Properties

Label 2175.2.a.w.1.2
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $1$
Dimension $5$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, 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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.15351\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.15351 q^{2} -1.00000 q^{3} +2.63760 q^{4} +2.15351 q^{6} -1.51591 q^{7} -1.37308 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.15351 q^{2} -1.00000 q^{3} +2.63760 q^{4} +2.15351 q^{6} -1.51591 q^{7} -1.37308 q^{8} +1.00000 q^{9} +1.88899 q^{11} -2.63760 q^{12} -0.484093 q^{13} +3.26452 q^{14} -2.31826 q^{16} -3.39257 q^{17} -2.15351 q^{18} +2.85121 q^{19} +1.51591 q^{21} -4.06795 q^{22} +1.28156 q^{23} +1.37308 q^{24} +1.04250 q^{26} -1.00000 q^{27} -3.99836 q^{28} +1.00000 q^{29} -5.47404 q^{31} +7.73856 q^{32} -1.88899 q^{33} +7.30594 q^{34} +2.63760 q^{36} -7.43995 q^{37} -6.14010 q^{38} +0.484093 q^{39} +8.28237 q^{41} -3.26452 q^{42} -1.43262 q^{43} +4.98240 q^{44} -2.75985 q^{46} +2.25328 q^{47} +2.31826 q^{48} -4.70203 q^{49} +3.39257 q^{51} -1.27684 q^{52} +5.62364 q^{53} +2.15351 q^{54} +2.08146 q^{56} -2.85121 q^{57} -2.15351 q^{58} -12.7907 q^{59} +7.40162 q^{61} +11.7884 q^{62} -1.51591 q^{63} -12.0285 q^{64} +4.06795 q^{66} +8.76646 q^{67} -8.94826 q^{68} -1.28156 q^{69} +14.5705 q^{71} -1.37308 q^{72} -2.80833 q^{73} +16.0220 q^{74} +7.52035 q^{76} -2.86353 q^{77} -1.04250 q^{78} -9.34617 q^{79} +1.00000 q^{81} -17.8362 q^{82} -1.79167 q^{83} +3.99836 q^{84} +3.08516 q^{86} -1.00000 q^{87} -2.59374 q^{88} +7.24703 q^{89} +0.733840 q^{91} +3.38025 q^{92} +5.47404 q^{93} -4.85246 q^{94} -7.73856 q^{96} -4.59510 q^{97} +10.1259 q^{98} +1.88899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{2} - 5 q^{3} + 5 q^{4} + 3 q^{6} - 8 q^{7} - 9 q^{8} + 5 q^{9} + 12 q^{11} - 5 q^{12} - 2 q^{13} + 6 q^{14} + q^{16} - 3 q^{18} - 2 q^{19} + 8 q^{21} - 14 q^{22} - 8 q^{23} + 9 q^{24} - 5 q^{27} + 6 q^{28} + 5 q^{29} + 2 q^{31} - q^{32} - 12 q^{33} + 4 q^{34} + 5 q^{36} - 16 q^{37} + 14 q^{38} + 2 q^{39} - 14 q^{41} - 6 q^{42} + 20 q^{44} - 6 q^{46} - 2 q^{47} - q^{48} - 7 q^{49} - 16 q^{52} - 26 q^{53} + 3 q^{54} - 2 q^{56} + 2 q^{57} - 3 q^{58} + 4 q^{59} - 12 q^{61} - 8 q^{63} - 9 q^{64} + 14 q^{66} - 12 q^{67} + 20 q^{68} + 8 q^{69} + 30 q^{71} - 9 q^{72} + 12 q^{73} + 2 q^{74} - 44 q^{76} - 18 q^{77} - 18 q^{79} + 5 q^{81} - 10 q^{82} - 2 q^{83} - 6 q^{84} + 30 q^{86} - 5 q^{87} - 42 q^{88} - 22 q^{89} - 12 q^{91} - 20 q^{92} - 2 q^{93} - 50 q^{94} + q^{96} - 20 q^{97} + 9 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15351 −1.52276 −0.761380 0.648305i \(-0.775478\pi\)
−0.761380 + 0.648305i \(0.775478\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.63760 1.31880
\(5\) 0 0
\(6\) 2.15351 0.879166
\(7\) −1.51591 −0.572959 −0.286480 0.958086i \(-0.592485\pi\)
−0.286480 + 0.958086i \(0.592485\pi\)
\(8\) −1.37308 −0.485458
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.88899 0.569552 0.284776 0.958594i \(-0.408081\pi\)
0.284776 + 0.958594i \(0.408081\pi\)
\(12\) −2.63760 −0.761410
\(13\) −0.484093 −0.134263 −0.0671316 0.997744i \(-0.521385\pi\)
−0.0671316 + 0.997744i \(0.521385\pi\)
\(14\) 3.26452 0.872480
\(15\) 0 0
\(16\) −2.31826 −0.579565
\(17\) −3.39257 −0.822820 −0.411410 0.911450i \(-0.634964\pi\)
−0.411410 + 0.911450i \(0.634964\pi\)
\(18\) −2.15351 −0.507587
\(19\) 2.85121 0.654112 0.327056 0.945005i \(-0.393943\pi\)
0.327056 + 0.945005i \(0.393943\pi\)
\(20\) 0 0
\(21\) 1.51591 0.330798
\(22\) −4.06795 −0.867291
\(23\) 1.28156 0.267224 0.133612 0.991034i \(-0.457342\pi\)
0.133612 + 0.991034i \(0.457342\pi\)
\(24\) 1.37308 0.280279
\(25\) 0 0
\(26\) 1.04250 0.204451
\(27\) −1.00000 −0.192450
\(28\) −3.99836 −0.755619
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −5.47404 −0.983166 −0.491583 0.870831i \(-0.663582\pi\)
−0.491583 + 0.870831i \(0.663582\pi\)
\(32\) 7.73856 1.36800
\(33\) −1.88899 −0.328831
\(34\) 7.30594 1.25296
\(35\) 0 0
\(36\) 2.63760 0.439600
\(37\) −7.43995 −1.22312 −0.611561 0.791198i \(-0.709458\pi\)
−0.611561 + 0.791198i \(0.709458\pi\)
\(38\) −6.14010 −0.996056
\(39\) 0.484093 0.0775169
\(40\) 0 0
\(41\) 8.28237 1.29349 0.646745 0.762707i \(-0.276130\pi\)
0.646745 + 0.762707i \(0.276130\pi\)
\(42\) −3.26452 −0.503726
\(43\) −1.43262 −0.218473 −0.109236 0.994016i \(-0.534841\pi\)
−0.109236 + 0.994016i \(0.534841\pi\)
\(44\) 4.98240 0.751125
\(45\) 0 0
\(46\) −2.75985 −0.406918
\(47\) 2.25328 0.328674 0.164337 0.986404i \(-0.447451\pi\)
0.164337 + 0.986404i \(0.447451\pi\)
\(48\) 2.31826 0.334612
\(49\) −4.70203 −0.671718
\(50\) 0 0
\(51\) 3.39257 0.475055
\(52\) −1.27684 −0.177066
\(53\) 5.62364 0.772466 0.386233 0.922401i \(-0.373776\pi\)
0.386233 + 0.922401i \(0.373776\pi\)
\(54\) 2.15351 0.293055
\(55\) 0 0
\(56\) 2.08146 0.278147
\(57\) −2.85121 −0.377652
\(58\) −2.15351 −0.282770
\(59\) −12.7907 −1.66520 −0.832601 0.553873i \(-0.813149\pi\)
−0.832601 + 0.553873i \(0.813149\pi\)
\(60\) 0 0
\(61\) 7.40162 0.947680 0.473840 0.880611i \(-0.342868\pi\)
0.473840 + 0.880611i \(0.342868\pi\)
\(62\) 11.7884 1.49713
\(63\) −1.51591 −0.190986
\(64\) −12.0285 −1.50357
\(65\) 0 0
\(66\) 4.06795 0.500731
\(67\) 8.76646 1.07099 0.535497 0.844537i \(-0.320124\pi\)
0.535497 + 0.844537i \(0.320124\pi\)
\(68\) −8.94826 −1.08514
\(69\) −1.28156 −0.154282
\(70\) 0 0
\(71\) 14.5705 1.72920 0.864598 0.502465i \(-0.167573\pi\)
0.864598 + 0.502465i \(0.167573\pi\)
\(72\) −1.37308 −0.161819
\(73\) −2.80833 −0.328691 −0.164345 0.986403i \(-0.552551\pi\)
−0.164345 + 0.986403i \(0.552551\pi\)
\(74\) 16.0220 1.86252
\(75\) 0 0
\(76\) 7.52035 0.862644
\(77\) −2.86353 −0.326330
\(78\) −1.04250 −0.118040
\(79\) −9.34617 −1.05153 −0.525763 0.850631i \(-0.676220\pi\)
−0.525763 + 0.850631i \(0.676220\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −17.8362 −1.96967
\(83\) −1.79167 −0.196661 −0.0983306 0.995154i \(-0.531350\pi\)
−0.0983306 + 0.995154i \(0.531350\pi\)
\(84\) 3.99836 0.436257
\(85\) 0 0
\(86\) 3.08516 0.332682
\(87\) −1.00000 −0.107211
\(88\) −2.59374 −0.276493
\(89\) 7.24703 0.768183 0.384092 0.923295i \(-0.374515\pi\)
0.384092 + 0.923295i \(0.374515\pi\)
\(90\) 0 0
\(91\) 0.733840 0.0769273
\(92\) 3.38025 0.352415
\(93\) 5.47404 0.567631
\(94\) −4.85246 −0.500493
\(95\) 0 0
\(96\) −7.73856 −0.789813
\(97\) −4.59510 −0.466562 −0.233281 0.972409i \(-0.574946\pi\)
−0.233281 + 0.972409i \(0.574946\pi\)
\(98\) 10.1259 1.02287
\(99\) 1.88899 0.189851
\(100\) 0 0
\(101\) −15.8611 −1.57823 −0.789117 0.614243i \(-0.789462\pi\)
−0.789117 + 0.614243i \(0.789462\pi\)
\(102\) −7.30594 −0.723396
\(103\) 8.46995 0.834569 0.417284 0.908776i \(-0.362982\pi\)
0.417284 + 0.908776i \(0.362982\pi\)
\(104\) 0.664699 0.0651791
\(105\) 0 0
\(106\) −12.1106 −1.17628
\(107\) −0.445502 −0.0430684 −0.0215342 0.999768i \(-0.506855\pi\)
−0.0215342 + 0.999768i \(0.506855\pi\)
\(108\) −2.63760 −0.253803
\(109\) −7.32651 −0.701752 −0.350876 0.936422i \(-0.614116\pi\)
−0.350876 + 0.936422i \(0.614116\pi\)
\(110\) 0 0
\(111\) 7.43995 0.706169
\(112\) 3.51427 0.332067
\(113\) 9.41884 0.886050 0.443025 0.896509i \(-0.353905\pi\)
0.443025 + 0.896509i \(0.353905\pi\)
\(114\) 6.14010 0.575073
\(115\) 0 0
\(116\) 2.63760 0.244895
\(117\) −0.484093 −0.0447544
\(118\) 27.5448 2.53570
\(119\) 5.14283 0.471442
\(120\) 0 0
\(121\) −7.43172 −0.675611
\(122\) −15.9394 −1.44309
\(123\) −8.28237 −0.746796
\(124\) −14.4383 −1.29660
\(125\) 0 0
\(126\) 3.26452 0.290827
\(127\) −2.22656 −0.197575 −0.0987875 0.995109i \(-0.531496\pi\)
−0.0987875 + 0.995109i \(0.531496\pi\)
\(128\) 10.4264 0.921576
\(129\) 1.43262 0.126135
\(130\) 0 0
\(131\) 15.3567 1.34172 0.670859 0.741585i \(-0.265925\pi\)
0.670859 + 0.741585i \(0.265925\pi\)
\(132\) −4.98240 −0.433662
\(133\) −4.32217 −0.374779
\(134\) −18.8787 −1.63087
\(135\) 0 0
\(136\) 4.65828 0.399444
\(137\) −1.72253 −0.147166 −0.0735828 0.997289i \(-0.523443\pi\)
−0.0735828 + 0.997289i \(0.523443\pi\)
\(138\) 2.75985 0.234934
\(139\) −6.68293 −0.566838 −0.283419 0.958996i \(-0.591469\pi\)
−0.283419 + 0.958996i \(0.591469\pi\)
\(140\) 0 0
\(141\) −2.25328 −0.189760
\(142\) −31.3776 −2.63315
\(143\) −0.914446 −0.0764698
\(144\) −2.31826 −0.193188
\(145\) 0 0
\(146\) 6.04777 0.500517
\(147\) 4.70203 0.387816
\(148\) −19.6236 −1.61305
\(149\) 0.474203 0.0388482 0.0194241 0.999811i \(-0.493817\pi\)
0.0194241 + 0.999811i \(0.493817\pi\)
\(150\) 0 0
\(151\) −12.5209 −1.01893 −0.509467 0.860490i \(-0.670157\pi\)
−0.509467 + 0.860490i \(0.670157\pi\)
\(152\) −3.91494 −0.317544
\(153\) −3.39257 −0.274273
\(154\) 6.16664 0.496922
\(155\) 0 0
\(156\) 1.27684 0.102229
\(157\) 23.8828 1.90606 0.953028 0.302881i \(-0.0979486\pi\)
0.953028 + 0.302881i \(0.0979486\pi\)
\(158\) 20.1271 1.60122
\(159\) −5.62364 −0.445984
\(160\) 0 0
\(161\) −1.94273 −0.153108
\(162\) −2.15351 −0.169196
\(163\) −14.7254 −1.15339 −0.576693 0.816961i \(-0.695657\pi\)
−0.576693 + 0.816961i \(0.695657\pi\)
\(164\) 21.8456 1.70585
\(165\) 0 0
\(166\) 3.85838 0.299468
\(167\) −2.95547 −0.228701 −0.114351 0.993440i \(-0.536479\pi\)
−0.114351 + 0.993440i \(0.536479\pi\)
\(168\) −2.08146 −0.160588
\(169\) −12.7657 −0.981973
\(170\) 0 0
\(171\) 2.85121 0.218037
\(172\) −3.77868 −0.288122
\(173\) 12.1974 0.927349 0.463675 0.886006i \(-0.346531\pi\)
0.463675 + 0.886006i \(0.346531\pi\)
\(174\) 2.15351 0.163257
\(175\) 0 0
\(176\) −4.37917 −0.330092
\(177\) 12.7907 0.961405
\(178\) −15.6065 −1.16976
\(179\) −12.3747 −0.924929 −0.462464 0.886638i \(-0.653035\pi\)
−0.462464 + 0.886638i \(0.653035\pi\)
\(180\) 0 0
\(181\) −13.1895 −0.980367 −0.490183 0.871619i \(-0.663070\pi\)
−0.490183 + 0.871619i \(0.663070\pi\)
\(182\) −1.58033 −0.117142
\(183\) −7.40162 −0.547143
\(184\) −1.75969 −0.129726
\(185\) 0 0
\(186\) −11.7884 −0.864367
\(187\) −6.40853 −0.468638
\(188\) 5.94325 0.433456
\(189\) 1.51591 0.110266
\(190\) 0 0
\(191\) −6.10636 −0.441841 −0.220920 0.975292i \(-0.570906\pi\)
−0.220920 + 0.975292i \(0.570906\pi\)
\(192\) 12.0285 0.868085
\(193\) −19.6580 −1.41501 −0.707507 0.706707i \(-0.750180\pi\)
−0.707507 + 0.706707i \(0.750180\pi\)
\(194\) 9.89560 0.710463
\(195\) 0 0
\(196\) −12.4021 −0.885862
\(197\) −24.2075 −1.72471 −0.862357 0.506301i \(-0.831013\pi\)
−0.862357 + 0.506301i \(0.831013\pi\)
\(198\) −4.06795 −0.289097
\(199\) 0.542480 0.0384554 0.0192277 0.999815i \(-0.493879\pi\)
0.0192277 + 0.999815i \(0.493879\pi\)
\(200\) 0 0
\(201\) −8.76646 −0.618339
\(202\) 34.1569 2.40327
\(203\) −1.51591 −0.106396
\(204\) 8.94826 0.626503
\(205\) 0 0
\(206\) −18.2401 −1.27085
\(207\) 1.28156 0.0890747
\(208\) 1.12225 0.0778142
\(209\) 5.38590 0.372551
\(210\) 0 0
\(211\) −1.20999 −0.0832989 −0.0416495 0.999132i \(-0.513261\pi\)
−0.0416495 + 0.999132i \(0.513261\pi\)
\(212\) 14.8329 1.01873
\(213\) −14.5705 −0.998351
\(214\) 0.959394 0.0655828
\(215\) 0 0
\(216\) 1.37308 0.0934264
\(217\) 8.29813 0.563314
\(218\) 15.7777 1.06860
\(219\) 2.80833 0.189770
\(220\) 0 0
\(221\) 1.64232 0.110474
\(222\) −16.0220 −1.07533
\(223\) −15.8267 −1.05983 −0.529917 0.848049i \(-0.677777\pi\)
−0.529917 + 0.848049i \(0.677777\pi\)
\(224\) −11.7309 −0.783806
\(225\) 0 0
\(226\) −20.2836 −1.34924
\(227\) 8.03996 0.533631 0.266815 0.963748i \(-0.414029\pi\)
0.266815 + 0.963748i \(0.414029\pi\)
\(228\) −7.52035 −0.498048
\(229\) −0.0946339 −0.00625359 −0.00312679 0.999995i \(-0.500995\pi\)
−0.00312679 + 0.999995i \(0.500995\pi\)
\(230\) 0 0
\(231\) 2.86353 0.188407
\(232\) −1.37308 −0.0901472
\(233\) −18.6579 −1.22232 −0.611159 0.791508i \(-0.709297\pi\)
−0.611159 + 0.791508i \(0.709297\pi\)
\(234\) 1.04250 0.0681502
\(235\) 0 0
\(236\) −33.7367 −2.19607
\(237\) 9.34617 0.607099
\(238\) −11.0751 −0.717894
\(239\) −25.3243 −1.63809 −0.819047 0.573726i \(-0.805497\pi\)
−0.819047 + 0.573726i \(0.805497\pi\)
\(240\) 0 0
\(241\) −28.5508 −1.83912 −0.919559 0.392951i \(-0.871454\pi\)
−0.919559 + 0.392951i \(0.871454\pi\)
\(242\) 16.0043 1.02879
\(243\) −1.00000 −0.0641500
\(244\) 19.5225 1.24980
\(245\) 0 0
\(246\) 17.8362 1.13719
\(247\) −1.38025 −0.0878232
\(248\) 7.51630 0.477285
\(249\) 1.79167 0.113542
\(250\) 0 0
\(251\) −16.4952 −1.04117 −0.520584 0.853810i \(-0.674286\pi\)
−0.520584 + 0.853810i \(0.674286\pi\)
\(252\) −3.99836 −0.251873
\(253\) 2.42086 0.152198
\(254\) 4.79491 0.300860
\(255\) 0 0
\(256\) 1.60362 0.100226
\(257\) −13.7659 −0.858694 −0.429347 0.903140i \(-0.641256\pi\)
−0.429347 + 0.903140i \(0.641256\pi\)
\(258\) −3.08516 −0.192074
\(259\) 11.2783 0.700798
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −33.0707 −2.04312
\(263\) 16.7039 1.03001 0.515004 0.857188i \(-0.327790\pi\)
0.515004 + 0.857188i \(0.327790\pi\)
\(264\) 2.59374 0.159633
\(265\) 0 0
\(266\) 9.30783 0.570700
\(267\) −7.24703 −0.443511
\(268\) 23.1224 1.41243
\(269\) −17.3518 −1.05796 −0.528979 0.848635i \(-0.677425\pi\)
−0.528979 + 0.848635i \(0.677425\pi\)
\(270\) 0 0
\(271\) 8.27747 0.502821 0.251410 0.967881i \(-0.419106\pi\)
0.251410 + 0.967881i \(0.419106\pi\)
\(272\) 7.86487 0.476878
\(273\) −0.733840 −0.0444140
\(274\) 3.70948 0.224098
\(275\) 0 0
\(276\) −3.38025 −0.203467
\(277\) −18.0560 −1.08488 −0.542439 0.840095i \(-0.682499\pi\)
−0.542439 + 0.840095i \(0.682499\pi\)
\(278\) 14.3917 0.863159
\(279\) −5.47404 −0.327722
\(280\) 0 0
\(281\) −18.6241 −1.11102 −0.555510 0.831510i \(-0.687477\pi\)
−0.555510 + 0.831510i \(0.687477\pi\)
\(282\) 4.85246 0.288960
\(283\) −16.6920 −0.992235 −0.496117 0.868256i \(-0.665241\pi\)
−0.496117 + 0.868256i \(0.665241\pi\)
\(284\) 38.4311 2.28046
\(285\) 0 0
\(286\) 1.96927 0.116445
\(287\) −12.5553 −0.741116
\(288\) 7.73856 0.455999
\(289\) −5.49045 −0.322968
\(290\) 0 0
\(291\) 4.59510 0.269370
\(292\) −7.40727 −0.433478
\(293\) −32.9866 −1.92710 −0.963549 0.267532i \(-0.913792\pi\)
−0.963549 + 0.267532i \(0.913792\pi\)
\(294\) −10.1259 −0.590552
\(295\) 0 0
\(296\) 10.2157 0.593773
\(297\) −1.88899 −0.109610
\(298\) −1.02120 −0.0591566
\(299\) −0.620395 −0.0358783
\(300\) 0 0
\(301\) 2.17172 0.125176
\(302\) 26.9638 1.55159
\(303\) 15.8611 0.911194
\(304\) −6.60984 −0.379100
\(305\) 0 0
\(306\) 7.30594 0.417653
\(307\) 5.20522 0.297077 0.148539 0.988907i \(-0.452543\pi\)
0.148539 + 0.988907i \(0.452543\pi\)
\(308\) −7.55286 −0.430364
\(309\) −8.46995 −0.481838
\(310\) 0 0
\(311\) 15.1777 0.860651 0.430325 0.902674i \(-0.358399\pi\)
0.430325 + 0.902674i \(0.358399\pi\)
\(312\) −0.664699 −0.0376312
\(313\) 6.91509 0.390864 0.195432 0.980717i \(-0.437389\pi\)
0.195432 + 0.980717i \(0.437389\pi\)
\(314\) −51.4319 −2.90247
\(315\) 0 0
\(316\) −24.6515 −1.38675
\(317\) −24.6694 −1.38557 −0.692786 0.721144i \(-0.743617\pi\)
−0.692786 + 0.721144i \(0.743617\pi\)
\(318\) 12.1106 0.679126
\(319\) 1.88899 0.105763
\(320\) 0 0
\(321\) 0.445502 0.0248655
\(322\) 4.18368 0.233148
\(323\) −9.67293 −0.538216
\(324\) 2.63760 0.146533
\(325\) 0 0
\(326\) 31.7114 1.75633
\(327\) 7.32651 0.405157
\(328\) −11.3724 −0.627934
\(329\) −3.41576 −0.188317
\(330\) 0 0
\(331\) −4.24031 −0.233069 −0.116534 0.993187i \(-0.537178\pi\)
−0.116534 + 0.993187i \(0.537178\pi\)
\(332\) −4.72571 −0.259357
\(333\) −7.43995 −0.407707
\(334\) 6.36463 0.348257
\(335\) 0 0
\(336\) −3.51427 −0.191719
\(337\) 26.2776 1.43143 0.715717 0.698390i \(-0.246100\pi\)
0.715717 + 0.698390i \(0.246100\pi\)
\(338\) 27.4910 1.49531
\(339\) −9.41884 −0.511561
\(340\) 0 0
\(341\) −10.3404 −0.559964
\(342\) −6.14010 −0.332019
\(343\) 17.7392 0.957826
\(344\) 1.96711 0.106059
\(345\) 0 0
\(346\) −26.2672 −1.41213
\(347\) −21.7745 −1.16892 −0.584458 0.811424i \(-0.698693\pi\)
−0.584458 + 0.811424i \(0.698693\pi\)
\(348\) −2.63760 −0.141390
\(349\) 8.94807 0.478979 0.239490 0.970899i \(-0.423020\pi\)
0.239490 + 0.970899i \(0.423020\pi\)
\(350\) 0 0
\(351\) 0.484093 0.0258390
\(352\) 14.6180 0.779145
\(353\) −7.14944 −0.380526 −0.190263 0.981733i \(-0.560934\pi\)
−0.190263 + 0.981733i \(0.560934\pi\)
\(354\) −27.5448 −1.46399
\(355\) 0 0
\(356\) 19.1148 1.01308
\(357\) −5.14283 −0.272187
\(358\) 26.6490 1.40845
\(359\) 31.4641 1.66061 0.830305 0.557309i \(-0.188166\pi\)
0.830305 + 0.557309i \(0.188166\pi\)
\(360\) 0 0
\(361\) −10.8706 −0.572137
\(362\) 28.4037 1.49286
\(363\) 7.43172 0.390064
\(364\) 1.93558 0.101452
\(365\) 0 0
\(366\) 15.9394 0.833168
\(367\) 27.7722 1.44970 0.724849 0.688908i \(-0.241910\pi\)
0.724849 + 0.688908i \(0.241910\pi\)
\(368\) −2.97099 −0.154874
\(369\) 8.28237 0.431163
\(370\) 0 0
\(371\) −8.52491 −0.442592
\(372\) 14.4383 0.748593
\(373\) 15.9293 0.824790 0.412395 0.911005i \(-0.364692\pi\)
0.412395 + 0.911005i \(0.364692\pi\)
\(374\) 13.8008 0.713624
\(375\) 0 0
\(376\) −3.09394 −0.159558
\(377\) −0.484093 −0.0249320
\(378\) −3.26452 −0.167909
\(379\) 18.4181 0.946073 0.473037 0.881043i \(-0.343158\pi\)
0.473037 + 0.881043i \(0.343158\pi\)
\(380\) 0 0
\(381\) 2.22656 0.114070
\(382\) 13.1501 0.672818
\(383\) 14.1973 0.725447 0.362724 0.931897i \(-0.381847\pi\)
0.362724 + 0.931897i \(0.381847\pi\)
\(384\) −10.4264 −0.532072
\(385\) 0 0
\(386\) 42.3337 2.15473
\(387\) −1.43262 −0.0728242
\(388\) −12.1201 −0.615303
\(389\) −25.0087 −1.26799 −0.633995 0.773337i \(-0.718586\pi\)
−0.633995 + 0.773337i \(0.718586\pi\)
\(390\) 0 0
\(391\) −4.34779 −0.219877
\(392\) 6.45626 0.326091
\(393\) −15.3567 −0.774641
\(394\) 52.1311 2.62633
\(395\) 0 0
\(396\) 4.98240 0.250375
\(397\) 3.67651 0.184519 0.0922595 0.995735i \(-0.470591\pi\)
0.0922595 + 0.995735i \(0.470591\pi\)
\(398\) −1.16824 −0.0585584
\(399\) 4.32217 0.216379
\(400\) 0 0
\(401\) −4.68500 −0.233958 −0.116979 0.993134i \(-0.537321\pi\)
−0.116979 + 0.993134i \(0.537321\pi\)
\(402\) 18.8787 0.941582
\(403\) 2.64994 0.132003
\(404\) −41.8352 −2.08138
\(405\) 0 0
\(406\) 3.26452 0.162015
\(407\) −14.0540 −0.696630
\(408\) −4.65828 −0.230619
\(409\) 36.8798 1.82359 0.911794 0.410647i \(-0.134697\pi\)
0.911794 + 0.410647i \(0.134697\pi\)
\(410\) 0 0
\(411\) 1.72253 0.0849661
\(412\) 22.3403 1.10063
\(413\) 19.3895 0.954092
\(414\) −2.75985 −0.135639
\(415\) 0 0
\(416\) −3.74618 −0.183672
\(417\) 6.68293 0.327264
\(418\) −11.5986 −0.567305
\(419\) 19.1400 0.935050 0.467525 0.883980i \(-0.345146\pi\)
0.467525 + 0.883980i \(0.345146\pi\)
\(420\) 0 0
\(421\) 29.5114 1.43830 0.719149 0.694856i \(-0.244532\pi\)
0.719149 + 0.694856i \(0.244532\pi\)
\(422\) 2.60572 0.126844
\(423\) 2.25328 0.109558
\(424\) −7.72171 −0.375000
\(425\) 0 0
\(426\) 31.3776 1.52025
\(427\) −11.2202 −0.542982
\(428\) −1.17506 −0.0567986
\(429\) 0.914446 0.0441499
\(430\) 0 0
\(431\) 13.6750 0.658703 0.329352 0.944207i \(-0.393170\pi\)
0.329352 + 0.944207i \(0.393170\pi\)
\(432\) 2.31826 0.111537
\(433\) −34.7905 −1.67192 −0.835961 0.548788i \(-0.815089\pi\)
−0.835961 + 0.548788i \(0.815089\pi\)
\(434\) −17.8701 −0.857793
\(435\) 0 0
\(436\) −19.3244 −0.925472
\(437\) 3.65400 0.174794
\(438\) −6.04777 −0.288974
\(439\) −17.0869 −0.815513 −0.407757 0.913091i \(-0.633689\pi\)
−0.407757 + 0.913091i \(0.633689\pi\)
\(440\) 0 0
\(441\) −4.70203 −0.223906
\(442\) −3.53675 −0.168226
\(443\) −10.2997 −0.489351 −0.244676 0.969605i \(-0.578682\pi\)
−0.244676 + 0.969605i \(0.578682\pi\)
\(444\) 19.6236 0.931297
\(445\) 0 0
\(446\) 34.0830 1.61388
\(447\) −0.474203 −0.0224290
\(448\) 18.2341 0.861482
\(449\) 15.3331 0.723615 0.361808 0.932253i \(-0.382160\pi\)
0.361808 + 0.932253i \(0.382160\pi\)
\(450\) 0 0
\(451\) 15.6453 0.736709
\(452\) 24.8431 1.16852
\(453\) 12.5209 0.588281
\(454\) −17.3141 −0.812592
\(455\) 0 0
\(456\) 3.91494 0.183334
\(457\) −10.1841 −0.476394 −0.238197 0.971217i \(-0.576556\pi\)
−0.238197 + 0.971217i \(0.576556\pi\)
\(458\) 0.203795 0.00952272
\(459\) 3.39257 0.158352
\(460\) 0 0
\(461\) −37.2567 −1.73522 −0.867609 0.497247i \(-0.834344\pi\)
−0.867609 + 0.497247i \(0.834344\pi\)
\(462\) −6.16664 −0.286898
\(463\) 24.2457 1.12680 0.563398 0.826186i \(-0.309494\pi\)
0.563398 + 0.826186i \(0.309494\pi\)
\(464\) −2.31826 −0.107623
\(465\) 0 0
\(466\) 40.1799 1.86130
\(467\) 28.5544 1.32134 0.660670 0.750676i \(-0.270272\pi\)
0.660670 + 0.750676i \(0.270272\pi\)
\(468\) −1.27684 −0.0590221
\(469\) −13.2891 −0.613636
\(470\) 0 0
\(471\) −23.8828 −1.10046
\(472\) 17.5626 0.808385
\(473\) −2.70620 −0.124431
\(474\) −20.1271 −0.924466
\(475\) 0 0
\(476\) 13.5647 0.621738
\(477\) 5.62364 0.257489
\(478\) 54.5362 2.49443
\(479\) 20.8455 0.952455 0.476227 0.879322i \(-0.342004\pi\)
0.476227 + 0.879322i \(0.342004\pi\)
\(480\) 0 0
\(481\) 3.60163 0.164220
\(482\) 61.4844 2.80054
\(483\) 1.94273 0.0883972
\(484\) −19.6019 −0.890996
\(485\) 0 0
\(486\) 2.15351 0.0976852
\(487\) 5.31743 0.240956 0.120478 0.992716i \(-0.461557\pi\)
0.120478 + 0.992716i \(0.461557\pi\)
\(488\) −10.1630 −0.460058
\(489\) 14.7254 0.665907
\(490\) 0 0
\(491\) 34.6711 1.56469 0.782343 0.622848i \(-0.214025\pi\)
0.782343 + 0.622848i \(0.214025\pi\)
\(492\) −21.8456 −0.984876
\(493\) −3.39257 −0.152794
\(494\) 2.97238 0.133734
\(495\) 0 0
\(496\) 12.6902 0.569809
\(497\) −22.0875 −0.990758
\(498\) −3.85838 −0.172898
\(499\) −37.2788 −1.66883 −0.834414 0.551138i \(-0.814194\pi\)
−0.834414 + 0.551138i \(0.814194\pi\)
\(500\) 0 0
\(501\) 2.95547 0.132041
\(502\) 35.5226 1.58545
\(503\) −10.7589 −0.479717 −0.239858 0.970808i \(-0.577101\pi\)
−0.239858 + 0.970808i \(0.577101\pi\)
\(504\) 2.08146 0.0927158
\(505\) 0 0
\(506\) −5.21333 −0.231761
\(507\) 12.7657 0.566943
\(508\) −5.87277 −0.260562
\(509\) −0.424406 −0.0188115 −0.00940573 0.999956i \(-0.502994\pi\)
−0.00940573 + 0.999956i \(0.502994\pi\)
\(510\) 0 0
\(511\) 4.25717 0.188326
\(512\) −24.3063 −1.07420
\(513\) −2.85121 −0.125884
\(514\) 29.6450 1.30759
\(515\) 0 0
\(516\) 3.77868 0.166347
\(517\) 4.25642 0.187197
\(518\) −24.2879 −1.06715
\(519\) −12.1974 −0.535405
\(520\) 0 0
\(521\) 6.39104 0.279996 0.139998 0.990152i \(-0.455290\pi\)
0.139998 + 0.990152i \(0.455290\pi\)
\(522\) −2.15351 −0.0942565
\(523\) −24.8564 −1.08689 −0.543446 0.839444i \(-0.682881\pi\)
−0.543446 + 0.839444i \(0.682881\pi\)
\(524\) 40.5048 1.76946
\(525\) 0 0
\(526\) −35.9721 −1.56846
\(527\) 18.5711 0.808968
\(528\) 4.37917 0.190579
\(529\) −21.3576 −0.928591
\(530\) 0 0
\(531\) −12.7907 −0.555067
\(532\) −11.4002 −0.494260
\(533\) −4.00944 −0.173668
\(534\) 15.6065 0.675361
\(535\) 0 0
\(536\) −12.0371 −0.519922
\(537\) 12.3747 0.534008
\(538\) 37.3673 1.61102
\(539\) −8.88207 −0.382578
\(540\) 0 0
\(541\) 13.2290 0.568759 0.284379 0.958712i \(-0.408212\pi\)
0.284379 + 0.958712i \(0.408212\pi\)
\(542\) −17.8256 −0.765676
\(543\) 13.1895 0.566015
\(544\) −26.2536 −1.12561
\(545\) 0 0
\(546\) 1.58033 0.0676319
\(547\) −43.7132 −1.86904 −0.934521 0.355908i \(-0.884172\pi\)
−0.934521 + 0.355908i \(0.884172\pi\)
\(548\) −4.54334 −0.194082
\(549\) 7.40162 0.315893
\(550\) 0 0
\(551\) 2.85121 0.121466
\(552\) 1.75969 0.0748973
\(553\) 14.1679 0.602481
\(554\) 38.8837 1.65201
\(555\) 0 0
\(556\) −17.6269 −0.747547
\(557\) −18.7407 −0.794067 −0.397033 0.917804i \(-0.629960\pi\)
−0.397033 + 0.917804i \(0.629960\pi\)
\(558\) 11.7884 0.499042
\(559\) 0.693521 0.0293328
\(560\) 0 0
\(561\) 6.40853 0.270568
\(562\) 40.1071 1.69182
\(563\) −6.52697 −0.275079 −0.137539 0.990496i \(-0.543919\pi\)
−0.137539 + 0.990496i \(0.543919\pi\)
\(564\) −5.94325 −0.250256
\(565\) 0 0
\(566\) 35.9463 1.51094
\(567\) −1.51591 −0.0636621
\(568\) −20.0064 −0.839451
\(569\) 24.4702 1.02585 0.512923 0.858435i \(-0.328563\pi\)
0.512923 + 0.858435i \(0.328563\pi\)
\(570\) 0 0
\(571\) 45.1164 1.88806 0.944031 0.329856i \(-0.107000\pi\)
0.944031 + 0.329856i \(0.107000\pi\)
\(572\) −2.41194 −0.100848
\(573\) 6.10636 0.255097
\(574\) 27.0380 1.12854
\(575\) 0 0
\(576\) −12.0285 −0.501189
\(577\) −25.4834 −1.06089 −0.530444 0.847720i \(-0.677975\pi\)
−0.530444 + 0.847720i \(0.677975\pi\)
\(578\) 11.8237 0.491803
\(579\) 19.6580 0.816958
\(580\) 0 0
\(581\) 2.71600 0.112679
\(582\) −9.89560 −0.410186
\(583\) 10.6230 0.439959
\(584\) 3.85607 0.159565
\(585\) 0 0
\(586\) 71.0370 2.93451
\(587\) 42.3930 1.74975 0.874873 0.484352i \(-0.160945\pi\)
0.874873 + 0.484352i \(0.160945\pi\)
\(588\) 12.4021 0.511453
\(589\) −15.6076 −0.643101
\(590\) 0 0
\(591\) 24.2075 0.995764
\(592\) 17.2477 0.708878
\(593\) −5.71303 −0.234606 −0.117303 0.993096i \(-0.537425\pi\)
−0.117303 + 0.993096i \(0.537425\pi\)
\(594\) 4.06795 0.166910
\(595\) 0 0
\(596\) 1.25076 0.0512331
\(597\) −0.542480 −0.0222022
\(598\) 1.33603 0.0546341
\(599\) −14.0026 −0.572132 −0.286066 0.958210i \(-0.592348\pi\)
−0.286066 + 0.958210i \(0.592348\pi\)
\(600\) 0 0
\(601\) −22.6834 −0.925277 −0.462639 0.886547i \(-0.653097\pi\)
−0.462639 + 0.886547i \(0.653097\pi\)
\(602\) −4.67682 −0.190613
\(603\) 8.76646 0.356998
\(604\) −33.0250 −1.34377
\(605\) 0 0
\(606\) −34.1569 −1.38753
\(607\) 27.7581 1.12667 0.563333 0.826230i \(-0.309519\pi\)
0.563333 + 0.826230i \(0.309519\pi\)
\(608\) 22.0642 0.894823
\(609\) 1.51591 0.0614277
\(610\) 0 0
\(611\) −1.09080 −0.0441289
\(612\) −8.94826 −0.361712
\(613\) 12.0619 0.487176 0.243588 0.969879i \(-0.421675\pi\)
0.243588 + 0.969879i \(0.421675\pi\)
\(614\) −11.2095 −0.452378
\(615\) 0 0
\(616\) 3.93186 0.158419
\(617\) −45.0988 −1.81561 −0.907804 0.419394i \(-0.862243\pi\)
−0.907804 + 0.419394i \(0.862243\pi\)
\(618\) 18.2401 0.733725
\(619\) −35.5206 −1.42769 −0.713846 0.700302i \(-0.753049\pi\)
−0.713846 + 0.700302i \(0.753049\pi\)
\(620\) 0 0
\(621\) −1.28156 −0.0514273
\(622\) −32.6854 −1.31057
\(623\) −10.9858 −0.440138
\(624\) −1.12225 −0.0449261
\(625\) 0 0
\(626\) −14.8917 −0.595192
\(627\) −5.38590 −0.215092
\(628\) 62.9934 2.51371
\(629\) 25.2406 1.00641
\(630\) 0 0
\(631\) −32.0402 −1.27550 −0.637750 0.770243i \(-0.720135\pi\)
−0.637750 + 0.770243i \(0.720135\pi\)
\(632\) 12.8330 0.510471
\(633\) 1.20999 0.0480927
\(634\) 53.1258 2.10989
\(635\) 0 0
\(636\) −14.8329 −0.588164
\(637\) 2.27622 0.0901870
\(638\) −4.06795 −0.161052
\(639\) 14.5705 0.576398
\(640\) 0 0
\(641\) −28.4963 −1.12554 −0.562769 0.826614i \(-0.690264\pi\)
−0.562769 + 0.826614i \(0.690264\pi\)
\(642\) −0.959394 −0.0378642
\(643\) 43.1625 1.70216 0.851081 0.525035i \(-0.175948\pi\)
0.851081 + 0.525035i \(0.175948\pi\)
\(644\) −5.12414 −0.201920
\(645\) 0 0
\(646\) 20.8307 0.819575
\(647\) −32.5875 −1.28115 −0.640573 0.767898i \(-0.721303\pi\)
−0.640573 + 0.767898i \(0.721303\pi\)
\(648\) −1.37308 −0.0539397
\(649\) −24.1614 −0.948418
\(650\) 0 0
\(651\) −8.29813 −0.325229
\(652\) −38.8398 −1.52109
\(653\) −28.1951 −1.10336 −0.551679 0.834057i \(-0.686013\pi\)
−0.551679 + 0.834057i \(0.686013\pi\)
\(654\) −15.7777 −0.616957
\(655\) 0 0
\(656\) −19.2007 −0.749661
\(657\) −2.80833 −0.109564
\(658\) 7.35587 0.286762
\(659\) 42.7564 1.66555 0.832776 0.553611i \(-0.186750\pi\)
0.832776 + 0.553611i \(0.186750\pi\)
\(660\) 0 0
\(661\) 17.4726 0.679606 0.339803 0.940497i \(-0.389640\pi\)
0.339803 + 0.940497i \(0.389640\pi\)
\(662\) 9.13155 0.354908
\(663\) −1.64232 −0.0637824
\(664\) 2.46011 0.0954707
\(665\) 0 0
\(666\) 16.0220 0.620840
\(667\) 1.28156 0.0496223
\(668\) −7.79535 −0.301611
\(669\) 15.8267 0.611896
\(670\) 0 0
\(671\) 13.9816 0.539753
\(672\) 11.7309 0.452531
\(673\) −12.5821 −0.485006 −0.242503 0.970151i \(-0.577968\pi\)
−0.242503 + 0.970151i \(0.577968\pi\)
\(674\) −56.5891 −2.17973
\(675\) 0 0
\(676\) −33.6707 −1.29503
\(677\) −27.7829 −1.06778 −0.533892 0.845553i \(-0.679271\pi\)
−0.533892 + 0.845553i \(0.679271\pi\)
\(678\) 20.2836 0.778985
\(679\) 6.96575 0.267321
\(680\) 0 0
\(681\) −8.03996 −0.308092
\(682\) 22.2681 0.852691
\(683\) −41.1587 −1.57489 −0.787446 0.616383i \(-0.788597\pi\)
−0.787446 + 0.616383i \(0.788597\pi\)
\(684\) 7.52035 0.287548
\(685\) 0 0
\(686\) −38.2015 −1.45854
\(687\) 0.0946339 0.00361051
\(688\) 3.32119 0.126619
\(689\) −2.72236 −0.103714
\(690\) 0 0
\(691\) −6.51859 −0.247979 −0.123989 0.992284i \(-0.539569\pi\)
−0.123989 + 0.992284i \(0.539569\pi\)
\(692\) 32.1718 1.22299
\(693\) −2.86353 −0.108777
\(694\) 46.8916 1.77998
\(695\) 0 0
\(696\) 1.37308 0.0520465
\(697\) −28.0985 −1.06431
\(698\) −19.2698 −0.729371
\(699\) 18.6579 0.705706
\(700\) 0 0
\(701\) 3.82210 0.144359 0.0721793 0.997392i \(-0.477005\pi\)
0.0721793 + 0.997392i \(0.477005\pi\)
\(702\) −1.04250 −0.0393466
\(703\) −21.2129 −0.800058
\(704\) −22.7218 −0.856359
\(705\) 0 0
\(706\) 15.3964 0.579450
\(707\) 24.0439 0.904264
\(708\) 33.7367 1.26790
\(709\) 4.67767 0.175673 0.0878367 0.996135i \(-0.472005\pi\)
0.0878367 + 0.996135i \(0.472005\pi\)
\(710\) 0 0
\(711\) −9.34617 −0.350509
\(712\) −9.95076 −0.372920
\(713\) −7.01532 −0.262726
\(714\) 11.0751 0.414476
\(715\) 0 0
\(716\) −32.6395 −1.21980
\(717\) 25.3243 0.945754
\(718\) −67.7582 −2.52871
\(719\) 26.0976 0.973276 0.486638 0.873604i \(-0.338223\pi\)
0.486638 + 0.873604i \(0.338223\pi\)
\(720\) 0 0
\(721\) −12.8397 −0.478174
\(722\) 23.4100 0.871228
\(723\) 28.5508 1.06182
\(724\) −34.7886 −1.29291
\(725\) 0 0
\(726\) −16.0043 −0.593975
\(727\) 42.0811 1.56070 0.780352 0.625341i \(-0.215040\pi\)
0.780352 + 0.625341i \(0.215040\pi\)
\(728\) −1.00762 −0.0373449
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.86027 0.179764
\(732\) −19.5225 −0.721573
\(733\) −5.06122 −0.186940 −0.0934702 0.995622i \(-0.529796\pi\)
−0.0934702 + 0.995622i \(0.529796\pi\)
\(734\) −59.8077 −2.20754
\(735\) 0 0
\(736\) 9.91744 0.365562
\(737\) 16.5598 0.609986
\(738\) −17.8362 −0.656558
\(739\) −26.3474 −0.969203 −0.484601 0.874735i \(-0.661035\pi\)
−0.484601 + 0.874735i \(0.661035\pi\)
\(740\) 0 0
\(741\) 1.38025 0.0507047
\(742\) 18.3585 0.673961
\(743\) −1.17798 −0.0432158 −0.0216079 0.999767i \(-0.506879\pi\)
−0.0216079 + 0.999767i \(0.506879\pi\)
\(744\) −7.51630 −0.275561
\(745\) 0 0
\(746\) −34.3040 −1.25596
\(747\) −1.79167 −0.0655537
\(748\) −16.9032 −0.618041
\(749\) 0.675340 0.0246764
\(750\) 0 0
\(751\) −18.1100 −0.660844 −0.330422 0.943833i \(-0.607191\pi\)
−0.330422 + 0.943833i \(0.607191\pi\)
\(752\) −5.22369 −0.190488
\(753\) 16.4952 0.601119
\(754\) 1.04250 0.0379655
\(755\) 0 0
\(756\) 3.99836 0.145419
\(757\) −4.02431 −0.146266 −0.0731331 0.997322i \(-0.523300\pi\)
−0.0731331 + 0.997322i \(0.523300\pi\)
\(758\) −39.6635 −1.44064
\(759\) −2.42086 −0.0878715
\(760\) 0 0
\(761\) 35.2780 1.27883 0.639414 0.768863i \(-0.279177\pi\)
0.639414 + 0.768863i \(0.279177\pi\)
\(762\) −4.79491 −0.173701
\(763\) 11.1063 0.402075
\(764\) −16.1062 −0.582700
\(765\) 0 0
\(766\) −30.5740 −1.10468
\(767\) 6.19186 0.223575
\(768\) −1.60362 −0.0578658
\(769\) 37.6829 1.35888 0.679440 0.733731i \(-0.262223\pi\)
0.679440 + 0.733731i \(0.262223\pi\)
\(770\) 0 0
\(771\) 13.7659 0.495767
\(772\) −51.8500 −1.86612
\(773\) −25.4551 −0.915558 −0.457779 0.889066i \(-0.651355\pi\)
−0.457779 + 0.889066i \(0.651355\pi\)
\(774\) 3.08516 0.110894
\(775\) 0 0
\(776\) 6.30945 0.226496
\(777\) −11.2783 −0.404606
\(778\) 53.8564 1.93085
\(779\) 23.6148 0.846087
\(780\) 0 0
\(781\) 27.5234 0.984866
\(782\) 9.36301 0.334820
\(783\) −1.00000 −0.0357371
\(784\) 10.9005 0.389304
\(785\) 0 0
\(786\) 33.0707 1.17959
\(787\) 27.0582 0.964521 0.482261 0.876028i \(-0.339816\pi\)
0.482261 + 0.876028i \(0.339816\pi\)
\(788\) −63.8498 −2.27455
\(789\) −16.7039 −0.594676
\(790\) 0 0
\(791\) −14.2781 −0.507670
\(792\) −2.59374 −0.0921644
\(793\) −3.58307 −0.127238
\(794\) −7.91741 −0.280978
\(795\) 0 0
\(796\) 1.43085 0.0507150
\(797\) 43.5501 1.54262 0.771312 0.636457i \(-0.219601\pi\)
0.771312 + 0.636457i \(0.219601\pi\)
\(798\) −9.30783 −0.329494
\(799\) −7.64441 −0.270440
\(800\) 0 0
\(801\) 7.24703 0.256061
\(802\) 10.0892 0.356262
\(803\) −5.30491 −0.187206
\(804\) −23.1224 −0.815466
\(805\) 0 0
\(806\) −5.70667 −0.201009
\(807\) 17.3518 0.610812
\(808\) 21.7785 0.766166
\(809\) 32.0173 1.12567 0.562834 0.826570i \(-0.309711\pi\)
0.562834 + 0.826570i \(0.309711\pi\)
\(810\) 0 0
\(811\) 4.19698 0.147376 0.0736880 0.997281i \(-0.476523\pi\)
0.0736880 + 0.997281i \(0.476523\pi\)
\(812\) −3.99836 −0.140315
\(813\) −8.27747 −0.290304
\(814\) 30.2654 1.06080
\(815\) 0 0
\(816\) −7.86487 −0.275325
\(817\) −4.08470 −0.142906
\(818\) −79.4210 −2.77689
\(819\) 0.733840 0.0256424
\(820\) 0 0
\(821\) 47.5461 1.65937 0.829686 0.558230i \(-0.188519\pi\)
0.829686 + 0.558230i \(0.188519\pi\)
\(822\) −3.70948 −0.129383
\(823\) −13.3520 −0.465422 −0.232711 0.972546i \(-0.574760\pi\)
−0.232711 + 0.972546i \(0.574760\pi\)
\(824\) −11.6299 −0.405148
\(825\) 0 0
\(826\) −41.7554 −1.45285
\(827\) 1.07408 0.0373493 0.0186747 0.999826i \(-0.494055\pi\)
0.0186747 + 0.999826i \(0.494055\pi\)
\(828\) 3.38025 0.117472
\(829\) −53.3629 −1.85337 −0.926684 0.375840i \(-0.877354\pi\)
−0.926684 + 0.375840i \(0.877354\pi\)
\(830\) 0 0
\(831\) 18.0560 0.626355
\(832\) 5.82293 0.201874
\(833\) 15.9520 0.552703
\(834\) −14.3917 −0.498345
\(835\) 0 0
\(836\) 14.2059 0.491320
\(837\) 5.47404 0.189210
\(838\) −41.2182 −1.42386
\(839\) 1.82410 0.0629751 0.0314875 0.999504i \(-0.489976\pi\)
0.0314875 + 0.999504i \(0.489976\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −63.5531 −2.19019
\(843\) 18.6241 0.641448
\(844\) −3.19146 −0.109855
\(845\) 0 0
\(846\) −4.85246 −0.166831
\(847\) 11.2658 0.387097
\(848\) −13.0371 −0.447694
\(849\) 16.6920 0.572867
\(850\) 0 0
\(851\) −9.53476 −0.326847
\(852\) −38.4311 −1.31663
\(853\) 26.3499 0.902203 0.451102 0.892473i \(-0.351031\pi\)
0.451102 + 0.892473i \(0.351031\pi\)
\(854\) 24.1627 0.826831
\(855\) 0 0
\(856\) 0.611711 0.0209079
\(857\) −39.7239 −1.35694 −0.678472 0.734627i \(-0.737357\pi\)
−0.678472 + 0.734627i \(0.737357\pi\)
\(858\) −1.96927 −0.0672297
\(859\) −39.1847 −1.33696 −0.668482 0.743729i \(-0.733055\pi\)
−0.668482 + 0.743729i \(0.733055\pi\)
\(860\) 0 0
\(861\) 12.5553 0.427884
\(862\) −29.4493 −1.00305
\(863\) −23.6601 −0.805399 −0.402699 0.915332i \(-0.631928\pi\)
−0.402699 + 0.915332i \(0.631928\pi\)
\(864\) −7.73856 −0.263271
\(865\) 0 0
\(866\) 74.9216 2.54594
\(867\) 5.49045 0.186465
\(868\) 21.8872 0.742899
\(869\) −17.6548 −0.598898
\(870\) 0 0
\(871\) −4.24378 −0.143795
\(872\) 10.0599 0.340671
\(873\) −4.59510 −0.155521
\(874\) −7.86892 −0.266170
\(875\) 0 0
\(876\) 7.40727 0.250268
\(877\) 52.7791 1.78222 0.891112 0.453783i \(-0.149926\pi\)
0.891112 + 0.453783i \(0.149926\pi\)
\(878\) 36.7968 1.24183
\(879\) 32.9866 1.11261
\(880\) 0 0
\(881\) −15.0520 −0.507115 −0.253558 0.967320i \(-0.581601\pi\)
−0.253558 + 0.967320i \(0.581601\pi\)
\(882\) 10.1259 0.340955
\(883\) −37.7972 −1.27198 −0.635989 0.771698i \(-0.719408\pi\)
−0.635989 + 0.771698i \(0.719408\pi\)
\(884\) 4.33179 0.145694
\(885\) 0 0
\(886\) 22.1804 0.745165
\(887\) 4.81353 0.161622 0.0808112 0.996729i \(-0.474249\pi\)
0.0808112 + 0.996729i \(0.474249\pi\)
\(888\) −10.2157 −0.342815
\(889\) 3.37526 0.113202
\(890\) 0 0
\(891\) 1.88899 0.0632835
\(892\) −41.7446 −1.39771
\(893\) 6.42457 0.214990
\(894\) 1.02120 0.0341541
\(895\) 0 0
\(896\) −15.8055 −0.528026
\(897\) 0.620395 0.0207144
\(898\) −33.0201 −1.10189
\(899\) −5.47404 −0.182569
\(900\) 0 0
\(901\) −19.0786 −0.635600
\(902\) −33.6923 −1.12183
\(903\) −2.17172 −0.0722703
\(904\) −12.9328 −0.430140
\(905\) 0 0
\(906\) −26.9638 −0.895812
\(907\) −43.5796 −1.44704 −0.723518 0.690305i \(-0.757476\pi\)
−0.723518 + 0.690305i \(0.757476\pi\)
\(908\) 21.2062 0.703753
\(909\) −15.8611 −0.526078
\(910\) 0 0
\(911\) −15.9296 −0.527770 −0.263885 0.964554i \(-0.585004\pi\)
−0.263885 + 0.964554i \(0.585004\pi\)
\(912\) 6.60984 0.218874
\(913\) −3.38444 −0.112009
\(914\) 21.9316 0.725434
\(915\) 0 0
\(916\) −0.249607 −0.00824724
\(917\) −23.2793 −0.768750
\(918\) −7.30594 −0.241132
\(919\) 44.8752 1.48030 0.740148 0.672444i \(-0.234755\pi\)
0.740148 + 0.672444i \(0.234755\pi\)
\(920\) 0 0
\(921\) −5.20522 −0.171518
\(922\) 80.2327 2.64232
\(923\) −7.05345 −0.232167
\(924\) 7.55286 0.248471
\(925\) 0 0
\(926\) −52.2134 −1.71584
\(927\) 8.46995 0.278190
\(928\) 7.73856 0.254031
\(929\) −34.3242 −1.12614 −0.563071 0.826409i \(-0.690380\pi\)
−0.563071 + 0.826409i \(0.690380\pi\)
\(930\) 0 0
\(931\) −13.4065 −0.439379
\(932\) −49.2121 −1.61200
\(933\) −15.1777 −0.496897
\(934\) −61.4922 −2.01209
\(935\) 0 0
\(936\) 0.664699 0.0217264
\(937\) −21.0494 −0.687653 −0.343827 0.939033i \(-0.611723\pi\)
−0.343827 + 0.939033i \(0.611723\pi\)
\(938\) 28.6183 0.934421
\(939\) −6.91509 −0.225665
\(940\) 0 0
\(941\) 46.2494 1.50769 0.753844 0.657053i \(-0.228197\pi\)
0.753844 + 0.657053i \(0.228197\pi\)
\(942\) 51.4319 1.67574
\(943\) 10.6144 0.345651
\(944\) 29.6521 0.965093
\(945\) 0 0
\(946\) 5.82784 0.189479
\(947\) 21.7174 0.705720 0.352860 0.935676i \(-0.385209\pi\)
0.352860 + 0.935676i \(0.385209\pi\)
\(948\) 24.6515 0.800642
\(949\) 1.35949 0.0441310
\(950\) 0 0
\(951\) 24.6694 0.799960
\(952\) −7.06152 −0.228865
\(953\) 16.6122 0.538120 0.269060 0.963123i \(-0.413287\pi\)
0.269060 + 0.963123i \(0.413287\pi\)
\(954\) −12.1106 −0.392094
\(955\) 0 0
\(956\) −66.7955 −2.16032
\(957\) −1.88899 −0.0610623
\(958\) −44.8910 −1.45036
\(959\) 2.61119 0.0843198
\(960\) 0 0
\(961\) −1.03492 −0.0333845
\(962\) −7.75614 −0.250068
\(963\) −0.445502 −0.0143561
\(964\) −75.3056 −2.42543
\(965\) 0 0
\(966\) −4.18368 −0.134608
\(967\) −12.1886 −0.391960 −0.195980 0.980608i \(-0.562789\pi\)
−0.195980 + 0.980608i \(0.562789\pi\)
\(968\) 10.2044 0.327981
\(969\) 9.67293 0.310739
\(970\) 0 0
\(971\) −10.0067 −0.321131 −0.160566 0.987025i \(-0.551332\pi\)
−0.160566 + 0.987025i \(0.551332\pi\)
\(972\) −2.63760 −0.0846011
\(973\) 10.1307 0.324775
\(974\) −11.4511 −0.366918
\(975\) 0 0
\(976\) −17.1589 −0.549242
\(977\) −23.0188 −0.736435 −0.368218 0.929740i \(-0.620032\pi\)
−0.368218 + 0.929740i \(0.620032\pi\)
\(978\) −31.7114 −1.01402
\(979\) 13.6895 0.437520
\(980\) 0 0
\(981\) −7.32651 −0.233917
\(982\) −74.6645 −2.38264
\(983\) −54.9480 −1.75257 −0.876285 0.481793i \(-0.839986\pi\)
−0.876285 + 0.481793i \(0.839986\pi\)
\(984\) 11.3724 0.362538
\(985\) 0 0
\(986\) 7.30594 0.232668
\(987\) 3.41576 0.108725
\(988\) −3.64055 −0.115821
\(989\) −1.83599 −0.0583811
\(990\) 0 0
\(991\) −8.69215 −0.276115 −0.138058 0.990424i \(-0.544086\pi\)
−0.138058 + 0.990424i \(0.544086\pi\)
\(992\) −42.3612 −1.34497
\(993\) 4.24031 0.134562
\(994\) 47.5656 1.50869
\(995\) 0 0
\(996\) 4.72571 0.149740
\(997\) 18.8275 0.596274 0.298137 0.954523i \(-0.403635\pi\)
0.298137 + 0.954523i \(0.403635\pi\)
\(998\) 80.2802 2.54123
\(999\) 7.43995 0.235390
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 2175.2.a.w.1.2 5
3.2 odd 2 6525.2.a.bs.1.4 5
5.2 odd 4 435.2.c.e.349.2 10
5.3 odd 4 435.2.c.e.349.9 yes 10
5.4 even 2 2175.2.a.z.1.4 5
15.2 even 4 1305.2.c.j.784.9 10
15.8 even 4 1305.2.c.j.784.2 10
15.14 odd 2 6525.2.a.bl.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.e.349.2 10 5.2 odd 4
435.2.c.e.349.9 yes 10 5.3 odd 4
1305.2.c.j.784.2 10 15.8 even 4
1305.2.c.j.784.9 10 15.2 even 4
2175.2.a.w.1.2 5 1.1 even 1 trivial
2175.2.a.z.1.4 5 5.4 even 2
6525.2.a.bl.1.2 5 15.14 odd 2
6525.2.a.bs.1.4 5 3.2 odd 2