\) 0 0
\(723\) −5.73792 0.463700i −0.213396 0.0172452i
\(724\) 0 0
\(725\) 10.8425 + 18.7797i 0.402680 + 0.697461i
\(726\) 0 0
\(727\) −6.95103 + 12.0395i −0.257799 + 0.446521i −0.965652 0.259839i \(-0.916331\pi\)
0.707853 + 0.706360i \(0.249664\pi\)
\(728\) 0 0
\(729\) 23.9010 + 12.5596i 0.885221 + 0.465171i
\(730\) 0 0
\(731\) 5.00154 8.66291i 0.184988 0.320409i
\(732\) 0 0
\(733\) −14.3180 24.7996i −0.528849 0.915993i −0.999434 0.0336388i \(-0.989290\pi\)
0.470585 0.882355i \(-0.344043\pi\)
\(734\) 0 0
\(735\) −6.08449 0.491707i −0.224430 0.0181369i
\(736\) 0 0
\(737\) −9.40089 −0.346286
\(738\) 0 0
\(739\) 16.2398 0.597389 0.298695 0.954349i \(-0.403449\pi\)
0.298695 + 0.954349i \(0.403449\pi\)
\(740\) 0 0
\(741\) 16.1757 23.4303i 0.594230 0.860735i
\(742\) 0 0
\(743\) −5.10227 8.83739i −0.187184 0.324213i 0.757126 0.653269i \(-0.226603\pi\)
−0.944310 + 0.329056i \(0.893269\pi\)
\(744\) 0 0
\(745\) 23.0696 39.9577i 0.845204 1.46394i
\(746\) 0 0
\(747\) 11.7461 + 30.9876i 0.429766 + 1.13378i
\(748\) 0 0
\(749\) 9.25359 16.0277i 0.338119 0.585639i
\(750\) 0 0
\(751\) −5.09533 8.82537i −0.185931 0.322042i 0.757959 0.652303i \(-0.226197\pi\)
−0.943890 + 0.330260i \(0.892863\pi\)
\(752\) 0 0
\(753\) 1.98936 + 4.19340i 0.0724963 + 0.152816i
\(754\) 0 0
\(755\) 50.1113 1.82374
\(756\) 0 0
\(757\) −50.9880 −1.85319 −0.926595 0.376061i \(-0.877278\pi\)
−0.926595 + 0.376061i \(0.877278\pi\)
\(758\) 0 0
\(759\) −27.7561 58.5075i −1.00748 2.12369i
\(760\) 0 0
\(761\) 4.54569 + 7.87336i 0.164781 + 0.285409i 0.936578 0.350460i \(-0.113975\pi\)
−0.771796 + 0.635870i \(0.780642\pi\)
\(762\) 0 0
\(763\) 1.33619 2.31434i 0.0483732 0.0837848i
\(764\) 0 0
\(765\) 39.6086 + 6.44388i 1.43205 + 0.232979i
\(766\) 0 0
\(767\) 1.71661 2.97325i 0.0619831 0.107358i
\(768\) 0 0
\(769\) −10.4748 18.1429i −0.377731 0.654250i 0.613001 0.790082i \(-0.289962\pi\)
−0.990732 + 0.135833i \(0.956629\pi\)
\(770\) 0 0
\(771\) 7.38899 10.7029i 0.266108 0.385455i
\(772\) 0 0
\(773\) −0.323229 −0.0116258 −0.00581288 0.999983i \(-0.501850\pi\)
−0.00581288 + 0.999983i \(0.501850\pi\)
\(774\) 0 0
\(775\) −45.8268 −1.64615
\(776\) 0 0
\(777\) −8.05634 0.651059i −0.289019 0.0233566i
\(778\) 0 0
\(779\) 9.93674 + 17.2109i 0.356021 + 0.616646i
\(780\) 0 0
\(781\) −20.6922 + 35.8400i −0.740426 + 1.28246i
\(782\) 0 0
\(783\) −10.5210 + 10.9480i −0.375990 + 0.391249i
\(784\) 0 0
\(785\) 17.1650 29.7307i 0.612646 1.06113i
\(786\) 0 0
\(787\) 27.3607 + 47.3901i 0.975302 + 1.68927i 0.678934 + 0.734200i \(0.262442\pi\)
0.296369 + 0.955074i \(0.404224\pi\)
\(788\) 0 0
\(789\) −32.2862 2.60916i −1.14942 0.0928884i
\(790\) 0 0
\(791\) −9.31211 −0.331100
\(792\) 0 0
\(793\) 36.7111 1.30365
\(794\) 0 0
\(795\) 39.6086 57.3726i 1.40477 2.03480i
\(796\) 0 0
\(797\) −15.0498 26.0670i −0.533091 0.923341i −0.999253 0.0386418i \(-0.987697\pi\)
0.466162 0.884699i \(-0.345636\pi\)
\(798\) 0 0
\(799\) −3.26606 + 5.65698i −0.115545 + 0.200130i
\(800\) 0 0
\(801\) 25.5619 31.2969i 0.903186 1.10582i
\(802\) 0 0
\(803\) 5.21185 9.02720i 0.183922 0.318563i
\(804\) 0 0
\(805\) 15.8080 + 27.3803i 0.557160 + 0.965030i
\(806\) 0 0
\(807\) −12.6231 26.6085i −0.444355 0.936662i
\(808\) 0 0
\(809\) 8.81348 0.309866 0.154933 0.987925i \(-0.450484\pi\)
0.154933 + 0.987925i \(0.450484\pi\)
\(810\) 0 0
\(811\) −42.3997 −1.48886 −0.744428 0.667702i \(-0.767278\pi\)
−0.744428 + 0.667702i \(0.767278\pi\)
\(812\) 0 0
\(813\) 17.1808 + 36.2157i 0.602557 + 1.27014i
\(814\) 0 0
\(815\) −32.0942 55.5888i −1.12421 1.94719i
\(816\) 0 0
\(817\) 7.12313 12.3376i 0.249207 0.431639i
\(818\) 0 0
\(819\) 5.77105 7.06581i 0.201657 0.246900i
\(820\) 0 0
\(821\) 9.67904 16.7646i 0.337801 0.585088i −0.646218 0.763153i \(-0.723650\pi\)
0.984019 + 0.178065i \(0.0569836\pi\)
\(822\) 0 0
\(823\) −12.2996 21.3035i −0.428737 0.742594i 0.568024 0.823012i \(-0.307708\pi\)
−0.996761 + 0.0804178i \(0.974375\pi\)
\(824\) 0 0
\(825\) −30.4345 + 44.0841i −1.05960 + 1.53481i
\(826\) 0 0
\(827\) −39.3747 −1.36919 −0.684597 0.728922i \(-0.740022\pi\)
−0.684597 + 0.728922i \(0.740022\pi\)
\(828\) 0 0
\(829\) 7.25597 0.252010 0.126005 0.992030i \(-0.459784\pi\)
0.126005 + 0.992030i \(0.459784\pi\)
\(830\) 0 0
\(831\) 23.9314 + 1.93397i 0.830170 + 0.0670888i
\(832\) 0 0
\(833\) −1.89773 3.28696i −0.0657524 0.113886i
\(834\) 0 0
\(835\) 27.5418 47.7038i 0.953123 1.65086i
\(836\) 0 0
\(837\) −8.91965 30.8234i −0.308308 1.06541i
\(838\) 0 0
\(839\) 5.72910 9.92309i 0.197790 0.342583i −0.750021 0.661414i \(-0.769957\pi\)
0.947812 + 0.318831i \(0.103290\pi\)
\(840\) 0 0
\(841\) 10.2306 + 17.7198i 0.352778 + 0.611029i
\(842\) 0 0
\(843\) −15.7769 1.27499i −0.543387 0.0439129i
\(844\) 0 0
\(845\) −13.2238 −0.454913
\(846\) 0 0
\(847\) −6.36982 −0.218869
\(848\) 0 0
\(849\) −6.79814 + 9.84703i −0.233312 + 0.337949i
\(850\) 0 0
\(851\) 20.9311 + 36.2537i 0.717509 + 1.24276i
\(852\) 0 0
\(853\) −25.1979 + 43.6440i −0.862758 + 1.49434i 0.00649778 + 0.999979i \(0.497932\pi\)
−0.869256 + 0.494362i \(0.835402\pi\)
\(854\) 0 0
\(855\) 56.4101 + 9.17730i 1.92918 + 0.313857i
\(856\) 0 0
\(857\) 24.0113 41.5888i 0.820211 1.42065i −0.0853131 0.996354i \(-0.527189\pi\)
0.905525 0.424294i \(-0.139478\pi\)
\(858\) 0 0
\(859\) 19.1422 + 33.1552i 0.653123 + 1.13124i 0.982361 + 0.186995i \(0.0598747\pi\)
−0.329238 + 0.944247i \(0.606792\pi\)
\(860\) 0 0
\(861\) 2.72943 + 5.75340i 0.0930187 + 0.196075i
\(862\) 0 0
\(863\) −13.2049 −0.449500 −0.224750 0.974416i \(-0.572157\pi\)
−0.224750 + 0.974416i \(0.572157\pi\)
\(864\) 0 0
\(865\) −36.9344 −1.25581
\(866\) 0 0
\(867\) −1.92613 4.06011i −0.0654147 0.137889i
\(868\) 0 0
\(869\) −13.2018 22.8662i −0.447840 0.775681i
\(870\) 0 0
\(871\) −3.42974 + 5.94049i −0.116212 + 0.201286i
\(872\) 0 0
\(873\) −20.1369 53.1238i −0.681532 1.79797i
\(874\) 0 0
\(875\) 4.26606 7.38904i 0.144219 0.249795i
\(876\) 0 0
\(877\) −11.9815 20.7526i −0.404586 0.700764i 0.589687 0.807632i \(-0.299251\pi\)
−0.994273 + 0.106868i \(0.965918\pi\)
\(878\) 0 0
\(879\) −18.3047 + 26.5142i −0.617403 + 0.894302i
\(880\) 0 0
\(881\) 39.3964 1.32730 0.663650 0.748044i \(-0.269007\pi\)
0.663650 + 0.748044i \(0.269007\pi\)
\(882\) 0 0
\(883\) 18.3593 0.617840 0.308920 0.951088i \(-0.400033\pi\)
0.308920 + 0.951088i \(0.400033\pi\)
\(884\) 0 0
\(885\) 6.86917 + 0.555120i 0.230905 + 0.0186602i
\(886\) 0 0
\(887\) −15.3324 26.5566i −0.514813 0.891682i −0.999852 0.0171897i \(-0.994528\pi\)
0.485039 0.874492i \(-0.338805\pi\)
\(888\) 0 0
\(889\) 8.38313 14.5200i 0.281161 0.486985i
\(890\) 0 0
\(891\) −35.5750 11.8900i −1.19181 0.398331i
\(892\) 0 0
\(893\) −4.65148 + 8.05660i −0.155656 + 0.269604i
\(894\) 0 0
\(895\) 31.8730 + 55.2057i 1.06540 + 1.84532i
\(896\) 0 0
\(897\) −47.0976 3.80611i −1.57254 0.127082i
\(898\) 0 0
\(899\) 18.0452 0.601842
\(900\) 0 0
\(901\) 43.3476 1.44412
\(902\) 0 0
\(903\) 2.59347 3.75661i 0.0863052 0.125012i
\(904\) 0 0
\(905\) −39.4761 68.3745i −1.31223 2.27285i
\(906\) 0 0
\(907\) 20.9899 36.3556i 0.696960 1.20717i −0.272556 0.962140i \(-0.587869\pi\)
0.969516 0.245030i \(-0.0787977\pi\)
\(908\) 0 0
\(909\) −10.2706 27.0952i −0.340655 0.898690i
\(910\) 0 0
\(911\) −18.5714 + 32.1667i −0.615299 + 1.06573i 0.375033 + 0.927012i \(0.377631\pi\)
−0.990332 + 0.138718i \(0.955702\pi\)
\(912\) 0 0
\(913\) −23.0191 39.8702i −0.761820 1.31951i
\(914\) 0 0
\(915\) 31.5852 + 66.5788i 1.04417 + 2.20103i
\(916\) 0 0
\(917\) 2.73131 0.0901960
\(918\) 0 0
\(919\) 16.5903 0.547263 0.273632 0.961835i \(-0.411775\pi\)
0.273632 + 0.961835i \(0.411775\pi\)
\(920\) 0 0
\(921\) 10.3602 + 21.8385i 0.341381 + 0.719602i
\(922\) 0 0
\(923\) 15.0984 + 26.1511i 0.496969 + 0.860775i
\(924\) 0 0
\(925\) 17.3148 29.9902i 0.569308 0.986071i
\(926\) 0 0
\(927\) −5.81308 0.945724i −0.190926 0.0310616i
\(928\) 0 0
\(929\) 18.5800 32.1814i 0.609589 1.05584i −0.381719 0.924278i \(-0.624668\pi\)
0.991308 0.131560i \(-0.0419987\pi\)
\(930\) 0 0
\(931\) −2.70272 4.68125i −0.0885781 0.153422i
\(932\) 0 0
\(933\) −17.4399 + 25.2615i −0.570956 + 0.827024i
\(934\) 0 0
\(935\) −55.7492 −1.82319
\(936\) 0 0
\(937\) −54.5323 −1.78149 −0.890746 0.454502i \(-0.849817\pi\)
−0.890746 + 0.454502i \(0.849817\pi\)
\(938\) 0 0
\(939\) −25.4925 2.06014i −0.831917 0.0672300i
\(940\) 0 0
\(941\) −20.9394 36.2682i −0.682606 1.18231i −0.974183 0.225761i \(-0.927513\pi\)
0.291577 0.956547i \(-0.405820\pi\)
\(942\) 0 0
\(943\) 16.4909 28.5630i 0.537017 0.930141i
\(944\) 0 0
\(945\) 17.7797 + 4.38708i 0.578374 + 0.142712i
\(946\) 0 0
\(947\) 6.71450 11.6299i 0.218192 0.377919i −0.736063 0.676913i \(-0.763317\pi\)
0.954255 + 0.298993i \(0.0966508\pi\)
\(948\) 0 0
\(949\) −3.80290 6.58682i −0.123447 0.213817i
\(950\) 0 0
\(951\) 4.44794 + 0.359453i 0.144234 + 0.0116560i
\(952\) 0 0
\(953\) −22.1328 −0.716950 −0.358475 0.933539i \(-0.616703\pi\)
−0.358475 + 0.933539i \(0.616703\pi\)
\(954\) 0 0
\(955\) −11.9468 −0.386590
\(956\) 0 0
\(957\) 11.9842 17.3590i 0.387395 0.561137i
\(958\) 0 0
\(959\) 8.12870 + 14.0793i 0.262489 + 0.454645i
\(960\) 0 0
\(961\) −3.56746 + 6.17902i −0.115079 + 0.199323i
\(962\) 0 0
\(963\) −35.1216 + 43.0013i −1.13178 + 1.38570i
\(964\) 0 0
\(965\) −28.1886 + 48.8241i −0.907423 + 1.57170i
\(966\) 0 0
\(967\) 14.9516 + 25.8969i 0.480811 + 0.832788i 0.999758 0.0220180i \(-0.00700911\pi\)
−0.518947 + 0.854806i \(0.673676\pi\)
\(968\) 0 0
\(969\) 15.2309 + 32.1054i 0.489286 + 1.03137i
\(970\) 0 0
\(971\) 17.0708 0.547827 0.273914 0.961754i \(-0.411682\pi\)
0.273914 + 0.961754i \(0.411682\pi\)
\(972\) 0 0
\(973\) −1.30811 −0.0419362
\(974\) 0 0
\(975\) 16.7536 + 35.3151i 0.536544 + 1.13099i
\(976\) 0 0
\(977\) 16.6299 + 28.8038i 0.532037 + 0.921515i 0.999300 + 0.0373970i \(0.0119066\pi\)
−0.467263 + 0.884118i \(0.654760\pi\)
\(978\) 0 0
\(979\) −28.0690 + 48.6169i −0.897089 + 1.55380i
\(980\) 0 0
\(981\) −5.07143 + 6.20924i −0.161918 + 0.198246i
\(982\) 0 0
\(983\) 13.2794 23.0006i 0.423548 0.733606i −0.572736 0.819740i \(-0.694118\pi\)
0.996284 + 0.0861339i \(0.0274513\pi\)
\(984\) 0 0
\(985\) 11.8961 + 20.6046i 0.379041 + 0.656518i
\(986\) 0 0
\(987\) −1.69356 + 2.45311i −0.0539067 + 0.0780833i
\(988\) 0 0
\(989\) −23.6429 −0.751800
\(990\) 0 0
\(991\) 48.1268 1.52880 0.764398 0.644744i \(-0.223036\pi\)
0.764398 + 0.644744i \(0.223036\pi\)
\(992\) 0 0
\(993\) 0.548232 + 0.0443044i 0.0173976 + 0.00140596i
\(994\) 0 0
\(995\) −20.1256 34.8585i −0.638023 1.10509i
\(996\) 0 0
\(997\) −9.67869 + 16.7640i −0.306527 + 0.530921i −0.977600 0.210471i \(-0.932500\pi\)
0.671073 + 0.741391i \(0.265834\pi\)
\(998\) 0 0
\(999\) 23.5417 + 5.80884i 0.744828 + 0.183783i
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 504.2.r.f.169.4 10
3.2 odd 2 1512.2.r.f.505.5 10
4.3 odd 2 1008.2.r.n.673.2 10
9.2 odd 6 4536.2.a.bc.1.1 5
9.4 even 3 inner 504.2.r.f.337.4 yes 10
9.5 odd 6 1512.2.r.f.1009.5 10
9.7 even 3 4536.2.a.bd.1.5 5
12.11 even 2 3024.2.r.n.2017.5 10
36.7 odd 6 9072.2.a.cn.1.5 5
36.11 even 6 9072.2.a.cm.1.1 5
36.23 even 6 3024.2.r.n.1009.5 10
36.31 odd 6 1008.2.r.n.337.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.f.169.4 10 1.1 even 1 trivial
504.2.r.f.337.4 yes 10 9.4 even 3 inner
1008.2.r.n.337.2 10 36.31 odd 6
1008.2.r.n.673.2 10 4.3 odd 2
1512.2.r.f.505.5 10 3.2 odd 2
1512.2.r.f.1009.5 10 9.5 odd 6
3024.2.r.n.1009.5 10 36.23 even 6
3024.2.r.n.2017.5 10 12.11 even 2
4536.2.a.bc.1.1 5 9.2 odd 6
4536.2.a.bd.1.5 5 9.7 even 3
9072.2.a.cm.1.1 5 36.11 even 6
9072.2.a.cn.1.5 5 36.7 odd 6