Properties

Label 2898.2.a.bf.1.1
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 1.1
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.14134 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.14134 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.14134 q^{10} +0.726656 q^{11} +2.36333 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.77801 q^{17} -1.50466 q^{19} -3.14134 q^{20} -0.726656 q^{22} +1.00000 q^{23} +4.86799 q^{25} -2.36333 q^{26} -1.00000 q^{28} -7.86799 q^{29} -0.778008 q^{31} -1.00000 q^{32} +2.77801 q^{34} +3.14134 q^{35} +5.86799 q^{37} +1.50466 q^{38} +3.14134 q^{40} -1.58532 q^{41} -8.15066 q^{43} +0.726656 q^{44} -1.00000 q^{46} +2.64600 q^{47} +1.00000 q^{49} -4.86799 q^{50} +2.36333 q^{52} -7.73599 q^{53} -2.28267 q^{55} +1.00000 q^{56} +7.86799 q^{58} +7.55602 q^{61} +0.778008 q^{62} +1.00000 q^{64} -7.42401 q^{65} -1.45331 q^{67} -2.77801 q^{68} -3.14134 q^{70} -9.29200 q^{71} -1.00933 q^{73} -5.86799 q^{74} -1.50466 q^{76} -0.726656 q^{77} +2.72666 q^{79} -3.14134 q^{80} +1.58532 q^{82} +7.78734 q^{83} +8.72666 q^{85} +8.15066 q^{86} -0.726656 q^{88} +17.5233 q^{89} -2.36333 q^{91} +1.00000 q^{92} -2.64600 q^{94} +4.72666 q^{95} +1.63667 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 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
\(5\) −3.14134 −1.40485 −0.702424 0.711759i \(-0.747899\pi\)
−0.702424 + 0.711759i \(0.747899\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.14134 0.993378
\(11\) 0.726656 0.219095 0.109548 0.993982i \(-0.465060\pi\)
0.109548 + 0.993982i \(0.465060\pi\)
\(12\) 0 0
\(13\) 2.36333 0.655469 0.327735 0.944770i \(-0.393715\pi\)
0.327735 + 0.944770i \(0.393715\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.77801 −0.673766 −0.336883 0.941547i \(-0.609373\pi\)
−0.336883 + 0.941547i \(0.609373\pi\)
\(18\) 0 0
\(19\) −1.50466 −0.345194 −0.172597 0.984993i \(-0.555216\pi\)
−0.172597 + 0.984993i \(0.555216\pi\)
\(20\) −3.14134 −0.702424
\(21\) 0 0
\(22\) −0.726656 −0.154924
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 4.86799 0.973599
\(26\) −2.36333 −0.463487
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −7.86799 −1.46105 −0.730525 0.682886i \(-0.760724\pi\)
−0.730525 + 0.682886i \(0.760724\pi\)
\(30\) 0 0
\(31\) −0.778008 −0.139734 −0.0698672 0.997556i \(-0.522258\pi\)
−0.0698672 + 0.997556i \(0.522258\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.77801 0.476424
\(35\) 3.14134 0.530983
\(36\) 0 0
\(37\) 5.86799 0.964692 0.482346 0.875981i \(-0.339785\pi\)
0.482346 + 0.875981i \(0.339785\pi\)
\(38\) 1.50466 0.244089
\(39\) 0 0
\(40\) 3.14134 0.496689
\(41\) −1.58532 −0.247585 −0.123793 0.992308i \(-0.539506\pi\)
−0.123793 + 0.992308i \(0.539506\pi\)
\(42\) 0 0
\(43\) −8.15066 −1.24296 −0.621482 0.783428i \(-0.713469\pi\)
−0.621482 + 0.783428i \(0.713469\pi\)
\(44\) 0.726656 0.109548
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 2.64600 0.385959 0.192979 0.981203i \(-0.438185\pi\)
0.192979 + 0.981203i \(0.438185\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.86799 −0.688438
\(51\) 0 0
\(52\) 2.36333 0.327735
\(53\) −7.73599 −1.06262 −0.531310 0.847178i \(-0.678300\pi\)
−0.531310 + 0.847178i \(0.678300\pi\)
\(54\) 0 0
\(55\) −2.28267 −0.307795
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 7.86799 1.03312
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.55602 0.967449 0.483724 0.875220i \(-0.339284\pi\)
0.483724 + 0.875220i \(0.339284\pi\)
\(62\) 0.778008 0.0988071
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.42401 −0.920835
\(66\) 0 0
\(67\) −1.45331 −0.177550 −0.0887752 0.996052i \(-0.528295\pi\)
−0.0887752 + 0.996052i \(0.528295\pi\)
\(68\) −2.77801 −0.336883
\(69\) 0 0
\(70\) −3.14134 −0.375461
\(71\) −9.29200 −1.10276 −0.551379 0.834255i \(-0.685898\pi\)
−0.551379 + 0.834255i \(0.685898\pi\)
\(72\) 0 0
\(73\) −1.00933 −0.118133 −0.0590665 0.998254i \(-0.518812\pi\)
−0.0590665 + 0.998254i \(0.518812\pi\)
\(74\) −5.86799 −0.682140
\(75\) 0 0
\(76\) −1.50466 −0.172597
\(77\) −0.726656 −0.0828102
\(78\) 0 0
\(79\) 2.72666 0.306773 0.153386 0.988166i \(-0.450982\pi\)
0.153386 + 0.988166i \(0.450982\pi\)
\(80\) −3.14134 −0.351212
\(81\) 0 0
\(82\) 1.58532 0.175069
\(83\) 7.78734 0.854771 0.427386 0.904069i \(-0.359435\pi\)
0.427386 + 0.904069i \(0.359435\pi\)
\(84\) 0 0
\(85\) 8.72666 0.946539
\(86\) 8.15066 0.878909
\(87\) 0 0
\(88\) −0.726656 −0.0774618
\(89\) 17.5233 1.85747 0.928734 0.370746i \(-0.120898\pi\)
0.928734 + 0.370746i \(0.120898\pi\)
\(90\) 0 0
\(91\) −2.36333 −0.247744
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −2.64600 −0.272914
\(95\) 4.72666 0.484945
\(96\) 0 0
\(97\) 1.63667 0.166179 0.0830894 0.996542i \(-0.473521\pi\)
0.0830894 + 0.996542i \(0.473521\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.86799 0.486799
\(101\) 7.24065 0.720472 0.360236 0.932861i \(-0.382696\pi\)
0.360236 + 0.932861i \(0.382696\pi\)
\(102\) 0 0
\(103\) 19.4240 1.91390 0.956952 0.290246i \(-0.0937370\pi\)
0.956952 + 0.290246i \(0.0937370\pi\)
\(104\) −2.36333 −0.231743
\(105\) 0 0
\(106\) 7.73599 0.751385
\(107\) 8.72666 0.843638 0.421819 0.906680i \(-0.361392\pi\)
0.421819 + 0.906680i \(0.361392\pi\)
\(108\) 0 0
\(109\) −2.41468 −0.231284 −0.115642 0.993291i \(-0.536893\pi\)
−0.115642 + 0.993291i \(0.536893\pi\)
\(110\) 2.28267 0.217644
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 3.14134 0.295512 0.147756 0.989024i \(-0.452795\pi\)
0.147756 + 0.989024i \(0.452795\pi\)
\(114\) 0 0
\(115\) −3.14134 −0.292931
\(116\) −7.86799 −0.730525
\(117\) 0 0
\(118\) 0 0
\(119\) 2.77801 0.254660
\(120\) 0 0
\(121\) −10.4720 −0.951997
\(122\) −7.55602 −0.684090
\(123\) 0 0
\(124\) −0.778008 −0.0698672
\(125\) 0.414680 0.0370901
\(126\) 0 0
\(127\) 2.85866 0.253665 0.126833 0.991924i \(-0.459519\pi\)
0.126833 + 0.991924i \(0.459519\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 7.42401 0.651129
\(131\) 0.264015 0.0230671 0.0115335 0.999933i \(-0.496329\pi\)
0.0115335 + 0.999933i \(0.496329\pi\)
\(132\) 0 0
\(133\) 1.50466 0.130471
\(134\) 1.45331 0.125547
\(135\) 0 0
\(136\) 2.77801 0.238212
\(137\) 2.15066 0.183744 0.0918718 0.995771i \(-0.470715\pi\)
0.0918718 + 0.995771i \(0.470715\pi\)
\(138\) 0 0
\(139\) 12.0480 1.02189 0.510947 0.859612i \(-0.329295\pi\)
0.510947 + 0.859612i \(0.329295\pi\)
\(140\) 3.14134 0.265491
\(141\) 0 0
\(142\) 9.29200 0.779767
\(143\) 1.71733 0.143610
\(144\) 0 0
\(145\) 24.7160 2.05255
\(146\) 1.00933 0.0835326
\(147\) 0 0
\(148\) 5.86799 0.482346
\(149\) 20.8480 1.70794 0.853968 0.520325i \(-0.174189\pi\)
0.853968 + 0.520325i \(0.174189\pi\)
\(150\) 0 0
\(151\) 3.42401 0.278642 0.139321 0.990247i \(-0.455508\pi\)
0.139321 + 0.990247i \(0.455508\pi\)
\(152\) 1.50466 0.122044
\(153\) 0 0
\(154\) 0.726656 0.0585556
\(155\) 2.44398 0.196306
\(156\) 0 0
\(157\) 14.5653 1.16244 0.581221 0.813746i \(-0.302575\pi\)
0.581221 + 0.813746i \(0.302575\pi\)
\(158\) −2.72666 −0.216921
\(159\) 0 0
\(160\) 3.14134 0.248344
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −13.5747 −1.06325 −0.531625 0.846980i \(-0.678419\pi\)
−0.531625 + 0.846980i \(0.678419\pi\)
\(164\) −1.58532 −0.123793
\(165\) 0 0
\(166\) −7.78734 −0.604415
\(167\) −3.22199 −0.249325 −0.124663 0.992199i \(-0.539785\pi\)
−0.124663 + 0.992199i \(0.539785\pi\)
\(168\) 0 0
\(169\) −7.41468 −0.570360
\(170\) −8.72666 −0.669304
\(171\) 0 0
\(172\) −8.15066 −0.621482
\(173\) 5.32469 0.404829 0.202415 0.979300i \(-0.435121\pi\)
0.202415 + 0.979300i \(0.435121\pi\)
\(174\) 0 0
\(175\) −4.86799 −0.367986
\(176\) 0.726656 0.0547738
\(177\) 0 0
\(178\) −17.5233 −1.31343
\(179\) −1.86799 −0.139620 −0.0698102 0.997560i \(-0.522239\pi\)
−0.0698102 + 0.997560i \(0.522239\pi\)
\(180\) 0 0
\(181\) −11.5560 −0.858952 −0.429476 0.903078i \(-0.641302\pi\)
−0.429476 + 0.903078i \(0.641302\pi\)
\(182\) 2.36333 0.175182
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −18.4333 −1.35525
\(186\) 0 0
\(187\) −2.01866 −0.147619
\(188\) 2.64600 0.192979
\(189\) 0 0
\(190\) −4.72666 −0.342908
\(191\) 21.8387 1.58019 0.790096 0.612983i \(-0.210031\pi\)
0.790096 + 0.612983i \(0.210031\pi\)
\(192\) 0 0
\(193\) 22.1693 1.59578 0.797891 0.602801i \(-0.205949\pi\)
0.797891 + 0.602801i \(0.205949\pi\)
\(194\) −1.63667 −0.117506
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −11.7067 −0.834066 −0.417033 0.908891i \(-0.636930\pi\)
−0.417033 + 0.908891i \(0.636930\pi\)
\(198\) 0 0
\(199\) 9.70668 0.688088 0.344044 0.938953i \(-0.388203\pi\)
0.344044 + 0.938953i \(0.388203\pi\)
\(200\) −4.86799 −0.344219
\(201\) 0 0
\(202\) −7.24065 −0.509450
\(203\) 7.86799 0.552225
\(204\) 0 0
\(205\) 4.98002 0.347820
\(206\) −19.4240 −1.35333
\(207\) 0 0
\(208\) 2.36333 0.163867
\(209\) −1.09337 −0.0756303
\(210\) 0 0
\(211\) 9.83869 0.677323 0.338662 0.940908i \(-0.390026\pi\)
0.338662 + 0.940908i \(0.390026\pi\)
\(212\) −7.73599 −0.531310
\(213\) 0 0
\(214\) −8.72666 −0.596542
\(215\) 25.6040 1.74618
\(216\) 0 0
\(217\) 0.778008 0.0528146
\(218\) 2.41468 0.163543
\(219\) 0 0
\(220\) −2.28267 −0.153898
\(221\) −6.56534 −0.441633
\(222\) 0 0
\(223\) −6.79667 −0.455138 −0.227569 0.973762i \(-0.573078\pi\)
−0.227569 + 0.973762i \(0.573078\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −3.14134 −0.208959
\(227\) 7.63667 0.506864 0.253432 0.967353i \(-0.418441\pi\)
0.253432 + 0.967353i \(0.418441\pi\)
\(228\) 0 0
\(229\) −5.57467 −0.368385 −0.184192 0.982890i \(-0.558967\pi\)
−0.184192 + 0.982890i \(0.558967\pi\)
\(230\) 3.14134 0.207134
\(231\) 0 0
\(232\) 7.86799 0.516559
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −8.31198 −0.542214
\(236\) 0 0
\(237\) 0 0
\(238\) −2.77801 −0.180072
\(239\) −9.29200 −0.601050 −0.300525 0.953774i \(-0.597162\pi\)
−0.300525 + 0.953774i \(0.597162\pi\)
\(240\) 0 0
\(241\) −6.09931 −0.392891 −0.196446 0.980515i \(-0.562940\pi\)
−0.196446 + 0.980515i \(0.562940\pi\)
\(242\) 10.4720 0.673164
\(243\) 0 0
\(244\) 7.55602 0.483724
\(245\) −3.14134 −0.200693
\(246\) 0 0
\(247\) −3.55602 −0.226264
\(248\) 0.778008 0.0494035
\(249\) 0 0
\(250\) −0.414680 −0.0262266
\(251\) −26.8446 −1.69442 −0.847209 0.531260i \(-0.821719\pi\)
−0.847209 + 0.531260i \(0.821719\pi\)
\(252\) 0 0
\(253\) 0.726656 0.0456845
\(254\) −2.85866 −0.179369
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.55602 0.221818 0.110909 0.993831i \(-0.464624\pi\)
0.110909 + 0.993831i \(0.464624\pi\)
\(258\) 0 0
\(259\) −5.86799 −0.364619
\(260\) −7.42401 −0.460417
\(261\) 0 0
\(262\) −0.264015 −0.0163109
\(263\) 16.1507 0.995893 0.497946 0.867208i \(-0.334088\pi\)
0.497946 + 0.867208i \(0.334088\pi\)
\(264\) 0 0
\(265\) 24.3013 1.49282
\(266\) −1.50466 −0.0922569
\(267\) 0 0
\(268\) −1.45331 −0.0887752
\(269\) 3.66598 0.223519 0.111759 0.993735i \(-0.464351\pi\)
0.111759 + 0.993735i \(0.464351\pi\)
\(270\) 0 0
\(271\) 9.40196 0.571128 0.285564 0.958360i \(-0.407819\pi\)
0.285564 + 0.958360i \(0.407819\pi\)
\(272\) −2.77801 −0.168441
\(273\) 0 0
\(274\) −2.15066 −0.129926
\(275\) 3.53736 0.213311
\(276\) 0 0
\(277\) −13.9414 −0.837657 −0.418828 0.908065i \(-0.637559\pi\)
−0.418828 + 0.908065i \(0.637559\pi\)
\(278\) −12.0480 −0.722589
\(279\) 0 0
\(280\) −3.14134 −0.187731
\(281\) 2.41468 0.144048 0.0720239 0.997403i \(-0.477054\pi\)
0.0720239 + 0.997403i \(0.477054\pi\)
\(282\) 0 0
\(283\) 24.1473 1.43541 0.717703 0.696349i \(-0.245193\pi\)
0.717703 + 0.696349i \(0.245193\pi\)
\(284\) −9.29200 −0.551379
\(285\) 0 0
\(286\) −1.71733 −0.101548
\(287\) 1.58532 0.0935785
\(288\) 0 0
\(289\) −9.28267 −0.546040
\(290\) −24.7160 −1.45137
\(291\) 0 0
\(292\) −1.00933 −0.0590665
\(293\) 14.4626 0.844917 0.422458 0.906382i \(-0.361167\pi\)
0.422458 + 0.906382i \(0.361167\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.86799 −0.341070
\(297\) 0 0
\(298\) −20.8480 −1.20769
\(299\) 2.36333 0.136675
\(300\) 0 0
\(301\) 8.15066 0.469797
\(302\) −3.42401 −0.197030
\(303\) 0 0
\(304\) −1.50466 −0.0862984
\(305\) −23.7360 −1.35912
\(306\) 0 0
\(307\) −30.0666 −1.71599 −0.857996 0.513656i \(-0.828291\pi\)
−0.857996 + 0.513656i \(0.828291\pi\)
\(308\) −0.726656 −0.0414051
\(309\) 0 0
\(310\) −2.44398 −0.138809
\(311\) −1.24065 −0.0703508 −0.0351754 0.999381i \(-0.511199\pi\)
−0.0351754 + 0.999381i \(0.511199\pi\)
\(312\) 0 0
\(313\) 14.7780 0.835302 0.417651 0.908607i \(-0.362853\pi\)
0.417651 + 0.908607i \(0.362853\pi\)
\(314\) −14.5653 −0.821970
\(315\) 0 0
\(316\) 2.72666 0.153386
\(317\) 7.76529 0.436142 0.218071 0.975933i \(-0.430024\pi\)
0.218071 + 0.975933i \(0.430024\pi\)
\(318\) 0 0
\(319\) −5.71733 −0.320109
\(320\) −3.14134 −0.175606
\(321\) 0 0
\(322\) 1.00000 0.0557278
\(323\) 4.17997 0.232580
\(324\) 0 0
\(325\) 11.5047 0.638164
\(326\) 13.5747 0.751832
\(327\) 0 0
\(328\) 1.58532 0.0875347
\(329\) −2.64600 −0.145879
\(330\) 0 0
\(331\) −23.7360 −1.30465 −0.652324 0.757940i \(-0.726206\pi\)
−0.652324 + 0.757940i \(0.726206\pi\)
\(332\) 7.78734 0.427386
\(333\) 0 0
\(334\) 3.22199 0.176300
\(335\) 4.56534 0.249431
\(336\) 0 0
\(337\) 4.64939 0.253268 0.126634 0.991949i \(-0.459583\pi\)
0.126634 + 0.991949i \(0.459583\pi\)
\(338\) 7.41468 0.403305
\(339\) 0 0
\(340\) 8.72666 0.473269
\(341\) −0.565344 −0.0306151
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.15066 0.439454
\(345\) 0 0
\(346\) −5.32469 −0.286257
\(347\) 11.7653 0.631594 0.315797 0.948827i \(-0.397728\pi\)
0.315797 + 0.948827i \(0.397728\pi\)
\(348\) 0 0
\(349\) 22.8153 1.22128 0.610638 0.791910i \(-0.290913\pi\)
0.610638 + 0.791910i \(0.290913\pi\)
\(350\) 4.86799 0.260205
\(351\) 0 0
\(352\) −0.726656 −0.0387309
\(353\) 19.8094 1.05435 0.527174 0.849758i \(-0.323252\pi\)
0.527174 + 0.849758i \(0.323252\pi\)
\(354\) 0 0
\(355\) 29.1893 1.54921
\(356\) 17.5233 0.928734
\(357\) 0 0
\(358\) 1.86799 0.0987265
\(359\) 11.1413 0.588017 0.294009 0.955803i \(-0.405011\pi\)
0.294009 + 0.955803i \(0.405011\pi\)
\(360\) 0 0
\(361\) −16.7360 −0.880841
\(362\) 11.5560 0.607371
\(363\) 0 0
\(364\) −2.36333 −0.123872
\(365\) 3.17064 0.165959
\(366\) 0 0
\(367\) 6.59465 0.344238 0.172119 0.985076i \(-0.444939\pi\)
0.172119 + 0.985076i \(0.444939\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 18.4333 0.958304
\(371\) 7.73599 0.401632
\(372\) 0 0
\(373\) 28.3013 1.46539 0.732694 0.680559i \(-0.238263\pi\)
0.732694 + 0.680559i \(0.238263\pi\)
\(374\) 2.01866 0.104382
\(375\) 0 0
\(376\) −2.64600 −0.136457
\(377\) −18.5946 −0.957673
\(378\) 0 0
\(379\) 15.1600 0.778717 0.389358 0.921086i \(-0.372697\pi\)
0.389358 + 0.921086i \(0.372697\pi\)
\(380\) 4.72666 0.242472
\(381\) 0 0
\(382\) −21.8387 −1.11736
\(383\) 5.71733 0.292142 0.146071 0.989274i \(-0.453337\pi\)
0.146071 + 0.989274i \(0.453337\pi\)
\(384\) 0 0
\(385\) 2.28267 0.116336
\(386\) −22.1693 −1.12839
\(387\) 0 0
\(388\) 1.63667 0.0830894
\(389\) −0.887968 −0.0450218 −0.0225109 0.999747i \(-0.507166\pi\)
−0.0225109 + 0.999747i \(0.507166\pi\)
\(390\) 0 0
\(391\) −2.77801 −0.140490
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 11.7067 0.589774
\(395\) −8.56534 −0.430969
\(396\) 0 0
\(397\) 7.76868 0.389899 0.194949 0.980813i \(-0.437546\pi\)
0.194949 + 0.980813i \(0.437546\pi\)
\(398\) −9.70668 −0.486552
\(399\) 0 0
\(400\) 4.86799 0.243400
\(401\) 22.5653 1.12686 0.563430 0.826164i \(-0.309482\pi\)
0.563430 + 0.826164i \(0.309482\pi\)
\(402\) 0 0
\(403\) −1.83869 −0.0915916
\(404\) 7.24065 0.360236
\(405\) 0 0
\(406\) −7.86799 −0.390482
\(407\) 4.26401 0.211359
\(408\) 0 0
\(409\) 13.1120 0.648348 0.324174 0.945997i \(-0.394914\pi\)
0.324174 + 0.945997i \(0.394914\pi\)
\(410\) −4.98002 −0.245946
\(411\) 0 0
\(412\) 19.4240 0.956952
\(413\) 0 0
\(414\) 0 0
\(415\) −24.4626 −1.20082
\(416\) −2.36333 −0.115872
\(417\) 0 0
\(418\) 1.09337 0.0534787
\(419\) 7.68463 0.375419 0.187709 0.982225i \(-0.439894\pi\)
0.187709 + 0.982225i \(0.439894\pi\)
\(420\) 0 0
\(421\) −24.9987 −1.21836 −0.609181 0.793032i \(-0.708502\pi\)
−0.609181 + 0.793032i \(0.708502\pi\)
\(422\) −9.83869 −0.478940
\(423\) 0 0
\(424\) 7.73599 0.375693
\(425\) −13.5233 −0.655977
\(426\) 0 0
\(427\) −7.55602 −0.365661
\(428\) 8.72666 0.421819
\(429\) 0 0
\(430\) −25.6040 −1.23473
\(431\) 0.678694 0.0326916 0.0163458 0.999866i \(-0.494797\pi\)
0.0163458 + 0.999866i \(0.494797\pi\)
\(432\) 0 0
\(433\) −18.5620 −0.892031 −0.446015 0.895025i \(-0.647157\pi\)
−0.446015 + 0.895025i \(0.647157\pi\)
\(434\) −0.778008 −0.0373456
\(435\) 0 0
\(436\) −2.41468 −0.115642
\(437\) −1.50466 −0.0719779
\(438\) 0 0
\(439\) 23.7287 1.13251 0.566255 0.824230i \(-0.308392\pi\)
0.566255 + 0.824230i \(0.308392\pi\)
\(440\) 2.28267 0.108822
\(441\) 0 0
\(442\) 6.56534 0.312282
\(443\) 9.15999 0.435204 0.217602 0.976038i \(-0.430176\pi\)
0.217602 + 0.976038i \(0.430176\pi\)
\(444\) 0 0
\(445\) −55.0466 −2.60946
\(446\) 6.79667 0.321831
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −3.92273 −0.185125 −0.0925626 0.995707i \(-0.529506\pi\)
−0.0925626 + 0.995707i \(0.529506\pi\)
\(450\) 0 0
\(451\) −1.15198 −0.0542448
\(452\) 3.14134 0.147756
\(453\) 0 0
\(454\) −7.63667 −0.358407
\(455\) 7.42401 0.348043
\(456\) 0 0
\(457\) −33.6774 −1.57536 −0.787681 0.616084i \(-0.788718\pi\)
−0.787681 + 0.616084i \(0.788718\pi\)
\(458\) 5.57467 0.260487
\(459\) 0 0
\(460\) −3.14134 −0.146466
\(461\) −34.5140 −1.60748 −0.803738 0.594983i \(-0.797159\pi\)
−0.803738 + 0.594983i \(0.797159\pi\)
\(462\) 0 0
\(463\) 26.6974 1.24073 0.620366 0.784313i \(-0.286984\pi\)
0.620366 + 0.784313i \(0.286984\pi\)
\(464\) −7.86799 −0.365262
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 17.9193 0.829208 0.414604 0.910002i \(-0.363920\pi\)
0.414604 + 0.910002i \(0.363920\pi\)
\(468\) 0 0
\(469\) 1.45331 0.0671078
\(470\) 8.31198 0.383403
\(471\) 0 0
\(472\) 0 0
\(473\) −5.92273 −0.272328
\(474\) 0 0
\(475\) −7.32469 −0.336080
\(476\) 2.77801 0.127330
\(477\) 0 0
\(478\) 9.29200 0.425006
\(479\) −15.3693 −0.702240 −0.351120 0.936331i \(-0.614199\pi\)
−0.351120 + 0.936331i \(0.614199\pi\)
\(480\) 0 0
\(481\) 13.8680 0.632326
\(482\) 6.09931 0.277816
\(483\) 0 0
\(484\) −10.4720 −0.475999
\(485\) −5.14134 −0.233456
\(486\) 0 0
\(487\) −13.3026 −0.602801 −0.301400 0.953498i \(-0.597454\pi\)
−0.301400 + 0.953498i \(0.597454\pi\)
\(488\) −7.55602 −0.342045
\(489\) 0 0
\(490\) 3.14134 0.141911
\(491\) 34.9694 1.57815 0.789073 0.614300i \(-0.210561\pi\)
0.789073 + 0.614300i \(0.210561\pi\)
\(492\) 0 0
\(493\) 21.8573 0.984405
\(494\) 3.55602 0.159993
\(495\) 0 0
\(496\) −0.778008 −0.0349336
\(497\) 9.29200 0.416803
\(498\) 0 0
\(499\) −37.5161 −1.67945 −0.839725 0.543012i \(-0.817284\pi\)
−0.839725 + 0.543012i \(0.817284\pi\)
\(500\) 0.414680 0.0185450
\(501\) 0 0
\(502\) 26.8446 1.19813
\(503\) 15.7360 0.701633 0.350816 0.936444i \(-0.385904\pi\)
0.350816 + 0.936444i \(0.385904\pi\)
\(504\) 0 0
\(505\) −22.7453 −1.01215
\(506\) −0.726656 −0.0323038
\(507\) 0 0
\(508\) 2.85866 0.126833
\(509\) 6.61670 0.293280 0.146640 0.989190i \(-0.453154\pi\)
0.146640 + 0.989190i \(0.453154\pi\)
\(510\) 0 0
\(511\) 1.00933 0.0446501
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.55602 −0.156849
\(515\) −61.0173 −2.68875
\(516\) 0 0
\(517\) 1.92273 0.0845617
\(518\) 5.86799 0.257825
\(519\) 0 0
\(520\) 7.42401 0.325564
\(521\) −6.57260 −0.287951 −0.143975 0.989581i \(-0.545989\pi\)
−0.143975 + 0.989581i \(0.545989\pi\)
\(522\) 0 0
\(523\) −17.6074 −0.769916 −0.384958 0.922934i \(-0.625784\pi\)
−0.384958 + 0.922934i \(0.625784\pi\)
\(524\) 0.264015 0.0115335
\(525\) 0 0
\(526\) −16.1507 −0.704202
\(527\) 2.16131 0.0941482
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −24.3013 −1.05558
\(531\) 0 0
\(532\) 1.50466 0.0652355
\(533\) −3.74663 −0.162285
\(534\) 0 0
\(535\) −27.4134 −1.18518
\(536\) 1.45331 0.0627736
\(537\) 0 0
\(538\) −3.66598 −0.158051
\(539\) 0.726656 0.0312993
\(540\) 0 0
\(541\) −11.9813 −0.515118 −0.257559 0.966263i \(-0.582918\pi\)
−0.257559 + 0.966263i \(0.582918\pi\)
\(542\) −9.40196 −0.403849
\(543\) 0 0
\(544\) 2.77801 0.119106
\(545\) 7.58532 0.324919
\(546\) 0 0
\(547\) 6.28267 0.268628 0.134314 0.990939i \(-0.457117\pi\)
0.134314 + 0.990939i \(0.457117\pi\)
\(548\) 2.15066 0.0918718
\(549\) 0 0
\(550\) −3.53736 −0.150833
\(551\) 11.8387 0.504345
\(552\) 0 0
\(553\) −2.72666 −0.115949
\(554\) 13.9414 0.592313
\(555\) 0 0
\(556\) 12.0480 0.510947
\(557\) 26.6027 1.12719 0.563595 0.826051i \(-0.309418\pi\)
0.563595 + 0.826051i \(0.309418\pi\)
\(558\) 0 0
\(559\) −19.2627 −0.814725
\(560\) 3.14134 0.132746
\(561\) 0 0
\(562\) −2.41468 −0.101857
\(563\) 23.6367 0.996167 0.498083 0.867129i \(-0.334037\pi\)
0.498083 + 0.867129i \(0.334037\pi\)
\(564\) 0 0
\(565\) −9.86799 −0.415150
\(566\) −24.1473 −1.01499
\(567\) 0 0
\(568\) 9.29200 0.389884
\(569\) −44.2066 −1.85324 −0.926619 0.376001i \(-0.877299\pi\)
−0.926619 + 0.376001i \(0.877299\pi\)
\(570\) 0 0
\(571\) −31.4720 −1.31706 −0.658530 0.752554i \(-0.728822\pi\)
−0.658530 + 0.752554i \(0.728822\pi\)
\(572\) 1.71733 0.0718051
\(573\) 0 0
\(574\) −1.58532 −0.0661700
\(575\) 4.86799 0.203009
\(576\) 0 0
\(577\) −20.1800 −0.840103 −0.420052 0.907500i \(-0.637988\pi\)
−0.420052 + 0.907500i \(0.637988\pi\)
\(578\) 9.28267 0.386108
\(579\) 0 0
\(580\) 24.7160 1.02628
\(581\) −7.78734 −0.323073
\(582\) 0 0
\(583\) −5.62140 −0.232815
\(584\) 1.00933 0.0417663
\(585\) 0 0
\(586\) −14.4626 −0.597446
\(587\) −16.9907 −0.701280 −0.350640 0.936510i \(-0.614036\pi\)
−0.350640 + 0.936510i \(0.614036\pi\)
\(588\) 0 0
\(589\) 1.17064 0.0482354
\(590\) 0 0
\(591\) 0 0
\(592\) 5.86799 0.241173
\(593\) 30.8773 1.26798 0.633990 0.773341i \(-0.281416\pi\)
0.633990 + 0.773341i \(0.281416\pi\)
\(594\) 0 0
\(595\) −8.72666 −0.357758
\(596\) 20.8480 0.853968
\(597\) 0 0
\(598\) −2.36333 −0.0966437
\(599\) 7.67738 0.313689 0.156845 0.987623i \(-0.449868\pi\)
0.156845 + 0.987623i \(0.449868\pi\)
\(600\) 0 0
\(601\) 12.8039 0.522283 0.261141 0.965301i \(-0.415901\pi\)
0.261141 + 0.965301i \(0.415901\pi\)
\(602\) −8.15066 −0.332196
\(603\) 0 0
\(604\) 3.42401 0.139321
\(605\) 32.8960 1.33741
\(606\) 0 0
\(607\) 9.50466 0.385782 0.192891 0.981220i \(-0.438214\pi\)
0.192891 + 0.981220i \(0.438214\pi\)
\(608\) 1.50466 0.0610222
\(609\) 0 0
\(610\) 23.7360 0.961042
\(611\) 6.25337 0.252984
\(612\) 0 0
\(613\) −10.1133 −0.408474 −0.204237 0.978921i \(-0.565471\pi\)
−0.204237 + 0.978921i \(0.565471\pi\)
\(614\) 30.0666 1.21339
\(615\) 0 0
\(616\) 0.726656 0.0292778
\(617\) 28.5840 1.15075 0.575374 0.817890i \(-0.304856\pi\)
0.575374 + 0.817890i \(0.304856\pi\)
\(618\) 0 0
\(619\) −37.9673 −1.52603 −0.763017 0.646378i \(-0.776283\pi\)
−0.763017 + 0.646378i \(0.776283\pi\)
\(620\) 2.44398 0.0981528
\(621\) 0 0
\(622\) 1.24065 0.0497455
\(623\) −17.5233 −0.702057
\(624\) 0 0
\(625\) −25.6426 −1.02570
\(626\) −14.7780 −0.590648
\(627\) 0 0
\(628\) 14.5653 0.581221
\(629\) −16.3013 −0.649977
\(630\) 0 0
\(631\) 24.6867 0.982762 0.491381 0.870945i \(-0.336492\pi\)
0.491381 + 0.870945i \(0.336492\pi\)
\(632\) −2.72666 −0.108461
\(633\) 0 0
\(634\) −7.76529 −0.308399
\(635\) −8.98002 −0.356361
\(636\) 0 0
\(637\) 2.36333 0.0936385
\(638\) 5.71733 0.226351
\(639\) 0 0
\(640\) 3.14134 0.124172
\(641\) −29.5853 −1.16855 −0.584275 0.811556i \(-0.698621\pi\)
−0.584275 + 0.811556i \(0.698621\pi\)
\(642\) 0 0
\(643\) −15.6260 −0.616230 −0.308115 0.951349i \(-0.599698\pi\)
−0.308115 + 0.951349i \(0.599698\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 0 0
\(646\) −4.17997 −0.164459
\(647\) 42.2686 1.66175 0.830876 0.556458i \(-0.187840\pi\)
0.830876 + 0.556458i \(0.187840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −11.5047 −0.451250
\(651\) 0 0
\(652\) −13.5747 −0.531625
\(653\) −1.32131 −0.0517067 −0.0258533 0.999666i \(-0.508230\pi\)
−0.0258533 + 0.999666i \(0.508230\pi\)
\(654\) 0 0
\(655\) −0.829359 −0.0324057
\(656\) −1.58532 −0.0618964
\(657\) 0 0
\(658\) 2.64600 0.103152
\(659\) 17.1307 0.667317 0.333658 0.942694i \(-0.391717\pi\)
0.333658 + 0.942694i \(0.391717\pi\)
\(660\) 0 0
\(661\) −34.7267 −1.35071 −0.675355 0.737493i \(-0.736010\pi\)
−0.675355 + 0.737493i \(0.736010\pi\)
\(662\) 23.7360 0.922525
\(663\) 0 0
\(664\) −7.78734 −0.302207
\(665\) −4.72666 −0.183292
\(666\) 0 0
\(667\) −7.86799 −0.304650
\(668\) −3.22199 −0.124663
\(669\) 0 0
\(670\) −4.56534 −0.176375
\(671\) 5.49063 0.211963
\(672\) 0 0
\(673\) −26.1107 −1.00649 −0.503247 0.864143i \(-0.667861\pi\)
−0.503247 + 0.864143i \(0.667861\pi\)
\(674\) −4.64939 −0.179088
\(675\) 0 0
\(676\) −7.41468 −0.285180
\(677\) −25.9346 −0.996748 −0.498374 0.866962i \(-0.666069\pi\)
−0.498374 + 0.866962i \(0.666069\pi\)
\(678\) 0 0
\(679\) −1.63667 −0.0628097
\(680\) −8.72666 −0.334652
\(681\) 0 0
\(682\) 0.565344 0.0216482
\(683\) 6.72666 0.257388 0.128694 0.991684i \(-0.458921\pi\)
0.128694 + 0.991684i \(0.458921\pi\)
\(684\) 0 0
\(685\) −6.75596 −0.258132
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −8.15066 −0.310741
\(689\) −18.2827 −0.696514
\(690\) 0 0
\(691\) −11.7253 −0.446053 −0.223026 0.974812i \(-0.571594\pi\)
−0.223026 + 0.974812i \(0.571594\pi\)
\(692\) 5.32469 0.202415
\(693\) 0 0
\(694\) −11.7653 −0.446604
\(695\) −37.8467 −1.43561
\(696\) 0 0
\(697\) 4.40403 0.166815
\(698\) −22.8153 −0.863573
\(699\) 0 0
\(700\) −4.86799 −0.183993
\(701\) 34.0373 1.28557 0.642786 0.766046i \(-0.277779\pi\)
0.642786 + 0.766046i \(0.277779\pi\)
\(702\) 0 0
\(703\) −8.82936 −0.333006
\(704\) 0.726656 0.0273869
\(705\) 0 0
\(706\) −19.8094 −0.745536
\(707\) −7.24065 −0.272313
\(708\) 0 0
\(709\) −46.5254 −1.74730 −0.873649 0.486557i \(-0.838253\pi\)
−0.873649 + 0.486557i \(0.838253\pi\)
\(710\) −29.1893 −1.09545
\(711\) 0 0
\(712\) −17.5233 −0.656714
\(713\) −0.778008 −0.0291366
\(714\) 0 0
\(715\) −5.39470 −0.201750
\(716\) −1.86799 −0.0698102
\(717\) 0 0
\(718\) −11.1413 −0.415791
\(719\) −10.0220 −0.373759 −0.186880 0.982383i \(-0.559837\pi\)
−0.186880 + 0.982383i \(0.559837\pi\)
\(720\) 0 0
\(721\) −19.4240 −0.723388
\(722\) 16.7360 0.622849
\(723\) 0 0
\(724\) −11.5560 −0.429476
\(725\) −38.3013 −1.42248
\(726\) 0 0
\(727\) 34.6867 1.28646 0.643229 0.765674i \(-0.277594\pi\)
0.643229 + 0.765674i \(0.277594\pi\)
\(728\) 2.36333 0.0875908
\(729\) 0 0
\(730\) −3.17064 −0.117351
\(731\) 22.6426 0.837467
\(732\) 0 0
\(733\) 13.3760 0.494056 0.247028 0.969008i \(-0.420546\pi\)
0.247028 + 0.969008i \(0.420546\pi\)
\(734\) −6.59465 −0.243413
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −1.05606 −0.0389004
\(738\) 0 0
\(739\) 16.6867 0.613830 0.306915 0.951737i \(-0.400703\pi\)
0.306915 + 0.951737i \(0.400703\pi\)
\(740\) −18.4333 −0.677623
\(741\) 0 0
\(742\) −7.73599 −0.283997
\(743\) 18.1613 0.666274 0.333137 0.942878i \(-0.391893\pi\)
0.333137 + 0.942878i \(0.391893\pi\)
\(744\) 0 0
\(745\) −65.4906 −2.39939
\(746\) −28.3013 −1.03619
\(747\) 0 0
\(748\) −2.01866 −0.0738094
\(749\) −8.72666 −0.318865
\(750\) 0 0
\(751\) 2.58400 0.0942916 0.0471458 0.998888i \(-0.484987\pi\)
0.0471458 + 0.998888i \(0.484987\pi\)
\(752\) 2.64600 0.0964897
\(753\) 0 0
\(754\) 18.5946 0.677177
\(755\) −10.7560 −0.391450
\(756\) 0 0
\(757\) 21.2666 0.772946 0.386473 0.922301i \(-0.373693\pi\)
0.386473 + 0.922301i \(0.373693\pi\)
\(758\) −15.1600 −0.550636
\(759\) 0 0
\(760\) −4.72666 −0.171454
\(761\) −29.7801 −1.07953 −0.539764 0.841817i \(-0.681486\pi\)
−0.539764 + 0.841817i \(0.681486\pi\)
\(762\) 0 0
\(763\) 2.41468 0.0874173
\(764\) 21.8387 0.790096
\(765\) 0 0
\(766\) −5.71733 −0.206575
\(767\) 0 0
\(768\) 0 0
\(769\) 16.0807 0.579883 0.289942 0.957044i \(-0.406364\pi\)
0.289942 + 0.957044i \(0.406364\pi\)
\(770\) −2.28267 −0.0822618
\(771\) 0 0
\(772\) 22.1693 0.797891
\(773\) −10.1507 −0.365094 −0.182547 0.983197i \(-0.558434\pi\)
−0.182547 + 0.983197i \(0.558434\pi\)
\(774\) 0 0
\(775\) −3.78734 −0.136045
\(776\) −1.63667 −0.0587531
\(777\) 0 0
\(778\) 0.887968 0.0318352
\(779\) 2.38538 0.0854649
\(780\) 0 0
\(781\) −6.75209 −0.241609
\(782\) 2.77801 0.0993414
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −45.7546 −1.63305
\(786\) 0 0
\(787\) 30.7967 1.09778 0.548891 0.835894i \(-0.315050\pi\)
0.548891 + 0.835894i \(0.315050\pi\)
\(788\) −11.7067 −0.417033
\(789\) 0 0
\(790\) 8.56534 0.304741
\(791\) −3.14134 −0.111693
\(792\) 0 0
\(793\) 17.8573 0.634133
\(794\) −7.76868 −0.275700
\(795\) 0 0
\(796\) 9.70668 0.344044
\(797\) −46.7160 −1.65477 −0.827383 0.561638i \(-0.810171\pi\)
−0.827383 + 0.561638i \(0.810171\pi\)
\(798\) 0 0
\(799\) −7.35061 −0.260046
\(800\) −4.86799 −0.172110
\(801\) 0 0
\(802\) −22.5653 −0.796810
\(803\) −0.733435 −0.0258824
\(804\) 0 0
\(805\) 3.14134 0.110718
\(806\) 1.83869 0.0647650
\(807\) 0 0
\(808\) −7.24065 −0.254725
\(809\) −38.9253 −1.36854 −0.684270 0.729229i \(-0.739879\pi\)
−0.684270 + 0.729229i \(0.739879\pi\)
\(810\) 0 0
\(811\) 17.0759 0.599618 0.299809 0.953999i \(-0.403077\pi\)
0.299809 + 0.953999i \(0.403077\pi\)
\(812\) 7.86799 0.276112
\(813\) 0 0
\(814\) −4.26401 −0.149454
\(815\) 42.6426 1.49371
\(816\) 0 0
\(817\) 12.2640 0.429064
\(818\) −13.1120 −0.458451
\(819\) 0 0
\(820\) 4.98002 0.173910
\(821\) 13.3693 0.466591 0.233295 0.972406i \(-0.425049\pi\)
0.233295 + 0.972406i \(0.425049\pi\)
\(822\) 0 0
\(823\) −4.93593 −0.172056 −0.0860279 0.996293i \(-0.527417\pi\)
−0.0860279 + 0.996293i \(0.527417\pi\)
\(824\) −19.4240 −0.676667
\(825\) 0 0
\(826\) 0 0
\(827\) −17.6519 −0.613818 −0.306909 0.951739i \(-0.599295\pi\)
−0.306909 + 0.951739i \(0.599295\pi\)
\(828\) 0 0
\(829\) −28.7967 −1.00015 −0.500075 0.865982i \(-0.666694\pi\)
−0.500075 + 0.865982i \(0.666694\pi\)
\(830\) 24.4626 0.849111
\(831\) 0 0
\(832\) 2.36333 0.0819337
\(833\) −2.77801 −0.0962523
\(834\) 0 0
\(835\) 10.1214 0.350264
\(836\) −1.09337 −0.0378151
\(837\) 0 0
\(838\) −7.68463 −0.265461
\(839\) −33.6960 −1.16332 −0.581658 0.813433i \(-0.697596\pi\)
−0.581658 + 0.813433i \(0.697596\pi\)
\(840\) 0 0
\(841\) 32.9053 1.13467
\(842\) 24.9987 0.861511
\(843\) 0 0
\(844\) 9.83869 0.338662
\(845\) 23.2920 0.801269
\(846\) 0 0
\(847\) 10.4720 0.359821
\(848\) −7.73599 −0.265655
\(849\) 0 0
\(850\) 13.5233 0.463846
\(851\) 5.86799 0.201152
\(852\) 0 0
\(853\) −3.28861 −0.112600 −0.0563000 0.998414i \(-0.517930\pi\)
−0.0563000 + 0.998414i \(0.517930\pi\)
\(854\) 7.55602 0.258562
\(855\) 0 0
\(856\) −8.72666 −0.298271
\(857\) −31.2440 −1.06728 −0.533638 0.845713i \(-0.679176\pi\)
−0.533638 + 0.845713i \(0.679176\pi\)
\(858\) 0 0
\(859\) 18.9546 0.646722 0.323361 0.946276i \(-0.395187\pi\)
0.323361 + 0.946276i \(0.395187\pi\)
\(860\) 25.6040 0.873088
\(861\) 0 0
\(862\) −0.678694 −0.0231164
\(863\) −5.91595 −0.201381 −0.100691 0.994918i \(-0.532105\pi\)
−0.100691 + 0.994918i \(0.532105\pi\)
\(864\) 0 0
\(865\) −16.7267 −0.568723
\(866\) 18.5620 0.630761
\(867\) 0 0
\(868\) 0.778008 0.0264073
\(869\) 1.98134 0.0672124
\(870\) 0 0
\(871\) −3.43466 −0.116379
\(872\) 2.41468 0.0817714
\(873\) 0 0
\(874\) 1.50466 0.0508960
\(875\) −0.414680 −0.0140187
\(876\) 0 0
\(877\) 7.82003 0.264064 0.132032 0.991245i \(-0.457850\pi\)
0.132032 + 0.991245i \(0.457850\pi\)
\(878\) −23.7287 −0.800806
\(879\) 0 0
\(880\) −2.28267 −0.0769489
\(881\) 4.43673 0.149477 0.0747386 0.997203i \(-0.476188\pi\)
0.0747386 + 0.997203i \(0.476188\pi\)
\(882\) 0 0
\(883\) 20.7453 0.698135 0.349068 0.937098i \(-0.386498\pi\)
0.349068 + 0.937098i \(0.386498\pi\)
\(884\) −6.56534 −0.220816
\(885\) 0 0
\(886\) −9.15999 −0.307736
\(887\) −28.1473 −0.945093 −0.472547 0.881306i \(-0.656665\pi\)
−0.472547 + 0.881306i \(0.656665\pi\)
\(888\) 0 0
\(889\) −2.85866 −0.0958765
\(890\) 55.0466 1.84517
\(891\) 0 0
\(892\) −6.79667 −0.227569
\(893\) −3.98134 −0.133231
\(894\) 0 0
\(895\) 5.86799 0.196145
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 3.92273 0.130903
\(899\) 6.12136 0.204159
\(900\) 0 0
\(901\) 21.4906 0.715957
\(902\) 1.15198 0.0383568
\(903\) 0 0
\(904\) −3.14134 −0.104479
\(905\) 36.3013 1.20670
\(906\) 0 0
\(907\) 26.2347 0.871109 0.435555 0.900162i \(-0.356552\pi\)
0.435555 + 0.900162i \(0.356552\pi\)
\(908\) 7.63667 0.253432
\(909\) 0 0
\(910\) −7.42401 −0.246103
\(911\) −15.4054 −0.510402 −0.255201 0.966888i \(-0.582142\pi\)
−0.255201 + 0.966888i \(0.582142\pi\)
\(912\) 0 0
\(913\) 5.65872 0.187276
\(914\) 33.6774 1.11395
\(915\) 0 0
\(916\) −5.57467 −0.184192
\(917\) −0.264015 −0.00871854
\(918\) 0 0
\(919\) −29.0466 −0.958160 −0.479080 0.877771i \(-0.659030\pi\)
−0.479080 + 0.877771i \(0.659030\pi\)
\(920\) 3.14134 0.103567
\(921\) 0 0
\(922\) 34.5140 1.13666
\(923\) −21.9600 −0.722824
\(924\) 0 0
\(925\) 28.5653 0.939223
\(926\) −26.6974 −0.877329
\(927\) 0 0
\(928\) 7.86799 0.258280
\(929\) −23.2440 −0.762612 −0.381306 0.924449i \(-0.624526\pi\)
−0.381306 + 0.924449i \(0.624526\pi\)
\(930\) 0 0
\(931\) −1.50466 −0.0493134
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) −17.9193 −0.586339
\(935\) 6.34128 0.207382
\(936\) 0 0
\(937\) −2.10609 −0.0688030 −0.0344015 0.999408i \(-0.510953\pi\)
−0.0344015 + 0.999408i \(0.510953\pi\)
\(938\) −1.45331 −0.0474523
\(939\) 0 0
\(940\) −8.31198 −0.271107
\(941\) −4.80005 −0.156477 −0.0782387 0.996935i \(-0.524930\pi\)
−0.0782387 + 0.996935i \(0.524930\pi\)
\(942\) 0 0
\(943\) −1.58532 −0.0516251
\(944\) 0 0
\(945\) 0 0
\(946\) 5.92273 0.192565
\(947\) −14.5174 −0.471752 −0.235876 0.971783i \(-0.575796\pi\)
−0.235876 + 0.971783i \(0.575796\pi\)
\(948\) 0 0
\(949\) −2.38538 −0.0774326
\(950\) 7.32469 0.237644
\(951\) 0 0
\(952\) −2.77801 −0.0900358
\(953\) 3.76142 0.121844 0.0609222 0.998143i \(-0.480596\pi\)
0.0609222 + 0.998143i \(0.480596\pi\)
\(954\) 0 0
\(955\) −68.6027 −2.21993
\(956\) −9.29200 −0.300525
\(957\) 0 0
\(958\) 15.3693 0.496558
\(959\) −2.15066 −0.0694486
\(960\) 0 0
\(961\) −30.3947 −0.980474
\(962\) −13.8680 −0.447122
\(963\) 0 0
\(964\) −6.09931 −0.196446
\(965\) −69.6413 −2.24183
\(966\) 0 0
\(967\) 46.2241 1.48647 0.743233 0.669033i \(-0.233291\pi\)
0.743233 + 0.669033i \(0.233291\pi\)
\(968\) 10.4720 0.336582
\(969\) 0 0
\(970\) 5.14134 0.165078
\(971\) −23.1566 −0.743131 −0.371565 0.928407i \(-0.621179\pi\)
−0.371565 + 0.928407i \(0.621179\pi\)
\(972\) 0 0
\(973\) −12.0480 −0.386240
\(974\) 13.3026 0.426244
\(975\) 0 0
\(976\) 7.55602 0.241862
\(977\) −34.2466 −1.09565 −0.547823 0.836594i \(-0.684543\pi\)
−0.547823 + 0.836594i \(0.684543\pi\)
\(978\) 0 0
\(979\) 12.7334 0.406962
\(980\) −3.14134 −0.100346
\(981\) 0 0
\(982\) −34.9694 −1.11592
\(983\) 35.4347 1.13019 0.565095 0.825026i \(-0.308840\pi\)
0.565095 + 0.825026i \(0.308840\pi\)
\(984\) 0 0
\(985\) 36.7746 1.17174
\(986\) −21.8573 −0.696080
\(987\) 0 0
\(988\) −3.55602 −0.113132
\(989\) −8.15066 −0.259176
\(990\) 0 0
\(991\) 53.6306 1.70363 0.851817 0.523840i \(-0.175501\pi\)
0.851817 + 0.523840i \(0.175501\pi\)
\(992\) 0.778008 0.0247018
\(993\) 0 0
\(994\) −9.29200 −0.294724
\(995\) −30.4919 −0.966660
\(996\) 0 0
\(997\) −40.3713 −1.27857 −0.639287 0.768969i \(-0.720770\pi\)
−0.639287 + 0.768969i \(0.720770\pi\)
\(998\) 37.5161 1.18755
\(999\) 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.1.1 3
3.2 odd 2 2898.2.a.bg.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2898.2.a.bf.1.1 3 1.1 even 1 trivial
2898.2.a.bg.1.3 yes 3 3.2 odd 2