Properties

Label 5225.2.a.p.1.5
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.682920\) 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)
 
\(f(q)\) \(=\) \(q-0.682920 q^{2} -2.48419 q^{3} -1.53362 q^{4} +1.69650 q^{6} -0.856948 q^{7} +2.41318 q^{8} +3.17118 q^{9} +O(q^{10})\) \(q-0.682920 q^{2} -2.48419 q^{3} -1.53362 q^{4} +1.69650 q^{6} -0.856948 q^{7} +2.41318 q^{8} +3.17118 q^{9} +1.00000 q^{11} +3.80980 q^{12} -5.50630 q^{13} +0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} -2.16566 q^{18} +1.00000 q^{19} +2.12882 q^{21} -0.682920 q^{22} -6.75237 q^{23} -5.99479 q^{24} +3.76036 q^{26} -0.425246 q^{27} +1.31423 q^{28} -0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.48419 q^{33} -2.39452 q^{34} -4.86339 q^{36} -0.420904 q^{37} -0.682920 q^{38} +13.6787 q^{39} +8.25141 q^{41} -1.45381 q^{42} +1.53320 q^{43} -1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -3.52563 q^{48} -6.26564 q^{49} -8.71030 q^{51} +8.44457 q^{52} +4.79753 q^{53} +0.290409 q^{54} -2.06797 q^{56} -2.48419 q^{57} +0.324169 q^{58} +1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} -2.71754 q^{63} +1.11946 q^{64} +1.69650 q^{66} +5.12424 q^{67} -5.37733 q^{68} +16.7742 q^{69} -12.1937 q^{71} +7.65263 q^{72} +12.7638 q^{73} +0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} -9.34144 q^{78} +10.8691 q^{79} -8.45715 q^{81} -5.63505 q^{82} -2.94669 q^{83} -3.26480 q^{84} -1.04705 q^{86} +1.17919 q^{87} +2.41318 q^{88} +3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +12.9114 q^{93} +1.69944 q^{94} +14.3973 q^{96} +8.88210 q^{97} +4.27893 q^{98} +3.17118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} - 3 q^{3} + 9 q^{4} - 13 q^{7} - 9 q^{8} + 12 q^{9} + 9 q^{11} + 5 q^{12} - 5 q^{13} - 2 q^{14} + q^{16} - 13 q^{17} - q^{18} + 9 q^{19} + q^{21} - 3 q^{22} - 8 q^{23} - 7 q^{24} + 8 q^{26} - 6 q^{27} - 10 q^{28} - 3 q^{29} - 9 q^{31} - 6 q^{32} - 3 q^{33} - 2 q^{34} - 15 q^{36} + 7 q^{37} - 3 q^{38} - 2 q^{39} + 9 q^{41} + 9 q^{42} - 23 q^{43} + 9 q^{44} - 32 q^{46} - 20 q^{47} + 18 q^{48} - 4 q^{49} + 8 q^{51} - 9 q^{52} + 5 q^{53} + 9 q^{54} + 4 q^{56} - 3 q^{57} - 3 q^{58} + 19 q^{59} + q^{61} - 18 q^{62} - 24 q^{63} + 23 q^{64} + 10 q^{67} - 9 q^{68} - 28 q^{69} - 24 q^{72} - 12 q^{73} + 5 q^{74} + 9 q^{76} - 13 q^{77} - 19 q^{78} - 21 q^{79} - 3 q^{81} + 14 q^{82} - 47 q^{83} - 11 q^{84} + 12 q^{86} + 2 q^{87} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 19 q^{92} + 2 q^{93} + 26 q^{94} - 55 q^{96} + 32 q^{97} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.682920 −0.482898 −0.241449 0.970414i \(-0.577623\pi\)
−0.241449 + 0.970414i \(0.577623\pi\)
\(3\) −2.48419 −1.43425 −0.717123 0.696947i \(-0.754541\pi\)
−0.717123 + 0.696947i \(0.754541\pi\)
\(4\) −1.53362 −0.766810
\(5\) 0 0
\(6\) 1.69650 0.692594
\(7\) −0.856948 −0.323896 −0.161948 0.986799i \(-0.551778\pi\)
−0.161948 + 0.986799i \(0.551778\pi\)
\(8\) 2.41318 0.853188
\(9\) 3.17118 1.05706
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 3.80980 1.09979
\(13\) −5.50630 −1.52717 −0.763586 0.645706i \(-0.776563\pi\)
−0.763586 + 0.645706i \(0.776563\pi\)
\(14\) 0.585228 0.156409
\(15\) 0 0
\(16\) 1.41923 0.354807
\(17\) 3.50630 0.850402 0.425201 0.905099i \(-0.360203\pi\)
0.425201 + 0.905099i \(0.360203\pi\)
\(18\) −2.16566 −0.510452
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.12882 0.464546
\(22\) −0.682920 −0.145599
\(23\) −6.75237 −1.40797 −0.703984 0.710216i \(-0.748597\pi\)
−0.703984 + 0.710216i \(0.748597\pi\)
\(24\) −5.99479 −1.22368
\(25\) 0 0
\(26\) 3.76036 0.737468
\(27\) −0.425246 −0.0818387
\(28\) 1.31423 0.248367
\(29\) −0.474680 −0.0881458 −0.0440729 0.999028i \(-0.514033\pi\)
−0.0440729 + 0.999028i \(0.514033\pi\)
\(30\) 0 0
\(31\) −5.19745 −0.933489 −0.466745 0.884392i \(-0.654573\pi\)
−0.466745 + 0.884392i \(0.654573\pi\)
\(32\) −5.79558 −1.02452
\(33\) −2.48419 −0.432441
\(34\) −2.39452 −0.410657
\(35\) 0 0
\(36\) −4.86339 −0.810564
\(37\) −0.420904 −0.0691961 −0.0345981 0.999401i \(-0.511015\pi\)
−0.0345981 + 0.999401i \(0.511015\pi\)
\(38\) −0.682920 −0.110784
\(39\) 13.6787 2.19034
\(40\) 0 0
\(41\) 8.25141 1.28865 0.644327 0.764750i \(-0.277138\pi\)
0.644327 + 0.764750i \(0.277138\pi\)
\(42\) −1.45381 −0.224328
\(43\) 1.53320 0.233811 0.116905 0.993143i \(-0.462703\pi\)
0.116905 + 0.993143i \(0.462703\pi\)
\(44\) −1.53362 −0.231202
\(45\) 0 0
\(46\) 4.61133 0.679904
\(47\) −2.48849 −0.362984 −0.181492 0.983392i \(-0.558093\pi\)
−0.181492 + 0.983392i \(0.558093\pi\)
\(48\) −3.52563 −0.508881
\(49\) −6.26564 −0.895091
\(50\) 0 0
\(51\) −8.71030 −1.21969
\(52\) 8.44457 1.17105
\(53\) 4.79753 0.658991 0.329495 0.944157i \(-0.393121\pi\)
0.329495 + 0.944157i \(0.393121\pi\)
\(54\) 0.290409 0.0395197
\(55\) 0 0
\(56\) −2.06797 −0.276344
\(57\) −2.48419 −0.329039
\(58\) 0.324169 0.0425654
\(59\) 1.75917 0.229025 0.114513 0.993422i \(-0.463469\pi\)
0.114513 + 0.993422i \(0.463469\pi\)
\(60\) 0 0
\(61\) 3.42433 0.438440 0.219220 0.975675i \(-0.429649\pi\)
0.219220 + 0.975675i \(0.429649\pi\)
\(62\) 3.54944 0.450780
\(63\) −2.71754 −0.342378
\(64\) 1.11946 0.139933
\(65\) 0 0
\(66\) 1.69650 0.208825
\(67\) 5.12424 0.626026 0.313013 0.949749i \(-0.398662\pi\)
0.313013 + 0.949749i \(0.398662\pi\)
\(68\) −5.37733 −0.652097
\(69\) 16.7742 2.01937
\(70\) 0 0
\(71\) −12.1937 −1.44713 −0.723563 0.690258i \(-0.757497\pi\)
−0.723563 + 0.690258i \(0.757497\pi\)
\(72\) 7.65263 0.901872
\(73\) 12.7638 1.49389 0.746945 0.664886i \(-0.231520\pi\)
0.746945 + 0.664886i \(0.231520\pi\)
\(74\) 0.287444 0.0334146
\(75\) 0 0
\(76\) −1.53362 −0.175918
\(77\) −0.856948 −0.0976583
\(78\) −9.34144 −1.05771
\(79\) 10.8691 1.22287 0.611437 0.791293i \(-0.290592\pi\)
0.611437 + 0.791293i \(0.290592\pi\)
\(80\) 0 0
\(81\) −8.45715 −0.939684
\(82\) −5.63505 −0.622288
\(83\) −2.94669 −0.323441 −0.161721 0.986837i \(-0.551704\pi\)
−0.161721 + 0.986837i \(0.551704\pi\)
\(84\) −3.26480 −0.356219
\(85\) 0 0
\(86\) −1.04705 −0.112907
\(87\) 1.17919 0.126423
\(88\) 2.41318 0.257246
\(89\) 3.08736 0.327259 0.163629 0.986522i \(-0.447680\pi\)
0.163629 + 0.986522i \(0.447680\pi\)
\(90\) 0 0
\(91\) 4.71861 0.494645
\(92\) 10.3556 1.07964
\(93\) 12.9114 1.33885
\(94\) 1.69944 0.175284
\(95\) 0 0
\(96\) 14.3973 1.46942
\(97\) 8.88210 0.901840 0.450920 0.892564i \(-0.351096\pi\)
0.450920 + 0.892564i \(0.351096\pi\)
\(98\) 4.27893 0.432238
\(99\) 3.17118 0.318716
\(100\) 0 0
\(101\) −2.09965 −0.208923 −0.104462 0.994529i \(-0.533312\pi\)
−0.104462 + 0.994529i \(0.533312\pi\)
\(102\) 5.94844 0.588983
\(103\) 3.91556 0.385812 0.192906 0.981217i \(-0.438209\pi\)
0.192906 + 0.981217i \(0.438209\pi\)
\(104\) −13.2877 −1.30297
\(105\) 0 0
\(106\) −3.27633 −0.318225
\(107\) 18.0439 1.74437 0.872184 0.489178i \(-0.162703\pi\)
0.872184 + 0.489178i \(0.162703\pi\)
\(108\) 0.652166 0.0627547
\(109\) 3.34558 0.320448 0.160224 0.987081i \(-0.448778\pi\)
0.160224 + 0.987081i \(0.448778\pi\)
\(110\) 0 0
\(111\) 1.04560 0.0992442
\(112\) −1.21621 −0.114921
\(113\) −7.26819 −0.683733 −0.341867 0.939748i \(-0.611059\pi\)
−0.341867 + 0.939748i \(0.611059\pi\)
\(114\) 1.69650 0.158892
\(115\) 0 0
\(116\) 0.727978 0.0675911
\(117\) −17.4615 −1.61431
\(118\) −1.20138 −0.110596
\(119\) −3.00472 −0.275442
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.33854 −0.211722
\(123\) −20.4980 −1.84825
\(124\) 7.97091 0.715809
\(125\) 0 0
\(126\) 1.85586 0.165333
\(127\) 16.9303 1.50232 0.751162 0.660118i \(-0.229494\pi\)
0.751162 + 0.660118i \(0.229494\pi\)
\(128\) 10.8267 0.956951
\(129\) −3.80875 −0.335342
\(130\) 0 0
\(131\) 5.26619 0.460109 0.230055 0.973178i \(-0.426109\pi\)
0.230055 + 0.973178i \(0.426109\pi\)
\(132\) 3.80980 0.331600
\(133\) −0.856948 −0.0743069
\(134\) −3.49945 −0.302307
\(135\) 0 0
\(136\) 8.46133 0.725553
\(137\) 20.6078 1.76064 0.880321 0.474378i \(-0.157327\pi\)
0.880321 + 0.474378i \(0.157327\pi\)
\(138\) −11.4554 −0.975149
\(139\) 1.26192 0.107034 0.0535171 0.998567i \(-0.482957\pi\)
0.0535171 + 0.998567i \(0.482957\pi\)
\(140\) 0 0
\(141\) 6.18188 0.520608
\(142\) 8.32733 0.698814
\(143\) −5.50630 −0.460460
\(144\) 4.50063 0.375053
\(145\) 0 0
\(146\) −8.71666 −0.721396
\(147\) 15.5650 1.28378
\(148\) 0.645506 0.0530603
\(149\) −11.3117 −0.926692 −0.463346 0.886177i \(-0.653351\pi\)
−0.463346 + 0.886177i \(0.653351\pi\)
\(150\) 0 0
\(151\) −3.70074 −0.301162 −0.150581 0.988598i \(-0.548114\pi\)
−0.150581 + 0.988598i \(0.548114\pi\)
\(152\) 2.41318 0.195735
\(153\) 11.1191 0.898927
\(154\) 0.585228 0.0471590
\(155\) 0 0
\(156\) −20.9779 −1.67957
\(157\) −2.91493 −0.232637 −0.116318 0.993212i \(-0.537109\pi\)
−0.116318 + 0.993212i \(0.537109\pi\)
\(158\) −7.42276 −0.590523
\(159\) −11.9179 −0.945155
\(160\) 0 0
\(161\) 5.78644 0.456035
\(162\) 5.77556 0.453771
\(163\) −18.4278 −1.44338 −0.721690 0.692216i \(-0.756634\pi\)
−0.721690 + 0.692216i \(0.756634\pi\)
\(164\) −12.6545 −0.988152
\(165\) 0 0
\(166\) 2.01235 0.156189
\(167\) −15.1433 −1.17182 −0.585912 0.810374i \(-0.699264\pi\)
−0.585912 + 0.810374i \(0.699264\pi\)
\(168\) 5.13723 0.396346
\(169\) 17.3193 1.33226
\(170\) 0 0
\(171\) 3.17118 0.242506
\(172\) −2.35135 −0.179288
\(173\) 0.110615 0.00840989 0.00420495 0.999991i \(-0.498662\pi\)
0.00420495 + 0.999991i \(0.498662\pi\)
\(174\) −0.805295 −0.0610493
\(175\) 0 0
\(176\) 1.41923 0.106978
\(177\) −4.37012 −0.328478
\(178\) −2.10842 −0.158033
\(179\) −9.42389 −0.704375 −0.352187 0.935930i \(-0.614562\pi\)
−0.352187 + 0.935930i \(0.614562\pi\)
\(180\) 0 0
\(181\) −12.2876 −0.913334 −0.456667 0.889638i \(-0.650957\pi\)
−0.456667 + 0.889638i \(0.650957\pi\)
\(182\) −3.22244 −0.238863
\(183\) −8.50667 −0.628831
\(184\) −16.2947 −1.20126
\(185\) 0 0
\(186\) −8.81748 −0.646529
\(187\) 3.50630 0.256406
\(188\) 3.81640 0.278340
\(189\) 0.364414 0.0265072
\(190\) 0 0
\(191\) 16.4558 1.19070 0.595351 0.803466i \(-0.297013\pi\)
0.595351 + 0.803466i \(0.297013\pi\)
\(192\) −2.78096 −0.200698
\(193\) −2.79958 −0.201518 −0.100759 0.994911i \(-0.532127\pi\)
−0.100759 + 0.994911i \(0.532127\pi\)
\(194\) −6.06577 −0.435497
\(195\) 0 0
\(196\) 9.60911 0.686365
\(197\) 7.19454 0.512589 0.256295 0.966599i \(-0.417498\pi\)
0.256295 + 0.966599i \(0.417498\pi\)
\(198\) −2.16566 −0.153907
\(199\) −11.9287 −0.845607 −0.422803 0.906221i \(-0.638954\pi\)
−0.422803 + 0.906221i \(0.638954\pi\)
\(200\) 0 0
\(201\) −12.7296 −0.897875
\(202\) 1.43390 0.100889
\(203\) 0.406776 0.0285501
\(204\) 13.3583 0.935267
\(205\) 0 0
\(206\) −2.67402 −0.186308
\(207\) −21.4130 −1.48831
\(208\) −7.81470 −0.541852
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 18.0430 1.24213 0.621065 0.783759i \(-0.286700\pi\)
0.621065 + 0.783759i \(0.286700\pi\)
\(212\) −7.35758 −0.505321
\(213\) 30.2914 2.07553
\(214\) −12.3225 −0.842351
\(215\) 0 0
\(216\) −1.02620 −0.0698238
\(217\) 4.45395 0.302354
\(218\) −2.28476 −0.154744
\(219\) −31.7077 −2.14261
\(220\) 0 0
\(221\) −19.3067 −1.29871
\(222\) −0.714064 −0.0479248
\(223\) 24.5001 1.64065 0.820323 0.571901i \(-0.193794\pi\)
0.820323 + 0.571901i \(0.193794\pi\)
\(224\) 4.96652 0.331839
\(225\) 0 0
\(226\) 4.96359 0.330173
\(227\) −16.6675 −1.10626 −0.553131 0.833094i \(-0.686567\pi\)
−0.553131 + 0.833094i \(0.686567\pi\)
\(228\) 3.80980 0.252310
\(229\) 14.8344 0.980282 0.490141 0.871643i \(-0.336945\pi\)
0.490141 + 0.871643i \(0.336945\pi\)
\(230\) 0 0
\(231\) 2.12882 0.140066
\(232\) −1.14549 −0.0752050
\(233\) −28.8509 −1.89009 −0.945043 0.326947i \(-0.893980\pi\)
−0.945043 + 0.326947i \(0.893980\pi\)
\(234\) 11.9248 0.779548
\(235\) 0 0
\(236\) −2.69790 −0.175619
\(237\) −27.0010 −1.75390
\(238\) 2.05198 0.133010
\(239\) 8.29173 0.536347 0.268174 0.963371i \(-0.413580\pi\)
0.268174 + 0.963371i \(0.413580\pi\)
\(240\) 0 0
\(241\) 3.48268 0.224339 0.112170 0.993689i \(-0.464220\pi\)
0.112170 + 0.993689i \(0.464220\pi\)
\(242\) −0.682920 −0.0438998
\(243\) 22.2849 1.42958
\(244\) −5.25162 −0.336200
\(245\) 0 0
\(246\) 13.9985 0.892513
\(247\) −5.50630 −0.350357
\(248\) −12.5424 −0.796442
\(249\) 7.32013 0.463894
\(250\) 0 0
\(251\) −28.8156 −1.81883 −0.909413 0.415895i \(-0.863468\pi\)
−0.909413 + 0.415895i \(0.863468\pi\)
\(252\) 4.16767 0.262539
\(253\) −6.75237 −0.424518
\(254\) −11.5621 −0.725469
\(255\) 0 0
\(256\) −9.63267 −0.602042
\(257\) 8.44177 0.526583 0.263292 0.964716i \(-0.415192\pi\)
0.263292 + 0.964716i \(0.415192\pi\)
\(258\) 2.60108 0.161936
\(259\) 0.360693 0.0224124
\(260\) 0 0
\(261\) −1.50530 −0.0931755
\(262\) −3.59639 −0.222186
\(263\) −21.4210 −1.32088 −0.660438 0.750881i \(-0.729629\pi\)
−0.660438 + 0.750881i \(0.729629\pi\)
\(264\) −5.99479 −0.368954
\(265\) 0 0
\(266\) 0.585228 0.0358826
\(267\) −7.66957 −0.469370
\(268\) −7.85864 −0.480043
\(269\) 10.3642 0.631916 0.315958 0.948773i \(-0.397674\pi\)
0.315958 + 0.948773i \(0.397674\pi\)
\(270\) 0 0
\(271\) −26.1079 −1.58594 −0.792972 0.609258i \(-0.791467\pi\)
−0.792972 + 0.609258i \(0.791467\pi\)
\(272\) 4.97624 0.301729
\(273\) −11.7219 −0.709443
\(274\) −14.0735 −0.850210
\(275\) 0 0
\(276\) −25.7252 −1.54847
\(277\) 18.7663 1.12756 0.563779 0.825925i \(-0.309347\pi\)
0.563779 + 0.825925i \(0.309347\pi\)
\(278\) −0.861788 −0.0516866
\(279\) −16.4821 −0.986755
\(280\) 0 0
\(281\) 24.6322 1.46944 0.734718 0.678373i \(-0.237315\pi\)
0.734718 + 0.678373i \(0.237315\pi\)
\(282\) −4.22173 −0.251400
\(283\) 13.7870 0.819550 0.409775 0.912187i \(-0.365607\pi\)
0.409775 + 0.912187i \(0.365607\pi\)
\(284\) 18.7005 1.10967
\(285\) 0 0
\(286\) 3.76036 0.222355
\(287\) −7.07103 −0.417390
\(288\) −18.3788 −1.08298
\(289\) −4.70587 −0.276816
\(290\) 0 0
\(291\) −22.0648 −1.29346
\(292\) −19.5748 −1.14553
\(293\) 3.53033 0.206244 0.103122 0.994669i \(-0.467117\pi\)
0.103122 + 0.994669i \(0.467117\pi\)
\(294\) −10.6297 −0.619935
\(295\) 0 0
\(296\) −1.01572 −0.0590373
\(297\) −0.425246 −0.0246753
\(298\) 7.72501 0.447498
\(299\) 37.1806 2.15021
\(300\) 0 0
\(301\) −1.31387 −0.0757304
\(302\) 2.52731 0.145430
\(303\) 5.21593 0.299647
\(304\) 1.41923 0.0813984
\(305\) 0 0
\(306\) −7.59347 −0.434090
\(307\) −5.59192 −0.319148 −0.159574 0.987186i \(-0.551012\pi\)
−0.159574 + 0.987186i \(0.551012\pi\)
\(308\) 1.31423 0.0748854
\(309\) −9.72699 −0.553349
\(310\) 0 0
\(311\) −3.38109 −0.191724 −0.0958621 0.995395i \(-0.530561\pi\)
−0.0958621 + 0.995395i \(0.530561\pi\)
\(312\) 33.0091 1.86877
\(313\) −13.5676 −0.766886 −0.383443 0.923565i \(-0.625262\pi\)
−0.383443 + 0.923565i \(0.625262\pi\)
\(314\) 1.99066 0.112340
\(315\) 0 0
\(316\) −16.6691 −0.937712
\(317\) 9.02947 0.507145 0.253573 0.967316i \(-0.418394\pi\)
0.253573 + 0.967316i \(0.418394\pi\)
\(318\) 8.13901 0.456413
\(319\) −0.474680 −0.0265770
\(320\) 0 0
\(321\) −44.8244 −2.50185
\(322\) −3.95167 −0.220218
\(323\) 3.50630 0.195096
\(324\) 12.9701 0.720559
\(325\) 0 0
\(326\) 12.5847 0.697005
\(327\) −8.31104 −0.459602
\(328\) 19.9121 1.09946
\(329\) 2.13251 0.117569
\(330\) 0 0
\(331\) −18.6966 −1.02766 −0.513829 0.857892i \(-0.671774\pi\)
−0.513829 + 0.857892i \(0.671774\pi\)
\(332\) 4.51910 0.248018
\(333\) −1.33476 −0.0731445
\(334\) 10.3417 0.565871
\(335\) 0 0
\(336\) 3.02128 0.164824
\(337\) −29.3264 −1.59751 −0.798755 0.601657i \(-0.794508\pi\)
−0.798755 + 0.601657i \(0.794508\pi\)
\(338\) −11.8277 −0.643343
\(339\) 18.0555 0.980642
\(340\) 0 0
\(341\) −5.19745 −0.281458
\(342\) −2.16566 −0.117106
\(343\) 11.3680 0.613813
\(344\) 3.69989 0.199485
\(345\) 0 0
\(346\) −0.0755411 −0.00406112
\(347\) −11.0048 −0.590768 −0.295384 0.955379i \(-0.595448\pi\)
−0.295384 + 0.955379i \(0.595448\pi\)
\(348\) −1.80843 −0.0969422
\(349\) −24.1430 −1.29235 −0.646174 0.763190i \(-0.723632\pi\)
−0.646174 + 0.763190i \(0.723632\pi\)
\(350\) 0 0
\(351\) 2.34153 0.124982
\(352\) −5.79558 −0.308906
\(353\) −13.1068 −0.697602 −0.348801 0.937197i \(-0.613411\pi\)
−0.348801 + 0.937197i \(0.613411\pi\)
\(354\) 2.98444 0.158621
\(355\) 0 0
\(356\) −4.73483 −0.250945
\(357\) 7.46428 0.395051
\(358\) 6.43577 0.340141
\(359\) −11.2347 −0.592942 −0.296471 0.955042i \(-0.595810\pi\)
−0.296471 + 0.955042i \(0.595810\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 8.39149 0.441047
\(363\) −2.48419 −0.130386
\(364\) −7.23656 −0.379299
\(365\) 0 0
\(366\) 5.80938 0.303661
\(367\) −0.796321 −0.0415676 −0.0207838 0.999784i \(-0.506616\pi\)
−0.0207838 + 0.999784i \(0.506616\pi\)
\(368\) −9.58316 −0.499557
\(369\) 26.1667 1.36218
\(370\) 0 0
\(371\) −4.11123 −0.213445
\(372\) −19.8012 −1.02665
\(373\) −4.22370 −0.218695 −0.109347 0.994004i \(-0.534876\pi\)
−0.109347 + 0.994004i \(0.534876\pi\)
\(374\) −2.39452 −0.123818
\(375\) 0 0
\(376\) −6.00518 −0.309694
\(377\) 2.61373 0.134614
\(378\) −0.248866 −0.0128003
\(379\) −22.5751 −1.15961 −0.579803 0.814757i \(-0.696870\pi\)
−0.579803 + 0.814757i \(0.696870\pi\)
\(380\) 0 0
\(381\) −42.0581 −2.15470
\(382\) −11.2380 −0.574987
\(383\) −32.5159 −1.66148 −0.830742 0.556658i \(-0.812083\pi\)
−0.830742 + 0.556658i \(0.812083\pi\)
\(384\) −26.8954 −1.37250
\(385\) 0 0
\(386\) 1.91189 0.0973128
\(387\) 4.86206 0.247152
\(388\) −13.6218 −0.691540
\(389\) −35.4867 −1.79925 −0.899624 0.436665i \(-0.856160\pi\)
−0.899624 + 0.436665i \(0.856160\pi\)
\(390\) 0 0
\(391\) −23.6758 −1.19734
\(392\) −15.1201 −0.763681
\(393\) −13.0822 −0.659910
\(394\) −4.91329 −0.247528
\(395\) 0 0
\(396\) −4.86339 −0.244394
\(397\) −11.7223 −0.588325 −0.294162 0.955755i \(-0.595041\pi\)
−0.294162 + 0.955755i \(0.595041\pi\)
\(398\) 8.14639 0.408341
\(399\) 2.12882 0.106574
\(400\) 0 0
\(401\) −33.5093 −1.67337 −0.836687 0.547681i \(-0.815511\pi\)
−0.836687 + 0.547681i \(0.815511\pi\)
\(402\) 8.69329 0.433582
\(403\) 28.6187 1.42560
\(404\) 3.22007 0.160205
\(405\) 0 0
\(406\) −0.277796 −0.0137868
\(407\) −0.420904 −0.0208634
\(408\) −21.0195 −1.04062
\(409\) −6.50358 −0.321581 −0.160791 0.986989i \(-0.551404\pi\)
−0.160791 + 0.986989i \(0.551404\pi\)
\(410\) 0 0
\(411\) −51.1936 −2.52519
\(412\) −6.00499 −0.295844
\(413\) −1.50752 −0.0741803
\(414\) 14.6234 0.718700
\(415\) 0 0
\(416\) 31.9122 1.56462
\(417\) −3.13483 −0.153513
\(418\) −0.682920 −0.0334027
\(419\) −7.49790 −0.366296 −0.183148 0.983085i \(-0.558629\pi\)
−0.183148 + 0.983085i \(0.558629\pi\)
\(420\) 0 0
\(421\) 11.5280 0.561840 0.280920 0.959731i \(-0.409360\pi\)
0.280920 + 0.959731i \(0.409360\pi\)
\(422\) −12.3219 −0.599822
\(423\) −7.89146 −0.383696
\(424\) 11.5773 0.562243
\(425\) 0 0
\(426\) −20.6866 −1.00227
\(427\) −2.93447 −0.142009
\(428\) −27.6724 −1.33760
\(429\) 13.6787 0.660412
\(430\) 0 0
\(431\) 22.4459 1.08118 0.540590 0.841286i \(-0.318201\pi\)
0.540590 + 0.841286i \(0.318201\pi\)
\(432\) −0.603522 −0.0290370
\(433\) 27.4999 1.32156 0.660781 0.750579i \(-0.270225\pi\)
0.660781 + 0.750579i \(0.270225\pi\)
\(434\) −3.04169 −0.146006
\(435\) 0 0
\(436\) −5.13084 −0.245723
\(437\) −6.75237 −0.323010
\(438\) 21.6538 1.03466
\(439\) 8.18489 0.390644 0.195322 0.980739i \(-0.437425\pi\)
0.195322 + 0.980739i \(0.437425\pi\)
\(440\) 0 0
\(441\) −19.8695 −0.946166
\(442\) 13.1850 0.627144
\(443\) 11.1175 0.528209 0.264104 0.964494i \(-0.414924\pi\)
0.264104 + 0.964494i \(0.414924\pi\)
\(444\) −1.60356 −0.0761015
\(445\) 0 0
\(446\) −16.7316 −0.792264
\(447\) 28.1004 1.32910
\(448\) −0.959323 −0.0453237
\(449\) 11.1426 0.525854 0.262927 0.964816i \(-0.415312\pi\)
0.262927 + 0.964816i \(0.415312\pi\)
\(450\) 0 0
\(451\) 8.25141 0.388544
\(452\) 11.1466 0.524294
\(453\) 9.19332 0.431940
\(454\) 11.3826 0.534212
\(455\) 0 0
\(456\) −5.99479 −0.280732
\(457\) −18.3241 −0.857164 −0.428582 0.903503i \(-0.640987\pi\)
−0.428582 + 0.903503i \(0.640987\pi\)
\(458\) −10.1307 −0.473376
\(459\) −1.49104 −0.0695958
\(460\) 0 0
\(461\) 31.6216 1.47276 0.736382 0.676566i \(-0.236533\pi\)
0.736382 + 0.676566i \(0.236533\pi\)
\(462\) −1.45381 −0.0676376
\(463\) 7.76077 0.360674 0.180337 0.983605i \(-0.442281\pi\)
0.180337 + 0.983605i \(0.442281\pi\)
\(464\) −0.673679 −0.0312748
\(465\) 0 0
\(466\) 19.7029 0.912718
\(467\) −31.2308 −1.44519 −0.722594 0.691273i \(-0.757050\pi\)
−0.722594 + 0.691273i \(0.757050\pi\)
\(468\) 26.7793 1.23787
\(469\) −4.39121 −0.202767
\(470\) 0 0
\(471\) 7.24122 0.333658
\(472\) 4.24521 0.195401
\(473\) 1.53320 0.0704966
\(474\) 18.4395 0.846955
\(475\) 0 0
\(476\) 4.60809 0.211212
\(477\) 15.2138 0.696593
\(478\) −5.66259 −0.259001
\(479\) −18.2169 −0.832353 −0.416177 0.909284i \(-0.636630\pi\)
−0.416177 + 0.909284i \(0.636630\pi\)
\(480\) 0 0
\(481\) 2.31762 0.105674
\(482\) −2.37840 −0.108333
\(483\) −14.3746 −0.654066
\(484\) −1.53362 −0.0697100
\(485\) 0 0
\(486\) −15.2188 −0.690339
\(487\) −39.2194 −1.77720 −0.888600 0.458683i \(-0.848321\pi\)
−0.888600 + 0.458683i \(0.848321\pi\)
\(488\) 8.26352 0.374072
\(489\) 45.7782 2.07016
\(490\) 0 0
\(491\) 36.5478 1.64938 0.824688 0.565587i \(-0.191350\pi\)
0.824688 + 0.565587i \(0.191350\pi\)
\(492\) 31.4362 1.41725
\(493\) −1.66437 −0.0749594
\(494\) 3.76036 0.169187
\(495\) 0 0
\(496\) −7.37637 −0.331209
\(497\) 10.4494 0.468719
\(498\) −4.99906 −0.224013
\(499\) 31.4475 1.40778 0.703892 0.710307i \(-0.251444\pi\)
0.703892 + 0.710307i \(0.251444\pi\)
\(500\) 0 0
\(501\) 37.6188 1.68068
\(502\) 19.6788 0.878307
\(503\) −20.7020 −0.923057 −0.461528 0.887125i \(-0.652699\pi\)
−0.461528 + 0.887125i \(0.652699\pi\)
\(504\) −6.55791 −0.292113
\(505\) 0 0
\(506\) 4.61133 0.204999
\(507\) −43.0244 −1.91078
\(508\) −25.9647 −1.15200
\(509\) 23.5196 1.04249 0.521245 0.853407i \(-0.325468\pi\)
0.521245 + 0.853407i \(0.325468\pi\)
\(510\) 0 0
\(511\) −10.9379 −0.483865
\(512\) −15.0750 −0.666226
\(513\) −0.425246 −0.0187751
\(514\) −5.76506 −0.254286
\(515\) 0 0
\(516\) 5.84118 0.257144
\(517\) −2.48849 −0.109444
\(518\) −0.246324 −0.0108229
\(519\) −0.274788 −0.0120619
\(520\) 0 0
\(521\) 28.6887 1.25687 0.628437 0.777860i \(-0.283695\pi\)
0.628437 + 0.777860i \(0.283695\pi\)
\(522\) 1.02800 0.0449942
\(523\) −5.70672 −0.249537 −0.124769 0.992186i \(-0.539819\pi\)
−0.124769 + 0.992186i \(0.539819\pi\)
\(524\) −8.07634 −0.352816
\(525\) 0 0
\(526\) 14.6288 0.637848
\(527\) −18.2238 −0.793842
\(528\) −3.52563 −0.153433
\(529\) 22.5945 0.982371
\(530\) 0 0
\(531\) 5.57866 0.242093
\(532\) 1.31423 0.0569792
\(533\) −45.4347 −1.96800
\(534\) 5.23770 0.226658
\(535\) 0 0
\(536\) 12.3657 0.534118
\(537\) 23.4107 1.01025
\(538\) −7.07792 −0.305151
\(539\) −6.26564 −0.269880
\(540\) 0 0
\(541\) 3.50382 0.150641 0.0753205 0.997159i \(-0.476002\pi\)
0.0753205 + 0.997159i \(0.476002\pi\)
\(542\) 17.8296 0.765849
\(543\) 30.5248 1.30994
\(544\) −20.3210 −0.871257
\(545\) 0 0
\(546\) 8.00514 0.342588
\(547\) −28.3788 −1.21339 −0.606696 0.794934i \(-0.707505\pi\)
−0.606696 + 0.794934i \(0.707505\pi\)
\(548\) −31.6045 −1.35008
\(549\) 10.8592 0.463458
\(550\) 0 0
\(551\) −0.474680 −0.0202220
\(552\) 40.4791 1.72290
\(553\) −9.31430 −0.396084
\(554\) −12.8159 −0.544495
\(555\) 0 0
\(556\) −1.93530 −0.0820749
\(557\) 0.440370 0.0186591 0.00932953 0.999956i \(-0.497030\pi\)
0.00932953 + 0.999956i \(0.497030\pi\)
\(558\) 11.2559 0.476502
\(559\) −8.44226 −0.357069
\(560\) 0 0
\(561\) −8.71030 −0.367749
\(562\) −16.8219 −0.709587
\(563\) −44.5147 −1.87607 −0.938036 0.346537i \(-0.887358\pi\)
−0.938036 + 0.346537i \(0.887358\pi\)
\(564\) −9.48065 −0.399207
\(565\) 0 0
\(566\) −9.41540 −0.395759
\(567\) 7.24734 0.304360
\(568\) −29.4256 −1.23467
\(569\) 5.32611 0.223282 0.111641 0.993749i \(-0.464389\pi\)
0.111641 + 0.993749i \(0.464389\pi\)
\(570\) 0 0
\(571\) 11.2850 0.472264 0.236132 0.971721i \(-0.424120\pi\)
0.236132 + 0.971721i \(0.424120\pi\)
\(572\) 8.44457 0.353085
\(573\) −40.8793 −1.70776
\(574\) 4.82895 0.201557
\(575\) 0 0
\(576\) 3.55002 0.147918
\(577\) −44.9805 −1.87256 −0.936281 0.351251i \(-0.885757\pi\)
−0.936281 + 0.351251i \(0.885757\pi\)
\(578\) 3.21374 0.133674
\(579\) 6.95469 0.289027
\(580\) 0 0
\(581\) 2.52516 0.104761
\(582\) 15.0685 0.624609
\(583\) 4.79753 0.198693
\(584\) 30.8014 1.27457
\(585\) 0 0
\(586\) −2.41094 −0.0995948
\(587\) 12.8765 0.531470 0.265735 0.964046i \(-0.414385\pi\)
0.265735 + 0.964046i \(0.414385\pi\)
\(588\) −23.8708 −0.984416
\(589\) −5.19745 −0.214157
\(590\) 0 0
\(591\) −17.8726 −0.735179
\(592\) −0.597359 −0.0245513
\(593\) 25.1716 1.03367 0.516837 0.856084i \(-0.327109\pi\)
0.516837 + 0.856084i \(0.327109\pi\)
\(594\) 0.290409 0.0119156
\(595\) 0 0
\(596\) 17.3479 0.710597
\(597\) 29.6332 1.21281
\(598\) −25.3914 −1.03833
\(599\) 30.2614 1.23645 0.618224 0.786002i \(-0.287853\pi\)
0.618224 + 0.786002i \(0.287853\pi\)
\(600\) 0 0
\(601\) 33.1672 1.35292 0.676458 0.736481i \(-0.263514\pi\)
0.676458 + 0.736481i \(0.263514\pi\)
\(602\) 0.897271 0.0365700
\(603\) 16.2499 0.661747
\(604\) 5.67552 0.230934
\(605\) 0 0
\(606\) −3.56207 −0.144699
\(607\) 10.4625 0.424660 0.212330 0.977198i \(-0.431895\pi\)
0.212330 + 0.977198i \(0.431895\pi\)
\(608\) −5.79558 −0.235042
\(609\) −1.01051 −0.0409478
\(610\) 0 0
\(611\) 13.7024 0.554339
\(612\) −17.0525 −0.689306
\(613\) 9.24317 0.373328 0.186664 0.982424i \(-0.440232\pi\)
0.186664 + 0.982424i \(0.440232\pi\)
\(614\) 3.81884 0.154116
\(615\) 0 0
\(616\) −2.06797 −0.0833210
\(617\) 1.95689 0.0787814 0.0393907 0.999224i \(-0.487458\pi\)
0.0393907 + 0.999224i \(0.487458\pi\)
\(618\) 6.64276 0.267211
\(619\) −48.3987 −1.94531 −0.972653 0.232263i \(-0.925387\pi\)
−0.972653 + 0.232263i \(0.925387\pi\)
\(620\) 0 0
\(621\) 2.87142 0.115226
\(622\) 2.30902 0.0925832
\(623\) −2.64570 −0.105998
\(624\) 19.4132 0.777149
\(625\) 0 0
\(626\) 9.26559 0.370327
\(627\) −2.48419 −0.0992088
\(628\) 4.47039 0.178388
\(629\) −1.47581 −0.0588445
\(630\) 0 0
\(631\) 30.8039 1.22628 0.613141 0.789973i \(-0.289906\pi\)
0.613141 + 0.789973i \(0.289906\pi\)
\(632\) 26.2292 1.04334
\(633\) −44.8222 −1.78152
\(634\) −6.16641 −0.244899
\(635\) 0 0
\(636\) 18.2776 0.724754
\(637\) 34.5005 1.36696
\(638\) 0.324169 0.0128340
\(639\) −38.6684 −1.52970
\(640\) 0 0
\(641\) −7.12059 −0.281246 −0.140623 0.990063i \(-0.544911\pi\)
−0.140623 + 0.990063i \(0.544911\pi\)
\(642\) 30.6115 1.20814
\(643\) 43.8135 1.72783 0.863917 0.503634i \(-0.168004\pi\)
0.863917 + 0.503634i \(0.168004\pi\)
\(644\) −8.87419 −0.349692
\(645\) 0 0
\(646\) −2.39452 −0.0942112
\(647\) −19.4584 −0.764988 −0.382494 0.923958i \(-0.624935\pi\)
−0.382494 + 0.923958i \(0.624935\pi\)
\(648\) −20.4086 −0.801727
\(649\) 1.75917 0.0690536
\(650\) 0 0
\(651\) −11.0644 −0.433649
\(652\) 28.2613 1.10680
\(653\) 35.4429 1.38699 0.693493 0.720463i \(-0.256071\pi\)
0.693493 + 0.720463i \(0.256071\pi\)
\(654\) 5.67578 0.221940
\(655\) 0 0
\(656\) 11.7106 0.457224
\(657\) 40.4763 1.57913
\(658\) −1.45633 −0.0567738
\(659\) −7.91185 −0.308202 −0.154101 0.988055i \(-0.549248\pi\)
−0.154101 + 0.988055i \(0.549248\pi\)
\(660\) 0 0
\(661\) −37.9285 −1.47525 −0.737625 0.675211i \(-0.764053\pi\)
−0.737625 + 0.675211i \(0.764053\pi\)
\(662\) 12.7683 0.496254
\(663\) 47.9615 1.86267
\(664\) −7.11090 −0.275956
\(665\) 0 0
\(666\) 0.911536 0.0353213
\(667\) 3.20522 0.124106
\(668\) 23.2241 0.898567
\(669\) −60.8627 −2.35309
\(670\) 0 0
\(671\) 3.42433 0.132195
\(672\) −12.3377 −0.475939
\(673\) 5.19834 0.200381 0.100191 0.994968i \(-0.468055\pi\)
0.100191 + 0.994968i \(0.468055\pi\)
\(674\) 20.0276 0.771434
\(675\) 0 0
\(676\) −26.5613 −1.02159
\(677\) 12.7010 0.488137 0.244069 0.969758i \(-0.421518\pi\)
0.244069 + 0.969758i \(0.421518\pi\)
\(678\) −12.3305 −0.473550
\(679\) −7.61150 −0.292103
\(680\) 0 0
\(681\) 41.4052 1.58665
\(682\) 3.54944 0.135915
\(683\) −45.3282 −1.73444 −0.867219 0.497928i \(-0.834095\pi\)
−0.867219 + 0.497928i \(0.834095\pi\)
\(684\) −4.86339 −0.185956
\(685\) 0 0
\(686\) −7.76342 −0.296409
\(687\) −36.8513 −1.40597
\(688\) 2.17596 0.0829578
\(689\) −26.4166 −1.00639
\(690\) 0 0
\(691\) 30.5149 1.16084 0.580421 0.814317i \(-0.302888\pi\)
0.580421 + 0.814317i \(0.302888\pi\)
\(692\) −0.169641 −0.00644879
\(693\) −2.71754 −0.103231
\(694\) 7.51539 0.285280
\(695\) 0 0
\(696\) 2.84561 0.107862
\(697\) 28.9319 1.09587
\(698\) 16.4878 0.624072
\(699\) 71.6710 2.71085
\(700\) 0 0
\(701\) 0.621590 0.0234771 0.0117386 0.999931i \(-0.496263\pi\)
0.0117386 + 0.999931i \(0.496263\pi\)
\(702\) −1.59908 −0.0603534
\(703\) −0.420904 −0.0158747
\(704\) 1.11946 0.0421914
\(705\) 0 0
\(706\) 8.95087 0.336870
\(707\) 1.79930 0.0676695
\(708\) 6.70210 0.251880
\(709\) −21.9142 −0.823004 −0.411502 0.911409i \(-0.634996\pi\)
−0.411502 + 0.911409i \(0.634996\pi\)
\(710\) 0 0
\(711\) 34.4680 1.29265
\(712\) 7.45035 0.279214
\(713\) 35.0951 1.31432
\(714\) −5.09751 −0.190769
\(715\) 0 0
\(716\) 14.4527 0.540121
\(717\) −20.5982 −0.769254
\(718\) 7.67238 0.286331
\(719\) −43.0880 −1.60691 −0.803455 0.595365i \(-0.797007\pi\)
−0.803455 + 0.595365i \(0.797007\pi\)
\(720\) 0 0
\(721\) −3.35544 −0.124963
\(722\) −0.682920 −0.0254157
\(723\) −8.65163 −0.321758
\(724\) 18.8446 0.700353
\(725\) 0 0
\(726\) 1.69650 0.0629631
\(727\) −30.8306 −1.14344 −0.571721 0.820448i \(-0.693724\pi\)
−0.571721 + 0.820448i \(0.693724\pi\)
\(728\) 11.3869 0.422025
\(729\) −29.9883 −1.11068
\(730\) 0 0
\(731\) 5.37586 0.198833
\(732\) 13.0460 0.482194
\(733\) 17.6688 0.652610 0.326305 0.945264i \(-0.394196\pi\)
0.326305 + 0.945264i \(0.394196\pi\)
\(734\) 0.543824 0.0200729
\(735\) 0 0
\(736\) 39.1339 1.44250
\(737\) 5.12424 0.188754
\(738\) −17.8698 −0.657796
\(739\) −6.72966 −0.247555 −0.123777 0.992310i \(-0.539501\pi\)
−0.123777 + 0.992310i \(0.539501\pi\)
\(740\) 0 0
\(741\) 13.6787 0.502499
\(742\) 2.80764 0.103072
\(743\) −48.4780 −1.77848 −0.889242 0.457437i \(-0.848767\pi\)
−0.889242 + 0.457437i \(0.848767\pi\)
\(744\) 31.1576 1.14229
\(745\) 0 0
\(746\) 2.88445 0.105607
\(747\) −9.34449 −0.341897
\(748\) −5.37733 −0.196615
\(749\) −15.4627 −0.564994
\(750\) 0 0
\(751\) −36.2684 −1.32345 −0.661726 0.749746i \(-0.730176\pi\)
−0.661726 + 0.749746i \(0.730176\pi\)
\(752\) −3.53174 −0.128789
\(753\) 71.5834 2.60864
\(754\) −1.78497 −0.0650047
\(755\) 0 0
\(756\) −0.558873 −0.0203260
\(757\) 5.13939 0.186794 0.0933972 0.995629i \(-0.470227\pi\)
0.0933972 + 0.995629i \(0.470227\pi\)
\(758\) 15.4170 0.559971
\(759\) 16.7742 0.608863
\(760\) 0 0
\(761\) 10.2119 0.370179 0.185090 0.982722i \(-0.440742\pi\)
0.185090 + 0.982722i \(0.440742\pi\)
\(762\) 28.7223 1.04050
\(763\) −2.86699 −0.103792
\(764\) −25.2370 −0.913042
\(765\) 0 0
\(766\) 22.2057 0.802326
\(767\) −9.68654 −0.349761
\(768\) 23.9294 0.863476
\(769\) −49.2015 −1.77425 −0.887126 0.461527i \(-0.847302\pi\)
−0.887126 + 0.461527i \(0.847302\pi\)
\(770\) 0 0
\(771\) −20.9709 −0.755249
\(772\) 4.29350 0.154526
\(773\) 40.7240 1.46474 0.732371 0.680906i \(-0.238414\pi\)
0.732371 + 0.680906i \(0.238414\pi\)
\(774\) −3.32040 −0.119349
\(775\) 0 0
\(776\) 21.4341 0.769440
\(777\) −0.896028 −0.0321448
\(778\) 24.2346 0.868853
\(779\) 8.25141 0.295637
\(780\) 0 0
\(781\) −12.1937 −0.436325
\(782\) 16.1687 0.578192
\(783\) 0.201856 0.00721374
\(784\) −8.89238 −0.317585
\(785\) 0 0
\(786\) 8.93410 0.318669
\(787\) −7.23417 −0.257870 −0.128935 0.991653i \(-0.541156\pi\)
−0.128935 + 0.991653i \(0.541156\pi\)
\(788\) −11.0337 −0.393059
\(789\) 53.2138 1.89446
\(790\) 0 0
\(791\) 6.22846 0.221459
\(792\) 7.65263 0.271925
\(793\) −18.8554 −0.669574
\(794\) 8.00539 0.284101
\(795\) 0 0
\(796\) 18.2942 0.648420
\(797\) −43.9784 −1.55779 −0.778897 0.627152i \(-0.784220\pi\)
−0.778897 + 0.627152i \(0.784220\pi\)
\(798\) −1.45381 −0.0514645
\(799\) −8.72540 −0.308682
\(800\) 0 0
\(801\) 9.79056 0.345933
\(802\) 22.8842 0.808068
\(803\) 12.7638 0.450425
\(804\) 19.5223 0.688500
\(805\) 0 0
\(806\) −19.5443 −0.688419
\(807\) −25.7466 −0.906323
\(808\) −5.06685 −0.178251
\(809\) 11.9798 0.421187 0.210593 0.977574i \(-0.432460\pi\)
0.210593 + 0.977574i \(0.432460\pi\)
\(810\) 0 0
\(811\) 30.0059 1.05365 0.526825 0.849974i \(-0.323382\pi\)
0.526825 + 0.849974i \(0.323382\pi\)
\(812\) −0.623840 −0.0218925
\(813\) 64.8570 2.27463
\(814\) 0.287444 0.0100749
\(815\) 0 0
\(816\) −12.3619 −0.432753
\(817\) 1.53320 0.0536399
\(818\) 4.44143 0.155291
\(819\) 14.9636 0.522870
\(820\) 0 0
\(821\) 26.5123 0.925284 0.462642 0.886545i \(-0.346901\pi\)
0.462642 + 0.886545i \(0.346901\pi\)
\(822\) 34.9611 1.21941
\(823\) −13.2244 −0.460972 −0.230486 0.973076i \(-0.574032\pi\)
−0.230486 + 0.973076i \(0.574032\pi\)
\(824\) 9.44896 0.329170
\(825\) 0 0
\(826\) 1.02952 0.0358215
\(827\) −25.4465 −0.884861 −0.442431 0.896803i \(-0.645884\pi\)
−0.442431 + 0.896803i \(0.645884\pi\)
\(828\) 32.8394 1.14125
\(829\) −28.6999 −0.996789 −0.498395 0.866950i \(-0.666077\pi\)
−0.498395 + 0.866950i \(0.666077\pi\)
\(830\) 0 0
\(831\) −46.6190 −1.61720
\(832\) −6.16410 −0.213702
\(833\) −21.9692 −0.761188
\(834\) 2.14084 0.0741313
\(835\) 0 0
\(836\) −1.53362 −0.0530413
\(837\) 2.21020 0.0763956
\(838\) 5.12047 0.176884
\(839\) −4.10582 −0.141749 −0.0708743 0.997485i \(-0.522579\pi\)
−0.0708743 + 0.997485i \(0.522579\pi\)
\(840\) 0 0
\(841\) −28.7747 −0.992230
\(842\) −7.87271 −0.271311
\(843\) −61.1911 −2.10753
\(844\) −27.6711 −0.952478
\(845\) 0 0
\(846\) 5.38924 0.185286
\(847\) −0.856948 −0.0294451
\(848\) 6.80879 0.233815
\(849\) −34.2494 −1.17544
\(850\) 0 0
\(851\) 2.84210 0.0974259
\(852\) −46.4555 −1.59154
\(853\) 51.2689 1.75541 0.877707 0.479198i \(-0.159072\pi\)
0.877707 + 0.479198i \(0.159072\pi\)
\(854\) 2.00401 0.0685758
\(855\) 0 0
\(856\) 43.5431 1.48827
\(857\) 34.2836 1.17110 0.585552 0.810635i \(-0.300878\pi\)
0.585552 + 0.810635i \(0.300878\pi\)
\(858\) −9.34144 −0.318912
\(859\) 39.5431 1.34919 0.674596 0.738187i \(-0.264318\pi\)
0.674596 + 0.738187i \(0.264318\pi\)
\(860\) 0 0
\(861\) 17.5658 0.598639
\(862\) −15.3288 −0.522100
\(863\) 13.1157 0.446462 0.223231 0.974766i \(-0.428340\pi\)
0.223231 + 0.974766i \(0.428340\pi\)
\(864\) 2.46455 0.0838457
\(865\) 0 0
\(866\) −18.7802 −0.638179
\(867\) 11.6903 0.397022
\(868\) −6.83066 −0.231848
\(869\) 10.8691 0.368711
\(870\) 0 0
\(871\) −28.2156 −0.956050
\(872\) 8.07349 0.273403
\(873\) 28.1667 0.953300
\(874\) 4.61133 0.155981
\(875\) 0 0
\(876\) 48.6275 1.64297
\(877\) 34.1036 1.15160 0.575798 0.817592i \(-0.304691\pi\)
0.575798 + 0.817592i \(0.304691\pi\)
\(878\) −5.58963 −0.188641
\(879\) −8.77000 −0.295805
\(880\) 0 0
\(881\) 39.1343 1.31847 0.659234 0.751938i \(-0.270881\pi\)
0.659234 + 0.751938i \(0.270881\pi\)
\(882\) 13.5693 0.456901
\(883\) −44.1281 −1.48503 −0.742515 0.669829i \(-0.766367\pi\)
−0.742515 + 0.669829i \(0.766367\pi\)
\(884\) 29.6092 0.995864
\(885\) 0 0
\(886\) −7.59238 −0.255071
\(887\) 48.8255 1.63940 0.819700 0.572793i \(-0.194140\pi\)
0.819700 + 0.572793i \(0.194140\pi\)
\(888\) 2.52323 0.0846740
\(889\) −14.5084 −0.486597
\(890\) 0 0
\(891\) −8.45715 −0.283325
\(892\) −37.5738 −1.25806
\(893\) −2.48849 −0.0832742
\(894\) −19.1904 −0.641821
\(895\) 0 0
\(896\) −9.27789 −0.309953
\(897\) −92.3635 −3.08393
\(898\) −7.60954 −0.253934
\(899\) 2.46712 0.0822832
\(900\) 0 0
\(901\) 16.8216 0.560407
\(902\) −5.63505 −0.187627
\(903\) 3.26391 0.108616
\(904\) −17.5394 −0.583353
\(905\) 0 0
\(906\) −6.27830 −0.208583
\(907\) 25.4676 0.845638 0.422819 0.906214i \(-0.361041\pi\)
0.422819 + 0.906214i \(0.361041\pi\)
\(908\) 25.5617 0.848293
\(909\) −6.65838 −0.220845
\(910\) 0 0
\(911\) −23.4140 −0.775740 −0.387870 0.921714i \(-0.626789\pi\)
−0.387870 + 0.921714i \(0.626789\pi\)
\(912\) −3.52563 −0.116745
\(913\) −2.94669 −0.0975212
\(914\) 12.5139 0.413923
\(915\) 0 0
\(916\) −22.7503 −0.751690
\(917\) −4.51286 −0.149028
\(918\) 1.01826 0.0336077
\(919\) −18.7511 −0.618542 −0.309271 0.950974i \(-0.600085\pi\)
−0.309271 + 0.950974i \(0.600085\pi\)
\(920\) 0 0
\(921\) 13.8914 0.457736
\(922\) −21.5950 −0.711194
\(923\) 67.1422 2.21001
\(924\) −3.26480 −0.107404
\(925\) 0 0
\(926\) −5.29999 −0.174169
\(927\) 12.4170 0.407827
\(928\) 2.75105 0.0903075
\(929\) 40.7701 1.33762 0.668812 0.743432i \(-0.266803\pi\)
0.668812 + 0.743432i \(0.266803\pi\)
\(930\) 0 0
\(931\) −6.26564 −0.205348
\(932\) 44.2463 1.44934
\(933\) 8.39927 0.274980
\(934\) 21.3281 0.697878
\(935\) 0 0
\(936\) −42.1377 −1.37731
\(937\) 6.54322 0.213758 0.106879 0.994272i \(-0.465914\pi\)
0.106879 + 0.994272i \(0.465914\pi\)
\(938\) 2.99885 0.0979159
\(939\) 33.7044 1.09990
\(940\) 0 0
\(941\) −34.5714 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(942\) −4.94518 −0.161123
\(943\) −55.7166 −1.81438
\(944\) 2.49667 0.0812597
\(945\) 0 0
\(946\) −1.04705 −0.0340427
\(947\) −17.6584 −0.573822 −0.286911 0.957957i \(-0.592628\pi\)
−0.286911 + 0.957957i \(0.592628\pi\)
\(948\) 41.4092 1.34491
\(949\) −70.2813 −2.28143
\(950\) 0 0
\(951\) −22.4309 −0.727371
\(952\) −7.25093 −0.235004
\(953\) 42.7340 1.38429 0.692145 0.721758i \(-0.256666\pi\)
0.692145 + 0.721758i \(0.256666\pi\)
\(954\) −10.3898 −0.336383
\(955\) 0 0
\(956\) −12.7164 −0.411276
\(957\) 1.17919 0.0381179
\(958\) 12.4407 0.401941
\(959\) −17.6598 −0.570265
\(960\) 0 0
\(961\) −3.98652 −0.128597
\(962\) −1.58275 −0.0510299
\(963\) 57.2204 1.84390
\(964\) −5.34111 −0.172026
\(965\) 0 0
\(966\) 9.81670 0.315847
\(967\) 3.62541 0.116585 0.0582927 0.998300i \(-0.481434\pi\)
0.0582927 + 0.998300i \(0.481434\pi\)
\(968\) 2.41318 0.0775626
\(969\) −8.71030 −0.279815
\(970\) 0 0
\(971\) 2.86101 0.0918142 0.0459071 0.998946i \(-0.485382\pi\)
0.0459071 + 0.998946i \(0.485382\pi\)
\(972\) −34.1765 −1.09621
\(973\) −1.08140 −0.0346680
\(974\) 26.7837 0.858206
\(975\) 0 0
\(976\) 4.85990 0.155562
\(977\) −29.4990 −0.943756 −0.471878 0.881664i \(-0.656424\pi\)
−0.471878 + 0.881664i \(0.656424\pi\)
\(978\) −31.2629 −0.999676
\(979\) 3.08736 0.0986723
\(980\) 0 0
\(981\) 10.6094 0.338733
\(982\) −24.9592 −0.796480
\(983\) −31.1737 −0.994287 −0.497144 0.867668i \(-0.665618\pi\)
−0.497144 + 0.867668i \(0.665618\pi\)
\(984\) −49.4655 −1.57690
\(985\) 0 0
\(986\) 1.13663 0.0361977
\(987\) −5.29755 −0.168623
\(988\) 8.44457 0.268657
\(989\) −10.3527 −0.329198
\(990\) 0 0
\(991\) −55.9145 −1.77618 −0.888092 0.459666i \(-0.847969\pi\)
−0.888092 + 0.459666i \(0.847969\pi\)
\(992\) 30.1222 0.956382
\(993\) 46.4459 1.47392
\(994\) −7.13609 −0.226343
\(995\) 0 0
\(996\) −11.2263 −0.355719
\(997\) −42.1768 −1.33575 −0.667877 0.744272i \(-0.732797\pi\)
−0.667877 + 0.744272i \(0.732797\pi\)
\(998\) −21.4762 −0.679816
\(999\) 0.178988 0.00566292
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.p.1.5 9
5.4 even 2 1045.2.a.k.1.5 9
15.14 odd 2 9405.2.a.bh.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.k.1.5 9 5.4 even 2
5225.2.a.p.1.5 9 1.1 even 1 trivial
9405.2.a.bh.1.5 9 15.14 odd 2