Properties

Label 2898.2.a.bf.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 \(-0.534579\pi\)
−0.108418 + 0.994105i \(0.534579\pi\)
\(6\) 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 \(-0.891235\pi\)
−0.942188 + 0.335084i \(0.891235\pi\)
\(12\) 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 \(-0.644223\pi\)
−0.437746 + 0.899099i \(0.644223\pi\)
\(18\) 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 \(-0.447603\pi\)
0.163868 + 0.986482i \(0.447603\pi\)
\(30\) 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 \(-0.431501\pi\)
0.213539 + 0.976935i \(0.431501\pi\)
\(42\) 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 \(-0.648188\pi\)
−0.448911 + 0.893576i \(0.648188\pi\)
\(48\) 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 \(-0.209107\pi\)
0.791871 + 0.610688i \(0.209107\pi\)
\(54\) 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 \(-0.335854\pi\)
0.493127 + 0.869957i \(0.335854\pi\)
\(72\) 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 \(-0.564560\pi\)
−0.201434 + 0.979502i \(0.564560\pi\)
\(84\) 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 \(-0.746641\pi\)
−0.699605 + 0.714530i \(0.746641\pi\)
\(90\) 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 \(-0.859371\pi\)
−0.903983 + 0.427568i \(0.859371\pi\)
\(102\) 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 \(-0.473039\pi\)
0.0846005 + 0.996415i \(0.473039\pi\)
\(108\) 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 \(-0.492740\pi\)
0.0228060 + 0.999740i \(0.492740\pi\)
\(114\) 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 \(-0.174681\pi\)
0.853164 + 0.521643i \(0.174681\pi\)
\(132\) 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 \(-0.684072\pi\)
−0.546583 + 0.837405i \(0.684072\pi\)
\(138\) 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 \(-0.435552\pi\)
0.201088 + 0.979573i \(0.435552\pi\)
\(150\) 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 \(-0.529480\pi\)
−0.0924817 + 0.995714i \(0.529480\pi\)
\(168\) 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 \(-0.223015\pi\)
0.764440 + 0.644694i \(0.223015\pi\)
\(174\) 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 \(-0.406282\pi\)
0.290188 + 0.956970i \(0.406282\pi\)
\(180\) 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 \(-0.271376\pi\)
0.658063 + 0.752963i \(0.271376\pi\)
\(192\) 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 \(-0.482123\pi\)
0.0561316 + 0.998423i \(0.482123\pi\)
\(198\) 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 \(-0.379634\pi\)
0.369192 + 0.929353i \(0.379634\pi\)
\(228\) 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 \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 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 \(-0.413381\pi\)
0.268775 + 0.963203i \(0.413381\pi\)
\(240\) 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 \(-0.166189\pi\)
0.866774 + 0.498701i \(0.166189\pi\)
\(252\) 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 \(-0.447950\pi\)
0.162792 + 0.986660i \(0.447950\pi\)
\(258\) 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 \(-0.488173\pi\)
0.0371460 + 0.999310i \(0.488173\pi\)
\(264\) 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 \(-0.488637\pi\)
0.0356912 + 0.999363i \(0.488637\pi\)
\(270\) 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 \(-0.435621\pi\)
0.200877 + 0.979616i \(0.435621\pi\)
\(282\) 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 \(-0.611811\pi\)
−0.344085 + 0.938938i \(0.611811\pi\)
\(294\) 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 \(-0.259683\pi\)
0.685273 + 0.728286i \(0.259683\pi\)
\(312\) 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 \(-0.663368\pi\)
−0.490998 + 0.871161i \(0.663368\pi\)
\(318\) 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 \(-0.617881\pi\)
−0.361928 + 0.932206i \(0.617881\pi\)
\(348\) 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 \(-0.299409\pi\)
0.589286 + 0.807925i \(0.299409\pi\)
\(354\) 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 \(-0.428119\pi\)
0.223907 + 0.974611i \(0.428119\pi\)
\(360\) 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 \(-0.409065\pi\)
0.281810 + 0.959470i \(0.409065\pi\)
\(384\) 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 \(-0.480306\pi\)
0.0618308 + 0.998087i \(0.480306\pi\)
\(390\) 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 \(-0.403643\pi\)
0.298114 + 0.954530i \(0.403643\pi\)
\(402\) 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 \(-0.657051\pi\)
−0.473616 + 0.880732i \(0.657051\pi\)
\(420\) 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 \(-0.301334\pi\)
0.584389 + 0.811474i \(0.301334\pi\)
\(432\) 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 \(-0.641271\pi\)
−0.429390 + 0.903119i \(0.641271\pi\)
\(444\) 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 \(-0.904048\pi\)
−0.954910 + 0.296896i \(0.904048\pi\)
\(450\) 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 \(-0.622171\pi\)
−0.374458 + 0.927244i \(0.622171\pi\)
\(462\) 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 \(-0.378540\pi\)
0.372385 + 0.928078i \(0.378540\pi\)
\(468\) 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 \(-0.153527\pi\)
0.885921 + 0.463835i \(0.153527\pi\)
\(480\) 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 \(-0.427048\pi\)
0.227184 + 0.973852i \(0.427048\pi\)
\(492\) 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 \(-0.525075\pi\)
−0.0786934 + 0.996899i \(0.525075\pi\)
\(504\) 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 \(-0.473169\pi\)
0.0841931 + 0.996449i \(0.473169\pi\)
\(510\) 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 \(-0.325198\pi\)
0.521969 + 0.852964i \(0.325198\pi\)
\(522\) 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 \(-0.748137\pi\)
−0.702957 + 0.711233i \(0.748137\pi\)
\(558\) 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 \(-0.306327\pi\)
0.571589 + 0.820540i \(0.306327\pi\)
\(564\) 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 \(-0.117839\pi\)
0.932254 + 0.361804i \(0.117839\pi\)
\(570\) 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 \(-0.706531\pi\)
−0.604258 + 0.796788i \(0.706531\pi\)
\(588\) 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 \(-0.441138\pi\)
0.183870 + 0.982951i \(0.441138\pi\)
\(594\) 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 \(-0.497539\pi\)
0.00773126 + 0.999970i \(0.497539\pi\)
\(600\) 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 \(-0.542547\pi\)
−0.133269 + 0.991080i \(0.542547\pi\)
\(618\) 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 \(-0.666285\pi\)
−0.498961 + 0.866624i \(0.666285\pi\)
\(642\) 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 \(-0.628679\pi\)
−0.393337 + 0.919394i \(0.628679\pi\)
\(648\) 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 \(-0.356523\pi\)
0.435638 + 0.900122i \(0.356523\pi\)
\(654\) 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 \(-0.525578\pi\)
−0.0802677 + 0.996773i \(0.525578\pi\)
\(660\) 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 \(-0.231802\pi\)
0.746356 + 0.665547i \(0.231802\pi\)
\(678\) 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 \(-0.501521\pi\)
−0.00477861 + 0.999989i \(0.501521\pi\)
\(684\) 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 \(-0.592173\pi\)
−0.285541 + 0.958367i \(0.592173\pi\)
\(702\) 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 \(-0.646459\pi\)
−0.444050 + 0.896002i \(0.646459\pi\)
\(720\) 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 \(-0.368982\pi\)
0.400080 + 0.916480i \(0.368982\pi\)
\(744\) 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 \(-0.742617\pi\)
−0.690516 + 0.723317i \(0.742617\pi\)
\(762\) 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 \(-0.472516\pi\)
0.0862354 + 0.996275i \(0.472516\pi\)
\(774\) 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 \(-0.622180\pi\)
−0.374484 + 0.927233i \(0.622180\pi\)
\(798\) 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 \(-0.423391\pi\)
0.238358 + 0.971177i \(0.423391\pi\)
\(810\) 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 \(-0.752026\pi\)
−0.711593 + 0.702592i \(0.752026\pi\)
\(822\) 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 \(-0.240951\pi\)
0.726920 + 0.686722i \(0.240951\pi\)
\(828\) 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 \(-0.509993\pi\)
−0.0313881 + 0.999507i \(0.509993\pi\)
\(840\) 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 \(-0.772357\pi\)
−0.754988 + 0.655738i \(0.772357\pi\)
\(858\) 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 \(-0.301707\pi\)
0.583438 + 0.812158i \(0.301707\pi\)
\(864\) 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 \(-0.375981\pi\)
0.379835 + 0.925054i \(0.375981\pi\)
\(882\) 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 \(-0.361137\pi\)
0.422545 + 0.906342i \(0.361137\pi\)
\(888\) 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 \(-0.677938\pi\)
−0.530347 + 0.847781i \(0.677938\pi\)
\(912\) 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 \(-0.702415\pi\)
−0.593906 + 0.804535i \(0.702415\pi\)
\(930\) 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 \(-0.602539\pi\)
−0.316592 + 0.948562i \(0.602539\pi\)
\(942\) 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 \(-0.689121\pi\)
−0.559797 + 0.828630i \(0.689121\pi\)
\(948\) 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 \(-0.297656\pi\)
0.593728 + 0.804666i \(0.297656\pi\)
\(954\) 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 \(-0.261491\pi\)
0.681126 + 0.732166i \(0.261491\pi\)
\(972\) 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 \(-0.266711\pi\)
0.669028 + 0.743237i \(0.266711\pi\)
\(978\) 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 \(-0.237392\pi\)
0.734552 + 0.678553i \(0.237392\pi\)
\(984\) 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.bf.1.2 3
3.2 odd 2 2898.2.a.bg.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2898.2.a.bf.1.2 3 1.1 even 1 trivial
2898.2.a.bg.1.2 yes 3 3.2 odd 2