Properties

Label 825.2.a.h.1.1
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $1$
Dimension $3$
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] = [825,2,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(6.58765816676\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.67513 q^{2} +1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} -2.80606 q^{7} -8.44358 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.67513 q^{2} +1.00000 q^{3} +5.15633 q^{4} -2.67513 q^{6} -2.80606 q^{7} -8.44358 q^{8} +1.00000 q^{9} -1.00000 q^{11} +5.15633 q^{12} +5.11871 q^{13} +7.50659 q^{14} +12.2750 q^{16} -4.54420 q^{17} -2.67513 q^{18} -4.57452 q^{19} -2.80606 q^{21} +2.67513 q^{22} -4.00000 q^{23} -8.44358 q^{24} -13.6932 q^{26} +1.00000 q^{27} -14.4690 q^{28} -2.38787 q^{29} -0.962389 q^{31} -15.9502 q^{32} -1.00000 q^{33} +12.1563 q^{34} +5.15633 q^{36} -1.61213 q^{37} +12.2374 q^{38} +5.11871 q^{39} -2.38787 q^{41} +7.50659 q^{42} -2.80606 q^{43} -5.15633 q^{44} +10.7005 q^{46} +4.31265 q^{47} +12.2750 q^{48} +0.873992 q^{49} -4.54420 q^{51} +26.3938 q^{52} +6.57452 q^{53} -2.67513 q^{54} +23.6932 q^{56} -4.57452 q^{57} +6.38787 q^{58} -13.2750 q^{59} +7.92478 q^{61} +2.57452 q^{62} -2.80606 q^{63} +18.1187 q^{64} +2.67513 q^{66} -10.7005 q^{67} -23.4314 q^{68} -4.00000 q^{69} -7.35026 q^{71} -8.44358 q^{72} -6.41819 q^{73} +4.31265 q^{74} -23.5877 q^{76} +2.80606 q^{77} -13.6932 q^{78} +1.35026 q^{79} +1.00000 q^{81} +6.38787 q^{82} +0.806063 q^{83} -14.4690 q^{84} +7.50659 q^{86} -2.38787 q^{87} +8.44358 q^{88} -2.96239 q^{89} -14.3634 q^{91} -20.6253 q^{92} -0.962389 q^{93} -11.5369 q^{94} -15.9502 q^{96} +9.92478 q^{97} -2.33804 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 5 q^{12} - 6 q^{13} + 2 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 2 q^{19} - 8 q^{21} + 3 q^{22} - 12 q^{23} - 9 q^{24} - 8 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} + 8 q^{31} - 11 q^{32} - 3 q^{33} + 26 q^{34} + 5 q^{36} - 4 q^{37} - 6 q^{38} - 6 q^{39} - 8 q^{41} + 2 q^{42} - 8 q^{43} - 5 q^{44} + 12 q^{46} - 8 q^{47} + 5 q^{48} + 11 q^{49} - 4 q^{51} + 26 q^{52} + 8 q^{53} - 3 q^{54} + 38 q^{56} - 2 q^{57} + 20 q^{58} - 8 q^{59} + 2 q^{61} - 4 q^{62} - 8 q^{63} + 33 q^{64} + 3 q^{66} - 12 q^{67} - 28 q^{68} - 12 q^{69} - 12 q^{71} - 9 q^{72} - 18 q^{73} - 8 q^{74} - 18 q^{76} + 8 q^{77} - 8 q^{78} - 6 q^{79} + 3 q^{81} + 20 q^{82} + 2 q^{83} - 12 q^{84} + 2 q^{86} - 8 q^{87} + 9 q^{88} + 2 q^{89} + 8 q^{91} - 20 q^{92} + 8 q^{93} - 12 q^{94} - 11 q^{96} + 8 q^{97} + 29 q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.67513 −1.89160 −0.945802 0.324745i \(-0.894721\pi\)
−0.945802 + 0.324745i \(0.894721\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.15633 2.57816
\(5\) 0 0
\(6\) −2.67513 −1.09212
\(7\) −2.80606 −1.06059 −0.530296 0.847812i \(-0.677919\pi\)
−0.530296 + 0.847812i \(0.677919\pi\)
\(8\) −8.44358 −2.98526
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 5.15633 1.48850
\(13\) 5.11871 1.41968 0.709838 0.704365i \(-0.248768\pi\)
0.709838 + 0.704365i \(0.248768\pi\)
\(14\) 7.50659 2.00622
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) −4.54420 −1.10213 −0.551065 0.834462i \(-0.685778\pi\)
−0.551065 + 0.834462i \(0.685778\pi\)
\(18\) −2.67513 −0.630534
\(19\) −4.57452 −1.04947 −0.524733 0.851267i \(-0.675835\pi\)
−0.524733 + 0.851267i \(0.675835\pi\)
\(20\) 0 0
\(21\) −2.80606 −0.612333
\(22\) 2.67513 0.570340
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −8.44358 −1.72354
\(25\) 0 0
\(26\) −13.6932 −2.68546
\(27\) 1.00000 0.192450
\(28\) −14.4690 −2.73438
\(29\) −2.38787 −0.443417 −0.221708 0.975113i \(-0.571163\pi\)
−0.221708 + 0.975113i \(0.571163\pi\)
\(30\) 0 0
\(31\) −0.962389 −0.172850 −0.0864250 0.996258i \(-0.527544\pi\)
−0.0864250 + 0.996258i \(0.527544\pi\)
\(32\) −15.9502 −2.81962
\(33\) −1.00000 −0.174078
\(34\) 12.1563 2.08479
\(35\) 0 0
\(36\) 5.15633 0.859388
\(37\) −1.61213 −0.265032 −0.132516 0.991181i \(-0.542306\pi\)
−0.132516 + 0.991181i \(0.542306\pi\)
\(38\) 12.2374 1.98517
\(39\) 5.11871 0.819650
\(40\) 0 0
\(41\) −2.38787 −0.372923 −0.186462 0.982462i \(-0.559702\pi\)
−0.186462 + 0.982462i \(0.559702\pi\)
\(42\) 7.50659 1.15829
\(43\) −2.80606 −0.427921 −0.213960 0.976842i \(-0.568636\pi\)
−0.213960 + 0.976842i \(0.568636\pi\)
\(44\) −5.15633 −0.777345
\(45\) 0 0
\(46\) 10.7005 1.57771
\(47\) 4.31265 0.629065 0.314532 0.949247i \(-0.398152\pi\)
0.314532 + 0.949247i \(0.398152\pi\)
\(48\) 12.2750 1.77175
\(49\) 0.873992 0.124856
\(50\) 0 0
\(51\) −4.54420 −0.636315
\(52\) 26.3938 3.66015
\(53\) 6.57452 0.903079 0.451540 0.892251i \(-0.350875\pi\)
0.451540 + 0.892251i \(0.350875\pi\)
\(54\) −2.67513 −0.364039
\(55\) 0 0
\(56\) 23.6932 3.16614
\(57\) −4.57452 −0.605909
\(58\) 6.38787 0.838769
\(59\) −13.2750 −1.72826 −0.864131 0.503266i \(-0.832132\pi\)
−0.864131 + 0.503266i \(0.832132\pi\)
\(60\) 0 0
\(61\) 7.92478 1.01466 0.507332 0.861751i \(-0.330632\pi\)
0.507332 + 0.861751i \(0.330632\pi\)
\(62\) 2.57452 0.326964
\(63\) −2.80606 −0.353531
\(64\) 18.1187 2.26484
\(65\) 0 0
\(66\) 2.67513 0.329286
\(67\) −10.7005 −1.30728 −0.653639 0.756807i \(-0.726758\pi\)
−0.653639 + 0.756807i \(0.726758\pi\)
\(68\) −23.4314 −2.84147
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −7.35026 −0.872316 −0.436158 0.899870i \(-0.643661\pi\)
−0.436158 + 0.899870i \(0.643661\pi\)
\(72\) −8.44358 −0.995086
\(73\) −6.41819 −0.751192 −0.375596 0.926783i \(-0.622562\pi\)
−0.375596 + 0.926783i \(0.622562\pi\)
\(74\) 4.31265 0.501335
\(75\) 0 0
\(76\) −23.5877 −2.70569
\(77\) 2.80606 0.319781
\(78\) −13.6932 −1.55045
\(79\) 1.35026 0.151916 0.0759582 0.997111i \(-0.475798\pi\)
0.0759582 + 0.997111i \(0.475798\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.38787 0.705423
\(83\) 0.806063 0.0884770 0.0442385 0.999021i \(-0.485914\pi\)
0.0442385 + 0.999021i \(0.485914\pi\)
\(84\) −14.4690 −1.57869
\(85\) 0 0
\(86\) 7.50659 0.809456
\(87\) −2.38787 −0.256007
\(88\) 8.44358 0.900089
\(89\) −2.96239 −0.314013 −0.157006 0.987598i \(-0.550184\pi\)
−0.157006 + 0.987598i \(0.550184\pi\)
\(90\) 0 0
\(91\) −14.3634 −1.50570
\(92\) −20.6253 −2.15034
\(93\) −0.962389 −0.0997950
\(94\) −11.5369 −1.18994
\(95\) 0 0
\(96\) −15.9502 −1.62791
\(97\) 9.92478 1.00771 0.503854 0.863789i \(-0.331915\pi\)
0.503854 + 0.863789i \(0.331915\pi\)
\(98\) −2.33804 −0.236178
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −13.6121 −1.35446 −0.677229 0.735773i \(-0.736819\pi\)
−0.677229 + 0.735773i \(0.736819\pi\)
\(102\) 12.1563 1.20366
\(103\) −16.3127 −1.60733 −0.803667 0.595080i \(-0.797120\pi\)
−0.803667 + 0.595080i \(0.797120\pi\)
\(104\) −43.2203 −4.23810
\(105\) 0 0
\(106\) −17.5877 −1.70827
\(107\) 9.43136 0.911764 0.455882 0.890040i \(-0.349324\pi\)
0.455882 + 0.890040i \(0.349324\pi\)
\(108\) 5.15633 0.496168
\(109\) −15.4010 −1.47515 −0.737576 0.675264i \(-0.764030\pi\)
−0.737576 + 0.675264i \(0.764030\pi\)
\(110\) 0 0
\(111\) −1.61213 −0.153016
\(112\) −34.4445 −3.25470
\(113\) 13.7381 1.29238 0.646188 0.763179i \(-0.276362\pi\)
0.646188 + 0.763179i \(0.276362\pi\)
\(114\) 12.2374 1.14614
\(115\) 0 0
\(116\) −12.3127 −1.14320
\(117\) 5.11871 0.473225
\(118\) 35.5125 3.26919
\(119\) 12.7513 1.16891
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −21.1998 −1.91934
\(123\) −2.38787 −0.215307
\(124\) −4.96239 −0.445636
\(125\) 0 0
\(126\) 7.50659 0.668740
\(127\) −12.4182 −1.10194 −0.550968 0.834526i \(-0.685741\pi\)
−0.550968 + 0.834526i \(0.685741\pi\)
\(128\) −16.5696 −1.46456
\(129\) −2.80606 −0.247060
\(130\) 0 0
\(131\) 5.92478 0.517650 0.258825 0.965924i \(-0.416665\pi\)
0.258825 + 0.965924i \(0.416665\pi\)
\(132\) −5.15633 −0.448800
\(133\) 12.8364 1.11306
\(134\) 28.6253 2.47285
\(135\) 0 0
\(136\) 38.3693 3.29014
\(137\) −9.79877 −0.837165 −0.418583 0.908179i \(-0.637473\pi\)
−0.418583 + 0.908179i \(0.637473\pi\)
\(138\) 10.7005 0.910889
\(139\) −2.12601 −0.180326 −0.0901628 0.995927i \(-0.528739\pi\)
−0.0901628 + 0.995927i \(0.528739\pi\)
\(140\) 0 0
\(141\) 4.31265 0.363191
\(142\) 19.6629 1.65007
\(143\) −5.11871 −0.428048
\(144\) 12.2750 1.02292
\(145\) 0 0
\(146\) 17.1695 1.42096
\(147\) 0.873992 0.0720856
\(148\) −8.31265 −0.683296
\(149\) 8.31265 0.680999 0.340499 0.940245i \(-0.389404\pi\)
0.340499 + 0.940245i \(0.389404\pi\)
\(150\) 0 0
\(151\) −15.2750 −1.24307 −0.621533 0.783388i \(-0.713490\pi\)
−0.621533 + 0.783388i \(0.713490\pi\)
\(152\) 38.6253 3.13293
\(153\) −4.54420 −0.367377
\(154\) −7.50659 −0.604898
\(155\) 0 0
\(156\) 26.3938 2.11319
\(157\) −7.01317 −0.559712 −0.279856 0.960042i \(-0.590287\pi\)
−0.279856 + 0.960042i \(0.590287\pi\)
\(158\) −3.61213 −0.287365
\(159\) 6.57452 0.521393
\(160\) 0 0
\(161\) 11.2243 0.884595
\(162\) −2.67513 −0.210178
\(163\) −6.76116 −0.529575 −0.264787 0.964307i \(-0.585302\pi\)
−0.264787 + 0.964307i \(0.585302\pi\)
\(164\) −12.3127 −0.961456
\(165\) 0 0
\(166\) −2.15633 −0.167363
\(167\) 11.8192 0.914600 0.457300 0.889312i \(-0.348817\pi\)
0.457300 + 0.889312i \(0.348817\pi\)
\(168\) 23.6932 1.82797
\(169\) 13.2012 1.01548
\(170\) 0 0
\(171\) −4.57452 −0.349822
\(172\) −14.4690 −1.10325
\(173\) −6.99271 −0.531646 −0.265823 0.964022i \(-0.585644\pi\)
−0.265823 + 0.964022i \(0.585644\pi\)
\(174\) 6.38787 0.484263
\(175\) 0 0
\(176\) −12.2750 −0.925266
\(177\) −13.2750 −0.997813
\(178\) 7.92478 0.593987
\(179\) −2.70052 −0.201847 −0.100923 0.994894i \(-0.532180\pi\)
−0.100923 + 0.994894i \(0.532180\pi\)
\(180\) 0 0
\(181\) −21.1998 −1.57577 −0.787885 0.615822i \(-0.788824\pi\)
−0.787885 + 0.615822i \(0.788824\pi\)
\(182\) 38.4241 2.84818
\(183\) 7.92478 0.585816
\(184\) 33.7743 2.48988
\(185\) 0 0
\(186\) 2.57452 0.188773
\(187\) 4.54420 0.332305
\(188\) 22.2374 1.62183
\(189\) −2.80606 −0.204111
\(190\) 0 0
\(191\) 13.1490 0.951430 0.475715 0.879599i \(-0.342189\pi\)
0.475715 + 0.879599i \(0.342189\pi\)
\(192\) 18.1187 1.30761
\(193\) −5.89446 −0.424293 −0.212146 0.977238i \(-0.568045\pi\)
−0.212146 + 0.977238i \(0.568045\pi\)
\(194\) −26.5501 −1.90618
\(195\) 0 0
\(196\) 4.50659 0.321899
\(197\) −20.3938 −1.45299 −0.726497 0.687169i \(-0.758853\pi\)
−0.726497 + 0.687169i \(0.758853\pi\)
\(198\) 2.67513 0.190113
\(199\) 17.4010 1.23353 0.616764 0.787148i \(-0.288443\pi\)
0.616764 + 0.787148i \(0.288443\pi\)
\(200\) 0 0
\(201\) −10.7005 −0.754757
\(202\) 36.4142 2.56210
\(203\) 6.70052 0.470285
\(204\) −23.4314 −1.64052
\(205\) 0 0
\(206\) 43.6385 3.04044
\(207\) −4.00000 −0.278019
\(208\) 62.8324 4.35664
\(209\) 4.57452 0.316426
\(210\) 0 0
\(211\) 18.1260 1.24785 0.623923 0.781486i \(-0.285538\pi\)
0.623923 + 0.781486i \(0.285538\pi\)
\(212\) 33.9003 2.32828
\(213\) −7.35026 −0.503632
\(214\) −25.2301 −1.72470
\(215\) 0 0
\(216\) −8.44358 −0.574513
\(217\) 2.70052 0.183323
\(218\) 41.1998 2.79040
\(219\) −6.41819 −0.433701
\(220\) 0 0
\(221\) −23.2605 −1.56467
\(222\) 4.31265 0.289446
\(223\) −23.6385 −1.58295 −0.791475 0.611202i \(-0.790686\pi\)
−0.791475 + 0.611202i \(0.790686\pi\)
\(224\) 44.7572 2.99047
\(225\) 0 0
\(226\) −36.7513 −2.44466
\(227\) 1.26916 0.0842371 0.0421185 0.999113i \(-0.486589\pi\)
0.0421185 + 0.999113i \(0.486589\pi\)
\(228\) −23.5877 −1.56213
\(229\) 16.1768 1.06899 0.534496 0.845171i \(-0.320501\pi\)
0.534496 + 0.845171i \(0.320501\pi\)
\(230\) 0 0
\(231\) 2.80606 0.184625
\(232\) 20.1622 1.32371
\(233\) 18.4690 1.20994 0.604971 0.796247i \(-0.293185\pi\)
0.604971 + 0.796247i \(0.293185\pi\)
\(234\) −13.6932 −0.895154
\(235\) 0 0
\(236\) −68.4504 −4.45574
\(237\) 1.35026 0.0877089
\(238\) −34.1114 −2.21111
\(239\) 19.3258 1.25008 0.625042 0.780591i \(-0.285082\pi\)
0.625042 + 0.780591i \(0.285082\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) −2.67513 −0.171964
\(243\) 1.00000 0.0641500
\(244\) 40.8627 2.61597
\(245\) 0 0
\(246\) 6.38787 0.407276
\(247\) −23.4156 −1.48990
\(248\) 8.12601 0.516002
\(249\) 0.806063 0.0510822
\(250\) 0 0
\(251\) 23.1998 1.46436 0.732180 0.681112i \(-0.238503\pi\)
0.732180 + 0.681112i \(0.238503\pi\)
\(252\) −14.4690 −0.911460
\(253\) 4.00000 0.251478
\(254\) 33.2203 2.08443
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) 10.8872 0.679123 0.339561 0.940584i \(-0.389721\pi\)
0.339561 + 0.940584i \(0.389721\pi\)
\(258\) 7.50659 0.467340
\(259\) 4.52373 0.281091
\(260\) 0 0
\(261\) −2.38787 −0.147806
\(262\) −15.8496 −0.979189
\(263\) −9.11871 −0.562284 −0.281142 0.959666i \(-0.590713\pi\)
−0.281142 + 0.959666i \(0.590713\pi\)
\(264\) 8.44358 0.519667
\(265\) 0 0
\(266\) −34.3390 −2.10546
\(267\) −2.96239 −0.181295
\(268\) −55.1754 −3.37037
\(269\) −11.2995 −0.688941 −0.344471 0.938797i \(-0.611942\pi\)
−0.344471 + 0.938797i \(0.611942\pi\)
\(270\) 0 0
\(271\) 25.1998 1.53078 0.765390 0.643567i \(-0.222546\pi\)
0.765390 + 0.643567i \(0.222546\pi\)
\(272\) −55.7802 −3.38217
\(273\) −14.3634 −0.869315
\(274\) 26.2130 1.58358
\(275\) 0 0
\(276\) −20.6253 −1.24150
\(277\) 2.41819 0.145295 0.0726475 0.997358i \(-0.476855\pi\)
0.0726475 + 0.997358i \(0.476855\pi\)
\(278\) 5.68735 0.341105
\(279\) −0.962389 −0.0576167
\(280\) 0 0
\(281\) 30.4894 1.81885 0.909424 0.415870i \(-0.136523\pi\)
0.909424 + 0.415870i \(0.136523\pi\)
\(282\) −11.5369 −0.687013
\(283\) −8.35756 −0.496805 −0.248403 0.968657i \(-0.579906\pi\)
−0.248403 + 0.968657i \(0.579906\pi\)
\(284\) −37.9003 −2.24897
\(285\) 0 0
\(286\) 13.6932 0.809698
\(287\) 6.70052 0.395519
\(288\) −15.9502 −0.939873
\(289\) 3.64974 0.214690
\(290\) 0 0
\(291\) 9.92478 0.581801
\(292\) −33.0943 −1.93670
\(293\) 23.0943 1.34918 0.674591 0.738192i \(-0.264320\pi\)
0.674591 + 0.738192i \(0.264320\pi\)
\(294\) −2.33804 −0.136357
\(295\) 0 0
\(296\) 13.6121 0.791189
\(297\) −1.00000 −0.0580259
\(298\) −22.2374 −1.28818
\(299\) −20.4749 −1.18409
\(300\) 0 0
\(301\) 7.87399 0.453849
\(302\) 40.8627 2.35139
\(303\) −13.6121 −0.781996
\(304\) −56.1524 −3.22056
\(305\) 0 0
\(306\) 12.1563 0.694931
\(307\) −2.65562 −0.151564 −0.0757821 0.997124i \(-0.524145\pi\)
−0.0757821 + 0.997124i \(0.524145\pi\)
\(308\) 14.4690 0.824446
\(309\) −16.3127 −0.927994
\(310\) 0 0
\(311\) −15.9756 −0.905891 −0.452946 0.891538i \(-0.649627\pi\)
−0.452946 + 0.891538i \(0.649627\pi\)
\(312\) −43.2203 −2.44687
\(313\) 0.0606343 0.00342726 0.00171363 0.999999i \(-0.499455\pi\)
0.00171363 + 0.999999i \(0.499455\pi\)
\(314\) 18.7612 1.05875
\(315\) 0 0
\(316\) 6.96239 0.391665
\(317\) −3.81336 −0.214180 −0.107090 0.994249i \(-0.534153\pi\)
−0.107090 + 0.994249i \(0.534153\pi\)
\(318\) −17.5877 −0.986269
\(319\) 2.38787 0.133695
\(320\) 0 0
\(321\) 9.43136 0.526407
\(322\) −30.0263 −1.67330
\(323\) 20.7875 1.15665
\(324\) 5.15633 0.286463
\(325\) 0 0
\(326\) 18.0870 1.00175
\(327\) −15.4010 −0.851680
\(328\) 20.1622 1.11327
\(329\) −12.1016 −0.667181
\(330\) 0 0
\(331\) 24.2882 1.33500 0.667500 0.744609i \(-0.267364\pi\)
0.667500 + 0.744609i \(0.267364\pi\)
\(332\) 4.15633 0.228108
\(333\) −1.61213 −0.0883440
\(334\) −31.6180 −1.73006
\(335\) 0 0
\(336\) −34.4445 −1.87910
\(337\) 22.5804 1.23003 0.615016 0.788514i \(-0.289149\pi\)
0.615016 + 0.788514i \(0.289149\pi\)
\(338\) −35.3150 −1.92088
\(339\) 13.7381 0.746153
\(340\) 0 0
\(341\) 0.962389 0.0521163
\(342\) 12.2374 0.661724
\(343\) 17.1900 0.928171
\(344\) 23.6932 1.27745
\(345\) 0 0
\(346\) 18.7064 1.00566
\(347\) −20.1925 −1.08399 −0.541996 0.840381i \(-0.682331\pi\)
−0.541996 + 0.840381i \(0.682331\pi\)
\(348\) −12.3127 −0.660027
\(349\) −27.2506 −1.45869 −0.729346 0.684145i \(-0.760175\pi\)
−0.729346 + 0.684145i \(0.760175\pi\)
\(350\) 0 0
\(351\) 5.11871 0.273217
\(352\) 15.9502 0.850147
\(353\) −5.02302 −0.267349 −0.133674 0.991025i \(-0.542678\pi\)
−0.133674 + 0.991025i \(0.542678\pi\)
\(354\) 35.5125 1.88747
\(355\) 0 0
\(356\) −15.2750 −0.809575
\(357\) 12.7513 0.674871
\(358\) 7.22425 0.381814
\(359\) −18.1768 −0.959334 −0.479667 0.877450i \(-0.659243\pi\)
−0.479667 + 0.877450i \(0.659243\pi\)
\(360\) 0 0
\(361\) 1.92619 0.101379
\(362\) 56.7123 2.98073
\(363\) 1.00000 0.0524864
\(364\) −74.0625 −3.88193
\(365\) 0 0
\(366\) −21.1998 −1.10813
\(367\) 8.56467 0.447072 0.223536 0.974696i \(-0.428240\pi\)
0.223536 + 0.974696i \(0.428240\pi\)
\(368\) −49.1002 −2.55952
\(369\) −2.38787 −0.124308
\(370\) 0 0
\(371\) −18.4485 −0.957799
\(372\) −4.96239 −0.257288
\(373\) −7.81924 −0.404865 −0.202432 0.979296i \(-0.564885\pi\)
−0.202432 + 0.979296i \(0.564885\pi\)
\(374\) −12.1563 −0.628589
\(375\) 0 0
\(376\) −36.4142 −1.87792
\(377\) −12.2228 −0.629508
\(378\) 7.50659 0.386097
\(379\) 1.14903 0.0590218 0.0295109 0.999564i \(-0.490605\pi\)
0.0295109 + 0.999564i \(0.490605\pi\)
\(380\) 0 0
\(381\) −12.4182 −0.636203
\(382\) −35.1754 −1.79973
\(383\) −34.0870 −1.74176 −0.870882 0.491493i \(-0.836451\pi\)
−0.870882 + 0.491493i \(0.836451\pi\)
\(384\) −16.5696 −0.845563
\(385\) 0 0
\(386\) 15.7685 0.802593
\(387\) −2.80606 −0.142640
\(388\) 51.1754 2.59804
\(389\) 23.7743 1.20541 0.602703 0.797965i \(-0.294090\pi\)
0.602703 + 0.797965i \(0.294090\pi\)
\(390\) 0 0
\(391\) 18.1768 0.919240
\(392\) −7.37962 −0.372727
\(393\) 5.92478 0.298865
\(394\) 54.5560 2.74849
\(395\) 0 0
\(396\) −5.15633 −0.259115
\(397\) 4.15045 0.208305 0.104152 0.994561i \(-0.466787\pi\)
0.104152 + 0.994561i \(0.466787\pi\)
\(398\) −46.5501 −2.33334
\(399\) 12.8364 0.642623
\(400\) 0 0
\(401\) −33.9149 −1.69363 −0.846815 0.531887i \(-0.821483\pi\)
−0.846815 + 0.531887i \(0.821483\pi\)
\(402\) 28.6253 1.42770
\(403\) −4.92619 −0.245391
\(404\) −70.1886 −3.49201
\(405\) 0 0
\(406\) −17.9248 −0.889592
\(407\) 1.61213 0.0799102
\(408\) 38.3693 1.89956
\(409\) 8.55008 0.422774 0.211387 0.977402i \(-0.432202\pi\)
0.211387 + 0.977402i \(0.432202\pi\)
\(410\) 0 0
\(411\) −9.79877 −0.483338
\(412\) −84.1133 −4.14397
\(413\) 37.2506 1.83298
\(414\) 10.7005 0.525902
\(415\) 0 0
\(416\) −81.6444 −4.00294
\(417\) −2.12601 −0.104111
\(418\) −12.2374 −0.598552
\(419\) 13.2995 0.649722 0.324861 0.945762i \(-0.394682\pi\)
0.324861 + 0.945762i \(0.394682\pi\)
\(420\) 0 0
\(421\) 33.3014 1.62301 0.811505 0.584345i \(-0.198649\pi\)
0.811505 + 0.584345i \(0.198649\pi\)
\(422\) −48.4894 −2.36043
\(423\) 4.31265 0.209688
\(424\) −55.5125 −2.69592
\(425\) 0 0
\(426\) 19.6629 0.952671
\(427\) −22.2374 −1.07614
\(428\) 48.6312 2.35068
\(429\) −5.11871 −0.247134
\(430\) 0 0
\(431\) 2.07522 0.0999600 0.0499800 0.998750i \(-0.484084\pi\)
0.0499800 + 0.998750i \(0.484084\pi\)
\(432\) 12.2750 0.590583
\(433\) 37.4010 1.79738 0.898690 0.438585i \(-0.144520\pi\)
0.898690 + 0.438585i \(0.144520\pi\)
\(434\) −7.22425 −0.346775
\(435\) 0 0
\(436\) −79.4128 −3.80318
\(437\) 18.2981 0.875315
\(438\) 17.1695 0.820390
\(439\) −6.02444 −0.287531 −0.143765 0.989612i \(-0.545921\pi\)
−0.143765 + 0.989612i \(0.545921\pi\)
\(440\) 0 0
\(441\) 0.873992 0.0416187
\(442\) 62.2247 2.95973
\(443\) 10.1359 0.481569 0.240785 0.970579i \(-0.422595\pi\)
0.240785 + 0.970579i \(0.422595\pi\)
\(444\) −8.31265 −0.394501
\(445\) 0 0
\(446\) 63.2360 2.99431
\(447\) 8.31265 0.393175
\(448\) −50.8423 −2.40207
\(449\) −11.4861 −0.542063 −0.271032 0.962570i \(-0.587365\pi\)
−0.271032 + 0.962570i \(0.587365\pi\)
\(450\) 0 0
\(451\) 2.38787 0.112441
\(452\) 70.8383 3.33195
\(453\) −15.2750 −0.717684
\(454\) −3.39517 −0.159343
\(455\) 0 0
\(456\) 38.6253 1.80880
\(457\) 15.0435 0.703705 0.351852 0.936055i \(-0.385552\pi\)
0.351852 + 0.936055i \(0.385552\pi\)
\(458\) −43.2750 −2.02211
\(459\) −4.54420 −0.212105
\(460\) 0 0
\(461\) −26.2374 −1.22200 −0.610999 0.791631i \(-0.709232\pi\)
−0.610999 + 0.791631i \(0.709232\pi\)
\(462\) −7.50659 −0.349238
\(463\) 5.46168 0.253826 0.126913 0.991914i \(-0.459493\pi\)
0.126913 + 0.991914i \(0.459493\pi\)
\(464\) −29.3112 −1.36074
\(465\) 0 0
\(466\) −49.4069 −2.28873
\(467\) 2.70052 0.124965 0.0624827 0.998046i \(-0.480098\pi\)
0.0624827 + 0.998046i \(0.480098\pi\)
\(468\) 26.3938 1.22005
\(469\) 30.0263 1.38649
\(470\) 0 0
\(471\) −7.01317 −0.323150
\(472\) 112.089 5.15931
\(473\) 2.80606 0.129023
\(474\) −3.61213 −0.165910
\(475\) 0 0
\(476\) 65.7499 3.01364
\(477\) 6.57452 0.301026
\(478\) −51.6991 −2.36466
\(479\) 16.7757 0.766503 0.383252 0.923644i \(-0.374804\pi\)
0.383252 + 0.923644i \(0.374804\pi\)
\(480\) 0 0
\(481\) −8.25202 −0.376260
\(482\) −76.3752 −3.47879
\(483\) 11.2243 0.510721
\(484\) 5.15633 0.234378
\(485\) 0 0
\(486\) −2.67513 −0.121346
\(487\) 34.3996 1.55880 0.779398 0.626529i \(-0.215525\pi\)
0.779398 + 0.626529i \(0.215525\pi\)
\(488\) −66.9135 −3.02903
\(489\) −6.76116 −0.305750
\(490\) 0 0
\(491\) 14.9525 0.674799 0.337399 0.941362i \(-0.390453\pi\)
0.337399 + 0.941362i \(0.390453\pi\)
\(492\) −12.3127 −0.555097
\(493\) 10.8510 0.488703
\(494\) 62.6399 2.81830
\(495\) 0 0
\(496\) −11.8134 −0.530435
\(497\) 20.6253 0.925171
\(498\) −2.15633 −0.0966272
\(499\) −2.85097 −0.127627 −0.0638135 0.997962i \(-0.520326\pi\)
−0.0638135 + 0.997962i \(0.520326\pi\)
\(500\) 0 0
\(501\) 11.8192 0.528045
\(502\) −62.0625 −2.76999
\(503\) −1.48024 −0.0660006 −0.0330003 0.999455i \(-0.510506\pi\)
−0.0330003 + 0.999455i \(0.510506\pi\)
\(504\) 23.6932 1.05538
\(505\) 0 0
\(506\) −10.7005 −0.475696
\(507\) 13.2012 0.586287
\(508\) −64.0322 −2.84097
\(509\) −23.2995 −1.03273 −0.516366 0.856368i \(-0.672715\pi\)
−0.516366 + 0.856368i \(0.672715\pi\)
\(510\) 0 0
\(511\) 18.0098 0.796709
\(512\) 11.5017 0.508306
\(513\) −4.57452 −0.201970
\(514\) −29.1246 −1.28463
\(515\) 0 0
\(516\) −14.4690 −0.636961
\(517\) −4.31265 −0.189670
\(518\) −12.1016 −0.531712
\(519\) −6.99271 −0.306946
\(520\) 0 0
\(521\) −6.81194 −0.298437 −0.149218 0.988804i \(-0.547676\pi\)
−0.149218 + 0.988804i \(0.547676\pi\)
\(522\) 6.38787 0.279590
\(523\) 16.0567 0.702109 0.351054 0.936355i \(-0.385823\pi\)
0.351054 + 0.936355i \(0.385823\pi\)
\(524\) 30.5501 1.33459
\(525\) 0 0
\(526\) 24.3938 1.06362
\(527\) 4.37328 0.190503
\(528\) −12.2750 −0.534203
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −13.2750 −0.576088
\(532\) 66.1886 2.86964
\(533\) −12.2228 −0.529430
\(534\) 7.92478 0.342939
\(535\) 0 0
\(536\) 90.3508 3.90256
\(537\) −2.70052 −0.116536
\(538\) 30.2276 1.30320
\(539\) −0.873992 −0.0376455
\(540\) 0 0
\(541\) 6.62530 0.284844 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(542\) −67.4128 −2.89563
\(543\) −21.1998 −0.909771
\(544\) 72.4807 3.10759
\(545\) 0 0
\(546\) 38.4241 1.64440
\(547\) 5.75860 0.246220 0.123110 0.992393i \(-0.460713\pi\)
0.123110 + 0.992393i \(0.460713\pi\)
\(548\) −50.5256 −2.15835
\(549\) 7.92478 0.338221
\(550\) 0 0
\(551\) 10.9234 0.465351
\(552\) 33.7743 1.43753
\(553\) −3.78892 −0.161121
\(554\) −6.46898 −0.274840
\(555\) 0 0
\(556\) −10.9624 −0.464909
\(557\) −19.8700 −0.841920 −0.420960 0.907079i \(-0.638307\pi\)
−0.420960 + 0.907079i \(0.638307\pi\)
\(558\) 2.57452 0.108988
\(559\) −14.3634 −0.607509
\(560\) 0 0
\(561\) 4.54420 0.191856
\(562\) −81.5633 −3.44054
\(563\) −5.83383 −0.245866 −0.122933 0.992415i \(-0.539230\pi\)
−0.122933 + 0.992415i \(0.539230\pi\)
\(564\) 22.2374 0.936365
\(565\) 0 0
\(566\) 22.3576 0.939758
\(567\) −2.80606 −0.117844
\(568\) 62.0625 2.60409
\(569\) 46.7123 1.95828 0.979140 0.203185i \(-0.0651293\pi\)
0.979140 + 0.203185i \(0.0651293\pi\)
\(570\) 0 0
\(571\) −24.4241 −1.02212 −0.511058 0.859546i \(-0.670746\pi\)
−0.511058 + 0.859546i \(0.670746\pi\)
\(572\) −26.3938 −1.10358
\(573\) 13.1490 0.549309
\(574\) −17.9248 −0.748166
\(575\) 0 0
\(576\) 18.1187 0.754946
\(577\) −16.5647 −0.689596 −0.344798 0.938677i \(-0.612053\pi\)
−0.344798 + 0.938677i \(0.612053\pi\)
\(578\) −9.76353 −0.406109
\(579\) −5.89446 −0.244965
\(580\) 0 0
\(581\) −2.26187 −0.0938380
\(582\) −26.5501 −1.10054
\(583\) −6.57452 −0.272289
\(584\) 54.1925 2.24250
\(585\) 0 0
\(586\) −61.7802 −2.55212
\(587\) −27.3258 −1.12786 −0.563929 0.825823i \(-0.690711\pi\)
−0.563929 + 0.825823i \(0.690711\pi\)
\(588\) 4.50659 0.185849
\(589\) 4.40246 0.181400
\(590\) 0 0
\(591\) −20.3938 −0.838887
\(592\) −19.7889 −0.813320
\(593\) 12.6048 0.517618 0.258809 0.965928i \(-0.416670\pi\)
0.258809 + 0.965928i \(0.416670\pi\)
\(594\) 2.67513 0.109762
\(595\) 0 0
\(596\) 42.8627 1.75573
\(597\) 17.4010 0.712177
\(598\) 54.7729 2.23983
\(599\) 10.5990 0.433061 0.216531 0.976276i \(-0.430526\pi\)
0.216531 + 0.976276i \(0.430526\pi\)
\(600\) 0 0
\(601\) −22.4749 −0.916768 −0.458384 0.888754i \(-0.651572\pi\)
−0.458384 + 0.888754i \(0.651572\pi\)
\(602\) −21.0640 −0.858503
\(603\) −10.7005 −0.435759
\(604\) −78.7631 −3.20482
\(605\) 0 0
\(606\) 36.4142 1.47923
\(607\) −33.5183 −1.36047 −0.680234 0.732995i \(-0.738122\pi\)
−0.680234 + 0.732995i \(0.738122\pi\)
\(608\) 72.9643 2.95909
\(609\) 6.70052 0.271519
\(610\) 0 0
\(611\) 22.0752 0.893068
\(612\) −23.4314 −0.947157
\(613\) −11.5672 −0.467196 −0.233598 0.972333i \(-0.575050\pi\)
−0.233598 + 0.972333i \(0.575050\pi\)
\(614\) 7.10413 0.286699
\(615\) 0 0
\(616\) −23.6932 −0.954627
\(617\) 33.3357 1.34204 0.671022 0.741438i \(-0.265856\pi\)
0.671022 + 0.741438i \(0.265856\pi\)
\(618\) 43.6385 1.75540
\(619\) 4.43866 0.178405 0.0892024 0.996014i \(-0.471568\pi\)
0.0892024 + 0.996014i \(0.471568\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 42.7367 1.71359
\(623\) 8.31265 0.333039
\(624\) 62.8324 2.51531
\(625\) 0 0
\(626\) −0.162205 −0.00648301
\(627\) 4.57452 0.182689
\(628\) −36.1622 −1.44303
\(629\) 7.32582 0.292100
\(630\) 0 0
\(631\) −39.8496 −1.58639 −0.793193 0.608971i \(-0.791583\pi\)
−0.793193 + 0.608971i \(0.791583\pi\)
\(632\) −11.4010 −0.453509
\(633\) 18.1260 0.720444
\(634\) 10.2012 0.405143
\(635\) 0 0
\(636\) 33.9003 1.34424
\(637\) 4.47371 0.177255
\(638\) −6.38787 −0.252898
\(639\) −7.35026 −0.290772
\(640\) 0 0
\(641\) −3.88858 −0.153590 −0.0767948 0.997047i \(-0.524469\pi\)
−0.0767948 + 0.997047i \(0.524469\pi\)
\(642\) −25.2301 −0.995754
\(643\) −40.9380 −1.61444 −0.807218 0.590254i \(-0.799028\pi\)
−0.807218 + 0.590254i \(0.799028\pi\)
\(644\) 57.8759 2.28063
\(645\) 0 0
\(646\) −55.6093 −2.18792
\(647\) −1.76257 −0.0692939 −0.0346469 0.999400i \(-0.511031\pi\)
−0.0346469 + 0.999400i \(0.511031\pi\)
\(648\) −8.44358 −0.331695
\(649\) 13.2750 0.521091
\(650\) 0 0
\(651\) 2.70052 0.105842
\(652\) −34.8627 −1.36533
\(653\) 7.03761 0.275403 0.137702 0.990474i \(-0.456029\pi\)
0.137702 + 0.990474i \(0.456029\pi\)
\(654\) 41.1998 1.61104
\(655\) 0 0
\(656\) −29.3112 −1.14441
\(657\) −6.41819 −0.250397
\(658\) 32.3733 1.26204
\(659\) −12.6253 −0.491812 −0.245906 0.969294i \(-0.579085\pi\)
−0.245906 + 0.969294i \(0.579085\pi\)
\(660\) 0 0
\(661\) 13.2243 0.514364 0.257182 0.966363i \(-0.417206\pi\)
0.257182 + 0.966363i \(0.417206\pi\)
\(662\) −64.9741 −2.52529
\(663\) −23.2605 −0.903361
\(664\) −6.80606 −0.264126
\(665\) 0 0
\(666\) 4.31265 0.167112
\(667\) 9.55149 0.369835
\(668\) 60.9438 2.35799
\(669\) −23.6385 −0.913916
\(670\) 0 0
\(671\) −7.92478 −0.305933
\(672\) 44.7572 1.72655
\(673\) −18.2677 −0.704170 −0.352085 0.935968i \(-0.614527\pi\)
−0.352085 + 0.935968i \(0.614527\pi\)
\(674\) −60.4055 −2.32673
\(675\) 0 0
\(676\) 68.0698 2.61807
\(677\) 33.0191 1.26903 0.634513 0.772912i \(-0.281201\pi\)
0.634513 + 0.772912i \(0.281201\pi\)
\(678\) −36.7513 −1.41143
\(679\) −27.8496 −1.06877
\(680\) 0 0
\(681\) 1.26916 0.0486343
\(682\) −2.57452 −0.0985833
\(683\) −30.8627 −1.18093 −0.590465 0.807063i \(-0.701056\pi\)
−0.590465 + 0.807063i \(0.701056\pi\)
\(684\) −23.5877 −0.901898
\(685\) 0 0
\(686\) −45.9854 −1.75573
\(687\) 16.1768 0.617183
\(688\) −34.4445 −1.31319
\(689\) 33.6531 1.28208
\(690\) 0 0
\(691\) 27.6991 1.05372 0.526862 0.849951i \(-0.323369\pi\)
0.526862 + 0.849951i \(0.323369\pi\)
\(692\) −36.0567 −1.37067
\(693\) 2.80606 0.106594
\(694\) 54.0176 2.05048
\(695\) 0 0
\(696\) 20.1622 0.764246
\(697\) 10.8510 0.411010
\(698\) 72.8989 2.75927
\(699\) 18.4690 0.698561
\(700\) 0 0
\(701\) −38.9643 −1.47166 −0.735831 0.677166i \(-0.763208\pi\)
−0.735831 + 0.677166i \(0.763208\pi\)
\(702\) −13.6932 −0.516818
\(703\) 7.37470 0.278142
\(704\) −18.1187 −0.682875
\(705\) 0 0
\(706\) 13.4372 0.505717
\(707\) 38.1965 1.43653
\(708\) −68.4504 −2.57252
\(709\) −4.32250 −0.162335 −0.0811674 0.996700i \(-0.525865\pi\)
−0.0811674 + 0.996700i \(0.525865\pi\)
\(710\) 0 0
\(711\) 1.35026 0.0506388
\(712\) 25.0132 0.937408
\(713\) 3.84955 0.144167
\(714\) −34.1114 −1.27659
\(715\) 0 0
\(716\) −13.9248 −0.520393
\(717\) 19.3258 0.721736
\(718\) 48.6253 1.81468
\(719\) −38.3996 −1.43206 −0.716032 0.698067i \(-0.754044\pi\)
−0.716032 + 0.698067i \(0.754044\pi\)
\(720\) 0 0
\(721\) 45.7743 1.70473
\(722\) −5.15282 −0.191768
\(723\) 28.5501 1.06179
\(724\) −109.313 −4.06259
\(725\) 0 0
\(726\) −2.67513 −0.0992834
\(727\) 21.6728 0.803798 0.401899 0.915684i \(-0.368350\pi\)
0.401899 + 0.915684i \(0.368350\pi\)
\(728\) 121.279 4.49489
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.7513 0.471624
\(732\) 40.8627 1.51033
\(733\) −25.0698 −0.925976 −0.462988 0.886365i \(-0.653223\pi\)
−0.462988 + 0.886365i \(0.653223\pi\)
\(734\) −22.9116 −0.845683
\(735\) 0 0
\(736\) 63.8007 2.35172
\(737\) 10.7005 0.394159
\(738\) 6.38787 0.235141
\(739\) −18.9018 −0.695312 −0.347656 0.937622i \(-0.613022\pi\)
−0.347656 + 0.937622i \(0.613022\pi\)
\(740\) 0 0
\(741\) −23.4156 −0.860195
\(742\) 49.3522 1.81178
\(743\) −43.6688 −1.60205 −0.801026 0.598629i \(-0.795712\pi\)
−0.801026 + 0.598629i \(0.795712\pi\)
\(744\) 8.12601 0.297914
\(745\) 0 0
\(746\) 20.9175 0.765843
\(747\) 0.806063 0.0294923
\(748\) 23.4314 0.856736
\(749\) −26.4650 −0.967010
\(750\) 0 0
\(751\) −47.0541 −1.71703 −0.858514 0.512789i \(-0.828612\pi\)
−0.858514 + 0.512789i \(0.828612\pi\)
\(752\) 52.9380 1.93045
\(753\) 23.1998 0.845448
\(754\) 32.6977 1.19078
\(755\) 0 0
\(756\) −14.4690 −0.526232
\(757\) 26.9116 0.978119 0.489059 0.872250i \(-0.337340\pi\)
0.489059 + 0.872250i \(0.337340\pi\)
\(758\) −3.07381 −0.111646
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 16.6859 0.604865 0.302432 0.953171i \(-0.402201\pi\)
0.302432 + 0.953171i \(0.402201\pi\)
\(762\) 33.2203 1.20344
\(763\) 43.2163 1.56454
\(764\) 67.8007 2.45294
\(765\) 0 0
\(766\) 91.1871 3.29473
\(767\) −67.9511 −2.45357
\(768\) 8.08840 0.291865
\(769\) 23.2995 0.840201 0.420100 0.907478i \(-0.361995\pi\)
0.420100 + 0.907478i \(0.361995\pi\)
\(770\) 0 0
\(771\) 10.8872 0.392092
\(772\) −30.3938 −1.09390
\(773\) 1.63656 0.0588631 0.0294316 0.999567i \(-0.490630\pi\)
0.0294316 + 0.999567i \(0.490630\pi\)
\(774\) 7.50659 0.269819
\(775\) 0 0
\(776\) −83.8007 −3.00827
\(777\) 4.52373 0.162288
\(778\) −63.5994 −2.28015
\(779\) 10.9234 0.391370
\(780\) 0 0
\(781\) 7.35026 0.263013
\(782\) −48.6253 −1.73884
\(783\) −2.38787 −0.0853356
\(784\) 10.7283 0.383153
\(785\) 0 0
\(786\) −15.8496 −0.565335
\(787\) 2.09095 0.0745344 0.0372672 0.999305i \(-0.488135\pi\)
0.0372672 + 0.999305i \(0.488135\pi\)
\(788\) −105.157 −3.74606
\(789\) −9.11871 −0.324635
\(790\) 0 0
\(791\) −38.5501 −1.37068
\(792\) 8.44358 0.300030
\(793\) 40.5647 1.44049
\(794\) −11.1030 −0.394030
\(795\) 0 0
\(796\) 89.7255 3.18023
\(797\) 26.8872 0.952392 0.476196 0.879339i \(-0.342015\pi\)
0.476196 + 0.879339i \(0.342015\pi\)
\(798\) −34.3390 −1.21559
\(799\) −19.5975 −0.693311
\(800\) 0 0
\(801\) −2.96239 −0.104671
\(802\) 90.7269 3.20368
\(803\) 6.41819 0.226493
\(804\) −55.1754 −1.94589
\(805\) 0 0
\(806\) 13.1782 0.464183
\(807\) −11.2995 −0.397760
\(808\) 114.935 4.04340
\(809\) −45.4128 −1.59663 −0.798315 0.602241i \(-0.794275\pi\)
−0.798315 + 0.602241i \(0.794275\pi\)
\(810\) 0 0
\(811\) 34.3488 1.20615 0.603076 0.797684i \(-0.293942\pi\)
0.603076 + 0.797684i \(0.293942\pi\)
\(812\) 34.5501 1.21247
\(813\) 25.1998 0.883796
\(814\) −4.31265 −0.151158
\(815\) 0 0
\(816\) −55.7802 −1.95270
\(817\) 12.8364 0.449088
\(818\) −22.8726 −0.799721
\(819\) −14.3634 −0.501899
\(820\) 0 0
\(821\) 18.7612 0.654769 0.327384 0.944891i \(-0.393833\pi\)
0.327384 + 0.944891i \(0.393833\pi\)
\(822\) 26.2130 0.914283
\(823\) 33.0592 1.15237 0.576186 0.817319i \(-0.304540\pi\)
0.576186 + 0.817319i \(0.304540\pi\)
\(824\) 137.737 4.79830
\(825\) 0 0
\(826\) −99.6502 −3.46728
\(827\) 10.8813 0.378379 0.189190 0.981941i \(-0.439414\pi\)
0.189190 + 0.981941i \(0.439414\pi\)
\(828\) −20.6253 −0.716779
\(829\) 42.7221 1.48380 0.741900 0.670510i \(-0.233925\pi\)
0.741900 + 0.670510i \(0.233925\pi\)
\(830\) 0 0
\(831\) 2.41819 0.0838861
\(832\) 92.7445 3.21534
\(833\) −3.97159 −0.137608
\(834\) 5.68735 0.196937
\(835\) 0 0
\(836\) 23.5877 0.815797
\(837\) −0.962389 −0.0332650
\(838\) −35.5778 −1.22902
\(839\) −21.1735 −0.730989 −0.365495 0.930813i \(-0.619100\pi\)
−0.365495 + 0.930813i \(0.619100\pi\)
\(840\) 0 0
\(841\) −23.2981 −0.803381
\(842\) −89.0856 −3.07009
\(843\) 30.4894 1.05011
\(844\) 93.4636 3.21715
\(845\) 0 0
\(846\) −11.5369 −0.396647
\(847\) −2.80606 −0.0964175
\(848\) 80.7024 2.77133
\(849\) −8.35756 −0.286831
\(850\) 0 0
\(851\) 6.44851 0.221052
\(852\) −37.9003 −1.29844
\(853\) −54.4709 −1.86505 −0.932524 0.361109i \(-0.882398\pi\)
−0.932524 + 0.361109i \(0.882398\pi\)
\(854\) 59.4880 2.03564
\(855\) 0 0
\(856\) −79.6345 −2.72185
\(857\) −38.5705 −1.31754 −0.658772 0.752342i \(-0.728924\pi\)
−0.658772 + 0.752342i \(0.728924\pi\)
\(858\) 13.6932 0.467479
\(859\) −18.5139 −0.631685 −0.315843 0.948812i \(-0.602287\pi\)
−0.315843 + 0.948812i \(0.602287\pi\)
\(860\) 0 0
\(861\) 6.70052 0.228353
\(862\) −5.55149 −0.189085
\(863\) −22.5383 −0.767213 −0.383607 0.923497i \(-0.625318\pi\)
−0.383607 + 0.923497i \(0.625318\pi\)
\(864\) −15.9502 −0.542636
\(865\) 0 0
\(866\) −100.053 −3.39993
\(867\) 3.64974 0.123952
\(868\) 13.9248 0.472638
\(869\) −1.35026 −0.0458045
\(870\) 0 0
\(871\) −54.7729 −1.85591
\(872\) 130.040 4.40371
\(873\) 9.92478 0.335903
\(874\) −48.9497 −1.65575
\(875\) 0 0
\(876\) −33.0943 −1.11815
\(877\) −18.9419 −0.639623 −0.319812 0.947481i \(-0.603620\pi\)
−0.319812 + 0.947481i \(0.603620\pi\)
\(878\) 16.1162 0.543894
\(879\) 23.0943 0.778951
\(880\) 0 0
\(881\) −20.0263 −0.674705 −0.337352 0.941378i \(-0.609531\pi\)
−0.337352 + 0.941378i \(0.609531\pi\)
\(882\) −2.33804 −0.0787260
\(883\) −22.2981 −0.750390 −0.375195 0.926946i \(-0.622424\pi\)
−0.375195 + 0.926946i \(0.622424\pi\)
\(884\) −119.938 −4.03397
\(885\) 0 0
\(886\) −27.1147 −0.910938
\(887\) −3.10413 −0.104226 −0.0521132 0.998641i \(-0.516596\pi\)
−0.0521132 + 0.998641i \(0.516596\pi\)
\(888\) 13.6121 0.456793
\(889\) 34.8462 1.16871
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −121.888 −4.08110
\(893\) −19.7283 −0.660182
\(894\) −22.2374 −0.743731
\(895\) 0 0
\(896\) 46.4953 1.55330
\(897\) −20.4749 −0.683636
\(898\) 30.7269 1.02537
\(899\) 2.29806 0.0766447
\(900\) 0 0
\(901\) −29.8759 −0.995311
\(902\) −6.38787 −0.212693
\(903\) 7.87399 0.262030
\(904\) −115.999 −3.85807
\(905\) 0 0
\(906\) 40.8627 1.35757
\(907\) −10.8627 −0.360691 −0.180345 0.983603i \(-0.557722\pi\)
−0.180345 + 0.983603i \(0.557722\pi\)
\(908\) 6.54420 0.217177
\(909\) −13.6121 −0.451486
\(910\) 0 0
\(911\) −5.82321 −0.192931 −0.0964657 0.995336i \(-0.530754\pi\)
−0.0964657 + 0.995336i \(0.530754\pi\)
\(912\) −56.1524 −1.85939
\(913\) −0.806063 −0.0266768
\(914\) −40.2433 −1.33113
\(915\) 0 0
\(916\) 83.4128 2.75604
\(917\) −16.6253 −0.549016
\(918\) 12.1563 0.401219
\(919\) −31.2750 −1.03167 −0.515834 0.856688i \(-0.672518\pi\)
−0.515834 + 0.856688i \(0.672518\pi\)
\(920\) 0 0
\(921\) −2.65562 −0.0875056
\(922\) 70.1886 2.31154
\(923\) −37.6239 −1.23841
\(924\) 14.4690 0.475994
\(925\) 0 0
\(926\) −14.6107 −0.480138
\(927\) −16.3127 −0.535778
\(928\) 38.0870 1.25027
\(929\) −21.2243 −0.696345 −0.348173 0.937430i \(-0.613198\pi\)
−0.348173 + 0.937430i \(0.613198\pi\)
\(930\) 0 0
\(931\) −3.99809 −0.131032
\(932\) 95.2320 3.11943
\(933\) −15.9756 −0.523016
\(934\) −7.22425 −0.236385
\(935\) 0 0
\(936\) −43.2203 −1.41270
\(937\) −16.4328 −0.536835 −0.268418 0.963303i \(-0.586501\pi\)
−0.268418 + 0.963303i \(0.586501\pi\)
\(938\) −80.3244 −2.62268
\(939\) 0.0606343 0.00197873
\(940\) 0 0
\(941\) 5.86414 0.191166 0.0955828 0.995421i \(-0.469529\pi\)
0.0955828 + 0.995421i \(0.469529\pi\)
\(942\) 18.7612 0.611272
\(943\) 9.55149 0.311039
\(944\) −162.952 −5.30362
\(945\) 0 0
\(946\) −7.50659 −0.244060
\(947\) 38.7415 1.25893 0.629464 0.777030i \(-0.283274\pi\)
0.629464 + 0.777030i \(0.283274\pi\)
\(948\) 6.96239 0.226128
\(949\) −32.8529 −1.06645
\(950\) 0 0
\(951\) −3.81336 −0.123657
\(952\) −107.667 −3.48950
\(953\) 6.09569 0.197459 0.0987294 0.995114i \(-0.468522\pi\)
0.0987294 + 0.995114i \(0.468522\pi\)
\(954\) −17.5877 −0.569422
\(955\) 0 0
\(956\) 99.6502 3.22292
\(957\) 2.38787 0.0771890
\(958\) −44.8773 −1.44992
\(959\) 27.4960 0.887891
\(960\) 0 0
\(961\) −30.0738 −0.970123
\(962\) 22.0752 0.711734
\(963\) 9.43136 0.303921
\(964\) 147.213 4.74143
\(965\) 0 0
\(966\) −30.0263 −0.966082
\(967\) −37.9697 −1.22102 −0.610511 0.792008i \(-0.709036\pi\)
−0.610511 + 0.792008i \(0.709036\pi\)
\(968\) −8.44358 −0.271387
\(969\) 20.7875 0.667791
\(970\) 0 0
\(971\) −60.8773 −1.95365 −0.976823 0.214049i \(-0.931335\pi\)
−0.976823 + 0.214049i \(0.931335\pi\)
\(972\) 5.15633 0.165389
\(973\) 5.96571 0.191252
\(974\) −92.0235 −2.94862
\(975\) 0 0
\(976\) 97.2769 3.11376
\(977\) −21.3963 −0.684529 −0.342264 0.939604i \(-0.611194\pi\)
−0.342264 + 0.939604i \(0.611194\pi\)
\(978\) 18.0870 0.578358
\(979\) 2.96239 0.0946784
\(980\) 0 0
\(981\) −15.4010 −0.491718
\(982\) −40.0000 −1.27645
\(983\) −18.8919 −0.602558 −0.301279 0.953536i \(-0.597414\pi\)
−0.301279 + 0.953536i \(0.597414\pi\)
\(984\) 20.1622 0.642748
\(985\) 0 0
\(986\) −29.0278 −0.924432
\(987\) −12.1016 −0.385197
\(988\) −120.739 −3.84121
\(989\) 11.2243 0.356911
\(990\) 0 0
\(991\) −1.33567 −0.0424291 −0.0212145 0.999775i \(-0.506753\pi\)
−0.0212145 + 0.999775i \(0.506753\pi\)
\(992\) 15.3503 0.487371
\(993\) 24.2882 0.770763
\(994\) −55.1754 −1.75006
\(995\) 0 0
\(996\) 4.15633 0.131698
\(997\) 48.9076 1.54892 0.774460 0.632623i \(-0.218022\pi\)
0.774460 + 0.632623i \(0.218022\pi\)
\(998\) 7.62672 0.241419
\(999\) −1.61213 −0.0510054
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 825.2.a.h.1.1 3
3.2 odd 2 2475.2.a.be.1.3 3
5.2 odd 4 165.2.c.a.34.1 6
5.3 odd 4 165.2.c.a.34.6 yes 6
5.4 even 2 825.2.a.n.1.3 3
11.10 odd 2 9075.2.a.ck.1.3 3
15.2 even 4 495.2.c.d.199.6 6
15.8 even 4 495.2.c.d.199.1 6
15.14 odd 2 2475.2.a.y.1.1 3
20.3 even 4 2640.2.d.i.529.1 6
20.7 even 4 2640.2.d.i.529.4 6
55.32 even 4 1815.2.c.d.364.6 6
55.43 even 4 1815.2.c.d.364.1 6
55.54 odd 2 9075.2.a.cc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.1 6 5.2 odd 4
165.2.c.a.34.6 yes 6 5.3 odd 4
495.2.c.d.199.1 6 15.8 even 4
495.2.c.d.199.6 6 15.2 even 4
825.2.a.h.1.1 3 1.1 even 1 trivial
825.2.a.n.1.3 3 5.4 even 2
1815.2.c.d.364.1 6 55.43 even 4
1815.2.c.d.364.6 6 55.32 even 4
2475.2.a.y.1.1 3 15.14 odd 2
2475.2.a.be.1.3 3 3.2 odd 2
2640.2.d.i.529.1 6 20.3 even 4
2640.2.d.i.529.4 6 20.7 even 4
9075.2.a.cc.1.1 3 55.54 odd 2
9075.2.a.ck.1.3 3 11.10 odd 2