Properties

Label 169.2.a.b.1.3
Level $169$
Weight $2$
Character 169.1
Self dual yes
Analytic conductor $1.349$
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] = [169,2,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(1.34947179416\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +O(q^{10})\) \(q+0.801938 q^{2} -2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} -2.35690 q^{7} -2.69202 q^{8} +2.04892 q^{9} +0.198062 q^{10} -4.24698 q^{11} +3.04892 q^{12} -1.89008 q^{14} -0.554958 q^{15} +0.554958 q^{16} +2.15883 q^{17} +1.64310 q^{18} -0.0881460 q^{19} -0.335126 q^{20} +5.29590 q^{21} -3.40581 q^{22} +1.49396 q^{23} +6.04892 q^{24} -4.93900 q^{25} +2.13706 q^{27} +3.19806 q^{28} +4.63102 q^{29} -0.445042 q^{30} -6.63102 q^{31} +5.82908 q^{32} +9.54288 q^{33} +1.73125 q^{34} -0.582105 q^{35} -2.78017 q^{36} +5.69202 q^{37} -0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} +4.24698 q^{42} -0.295897 q^{43} +5.76271 q^{44} +0.506041 q^{45} +1.19806 q^{46} -7.35690 q^{47} -1.24698 q^{48} -1.44504 q^{49} -3.96077 q^{50} -4.85086 q^{51} -10.3937 q^{53} +1.71379 q^{54} -1.04892 q^{55} +6.34481 q^{56} +0.198062 q^{57} +3.71379 q^{58} -6.78017 q^{59} +0.753020 q^{60} +3.47219 q^{61} -5.31767 q^{62} -4.82908 q^{63} +3.56465 q^{64} +7.65279 q^{66} +7.67994 q^{67} -2.92931 q^{68} -3.35690 q^{69} -0.466812 q^{70} -8.66487 q^{71} -5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} +11.0978 q^{75} +0.119605 q^{76} +10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -10.9487 q^{81} -9.29590 q^{82} +1.60925 q^{83} -7.18598 q^{84} +0.533188 q^{85} -0.237291 q^{86} -10.4058 q^{87} +11.4330 q^{88} -2.88471 q^{89} +0.405813 q^{90} -2.02715 q^{92} +14.8998 q^{93} -5.89977 q^{94} -0.0217703 q^{95} -13.0978 q^{96} -8.05861 q^{97} -1.15883 q^{98} -8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} + q^{27} + 14 q^{28} - q^{29} - q^{30} - 5 q^{31} + 7 q^{32} + 10 q^{33} + 13 q^{34} + 4 q^{35} - 7 q^{36} + 12 q^{37} + 12 q^{38} - 3 q^{40} - 7 q^{41} + 8 q^{42} + 13 q^{43} + 11 q^{45} + 8 q^{46} - 18 q^{47} + q^{48} - 4 q^{49} + q^{50} - q^{51} + q^{53} - 3 q^{54} + 6 q^{55} - 4 q^{56} + 5 q^{57} + 3 q^{58} - 19 q^{59} + 7 q^{60} + 4 q^{61} + q^{62} - 4 q^{63} - 11 q^{64} + 5 q^{66} - q^{67} - 21 q^{68} - 6 q^{69} + 2 q^{70} - 27 q^{71} - 4 q^{72} + 9 q^{73} - 8 q^{74} + 15 q^{75} - 21 q^{76} + 8 q^{77} - 5 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 7 q^{83} - 7 q^{84} + 5 q^{85} - 18 q^{86} - 18 q^{87} + 15 q^{88} - 11 q^{89} - 12 q^{90} + 22 q^{93} + 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.801938 0.567056 0.283528 0.958964i \(-0.408495\pi\)
0.283528 + 0.958964i \(0.408495\pi\)
\(3\) −2.24698 −1.29729 −0.648647 0.761089i \(-0.724665\pi\)
−0.648647 + 0.761089i \(0.724665\pi\)
\(4\) −1.35690 −0.678448
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) −1.80194 −0.735638
\(7\) −2.35690 −0.890823 −0.445411 0.895326i \(-0.646943\pi\)
−0.445411 + 0.895326i \(0.646943\pi\)
\(8\) −2.69202 −0.951773
\(9\) 2.04892 0.682972
\(10\) 0.198062 0.0626328
\(11\) −4.24698 −1.28051 −0.640256 0.768161i \(-0.721172\pi\)
−0.640256 + 0.768161i \(0.721172\pi\)
\(12\) 3.04892 0.880147
\(13\) 0 0
\(14\) −1.89008 −0.505146
\(15\) −0.554958 −0.143290
\(16\) 0.554958 0.138740
\(17\) 2.15883 0.523594 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) 1.64310 0.387283
\(19\) −0.0881460 −0.0202221 −0.0101110 0.999949i \(-0.503218\pi\)
−0.0101110 + 0.999949i \(0.503218\pi\)
\(20\) −0.335126 −0.0749364
\(21\) 5.29590 1.15566
\(22\) −3.40581 −0.726122
\(23\) 1.49396 0.311512 0.155756 0.987796i \(-0.450219\pi\)
0.155756 + 0.987796i \(0.450219\pi\)
\(24\) 6.04892 1.23473
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) 2.13706 0.411278
\(28\) 3.19806 0.604377
\(29\) 4.63102 0.859959 0.429980 0.902839i \(-0.358521\pi\)
0.429980 + 0.902839i \(0.358521\pi\)
\(30\) −0.445042 −0.0812532
\(31\) −6.63102 −1.19097 −0.595483 0.803368i \(-0.703039\pi\)
−0.595483 + 0.803368i \(0.703039\pi\)
\(32\) 5.82908 1.03045
\(33\) 9.54288 1.66120
\(34\) 1.73125 0.296907
\(35\) −0.582105 −0.0983937
\(36\) −2.78017 −0.463361
\(37\) 5.69202 0.935763 0.467881 0.883791i \(-0.345017\pi\)
0.467881 + 0.883791i \(0.345017\pi\)
\(38\) −0.0706876 −0.0114670
\(39\) 0 0
\(40\) −0.664874 −0.105126
\(41\) −11.5918 −1.81033 −0.905167 0.425056i \(-0.860254\pi\)
−0.905167 + 0.425056i \(0.860254\pi\)
\(42\) 4.24698 0.655323
\(43\) −0.295897 −0.0451239 −0.0225619 0.999745i \(-0.507182\pi\)
−0.0225619 + 0.999745i \(0.507182\pi\)
\(44\) 5.76271 0.868761
\(45\) 0.506041 0.0754361
\(46\) 1.19806 0.176645
\(47\) −7.35690 −1.07311 −0.536557 0.843864i \(-0.680275\pi\)
−0.536557 + 0.843864i \(0.680275\pi\)
\(48\) −1.24698 −0.179986
\(49\) −1.44504 −0.206435
\(50\) −3.96077 −0.560138
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) −10.3937 −1.42769 −0.713844 0.700304i \(-0.753048\pi\)
−0.713844 + 0.700304i \(0.753048\pi\)
\(54\) 1.71379 0.233218
\(55\) −1.04892 −0.141436
\(56\) 6.34481 0.847861
\(57\) 0.198062 0.0262340
\(58\) 3.71379 0.487645
\(59\) −6.78017 −0.882703 −0.441351 0.897334i \(-0.645501\pi\)
−0.441351 + 0.897334i \(0.645501\pi\)
\(60\) 0.753020 0.0972145
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) −5.31767 −0.675344
\(63\) −4.82908 −0.608407
\(64\) 3.56465 0.445581
\(65\) 0 0
\(66\) 7.65279 0.941994
\(67\) 7.67994 0.938254 0.469127 0.883131i \(-0.344569\pi\)
0.469127 + 0.883131i \(0.344569\pi\)
\(68\) −2.92931 −0.355231
\(69\) −3.35690 −0.404123
\(70\) −0.466812 −0.0557947
\(71\) −8.66487 −1.02833 −0.514166 0.857691i \(-0.671898\pi\)
−0.514166 + 0.857691i \(0.671898\pi\)
\(72\) −5.51573 −0.650035
\(73\) 6.73556 0.788338 0.394169 0.919038i \(-0.371032\pi\)
0.394169 + 0.919038i \(0.371032\pi\)
\(74\) 4.56465 0.530629
\(75\) 11.0978 1.28147
\(76\) 0.119605 0.0137196
\(77\) 10.0097 1.14071
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0.137063 0.0153241
\(81\) −10.9487 −1.21652
\(82\) −9.29590 −1.02656
\(83\) 1.60925 0.176638 0.0883192 0.996092i \(-0.471850\pi\)
0.0883192 + 0.996092i \(0.471850\pi\)
\(84\) −7.18598 −0.784055
\(85\) 0.533188 0.0578323
\(86\) −0.237291 −0.0255877
\(87\) −10.4058 −1.11562
\(88\) 11.4330 1.21876
\(89\) −2.88471 −0.305778 −0.152889 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(90\) 0.405813 0.0427765
\(91\) 0 0
\(92\) −2.02715 −0.211345
\(93\) 14.8998 1.54503
\(94\) −5.89977 −0.608515
\(95\) −0.0217703 −0.00223358
\(96\) −13.0978 −1.33679
\(97\) −8.05861 −0.818227 −0.409114 0.912483i \(-0.634162\pi\)
−0.409114 + 0.912483i \(0.634162\pi\)
\(98\) −1.15883 −0.117060
\(99\) −8.70171 −0.874555
\(100\) 6.70171 0.670171
\(101\) −13.3545 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(102\) −3.89008 −0.385176
\(103\) 1.36227 0.134229 0.0671144 0.997745i \(-0.478621\pi\)
0.0671144 + 0.997745i \(0.478621\pi\)
\(104\) 0 0
\(105\) 1.30798 0.127646
\(106\) −8.33513 −0.809579
\(107\) 3.26875 0.316002 0.158001 0.987439i \(-0.449495\pi\)
0.158001 + 0.987439i \(0.449495\pi\)
\(108\) −2.89977 −0.279031
\(109\) 15.7017 1.50395 0.751976 0.659191i \(-0.229101\pi\)
0.751976 + 0.659191i \(0.229101\pi\)
\(110\) −0.841166 −0.0802021
\(111\) −12.7899 −1.21396
\(112\) −1.30798 −0.123592
\(113\) 12.0489 1.13347 0.566733 0.823901i \(-0.308207\pi\)
0.566733 + 0.823901i \(0.308207\pi\)
\(114\) 0.158834 0.0148761
\(115\) 0.368977 0.0344073
\(116\) −6.28382 −0.583438
\(117\) 0 0
\(118\) −5.43727 −0.500541
\(119\) −5.08815 −0.466430
\(120\) 1.49396 0.136379
\(121\) 7.03684 0.639712
\(122\) 2.78448 0.252095
\(123\) 26.0465 2.34854
\(124\) 8.99761 0.808009
\(125\) −2.45473 −0.219558
\(126\) −3.87263 −0.345001
\(127\) −9.80731 −0.870258 −0.435129 0.900368i \(-0.643297\pi\)
−0.435129 + 0.900368i \(0.643297\pi\)
\(128\) −8.79954 −0.777777
\(129\) 0.664874 0.0585389
\(130\) 0 0
\(131\) −6.57673 −0.574611 −0.287306 0.957839i \(-0.592760\pi\)
−0.287306 + 0.957839i \(0.592760\pi\)
\(132\) −12.9487 −1.12704
\(133\) 0.207751 0.0180143
\(134\) 6.15883 0.532042
\(135\) 0.527811 0.0454267
\(136\) −5.81163 −0.498343
\(137\) 6.21983 0.531396 0.265698 0.964056i \(-0.414398\pi\)
0.265698 + 0.964056i \(0.414398\pi\)
\(138\) −2.69202 −0.229160
\(139\) −14.7071 −1.24744 −0.623719 0.781648i \(-0.714379\pi\)
−0.623719 + 0.781648i \(0.714379\pi\)
\(140\) 0.789856 0.0667550
\(141\) 16.5308 1.39214
\(142\) −6.94869 −0.583121
\(143\) 0 0
\(144\) 1.13706 0.0947553
\(145\) 1.14377 0.0949848
\(146\) 5.40150 0.447031
\(147\) 3.24698 0.267806
\(148\) −7.72348 −0.634866
\(149\) 4.33513 0.355147 0.177574 0.984108i \(-0.443175\pi\)
0.177574 + 0.984108i \(0.443175\pi\)
\(150\) 8.89977 0.726663
\(151\) 3.94438 0.320989 0.160494 0.987037i \(-0.448691\pi\)
0.160494 + 0.987037i \(0.448691\pi\)
\(152\) 0.237291 0.0192468
\(153\) 4.42327 0.357600
\(154\) 8.02715 0.646846
\(155\) −1.63773 −0.131545
\(156\) 0 0
\(157\) 4.45473 0.355526 0.177763 0.984073i \(-0.443114\pi\)
0.177763 + 0.984073i \(0.443114\pi\)
\(158\) 7.99569 0.636103
\(159\) 23.3545 1.85213
\(160\) 1.43967 0.113816
\(161\) −3.52111 −0.277502
\(162\) −8.78017 −0.689835
\(163\) 16.1588 1.26566 0.632829 0.774292i \(-0.281894\pi\)
0.632829 + 0.774292i \(0.281894\pi\)
\(164\) 15.7289 1.22822
\(165\) 2.35690 0.183484
\(166\) 1.29052 0.100164
\(167\) 16.1172 1.24719 0.623594 0.781749i \(-0.285672\pi\)
0.623594 + 0.781749i \(0.285672\pi\)
\(168\) −14.2567 −1.09993
\(169\) 0 0
\(170\) 0.427583 0.0327942
\(171\) −0.180604 −0.0138111
\(172\) 0.401501 0.0306142
\(173\) −21.5362 −1.63736 −0.818682 0.574247i \(-0.805295\pi\)
−0.818682 + 0.574247i \(0.805295\pi\)
\(174\) −8.34481 −0.632619
\(175\) 11.6407 0.879955
\(176\) −2.35690 −0.177658
\(177\) 15.2349 1.14513
\(178\) −2.31336 −0.173393
\(179\) 11.4330 0.854540 0.427270 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(180\) −0.686645 −0.0511795
\(181\) 20.9705 1.55872 0.779361 0.626575i \(-0.215544\pi\)
0.779361 + 0.626575i \(0.215544\pi\)
\(182\) 0 0
\(183\) −7.80194 −0.576736
\(184\) −4.02177 −0.296489
\(185\) 1.40581 0.103357
\(186\) 11.9487 0.876120
\(187\) −9.16852 −0.670469
\(188\) 9.98254 0.728052
\(189\) −5.03684 −0.366376
\(190\) −0.0174584 −0.00126657
\(191\) −14.4373 −1.04464 −0.522322 0.852748i \(-0.674934\pi\)
−0.522322 + 0.852748i \(0.674934\pi\)
\(192\) −8.00969 −0.578049
\(193\) −13.5797 −0.977489 −0.488745 0.872427i \(-0.662545\pi\)
−0.488745 + 0.872427i \(0.662545\pi\)
\(194\) −6.46250 −0.463980
\(195\) 0 0
\(196\) 1.96077 0.140055
\(197\) −0.560335 −0.0399222 −0.0199611 0.999801i \(-0.506354\pi\)
−0.0199611 + 0.999801i \(0.506354\pi\)
\(198\) −6.97823 −0.495921
\(199\) 11.4916 0.814616 0.407308 0.913291i \(-0.366468\pi\)
0.407308 + 0.913291i \(0.366468\pi\)
\(200\) 13.2959 0.940162
\(201\) −17.2567 −1.21719
\(202\) −10.7095 −0.753516
\(203\) −10.9148 −0.766071
\(204\) 6.58211 0.460840
\(205\) −2.86294 −0.199956
\(206\) 1.09246 0.0761151
\(207\) 3.06100 0.212754
\(208\) 0 0
\(209\) 0.374354 0.0258946
\(210\) 1.04892 0.0723822
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) 14.1032 0.968613
\(213\) 19.4698 1.33405
\(214\) 2.62133 0.179191
\(215\) −0.0730805 −0.00498405
\(216\) −5.75302 −0.391443
\(217\) 15.6286 1.06094
\(218\) 12.5918 0.852824
\(219\) −15.1347 −1.02271
\(220\) 1.42327 0.0959570
\(221\) 0 0
\(222\) −10.2567 −0.688383
\(223\) −2.25906 −0.151278 −0.0756390 0.997135i \(-0.524100\pi\)
−0.0756390 + 0.997135i \(0.524100\pi\)
\(224\) −13.7385 −0.917945
\(225\) −10.1196 −0.674640
\(226\) 9.66248 0.642739
\(227\) −6.96615 −0.462359 −0.231180 0.972911i \(-0.574259\pi\)
−0.231180 + 0.972911i \(0.574259\pi\)
\(228\) −0.268750 −0.0177984
\(229\) −24.1739 −1.59746 −0.798728 0.601692i \(-0.794493\pi\)
−0.798728 + 0.601692i \(0.794493\pi\)
\(230\) 0.295897 0.0195109
\(231\) −22.4916 −1.47984
\(232\) −12.4668 −0.818486
\(233\) −3.06100 −0.200533 −0.100266 0.994961i \(-0.531969\pi\)
−0.100266 + 0.994961i \(0.531969\pi\)
\(234\) 0 0
\(235\) −1.81700 −0.118528
\(236\) 9.19998 0.598868
\(237\) −22.4034 −1.45526
\(238\) −4.08038 −0.264492
\(239\) −25.1468 −1.62661 −0.813304 0.581839i \(-0.802333\pi\)
−0.813304 + 0.581839i \(0.802333\pi\)
\(240\) −0.307979 −0.0198799
\(241\) −20.2664 −1.30547 −0.652735 0.757586i \(-0.726379\pi\)
−0.652735 + 0.757586i \(0.726379\pi\)
\(242\) 5.64310 0.362752
\(243\) 18.1903 1.16691
\(244\) −4.71140 −0.301616
\(245\) −0.356896 −0.0228012
\(246\) 20.8877 1.33175
\(247\) 0 0
\(248\) 17.8509 1.13353
\(249\) −3.61596 −0.229152
\(250\) −1.96854 −0.124501
\(251\) −23.7211 −1.49726 −0.748631 0.662987i \(-0.769288\pi\)
−0.748631 + 0.662987i \(0.769288\pi\)
\(252\) 6.55257 0.412773
\(253\) −6.34481 −0.398895
\(254\) −7.86486 −0.493485
\(255\) −1.19806 −0.0750256
\(256\) −14.1860 −0.886624
\(257\) 14.2241 0.887278 0.443639 0.896206i \(-0.353687\pi\)
0.443639 + 0.896206i \(0.353687\pi\)
\(258\) 0.533188 0.0331948
\(259\) −13.4155 −0.833599
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) −5.27413 −0.325837
\(263\) −17.0954 −1.05415 −0.527075 0.849819i \(-0.676711\pi\)
−0.527075 + 0.849819i \(0.676711\pi\)
\(264\) −25.6896 −1.58109
\(265\) −2.56704 −0.157692
\(266\) 0.166603 0.0102151
\(267\) 6.48188 0.396684
\(268\) −10.4209 −0.636556
\(269\) −6.46681 −0.394288 −0.197144 0.980374i \(-0.563167\pi\)
−0.197144 + 0.980374i \(0.563167\pi\)
\(270\) 0.423272 0.0257595
\(271\) 6.44803 0.391690 0.195845 0.980635i \(-0.437255\pi\)
0.195845 + 0.980635i \(0.437255\pi\)
\(272\) 1.19806 0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) 20.9758 1.26489
\(276\) 4.55496 0.274176
\(277\) 13.4601 0.808739 0.404370 0.914596i \(-0.367491\pi\)
0.404370 + 0.914596i \(0.367491\pi\)
\(278\) −11.7942 −0.707367
\(279\) −13.5864 −0.813398
\(280\) 1.56704 0.0936485
\(281\) 5.03684 0.300472 0.150236 0.988650i \(-0.451997\pi\)
0.150236 + 0.988650i \(0.451997\pi\)
\(282\) 13.2567 0.789423
\(283\) 22.1280 1.31537 0.657686 0.753293i \(-0.271536\pi\)
0.657686 + 0.753293i \(0.271536\pi\)
\(284\) 11.7573 0.697669
\(285\) 0.0489173 0.00289761
\(286\) 0 0
\(287\) 27.3207 1.61269
\(288\) 11.9433 0.703766
\(289\) −12.3394 −0.725849
\(290\) 0.917231 0.0538616
\(291\) 18.1075 1.06148
\(292\) −9.13946 −0.534846
\(293\) 14.9463 0.873172 0.436586 0.899663i \(-0.356187\pi\)
0.436586 + 0.899663i \(0.356187\pi\)
\(294\) 2.60388 0.151861
\(295\) −1.67456 −0.0974968
\(296\) −15.3230 −0.890634
\(297\) −9.07606 −0.526647
\(298\) 3.47650 0.201388
\(299\) 0 0
\(300\) −15.0586 −0.869409
\(301\) 0.697398 0.0401974
\(302\) 3.16315 0.182019
\(303\) 30.0073 1.72387
\(304\) −0.0489173 −0.00280560
\(305\) 0.857560 0.0491037
\(306\) 3.54719 0.202779
\(307\) 19.1293 1.09177 0.545883 0.837861i \(-0.316194\pi\)
0.545883 + 0.837861i \(0.316194\pi\)
\(308\) −13.5821 −0.773912
\(309\) −3.06100 −0.174134
\(310\) −1.31336 −0.0745936
\(311\) −0.269815 −0.0152998 −0.00764990 0.999971i \(-0.502435\pi\)
−0.00764990 + 0.999971i \(0.502435\pi\)
\(312\) 0 0
\(313\) −23.3937 −1.32229 −0.661146 0.750257i \(-0.729930\pi\)
−0.661146 + 0.750257i \(0.729930\pi\)
\(314\) 3.57242 0.201603
\(315\) −1.19269 −0.0672002
\(316\) −13.5289 −0.761059
\(317\) −13.9952 −0.786050 −0.393025 0.919528i \(-0.628571\pi\)
−0.393025 + 0.919528i \(0.628571\pi\)
\(318\) 18.7289 1.05026
\(319\) −19.6679 −1.10119
\(320\) 0.880395 0.0492156
\(321\) −7.34481 −0.409948
\(322\) −2.82371 −0.157359
\(323\) −0.190293 −0.0105882
\(324\) 14.8562 0.825346
\(325\) 0 0
\(326\) 12.9584 0.717698
\(327\) −35.2814 −1.95107
\(328\) 31.2054 1.72303
\(329\) 17.3394 0.955954
\(330\) 1.89008 0.104046
\(331\) 17.8213 0.979548 0.489774 0.871849i \(-0.337079\pi\)
0.489774 + 0.871849i \(0.337079\pi\)
\(332\) −2.18359 −0.119840
\(333\) 11.6625 0.639100
\(334\) 12.9250 0.707225
\(335\) 1.89679 0.103633
\(336\) 2.93900 0.160336
\(337\) −27.8485 −1.51700 −0.758501 0.651672i \(-0.774068\pi\)
−0.758501 + 0.651672i \(0.774068\pi\)
\(338\) 0 0
\(339\) −27.0737 −1.47044
\(340\) −0.723480 −0.0392362
\(341\) 28.1618 1.52505
\(342\) −0.144833 −0.00783167
\(343\) 19.9041 1.07472
\(344\) 0.796561 0.0429477
\(345\) −0.829085 −0.0446364
\(346\) −17.2707 −0.928477
\(347\) 1.50365 0.0807200 0.0403600 0.999185i \(-0.487150\pi\)
0.0403600 + 0.999185i \(0.487150\pi\)
\(348\) 14.1196 0.756890
\(349\) −14.1860 −0.759358 −0.379679 0.925118i \(-0.623966\pi\)
−0.379679 + 0.925118i \(0.623966\pi\)
\(350\) 9.33513 0.498983
\(351\) 0 0
\(352\) −24.7560 −1.31950
\(353\) −7.16852 −0.381542 −0.190771 0.981635i \(-0.561099\pi\)
−0.190771 + 0.981635i \(0.561099\pi\)
\(354\) 12.2174 0.649350
\(355\) −2.14005 −0.113582
\(356\) 3.91425 0.207455
\(357\) 11.4330 0.605096
\(358\) 9.16852 0.484571
\(359\) 19.8853 1.04951 0.524753 0.851255i \(-0.324158\pi\)
0.524753 + 0.851255i \(0.324158\pi\)
\(360\) −1.36227 −0.0717981
\(361\) −18.9922 −0.999591
\(362\) 16.8170 0.883882
\(363\) −15.8116 −0.829895
\(364\) 0 0
\(365\) 1.66355 0.0870740
\(366\) −6.25667 −0.327041
\(367\) 1.08383 0.0565757 0.0282878 0.999600i \(-0.490994\pi\)
0.0282878 + 0.999600i \(0.490994\pi\)
\(368\) 0.829085 0.0432190
\(369\) −23.7506 −1.23641
\(370\) 1.12737 0.0586094
\(371\) 24.4969 1.27182
\(372\) −20.2174 −1.04823
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) −7.35258 −0.380193
\(375\) 5.51573 0.284831
\(376\) 19.8049 1.02136
\(377\) 0 0
\(378\) −4.03923 −0.207756
\(379\) −2.40880 −0.123732 −0.0618658 0.998084i \(-0.519705\pi\)
−0.0618658 + 0.998084i \(0.519705\pi\)
\(380\) 0.0295400 0.00151537
\(381\) 22.0368 1.12898
\(382\) −11.5778 −0.592371
\(383\) −30.3913 −1.55292 −0.776462 0.630164i \(-0.782988\pi\)
−0.776462 + 0.630164i \(0.782988\pi\)
\(384\) 19.7724 1.00901
\(385\) 2.47219 0.125994
\(386\) −10.8901 −0.554291
\(387\) −0.606268 −0.0308184
\(388\) 10.9347 0.555125
\(389\) −15.9409 −0.808237 −0.404118 0.914707i \(-0.632422\pi\)
−0.404118 + 0.914707i \(0.632422\pi\)
\(390\) 0 0
\(391\) 3.22521 0.163106
\(392\) 3.89008 0.196479
\(393\) 14.7778 0.745440
\(394\) −0.449354 −0.0226381
\(395\) 2.46250 0.123902
\(396\) 11.8073 0.593340
\(397\) 16.9148 0.848931 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(398\) 9.21552 0.461932
\(399\) −0.466812 −0.0233698
\(400\) −2.74094 −0.137047
\(401\) 26.6625 1.33146 0.665730 0.746192i \(-0.268120\pi\)
0.665730 + 0.746192i \(0.268120\pi\)
\(402\) −13.8388 −0.690215
\(403\) 0 0
\(404\) 18.1207 0.901537
\(405\) −2.70410 −0.134368
\(406\) −8.75302 −0.434405
\(407\) −24.1739 −1.19826
\(408\) 13.0586 0.646497
\(409\) 28.5163 1.41004 0.705021 0.709187i \(-0.250938\pi\)
0.705021 + 0.709187i \(0.250938\pi\)
\(410\) −2.29590 −0.113386
\(411\) −13.9758 −0.689377
\(412\) −1.84846 −0.0910672
\(413\) 15.9801 0.786332
\(414\) 2.45473 0.120643
\(415\) 0.397452 0.0195102
\(416\) 0 0
\(417\) 33.0465 1.61830
\(418\) 0.300209 0.0146837
\(419\) −29.6093 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(420\) −1.77479 −0.0866009
\(421\) −11.6606 −0.568301 −0.284151 0.958780i \(-0.591712\pi\)
−0.284151 + 0.958780i \(0.591712\pi\)
\(422\) 7.04461 0.342926
\(423\) −15.0737 −0.732907
\(424\) 27.9801 1.35884
\(425\) −10.6625 −0.517206
\(426\) 15.6136 0.756480
\(427\) −8.18359 −0.396032
\(428\) −4.43535 −0.214391
\(429\) 0 0
\(430\) −0.0586060 −0.00282623
\(431\) 4.34913 0.209490 0.104745 0.994499i \(-0.466597\pi\)
0.104745 + 0.994499i \(0.466597\pi\)
\(432\) 1.18598 0.0570605
\(433\) −14.3884 −0.691460 −0.345730 0.938334i \(-0.612369\pi\)
−0.345730 + 0.938334i \(0.612369\pi\)
\(434\) 12.5332 0.601612
\(435\) −2.57002 −0.123223
\(436\) −21.3056 −1.02035
\(437\) −0.131687 −0.00629942
\(438\) −12.1371 −0.579931
\(439\) −20.2325 −0.965645 −0.482822 0.875718i \(-0.660388\pi\)
−0.482822 + 0.875718i \(0.660388\pi\)
\(440\) 2.82371 0.134615
\(441\) −2.96077 −0.140989
\(442\) 0 0
\(443\) 8.12200 0.385888 0.192944 0.981210i \(-0.438196\pi\)
0.192944 + 0.981210i \(0.438196\pi\)
\(444\) 17.3545 0.823608
\(445\) −0.712464 −0.0337740
\(446\) −1.81163 −0.0857830
\(447\) −9.74094 −0.460731
\(448\) −8.40150 −0.396934
\(449\) 12.4916 0.589513 0.294757 0.955572i \(-0.404761\pi\)
0.294757 + 0.955572i \(0.404761\pi\)
\(450\) −8.11529 −0.382559
\(451\) 49.2301 2.31816
\(452\) −16.3491 −0.768998
\(453\) −8.86294 −0.416417
\(454\) −5.58642 −0.262184
\(455\) 0 0
\(456\) −0.533188 −0.0249688
\(457\) 5.98121 0.279789 0.139895 0.990166i \(-0.455324\pi\)
0.139895 + 0.990166i \(0.455324\pi\)
\(458\) −19.3860 −0.905847
\(459\) 4.61356 0.215343
\(460\) −0.500664 −0.0233436
\(461\) −2.05669 −0.0957895 −0.0478947 0.998852i \(-0.515251\pi\)
−0.0478947 + 0.998852i \(0.515251\pi\)
\(462\) −18.0368 −0.839150
\(463\) −8.44935 −0.392675 −0.196337 0.980536i \(-0.562905\pi\)
−0.196337 + 0.980536i \(0.562905\pi\)
\(464\) 2.57002 0.119310
\(465\) 3.67994 0.170653
\(466\) −2.45473 −0.113713
\(467\) 33.5139 1.55084 0.775420 0.631446i \(-0.217538\pi\)
0.775420 + 0.631446i \(0.217538\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) −1.45712 −0.0672121
\(471\) −10.0097 −0.461222
\(472\) 18.2524 0.840133
\(473\) 1.25667 0.0577817
\(474\) −17.9661 −0.825213
\(475\) 0.435353 0.0199754
\(476\) 6.90408 0.316448
\(477\) −21.2959 −0.975072
\(478\) −20.1661 −0.922377
\(479\) −24.7313 −1.13000 −0.565000 0.825091i \(-0.691124\pi\)
−0.565000 + 0.825091i \(0.691124\pi\)
\(480\) −3.23490 −0.147652
\(481\) 0 0
\(482\) −16.2524 −0.740275
\(483\) 7.91185 0.360002
\(484\) −9.54825 −0.434012
\(485\) −1.99031 −0.0903754
\(486\) 14.5875 0.661702
\(487\) −37.7555 −1.71087 −0.855433 0.517913i \(-0.826709\pi\)
−0.855433 + 0.517913i \(0.826709\pi\)
\(488\) −9.34721 −0.423128
\(489\) −36.3086 −1.64193
\(490\) −0.286208 −0.0129296
\(491\) 31.3110 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(492\) −35.3424 −1.59336
\(493\) 9.99761 0.450270
\(494\) 0 0
\(495\) −2.14914 −0.0965969
\(496\) −3.67994 −0.165234
\(497\) 20.4222 0.916061
\(498\) −2.89977 −0.129942
\(499\) 21.4873 0.961902 0.480951 0.876748i \(-0.340292\pi\)
0.480951 + 0.876748i \(0.340292\pi\)
\(500\) 3.33081 0.148959
\(501\) −36.2150 −1.61797
\(502\) −19.0228 −0.849031
\(503\) 37.5924 1.67616 0.838081 0.545546i \(-0.183678\pi\)
0.838081 + 0.545546i \(0.183678\pi\)
\(504\) 13.0000 0.579066
\(505\) −3.29829 −0.146772
\(506\) −5.08815 −0.226196
\(507\) 0 0
\(508\) 13.3075 0.590425
\(509\) −17.1075 −0.758278 −0.379139 0.925340i \(-0.623780\pi\)
−0.379139 + 0.925340i \(0.623780\pi\)
\(510\) −0.960771 −0.0425437
\(511\) −15.8750 −0.702269
\(512\) 6.22282 0.275012
\(513\) −0.188374 −0.00831690
\(514\) 11.4069 0.503136
\(515\) 0.336454 0.0148259
\(516\) −0.902165 −0.0397156
\(517\) 31.2446 1.37414
\(518\) −10.7584 −0.472697
\(519\) 48.3913 2.12414
\(520\) 0 0
\(521\) −19.8465 −0.869493 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(522\) 7.60925 0.333048
\(523\) −11.4300 −0.499798 −0.249899 0.968272i \(-0.580397\pi\)
−0.249899 + 0.968272i \(0.580397\pi\)
\(524\) 8.92394 0.389844
\(525\) −26.1564 −1.14156
\(526\) −13.7095 −0.597762
\(527\) −14.3153 −0.623583
\(528\) 5.29590 0.230474
\(529\) −20.7681 −0.902960
\(530\) −2.05861 −0.0894201
\(531\) −13.8920 −0.602862
\(532\) −0.281896 −0.0122218
\(533\) 0 0
\(534\) 5.19806 0.224942
\(535\) 0.807315 0.0349033
\(536\) −20.6746 −0.893005
\(537\) −25.6896 −1.10859
\(538\) −5.18598 −0.223584
\(539\) 6.13706 0.264342
\(540\) −0.716185 −0.0308197
\(541\) 16.1884 0.695993 0.347996 0.937496i \(-0.386862\pi\)
0.347996 + 0.937496i \(0.386862\pi\)
\(542\) 5.17092 0.222110
\(543\) −47.1202 −2.02212
\(544\) 12.5840 0.539536
\(545\) 3.87800 0.166115
\(546\) 0 0
\(547\) 5.33081 0.227929 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(548\) −8.43967 −0.360525
\(549\) 7.11423 0.303628
\(550\) 16.8213 0.717263
\(551\) −0.408206 −0.0173902
\(552\) 9.03684 0.384633
\(553\) −23.4993 −0.999293
\(554\) 10.7942 0.458600
\(555\) −3.15883 −0.134085
\(556\) 19.9560 0.846322
\(557\) 7.39075 0.313156 0.156578 0.987666i \(-0.449954\pi\)
0.156578 + 0.987666i \(0.449954\pi\)
\(558\) −10.8955 −0.461242
\(559\) 0 0
\(560\) −0.323044 −0.0136511
\(561\) 20.6015 0.869795
\(562\) 4.03923 0.170385
\(563\) −9.47889 −0.399488 −0.199744 0.979848i \(-0.564011\pi\)
−0.199744 + 0.979848i \(0.564011\pi\)
\(564\) −22.4306 −0.944497
\(565\) 2.97584 0.125194
\(566\) 17.7453 0.745889
\(567\) 25.8049 1.08370
\(568\) 23.3260 0.978738
\(569\) −10.1438 −0.425249 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(570\) 0.0392287 0.00164311
\(571\) −14.0925 −0.589751 −0.294876 0.955536i \(-0.595278\pi\)
−0.294876 + 0.955536i \(0.595278\pi\)
\(572\) 0 0
\(573\) 32.4403 1.35521
\(574\) 21.9095 0.914483
\(575\) −7.37867 −0.307712
\(576\) 7.30367 0.304319
\(577\) 25.1545 1.04720 0.523598 0.851965i \(-0.324589\pi\)
0.523598 + 0.851965i \(0.324589\pi\)
\(578\) −9.89546 −0.411597
\(579\) 30.5133 1.26809
\(580\) −1.55197 −0.0644422
\(581\) −3.79284 −0.157354
\(582\) 14.5211 0.601919
\(583\) 44.1420 1.82817
\(584\) −18.1323 −0.750319
\(585\) 0 0
\(586\) 11.9860 0.495137
\(587\) 43.8353 1.80928 0.904639 0.426180i \(-0.140141\pi\)
0.904639 + 0.426180i \(0.140141\pi\)
\(588\) −4.40581 −0.181693
\(589\) 0.584498 0.0240838
\(590\) −1.34290 −0.0552861
\(591\) 1.25906 0.0517909
\(592\) 3.15883 0.129827
\(593\) −24.9965 −1.02648 −0.513242 0.858244i \(-0.671556\pi\)
−0.513242 + 0.858244i \(0.671556\pi\)
\(594\) −7.27844 −0.298638
\(595\) −1.25667 −0.0515184
\(596\) −5.88231 −0.240949
\(597\) −25.8213 −1.05680
\(598\) 0 0
\(599\) −6.24027 −0.254971 −0.127485 0.991840i \(-0.540691\pi\)
−0.127485 + 0.991840i \(0.540691\pi\)
\(600\) −29.8756 −1.21967
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) 0.559270 0.0227941
\(603\) 15.7356 0.640802
\(604\) −5.35211 −0.217774
\(605\) 1.73795 0.0706579
\(606\) 24.0640 0.977532
\(607\) −43.6480 −1.77162 −0.885809 0.464050i \(-0.846396\pi\)
−0.885809 + 0.464050i \(0.846396\pi\)
\(608\) −0.513811 −0.0208378
\(609\) 24.5254 0.993820
\(610\) 0.687710 0.0278445
\(611\) 0 0
\(612\) −6.00192 −0.242613
\(613\) −25.9541 −1.04827 −0.524137 0.851634i \(-0.675612\pi\)
−0.524137 + 0.851634i \(0.675612\pi\)
\(614\) 15.3405 0.619092
\(615\) 6.43296 0.259402
\(616\) −26.9463 −1.08570
\(617\) 45.9396 1.84946 0.924729 0.380626i \(-0.124291\pi\)
0.924729 + 0.380626i \(0.124291\pi\)
\(618\) −2.45473 −0.0987437
\(619\) 6.73556 0.270725 0.135363 0.990796i \(-0.456780\pi\)
0.135363 + 0.990796i \(0.456780\pi\)
\(620\) 2.22223 0.0892467
\(621\) 3.19269 0.128118
\(622\) −0.216375 −0.00867583
\(623\) 6.79895 0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) −18.7603 −0.749813
\(627\) −0.841166 −0.0335930
\(628\) −6.04461 −0.241206
\(629\) 12.2881 0.489960
\(630\) −0.956459 −0.0381063
\(631\) 45.0998 1.79539 0.897696 0.440614i \(-0.145239\pi\)
0.897696 + 0.440614i \(0.145239\pi\)
\(632\) −26.8407 −1.06767
\(633\) −19.7385 −0.784537
\(634\) −11.2233 −0.445734
\(635\) −2.42221 −0.0961223
\(636\) −31.6896 −1.25658
\(637\) 0 0
\(638\) −15.7724 −0.624435
\(639\) −17.7536 −0.702322
\(640\) −2.17331 −0.0859075
\(641\) 32.5821 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(642\) −5.89008 −0.232463
\(643\) 25.5754 1.00860 0.504298 0.863530i \(-0.331751\pi\)
0.504298 + 0.863530i \(0.331751\pi\)
\(644\) 4.77777 0.188271
\(645\) 0.164210 0.00646578
\(646\) −0.152603 −0.00600408
\(647\) −30.1715 −1.18616 −0.593082 0.805142i \(-0.702089\pi\)
−0.593082 + 0.805142i \(0.702089\pi\)
\(648\) 29.4741 1.15785
\(649\) 28.7952 1.13031
\(650\) 0 0
\(651\) −35.1172 −1.37635
\(652\) −21.9259 −0.858683
\(653\) 36.9028 1.44412 0.722058 0.691832i \(-0.243196\pi\)
0.722058 + 0.691832i \(0.243196\pi\)
\(654\) −28.2935 −1.10636
\(655\) −1.62432 −0.0634673
\(656\) −6.43296 −0.251165
\(657\) 13.8006 0.538413
\(658\) 13.9051 0.542079
\(659\) 23.6866 0.922701 0.461350 0.887218i \(-0.347365\pi\)
0.461350 + 0.887218i \(0.347365\pi\)
\(660\) −3.19806 −0.124484
\(661\) −31.7590 −1.23528 −0.617641 0.786460i \(-0.711911\pi\)
−0.617641 + 0.786460i \(0.711911\pi\)
\(662\) 14.2916 0.555458
\(663\) 0 0
\(664\) −4.33214 −0.168120
\(665\) 0.0513102 0.00198973
\(666\) 9.35258 0.362405
\(667\) 6.91856 0.267888
\(668\) −21.8694 −0.846152
\(669\) 5.07606 0.196252
\(670\) 1.52111 0.0587655
\(671\) −14.7463 −0.569275
\(672\) 30.8702 1.19085
\(673\) −7.50232 −0.289193 −0.144597 0.989491i \(-0.546188\pi\)
−0.144597 + 0.989491i \(0.546188\pi\)
\(674\) −22.3327 −0.860225
\(675\) −10.5550 −0.406261
\(676\) 0 0
\(677\) −35.0315 −1.34637 −0.673184 0.739475i \(-0.735074\pi\)
−0.673184 + 0.739475i \(0.735074\pi\)
\(678\) −21.7114 −0.833821
\(679\) 18.9933 0.728896
\(680\) −1.43535 −0.0550433
\(681\) 15.6528 0.599816
\(682\) 22.5840 0.864787
\(683\) −24.0834 −0.921524 −0.460762 0.887524i \(-0.652424\pi\)
−0.460762 + 0.887524i \(0.652424\pi\)
\(684\) 0.245061 0.00937013
\(685\) 1.53617 0.0586941
\(686\) 15.9618 0.609426
\(687\) 54.3183 2.07237
\(688\) −0.164210 −0.00626046
\(689\) 0 0
\(690\) −0.664874 −0.0253113
\(691\) 2.01447 0.0766342 0.0383171 0.999266i \(-0.487800\pi\)
0.0383171 + 0.999266i \(0.487800\pi\)
\(692\) 29.2223 1.11087
\(693\) 20.5090 0.779073
\(694\) 1.20583 0.0457728
\(695\) −3.63235 −0.137783
\(696\) 28.0127 1.06182
\(697\) −25.0248 −0.947880
\(698\) −11.3763 −0.430598
\(699\) 6.87800 0.260150
\(700\) −15.7952 −0.597004
\(701\) −48.8189 −1.84387 −0.921933 0.387350i \(-0.873390\pi\)
−0.921933 + 0.387350i \(0.873390\pi\)
\(702\) 0 0
\(703\) −0.501729 −0.0189231
\(704\) −15.1390 −0.570572
\(705\) 4.08277 0.153766
\(706\) −5.74871 −0.216355
\(707\) 31.4752 1.18375
\(708\) −20.6722 −0.776908
\(709\) −20.8060 −0.781385 −0.390693 0.920521i \(-0.627764\pi\)
−0.390693 + 0.920521i \(0.627764\pi\)
\(710\) −1.71618 −0.0644073
\(711\) 20.4286 0.766134
\(712\) 7.76569 0.291032
\(713\) −9.90648 −0.371000
\(714\) 9.16852 0.343123
\(715\) 0 0
\(716\) −15.5133 −0.579761
\(717\) 56.5042 2.11019
\(718\) 15.9468 0.595128
\(719\) 21.4306 0.799225 0.399613 0.916684i \(-0.369145\pi\)
0.399613 + 0.916684i \(0.369145\pi\)
\(720\) 0.280831 0.0104660
\(721\) −3.21073 −0.119574
\(722\) −15.2306 −0.566824
\(723\) 45.5381 1.69358
\(724\) −28.4547 −1.05751
\(725\) −22.8726 −0.849468
\(726\) −12.6799 −0.470597
\(727\) 13.4862 0.500175 0.250088 0.968223i \(-0.419541\pi\)
0.250088 + 0.968223i \(0.419541\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) 1.33406 0.0493758
\(731\) −0.638792 −0.0236266
\(732\) 10.5864 0.391285
\(733\) −43.5424 −1.60828 −0.804138 0.594443i \(-0.797373\pi\)
−0.804138 + 0.594443i \(0.797373\pi\)
\(734\) 0.869167 0.0320816
\(735\) 0.801938 0.0295799
\(736\) 8.70841 0.320996
\(737\) −32.6165 −1.20145
\(738\) −19.0465 −0.701112
\(739\) 20.0543 0.737709 0.368855 0.929487i \(-0.379750\pi\)
0.368855 + 0.929487i \(0.379750\pi\)
\(740\) −1.90754 −0.0701226
\(741\) 0 0
\(742\) 19.6450 0.721191
\(743\) −33.1685 −1.21684 −0.608418 0.793617i \(-0.708195\pi\)
−0.608418 + 0.793617i \(0.708195\pi\)
\(744\) −40.1105 −1.47052
\(745\) 1.07069 0.0392270
\(746\) −4.91617 −0.179994
\(747\) 3.29722 0.120639
\(748\) 12.4407 0.454878
\(749\) −7.70410 −0.281502
\(750\) 4.42327 0.161515
\(751\) 39.2814 1.43340 0.716700 0.697382i \(-0.245652\pi\)
0.716700 + 0.697382i \(0.245652\pi\)
\(752\) −4.08277 −0.148883
\(753\) 53.3008 1.94239
\(754\) 0 0
\(755\) 0.974181 0.0354541
\(756\) 6.83446 0.248567
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) −1.93171 −0.0701627
\(759\) 14.2567 0.517484
\(760\) 0.0586060 0.00212586
\(761\) −21.8984 −0.793818 −0.396909 0.917858i \(-0.629917\pi\)
−0.396909 + 0.917858i \(0.629917\pi\)
\(762\) 17.6722 0.640195
\(763\) −37.0073 −1.33975
\(764\) 19.5899 0.708737
\(765\) 1.09246 0.0394979
\(766\) −24.3720 −0.880595
\(767\) 0 0
\(768\) 31.8756 1.15021
\(769\) 46.7096 1.68439 0.842196 0.539172i \(-0.181263\pi\)
0.842196 + 0.539172i \(0.181263\pi\)
\(770\) 1.98254 0.0714458
\(771\) −31.9614 −1.15106
\(772\) 18.4263 0.663175
\(773\) −30.2416 −1.08771 −0.543857 0.839178i \(-0.683037\pi\)
−0.543857 + 0.839178i \(0.683037\pi\)
\(774\) −0.486189 −0.0174757
\(775\) 32.7506 1.17644
\(776\) 21.6939 0.778767
\(777\) 30.1444 1.08142
\(778\) −12.7836 −0.458315
\(779\) 1.02177 0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 2.58642 0.0924901
\(783\) 9.89679 0.353682
\(784\) −0.801938 −0.0286406
\(785\) 1.10023 0.0392688
\(786\) 11.8509 0.422706
\(787\) 28.7023 1.02313 0.511563 0.859246i \(-0.329067\pi\)
0.511563 + 0.859246i \(0.329067\pi\)
\(788\) 0.760316 0.0270851
\(789\) 38.4131 1.36754
\(790\) 1.97477 0.0702592
\(791\) −28.3980 −1.00972
\(792\) 23.4252 0.832378
\(793\) 0 0
\(794\) 13.5646 0.481391
\(795\) 5.76809 0.204573
\(796\) −15.5929 −0.552674
\(797\) −18.5418 −0.656785 −0.328392 0.944541i \(-0.606507\pi\)
−0.328392 + 0.944541i \(0.606507\pi\)
\(798\) −0.374354 −0.0132520
\(799\) −15.8823 −0.561876
\(800\) −28.7899 −1.01788
\(801\) −5.91053 −0.208838
\(802\) 21.3817 0.755012
\(803\) −28.6058 −1.00948
\(804\) 23.4155 0.825801
\(805\) −0.869641 −0.0306508
\(806\) 0 0
\(807\) 14.5308 0.511508
\(808\) 35.9506 1.26474
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) −2.16852 −0.0761941
\(811\) −10.0285 −0.352147 −0.176074 0.984377i \(-0.556340\pi\)
−0.176074 + 0.984377i \(0.556340\pi\)
\(812\) 14.8103 0.519740
\(813\) −14.4886 −0.508137
\(814\) −19.3860 −0.679478
\(815\) 3.99090 0.139795
\(816\) −2.69202 −0.0942396
\(817\) 0.0260821 0.000912498 0
\(818\) 22.8683 0.799572
\(819\) 0 0
\(820\) 3.88471 0.135660
\(821\) −26.1704 −0.913355 −0.456677 0.889632i \(-0.650961\pi\)
−0.456677 + 0.889632i \(0.650961\pi\)
\(822\) −11.2078 −0.390915
\(823\) 1.82238 0.0635242 0.0317621 0.999495i \(-0.489888\pi\)
0.0317621 + 0.999495i \(0.489888\pi\)
\(824\) −3.66727 −0.127755
\(825\) −47.1323 −1.64094
\(826\) 12.8151 0.445894
\(827\) −32.2941 −1.12298 −0.561488 0.827485i \(-0.689771\pi\)
−0.561488 + 0.827485i \(0.689771\pi\)
\(828\) −4.15346 −0.144343
\(829\) 15.1002 0.524453 0.262226 0.965006i \(-0.415543\pi\)
0.262226 + 0.965006i \(0.415543\pi\)
\(830\) 0.318732 0.0110634
\(831\) −30.2446 −1.04917
\(832\) 0 0
\(833\) −3.11960 −0.108088
\(834\) 26.5013 0.917663
\(835\) 3.98062 0.137755
\(836\) −0.507960 −0.0175682
\(837\) −14.1709 −0.489818
\(838\) −23.7448 −0.820250
\(839\) −32.9965 −1.13917 −0.569584 0.821933i \(-0.692896\pi\)
−0.569584 + 0.821933i \(0.692896\pi\)
\(840\) −3.52111 −0.121490
\(841\) −7.55363 −0.260470
\(842\) −9.35105 −0.322258
\(843\) −11.3177 −0.389801
\(844\) −11.9196 −0.410290
\(845\) 0 0
\(846\) −12.0881 −0.415599
\(847\) −16.5851 −0.569870
\(848\) −5.76809 −0.198077
\(849\) −49.7211 −1.70642
\(850\) −8.55065 −0.293285
\(851\) 8.50365 0.291501
\(852\) −26.4185 −0.905082
\(853\) −37.7802 −1.29357 −0.646784 0.762673i \(-0.723887\pi\)
−0.646784 + 0.762673i \(0.723887\pi\)
\(854\) −6.56273 −0.224572
\(855\) −0.0446055 −0.00152547
\(856\) −8.79954 −0.300762
\(857\) 27.3623 0.934677 0.467339 0.884078i \(-0.345213\pi\)
0.467339 + 0.884078i \(0.345213\pi\)
\(858\) 0 0
\(859\) −20.0629 −0.684538 −0.342269 0.939602i \(-0.611195\pi\)
−0.342269 + 0.939602i \(0.611195\pi\)
\(860\) 0.0991626 0.00338142
\(861\) −61.3889 −2.09213
\(862\) 3.48773 0.118792
\(863\) −6.14483 −0.209173 −0.104586 0.994516i \(-0.533352\pi\)
−0.104586 + 0.994516i \(0.533352\pi\)
\(864\) 12.4571 0.423800
\(865\) −5.31900 −0.180851
\(866\) −11.5386 −0.392096
\(867\) 27.7265 0.941640
\(868\) −21.2064 −0.719793
\(869\) −42.3443 −1.43643
\(870\) −2.06100 −0.0698744
\(871\) 0 0
\(872\) −42.2693 −1.43142
\(873\) −16.5114 −0.558827
\(874\) −0.105604 −0.00357212
\(875\) 5.78554 0.195587
\(876\) 20.5362 0.693853
\(877\) −13.5077 −0.456123 −0.228061 0.973647i \(-0.573239\pi\)
−0.228061 + 0.973647i \(0.573239\pi\)
\(878\) −16.2252 −0.547574
\(879\) −33.5840 −1.13276
\(880\) −0.582105 −0.0196228
\(881\) −5.23431 −0.176348 −0.0881741 0.996105i \(-0.528103\pi\)
−0.0881741 + 0.996105i \(0.528103\pi\)
\(882\) −2.37435 −0.0799487
\(883\) −4.57301 −0.153894 −0.0769470 0.997035i \(-0.524517\pi\)
−0.0769470 + 0.997035i \(0.524517\pi\)
\(884\) 0 0
\(885\) 3.76271 0.126482
\(886\) 6.51334 0.218820
\(887\) −1.64071 −0.0550897 −0.0275448 0.999621i \(-0.508769\pi\)
−0.0275448 + 0.999621i \(0.508769\pi\)
\(888\) 34.4306 1.15541
\(889\) 23.1148 0.775246
\(890\) −0.571352 −0.0191517
\(891\) 46.4989 1.55777
\(892\) 3.06531 0.102634
\(893\) 0.648481 0.0217006
\(894\) −7.81163 −0.261260
\(895\) 2.82371 0.0943861
\(896\) 20.7396 0.692862
\(897\) 0 0
\(898\) 10.0175 0.334287
\(899\) −30.7084 −1.02418
\(900\) 13.7313 0.457708
\(901\) −22.4383 −0.747529
\(902\) 39.4795 1.31452
\(903\) −1.56704 −0.0521478
\(904\) −32.4359 −1.07880
\(905\) 5.17928 0.172165
\(906\) −7.10752 −0.236132
\(907\) 8.10215 0.269027 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(908\) 9.45234 0.313687
\(909\) −27.3623 −0.907549
\(910\) 0 0
\(911\) −9.18119 −0.304187 −0.152093 0.988366i \(-0.548601\pi\)
−0.152093 + 0.988366i \(0.548601\pi\)
\(912\) 0.109916 0.00363969
\(913\) −6.83446 −0.226188
\(914\) 4.79656 0.158656
\(915\) −1.92692 −0.0637020
\(916\) 32.8015 1.08379
\(917\) 15.5007 0.511877
\(918\) 3.69979 0.122111
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) −0.993295 −0.0327480
\(921\) −42.9831 −1.41634
\(922\) −1.64933 −0.0543180
\(923\) 0 0
\(924\) 30.5187 1.00399
\(925\) −28.1129 −0.924346
\(926\) −6.77586 −0.222668
\(927\) 2.79118 0.0916745
\(928\) 26.9946 0.886142
\(929\) 24.2131 0.794407 0.397203 0.917731i \(-0.369981\pi\)
0.397203 + 0.917731i \(0.369981\pi\)
\(930\) 2.95108 0.0967698
\(931\) 0.127375 0.00417454
\(932\) 4.15346 0.136051
\(933\) 0.606268 0.0198483
\(934\) 26.8761 0.879412
\(935\) −2.26444 −0.0740550
\(936\) 0 0
\(937\) 11.1830 0.365333 0.182666 0.983175i \(-0.441527\pi\)
0.182666 + 0.983175i \(0.441527\pi\)
\(938\) −14.5157 −0.473955
\(939\) 52.5652 1.71540
\(940\) 2.46548 0.0804152
\(941\) −15.9638 −0.520404 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(942\) −8.02715 −0.261539
\(943\) −17.3177 −0.563941
\(944\) −3.76271 −0.122466
\(945\) −1.24400 −0.0404672
\(946\) 1.00777 0.0327654
\(947\) 6.51466 0.211698 0.105849 0.994382i \(-0.466244\pi\)
0.105849 + 0.994382i \(0.466244\pi\)
\(948\) 30.3991 0.987317
\(949\) 0 0
\(950\) 0.349126 0.0113271
\(951\) 31.4470 1.01974
\(952\) 13.6974 0.443935
\(953\) 47.6469 1.54344 0.771718 0.635965i \(-0.219398\pi\)
0.771718 + 0.635965i \(0.219398\pi\)
\(954\) −17.0780 −0.552920
\(955\) −3.56571 −0.115384
\(956\) 34.1215 1.10357
\(957\) 44.1933 1.42857
\(958\) −19.8329 −0.640773
\(959\) −14.6595 −0.473380
\(960\) −1.97823 −0.0638471
\(961\) 12.9705 0.418402
\(962\) 0 0
\(963\) 6.69740 0.215821
\(964\) 27.4993 0.885694
\(965\) −3.35391 −0.107966
\(966\) 6.34481 0.204141
\(967\) −43.8122 −1.40891 −0.704453 0.709751i \(-0.748808\pi\)
−0.704453 + 0.709751i \(0.748808\pi\)
\(968\) −18.9433 −0.608861
\(969\) 0.427583 0.0137360
\(970\) −1.59611 −0.0512479
\(971\) 4.29483 0.137828 0.0689139 0.997623i \(-0.478047\pi\)
0.0689139 + 0.997623i \(0.478047\pi\)
\(972\) −24.6823 −0.791686
\(973\) 34.6631 1.11125
\(974\) −30.2776 −0.970156
\(975\) 0 0
\(976\) 1.92692 0.0616792
\(977\) 26.8019 0.857470 0.428735 0.903430i \(-0.358959\pi\)
0.428735 + 0.903430i \(0.358959\pi\)
\(978\) −29.1172 −0.931066
\(979\) 12.2513 0.391553
\(980\) 0.484271 0.0154695
\(981\) 32.1715 1.02716
\(982\) 25.1094 0.801275
\(983\) 27.2495 0.869124 0.434562 0.900642i \(-0.356903\pi\)
0.434562 + 0.900642i \(0.356903\pi\)
\(984\) −70.1178 −2.23527
\(985\) −0.138391 −0.00440951
\(986\) 8.01746 0.255328
\(987\) −38.9614 −1.24015
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) −1.72348 −0.0547758
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) −38.6528 −1.22723
\(993\) −40.0441 −1.27076
\(994\) 16.3773 0.519458
\(995\) 2.83818 0.0899764
\(996\) 4.90648 0.155468
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) 17.2314 0.545452
\(999\) 12.1642 0.384859
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 169.2.a.b.1.3 3
3.2 odd 2 1521.2.a.r.1.1 3
4.3 odd 2 2704.2.a.z.1.3 3
5.4 even 2 4225.2.a.bg.1.1 3
7.6 odd 2 8281.2.a.bf.1.3 3
13.2 odd 12 169.2.e.b.147.5 12
13.3 even 3 169.2.c.c.22.1 6
13.4 even 6 169.2.c.b.146.3 6
13.5 odd 4 169.2.b.b.168.2 6
13.6 odd 12 169.2.e.b.23.2 12
13.7 odd 12 169.2.e.b.23.5 12
13.8 odd 4 169.2.b.b.168.5 6
13.9 even 3 169.2.c.c.146.1 6
13.10 even 6 169.2.c.b.22.3 6
13.11 odd 12 169.2.e.b.147.2 12
13.12 even 2 169.2.a.c.1.1 yes 3
39.5 even 4 1521.2.b.l.1351.5 6
39.8 even 4 1521.2.b.l.1351.2 6
39.38 odd 2 1521.2.a.o.1.3 3
52.31 even 4 2704.2.f.o.337.5 6
52.47 even 4 2704.2.f.o.337.6 6
52.51 odd 2 2704.2.a.ba.1.3 3
65.64 even 2 4225.2.a.bb.1.3 3
91.90 odd 2 8281.2.a.bj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 1.1 even 1 trivial
169.2.a.c.1.1 yes 3 13.12 even 2
169.2.b.b.168.2 6 13.5 odd 4
169.2.b.b.168.5 6 13.8 odd 4
169.2.c.b.22.3 6 13.10 even 6
169.2.c.b.146.3 6 13.4 even 6
169.2.c.c.22.1 6 13.3 even 3
169.2.c.c.146.1 6 13.9 even 3
169.2.e.b.23.2 12 13.6 odd 12
169.2.e.b.23.5 12 13.7 odd 12
169.2.e.b.147.2 12 13.11 odd 12
169.2.e.b.147.5 12 13.2 odd 12
1521.2.a.o.1.3 3 39.38 odd 2
1521.2.a.r.1.1 3 3.2 odd 2
1521.2.b.l.1351.2 6 39.8 even 4
1521.2.b.l.1351.5 6 39.5 even 4
2704.2.a.z.1.3 3 4.3 odd 2
2704.2.a.ba.1.3 3 52.51 odd 2
2704.2.f.o.337.5 6 52.31 even 4
2704.2.f.o.337.6 6 52.47 even 4
4225.2.a.bb.1.3 3 65.64 even 2
4225.2.a.bg.1.1 3 5.4 even 2
8281.2.a.bf.1.3 3 7.6 odd 2
8281.2.a.bj.1.1 3 91.90 odd 2