Properties

Label 2009.2.a.s.1.7
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $17$
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] = [2009,2,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(16.0419457661\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + \cdots - 464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.902479\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.902479 q^{2} -3.32518 q^{3} -1.18553 q^{4} -0.938362 q^{5} +3.00091 q^{6} +2.87488 q^{8} +8.05685 q^{9} +O(q^{10})\) \(q-0.902479 q^{2} -3.32518 q^{3} -1.18553 q^{4} -0.938362 q^{5} +3.00091 q^{6} +2.87488 q^{8} +8.05685 q^{9} +0.846852 q^{10} +5.46037 q^{11} +3.94211 q^{12} -2.52551 q^{13} +3.12023 q^{15} -0.223453 q^{16} +6.71133 q^{17} -7.27114 q^{18} +0.857408 q^{19} +1.11246 q^{20} -4.92787 q^{22} +4.76683 q^{23} -9.55949 q^{24} -4.11948 q^{25} +2.27922 q^{26} -16.8150 q^{27} +3.28519 q^{29} -2.81594 q^{30} +0.119237 q^{31} -5.54809 q^{32} -18.1567 q^{33} -6.05684 q^{34} -9.55165 q^{36} -3.81897 q^{37} -0.773793 q^{38} +8.39777 q^{39} -2.69767 q^{40} +1.00000 q^{41} +1.64508 q^{43} -6.47344 q^{44} -7.56024 q^{45} -4.30197 q^{46} +4.73681 q^{47} +0.743022 q^{48} +3.71774 q^{50} -22.3164 q^{51} +2.99407 q^{52} +2.24391 q^{53} +15.1752 q^{54} -5.12380 q^{55} -2.85104 q^{57} -2.96482 q^{58} +4.44654 q^{59} -3.69913 q^{60} +11.9605 q^{61} -0.107609 q^{62} +5.45394 q^{64} +2.36984 q^{65} +16.3861 q^{66} -11.7311 q^{67} -7.95649 q^{68} -15.8506 q^{69} +9.78432 q^{71} +23.1624 q^{72} -3.71532 q^{73} +3.44654 q^{74} +13.6980 q^{75} -1.01648 q^{76} -7.57881 q^{78} -2.81292 q^{79} +0.209680 q^{80} +31.7423 q^{81} -0.902479 q^{82} +3.41464 q^{83} -6.29766 q^{85} -1.48465 q^{86} -10.9239 q^{87} +15.6979 q^{88} -14.2417 q^{89} +6.82296 q^{90} -5.65123 q^{92} -0.396486 q^{93} -4.27487 q^{94} -0.804559 q^{95} +18.4484 q^{96} -8.40546 q^{97} +43.9934 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} + q^{3} + 25 q^{4} - q^{5} + 2 q^{6} + 9 q^{8} + 26 q^{9} - 2 q^{10} + 15 q^{11} + 4 q^{12} - 5 q^{13} + 24 q^{15} + 33 q^{16} + 4 q^{17} + 10 q^{18} + 5 q^{19} - 26 q^{20} + 16 q^{22} + 12 q^{23} + 16 q^{24} + 24 q^{25} + 31 q^{26} - 11 q^{27} + 14 q^{29} - 33 q^{30} - 3 q^{31} + 16 q^{32} + 4 q^{33} - 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} + 36 q^{40} + 17 q^{41} + 14 q^{43} - 9 q^{44} - 21 q^{45} + 44 q^{46} + 19 q^{47} - 60 q^{48} - 4 q^{50} + 2 q^{51} - 25 q^{52} + 4 q^{53} + 68 q^{54} + 9 q^{55} - 12 q^{57} - q^{58} - 27 q^{59} + 66 q^{60} - q^{61} - 23 q^{62} + 75 q^{64} + 22 q^{65} - 16 q^{66} + 49 q^{67} + 45 q^{68} + 12 q^{69} + 40 q^{71} - 23 q^{72} - 14 q^{73} + 33 q^{74} + 27 q^{75} - 9 q^{76} - 12 q^{78} + 61 q^{79} - 82 q^{80} + 53 q^{81} + 3 q^{82} - 18 q^{83} - 13 q^{85} - 4 q^{86} - 17 q^{87} + 74 q^{88} + 18 q^{89} - 20 q^{90} + 28 q^{92} - 36 q^{93} - 5 q^{94} + 20 q^{95} + 148 q^{96} + 26 q^{97} + 38 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.902479 −0.638149 −0.319075 0.947730i \(-0.603372\pi\)
−0.319075 + 0.947730i \(0.603372\pi\)
\(3\) −3.32518 −1.91980 −0.959898 0.280349i \(-0.909550\pi\)
−0.959898 + 0.280349i \(0.909550\pi\)
\(4\) −1.18553 −0.592766
\(5\) −0.938362 −0.419648 −0.209824 0.977739i \(-0.567289\pi\)
−0.209824 + 0.977739i \(0.567289\pi\)
\(6\) 3.00091 1.22512
\(7\) 0 0
\(8\) 2.87488 1.01642
\(9\) 8.05685 2.68562
\(10\) 0.846852 0.267798
\(11\) 5.46037 1.64636 0.823181 0.567779i \(-0.192197\pi\)
0.823181 + 0.567779i \(0.192197\pi\)
\(12\) 3.94211 1.13799
\(13\) −2.52551 −0.700449 −0.350225 0.936666i \(-0.613895\pi\)
−0.350225 + 0.936666i \(0.613895\pi\)
\(14\) 0 0
\(15\) 3.12023 0.805639
\(16\) −0.223453 −0.0558632
\(17\) 6.71133 1.62774 0.813868 0.581049i \(-0.197358\pi\)
0.813868 + 0.581049i \(0.197358\pi\)
\(18\) −7.27114 −1.71382
\(19\) 0.857408 0.196703 0.0983514 0.995152i \(-0.468643\pi\)
0.0983514 + 0.995152i \(0.468643\pi\)
\(20\) 1.11246 0.248753
\(21\) 0 0
\(22\) −4.92787 −1.05062
\(23\) 4.76683 0.993953 0.496977 0.867764i \(-0.334443\pi\)
0.496977 + 0.867764i \(0.334443\pi\)
\(24\) −9.55949 −1.95132
\(25\) −4.11948 −0.823895
\(26\) 2.27922 0.446991
\(27\) −16.8150 −3.23604
\(28\) 0 0
\(29\) 3.28519 0.610045 0.305023 0.952345i \(-0.401336\pi\)
0.305023 + 0.952345i \(0.401336\pi\)
\(30\) −2.81594 −0.514118
\(31\) 0.119237 0.0214157 0.0107078 0.999943i \(-0.496592\pi\)
0.0107078 + 0.999943i \(0.496592\pi\)
\(32\) −5.54809 −0.980773
\(33\) −18.1567 −3.16068
\(34\) −6.05684 −1.03874
\(35\) 0 0
\(36\) −9.55165 −1.59194
\(37\) −3.81897 −0.627835 −0.313917 0.949450i \(-0.601642\pi\)
−0.313917 + 0.949450i \(0.601642\pi\)
\(38\) −0.773793 −0.125526
\(39\) 8.39777 1.34472
\(40\) −2.69767 −0.426540
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 1.64508 0.250873 0.125436 0.992102i \(-0.459967\pi\)
0.125436 + 0.992102i \(0.459967\pi\)
\(44\) −6.47344 −0.975907
\(45\) −7.56024 −1.12701
\(46\) −4.30197 −0.634290
\(47\) 4.73681 0.690935 0.345468 0.938431i \(-0.387720\pi\)
0.345468 + 0.938431i \(0.387720\pi\)
\(48\) 0.743022 0.107246
\(49\) 0 0
\(50\) 3.71774 0.525768
\(51\) −22.3164 −3.12492
\(52\) 2.99407 0.415202
\(53\) 2.24391 0.308224 0.154112 0.988053i \(-0.450748\pi\)
0.154112 + 0.988053i \(0.450748\pi\)
\(54\) 15.1752 2.06508
\(55\) −5.12380 −0.690893
\(56\) 0 0
\(57\) −2.85104 −0.377629
\(58\) −2.96482 −0.389300
\(59\) 4.44654 0.578890 0.289445 0.957195i \(-0.406529\pi\)
0.289445 + 0.957195i \(0.406529\pi\)
\(60\) −3.69913 −0.477555
\(61\) 11.9605 1.53138 0.765690 0.643210i \(-0.222398\pi\)
0.765690 + 0.643210i \(0.222398\pi\)
\(62\) −0.107609 −0.0136664
\(63\) 0 0
\(64\) 5.45394 0.681743
\(65\) 2.36984 0.293942
\(66\) 16.3861 2.01699
\(67\) −11.7311 −1.43318 −0.716591 0.697493i \(-0.754299\pi\)
−0.716591 + 0.697493i \(0.754299\pi\)
\(68\) −7.95649 −0.964867
\(69\) −15.8506 −1.90819
\(70\) 0 0
\(71\) 9.78432 1.16119 0.580593 0.814194i \(-0.302821\pi\)
0.580593 + 0.814194i \(0.302821\pi\)
\(72\) 23.1624 2.72972
\(73\) −3.71532 −0.434845 −0.217422 0.976078i \(-0.569765\pi\)
−0.217422 + 0.976078i \(0.569765\pi\)
\(74\) 3.44654 0.400652
\(75\) 13.6980 1.58171
\(76\) −1.01648 −0.116599
\(77\) 0 0
\(78\) −7.57881 −0.858132
\(79\) −2.81292 −0.316478 −0.158239 0.987401i \(-0.550582\pi\)
−0.158239 + 0.987401i \(0.550582\pi\)
\(80\) 0.209680 0.0234429
\(81\) 31.7423 3.52692
\(82\) −0.902479 −0.0996622
\(83\) 3.41464 0.374805 0.187403 0.982283i \(-0.439993\pi\)
0.187403 + 0.982283i \(0.439993\pi\)
\(84\) 0 0
\(85\) −6.29766 −0.683077
\(86\) −1.48465 −0.160094
\(87\) −10.9239 −1.17116
\(88\) 15.6979 1.67340
\(89\) −14.2417 −1.50962 −0.754808 0.655946i \(-0.772270\pi\)
−0.754808 + 0.655946i \(0.772270\pi\)
\(90\) 6.82296 0.719203
\(91\) 0 0
\(92\) −5.65123 −0.589181
\(93\) −0.396486 −0.0411137
\(94\) −4.27487 −0.440920
\(95\) −0.804559 −0.0825460
\(96\) 18.4484 1.88288
\(97\) −8.40546 −0.853446 −0.426723 0.904382i \(-0.640332\pi\)
−0.426723 + 0.904382i \(0.640332\pi\)
\(98\) 0 0
\(99\) 43.9934 4.42150
\(100\) 4.88377 0.488377
\(101\) −6.27309 −0.624196 −0.312098 0.950050i \(-0.601032\pi\)
−0.312098 + 0.950050i \(0.601032\pi\)
\(102\) 20.1401 1.99417
\(103\) 8.74565 0.861735 0.430867 0.902415i \(-0.358208\pi\)
0.430867 + 0.902415i \(0.358208\pi\)
\(104\) −7.26051 −0.711952
\(105\) 0 0
\(106\) −2.02508 −0.196693
\(107\) −10.5229 −1.01729 −0.508643 0.860978i \(-0.669853\pi\)
−0.508643 + 0.860978i \(0.669853\pi\)
\(108\) 19.9347 1.91821
\(109\) −3.24852 −0.311152 −0.155576 0.987824i \(-0.549723\pi\)
−0.155576 + 0.987824i \(0.549723\pi\)
\(110\) 4.62412 0.440893
\(111\) 12.6988 1.20531
\(112\) 0 0
\(113\) 4.20493 0.395566 0.197783 0.980246i \(-0.436626\pi\)
0.197783 + 0.980246i \(0.436626\pi\)
\(114\) 2.57300 0.240984
\(115\) −4.47301 −0.417111
\(116\) −3.89470 −0.361614
\(117\) −20.3476 −1.88114
\(118\) −4.01291 −0.369418
\(119\) 0 0
\(120\) 8.97026 0.818869
\(121\) 18.8156 1.71051
\(122\) −10.7941 −0.977248
\(123\) −3.32518 −0.299822
\(124\) −0.141360 −0.0126945
\(125\) 8.55737 0.765394
\(126\) 0 0
\(127\) 6.36861 0.565123 0.282561 0.959249i \(-0.408816\pi\)
0.282561 + 0.959249i \(0.408816\pi\)
\(128\) 6.17411 0.545720
\(129\) −5.47020 −0.481624
\(130\) −2.13873 −0.187579
\(131\) 6.50705 0.568524 0.284262 0.958747i \(-0.408251\pi\)
0.284262 + 0.958747i \(0.408251\pi\)
\(132\) 21.5254 1.87354
\(133\) 0 0
\(134\) 10.5871 0.914584
\(135\) 15.7785 1.35800
\(136\) 19.2942 1.65447
\(137\) 13.2635 1.13318 0.566588 0.824001i \(-0.308263\pi\)
0.566588 + 0.824001i \(0.308263\pi\)
\(138\) 14.3048 1.21771
\(139\) 5.17790 0.439184 0.219592 0.975592i \(-0.429527\pi\)
0.219592 + 0.975592i \(0.429527\pi\)
\(140\) 0 0
\(141\) −15.7508 −1.32645
\(142\) −8.83015 −0.741010
\(143\) −13.7902 −1.15319
\(144\) −1.80033 −0.150027
\(145\) −3.08270 −0.256004
\(146\) 3.35299 0.277496
\(147\) 0 0
\(148\) 4.52751 0.372159
\(149\) 3.67578 0.301132 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(150\) −12.3622 −1.00937
\(151\) 14.2667 1.16101 0.580505 0.814257i \(-0.302855\pi\)
0.580505 + 0.814257i \(0.302855\pi\)
\(152\) 2.46494 0.199933
\(153\) 54.0722 4.37148
\(154\) 0 0
\(155\) −0.111888 −0.00898705
\(156\) −9.95582 −0.797104
\(157\) 9.04941 0.722222 0.361111 0.932523i \(-0.382398\pi\)
0.361111 + 0.932523i \(0.382398\pi\)
\(158\) 2.53860 0.201960
\(159\) −7.46140 −0.591727
\(160\) 5.20612 0.411580
\(161\) 0 0
\(162\) −28.6468 −2.25070
\(163\) −8.89264 −0.696525 −0.348263 0.937397i \(-0.613228\pi\)
−0.348263 + 0.937397i \(0.613228\pi\)
\(164\) −1.18553 −0.0925744
\(165\) 17.0376 1.32637
\(166\) −3.08164 −0.239182
\(167\) 7.46298 0.577503 0.288751 0.957404i \(-0.406760\pi\)
0.288751 + 0.957404i \(0.406760\pi\)
\(168\) 0 0
\(169\) −6.62182 −0.509371
\(170\) 5.68350 0.435905
\(171\) 6.90801 0.528269
\(172\) −1.95030 −0.148709
\(173\) −15.0207 −1.14200 −0.571001 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(174\) 9.85857 0.747377
\(175\) 0 0
\(176\) −1.22013 −0.0919711
\(177\) −14.7856 −1.11135
\(178\) 12.8528 0.963360
\(179\) −17.9193 −1.33935 −0.669677 0.742653i \(-0.733567\pi\)
−0.669677 + 0.742653i \(0.733567\pi\)
\(180\) 8.96290 0.668055
\(181\) −24.9578 −1.85510 −0.927548 0.373705i \(-0.878087\pi\)
−0.927548 + 0.373705i \(0.878087\pi\)
\(182\) 0 0
\(183\) −39.7707 −2.93994
\(184\) 13.7041 1.01028
\(185\) 3.58358 0.263470
\(186\) 0.357821 0.0262367
\(187\) 36.6463 2.67984
\(188\) −5.61564 −0.409563
\(189\) 0 0
\(190\) 0.726098 0.0526767
\(191\) 16.3014 1.17952 0.589762 0.807577i \(-0.299221\pi\)
0.589762 + 0.807577i \(0.299221\pi\)
\(192\) −18.1354 −1.30881
\(193\) −9.74950 −0.701784 −0.350892 0.936416i \(-0.614122\pi\)
−0.350892 + 0.936416i \(0.614122\pi\)
\(194\) 7.58576 0.544626
\(195\) −7.88015 −0.564309
\(196\) 0 0
\(197\) −24.4370 −1.74106 −0.870531 0.492114i \(-0.836224\pi\)
−0.870531 + 0.492114i \(0.836224\pi\)
\(198\) −39.7031 −2.82158
\(199\) −4.93046 −0.349511 −0.174756 0.984612i \(-0.555914\pi\)
−0.174756 + 0.984612i \(0.555914\pi\)
\(200\) −11.8430 −0.837425
\(201\) 39.0081 2.75142
\(202\) 5.66133 0.398330
\(203\) 0 0
\(204\) 26.4568 1.85235
\(205\) −0.938362 −0.0655380
\(206\) −7.89277 −0.549915
\(207\) 38.4057 2.66938
\(208\) 0.564331 0.0391293
\(209\) 4.68176 0.323844
\(210\) 0 0
\(211\) 27.1561 1.86950 0.934751 0.355303i \(-0.115622\pi\)
0.934751 + 0.355303i \(0.115622\pi\)
\(212\) −2.66022 −0.182705
\(213\) −32.5347 −2.22924
\(214\) 9.49668 0.649180
\(215\) −1.54368 −0.105278
\(216\) −48.3409 −3.28918
\(217\) 0 0
\(218\) 2.93172 0.198561
\(219\) 12.3541 0.834813
\(220\) 6.07442 0.409538
\(221\) −16.9495 −1.14015
\(222\) −11.4604 −0.769171
\(223\) −8.74237 −0.585432 −0.292716 0.956199i \(-0.594559\pi\)
−0.292716 + 0.956199i \(0.594559\pi\)
\(224\) 0 0
\(225\) −33.1900 −2.21267
\(226\) −3.79486 −0.252430
\(227\) 14.6898 0.975000 0.487500 0.873123i \(-0.337909\pi\)
0.487500 + 0.873123i \(0.337909\pi\)
\(228\) 3.38000 0.223846
\(229\) −5.30412 −0.350506 −0.175253 0.984523i \(-0.556074\pi\)
−0.175253 + 0.984523i \(0.556074\pi\)
\(230\) 4.03680 0.266179
\(231\) 0 0
\(232\) 9.44453 0.620064
\(233\) −4.42564 −0.289933 −0.144967 0.989437i \(-0.546307\pi\)
−0.144967 + 0.989437i \(0.546307\pi\)
\(234\) 18.3633 1.20045
\(235\) −4.44484 −0.289950
\(236\) −5.27151 −0.343146
\(237\) 9.35347 0.607573
\(238\) 0 0
\(239\) −0.685835 −0.0443630 −0.0221815 0.999754i \(-0.507061\pi\)
−0.0221815 + 0.999754i \(0.507061\pi\)
\(240\) −0.697223 −0.0450056
\(241\) 8.86786 0.571229 0.285614 0.958345i \(-0.407802\pi\)
0.285614 + 0.958345i \(0.407802\pi\)
\(242\) −16.9807 −1.09156
\(243\) −55.1041 −3.53493
\(244\) −14.1795 −0.907749
\(245\) 0 0
\(246\) 3.00091 0.191331
\(247\) −2.16539 −0.137780
\(248\) 0.342793 0.0217674
\(249\) −11.3543 −0.719550
\(250\) −7.72285 −0.488436
\(251\) 3.42941 0.216463 0.108231 0.994126i \(-0.465481\pi\)
0.108231 + 0.994126i \(0.465481\pi\)
\(252\) 0 0
\(253\) 26.0287 1.63641
\(254\) −5.74754 −0.360633
\(255\) 20.9409 1.31137
\(256\) −16.4799 −1.02999
\(257\) 15.2304 0.950048 0.475024 0.879973i \(-0.342439\pi\)
0.475024 + 0.879973i \(0.342439\pi\)
\(258\) 4.93674 0.307348
\(259\) 0 0
\(260\) −2.80952 −0.174239
\(261\) 26.4683 1.63835
\(262\) −5.87248 −0.362803
\(263\) −12.2741 −0.756852 −0.378426 0.925632i \(-0.623535\pi\)
−0.378426 + 0.925632i \(0.623535\pi\)
\(264\) −52.1983 −3.21258
\(265\) −2.10560 −0.129346
\(266\) 0 0
\(267\) 47.3562 2.89815
\(268\) 13.9076 0.849541
\(269\) 12.8300 0.782258 0.391129 0.920336i \(-0.372085\pi\)
0.391129 + 0.920336i \(0.372085\pi\)
\(270\) −14.2398 −0.866606
\(271\) 17.9718 1.09171 0.545854 0.837881i \(-0.316205\pi\)
0.545854 + 0.837881i \(0.316205\pi\)
\(272\) −1.49967 −0.0909306
\(273\) 0 0
\(274\) −11.9700 −0.723135
\(275\) −22.4939 −1.35643
\(276\) 18.7914 1.13111
\(277\) −0.834736 −0.0501544 −0.0250772 0.999686i \(-0.507983\pi\)
−0.0250772 + 0.999686i \(0.507983\pi\)
\(278\) −4.67295 −0.280265
\(279\) 0.960678 0.0575143
\(280\) 0 0
\(281\) 5.96703 0.355963 0.177982 0.984034i \(-0.443043\pi\)
0.177982 + 0.984034i \(0.443043\pi\)
\(282\) 14.2147 0.846476
\(283\) 18.4158 1.09470 0.547352 0.836902i \(-0.315636\pi\)
0.547352 + 0.836902i \(0.315636\pi\)
\(284\) −11.5996 −0.688311
\(285\) 2.67531 0.158472
\(286\) 12.4454 0.735909
\(287\) 0 0
\(288\) −44.7001 −2.63398
\(289\) 28.0420 1.64953
\(290\) 2.78207 0.163369
\(291\) 27.9497 1.63844
\(292\) 4.40462 0.257761
\(293\) −18.3646 −1.07287 −0.536435 0.843941i \(-0.680229\pi\)
−0.536435 + 0.843941i \(0.680229\pi\)
\(294\) 0 0
\(295\) −4.17246 −0.242930
\(296\) −10.9791 −0.638145
\(297\) −91.8158 −5.32770
\(298\) −3.31732 −0.192167
\(299\) −12.0387 −0.696214
\(300\) −16.2394 −0.937584
\(301\) 0 0
\(302\) −12.8754 −0.740897
\(303\) 20.8592 1.19833
\(304\) −0.191590 −0.0109885
\(305\) −11.2232 −0.642640
\(306\) −48.7990 −2.78965
\(307\) 16.8758 0.963153 0.481577 0.876404i \(-0.340064\pi\)
0.481577 + 0.876404i \(0.340064\pi\)
\(308\) 0 0
\(309\) −29.0809 −1.65436
\(310\) 0.100976 0.00573508
\(311\) 10.4537 0.592775 0.296387 0.955068i \(-0.404218\pi\)
0.296387 + 0.955068i \(0.404218\pi\)
\(312\) 24.1425 1.36680
\(313\) −6.26929 −0.354361 −0.177180 0.984178i \(-0.556698\pi\)
−0.177180 + 0.984178i \(0.556698\pi\)
\(314\) −8.16691 −0.460885
\(315\) 0 0
\(316\) 3.33480 0.187597
\(317\) −26.1750 −1.47013 −0.735066 0.677995i \(-0.762849\pi\)
−0.735066 + 0.677995i \(0.762849\pi\)
\(318\) 6.73376 0.377610
\(319\) 17.9384 1.00436
\(320\) −5.11777 −0.286092
\(321\) 34.9905 1.95298
\(322\) 0 0
\(323\) 5.75435 0.320181
\(324\) −37.6315 −2.09064
\(325\) 10.4038 0.577097
\(326\) 8.02542 0.444487
\(327\) 10.8019 0.597349
\(328\) 2.87488 0.158738
\(329\) 0 0
\(330\) −15.3761 −0.846424
\(331\) 7.35150 0.404075 0.202037 0.979378i \(-0.435244\pi\)
0.202037 + 0.979378i \(0.435244\pi\)
\(332\) −4.04816 −0.222172
\(333\) −30.7689 −1.68612
\(334\) −6.73518 −0.368533
\(335\) 11.0080 0.601432
\(336\) 0 0
\(337\) −1.58075 −0.0861092 −0.0430546 0.999073i \(-0.513709\pi\)
−0.0430546 + 0.999073i \(0.513709\pi\)
\(338\) 5.97606 0.325055
\(339\) −13.9822 −0.759407
\(340\) 7.46607 0.404904
\(341\) 0.651080 0.0352580
\(342\) −6.23433 −0.337114
\(343\) 0 0
\(344\) 4.72940 0.254992
\(345\) 14.8736 0.800767
\(346\) 13.5559 0.728768
\(347\) 21.8834 1.17476 0.587382 0.809310i \(-0.300159\pi\)
0.587382 + 0.809310i \(0.300159\pi\)
\(348\) 12.9506 0.694225
\(349\) 5.31231 0.284362 0.142181 0.989841i \(-0.454589\pi\)
0.142181 + 0.989841i \(0.454589\pi\)
\(350\) 0 0
\(351\) 42.4663 2.26668
\(352\) −30.2946 −1.61471
\(353\) 25.5285 1.35874 0.679371 0.733795i \(-0.262253\pi\)
0.679371 + 0.733795i \(0.262253\pi\)
\(354\) 13.3437 0.709207
\(355\) −9.18124 −0.487289
\(356\) 16.8840 0.894848
\(357\) 0 0
\(358\) 16.1718 0.854708
\(359\) 20.2278 1.06758 0.533792 0.845616i \(-0.320766\pi\)
0.533792 + 0.845616i \(0.320766\pi\)
\(360\) −21.7348 −1.14552
\(361\) −18.2649 −0.961308
\(362\) 22.5239 1.18383
\(363\) −62.5653 −3.28383
\(364\) 0 0
\(365\) 3.48631 0.182482
\(366\) 35.8922 1.87612
\(367\) −32.6414 −1.70387 −0.851934 0.523649i \(-0.824570\pi\)
−0.851934 + 0.523649i \(0.824570\pi\)
\(368\) −1.06516 −0.0555254
\(369\) 8.05685 0.419423
\(370\) −3.23410 −0.168133
\(371\) 0 0
\(372\) 0.470047 0.0243708
\(373\) −8.84603 −0.458030 −0.229015 0.973423i \(-0.573550\pi\)
−0.229015 + 0.973423i \(0.573550\pi\)
\(374\) −33.0725 −1.71014
\(375\) −28.4548 −1.46940
\(376\) 13.6177 0.702282
\(377\) −8.29678 −0.427306
\(378\) 0 0
\(379\) 10.5565 0.542252 0.271126 0.962544i \(-0.412604\pi\)
0.271126 + 0.962544i \(0.412604\pi\)
\(380\) 0.953830 0.0489304
\(381\) −21.1768 −1.08492
\(382\) −14.7116 −0.752713
\(383\) 24.8543 1.27000 0.634998 0.772514i \(-0.281001\pi\)
0.634998 + 0.772514i \(0.281001\pi\)
\(384\) −20.5301 −1.04767
\(385\) 0 0
\(386\) 8.79872 0.447843
\(387\) 13.2542 0.673747
\(388\) 9.96494 0.505893
\(389\) 7.24246 0.367207 0.183604 0.983000i \(-0.441224\pi\)
0.183604 + 0.983000i \(0.441224\pi\)
\(390\) 7.11167 0.360113
\(391\) 31.9918 1.61789
\(392\) 0 0
\(393\) −21.6372 −1.09145
\(394\) 22.0538 1.11106
\(395\) 2.63953 0.132809
\(396\) −52.1555 −2.62091
\(397\) −34.7019 −1.74164 −0.870819 0.491603i \(-0.836411\pi\)
−0.870819 + 0.491603i \(0.836411\pi\)
\(398\) 4.44964 0.223040
\(399\) 0 0
\(400\) 0.920509 0.0460254
\(401\) 18.7650 0.937080 0.468540 0.883442i \(-0.344780\pi\)
0.468540 + 0.883442i \(0.344780\pi\)
\(402\) −35.2040 −1.75582
\(403\) −0.301135 −0.0150006
\(404\) 7.43694 0.370002
\(405\) −29.7858 −1.48007
\(406\) 0 0
\(407\) −20.8530 −1.03364
\(408\) −64.1569 −3.17624
\(409\) 29.6476 1.46598 0.732989 0.680241i \(-0.238125\pi\)
0.732989 + 0.680241i \(0.238125\pi\)
\(410\) 0.846852 0.0418230
\(411\) −44.1035 −2.17547
\(412\) −10.3682 −0.510807
\(413\) 0 0
\(414\) −34.6603 −1.70346
\(415\) −3.20417 −0.157286
\(416\) 14.0117 0.686982
\(417\) −17.2175 −0.843144
\(418\) −4.22519 −0.206661
\(419\) −8.62323 −0.421273 −0.210636 0.977564i \(-0.567554\pi\)
−0.210636 + 0.977564i \(0.567554\pi\)
\(420\) 0 0
\(421\) 11.6599 0.568267 0.284134 0.958785i \(-0.408294\pi\)
0.284134 + 0.958785i \(0.408294\pi\)
\(422\) −24.5078 −1.19302
\(423\) 38.1638 1.85559
\(424\) 6.45095 0.313286
\(425\) −27.6472 −1.34108
\(426\) 29.3619 1.42259
\(427\) 0 0
\(428\) 12.4752 0.603012
\(429\) 45.8549 2.21390
\(430\) 1.39314 0.0671832
\(431\) −28.5212 −1.37382 −0.686910 0.726743i \(-0.741033\pi\)
−0.686910 + 0.726743i \(0.741033\pi\)
\(432\) 3.75735 0.180776
\(433\) −6.13071 −0.294623 −0.147312 0.989090i \(-0.547062\pi\)
−0.147312 + 0.989090i \(0.547062\pi\)
\(434\) 0 0
\(435\) 10.2506 0.491476
\(436\) 3.85123 0.184440
\(437\) 4.08712 0.195514
\(438\) −11.1493 −0.532735
\(439\) 23.3280 1.11338 0.556692 0.830719i \(-0.312070\pi\)
0.556692 + 0.830719i \(0.312070\pi\)
\(440\) −14.7303 −0.702239
\(441\) 0 0
\(442\) 15.2966 0.727584
\(443\) −22.2285 −1.05611 −0.528053 0.849211i \(-0.677078\pi\)
−0.528053 + 0.849211i \(0.677078\pi\)
\(444\) −15.0548 −0.714469
\(445\) 13.3639 0.633507
\(446\) 7.88981 0.373593
\(447\) −12.2227 −0.578112
\(448\) 0 0
\(449\) −11.6146 −0.548126 −0.274063 0.961712i \(-0.588368\pi\)
−0.274063 + 0.961712i \(0.588368\pi\)
\(450\) 29.9533 1.41201
\(451\) 5.46037 0.257119
\(452\) −4.98507 −0.234478
\(453\) −47.4395 −2.22890
\(454\) −13.2573 −0.622195
\(455\) 0 0
\(456\) −8.19639 −0.383831
\(457\) −3.42393 −0.160165 −0.0800823 0.996788i \(-0.525518\pi\)
−0.0800823 + 0.996788i \(0.525518\pi\)
\(458\) 4.78685 0.223675
\(459\) −112.851 −5.26742
\(460\) 5.30290 0.247249
\(461\) 9.15435 0.426360 0.213180 0.977013i \(-0.431618\pi\)
0.213180 + 0.977013i \(0.431618\pi\)
\(462\) 0 0
\(463\) 4.14841 0.192793 0.0963965 0.995343i \(-0.469268\pi\)
0.0963965 + 0.995343i \(0.469268\pi\)
\(464\) −0.734086 −0.0340791
\(465\) 0.372048 0.0172533
\(466\) 3.99405 0.185021
\(467\) −3.34964 −0.155003 −0.0775014 0.996992i \(-0.524694\pi\)
−0.0775014 + 0.996992i \(0.524694\pi\)
\(468\) 24.1227 1.11507
\(469\) 0 0
\(470\) 4.01138 0.185031
\(471\) −30.0910 −1.38652
\(472\) 12.7832 0.588396
\(473\) 8.98275 0.413027
\(474\) −8.44131 −0.387722
\(475\) −3.53207 −0.162063
\(476\) 0 0
\(477\) 18.0788 0.827772
\(478\) 0.618952 0.0283102
\(479\) −2.03920 −0.0931733 −0.0465866 0.998914i \(-0.514834\pi\)
−0.0465866 + 0.998914i \(0.514834\pi\)
\(480\) −17.3113 −0.790149
\(481\) 9.64483 0.439766
\(482\) −8.00306 −0.364529
\(483\) 0 0
\(484\) −22.3065 −1.01393
\(485\) 7.88737 0.358147
\(486\) 49.7303 2.25581
\(487\) 26.9204 1.21988 0.609941 0.792447i \(-0.291193\pi\)
0.609941 + 0.792447i \(0.291193\pi\)
\(488\) 34.3848 1.55653
\(489\) 29.5697 1.33719
\(490\) 0 0
\(491\) −29.1260 −1.31444 −0.657218 0.753700i \(-0.728267\pi\)
−0.657218 + 0.753700i \(0.728267\pi\)
\(492\) 3.94211 0.177724
\(493\) 22.0480 0.992993
\(494\) 1.95422 0.0879244
\(495\) −41.2817 −1.85547
\(496\) −0.0266439 −0.00119635
\(497\) 0 0
\(498\) 10.2470 0.459180
\(499\) −19.9991 −0.895283 −0.447642 0.894213i \(-0.647736\pi\)
−0.447642 + 0.894213i \(0.647736\pi\)
\(500\) −10.1450 −0.453700
\(501\) −24.8158 −1.10869
\(502\) −3.09497 −0.138135
\(503\) 41.5151 1.85107 0.925534 0.378663i \(-0.123616\pi\)
0.925534 + 0.378663i \(0.123616\pi\)
\(504\) 0 0
\(505\) 5.88643 0.261943
\(506\) −23.4903 −1.04427
\(507\) 22.0188 0.977888
\(508\) −7.55019 −0.334985
\(509\) 29.0128 1.28597 0.642985 0.765879i \(-0.277696\pi\)
0.642985 + 0.765879i \(0.277696\pi\)
\(510\) −18.8987 −0.836848
\(511\) 0 0
\(512\) 2.52453 0.111570
\(513\) −14.4173 −0.636539
\(514\) −13.7451 −0.606272
\(515\) −8.20659 −0.361625
\(516\) 6.48509 0.285490
\(517\) 25.8647 1.13753
\(518\) 0 0
\(519\) 49.9466 2.19241
\(520\) 6.81299 0.298769
\(521\) 11.3478 0.497155 0.248578 0.968612i \(-0.420037\pi\)
0.248578 + 0.968612i \(0.420037\pi\)
\(522\) −23.8871 −1.04551
\(523\) −25.2211 −1.10284 −0.551420 0.834227i \(-0.685914\pi\)
−0.551420 + 0.834227i \(0.685914\pi\)
\(524\) −7.71432 −0.337002
\(525\) 0 0
\(526\) 11.0771 0.482984
\(527\) 0.800242 0.0348591
\(528\) 4.05717 0.176566
\(529\) −0.277306 −0.0120568
\(530\) 1.90026 0.0825418
\(531\) 35.8251 1.55468
\(532\) 0 0
\(533\) −2.52551 −0.109392
\(534\) −42.7380 −1.84945
\(535\) 9.87427 0.426902
\(536\) −33.7255 −1.45672
\(537\) 59.5851 2.57129
\(538\) −11.5788 −0.499197
\(539\) 0 0
\(540\) −18.7059 −0.804975
\(541\) −35.2542 −1.51570 −0.757848 0.652431i \(-0.773749\pi\)
−0.757848 + 0.652431i \(0.773749\pi\)
\(542\) −16.2191 −0.696672
\(543\) 82.9891 3.56140
\(544\) −37.2351 −1.59644
\(545\) 3.04829 0.130574
\(546\) 0 0
\(547\) 31.7557 1.35778 0.678889 0.734241i \(-0.262462\pi\)
0.678889 + 0.734241i \(0.262462\pi\)
\(548\) −15.7243 −0.671708
\(549\) 96.3636 4.11270
\(550\) 20.3002 0.865605
\(551\) 2.81675 0.119998
\(552\) −45.5685 −1.93952
\(553\) 0 0
\(554\) 0.753332 0.0320060
\(555\) −11.9161 −0.505808
\(556\) −6.13857 −0.260333
\(557\) 34.4676 1.46044 0.730219 0.683213i \(-0.239418\pi\)
0.730219 + 0.683213i \(0.239418\pi\)
\(558\) −0.866992 −0.0367027
\(559\) −4.15466 −0.175723
\(560\) 0 0
\(561\) −121.856 −5.14476
\(562\) −5.38512 −0.227158
\(563\) 12.2696 0.517103 0.258552 0.965997i \(-0.416755\pi\)
0.258552 + 0.965997i \(0.416755\pi\)
\(564\) 18.6730 0.786277
\(565\) −3.94574 −0.165999
\(566\) −16.6199 −0.698584
\(567\) 0 0
\(568\) 28.1287 1.18025
\(569\) −13.4078 −0.562083 −0.281041 0.959696i \(-0.590680\pi\)
−0.281041 + 0.959696i \(0.590680\pi\)
\(570\) −2.41441 −0.101128
\(571\) 10.4275 0.436376 0.218188 0.975907i \(-0.429985\pi\)
0.218188 + 0.975907i \(0.429985\pi\)
\(572\) 16.3487 0.683573
\(573\) −54.2050 −2.26445
\(574\) 0 0
\(575\) −19.6369 −0.818914
\(576\) 43.9416 1.83090
\(577\) 1.40263 0.0583921 0.0291960 0.999574i \(-0.490705\pi\)
0.0291960 + 0.999574i \(0.490705\pi\)
\(578\) −25.3073 −1.05264
\(579\) 32.4189 1.34728
\(580\) 3.65464 0.151751
\(581\) 0 0
\(582\) −25.2240 −1.04557
\(583\) 12.2525 0.507449
\(584\) −10.6811 −0.441986
\(585\) 19.0934 0.789416
\(586\) 16.5737 0.684651
\(587\) 40.2338 1.66063 0.830313 0.557298i \(-0.188162\pi\)
0.830313 + 0.557298i \(0.188162\pi\)
\(588\) 0 0
\(589\) 0.102235 0.00421253
\(590\) 3.76556 0.155026
\(591\) 81.2574 3.34248
\(592\) 0.853360 0.0350729
\(593\) 15.2417 0.625903 0.312952 0.949769i \(-0.398682\pi\)
0.312952 + 0.949769i \(0.398682\pi\)
\(594\) 82.8619 3.39986
\(595\) 0 0
\(596\) −4.35776 −0.178501
\(597\) 16.3947 0.670990
\(598\) 10.8646 0.444288
\(599\) −2.71702 −0.111014 −0.0555071 0.998458i \(-0.517678\pi\)
−0.0555071 + 0.998458i \(0.517678\pi\)
\(600\) 39.3801 1.60769
\(601\) 22.9588 0.936511 0.468255 0.883593i \(-0.344883\pi\)
0.468255 + 0.883593i \(0.344883\pi\)
\(602\) 0 0
\(603\) −94.5158 −3.84898
\(604\) −16.9136 −0.688206
\(605\) −17.6558 −0.717812
\(606\) −18.8250 −0.764712
\(607\) −44.2606 −1.79648 −0.898241 0.439502i \(-0.855155\pi\)
−0.898241 + 0.439502i \(0.855155\pi\)
\(608\) −4.75698 −0.192921
\(609\) 0 0
\(610\) 10.1287 0.410100
\(611\) −11.9628 −0.483965
\(612\) −64.1043 −2.59126
\(613\) 21.0919 0.851894 0.425947 0.904748i \(-0.359941\pi\)
0.425947 + 0.904748i \(0.359941\pi\)
\(614\) −15.2301 −0.614635
\(615\) 3.12023 0.125820
\(616\) 0 0
\(617\) −22.5480 −0.907750 −0.453875 0.891065i \(-0.649959\pi\)
−0.453875 + 0.891065i \(0.649959\pi\)
\(618\) 26.2449 1.05573
\(619\) −33.7969 −1.35841 −0.679205 0.733948i \(-0.737675\pi\)
−0.679205 + 0.733948i \(0.737675\pi\)
\(620\) 0.132647 0.00532721
\(621\) −80.1541 −3.21647
\(622\) −9.43424 −0.378279
\(623\) 0 0
\(624\) −1.87651 −0.0751203
\(625\) 12.5675 0.502699
\(626\) 5.65790 0.226135
\(627\) −15.5677 −0.621715
\(628\) −10.7284 −0.428108
\(629\) −25.6304 −1.02195
\(630\) 0 0
\(631\) 34.1027 1.35761 0.678803 0.734321i \(-0.262499\pi\)
0.678803 + 0.734321i \(0.262499\pi\)
\(632\) −8.08679 −0.321675
\(633\) −90.2990 −3.58906
\(634\) 23.6224 0.938164
\(635\) −5.97606 −0.237153
\(636\) 8.84572 0.350756
\(637\) 0 0
\(638\) −16.1890 −0.640929
\(639\) 78.8308 3.11850
\(640\) −5.79355 −0.229010
\(641\) 11.0824 0.437728 0.218864 0.975755i \(-0.429765\pi\)
0.218864 + 0.975755i \(0.429765\pi\)
\(642\) −31.5782 −1.24629
\(643\) 33.9767 1.33991 0.669954 0.742402i \(-0.266314\pi\)
0.669954 + 0.742402i \(0.266314\pi\)
\(644\) 0 0
\(645\) 5.13302 0.202113
\(646\) −5.19318 −0.204323
\(647\) 7.89261 0.310291 0.155145 0.987892i \(-0.450415\pi\)
0.155145 + 0.987892i \(0.450415\pi\)
\(648\) 91.2551 3.58484
\(649\) 24.2797 0.953062
\(650\) −9.38918 −0.368274
\(651\) 0 0
\(652\) 10.5425 0.412876
\(653\) 34.3369 1.34370 0.671852 0.740685i \(-0.265499\pi\)
0.671852 + 0.740685i \(0.265499\pi\)
\(654\) −9.74853 −0.381198
\(655\) −6.10597 −0.238580
\(656\) −0.223453 −0.00872437
\(657\) −29.9337 −1.16783
\(658\) 0 0
\(659\) −23.9977 −0.934819 −0.467409 0.884041i \(-0.654813\pi\)
−0.467409 + 0.884041i \(0.654813\pi\)
\(660\) −20.1986 −0.786229
\(661\) −9.55555 −0.371668 −0.185834 0.982581i \(-0.559499\pi\)
−0.185834 + 0.982581i \(0.559499\pi\)
\(662\) −6.63457 −0.257860
\(663\) 56.3602 2.18885
\(664\) 9.81667 0.380961
\(665\) 0 0
\(666\) 27.7683 1.07600
\(667\) 15.6600 0.606357
\(668\) −8.84759 −0.342324
\(669\) 29.0700 1.12391
\(670\) −9.93451 −0.383804
\(671\) 65.3085 2.52120
\(672\) 0 0
\(673\) 3.20764 0.123645 0.0618227 0.998087i \(-0.480309\pi\)
0.0618227 + 0.998087i \(0.480309\pi\)
\(674\) 1.42660 0.0549505
\(675\) 69.2688 2.66616
\(676\) 7.85038 0.301938
\(677\) 1.62436 0.0624293 0.0312147 0.999513i \(-0.490062\pi\)
0.0312147 + 0.999513i \(0.490062\pi\)
\(678\) 12.6186 0.484615
\(679\) 0 0
\(680\) −18.1050 −0.694294
\(681\) −48.8465 −1.87180
\(682\) −0.587586 −0.0224998
\(683\) 37.7267 1.44357 0.721786 0.692117i \(-0.243322\pi\)
0.721786 + 0.692117i \(0.243322\pi\)
\(684\) −8.18966 −0.313140
\(685\) −12.4459 −0.475535
\(686\) 0 0
\(687\) 17.6372 0.672900
\(688\) −0.367598 −0.0140145
\(689\) −5.66699 −0.215895
\(690\) −13.4231 −0.511009
\(691\) 38.9512 1.48177 0.740886 0.671631i \(-0.234406\pi\)
0.740886 + 0.671631i \(0.234406\pi\)
\(692\) 17.8075 0.676940
\(693\) 0 0
\(694\) −19.7493 −0.749674
\(695\) −4.85875 −0.184303
\(696\) −31.4048 −1.19040
\(697\) 6.71133 0.254210
\(698\) −4.79425 −0.181465
\(699\) 14.7161 0.556613
\(700\) 0 0
\(701\) −8.29770 −0.313400 −0.156700 0.987646i \(-0.550086\pi\)
−0.156700 + 0.987646i \(0.550086\pi\)
\(702\) −38.3249 −1.44648
\(703\) −3.27442 −0.123497
\(704\) 29.7805 1.12240
\(705\) 14.7799 0.556644
\(706\) −23.0389 −0.867081
\(707\) 0 0
\(708\) 17.5287 0.658770
\(709\) 21.9274 0.823502 0.411751 0.911296i \(-0.364917\pi\)
0.411751 + 0.911296i \(0.364917\pi\)
\(710\) 8.28588 0.310963
\(711\) −22.6633 −0.849939
\(712\) −40.9431 −1.53441
\(713\) 0.568385 0.0212862
\(714\) 0 0
\(715\) 12.9402 0.483935
\(716\) 21.2439 0.793923
\(717\) 2.28053 0.0851679
\(718\) −18.2552 −0.681278
\(719\) −12.1598 −0.453485 −0.226742 0.973955i \(-0.572808\pi\)
−0.226742 + 0.973955i \(0.572808\pi\)
\(720\) 1.68936 0.0629586
\(721\) 0 0
\(722\) 16.4836 0.613458
\(723\) −29.4873 −1.09664
\(724\) 29.5882 1.09964
\(725\) −13.5333 −0.502614
\(726\) 56.4639 2.09557
\(727\) 5.65386 0.209690 0.104845 0.994489i \(-0.466565\pi\)
0.104845 + 0.994489i \(0.466565\pi\)
\(728\) 0 0
\(729\) 88.0044 3.25942
\(730\) −3.14632 −0.116451
\(731\) 11.0407 0.408354
\(732\) 47.1494 1.74269
\(733\) −24.3253 −0.898475 −0.449238 0.893412i \(-0.648304\pi\)
−0.449238 + 0.893412i \(0.648304\pi\)
\(734\) 29.4582 1.08732
\(735\) 0 0
\(736\) −26.4468 −0.974843
\(737\) −64.0561 −2.35954
\(738\) −7.27114 −0.267654
\(739\) −38.7324 −1.42480 −0.712398 0.701776i \(-0.752391\pi\)
−0.712398 + 0.701776i \(0.752391\pi\)
\(740\) −4.24844 −0.156176
\(741\) 7.20032 0.264510
\(742\) 0 0
\(743\) −5.26170 −0.193033 −0.0965165 0.995331i \(-0.530770\pi\)
−0.0965165 + 0.995331i \(0.530770\pi\)
\(744\) −1.13985 −0.0417889
\(745\) −3.44921 −0.126369
\(746\) 7.98336 0.292291
\(747\) 27.5112 1.00658
\(748\) −43.4454 −1.58852
\(749\) 0 0
\(750\) 25.6799 0.937697
\(751\) 52.3230 1.90929 0.954646 0.297743i \(-0.0962340\pi\)
0.954646 + 0.297743i \(0.0962340\pi\)
\(752\) −1.05845 −0.0385978
\(753\) −11.4034 −0.415564
\(754\) 7.48767 0.272685
\(755\) −13.3873 −0.487215
\(756\) 0 0
\(757\) 21.9399 0.797421 0.398710 0.917077i \(-0.369458\pi\)
0.398710 + 0.917077i \(0.369458\pi\)
\(758\) −9.52704 −0.346038
\(759\) −86.5501 −3.14157
\(760\) −2.31301 −0.0839016
\(761\) −9.81830 −0.355913 −0.177957 0.984038i \(-0.556949\pi\)
−0.177957 + 0.984038i \(0.556949\pi\)
\(762\) 19.1116 0.692341
\(763\) 0 0
\(764\) −19.3258 −0.699182
\(765\) −50.7393 −1.83448
\(766\) −22.4305 −0.810447
\(767\) −11.2297 −0.405483
\(768\) 54.7987 1.97738
\(769\) 3.37717 0.121784 0.0608920 0.998144i \(-0.480605\pi\)
0.0608920 + 0.998144i \(0.480605\pi\)
\(770\) 0 0
\(771\) −50.6440 −1.82390
\(772\) 11.5583 0.415994
\(773\) −38.3502 −1.37936 −0.689681 0.724114i \(-0.742249\pi\)
−0.689681 + 0.724114i \(0.742249\pi\)
\(774\) −11.9616 −0.429951
\(775\) −0.491196 −0.0176443
\(776\) −24.1647 −0.867461
\(777\) 0 0
\(778\) −6.53617 −0.234333
\(779\) 0.857408 0.0307198
\(780\) 9.34216 0.334503
\(781\) 53.4260 1.91173
\(782\) −28.8719 −1.03246
\(783\) −55.2404 −1.97413
\(784\) 0 0
\(785\) −8.49162 −0.303079
\(786\) 19.5271 0.696508
\(787\) −17.2611 −0.615291 −0.307645 0.951501i \(-0.599541\pi\)
−0.307645 + 0.951501i \(0.599541\pi\)
\(788\) 28.9708 1.03204
\(789\) 40.8136 1.45300
\(790\) −2.38213 −0.0847522
\(791\) 0 0
\(792\) 126.475 4.49411
\(793\) −30.2062 −1.07265
\(794\) 31.3177 1.11143
\(795\) 7.00149 0.248317
\(796\) 5.84521 0.207178
\(797\) 19.8669 0.703722 0.351861 0.936052i \(-0.385549\pi\)
0.351861 + 0.936052i \(0.385549\pi\)
\(798\) 0 0
\(799\) 31.7903 1.12466
\(800\) 22.8552 0.808054
\(801\) −114.743 −4.05425
\(802\) −16.9350 −0.597997
\(803\) −20.2870 −0.715912
\(804\) −46.2453 −1.63095
\(805\) 0 0
\(806\) 0.271768 0.00957262
\(807\) −42.6621 −1.50178
\(808\) −18.0344 −0.634446
\(809\) −30.8800 −1.08568 −0.542842 0.839835i \(-0.682652\pi\)
−0.542842 + 0.839835i \(0.682652\pi\)
\(810\) 26.8810 0.944503
\(811\) 22.6923 0.796834 0.398417 0.917204i \(-0.369560\pi\)
0.398417 + 0.917204i \(0.369560\pi\)
\(812\) 0 0
\(813\) −59.7594 −2.09586
\(814\) 18.8194 0.659619
\(815\) 8.34452 0.292296
\(816\) 4.98666 0.174568
\(817\) 1.41051 0.0493474
\(818\) −26.7563 −0.935512
\(819\) 0 0
\(820\) 1.11246 0.0388487
\(821\) 30.3921 1.06069 0.530345 0.847782i \(-0.322062\pi\)
0.530345 + 0.847782i \(0.322062\pi\)
\(822\) 39.8025 1.38827
\(823\) −51.0855 −1.78073 −0.890364 0.455249i \(-0.849550\pi\)
−0.890364 + 0.455249i \(0.849550\pi\)
\(824\) 25.1427 0.875886
\(825\) 74.7962 2.60407
\(826\) 0 0
\(827\) −53.5924 −1.86359 −0.931794 0.362987i \(-0.881757\pi\)
−0.931794 + 0.362987i \(0.881757\pi\)
\(828\) −45.5311 −1.58232
\(829\) −4.38093 −0.152156 −0.0760780 0.997102i \(-0.524240\pi\)
−0.0760780 + 0.997102i \(0.524240\pi\)
\(830\) 2.89169 0.100372
\(831\) 2.77565 0.0962863
\(832\) −13.7740 −0.477526
\(833\) 0 0
\(834\) 15.5384 0.538051
\(835\) −7.00297 −0.242348
\(836\) −5.55038 −0.191964
\(837\) −2.00497 −0.0693020
\(838\) 7.78229 0.268835
\(839\) −5.15856 −0.178093 −0.0890466 0.996027i \(-0.528382\pi\)
−0.0890466 + 0.996027i \(0.528382\pi\)
\(840\) 0 0
\(841\) −18.2075 −0.627845
\(842\) −10.5228 −0.362639
\(843\) −19.8415 −0.683376
\(844\) −32.1944 −1.10818
\(845\) 6.21367 0.213757
\(846\) −34.4420 −1.18414
\(847\) 0 0
\(848\) −0.501407 −0.0172184
\(849\) −61.2358 −2.10161
\(850\) 24.9510 0.855812
\(851\) −18.2044 −0.624039
\(852\) 38.5709 1.32142
\(853\) 16.7568 0.573744 0.286872 0.957969i \(-0.407385\pi\)
0.286872 + 0.957969i \(0.407385\pi\)
\(854\) 0 0
\(855\) −6.48221 −0.221687
\(856\) −30.2520 −1.03399
\(857\) −32.2109 −1.10030 −0.550152 0.835064i \(-0.685430\pi\)
−0.550152 + 0.835064i \(0.685430\pi\)
\(858\) −41.3831 −1.41280
\(859\) −25.0365 −0.854234 −0.427117 0.904196i \(-0.640471\pi\)
−0.427117 + 0.904196i \(0.640471\pi\)
\(860\) 1.83008 0.0624053
\(861\) 0 0
\(862\) 25.7398 0.876702
\(863\) 5.24286 0.178469 0.0892346 0.996011i \(-0.471558\pi\)
0.0892346 + 0.996011i \(0.471558\pi\)
\(864\) 93.2909 3.17382
\(865\) 14.0948 0.479239
\(866\) 5.53284 0.188013
\(867\) −93.2447 −3.16676
\(868\) 0 0
\(869\) −15.3596 −0.521037
\(870\) −9.25091 −0.313635
\(871\) 29.6270 1.00387
\(872\) −9.33910 −0.316262
\(873\) −67.7216 −2.29203
\(874\) −3.68854 −0.124767
\(875\) 0 0
\(876\) −14.6462 −0.494849
\(877\) 49.7732 1.68072 0.840361 0.542028i \(-0.182343\pi\)
0.840361 + 0.542028i \(0.182343\pi\)
\(878\) −21.0530 −0.710505
\(879\) 61.0656 2.05969
\(880\) 1.14493 0.0385955
\(881\) 4.99315 0.168223 0.0841117 0.996456i \(-0.473195\pi\)
0.0841117 + 0.996456i \(0.473195\pi\)
\(882\) 0 0
\(883\) −44.6535 −1.50271 −0.751354 0.659899i \(-0.770599\pi\)
−0.751354 + 0.659899i \(0.770599\pi\)
\(884\) 20.0942 0.675840
\(885\) 13.8742 0.466376
\(886\) 20.0607 0.673954
\(887\) 36.4120 1.22259 0.611297 0.791401i \(-0.290648\pi\)
0.611297 + 0.791401i \(0.290648\pi\)
\(888\) 36.5074 1.22511
\(889\) 0 0
\(890\) −12.0606 −0.404272
\(891\) 173.325 5.80659
\(892\) 10.3644 0.347024
\(893\) 4.06138 0.135909
\(894\) 11.0307 0.368922
\(895\) 16.8148 0.562057
\(896\) 0 0
\(897\) 40.0308 1.33659
\(898\) 10.4819 0.349786
\(899\) 0.391718 0.0130645
\(900\) 39.3478 1.31159
\(901\) 15.0596 0.501708
\(902\) −4.92787 −0.164080
\(903\) 0 0
\(904\) 12.0886 0.402062
\(905\) 23.4194 0.778487
\(906\) 42.8131 1.42237
\(907\) −2.20992 −0.0733791 −0.0366896 0.999327i \(-0.511681\pi\)
−0.0366896 + 0.999327i \(0.511681\pi\)
\(908\) −17.4153 −0.577946
\(909\) −50.5413 −1.67635
\(910\) 0 0
\(911\) 46.7251 1.54807 0.774035 0.633142i \(-0.218235\pi\)
0.774035 + 0.633142i \(0.218235\pi\)
\(912\) 0.637073 0.0210956
\(913\) 18.6452 0.617066
\(914\) 3.09002 0.102209
\(915\) 37.3193 1.23374
\(916\) 6.28820 0.207768
\(917\) 0 0
\(918\) 101.845 3.36140
\(919\) 24.9123 0.821780 0.410890 0.911685i \(-0.365218\pi\)
0.410890 + 0.911685i \(0.365218\pi\)
\(920\) −12.8594 −0.423961
\(921\) −56.1152 −1.84906
\(922\) −8.26161 −0.272082
\(923\) −24.7104 −0.813352
\(924\) 0 0
\(925\) 15.7322 0.517270
\(926\) −3.74385 −0.123031
\(927\) 70.4624 2.31429
\(928\) −18.2266 −0.598316
\(929\) −1.10615 −0.0362915 −0.0181458 0.999835i \(-0.505776\pi\)
−0.0181458 + 0.999835i \(0.505776\pi\)
\(930\) −0.335765 −0.0110102
\(931\) 0 0
\(932\) 5.24673 0.171862
\(933\) −34.7605 −1.13801
\(934\) 3.02298 0.0989149
\(935\) −34.3875 −1.12459
\(936\) −58.4969 −1.91203
\(937\) −4.28847 −0.140098 −0.0700491 0.997544i \(-0.522316\pi\)
−0.0700491 + 0.997544i \(0.522316\pi\)
\(938\) 0 0
\(939\) 20.8465 0.680301
\(940\) 5.26950 0.171872
\(941\) −20.9465 −0.682836 −0.341418 0.939912i \(-0.610907\pi\)
−0.341418 + 0.939912i \(0.610907\pi\)
\(942\) 27.1565 0.884806
\(943\) 4.76683 0.155229
\(944\) −0.993591 −0.0323386
\(945\) 0 0
\(946\) −8.10674 −0.263573
\(947\) 7.51231 0.244117 0.122059 0.992523i \(-0.461050\pi\)
0.122059 + 0.992523i \(0.461050\pi\)
\(948\) −11.0888 −0.360149
\(949\) 9.38305 0.304587
\(950\) 3.18762 0.103420
\(951\) 87.0366 2.82236
\(952\) 0 0
\(953\) −30.4377 −0.985974 −0.492987 0.870037i \(-0.664095\pi\)
−0.492987 + 0.870037i \(0.664095\pi\)
\(954\) −16.3157 −0.528242
\(955\) −15.2966 −0.494985
\(956\) 0.813079 0.0262969
\(957\) −59.6484 −1.92816
\(958\) 1.84033 0.0594585
\(959\) 0 0
\(960\) 17.0175 0.549238
\(961\) −30.9858 −0.999541
\(962\) −8.70426 −0.280637
\(963\) −84.7813 −2.73204
\(964\) −10.5131 −0.338605
\(965\) 9.14856 0.294503
\(966\) 0 0
\(967\) 43.8093 1.40881 0.704405 0.709798i \(-0.251214\pi\)
0.704405 + 0.709798i \(0.251214\pi\)
\(968\) 54.0925 1.73860
\(969\) −19.1343 −0.614681
\(970\) −7.11818 −0.228551
\(971\) 7.16246 0.229854 0.114927 0.993374i \(-0.463337\pi\)
0.114927 + 0.993374i \(0.463337\pi\)
\(972\) 65.3276 2.09538
\(973\) 0 0
\(974\) −24.2951 −0.778467
\(975\) −34.5944 −1.10791
\(976\) −2.67260 −0.0855477
\(977\) 42.0266 1.34455 0.672276 0.740301i \(-0.265317\pi\)
0.672276 + 0.740301i \(0.265317\pi\)
\(978\) −26.6860 −0.853325
\(979\) −77.7648 −2.48537
\(980\) 0 0
\(981\) −26.1729 −0.835635
\(982\) 26.2856 0.838806
\(983\) 13.2127 0.421421 0.210710 0.977549i \(-0.432422\pi\)
0.210710 + 0.977549i \(0.432422\pi\)
\(984\) −9.55949 −0.304745
\(985\) 22.9307 0.730633
\(986\) −19.8979 −0.633678
\(987\) 0 0
\(988\) 2.56714 0.0816715
\(989\) 7.84183 0.249356
\(990\) 37.2559 1.18407
\(991\) 26.6369 0.846150 0.423075 0.906095i \(-0.360951\pi\)
0.423075 + 0.906095i \(0.360951\pi\)
\(992\) −0.661540 −0.0210039
\(993\) −24.4451 −0.775741
\(994\) 0 0
\(995\) 4.62656 0.146672
\(996\) 13.4609 0.426525
\(997\) 55.5759 1.76011 0.880054 0.474874i \(-0.157506\pi\)
0.880054 + 0.474874i \(0.157506\pi\)
\(998\) 18.0488 0.571324
\(999\) 64.2158 2.03170
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 2009.2.a.s.1.7 17
7.2 even 3 287.2.e.d.165.11 34
7.4 even 3 287.2.e.d.247.11 yes 34
7.6 odd 2 2009.2.a.r.1.7 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.e.d.165.11 34 7.2 even 3
287.2.e.d.247.11 yes 34 7.4 even 3
2009.2.a.r.1.7 17 7.6 odd 2
2009.2.a.s.1.7 17 1.1 even 1 trivial