Properties

Label 504.2.r.d.337.1
Level $504$
Weight $2$
Character 504.337
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.1
Root \(0.947217 - 0.807294i\) of defining polynomial
Character \(\chi\) \(=\) 504.337
Dual form 504.2.r.d.169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.67275 - 0.449358i) q^{3} +(1.87447 - 3.24667i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(2.59615 + 1.50332i) q^{9} +O(q^{10})\) \(q+(-1.67275 - 0.449358i) q^{3} +(1.87447 - 3.24667i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(2.59615 + 1.50332i) q^{9} +(1.82552 + 3.16190i) q^{11} +(2.77274 - 4.80253i) q^{13} +(-4.59442 + 4.58854i) q^{15} -7.20767 q^{17} -3.30555 q^{19} +(0.447217 + 1.67332i) q^{21} +(2.49443 - 4.32048i) q^{23} +(-4.52724 - 7.84141i) q^{25} +(-3.66717 - 3.68128i) q^{27} +(-0.245497 - 0.425213i) q^{29} +(1.94722 - 3.37268i) q^{31} +(-1.63281 - 6.10936i) q^{33} -3.74893 q^{35} +7.89443 q^{37} +(-6.79614 + 6.78745i) q^{39} +(-2.38215 + 4.12600i) q^{41} +(0.801714 + 1.38861i) q^{43} +(9.74720 - 5.61092i) q^{45} +(-4.81995 - 8.34840i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(12.0566 + 3.23883i) q^{51} -8.03647 q^{53} +13.6875 q^{55} +(5.52935 + 1.48538i) q^{57} +(0.754503 - 1.30684i) q^{59} +(1.04337 + 1.80717i) q^{61} +(0.00384004 - 3.00000i) q^{63} +(-10.3948 - 18.0043i) q^{65} +(-1.70172 + 2.94747i) q^{67} +(-6.11398 + 6.10616i) q^{69} -10.7301 q^{71} +9.83567 q^{73} +(4.04932 + 15.1510i) q^{75} +(1.82552 - 3.16190i) q^{77} +(-1.86391 - 3.22839i) q^{79} +(4.48003 + 7.80572i) q^{81} +(5.69058 + 9.85637i) q^{83} +(-13.5105 + 23.4009i) q^{85} +(0.219580 + 0.821588i) q^{87} +7.14891 q^{89} -5.54548 q^{91} +(-4.77274 + 4.76663i) q^{93} +(-6.19615 + 10.7320i) q^{95} +(5.45316 + 9.44516i) q^{97} +(-0.0140201 + 10.9531i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 10 q^{9} - 6 q^{11} - 3 q^{13} + 4 q^{15} - 16 q^{17} - 4 q^{19} - q^{21} - 5 q^{23} - 14 q^{25} + 5 q^{27} + q^{29} + 11 q^{31} - 8 q^{35} + 54 q^{37} - 12 q^{39} + 2 q^{41} - 11 q^{43} + 26 q^{45} + 7 q^{47} - 4 q^{49} + 17 q^{51} - 8 q^{53} + 12 q^{55} - 13 q^{57} + 9 q^{59} - 7 q^{61} - 5 q^{63} - 9 q^{65} - 12 q^{67} + 4 q^{69} - 24 q^{71} + 26 q^{73} - 23 q^{75} - 6 q^{77} - 22 q^{79} + 34 q^{81} - 6 q^{83} - 11 q^{85} + 37 q^{87} - 28 q^{89} + 6 q^{91} - 13 q^{93} - 23 q^{95} - q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.67275 0.449358i −0.965760 0.259437i
\(4\) 0 0
\(5\) 1.87447 3.24667i 0.838287 1.45195i −0.0530397 0.998592i \(-0.516891\pi\)
0.891326 0.453362i \(-0.149776\pi\)
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.188982 0.327327i
\(8\) 0 0
\(9\) 2.59615 + 1.50332i 0.865385 + 0.501108i
\(10\) 0 0
\(11\) 1.82552 + 3.16190i 0.550416 + 0.953348i 0.998244 + 0.0592287i \(0.0188641\pi\)
−0.447829 + 0.894119i \(0.647803\pi\)
\(12\) 0 0
\(13\) 2.77274 4.80253i 0.769020 1.33198i −0.169076 0.985603i \(-0.554078\pi\)
0.938095 0.346378i \(-0.112588\pi\)
\(14\) 0 0
\(15\) −4.59442 + 4.58854i −1.18627 + 1.18476i
\(16\) 0 0
\(17\) −7.20767 −1.74812 −0.874058 0.485821i \(-0.838521\pi\)
−0.874058 + 0.485821i \(0.838521\pi\)
\(18\) 0 0
\(19\) −3.30555 −0.758346 −0.379173 0.925326i \(-0.623792\pi\)
−0.379173 + 0.925326i \(0.623792\pi\)
\(20\) 0 0
\(21\) 0.447217 + 1.67332i 0.0975907 + 0.365148i
\(22\) 0 0
\(23\) 2.49443 4.32048i 0.520124 0.900881i −0.479602 0.877486i \(-0.659219\pi\)
0.999726 0.0233954i \(-0.00744767\pi\)
\(24\) 0 0
\(25\) −4.52724 7.84141i −0.905449 1.56828i
\(26\) 0 0
\(27\) −3.66717 3.68128i −0.705748 0.708463i
\(28\) 0 0
\(29\) −0.245497 0.425213i −0.0455876 0.0789600i 0.842331 0.538960i \(-0.181183\pi\)
−0.887919 + 0.460000i \(0.847849\pi\)
\(30\) 0 0
\(31\) 1.94722 3.37268i 0.349730 0.605751i −0.636471 0.771301i \(-0.719606\pi\)
0.986201 + 0.165550i \(0.0529398\pi\)
\(32\) 0 0
\(33\) −1.63281 6.10936i −0.284236 1.06350i
\(34\) 0 0
\(35\) −3.74893 −0.633685
\(36\) 0 0
\(37\) 7.89443 1.29784 0.648918 0.760858i \(-0.275222\pi\)
0.648918 + 0.760858i \(0.275222\pi\)
\(38\) 0 0
\(39\) −6.79614 + 6.78745i −1.08825 + 1.08686i
\(40\) 0 0
\(41\) −2.38215 + 4.12600i −0.372029 + 0.644373i −0.989878 0.141924i \(-0.954671\pi\)
0.617849 + 0.786297i \(0.288004\pi\)
\(42\) 0 0
\(43\) 0.801714 + 1.38861i 0.122260 + 0.211761i 0.920659 0.390369i \(-0.127652\pi\)
−0.798398 + 0.602130i \(0.794319\pi\)
\(44\) 0 0
\(45\) 9.74720 5.61092i 1.45303 0.836427i
\(46\) 0 0
\(47\) −4.81995 8.34840i −0.703062 1.21774i −0.967386 0.253305i \(-0.918482\pi\)
0.264324 0.964434i \(-0.414851\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 12.0566 + 3.23883i 1.68826 + 0.453526i
\(52\) 0 0
\(53\) −8.03647 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(54\) 0 0
\(55\) 13.6875 1.84562
\(56\) 0 0
\(57\) 5.52935 + 1.48538i 0.732380 + 0.196743i
\(58\) 0 0
\(59\) 0.754503 1.30684i 0.0982280 0.170136i −0.812723 0.582650i \(-0.802016\pi\)
0.910951 + 0.412514i \(0.135349\pi\)
\(60\) 0 0
\(61\) 1.04337 + 1.80717i 0.133590 + 0.231385i 0.925058 0.379826i \(-0.124016\pi\)
−0.791468 + 0.611211i \(0.790683\pi\)
\(62\) 0 0
\(63\) 0.00384004 3.00000i 0.000483799 0.377964i
\(64\) 0 0
\(65\) −10.3948 18.0043i −1.28932 2.23316i
\(66\) 0 0
\(67\) −1.70172 + 2.94747i −0.207898 + 0.360090i −0.951052 0.309030i \(-0.899996\pi\)
0.743154 + 0.669120i \(0.233329\pi\)
\(68\) 0 0
\(69\) −6.11398 + 6.10616i −0.736037 + 0.735096i
\(70\) 0 0
\(71\) −10.7301 −1.27343 −0.636715 0.771099i \(-0.719707\pi\)
−0.636715 + 0.771099i \(0.719707\pi\)
\(72\) 0 0
\(73\) 9.83567 1.15118 0.575589 0.817739i \(-0.304773\pi\)
0.575589 + 0.817739i \(0.304773\pi\)
\(74\) 0 0
\(75\) 4.04932 + 15.1510i 0.467575 + 1.74949i
\(76\) 0 0
\(77\) 1.82552 3.16190i 0.208038 0.360332i
\(78\) 0 0
\(79\) −1.86391 3.22839i −0.209706 0.363222i 0.741916 0.670493i \(-0.233917\pi\)
−0.951622 + 0.307271i \(0.900584\pi\)
\(80\) 0 0
\(81\) 4.48003 + 7.80572i 0.497781 + 0.867303i
\(82\) 0 0
\(83\) 5.69058 + 9.85637i 0.624622 + 1.08188i 0.988614 + 0.150475i \(0.0480802\pi\)
−0.363992 + 0.931402i \(0.618586\pi\)
\(84\) 0 0
\(85\) −13.5105 + 23.4009i −1.46542 + 2.53819i
\(86\) 0 0
\(87\) 0.219580 + 0.821588i 0.0235415 + 0.0880835i
\(88\) 0 0
\(89\) 7.14891 0.757783 0.378891 0.925441i \(-0.376305\pi\)
0.378891 + 0.925441i \(0.376305\pi\)
\(90\) 0 0
\(91\) −5.54548 −0.581324
\(92\) 0 0
\(93\) −4.77274 + 4.76663i −0.494910 + 0.494277i
\(94\) 0 0
\(95\) −6.19615 + 10.7320i −0.635711 + 1.10108i
\(96\) 0 0
\(97\) 5.45316 + 9.44516i 0.553685 + 0.959011i 0.998005 + 0.0631422i \(0.0201122\pi\)
−0.444320 + 0.895868i \(0.646555\pi\)
\(98\) 0 0
\(99\) −0.0140201 + 10.9531i −0.00140908 + 1.10083i
\(100\) 0 0
\(101\) 6.40171 + 11.0881i 0.636994 + 1.10331i 0.986089 + 0.166218i \(0.0531557\pi\)
−0.349095 + 0.937087i \(0.613511\pi\)
\(102\) 0 0
\(103\) 4.76717 8.25698i 0.469723 0.813584i −0.529678 0.848199i \(-0.677687\pi\)
0.999401 + 0.0346149i \(0.0110205\pi\)
\(104\) 0 0
\(105\) 6.27101 + 1.68461i 0.611988 + 0.164401i
\(106\) 0 0
\(107\) 8.43223 0.815175 0.407587 0.913166i \(-0.366370\pi\)
0.407587 + 0.913166i \(0.366370\pi\)
\(108\) 0 0
\(109\) 16.5574 1.58591 0.792954 0.609281i \(-0.208542\pi\)
0.792954 + 0.609281i \(0.208542\pi\)
\(110\) 0 0
\(111\) −13.2054 3.54743i −1.25340 0.336707i
\(112\) 0 0
\(113\) −0.0328150 + 0.0568372i −0.00308697 + 0.00534679i −0.867565 0.497324i \(-0.834316\pi\)
0.864478 + 0.502671i \(0.167649\pi\)
\(114\) 0 0
\(115\) −9.35144 16.1972i −0.872026 1.51039i
\(116\) 0 0
\(117\) 14.4182 8.29977i 1.33296 0.767314i
\(118\) 0 0
\(119\) 3.60383 + 6.24202i 0.330363 + 0.572205i
\(120\) 0 0
\(121\) −1.16507 + 2.01795i −0.105915 + 0.183450i
\(122\) 0 0
\(123\) 5.83877 5.83131i 0.526465 0.525791i
\(124\) 0 0
\(125\) −15.2000 −1.35953
\(126\) 0 0
\(127\) 3.20080 0.284025 0.142012 0.989865i \(-0.454643\pi\)
0.142012 + 0.989865i \(0.454643\pi\)
\(128\) 0 0
\(129\) −0.717080 2.68305i −0.0631354 0.236229i
\(130\) 0 0
\(131\) −7.67061 + 13.2859i −0.670184 + 1.16079i 0.307668 + 0.951494i \(0.400452\pi\)
−0.977852 + 0.209299i \(0.932882\pi\)
\(132\) 0 0
\(133\) 1.65278 + 2.86269i 0.143314 + 0.248227i
\(134\) 0 0
\(135\) −18.8259 + 5.00566i −1.62028 + 0.430819i
\(136\) 0 0
\(137\) −0.342977 0.594054i −0.0293025 0.0507535i 0.851002 0.525162i \(-0.175995\pi\)
−0.880305 + 0.474409i \(0.842662\pi\)
\(138\) 0 0
\(139\) 10.1004 17.4944i 0.856702 1.48385i −0.0183546 0.999832i \(-0.505843\pi\)
0.875057 0.484020i \(-0.160824\pi\)
\(140\) 0 0
\(141\) 4.31113 + 16.1306i 0.363062 + 1.35844i
\(142\) 0 0
\(143\) 20.2468 1.69312
\(144\) 0 0
\(145\) −1.84070 −0.152862
\(146\) 0 0
\(147\) 1.22553 1.22396i 0.101080 0.100951i
\(148\) 0 0
\(149\) 4.31652 7.47642i 0.353623 0.612493i −0.633259 0.773940i \(-0.718283\pi\)
0.986881 + 0.161448i \(0.0516163\pi\)
\(150\) 0 0
\(151\) 5.43836 + 9.41952i 0.442568 + 0.766550i 0.997879 0.0650926i \(-0.0207343\pi\)
−0.555311 + 0.831642i \(0.687401\pi\)
\(152\) 0 0
\(153\) −18.7122 10.8355i −1.51279 0.875995i
\(154\) 0 0
\(155\) −7.29998 12.6439i −0.586349 1.01559i
\(156\) 0 0
\(157\) 5.49613 9.51958i 0.438639 0.759745i −0.558946 0.829204i \(-0.688794\pi\)
0.997585 + 0.0694592i \(0.0221274\pi\)
\(158\) 0 0
\(159\) 13.4430 + 3.61126i 1.06610 + 0.286391i
\(160\) 0 0
\(161\) −4.98886 −0.393177
\(162\) 0 0
\(163\) 4.79233 0.375364 0.187682 0.982230i \(-0.439902\pi\)
0.187682 + 0.982230i \(0.439902\pi\)
\(164\) 0 0
\(165\) −22.8957 6.15060i −1.78243 0.478824i
\(166\) 0 0
\(167\) −1.46719 + 2.54124i −0.113534 + 0.196647i −0.917193 0.398444i \(-0.869550\pi\)
0.803659 + 0.595091i \(0.202884\pi\)
\(168\) 0 0
\(169\) −8.87617 15.3740i −0.682782 1.18261i
\(170\) 0 0
\(171\) −8.58173 4.96932i −0.656261 0.380013i
\(172\) 0 0
\(173\) 5.99483 + 10.3834i 0.455779 + 0.789432i 0.998733 0.0503306i \(-0.0160275\pi\)
−0.542954 + 0.839763i \(0.682694\pi\)
\(174\) 0 0
\(175\) −4.52724 + 7.84141i −0.342227 + 0.592755i
\(176\) 0 0
\(177\) −1.84933 + 1.84697i −0.139004 + 0.138826i
\(178\) 0 0
\(179\) 24.9266 1.86310 0.931552 0.363608i \(-0.118455\pi\)
0.931552 + 0.363608i \(0.118455\pi\)
\(180\) 0 0
\(181\) −21.4203 −1.59216 −0.796078 0.605194i \(-0.793096\pi\)
−0.796078 + 0.605194i \(0.793096\pi\)
\(182\) 0 0
\(183\) −0.933226 3.49179i −0.0689861 0.258120i
\(184\) 0 0
\(185\) 14.7978 25.6306i 1.08796 1.88440i
\(186\) 0 0
\(187\) −13.1578 22.7899i −0.962191 1.66656i
\(188\) 0 0
\(189\) −1.35450 + 5.01651i −0.0985252 + 0.364897i
\(190\) 0 0
\(191\) −8.98003 15.5539i −0.649772 1.12544i −0.983177 0.182655i \(-0.941531\pi\)
0.333405 0.942784i \(-0.391802\pi\)
\(192\) 0 0
\(193\) −4.68943 + 8.12233i −0.337553 + 0.584658i −0.983972 0.178324i \(-0.942932\pi\)
0.646419 + 0.762983i \(0.276266\pi\)
\(194\) 0 0
\(195\) 9.29747 + 34.7877i 0.665806 + 2.49120i
\(196\) 0 0
\(197\) −0.206917 −0.0147422 −0.00737110 0.999973i \(-0.502346\pi\)
−0.00737110 + 0.999973i \(0.502346\pi\)
\(198\) 0 0
\(199\) 12.6169 0.894386 0.447193 0.894438i \(-0.352424\pi\)
0.447193 + 0.894438i \(0.352424\pi\)
\(200\) 0 0
\(201\) 4.17101 4.16568i 0.294201 0.293824i
\(202\) 0 0
\(203\) −0.245497 + 0.425213i −0.0172305 + 0.0298441i
\(204\) 0 0
\(205\) 8.93050 + 15.4681i 0.623733 + 1.08034i
\(206\) 0 0
\(207\) 12.9710 7.46669i 0.901547 0.518971i
\(208\) 0 0
\(209\) −6.03436 10.4518i −0.417406 0.722968i
\(210\) 0 0
\(211\) 2.96505 5.13561i 0.204122 0.353550i −0.745730 0.666248i \(-0.767899\pi\)
0.949853 + 0.312698i \(0.101233\pi\)
\(212\) 0 0
\(213\) 17.9487 + 4.82166i 1.22983 + 0.330375i
\(214\) 0 0
\(215\) 6.01114 0.409957
\(216\) 0 0
\(217\) −3.89443 −0.264371
\(218\) 0 0
\(219\) −16.4526 4.41974i −1.11176 0.298659i
\(220\) 0 0
\(221\) −19.9850 + 34.6150i −1.34434 + 2.32846i
\(222\) 0 0
\(223\) 1.80500 + 3.12634i 0.120871 + 0.209355i 0.920112 0.391656i \(-0.128098\pi\)
−0.799240 + 0.601012i \(0.794764\pi\)
\(224\) 0 0
\(225\) 0.0347696 27.1634i 0.00231797 1.81090i
\(226\) 0 0
\(227\) 9.30710 + 16.1204i 0.617734 + 1.06995i 0.989898 + 0.141780i \(0.0452826\pi\)
−0.372164 + 0.928167i \(0.621384\pi\)
\(228\) 0 0
\(229\) −11.4965 + 19.9126i −0.759712 + 1.31586i 0.183285 + 0.983060i \(0.441327\pi\)
−0.942997 + 0.332801i \(0.892006\pi\)
\(230\) 0 0
\(231\) −4.47446 + 4.46874i −0.294398 + 0.294021i
\(232\) 0 0
\(233\) −4.07066 −0.266678 −0.133339 0.991071i \(-0.542570\pi\)
−0.133339 + 0.991071i \(0.542570\pi\)
\(234\) 0 0
\(235\) −36.1393 −2.35747
\(236\) 0 0
\(237\) 1.66714 + 6.23783i 0.108293 + 0.405191i
\(238\) 0 0
\(239\) −2.50000 + 4.33013i −0.161712 + 0.280093i −0.935483 0.353373i \(-0.885035\pi\)
0.773771 + 0.633465i \(0.218368\pi\)
\(240\) 0 0
\(241\) −7.30843 12.6586i −0.470777 0.815410i 0.528664 0.848831i \(-0.322693\pi\)
−0.999441 + 0.0334208i \(0.989360\pi\)
\(242\) 0 0
\(243\) −3.98639 15.0701i −0.255727 0.966749i
\(244\) 0 0
\(245\) 1.87447 + 3.24667i 0.119755 + 0.207422i
\(246\) 0 0
\(247\) −9.16544 + 15.8750i −0.583183 + 1.01010i
\(248\) 0 0
\(249\) −5.08984 19.0443i −0.322556 1.20688i
\(250\) 0 0
\(251\) 8.74206 0.551794 0.275897 0.961187i \(-0.411025\pi\)
0.275897 + 0.961187i \(0.411025\pi\)
\(252\) 0 0
\(253\) 18.2145 1.14514
\(254\) 0 0
\(255\) 33.1151 33.0727i 2.07375 2.07109i
\(256\) 0 0
\(257\) −10.6399 + 18.4288i −0.663699 + 1.14956i 0.315938 + 0.948780i \(0.397681\pi\)
−0.979636 + 0.200780i \(0.935652\pi\)
\(258\) 0 0
\(259\) −3.94722 6.83678i −0.245268 0.424817i
\(260\) 0 0
\(261\) 0.00188543 1.47298i 0.000116705 0.0911751i
\(262\) 0 0
\(263\) −6.80515 11.7869i −0.419623 0.726809i 0.576278 0.817254i \(-0.304505\pi\)
−0.995901 + 0.0904446i \(0.971171\pi\)
\(264\) 0 0
\(265\) −15.0641 + 26.0918i −0.925380 + 1.60280i
\(266\) 0 0
\(267\) −11.9583 3.21242i −0.731836 0.196597i
\(268\) 0 0
\(269\) −18.9343 −1.15445 −0.577223 0.816587i \(-0.695864\pi\)
−0.577223 + 0.816587i \(0.695864\pi\)
\(270\) 0 0
\(271\) 14.1669 0.860579 0.430290 0.902691i \(-0.358411\pi\)
0.430290 + 0.902691i \(0.358411\pi\)
\(272\) 0 0
\(273\) 9.27617 + 2.49191i 0.561420 + 0.150817i
\(274\) 0 0
\(275\) 16.5292 28.6294i 0.996746 1.72642i
\(276\) 0 0
\(277\) −0.636090 1.10174i −0.0382190 0.0661972i 0.846283 0.532733i \(-0.178835\pi\)
−0.884502 + 0.466536i \(0.845502\pi\)
\(278\) 0 0
\(279\) 10.1255 5.82869i 0.606198 0.348955i
\(280\) 0 0
\(281\) 4.66891 + 8.08678i 0.278524 + 0.482417i 0.971018 0.239006i \(-0.0768216\pi\)
−0.692494 + 0.721423i \(0.743488\pi\)
\(282\) 0 0
\(283\) −4.88983 + 8.46943i −0.290670 + 0.503455i −0.973968 0.226684i \(-0.927212\pi\)
0.683298 + 0.730139i \(0.260545\pi\)
\(284\) 0 0
\(285\) 15.1871 15.1677i 0.899607 0.898456i
\(286\) 0 0
\(287\) 4.76429 0.281227
\(288\) 0 0
\(289\) 34.9505 2.05591
\(290\) 0 0
\(291\) −4.87749 18.2498i −0.285924 1.06982i
\(292\) 0 0
\(293\) 12.0672 20.9010i 0.704972 1.22105i −0.261729 0.965141i \(-0.584293\pi\)
0.966702 0.255906i \(-0.0823738\pi\)
\(294\) 0 0
\(295\) −2.82858 4.89925i −0.164686 0.285245i
\(296\) 0 0
\(297\) 4.94533 18.3155i 0.286957 1.06277i
\(298\) 0 0
\(299\) −13.8328 23.9591i −0.799971 1.38559i
\(300\) 0 0
\(301\) 0.801714 1.38861i 0.0462100 0.0800382i
\(302\) 0 0
\(303\) −5.72590 21.4242i −0.328945 1.23079i
\(304\) 0 0
\(305\) 7.82305 0.447947
\(306\) 0 0
\(307\) 20.0425 1.14389 0.571944 0.820293i \(-0.306190\pi\)
0.571944 + 0.820293i \(0.306190\pi\)
\(308\) 0 0
\(309\) −11.6846 + 11.6697i −0.664714 + 0.663863i
\(310\) 0 0
\(311\) −6.56758 + 11.3754i −0.372414 + 0.645039i −0.989936 0.141514i \(-0.954803\pi\)
0.617523 + 0.786553i \(0.288136\pi\)
\(312\) 0 0
\(313\) −6.29311 10.9000i −0.355708 0.616104i 0.631531 0.775351i \(-0.282427\pi\)
−0.987239 + 0.159247i \(0.949093\pi\)
\(314\) 0 0
\(315\) −9.73280 5.63586i −0.548381 0.317545i
\(316\) 0 0
\(317\) 5.36679 + 9.29555i 0.301429 + 0.522090i 0.976460 0.215699i \(-0.0692031\pi\)
−0.675031 + 0.737789i \(0.735870\pi\)
\(318\) 0 0
\(319\) 0.896319 1.55247i 0.0501842 0.0869217i
\(320\) 0 0
\(321\) −14.1050 3.78909i −0.787263 0.211487i
\(322\) 0 0
\(323\) 23.8253 1.32568
\(324\) 0 0
\(325\) −50.2115 −2.78523
\(326\) 0 0
\(327\) −27.6963 7.44020i −1.53161 0.411444i
\(328\) 0 0
\(329\) −4.81995 + 8.34840i −0.265732 + 0.460262i
\(330\) 0 0
\(331\) 5.61919 + 9.73273i 0.308859 + 0.534959i 0.978113 0.208074i \(-0.0667196\pi\)
−0.669254 + 0.743034i \(0.733386\pi\)
\(332\) 0 0
\(333\) 20.4952 + 11.8679i 1.12313 + 0.650357i
\(334\) 0 0
\(335\) 6.37963 + 11.0498i 0.348557 + 0.603718i
\(336\) 0 0
\(337\) −1.35453 + 2.34611i −0.0737858 + 0.127801i −0.900558 0.434737i \(-0.856841\pi\)
0.826772 + 0.562537i \(0.190175\pi\)
\(338\) 0 0
\(339\) 0.0804313 0.0803284i 0.00436843 0.00436284i
\(340\) 0 0
\(341\) 14.2188 0.769989
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 8.36424 + 31.2959i 0.450316 + 1.68491i
\(346\) 0 0
\(347\) 4.77314 8.26733i 0.256236 0.443813i −0.708995 0.705214i \(-0.750851\pi\)
0.965230 + 0.261400i \(0.0841843\pi\)
\(348\) 0 0
\(349\) 1.01114 + 1.75135i 0.0541253 + 0.0937478i 0.891819 0.452393i \(-0.149430\pi\)
−0.837693 + 0.546141i \(0.816096\pi\)
\(350\) 0 0
\(351\) −27.8476 + 7.40446i −1.48639 + 0.395221i
\(352\) 0 0
\(353\) −5.93836 10.2855i −0.316067 0.547444i 0.663597 0.748091i \(-0.269029\pi\)
−0.979664 + 0.200646i \(0.935696\pi\)
\(354\) 0 0
\(355\) −20.1132 + 34.8371i −1.06750 + 1.84896i
\(356\) 0 0
\(357\) −3.22339 12.0607i −0.170600 0.638321i
\(358\) 0 0
\(359\) −13.9796 −0.737817 −0.368909 0.929466i \(-0.620268\pi\)
−0.368909 + 0.929466i \(0.620268\pi\)
\(360\) 0 0
\(361\) −8.07331 −0.424911
\(362\) 0 0
\(363\) 2.85564 2.85199i 0.149882 0.149691i
\(364\) 0 0
\(365\) 18.4366 31.9332i 0.965017 1.67146i
\(366\) 0 0
\(367\) 1.14299 + 1.97972i 0.0596635 + 0.103340i 0.894314 0.447439i \(-0.147664\pi\)
−0.834651 + 0.550779i \(0.814331\pi\)
\(368\) 0 0
\(369\) −12.3871 + 7.13059i −0.644848 + 0.371204i
\(370\) 0 0
\(371\) 4.01824 + 6.95979i 0.208616 + 0.361334i
\(372\) 0 0
\(373\) 17.0527 29.5362i 0.882957 1.52933i 0.0349186 0.999390i \(-0.488883\pi\)
0.848038 0.529936i \(-0.177784\pi\)
\(374\) 0 0
\(375\) 25.4257 + 6.83024i 1.31298 + 0.352712i
\(376\) 0 0
\(377\) −2.72279 −0.140231
\(378\) 0 0
\(379\) −6.50473 −0.334126 −0.167063 0.985946i \(-0.553428\pi\)
−0.167063 + 0.985946i \(0.553428\pi\)
\(380\) 0 0
\(381\) −5.35412 1.43831i −0.274300 0.0736866i
\(382\) 0 0
\(383\) −11.2348 + 19.4592i −0.574069 + 0.994317i 0.422073 + 0.906562i \(0.361303\pi\)
−0.996142 + 0.0877552i \(0.972031\pi\)
\(384\) 0 0
\(385\) −6.84376 11.8537i −0.348790 0.604122i
\(386\) 0 0
\(387\) −0.00615723 + 4.81028i −0.000312989 + 0.244520i
\(388\) 0 0
\(389\) 2.57316 + 4.45684i 0.130464 + 0.225971i 0.923856 0.382741i \(-0.125020\pi\)
−0.793391 + 0.608712i \(0.791687\pi\)
\(390\) 0 0
\(391\) −17.9790 + 31.1406i −0.909238 + 1.57485i
\(392\) 0 0
\(393\) 18.8011 18.7770i 0.948390 0.947177i
\(394\) 0 0
\(395\) −13.9753 −0.703176
\(396\) 0 0
\(397\) −21.0214 −1.05503 −0.527517 0.849544i \(-0.676877\pi\)
−0.527517 + 0.849544i \(0.676877\pi\)
\(398\) 0 0
\(399\) −1.47830 5.53125i −0.0740076 0.276909i
\(400\) 0 0
\(401\) 0.803852 1.39231i 0.0401424 0.0695288i −0.845256 0.534361i \(-0.820552\pi\)
0.885399 + 0.464833i \(0.153885\pi\)
\(402\) 0 0
\(403\) −10.7983 18.7031i −0.537899 0.931669i
\(404\) 0 0
\(405\) 33.7403 + 0.0863761i 1.67657 + 0.00429207i
\(406\) 0 0
\(407\) 14.4115 + 24.9614i 0.714350 + 1.23729i
\(408\) 0 0
\(409\) −10.7055 + 18.5425i −0.529354 + 0.916869i 0.470059 + 0.882635i \(0.344232\pi\)
−0.999414 + 0.0342340i \(0.989101\pi\)
\(410\) 0 0
\(411\) 0.306771 + 1.14782i 0.0151319 + 0.0566178i
\(412\) 0 0
\(413\) −1.50901 −0.0742534
\(414\) 0 0
\(415\) 42.6672 2.09445
\(416\) 0 0
\(417\) −24.7566 + 24.7249i −1.21234 + 1.21078i
\(418\) 0 0
\(419\) 17.3497 30.0505i 0.847587 1.46806i −0.0357681 0.999360i \(-0.511388\pi\)
0.883355 0.468704i \(-0.155279\pi\)
\(420\) 0 0
\(421\) 11.6234 + 20.1323i 0.566488 + 0.981186i 0.996910 + 0.0785583i \(0.0250317\pi\)
−0.430421 + 0.902628i \(0.641635\pi\)
\(422\) 0 0
\(423\) 0.0370176 28.9197i 0.00179986 1.40612i
\(424\) 0 0
\(425\) 32.6309 + 56.5183i 1.58283 + 2.74154i
\(426\) 0 0
\(427\) 1.04337 1.80717i 0.0504923 0.0874552i
\(428\) 0 0
\(429\) −33.8677 9.09807i −1.63515 0.439259i
\(430\) 0 0
\(431\) 10.5201 0.506735 0.253368 0.967370i \(-0.418462\pi\)
0.253368 + 0.967370i \(0.418462\pi\)
\(432\) 0 0
\(433\) −29.8296 −1.43352 −0.716758 0.697322i \(-0.754375\pi\)
−0.716758 + 0.697322i \(0.754375\pi\)
\(434\) 0 0
\(435\) 3.07902 + 0.827134i 0.147628 + 0.0396580i
\(436\) 0 0
\(437\) −8.24547 + 14.2816i −0.394434 + 0.683180i
\(438\) 0 0
\(439\) −2.28462 3.95708i −0.109039 0.188861i 0.806342 0.591449i \(-0.201444\pi\)
−0.915381 + 0.402588i \(0.868111\pi\)
\(440\) 0 0
\(441\) −2.59999 + 1.49667i −0.123809 + 0.0712702i
\(442\) 0 0
\(443\) 5.01455 + 8.68545i 0.238248 + 0.412658i 0.960212 0.279273i \(-0.0900934\pi\)
−0.721963 + 0.691931i \(0.756760\pi\)
\(444\) 0 0
\(445\) 13.4004 23.2101i 0.635239 1.10027i
\(446\) 0 0
\(447\) −10.5800 + 10.5665i −0.500418 + 0.499778i
\(448\) 0 0
\(449\) −18.1908 −0.858476 −0.429238 0.903192i \(-0.641218\pi\)
−0.429238 + 0.903192i \(0.641218\pi\)
\(450\) 0 0
\(451\) −17.3946 −0.819082
\(452\) 0 0
\(453\) −4.86426 18.2002i −0.228543 0.855122i
\(454\) 0 0
\(455\) −10.3948 + 18.0043i −0.487316 + 0.844056i
\(456\) 0 0
\(457\) 2.62708 + 4.55024i 0.122890 + 0.212851i 0.920906 0.389784i \(-0.127451\pi\)
−0.798016 + 0.602636i \(0.794117\pi\)
\(458\) 0 0
\(459\) 26.4318 + 26.5335i 1.23373 + 1.23848i
\(460\) 0 0
\(461\) −4.32763 7.49568i −0.201558 0.349108i 0.747473 0.664293i \(-0.231267\pi\)
−0.949031 + 0.315184i \(0.897934\pi\)
\(462\) 0 0
\(463\) 0.509788 0.882979i 0.0236919 0.0410355i −0.853936 0.520377i \(-0.825791\pi\)
0.877628 + 0.479342i \(0.159125\pi\)
\(464\) 0 0
\(465\) 6.52935 + 24.4304i 0.302791 + 1.13293i
\(466\) 0 0
\(467\) 18.7167 0.866104 0.433052 0.901369i \(-0.357437\pi\)
0.433052 + 0.901369i \(0.357437\pi\)
\(468\) 0 0
\(469\) 3.40344 0.157156
\(470\) 0 0
\(471\) −13.4713 + 13.4541i −0.620726 + 0.619932i
\(472\) 0 0
\(473\) −2.92710 + 5.06988i −0.134588 + 0.233113i
\(474\) 0 0
\(475\) 14.9650 + 25.9202i 0.686643 + 1.18930i
\(476\) 0 0
\(477\) −20.8639 12.0814i −0.955293 0.553170i
\(478\) 0 0
\(479\) 18.3255 + 31.7407i 0.837313 + 1.45027i 0.892133 + 0.451773i \(0.149208\pi\)
−0.0548199 + 0.998496i \(0.517458\pi\)
\(480\) 0 0
\(481\) 21.8892 37.9132i 0.998062 1.72869i
\(482\) 0 0
\(483\) 8.34509 + 2.24178i 0.379715 + 0.102005i
\(484\) 0 0
\(485\) 40.8871 1.85659
\(486\) 0 0
\(487\) −35.1877 −1.59451 −0.797253 0.603646i \(-0.793714\pi\)
−0.797253 + 0.603646i \(0.793714\pi\)
\(488\) 0 0
\(489\) −8.01635 2.15347i −0.362512 0.0973835i
\(490\) 0 0
\(491\) −9.15489 + 15.8567i −0.413154 + 0.715604i −0.995233 0.0975285i \(-0.968906\pi\)
0.582079 + 0.813133i \(0.302240\pi\)
\(492\) 0 0
\(493\) 1.76946 + 3.06479i 0.0796924 + 0.138031i
\(494\) 0 0
\(495\) 35.5349 + 20.5768i 1.59718 + 0.924857i
\(496\) 0 0
\(497\) 5.36505 + 9.29255i 0.240656 + 0.416828i
\(498\) 0 0
\(499\) −12.2791 + 21.2679i −0.549686 + 0.952084i 0.448610 + 0.893728i \(0.351919\pi\)
−0.998296 + 0.0583561i \(0.981414\pi\)
\(500\) 0 0
\(501\) 3.59615 3.59155i 0.160664 0.160459i
\(502\) 0 0
\(503\) −17.0176 −0.758779 −0.379390 0.925237i \(-0.623866\pi\)
−0.379390 + 0.925237i \(0.623866\pi\)
\(504\) 0 0
\(505\) 47.9991 2.13593
\(506\) 0 0
\(507\) 7.93914 + 29.7053i 0.352590 + 1.31926i
\(508\) 0 0
\(509\) 10.3817 17.9817i 0.460163 0.797025i −0.538806 0.842430i \(-0.681124\pi\)
0.998969 + 0.0454049i \(0.0144578\pi\)
\(510\) 0 0
\(511\) −4.91784 8.51794i −0.217552 0.376812i
\(512\) 0 0
\(513\) 12.1220 + 12.1687i 0.535201 + 0.537260i
\(514\) 0 0
\(515\) −17.8718 30.9548i −0.787525 1.36403i
\(516\) 0 0
\(517\) 17.5979 30.4804i 0.773953 1.34053i
\(518\) 0 0
\(519\) −5.36198 20.0625i −0.235365 0.880648i
\(520\) 0 0
\(521\) 29.9915 1.31395 0.656977 0.753911i \(-0.271835\pi\)
0.656977 + 0.753911i \(0.271835\pi\)
\(522\) 0 0
\(523\) 2.57036 0.112394 0.0561970 0.998420i \(-0.482102\pi\)
0.0561970 + 0.998420i \(0.482102\pi\)
\(524\) 0 0
\(525\) 11.0965 11.0823i 0.484292 0.483673i
\(526\) 0 0
\(527\) −14.0349 + 24.3091i −0.611370 + 1.05892i
\(528\) 0 0
\(529\) −0.944341 1.63565i −0.0410583 0.0711151i
\(530\) 0 0
\(531\) 3.92341 2.25849i 0.170261 0.0980101i
\(532\) 0 0
\(533\) 13.2101 + 22.8806i 0.572195 + 0.991070i
\(534\) 0 0
\(535\) 15.8059 27.3767i 0.683350 1.18360i
\(536\) 0 0
\(537\) −41.6959 11.2010i −1.79931 0.483358i
\(538\) 0 0
\(539\) −3.65105 −0.157262
\(540\) 0 0
\(541\) 24.7962 1.06607 0.533036 0.846093i \(-0.321051\pi\)
0.533036 + 0.846093i \(0.321051\pi\)
\(542\) 0 0
\(543\) 35.8307 + 9.62538i 1.53764 + 0.413065i
\(544\) 0 0
\(545\) 31.0362 53.7563i 1.32945 2.30267i
\(546\) 0 0
\(547\) 5.71955 + 9.90655i 0.244550 + 0.423574i 0.962005 0.273031i \(-0.0880263\pi\)
−0.717455 + 0.696605i \(0.754693\pi\)
\(548\) 0 0
\(549\) −0.00801317 + 6.26022i −0.000341994 + 0.267180i
\(550\) 0 0
\(551\) 0.811502 + 1.40556i 0.0345712 + 0.0598790i
\(552\) 0 0
\(553\) −1.86391 + 3.22839i −0.0792615 + 0.137285i
\(554\) 0 0
\(555\) −36.2704 + 36.2240i −1.53959 + 1.53762i
\(556\) 0 0
\(557\) 13.8445 0.586609 0.293305 0.956019i \(-0.405245\pi\)
0.293305 + 0.956019i \(0.405245\pi\)
\(558\) 0 0
\(559\) 8.89178 0.376082
\(560\) 0 0
\(561\) 11.7687 + 44.0343i 0.496877 + 1.85913i
\(562\) 0 0
\(563\) −17.5194 + 30.3444i −0.738353 + 1.27886i 0.214884 + 0.976640i \(0.431063\pi\)
−0.953237 + 0.302225i \(0.902271\pi\)
\(564\) 0 0
\(565\) 0.123021 + 0.213079i 0.00517553 + 0.00896429i
\(566\) 0 0
\(567\) 4.51994 7.78268i 0.189820 0.326842i
\(568\) 0 0
\(569\) −3.72940 6.45951i −0.156344 0.270797i 0.777203 0.629250i \(-0.216638\pi\)
−0.933548 + 0.358453i \(0.883304\pi\)
\(570\) 0 0
\(571\) −21.4174 + 37.0961i −0.896292 + 1.55242i −0.0640949 + 0.997944i \(0.520416\pi\)
−0.832197 + 0.554480i \(0.812917\pi\)
\(572\) 0 0
\(573\) 8.03204 + 30.0529i 0.335543 + 1.25548i
\(574\) 0 0
\(575\) −45.1715 −1.88378
\(576\) 0 0
\(577\) −16.1386 −0.671860 −0.335930 0.941887i \(-0.609051\pi\)
−0.335930 + 0.941887i \(0.609051\pi\)
\(578\) 0 0
\(579\) 11.4941 11.4794i 0.477677 0.477066i
\(580\) 0 0
\(581\) 5.69058 9.85637i 0.236085 0.408911i
\(582\) 0 0
\(583\) −14.6708 25.4105i −0.607601 1.05240i
\(584\) 0 0
\(585\) 0.0798329 62.3688i 0.00330068 2.57863i
\(586\) 0 0
\(587\) −22.9094 39.6802i −0.945570 1.63778i −0.754605 0.656179i \(-0.772171\pi\)
−0.190965 0.981597i \(-0.561162\pi\)
\(588\) 0 0
\(589\) −6.43663 + 11.1486i −0.265217 + 0.459369i
\(590\) 0 0
\(591\) 0.346119 + 0.0929797i 0.0142374 + 0.00382467i
\(592\) 0 0
\(593\) −22.4176 −0.920579 −0.460290 0.887769i \(-0.652254\pi\)
−0.460290 + 0.887769i \(0.652254\pi\)
\(594\) 0 0
\(595\) 27.0211 1.10776
\(596\) 0 0
\(597\) −21.1048 5.66949i −0.863762 0.232037i
\(598\) 0 0
\(599\) −0.622253 + 1.07777i −0.0254246 + 0.0440366i −0.878458 0.477820i \(-0.841427\pi\)
0.853033 + 0.521857i \(0.174760\pi\)
\(600\) 0 0
\(601\) 18.9011 + 32.7377i 0.770993 + 1.33540i 0.937019 + 0.349277i \(0.113573\pi\)
−0.166027 + 0.986121i \(0.553094\pi\)
\(602\) 0 0
\(603\) −8.84893 + 5.09384i −0.360356 + 0.207437i
\(604\) 0 0
\(605\) 4.36775 + 7.56516i 0.177574 + 0.307568i
\(606\) 0 0
\(607\) −6.01266 + 10.4142i −0.244047 + 0.422701i −0.961863 0.273531i \(-0.911808\pi\)
0.717817 + 0.696232i \(0.245142\pi\)
\(608\) 0 0
\(609\) 0.601726 0.600956i 0.0243832 0.0243520i
\(610\) 0 0
\(611\) −53.4579 −2.16267
\(612\) 0 0
\(613\) −29.5048 −1.19169 −0.595843 0.803101i \(-0.703182\pi\)
−0.595843 + 0.803101i \(0.703182\pi\)
\(614\) 0 0
\(615\) −7.98774 29.8872i −0.322097 1.20517i
\(616\) 0 0
\(617\) 6.26890 10.8581i 0.252376 0.437129i −0.711803 0.702379i \(-0.752121\pi\)
0.964180 + 0.265250i \(0.0854546\pi\)
\(618\) 0 0
\(619\) 15.2153 + 26.3537i 0.611555 + 1.05924i 0.990978 + 0.134022i \(0.0427891\pi\)
−0.379423 + 0.925223i \(0.623878\pi\)
\(620\) 0 0
\(621\) −25.0524 + 6.66124i −1.00532 + 0.267306i
\(622\) 0 0
\(623\) −3.57445 6.19114i −0.143207 0.248043i
\(624\) 0 0
\(625\) −5.85564 + 10.1423i −0.234226 + 0.405691i
\(626\) 0 0
\(627\) 5.39734 + 20.1948i 0.215549 + 0.806504i
\(628\) 0 0
\(629\) −56.9005 −2.26877
\(630\) 0 0
\(631\) −20.6901 −0.823660 −0.411830 0.911261i \(-0.635110\pi\)
−0.411830 + 0.911261i \(0.635110\pi\)
\(632\) 0 0
\(633\) −7.26750 + 7.25820i −0.288857 + 0.288488i
\(634\) 0 0
\(635\) 5.99979 10.3919i 0.238094 0.412391i
\(636\) 0 0
\(637\) 2.77274 + 4.80253i 0.109860 + 0.190283i
\(638\) 0 0
\(639\) −27.8570 16.1308i −1.10201 0.638126i
\(640\) 0 0
\(641\) 5.24417 + 9.08317i 0.207132 + 0.358764i 0.950810 0.309775i \(-0.100254\pi\)
−0.743678 + 0.668538i \(0.766920\pi\)
\(642\) 0 0
\(643\) 20.2056 34.9971i 0.796830 1.38015i −0.124841 0.992177i \(-0.539842\pi\)
0.921671 0.387972i \(-0.126824\pi\)
\(644\) 0 0
\(645\) −10.0551 2.70116i −0.395920 0.106358i
\(646\) 0 0
\(647\) −16.8906 −0.664038 −0.332019 0.943273i \(-0.607730\pi\)
−0.332019 + 0.943273i \(0.607730\pi\)
\(648\) 0 0
\(649\) 5.50945 0.216265
\(650\) 0 0
\(651\) 6.51440 + 1.75000i 0.255319 + 0.0685878i
\(652\) 0 0
\(653\) 19.0982 33.0791i 0.747372 1.29449i −0.201707 0.979446i \(-0.564649\pi\)
0.949079 0.315040i \(-0.102018\pi\)
\(654\) 0 0
\(655\) 28.7566 + 49.8079i 1.12361 + 1.94615i
\(656\) 0 0
\(657\) 25.5349 + 14.7862i 0.996212 + 0.576865i
\(658\) 0 0
\(659\) −15.2864 26.4768i −0.595472 1.03139i −0.993480 0.114006i \(-0.963632\pi\)
0.398008 0.917382i \(-0.369702\pi\)
\(660\) 0 0
\(661\) −12.0894 + 20.9395i −0.470223 + 0.814451i −0.999420 0.0340482i \(-0.989160\pi\)
0.529197 + 0.848499i \(0.322493\pi\)
\(662\) 0 0
\(663\) 48.9843 48.9217i 1.90239 1.89996i
\(664\) 0 0
\(665\) 12.3923 0.480553
\(666\) 0 0
\(667\) −2.44949 −0.0948448
\(668\) 0 0
\(669\) −1.61445 6.04067i −0.0624182 0.233546i
\(670\) 0 0
\(671\) −3.80939 + 6.59806i −0.147060 + 0.254715i
\(672\) 0 0
\(673\) −13.5485 23.4667i −0.522257 0.904575i −0.999665 0.0258933i \(-0.991757\pi\)
0.477408 0.878682i \(-0.341576\pi\)
\(674\) 0 0
\(675\) −12.2643 + 45.4219i −0.472052 + 1.74829i
\(676\) 0 0
\(677\) −6.72453 11.6472i −0.258445 0.447640i 0.707381 0.706833i \(-0.249877\pi\)
−0.965825 + 0.259193i \(0.916543\pi\)
\(678\) 0 0
\(679\) 5.45316 9.44516i 0.209273 0.362472i
\(680\) 0 0
\(681\) −8.32459 31.1475i −0.318999 1.19358i
\(682\) 0 0
\(683\) 11.4449 0.437925 0.218963 0.975733i \(-0.429733\pi\)
0.218963 + 0.975733i \(0.429733\pi\)
\(684\) 0 0
\(685\) −2.57160 −0.0982557
\(686\) 0 0
\(687\) 28.1787 28.1426i 1.07508 1.07371i
\(688\) 0 0
\(689\) −22.2830 + 38.5954i −0.848916 + 1.47037i
\(690\) 0 0
\(691\) −18.1416 31.4222i −0.690139 1.19536i −0.971792 0.235840i \(-0.924216\pi\)
0.281653 0.959516i \(-0.409117\pi\)
\(692\) 0 0
\(693\) 9.49270 5.46442i 0.360598 0.207576i
\(694\) 0 0
\(695\) −37.8656 65.5851i −1.43632 2.48779i
\(696\) 0 0
\(697\) 17.1697 29.7388i 0.650349 1.12644i
\(698\) 0 0
\(699\) 6.80917 + 1.82918i 0.257547 + 0.0691861i
\(700\) 0 0
\(701\) 4.80688 0.181553 0.0907767 0.995871i \(-0.471065\pi\)
0.0907767 + 0.995871i \(0.471065\pi\)
\(702\) 0 0
\(703\) −26.0955 −0.984210
\(704\) 0 0
\(705\) 60.4519 + 16.2395i 2.27675 + 0.611615i
\(706\) 0 0
\(707\) 6.40171 11.0881i 0.240761 0.417010i
\(708\) 0 0
\(709\) 6.17770 + 10.7001i 0.232008 + 0.401850i 0.958399 0.285432i \(-0.0921370\pi\)
−0.726391 + 0.687282i \(0.758804\pi\)
\(710\) 0 0
\(711\) 0.0143150 11.1834i 0.000536853 0.419412i
\(712\) 0 0
\(713\) −9.71438 16.8258i −0.363807 0.630131i
\(714\) 0 0
\(715\) 37.9519 65.7347i 1.41932 2.45834i
\(716\) 0 0
\(717\) 6.12764 6.11980i 0.228841 0.228548i
\(718\) 0 0
\(719\) −10.4488 −0.389673 −0.194837 0.980836i \(-0.562418\pi\)
−0.194837 + 0.980836i \(0.562418\pi\)
\(720\) 0 0
\(721\) −9.53433 −0.355077
\(722\) 0 0
\(723\) 6.53691 + 24.4587i 0.243110 + 0.909628i
\(724\) 0 0
\(725\) −2.22285 + 3.85008i −0.0825544 + 0.142988i
\(726\) 0 0
\(727\) 9.13266 + 15.8182i 0.338711 + 0.586665i 0.984191 0.177112i \(-0.0566756\pi\)
−0.645479 + 0.763778i \(0.723342\pi\)
\(728\) 0 0
\(729\) −0.103681 + 26.9998i −0.00384003 + 0.999993i
\(730\) 0 0
\(731\) −5.77849 10.0086i −0.213725 0.370183i
\(732\) 0 0
\(733\) 24.3245 42.1313i 0.898447 1.55616i 0.0689673 0.997619i \(-0.478030\pi\)
0.829480 0.558537i \(-0.188637\pi\)
\(734\) 0 0
\(735\) −1.67659 6.27316i −0.0618418 0.231389i
\(736\) 0 0
\(737\) −12.4261 −0.457722
\(738\) 0 0
\(739\) 52.2935 1.92365 0.961823 0.273671i \(-0.0882379\pi\)
0.961823 + 0.273671i \(0.0882379\pi\)
\(740\) 0 0
\(741\) 22.4650 22.4363i 0.825273 0.824217i
\(742\) 0 0
\(743\) −5.21154 + 9.02665i −0.191193 + 0.331156i −0.945646 0.325198i \(-0.894569\pi\)
0.754453 + 0.656354i \(0.227902\pi\)
\(744\) 0 0
\(745\) −16.1823 28.0286i −0.592874 1.02689i
\(746\) 0 0
\(747\) −0.0437040 + 34.1434i −0.00159905 + 1.24924i
\(748\) 0 0
\(749\) −4.21612 7.30253i −0.154054 0.266829i
\(750\) 0 0
\(751\) −5.10383 + 8.84010i −0.186242 + 0.322580i −0.943994 0.329962i \(-0.892964\pi\)
0.757753 + 0.652542i \(0.226297\pi\)
\(752\) 0 0
\(753\) −14.6232 3.92832i −0.532901 0.143156i
\(754\) 0 0
\(755\) 40.7761 1.48399
\(756\) 0 0
\(757\) −6.13207 −0.222874 −0.111437 0.993772i \(-0.535545\pi\)
−0.111437 + 0.993772i \(0.535545\pi\)
\(758\) 0 0
\(759\) −30.4683 8.18486i −1.10593 0.297091i
\(760\) 0 0
\(761\) −16.0967 + 27.8803i −0.583505 + 1.01066i 0.411555 + 0.911385i \(0.364986\pi\)
−0.995060 + 0.0992757i \(0.968347\pi\)
\(762\) 0 0
\(763\) −8.27869 14.3391i −0.299709 0.519111i
\(764\) 0 0
\(765\) −70.2546 + 40.4417i −2.54006 + 1.46217i
\(766\) 0 0
\(767\) −4.18408 7.24704i −0.151078 0.261676i
\(768\) 0 0
\(769\) 5.38520 9.32745i 0.194195 0.336356i −0.752441 0.658660i \(-0.771124\pi\)
0.946636 + 0.322303i \(0.104457\pi\)
\(770\) 0 0
\(771\) 26.0790 26.0456i 0.939212 0.938011i
\(772\) 0 0
\(773\) −26.6154 −0.957289 −0.478644 0.878009i \(-0.658872\pi\)
−0.478644 + 0.878009i \(0.658872\pi\)
\(774\) 0 0
\(775\) −35.2621 −1.26665
\(776\) 0 0
\(777\) 3.53052 + 13.2099i 0.126657 + 0.473903i
\(778\) 0 0
\(779\) 7.87431 13.6387i 0.282127 0.488657i
\(780\) 0 0
\(781\) −19.5881 33.9275i −0.700916 1.21402i
\(782\) 0 0
\(783\) −0.665049 + 2.46307i −0.0237669 + 0.0880230i
\(784\) 0 0
\(785\) −20.6046 35.6882i −0.735410 1.27377i
\(786\) 0 0
\(787\) 4.00251 6.93256i 0.142674 0.247119i −0.785829 0.618444i \(-0.787763\pi\)
0.928503 + 0.371325i \(0.121097\pi\)
\(788\) 0 0
\(789\) 6.08676 + 22.7744i 0.216694 + 0.810789i
\(790\) 0 0
\(791\) 0.0656299 0.00233353
\(792\) 0 0
\(793\) 11.5720 0.410933
\(794\) 0 0
\(795\) 36.9229 36.8757i 1.30952 1.30785i
\(796\) 0 0
\(797\) −11.9675 + 20.7284i −0.423912 + 0.734237i −0.996318 0.0857334i \(-0.972677\pi\)
0.572406 + 0.819970i \(0.306010\pi\)
\(798\) 0 0
\(799\) 34.7406 + 60.1725i 1.22903 + 2.12875i
\(800\) 0 0
\(801\) 18.5597 + 10.7471i 0.655774 + 0.379731i
\(802\) 0 0
\(803\) 17.9552 + 31.0994i 0.633627 + 1.09747i
\(804\) 0 0
\(805\) −9.35144 + 16.1972i −0.329595 + 0.570875i
\(806\) 0 0
\(807\) 31.6723 + 8.50829i 1.11492 + 0.299506i
\(808\) 0 0
\(809\) 26.9965 0.949148 0.474574 0.880216i \(-0.342602\pi\)
0.474574 + 0.880216i \(0.342602\pi\)
\(810\) 0 0
\(811\) −41.6011 −1.46081 −0.730406 0.683013i \(-0.760669\pi\)
−0.730406 + 0.683013i \(0.760669\pi\)
\(812\) 0 0
\(813\) −23.6977 6.36602i −0.831113 0.223266i
\(814\) 0 0
\(815\) 8.98306 15.5591i 0.314663 0.545012i
\(816\) 0 0
\(817\) −2.65011 4.59013i −0.0927156 0.160588i
\(818\) 0 0
\(819\) −14.3969 8.33665i −0.503069 0.291306i
\(820\) 0 0
\(821\) −9.33583 16.1701i −0.325823 0.564341i 0.655856 0.754886i \(-0.272308\pi\)
−0.981678 + 0.190545i \(0.938975\pi\)
\(822\) 0 0
\(823\) 21.4756 37.1968i 0.748591 1.29660i −0.199906 0.979815i \(-0.564064\pi\)
0.948498 0.316783i \(-0.102603\pi\)
\(824\) 0 0
\(825\) −40.5139 + 40.4621i −1.41051 + 1.40871i
\(826\) 0 0
\(827\) 28.4602 0.989657 0.494828 0.868991i \(-0.335231\pi\)
0.494828 + 0.868991i \(0.335231\pi\)
\(828\) 0 0
\(829\) 32.0061 1.11162 0.555809 0.831310i \(-0.312409\pi\)
0.555809 + 0.831310i \(0.312409\pi\)
\(830\) 0 0
\(831\) 0.568941 + 2.12876i 0.0197363 + 0.0738460i
\(832\) 0 0
\(833\) 3.60383 6.24202i 0.124865 0.216273i
\(834\) 0 0
\(835\) 5.50038 + 9.52693i 0.190348 + 0.329693i
\(836\) 0 0
\(837\) −19.5566 + 5.19994i −0.675974 + 0.179736i
\(838\) 0 0
\(839\) −2.01191 3.48473i −0.0694589 0.120306i 0.829204 0.558946i \(-0.188794\pi\)
−0.898663 + 0.438639i \(0.855461\pi\)
\(840\) 0 0
\(841\) 14.3795 24.9060i 0.495844 0.858826i
\(842\) 0 0
\(843\) −4.17603 15.6251i −0.143830 0.538158i
\(844\) 0 0
\(845\) −66.5523 −2.28947
\(846\) 0 0
\(847\) 2.33013 0.0800642
\(848\) 0 0
\(849\) 11.9852 11.9699i 0.411332 0.410806i
\(850\) 0 0
\(851\) 19.6921 34.1077i 0.675036 1.16920i
\(852\) 0 0
\(853\) −25.1214 43.5116i −0.860141 1.48981i −0.871792 0.489876i \(-0.837042\pi\)
0.0116510 0.999932i \(-0.496291\pi\)
\(854\) 0 0
\(855\) −32.2199 + 18.5472i −1.10190 + 0.634301i
\(856\) 0 0
\(857\) 5.37909 + 9.31685i 0.183746 + 0.318258i 0.943153 0.332358i \(-0.107844\pi\)
−0.759407 + 0.650616i \(0.774511\pi\)
\(858\) 0 0
\(859\) −21.6126 + 37.4342i −0.737413 + 1.27724i 0.216243 + 0.976340i \(0.430620\pi\)
−0.953656 + 0.300898i \(0.902714\pi\)
\(860\) 0 0
\(861\) −7.96945 2.14087i −0.271598 0.0729608i
\(862\) 0 0
\(863\) 44.1446 1.50270 0.751349 0.659905i \(-0.229403\pi\)
0.751349 + 0.659905i \(0.229403\pi\)
\(864\) 0 0
\(865\) 44.9484 1.52829
\(866\) 0 0
\(867\) −58.4633 15.7053i −1.98552 0.533380i
\(868\) 0 0
\(869\) 6.80522 11.7870i 0.230851 0.399846i
\(870\) 0 0
\(871\) 9.43685 + 16.3451i 0.319756 + 0.553833i
\(872\) 0 0
\(873\) −0.0418807 + 32.7190i −0.00141745 + 1.10737i
\(874\) 0 0
\(875\) 7.59999 + 13.1636i 0.256927 + 0.445010i
\(876\) 0 0
\(877\) −10.9631 + 18.9886i −0.370196 + 0.641199i −0.989596 0.143877i \(-0.954043\pi\)
0.619399 + 0.785076i \(0.287376\pi\)
\(878\) 0 0
\(879\) −29.5773 + 29.5395i −0.997619 + 0.996343i
\(880\) 0 0
\(881\) 49.6969 1.67433 0.837166 0.546949i \(-0.184211\pi\)
0.837166 + 0.546949i \(0.184211\pi\)
\(882\) 0 0
\(883\) 16.8441 0.566848 0.283424 0.958995i \(-0.408530\pi\)
0.283424 + 0.958995i \(0.408530\pi\)
\(884\) 0 0
\(885\) 2.52998 + 9.46624i 0.0850443 + 0.318204i
\(886\) 0 0
\(887\) −10.4950 + 18.1779i −0.352387 + 0.610352i −0.986667 0.162751i \(-0.947963\pi\)
0.634280 + 0.773103i \(0.281297\pi\)
\(888\) 0 0
\(889\) −1.60040 2.77197i −0.0536757 0.0929690i
\(890\) 0 0
\(891\) −16.5025 + 28.4149i −0.552855 + 0.951936i
\(892\) 0 0
\(893\) 15.9326 + 27.5961i 0.533164 + 0.923468i
\(894\) 0 0
\(895\) 46.7241 80.9285i 1.56181 2.70514i
\(896\) 0 0
\(897\) 12.3725 + 46.2934i 0.413106 + 1.54569i
\(898\) 0 0
\(899\) −1.91214 −0.0637735
\(900\) 0 0
\(901\) 57.9242 1.92974
\(902\) 0 0
\(903\) −1.96505 + 1.96253i −0.0653927 + 0.0653090i
\(904\) 0 0
\(905\) −40.1516 + 69.5446i −1.33468 + 2.31174i
\(906\) 0 0
\(907\) −4.55814 7.89494i −0.151351 0.262147i 0.780374 0.625314i \(-0.215029\pi\)
−0.931724 + 0.363167i \(0.881696\pi\)
\(908\) 0 0
\(909\) −0.0491656 + 38.4102i −0.00163072 + 1.27399i
\(910\) 0 0
\(911\) −9.16079 15.8670i −0.303511 0.525696i 0.673418 0.739262i \(-0.264825\pi\)
−0.976929 + 0.213566i \(0.931492\pi\)
\(912\) 0 0
\(913\) −20.7766 + 35.9860i −0.687603 + 1.19096i
\(914\) 0 0
\(915\) −13.0860 3.51535i −0.432609 0.116214i
\(916\) 0 0
\(917\) 15.3412 0.506612
\(918\) 0 0
\(919\) 55.7097 1.83769 0.918847 0.394613i \(-0.129121\pi\)
0.918847 + 0.394613i \(0.129121\pi\)
\(920\) 0 0
\(921\) −33.5261 9.00628i −1.10472 0.296767i
\(922\) 0 0
\(923\) −29.7518 + 51.5316i −0.979292 + 1.69618i
\(924\) 0 0
\(925\) −35.7400 61.9035i −1.17512 2.03538i
\(926\) 0 0
\(927\) 24.7892 14.2698i 0.814185 0.468681i
\(928\) 0 0
\(929\) −10.4420 18.0861i −0.342592 0.593386i 0.642322 0.766435i \(-0.277971\pi\)
−0.984913 + 0.173049i \(0.944638\pi\)
\(930\) 0 0
\(931\) 1.65278 2.86269i 0.0541676 0.0938210i
\(932\) 0 0
\(933\) 16.0975 16.0769i 0.527009 0.526335i
\(934\) 0 0
\(935\) −98.6551 −3.22637
\(936\) 0 0
\(937\) −6.56584 −0.214497 −0.107248 0.994232i \(-0.534204\pi\)
−0.107248 + 0.994232i \(0.534204\pi\)
\(938\) 0 0
\(939\) 5.62877 + 21.0608i 0.183688 + 0.687292i
\(940\) 0 0
\(941\) −5.21649 + 9.03523i −0.170053 + 0.294540i −0.938438 0.345447i \(-0.887727\pi\)
0.768385 + 0.639988i \(0.221060\pi\)
\(942\) 0 0
\(943\) 11.8842 + 20.5840i 0.387002 + 0.670307i
\(944\) 0 0
\(945\) 13.7480 + 13.8009i 0.447222 + 0.448942i
\(946\) 0 0
\(947\) −15.5391 26.9146i −0.504954 0.874606i −0.999984 0.00573005i \(-0.998176\pi\)
0.495029 0.868876i \(-0.335157\pi\)
\(948\) 0 0
\(949\) 27.2718 47.2361i 0.885279 1.53335i
\(950\) 0 0
\(951\) −4.80023 17.9607i −0.155658 0.582415i
\(952\) 0 0
\(953\) 27.2222 0.881814 0.440907 0.897553i \(-0.354657\pi\)
0.440907 + 0.897553i \(0.354657\pi\)
\(954\) 0 0
\(955\) −67.3310 −2.17878
\(956\) 0 0
\(957\) −2.19693 + 2.19412i −0.0710166 + 0.0709258i
\(958\) 0 0
\(959\) −0.342977 + 0.594054i −0.0110753 + 0.0191830i
\(960\) 0 0
\(961\) 7.91669 + 13.7121i 0.255377 + 0.442326i
\(962\) 0 0
\(963\) 21.8914 + 12.6764i 0.705440 + 0.408491i
\(964\) 0 0
\(965\) 17.5804 + 30.4501i 0.565932 + 0.980223i
\(966\) 0 0
\(967\) −21.5710 + 37.3620i −0.693675 + 1.20148i 0.276950 + 0.960884i \(0.410676\pi\)
−0.970625 + 0.240597i \(0.922657\pi\)
\(968\) 0 0
\(969\) −39.8537 10.7061i −1.28029 0.343930i
\(970\) 0 0
\(971\) 18.0357 0.578794 0.289397 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(972\) 0 0
\(973\) −20.2007 −0.647606
\(974\) 0 0
\(975\) 83.9910 + 22.5629i 2.68986 + 0.722592i
\(976\) 0 0
\(977\) −13.6015 + 23.5586i −0.435152 + 0.753705i −0.997308 0.0733259i \(-0.976639\pi\)
0.562156 + 0.827031i \(0.309972\pi\)
\(978\) 0 0
\(979\) 13.0505 + 22.6041i 0.417096 + 0.722431i
\(980\) 0 0
\(981\) 42.9855 + 24.8911i 1.37242 + 0.794712i
\(982\) 0 0
\(983\) −9.03965 15.6571i −0.288320 0.499385i 0.685089 0.728460i \(-0.259763\pi\)
−0.973409 + 0.229074i \(0.926430\pi\)
\(984\) 0 0
\(985\) −0.387858 + 0.671790i −0.0123582 + 0.0214050i
\(986\) 0 0
\(987\) 11.8140 11.7989i 0.376043 0.375562i
\(988\) 0 0
\(989\) 7.99927 0.254362
\(990\) 0 0
\(991\) −50.2203 −1.59530 −0.797650 0.603121i \(-0.793924\pi\)
−0.797650 + 0.603121i \(0.793924\pi\)
\(992\) 0 0
\(993\) −5.02600 18.8054i −0.159495 0.596772i
\(994\) 0 0
\(995\) 23.6499 40.9628i 0.749751 1.29861i
\(996\) 0 0
\(997\) 13.3757 + 23.1673i 0.423611 + 0.733716i 0.996290 0.0860640i \(-0.0274290\pi\)
−0.572678 + 0.819780i \(0.694096\pi\)
\(998\) 0 0
\(999\) −28.9503 29.0616i −0.915946 0.919470i
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.d.337.1 yes 8
3.2 odd 2 1512.2.r.d.1009.1 8
4.3 odd 2 1008.2.r.m.337.4 8
9.2 odd 6 1512.2.r.d.505.1 8
9.4 even 3 4536.2.a.x.1.1 4
9.5 odd 6 4536.2.a.ba.1.4 4
9.7 even 3 inner 504.2.r.d.169.1 8
12.11 even 2 3024.2.r.l.1009.1 8
36.7 odd 6 1008.2.r.m.673.4 8
36.11 even 6 3024.2.r.l.2017.1 8
36.23 even 6 9072.2.a.cl.1.4 4
36.31 odd 6 9072.2.a.ce.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.d.169.1 8 9.7 even 3 inner
504.2.r.d.337.1 yes 8 1.1 even 1 trivial
1008.2.r.m.337.4 8 4.3 odd 2
1008.2.r.m.673.4 8 36.7 odd 6
1512.2.r.d.505.1 8 9.2 odd 6
1512.2.r.d.1009.1 8 3.2 odd 2
3024.2.r.l.1009.1 8 12.11 even 2
3024.2.r.l.2017.1 8 36.11 even 6
4536.2.a.x.1.1 4 9.4 even 3
4536.2.a.ba.1.4 4 9.5 odd 6
9072.2.a.ce.1.1 4 36.31 odd 6
9072.2.a.cl.1.4 4 36.23 even 6