Properties

Label 8281.2.a.cm.1.8
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 32x^{10} + 393x^{8} - 2334x^{6} + 6955x^{4} - 9591x^{2} + 4537 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.55274\) 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)
 
\(f(q)\) \(=\) \(q+0.380581 q^{2} +1.55274 q^{3} -1.85516 q^{4} -3.06093 q^{5} +0.590942 q^{6} -1.46720 q^{8} -0.589012 q^{9} +O(q^{10})\) \(q+0.380581 q^{2} +1.55274 q^{3} -1.85516 q^{4} -3.06093 q^{5} +0.590942 q^{6} -1.46720 q^{8} -0.589012 q^{9} -1.16493 q^{10} -3.73748 q^{11} -2.88057 q^{12} -4.75281 q^{15} +3.15193 q^{16} +7.73178 q^{17} -0.224167 q^{18} +1.81919 q^{19} +5.67850 q^{20} -1.42241 q^{22} +7.93036 q^{23} -2.27817 q^{24} +4.36927 q^{25} -5.57279 q^{27} +1.73572 q^{29} -1.80883 q^{30} -7.54905 q^{31} +4.13396 q^{32} -5.80331 q^{33} +2.94257 q^{34} +1.09271 q^{36} -0.792076 q^{37} +0.692348 q^{38} +4.49099 q^{40} -0.638943 q^{41} +6.02523 q^{43} +6.93361 q^{44} +1.80292 q^{45} +3.01815 q^{46} -4.04419 q^{47} +4.89411 q^{48} +1.66286 q^{50} +12.0054 q^{51} -1.35719 q^{53} -2.12090 q^{54} +11.4401 q^{55} +2.82472 q^{57} +0.660583 q^{58} +9.65105 q^{59} +8.81722 q^{60} -13.1507 q^{61} -2.87303 q^{62} -4.73055 q^{64} -2.20863 q^{66} +9.49985 q^{67} -14.3437 q^{68} +12.3138 q^{69} -3.89385 q^{71} +0.864198 q^{72} +13.7495 q^{73} -0.301449 q^{74} +6.78433 q^{75} -3.37488 q^{76} -11.3589 q^{79} -9.64782 q^{80} -6.88603 q^{81} -0.243170 q^{82} -0.0617751 q^{83} -23.6664 q^{85} +2.29309 q^{86} +2.69512 q^{87} +5.48363 q^{88} -13.4132 q^{89} +0.686158 q^{90} -14.7121 q^{92} -11.7217 q^{93} -1.53914 q^{94} -5.56840 q^{95} +6.41895 q^{96} -12.0577 q^{97} +2.20142 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} - 18 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} - 18 q^{8} + 28 q^{9} - 20 q^{11} - 30 q^{15} + 16 q^{16} - 32 q^{18} - 16 q^{22} + 40 q^{25} + 2 q^{29} + 52 q^{30} - 20 q^{32} + 40 q^{36} - 10 q^{37} + 28 q^{43} + 12 q^{44} + 36 q^{46} - 24 q^{50} - 20 q^{51} - 30 q^{53} - 32 q^{57} + 30 q^{58} + 50 q^{64} + 32 q^{67} - 86 q^{71} - 94 q^{72} + 28 q^{79} + 8 q^{81} - 30 q^{85} - 46 q^{86} + 48 q^{88} - 24 q^{92} - 18 q^{93} - 30 q^{95} - 66 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.380581 0.269111 0.134556 0.990906i \(-0.457039\pi\)
0.134556 + 0.990906i \(0.457039\pi\)
\(3\) 1.55274 0.896472 0.448236 0.893915i \(-0.352052\pi\)
0.448236 + 0.893915i \(0.352052\pi\)
\(4\) −1.85516 −0.927579
\(5\) −3.06093 −1.36889 −0.684444 0.729065i \(-0.739955\pi\)
−0.684444 + 0.729065i \(0.739955\pi\)
\(6\) 0.590942 0.241251
\(7\) 0 0
\(8\) −1.46720 −0.518734
\(9\) −0.589012 −0.196337
\(10\) −1.16493 −0.368384
\(11\) −3.73748 −1.12689 −0.563446 0.826153i \(-0.690525\pi\)
−0.563446 + 0.826153i \(0.690525\pi\)
\(12\) −2.88057 −0.831549
\(13\) 0 0
\(14\) 0 0
\(15\) −4.75281 −1.22717
\(16\) 3.15193 0.787982
\(17\) 7.73178 1.87523 0.937616 0.347674i \(-0.113028\pi\)
0.937616 + 0.347674i \(0.113028\pi\)
\(18\) −0.224167 −0.0528366
\(19\) 1.81919 0.417350 0.208675 0.977985i \(-0.433085\pi\)
0.208675 + 0.977985i \(0.433085\pi\)
\(20\) 5.67850 1.26975
\(21\) 0 0
\(22\) −1.42241 −0.303260
\(23\) 7.93036 1.65360 0.826798 0.562500i \(-0.190160\pi\)
0.826798 + 0.562500i \(0.190160\pi\)
\(24\) −2.27817 −0.465030
\(25\) 4.36927 0.873855
\(26\) 0 0
\(27\) −5.57279 −1.07248
\(28\) 0 0
\(29\) 1.73572 0.322315 0.161158 0.986929i \(-0.448477\pi\)
0.161158 + 0.986929i \(0.448477\pi\)
\(30\) −1.80883 −0.330246
\(31\) −7.54905 −1.35585 −0.677925 0.735131i \(-0.737120\pi\)
−0.677925 + 0.735131i \(0.737120\pi\)
\(32\) 4.13396 0.730789
\(33\) −5.80331 −1.01023
\(34\) 2.94257 0.504646
\(35\) 0 0
\(36\) 1.09271 0.182118
\(37\) −0.792076 −0.130217 −0.0651083 0.997878i \(-0.520739\pi\)
−0.0651083 + 0.997878i \(0.520739\pi\)
\(38\) 0.692348 0.112314
\(39\) 0 0
\(40\) 4.49099 0.710088
\(41\) −0.638943 −0.0997862 −0.0498931 0.998755i \(-0.515888\pi\)
−0.0498931 + 0.998755i \(0.515888\pi\)
\(42\) 0 0
\(43\) 6.02523 0.918839 0.459420 0.888219i \(-0.348057\pi\)
0.459420 + 0.888219i \(0.348057\pi\)
\(44\) 6.93361 1.04528
\(45\) 1.80292 0.268764
\(46\) 3.01815 0.445001
\(47\) −4.04419 −0.589905 −0.294953 0.955512i \(-0.595304\pi\)
−0.294953 + 0.955512i \(0.595304\pi\)
\(48\) 4.89411 0.706404
\(49\) 0 0
\(50\) 1.66286 0.235164
\(51\) 12.0054 1.68109
\(52\) 0 0
\(53\) −1.35719 −0.186424 −0.0932120 0.995646i \(-0.529713\pi\)
−0.0932120 + 0.995646i \(0.529713\pi\)
\(54\) −2.12090 −0.288618
\(55\) 11.4401 1.54259
\(56\) 0 0
\(57\) 2.82472 0.374143
\(58\) 0.660583 0.0867387
\(59\) 9.65105 1.25646 0.628230 0.778028i \(-0.283780\pi\)
0.628230 + 0.778028i \(0.283780\pi\)
\(60\) 8.81722 1.13830
\(61\) −13.1507 −1.68378 −0.841890 0.539649i \(-0.818557\pi\)
−0.841890 + 0.539649i \(0.818557\pi\)
\(62\) −2.87303 −0.364875
\(63\) 0 0
\(64\) −4.73055 −0.591318
\(65\) 0 0
\(66\) −2.20863 −0.271864
\(67\) 9.49985 1.16059 0.580295 0.814406i \(-0.302937\pi\)
0.580295 + 0.814406i \(0.302937\pi\)
\(68\) −14.3437 −1.73943
\(69\) 12.3138 1.48240
\(70\) 0 0
\(71\) −3.89385 −0.462115 −0.231058 0.972940i \(-0.574219\pi\)
−0.231058 + 0.972940i \(0.574219\pi\)
\(72\) 0.864198 0.101847
\(73\) 13.7495 1.60925 0.804626 0.593782i \(-0.202366\pi\)
0.804626 + 0.593782i \(0.202366\pi\)
\(74\) −0.301449 −0.0350428
\(75\) 6.78433 0.783387
\(76\) −3.37488 −0.387125
\(77\) 0 0
\(78\) 0 0
\(79\) −11.3589 −1.27797 −0.638987 0.769217i \(-0.720646\pi\)
−0.638987 + 0.769217i \(0.720646\pi\)
\(80\) −9.64782 −1.07866
\(81\) −6.88603 −0.765115
\(82\) −0.243170 −0.0268536
\(83\) −0.0617751 −0.00678070 −0.00339035 0.999994i \(-0.501079\pi\)
−0.00339035 + 0.999994i \(0.501079\pi\)
\(84\) 0 0
\(85\) −23.6664 −2.56698
\(86\) 2.29309 0.247270
\(87\) 2.69512 0.288947
\(88\) 5.48363 0.584557
\(89\) −13.4132 −1.42179 −0.710896 0.703298i \(-0.751710\pi\)
−0.710896 + 0.703298i \(0.751710\pi\)
\(90\) 0.686158 0.0723274
\(91\) 0 0
\(92\) −14.7121 −1.53384
\(93\) −11.7217 −1.21548
\(94\) −1.53914 −0.158750
\(95\) −5.56840 −0.571306
\(96\) 6.41895 0.655132
\(97\) −12.0577 −1.22428 −0.612138 0.790751i \(-0.709690\pi\)
−0.612138 + 0.790751i \(0.709690\pi\)
\(98\) 0 0
\(99\) 2.20142 0.221251
\(100\) −8.10570 −0.810570
\(101\) 11.4174 1.13608 0.568039 0.823001i \(-0.307702\pi\)
0.568039 + 0.823001i \(0.307702\pi\)
\(102\) 4.56903 0.452401
\(103\) 10.8166 1.06579 0.532894 0.846182i \(-0.321104\pi\)
0.532894 + 0.846182i \(0.321104\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −0.516520 −0.0501688
\(107\) −5.94982 −0.575191 −0.287595 0.957752i \(-0.592856\pi\)
−0.287595 + 0.957752i \(0.592856\pi\)
\(108\) 10.3384 0.994813
\(109\) 13.2566 1.26975 0.634874 0.772616i \(-0.281052\pi\)
0.634874 + 0.772616i \(0.281052\pi\)
\(110\) 4.35390 0.415128
\(111\) −1.22989 −0.116736
\(112\) 0 0
\(113\) −3.61581 −0.340147 −0.170074 0.985431i \(-0.554401\pi\)
−0.170074 + 0.985431i \(0.554401\pi\)
\(114\) 1.07503 0.100686
\(115\) −24.2743 −2.26359
\(116\) −3.22004 −0.298973
\(117\) 0 0
\(118\) 3.67301 0.338128
\(119\) 0 0
\(120\) 6.97333 0.636575
\(121\) 2.96873 0.269885
\(122\) −5.00492 −0.453125
\(123\) −0.992110 −0.0894556
\(124\) 14.0047 1.25766
\(125\) 1.93060 0.172678
\(126\) 0 0
\(127\) −13.1895 −1.17038 −0.585190 0.810896i \(-0.698980\pi\)
−0.585190 + 0.810896i \(0.698980\pi\)
\(128\) −10.0683 −0.889919
\(129\) 9.35559 0.823714
\(130\) 0 0
\(131\) 4.72057 0.412438 0.206219 0.978506i \(-0.433884\pi\)
0.206219 + 0.978506i \(0.433884\pi\)
\(132\) 10.7661 0.937066
\(133\) 0 0
\(134\) 3.61546 0.312328
\(135\) 17.0579 1.46811
\(136\) −11.3441 −0.972746
\(137\) 18.5298 1.58311 0.791556 0.611097i \(-0.209271\pi\)
0.791556 + 0.611097i \(0.209271\pi\)
\(138\) 4.68638 0.398932
\(139\) 10.4140 0.883302 0.441651 0.897187i \(-0.354393\pi\)
0.441651 + 0.897187i \(0.354393\pi\)
\(140\) 0 0
\(141\) −6.27955 −0.528834
\(142\) −1.48193 −0.124360
\(143\) 0 0
\(144\) −1.85652 −0.154710
\(145\) −5.31292 −0.441214
\(146\) 5.23278 0.433068
\(147\) 0 0
\(148\) 1.46943 0.120786
\(149\) −19.7675 −1.61942 −0.809708 0.586833i \(-0.800374\pi\)
−0.809708 + 0.586833i \(0.800374\pi\)
\(150\) 2.58199 0.210818
\(151\) −3.27167 −0.266245 −0.133122 0.991100i \(-0.542500\pi\)
−0.133122 + 0.991100i \(0.542500\pi\)
\(152\) −2.66911 −0.216494
\(153\) −4.55411 −0.368178
\(154\) 0 0
\(155\) 23.1071 1.85601
\(156\) 0 0
\(157\) 7.05962 0.563419 0.281709 0.959500i \(-0.409099\pi\)
0.281709 + 0.959500i \(0.409099\pi\)
\(158\) −4.32298 −0.343918
\(159\) −2.10735 −0.167124
\(160\) −12.6538 −1.00037
\(161\) 0 0
\(162\) −2.62069 −0.205901
\(163\) −11.5801 −0.907020 −0.453510 0.891251i \(-0.649828\pi\)
−0.453510 + 0.891251i \(0.649828\pi\)
\(164\) 1.18534 0.0925596
\(165\) 17.7635 1.38289
\(166\) −0.0235104 −0.00182476
\(167\) 6.42722 0.497353 0.248677 0.968587i \(-0.420004\pi\)
0.248677 + 0.968587i \(0.420004\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −9.00699 −0.690804
\(171\) −1.07152 −0.0819413
\(172\) −11.1778 −0.852296
\(173\) 2.00814 0.152676 0.0763381 0.997082i \(-0.475677\pi\)
0.0763381 + 0.997082i \(0.475677\pi\)
\(174\) 1.02571 0.0777589
\(175\) 0 0
\(176\) −11.7803 −0.887970
\(177\) 14.9855 1.12638
\(178\) −5.10479 −0.382620
\(179\) −16.5700 −1.23850 −0.619251 0.785193i \(-0.712563\pi\)
−0.619251 + 0.785193i \(0.712563\pi\)
\(180\) −3.34470 −0.249300
\(181\) −16.6654 −1.23873 −0.619364 0.785104i \(-0.712609\pi\)
−0.619364 + 0.785104i \(0.712609\pi\)
\(182\) 0 0
\(183\) −20.4196 −1.50946
\(184\) −11.6354 −0.857775
\(185\) 2.42449 0.178252
\(186\) −4.46105 −0.327100
\(187\) −28.8973 −2.11318
\(188\) 7.50260 0.547184
\(189\) 0 0
\(190\) −2.11923 −0.153745
\(191\) −23.2729 −1.68397 −0.841983 0.539504i \(-0.818612\pi\)
−0.841983 + 0.539504i \(0.818612\pi\)
\(192\) −7.34529 −0.530100
\(193\) −18.7077 −1.34661 −0.673304 0.739366i \(-0.735125\pi\)
−0.673304 + 0.739366i \(0.735125\pi\)
\(194\) −4.58894 −0.329467
\(195\) 0 0
\(196\) 0 0
\(197\) −24.5601 −1.74983 −0.874916 0.484274i \(-0.839084\pi\)
−0.874916 + 0.484274i \(0.839084\pi\)
\(198\) 0.837818 0.0595411
\(199\) −13.0279 −0.923521 −0.461761 0.887005i \(-0.652782\pi\)
−0.461761 + 0.887005i \(0.652782\pi\)
\(200\) −6.41060 −0.453298
\(201\) 14.7508 1.04044
\(202\) 4.34527 0.305732
\(203\) 0 0
\(204\) −22.2719 −1.55935
\(205\) 1.95576 0.136596
\(206\) 4.11658 0.286816
\(207\) −4.67108 −0.324662
\(208\) 0 0
\(209\) −6.79917 −0.470308
\(210\) 0 0
\(211\) 16.3026 1.12232 0.561160 0.827708i \(-0.310355\pi\)
0.561160 + 0.827708i \(0.310355\pi\)
\(212\) 2.51780 0.172923
\(213\) −6.04612 −0.414273
\(214\) −2.26439 −0.154790
\(215\) −18.4428 −1.25779
\(216\) 8.17639 0.556333
\(217\) 0 0
\(218\) 5.04520 0.341704
\(219\) 21.3493 1.44265
\(220\) −21.2233 −1.43087
\(221\) 0 0
\(222\) −0.468071 −0.0314149
\(223\) −13.1556 −0.880962 −0.440481 0.897762i \(-0.645192\pi\)
−0.440481 + 0.897762i \(0.645192\pi\)
\(224\) 0 0
\(225\) −2.57355 −0.171570
\(226\) −1.37611 −0.0915375
\(227\) 19.8288 1.31608 0.658041 0.752982i \(-0.271385\pi\)
0.658041 + 0.752982i \(0.271385\pi\)
\(228\) −5.24029 −0.347047
\(229\) −12.3921 −0.818895 −0.409447 0.912334i \(-0.634278\pi\)
−0.409447 + 0.912334i \(0.634278\pi\)
\(230\) −9.23833 −0.609157
\(231\) 0 0
\(232\) −2.54665 −0.167196
\(233\) 7.21563 0.472711 0.236356 0.971667i \(-0.424047\pi\)
0.236356 + 0.971667i \(0.424047\pi\)
\(234\) 0 0
\(235\) 12.3790 0.807514
\(236\) −17.9042 −1.16547
\(237\) −17.6374 −1.14567
\(238\) 0 0
\(239\) 2.77492 0.179495 0.0897473 0.995965i \(-0.471394\pi\)
0.0897473 + 0.995965i \(0.471394\pi\)
\(240\) −14.9805 −0.966988
\(241\) −25.8988 −1.66829 −0.834143 0.551548i \(-0.814037\pi\)
−0.834143 + 0.551548i \(0.814037\pi\)
\(242\) 1.12984 0.0726291
\(243\) 6.02617 0.386579
\(244\) 24.3967 1.56184
\(245\) 0 0
\(246\) −0.377578 −0.0240735
\(247\) 0 0
\(248\) 11.0760 0.703325
\(249\) −0.0959204 −0.00607871
\(250\) 0.734751 0.0464698
\(251\) −12.7038 −0.801859 −0.400929 0.916109i \(-0.631313\pi\)
−0.400929 + 0.916109i \(0.631313\pi\)
\(252\) 0 0
\(253\) −29.6396 −1.86342
\(254\) −5.01968 −0.314963
\(255\) −36.7477 −2.30123
\(256\) 5.62929 0.351831
\(257\) −8.15704 −0.508822 −0.254411 0.967096i \(-0.581882\pi\)
−0.254411 + 0.967096i \(0.581882\pi\)
\(258\) 3.56056 0.221671
\(259\) 0 0
\(260\) 0 0
\(261\) −1.02236 −0.0632825
\(262\) 1.79656 0.110992
\(263\) −15.3357 −0.945637 −0.472819 0.881160i \(-0.656763\pi\)
−0.472819 + 0.881160i \(0.656763\pi\)
\(264\) 8.51462 0.524039
\(265\) 4.15425 0.255194
\(266\) 0 0
\(267\) −20.8271 −1.27460
\(268\) −17.6237 −1.07654
\(269\) −16.8969 −1.03022 −0.515110 0.857124i \(-0.672249\pi\)
−0.515110 + 0.857124i \(0.672249\pi\)
\(270\) 6.49191 0.395085
\(271\) 8.83794 0.536866 0.268433 0.963298i \(-0.413494\pi\)
0.268433 + 0.963298i \(0.413494\pi\)
\(272\) 24.3700 1.47765
\(273\) 0 0
\(274\) 7.05211 0.426033
\(275\) −16.3301 −0.984740
\(276\) −22.8440 −1.37505
\(277\) 2.99750 0.180102 0.0900511 0.995937i \(-0.471297\pi\)
0.0900511 + 0.995937i \(0.471297\pi\)
\(278\) 3.96337 0.237707
\(279\) 4.44648 0.266204
\(280\) 0 0
\(281\) −4.22958 −0.252316 −0.126158 0.992010i \(-0.540265\pi\)
−0.126158 + 0.992010i \(0.540265\pi\)
\(282\) −2.38988 −0.142315
\(283\) −3.94077 −0.234255 −0.117127 0.993117i \(-0.537369\pi\)
−0.117127 + 0.993117i \(0.537369\pi\)
\(284\) 7.22371 0.428648
\(285\) −8.64625 −0.512160
\(286\) 0 0
\(287\) 0 0
\(288\) −2.43495 −0.143481
\(289\) 42.7804 2.51649
\(290\) −2.02200 −0.118736
\(291\) −18.7225 −1.09753
\(292\) −25.5074 −1.49271
\(293\) −17.4163 −1.01747 −0.508735 0.860923i \(-0.669887\pi\)
−0.508735 + 0.860923i \(0.669887\pi\)
\(294\) 0 0
\(295\) −29.5412 −1.71995
\(296\) 1.16213 0.0675477
\(297\) 20.8282 1.20857
\(298\) −7.52313 −0.435803
\(299\) 0 0
\(300\) −12.5860 −0.726653
\(301\) 0 0
\(302\) −1.24514 −0.0716495
\(303\) 17.7283 1.01846
\(304\) 5.73394 0.328864
\(305\) 40.2535 2.30491
\(306\) −1.73321 −0.0990808
\(307\) −24.2434 −1.38365 −0.691823 0.722067i \(-0.743192\pi\)
−0.691823 + 0.722067i \(0.743192\pi\)
\(308\) 0 0
\(309\) 16.7953 0.955450
\(310\) 8.79412 0.499473
\(311\) 10.1457 0.575309 0.287655 0.957734i \(-0.407124\pi\)
0.287655 + 0.957734i \(0.407124\pi\)
\(312\) 0 0
\(313\) 7.32223 0.413877 0.206938 0.978354i \(-0.433650\pi\)
0.206938 + 0.978354i \(0.433650\pi\)
\(314\) 2.68676 0.151622
\(315\) 0 0
\(316\) 21.0725 1.18542
\(317\) −9.20624 −0.517074 −0.258537 0.966001i \(-0.583240\pi\)
−0.258537 + 0.966001i \(0.583240\pi\)
\(318\) −0.802019 −0.0449750
\(319\) −6.48722 −0.363214
\(320\) 14.4799 0.809449
\(321\) −9.23849 −0.515642
\(322\) 0 0
\(323\) 14.0655 0.782628
\(324\) 12.7747 0.709704
\(325\) 0 0
\(326\) −4.40715 −0.244090
\(327\) 20.5839 1.13829
\(328\) 0.937458 0.0517624
\(329\) 0 0
\(330\) 6.76046 0.372151
\(331\) −1.39303 −0.0765677 −0.0382839 0.999267i \(-0.512189\pi\)
−0.0382839 + 0.999267i \(0.512189\pi\)
\(332\) 0.114603 0.00628964
\(333\) 0.466542 0.0255663
\(334\) 2.44608 0.133844
\(335\) −29.0783 −1.58872
\(336\) 0 0
\(337\) 14.7496 0.803463 0.401731 0.915758i \(-0.368409\pi\)
0.401731 + 0.915758i \(0.368409\pi\)
\(338\) 0 0
\(339\) −5.61440 −0.304933
\(340\) 43.9049 2.38108
\(341\) 28.2144 1.52790
\(342\) −0.407801 −0.0220514
\(343\) 0 0
\(344\) −8.84022 −0.476633
\(345\) −37.6915 −2.02924
\(346\) 0.764261 0.0410869
\(347\) −1.76919 −0.0949751 −0.0474876 0.998872i \(-0.515121\pi\)
−0.0474876 + 0.998872i \(0.515121\pi\)
\(348\) −4.99987 −0.268021
\(349\) 22.7396 1.21722 0.608611 0.793468i \(-0.291727\pi\)
0.608611 + 0.793468i \(0.291727\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.4506 −0.823520
\(353\) 10.6199 0.565238 0.282619 0.959232i \(-0.408797\pi\)
0.282619 + 0.959232i \(0.408797\pi\)
\(354\) 5.70321 0.303122
\(355\) 11.9188 0.632584
\(356\) 24.8835 1.31882
\(357\) 0 0
\(358\) −6.30624 −0.333295
\(359\) −22.5223 −1.18868 −0.594341 0.804213i \(-0.702587\pi\)
−0.594341 + 0.804213i \(0.702587\pi\)
\(360\) −2.64525 −0.139417
\(361\) −15.6906 −0.825819
\(362\) −6.34253 −0.333356
\(363\) 4.60966 0.241944
\(364\) 0 0
\(365\) −42.0861 −2.20289
\(366\) −7.77133 −0.406214
\(367\) −4.19604 −0.219032 −0.109516 0.993985i \(-0.534930\pi\)
−0.109516 + 0.993985i \(0.534930\pi\)
\(368\) 24.9959 1.30300
\(369\) 0.376345 0.0195917
\(370\) 0.922714 0.0479696
\(371\) 0 0
\(372\) 21.7456 1.12746
\(373\) −23.0008 −1.19093 −0.595467 0.803380i \(-0.703033\pi\)
−0.595467 + 0.803380i \(0.703033\pi\)
\(374\) −10.9978 −0.568682
\(375\) 2.99772 0.154801
\(376\) 5.93363 0.306004
\(377\) 0 0
\(378\) 0 0
\(379\) −36.1835 −1.85862 −0.929312 0.369296i \(-0.879599\pi\)
−0.929312 + 0.369296i \(0.879599\pi\)
\(380\) 10.3303 0.529931
\(381\) −20.4798 −1.04921
\(382\) −8.85722 −0.453175
\(383\) −5.13170 −0.262218 −0.131109 0.991368i \(-0.541854\pi\)
−0.131109 + 0.991368i \(0.541854\pi\)
\(384\) −15.6334 −0.797788
\(385\) 0 0
\(386\) −7.11979 −0.362388
\(387\) −3.54893 −0.180402
\(388\) 22.3690 1.13561
\(389\) −6.14146 −0.311385 −0.155692 0.987806i \(-0.549761\pi\)
−0.155692 + 0.987806i \(0.549761\pi\)
\(390\) 0 0
\(391\) 61.3158 3.10087
\(392\) 0 0
\(393\) 7.32979 0.369739
\(394\) −9.34710 −0.470900
\(395\) 34.7687 1.74940
\(396\) −4.08398 −0.205228
\(397\) 27.2224 1.36625 0.683126 0.730300i \(-0.260620\pi\)
0.683126 + 0.730300i \(0.260620\pi\)
\(398\) −4.95816 −0.248530
\(399\) 0 0
\(400\) 13.7716 0.688582
\(401\) −19.2585 −0.961724 −0.480862 0.876796i \(-0.659676\pi\)
−0.480862 + 0.876796i \(0.659676\pi\)
\(402\) 5.61386 0.279994
\(403\) 0 0
\(404\) −21.1812 −1.05380
\(405\) 21.0776 1.04736
\(406\) 0 0
\(407\) 2.96037 0.146740
\(408\) −17.6143 −0.872040
\(409\) 14.3384 0.708988 0.354494 0.935058i \(-0.384653\pi\)
0.354494 + 0.935058i \(0.384653\pi\)
\(410\) 0.744325 0.0367596
\(411\) 28.7720 1.41922
\(412\) −20.0664 −0.988603
\(413\) 0 0
\(414\) −1.77772 −0.0873703
\(415\) 0.189089 0.00928202
\(416\) 0 0
\(417\) 16.1702 0.791856
\(418\) −2.58763 −0.126565
\(419\) −29.9476 −1.46304 −0.731518 0.681822i \(-0.761188\pi\)
−0.731518 + 0.681822i \(0.761188\pi\)
\(420\) 0 0
\(421\) 38.0011 1.85206 0.926030 0.377450i \(-0.123199\pi\)
0.926030 + 0.377450i \(0.123199\pi\)
\(422\) 6.20447 0.302029
\(423\) 2.38207 0.115820
\(424\) 1.99127 0.0967044
\(425\) 33.7823 1.63868
\(426\) −2.30104 −0.111486
\(427\) 0 0
\(428\) 11.0379 0.533535
\(429\) 0 0
\(430\) −7.01898 −0.338485
\(431\) −4.41887 −0.212850 −0.106425 0.994321i \(-0.533940\pi\)
−0.106425 + 0.994321i \(0.533940\pi\)
\(432\) −17.5650 −0.845097
\(433\) 1.64351 0.0789819 0.0394910 0.999220i \(-0.487426\pi\)
0.0394910 + 0.999220i \(0.487426\pi\)
\(434\) 0 0
\(435\) −8.24955 −0.395536
\(436\) −24.5930 −1.17779
\(437\) 14.4268 0.690128
\(438\) 8.12513 0.388234
\(439\) 22.9235 1.09408 0.547040 0.837106i \(-0.315754\pi\)
0.547040 + 0.837106i \(0.315754\pi\)
\(440\) −16.7850 −0.800193
\(441\) 0 0
\(442\) 0 0
\(443\) −3.19065 −0.151593 −0.0757963 0.997123i \(-0.524150\pi\)
−0.0757963 + 0.997123i \(0.524150\pi\)
\(444\) 2.28163 0.108281
\(445\) 41.0567 1.94627
\(446\) −5.00676 −0.237077
\(447\) −30.6937 −1.45176
\(448\) 0 0
\(449\) −15.8788 −0.749368 −0.374684 0.927153i \(-0.622249\pi\)
−0.374684 + 0.927153i \(0.622249\pi\)
\(450\) −0.979446 −0.0461715
\(451\) 2.38804 0.112448
\(452\) 6.70791 0.315513
\(453\) −5.08004 −0.238681
\(454\) 7.54646 0.354173
\(455\) 0 0
\(456\) −4.14442 −0.194080
\(457\) 7.21120 0.337325 0.168663 0.985674i \(-0.446055\pi\)
0.168663 + 0.985674i \(0.446055\pi\)
\(458\) −4.71621 −0.220374
\(459\) −43.0875 −2.01115
\(460\) 45.0326 2.09966
\(461\) 26.5787 1.23789 0.618946 0.785434i \(-0.287560\pi\)
0.618946 + 0.785434i \(0.287560\pi\)
\(462\) 0 0
\(463\) −5.65498 −0.262809 −0.131405 0.991329i \(-0.541949\pi\)
−0.131405 + 0.991329i \(0.541949\pi\)
\(464\) 5.47087 0.253979
\(465\) 35.8792 1.66386
\(466\) 2.74613 0.127212
\(467\) 12.0821 0.559092 0.279546 0.960132i \(-0.409816\pi\)
0.279546 + 0.960132i \(0.409816\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.71120 0.217311
\(471\) 10.9617 0.505089
\(472\) −14.1600 −0.651768
\(473\) −22.5192 −1.03543
\(474\) −6.71244 −0.308313
\(475\) 7.94853 0.364703
\(476\) 0 0
\(477\) 0.799399 0.0366020
\(478\) 1.05608 0.0483041
\(479\) 12.5135 0.571754 0.285877 0.958266i \(-0.407715\pi\)
0.285877 + 0.958266i \(0.407715\pi\)
\(480\) −19.6480 −0.896802
\(481\) 0 0
\(482\) −9.85658 −0.448955
\(483\) 0 0
\(484\) −5.50747 −0.250340
\(485\) 36.9078 1.67590
\(486\) 2.29345 0.104033
\(487\) −17.3599 −0.786652 −0.393326 0.919399i \(-0.628676\pi\)
−0.393326 + 0.919399i \(0.628676\pi\)
\(488\) 19.2948 0.873433
\(489\) −17.9808 −0.813119
\(490\) 0 0
\(491\) −22.6384 −1.02166 −0.510829 0.859683i \(-0.670661\pi\)
−0.510829 + 0.859683i \(0.670661\pi\)
\(492\) 1.84052 0.0829771
\(493\) 13.4202 0.604416
\(494\) 0 0
\(495\) −6.73838 −0.302868
\(496\) −23.7941 −1.06838
\(497\) 0 0
\(498\) −0.0365055 −0.00163585
\(499\) 8.34265 0.373468 0.186734 0.982410i \(-0.440210\pi\)
0.186734 + 0.982410i \(0.440210\pi\)
\(500\) −3.58157 −0.160173
\(501\) 9.97978 0.445864
\(502\) −4.83484 −0.215789
\(503\) 18.2399 0.813275 0.406638 0.913590i \(-0.366701\pi\)
0.406638 + 0.913590i \(0.366701\pi\)
\(504\) 0 0
\(505\) −34.9480 −1.55516
\(506\) −11.2803 −0.501468
\(507\) 0 0
\(508\) 24.4686 1.08562
\(509\) 41.3455 1.83261 0.916304 0.400483i \(-0.131158\pi\)
0.916304 + 0.400483i \(0.131158\pi\)
\(510\) −13.9855 −0.619287
\(511\) 0 0
\(512\) 22.2790 0.984601
\(513\) −10.1379 −0.447601
\(514\) −3.10442 −0.136930
\(515\) −33.1087 −1.45895
\(516\) −17.3561 −0.764060
\(517\) 15.1151 0.664759
\(518\) 0 0
\(519\) 3.11812 0.136870
\(520\) 0 0
\(521\) −13.5390 −0.593155 −0.296577 0.955009i \(-0.595845\pi\)
−0.296577 + 0.955009i \(0.595845\pi\)
\(522\) −0.389091 −0.0170300
\(523\) 33.7576 1.47612 0.738058 0.674738i \(-0.235743\pi\)
0.738058 + 0.674738i \(0.235743\pi\)
\(524\) −8.75740 −0.382569
\(525\) 0 0
\(526\) −5.83646 −0.254482
\(527\) −58.3676 −2.54253
\(528\) −18.2916 −0.796041
\(529\) 39.8907 1.73438
\(530\) 1.58103 0.0686755
\(531\) −5.68458 −0.246690
\(532\) 0 0
\(533\) 0 0
\(534\) −7.92639 −0.343009
\(535\) 18.2120 0.787372
\(536\) −13.9382 −0.602038
\(537\) −25.7289 −1.11028
\(538\) −6.43063 −0.277244
\(539\) 0 0
\(540\) −31.6451 −1.36179
\(541\) 9.62332 0.413739 0.206870 0.978369i \(-0.433672\pi\)
0.206870 + 0.978369i \(0.433672\pi\)
\(542\) 3.36355 0.144477
\(543\) −25.8769 −1.11049
\(544\) 31.9629 1.37040
\(545\) −40.5774 −1.73814
\(546\) 0 0
\(547\) −27.2943 −1.16702 −0.583509 0.812106i \(-0.698321\pi\)
−0.583509 + 0.812106i \(0.698321\pi\)
\(548\) −34.3758 −1.46846
\(549\) 7.74594 0.330589
\(550\) −6.21491 −0.265005
\(551\) 3.15760 0.134518
\(552\) −18.0668 −0.768972
\(553\) 0 0
\(554\) 1.14079 0.0484676
\(555\) 3.76459 0.159798
\(556\) −19.3196 −0.819333
\(557\) 14.8829 0.630609 0.315305 0.948991i \(-0.397893\pi\)
0.315305 + 0.948991i \(0.397893\pi\)
\(558\) 1.69225 0.0716385
\(559\) 0 0
\(560\) 0 0
\(561\) −44.8699 −1.89441
\(562\) −1.60970 −0.0679011
\(563\) −21.2764 −0.896692 −0.448346 0.893860i \(-0.647987\pi\)
−0.448346 + 0.893860i \(0.647987\pi\)
\(564\) 11.6496 0.490535
\(565\) 11.0677 0.465623
\(566\) −1.49978 −0.0630406
\(567\) 0 0
\(568\) 5.71306 0.239715
\(569\) −5.23482 −0.219455 −0.109728 0.993962i \(-0.534998\pi\)
−0.109728 + 0.993962i \(0.534998\pi\)
\(570\) −3.29060 −0.137828
\(571\) 4.71034 0.197122 0.0985609 0.995131i \(-0.468576\pi\)
0.0985609 + 0.995131i \(0.468576\pi\)
\(572\) 0 0
\(573\) −36.1366 −1.50963
\(574\) 0 0
\(575\) 34.6499 1.44500
\(576\) 2.78635 0.116098
\(577\) 24.4488 1.01782 0.508908 0.860821i \(-0.330049\pi\)
0.508908 + 0.860821i \(0.330049\pi\)
\(578\) 16.2814 0.677217
\(579\) −29.0481 −1.20720
\(580\) 9.85630 0.409260
\(581\) 0 0
\(582\) −7.12542 −0.295358
\(583\) 5.07246 0.210080
\(584\) −20.1732 −0.834773
\(585\) 0 0
\(586\) −6.62830 −0.273813
\(587\) −15.7681 −0.650819 −0.325409 0.945573i \(-0.605502\pi\)
−0.325409 + 0.945573i \(0.605502\pi\)
\(588\) 0 0
\(589\) −13.7331 −0.565864
\(590\) −11.2428 −0.462859
\(591\) −38.1353 −1.56868
\(592\) −2.49657 −0.102608
\(593\) −27.2990 −1.12104 −0.560518 0.828142i \(-0.689398\pi\)
−0.560518 + 0.828142i \(0.689398\pi\)
\(594\) 7.92681 0.325241
\(595\) 0 0
\(596\) 36.6718 1.50214
\(597\) −20.2288 −0.827911
\(598\) 0 0
\(599\) 10.1193 0.413462 0.206731 0.978398i \(-0.433718\pi\)
0.206731 + 0.978398i \(0.433718\pi\)
\(600\) −9.95397 −0.406369
\(601\) 42.8861 1.74936 0.874680 0.484701i \(-0.161071\pi\)
0.874680 + 0.484701i \(0.161071\pi\)
\(602\) 0 0
\(603\) −5.59552 −0.227867
\(604\) 6.06946 0.246963
\(605\) −9.08708 −0.369442
\(606\) 6.74705 0.274080
\(607\) 17.5891 0.713920 0.356960 0.934120i \(-0.383813\pi\)
0.356960 + 0.934120i \(0.383813\pi\)
\(608\) 7.52045 0.304995
\(609\) 0 0
\(610\) 15.3197 0.620277
\(611\) 0 0
\(612\) 8.44859 0.341514
\(613\) −26.3045 −1.06243 −0.531214 0.847238i \(-0.678264\pi\)
−0.531214 + 0.847238i \(0.678264\pi\)
\(614\) −9.22660 −0.372355
\(615\) 3.03678 0.122455
\(616\) 0 0
\(617\) 29.7580 1.19801 0.599007 0.800744i \(-0.295562\pi\)
0.599007 + 0.800744i \(0.295562\pi\)
\(618\) 6.39196 0.257123
\(619\) 15.4684 0.621727 0.310864 0.950455i \(-0.399382\pi\)
0.310864 + 0.950455i \(0.399382\pi\)
\(620\) −42.8673 −1.72159
\(621\) −44.1942 −1.77345
\(622\) 3.86126 0.154822
\(623\) 0 0
\(624\) 0 0
\(625\) −27.7558 −1.11023
\(626\) 2.78670 0.111379
\(627\) −10.5573 −0.421618
\(628\) −13.0967 −0.522615
\(629\) −6.12416 −0.244186
\(630\) 0 0
\(631\) 6.00930 0.239227 0.119613 0.992821i \(-0.461835\pi\)
0.119613 + 0.992821i \(0.461835\pi\)
\(632\) 16.6658 0.662929
\(633\) 25.3137 1.00613
\(634\) −3.50372 −0.139151
\(635\) 40.3721 1.60212
\(636\) 3.90947 0.155021
\(637\) 0 0
\(638\) −2.46891 −0.0977452
\(639\) 2.29352 0.0907304
\(640\) 30.8183 1.21820
\(641\) 22.5972 0.892534 0.446267 0.894900i \(-0.352753\pi\)
0.446267 + 0.894900i \(0.352753\pi\)
\(642\) −3.51600 −0.138765
\(643\) −5.53812 −0.218402 −0.109201 0.994020i \(-0.534829\pi\)
−0.109201 + 0.994020i \(0.534829\pi\)
\(644\) 0 0
\(645\) −28.6368 −1.12757
\(646\) 5.35308 0.210614
\(647\) −18.2801 −0.718666 −0.359333 0.933209i \(-0.616996\pi\)
−0.359333 + 0.933209i \(0.616996\pi\)
\(648\) 10.1032 0.396891
\(649\) −36.0706 −1.41589
\(650\) 0 0
\(651\) 0 0
\(652\) 21.4828 0.841333
\(653\) −23.2707 −0.910654 −0.455327 0.890324i \(-0.650478\pi\)
−0.455327 + 0.890324i \(0.650478\pi\)
\(654\) 7.83386 0.306328
\(655\) −14.4493 −0.564581
\(656\) −2.01390 −0.0786297
\(657\) −8.09859 −0.315956
\(658\) 0 0
\(659\) −25.3408 −0.987137 −0.493569 0.869707i \(-0.664308\pi\)
−0.493569 + 0.869707i \(0.664308\pi\)
\(660\) −32.9541 −1.28274
\(661\) −0.199005 −0.00774041 −0.00387020 0.999993i \(-0.501232\pi\)
−0.00387020 + 0.999993i \(0.501232\pi\)
\(662\) −0.530160 −0.0206053
\(663\) 0 0
\(664\) 0.0906365 0.00351738
\(665\) 0 0
\(666\) 0.177557 0.00688020
\(667\) 13.7649 0.532979
\(668\) −11.9235 −0.461335
\(669\) −20.4271 −0.789758
\(670\) −11.0667 −0.427543
\(671\) 49.1506 1.89744
\(672\) 0 0
\(673\) 35.4410 1.36615 0.683076 0.730347i \(-0.260642\pi\)
0.683076 + 0.730347i \(0.260642\pi\)
\(674\) 5.61342 0.216221
\(675\) −24.3490 −0.937195
\(676\) 0 0
\(677\) −26.4391 −1.01614 −0.508069 0.861316i \(-0.669641\pi\)
−0.508069 + 0.861316i \(0.669641\pi\)
\(678\) −2.13674 −0.0820608
\(679\) 0 0
\(680\) 34.7234 1.33158
\(681\) 30.7888 1.17983
\(682\) 10.7379 0.411174
\(683\) 36.6530 1.40249 0.701244 0.712921i \(-0.252628\pi\)
0.701244 + 0.712921i \(0.252628\pi\)
\(684\) 1.98784 0.0760071
\(685\) −56.7185 −2.16710
\(686\) 0 0
\(687\) −19.2417 −0.734117
\(688\) 18.9911 0.724029
\(689\) 0 0
\(690\) −14.3447 −0.546093
\(691\) −28.9855 −1.10266 −0.551330 0.834287i \(-0.685880\pi\)
−0.551330 + 0.834287i \(0.685880\pi\)
\(692\) −3.72542 −0.141619
\(693\) 0 0
\(694\) −0.673321 −0.0255589
\(695\) −31.8764 −1.20914
\(696\) −3.95428 −0.149886
\(697\) −4.94017 −0.187122
\(698\) 8.65426 0.327569
\(699\) 11.2040 0.423773
\(700\) 0 0
\(701\) 3.59251 0.135687 0.0678436 0.997696i \(-0.478388\pi\)
0.0678436 + 0.997696i \(0.478388\pi\)
\(702\) 0 0
\(703\) −1.44093 −0.0543459
\(704\) 17.6803 0.666352
\(705\) 19.2213 0.723914
\(706\) 4.04172 0.152112
\(707\) 0 0
\(708\) −27.8005 −1.04481
\(709\) 33.8684 1.27196 0.635978 0.771707i \(-0.280597\pi\)
0.635978 + 0.771707i \(0.280597\pi\)
\(710\) 4.53607 0.170236
\(711\) 6.69052 0.250914
\(712\) 19.6798 0.737531
\(713\) −59.8667 −2.24203
\(714\) 0 0
\(715\) 0 0
\(716\) 30.7400 1.14881
\(717\) 4.30872 0.160912
\(718\) −8.57157 −0.319888
\(719\) 4.19141 0.156313 0.0781565 0.996941i \(-0.475097\pi\)
0.0781565 + 0.996941i \(0.475097\pi\)
\(720\) 5.68268 0.211781
\(721\) 0 0
\(722\) −5.97153 −0.222237
\(723\) −40.2139 −1.49557
\(724\) 30.9169 1.14902
\(725\) 7.58384 0.281657
\(726\) 1.75435 0.0651100
\(727\) 16.0777 0.596289 0.298145 0.954521i \(-0.403632\pi\)
0.298145 + 0.954521i \(0.403632\pi\)
\(728\) 0 0
\(729\) 30.0151 1.11167
\(730\) −16.0172 −0.592822
\(731\) 46.5857 1.72304
\(732\) 37.8816 1.40015
\(733\) −5.57149 −0.205788 −0.102894 0.994692i \(-0.532810\pi\)
−0.102894 + 0.994692i \(0.532810\pi\)
\(734\) −1.59693 −0.0589439
\(735\) 0 0
\(736\) 32.7838 1.20843
\(737\) −35.5055 −1.30786
\(738\) 0.143230 0.00527236
\(739\) −5.57700 −0.205153 −0.102577 0.994725i \(-0.532709\pi\)
−0.102577 + 0.994725i \(0.532709\pi\)
\(740\) −4.49781 −0.165343
\(741\) 0 0
\(742\) 0 0
\(743\) −3.49100 −0.128073 −0.0640363 0.997948i \(-0.520397\pi\)
−0.0640363 + 0.997948i \(0.520397\pi\)
\(744\) 17.1981 0.630511
\(745\) 60.5068 2.21680
\(746\) −8.75366 −0.320494
\(747\) 0.0363863 0.00133130
\(748\) 53.6091 1.96014
\(749\) 0 0
\(750\) 1.14087 0.0416589
\(751\) 13.8176 0.504213 0.252106 0.967700i \(-0.418877\pi\)
0.252106 + 0.967700i \(0.418877\pi\)
\(752\) −12.7470 −0.464835
\(753\) −19.7257 −0.718844
\(754\) 0 0
\(755\) 10.0143 0.364459
\(756\) 0 0
\(757\) 2.70505 0.0983166 0.0491583 0.998791i \(-0.484346\pi\)
0.0491583 + 0.998791i \(0.484346\pi\)
\(758\) −13.7708 −0.500177
\(759\) −46.0224 −1.67051
\(760\) 8.16995 0.296355
\(761\) −11.4906 −0.416533 −0.208266 0.978072i \(-0.566782\pi\)
−0.208266 + 0.978072i \(0.566782\pi\)
\(762\) −7.79423 −0.282355
\(763\) 0 0
\(764\) 43.1749 1.56201
\(765\) 13.9398 0.503994
\(766\) −1.95303 −0.0705658
\(767\) 0 0
\(768\) 8.74080 0.315407
\(769\) 3.84729 0.138737 0.0693684 0.997591i \(-0.477902\pi\)
0.0693684 + 0.997591i \(0.477902\pi\)
\(770\) 0 0
\(771\) −12.6657 −0.456145
\(772\) 34.7057 1.24908
\(773\) −39.3566 −1.41556 −0.707779 0.706434i \(-0.750303\pi\)
−0.707779 + 0.706434i \(0.750303\pi\)
\(774\) −1.35066 −0.0485483
\(775\) −32.9839 −1.18482
\(776\) 17.6911 0.635073
\(777\) 0 0
\(778\) −2.33732 −0.0837972
\(779\) −1.16236 −0.0416458
\(780\) 0 0
\(781\) 14.5532 0.520754
\(782\) 23.3356 0.834481
\(783\) −9.67280 −0.345678
\(784\) 0 0
\(785\) −21.6090 −0.771257
\(786\) 2.78958 0.0995011
\(787\) 45.6015 1.62552 0.812759 0.582600i \(-0.197965\pi\)
0.812759 + 0.582600i \(0.197965\pi\)
\(788\) 45.5628 1.62311
\(789\) −23.8122 −0.847738
\(790\) 13.2323 0.470785
\(791\) 0 0
\(792\) −3.22992 −0.114770
\(793\) 0 0
\(794\) 10.3603 0.367674
\(795\) 6.45045 0.228774
\(796\) 24.1688 0.856639
\(797\) 43.4464 1.53895 0.769474 0.638678i \(-0.220518\pi\)
0.769474 + 0.638678i \(0.220518\pi\)
\(798\) 0 0
\(799\) −31.2687 −1.10621
\(800\) 18.0624 0.638603
\(801\) 7.90050 0.279150
\(802\) −7.32942 −0.258811
\(803\) −51.3883 −1.81345
\(804\) −27.3650 −0.965088
\(805\) 0 0
\(806\) 0 0
\(807\) −26.2364 −0.923564
\(808\) −16.7517 −0.589322
\(809\) −38.8316 −1.36525 −0.682623 0.730771i \(-0.739161\pi\)
−0.682623 + 0.730771i \(0.739161\pi\)
\(810\) 8.02175 0.281856
\(811\) −12.0920 −0.424608 −0.212304 0.977204i \(-0.568097\pi\)
−0.212304 + 0.977204i \(0.568097\pi\)
\(812\) 0 0
\(813\) 13.7230 0.481286
\(814\) 1.12666 0.0394894
\(815\) 35.4457 1.24161
\(816\) 37.8402 1.32467
\(817\) 10.9610 0.383478
\(818\) 5.45692 0.190797
\(819\) 0 0
\(820\) −3.62824 −0.126704
\(821\) −17.9409 −0.626141 −0.313070 0.949730i \(-0.601358\pi\)
−0.313070 + 0.949730i \(0.601358\pi\)
\(822\) 10.9501 0.381927
\(823\) −24.5609 −0.856137 −0.428069 0.903746i \(-0.640806\pi\)
−0.428069 + 0.903746i \(0.640806\pi\)
\(824\) −15.8701 −0.552860
\(825\) −25.3563 −0.882792
\(826\) 0 0
\(827\) −12.5984 −0.438088 −0.219044 0.975715i \(-0.570294\pi\)
−0.219044 + 0.975715i \(0.570294\pi\)
\(828\) 8.66558 0.301150
\(829\) −39.5002 −1.37190 −0.685950 0.727649i \(-0.740613\pi\)
−0.685950 + 0.727649i \(0.740613\pi\)
\(830\) 0.0719638 0.00249790
\(831\) 4.65432 0.161457
\(832\) 0 0
\(833\) 0 0
\(834\) 6.15406 0.213098
\(835\) −19.6733 −0.680821
\(836\) 12.6135 0.436248
\(837\) 42.0692 1.45413
\(838\) −11.3975 −0.393720
\(839\) −8.12030 −0.280344 −0.140172 0.990127i \(-0.544766\pi\)
−0.140172 + 0.990127i \(0.544766\pi\)
\(840\) 0 0
\(841\) −25.9873 −0.896113
\(842\) 14.4625 0.498411
\(843\) −6.56742 −0.226194
\(844\) −30.2440 −1.04104
\(845\) 0 0
\(846\) 0.906572 0.0311686
\(847\) 0 0
\(848\) −4.27776 −0.146899
\(849\) −6.11898 −0.210003
\(850\) 12.8569 0.440988
\(851\) −6.28145 −0.215325
\(852\) 11.2165 0.384271
\(853\) −17.7841 −0.608914 −0.304457 0.952526i \(-0.598475\pi\)
−0.304457 + 0.952526i \(0.598475\pi\)
\(854\) 0 0
\(855\) 3.27985 0.112169
\(856\) 8.72957 0.298371
\(857\) 6.33988 0.216566 0.108283 0.994120i \(-0.465465\pi\)
0.108283 + 0.994120i \(0.465465\pi\)
\(858\) 0 0
\(859\) 37.0394 1.26377 0.631883 0.775063i \(-0.282282\pi\)
0.631883 + 0.775063i \(0.282282\pi\)
\(860\) 34.2143 1.16670
\(861\) 0 0
\(862\) −1.68174 −0.0572803
\(863\) −29.5338 −1.00534 −0.502672 0.864477i \(-0.667650\pi\)
−0.502672 + 0.864477i \(0.667650\pi\)
\(864\) −23.0377 −0.783759
\(865\) −6.14678 −0.208997
\(866\) 0.625488 0.0212549
\(867\) 66.4266 2.25597
\(868\) 0 0
\(869\) 42.4536 1.44014
\(870\) −3.13962 −0.106443
\(871\) 0 0
\(872\) −19.4500 −0.658661
\(873\) 7.10214 0.240371
\(874\) 5.49057 0.185721
\(875\) 0 0
\(876\) −39.6063 −1.33817
\(877\) −1.60539 −0.0542101 −0.0271050 0.999633i \(-0.508629\pi\)
−0.0271050 + 0.999633i \(0.508629\pi\)
\(878\) 8.72426 0.294430
\(879\) −27.0429 −0.912133
\(880\) 36.0585 1.21553
\(881\) −24.3581 −0.820645 −0.410322 0.911941i \(-0.634584\pi\)
−0.410322 + 0.911941i \(0.634584\pi\)
\(882\) 0 0
\(883\) −14.5955 −0.491176 −0.245588 0.969374i \(-0.578981\pi\)
−0.245588 + 0.969374i \(0.578981\pi\)
\(884\) 0 0
\(885\) −45.8696 −1.54189
\(886\) −1.21430 −0.0407953
\(887\) −37.0966 −1.24558 −0.622790 0.782389i \(-0.714001\pi\)
−0.622790 + 0.782389i \(0.714001\pi\)
\(888\) 1.80449 0.0605547
\(889\) 0 0
\(890\) 15.6254 0.523764
\(891\) 25.7364 0.862201
\(892\) 24.4056 0.817161
\(893\) −7.35713 −0.246197
\(894\) −11.6814 −0.390686
\(895\) 50.7196 1.69537
\(896\) 0 0
\(897\) 0 0
\(898\) −6.04318 −0.201663
\(899\) −13.1030 −0.437011
\(900\) 4.77435 0.159145
\(901\) −10.4935 −0.349588
\(902\) 0.908841 0.0302611
\(903\) 0 0
\(904\) 5.30512 0.176446
\(905\) 51.0115 1.69568
\(906\) −1.93337 −0.0642318
\(907\) 18.7039 0.621053 0.310526 0.950565i \(-0.399495\pi\)
0.310526 + 0.950565i \(0.399495\pi\)
\(908\) −36.7855 −1.22077
\(909\) −6.72501 −0.223054
\(910\) 0 0
\(911\) 37.0911 1.22888 0.614442 0.788962i \(-0.289381\pi\)
0.614442 + 0.788962i \(0.289381\pi\)
\(912\) 8.90330 0.294818
\(913\) 0.230883 0.00764112
\(914\) 2.74444 0.0907782
\(915\) 62.5030 2.06629
\(916\) 22.9894 0.759590
\(917\) 0 0
\(918\) −16.3983 −0.541225
\(919\) 15.8724 0.523581 0.261791 0.965125i \(-0.415687\pi\)
0.261791 + 0.965125i \(0.415687\pi\)
\(920\) 35.6152 1.17420
\(921\) −37.6437 −1.24040
\(922\) 10.1153 0.333131
\(923\) 0 0
\(924\) 0 0
\(925\) −3.46080 −0.113790
\(926\) −2.15218 −0.0707250
\(927\) −6.37108 −0.209254
\(928\) 7.17541 0.235544
\(929\) −18.0818 −0.593244 −0.296622 0.954995i \(-0.595860\pi\)
−0.296622 + 0.954995i \(0.595860\pi\)
\(930\) 13.6549 0.447763
\(931\) 0 0
\(932\) −13.3861 −0.438477
\(933\) 15.7536 0.515749
\(934\) 4.59821 0.150458
\(935\) 88.4526 2.89271
\(936\) 0 0
\(937\) −7.07980 −0.231287 −0.115643 0.993291i \(-0.536893\pi\)
−0.115643 + 0.993291i \(0.536893\pi\)
\(938\) 0 0
\(939\) 11.3695 0.371029
\(940\) −22.9649 −0.749033
\(941\) −5.13090 −0.167262 −0.0836312 0.996497i \(-0.526652\pi\)
−0.0836312 + 0.996497i \(0.526652\pi\)
\(942\) 4.17182 0.135925
\(943\) −5.06705 −0.165006
\(944\) 30.4194 0.990068
\(945\) 0 0
\(946\) −8.57037 −0.278647
\(947\) −51.4951 −1.67336 −0.836682 0.547688i \(-0.815508\pi\)
−0.836682 + 0.547688i \(0.815508\pi\)
\(948\) 32.7201 1.06270
\(949\) 0 0
\(950\) 3.02506 0.0981459
\(951\) −14.2949 −0.463543
\(952\) 0 0
\(953\) 11.2520 0.364487 0.182243 0.983253i \(-0.441664\pi\)
0.182243 + 0.983253i \(0.441664\pi\)
\(954\) 0.304236 0.00985001
\(955\) 71.2366 2.30516
\(956\) −5.14791 −0.166495
\(957\) −10.0729 −0.325612
\(958\) 4.76239 0.153866
\(959\) 0 0
\(960\) 22.4834 0.725648
\(961\) 25.9881 0.838327
\(962\) 0 0
\(963\) 3.50451 0.112931
\(964\) 48.0463 1.54747
\(965\) 57.2628 1.84336
\(966\) 0 0
\(967\) −5.02854 −0.161707 −0.0808535 0.996726i \(-0.525765\pi\)
−0.0808535 + 0.996726i \(0.525765\pi\)
\(968\) −4.35573 −0.139998
\(969\) 21.8401 0.701604
\(970\) 14.0464 0.451003
\(971\) −32.9918 −1.05876 −0.529378 0.848386i \(-0.677575\pi\)
−0.529378 + 0.848386i \(0.677575\pi\)
\(972\) −11.1795 −0.358583
\(973\) 0 0
\(974\) −6.60685 −0.211697
\(975\) 0 0
\(976\) −41.4502 −1.32679
\(977\) −45.7585 −1.46394 −0.731972 0.681335i \(-0.761400\pi\)
−0.731972 + 0.681335i \(0.761400\pi\)
\(978\) −6.84314 −0.218820
\(979\) 50.1313 1.60220
\(980\) 0 0
\(981\) −7.80827 −0.249299
\(982\) −8.61575 −0.274940
\(983\) 37.4003 1.19288 0.596442 0.802656i \(-0.296581\pi\)
0.596442 + 0.802656i \(0.296581\pi\)
\(984\) 1.45562 0.0464036
\(985\) 75.1766 2.39533
\(986\) 5.10748 0.162655
\(987\) 0 0
\(988\) 0 0
\(989\) 47.7823 1.51939
\(990\) −2.56450 −0.0815051
\(991\) 27.9614 0.888222 0.444111 0.895972i \(-0.353520\pi\)
0.444111 + 0.895972i \(0.353520\pi\)
\(992\) −31.2075 −0.990839
\(993\) −2.16300 −0.0686409
\(994\) 0 0
\(995\) 39.8774 1.26420
\(996\) 0.177948 0.00563848
\(997\) −20.0402 −0.634678 −0.317339 0.948312i \(-0.602789\pi\)
−0.317339 + 0.948312i \(0.602789\pi\)
\(998\) 3.17505 0.100505
\(999\) 4.41407 0.139655
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.cm.1.8 yes 12
7.6 odd 2 inner 8281.2.a.cm.1.7 12
13.12 even 2 8281.2.a.cr.1.6 yes 12
91.90 odd 2 8281.2.a.cr.1.5 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8281.2.a.cm.1.7 12 7.6 odd 2 inner
8281.2.a.cm.1.8 yes 12 1.1 even 1 trivial
8281.2.a.cr.1.5 yes 12 91.90 odd 2
8281.2.a.cr.1.6 yes 12 13.12 even 2