Properties

Label 8281.2.a.cs.1.11
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 42x^{14} + 641x^{12} - 4448x^{10} + 14076x^{8} - 17900x^{6} + 6960x^{4} - 416x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.93761\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.52340 q^{2} -2.99510 q^{3} +0.320733 q^{4} +2.94606 q^{5} -4.56272 q^{6} -2.55819 q^{8} +5.97063 q^{9} +O(q^{10})\) \(q+1.52340 q^{2} -2.99510 q^{3} +0.320733 q^{4} +2.94606 q^{5} -4.56272 q^{6} -2.55819 q^{8} +5.97063 q^{9} +4.48801 q^{10} -2.37108 q^{11} -0.960628 q^{12} -8.82374 q^{15} -4.53860 q^{16} -5.36994 q^{17} +9.09563 q^{18} +5.35438 q^{19} +0.944899 q^{20} -3.61209 q^{22} +2.79056 q^{23} +7.66203 q^{24} +3.67927 q^{25} -8.89733 q^{27} +0.585818 q^{29} -13.4421 q^{30} -8.33379 q^{31} -1.79770 q^{32} +7.10163 q^{33} -8.18054 q^{34} +1.91498 q^{36} +0.675710 q^{37} +8.15683 q^{38} -7.53657 q^{40} -11.2799 q^{41} +10.5271 q^{43} -0.760484 q^{44} +17.5898 q^{45} +4.25113 q^{46} -0.537744 q^{47} +13.5936 q^{48} +5.60498 q^{50} +16.0835 q^{51} +7.90345 q^{53} -13.5541 q^{54} -6.98535 q^{55} -16.0369 q^{57} +0.892432 q^{58} +6.14603 q^{59} -2.83007 q^{60} +2.78689 q^{61} -12.6957 q^{62} +6.33858 q^{64} +10.8186 q^{66} +4.21353 q^{67} -1.72232 q^{68} -8.35802 q^{69} +1.25151 q^{71} -15.2740 q^{72} -10.8620 q^{73} +1.02937 q^{74} -11.0198 q^{75} +1.71733 q^{76} +6.29136 q^{79} -13.3710 q^{80} +8.73651 q^{81} -17.1837 q^{82} +1.74725 q^{83} -15.8202 q^{85} +16.0369 q^{86} -1.75458 q^{87} +6.06567 q^{88} -10.2343 q^{89} +26.7963 q^{90} +0.895027 q^{92} +24.9605 q^{93} -0.819196 q^{94} +15.7743 q^{95} +5.38430 q^{96} +9.90640 q^{97} -14.1568 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92} + 84 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.52340 1.07720 0.538602 0.842561i \(-0.318953\pi\)
0.538602 + 0.842561i \(0.318953\pi\)
\(3\) −2.99510 −1.72922 −0.864611 0.502442i \(-0.832435\pi\)
−0.864611 + 0.502442i \(0.832435\pi\)
\(4\) 0.320733 0.160367
\(5\) 2.94606 1.31752 0.658759 0.752354i \(-0.271082\pi\)
0.658759 + 0.752354i \(0.271082\pi\)
\(6\) −4.56272 −1.86272
\(7\) 0 0
\(8\) −2.55819 −0.904456
\(9\) 5.97063 1.99021
\(10\) 4.48801 1.41923
\(11\) −2.37108 −0.714908 −0.357454 0.933931i \(-0.616355\pi\)
−0.357454 + 0.933931i \(0.616355\pi\)
\(12\) −0.960628 −0.277309
\(13\) 0 0
\(14\) 0 0
\(15\) −8.82374 −2.27828
\(16\) −4.53860 −1.13465
\(17\) −5.36994 −1.30240 −0.651201 0.758905i \(-0.725735\pi\)
−0.651201 + 0.758905i \(0.725735\pi\)
\(18\) 9.09563 2.14386
\(19\) 5.35438 1.22838 0.614189 0.789159i \(-0.289483\pi\)
0.614189 + 0.789159i \(0.289483\pi\)
\(20\) 0.944899 0.211286
\(21\) 0 0
\(22\) −3.61209 −0.770101
\(23\) 2.79056 0.581873 0.290936 0.956742i \(-0.406033\pi\)
0.290936 + 0.956742i \(0.406033\pi\)
\(24\) 7.66203 1.56400
\(25\) 3.67927 0.735853
\(26\) 0 0
\(27\) −8.89733 −1.71229
\(28\) 0 0
\(29\) 0.585818 0.108784 0.0543918 0.998520i \(-0.482678\pi\)
0.0543918 + 0.998520i \(0.482678\pi\)
\(30\) −13.4421 −2.45417
\(31\) −8.33379 −1.49679 −0.748397 0.663252i \(-0.769176\pi\)
−0.748397 + 0.663252i \(0.769176\pi\)
\(32\) −1.79770 −0.317792
\(33\) 7.10163 1.23623
\(34\) −8.18054 −1.40295
\(35\) 0 0
\(36\) 1.91498 0.319163
\(37\) 0.675710 0.111086 0.0555430 0.998456i \(-0.482311\pi\)
0.0555430 + 0.998456i \(0.482311\pi\)
\(38\) 8.15683 1.32321
\(39\) 0 0
\(40\) −7.53657 −1.19164
\(41\) −11.2799 −1.76162 −0.880808 0.473473i \(-0.843000\pi\)
−0.880808 + 0.473473i \(0.843000\pi\)
\(42\) 0 0
\(43\) 10.5271 1.60536 0.802682 0.596408i \(-0.203406\pi\)
0.802682 + 0.596408i \(0.203406\pi\)
\(44\) −0.760484 −0.114647
\(45\) 17.5898 2.62214
\(46\) 4.25113 0.626795
\(47\) −0.537744 −0.0784380 −0.0392190 0.999231i \(-0.512487\pi\)
−0.0392190 + 0.999231i \(0.512487\pi\)
\(48\) 13.5936 1.96206
\(49\) 0 0
\(50\) 5.60498 0.792664
\(51\) 16.0835 2.25214
\(52\) 0 0
\(53\) 7.90345 1.08562 0.542811 0.839855i \(-0.317360\pi\)
0.542811 + 0.839855i \(0.317360\pi\)
\(54\) −13.5541 −1.84449
\(55\) −6.98535 −0.941904
\(56\) 0 0
\(57\) −16.0369 −2.12414
\(58\) 0.892432 0.117182
\(59\) 6.14603 0.800145 0.400072 0.916484i \(-0.368985\pi\)
0.400072 + 0.916484i \(0.368985\pi\)
\(60\) −2.83007 −0.365360
\(61\) 2.78689 0.356824 0.178412 0.983956i \(-0.442904\pi\)
0.178412 + 0.983956i \(0.442904\pi\)
\(62\) −12.6957 −1.61235
\(63\) 0 0
\(64\) 6.33858 0.792323
\(65\) 0 0
\(66\) 10.8186 1.33168
\(67\) 4.21353 0.514765 0.257382 0.966310i \(-0.417140\pi\)
0.257382 + 0.966310i \(0.417140\pi\)
\(68\) −1.72232 −0.208862
\(69\) −8.35802 −1.00619
\(70\) 0 0
\(71\) 1.25151 0.148527 0.0742637 0.997239i \(-0.476339\pi\)
0.0742637 + 0.997239i \(0.476339\pi\)
\(72\) −15.2740 −1.80006
\(73\) −10.8620 −1.27130 −0.635650 0.771977i \(-0.719268\pi\)
−0.635650 + 0.771977i \(0.719268\pi\)
\(74\) 1.02937 0.119662
\(75\) −11.0198 −1.27245
\(76\) 1.71733 0.196991
\(77\) 0 0
\(78\) 0 0
\(79\) 6.29136 0.707833 0.353917 0.935277i \(-0.384850\pi\)
0.353917 + 0.935277i \(0.384850\pi\)
\(80\) −13.3710 −1.49492
\(81\) 8.73651 0.970723
\(82\) −17.1837 −1.89762
\(83\) 1.74725 0.191786 0.0958928 0.995392i \(-0.469429\pi\)
0.0958928 + 0.995392i \(0.469429\pi\)
\(84\) 0 0
\(85\) −15.8202 −1.71594
\(86\) 16.0369 1.72930
\(87\) −1.75458 −0.188111
\(88\) 6.06567 0.646602
\(89\) −10.2343 −1.08483 −0.542416 0.840110i \(-0.682490\pi\)
−0.542416 + 0.840110i \(0.682490\pi\)
\(90\) 26.7963 2.82457
\(91\) 0 0
\(92\) 0.895027 0.0933130
\(93\) 24.9605 2.58829
\(94\) −0.819196 −0.0844936
\(95\) 15.7743 1.61841
\(96\) 5.38430 0.549533
\(97\) 9.90640 1.00584 0.502921 0.864332i \(-0.332259\pi\)
0.502921 + 0.864332i \(0.332259\pi\)
\(98\) 0 0
\(99\) −14.1568 −1.42282
\(100\) 1.18006 0.118006
\(101\) 2.05077 0.204059 0.102030 0.994781i \(-0.467466\pi\)
0.102030 + 0.994781i \(0.467466\pi\)
\(102\) 24.5016 2.42602
\(103\) 3.02962 0.298517 0.149259 0.988798i \(-0.452311\pi\)
0.149259 + 0.988798i \(0.452311\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0401 1.16944
\(107\) −12.4305 −1.20170 −0.600852 0.799360i \(-0.705172\pi\)
−0.600852 + 0.799360i \(0.705172\pi\)
\(108\) −2.85367 −0.274594
\(109\) −3.53782 −0.338862 −0.169431 0.985542i \(-0.554193\pi\)
−0.169431 + 0.985542i \(0.554193\pi\)
\(110\) −10.6414 −1.01462
\(111\) −2.02382 −0.192092
\(112\) 0 0
\(113\) 10.3848 0.976921 0.488460 0.872586i \(-0.337559\pi\)
0.488460 + 0.872586i \(0.337559\pi\)
\(114\) −24.4305 −2.28813
\(115\) 8.22117 0.766628
\(116\) 0.187891 0.0174453
\(117\) 0 0
\(118\) 9.36283 0.861918
\(119\) 0 0
\(120\) 22.5728 2.06060
\(121\) −5.37797 −0.488907
\(122\) 4.24553 0.384372
\(123\) 33.7843 3.04623
\(124\) −2.67292 −0.240036
\(125\) −3.89096 −0.348018
\(126\) 0 0
\(127\) 5.09504 0.452112 0.226056 0.974114i \(-0.427417\pi\)
0.226056 + 0.974114i \(0.427417\pi\)
\(128\) 13.2516 1.17128
\(129\) −31.5296 −2.77603
\(130\) 0 0
\(131\) 22.5443 1.96971 0.984854 0.173386i \(-0.0554707\pi\)
0.984854 + 0.173386i \(0.0554707\pi\)
\(132\) 2.27773 0.198251
\(133\) 0 0
\(134\) 6.41887 0.554506
\(135\) −26.2121 −2.25597
\(136\) 13.7373 1.17797
\(137\) 1.03479 0.0884082 0.0442041 0.999023i \(-0.485925\pi\)
0.0442041 + 0.999023i \(0.485925\pi\)
\(138\) −12.7326 −1.08387
\(139\) 8.49456 0.720499 0.360249 0.932856i \(-0.382692\pi\)
0.360249 + 0.932856i \(0.382692\pi\)
\(140\) 0 0
\(141\) 1.61060 0.135637
\(142\) 1.90655 0.159994
\(143\) 0 0
\(144\) −27.0983 −2.25819
\(145\) 1.72585 0.143324
\(146\) −16.5471 −1.36945
\(147\) 0 0
\(148\) 0.216723 0.0178145
\(149\) 4.31343 0.353370 0.176685 0.984267i \(-0.443463\pi\)
0.176685 + 0.984267i \(0.443463\pi\)
\(150\) −16.7875 −1.37069
\(151\) 12.2447 0.996463 0.498232 0.867044i \(-0.333983\pi\)
0.498232 + 0.867044i \(0.333983\pi\)
\(152\) −13.6975 −1.11101
\(153\) −32.0619 −2.59205
\(154\) 0 0
\(155\) −24.5518 −1.97205
\(156\) 0 0
\(157\) −22.2040 −1.77207 −0.886035 0.463619i \(-0.846551\pi\)
−0.886035 + 0.463619i \(0.846551\pi\)
\(158\) 9.58423 0.762480
\(159\) −23.6716 −1.87728
\(160\) −5.29614 −0.418697
\(161\) 0 0
\(162\) 13.3092 1.04567
\(163\) 11.3644 0.890127 0.445063 0.895499i \(-0.353181\pi\)
0.445063 + 0.895499i \(0.353181\pi\)
\(164\) −3.61782 −0.282505
\(165\) 20.9218 1.62876
\(166\) 2.66175 0.206592
\(167\) −1.88127 −0.145577 −0.0727885 0.997347i \(-0.523190\pi\)
−0.0727885 + 0.997347i \(0.523190\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −24.1004 −1.84841
\(171\) 31.9690 2.44473
\(172\) 3.37638 0.257447
\(173\) −5.57113 −0.423565 −0.211783 0.977317i \(-0.567927\pi\)
−0.211783 + 0.977317i \(0.567927\pi\)
\(174\) −2.67292 −0.202634
\(175\) 0 0
\(176\) 10.7614 0.811170
\(177\) −18.4080 −1.38363
\(178\) −15.5909 −1.16858
\(179\) −4.66142 −0.348411 −0.174205 0.984709i \(-0.555736\pi\)
−0.174205 + 0.984709i \(0.555736\pi\)
\(180\) 5.64164 0.420503
\(181\) 12.5360 0.931794 0.465897 0.884839i \(-0.345732\pi\)
0.465897 + 0.884839i \(0.345732\pi\)
\(182\) 0 0
\(183\) −8.34701 −0.617029
\(184\) −7.13879 −0.526278
\(185\) 1.99068 0.146358
\(186\) 38.0248 2.78811
\(187\) 12.7326 0.931098
\(188\) −0.172472 −0.0125788
\(189\) 0 0
\(190\) 24.0305 1.74336
\(191\) −1.31764 −0.0953408 −0.0476704 0.998863i \(-0.515180\pi\)
−0.0476704 + 0.998863i \(0.515180\pi\)
\(192\) −18.9847 −1.37010
\(193\) −15.1445 −1.09012 −0.545061 0.838396i \(-0.683494\pi\)
−0.545061 + 0.838396i \(0.683494\pi\)
\(194\) 15.0914 1.08350
\(195\) 0 0
\(196\) 0 0
\(197\) 9.90571 0.705753 0.352876 0.935670i \(-0.385204\pi\)
0.352876 + 0.935670i \(0.385204\pi\)
\(198\) −21.5665 −1.53266
\(199\) −8.57326 −0.607742 −0.303871 0.952713i \(-0.598279\pi\)
−0.303871 + 0.952713i \(0.598279\pi\)
\(200\) −9.41225 −0.665547
\(201\) −12.6199 −0.890142
\(202\) 3.12413 0.219813
\(203\) 0 0
\(204\) 5.15852 0.361169
\(205\) −33.2311 −2.32096
\(206\) 4.61530 0.321564
\(207\) 16.6614 1.15805
\(208\) 0 0
\(209\) −12.6957 −0.878177
\(210\) 0 0
\(211\) −1.44515 −0.0994881 −0.0497440 0.998762i \(-0.515841\pi\)
−0.0497440 + 0.998762i \(0.515841\pi\)
\(212\) 2.53490 0.174098
\(213\) −3.74841 −0.256837
\(214\) −18.9366 −1.29448
\(215\) 31.0134 2.11510
\(216\) 22.7610 1.54869
\(217\) 0 0
\(218\) −5.38950 −0.365023
\(219\) 32.5328 2.19836
\(220\) −2.24043 −0.151050
\(221\) 0 0
\(222\) −3.08307 −0.206922
\(223\) 14.9949 1.00413 0.502067 0.864829i \(-0.332573\pi\)
0.502067 + 0.864829i \(0.332573\pi\)
\(224\) 0 0
\(225\) 21.9675 1.46450
\(226\) 15.8202 1.05234
\(227\) 1.14220 0.0758105 0.0379052 0.999281i \(-0.487932\pi\)
0.0379052 + 0.999281i \(0.487932\pi\)
\(228\) −5.14356 −0.340641
\(229\) 6.26935 0.414290 0.207145 0.978310i \(-0.433583\pi\)
0.207145 + 0.978310i \(0.433583\pi\)
\(230\) 12.5241 0.825814
\(231\) 0 0
\(232\) −1.49863 −0.0983900
\(233\) −6.16695 −0.404010 −0.202005 0.979384i \(-0.564746\pi\)
−0.202005 + 0.979384i \(0.564746\pi\)
\(234\) 0 0
\(235\) −1.58423 −0.103343
\(236\) 1.97124 0.128317
\(237\) −18.8433 −1.22400
\(238\) 0 0
\(239\) 27.4589 1.77617 0.888083 0.459684i \(-0.152037\pi\)
0.888083 + 0.459684i \(0.152037\pi\)
\(240\) 40.0474 2.58505
\(241\) −9.79378 −0.630873 −0.315436 0.948947i \(-0.602151\pi\)
−0.315436 + 0.948947i \(0.602151\pi\)
\(242\) −8.19278 −0.526652
\(243\) 0.525262 0.0336956
\(244\) 0.893848 0.0572227
\(245\) 0 0
\(246\) 51.4668 3.28140
\(247\) 0 0
\(248\) 21.3194 1.35378
\(249\) −5.23319 −0.331640
\(250\) −5.92747 −0.374886
\(251\) 12.4945 0.788644 0.394322 0.918972i \(-0.370979\pi\)
0.394322 + 0.918972i \(0.370979\pi\)
\(252\) 0 0
\(253\) −6.61665 −0.415985
\(254\) 7.76176 0.487016
\(255\) 47.3830 2.96724
\(256\) 7.51022 0.469389
\(257\) 19.1929 1.19722 0.598610 0.801040i \(-0.295720\pi\)
0.598610 + 0.801040i \(0.295720\pi\)
\(258\) −48.0321 −2.99035
\(259\) 0 0
\(260\) 0 0
\(261\) 3.49770 0.216502
\(262\) 34.3439 2.12178
\(263\) 28.8153 1.77683 0.888415 0.459041i \(-0.151807\pi\)
0.888415 + 0.459041i \(0.151807\pi\)
\(264\) −18.1673 −1.11812
\(265\) 23.2840 1.43033
\(266\) 0 0
\(267\) 30.6527 1.87592
\(268\) 1.35142 0.0825511
\(269\) 8.15419 0.497170 0.248585 0.968610i \(-0.420035\pi\)
0.248585 + 0.968610i \(0.420035\pi\)
\(270\) −39.9313 −2.43014
\(271\) −18.5788 −1.12858 −0.564290 0.825577i \(-0.690850\pi\)
−0.564290 + 0.825577i \(0.690850\pi\)
\(272\) 24.3720 1.47777
\(273\) 0 0
\(274\) 1.57640 0.0952336
\(275\) −8.72384 −0.526067
\(276\) −2.68069 −0.161359
\(277\) −17.2342 −1.03550 −0.517752 0.855531i \(-0.673231\pi\)
−0.517752 + 0.855531i \(0.673231\pi\)
\(278\) 12.9406 0.776124
\(279\) −49.7580 −2.97893
\(280\) 0 0
\(281\) 11.4691 0.684191 0.342096 0.939665i \(-0.388863\pi\)
0.342096 + 0.939665i \(0.388863\pi\)
\(282\) 2.45358 0.146108
\(283\) 1.45575 0.0865355 0.0432678 0.999064i \(-0.486223\pi\)
0.0432678 + 0.999064i \(0.486223\pi\)
\(284\) 0.401402 0.0238188
\(285\) −47.2456 −2.79859
\(286\) 0 0
\(287\) 0 0
\(288\) −10.7334 −0.632472
\(289\) 11.8363 0.696252
\(290\) 2.62916 0.154389
\(291\) −29.6707 −1.73932
\(292\) −3.48380 −0.203874
\(293\) 30.6962 1.79329 0.896645 0.442750i \(-0.145997\pi\)
0.896645 + 0.442750i \(0.145997\pi\)
\(294\) 0 0
\(295\) 18.1066 1.05421
\(296\) −1.72859 −0.100472
\(297\) 21.0963 1.22413
\(298\) 6.57107 0.380652
\(299\) 0 0
\(300\) −3.53441 −0.204059
\(301\) 0 0
\(302\) 18.6536 1.07339
\(303\) −6.14226 −0.352864
\(304\) −24.3013 −1.39378
\(305\) 8.21034 0.470123
\(306\) −48.8430 −2.79217
\(307\) 21.2602 1.21339 0.606693 0.794936i \(-0.292496\pi\)
0.606693 + 0.794936i \(0.292496\pi\)
\(308\) 0 0
\(309\) −9.07401 −0.516202
\(310\) −37.4022 −2.12430
\(311\) 4.44647 0.252136 0.126068 0.992022i \(-0.459764\pi\)
0.126068 + 0.992022i \(0.459764\pi\)
\(312\) 0 0
\(313\) −2.28687 −0.129261 −0.0646307 0.997909i \(-0.520587\pi\)
−0.0646307 + 0.997909i \(0.520587\pi\)
\(314\) −33.8254 −1.90888
\(315\) 0 0
\(316\) 2.01785 0.113513
\(317\) 8.10086 0.454990 0.227495 0.973779i \(-0.426947\pi\)
0.227495 + 0.973779i \(0.426947\pi\)
\(318\) −36.0613 −2.02222
\(319\) −1.38902 −0.0777703
\(320\) 18.6738 1.04390
\(321\) 37.2307 2.07801
\(322\) 0 0
\(323\) −28.7527 −1.59984
\(324\) 2.80209 0.155672
\(325\) 0 0
\(326\) 17.3124 0.958847
\(327\) 10.5961 0.585967
\(328\) 28.8560 1.59330
\(329\) 0 0
\(330\) 31.8722 1.75451
\(331\) 11.7858 0.647804 0.323902 0.946091i \(-0.395005\pi\)
0.323902 + 0.946091i \(0.395005\pi\)
\(332\) 0.560401 0.0307560
\(333\) 4.03441 0.221084
\(334\) −2.86592 −0.156816
\(335\) 12.4133 0.678212
\(336\) 0 0
\(337\) 2.63045 0.143290 0.0716450 0.997430i \(-0.477175\pi\)
0.0716450 + 0.997430i \(0.477175\pi\)
\(338\) 0 0
\(339\) −31.1035 −1.68931
\(340\) −5.07405 −0.275179
\(341\) 19.7601 1.07007
\(342\) 48.7014 2.63347
\(343\) 0 0
\(344\) −26.9302 −1.45198
\(345\) −24.6232 −1.32567
\(346\) −8.48704 −0.456266
\(347\) 4.82463 0.259000 0.129500 0.991579i \(-0.458663\pi\)
0.129500 + 0.991579i \(0.458663\pi\)
\(348\) −0.562753 −0.0301667
\(349\) 17.5231 0.937991 0.468995 0.883201i \(-0.344616\pi\)
0.468995 + 0.883201i \(0.344616\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.26250 0.227192
\(353\) 6.59724 0.351136 0.175568 0.984467i \(-0.443824\pi\)
0.175568 + 0.984467i \(0.443824\pi\)
\(354\) −28.0426 −1.49045
\(355\) 3.68704 0.195688
\(356\) −3.28248 −0.173971
\(357\) 0 0
\(358\) −7.10118 −0.375309
\(359\) −23.0484 −1.21645 −0.608225 0.793764i \(-0.708118\pi\)
−0.608225 + 0.793764i \(0.708118\pi\)
\(360\) −44.9981 −2.37161
\(361\) 9.66934 0.508912
\(362\) 19.0973 1.00373
\(363\) 16.1076 0.845428
\(364\) 0 0
\(365\) −32.0001 −1.67496
\(366\) −12.7158 −0.664665
\(367\) −15.9318 −0.831631 −0.415815 0.909449i \(-0.636504\pi\)
−0.415815 + 0.909449i \(0.636504\pi\)
\(368\) −12.6652 −0.660222
\(369\) −67.3478 −3.50599
\(370\) 3.03259 0.157657
\(371\) 0 0
\(372\) 8.00568 0.415075
\(373\) 23.4289 1.21310 0.606552 0.795044i \(-0.292552\pi\)
0.606552 + 0.795044i \(0.292552\pi\)
\(374\) 19.3967 1.00298
\(375\) 11.6538 0.601800
\(376\) 1.37565 0.0709437
\(377\) 0 0
\(378\) 0 0
\(379\) 30.1041 1.54634 0.773172 0.634196i \(-0.218669\pi\)
0.773172 + 0.634196i \(0.218669\pi\)
\(380\) 5.05934 0.259539
\(381\) −15.2602 −0.781802
\(382\) −2.00728 −0.102701
\(383\) −12.8597 −0.657101 −0.328551 0.944486i \(-0.606560\pi\)
−0.328551 + 0.944486i \(0.606560\pi\)
\(384\) −39.6898 −2.02541
\(385\) 0 0
\(386\) −23.0710 −1.17428
\(387\) 62.8532 3.19501
\(388\) 3.17731 0.161304
\(389\) 36.8531 1.86853 0.934264 0.356582i \(-0.116058\pi\)
0.934264 + 0.356582i \(0.116058\pi\)
\(390\) 0 0
\(391\) −14.9852 −0.757833
\(392\) 0 0
\(393\) −67.5226 −3.40606
\(394\) 15.0903 0.760239
\(395\) 18.5347 0.932583
\(396\) −4.54057 −0.228172
\(397\) −20.3087 −1.01926 −0.509631 0.860393i \(-0.670218\pi\)
−0.509631 + 0.860393i \(0.670218\pi\)
\(398\) −13.0605 −0.654662
\(399\) 0 0
\(400\) −16.6987 −0.834935
\(401\) 19.6699 0.982268 0.491134 0.871084i \(-0.336583\pi\)
0.491134 + 0.871084i \(0.336583\pi\)
\(402\) −19.2252 −0.958864
\(403\) 0 0
\(404\) 0.657750 0.0327243
\(405\) 25.7383 1.27895
\(406\) 0 0
\(407\) −1.60216 −0.0794162
\(408\) −41.1446 −2.03696
\(409\) 23.1755 1.14595 0.572976 0.819572i \(-0.305789\pi\)
0.572976 + 0.819572i \(0.305789\pi\)
\(410\) −50.6241 −2.50015
\(411\) −3.09931 −0.152877
\(412\) 0.971699 0.0478722
\(413\) 0 0
\(414\) 25.3819 1.24745
\(415\) 5.14750 0.252681
\(416\) 0 0
\(417\) −25.4421 −1.24590
\(418\) −19.3405 −0.945975
\(419\) 28.6478 1.39953 0.699767 0.714371i \(-0.253287\pi\)
0.699767 + 0.714371i \(0.253287\pi\)
\(420\) 0 0
\(421\) −36.3249 −1.77037 −0.885184 0.465241i \(-0.845968\pi\)
−0.885184 + 0.465241i \(0.845968\pi\)
\(422\) −2.20153 −0.107169
\(423\) −3.21067 −0.156108
\(424\) −20.2185 −0.981898
\(425\) −19.7575 −0.958377
\(426\) −5.71031 −0.276666
\(427\) 0 0
\(428\) −3.98688 −0.192713
\(429\) 0 0
\(430\) 47.2456 2.27839
\(431\) 7.93131 0.382038 0.191019 0.981586i \(-0.438821\pi\)
0.191019 + 0.981586i \(0.438821\pi\)
\(432\) 40.3814 1.94285
\(433\) 9.60994 0.461824 0.230912 0.972975i \(-0.425829\pi\)
0.230912 + 0.972975i \(0.425829\pi\)
\(434\) 0 0
\(435\) −5.16911 −0.247840
\(436\) −1.13470 −0.0543421
\(437\) 14.9417 0.714760
\(438\) 49.5603 2.36808
\(439\) 3.46690 0.165466 0.0827332 0.996572i \(-0.473635\pi\)
0.0827332 + 0.996572i \(0.473635\pi\)
\(440\) 17.8698 0.851910
\(441\) 0 0
\(442\) 0 0
\(443\) 2.45667 0.116720 0.0583600 0.998296i \(-0.481413\pi\)
0.0583600 + 0.998296i \(0.481413\pi\)
\(444\) −0.649106 −0.0308052
\(445\) −30.1508 −1.42929
\(446\) 22.8432 1.08166
\(447\) −12.9192 −0.611056
\(448\) 0 0
\(449\) −22.7487 −1.07358 −0.536790 0.843716i \(-0.680363\pi\)
−0.536790 + 0.843716i \(0.680363\pi\)
\(450\) 33.4652 1.57757
\(451\) 26.7454 1.25939
\(452\) 3.33075 0.156665
\(453\) −36.6742 −1.72311
\(454\) 1.74002 0.0816633
\(455\) 0 0
\(456\) 41.0254 1.92119
\(457\) 12.2568 0.573349 0.286675 0.958028i \(-0.407450\pi\)
0.286675 + 0.958028i \(0.407450\pi\)
\(458\) 9.55070 0.446275
\(459\) 47.7781 2.23009
\(460\) 2.63680 0.122942
\(461\) 24.0650 1.12082 0.560409 0.828216i \(-0.310644\pi\)
0.560409 + 0.828216i \(0.310644\pi\)
\(462\) 0 0
\(463\) 19.3956 0.901390 0.450695 0.892678i \(-0.351176\pi\)
0.450695 + 0.892678i \(0.351176\pi\)
\(464\) −2.65879 −0.123431
\(465\) 73.5352 3.41012
\(466\) −9.39470 −0.435201
\(467\) 17.8782 0.827307 0.413653 0.910434i \(-0.364253\pi\)
0.413653 + 0.910434i \(0.364253\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2.41340 −0.111322
\(471\) 66.5031 3.06430
\(472\) −15.7227 −0.723696
\(473\) −24.9605 −1.14769
\(474\) −28.7057 −1.31850
\(475\) 19.7002 0.903906
\(476\) 0 0
\(477\) 47.1886 2.16062
\(478\) 41.8307 1.91329
\(479\) −20.7671 −0.948875 −0.474437 0.880289i \(-0.657348\pi\)
−0.474437 + 0.880289i \(0.657348\pi\)
\(480\) 15.8625 0.724019
\(481\) 0 0
\(482\) −14.9198 −0.679578
\(483\) 0 0
\(484\) −1.72490 −0.0784043
\(485\) 29.1848 1.32522
\(486\) 0.800181 0.0362970
\(487\) −19.1987 −0.869977 −0.434988 0.900436i \(-0.643248\pi\)
−0.434988 + 0.900436i \(0.643248\pi\)
\(488\) −7.12938 −0.322732
\(489\) −34.0374 −1.53923
\(490\) 0 0
\(491\) 7.91648 0.357266 0.178633 0.983916i \(-0.442833\pi\)
0.178633 + 0.983916i \(0.442833\pi\)
\(492\) 10.8357 0.488513
\(493\) −3.14581 −0.141680
\(494\) 0 0
\(495\) −41.7069 −1.87459
\(496\) 37.8237 1.69833
\(497\) 0 0
\(498\) −7.97222 −0.357244
\(499\) 39.1750 1.75371 0.876856 0.480753i \(-0.159637\pi\)
0.876856 + 0.480753i \(0.159637\pi\)
\(500\) −1.24796 −0.0558105
\(501\) 5.63459 0.251735
\(502\) 19.0340 0.849530
\(503\) −16.1533 −0.720239 −0.360120 0.932906i \(-0.617264\pi\)
−0.360120 + 0.932906i \(0.617264\pi\)
\(504\) 0 0
\(505\) 6.04169 0.268852
\(506\) −10.0798 −0.448101
\(507\) 0 0
\(508\) 1.63415 0.0725036
\(509\) 24.3177 1.07786 0.538930 0.842350i \(-0.318829\pi\)
0.538930 + 0.842350i \(0.318829\pi\)
\(510\) 72.1830 3.19632
\(511\) 0 0
\(512\) −15.0621 −0.665658
\(513\) −47.6396 −2.10334
\(514\) 29.2384 1.28965
\(515\) 8.92543 0.393302
\(516\) −10.1126 −0.445183
\(517\) 1.27503 0.0560759
\(518\) 0 0
\(519\) 16.6861 0.732439
\(520\) 0 0
\(521\) 34.4550 1.50950 0.754750 0.656012i \(-0.227758\pi\)
0.754750 + 0.656012i \(0.227758\pi\)
\(522\) 5.32838 0.233217
\(523\) −34.6865 −1.51674 −0.758368 0.651827i \(-0.774003\pi\)
−0.758368 + 0.651827i \(0.774003\pi\)
\(524\) 7.23072 0.315875
\(525\) 0 0
\(526\) 43.8971 1.91401
\(527\) 44.7520 1.94943
\(528\) −32.2314 −1.40269
\(529\) −15.2128 −0.661424
\(530\) 35.4708 1.54075
\(531\) 36.6956 1.59246
\(532\) 0 0
\(533\) 0 0
\(534\) 46.6962 2.02074
\(535\) −36.6211 −1.58327
\(536\) −10.7790 −0.465582
\(537\) 13.9614 0.602480
\(538\) 12.4221 0.535553
\(539\) 0 0
\(540\) −8.40708 −0.361783
\(541\) −12.2568 −0.526961 −0.263481 0.964665i \(-0.584871\pi\)
−0.263481 + 0.964665i \(0.584871\pi\)
\(542\) −28.3028 −1.21571
\(543\) −37.5466 −1.61128
\(544\) 9.65356 0.413893
\(545\) −10.4226 −0.446456
\(546\) 0 0
\(547\) −16.4110 −0.701686 −0.350843 0.936434i \(-0.614105\pi\)
−0.350843 + 0.936434i \(0.614105\pi\)
\(548\) 0.331892 0.0141777
\(549\) 16.6395 0.710155
\(550\) −13.2899 −0.566681
\(551\) 3.13669 0.133627
\(552\) 21.3814 0.910052
\(553\) 0 0
\(554\) −26.2545 −1.11545
\(555\) −5.96229 −0.253085
\(556\) 2.72449 0.115544
\(557\) 35.6541 1.51071 0.755356 0.655315i \(-0.227464\pi\)
0.755356 + 0.655315i \(0.227464\pi\)
\(558\) −75.8010 −3.20891
\(559\) 0 0
\(560\) 0 0
\(561\) −38.1353 −1.61007
\(562\) 17.4720 0.737013
\(563\) −10.7748 −0.454103 −0.227052 0.973883i \(-0.572909\pi\)
−0.227052 + 0.973883i \(0.572909\pi\)
\(564\) 0.516572 0.0217516
\(565\) 30.5943 1.28711
\(566\) 2.21769 0.0932163
\(567\) 0 0
\(568\) −3.20161 −0.134336
\(569\) 20.9328 0.877550 0.438775 0.898597i \(-0.355413\pi\)
0.438775 + 0.898597i \(0.355413\pi\)
\(570\) −71.9738 −3.01465
\(571\) 32.2668 1.35032 0.675161 0.737670i \(-0.264074\pi\)
0.675161 + 0.737670i \(0.264074\pi\)
\(572\) 0 0
\(573\) 3.94645 0.164865
\(574\) 0 0
\(575\) 10.2672 0.428173
\(576\) 37.8453 1.57689
\(577\) −0.197896 −0.00823851 −0.00411925 0.999992i \(-0.501311\pi\)
−0.00411925 + 0.999992i \(0.501311\pi\)
\(578\) 18.0313 0.750005
\(579\) 45.3592 1.88506
\(580\) 0.553539 0.0229844
\(581\) 0 0
\(582\) −45.2001 −1.87361
\(583\) −18.7397 −0.776120
\(584\) 27.7870 1.14984
\(585\) 0 0
\(586\) 46.7624 1.93174
\(587\) −10.1251 −0.417907 −0.208954 0.977926i \(-0.567006\pi\)
−0.208954 + 0.977926i \(0.567006\pi\)
\(588\) 0 0
\(589\) −44.6222 −1.83863
\(590\) 27.5835 1.13559
\(591\) −29.6686 −1.22040
\(592\) −3.06677 −0.126044
\(593\) −29.9345 −1.22926 −0.614631 0.788815i \(-0.710695\pi\)
−0.614631 + 0.788815i \(0.710695\pi\)
\(594\) 32.1380 1.31864
\(595\) 0 0
\(596\) 1.38346 0.0566688
\(597\) 25.6778 1.05092
\(598\) 0 0
\(599\) 26.8872 1.09858 0.549291 0.835631i \(-0.314898\pi\)
0.549291 + 0.835631i \(0.314898\pi\)
\(600\) 28.1906 1.15088
\(601\) −24.7432 −1.00930 −0.504649 0.863325i \(-0.668378\pi\)
−0.504649 + 0.863325i \(0.668378\pi\)
\(602\) 0 0
\(603\) 25.1574 1.02449
\(604\) 3.92730 0.159799
\(605\) −15.8438 −0.644143
\(606\) −9.35709 −0.380106
\(607\) 3.09235 0.125515 0.0627573 0.998029i \(-0.480011\pi\)
0.0627573 + 0.998029i \(0.480011\pi\)
\(608\) −9.62557 −0.390369
\(609\) 0 0
\(610\) 12.5076 0.506417
\(611\) 0 0
\(612\) −10.2833 −0.415679
\(613\) 31.3605 1.26664 0.633319 0.773891i \(-0.281692\pi\)
0.633319 + 0.773891i \(0.281692\pi\)
\(614\) 32.3877 1.30706
\(615\) 99.5305 4.01346
\(616\) 0 0
\(617\) 12.5311 0.504484 0.252242 0.967664i \(-0.418832\pi\)
0.252242 + 0.967664i \(0.418832\pi\)
\(618\) −13.8233 −0.556055
\(619\) −11.6112 −0.466692 −0.233346 0.972394i \(-0.574968\pi\)
−0.233346 + 0.972394i \(0.574968\pi\)
\(620\) −7.87459 −0.316251
\(621\) −24.8286 −0.996336
\(622\) 6.77374 0.271602
\(623\) 0 0
\(624\) 0 0
\(625\) −29.8593 −1.19437
\(626\) −3.48380 −0.139241
\(627\) 38.0248 1.51856
\(628\) −7.12155 −0.284181
\(629\) −3.62852 −0.144679
\(630\) 0 0
\(631\) −12.7037 −0.505728 −0.252864 0.967502i \(-0.581372\pi\)
−0.252864 + 0.967502i \(0.581372\pi\)
\(632\) −16.0945 −0.640204
\(633\) 4.32836 0.172037
\(634\) 12.3408 0.490116
\(635\) 15.0103 0.595665
\(636\) −7.59228 −0.301054
\(637\) 0 0
\(638\) −2.11603 −0.0837744
\(639\) 7.47233 0.295601
\(640\) 39.0399 1.54319
\(641\) −40.3015 −1.59181 −0.795906 0.605420i \(-0.793005\pi\)
−0.795906 + 0.605420i \(0.793005\pi\)
\(642\) 56.7170 2.23844
\(643\) 11.6643 0.459997 0.229998 0.973191i \(-0.426128\pi\)
0.229998 + 0.973191i \(0.426128\pi\)
\(644\) 0 0
\(645\) −92.8882 −3.65747
\(646\) −43.8017 −1.72336
\(647\) −11.0387 −0.433977 −0.216988 0.976174i \(-0.569623\pi\)
−0.216988 + 0.976174i \(0.569623\pi\)
\(648\) −22.3496 −0.877976
\(649\) −14.5727 −0.572030
\(650\) 0 0
\(651\) 0 0
\(652\) 3.64493 0.142747
\(653\) 19.1260 0.748458 0.374229 0.927336i \(-0.377907\pi\)
0.374229 + 0.927336i \(0.377907\pi\)
\(654\) 16.1421 0.631206
\(655\) 66.4170 2.59513
\(656\) 51.1947 1.99882
\(657\) −64.8529 −2.53015
\(658\) 0 0
\(659\) 28.8233 1.12279 0.561397 0.827546i \(-0.310264\pi\)
0.561397 + 0.827546i \(0.310264\pi\)
\(660\) 6.71032 0.261199
\(661\) 48.2385 1.87626 0.938131 0.346281i \(-0.112556\pi\)
0.938131 + 0.346281i \(0.112556\pi\)
\(662\) 17.9544 0.697816
\(663\) 0 0
\(664\) −4.46979 −0.173462
\(665\) 0 0
\(666\) 6.14600 0.238153
\(667\) 1.63476 0.0632982
\(668\) −0.603386 −0.0233457
\(669\) −44.9112 −1.73637
\(670\) 18.9104 0.730572
\(671\) −6.60794 −0.255097
\(672\) 0 0
\(673\) −33.0426 −1.27370 −0.636849 0.770989i \(-0.719762\pi\)
−0.636849 + 0.770989i \(0.719762\pi\)
\(674\) 4.00722 0.154352
\(675\) −32.7356 −1.26000
\(676\) 0 0
\(677\) 15.7186 0.604117 0.302058 0.953289i \(-0.402326\pi\)
0.302058 + 0.953289i \(0.402326\pi\)
\(678\) −47.3830 −1.81973
\(679\) 0 0
\(680\) 40.4710 1.55199
\(681\) −3.42100 −0.131093
\(682\) 30.1024 1.15268
\(683\) 33.3629 1.27660 0.638298 0.769789i \(-0.279639\pi\)
0.638298 + 0.769789i \(0.279639\pi\)
\(684\) 10.2535 0.392053
\(685\) 3.04856 0.116479
\(686\) 0 0
\(687\) −18.7773 −0.716400
\(688\) −47.7781 −1.82152
\(689\) 0 0
\(690\) −37.5109 −1.42802
\(691\) −45.3507 −1.72522 −0.862610 0.505869i \(-0.831172\pi\)
−0.862610 + 0.505869i \(0.831172\pi\)
\(692\) −1.78685 −0.0679257
\(693\) 0 0
\(694\) 7.34981 0.278995
\(695\) 25.0255 0.949270
\(696\) 4.48855 0.170138
\(697\) 60.5721 2.29433
\(698\) 26.6946 1.01041
\(699\) 18.4706 0.698623
\(700\) 0 0
\(701\) 23.5681 0.890155 0.445077 0.895492i \(-0.353176\pi\)
0.445077 + 0.895492i \(0.353176\pi\)
\(702\) 0 0
\(703\) 3.61800 0.136456
\(704\) −15.0293 −0.566438
\(705\) 4.74491 0.178704
\(706\) 10.0502 0.378244
\(707\) 0 0
\(708\) −5.90405 −0.221888
\(709\) 8.66932 0.325583 0.162792 0.986660i \(-0.447950\pi\)
0.162792 + 0.986660i \(0.447950\pi\)
\(710\) 5.61681 0.210795
\(711\) 37.5634 1.40874
\(712\) 26.1812 0.981183
\(713\) −23.2560 −0.870943
\(714\) 0 0
\(715\) 0 0
\(716\) −1.49507 −0.0558735
\(717\) −82.2420 −3.07138
\(718\) −35.1119 −1.31036
\(719\) 11.0038 0.410373 0.205186 0.978723i \(-0.434220\pi\)
0.205186 + 0.978723i \(0.434220\pi\)
\(720\) −79.8331 −2.97520
\(721\) 0 0
\(722\) 14.7302 0.548202
\(723\) 29.3333 1.09092
\(724\) 4.02071 0.149429
\(725\) 2.15538 0.0800488
\(726\) 24.5382 0.910698
\(727\) −40.7049 −1.50966 −0.754831 0.655919i \(-0.772281\pi\)
−0.754831 + 0.655919i \(0.772281\pi\)
\(728\) 0 0
\(729\) −27.7827 −1.02899
\(730\) −48.7488 −1.80427
\(731\) −56.5298 −2.09083
\(732\) −2.67716 −0.0989508
\(733\) −10.3536 −0.382418 −0.191209 0.981549i \(-0.561241\pi\)
−0.191209 + 0.981549i \(0.561241\pi\)
\(734\) −24.2704 −0.895835
\(735\) 0 0
\(736\) −5.01660 −0.184914
\(737\) −9.99062 −0.368009
\(738\) −102.597 −3.77666
\(739\) −48.5321 −1.78528 −0.892640 0.450770i \(-0.851150\pi\)
−0.892640 + 0.450770i \(0.851150\pi\)
\(740\) 0.638477 0.0234709
\(741\) 0 0
\(742\) 0 0
\(743\) −24.5215 −0.899608 −0.449804 0.893127i \(-0.648506\pi\)
−0.449804 + 0.893127i \(0.648506\pi\)
\(744\) −63.8537 −2.34099
\(745\) 12.7076 0.465572
\(746\) 35.6915 1.30676
\(747\) 10.4322 0.381694
\(748\) 4.08376 0.149317
\(749\) 0 0
\(750\) 17.7534 0.648261
\(751\) −28.0663 −1.02415 −0.512077 0.858939i \(-0.671124\pi\)
−0.512077 + 0.858939i \(0.671124\pi\)
\(752\) 2.44060 0.0889996
\(753\) −37.4222 −1.36374
\(754\) 0 0
\(755\) 36.0738 1.31286
\(756\) 0 0
\(757\) 5.70864 0.207484 0.103742 0.994604i \(-0.466918\pi\)
0.103742 + 0.994604i \(0.466918\pi\)
\(758\) 45.8605 1.66573
\(759\) 19.8175 0.719331
\(760\) −40.3536 −1.46378
\(761\) 7.69598 0.278979 0.139490 0.990224i \(-0.455454\pi\)
0.139490 + 0.990224i \(0.455454\pi\)
\(762\) −23.2473 −0.842159
\(763\) 0 0
\(764\) −0.422610 −0.0152895
\(765\) −94.4563 −3.41508
\(766\) −19.5904 −0.707832
\(767\) 0 0
\(768\) −22.4939 −0.811677
\(769\) −15.0754 −0.543634 −0.271817 0.962349i \(-0.587625\pi\)
−0.271817 + 0.962349i \(0.587625\pi\)
\(770\) 0 0
\(771\) −57.4847 −2.07026
\(772\) −4.85733 −0.174819
\(773\) −21.2624 −0.764755 −0.382378 0.924006i \(-0.624895\pi\)
−0.382378 + 0.924006i \(0.624895\pi\)
\(774\) 95.7503 3.44167
\(775\) −30.6622 −1.10142
\(776\) −25.3424 −0.909740
\(777\) 0 0
\(778\) 56.1419 2.01278
\(779\) −60.3966 −2.16393
\(780\) 0 0
\(781\) −2.96744 −0.106183
\(782\) −22.8283 −0.816340
\(783\) −5.21221 −0.186269
\(784\) 0 0
\(785\) −65.4142 −2.33473
\(786\) −102.864 −3.66902
\(787\) −9.28939 −0.331131 −0.165565 0.986199i \(-0.552945\pi\)
−0.165565 + 0.986199i \(0.552945\pi\)
\(788\) 3.17709 0.113179
\(789\) −86.3048 −3.07253
\(790\) 28.2357 1.00458
\(791\) 0 0
\(792\) 36.2158 1.28687
\(793\) 0 0
\(794\) −30.9381 −1.09795
\(795\) −69.7381 −2.47335
\(796\) −2.74973 −0.0974615
\(797\) −49.7686 −1.76289 −0.881447 0.472283i \(-0.843430\pi\)
−0.881447 + 0.472283i \(0.843430\pi\)
\(798\) 0 0
\(799\) 2.88765 0.102158
\(800\) −6.61423 −0.233848
\(801\) −61.1051 −2.15904
\(802\) 29.9650 1.05810
\(803\) 25.7547 0.908863
\(804\) −4.04764 −0.142749
\(805\) 0 0
\(806\) 0 0
\(807\) −24.4226 −0.859717
\(808\) −5.24625 −0.184563
\(809\) −9.59433 −0.337319 −0.168659 0.985674i \(-0.553944\pi\)
−0.168659 + 0.985674i \(0.553944\pi\)
\(810\) 39.2096 1.37768
\(811\) 35.6060 1.25030 0.625148 0.780506i \(-0.285038\pi\)
0.625148 + 0.780506i \(0.285038\pi\)
\(812\) 0 0
\(813\) 55.6453 1.95157
\(814\) −2.44073 −0.0855474
\(815\) 33.4801 1.17276
\(816\) −72.9966 −2.55539
\(817\) 56.3659 1.97199
\(818\) 35.3054 1.23442
\(819\) 0 0
\(820\) −10.6583 −0.372205
\(821\) 0.296780 0.0103577 0.00517884 0.999987i \(-0.498352\pi\)
0.00517884 + 0.999987i \(0.498352\pi\)
\(822\) −4.72147 −0.164680
\(823\) 22.9119 0.798658 0.399329 0.916808i \(-0.369243\pi\)
0.399329 + 0.916808i \(0.369243\pi\)
\(824\) −7.75033 −0.269995
\(825\) 26.1288 0.909687
\(826\) 0 0
\(827\) −8.50142 −0.295623 −0.147812 0.989016i \(-0.547223\pi\)
−0.147812 + 0.989016i \(0.547223\pi\)
\(828\) 5.34387 0.185712
\(829\) 54.3605 1.88802 0.944009 0.329919i \(-0.107022\pi\)
0.944009 + 0.329919i \(0.107022\pi\)
\(830\) 7.84168 0.272189
\(831\) 51.6182 1.79061
\(832\) 0 0
\(833\) 0 0
\(834\) −38.7583 −1.34209
\(835\) −5.54233 −0.191800
\(836\) −4.07192 −0.140830
\(837\) 74.1485 2.56295
\(838\) 43.6419 1.50758
\(839\) −17.0486 −0.588585 −0.294292 0.955715i \(-0.595084\pi\)
−0.294292 + 0.955715i \(0.595084\pi\)
\(840\) 0 0
\(841\) −28.6568 −0.988166
\(842\) −55.3372 −1.90705
\(843\) −34.3512 −1.18312
\(844\) −0.463507 −0.0159546
\(845\) 0 0
\(846\) −4.89112 −0.168160
\(847\) 0 0
\(848\) −35.8706 −1.23180
\(849\) −4.36013 −0.149639
\(850\) −30.0984 −1.03237
\(851\) 1.88561 0.0646379
\(852\) −1.20224 −0.0411881
\(853\) 45.0401 1.54214 0.771072 0.636747i \(-0.219721\pi\)
0.771072 + 0.636747i \(0.219721\pi\)
\(854\) 0 0
\(855\) 94.1825 3.22097
\(856\) 31.7996 1.08689
\(857\) −39.7927 −1.35929 −0.679646 0.733541i \(-0.737866\pi\)
−0.679646 + 0.733541i \(0.737866\pi\)
\(858\) 0 0
\(859\) 22.3379 0.762158 0.381079 0.924542i \(-0.375553\pi\)
0.381079 + 0.924542i \(0.375553\pi\)
\(860\) 9.94702 0.339191
\(861\) 0 0
\(862\) 12.0825 0.411532
\(863\) −14.1624 −0.482094 −0.241047 0.970513i \(-0.577491\pi\)
−0.241047 + 0.970513i \(0.577491\pi\)
\(864\) 15.9947 0.544152
\(865\) −16.4129 −0.558055
\(866\) 14.6397 0.497479
\(867\) −35.4509 −1.20397
\(868\) 0 0
\(869\) −14.9173 −0.506036
\(870\) −7.87459 −0.266974
\(871\) 0 0
\(872\) 9.05041 0.306485
\(873\) 59.1474 2.00184
\(874\) 22.7622 0.769941
\(875\) 0 0
\(876\) 10.4343 0.352544
\(877\) −43.9546 −1.48424 −0.742121 0.670266i \(-0.766180\pi\)
−0.742121 + 0.670266i \(0.766180\pi\)
\(878\) 5.28147 0.178241
\(879\) −91.9382 −3.10100
\(880\) 31.7037 1.06873
\(881\) 15.3849 0.518331 0.259165 0.965833i \(-0.416553\pi\)
0.259165 + 0.965833i \(0.416553\pi\)
\(882\) 0 0
\(883\) −44.5262 −1.49842 −0.749212 0.662330i \(-0.769568\pi\)
−0.749212 + 0.662330i \(0.769568\pi\)
\(884\) 0 0
\(885\) −54.2310 −1.82295
\(886\) 3.74248 0.125731
\(887\) 16.1072 0.540826 0.270413 0.962744i \(-0.412840\pi\)
0.270413 + 0.962744i \(0.412840\pi\)
\(888\) 5.17731 0.173739
\(889\) 0 0
\(890\) −45.9316 −1.53963
\(891\) −20.7150 −0.693978
\(892\) 4.80936 0.161029
\(893\) −2.87928 −0.0963515
\(894\) −19.6810 −0.658231
\(895\) −13.7328 −0.459037
\(896\) 0 0
\(897\) 0 0
\(898\) −34.6553 −1.15646
\(899\) −4.88208 −0.162827
\(900\) 7.04572 0.234857
\(901\) −42.4411 −1.41392
\(902\) 40.7439 1.35662
\(903\) 0 0
\(904\) −26.5663 −0.883582
\(905\) 36.9318 1.22766
\(906\) −55.8694 −1.85614
\(907\) 18.6037 0.617725 0.308862 0.951107i \(-0.400052\pi\)
0.308862 + 0.951107i \(0.400052\pi\)
\(908\) 0.366341 0.0121575
\(909\) 12.2444 0.406121
\(910\) 0 0
\(911\) 38.1801 1.26496 0.632481 0.774576i \(-0.282037\pi\)
0.632481 + 0.774576i \(0.282037\pi\)
\(912\) 72.7850 2.41015
\(913\) −4.14287 −0.137109
\(914\) 18.6720 0.617614
\(915\) −24.5908 −0.812946
\(916\) 2.01079 0.0664383
\(917\) 0 0
\(918\) 72.7850 2.40226
\(919\) −16.7629 −0.552956 −0.276478 0.961020i \(-0.589167\pi\)
−0.276478 + 0.961020i \(0.589167\pi\)
\(920\) −21.0313 −0.693381
\(921\) −63.6765 −2.09821
\(922\) 36.6605 1.20735
\(923\) 0 0
\(924\) 0 0
\(925\) 2.48612 0.0817430
\(926\) 29.5472 0.970980
\(927\) 18.0887 0.594111
\(928\) −1.05313 −0.0345706
\(929\) −6.58871 −0.216169 −0.108084 0.994142i \(-0.534472\pi\)
−0.108084 + 0.994142i \(0.534472\pi\)
\(930\) 112.023 3.67339
\(931\) 0 0
\(932\) −1.97794 −0.0647897
\(933\) −13.3176 −0.436000
\(934\) 27.2356 0.891177
\(935\) 37.5109 1.22674
\(936\) 0 0
\(937\) 39.0958 1.27721 0.638603 0.769537i \(-0.279513\pi\)
0.638603 + 0.769537i \(0.279513\pi\)
\(938\) 0 0
\(939\) 6.84940 0.223522
\(940\) −0.508114 −0.0165728
\(941\) 13.2351 0.431452 0.215726 0.976454i \(-0.430788\pi\)
0.215726 + 0.976454i \(0.430788\pi\)
\(942\) 101.311 3.30088
\(943\) −31.4771 −1.02504
\(944\) −27.8943 −0.907884
\(945\) 0 0
\(946\) −38.0248 −1.23629
\(947\) −50.1427 −1.62942 −0.814709 0.579870i \(-0.803103\pi\)
−0.814709 + 0.579870i \(0.803103\pi\)
\(948\) −6.04366 −0.196289
\(949\) 0 0
\(950\) 30.0112 0.973690
\(951\) −24.2629 −0.786778
\(952\) 0 0
\(953\) 29.9832 0.971252 0.485626 0.874167i \(-0.338592\pi\)
0.485626 + 0.874167i \(0.338592\pi\)
\(954\) 71.8869 2.32742
\(955\) −3.88184 −0.125613
\(956\) 8.80697 0.284838
\(957\) 4.16026 0.134482
\(958\) −31.6366 −1.02213
\(959\) 0 0
\(960\) −55.9300 −1.80513
\(961\) 38.4521 1.24039
\(962\) 0 0
\(963\) −74.2180 −2.39164
\(964\) −3.14119 −0.101171
\(965\) −44.6165 −1.43626
\(966\) 0 0
\(967\) −37.8356 −1.21671 −0.608356 0.793665i \(-0.708171\pi\)
−0.608356 + 0.793665i \(0.708171\pi\)
\(968\) 13.7579 0.442195
\(969\) 86.1172 2.76648
\(970\) 44.4600 1.42753
\(971\) −15.3432 −0.492388 −0.246194 0.969221i \(-0.579180\pi\)
−0.246194 + 0.969221i \(0.579180\pi\)
\(972\) 0.168469 0.00540364
\(973\) 0 0
\(974\) −29.2472 −0.937142
\(975\) 0 0
\(976\) −12.6486 −0.404871
\(977\) −26.5487 −0.849367 −0.424684 0.905342i \(-0.639615\pi\)
−0.424684 + 0.905342i \(0.639615\pi\)
\(978\) −51.8525 −1.65806
\(979\) 24.2663 0.775555
\(980\) 0 0
\(981\) −21.1230 −0.674406
\(982\) 12.0599 0.384848
\(983\) −8.02003 −0.255799 −0.127900 0.991787i \(-0.540824\pi\)
−0.127900 + 0.991787i \(0.540824\pi\)
\(984\) −86.4265 −2.75518
\(985\) 29.1828 0.929842
\(986\) −4.79231 −0.152618
\(987\) 0 0
\(988\) 0 0
\(989\) 29.3765 0.934117
\(990\) −63.5361 −2.01931
\(991\) 49.0958 1.55958 0.779789 0.626042i \(-0.215326\pi\)
0.779789 + 0.626042i \(0.215326\pi\)
\(992\) 14.9817 0.475669
\(993\) −35.2995 −1.12020
\(994\) 0 0
\(995\) −25.2573 −0.800711
\(996\) −1.67846 −0.0531840
\(997\) −19.9476 −0.631746 −0.315873 0.948801i \(-0.602297\pi\)
−0.315873 + 0.948801i \(0.602297\pi\)
\(998\) 59.6790 1.88910
\(999\) −6.01201 −0.190212
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 8281.2.a.cs.1.11 16
7.6 odd 2 inner 8281.2.a.cs.1.12 16
13.5 odd 4 637.2.c.g.246.5 16
13.8 odd 4 637.2.c.g.246.11 yes 16
13.12 even 2 inner 8281.2.a.cs.1.5 16
91.5 even 12 637.2.r.g.116.5 32
91.18 odd 12 637.2.r.g.324.12 32
91.31 even 12 637.2.r.g.324.11 32
91.34 even 4 637.2.c.g.246.12 yes 16
91.44 odd 12 637.2.r.g.116.6 32
91.47 even 12 637.2.r.g.116.11 32
91.60 odd 12 637.2.r.g.324.6 32
91.73 even 12 637.2.r.g.324.5 32
91.83 even 4 637.2.c.g.246.6 yes 16
91.86 odd 12 637.2.r.g.116.12 32
91.90 odd 2 inner 8281.2.a.cs.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.c.g.246.5 16 13.5 odd 4
637.2.c.g.246.6 yes 16 91.83 even 4
637.2.c.g.246.11 yes 16 13.8 odd 4
637.2.c.g.246.12 yes 16 91.34 even 4
637.2.r.g.116.5 32 91.5 even 12
637.2.r.g.116.6 32 91.44 odd 12
637.2.r.g.116.11 32 91.47 even 12
637.2.r.g.116.12 32 91.86 odd 12
637.2.r.g.324.5 32 91.73 even 12
637.2.r.g.324.6 32 91.60 odd 12
637.2.r.g.324.11 32 91.31 even 12
637.2.r.g.324.12 32 91.18 odd 12
8281.2.a.cs.1.5 16 13.12 even 2 inner
8281.2.a.cs.1.6 16 91.90 odd 2 inner
8281.2.a.cs.1.11 16 1.1 even 1 trivial
8281.2.a.cs.1.12 16 7.6 odd 2 inner