Properties

Label 1521.2.a.o.1.3
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
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] = [1521,2,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(12.1452461474\)
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: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.801938 q^{2} -1.35690 q^{4} +0.246980 q^{5} +2.35690 q^{7} -2.69202 q^{8} +O(q^{10})\) \(q+0.801938 q^{2} -1.35690 q^{4} +0.246980 q^{5} +2.35690 q^{7} -2.69202 q^{8} +0.198062 q^{10} -4.24698 q^{11} +1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} +0.0881460 q^{19} -0.335126 q^{20} -3.40581 q^{22} -1.49396 q^{23} -4.93900 q^{25} -3.19806 q^{28} -4.63102 q^{29} +6.63102 q^{31} +5.82908 q^{32} -1.73125 q^{34} +0.582105 q^{35} -5.69202 q^{37} +0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} -0.295897 q^{43} +5.76271 q^{44} -1.19806 q^{46} -7.35690 q^{47} -1.44504 q^{49} -3.96077 q^{50} +10.3937 q^{53} -1.04892 q^{55} -6.34481 q^{56} -3.71379 q^{58} -6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} +3.56465 q^{64} -7.67994 q^{67} +2.92931 q^{68} +0.466812 q^{70} -8.66487 q^{71} -6.73556 q^{73} -4.56465 q^{74} -0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -9.29590 q^{82} +1.60925 q^{83} -0.533188 q^{85} -0.237291 q^{86} +11.4330 q^{88} -2.88471 q^{89} +2.02715 q^{92} -5.89977 q^{94} +0.0217703 q^{95} +8.05861 q^{97} -1.15883 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35} - 12 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 13 q^{43} - 8 q^{46} - 18 q^{47} - 4 q^{49} + q^{50} - q^{53} + 6 q^{55} + 4 q^{56} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} + q^{67} + 21 q^{68} - 2 q^{70} - 27 q^{71} - 9 q^{73} + 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - 5 q^{80} - 14 q^{82} - 7 q^{83} - 5 q^{85} - 18 q^{86} + 15 q^{88} - 11 q^{89} + 5 q^{94} - 3 q^{95} - 7 q^{97} + 5 q^{98}+O(q^{100}) \) 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 \(0.408495\pi\)
\(3\) 0 0
\(4\) −1.35690 −0.678448
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) 0 0
\(7\) 2.35690 0.890823 0.445411 0.895326i \(-0.353057\pi\)
0.445411 + 0.895326i \(0.353057\pi\)
\(8\) −2.69202 −0.951773
\(9\) 0 0
\(10\) 0.198062 0.0626328
\(11\) −4.24698 −1.28051 −0.640256 0.768161i \(-0.721172\pi\)
−0.640256 + 0.768161i \(0.721172\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.89008 0.505146
\(15\) 0 0
\(16\) 0.554958 0.138740
\(17\) −2.15883 −0.523594 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) 0 0
\(19\) 0.0881460 0.0202221 0.0101110 0.999949i \(-0.496782\pi\)
0.0101110 + 0.999949i \(0.496782\pi\)
\(20\) −0.335126 −0.0749364
\(21\) 0 0
\(22\) −3.40581 −0.726122
\(23\) −1.49396 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(24\) 0 0
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) 0 0
\(28\) −3.19806 −0.604377
\(29\) −4.63102 −0.859959 −0.429980 0.902839i \(-0.641479\pi\)
−0.429980 + 0.902839i \(0.641479\pi\)
\(30\) 0 0
\(31\) 6.63102 1.19097 0.595483 0.803368i \(-0.296961\pi\)
0.595483 + 0.803368i \(0.296961\pi\)
\(32\) 5.82908 1.03045
\(33\) 0 0
\(34\) −1.73125 −0.296907
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) −5.69202 −0.935763 −0.467881 0.883791i \(-0.654983\pi\)
−0.467881 + 0.883791i \(0.654983\pi\)
\(38\) 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 \(0.860254\pi\)
\(42\) 0 0
\(43\) −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 0
\(46\) −1.19806 −0.176645
\(47\) −7.35690 −1.07311 −0.536557 0.843864i \(-0.680275\pi\)
−0.536557 + 0.843864i \(0.680275\pi\)
\(48\) 0 0
\(49\) −1.44504 −0.206435
\(50\) −3.96077 −0.560138
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3937 1.42769 0.713844 0.700304i \(-0.246952\pi\)
0.713844 + 0.700304i \(0.246952\pi\)
\(54\) 0 0
\(55\) −1.04892 −0.141436
\(56\) −6.34481 −0.847861
\(57\) 0 0
\(58\) −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 0
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 5.31767 0.675344
\(63\) 0 0
\(64\) 3.56465 0.445581
\(65\) 0 0
\(66\) 0 0
\(67\) −7.67994 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(68\) 2.92931 0.355231
\(69\) 0 0
\(70\) 0.466812 0.0557947
\(71\) −8.66487 −1.02833 −0.514166 0.857691i \(-0.671898\pi\)
−0.514166 + 0.857691i \(0.671898\pi\)
\(72\) 0 0
\(73\) −6.73556 −0.788338 −0.394169 0.919038i \(-0.628968\pi\)
−0.394169 + 0.919038i \(0.628968\pi\)
\(74\) −4.56465 −0.530629
\(75\) 0 0
\(76\) −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
\(81\) 0 0
\(82\) −9.29590 −1.02656
\(83\) 1.60925 0.176638 0.0883192 0.996092i \(-0.471850\pi\)
0.0883192 + 0.996092i \(0.471850\pi\)
\(84\) 0 0
\(85\) −0.533188 −0.0578323
\(86\) −0.237291 −0.0255877
\(87\) 0 0
\(88\) 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 0
\(91\) 0 0
\(92\) 2.02715 0.211345
\(93\) 0 0
\(94\) −5.89977 −0.608515
\(95\) 0.0217703 0.00223358
\(96\) 0 0
\(97\) 8.05861 0.818227 0.409114 0.912483i \(-0.365838\pi\)
0.409114 + 0.912483i \(0.365838\pi\)
\(98\) −1.15883 −0.117060
\(99\) 0 0
\(100\) 6.70171 0.670171
\(101\) 13.3545 1.32882 0.664411 0.747367i \(-0.268682\pi\)
0.664411 + 0.747367i \(0.268682\pi\)
\(102\) 0 0
\(103\) 1.36227 0.134229 0.0671144 0.997745i \(-0.478621\pi\)
0.0671144 + 0.997745i \(0.478621\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.33513 0.809579
\(107\) −3.26875 −0.316002 −0.158001 0.987439i \(-0.550505\pi\)
−0.158001 + 0.987439i \(0.550505\pi\)
\(108\) 0 0
\(109\) −15.7017 −1.50395 −0.751976 0.659191i \(-0.770899\pi\)
−0.751976 + 0.659191i \(0.770899\pi\)
\(110\) −0.841166 −0.0802021
\(111\) 0 0
\(112\) 1.30798 0.123592
\(113\) −12.0489 −1.13347 −0.566733 0.823901i \(-0.691793\pi\)
−0.566733 + 0.823901i \(0.691793\pi\)
\(114\) 0 0
\(115\) −0.368977 −0.0344073
\(116\) 6.28382 0.583438
\(117\) 0 0
\(118\) −5.43727 −0.500541
\(119\) −5.08815 −0.466430
\(120\) 0 0
\(121\) 7.03684 0.639712
\(122\) 2.78448 0.252095
\(123\) 0 0
\(124\) −8.99761 −0.808009
\(125\) −2.45473 −0.219558
\(126\) 0 0
\(127\) −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 0
\(130\) 0 0
\(131\) 6.57673 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(132\) 0 0
\(133\) 0.207751 0.0180143
\(134\) −6.15883 −0.532042
\(135\) 0 0
\(136\) 5.81163 0.498343
\(137\) 6.21983 0.531396 0.265698 0.964056i \(-0.414398\pi\)
0.265698 + 0.964056i \(0.414398\pi\)
\(138\) 0 0
\(139\) −14.7071 −1.24744 −0.623719 0.781648i \(-0.714379\pi\)
−0.623719 + 0.781648i \(0.714379\pi\)
\(140\) −0.789856 −0.0667550
\(141\) 0 0
\(142\) −6.94869 −0.583121
\(143\) 0 0
\(144\) 0 0
\(145\) −1.14377 −0.0949848
\(146\) −5.40150 −0.447031
\(147\) 0 0
\(148\) 7.72348 0.634866
\(149\) 4.33513 0.355147 0.177574 0.984108i \(-0.443175\pi\)
0.177574 + 0.984108i \(0.443175\pi\)
\(150\) 0 0
\(151\) −3.94438 −0.320989 −0.160494 0.987037i \(-0.551309\pi\)
−0.160494 + 0.987037i \(0.551309\pi\)
\(152\) −0.237291 −0.0192468
\(153\) 0 0
\(154\) −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
\(159\) 0 0
\(160\) 1.43967 0.113816
\(161\) −3.52111 −0.277502
\(162\) 0 0
\(163\) −16.1588 −1.26566 −0.632829 0.774292i \(-0.718106\pi\)
−0.632829 + 0.774292i \(0.718106\pi\)
\(164\) 15.7289 1.22822
\(165\) 0 0
\(166\) 1.29052 0.100164
\(167\) 16.1172 1.24719 0.623594 0.781749i \(-0.285672\pi\)
0.623594 + 0.781749i \(0.285672\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.427583 −0.0327942
\(171\) 0 0
\(172\) 0.401501 0.0306142
\(173\) 21.5362 1.63736 0.818682 0.574247i \(-0.194705\pi\)
0.818682 + 0.574247i \(0.194705\pi\)
\(174\) 0 0
\(175\) −11.6407 −0.879955
\(176\) −2.35690 −0.177658
\(177\) 0 0
\(178\) −2.31336 −0.173393
\(179\) −11.4330 −0.854540 −0.427270 0.904124i \(-0.640525\pi\)
−0.427270 + 0.904124i \(0.640525\pi\)
\(180\) 0 0
\(181\) 20.9705 1.55872 0.779361 0.626575i \(-0.215544\pi\)
0.779361 + 0.626575i \(0.215544\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.02177 0.296489
\(185\) −1.40581 −0.103357
\(186\) 0 0
\(187\) 9.16852 0.670469
\(188\) 9.98254 0.728052
\(189\) 0 0
\(190\) 0.0174584 0.00126657
\(191\) 14.4373 1.04464 0.522322 0.852748i \(-0.325066\pi\)
0.522322 + 0.852748i \(0.325066\pi\)
\(192\) 0 0
\(193\) 13.5797 0.977489 0.488745 0.872427i \(-0.337455\pi\)
0.488745 + 0.872427i \(0.337455\pi\)
\(194\) 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 \(0.506354\pi\)
\(198\) 0 0
\(199\) 11.4916 0.814616 0.407308 0.913291i \(-0.366468\pi\)
0.407308 + 0.913291i \(0.366468\pi\)
\(200\) 13.2959 0.940162
\(201\) 0 0
\(202\) 10.7095 0.753516
\(203\) −10.9148 −0.766071
\(204\) 0 0
\(205\) −2.86294 −0.199956
\(206\) 1.09246 0.0761151
\(207\) 0 0
\(208\) 0 0
\(209\) −0.374354 −0.0258946
\(210\) 0 0
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) −14.1032 −0.968613
\(213\) 0 0
\(214\) −2.62133 −0.179191
\(215\) −0.0730805 −0.00498405
\(216\) 0 0
\(217\) 15.6286 1.06094
\(218\) −12.5918 −0.852824
\(219\) 0 0
\(220\) 1.42327 0.0959570
\(221\) 0 0
\(222\) 0 0
\(223\) 2.25906 0.151278 0.0756390 0.997135i \(-0.475900\pi\)
0.0756390 + 0.997135i \(0.475900\pi\)
\(224\) 13.7385 0.917945
\(225\) 0 0
\(226\) −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 0
\(229\) 24.1739 1.59746 0.798728 0.601692i \(-0.205507\pi\)
0.798728 + 0.601692i \(0.205507\pi\)
\(230\) −0.295897 −0.0195109
\(231\) 0 0
\(232\) 12.4668 0.818486
\(233\) 3.06100 0.200533 0.100266 0.994961i \(-0.468031\pi\)
0.100266 + 0.994961i \(0.468031\pi\)
\(234\) 0 0
\(235\) −1.81700 −0.118528
\(236\) 9.19998 0.598868
\(237\) 0 0
\(238\) −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 0
\(241\) 20.2664 1.30547 0.652735 0.757586i \(-0.273621\pi\)
0.652735 + 0.757586i \(0.273621\pi\)
\(242\) 5.64310 0.362752
\(243\) 0 0
\(244\) −4.71140 −0.301616
\(245\) −0.356896 −0.0228012
\(246\) 0 0
\(247\) 0 0
\(248\) −17.8509 −1.13353
\(249\) 0 0
\(250\) −1.96854 −0.124501
\(251\) 23.7211 1.49726 0.748631 0.662987i \(-0.230712\pi\)
0.748631 + 0.662987i \(0.230712\pi\)
\(252\) 0 0
\(253\) 6.34481 0.398895
\(254\) −7.86486 −0.493485
\(255\) 0 0
\(256\) −14.1860 −0.886624
\(257\) −14.2241 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(258\) 0 0
\(259\) −13.4155 −0.833599
\(260\) 0 0
\(261\) 0 0
\(262\) 5.27413 0.325837
\(263\) 17.0954 1.05415 0.527075 0.849819i \(-0.323289\pi\)
0.527075 + 0.849819i \(0.323289\pi\)
\(264\) 0 0
\(265\) 2.56704 0.157692
\(266\) 0.166603 0.0102151
\(267\) 0 0
\(268\) 10.4209 0.636556
\(269\) 6.46681 0.394288 0.197144 0.980374i \(-0.436833\pi\)
0.197144 + 0.980374i \(0.436833\pi\)
\(270\) 0 0
\(271\) −6.44803 −0.391690 −0.195845 0.980635i \(-0.562745\pi\)
−0.195845 + 0.980635i \(0.562745\pi\)
\(272\) −1.19806 −0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) 20.9758 1.26489
\(276\) 0 0
\(277\) 13.4601 0.808739 0.404370 0.914596i \(-0.367491\pi\)
0.404370 + 0.914596i \(0.367491\pi\)
\(278\) −11.7942 −0.707367
\(279\) 0 0
\(280\) −1.56704 −0.0936485
\(281\) 5.03684 0.300472 0.150236 0.988650i \(-0.451997\pi\)
0.150236 + 0.988650i \(0.451997\pi\)
\(282\) 0 0
\(283\) 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 0
\(286\) 0 0
\(287\) −27.3207 −1.61269
\(288\) 0 0
\(289\) −12.3394 −0.725849
\(290\) −0.917231 −0.0538616
\(291\) 0 0
\(292\) 9.13946 0.534846
\(293\) 14.9463 0.873172 0.436586 0.899663i \(-0.356187\pi\)
0.436586 + 0.899663i \(0.356187\pi\)
\(294\) 0 0
\(295\) −1.67456 −0.0974968
\(296\) 15.3230 0.890634
\(297\) 0 0
\(298\) 3.47650 0.201388
\(299\) 0 0
\(300\) 0 0
\(301\) −0.697398 −0.0401974
\(302\) −3.16315 −0.182019
\(303\) 0 0
\(304\) 0.0489173 0.00280560
\(305\) 0.857560 0.0491037
\(306\) 0 0
\(307\) −19.1293 −1.09177 −0.545883 0.837861i \(-0.683806\pi\)
−0.545883 + 0.837861i \(0.683806\pi\)
\(308\) 13.5821 0.773912
\(309\) 0 0
\(310\) 1.31336 0.0745936
\(311\) 0.269815 0.0152998 0.00764990 0.999971i \(-0.497565\pi\)
0.00764990 + 0.999971i \(0.497565\pi\)
\(312\) 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
\(315\) 0 0
\(316\) −13.5289 −0.761059
\(317\) −13.9952 −0.786050 −0.393025 0.919528i \(-0.628571\pi\)
−0.393025 + 0.919528i \(0.628571\pi\)
\(318\) 0 0
\(319\) 19.6679 1.10119
\(320\) 0.880395 0.0492156
\(321\) 0 0
\(322\) −2.82371 −0.157359
\(323\) −0.190293 −0.0105882
\(324\) 0 0
\(325\) 0 0
\(326\) −12.9584 −0.717698
\(327\) 0 0
\(328\) 31.2054 1.72303
\(329\) −17.3394 −0.955954
\(330\) 0 0
\(331\) −17.8213 −0.979548 −0.489774 0.871849i \(-0.662921\pi\)
−0.489774 + 0.871849i \(0.662921\pi\)
\(332\) −2.18359 −0.119840
\(333\) 0 0
\(334\) 12.9250 0.707225
\(335\) −1.89679 −0.103633
\(336\) 0 0
\(337\) −27.8485 −1.51700 −0.758501 0.651672i \(-0.774068\pi\)
−0.758501 + 0.651672i \(0.774068\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0.723480 0.0392362
\(341\) −28.1618 −1.52505
\(342\) 0 0
\(343\) −19.9041 −1.07472
\(344\) 0.796561 0.0429477
\(345\) 0 0
\(346\) 17.2707 0.928477
\(347\) −1.50365 −0.0807200 −0.0403600 0.999185i \(-0.512850\pi\)
−0.0403600 + 0.999185i \(0.512850\pi\)
\(348\) 0 0
\(349\) 14.1860 0.759358 0.379679 0.925118i \(-0.376034\pi\)
0.379679 + 0.925118i \(0.376034\pi\)
\(350\) −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 \(0.561099\pi\)
\(354\) 0 0
\(355\) −2.14005 −0.113582
\(356\) 3.91425 0.207455
\(357\) 0 0
\(358\) −9.16852 −0.484571
\(359\) 19.8853 1.04951 0.524753 0.851255i \(-0.324158\pi\)
0.524753 + 0.851255i \(0.324158\pi\)
\(360\) 0 0
\(361\) −18.9922 −0.999591
\(362\) 16.8170 0.883882
\(363\) 0 0
\(364\) 0 0
\(365\) −1.66355 −0.0870740
\(366\) 0 0
\(367\) 1.08383 0.0565757 0.0282878 0.999600i \(-0.490994\pi\)
0.0282878 + 0.999600i \(0.490994\pi\)
\(368\) −0.829085 −0.0432190
\(369\) 0 0
\(370\) −1.12737 −0.0586094
\(371\) 24.4969 1.27182
\(372\) 0 0
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) 7.35258 0.380193
\(375\) 0 0
\(376\) 19.8049 1.02136
\(377\) 0 0
\(378\) 0 0
\(379\) 2.40880 0.123732 0.0618658 0.998084i \(-0.480295\pi\)
0.0618658 + 0.998084i \(0.480295\pi\)
\(380\) −0.0295400 −0.00151537
\(381\) 0 0
\(382\) 11.5778 0.592371
\(383\) −30.3913 −1.55292 −0.776462 0.630164i \(-0.782988\pi\)
−0.776462 + 0.630164i \(0.782988\pi\)
\(384\) 0 0
\(385\) −2.47219 −0.125994
\(386\) 10.8901 0.554291
\(387\) 0 0
\(388\) −10.9347 −0.555125
\(389\) 15.9409 0.808237 0.404118 0.914707i \(-0.367578\pi\)
0.404118 + 0.914707i \(0.367578\pi\)
\(390\) 0 0
\(391\) 3.22521 0.163106
\(392\) 3.89008 0.196479
\(393\) 0 0
\(394\) −0.449354 −0.0226381
\(395\) 2.46250 0.123902
\(396\) 0 0
\(397\) −16.9148 −0.848931 −0.424466 0.905444i \(-0.639538\pi\)
−0.424466 + 0.905444i \(0.639538\pi\)
\(398\) 9.21552 0.461932
\(399\) 0 0
\(400\) −2.74094 −0.137047
\(401\) 26.6625 1.33146 0.665730 0.746192i \(-0.268120\pi\)
0.665730 + 0.746192i \(0.268120\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −18.1207 −0.901537
\(405\) 0 0
\(406\) −8.75302 −0.434405
\(407\) 24.1739 1.19826
\(408\) 0 0
\(409\) −28.5163 −1.41004 −0.705021 0.709187i \(-0.749062\pi\)
−0.705021 + 0.709187i \(0.749062\pi\)
\(410\) −2.29590 −0.113386
\(411\) 0 0
\(412\) −1.84846 −0.0910672
\(413\) −15.9801 −0.786332
\(414\) 0 0
\(415\) 0.397452 0.0195102
\(416\) 0 0
\(417\) 0 0
\(418\) −0.300209 −0.0146837
\(419\) 29.6093 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(420\) 0 0
\(421\) 11.6606 0.568301 0.284151 0.958780i \(-0.408288\pi\)
0.284151 + 0.958780i \(0.408288\pi\)
\(422\) 7.04461 0.342926
\(423\) 0 0
\(424\) −27.9801 −1.35884
\(425\) 10.6625 0.517206
\(426\) 0 0
\(427\) 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 \(0.466597\pi\)
\(432\) 0 0
\(433\) −14.3884 −0.691460 −0.345730 0.938334i \(-0.612369\pi\)
−0.345730 + 0.938334i \(0.612369\pi\)
\(434\) 12.5332 0.601612
\(435\) 0 0
\(436\) 21.3056 1.02035
\(437\) −0.131687 −0.00629942
\(438\) 0 0
\(439\) −20.2325 −0.965645 −0.482822 0.875718i \(-0.660388\pi\)
−0.482822 + 0.875718i \(0.660388\pi\)
\(440\) 2.82371 0.134615
\(441\) 0 0
\(442\) 0 0
\(443\) −8.12200 −0.385888 −0.192944 0.981210i \(-0.561804\pi\)
−0.192944 + 0.981210i \(0.561804\pi\)
\(444\) 0 0
\(445\) −0.712464 −0.0337740
\(446\) 1.81163 0.0857830
\(447\) 0 0
\(448\) 8.40150 0.396934
\(449\) 12.4916 0.589513 0.294757 0.955572i \(-0.404761\pi\)
0.294757 + 0.955572i \(0.404761\pi\)
\(450\) 0 0
\(451\) 49.2301 2.31816
\(452\) 16.3491 0.768998
\(453\) 0 0
\(454\) −5.58642 −0.262184
\(455\) 0 0
\(456\) 0 0
\(457\) −5.98121 −0.279789 −0.139895 0.990166i \(-0.544676\pi\)
−0.139895 + 0.990166i \(0.544676\pi\)
\(458\) 19.3860 0.905847
\(459\) 0 0
\(460\) 0.500664 0.0233436
\(461\) −2.05669 −0.0957895 −0.0478947 0.998852i \(-0.515251\pi\)
−0.0478947 + 0.998852i \(0.515251\pi\)
\(462\) 0 0
\(463\) 8.44935 0.392675 0.196337 0.980536i \(-0.437095\pi\)
0.196337 + 0.980536i \(0.437095\pi\)
\(464\) −2.57002 −0.119310
\(465\) 0 0
\(466\) 2.45473 0.113713
\(467\) −33.5139 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) −1.45712 −0.0672121
\(471\) 0 0
\(472\) 18.2524 0.840133
\(473\) 1.25667 0.0577817
\(474\) 0 0
\(475\) −0.435353 −0.0199754
\(476\) 6.90408 0.316448
\(477\) 0 0
\(478\) −20.1661 −0.922377
\(479\) −24.7313 −1.13000 −0.565000 0.825091i \(-0.691124\pi\)
−0.565000 + 0.825091i \(0.691124\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 16.2524 0.740275
\(483\) 0 0
\(484\) −9.54825 −0.434012
\(485\) 1.99031 0.0903754
\(486\) 0 0
\(487\) 37.7555 1.71087 0.855433 0.517913i \(-0.173291\pi\)
0.855433 + 0.517913i \(0.173291\pi\)
\(488\) −9.34721 −0.423128
\(489\) 0 0
\(490\) −0.286208 −0.0129296
\(491\) −31.3110 −1.41304 −0.706522 0.707691i \(-0.749737\pi\)
−0.706522 + 0.707691i \(0.749737\pi\)
\(492\) 0 0
\(493\) 9.99761 0.450270
\(494\) 0 0
\(495\) 0 0
\(496\) 3.67994 0.165234
\(497\) −20.4222 −0.916061
\(498\) 0 0
\(499\) −21.4873 −0.961902 −0.480951 0.876748i \(-0.659708\pi\)
−0.480951 + 0.876748i \(0.659708\pi\)
\(500\) 3.33081 0.148959
\(501\) 0 0
\(502\) 19.0228 0.849031
\(503\) −37.5924 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(504\) 0 0
\(505\) 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 0
\(511\) −15.8750 −0.702269
\(512\) 6.22282 0.275012
\(513\) 0 0
\(514\) −11.4069 −0.503136
\(515\) 0.336454 0.0148259
\(516\) 0 0
\(517\) 31.2446 1.37414
\(518\) −10.7584 −0.472697
\(519\) 0 0
\(520\) 0 0
\(521\) 19.8465 0.869493 0.434746 0.900553i \(-0.356838\pi\)
0.434746 + 0.900553i \(0.356838\pi\)
\(522\) 0 0
\(523\) −11.4300 −0.499798 −0.249899 0.968272i \(-0.580397\pi\)
−0.249899 + 0.968272i \(0.580397\pi\)
\(524\) −8.92394 −0.389844
\(525\) 0 0
\(526\) 13.7095 0.597762
\(527\) −14.3153 −0.623583
\(528\) 0 0
\(529\) −20.7681 −0.902960
\(530\) 2.05861 0.0894201
\(531\) 0 0
\(532\) −0.281896 −0.0122218
\(533\) 0 0
\(534\) 0 0
\(535\) −0.807315 −0.0349033
\(536\) 20.6746 0.893005
\(537\) 0 0
\(538\) 5.18598 0.223584
\(539\) 6.13706 0.264342
\(540\) 0 0
\(541\) −16.1884 −0.695993 −0.347996 0.937496i \(-0.613138\pi\)
−0.347996 + 0.937496i \(0.613138\pi\)
\(542\) −5.17092 −0.222110
\(543\) 0 0
\(544\) −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
\(549\) 0 0
\(550\) 16.8213 0.717263
\(551\) −0.408206 −0.0173902
\(552\) 0 0
\(553\) 23.4993 0.999293
\(554\) 10.7942 0.458600
\(555\) 0 0
\(556\) 19.9560 0.846322
\(557\) 7.39075 0.313156 0.156578 0.987666i \(-0.449954\pi\)
0.156578 + 0.987666i \(0.449954\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.323044 0.0136511
\(561\) 0 0
\(562\) 4.03923 0.170385
\(563\) 9.47889 0.399488 0.199744 0.979848i \(-0.435989\pi\)
0.199744 + 0.979848i \(0.435989\pi\)
\(564\) 0 0
\(565\) −2.97584 −0.125194
\(566\) 17.7453 0.745889
\(567\) 0 0
\(568\) 23.3260 0.978738
\(569\) 10.1438 0.425249 0.212624 0.977134i \(-0.431799\pi\)
0.212624 + 0.977134i \(0.431799\pi\)
\(570\) 0 0
\(571\) −14.0925 −0.589751 −0.294876 0.955536i \(-0.595278\pi\)
−0.294876 + 0.955536i \(0.595278\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −21.9095 −0.914483
\(575\) 7.37867 0.307712
\(576\) 0 0
\(577\) −25.1545 −1.04720 −0.523598 0.851965i \(-0.675411\pi\)
−0.523598 + 0.851965i \(0.675411\pi\)
\(578\) −9.89546 −0.411597
\(579\) 0 0
\(580\) 1.55197 0.0644422
\(581\) 3.79284 0.157354
\(582\) 0 0
\(583\) −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 \(0.140141\pi\)
\(588\) 0 0
\(589\) 0.584498 0.0240838
\(590\) −1.34290 −0.0552861
\(591\) 0 0
\(592\) −3.15883 −0.129827
\(593\) −24.9965 −1.02648 −0.513242 0.858244i \(-0.671556\pi\)
−0.513242 + 0.858244i \(0.671556\pi\)
\(594\) 0 0
\(595\) −1.25667 −0.0515184
\(596\) −5.88231 −0.240949
\(597\) 0 0
\(598\) 0 0
\(599\) 6.24027 0.254971 0.127485 0.991840i \(-0.459309\pi\)
0.127485 + 0.991840i \(0.459309\pi\)
\(600\) 0 0
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) −0.559270 −0.0227941
\(603\) 0 0
\(604\) 5.35211 0.217774
\(605\) 1.73795 0.0706579
\(606\) 0 0
\(607\) −43.6480 −1.77162 −0.885809 0.464050i \(-0.846396\pi\)
−0.885809 + 0.464050i \(0.846396\pi\)
\(608\) 0.513811 0.0208378
\(609\) 0 0
\(610\) 0.687710 0.0278445
\(611\) 0 0
\(612\) 0 0
\(613\) 25.9541 1.04827 0.524137 0.851634i \(-0.324388\pi\)
0.524137 + 0.851634i \(0.324388\pi\)
\(614\) −15.3405 −0.619092
\(615\) 0 0
\(616\) 26.9463 1.08570
\(617\) 45.9396 1.84946 0.924729 0.380626i \(-0.124291\pi\)
0.924729 + 0.380626i \(0.124291\pi\)
\(618\) 0 0
\(619\) −6.73556 −0.270725 −0.135363 0.990796i \(-0.543220\pi\)
−0.135363 + 0.990796i \(0.543220\pi\)
\(620\) −2.22223 −0.0892467
\(621\) 0 0
\(622\) 0.216375 0.00867583
\(623\) −6.79895 −0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) −18.7603 −0.749813
\(627\) 0 0
\(628\) −6.04461 −0.241206
\(629\) 12.2881 0.489960
\(630\) 0 0
\(631\) −45.0998 −1.79539 −0.897696 0.440614i \(-0.854761\pi\)
−0.897696 + 0.440614i \(0.854761\pi\)
\(632\) −26.8407 −1.06767
\(633\) 0 0
\(634\) −11.2233 −0.445734
\(635\) −2.42221 −0.0961223
\(636\) 0 0
\(637\) 0 0
\(638\) 15.7724 0.624435
\(639\) 0 0
\(640\) −2.17331 −0.0859075
\(641\) −32.5821 −1.28692 −0.643458 0.765482i \(-0.722501\pi\)
−0.643458 + 0.765482i \(0.722501\pi\)
\(642\) 0 0
\(643\) −25.5754 −1.00860 −0.504298 0.863530i \(-0.668249\pi\)
−0.504298 + 0.863530i \(0.668249\pi\)
\(644\) 4.77777 0.188271
\(645\) 0 0
\(646\) −0.152603 −0.00600408
\(647\) 30.1715 1.18616 0.593082 0.805142i \(-0.297911\pi\)
0.593082 + 0.805142i \(0.297911\pi\)
\(648\) 0 0
\(649\) 28.7952 1.13031
\(650\) 0 0
\(651\) 0 0
\(652\) 21.9259 0.858683
\(653\) −36.9028 −1.44412 −0.722058 0.691832i \(-0.756804\pi\)
−0.722058 + 0.691832i \(0.756804\pi\)
\(654\) 0 0
\(655\) 1.62432 0.0634673
\(656\) −6.43296 −0.251165
\(657\) 0 0
\(658\) −13.9051 −0.542079
\(659\) −23.6866 −0.922701 −0.461350 0.887218i \(-0.652635\pi\)
−0.461350 + 0.887218i \(0.652635\pi\)
\(660\) 0 0
\(661\) 31.7590 1.23528 0.617641 0.786460i \(-0.288089\pi\)
0.617641 + 0.786460i \(0.288089\pi\)
\(662\) −14.2916 −0.555458
\(663\) 0 0
\(664\) −4.33214 −0.168120
\(665\) 0.0513102 0.00198973
\(666\) 0 0
\(667\) 6.91856 0.267888
\(668\) −21.8694 −0.846152
\(669\) 0 0
\(670\) −1.52111 −0.0587655
\(671\) −14.7463 −0.569275
\(672\) 0 0
\(673\) −7.50232 −0.289193 −0.144597 0.989491i \(-0.546188\pi\)
−0.144597 + 0.989491i \(0.546188\pi\)
\(674\) −22.3327 −0.860225
\(675\) 0 0
\(676\) 0 0
\(677\) 35.0315 1.34637 0.673184 0.739475i \(-0.264926\pi\)
0.673184 + 0.739475i \(0.264926\pi\)
\(678\) 0 0
\(679\) 18.9933 0.728896
\(680\) 1.43535 0.0550433
\(681\) 0 0
\(682\) −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 0
\(685\) 1.53617 0.0586941
\(686\) −15.9618 −0.609426
\(687\) 0 0
\(688\) −0.164210 −0.00626046
\(689\) 0 0
\(690\) 0 0
\(691\) −2.01447 −0.0766342 −0.0383171 0.999266i \(-0.512200\pi\)
−0.0383171 + 0.999266i \(0.512200\pi\)
\(692\) −29.2223 −1.11087
\(693\) 0 0
\(694\) −1.20583 −0.0457728
\(695\) −3.63235 −0.137783
\(696\) 0 0
\(697\) 25.0248 0.947880
\(698\) 11.3763 0.430598
\(699\) 0 0
\(700\) 15.7952 0.597004
\(701\) 48.8189 1.84387 0.921933 0.387350i \(-0.126610\pi\)
0.921933 + 0.387350i \(0.126610\pi\)
\(702\) 0 0
\(703\) −0.501729 −0.0189231
\(704\) −15.1390 −0.570572
\(705\) 0 0
\(706\) −5.74871 −0.216355
\(707\) 31.4752 1.18375
\(708\) 0 0
\(709\) 20.8060 0.781385 0.390693 0.920521i \(-0.372236\pi\)
0.390693 + 0.920521i \(0.372236\pi\)
\(710\) −1.71618 −0.0644073
\(711\) 0 0
\(712\) 7.76569 0.291032
\(713\) −9.90648 −0.371000
\(714\) 0 0
\(715\) 0 0
\(716\) 15.5133 0.579761
\(717\) 0 0
\(718\) 15.9468 0.595128
\(719\) −21.4306 −0.799225 −0.399613 0.916684i \(-0.630855\pi\)
−0.399613 + 0.916684i \(0.630855\pi\)
\(720\) 0 0
\(721\) 3.21073 0.119574
\(722\) −15.2306 −0.566824
\(723\) 0 0
\(724\) −28.4547 −1.05751
\(725\) 22.8726 0.849468
\(726\) 0 0
\(727\) 13.4862 0.500175 0.250088 0.968223i \(-0.419541\pi\)
0.250088 + 0.968223i \(0.419541\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.33406 −0.0493758
\(731\) 0.638792 0.0236266
\(732\) 0 0
\(733\) 43.5424 1.60828 0.804138 0.594443i \(-0.202627\pi\)
0.804138 + 0.594443i \(0.202627\pi\)
\(734\) 0.869167 0.0320816
\(735\) 0 0
\(736\) −8.70841 −0.320996
\(737\) 32.6165 1.20145
\(738\) 0 0
\(739\) −20.0543 −0.737709 −0.368855 0.929487i \(-0.620250\pi\)
−0.368855 + 0.929487i \(0.620250\pi\)
\(740\) 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 \(0.708195\pi\)
\(744\) 0 0
\(745\) 1.07069 0.0392270
\(746\) −4.91617 −0.179994
\(747\) 0 0
\(748\) −12.4407 −0.454878
\(749\) −7.70410 −0.281502
\(750\) 0 0
\(751\) 39.2814 1.43340 0.716700 0.697382i \(-0.245652\pi\)
0.716700 + 0.697382i \(0.245652\pi\)
\(752\) −4.08277 −0.148883
\(753\) 0 0
\(754\) 0 0
\(755\) −0.974181 −0.0354541
\(756\) 0 0
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) 1.93171 0.0701627
\(759\) 0 0
\(760\) −0.0586060 −0.00212586
\(761\) −21.8984 −0.793818 −0.396909 0.917858i \(-0.629917\pi\)
−0.396909 + 0.917858i \(0.629917\pi\)
\(762\) 0 0
\(763\) −37.0073 −1.33975
\(764\) −19.5899 −0.708737
\(765\) 0 0
\(766\) −24.3720 −0.880595
\(767\) 0 0
\(768\) 0 0
\(769\) −46.7096 −1.68439 −0.842196 0.539172i \(-0.818737\pi\)
−0.842196 + 0.539172i \(0.818737\pi\)
\(770\) −1.98254 −0.0714458
\(771\) 0 0
\(772\) −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 0
\(775\) −32.7506 −1.17644
\(776\) −21.6939 −0.778767
\(777\) 0 0
\(778\) 12.7836 0.458315
\(779\) −1.02177 −0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 2.58642 0.0924901
\(783\) 0 0
\(784\) −0.801938 −0.0286406
\(785\) 1.10023 0.0392688
\(786\) 0 0
\(787\) −28.7023 −1.02313 −0.511563 0.859246i \(-0.670933\pi\)
−0.511563 + 0.859246i \(0.670933\pi\)
\(788\) 0.760316 0.0270851
\(789\) 0 0
\(790\) 1.97477 0.0702592
\(791\) −28.3980 −1.00972
\(792\) 0 0
\(793\) 0 0
\(794\) −13.5646 −0.481391
\(795\) 0 0
\(796\) −15.5929 −0.552674
\(797\) 18.5418 0.656785 0.328392 0.944541i \(-0.393493\pi\)
0.328392 + 0.944541i \(0.393493\pi\)
\(798\) 0 0
\(799\) 15.8823 0.561876
\(800\) −28.7899 −1.01788
\(801\) 0 0
\(802\) 21.3817 0.755012
\(803\) 28.6058 1.00948
\(804\) 0 0
\(805\) −0.869641 −0.0306508
\(806\) 0 0
\(807\) 0 0
\(808\) −35.9506 −1.26474
\(809\) 10.0677 0.353962 0.176981 0.984214i \(-0.443367\pi\)
0.176981 + 0.984214i \(0.443367\pi\)
\(810\) 0 0
\(811\) 10.0285 0.352147 0.176074 0.984377i \(-0.443660\pi\)
0.176074 + 0.984377i \(0.443660\pi\)
\(812\) 14.8103 0.519740
\(813\) 0 0
\(814\) 19.3860 0.679478
\(815\) −3.99090 −0.139795
\(816\) 0 0
\(817\) −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 \(0.650961\pi\)
\(822\) 0 0
\(823\) 1.82238 0.0635242 0.0317621 0.999495i \(-0.489888\pi\)
0.0317621 + 0.999495i \(0.489888\pi\)
\(824\) −3.66727 −0.127755
\(825\) 0 0
\(826\) −12.8151 −0.445894
\(827\) −32.2941 −1.12298 −0.561488 0.827485i \(-0.689771\pi\)
−0.561488 + 0.827485i \(0.689771\pi\)
\(828\) 0 0
\(829\) 15.1002 0.524453 0.262226 0.965006i \(-0.415543\pi\)
0.262226 + 0.965006i \(0.415543\pi\)
\(830\) 0.318732 0.0110634
\(831\) 0 0
\(832\) 0 0
\(833\) 3.11960 0.108088
\(834\) 0 0
\(835\) 3.98062 0.137755
\(836\) 0.507960 0.0175682
\(837\) 0 0
\(838\) 23.7448 0.820250
\(839\) −32.9965 −1.13917 −0.569584 0.821933i \(-0.692896\pi\)
−0.569584 + 0.821933i \(0.692896\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) 9.35105 0.322258
\(843\) 0 0
\(844\) −11.9196 −0.410290
\(845\) 0 0
\(846\) 0 0
\(847\) 16.5851 0.569870
\(848\) 5.76809 0.198077
\(849\) 0 0
\(850\) 8.55065 0.293285
\(851\) 8.50365 0.291501
\(852\) 0 0
\(853\) 37.7802 1.29357 0.646784 0.762673i \(-0.276113\pi\)
0.646784 + 0.762673i \(0.276113\pi\)
\(854\) 6.56273 0.224572
\(855\) 0 0
\(856\) 8.79954 0.300762
\(857\) −27.3623 −0.934677 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(858\) 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
\(861\) 0 0
\(862\) 3.48773 0.118792
\(863\) −6.14483 −0.209173 −0.104586 0.994516i \(-0.533352\pi\)
−0.104586 + 0.994516i \(0.533352\pi\)
\(864\) 0 0
\(865\) 5.31900 0.180851
\(866\) −11.5386 −0.392096
\(867\) 0 0
\(868\) −21.2064 −0.719793
\(869\) −42.3443 −1.43643
\(870\) 0 0
\(871\) 0 0
\(872\) 42.2693 1.43142
\(873\) 0 0
\(874\) −0.105604 −0.00357212
\(875\) −5.78554 −0.195587
\(876\) 0 0
\(877\) 13.5077 0.456123 0.228061 0.973647i \(-0.426761\pi\)
0.228061 + 0.973647i \(0.426761\pi\)
\(878\) −16.2252 −0.547574
\(879\) 0 0
\(880\) −0.582105 −0.0196228
\(881\) 5.23431 0.176348 0.0881741 0.996105i \(-0.471897\pi\)
0.0881741 + 0.996105i \(0.471897\pi\)
\(882\) 0 0
\(883\) −4.57301 −0.153894 −0.0769470 0.997035i \(-0.524517\pi\)
−0.0769470 + 0.997035i \(0.524517\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −6.51334 −0.218820
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) 0 0
\(889\) −23.1148 −0.775246
\(890\) −0.571352 −0.0191517
\(891\) 0 0
\(892\) −3.06531 −0.102634
\(893\) −0.648481 −0.0217006
\(894\) 0 0
\(895\) −2.82371 −0.0943861
\(896\) −20.7396 −0.692862
\(897\) 0 0
\(898\) 10.0175 0.334287
\(899\) −30.7084 −1.02418
\(900\) 0 0
\(901\) −22.4383 −0.747529
\(902\) 39.4795 1.31452
\(903\) 0 0
\(904\) 32.4359 1.07880
\(905\) 5.17928 0.172165
\(906\) 0 0
\(907\) 8.10215 0.269027 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(908\) 9.45234 0.313687
\(909\) 0 0
\(910\) 0 0
\(911\) 9.18119 0.304187 0.152093 0.988366i \(-0.451399\pi\)
0.152093 + 0.988366i \(0.451399\pi\)
\(912\) 0 0
\(913\) −6.83446 −0.226188
\(914\) −4.79656 −0.158656
\(915\) 0 0
\(916\) −32.8015 −1.08379
\(917\) 15.5007 0.511877
\(918\) 0 0
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) 0.993295 0.0327480
\(921\) 0 0
\(922\) −1.64933 −0.0543180
\(923\) 0 0
\(924\) 0 0
\(925\) 28.1129 0.924346
\(926\) 6.77586 0.222668
\(927\) 0 0
\(928\) −26.9946 −0.886142
\(929\) 24.2131 0.794407 0.397203 0.917731i \(-0.369981\pi\)
0.397203 + 0.917731i \(0.369981\pi\)
\(930\) 0 0
\(931\) −0.127375 −0.00417454
\(932\) −4.15346 −0.136051
\(933\) 0 0
\(934\) −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
\(939\) 0 0
\(940\) 2.46548 0.0804152
\(941\) −15.9638 −0.520404 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(942\) 0 0
\(943\) 17.3177 0.563941
\(944\) −3.76271 −0.122466
\(945\) 0 0
\(946\) 1.00777 0.0327654
\(947\) 6.51466 0.211698 0.105849 0.994382i \(-0.466244\pi\)
0.105849 + 0.994382i \(0.466244\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −0.349126 −0.0113271
\(951\) 0 0
\(952\) 13.6974 0.443935
\(953\) −47.6469 −1.54344 −0.771718 0.635965i \(-0.780602\pi\)
−0.771718 + 0.635965i \(0.780602\pi\)
\(954\) 0 0
\(955\) 3.56571 0.115384
\(956\) 34.1215 1.10357
\(957\) 0 0
\(958\) −19.8329 −0.640773
\(959\) 14.6595 0.473380
\(960\) 0 0
\(961\) 12.9705 0.418402
\(962\) 0 0
\(963\) 0 0
\(964\) −27.4993 −0.885694
\(965\) 3.35391 0.107966
\(966\) 0 0
\(967\) 43.8122 1.40891 0.704453 0.709751i \(-0.251192\pi\)
0.704453 + 0.709751i \(0.251192\pi\)
\(968\) −18.9433 −0.608861
\(969\) 0 0
\(970\) 1.59611 0.0512479
\(971\) −4.29483 −0.137828 −0.0689139 0.997623i \(-0.521953\pi\)
−0.0689139 + 0.997623i \(0.521953\pi\)
\(972\) 0 0
\(973\) −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 \(0.358959\pi\)
\(978\) 0 0
\(979\) 12.2513 0.391553
\(980\) 0.484271 0.0154695
\(981\) 0 0
\(982\) −25.1094 −0.801275
\(983\) 27.2495 0.869124 0.434562 0.900642i \(-0.356903\pi\)
0.434562 + 0.900642i \(0.356903\pi\)
\(984\) 0 0
\(985\) −0.138391 −0.00440951
\(986\) 8.01746 0.255328
\(987\) 0 0
\(988\) 0 0
\(989\) 0.442058 0.0140566
\(990\) 0 0
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) 38.6528 1.22723
\(993\) 0 0
\(994\) −16.3773 −0.519458
\(995\) 2.83818 0.0899764
\(996\) 0 0
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) −17.2314 −0.545452
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1521.2.a.o.1.3 3
3.2 odd 2 169.2.a.c.1.1 yes 3
12.11 even 2 2704.2.a.ba.1.3 3
13.5 odd 4 1521.2.b.l.1351.2 6
13.8 odd 4 1521.2.b.l.1351.5 6
13.12 even 2 1521.2.a.r.1.1 3
15.14 odd 2 4225.2.a.bb.1.3 3
21.20 even 2 8281.2.a.bj.1.1 3
39.2 even 12 169.2.e.b.147.2 12
39.5 even 4 169.2.b.b.168.5 6
39.8 even 4 169.2.b.b.168.2 6
39.11 even 12 169.2.e.b.147.5 12
39.17 odd 6 169.2.c.c.146.1 6
39.20 even 12 169.2.e.b.23.2 12
39.23 odd 6 169.2.c.c.22.1 6
39.29 odd 6 169.2.c.b.22.3 6
39.32 even 12 169.2.e.b.23.5 12
39.35 odd 6 169.2.c.b.146.3 6
39.38 odd 2 169.2.a.b.1.3 3
156.47 odd 4 2704.2.f.o.337.5 6
156.83 odd 4 2704.2.f.o.337.6 6
156.155 even 2 2704.2.a.z.1.3 3
195.194 odd 2 4225.2.a.bg.1.1 3
273.272 even 2 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 39.38 odd 2
169.2.a.c.1.1 yes 3 3.2 odd 2
169.2.b.b.168.2 6 39.8 even 4
169.2.b.b.168.5 6 39.5 even 4
169.2.c.b.22.3 6 39.29 odd 6
169.2.c.b.146.3 6 39.35 odd 6
169.2.c.c.22.1 6 39.23 odd 6
169.2.c.c.146.1 6 39.17 odd 6
169.2.e.b.23.2 12 39.20 even 12
169.2.e.b.23.5 12 39.32 even 12
169.2.e.b.147.2 12 39.2 even 12
169.2.e.b.147.5 12 39.11 even 12
1521.2.a.o.1.3 3 1.1 even 1 trivial
1521.2.a.r.1.1 3 13.12 even 2
1521.2.b.l.1351.2 6 13.5 odd 4
1521.2.b.l.1351.5 6 13.8 odd 4
2704.2.a.z.1.3 3 156.155 even 2
2704.2.a.ba.1.3 3 12.11 even 2
2704.2.f.o.337.5 6 156.47 odd 4
2704.2.f.o.337.6 6 156.83 odd 4
4225.2.a.bb.1.3 3 15.14 odd 2
4225.2.a.bg.1.1 3 195.194 odd 2
8281.2.a.bf.1.3 3 273.272 even 2
8281.2.a.bj.1.1 3 21.20 even 2