Properties

Label 2009.2.a.p.1.3
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 10x^{5} - x^{4} + 30x^{3} + 7x^{2} - 25x - 11 \) 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 1.3
Root \(1.28109\) 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)
 
\(f(q)\) \(=\) \(q-1.35880 q^{2} +0.281095 q^{3} -0.153673 q^{4} -3.77273 q^{5} -0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} +O(q^{10})\) \(q-1.35880 q^{2} +0.281095 q^{3} -0.153673 q^{4} -3.77273 q^{5} -0.381951 q^{6} +2.92640 q^{8} -2.92099 q^{9} +5.12637 q^{10} -1.53021 q^{11} -0.0431968 q^{12} +0.587073 q^{13} -1.06049 q^{15} -3.66904 q^{16} +6.56030 q^{17} +3.96902 q^{18} +0.782214 q^{19} +0.579768 q^{20} +2.07924 q^{22} +3.72301 q^{23} +0.822597 q^{24} +9.23349 q^{25} -0.797713 q^{26} -1.66436 q^{27} +7.82177 q^{29} +1.44100 q^{30} -5.80820 q^{31} -0.867331 q^{32} -0.430133 q^{33} -8.91411 q^{34} +0.448877 q^{36} +8.38915 q^{37} -1.06287 q^{38} +0.165023 q^{39} -11.0405 q^{40} +1.00000 q^{41} -9.36609 q^{43} +0.235152 q^{44} +11.0201 q^{45} -5.05881 q^{46} -11.9226 q^{47} -1.03135 q^{48} -12.5464 q^{50} +1.84407 q^{51} -0.0902175 q^{52} +1.92403 q^{53} +2.26152 q^{54} +5.77306 q^{55} +0.219876 q^{57} -10.6282 q^{58} -4.28746 q^{59} +0.162970 q^{60} +8.46897 q^{61} +7.89217 q^{62} +8.51660 q^{64} -2.21487 q^{65} +0.584463 q^{66} -11.5220 q^{67} -1.00814 q^{68} +1.04652 q^{69} +8.18700 q^{71} -8.54798 q^{72} +1.05736 q^{73} -11.3991 q^{74} +2.59549 q^{75} -0.120205 q^{76} -0.224233 q^{78} +7.19339 q^{79} +13.8423 q^{80} +8.29511 q^{81} -1.35880 q^{82} -8.44048 q^{83} -24.7502 q^{85} +12.7266 q^{86} +2.19866 q^{87} -4.47800 q^{88} -7.83130 q^{89} -14.9741 q^{90} -0.572127 q^{92} -1.63266 q^{93} +16.2004 q^{94} -2.95108 q^{95} -0.243802 q^{96} -13.2606 q^{97} +4.46971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} - 7 q^{3} + 9 q^{4} - 4 q^{5} + 4 q^{6} - 12 q^{8} + 6 q^{9} - 8 q^{10} - 12 q^{12} - q^{13} - 4 q^{15} + 5 q^{16} + 11 q^{17} + 16 q^{18} - 9 q^{19} - 12 q^{20} + 3 q^{23} + 23 q^{24} + 31 q^{25} - 10 q^{26} - 22 q^{27} - 14 q^{29} + 6 q^{30} - 34 q^{31} - 20 q^{32} - 12 q^{33} - 15 q^{34} + q^{36} + 7 q^{37} + 39 q^{38} - 22 q^{39} - 50 q^{40} + 7 q^{41} - 3 q^{43} + 26 q^{44} + 4 q^{45} - 8 q^{46} - 17 q^{47} - 17 q^{48} + 11 q^{50} + 8 q^{51} + 25 q^{52} + 24 q^{53} - 68 q^{54} - 48 q^{55} + 22 q^{57} - 38 q^{58} - 4 q^{59} - 6 q^{60} - 16 q^{61} - 24 q^{62} + 8 q^{64} - 6 q^{65} + 12 q^{66} - 24 q^{67} + 10 q^{68} - 35 q^{69} - 12 q^{71} - 11 q^{72} - 14 q^{73} - 6 q^{74} + 19 q^{75} - 42 q^{76} - 29 q^{78} - 8 q^{79} + 92 q^{80} + 15 q^{81} - q^{82} - 14 q^{83} + 16 q^{85} + 35 q^{86} + 20 q^{87} + 22 q^{88} + 19 q^{89} - 24 q^{90} - 10 q^{92} + 2 q^{93} + 20 q^{94} - 8 q^{95} + 25 q^{96} - 23 q^{97} + 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35880 −0.960814 −0.480407 0.877046i \(-0.659511\pi\)
−0.480407 + 0.877046i \(0.659511\pi\)
\(3\) 0.281095 0.162290 0.0811451 0.996702i \(-0.474142\pi\)
0.0811451 + 0.996702i \(0.474142\pi\)
\(4\) −0.153673 −0.0768366
\(5\) −3.77273 −1.68722 −0.843608 0.536960i \(-0.819573\pi\)
−0.843608 + 0.536960i \(0.819573\pi\)
\(6\) −0.381951 −0.155931
\(7\) 0 0
\(8\) 2.92640 1.03464
\(9\) −2.92099 −0.973662
\(10\) 5.12637 1.62110
\(11\) −1.53021 −0.461375 −0.230687 0.973028i \(-0.574097\pi\)
−0.230687 + 0.973028i \(0.574097\pi\)
\(12\) −0.0431968 −0.0124698
\(13\) 0.587073 0.162825 0.0814124 0.996680i \(-0.474057\pi\)
0.0814124 + 0.996680i \(0.474057\pi\)
\(14\) 0 0
\(15\) −1.06049 −0.273819
\(16\) −3.66904 −0.917260
\(17\) 6.56030 1.59111 0.795553 0.605884i \(-0.207180\pi\)
0.795553 + 0.605884i \(0.207180\pi\)
\(18\) 3.96902 0.935508
\(19\) 0.782214 0.179452 0.0897261 0.995966i \(-0.471401\pi\)
0.0897261 + 0.995966i \(0.471401\pi\)
\(20\) 0.579768 0.129640
\(21\) 0 0
\(22\) 2.07924 0.443295
\(23\) 3.72301 0.776301 0.388151 0.921596i \(-0.373114\pi\)
0.388151 + 0.921596i \(0.373114\pi\)
\(24\) 0.822597 0.167912
\(25\) 9.23349 1.84670
\(26\) −0.797713 −0.156444
\(27\) −1.66436 −0.320306
\(28\) 0 0
\(29\) 7.82177 1.45247 0.726233 0.687449i \(-0.241269\pi\)
0.726233 + 0.687449i \(0.241269\pi\)
\(30\) 1.44100 0.263089
\(31\) −5.80820 −1.04318 −0.521592 0.853195i \(-0.674662\pi\)
−0.521592 + 0.853195i \(0.674662\pi\)
\(32\) −0.867331 −0.153324
\(33\) −0.430133 −0.0748766
\(34\) −8.91411 −1.52876
\(35\) 0 0
\(36\) 0.448877 0.0748129
\(37\) 8.38915 1.37917 0.689584 0.724206i \(-0.257793\pi\)
0.689584 + 0.724206i \(0.257793\pi\)
\(38\) −1.06287 −0.172420
\(39\) 0.165023 0.0264249
\(40\) −11.0405 −1.74566
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −9.36609 −1.42831 −0.714157 0.699985i \(-0.753190\pi\)
−0.714157 + 0.699985i \(0.753190\pi\)
\(44\) 0.235152 0.0354505
\(45\) 11.0201 1.64278
\(46\) −5.05881 −0.745881
\(47\) −11.9226 −1.73909 −0.869547 0.493851i \(-0.835589\pi\)
−0.869547 + 0.493851i \(0.835589\pi\)
\(48\) −1.03135 −0.148862
\(49\) 0 0
\(50\) −12.5464 −1.77433
\(51\) 1.84407 0.258221
\(52\) −0.0902175 −0.0125109
\(53\) 1.92403 0.264286 0.132143 0.991231i \(-0.457814\pi\)
0.132143 + 0.991231i \(0.457814\pi\)
\(54\) 2.26152 0.307754
\(55\) 5.77306 0.778439
\(56\) 0 0
\(57\) 0.219876 0.0291233
\(58\) −10.6282 −1.39555
\(59\) −4.28746 −0.558180 −0.279090 0.960265i \(-0.590033\pi\)
−0.279090 + 0.960265i \(0.590033\pi\)
\(60\) 0.162970 0.0210393
\(61\) 8.46897 1.08434 0.542170 0.840269i \(-0.317603\pi\)
0.542170 + 0.840269i \(0.317603\pi\)
\(62\) 7.89217 1.00231
\(63\) 0 0
\(64\) 8.51660 1.06458
\(65\) −2.21487 −0.274721
\(66\) 0.584463 0.0719425
\(67\) −11.5220 −1.40764 −0.703818 0.710381i \(-0.748523\pi\)
−0.703818 + 0.710381i \(0.748523\pi\)
\(68\) −1.00814 −0.122255
\(69\) 1.04652 0.125986
\(70\) 0 0
\(71\) 8.18700 0.971618 0.485809 0.874065i \(-0.338525\pi\)
0.485809 + 0.874065i \(0.338525\pi\)
\(72\) −8.54798 −1.00739
\(73\) 1.05736 0.123755 0.0618774 0.998084i \(-0.480291\pi\)
0.0618774 + 0.998084i \(0.480291\pi\)
\(74\) −11.3991 −1.32512
\(75\) 2.59549 0.299701
\(76\) −0.120205 −0.0137885
\(77\) 0 0
\(78\) −0.224233 −0.0253894
\(79\) 7.19339 0.809319 0.404660 0.914467i \(-0.367390\pi\)
0.404660 + 0.914467i \(0.367390\pi\)
\(80\) 13.8423 1.54761
\(81\) 8.29511 0.921679
\(82\) −1.35880 −0.150054
\(83\) −8.44048 −0.926463 −0.463231 0.886237i \(-0.653310\pi\)
−0.463231 + 0.886237i \(0.653310\pi\)
\(84\) 0 0
\(85\) −24.7502 −2.68454
\(86\) 12.7266 1.37234
\(87\) 2.19866 0.235721
\(88\) −4.47800 −0.477357
\(89\) −7.83130 −0.830116 −0.415058 0.909795i \(-0.636239\pi\)
−0.415058 + 0.909795i \(0.636239\pi\)
\(90\) −14.9741 −1.57840
\(91\) 0 0
\(92\) −0.572127 −0.0596484
\(93\) −1.63266 −0.169299
\(94\) 16.2004 1.67095
\(95\) −2.95108 −0.302775
\(96\) −0.243802 −0.0248830
\(97\) −13.2606 −1.34641 −0.673204 0.739457i \(-0.735083\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(98\) 0 0
\(99\) 4.46971 0.449223
\(100\) −1.41894 −0.141894
\(101\) 6.44946 0.641746 0.320873 0.947122i \(-0.396024\pi\)
0.320873 + 0.947122i \(0.396024\pi\)
\(102\) −2.50571 −0.248102
\(103\) −18.0323 −1.77678 −0.888388 0.459094i \(-0.848174\pi\)
−0.888388 + 0.459094i \(0.848174\pi\)
\(104\) 1.71801 0.168465
\(105\) 0 0
\(106\) −2.61436 −0.253929
\(107\) 13.0048 1.25722 0.628609 0.777721i \(-0.283625\pi\)
0.628609 + 0.777721i \(0.283625\pi\)
\(108\) 0.255767 0.0246112
\(109\) −10.5419 −1.00973 −0.504866 0.863198i \(-0.668458\pi\)
−0.504866 + 0.863198i \(0.668458\pi\)
\(110\) −7.84441 −0.747935
\(111\) 2.35815 0.223825
\(112\) 0 0
\(113\) −0.782350 −0.0735973 −0.0367987 0.999323i \(-0.511716\pi\)
−0.0367987 + 0.999323i \(0.511716\pi\)
\(114\) −0.298767 −0.0279821
\(115\) −14.0459 −1.30979
\(116\) −1.20200 −0.111603
\(117\) −1.71483 −0.158536
\(118\) 5.82578 0.536307
\(119\) 0 0
\(120\) −3.10343 −0.283304
\(121\) −8.65847 −0.787133
\(122\) −11.5076 −1.04185
\(123\) 0.281095 0.0253455
\(124\) 0.892566 0.0801548
\(125\) −15.9718 −1.42856
\(126\) 0 0
\(127\) −3.17842 −0.282039 −0.141020 0.990007i \(-0.545038\pi\)
−0.141020 + 0.990007i \(0.545038\pi\)
\(128\) −9.83766 −0.869535
\(129\) −2.63276 −0.231801
\(130\) 3.00956 0.263955
\(131\) 14.7349 1.28739 0.643695 0.765282i \(-0.277401\pi\)
0.643695 + 0.765282i \(0.277401\pi\)
\(132\) 0.0661000 0.00575326
\(133\) 0 0
\(134\) 15.6560 1.35248
\(135\) 6.27917 0.540425
\(136\) 19.1981 1.64622
\(137\) 7.83949 0.669772 0.334886 0.942259i \(-0.391302\pi\)
0.334886 + 0.942259i \(0.391302\pi\)
\(138\) −1.42201 −0.121049
\(139\) −10.9730 −0.930716 −0.465358 0.885123i \(-0.654074\pi\)
−0.465358 + 0.885123i \(0.654074\pi\)
\(140\) 0 0
\(141\) −3.35139 −0.282238
\(142\) −11.1245 −0.933544
\(143\) −0.898344 −0.0751233
\(144\) 10.7172 0.893101
\(145\) −29.5094 −2.45062
\(146\) −1.43674 −0.118905
\(147\) 0 0
\(148\) −1.28919 −0.105971
\(149\) −6.34397 −0.519718 −0.259859 0.965647i \(-0.583676\pi\)
−0.259859 + 0.965647i \(0.583676\pi\)
\(150\) −3.52674 −0.287957
\(151\) −0.708131 −0.0576269 −0.0288134 0.999585i \(-0.509173\pi\)
−0.0288134 + 0.999585i \(0.509173\pi\)
\(152\) 2.28907 0.185668
\(153\) −19.1625 −1.54920
\(154\) 0 0
\(155\) 21.9128 1.76008
\(156\) −0.0253597 −0.00203040
\(157\) −12.2054 −0.974096 −0.487048 0.873375i \(-0.661926\pi\)
−0.487048 + 0.873375i \(0.661926\pi\)
\(158\) −9.77435 −0.777605
\(159\) 0.540834 0.0428909
\(160\) 3.27221 0.258691
\(161\) 0 0
\(162\) −11.2714 −0.885562
\(163\) 1.01447 0.0794595 0.0397297 0.999210i \(-0.487350\pi\)
0.0397297 + 0.999210i \(0.487350\pi\)
\(164\) −0.153673 −0.0119999
\(165\) 1.62278 0.126333
\(166\) 11.4689 0.890158
\(167\) 5.31016 0.410913 0.205456 0.978666i \(-0.434132\pi\)
0.205456 + 0.978666i \(0.434132\pi\)
\(168\) 0 0
\(169\) −12.6553 −0.973488
\(170\) 33.6305 2.57934
\(171\) −2.28484 −0.174726
\(172\) 1.43932 0.109747
\(173\) 2.51787 0.191430 0.0957151 0.995409i \(-0.469486\pi\)
0.0957151 + 0.995409i \(0.469486\pi\)
\(174\) −2.98753 −0.226484
\(175\) 0 0
\(176\) 5.61439 0.423200
\(177\) −1.20518 −0.0905871
\(178\) 10.6411 0.797587
\(179\) 13.8227 1.03316 0.516580 0.856239i \(-0.327205\pi\)
0.516580 + 0.856239i \(0.327205\pi\)
\(180\) −1.69349 −0.126226
\(181\) 3.90137 0.289987 0.144993 0.989433i \(-0.453684\pi\)
0.144993 + 0.989433i \(0.453684\pi\)
\(182\) 0 0
\(183\) 2.38058 0.175978
\(184\) 10.8950 0.803192
\(185\) −31.6500 −2.32695
\(186\) 2.21845 0.162664
\(187\) −10.0386 −0.734096
\(188\) 1.83219 0.133626
\(189\) 0 0
\(190\) 4.00992 0.290910
\(191\) −15.2024 −1.10001 −0.550003 0.835163i \(-0.685373\pi\)
−0.550003 + 0.835163i \(0.685373\pi\)
\(192\) 2.39397 0.172770
\(193\) 14.7029 1.05834 0.529168 0.848517i \(-0.322504\pi\)
0.529168 + 0.848517i \(0.322504\pi\)
\(194\) 18.0184 1.29365
\(195\) −0.622588 −0.0445845
\(196\) 0 0
\(197\) 11.4673 0.817010 0.408505 0.912756i \(-0.366050\pi\)
0.408505 + 0.912756i \(0.366050\pi\)
\(198\) −6.07343 −0.431620
\(199\) −16.6817 −1.18253 −0.591266 0.806476i \(-0.701372\pi\)
−0.591266 + 0.806476i \(0.701372\pi\)
\(200\) 27.0209 1.91067
\(201\) −3.23877 −0.228445
\(202\) −8.76351 −0.616598
\(203\) 0 0
\(204\) −0.283384 −0.0198408
\(205\) −3.77273 −0.263499
\(206\) 24.5022 1.70715
\(207\) −10.8749 −0.755855
\(208\) −2.15399 −0.149353
\(209\) −1.19695 −0.0827947
\(210\) 0 0
\(211\) 9.81295 0.675551 0.337776 0.941227i \(-0.390325\pi\)
0.337776 + 0.941227i \(0.390325\pi\)
\(212\) −0.295672 −0.0203068
\(213\) 2.30132 0.157684
\(214\) −17.6708 −1.20795
\(215\) 35.3357 2.40988
\(216\) −4.87058 −0.331401
\(217\) 0 0
\(218\) 14.3243 0.970164
\(219\) 0.297219 0.0200842
\(220\) −0.887164 −0.0598126
\(221\) 3.85138 0.259072
\(222\) −3.20424 −0.215055
\(223\) −7.71539 −0.516661 −0.258330 0.966057i \(-0.583172\pi\)
−0.258330 + 0.966057i \(0.583172\pi\)
\(224\) 0 0
\(225\) −26.9709 −1.79806
\(226\) 1.06305 0.0707133
\(227\) −26.8002 −1.77879 −0.889396 0.457137i \(-0.848875\pi\)
−0.889396 + 0.457137i \(0.848875\pi\)
\(228\) −0.0337891 −0.00223774
\(229\) −10.8206 −0.715049 −0.357524 0.933904i \(-0.616379\pi\)
−0.357524 + 0.933904i \(0.616379\pi\)
\(230\) 19.0855 1.25846
\(231\) 0 0
\(232\) 22.8896 1.50278
\(233\) −9.09982 −0.596149 −0.298075 0.954543i \(-0.596344\pi\)
−0.298075 + 0.954543i \(0.596344\pi\)
\(234\) 2.33011 0.152324
\(235\) 44.9808 2.93423
\(236\) 0.658868 0.0428886
\(237\) 2.02202 0.131345
\(238\) 0 0
\(239\) −3.68484 −0.238353 −0.119176 0.992873i \(-0.538025\pi\)
−0.119176 + 0.992873i \(0.538025\pi\)
\(240\) 3.89100 0.251163
\(241\) −8.01651 −0.516389 −0.258194 0.966093i \(-0.583127\pi\)
−0.258194 + 0.966093i \(0.583127\pi\)
\(242\) 11.7651 0.756289
\(243\) 7.32479 0.469885
\(244\) −1.30145 −0.0833170
\(245\) 0 0
\(246\) −0.381951 −0.0243523
\(247\) 0.459217 0.0292193
\(248\) −16.9971 −1.07932
\(249\) −2.37257 −0.150356
\(250\) 21.7024 1.37258
\(251\) 1.60958 0.101596 0.0507979 0.998709i \(-0.483824\pi\)
0.0507979 + 0.998709i \(0.483824\pi\)
\(252\) 0 0
\(253\) −5.69697 −0.358166
\(254\) 4.31882 0.270987
\(255\) −6.95716 −0.435675
\(256\) −3.66583 −0.229114
\(257\) 20.1668 1.25797 0.628986 0.777417i \(-0.283470\pi\)
0.628986 + 0.777417i \(0.283470\pi\)
\(258\) 3.57738 0.222718
\(259\) 0 0
\(260\) 0.340366 0.0211086
\(261\) −22.8473 −1.41421
\(262\) −20.0217 −1.23694
\(263\) −24.8290 −1.53102 −0.765512 0.643422i \(-0.777514\pi\)
−0.765512 + 0.643422i \(0.777514\pi\)
\(264\) −1.25874 −0.0774703
\(265\) −7.25884 −0.445907
\(266\) 0 0
\(267\) −2.20134 −0.134720
\(268\) 1.77062 0.108158
\(269\) −30.7089 −1.87235 −0.936177 0.351530i \(-0.885662\pi\)
−0.936177 + 0.351530i \(0.885662\pi\)
\(270\) −8.53212 −0.519248
\(271\) −18.2934 −1.11125 −0.555624 0.831434i \(-0.687520\pi\)
−0.555624 + 0.831434i \(0.687520\pi\)
\(272\) −24.0700 −1.45946
\(273\) 0 0
\(274\) −10.6523 −0.643527
\(275\) −14.1291 −0.852020
\(276\) −0.160822 −0.00968034
\(277\) 1.06650 0.0640798 0.0320399 0.999487i \(-0.489800\pi\)
0.0320399 + 0.999487i \(0.489800\pi\)
\(278\) 14.9100 0.894245
\(279\) 16.9657 1.01571
\(280\) 0 0
\(281\) 1.32567 0.0790828 0.0395414 0.999218i \(-0.487410\pi\)
0.0395414 + 0.999218i \(0.487410\pi\)
\(282\) 4.55385 0.271178
\(283\) 21.5009 1.27809 0.639047 0.769168i \(-0.279329\pi\)
0.639047 + 0.769168i \(0.279329\pi\)
\(284\) −1.25812 −0.0746558
\(285\) −0.829534 −0.0491373
\(286\) 1.22067 0.0721795
\(287\) 0 0
\(288\) 2.53346 0.149286
\(289\) 26.0375 1.53162
\(290\) 40.0973 2.35459
\(291\) −3.72748 −0.218509
\(292\) −0.162488 −0.00950890
\(293\) 19.4483 1.13618 0.568092 0.822965i \(-0.307682\pi\)
0.568092 + 0.822965i \(0.307682\pi\)
\(294\) 0 0
\(295\) 16.1754 0.941770
\(296\) 24.5500 1.42694
\(297\) 2.54681 0.147781
\(298\) 8.62017 0.499353
\(299\) 2.18568 0.126401
\(300\) −0.398857 −0.0230280
\(301\) 0 0
\(302\) 0.962205 0.0553687
\(303\) 1.81291 0.104149
\(304\) −2.86997 −0.164604
\(305\) −31.9511 −1.82952
\(306\) 26.0380 1.48849
\(307\) −28.5378 −1.62874 −0.814369 0.580348i \(-0.802917\pi\)
−0.814369 + 0.580348i \(0.802917\pi\)
\(308\) 0 0
\(309\) −5.06879 −0.288353
\(310\) −29.7750 −1.69111
\(311\) −9.63544 −0.546375 −0.273188 0.961961i \(-0.588078\pi\)
−0.273188 + 0.961961i \(0.588078\pi\)
\(312\) 0.482925 0.0273402
\(313\) 33.6028 1.89934 0.949671 0.313248i \(-0.101417\pi\)
0.949671 + 0.313248i \(0.101417\pi\)
\(314\) 16.5846 0.935925
\(315\) 0 0
\(316\) −1.10543 −0.0621854
\(317\) −30.9417 −1.73786 −0.868931 0.494934i \(-0.835192\pi\)
−0.868931 + 0.494934i \(0.835192\pi\)
\(318\) −0.734884 −0.0412102
\(319\) −11.9689 −0.670131
\(320\) −32.1308 −1.79617
\(321\) 3.65557 0.204034
\(322\) 0 0
\(323\) 5.13156 0.285528
\(324\) −1.27474 −0.0708187
\(325\) 5.42073 0.300688
\(326\) −1.37846 −0.0763458
\(327\) −2.96328 −0.163870
\(328\) 2.92640 0.161584
\(329\) 0 0
\(330\) −2.20502 −0.121382
\(331\) −10.0239 −0.550962 −0.275481 0.961306i \(-0.588837\pi\)
−0.275481 + 0.961306i \(0.588837\pi\)
\(332\) 1.29708 0.0711863
\(333\) −24.5046 −1.34284
\(334\) −7.21543 −0.394811
\(335\) 43.4694 2.37498
\(336\) 0 0
\(337\) 5.00576 0.272681 0.136341 0.990662i \(-0.456466\pi\)
0.136341 + 0.990662i \(0.456466\pi\)
\(338\) 17.1960 0.935341
\(339\) −0.219915 −0.0119441
\(340\) 3.80345 0.206271
\(341\) 8.88775 0.481299
\(342\) 3.10463 0.167879
\(343\) 0 0
\(344\) −27.4089 −1.47779
\(345\) −3.94823 −0.212566
\(346\) −3.42127 −0.183929
\(347\) −11.0497 −0.593177 −0.296589 0.955005i \(-0.595849\pi\)
−0.296589 + 0.955005i \(0.595849\pi\)
\(348\) −0.337875 −0.0181120
\(349\) −10.7923 −0.577698 −0.288849 0.957375i \(-0.593273\pi\)
−0.288849 + 0.957375i \(0.593273\pi\)
\(350\) 0 0
\(351\) −0.977100 −0.0521538
\(352\) 1.32720 0.0707398
\(353\) 21.4959 1.14411 0.572057 0.820214i \(-0.306146\pi\)
0.572057 + 0.820214i \(0.306146\pi\)
\(354\) 1.63760 0.0870373
\(355\) −30.8873 −1.63933
\(356\) 1.20346 0.0637833
\(357\) 0 0
\(358\) −18.7823 −0.992674
\(359\) −9.86289 −0.520543 −0.260272 0.965535i \(-0.583812\pi\)
−0.260272 + 0.965535i \(0.583812\pi\)
\(360\) 32.2492 1.69968
\(361\) −18.3881 −0.967797
\(362\) −5.30117 −0.278623
\(363\) −2.43385 −0.127744
\(364\) 0 0
\(365\) −3.98914 −0.208801
\(366\) −3.23473 −0.169082
\(367\) −27.1347 −1.41642 −0.708210 0.706001i \(-0.750497\pi\)
−0.708210 + 0.706001i \(0.750497\pi\)
\(368\) −13.6599 −0.712070
\(369\) −2.92099 −0.152060
\(370\) 43.0059 2.23577
\(371\) 0 0
\(372\) 0.250896 0.0130083
\(373\) −2.74716 −0.142242 −0.0711212 0.997468i \(-0.522658\pi\)
−0.0711212 + 0.997468i \(0.522658\pi\)
\(374\) 13.6404 0.705330
\(375\) −4.48959 −0.231842
\(376\) −34.8904 −1.79934
\(377\) 4.59195 0.236498
\(378\) 0 0
\(379\) 29.3984 1.51010 0.755048 0.655670i \(-0.227614\pi\)
0.755048 + 0.655670i \(0.227614\pi\)
\(380\) 0.453502 0.0232642
\(381\) −0.893437 −0.0457722
\(382\) 20.6569 1.05690
\(383\) 6.69629 0.342164 0.171082 0.985257i \(-0.445274\pi\)
0.171082 + 0.985257i \(0.445274\pi\)
\(384\) −2.76532 −0.141117
\(385\) 0 0
\(386\) −19.9782 −1.01686
\(387\) 27.3582 1.39070
\(388\) 2.03780 0.103453
\(389\) 0.478863 0.0242793 0.0121397 0.999926i \(-0.496136\pi\)
0.0121397 + 0.999926i \(0.496136\pi\)
\(390\) 0.845970 0.0428374
\(391\) 24.4241 1.23518
\(392\) 0 0
\(393\) 4.14189 0.208931
\(394\) −15.5817 −0.784995
\(395\) −27.1387 −1.36550
\(396\) −0.686875 −0.0345168
\(397\) 24.2293 1.21603 0.608016 0.793925i \(-0.291966\pi\)
0.608016 + 0.793925i \(0.291966\pi\)
\(398\) 22.6670 1.13619
\(399\) 0 0
\(400\) −33.8780 −1.69390
\(401\) 23.2843 1.16276 0.581381 0.813631i \(-0.302512\pi\)
0.581381 + 0.813631i \(0.302512\pi\)
\(402\) 4.40083 0.219493
\(403\) −3.40984 −0.169856
\(404\) −0.991110 −0.0493096
\(405\) −31.2952 −1.55507
\(406\) 0 0
\(407\) −12.8371 −0.636313
\(408\) 5.39648 0.267166
\(409\) −30.6511 −1.51560 −0.757800 0.652487i \(-0.773726\pi\)
−0.757800 + 0.652487i \(0.773726\pi\)
\(410\) 5.12637 0.253173
\(411\) 2.20364 0.108697
\(412\) 2.77108 0.136521
\(413\) 0 0
\(414\) 14.7767 0.726236
\(415\) 31.8436 1.56314
\(416\) −0.509187 −0.0249649
\(417\) −3.08445 −0.151046
\(418\) 1.62641 0.0795503
\(419\) −5.54908 −0.271090 −0.135545 0.990771i \(-0.543279\pi\)
−0.135545 + 0.990771i \(0.543279\pi\)
\(420\) 0 0
\(421\) 11.8783 0.578913 0.289457 0.957191i \(-0.406525\pi\)
0.289457 + 0.957191i \(0.406525\pi\)
\(422\) −13.3338 −0.649079
\(423\) 34.8258 1.69329
\(424\) 5.63048 0.273440
\(425\) 60.5745 2.93829
\(426\) −3.12703 −0.151505
\(427\) 0 0
\(428\) −1.99849 −0.0966004
\(429\) −0.252520 −0.0121918
\(430\) −48.0140 −2.31544
\(431\) −15.8059 −0.761345 −0.380673 0.924710i \(-0.624307\pi\)
−0.380673 + 0.924710i \(0.624307\pi\)
\(432\) 6.10659 0.293804
\(433\) 17.8949 0.859976 0.429988 0.902835i \(-0.358518\pi\)
0.429988 + 0.902835i \(0.358518\pi\)
\(434\) 0 0
\(435\) −8.29495 −0.397712
\(436\) 1.62001 0.0775844
\(437\) 2.91219 0.139309
\(438\) −0.403860 −0.0192972
\(439\) −39.1919 −1.87053 −0.935263 0.353955i \(-0.884837\pi\)
−0.935263 + 0.353955i \(0.884837\pi\)
\(440\) 16.8943 0.805404
\(441\) 0 0
\(442\) −5.23324 −0.248920
\(443\) 37.8749 1.79949 0.899746 0.436414i \(-0.143752\pi\)
0.899746 + 0.436414i \(0.143752\pi\)
\(444\) −0.362384 −0.0171980
\(445\) 29.5454 1.40058
\(446\) 10.4836 0.496415
\(447\) −1.78326 −0.0843452
\(448\) 0 0
\(449\) 22.7250 1.07246 0.536230 0.844072i \(-0.319848\pi\)
0.536230 + 0.844072i \(0.319848\pi\)
\(450\) 36.6479 1.72760
\(451\) −1.53021 −0.0720546
\(452\) 0.120226 0.00565497
\(453\) −0.199052 −0.00935227
\(454\) 36.4160 1.70909
\(455\) 0 0
\(456\) 0.643447 0.0301322
\(457\) −24.3084 −1.13710 −0.568550 0.822649i \(-0.692495\pi\)
−0.568550 + 0.822649i \(0.692495\pi\)
\(458\) 14.7031 0.687029
\(459\) −10.9187 −0.509641
\(460\) 2.15848 0.100640
\(461\) −27.0463 −1.25967 −0.629837 0.776727i \(-0.716878\pi\)
−0.629837 + 0.776727i \(0.716878\pi\)
\(462\) 0 0
\(463\) 9.77747 0.454398 0.227199 0.973848i \(-0.427043\pi\)
0.227199 + 0.973848i \(0.427043\pi\)
\(464\) −28.6984 −1.33229
\(465\) 6.15957 0.285643
\(466\) 12.3648 0.572788
\(467\) 25.1283 1.16280 0.581400 0.813618i \(-0.302505\pi\)
0.581400 + 0.813618i \(0.302505\pi\)
\(468\) 0.263524 0.0121814
\(469\) 0 0
\(470\) −61.1198 −2.81925
\(471\) −3.43087 −0.158086
\(472\) −12.5468 −0.577515
\(473\) 14.3320 0.658988
\(474\) −2.74752 −0.126198
\(475\) 7.22256 0.331394
\(476\) 0 0
\(477\) −5.62006 −0.257325
\(478\) 5.00695 0.229012
\(479\) 35.1189 1.60462 0.802311 0.596906i \(-0.203603\pi\)
0.802311 + 0.596906i \(0.203603\pi\)
\(480\) 0.919800 0.0419829
\(481\) 4.92505 0.224563
\(482\) 10.8928 0.496153
\(483\) 0 0
\(484\) 1.33057 0.0604807
\(485\) 50.0286 2.27168
\(486\) −9.95289 −0.451472
\(487\) −19.5496 −0.885876 −0.442938 0.896552i \(-0.646064\pi\)
−0.442938 + 0.896552i \(0.646064\pi\)
\(488\) 24.7836 1.12190
\(489\) 0.285162 0.0128955
\(490\) 0 0
\(491\) 10.2980 0.464742 0.232371 0.972627i \(-0.425352\pi\)
0.232371 + 0.972627i \(0.425352\pi\)
\(492\) −0.0431968 −0.00194746
\(493\) 51.3132 2.31103
\(494\) −0.623982 −0.0280743
\(495\) −16.8630 −0.757936
\(496\) 21.3105 0.956871
\(497\) 0 0
\(498\) 3.22384 0.144464
\(499\) −21.2111 −0.949541 −0.474770 0.880110i \(-0.657469\pi\)
−0.474770 + 0.880110i \(0.657469\pi\)
\(500\) 2.45444 0.109766
\(501\) 1.49266 0.0666871
\(502\) −2.18709 −0.0976146
\(503\) −44.6725 −1.99185 −0.995923 0.0902029i \(-0.971248\pi\)
−0.995923 + 0.0902029i \(0.971248\pi\)
\(504\) 0 0
\(505\) −24.3321 −1.08276
\(506\) 7.74103 0.344131
\(507\) −3.55735 −0.157988
\(508\) 0.488438 0.0216709
\(509\) 23.3618 1.03549 0.517747 0.855534i \(-0.326771\pi\)
0.517747 + 0.855534i \(0.326771\pi\)
\(510\) 9.45337 0.418602
\(511\) 0 0
\(512\) 24.6564 1.08967
\(513\) −1.30188 −0.0574796
\(514\) −27.4026 −1.20868
\(515\) 68.0310 2.99780
\(516\) 0.404585 0.0178108
\(517\) 18.2441 0.802374
\(518\) 0 0
\(519\) 0.707761 0.0310672
\(520\) −6.48160 −0.284237
\(521\) −2.33762 −0.102413 −0.0512064 0.998688i \(-0.516307\pi\)
−0.0512064 + 0.998688i \(0.516307\pi\)
\(522\) 31.0448 1.35879
\(523\) 1.57772 0.0689888 0.0344944 0.999405i \(-0.489018\pi\)
0.0344944 + 0.999405i \(0.489018\pi\)
\(524\) −2.26435 −0.0989187
\(525\) 0 0
\(526\) 33.7376 1.47103
\(527\) −38.1036 −1.65982
\(528\) 1.57818 0.0686813
\(529\) −9.13920 −0.397357
\(530\) 9.86328 0.428433
\(531\) 12.5236 0.543478
\(532\) 0 0
\(533\) 0.587073 0.0254290
\(534\) 2.99117 0.129441
\(535\) −49.0635 −2.12120
\(536\) −33.7180 −1.45640
\(537\) 3.88550 0.167672
\(538\) 41.7271 1.79898
\(539\) 0 0
\(540\) −0.964941 −0.0415245
\(541\) −5.19622 −0.223403 −0.111702 0.993742i \(-0.535630\pi\)
−0.111702 + 0.993742i \(0.535630\pi\)
\(542\) 24.8571 1.06770
\(543\) 1.09666 0.0470620
\(544\) −5.68995 −0.243955
\(545\) 39.7718 1.70364
\(546\) 0 0
\(547\) 10.6657 0.456031 0.228016 0.973657i \(-0.426776\pi\)
0.228016 + 0.973657i \(0.426776\pi\)
\(548\) −1.20472 −0.0514631
\(549\) −24.7377 −1.05578
\(550\) 19.1986 0.818632
\(551\) 6.11830 0.260648
\(552\) 3.06253 0.130350
\(553\) 0 0
\(554\) −1.44916 −0.0615688
\(555\) −8.89665 −0.377642
\(556\) 1.68625 0.0715131
\(557\) −25.3290 −1.07322 −0.536611 0.843830i \(-0.680296\pi\)
−0.536611 + 0.843830i \(0.680296\pi\)
\(558\) −23.0529 −0.975907
\(559\) −5.49858 −0.232565
\(560\) 0 0
\(561\) −2.82180 −0.119137
\(562\) −1.80131 −0.0759839
\(563\) 2.64057 0.111287 0.0556433 0.998451i \(-0.482279\pi\)
0.0556433 + 0.998451i \(0.482279\pi\)
\(564\) 0.515019 0.0216862
\(565\) 2.95160 0.124175
\(566\) −29.2153 −1.22801
\(567\) 0 0
\(568\) 23.9585 1.00527
\(569\) −29.7217 −1.24600 −0.622998 0.782223i \(-0.714086\pi\)
−0.622998 + 0.782223i \(0.714086\pi\)
\(570\) 1.12717 0.0472118
\(571\) 14.4005 0.602640 0.301320 0.953523i \(-0.402573\pi\)
0.301320 + 0.953523i \(0.402573\pi\)
\(572\) 0.138051 0.00577222
\(573\) −4.27331 −0.178520
\(574\) 0 0
\(575\) 34.3764 1.43359
\(576\) −24.8769 −1.03654
\(577\) 7.51404 0.312813 0.156407 0.987693i \(-0.450009\pi\)
0.156407 + 0.987693i \(0.450009\pi\)
\(578\) −35.3797 −1.47160
\(579\) 4.13290 0.171758
\(580\) 4.53481 0.188298
\(581\) 0 0
\(582\) 5.06489 0.209946
\(583\) −2.94416 −0.121935
\(584\) 3.09426 0.128042
\(585\) 6.46960 0.267485
\(586\) −26.4263 −1.09166
\(587\) 45.6741 1.88517 0.942585 0.333965i \(-0.108387\pi\)
0.942585 + 0.333965i \(0.108387\pi\)
\(588\) 0 0
\(589\) −4.54326 −0.187202
\(590\) −21.9791 −0.904865
\(591\) 3.22339 0.132593
\(592\) −30.7801 −1.26506
\(593\) −24.2450 −0.995624 −0.497812 0.867285i \(-0.665863\pi\)
−0.497812 + 0.867285i \(0.665863\pi\)
\(594\) −3.46060 −0.141990
\(595\) 0 0
\(596\) 0.974899 0.0399334
\(597\) −4.68913 −0.191913
\(598\) −2.96989 −0.121448
\(599\) 6.56723 0.268330 0.134165 0.990959i \(-0.457165\pi\)
0.134165 + 0.990959i \(0.457165\pi\)
\(600\) 7.59544 0.310082
\(601\) 22.8645 0.932661 0.466331 0.884610i \(-0.345576\pi\)
0.466331 + 0.884610i \(0.345576\pi\)
\(602\) 0 0
\(603\) 33.6556 1.37056
\(604\) 0.108821 0.00442785
\(605\) 32.6661 1.32806
\(606\) −2.46338 −0.100068
\(607\) −2.13050 −0.0864744 −0.0432372 0.999065i \(-0.513767\pi\)
−0.0432372 + 0.999065i \(0.513767\pi\)
\(608\) −0.678439 −0.0275143
\(609\) 0 0
\(610\) 43.4151 1.75782
\(611\) −6.99946 −0.283168
\(612\) 2.94477 0.119035
\(613\) −13.9256 −0.562448 −0.281224 0.959642i \(-0.590740\pi\)
−0.281224 + 0.959642i \(0.590740\pi\)
\(614\) 38.7770 1.56491
\(615\) −1.06049 −0.0427633
\(616\) 0 0
\(617\) −34.8843 −1.40439 −0.702195 0.711985i \(-0.747796\pi\)
−0.702195 + 0.711985i \(0.747796\pi\)
\(618\) 6.88745 0.277054
\(619\) 31.8727 1.28107 0.640537 0.767928i \(-0.278712\pi\)
0.640537 + 0.767928i \(0.278712\pi\)
\(620\) −3.36741 −0.135238
\(621\) −6.19642 −0.248654
\(622\) 13.0926 0.524965
\(623\) 0 0
\(624\) −0.605477 −0.0242385
\(625\) 14.0899 0.563594
\(626\) −45.6594 −1.82492
\(627\) −0.336456 −0.0134368
\(628\) 1.87564 0.0748463
\(629\) 55.0354 2.19440
\(630\) 0 0
\(631\) 9.96929 0.396871 0.198436 0.980114i \(-0.436414\pi\)
0.198436 + 0.980114i \(0.436414\pi\)
\(632\) 21.0508 0.837354
\(633\) 2.75837 0.109635
\(634\) 42.0435 1.66976
\(635\) 11.9913 0.475861
\(636\) −0.0831118 −0.00329560
\(637\) 0 0
\(638\) 16.2633 0.643871
\(639\) −23.9141 −0.946027
\(640\) 37.1148 1.46709
\(641\) 38.8172 1.53319 0.766594 0.642133i \(-0.221950\pi\)
0.766594 + 0.642133i \(0.221950\pi\)
\(642\) −4.96718 −0.196039
\(643\) 17.9051 0.706110 0.353055 0.935603i \(-0.385143\pi\)
0.353055 + 0.935603i \(0.385143\pi\)
\(644\) 0 0
\(645\) 9.93268 0.391099
\(646\) −6.97274 −0.274339
\(647\) 6.22800 0.244848 0.122424 0.992478i \(-0.460933\pi\)
0.122424 + 0.992478i \(0.460933\pi\)
\(648\) 24.2748 0.953606
\(649\) 6.56070 0.257530
\(650\) −7.36567 −0.288905
\(651\) 0 0
\(652\) −0.155897 −0.00610540
\(653\) −44.2209 −1.73050 −0.865248 0.501344i \(-0.832839\pi\)
−0.865248 + 0.501344i \(0.832839\pi\)
\(654\) 4.02649 0.157448
\(655\) −55.5906 −2.17211
\(656\) −3.66904 −0.143252
\(657\) −3.08854 −0.120495
\(658\) 0 0
\(659\) −25.3675 −0.988178 −0.494089 0.869411i \(-0.664498\pi\)
−0.494089 + 0.869411i \(0.664498\pi\)
\(660\) −0.249377 −0.00970700
\(661\) 27.2889 1.06141 0.530707 0.847556i \(-0.321927\pi\)
0.530707 + 0.847556i \(0.321927\pi\)
\(662\) 13.6204 0.529372
\(663\) 1.08260 0.0420448
\(664\) −24.7002 −0.958555
\(665\) 0 0
\(666\) 33.2967 1.29022
\(667\) 29.1205 1.12755
\(668\) −0.816030 −0.0315731
\(669\) −2.16876 −0.0838489
\(670\) −59.0660 −2.28192
\(671\) −12.9593 −0.500287
\(672\) 0 0
\(673\) 2.43120 0.0937158 0.0468579 0.998902i \(-0.485079\pi\)
0.0468579 + 0.998902i \(0.485079\pi\)
\(674\) −6.80181 −0.261996
\(675\) −15.3678 −0.591508
\(676\) 1.94479 0.0747995
\(677\) 3.98914 0.153315 0.0766576 0.997057i \(-0.475575\pi\)
0.0766576 + 0.997057i \(0.475575\pi\)
\(678\) 0.298819 0.0114761
\(679\) 0 0
\(680\) −72.4292 −2.77753
\(681\) −7.53340 −0.288681
\(682\) −12.0766 −0.462439
\(683\) 29.9721 1.14685 0.573426 0.819257i \(-0.305614\pi\)
0.573426 + 0.819257i \(0.305614\pi\)
\(684\) 0.351118 0.0134253
\(685\) −29.5763 −1.13005
\(686\) 0 0
\(687\) −3.04163 −0.116045
\(688\) 34.3645 1.31014
\(689\) 1.12955 0.0430323
\(690\) 5.36484 0.204236
\(691\) 13.4964 0.513425 0.256713 0.966488i \(-0.417361\pi\)
0.256713 + 0.966488i \(0.417361\pi\)
\(692\) −0.386929 −0.0147089
\(693\) 0 0
\(694\) 15.0142 0.569933
\(695\) 41.3981 1.57032
\(696\) 6.43416 0.243886
\(697\) 6.56030 0.248489
\(698\) 14.6645 0.555061
\(699\) −2.55791 −0.0967491
\(700\) 0 0
\(701\) −42.7642 −1.61518 −0.807591 0.589743i \(-0.799229\pi\)
−0.807591 + 0.589743i \(0.799229\pi\)
\(702\) 1.32768 0.0501101
\(703\) 6.56211 0.247495
\(704\) −13.0322 −0.491168
\(705\) 12.6439 0.476196
\(706\) −29.2086 −1.09928
\(707\) 0 0
\(708\) 0.185204 0.00696040
\(709\) −9.91620 −0.372411 −0.186205 0.982511i \(-0.559619\pi\)
−0.186205 + 0.982511i \(0.559619\pi\)
\(710\) 41.9696 1.57509
\(711\) −21.0118 −0.788003
\(712\) −22.9175 −0.858871
\(713\) −21.6240 −0.809825
\(714\) 0 0
\(715\) 3.38921 0.126749
\(716\) −2.12418 −0.0793845
\(717\) −1.03579 −0.0386823
\(718\) 13.4017 0.500145
\(719\) −17.3198 −0.645919 −0.322960 0.946413i \(-0.604678\pi\)
−0.322960 + 0.946413i \(0.604678\pi\)
\(720\) −40.4331 −1.50685
\(721\) 0 0
\(722\) 24.9857 0.929873
\(723\) −2.25340 −0.0838048
\(724\) −0.599537 −0.0222816
\(725\) 72.2222 2.68227
\(726\) 3.30711 0.122738
\(727\) −40.4385 −1.49978 −0.749891 0.661561i \(-0.769894\pi\)
−0.749891 + 0.661561i \(0.769894\pi\)
\(728\) 0 0
\(729\) −22.8264 −0.845422
\(730\) 5.42042 0.200619
\(731\) −61.4443 −2.27260
\(732\) −0.365832 −0.0135215
\(733\) −5.25881 −0.194238 −0.0971192 0.995273i \(-0.530963\pi\)
−0.0971192 + 0.995273i \(0.530963\pi\)
\(734\) 36.8706 1.36092
\(735\) 0 0
\(736\) −3.22908 −0.119026
\(737\) 17.6310 0.649447
\(738\) 3.96902 0.146102
\(739\) 35.8773 1.31977 0.659884 0.751368i \(-0.270605\pi\)
0.659884 + 0.751368i \(0.270605\pi\)
\(740\) 4.86376 0.178795
\(741\) 0.129084 0.00474200
\(742\) 0 0
\(743\) −30.2113 −1.10835 −0.554173 0.832402i \(-0.686965\pi\)
−0.554173 + 0.832402i \(0.686965\pi\)
\(744\) −4.77781 −0.175163
\(745\) 23.9341 0.876877
\(746\) 3.73282 0.136668
\(747\) 24.6545 0.902061
\(748\) 1.54267 0.0564055
\(749\) 0 0
\(750\) 6.10044 0.222757
\(751\) 41.2217 1.50420 0.752101 0.659047i \(-0.229040\pi\)
0.752101 + 0.659047i \(0.229040\pi\)
\(752\) 43.7446 1.59520
\(753\) 0.452444 0.0164880
\(754\) −6.23953 −0.227230
\(755\) 2.67159 0.0972290
\(756\) 0 0
\(757\) −29.1279 −1.05867 −0.529335 0.848413i \(-0.677559\pi\)
−0.529335 + 0.848413i \(0.677559\pi\)
\(758\) −39.9465 −1.45092
\(759\) −1.60139 −0.0581268
\(760\) −8.63606 −0.313263
\(761\) 10.2126 0.370207 0.185104 0.982719i \(-0.440738\pi\)
0.185104 + 0.982719i \(0.440738\pi\)
\(762\) 1.21400 0.0439785
\(763\) 0 0
\(764\) 2.33620 0.0845208
\(765\) 72.2951 2.61383
\(766\) −9.09889 −0.328756
\(767\) −2.51705 −0.0908855
\(768\) −1.03044 −0.0371830
\(769\) −44.8525 −1.61742 −0.808711 0.588206i \(-0.799834\pi\)
−0.808711 + 0.588206i \(0.799834\pi\)
\(770\) 0 0
\(771\) 5.66879 0.204157
\(772\) −2.25944 −0.0813190
\(773\) −6.80825 −0.244876 −0.122438 0.992476i \(-0.539071\pi\)
−0.122438 + 0.992476i \(0.539071\pi\)
\(774\) −37.1742 −1.33620
\(775\) −53.6300 −1.92645
\(776\) −38.8058 −1.39305
\(777\) 0 0
\(778\) −0.650677 −0.0233279
\(779\) 0.782214 0.0280257
\(780\) 0.0956752 0.00342572
\(781\) −12.5278 −0.448280
\(782\) −33.1873 −1.18678
\(783\) −13.0182 −0.465234
\(784\) 0 0
\(785\) 46.0476 1.64351
\(786\) −5.62799 −0.200744
\(787\) −56.0803 −1.99905 −0.999524 0.0308539i \(-0.990177\pi\)
−0.999524 + 0.0308539i \(0.990177\pi\)
\(788\) −1.76221 −0.0627763
\(789\) −6.97931 −0.248470
\(790\) 36.8760 1.31199
\(791\) 0 0
\(792\) 13.0802 0.464784
\(793\) 4.97190 0.176557
\(794\) −32.9226 −1.16838
\(795\) −2.04042 −0.0723663
\(796\) 2.56353 0.0908618
\(797\) −48.1716 −1.70632 −0.853162 0.521646i \(-0.825318\pi\)
−0.853162 + 0.521646i \(0.825318\pi\)
\(798\) 0 0
\(799\) −78.2160 −2.76708
\(800\) −8.00849 −0.283143
\(801\) 22.8751 0.808252
\(802\) −31.6386 −1.11720
\(803\) −1.61798 −0.0570973
\(804\) 0.497713 0.0175530
\(805\) 0 0
\(806\) 4.63328 0.163200
\(807\) −8.63211 −0.303865
\(808\) 18.8737 0.663975
\(809\) −42.4935 −1.49399 −0.746997 0.664828i \(-0.768505\pi\)
−0.746997 + 0.664828i \(0.768505\pi\)
\(810\) 42.5238 1.49413
\(811\) −2.95213 −0.103663 −0.0518317 0.998656i \(-0.516506\pi\)
−0.0518317 + 0.998656i \(0.516506\pi\)
\(812\) 0 0
\(813\) −5.14219 −0.180345
\(814\) 17.4431 0.611379
\(815\) −3.82732 −0.134065
\(816\) −6.76595 −0.236856
\(817\) −7.32628 −0.256314
\(818\) 41.6486 1.45621
\(819\) 0 0
\(820\) 0.579768 0.0202464
\(821\) −0.811655 −0.0283269 −0.0141635 0.999900i \(-0.504509\pi\)
−0.0141635 + 0.999900i \(0.504509\pi\)
\(822\) −2.99430 −0.104438
\(823\) −4.07214 −0.141946 −0.0709729 0.997478i \(-0.522610\pi\)
−0.0709729 + 0.997478i \(0.522610\pi\)
\(824\) −52.7698 −1.83832
\(825\) −3.97163 −0.138274
\(826\) 0 0
\(827\) −20.3822 −0.708758 −0.354379 0.935102i \(-0.615308\pi\)
−0.354379 + 0.935102i \(0.615308\pi\)
\(828\) 1.67117 0.0580773
\(829\) −29.3639 −1.01985 −0.509925 0.860219i \(-0.670327\pi\)
−0.509925 + 0.860219i \(0.670327\pi\)
\(830\) −43.2690 −1.50189
\(831\) 0.299788 0.0103995
\(832\) 4.99987 0.173339
\(833\) 0 0
\(834\) 4.19114 0.145127
\(835\) −20.0338 −0.693298
\(836\) 0.183939 0.00636167
\(837\) 9.66693 0.334138
\(838\) 7.54007 0.260468
\(839\) −9.95705 −0.343756 −0.171878 0.985118i \(-0.554983\pi\)
−0.171878 + 0.985118i \(0.554983\pi\)
\(840\) 0 0
\(841\) 32.1801 1.10966
\(842\) −16.1402 −0.556228
\(843\) 0.372639 0.0128344
\(844\) −1.50799 −0.0519071
\(845\) 47.7452 1.64248
\(846\) −47.3212 −1.62694
\(847\) 0 0
\(848\) −7.05933 −0.242418
\(849\) 6.04378 0.207422
\(850\) −82.3083 −2.82315
\(851\) 31.2329 1.07065
\(852\) −0.353652 −0.0121159
\(853\) 12.4050 0.424740 0.212370 0.977189i \(-0.431882\pi\)
0.212370 + 0.977189i \(0.431882\pi\)
\(854\) 0 0
\(855\) 8.62007 0.294800
\(856\) 38.0572 1.30077
\(857\) −16.1448 −0.551494 −0.275747 0.961230i \(-0.588925\pi\)
−0.275747 + 0.961230i \(0.588925\pi\)
\(858\) 0.343123 0.0117140
\(859\) −43.8520 −1.49621 −0.748106 0.663580i \(-0.769036\pi\)
−0.748106 + 0.663580i \(0.769036\pi\)
\(860\) −5.43015 −0.185167
\(861\) 0 0
\(862\) 21.4770 0.731511
\(863\) −29.4794 −1.00349 −0.501745 0.865016i \(-0.667308\pi\)
−0.501745 + 0.865016i \(0.667308\pi\)
\(864\) 1.44355 0.0491106
\(865\) −9.49925 −0.322984
\(866\) −24.3156 −0.826277
\(867\) 7.31902 0.248567
\(868\) 0 0
\(869\) −11.0074 −0.373399
\(870\) 11.2711 0.382127
\(871\) −6.76425 −0.229198
\(872\) −30.8499 −1.04471
\(873\) 38.7340 1.31095
\(874\) −3.95707 −0.133850
\(875\) 0 0
\(876\) −0.0456746 −0.00154320
\(877\) 33.3689 1.12679 0.563393 0.826189i \(-0.309496\pi\)
0.563393 + 0.826189i \(0.309496\pi\)
\(878\) 53.2537 1.79723
\(879\) 5.46683 0.184391
\(880\) −21.1816 −0.714030
\(881\) −35.4625 −1.19476 −0.597381 0.801958i \(-0.703792\pi\)
−0.597381 + 0.801958i \(0.703792\pi\)
\(882\) 0 0
\(883\) −55.1635 −1.85640 −0.928200 0.372082i \(-0.878644\pi\)
−0.928200 + 0.372082i \(0.878644\pi\)
\(884\) −0.591854 −0.0199062
\(885\) 4.54683 0.152840
\(886\) −51.4643 −1.72898
\(887\) 10.4949 0.352383 0.176192 0.984356i \(-0.443622\pi\)
0.176192 + 0.984356i \(0.443622\pi\)
\(888\) 6.90089 0.231579
\(889\) 0 0
\(890\) −40.1461 −1.34570
\(891\) −12.6932 −0.425240
\(892\) 1.18565 0.0396985
\(893\) −9.32605 −0.312084
\(894\) 2.42308 0.0810400
\(895\) −52.1494 −1.74316
\(896\) 0 0
\(897\) 0.614383 0.0205137
\(898\) −30.8786 −1.03043
\(899\) −45.4304 −1.51519
\(900\) 4.14470 0.138157
\(901\) 12.6222 0.420507
\(902\) 2.07924 0.0692311
\(903\) 0 0
\(904\) −2.28947 −0.0761467
\(905\) −14.7188 −0.489270
\(906\) 0.270471 0.00898579
\(907\) 12.5409 0.416415 0.208207 0.978085i \(-0.433237\pi\)
0.208207 + 0.978085i \(0.433237\pi\)
\(908\) 4.11848 0.136676
\(909\) −18.8388 −0.624843
\(910\) 0 0
\(911\) −45.5700 −1.50980 −0.754901 0.655839i \(-0.772315\pi\)
−0.754901 + 0.655839i \(0.772315\pi\)
\(912\) −0.806735 −0.0267137
\(913\) 12.9157 0.427446
\(914\) 33.0302 1.09254
\(915\) −8.98129 −0.296912
\(916\) 1.66284 0.0549419
\(917\) 0 0
\(918\) 14.8363 0.489670
\(919\) 52.3114 1.72559 0.862797 0.505550i \(-0.168711\pi\)
0.862797 + 0.505550i \(0.168711\pi\)
\(920\) −41.1040 −1.35516
\(921\) −8.02182 −0.264328
\(922\) 36.7505 1.21031
\(923\) 4.80637 0.158204
\(924\) 0 0
\(925\) 77.4611 2.54691
\(926\) −13.2856 −0.436592
\(927\) 52.6721 1.72998
\(928\) −6.78406 −0.222698
\(929\) 11.0041 0.361034 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(930\) −8.36960 −0.274450
\(931\) 0 0
\(932\) 1.39840 0.0458061
\(933\) −2.70847 −0.0886714
\(934\) −34.1442 −1.11723
\(935\) 37.8730 1.23858
\(936\) −5.01829 −0.164028
\(937\) 8.03703 0.262558 0.131279 0.991345i \(-0.458092\pi\)
0.131279 + 0.991345i \(0.458092\pi\)
\(938\) 0 0
\(939\) 9.44557 0.308245
\(940\) −6.91235 −0.225456
\(941\) 23.7694 0.774859 0.387429 0.921899i \(-0.373363\pi\)
0.387429 + 0.921899i \(0.373363\pi\)
\(942\) 4.66186 0.151891
\(943\) 3.72301 0.121238
\(944\) 15.7308 0.511996
\(945\) 0 0
\(946\) −19.4743 −0.633165
\(947\) −33.2635 −1.08092 −0.540459 0.841370i \(-0.681750\pi\)
−0.540459 + 0.841370i \(0.681750\pi\)
\(948\) −0.310731 −0.0100921
\(949\) 0.620748 0.0201503
\(950\) −9.81399 −0.318408
\(951\) −8.69756 −0.282038
\(952\) 0 0
\(953\) 29.0945 0.942463 0.471232 0.882009i \(-0.343810\pi\)
0.471232 + 0.882009i \(0.343810\pi\)
\(954\) 7.63651 0.247241
\(955\) 57.3545 1.85595
\(956\) 0.566261 0.0183142
\(957\) −3.36440 −0.108756
\(958\) −47.7194 −1.54174
\(959\) 0 0
\(960\) −9.03181 −0.291500
\(961\) 2.73524 0.0882335
\(962\) −6.69213 −0.215763
\(963\) −37.9867 −1.22411
\(964\) 1.23192 0.0396776
\(965\) −55.4700 −1.78564
\(966\) 0 0
\(967\) −52.2398 −1.67992 −0.839959 0.542650i \(-0.817421\pi\)
−0.839959 + 0.542650i \(0.817421\pi\)
\(968\) −25.3382 −0.814399
\(969\) 1.44245 0.0463383
\(970\) −67.9787 −2.18266
\(971\) 1.75797 0.0564159 0.0282080 0.999602i \(-0.491020\pi\)
0.0282080 + 0.999602i \(0.491020\pi\)
\(972\) −1.12562 −0.0361044
\(973\) 0 0
\(974\) 26.5639 0.851162
\(975\) 1.52374 0.0487987
\(976\) −31.0730 −0.994621
\(977\) −37.5188 −1.20033 −0.600167 0.799875i \(-0.704899\pi\)
−0.600167 + 0.799875i \(0.704899\pi\)
\(978\) −0.387478 −0.0123902
\(979\) 11.9835 0.382995
\(980\) 0 0
\(981\) 30.7928 0.983137
\(982\) −13.9929 −0.446531
\(983\) 32.6951 1.04281 0.521405 0.853309i \(-0.325408\pi\)
0.521405 + 0.853309i \(0.325408\pi\)
\(984\) 0.822597 0.0262234
\(985\) −43.2630 −1.37847
\(986\) −69.7241 −2.22047
\(987\) 0 0
\(988\) −0.0705694 −0.00224511
\(989\) −34.8700 −1.10880
\(990\) 22.9134 0.728236
\(991\) −39.7198 −1.26174 −0.630870 0.775889i \(-0.717302\pi\)
−0.630870 + 0.775889i \(0.717302\pi\)
\(992\) 5.03764 0.159945
\(993\) −2.81766 −0.0894158
\(994\) 0 0
\(995\) 62.9354 1.99519
\(996\) 0.364601 0.0115528
\(997\) 8.11982 0.257157 0.128579 0.991699i \(-0.458959\pi\)
0.128579 + 0.991699i \(0.458959\pi\)
\(998\) 28.8216 0.912332
\(999\) −13.9626 −0.441756
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.p.1.3 7
7.6 odd 2 2009.2.a.q.1.3 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2009.2.a.p.1.3 7 1.1 even 1 trivial
2009.2.a.q.1.3 yes 7 7.6 odd 2