Properties

Label 1014.2.a.k.1.1
Level $1014$
Weight $2$
Character 1014.1
Self dual yes
Analytic conductor $8.097$
Analytic rank $0$
Dimension $2$
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] = [1014,2,Mod(1,1014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.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: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1014.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.267949 q^{5} +1.00000 q^{6} +0.732051 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.267949 q^{5} +1.00000 q^{6} +0.732051 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.267949 q^{10} +4.73205 q^{11} +1.00000 q^{12} +0.732051 q^{14} +0.267949 q^{15} +1.00000 q^{16} -2.26795 q^{17} +1.00000 q^{18} +1.26795 q^{19} +0.267949 q^{20} +0.732051 q^{21} +4.73205 q^{22} -6.19615 q^{23} +1.00000 q^{24} -4.92820 q^{25} +1.00000 q^{27} +0.732051 q^{28} +2.46410 q^{29} +0.267949 q^{30} -5.46410 q^{31} +1.00000 q^{32} +4.73205 q^{33} -2.26795 q^{34} +0.196152 q^{35} +1.00000 q^{36} +10.4641 q^{37} +1.26795 q^{38} +0.267949 q^{40} +11.3923 q^{41} +0.732051 q^{42} +7.66025 q^{43} +4.73205 q^{44} +0.267949 q^{45} -6.19615 q^{46} -8.19615 q^{47} +1.00000 q^{48} -6.46410 q^{49} -4.92820 q^{50} -2.26795 q^{51} +0.464102 q^{53} +1.00000 q^{54} +1.26795 q^{55} +0.732051 q^{56} +1.26795 q^{57} +2.46410 q^{58} +8.00000 q^{59} +0.267949 q^{60} +1.19615 q^{61} -5.46410 q^{62} +0.732051 q^{63} +1.00000 q^{64} +4.73205 q^{66} -11.1244 q^{67} -2.26795 q^{68} -6.19615 q^{69} +0.196152 q^{70} +1.26795 q^{71} +1.00000 q^{72} -9.73205 q^{73} +10.4641 q^{74} -4.92820 q^{75} +1.26795 q^{76} +3.46410 q^{77} -9.46410 q^{79} +0.267949 q^{80} +1.00000 q^{81} +11.3923 q^{82} +10.1962 q^{83} +0.732051 q^{84} -0.607695 q^{85} +7.66025 q^{86} +2.46410 q^{87} +4.73205 q^{88} -2.53590 q^{89} +0.267949 q^{90} -6.19615 q^{92} -5.46410 q^{93} -8.19615 q^{94} +0.339746 q^{95} +1.00000 q^{96} -6.00000 q^{97} -6.46410 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 6 q^{11} + 2 q^{12} - 2 q^{14} + 4 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} + 6 q^{19} + 4 q^{20} - 2 q^{21} + 6 q^{22} - 2 q^{23} + 2 q^{24} + 4 q^{25} + 2 q^{27} - 2 q^{28} - 2 q^{29} + 4 q^{30} - 4 q^{31} + 2 q^{32} + 6 q^{33} - 8 q^{34} - 10 q^{35} + 2 q^{36} + 14 q^{37} + 6 q^{38} + 4 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{43} + 6 q^{44} + 4 q^{45} - 2 q^{46} - 6 q^{47} + 2 q^{48} - 6 q^{49} + 4 q^{50} - 8 q^{51} - 6 q^{53} + 2 q^{54} + 6 q^{55} - 2 q^{56} + 6 q^{57} - 2 q^{58} + 16 q^{59} + 4 q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 6 q^{66} + 2 q^{67} - 8 q^{68} - 2 q^{69} - 10 q^{70} + 6 q^{71} + 2 q^{72} - 16 q^{73} + 14 q^{74} + 4 q^{75} + 6 q^{76} - 12 q^{79} + 4 q^{80} + 2 q^{81} + 2 q^{82} + 10 q^{83} - 2 q^{84} - 22 q^{85} - 2 q^{86} - 2 q^{87} + 6 q^{88} - 12 q^{89} + 4 q^{90} - 2 q^{92} - 4 q^{93} - 6 q^{94} + 18 q^{95} + 2 q^{96} - 12 q^{97} - 6 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.267949 0.119831 0.0599153 0.998203i \(-0.480917\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 1.00000 0.408248
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.267949 0.0847330
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 0.732051 0.195649
\(15\) 0.267949 0.0691842
\(16\) 1.00000 0.250000
\(17\) −2.26795 −0.550058 −0.275029 0.961436i \(-0.588688\pi\)
−0.275029 + 0.961436i \(0.588688\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.26795 0.290887 0.145444 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) 0.267949 0.0599153
\(21\) 0.732051 0.159747
\(22\) 4.73205 1.00888
\(23\) −6.19615 −1.29199 −0.645994 0.763343i \(-0.723557\pi\)
−0.645994 + 0.763343i \(0.723557\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.92820 −0.985641
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.732051 0.138345
\(29\) 2.46410 0.457572 0.228786 0.973477i \(-0.426524\pi\)
0.228786 + 0.973477i \(0.426524\pi\)
\(30\) 0.267949 0.0489206
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.73205 0.823744
\(34\) −2.26795 −0.388950
\(35\) 0.196152 0.0331558
\(36\) 1.00000 0.166667
\(37\) 10.4641 1.72029 0.860144 0.510052i \(-0.170374\pi\)
0.860144 + 0.510052i \(0.170374\pi\)
\(38\) 1.26795 0.205689
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) 11.3923 1.77918 0.889590 0.456761i \(-0.150990\pi\)
0.889590 + 0.456761i \(0.150990\pi\)
\(42\) 0.732051 0.112958
\(43\) 7.66025 1.16818 0.584089 0.811690i \(-0.301452\pi\)
0.584089 + 0.811690i \(0.301452\pi\)
\(44\) 4.73205 0.713384
\(45\) 0.267949 0.0399435
\(46\) −6.19615 −0.913573
\(47\) −8.19615 −1.19553 −0.597766 0.801671i \(-0.703945\pi\)
−0.597766 + 0.801671i \(0.703945\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.46410 −0.923443
\(50\) −4.92820 −0.696953
\(51\) −2.26795 −0.317576
\(52\) 0 0
\(53\) 0.464102 0.0637493 0.0318746 0.999492i \(-0.489852\pi\)
0.0318746 + 0.999492i \(0.489852\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.26795 0.170970
\(56\) 0.732051 0.0978244
\(57\) 1.26795 0.167944
\(58\) 2.46410 0.323552
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0.267949 0.0345921
\(61\) 1.19615 0.153152 0.0765758 0.997064i \(-0.475601\pi\)
0.0765758 + 0.997064i \(0.475601\pi\)
\(62\) −5.46410 −0.693942
\(63\) 0.732051 0.0922297
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.73205 0.582475
\(67\) −11.1244 −1.35906 −0.679528 0.733649i \(-0.737815\pi\)
−0.679528 + 0.733649i \(0.737815\pi\)
\(68\) −2.26795 −0.275029
\(69\) −6.19615 −0.745929
\(70\) 0.196152 0.0234447
\(71\) 1.26795 0.150478 0.0752389 0.997166i \(-0.476028\pi\)
0.0752389 + 0.997166i \(0.476028\pi\)
\(72\) 1.00000 0.117851
\(73\) −9.73205 −1.13905 −0.569525 0.821974i \(-0.692873\pi\)
−0.569525 + 0.821974i \(0.692873\pi\)
\(74\) 10.4641 1.21643
\(75\) −4.92820 −0.569060
\(76\) 1.26795 0.145444
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) −9.46410 −1.06479 −0.532397 0.846495i \(-0.678709\pi\)
−0.532397 + 0.846495i \(0.678709\pi\)
\(80\) 0.267949 0.0299576
\(81\) 1.00000 0.111111
\(82\) 11.3923 1.25807
\(83\) 10.1962 1.11917 0.559587 0.828772i \(-0.310960\pi\)
0.559587 + 0.828772i \(0.310960\pi\)
\(84\) 0.732051 0.0798733
\(85\) −0.607695 −0.0659138
\(86\) 7.66025 0.826026
\(87\) 2.46410 0.264179
\(88\) 4.73205 0.504438
\(89\) −2.53590 −0.268805 −0.134402 0.990927i \(-0.542911\pi\)
−0.134402 + 0.990927i \(0.542911\pi\)
\(90\) 0.267949 0.0282443
\(91\) 0 0
\(92\) −6.19615 −0.645994
\(93\) −5.46410 −0.566601
\(94\) −8.19615 −0.845369
\(95\) 0.339746 0.0348572
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −6.46410 −0.652973
\(99\) 4.73205 0.475589
\(100\) −4.92820 −0.492820
\(101\) −11.9282 −1.18690 −0.593450 0.804871i \(-0.702235\pi\)
−0.593450 + 0.804871i \(0.702235\pi\)
\(102\) −2.26795 −0.224560
\(103\) −18.7321 −1.84572 −0.922862 0.385131i \(-0.874156\pi\)
−0.922862 + 0.385131i \(0.874156\pi\)
\(104\) 0 0
\(105\) 0.196152 0.0191425
\(106\) 0.464102 0.0450775
\(107\) −0.196152 −0.0189628 −0.00948139 0.999955i \(-0.503018\pi\)
−0.00948139 + 0.999955i \(0.503018\pi\)
\(108\) 1.00000 0.0962250
\(109\) −5.46410 −0.523366 −0.261683 0.965154i \(-0.584277\pi\)
−0.261683 + 0.965154i \(0.584277\pi\)
\(110\) 1.26795 0.120894
\(111\) 10.4641 0.993209
\(112\) 0.732051 0.0691723
\(113\) −18.6603 −1.75541 −0.877705 0.479202i \(-0.840926\pi\)
−0.877705 + 0.479202i \(0.840926\pi\)
\(114\) 1.26795 0.118754
\(115\) −1.66025 −0.154819
\(116\) 2.46410 0.228786
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) −1.66025 −0.152195
\(120\) 0.267949 0.0244603
\(121\) 11.3923 1.03566
\(122\) 1.19615 0.108295
\(123\) 11.3923 1.02721
\(124\) −5.46410 −0.490691
\(125\) −2.66025 −0.237940
\(126\) 0.732051 0.0652163
\(127\) 17.8564 1.58450 0.792250 0.610197i \(-0.208910\pi\)
0.792250 + 0.610197i \(0.208910\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.66025 0.674448
\(130\) 0 0
\(131\) 13.4641 1.17636 0.588182 0.808729i \(-0.299844\pi\)
0.588182 + 0.808729i \(0.299844\pi\)
\(132\) 4.73205 0.411872
\(133\) 0.928203 0.0804854
\(134\) −11.1244 −0.960998
\(135\) 0.267949 0.0230614
\(136\) −2.26795 −0.194475
\(137\) −1.92820 −0.164738 −0.0823688 0.996602i \(-0.526249\pi\)
−0.0823688 + 0.996602i \(0.526249\pi\)
\(138\) −6.19615 −0.527452
\(139\) −9.85641 −0.836009 −0.418005 0.908445i \(-0.637270\pi\)
−0.418005 + 0.908445i \(0.637270\pi\)
\(140\) 0.196152 0.0165779
\(141\) −8.19615 −0.690241
\(142\) 1.26795 0.106404
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0.660254 0.0548311
\(146\) −9.73205 −0.805430
\(147\) −6.46410 −0.533150
\(148\) 10.4641 0.860144
\(149\) 2.80385 0.229700 0.114850 0.993383i \(-0.463361\pi\)
0.114850 + 0.993383i \(0.463361\pi\)
\(150\) −4.92820 −0.402386
\(151\) 3.26795 0.265942 0.132971 0.991120i \(-0.457548\pi\)
0.132971 + 0.991120i \(0.457548\pi\)
\(152\) 1.26795 0.102844
\(153\) −2.26795 −0.183353
\(154\) 3.46410 0.279145
\(155\) −1.46410 −0.117599
\(156\) 0 0
\(157\) −23.5885 −1.88256 −0.941282 0.337622i \(-0.890378\pi\)
−0.941282 + 0.337622i \(0.890378\pi\)
\(158\) −9.46410 −0.752923
\(159\) 0.464102 0.0368057
\(160\) 0.267949 0.0211832
\(161\) −4.53590 −0.357479
\(162\) 1.00000 0.0785674
\(163\) −6.53590 −0.511931 −0.255966 0.966686i \(-0.582393\pi\)
−0.255966 + 0.966686i \(0.582393\pi\)
\(164\) 11.3923 0.889590
\(165\) 1.26795 0.0987097
\(166\) 10.1962 0.791375
\(167\) −2.53590 −0.196234 −0.0981169 0.995175i \(-0.531282\pi\)
−0.0981169 + 0.995175i \(0.531282\pi\)
\(168\) 0.732051 0.0564789
\(169\) 0 0
\(170\) −0.607695 −0.0466081
\(171\) 1.26795 0.0969625
\(172\) 7.66025 0.584089
\(173\) −16.3923 −1.24628 −0.623142 0.782109i \(-0.714144\pi\)
−0.623142 + 0.782109i \(0.714144\pi\)
\(174\) 2.46410 0.186803
\(175\) −3.60770 −0.272716
\(176\) 4.73205 0.356692
\(177\) 8.00000 0.601317
\(178\) −2.53590 −0.190074
\(179\) 22.0526 1.64829 0.824143 0.566382i \(-0.191657\pi\)
0.824143 + 0.566382i \(0.191657\pi\)
\(180\) 0.267949 0.0199718
\(181\) 8.80385 0.654385 0.327192 0.944958i \(-0.393897\pi\)
0.327192 + 0.944958i \(0.393897\pi\)
\(182\) 0 0
\(183\) 1.19615 0.0884221
\(184\) −6.19615 −0.456786
\(185\) 2.80385 0.206143
\(186\) −5.46410 −0.400647
\(187\) −10.7321 −0.784805
\(188\) −8.19615 −0.597766
\(189\) 0.732051 0.0532489
\(190\) 0.339746 0.0246478
\(191\) 6.92820 0.501307 0.250654 0.968077i \(-0.419354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.26795 −0.595140 −0.297570 0.954700i \(-0.596176\pi\)
−0.297570 + 0.954700i \(0.596176\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) 9.85641 0.702240 0.351120 0.936330i \(-0.385801\pi\)
0.351120 + 0.936330i \(0.385801\pi\)
\(198\) 4.73205 0.336292
\(199\) −3.80385 −0.269648 −0.134824 0.990870i \(-0.543047\pi\)
−0.134824 + 0.990870i \(0.543047\pi\)
\(200\) −4.92820 −0.348477
\(201\) −11.1244 −0.784652
\(202\) −11.9282 −0.839265
\(203\) 1.80385 0.126605
\(204\) −2.26795 −0.158788
\(205\) 3.05256 0.213200
\(206\) −18.7321 −1.30512
\(207\) −6.19615 −0.430662
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0.196152 0.0135358
\(211\) −4.39230 −0.302379 −0.151189 0.988505i \(-0.548310\pi\)
−0.151189 + 0.988505i \(0.548310\pi\)
\(212\) 0.464102 0.0318746
\(213\) 1.26795 0.0868784
\(214\) −0.196152 −0.0134087
\(215\) 2.05256 0.139983
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) −5.46410 −0.370076
\(219\) −9.73205 −0.657631
\(220\) 1.26795 0.0854851
\(221\) 0 0
\(222\) 10.4641 0.702305
\(223\) −13.0718 −0.875352 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(224\) 0.732051 0.0489122
\(225\) −4.92820 −0.328547
\(226\) −18.6603 −1.24126
\(227\) −1.80385 −0.119726 −0.0598628 0.998207i \(-0.519066\pi\)
−0.0598628 + 0.998207i \(0.519066\pi\)
\(228\) 1.26795 0.0839720
\(229\) 15.8564 1.04782 0.523910 0.851773i \(-0.324473\pi\)
0.523910 + 0.851773i \(0.324473\pi\)
\(230\) −1.66025 −0.109474
\(231\) 3.46410 0.227921
\(232\) 2.46410 0.161776
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) −2.19615 −0.143261
\(236\) 8.00000 0.520756
\(237\) −9.46410 −0.614759
\(238\) −1.66025 −0.107618
\(239\) −9.66025 −0.624870 −0.312435 0.949939i \(-0.601145\pi\)
−0.312435 + 0.949939i \(0.601145\pi\)
\(240\) 0.267949 0.0172960
\(241\) −17.5885 −1.13297 −0.566486 0.824071i \(-0.691698\pi\)
−0.566486 + 0.824071i \(0.691698\pi\)
\(242\) 11.3923 0.732325
\(243\) 1.00000 0.0641500
\(244\) 1.19615 0.0765758
\(245\) −1.73205 −0.110657
\(246\) 11.3923 0.726347
\(247\) 0 0
\(248\) −5.46410 −0.346971
\(249\) 10.1962 0.646155
\(250\) −2.66025 −0.168249
\(251\) 6.53590 0.412542 0.206271 0.978495i \(-0.433867\pi\)
0.206271 + 0.978495i \(0.433867\pi\)
\(252\) 0.732051 0.0461149
\(253\) −29.3205 −1.84336
\(254\) 17.8564 1.12041
\(255\) −0.607695 −0.0380553
\(256\) 1.00000 0.0625000
\(257\) 26.6603 1.66302 0.831510 0.555509i \(-0.187477\pi\)
0.831510 + 0.555509i \(0.187477\pi\)
\(258\) 7.66025 0.476907
\(259\) 7.66025 0.475985
\(260\) 0 0
\(261\) 2.46410 0.152524
\(262\) 13.4641 0.831815
\(263\) 28.0526 1.72979 0.864897 0.501949i \(-0.167383\pi\)
0.864897 + 0.501949i \(0.167383\pi\)
\(264\) 4.73205 0.291238
\(265\) 0.124356 0.00763911
\(266\) 0.928203 0.0569118
\(267\) −2.53590 −0.155194
\(268\) −11.1244 −0.679528
\(269\) −1.46410 −0.0892679 −0.0446339 0.999003i \(-0.514212\pi\)
−0.0446339 + 0.999003i \(0.514212\pi\)
\(270\) 0.267949 0.0163069
\(271\) 5.85641 0.355751 0.177876 0.984053i \(-0.443078\pi\)
0.177876 + 0.984053i \(0.443078\pi\)
\(272\) −2.26795 −0.137515
\(273\) 0 0
\(274\) −1.92820 −0.116487
\(275\) −23.3205 −1.40628
\(276\) −6.19615 −0.372965
\(277\) −2.26795 −0.136268 −0.0681339 0.997676i \(-0.521705\pi\)
−0.0681339 + 0.997676i \(0.521705\pi\)
\(278\) −9.85641 −0.591148
\(279\) −5.46410 −0.327127
\(280\) 0.196152 0.0117223
\(281\) −22.3205 −1.33153 −0.665765 0.746162i \(-0.731895\pi\)
−0.665765 + 0.746162i \(0.731895\pi\)
\(282\) −8.19615 −0.488074
\(283\) 8.33975 0.495746 0.247873 0.968792i \(-0.420268\pi\)
0.247873 + 0.968792i \(0.420268\pi\)
\(284\) 1.26795 0.0752389
\(285\) 0.339746 0.0201248
\(286\) 0 0
\(287\) 8.33975 0.492280
\(288\) 1.00000 0.0589256
\(289\) −11.8564 −0.697436
\(290\) 0.660254 0.0387715
\(291\) −6.00000 −0.351726
\(292\) −9.73205 −0.569525
\(293\) −14.5167 −0.848072 −0.424036 0.905645i \(-0.639387\pi\)
−0.424036 + 0.905645i \(0.639387\pi\)
\(294\) −6.46410 −0.376994
\(295\) 2.14359 0.124805
\(296\) 10.4641 0.608214
\(297\) 4.73205 0.274581
\(298\) 2.80385 0.162423
\(299\) 0 0
\(300\) −4.92820 −0.284530
\(301\) 5.60770 0.323222
\(302\) 3.26795 0.188049
\(303\) −11.9282 −0.685257
\(304\) 1.26795 0.0727219
\(305\) 0.320508 0.0183522
\(306\) −2.26795 −0.129650
\(307\) 8.58846 0.490169 0.245085 0.969502i \(-0.421184\pi\)
0.245085 + 0.969502i \(0.421184\pi\)
\(308\) 3.46410 0.197386
\(309\) −18.7321 −1.06563
\(310\) −1.46410 −0.0831554
\(311\) 15.6603 0.888012 0.444006 0.896024i \(-0.353557\pi\)
0.444006 + 0.896024i \(0.353557\pi\)
\(312\) 0 0
\(313\) 13.4641 0.761036 0.380518 0.924774i \(-0.375746\pi\)
0.380518 + 0.924774i \(0.375746\pi\)
\(314\) −23.5885 −1.33117
\(315\) 0.196152 0.0110519
\(316\) −9.46410 −0.532397
\(317\) 3.33975 0.187579 0.0937894 0.995592i \(-0.470102\pi\)
0.0937894 + 0.995592i \(0.470102\pi\)
\(318\) 0.464102 0.0260255
\(319\) 11.6603 0.652849
\(320\) 0.267949 0.0149788
\(321\) −0.196152 −0.0109482
\(322\) −4.53590 −0.252776
\(323\) −2.87564 −0.160005
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.53590 −0.361990
\(327\) −5.46410 −0.302166
\(328\) 11.3923 0.629035
\(329\) −6.00000 −0.330791
\(330\) 1.26795 0.0697983
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 10.1962 0.559587
\(333\) 10.4641 0.573429
\(334\) −2.53590 −0.138758
\(335\) −2.98076 −0.162856
\(336\) 0.732051 0.0399366
\(337\) 6.85641 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(338\) 0 0
\(339\) −18.6603 −1.01349
\(340\) −0.607695 −0.0329569
\(341\) −25.8564 −1.40020
\(342\) 1.26795 0.0685628
\(343\) −9.85641 −0.532196
\(344\) 7.66025 0.413013
\(345\) −1.66025 −0.0893851
\(346\) −16.3923 −0.881256
\(347\) 8.87564 0.476470 0.238235 0.971208i \(-0.423431\pi\)
0.238235 + 0.971208i \(0.423431\pi\)
\(348\) 2.46410 0.132090
\(349\) −19.3205 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(350\) −3.60770 −0.192839
\(351\) 0 0
\(352\) 4.73205 0.252219
\(353\) 19.7846 1.05303 0.526514 0.850166i \(-0.323499\pi\)
0.526514 + 0.850166i \(0.323499\pi\)
\(354\) 8.00000 0.425195
\(355\) 0.339746 0.0180318
\(356\) −2.53590 −0.134402
\(357\) −1.66025 −0.0878700
\(358\) 22.0526 1.16551
\(359\) 23.1244 1.22046 0.610228 0.792226i \(-0.291078\pi\)
0.610228 + 0.792226i \(0.291078\pi\)
\(360\) 0.267949 0.0141222
\(361\) −17.3923 −0.915384
\(362\) 8.80385 0.462720
\(363\) 11.3923 0.597941
\(364\) 0 0
\(365\) −2.60770 −0.136493
\(366\) 1.19615 0.0625239
\(367\) −14.7321 −0.769007 −0.384503 0.923124i \(-0.625627\pi\)
−0.384503 + 0.923124i \(0.625627\pi\)
\(368\) −6.19615 −0.322997
\(369\) 11.3923 0.593060
\(370\) 2.80385 0.145765
\(371\) 0.339746 0.0176387
\(372\) −5.46410 −0.283300
\(373\) −10.2679 −0.531654 −0.265827 0.964021i \(-0.585645\pi\)
−0.265827 + 0.964021i \(0.585645\pi\)
\(374\) −10.7321 −0.554941
\(375\) −2.66025 −0.137375
\(376\) −8.19615 −0.422684
\(377\) 0 0
\(378\) 0.732051 0.0376526
\(379\) 1.46410 0.0752058 0.0376029 0.999293i \(-0.488028\pi\)
0.0376029 + 0.999293i \(0.488028\pi\)
\(380\) 0.339746 0.0174286
\(381\) 17.8564 0.914811
\(382\) 6.92820 0.354478
\(383\) 5.46410 0.279203 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.928203 0.0473056
\(386\) −8.26795 −0.420828
\(387\) 7.66025 0.389393
\(388\) −6.00000 −0.304604
\(389\) −29.7846 −1.51014 −0.755070 0.655644i \(-0.772397\pi\)
−0.755070 + 0.655644i \(0.772397\pi\)
\(390\) 0 0
\(391\) 14.0526 0.710668
\(392\) −6.46410 −0.326486
\(393\) 13.4641 0.679174
\(394\) 9.85641 0.496559
\(395\) −2.53590 −0.127595
\(396\) 4.73205 0.237795
\(397\) −0.392305 −0.0196892 −0.00984461 0.999952i \(-0.503134\pi\)
−0.00984461 + 0.999952i \(0.503134\pi\)
\(398\) −3.80385 −0.190670
\(399\) 0.928203 0.0464683
\(400\) −4.92820 −0.246410
\(401\) 21.9282 1.09504 0.547521 0.836792i \(-0.315572\pi\)
0.547521 + 0.836792i \(0.315572\pi\)
\(402\) −11.1244 −0.554832
\(403\) 0 0
\(404\) −11.9282 −0.593450
\(405\) 0.267949 0.0133145
\(406\) 1.80385 0.0895235
\(407\) 49.5167 2.45445
\(408\) −2.26795 −0.112280
\(409\) 14.2679 0.705505 0.352752 0.935717i \(-0.385246\pi\)
0.352752 + 0.935717i \(0.385246\pi\)
\(410\) 3.05256 0.150755
\(411\) −1.92820 −0.0951113
\(412\) −18.7321 −0.922862
\(413\) 5.85641 0.288175
\(414\) −6.19615 −0.304524
\(415\) 2.73205 0.134111
\(416\) 0 0
\(417\) −9.85641 −0.482670
\(418\) 6.00000 0.293470
\(419\) −10.5359 −0.514712 −0.257356 0.966317i \(-0.582851\pi\)
−0.257356 + 0.966317i \(0.582851\pi\)
\(420\) 0.196152 0.00957126
\(421\) 32.7128 1.59432 0.797162 0.603765i \(-0.206333\pi\)
0.797162 + 0.603765i \(0.206333\pi\)
\(422\) −4.39230 −0.213814
\(423\) −8.19615 −0.398511
\(424\) 0.464102 0.0225388
\(425\) 11.1769 0.542160
\(426\) 1.26795 0.0614323
\(427\) 0.875644 0.0423754
\(428\) −0.196152 −0.00948139
\(429\) 0 0
\(430\) 2.05256 0.0989832
\(431\) −11.1244 −0.535841 −0.267921 0.963441i \(-0.586337\pi\)
−0.267921 + 0.963441i \(0.586337\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.8564 −0.713953 −0.356977 0.934113i \(-0.616192\pi\)
−0.356977 + 0.934113i \(0.616192\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0.660254 0.0316568
\(436\) −5.46410 −0.261683
\(437\) −7.85641 −0.375823
\(438\) −9.73205 −0.465015
\(439\) −17.6603 −0.842878 −0.421439 0.906857i \(-0.638475\pi\)
−0.421439 + 0.906857i \(0.638475\pi\)
\(440\) 1.26795 0.0604471
\(441\) −6.46410 −0.307814
\(442\) 0 0
\(443\) 36.3923 1.72905 0.864525 0.502589i \(-0.167619\pi\)
0.864525 + 0.502589i \(0.167619\pi\)
\(444\) 10.4641 0.496604
\(445\) −0.679492 −0.0322110
\(446\) −13.0718 −0.618968
\(447\) 2.80385 0.132617
\(448\) 0.732051 0.0345861
\(449\) −23.3205 −1.10056 −0.550281 0.834979i \(-0.685480\pi\)
−0.550281 + 0.834979i \(0.685480\pi\)
\(450\) −4.92820 −0.232318
\(451\) 53.9090 2.53847
\(452\) −18.6603 −0.877705
\(453\) 3.26795 0.153542
\(454\) −1.80385 −0.0846588
\(455\) 0 0
\(456\) 1.26795 0.0593772
\(457\) 18.6603 0.872890 0.436445 0.899731i \(-0.356237\pi\)
0.436445 + 0.899731i \(0.356237\pi\)
\(458\) 15.8564 0.740921
\(459\) −2.26795 −0.105859
\(460\) −1.66025 −0.0774097
\(461\) 25.7321 1.19846 0.599231 0.800577i \(-0.295473\pi\)
0.599231 + 0.800577i \(0.295473\pi\)
\(462\) 3.46410 0.161165
\(463\) −28.0526 −1.30371 −0.651856 0.758342i \(-0.726010\pi\)
−0.651856 + 0.758342i \(0.726010\pi\)
\(464\) 2.46410 0.114393
\(465\) −1.46410 −0.0678961
\(466\) 19.8564 0.919830
\(467\) −12.5885 −0.582524 −0.291262 0.956643i \(-0.594075\pi\)
−0.291262 + 0.956643i \(0.594075\pi\)
\(468\) 0 0
\(469\) −8.14359 −0.376036
\(470\) −2.19615 −0.101301
\(471\) −23.5885 −1.08690
\(472\) 8.00000 0.368230
\(473\) 36.2487 1.66672
\(474\) −9.46410 −0.434701
\(475\) −6.24871 −0.286711
\(476\) −1.66025 −0.0760976
\(477\) 0.464102 0.0212498
\(478\) −9.66025 −0.441850
\(479\) 26.5359 1.21246 0.606228 0.795291i \(-0.292682\pi\)
0.606228 + 0.795291i \(0.292682\pi\)
\(480\) 0.267949 0.0122302
\(481\) 0 0
\(482\) −17.5885 −0.801132
\(483\) −4.53590 −0.206391
\(484\) 11.3923 0.517832
\(485\) −1.60770 −0.0730017
\(486\) 1.00000 0.0453609
\(487\) 21.1244 0.957236 0.478618 0.878023i \(-0.341138\pi\)
0.478618 + 0.878023i \(0.341138\pi\)
\(488\) 1.19615 0.0541473
\(489\) −6.53590 −0.295564
\(490\) −1.73205 −0.0782461
\(491\) 5.26795 0.237739 0.118870 0.992910i \(-0.462073\pi\)
0.118870 + 0.992910i \(0.462073\pi\)
\(492\) 11.3923 0.513605
\(493\) −5.58846 −0.251691
\(494\) 0 0
\(495\) 1.26795 0.0569901
\(496\) −5.46410 −0.245345
\(497\) 0.928203 0.0416356
\(498\) 10.1962 0.456901
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −2.66025 −0.118970
\(501\) −2.53590 −0.113296
\(502\) 6.53590 0.291711
\(503\) −10.9808 −0.489608 −0.244804 0.969573i \(-0.578724\pi\)
−0.244804 + 0.969573i \(0.578724\pi\)
\(504\) 0.732051 0.0326081
\(505\) −3.19615 −0.142227
\(506\) −29.3205 −1.30346
\(507\) 0 0
\(508\) 17.8564 0.792250
\(509\) 10.2679 0.455119 0.227559 0.973764i \(-0.426925\pi\)
0.227559 + 0.973764i \(0.426925\pi\)
\(510\) −0.607695 −0.0269092
\(511\) −7.12436 −0.315163
\(512\) 1.00000 0.0441942
\(513\) 1.26795 0.0559813
\(514\) 26.6603 1.17593
\(515\) −5.01924 −0.221174
\(516\) 7.66025 0.337224
\(517\) −38.7846 −1.70575
\(518\) 7.66025 0.336572
\(519\) −16.3923 −0.719542
\(520\) 0 0
\(521\) −17.4449 −0.764273 −0.382137 0.924106i \(-0.624812\pi\)
−0.382137 + 0.924106i \(0.624812\pi\)
\(522\) 2.46410 0.107851
\(523\) 36.4449 1.59362 0.796811 0.604228i \(-0.206518\pi\)
0.796811 + 0.604228i \(0.206518\pi\)
\(524\) 13.4641 0.588182
\(525\) −3.60770 −0.157453
\(526\) 28.0526 1.22315
\(527\) 12.3923 0.539817
\(528\) 4.73205 0.205936
\(529\) 15.3923 0.669231
\(530\) 0.124356 0.00540166
\(531\) 8.00000 0.347170
\(532\) 0.928203 0.0402427
\(533\) 0 0
\(534\) −2.53590 −0.109739
\(535\) −0.0525589 −0.00227232
\(536\) −11.1244 −0.480499
\(537\) 22.0526 0.951638
\(538\) −1.46410 −0.0631219
\(539\) −30.5885 −1.31754
\(540\) 0.267949 0.0115307
\(541\) 40.3205 1.73351 0.866757 0.498731i \(-0.166200\pi\)
0.866757 + 0.498731i \(0.166200\pi\)
\(542\) 5.85641 0.251554
\(543\) 8.80385 0.377809
\(544\) −2.26795 −0.0972375
\(545\) −1.46410 −0.0627152
\(546\) 0 0
\(547\) 6.19615 0.264928 0.132464 0.991188i \(-0.457711\pi\)
0.132464 + 0.991188i \(0.457711\pi\)
\(548\) −1.92820 −0.0823688
\(549\) 1.19615 0.0510505
\(550\) −23.3205 −0.994390
\(551\) 3.12436 0.133102
\(552\) −6.19615 −0.263726
\(553\) −6.92820 −0.294617
\(554\) −2.26795 −0.0963559
\(555\) 2.80385 0.119017
\(556\) −9.85641 −0.418005
\(557\) −30.3731 −1.28695 −0.643474 0.765468i \(-0.722508\pi\)
−0.643474 + 0.765468i \(0.722508\pi\)
\(558\) −5.46410 −0.231314
\(559\) 0 0
\(560\) 0.196152 0.00828895
\(561\) −10.7321 −0.453108
\(562\) −22.3205 −0.941534
\(563\) 21.0718 0.888070 0.444035 0.896009i \(-0.353547\pi\)
0.444035 + 0.896009i \(0.353547\pi\)
\(564\) −8.19615 −0.345120
\(565\) −5.00000 −0.210352
\(566\) 8.33975 0.350546
\(567\) 0.732051 0.0307432
\(568\) 1.26795 0.0532020
\(569\) −38.6410 −1.61992 −0.809958 0.586488i \(-0.800510\pi\)
−0.809958 + 0.586488i \(0.800510\pi\)
\(570\) 0.339746 0.0142304
\(571\) −24.0526 −1.00657 −0.503284 0.864121i \(-0.667875\pi\)
−0.503284 + 0.864121i \(0.667875\pi\)
\(572\) 0 0
\(573\) 6.92820 0.289430
\(574\) 8.33975 0.348094
\(575\) 30.5359 1.27343
\(576\) 1.00000 0.0416667
\(577\) 0.267949 0.0111549 0.00557744 0.999984i \(-0.498225\pi\)
0.00557744 + 0.999984i \(0.498225\pi\)
\(578\) −11.8564 −0.493161
\(579\) −8.26795 −0.343604
\(580\) 0.660254 0.0274156
\(581\) 7.46410 0.309663
\(582\) −6.00000 −0.248708
\(583\) 2.19615 0.0909553
\(584\) −9.73205 −0.402715
\(585\) 0 0
\(586\) −14.5167 −0.599678
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) −6.46410 −0.266575
\(589\) −6.92820 −0.285472
\(590\) 2.14359 0.0882503
\(591\) 9.85641 0.405438
\(592\) 10.4641 0.430072
\(593\) −36.8564 −1.51351 −0.756756 0.653698i \(-0.773217\pi\)
−0.756756 + 0.653698i \(0.773217\pi\)
\(594\) 4.73205 0.194158
\(595\) −0.444864 −0.0182376
\(596\) 2.80385 0.114850
\(597\) −3.80385 −0.155681
\(598\) 0 0
\(599\) −9.46410 −0.386693 −0.193346 0.981131i \(-0.561934\pi\)
−0.193346 + 0.981131i \(0.561934\pi\)
\(600\) −4.92820 −0.201193
\(601\) −5.92820 −0.241816 −0.120908 0.992664i \(-0.538581\pi\)
−0.120908 + 0.992664i \(0.538581\pi\)
\(602\) 5.60770 0.228553
\(603\) −11.1244 −0.453019
\(604\) 3.26795 0.132971
\(605\) 3.05256 0.124104
\(606\) −11.9282 −0.484550
\(607\) 0.784610 0.0318463 0.0159232 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(608\) 1.26795 0.0514221
\(609\) 1.80385 0.0730956
\(610\) 0.320508 0.0129770
\(611\) 0 0
\(612\) −2.26795 −0.0916764
\(613\) 11.3923 0.460131 0.230065 0.973175i \(-0.426106\pi\)
0.230065 + 0.973175i \(0.426106\pi\)
\(614\) 8.58846 0.346602
\(615\) 3.05256 0.123091
\(616\) 3.46410 0.139573
\(617\) 35.2487 1.41906 0.709530 0.704675i \(-0.248907\pi\)
0.709530 + 0.704675i \(0.248907\pi\)
\(618\) −18.7321 −0.753514
\(619\) 10.5359 0.423474 0.211737 0.977327i \(-0.432088\pi\)
0.211737 + 0.977327i \(0.432088\pi\)
\(620\) −1.46410 −0.0587997
\(621\) −6.19615 −0.248643
\(622\) 15.6603 0.627919
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 13.4641 0.538134
\(627\) 6.00000 0.239617
\(628\) −23.5885 −0.941282
\(629\) −23.7321 −0.946259
\(630\) 0.196152 0.00781490
\(631\) 47.7128 1.89942 0.949709 0.313135i \(-0.101379\pi\)
0.949709 + 0.313135i \(0.101379\pi\)
\(632\) −9.46410 −0.376462
\(633\) −4.39230 −0.174578
\(634\) 3.33975 0.132638
\(635\) 4.78461 0.189871
\(636\) 0.464102 0.0184028
\(637\) 0 0
\(638\) 11.6603 0.461634
\(639\) 1.26795 0.0501593
\(640\) 0.267949 0.0105916
\(641\) 25.9808 1.02618 0.513089 0.858335i \(-0.328501\pi\)
0.513089 + 0.858335i \(0.328501\pi\)
\(642\) −0.196152 −0.00774152
\(643\) 13.8564 0.546443 0.273222 0.961951i \(-0.411911\pi\)
0.273222 + 0.961951i \(0.411911\pi\)
\(644\) −4.53590 −0.178739
\(645\) 2.05256 0.0808194
\(646\) −2.87564 −0.113141
\(647\) 26.2487 1.03194 0.515972 0.856606i \(-0.327431\pi\)
0.515972 + 0.856606i \(0.327431\pi\)
\(648\) 1.00000 0.0392837
\(649\) 37.8564 1.48599
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) −6.53590 −0.255966
\(653\) −10.5359 −0.412302 −0.206151 0.978520i \(-0.566094\pi\)
−0.206151 + 0.978520i \(0.566094\pi\)
\(654\) −5.46410 −0.213663
\(655\) 3.60770 0.140964
\(656\) 11.3923 0.444795
\(657\) −9.73205 −0.379683
\(658\) −6.00000 −0.233904
\(659\) 38.2487 1.48996 0.744979 0.667088i \(-0.232459\pi\)
0.744979 + 0.667088i \(0.232459\pi\)
\(660\) 1.26795 0.0493549
\(661\) −9.39230 −0.365318 −0.182659 0.983176i \(-0.558470\pi\)
−0.182659 + 0.983176i \(0.558470\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 10.1962 0.395687
\(665\) 0.248711 0.00964461
\(666\) 10.4641 0.405476
\(667\) −15.2679 −0.591177
\(668\) −2.53590 −0.0981169
\(669\) −13.0718 −0.505385
\(670\) −2.98076 −0.115157
\(671\) 5.66025 0.218512
\(672\) 0.732051 0.0282395
\(673\) 14.0718 0.542428 0.271214 0.962519i \(-0.412575\pi\)
0.271214 + 0.962519i \(0.412575\pi\)
\(674\) 6.85641 0.264099
\(675\) −4.92820 −0.189687
\(676\) 0 0
\(677\) −38.5359 −1.48105 −0.740527 0.672026i \(-0.765424\pi\)
−0.740527 + 0.672026i \(0.765424\pi\)
\(678\) −18.6603 −0.716643
\(679\) −4.39230 −0.168561
\(680\) −0.607695 −0.0233040
\(681\) −1.80385 −0.0691236
\(682\) −25.8564 −0.990093
\(683\) −37.8564 −1.44854 −0.724268 0.689519i \(-0.757822\pi\)
−0.724268 + 0.689519i \(0.757822\pi\)
\(684\) 1.26795 0.0484812
\(685\) −0.516660 −0.0197406
\(686\) −9.85641 −0.376319
\(687\) 15.8564 0.604960
\(688\) 7.66025 0.292044
\(689\) 0 0
\(690\) −1.66025 −0.0632048
\(691\) 26.3397 1.00201 0.501006 0.865444i \(-0.332964\pi\)
0.501006 + 0.865444i \(0.332964\pi\)
\(692\) −16.3923 −0.623142
\(693\) 3.46410 0.131590
\(694\) 8.87564 0.336915
\(695\) −2.64102 −0.100179
\(696\) 2.46410 0.0934015
\(697\) −25.8372 −0.978653
\(698\) −19.3205 −0.731292
\(699\) 19.8564 0.751038
\(700\) −3.60770 −0.136358
\(701\) 31.3205 1.18296 0.591480 0.806320i \(-0.298544\pi\)
0.591480 + 0.806320i \(0.298544\pi\)
\(702\) 0 0
\(703\) 13.2679 0.500410
\(704\) 4.73205 0.178346
\(705\) −2.19615 −0.0827119
\(706\) 19.7846 0.744604
\(707\) −8.73205 −0.328403
\(708\) 8.00000 0.300658
\(709\) 40.8564 1.53439 0.767197 0.641411i \(-0.221651\pi\)
0.767197 + 0.641411i \(0.221651\pi\)
\(710\) 0.339746 0.0127504
\(711\) −9.46410 −0.354932
\(712\) −2.53590 −0.0950368
\(713\) 33.8564 1.26793
\(714\) −1.66025 −0.0621334
\(715\) 0 0
\(716\) 22.0526 0.824143
\(717\) −9.66025 −0.360769
\(718\) 23.1244 0.862993
\(719\) −22.5359 −0.840447 −0.420224 0.907421i \(-0.638048\pi\)
−0.420224 + 0.907421i \(0.638048\pi\)
\(720\) 0.267949 0.00998588
\(721\) −13.7128 −0.510692
\(722\) −17.3923 −0.647275
\(723\) −17.5885 −0.654122
\(724\) 8.80385 0.327192
\(725\) −12.1436 −0.451002
\(726\) 11.3923 0.422808
\(727\) 20.9808 0.778133 0.389067 0.921210i \(-0.372798\pi\)
0.389067 + 0.921210i \(0.372798\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.60770 −0.0965151
\(731\) −17.3731 −0.642566
\(732\) 1.19615 0.0442111
\(733\) 19.0000 0.701781 0.350891 0.936416i \(-0.385879\pi\)
0.350891 + 0.936416i \(0.385879\pi\)
\(734\) −14.7321 −0.543770
\(735\) −1.73205 −0.0638877
\(736\) −6.19615 −0.228393
\(737\) −52.6410 −1.93906
\(738\) 11.3923 0.419357
\(739\) −10.9282 −0.402000 −0.201000 0.979591i \(-0.564419\pi\)
−0.201000 + 0.979591i \(0.564419\pi\)
\(740\) 2.80385 0.103071
\(741\) 0 0
\(742\) 0.339746 0.0124725
\(743\) −27.6077 −1.01283 −0.506414 0.862290i \(-0.669029\pi\)
−0.506414 + 0.862290i \(0.669029\pi\)
\(744\) −5.46410 −0.200324
\(745\) 0.751289 0.0275251
\(746\) −10.2679 −0.375936
\(747\) 10.1962 0.373058
\(748\) −10.7321 −0.392403
\(749\) −0.143594 −0.00524679
\(750\) −2.66025 −0.0971387
\(751\) 15.9090 0.580526 0.290263 0.956947i \(-0.406257\pi\)
0.290263 + 0.956947i \(0.406257\pi\)
\(752\) −8.19615 −0.298883
\(753\) 6.53590 0.238181
\(754\) 0 0
\(755\) 0.875644 0.0318680
\(756\) 0.732051 0.0266244
\(757\) 7.07180 0.257029 0.128514 0.991708i \(-0.458979\pi\)
0.128514 + 0.991708i \(0.458979\pi\)
\(758\) 1.46410 0.0531786
\(759\) −29.3205 −1.06427
\(760\) 0.339746 0.0123239
\(761\) −23.3205 −0.845368 −0.422684 0.906277i \(-0.638912\pi\)
−0.422684 + 0.906277i \(0.638912\pi\)
\(762\) 17.8564 0.646869
\(763\) −4.00000 −0.144810
\(764\) 6.92820 0.250654
\(765\) −0.607695 −0.0219713
\(766\) 5.46410 0.197426
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −16.1436 −0.582153 −0.291076 0.956700i \(-0.594013\pi\)
−0.291076 + 0.956700i \(0.594013\pi\)
\(770\) 0.928203 0.0334501
\(771\) 26.6603 0.960146
\(772\) −8.26795 −0.297570
\(773\) −35.0718 −1.26144 −0.630722 0.776009i \(-0.717241\pi\)
−0.630722 + 0.776009i \(0.717241\pi\)
\(774\) 7.66025 0.275342
\(775\) 26.9282 0.967290
\(776\) −6.00000 −0.215387
\(777\) 7.66025 0.274810
\(778\) −29.7846 −1.06783
\(779\) 14.4449 0.517541
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 14.0526 0.502518
\(783\) 2.46410 0.0880598
\(784\) −6.46410 −0.230861
\(785\) −6.32051 −0.225589
\(786\) 13.4641 0.480249
\(787\) 39.3205 1.40162 0.700812 0.713346i \(-0.252821\pi\)
0.700812 + 0.713346i \(0.252821\pi\)
\(788\) 9.85641 0.351120
\(789\) 28.0526 0.998698
\(790\) −2.53590 −0.0902232
\(791\) −13.6603 −0.485703
\(792\) 4.73205 0.168146
\(793\) 0 0
\(794\) −0.392305 −0.0139224
\(795\) 0.124356 0.00441044
\(796\) −3.80385 −0.134824
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0.928203 0.0328580
\(799\) 18.5885 0.657612
\(800\) −4.92820 −0.174238
\(801\) −2.53590 −0.0896016
\(802\) 21.9282 0.774312
\(803\) −46.0526 −1.62516
\(804\) −11.1244 −0.392326
\(805\) −1.21539 −0.0428369
\(806\) 0 0
\(807\) −1.46410 −0.0515388
\(808\) −11.9282 −0.419633
\(809\) −22.4115 −0.787948 −0.393974 0.919122i \(-0.628900\pi\)
−0.393974 + 0.919122i \(0.628900\pi\)
\(810\) 0.267949 0.00941477
\(811\) −45.1769 −1.58638 −0.793188 0.608977i \(-0.791580\pi\)
−0.793188 + 0.608977i \(0.791580\pi\)
\(812\) 1.80385 0.0633026
\(813\) 5.85641 0.205393
\(814\) 49.5167 1.73556
\(815\) −1.75129 −0.0613450
\(816\) −2.26795 −0.0793941
\(817\) 9.71281 0.339808
\(818\) 14.2679 0.498867
\(819\) 0 0
\(820\) 3.05256 0.106600
\(821\) 12.9282 0.451197 0.225599 0.974220i \(-0.427566\pi\)
0.225599 + 0.974220i \(0.427566\pi\)
\(822\) −1.92820 −0.0672538
\(823\) −41.5692 −1.44901 −0.724506 0.689269i \(-0.757932\pi\)
−0.724506 + 0.689269i \(0.757932\pi\)
\(824\) −18.7321 −0.652562
\(825\) −23.3205 −0.811916
\(826\) 5.85641 0.203770
\(827\) 33.4641 1.16366 0.581830 0.813310i \(-0.302337\pi\)
0.581830 + 0.813310i \(0.302337\pi\)
\(828\) −6.19615 −0.215331
\(829\) 12.1244 0.421096 0.210548 0.977583i \(-0.432475\pi\)
0.210548 + 0.977583i \(0.432475\pi\)
\(830\) 2.73205 0.0948309
\(831\) −2.26795 −0.0786743
\(832\) 0 0
\(833\) 14.6603 0.507948
\(834\) −9.85641 −0.341299
\(835\) −0.679492 −0.0235148
\(836\) 6.00000 0.207514
\(837\) −5.46410 −0.188867
\(838\) −10.5359 −0.363957
\(839\) 14.1436 0.488291 0.244146 0.969739i \(-0.421493\pi\)
0.244146 + 0.969739i \(0.421493\pi\)
\(840\) 0.196152 0.00676790
\(841\) −22.9282 −0.790628
\(842\) 32.7128 1.12736
\(843\) −22.3205 −0.768759
\(844\) −4.39230 −0.151189
\(845\) 0 0
\(846\) −8.19615 −0.281790
\(847\) 8.33975 0.286557
\(848\) 0.464102 0.0159373
\(849\) 8.33975 0.286219
\(850\) 11.1769 0.383365
\(851\) −64.8372 −2.22259
\(852\) 1.26795 0.0434392
\(853\) 8.17691 0.279972 0.139986 0.990153i \(-0.455294\pi\)
0.139986 + 0.990153i \(0.455294\pi\)
\(854\) 0.875644 0.0299639
\(855\) 0.339746 0.0116191
\(856\) −0.196152 −0.00670435
\(857\) −19.4449 −0.664224 −0.332112 0.943240i \(-0.607761\pi\)
−0.332112 + 0.943240i \(0.607761\pi\)
\(858\) 0 0
\(859\) −22.8756 −0.780507 −0.390253 0.920707i \(-0.627613\pi\)
−0.390253 + 0.920707i \(0.627613\pi\)
\(860\) 2.05256 0.0699917
\(861\) 8.33975 0.284218
\(862\) −11.1244 −0.378897
\(863\) 7.12436 0.242516 0.121258 0.992621i \(-0.461307\pi\)
0.121258 + 0.992621i \(0.461307\pi\)
\(864\) 1.00000 0.0340207
\(865\) −4.39230 −0.149343
\(866\) −14.8564 −0.504841
\(867\) −11.8564 −0.402665
\(868\) −4.00000 −0.135769
\(869\) −44.7846 −1.51921
\(870\) 0.660254 0.0223847
\(871\) 0 0
\(872\) −5.46410 −0.185038
\(873\) −6.00000 −0.203069
\(874\) −7.85641 −0.265747
\(875\) −1.94744 −0.0658355
\(876\) −9.73205 −0.328816
\(877\) 10.0718 0.340100 0.170050 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(878\) −17.6603 −0.596005
\(879\) −14.5167 −0.489635
\(880\) 1.26795 0.0427426
\(881\) 51.8372 1.74644 0.873219 0.487327i \(-0.162028\pi\)
0.873219 + 0.487327i \(0.162028\pi\)
\(882\) −6.46410 −0.217658
\(883\) 29.0718 0.978344 0.489172 0.872187i \(-0.337299\pi\)
0.489172 + 0.872187i \(0.337299\pi\)
\(884\) 0 0
\(885\) 2.14359 0.0720561
\(886\) 36.3923 1.22262
\(887\) 10.1436 0.340589 0.170294 0.985393i \(-0.445528\pi\)
0.170294 + 0.985393i \(0.445528\pi\)
\(888\) 10.4641 0.351152
\(889\) 13.0718 0.438414
\(890\) −0.679492 −0.0227766
\(891\) 4.73205 0.158530
\(892\) −13.0718 −0.437676
\(893\) −10.3923 −0.347765
\(894\) 2.80385 0.0937747
\(895\) 5.90897 0.197515
\(896\) 0.732051 0.0244561
\(897\) 0 0
\(898\) −23.3205 −0.778215
\(899\) −13.4641 −0.449053
\(900\) −4.92820 −0.164273
\(901\) −1.05256 −0.0350658
\(902\) 53.9090 1.79497
\(903\) 5.60770 0.186612
\(904\) −18.6603 −0.620631
\(905\) 2.35898 0.0784153
\(906\) 3.26795 0.108570
\(907\) 15.6077 0.518245 0.259123 0.965844i \(-0.416567\pi\)
0.259123 + 0.965844i \(0.416567\pi\)
\(908\) −1.80385 −0.0598628
\(909\) −11.9282 −0.395634
\(910\) 0 0
\(911\) −9.46410 −0.313560 −0.156780 0.987634i \(-0.550111\pi\)
−0.156780 + 0.987634i \(0.550111\pi\)
\(912\) 1.26795 0.0419860
\(913\) 48.2487 1.59680
\(914\) 18.6603 0.617226
\(915\) 0.320508 0.0105957
\(916\) 15.8564 0.523910
\(917\) 9.85641 0.325487
\(918\) −2.26795 −0.0748535
\(919\) −57.9615 −1.91197 −0.955987 0.293409i \(-0.905210\pi\)
−0.955987 + 0.293409i \(0.905210\pi\)
\(920\) −1.66025 −0.0547370
\(921\) 8.58846 0.282999
\(922\) 25.7321 0.847440
\(923\) 0 0
\(924\) 3.46410 0.113961
\(925\) −51.5692 −1.69559
\(926\) −28.0526 −0.921864
\(927\) −18.7321 −0.615241
\(928\) 2.46410 0.0808881
\(929\) 9.24871 0.303440 0.151720 0.988423i \(-0.451519\pi\)
0.151720 + 0.988423i \(0.451519\pi\)
\(930\) −1.46410 −0.0480098
\(931\) −8.19615 −0.268618
\(932\) 19.8564 0.650418
\(933\) 15.6603 0.512694
\(934\) −12.5885 −0.411907
\(935\) −2.87564 −0.0940436
\(936\) 0 0
\(937\) 43.2487 1.41287 0.706437 0.707776i \(-0.250301\pi\)
0.706437 + 0.707776i \(0.250301\pi\)
\(938\) −8.14359 −0.265898
\(939\) 13.4641 0.439384
\(940\) −2.19615 −0.0716306
\(941\) −56.6410 −1.84644 −0.923222 0.384267i \(-0.874454\pi\)
−0.923222 + 0.384267i \(0.874454\pi\)
\(942\) −23.5885 −0.768553
\(943\) −70.5885 −2.29868
\(944\) 8.00000 0.260378
\(945\) 0.196152 0.00638084
\(946\) 36.2487 1.17855
\(947\) 34.9282 1.13501 0.567507 0.823369i \(-0.307908\pi\)
0.567507 + 0.823369i \(0.307908\pi\)
\(948\) −9.46410 −0.307380
\(949\) 0 0
\(950\) −6.24871 −0.202735
\(951\) 3.33975 0.108299
\(952\) −1.66025 −0.0538091
\(953\) −41.5692 −1.34656 −0.673280 0.739388i \(-0.735115\pi\)
−0.673280 + 0.739388i \(0.735115\pi\)
\(954\) 0.464102 0.0150258
\(955\) 1.85641 0.0600719
\(956\) −9.66025 −0.312435
\(957\) 11.6603 0.376922
\(958\) 26.5359 0.857336
\(959\) −1.41154 −0.0455811
\(960\) 0.267949 0.00864802
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −0.196152 −0.00632092
\(964\) −17.5885 −0.566486
\(965\) −2.21539 −0.0713159
\(966\) −4.53590 −0.145940
\(967\) −18.8756 −0.607000 −0.303500 0.952831i \(-0.598155\pi\)
−0.303500 + 0.952831i \(0.598155\pi\)
\(968\) 11.3923 0.366163
\(969\) −2.87564 −0.0923790
\(970\) −1.60770 −0.0516200
\(971\) −18.2487 −0.585629 −0.292815 0.956169i \(-0.594592\pi\)
−0.292815 + 0.956169i \(0.594592\pi\)
\(972\) 1.00000 0.0320750
\(973\) −7.21539 −0.231315
\(974\) 21.1244 0.676868
\(975\) 0 0
\(976\) 1.19615 0.0382879
\(977\) 32.0718 1.02607 0.513034 0.858368i \(-0.328522\pi\)
0.513034 + 0.858368i \(0.328522\pi\)
\(978\) −6.53590 −0.208995
\(979\) −12.0000 −0.383522
\(980\) −1.73205 −0.0553283
\(981\) −5.46410 −0.174455
\(982\) 5.26795 0.168107
\(983\) −20.7846 −0.662926 −0.331463 0.943468i \(-0.607542\pi\)
−0.331463 + 0.943468i \(0.607542\pi\)
\(984\) 11.3923 0.363173
\(985\) 2.64102 0.0841498
\(986\) −5.58846 −0.177973
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) −47.4641 −1.50927
\(990\) 1.26795 0.0402981
\(991\) −8.58846 −0.272821 −0.136411 0.990652i \(-0.543557\pi\)
−0.136411 + 0.990652i \(0.543557\pi\)
\(992\) −5.46410 −0.173485
\(993\) 20.0000 0.634681
\(994\) 0.928203 0.0294408
\(995\) −1.01924 −0.0323120
\(996\) 10.1962 0.323077
\(997\) 38.6603 1.22438 0.612191 0.790710i \(-0.290288\pi\)
0.612191 + 0.790710i \(0.290288\pi\)
\(998\) 32.0000 1.01294
\(999\) 10.4641 0.331070
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 1014.2.a.k.1.1 2
3.2 odd 2 3042.2.a.p.1.2 2
4.3 odd 2 8112.2.a.bp.1.1 2
13.2 odd 12 78.2.i.a.43.2 4
13.3 even 3 1014.2.e.g.529.1 4
13.4 even 6 1014.2.e.i.991.2 4
13.5 odd 4 1014.2.b.e.337.2 4
13.6 odd 12 1014.2.i.a.361.1 4
13.7 odd 12 78.2.i.a.49.2 yes 4
13.8 odd 4 1014.2.b.e.337.3 4
13.9 even 3 1014.2.e.g.991.1 4
13.10 even 6 1014.2.e.i.529.2 4
13.11 odd 12 1014.2.i.a.823.1 4
13.12 even 2 1014.2.a.i.1.2 2
39.2 even 12 234.2.l.c.199.1 4
39.5 even 4 3042.2.b.i.1351.3 4
39.8 even 4 3042.2.b.i.1351.2 4
39.20 even 12 234.2.l.c.127.1 4
39.38 odd 2 3042.2.a.y.1.1 2
52.7 even 12 624.2.bv.e.49.2 4
52.15 even 12 624.2.bv.e.433.1 4
52.51 odd 2 8112.2.a.bj.1.2 2
65.2 even 12 1950.2.y.b.199.2 4
65.7 even 12 1950.2.y.g.49.1 4
65.28 even 12 1950.2.y.g.199.1 4
65.33 even 12 1950.2.y.b.49.2 4
65.54 odd 12 1950.2.bc.d.901.1 4
65.59 odd 12 1950.2.bc.d.751.1 4
156.59 odd 12 1872.2.by.h.1297.1 4
156.119 odd 12 1872.2.by.h.433.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 4 13.2 odd 12
78.2.i.a.49.2 yes 4 13.7 odd 12
234.2.l.c.127.1 4 39.20 even 12
234.2.l.c.199.1 4 39.2 even 12
624.2.bv.e.49.2 4 52.7 even 12
624.2.bv.e.433.1 4 52.15 even 12
1014.2.a.i.1.2 2 13.12 even 2
1014.2.a.k.1.1 2 1.1 even 1 trivial
1014.2.b.e.337.2 4 13.5 odd 4
1014.2.b.e.337.3 4 13.8 odd 4
1014.2.e.g.529.1 4 13.3 even 3
1014.2.e.g.991.1 4 13.9 even 3
1014.2.e.i.529.2 4 13.10 even 6
1014.2.e.i.991.2 4 13.4 even 6
1014.2.i.a.361.1 4 13.6 odd 12
1014.2.i.a.823.1 4 13.11 odd 12
1872.2.by.h.433.2 4 156.119 odd 12
1872.2.by.h.1297.1 4 156.59 odd 12
1950.2.y.b.49.2 4 65.33 even 12
1950.2.y.b.199.2 4 65.2 even 12
1950.2.y.g.49.1 4 65.7 even 12
1950.2.y.g.199.1 4 65.28 even 12
1950.2.bc.d.751.1 4 65.59 odd 12
1950.2.bc.d.901.1 4 65.54 odd 12
3042.2.a.p.1.2 2 3.2 odd 2
3042.2.a.y.1.1 2 39.38 odd 2
3042.2.b.i.1351.2 4 39.8 even 4
3042.2.b.i.1351.3 4 39.5 even 4
8112.2.a.bj.1.2 2 52.51 odd 2
8112.2.a.bp.1.1 2 4.3 odd 2