Properties

Label 9025.2.a.ck.1.9
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

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: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 22x^{14} + 190x^{12} - 820x^{10} + 1862x^{8} - 2154x^{6} + 1163x^{4} - 256x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(0.317290\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.317290 q^{2} -2.04300 q^{3} -1.89933 q^{4} -0.648223 q^{6} -3.69961 q^{7} -1.23722 q^{8} +1.17385 q^{9} +O(q^{10})\) \(q+0.317290 q^{2} -2.04300 q^{3} -1.89933 q^{4} -0.648223 q^{6} -3.69961 q^{7} -1.23722 q^{8} +1.17385 q^{9} -5.01076 q^{11} +3.88033 q^{12} -4.59159 q^{13} -1.17385 q^{14} +3.40610 q^{16} +3.21112 q^{17} +0.372450 q^{18} +7.55831 q^{21} -1.58986 q^{22} -0.220987 q^{23} +2.52763 q^{24} -1.45686 q^{26} +3.73083 q^{27} +7.02678 q^{28} -3.42114 q^{29} -7.89725 q^{31} +3.55515 q^{32} +10.2370 q^{33} +1.01885 q^{34} -2.22952 q^{36} +2.98622 q^{37} +9.38061 q^{39} +0.183247 q^{41} +2.39817 q^{42} -7.30424 q^{43} +9.51707 q^{44} -0.0701168 q^{46} +10.8876 q^{47} -6.95866 q^{48} +6.68714 q^{49} -6.56032 q^{51} +8.72092 q^{52} -1.93670 q^{53} +1.18375 q^{54} +4.57722 q^{56} -1.08549 q^{58} +6.15338 q^{59} +5.02293 q^{61} -2.50572 q^{62} -4.34279 q^{63} -5.68419 q^{64} +3.24809 q^{66} +11.2753 q^{67} -6.09897 q^{68} +0.451476 q^{69} -0.851223 q^{71} -1.45230 q^{72} +5.15625 q^{73} +0.947498 q^{74} +18.5379 q^{77} +2.97637 q^{78} -9.19657 q^{79} -11.1436 q^{81} +0.0581424 q^{82} +4.53528 q^{83} -14.3557 q^{84} -2.31756 q^{86} +6.98938 q^{87} +6.19939 q^{88} +14.8752 q^{89} +16.9871 q^{91} +0.419726 q^{92} +16.1341 q^{93} +3.45452 q^{94} -7.26317 q^{96} +1.36602 q^{97} +2.12176 q^{98} -5.88187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 10 q^{6} + 6 q^{9} - 22 q^{11} - 6 q^{14} + 8 q^{16} + 20 q^{21} - 14 q^{24} - 16 q^{26} - 2 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} - 36 q^{39} - 26 q^{41} - 64 q^{44} - 2 q^{46} - 20 q^{49} + 38 q^{51} - 12 q^{54} - 6 q^{56} - 10 q^{59} - 30 q^{61} - 16 q^{64} + 4 q^{66} - 68 q^{69} + 20 q^{71} - 40 q^{74} - 12 q^{79} - 48 q^{81} + 2 q^{84} + 20 q^{86} + 86 q^{91} + 38 q^{94} + 22 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.317290 0.224358 0.112179 0.993688i \(-0.464217\pi\)
0.112179 + 0.993688i \(0.464217\pi\)
\(3\) −2.04300 −1.17953 −0.589763 0.807576i \(-0.700779\pi\)
−0.589763 + 0.807576i \(0.700779\pi\)
\(4\) −1.89933 −0.949664
\(5\) 0 0
\(6\) −0.648223 −0.264636
\(7\) −3.69961 −1.39832 −0.699161 0.714964i \(-0.746443\pi\)
−0.699161 + 0.714964i \(0.746443\pi\)
\(8\) −1.23722 −0.437422
\(9\) 1.17385 0.391283
\(10\) 0 0
\(11\) −5.01076 −1.51080 −0.755400 0.655263i \(-0.772558\pi\)
−0.755400 + 0.655263i \(0.772558\pi\)
\(12\) 3.88033 1.12015
\(13\) −4.59159 −1.27348 −0.636738 0.771080i \(-0.719717\pi\)
−0.636738 + 0.771080i \(0.719717\pi\)
\(14\) −1.17385 −0.313724
\(15\) 0 0
\(16\) 3.40610 0.851525
\(17\) 3.21112 0.778811 0.389405 0.921066i \(-0.372681\pi\)
0.389405 + 0.921066i \(0.372681\pi\)
\(18\) 0.372450 0.0877873
\(19\) 0 0
\(20\) 0 0
\(21\) 7.55831 1.64936
\(22\) −1.58986 −0.338960
\(23\) −0.220987 −0.0460789 −0.0230395 0.999735i \(-0.507334\pi\)
−0.0230395 + 0.999735i \(0.507334\pi\)
\(24\) 2.52763 0.515951
\(25\) 0 0
\(26\) −1.45686 −0.285714
\(27\) 3.73083 0.717998
\(28\) 7.02678 1.32794
\(29\) −3.42114 −0.635289 −0.317644 0.948210i \(-0.602892\pi\)
−0.317644 + 0.948210i \(0.602892\pi\)
\(30\) 0 0
\(31\) −7.89725 −1.41839 −0.709194 0.705013i \(-0.750941\pi\)
−0.709194 + 0.705013i \(0.750941\pi\)
\(32\) 3.55515 0.628468
\(33\) 10.2370 1.78203
\(34\) 1.01885 0.174732
\(35\) 0 0
\(36\) −2.22952 −0.371587
\(37\) 2.98622 0.490932 0.245466 0.969405i \(-0.421059\pi\)
0.245466 + 0.969405i \(0.421059\pi\)
\(38\) 0 0
\(39\) 9.38061 1.50210
\(40\) 0 0
\(41\) 0.183247 0.0286184 0.0143092 0.999898i \(-0.495445\pi\)
0.0143092 + 0.999898i \(0.495445\pi\)
\(42\) 2.39817 0.370046
\(43\) −7.30424 −1.11389 −0.556943 0.830551i \(-0.688026\pi\)
−0.556943 + 0.830551i \(0.688026\pi\)
\(44\) 9.51707 1.43475
\(45\) 0 0
\(46\) −0.0701168 −0.0103382
\(47\) 10.8876 1.58812 0.794060 0.607840i \(-0.207964\pi\)
0.794060 + 0.607840i \(0.207964\pi\)
\(48\) −6.95866 −1.00440
\(49\) 6.68714 0.955306
\(50\) 0 0
\(51\) −6.56032 −0.918628
\(52\) 8.72092 1.20937
\(53\) −1.93670 −0.266026 −0.133013 0.991114i \(-0.542465\pi\)
−0.133013 + 0.991114i \(0.542465\pi\)
\(54\) 1.18375 0.161088
\(55\) 0 0
\(56\) 4.57722 0.611657
\(57\) 0 0
\(58\) −1.08549 −0.142532
\(59\) 6.15338 0.801102 0.400551 0.916275i \(-0.368819\pi\)
0.400551 + 0.916275i \(0.368819\pi\)
\(60\) 0 0
\(61\) 5.02293 0.643121 0.321560 0.946889i \(-0.395793\pi\)
0.321560 + 0.946889i \(0.395793\pi\)
\(62\) −2.50572 −0.318226
\(63\) −4.34279 −0.547140
\(64\) −5.68419 −0.710523
\(65\) 0 0
\(66\) 3.24809 0.399812
\(67\) 11.2753 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(68\) −6.09897 −0.739608
\(69\) 0.451476 0.0543513
\(70\) 0 0
\(71\) −0.851223 −0.101022 −0.0505108 0.998724i \(-0.516085\pi\)
−0.0505108 + 0.998724i \(0.516085\pi\)
\(72\) −1.45230 −0.171156
\(73\) 5.15625 0.603494 0.301747 0.953388i \(-0.402430\pi\)
0.301747 + 0.953388i \(0.402430\pi\)
\(74\) 0.947498 0.110144
\(75\) 0 0
\(76\) 0 0
\(77\) 18.5379 2.11259
\(78\) 2.97637 0.337007
\(79\) −9.19657 −1.03469 −0.517347 0.855776i \(-0.673080\pi\)
−0.517347 + 0.855776i \(0.673080\pi\)
\(80\) 0 0
\(81\) −11.1436 −1.23818
\(82\) 0.0581424 0.00642076
\(83\) 4.53528 0.497811 0.248906 0.968528i \(-0.419929\pi\)
0.248906 + 0.968528i \(0.419929\pi\)
\(84\) −14.3557 −1.56634
\(85\) 0 0
\(86\) −2.31756 −0.249909
\(87\) 6.98938 0.749340
\(88\) 6.19939 0.660857
\(89\) 14.8752 1.57677 0.788385 0.615183i \(-0.210918\pi\)
0.788385 + 0.615183i \(0.210918\pi\)
\(90\) 0 0
\(91\) 16.9871 1.78073
\(92\) 0.419726 0.0437595
\(93\) 16.1341 1.67303
\(94\) 3.45452 0.356307
\(95\) 0 0
\(96\) −7.26317 −0.741295
\(97\) 1.36602 0.138698 0.0693490 0.997592i \(-0.477908\pi\)
0.0693490 + 0.997592i \(0.477908\pi\)
\(98\) 2.12176 0.214330
\(99\) −5.88187 −0.591151
\(100\) 0 0
\(101\) −6.08643 −0.605622 −0.302811 0.953051i \(-0.597925\pi\)
−0.302811 + 0.953051i \(0.597925\pi\)
\(102\) −2.08152 −0.206101
\(103\) 13.7063 1.35052 0.675261 0.737579i \(-0.264031\pi\)
0.675261 + 0.737579i \(0.264031\pi\)
\(104\) 5.68078 0.557047
\(105\) 0 0
\(106\) −0.614494 −0.0596850
\(107\) 17.3036 1.67280 0.836400 0.548119i \(-0.184656\pi\)
0.836400 + 0.548119i \(0.184656\pi\)
\(108\) −7.08606 −0.681857
\(109\) −13.7511 −1.31712 −0.658561 0.752528i \(-0.728834\pi\)
−0.658561 + 0.752528i \(0.728834\pi\)
\(110\) 0 0
\(111\) −6.10086 −0.579068
\(112\) −12.6012 −1.19071
\(113\) −4.15573 −0.390938 −0.195469 0.980710i \(-0.562623\pi\)
−0.195469 + 0.980710i \(0.562623\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.49786 0.603311
\(117\) −5.38983 −0.498290
\(118\) 1.95240 0.179733
\(119\) −11.8799 −1.08903
\(120\) 0 0
\(121\) 14.1077 1.28252
\(122\) 1.59372 0.144289
\(123\) −0.374374 −0.0337562
\(124\) 14.9995 1.34699
\(125\) 0 0
\(126\) −1.37792 −0.122755
\(127\) −11.9185 −1.05760 −0.528799 0.848747i \(-0.677357\pi\)
−0.528799 + 0.848747i \(0.677357\pi\)
\(128\) −8.91384 −0.787879
\(129\) 14.9226 1.31386
\(130\) 0 0
\(131\) −7.58093 −0.662349 −0.331175 0.943569i \(-0.607445\pi\)
−0.331175 + 0.943569i \(0.607445\pi\)
\(132\) −19.4434 −1.69233
\(133\) 0 0
\(134\) 3.57755 0.309053
\(135\) 0 0
\(136\) −3.97285 −0.340669
\(137\) −7.98581 −0.682274 −0.341137 0.940014i \(-0.610812\pi\)
−0.341137 + 0.940014i \(0.610812\pi\)
\(138\) 0.143249 0.0121941
\(139\) 2.65651 0.225322 0.112661 0.993633i \(-0.464063\pi\)
0.112661 + 0.993633i \(0.464063\pi\)
\(140\) 0 0
\(141\) −22.2434 −1.87323
\(142\) −0.270084 −0.0226650
\(143\) 23.0073 1.92397
\(144\) 3.99825 0.333187
\(145\) 0 0
\(146\) 1.63603 0.135398
\(147\) −13.6618 −1.12681
\(148\) −5.67182 −0.466221
\(149\) 8.35153 0.684184 0.342092 0.939666i \(-0.388865\pi\)
0.342092 + 0.939666i \(0.388865\pi\)
\(150\) 0 0
\(151\) −8.56471 −0.696986 −0.348493 0.937311i \(-0.613306\pi\)
−0.348493 + 0.937311i \(0.613306\pi\)
\(152\) 0 0
\(153\) 3.76937 0.304735
\(154\) 5.88187 0.473975
\(155\) 0 0
\(156\) −17.8168 −1.42649
\(157\) −11.3335 −0.904510 −0.452255 0.891889i \(-0.649380\pi\)
−0.452255 + 0.891889i \(0.649380\pi\)
\(158\) −2.91797 −0.232142
\(159\) 3.95668 0.313785
\(160\) 0 0
\(161\) 0.817566 0.0644332
\(162\) −3.53576 −0.277795
\(163\) −4.65048 −0.364254 −0.182127 0.983275i \(-0.558298\pi\)
−0.182127 + 0.983275i \(0.558298\pi\)
\(164\) −0.348046 −0.0271779
\(165\) 0 0
\(166\) 1.43900 0.111688
\(167\) −8.89485 −0.688304 −0.344152 0.938914i \(-0.611834\pi\)
−0.344152 + 0.938914i \(0.611834\pi\)
\(168\) −9.35126 −0.721466
\(169\) 8.08265 0.621743
\(170\) 0 0
\(171\) 0 0
\(172\) 13.8731 1.05782
\(173\) 18.6570 1.41847 0.709233 0.704974i \(-0.249042\pi\)
0.709233 + 0.704974i \(0.249042\pi\)
\(174\) 2.21766 0.168120
\(175\) 0 0
\(176\) −17.0671 −1.28648
\(177\) −12.5713 −0.944921
\(178\) 4.71975 0.353760
\(179\) 0.224791 0.0168017 0.00840083 0.999965i \(-0.497326\pi\)
0.00840083 + 0.999965i \(0.497326\pi\)
\(180\) 0 0
\(181\) 24.8780 1.84917 0.924583 0.380981i \(-0.124414\pi\)
0.924583 + 0.380981i \(0.124414\pi\)
\(182\) 5.38983 0.399521
\(183\) −10.2618 −0.758578
\(184\) 0.273408 0.0201559
\(185\) 0 0
\(186\) 5.11918 0.375356
\(187\) −16.0901 −1.17663
\(188\) −20.6791 −1.50818
\(189\) −13.8026 −1.00399
\(190\) 0 0
\(191\) 8.43889 0.610616 0.305308 0.952254i \(-0.401241\pi\)
0.305308 + 0.952254i \(0.401241\pi\)
\(192\) 11.6128 0.838081
\(193\) 1.60346 0.115420 0.0577098 0.998333i \(-0.481620\pi\)
0.0577098 + 0.998333i \(0.481620\pi\)
\(194\) 0.433423 0.0311180
\(195\) 0 0
\(196\) −12.7011 −0.907219
\(197\) 2.01284 0.143409 0.0717043 0.997426i \(-0.477156\pi\)
0.0717043 + 0.997426i \(0.477156\pi\)
\(198\) −1.86626 −0.132629
\(199\) 5.03693 0.357059 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(200\) 0 0
\(201\) −23.0355 −1.62480
\(202\) −1.93116 −0.135876
\(203\) 12.6569 0.888339
\(204\) 12.4602 0.872388
\(205\) 0 0
\(206\) 4.34886 0.303000
\(207\) −0.259405 −0.0180299
\(208\) −15.6394 −1.08440
\(209\) 0 0
\(210\) 0 0
\(211\) −4.18849 −0.288348 −0.144174 0.989552i \(-0.546052\pi\)
−0.144174 + 0.989552i \(0.546052\pi\)
\(212\) 3.67842 0.252635
\(213\) 1.73905 0.119158
\(214\) 5.49025 0.375306
\(215\) 0 0
\(216\) −4.61584 −0.314068
\(217\) 29.2168 1.98336
\(218\) −4.36310 −0.295506
\(219\) −10.5342 −0.711837
\(220\) 0 0
\(221\) −14.7441 −0.991797
\(222\) −1.93574 −0.129918
\(223\) −5.32967 −0.356901 −0.178451 0.983949i \(-0.557108\pi\)
−0.178451 + 0.983949i \(0.557108\pi\)
\(224\) −13.1527 −0.878801
\(225\) 0 0
\(226\) −1.31857 −0.0877099
\(227\) 5.17899 0.343741 0.171871 0.985120i \(-0.445019\pi\)
0.171871 + 0.985120i \(0.445019\pi\)
\(228\) 0 0
\(229\) −17.0790 −1.12861 −0.564306 0.825566i \(-0.690856\pi\)
−0.564306 + 0.825566i \(0.690856\pi\)
\(230\) 0 0
\(231\) −37.8729 −2.49185
\(232\) 4.23268 0.277889
\(233\) 15.8503 1.03839 0.519195 0.854656i \(-0.326232\pi\)
0.519195 + 0.854656i \(0.326232\pi\)
\(234\) −1.71014 −0.111795
\(235\) 0 0
\(236\) −11.6873 −0.760777
\(237\) 18.7886 1.22045
\(238\) −3.76937 −0.244332
\(239\) −17.7688 −1.14937 −0.574685 0.818375i \(-0.694875\pi\)
−0.574685 + 0.818375i \(0.694875\pi\)
\(240\) 0 0
\(241\) 27.3823 1.76385 0.881925 0.471390i \(-0.156247\pi\)
0.881925 + 0.471390i \(0.156247\pi\)
\(242\) 4.47623 0.287743
\(243\) 11.5739 0.742469
\(244\) −9.54019 −0.610748
\(245\) 0 0
\(246\) −0.118785 −0.00757345
\(247\) 0 0
\(248\) 9.77061 0.620434
\(249\) −9.26557 −0.587182
\(250\) 0 0
\(251\) −8.50216 −0.536651 −0.268326 0.963328i \(-0.586470\pi\)
−0.268326 + 0.963328i \(0.586470\pi\)
\(252\) 8.24837 0.519599
\(253\) 1.10731 0.0696161
\(254\) −3.78162 −0.237280
\(255\) 0 0
\(256\) 8.54010 0.533756
\(257\) 20.9865 1.30910 0.654552 0.756017i \(-0.272857\pi\)
0.654552 + 0.756017i \(0.272857\pi\)
\(258\) 4.73477 0.294774
\(259\) −11.0479 −0.686482
\(260\) 0 0
\(261\) −4.01590 −0.248578
\(262\) −2.40535 −0.148603
\(263\) −11.7173 −0.722522 −0.361261 0.932465i \(-0.617654\pi\)
−0.361261 + 0.932465i \(0.617654\pi\)
\(264\) −12.6654 −0.779499
\(265\) 0 0
\(266\) 0 0
\(267\) −30.3901 −1.85984
\(268\) −21.4156 −1.30816
\(269\) 13.1743 0.803253 0.401626 0.915804i \(-0.368445\pi\)
0.401626 + 0.915804i \(0.368445\pi\)
\(270\) 0 0
\(271\) −25.7363 −1.56337 −0.781683 0.623676i \(-0.785639\pi\)
−0.781683 + 0.623676i \(0.785639\pi\)
\(272\) 10.9374 0.663177
\(273\) −34.7046 −2.10042
\(274\) −2.53381 −0.153073
\(275\) 0 0
\(276\) −0.857501 −0.0516155
\(277\) −5.11697 −0.307449 −0.153724 0.988114i \(-0.549127\pi\)
−0.153724 + 0.988114i \(0.549127\pi\)
\(278\) 0.842882 0.0505527
\(279\) −9.27018 −0.554991
\(280\) 0 0
\(281\) 1.80790 0.107850 0.0539251 0.998545i \(-0.482827\pi\)
0.0539251 + 0.998545i \(0.482827\pi\)
\(282\) −7.05759 −0.420273
\(283\) −18.6741 −1.11006 −0.555029 0.831831i \(-0.687293\pi\)
−0.555029 + 0.831831i \(0.687293\pi\)
\(284\) 1.61675 0.0959366
\(285\) 0 0
\(286\) 7.29999 0.431657
\(287\) −0.677944 −0.0400178
\(288\) 4.17321 0.245909
\(289\) −6.68871 −0.393454
\(290\) 0 0
\(291\) −2.79077 −0.163598
\(292\) −9.79341 −0.573116
\(293\) 0.150745 0.00880664 0.00440332 0.999990i \(-0.498598\pi\)
0.00440332 + 0.999990i \(0.498598\pi\)
\(294\) −4.33475 −0.252808
\(295\) 0 0
\(296\) −3.69461 −0.214745
\(297\) −18.6943 −1.08475
\(298\) 2.64985 0.153502
\(299\) 1.01468 0.0586804
\(300\) 0 0
\(301\) 27.0228 1.55757
\(302\) −2.71749 −0.156374
\(303\) 12.4346 0.714347
\(304\) 0 0
\(305\) 0 0
\(306\) 1.19598 0.0683697
\(307\) 4.94936 0.282475 0.141237 0.989976i \(-0.454892\pi\)
0.141237 + 0.989976i \(0.454892\pi\)
\(308\) −35.2095 −2.00625
\(309\) −28.0020 −1.59298
\(310\) 0 0
\(311\) −23.6262 −1.33972 −0.669859 0.742489i \(-0.733645\pi\)
−0.669859 + 0.742489i \(0.733645\pi\)
\(312\) −11.6058 −0.657051
\(313\) 8.89248 0.502632 0.251316 0.967905i \(-0.419137\pi\)
0.251316 + 0.967905i \(0.419137\pi\)
\(314\) −3.59600 −0.202934
\(315\) 0 0
\(316\) 17.4673 0.982612
\(317\) 29.5976 1.66237 0.831185 0.555996i \(-0.187663\pi\)
0.831185 + 0.555996i \(0.187663\pi\)
\(318\) 1.25541 0.0704000
\(319\) 17.1425 0.959795
\(320\) 0 0
\(321\) −35.3512 −1.97311
\(322\) 0.259405 0.0144561
\(323\) 0 0
\(324\) 21.1654 1.17586
\(325\) 0 0
\(326\) −1.47555 −0.0817232
\(327\) 28.0936 1.55358
\(328\) −0.226716 −0.0125183
\(329\) −40.2799 −2.22070
\(330\) 0 0
\(331\) −25.8866 −1.42285 −0.711427 0.702760i \(-0.751951\pi\)
−0.711427 + 0.702760i \(0.751951\pi\)
\(332\) −8.61397 −0.472753
\(333\) 3.50538 0.192093
\(334\) −2.82224 −0.154426
\(335\) 0 0
\(336\) 25.7444 1.40447
\(337\) 6.08960 0.331721 0.165861 0.986149i \(-0.446960\pi\)
0.165861 + 0.986149i \(0.446960\pi\)
\(338\) 2.56454 0.139493
\(339\) 8.49015 0.461122
\(340\) 0 0
\(341\) 39.5712 2.14290
\(342\) 0 0
\(343\) 1.15747 0.0624972
\(344\) 9.03692 0.487238
\(345\) 0 0
\(346\) 5.91967 0.318244
\(347\) 6.60514 0.354582 0.177291 0.984158i \(-0.443267\pi\)
0.177291 + 0.984158i \(0.443267\pi\)
\(348\) −13.2751 −0.711621
\(349\) 19.6123 1.04982 0.524912 0.851156i \(-0.324098\pi\)
0.524912 + 0.851156i \(0.324098\pi\)
\(350\) 0 0
\(351\) −17.1304 −0.914354
\(352\) −17.8140 −0.949490
\(353\) 19.2599 1.02510 0.512551 0.858657i \(-0.328701\pi\)
0.512551 + 0.858657i \(0.328701\pi\)
\(354\) −3.98876 −0.212000
\(355\) 0 0
\(356\) −28.2529 −1.49740
\(357\) 24.2706 1.28454
\(358\) 0.0713238 0.00376958
\(359\) −35.7362 −1.88609 −0.943043 0.332671i \(-0.892050\pi\)
−0.943043 + 0.332671i \(0.892050\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 7.89352 0.414874
\(363\) −28.8221 −1.51277
\(364\) −32.2640 −1.69110
\(365\) 0 0
\(366\) −3.25598 −0.170193
\(367\) −13.8593 −0.723450 −0.361725 0.932285i \(-0.617812\pi\)
−0.361725 + 0.932285i \(0.617812\pi\)
\(368\) −0.752703 −0.0392373
\(369\) 0.215105 0.0111979
\(370\) 0 0
\(371\) 7.16504 0.371990
\(372\) −30.6439 −1.58881
\(373\) −23.6819 −1.22620 −0.613101 0.790005i \(-0.710078\pi\)
−0.613101 + 0.790005i \(0.710078\pi\)
\(374\) −5.10524 −0.263985
\(375\) 0 0
\(376\) −13.4703 −0.694678
\(377\) 15.7084 0.809026
\(378\) −4.37943 −0.225253
\(379\) 33.7504 1.73364 0.866820 0.498622i \(-0.166160\pi\)
0.866820 + 0.498622i \(0.166160\pi\)
\(380\) 0 0
\(381\) 24.3495 1.24746
\(382\) 2.67757 0.136996
\(383\) −15.9027 −0.812588 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(384\) 18.2110 0.929325
\(385\) 0 0
\(386\) 0.508761 0.0258953
\(387\) −8.57407 −0.435844
\(388\) −2.59451 −0.131716
\(389\) −0.715535 −0.0362791 −0.0181395 0.999835i \(-0.505774\pi\)
−0.0181395 + 0.999835i \(0.505774\pi\)
\(390\) 0 0
\(391\) −0.709615 −0.0358868
\(392\) −8.27344 −0.417872
\(393\) 15.4878 0.781259
\(394\) 0.638652 0.0321748
\(395\) 0 0
\(396\) 11.1716 0.561394
\(397\) 14.3593 0.720671 0.360336 0.932823i \(-0.382662\pi\)
0.360336 + 0.932823i \(0.382662\pi\)
\(398\) 1.59817 0.0801088
\(399\) 0 0
\(400\) 0 0
\(401\) 34.5791 1.72680 0.863398 0.504524i \(-0.168332\pi\)
0.863398 + 0.504524i \(0.168332\pi\)
\(402\) −7.30893 −0.364536
\(403\) 36.2609 1.80628
\(404\) 11.5601 0.575137
\(405\) 0 0
\(406\) 4.01590 0.199306
\(407\) −14.9633 −0.741701
\(408\) 8.11653 0.401828
\(409\) 18.4355 0.911575 0.455788 0.890089i \(-0.349358\pi\)
0.455788 + 0.890089i \(0.349358\pi\)
\(410\) 0 0
\(411\) 16.3150 0.804760
\(412\) −26.0327 −1.28254
\(413\) −22.7651 −1.12020
\(414\) −0.0823065 −0.00404514
\(415\) 0 0
\(416\) −16.3238 −0.800339
\(417\) −5.42725 −0.265773
\(418\) 0 0
\(419\) −34.3437 −1.67780 −0.838901 0.544285i \(-0.816801\pi\)
−0.838901 + 0.544285i \(0.816801\pi\)
\(420\) 0 0
\(421\) 0.225020 0.0109668 0.00548340 0.999985i \(-0.498255\pi\)
0.00548340 + 0.999985i \(0.498255\pi\)
\(422\) −1.32897 −0.0646930
\(423\) 12.7804 0.621404
\(424\) 2.39611 0.116366
\(425\) 0 0
\(426\) 0.551782 0.0267339
\(427\) −18.5829 −0.899290
\(428\) −32.8652 −1.58860
\(429\) −47.0040 −2.26937
\(430\) 0 0
\(431\) −13.3531 −0.643198 −0.321599 0.946876i \(-0.604220\pi\)
−0.321599 + 0.946876i \(0.604220\pi\)
\(432\) 12.7076 0.611393
\(433\) −21.4161 −1.02919 −0.514595 0.857433i \(-0.672058\pi\)
−0.514595 + 0.857433i \(0.672058\pi\)
\(434\) 9.27018 0.444983
\(435\) 0 0
\(436\) 26.1179 1.25082
\(437\) 0 0
\(438\) −3.34240 −0.159706
\(439\) 7.21678 0.344438 0.172219 0.985059i \(-0.444906\pi\)
0.172219 + 0.985059i \(0.444906\pi\)
\(440\) 0 0
\(441\) 7.84969 0.373795
\(442\) −4.67816 −0.222517
\(443\) 34.6591 1.64671 0.823353 0.567530i \(-0.192101\pi\)
0.823353 + 0.567530i \(0.192101\pi\)
\(444\) 11.5875 0.549920
\(445\) 0 0
\(446\) −1.69105 −0.0800735
\(447\) −17.0622 −0.807013
\(448\) 21.0293 0.993540
\(449\) −5.09006 −0.240215 −0.120107 0.992761i \(-0.538324\pi\)
−0.120107 + 0.992761i \(0.538324\pi\)
\(450\) 0 0
\(451\) −0.918208 −0.0432367
\(452\) 7.89309 0.371260
\(453\) 17.4977 0.822113
\(454\) 1.64324 0.0771210
\(455\) 0 0
\(456\) 0 0
\(457\) −39.9916 −1.87073 −0.935364 0.353686i \(-0.884928\pi\)
−0.935364 + 0.353686i \(0.884928\pi\)
\(458\) −5.41899 −0.253213
\(459\) 11.9801 0.559185
\(460\) 0 0
\(461\) 26.4955 1.23402 0.617009 0.786956i \(-0.288344\pi\)
0.617009 + 0.786956i \(0.288344\pi\)
\(462\) −12.0167 −0.559066
\(463\) 9.68844 0.450260 0.225130 0.974329i \(-0.427719\pi\)
0.225130 + 0.974329i \(0.427719\pi\)
\(464\) −11.6527 −0.540964
\(465\) 0 0
\(466\) 5.02915 0.232971
\(467\) 5.78310 0.267610 0.133805 0.991008i \(-0.457280\pi\)
0.133805 + 0.991008i \(0.457280\pi\)
\(468\) 10.2370 0.473208
\(469\) −41.7144 −1.92619
\(470\) 0 0
\(471\) 23.1543 1.06689
\(472\) −7.61306 −0.350419
\(473\) 36.5998 1.68286
\(474\) 5.96142 0.273817
\(475\) 0 0
\(476\) 22.5638 1.03421
\(477\) −2.27339 −0.104091
\(478\) −5.63786 −0.257870
\(479\) 1.77194 0.0809620 0.0404810 0.999180i \(-0.487111\pi\)
0.0404810 + 0.999180i \(0.487111\pi\)
\(480\) 0 0
\(481\) −13.7115 −0.625191
\(482\) 8.68812 0.395733
\(483\) −1.67029 −0.0760007
\(484\) −26.7952 −1.21796
\(485\) 0 0
\(486\) 3.67229 0.166579
\(487\) −16.2131 −0.734688 −0.367344 0.930085i \(-0.619733\pi\)
−0.367344 + 0.930085i \(0.619733\pi\)
\(488\) −6.21445 −0.281315
\(489\) 9.50094 0.429647
\(490\) 0 0
\(491\) 18.4869 0.834300 0.417150 0.908838i \(-0.363029\pi\)
0.417150 + 0.908838i \(0.363029\pi\)
\(492\) 0.711059 0.0320570
\(493\) −10.9857 −0.494770
\(494\) 0 0
\(495\) 0 0
\(496\) −26.8988 −1.20779
\(497\) 3.14920 0.141261
\(498\) −2.93987 −0.131739
\(499\) −17.2843 −0.773751 −0.386875 0.922132i \(-0.626446\pi\)
−0.386875 + 0.922132i \(0.626446\pi\)
\(500\) 0 0
\(501\) 18.1722 0.811873
\(502\) −2.69765 −0.120402
\(503\) −40.7325 −1.81617 −0.908087 0.418782i \(-0.862457\pi\)
−0.908087 + 0.418782i \(0.862457\pi\)
\(504\) 5.37297 0.239331
\(505\) 0 0
\(506\) 0.351338 0.0156189
\(507\) −16.5129 −0.733362
\(508\) 22.6372 1.00436
\(509\) −21.2017 −0.939747 −0.469873 0.882734i \(-0.655700\pi\)
−0.469873 + 0.882734i \(0.655700\pi\)
\(510\) 0 0
\(511\) −19.0761 −0.843879
\(512\) 20.5374 0.907632
\(513\) 0 0
\(514\) 6.65881 0.293707
\(515\) 0 0
\(516\) −28.3428 −1.24772
\(517\) −54.5551 −2.39933
\(518\) −3.50538 −0.154017
\(519\) −38.1163 −1.67312
\(520\) 0 0
\(521\) −19.2704 −0.844251 −0.422125 0.906538i \(-0.638716\pi\)
−0.422125 + 0.906538i \(0.638716\pi\)
\(522\) −1.27420 −0.0557703
\(523\) 24.4971 1.07118 0.535591 0.844477i \(-0.320089\pi\)
0.535591 + 0.844477i \(0.320089\pi\)
\(524\) 14.3987 0.629009
\(525\) 0 0
\(526\) −3.71779 −0.162103
\(527\) −25.3590 −1.10466
\(528\) 34.8682 1.51744
\(529\) −22.9512 −0.997877
\(530\) 0 0
\(531\) 7.22313 0.313457
\(532\) 0 0
\(533\) −0.841395 −0.0364449
\(534\) −9.64245 −0.417270
\(535\) 0 0
\(536\) −13.9500 −0.602550
\(537\) −0.459248 −0.0198180
\(538\) 4.18008 0.180216
\(539\) −33.5076 −1.44328
\(540\) 0 0
\(541\) 30.5029 1.31142 0.655711 0.755012i \(-0.272369\pi\)
0.655711 + 0.755012i \(0.272369\pi\)
\(542\) −8.16585 −0.350753
\(543\) −50.8257 −2.18114
\(544\) 11.4160 0.489458
\(545\) 0 0
\(546\) −11.0114 −0.471245
\(547\) 3.16224 0.135207 0.0676037 0.997712i \(-0.478465\pi\)
0.0676037 + 0.997712i \(0.478465\pi\)
\(548\) 15.1677 0.647931
\(549\) 5.89616 0.251642
\(550\) 0 0
\(551\) 0 0
\(552\) −0.558573 −0.0237745
\(553\) 34.0237 1.44684
\(554\) −1.62356 −0.0689785
\(555\) 0 0
\(556\) −5.04558 −0.213980
\(557\) 33.8886 1.43591 0.717953 0.696092i \(-0.245079\pi\)
0.717953 + 0.696092i \(0.245079\pi\)
\(558\) −2.94133 −0.124517
\(559\) 33.5380 1.41851
\(560\) 0 0
\(561\) 32.8722 1.38786
\(562\) 0.573627 0.0241970
\(563\) 36.5457 1.54022 0.770110 0.637911i \(-0.220201\pi\)
0.770110 + 0.637911i \(0.220201\pi\)
\(564\) 42.2474 1.77894
\(565\) 0 0
\(566\) −5.92509 −0.249050
\(567\) 41.2271 1.73138
\(568\) 1.05315 0.0441891
\(569\) 1.31802 0.0552543 0.0276272 0.999618i \(-0.491205\pi\)
0.0276272 + 0.999618i \(0.491205\pi\)
\(570\) 0 0
\(571\) −17.0771 −0.714656 −0.357328 0.933979i \(-0.616312\pi\)
−0.357328 + 0.933979i \(0.616312\pi\)
\(572\) −43.6985 −1.82712
\(573\) −17.2406 −0.720238
\(574\) −0.215105 −0.00897829
\(575\) 0 0
\(576\) −6.67237 −0.278016
\(577\) 47.2699 1.96787 0.983937 0.178517i \(-0.0571299\pi\)
0.983937 + 0.178517i \(0.0571299\pi\)
\(578\) −2.12226 −0.0882744
\(579\) −3.27587 −0.136140
\(580\) 0 0
\(581\) −16.7788 −0.696101
\(582\) −0.885483 −0.0367045
\(583\) 9.70433 0.401912
\(584\) −6.37940 −0.263981
\(585\) 0 0
\(586\) 0.0478300 0.00197584
\(587\) −0.620087 −0.0255937 −0.0127969 0.999918i \(-0.504073\pi\)
−0.0127969 + 0.999918i \(0.504073\pi\)
\(588\) 25.9483 1.07009
\(589\) 0 0
\(590\) 0 0
\(591\) −4.11222 −0.169154
\(592\) 10.1714 0.418041
\(593\) 8.96071 0.367972 0.183986 0.982929i \(-0.441100\pi\)
0.183986 + 0.982929i \(0.441100\pi\)
\(594\) −5.93150 −0.243372
\(595\) 0 0
\(596\) −15.8623 −0.649745
\(597\) −10.2905 −0.421160
\(598\) 0.321947 0.0131654
\(599\) −0.338212 −0.0138190 −0.00690948 0.999976i \(-0.502199\pi\)
−0.00690948 + 0.999976i \(0.502199\pi\)
\(600\) 0 0
\(601\) −39.7523 −1.62153 −0.810766 0.585370i \(-0.800949\pi\)
−0.810766 + 0.585370i \(0.800949\pi\)
\(602\) 8.57407 0.349453
\(603\) 13.2355 0.538993
\(604\) 16.2672 0.661902
\(605\) 0 0
\(606\) 3.94536 0.160269
\(607\) 22.5614 0.915738 0.457869 0.889020i \(-0.348613\pi\)
0.457869 + 0.889020i \(0.348613\pi\)
\(608\) 0 0
\(609\) −25.8580 −1.04782
\(610\) 0 0
\(611\) −49.9913 −2.02243
\(612\) −7.15926 −0.289396
\(613\) −32.5953 −1.31651 −0.658257 0.752794i \(-0.728706\pi\)
−0.658257 + 0.752794i \(0.728706\pi\)
\(614\) 1.57038 0.0633754
\(615\) 0 0
\(616\) −22.9354 −0.924092
\(617\) −29.4270 −1.18469 −0.592343 0.805686i \(-0.701797\pi\)
−0.592343 + 0.805686i \(0.701797\pi\)
\(618\) −8.88473 −0.357396
\(619\) −24.9940 −1.00459 −0.502297 0.864695i \(-0.667512\pi\)
−0.502297 + 0.864695i \(0.667512\pi\)
\(620\) 0 0
\(621\) −0.824463 −0.0330846
\(622\) −7.49634 −0.300576
\(623\) −55.0325 −2.20483
\(624\) 31.9513 1.27907
\(625\) 0 0
\(626\) 2.82149 0.112769
\(627\) 0 0
\(628\) 21.5260 0.858981
\(629\) 9.58912 0.382343
\(630\) 0 0
\(631\) −28.3101 −1.12701 −0.563504 0.826113i \(-0.690547\pi\)
−0.563504 + 0.826113i \(0.690547\pi\)
\(632\) 11.3781 0.452598
\(633\) 8.55709 0.340114
\(634\) 9.39103 0.372965
\(635\) 0 0
\(636\) −7.51502 −0.297990
\(637\) −30.7046 −1.21656
\(638\) 5.43913 0.215337
\(639\) −0.999208 −0.0395280
\(640\) 0 0
\(641\) 28.7741 1.13651 0.568254 0.822853i \(-0.307619\pi\)
0.568254 + 0.822853i \(0.307619\pi\)
\(642\) −11.2166 −0.442683
\(643\) 10.0599 0.396722 0.198361 0.980129i \(-0.436438\pi\)
0.198361 + 0.980129i \(0.436438\pi\)
\(644\) −1.55282 −0.0611899
\(645\) 0 0
\(646\) 0 0
\(647\) 18.9903 0.746586 0.373293 0.927714i \(-0.378229\pi\)
0.373293 + 0.927714i \(0.378229\pi\)
\(648\) 13.7871 0.541607
\(649\) −30.8331 −1.21030
\(650\) 0 0
\(651\) −59.6899 −2.33943
\(652\) 8.83279 0.345919
\(653\) 42.6940 1.67074 0.835372 0.549685i \(-0.185252\pi\)
0.835372 + 0.549685i \(0.185252\pi\)
\(654\) 8.91381 0.348557
\(655\) 0 0
\(656\) 0.624158 0.0243693
\(657\) 6.05266 0.236137
\(658\) −12.7804 −0.498232
\(659\) −14.4043 −0.561111 −0.280556 0.959838i \(-0.590519\pi\)
−0.280556 + 0.959838i \(0.590519\pi\)
\(660\) 0 0
\(661\) 8.24176 0.320567 0.160284 0.987071i \(-0.448759\pi\)
0.160284 + 0.987071i \(0.448759\pi\)
\(662\) −8.21354 −0.319228
\(663\) 30.1222 1.16985
\(664\) −5.61112 −0.217754
\(665\) 0 0
\(666\) 1.11222 0.0430976
\(667\) 0.756026 0.0292734
\(668\) 16.8942 0.653657
\(669\) 10.8885 0.420974
\(670\) 0 0
\(671\) −25.1687 −0.971627
\(672\) 26.8709 1.03657
\(673\) −37.9246 −1.46189 −0.730944 0.682438i \(-0.760920\pi\)
−0.730944 + 0.682438i \(0.760920\pi\)
\(674\) 1.93217 0.0744242
\(675\) 0 0
\(676\) −15.3516 −0.590446
\(677\) −8.17207 −0.314078 −0.157039 0.987592i \(-0.550195\pi\)
−0.157039 + 0.987592i \(0.550195\pi\)
\(678\) 2.69384 0.103456
\(679\) −5.05374 −0.193945
\(680\) 0 0
\(681\) −10.5807 −0.405452
\(682\) 12.5555 0.480777
\(683\) 16.4611 0.629868 0.314934 0.949114i \(-0.398018\pi\)
0.314934 + 0.949114i \(0.398018\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.367252 0.0140217
\(687\) 34.8924 1.33123
\(688\) −24.8789 −0.948501
\(689\) 8.89252 0.338778
\(690\) 0 0
\(691\) 11.9386 0.454166 0.227083 0.973875i \(-0.427081\pi\)
0.227083 + 0.973875i \(0.427081\pi\)
\(692\) −35.4358 −1.34707
\(693\) 21.7607 0.826619
\(694\) 2.09574 0.0795532
\(695\) 0 0
\(696\) −8.64737 −0.327778
\(697\) 0.588429 0.0222883
\(698\) 6.22279 0.235536
\(699\) −32.3822 −1.22481
\(700\) 0 0
\(701\) −15.7112 −0.593404 −0.296702 0.954970i \(-0.595887\pi\)
−0.296702 + 0.954970i \(0.595887\pi\)
\(702\) −5.43530 −0.205142
\(703\) 0 0
\(704\) 28.4821 1.07346
\(705\) 0 0
\(706\) 6.11097 0.229989
\(707\) 22.5174 0.846855
\(708\) 23.8771 0.897357
\(709\) −0.810620 −0.0304435 −0.0152217 0.999884i \(-0.504845\pi\)
−0.0152217 + 0.999884i \(0.504845\pi\)
\(710\) 0 0
\(711\) −10.7954 −0.404858
\(712\) −18.4039 −0.689713
\(713\) 1.74519 0.0653578
\(714\) 7.70082 0.288196
\(715\) 0 0
\(716\) −0.426951 −0.0159559
\(717\) 36.3017 1.35571
\(718\) −11.3387 −0.423158
\(719\) −13.4593 −0.501948 −0.250974 0.967994i \(-0.580751\pi\)
−0.250974 + 0.967994i \(0.580751\pi\)
\(720\) 0 0
\(721\) −50.7080 −1.88846
\(722\) 0 0
\(723\) −55.9421 −2.08051
\(724\) −47.2514 −1.75609
\(725\) 0 0
\(726\) −9.14494 −0.339401
\(727\) 2.13941 0.0793464 0.0396732 0.999213i \(-0.487368\pi\)
0.0396732 + 0.999213i \(0.487368\pi\)
\(728\) −21.0167 −0.778931
\(729\) 9.78531 0.362419
\(730\) 0 0
\(731\) −23.4548 −0.867506
\(732\) 19.4906 0.720394
\(733\) 0.353890 0.0130712 0.00653561 0.999979i \(-0.497920\pi\)
0.00653561 + 0.999979i \(0.497920\pi\)
\(734\) −4.39741 −0.162311
\(735\) 0 0
\(736\) −0.785641 −0.0289591
\(737\) −56.4980 −2.08113
\(738\) 0.0682504 0.00251233
\(739\) 0.530279 0.0195066 0.00975332 0.999952i \(-0.496895\pi\)
0.00975332 + 0.999952i \(0.496895\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.27339 0.0834588
\(743\) −10.9210 −0.400654 −0.200327 0.979729i \(-0.564200\pi\)
−0.200327 + 0.979729i \(0.564200\pi\)
\(744\) −19.9614 −0.731819
\(745\) 0 0
\(746\) −7.51402 −0.275108
\(747\) 5.32373 0.194785
\(748\) 30.5605 1.11740
\(749\) −64.0166 −2.33911
\(750\) 0 0
\(751\) 8.00460 0.292092 0.146046 0.989278i \(-0.453345\pi\)
0.146046 + 0.989278i \(0.453345\pi\)
\(752\) 37.0842 1.35232
\(753\) 17.3699 0.632995
\(754\) 4.98412 0.181511
\(755\) 0 0
\(756\) 26.2157 0.953455
\(757\) 30.1904 1.09729 0.548645 0.836056i \(-0.315144\pi\)
0.548645 + 0.836056i \(0.315144\pi\)
\(758\) 10.7086 0.388955
\(759\) −2.26224 −0.0821140
\(760\) 0 0
\(761\) −0.399315 −0.0144751 −0.00723757 0.999974i \(-0.502304\pi\)
−0.00723757 + 0.999974i \(0.502304\pi\)
\(762\) 7.72585 0.279878
\(763\) 50.8739 1.84176
\(764\) −16.0282 −0.579880
\(765\) 0 0
\(766\) −5.04575 −0.182310
\(767\) −28.2538 −1.02018
\(768\) −17.4474 −0.629580
\(769\) −6.52791 −0.235403 −0.117701 0.993049i \(-0.537553\pi\)
−0.117701 + 0.993049i \(0.537553\pi\)
\(770\) 0 0
\(771\) −42.8755 −1.54412
\(772\) −3.04550 −0.109610
\(773\) −29.2223 −1.05105 −0.525527 0.850777i \(-0.676132\pi\)
−0.525527 + 0.850777i \(0.676132\pi\)
\(774\) −2.72046 −0.0977850
\(775\) 0 0
\(776\) −1.69006 −0.0606696
\(777\) 22.5708 0.809723
\(778\) −0.227032 −0.00813949
\(779\) 0 0
\(780\) 0 0
\(781\) 4.26528 0.152624
\(782\) −0.225153 −0.00805147
\(783\) −12.7637 −0.456136
\(784\) 22.7771 0.813466
\(785\) 0 0
\(786\) 4.91413 0.175281
\(787\) −43.2638 −1.54219 −0.771093 0.636722i \(-0.780290\pi\)
−0.771093 + 0.636722i \(0.780290\pi\)
\(788\) −3.82303 −0.136190
\(789\) 23.9385 0.852234
\(790\) 0 0
\(791\) 15.3746 0.546657
\(792\) 7.27715 0.258582
\(793\) −23.0632 −0.818999
\(794\) 4.55605 0.161688
\(795\) 0 0
\(796\) −9.56678 −0.339086
\(797\) −27.5833 −0.977050 −0.488525 0.872550i \(-0.662465\pi\)
−0.488525 + 0.872550i \(0.662465\pi\)
\(798\) 0 0
\(799\) 34.9614 1.23684
\(800\) 0 0
\(801\) 17.4613 0.616963
\(802\) 10.9716 0.387420
\(803\) −25.8368 −0.911759
\(804\) 43.7520 1.54301
\(805\) 0 0
\(806\) 11.5052 0.405254
\(807\) −26.9151 −0.947458
\(808\) 7.53023 0.264912
\(809\) 47.2870 1.66252 0.831262 0.555881i \(-0.187619\pi\)
0.831262 + 0.555881i \(0.187619\pi\)
\(810\) 0 0
\(811\) 35.1587 1.23459 0.617294 0.786733i \(-0.288229\pi\)
0.617294 + 0.786733i \(0.288229\pi\)
\(812\) −24.0396 −0.843623
\(813\) 52.5792 1.84403
\(814\) −4.74769 −0.166406
\(815\) 0 0
\(816\) −22.3451 −0.782234
\(817\) 0 0
\(818\) 5.84938 0.204519
\(819\) 19.9403 0.696770
\(820\) 0 0
\(821\) 32.1078 1.12057 0.560285 0.828300i \(-0.310692\pi\)
0.560285 + 0.828300i \(0.310692\pi\)
\(822\) 5.17658 0.180554
\(823\) −24.6481 −0.859180 −0.429590 0.903024i \(-0.641342\pi\)
−0.429590 + 0.903024i \(0.641342\pi\)
\(824\) −16.9576 −0.590748
\(825\) 0 0
\(826\) −7.22313 −0.251325
\(827\) −4.29867 −0.149479 −0.0747397 0.997203i \(-0.523813\pi\)
−0.0747397 + 0.997203i \(0.523813\pi\)
\(828\) 0.492695 0.0171223
\(829\) 11.2729 0.391524 0.195762 0.980651i \(-0.437282\pi\)
0.195762 + 0.980651i \(0.437282\pi\)
\(830\) 0 0
\(831\) 10.4540 0.362644
\(832\) 26.0994 0.904835
\(833\) 21.4732 0.744002
\(834\) −1.72201 −0.0596283
\(835\) 0 0
\(836\) 0 0
\(837\) −29.4633 −1.01840
\(838\) −10.8969 −0.376427
\(839\) 36.5388 1.26146 0.630730 0.776003i \(-0.282756\pi\)
0.630730 + 0.776003i \(0.282756\pi\)
\(840\) 0 0
\(841\) −17.2958 −0.596408
\(842\) 0.0713965 0.00246049
\(843\) −3.69353 −0.127212
\(844\) 7.95532 0.273833
\(845\) 0 0
\(846\) 4.05509 0.139417
\(847\) −52.1931 −1.79338
\(848\) −6.59659 −0.226528
\(849\) 38.1511 1.30934
\(850\) 0 0
\(851\) −0.659916 −0.0226216
\(852\) −3.30302 −0.113160
\(853\) −13.5898 −0.465306 −0.232653 0.972560i \(-0.574741\pi\)
−0.232653 + 0.972560i \(0.574741\pi\)
\(854\) −5.89616 −0.201763
\(855\) 0 0
\(856\) −21.4083 −0.731720
\(857\) 5.85278 0.199927 0.0999637 0.994991i \(-0.468127\pi\)
0.0999637 + 0.994991i \(0.468127\pi\)
\(858\) −14.9139 −0.509151
\(859\) −20.2569 −0.691157 −0.345579 0.938390i \(-0.612317\pi\)
−0.345579 + 0.938390i \(0.612317\pi\)
\(860\) 0 0
\(861\) 1.38504 0.0472020
\(862\) −4.23681 −0.144306
\(863\) 5.65447 0.192481 0.0962403 0.995358i \(-0.469318\pi\)
0.0962403 + 0.995358i \(0.469318\pi\)
\(864\) 13.2637 0.451239
\(865\) 0 0
\(866\) −6.79509 −0.230907
\(867\) 13.6650 0.464089
\(868\) −55.4922 −1.88353
\(869\) 46.0818 1.56322
\(870\) 0 0
\(871\) −51.7717 −1.75422
\(872\) 17.0131 0.576138
\(873\) 1.60350 0.0542702
\(874\) 0 0
\(875\) 0 0
\(876\) 20.0079 0.676006
\(877\) −57.4540 −1.94008 −0.970041 0.242939i \(-0.921888\pi\)
−0.970041 + 0.242939i \(0.921888\pi\)
\(878\) 2.28981 0.0772773
\(879\) −0.307973 −0.0103877
\(880\) 0 0
\(881\) 8.46121 0.285065 0.142533 0.989790i \(-0.454475\pi\)
0.142533 + 0.989790i \(0.454475\pi\)
\(882\) 2.49063 0.0838637
\(883\) −12.8839 −0.433578 −0.216789 0.976218i \(-0.569558\pi\)
−0.216789 + 0.976218i \(0.569558\pi\)
\(884\) 28.0039 0.941874
\(885\) 0 0
\(886\) 10.9970 0.369451
\(887\) 32.9290 1.10565 0.552823 0.833298i \(-0.313550\pi\)
0.552823 + 0.833298i \(0.313550\pi\)
\(888\) 7.54808 0.253297
\(889\) 44.0939 1.47886
\(890\) 0 0
\(891\) 55.8380 1.87064
\(892\) 10.1228 0.338936
\(893\) 0 0
\(894\) −5.41365 −0.181060
\(895\) 0 0
\(896\) 32.9777 1.10171
\(897\) −2.07299 −0.0692151
\(898\) −1.61502 −0.0538940
\(899\) 27.0176 0.901087
\(900\) 0 0
\(901\) −6.21897 −0.207184
\(902\) −0.291338 −0.00970049
\(903\) −55.2077 −1.83720
\(904\) 5.14153 0.171005
\(905\) 0 0
\(906\) 5.55184 0.184447
\(907\) −34.1251 −1.13311 −0.566553 0.824025i \(-0.691723\pi\)
−0.566553 + 0.824025i \(0.691723\pi\)
\(908\) −9.83659 −0.326439
\(909\) −7.14455 −0.236970
\(910\) 0 0
\(911\) 27.5938 0.914222 0.457111 0.889410i \(-0.348884\pi\)
0.457111 + 0.889410i \(0.348884\pi\)
\(912\) 0 0
\(913\) −22.7252 −0.752094
\(914\) −12.6889 −0.419712
\(915\) 0 0
\(916\) 32.4386 1.07180
\(917\) 28.0465 0.926178
\(918\) 3.80117 0.125457
\(919\) 13.1572 0.434017 0.217009 0.976170i \(-0.430370\pi\)
0.217009 + 0.976170i \(0.430370\pi\)
\(920\) 0 0
\(921\) −10.1115 −0.333186
\(922\) 8.40674 0.276861
\(923\) 3.90846 0.128649
\(924\) 71.9330 2.36642
\(925\) 0 0
\(926\) 3.07404 0.101019
\(927\) 16.0891 0.528436
\(928\) −12.1627 −0.399259
\(929\) −23.5599 −0.772975 −0.386487 0.922295i \(-0.626312\pi\)
−0.386487 + 0.922295i \(0.626312\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −30.1050 −0.986121
\(933\) 48.2683 1.58023
\(934\) 1.83492 0.0600403
\(935\) 0 0
\(936\) 6.66838 0.217963
\(937\) 35.9997 1.17606 0.588030 0.808839i \(-0.299904\pi\)
0.588030 + 0.808839i \(0.299904\pi\)
\(938\) −13.2355 −0.432156
\(939\) −18.1673 −0.592868
\(940\) 0 0
\(941\) −41.8515 −1.36432 −0.682160 0.731203i \(-0.738959\pi\)
−0.682160 + 0.731203i \(0.738959\pi\)
\(942\) 7.34662 0.239366
\(943\) −0.0404952 −0.00131871
\(944\) 20.9590 0.682158
\(945\) 0 0
\(946\) 11.6127 0.377562
\(947\) −3.63656 −0.118172 −0.0590861 0.998253i \(-0.518819\pi\)
−0.0590861 + 0.998253i \(0.518819\pi\)
\(948\) −35.6857 −1.15902
\(949\) −23.6754 −0.768535
\(950\) 0 0
\(951\) −60.4680 −1.96081
\(952\) 14.6980 0.476365
\(953\) −10.7543 −0.348366 −0.174183 0.984713i \(-0.555728\pi\)
−0.174183 + 0.984713i \(0.555728\pi\)
\(954\) −0.721324 −0.0233537
\(955\) 0 0
\(956\) 33.7488 1.09151
\(957\) −35.0221 −1.13210
\(958\) 0.562218 0.0181644
\(959\) 29.5444 0.954039
\(960\) 0 0
\(961\) 31.3666 1.01183
\(962\) −4.35052 −0.140266
\(963\) 20.3118 0.654538
\(964\) −52.0080 −1.67506
\(965\) 0 0
\(966\) −0.529964 −0.0170513
\(967\) −18.7666 −0.603493 −0.301747 0.953388i \(-0.597570\pi\)
−0.301747 + 0.953388i \(0.597570\pi\)
\(968\) −17.4543 −0.561002
\(969\) 0 0
\(970\) 0 0
\(971\) 38.4930 1.23530 0.617650 0.786453i \(-0.288085\pi\)
0.617650 + 0.786453i \(0.288085\pi\)
\(972\) −21.9827 −0.705096
\(973\) −9.82805 −0.315073
\(974\) −5.14426 −0.164833
\(975\) 0 0
\(976\) 17.1086 0.547633
\(977\) 21.7651 0.696328 0.348164 0.937434i \(-0.386805\pi\)
0.348164 + 0.937434i \(0.386805\pi\)
\(978\) 3.01455 0.0963946
\(979\) −74.5361 −2.38218
\(980\) 0 0
\(981\) −16.1418 −0.515367
\(982\) 5.86569 0.187182
\(983\) 32.9796 1.05189 0.525943 0.850520i \(-0.323712\pi\)
0.525943 + 0.850520i \(0.323712\pi\)
\(984\) 0.463182 0.0147657
\(985\) 0 0
\(986\) −3.48564 −0.111005
\(987\) 82.2918 2.61938
\(988\) 0 0
\(989\) 1.61414 0.0513266
\(990\) 0 0
\(991\) 44.9400 1.42757 0.713783 0.700367i \(-0.246980\pi\)
0.713783 + 0.700367i \(0.246980\pi\)
\(992\) −28.0759 −0.891412
\(993\) 52.8862 1.67829
\(994\) 0.999208 0.0316929
\(995\) 0 0
\(996\) 17.5983 0.557625
\(997\) −1.52970 −0.0484462 −0.0242231 0.999707i \(-0.507711\pi\)
−0.0242231 + 0.999707i \(0.507711\pi\)
\(998\) −5.48412 −0.173597
\(999\) 11.1411 0.352488
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 9025.2.a.ck.1.9 16
5.2 odd 4 1805.2.b.j.1084.9 yes 16
5.3 odd 4 1805.2.b.j.1084.8 yes 16
5.4 even 2 inner 9025.2.a.ck.1.8 16
19.18 odd 2 9025.2.a.cl.1.8 16
95.18 even 4 1805.2.b.i.1084.9 yes 16
95.37 even 4 1805.2.b.i.1084.8 16
95.94 odd 2 9025.2.a.cl.1.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.b.i.1084.8 16 95.37 even 4
1805.2.b.i.1084.9 yes 16 95.18 even 4
1805.2.b.j.1084.8 yes 16 5.3 odd 4
1805.2.b.j.1084.9 yes 16 5.2 odd 4
9025.2.a.ck.1.8 16 5.4 even 2 inner
9025.2.a.ck.1.9 16 1.1 even 1 trivial
9025.2.a.cl.1.8 16 19.18 odd 2
9025.2.a.cl.1.9 16 95.94 odd 2