Properties

Label 7098.2.a.cd.1.3
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7098,2,Mod(1,7098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7098, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7098.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.04892 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.04892 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.04892 q^{10} -2.13706 q^{11} -1.00000 q^{12} -1.00000 q^{14} -3.04892 q^{15} +1.00000 q^{16} -1.40581 q^{17} -1.00000 q^{18} -2.13706 q^{19} +3.04892 q^{20} -1.00000 q^{21} +2.13706 q^{22} -2.26875 q^{23} +1.00000 q^{24} +4.29590 q^{25} -1.00000 q^{27} +1.00000 q^{28} -2.58211 q^{29} +3.04892 q^{30} +6.34481 q^{31} -1.00000 q^{32} +2.13706 q^{33} +1.40581 q^{34} +3.04892 q^{35} +1.00000 q^{36} -10.6528 q^{37} +2.13706 q^{38} -3.04892 q^{40} +4.43296 q^{41} +1.00000 q^{42} -9.55496 q^{43} -2.13706 q^{44} +3.04892 q^{45} +2.26875 q^{46} +3.98792 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.29590 q^{50} +1.40581 q^{51} +6.46681 q^{53} +1.00000 q^{54} -6.51573 q^{55} -1.00000 q^{56} +2.13706 q^{57} +2.58211 q^{58} +3.44504 q^{59} -3.04892 q^{60} +9.98792 q^{61} -6.34481 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.13706 q^{66} -11.3937 q^{67} -1.40581 q^{68} +2.26875 q^{69} -3.04892 q^{70} -4.21983 q^{71} -1.00000 q^{72} -5.84548 q^{73} +10.6528 q^{74} -4.29590 q^{75} -2.13706 q^{76} -2.13706 q^{77} -10.7071 q^{79} +3.04892 q^{80} +1.00000 q^{81} -4.43296 q^{82} -8.94869 q^{83} -1.00000 q^{84} -4.28621 q^{85} +9.55496 q^{86} +2.58211 q^{87} +2.13706 q^{88} +3.73556 q^{89} -3.04892 q^{90} -2.26875 q^{92} -6.34481 q^{93} -3.98792 q^{94} -6.51573 q^{95} +1.00000 q^{96} +5.10321 q^{97} -1.00000 q^{98} -2.13706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - q^{11} - 3 q^{12} - 3 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{18} - q^{19} - 3 q^{21} + q^{22} + q^{23} + 3 q^{24} - q^{25} - 3 q^{27} + 3 q^{28} - 2 q^{29} - 4 q^{31} - 3 q^{32} + q^{33} - 9 q^{34} + 3 q^{36} - 14 q^{37} + q^{38} - 6 q^{41} + 3 q^{42} - 29 q^{43} - q^{44} - q^{46} - 7 q^{47} - 3 q^{48} + 3 q^{49} + q^{50} - 9 q^{51} + 16 q^{53} + 3 q^{54} - 7 q^{55} - 3 q^{56} + q^{57} + 2 q^{58} + 10 q^{59} + 11 q^{61} + 4 q^{62} + 3 q^{63} + 3 q^{64} - q^{66} - 2 q^{67} + 9 q^{68} - q^{69} - 14 q^{71} - 3 q^{72} - 7 q^{73} + 14 q^{74} + q^{75} - q^{76} - q^{77} - 2 q^{79} + 3 q^{81} + 6 q^{82} + 5 q^{83} - 3 q^{84} - 21 q^{85} + 29 q^{86} + 2 q^{87} + q^{88} + q^{92} + 4 q^{93} + 7 q^{94} - 7 q^{95} + 3 q^{96} - 6 q^{97} - 3 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.04892 1.36352 0.681759 0.731577i \(-0.261215\pi\)
0.681759 + 0.731577i \(0.261215\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.04892 −0.964152
\(11\) −2.13706 −0.644349 −0.322174 0.946680i \(-0.604414\pi\)
−0.322174 + 0.946680i \(0.604414\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −3.04892 −0.787227
\(16\) 1.00000 0.250000
\(17\) −1.40581 −0.340960 −0.170480 0.985361i \(-0.554532\pi\)
−0.170480 + 0.985361i \(0.554532\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.13706 −0.490276 −0.245138 0.969488i \(-0.578833\pi\)
−0.245138 + 0.969488i \(0.578833\pi\)
\(20\) 3.04892 0.681759
\(21\) −1.00000 −0.218218
\(22\) 2.13706 0.455623
\(23\) −2.26875 −0.473067 −0.236534 0.971623i \(-0.576011\pi\)
−0.236534 + 0.971623i \(0.576011\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.29590 0.859179
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −2.58211 −0.479485 −0.239742 0.970837i \(-0.577063\pi\)
−0.239742 + 0.970837i \(0.577063\pi\)
\(30\) 3.04892 0.556654
\(31\) 6.34481 1.13956 0.569781 0.821796i \(-0.307028\pi\)
0.569781 + 0.821796i \(0.307028\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.13706 0.372015
\(34\) 1.40581 0.241095
\(35\) 3.04892 0.515361
\(36\) 1.00000 0.166667
\(37\) −10.6528 −1.75131 −0.875654 0.482939i \(-0.839569\pi\)
−0.875654 + 0.482939i \(0.839569\pi\)
\(38\) 2.13706 0.346677
\(39\) 0 0
\(40\) −3.04892 −0.482076
\(41\) 4.43296 0.692312 0.346156 0.938177i \(-0.387487\pi\)
0.346156 + 0.938177i \(0.387487\pi\)
\(42\) 1.00000 0.154303
\(43\) −9.55496 −1.45712 −0.728559 0.684983i \(-0.759809\pi\)
−0.728559 + 0.684983i \(0.759809\pi\)
\(44\) −2.13706 −0.322174
\(45\) 3.04892 0.454506
\(46\) 2.26875 0.334509
\(47\) 3.98792 0.581698 0.290849 0.956769i \(-0.406062\pi\)
0.290849 + 0.956769i \(0.406062\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −4.29590 −0.607532
\(51\) 1.40581 0.196853
\(52\) 0 0
\(53\) 6.46681 0.888285 0.444142 0.895956i \(-0.353508\pi\)
0.444142 + 0.895956i \(0.353508\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.51573 −0.878581
\(56\) −1.00000 −0.133631
\(57\) 2.13706 0.283061
\(58\) 2.58211 0.339047
\(59\) 3.44504 0.448506 0.224253 0.974531i \(-0.428006\pi\)
0.224253 + 0.974531i \(0.428006\pi\)
\(60\) −3.04892 −0.393614
\(61\) 9.98792 1.27882 0.639411 0.768865i \(-0.279178\pi\)
0.639411 + 0.768865i \(0.279178\pi\)
\(62\) −6.34481 −0.805792
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.13706 −0.263054
\(67\) −11.3937 −1.39197 −0.695983 0.718058i \(-0.745031\pi\)
−0.695983 + 0.718058i \(0.745031\pi\)
\(68\) −1.40581 −0.170480
\(69\) 2.26875 0.273125
\(70\) −3.04892 −0.364415
\(71\) −4.21983 −0.500802 −0.250401 0.968142i \(-0.580562\pi\)
−0.250401 + 0.968142i \(0.580562\pi\)
\(72\) −1.00000 −0.117851
\(73\) −5.84548 −0.684161 −0.342081 0.939671i \(-0.611132\pi\)
−0.342081 + 0.939671i \(0.611132\pi\)
\(74\) 10.6528 1.23836
\(75\) −4.29590 −0.496047
\(76\) −2.13706 −0.245138
\(77\) −2.13706 −0.243541
\(78\) 0 0
\(79\) −10.7071 −1.20464 −0.602321 0.798254i \(-0.705757\pi\)
−0.602321 + 0.798254i \(0.705757\pi\)
\(80\) 3.04892 0.340879
\(81\) 1.00000 0.111111
\(82\) −4.43296 −0.489539
\(83\) −8.94869 −0.982246 −0.491123 0.871090i \(-0.663414\pi\)
−0.491123 + 0.871090i \(0.663414\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.28621 −0.464905
\(86\) 9.55496 1.03034
\(87\) 2.58211 0.276831
\(88\) 2.13706 0.227812
\(89\) 3.73556 0.395969 0.197984 0.980205i \(-0.436560\pi\)
0.197984 + 0.980205i \(0.436560\pi\)
\(90\) −3.04892 −0.321384
\(91\) 0 0
\(92\) −2.26875 −0.236534
\(93\) −6.34481 −0.657927
\(94\) −3.98792 −0.411322
\(95\) −6.51573 −0.668500
\(96\) 1.00000 0.102062
\(97\) 5.10321 0.518153 0.259076 0.965857i \(-0.416582\pi\)
0.259076 + 0.965857i \(0.416582\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.13706 −0.214783
\(100\) 4.29590 0.429590
\(101\) −12.2174 −1.21568 −0.607840 0.794059i \(-0.707964\pi\)
−0.607840 + 0.794059i \(0.707964\pi\)
\(102\) −1.40581 −0.139196
\(103\) −4.53319 −0.446668 −0.223334 0.974742i \(-0.571694\pi\)
−0.223334 + 0.974742i \(0.571694\pi\)
\(104\) 0 0
\(105\) −3.04892 −0.297544
\(106\) −6.46681 −0.628112
\(107\) −12.5254 −1.21088 −0.605439 0.795892i \(-0.707002\pi\)
−0.605439 + 0.795892i \(0.707002\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 1.15883 0.110996 0.0554981 0.998459i \(-0.482325\pi\)
0.0554981 + 0.998459i \(0.482325\pi\)
\(110\) 6.51573 0.621250
\(111\) 10.6528 1.01112
\(112\) 1.00000 0.0944911
\(113\) 4.34721 0.408951 0.204475 0.978872i \(-0.434451\pi\)
0.204475 + 0.978872i \(0.434451\pi\)
\(114\) −2.13706 −0.200154
\(115\) −6.91723 −0.645035
\(116\) −2.58211 −0.239742
\(117\) 0 0
\(118\) −3.44504 −0.317142
\(119\) −1.40581 −0.128871
\(120\) 3.04892 0.278327
\(121\) −6.43296 −0.584815
\(122\) −9.98792 −0.904264
\(123\) −4.43296 −0.399707
\(124\) 6.34481 0.569781
\(125\) −2.14675 −0.192011
\(126\) −1.00000 −0.0890871
\(127\) 18.2959 1.62350 0.811749 0.584006i \(-0.198516\pi\)
0.811749 + 0.584006i \(0.198516\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 9.55496 0.841267
\(130\) 0 0
\(131\) 3.52781 0.308226 0.154113 0.988053i \(-0.450748\pi\)
0.154113 + 0.988053i \(0.450748\pi\)
\(132\) 2.13706 0.186007
\(133\) −2.13706 −0.185307
\(134\) 11.3937 0.984268
\(135\) −3.04892 −0.262409
\(136\) 1.40581 0.120547
\(137\) −0.249373 −0.0213053 −0.0106527 0.999943i \(-0.503391\pi\)
−0.0106527 + 0.999943i \(0.503391\pi\)
\(138\) −2.26875 −0.193129
\(139\) 0.168522 0.0142939 0.00714694 0.999974i \(-0.497725\pi\)
0.00714694 + 0.999974i \(0.497725\pi\)
\(140\) 3.04892 0.257681
\(141\) −3.98792 −0.335843
\(142\) 4.21983 0.354120
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −7.87263 −0.653786
\(146\) 5.84548 0.483775
\(147\) −1.00000 −0.0824786
\(148\) −10.6528 −0.875654
\(149\) −16.3327 −1.33803 −0.669015 0.743249i \(-0.733284\pi\)
−0.669015 + 0.743249i \(0.733284\pi\)
\(150\) 4.29590 0.350759
\(151\) −24.3448 −1.98115 −0.990576 0.136961i \(-0.956267\pi\)
−0.990576 + 0.136961i \(0.956267\pi\)
\(152\) 2.13706 0.173339
\(153\) −1.40581 −0.113653
\(154\) 2.13706 0.172209
\(155\) 19.3448 1.55381
\(156\) 0 0
\(157\) −12.0978 −0.965512 −0.482756 0.875755i \(-0.660364\pi\)
−0.482756 + 0.875755i \(0.660364\pi\)
\(158\) 10.7071 0.851810
\(159\) −6.46681 −0.512852
\(160\) −3.04892 −0.241038
\(161\) −2.26875 −0.178803
\(162\) −1.00000 −0.0785674
\(163\) −14.0707 −1.10210 −0.551051 0.834472i \(-0.685773\pi\)
−0.551051 + 0.834472i \(0.685773\pi\)
\(164\) 4.43296 0.346156
\(165\) 6.51573 0.507249
\(166\) 8.94869 0.694553
\(167\) −19.9541 −1.54409 −0.772046 0.635567i \(-0.780767\pi\)
−0.772046 + 0.635567i \(0.780767\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 4.28621 0.328737
\(171\) −2.13706 −0.163425
\(172\) −9.55496 −0.728559
\(173\) −6.61596 −0.503002 −0.251501 0.967857i \(-0.580924\pi\)
−0.251501 + 0.967857i \(0.580924\pi\)
\(174\) −2.58211 −0.195749
\(175\) 4.29590 0.324739
\(176\) −2.13706 −0.161087
\(177\) −3.44504 −0.258945
\(178\) −3.73556 −0.279992
\(179\) −7.49396 −0.560125 −0.280062 0.959982i \(-0.590355\pi\)
−0.280062 + 0.959982i \(0.590355\pi\)
\(180\) 3.04892 0.227253
\(181\) 19.1226 1.42137 0.710685 0.703510i \(-0.248385\pi\)
0.710685 + 0.703510i \(0.248385\pi\)
\(182\) 0 0
\(183\) −9.98792 −0.738328
\(184\) 2.26875 0.167254
\(185\) −32.4795 −2.38794
\(186\) 6.34481 0.465224
\(187\) 3.00431 0.219697
\(188\) 3.98792 0.290849
\(189\) −1.00000 −0.0727393
\(190\) 6.51573 0.472701
\(191\) 23.2403 1.68161 0.840804 0.541340i \(-0.182083\pi\)
0.840804 + 0.541340i \(0.182083\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.00538 −0.288313 −0.144157 0.989555i \(-0.546047\pi\)
−0.144157 + 0.989555i \(0.546047\pi\)
\(194\) −5.10321 −0.366389
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 7.75840 0.552763 0.276381 0.961048i \(-0.410865\pi\)
0.276381 + 0.961048i \(0.410865\pi\)
\(198\) 2.13706 0.151874
\(199\) 14.5526 1.03160 0.515802 0.856708i \(-0.327494\pi\)
0.515802 + 0.856708i \(0.327494\pi\)
\(200\) −4.29590 −0.303766
\(201\) 11.3937 0.803652
\(202\) 12.2174 0.859616
\(203\) −2.58211 −0.181228
\(204\) 1.40581 0.0984266
\(205\) 13.5157 0.943979
\(206\) 4.53319 0.315842
\(207\) −2.26875 −0.157689
\(208\) 0 0
\(209\) 4.56704 0.315909
\(210\) 3.04892 0.210395
\(211\) 3.80194 0.261736 0.130868 0.991400i \(-0.458224\pi\)
0.130868 + 0.991400i \(0.458224\pi\)
\(212\) 6.46681 0.444142
\(213\) 4.21983 0.289138
\(214\) 12.5254 0.856220
\(215\) −29.1323 −1.98680
\(216\) 1.00000 0.0680414
\(217\) 6.34481 0.430714
\(218\) −1.15883 −0.0784861
\(219\) 5.84548 0.395001
\(220\) −6.51573 −0.439290
\(221\) 0 0
\(222\) −10.6528 −0.714969
\(223\) 23.1347 1.54921 0.774606 0.632444i \(-0.217948\pi\)
0.774606 + 0.632444i \(0.217948\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 4.29590 0.286393
\(226\) −4.34721 −0.289172
\(227\) 2.72348 0.180764 0.0903819 0.995907i \(-0.471191\pi\)
0.0903819 + 0.995907i \(0.471191\pi\)
\(228\) 2.13706 0.141530
\(229\) −11.7385 −0.775705 −0.387852 0.921721i \(-0.626783\pi\)
−0.387852 + 0.921721i \(0.626783\pi\)
\(230\) 6.91723 0.456109
\(231\) 2.13706 0.140608
\(232\) 2.58211 0.169524
\(233\) 9.92931 0.650491 0.325245 0.945630i \(-0.394553\pi\)
0.325245 + 0.945630i \(0.394553\pi\)
\(234\) 0 0
\(235\) 12.1588 0.793155
\(236\) 3.44504 0.224253
\(237\) 10.7071 0.695500
\(238\) 1.40581 0.0911253
\(239\) −22.1172 −1.43064 −0.715322 0.698795i \(-0.753720\pi\)
−0.715322 + 0.698795i \(0.753720\pi\)
\(240\) −3.04892 −0.196807
\(241\) −24.5429 −1.58095 −0.790473 0.612497i \(-0.790165\pi\)
−0.790473 + 0.612497i \(0.790165\pi\)
\(242\) 6.43296 0.413526
\(243\) −1.00000 −0.0641500
\(244\) 9.98792 0.639411
\(245\) 3.04892 0.194788
\(246\) 4.43296 0.282635
\(247\) 0 0
\(248\) −6.34481 −0.402896
\(249\) 8.94869 0.567100
\(250\) 2.14675 0.135773
\(251\) 27.2489 1.71994 0.859968 0.510349i \(-0.170484\pi\)
0.859968 + 0.510349i \(0.170484\pi\)
\(252\) 1.00000 0.0629941
\(253\) 4.84846 0.304820
\(254\) −18.2959 −1.14799
\(255\) 4.28621 0.268413
\(256\) 1.00000 0.0625000
\(257\) 17.2349 1.07508 0.537542 0.843237i \(-0.319353\pi\)
0.537542 + 0.843237i \(0.319353\pi\)
\(258\) −9.55496 −0.594866
\(259\) −10.6528 −0.661932
\(260\) 0 0
\(261\) −2.58211 −0.159828
\(262\) −3.52781 −0.217949
\(263\) 8.46011 0.521673 0.260836 0.965383i \(-0.416002\pi\)
0.260836 + 0.965383i \(0.416002\pi\)
\(264\) −2.13706 −0.131527
\(265\) 19.7168 1.21119
\(266\) 2.13706 0.131032
\(267\) −3.73556 −0.228613
\(268\) −11.3937 −0.695983
\(269\) 1.29590 0.0790122 0.0395061 0.999219i \(-0.487422\pi\)
0.0395061 + 0.999219i \(0.487422\pi\)
\(270\) 3.04892 0.185551
\(271\) 5.46250 0.331823 0.165912 0.986141i \(-0.446943\pi\)
0.165912 + 0.986141i \(0.446943\pi\)
\(272\) −1.40581 −0.0852399
\(273\) 0 0
\(274\) 0.249373 0.0150651
\(275\) −9.18060 −0.553611
\(276\) 2.26875 0.136563
\(277\) 1.01746 0.0611332 0.0305666 0.999533i \(-0.490269\pi\)
0.0305666 + 0.999533i \(0.490269\pi\)
\(278\) −0.168522 −0.0101073
\(279\) 6.34481 0.379854
\(280\) −3.04892 −0.182208
\(281\) 23.7482 1.41670 0.708350 0.705861i \(-0.249440\pi\)
0.708350 + 0.705861i \(0.249440\pi\)
\(282\) 3.98792 0.237477
\(283\) −19.9148 −1.18381 −0.591907 0.806006i \(-0.701625\pi\)
−0.591907 + 0.806006i \(0.701625\pi\)
\(284\) −4.21983 −0.250401
\(285\) 6.51573 0.385959
\(286\) 0 0
\(287\) 4.43296 0.261669
\(288\) −1.00000 −0.0589256
\(289\) −15.0237 −0.883746
\(290\) 7.87263 0.462296
\(291\) −5.10321 −0.299156
\(292\) −5.84548 −0.342081
\(293\) −9.15213 −0.534673 −0.267337 0.963603i \(-0.586144\pi\)
−0.267337 + 0.963603i \(0.586144\pi\)
\(294\) 1.00000 0.0583212
\(295\) 10.5036 0.611546
\(296\) 10.6528 0.619181
\(297\) 2.13706 0.124005
\(298\) 16.3327 0.946130
\(299\) 0 0
\(300\) −4.29590 −0.248024
\(301\) −9.55496 −0.550739
\(302\) 24.3448 1.40089
\(303\) 12.2174 0.701874
\(304\) −2.13706 −0.122569
\(305\) 30.4523 1.74370
\(306\) 1.40581 0.0803650
\(307\) −22.2784 −1.27150 −0.635749 0.771896i \(-0.719309\pi\)
−0.635749 + 0.771896i \(0.719309\pi\)
\(308\) −2.13706 −0.121770
\(309\) 4.53319 0.257884
\(310\) −19.3448 −1.09871
\(311\) −0.254749 −0.0144455 −0.00722275 0.999974i \(-0.502299\pi\)
−0.00722275 + 0.999974i \(0.502299\pi\)
\(312\) 0 0
\(313\) 24.1129 1.36294 0.681471 0.731845i \(-0.261341\pi\)
0.681471 + 0.731845i \(0.261341\pi\)
\(314\) 12.0978 0.682720
\(315\) 3.04892 0.171787
\(316\) −10.7071 −0.602321
\(317\) 12.8321 0.720721 0.360360 0.932813i \(-0.382654\pi\)
0.360360 + 0.932813i \(0.382654\pi\)
\(318\) 6.46681 0.362641
\(319\) 5.51812 0.308956
\(320\) 3.04892 0.170440
\(321\) 12.5254 0.699101
\(322\) 2.26875 0.126432
\(323\) 3.00431 0.167164
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.0707 0.779303
\(327\) −1.15883 −0.0640836
\(328\) −4.43296 −0.244769
\(329\) 3.98792 0.219861
\(330\) −6.51573 −0.358679
\(331\) −29.0707 −1.59787 −0.798935 0.601418i \(-0.794603\pi\)
−0.798935 + 0.601418i \(0.794603\pi\)
\(332\) −8.94869 −0.491123
\(333\) −10.6528 −0.583769
\(334\) 19.9541 1.09184
\(335\) −34.7385 −1.89797
\(336\) −1.00000 −0.0545545
\(337\) 17.7821 0.968652 0.484326 0.874888i \(-0.339065\pi\)
0.484326 + 0.874888i \(0.339065\pi\)
\(338\) 0 0
\(339\) −4.34721 −0.236108
\(340\) −4.28621 −0.232452
\(341\) −13.5593 −0.734276
\(342\) 2.13706 0.115559
\(343\) 1.00000 0.0539949
\(344\) 9.55496 0.515169
\(345\) 6.91723 0.372411
\(346\) 6.61596 0.355676
\(347\) −10.5144 −0.564443 −0.282221 0.959349i \(-0.591071\pi\)
−0.282221 + 0.959349i \(0.591071\pi\)
\(348\) 2.58211 0.138415
\(349\) −18.1274 −0.970336 −0.485168 0.874421i \(-0.661242\pi\)
−0.485168 + 0.874421i \(0.661242\pi\)
\(350\) −4.29590 −0.229625
\(351\) 0 0
\(352\) 2.13706 0.113906
\(353\) 16.1715 0.860722 0.430361 0.902657i \(-0.358386\pi\)
0.430361 + 0.902657i \(0.358386\pi\)
\(354\) 3.44504 0.183102
\(355\) −12.8659 −0.682852
\(356\) 3.73556 0.197984
\(357\) 1.40581 0.0744035
\(358\) 7.49396 0.396068
\(359\) −7.77240 −0.410211 −0.205106 0.978740i \(-0.565754\pi\)
−0.205106 + 0.978740i \(0.565754\pi\)
\(360\) −3.04892 −0.160692
\(361\) −14.4330 −0.759629
\(362\) −19.1226 −1.00506
\(363\) 6.43296 0.337643
\(364\) 0 0
\(365\) −17.8224 −0.932866
\(366\) 9.98792 0.522077
\(367\) −2.07846 −0.108495 −0.0542473 0.998528i \(-0.517276\pi\)
−0.0542473 + 0.998528i \(0.517276\pi\)
\(368\) −2.26875 −0.118267
\(369\) 4.43296 0.230771
\(370\) 32.4795 1.68853
\(371\) 6.46681 0.335740
\(372\) −6.34481 −0.328963
\(373\) 8.10752 0.419792 0.209896 0.977724i \(-0.432688\pi\)
0.209896 + 0.977724i \(0.432688\pi\)
\(374\) −3.00431 −0.155349
\(375\) 2.14675 0.110858
\(376\) −3.98792 −0.205661
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −13.4222 −0.689452 −0.344726 0.938703i \(-0.612028\pi\)
−0.344726 + 0.938703i \(0.612028\pi\)
\(380\) −6.51573 −0.334250
\(381\) −18.2959 −0.937327
\(382\) −23.2403 −1.18908
\(383\) −12.4209 −0.634677 −0.317339 0.948312i \(-0.602789\pi\)
−0.317339 + 0.948312i \(0.602789\pi\)
\(384\) 1.00000 0.0510310
\(385\) −6.51573 −0.332072
\(386\) 4.00538 0.203868
\(387\) −9.55496 −0.485706
\(388\) 5.10321 0.259076
\(389\) 37.5666 1.90470 0.952350 0.305007i \(-0.0986587\pi\)
0.952350 + 0.305007i \(0.0986587\pi\)
\(390\) 0 0
\(391\) 3.18944 0.161297
\(392\) −1.00000 −0.0505076
\(393\) −3.52781 −0.177955
\(394\) −7.75840 −0.390862
\(395\) −32.6450 −1.64255
\(396\) −2.13706 −0.107391
\(397\) −36.1159 −1.81260 −0.906302 0.422630i \(-0.861107\pi\)
−0.906302 + 0.422630i \(0.861107\pi\)
\(398\) −14.5526 −0.729454
\(399\) 2.13706 0.106987
\(400\) 4.29590 0.214795
\(401\) 9.82610 0.490692 0.245346 0.969436i \(-0.421098\pi\)
0.245346 + 0.969436i \(0.421098\pi\)
\(402\) −11.3937 −0.568268
\(403\) 0 0
\(404\) −12.2174 −0.607840
\(405\) 3.04892 0.151502
\(406\) 2.58211 0.128148
\(407\) 22.7657 1.12845
\(408\) −1.40581 −0.0695981
\(409\) 3.09352 0.152965 0.0764824 0.997071i \(-0.475631\pi\)
0.0764824 + 0.997071i \(0.475631\pi\)
\(410\) −13.5157 −0.667494
\(411\) 0.249373 0.0123006
\(412\) −4.53319 −0.223334
\(413\) 3.44504 0.169519
\(414\) 2.26875 0.111503
\(415\) −27.2838 −1.33931
\(416\) 0 0
\(417\) −0.168522 −0.00825257
\(418\) −4.56704 −0.223381
\(419\) −20.1535 −0.984561 −0.492280 0.870437i \(-0.663837\pi\)
−0.492280 + 0.870437i \(0.663837\pi\)
\(420\) −3.04892 −0.148772
\(421\) −11.2034 −0.546022 −0.273011 0.962011i \(-0.588020\pi\)
−0.273011 + 0.962011i \(0.588020\pi\)
\(422\) −3.80194 −0.185075
\(423\) 3.98792 0.193899
\(424\) −6.46681 −0.314056
\(425\) −6.03923 −0.292946
\(426\) −4.21983 −0.204452
\(427\) 9.98792 0.483349
\(428\) −12.5254 −0.605439
\(429\) 0 0
\(430\) 29.1323 1.40488
\(431\) 38.9898 1.87807 0.939037 0.343816i \(-0.111720\pi\)
0.939037 + 0.343816i \(0.111720\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 23.4426 1.12658 0.563291 0.826259i \(-0.309535\pi\)
0.563291 + 0.826259i \(0.309535\pi\)
\(434\) −6.34481 −0.304561
\(435\) 7.87263 0.377463
\(436\) 1.15883 0.0554981
\(437\) 4.84846 0.231933
\(438\) −5.84548 −0.279308
\(439\) −26.8592 −1.28192 −0.640960 0.767574i \(-0.721464\pi\)
−0.640960 + 0.767574i \(0.721464\pi\)
\(440\) 6.51573 0.310625
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −11.1347 −0.529024 −0.264512 0.964382i \(-0.585211\pi\)
−0.264512 + 0.964382i \(0.585211\pi\)
\(444\) 10.6528 0.505559
\(445\) 11.3894 0.539910
\(446\) −23.1347 −1.09546
\(447\) 16.3327 0.772512
\(448\) 1.00000 0.0472456
\(449\) −27.0858 −1.27826 −0.639128 0.769100i \(-0.720705\pi\)
−0.639128 + 0.769100i \(0.720705\pi\)
\(450\) −4.29590 −0.202511
\(451\) −9.47352 −0.446090
\(452\) 4.34721 0.204475
\(453\) 24.3448 1.14382
\(454\) −2.72348 −0.127819
\(455\) 0 0
\(456\) −2.13706 −0.100077
\(457\) −3.40880 −0.159457 −0.0797284 0.996817i \(-0.525405\pi\)
−0.0797284 + 0.996817i \(0.525405\pi\)
\(458\) 11.7385 0.548506
\(459\) 1.40581 0.0656177
\(460\) −6.91723 −0.322518
\(461\) 30.9178 1.43999 0.719993 0.693981i \(-0.244145\pi\)
0.719993 + 0.693981i \(0.244145\pi\)
\(462\) −2.13706 −0.0994252
\(463\) −35.0465 −1.62875 −0.814375 0.580339i \(-0.802920\pi\)
−0.814375 + 0.580339i \(0.802920\pi\)
\(464\) −2.58211 −0.119871
\(465\) −19.3448 −0.897094
\(466\) −9.92931 −0.459967
\(467\) 4.04652 0.187251 0.0936254 0.995607i \(-0.470154\pi\)
0.0936254 + 0.995607i \(0.470154\pi\)
\(468\) 0 0
\(469\) −11.3937 −0.526114
\(470\) −12.1588 −0.560845
\(471\) 12.0978 0.557439
\(472\) −3.44504 −0.158571
\(473\) 20.4196 0.938892
\(474\) −10.7071 −0.491793
\(475\) −9.18060 −0.421235
\(476\) −1.40581 −0.0644353
\(477\) 6.46681 0.296095
\(478\) 22.1172 1.01162
\(479\) 16.0411 0.732939 0.366469 0.930430i \(-0.380566\pi\)
0.366469 + 0.930430i \(0.380566\pi\)
\(480\) 3.04892 0.139163
\(481\) 0 0
\(482\) 24.5429 1.11790
\(483\) 2.26875 0.103232
\(484\) −6.43296 −0.292407
\(485\) 15.5593 0.706510
\(486\) 1.00000 0.0453609
\(487\) −36.7724 −1.66632 −0.833158 0.553035i \(-0.813470\pi\)
−0.833158 + 0.553035i \(0.813470\pi\)
\(488\) −9.98792 −0.452132
\(489\) 14.0707 0.636298
\(490\) −3.04892 −0.137736
\(491\) −7.69069 −0.347076 −0.173538 0.984827i \(-0.555520\pi\)
−0.173538 + 0.984827i \(0.555520\pi\)
\(492\) −4.43296 −0.199853
\(493\) 3.62996 0.163485
\(494\) 0 0
\(495\) −6.51573 −0.292860
\(496\) 6.34481 0.284891
\(497\) −4.21983 −0.189285
\(498\) −8.94869 −0.401000
\(499\) 21.8213 0.976856 0.488428 0.872604i \(-0.337570\pi\)
0.488428 + 0.872604i \(0.337570\pi\)
\(500\) −2.14675 −0.0960057
\(501\) 19.9541 0.891482
\(502\) −27.2489 −1.21618
\(503\) 0.875018 0.0390151 0.0195076 0.999810i \(-0.493790\pi\)
0.0195076 + 0.999810i \(0.493790\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −37.2500 −1.65760
\(506\) −4.84846 −0.215540
\(507\) 0 0
\(508\) 18.2959 0.811749
\(509\) 40.2043 1.78202 0.891012 0.453980i \(-0.149996\pi\)
0.891012 + 0.453980i \(0.149996\pi\)
\(510\) −4.28621 −0.189796
\(511\) −5.84548 −0.258589
\(512\) −1.00000 −0.0441942
\(513\) 2.13706 0.0943537
\(514\) −17.2349 −0.760199
\(515\) −13.8213 −0.609040
\(516\) 9.55496 0.420634
\(517\) −8.52243 −0.374816
\(518\) 10.6528 0.468057
\(519\) 6.61596 0.290408
\(520\) 0 0
\(521\) −4.36898 −0.191408 −0.0957042 0.995410i \(-0.530510\pi\)
−0.0957042 + 0.995410i \(0.530510\pi\)
\(522\) 2.58211 0.113016
\(523\) −9.54958 −0.417574 −0.208787 0.977961i \(-0.566952\pi\)
−0.208787 + 0.977961i \(0.566952\pi\)
\(524\) 3.52781 0.154113
\(525\) −4.29590 −0.187488
\(526\) −8.46011 −0.368878
\(527\) −8.91962 −0.388545
\(528\) 2.13706 0.0930037
\(529\) −17.8528 −0.776208
\(530\) −19.7168 −0.856442
\(531\) 3.44504 0.149502
\(532\) −2.13706 −0.0926534
\(533\) 0 0
\(534\) 3.73556 0.161654
\(535\) −38.1890 −1.65105
\(536\) 11.3937 0.492134
\(537\) 7.49396 0.323388
\(538\) −1.29590 −0.0558701
\(539\) −2.13706 −0.0920498
\(540\) −3.04892 −0.131205
\(541\) −4.13813 −0.177912 −0.0889560 0.996036i \(-0.528353\pi\)
−0.0889560 + 0.996036i \(0.528353\pi\)
\(542\) −5.46250 −0.234634
\(543\) −19.1226 −0.820629
\(544\) 1.40581 0.0602737
\(545\) 3.53319 0.151345
\(546\) 0 0
\(547\) −17.0242 −0.727901 −0.363950 0.931418i \(-0.618572\pi\)
−0.363950 + 0.931418i \(0.618572\pi\)
\(548\) −0.249373 −0.0106527
\(549\) 9.98792 0.426274
\(550\) 9.18060 0.391462
\(551\) 5.51812 0.235080
\(552\) −2.26875 −0.0965644
\(553\) −10.7071 −0.455312
\(554\) −1.01746 −0.0432277
\(555\) 32.4795 1.37868
\(556\) 0.168522 0.00714694
\(557\) 18.9778 0.804113 0.402057 0.915615i \(-0.368295\pi\)
0.402057 + 0.915615i \(0.368295\pi\)
\(558\) −6.34481 −0.268597
\(559\) 0 0
\(560\) 3.04892 0.128840
\(561\) −3.00431 −0.126842
\(562\) −23.7482 −1.00176
\(563\) −31.3840 −1.32268 −0.661340 0.750086i \(-0.730012\pi\)
−0.661340 + 0.750086i \(0.730012\pi\)
\(564\) −3.98792 −0.167922
\(565\) 13.2543 0.557612
\(566\) 19.9148 0.837083
\(567\) 1.00000 0.0419961
\(568\) 4.21983 0.177060
\(569\) 30.4196 1.27525 0.637627 0.770345i \(-0.279916\pi\)
0.637627 + 0.770345i \(0.279916\pi\)
\(570\) −6.51573 −0.272914
\(571\) −26.4534 −1.10704 −0.553520 0.832836i \(-0.686716\pi\)
−0.553520 + 0.832836i \(0.686716\pi\)
\(572\) 0 0
\(573\) −23.2403 −0.970876
\(574\) −4.43296 −0.185028
\(575\) −9.74632 −0.406449
\(576\) 1.00000 0.0416667
\(577\) 6.25667 0.260469 0.130234 0.991483i \(-0.458427\pi\)
0.130234 + 0.991483i \(0.458427\pi\)
\(578\) 15.0237 0.624903
\(579\) 4.00538 0.166458
\(580\) −7.87263 −0.326893
\(581\) −8.94869 −0.371254
\(582\) 5.10321 0.211535
\(583\) −13.8200 −0.572365
\(584\) 5.84548 0.241888
\(585\) 0 0
\(586\) 9.15213 0.378071
\(587\) 16.9028 0.697651 0.348826 0.937188i \(-0.386581\pi\)
0.348826 + 0.937188i \(0.386581\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −13.5593 −0.558700
\(590\) −10.5036 −0.432428
\(591\) −7.75840 −0.319138
\(592\) −10.6528 −0.437827
\(593\) −8.60089 −0.353196 −0.176598 0.984283i \(-0.556509\pi\)
−0.176598 + 0.984283i \(0.556509\pi\)
\(594\) −2.13706 −0.0876848
\(595\) −4.28621 −0.175717
\(596\) −16.3327 −0.669015
\(597\) −14.5526 −0.595597
\(598\) 0 0
\(599\) −28.3744 −1.15934 −0.579672 0.814850i \(-0.696819\pi\)
−0.579672 + 0.814850i \(0.696819\pi\)
\(600\) 4.29590 0.175379
\(601\) 30.3394 1.23757 0.618786 0.785560i \(-0.287625\pi\)
0.618786 + 0.785560i \(0.287625\pi\)
\(602\) 9.55496 0.389431
\(603\) −11.3937 −0.463989
\(604\) −24.3448 −0.990576
\(605\) −19.6136 −0.797405
\(606\) −12.2174 −0.496300
\(607\) 9.64609 0.391523 0.195761 0.980652i \(-0.437282\pi\)
0.195761 + 0.980652i \(0.437282\pi\)
\(608\) 2.13706 0.0866694
\(609\) 2.58211 0.104632
\(610\) −30.4523 −1.23298
\(611\) 0 0
\(612\) −1.40581 −0.0568266
\(613\) −27.1444 −1.09635 −0.548175 0.836364i \(-0.684677\pi\)
−0.548175 + 0.836364i \(0.684677\pi\)
\(614\) 22.2784 0.899085
\(615\) −13.5157 −0.545007
\(616\) 2.13706 0.0861047
\(617\) −30.4510 −1.22591 −0.612956 0.790117i \(-0.710020\pi\)
−0.612956 + 0.790117i \(0.710020\pi\)
\(618\) −4.53319 −0.182352
\(619\) −3.06578 −0.123224 −0.0616121 0.998100i \(-0.519624\pi\)
−0.0616121 + 0.998100i \(0.519624\pi\)
\(620\) 19.3448 0.776906
\(621\) 2.26875 0.0910418
\(622\) 0.254749 0.0102145
\(623\) 3.73556 0.149662
\(624\) 0 0
\(625\) −28.0248 −1.12099
\(626\) −24.1129 −0.963745
\(627\) −4.56704 −0.182390
\(628\) −12.0978 −0.482756
\(629\) 14.9758 0.597126
\(630\) −3.04892 −0.121472
\(631\) −17.7791 −0.707775 −0.353887 0.935288i \(-0.615140\pi\)
−0.353887 + 0.935288i \(0.615140\pi\)
\(632\) 10.7071 0.425905
\(633\) −3.80194 −0.151113
\(634\) −12.8321 −0.509627
\(635\) 55.7827 2.21367
\(636\) −6.46681 −0.256426
\(637\) 0 0
\(638\) −5.51812 −0.218465
\(639\) −4.21983 −0.166934
\(640\) −3.04892 −0.120519
\(641\) −41.9754 −1.65793 −0.828964 0.559303i \(-0.811069\pi\)
−0.828964 + 0.559303i \(0.811069\pi\)
\(642\) −12.5254 −0.494339
\(643\) 26.7222 1.05382 0.526909 0.849921i \(-0.323351\pi\)
0.526909 + 0.849921i \(0.323351\pi\)
\(644\) −2.26875 −0.0894013
\(645\) 29.1323 1.14708
\(646\) −3.00431 −0.118203
\(647\) −33.7399 −1.32645 −0.663226 0.748419i \(-0.730813\pi\)
−0.663226 + 0.748419i \(0.730813\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −7.36227 −0.288994
\(650\) 0 0
\(651\) −6.34481 −0.248673
\(652\) −14.0707 −0.551051
\(653\) −14.8374 −0.580634 −0.290317 0.956931i \(-0.593761\pi\)
−0.290317 + 0.956931i \(0.593761\pi\)
\(654\) 1.15883 0.0453140
\(655\) 10.7560 0.420272
\(656\) 4.43296 0.173078
\(657\) −5.84548 −0.228054
\(658\) −3.98792 −0.155465
\(659\) −25.0508 −0.975842 −0.487921 0.872888i \(-0.662245\pi\)
−0.487921 + 0.872888i \(0.662245\pi\)
\(660\) 6.51573 0.253624
\(661\) −34.5816 −1.34507 −0.672535 0.740066i \(-0.734794\pi\)
−0.672535 + 0.740066i \(0.734794\pi\)
\(662\) 29.0707 1.12986
\(663\) 0 0
\(664\) 8.94869 0.347277
\(665\) −6.51573 −0.252669
\(666\) 10.6528 0.412787
\(667\) 5.85815 0.226829
\(668\) −19.9541 −0.772046
\(669\) −23.1347 −0.894438
\(670\) 34.7385 1.34207
\(671\) −21.3448 −0.824007
\(672\) 1.00000 0.0385758
\(673\) 24.7791 0.955164 0.477582 0.878587i \(-0.341513\pi\)
0.477582 + 0.878587i \(0.341513\pi\)
\(674\) −17.7821 −0.684940
\(675\) −4.29590 −0.165349
\(676\) 0 0
\(677\) −38.1347 −1.46563 −0.732817 0.680426i \(-0.761795\pi\)
−0.732817 + 0.680426i \(0.761795\pi\)
\(678\) 4.34721 0.166953
\(679\) 5.10321 0.195843
\(680\) 4.28621 0.164369
\(681\) −2.72348 −0.104364
\(682\) 13.5593 0.519211
\(683\) 38.6407 1.47855 0.739273 0.673406i \(-0.235169\pi\)
0.739273 + 0.673406i \(0.235169\pi\)
\(684\) −2.13706 −0.0817127
\(685\) −0.760316 −0.0290502
\(686\) −1.00000 −0.0381802
\(687\) 11.7385 0.447853
\(688\) −9.55496 −0.364279
\(689\) 0 0
\(690\) −6.91723 −0.263334
\(691\) −6.00969 −0.228619 −0.114310 0.993445i \(-0.536466\pi\)
−0.114310 + 0.993445i \(0.536466\pi\)
\(692\) −6.61596 −0.251501
\(693\) −2.13706 −0.0811803
\(694\) 10.5144 0.399121
\(695\) 0.513811 0.0194899
\(696\) −2.58211 −0.0978744
\(697\) −6.23191 −0.236051
\(698\) 18.1274 0.686131
\(699\) −9.92931 −0.375561
\(700\) 4.29590 0.162370
\(701\) 24.3642 0.920223 0.460111 0.887861i \(-0.347809\pi\)
0.460111 + 0.887861i \(0.347809\pi\)
\(702\) 0 0
\(703\) 22.7657 0.858624
\(704\) −2.13706 −0.0805436
\(705\) −12.1588 −0.457928
\(706\) −16.1715 −0.608623
\(707\) −12.2174 −0.459484
\(708\) −3.44504 −0.129473
\(709\) 26.1239 0.981104 0.490552 0.871412i \(-0.336795\pi\)
0.490552 + 0.871412i \(0.336795\pi\)
\(710\) 12.8659 0.482849
\(711\) −10.7071 −0.401547
\(712\) −3.73556 −0.139996
\(713\) −14.3948 −0.539089
\(714\) −1.40581 −0.0526112
\(715\) 0 0
\(716\) −7.49396 −0.280062
\(717\) 22.1172 0.825982
\(718\) 7.77240 0.290063
\(719\) −4.64981 −0.173409 −0.0867043 0.996234i \(-0.527634\pi\)
−0.0867043 + 0.996234i \(0.527634\pi\)
\(720\) 3.04892 0.113626
\(721\) −4.53319 −0.168825
\(722\) 14.4330 0.537139
\(723\) 24.5429 0.912759
\(724\) 19.1226 0.710685
\(725\) −11.0925 −0.411964
\(726\) −6.43296 −0.238750
\(727\) −28.4292 −1.05438 −0.527191 0.849747i \(-0.676755\pi\)
−0.527191 + 0.849747i \(0.676755\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 17.8224 0.659636
\(731\) 13.4325 0.496818
\(732\) −9.98792 −0.369164
\(733\) −9.99356 −0.369120 −0.184560 0.982821i \(-0.559086\pi\)
−0.184560 + 0.982821i \(0.559086\pi\)
\(734\) 2.07846 0.0767173
\(735\) −3.04892 −0.112461
\(736\) 2.26875 0.0836272
\(737\) 24.3491 0.896912
\(738\) −4.43296 −0.163180
\(739\) 13.5101 0.496977 0.248488 0.968635i \(-0.420066\pi\)
0.248488 + 0.968635i \(0.420066\pi\)
\(740\) −32.4795 −1.19397
\(741\) 0 0
\(742\) −6.46681 −0.237404
\(743\) −38.1540 −1.39974 −0.699868 0.714272i \(-0.746758\pi\)
−0.699868 + 0.714272i \(0.746758\pi\)
\(744\) 6.34481 0.232612
\(745\) −49.7972 −1.82443
\(746\) −8.10752 −0.296838
\(747\) −8.94869 −0.327415
\(748\) 3.00431 0.109849
\(749\) −12.5254 −0.457669
\(750\) −2.14675 −0.0783883
\(751\) −24.1473 −0.881149 −0.440575 0.897716i \(-0.645225\pi\)
−0.440575 + 0.897716i \(0.645225\pi\)
\(752\) 3.98792 0.145424
\(753\) −27.2489 −0.993005
\(754\) 0 0
\(755\) −74.2253 −2.70134
\(756\) −1.00000 −0.0363696
\(757\) −7.27173 −0.264296 −0.132148 0.991230i \(-0.542187\pi\)
−0.132148 + 0.991230i \(0.542187\pi\)
\(758\) 13.4222 0.487516
\(759\) −4.84846 −0.175988
\(760\) 6.51573 0.236350
\(761\) −27.7894 −1.00736 −0.503682 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(762\) 18.2959 0.662790
\(763\) 1.15883 0.0419526
\(764\) 23.2403 0.840804
\(765\) −4.28621 −0.154968
\(766\) 12.4209 0.448785
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 12.5894 0.453985 0.226993 0.973896i \(-0.427111\pi\)
0.226993 + 0.973896i \(0.427111\pi\)
\(770\) 6.51573 0.234811
\(771\) −17.2349 −0.620700
\(772\) −4.00538 −0.144157
\(773\) −42.4935 −1.52838 −0.764192 0.644989i \(-0.776862\pi\)
−0.764192 + 0.644989i \(0.776862\pi\)
\(774\) 9.55496 0.343446
\(775\) 27.2567 0.979088
\(776\) −5.10321 −0.183195
\(777\) 10.6528 0.382167
\(778\) −37.5666 −1.34683
\(779\) −9.47352 −0.339424
\(780\) 0 0
\(781\) 9.01805 0.322691
\(782\) −3.18944 −0.114054
\(783\) 2.58211 0.0922769
\(784\) 1.00000 0.0357143
\(785\) −36.8853 −1.31649
\(786\) 3.52781 0.125833
\(787\) 30.9017 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(788\) 7.75840 0.276381
\(789\) −8.46011 −0.301188
\(790\) 32.6450 1.16146
\(791\) 4.34721 0.154569
\(792\) 2.13706 0.0759372
\(793\) 0 0
\(794\) 36.1159 1.28170
\(795\) −19.7168 −0.699282
\(796\) 14.5526 0.515802
\(797\) 52.3038 1.85270 0.926348 0.376670i \(-0.122931\pi\)
0.926348 + 0.376670i \(0.122931\pi\)
\(798\) −2.13706 −0.0756512
\(799\) −5.60627 −0.198336
\(800\) −4.29590 −0.151883
\(801\) 3.73556 0.131990
\(802\) −9.82610 −0.346972
\(803\) 12.4922 0.440839
\(804\) 11.3937 0.401826
\(805\) −6.91723 −0.243800
\(806\) 0 0
\(807\) −1.29590 −0.0456177
\(808\) 12.2174 0.429808
\(809\) 31.7942 1.11782 0.558912 0.829227i \(-0.311219\pi\)
0.558912 + 0.829227i \(0.311219\pi\)
\(810\) −3.04892 −0.107128
\(811\) 12.3196 0.432599 0.216300 0.976327i \(-0.430601\pi\)
0.216300 + 0.976327i \(0.430601\pi\)
\(812\) −2.58211 −0.0906141
\(813\) −5.46250 −0.191578
\(814\) −22.7657 −0.797937
\(815\) −42.9004 −1.50273
\(816\) 1.40581 0.0492133
\(817\) 20.4196 0.714390
\(818\) −3.09352 −0.108162
\(819\) 0 0
\(820\) 13.5157 0.471990
\(821\) 41.0901 1.43405 0.717027 0.697046i \(-0.245503\pi\)
0.717027 + 0.697046i \(0.245503\pi\)
\(822\) −0.249373 −0.00869787
\(823\) −2.76164 −0.0962649 −0.0481324 0.998841i \(-0.515327\pi\)
−0.0481324 + 0.998841i \(0.515327\pi\)
\(824\) 4.53319 0.157921
\(825\) 9.18060 0.319628
\(826\) −3.44504 −0.119868
\(827\) −28.6098 −0.994862 −0.497431 0.867504i \(-0.665723\pi\)
−0.497431 + 0.867504i \(0.665723\pi\)
\(828\) −2.26875 −0.0788445
\(829\) −50.4741 −1.75304 −0.876519 0.481367i \(-0.840140\pi\)
−0.876519 + 0.481367i \(0.840140\pi\)
\(830\) 27.2838 0.947035
\(831\) −1.01746 −0.0352952
\(832\) 0 0
\(833\) −1.40581 −0.0487085
\(834\) 0.168522 0.00583545
\(835\) −60.8383 −2.10540
\(836\) 4.56704 0.157954
\(837\) −6.34481 −0.219309
\(838\) 20.1535 0.696190
\(839\) 38.4185 1.32635 0.663177 0.748463i \(-0.269208\pi\)
0.663177 + 0.748463i \(0.269208\pi\)
\(840\) 3.04892 0.105198
\(841\) −22.3327 −0.770094
\(842\) 11.2034 0.386096
\(843\) −23.7482 −0.817933
\(844\) 3.80194 0.130868
\(845\) 0 0
\(846\) −3.98792 −0.137107
\(847\) −6.43296 −0.221039
\(848\) 6.46681 0.222071
\(849\) 19.9148 0.683475
\(850\) 6.03923 0.207144
\(851\) 24.1685 0.828486
\(852\) 4.21983 0.144569
\(853\) 7.09677 0.242989 0.121494 0.992592i \(-0.461231\pi\)
0.121494 + 0.992592i \(0.461231\pi\)
\(854\) −9.98792 −0.341780
\(855\) −6.51573 −0.222833
\(856\) 12.5254 0.428110
\(857\) −37.8901 −1.29430 −0.647150 0.762362i \(-0.724039\pi\)
−0.647150 + 0.762362i \(0.724039\pi\)
\(858\) 0 0
\(859\) 32.9329 1.12366 0.561828 0.827254i \(-0.310098\pi\)
0.561828 + 0.827254i \(0.310098\pi\)
\(860\) −29.1323 −0.993402
\(861\) −4.43296 −0.151075
\(862\) −38.9898 −1.32800
\(863\) 27.3400 0.930665 0.465333 0.885136i \(-0.345935\pi\)
0.465333 + 0.885136i \(0.345935\pi\)
\(864\) 1.00000 0.0340207
\(865\) −20.1715 −0.685852
\(866\) −23.4426 −0.796614
\(867\) 15.0237 0.510231
\(868\) 6.34481 0.215357
\(869\) 22.8817 0.776209
\(870\) −7.87263 −0.266907
\(871\) 0 0
\(872\) −1.15883 −0.0392431
\(873\) 5.10321 0.172718
\(874\) −4.84846 −0.164002
\(875\) −2.14675 −0.0725735
\(876\) 5.84548 0.197500
\(877\) −7.54852 −0.254895 −0.127448 0.991845i \(-0.540678\pi\)
−0.127448 + 0.991845i \(0.540678\pi\)
\(878\) 26.8592 0.906455
\(879\) 9.15213 0.308694
\(880\) −6.51573 −0.219645
\(881\) −9.14138 −0.307981 −0.153990 0.988072i \(-0.549212\pi\)
−0.153990 + 0.988072i \(0.549212\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 31.5838 1.06288 0.531439 0.847097i \(-0.321651\pi\)
0.531439 + 0.847097i \(0.321651\pi\)
\(884\) 0 0
\(885\) −10.5036 −0.353076
\(886\) 11.1347 0.374077
\(887\) 57.3159 1.92448 0.962239 0.272205i \(-0.0877530\pi\)
0.962239 + 0.272205i \(0.0877530\pi\)
\(888\) −10.6528 −0.357484
\(889\) 18.2959 0.613625
\(890\) −11.3894 −0.381774
\(891\) −2.13706 −0.0715943
\(892\) 23.1347 0.774606
\(893\) −8.52243 −0.285192
\(894\) −16.3327 −0.546248
\(895\) −22.8485 −0.763740
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 27.0858 0.903863
\(899\) −16.3830 −0.546403
\(900\) 4.29590 0.143197
\(901\) −9.09113 −0.302869
\(902\) 9.47352 0.315434
\(903\) 9.55496 0.317969
\(904\) −4.34721 −0.144586
\(905\) 58.3032 1.93806
\(906\) −24.3448 −0.808802
\(907\) −35.9963 −1.19524 −0.597618 0.801781i \(-0.703886\pi\)
−0.597618 + 0.801781i \(0.703886\pi\)
\(908\) 2.72348 0.0903819
\(909\) −12.2174 −0.405227
\(910\) 0 0
\(911\) −8.76569 −0.290420 −0.145210 0.989401i \(-0.546386\pi\)
−0.145210 + 0.989401i \(0.546386\pi\)
\(912\) 2.13706 0.0707652
\(913\) 19.1239 0.632909
\(914\) 3.40880 0.112753
\(915\) −30.4523 −1.00672
\(916\) −11.7385 −0.387852
\(917\) 3.52781 0.116499
\(918\) −1.40581 −0.0463987
\(919\) 28.5695 0.942422 0.471211 0.882020i \(-0.343817\pi\)
0.471211 + 0.882020i \(0.343817\pi\)
\(920\) 6.91723 0.228054
\(921\) 22.2784 0.734100
\(922\) −30.9178 −1.01822
\(923\) 0 0
\(924\) 2.13706 0.0703042
\(925\) −45.7633 −1.50469
\(926\) 35.0465 1.15170
\(927\) −4.53319 −0.148889
\(928\) 2.58211 0.0847618
\(929\) 23.2644 0.763281 0.381641 0.924311i \(-0.375359\pi\)
0.381641 + 0.924311i \(0.375359\pi\)
\(930\) 19.3448 0.634341
\(931\) −2.13706 −0.0700394
\(932\) 9.92931 0.325245
\(933\) 0.254749 0.00834012
\(934\) −4.04652 −0.132406
\(935\) 9.15990 0.299561
\(936\) 0 0
\(937\) 57.3491 1.87351 0.936757 0.349980i \(-0.113812\pi\)
0.936757 + 0.349980i \(0.113812\pi\)
\(938\) 11.3937 0.372019
\(939\) −24.1129 −0.786895
\(940\) 12.1588 0.396577
\(941\) −50.8840 −1.65877 −0.829385 0.558677i \(-0.811309\pi\)
−0.829385 + 0.558677i \(0.811309\pi\)
\(942\) −12.0978 −0.394169
\(943\) −10.0573 −0.327510
\(944\) 3.44504 0.112127
\(945\) −3.04892 −0.0991813
\(946\) −20.4196 −0.663897
\(947\) 9.63593 0.313126 0.156563 0.987668i \(-0.449959\pi\)
0.156563 + 0.987668i \(0.449959\pi\)
\(948\) 10.7071 0.347750
\(949\) 0 0
\(950\) 9.18060 0.297858
\(951\) −12.8321 −0.416108
\(952\) 1.40581 0.0455627
\(953\) −6.38345 −0.206780 −0.103390 0.994641i \(-0.532969\pi\)
−0.103390 + 0.994641i \(0.532969\pi\)
\(954\) −6.46681 −0.209371
\(955\) 70.8577 2.29290
\(956\) −22.1172 −0.715322
\(957\) −5.51812 −0.178376
\(958\) −16.0411 −0.518266
\(959\) −0.249373 −0.00805266
\(960\) −3.04892 −0.0984034
\(961\) 9.25667 0.298602
\(962\) 0 0
\(963\) −12.5254 −0.403626
\(964\) −24.5429 −0.790473
\(965\) −12.2121 −0.393120
\(966\) −2.26875 −0.0729958
\(967\) 28.8692 0.928370 0.464185 0.885738i \(-0.346347\pi\)
0.464185 + 0.885738i \(0.346347\pi\)
\(968\) 6.43296 0.206763
\(969\) −3.00431 −0.0965124
\(970\) −15.5593 −0.499578
\(971\) −35.9661 −1.15421 −0.577104 0.816670i \(-0.695817\pi\)
−0.577104 + 0.816670i \(0.695817\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0.168522 0.00540258
\(974\) 36.7724 1.17826
\(975\) 0 0
\(976\) 9.98792 0.319705
\(977\) −42.2664 −1.35222 −0.676110 0.736800i \(-0.736336\pi\)
−0.676110 + 0.736800i \(0.736336\pi\)
\(978\) −14.0707 −0.449931
\(979\) −7.98313 −0.255142
\(980\) 3.04892 0.0973941
\(981\) 1.15883 0.0369987
\(982\) 7.69069 0.245420
\(983\) −39.2435 −1.25167 −0.625837 0.779954i \(-0.715242\pi\)
−0.625837 + 0.779954i \(0.715242\pi\)
\(984\) 4.43296 0.141318
\(985\) 23.6547 0.753702
\(986\) −3.62996 −0.115601
\(987\) −3.98792 −0.126937
\(988\) 0 0
\(989\) 21.6778 0.689314
\(990\) 6.51573 0.207083
\(991\) 31.1892 0.990758 0.495379 0.868677i \(-0.335029\pi\)
0.495379 + 0.868677i \(0.335029\pi\)
\(992\) −6.34481 −0.201448
\(993\) 29.0707 0.922530
\(994\) 4.21983 0.133845
\(995\) 44.3696 1.40661
\(996\) 8.94869 0.283550
\(997\) −23.7275 −0.751458 −0.375729 0.926730i \(-0.622608\pi\)
−0.375729 + 0.926730i \(0.622608\pi\)
\(998\) −21.8213 −0.690742
\(999\) 10.6528 0.337039
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 7098.2.a.cd.1.3 3
13.12 even 2 7098.2.a.ci.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cd.1.3 3 1.1 even 1 trivial
7098.2.a.ci.1.1 yes 3 13.12 even 2