Properties

Label 504.2.t.d.193.6
Level $504$
Weight $2$
Character 504.193
Analytic conductor $4.024$
Analytic rank $0$
Dimension $22$
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(193,504)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.t (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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.6
Character \(\chi\) \(=\) 504.193
Dual form 504.2.t.d.457.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.371921 + 1.69165i) q^{3} +1.68316 q^{5} +(0.960133 + 2.46539i) q^{7} +(-2.72335 + 1.25832i) q^{9} +O(q^{10})\) \(q+(0.371921 + 1.69165i) q^{3} +1.68316 q^{5} +(0.960133 + 2.46539i) q^{7} +(-2.72335 + 1.25832i) q^{9} +1.24498 q^{11} +(1.96039 - 3.39550i) q^{13} +(0.626001 + 2.84731i) q^{15} +(-1.62691 + 2.81788i) q^{17} +(2.36192 + 4.09097i) q^{19} +(-3.81348 + 2.54114i) q^{21} -0.398135 q^{23} -2.16699 q^{25} +(-3.14150 - 4.13895i) q^{27} +(-3.19896 - 5.54076i) q^{29} +(0.289184 + 0.500881i) q^{31} +(0.463034 + 2.10607i) q^{33} +(1.61605 + 4.14963i) q^{35} +(2.72146 + 4.71371i) q^{37} +(6.47311 + 2.05344i) q^{39} +(4.20216 - 7.27836i) q^{41} +(2.46299 + 4.26603i) q^{43} +(-4.58382 + 2.11795i) q^{45} +(0.212595 - 0.368225i) q^{47} +(-5.15629 + 4.73420i) q^{49} +(-5.37195 - 1.70412i) q^{51} +(-0.466315 + 0.807681i) q^{53} +2.09550 q^{55} +(-6.04204 + 5.51707i) q^{57} +(-3.02527 - 5.23992i) q^{59} +(-5.10459 + 8.84140i) q^{61} +(-5.71702 - 5.50596i) q^{63} +(3.29965 - 5.71516i) q^{65} +(4.70976 + 8.15754i) q^{67} +(-0.148075 - 0.673505i) q^{69} +8.46617 q^{71} +(6.82340 - 11.8185i) q^{73} +(-0.805948 - 3.66578i) q^{75} +(1.19535 + 3.06936i) q^{77} +(2.76670 - 4.79207i) q^{79} +(5.83327 - 6.85369i) q^{81} +(-8.03669 - 13.9199i) q^{83} +(-2.73833 + 4.74293i) q^{85} +(8.18325 - 7.47224i) q^{87} +(-6.03776 - 10.4577i) q^{89} +(10.2535 + 1.57300i) q^{91} +(-0.739760 + 0.675485i) q^{93} +(3.97549 + 6.88575i) q^{95} +(-5.86046 - 10.1506i) q^{97} +(-3.39052 + 1.56658i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{3} - 6 q^{5} + 7 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 2 q^{3} - 6 q^{5} + 7 q^{7} - 8 q^{9} + 6 q^{11} - 3 q^{13} - q^{15} + 7 q^{17} - q^{19} - 15 q^{21} - 4 q^{23} + 20 q^{25} - 4 q^{27} + 9 q^{29} - 4 q^{31} - 31 q^{33} + 14 q^{35} + 2 q^{37} + 8 q^{39} + 16 q^{41} + 22 q^{45} + 5 q^{47} - 15 q^{49} + 7 q^{51} + 11 q^{53} + 22 q^{55} + 7 q^{57} - 19 q^{59} - 13 q^{61} + 21 q^{63} + 13 q^{65} + 26 q^{67} - 4 q^{69} - 48 q^{71} - 35 q^{73} - 8 q^{75} - 4 q^{77} + 10 q^{79} - 8 q^{81} - 28 q^{83} - 20 q^{85} + 9 q^{87} + 6 q^{89} - 37 q^{91} - 32 q^{93} + 12 q^{95} - 29 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

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

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
\(3\) 0.371921 + 1.69165i 0.214729 + 0.976674i
\(4\) 0 0
\(5\) 1.68316 0.752730 0.376365 0.926471i \(-0.377174\pi\)
0.376365 + 0.926471i \(0.377174\pi\)
\(6\) 0 0
\(7\) 0.960133 + 2.46539i 0.362896 + 0.931830i
\(8\) 0 0
\(9\) −2.72335 + 1.25832i −0.907783 + 0.419440i
\(10\) 0 0
\(11\) 1.24498 0.375376 0.187688 0.982229i \(-0.439901\pi\)
0.187688 + 0.982229i \(0.439901\pi\)
\(12\) 0 0
\(13\) 1.96039 3.39550i 0.543715 0.941743i −0.454971 0.890506i \(-0.650350\pi\)
0.998687 0.0512366i \(-0.0163162\pi\)
\(14\) 0 0
\(15\) 0.626001 + 2.84731i 0.161633 + 0.735172i
\(16\) 0 0
\(17\) −1.62691 + 2.81788i −0.394582 + 0.683437i −0.993048 0.117712i \(-0.962444\pi\)
0.598465 + 0.801149i \(0.295777\pi\)
\(18\) 0 0
\(19\) 2.36192 + 4.09097i 0.541863 + 0.938534i 0.998797 + 0.0490333i \(0.0156140\pi\)
−0.456935 + 0.889500i \(0.651053\pi\)
\(20\) 0 0
\(21\) −3.81348 + 2.54114i −0.832169 + 0.554522i
\(22\) 0 0
\(23\) −0.398135 −0.0830170 −0.0415085 0.999138i \(-0.513216\pi\)
−0.0415085 + 0.999138i \(0.513216\pi\)
\(24\) 0 0
\(25\) −2.16699 −0.433397
\(26\) 0 0
\(27\) −3.14150 4.13895i −0.604583 0.796542i
\(28\) 0 0
\(29\) −3.19896 5.54076i −0.594032 1.02889i −0.993683 0.112226i \(-0.964202\pi\)
0.399651 0.916667i \(-0.369131\pi\)
\(30\) 0 0
\(31\) 0.289184 + 0.500881i 0.0519389 + 0.0899608i 0.890826 0.454345i \(-0.150127\pi\)
−0.838887 + 0.544306i \(0.816793\pi\)
\(32\) 0 0
\(33\) 0.463034 + 2.10607i 0.0806039 + 0.366620i
\(34\) 0 0
\(35\) 1.61605 + 4.14963i 0.273163 + 0.701416i
\(36\) 0 0
\(37\) 2.72146 + 4.71371i 0.447405 + 0.774928i 0.998216 0.0597015i \(-0.0190149\pi\)
−0.550811 + 0.834630i \(0.685682\pi\)
\(38\) 0 0
\(39\) 6.47311 + 2.05344i 1.03653 + 0.328813i
\(40\) 0 0
\(41\) 4.20216 7.27836i 0.656267 1.13669i −0.325307 0.945608i \(-0.605468\pi\)
0.981574 0.191080i \(-0.0611990\pi\)
\(42\) 0 0
\(43\) 2.46299 + 4.26603i 0.375603 + 0.650563i 0.990417 0.138109i \(-0.0441024\pi\)
−0.614814 + 0.788672i \(0.710769\pi\)
\(44\) 0 0
\(45\) −4.58382 + 2.11795i −0.683316 + 0.315725i
\(46\) 0 0
\(47\) 0.212595 0.368225i 0.0310101 0.0537112i −0.850104 0.526615i \(-0.823461\pi\)
0.881114 + 0.472904i \(0.156794\pi\)
\(48\) 0 0
\(49\) −5.15629 + 4.73420i −0.736613 + 0.676315i
\(50\) 0 0
\(51\) −5.37195 1.70412i −0.752223 0.238625i
\(52\) 0 0
\(53\) −0.466315 + 0.807681i −0.0640533 + 0.110944i −0.896274 0.443501i \(-0.853736\pi\)
0.832220 + 0.554445i \(0.187069\pi\)
\(54\) 0 0
\(55\) 2.09550 0.282557
\(56\) 0 0
\(57\) −6.04204 + 5.51707i −0.800288 + 0.730753i
\(58\) 0 0
\(59\) −3.02527 5.23992i −0.393856 0.682179i 0.599098 0.800676i \(-0.295526\pi\)
−0.992954 + 0.118496i \(0.962193\pi\)
\(60\) 0 0
\(61\) −5.10459 + 8.84140i −0.653575 + 1.13203i 0.328674 + 0.944444i \(0.393398\pi\)
−0.982249 + 0.187582i \(0.939935\pi\)
\(62\) 0 0
\(63\) −5.71702 5.50596i −0.720277 0.693686i
\(64\) 0 0
\(65\) 3.29965 5.71516i 0.409271 0.708878i
\(66\) 0 0
\(67\) 4.70976 + 8.15754i 0.575389 + 0.996602i 0.995999 + 0.0893612i \(0.0284825\pi\)
−0.420611 + 0.907241i \(0.638184\pi\)
\(68\) 0 0
\(69\) −0.148075 0.673505i −0.0178261 0.0810805i
\(70\) 0 0
\(71\) 8.46617 1.00475 0.502375 0.864650i \(-0.332460\pi\)
0.502375 + 0.864650i \(0.332460\pi\)
\(72\) 0 0
\(73\) 6.82340 11.8185i 0.798619 1.38325i −0.121897 0.992543i \(-0.538898\pi\)
0.920516 0.390705i \(-0.127769\pi\)
\(74\) 0 0
\(75\) −0.805948 3.66578i −0.0930628 0.423288i
\(76\) 0 0
\(77\) 1.19535 + 3.06936i 0.136222 + 0.349786i
\(78\) 0 0
\(79\) 2.76670 4.79207i 0.311278 0.539149i −0.667361 0.744734i \(-0.732576\pi\)
0.978639 + 0.205585i \(0.0659096\pi\)
\(80\) 0 0
\(81\) 5.83327 6.85369i 0.648141 0.761521i
\(82\) 0 0
\(83\) −8.03669 13.9199i −0.882141 1.52791i −0.848956 0.528463i \(-0.822768\pi\)
−0.0331848 0.999449i \(-0.510565\pi\)
\(84\) 0 0
\(85\) −2.73833 + 4.74293i −0.297014 + 0.514444i
\(86\) 0 0
\(87\) 8.18325 7.47224i 0.877337 0.801108i
\(88\) 0 0
\(89\) −6.03776 10.4577i −0.640001 1.10851i −0.985432 0.170070i \(-0.945601\pi\)
0.345431 0.938444i \(-0.387733\pi\)
\(90\) 0 0
\(91\) 10.2535 + 1.57300i 1.07486 + 0.164895i
\(92\) 0 0
\(93\) −0.739760 + 0.675485i −0.0767096 + 0.0700445i
\(94\) 0 0
\(95\) 3.97549 + 6.88575i 0.407876 + 0.706463i
\(96\) 0 0
\(97\) −5.86046 10.1506i −0.595040 1.03064i −0.993541 0.113472i \(-0.963803\pi\)
0.398501 0.917168i \(-0.369530\pi\)
\(98\) 0 0
\(99\) −3.39052 + 1.56658i −0.340760 + 0.157447i
\(100\) 0 0
\(101\) 5.40605 0.537922 0.268961 0.963151i \(-0.413320\pi\)
0.268961 + 0.963151i \(0.413320\pi\)
\(102\) 0 0
\(103\) 14.6204 1.44059 0.720294 0.693669i \(-0.244007\pi\)
0.720294 + 0.693669i \(0.244007\pi\)
\(104\) 0 0
\(105\) −6.41868 + 4.27713i −0.626399 + 0.417405i
\(106\) 0 0
\(107\) −3.40209 5.89259i −0.328892 0.569658i 0.653400 0.757013i \(-0.273342\pi\)
−0.982292 + 0.187354i \(0.940009\pi\)
\(108\) 0 0
\(109\) 8.37636 14.5083i 0.802310 1.38964i −0.115783 0.993275i \(-0.536938\pi\)
0.918092 0.396367i \(-0.129729\pi\)
\(110\) 0 0
\(111\) −6.96176 + 6.35688i −0.660781 + 0.603368i
\(112\) 0 0
\(113\) 6.77154 11.7287i 0.637013 1.10334i −0.349072 0.937096i \(-0.613503\pi\)
0.986085 0.166243i \(-0.0531636\pi\)
\(114\) 0 0
\(115\) −0.670124 −0.0624894
\(116\) 0 0
\(117\) −1.06621 + 11.7139i −0.0985714 + 1.08295i
\(118\) 0 0
\(119\) −8.50922 1.30541i −0.780039 0.119667i
\(120\) 0 0
\(121\) −9.45002 −0.859093
\(122\) 0 0
\(123\) 13.8753 + 4.40161i 1.25109 + 0.396880i
\(124\) 0 0
\(125\) −12.0632 −1.07896
\(126\) 0 0
\(127\) 10.5904 0.939748 0.469874 0.882734i \(-0.344299\pi\)
0.469874 + 0.882734i \(0.344299\pi\)
\(128\) 0 0
\(129\) −6.30058 + 5.75314i −0.554735 + 0.506536i
\(130\) 0 0
\(131\) −22.3638 −1.95394 −0.976968 0.213387i \(-0.931551\pi\)
−0.976968 + 0.213387i \(0.931551\pi\)
\(132\) 0 0
\(133\) −7.81808 + 9.75094i −0.677914 + 0.845514i
\(134\) 0 0
\(135\) −5.28764 6.96651i −0.455088 0.599581i
\(136\) 0 0
\(137\) 17.9540 1.53391 0.766957 0.641699i \(-0.221770\pi\)
0.766957 + 0.641699i \(0.221770\pi\)
\(138\) 0 0
\(139\) 0.570825 0.988699i 0.0484168 0.0838603i −0.840801 0.541344i \(-0.817916\pi\)
0.889218 + 0.457483i \(0.151249\pi\)
\(140\) 0 0
\(141\) 0.701976 + 0.222685i 0.0591170 + 0.0187535i
\(142\) 0 0
\(143\) 2.44065 4.22733i 0.204098 0.353507i
\(144\) 0 0
\(145\) −5.38434 9.32596i −0.447145 0.774479i
\(146\) 0 0
\(147\) −9.92634 6.96188i −0.818711 0.574206i
\(148\) 0 0
\(149\) 17.6583 1.44663 0.723313 0.690521i \(-0.242619\pi\)
0.723313 + 0.690521i \(0.242619\pi\)
\(150\) 0 0
\(151\) −15.2354 −1.23984 −0.619919 0.784666i \(-0.712834\pi\)
−0.619919 + 0.784666i \(0.712834\pi\)
\(152\) 0 0
\(153\) 0.884835 9.72124i 0.0715347 0.785916i
\(154\) 0 0
\(155\) 0.486741 + 0.843060i 0.0390960 + 0.0677162i
\(156\) 0 0
\(157\) 6.81439 + 11.8029i 0.543847 + 0.941971i 0.998678 + 0.0513933i \(0.0163662\pi\)
−0.454831 + 0.890578i \(0.650300\pi\)
\(158\) 0 0
\(159\) −1.53974 0.488447i −0.122110 0.0387364i
\(160\) 0 0
\(161\) −0.382263 0.981559i −0.0301265 0.0773577i
\(162\) 0 0
\(163\) −4.04726 7.01005i −0.317006 0.549070i 0.662856 0.748747i \(-0.269344\pi\)
−0.979862 + 0.199677i \(0.936011\pi\)
\(164\) 0 0
\(165\) 0.779359 + 3.54484i 0.0606730 + 0.275966i
\(166\) 0 0
\(167\) −2.07739 + 3.59814i −0.160753 + 0.278433i −0.935139 0.354281i \(-0.884726\pi\)
0.774386 + 0.632714i \(0.218059\pi\)
\(168\) 0 0
\(169\) −1.18629 2.05471i −0.0912529 0.158055i
\(170\) 0 0
\(171\) −11.5801 8.16909i −0.885552 0.624706i
\(172\) 0 0
\(173\) −6.91730 + 11.9811i −0.525913 + 0.910907i 0.473632 + 0.880723i \(0.342943\pi\)
−0.999544 + 0.0301845i \(0.990391\pi\)
\(174\) 0 0
\(175\) −2.08059 5.34247i −0.157278 0.403852i
\(176\) 0 0
\(177\) 7.73894 7.06653i 0.581694 0.531153i
\(178\) 0 0
\(179\) −4.71167 + 8.16084i −0.352166 + 0.609970i −0.986629 0.162984i \(-0.947888\pi\)
0.634462 + 0.772954i \(0.281222\pi\)
\(180\) 0 0
\(181\) 1.32133 0.0982136 0.0491068 0.998794i \(-0.484363\pi\)
0.0491068 + 0.998794i \(0.484363\pi\)
\(182\) 0 0
\(183\) −16.8550 5.34686i −1.24596 0.395251i
\(184\) 0 0
\(185\) 4.58064 + 7.93390i 0.336775 + 0.583312i
\(186\) 0 0
\(187\) −2.02546 + 3.50821i −0.148117 + 0.256546i
\(188\) 0 0
\(189\) 7.18787 11.7190i 0.522841 0.852430i
\(190\) 0 0
\(191\) −8.14271 + 14.1036i −0.589186 + 1.02050i 0.405153 + 0.914249i \(0.367218\pi\)
−0.994339 + 0.106251i \(0.966115\pi\)
\(192\) 0 0
\(193\) −1.28077 2.21837i −0.0921921 0.159681i 0.816241 0.577711i \(-0.196054\pi\)
−0.908433 + 0.418030i \(0.862721\pi\)
\(194\) 0 0
\(195\) 10.8952 + 3.45626i 0.780225 + 0.247508i
\(196\) 0 0
\(197\) 21.6916 1.54546 0.772730 0.634735i \(-0.218891\pi\)
0.772730 + 0.634735i \(0.218891\pi\)
\(198\) 0 0
\(199\) −5.59684 + 9.69402i −0.396750 + 0.687191i −0.993323 0.115368i \(-0.963195\pi\)
0.596573 + 0.802559i \(0.296529\pi\)
\(200\) 0 0
\(201\) −12.0480 + 11.0012i −0.849803 + 0.775966i
\(202\) 0 0
\(203\) 10.5887 13.2065i 0.743181 0.926917i
\(204\) 0 0
\(205\) 7.07289 12.2506i 0.493992 0.855620i
\(206\) 0 0
\(207\) 1.08426 0.500981i 0.0753614 0.0348206i
\(208\) 0 0
\(209\) 2.94055 + 5.09318i 0.203402 + 0.352303i
\(210\) 0 0
\(211\) −14.1807 + 24.5616i −0.976237 + 1.69089i −0.300444 + 0.953799i \(0.597135\pi\)
−0.675793 + 0.737092i \(0.736199\pi\)
\(212\) 0 0
\(213\) 3.14875 + 14.3218i 0.215749 + 0.981313i
\(214\) 0 0
\(215\) 4.14560 + 7.18039i 0.282728 + 0.489699i
\(216\) 0 0
\(217\) −0.957211 + 1.19386i −0.0649797 + 0.0810446i
\(218\) 0 0
\(219\) 22.5305 + 7.14726i 1.52247 + 0.482967i
\(220\) 0 0
\(221\) 6.37875 + 11.0483i 0.429081 + 0.743190i
\(222\) 0 0
\(223\) −12.6962 21.9905i −0.850202 1.47259i −0.881026 0.473068i \(-0.843147\pi\)
0.0308242 0.999525i \(-0.490187\pi\)
\(224\) 0 0
\(225\) 5.90146 2.72676i 0.393431 0.181784i
\(226\) 0 0
\(227\) 4.62860 0.307211 0.153606 0.988132i \(-0.450911\pi\)
0.153606 + 0.988132i \(0.450911\pi\)
\(228\) 0 0
\(229\) 2.32592 0.153701 0.0768506 0.997043i \(-0.475514\pi\)
0.0768506 + 0.997043i \(0.475514\pi\)
\(230\) 0 0
\(231\) −4.74771 + 3.16367i −0.312376 + 0.208154i
\(232\) 0 0
\(233\) −6.37989 11.0503i −0.417960 0.723929i 0.577774 0.816197i \(-0.303922\pi\)
−0.995734 + 0.0922683i \(0.970588\pi\)
\(234\) 0 0
\(235\) 0.357830 0.619780i 0.0233423 0.0404300i
\(236\) 0 0
\(237\) 9.13548 + 2.89801i 0.593413 + 0.188246i
\(238\) 0 0
\(239\) −11.0492 + 19.1378i −0.714714 + 1.23792i 0.248355 + 0.968669i \(0.420110\pi\)
−0.963070 + 0.269252i \(0.913223\pi\)
\(240\) 0 0
\(241\) 20.0177 1.28945 0.644726 0.764414i \(-0.276972\pi\)
0.644726 + 0.764414i \(0.276972\pi\)
\(242\) 0 0
\(243\) 13.7635 + 7.31880i 0.882932 + 0.469502i
\(244\) 0 0
\(245\) −8.67884 + 7.96840i −0.554471 + 0.509082i
\(246\) 0 0
\(247\) 18.5212 1.17848
\(248\) 0 0
\(249\) 20.5586 18.7724i 1.30285 1.18965i
\(250\) 0 0
\(251\) −2.12390 −0.134059 −0.0670297 0.997751i \(-0.521352\pi\)
−0.0670297 + 0.997751i \(0.521352\pi\)
\(252\) 0 0
\(253\) −0.495671 −0.0311625
\(254\) 0 0
\(255\) −9.04182 2.86830i −0.566221 0.179620i
\(256\) 0 0
\(257\) −12.7630 −0.796134 −0.398067 0.917356i \(-0.630319\pi\)
−0.398067 + 0.917356i \(0.630319\pi\)
\(258\) 0 0
\(259\) −9.00816 + 11.2352i −0.559740 + 0.698124i
\(260\) 0 0
\(261\) 15.6839 + 11.0641i 0.970810 + 0.684851i
\(262\) 0 0
\(263\) −11.0686 −0.682522 −0.341261 0.939969i \(-0.610854\pi\)
−0.341261 + 0.939969i \(0.610854\pi\)
\(264\) 0 0
\(265\) −0.784881 + 1.35945i −0.0482148 + 0.0835105i
\(266\) 0 0
\(267\) 15.4452 14.1032i 0.945230 0.863102i
\(268\) 0 0
\(269\) −1.77479 + 3.07403i −0.108211 + 0.187427i −0.915046 0.403351i \(-0.867846\pi\)
0.806835 + 0.590777i \(0.201179\pi\)
\(270\) 0 0
\(271\) −0.687666 1.19107i −0.0417727 0.0723525i 0.844383 0.535740i \(-0.179967\pi\)
−0.886156 + 0.463387i \(0.846634\pi\)
\(272\) 0 0
\(273\) 1.15252 + 17.9303i 0.0697534 + 1.08519i
\(274\) 0 0
\(275\) −2.69786 −0.162687
\(276\) 0 0
\(277\) −29.1617 −1.75216 −0.876079 0.482168i \(-0.839849\pi\)
−0.876079 + 0.482168i \(0.839849\pi\)
\(278\) 0 0
\(279\) −1.41782 1.00019i −0.0848824 0.0598797i
\(280\) 0 0
\(281\) 6.29603 + 10.9050i 0.375590 + 0.650540i 0.990415 0.138123i \(-0.0441069\pi\)
−0.614826 + 0.788663i \(0.710774\pi\)
\(282\) 0 0
\(283\) 4.73028 + 8.19309i 0.281186 + 0.487029i 0.971677 0.236312i \(-0.0759389\pi\)
−0.690491 + 0.723341i \(0.742606\pi\)
\(284\) 0 0
\(285\) −10.1697 + 9.28608i −0.602401 + 0.550060i
\(286\) 0 0
\(287\) 21.9786 + 3.37178i 1.29736 + 0.199030i
\(288\) 0 0
\(289\) 3.20636 + 5.55358i 0.188609 + 0.326681i
\(290\) 0 0
\(291\) 14.9917 13.6891i 0.878826 0.802468i
\(292\) 0 0
\(293\) −8.32726 + 14.4232i −0.486484 + 0.842614i −0.999879 0.0155376i \(-0.995054\pi\)
0.513396 + 0.858152i \(0.328387\pi\)
\(294\) 0 0
\(295\) −5.09200 8.81960i −0.296468 0.513497i
\(296\) 0 0
\(297\) −3.91111 5.15292i −0.226946 0.299003i
\(298\) 0 0
\(299\) −0.780502 + 1.35187i −0.0451376 + 0.0781806i
\(300\) 0 0
\(301\) −8.15262 + 10.1682i −0.469909 + 0.586085i
\(302\) 0 0
\(303\) 2.01062 + 9.14513i 0.115507 + 0.525374i
\(304\) 0 0
\(305\) −8.59181 + 14.8815i −0.491966 + 0.852110i
\(306\) 0 0
\(307\) −9.55966 −0.545599 −0.272799 0.962071i \(-0.587949\pi\)
−0.272799 + 0.962071i \(0.587949\pi\)
\(308\) 0 0
\(309\) 5.43762 + 24.7325i 0.309335 + 1.40698i
\(310\) 0 0
\(311\) −13.1851 22.8373i −0.747658 1.29498i −0.948943 0.315449i \(-0.897845\pi\)
0.201284 0.979533i \(-0.435488\pi\)
\(312\) 0 0
\(313\) 6.35091 11.0001i 0.358975 0.621762i −0.628815 0.777555i \(-0.716460\pi\)
0.987790 + 0.155792i \(0.0497931\pi\)
\(314\) 0 0
\(315\) −9.62264 9.26739i −0.542174 0.522159i
\(316\) 0 0
\(317\) −0.0165768 + 0.0287119i −0.000931047 + 0.00161262i −0.866491 0.499193i \(-0.833630\pi\)
0.865560 + 0.500806i \(0.166963\pi\)
\(318\) 0 0
\(319\) −3.98264 6.89813i −0.222985 0.386221i
\(320\) 0 0
\(321\) 8.70289 7.94672i 0.485748 0.443543i
\(322\) 0 0
\(323\) −15.3705 −0.855238
\(324\) 0 0
\(325\) −4.24815 + 7.35801i −0.235645 + 0.408149i
\(326\) 0 0
\(327\) 27.6583 + 8.77392i 1.52951 + 0.485199i
\(328\) 0 0
\(329\) 1.11194 + 0.170584i 0.0613031 + 0.00940461i
\(330\) 0 0
\(331\) 2.42694 4.20358i 0.133397 0.231050i −0.791587 0.611056i \(-0.790745\pi\)
0.924984 + 0.380006i \(0.124078\pi\)
\(332\) 0 0
\(333\) −13.3428 9.41260i −0.731183 0.515807i
\(334\) 0 0
\(335\) 7.92726 + 13.7304i 0.433112 + 0.750173i
\(336\) 0 0
\(337\) −4.32200 + 7.48592i −0.235434 + 0.407784i −0.959399 0.282053i \(-0.908985\pi\)
0.723965 + 0.689837i \(0.242318\pi\)
\(338\) 0 0
\(339\) 22.3592 + 7.09293i 1.21439 + 0.385235i
\(340\) 0 0
\(341\) 0.360028 + 0.623586i 0.0194966 + 0.0337691i
\(342\) 0 0
\(343\) −16.6224 8.16680i −0.897524 0.440966i
\(344\) 0 0
\(345\) −0.249233 1.13361i −0.0134183 0.0610317i
\(346\) 0 0
\(347\) 11.6116 + 20.1119i 0.623344 + 1.07966i 0.988859 + 0.148858i \(0.0475596\pi\)
−0.365515 + 0.930806i \(0.619107\pi\)
\(348\) 0 0
\(349\) −3.76025 6.51295i −0.201282 0.348630i 0.747660 0.664082i \(-0.231177\pi\)
−0.948942 + 0.315452i \(0.897844\pi\)
\(350\) 0 0
\(351\) −20.2124 + 2.55300i −1.07886 + 0.136269i
\(352\) 0 0
\(353\) −0.919056 −0.0489164 −0.0244582 0.999701i \(-0.507786\pi\)
−0.0244582 + 0.999701i \(0.507786\pi\)
\(354\) 0 0
\(355\) 14.2499 0.756306
\(356\) 0 0
\(357\) −0.956458 14.8801i −0.0506211 0.787540i
\(358\) 0 0
\(359\) 8.24300 + 14.2773i 0.435049 + 0.753527i 0.997300 0.0734398i \(-0.0233977\pi\)
−0.562251 + 0.826967i \(0.690064\pi\)
\(360\) 0 0
\(361\) −1.65737 + 2.87066i −0.0872302 + 0.151087i
\(362\) 0 0
\(363\) −3.51466 15.9861i −0.184472 0.839054i
\(364\) 0 0
\(365\) 11.4848 19.8923i 0.601144 1.04121i
\(366\) 0 0
\(367\) −12.6784 −0.661808 −0.330904 0.943664i \(-0.607354\pi\)
−0.330904 + 0.943664i \(0.607354\pi\)
\(368\) 0 0
\(369\) −2.28546 + 25.1092i −0.118976 + 1.30713i
\(370\) 0 0
\(371\) −2.43897 0.374167i −0.126625 0.0194258i
\(372\) 0 0
\(373\) 22.6821 1.17443 0.587217 0.809430i \(-0.300224\pi\)
0.587217 + 0.809430i \(0.300224\pi\)
\(374\) 0 0
\(375\) −4.48654 20.4066i −0.231684 1.05379i
\(376\) 0 0
\(377\) −25.0849 −1.29194
\(378\) 0 0
\(379\) 19.0925 0.980717 0.490358 0.871521i \(-0.336866\pi\)
0.490358 + 0.871521i \(0.336866\pi\)
\(380\) 0 0
\(381\) 3.93880 + 17.9153i 0.201791 + 0.917827i
\(382\) 0 0
\(383\) −6.07925 −0.310635 −0.155318 0.987865i \(-0.549640\pi\)
−0.155318 + 0.987865i \(0.549640\pi\)
\(384\) 0 0
\(385\) 2.01195 + 5.16621i 0.102539 + 0.263295i
\(386\) 0 0
\(387\) −12.0756 8.51865i −0.613838 0.433028i
\(388\) 0 0
\(389\) −0.631562 −0.0320214 −0.0160107 0.999872i \(-0.505097\pi\)
−0.0160107 + 0.999872i \(0.505097\pi\)
\(390\) 0 0
\(391\) 0.647728 1.12190i 0.0327570 0.0567368i
\(392\) 0 0
\(393\) −8.31757 37.8317i −0.419566 1.90836i
\(394\) 0 0
\(395\) 4.65679 8.06579i 0.234308 0.405834i
\(396\) 0 0
\(397\) −18.1830 31.4939i −0.912578 1.58063i −0.810408 0.585865i \(-0.800755\pi\)
−0.102170 0.994767i \(-0.532579\pi\)
\(398\) 0 0
\(399\) −19.4029 9.59887i −0.971359 0.480544i
\(400\) 0 0
\(401\) −31.6137 −1.57871 −0.789357 0.613934i \(-0.789586\pi\)
−0.789357 + 0.613934i \(0.789586\pi\)
\(402\) 0 0
\(403\) 2.26765 0.112960
\(404\) 0 0
\(405\) 9.81829 11.5358i 0.487875 0.573220i
\(406\) 0 0
\(407\) 3.38816 + 5.86847i 0.167945 + 0.290889i
\(408\) 0 0
\(409\) −10.0906 17.4774i −0.498948 0.864203i 0.501051 0.865418i \(-0.332947\pi\)
−0.999999 + 0.00121422i \(0.999614\pi\)
\(410\) 0 0
\(411\) 6.67747 + 30.3718i 0.329375 + 1.49813i
\(412\) 0 0
\(413\) 10.0138 12.4895i 0.492746 0.614567i
\(414\) 0 0
\(415\) −13.5270 23.4294i −0.664014 1.15011i
\(416\) 0 0
\(417\) 1.88483 + 0.597918i 0.0923006 + 0.0292802i
\(418\) 0 0
\(419\) −12.4159 + 21.5049i −0.606555 + 1.05058i 0.385248 + 0.922813i \(0.374116\pi\)
−0.991804 + 0.127772i \(0.959217\pi\)
\(420\) 0 0
\(421\) −5.71841 9.90458i −0.278698 0.482720i 0.692363 0.721549i \(-0.256570\pi\)
−0.971062 + 0.238829i \(0.923236\pi\)
\(422\) 0 0
\(423\) −0.115625 + 1.27032i −0.00562190 + 0.0617650i
\(424\) 0 0
\(425\) 3.52548 6.10631i 0.171011 0.296200i
\(426\) 0 0
\(427\) −26.6986 4.09587i −1.29203 0.198213i
\(428\) 0 0
\(429\) 8.05889 + 2.55649i 0.389087 + 0.123429i
\(430\) 0 0
\(431\) 2.80157 4.85246i 0.134947 0.233735i −0.790630 0.612294i \(-0.790247\pi\)
0.925577 + 0.378559i \(0.123580\pi\)
\(432\) 0 0
\(433\) −4.22555 −0.203067 −0.101534 0.994832i \(-0.532375\pi\)
−0.101534 + 0.994832i \(0.532375\pi\)
\(434\) 0 0
\(435\) 13.7737 12.5769i 0.660398 0.603018i
\(436\) 0 0
\(437\) −0.940366 1.62876i −0.0449838 0.0779142i
\(438\) 0 0
\(439\) 17.7316 30.7120i 0.846281 1.46580i −0.0382233 0.999269i \(-0.512170\pi\)
0.884504 0.466532i \(-0.154497\pi\)
\(440\) 0 0
\(441\) 8.08524 19.3811i 0.385012 0.922912i
\(442\) 0 0
\(443\) 11.4658 19.8593i 0.544755 0.943543i −0.453867 0.891069i \(-0.649956\pi\)
0.998622 0.0524740i \(-0.0167107\pi\)
\(444\) 0 0
\(445\) −10.1625 17.6019i −0.481748 0.834412i
\(446\) 0 0
\(447\) 6.56750 + 29.8717i 0.310632 + 1.41288i
\(448\) 0 0
\(449\) 20.4850 0.966747 0.483373 0.875414i \(-0.339411\pi\)
0.483373 + 0.875414i \(0.339411\pi\)
\(450\) 0 0
\(451\) 5.23161 9.06141i 0.246347 0.426685i
\(452\) 0 0
\(453\) −5.66636 25.7729i −0.266229 1.21092i
\(454\) 0 0
\(455\) 17.2582 + 2.64761i 0.809076 + 0.124122i
\(456\) 0 0
\(457\) −7.72677 + 13.3832i −0.361443 + 0.626038i −0.988199 0.153178i \(-0.951049\pi\)
0.626755 + 0.779216i \(0.284383\pi\)
\(458\) 0 0
\(459\) 16.7740 2.11870i 0.782944 0.0988927i
\(460\) 0 0
\(461\) 14.0815 + 24.3898i 0.655839 + 1.13595i 0.981683 + 0.190523i \(0.0610183\pi\)
−0.325844 + 0.945424i \(0.605648\pi\)
\(462\) 0 0
\(463\) −15.3193 + 26.5338i −0.711948 + 1.23313i 0.252177 + 0.967681i \(0.418854\pi\)
−0.964125 + 0.265449i \(0.914480\pi\)
\(464\) 0 0
\(465\) −1.24513 + 1.13695i −0.0577416 + 0.0527246i
\(466\) 0 0
\(467\) −6.61798 11.4627i −0.306244 0.530429i 0.671294 0.741191i \(-0.265739\pi\)
−0.977537 + 0.210762i \(0.932406\pi\)
\(468\) 0 0
\(469\) −15.5895 + 19.4437i −0.719857 + 0.897827i
\(470\) 0 0
\(471\) −17.4319 + 15.9173i −0.803219 + 0.733429i
\(472\) 0 0
\(473\) 3.06638 + 5.31112i 0.140992 + 0.244206i
\(474\) 0 0
\(475\) −5.11826 8.86508i −0.234842 0.406758i
\(476\) 0 0
\(477\) 0.253618 2.78637i 0.0116124 0.127579i
\(478\) 0 0
\(479\) 14.0872 0.643658 0.321829 0.946798i \(-0.395702\pi\)
0.321829 + 0.946798i \(0.395702\pi\)
\(480\) 0 0
\(481\) 21.3405 0.973044
\(482\) 0 0
\(483\) 1.51828 1.01172i 0.0690842 0.0460347i
\(484\) 0 0
\(485\) −9.86407 17.0851i −0.447904 0.775793i
\(486\) 0 0
\(487\) 3.45654 5.98690i 0.156631 0.271292i −0.777021 0.629475i \(-0.783270\pi\)
0.933652 + 0.358183i \(0.116603\pi\)
\(488\) 0 0
\(489\) 10.3533 9.45372i 0.468192 0.427512i
\(490\) 0 0
\(491\) −15.3481 + 26.5837i −0.692651 + 1.19971i 0.278315 + 0.960490i \(0.410224\pi\)
−0.970966 + 0.239217i \(0.923109\pi\)
\(492\) 0 0
\(493\) 20.8176 0.937578
\(494\) 0 0
\(495\) −5.70677 + 2.63680i −0.256500 + 0.118515i
\(496\) 0 0
\(497\) 8.12865 + 20.8724i 0.364620 + 0.936256i
\(498\) 0 0
\(499\) −22.0371 −0.986518 −0.493259 0.869882i \(-0.664195\pi\)
−0.493259 + 0.869882i \(0.664195\pi\)
\(500\) 0 0
\(501\) −6.85942 2.17599i −0.306456 0.0972159i
\(502\) 0 0
\(503\) 38.9653 1.73737 0.868687 0.495361i \(-0.164964\pi\)
0.868687 + 0.495361i \(0.164964\pi\)
\(504\) 0 0
\(505\) 9.09922 0.404910
\(506\) 0 0
\(507\) 3.03464 2.77097i 0.134773 0.123063i
\(508\) 0 0
\(509\) −28.2941 −1.25412 −0.627058 0.778973i \(-0.715741\pi\)
−0.627058 + 0.778973i \(0.715741\pi\)
\(510\) 0 0
\(511\) 35.6885 + 5.47503i 1.57877 + 0.242201i
\(512\) 0 0
\(513\) 9.51236 22.6277i 0.419981 0.999038i
\(514\) 0 0
\(515\) 24.6084 1.08437
\(516\) 0 0
\(517\) 0.264676 0.458433i 0.0116405 0.0201619i
\(518\) 0 0
\(519\) −22.8405 7.24561i −1.00259 0.318047i
\(520\) 0 0
\(521\) −5.98150 + 10.3603i −0.262054 + 0.453892i −0.966788 0.255581i \(-0.917733\pi\)
0.704733 + 0.709472i \(0.251067\pi\)
\(522\) 0 0
\(523\) −3.15056 5.45693i −0.137764 0.238615i 0.788886 0.614540i \(-0.210658\pi\)
−0.926650 + 0.375925i \(0.877325\pi\)
\(524\) 0 0
\(525\) 8.26376 5.50661i 0.360660 0.240328i
\(526\) 0 0
\(527\) −1.88190 −0.0819767
\(528\) 0 0
\(529\) −22.8415 −0.993108
\(530\) 0 0
\(531\) 14.8324 + 10.4634i 0.643669 + 0.454072i
\(532\) 0 0
\(533\) −16.4758 28.5369i −0.713645 1.23607i
\(534\) 0 0
\(535\) −5.72625 9.91815i −0.247567 0.428799i
\(536\) 0 0
\(537\) −15.5576 4.93529i −0.671362 0.212974i
\(538\) 0 0
\(539\) −6.41948 + 5.89399i −0.276507 + 0.253872i
\(540\) 0 0
\(541\) 9.88191 + 17.1160i 0.424857 + 0.735873i 0.996407 0.0846937i \(-0.0269912\pi\)
−0.571550 + 0.820567i \(0.693658\pi\)
\(542\) 0 0
\(543\) 0.491430 + 2.23522i 0.0210893 + 0.0959227i
\(544\) 0 0
\(545\) 14.0987 24.4197i 0.603923 1.04602i
\(546\) 0 0
\(547\) 21.6125 + 37.4340i 0.924085 + 1.60056i 0.793026 + 0.609188i \(0.208504\pi\)
0.131059 + 0.991375i \(0.458162\pi\)
\(548\) 0 0
\(549\) 2.77626 30.5014i 0.118488 1.30177i
\(550\) 0 0
\(551\) 15.1114 26.1737i 0.643767 1.11504i
\(552\) 0 0
\(553\) 14.4707 + 2.21997i 0.615357 + 0.0944029i
\(554\) 0 0
\(555\) −11.7177 + 10.6996i −0.497390 + 0.454173i
\(556\) 0 0
\(557\) −14.0838 + 24.3938i −0.596748 + 1.03360i 0.396550 + 0.918013i \(0.370207\pi\)
−0.993298 + 0.115584i \(0.963126\pi\)
\(558\) 0 0
\(559\) 19.3137 0.816884
\(560\) 0 0
\(561\) −6.68797 2.12160i −0.282366 0.0895739i
\(562\) 0 0
\(563\) −12.8472 22.2520i −0.541445 0.937810i −0.998821 0.0485366i \(-0.984544\pi\)
0.457377 0.889273i \(-0.348789\pi\)
\(564\) 0 0
\(565\) 11.3976 19.7411i 0.479499 0.830516i
\(566\) 0 0
\(567\) 22.4977 + 7.80082i 0.944815 + 0.327604i
\(568\) 0 0
\(569\) −17.7441 + 30.7336i −0.743870 + 1.28842i 0.206851 + 0.978373i \(0.433679\pi\)
−0.950721 + 0.310048i \(0.899655\pi\)
\(570\) 0 0
\(571\) −10.8412 18.7775i −0.453689 0.785813i 0.544923 0.838486i \(-0.316559\pi\)
−0.998612 + 0.0526737i \(0.983226\pi\)
\(572\) 0 0
\(573\) −26.8868 8.52919i −1.12321 0.356312i
\(574\) 0 0
\(575\) 0.862754 0.0359793
\(576\) 0 0
\(577\) −7.60727 + 13.1762i −0.316695 + 0.548531i −0.979796 0.199998i \(-0.935906\pi\)
0.663102 + 0.748529i \(0.269240\pi\)
\(578\) 0 0
\(579\) 3.27635 2.99168i 0.136160 0.124330i
\(580\) 0 0
\(581\) 26.6018 33.1786i 1.10363 1.37648i
\(582\) 0 0
\(583\) −0.580553 + 1.00555i −0.0240440 + 0.0416455i
\(584\) 0 0
\(585\) −1.79460 + 19.7164i −0.0741976 + 0.815172i
\(586\) 0 0
\(587\) −19.1924 33.2423i −0.792157 1.37206i −0.924629 0.380869i \(-0.875625\pi\)
0.132472 0.991187i \(-0.457708\pi\)
\(588\) 0 0
\(589\) −1.36606 + 2.36608i −0.0562875 + 0.0974928i
\(590\) 0 0
\(591\) 8.06754 + 36.6945i 0.331854 + 1.50941i
\(592\) 0 0
\(593\) −9.25559 16.0311i −0.380081 0.658320i 0.610992 0.791637i \(-0.290771\pi\)
−0.991074 + 0.133316i \(0.957437\pi\)
\(594\) 0 0
\(595\) −14.3223 2.19721i −0.587159 0.0900770i
\(596\) 0 0
\(597\) −18.4805 5.86248i −0.756355 0.239935i
\(598\) 0 0
\(599\) 10.8413 + 18.7776i 0.442962 + 0.767233i 0.997908 0.0646536i \(-0.0205942\pi\)
−0.554946 + 0.831887i \(0.687261\pi\)
\(600\) 0 0
\(601\) 3.95776 + 6.85505i 0.161441 + 0.279623i 0.935386 0.353630i \(-0.115053\pi\)
−0.773945 + 0.633253i \(0.781719\pi\)
\(602\) 0 0
\(603\) −23.0911 16.2895i −0.940343 0.663358i
\(604\) 0 0
\(605\) −15.9059 −0.646665
\(606\) 0 0
\(607\) 35.9247 1.45814 0.729068 0.684441i \(-0.239954\pi\)
0.729068 + 0.684441i \(0.239954\pi\)
\(608\) 0 0
\(609\) 26.2790 + 13.0006i 1.06488 + 0.526810i
\(610\) 0 0
\(611\) −0.833539 1.44373i −0.0337214 0.0584072i
\(612\) 0 0
\(613\) −1.60252 + 2.77565i −0.0647253 + 0.112108i −0.896572 0.442898i \(-0.853950\pi\)
0.831847 + 0.555005i \(0.187284\pi\)
\(614\) 0 0
\(615\) 23.3543 + 7.40859i 0.941736 + 0.298743i
\(616\) 0 0
\(617\) 15.9357 27.6015i 0.641549 1.11120i −0.343538 0.939139i \(-0.611626\pi\)
0.985087 0.172056i \(-0.0550411\pi\)
\(618\) 0 0
\(619\) −20.9726 −0.842959 −0.421480 0.906838i \(-0.638489\pi\)
−0.421480 + 0.906838i \(0.638489\pi\)
\(620\) 0 0
\(621\) 1.25074 + 1.64786i 0.0501906 + 0.0661265i
\(622\) 0 0
\(623\) 19.9853 24.9262i 0.800692 0.998647i
\(624\) 0 0
\(625\) −9.46924 −0.378769
\(626\) 0 0
\(627\) −7.52222 + 6.86864i −0.300409 + 0.274307i
\(628\) 0 0
\(629\) −17.7102 −0.706153
\(630\) 0 0
\(631\) 26.4435 1.05270 0.526349 0.850268i \(-0.323560\pi\)
0.526349 + 0.850268i \(0.323560\pi\)
\(632\) 0 0
\(633\) −46.8237 14.8537i −1.86108 0.590382i
\(634\) 0 0
\(635\) 17.8253 0.707377
\(636\) 0 0
\(637\) 5.96663 + 26.7891i 0.236407 + 1.06142i
\(638\) 0 0
\(639\) −23.0564 + 10.6532i −0.912095 + 0.421432i
\(640\) 0 0
\(641\) −27.9618 −1.10443 −0.552213 0.833703i \(-0.686217\pi\)
−0.552213 + 0.833703i \(0.686217\pi\)
\(642\) 0 0
\(643\) −6.12936 + 10.6164i −0.241718 + 0.418669i −0.961204 0.275839i \(-0.911044\pi\)
0.719485 + 0.694508i \(0.244378\pi\)
\(644\) 0 0
\(645\) −10.6049 + 9.68344i −0.417566 + 0.381285i
\(646\) 0 0
\(647\) 4.49923 7.79290i 0.176883 0.306371i −0.763928 0.645301i \(-0.776732\pi\)
0.940811 + 0.338931i \(0.110065\pi\)
\(648\) 0 0
\(649\) −3.76640 6.52360i −0.147844 0.256074i
\(650\) 0 0
\(651\) −2.37560 1.17524i −0.0931072 0.0460614i
\(652\) 0 0
\(653\) −22.7901 −0.891844 −0.445922 0.895072i \(-0.647124\pi\)
−0.445922 + 0.895072i \(0.647124\pi\)
\(654\) 0 0
\(655\) −37.6418 −1.47079
\(656\) 0 0
\(657\) −3.71109 + 40.7719i −0.144783 + 1.59066i
\(658\) 0 0
\(659\) −19.3311 33.4824i −0.753033 1.30429i −0.946347 0.323154i \(-0.895257\pi\)
0.193314 0.981137i \(-0.438076\pi\)
\(660\) 0 0
\(661\) 5.75399 + 9.96621i 0.223804 + 0.387641i 0.955960 0.293497i \(-0.0948190\pi\)
−0.732156 + 0.681137i \(0.761486\pi\)
\(662\) 0 0
\(663\) −16.3175 + 14.8997i −0.633718 + 0.578657i
\(664\) 0 0
\(665\) −13.1590 + 16.4124i −0.510286 + 0.636444i
\(666\) 0 0
\(667\) 1.27362 + 2.20597i 0.0493147 + 0.0854156i
\(668\) 0 0
\(669\) 32.4782 29.6563i 1.25568 1.14658i
\(670\) 0 0
\(671\) −6.35511 + 11.0074i −0.245336 + 0.424935i
\(672\) 0 0
\(673\) −10.6642 18.4709i −0.411075 0.712002i 0.583933 0.811802i \(-0.301513\pi\)
−0.995008 + 0.0997997i \(0.968180\pi\)
\(674\) 0 0
\(675\) 6.80760 + 8.96906i 0.262025 + 0.345219i
\(676\) 0 0
\(677\) −13.8799 + 24.0407i −0.533449 + 0.923961i 0.465788 + 0.884896i \(0.345771\pi\)
−0.999237 + 0.0390641i \(0.987562\pi\)
\(678\) 0 0
\(679\) 19.3984 24.1943i 0.744443 0.928491i
\(680\) 0 0
\(681\) 1.72147 + 7.82997i 0.0659670 + 0.300045i
\(682\) 0 0
\(683\) −8.66854 + 15.0143i −0.331692 + 0.574508i −0.982844 0.184440i \(-0.940953\pi\)
0.651151 + 0.758948i \(0.274286\pi\)
\(684\) 0 0
\(685\) 30.2194 1.15462
\(686\) 0 0
\(687\) 0.865059 + 3.93464i 0.0330041 + 0.150116i
\(688\) 0 0
\(689\) 1.82832 + 3.16675i 0.0696535 + 0.120643i
\(690\) 0 0
\(691\) −19.8023 + 34.2986i −0.753315 + 1.30478i 0.192892 + 0.981220i \(0.438213\pi\)
−0.946207 + 0.323560i \(0.895120\pi\)
\(692\) 0 0
\(693\) −7.11758 6.85482i −0.270375 0.260393i
\(694\) 0 0
\(695\) 0.960788 1.66413i 0.0364448 0.0631242i
\(696\) 0 0
\(697\) 13.6730 + 23.6824i 0.517903 + 0.897035i
\(698\) 0 0
\(699\) 16.3204 14.9024i 0.617294 0.563659i
\(700\) 0 0
\(701\) 7.28469 0.275139 0.137570 0.990492i \(-0.456071\pi\)
0.137570 + 0.990492i \(0.456071\pi\)
\(702\) 0 0
\(703\) −12.8558 + 22.2668i −0.484864 + 0.839809i
\(704\) 0 0
\(705\) 1.18153 + 0.374814i 0.0444992 + 0.0141163i
\(706\) 0 0
\(707\) 5.19052 + 13.3280i 0.195210 + 0.501251i
\(708\) 0 0
\(709\) −6.00541 + 10.4017i −0.225538 + 0.390643i −0.956481 0.291796i \(-0.905747\pi\)
0.730943 + 0.682439i \(0.239081\pi\)
\(710\) 0 0
\(711\) −1.50474 + 16.5319i −0.0564323 + 0.619993i
\(712\) 0 0
\(713\) −0.115134 0.199418i −0.00431181 0.00746827i
\(714\) 0 0
\(715\) 4.10800 7.11526i 0.153630 0.266096i
\(716\) 0 0
\(717\) −36.4839 11.5736i −1.36252 0.432225i
\(718\) 0 0
\(719\) −2.15819 3.73809i −0.0804868 0.139407i 0.822972 0.568081i \(-0.192314\pi\)
−0.903459 + 0.428674i \(0.858981\pi\)
\(720\) 0 0
\(721\) 14.0375 + 36.0449i 0.522783 + 1.34238i
\(722\) 0 0
\(723\) 7.44499 + 33.8629i 0.276882 + 1.25937i
\(724\) 0 0
\(725\) 6.93210 + 12.0067i 0.257452 + 0.445919i
\(726\) 0 0
\(727\) 10.2483 + 17.7506i 0.380090 + 0.658334i 0.991075 0.133308i \(-0.0425598\pi\)
−0.610985 + 0.791642i \(0.709226\pi\)
\(728\) 0 0
\(729\) −7.26189 + 26.0051i −0.268959 + 0.963152i
\(730\) 0 0
\(731\) −16.0282 −0.592825
\(732\) 0 0
\(733\) 25.3322 0.935666 0.467833 0.883817i \(-0.345035\pi\)
0.467833 + 0.883817i \(0.345035\pi\)
\(734\) 0 0
\(735\) −16.7076 11.7179i −0.616268 0.432222i
\(736\) 0 0
\(737\) 5.86356 + 10.1560i 0.215987 + 0.374100i
\(738\) 0 0
\(739\) −6.63391 + 11.4903i −0.244032 + 0.422676i −0.961859 0.273545i \(-0.911804\pi\)
0.717827 + 0.696222i \(0.245137\pi\)
\(740\) 0 0
\(741\) 6.88843 + 31.3314i 0.253053 + 1.15099i
\(742\) 0 0
\(743\) 22.6116 39.1644i 0.829538 1.43680i −0.0688624 0.997626i \(-0.521937\pi\)
0.898401 0.439176i \(-0.144730\pi\)
\(744\) 0 0
\(745\) 29.7217 1.08892
\(746\) 0 0
\(747\) 39.4024 + 27.7962i 1.44166 + 1.01701i
\(748\) 0 0
\(749\) 11.2611 14.0451i 0.411471 0.513199i
\(750\) 0 0
\(751\) −28.3797 −1.03559 −0.517795 0.855505i \(-0.673247\pi\)
−0.517795 + 0.855505i \(0.673247\pi\)
\(752\) 0 0
\(753\) −0.789924 3.59289i −0.0287864 0.130932i
\(754\) 0 0
\(755\) −25.6435 −0.933263
\(756\) 0 0
\(757\) 5.08483 0.184811 0.0924056 0.995721i \(-0.470544\pi\)
0.0924056 + 0.995721i \(0.470544\pi\)
\(758\) 0 0
\(759\) −0.184350 0.838500i −0.00669149 0.0304356i
\(760\) 0 0
\(761\) 37.7225 1.36744 0.683720 0.729745i \(-0.260361\pi\)
0.683720 + 0.729745i \(0.260361\pi\)
\(762\) 0 0
\(763\) 43.8110 + 6.72111i 1.58606 + 0.243321i
\(764\) 0 0
\(765\) 1.48932 16.3624i 0.0538463 0.591583i
\(766\) 0 0
\(767\) −23.7229 −0.856583
\(768\) 0 0
\(769\) 11.8729 20.5644i 0.428147 0.741572i −0.568562 0.822641i \(-0.692500\pi\)
0.996709 + 0.0810688i \(0.0258333\pi\)
\(770\) 0 0
\(771\) −4.74683 21.5905i −0.170953 0.777563i
\(772\) 0 0
\(773\) 19.5347 33.8351i 0.702614 1.21696i −0.264932 0.964267i \(-0.585350\pi\)
0.967546 0.252696i \(-0.0813172\pi\)
\(774\) 0 0
\(775\) −0.626657 1.08540i −0.0225102 0.0389888i
\(776\) 0 0
\(777\) −22.3564 11.0600i −0.802031 0.396776i
\(778\) 0 0
\(779\) 39.7008 1.42243
\(780\) 0 0
\(781\) 10.5402 0.377159
\(782\) 0 0
\(783\) −12.8834 + 30.6467i −0.460415 + 1.09522i
\(784\) 0 0
\(785\) 11.4697 + 19.8661i 0.409370 + 0.709050i
\(786\) 0 0
\(787\) −1.90458 3.29882i −0.0678908 0.117590i 0.830082 0.557642i \(-0.188294\pi\)
−0.897973 + 0.440051i \(0.854960\pi\)
\(788\) 0 0
\(789\) −4.11666 18.7242i −0.146557 0.666601i
\(790\) 0 0
\(791\) 35.4173 + 5.43342i 1.25929 + 0.193190i
\(792\) 0 0
\(793\) 20.0140 + 34.6653i 0.710718 + 1.23100i
\(794\) 0 0
\(795\) −2.59163 0.822133i −0.0919157 0.0291581i
\(796\) 0 0
\(797\) −10.8239 + 18.7476i −0.383404 + 0.664075i −0.991546 0.129753i \(-0.958582\pi\)
0.608143 + 0.793828i \(0.291915\pi\)
\(798\) 0 0
\(799\) 0.691743 + 1.19813i 0.0244721 + 0.0423870i
\(800\) 0 0
\(801\) 29.6020 + 20.8825i 1.04594 + 0.737848i
\(802\) 0 0
\(803\) 8.49500 14.7138i 0.299782 0.519238i
\(804\) 0 0
\(805\) −0.643408 1.65212i −0.0226771 0.0582294i
\(806\) 0 0
\(807\) −5.86026 1.85903i −0.206291 0.0654408i
\(808\) 0 0
\(809\) −18.5128 + 32.0652i −0.650877 + 1.12735i 0.332034 + 0.943268i \(0.392265\pi\)
−0.982910 + 0.184084i \(0.941068\pi\)
\(810\) 0 0
\(811\) 5.37416 0.188712 0.0943561 0.995539i \(-0.469921\pi\)
0.0943561 + 0.995539i \(0.469921\pi\)
\(812\) 0 0
\(813\) 1.75912 1.60627i 0.0616950 0.0563345i
\(814\) 0 0
\(815\) −6.81216 11.7990i −0.238620 0.413301i
\(816\) 0 0
\(817\) −11.6348 + 20.1521i −0.407050 + 0.705032i
\(818\) 0 0
\(819\) −29.9031 + 8.61831i −1.04490 + 0.301148i
\(820\) 0 0
\(821\) −1.11119 + 1.92464i −0.0387809 + 0.0671706i −0.884764 0.466039i \(-0.845681\pi\)
0.845983 + 0.533209i \(0.179014\pi\)
\(822\) 0 0
\(823\) −18.5537 32.1359i −0.646740 1.12019i −0.983897 0.178738i \(-0.942799\pi\)
0.337157 0.941449i \(-0.390535\pi\)
\(824\) 0 0
\(825\) −1.00339 4.56382i −0.0349335 0.158892i
\(826\) 0 0
\(827\) 19.4790 0.677351 0.338676 0.940903i \(-0.390021\pi\)
0.338676 + 0.940903i \(0.390021\pi\)
\(828\) 0 0
\(829\) 0.137129 0.237514i 0.00476267 0.00824919i −0.863634 0.504119i \(-0.831817\pi\)
0.868397 + 0.495870i \(0.165151\pi\)
\(830\) 0 0
\(831\) −10.8459 49.3314i −0.376238 1.71129i
\(832\) 0 0
\(833\) −4.95163 22.2319i −0.171564 0.770290i
\(834\) 0 0
\(835\) −3.49657 + 6.05623i −0.121004 + 0.209585i
\(836\) 0 0
\(837\) 1.16465 2.77044i 0.0402562 0.0957603i
\(838\) 0 0
\(839\) 21.0711 + 36.4962i 0.727455 + 1.25999i 0.957956 + 0.286917i \(0.0926303\pi\)
−0.230501 + 0.973072i \(0.574036\pi\)
\(840\) 0 0
\(841\) −5.96666 + 10.3346i −0.205747 + 0.356364i
\(842\) 0 0
\(843\) −16.1059 + 14.7065i −0.554716 + 0.506518i
\(844\) 0 0
\(845\) −1.99671 3.45840i −0.0686888 0.118973i
\(846\) 0 0
\(847\) −9.07328 23.2980i −0.311762 0.800528i
\(848\) 0 0
\(849\) −12.1005 + 11.0492i −0.415289 + 0.379206i
\(850\) 0 0
\(851\) −1.08351 1.87669i −0.0371422 0.0643322i
\(852\) 0 0
\(853\) 22.3086 + 38.6397i 0.763833 + 1.32300i 0.940862 + 0.338791i \(0.110018\pi\)
−0.177029 + 0.984206i \(0.556649\pi\)
\(854\) 0 0
\(855\) −19.4911 13.7499i −0.666582 0.470235i
\(856\) 0 0
\(857\) −7.33227 −0.250466 −0.125233 0.992127i \(-0.539968\pi\)
−0.125233 + 0.992127i \(0.539968\pi\)
\(858\) 0 0
\(859\) −2.70146 −0.0921725 −0.0460863 0.998937i \(-0.514675\pi\)
−0.0460863 + 0.998937i \(0.514675\pi\)
\(860\) 0 0
\(861\) 2.47045 + 38.4341i 0.0841928 + 1.30983i
\(862\) 0 0
\(863\) 7.40188 + 12.8204i 0.251963 + 0.436413i 0.964066 0.265662i \(-0.0855905\pi\)
−0.712103 + 0.702075i \(0.752257\pi\)
\(864\) 0 0
\(865\) −11.6429 + 20.1661i −0.395870 + 0.685668i
\(866\) 0 0
\(867\) −8.20219 + 7.48953i −0.278561 + 0.254358i
\(868\) 0 0
\(869\) 3.44449 5.96603i 0.116846 0.202384i
\(870\) 0 0
\(871\) 36.9319 1.25139
\(872\) 0 0
\(873\) 28.7328 + 20.2694i 0.972458 + 0.686014i
\(874\) 0 0
\(875\) −11.5822 29.7404i −0.391551 1.00541i
\(876\) 0 0
\(877\) −11.0961 −0.374690 −0.187345 0.982294i \(-0.559988\pi\)
−0.187345 + 0.982294i \(0.559988\pi\)
\(878\) 0 0
\(879\) −27.4961 8.72249i −0.927421 0.294202i
\(880\) 0 0
\(881\) 15.0536 0.507168 0.253584 0.967313i \(-0.418391\pi\)
0.253584 + 0.967313i \(0.418391\pi\)
\(882\) 0 0
\(883\) 2.39418 0.0805704 0.0402852 0.999188i \(-0.487173\pi\)
0.0402852 + 0.999188i \(0.487173\pi\)
\(884\) 0 0
\(885\) 13.0258 11.8941i 0.437859 0.399815i
\(886\) 0 0
\(887\) −4.93394 −0.165666 −0.0828328 0.996563i \(-0.526397\pi\)
−0.0828328 + 0.996563i \(0.526397\pi\)
\(888\) 0 0
\(889\) 10.1682 + 26.1095i 0.341031 + 0.875685i
\(890\) 0 0
\(891\) 7.26230 8.53270i 0.243296 0.285856i
\(892\) 0 0
\(893\) 2.00853 0.0672130
\(894\) 0 0
\(895\) −7.93047 + 13.7360i −0.265086 + 0.459143i
\(896\) 0 0
\(897\) −2.57717 0.817547i −0.0860493 0.0272971i
\(898\) 0 0
\(899\) 1.85017 3.20459i 0.0617067 0.106879i
\(900\) 0 0
\(901\) −1.51730 2.62804i −0.0505486 0.0875527i
\(902\) 0 0
\(903\) −20.2331 10.0096i −0.673316 0.333099i
\(904\) 0 0
\(905\) 2.22400 0.0739284
\(906\) 0 0
\(907\) −3.39631 −0.112773 −0.0563863 0.998409i \(-0.517958\pi\)
−0.0563863 + 0.998409i \(0.517958\pi\)
\(908\) 0 0
\(909\) −14.7226 + 6.80253i −0.488316 + 0.225626i
\(910\) 0 0
\(911\) 4.65142 + 8.05649i 0.154108 + 0.266924i 0.932734 0.360565i \(-0.117416\pi\)
−0.778626 + 0.627489i \(0.784083\pi\)
\(912\) 0 0
\(913\) −10.0055 17.3301i −0.331134 0.573541i
\(914\) 0 0
\(915\) −28.3697 8.99960i −0.937872 0.297518i
\(916\) 0 0
\(917\) −21.4722 55.1355i −0.709076 1.82073i
\(918\) 0 0
\(919\) 8.92656 + 15.4613i 0.294460 + 0.510020i 0.974859 0.222822i \(-0.0715268\pi\)
−0.680399 + 0.732842i \(0.738194\pi\)
\(920\) 0 0
\(921\) −3.55544 16.1716i −0.117156 0.532872i
\(922\) 0 0
\(923\) 16.5970 28.7469i 0.546298 0.946216i
\(924\) 0 0
\(925\) −5.89737 10.2145i −0.193904 0.335852i
\(926\) 0 0
\(927\) −39.8164 + 18.3971i −1.30774 + 0.604240i
\(928\) 0 0
\(929\) 23.6430 40.9510i 0.775703 1.34356i −0.158695 0.987328i \(-0.550729\pi\)
0.934398 0.356230i \(-0.115938\pi\)
\(930\) 0 0
\(931\) −31.5463 9.91242i −1.03389 0.324866i
\(932\) 0 0
\(933\) 33.7288 30.7982i 1.10423 1.00829i
\(934\) 0 0
\(935\) −3.40917 + 5.90486i −0.111492 + 0.193110i
\(936\) 0 0
\(937\) 21.2493 0.694183 0.347092 0.937831i \(-0.387169\pi\)
0.347092 + 0.937831i \(0.387169\pi\)
\(938\) 0 0
\(939\) 20.9703 + 6.65234i 0.684341 + 0.217091i
\(940\) 0 0
\(941\) 30.0405 + 52.0316i 0.979292 + 1.69618i 0.664977 + 0.746864i \(0.268441\pi\)
0.314315 + 0.949319i \(0.398225\pi\)
\(942\) 0 0
\(943\) −1.67303 + 2.89777i −0.0544813 + 0.0943644i
\(944\) 0 0
\(945\) 12.0983 19.7249i 0.393558 0.641650i
\(946\) 0 0
\(947\) −1.83468 + 3.17775i −0.0596189 + 0.103263i −0.894294 0.447479i \(-0.852322\pi\)
0.834675 + 0.550742i \(0.185655\pi\)
\(948\) 0 0
\(949\) −26.7531 46.3377i −0.868442 1.50419i
\(950\) 0 0
\(951\) −0.0547357 0.0173636i −0.00177493 0.000563053i
\(952\) 0 0
\(953\) 0.837421 0.0271267 0.0135634 0.999908i \(-0.495683\pi\)
0.0135634 + 0.999908i \(0.495683\pi\)
\(954\) 0 0
\(955\) −13.7055 + 23.7385i −0.443498 + 0.768161i
\(956\) 0 0
\(957\) 10.1880 9.30279i 0.329331 0.300716i
\(958\) 0 0
\(959\) 17.2382 + 44.2636i 0.556651 + 1.42935i
\(960\) 0 0
\(961\) 15.3327 26.5571i 0.494605 0.856680i
\(962\) 0 0
\(963\) 16.6798 + 11.7667i 0.537500 + 0.379176i
\(964\) 0 0
\(965\) −2.15574 3.73386i −0.0693958 0.120197i
\(966\) 0 0
\(967\) 21.8856 37.9070i 0.703795 1.21901i −0.263330 0.964706i \(-0.584821\pi\)
0.967125 0.254302i \(-0.0818458\pi\)
\(968\) 0 0
\(969\) −5.71661 26.0015i −0.183644 0.835288i
\(970\) 0 0
\(971\) −20.3059 35.1709i −0.651648 1.12869i −0.982723 0.185083i \(-0.940745\pi\)
0.331075 0.943605i \(-0.392589\pi\)
\(972\) 0 0
\(973\) 2.98560 + 0.458025i 0.0957138 + 0.0146836i
\(974\) 0 0
\(975\) −14.0271 4.44977i −0.449228 0.142507i
\(976\) 0 0
\(977\) −28.7479 49.7929i −0.919728 1.59302i −0.799827 0.600230i \(-0.795076\pi\)
−0.119901 0.992786i \(-0.538258\pi\)
\(978\) 0 0
\(979\) −7.51689 13.0196i −0.240241 0.416109i
\(980\) 0 0
\(981\) −4.55571 + 50.0513i −0.145452 + 1.59801i
\(982\) 0 0
\(983\) −13.4594 −0.429289 −0.214644 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(984\) 0 0
\(985\) 36.5103 1.16331
\(986\) 0 0
\(987\) 0.124985 + 1.94445i 0.00397830 + 0.0618926i
\(988\) 0 0
\(989\) −0.980604 1.69846i −0.0311814 0.0540078i
\(990\) 0 0
\(991\) −17.7821 + 30.7995i −0.564867 + 0.978379i 0.432195 + 0.901780i \(0.357739\pi\)
−0.997062 + 0.0765983i \(0.975594\pi\)
\(992\) 0 0
\(993\) 8.01362 + 2.54213i 0.254304 + 0.0806720i
\(994\) 0 0
\(995\) −9.42036 + 16.3165i −0.298645 + 0.517269i
\(996\) 0 0
\(997\) 32.7289 1.03653 0.518267 0.855219i \(-0.326577\pi\)
0.518267 + 0.855219i \(0.326577\pi\)
\(998\) 0 0
\(999\) 10.9603 26.0721i 0.346770 0.824885i
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.t.d.193.6 yes 22
3.2 odd 2 1512.2.t.d.361.4 22
4.3 odd 2 1008.2.t.k.193.6 22
7.2 even 3 504.2.q.d.121.1 yes 22
9.2 odd 6 1512.2.q.c.1369.8 22
9.7 even 3 504.2.q.d.25.1 22
12.11 even 2 3024.2.t.l.1873.4 22
21.2 odd 6 1512.2.q.c.793.8 22
28.23 odd 6 1008.2.q.k.625.11 22
36.7 odd 6 1008.2.q.k.529.11 22
36.11 even 6 3024.2.q.k.2881.8 22
63.2 odd 6 1512.2.t.d.289.4 22
63.16 even 3 inner 504.2.t.d.457.6 yes 22
84.23 even 6 3024.2.q.k.2305.8 22
252.79 odd 6 1008.2.t.k.961.6 22
252.191 even 6 3024.2.t.l.289.4 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.d.25.1 22 9.7 even 3
504.2.q.d.121.1 yes 22 7.2 even 3
504.2.t.d.193.6 yes 22 1.1 even 1 trivial
504.2.t.d.457.6 yes 22 63.16 even 3 inner
1008.2.q.k.529.11 22 36.7 odd 6
1008.2.q.k.625.11 22 28.23 odd 6
1008.2.t.k.193.6 22 4.3 odd 2
1008.2.t.k.961.6 22 252.79 odd 6
1512.2.q.c.793.8 22 21.2 odd 6
1512.2.q.c.1369.8 22 9.2 odd 6
1512.2.t.d.289.4 22 63.2 odd 6
1512.2.t.d.361.4 22 3.2 odd 2
3024.2.q.k.2305.8 22 84.23 even 6
3024.2.q.k.2881.8 22 36.11 even 6
3024.2.t.l.289.4 22 252.191 even 6
3024.2.t.l.1873.4 22 12.11 even 2