Properties

Label 572.2.p.a.309.8
Level $572$
Weight $2$
Character 572.309
Analytic conductor $4.567$
Analytic rank $0$
Dimension $24$
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] = [572,2,Mod(309,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.p (of order \(6\), 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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 309.8
Character \(\chi\) \(=\) 572.309
Dual form 572.2.p.a.485.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.196667 + 0.340638i) q^{3} +4.15244i q^{5} +(2.01581 + 1.16383i) q^{7} +(1.42264 - 2.46409i) q^{9} +O(q^{10})\) \(q+(0.196667 + 0.340638i) q^{3} +4.15244i q^{5} +(2.01581 + 1.16383i) q^{7} +(1.42264 - 2.46409i) q^{9} +(-0.866025 + 0.500000i) q^{11} +(-2.74448 + 2.33834i) q^{13} +(-1.41448 + 0.816649i) q^{15} +(0.0827699 - 0.143362i) q^{17} +(1.59019 + 0.918097i) q^{19} +0.915549i q^{21} +(-0.418176 - 0.724302i) q^{23} -12.2427 q^{25} +2.29915 q^{27} +(0.469194 + 0.812668i) q^{29} +1.37801i q^{31} +(-0.340638 - 0.196667i) q^{33} +(-4.83273 + 8.37054i) q^{35} +(-0.644128 + 0.371887i) q^{37} +(-1.33628 - 0.475000i) q^{39} +(-7.53339 + 4.34940i) q^{41} +(4.41962 - 7.65501i) q^{43} +(10.2320 + 5.90744i) q^{45} +9.78488i q^{47} +(-0.790999 - 1.37005i) q^{49} +0.0651125 q^{51} +13.1251 q^{53} +(-2.07622 - 3.59612i) q^{55} +0.722239i q^{57} +(6.81016 + 3.93185i) q^{59} +(-1.13275 + 1.96197i) q^{61} +(5.73557 - 3.31143i) q^{63} +(-9.70980 - 11.3963i) q^{65} +(10.7974 - 6.23386i) q^{67} +(0.164483 - 0.284893i) q^{69} +(7.71084 + 4.45185i) q^{71} -14.7918i q^{73} +(-2.40775 - 4.17034i) q^{75} -2.32766 q^{77} -1.93399 q^{79} +(-3.81576 - 6.60910i) q^{81} +12.5351i q^{83} +(0.595300 + 0.343697i) q^{85} +(-0.184550 + 0.319651i) q^{87} +(-2.14413 + 1.23791i) q^{89} +(-8.25379 + 1.51954i) q^{91} +(-0.469401 + 0.271009i) q^{93} +(-3.81234 + 6.60317i) q^{95} +(-3.09963 - 1.78957i) q^{97} +2.84529i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{3} + 6 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{3} + 6 q^{7} - 14 q^{9} - 2 q^{13} - 6 q^{19} + 10 q^{23} - 40 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{35} + 18 q^{37} + 36 q^{41} + 10 q^{43} - 30 q^{45} + 14 q^{49} + 44 q^{51} + 16 q^{53} - 24 q^{59} + 6 q^{61} - 6 q^{63} - 24 q^{65} - 54 q^{67} + 10 q^{69} + 18 q^{71} + 6 q^{75} - 16 q^{77} - 32 q^{79} - 4 q^{81} + 52 q^{87} - 18 q^{89} - 18 q^{91} + 30 q^{93} - 12 q^{95} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.196667 + 0.340638i 0.113546 + 0.196667i 0.917198 0.398433i \(-0.130446\pi\)
−0.803652 + 0.595100i \(0.797112\pi\)
\(4\) 0 0
\(5\) 4.15244i 1.85703i 0.371299 + 0.928513i \(0.378913\pi\)
−0.371299 + 0.928513i \(0.621087\pi\)
\(6\) 0 0
\(7\) 2.01581 + 1.16383i 0.761906 + 0.439886i 0.829980 0.557794i \(-0.188352\pi\)
−0.0680739 + 0.997680i \(0.521685\pi\)
\(8\) 0 0
\(9\) 1.42264 2.46409i 0.474215 0.821364i
\(10\) 0 0
\(11\) −0.866025 + 0.500000i −0.261116 + 0.150756i
\(12\) 0 0
\(13\) −2.74448 + 2.33834i −0.761182 + 0.648538i
\(14\) 0 0
\(15\) −1.41448 + 0.816649i −0.365216 + 0.210858i
\(16\) 0 0
\(17\) 0.0827699 0.143362i 0.0200746 0.0347703i −0.855814 0.517284i \(-0.826943\pi\)
0.875888 + 0.482514i \(0.160276\pi\)
\(18\) 0 0
\(19\) 1.59019 + 0.918097i 0.364815 + 0.210626i 0.671191 0.741285i \(-0.265783\pi\)
−0.306376 + 0.951911i \(0.599116\pi\)
\(20\) 0 0
\(21\) 0.915549i 0.199789i
\(22\) 0 0
\(23\) −0.418176 0.724302i −0.0871958 0.151027i 0.819129 0.573609i \(-0.194457\pi\)
−0.906325 + 0.422582i \(0.861124\pi\)
\(24\) 0 0
\(25\) −12.2427 −2.44855
\(26\) 0 0
\(27\) 2.29915 0.442472
\(28\) 0 0
\(29\) 0.469194 + 0.812668i 0.0871272 + 0.150909i 0.906296 0.422644i \(-0.138898\pi\)
−0.819168 + 0.573553i \(0.805565\pi\)
\(30\) 0 0
\(31\) 1.37801i 0.247497i 0.992314 + 0.123749i \(0.0394916\pi\)
−0.992314 + 0.123749i \(0.960508\pi\)
\(32\) 0 0
\(33\) −0.340638 0.196667i −0.0592974 0.0342354i
\(34\) 0 0
\(35\) −4.83273 + 8.37054i −0.816881 + 1.41488i
\(36\) 0 0
\(37\) −0.644128 + 0.371887i −0.105894 + 0.0611379i −0.552012 0.833836i \(-0.686140\pi\)
0.446118 + 0.894974i \(0.352806\pi\)
\(38\) 0 0
\(39\) −1.33628 0.475000i −0.213975 0.0760608i
\(40\) 0 0
\(41\) −7.53339 + 4.34940i −1.17652 + 0.679263i −0.955206 0.295941i \(-0.904367\pi\)
−0.221311 + 0.975203i \(0.571034\pi\)
\(42\) 0 0
\(43\) 4.41962 7.65501i 0.673986 1.16738i −0.302778 0.953061i \(-0.597914\pi\)
0.976764 0.214317i \(-0.0687527\pi\)
\(44\) 0 0
\(45\) 10.2320 + 5.90744i 1.52529 + 0.880629i
\(46\) 0 0
\(47\) 9.78488i 1.42727i 0.700517 + 0.713636i \(0.252953\pi\)
−0.700517 + 0.713636i \(0.747047\pi\)
\(48\) 0 0
\(49\) −0.790999 1.37005i −0.113000 0.195722i
\(50\) 0 0
\(51\) 0.0651125 0.00911757
\(52\) 0 0
\(53\) 13.1251 1.80288 0.901439 0.432907i \(-0.142512\pi\)
0.901439 + 0.432907i \(0.142512\pi\)
\(54\) 0 0
\(55\) −2.07622 3.59612i −0.279957 0.484900i
\(56\) 0 0
\(57\) 0.722239i 0.0956628i
\(58\) 0 0
\(59\) 6.81016 + 3.93185i 0.886607 + 0.511883i 0.872831 0.488022i \(-0.162281\pi\)
0.0137761 + 0.999905i \(0.495615\pi\)
\(60\) 0 0
\(61\) −1.13275 + 1.96197i −0.145033 + 0.251205i −0.929385 0.369111i \(-0.879662\pi\)
0.784352 + 0.620316i \(0.212996\pi\)
\(62\) 0 0
\(63\) 5.73557 3.31143i 0.722614 0.417201i
\(64\) 0 0
\(65\) −9.70980 11.3963i −1.20435 1.41354i
\(66\) 0 0
\(67\) 10.7974 6.23386i 1.31911 0.761587i 0.335523 0.942032i \(-0.391087\pi\)
0.983585 + 0.180445i \(0.0577537\pi\)
\(68\) 0 0
\(69\) 0.164483 0.284893i 0.0198014 0.0342971i
\(70\) 0 0
\(71\) 7.71084 + 4.45185i 0.915108 + 0.528338i 0.882071 0.471116i \(-0.156149\pi\)
0.0330368 + 0.999454i \(0.489482\pi\)
\(72\) 0 0
\(73\) 14.7918i 1.73125i −0.500693 0.865625i \(-0.666921\pi\)
0.500693 0.865625i \(-0.333079\pi\)
\(74\) 0 0
\(75\) −2.40775 4.17034i −0.278023 0.481549i
\(76\) 0 0
\(77\) −2.32766 −0.265261
\(78\) 0 0
\(79\) −1.93399 −0.217591 −0.108795 0.994064i \(-0.534699\pi\)
−0.108795 + 0.994064i \(0.534699\pi\)
\(80\) 0 0
\(81\) −3.81576 6.60910i −0.423974 0.734344i
\(82\) 0 0
\(83\) 12.5351i 1.37591i 0.725753 + 0.687956i \(0.241492\pi\)
−0.725753 + 0.687956i \(0.758508\pi\)
\(84\) 0 0
\(85\) 0.595300 + 0.343697i 0.0645694 + 0.0372792i
\(86\) 0 0
\(87\) −0.184550 + 0.319651i −0.0197859 + 0.0342701i
\(88\) 0 0
\(89\) −2.14413 + 1.23791i −0.227277 + 0.131219i −0.609315 0.792928i \(-0.708556\pi\)
0.382038 + 0.924147i \(0.375222\pi\)
\(90\) 0 0
\(91\) −8.25379 + 1.51954i −0.865232 + 0.159291i
\(92\) 0 0
\(93\) −0.469401 + 0.271009i −0.0486746 + 0.0281023i
\(94\) 0 0
\(95\) −3.81234 + 6.60317i −0.391138 + 0.677471i
\(96\) 0 0
\(97\) −3.09963 1.78957i −0.314720 0.181704i 0.334317 0.942461i \(-0.391495\pi\)
−0.649037 + 0.760757i \(0.724828\pi\)
\(98\) 0 0
\(99\) 2.84529i 0.285962i
\(100\) 0 0
\(101\) −5.52574 9.57087i −0.549832 0.952337i −0.998286 0.0585308i \(-0.981358\pi\)
0.448454 0.893806i \(-0.351975\pi\)
\(102\) 0 0
\(103\) −5.64085 −0.555809 −0.277905 0.960609i \(-0.589640\pi\)
−0.277905 + 0.960609i \(0.589640\pi\)
\(104\) 0 0
\(105\) −3.80176 −0.371014
\(106\) 0 0
\(107\) −6.06950 10.5127i −0.586760 1.01630i −0.994653 0.103269i \(-0.967070\pi\)
0.407893 0.913030i \(-0.366264\pi\)
\(108\) 0 0
\(109\) 9.58233i 0.917821i −0.888483 0.458910i \(-0.848240\pi\)
0.888483 0.458910i \(-0.151760\pi\)
\(110\) 0 0
\(111\) −0.253358 0.146276i −0.0240477 0.0138839i
\(112\) 0 0
\(113\) 5.30405 9.18689i 0.498963 0.864230i −0.501036 0.865426i \(-0.667048\pi\)
0.999999 + 0.00119673i \(0.000380930\pi\)
\(114\) 0 0
\(115\) 3.00762 1.73645i 0.280462 0.161925i
\(116\) 0 0
\(117\) 1.85746 + 10.0893i 0.171722 + 0.932754i
\(118\) 0 0
\(119\) 0.333697 0.192660i 0.0305900 0.0176611i
\(120\) 0 0
\(121\) 0.500000 0.866025i 0.0454545 0.0787296i
\(122\) 0 0
\(123\) −2.96314 1.71077i −0.267177 0.154255i
\(124\) 0 0
\(125\) 30.0750i 2.68999i
\(126\) 0 0
\(127\) 2.29257 + 3.97084i 0.203432 + 0.352355i 0.949632 0.313367i \(-0.101457\pi\)
−0.746200 + 0.665722i \(0.768124\pi\)
\(128\) 0 0
\(129\) 3.47678 0.306114
\(130\) 0 0
\(131\) 19.1080 1.66947 0.834736 0.550650i \(-0.185620\pi\)
0.834736 + 0.550650i \(0.185620\pi\)
\(132\) 0 0
\(133\) 2.13702 + 3.70142i 0.185303 + 0.320954i
\(134\) 0 0
\(135\) 9.54709i 0.821683i
\(136\) 0 0
\(137\) 6.20502 + 3.58247i 0.530130 + 0.306071i 0.741070 0.671428i \(-0.234319\pi\)
−0.210939 + 0.977499i \(0.567652\pi\)
\(138\) 0 0
\(139\) −3.49453 + 6.05270i −0.296402 + 0.513383i −0.975310 0.220840i \(-0.929120\pi\)
0.678908 + 0.734223i \(0.262453\pi\)
\(140\) 0 0
\(141\) −3.33310 + 1.92437i −0.280698 + 0.162061i
\(142\) 0 0
\(143\) 1.20762 3.39730i 0.100986 0.284097i
\(144\) 0 0
\(145\) −3.37455 + 1.94830i −0.280242 + 0.161798i
\(146\) 0 0
\(147\) 0.311127 0.538888i 0.0256613 0.0444467i
\(148\) 0 0
\(149\) −2.81034 1.62255i −0.230232 0.132925i 0.380447 0.924803i \(-0.375770\pi\)
−0.610679 + 0.791878i \(0.709103\pi\)
\(150\) 0 0
\(151\) 19.7292i 1.60554i −0.596290 0.802769i \(-0.703359\pi\)
0.596290 0.802769i \(-0.296641\pi\)
\(152\) 0 0
\(153\) −0.235504 0.407905i −0.0190394 0.0329772i
\(154\) 0 0
\(155\) −5.72208 −0.459609
\(156\) 0 0
\(157\) 18.8959 1.50806 0.754029 0.656841i \(-0.228108\pi\)
0.754029 + 0.656841i \(0.228108\pi\)
\(158\) 0 0
\(159\) 2.58129 + 4.47092i 0.204709 + 0.354567i
\(160\) 0 0
\(161\) 1.94674i 0.153425i
\(162\) 0 0
\(163\) 1.52129 + 0.878315i 0.119156 + 0.0687949i 0.558394 0.829576i \(-0.311418\pi\)
−0.439237 + 0.898371i \(0.644751\pi\)
\(164\) 0 0
\(165\) 0.816649 1.41448i 0.0635760 0.110117i
\(166\) 0 0
\(167\) 21.0909 12.1769i 1.63207 0.942274i 0.648611 0.761120i \(-0.275350\pi\)
0.983455 0.181154i \(-0.0579833\pi\)
\(168\) 0 0
\(169\) 2.06436 12.8350i 0.158797 0.987311i
\(170\) 0 0
\(171\) 4.52455 2.61225i 0.346001 0.199764i
\(172\) 0 0
\(173\) −6.49079 + 11.2424i −0.493485 + 0.854742i −0.999972 0.00750605i \(-0.997611\pi\)
0.506486 + 0.862248i \(0.330944\pi\)
\(174\) 0 0
\(175\) −24.6791 14.2485i −1.86556 1.07708i
\(176\) 0 0
\(177\) 3.09306i 0.232489i
\(178\) 0 0
\(179\) 8.12703 + 14.0764i 0.607443 + 1.05212i 0.991660 + 0.128879i \(0.0411379\pi\)
−0.384218 + 0.923243i \(0.625529\pi\)
\(180\) 0 0
\(181\) 1.68869 0.125519 0.0627597 0.998029i \(-0.480010\pi\)
0.0627597 + 0.998029i \(0.480010\pi\)
\(182\) 0 0
\(183\) −0.891096 −0.0658717
\(184\) 0 0
\(185\) −1.54424 2.67470i −0.113535 0.196648i
\(186\) 0 0
\(187\) 0.165540i 0.0121055i
\(188\) 0 0
\(189\) 4.63466 + 2.67582i 0.337122 + 0.194638i
\(190\) 0 0
\(191\) 3.85529 6.67756i 0.278959 0.483171i −0.692167 0.721737i \(-0.743344\pi\)
0.971126 + 0.238566i \(0.0766773\pi\)
\(192\) 0 0
\(193\) −12.1602 + 7.02071i −0.875312 + 0.505362i −0.869110 0.494619i \(-0.835308\pi\)
−0.00620235 + 0.999981i \(0.501974\pi\)
\(194\) 0 0
\(195\) 1.97241 5.54880i 0.141247 0.397358i
\(196\) 0 0
\(197\) −6.56309 + 3.78920i −0.467601 + 0.269970i −0.715235 0.698884i \(-0.753680\pi\)
0.247634 + 0.968854i \(0.420347\pi\)
\(198\) 0 0
\(199\) −11.0842 + 19.1983i −0.785736 + 1.36093i 0.142823 + 0.989748i \(0.454382\pi\)
−0.928558 + 0.371186i \(0.878951\pi\)
\(200\) 0 0
\(201\) 4.24698 + 2.45199i 0.299559 + 0.172950i
\(202\) 0 0
\(203\) 2.18425i 0.153304i
\(204\) 0 0
\(205\) −18.0606 31.2819i −1.26141 2.18482i
\(206\) 0 0
\(207\) −2.37966 −0.165398
\(208\) 0 0
\(209\) −1.83619 −0.127012
\(210\) 0 0
\(211\) 6.58671 + 11.4085i 0.453448 + 0.785394i 0.998597 0.0529442i \(-0.0168605\pi\)
−0.545150 + 0.838339i \(0.683527\pi\)
\(212\) 0 0
\(213\) 3.50214i 0.239962i
\(214\) 0 0
\(215\) 31.7870 + 18.3522i 2.16785 + 1.25161i
\(216\) 0 0
\(217\) −1.60376 + 2.77780i −0.108871 + 0.188570i
\(218\) 0 0
\(219\) 5.03865 2.90906i 0.340480 0.196576i
\(220\) 0 0
\(221\) 0.108067 + 0.586997i 0.00726940 + 0.0394857i
\(222\) 0 0
\(223\) 20.7194 11.9624i 1.38747 0.801059i 0.394445 0.918920i \(-0.370937\pi\)
0.993030 + 0.117861i \(0.0376037\pi\)
\(224\) 0 0
\(225\) −17.4171 + 30.1672i −1.16114 + 2.01115i
\(226\) 0 0
\(227\) −10.3710 5.98771i −0.688349 0.397419i 0.114644 0.993407i \(-0.463427\pi\)
−0.802993 + 0.595988i \(0.796761\pi\)
\(228\) 0 0
\(229\) 19.0257i 1.25725i 0.777707 + 0.628627i \(0.216383\pi\)
−0.777707 + 0.628627i \(0.783617\pi\)
\(230\) 0 0
\(231\) −0.457775 0.792889i −0.0301194 0.0521683i
\(232\) 0 0
\(233\) 14.3767 0.941850 0.470925 0.882173i \(-0.343920\pi\)
0.470925 + 0.882173i \(0.343920\pi\)
\(234\) 0 0
\(235\) −40.6311 −2.65048
\(236\) 0 0
\(237\) −0.380352 0.658789i −0.0247065 0.0427930i
\(238\) 0 0
\(239\) 15.6839i 1.01450i −0.861798 0.507252i \(-0.830661\pi\)
0.861798 0.507252i \(-0.169339\pi\)
\(240\) 0 0
\(241\) −14.3417 8.28018i −0.923830 0.533373i −0.0389750 0.999240i \(-0.512409\pi\)
−0.884855 + 0.465867i \(0.845743\pi\)
\(242\) 0 0
\(243\) 4.94960 8.57296i 0.317517 0.549956i
\(244\) 0 0
\(245\) 5.68905 3.28457i 0.363460 0.209844i
\(246\) 0 0
\(247\) −6.51107 + 1.19870i −0.414290 + 0.0762716i
\(248\) 0 0
\(249\) −4.26994 + 2.46525i −0.270597 + 0.156229i
\(250\) 0 0
\(251\) −5.82612 + 10.0911i −0.367741 + 0.636947i −0.989212 0.146491i \(-0.953202\pi\)
0.621471 + 0.783437i \(0.286535\pi\)
\(252\) 0 0
\(253\) 0.724302 + 0.418176i 0.0455365 + 0.0262905i
\(254\) 0 0
\(255\) 0.270376i 0.0169316i
\(256\) 0 0
\(257\) 3.92113 + 6.79159i 0.244593 + 0.423648i 0.962017 0.272989i \(-0.0880122\pi\)
−0.717424 + 0.696637i \(0.754679\pi\)
\(258\) 0 0
\(259\) −1.73126 −0.107575
\(260\) 0 0
\(261\) 2.66999 0.165268
\(262\) 0 0
\(263\) −8.95584 15.5120i −0.552241 0.956509i −0.998112 0.0614125i \(-0.980439\pi\)
0.445871 0.895097i \(-0.352894\pi\)
\(264\) 0 0
\(265\) 54.5013i 3.34799i
\(266\) 0 0
\(267\) −0.843360 0.486914i −0.0516128 0.0297986i
\(268\) 0 0
\(269\) −10.5579 + 18.2868i −0.643727 + 1.11497i 0.340867 + 0.940111i \(0.389279\pi\)
−0.984594 + 0.174856i \(0.944054\pi\)
\(270\) 0 0
\(271\) 21.6463 12.4975i 1.31492 0.759169i 0.332013 0.943275i \(-0.392272\pi\)
0.982906 + 0.184106i \(0.0589390\pi\)
\(272\) 0 0
\(273\) −2.14086 2.51271i −0.129571 0.152076i
\(274\) 0 0
\(275\) 10.6025 6.12137i 0.639356 0.369133i
\(276\) 0 0
\(277\) −11.5789 + 20.0552i −0.695708 + 1.20500i 0.274233 + 0.961663i \(0.411576\pi\)
−0.969941 + 0.243339i \(0.921757\pi\)
\(278\) 0 0
\(279\) 3.39553 + 1.96041i 0.203285 + 0.117367i
\(280\) 0 0
\(281\) 4.30130i 0.256594i 0.991736 + 0.128297i \(0.0409511\pi\)
−0.991736 + 0.128297i \(0.959049\pi\)
\(282\) 0 0
\(283\) −1.53163 2.65286i −0.0910459 0.157696i 0.816906 0.576771i \(-0.195688\pi\)
−0.907951 + 0.419075i \(0.862354\pi\)
\(284\) 0 0
\(285\) −2.99905 −0.177648
\(286\) 0 0
\(287\) −20.2479 −1.19519
\(288\) 0 0
\(289\) 8.48630 + 14.6987i 0.499194 + 0.864629i
\(290\) 0 0
\(291\) 1.40780i 0.0825268i
\(292\) 0 0
\(293\) −1.45792 0.841730i −0.0851726 0.0491744i 0.456809 0.889565i \(-0.348992\pi\)
−0.541981 + 0.840390i \(0.682326\pi\)
\(294\) 0 0
\(295\) −16.3268 + 28.2788i −0.950581 + 1.64645i
\(296\) 0 0
\(297\) −1.99113 + 1.14958i −0.115537 + 0.0667052i
\(298\) 0 0
\(299\) 2.84134 + 1.01000i 0.164319 + 0.0584097i
\(300\) 0 0
\(301\) 17.8183 10.2874i 1.02703 0.592955i
\(302\) 0 0
\(303\) 2.17347 3.76455i 0.124862 0.216268i
\(304\) 0 0
\(305\) −8.14698 4.70366i −0.466494 0.269331i
\(306\) 0 0
\(307\) 3.06839i 0.175122i −0.996159 0.0875611i \(-0.972093\pi\)
0.996159 0.0875611i \(-0.0279073\pi\)
\(308\) 0 0
\(309\) −1.10937 1.92148i −0.0631098 0.109309i
\(310\) 0 0
\(311\) 12.2084 0.692273 0.346136 0.938184i \(-0.387493\pi\)
0.346136 + 0.938184i \(0.387493\pi\)
\(312\) 0 0
\(313\) −5.42766 −0.306789 −0.153395 0.988165i \(-0.549021\pi\)
−0.153395 + 0.988165i \(0.549021\pi\)
\(314\) 0 0
\(315\) 13.7505 + 23.8166i 0.774754 + 1.34191i
\(316\) 0 0
\(317\) 28.6945i 1.61164i −0.592160 0.805821i \(-0.701725\pi\)
0.592160 0.805821i \(-0.298275\pi\)
\(318\) 0 0
\(319\) −0.812668 0.469194i −0.0455007 0.0262698i
\(320\) 0 0
\(321\) 2.38734 4.13500i 0.133248 0.230793i
\(322\) 0 0
\(323\) 0.263240 0.151982i 0.0146471 0.00845648i
\(324\) 0 0
\(325\) 33.6000 28.6277i 1.86379 1.58798i
\(326\) 0 0
\(327\) 3.26410 1.88453i 0.180505 0.104215i
\(328\) 0 0
\(329\) −11.3879 + 19.7245i −0.627837 + 1.08745i
\(330\) 0 0
\(331\) −18.3596 10.5999i −1.00913 0.582624i −0.0981962 0.995167i \(-0.531307\pi\)
−0.910938 + 0.412543i \(0.864641\pi\)
\(332\) 0 0
\(333\) 2.11625i 0.115970i
\(334\) 0 0
\(335\) 25.8857 + 44.8354i 1.41429 + 2.44962i
\(336\) 0 0
\(337\) −2.84607 −0.155035 −0.0775177 0.996991i \(-0.524699\pi\)
−0.0775177 + 0.996991i \(0.524699\pi\)
\(338\) 0 0
\(339\) 4.17253 0.226621
\(340\) 0 0
\(341\) −0.689003 1.19339i −0.0373116 0.0646256i
\(342\) 0 0
\(343\) 19.9760i 1.07860i
\(344\) 0 0
\(345\) 1.18300 + 0.683006i 0.0636906 + 0.0367718i
\(346\) 0 0
\(347\) −2.10902 + 3.65293i −0.113218 + 0.196099i −0.917066 0.398735i \(-0.869449\pi\)
0.803848 + 0.594835i \(0.202783\pi\)
\(348\) 0 0
\(349\) −2.47314 + 1.42787i −0.132384 + 0.0764320i −0.564729 0.825276i \(-0.691019\pi\)
0.432345 + 0.901708i \(0.357686\pi\)
\(350\) 0 0
\(351\) −6.30998 + 5.37620i −0.336802 + 0.286960i
\(352\) 0 0
\(353\) −4.38074 + 2.52922i −0.233163 + 0.134617i −0.612030 0.790834i \(-0.709647\pi\)
0.378867 + 0.925451i \(0.376314\pi\)
\(354\) 0 0
\(355\) −18.4860 + 32.0188i −0.981137 + 1.69938i
\(356\) 0 0
\(357\) 0.131255 + 0.0757799i 0.00694673 + 0.00401070i
\(358\) 0 0
\(359\) 30.8849i 1.63004i 0.579431 + 0.815021i \(0.303275\pi\)
−0.579431 + 0.815021i \(0.696725\pi\)
\(360\) 0 0
\(361\) −7.81420 13.5346i −0.411273 0.712346i
\(362\) 0 0
\(363\) 0.393335 0.0206447
\(364\) 0 0
\(365\) 61.4221 3.21498
\(366\) 0 0
\(367\) −9.24324 16.0098i −0.482493 0.835703i 0.517305 0.855801i \(-0.326935\pi\)
−0.999798 + 0.0200985i \(0.993602\pi\)
\(368\) 0 0
\(369\) 24.7506i 1.28847i
\(370\) 0 0
\(371\) 26.4578 + 15.2754i 1.37362 + 0.793061i
\(372\) 0 0
\(373\) 8.68112 15.0361i 0.449491 0.778541i −0.548862 0.835913i \(-0.684939\pi\)
0.998353 + 0.0573717i \(0.0182720\pi\)
\(374\) 0 0
\(375\) 10.2447 5.91478i 0.529034 0.305438i
\(376\) 0 0
\(377\) −3.18799 1.13322i −0.164190 0.0583637i
\(378\) 0 0
\(379\) −1.69614 + 0.979265i −0.0871247 + 0.0503015i −0.542929 0.839778i \(-0.682685\pi\)
0.455805 + 0.890080i \(0.349352\pi\)
\(380\) 0 0
\(381\) −0.901746 + 1.56187i −0.0461978 + 0.0800170i
\(382\) 0 0
\(383\) −26.2399 15.1496i −1.34080 0.774109i −0.353871 0.935294i \(-0.615135\pi\)
−0.986924 + 0.161185i \(0.948468\pi\)
\(384\) 0 0
\(385\) 9.66546i 0.492598i
\(386\) 0 0
\(387\) −12.5751 21.7807i −0.639228 1.10718i
\(388\) 0 0
\(389\) −17.7855 −0.901763 −0.450882 0.892584i \(-0.648890\pi\)
−0.450882 + 0.892584i \(0.648890\pi\)
\(390\) 0 0
\(391\) −0.138450 −0.00700170
\(392\) 0 0
\(393\) 3.75791 + 6.50890i 0.189562 + 0.328330i
\(394\) 0 0
\(395\) 8.03076i 0.404072i
\(396\) 0 0
\(397\) 11.0539 + 6.38194i 0.554777 + 0.320301i 0.751046 0.660249i \(-0.229549\pi\)
−0.196270 + 0.980550i \(0.562883\pi\)
\(398\) 0 0
\(399\) −0.840563 + 1.45590i −0.0420808 + 0.0728861i
\(400\) 0 0
\(401\) 2.62185 1.51372i 0.130929 0.0755917i −0.433105 0.901343i \(-0.642582\pi\)
0.564034 + 0.825752i \(0.309249\pi\)
\(402\) 0 0
\(403\) −3.22224 3.78191i −0.160511 0.188390i
\(404\) 0 0
\(405\) 27.4439 15.8447i 1.36370 0.787331i
\(406\) 0 0
\(407\) 0.371887 0.644128i 0.0184338 0.0319282i
\(408\) 0 0
\(409\) −10.5147 6.07065i −0.519917 0.300174i 0.216983 0.976175i \(-0.430378\pi\)
−0.736901 + 0.676001i \(0.763712\pi\)
\(410\) 0 0
\(411\) 2.81822i 0.139012i
\(412\) 0 0
\(413\) 9.15201 + 15.8517i 0.450341 + 0.780013i
\(414\) 0 0
\(415\) −52.0514 −2.55510
\(416\) 0 0
\(417\) −2.74904 −0.134621
\(418\) 0 0
\(419\) 1.45520 + 2.52048i 0.0710913 + 0.123134i 0.899380 0.437168i \(-0.144019\pi\)
−0.828289 + 0.560302i \(0.810685\pi\)
\(420\) 0 0
\(421\) 11.6503i 0.567800i −0.958854 0.283900i \(-0.908372\pi\)
0.958854 0.283900i \(-0.0916284\pi\)
\(422\) 0 0
\(423\) 24.1108 + 13.9204i 1.17231 + 0.676833i
\(424\) 0 0
\(425\) −1.01333 + 1.75514i −0.0491537 + 0.0851368i
\(426\) 0 0
\(427\) −4.56681 + 2.63665i −0.221003 + 0.127596i
\(428\) 0 0
\(429\) 1.39475 0.256776i 0.0673391 0.0123973i
\(430\) 0 0
\(431\) −20.6602 + 11.9282i −0.995167 + 0.574560i −0.906815 0.421530i \(-0.861493\pi\)
−0.0883520 + 0.996089i \(0.528160\pi\)
\(432\) 0 0
\(433\) −3.46406 + 5.99992i −0.166472 + 0.288338i −0.937177 0.348854i \(-0.886571\pi\)
0.770705 + 0.637192i \(0.219904\pi\)
\(434\) 0 0
\(435\) −1.32733 0.766334i −0.0636406 0.0367429i
\(436\) 0 0
\(437\) 1.53571i 0.0734628i
\(438\) 0 0
\(439\) −20.1741 34.9425i −0.962856 1.66771i −0.715269 0.698849i \(-0.753696\pi\)
−0.247586 0.968866i \(-0.579637\pi\)
\(440\) 0 0
\(441\) −4.50124 −0.214345
\(442\) 0 0
\(443\) −15.1853 −0.721476 −0.360738 0.932667i \(-0.617475\pi\)
−0.360738 + 0.932667i \(0.617475\pi\)
\(444\) 0 0
\(445\) −5.14036 8.90336i −0.243676 0.422060i
\(446\) 0 0
\(447\) 1.27641i 0.0603721i
\(448\) 0 0
\(449\) 10.0286 + 5.79004i 0.473281 + 0.273249i 0.717612 0.696443i \(-0.245235\pi\)
−0.244331 + 0.969692i \(0.578568\pi\)
\(450\) 0 0
\(451\) 4.34940 7.53339i 0.204805 0.354733i
\(452\) 0 0
\(453\) 6.72050 3.88008i 0.315757 0.182302i
\(454\) 0 0
\(455\) −6.30979 34.2733i −0.295808 1.60676i
\(456\) 0 0
\(457\) −8.05179 + 4.64870i −0.376647 + 0.217457i −0.676358 0.736573i \(-0.736443\pi\)
0.299712 + 0.954030i \(0.403110\pi\)
\(458\) 0 0
\(459\) 0.190301 0.329610i 0.00888247 0.0153849i
\(460\) 0 0
\(461\) −22.8146 13.1720i −1.06258 0.613481i −0.136435 0.990649i \(-0.543564\pi\)
−0.926145 + 0.377168i \(0.876898\pi\)
\(462\) 0 0
\(463\) 10.2989i 0.478630i 0.970942 + 0.239315i \(0.0769228\pi\)
−0.970942 + 0.239315i \(0.923077\pi\)
\(464\) 0 0
\(465\) −1.12535 1.94916i −0.0521867 0.0903900i
\(466\) 0 0
\(467\) −15.9502 −0.738085 −0.369043 0.929413i \(-0.620314\pi\)
−0.369043 + 0.929413i \(0.620314\pi\)
\(468\) 0 0
\(469\) 29.0206 1.34005
\(470\) 0 0
\(471\) 3.71621 + 6.43666i 0.171234 + 0.296586i
\(472\) 0 0
\(473\) 8.83925i 0.406429i
\(474\) 0 0
\(475\) −19.4683 11.2400i −0.893267 0.515728i
\(476\) 0 0
\(477\) 18.6724 32.3416i 0.854951 1.48082i
\(478\) 0 0
\(479\) −32.9498 + 19.0236i −1.50552 + 0.869210i −0.505536 + 0.862805i \(0.668705\pi\)
−0.999979 + 0.00640464i \(0.997961\pi\)
\(480\) 0 0
\(481\) 0.898199 2.52683i 0.0409544 0.115213i
\(482\) 0 0
\(483\) 0.663134 0.382861i 0.0301737 0.0174208i
\(484\) 0 0
\(485\) 7.43110 12.8710i 0.337429 0.584444i
\(486\) 0 0
\(487\) −11.0666 6.38928i −0.501474 0.289526i 0.227848 0.973697i \(-0.426831\pi\)
−0.729322 + 0.684171i \(0.760164\pi\)
\(488\) 0 0
\(489\) 0.690943i 0.0312455i
\(490\) 0 0
\(491\) −8.64322 14.9705i −0.390063 0.675609i 0.602394 0.798199i \(-0.294214\pi\)
−0.992457 + 0.122589i \(0.960880\pi\)
\(492\) 0 0
\(493\) 0.155341 0.00699619
\(494\) 0 0
\(495\) −11.8149 −0.531039
\(496\) 0 0
\(497\) 10.3624 + 17.9482i 0.464817 + 0.805087i
\(498\) 0 0
\(499\) 40.3670i 1.80708i 0.428508 + 0.903538i \(0.359039\pi\)
−0.428508 + 0.903538i \(0.640961\pi\)
\(500\) 0 0
\(501\) 8.29580 + 4.78958i 0.370629 + 0.213983i
\(502\) 0 0
\(503\) 9.63045 16.6804i 0.429401 0.743744i −0.567419 0.823429i \(-0.692058\pi\)
0.996820 + 0.0796851i \(0.0253915\pi\)
\(504\) 0 0
\(505\) 39.7424 22.9453i 1.76852 1.02105i
\(506\) 0 0
\(507\) 4.77809 1.82104i 0.212203 0.0808750i
\(508\) 0 0
\(509\) −27.0826 + 15.6361i −1.20041 + 0.693059i −0.960647 0.277771i \(-0.910404\pi\)
−0.239767 + 0.970831i \(0.577071\pi\)
\(510\) 0 0
\(511\) 17.2152 29.8175i 0.761554 1.31905i
\(512\) 0 0
\(513\) 3.65609 + 2.11085i 0.161420 + 0.0931961i
\(514\) 0 0
\(515\) 23.4233i 1.03215i
\(516\) 0 0
\(517\) −4.89244 8.47395i −0.215169 0.372684i
\(518\) 0 0
\(519\) −5.10610 −0.224133
\(520\) 0 0
\(521\) −24.6133 −1.07833 −0.539165 0.842200i \(-0.681260\pi\)
−0.539165 + 0.842200i \(0.681260\pi\)
\(522\) 0 0
\(523\) −20.1978 34.9836i −0.883187 1.52973i −0.847777 0.530353i \(-0.822060\pi\)
−0.0354102 0.999373i \(-0.511274\pi\)
\(524\) 0 0
\(525\) 11.2088i 0.489194i
\(526\) 0 0
\(527\) 0.197553 + 0.114057i 0.00860555 + 0.00496842i
\(528\) 0 0
\(529\) 11.1503 19.3128i 0.484794 0.839687i
\(530\) 0 0
\(531\) 19.3769 11.1872i 0.840884 0.485485i
\(532\) 0 0
\(533\) 10.5049 29.5525i 0.455016 1.28006i
\(534\) 0 0
\(535\) 43.6532 25.2032i 1.88729 1.08963i
\(536\) 0 0
\(537\) −3.19664 + 5.53675i −0.137945 + 0.238928i
\(538\) 0 0
\(539\) 1.37005 + 0.790999i 0.0590123 + 0.0340707i
\(540\) 0 0
\(541\) 44.8068i 1.92639i 0.268797 + 0.963197i \(0.413374\pi\)
−0.268797 + 0.963197i \(0.586626\pi\)
\(542\) 0 0
\(543\) 0.332110 + 0.575232i 0.0142522 + 0.0246856i
\(544\) 0 0
\(545\) 39.7900 1.70442
\(546\) 0 0
\(547\) 26.7801 1.14503 0.572517 0.819893i \(-0.305967\pi\)
0.572517 + 0.819893i \(0.305967\pi\)
\(548\) 0 0
\(549\) 3.22299 + 5.58238i 0.137554 + 0.238250i
\(550\) 0 0
\(551\) 1.72306i 0.0734050i
\(552\) 0 0
\(553\) −3.89856 2.25083i −0.165784 0.0957152i
\(554\) 0 0
\(555\) 0.607403 1.05205i 0.0257828 0.0446571i
\(556\) 0 0
\(557\) −16.9164 + 9.76666i −0.716769 + 0.413827i −0.813562 0.581478i \(-0.802475\pi\)
0.0967934 + 0.995304i \(0.469141\pi\)
\(558\) 0 0
\(559\) 5.77042 + 31.3436i 0.244063 + 1.32569i
\(560\) 0 0
\(561\) −0.0563891 + 0.0325563i −0.00238075 + 0.00137453i
\(562\) 0 0
\(563\) 8.32028 14.4111i 0.350658 0.607357i −0.635707 0.771930i \(-0.719291\pi\)
0.986365 + 0.164573i \(0.0526247\pi\)
\(564\) 0 0
\(565\) 38.1480 + 22.0247i 1.60490 + 0.926588i
\(566\) 0 0
\(567\) 17.7636i 0.746001i
\(568\) 0 0
\(569\) 15.7339 + 27.2518i 0.659597 + 1.14246i 0.980720 + 0.195419i \(0.0626065\pi\)
−0.321123 + 0.947038i \(0.604060\pi\)
\(570\) 0 0
\(571\) 28.8120 1.20574 0.602871 0.797838i \(-0.294023\pi\)
0.602871 + 0.797838i \(0.294023\pi\)
\(572\) 0 0
\(573\) 3.03284 0.126699
\(574\) 0 0
\(575\) 5.11962 + 8.86745i 0.213503 + 0.369798i
\(576\) 0 0
\(577\) 3.99850i 0.166460i −0.996530 0.0832299i \(-0.973476\pi\)
0.996530 0.0832299i \(-0.0265236\pi\)
\(578\) 0 0
\(579\) −4.78304 2.76149i −0.198776 0.114764i
\(580\) 0 0
\(581\) −14.5888 + 25.2685i −0.605245 + 1.04831i
\(582\) 0 0
\(583\) −11.3667 + 6.56257i −0.470761 + 0.271794i
\(584\) 0 0
\(585\) −41.8951 + 7.71297i −1.73215 + 0.318892i
\(586\) 0 0
\(587\) 21.6537 12.5018i 0.893743 0.516003i 0.0185782 0.999827i \(-0.494086\pi\)
0.875165 + 0.483825i \(0.160753\pi\)
\(588\) 0 0
\(589\) −1.26514 + 2.19129i −0.0521293 + 0.0902906i
\(590\) 0 0
\(591\) −2.58149 1.49042i −0.106188 0.0613079i
\(592\) 0 0
\(593\) 10.2878i 0.422468i −0.977436 0.211234i \(-0.932252\pi\)
0.977436 0.211234i \(-0.0677481\pi\)
\(594\) 0 0
\(595\) 0.800009 + 1.38566i 0.0327972 + 0.0568064i
\(596\) 0 0
\(597\) −8.71957 −0.356868
\(598\) 0 0
\(599\) 31.4134 1.28352 0.641759 0.766906i \(-0.278205\pi\)
0.641759 + 0.766906i \(0.278205\pi\)
\(600\) 0 0
\(601\) −15.4959 26.8398i −0.632093 1.09482i −0.987123 0.159962i \(-0.948863\pi\)
0.355030 0.934855i \(-0.384471\pi\)
\(602\) 0 0
\(603\) 35.4743i 1.44462i
\(604\) 0 0
\(605\) 3.59612 + 2.07622i 0.146203 + 0.0844103i
\(606\) 0 0
\(607\) −18.4065 + 31.8811i −0.747099 + 1.29401i 0.202109 + 0.979363i \(0.435220\pi\)
−0.949208 + 0.314650i \(0.898113\pi\)
\(608\) 0 0
\(609\) −0.744038 + 0.429570i −0.0301499 + 0.0174071i
\(610\) 0 0
\(611\) −22.8803 26.8544i −0.925640 1.08641i
\(612\) 0 0
\(613\) 13.0921 7.55872i 0.528784 0.305294i −0.211737 0.977327i \(-0.567912\pi\)
0.740521 + 0.672033i \(0.234579\pi\)
\(614\) 0 0
\(615\) 7.10387 12.3043i 0.286456 0.496156i
\(616\) 0 0
\(617\) 29.9492 + 17.2912i 1.20571 + 0.696117i 0.961819 0.273686i \(-0.0882428\pi\)
0.243891 + 0.969803i \(0.421576\pi\)
\(618\) 0 0
\(619\) 45.0400i 1.81031i −0.425083 0.905154i \(-0.639755\pi\)
0.425083 0.905154i \(-0.360245\pi\)
\(620\) 0 0
\(621\) −0.961451 1.66528i −0.0385817 0.0668255i
\(622\) 0 0
\(623\) −5.76288 −0.230885
\(624\) 0 0
\(625\) 63.6710 2.54684
\(626\) 0 0
\(627\) −0.361119 0.625477i −0.0144217 0.0249791i
\(628\) 0 0
\(629\) 0.123124i 0.00490929i
\(630\) 0 0
\(631\) −6.78431 3.91692i −0.270079 0.155930i 0.358845 0.933397i \(-0.383171\pi\)
−0.628924 + 0.777467i \(0.716504\pi\)
\(632\) 0 0
\(633\) −2.59078 + 4.48736i −0.102974 + 0.178357i
\(634\) 0 0
\(635\) −16.4887 + 9.51974i −0.654333 + 0.377780i
\(636\) 0 0
\(637\) 5.37452 + 1.91046i 0.212946 + 0.0756950i
\(638\) 0 0
\(639\) 21.9395 12.6668i 0.867915 0.501091i
\(640\) 0 0
\(641\) −10.2601 + 17.7710i −0.405250 + 0.701914i −0.994351 0.106146i \(-0.966149\pi\)
0.589100 + 0.808060i \(0.299482\pi\)
\(642\) 0 0
\(643\) 18.0680 + 10.4315i 0.712531 + 0.411380i 0.811998 0.583661i \(-0.198380\pi\)
−0.0994663 + 0.995041i \(0.531714\pi\)
\(644\) 0 0
\(645\) 14.4371i 0.568461i
\(646\) 0 0
\(647\) −16.3504 28.3197i −0.642801 1.11336i −0.984805 0.173665i \(-0.944439\pi\)
0.342004 0.939698i \(-0.388894\pi\)
\(648\) 0 0
\(649\) −7.86370 −0.308677
\(650\) 0 0
\(651\) −1.26163 −0.0494473
\(652\) 0 0
\(653\) −17.6070 30.4962i −0.689016 1.19341i −0.972157 0.234332i \(-0.924710\pi\)
0.283141 0.959078i \(-0.408624\pi\)
\(654\) 0 0
\(655\) 79.3447i 3.10025i
\(656\) 0 0
\(657\) −36.4484 21.0435i −1.42199 0.820984i
\(658\) 0 0
\(659\) −14.3910 + 24.9259i −0.560594 + 0.970977i 0.436851 + 0.899534i \(0.356094\pi\)
−0.997445 + 0.0714427i \(0.977240\pi\)
\(660\) 0 0
\(661\) 5.68713 3.28346i 0.221204 0.127712i −0.385304 0.922790i \(-0.625903\pi\)
0.606507 + 0.795078i \(0.292570\pi\)
\(662\) 0 0
\(663\) −0.178700 + 0.152255i −0.00694014 + 0.00591309i
\(664\) 0 0
\(665\) −15.3699 + 8.87384i −0.596020 + 0.344113i
\(666\) 0 0
\(667\) 0.392412 0.679677i 0.0151942 0.0263172i
\(668\) 0 0
\(669\) 8.14966 + 4.70521i 0.315084 + 0.181914i
\(670\) 0 0
\(671\) 2.26549i 0.0874584i
\(672\) 0 0
\(673\) −15.2655 26.4406i −0.588442 1.01921i −0.994437 0.105336i \(-0.966408\pi\)
0.405995 0.913875i \(-0.366925\pi\)
\(674\) 0 0
\(675\) −28.1479 −1.08341
\(676\) 0 0
\(677\) −51.0976 −1.96384 −0.981919 0.189300i \(-0.939378\pi\)
−0.981919 + 0.189300i \(0.939378\pi\)
\(678\) 0 0
\(679\) −4.16552 7.21489i −0.159858 0.276882i
\(680\) 0 0
\(681\) 4.71035i 0.180501i
\(682\) 0 0
\(683\) 7.18785 + 4.14991i 0.275036 + 0.158792i 0.631174 0.775641i \(-0.282573\pi\)
−0.356138 + 0.934433i \(0.615907\pi\)
\(684\) 0 0
\(685\) −14.8760 + 25.7660i −0.568382 + 0.984466i
\(686\) 0 0
\(687\) −6.48087 + 3.74173i −0.247261 + 0.142756i
\(688\) 0 0
\(689\) −36.0217 + 30.6910i −1.37232 + 1.16923i
\(690\) 0 0
\(691\) −20.6477 + 11.9209i −0.785474 + 0.453494i −0.838367 0.545107i \(-0.816489\pi\)
0.0528928 + 0.998600i \(0.483156\pi\)
\(692\) 0 0
\(693\) −3.31143 + 5.73557i −0.125791 + 0.217876i
\(694\) 0 0
\(695\) −25.1335 14.5108i −0.953366 0.550426i
\(696\) 0 0
\(697\) 1.44000i 0.0545438i
\(698\) 0 0
\(699\) 2.82743 + 4.89725i 0.106943 + 0.185231i
\(700\) 0 0
\(701\) 6.86402 0.259250 0.129625 0.991563i \(-0.458623\pi\)
0.129625 + 0.991563i \(0.458623\pi\)
\(702\) 0 0
\(703\) −1.36572 −0.0515089
\(704\) 0 0
\(705\) −7.99081 13.8405i −0.300951 0.521263i
\(706\) 0 0
\(707\) 25.7241i 0.967454i
\(708\) 0 0
\(709\) 21.6102 + 12.4767i 0.811589 + 0.468571i 0.847508 0.530783i \(-0.178102\pi\)
−0.0359182 + 0.999355i \(0.511436\pi\)
\(710\) 0 0
\(711\) −2.75138 + 4.76552i −0.103185 + 0.178721i
\(712\) 0 0
\(713\) 0.998093 0.576249i 0.0373789 0.0215807i
\(714\) 0 0
\(715\) 14.1071 + 5.01458i 0.527575 + 0.187535i
\(716\) 0 0
\(717\) 5.34251 3.08450i 0.199520 0.115193i
\(718\) 0 0
\(719\) 15.3833 26.6446i 0.573700 0.993678i −0.422481 0.906372i \(-0.638841\pi\)
0.996182 0.0873061i \(-0.0278258\pi\)
\(720\) 0 0
\(721\) −11.3709 6.56499i −0.423474 0.244493i
\(722\) 0 0
\(723\) 6.51376i 0.242249i
\(724\) 0 0
\(725\) −5.74422 9.94929i −0.213335 0.369507i
\(726\) 0 0
\(727\) 25.4796 0.944986 0.472493 0.881335i \(-0.343354\pi\)
0.472493 + 0.881335i \(0.343354\pi\)
\(728\) 0 0
\(729\) −19.0009 −0.703736
\(730\) 0 0
\(731\) −0.731623 1.26721i −0.0270601 0.0468694i
\(732\) 0 0
\(733\) 8.13794i 0.300582i −0.988642 0.150291i \(-0.951979\pi\)
0.988642 0.150291i \(-0.0480210\pi\)
\(734\) 0 0
\(735\) 2.23770 + 1.29194i 0.0825388 + 0.0476538i
\(736\) 0 0
\(737\) −6.23386 + 10.7974i −0.229627 + 0.397726i
\(738\) 0 0
\(739\) −31.4183 + 18.1394i −1.15574 + 0.667267i −0.950280 0.311398i \(-0.899203\pi\)
−0.205461 + 0.978665i \(0.565869\pi\)
\(740\) 0 0
\(741\) −1.68884 1.98217i −0.0620410 0.0728169i
\(742\) 0 0
\(743\) −3.32307 + 1.91857i −0.121911 + 0.0703856i −0.559716 0.828685i \(-0.689090\pi\)
0.437804 + 0.899070i \(0.355756\pi\)
\(744\) 0 0
\(745\) 6.73754 11.6698i 0.246844 0.427547i
\(746\) 0 0
\(747\) 30.8878 + 17.8331i 1.13012 + 0.652477i
\(748\) 0 0
\(749\) 28.2555i 1.03243i
\(750\) 0 0
\(751\) −4.99922 8.65891i −0.182424 0.315968i 0.760281 0.649594i \(-0.225061\pi\)
−0.942706 + 0.333626i \(0.891728\pi\)
\(752\) 0 0
\(753\) −4.58323 −0.167022
\(754\) 0 0
\(755\) 81.9242 2.98153
\(756\) 0 0
\(757\) −5.97749 10.3533i −0.217256 0.376298i 0.736712 0.676206i \(-0.236377\pi\)
−0.953968 + 0.299908i \(0.903044\pi\)
\(758\) 0 0
\(759\) 0.328966i 0.0119407i
\(760\) 0 0
\(761\) −14.6310 8.44722i −0.530374 0.306212i 0.210795 0.977530i \(-0.432395\pi\)
−0.741169 + 0.671319i \(0.765728\pi\)
\(762\) 0 0
\(763\) 11.1522 19.3162i 0.403737 0.699293i
\(764\) 0 0
\(765\) 1.69380 0.977916i 0.0612395 0.0353566i
\(766\) 0 0
\(767\) −27.8843 + 5.13357i −1.00685 + 0.185362i
\(768\) 0 0
\(769\) −28.3755 + 16.3826i −1.02325 + 0.590772i −0.915043 0.403357i \(-0.867843\pi\)
−0.108204 + 0.994129i \(0.534510\pi\)
\(770\) 0 0
\(771\) −1.54232 + 2.67137i −0.0555451 + 0.0962070i
\(772\) 0 0
\(773\) 29.9048 + 17.2655i 1.07560 + 0.620997i 0.929706 0.368303i \(-0.120061\pi\)
0.145893 + 0.989300i \(0.453394\pi\)
\(774\) 0 0
\(775\) 16.8706i 0.606009i
\(776\) 0 0
\(777\) −0.340481 0.589731i −0.0122147 0.0211565i
\(778\) 0 0
\(779\) −15.9727 −0.572281
\(780\) 0 0
\(781\) −8.90371 −0.318600
\(782\) 0 0
\(783\) 1.07875 + 1.86845i 0.0385514 + 0.0667729i
\(784\) 0 0
\(785\) 78.4641i 2.80051i
\(786\) 0 0
\(787\) 23.0764 + 13.3232i 0.822586 + 0.474920i 0.851307 0.524667i \(-0.175810\pi\)
−0.0287213 + 0.999587i \(0.509144\pi\)
\(788\) 0 0
\(789\) 3.52264 6.10140i 0.125409 0.217215i
\(790\) 0 0
\(791\) 21.3840 12.3460i 0.760326 0.438974i
\(792\) 0 0
\(793\) −1.47896 8.03334i −0.0525193 0.285272i
\(794\) 0 0
\(795\) −18.5652 + 10.7186i −0.658440 + 0.380151i
\(796\) 0 0
\(797\) −9.59793 + 16.6241i −0.339976 + 0.588856i −0.984428 0.175789i \(-0.943752\pi\)
0.644452 + 0.764645i \(0.277086\pi\)
\(798\) 0 0
\(799\) 1.40278 + 0.809893i 0.0496266 + 0.0286520i
\(800\) 0 0
\(801\) 7.04444i 0.248903i
\(802\) 0 0
\(803\) 7.39590 + 12.8101i 0.260996 + 0.452058i
\(804\) 0 0
\(805\) 8.08373 0.284914
\(806\) 0 0
\(807\) −8.30558 −0.292370
\(808\) 0 0
\(809\) −1.98755 3.44254i −0.0698787 0.121033i 0.828969 0.559294i \(-0.188928\pi\)
−0.898848 + 0.438261i \(0.855595\pi\)
\(810\) 0 0
\(811\) 16.0060i 0.562046i 0.959701 + 0.281023i \(0.0906738\pi\)
−0.959701 + 0.281023i \(0.909326\pi\)
\(812\) 0 0
\(813\) 8.51424 + 4.91570i 0.298607 + 0.172401i
\(814\) 0 0
\(815\) −3.64715 + 6.31704i −0.127754 + 0.221276i
\(816\) 0 0
\(817\) 14.0561 8.11529i 0.491760 0.283918i
\(818\) 0 0
\(819\) −7.99792 + 22.4999i −0.279470 + 0.786209i
\(820\) 0 0
\(821\) −26.4599 + 15.2766i −0.923457 + 0.533158i −0.884736 0.466092i \(-0.845662\pi\)
−0.0387209 + 0.999250i \(0.512328\pi\)
\(822\) 0 0
\(823\) −2.51382 + 4.35406i −0.0876262 + 0.151773i −0.906507 0.422190i \(-0.861261\pi\)
0.818881 + 0.573963i \(0.194595\pi\)
\(824\) 0 0
\(825\) 4.17034 + 2.40775i 0.145193 + 0.0838270i
\(826\) 0 0
\(827\) 14.8249i 0.515513i −0.966210 0.257757i \(-0.917017\pi\)
0.966210 0.257757i \(-0.0829833\pi\)
\(828\) 0 0
\(829\) 19.1458 + 33.1615i 0.664962 + 1.15175i 0.979296 + 0.202436i \(0.0648857\pi\)
−0.314334 + 0.949313i \(0.601781\pi\)
\(830\) 0 0
\(831\) −9.10876 −0.315979
\(832\) 0 0
\(833\) −0.261884 −0.00907373
\(834\) 0 0
\(835\) 50.5637 + 87.5788i 1.74983 + 3.03079i
\(836\) 0 0
\(837\) 3.16825i 0.109511i
\(838\) 0 0
\(839\) 37.6753 + 21.7518i 1.30069 + 0.750956i 0.980523 0.196404i \(-0.0629262\pi\)
0.320171 + 0.947360i \(0.396260\pi\)
\(840\) 0 0
\(841\) 14.0597 24.3521i 0.484818 0.839729i
\(842\) 0 0
\(843\) −1.46519 + 0.845926i −0.0504637 + 0.0291352i
\(844\) 0 0
\(845\) 53.2967 + 8.57212i 1.83346 + 0.294890i
\(846\) 0 0
\(847\) 2.01581 1.16383i 0.0692641 0.0399897i
\(848\) 0 0
\(849\) 0.602442 1.04346i 0.0206758 0.0358115i
\(850\) 0 0
\(851\) 0.538718 + 0.311029i 0.0184670 + 0.0106619i
\(852\) 0 0
\(853\) 0.229155i 0.00784612i −0.999992 0.00392306i \(-0.998751\pi\)
0.999992 0.00392306i \(-0.00124875\pi\)
\(854\) 0 0
\(855\) 10.8472 + 18.7879i 0.370967 + 0.642533i
\(856\) 0 0
\(857\) 37.2958 1.27400 0.636999 0.770864i \(-0.280175\pi\)
0.636999 + 0.770864i \(0.280175\pi\)
\(858\) 0 0
\(859\) −35.4892 −1.21088 −0.605438 0.795893i \(-0.707002\pi\)
−0.605438 + 0.795893i \(0.707002\pi\)
\(860\) 0 0
\(861\) −3.98209 6.89719i −0.135709 0.235055i
\(862\) 0 0
\(863\) 3.81967i 0.130023i −0.997885 0.0650115i \(-0.979292\pi\)
0.997885 0.0650115i \(-0.0207084\pi\)
\(864\) 0 0
\(865\) −46.6833 26.9526i −1.58728 0.916416i
\(866\) 0 0
\(867\) −3.33795 + 5.78151i −0.113363 + 0.196350i
\(868\) 0 0
\(869\) 1.67488 0.966994i 0.0568165 0.0328030i
\(870\) 0 0
\(871\) −15.0563 + 42.3566i −0.510163 + 1.43520i
\(872\) 0 0
\(873\) −8.81935 + 5.09185i −0.298490 + 0.172333i
\(874\) 0 0
\(875\) 35.0022 60.6256i 1.18329 2.04952i
\(876\) 0 0
\(877\) 44.4636 + 25.6710i 1.50143 + 0.866850i 0.999999 + 0.00165085i \(0.000525482\pi\)
0.501429 + 0.865199i \(0.332808\pi\)
\(878\) 0 0
\(879\) 0.662163i 0.0223342i
\(880\) 0 0
\(881\) −6.86059 11.8829i −0.231139 0.400345i 0.727004 0.686633i \(-0.240912\pi\)
−0.958144 + 0.286288i \(0.907579\pi\)
\(882\) 0 0
\(883\) −38.0437 −1.28027 −0.640136 0.768261i \(-0.721122\pi\)
−0.640136 + 0.768261i \(0.721122\pi\)
\(884\) 0 0
\(885\) −12.8438 −0.431738
\(886\) 0 0
\(887\) 3.78761 + 6.56034i 0.127176 + 0.220275i 0.922581 0.385803i \(-0.126076\pi\)
−0.795406 + 0.606077i \(0.792742\pi\)
\(888\) 0 0
\(889\) 10.6726i 0.357949i
\(890\) 0 0
\(891\) 6.60910 + 3.81576i 0.221413 + 0.127833i
\(892\) 0 0
\(893\) −8.98347 + 15.5598i −0.300620 + 0.520690i
\(894\) 0 0
\(895\) −58.4515 + 33.7470i −1.95382 + 1.12804i
\(896\) 0 0
\(897\) 0.214755 + 1.16650i 0.00717047 + 0.0389483i
\(898\) 0 0
\(899\) −1.11986 + 0.646553i −0.0373495 + 0.0215637i
\(900\) 0 0
\(901\) 1.08637 1.88164i 0.0361921 0.0626866i
\(902\) 0 0
\(903\) 7.00854 + 4.04638i 0.233230 + 0.134655i
\(904\) 0 0
\(905\) 7.01219i 0.233093i
\(906\) 0 0
\(907\) −5.98922 10.3736i −0.198869 0.344451i 0.749293 0.662239i \(-0.230393\pi\)
−0.948162 + 0.317788i \(0.897060\pi\)
\(908\) 0 0
\(909\) −31.4447 −1.04295
\(910\) 0 0
\(911\) −0.982872 −0.0325640 −0.0162820 0.999867i \(-0.505183\pi\)
−0.0162820 + 0.999867i \(0.505183\pi\)
\(912\) 0 0
\(913\) −6.26757 10.8558i −0.207426 0.359273i
\(914\) 0 0
\(915\) 3.70022i 0.122326i
\(916\) 0 0
\(917\) 38.5181 + 22.2384i 1.27198 + 0.734378i
\(918\) 0 0
\(919\) 10.1135 17.5172i 0.333615 0.577838i −0.649603 0.760274i \(-0.725065\pi\)
0.983218 + 0.182436i \(0.0583982\pi\)
\(920\) 0 0
\(921\) 1.04521 0.603452i 0.0344408 0.0198844i
\(922\) 0 0
\(923\) −31.5722 + 5.81250i −1.03921 + 0.191321i
\(924\) 0 0
\(925\) 7.88589 4.55292i 0.259287 0.149699i
\(926\) 0 0
\(927\) −8.02491 + 13.8996i −0.263573 + 0.456521i
\(928\) 0 0
\(929\) −29.9575 17.2960i −0.982874 0.567463i −0.0797377 0.996816i \(-0.525408\pi\)
−0.903137 + 0.429353i \(0.858742\pi\)
\(930\) 0 0
\(931\) 2.90486i 0.0952028i
\(932\) 0 0
\(933\) 2.40099 + 4.15863i 0.0786047 + 0.136147i
\(934\) 0 0
\(935\) −0.687394 −0.0224802
\(936\) 0 0
\(937\) 56.2497 1.83760 0.918798 0.394728i \(-0.129161\pi\)
0.918798 + 0.394728i \(0.129161\pi\)
\(938\) 0 0
\(939\) −1.06744 1.84887i −0.0348347 0.0603354i
\(940\) 0 0
\(941\) 5.03747i 0.164217i −0.996623 0.0821083i \(-0.973835\pi\)
0.996623 0.0821083i \(-0.0261654\pi\)
\(942\) 0 0
\(943\) 6.30057 + 3.63763i 0.205175 + 0.118458i
\(944\) 0 0
\(945\) −11.1112 + 19.2452i −0.361447 + 0.626045i
\(946\) 0 0
\(947\) −33.7612 + 19.4921i −1.09709 + 0.633407i −0.935456 0.353444i \(-0.885011\pi\)
−0.161637 + 0.986850i \(0.551677\pi\)
\(948\) 0 0
\(949\) 34.5882 + 40.5958i 1.12278 + 1.31780i
\(950\) 0 0
\(951\) 9.77441 5.64326i 0.316957 0.182995i
\(952\) 0 0
\(953\) 7.68128 13.3044i 0.248821 0.430970i −0.714378 0.699760i \(-0.753290\pi\)
0.963199 + 0.268790i \(0.0866236\pi\)
\(954\) 0 0
\(955\) 27.7281 + 16.0088i 0.897262 + 0.518034i
\(956\) 0 0
\(957\) 0.369101i 0.0119313i
\(958\) 0 0
\(959\) 8.33877 + 14.4432i 0.269273 + 0.466394i
\(960\) 0 0
\(961\) 29.1011 0.938745
\(962\) 0 0
\(963\) −34.5389 −1.11300
\(964\) 0 0
\(965\) −29.1531 50.4946i −0.938470 1.62548i
\(966\) 0 0
\(967\) 10.8969i 0.350420i 0.984531 + 0.175210i \(0.0560604\pi\)
−0.984531 + 0.175210i \(0.943940\pi\)
\(968\) 0 0
\(969\) 0.103541 + 0.0597796i 0.00332623 + 0.00192040i
\(970\) 0 0
\(971\) 19.0861 33.0581i 0.612502 1.06088i −0.378315 0.925677i \(-0.623496\pi\)
0.990817 0.135208i \(-0.0431702\pi\)
\(972\) 0 0
\(973\) −14.0886 + 8.13407i −0.451660 + 0.260766i
\(974\) 0 0
\(975\) 16.3597 + 5.81530i 0.523929 + 0.186239i
\(976\) 0 0
\(977\) −16.7803 + 9.68811i −0.536849 + 0.309950i −0.743801 0.668401i \(-0.766979\pi\)
0.206952 + 0.978351i \(0.433646\pi\)
\(978\) 0 0
\(979\) 1.23791 2.14413i 0.0395639 0.0685266i
\(980\) 0 0
\(981\) −23.6117 13.6322i −0.753865 0.435244i
\(982\) 0 0
\(983\) 42.9882i 1.37111i −0.728021 0.685555i \(-0.759560\pi\)
0.728021 0.685555i \(-0.240440\pi\)
\(984\) 0 0
\(985\) −15.7344 27.2528i −0.501341 0.868348i
\(986\) 0 0
\(987\) −8.95854 −0.285153
\(988\) 0 0
\(989\) −7.39273 −0.235075
\(990\) 0 0
\(991\) 24.5387 + 42.5022i 0.779496 + 1.35013i 0.932233 + 0.361860i \(0.117858\pi\)
−0.152737 + 0.988267i \(0.548809\pi\)
\(992\) 0 0
\(993\) 8.33862i 0.264618i
\(994\) 0 0
\(995\) −79.7199 46.0263i −2.52729 1.45913i
\(996\) 0 0
\(997\) −16.4390 + 28.4733i −0.520630 + 0.901757i 0.479082 + 0.877770i \(0.340969\pi\)
−0.999712 + 0.0239873i \(0.992364\pi\)
\(998\) 0 0
\(999\) −1.48095 + 0.855027i −0.0468552 + 0.0270518i
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 572.2.p.a.309.8 24
13.2 odd 12 7436.2.a.v.1.5 12
13.4 even 6 inner 572.2.p.a.485.8 yes 24
13.11 odd 12 7436.2.a.u.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
572.2.p.a.309.8 24 1.1 even 1 trivial
572.2.p.a.485.8 yes 24 13.4 even 6 inner
7436.2.a.u.1.5 12 13.11 odd 12
7436.2.a.v.1.5 12 13.2 odd 12