Properties

Label 2535.2.a.bd.1.1
Level $2535$
Weight $2$
Character 2535.1
Self dual yes
Analytic conductor $20.242$
Analytic rank $1$
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] = [2535,2,Mod(1,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, 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("2535.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.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: \(20.2420769124\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 2535.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.24698 q^{2} -1.00000 q^{3} -0.445042 q^{4} -1.00000 q^{5} +1.24698 q^{6} -2.24698 q^{7} +3.04892 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.24698 q^{2} -1.00000 q^{3} -0.445042 q^{4} -1.00000 q^{5} +1.24698 q^{6} -2.24698 q^{7} +3.04892 q^{8} +1.00000 q^{9} +1.24698 q^{10} +1.00000 q^{11} +0.445042 q^{12} +2.80194 q^{14} +1.00000 q^{15} -2.91185 q^{16} -6.49396 q^{17} -1.24698 q^{18} +2.33513 q^{19} +0.445042 q^{20} +2.24698 q^{21} -1.24698 q^{22} +0.198062 q^{23} -3.04892 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +3.82908 q^{29} -1.24698 q^{30} +1.71379 q^{31} -2.46681 q^{32} -1.00000 q^{33} +8.09783 q^{34} +2.24698 q^{35} -0.445042 q^{36} +4.54288 q^{37} -2.91185 q^{38} -3.04892 q^{40} +5.75302 q^{41} -2.80194 q^{42} -3.75302 q^{43} -0.445042 q^{44} -1.00000 q^{45} -0.246980 q^{46} -1.97285 q^{47} +2.91185 q^{48} -1.95108 q^{49} -1.24698 q^{50} +6.49396 q^{51} +12.0586 q^{53} +1.24698 q^{54} -1.00000 q^{55} -6.85086 q^{56} -2.33513 q^{57} -4.77479 q^{58} +9.83877 q^{59} -0.445042 q^{60} -13.6136 q^{61} -2.13706 q^{62} -2.24698 q^{63} +8.89977 q^{64} +1.24698 q^{66} +7.91185 q^{67} +2.89008 q^{68} -0.198062 q^{69} -2.80194 q^{70} +7.14675 q^{71} +3.04892 q^{72} +14.3545 q^{73} -5.66487 q^{74} -1.00000 q^{75} -1.03923 q^{76} -2.24698 q^{77} +10.6189 q^{79} +2.91185 q^{80} +1.00000 q^{81} -7.17390 q^{82} -12.7192 q^{83} -1.00000 q^{84} +6.49396 q^{85} +4.67994 q^{86} -3.82908 q^{87} +3.04892 q^{88} -6.87263 q^{89} +1.24698 q^{90} -0.0881460 q^{92} -1.71379 q^{93} +2.46011 q^{94} -2.33513 q^{95} +2.46681 q^{96} -11.1414 q^{97} +2.43296 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 3 q^{3} - q^{4} - 3 q^{5} - q^{6} - 2 q^{7} + 3 q^{9} - q^{10} + 3 q^{11} + q^{12} + 4 q^{14} + 3 q^{15} - 5 q^{16} - 10 q^{17} + q^{18} + 6 q^{19} + q^{20} + 2 q^{21} + q^{22} + 5 q^{23} + 3 q^{25} - 3 q^{27} + 3 q^{28} + q^{29} + q^{30} - 3 q^{31} - 4 q^{32} - 3 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} - 5 q^{37} - 5 q^{38} + 22 q^{41} - 4 q^{42} - 16 q^{43} - q^{44} - 3 q^{45} + 4 q^{46} - 12 q^{47} + 5 q^{48} - 15 q^{49} + q^{50} + 10 q^{51} + 5 q^{53} - q^{54} - 3 q^{55} - 7 q^{56} - 6 q^{57} - 16 q^{58} - 3 q^{59} - q^{60} - 10 q^{61} - q^{62} - 2 q^{63} + 4 q^{64} - q^{66} + 20 q^{67} + 8 q^{68} - 5 q^{69} - 4 q^{70} - 6 q^{71} - 2 q^{73} - 18 q^{74} - 3 q^{75} - 16 q^{76} - 2 q^{77} - 2 q^{79} + 5 q^{80} + 3 q^{81} + 12 q^{82} - 27 q^{83} - 3 q^{84} + 10 q^{85} - 10 q^{86} - q^{87} - 4 q^{89} - q^{90} - 4 q^{92} + 3 q^{93} - 18 q^{94} - 6 q^{95} + 4 q^{96} - 9 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.24698 −0.881748 −0.440874 0.897569i \(-0.645331\pi\)
−0.440874 + 0.897569i \(0.645331\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.445042 −0.222521
\(5\) −1.00000 −0.447214
\(6\) 1.24698 0.509077
\(7\) −2.24698 −0.849278 −0.424639 0.905363i \(-0.639599\pi\)
−0.424639 + 0.905363i \(0.639599\pi\)
\(8\) 3.04892 1.07796
\(9\) 1.00000 0.333333
\(10\) 1.24698 0.394330
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0.445042 0.128473
\(13\) 0 0
\(14\) 2.80194 0.748849
\(15\) 1.00000 0.258199
\(16\) −2.91185 −0.727963
\(17\) −6.49396 −1.57502 −0.787508 0.616304i \(-0.788629\pi\)
−0.787508 + 0.616304i \(0.788629\pi\)
\(18\) −1.24698 −0.293916
\(19\) 2.33513 0.535715 0.267857 0.963459i \(-0.413684\pi\)
0.267857 + 0.963459i \(0.413684\pi\)
\(20\) 0.445042 0.0995144
\(21\) 2.24698 0.490331
\(22\) −1.24698 −0.265857
\(23\) 0.198062 0.0412988 0.0206494 0.999787i \(-0.493427\pi\)
0.0206494 + 0.999787i \(0.493427\pi\)
\(24\) −3.04892 −0.622358
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 3.82908 0.711043 0.355522 0.934668i \(-0.384303\pi\)
0.355522 + 0.934668i \(0.384303\pi\)
\(30\) −1.24698 −0.227666
\(31\) 1.71379 0.307806 0.153903 0.988086i \(-0.450816\pi\)
0.153903 + 0.988086i \(0.450816\pi\)
\(32\) −2.46681 −0.436075
\(33\) −1.00000 −0.174078
\(34\) 8.09783 1.38877
\(35\) 2.24698 0.379809
\(36\) −0.445042 −0.0741736
\(37\) 4.54288 0.746844 0.373422 0.927662i \(-0.378184\pi\)
0.373422 + 0.927662i \(0.378184\pi\)
\(38\) −2.91185 −0.472365
\(39\) 0 0
\(40\) −3.04892 −0.482076
\(41\) 5.75302 0.898471 0.449235 0.893413i \(-0.351696\pi\)
0.449235 + 0.893413i \(0.351696\pi\)
\(42\) −2.80194 −0.432348
\(43\) −3.75302 −0.572330 −0.286165 0.958180i \(-0.592381\pi\)
−0.286165 + 0.958180i \(0.592381\pi\)
\(44\) −0.445042 −0.0670926
\(45\) −1.00000 −0.149071
\(46\) −0.246980 −0.0364152
\(47\) −1.97285 −0.287770 −0.143885 0.989594i \(-0.545960\pi\)
−0.143885 + 0.989594i \(0.545960\pi\)
\(48\) 2.91185 0.420290
\(49\) −1.95108 −0.278726
\(50\) −1.24698 −0.176350
\(51\) 6.49396 0.909336
\(52\) 0 0
\(53\) 12.0586 1.65638 0.828188 0.560450i \(-0.189372\pi\)
0.828188 + 0.560450i \(0.189372\pi\)
\(54\) 1.24698 0.169692
\(55\) −1.00000 −0.134840
\(56\) −6.85086 −0.915484
\(57\) −2.33513 −0.309295
\(58\) −4.77479 −0.626961
\(59\) 9.83877 1.28090 0.640450 0.768000i \(-0.278748\pi\)
0.640450 + 0.768000i \(0.278748\pi\)
\(60\) −0.445042 −0.0574547
\(61\) −13.6136 −1.74304 −0.871519 0.490361i \(-0.836865\pi\)
−0.871519 + 0.490361i \(0.836865\pi\)
\(62\) −2.13706 −0.271407
\(63\) −2.24698 −0.283093
\(64\) 8.89977 1.11247
\(65\) 0 0
\(66\) 1.24698 0.153493
\(67\) 7.91185 0.966587 0.483293 0.875458i \(-0.339440\pi\)
0.483293 + 0.875458i \(0.339440\pi\)
\(68\) 2.89008 0.350474
\(69\) −0.198062 −0.0238439
\(70\) −2.80194 −0.334896
\(71\) 7.14675 0.848163 0.424082 0.905624i \(-0.360597\pi\)
0.424082 + 0.905624i \(0.360597\pi\)
\(72\) 3.04892 0.359318
\(73\) 14.3545 1.68007 0.840034 0.542534i \(-0.182535\pi\)
0.840034 + 0.542534i \(0.182535\pi\)
\(74\) −5.66487 −0.658528
\(75\) −1.00000 −0.115470
\(76\) −1.03923 −0.119208
\(77\) −2.24698 −0.256067
\(78\) 0 0
\(79\) 10.6189 1.19472 0.597362 0.801972i \(-0.296216\pi\)
0.597362 + 0.801972i \(0.296216\pi\)
\(80\) 2.91185 0.325555
\(81\) 1.00000 0.111111
\(82\) −7.17390 −0.792225
\(83\) −12.7192 −1.39611 −0.698055 0.716044i \(-0.745951\pi\)
−0.698055 + 0.716044i \(0.745951\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.49396 0.704369
\(86\) 4.67994 0.504651
\(87\) −3.82908 −0.410521
\(88\) 3.04892 0.325016
\(89\) −6.87263 −0.728497 −0.364248 0.931302i \(-0.618674\pi\)
−0.364248 + 0.931302i \(0.618674\pi\)
\(90\) 1.24698 0.131443
\(91\) 0 0
\(92\) −0.0881460 −0.00918986
\(93\) −1.71379 −0.177712
\(94\) 2.46011 0.253741
\(95\) −2.33513 −0.239579
\(96\) 2.46681 0.251768
\(97\) −11.1414 −1.13124 −0.565618 0.824668i \(-0.691362\pi\)
−0.565618 + 0.824668i \(0.691362\pi\)
\(98\) 2.43296 0.245766
\(99\) 1.00000 0.100504
\(100\) −0.445042 −0.0445042
\(101\) −8.19136 −0.815071 −0.407535 0.913189i \(-0.633612\pi\)
−0.407535 + 0.913189i \(0.633612\pi\)
\(102\) −8.09783 −0.801805
\(103\) −9.52111 −0.938142 −0.469071 0.883160i \(-0.655411\pi\)
−0.469071 + 0.883160i \(0.655411\pi\)
\(104\) 0 0
\(105\) −2.24698 −0.219283
\(106\) −15.0368 −1.46051
\(107\) −14.1836 −1.37118 −0.685590 0.727988i \(-0.740456\pi\)
−0.685590 + 0.727988i \(0.740456\pi\)
\(108\) 0.445042 0.0428242
\(109\) −10.1957 −0.976568 −0.488284 0.872685i \(-0.662377\pi\)
−0.488284 + 0.872685i \(0.662377\pi\)
\(110\) 1.24698 0.118895
\(111\) −4.54288 −0.431191
\(112\) 6.54288 0.618244
\(113\) 6.88769 0.647939 0.323970 0.946067i \(-0.394982\pi\)
0.323970 + 0.946067i \(0.394982\pi\)
\(114\) 2.91185 0.272720
\(115\) −0.198062 −0.0184694
\(116\) −1.70410 −0.158222
\(117\) 0 0
\(118\) −12.2687 −1.12943
\(119\) 14.5918 1.33763
\(120\) 3.04892 0.278327
\(121\) −10.0000 −0.909091
\(122\) 16.9758 1.53692
\(123\) −5.75302 −0.518732
\(124\) −0.762709 −0.0684933
\(125\) −1.00000 −0.0894427
\(126\) 2.80194 0.249616
\(127\) −18.4547 −1.63759 −0.818796 0.574084i \(-0.805358\pi\)
−0.818796 + 0.574084i \(0.805358\pi\)
\(128\) −6.16421 −0.544844
\(129\) 3.75302 0.330435
\(130\) 0 0
\(131\) 8.11960 0.709413 0.354707 0.934978i \(-0.384581\pi\)
0.354707 + 0.934978i \(0.384581\pi\)
\(132\) 0.445042 0.0387359
\(133\) −5.24698 −0.454971
\(134\) −9.86592 −0.852286
\(135\) 1.00000 0.0860663
\(136\) −19.7995 −1.69780
\(137\) 21.7778 1.86060 0.930300 0.366798i \(-0.119546\pi\)
0.930300 + 0.366798i \(0.119546\pi\)
\(138\) 0.246980 0.0210243
\(139\) −19.4058 −1.64598 −0.822990 0.568056i \(-0.807696\pi\)
−0.822990 + 0.568056i \(0.807696\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 1.97285 0.166144
\(142\) −8.91185 −0.747866
\(143\) 0 0
\(144\) −2.91185 −0.242654
\(145\) −3.82908 −0.317988
\(146\) −17.8998 −1.48140
\(147\) 1.95108 0.160923
\(148\) −2.02177 −0.166188
\(149\) 11.4668 0.939398 0.469699 0.882827i \(-0.344362\pi\)
0.469699 + 0.882827i \(0.344362\pi\)
\(150\) 1.24698 0.101815
\(151\) 6.17092 0.502182 0.251091 0.967963i \(-0.419211\pi\)
0.251091 + 0.967963i \(0.419211\pi\)
\(152\) 7.11960 0.577476
\(153\) −6.49396 −0.525005
\(154\) 2.80194 0.225787
\(155\) −1.71379 −0.137655
\(156\) 0 0
\(157\) −15.3080 −1.22171 −0.610855 0.791743i \(-0.709174\pi\)
−0.610855 + 0.791743i \(0.709174\pi\)
\(158\) −13.2416 −1.05345
\(159\) −12.0586 −0.956310
\(160\) 2.46681 0.195019
\(161\) −0.445042 −0.0350742
\(162\) −1.24698 −0.0979720
\(163\) −19.5918 −1.53455 −0.767274 0.641320i \(-0.778387\pi\)
−0.767274 + 0.641320i \(0.778387\pi\)
\(164\) −2.56033 −0.199929
\(165\) 1.00000 0.0778499
\(166\) 15.8605 1.23102
\(167\) 9.05429 0.700642 0.350321 0.936630i \(-0.386072\pi\)
0.350321 + 0.936630i \(0.386072\pi\)
\(168\) 6.85086 0.528555
\(169\) 0 0
\(170\) −8.09783 −0.621076
\(171\) 2.33513 0.178572
\(172\) 1.67025 0.127355
\(173\) −13.4547 −1.02294 −0.511472 0.859300i \(-0.670900\pi\)
−0.511472 + 0.859300i \(0.670900\pi\)
\(174\) 4.77479 0.361976
\(175\) −2.24698 −0.169856
\(176\) −2.91185 −0.219489
\(177\) −9.83877 −0.739528
\(178\) 8.57002 0.642350
\(179\) 21.3980 1.59936 0.799682 0.600423i \(-0.205001\pi\)
0.799682 + 0.600423i \(0.205001\pi\)
\(180\) 0.445042 0.0331715
\(181\) −14.1032 −1.04828 −0.524142 0.851631i \(-0.675614\pi\)
−0.524142 + 0.851631i \(0.675614\pi\)
\(182\) 0 0
\(183\) 13.6136 1.00634
\(184\) 0.603875 0.0445183
\(185\) −4.54288 −0.333999
\(186\) 2.13706 0.156697
\(187\) −6.49396 −0.474885
\(188\) 0.878002 0.0640349
\(189\) 2.24698 0.163444
\(190\) 2.91185 0.211248
\(191\) 7.19136 0.520348 0.260174 0.965562i \(-0.416220\pi\)
0.260174 + 0.965562i \(0.416220\pi\)
\(192\) −8.89977 −0.642286
\(193\) −11.3013 −0.813483 −0.406742 0.913543i \(-0.633335\pi\)
−0.406742 + 0.913543i \(0.633335\pi\)
\(194\) 13.8931 0.997464
\(195\) 0 0
\(196\) 0.868313 0.0620224
\(197\) −20.5066 −1.46104 −0.730519 0.682893i \(-0.760722\pi\)
−0.730519 + 0.682893i \(0.760722\pi\)
\(198\) −1.24698 −0.0886190
\(199\) 19.9681 1.41550 0.707749 0.706464i \(-0.249711\pi\)
0.707749 + 0.706464i \(0.249711\pi\)
\(200\) 3.04892 0.215591
\(201\) −7.91185 −0.558059
\(202\) 10.2145 0.718687
\(203\) −8.60388 −0.603874
\(204\) −2.89008 −0.202346
\(205\) −5.75302 −0.401808
\(206\) 11.8726 0.827205
\(207\) 0.198062 0.0137663
\(208\) 0 0
\(209\) 2.33513 0.161524
\(210\) 2.80194 0.193352
\(211\) −9.74632 −0.670964 −0.335482 0.942047i \(-0.608899\pi\)
−0.335482 + 0.942047i \(0.608899\pi\)
\(212\) −5.36658 −0.368579
\(213\) −7.14675 −0.489687
\(214\) 17.6866 1.20903
\(215\) 3.75302 0.255954
\(216\) −3.04892 −0.207453
\(217\) −3.85086 −0.261413
\(218\) 12.7138 0.861087
\(219\) −14.3545 −0.969987
\(220\) 0.445042 0.0300047
\(221\) 0 0
\(222\) 5.66487 0.380201
\(223\) −24.5351 −1.64299 −0.821496 0.570214i \(-0.806860\pi\)
−0.821496 + 0.570214i \(0.806860\pi\)
\(224\) 5.54288 0.370349
\(225\) 1.00000 0.0666667
\(226\) −8.58881 −0.571319
\(227\) 0.0217703 0.00144494 0.000722472 1.00000i \(-0.499770\pi\)
0.000722472 1.00000i \(0.499770\pi\)
\(228\) 1.03923 0.0688246
\(229\) 7.36898 0.486956 0.243478 0.969906i \(-0.421712\pi\)
0.243478 + 0.969906i \(0.421712\pi\)
\(230\) 0.246980 0.0162854
\(231\) 2.24698 0.147840
\(232\) 11.6746 0.766473
\(233\) −8.10992 −0.531298 −0.265649 0.964070i \(-0.585586\pi\)
−0.265649 + 0.964070i \(0.585586\pi\)
\(234\) 0 0
\(235\) 1.97285 0.128695
\(236\) −4.37867 −0.285027
\(237\) −10.6189 −0.689774
\(238\) −18.1957 −1.17945
\(239\) −6.38165 −0.412795 −0.206397 0.978468i \(-0.566174\pi\)
−0.206397 + 0.978468i \(0.566174\pi\)
\(240\) −2.91185 −0.187959
\(241\) −1.30127 −0.0838224 −0.0419112 0.999121i \(-0.513345\pi\)
−0.0419112 + 0.999121i \(0.513345\pi\)
\(242\) 12.4698 0.801589
\(243\) −1.00000 −0.0641500
\(244\) 6.05861 0.387863
\(245\) 1.95108 0.124650
\(246\) 7.17390 0.457391
\(247\) 0 0
\(248\) 5.22521 0.331801
\(249\) 12.7192 0.806045
\(250\) 1.24698 0.0788659
\(251\) 9.51142 0.600355 0.300178 0.953883i \(-0.402954\pi\)
0.300178 + 0.953883i \(0.402954\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0.198062 0.0124521
\(254\) 23.0127 1.44394
\(255\) −6.49396 −0.406668
\(256\) −10.1129 −0.632056
\(257\) 7.38942 0.460939 0.230470 0.973080i \(-0.425974\pi\)
0.230470 + 0.973080i \(0.425974\pi\)
\(258\) −4.67994 −0.291360
\(259\) −10.2078 −0.634279
\(260\) 0 0
\(261\) 3.82908 0.237014
\(262\) −10.1250 −0.625523
\(263\) −3.06398 −0.188933 −0.0944666 0.995528i \(-0.530115\pi\)
−0.0944666 + 0.995528i \(0.530115\pi\)
\(264\) −3.04892 −0.187648
\(265\) −12.0586 −0.740754
\(266\) 6.54288 0.401170
\(267\) 6.87263 0.420598
\(268\) −3.52111 −0.215086
\(269\) −18.2687 −1.11387 −0.556933 0.830558i \(-0.688022\pi\)
−0.556933 + 0.830558i \(0.688022\pi\)
\(270\) −1.24698 −0.0758888
\(271\) −4.04892 −0.245954 −0.122977 0.992410i \(-0.539244\pi\)
−0.122977 + 0.992410i \(0.539244\pi\)
\(272\) 18.9095 1.14655
\(273\) 0 0
\(274\) −27.1564 −1.64058
\(275\) 1.00000 0.0603023
\(276\) 0.0881460 0.00530577
\(277\) 28.1608 1.69202 0.846008 0.533170i \(-0.179000\pi\)
0.846008 + 0.533170i \(0.179000\pi\)
\(278\) 24.1987 1.45134
\(279\) 1.71379 0.102602
\(280\) 6.85086 0.409417
\(281\) −24.3937 −1.45521 −0.727604 0.685997i \(-0.759366\pi\)
−0.727604 + 0.685997i \(0.759366\pi\)
\(282\) −2.46011 −0.146497
\(283\) −7.16315 −0.425805 −0.212902 0.977073i \(-0.568292\pi\)
−0.212902 + 0.977073i \(0.568292\pi\)
\(284\) −3.18060 −0.188734
\(285\) 2.33513 0.138321
\(286\) 0 0
\(287\) −12.9269 −0.763052
\(288\) −2.46681 −0.145358
\(289\) 25.1715 1.48068
\(290\) 4.77479 0.280385
\(291\) 11.1414 0.653119
\(292\) −6.38835 −0.373850
\(293\) −5.01075 −0.292731 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(294\) −2.43296 −0.141893
\(295\) −9.83877 −0.572836
\(296\) 13.8509 0.805065
\(297\) −1.00000 −0.0580259
\(298\) −14.2989 −0.828312
\(299\) 0 0
\(300\) 0.445042 0.0256945
\(301\) 8.43296 0.486068
\(302\) −7.69501 −0.442798
\(303\) 8.19136 0.470581
\(304\) −6.79954 −0.389981
\(305\) 13.6136 0.779510
\(306\) 8.09783 0.462922
\(307\) −29.8213 −1.70199 −0.850996 0.525172i \(-0.824001\pi\)
−0.850996 + 0.525172i \(0.824001\pi\)
\(308\) 1.00000 0.0569803
\(309\) 9.52111 0.541637
\(310\) 2.13706 0.121377
\(311\) −6.22952 −0.353244 −0.176622 0.984279i \(-0.556517\pi\)
−0.176622 + 0.984279i \(0.556517\pi\)
\(312\) 0 0
\(313\) 8.22713 0.465025 0.232512 0.972593i \(-0.425305\pi\)
0.232512 + 0.972593i \(0.425305\pi\)
\(314\) 19.0887 1.07724
\(315\) 2.24698 0.126603
\(316\) −4.72587 −0.265851
\(317\) −20.2010 −1.13460 −0.567302 0.823510i \(-0.692013\pi\)
−0.567302 + 0.823510i \(0.692013\pi\)
\(318\) 15.0368 0.843224
\(319\) 3.82908 0.214388
\(320\) −8.89977 −0.497512
\(321\) 14.1836 0.791651
\(322\) 0.554958 0.0309266
\(323\) −15.1642 −0.843759
\(324\) −0.445042 −0.0247245
\(325\) 0 0
\(326\) 24.4306 1.35308
\(327\) 10.1957 0.563822
\(328\) 17.5405 0.968511
\(329\) 4.43296 0.244397
\(330\) −1.24698 −0.0686440
\(331\) −11.2771 −0.619846 −0.309923 0.950762i \(-0.600303\pi\)
−0.309923 + 0.950762i \(0.600303\pi\)
\(332\) 5.66056 0.310664
\(333\) 4.54288 0.248948
\(334\) −11.2905 −0.617790
\(335\) −7.91185 −0.432271
\(336\) −6.54288 −0.356943
\(337\) −21.1075 −1.14980 −0.574900 0.818224i \(-0.694959\pi\)
−0.574900 + 0.818224i \(0.694959\pi\)
\(338\) 0 0
\(339\) −6.88769 −0.374088
\(340\) −2.89008 −0.156737
\(341\) 1.71379 0.0928070
\(342\) −2.91185 −0.157455
\(343\) 20.1129 1.08599
\(344\) −11.4426 −0.616946
\(345\) 0.198062 0.0106633
\(346\) 16.7778 0.901979
\(347\) 21.3744 1.14744 0.573718 0.819053i \(-0.305501\pi\)
0.573718 + 0.819053i \(0.305501\pi\)
\(348\) 1.70410 0.0913495
\(349\) −4.31527 −0.230991 −0.115496 0.993308i \(-0.536846\pi\)
−0.115496 + 0.993308i \(0.536846\pi\)
\(350\) 2.80194 0.149770
\(351\) 0 0
\(352\) −2.46681 −0.131482
\(353\) −10.2198 −0.543947 −0.271973 0.962305i \(-0.587676\pi\)
−0.271973 + 0.962305i \(0.587676\pi\)
\(354\) 12.2687 0.652077
\(355\) −7.14675 −0.379310
\(356\) 3.05861 0.162106
\(357\) −14.5918 −0.772280
\(358\) −26.6829 −1.41024
\(359\) −8.71678 −0.460054 −0.230027 0.973184i \(-0.573881\pi\)
−0.230027 + 0.973184i \(0.573881\pi\)
\(360\) −3.04892 −0.160692
\(361\) −13.5472 −0.713010
\(362\) 17.5864 0.924322
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −14.3545 −0.751349
\(366\) −16.9758 −0.887341
\(367\) −5.37973 −0.280820 −0.140410 0.990093i \(-0.544842\pi\)
−0.140410 + 0.990093i \(0.544842\pi\)
\(368\) −0.576728 −0.0300640
\(369\) 5.75302 0.299490
\(370\) 5.66487 0.294503
\(371\) −27.0954 −1.40673
\(372\) 0.762709 0.0395446
\(373\) −12.6843 −0.656766 −0.328383 0.944545i \(-0.606504\pi\)
−0.328383 + 0.944545i \(0.606504\pi\)
\(374\) 8.09783 0.418729
\(375\) 1.00000 0.0516398
\(376\) −6.01507 −0.310203
\(377\) 0 0
\(378\) −2.80194 −0.144116
\(379\) 7.99223 0.410533 0.205267 0.978706i \(-0.434194\pi\)
0.205267 + 0.978706i \(0.434194\pi\)
\(380\) 1.03923 0.0533113
\(381\) 18.4547 0.945464
\(382\) −8.96748 −0.458816
\(383\) −0.763774 −0.0390270 −0.0195135 0.999810i \(-0.506212\pi\)
−0.0195135 + 0.999810i \(0.506212\pi\)
\(384\) 6.16421 0.314566
\(385\) 2.24698 0.114517
\(386\) 14.0925 0.717287
\(387\) −3.75302 −0.190777
\(388\) 4.95838 0.251724
\(389\) −28.8799 −1.46427 −0.732135 0.681159i \(-0.761476\pi\)
−0.732135 + 0.681159i \(0.761476\pi\)
\(390\) 0 0
\(391\) −1.28621 −0.0650463
\(392\) −5.94869 −0.300454
\(393\) −8.11960 −0.409580
\(394\) 25.5714 1.28827
\(395\) −10.6189 −0.534297
\(396\) −0.445042 −0.0223642
\(397\) −33.1347 −1.66298 −0.831491 0.555539i \(-0.812512\pi\)
−0.831491 + 0.555539i \(0.812512\pi\)
\(398\) −24.8998 −1.24811
\(399\) 5.24698 0.262678
\(400\) −2.91185 −0.145593
\(401\) −11.0435 −0.551488 −0.275744 0.961231i \(-0.588924\pi\)
−0.275744 + 0.961231i \(0.588924\pi\)
\(402\) 9.86592 0.492067
\(403\) 0 0
\(404\) 3.64550 0.181370
\(405\) −1.00000 −0.0496904
\(406\) 10.7289 0.532464
\(407\) 4.54288 0.225182
\(408\) 19.7995 0.980224
\(409\) 33.3424 1.64868 0.824338 0.566097i \(-0.191547\pi\)
0.824338 + 0.566097i \(0.191547\pi\)
\(410\) 7.17390 0.354294
\(411\) −21.7778 −1.07422
\(412\) 4.23729 0.208756
\(413\) −22.1075 −1.08784
\(414\) −0.246980 −0.0121384
\(415\) 12.7192 0.624359
\(416\) 0 0
\(417\) 19.4058 0.950307
\(418\) −2.91185 −0.142423
\(419\) 39.9071 1.94959 0.974794 0.223107i \(-0.0716200\pi\)
0.974794 + 0.223107i \(0.0716200\pi\)
\(420\) 1.00000 0.0487950
\(421\) 9.41657 0.458935 0.229468 0.973316i \(-0.426301\pi\)
0.229468 + 0.973316i \(0.426301\pi\)
\(422\) 12.1535 0.591621
\(423\) −1.97285 −0.0959234
\(424\) 36.7657 1.78550
\(425\) −6.49396 −0.315003
\(426\) 8.91185 0.431781
\(427\) 30.5894 1.48032
\(428\) 6.31229 0.305116
\(429\) 0 0
\(430\) −4.67994 −0.225687
\(431\) 25.9976 1.25226 0.626130 0.779718i \(-0.284638\pi\)
0.626130 + 0.779718i \(0.284638\pi\)
\(432\) 2.91185 0.140097
\(433\) 6.93661 0.333352 0.166676 0.986012i \(-0.446697\pi\)
0.166676 + 0.986012i \(0.446697\pi\)
\(434\) 4.80194 0.230500
\(435\) 3.82908 0.183591
\(436\) 4.53750 0.217307
\(437\) 0.462500 0.0221244
\(438\) 17.8998 0.855284
\(439\) 18.6775 0.891431 0.445715 0.895175i \(-0.352949\pi\)
0.445715 + 0.895175i \(0.352949\pi\)
\(440\) −3.04892 −0.145351
\(441\) −1.95108 −0.0929087
\(442\) 0 0
\(443\) 36.9245 1.75434 0.877169 0.480183i \(-0.159430\pi\)
0.877169 + 0.480183i \(0.159430\pi\)
\(444\) 2.02177 0.0959490
\(445\) 6.87263 0.325794
\(446\) 30.5948 1.44870
\(447\) −11.4668 −0.542362
\(448\) −19.9976 −0.944798
\(449\) −15.3884 −0.726221 −0.363111 0.931746i \(-0.618285\pi\)
−0.363111 + 0.931746i \(0.618285\pi\)
\(450\) −1.24698 −0.0587832
\(451\) 5.75302 0.270899
\(452\) −3.06531 −0.144180
\(453\) −6.17092 −0.289935
\(454\) −0.0271471 −0.00127408
\(455\) 0 0
\(456\) −7.11960 −0.333406
\(457\) −25.2946 −1.18323 −0.591615 0.806221i \(-0.701509\pi\)
−0.591615 + 0.806221i \(0.701509\pi\)
\(458\) −9.18896 −0.429372
\(459\) 6.49396 0.303112
\(460\) 0.0881460 0.00410983
\(461\) −8.68664 −0.404577 −0.202289 0.979326i \(-0.564838\pi\)
−0.202289 + 0.979326i \(0.564838\pi\)
\(462\) −2.80194 −0.130358
\(463\) 1.73663 0.0807079 0.0403539 0.999185i \(-0.487151\pi\)
0.0403539 + 0.999185i \(0.487151\pi\)
\(464\) −11.1497 −0.517613
\(465\) 1.71379 0.0794752
\(466\) 10.1129 0.468471
\(467\) 22.3177 1.03274 0.516369 0.856366i \(-0.327283\pi\)
0.516369 + 0.856366i \(0.327283\pi\)
\(468\) 0 0
\(469\) −17.7778 −0.820901
\(470\) −2.46011 −0.113476
\(471\) 15.3080 0.705354
\(472\) 29.9976 1.38075
\(473\) −3.75302 −0.172564
\(474\) 13.2416 0.608207
\(475\) 2.33513 0.107143
\(476\) −6.49396 −0.297650
\(477\) 12.0586 0.552126
\(478\) 7.95779 0.363981
\(479\) −26.6765 −1.21888 −0.609440 0.792832i \(-0.708606\pi\)
−0.609440 + 0.792832i \(0.708606\pi\)
\(480\) −2.46681 −0.112594
\(481\) 0 0
\(482\) 1.62266 0.0739102
\(483\) 0.445042 0.0202501
\(484\) 4.45042 0.202292
\(485\) 11.1414 0.505904
\(486\) 1.24698 0.0565641
\(487\) 8.61655 0.390453 0.195227 0.980758i \(-0.437456\pi\)
0.195227 + 0.980758i \(0.437456\pi\)
\(488\) −41.5066 −1.87892
\(489\) 19.5918 0.885972
\(490\) −2.43296 −0.109910
\(491\) −0.992230 −0.0447787 −0.0223894 0.999749i \(-0.507127\pi\)
−0.0223894 + 0.999749i \(0.507127\pi\)
\(492\) 2.56033 0.115429
\(493\) −24.8659 −1.11990
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) −4.99031 −0.224072
\(497\) −16.0586 −0.720327
\(498\) −15.8605 −0.710728
\(499\) 16.7138 0.748212 0.374106 0.927386i \(-0.377950\pi\)
0.374106 + 0.927386i \(0.377950\pi\)
\(500\) 0.445042 0.0199029
\(501\) −9.05429 −0.404516
\(502\) −11.8605 −0.529362
\(503\) 30.6547 1.36683 0.683413 0.730032i \(-0.260495\pi\)
0.683413 + 0.730032i \(0.260495\pi\)
\(504\) −6.85086 −0.305161
\(505\) 8.19136 0.364511
\(506\) −0.246980 −0.0109796
\(507\) 0 0
\(508\) 8.21313 0.364399
\(509\) 25.9506 1.15024 0.575120 0.818069i \(-0.304955\pi\)
0.575120 + 0.818069i \(0.304955\pi\)
\(510\) 8.09783 0.358578
\(511\) −32.2543 −1.42685
\(512\) 24.9390 1.10216
\(513\) −2.33513 −0.103098
\(514\) −9.21446 −0.406432
\(515\) 9.52111 0.419550
\(516\) −1.67025 −0.0735287
\(517\) −1.97285 −0.0867660
\(518\) 12.7289 0.559274
\(519\) 13.4547 0.590597
\(520\) 0 0
\(521\) −12.3375 −0.540516 −0.270258 0.962788i \(-0.587109\pi\)
−0.270258 + 0.962788i \(0.587109\pi\)
\(522\) −4.77479 −0.208987
\(523\) −20.7748 −0.908418 −0.454209 0.890895i \(-0.650078\pi\)
−0.454209 + 0.890895i \(0.650078\pi\)
\(524\) −3.61356 −0.157859
\(525\) 2.24698 0.0980662
\(526\) 3.82072 0.166591
\(527\) −11.1293 −0.484800
\(528\) 2.91185 0.126722
\(529\) −22.9608 −0.998294
\(530\) 15.0368 0.653158
\(531\) 9.83877 0.426966
\(532\) 2.33513 0.101241
\(533\) 0 0
\(534\) −8.57002 −0.370861
\(535\) 14.1836 0.613210
\(536\) 24.1226 1.04194
\(537\) −21.3980 −0.923394
\(538\) 22.7808 0.982148
\(539\) −1.95108 −0.0840391
\(540\) −0.445042 −0.0191516
\(541\) 45.5797 1.95962 0.979812 0.199919i \(-0.0640679\pi\)
0.979812 + 0.199919i \(0.0640679\pi\)
\(542\) 5.04892 0.216870
\(543\) 14.1032 0.605227
\(544\) 16.0194 0.686825
\(545\) 10.1957 0.436734
\(546\) 0 0
\(547\) −14.0194 −0.599425 −0.299713 0.954030i \(-0.596891\pi\)
−0.299713 + 0.954030i \(0.596891\pi\)
\(548\) −9.69202 −0.414023
\(549\) −13.6136 −0.581013
\(550\) −1.24698 −0.0531714
\(551\) 8.94139 0.380916
\(552\) −0.603875 −0.0257026
\(553\) −23.8605 −1.01465
\(554\) −35.1159 −1.49193
\(555\) 4.54288 0.192834
\(556\) 8.63640 0.366265
\(557\) −21.2591 −0.900775 −0.450388 0.892833i \(-0.648714\pi\)
−0.450388 + 0.892833i \(0.648714\pi\)
\(558\) −2.13706 −0.0904691
\(559\) 0 0
\(560\) −6.54288 −0.276487
\(561\) 6.49396 0.274175
\(562\) 30.4185 1.28313
\(563\) 13.5198 0.569791 0.284895 0.958559i \(-0.408041\pi\)
0.284895 + 0.958559i \(0.408041\pi\)
\(564\) −0.878002 −0.0369706
\(565\) −6.88769 −0.289767
\(566\) 8.93230 0.375452
\(567\) −2.24698 −0.0943643
\(568\) 21.7899 0.914282
\(569\) −30.9202 −1.29624 −0.648121 0.761537i \(-0.724445\pi\)
−0.648121 + 0.761537i \(0.724445\pi\)
\(570\) −2.91185 −0.121964
\(571\) 5.33645 0.223324 0.111662 0.993746i \(-0.464383\pi\)
0.111662 + 0.993746i \(0.464383\pi\)
\(572\) 0 0
\(573\) −7.19136 −0.300423
\(574\) 16.1196 0.672819
\(575\) 0.198062 0.00825977
\(576\) 8.89977 0.370824
\(577\) 17.3889 0.723911 0.361956 0.932195i \(-0.382109\pi\)
0.361956 + 0.932195i \(0.382109\pi\)
\(578\) −31.3884 −1.30558
\(579\) 11.3013 0.469665
\(580\) 1.70410 0.0707590
\(581\) 28.5797 1.18569
\(582\) −13.8931 −0.575886
\(583\) 12.0586 0.499416
\(584\) 43.7657 1.81104
\(585\) 0 0
\(586\) 6.24831 0.258115
\(587\) 21.4668 0.886030 0.443015 0.896514i \(-0.353909\pi\)
0.443015 + 0.896514i \(0.353909\pi\)
\(588\) −0.868313 −0.0358086
\(589\) 4.00192 0.164896
\(590\) 12.2687 0.505096
\(591\) 20.5066 0.843530
\(592\) −13.2282 −0.543675
\(593\) −4.27605 −0.175596 −0.0877981 0.996138i \(-0.527983\pi\)
−0.0877981 + 0.996138i \(0.527983\pi\)
\(594\) 1.24698 0.0511642
\(595\) −14.5918 −0.598205
\(596\) −5.10321 −0.209036
\(597\) −19.9681 −0.817239
\(598\) 0 0
\(599\) −23.2180 −0.948663 −0.474332 0.880346i \(-0.657310\pi\)
−0.474332 + 0.880346i \(0.657310\pi\)
\(600\) −3.04892 −0.124472
\(601\) −32.2704 −1.31634 −0.658169 0.752871i \(-0.728669\pi\)
−0.658169 + 0.752871i \(0.728669\pi\)
\(602\) −10.5157 −0.428589
\(603\) 7.91185 0.322196
\(604\) −2.74632 −0.111746
\(605\) 10.0000 0.406558
\(606\) −10.2145 −0.414934
\(607\) 13.8920 0.563859 0.281929 0.959435i \(-0.409026\pi\)
0.281929 + 0.959435i \(0.409026\pi\)
\(608\) −5.76032 −0.233612
\(609\) 8.60388 0.348647
\(610\) −16.9758 −0.687332
\(611\) 0 0
\(612\) 2.89008 0.116825
\(613\) 27.6668 1.11745 0.558726 0.829353i \(-0.311290\pi\)
0.558726 + 0.829353i \(0.311290\pi\)
\(614\) 37.1866 1.50073
\(615\) 5.75302 0.231984
\(616\) −6.85086 −0.276029
\(617\) 10.3709 0.417517 0.208758 0.977967i \(-0.433058\pi\)
0.208758 + 0.977967i \(0.433058\pi\)
\(618\) −11.8726 −0.477587
\(619\) −43.1473 −1.73424 −0.867119 0.498101i \(-0.834031\pi\)
−0.867119 + 0.498101i \(0.834031\pi\)
\(620\) 0.762709 0.0306311
\(621\) −0.198062 −0.00794796
\(622\) 7.76809 0.311472
\(623\) 15.4426 0.618697
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.2591 −0.410035
\(627\) −2.33513 −0.0932559
\(628\) 6.81269 0.271856
\(629\) −29.5013 −1.17629
\(630\) −2.80194 −0.111632
\(631\) −22.6987 −0.903622 −0.451811 0.892114i \(-0.649222\pi\)
−0.451811 + 0.892114i \(0.649222\pi\)
\(632\) 32.3763 1.28786
\(633\) 9.74632 0.387381
\(634\) 25.1903 1.00043
\(635\) 18.4547 0.732354
\(636\) 5.36658 0.212799
\(637\) 0 0
\(638\) −4.77479 −0.189036
\(639\) 7.14675 0.282721
\(640\) 6.16421 0.243662
\(641\) −16.1075 −0.636209 −0.318104 0.948056i \(-0.603046\pi\)
−0.318104 + 0.948056i \(0.603046\pi\)
\(642\) −17.6866 −0.698036
\(643\) 32.8732 1.29639 0.648197 0.761473i \(-0.275524\pi\)
0.648197 + 0.761473i \(0.275524\pi\)
\(644\) 0.198062 0.00780475
\(645\) −3.75302 −0.147775
\(646\) 18.9095 0.743983
\(647\) −10.2814 −0.404204 −0.202102 0.979364i \(-0.564777\pi\)
−0.202102 + 0.979364i \(0.564777\pi\)
\(648\) 3.04892 0.119773
\(649\) 9.83877 0.386206
\(650\) 0 0
\(651\) 3.85086 0.150927
\(652\) 8.71917 0.341469
\(653\) 14.3575 0.561852 0.280926 0.959729i \(-0.409358\pi\)
0.280926 + 0.959729i \(0.409358\pi\)
\(654\) −12.7138 −0.497149
\(655\) −8.11960 −0.317259
\(656\) −16.7520 −0.654054
\(657\) 14.3545 0.560022
\(658\) −5.52781 −0.215497
\(659\) −16.0258 −0.624277 −0.312139 0.950037i \(-0.601045\pi\)
−0.312139 + 0.950037i \(0.601045\pi\)
\(660\) −0.445042 −0.0173232
\(661\) 27.4282 1.06683 0.533416 0.845853i \(-0.320908\pi\)
0.533416 + 0.845853i \(0.320908\pi\)
\(662\) 14.0623 0.546548
\(663\) 0 0
\(664\) −38.7797 −1.50494
\(665\) 5.24698 0.203469
\(666\) −5.66487 −0.219509
\(667\) 0.758397 0.0293653
\(668\) −4.02954 −0.155908
\(669\) 24.5351 0.948582
\(670\) 9.86592 0.381154
\(671\) −13.6136 −0.525546
\(672\) −5.54288 −0.213821
\(673\) 1.10321 0.0425257 0.0212628 0.999774i \(-0.493231\pi\)
0.0212628 + 0.999774i \(0.493231\pi\)
\(674\) 26.3207 1.01383
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −31.7754 −1.22123 −0.610614 0.791929i \(-0.709077\pi\)
−0.610614 + 0.791929i \(0.709077\pi\)
\(678\) 8.58881 0.329851
\(679\) 25.0344 0.960734
\(680\) 19.7995 0.759278
\(681\) −0.0217703 −0.000834238 0
\(682\) −2.13706 −0.0818324
\(683\) 4.45606 0.170506 0.0852532 0.996359i \(-0.472830\pi\)
0.0852532 + 0.996359i \(0.472830\pi\)
\(684\) −1.03923 −0.0397359
\(685\) −21.7778 −0.832086
\(686\) −25.0804 −0.957573
\(687\) −7.36898 −0.281144
\(688\) 10.9282 0.416636
\(689\) 0 0
\(690\) −0.246980 −0.00940235
\(691\) −14.1438 −0.538054 −0.269027 0.963133i \(-0.586702\pi\)
−0.269027 + 0.963133i \(0.586702\pi\)
\(692\) 5.98792 0.227627
\(693\) −2.24698 −0.0853557
\(694\) −26.6534 −1.01175
\(695\) 19.4058 0.736104
\(696\) −11.6746 −0.442523
\(697\) −37.3599 −1.41511
\(698\) 5.38106 0.203676
\(699\) 8.10992 0.306745
\(700\) 1.00000 0.0377964
\(701\) −4.74632 −0.179266 −0.0896329 0.995975i \(-0.528569\pi\)
−0.0896329 + 0.995975i \(0.528569\pi\)
\(702\) 0 0
\(703\) 10.6082 0.400095
\(704\) 8.89977 0.335423
\(705\) −1.97285 −0.0743019
\(706\) 12.7439 0.479624
\(707\) 18.4058 0.692222
\(708\) 4.37867 0.164560
\(709\) 15.4112 0.578779 0.289390 0.957211i \(-0.406548\pi\)
0.289390 + 0.957211i \(0.406548\pi\)
\(710\) 8.91185 0.334456
\(711\) 10.6189 0.398241
\(712\) −20.9541 −0.785287
\(713\) 0.339437 0.0127120
\(714\) 18.1957 0.680956
\(715\) 0 0
\(716\) −9.52303 −0.355892
\(717\) 6.38165 0.238327
\(718\) 10.8696 0.405651
\(719\) 11.8194 0.440789 0.220395 0.975411i \(-0.429266\pi\)
0.220395 + 0.975411i \(0.429266\pi\)
\(720\) 2.91185 0.108518
\(721\) 21.3937 0.796744
\(722\) 16.8931 0.628695
\(723\) 1.30127 0.0483949
\(724\) 6.27652 0.233265
\(725\) 3.82908 0.142209
\(726\) −12.4698 −0.462798
\(727\) −21.9675 −0.814729 −0.407364 0.913266i \(-0.633552\pi\)
−0.407364 + 0.913266i \(0.633552\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 17.8998 0.662500
\(731\) 24.3720 0.901430
\(732\) −6.05861 −0.223933
\(733\) −36.0586 −1.33186 −0.665928 0.746016i \(-0.731964\pi\)
−0.665928 + 0.746016i \(0.731964\pi\)
\(734\) 6.70841 0.247612
\(735\) −1.95108 −0.0719668
\(736\) −0.488582 −0.0180094
\(737\) 7.91185 0.291437
\(738\) −7.17390 −0.264075
\(739\) 36.9842 1.36049 0.680243 0.732987i \(-0.261874\pi\)
0.680243 + 0.732987i \(0.261874\pi\)
\(740\) 2.02177 0.0743218
\(741\) 0 0
\(742\) 33.7875 1.24038
\(743\) 1.42865 0.0524120 0.0262060 0.999657i \(-0.491657\pi\)
0.0262060 + 0.999657i \(0.491657\pi\)
\(744\) −5.22521 −0.191565
\(745\) −11.4668 −0.420111
\(746\) 15.8170 0.579102
\(747\) −12.7192 −0.465370
\(748\) 2.89008 0.105672
\(749\) 31.8702 1.16451
\(750\) −1.24698 −0.0455333
\(751\) −37.9874 −1.38618 −0.693091 0.720850i \(-0.743752\pi\)
−0.693091 + 0.720850i \(0.743752\pi\)
\(752\) 5.74466 0.209486
\(753\) −9.51142 −0.346615
\(754\) 0 0
\(755\) −6.17092 −0.224583
\(756\) −1.00000 −0.0363696
\(757\) 30.6805 1.11510 0.557551 0.830142i \(-0.311741\pi\)
0.557551 + 0.830142i \(0.311741\pi\)
\(758\) −9.96615 −0.361987
\(759\) −0.198062 −0.00718920
\(760\) −7.11960 −0.258255
\(761\) −52.1213 −1.88939 −0.944697 0.327944i \(-0.893644\pi\)
−0.944697 + 0.327944i \(0.893644\pi\)
\(762\) −23.0127 −0.833661
\(763\) 22.9095 0.829378
\(764\) −3.20046 −0.115788
\(765\) 6.49396 0.234790
\(766\) 0.952411 0.0344120
\(767\) 0 0
\(768\) 10.1129 0.364918
\(769\) 30.4222 1.09705 0.548526 0.836133i \(-0.315189\pi\)
0.548526 + 0.836133i \(0.315189\pi\)
\(770\) −2.80194 −0.100975
\(771\) −7.38942 −0.266123
\(772\) 5.02954 0.181017
\(773\) −23.1118 −0.831275 −0.415637 0.909530i \(-0.636441\pi\)
−0.415637 + 0.909530i \(0.636441\pi\)
\(774\) 4.67994 0.168217
\(775\) 1.71379 0.0615612
\(776\) −33.9691 −1.21942
\(777\) 10.2078 0.366201
\(778\) 36.0127 1.29112
\(779\) 13.4340 0.481324
\(780\) 0 0
\(781\) 7.14675 0.255731
\(782\) 1.60388 0.0573545
\(783\) −3.82908 −0.136840
\(784\) 5.68127 0.202902
\(785\) 15.3080 0.546365
\(786\) 10.1250 0.361146
\(787\) −16.2929 −0.580780 −0.290390 0.956908i \(-0.593785\pi\)
−0.290390 + 0.956908i \(0.593785\pi\)
\(788\) 9.12631 0.325111
\(789\) 3.06398 0.109081
\(790\) 13.2416 0.471115
\(791\) −15.4765 −0.550281
\(792\) 3.04892 0.108339
\(793\) 0 0
\(794\) 41.3183 1.46633
\(795\) 12.0586 0.427675
\(796\) −8.88663 −0.314978
\(797\) 18.2573 0.646705 0.323353 0.946279i \(-0.395190\pi\)
0.323353 + 0.946279i \(0.395190\pi\)
\(798\) −6.54288 −0.231615
\(799\) 12.8116 0.453243
\(800\) −2.46681 −0.0872150
\(801\) −6.87263 −0.242832
\(802\) 13.7711 0.486273
\(803\) 14.3545 0.506559
\(804\) 3.52111 0.124180
\(805\) 0.445042 0.0156857
\(806\) 0 0
\(807\) 18.2687 0.643090
\(808\) −24.9748 −0.878609
\(809\) −24.7259 −0.869315 −0.434658 0.900596i \(-0.643131\pi\)
−0.434658 + 0.900596i \(0.643131\pi\)
\(810\) 1.24698 0.0438144
\(811\) 3.05429 0.107251 0.0536254 0.998561i \(-0.482922\pi\)
0.0536254 + 0.998561i \(0.482922\pi\)
\(812\) 3.82908 0.134375
\(813\) 4.04892 0.142002
\(814\) −5.66487 −0.198554
\(815\) 19.5918 0.686271
\(816\) −18.9095 −0.661964
\(817\) −8.76377 −0.306606
\(818\) −41.5773 −1.45372
\(819\) 0 0
\(820\) 2.56033 0.0894108
\(821\) −17.7646 −0.619990 −0.309995 0.950738i \(-0.600327\pi\)
−0.309995 + 0.950738i \(0.600327\pi\)
\(822\) 27.1564 0.947190
\(823\) 32.1333 1.12010 0.560049 0.828460i \(-0.310782\pi\)
0.560049 + 0.828460i \(0.310782\pi\)
\(824\) −29.0291 −1.01128
\(825\) −1.00000 −0.0348155
\(826\) 27.5676 0.959201
\(827\) 24.5724 0.854467 0.427233 0.904141i \(-0.359488\pi\)
0.427233 + 0.904141i \(0.359488\pi\)
\(828\) −0.0881460 −0.00306329
\(829\) −9.57135 −0.332427 −0.166213 0.986090i \(-0.553154\pi\)
−0.166213 + 0.986090i \(0.553154\pi\)
\(830\) −15.8605 −0.550528
\(831\) −28.1608 −0.976886
\(832\) 0 0
\(833\) 12.6703 0.438998
\(834\) −24.1987 −0.837931
\(835\) −9.05429 −0.313337
\(836\) −1.03923 −0.0359425
\(837\) −1.71379 −0.0592373
\(838\) −49.7633 −1.71904
\(839\) 48.3510 1.66926 0.834632 0.550808i \(-0.185680\pi\)
0.834632 + 0.550808i \(0.185680\pi\)
\(840\) −6.85086 −0.236377
\(841\) −14.3381 −0.494418
\(842\) −11.7423 −0.404665
\(843\) 24.3937 0.840165
\(844\) 4.33752 0.149304
\(845\) 0 0
\(846\) 2.46011 0.0845802
\(847\) 22.4698 0.772071
\(848\) −35.1129 −1.20578
\(849\) 7.16315 0.245839
\(850\) 8.09783 0.277753
\(851\) 0.899772 0.0308438
\(852\) 3.18060 0.108966
\(853\) 0.524090 0.0179445 0.00897225 0.999960i \(-0.497144\pi\)
0.00897225 + 0.999960i \(0.497144\pi\)
\(854\) −38.1444 −1.30527
\(855\) −2.33513 −0.0798596
\(856\) −43.2446 −1.47807
\(857\) −19.9377 −0.681058 −0.340529 0.940234i \(-0.610606\pi\)
−0.340529 + 0.940234i \(0.610606\pi\)
\(858\) 0 0
\(859\) −34.8437 −1.18885 −0.594425 0.804151i \(-0.702620\pi\)
−0.594425 + 0.804151i \(0.702620\pi\)
\(860\) −1.67025 −0.0569551
\(861\) 12.9269 0.440548
\(862\) −32.4185 −1.10418
\(863\) 15.4416 0.525638 0.262819 0.964845i \(-0.415348\pi\)
0.262819 + 0.964845i \(0.415348\pi\)
\(864\) 2.46681 0.0839227
\(865\) 13.4547 0.457475
\(866\) −8.64981 −0.293932
\(867\) −25.1715 −0.854869
\(868\) 1.71379 0.0581699
\(869\) 10.6189 0.360223
\(870\) −4.77479 −0.161881
\(871\) 0 0
\(872\) −31.0858 −1.05270
\(873\) −11.1414 −0.377078
\(874\) −0.576728 −0.0195081
\(875\) 2.24698 0.0759618
\(876\) 6.38835 0.215843
\(877\) 7.15751 0.241692 0.120846 0.992671i \(-0.461439\pi\)
0.120846 + 0.992671i \(0.461439\pi\)
\(878\) −23.2905 −0.786017
\(879\) 5.01075 0.169009
\(880\) 2.91185 0.0981586
\(881\) −29.9420 −1.00877 −0.504386 0.863479i \(-0.668281\pi\)
−0.504386 + 0.863479i \(0.668281\pi\)
\(882\) 2.43296 0.0819220
\(883\) 18.5139 0.623043 0.311522 0.950239i \(-0.399161\pi\)
0.311522 + 0.950239i \(0.399161\pi\)
\(884\) 0 0
\(885\) 9.83877 0.330727
\(886\) −46.0441 −1.54688
\(887\) −46.5991 −1.56464 −0.782322 0.622874i \(-0.785965\pi\)
−0.782322 + 0.622874i \(0.785965\pi\)
\(888\) −13.8509 −0.464804
\(889\) 41.4674 1.39077
\(890\) −8.57002 −0.287268
\(891\) 1.00000 0.0335013
\(892\) 10.9191 0.365600
\(893\) −4.60686 −0.154163
\(894\) 14.2989 0.478226
\(895\) −21.3980 −0.715258
\(896\) 13.8509 0.462725
\(897\) 0 0
\(898\) 19.1890 0.640344
\(899\) 6.56225 0.218863
\(900\) −0.445042 −0.0148347
\(901\) −78.3081 −2.60882
\(902\) −7.17390 −0.238865
\(903\) −8.43296 −0.280631
\(904\) 21.0000 0.698450
\(905\) 14.1032 0.468807
\(906\) 7.69501 0.255649
\(907\) 25.6004 0.850048 0.425024 0.905182i \(-0.360266\pi\)
0.425024 + 0.905182i \(0.360266\pi\)
\(908\) −0.00968868 −0.000321530 0
\(909\) −8.19136 −0.271690
\(910\) 0 0
\(911\) −24.9782 −0.827566 −0.413783 0.910376i \(-0.635793\pi\)
−0.413783 + 0.910376i \(0.635793\pi\)
\(912\) 6.79954 0.225155
\(913\) −12.7192 −0.420943
\(914\) 31.5418 1.04331
\(915\) −13.6136 −0.450051
\(916\) −3.27950 −0.108358
\(917\) −18.2446 −0.602489
\(918\) −8.09783 −0.267268
\(919\) 38.5338 1.27111 0.635556 0.772055i \(-0.280771\pi\)
0.635556 + 0.772055i \(0.280771\pi\)
\(920\) −0.603875 −0.0199092
\(921\) 29.8213 0.982646
\(922\) 10.8321 0.356735
\(923\) 0 0
\(924\) −1.00000 −0.0328976
\(925\) 4.54288 0.149369
\(926\) −2.16554 −0.0711640
\(927\) −9.52111 −0.312714
\(928\) −9.44563 −0.310068
\(929\) −36.7840 −1.20684 −0.603422 0.797422i \(-0.706197\pi\)
−0.603422 + 0.797422i \(0.706197\pi\)
\(930\) −2.13706 −0.0700771
\(931\) −4.55602 −0.149318
\(932\) 3.60925 0.118225
\(933\) 6.22952 0.203945
\(934\) −27.8297 −0.910615
\(935\) 6.49396 0.212375
\(936\) 0 0
\(937\) −10.3244 −0.337283 −0.168641 0.985677i \(-0.553938\pi\)
−0.168641 + 0.985677i \(0.553938\pi\)
\(938\) 22.1685 0.723828
\(939\) −8.22713 −0.268482
\(940\) −0.878002 −0.0286373
\(941\) 21.8170 0.711214 0.355607 0.934636i \(-0.384274\pi\)
0.355607 + 0.934636i \(0.384274\pi\)
\(942\) −19.0887 −0.621945
\(943\) 1.13946 0.0371058
\(944\) −28.6491 −0.932448
\(945\) −2.24698 −0.0730943
\(946\) 4.67994 0.152158
\(947\) −59.4476 −1.93179 −0.965893 0.258942i \(-0.916626\pi\)
−0.965893 + 0.258942i \(0.916626\pi\)
\(948\) 4.72587 0.153489
\(949\) 0 0
\(950\) −2.91185 −0.0944730
\(951\) 20.2010 0.655064
\(952\) 44.4892 1.44190
\(953\) 21.4601 0.695161 0.347581 0.937650i \(-0.387003\pi\)
0.347581 + 0.937650i \(0.387003\pi\)
\(954\) −15.0368 −0.486836
\(955\) −7.19136 −0.232707
\(956\) 2.84010 0.0918554
\(957\) −3.82908 −0.123777
\(958\) 33.2650 1.07474
\(959\) −48.9342 −1.58017
\(960\) 8.89977 0.287239
\(961\) −28.0629 −0.905255
\(962\) 0 0
\(963\) −14.1836 −0.457060
\(964\) 0.579121 0.0186522
\(965\) 11.3013 0.363801
\(966\) −0.554958 −0.0178555
\(967\) 8.09379 0.260279 0.130139 0.991496i \(-0.458458\pi\)
0.130139 + 0.991496i \(0.458458\pi\)
\(968\) −30.4892 −0.979959
\(969\) 15.1642 0.487145
\(970\) −13.8931 −0.446080
\(971\) 55.6238 1.78505 0.892527 0.450994i \(-0.148930\pi\)
0.892527 + 0.450994i \(0.148930\pi\)
\(972\) 0.445042 0.0142747
\(973\) 43.6045 1.39790
\(974\) −10.7447 −0.344281
\(975\) 0 0
\(976\) 39.6407 1.26887
\(977\) −38.0847 −1.21844 −0.609219 0.793002i \(-0.708517\pi\)
−0.609219 + 0.793002i \(0.708517\pi\)
\(978\) −24.4306 −0.781203
\(979\) −6.87263 −0.219650
\(980\) −0.868313 −0.0277373
\(981\) −10.1957 −0.325523
\(982\) 1.23729 0.0394835
\(983\) 7.66786 0.244567 0.122283 0.992495i \(-0.460978\pi\)
0.122283 + 0.992495i \(0.460978\pi\)
\(984\) −17.5405 −0.559170
\(985\) 20.5066 0.653396
\(986\) 31.0073 0.987473
\(987\) −4.43296 −0.141103
\(988\) 0 0
\(989\) −0.743332 −0.0236366
\(990\) 1.24698 0.0396316
\(991\) −34.7784 −1.10477 −0.552386 0.833589i \(-0.686282\pi\)
−0.552386 + 0.833589i \(0.686282\pi\)
\(992\) −4.22760 −0.134227
\(993\) 11.2771 0.357868
\(994\) 20.0248 0.635147
\(995\) −19.9681 −0.633030
\(996\) −5.66056 −0.179362
\(997\) −59.5220 −1.88508 −0.942540 0.334094i \(-0.891570\pi\)
−0.942540 + 0.334094i \(0.891570\pi\)
\(998\) −20.8418 −0.659734
\(999\) −4.54288 −0.143730
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 2535.2.a.bd.1.1 yes 3
3.2 odd 2 7605.2.a.bq.1.3 3
13.12 even 2 2535.2.a.v.1.3 3
39.38 odd 2 7605.2.a.bz.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2535.2.a.v.1.3 3 13.12 even 2
2535.2.a.bd.1.1 yes 3 1.1 even 1 trivial
7605.2.a.bq.1.3 3 3.2 odd 2
7605.2.a.bz.1.1 3 39.38 odd 2