Properties

Label 77.2.l.a.13.2
Level $77$
Weight $2$
Character 77.13
Analytic conductor $0.615$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,2,Mod(6,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.l (of order \(10\), degree \(4\), 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: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.37515625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - x^{6} + 3x^{5} - x^{4} + 6x^{3} - 4x^{2} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label 13.2
Root \(-1.41264 + 0.0667372i\) of defining polynomial
Character \(\chi\) \(=\) 77.13
Dual form 77.2.l.a.6.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.47668 + 0.479802i) q^{2} +(0.332338 + 0.241457i) q^{4} +(-1.55513 + 2.14046i) q^{7} +(-1.45037 - 1.99627i) q^{8} +(0.927051 - 2.85317i) q^{9} +O(q^{10})\) \(q+(1.47668 + 0.479802i) q^{2} +(0.332338 + 0.241457i) q^{4} +(-1.55513 + 2.14046i) q^{7} +(-1.45037 - 1.99627i) q^{8} +(0.927051 - 2.85317i) q^{9} +(-1.89823 + 2.71970i) q^{11} +(-3.32343 + 2.41461i) q^{14} +(-1.43780 - 4.42511i) q^{16} +(2.73791 - 3.76842i) q^{18} +(-4.10799 + 3.10535i) q^{22} +9.58240 q^{23} +(-4.04508 + 2.93893i) q^{25} +(-1.03366 + 0.335856i) q^{28} +(0.804217 - 1.10691i) q^{29} -2.28929i q^{32} +(0.997013 - 0.724372i) q^{36} +(1.10547 + 0.803169i) q^{37} +9.77751i q^{43} +(-1.28754 + 0.445517i) q^{44} +(14.1501 + 4.59766i) q^{46} +(-2.16312 - 6.65740i) q^{49} +(-7.38340 + 2.39901i) q^{50} +(4.06518 - 12.5113i) q^{53} +6.52844 q^{56} +(1.71867 - 1.24869i) q^{58} +(4.66540 + 6.42137i) q^{63} +(-1.77720 + 5.46967i) q^{64} -16.3336 q^{67} +(-3.08300 - 9.48849i) q^{71} +(-7.04025 + 2.28752i) q^{72} +(1.24706 + 1.71643i) q^{74} +(-2.86940 - 8.29256i) q^{77} +(-11.9015 - 3.86702i) q^{79} +(-7.28115 - 5.29007i) q^{81} +(-4.69127 + 14.4383i) q^{86} +(8.18236 - 0.155208i) q^{88} +(3.18459 + 2.31374i) q^{92} -10.8687i q^{98} +(6.00000 + 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{4} - 10 q^{8} - 6 q^{9} - 4 q^{11} - 21 q^{14} + 8 q^{16} - 15 q^{18} - 14 q^{22} + 16 q^{23} - 10 q^{25} + 35 q^{28} + 30 q^{36} - 18 q^{37} + 25 q^{44} + 15 q^{46} + 14 q^{49} + 30 q^{53} - 42 q^{56} + 19 q^{58} - 34 q^{64} + 8 q^{67} - 48 q^{71} - 75 q^{72} - 14 q^{77} - 40 q^{79} - 18 q^{81} + 23 q^{86} - 8 q^{88} + 25 q^{92} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.47668 + 0.479802i 1.04417 + 0.339271i 0.780378 0.625308i \(-0.215027\pi\)
0.263792 + 0.964580i \(0.415027\pi\)
\(3\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(4\) 0.332338 + 0.241457i 0.166169 + 0.120729i
\(5\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(6\) 0 0
\(7\) −1.55513 + 2.14046i −0.587785 + 0.809017i
\(8\) −1.45037 1.99627i −0.512784 0.705786i
\(9\) 0.927051 2.85317i 0.309017 0.951057i
\(10\) 0 0
\(11\) −1.89823 + 2.71970i −0.572336 + 0.820019i
\(12\) 0 0
\(13\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(14\) −3.32343 + 2.41461i −0.888224 + 0.645333i
\(15\) 0 0
\(16\) −1.43780 4.42511i −0.359451 1.10628i
\(17\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(18\) 2.73791 3.76842i 0.645333 0.888224i
\(19\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.10799 + 3.10535i −0.875825 + 0.662062i
\(23\) 9.58240 1.99807 0.999035 0.0439305i \(-0.0139880\pi\)
0.999035 + 0.0439305i \(0.0139880\pi\)
\(24\) 0 0
\(25\) −4.04508 + 2.93893i −0.809017 + 0.587785i
\(26\) 0 0
\(27\) 0 0
\(28\) −1.03366 + 0.335856i −0.195343 + 0.0634708i
\(29\) 0.804217 1.10691i 0.149339 0.205548i −0.727793 0.685797i \(-0.759454\pi\)
0.877132 + 0.480249i \(0.159454\pi\)
\(30\) 0 0
\(31\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(32\) 2.28929i 0.404693i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.997013 0.724372i 0.166169 0.120729i
\(37\) 1.10547 + 0.803169i 0.181738 + 0.132040i 0.674935 0.737878i \(-0.264172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(42\) 0 0
\(43\) 9.77751i 1.49106i 0.666474 + 0.745528i \(0.267803\pi\)
−0.666474 + 0.745528i \(0.732197\pi\)
\(44\) −1.28754 + 0.445517i −0.194104 + 0.0671642i
\(45\) 0 0
\(46\) 14.1501 + 4.59766i 2.08632 + 0.677888i
\(47\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(48\) 0 0
\(49\) −2.16312 6.65740i −0.309017 0.951057i
\(50\) −7.38340 + 2.39901i −1.04417 + 0.339271i
\(51\) 0 0
\(52\) 0 0
\(53\) 4.06518 12.5113i 0.558396 1.71857i −0.128407 0.991722i \(-0.540986\pi\)
0.686803 0.726844i \(-0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.52844 0.872400
\(57\) 0 0
\(58\) 1.71867 1.24869i 0.225672 0.163960i
\(59\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(60\) 0 0
\(61\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) 4.66540 + 6.42137i 0.587785 + 0.809017i
\(64\) −1.77720 + 5.46967i −0.222151 + 0.683709i
\(65\) 0 0
\(66\) 0 0
\(67\) −16.3336 −1.99547 −0.997735 0.0672706i \(-0.978571\pi\)
−0.997735 + 0.0672706i \(0.978571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.08300 9.48849i −0.365885 1.12608i −0.949425 0.313993i \(-0.898333\pi\)
0.583541 0.812084i \(-0.301667\pi\)
\(72\) −7.04025 + 2.28752i −0.829702 + 0.269586i
\(73\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(74\) 1.24706 + 1.71643i 0.144968 + 0.199531i
\(75\) 0 0
\(76\) 0 0
\(77\) −2.86940 8.29256i −0.326998 0.945025i
\(78\) 0 0
\(79\) −11.9015 3.86702i −1.33902 0.435074i −0.450035 0.893011i \(-0.648589\pi\)
−0.888985 + 0.457937i \(0.848589\pi\)
\(80\) 0 0
\(81\) −7.28115 5.29007i −0.809017 0.587785i
\(82\) 0 0
\(83\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.69127 + 14.4383i −0.505873 + 1.55692i
\(87\) 0 0
\(88\) 8.18236 0.155208i 0.872243 0.0165452i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.18459 + 2.31374i 0.332017 + 0.241224i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(98\) 10.8687i 1.09791i
\(99\) 6.00000 + 7.93725i 0.603023 + 0.797724i
\(100\) −2.05396 −0.205396
\(101\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0059 16.5248i 1.16612 1.60503i
\(107\) 9.42609 + 12.9739i 0.911254 + 1.25423i 0.966736 + 0.255774i \(0.0823304\pi\)
−0.0554821 + 0.998460i \(0.517670\pi\)
\(108\) 0 0
\(109\) 19.1420i 1.83347i 0.399498 + 0.916734i \(0.369184\pi\)
−0.399498 + 0.916734i \(0.630816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 11.7077 + 3.80407i 1.10628 + 0.359451i
\(113\) 15.7856 11.4689i 1.48498 1.07890i 0.509073 0.860724i \(-0.329988\pi\)
0.975909 0.218179i \(-0.0700116\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.534543 0.173684i 0.0496311 0.0161261i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.79348 10.3252i −0.344862 0.938653i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 3.80831 + 11.7208i 0.339271 + 1.04417i
\(127\) 21.1584 6.87480i 1.87751 0.610040i 0.889212 0.457495i \(-0.151253\pi\)
0.988297 0.152545i \(-0.0487468\pi\)
\(128\) −7.93994 + 10.9284i −0.701798 + 0.965942i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −24.1195 7.83691i −2.08361 0.677006i
\(135\) 0 0
\(136\) 0 0
\(137\) 6.34450 + 19.5264i 0.542047 + 1.66825i 0.727909 + 0.685674i \(0.240493\pi\)
−0.185861 + 0.982576i \(0.559507\pi\)
\(138\) 0 0
\(139\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.4907i 1.29995i
\(143\) 0 0
\(144\) −13.9585 −1.16321
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0.173457 + 0.533847i 0.0142581 + 0.0438819i
\(149\) −10.0650 + 3.27033i −0.824560 + 0.267916i −0.690752 0.723092i \(-0.742720\pi\)
−0.133808 + 0.991007i \(0.542720\pi\)
\(150\) 0 0
\(151\) −5.77554 7.94934i −0.470007 0.646909i 0.506540 0.862217i \(-0.330924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.258394 13.6222i −0.0208220 1.09771i
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(158\) −15.7192 11.4207i −1.25056 0.908582i
\(159\) 0 0
\(160\) 0 0
\(161\) −14.9019 + 20.5107i −1.17444 + 1.61647i
\(162\) −8.21374 11.3052i −0.645333 0.888224i
\(163\) −2.11662 + 6.51430i −0.165787 + 0.510239i −0.999093 0.0425718i \(-0.986445\pi\)
0.833307 + 0.552811i \(0.186445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(168\) 0 0
\(169\) 10.5172 + 7.64121i 0.809017 + 0.587785i
\(170\) 0 0
\(171\) 0 0
\(172\) −2.36085 + 3.24944i −0.180013 + 0.247767i
\(173\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(174\) 0 0
\(175\) 13.2288i 1.00000i
\(176\) 14.7642 + 4.48946i 1.11290 + 0.338406i
\(177\) 0 0
\(178\) 0 0
\(179\) −9.96326 + 7.23873i −0.744689 + 0.541048i −0.894176 0.447715i \(-0.852238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −13.8980 19.1290i −1.02458 1.41021i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.3570 + 13.3371i 1.32826 + 0.965040i 0.999789 + 0.0205267i \(0.00653431\pi\)
0.328474 + 0.944513i \(0.393466\pi\)
\(192\) 0 0
\(193\) 22.5017 7.31124i 1.61971 0.526275i 0.647834 0.761781i \(-0.275675\pi\)
0.971873 + 0.235507i \(0.0756750\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.888592 2.73480i 0.0634708 0.195343i
\(197\) 6.72059i 0.478822i −0.970918 0.239411i \(-0.923046\pi\)
0.970918 0.239411i \(-0.0769543\pi\)
\(198\) 5.05176 + 14.5996i 0.359013 + 1.03755i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 11.7338 + 3.81253i 0.829702 + 0.269586i
\(201\) 0 0
\(202\) 0 0
\(203\) 1.11863 + 3.44278i 0.0785123 + 0.241636i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.88338 27.3402i 0.617437 1.90028i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −27.0652 8.79401i −1.86324 0.605404i −0.993774 0.111417i \(-0.964461\pi\)
−0.869469 0.493987i \(-0.835539\pi\)
\(212\) 4.37197 3.17642i 0.300268 0.218158i
\(213\) 0 0
\(214\) 7.69441 + 23.6809i 0.525979 + 1.61880i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −9.18436 + 28.2665i −0.622043 + 1.91445i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(224\) 4.90012 + 3.56015i 0.327403 + 0.237872i
\(225\) 4.63525 + 14.2658i 0.309017 + 0.951057i
\(226\) 28.8130 9.36192i 1.91661 0.622746i
\(227\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(228\) 0 0
\(229\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.37610 −0.221652
\(233\) −20.1301 6.54066i −1.31876 0.428493i −0.436694 0.899610i \(-0.643851\pi\)
−0.882071 + 0.471117i \(0.843851\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.13864 5.69636i −0.267707 0.368467i 0.653907 0.756575i \(-0.273129\pi\)
−0.921614 + 0.388108i \(0.873129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −0.647711 17.0671i −0.0416365 1.09712i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(252\) 3.26056i 0.205396i
\(253\) −18.1896 + 26.0612i −1.14357 + 1.63845i
\(254\) 34.5428 2.16741
\(255\) 0 0
\(256\) −7.66265 + 5.56724i −0.478916 + 0.347953i
\(257\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(258\) 0 0
\(259\) −3.43830 + 1.11717i −0.213645 + 0.0694176i
\(260\) 0 0
\(261\) −2.41265 3.32073i −0.149339 0.205548i
\(262\) 0 0
\(263\) 32.0690i 1.97746i −0.149718 0.988729i \(-0.547836\pi\)
0.149718 0.988729i \(-0.452164\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −5.42828 3.94387i −0.331585 0.240910i
\(269\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(270\) 0 0
\(271\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 31.8783i 1.92584i
\(275\) −0.314502 16.5801i −0.0189652 0.999820i
\(276\) 0 0
\(277\) 30.0185 + 9.75361i 1.80364 + 0.586038i 0.999959 0.00902525i \(-0.00287287\pi\)
0.803679 + 0.595063i \(0.202873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −17.2967 + 5.62003i −1.03183 + 0.335263i −0.775515 0.631329i \(-0.782510\pi\)
−0.256319 + 0.966592i \(0.582510\pi\)
\(282\) 0 0
\(283\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(284\) 1.26647 3.89780i 0.0751512 0.231292i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −6.53173 2.12229i −0.384886 0.125057i
\(289\) 13.7533 9.99235i 0.809017 0.587785i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.37170i 0.195976i
\(297\) 0 0
\(298\) −16.4319 −0.951877
\(299\) 0 0
\(300\) 0 0
\(301\) −20.9284 15.2053i −1.20629 0.876421i
\(302\) −4.71450 14.5097i −0.271289 0.834942i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 1.04869 3.44877i 0.0597547 0.196512i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −3.02159 4.15886i −0.169977 0.233954i
\(317\) −0.136444 + 0.419931i −0.00766345 + 0.0235857i −0.954815 0.297200i \(-0.903947\pi\)
0.947152 + 0.320786i \(0.103947\pi\)
\(318\) 0 0
\(319\) 1.48387 + 4.28839i 0.0830809 + 0.240104i
\(320\) 0 0
\(321\) 0 0
\(322\) −31.8464 + 23.1378i −1.77473 + 1.28942i
\(323\) 0 0
\(324\) −1.14248 3.51618i −0.0634708 0.195343i
\(325\) 0 0
\(326\) −6.25115 + 8.60397i −0.346219 + 0.476530i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −26.0143 −1.42988 −0.714939 0.699187i \(-0.753545\pi\)
−0.714939 + 0.699187i \(0.753545\pi\)
\(332\) 0 0
\(333\) 3.31640 2.40951i 0.181738 0.132040i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.299709 0.412514i 0.0163262 0.0224711i −0.800776 0.598964i \(-0.795579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 11.8643 + 16.3298i 0.645333 + 0.888224i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 17.6138 + 5.72307i 0.951057 + 0.309017i
\(344\) 19.5185 14.1810i 1.05237 0.764590i
\(345\) 0 0
\(346\) 0 0
\(347\) −14.5040 + 4.71263i −0.778614 + 0.252987i −0.671248 0.741233i \(-0.734242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(350\) 6.34719 19.5346i 0.339271 1.04417i
\(351\) 0 0
\(352\) 6.22617 + 4.34558i 0.331856 + 0.231620i
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −18.1857 + 5.90889i −0.961144 + 0.312295i
\(359\) 20.3777 28.0476i 1.07550 1.48029i 0.211112 0.977462i \(-0.432292\pi\)
0.864384 0.502832i \(-0.167708\pi\)
\(360\) 0 0
\(361\) −5.87132 + 18.0701i −0.309017 + 0.951057i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(368\) −13.7776 42.4032i −0.718208 2.21042i
\(369\) 0 0
\(370\) 0 0
\(371\) 20.4581 + 28.1582i 1.06213 + 1.46190i
\(372\) 0 0
\(373\) 31.7490i 1.64390i 0.569558 + 0.821951i \(0.307114\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.72788 29.9393i −0.499688 1.53788i −0.809522 0.587090i \(-0.800274\pi\)
0.309834 0.950791i \(-0.399726\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 20.7082 + 28.5024i 1.05952 + 1.45831i
\(383\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 36.7357 1.86980
\(387\) 27.8969 + 9.06425i 1.41808 + 0.460762i
\(388\) 0 0
\(389\) 29.9038 + 21.7264i 1.51618 + 1.10157i 0.963338 + 0.268290i \(0.0864585\pi\)
0.552847 + 0.833283i \(0.313542\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −10.1526 + 13.9739i −0.512784 + 0.705786i
\(393\) 0 0
\(394\) 3.22455 9.92415i 0.162451 0.499971i
\(395\) 0 0
\(396\) 0.0775169 + 4.08659i 0.00389537 + 0.205359i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.8211 + 13.6743i 0.941055 + 0.683717i
\(401\) −4.65550 14.3282i −0.232484 0.715514i −0.997445 0.0714367i \(-0.977242\pi\)
0.764961 0.644077i \(-0.222758\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 5.62061i 0.278946i
\(407\) −4.28280 + 1.48194i −0.212290 + 0.0734570i
\(408\) 0 0
\(409\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 26.2358 36.1105i 1.28942 1.77473i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 32.1148 23.3327i 1.56518 1.13717i 0.633581 0.773676i \(-0.281584\pi\)
0.931597 0.363492i \(-0.118416\pi\)
\(422\) −35.7472 25.9719i −1.74015 1.26429i
\(423\) 0 0
\(424\) −30.8720 + 10.0309i −1.49928 + 0.487144i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 6.58771i 0.318429i
\(429\) 0 0
\(430\) 0 0
\(431\) −2.46842 0.802038i −0.118900 0.0386328i 0.248963 0.968513i \(-0.419910\pi\)
−0.367862 + 0.929880i \(0.619910\pi\)
\(432\) 0 0
\(433\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.62197 + 6.36160i −0.221352 + 0.304665i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −21.0000 −1.00000
\(442\) 0 0
\(443\) −23.4997 + 17.0736i −1.11651 + 0.811190i −0.983676 0.179949i \(-0.942407\pi\)
−0.132831 + 0.991139i \(0.542407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −8.94381 12.3101i −0.422555 0.581598i
\(449\) −7.18900 + 22.1255i −0.339270 + 1.04417i 0.625310 + 0.780376i \(0.284972\pi\)
−0.964580 + 0.263790i \(0.915028\pi\)
\(450\) 23.2901i 1.09791i
\(451\) 0 0
\(452\) 8.01539 0.377012
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.9252 + 11.6728i −1.68051 + 0.546031i −0.985011 0.172493i \(-0.944818\pi\)
−0.695501 + 0.718525i \(0.744818\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −2.73687 −0.127193 −0.0635967 0.997976i \(-0.520257\pi\)
−0.0635967 + 0.997976i \(0.520257\pi\)
\(464\) −6.05450 1.96723i −0.281073 0.0913262i
\(465\) 0 0
\(466\) −26.5874 19.3169i −1.23164 0.894838i
\(467\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(468\) 0 0
\(469\) 25.4010 34.9614i 1.17291 1.61437i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −26.5919 18.5599i −1.22269 0.853386i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −31.9284 23.1973i −1.46190 1.06213i
\(478\) −3.37833 10.3974i −0.154521 0.475567i
\(479\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.23238 4.34741i 0.0560171 0.197610i
\(485\) 0 0
\(486\) 0 0
\(487\) −34.4997 + 25.0655i −1.56333 + 1.13583i −0.630123 + 0.776495i \(0.716996\pi\)
−0.933210 + 0.359333i \(0.883004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.11027 + 4.28092i −0.140364 + 0.193195i −0.873412 0.486983i \(-0.838097\pi\)
0.733047 + 0.680178i \(0.238097\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.1042 + 8.15685i 1.12608 + 0.365885i
\(498\) 0 0
\(499\) 29.3570 + 21.3291i 1.31420 + 0.954821i 0.999985 + 0.00546838i \(0.00174065\pi\)
0.314213 + 0.949352i \(0.398259\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(504\) 6.05220 18.6268i 0.269586 0.829702i
\(505\) 0 0
\(506\) −39.3644 + 29.7567i −1.74996 + 1.32285i
\(507\) 0 0
\(508\) 8.69172 + 2.82411i 0.385633 + 0.125300i
\(509\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.7077 3.80407i 0.517414 0.168118i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −5.61328 −0.246634
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(522\) −1.96942 6.06125i −0.0861991 0.265294i
\(523\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 15.3868 47.3556i 0.670895 2.06480i
\(527\) 0 0
\(528\) 0 0
\(529\) 68.8225 2.99228
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 23.6898 + 32.6062i 1.02324 + 1.40838i
\(537\) 0 0
\(538\) 0 0
\(539\) 22.2122 + 6.75421i 0.956746 + 0.290924i
\(540\) 0 0
\(541\) 5.42178 + 1.76164i 0.233100 + 0.0757389i 0.423238 0.906019i \(-0.360893\pi\)
−0.190138 + 0.981757i \(0.560893\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.33080 + 12.8427i 0.398956 + 0.549116i 0.960482 0.278343i \(-0.0897853\pi\)
−0.561525 + 0.827460i \(0.689785\pi\)
\(548\) −2.60627 + 8.02128i −0.111334 + 0.342652i
\(549\) 0 0
\(550\) 7.49077 24.6345i 0.319407 1.05042i
\(551\) 0 0
\(552\) 0 0
\(553\) 26.7856 19.4609i 1.13904 0.827560i
\(554\) 39.6479 + 28.8059i 1.68448 + 1.22385i
\(555\) 0 0
\(556\) 0 0
\(557\) −23.7925 + 32.7476i −1.00812 + 1.38756i −0.0879152 + 0.996128i \(0.528020\pi\)
−0.920207 + 0.391433i \(0.871980\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −28.2382 −1.19116
\(563\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.6463 7.35824i 0.951057 0.309017i
\(568\) −14.4701 + 19.9163i −0.607150 + 0.835670i
\(569\) −24.8821 34.2473i −1.04311 1.43572i −0.894630 0.446808i \(-0.852561\pi\)
−0.148483 0.988915i \(-0.547439\pi\)
\(570\) 0 0
\(571\) 40.8794i 1.71075i −0.518010 0.855374i \(-0.673327\pi\)
0.518010 0.855374i \(-0.326673\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −38.7616 + 28.1620i −1.61647 + 1.17444i
\(576\) 13.9583 + 10.1413i 0.581598 + 0.422555i
\(577\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(578\) 25.1035 8.15664i 1.04417 0.339271i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 26.3104 + 34.8054i 1.08967 + 1.44149i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.96466 6.04661i 0.0807472 0.248514i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.13464 1.34342i −0.169361 0.0550288i
\(597\) 0 0
\(598\) 0 0
\(599\) 1.27212 + 3.91519i 0.0519775 + 0.159970i 0.973676 0.227937i \(-0.0731980\pi\)
−0.921698 + 0.387907i \(0.873198\pi\)
\(600\) 0 0
\(601\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(602\) −23.6089 32.4949i −0.962228 1.32439i
\(603\) −15.1421 + 46.6026i −0.616634 + 1.89780i
\(604\) 4.03641i 0.164239i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −27.0094 37.1752i −1.09090 1.50149i −0.846925 0.531712i \(-0.821549\pi\)
−0.243974 0.969782i \(-0.578451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −12.3925 + 17.7554i −0.499306 + 0.715384i
\(617\) 32.2257 1.29736 0.648679 0.761062i \(-0.275322\pi\)
0.648679 + 0.761062i \(0.275322\pi\)
\(618\) 0 0
\(619\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.72542 23.7764i 0.309017 0.951057i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −12.1742 8.84507i −0.484647 0.352117i 0.318475 0.947931i \(-0.396829\pi\)
−0.803122 + 0.595815i \(0.796829\pi\)
\(632\) 9.54195 + 29.3671i 0.379558 + 1.16816i
\(633\) 0 0
\(634\) −0.402968 + 0.554638i −0.0160039 + 0.0220275i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0.133625 + 7.04454i 0.00529026 + 0.278896i
\(639\) −29.9304 −1.18403
\(640\) 0 0
\(641\) 4.78557 3.47692i 0.189019 0.137330i −0.489251 0.872143i \(-0.662730\pi\)
0.678270 + 0.734813i \(0.262730\pi\)
\(642\) 0 0
\(643\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(644\) −9.90494 + 3.21831i −0.390309 + 0.126819i
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(648\) 22.2077i 0.872400i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −2.27636 + 1.65387i −0.0891491 + 0.0647706i
\(653\) −27.6929 20.1201i −1.08371 0.787359i −0.105382 0.994432i \(-0.533606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 26.4575i 1.03064i −0.856998 0.515319i \(-0.827673\pi\)
0.856998 0.515319i \(-0.172327\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −38.4148 12.4817i −1.49304 0.485117i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 6.05335 1.96685i 0.234562 0.0762140i
\(667\) 7.70633 10.6069i 0.298390 0.410699i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.6942 4.77443i −0.566419 0.184041i 0.0117883 0.999931i \(-0.496248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0.640499 0.465350i 0.0246711 0.0179246i
\(675\) 0 0
\(676\) 1.65024 + 5.07892i 0.0634708 + 0.195343i
\(677\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 45.1792 1.72873 0.864366 0.502863i \(-0.167720\pi\)
0.864366 + 0.502863i \(0.167720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 23.2640 + 16.9023i 0.888224 + 0.645333i
\(687\) 0 0
\(688\) 43.2666 14.0582i 1.64952 0.535962i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(692\) 0 0
\(693\) −26.3202 + 0.499256i −0.999820 + 0.0189652i
\(694\) −23.6788 −0.898836
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 3.19418 4.39641i 0.120729 0.166169i
\(701\) −25.8536 35.5844i −0.976476 1.34400i −0.938707 0.344717i \(-0.887975\pi\)
−0.0377695 0.999286i \(-0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −11.5023 15.2161i −0.433509 0.573479i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16.1243 + 49.6254i 0.605560 + 1.86372i 0.492893 + 0.870090i \(0.335939\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −22.0665 + 30.3720i −0.827560 + 1.13904i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −5.05901 −0.189064
\(717\) 0 0
\(718\) 43.5487 31.6400i 1.62522 1.18079i
\(719\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.3401 + 23.8666i −0.645333 + 0.888224i
\(723\) 0 0
\(724\) 0 0
\(725\) 6.84108i 0.254071i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −21.8435 + 15.8702i −0.809017 + 0.587785i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 21.9369i 0.808604i
\(737\) 31.0049 44.4225i 1.14208 1.63632i
\(738\) 0 0
\(739\) −51.6998 16.7983i −1.90181 0.617935i −0.956425 0.291977i \(-0.905687\pi\)
−0.945384 0.325959i \(-0.894313\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 16.6997 + 51.3964i 0.613066 + 1.88682i
\(743\) 25.2944 8.21865i 0.927962 0.301513i 0.194233 0.980955i \(-0.437778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −15.2332 + 46.8831i −0.557729 + 1.71651i
\(747\) 0 0
\(748\) 0 0
\(749\) −42.4289 −1.55032
\(750\) 0 0
\(751\) 18.8351 13.6845i 0.687303 0.499355i −0.188469 0.982079i \(-0.560353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.26681 25.4426i 0.300462 0.924728i −0.680869 0.732405i \(-0.738398\pi\)
0.981332 0.192323i \(-0.0616021\pi\)
\(758\) 48.8782i 1.77534i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(762\) 0 0
\(763\) −40.9726 29.7683i −1.48331 1.07769i
\(764\) 2.88036 + 8.86485i 0.104208 + 0.320719i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.24351 + 3.00340i 0.332681 + 0.108095i
\(773\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(774\) 36.8457 + 26.7700i 1.32439 + 0.962228i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 33.7340 + 46.4309i 1.20942 + 1.66463i
\(779\) 0 0
\(780\) 0 0
\(781\) 31.6580 + 9.62648i 1.13281 + 0.344462i
\(782\) 0 0
\(783\) 0 0
\(784\) −26.3496 + 19.1441i −0.941055 + 0.683717i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(788\) 1.62274 2.23350i 0.0578076 0.0795653i
\(789\) 0 0
\(790\) 0 0
\(791\) 51.6240i 1.83554i
\(792\) 7.14263 23.4896i 0.253802 0.834665i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 6.72805 + 9.26036i 0.237872 + 0.327403i
\(801\) 0 0
\(802\) 23.3918i 0.825993i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −11.3285 + 3.68085i −0.398289 + 0.129412i −0.501311 0.865267i \(-0.667149\pi\)
0.103022 + 0.994679i \(0.467149\pi\)
\(810\) 0 0
\(811\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(812\) −0.459524 + 1.41427i −0.0161261 + 0.0496311i
\(813\) 0 0
\(814\) −7.03536 + 0.133451i −0.246589 + 0.00467745i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31.1027 42.8092i 1.08549 1.49405i 0.232162 0.972677i \(-0.425420\pi\)
0.853329 0.521373i \(-0.174580\pi\)
\(822\) 0 0
\(823\) −10.9405 + 33.6713i −0.381361 + 1.17371i 0.557725 + 0.830026i \(0.311674\pi\)
−0.939086 + 0.343683i \(0.888326\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.2276 + 11.4461i 1.22498 + 0.398022i 0.848895 0.528562i \(-0.177268\pi\)
0.376090 + 0.926583i \(0.377268\pi\)
\(828\) 9.55378 6.94123i 0.332017 0.241224i
\(829\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(840\) 0 0
\(841\) 8.38301 + 25.8002i 0.289069 + 0.889664i
\(842\) 58.6183 19.0462i 2.02012 0.656377i
\(843\) 0 0
\(844\) −6.87140 9.45767i −0.236523 0.325546i
\(845\) 0 0
\(846\) 0 0
\(847\) 28.0000 + 7.93725i 0.962091 + 0.272727i
\(848\) −61.2090 −2.10193
\(849\) 0 0
\(850\) 0 0
\(851\) 10.5930 + 7.69629i 0.363124 + 0.263825i
\(852\) 0 0
\(853\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.2280 37.6339i 0.417945 1.28630i
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −3.26024 2.36871i −0.111044 0.0806784i
\(863\) 2.47214 + 7.60845i 0.0841525 + 0.258995i 0.984275 0.176642i \(-0.0565234\pi\)
−0.900123 + 0.435636i \(0.856523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 33.1088 25.0279i 1.12314 0.849013i
\(870\) 0 0
\(871\) 0 0
\(872\) 38.2124 27.7630i 1.29404 0.940173i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.1841 + 30.5338i 0.749104 + 1.03105i 0.998043 + 0.0625337i \(0.0199181\pi\)
−0.248939 + 0.968519i \(0.580082\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −31.0103 10.0758i −1.04417 0.339271i
\(883\) 9.70820 7.05342i 0.326707 0.237367i −0.412325 0.911037i \(-0.635283\pi\)
0.739032 + 0.673670i \(0.235283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −42.8935 + 13.9370i −1.44104 + 0.468221i
\(887\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(888\) 0 0
\(889\) −18.1890 + 55.9800i −0.610040 + 1.87751i
\(890\) 0 0
\(891\) 28.2086 9.76078i 0.945025 0.326998i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −11.0441 33.9902i −0.368957 1.13553i
\(897\) 0 0
\(898\) −21.2317 + 29.2229i −0.708511 + 0.975182i
\(899\) 0 0
\(900\) −1.90413 + 5.86030i −0.0634708 + 0.195343i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −45.7899 14.8780i −1.52295 0.494836i
\(905\) 0 0
\(906\) 0 0
\(907\) −7.28462 22.4198i −0.241882 0.744436i −0.996134 0.0878507i \(-0.972000\pi\)
0.754252 0.656585i \(-0.228000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.94427 + 15.2169i −0.163811 + 0.504159i −0.998947 0.0458855i \(-0.985389\pi\)
0.835136 + 0.550044i \(0.185389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −58.6507 −1.93999
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −54.3023 + 17.6439i −1.79127 + 0.582019i −0.999582 0.0289084i \(-0.990797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.83216 −0.224640
\(926\) −4.04149 1.31316i −0.132811 0.0431531i
\(927\) 0 0
\(928\) −2.53403 1.84108i −0.0831838 0.0604365i
\(929\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −5.11069 7.03426i −0.167406 0.230415i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(938\) 54.2836 39.4394i 1.77242 1.28774i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −30.3626 40.1659i −0.987172 1.30591i
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30.1035 + 41.4340i −0.975149 + 1.34218i −0.0357473 + 0.999361i \(0.511381\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) −36.0178 49.5743i −1.16612 1.60503i
\(955\) 0 0
\(956\) 2.89242i 0.0935476i
\(957\) 0 0
\(958\) 0 0
\(959\) −51.6619 16.7860i −1.66825 0.542047i
\(960\) 0 0
\(961\) −25.0795 18.2213i −0.809017 0.587785i
\(962\) 0 0
\(963\) 45.7552 14.8668i 1.47444 0.479075i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 15.0168i 0.482909i −0.970412 0.241454i \(-0.922376\pi\)
0.970412 0.241454i \(-0.0776244\pi\)
\(968\) −15.1099 + 22.5482i −0.485649 + 0.724725i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −62.9716 + 20.4607i −2.01774 + 0.655603i
\(975\) 0 0
\(976\) 0 0
\(977\) 3.81100 11.7290i 0.121925 0.375245i −0.871404 0.490567i \(-0.836790\pi\)
0.993328 + 0.115321i \(0.0367898\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 54.6153 + 17.7456i 1.74373 + 0.566573i
\(982\) −6.64686 + 4.82923i −0.212110 + 0.154107i
\(983\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 93.6921i 2.97923i
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 33.1572 + 24.0901i 1.05168 + 0.764091i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.587785 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(998\) 33.1171 + 45.5817i 1.04830 + 1.44287i
\(999\) 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 77.2.l.a.13.2 yes 8
3.2 odd 2 693.2.bu.a.244.1 8
7.2 even 3 539.2.s.a.178.1 16
7.3 odd 6 539.2.s.a.68.1 16
7.4 even 3 539.2.s.a.68.1 16
7.5 odd 6 539.2.s.a.178.1 16
7.6 odd 2 CM 77.2.l.a.13.2 yes 8
11.2 odd 10 847.2.l.a.118.2 8
11.3 even 5 847.2.l.a.524.2 8
11.4 even 5 847.2.b.b.846.6 8
11.5 even 5 847.2.l.c.699.1 8
11.6 odd 10 inner 77.2.l.a.6.2 8
11.7 odd 10 847.2.b.b.846.3 8
11.8 odd 10 847.2.l.d.524.1 8
11.9 even 5 847.2.l.d.118.1 8
11.10 odd 2 847.2.l.c.475.1 8
21.20 even 2 693.2.bu.a.244.1 8
33.17 even 10 693.2.bu.a.622.1 8
77.6 even 10 inner 77.2.l.a.6.2 8
77.13 even 10 847.2.l.a.118.2 8
77.17 even 30 539.2.s.a.215.1 16
77.20 odd 10 847.2.l.d.118.1 8
77.27 odd 10 847.2.l.c.699.1 8
77.39 odd 30 539.2.s.a.215.1 16
77.41 even 10 847.2.l.d.524.1 8
77.48 odd 10 847.2.b.b.846.6 8
77.61 even 30 539.2.s.a.325.1 16
77.62 even 10 847.2.b.b.846.3 8
77.69 odd 10 847.2.l.a.524.2 8
77.72 odd 30 539.2.s.a.325.1 16
77.76 even 2 847.2.l.c.475.1 8
231.83 odd 10 693.2.bu.a.622.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.l.a.6.2 8 11.6 odd 10 inner
77.2.l.a.6.2 8 77.6 even 10 inner
77.2.l.a.13.2 yes 8 1.1 even 1 trivial
77.2.l.a.13.2 yes 8 7.6 odd 2 CM
539.2.s.a.68.1 16 7.3 odd 6
539.2.s.a.68.1 16 7.4 even 3
539.2.s.a.178.1 16 7.2 even 3
539.2.s.a.178.1 16 7.5 odd 6
539.2.s.a.215.1 16 77.17 even 30
539.2.s.a.215.1 16 77.39 odd 30
539.2.s.a.325.1 16 77.61 even 30
539.2.s.a.325.1 16 77.72 odd 30
693.2.bu.a.244.1 8 3.2 odd 2
693.2.bu.a.244.1 8 21.20 even 2
693.2.bu.a.622.1 8 33.17 even 10
693.2.bu.a.622.1 8 231.83 odd 10
847.2.b.b.846.3 8 11.7 odd 10
847.2.b.b.846.3 8 77.62 even 10
847.2.b.b.846.6 8 11.4 even 5
847.2.b.b.846.6 8 77.48 odd 10
847.2.l.a.118.2 8 11.2 odd 10
847.2.l.a.118.2 8 77.13 even 10
847.2.l.a.524.2 8 11.3 even 5
847.2.l.a.524.2 8 77.69 odd 10
847.2.l.c.475.1 8 11.10 odd 2
847.2.l.c.475.1 8 77.76 even 2
847.2.l.c.699.1 8 11.5 even 5
847.2.l.c.699.1 8 77.27 odd 10
847.2.l.d.118.1 8 11.9 even 5
847.2.l.d.118.1 8 77.20 odd 10
847.2.l.d.524.1 8 11.8 odd 10
847.2.l.d.524.1 8 77.41 even 10