Properties

Label 2898.2.a.bg.1.2
Level $2898$
Weight $2$
Character 2898.1
Self dual yes
Analytic conductor $23.141$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2898,2,Mod(1,2898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2898, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2898.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.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: \(23.1406465058\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 2898.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.484862 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.484862 q^{5} -1.00000 q^{7} +1.00000 q^{8} +0.484862 q^{10} +6.24977 q^{11} -1.12489 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.60975 q^{17} +4.64002 q^{19} +0.484862 q^{20} +6.24977 q^{22} -1.00000 q^{23} -4.76491 q^{25} -1.12489 q^{26} -1.00000 q^{28} -1.76491 q^{29} -1.60975 q^{31} +1.00000 q^{32} +3.60975 q^{34} -0.484862 q^{35} -3.76491 q^{37} +4.64002 q^{38} +0.484862 q^{40} -2.73463 q^{41} +6.79518 q^{43} +6.24977 q^{44} -1.00000 q^{46} +6.15516 q^{47} +1.00000 q^{49} -4.76491 q^{50} -1.12489 q^{52} -11.5298 q^{53} +3.03028 q^{55} -1.00000 q^{56} -1.76491 q^{58} +9.21949 q^{61} -1.60975 q^{62} +1.00000 q^{64} -0.545414 q^{65} +12.4995 q^{67} +3.60975 q^{68} -0.484862 q^{70} -8.31032 q^{71} +11.2800 q^{73} -3.76491 q^{74} +4.64002 q^{76} -6.24977 q^{77} -4.24977 q^{79} +0.484862 q^{80} -2.73463 q^{82} +3.67030 q^{83} +1.75023 q^{85} +6.79518 q^{86} +6.24977 q^{88} +13.2001 q^{89} +1.12489 q^{91} -1.00000 q^{92} +6.15516 q^{94} +2.24977 q^{95} +5.12489 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8} + q^{10} + 2 q^{11} + 5 q^{13} - 3 q^{14} + 3 q^{16} + 2 q^{17} + 6 q^{19} + q^{20} + 2 q^{22} - 3 q^{23} + 2 q^{25} + 5 q^{26} - 3 q^{28} + 11 q^{29} + 4 q^{31} + 3 q^{32} + 2 q^{34} - q^{35} + 5 q^{37} + 6 q^{38} + q^{40} + 9 q^{41} + 5 q^{43} + 2 q^{44} - 3 q^{46} + 11 q^{47} + 3 q^{49} + 2 q^{50} + 5 q^{52} - 2 q^{53} + 10 q^{55} - 3 q^{56} + 11 q^{58} + 10 q^{61} + 4 q^{62} + 3 q^{64} - 3 q^{65} + 4 q^{67} + 2 q^{68} - q^{70} - 10 q^{71} + 18 q^{73} + 5 q^{74} + 6 q^{76} - 2 q^{77} + 4 q^{79} + q^{80} + 9 q^{82} + 4 q^{83} + 22 q^{85} + 5 q^{86} + 2 q^{88} - 5 q^{91} - 3 q^{92} + 11 q^{94} - 10 q^{95} + 7 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.484862 0.216837 0.108418 0.994105i \(-0.465421\pi\)
0.108418 + 0.994105i \(0.465421\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0.484862 0.153327
\(11\) 6.24977 1.88438 0.942188 0.335084i \(-0.108765\pi\)
0.942188 + 0.335084i \(0.108765\pi\)
\(12\) 0 0
\(13\) −1.12489 −0.311987 −0.155994 0.987758i \(-0.549858\pi\)
−0.155994 + 0.987758i \(0.549858\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.60975 0.875492 0.437746 0.899099i \(-0.355777\pi\)
0.437746 + 0.899099i \(0.355777\pi\)
\(18\) 0 0
\(19\) 4.64002 1.06449 0.532247 0.846589i \(-0.321348\pi\)
0.532247 + 0.846589i \(0.321348\pi\)
\(20\) 0.484862 0.108418
\(21\) 0 0
\(22\) 6.24977 1.33246
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.76491 −0.952982
\(26\) −1.12489 −0.220608
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −1.76491 −0.327735 −0.163868 0.986482i \(-0.552397\pi\)
−0.163868 + 0.986482i \(0.552397\pi\)
\(30\) 0 0
\(31\) −1.60975 −0.289119 −0.144560 0.989496i \(-0.546177\pi\)
−0.144560 + 0.989496i \(0.546177\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 3.60975 0.619067
\(35\) −0.484862 −0.0819566
\(36\) 0 0
\(37\) −3.76491 −0.618947 −0.309474 0.950908i \(-0.600153\pi\)
−0.309474 + 0.950908i \(0.600153\pi\)
\(38\) 4.64002 0.752711
\(39\) 0 0
\(40\) 0.484862 0.0766634
\(41\) −2.73463 −0.427078 −0.213539 0.976935i \(-0.568499\pi\)
−0.213539 + 0.976935i \(0.568499\pi\)
\(42\) 0 0
\(43\) 6.79518 1.03626 0.518128 0.855303i \(-0.326629\pi\)
0.518128 + 0.855303i \(0.326629\pi\)
\(44\) 6.24977 0.942188
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.15516 0.897823 0.448911 0.893576i \(-0.351812\pi\)
0.448911 + 0.893576i \(0.351812\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.76491 −0.673860
\(51\) 0 0
\(52\) −1.12489 −0.155994
\(53\) −11.5298 −1.58374 −0.791871 0.610688i \(-0.790893\pi\)
−0.791871 + 0.610688i \(0.790893\pi\)
\(54\) 0 0
\(55\) 3.03028 0.408602
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −1.76491 −0.231744
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 9.21949 1.18044 0.590218 0.807244i \(-0.299042\pi\)
0.590218 + 0.807244i \(0.299042\pi\)
\(62\) −1.60975 −0.204438
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.545414 −0.0676503
\(66\) 0 0
\(67\) 12.4995 1.52706 0.763531 0.645771i \(-0.223464\pi\)
0.763531 + 0.645771i \(0.223464\pi\)
\(68\) 3.60975 0.437746
\(69\) 0 0
\(70\) −0.484862 −0.0579521
\(71\) −8.31032 −0.986254 −0.493127 0.869957i \(-0.664146\pi\)
−0.493127 + 0.869957i \(0.664146\pi\)
\(72\) 0 0
\(73\) 11.2800 1.32023 0.660115 0.751165i \(-0.270508\pi\)
0.660115 + 0.751165i \(0.270508\pi\)
\(74\) −3.76491 −0.437662
\(75\) 0 0
\(76\) 4.64002 0.532247
\(77\) −6.24977 −0.712227
\(78\) 0 0
\(79\) −4.24977 −0.478137 −0.239068 0.971003i \(-0.576842\pi\)
−0.239068 + 0.971003i \(0.576842\pi\)
\(80\) 0.484862 0.0542092
\(81\) 0 0
\(82\) −2.73463 −0.301990
\(83\) 3.67030 0.402868 0.201434 0.979502i \(-0.435440\pi\)
0.201434 + 0.979502i \(0.435440\pi\)
\(84\) 0 0
\(85\) 1.75023 0.189839
\(86\) 6.79518 0.732744
\(87\) 0 0
\(88\) 6.24977 0.666228
\(89\) 13.2001 1.39921 0.699605 0.714530i \(-0.253359\pi\)
0.699605 + 0.714530i \(0.253359\pi\)
\(90\) 0 0
\(91\) 1.12489 0.117920
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 6.15516 0.634856
\(95\) 2.24977 0.230822
\(96\) 0 0
\(97\) 5.12489 0.520353 0.260177 0.965561i \(-0.416219\pi\)
0.260177 + 0.965561i \(0.416219\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −4.76491 −0.476491
\(101\) 18.1698 1.80797 0.903983 0.427568i \(-0.140629\pi\)
0.903983 + 0.427568i \(0.140629\pi\)
\(102\) 0 0
\(103\) 11.4546 1.12865 0.564327 0.825551i \(-0.309136\pi\)
0.564327 + 0.825551i \(0.309136\pi\)
\(104\) −1.12489 −0.110304
\(105\) 0 0
\(106\) −11.5298 −1.11987
\(107\) −1.75023 −0.169201 −0.0846005 0.996415i \(-0.526961\pi\)
−0.0846005 + 0.996415i \(0.526961\pi\)
\(108\) 0 0
\(109\) −6.73463 −0.645061 −0.322530 0.946559i \(-0.604533\pi\)
−0.322530 + 0.946559i \(0.604533\pi\)
\(110\) 3.03028 0.288925
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −0.484862 −0.0456120 −0.0228060 0.999740i \(-0.507260\pi\)
−0.0228060 + 0.999740i \(0.507260\pi\)
\(114\) 0 0
\(115\) −0.484862 −0.0452136
\(116\) −1.76491 −0.163868
\(117\) 0 0
\(118\) 0 0
\(119\) −3.60975 −0.330905
\(120\) 0 0
\(121\) 28.0596 2.55088
\(122\) 9.21949 0.834694
\(123\) 0 0
\(124\) −1.60975 −0.144560
\(125\) −4.73463 −0.423478
\(126\) 0 0
\(127\) 5.51514 0.489389 0.244695 0.969600i \(-0.421312\pi\)
0.244695 + 0.969600i \(0.421312\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −0.545414 −0.0478360
\(131\) −19.5298 −1.70633 −0.853164 0.521643i \(-0.825319\pi\)
−0.853164 + 0.521643i \(0.825319\pi\)
\(132\) 0 0
\(133\) −4.64002 −0.402341
\(134\) 12.4995 1.07980
\(135\) 0 0
\(136\) 3.60975 0.309533
\(137\) 12.7952 1.09317 0.546583 0.837405i \(-0.315928\pi\)
0.546583 + 0.837405i \(0.315928\pi\)
\(138\) 0 0
\(139\) −18.5142 −1.57036 −0.785178 0.619270i \(-0.787429\pi\)
−0.785178 + 0.619270i \(0.787429\pi\)
\(140\) −0.484862 −0.0409783
\(141\) 0 0
\(142\) −8.31032 −0.697387
\(143\) −7.03028 −0.587901
\(144\) 0 0
\(145\) −0.855737 −0.0710651
\(146\) 11.2800 0.933543
\(147\) 0 0
\(148\) −3.76491 −0.309474
\(149\) −4.90917 −0.402175 −0.201088 0.979573i \(-0.564448\pi\)
−0.201088 + 0.979573i \(0.564448\pi\)
\(150\) 0 0
\(151\) −4.54541 −0.369901 −0.184950 0.982748i \(-0.559212\pi\)
−0.184950 + 0.982748i \(0.559212\pi\)
\(152\) 4.64002 0.376356
\(153\) 0 0
\(154\) −6.24977 −0.503621
\(155\) −0.780505 −0.0626917
\(156\) 0 0
\(157\) 3.93945 0.314402 0.157201 0.987567i \(-0.449753\pi\)
0.157201 + 0.987567i \(0.449753\pi\)
\(158\) −4.24977 −0.338094
\(159\) 0 0
\(160\) 0.484862 0.0383317
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 9.34060 0.731612 0.365806 0.930691i \(-0.380793\pi\)
0.365806 + 0.930691i \(0.380793\pi\)
\(164\) −2.73463 −0.213539
\(165\) 0 0
\(166\) 3.67030 0.284870
\(167\) 2.39025 0.184963 0.0924817 0.995714i \(-0.470520\pi\)
0.0924817 + 0.995714i \(0.470520\pi\)
\(168\) 0 0
\(169\) −11.7346 −0.902664
\(170\) 1.75023 0.134236
\(171\) 0 0
\(172\) 6.79518 0.518128
\(173\) −20.1093 −1.52888 −0.764440 0.644694i \(-0.776985\pi\)
−0.764440 + 0.644694i \(0.776985\pi\)
\(174\) 0 0
\(175\) 4.76491 0.360193
\(176\) 6.24977 0.471094
\(177\) 0 0
\(178\) 13.2001 0.989391
\(179\) −7.76491 −0.580377 −0.290188 0.956970i \(-0.593718\pi\)
−0.290188 + 0.956970i \(0.593718\pi\)
\(180\) 0 0
\(181\) −13.2195 −0.982597 −0.491299 0.870991i \(-0.663478\pi\)
−0.491299 + 0.870991i \(0.663478\pi\)
\(182\) 1.12489 0.0833821
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −1.82546 −0.134211
\(186\) 0 0
\(187\) 22.5601 1.64976
\(188\) 6.15516 0.448911
\(189\) 0 0
\(190\) 2.24977 0.163216
\(191\) −18.1892 −1.31613 −0.658063 0.752963i \(-0.728624\pi\)
−0.658063 + 0.752963i \(0.728624\pi\)
\(192\) 0 0
\(193\) −17.3553 −1.24926 −0.624630 0.780921i \(-0.714750\pi\)
−0.624630 + 0.780921i \(0.714750\pi\)
\(194\) 5.12489 0.367945
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −1.57569 −0.112263 −0.0561316 0.998423i \(-0.517877\pi\)
−0.0561316 + 0.998423i \(0.517877\pi\)
\(198\) 0 0
\(199\) −3.57569 −0.253474 −0.126737 0.991936i \(-0.540450\pi\)
−0.126737 + 0.991936i \(0.540450\pi\)
\(200\) −4.76491 −0.336930
\(201\) 0 0
\(202\) 18.1698 1.27843
\(203\) 1.76491 0.123872
\(204\) 0 0
\(205\) −1.32592 −0.0926062
\(206\) 11.4546 0.798079
\(207\) 0 0
\(208\) −1.12489 −0.0779968
\(209\) 28.9991 2.00591
\(210\) 0 0
\(211\) 6.18922 0.426083 0.213042 0.977043i \(-0.431663\pi\)
0.213042 + 0.977043i \(0.431663\pi\)
\(212\) −11.5298 −0.791871
\(213\) 0 0
\(214\) −1.75023 −0.119643
\(215\) 3.29473 0.224698
\(216\) 0 0
\(217\) 1.60975 0.109277
\(218\) −6.73463 −0.456127
\(219\) 0 0
\(220\) 3.03028 0.204301
\(221\) −4.06055 −0.273142
\(222\) 0 0
\(223\) 16.9503 1.13508 0.567540 0.823346i \(-0.307895\pi\)
0.567540 + 0.823346i \(0.307895\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −0.484862 −0.0322525
\(227\) −11.1249 −0.738385 −0.369192 0.929353i \(-0.620366\pi\)
−0.369192 + 0.929353i \(0.620366\pi\)
\(228\) 0 0
\(229\) 17.3406 1.14590 0.572950 0.819591i \(-0.305799\pi\)
0.572950 + 0.819591i \(0.305799\pi\)
\(230\) −0.484862 −0.0319709
\(231\) 0 0
\(232\) −1.76491 −0.115872
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 2.98440 0.194681
\(236\) 0 0
\(237\) 0 0
\(238\) −3.60975 −0.233985
\(239\) −8.31032 −0.537550 −0.268775 0.963203i \(-0.586619\pi\)
−0.268775 + 0.963203i \(0.586619\pi\)
\(240\) 0 0
\(241\) 16.6547 1.07282 0.536412 0.843956i \(-0.319779\pi\)
0.536412 + 0.843956i \(0.319779\pi\)
\(242\) 28.0596 1.80374
\(243\) 0 0
\(244\) 9.21949 0.590218
\(245\) 0.484862 0.0309767
\(246\) 0 0
\(247\) −5.21949 −0.332108
\(248\) −1.60975 −0.102219
\(249\) 0 0
\(250\) −4.73463 −0.299444
\(251\) −27.4646 −1.73355 −0.866774 0.498701i \(-0.833811\pi\)
−0.866774 + 0.498701i \(0.833811\pi\)
\(252\) 0 0
\(253\) −6.24977 −0.392920
\(254\) 5.51514 0.346051
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.21949 −0.325583 −0.162792 0.986660i \(-0.552050\pi\)
−0.162792 + 0.986660i \(0.552050\pi\)
\(258\) 0 0
\(259\) 3.76491 0.233940
\(260\) −0.545414 −0.0338251
\(261\) 0 0
\(262\) −19.5298 −1.20656
\(263\) −1.20482 −0.0742921 −0.0371460 0.999310i \(-0.511827\pi\)
−0.0371460 + 0.999310i \(0.511827\pi\)
\(264\) 0 0
\(265\) −5.59037 −0.343414
\(266\) −4.64002 −0.284498
\(267\) 0 0
\(268\) 12.4995 0.763531
\(269\) −1.17076 −0.0713824 −0.0356912 0.999363i \(-0.511363\pi\)
−0.0356912 + 0.999363i \(0.511363\pi\)
\(270\) 0 0
\(271\) −12.3591 −0.750759 −0.375380 0.926871i \(-0.622488\pi\)
−0.375380 + 0.926871i \(0.622488\pi\)
\(272\) 3.60975 0.218873
\(273\) 0 0
\(274\) 12.7952 0.772985
\(275\) −29.7796 −1.79578
\(276\) 0 0
\(277\) −25.9083 −1.55668 −0.778338 0.627845i \(-0.783937\pi\)
−0.778338 + 0.627845i \(0.783937\pi\)
\(278\) −18.5142 −1.11041
\(279\) 0 0
\(280\) −0.484862 −0.0289760
\(281\) −6.73463 −0.401755 −0.200877 0.979616i \(-0.564379\pi\)
−0.200877 + 0.979616i \(0.564379\pi\)
\(282\) 0 0
\(283\) −29.1689 −1.73391 −0.866956 0.498384i \(-0.833927\pi\)
−0.866956 + 0.498384i \(0.833927\pi\)
\(284\) −8.31032 −0.493127
\(285\) 0 0
\(286\) −7.03028 −0.415709
\(287\) 2.73463 0.161420
\(288\) 0 0
\(289\) −3.96972 −0.233513
\(290\) −0.855737 −0.0502506
\(291\) 0 0
\(292\) 11.2800 0.660115
\(293\) 11.7796 0.688171 0.344085 0.938938i \(-0.388189\pi\)
0.344085 + 0.938938i \(0.388189\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.76491 −0.218831
\(297\) 0 0
\(298\) −4.90917 −0.284381
\(299\) 1.12489 0.0650538
\(300\) 0 0
\(301\) −6.79518 −0.391668
\(302\) −4.54541 −0.261559
\(303\) 0 0
\(304\) 4.64002 0.266124
\(305\) 4.47018 0.255962
\(306\) 0 0
\(307\) 25.0743 1.43107 0.715533 0.698579i \(-0.246184\pi\)
0.715533 + 0.698579i \(0.246184\pi\)
\(308\) −6.24977 −0.356114
\(309\) 0 0
\(310\) −0.780505 −0.0443297
\(311\) −24.1698 −1.37055 −0.685273 0.728286i \(-0.740317\pi\)
−0.685273 + 0.728286i \(0.740317\pi\)
\(312\) 0 0
\(313\) 15.6097 0.882315 0.441158 0.897430i \(-0.354568\pi\)
0.441158 + 0.897430i \(0.354568\pi\)
\(314\) 3.93945 0.222316
\(315\) 0 0
\(316\) −4.24977 −0.239068
\(317\) 17.4839 0.981996 0.490998 0.871161i \(-0.336632\pi\)
0.490998 + 0.871161i \(0.336632\pi\)
\(318\) 0 0
\(319\) −11.0303 −0.617577
\(320\) 0.484862 0.0271046
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 16.7493 0.931957
\(324\) 0 0
\(325\) 5.35998 0.297318
\(326\) 9.34060 0.517328
\(327\) 0 0
\(328\) −2.73463 −0.150995
\(329\) −6.15516 −0.339345
\(330\) 0 0
\(331\) −4.47018 −0.245703 −0.122852 0.992425i \(-0.539204\pi\)
−0.122852 + 0.992425i \(0.539204\pi\)
\(332\) 3.67030 0.201434
\(333\) 0 0
\(334\) 2.39025 0.130789
\(335\) 6.06055 0.331123
\(336\) 0 0
\(337\) 34.2186 1.86400 0.932002 0.362452i \(-0.118060\pi\)
0.932002 + 0.362452i \(0.118060\pi\)
\(338\) −11.7346 −0.638280
\(339\) 0 0
\(340\) 1.75023 0.0949195
\(341\) −10.0606 −0.544809
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 6.79518 0.366372
\(345\) 0 0
\(346\) −20.1093 −1.08108
\(347\) 13.4839 0.723856 0.361928 0.932206i \(-0.382119\pi\)
0.361928 + 0.932206i \(0.382119\pi\)
\(348\) 0 0
\(349\) −25.5104 −1.36554 −0.682771 0.730632i \(-0.739225\pi\)
−0.682771 + 0.730632i \(0.739225\pi\)
\(350\) 4.76491 0.254695
\(351\) 0 0
\(352\) 6.24977 0.333114
\(353\) −22.1433 −1.17857 −0.589286 0.807925i \(-0.700591\pi\)
−0.589286 + 0.807925i \(0.700591\pi\)
\(354\) 0 0
\(355\) −4.02936 −0.213856
\(356\) 13.2001 0.699605
\(357\) 0 0
\(358\) −7.76491 −0.410388
\(359\) −8.48486 −0.447814 −0.223907 0.974611i \(-0.571881\pi\)
−0.223907 + 0.974611i \(0.571881\pi\)
\(360\) 0 0
\(361\) 2.52982 0.133148
\(362\) −13.2195 −0.694801
\(363\) 0 0
\(364\) 1.12489 0.0589600
\(365\) 5.46927 0.286274
\(366\) 0 0
\(367\) −10.0147 −0.522762 −0.261381 0.965236i \(-0.584178\pi\)
−0.261381 + 0.965236i \(0.584178\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) −1.82546 −0.0949012
\(371\) 11.5298 0.598598
\(372\) 0 0
\(373\) −1.59037 −0.0823462 −0.0411731 0.999152i \(-0.513110\pi\)
−0.0411731 + 0.999152i \(0.513110\pi\)
\(374\) 22.5601 1.16655
\(375\) 0 0
\(376\) 6.15516 0.317428
\(377\) 1.98532 0.102249
\(378\) 0 0
\(379\) −12.0752 −0.620263 −0.310132 0.950694i \(-0.600373\pi\)
−0.310132 + 0.950694i \(0.600373\pi\)
\(380\) 2.24977 0.115411
\(381\) 0 0
\(382\) −18.1892 −0.930641
\(383\) −11.0303 −0.563621 −0.281810 0.959470i \(-0.590935\pi\)
−0.281810 + 0.959470i \(0.590935\pi\)
\(384\) 0 0
\(385\) −3.03028 −0.154437
\(386\) −17.3553 −0.883360
\(387\) 0 0
\(388\) 5.12489 0.260177
\(389\) −2.43899 −0.123662 −0.0618308 0.998087i \(-0.519694\pi\)
−0.0618308 + 0.998087i \(0.519694\pi\)
\(390\) 0 0
\(391\) −3.60975 −0.182553
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −1.57569 −0.0793821
\(395\) −2.06055 −0.103678
\(396\) 0 0
\(397\) 20.8898 1.04843 0.524214 0.851586i \(-0.324359\pi\)
0.524214 + 0.851586i \(0.324359\pi\)
\(398\) −3.57569 −0.179233
\(399\) 0 0
\(400\) −4.76491 −0.238245
\(401\) −11.9394 −0.596228 −0.298114 0.954530i \(-0.596357\pi\)
−0.298114 + 0.954530i \(0.596357\pi\)
\(402\) 0 0
\(403\) 1.81078 0.0902014
\(404\) 18.1698 0.903983
\(405\) 0 0
\(406\) 1.76491 0.0875910
\(407\) −23.5298 −1.16633
\(408\) 0 0
\(409\) 16.4390 0.812856 0.406428 0.913683i \(-0.366774\pi\)
0.406428 + 0.913683i \(0.366774\pi\)
\(410\) −1.32592 −0.0654825
\(411\) 0 0
\(412\) 11.4546 0.564327
\(413\) 0 0
\(414\) 0 0
\(415\) 1.77959 0.0873566
\(416\) −1.12489 −0.0551520
\(417\) 0 0
\(418\) 28.9991 1.41839
\(419\) 19.3893 0.947231 0.473616 0.880732i \(-0.342949\pi\)
0.473616 + 0.880732i \(0.342949\pi\)
\(420\) 0 0
\(421\) 5.88601 0.286867 0.143433 0.989660i \(-0.454186\pi\)
0.143433 + 0.989660i \(0.454186\pi\)
\(422\) 6.18922 0.301286
\(423\) 0 0
\(424\) −11.5298 −0.559937
\(425\) −17.2001 −0.834328
\(426\) 0 0
\(427\) −9.21949 −0.446163
\(428\) −1.75023 −0.0846005
\(429\) 0 0
\(430\) 3.29473 0.158886
\(431\) −24.2645 −1.16878 −0.584389 0.811474i \(-0.698666\pi\)
−0.584389 + 0.811474i \(0.698666\pi\)
\(432\) 0 0
\(433\) 30.4343 1.46258 0.731289 0.682067i \(-0.238919\pi\)
0.731289 + 0.682067i \(0.238919\pi\)
\(434\) 1.60975 0.0772703
\(435\) 0 0
\(436\) −6.73463 −0.322530
\(437\) −4.64002 −0.221962
\(438\) 0 0
\(439\) 24.2380 1.15681 0.578407 0.815748i \(-0.303674\pi\)
0.578407 + 0.815748i \(0.303674\pi\)
\(440\) 3.03028 0.144463
\(441\) 0 0
\(442\) −4.06055 −0.193141
\(443\) 18.0752 0.858780 0.429390 0.903119i \(-0.358729\pi\)
0.429390 + 0.903119i \(0.358729\pi\)
\(444\) 0 0
\(445\) 6.40023 0.303400
\(446\) 16.9503 0.802622
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 40.4683 1.90982 0.954910 0.296896i \(-0.0959516\pi\)
0.954910 + 0.296896i \(0.0959516\pi\)
\(450\) 0 0
\(451\) −17.0908 −0.804776
\(452\) −0.484862 −0.0228060
\(453\) 0 0
\(454\) −11.1249 −0.522117
\(455\) 0.545414 0.0255694
\(456\) 0 0
\(457\) −26.3784 −1.23393 −0.616966 0.786990i \(-0.711638\pi\)
−0.616966 + 0.786990i \(0.711638\pi\)
\(458\) 17.3406 0.810273
\(459\) 0 0
\(460\) −0.484862 −0.0226068
\(461\) 16.0799 0.748917 0.374458 0.927244i \(-0.377829\pi\)
0.374458 + 0.927244i \(0.377829\pi\)
\(462\) 0 0
\(463\) 25.7044 1.19458 0.597291 0.802024i \(-0.296244\pi\)
0.597291 + 0.802024i \(0.296244\pi\)
\(464\) −1.76491 −0.0819338
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −16.0946 −0.744770 −0.372385 0.928078i \(-0.621460\pi\)
−0.372385 + 0.928078i \(0.621460\pi\)
\(468\) 0 0
\(469\) −12.4995 −0.577175
\(470\) 2.98440 0.137660
\(471\) 0 0
\(472\) 0 0
\(473\) 42.4683 1.95270
\(474\) 0 0
\(475\) −22.1093 −1.01444
\(476\) −3.60975 −0.165452
\(477\) 0 0
\(478\) −8.31032 −0.380105
\(479\) −38.7787 −1.77184 −0.885921 0.463835i \(-0.846473\pi\)
−0.885921 + 0.463835i \(0.846473\pi\)
\(480\) 0 0
\(481\) 4.23509 0.193104
\(482\) 16.6547 0.758601
\(483\) 0 0
\(484\) 28.0596 1.27544
\(485\) 2.48486 0.112832
\(486\) 0 0
\(487\) −14.2956 −0.647797 −0.323899 0.946092i \(-0.604994\pi\)
−0.323899 + 0.946092i \(0.604994\pi\)
\(488\) 9.21949 0.417347
\(489\) 0 0
\(490\) 0.484862 0.0219038
\(491\) −10.0681 −0.454368 −0.227184 0.973852i \(-0.572952\pi\)
−0.227184 + 0.973852i \(0.572952\pi\)
\(492\) 0 0
\(493\) −6.37088 −0.286930
\(494\) −5.21949 −0.234836
\(495\) 0 0
\(496\) −1.60975 −0.0722798
\(497\) 8.31032 0.372769
\(498\) 0 0
\(499\) −26.5677 −1.18933 −0.594666 0.803973i \(-0.702716\pi\)
−0.594666 + 0.803973i \(0.702716\pi\)
\(500\) −4.73463 −0.211739
\(501\) 0 0
\(502\) −27.4646 −1.22580
\(503\) 3.52982 0.157387 0.0786934 0.996899i \(-0.474925\pi\)
0.0786934 + 0.996899i \(0.474925\pi\)
\(504\) 0 0
\(505\) 8.80986 0.392034
\(506\) −6.24977 −0.277836
\(507\) 0 0
\(508\) 5.51514 0.244695
\(509\) −3.79897 −0.168386 −0.0841931 0.996449i \(-0.526831\pi\)
−0.0841931 + 0.996449i \(0.526831\pi\)
\(510\) 0 0
\(511\) −11.2800 −0.499000
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −5.21949 −0.230222
\(515\) 5.55389 0.244734
\(516\) 0 0
\(517\) 38.4683 1.69184
\(518\) 3.76491 0.165421
\(519\) 0 0
\(520\) −0.545414 −0.0239180
\(521\) −23.8283 −1.04394 −0.521969 0.852964i \(-0.674802\pi\)
−0.521969 + 0.852964i \(0.674802\pi\)
\(522\) 0 0
\(523\) −27.0790 −1.18408 −0.592041 0.805908i \(-0.701678\pi\)
−0.592041 + 0.805908i \(0.701678\pi\)
\(524\) −19.5298 −0.853164
\(525\) 0 0
\(526\) −1.20482 −0.0525324
\(527\) −5.81078 −0.253122
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −5.59037 −0.242830
\(531\) 0 0
\(532\) −4.64002 −0.201171
\(533\) 3.07615 0.133243
\(534\) 0 0
\(535\) −0.848620 −0.0366890
\(536\) 12.4995 0.539898
\(537\) 0 0
\(538\) −1.17076 −0.0504750
\(539\) 6.24977 0.269197
\(540\) 0 0
\(541\) −36.5601 −1.57184 −0.785921 0.618327i \(-0.787811\pi\)
−0.785921 + 0.618327i \(0.787811\pi\)
\(542\) −12.3591 −0.530867
\(543\) 0 0
\(544\) 3.60975 0.154767
\(545\) −3.26537 −0.139873
\(546\) 0 0
\(547\) 0.969724 0.0414624 0.0207312 0.999785i \(-0.493401\pi\)
0.0207312 + 0.999785i \(0.493401\pi\)
\(548\) 12.7952 0.546583
\(549\) 0 0
\(550\) −29.7796 −1.26981
\(551\) −8.18922 −0.348872
\(552\) 0 0
\(553\) 4.24977 0.180719
\(554\) −25.9083 −1.10074
\(555\) 0 0
\(556\) −18.5142 −0.785178
\(557\) 33.1807 1.40591 0.702957 0.711233i \(-0.251863\pi\)
0.702957 + 0.711233i \(0.251863\pi\)
\(558\) 0 0
\(559\) −7.64380 −0.323298
\(560\) −0.484862 −0.0204892
\(561\) 0 0
\(562\) −6.73463 −0.284083
\(563\) −27.1249 −1.14318 −0.571589 0.820540i \(-0.693673\pi\)
−0.571589 + 0.820540i \(0.693673\pi\)
\(564\) 0 0
\(565\) −0.235091 −0.00989036
\(566\) −29.1689 −1.22606
\(567\) 0 0
\(568\) −8.31032 −0.348693
\(569\) −44.4755 −1.86451 −0.932254 0.361804i \(-0.882161\pi\)
−0.932254 + 0.361804i \(0.882161\pi\)
\(570\) 0 0
\(571\) 7.05964 0.295437 0.147718 0.989029i \(-0.452807\pi\)
0.147718 + 0.989029i \(0.452807\pi\)
\(572\) −7.03028 −0.293951
\(573\) 0 0
\(574\) 2.73463 0.114141
\(575\) 4.76491 0.198710
\(576\) 0 0
\(577\) 0.749313 0.0311943 0.0155971 0.999878i \(-0.495035\pi\)
0.0155971 + 0.999878i \(0.495035\pi\)
\(578\) −3.96972 −0.165119
\(579\) 0 0
\(580\) −0.855737 −0.0355326
\(581\) −3.67030 −0.152270
\(582\) 0 0
\(583\) −72.0587 −2.98437
\(584\) 11.2800 0.466772
\(585\) 0 0
\(586\) 11.7796 0.486610
\(587\) 29.2800 1.20852 0.604258 0.796788i \(-0.293469\pi\)
0.604258 + 0.796788i \(0.293469\pi\)
\(588\) 0 0
\(589\) −7.46927 −0.307766
\(590\) 0 0
\(591\) 0 0
\(592\) −3.76491 −0.154737
\(593\) −8.95504 −0.367740 −0.183870 0.982951i \(-0.558862\pi\)
−0.183870 + 0.982951i \(0.558862\pi\)
\(594\) 0 0
\(595\) −1.75023 −0.0717524
\(596\) −4.90917 −0.201088
\(597\) 0 0
\(598\) 1.12489 0.0460000
\(599\) −0.378437 −0.0154625 −0.00773126 0.999970i \(-0.502461\pi\)
−0.00773126 + 0.999970i \(0.502461\pi\)
\(600\) 0 0
\(601\) −30.7181 −1.25302 −0.626509 0.779414i \(-0.715517\pi\)
−0.626509 + 0.779414i \(0.715517\pi\)
\(602\) −6.79518 −0.276951
\(603\) 0 0
\(604\) −4.54541 −0.184950
\(605\) 13.6050 0.553124
\(606\) 0 0
\(607\) 3.35998 0.136377 0.0681886 0.997672i \(-0.478278\pi\)
0.0681886 + 0.997672i \(0.478278\pi\)
\(608\) 4.64002 0.188178
\(609\) 0 0
\(610\) 4.47018 0.180992
\(611\) −6.92385 −0.280109
\(612\) 0 0
\(613\) −44.3250 −1.79027 −0.895135 0.445795i \(-0.852921\pi\)
−0.895135 + 0.445795i \(0.852921\pi\)
\(614\) 25.0743 1.01192
\(615\) 0 0
\(616\) −6.24977 −0.251810
\(617\) 6.62065 0.266537 0.133269 0.991080i \(-0.457453\pi\)
0.133269 + 0.991080i \(0.457453\pi\)
\(618\) 0 0
\(619\) −5.58039 −0.224295 −0.112147 0.993692i \(-0.535773\pi\)
−0.112147 + 0.993692i \(0.535773\pi\)
\(620\) −0.780505 −0.0313458
\(621\) 0 0
\(622\) −24.1698 −0.969122
\(623\) −13.2001 −0.528852
\(624\) 0 0
\(625\) 21.5289 0.861156
\(626\) 15.6097 0.623891
\(627\) 0 0
\(628\) 3.93945 0.157201
\(629\) −13.5904 −0.541884
\(630\) 0 0
\(631\) 5.09839 0.202964 0.101482 0.994837i \(-0.467642\pi\)
0.101482 + 0.994837i \(0.467642\pi\)
\(632\) −4.24977 −0.169047
\(633\) 0 0
\(634\) 17.4839 0.694376
\(635\) 2.67408 0.106118
\(636\) 0 0
\(637\) −1.12489 −0.0445696
\(638\) −11.0303 −0.436693
\(639\) 0 0
\(640\) 0.484862 0.0191659
\(641\) 25.2654 0.997922 0.498961 0.866624i \(-0.333715\pi\)
0.498961 + 0.866624i \(0.333715\pi\)
\(642\) 0 0
\(643\) −0.518919 −0.0204642 −0.0102321 0.999948i \(-0.503257\pi\)
−0.0102321 + 0.999948i \(0.503257\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 16.7493 0.658993
\(647\) 20.0100 0.786674 0.393337 0.919394i \(-0.371321\pi\)
0.393337 + 0.919394i \(0.371321\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 5.35998 0.210236
\(651\) 0 0
\(652\) 9.34060 0.365806
\(653\) −22.2645 −0.871275 −0.435638 0.900122i \(-0.643477\pi\)
−0.435638 + 0.900122i \(0.643477\pi\)
\(654\) 0 0
\(655\) −9.46927 −0.369995
\(656\) −2.73463 −0.106769
\(657\) 0 0
\(658\) −6.15516 −0.239953
\(659\) 4.12110 0.160535 0.0802677 0.996773i \(-0.474422\pi\)
0.0802677 + 0.996773i \(0.474422\pi\)
\(660\) 0 0
\(661\) −27.7502 −1.07936 −0.539679 0.841871i \(-0.681455\pi\)
−0.539679 + 0.841871i \(0.681455\pi\)
\(662\) −4.47018 −0.173739
\(663\) 0 0
\(664\) 3.67030 0.142435
\(665\) −2.24977 −0.0872424
\(666\) 0 0
\(667\) 1.76491 0.0683375
\(668\) 2.39025 0.0924817
\(669\) 0 0
\(670\) 6.06055 0.234140
\(671\) 57.6197 2.22438
\(672\) 0 0
\(673\) 1.44702 0.0557787 0.0278893 0.999611i \(-0.491121\pi\)
0.0278893 + 0.999611i \(0.491121\pi\)
\(674\) 34.2186 1.31805
\(675\) 0 0
\(676\) −11.7346 −0.451332
\(677\) −38.8392 −1.49271 −0.746356 0.665547i \(-0.768198\pi\)
−0.746356 + 0.665547i \(0.768198\pi\)
\(678\) 0 0
\(679\) −5.12489 −0.196675
\(680\) 1.75023 0.0671182
\(681\) 0 0
\(682\) −10.0606 −0.385238
\(683\) 0.249771 0.00955722 0.00477861 0.999989i \(-0.498479\pi\)
0.00477861 + 0.999989i \(0.498479\pi\)
\(684\) 0 0
\(685\) 6.20390 0.237039
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 6.79518 0.259064
\(689\) 12.9697 0.494107
\(690\) 0 0
\(691\) 26.1358 0.994252 0.497126 0.867678i \(-0.334389\pi\)
0.497126 + 0.867678i \(0.334389\pi\)
\(692\) −20.1093 −0.764440
\(693\) 0 0
\(694\) 13.4839 0.511844
\(695\) −8.97684 −0.340511
\(696\) 0 0
\(697\) −9.87133 −0.373903
\(698\) −25.5104 −0.965584
\(699\) 0 0
\(700\) 4.76491 0.180097
\(701\) 15.1202 0.571082 0.285541 0.958367i \(-0.407827\pi\)
0.285541 + 0.958367i \(0.407827\pi\)
\(702\) 0 0
\(703\) −17.4693 −0.658866
\(704\) 6.24977 0.235547
\(705\) 0 0
\(706\) −22.1433 −0.833376
\(707\) −18.1698 −0.683347
\(708\) 0 0
\(709\) −23.2876 −0.874585 −0.437292 0.899319i \(-0.644062\pi\)
−0.437292 + 0.899319i \(0.644062\pi\)
\(710\) −4.02936 −0.151219
\(711\) 0 0
\(712\) 13.2001 0.494695
\(713\) 1.60975 0.0602855
\(714\) 0 0
\(715\) −3.40871 −0.127479
\(716\) −7.76491 −0.290188
\(717\) 0 0
\(718\) −8.48486 −0.316652
\(719\) 23.8136 0.888099 0.444050 0.896002i \(-0.353541\pi\)
0.444050 + 0.896002i \(0.353541\pi\)
\(720\) 0 0
\(721\) −11.4546 −0.426591
\(722\) 2.52982 0.0941501
\(723\) 0 0
\(724\) −13.2195 −0.491299
\(725\) 8.40963 0.312326
\(726\) 0 0
\(727\) 15.0984 0.559968 0.279984 0.960005i \(-0.409671\pi\)
0.279984 + 0.960005i \(0.409671\pi\)
\(728\) 1.12489 0.0416910
\(729\) 0 0
\(730\) 5.46927 0.202427
\(731\) 24.5289 0.907234
\(732\) 0 0
\(733\) 35.9688 1.32854 0.664269 0.747494i \(-0.268743\pi\)
0.664269 + 0.747494i \(0.268743\pi\)
\(734\) −10.0147 −0.369649
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 78.1193 2.87756
\(738\) 0 0
\(739\) −2.90161 −0.106737 −0.0533687 0.998575i \(-0.516996\pi\)
−0.0533687 + 0.998575i \(0.516996\pi\)
\(740\) −1.82546 −0.0671053
\(741\) 0 0
\(742\) 11.5298 0.423273
\(743\) −21.8108 −0.800160 −0.400080 0.916480i \(-0.631018\pi\)
−0.400080 + 0.916480i \(0.631018\pi\)
\(744\) 0 0
\(745\) −2.38027 −0.0872064
\(746\) −1.59037 −0.0582276
\(747\) 0 0
\(748\) 22.5601 0.824879
\(749\) 1.75023 0.0639520
\(750\) 0 0
\(751\) −32.6206 −1.19034 −0.595172 0.803598i \(-0.702916\pi\)
−0.595172 + 0.803598i \(0.702916\pi\)
\(752\) 6.15516 0.224456
\(753\) 0 0
\(754\) 1.98532 0.0723011
\(755\) −2.20390 −0.0802081
\(756\) 0 0
\(757\) −48.4977 −1.76268 −0.881340 0.472483i \(-0.843358\pi\)
−0.881340 + 0.472483i \(0.843358\pi\)
\(758\) −12.0752 −0.438592
\(759\) 0 0
\(760\) 2.24977 0.0816078
\(761\) 38.0975 1.38103 0.690516 0.723317i \(-0.257383\pi\)
0.690516 + 0.723317i \(0.257383\pi\)
\(762\) 0 0
\(763\) 6.73463 0.243810
\(764\) −18.1892 −0.658063
\(765\) 0 0
\(766\) −11.0303 −0.398540
\(767\) 0 0
\(768\) 0 0
\(769\) 17.9054 0.645685 0.322842 0.946453i \(-0.395362\pi\)
0.322842 + 0.946453i \(0.395362\pi\)
\(770\) −3.03028 −0.109204
\(771\) 0 0
\(772\) −17.3553 −0.624630
\(773\) −4.79518 −0.172471 −0.0862354 0.996275i \(-0.527484\pi\)
−0.0862354 + 0.996275i \(0.527484\pi\)
\(774\) 0 0
\(775\) 7.67030 0.275525
\(776\) 5.12489 0.183973
\(777\) 0 0
\(778\) −2.43899 −0.0874420
\(779\) −12.6888 −0.454622
\(780\) 0 0
\(781\) −51.9376 −1.85847
\(782\) −3.60975 −0.129084
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 1.91009 0.0681740
\(786\) 0 0
\(787\) 7.04965 0.251293 0.125647 0.992075i \(-0.459899\pi\)
0.125647 + 0.992075i \(0.459899\pi\)
\(788\) −1.57569 −0.0561316
\(789\) 0 0
\(790\) −2.06055 −0.0733112
\(791\) 0.484862 0.0172397
\(792\) 0 0
\(793\) −10.3709 −0.368281
\(794\) 20.8898 0.741351
\(795\) 0 0
\(796\) −3.57569 −0.126737
\(797\) 21.1443 0.748968 0.374484 0.927233i \(-0.377820\pi\)
0.374484 + 0.927233i \(0.377820\pi\)
\(798\) 0 0
\(799\) 22.2186 0.786037
\(800\) −4.76491 −0.168465
\(801\) 0 0
\(802\) −11.9394 −0.421597
\(803\) 70.4977 2.48781
\(804\) 0 0
\(805\) 0.484862 0.0170891
\(806\) 1.81078 0.0637821
\(807\) 0 0
\(808\) 18.1698 0.639213
\(809\) −13.5592 −0.476715 −0.238358 0.971177i \(-0.576609\pi\)
−0.238358 + 0.971177i \(0.576609\pi\)
\(810\) 0 0
\(811\) −50.3544 −1.76818 −0.884090 0.467316i \(-0.845221\pi\)
−0.884090 + 0.467316i \(0.845221\pi\)
\(812\) 1.76491 0.0619362
\(813\) 0 0
\(814\) −23.5298 −0.824720
\(815\) 4.52890 0.158640
\(816\) 0 0
\(817\) 31.5298 1.10309
\(818\) 16.4390 0.574776
\(819\) 0 0
\(820\) −1.32592 −0.0463031
\(821\) 40.7787 1.42319 0.711593 0.702592i \(-0.247974\pi\)
0.711593 + 0.702592i \(0.247974\pi\)
\(822\) 0 0
\(823\) 28.9532 1.00925 0.504623 0.863340i \(-0.331632\pi\)
0.504623 + 0.863340i \(0.331632\pi\)
\(824\) 11.4546 0.399039
\(825\) 0 0
\(826\) 0 0
\(827\) −41.8089 −1.45384 −0.726920 0.686722i \(-0.759049\pi\)
−0.726920 + 0.686722i \(0.759049\pi\)
\(828\) 0 0
\(829\) −5.04965 −0.175382 −0.0876909 0.996148i \(-0.527949\pi\)
−0.0876909 + 0.996148i \(0.527949\pi\)
\(830\) 1.77959 0.0617704
\(831\) 0 0
\(832\) −1.12489 −0.0389984
\(833\) 3.60975 0.125070
\(834\) 0 0
\(835\) 1.15894 0.0401069
\(836\) 28.9991 1.00295
\(837\) 0 0
\(838\) 19.3893 0.669793
\(839\) 1.81834 0.0627762 0.0313881 0.999507i \(-0.490007\pi\)
0.0313881 + 0.999507i \(0.490007\pi\)
\(840\) 0 0
\(841\) −25.8851 −0.892590
\(842\) 5.88601 0.202845
\(843\) 0 0
\(844\) 6.18922 0.213042
\(845\) −5.68968 −0.195731
\(846\) 0 0
\(847\) −28.0596 −0.964140
\(848\) −11.5298 −0.395936
\(849\) 0 0
\(850\) −17.2001 −0.589959
\(851\) 3.76491 0.129059
\(852\) 0 0
\(853\) 52.6841 1.80387 0.901934 0.431874i \(-0.142147\pi\)
0.901934 + 0.431874i \(0.142147\pi\)
\(854\) −9.21949 −0.315485
\(855\) 0 0
\(856\) −1.75023 −0.0598216
\(857\) 44.2039 1.50998 0.754988 0.655738i \(-0.227643\pi\)
0.754988 + 0.655738i \(0.227643\pi\)
\(858\) 0 0
\(859\) −39.5133 −1.34818 −0.674088 0.738651i \(-0.735463\pi\)
−0.674088 + 0.738651i \(0.735463\pi\)
\(860\) 3.29473 0.112349
\(861\) 0 0
\(862\) −24.2645 −0.826450
\(863\) −34.2791 −1.16688 −0.583438 0.812158i \(-0.698293\pi\)
−0.583438 + 0.812158i \(0.698293\pi\)
\(864\) 0 0
\(865\) −9.75023 −0.331518
\(866\) 30.4343 1.03420
\(867\) 0 0
\(868\) 1.60975 0.0546384
\(869\) −26.5601 −0.900989
\(870\) 0 0
\(871\) −14.0606 −0.476424
\(872\) −6.73463 −0.228063
\(873\) 0 0
\(874\) −4.64002 −0.156951
\(875\) 4.73463 0.160060
\(876\) 0 0
\(877\) 28.7493 0.970795 0.485398 0.874293i \(-0.338675\pi\)
0.485398 + 0.874293i \(0.338675\pi\)
\(878\) 24.2380 0.817991
\(879\) 0 0
\(880\) 3.03028 0.102151
\(881\) −22.5483 −0.759671 −0.379835 0.925054i \(-0.624019\pi\)
−0.379835 + 0.925054i \(0.624019\pi\)
\(882\) 0 0
\(883\) −10.8099 −0.363781 −0.181890 0.983319i \(-0.558222\pi\)
−0.181890 + 0.983319i \(0.558222\pi\)
\(884\) −4.06055 −0.136571
\(885\) 0 0
\(886\) 18.0752 0.607249
\(887\) −25.1689 −0.845090 −0.422545 0.906342i \(-0.638863\pi\)
−0.422545 + 0.906342i \(0.638863\pi\)
\(888\) 0 0
\(889\) −5.51514 −0.184972
\(890\) 6.40023 0.214536
\(891\) 0 0
\(892\) 16.9503 0.567540
\(893\) 28.5601 0.955727
\(894\) 0 0
\(895\) −3.76491 −0.125847
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 40.4683 1.35045
\(899\) 2.84106 0.0947546
\(900\) 0 0
\(901\) −41.6197 −1.38655
\(902\) −17.0908 −0.569062
\(903\) 0 0
\(904\) −0.484862 −0.0161263
\(905\) −6.40963 −0.213063
\(906\) 0 0
\(907\) 51.4839 1.70950 0.854748 0.519043i \(-0.173712\pi\)
0.854748 + 0.519043i \(0.173712\pi\)
\(908\) −11.1249 −0.369192
\(909\) 0 0
\(910\) 0.545414 0.0180803
\(911\) 32.0147 1.06069 0.530347 0.847781i \(-0.322062\pi\)
0.530347 + 0.847781i \(0.322062\pi\)
\(912\) 0 0
\(913\) 22.9385 0.759155
\(914\) −26.3784 −0.872521
\(915\) 0 0
\(916\) 17.3406 0.572950
\(917\) 19.5298 0.644931
\(918\) 0 0
\(919\) 32.4002 1.06879 0.534393 0.845236i \(-0.320540\pi\)
0.534393 + 0.845236i \(0.320540\pi\)
\(920\) −0.484862 −0.0159854
\(921\) 0 0
\(922\) 16.0799 0.529564
\(923\) 9.34816 0.307698
\(924\) 0 0
\(925\) 17.9394 0.589845
\(926\) 25.7044 0.844698
\(927\) 0 0
\(928\) −1.76491 −0.0579360
\(929\) 36.2039 1.18781 0.593906 0.804535i \(-0.297585\pi\)
0.593906 + 0.804535i \(0.297585\pi\)
\(930\) 0 0
\(931\) 4.64002 0.152071
\(932\) −10.0000 −0.327561
\(933\) 0 0
\(934\) −16.0946 −0.526632
\(935\) 10.9385 0.357728
\(936\) 0 0
\(937\) −56.0928 −1.83247 −0.916236 0.400640i \(-0.868788\pi\)
−0.916236 + 0.400640i \(0.868788\pi\)
\(938\) −12.4995 −0.408125
\(939\) 0 0
\(940\) 2.98440 0.0973405
\(941\) 19.4234 0.633185 0.316592 0.948562i \(-0.397461\pi\)
0.316592 + 0.948562i \(0.397461\pi\)
\(942\) 0 0
\(943\) 2.73463 0.0890519
\(944\) 0 0
\(945\) 0 0
\(946\) 42.4683 1.38077
\(947\) 34.4537 1.11959 0.559797 0.828630i \(-0.310879\pi\)
0.559797 + 0.828630i \(0.310879\pi\)
\(948\) 0 0
\(949\) −12.6888 −0.411895
\(950\) −22.1093 −0.717320
\(951\) 0 0
\(952\) −3.60975 −0.116993
\(953\) −36.6576 −1.18746 −0.593728 0.804666i \(-0.702344\pi\)
−0.593728 + 0.804666i \(0.702344\pi\)
\(954\) 0 0
\(955\) −8.81926 −0.285385
\(956\) −8.31032 −0.268775
\(957\) 0 0
\(958\) −38.7787 −1.25288
\(959\) −12.7952 −0.413178
\(960\) 0 0
\(961\) −28.4087 −0.916410
\(962\) 4.23509 0.136545
\(963\) 0 0
\(964\) 16.6547 0.536412
\(965\) −8.41491 −0.270886
\(966\) 0 0
\(967\) 52.8780 1.70044 0.850221 0.526427i \(-0.176469\pi\)
0.850221 + 0.526427i \(0.176469\pi\)
\(968\) 28.0596 0.901871
\(969\) 0 0
\(970\) 2.48486 0.0797841
\(971\) −42.4490 −1.36225 −0.681126 0.732166i \(-0.738509\pi\)
−0.681126 + 0.732166i \(0.738509\pi\)
\(972\) 0 0
\(973\) 18.5142 0.593539
\(974\) −14.2956 −0.458062
\(975\) 0 0
\(976\) 9.21949 0.295109
\(977\) −41.8236 −1.33806 −0.669028 0.743237i \(-0.733289\pi\)
−0.669028 + 0.743237i \(0.733289\pi\)
\(978\) 0 0
\(979\) 82.4977 2.63664
\(980\) 0.484862 0.0154883
\(981\) 0 0
\(982\) −10.0681 −0.321286
\(983\) −46.0606 −1.46910 −0.734552 0.678553i \(-0.762608\pi\)
−0.734552 + 0.678553i \(0.762608\pi\)
\(984\) 0 0
\(985\) −0.763992 −0.0243428
\(986\) −6.37088 −0.202890
\(987\) 0 0
\(988\) −5.21949 −0.166054
\(989\) −6.79518 −0.216074
\(990\) 0 0
\(991\) −43.0209 −1.36660 −0.683302 0.730136i \(-0.739457\pi\)
−0.683302 + 0.730136i \(0.739457\pi\)
\(992\) −1.60975 −0.0511095
\(993\) 0 0
\(994\) 8.31032 0.263587
\(995\) −1.73372 −0.0549625
\(996\) 0 0
\(997\) 6.29095 0.199236 0.0996181 0.995026i \(-0.468238\pi\)
0.0996181 + 0.995026i \(0.468238\pi\)
\(998\) −26.5677 −0.840984
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2898.2.a.bg.1.2 yes 3
3.2 odd 2 2898.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2898.2.a.bf.1.2 3 3.2 odd 2
2898.2.a.bg.1.2 yes 3 1.1 even 1 trivial