Properties

Label 5225.2.a.k.1.6
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.36323\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.36323 q^{2} -2.87321 q^{3} +3.58485 q^{4} -6.79006 q^{6} +0.269449 q^{7} +3.74537 q^{8} +5.25535 q^{9} +O(q^{10})\) \(q+2.36323 q^{2} -2.87321 q^{3} +3.58485 q^{4} -6.79006 q^{6} +0.269449 q^{7} +3.74537 q^{8} +5.25535 q^{9} -1.00000 q^{11} -10.3000 q^{12} +1.50998 q^{13} +0.636771 q^{14} +1.68146 q^{16} -3.65972 q^{17} +12.4196 q^{18} -1.00000 q^{19} -0.774186 q^{21} -2.36323 q^{22} -2.55781 q^{23} -10.7612 q^{24} +3.56844 q^{26} -6.48011 q^{27} +0.965936 q^{28} +0.572530 q^{29} +5.82024 q^{31} -3.51706 q^{32} +2.87321 q^{33} -8.64876 q^{34} +18.8397 q^{36} -7.49463 q^{37} -2.36323 q^{38} -4.33851 q^{39} +11.2021 q^{41} -1.82958 q^{42} +3.61858 q^{43} -3.58485 q^{44} -6.04469 q^{46} -13.6320 q^{47} -4.83120 q^{48} -6.92740 q^{49} +10.5152 q^{51} +5.41307 q^{52} +3.41181 q^{53} -15.3140 q^{54} +1.00919 q^{56} +2.87321 q^{57} +1.35302 q^{58} +1.76356 q^{59} -11.4676 q^{61} +13.7546 q^{62} +1.41605 q^{63} -11.6745 q^{64} +6.79006 q^{66} -3.50098 q^{67} -13.1196 q^{68} +7.34913 q^{69} -10.1638 q^{71} +19.6832 q^{72} +0.482272 q^{73} -17.7115 q^{74} -3.58485 q^{76} -0.269449 q^{77} -10.2529 q^{78} -1.93576 q^{79} +2.85268 q^{81} +26.4731 q^{82} +4.76907 q^{83} -2.77534 q^{84} +8.55154 q^{86} -1.64500 q^{87} -3.74537 q^{88} +0.150946 q^{89} +0.406864 q^{91} -9.16937 q^{92} -16.7228 q^{93} -32.2156 q^{94} +10.1053 q^{96} +1.43635 q^{97} -16.3710 q^{98} -5.25535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.36323 1.67106 0.835528 0.549448i \(-0.185162\pi\)
0.835528 + 0.549448i \(0.185162\pi\)
\(3\) −2.87321 −1.65885 −0.829425 0.558618i \(-0.811332\pi\)
−0.829425 + 0.558618i \(0.811332\pi\)
\(4\) 3.58485 1.79243
\(5\) 0 0
\(6\) −6.79006 −2.77203
\(7\) 0.269449 0.101842 0.0509212 0.998703i \(-0.483784\pi\)
0.0509212 + 0.998703i \(0.483784\pi\)
\(8\) 3.74537 1.32419
\(9\) 5.25535 1.75178
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −10.3000 −2.97337
\(13\) 1.50998 0.418794 0.209397 0.977831i \(-0.432850\pi\)
0.209397 + 0.977831i \(0.432850\pi\)
\(14\) 0.636771 0.170184
\(15\) 0 0
\(16\) 1.68146 0.420366
\(17\) −3.65972 −0.887613 −0.443806 0.896123i \(-0.646372\pi\)
−0.443806 + 0.896123i \(0.646372\pi\)
\(18\) 12.4196 2.92733
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.774186 −0.168941
\(22\) −2.36323 −0.503842
\(23\) −2.55781 −0.533340 −0.266670 0.963788i \(-0.585923\pi\)
−0.266670 + 0.963788i \(0.585923\pi\)
\(24\) −10.7612 −2.19663
\(25\) 0 0
\(26\) 3.56844 0.699828
\(27\) −6.48011 −1.24710
\(28\) 0.965936 0.182545
\(29\) 0.572530 0.106316 0.0531581 0.998586i \(-0.483071\pi\)
0.0531581 + 0.998586i \(0.483071\pi\)
\(30\) 0 0
\(31\) 5.82024 1.04535 0.522673 0.852533i \(-0.324935\pi\)
0.522673 + 0.852533i \(0.324935\pi\)
\(32\) −3.51706 −0.621734
\(33\) 2.87321 0.500162
\(34\) −8.64876 −1.48325
\(35\) 0 0
\(36\) 18.8397 3.13994
\(37\) −7.49463 −1.23211 −0.616054 0.787704i \(-0.711270\pi\)
−0.616054 + 0.787704i \(0.711270\pi\)
\(38\) −2.36323 −0.383366
\(39\) −4.33851 −0.694717
\(40\) 0 0
\(41\) 11.2021 1.74947 0.874735 0.484602i \(-0.161035\pi\)
0.874735 + 0.484602i \(0.161035\pi\)
\(42\) −1.82958 −0.282310
\(43\) 3.61858 0.551829 0.275914 0.961182i \(-0.411019\pi\)
0.275914 + 0.961182i \(0.411019\pi\)
\(44\) −3.58485 −0.540437
\(45\) 0 0
\(46\) −6.04469 −0.891241
\(47\) −13.6320 −1.98844 −0.994219 0.107373i \(-0.965756\pi\)
−0.994219 + 0.107373i \(0.965756\pi\)
\(48\) −4.83120 −0.697324
\(49\) −6.92740 −0.989628
\(50\) 0 0
\(51\) 10.5152 1.47242
\(52\) 5.41307 0.750658
\(53\) 3.41181 0.468648 0.234324 0.972159i \(-0.424712\pi\)
0.234324 + 0.972159i \(0.424712\pi\)
\(54\) −15.3140 −2.08397
\(55\) 0 0
\(56\) 1.00919 0.134858
\(57\) 2.87321 0.380566
\(58\) 1.35302 0.177660
\(59\) 1.76356 0.229596 0.114798 0.993389i \(-0.463378\pi\)
0.114798 + 0.993389i \(0.463378\pi\)
\(60\) 0 0
\(61\) −11.4676 −1.46828 −0.734140 0.678998i \(-0.762415\pi\)
−0.734140 + 0.678998i \(0.762415\pi\)
\(62\) 13.7546 1.74683
\(63\) 1.41605 0.178406
\(64\) −11.6745 −1.45932
\(65\) 0 0
\(66\) 6.79006 0.835799
\(67\) −3.50098 −0.427712 −0.213856 0.976865i \(-0.568602\pi\)
−0.213856 + 0.976865i \(0.568602\pi\)
\(68\) −13.1196 −1.59098
\(69\) 7.34913 0.884732
\(70\) 0 0
\(71\) −10.1638 −1.20622 −0.603109 0.797659i \(-0.706071\pi\)
−0.603109 + 0.797659i \(0.706071\pi\)
\(72\) 19.6832 2.31969
\(73\) 0.482272 0.0564456 0.0282228 0.999602i \(-0.491015\pi\)
0.0282228 + 0.999602i \(0.491015\pi\)
\(74\) −17.7115 −2.05892
\(75\) 0 0
\(76\) −3.58485 −0.411211
\(77\) −0.269449 −0.0307066
\(78\) −10.2529 −1.16091
\(79\) −1.93576 −0.217790 −0.108895 0.994053i \(-0.534731\pi\)
−0.108895 + 0.994053i \(0.534731\pi\)
\(80\) 0 0
\(81\) 2.85268 0.316964
\(82\) 26.4731 2.92346
\(83\) 4.76907 0.523474 0.261737 0.965139i \(-0.415705\pi\)
0.261737 + 0.965139i \(0.415705\pi\)
\(84\) −2.77534 −0.302815
\(85\) 0 0
\(86\) 8.55154 0.922136
\(87\) −1.64500 −0.176363
\(88\) −3.74537 −0.399258
\(89\) 0.150946 0.0160002 0.00800011 0.999968i \(-0.497453\pi\)
0.00800011 + 0.999968i \(0.497453\pi\)
\(90\) 0 0
\(91\) 0.406864 0.0426510
\(92\) −9.16937 −0.955973
\(93\) −16.7228 −1.73407
\(94\) −32.2156 −3.32279
\(95\) 0 0
\(96\) 10.1053 1.03136
\(97\) 1.43635 0.145839 0.0729196 0.997338i \(-0.476768\pi\)
0.0729196 + 0.997338i \(0.476768\pi\)
\(98\) −16.3710 −1.65372
\(99\) −5.25535 −0.528183
\(100\) 0 0
\(101\) −5.48472 −0.545750 −0.272875 0.962049i \(-0.587975\pi\)
−0.272875 + 0.962049i \(0.587975\pi\)
\(102\) 24.8497 2.46049
\(103\) 10.2028 1.00531 0.502656 0.864487i \(-0.332356\pi\)
0.502656 + 0.864487i \(0.332356\pi\)
\(104\) 5.65545 0.554562
\(105\) 0 0
\(106\) 8.06288 0.783136
\(107\) −12.3928 −1.19805 −0.599027 0.800729i \(-0.704446\pi\)
−0.599027 + 0.800729i \(0.704446\pi\)
\(108\) −23.2302 −2.23533
\(109\) −3.96699 −0.379969 −0.189985 0.981787i \(-0.560844\pi\)
−0.189985 + 0.981787i \(0.560844\pi\)
\(110\) 0 0
\(111\) 21.5337 2.04388
\(112\) 0.453069 0.0428110
\(113\) −2.08887 −0.196504 −0.0982520 0.995162i \(-0.531325\pi\)
−0.0982520 + 0.995162i \(0.531325\pi\)
\(114\) 6.79006 0.635948
\(115\) 0 0
\(116\) 2.05244 0.190564
\(117\) 7.93550 0.733637
\(118\) 4.16769 0.383667
\(119\) −0.986110 −0.0903965
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −27.1007 −2.45358
\(123\) −32.1859 −2.90211
\(124\) 20.8647 1.87370
\(125\) 0 0
\(126\) 3.34646 0.298126
\(127\) −11.6605 −1.03470 −0.517350 0.855774i \(-0.673081\pi\)
−0.517350 + 0.855774i \(0.673081\pi\)
\(128\) −20.5555 −1.81687
\(129\) −10.3970 −0.915401
\(130\) 0 0
\(131\) 2.52155 0.220309 0.110155 0.993914i \(-0.464865\pi\)
0.110155 + 0.993914i \(0.464865\pi\)
\(132\) 10.3000 0.896504
\(133\) −0.269449 −0.0233642
\(134\) −8.27361 −0.714731
\(135\) 0 0
\(136\) −13.7070 −1.17537
\(137\) 9.53639 0.814749 0.407374 0.913261i \(-0.366444\pi\)
0.407374 + 0.913261i \(0.366444\pi\)
\(138\) 17.3677 1.47844
\(139\) 17.0241 1.44396 0.721982 0.691911i \(-0.243231\pi\)
0.721982 + 0.691911i \(0.243231\pi\)
\(140\) 0 0
\(141\) 39.1678 3.29852
\(142\) −24.0193 −2.01566
\(143\) −1.50998 −0.126271
\(144\) 8.83668 0.736390
\(145\) 0 0
\(146\) 1.13972 0.0943238
\(147\) 19.9039 1.64165
\(148\) −26.8671 −2.20846
\(149\) 5.02421 0.411600 0.205800 0.978594i \(-0.434020\pi\)
0.205800 + 0.978594i \(0.434020\pi\)
\(150\) 0 0
\(151\) −22.1631 −1.80361 −0.901806 0.432142i \(-0.857758\pi\)
−0.901806 + 0.432142i \(0.857758\pi\)
\(152\) −3.74537 −0.303790
\(153\) −19.2331 −1.55491
\(154\) −0.636771 −0.0513124
\(155\) 0 0
\(156\) −15.5529 −1.24523
\(157\) 13.2234 1.05534 0.527671 0.849449i \(-0.323065\pi\)
0.527671 + 0.849449i \(0.323065\pi\)
\(158\) −4.57464 −0.363939
\(159\) −9.80285 −0.777417
\(160\) 0 0
\(161\) −0.689200 −0.0543166
\(162\) 6.74154 0.529665
\(163\) −17.0772 −1.33759 −0.668795 0.743447i \(-0.733190\pi\)
−0.668795 + 0.743447i \(0.733190\pi\)
\(164\) 40.1578 3.13580
\(165\) 0 0
\(166\) 11.2704 0.874753
\(167\) −8.08784 −0.625856 −0.312928 0.949777i \(-0.601310\pi\)
−0.312928 + 0.949777i \(0.601310\pi\)
\(168\) −2.89961 −0.223710
\(169\) −10.7199 −0.824611
\(170\) 0 0
\(171\) −5.25535 −0.401887
\(172\) 12.9721 0.989112
\(173\) −1.93735 −0.147294 −0.0736472 0.997284i \(-0.523464\pi\)
−0.0736472 + 0.997284i \(0.523464\pi\)
\(174\) −3.88751 −0.294712
\(175\) 0 0
\(176\) −1.68146 −0.126745
\(177\) −5.06708 −0.380865
\(178\) 0.356719 0.0267373
\(179\) 4.24521 0.317302 0.158651 0.987335i \(-0.449286\pi\)
0.158651 + 0.987335i \(0.449286\pi\)
\(180\) 0 0
\(181\) −15.7971 −1.17419 −0.587096 0.809517i \(-0.699729\pi\)
−0.587096 + 0.809517i \(0.699729\pi\)
\(182\) 0.961514 0.0712721
\(183\) 32.9490 2.43566
\(184\) −9.57994 −0.706243
\(185\) 0 0
\(186\) −39.5198 −2.89773
\(187\) 3.65972 0.267625
\(188\) −48.8689 −3.56413
\(189\) −1.74606 −0.127007
\(190\) 0 0
\(191\) −17.6702 −1.27857 −0.639285 0.768969i \(-0.720770\pi\)
−0.639285 + 0.768969i \(0.720770\pi\)
\(192\) 33.5434 2.42079
\(193\) 2.61056 0.187912 0.0939561 0.995576i \(-0.470049\pi\)
0.0939561 + 0.995576i \(0.470049\pi\)
\(194\) 3.39442 0.243705
\(195\) 0 0
\(196\) −24.8337 −1.77384
\(197\) −8.58931 −0.611963 −0.305982 0.952037i \(-0.598985\pi\)
−0.305982 + 0.952037i \(0.598985\pi\)
\(198\) −12.4196 −0.882623
\(199\) −26.1761 −1.85557 −0.927786 0.373113i \(-0.878290\pi\)
−0.927786 + 0.373113i \(0.878290\pi\)
\(200\) 0 0
\(201\) 10.0591 0.709511
\(202\) −12.9617 −0.911979
\(203\) 0.154268 0.0108275
\(204\) 37.6953 2.63920
\(205\) 0 0
\(206\) 24.1115 1.67993
\(207\) −13.4422 −0.934297
\(208\) 2.53898 0.176047
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −17.4084 −1.19845 −0.599223 0.800582i \(-0.704524\pi\)
−0.599223 + 0.800582i \(0.704524\pi\)
\(212\) 12.2308 0.840017
\(213\) 29.2027 2.00093
\(214\) −29.2870 −2.00201
\(215\) 0 0
\(216\) −24.2704 −1.65139
\(217\) 1.56826 0.106460
\(218\) −9.37491 −0.634949
\(219\) −1.38567 −0.0936349
\(220\) 0 0
\(221\) −5.52612 −0.371727
\(222\) 50.8890 3.41544
\(223\) −13.6764 −0.915840 −0.457920 0.888993i \(-0.651405\pi\)
−0.457920 + 0.888993i \(0.651405\pi\)
\(224\) −0.947669 −0.0633188
\(225\) 0 0
\(226\) −4.93647 −0.328369
\(227\) −11.8910 −0.789232 −0.394616 0.918846i \(-0.629122\pi\)
−0.394616 + 0.918846i \(0.629122\pi\)
\(228\) 10.3000 0.682137
\(229\) 9.56459 0.632046 0.316023 0.948752i \(-0.397652\pi\)
0.316023 + 0.948752i \(0.397652\pi\)
\(230\) 0 0
\(231\) 0.774186 0.0509377
\(232\) 2.14434 0.140783
\(233\) −2.75254 −0.180325 −0.0901626 0.995927i \(-0.528739\pi\)
−0.0901626 + 0.995927i \(0.528739\pi\)
\(234\) 18.7534 1.22595
\(235\) 0 0
\(236\) 6.32209 0.411533
\(237\) 5.56185 0.361281
\(238\) −2.33040 −0.151058
\(239\) −12.8426 −0.830716 −0.415358 0.909658i \(-0.636344\pi\)
−0.415358 + 0.909658i \(0.636344\pi\)
\(240\) 0 0
\(241\) 2.50671 0.161471 0.0807356 0.996736i \(-0.474273\pi\)
0.0807356 + 0.996736i \(0.474273\pi\)
\(242\) 2.36323 0.151914
\(243\) 11.2440 0.721301
\(244\) −41.1098 −2.63178
\(245\) 0 0
\(246\) −76.0628 −4.84959
\(247\) −1.50998 −0.0960780
\(248\) 21.7989 1.38423
\(249\) −13.7026 −0.868364
\(250\) 0 0
\(251\) −10.0939 −0.637123 −0.318562 0.947902i \(-0.603200\pi\)
−0.318562 + 0.947902i \(0.603200\pi\)
\(252\) 5.07634 0.319779
\(253\) 2.55781 0.160808
\(254\) −27.5564 −1.72904
\(255\) 0 0
\(256\) −25.2283 −1.57677
\(257\) 17.4000 1.08538 0.542691 0.839932i \(-0.317405\pi\)
0.542691 + 0.839932i \(0.317405\pi\)
\(258\) −24.5704 −1.52969
\(259\) −2.01942 −0.125481
\(260\) 0 0
\(261\) 3.00885 0.186243
\(262\) 5.95901 0.368149
\(263\) −4.18923 −0.258319 −0.129159 0.991624i \(-0.541228\pi\)
−0.129159 + 0.991624i \(0.541228\pi\)
\(264\) 10.7612 0.662309
\(265\) 0 0
\(266\) −0.636771 −0.0390429
\(267\) −0.433699 −0.0265420
\(268\) −12.5505 −0.766643
\(269\) −3.64105 −0.221999 −0.110999 0.993820i \(-0.535405\pi\)
−0.110999 + 0.993820i \(0.535405\pi\)
\(270\) 0 0
\(271\) −22.1472 −1.34535 −0.672674 0.739939i \(-0.734854\pi\)
−0.672674 + 0.739939i \(0.734854\pi\)
\(272\) −6.15368 −0.373122
\(273\) −1.16901 −0.0707516
\(274\) 22.5367 1.36149
\(275\) 0 0
\(276\) 26.3456 1.58582
\(277\) −3.32919 −0.200032 −0.100016 0.994986i \(-0.531889\pi\)
−0.100016 + 0.994986i \(0.531889\pi\)
\(278\) 40.2318 2.41294
\(279\) 30.5874 1.83122
\(280\) 0 0
\(281\) 26.8500 1.60174 0.800868 0.598841i \(-0.204372\pi\)
0.800868 + 0.598841i \(0.204372\pi\)
\(282\) 92.5624 5.51201
\(283\) 10.7579 0.639489 0.319745 0.947504i \(-0.396403\pi\)
0.319745 + 0.947504i \(0.396403\pi\)
\(284\) −36.4356 −2.16206
\(285\) 0 0
\(286\) −3.56844 −0.211006
\(287\) 3.01839 0.178170
\(288\) −18.4834 −1.08914
\(289\) −3.60644 −0.212144
\(290\) 0 0
\(291\) −4.12694 −0.241925
\(292\) 1.72887 0.101175
\(293\) 19.3755 1.13193 0.565964 0.824430i \(-0.308504\pi\)
0.565964 + 0.824430i \(0.308504\pi\)
\(294\) 47.0375 2.74328
\(295\) 0 0
\(296\) −28.0701 −1.63154
\(297\) 6.48011 0.376014
\(298\) 11.8734 0.687806
\(299\) −3.86225 −0.223360
\(300\) 0 0
\(301\) 0.975025 0.0561995
\(302\) −52.3766 −3.01393
\(303\) 15.7588 0.905318
\(304\) −1.68146 −0.0964385
\(305\) 0 0
\(306\) −45.4523 −2.59833
\(307\) −26.8951 −1.53499 −0.767493 0.641058i \(-0.778496\pi\)
−0.767493 + 0.641058i \(0.778496\pi\)
\(308\) −0.965936 −0.0550393
\(309\) −29.3148 −1.66766
\(310\) 0 0
\(311\) 7.36426 0.417589 0.208794 0.977960i \(-0.433046\pi\)
0.208794 + 0.977960i \(0.433046\pi\)
\(312\) −16.2493 −0.919936
\(313\) 17.3778 0.982254 0.491127 0.871088i \(-0.336585\pi\)
0.491127 + 0.871088i \(0.336585\pi\)
\(314\) 31.2499 1.76354
\(315\) 0 0
\(316\) −6.93941 −0.390372
\(317\) 15.4600 0.868321 0.434161 0.900835i \(-0.357045\pi\)
0.434161 + 0.900835i \(0.357045\pi\)
\(318\) −23.1664 −1.29911
\(319\) −0.572530 −0.0320555
\(320\) 0 0
\(321\) 35.6071 1.98739
\(322\) −1.62874 −0.0907661
\(323\) 3.65972 0.203632
\(324\) 10.2264 0.568135
\(325\) 0 0
\(326\) −40.3574 −2.23519
\(327\) 11.3980 0.630312
\(328\) 41.9559 2.31663
\(329\) −3.67315 −0.202507
\(330\) 0 0
\(331\) 35.6299 1.95840 0.979198 0.202905i \(-0.0650382\pi\)
0.979198 + 0.202905i \(0.0650382\pi\)
\(332\) 17.0964 0.938288
\(333\) −39.3869 −2.15839
\(334\) −19.1134 −1.04584
\(335\) 0 0
\(336\) −1.30176 −0.0710170
\(337\) 0.207036 0.0112780 0.00563898 0.999984i \(-0.498205\pi\)
0.00563898 + 0.999984i \(0.498205\pi\)
\(338\) −25.3337 −1.37797
\(339\) 6.00176 0.325971
\(340\) 0 0
\(341\) −5.82024 −0.315184
\(342\) −12.4196 −0.671575
\(343\) −3.75273 −0.202628
\(344\) 13.5529 0.730725
\(345\) 0 0
\(346\) −4.57841 −0.246137
\(347\) −7.01279 −0.376467 −0.188233 0.982124i \(-0.560276\pi\)
−0.188233 + 0.982124i \(0.560276\pi\)
\(348\) −5.89709 −0.316117
\(349\) −18.5946 −0.995348 −0.497674 0.867364i \(-0.665812\pi\)
−0.497674 + 0.867364i \(0.665812\pi\)
\(350\) 0 0
\(351\) −9.78486 −0.522277
\(352\) 3.51706 0.187460
\(353\) 28.2404 1.50308 0.751541 0.659686i \(-0.229311\pi\)
0.751541 + 0.659686i \(0.229311\pi\)
\(354\) −11.9747 −0.636446
\(355\) 0 0
\(356\) 0.541118 0.0286792
\(357\) 2.83330 0.149954
\(358\) 10.0324 0.530229
\(359\) −29.2222 −1.54229 −0.771143 0.636661i \(-0.780315\pi\)
−0.771143 + 0.636661i \(0.780315\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −37.3323 −1.96214
\(363\) −2.87321 −0.150805
\(364\) 1.45855 0.0764487
\(365\) 0 0
\(366\) 77.8660 4.07012
\(367\) 20.0779 1.04806 0.524028 0.851701i \(-0.324429\pi\)
0.524028 + 0.851701i \(0.324429\pi\)
\(368\) −4.30086 −0.224198
\(369\) 58.8709 3.06469
\(370\) 0 0
\(371\) 0.919310 0.0477282
\(372\) −59.9487 −3.10820
\(373\) −32.7276 −1.69457 −0.847284 0.531139i \(-0.821764\pi\)
−0.847284 + 0.531139i \(0.821764\pi\)
\(374\) 8.64876 0.447217
\(375\) 0 0
\(376\) −51.0570 −2.63307
\(377\) 0.864511 0.0445246
\(378\) −4.12635 −0.212236
\(379\) 20.4080 1.04829 0.524144 0.851629i \(-0.324385\pi\)
0.524144 + 0.851629i \(0.324385\pi\)
\(380\) 0 0
\(381\) 33.5030 1.71641
\(382\) −41.7587 −2.13656
\(383\) 32.7540 1.67365 0.836824 0.547471i \(-0.184410\pi\)
0.836824 + 0.547471i \(0.184410\pi\)
\(384\) 59.0603 3.01391
\(385\) 0 0
\(386\) 6.16935 0.314012
\(387\) 19.0169 0.966685
\(388\) 5.14910 0.261406
\(389\) 30.7107 1.55710 0.778548 0.627585i \(-0.215956\pi\)
0.778548 + 0.627585i \(0.215956\pi\)
\(390\) 0 0
\(391\) 9.36087 0.473400
\(392\) −25.9457 −1.31045
\(393\) −7.24496 −0.365460
\(394\) −20.2985 −1.02262
\(395\) 0 0
\(396\) −18.8397 −0.946729
\(397\) −14.5208 −0.728779 −0.364390 0.931247i \(-0.618722\pi\)
−0.364390 + 0.931247i \(0.618722\pi\)
\(398\) −61.8600 −3.10076
\(399\) 0.774186 0.0387578
\(400\) 0 0
\(401\) −9.20455 −0.459653 −0.229827 0.973232i \(-0.573816\pi\)
−0.229827 + 0.973232i \(0.573816\pi\)
\(402\) 23.7719 1.18563
\(403\) 8.78847 0.437785
\(404\) −19.6619 −0.978217
\(405\) 0 0
\(406\) 0.364570 0.0180933
\(407\) 7.49463 0.371495
\(408\) 39.3832 1.94976
\(409\) 23.8193 1.17779 0.588894 0.808210i \(-0.299563\pi\)
0.588894 + 0.808210i \(0.299563\pi\)
\(410\) 0 0
\(411\) −27.4001 −1.35155
\(412\) 36.5755 1.80195
\(413\) 0.475190 0.0233826
\(414\) −31.7670 −1.56126
\(415\) 0 0
\(416\) −5.31070 −0.260379
\(417\) −48.9138 −2.39532
\(418\) 2.36323 0.115589
\(419\) 6.91381 0.337761 0.168881 0.985636i \(-0.445985\pi\)
0.168881 + 0.985636i \(0.445985\pi\)
\(420\) 0 0
\(421\) 4.96199 0.241833 0.120916 0.992663i \(-0.461417\pi\)
0.120916 + 0.992663i \(0.461417\pi\)
\(422\) −41.1401 −2.00267
\(423\) −71.6412 −3.48331
\(424\) 12.7785 0.620578
\(425\) 0 0
\(426\) 69.0126 3.34367
\(427\) −3.08995 −0.149533
\(428\) −44.4262 −2.14742
\(429\) 4.33851 0.209465
\(430\) 0 0
\(431\) −29.4702 −1.41953 −0.709764 0.704439i \(-0.751198\pi\)
−0.709764 + 0.704439i \(0.751198\pi\)
\(432\) −10.8961 −0.524237
\(433\) −15.1549 −0.728299 −0.364149 0.931341i \(-0.618640\pi\)
−0.364149 + 0.931341i \(0.618640\pi\)
\(434\) 3.70616 0.177901
\(435\) 0 0
\(436\) −14.2211 −0.681066
\(437\) 2.55781 0.122357
\(438\) −3.27465 −0.156469
\(439\) −26.0950 −1.24545 −0.622724 0.782442i \(-0.713974\pi\)
−0.622724 + 0.782442i \(0.713974\pi\)
\(440\) 0 0
\(441\) −36.4059 −1.73362
\(442\) −13.0595 −0.621176
\(443\) 39.2407 1.86438 0.932191 0.361966i \(-0.117894\pi\)
0.932191 + 0.361966i \(0.117894\pi\)
\(444\) 77.1950 3.66351
\(445\) 0 0
\(446\) −32.3205 −1.53042
\(447\) −14.4356 −0.682782
\(448\) −3.14570 −0.148620
\(449\) −15.0258 −0.709112 −0.354556 0.935035i \(-0.615368\pi\)
−0.354556 + 0.935035i \(0.615368\pi\)
\(450\) 0 0
\(451\) −11.2021 −0.527485
\(452\) −7.48827 −0.352219
\(453\) 63.6794 2.99192
\(454\) −28.1011 −1.31885
\(455\) 0 0
\(456\) 10.7612 0.503941
\(457\) 38.5344 1.80256 0.901282 0.433233i \(-0.142627\pi\)
0.901282 + 0.433233i \(0.142627\pi\)
\(458\) 22.6033 1.05618
\(459\) 23.7154 1.10694
\(460\) 0 0
\(461\) 33.5770 1.56384 0.781918 0.623381i \(-0.214242\pi\)
0.781918 + 0.623381i \(0.214242\pi\)
\(462\) 1.82958 0.0851197
\(463\) 22.5403 1.04753 0.523767 0.851861i \(-0.324526\pi\)
0.523767 + 0.851861i \(0.324526\pi\)
\(464\) 0.962688 0.0446917
\(465\) 0 0
\(466\) −6.50489 −0.301333
\(467\) −20.5656 −0.951664 −0.475832 0.879536i \(-0.657853\pi\)
−0.475832 + 0.879536i \(0.657853\pi\)
\(468\) 28.4476 1.31499
\(469\) −0.943336 −0.0435592
\(470\) 0 0
\(471\) −37.9937 −1.75066
\(472\) 6.60518 0.304028
\(473\) −3.61858 −0.166383
\(474\) 13.1439 0.603720
\(475\) 0 0
\(476\) −3.53506 −0.162029
\(477\) 17.9303 0.820970
\(478\) −30.3499 −1.38817
\(479\) −12.6051 −0.575942 −0.287971 0.957639i \(-0.592981\pi\)
−0.287971 + 0.957639i \(0.592981\pi\)
\(480\) 0 0
\(481\) −11.3168 −0.516000
\(482\) 5.92393 0.269827
\(483\) 1.98022 0.0901031
\(484\) 3.58485 0.162948
\(485\) 0 0
\(486\) 26.5721 1.20533
\(487\) −3.20539 −0.145250 −0.0726250 0.997359i \(-0.523138\pi\)
−0.0726250 + 0.997359i \(0.523138\pi\)
\(488\) −42.9505 −1.94428
\(489\) 49.0665 2.21886
\(490\) 0 0
\(491\) 44.0578 1.98830 0.994151 0.107999i \(-0.0344444\pi\)
0.994151 + 0.107999i \(0.0344444\pi\)
\(492\) −115.382 −5.20182
\(493\) −2.09530 −0.0943676
\(494\) −3.56844 −0.160552
\(495\) 0 0
\(496\) 9.78651 0.439427
\(497\) −2.73862 −0.122844
\(498\) −32.3823 −1.45109
\(499\) 2.95628 0.132341 0.0661706 0.997808i \(-0.478922\pi\)
0.0661706 + 0.997808i \(0.478922\pi\)
\(500\) 0 0
\(501\) 23.2381 1.03820
\(502\) −23.8543 −1.06467
\(503\) 22.3611 0.997034 0.498517 0.866880i \(-0.333878\pi\)
0.498517 + 0.866880i \(0.333878\pi\)
\(504\) 5.30364 0.236243
\(505\) 0 0
\(506\) 6.04469 0.268719
\(507\) 30.8007 1.36791
\(508\) −41.8011 −1.85462
\(509\) −22.6466 −1.00379 −0.501897 0.864927i \(-0.667364\pi\)
−0.501897 + 0.864927i \(0.667364\pi\)
\(510\) 0 0
\(511\) 0.129948 0.00574855
\(512\) −18.5092 −0.817999
\(513\) 6.48011 0.286104
\(514\) 41.1202 1.81373
\(515\) 0 0
\(516\) −37.2716 −1.64079
\(517\) 13.6320 0.599536
\(518\) −4.77236 −0.209685
\(519\) 5.56643 0.244339
\(520\) 0 0
\(521\) 8.55935 0.374992 0.187496 0.982265i \(-0.439963\pi\)
0.187496 + 0.982265i \(0.439963\pi\)
\(522\) 7.11060 0.311222
\(523\) 24.2895 1.06210 0.531052 0.847339i \(-0.321797\pi\)
0.531052 + 0.847339i \(0.321797\pi\)
\(524\) 9.03940 0.394888
\(525\) 0 0
\(526\) −9.90010 −0.431665
\(527\) −21.3004 −0.927862
\(528\) 4.83120 0.210251
\(529\) −16.4576 −0.715548
\(530\) 0 0
\(531\) 9.26812 0.402202
\(532\) −0.965936 −0.0418787
\(533\) 16.9150 0.732668
\(534\) −1.02493 −0.0443531
\(535\) 0 0
\(536\) −13.1125 −0.566372
\(537\) −12.1974 −0.526357
\(538\) −8.60463 −0.370972
\(539\) 6.92740 0.298384
\(540\) 0 0
\(541\) −16.3282 −0.702003 −0.351001 0.936375i \(-0.614159\pi\)
−0.351001 + 0.936375i \(0.614159\pi\)
\(542\) −52.3390 −2.24815
\(543\) 45.3885 1.94781
\(544\) 12.8715 0.551859
\(545\) 0 0
\(546\) −2.76263 −0.118230
\(547\) 30.9668 1.32404 0.662022 0.749485i \(-0.269699\pi\)
0.662022 + 0.749485i \(0.269699\pi\)
\(548\) 34.1865 1.46038
\(549\) −60.2665 −2.57211
\(550\) 0 0
\(551\) −0.572530 −0.0243906
\(552\) 27.5252 1.17155
\(553\) −0.521589 −0.0221802
\(554\) −7.86764 −0.334264
\(555\) 0 0
\(556\) 61.0289 2.58820
\(557\) 43.5226 1.84411 0.922055 0.387058i \(-0.126509\pi\)
0.922055 + 0.387058i \(0.126509\pi\)
\(558\) 72.2851 3.06007
\(559\) 5.46400 0.231103
\(560\) 0 0
\(561\) −10.5152 −0.443950
\(562\) 63.4527 2.67659
\(563\) 16.5384 0.697010 0.348505 0.937307i \(-0.386689\pi\)
0.348505 + 0.937307i \(0.386689\pi\)
\(564\) 140.411 5.91235
\(565\) 0 0
\(566\) 25.4233 1.06862
\(567\) 0.768653 0.0322804
\(568\) −38.0671 −1.59726
\(569\) 44.4882 1.86504 0.932522 0.361113i \(-0.117603\pi\)
0.932522 + 0.361113i \(0.117603\pi\)
\(570\) 0 0
\(571\) 40.2470 1.68429 0.842143 0.539255i \(-0.181294\pi\)
0.842143 + 0.539255i \(0.181294\pi\)
\(572\) −5.41307 −0.226332
\(573\) 50.7703 2.12096
\(574\) 7.13315 0.297732
\(575\) 0 0
\(576\) −61.3538 −2.55641
\(577\) 7.74126 0.322273 0.161137 0.986932i \(-0.448484\pi\)
0.161137 + 0.986932i \(0.448484\pi\)
\(578\) −8.52285 −0.354504
\(579\) −7.50069 −0.311718
\(580\) 0 0
\(581\) 1.28502 0.0533118
\(582\) −9.75290 −0.404271
\(583\) −3.41181 −0.141303
\(584\) 1.80629 0.0747446
\(585\) 0 0
\(586\) 45.7887 1.89151
\(587\) −14.7960 −0.610697 −0.305349 0.952241i \(-0.598773\pi\)
−0.305349 + 0.952241i \(0.598773\pi\)
\(588\) 71.3525 2.94253
\(589\) −5.82024 −0.239819
\(590\) 0 0
\(591\) 24.6789 1.01516
\(592\) −12.6019 −0.517936
\(593\) 27.4949 1.12908 0.564540 0.825406i \(-0.309054\pi\)
0.564540 + 0.825406i \(0.309054\pi\)
\(594\) 15.3140 0.628341
\(595\) 0 0
\(596\) 18.0111 0.737762
\(597\) 75.2094 3.07812
\(598\) −9.12739 −0.373247
\(599\) 22.3573 0.913496 0.456748 0.889596i \(-0.349014\pi\)
0.456748 + 0.889596i \(0.349014\pi\)
\(600\) 0 0
\(601\) 4.91993 0.200688 0.100344 0.994953i \(-0.468006\pi\)
0.100344 + 0.994953i \(0.468006\pi\)
\(602\) 2.30421 0.0939125
\(603\) −18.3989 −0.749260
\(604\) −79.4516 −3.23284
\(605\) 0 0
\(606\) 37.2416 1.51284
\(607\) −25.5414 −1.03669 −0.518346 0.855171i \(-0.673452\pi\)
−0.518346 + 0.855171i \(0.673452\pi\)
\(608\) 3.51706 0.142636
\(609\) −0.443245 −0.0179612
\(610\) 0 0
\(611\) −20.5842 −0.832746
\(612\) −68.9479 −2.78705
\(613\) 34.4326 1.39072 0.695359 0.718663i \(-0.255245\pi\)
0.695359 + 0.718663i \(0.255245\pi\)
\(614\) −63.5593 −2.56505
\(615\) 0 0
\(616\) −1.00919 −0.0406613
\(617\) 20.3399 0.818855 0.409427 0.912343i \(-0.365729\pi\)
0.409427 + 0.912343i \(0.365729\pi\)
\(618\) −69.2776 −2.78675
\(619\) 36.8315 1.48038 0.740192 0.672395i \(-0.234734\pi\)
0.740192 + 0.672395i \(0.234734\pi\)
\(620\) 0 0
\(621\) 16.5749 0.665128
\(622\) 17.4034 0.697814
\(623\) 0.0406723 0.00162950
\(624\) −7.29503 −0.292035
\(625\) 0 0
\(626\) 41.0678 1.64140
\(627\) −2.87321 −0.114745
\(628\) 47.4039 1.89162
\(629\) 27.4282 1.09364
\(630\) 0 0
\(631\) −29.8615 −1.18877 −0.594384 0.804181i \(-0.702604\pi\)
−0.594384 + 0.804181i \(0.702604\pi\)
\(632\) −7.25013 −0.288395
\(633\) 50.0182 1.98804
\(634\) 36.5356 1.45101
\(635\) 0 0
\(636\) −35.1418 −1.39346
\(637\) −10.4603 −0.414450
\(638\) −1.35302 −0.0535666
\(639\) −53.4142 −2.11303
\(640\) 0 0
\(641\) 8.93034 0.352727 0.176364 0.984325i \(-0.443567\pi\)
0.176364 + 0.984325i \(0.443567\pi\)
\(642\) 84.1477 3.32104
\(643\) −35.6606 −1.40632 −0.703158 0.711034i \(-0.748227\pi\)
−0.703158 + 0.711034i \(0.748227\pi\)
\(644\) −2.47068 −0.0973585
\(645\) 0 0
\(646\) 8.64876 0.340281
\(647\) −44.1953 −1.73750 −0.868749 0.495253i \(-0.835075\pi\)
−0.868749 + 0.495253i \(0.835075\pi\)
\(648\) 10.6843 0.419721
\(649\) −1.76356 −0.0692257
\(650\) 0 0
\(651\) −4.50594 −0.176602
\(652\) −61.2193 −2.39753
\(653\) −34.4397 −1.34773 −0.673864 0.738855i \(-0.735367\pi\)
−0.673864 + 0.738855i \(0.735367\pi\)
\(654\) 26.9361 1.05329
\(655\) 0 0
\(656\) 18.8359 0.735417
\(657\) 2.53451 0.0988806
\(658\) −8.68048 −0.338401
\(659\) 16.4152 0.639445 0.319722 0.947511i \(-0.396410\pi\)
0.319722 + 0.947511i \(0.396410\pi\)
\(660\) 0 0
\(661\) 16.6165 0.646307 0.323153 0.946347i \(-0.395257\pi\)
0.323153 + 0.946347i \(0.395257\pi\)
\(662\) 84.2016 3.27259
\(663\) 15.8777 0.616640
\(664\) 17.8619 0.693178
\(665\) 0 0
\(666\) −93.0803 −3.60679
\(667\) −1.46442 −0.0567027
\(668\) −28.9937 −1.12180
\(669\) 39.2952 1.51924
\(670\) 0 0
\(671\) 11.4676 0.442703
\(672\) 2.72286 0.105036
\(673\) −24.7365 −0.953522 −0.476761 0.879033i \(-0.658189\pi\)
−0.476761 + 0.879033i \(0.658189\pi\)
\(674\) 0.489273 0.0188461
\(675\) 0 0
\(676\) −38.4294 −1.47806
\(677\) 2.57708 0.0990452 0.0495226 0.998773i \(-0.484230\pi\)
0.0495226 + 0.998773i \(0.484230\pi\)
\(678\) 14.1835 0.544715
\(679\) 0.387024 0.0148526
\(680\) 0 0
\(681\) 34.1653 1.30922
\(682\) −13.7546 −0.526689
\(683\) 23.7289 0.907961 0.453980 0.891012i \(-0.350004\pi\)
0.453980 + 0.891012i \(0.350004\pi\)
\(684\) −18.8397 −0.720353
\(685\) 0 0
\(686\) −8.86856 −0.338603
\(687\) −27.4811 −1.04847
\(688\) 6.08451 0.231970
\(689\) 5.15177 0.196267
\(690\) 0 0
\(691\) 49.0740 1.86686 0.933432 0.358755i \(-0.116799\pi\)
0.933432 + 0.358755i \(0.116799\pi\)
\(692\) −6.94513 −0.264014
\(693\) −1.41605 −0.0537914
\(694\) −16.5728 −0.629096
\(695\) 0 0
\(696\) −6.16114 −0.233537
\(697\) −40.9965 −1.55285
\(698\) −43.9434 −1.66328
\(699\) 7.90864 0.299132
\(700\) 0 0
\(701\) −38.2763 −1.44568 −0.722838 0.691017i \(-0.757163\pi\)
−0.722838 + 0.691017i \(0.757163\pi\)
\(702\) −23.1239 −0.872754
\(703\) 7.49463 0.282665
\(704\) 11.6745 0.440001
\(705\) 0 0
\(706\) 66.7384 2.51173
\(707\) −1.47785 −0.0555805
\(708\) −18.1647 −0.682672
\(709\) −35.1393 −1.31969 −0.659843 0.751404i \(-0.729377\pi\)
−0.659843 + 0.751404i \(0.729377\pi\)
\(710\) 0 0
\(711\) −10.1731 −0.381521
\(712\) 0.565348 0.0211873
\(713\) −14.8871 −0.557525
\(714\) 6.69575 0.250582
\(715\) 0 0
\(716\) 15.2185 0.568740
\(717\) 36.8994 1.37803
\(718\) −69.0587 −2.57725
\(719\) 50.4256 1.88056 0.940279 0.340405i \(-0.110564\pi\)
0.940279 + 0.340405i \(0.110564\pi\)
\(720\) 0 0
\(721\) 2.74914 0.102383
\(722\) 2.36323 0.0879503
\(723\) −7.20231 −0.267857
\(724\) −56.6304 −2.10465
\(725\) 0 0
\(726\) −6.79006 −0.252003
\(727\) 29.9437 1.11055 0.555275 0.831667i \(-0.312613\pi\)
0.555275 + 0.831667i \(0.312613\pi\)
\(728\) 1.52386 0.0564779
\(729\) −40.8644 −1.51350
\(730\) 0 0
\(731\) −13.2430 −0.489810
\(732\) 118.117 4.36574
\(733\) −10.5029 −0.387934 −0.193967 0.981008i \(-0.562135\pi\)
−0.193967 + 0.981008i \(0.562135\pi\)
\(734\) 47.4486 1.75136
\(735\) 0 0
\(736\) 8.99597 0.331596
\(737\) 3.50098 0.128960
\(738\) 139.125 5.12127
\(739\) −26.3721 −0.970114 −0.485057 0.874482i \(-0.661201\pi\)
−0.485057 + 0.874482i \(0.661201\pi\)
\(740\) 0 0
\(741\) 4.33851 0.159379
\(742\) 2.17254 0.0797564
\(743\) −19.2435 −0.705975 −0.352988 0.935628i \(-0.614834\pi\)
−0.352988 + 0.935628i \(0.614834\pi\)
\(744\) −62.6330 −2.29624
\(745\) 0 0
\(746\) −77.3427 −2.83172
\(747\) 25.0632 0.917013
\(748\) 13.1196 0.479699
\(749\) −3.33922 −0.122013
\(750\) 0 0
\(751\) 24.9991 0.912229 0.456114 0.889921i \(-0.349241\pi\)
0.456114 + 0.889921i \(0.349241\pi\)
\(752\) −22.9218 −0.835871
\(753\) 29.0020 1.05689
\(754\) 2.04304 0.0744031
\(755\) 0 0
\(756\) −6.25938 −0.227651
\(757\) −8.23644 −0.299359 −0.149679 0.988735i \(-0.547824\pi\)
−0.149679 + 0.988735i \(0.547824\pi\)
\(758\) 48.2288 1.75175
\(759\) −7.34913 −0.266757
\(760\) 0 0
\(761\) 9.52256 0.345192 0.172596 0.984993i \(-0.444784\pi\)
0.172596 + 0.984993i \(0.444784\pi\)
\(762\) 79.1753 2.86822
\(763\) −1.06890 −0.0386969
\(764\) −63.3451 −2.29174
\(765\) 0 0
\(766\) 77.4051 2.79676
\(767\) 2.66294 0.0961533
\(768\) 72.4862 2.61562
\(769\) −20.5017 −0.739309 −0.369654 0.929169i \(-0.620524\pi\)
−0.369654 + 0.929169i \(0.620524\pi\)
\(770\) 0 0
\(771\) −49.9939 −1.80049
\(772\) 9.35847 0.336819
\(773\) −14.0912 −0.506824 −0.253412 0.967359i \(-0.581553\pi\)
−0.253412 + 0.967359i \(0.581553\pi\)
\(774\) 44.9414 1.61538
\(775\) 0 0
\(776\) 5.37966 0.193119
\(777\) 5.80223 0.208154
\(778\) 72.5765 2.60200
\(779\) −11.2021 −0.401356
\(780\) 0 0
\(781\) 10.1638 0.363688
\(782\) 22.1219 0.791077
\(783\) −3.71006 −0.132587
\(784\) −11.6482 −0.416006
\(785\) 0 0
\(786\) −17.1215 −0.610704
\(787\) −11.7044 −0.417218 −0.208609 0.977999i \(-0.566894\pi\)
−0.208609 + 0.977999i \(0.566894\pi\)
\(788\) −30.7914 −1.09690
\(789\) 12.0365 0.428512
\(790\) 0 0
\(791\) −0.562844 −0.0200124
\(792\) −19.6832 −0.699414
\(793\) −17.3159 −0.614907
\(794\) −34.3160 −1.21783
\(795\) 0 0
\(796\) −93.8373 −3.32598
\(797\) −10.7297 −0.380066 −0.190033 0.981778i \(-0.560859\pi\)
−0.190033 + 0.981778i \(0.560859\pi\)
\(798\) 1.82958 0.0647664
\(799\) 49.8895 1.76496
\(800\) 0 0
\(801\) 0.793273 0.0280289
\(802\) −21.7525 −0.768106
\(803\) −0.482272 −0.0170190
\(804\) 36.0602 1.27175
\(805\) 0 0
\(806\) 20.7692 0.731562
\(807\) 10.4615 0.368262
\(808\) −20.5423 −0.722676
\(809\) −29.9973 −1.05465 −0.527324 0.849664i \(-0.676805\pi\)
−0.527324 + 0.849664i \(0.676805\pi\)
\(810\) 0 0
\(811\) −34.0557 −1.19586 −0.597929 0.801549i \(-0.704009\pi\)
−0.597929 + 0.801549i \(0.704009\pi\)
\(812\) 0.553028 0.0194075
\(813\) 63.6337 2.23173
\(814\) 17.7115 0.620788
\(815\) 0 0
\(816\) 17.6808 0.618953
\(817\) −3.61858 −0.126598
\(818\) 56.2905 1.96815
\(819\) 2.13822 0.0747153
\(820\) 0 0
\(821\) −16.9488 −0.591518 −0.295759 0.955263i \(-0.595573\pi\)
−0.295759 + 0.955263i \(0.595573\pi\)
\(822\) −64.7527 −2.25851
\(823\) −46.6538 −1.62625 −0.813124 0.582090i \(-0.802235\pi\)
−0.813124 + 0.582090i \(0.802235\pi\)
\(824\) 38.2132 1.33122
\(825\) 0 0
\(826\) 1.12298 0.0390735
\(827\) −10.0896 −0.350849 −0.175425 0.984493i \(-0.556130\pi\)
−0.175425 + 0.984493i \(0.556130\pi\)
\(828\) −48.1883 −1.67466
\(829\) −4.61277 −0.160208 −0.0801040 0.996787i \(-0.525525\pi\)
−0.0801040 + 0.996787i \(0.525525\pi\)
\(830\) 0 0
\(831\) 9.56548 0.331823
\(832\) −17.6284 −0.611154
\(833\) 25.3523 0.878407
\(834\) −115.595 −4.00271
\(835\) 0 0
\(836\) 3.58485 0.123985
\(837\) −37.7158 −1.30365
\(838\) 16.3389 0.564418
\(839\) −36.5758 −1.26274 −0.631368 0.775484i \(-0.717506\pi\)
−0.631368 + 0.775484i \(0.717506\pi\)
\(840\) 0 0
\(841\) −28.6722 −0.988697
\(842\) 11.7263 0.404116
\(843\) −77.1458 −2.65704
\(844\) −62.4067 −2.14813
\(845\) 0 0
\(846\) −169.305 −5.82081
\(847\) 0.269449 0.00925839
\(848\) 5.73683 0.197003
\(849\) −30.9097 −1.06082
\(850\) 0 0
\(851\) 19.1698 0.657133
\(852\) 104.687 3.58653
\(853\) 24.3932 0.835206 0.417603 0.908630i \(-0.362870\pi\)
0.417603 + 0.908630i \(0.362870\pi\)
\(854\) −7.30226 −0.249878
\(855\) 0 0
\(856\) −46.4155 −1.58645
\(857\) −31.4495 −1.07429 −0.537147 0.843489i \(-0.680498\pi\)
−0.537147 + 0.843489i \(0.680498\pi\)
\(858\) 10.2529 0.350028
\(859\) 1.06992 0.0365051 0.0182526 0.999833i \(-0.494190\pi\)
0.0182526 + 0.999833i \(0.494190\pi\)
\(860\) 0 0
\(861\) −8.67248 −0.295557
\(862\) −69.6448 −2.37211
\(863\) −35.8665 −1.22091 −0.610455 0.792051i \(-0.709014\pi\)
−0.610455 + 0.792051i \(0.709014\pi\)
\(864\) 22.7909 0.775363
\(865\) 0 0
\(866\) −35.8145 −1.21703
\(867\) 10.3621 0.351915
\(868\) 5.62198 0.190822
\(869\) 1.93576 0.0656661
\(870\) 0 0
\(871\) −5.28642 −0.179123
\(872\) −14.8579 −0.503150
\(873\) 7.54853 0.255479
\(874\) 6.04469 0.204465
\(875\) 0 0
\(876\) −4.96742 −0.167834
\(877\) −18.2115 −0.614959 −0.307479 0.951555i \(-0.599486\pi\)
−0.307479 + 0.951555i \(0.599486\pi\)
\(878\) −61.6685 −2.08121
\(879\) −55.6699 −1.87770
\(880\) 0 0
\(881\) 21.4148 0.721482 0.360741 0.932666i \(-0.382524\pi\)
0.360741 + 0.932666i \(0.382524\pi\)
\(882\) −86.0355 −2.89697
\(883\) 26.6128 0.895593 0.447797 0.894135i \(-0.352209\pi\)
0.447797 + 0.894135i \(0.352209\pi\)
\(884\) −19.8103 −0.666293
\(885\) 0 0
\(886\) 92.7348 3.11549
\(887\) −11.9257 −0.400424 −0.200212 0.979753i \(-0.564163\pi\)
−0.200212 + 0.979753i \(0.564163\pi\)
\(888\) 80.6515 2.70649
\(889\) −3.14191 −0.105376
\(890\) 0 0
\(891\) −2.85268 −0.0955684
\(892\) −49.0279 −1.64158
\(893\) 13.6320 0.456179
\(894\) −34.1147 −1.14097
\(895\) 0 0
\(896\) −5.53867 −0.185034
\(897\) 11.0971 0.370520
\(898\) −35.5094 −1.18497
\(899\) 3.33226 0.111137
\(900\) 0 0
\(901\) −12.4863 −0.415978
\(902\) −26.4731 −0.881457
\(903\) −2.80145 −0.0932266
\(904\) −7.82357 −0.260208
\(905\) 0 0
\(906\) 150.489 4.99967
\(907\) −34.7821 −1.15492 −0.577460 0.816419i \(-0.695956\pi\)
−0.577460 + 0.816419i \(0.695956\pi\)
\(908\) −42.6274 −1.41464
\(909\) −28.8241 −0.956037
\(910\) 0 0
\(911\) 2.18025 0.0722351 0.0361175 0.999348i \(-0.488501\pi\)
0.0361175 + 0.999348i \(0.488501\pi\)
\(912\) 4.83120 0.159977
\(913\) −4.76907 −0.157833
\(914\) 91.0657 3.01218
\(915\) 0 0
\(916\) 34.2876 1.13290
\(917\) 0.679431 0.0224368
\(918\) 56.0449 1.84976
\(919\) 34.9180 1.15184 0.575920 0.817506i \(-0.304644\pi\)
0.575920 + 0.817506i \(0.304644\pi\)
\(920\) 0 0
\(921\) 77.2754 2.54631
\(922\) 79.3501 2.61326
\(923\) −15.3471 −0.505157
\(924\) 2.77534 0.0913020
\(925\) 0 0
\(926\) 53.2678 1.75049
\(927\) 53.6193 1.76109
\(928\) −2.01362 −0.0661004
\(929\) −26.9768 −0.885080 −0.442540 0.896749i \(-0.645922\pi\)
−0.442540 + 0.896749i \(0.645922\pi\)
\(930\) 0 0
\(931\) 6.92740 0.227036
\(932\) −9.86746 −0.323219
\(933\) −21.1591 −0.692717
\(934\) −48.6013 −1.59028
\(935\) 0 0
\(936\) 29.7214 0.971474
\(937\) 19.3062 0.630707 0.315354 0.948974i \(-0.397877\pi\)
0.315354 + 0.948974i \(0.397877\pi\)
\(938\) −2.22932 −0.0727899
\(939\) −49.9302 −1.62941
\(940\) 0 0
\(941\) −0.207907 −0.00677756 −0.00338878 0.999994i \(-0.501079\pi\)
−0.00338878 + 0.999994i \(0.501079\pi\)
\(942\) −89.7877 −2.92544
\(943\) −28.6528 −0.933063
\(944\) 2.96536 0.0965141
\(945\) 0 0
\(946\) −8.55154 −0.278035
\(947\) −51.9687 −1.68876 −0.844378 0.535748i \(-0.820030\pi\)
−0.844378 + 0.535748i \(0.820030\pi\)
\(948\) 19.9384 0.647569
\(949\) 0.728222 0.0236391
\(950\) 0 0
\(951\) −44.4199 −1.44042
\(952\) −3.69335 −0.119702
\(953\) 16.0553 0.520081 0.260040 0.965598i \(-0.416264\pi\)
0.260040 + 0.965598i \(0.416264\pi\)
\(954\) 42.3733 1.37189
\(955\) 0 0
\(956\) −46.0386 −1.48900
\(957\) 1.64500 0.0531753
\(958\) −29.7888 −0.962431
\(959\) 2.56957 0.0829759
\(960\) 0 0
\(961\) 2.87517 0.0927476
\(962\) −26.7441 −0.862265
\(963\) −65.1284 −2.09873
\(964\) 8.98618 0.289425
\(965\) 0 0
\(966\) 4.67971 0.150567
\(967\) 10.4378 0.335656 0.167828 0.985816i \(-0.446325\pi\)
0.167828 + 0.985816i \(0.446325\pi\)
\(968\) 3.74537 0.120381
\(969\) −10.5152 −0.337796
\(970\) 0 0
\(971\) −55.0457 −1.76650 −0.883250 0.468903i \(-0.844649\pi\)
−0.883250 + 0.468903i \(0.844649\pi\)
\(972\) 40.3080 1.29288
\(973\) 4.58713 0.147057
\(974\) −7.57507 −0.242721
\(975\) 0 0
\(976\) −19.2824 −0.617215
\(977\) 55.8927 1.78817 0.894083 0.447902i \(-0.147828\pi\)
0.894083 + 0.447902i \(0.147828\pi\)
\(978\) 115.955 3.70784
\(979\) −0.150946 −0.00482425
\(980\) 0 0
\(981\) −20.8480 −0.665624
\(982\) 104.119 3.32256
\(983\) −35.4032 −1.12919 −0.564593 0.825370i \(-0.690967\pi\)
−0.564593 + 0.825370i \(0.690967\pi\)
\(984\) −120.548 −3.84294
\(985\) 0 0
\(986\) −4.95168 −0.157693
\(987\) 10.5537 0.335929
\(988\) −5.41307 −0.172213
\(989\) −9.25565 −0.294312
\(990\) 0 0
\(991\) −53.6641 −1.70470 −0.852349 0.522974i \(-0.824823\pi\)
−0.852349 + 0.522974i \(0.824823\pi\)
\(992\) −20.4701 −0.649927
\(993\) −102.372 −3.24869
\(994\) −6.47199 −0.205279
\(995\) 0 0
\(996\) −49.1217 −1.55648
\(997\) 46.1890 1.46282 0.731410 0.681938i \(-0.238863\pi\)
0.731410 + 0.681938i \(0.238863\pi\)
\(998\) 6.98636 0.221149
\(999\) 48.5660 1.53656
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 5225.2.a.k.1.6 6
5.4 even 2 1045.2.a.g.1.1 6
15.14 odd 2 9405.2.a.w.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.1 6 5.4 even 2
5225.2.a.k.1.6 6 1.1 even 1 trivial
9405.2.a.w.1.6 6 15.14 odd 2