Properties

Label 1512.2.j.d.323.29
Level $1512$
Weight $2$
Character 1512.323
Analytic conductor $12.073$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (of order \(2\), degree \(1\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.29
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.d.323.30

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.508876 - 1.31949i) q^{2} +(-1.48209 - 1.34291i) q^{4} +2.29653 q^{5} -1.00000i q^{7} +(-2.52615 + 1.27222i) q^{8} +O(q^{10})\) \(q+(0.508876 - 1.31949i) q^{2} +(-1.48209 - 1.34291i) q^{4} +2.29653 q^{5} -1.00000i q^{7} +(-2.52615 + 1.27222i) q^{8} +(1.16865 - 3.03024i) q^{10} -4.02419i q^{11} -1.82461i q^{13} +(-1.31949 - 0.508876i) q^{14} +(0.393186 + 3.98063i) q^{16} -0.430472i q^{17} +5.01046 q^{19} +(-3.40366 - 3.08403i) q^{20} +(-5.30987 - 2.04781i) q^{22} -3.49479 q^{23} +0.274031 q^{25} +(-2.40754 - 0.928498i) q^{26} +(-1.34291 + 1.48209i) q^{28} +2.16016 q^{29} -2.10635i q^{31} +(5.45247 + 1.50684i) q^{32} +(-0.568002 - 0.219057i) q^{34} -2.29653i q^{35} -2.19775i q^{37} +(2.54970 - 6.61124i) q^{38} +(-5.80137 + 2.92170i) q^{40} -4.35032i q^{41} -12.0354 q^{43} +(-5.40413 + 5.96422i) q^{44} +(-1.77842 + 4.61133i) q^{46} +1.72531 q^{47} -1.00000 q^{49} +(0.139448 - 0.361580i) q^{50} +(-2.45028 + 2.70423i) q^{52} +10.0938 q^{53} -9.24166i q^{55} +(1.27222 + 2.52615i) q^{56} +(1.09925 - 2.85030i) q^{58} -10.6175i q^{59} +8.44575i q^{61} +(-2.77930 - 1.07187i) q^{62} +(4.76289 - 6.42767i) q^{64} -4.19026i q^{65} -12.6358 q^{67} +(-0.578085 + 0.637998i) q^{68} +(-3.03024 - 1.16865i) q^{70} +8.95532 q^{71} -7.16051 q^{73} +(-2.89991 - 1.11838i) q^{74} +(-7.42596 - 6.72860i) q^{76} -4.02419 q^{77} -15.2405i q^{79} +(0.902963 + 9.14162i) q^{80} +(-5.74018 - 2.21377i) q^{82} -13.4507i q^{83} -0.988589i q^{85} +(-6.12452 + 15.8805i) q^{86} +(5.11968 + 10.1657i) q^{88} +7.44987i q^{89} -1.82461 q^{91} +(5.17960 + 4.69319i) q^{92} +(0.877970 - 2.27653i) q^{94} +11.5067 q^{95} +1.68878 q^{97} +(-0.508876 + 1.31949i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{4} - 12 q^{10} - 12 q^{16} + 16 q^{19} - 24 q^{22} + 48 q^{25} + 8 q^{28} + 12 q^{34} + 32 q^{40} + 64 q^{43} + 60 q^{46} - 48 q^{49} + 16 q^{52} + 36 q^{58} - 4 q^{64} + 32 q^{67} + 12 q^{70} - 32 q^{76} + 36 q^{82} + 24 q^{88} - 60 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.508876 1.31949i 0.359830 0.933018i
\(3\) 0 0
\(4\) −1.48209 1.34291i −0.741045 0.671455i
\(5\) 2.29653 1.02704 0.513519 0.858078i \(-0.328342\pi\)
0.513519 + 0.858078i \(0.328342\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −2.52615 + 1.27222i −0.893130 + 0.449799i
\(9\) 0 0
\(10\) 1.16865 3.03024i 0.369558 0.958245i
\(11\) 4.02419i 1.21334i −0.794954 0.606670i \(-0.792505\pi\)
0.794954 0.606670i \(-0.207495\pi\)
\(12\) 0 0
\(13\) 1.82461i 0.506055i −0.967459 0.253027i \(-0.918574\pi\)
0.967459 0.253027i \(-0.0814263\pi\)
\(14\) −1.31949 0.508876i −0.352648 0.136003i
\(15\) 0 0
\(16\) 0.393186 + 3.98063i 0.0982966 + 0.995157i
\(17\) 0.430472i 0.104405i −0.998637 0.0522024i \(-0.983376\pi\)
0.998637 0.0522024i \(-0.0166241\pi\)
\(18\) 0 0
\(19\) 5.01046 1.14948 0.574739 0.818336i \(-0.305103\pi\)
0.574739 + 0.818336i \(0.305103\pi\)
\(20\) −3.40366 3.08403i −0.761081 0.689609i
\(21\) 0 0
\(22\) −5.30987 2.04781i −1.13207 0.436595i
\(23\) −3.49479 −0.728715 −0.364357 0.931259i \(-0.618711\pi\)
−0.364357 + 0.931259i \(0.618711\pi\)
\(24\) 0 0
\(25\) 0.274031 0.0548062
\(26\) −2.40754 0.928498i −0.472158 0.182093i
\(27\) 0 0
\(28\) −1.34291 + 1.48209i −0.253786 + 0.280089i
\(29\) 2.16016 0.401131 0.200565 0.979680i \(-0.435722\pi\)
0.200565 + 0.979680i \(0.435722\pi\)
\(30\) 0 0
\(31\) 2.10635i 0.378311i −0.981947 0.189156i \(-0.939425\pi\)
0.981947 0.189156i \(-0.0605750\pi\)
\(32\) 5.45247 + 1.50684i 0.963870 + 0.266374i
\(33\) 0 0
\(34\) −0.568002 0.219057i −0.0974115 0.0375679i
\(35\) 2.29653i 0.388184i
\(36\) 0 0
\(37\) 2.19775i 0.361309i −0.983547 0.180654i \(-0.942178\pi\)
0.983547 0.180654i \(-0.0578215\pi\)
\(38\) 2.54970 6.61124i 0.413616 1.07248i
\(39\) 0 0
\(40\) −5.80137 + 2.92170i −0.917278 + 0.461961i
\(41\) 4.35032i 0.679405i −0.940533 0.339703i \(-0.889674\pi\)
0.940533 0.339703i \(-0.110326\pi\)
\(42\) 0 0
\(43\) −12.0354 −1.83538 −0.917690 0.397298i \(-0.869948\pi\)
−0.917690 + 0.397298i \(0.869948\pi\)
\(44\) −5.40413 + 5.96422i −0.814703 + 0.899140i
\(45\) 0 0
\(46\) −1.77842 + 4.61133i −0.262213 + 0.679904i
\(47\) 1.72531 0.251663 0.125831 0.992052i \(-0.459840\pi\)
0.125831 + 0.992052i \(0.459840\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0.139448 0.361580i 0.0197209 0.0511351i
\(51\) 0 0
\(52\) −2.45028 + 2.70423i −0.339793 + 0.375010i
\(53\) 10.0938 1.38649 0.693246 0.720701i \(-0.256180\pi\)
0.693246 + 0.720701i \(0.256180\pi\)
\(54\) 0 0
\(55\) 9.24166i 1.24615i
\(56\) 1.27222 + 2.52615i 0.170008 + 0.337571i
\(57\) 0 0
\(58\) 1.09925 2.85030i 0.144339 0.374262i
\(59\) 10.6175i 1.38228i −0.722721 0.691140i \(-0.757109\pi\)
0.722721 0.691140i \(-0.242891\pi\)
\(60\) 0 0
\(61\) 8.44575i 1.08137i 0.841226 + 0.540684i \(0.181834\pi\)
−0.841226 + 0.540684i \(0.818166\pi\)
\(62\) −2.77930 1.07187i −0.352971 0.136127i
\(63\) 0 0
\(64\) 4.76289 6.42767i 0.595361 0.803458i
\(65\) 4.19026i 0.519737i
\(66\) 0 0
\(67\) −12.6358 −1.54370 −0.771852 0.635802i \(-0.780669\pi\)
−0.771852 + 0.635802i \(0.780669\pi\)
\(68\) −0.578085 + 0.637998i −0.0701031 + 0.0773686i
\(69\) 0 0
\(70\) −3.03024 1.16865i −0.362182 0.139680i
\(71\) 8.95532 1.06280 0.531401 0.847121i \(-0.321666\pi\)
0.531401 + 0.847121i \(0.321666\pi\)
\(72\) 0 0
\(73\) −7.16051 −0.838075 −0.419037 0.907969i \(-0.637632\pi\)
−0.419037 + 0.907969i \(0.637632\pi\)
\(74\) −2.89991 1.11838i −0.337108 0.130010i
\(75\) 0 0
\(76\) −7.42596 6.72860i −0.851816 0.771823i
\(77\) −4.02419 −0.458599
\(78\) 0 0
\(79\) 15.2405i 1.71468i −0.514747 0.857342i \(-0.672114\pi\)
0.514747 0.857342i \(-0.327886\pi\)
\(80\) 0.902963 + 9.14162i 0.100954 + 1.02206i
\(81\) 0 0
\(82\) −5.74018 2.21377i −0.633897 0.244470i
\(83\) 13.4507i 1.47640i −0.674581 0.738201i \(-0.735676\pi\)
0.674581 0.738201i \(-0.264324\pi\)
\(84\) 0 0
\(85\) 0.988589i 0.107228i
\(86\) −6.12452 + 15.8805i −0.660424 + 1.71244i
\(87\) 0 0
\(88\) 5.11968 + 10.1657i 0.545760 + 1.08367i
\(89\) 7.44987i 0.789684i 0.918749 + 0.394842i \(0.129201\pi\)
−0.918749 + 0.394842i \(0.870799\pi\)
\(90\) 0 0
\(91\) −1.82461 −0.191271
\(92\) 5.17960 + 4.69319i 0.540011 + 0.489299i
\(93\) 0 0
\(94\) 0.877970 2.27653i 0.0905556 0.234806i
\(95\) 11.5067 1.18056
\(96\) 0 0
\(97\) 1.68878 0.171469 0.0857347 0.996318i \(-0.472676\pi\)
0.0857347 + 0.996318i \(0.472676\pi\)
\(98\) −0.508876 + 1.31949i −0.0514042 + 0.133288i
\(99\) 0 0
\(100\) −0.406139 0.367999i −0.0406139 0.0367999i
\(101\) 13.3668 1.33005 0.665024 0.746822i \(-0.268421\pi\)
0.665024 + 0.746822i \(0.268421\pi\)
\(102\) 0 0
\(103\) 4.59723i 0.452978i 0.974014 + 0.226489i \(0.0727248\pi\)
−0.974014 + 0.226489i \(0.927275\pi\)
\(104\) 2.32131 + 4.60923i 0.227623 + 0.451973i
\(105\) 0 0
\(106\) 5.13650 13.3186i 0.498901 1.29362i
\(107\) 15.7048i 1.51824i 0.650951 + 0.759120i \(0.274370\pi\)
−0.650951 + 0.759120i \(0.725630\pi\)
\(108\) 0 0
\(109\) 16.0140i 1.53386i 0.641728 + 0.766932i \(0.278218\pi\)
−0.641728 + 0.766932i \(0.721782\pi\)
\(110\) −12.1943 4.70286i −1.16268 0.448400i
\(111\) 0 0
\(112\) 3.98063 0.393186i 0.376134 0.0371526i
\(113\) 1.46858i 0.138152i −0.997611 0.0690760i \(-0.977995\pi\)
0.997611 0.0690760i \(-0.0220051\pi\)
\(114\) 0 0
\(115\) −8.02588 −0.748417
\(116\) −3.20155 2.90089i −0.297256 0.269341i
\(117\) 0 0
\(118\) −14.0096 5.40298i −1.28969 0.497385i
\(119\) −0.430472 −0.0394613
\(120\) 0 0
\(121\) −5.19413 −0.472193
\(122\) 11.1441 + 4.29784i 1.00893 + 0.389108i
\(123\) 0 0
\(124\) −2.82863 + 3.12180i −0.254019 + 0.280346i
\(125\) −10.8533 −0.970750
\(126\) 0 0
\(127\) 6.35551i 0.563960i 0.959420 + 0.281980i \(0.0909913\pi\)
−0.959420 + 0.281980i \(0.909009\pi\)
\(128\) −6.05750 9.55545i −0.535413 0.844591i
\(129\) 0 0
\(130\) −5.52899 2.13232i −0.484924 0.187017i
\(131\) 7.75532i 0.677585i −0.940861 0.338793i \(-0.889982\pi\)
0.940861 0.338793i \(-0.110018\pi\)
\(132\) 0 0
\(133\) 5.01046i 0.434462i
\(134\) −6.43003 + 16.6727i −0.555470 + 1.44030i
\(135\) 0 0
\(136\) 0.547657 + 1.08744i 0.0469612 + 0.0932469i
\(137\) 3.09831i 0.264706i 0.991203 + 0.132353i \(0.0422533\pi\)
−0.991203 + 0.132353i \(0.957747\pi\)
\(138\) 0 0
\(139\) −3.76422 −0.319277 −0.159638 0.987176i \(-0.551033\pi\)
−0.159638 + 0.987176i \(0.551033\pi\)
\(140\) −3.08403 + 3.40366i −0.260648 + 0.287662i
\(141\) 0 0
\(142\) 4.55715 11.8164i 0.382427 0.991613i
\(143\) −7.34257 −0.614016
\(144\) 0 0
\(145\) 4.96085 0.411976
\(146\) −3.64381 + 9.44820i −0.301564 + 0.781939i
\(147\) 0 0
\(148\) −2.95139 + 3.25727i −0.242602 + 0.267746i
\(149\) −6.57894 −0.538968 −0.269484 0.963005i \(-0.586853\pi\)
−0.269484 + 0.963005i \(0.586853\pi\)
\(150\) 0 0
\(151\) 1.08336i 0.0881627i −0.999028 0.0440814i \(-0.985964\pi\)
0.999028 0.0440814i \(-0.0140361\pi\)
\(152\) −12.6572 + 6.37444i −1.02663 + 0.517035i
\(153\) 0 0
\(154\) −2.04781 + 5.30987i −0.165018 + 0.427881i
\(155\) 4.83728i 0.388540i
\(156\) 0 0
\(157\) 16.6130i 1.32586i 0.748680 + 0.662932i \(0.230688\pi\)
−0.748680 + 0.662932i \(0.769312\pi\)
\(158\) −20.1096 7.75550i −1.59983 0.616994i
\(159\) 0 0
\(160\) 12.5217 + 3.46050i 0.989930 + 0.273577i
\(161\) 3.49479i 0.275428i
\(162\) 0 0
\(163\) 11.4881 0.899818 0.449909 0.893074i \(-0.351456\pi\)
0.449909 + 0.893074i \(0.351456\pi\)
\(164\) −5.84208 + 6.44756i −0.456190 + 0.503470i
\(165\) 0 0
\(166\) −17.7480 6.84472i −1.37751 0.531253i
\(167\) 13.4761 1.04281 0.521405 0.853309i \(-0.325408\pi\)
0.521405 + 0.853309i \(0.325408\pi\)
\(168\) 0 0
\(169\) 9.67081 0.743909
\(170\) −1.30443 0.503069i −0.100045 0.0385836i
\(171\) 0 0
\(172\) 17.8375 + 16.1624i 1.36010 + 1.23237i
\(173\) 4.98344 0.378884 0.189442 0.981892i \(-0.439332\pi\)
0.189442 + 0.981892i \(0.439332\pi\)
\(174\) 0 0
\(175\) 0.274031i 0.0207148i
\(176\) 16.0188 1.58226i 1.20746 0.119267i
\(177\) 0 0
\(178\) 9.83000 + 3.79106i 0.736790 + 0.284152i
\(179\) 3.45929i 0.258559i 0.991608 + 0.129280i \(0.0412665\pi\)
−0.991608 + 0.129280i \(0.958733\pi\)
\(180\) 0 0
\(181\) 10.1409i 0.753768i 0.926260 + 0.376884i \(0.123004\pi\)
−0.926260 + 0.376884i \(0.876996\pi\)
\(182\) −0.928498 + 2.40754i −0.0688249 + 0.178459i
\(183\) 0 0
\(184\) 8.82838 4.44616i 0.650837 0.327775i
\(185\) 5.04720i 0.371078i
\(186\) 0 0
\(187\) −1.73230 −0.126678
\(188\) −2.55707 2.31694i −0.186493 0.168980i
\(189\) 0 0
\(190\) 5.85546 15.1829i 0.424800 1.10148i
\(191\) 10.9743 0.794074 0.397037 0.917803i \(-0.370038\pi\)
0.397037 + 0.917803i \(0.370038\pi\)
\(192\) 0 0
\(193\) 2.07226 0.149165 0.0745824 0.997215i \(-0.476238\pi\)
0.0745824 + 0.997215i \(0.476238\pi\)
\(194\) 0.859378 2.22832i 0.0616998 0.159984i
\(195\) 0 0
\(196\) 1.48209 + 1.34291i 0.105864 + 0.0959221i
\(197\) 4.69547 0.334539 0.167269 0.985911i \(-0.446505\pi\)
0.167269 + 0.985911i \(0.446505\pi\)
\(198\) 0 0
\(199\) 16.7556i 1.18777i −0.804549 0.593887i \(-0.797593\pi\)
0.804549 0.593887i \(-0.202407\pi\)
\(200\) −0.692244 + 0.348629i −0.0489490 + 0.0246518i
\(201\) 0 0
\(202\) 6.80205 17.6373i 0.478591 1.24096i
\(203\) 2.16016i 0.151613i
\(204\) 0 0
\(205\) 9.99061i 0.697775i
\(206\) 6.06598 + 2.33942i 0.422637 + 0.162995i
\(207\) 0 0
\(208\) 7.26308 0.717410i 0.503604 0.0497435i
\(209\) 20.1631i 1.39471i
\(210\) 0 0
\(211\) 15.2955 1.05299 0.526494 0.850179i \(-0.323506\pi\)
0.526494 + 0.850179i \(0.323506\pi\)
\(212\) −14.9599 13.5551i −1.02745 0.930967i
\(213\) 0 0
\(214\) 20.7223 + 7.99179i 1.41654 + 0.546307i
\(215\) −27.6396 −1.88500
\(216\) 0 0
\(217\) −2.10635 −0.142988
\(218\) 21.1303 + 8.14914i 1.43112 + 0.551930i
\(219\) 0 0
\(220\) −12.4107 + 13.6970i −0.836731 + 0.923450i
\(221\) −0.785442 −0.0528345
\(222\) 0 0
\(223\) 23.5657i 1.57808i −0.614343 0.789039i \(-0.710579\pi\)
0.614343 0.789039i \(-0.289421\pi\)
\(224\) 1.50684 5.45247i 0.100680 0.364308i
\(225\) 0 0
\(226\) −1.93777 0.747323i −0.128898 0.0497112i
\(227\) 16.6514i 1.10519i 0.833450 + 0.552595i \(0.186362\pi\)
−0.833450 + 0.552595i \(0.813638\pi\)
\(228\) 0 0
\(229\) 24.4291i 1.61432i 0.590332 + 0.807161i \(0.298997\pi\)
−0.590332 + 0.807161i \(0.701003\pi\)
\(230\) −4.08418 + 10.5900i −0.269303 + 0.698287i
\(231\) 0 0
\(232\) −5.45688 + 2.74820i −0.358262 + 0.180428i
\(233\) 18.3470i 1.20195i −0.799268 0.600975i \(-0.794779\pi\)
0.799268 0.600975i \(-0.205221\pi\)
\(234\) 0 0
\(235\) 3.96222 0.258467
\(236\) −14.2583 + 15.7361i −0.928138 + 1.02433i
\(237\) 0 0
\(238\) −0.219057 + 0.568002i −0.0141993 + 0.0368181i
\(239\) 30.1173 1.94812 0.974062 0.226280i \(-0.0726563\pi\)
0.974062 + 0.226280i \(0.0726563\pi\)
\(240\) 0 0
\(241\) −5.51878 −0.355496 −0.177748 0.984076i \(-0.556881\pi\)
−0.177748 + 0.984076i \(0.556881\pi\)
\(242\) −2.64317 + 6.85358i −0.169909 + 0.440565i
\(243\) 0 0
\(244\) 11.3419 12.5174i 0.726089 0.801342i
\(245\) −2.29653 −0.146720
\(246\) 0 0
\(247\) 9.14212i 0.581699i
\(248\) 2.67975 + 5.32095i 0.170164 + 0.337881i
\(249\) 0 0
\(250\) −5.52299 + 14.3208i −0.349304 + 0.905727i
\(251\) 18.2010i 1.14884i 0.818561 + 0.574420i \(0.194772\pi\)
−0.818561 + 0.574420i \(0.805228\pi\)
\(252\) 0 0
\(253\) 14.0637i 0.884178i
\(254\) 8.38601 + 3.23417i 0.526185 + 0.202930i
\(255\) 0 0
\(256\) −15.6908 + 3.13026i −0.980676 + 0.195641i
\(257\) 5.91116i 0.368728i −0.982858 0.184364i \(-0.940977\pi\)
0.982858 0.184364i \(-0.0590226\pi\)
\(258\) 0 0
\(259\) −2.19775 −0.136562
\(260\) −5.62714 + 6.21034i −0.348980 + 0.385149i
\(261\) 0 0
\(262\) −10.2330 3.94649i −0.632199 0.243815i
\(263\) 15.7132 0.968916 0.484458 0.874815i \(-0.339017\pi\)
0.484458 + 0.874815i \(0.339017\pi\)
\(264\) 0 0
\(265\) 23.1807 1.42398
\(266\) −6.61124 2.54970i −0.405361 0.156332i
\(267\) 0 0
\(268\) 18.7273 + 16.9687i 1.14395 + 1.03653i
\(269\) 26.9473 1.64301 0.821504 0.570202i \(-0.193135\pi\)
0.821504 + 0.570202i \(0.193135\pi\)
\(270\) 0 0
\(271\) 5.68443i 0.345305i 0.984983 + 0.172652i \(0.0552337\pi\)
−0.984983 + 0.172652i \(0.944766\pi\)
\(272\) 1.71355 0.169256i 0.103899 0.0102626i
\(273\) 0 0
\(274\) 4.08818 + 1.57665i 0.246976 + 0.0952491i
\(275\) 1.10275i 0.0664985i
\(276\) 0 0
\(277\) 7.86390i 0.472496i −0.971693 0.236248i \(-0.924082\pi\)
0.971693 0.236248i \(-0.0759178\pi\)
\(278\) −1.91552 + 4.96683i −0.114885 + 0.297891i
\(279\) 0 0
\(280\) 2.92170 + 5.80137i 0.174605 + 0.346698i
\(281\) 0.186660i 0.0111352i 0.999985 + 0.00556760i \(0.00177223\pi\)
−0.999985 + 0.00556760i \(0.998228\pi\)
\(282\) 0 0
\(283\) −12.4545 −0.740343 −0.370171 0.928963i \(-0.620701\pi\)
−0.370171 + 0.928963i \(0.620701\pi\)
\(284\) −13.2726 12.0262i −0.787584 0.713623i
\(285\) 0 0
\(286\) −3.73646 + 9.68842i −0.220941 + 0.572888i
\(287\) −4.35032 −0.256791
\(288\) 0 0
\(289\) 16.8147 0.989100
\(290\) 2.52446 6.54578i 0.148241 0.384381i
\(291\) 0 0
\(292\) 10.6125 + 9.61592i 0.621051 + 0.562729i
\(293\) −8.52169 −0.497843 −0.248921 0.968524i \(-0.580076\pi\)
−0.248921 + 0.968524i \(0.580076\pi\)
\(294\) 0 0
\(295\) 24.3833i 1.41965i
\(296\) 2.79604 + 5.55186i 0.162516 + 0.322695i
\(297\) 0 0
\(298\) −3.34787 + 8.68083i −0.193937 + 0.502867i
\(299\) 6.37662i 0.368770i
\(300\) 0 0
\(301\) 12.0354i 0.693708i
\(302\) −1.42948 0.551296i −0.0822574 0.0317235i
\(303\) 0 0
\(304\) 1.97005 + 19.9448i 0.112990 + 1.14391i
\(305\) 19.3959i 1.11060i
\(306\) 0 0
\(307\) 23.4380 1.33768 0.668838 0.743408i \(-0.266792\pi\)
0.668838 + 0.743408i \(0.266792\pi\)
\(308\) 5.96422 + 5.40413i 0.339843 + 0.307929i
\(309\) 0 0
\(310\) −6.38273 2.46157i −0.362515 0.139808i
\(311\) −12.1365 −0.688199 −0.344099 0.938933i \(-0.611816\pi\)
−0.344099 + 0.938933i \(0.611816\pi\)
\(312\) 0 0
\(313\) −7.54204 −0.426301 −0.213151 0.977019i \(-0.568372\pi\)
−0.213151 + 0.977019i \(0.568372\pi\)
\(314\) 21.9207 + 8.45396i 1.23705 + 0.477085i
\(315\) 0 0
\(316\) −20.4666 + 22.5877i −1.15133 + 1.27066i
\(317\) 29.7545 1.67118 0.835590 0.549354i \(-0.185126\pi\)
0.835590 + 0.549354i \(0.185126\pi\)
\(318\) 0 0
\(319\) 8.69288i 0.486708i
\(320\) 10.9381 14.7613i 0.611458 0.825182i
\(321\) 0 0
\(322\) 4.61133 + 1.77842i 0.256980 + 0.0991072i
\(323\) 2.15686i 0.120011i
\(324\) 0 0
\(325\) 0.499998i 0.0277349i
\(326\) 5.84602 15.1584i 0.323781 0.839547i
\(327\) 0 0
\(328\) 5.53458 + 10.9896i 0.305596 + 0.606797i
\(329\) 1.72531i 0.0951195i
\(330\) 0 0
\(331\) −16.2670 −0.894117 −0.447058 0.894505i \(-0.647528\pi\)
−0.447058 + 0.894505i \(0.647528\pi\)
\(332\) −18.0630 + 19.9351i −0.991337 + 1.09408i
\(333\) 0 0
\(334\) 6.85765 17.7815i 0.375234 0.972961i
\(335\) −29.0184 −1.58544
\(336\) 0 0
\(337\) 32.0056 1.74345 0.871727 0.489992i \(-0.163000\pi\)
0.871727 + 0.489992i \(0.163000\pi\)
\(338\) 4.92124 12.7605i 0.267680 0.694080i
\(339\) 0 0
\(340\) −1.32759 + 1.46518i −0.0719985 + 0.0794605i
\(341\) −8.47634 −0.459020
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 30.4032 15.3117i 1.63923 0.825553i
\(345\) 0 0
\(346\) 2.53595 6.57559i 0.136334 0.353506i
\(347\) 15.3562i 0.824366i 0.911101 + 0.412183i \(0.135234\pi\)
−0.911101 + 0.412183i \(0.864766\pi\)
\(348\) 0 0
\(349\) 10.7044i 0.572996i 0.958081 + 0.286498i \(0.0924912\pi\)
−0.958081 + 0.286498i \(0.907509\pi\)
\(350\) −0.361580 0.139448i −0.0193273 0.00745379i
\(351\) 0 0
\(352\) 6.06382 21.9418i 0.323203 1.16950i
\(353\) 17.0572i 0.907866i −0.891036 0.453933i \(-0.850021\pi\)
0.891036 0.453933i \(-0.149979\pi\)
\(354\) 0 0
\(355\) 20.5661 1.09154
\(356\) 10.0045 11.0414i 0.530237 0.585192i
\(357\) 0 0
\(358\) 4.56449 + 1.76035i 0.241241 + 0.0930373i
\(359\) −17.2174 −0.908697 −0.454349 0.890824i \(-0.650128\pi\)
−0.454349 + 0.890824i \(0.650128\pi\)
\(360\) 0 0
\(361\) 6.10474 0.321302
\(362\) 13.3808 + 5.16047i 0.703279 + 0.271228i
\(363\) 0 0
\(364\) 2.70423 + 2.45028i 0.141740 + 0.128430i
\(365\) −16.4443 −0.860734
\(366\) 0 0
\(367\) 25.6116i 1.33692i 0.743750 + 0.668458i \(0.233045\pi\)
−0.743750 + 0.668458i \(0.766955\pi\)
\(368\) −1.37410 13.9115i −0.0716302 0.725186i
\(369\) 0 0
\(370\) −6.65972 2.56840i −0.346222 0.133525i
\(371\) 10.0938i 0.524045i
\(372\) 0 0
\(373\) 3.99889i 0.207054i 0.994627 + 0.103527i \(0.0330129\pi\)
−0.994627 + 0.103527i \(0.966987\pi\)
\(374\) −0.881526 + 2.28575i −0.0455826 + 0.118193i
\(375\) 0 0
\(376\) −4.35840 + 2.19498i −0.224767 + 0.113198i
\(377\) 3.94143i 0.202994i
\(378\) 0 0
\(379\) −21.0603 −1.08180 −0.540898 0.841088i \(-0.681915\pi\)
−0.540898 + 0.841088i \(0.681915\pi\)
\(380\) −17.0539 15.4524i −0.874847 0.792692i
\(381\) 0 0
\(382\) 5.58457 14.4805i 0.285731 0.740885i
\(383\) 27.1156 1.38554 0.692771 0.721158i \(-0.256390\pi\)
0.692771 + 0.721158i \(0.256390\pi\)
\(384\) 0 0
\(385\) −9.24166 −0.470999
\(386\) 1.05452 2.73432i 0.0536739 0.139173i
\(387\) 0 0
\(388\) −2.50292 2.26788i −0.127067 0.115134i
\(389\) −8.59148 −0.435605 −0.217803 0.975993i \(-0.569889\pi\)
−0.217803 + 0.975993i \(0.569889\pi\)
\(390\) 0 0
\(391\) 1.50441i 0.0760813i
\(392\) 2.52615 1.27222i 0.127590 0.0642571i
\(393\) 0 0
\(394\) 2.38941 6.19561i 0.120377 0.312130i
\(395\) 35.0001i 1.76105i
\(396\) 0 0
\(397\) 33.3145i 1.67201i 0.548725 + 0.836003i \(0.315113\pi\)
−0.548725 + 0.836003i \(0.684887\pi\)
\(398\) −22.1088 8.52653i −1.10821 0.427396i
\(399\) 0 0
\(400\) 0.107745 + 1.09082i 0.00538726 + 0.0545408i
\(401\) 5.42643i 0.270983i −0.990779 0.135491i \(-0.956739\pi\)
0.990779 0.135491i \(-0.0432613\pi\)
\(402\) 0 0
\(403\) −3.84325 −0.191446
\(404\) −19.8108 17.9504i −0.985626 0.893068i
\(405\) 0 0
\(406\) −2.85030 1.09925i −0.141458 0.0545549i
\(407\) −8.84419 −0.438390
\(408\) 0 0
\(409\) 37.1175 1.83534 0.917672 0.397339i \(-0.130066\pi\)
0.917672 + 0.397339i \(0.130066\pi\)
\(410\) −13.1825 5.08398i −0.651036 0.251080i
\(411\) 0 0
\(412\) 6.17366 6.81351i 0.304155 0.335678i
\(413\) −10.6175 −0.522452
\(414\) 0 0
\(415\) 30.8898i 1.51632i
\(416\) 2.74939 9.94861i 0.134800 0.487771i
\(417\) 0 0
\(418\) −26.6049 10.2605i −1.30129 0.501857i
\(419\) 19.7739i 0.966016i −0.875616 0.483008i \(-0.839544\pi\)
0.875616 0.483008i \(-0.160456\pi\)
\(420\) 0 0
\(421\) 19.4861i 0.949692i −0.880069 0.474846i \(-0.842504\pi\)
0.880069 0.474846i \(-0.157496\pi\)
\(422\) 7.78353 20.1823i 0.378896 0.982457i
\(423\) 0 0
\(424\) −25.4985 + 12.8416i −1.23832 + 0.623643i
\(425\) 0.117963i 0.00572202i
\(426\) 0 0
\(427\) 8.44575 0.408718
\(428\) 21.0901 23.2759i 1.01943 1.12508i
\(429\) 0 0
\(430\) −14.0651 + 36.4701i −0.678280 + 1.75874i
\(431\) 33.0237 1.59069 0.795347 0.606154i \(-0.207288\pi\)
0.795347 + 0.606154i \(0.207288\pi\)
\(432\) 0 0
\(433\) −33.2428 −1.59755 −0.798773 0.601632i \(-0.794517\pi\)
−0.798773 + 0.601632i \(0.794517\pi\)
\(434\) −1.07187 + 2.77930i −0.0514514 + 0.133411i
\(435\) 0 0
\(436\) 21.5054 23.7342i 1.02992 1.13666i
\(437\) −17.5105 −0.837642
\(438\) 0 0
\(439\) 9.28199i 0.443005i 0.975160 + 0.221503i \(0.0710962\pi\)
−0.975160 + 0.221503i \(0.928904\pi\)
\(440\) 11.7575 + 23.3458i 0.560516 + 1.11297i
\(441\) 0 0
\(442\) −0.399692 + 1.03638i −0.0190114 + 0.0492956i
\(443\) 2.61270i 0.124133i 0.998072 + 0.0620667i \(0.0197691\pi\)
−0.998072 + 0.0620667i \(0.980231\pi\)
\(444\) 0 0
\(445\) 17.1088i 0.811035i
\(446\) −31.0947 11.9920i −1.47238 0.567839i
\(447\) 0 0
\(448\) −6.42767 4.76289i −0.303679 0.225025i
\(449\) 30.3318i 1.43145i −0.698384 0.715723i \(-0.746097\pi\)
0.698384 0.715723i \(-0.253903\pi\)
\(450\) 0 0
\(451\) −17.5065 −0.824349
\(452\) −1.97217 + 2.17656i −0.0927629 + 0.102377i
\(453\) 0 0
\(454\) 21.9713 + 8.47348i 1.03116 + 0.397680i
\(455\) −4.19026 −0.196442
\(456\) 0 0
\(457\) 6.64077 0.310642 0.155321 0.987864i \(-0.450359\pi\)
0.155321 + 0.987864i \(0.450359\pi\)
\(458\) 32.2339 + 12.4314i 1.50619 + 0.580881i
\(459\) 0 0
\(460\) 11.8951 + 10.7780i 0.554611 + 0.502528i
\(461\) −35.1189 −1.63565 −0.817826 0.575465i \(-0.804821\pi\)
−0.817826 + 0.575465i \(0.804821\pi\)
\(462\) 0 0
\(463\) 2.53401i 0.117765i −0.998265 0.0588826i \(-0.981246\pi\)
0.998265 0.0588826i \(-0.0187538\pi\)
\(464\) 0.849343 + 8.59878i 0.0394298 + 0.399188i
\(465\) 0 0
\(466\) −24.2086 9.33633i −1.12144 0.432497i
\(467\) 16.0379i 0.742145i 0.928604 + 0.371072i \(0.121010\pi\)
−0.928604 + 0.371072i \(0.878990\pi\)
\(468\) 0 0
\(469\) 12.6358i 0.583465i
\(470\) 2.01628 5.22810i 0.0930040 0.241154i
\(471\) 0 0
\(472\) 13.5078 + 26.8214i 0.621748 + 1.23455i
\(473\) 48.4327i 2.22694i
\(474\) 0 0
\(475\) 1.37302 0.0629985
\(476\) 0.637998 + 0.578085i 0.0292426 + 0.0264965i
\(477\) 0 0
\(478\) 15.3260 39.7394i 0.700993 1.81764i
\(479\) −29.2616 −1.33700 −0.668498 0.743714i \(-0.733062\pi\)
−0.668498 + 0.743714i \(0.733062\pi\)
\(480\) 0 0
\(481\) −4.01004 −0.182842
\(482\) −2.80837 + 7.28196i −0.127918 + 0.331684i
\(483\) 0 0
\(484\) 7.69817 + 6.97524i 0.349917 + 0.317057i
\(485\) 3.87832 0.176106
\(486\) 0 0
\(487\) 2.81399i 0.127514i 0.997965 + 0.0637570i \(0.0203083\pi\)
−0.997965 + 0.0637570i \(0.979692\pi\)
\(488\) −10.7449 21.3352i −0.486398 0.965801i
\(489\) 0 0
\(490\) −1.16865 + 3.03024i −0.0527941 + 0.136892i
\(491\) 19.2668i 0.869497i −0.900552 0.434748i \(-0.856837\pi\)
0.900552 0.434748i \(-0.143163\pi\)
\(492\) 0 0
\(493\) 0.929886i 0.0418799i
\(494\) −12.0629 4.65221i −0.542736 0.209313i
\(495\) 0 0
\(496\) 8.38458 0.828187i 0.376479 0.0371867i
\(497\) 8.95532i 0.401701i
\(498\) 0 0
\(499\) 39.4436 1.76574 0.882869 0.469620i \(-0.155609\pi\)
0.882869 + 0.469620i \(0.155609\pi\)
\(500\) 16.0856 + 14.5750i 0.719370 + 0.651815i
\(501\) 0 0
\(502\) 24.0160 + 9.26207i 1.07189 + 0.413387i
\(503\) −33.3860 −1.48861 −0.744304 0.667841i \(-0.767219\pi\)
−0.744304 + 0.667841i \(0.767219\pi\)
\(504\) 0 0
\(505\) 30.6973 1.36601
\(506\) 18.5569 + 7.15669i 0.824954 + 0.318154i
\(507\) 0 0
\(508\) 8.53488 9.41945i 0.378674 0.417920i
\(509\) 9.16584 0.406269 0.203134 0.979151i \(-0.434887\pi\)
0.203134 + 0.979151i \(0.434887\pi\)
\(510\) 0 0
\(511\) 7.16051i 0.316762i
\(512\) −3.85434 + 22.2967i −0.170339 + 0.985385i
\(513\) 0 0
\(514\) −7.79970 3.00805i −0.344030 0.132679i
\(515\) 10.5577i 0.465226i
\(516\) 0 0
\(517\) 6.94299i 0.305352i
\(518\) −1.11838 + 2.89991i −0.0491390 + 0.127415i
\(519\) 0 0
\(520\) 5.33095 + 10.5852i 0.233778 + 0.464193i
\(521\) 1.14191i 0.0500280i 0.999687 + 0.0250140i \(0.00796304\pi\)
−0.999687 + 0.0250140i \(0.992037\pi\)
\(522\) 0 0
\(523\) −19.5606 −0.855323 −0.427662 0.903939i \(-0.640663\pi\)
−0.427662 + 0.903939i \(0.640663\pi\)
\(524\) −10.4147 + 11.4941i −0.454968 + 0.502121i
\(525\) 0 0
\(526\) 7.99605 20.7333i 0.348645 0.904016i
\(527\) −0.906723 −0.0394975
\(528\) 0 0
\(529\) −10.7864 −0.468975
\(530\) 11.7961 30.5866i 0.512390 1.32860i
\(531\) 0 0
\(532\) −6.72860 + 7.42596i −0.291722 + 0.321956i
\(533\) −7.93762 −0.343816
\(534\) 0 0
\(535\) 36.0665i 1.55929i
\(536\) 31.9199 16.0755i 1.37873 0.694357i
\(537\) 0 0
\(538\) 13.7129 35.5567i 0.591203 1.53296i
\(539\) 4.02419i 0.173334i
\(540\) 0 0
\(541\) 35.9815i 1.54696i −0.633818 0.773482i \(-0.718513\pi\)
0.633818 0.773482i \(-0.281487\pi\)
\(542\) 7.50053 + 2.89267i 0.322175 + 0.124251i
\(543\) 0 0
\(544\) 0.648653 2.34713i 0.0278108 0.100633i
\(545\) 36.7766i 1.57534i
\(546\) 0 0
\(547\) −19.0634 −0.815092 −0.407546 0.913185i \(-0.633615\pi\)
−0.407546 + 0.913185i \(0.633615\pi\)
\(548\) 4.16075 4.59197i 0.177738 0.196159i
\(549\) 0 0
\(550\) −1.45507 0.561164i −0.0620443 0.0239281i
\(551\) 10.8234 0.461091
\(552\) 0 0
\(553\) −15.2405 −0.648090
\(554\) −10.3763 4.00175i −0.440847 0.170018i
\(555\) 0 0
\(556\) 5.57891 + 5.05500i 0.236598 + 0.214380i
\(557\) 18.4543 0.781934 0.390967 0.920405i \(-0.372141\pi\)
0.390967 + 0.920405i \(0.372141\pi\)
\(558\) 0 0
\(559\) 21.9598i 0.928803i
\(560\) 9.14162 0.902963i 0.386304 0.0381571i
\(561\) 0 0
\(562\) 0.246295 + 0.0949867i 0.0103893 + 0.00400677i
\(563\) 31.6916i 1.33564i 0.744322 + 0.667821i \(0.232773\pi\)
−0.744322 + 0.667821i \(0.767227\pi\)
\(564\) 0 0
\(565\) 3.37262i 0.141887i
\(566\) −6.33779 + 16.4335i −0.266397 + 0.690753i
\(567\) 0 0
\(568\) −22.6225 + 11.3932i −0.949219 + 0.478047i
\(569\) 27.5168i 1.15357i 0.816897 + 0.576783i \(0.195692\pi\)
−0.816897 + 0.576783i \(0.804308\pi\)
\(570\) 0 0
\(571\) −27.7326 −1.16057 −0.580287 0.814412i \(-0.697060\pi\)
−0.580287 + 0.814412i \(0.697060\pi\)
\(572\) 10.8824 + 9.86041i 0.455014 + 0.412284i
\(573\) 0 0
\(574\) −2.21377 + 5.74018i −0.0924010 + 0.239591i
\(575\) −0.957681 −0.0399381
\(576\) 0 0
\(577\) 38.3530 1.59666 0.798328 0.602223i \(-0.205718\pi\)
0.798328 + 0.602223i \(0.205718\pi\)
\(578\) 8.55659 22.1868i 0.355907 0.922848i
\(579\) 0 0
\(580\) −7.35243 6.66198i −0.305293 0.276624i
\(581\) −13.4507 −0.558027
\(582\) 0 0
\(583\) 40.6194i 1.68229i
\(584\) 18.0885 9.10978i 0.748509 0.376966i
\(585\) 0 0
\(586\) −4.33648 + 11.2443i −0.179138 + 0.464496i
\(587\) 13.7550i 0.567731i −0.958864 0.283865i \(-0.908383\pi\)
0.958864 0.283865i \(-0.0916169\pi\)
\(588\) 0 0
\(589\) 10.5538i 0.434861i
\(590\) −32.1735 12.4081i −1.32456 0.510833i
\(591\) 0 0
\(592\) 8.74845 0.864127i 0.359559 0.0355154i
\(593\) 31.5293i 1.29475i 0.762171 + 0.647376i \(0.224134\pi\)
−0.762171 + 0.647376i \(0.775866\pi\)
\(594\) 0 0
\(595\) −0.988589 −0.0405282
\(596\) 9.75059 + 8.83493i 0.399400 + 0.361893i
\(597\) 0 0
\(598\) 8.41387 + 3.24491i 0.344069 + 0.132694i
\(599\) 14.6650 0.599196 0.299598 0.954066i \(-0.403147\pi\)
0.299598 + 0.954066i \(0.403147\pi\)
\(600\) 0 0
\(601\) −40.8677 −1.66703 −0.833515 0.552497i \(-0.813675\pi\)
−0.833515 + 0.552497i \(0.813675\pi\)
\(602\) 15.8805 + 6.12452i 0.647242 + 0.249617i
\(603\) 0 0
\(604\) −1.45486 + 1.60564i −0.0591973 + 0.0653326i
\(605\) −11.9284 −0.484960
\(606\) 0 0
\(607\) 40.0141i 1.62412i −0.583572 0.812061i \(-0.698345\pi\)
0.583572 0.812061i \(-0.301655\pi\)
\(608\) 27.3194 + 7.54997i 1.10795 + 0.306192i
\(609\) 0 0
\(610\) 25.5926 + 9.87009i 1.03621 + 0.399628i
\(611\) 3.14802i 0.127355i
\(612\) 0 0
\(613\) 3.46053i 0.139770i −0.997555 0.0698848i \(-0.977737\pi\)
0.997555 0.0698848i \(-0.0222632\pi\)
\(614\) 11.9270 30.9261i 0.481335 1.24808i
\(615\) 0 0
\(616\) 10.1657 5.11968i 0.409589 0.206278i
\(617\) 36.6562i 1.47572i 0.674952 + 0.737861i \(0.264164\pi\)
−0.674952 + 0.737861i \(0.735836\pi\)
\(618\) 0 0
\(619\) −10.0770 −0.405029 −0.202514 0.979279i \(-0.564911\pi\)
−0.202514 + 0.979279i \(0.564911\pi\)
\(620\) −6.49603 + 7.16929i −0.260887 + 0.287926i
\(621\) 0 0
\(622\) −6.17598 + 16.0140i −0.247634 + 0.642102i
\(623\) 7.44987 0.298473
\(624\) 0 0
\(625\) −26.2951 −1.05180
\(626\) −3.83796 + 9.95162i −0.153396 + 0.397747i
\(627\) 0 0
\(628\) 22.3098 24.6220i 0.890257 0.982525i
\(629\) −0.946071 −0.0377223
\(630\) 0 0
\(631\) 28.1634i 1.12117i 0.828098 + 0.560583i \(0.189423\pi\)
−0.828098 + 0.560583i \(0.810577\pi\)
\(632\) 19.3893 + 38.4997i 0.771264 + 1.53144i
\(633\) 0 0
\(634\) 15.1414 39.2607i 0.601340 1.55924i
\(635\) 14.5956i 0.579209i
\(636\) 0 0
\(637\) 1.82461i 0.0722935i
\(638\) −11.4701 4.42360i −0.454107 0.175132i
\(639\) 0 0
\(640\) −13.9112 21.9443i −0.549889 0.867426i
\(641\) 12.3588i 0.488142i 0.969757 + 0.244071i \(0.0784830\pi\)
−0.969757 + 0.244071i \(0.921517\pi\)
\(642\) 0 0
\(643\) 28.0875 1.10766 0.553831 0.832629i \(-0.313165\pi\)
0.553831 + 0.832629i \(0.313165\pi\)
\(644\) 4.69319 5.17960i 0.184938 0.204105i
\(645\) 0 0
\(646\) −2.84595 1.09758i −0.111972 0.0431835i
\(647\) −43.3350 −1.70368 −0.851838 0.523806i \(-0.824512\pi\)
−0.851838 + 0.523806i \(0.824512\pi\)
\(648\) 0 0
\(649\) −42.7268 −1.67717
\(650\) −0.659741 0.254437i −0.0258772 0.00997985i
\(651\) 0 0
\(652\) −17.0264 15.4275i −0.666806 0.604187i
\(653\) −32.0509 −1.25425 −0.627125 0.778919i \(-0.715768\pi\)
−0.627125 + 0.778919i \(0.715768\pi\)
\(654\) 0 0
\(655\) 17.8103i 0.695905i
\(656\) 17.3170 1.71048i 0.676115 0.0667832i
\(657\) 0 0
\(658\) −2.27653 0.877970i −0.0887482 0.0342268i
\(659\) 6.88401i 0.268163i 0.990970 + 0.134081i \(0.0428084\pi\)
−0.990970 + 0.134081i \(0.957192\pi\)
\(660\) 0 0
\(661\) 47.2652i 1.83840i 0.393788 + 0.919201i \(0.371164\pi\)
−0.393788 + 0.919201i \(0.628836\pi\)
\(662\) −8.27790 + 21.4641i −0.321730 + 0.834227i
\(663\) 0 0
\(664\) 17.1123 + 33.9784i 0.664085 + 1.31862i
\(665\) 11.5067i 0.446209i
\(666\) 0 0
\(667\) −7.54929 −0.292310
\(668\) −19.9728 18.0972i −0.772770 0.700200i
\(669\) 0 0
\(670\) −14.7667 + 38.2893i −0.570489 + 1.47925i
\(671\) 33.9873 1.31207
\(672\) 0 0
\(673\) 6.89911 0.265941 0.132971 0.991120i \(-0.457548\pi\)
0.132971 + 0.991120i \(0.457548\pi\)
\(674\) 16.2869 42.2309i 0.627346 1.62667i
\(675\) 0 0
\(676\) −14.3330 12.9870i −0.551270 0.499501i
\(677\) 23.7479 0.912707 0.456353 0.889799i \(-0.349155\pi\)
0.456353 + 0.889799i \(0.349155\pi\)
\(678\) 0 0
\(679\) 1.68878i 0.0648094i
\(680\) 1.25771 + 2.49733i 0.0482309 + 0.0957681i
\(681\) 0 0
\(682\) −4.31341 + 11.1844i −0.165169 + 0.428274i
\(683\) 41.5546i 1.59004i 0.606583 + 0.795020i \(0.292540\pi\)
−0.606583 + 0.795020i \(0.707460\pi\)
\(684\) 0 0
\(685\) 7.11534i 0.271863i
\(686\) 1.31949 + 0.508876i 0.0503782 + 0.0194290i
\(687\) 0 0
\(688\) −4.73215 47.9084i −0.180412 1.82649i
\(689\) 18.4172i 0.701641i
\(690\) 0 0
\(691\) −17.5146 −0.666285 −0.333142 0.942876i \(-0.608109\pi\)
−0.333142 + 0.942876i \(0.608109\pi\)
\(692\) −7.38592 6.69232i −0.280770 0.254404i
\(693\) 0 0
\(694\) 20.2624 + 7.81442i 0.769149 + 0.296631i
\(695\) −8.64462 −0.327909
\(696\) 0 0
\(697\) −1.87269 −0.0709331
\(698\) 14.1244 + 5.44724i 0.534616 + 0.206181i
\(699\) 0 0
\(700\) −0.367999 + 0.406139i −0.0139090 + 0.0153506i
\(701\) −1.39361 −0.0526361 −0.0263180 0.999654i \(-0.508378\pi\)
−0.0263180 + 0.999654i \(0.508378\pi\)
\(702\) 0 0
\(703\) 11.0118i 0.415317i
\(704\) −25.8662 19.1668i −0.974868 0.722375i
\(705\) 0 0
\(706\) −22.5068 8.68002i −0.847055 0.326677i
\(707\) 13.3668i 0.502711i
\(708\) 0 0
\(709\) 42.1733i 1.58385i 0.610616 + 0.791927i \(0.290922\pi\)
−0.610616 + 0.791927i \(0.709078\pi\)
\(710\) 10.4656 27.1367i 0.392767 1.01842i
\(711\) 0 0
\(712\) −9.47790 18.8195i −0.355199 0.705290i
\(713\) 7.36125i 0.275681i
\(714\) 0 0
\(715\) −16.8624 −0.630618
\(716\) 4.64551 5.12698i 0.173611 0.191604i
\(717\) 0 0
\(718\) −8.76149 + 22.7181i −0.326976 + 0.847831i
\(719\) −11.8108 −0.440468 −0.220234 0.975447i \(-0.570682\pi\)
−0.220234 + 0.975447i \(0.570682\pi\)
\(720\) 0 0
\(721\) 4.59723 0.171210
\(722\) 3.10655 8.05512i 0.115614 0.299781i
\(723\) 0 0
\(724\) 13.6183 15.0298i 0.506121 0.558576i
\(725\) 0.591949 0.0219844
\(726\) 0 0
\(727\) 45.5504i 1.68937i −0.535264 0.844685i \(-0.679788\pi\)
0.535264 0.844685i \(-0.320212\pi\)
\(728\) 4.60923 2.32131i 0.170830 0.0860335i
\(729\) 0 0
\(730\) −8.36811 + 21.6980i −0.309718 + 0.803081i
\(731\) 5.18089i 0.191622i
\(732\) 0 0
\(733\) 38.4714i 1.42097i −0.703711 0.710487i \(-0.748475\pi\)
0.703711 0.710487i \(-0.251525\pi\)
\(734\) 33.7942 + 13.0331i 1.24737 + 0.481062i
\(735\) 0 0
\(736\) −19.0553 5.26610i −0.702386 0.194111i
\(737\) 50.8487i 1.87304i
\(738\) 0 0
\(739\) 12.1292 0.446179 0.223090 0.974798i \(-0.428386\pi\)
0.223090 + 0.974798i \(0.428386\pi\)
\(740\) −6.77794 + 7.48041i −0.249162 + 0.274985i
\(741\) 0 0
\(742\) −13.3186 5.13650i −0.488943 0.188567i
\(743\) −46.7546 −1.71526 −0.857630 0.514267i \(-0.828064\pi\)
−0.857630 + 0.514267i \(0.828064\pi\)
\(744\) 0 0
\(745\) −15.1087 −0.553540
\(746\) 5.27648 + 2.03494i 0.193186 + 0.0745043i
\(747\) 0 0
\(748\) 2.56743 + 2.32632i 0.0938744 + 0.0850588i
\(749\) 15.7048 0.573841
\(750\) 0 0
\(751\) 29.7196i 1.08448i 0.840222 + 0.542242i \(0.182424\pi\)
−0.840222 + 0.542242i \(0.817576\pi\)
\(752\) 0.678369 + 6.86783i 0.0247376 + 0.250444i
\(753\) 0 0
\(754\) −5.20067 2.00570i −0.189397 0.0730433i
\(755\) 2.48797i 0.0905464i
\(756\) 0 0
\(757\) 16.0932i 0.584919i −0.956278 0.292459i \(-0.905526\pi\)
0.956278 0.292459i \(-0.0944737\pi\)
\(758\) −10.7171 + 27.7888i −0.389262 + 1.00933i
\(759\) 0 0
\(760\) −29.0676 + 14.6391i −1.05439 + 0.531014i
\(761\) 5.21482i 0.189037i 0.995523 + 0.0945186i \(0.0301312\pi\)
−0.995523 + 0.0945186i \(0.969869\pi\)
\(762\) 0 0
\(763\) 16.0140 0.579746
\(764\) −16.2649 14.7375i −0.588445 0.533185i
\(765\) 0 0
\(766\) 13.7985 35.7787i 0.498559 1.29274i
\(767\) −19.3727 −0.699509
\(768\) 0 0
\(769\) 17.7856 0.641367 0.320683 0.947186i \(-0.396087\pi\)
0.320683 + 0.947186i \(0.396087\pi\)
\(770\) −4.70286 + 12.1943i −0.169479 + 0.439450i
\(771\) 0 0
\(772\) −3.07128 2.78286i −0.110538 0.100157i
\(773\) 19.9688 0.718228 0.359114 0.933294i \(-0.383079\pi\)
0.359114 + 0.933294i \(0.383079\pi\)
\(774\) 0 0
\(775\) 0.577204i 0.0207338i
\(776\) −4.26611 + 2.14851i −0.153144 + 0.0771269i
\(777\) 0 0
\(778\) −4.37200 + 11.3363i −0.156744 + 0.406428i
\(779\) 21.7971i 0.780962i
\(780\) 0 0
\(781\) 36.0379i 1.28954i
\(782\) 1.98505 + 0.765558i 0.0709852 + 0.0273763i
\(783\) 0 0
\(784\) −0.393186 3.98063i −0.0140424 0.142165i
\(785\) 38.1522i 1.36171i
\(786\) 0 0
\(787\) −32.1178 −1.14487 −0.572437 0.819948i \(-0.694002\pi\)
−0.572437 + 0.819948i \(0.694002\pi\)
\(788\) −6.95912 6.30560i −0.247908 0.224628i
\(789\) 0 0
\(790\) −46.1822 17.8107i −1.64309 0.633676i
\(791\) −1.46858 −0.0522166
\(792\) 0 0
\(793\) 15.4102 0.547231
\(794\) 43.9580 + 16.9529i 1.56001 + 0.601637i
\(795\) 0 0
\(796\) −22.5013 + 24.8333i −0.797537 + 0.880194i
\(797\) −10.4964 −0.371800 −0.185900 0.982569i \(-0.559520\pi\)
−0.185900 + 0.982569i \(0.559520\pi\)
\(798\) 0 0
\(799\) 0.742698i 0.0262748i
\(800\) 1.49414 + 0.412921i 0.0528260 + 0.0145990i
\(801\) 0 0
\(802\) −7.16010 2.76138i −0.252832 0.0975077i
\(803\) 28.8153i 1.01687i
\(804\) 0 0
\(805\) 8.02588i 0.282875i
\(806\) −1.95574 + 5.07112i −0.0688880 + 0.178623i
\(807\) 0 0
\(808\) −33.7666 + 17.0056i −1.18791 + 0.598255i
\(809\) 31.8333i 1.11920i −0.828763 0.559599i \(-0.810955\pi\)
0.828763 0.559599i \(-0.189045\pi\)
\(810\) 0 0
\(811\) −1.39041 −0.0488238 −0.0244119 0.999702i \(-0.507771\pi\)
−0.0244119 + 0.999702i \(0.507771\pi\)
\(812\) −2.90089 + 3.20155i −0.101801 + 0.112352i
\(813\) 0 0
\(814\) −4.50059 + 11.6698i −0.157746 + 0.409026i
\(815\) 26.3827 0.924147
\(816\) 0 0
\(817\) −60.3029 −2.10973
\(818\) 18.8882 48.9761i 0.660411 1.71241i
\(819\) 0 0
\(820\) −13.4165 + 14.8070i −0.468524 + 0.517083i
\(821\) 18.4554 0.644097 0.322048 0.946723i \(-0.395629\pi\)
0.322048 + 0.946723i \(0.395629\pi\)
\(822\) 0 0
\(823\) 41.1895i 1.43578i 0.696159 + 0.717888i \(0.254891\pi\)
−0.696159 + 0.717888i \(0.745109\pi\)
\(824\) −5.84871 11.6133i −0.203749 0.404568i
\(825\) 0 0
\(826\) −5.40298 + 14.0096i −0.187994 + 0.487457i
\(827\) 6.08227i 0.211501i −0.994393 0.105751i \(-0.966275\pi\)
0.994393 0.105751i \(-0.0337245\pi\)
\(828\) 0 0
\(829\) 44.2474i 1.53677i −0.639985 0.768387i \(-0.721060\pi\)
0.639985 0.768387i \(-0.278940\pi\)
\(830\) −40.7587 15.7191i −1.41475 0.545617i
\(831\) 0 0
\(832\) −11.7280 8.69040i −0.406594 0.301285i
\(833\) 0.430472i 0.0149150i
\(834\) 0 0
\(835\) 30.9482 1.07101
\(836\) −27.0772 + 29.8835i −0.936484 + 1.03354i
\(837\) 0 0
\(838\) −26.0914 10.0624i −0.901311 0.347601i
\(839\) −45.9488 −1.58633 −0.793164 0.609008i \(-0.791568\pi\)
−0.793164 + 0.609008i \(0.791568\pi\)
\(840\) 0 0
\(841\) −24.3337 −0.839094
\(842\) −25.7116 9.91598i −0.886080 0.341727i
\(843\) 0 0
\(844\) −22.6694 20.5405i −0.780312 0.707034i
\(845\) 22.2093 0.764022
\(846\) 0 0
\(847\) 5.19413i 0.178472i
\(848\) 3.96875 + 40.1797i 0.136287 + 1.37978i
\(849\) 0 0
\(850\) −0.155650 0.0600283i −0.00533875 0.00205895i
\(851\) 7.68070i 0.263291i
\(852\) 0 0
\(853\) 22.7862i 0.780185i −0.920776 0.390093i \(-0.872443\pi\)
0.920776 0.390093i \(-0.127557\pi\)
\(854\) 4.29784 11.1441i 0.147069 0.381342i
\(855\) 0 0
\(856\) −19.9800 39.6727i −0.682903 1.35598i
\(857\) 28.8660i 0.986042i −0.870017 0.493021i \(-0.835892\pi\)
0.870017 0.493021i \(-0.164108\pi\)
\(858\) 0 0
\(859\) 53.1350 1.81294 0.906471 0.422268i \(-0.138766\pi\)
0.906471 + 0.422268i \(0.138766\pi\)
\(860\) 40.9644 + 37.1175i 1.39687 + 1.26569i
\(861\) 0 0
\(862\) 16.8050 43.5743i 0.572379 1.48415i
\(863\) −27.2397 −0.927250 −0.463625 0.886032i \(-0.653451\pi\)
−0.463625 + 0.886032i \(0.653451\pi\)
\(864\) 0 0
\(865\) 11.4446 0.389128
\(866\) −16.9165 + 43.8634i −0.574845 + 1.49054i
\(867\) 0 0
\(868\) 3.12180 + 2.82863i 0.105961 + 0.0960101i
\(869\) −61.3305 −2.08050
\(870\) 0 0
\(871\) 23.0553i 0.781199i
\(872\) −20.3734 40.4538i −0.689931 1.36994i
\(873\) 0 0
\(874\) −8.91068 + 23.1049i −0.301408 + 0.781535i
\(875\) 10.8533i 0.366909i
\(876\) 0 0
\(877\) 24.5161i 0.827850i −0.910311 0.413925i \(-0.864158\pi\)
0.910311 0.413925i \(-0.135842\pi\)
\(878\) 12.2475 + 4.72338i 0.413332 + 0.159406i
\(879\) 0 0
\(880\) 36.7876 3.63370i 1.24011 0.122492i
\(881\) 18.5560i 0.625167i −0.949890 0.312584i \(-0.898806\pi\)
0.949890 0.312584i \(-0.101194\pi\)
\(882\) 0 0
\(883\) −48.1013 −1.61874 −0.809369 0.587300i \(-0.800191\pi\)
−0.809369 + 0.587300i \(0.800191\pi\)
\(884\) 1.16410 + 1.05478i 0.0391528 + 0.0354760i
\(885\) 0 0
\(886\) 3.44743 + 1.32954i 0.115819 + 0.0446668i
\(887\) 5.36841 0.180254 0.0901268 0.995930i \(-0.471273\pi\)
0.0901268 + 0.995930i \(0.471273\pi\)
\(888\) 0 0
\(889\) 6.35551 0.213157
\(890\) 22.5748 + 8.70626i 0.756711 + 0.291834i
\(891\) 0 0
\(892\) −31.6467 + 34.9266i −1.05961 + 1.16943i
\(893\) 8.64461 0.289281
\(894\) 0 0
\(895\) 7.94435i 0.265550i
\(896\) −9.55545 + 6.05750i −0.319225 + 0.202367i
\(897\) 0 0
\(898\) −40.0224 15.4351i −1.33556 0.515076i
\(899\) 4.55004i 0.151752i
\(900\) 0 0
\(901\) 4.34510i 0.144756i
\(902\) −8.90864 + 23.0996i −0.296625 + 0.769133i
\(903\) 0 0
\(904\) 1.86836 + 3.70985i 0.0621407 + 0.123388i
\(905\) 23.2889i 0.774148i
\(906\) 0 0
\(907\) 4.80217 0.159453 0.0797267 0.996817i \(-0.474595\pi\)
0.0797267 + 0.996817i \(0.474595\pi\)
\(908\) 22.3613 24.6788i 0.742085 0.818996i
\(909\) 0 0
\(910\) −2.13232 + 5.52899i −0.0706857 + 0.183284i
\(911\) −2.79257 −0.0925219 −0.0462609 0.998929i \(-0.514731\pi\)
−0.0462609 + 0.998929i \(0.514731\pi\)
\(912\) 0 0
\(913\) −54.1280 −1.79138
\(914\) 3.37933 8.76241i 0.111778 0.289835i
\(915\) 0 0
\(916\) 32.8061 36.2062i 1.08394 1.19629i
\(917\) −7.75532 −0.256103
\(918\) 0 0
\(919\) 7.90433i 0.260740i 0.991465 + 0.130370i \(0.0416165\pi\)
−0.991465 + 0.130370i \(0.958384\pi\)
\(920\) 20.2746 10.2107i 0.668434 0.336638i
\(921\) 0 0
\(922\) −17.8712 + 46.3390i −0.588556 + 1.52609i
\(923\) 16.3399i 0.537836i
\(924\) 0 0
\(925\) 0.602253i 0.0198019i
\(926\) −3.34359 1.28949i −0.109877 0.0423754i
\(927\) 0 0
\(928\) 11.7782 + 3.25501i 0.386638 + 0.106851i
\(929\) 4.24927i 0.139414i −0.997568 0.0697070i \(-0.977794\pi\)
0.997568 0.0697070i \(-0.0222064\pi\)
\(930\) 0 0
\(931\) −5.01046 −0.164211
\(932\) −24.6383 + 27.1919i −0.807055 + 0.890699i
\(933\) 0 0
\(934\) 21.1618 + 8.16129i 0.692434 + 0.267046i
\(935\) −3.97827 −0.130103
\(936\) 0 0
\(937\) 1.70393 0.0556649 0.0278324 0.999613i \(-0.491140\pi\)
0.0278324 + 0.999613i \(0.491140\pi\)
\(938\) 16.6727 + 6.43003i 0.544384 + 0.209948i
\(939\) 0 0
\(940\) −5.87237 5.32091i −0.191536 0.173549i
\(941\) −53.7297 −1.75154 −0.875769 0.482730i \(-0.839645\pi\)
−0.875769 + 0.482730i \(0.839645\pi\)
\(942\) 0 0
\(943\) 15.2035i 0.495093i
\(944\) 42.2643 4.17465i 1.37558 0.135873i
\(945\) 0 0
\(946\) 63.9063 + 24.6462i 2.07777 + 0.801318i
\(947\) 48.4816i 1.57544i 0.616033 + 0.787720i \(0.288739\pi\)
−0.616033 + 0.787720i \(0.711261\pi\)
\(948\) 0 0
\(949\) 13.0651i 0.424112i
\(950\) 0.698697 1.81168i 0.0226687 0.0587788i
\(951\) 0 0
\(952\) 1.08744 0.547657i 0.0352440 0.0177497i
\(953\) 51.7471i 1.67625i −0.545476 0.838126i \(-0.683651\pi\)
0.545476 0.838126i \(-0.316349\pi\)
\(954\) 0 0
\(955\) 25.2028 0.815544
\(956\) −44.6365 40.4448i −1.44365 1.30808i
\(957\) 0 0
\(958\) −14.8905 + 38.6103i −0.481090 + 1.24744i
\(959\) 3.09831 0.100050
\(960\) 0 0
\(961\) 26.5633 0.856881
\(962\) −2.04061 + 5.29119i −0.0657920 + 0.170595i
\(963\) 0 0
\(964\) 8.17933 + 7.41122i 0.263439 + 0.238699i
\(965\) 4.75901 0.153198
\(966\) 0 0
\(967\) 7.89472i 0.253877i −0.991911 0.126939i \(-0.959485\pi\)
0.991911 0.126939i \(-0.0405151\pi\)
\(968\) 13.1212 6.60810i 0.421730 0.212392i
\(969\) 0 0
\(970\) 1.97358 5.11740i 0.0633680 0.164310i
\(971\) 15.1077i 0.484829i 0.970173 + 0.242414i \(0.0779393\pi\)
−0.970173 + 0.242414i \(0.922061\pi\)
\(972\) 0 0
\(973\) 3.76422i 0.120675i
\(974\) 3.71302 + 1.43197i 0.118973 + 0.0458833i
\(975\) 0 0
\(976\) −33.6194 + 3.32075i −1.07613 + 0.106295i
\(977\) 11.3398i 0.362792i 0.983410 + 0.181396i \(0.0580616\pi\)
−0.983410 + 0.181396i \(0.941938\pi\)
\(978\) 0 0
\(979\) 29.9797 0.958155
\(980\) 3.40366 + 3.08403i 0.108726 + 0.0985156i
\(981\) 0 0
\(982\) −25.4222 9.80439i −0.811256 0.312871i
\(983\) −34.3365 −1.09517 −0.547583 0.836751i \(-0.684452\pi\)
−0.547583 + 0.836751i \(0.684452\pi\)
\(984\) 0 0
\(985\) 10.7833 0.343584
\(986\) −1.22697 0.473196i −0.0390747 0.0150696i
\(987\) 0 0
\(988\) −12.2770 + 13.5495i −0.390585 + 0.431066i
\(989\) 42.0612 1.33747
\(990\) 0 0
\(991\) 20.9504i 0.665510i 0.943013 + 0.332755i \(0.107978\pi\)
−0.943013 + 0.332755i \(0.892022\pi\)
\(992\) 3.17393 11.4848i 0.100772 0.364643i
\(993\) 0 0
\(994\) −11.8164 4.55715i −0.374794 0.144544i
\(995\) 38.4797i 1.21989i
\(996\) 0 0
\(997\) 8.15182i 0.258171i 0.991633 + 0.129085i \(0.0412041\pi\)
−0.991633 + 0.129085i \(0.958796\pi\)
\(998\) 20.0719 52.0453i 0.635365 1.64747i
\(999\) 0 0
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 1512.2.j.d.323.29 yes 48
3.2 odd 2 inner 1512.2.j.d.323.20 yes 48
4.3 odd 2 6048.2.j.d.5615.35 48
8.3 odd 2 inner 1512.2.j.d.323.19 48
8.5 even 2 6048.2.j.d.5615.13 48
12.11 even 2 6048.2.j.d.5615.14 48
24.5 odd 2 6048.2.j.d.5615.36 48
24.11 even 2 inner 1512.2.j.d.323.30 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.d.323.19 48 8.3 odd 2 inner
1512.2.j.d.323.20 yes 48 3.2 odd 2 inner
1512.2.j.d.323.29 yes 48 1.1 even 1 trivial
1512.2.j.d.323.30 yes 48 24.11 even 2 inner
6048.2.j.d.5615.13 48 8.5 even 2
6048.2.j.d.5615.14 48 12.11 even 2
6048.2.j.d.5615.35 48 4.3 odd 2
6048.2.j.d.5615.36 48 24.5 odd 2