Properties

Label 2366.2.d.p.337.6
Level $2366$
Weight $2$
Character 2366.337
Analytic conductor $18.893$
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] = [2366,2,Mod(337,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2366.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.d (of order \(2\), degree \(1\), not 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: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.6
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 2366.337
Dual form 2366.2.d.p.337.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} +1.69202i q^{5} +2.24698i q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.04892 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} +1.69202i q^{5} +2.24698i q^{6} +1.00000i q^{7} -1.00000i q^{8} +2.04892 q^{9} -1.69202 q^{10} -0.445042i q^{11} -2.24698 q^{12} -1.00000 q^{14} +3.80194i q^{15} +1.00000 q^{16} +2.15883 q^{17} +2.04892i q^{18} +6.35690i q^{19} -1.69202i q^{20} +2.24698i q^{21} +0.445042 q^{22} +0.911854 q^{23} -2.24698i q^{24} +2.13706 q^{25} -2.13706 q^{27} -1.00000i q^{28} +3.58211 q^{29} -3.80194 q^{30} +8.89977i q^{31} +1.00000i q^{32} -1.00000i q^{33} +2.15883i q^{34} -1.69202 q^{35} -2.04892 q^{36} -10.8019i q^{37} -6.35690 q^{38} +1.69202 q^{40} +2.41789i q^{41} -2.24698 q^{42} -4.63102 q^{43} +0.445042i q^{44} +3.46681i q^{45} +0.911854i q^{46} +9.75063i q^{47} +2.24698 q^{48} -1.00000 q^{49} +2.13706i q^{50} +4.85086 q^{51} -8.74094 q^{53} -2.13706i q^{54} +0.753020 q^{55} +1.00000 q^{56} +14.2838i q^{57} +3.58211i q^{58} +10.1468i q^{59} -3.80194i q^{60} +1.37867 q^{61} -8.89977 q^{62} +2.04892i q^{63} -1.00000 q^{64} +1.00000 q^{66} +6.23490i q^{67} -2.15883 q^{68} +2.04892 q^{69} -1.69202i q^{70} -5.76271i q^{71} -2.04892i q^{72} +9.93661i q^{73} +10.8019 q^{74} +4.80194 q^{75} -6.35690i q^{76} +0.445042 q^{77} -6.30127 q^{79} +1.69202i q^{80} -10.9487 q^{81} -2.41789 q^{82} -2.91185i q^{83} -2.24698i q^{84} +3.65279i q^{85} -4.63102i q^{86} +8.04892 q^{87} -0.445042 q^{88} -18.1075i q^{89} -3.46681 q^{90} -0.911854 q^{92} +19.9976i q^{93} -9.75063 q^{94} -10.7560 q^{95} +2.24698i q^{96} -3.30798i q^{97} -1.00000i q^{98} -0.911854i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9} - 4 q^{12} - 6 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 10 q^{29} - 14 q^{30} + 6 q^{36} - 30 q^{38} - 4 q^{42} + 2 q^{43} + 4 q^{48} - 6 q^{49} + 2 q^{51} - 24 q^{53} + 14 q^{55} + 6 q^{56} - 6 q^{61} - 8 q^{62} - 6 q^{64} + 6 q^{66} + 4 q^{68} - 6 q^{69} + 56 q^{74} + 20 q^{75} + 2 q^{77} - 4 q^{79} - 2 q^{81} - 26 q^{82} + 30 q^{87} - 2 q^{88} - 14 q^{90} + 2 q^{92} + 14 q^{94} + 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(2199\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 2.24698 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.69202i 0.756695i 0.925664 + 0.378348i \(0.123508\pi\)
−0.925664 + 0.378348i \(0.876492\pi\)
\(6\) 2.24698i 0.917326i
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) 2.04892 0.682972
\(10\) −1.69202 −0.535064
\(11\) − 0.445042i − 0.134185i −0.997747 0.0670926i \(-0.978628\pi\)
0.997747 0.0670926i \(-0.0213723\pi\)
\(12\) −2.24698 −0.648647
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 3.80194i 0.981656i
\(16\) 1.00000 0.250000
\(17\) 2.15883 0.523594 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) 2.04892i 0.482934i
\(19\) 6.35690i 1.45837i 0.684316 + 0.729186i \(0.260101\pi\)
−0.684316 + 0.729186i \(0.739899\pi\)
\(20\) − 1.69202i − 0.378348i
\(21\) 2.24698i 0.490331i
\(22\) 0.445042 0.0948832
\(23\) 0.911854 0.190135 0.0950674 0.995471i \(-0.469693\pi\)
0.0950674 + 0.995471i \(0.469693\pi\)
\(24\) − 2.24698i − 0.458663i
\(25\) 2.13706 0.427413
\(26\) 0 0
\(27\) −2.13706 −0.411278
\(28\) − 1.00000i − 0.188982i
\(29\) 3.58211 0.665180 0.332590 0.943071i \(-0.392077\pi\)
0.332590 + 0.943071i \(0.392077\pi\)
\(30\) −3.80194 −0.694136
\(31\) 8.89977i 1.59845i 0.601034 + 0.799223i \(0.294755\pi\)
−0.601034 + 0.799223i \(0.705245\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.00000i − 0.174078i
\(34\) 2.15883i 0.370237i
\(35\) −1.69202 −0.286004
\(36\) −2.04892 −0.341486
\(37\) − 10.8019i − 1.77583i −0.460010 0.887914i \(-0.652154\pi\)
0.460010 0.887914i \(-0.347846\pi\)
\(38\) −6.35690 −1.03122
\(39\) 0 0
\(40\) 1.69202 0.267532
\(41\) 2.41789i 0.377612i 0.982014 + 0.188806i \(0.0604617\pi\)
−0.982014 + 0.188806i \(0.939538\pi\)
\(42\) −2.24698 −0.346716
\(43\) −4.63102 −0.706224 −0.353112 0.935581i \(-0.614877\pi\)
−0.353112 + 0.935581i \(0.614877\pi\)
\(44\) 0.445042i 0.0670926i
\(45\) 3.46681i 0.516802i
\(46\) 0.911854i 0.134446i
\(47\) 9.75063i 1.42228i 0.703053 + 0.711138i \(0.251820\pi\)
−0.703053 + 0.711138i \(0.748180\pi\)
\(48\) 2.24698 0.324324
\(49\) −1.00000 −0.142857
\(50\) 2.13706i 0.302226i
\(51\) 4.85086 0.679256
\(52\) 0 0
\(53\) −8.74094 −1.20066 −0.600330 0.799752i \(-0.704964\pi\)
−0.600330 + 0.799752i \(0.704964\pi\)
\(54\) − 2.13706i − 0.290817i
\(55\) 0.753020 0.101537
\(56\) 1.00000 0.133631
\(57\) 14.2838i 1.89194i
\(58\) 3.58211i 0.470353i
\(59\) 10.1468i 1.32099i 0.750828 + 0.660497i \(0.229655\pi\)
−0.750828 + 0.660497i \(0.770345\pi\)
\(60\) − 3.80194i − 0.490828i
\(61\) 1.37867 0.176520 0.0882601 0.996097i \(-0.471869\pi\)
0.0882601 + 0.996097i \(0.471869\pi\)
\(62\) −8.89977 −1.13027
\(63\) 2.04892i 0.258139i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 6.23490i 0.761714i 0.924634 + 0.380857i \(0.124371\pi\)
−0.924634 + 0.380857i \(0.875629\pi\)
\(68\) −2.15883 −0.261797
\(69\) 2.04892 0.246661
\(70\) − 1.69202i − 0.202235i
\(71\) − 5.76271i − 0.683908i −0.939717 0.341954i \(-0.888911\pi\)
0.939717 0.341954i \(-0.111089\pi\)
\(72\) − 2.04892i − 0.241467i
\(73\) 9.93661i 1.16299i 0.813549 + 0.581496i \(0.197532\pi\)
−0.813549 + 0.581496i \(0.802468\pi\)
\(74\) 10.8019 1.25570
\(75\) 4.80194 0.554480
\(76\) − 6.35690i − 0.729186i
\(77\) 0.445042 0.0507172
\(78\) 0 0
\(79\) −6.30127 −0.708949 −0.354474 0.935066i \(-0.615340\pi\)
−0.354474 + 0.935066i \(0.615340\pi\)
\(80\) 1.69202i 0.189174i
\(81\) −10.9487 −1.21652
\(82\) −2.41789 −0.267012
\(83\) − 2.91185i − 0.319617i −0.987148 0.159809i \(-0.948912\pi\)
0.987148 0.159809i \(-0.0510878\pi\)
\(84\) − 2.24698i − 0.245166i
\(85\) 3.65279i 0.396201i
\(86\) − 4.63102i − 0.499376i
\(87\) 8.04892 0.862935
\(88\) −0.445042 −0.0474416
\(89\) − 18.1075i − 1.91939i −0.281037 0.959697i \(-0.590678\pi\)
0.281037 0.959697i \(-0.409322\pi\)
\(90\) −3.46681 −0.365434
\(91\) 0 0
\(92\) −0.911854 −0.0950674
\(93\) 19.9976i 2.07366i
\(94\) −9.75063 −1.00570
\(95\) −10.7560 −1.10354
\(96\) 2.24698i 0.229331i
\(97\) − 3.30798i − 0.335874i −0.985798 0.167937i \(-0.946289\pi\)
0.985798 0.167937i \(-0.0537106\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) − 0.911854i − 0.0916448i
\(100\) −2.13706 −0.213706
\(101\) −2.08815 −0.207778 −0.103889 0.994589i \(-0.533129\pi\)
−0.103889 + 0.994589i \(0.533129\pi\)
\(102\) 4.85086i 0.480306i
\(103\) 11.3013 1.11355 0.556774 0.830664i \(-0.312039\pi\)
0.556774 + 0.830664i \(0.312039\pi\)
\(104\) 0 0
\(105\) −3.80194 −0.371031
\(106\) − 8.74094i − 0.848995i
\(107\) −7.13467 −0.689735 −0.344867 0.938651i \(-0.612076\pi\)
−0.344867 + 0.938651i \(0.612076\pi\)
\(108\) 2.13706 0.205639
\(109\) − 6.57002i − 0.629294i −0.949209 0.314647i \(-0.898114\pi\)
0.949209 0.314647i \(-0.101886\pi\)
\(110\) 0.753020i 0.0717977i
\(111\) − 24.2717i − 2.30377i
\(112\) 1.00000i 0.0944911i
\(113\) 10.6015 0.997304 0.498652 0.866802i \(-0.333829\pi\)
0.498652 + 0.866802i \(0.333829\pi\)
\(114\) −14.2838 −1.33780
\(115\) 1.54288i 0.143874i
\(116\) −3.58211 −0.332590
\(117\) 0 0
\(118\) −10.1468 −0.934084
\(119\) 2.15883i 0.197900i
\(120\) 3.80194 0.347068
\(121\) 10.8019 0.981994
\(122\) 1.37867i 0.124819i
\(123\) 5.43296i 0.489874i
\(124\) − 8.89977i − 0.799223i
\(125\) 12.0761i 1.08012i
\(126\) −2.04892 −0.182532
\(127\) 21.5646 1.91355 0.956776 0.290824i \(-0.0939295\pi\)
0.956776 + 0.290824i \(0.0939295\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −10.4058 −0.916181
\(130\) 0 0
\(131\) 3.14675 0.274933 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) −6.35690 −0.551213
\(134\) −6.23490 −0.538613
\(135\) − 3.61596i − 0.311212i
\(136\) − 2.15883i − 0.185118i
\(137\) 3.93362i 0.336072i 0.985781 + 0.168036i \(0.0537425\pi\)
−0.985781 + 0.168036i \(0.946257\pi\)
\(138\) 2.04892i 0.174415i
\(139\) −11.2078 −0.950629 −0.475315 0.879816i \(-0.657666\pi\)
−0.475315 + 0.879816i \(0.657666\pi\)
\(140\) 1.69202 0.143002
\(141\) 21.9095i 1.84511i
\(142\) 5.76271 0.483596
\(143\) 0 0
\(144\) 2.04892 0.170743
\(145\) 6.06100i 0.503339i
\(146\) −9.93661 −0.822360
\(147\) −2.24698 −0.185328
\(148\) 10.8019i 0.887914i
\(149\) − 21.8726i − 1.79188i −0.444180 0.895938i \(-0.646505\pi\)
0.444180 0.895938i \(-0.353495\pi\)
\(150\) 4.80194i 0.392077i
\(151\) − 18.3002i − 1.48925i −0.667483 0.744625i \(-0.732628\pi\)
0.667483 0.744625i \(-0.267372\pi\)
\(152\) 6.35690 0.515612
\(153\) 4.42327 0.357600
\(154\) 0.445042i 0.0358625i
\(155\) −15.0586 −1.20954
\(156\) 0 0
\(157\) −10.8442 −0.865457 −0.432729 0.901524i \(-0.642449\pi\)
−0.432729 + 0.901524i \(0.642449\pi\)
\(158\) − 6.30127i − 0.501302i
\(159\) −19.6407 −1.55761
\(160\) −1.69202 −0.133766
\(161\) 0.911854i 0.0718642i
\(162\) − 10.9487i − 0.860210i
\(163\) − 14.4058i − 1.12835i −0.825655 0.564175i \(-0.809194\pi\)
0.825655 0.564175i \(-0.190806\pi\)
\(164\) − 2.41789i − 0.188806i
\(165\) 1.69202 0.131724
\(166\) 2.91185 0.226004
\(167\) 13.6963i 1.05985i 0.848043 + 0.529927i \(0.177781\pi\)
−0.848043 + 0.529927i \(0.822219\pi\)
\(168\) 2.24698 0.173358
\(169\) 0 0
\(170\) −3.65279 −0.280156
\(171\) 13.0248i 0.996028i
\(172\) 4.63102 0.353112
\(173\) 4.47650 0.340342 0.170171 0.985415i \(-0.445568\pi\)
0.170171 + 0.985415i \(0.445568\pi\)
\(174\) 8.04892i 0.610187i
\(175\) 2.13706i 0.161547i
\(176\) − 0.445042i − 0.0335463i
\(177\) 22.7995i 1.71372i
\(178\) 18.1075 1.35722
\(179\) 23.0737 1.72461 0.862304 0.506392i \(-0.169021\pi\)
0.862304 + 0.506392i \(0.169021\pi\)
\(180\) − 3.46681i − 0.258401i
\(181\) −8.46980 −0.629555 −0.314777 0.949165i \(-0.601930\pi\)
−0.314777 + 0.949165i \(0.601930\pi\)
\(182\) 0 0
\(183\) 3.09783 0.228999
\(184\) − 0.911854i − 0.0672228i
\(185\) 18.2771 1.34376
\(186\) −19.9976 −1.46630
\(187\) − 0.960771i − 0.0702586i
\(188\) − 9.75063i − 0.711138i
\(189\) − 2.13706i − 0.155448i
\(190\) − 10.7560i − 0.780323i
\(191\) 19.7235 1.42714 0.713570 0.700583i \(-0.247077\pi\)
0.713570 + 0.700583i \(0.247077\pi\)
\(192\) −2.24698 −0.162162
\(193\) 9.15883i 0.659267i 0.944109 + 0.329634i \(0.106925\pi\)
−0.944109 + 0.329634i \(0.893075\pi\)
\(194\) 3.30798 0.237499
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 3.73019i 0.265765i 0.991132 + 0.132882i \(0.0424232\pi\)
−0.991132 + 0.132882i \(0.957577\pi\)
\(198\) 0.911854 0.0648026
\(199\) −19.5918 −1.38883 −0.694413 0.719577i \(-0.744336\pi\)
−0.694413 + 0.719577i \(0.744336\pi\)
\(200\) − 2.13706i − 0.151113i
\(201\) 14.0097i 0.988167i
\(202\) − 2.08815i − 0.146921i
\(203\) 3.58211i 0.251414i
\(204\) −4.85086 −0.339628
\(205\) −4.09113 −0.285737
\(206\) 11.3013i 0.787397i
\(207\) 1.86831 0.129857
\(208\) 0 0
\(209\) 2.82908 0.195692
\(210\) − 3.80194i − 0.262359i
\(211\) 23.3860 1.60996 0.804978 0.593305i \(-0.202177\pi\)
0.804978 + 0.593305i \(0.202177\pi\)
\(212\) 8.74094 0.600330
\(213\) − 12.9487i − 0.887230i
\(214\) − 7.13467i − 0.487716i
\(215\) − 7.83579i − 0.534396i
\(216\) 2.13706i 0.145409i
\(217\) −8.89977 −0.604156
\(218\) 6.57002 0.444978
\(219\) 22.3274i 1.50874i
\(220\) −0.753020 −0.0507686
\(221\) 0 0
\(222\) 24.2717 1.62901
\(223\) − 21.7482i − 1.45637i −0.685381 0.728185i \(-0.740364\pi\)
0.685381 0.728185i \(-0.259636\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.37867 0.291911
\(226\) 10.6015i 0.705200i
\(227\) 6.55927i 0.435354i 0.976021 + 0.217677i \(0.0698479\pi\)
−0.976021 + 0.217677i \(0.930152\pi\)
\(228\) − 14.2838i − 0.945969i
\(229\) 0.948690i 0.0626912i 0.999509 + 0.0313456i \(0.00997925\pi\)
−0.999509 + 0.0313456i \(0.990021\pi\)
\(230\) −1.54288 −0.101734
\(231\) 1.00000 0.0657952
\(232\) − 3.58211i − 0.235177i
\(233\) 15.4765 1.01390 0.506950 0.861976i \(-0.330773\pi\)
0.506950 + 0.861976i \(0.330773\pi\)
\(234\) 0 0
\(235\) −16.4983 −1.07623
\(236\) − 10.1468i − 0.660497i
\(237\) −14.1588 −0.919715
\(238\) −2.15883 −0.139936
\(239\) 25.3448i 1.63942i 0.572779 + 0.819710i \(0.305865\pi\)
−0.572779 + 0.819710i \(0.694135\pi\)
\(240\) 3.80194i 0.245414i
\(241\) 28.2664i 1.82080i 0.413733 + 0.910398i \(0.364225\pi\)
−0.413733 + 0.910398i \(0.635775\pi\)
\(242\) 10.8019i 0.694375i
\(243\) −18.1903 −1.16691
\(244\) −1.37867 −0.0882601
\(245\) − 1.69202i − 0.108099i
\(246\) −5.43296 −0.346393
\(247\) 0 0
\(248\) 8.89977 0.565136
\(249\) − 6.54288i − 0.414638i
\(250\) −12.0761 −0.763757
\(251\) 13.7235 0.866218 0.433109 0.901341i \(-0.357416\pi\)
0.433109 + 0.901341i \(0.357416\pi\)
\(252\) − 2.04892i − 0.129070i
\(253\) − 0.405813i − 0.0255133i
\(254\) 21.5646i 1.35309i
\(255\) 8.20775i 0.513989i
\(256\) 1.00000 0.0625000
\(257\) 13.3502 0.832762 0.416381 0.909190i \(-0.363298\pi\)
0.416381 + 0.909190i \(0.363298\pi\)
\(258\) − 10.4058i − 0.647838i
\(259\) 10.8019 0.671200
\(260\) 0 0
\(261\) 7.33944 0.454300
\(262\) 3.14675i 0.194407i
\(263\) 19.3153 1.19103 0.595515 0.803344i \(-0.296948\pi\)
0.595515 + 0.803344i \(0.296948\pi\)
\(264\) −1.00000 −0.0615457
\(265\) − 14.7899i − 0.908534i
\(266\) − 6.35690i − 0.389766i
\(267\) − 40.6872i − 2.49002i
\(268\) − 6.23490i − 0.380857i
\(269\) 24.2935 1.48120 0.740601 0.671946i \(-0.234541\pi\)
0.740601 + 0.671946i \(0.234541\pi\)
\(270\) 3.61596 0.220060
\(271\) − 29.0368i − 1.76386i −0.471378 0.881931i \(-0.656243\pi\)
0.471378 0.881931i \(-0.343757\pi\)
\(272\) 2.15883 0.130899
\(273\) 0 0
\(274\) −3.93362 −0.237639
\(275\) − 0.951083i − 0.0573524i
\(276\) −2.04892 −0.123330
\(277\) −1.19029 −0.0715177 −0.0357589 0.999360i \(-0.511385\pi\)
−0.0357589 + 0.999360i \(0.511385\pi\)
\(278\) − 11.2078i − 0.672196i
\(279\) 18.2349i 1.09169i
\(280\) 1.69202i 0.101118i
\(281\) − 21.2325i − 1.26663i −0.773896 0.633313i \(-0.781695\pi\)
0.773896 0.633313i \(-0.218305\pi\)
\(282\) −21.9095 −1.30469
\(283\) −11.6756 −0.694044 −0.347022 0.937857i \(-0.612807\pi\)
−0.347022 + 0.937857i \(0.612807\pi\)
\(284\) 5.76271i 0.341954i
\(285\) −24.1685 −1.43162
\(286\) 0 0
\(287\) −2.41789 −0.142724
\(288\) 2.04892i 0.120734i
\(289\) −12.3394 −0.725849
\(290\) −6.06100 −0.355914
\(291\) − 7.43296i − 0.435728i
\(292\) − 9.93661i − 0.581496i
\(293\) − 14.4101i − 0.841848i −0.907096 0.420924i \(-0.861706\pi\)
0.907096 0.420924i \(-0.138294\pi\)
\(294\) − 2.24698i − 0.131047i
\(295\) −17.1685 −0.999590
\(296\) −10.8019 −0.627850
\(297\) 0.951083i 0.0551874i
\(298\) 21.8726 1.26705
\(299\) 0 0
\(300\) −4.80194 −0.277240
\(301\) − 4.63102i − 0.266928i
\(302\) 18.3002 1.05306
\(303\) −4.69202 −0.269550
\(304\) 6.35690i 0.364593i
\(305\) 2.33273i 0.133572i
\(306\) 4.42327i 0.252862i
\(307\) − 2.55257i − 0.145683i −0.997344 0.0728413i \(-0.976793\pi\)
0.997344 0.0728413i \(-0.0232067\pi\)
\(308\) −0.445042 −0.0253586
\(309\) 25.3937 1.44460
\(310\) − 15.0586i − 0.855271i
\(311\) 1.63533 0.0927313 0.0463657 0.998925i \(-0.485236\pi\)
0.0463657 + 0.998925i \(0.485236\pi\)
\(312\) 0 0
\(313\) −12.7724 −0.721939 −0.360969 0.932578i \(-0.617554\pi\)
−0.360969 + 0.932578i \(0.617554\pi\)
\(314\) − 10.8442i − 0.611971i
\(315\) −3.46681 −0.195333
\(316\) 6.30127 0.354474
\(317\) − 10.9041i − 0.612434i −0.951962 0.306217i \(-0.900937\pi\)
0.951962 0.306217i \(-0.0990634\pi\)
\(318\) − 19.6407i − 1.10140i
\(319\) − 1.59419i − 0.0892573i
\(320\) − 1.69202i − 0.0945869i
\(321\) −16.0315 −0.894789
\(322\) −0.911854 −0.0508156
\(323\) 13.7235i 0.763595i
\(324\) 10.9487 0.608261
\(325\) 0 0
\(326\) 14.4058 0.797864
\(327\) − 14.7627i − 0.816380i
\(328\) 2.41789 0.133506
\(329\) −9.75063 −0.537569
\(330\) 1.69202i 0.0931427i
\(331\) 9.94869i 0.546829i 0.961896 + 0.273415i \(0.0881531\pi\)
−0.961896 + 0.273415i \(0.911847\pi\)
\(332\) 2.91185i 0.159809i
\(333\) − 22.1323i − 1.21284i
\(334\) −13.6963 −0.749430
\(335\) −10.5496 −0.576385
\(336\) 2.24698i 0.122583i
\(337\) −8.06829 −0.439508 −0.219754 0.975555i \(-0.570525\pi\)
−0.219754 + 0.975555i \(0.570525\pi\)
\(338\) 0 0
\(339\) 23.8213 1.29380
\(340\) − 3.65279i − 0.198101i
\(341\) 3.96077 0.214488
\(342\) −13.0248 −0.704298
\(343\) − 1.00000i − 0.0539949i
\(344\) 4.63102i 0.249688i
\(345\) 3.46681i 0.186647i
\(346\) 4.47650i 0.240658i
\(347\) −33.1769 −1.78103 −0.890514 0.454955i \(-0.849655\pi\)
−0.890514 + 0.454955i \(0.849655\pi\)
\(348\) −8.04892 −0.431467
\(349\) − 24.6872i − 1.32148i −0.750616 0.660739i \(-0.770243\pi\)
0.750616 0.660739i \(-0.229757\pi\)
\(350\) −2.13706 −0.114231
\(351\) 0 0
\(352\) 0.445042 0.0237208
\(353\) 22.5754i 1.20157i 0.799412 + 0.600784i \(0.205145\pi\)
−0.799412 + 0.600784i \(0.794855\pi\)
\(354\) −22.7995 −1.21178
\(355\) 9.75063 0.517510
\(356\) 18.1075i 0.959697i
\(357\) 4.85086i 0.256734i
\(358\) 23.0737i 1.21948i
\(359\) − 2.30798i − 0.121810i −0.998144 0.0609052i \(-0.980601\pi\)
0.998144 0.0609052i \(-0.0193987\pi\)
\(360\) 3.46681 0.182717
\(361\) −21.4101 −1.12685
\(362\) − 8.46980i − 0.445163i
\(363\) 24.2717 1.27394
\(364\) 0 0
\(365\) −16.8130 −0.880030
\(366\) 3.09783i 0.161926i
\(367\) 25.1685 1.31379 0.656893 0.753984i \(-0.271870\pi\)
0.656893 + 0.753984i \(0.271870\pi\)
\(368\) 0.911854 0.0475337
\(369\) 4.95407i 0.257898i
\(370\) 18.2771i 0.950182i
\(371\) − 8.74094i − 0.453807i
\(372\) − 19.9976i − 1.03683i
\(373\) 11.6407 0.602733 0.301367 0.953508i \(-0.402557\pi\)
0.301367 + 0.953508i \(0.402557\pi\)
\(374\) 0.960771 0.0496803
\(375\) 27.1347i 1.40123i
\(376\) 9.75063 0.502850
\(377\) 0 0
\(378\) 2.13706 0.109919
\(379\) − 16.5623i − 0.850746i −0.905018 0.425373i \(-0.860143\pi\)
0.905018 0.425373i \(-0.139857\pi\)
\(380\) 10.7560 0.551771
\(381\) 48.4553 2.48244
\(382\) 19.7235i 1.00914i
\(383\) − 7.39506i − 0.377870i −0.981990 0.188935i \(-0.939496\pi\)
0.981990 0.188935i \(-0.0605035\pi\)
\(384\) − 2.24698i − 0.114666i
\(385\) 0.753020i 0.0383775i
\(386\) −9.15883 −0.466172
\(387\) −9.48858 −0.482332
\(388\) 3.30798i 0.167937i
\(389\) −23.4674 −1.18984 −0.594922 0.803783i \(-0.702817\pi\)
−0.594922 + 0.803783i \(0.702817\pi\)
\(390\) 0 0
\(391\) 1.96854 0.0995534
\(392\) 1.00000i 0.0505076i
\(393\) 7.07069 0.356669
\(394\) −3.73019 −0.187924
\(395\) − 10.6619i − 0.536458i
\(396\) 0.911854i 0.0458224i
\(397\) 0.0499823i 0.00250854i 0.999999 + 0.00125427i \(0.000399246\pi\)
−0.999999 + 0.00125427i \(0.999601\pi\)
\(398\) − 19.5918i − 0.982048i
\(399\) −14.2838 −0.715085
\(400\) 2.13706 0.106853
\(401\) − 4.84010i − 0.241703i −0.992671 0.120852i \(-0.961437\pi\)
0.992671 0.120852i \(-0.0385625\pi\)
\(402\) −14.0097 −0.698740
\(403\) 0 0
\(404\) 2.08815 0.103889
\(405\) − 18.5254i − 0.920535i
\(406\) −3.58211 −0.177777
\(407\) −4.80731 −0.238290
\(408\) − 4.85086i − 0.240153i
\(409\) 29.8866i 1.47780i 0.673816 + 0.738899i \(0.264654\pi\)
−0.673816 + 0.738899i \(0.735346\pi\)
\(410\) − 4.09113i − 0.202047i
\(411\) 8.83877i 0.435985i
\(412\) −11.3013 −0.556774
\(413\) −10.1468 −0.499289
\(414\) 1.86831i 0.0918226i
\(415\) 4.92692 0.241853
\(416\) 0 0
\(417\) −25.1836 −1.23325
\(418\) 2.82908i 0.138375i
\(419\) 8.13898 0.397615 0.198808 0.980039i \(-0.436293\pi\)
0.198808 + 0.980039i \(0.436293\pi\)
\(420\) 3.80194 0.185516
\(421\) − 17.0398i − 0.830470i −0.909714 0.415235i \(-0.863699\pi\)
0.909714 0.415235i \(-0.136301\pi\)
\(422\) 23.3860i 1.13841i
\(423\) 19.9782i 0.971375i
\(424\) 8.74094i 0.424498i
\(425\) 4.61356 0.223791
\(426\) 12.9487 0.627366
\(427\) 1.37867i 0.0667183i
\(428\) 7.13467 0.344867
\(429\) 0 0
\(430\) 7.83579 0.377875
\(431\) − 6.95779i − 0.335145i −0.985860 0.167572i \(-0.946407\pi\)
0.985860 0.167572i \(-0.0535928\pi\)
\(432\) −2.13706 −0.102820
\(433\) 6.80731 0.327139 0.163569 0.986532i \(-0.447699\pi\)
0.163569 + 0.986532i \(0.447699\pi\)
\(434\) − 8.89977i − 0.427203i
\(435\) 13.6189i 0.652978i
\(436\) 6.57002i 0.314647i
\(437\) 5.79656i 0.277287i
\(438\) −22.3274 −1.06684
\(439\) −26.3351 −1.25691 −0.628453 0.777847i \(-0.716312\pi\)
−0.628453 + 0.777847i \(0.716312\pi\)
\(440\) − 0.753020i − 0.0358988i
\(441\) −2.04892 −0.0975675
\(442\) 0 0
\(443\) 7.32437 0.347991 0.173996 0.984746i \(-0.444332\pi\)
0.173996 + 0.984746i \(0.444332\pi\)
\(444\) 24.2717i 1.15189i
\(445\) 30.6383 1.45240
\(446\) 21.7482 1.02981
\(447\) − 49.1473i − 2.32459i
\(448\) − 1.00000i − 0.0472456i
\(449\) 41.4228i 1.95486i 0.211253 + 0.977431i \(0.432245\pi\)
−0.211253 + 0.977431i \(0.567755\pi\)
\(450\) 4.37867i 0.206412i
\(451\) 1.07606 0.0506699
\(452\) −10.6015 −0.498652
\(453\) − 41.1202i − 1.93200i
\(454\) −6.55927 −0.307842
\(455\) 0 0
\(456\) 14.2838 0.668901
\(457\) 11.7125i 0.547886i 0.961746 + 0.273943i \(0.0883280\pi\)
−0.961746 + 0.273943i \(0.911672\pi\)
\(458\) −0.948690 −0.0443294
\(459\) −4.61356 −0.215343
\(460\) − 1.54288i − 0.0719370i
\(461\) − 20.6461i − 0.961584i −0.876835 0.480792i \(-0.840349\pi\)
0.876835 0.480792i \(-0.159651\pi\)
\(462\) 1.00000i 0.0465242i
\(463\) − 10.9312i − 0.508017i −0.967202 0.254009i \(-0.918251\pi\)
0.967202 0.254009i \(-0.0817492\pi\)
\(464\) 3.58211 0.166295
\(465\) −33.8364 −1.56912
\(466\) 15.4765i 0.716935i
\(467\) −15.4450 −0.714711 −0.357356 0.933968i \(-0.616322\pi\)
−0.357356 + 0.933968i \(0.616322\pi\)
\(468\) 0 0
\(469\) −6.23490 −0.287901
\(470\) − 16.4983i − 0.761008i
\(471\) −24.3666 −1.12275
\(472\) 10.1468 0.467042
\(473\) 2.06100i 0.0947648i
\(474\) − 14.1588i − 0.650337i
\(475\) 13.5851i 0.623327i
\(476\) − 2.15883i − 0.0989500i
\(477\) −17.9095 −0.820018
\(478\) −25.3448 −1.15924
\(479\) − 25.2828i − 1.15520i −0.816321 0.577599i \(-0.803990\pi\)
0.816321 0.577599i \(-0.196010\pi\)
\(480\) −3.80194 −0.173534
\(481\) 0 0
\(482\) −28.2664 −1.28750
\(483\) 2.04892i 0.0932290i
\(484\) −10.8019 −0.490997
\(485\) 5.59717 0.254154
\(486\) − 18.1903i − 0.825128i
\(487\) 1.01938i 0.0461924i 0.999733 + 0.0230962i \(0.00735240\pi\)
−0.999733 + 0.0230962i \(0.992648\pi\)
\(488\) − 1.37867i − 0.0624093i
\(489\) − 32.3696i − 1.46380i
\(490\) 1.69202 0.0764377
\(491\) 36.7198 1.65714 0.828570 0.559886i \(-0.189155\pi\)
0.828570 + 0.559886i \(0.189155\pi\)
\(492\) − 5.43296i − 0.244937i
\(493\) 7.73317 0.348284
\(494\) 0 0
\(495\) 1.54288 0.0693471
\(496\) 8.89977i 0.399612i
\(497\) 5.76271 0.258493
\(498\) 6.54288 0.293193
\(499\) − 41.4685i − 1.85638i −0.372102 0.928192i \(-0.621363\pi\)
0.372102 0.928192i \(-0.378637\pi\)
\(500\) − 12.0761i − 0.540058i
\(501\) 30.7754i 1.37494i
\(502\) 13.7235i 0.612509i
\(503\) −2.15346 −0.0960179 −0.0480089 0.998847i \(-0.515288\pi\)
−0.0480089 + 0.998847i \(0.515288\pi\)
\(504\) 2.04892 0.0912660
\(505\) − 3.53319i − 0.157225i
\(506\) 0.405813 0.0180406
\(507\) 0 0
\(508\) −21.5646 −0.956776
\(509\) − 7.62565i − 0.338001i −0.985616 0.169000i \(-0.945946\pi\)
0.985616 0.169000i \(-0.0540539\pi\)
\(510\) −8.20775 −0.363445
\(511\) −9.93661 −0.439570
\(512\) 1.00000i 0.0441942i
\(513\) − 13.5851i − 0.599796i
\(514\) 13.3502i 0.588852i
\(515\) 19.1220i 0.842616i
\(516\) 10.4058 0.458090
\(517\) 4.33944 0.190848
\(518\) 10.8019i 0.474610i
\(519\) 10.0586 0.441524
\(520\) 0 0
\(521\) 20.8377 0.912917 0.456458 0.889745i \(-0.349118\pi\)
0.456458 + 0.889745i \(0.349118\pi\)
\(522\) 7.33944i 0.321238i
\(523\) 38.3491 1.67689 0.838445 0.544986i \(-0.183465\pi\)
0.838445 + 0.544986i \(0.183465\pi\)
\(524\) −3.14675 −0.137467
\(525\) 4.80194i 0.209574i
\(526\) 19.3153i 0.842186i
\(527\) 19.2131i 0.836937i
\(528\) − 1.00000i − 0.0435194i
\(529\) −22.1685 −0.963849
\(530\) 14.7899 0.642430
\(531\) 20.7899i 0.902203i
\(532\) 6.35690 0.275606
\(533\) 0 0
\(534\) 40.6872 1.76071
\(535\) − 12.0720i − 0.521919i
\(536\) 6.23490 0.269307
\(537\) 51.8461 2.23732
\(538\) 24.2935i 1.04737i
\(539\) 0.445042i 0.0191693i
\(540\) 3.61596i 0.155606i
\(541\) 5.46740i 0.235062i 0.993069 + 0.117531i \(0.0374980\pi\)
−0.993069 + 0.117531i \(0.962502\pi\)
\(542\) 29.0368 1.24724
\(543\) −19.0315 −0.816718
\(544\) 2.15883i 0.0925592i
\(545\) 11.1166 0.476184
\(546\) 0 0
\(547\) −6.26636 −0.267930 −0.133965 0.990986i \(-0.542771\pi\)
−0.133965 + 0.990986i \(0.542771\pi\)
\(548\) − 3.93362i − 0.168036i
\(549\) 2.82477 0.120558
\(550\) 0.951083 0.0405543
\(551\) 22.7711i 0.970080i
\(552\) − 2.04892i − 0.0872077i
\(553\) − 6.30127i − 0.267957i
\(554\) − 1.19029i − 0.0505707i
\(555\) 41.0683 1.74325
\(556\) 11.2078 0.475315
\(557\) 23.1371i 0.980349i 0.871624 + 0.490174i \(0.163067\pi\)
−0.871624 + 0.490174i \(0.836933\pi\)
\(558\) −18.2349 −0.771945
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) − 2.15883i − 0.0911460i
\(562\) 21.2325 0.895639
\(563\) −1.21685 −0.0512841 −0.0256420 0.999671i \(-0.508163\pi\)
−0.0256420 + 0.999671i \(0.508163\pi\)
\(564\) − 21.9095i − 0.922555i
\(565\) 17.9379i 0.754655i
\(566\) − 11.6756i − 0.490763i
\(567\) − 10.9487i − 0.459802i
\(568\) −5.76271 −0.241798
\(569\) 28.5816 1.19820 0.599102 0.800673i \(-0.295524\pi\)
0.599102 + 0.800673i \(0.295524\pi\)
\(570\) − 24.1685i − 1.01231i
\(571\) 32.1825 1.34680 0.673398 0.739280i \(-0.264834\pi\)
0.673398 + 0.739280i \(0.264834\pi\)
\(572\) 0 0
\(573\) 44.3183 1.85142
\(574\) − 2.41789i − 0.100921i
\(575\) 1.94869 0.0812660
\(576\) −2.04892 −0.0853716
\(577\) − 2.94810i − 0.122731i −0.998115 0.0613655i \(-0.980454\pi\)
0.998115 0.0613655i \(-0.0195455\pi\)
\(578\) − 12.3394i − 0.513253i
\(579\) 20.5797i 0.855264i
\(580\) − 6.06100i − 0.251669i
\(581\) 2.91185 0.120804
\(582\) 7.43296 0.308106
\(583\) 3.89008i 0.161111i
\(584\) 9.93661 0.411180
\(585\) 0 0
\(586\) 14.4101 0.595277
\(587\) 2.77538i 0.114552i 0.998358 + 0.0572761i \(0.0182415\pi\)
−0.998358 + 0.0572761i \(0.981758\pi\)
\(588\) 2.24698 0.0926639
\(589\) −56.5749 −2.33113
\(590\) − 17.1685i − 0.706817i
\(591\) 8.38165i 0.344775i
\(592\) − 10.8019i − 0.443957i
\(593\) − 21.5241i − 0.883888i −0.897043 0.441944i \(-0.854289\pi\)
0.897043 0.441944i \(-0.145711\pi\)
\(594\) −0.951083 −0.0390234
\(595\) −3.65279 −0.149750
\(596\) 21.8726i 0.895938i
\(597\) −44.0224 −1.80172
\(598\) 0 0
\(599\) 23.3230 0.952954 0.476477 0.879187i \(-0.341914\pi\)
0.476477 + 0.879187i \(0.341914\pi\)
\(600\) − 4.80194i − 0.196038i
\(601\) 15.7429 0.642165 0.321082 0.947051i \(-0.395953\pi\)
0.321082 + 0.947051i \(0.395953\pi\)
\(602\) 4.63102 0.188746
\(603\) 12.7748i 0.520230i
\(604\) 18.3002i 0.744625i
\(605\) 18.2771i 0.743070i
\(606\) − 4.69202i − 0.190600i
\(607\) −19.7845 −0.803027 −0.401514 0.915853i \(-0.631516\pi\)
−0.401514 + 0.915853i \(0.631516\pi\)
\(608\) −6.35690 −0.257806
\(609\) 8.04892i 0.326159i
\(610\) −2.33273 −0.0944496
\(611\) 0 0
\(612\) −4.42327 −0.178800
\(613\) − 7.24996i − 0.292823i −0.989224 0.146412i \(-0.953228\pi\)
0.989224 0.146412i \(-0.0467724\pi\)
\(614\) 2.55257 0.103013
\(615\) −9.19269 −0.370685
\(616\) − 0.445042i − 0.0179312i
\(617\) − 36.7415i − 1.47916i −0.673070 0.739579i \(-0.735025\pi\)
0.673070 0.739579i \(-0.264975\pi\)
\(618\) 25.3937i 1.02149i
\(619\) 17.4959i 0.703219i 0.936147 + 0.351609i \(0.114365\pi\)
−0.936147 + 0.351609i \(0.885635\pi\)
\(620\) 15.0586 0.604768
\(621\) −1.94869 −0.0781982
\(622\) 1.63533i 0.0655709i
\(623\) 18.1075 0.725463
\(624\) 0 0
\(625\) −9.74764 −0.389906
\(626\) − 12.7724i − 0.510488i
\(627\) 6.35690 0.253870
\(628\) 10.8442 0.432729
\(629\) − 23.3196i − 0.929813i
\(630\) − 3.46681i − 0.138121i
\(631\) − 31.4644i − 1.25258i −0.779591 0.626289i \(-0.784573\pi\)
0.779591 0.626289i \(-0.215427\pi\)
\(632\) 6.30127i 0.250651i
\(633\) 52.5478 2.08859
\(634\) 10.9041 0.433057
\(635\) 36.4878i 1.44798i
\(636\) 19.6407 0.778805
\(637\) 0 0
\(638\) 1.59419 0.0631145
\(639\) − 11.8073i − 0.467090i
\(640\) 1.69202 0.0668830
\(641\) 24.6799 0.974799 0.487400 0.873179i \(-0.337946\pi\)
0.487400 + 0.873179i \(0.337946\pi\)
\(642\) − 16.0315i − 0.632711i
\(643\) − 11.1588i − 0.440061i −0.975493 0.220031i \(-0.929384\pi\)
0.975493 0.220031i \(-0.0706158\pi\)
\(644\) − 0.911854i − 0.0359321i
\(645\) − 17.6069i − 0.693269i
\(646\) −13.7235 −0.539943
\(647\) −14.8791 −0.584956 −0.292478 0.956272i \(-0.594480\pi\)
−0.292478 + 0.956272i \(0.594480\pi\)
\(648\) 10.9487i 0.430105i
\(649\) 4.51573 0.177258
\(650\) 0 0
\(651\) −19.9976 −0.783768
\(652\) 14.4058i 0.564175i
\(653\) −5.13600 −0.200987 −0.100494 0.994938i \(-0.532042\pi\)
−0.100494 + 0.994938i \(0.532042\pi\)
\(654\) 14.7627 0.577268
\(655\) 5.32437i 0.208040i
\(656\) 2.41789i 0.0944029i
\(657\) 20.3593i 0.794292i
\(658\) − 9.75063i − 0.380119i
\(659\) 8.42998 0.328385 0.164193 0.986428i \(-0.447498\pi\)
0.164193 + 0.986428i \(0.447498\pi\)
\(660\) −1.69202 −0.0658618
\(661\) − 20.3793i − 0.792661i −0.918108 0.396331i \(-0.870283\pi\)
0.918108 0.396331i \(-0.129717\pi\)
\(662\) −9.94869 −0.386667
\(663\) 0 0
\(664\) −2.91185 −0.113002
\(665\) − 10.7560i − 0.417100i
\(666\) 22.1323 0.857608
\(667\) 3.26636 0.126474
\(668\) − 13.6963i − 0.529927i
\(669\) − 48.8678i − 1.88934i
\(670\) − 10.5496i − 0.407566i
\(671\) − 0.613564i − 0.0236864i
\(672\) −2.24698 −0.0866791
\(673\) −21.6383 −0.834096 −0.417048 0.908884i \(-0.636935\pi\)
−0.417048 + 0.908884i \(0.636935\pi\)
\(674\) − 8.06829i − 0.310779i
\(675\) −4.56704 −0.175785
\(676\) 0 0
\(677\) 40.5351 1.55789 0.778945 0.627092i \(-0.215755\pi\)
0.778945 + 0.627092i \(0.215755\pi\)
\(678\) 23.8213i 0.914852i
\(679\) 3.30798 0.126949
\(680\) 3.65279 0.140078
\(681\) 14.7385i 0.564782i
\(682\) 3.96077i 0.151666i
\(683\) − 21.6969i − 0.830210i −0.909774 0.415105i \(-0.863745\pi\)
0.909774 0.415105i \(-0.136255\pi\)
\(684\) − 13.0248i − 0.498014i
\(685\) −6.65578 −0.254304
\(686\) 1.00000 0.0381802
\(687\) 2.13169i 0.0813289i
\(688\) −4.63102 −0.176556
\(689\) 0 0
\(690\) −3.46681 −0.131979
\(691\) 28.0610i 1.06749i 0.845645 + 0.533745i \(0.179216\pi\)
−0.845645 + 0.533745i \(0.820784\pi\)
\(692\) −4.47650 −0.170171
\(693\) 0.911854 0.0346385
\(694\) − 33.1769i − 1.25938i
\(695\) − 18.9638i − 0.719336i
\(696\) − 8.04892i − 0.305093i
\(697\) 5.21983i 0.197715i
\(698\) 24.6872 0.934426
\(699\) 34.7754 1.31533
\(700\) − 2.13706i − 0.0807734i
\(701\) 13.7380 0.518875 0.259438 0.965760i \(-0.416463\pi\)
0.259438 + 0.965760i \(0.416463\pi\)
\(702\) 0 0
\(703\) 68.6668 2.58982
\(704\) 0.445042i 0.0167731i
\(705\) −37.0713 −1.39619
\(706\) −22.5754 −0.849636
\(707\) − 2.08815i − 0.0785328i
\(708\) − 22.7995i − 0.856859i
\(709\) 23.0586i 0.865984i 0.901398 + 0.432992i \(0.142542\pi\)
−0.901398 + 0.432992i \(0.857458\pi\)
\(710\) 9.75063i 0.365935i
\(711\) −12.9108 −0.484192
\(712\) −18.1075 −0.678608
\(713\) 8.11529i 0.303920i
\(714\) −4.85086 −0.181539
\(715\) 0 0
\(716\) −23.0737 −0.862304
\(717\) 56.9493i 2.12681i
\(718\) 2.30798 0.0861330
\(719\) −22.8750 −0.853094 −0.426547 0.904465i \(-0.640270\pi\)
−0.426547 + 0.904465i \(0.640270\pi\)
\(720\) 3.46681i 0.129200i
\(721\) 11.3013i 0.420881i
\(722\) − 21.4101i − 0.796802i
\(723\) 63.5139i 2.36211i
\(724\) 8.46980 0.314777
\(725\) 7.65519 0.284306
\(726\) 24.2717i 0.900809i
\(727\) −9.76377 −0.362118 −0.181059 0.983472i \(-0.557953\pi\)
−0.181059 + 0.983472i \(0.557953\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) − 16.8130i − 0.622275i
\(731\) −9.99761 −0.369775
\(732\) −3.09783 −0.114499
\(733\) 2.65519i 0.0980715i 0.998797 + 0.0490358i \(0.0156148\pi\)
−0.998797 + 0.0490358i \(0.984385\pi\)
\(734\) 25.1685i 0.928987i
\(735\) − 3.80194i − 0.140237i
\(736\) 0.911854i 0.0336114i
\(737\) 2.77479 0.102211
\(738\) −4.95407 −0.182362
\(739\) − 0.135144i − 0.00497137i −0.999997 0.00248568i \(-0.999209\pi\)
0.999997 0.00248568i \(-0.000791218\pi\)
\(740\) −18.2771 −0.671880
\(741\) 0 0
\(742\) 8.74094 0.320890
\(743\) − 38.1943i − 1.40121i −0.713547 0.700607i \(-0.752913\pi\)
0.713547 0.700607i \(-0.247087\pi\)
\(744\) 19.9976 0.733148
\(745\) 37.0090 1.35590
\(746\) 11.6407i 0.426197i
\(747\) − 5.96615i − 0.218290i
\(748\) 0.960771i 0.0351293i
\(749\) − 7.13467i − 0.260695i
\(750\) −27.1347 −0.990818
\(751\) −31.9614 −1.16629 −0.583143 0.812369i \(-0.698177\pi\)
−0.583143 + 0.812369i \(0.698177\pi\)
\(752\) 9.75063i 0.355569i
\(753\) 30.8364 1.12374
\(754\) 0 0
\(755\) 30.9643 1.12691
\(756\) 2.13706i 0.0777242i
\(757\) −51.4258 −1.86910 −0.934551 0.355829i \(-0.884198\pi\)
−0.934551 + 0.355829i \(0.884198\pi\)
\(758\) 16.5623 0.601568
\(759\) − 0.911854i − 0.0330982i
\(760\) 10.7560i 0.390161i
\(761\) − 29.4782i − 1.06858i −0.845301 0.534291i \(-0.820579\pi\)
0.845301 0.534291i \(-0.179421\pi\)
\(762\) 48.4553i 1.75535i
\(763\) 6.57002 0.237851
\(764\) −19.7235 −0.713570
\(765\) 7.48427i 0.270594i
\(766\) 7.39506 0.267194
\(767\) 0 0
\(768\) 2.24698 0.0810809
\(769\) − 24.7114i − 0.891116i −0.895253 0.445558i \(-0.853005\pi\)
0.895253 0.445558i \(-0.146995\pi\)
\(770\) −0.753020 −0.0271370
\(771\) 29.9976 1.08034
\(772\) − 9.15883i − 0.329634i
\(773\) − 22.3032i − 0.802190i −0.916036 0.401095i \(-0.868630\pi\)
0.916036 0.401095i \(-0.131370\pi\)
\(774\) − 9.48858i − 0.341060i
\(775\) 19.0194i 0.683196i
\(776\) −3.30798 −0.118750
\(777\) 24.2717 0.870744
\(778\) − 23.4674i − 0.841347i
\(779\) −15.3703 −0.550698
\(780\) 0 0
\(781\) −2.56465 −0.0917703
\(782\) 1.96854i 0.0703949i
\(783\) −7.65519 −0.273574
\(784\) −1.00000 −0.0357143
\(785\) − 18.3485i − 0.654887i
\(786\) 7.07069i 0.252203i
\(787\) 16.4896i 0.587792i 0.955837 + 0.293896i \(0.0949520\pi\)
−0.955837 + 0.293896i \(0.905048\pi\)
\(788\) − 3.73019i − 0.132882i
\(789\) 43.4010 1.54512
\(790\) 10.6619 0.379333
\(791\) 10.6015i 0.376945i
\(792\) −0.911854 −0.0324013
\(793\) 0 0
\(794\) −0.0499823 −0.00177380
\(795\) − 33.2325i − 1.17864i
\(796\) 19.5918 0.694413
\(797\) 1.00106 0.0354595 0.0177298 0.999843i \(-0.494356\pi\)
0.0177298 + 0.999843i \(0.494356\pi\)
\(798\) − 14.2838i − 0.505642i
\(799\) 21.0500i 0.744695i
\(800\) 2.13706i 0.0755566i
\(801\) − 37.1008i − 1.31089i
\(802\) 4.84010 0.170910
\(803\) 4.42221 0.156056
\(804\) − 14.0097i − 0.494084i
\(805\) −1.54288 −0.0543793
\(806\) 0 0
\(807\) 54.5870 1.92155
\(808\) 2.08815i 0.0734607i
\(809\) −49.9172 −1.75500 −0.877498 0.479580i \(-0.840789\pi\)
−0.877498 + 0.479580i \(0.840789\pi\)
\(810\) 18.5254 0.650917
\(811\) 32.9554i 1.15722i 0.815604 + 0.578610i \(0.196405\pi\)
−0.815604 + 0.578610i \(0.803595\pi\)
\(812\) − 3.58211i − 0.125707i
\(813\) − 65.2452i − 2.28825i
\(814\) − 4.80731i − 0.168496i
\(815\) 24.3749 0.853817
\(816\) 4.85086 0.169814
\(817\) − 29.4389i − 1.02994i
\(818\) −29.8866 −1.04496
\(819\) 0 0
\(820\) 4.09113 0.142868
\(821\) 29.3129i 1.02303i 0.859275 + 0.511513i \(0.170915\pi\)
−0.859275 + 0.511513i \(0.829085\pi\)
\(822\) −8.83877 −0.308288
\(823\) 11.3013 0.393938 0.196969 0.980410i \(-0.436890\pi\)
0.196969 + 0.980410i \(0.436890\pi\)
\(824\) − 11.3013i − 0.393699i
\(825\) − 2.13706i − 0.0744030i
\(826\) − 10.1468i − 0.353051i
\(827\) 32.8568i 1.14254i 0.820761 + 0.571272i \(0.193550\pi\)
−0.820761 + 0.571272i \(0.806450\pi\)
\(828\) −1.86831 −0.0649284
\(829\) −36.6853 −1.27413 −0.637067 0.770809i \(-0.719852\pi\)
−0.637067 + 0.770809i \(0.719852\pi\)
\(830\) 4.92692i 0.171016i
\(831\) −2.67456 −0.0927796
\(832\) 0 0
\(833\) −2.15883 −0.0747992
\(834\) − 25.1836i − 0.872036i
\(835\) −23.1745 −0.801986
\(836\) −2.82908 −0.0978459
\(837\) − 19.0194i − 0.657406i
\(838\) 8.13898i 0.281156i
\(839\) 23.7687i 0.820586i 0.911954 + 0.410293i \(0.134574\pi\)
−0.911954 + 0.410293i \(0.865426\pi\)
\(840\) 3.80194i 0.131179i
\(841\) −16.1685 −0.557535
\(842\) 17.0398 0.587231
\(843\) − 47.7090i − 1.64319i
\(844\) −23.3860 −0.804978
\(845\) 0 0
\(846\) −19.9782 −0.686866
\(847\) 10.8019i 0.371159i
\(848\) −8.74094 −0.300165
\(849\) −26.2349 −0.900379
\(850\) 4.61356i 0.158244i
\(851\) − 9.84979i − 0.337646i
\(852\) 12.9487i 0.443615i
\(853\) − 33.6437i − 1.15194i −0.817471 0.575969i \(-0.804625\pi\)
0.817471 0.575969i \(-0.195375\pi\)
\(854\) −1.37867 −0.0471770
\(855\) −22.0382 −0.753689
\(856\) 7.13467i 0.243858i
\(857\) 35.2646 1.20461 0.602307 0.798264i \(-0.294248\pi\)
0.602307 + 0.798264i \(0.294248\pi\)
\(858\) 0 0
\(859\) 9.80433 0.334519 0.167260 0.985913i \(-0.446508\pi\)
0.167260 + 0.985913i \(0.446508\pi\)
\(860\) 7.83579i 0.267198i
\(861\) −5.43296 −0.185155
\(862\) 6.95779 0.236983
\(863\) 34.2586i 1.16618i 0.812409 + 0.583088i \(0.198156\pi\)
−0.812409 + 0.583088i \(0.801844\pi\)
\(864\) − 2.13706i − 0.0727044i
\(865\) 7.57434i 0.257535i
\(866\) 6.80731i 0.231322i
\(867\) −27.7265 −0.941640
\(868\) 8.89977 0.302078
\(869\) 2.80433i 0.0951304i
\(870\) −13.6189 −0.461725
\(871\) 0 0
\(872\) −6.57002 −0.222489
\(873\) − 6.77777i − 0.229393i
\(874\) −5.79656 −0.196072
\(875\) −12.0761 −0.408245
\(876\) − 22.3274i − 0.754371i
\(877\) − 28.5174i − 0.962964i −0.876456 0.481482i \(-0.840099\pi\)
0.876456 0.481482i \(-0.159901\pi\)
\(878\) − 26.3351i − 0.888767i
\(879\) − 32.3793i − 1.09213i
\(880\) 0.753020 0.0253843
\(881\) 55.3193 1.86376 0.931878 0.362773i \(-0.118170\pi\)
0.931878 + 0.362773i \(0.118170\pi\)
\(882\) − 2.04892i − 0.0689906i
\(883\) −0.815938 −0.0274585 −0.0137293 0.999906i \(-0.504370\pi\)
−0.0137293 + 0.999906i \(0.504370\pi\)
\(884\) 0 0
\(885\) −38.5773 −1.29676
\(886\) 7.32437i 0.246067i
\(887\) 39.8713 1.33875 0.669374 0.742926i \(-0.266563\pi\)
0.669374 + 0.742926i \(0.266563\pi\)
\(888\) −24.2717 −0.814506
\(889\) 21.5646i 0.723255i
\(890\) 30.6383i 1.02700i
\(891\) 4.87263i 0.163239i
\(892\) 21.7482i 0.728185i
\(893\) −61.9837 −2.07421
\(894\) 49.1473 1.64373
\(895\) 39.0411i 1.30500i
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −41.4228 −1.38230
\(899\) 31.8799i 1.06325i
\(900\) −4.37867 −0.145956
\(901\) −18.8702 −0.628659
\(902\) 1.07606i 0.0358290i
\(903\) − 10.4058i − 0.346284i
\(904\) − 10.6015i − 0.352600i
\(905\) − 14.3311i − 0.476381i
\(906\) 41.1202 1.36613
\(907\) −5.54719 −0.184191 −0.0920957 0.995750i \(-0.529357\pi\)
−0.0920957 + 0.995750i \(0.529357\pi\)
\(908\) − 6.55927i − 0.217677i
\(909\) −4.27844 −0.141907
\(910\) 0 0
\(911\) 12.3357 0.408701 0.204350 0.978898i \(-0.434492\pi\)
0.204350 + 0.978898i \(0.434492\pi\)
\(912\) 14.2838i 0.472984i
\(913\) −1.29590 −0.0428879
\(914\) −11.7125 −0.387414
\(915\) 5.24160i 0.173282i
\(916\) − 0.948690i − 0.0313456i
\(917\) 3.14675i 0.103915i
\(918\) − 4.61356i − 0.152270i
\(919\) 5.22223 0.172265 0.0861327 0.996284i \(-0.472549\pi\)
0.0861327 + 0.996284i \(0.472549\pi\)
\(920\) 1.54288 0.0508671
\(921\) − 5.73556i − 0.188993i
\(922\) 20.6461 0.679943
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) − 23.0844i − 0.759011i
\(926\) 10.9312 0.359223
\(927\) 23.1554 0.760522
\(928\) 3.58211i 0.117588i
\(929\) 10.9957i 0.360757i 0.983597 + 0.180378i \(0.0577322\pi\)
−0.983597 + 0.180378i \(0.942268\pi\)
\(930\) − 33.8364i − 1.10954i
\(931\) − 6.35690i − 0.208339i
\(932\) −15.4765 −0.506950
\(933\) 3.67456 0.120300
\(934\) − 15.4450i − 0.505377i
\(935\) 1.62565 0.0531643
\(936\) 0 0
\(937\) −21.5190 −0.702994 −0.351497 0.936189i \(-0.614327\pi\)
−0.351497 + 0.936189i \(0.614327\pi\)
\(938\) − 6.23490i − 0.203577i
\(939\) −28.6993 −0.936567
\(940\) 16.4983 0.538114
\(941\) − 11.2784i − 0.367667i −0.982957 0.183833i \(-0.941149\pi\)
0.982957 0.183833i \(-0.0588507\pi\)
\(942\) − 24.3666i − 0.793906i
\(943\) 2.20477i 0.0717971i
\(944\) 10.1468i 0.330249i
\(945\) 3.61596 0.117627
\(946\) −2.06100 −0.0670089
\(947\) 1.92154i 0.0624417i 0.999513 + 0.0312209i \(0.00993953\pi\)
−0.999513 + 0.0312209i \(0.990060\pi\)
\(948\) 14.1588 0.459858
\(949\) 0 0
\(950\) −13.5851 −0.440758
\(951\) − 24.5013i − 0.794508i
\(952\) 2.15883 0.0699682
\(953\) −32.0538 −1.03833 −0.519163 0.854676i \(-0.673756\pi\)
−0.519163 + 0.854676i \(0.673756\pi\)
\(954\) − 17.9095i − 0.579840i
\(955\) 33.3726i 1.07991i
\(956\) − 25.3448i − 0.819710i
\(957\) − 3.58211i − 0.115793i
\(958\) 25.2828 0.816849
\(959\) −3.93362 −0.127023
\(960\) − 3.80194i − 0.122707i
\(961\) −48.2059 −1.55503
\(962\) 0 0
\(963\) −14.6183 −0.471070
\(964\) − 28.2664i − 0.910398i
\(965\) −15.4969 −0.498864
\(966\) −2.04892 −0.0659228
\(967\) 47.5080i 1.52775i 0.645362 + 0.763876i \(0.276706\pi\)
−0.645362 + 0.763876i \(0.723294\pi\)
\(968\) − 10.8019i − 0.347187i
\(969\) 30.8364i 0.990607i
\(970\) 5.59717i 0.179714i
\(971\) 36.0224 1.15601 0.578006 0.816032i \(-0.303831\pi\)
0.578006 + 0.816032i \(0.303831\pi\)
\(972\) 18.1903 0.583454
\(973\) − 11.2078i − 0.359304i
\(974\) −1.01938 −0.0326630
\(975\) 0 0
\(976\) 1.37867 0.0441300
\(977\) − 28.2620i − 0.904183i −0.891972 0.452091i \(-0.850678\pi\)
0.891972 0.452091i \(-0.149322\pi\)
\(978\) 32.3696 1.03506
\(979\) −8.05861 −0.257554
\(980\) 1.69202i 0.0540496i
\(981\) − 13.4614i − 0.429791i
\(982\) 36.7198i 1.17177i
\(983\) − 8.60196i − 0.274360i −0.990546 0.137180i \(-0.956196\pi\)
0.990546 0.137180i \(-0.0438038\pi\)
\(984\) 5.43296 0.173196
\(985\) −6.31155 −0.201103
\(986\) 7.73317i 0.246274i
\(987\) −21.9095 −0.697386
\(988\) 0 0
\(989\) −4.22282 −0.134278
\(990\) 1.54288i 0.0490358i
\(991\) −17.8549 −0.567180 −0.283590 0.958946i \(-0.591525\pi\)
−0.283590 + 0.958946i \(0.591525\pi\)
\(992\) −8.89977 −0.282568
\(993\) 22.3545i 0.709399i
\(994\) 5.76271i 0.182782i
\(995\) − 33.1497i − 1.05092i
\(996\) 6.54288i 0.207319i
\(997\) −36.3247 −1.15041 −0.575207 0.818008i \(-0.695079\pi\)
−0.575207 + 0.818008i \(0.695079\pi\)
\(998\) 41.4685 1.31266
\(999\) 23.0844i 0.730359i
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 2366.2.d.p.337.6 6
13.5 odd 4 2366.2.a.bd.1.3 yes 3
13.8 odd 4 2366.2.a.y.1.3 3
13.12 even 2 inner 2366.2.d.p.337.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2366.2.a.y.1.3 3 13.8 odd 4
2366.2.a.bd.1.3 yes 3 13.5 odd 4
2366.2.d.p.337.3 6 13.12 even 2 inner
2366.2.d.p.337.6 6 1.1 even 1 trivial