Properties

Label 495.2.c.d.199.1
Level $495$
Weight $2$
Character 495.199
Analytic conductor $3.953$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(199,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 495.199
Dual form 495.2.c.d.199.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.67513i q^{2} -5.15633 q^{4} +(1.48119 + 1.67513i) q^{5} +2.80606i q^{7} +8.44358i q^{8} +O(q^{10})\) \(q-2.67513i q^{2} -5.15633 q^{4} +(1.48119 + 1.67513i) q^{5} +2.80606i q^{7} +8.44358i q^{8} +(4.48119 - 3.96239i) q^{10} +1.00000 q^{11} +5.11871i q^{13} +7.50659 q^{14} +12.2750 q^{16} -4.54420i q^{17} +4.57452 q^{19} +(-7.63752 - 8.63752i) q^{20} -2.67513i q^{22} +4.00000i q^{23} +(-0.612127 + 4.96239i) q^{25} +13.6932 q^{26} -14.4690i q^{28} -2.38787 q^{29} -0.962389 q^{31} -15.9502i q^{32} -12.1563 q^{34} +(-4.70052 + 4.15633i) q^{35} +1.61213i q^{37} -12.2374i q^{38} +(-14.1441 + 12.5066i) q^{40} +2.38787 q^{41} -2.80606i q^{43} -5.15633 q^{44} +10.7005 q^{46} +4.31265i q^{47} -0.873992 q^{49} +(13.2750 + 1.63752i) q^{50} -26.3938i q^{52} -6.57452i q^{53} +(1.48119 + 1.67513i) q^{55} -23.6932 q^{56} +6.38787i q^{58} -13.2750 q^{59} +7.92478 q^{61} +2.57452i q^{62} -18.1187 q^{64} +(-8.57452 + 7.58181i) q^{65} +10.7005i q^{67} +23.4314i q^{68} +(11.1187 + 12.5745i) q^{70} +7.35026 q^{71} -6.41819i q^{73} +4.31265 q^{74} -23.5877 q^{76} +2.80606i q^{77} -1.35026 q^{79} +(18.1817 + 20.5623i) q^{80} -6.38787i q^{82} -0.806063i q^{83} +(7.61213 - 6.73084i) q^{85} -7.50659 q^{86} +8.44358i q^{88} -2.96239 q^{89} -14.3634 q^{91} -20.6253i q^{92} +11.5369 q^{94} +(6.77575 + 7.66291i) q^{95} -9.92478i q^{97} +2.33804i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{5} + 16 q^{10} + 6 q^{11} + 4 q^{14} + 10 q^{16} + 4 q^{19} - 14 q^{20} - 2 q^{25} + 16 q^{26} - 16 q^{29} + 16 q^{31} - 52 q^{34} + 12 q^{35} - 12 q^{40} + 16 q^{41} - 10 q^{44} + 24 q^{46} - 22 q^{49} + 16 q^{50} - 2 q^{55} - 76 q^{56} - 16 q^{59} + 4 q^{61} - 66 q^{64} - 28 q^{65} + 24 q^{70} + 24 q^{71} - 16 q^{74} - 36 q^{76} + 12 q^{79} + 58 q^{80} + 44 q^{85} - 4 q^{86} + 4 q^{89} + 16 q^{91} + 24 q^{94} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513i 1.89160i −0.324745 0.945802i \(-0.605279\pi\)
0.324745 0.945802i \(-0.394721\pi\)
\(3\) 0 0
\(4\) −5.15633 −2.57816
\(5\) 1.48119 + 1.67513i 0.662410 + 0.749141i
\(6\) 0 0
\(7\) 2.80606i 1.06059i 0.847812 + 0.530296i \(0.177919\pi\)
−0.847812 + 0.530296i \(0.822081\pi\)
\(8\) 8.44358i 2.98526i
\(9\) 0 0
\(10\) 4.48119 3.96239i 1.41708 1.25302i
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.11871i 1.41968i 0.704365 + 0.709838i \(0.251232\pi\)
−0.704365 + 0.709838i \(0.748768\pi\)
\(14\) 7.50659 2.00622
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 4.54420i 1.10213i −0.834462 0.551065i \(-0.814222\pi\)
0.834462 0.551065i \(-0.185778\pi\)
\(18\) 0 0
\(19\) 4.57452 1.04947 0.524733 0.851267i \(-0.324165\pi\)
0.524733 + 0.851267i \(0.324165\pi\)
\(20\) −7.63752 8.63752i −1.70780 1.93141i
\(21\) 0 0
\(22\) 2.67513i 0.570340i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) −0.612127 + 4.96239i −0.122425 + 0.992478i
\(26\) 13.6932 2.68546
\(27\) 0 0
\(28\) 14.4690i 2.73438i
\(29\) −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 \(0.527544\pi\)
\(32\) 15.9502i 2.81962i
\(33\) 0 0
\(34\) −12.1563 −2.08479
\(35\) −4.70052 + 4.15633i −0.794533 + 0.702547i
\(36\) 0 0
\(37\) 1.61213i 0.265032i 0.991181 + 0.132516i \(0.0423056\pi\)
−0.991181 + 0.132516i \(0.957694\pi\)
\(38\) 12.2374i 1.98517i
\(39\) 0 0
\(40\) −14.1441 + 12.5066i −2.23638 + 1.97747i
\(41\) 2.38787 0.372923 0.186462 0.982462i \(-0.440298\pi\)
0.186462 + 0.982462i \(0.440298\pi\)
\(42\) 0 0
\(43\) 2.80606i 0.427921i −0.976842 0.213960i \(-0.931364\pi\)
0.976842 0.213960i \(-0.0686363\pi\)
\(44\) −5.15633 −0.777345
\(45\) 0 0
\(46\) 10.7005 1.57771
\(47\) 4.31265i 0.629065i 0.949247 + 0.314532i \(0.101848\pi\)
−0.949247 + 0.314532i \(0.898152\pi\)
\(48\) 0 0
\(49\) −0.873992 −0.124856
\(50\) 13.2750 + 1.63752i 1.87737 + 0.231580i
\(51\) 0 0
\(52\) 26.3938i 3.66015i
\(53\) 6.57452i 0.903079i −0.892251 0.451540i \(-0.850875\pi\)
0.892251 0.451540i \(-0.149125\pi\)
\(54\) 0 0
\(55\) 1.48119 + 1.67513i 0.199724 + 0.225875i
\(56\) −23.6932 −3.16614
\(57\) 0 0
\(58\) 6.38787i 0.838769i
\(59\) −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 \(0.330632\pi\)
\(62\) 2.57452i 0.326964i
\(63\) 0 0
\(64\) −18.1187 −2.26484
\(65\) −8.57452 + 7.58181i −1.06354 + 0.940408i
\(66\) 0 0
\(67\) 10.7005i 1.30728i 0.756807 + 0.653639i \(0.226758\pi\)
−0.756807 + 0.653639i \(0.773242\pi\)
\(68\) 23.4314i 2.84147i
\(69\) 0 0
\(70\) 11.1187 + 12.5745i 1.32894 + 1.50294i
\(71\) 7.35026 0.872316 0.436158 0.899870i \(-0.356339\pi\)
0.436158 + 0.899870i \(0.356339\pi\)
\(72\) 0 0
\(73\) 6.41819i 0.751192i −0.926783 0.375596i \(-0.877438\pi\)
0.926783 0.375596i \(-0.122562\pi\)
\(74\) 4.31265 0.501335
\(75\) 0 0
\(76\) −23.5877 −2.70569
\(77\) 2.80606i 0.319781i
\(78\) 0 0
\(79\) −1.35026 −0.151916 −0.0759582 0.997111i \(-0.524202\pi\)
−0.0759582 + 0.997111i \(0.524202\pi\)
\(80\) 18.1817 + 20.5623i 2.03278 + 2.29893i
\(81\) 0 0
\(82\) 6.38787i 0.705423i
\(83\) 0.806063i 0.0884770i −0.999021 0.0442385i \(-0.985914\pi\)
0.999021 0.0442385i \(-0.0140861\pi\)
\(84\) 0 0
\(85\) 7.61213 6.73084i 0.825651 0.730062i
\(86\) −7.50659 −0.809456
\(87\) 0 0
\(88\) 8.44358i 0.900089i
\(89\) −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
\(92\) 20.6253i 2.15034i
\(93\) 0 0
\(94\) 11.5369 1.18994
\(95\) 6.77575 + 7.66291i 0.695177 + 0.786198i
\(96\) 0 0
\(97\) 9.92478i 1.00771i −0.863789 0.503854i \(-0.831915\pi\)
0.863789 0.503854i \(-0.168085\pi\)
\(98\) 2.33804i 0.236178i
\(99\) 0 0
\(100\) 3.15633 25.5877i 0.315633 2.55877i
\(101\) 13.6121 1.35446 0.677229 0.735773i \(-0.263181\pi\)
0.677229 + 0.735773i \(0.263181\pi\)
\(102\) 0 0
\(103\) 16.3127i 1.60733i −0.595080 0.803667i \(-0.702880\pi\)
0.595080 0.803667i \(-0.297120\pi\)
\(104\) −43.2203 −4.23810
\(105\) 0 0
\(106\) −17.5877 −1.70827
\(107\) 9.43136i 0.911764i 0.890040 + 0.455882i \(0.150676\pi\)
−0.890040 + 0.455882i \(0.849324\pi\)
\(108\) 0 0
\(109\) 15.4010 1.47515 0.737576 0.675264i \(-0.235970\pi\)
0.737576 + 0.675264i \(0.235970\pi\)
\(110\) 4.48119 3.96239i 0.427265 0.377799i
\(111\) 0 0
\(112\) 34.4445i 3.25470i
\(113\) 13.7381i 1.29238i −0.763179 0.646188i \(-0.776362\pi\)
0.763179 0.646188i \(-0.223638\pi\)
\(114\) 0 0
\(115\) −6.70052 + 5.92478i −0.624827 + 0.552488i
\(116\) 12.3127 1.14320
\(117\) 0 0
\(118\) 35.5125i 3.26919i
\(119\) 12.7513 1.16891
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 21.1998i 1.91934i
\(123\) 0 0
\(124\) 4.96239 0.445636
\(125\) −9.21933 + 6.32487i −0.824602 + 0.565713i
\(126\) 0 0
\(127\) 12.4182i 1.10194i 0.834526 + 0.550968i \(0.185741\pi\)
−0.834526 + 0.550968i \(0.814259\pi\)
\(128\) 16.5696i 1.46456i
\(129\) 0 0
\(130\) 20.2823 + 22.9380i 1.77888 + 2.01179i
\(131\) −5.92478 −0.517650 −0.258825 0.965924i \(-0.583335\pi\)
−0.258825 + 0.965924i \(0.583335\pi\)
\(132\) 0 0
\(133\) 12.8364i 1.11306i
\(134\) 28.6253 2.47285
\(135\) 0 0
\(136\) 38.3693 3.29014
\(137\) 9.79877i 0.837165i −0.908179 0.418583i \(-0.862527\pi\)
0.908179 0.418583i \(-0.137473\pi\)
\(138\) 0 0
\(139\) 2.12601 0.180326 0.0901628 0.995927i \(-0.471261\pi\)
0.0901628 + 0.995927i \(0.471261\pi\)
\(140\) 24.2374 21.4314i 2.04844 1.81128i
\(141\) 0 0
\(142\) 19.6629i 1.65007i
\(143\) 5.11871i 0.428048i
\(144\) 0 0
\(145\) −3.53690 4.00000i −0.293724 0.332182i
\(146\) −17.1695 −1.42096
\(147\) 0 0
\(148\) 8.31265i 0.683296i
\(149\) 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 \(0.713490\pi\)
\(152\) 38.6253i 3.13293i
\(153\) 0 0
\(154\) 7.50659 0.604898
\(155\) −1.42548 1.61213i −0.114498 0.129489i
\(156\) 0 0
\(157\) 7.01317i 0.559712i 0.960042 + 0.279856i \(0.0902868\pi\)
−0.960042 + 0.279856i \(0.909713\pi\)
\(158\) 3.61213i 0.287365i
\(159\) 0 0
\(160\) 26.7186 23.6253i 2.11229 1.86774i
\(161\) −11.2243 −0.884595
\(162\) 0 0
\(163\) 6.76116i 0.529575i −0.964307 0.264787i \(-0.914698\pi\)
0.964307 0.264787i \(-0.0853018\pi\)
\(164\) −12.3127 −0.961456
\(165\) 0 0
\(166\) −2.15633 −0.167363
\(167\) 11.8192i 0.914600i 0.889312 + 0.457300i \(0.151183\pi\)
−0.889312 + 0.457300i \(0.848817\pi\)
\(168\) 0 0
\(169\) −13.2012 −1.01548
\(170\) −18.0059 20.3634i −1.38099 1.56180i
\(171\) 0 0
\(172\) 14.4690i 1.10325i
\(173\) 6.99271i 0.531646i 0.964022 + 0.265823i \(0.0856436\pi\)
−0.964022 + 0.265823i \(0.914356\pi\)
\(174\) 0 0
\(175\) −13.9248 1.71767i −1.05261 0.129843i
\(176\) 12.2750 0.925266
\(177\) 0 0
\(178\) 7.92478i 0.593987i
\(179\) −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 \(0.788824\pi\)
\(182\) 38.4241i 2.84818i
\(183\) 0 0
\(184\) −33.7743 −2.48988
\(185\) −2.70052 + 2.38787i −0.198546 + 0.175560i
\(186\) 0 0
\(187\) 4.54420i 0.332305i
\(188\) 22.2374i 1.62183i
\(189\) 0 0
\(190\) 20.4993 18.1260i 1.48717 1.31500i
\(191\) −13.1490 −0.951430 −0.475715 0.879599i \(-0.657811\pi\)
−0.475715 + 0.879599i \(0.657811\pi\)
\(192\) 0 0
\(193\) 5.89446i 0.424293i −0.977238 0.212146i \(-0.931955\pi\)
0.977238 0.212146i \(-0.0680453\pi\)
\(194\) −26.5501 −1.90618
\(195\) 0 0
\(196\) 4.50659 0.321899
\(197\) 20.3938i 1.45299i −0.687169 0.726497i \(-0.741147\pi\)
0.687169 0.726497i \(-0.258853\pi\)
\(198\) 0 0
\(199\) −17.4010 −1.23353 −0.616764 0.787148i \(-0.711557\pi\)
−0.616764 + 0.787148i \(0.711557\pi\)
\(200\) −41.9003 5.16854i −2.96280 0.365471i
\(201\) 0 0
\(202\) 36.4142i 2.56210i
\(203\) 6.70052i 0.470285i
\(204\) 0 0
\(205\) 3.53690 + 4.00000i 0.247028 + 0.279372i
\(206\) −43.6385 −3.04044
\(207\) 0 0
\(208\) 62.8324i 4.35664i
\(209\) 4.57452 0.316426
\(210\) 0 0
\(211\) 18.1260 1.24785 0.623923 0.781486i \(-0.285538\pi\)
0.623923 + 0.781486i \(0.285538\pi\)
\(212\) 33.9003i 2.32828i
\(213\) 0 0
\(214\) 25.2301 1.72470
\(215\) 4.70052 4.15633i 0.320573 0.283459i
\(216\) 0 0
\(217\) 2.70052i 0.183323i
\(218\) 41.1998i 2.79040i
\(219\) 0 0
\(220\) −7.63752 8.63752i −0.514921 0.582341i
\(221\) 23.2605 1.56467
\(222\) 0 0
\(223\) 23.6385i 1.58295i −0.611202 0.791475i \(-0.709314\pi\)
0.611202 0.791475i \(-0.290686\pi\)
\(224\) 44.7572 2.99047
\(225\) 0 0
\(226\) −36.7513 −2.44466
\(227\) 1.26916i 0.0842371i 0.999113 + 0.0421185i \(0.0134107\pi\)
−0.999113 + 0.0421185i \(0.986589\pi\)
\(228\) 0 0
\(229\) −16.1768 −1.06899 −0.534496 0.845171i \(-0.679499\pi\)
−0.534496 + 0.845171i \(0.679499\pi\)
\(230\) 15.8496 + 17.9248i 1.04509 + 1.18192i
\(231\) 0 0
\(232\) 20.1622i 1.32371i
\(233\) 18.4690i 1.20994i −0.796247 0.604971i \(-0.793185\pi\)
0.796247 0.604971i \(-0.206815\pi\)
\(234\) 0 0
\(235\) −7.22425 + 6.38787i −0.471258 + 0.416699i
\(236\) 68.4504 4.45574
\(237\) 0 0
\(238\) 34.1114i 2.21111i
\(239\) 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 \(0.128565\pi\)
\(242\) 2.67513i 0.171964i
\(243\) 0 0
\(244\) −40.8627 −2.61597
\(245\) −1.29455 1.46405i −0.0827059 0.0935348i
\(246\) 0 0
\(247\) 23.4156i 1.48990i
\(248\) 8.12601i 0.516002i
\(249\) 0 0
\(250\) 16.9199 + 24.6629i 1.07011 + 1.55982i
\(251\) −23.1998 −1.46436 −0.732180 0.681112i \(-0.761497\pi\)
−0.732180 + 0.681112i \(0.761497\pi\)
\(252\) 0 0
\(253\) 4.00000i 0.251478i
\(254\) 33.2203 2.08443
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) 10.8872i 0.679123i 0.940584 + 0.339561i \(0.110279\pi\)
−0.940584 + 0.339561i \(0.889721\pi\)
\(258\) 0 0
\(259\) −4.52373 −0.281091
\(260\) 44.2130 39.0943i 2.74197 2.42452i
\(261\) 0 0
\(262\) 15.8496i 0.979189i
\(263\) 9.11871i 0.562284i 0.959666 + 0.281142i \(0.0907132\pi\)
−0.959666 + 0.281142i \(0.909287\pi\)
\(264\) 0 0
\(265\) 11.0132 9.73813i 0.676534 0.598209i
\(266\) 34.3390 2.10546
\(267\) 0 0
\(268\) 55.1754i 3.37037i
\(269\) −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 \(0.222546\pi\)
\(272\) 55.7802i 3.38217i
\(273\) 0 0
\(274\) −26.2130 −1.58358
\(275\) −0.612127 + 4.96239i −0.0369126 + 0.299243i
\(276\) 0 0
\(277\) 2.41819i 0.145295i −0.997358 0.0726475i \(-0.976855\pi\)
0.997358 0.0726475i \(-0.0231448\pi\)
\(278\) 5.68735i 0.341105i
\(279\) 0 0
\(280\) −35.0943 39.6893i −2.09728 2.37189i
\(281\) −30.4894 −1.81885 −0.909424 0.415870i \(-0.863477\pi\)
−0.909424 + 0.415870i \(0.863477\pi\)
\(282\) 0 0
\(283\) 8.35756i 0.496805i −0.968657 0.248403i \(-0.920094\pi\)
0.968657 0.248403i \(-0.0799056\pi\)
\(284\) −37.9003 −2.24897
\(285\) 0 0
\(286\) 13.6932 0.809698
\(287\) 6.70052i 0.395519i
\(288\) 0 0
\(289\) −3.64974 −0.214690
\(290\) −10.7005 + 9.46168i −0.628356 + 0.555609i
\(291\) 0 0
\(292\) 33.0943i 1.93670i
\(293\) 23.0943i 1.34918i −0.738192 0.674591i \(-0.764320\pi\)
0.738192 0.674591i \(-0.235680\pi\)
\(294\) 0 0
\(295\) −19.6629 22.2374i −1.14482 1.29471i
\(296\) −13.6121 −0.791189
\(297\) 0 0
\(298\) 22.2374i 1.28818i
\(299\) −20.4749 −1.18409
\(300\) 0 0
\(301\) 7.87399 0.453849
\(302\) 40.8627i 2.35139i
\(303\) 0 0
\(304\) 56.1524 3.22056
\(305\) 11.7381 + 13.2750i 0.672124 + 0.760127i
\(306\) 0 0
\(307\) 2.65562i 0.151564i 0.997124 + 0.0757821i \(0.0241453\pi\)
−0.997124 + 0.0757821i \(0.975855\pi\)
\(308\) 14.4690i 0.824446i
\(309\) 0 0
\(310\) −4.31265 + 3.81336i −0.244942 + 0.216584i
\(311\) 15.9756 0.905891 0.452946 0.891538i \(-0.350373\pi\)
0.452946 + 0.891538i \(0.350373\pi\)
\(312\) 0 0
\(313\) 0.0606343i 0.00342726i 0.999999 + 0.00171363i \(0.000545465\pi\)
−0.999999 + 0.00171363i \(0.999455\pi\)
\(314\) 18.7612 1.05875
\(315\) 0 0
\(316\) 6.96239 0.391665
\(317\) 3.81336i 0.214180i −0.994249 0.107090i \(-0.965847\pi\)
0.994249 0.107090i \(-0.0341532\pi\)
\(318\) 0 0
\(319\) −2.38787 −0.133695
\(320\) −26.8373 30.3512i −1.50025 1.69668i
\(321\) 0 0
\(322\) 30.0263i 1.67330i
\(323\) 20.7875i 1.15665i
\(324\) 0 0
\(325\) −25.4010 3.13330i −1.40900 0.173804i
\(326\) −18.0870 −1.00175
\(327\) 0 0
\(328\) 20.1622i 1.11327i
\(329\) −12.1016 −0.667181
\(330\) 0 0
\(331\) 24.2882 1.33500 0.667500 0.744609i \(-0.267364\pi\)
0.667500 + 0.744609i \(0.267364\pi\)
\(332\) 4.15633i 0.228108i
\(333\) 0 0
\(334\) 31.6180 1.73006
\(335\) −17.9248 + 15.8496i −0.979335 + 0.865954i
\(336\) 0 0
\(337\) 22.5804i 1.23003i −0.788514 0.615016i \(-0.789149\pi\)
0.788514 0.615016i \(-0.210851\pi\)
\(338\) 35.3150i 1.92088i
\(339\) 0 0
\(340\) −39.2506 + 34.7064i −2.12866 + 1.88222i
\(341\) −0.962389 −0.0521163
\(342\) 0 0
\(343\) 17.1900i 0.928171i
\(344\) 23.6932 1.27745
\(345\) 0 0
\(346\) 18.7064 1.00566
\(347\) 20.1925i 1.08399i −0.840381 0.541996i \(-0.817669\pi\)
0.840381 0.541996i \(-0.182331\pi\)
\(348\) 0 0
\(349\) 27.2506 1.45869 0.729346 0.684145i \(-0.239825\pi\)
0.729346 + 0.684145i \(0.239825\pi\)
\(350\) −4.59498 + 37.2506i −0.245612 + 1.99113i
\(351\) 0 0
\(352\) 15.9502i 0.850147i
\(353\) 5.02302i 0.267349i 0.991025 + 0.133674i \(0.0426776\pi\)
−0.991025 + 0.133674i \(0.957322\pi\)
\(354\) 0 0
\(355\) 10.8872 + 12.3127i 0.577831 + 0.653488i
\(356\) 15.2750 0.809575
\(357\) 0 0
\(358\) 7.22425i 0.381814i
\(359\) −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
\(362\) 56.7123i 2.98073i
\(363\) 0 0
\(364\) 74.0625 3.88193
\(365\) 10.7513 9.50659i 0.562749 0.497598i
\(366\) 0 0
\(367\) 8.56467i 0.447072i −0.974696 0.223536i \(-0.928240\pi\)
0.974696 0.223536i \(-0.0717600\pi\)
\(368\) 49.1002i 2.55952i
\(369\) 0 0
\(370\) 6.38787 + 7.22425i 0.332090 + 0.375571i
\(371\) 18.4485 0.957799
\(372\) 0 0
\(373\) 7.81924i 0.404865i −0.979296 0.202432i \(-0.935115\pi\)
0.979296 0.202432i \(-0.0648846\pi\)
\(374\) −12.1563 −0.628589
\(375\) 0 0
\(376\) −36.4142 −1.87792
\(377\) 12.2228i 0.629508i
\(378\) 0 0
\(379\) −1.14903 −0.0590218 −0.0295109 0.999564i \(-0.509395\pi\)
−0.0295109 + 0.999564i \(0.509395\pi\)
\(380\) −34.9380 39.5125i −1.79228 2.02695i
\(381\) 0 0
\(382\) 35.1754i 1.79973i
\(383\) 34.0870i 1.74176i 0.491493 + 0.870882i \(0.336451\pi\)
−0.491493 + 0.870882i \(0.663549\pi\)
\(384\) 0 0
\(385\) −4.70052 + 4.15633i −0.239561 + 0.211826i
\(386\) −15.7685 −0.802593
\(387\) 0 0
\(388\) 51.1754i 2.59804i
\(389\) 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
\(392\) 7.37962i 0.372727i
\(393\) 0 0
\(394\) −54.5560 −2.74849
\(395\) −2.00000 2.26187i −0.100631 0.113807i
\(396\) 0 0
\(397\) 4.15045i 0.208305i −0.994561 0.104152i \(-0.966787\pi\)
0.994561 0.104152i \(-0.0332130\pi\)
\(398\) 46.5501i 2.33334i
\(399\) 0 0
\(400\) −7.51388 + 60.9135i −0.375694 + 3.04568i
\(401\) 33.9149 1.69363 0.846815 0.531887i \(-0.178517\pi\)
0.846815 + 0.531887i \(0.178517\pi\)
\(402\) 0 0
\(403\) 4.92619i 0.245391i
\(404\) −70.1886 −3.49201
\(405\) 0 0
\(406\) −17.9248 −0.889592
\(407\) 1.61213i 0.0799102i
\(408\) 0 0
\(409\) −8.55008 −0.422774 −0.211387 0.977402i \(-0.567798\pi\)
−0.211387 + 0.977402i \(0.567798\pi\)
\(410\) 10.7005 9.46168i 0.528461 0.467279i
\(411\) 0 0
\(412\) 84.1133i 4.14397i
\(413\) 37.2506i 1.83298i
\(414\) 0 0
\(415\) 1.35026 1.19394i 0.0662817 0.0586080i
\(416\) 81.6444 4.00294
\(417\) 0 0
\(418\) 12.2374i 0.598552i
\(419\) 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 \(0.198649\pi\)
\(422\) 48.4894i 2.36043i
\(423\) 0 0
\(424\) 55.5125 2.69592
\(425\) 22.5501 + 2.78163i 1.09384 + 0.134929i
\(426\) 0 0
\(427\) 22.2374i 1.07614i
\(428\) 48.6312i 2.35068i
\(429\) 0 0
\(430\) −11.1187 12.5745i −0.536192 0.606397i
\(431\) −2.07522 −0.0999600 −0.0499800 0.998750i \(-0.515916\pi\)
−0.0499800 + 0.998750i \(0.515916\pi\)
\(432\) 0 0
\(433\) 37.4010i 1.79738i 0.438585 + 0.898690i \(0.355480\pi\)
−0.438585 + 0.898690i \(0.644520\pi\)
\(434\) −7.22425 −0.346775
\(435\) 0 0
\(436\) −79.4128 −3.80318
\(437\) 18.2981i 0.875315i
\(438\) 0 0
\(439\) 6.02444 0.287531 0.143765 0.989612i \(-0.454079\pi\)
0.143765 + 0.989612i \(0.454079\pi\)
\(440\) −14.1441 + 12.5066i −0.674294 + 0.596228i
\(441\) 0 0
\(442\) 62.2247i 2.95973i
\(443\) 10.1359i 0.481569i −0.970579 0.240785i \(-0.922595\pi\)
0.970579 0.240785i \(-0.0774047\pi\)
\(444\) 0 0
\(445\) −4.38787 4.96239i −0.208005 0.235240i
\(446\) −63.2360 −2.99431
\(447\) 0 0
\(448\) 50.8423i 2.40207i
\(449\) −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
\(452\) 70.8383i 3.33195i
\(453\) 0 0
\(454\) 3.39517 0.159343
\(455\) −21.2750 24.0606i −0.997389 1.12798i
\(456\) 0 0
\(457\) 15.0435i 0.703705i −0.936055 0.351852i \(-0.885552\pi\)
0.936055 0.351852i \(-0.114448\pi\)
\(458\) 43.2750i 2.02211i
\(459\) 0 0
\(460\) 34.5501 30.5501i 1.61091 1.42440i
\(461\) 26.2374 1.22200 0.610999 0.791631i \(-0.290768\pi\)
0.610999 + 0.791631i \(0.290768\pi\)
\(462\) 0 0
\(463\) 5.46168i 0.253826i 0.991914 + 0.126913i \(0.0405069\pi\)
−0.991914 + 0.126913i \(0.959493\pi\)
\(464\) −29.3112 −1.36074
\(465\) 0 0
\(466\) −49.4069 −2.28873
\(467\) 2.70052i 0.124965i 0.998046 + 0.0624827i \(0.0199018\pi\)
−0.998046 + 0.0624827i \(0.980098\pi\)
\(468\) 0 0
\(469\) −30.0263 −1.38649
\(470\) 17.0884 + 19.3258i 0.788229 + 0.891434i
\(471\) 0 0
\(472\) 112.089i 5.15931i
\(473\) 2.80606i 0.129023i
\(474\) 0 0
\(475\) −2.80018 + 22.7005i −0.128481 + 1.04157i
\(476\) −65.7499 −3.01364
\(477\) 0 0
\(478\) 51.6991i 2.36466i
\(479\) 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
\(482\) 76.3752i 3.47879i
\(483\) 0 0
\(484\) −5.15633 −0.234378
\(485\) 16.6253 14.7005i 0.754916 0.667516i
\(486\) 0 0
\(487\) 34.3996i 1.55880i −0.626529 0.779398i \(-0.715525\pi\)
0.626529 0.779398i \(-0.284475\pi\)
\(488\) 66.9135i 3.02903i
\(489\) 0 0
\(490\) −3.91653 + 3.46310i −0.176931 + 0.156447i
\(491\) −14.9525 −0.674799 −0.337399 0.941362i \(-0.609547\pi\)
−0.337399 + 0.941362i \(0.609547\pi\)
\(492\) 0 0
\(493\) 10.8510i 0.488703i
\(494\) 62.6399 2.81830
\(495\) 0 0
\(496\) −11.8134 −0.530435
\(497\) 20.6253i 0.925171i
\(498\) 0 0
\(499\) 2.85097 0.127627 0.0638135 0.997962i \(-0.479674\pi\)
0.0638135 + 0.997962i \(0.479674\pi\)
\(500\) 47.5379 32.6131i 2.12596 1.45850i
\(501\) 0 0
\(502\) 62.0625i 2.76999i
\(503\) 1.48024i 0.0660006i 0.999455 + 0.0330003i \(0.0105062\pi\)
−0.999455 + 0.0330003i \(0.989494\pi\)
\(504\) 0 0
\(505\) 20.1622 + 22.8021i 0.897206 + 1.01468i
\(506\) 10.7005 0.475696
\(507\) 0 0
\(508\) 64.0322i 2.84097i
\(509\) −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
\(512\) 11.5017i 0.508306i
\(513\) 0 0
\(514\) 29.1246 1.28463
\(515\) 27.3258 24.1622i 1.20412 1.06471i
\(516\) 0 0
\(517\) 4.31265i 0.189670i
\(518\) 12.1016i 0.531712i
\(519\) 0 0
\(520\) −64.0176 72.3996i −2.80736 3.17493i
\(521\) 6.81194 0.298437 0.149218 0.988804i \(-0.452324\pi\)
0.149218 + 0.988804i \(0.452324\pi\)
\(522\) 0 0
\(523\) 16.0567i 0.702109i 0.936355 + 0.351054i \(0.114177\pi\)
−0.936355 + 0.351054i \(0.885823\pi\)
\(524\) 30.5501 1.33459
\(525\) 0 0
\(526\) 24.3938 1.06362
\(527\) 4.37328i 0.190503i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) −26.0508 29.4617i −1.13157 1.27973i
\(531\) 0 0
\(532\) 66.1886i 2.86964i
\(533\) 12.2228i 0.529430i
\(534\) 0 0
\(535\) −15.7988 + 13.9697i −0.683040 + 0.603962i
\(536\) −90.3508 −3.90256
\(537\) 0 0
\(538\) 30.2276i 1.30320i
\(539\) −0.873992 −0.0376455
\(540\) 0 0
\(541\) 6.62530 0.284844 0.142422 0.989806i \(-0.454511\pi\)
0.142422 + 0.989806i \(0.454511\pi\)
\(542\) 67.4128i 2.89563i
\(543\) 0 0
\(544\) −72.4807 −3.10759
\(545\) 22.8119 + 25.7988i 0.977156 + 1.10510i
\(546\) 0 0
\(547\) 5.75860i 0.246220i −0.992393 0.123110i \(-0.960713\pi\)
0.992393 0.123110i \(-0.0392868\pi\)
\(548\) 50.5256i 2.15835i
\(549\) 0 0
\(550\) 13.2750 + 1.63752i 0.566050 + 0.0698241i
\(551\) −10.9234 −0.465351
\(552\) 0 0
\(553\) 3.78892i 0.161121i
\(554\) −6.46898 −0.274840
\(555\) 0 0
\(556\) −10.9624 −0.464909
\(557\) 19.8700i 0.841920i −0.907079 0.420960i \(-0.861693\pi\)
0.907079 0.420960i \(-0.138307\pi\)
\(558\) 0 0
\(559\) 14.3634 0.607509
\(560\) −57.6991 + 51.0191i −2.43823 + 2.15595i
\(561\) 0 0
\(562\) 81.5633i 3.44054i
\(563\) 5.83383i 0.245866i 0.992415 + 0.122933i \(0.0392301\pi\)
−0.992415 + 0.122933i \(0.960770\pi\)
\(564\) 0 0
\(565\) 23.0132 20.3488i 0.968171 0.856082i
\(566\) −22.3576 −0.939758
\(567\) 0 0
\(568\) 62.0625i 2.60409i
\(569\) 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 \(0.670746\pi\)
\(572\) 26.3938i 1.10358i
\(573\) 0 0
\(574\) 17.9248 0.748166
\(575\) −19.8496 2.44851i −0.827784 0.102110i
\(576\) 0 0
\(577\) 16.5647i 0.689596i 0.938677 + 0.344798i \(0.112053\pi\)
−0.938677 + 0.344798i \(0.887947\pi\)
\(578\) 9.76353i 0.406109i
\(579\) 0 0
\(580\) 18.2374 + 20.6253i 0.757268 + 0.856419i
\(581\) 2.26187 0.0938380
\(582\) 0 0
\(583\) 6.57452i 0.272289i
\(584\) 54.1925 2.24250
\(585\) 0 0
\(586\) −61.7802 −2.55212
\(587\) 27.3258i 1.12786i −0.825823 0.563929i \(-0.809289\pi\)
0.825823 0.563929i \(-0.190711\pi\)
\(588\) 0 0
\(589\) −4.40246 −0.181400
\(590\) −59.4880 + 52.6009i −2.44908 + 2.16554i
\(591\) 0 0
\(592\) 19.7889i 0.813320i
\(593\) 12.6048i 0.517618i −0.965928 0.258809i \(-0.916670\pi\)
0.965928 0.258809i \(-0.0833301\pi\)
\(594\) 0 0
\(595\) 18.8872 + 21.3601i 0.774298 + 0.875679i
\(596\) −42.8627 −1.75573
\(597\) 0 0
\(598\) 54.7729i 2.23983i
\(599\) 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 \(0.651572\pi\)
\(602\) 21.0640i 0.858503i
\(603\) 0 0
\(604\) 78.7631 3.20482
\(605\) 1.48119 + 1.67513i 0.0602191 + 0.0681038i
\(606\) 0 0
\(607\) 33.5183i 1.36047i 0.732995 + 0.680234i \(0.238122\pi\)
−0.732995 + 0.680234i \(0.761878\pi\)
\(608\) 72.9643i 2.95909i
\(609\) 0 0
\(610\) 35.5125 31.4010i 1.43786 1.27139i
\(611\) −22.0752 −0.893068
\(612\) 0 0
\(613\) 11.5672i 0.467196i −0.972333 0.233598i \(-0.924950\pi\)
0.972333 0.233598i \(-0.0750499\pi\)
\(614\) 7.10413 0.286699
\(615\) 0 0
\(616\) −23.6932 −0.954627
\(617\) 33.3357i 1.34204i 0.741438 + 0.671022i \(0.234144\pi\)
−0.741438 + 0.671022i \(0.765856\pi\)
\(618\) 0 0
\(619\) −4.43866 −0.178405 −0.0892024 0.996014i \(-0.528432\pi\)
−0.0892024 + 0.996014i \(0.528432\pi\)
\(620\) 7.35026 + 8.31265i 0.295194 + 0.333844i
\(621\) 0 0
\(622\) 42.7367i 1.71359i
\(623\) 8.31265i 0.333039i
\(624\) 0 0
\(625\) −24.2506 6.07522i −0.970024 0.243009i
\(626\) 0.162205 0.00648301
\(627\) 0 0
\(628\) 36.1622i 1.44303i
\(629\) 7.32582 0.292100
\(630\) 0 0
\(631\) −39.8496 −1.58639 −0.793193 0.608971i \(-0.791583\pi\)
−0.793193 + 0.608971i \(0.791583\pi\)
\(632\) 11.4010i 0.453509i
\(633\) 0 0
\(634\) −10.2012 −0.405143
\(635\) −20.8021 + 18.3938i −0.825506 + 0.729934i
\(636\) 0 0
\(637\) 4.47371i 0.177255i
\(638\) 6.38787i 0.252898i
\(639\) 0 0
\(640\) −27.7562 + 24.5428i −1.09716 + 0.970139i
\(641\) 3.88858 0.153590 0.0767948 0.997047i \(-0.475531\pi\)
0.0767948 + 0.997047i \(0.475531\pi\)
\(642\) 0 0
\(643\) 40.9380i 1.61444i −0.590254 0.807218i \(-0.700972\pi\)
0.590254 0.807218i \(-0.299028\pi\)
\(644\) 57.8759 2.28063
\(645\) 0 0
\(646\) −55.6093 −2.18792
\(647\) 1.76257i 0.0692939i −0.999400 0.0346469i \(-0.988969\pi\)
0.999400 0.0346469i \(-0.0110307\pi\)
\(648\) 0 0
\(649\) −13.2750 −0.521091
\(650\) −8.38199 + 67.9511i −0.328769 + 2.66526i
\(651\) 0 0
\(652\) 34.8627i 1.36533i
\(653\) 7.03761i 0.275403i −0.990474 0.137702i \(-0.956029\pi\)
0.990474 0.137702i \(-0.0439715\pi\)
\(654\) 0 0
\(655\) −8.77575 9.92478i −0.342897 0.387793i
\(656\) 29.3112 1.14441
\(657\) 0 0
\(658\) 32.3733i 1.26204i
\(659\) −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 \(0.417206\pi\)
\(662\) 64.9741i 2.52529i
\(663\) 0 0
\(664\) 6.80606 0.264126
\(665\) −21.5026 + 19.0132i −0.833836 + 0.737299i
\(666\) 0 0
\(667\) 9.55149i 0.369835i
\(668\) 60.9438i 2.35799i
\(669\) 0 0
\(670\) 42.3996 + 47.9511i 1.63804 + 1.85251i
\(671\) 7.92478 0.305933
\(672\) 0 0
\(673\) 18.2677i 0.704170i −0.935968 0.352085i \(-0.885473\pi\)
0.935968 0.352085i \(-0.114527\pi\)
\(674\) −60.4055 −2.32673
\(675\) 0 0
\(676\) 68.0698 2.61807
\(677\) 33.0191i 1.26903i 0.772912 + 0.634513i \(0.218799\pi\)
−0.772912 + 0.634513i \(0.781201\pi\)
\(678\) 0 0
\(679\) 27.8496 1.06877
\(680\) 56.8324 + 64.2736i 2.17942 + 2.46478i
\(681\) 0 0
\(682\) 2.57452i 0.0985833i
\(683\) 30.8627i 1.18093i 0.807063 + 0.590465i \(0.201056\pi\)
−0.807063 + 0.590465i \(0.798944\pi\)
\(684\) 0 0
\(685\) 16.4142 14.5139i 0.627155 0.554547i
\(686\) 45.9854 1.75573
\(687\) 0 0
\(688\) 34.4445i 1.31319i
\(689\) 33.6531 1.28208
\(690\) 0 0
\(691\) 27.6991 1.05372 0.526862 0.849951i \(-0.323369\pi\)
0.526862 + 0.849951i \(0.323369\pi\)
\(692\) 36.0567i 1.37067i
\(693\) 0 0
\(694\) −54.0176 −2.05048
\(695\) 3.14903 + 3.56134i 0.119450 + 0.135089i
\(696\) 0 0
\(697\) 10.8510i 0.411010i
\(698\) 72.8989i 2.75927i
\(699\) 0 0
\(700\) 71.8007 + 8.85685i 2.71381 + 0.334757i
\(701\) 38.9643 1.47166 0.735831 0.677166i \(-0.236792\pi\)
0.735831 + 0.677166i \(0.236792\pi\)
\(702\) 0 0
\(703\) 7.37470i 0.278142i
\(704\) −18.1187 −0.682875
\(705\) 0 0
\(706\) 13.4372 0.505717
\(707\) 38.1965i 1.43653i
\(708\) 0 0
\(709\) 4.32250 0.162335 0.0811674 0.996700i \(-0.474135\pi\)
0.0811674 + 0.996700i \(0.474135\pi\)
\(710\) 32.9380 29.1246i 1.23614 1.09303i
\(711\) 0 0
\(712\) 25.0132i 0.937408i
\(713\) 3.84955i 0.144167i
\(714\) 0 0
\(715\) −8.57452 + 7.58181i −0.320669 + 0.283544i
\(716\) 13.9248 0.520393
\(717\) 0 0
\(718\) 48.6253i 1.81468i
\(719\) −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
\(722\) 5.15282i 0.191768i
\(723\) 0 0
\(724\) 109.313 4.06259
\(725\) 1.46168 11.8496i 0.0542855 0.440081i
\(726\) 0 0
\(727\) 21.6728i 0.803798i −0.915684 0.401899i \(-0.868350\pi\)
0.915684 0.401899i \(-0.131650\pi\)
\(728\) 121.279i 4.49489i
\(729\) 0 0
\(730\) −25.4314 28.7612i −0.941257 1.06450i
\(731\) −12.7513 −0.471624
\(732\) 0 0
\(733\) 25.0698i 0.925976i −0.886365 0.462988i \(-0.846777\pi\)
0.886365 0.462988i \(-0.153223\pi\)
\(734\) −22.9116 −0.845683
\(735\) 0 0
\(736\) 63.8007 2.35172
\(737\) 10.7005i 0.394159i
\(738\) 0 0
\(739\) 18.9018 0.695312 0.347656 0.937622i \(-0.386978\pi\)
0.347656 + 0.937622i \(0.386978\pi\)
\(740\) 13.9248 12.3127i 0.511885 0.452622i
\(741\) 0 0
\(742\) 49.3522i 1.81178i
\(743\) 43.6688i 1.60205i 0.598629 + 0.801026i \(0.295712\pi\)
−0.598629 + 0.801026i \(0.704288\pi\)
\(744\) 0 0
\(745\) 12.3127 + 13.9248i 0.451101 + 0.510164i
\(746\) −20.9175 −0.765843
\(747\) 0 0
\(748\) 23.4314i 0.856736i
\(749\) −26.4650 −0.967010
\(750\) 0 0
\(751\) −47.0541 −1.71703 −0.858514 0.512789i \(-0.828612\pi\)
−0.858514 + 0.512789i \(0.828612\pi\)
\(752\) 52.9380i 1.93045i
\(753\) 0 0
\(754\) −32.6977 −1.19078
\(755\) −22.6253 25.5877i −0.823419 0.931231i
\(756\) 0 0
\(757\) 26.9116i 0.978119i −0.872250 0.489059i \(-0.837340\pi\)
0.872250 0.489059i \(-0.162660\pi\)
\(758\) 3.07381i 0.111646i
\(759\) 0 0
\(760\) −64.7024 + 57.2116i −2.34700 + 2.07528i
\(761\) −16.6859 −0.604865 −0.302432 0.953171i \(-0.597799\pi\)
−0.302432 + 0.953171i \(0.597799\pi\)
\(762\) 0 0
\(763\) 43.2163i 1.56454i
\(764\) 67.8007 2.45294
\(765\) 0 0
\(766\) 91.1871 3.29473
\(767\) 67.9511i 2.45357i
\(768\) 0 0
\(769\) −23.2995 −0.840201 −0.420100 0.907478i \(-0.638005\pi\)
−0.420100 + 0.907478i \(0.638005\pi\)
\(770\) 11.1187 + 12.5745i 0.400691 + 0.453154i
\(771\) 0 0
\(772\) 30.3938i 1.09390i
\(773\) 1.63656i 0.0588631i −0.999567 0.0294316i \(-0.990630\pi\)
0.999567 0.0294316i \(-0.00936971\pi\)
\(774\) 0 0
\(775\) 0.589104 4.77575i 0.0211612 0.171550i
\(776\) 83.8007 3.00827
\(777\) 0 0
\(778\) 63.5994i 2.28015i
\(779\) 10.9234 0.391370
\(780\) 0 0
\(781\) 7.35026 0.263013
\(782\) 48.6253i 1.73884i
\(783\) 0 0
\(784\) −10.7283 −0.383153
\(785\) −11.7480 + 10.3879i −0.419304 + 0.370759i
\(786\) 0 0
\(787\) 2.09095i 0.0745344i −0.999305 0.0372672i \(-0.988135\pi\)
0.999305 0.0372672i \(-0.0118653\pi\)
\(788\) 105.157i 3.74606i
\(789\) 0 0
\(790\) −6.05079 + 5.35026i −0.215277 + 0.190354i
\(791\) 38.5501 1.37068
\(792\) 0 0
\(793\) 40.5647i 1.44049i
\(794\) −11.1030 −0.394030
\(795\) 0 0
\(796\) 89.7255 3.18023
\(797\) 26.8872i 0.952392i 0.879339 + 0.476196i \(0.157985\pi\)
−0.879339 + 0.476196i \(0.842015\pi\)
\(798\) 0 0
\(799\) 19.5975 0.693311
\(800\) 79.1509 + 9.76353i 2.79841 + 0.345193i
\(801\) 0 0
\(802\) 90.7269i 3.20368i
\(803\) 6.41819i 0.226493i
\(804\) 0 0
\(805\) −16.6253 18.8021i −0.585965 0.662687i
\(806\) −13.1782 −0.464183
\(807\) 0 0
\(808\) 114.935i 4.04340i
\(809\) −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 \(0.293942\pi\)
\(812\) 34.5501i 1.21247i
\(813\) 0 0
\(814\) 4.31265 0.151158
\(815\) 11.3258 10.0146i 0.396726 0.350796i
\(816\) 0 0
\(817\) 12.8364i 0.449088i
\(818\) 22.8726i 0.799721i
\(819\) 0 0
\(820\) −18.2374 20.6253i −0.636879 0.720267i
\(821\) −18.7612 −0.654769 −0.327384 0.944891i \(-0.606167\pi\)
−0.327384 + 0.944891i \(0.606167\pi\)
\(822\) 0 0
\(823\) 33.0592i 1.15237i 0.817319 + 0.576186i \(0.195460\pi\)
−0.817319 + 0.576186i \(0.804540\pi\)
\(824\) 137.737 4.79830
\(825\) 0 0
\(826\) −99.6502 −3.46728
\(827\) 10.8813i 0.378379i 0.981941 + 0.189190i \(0.0605861\pi\)
−0.981941 + 0.189190i \(0.939414\pi\)
\(828\) 0 0
\(829\) −42.7221 −1.48380 −0.741900 0.670510i \(-0.766075\pi\)
−0.741900 + 0.670510i \(0.766075\pi\)
\(830\) −3.19394 3.61213i −0.110863 0.125379i
\(831\) 0 0
\(832\) 92.7445i 3.21534i
\(833\) 3.97159i 0.137608i
\(834\) 0 0
\(835\) −19.7988 + 17.5066i −0.685165 + 0.605840i
\(836\) −23.5877 −0.815797
\(837\) 0 0
\(838\) 35.5778i 1.22902i
\(839\) −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
\(842\) 89.0856i 3.07009i
\(843\) 0 0
\(844\) −93.4636 −3.21715
\(845\) −19.5536 22.1138i −0.672664 0.760737i
\(846\) 0 0
\(847\) 2.80606i 0.0964175i
\(848\) 80.7024i 2.77133i
\(849\) 0 0
\(850\) 7.44121 60.3244i 0.255232 2.06911i
\(851\) −6.44851 −0.221052
\(852\) 0 0
\(853\) 54.4709i 1.86505i −0.361109 0.932524i \(-0.617602\pi\)
0.361109 0.932524i \(-0.382398\pi\)
\(854\) 59.4880 2.03564
\(855\) 0 0
\(856\) −79.6345 −2.72185
\(857\) 38.5705i 1.31754i −0.752342 0.658772i \(-0.771076\pi\)
0.752342 0.658772i \(-0.228924\pi\)
\(858\) 0 0
\(859\) 18.5139 0.631685 0.315843 0.948812i \(-0.397713\pi\)
0.315843 + 0.948812i \(0.397713\pi\)
\(860\) −24.2374 + 21.4314i −0.826489 + 0.730803i
\(861\) 0 0
\(862\) 5.55149i 0.189085i
\(863\) 22.5383i 0.767213i 0.923497 + 0.383607i \(0.125318\pi\)
−0.923497 + 0.383607i \(0.874682\pi\)
\(864\) 0 0
\(865\) −11.7137 + 10.3576i −0.398278 + 0.352167i
\(866\) 100.053 3.39993
\(867\) 0 0
\(868\) 13.9248i 0.472638i
\(869\) −1.35026 −0.0458045
\(870\) 0 0
\(871\) −54.7729 −1.85591
\(872\) 130.040i 4.40371i
\(873\) 0 0
\(874\) 48.9497 1.65575
\(875\) −17.7480 25.8700i −0.599991 0.874566i
\(876\) 0 0
\(877\) 18.9419i 0.639623i 0.947481 + 0.319812i \(0.103620\pi\)
−0.947481 + 0.319812i \(0.896380\pi\)
\(878\) 16.1162i 0.543894i
\(879\) 0 0
\(880\) 18.1817 + 20.5623i 0.612906 + 0.693155i
\(881\) 20.0263 0.674705 0.337352 0.941378i \(-0.390469\pi\)
0.337352 + 0.941378i \(0.390469\pi\)
\(882\) 0 0
\(883\) 22.2981i 0.750390i −0.926946 0.375195i \(-0.877576\pi\)
0.926946 0.375195i \(-0.122424\pi\)
\(884\) −119.938 −4.03397
\(885\) 0 0
\(886\) −27.1147 −0.910938
\(887\) 3.10413i 0.104226i −0.998641 0.0521132i \(-0.983404\pi\)
0.998641 0.0521132i \(-0.0165957\pi\)
\(888\) 0 0
\(889\) −34.8462 −1.16871
\(890\) −13.2750 + 11.7381i −0.444980 + 0.393463i
\(891\) 0 0
\(892\) 121.888i 4.08110i
\(893\) 19.7283i 0.660182i
\(894\) 0 0
\(895\) −4.00000 4.52373i −0.133705 0.151212i
\(896\) −46.4953 −1.55330
\(897\) 0 0
\(898\) 30.7269i 1.02537i
\(899\) 2.29806 0.0766447
\(900\) 0 0
\(901\) −29.8759 −0.995311
\(902\) 6.38787i 0.212693i
\(903\) 0 0
\(904\) 115.999 3.85807
\(905\) −31.4010 35.5125i −1.04381 1.18047i
\(906\) 0 0
\(907\) 10.8627i 0.360691i 0.983603 + 0.180345i \(0.0577216\pi\)
−0.983603 + 0.180345i \(0.942278\pi\)
\(908\) 6.54420i 0.217177i
\(909\) 0 0
\(910\) −64.3653 + 56.9135i −2.13369 + 1.88666i
\(911\) 5.82321 0.192931 0.0964657 0.995336i \(-0.469246\pi\)
0.0964657 + 0.995336i \(0.469246\pi\)
\(912\) 0 0
\(913\) 0.806063i 0.0266768i
\(914\) −40.2433 −1.33113
\(915\) 0 0
\(916\) 83.4128 2.75604
\(917\) 16.6253i 0.549016i
\(918\) 0 0
\(919\) 31.2750 1.03167 0.515834 0.856688i \(-0.327482\pi\)
0.515834 + 0.856688i \(0.327482\pi\)
\(920\) −50.0263 56.5764i −1.64932 1.86527i
\(921\) 0 0
\(922\) 70.1886i 2.31154i
\(923\) 37.6239i 1.23841i
\(924\) 0 0
\(925\) −8.00000 0.986826i −0.263038 0.0324466i
\(926\) 14.6107 0.480138
\(927\) 0 0
\(928\) 38.0870i 1.25027i
\(929\) −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
\(932\) 95.2320i 3.11943i
\(933\) 0 0
\(934\) 7.22425 0.236385
\(935\) 7.61213 6.73084i 0.248943 0.220122i
\(936\) 0 0
\(937\) 16.4328i 0.536835i 0.963303 + 0.268418i \(0.0865007\pi\)
−0.963303 + 0.268418i \(0.913499\pi\)
\(938\) 80.3244i 2.62268i
\(939\) 0 0
\(940\) 37.2506 32.9380i 1.21498 1.07432i
\(941\) −5.86414 −0.191166 −0.0955828 0.995421i \(-0.530471\pi\)
−0.0955828 + 0.995421i \(0.530471\pi\)
\(942\) 0 0
\(943\) 9.55149i 0.311039i
\(944\) −162.952 −5.30362
\(945\) 0 0
\(946\) −7.50659 −0.244060
\(947\) 38.7415i 1.25893i 0.777030 + 0.629464i \(0.216726\pi\)
−0.777030 + 0.629464i \(0.783274\pi\)
\(948\) 0 0
\(949\) 32.8529 1.06645
\(950\) 60.7269 + 7.49086i 1.97024 + 0.243036i
\(951\) 0 0
\(952\) 107.667i 3.48950i
\(953\) 6.09569i 0.197459i −0.995114 0.0987294i \(-0.968522\pi\)
0.995114 0.0987294i \(-0.0314778\pi\)
\(954\) 0 0
\(955\) −19.4763 22.0263i −0.630237 0.712756i
\(956\) −99.6502 −3.22292
\(957\) 0 0
\(958\) 44.8773i 1.44992i
\(959\) 27.4960 0.887891
\(960\) 0 0
\(961\) −30.0738 −0.970123
\(962\) 22.0752i 0.711734i
\(963\) 0 0
\(964\) −147.213 −4.74143
\(965\) 9.87399 8.73084i 0.317855 0.281056i
\(966\) 0 0
\(967\) 37.9697i 1.22102i 0.792008 + 0.610511i \(0.209036\pi\)
−0.792008 + 0.610511i \(0.790964\pi\)
\(968\) 8.44358i 0.271387i
\(969\) 0 0
\(970\) −39.3258 44.4749i −1.26268 1.42800i
\(971\) 60.8773 1.95365 0.976823 0.214049i \(-0.0686651\pi\)
0.976823 + 0.214049i \(0.0686651\pi\)
\(972\) 0 0
\(973\) 5.96571i 0.191252i
\(974\) −92.0235 −2.94862
\(975\) 0 0
\(976\) 97.2769 3.11376
\(977\) 21.3963i 0.684529i −0.939604 0.342264i \(-0.888806\pi\)
0.939604 0.342264i \(-0.111194\pi\)
\(978\) 0 0
\(979\) −2.96239 −0.0946784
\(980\) 6.67513 + 7.54912i 0.213229 + 0.241148i
\(981\) 0 0
\(982\) 40.0000i 1.27645i
\(983\) 18.8919i 0.602558i 0.953536 + 0.301279i \(0.0974136\pi\)
−0.953536 + 0.301279i \(0.902586\pi\)
\(984\) 0 0
\(985\) 34.1622 30.2071i 1.08850 0.962479i
\(986\) 29.0278 0.924432
\(987\) 0 0
\(988\) 120.739i 3.84121i
\(989\) 11.2243 0.356911
\(990\) 0 0
\(991\) −1.33567 −0.0424291 −0.0212145 0.999775i \(-0.506753\pi\)
−0.0212145 + 0.999775i \(0.506753\pi\)
\(992\) 15.3503i 0.487371i
\(993\) 0 0
\(994\) 55.1754 1.75006
\(995\) −25.7743 29.1490i −0.817101 0.924086i
\(996\) 0 0
\(997\) 48.9076i 1.54892i −0.632623 0.774460i \(-0.718022\pi\)
0.632623 0.774460i \(-0.281978\pi\)
\(998\) 7.62672i 0.241419i
\(999\) 0 0
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 495.2.c.d.199.1 6
3.2 odd 2 165.2.c.a.34.6 yes 6
5.2 odd 4 2475.2.a.be.1.3 3
5.3 odd 4 2475.2.a.y.1.1 3
5.4 even 2 inner 495.2.c.d.199.6 6
12.11 even 2 2640.2.d.i.529.1 6
15.2 even 4 825.2.a.h.1.1 3
15.8 even 4 825.2.a.n.1.3 3
15.14 odd 2 165.2.c.a.34.1 6
33.32 even 2 1815.2.c.d.364.1 6
60.59 even 2 2640.2.d.i.529.4 6
165.32 odd 4 9075.2.a.ck.1.3 3
165.98 odd 4 9075.2.a.cc.1.1 3
165.164 even 2 1815.2.c.d.364.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.1 6 15.14 odd 2
165.2.c.a.34.6 yes 6 3.2 odd 2
495.2.c.d.199.1 6 1.1 even 1 trivial
495.2.c.d.199.6 6 5.4 even 2 inner
825.2.a.h.1.1 3 15.2 even 4
825.2.a.n.1.3 3 15.8 even 4
1815.2.c.d.364.1 6 33.32 even 2
1815.2.c.d.364.6 6 165.164 even 2
2475.2.a.y.1.1 3 5.3 odd 4
2475.2.a.be.1.3 3 5.2 odd 4
2640.2.d.i.529.1 6 12.11 even 2
2640.2.d.i.529.4 6 60.59 even 2
9075.2.a.cc.1.1 3 165.98 odd 4
9075.2.a.ck.1.3 3 165.32 odd 4