Properties

Label 4025.2.a.z.1.13
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 18 x^{12} + 58 x^{11} + 111 x^{10} - 414 x^{9} - 244 x^{8} + 1330 x^{7} - 27 x^{6} - 1853 x^{5} + 539 x^{4} + 891 x^{3} - 218 x^{2} - 133 x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-2.33696\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.33696 q^{2} -0.00757499 q^{3} +3.46136 q^{4} -0.0177024 q^{6} -1.00000 q^{7} +3.41514 q^{8} -2.99994 q^{9} +O(q^{10})\) \(q+2.33696 q^{2} -0.00757499 q^{3} +3.46136 q^{4} -0.0177024 q^{6} -1.00000 q^{7} +3.41514 q^{8} -2.99994 q^{9} -4.99434 q^{11} -0.0262198 q^{12} +3.87824 q^{13} -2.33696 q^{14} +1.05832 q^{16} +0.475637 q^{17} -7.01073 q^{18} -1.54155 q^{19} +0.00757499 q^{21} -11.6715 q^{22} -1.00000 q^{23} -0.0258697 q^{24} +9.06327 q^{26} +0.0454495 q^{27} -3.46136 q^{28} -9.38110 q^{29} -11.0814 q^{31} -4.35705 q^{32} +0.0378321 q^{33} +1.11154 q^{34} -10.3839 q^{36} +4.42722 q^{37} -3.60253 q^{38} -0.0293776 q^{39} +3.21792 q^{41} +0.0177024 q^{42} -3.40274 q^{43} -17.2872 q^{44} -2.33696 q^{46} +8.07320 q^{47} -0.00801673 q^{48} +1.00000 q^{49} -0.00360295 q^{51} +13.4240 q^{52} +1.46842 q^{53} +0.106214 q^{54} -3.41514 q^{56} +0.0116772 q^{57} -21.9232 q^{58} -11.8954 q^{59} +8.24535 q^{61} -25.8968 q^{62} +2.99994 q^{63} -12.2989 q^{64} +0.0884119 q^{66} -4.54321 q^{67} +1.64635 q^{68} +0.00757499 q^{69} +11.0031 q^{71} -10.2452 q^{72} -13.6586 q^{73} +10.3462 q^{74} -5.33586 q^{76} +4.99434 q^{77} -0.0686542 q^{78} +13.1029 q^{79} +8.99948 q^{81} +7.52014 q^{82} -14.7630 q^{83} +0.0262198 q^{84} -7.95205 q^{86} +0.0710618 q^{87} -17.0564 q^{88} -6.08219 q^{89} -3.87824 q^{91} -3.46136 q^{92} +0.0839417 q^{93} +18.8667 q^{94} +0.0330047 q^{96} -3.91724 q^{97} +2.33696 q^{98} +14.9827 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 3 q^{2} - 4 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 3 q^{2} - 4 q^{3} + 17 q^{4} - 4 q^{6} - 14 q^{7} - 3 q^{8} + 18 q^{9} + 5 q^{11} - 17 q^{12} - 11 q^{13} + 3 q^{14} + 23 q^{16} - 3 q^{17} - 15 q^{18} - 2 q^{19} + 4 q^{21} - 23 q^{22} - 14 q^{23} - 12 q^{24} - 9 q^{26} - 25 q^{27} - 17 q^{28} + 7 q^{29} - 3 q^{31} - 24 q^{32} - 6 q^{33} - 14 q^{34} + 13 q^{36} - 22 q^{37} - 20 q^{38} - 10 q^{39} - 17 q^{41} + 4 q^{42} - 18 q^{43} + 28 q^{44} + 3 q^{46} - 30 q^{47} - 8 q^{48} + 14 q^{49} + 4 q^{51} - 8 q^{52} - 11 q^{53} + 20 q^{54} + 3 q^{56} - 18 q^{57} - 38 q^{58} - 22 q^{59} - 8 q^{61} + 22 q^{62} - 18 q^{63} + 29 q^{64} - 9 q^{66} - 39 q^{67} - q^{68} + 4 q^{69} - 5 q^{71} + 24 q^{72} - 18 q^{73} + 35 q^{74} - 41 q^{76} - 5 q^{77} + 22 q^{78} + 10 q^{79} + 2 q^{81} + 8 q^{82} - 24 q^{83} + 17 q^{84} - 26 q^{86} - 5 q^{87} - 58 q^{88} + 25 q^{89} + 11 q^{91} - 17 q^{92} - 47 q^{93} - 2 q^{94} - 117 q^{96} - 43 q^{97} - 3 q^{98} + 55 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.33696 1.65248 0.826239 0.563320i \(-0.190476\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(3\) −0.00757499 −0.00437342 −0.00218671 0.999998i \(-0.500696\pi\)
−0.00218671 + 0.999998i \(0.500696\pi\)
\(4\) 3.46136 1.73068
\(5\) 0 0
\(6\) −0.0177024 −0.00722699
\(7\) −1.00000 −0.377964
\(8\) 3.41514 1.20744
\(9\) −2.99994 −0.999981
\(10\) 0 0
\(11\) −4.99434 −1.50585 −0.752924 0.658107i \(-0.771357\pi\)
−0.752924 + 0.658107i \(0.771357\pi\)
\(12\) −0.0262198 −0.00756901
\(13\) 3.87824 1.07563 0.537815 0.843063i \(-0.319250\pi\)
0.537815 + 0.843063i \(0.319250\pi\)
\(14\) −2.33696 −0.624578
\(15\) 0 0
\(16\) 1.05832 0.264579
\(17\) 0.475637 0.115359 0.0576794 0.998335i \(-0.481630\pi\)
0.0576794 + 0.998335i \(0.481630\pi\)
\(18\) −7.01073 −1.65245
\(19\) −1.54155 −0.353655 −0.176828 0.984242i \(-0.556583\pi\)
−0.176828 + 0.984242i \(0.556583\pi\)
\(20\) 0 0
\(21\) 0.00757499 0.00165300
\(22\) −11.6715 −2.48838
\(23\) −1.00000 −0.208514
\(24\) −0.0258697 −0.00528063
\(25\) 0 0
\(26\) 9.06327 1.77745
\(27\) 0.0454495 0.00874677
\(28\) −3.46136 −0.654136
\(29\) −9.38110 −1.74203 −0.871013 0.491259i \(-0.836537\pi\)
−0.871013 + 0.491259i \(0.836537\pi\)
\(30\) 0 0
\(31\) −11.0814 −1.99028 −0.995141 0.0984581i \(-0.968609\pi\)
−0.995141 + 0.0984581i \(0.968609\pi\)
\(32\) −4.35705 −0.770226
\(33\) 0.0378321 0.00658572
\(34\) 1.11154 0.190628
\(35\) 0 0
\(36\) −10.3839 −1.73065
\(37\) 4.42722 0.727830 0.363915 0.931432i \(-0.381440\pi\)
0.363915 + 0.931432i \(0.381440\pi\)
\(38\) −3.60253 −0.584407
\(39\) −0.0293776 −0.00470419
\(40\) 0 0
\(41\) 3.21792 0.502555 0.251278 0.967915i \(-0.419149\pi\)
0.251278 + 0.967915i \(0.419149\pi\)
\(42\) 0.0177024 0.00273154
\(43\) −3.40274 −0.518912 −0.259456 0.965755i \(-0.583543\pi\)
−0.259456 + 0.965755i \(0.583543\pi\)
\(44\) −17.2872 −2.60615
\(45\) 0 0
\(46\) −2.33696 −0.344565
\(47\) 8.07320 1.17760 0.588799 0.808280i \(-0.299601\pi\)
0.588799 + 0.808280i \(0.299601\pi\)
\(48\) −0.00801673 −0.00115712
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.00360295 −0.000504513 0
\(52\) 13.4240 1.86157
\(53\) 1.46842 0.201703 0.100852 0.994901i \(-0.467843\pi\)
0.100852 + 0.994901i \(0.467843\pi\)
\(54\) 0.106214 0.0144538
\(55\) 0 0
\(56\) −3.41514 −0.456368
\(57\) 0.0116772 0.00154668
\(58\) −21.9232 −2.87866
\(59\) −11.8954 −1.54865 −0.774326 0.632787i \(-0.781911\pi\)
−0.774326 + 0.632787i \(0.781911\pi\)
\(60\) 0 0
\(61\) 8.24535 1.05571 0.527854 0.849335i \(-0.322997\pi\)
0.527854 + 0.849335i \(0.322997\pi\)
\(62\) −25.8968 −3.28890
\(63\) 2.99994 0.377957
\(64\) −12.2989 −1.53736
\(65\) 0 0
\(66\) 0.0884119 0.0108827
\(67\) −4.54321 −0.555041 −0.277521 0.960720i \(-0.589513\pi\)
−0.277521 + 0.960720i \(0.589513\pi\)
\(68\) 1.64635 0.199650
\(69\) 0.00757499 0.000911922 0
\(70\) 0 0
\(71\) 11.0031 1.30583 0.652916 0.757430i \(-0.273545\pi\)
0.652916 + 0.757430i \(0.273545\pi\)
\(72\) −10.2452 −1.20741
\(73\) −13.6586 −1.59862 −0.799308 0.600921i \(-0.794800\pi\)
−0.799308 + 0.600921i \(0.794800\pi\)
\(74\) 10.3462 1.20272
\(75\) 0 0
\(76\) −5.33586 −0.612065
\(77\) 4.99434 0.569157
\(78\) −0.0686542 −0.00777356
\(79\) 13.1029 1.47419 0.737096 0.675788i \(-0.236197\pi\)
0.737096 + 0.675788i \(0.236197\pi\)
\(80\) 0 0
\(81\) 8.99948 0.999943
\(82\) 7.52014 0.830461
\(83\) −14.7630 −1.62045 −0.810225 0.586118i \(-0.800655\pi\)
−0.810225 + 0.586118i \(0.800655\pi\)
\(84\) 0.0262198 0.00286082
\(85\) 0 0
\(86\) −7.95205 −0.857491
\(87\) 0.0710618 0.00761862
\(88\) −17.0564 −1.81822
\(89\) −6.08219 −0.644710 −0.322355 0.946619i \(-0.604475\pi\)
−0.322355 + 0.946619i \(0.604475\pi\)
\(90\) 0 0
\(91\) −3.87824 −0.406550
\(92\) −3.46136 −0.360872
\(93\) 0.0839417 0.00870435
\(94\) 18.8667 1.94595
\(95\) 0 0
\(96\) 0.0330047 0.00336852
\(97\) −3.91724 −0.397735 −0.198867 0.980026i \(-0.563726\pi\)
−0.198867 + 0.980026i \(0.563726\pi\)
\(98\) 2.33696 0.236068
\(99\) 14.9827 1.50582
\(100\) 0 0
\(101\) 2.79080 0.277695 0.138848 0.990314i \(-0.455660\pi\)
0.138848 + 0.990314i \(0.455660\pi\)
\(102\) −0.00841993 −0.000833697 0
\(103\) −13.1299 −1.29373 −0.646863 0.762606i \(-0.723919\pi\)
−0.646863 + 0.762606i \(0.723919\pi\)
\(104\) 13.2447 1.29875
\(105\) 0 0
\(106\) 3.43164 0.333310
\(107\) 18.1208 1.75181 0.875903 0.482488i \(-0.160267\pi\)
0.875903 + 0.482488i \(0.160267\pi\)
\(108\) 0.157317 0.0151379
\(109\) −8.86577 −0.849186 −0.424593 0.905384i \(-0.639583\pi\)
−0.424593 + 0.905384i \(0.639583\pi\)
\(110\) 0 0
\(111\) −0.0335361 −0.00318311
\(112\) −1.05832 −0.100001
\(113\) 5.71585 0.537702 0.268851 0.963182i \(-0.413356\pi\)
0.268851 + 0.963182i \(0.413356\pi\)
\(114\) 0.0272891 0.00255586
\(115\) 0 0
\(116\) −32.4714 −3.01489
\(117\) −11.6345 −1.07561
\(118\) −27.7991 −2.55911
\(119\) −0.475637 −0.0436016
\(120\) 0 0
\(121\) 13.9434 1.26758
\(122\) 19.2690 1.74454
\(123\) −0.0243757 −0.00219789
\(124\) −38.3568 −3.44455
\(125\) 0 0
\(126\) 7.01073 0.624566
\(127\) −14.5563 −1.29167 −0.645833 0.763479i \(-0.723490\pi\)
−0.645833 + 0.763479i \(0.723490\pi\)
\(128\) −20.0278 −1.77023
\(129\) 0.0257757 0.00226942
\(130\) 0 0
\(131\) −6.58099 −0.574984 −0.287492 0.957783i \(-0.592821\pi\)
−0.287492 + 0.957783i \(0.592821\pi\)
\(132\) 0.130951 0.0113978
\(133\) 1.54155 0.133669
\(134\) −10.6173 −0.917193
\(135\) 0 0
\(136\) 1.62437 0.139288
\(137\) 7.37371 0.629978 0.314989 0.949095i \(-0.397999\pi\)
0.314989 + 0.949095i \(0.397999\pi\)
\(138\) 0.0177024 0.00150693
\(139\) −11.7472 −0.996384 −0.498192 0.867067i \(-0.666003\pi\)
−0.498192 + 0.867067i \(0.666003\pi\)
\(140\) 0 0
\(141\) −0.0611545 −0.00515014
\(142\) 25.7138 2.15786
\(143\) −19.3692 −1.61974
\(144\) −3.17488 −0.264574
\(145\) 0 0
\(146\) −31.9195 −2.64168
\(147\) −0.00757499 −0.000624775 0
\(148\) 15.3242 1.25964
\(149\) 0.282601 0.0231516 0.0115758 0.999933i \(-0.496315\pi\)
0.0115758 + 0.999933i \(0.496315\pi\)
\(150\) 0 0
\(151\) 19.5423 1.59033 0.795166 0.606391i \(-0.207384\pi\)
0.795166 + 0.606391i \(0.207384\pi\)
\(152\) −5.26461 −0.427016
\(153\) −1.42688 −0.115357
\(154\) 11.6715 0.940520
\(155\) 0 0
\(156\) −0.101687 −0.00814145
\(157\) 23.7122 1.89244 0.946220 0.323523i \(-0.104867\pi\)
0.946220 + 0.323523i \(0.104867\pi\)
\(158\) 30.6209 2.43607
\(159\) −0.0111233 −0.000882134 0
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 21.0314 1.65238
\(163\) −0.0532661 −0.00417212 −0.00208606 0.999998i \(-0.500664\pi\)
−0.00208606 + 0.999998i \(0.500664\pi\)
\(164\) 11.1384 0.869763
\(165\) 0 0
\(166\) −34.5005 −2.67776
\(167\) −6.63072 −0.513101 −0.256550 0.966531i \(-0.582586\pi\)
−0.256550 + 0.966531i \(0.582586\pi\)
\(168\) 0.0258697 0.00199589
\(169\) 2.04074 0.156980
\(170\) 0 0
\(171\) 4.62455 0.353648
\(172\) −11.7781 −0.898073
\(173\) 6.76778 0.514545 0.257273 0.966339i \(-0.417176\pi\)
0.257273 + 0.966339i \(0.417176\pi\)
\(174\) 0.166068 0.0125896
\(175\) 0 0
\(176\) −5.28558 −0.398416
\(177\) 0.0901077 0.00677291
\(178\) −14.2138 −1.06537
\(179\) 9.48534 0.708967 0.354484 0.935062i \(-0.384657\pi\)
0.354484 + 0.935062i \(0.384657\pi\)
\(180\) 0 0
\(181\) 18.0503 1.34167 0.670834 0.741607i \(-0.265936\pi\)
0.670834 + 0.741607i \(0.265936\pi\)
\(182\) −9.06327 −0.671815
\(183\) −0.0624585 −0.00461706
\(184\) −3.41514 −0.251768
\(185\) 0 0
\(186\) 0.196168 0.0143837
\(187\) −2.37549 −0.173713
\(188\) 27.9443 2.03805
\(189\) −0.0454495 −0.00330597
\(190\) 0 0
\(191\) −23.6976 −1.71470 −0.857349 0.514735i \(-0.827890\pi\)
−0.857349 + 0.514735i \(0.827890\pi\)
\(192\) 0.0931639 0.00672352
\(193\) −14.5036 −1.04399 −0.521995 0.852948i \(-0.674812\pi\)
−0.521995 + 0.852948i \(0.674812\pi\)
\(194\) −9.15441 −0.657248
\(195\) 0 0
\(196\) 3.46136 0.247240
\(197\) −2.63536 −0.187762 −0.0938809 0.995583i \(-0.529927\pi\)
−0.0938809 + 0.995583i \(0.529927\pi\)
\(198\) 35.0140 2.48833
\(199\) −19.5257 −1.38414 −0.692070 0.721830i \(-0.743301\pi\)
−0.692070 + 0.721830i \(0.743301\pi\)
\(200\) 0 0
\(201\) 0.0344148 0.00242743
\(202\) 6.52198 0.458885
\(203\) 9.38110 0.658424
\(204\) −0.0124711 −0.000873152 0
\(205\) 0 0
\(206\) −30.6840 −2.13785
\(207\) 2.99994 0.208510
\(208\) 4.10440 0.284589
\(209\) 7.69900 0.532551
\(210\) 0 0
\(211\) −10.5337 −0.725168 −0.362584 0.931951i \(-0.618105\pi\)
−0.362584 + 0.931951i \(0.618105\pi\)
\(212\) 5.08274 0.349084
\(213\) −0.0833487 −0.00571096
\(214\) 42.3476 2.89482
\(215\) 0 0
\(216\) 0.155217 0.0105612
\(217\) 11.0814 0.752256
\(218\) −20.7189 −1.40326
\(219\) 0.103464 0.00699143
\(220\) 0 0
\(221\) 1.84463 0.124083
\(222\) −0.0783725 −0.00526002
\(223\) −15.2356 −1.02025 −0.510127 0.860099i \(-0.670402\pi\)
−0.510127 + 0.860099i \(0.670402\pi\)
\(224\) 4.35705 0.291118
\(225\) 0 0
\(226\) 13.3577 0.888540
\(227\) 27.4434 1.82148 0.910740 0.412980i \(-0.135512\pi\)
0.910740 + 0.412980i \(0.135512\pi\)
\(228\) 0.0404191 0.00267682
\(229\) 29.8273 1.97105 0.985523 0.169540i \(-0.0542283\pi\)
0.985523 + 0.169540i \(0.0542283\pi\)
\(230\) 0 0
\(231\) −0.0378321 −0.00248917
\(232\) −32.0378 −2.10339
\(233\) 22.9219 1.50166 0.750832 0.660494i \(-0.229653\pi\)
0.750832 + 0.660494i \(0.229653\pi\)
\(234\) −27.1893 −1.77742
\(235\) 0 0
\(236\) −41.1744 −2.68022
\(237\) −0.0992544 −0.00644727
\(238\) −1.11154 −0.0720506
\(239\) 15.2760 0.988122 0.494061 0.869427i \(-0.335512\pi\)
0.494061 + 0.869427i \(0.335512\pi\)
\(240\) 0 0
\(241\) 5.89087 0.379464 0.189732 0.981836i \(-0.439238\pi\)
0.189732 + 0.981836i \(0.439238\pi\)
\(242\) 32.5851 2.09465
\(243\) −0.204520 −0.0131199
\(244\) 28.5402 1.82710
\(245\) 0 0
\(246\) −0.0569650 −0.00363196
\(247\) −5.97849 −0.380402
\(248\) −37.8447 −2.40314
\(249\) 0.111830 0.00708692
\(250\) 0 0
\(251\) 4.08671 0.257951 0.128975 0.991648i \(-0.458831\pi\)
0.128975 + 0.991648i \(0.458831\pi\)
\(252\) 10.3839 0.654124
\(253\) 4.99434 0.313991
\(254\) −34.0175 −2.13445
\(255\) 0 0
\(256\) −22.2064 −1.38790
\(257\) −19.2722 −1.20216 −0.601082 0.799187i \(-0.705264\pi\)
−0.601082 + 0.799187i \(0.705264\pi\)
\(258\) 0.0602367 0.00375017
\(259\) −4.42722 −0.275094
\(260\) 0 0
\(261\) 28.1428 1.74199
\(262\) −15.3795 −0.950148
\(263\) 25.2660 1.55797 0.778984 0.627044i \(-0.215736\pi\)
0.778984 + 0.627044i \(0.215736\pi\)
\(264\) 0.129202 0.00795183
\(265\) 0 0
\(266\) 3.60253 0.220885
\(267\) 0.0460725 0.00281959
\(268\) −15.7257 −0.960600
\(269\) 7.84347 0.478225 0.239112 0.970992i \(-0.423144\pi\)
0.239112 + 0.970992i \(0.423144\pi\)
\(270\) 0 0
\(271\) −19.5068 −1.18495 −0.592477 0.805588i \(-0.701850\pi\)
−0.592477 + 0.805588i \(0.701850\pi\)
\(272\) 0.503374 0.0305215
\(273\) 0.0293776 0.00177802
\(274\) 17.2320 1.04103
\(275\) 0 0
\(276\) 0.0262198 0.00157825
\(277\) −24.9162 −1.49707 −0.748534 0.663097i \(-0.769242\pi\)
−0.748534 + 0.663097i \(0.769242\pi\)
\(278\) −27.4527 −1.64650
\(279\) 33.2436 1.99024
\(280\) 0 0
\(281\) −2.78384 −0.166070 −0.0830349 0.996547i \(-0.526461\pi\)
−0.0830349 + 0.996547i \(0.526461\pi\)
\(282\) −0.142915 −0.00851048
\(283\) 12.1104 0.719891 0.359946 0.932973i \(-0.382795\pi\)
0.359946 + 0.932973i \(0.382795\pi\)
\(284\) 38.0859 2.25998
\(285\) 0 0
\(286\) −45.2650 −2.67658
\(287\) −3.21792 −0.189948
\(288\) 13.0709 0.770211
\(289\) −16.7738 −0.986692
\(290\) 0 0
\(291\) 0.0296730 0.00173946
\(292\) −47.2773 −2.76670
\(293\) 0.650821 0.0380214 0.0190107 0.999819i \(-0.493948\pi\)
0.0190107 + 0.999819i \(0.493948\pi\)
\(294\) −0.0177024 −0.00103243
\(295\) 0 0
\(296\) 15.1196 0.878808
\(297\) −0.226990 −0.0131713
\(298\) 0.660427 0.0382575
\(299\) −3.87824 −0.224284
\(300\) 0 0
\(301\) 3.40274 0.196130
\(302\) 45.6696 2.62799
\(303\) −0.0211403 −0.00121448
\(304\) −1.63144 −0.0935697
\(305\) 0 0
\(306\) −3.33456 −0.190624
\(307\) −22.5537 −1.28721 −0.643603 0.765359i \(-0.722561\pi\)
−0.643603 + 0.765359i \(0.722561\pi\)
\(308\) 17.2872 0.985031
\(309\) 0.0994587 0.00565801
\(310\) 0 0
\(311\) 25.7851 1.46214 0.731069 0.682303i \(-0.239022\pi\)
0.731069 + 0.682303i \(0.239022\pi\)
\(312\) −0.100329 −0.00568000
\(313\) −5.58377 −0.315613 −0.157807 0.987470i \(-0.550442\pi\)
−0.157807 + 0.987470i \(0.550442\pi\)
\(314\) 55.4144 3.12722
\(315\) 0 0
\(316\) 45.3539 2.55136
\(317\) 4.35295 0.244486 0.122243 0.992500i \(-0.460991\pi\)
0.122243 + 0.992500i \(0.460991\pi\)
\(318\) −0.0259946 −0.00145771
\(319\) 46.8524 2.62323
\(320\) 0 0
\(321\) −0.137265 −0.00766139
\(322\) 2.33696 0.130233
\(323\) −0.733217 −0.0407973
\(324\) 31.1505 1.73058
\(325\) 0 0
\(326\) −0.124480 −0.00689434
\(327\) 0.0671581 0.00371385
\(328\) 10.9897 0.606803
\(329\) −8.07320 −0.445090
\(330\) 0 0
\(331\) −10.7648 −0.591685 −0.295842 0.955237i \(-0.595600\pi\)
−0.295842 + 0.955237i \(0.595600\pi\)
\(332\) −51.1002 −2.80449
\(333\) −13.2814 −0.727816
\(334\) −15.4957 −0.847888
\(335\) 0 0
\(336\) 0.00801673 0.000437349 0
\(337\) 0.415084 0.0226111 0.0113055 0.999936i \(-0.496401\pi\)
0.0113055 + 0.999936i \(0.496401\pi\)
\(338\) 4.76911 0.259405
\(339\) −0.0432975 −0.00235160
\(340\) 0 0
\(341\) 55.3443 2.99706
\(342\) 10.8074 0.584396
\(343\) −1.00000 −0.0539949
\(344\) −11.6208 −0.626554
\(345\) 0 0
\(346\) 15.8160 0.850274
\(347\) 4.09489 0.219825 0.109913 0.993941i \(-0.464943\pi\)
0.109913 + 0.993941i \(0.464943\pi\)
\(348\) 0.245971 0.0131854
\(349\) −4.93152 −0.263978 −0.131989 0.991251i \(-0.542136\pi\)
−0.131989 + 0.991251i \(0.542136\pi\)
\(350\) 0 0
\(351\) 0.176264 0.00940828
\(352\) 21.7606 1.15984
\(353\) −14.1977 −0.755667 −0.377833 0.925874i \(-0.623331\pi\)
−0.377833 + 0.925874i \(0.623331\pi\)
\(354\) 0.210578 0.0111921
\(355\) 0 0
\(356\) −21.0527 −1.11579
\(357\) 0.00360295 0.000190688 0
\(358\) 22.1668 1.17155
\(359\) −24.5208 −1.29416 −0.647078 0.762424i \(-0.724009\pi\)
−0.647078 + 0.762424i \(0.724009\pi\)
\(360\) 0 0
\(361\) −16.6236 −0.874928
\(362\) 42.1828 2.21708
\(363\) −0.105621 −0.00554367
\(364\) −13.4240 −0.703609
\(365\) 0 0
\(366\) −0.145963 −0.00762959
\(367\) 16.4023 0.856191 0.428096 0.903733i \(-0.359185\pi\)
0.428096 + 0.903733i \(0.359185\pi\)
\(368\) −1.05832 −0.0551685
\(369\) −9.65358 −0.502546
\(370\) 0 0
\(371\) −1.46842 −0.0762367
\(372\) 0.290553 0.0150645
\(373\) 3.44368 0.178307 0.0891535 0.996018i \(-0.471584\pi\)
0.0891535 + 0.996018i \(0.471584\pi\)
\(374\) −5.55142 −0.287057
\(375\) 0 0
\(376\) 27.5712 1.42187
\(377\) −36.3822 −1.87378
\(378\) −0.106214 −0.00546304
\(379\) −3.28827 −0.168907 −0.0844536 0.996427i \(-0.526914\pi\)
−0.0844536 + 0.996427i \(0.526914\pi\)
\(380\) 0 0
\(381\) 0.110264 0.00564900
\(382\) −55.3803 −2.83350
\(383\) 18.1838 0.929151 0.464576 0.885533i \(-0.346207\pi\)
0.464576 + 0.885533i \(0.346207\pi\)
\(384\) 0.151711 0.00774195
\(385\) 0 0
\(386\) −33.8942 −1.72517
\(387\) 10.2080 0.518903
\(388\) −13.5590 −0.688353
\(389\) −27.9376 −1.41649 −0.708246 0.705966i \(-0.750513\pi\)
−0.708246 + 0.705966i \(0.750513\pi\)
\(390\) 0 0
\(391\) −0.475637 −0.0240540
\(392\) 3.41514 0.172491
\(393\) 0.0498510 0.00251465
\(394\) −6.15872 −0.310272
\(395\) 0 0
\(396\) 51.8607 2.60610
\(397\) 20.5903 1.03340 0.516699 0.856167i \(-0.327161\pi\)
0.516699 + 0.856167i \(0.327161\pi\)
\(398\) −45.6307 −2.28726
\(399\) −0.0116772 −0.000584592 0
\(400\) 0 0
\(401\) 0.448572 0.0224006 0.0112003 0.999937i \(-0.496435\pi\)
0.0112003 + 0.999937i \(0.496435\pi\)
\(402\) 0.0804258 0.00401127
\(403\) −42.9764 −2.14081
\(404\) 9.65998 0.480602
\(405\) 0 0
\(406\) 21.9232 1.08803
\(407\) −22.1110 −1.09600
\(408\) −0.0123046 −0.000609168 0
\(409\) −6.40103 −0.316511 −0.158255 0.987398i \(-0.550587\pi\)
−0.158255 + 0.987398i \(0.550587\pi\)
\(410\) 0 0
\(411\) −0.0558558 −0.00275516
\(412\) −45.4473 −2.23903
\(413\) 11.8954 0.585335
\(414\) 7.01073 0.344559
\(415\) 0 0
\(416\) −16.8977 −0.828478
\(417\) 0.0889850 0.00435761
\(418\) 17.9922 0.880029
\(419\) −8.49010 −0.414768 −0.207384 0.978260i \(-0.566495\pi\)
−0.207384 + 0.978260i \(0.566495\pi\)
\(420\) 0 0
\(421\) −25.3915 −1.23751 −0.618753 0.785585i \(-0.712362\pi\)
−0.618753 + 0.785585i \(0.712362\pi\)
\(422\) −24.6167 −1.19832
\(423\) −24.2192 −1.17758
\(424\) 5.01487 0.243544
\(425\) 0 0
\(426\) −0.194782 −0.00943723
\(427\) −8.24535 −0.399020
\(428\) 62.7227 3.03182
\(429\) 0.146722 0.00708379
\(430\) 0 0
\(431\) 19.3251 0.930857 0.465428 0.885086i \(-0.345900\pi\)
0.465428 + 0.885086i \(0.345900\pi\)
\(432\) 0.0480999 0.00231421
\(433\) −35.6715 −1.71426 −0.857132 0.515097i \(-0.827756\pi\)
−0.857132 + 0.515097i \(0.827756\pi\)
\(434\) 25.8968 1.24309
\(435\) 0 0
\(436\) −30.6876 −1.46967
\(437\) 1.54155 0.0737422
\(438\) 0.241790 0.0115532
\(439\) −18.2969 −0.873261 −0.436631 0.899641i \(-0.643828\pi\)
−0.436631 + 0.899641i \(0.643828\pi\)
\(440\) 0 0
\(441\) −2.99994 −0.142854
\(442\) 4.31083 0.205045
\(443\) 4.02535 0.191250 0.0956251 0.995417i \(-0.469515\pi\)
0.0956251 + 0.995417i \(0.469515\pi\)
\(444\) −0.116081 −0.00550895
\(445\) 0 0
\(446\) −35.6050 −1.68595
\(447\) −0.00214070 −0.000101252 0
\(448\) 12.2989 0.581067
\(449\) 29.7863 1.40570 0.702851 0.711337i \(-0.251910\pi\)
0.702851 + 0.711337i \(0.251910\pi\)
\(450\) 0 0
\(451\) −16.0714 −0.756772
\(452\) 19.7846 0.930591
\(453\) −0.148033 −0.00695520
\(454\) 64.1339 3.00996
\(455\) 0 0
\(456\) 0.0398794 0.00186752
\(457\) 19.5513 0.914572 0.457286 0.889320i \(-0.348822\pi\)
0.457286 + 0.889320i \(0.348822\pi\)
\(458\) 69.7052 3.25711
\(459\) 0.0216175 0.00100902
\(460\) 0 0
\(461\) 0.191900 0.00893768 0.00446884 0.999990i \(-0.498578\pi\)
0.00446884 + 0.999990i \(0.498578\pi\)
\(462\) −0.0884119 −0.00411329
\(463\) −26.2178 −1.21845 −0.609223 0.792999i \(-0.708519\pi\)
−0.609223 + 0.792999i \(0.708519\pi\)
\(464\) −9.92816 −0.460903
\(465\) 0 0
\(466\) 53.5675 2.48147
\(467\) −41.2187 −1.90737 −0.953687 0.300799i \(-0.902747\pi\)
−0.953687 + 0.300799i \(0.902747\pi\)
\(468\) −40.2712 −1.86154
\(469\) 4.54321 0.209786
\(470\) 0 0
\(471\) −0.179620 −0.00827645
\(472\) −40.6246 −1.86990
\(473\) 16.9944 0.781404
\(474\) −0.231953 −0.0106540
\(475\) 0 0
\(476\) −1.64635 −0.0754604
\(477\) −4.40518 −0.201699
\(478\) 35.6993 1.63285
\(479\) −4.54688 −0.207752 −0.103876 0.994590i \(-0.533125\pi\)
−0.103876 + 0.994590i \(0.533125\pi\)
\(480\) 0 0
\(481\) 17.1698 0.782876
\(482\) 13.7667 0.627056
\(483\) −0.00757499 −0.000344674 0
\(484\) 48.2632 2.19378
\(485\) 0 0
\(486\) −0.477953 −0.0216804
\(487\) −9.38327 −0.425197 −0.212598 0.977140i \(-0.568193\pi\)
−0.212598 + 0.977140i \(0.568193\pi\)
\(488\) 28.1591 1.27470
\(489\) 0.000403490 0 1.82465e−5 0
\(490\) 0 0
\(491\) 24.4624 1.10397 0.551987 0.833853i \(-0.313870\pi\)
0.551987 + 0.833853i \(0.313870\pi\)
\(492\) −0.0843733 −0.00380384
\(493\) −4.46200 −0.200958
\(494\) −13.9715 −0.628606
\(495\) 0 0
\(496\) −11.7276 −0.526587
\(497\) −11.0031 −0.493558
\(498\) 0.261341 0.0117110
\(499\) 9.26057 0.414560 0.207280 0.978282i \(-0.433539\pi\)
0.207280 + 0.978282i \(0.433539\pi\)
\(500\) 0 0
\(501\) 0.0502277 0.00224401
\(502\) 9.55045 0.426258
\(503\) −5.76022 −0.256836 −0.128418 0.991720i \(-0.540990\pi\)
−0.128418 + 0.991720i \(0.540990\pi\)
\(504\) 10.2452 0.456359
\(505\) 0 0
\(506\) 11.6715 0.518863
\(507\) −0.0154586 −0.000686539 0
\(508\) −50.3848 −2.23546
\(509\) 10.7526 0.476600 0.238300 0.971192i \(-0.423410\pi\)
0.238300 + 0.971192i \(0.423410\pi\)
\(510\) 0 0
\(511\) 13.6586 0.604220
\(512\) −11.8397 −0.523247
\(513\) −0.0700626 −0.00309334
\(514\) −45.0382 −1.98655
\(515\) 0 0
\(516\) 0.0892191 0.00392765
\(517\) −40.3203 −1.77328
\(518\) −10.3462 −0.454586
\(519\) −0.0512659 −0.00225032
\(520\) 0 0
\(521\) 1.29984 0.0569470 0.0284735 0.999595i \(-0.490935\pi\)
0.0284735 + 0.999595i \(0.490935\pi\)
\(522\) 65.7684 2.87861
\(523\) 4.59165 0.200779 0.100390 0.994948i \(-0.467991\pi\)
0.100390 + 0.994948i \(0.467991\pi\)
\(524\) −22.7792 −0.995114
\(525\) 0 0
\(526\) 59.0455 2.57451
\(527\) −5.27073 −0.229597
\(528\) 0.0400382 0.00174244
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 35.6856 1.54862
\(532\) 5.33586 0.231339
\(533\) 12.4799 0.540563
\(534\) 0.107669 0.00465931
\(535\) 0 0
\(536\) −15.5157 −0.670177
\(537\) −0.0718514 −0.00310062
\(538\) 18.3298 0.790256
\(539\) −4.99434 −0.215121
\(540\) 0 0
\(541\) 34.9259 1.50158 0.750791 0.660540i \(-0.229673\pi\)
0.750791 + 0.660540i \(0.229673\pi\)
\(542\) −45.5865 −1.95811
\(543\) −0.136731 −0.00586769
\(544\) −2.07238 −0.0888524
\(545\) 0 0
\(546\) 0.0686542 0.00293813
\(547\) −27.0941 −1.15846 −0.579229 0.815165i \(-0.696646\pi\)
−0.579229 + 0.815165i \(0.696646\pi\)
\(548\) 25.5231 1.09029
\(549\) −24.7356 −1.05569
\(550\) 0 0
\(551\) 14.4614 0.616077
\(552\) 0.0258697 0.00110109
\(553\) −13.1029 −0.557192
\(554\) −58.2280 −2.47387
\(555\) 0 0
\(556\) −40.6613 −1.72442
\(557\) −23.9676 −1.01554 −0.507770 0.861492i \(-0.669530\pi\)
−0.507770 + 0.861492i \(0.669530\pi\)
\(558\) 77.6889 3.28883
\(559\) −13.1966 −0.558158
\(560\) 0 0
\(561\) 0.0179943 0.000759721 0
\(562\) −6.50571 −0.274427
\(563\) −33.5325 −1.41323 −0.706613 0.707601i \(-0.749778\pi\)
−0.706613 + 0.707601i \(0.749778\pi\)
\(564\) −0.211678 −0.00891325
\(565\) 0 0
\(566\) 28.3016 1.18960
\(567\) −8.99948 −0.377943
\(568\) 37.5773 1.57671
\(569\) −8.02616 −0.336474 −0.168237 0.985747i \(-0.553807\pi\)
−0.168237 + 0.985747i \(0.553807\pi\)
\(570\) 0 0
\(571\) 9.76773 0.408767 0.204383 0.978891i \(-0.434481\pi\)
0.204383 + 0.978891i \(0.434481\pi\)
\(572\) −67.0440 −2.80325
\(573\) 0.179509 0.00749911
\(574\) −7.52014 −0.313885
\(575\) 0 0
\(576\) 36.8959 1.53733
\(577\) −4.38607 −0.182595 −0.0912973 0.995824i \(-0.529101\pi\)
−0.0912973 + 0.995824i \(0.529101\pi\)
\(578\) −39.1996 −1.63049
\(579\) 0.109865 0.00456581
\(580\) 0 0
\(581\) 14.7630 0.612473
\(582\) 0.0693446 0.00287442
\(583\) −7.33379 −0.303735
\(584\) −46.6460 −1.93023
\(585\) 0 0
\(586\) 1.52094 0.0628295
\(587\) 18.7391 0.773445 0.386723 0.922196i \(-0.373607\pi\)
0.386723 + 0.922196i \(0.373607\pi\)
\(588\) −0.0262198 −0.00108129
\(589\) 17.0825 0.703874
\(590\) 0 0
\(591\) 0.0199628 0.000821162 0
\(592\) 4.68539 0.192568
\(593\) 22.0316 0.904731 0.452366 0.891833i \(-0.350580\pi\)
0.452366 + 0.891833i \(0.350580\pi\)
\(594\) −0.530466 −0.0217653
\(595\) 0 0
\(596\) 0.978186 0.0400681
\(597\) 0.147907 0.00605343
\(598\) −9.06327 −0.370625
\(599\) 0.514129 0.0210067 0.0105034 0.999945i \(-0.496657\pi\)
0.0105034 + 0.999945i \(0.496657\pi\)
\(600\) 0 0
\(601\) −37.8191 −1.54267 −0.771337 0.636427i \(-0.780412\pi\)
−0.771337 + 0.636427i \(0.780412\pi\)
\(602\) 7.95205 0.324101
\(603\) 13.6294 0.555030
\(604\) 67.6431 2.75236
\(605\) 0 0
\(606\) −0.0494040 −0.00200690
\(607\) 41.2268 1.67334 0.836672 0.547704i \(-0.184498\pi\)
0.836672 + 0.547704i \(0.184498\pi\)
\(608\) 6.71660 0.272394
\(609\) −0.0710618 −0.00287957
\(610\) 0 0
\(611\) 31.3098 1.26666
\(612\) −4.93896 −0.199646
\(613\) −0.810033 −0.0327169 −0.0163585 0.999866i \(-0.505207\pi\)
−0.0163585 + 0.999866i \(0.505207\pi\)
\(614\) −52.7069 −2.12708
\(615\) 0 0
\(616\) 17.0564 0.687221
\(617\) −21.8594 −0.880025 −0.440012 0.897992i \(-0.645026\pi\)
−0.440012 + 0.897992i \(0.645026\pi\)
\(618\) 0.232431 0.00934974
\(619\) −1.80602 −0.0725901 −0.0362950 0.999341i \(-0.511556\pi\)
−0.0362950 + 0.999341i \(0.511556\pi\)
\(620\) 0 0
\(621\) −0.0454495 −0.00182383
\(622\) 60.2586 2.41615
\(623\) 6.08219 0.243678
\(624\) −0.0310908 −0.00124463
\(625\) 0 0
\(626\) −13.0490 −0.521544
\(627\) −0.0583199 −0.00232907
\(628\) 82.0766 3.27521
\(629\) 2.10575 0.0839616
\(630\) 0 0
\(631\) 42.9474 1.70971 0.854854 0.518869i \(-0.173647\pi\)
0.854854 + 0.518869i \(0.173647\pi\)
\(632\) 44.7483 1.77999
\(633\) 0.0797925 0.00317147
\(634\) 10.1726 0.404007
\(635\) 0 0
\(636\) −0.0385018 −0.00152669
\(637\) 3.87824 0.153661
\(638\) 109.492 4.33483
\(639\) −33.0088 −1.30581
\(640\) 0 0
\(641\) 40.7586 1.60987 0.804933 0.593365i \(-0.202201\pi\)
0.804933 + 0.593365i \(0.202201\pi\)
\(642\) −0.320782 −0.0126603
\(643\) −40.0053 −1.57765 −0.788827 0.614615i \(-0.789311\pi\)
−0.788827 + 0.614615i \(0.789311\pi\)
\(644\) 3.46136 0.136397
\(645\) 0 0
\(646\) −1.71350 −0.0674166
\(647\) −12.2909 −0.483207 −0.241604 0.970375i \(-0.577673\pi\)
−0.241604 + 0.970375i \(0.577673\pi\)
\(648\) 30.7345 1.20737
\(649\) 59.4097 2.33203
\(650\) 0 0
\(651\) −0.0839417 −0.00328993
\(652\) −0.184373 −0.00722061
\(653\) −18.1745 −0.711221 −0.355611 0.934634i \(-0.615727\pi\)
−0.355611 + 0.934634i \(0.615727\pi\)
\(654\) 0.156946 0.00613706
\(655\) 0 0
\(656\) 3.40558 0.132965
\(657\) 40.9750 1.59859
\(658\) −18.8667 −0.735502
\(659\) 4.72077 0.183895 0.0919476 0.995764i \(-0.470691\pi\)
0.0919476 + 0.995764i \(0.470691\pi\)
\(660\) 0 0
\(661\) −32.3257 −1.25732 −0.628662 0.777679i \(-0.716397\pi\)
−0.628662 + 0.777679i \(0.716397\pi\)
\(662\) −25.1568 −0.977746
\(663\) −0.0139731 −0.000542670 0
\(664\) −50.4178 −1.95659
\(665\) 0 0
\(666\) −31.0380 −1.20270
\(667\) 9.38110 0.363238
\(668\) −22.9514 −0.888014
\(669\) 0.115410 0.00446200
\(670\) 0 0
\(671\) −41.1800 −1.58974
\(672\) −0.0330047 −0.00127318
\(673\) 24.3701 0.939398 0.469699 0.882827i \(-0.344363\pi\)
0.469699 + 0.882827i \(0.344363\pi\)
\(674\) 0.970033 0.0373643
\(675\) 0 0
\(676\) 7.06373 0.271682
\(677\) −4.98060 −0.191420 −0.0957101 0.995409i \(-0.530512\pi\)
−0.0957101 + 0.995409i \(0.530512\pi\)
\(678\) −0.101184 −0.00388596
\(679\) 3.91724 0.150330
\(680\) 0 0
\(681\) −0.207883 −0.00796611
\(682\) 129.337 4.95258
\(683\) −46.8124 −1.79122 −0.895612 0.444835i \(-0.853262\pi\)
−0.895612 + 0.444835i \(0.853262\pi\)
\(684\) 16.0073 0.612053
\(685\) 0 0
\(686\) −2.33696 −0.0892254
\(687\) −0.225942 −0.00862022
\(688\) −3.60117 −0.137293
\(689\) 5.69489 0.216958
\(690\) 0 0
\(691\) −33.7972 −1.28571 −0.642853 0.765990i \(-0.722249\pi\)
−0.642853 + 0.765990i \(0.722249\pi\)
\(692\) 23.4258 0.890514
\(693\) −14.9827 −0.569147
\(694\) 9.56957 0.363256
\(695\) 0 0
\(696\) 0.242686 0.00919900
\(697\) 1.53056 0.0579742
\(698\) −11.5248 −0.436218
\(699\) −0.173633 −0.00656741
\(700\) 0 0
\(701\) 3.84137 0.145087 0.0725433 0.997365i \(-0.476888\pi\)
0.0725433 + 0.997365i \(0.476888\pi\)
\(702\) 0.411922 0.0155470
\(703\) −6.82476 −0.257401
\(704\) 61.4247 2.31503
\(705\) 0 0
\(706\) −33.1794 −1.24872
\(707\) −2.79080 −0.104959
\(708\) 0.311896 0.0117218
\(709\) −16.3215 −0.612966 −0.306483 0.951876i \(-0.599152\pi\)
−0.306483 + 0.951876i \(0.599152\pi\)
\(710\) 0 0
\(711\) −39.3080 −1.47416
\(712\) −20.7715 −0.778447
\(713\) 11.0814 0.415003
\(714\) 0.00841993 0.000315108 0
\(715\) 0 0
\(716\) 32.8322 1.22700
\(717\) −0.115716 −0.00432148
\(718\) −57.3039 −2.13856
\(719\) 22.4621 0.837694 0.418847 0.908057i \(-0.362434\pi\)
0.418847 + 0.908057i \(0.362434\pi\)
\(720\) 0 0
\(721\) 13.1299 0.488982
\(722\) −38.8487 −1.44580
\(723\) −0.0446233 −0.00165956
\(724\) 62.4787 2.32200
\(725\) 0 0
\(726\) −0.246832 −0.00916079
\(727\) −22.6321 −0.839378 −0.419689 0.907668i \(-0.637861\pi\)
−0.419689 + 0.907668i \(0.637861\pi\)
\(728\) −13.2447 −0.490883
\(729\) −26.9969 −0.999885
\(730\) 0 0
\(731\) −1.61847 −0.0598612
\(732\) −0.216192 −0.00799067
\(733\) −39.9481 −1.47552 −0.737758 0.675065i \(-0.764116\pi\)
−0.737758 + 0.675065i \(0.764116\pi\)
\(734\) 38.3314 1.41484
\(735\) 0 0
\(736\) 4.35705 0.160603
\(737\) 22.6903 0.835808
\(738\) −22.5600 −0.830445
\(739\) −30.1882 −1.11049 −0.555246 0.831686i \(-0.687376\pi\)
−0.555246 + 0.831686i \(0.687376\pi\)
\(740\) 0 0
\(741\) 0.0452870 0.00166366
\(742\) −3.43164 −0.125979
\(743\) 31.0282 1.13831 0.569156 0.822229i \(-0.307270\pi\)
0.569156 + 0.822229i \(0.307270\pi\)
\(744\) 0.286673 0.0105099
\(745\) 0 0
\(746\) 8.04773 0.294648
\(747\) 44.2882 1.62042
\(748\) −8.22244 −0.300642
\(749\) −18.1208 −0.662120
\(750\) 0 0
\(751\) 39.9950 1.45944 0.729720 0.683746i \(-0.239651\pi\)
0.729720 + 0.683746i \(0.239651\pi\)
\(752\) 8.54400 0.311567
\(753\) −0.0309568 −0.00112813
\(754\) −85.0235 −3.09637
\(755\) 0 0
\(756\) −0.157317 −0.00572158
\(757\) 35.1223 1.27654 0.638270 0.769812i \(-0.279650\pi\)
0.638270 + 0.769812i \(0.279650\pi\)
\(758\) −7.68455 −0.279115
\(759\) −0.0378321 −0.00137322
\(760\) 0 0
\(761\) 5.62354 0.203853 0.101927 0.994792i \(-0.467499\pi\)
0.101927 + 0.994792i \(0.467499\pi\)
\(762\) 0.257682 0.00933485
\(763\) 8.86577 0.320962
\(764\) −82.0261 −2.96760
\(765\) 0 0
\(766\) 42.4949 1.53540
\(767\) −46.1333 −1.66578
\(768\) 0.168213 0.00606988
\(769\) −29.0696 −1.04828 −0.524138 0.851634i \(-0.675612\pi\)
−0.524138 + 0.851634i \(0.675612\pi\)
\(770\) 0 0
\(771\) 0.145986 0.00525758
\(772\) −50.2022 −1.80682
\(773\) 32.3579 1.16383 0.581917 0.813248i \(-0.302303\pi\)
0.581917 + 0.813248i \(0.302303\pi\)
\(774\) 23.8557 0.857475
\(775\) 0 0
\(776\) −13.3779 −0.480240
\(777\) 0.0335361 0.00120310
\(778\) −65.2889 −2.34072
\(779\) −4.96058 −0.177731
\(780\) 0 0
\(781\) −54.9533 −1.96639
\(782\) −1.11154 −0.0397487
\(783\) −0.426367 −0.0152371
\(784\) 1.05832 0.0377970
\(785\) 0 0
\(786\) 0.116500 0.00415540
\(787\) −12.6066 −0.449378 −0.224689 0.974431i \(-0.572137\pi\)
−0.224689 + 0.974431i \(0.572137\pi\)
\(788\) −9.12195 −0.324956
\(789\) −0.191390 −0.00681365
\(790\) 0 0
\(791\) −5.71585 −0.203232
\(792\) 51.1682 1.81818
\(793\) 31.9774 1.13555
\(794\) 48.1187 1.70767
\(795\) 0 0
\(796\) −67.5856 −2.39551
\(797\) 31.7455 1.12448 0.562242 0.826973i \(-0.309939\pi\)
0.562242 + 0.826973i \(0.309939\pi\)
\(798\) −0.0272891 −0.000966025 0
\(799\) 3.83991 0.135846
\(800\) 0 0
\(801\) 18.2462 0.644698
\(802\) 1.04829 0.0370165
\(803\) 68.2155 2.40727
\(804\) 0.119122 0.00420111
\(805\) 0 0
\(806\) −100.434 −3.53764
\(807\) −0.0594142 −0.00209148
\(808\) 9.53099 0.335299
\(809\) −21.5831 −0.758821 −0.379411 0.925228i \(-0.623873\pi\)
−0.379411 + 0.925228i \(0.623873\pi\)
\(810\) 0 0
\(811\) −6.27428 −0.220320 −0.110160 0.993914i \(-0.535136\pi\)
−0.110160 + 0.993914i \(0.535136\pi\)
\(812\) 32.4714 1.13952
\(813\) 0.147764 0.00518230
\(814\) −51.6724 −1.81112
\(815\) 0 0
\(816\) −0.00381305 −0.000133484 0
\(817\) 5.24548 0.183516
\(818\) −14.9589 −0.523027
\(819\) 11.6345 0.406542
\(820\) 0 0
\(821\) −50.7376 −1.77075 −0.885377 0.464874i \(-0.846100\pi\)
−0.885377 + 0.464874i \(0.846100\pi\)
\(822\) −0.130533 −0.00455285
\(823\) 20.6689 0.720473 0.360236 0.932861i \(-0.382696\pi\)
0.360236 + 0.932861i \(0.382696\pi\)
\(824\) −44.8404 −1.56209
\(825\) 0 0
\(826\) 27.7991 0.967253
\(827\) −9.51670 −0.330928 −0.165464 0.986216i \(-0.552912\pi\)
−0.165464 + 0.986216i \(0.552912\pi\)
\(828\) 10.3839 0.360865
\(829\) 43.2509 1.50217 0.751083 0.660208i \(-0.229532\pi\)
0.751083 + 0.660208i \(0.229532\pi\)
\(830\) 0 0
\(831\) 0.188740 0.00654731
\(832\) −47.6980 −1.65363
\(833\) 0.475637 0.0164798
\(834\) 0.207954 0.00720086
\(835\) 0 0
\(836\) 26.6491 0.921677
\(837\) −0.503645 −0.0174085
\(838\) −19.8410 −0.685396
\(839\) 12.6656 0.437264 0.218632 0.975807i \(-0.429841\pi\)
0.218632 + 0.975807i \(0.429841\pi\)
\(840\) 0 0
\(841\) 59.0051 2.03466
\(842\) −59.3389 −2.04495
\(843\) 0.0210876 0.000726294 0
\(844\) −36.4609 −1.25504
\(845\) 0 0
\(846\) −56.5991 −1.94592
\(847\) −13.9434 −0.479101
\(848\) 1.55405 0.0533664
\(849\) −0.0917365 −0.00314839
\(850\) 0 0
\(851\) −4.42722 −0.151763
\(852\) −0.288500 −0.00988385
\(853\) −5.14537 −0.176174 −0.0880871 0.996113i \(-0.528075\pi\)
−0.0880871 + 0.996113i \(0.528075\pi\)
\(854\) −19.2690 −0.659372
\(855\) 0 0
\(856\) 61.8852 2.11519
\(857\) 13.1648 0.449702 0.224851 0.974393i \(-0.427810\pi\)
0.224851 + 0.974393i \(0.427810\pi\)
\(858\) 0.342882 0.0117058
\(859\) −27.8506 −0.950249 −0.475124 0.879919i \(-0.657597\pi\)
−0.475124 + 0.879919i \(0.657597\pi\)
\(860\) 0 0
\(861\) 0.0243757 0.000830723 0
\(862\) 45.1619 1.53822
\(863\) −36.4254 −1.23994 −0.619968 0.784627i \(-0.712854\pi\)
−0.619968 + 0.784627i \(0.712854\pi\)
\(864\) −0.198026 −0.00673698
\(865\) 0 0
\(866\) −83.3628 −2.83278
\(867\) 0.127061 0.00431522
\(868\) 38.3568 1.30192
\(869\) −65.4403 −2.21991
\(870\) 0 0
\(871\) −17.6196 −0.597019
\(872\) −30.2779 −1.02534
\(873\) 11.7515 0.397727
\(874\) 3.60253 0.121857
\(875\) 0 0
\(876\) 0.358125 0.0120999
\(877\) 32.5408 1.09882 0.549412 0.835551i \(-0.314852\pi\)
0.549412 + 0.835551i \(0.314852\pi\)
\(878\) −42.7589 −1.44304
\(879\) −0.00492997 −0.000166284 0
\(880\) 0 0
\(881\) −22.7388 −0.766088 −0.383044 0.923730i \(-0.625124\pi\)
−0.383044 + 0.923730i \(0.625124\pi\)
\(882\) −7.01073 −0.236064
\(883\) 13.1029 0.440946 0.220473 0.975393i \(-0.429240\pi\)
0.220473 + 0.975393i \(0.429240\pi\)
\(884\) 6.38495 0.214749
\(885\) 0 0
\(886\) 9.40707 0.316037
\(887\) 16.4374 0.551914 0.275957 0.961170i \(-0.411005\pi\)
0.275957 + 0.961170i \(0.411005\pi\)
\(888\) −0.114531 −0.00384340
\(889\) 14.5563 0.488204
\(890\) 0 0
\(891\) −44.9464 −1.50576
\(892\) −52.7361 −1.76574
\(893\) −12.4452 −0.416464
\(894\) −0.00500273 −0.000167316 0
\(895\) 0 0
\(896\) 20.0278 0.669083
\(897\) 0.0293776 0.000980891 0
\(898\) 69.6093 2.32289
\(899\) 103.956 3.46713
\(900\) 0 0
\(901\) 0.698436 0.0232683
\(902\) −37.5581 −1.25055
\(903\) −0.0257757 −0.000857762 0
\(904\) 19.5205 0.649241
\(905\) 0 0
\(906\) −0.345947 −0.0114933
\(907\) 49.2654 1.63583 0.817916 0.575338i \(-0.195129\pi\)
0.817916 + 0.575338i \(0.195129\pi\)
\(908\) 94.9915 3.15240
\(909\) −8.37225 −0.277690
\(910\) 0 0
\(911\) 7.82158 0.259140 0.129570 0.991570i \(-0.458640\pi\)
0.129570 + 0.991570i \(0.458640\pi\)
\(912\) 0.0123582 0.000409220 0
\(913\) 73.7314 2.44015
\(914\) 45.6906 1.51131
\(915\) 0 0
\(916\) 103.243 3.41126
\(917\) 6.58099 0.217323
\(918\) 0.0505191 0.00166738
\(919\) 47.8000 1.57678 0.788388 0.615178i \(-0.210916\pi\)
0.788388 + 0.615178i \(0.210916\pi\)
\(920\) 0 0
\(921\) 0.170844 0.00562950
\(922\) 0.448462 0.0147693
\(923\) 42.6728 1.40459
\(924\) −0.130951 −0.00430796
\(925\) 0 0
\(926\) −61.2699 −2.01345
\(927\) 39.3889 1.29370
\(928\) 40.8740 1.34175
\(929\) −22.2564 −0.730210 −0.365105 0.930966i \(-0.618967\pi\)
−0.365105 + 0.930966i \(0.618967\pi\)
\(930\) 0 0
\(931\) −1.54155 −0.0505222
\(932\) 79.3410 2.59890
\(933\) −0.195322 −0.00639455
\(934\) −96.3263 −3.15189
\(935\) 0 0
\(936\) −39.7335 −1.29873
\(937\) 3.52344 0.115106 0.0575529 0.998342i \(-0.481670\pi\)
0.0575529 + 0.998342i \(0.481670\pi\)
\(938\) 10.6173 0.346666
\(939\) 0.0422970 0.00138031
\(940\) 0 0
\(941\) −21.9747 −0.716355 −0.358178 0.933653i \(-0.616602\pi\)
−0.358178 + 0.933653i \(0.616602\pi\)
\(942\) −0.419764 −0.0136766
\(943\) −3.21792 −0.104790
\(944\) −12.5891 −0.409740
\(945\) 0 0
\(946\) 39.7152 1.29125
\(947\) −8.71337 −0.283146 −0.141573 0.989928i \(-0.545216\pi\)
−0.141573 + 0.989928i \(0.545216\pi\)
\(948\) −0.343556 −0.0111582
\(949\) −52.9712 −1.71952
\(950\) 0 0
\(951\) −0.0329736 −0.00106924
\(952\) −1.62437 −0.0526461
\(953\) −22.0600 −0.714594 −0.357297 0.933991i \(-0.616302\pi\)
−0.357297 + 0.933991i \(0.616302\pi\)
\(954\) −10.2947 −0.333304
\(955\) 0 0
\(956\) 52.8758 1.71013
\(957\) −0.354906 −0.0114725
\(958\) −10.6259 −0.343306
\(959\) −7.37371 −0.238109
\(960\) 0 0
\(961\) 91.7979 2.96122
\(962\) 40.1251 1.29368
\(963\) −54.3614 −1.75177
\(964\) 20.3904 0.656732
\(965\) 0 0
\(966\) −0.0177024 −0.000569566 0
\(967\) −14.8633 −0.477973 −0.238986 0.971023i \(-0.576815\pi\)
−0.238986 + 0.971023i \(0.576815\pi\)
\(968\) 47.6187 1.53052
\(969\) 0.00555411 0.000178424 0
\(970\) 0 0
\(971\) −32.4718 −1.04207 −0.521036 0.853535i \(-0.674454\pi\)
−0.521036 + 0.853535i \(0.674454\pi\)
\(972\) −0.707917 −0.0227064
\(973\) 11.7472 0.376598
\(974\) −21.9283 −0.702628
\(975\) 0 0
\(976\) 8.72618 0.279318
\(977\) −17.9922 −0.575622 −0.287811 0.957687i \(-0.592928\pi\)
−0.287811 + 0.957687i \(0.592928\pi\)
\(978\) 0.000942939 0 3.01519e−5 0
\(979\) 30.3765 0.970836
\(980\) 0 0
\(981\) 26.5968 0.849170
\(982\) 57.1677 1.82429
\(983\) −7.91109 −0.252325 −0.126162 0.992010i \(-0.540266\pi\)
−0.126162 + 0.992010i \(0.540266\pi\)
\(984\) −0.0832467 −0.00265381
\(985\) 0 0
\(986\) −10.4275 −0.332079
\(987\) 0.0611545 0.00194657
\(988\) −20.6937 −0.658355
\(989\) 3.40274 0.108201
\(990\) 0 0
\(991\) 23.8214 0.756712 0.378356 0.925660i \(-0.376490\pi\)
0.378356 + 0.925660i \(0.376490\pi\)
\(992\) 48.2824 1.53297
\(993\) 0.0815430 0.00258769
\(994\) −25.7138 −0.815594
\(995\) 0 0
\(996\) 0.387083 0.0122652
\(997\) −3.86775 −0.122493 −0.0612464 0.998123i \(-0.519508\pi\)
−0.0612464 + 0.998123i \(0.519508\pi\)
\(998\) 21.6415 0.685051
\(999\) 0.201215 0.00636616
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 4025.2.a.z.1.13 14
5.4 even 2 4025.2.a.bc.1.2 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4025.2.a.z.1.13 14 1.1 even 1 trivial
4025.2.a.bc.1.2 yes 14 5.4 even 2