Properties

Label 3610.2.a.bj.1.5
Level $3610$
Weight $2$
Character 3610.1
Self dual yes
Analytic conductor $28.826$
Analytic rank $0$
Dimension $9$
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] = [3610,2,Mod(1,3610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3610, 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("3610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.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: \(28.8259951297\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 24x^{7} - 6x^{6} + 183x^{5} + 78x^{4} - 455x^{3} - 168x^{2} + 228x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.0361439\) of defining polynomial
Character \(\chi\) \(=\) 3610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.0361439 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.0361439 q^{6} +1.83741 q^{7} +1.00000 q^{8} -2.99869 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.0361439 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.0361439 q^{6} +1.83741 q^{7} +1.00000 q^{8} -2.99869 q^{9} +1.00000 q^{10} -2.46576 q^{11} -0.0361439 q^{12} -2.39309 q^{13} +1.83741 q^{14} -0.0361439 q^{15} +1.00000 q^{16} +6.74198 q^{17} -2.99869 q^{18} +1.00000 q^{20} -0.0664112 q^{21} -2.46576 q^{22} +1.62092 q^{23} -0.0361439 q^{24} +1.00000 q^{25} -2.39309 q^{26} +0.216816 q^{27} +1.83741 q^{28} -3.30157 q^{29} -0.0361439 q^{30} +3.50384 q^{31} +1.00000 q^{32} +0.0891221 q^{33} +6.74198 q^{34} +1.83741 q^{35} -2.99869 q^{36} +6.00888 q^{37} +0.0864957 q^{39} +1.00000 q^{40} +7.98063 q^{41} -0.0664112 q^{42} +5.71443 q^{43} -2.46576 q^{44} -2.99869 q^{45} +1.62092 q^{46} +12.4926 q^{47} -0.0361439 q^{48} -3.62392 q^{49} +1.00000 q^{50} -0.243682 q^{51} -2.39309 q^{52} +1.92136 q^{53} +0.216816 q^{54} -2.46576 q^{55} +1.83741 q^{56} -3.30157 q^{58} +4.75396 q^{59} -0.0361439 q^{60} +13.6642 q^{61} +3.50384 q^{62} -5.50983 q^{63} +1.00000 q^{64} -2.39309 q^{65} +0.0891221 q^{66} +3.71765 q^{67} +6.74198 q^{68} -0.0585864 q^{69} +1.83741 q^{70} -5.83313 q^{71} -2.99869 q^{72} -0.510408 q^{73} +6.00888 q^{74} -0.0361439 q^{75} -4.53061 q^{77} +0.0864957 q^{78} -12.1354 q^{79} +1.00000 q^{80} +8.98824 q^{81} +7.98063 q^{82} +6.23256 q^{83} -0.0664112 q^{84} +6.74198 q^{85} +5.71443 q^{86} +0.119332 q^{87} -2.46576 q^{88} -17.0020 q^{89} -2.99869 q^{90} -4.39709 q^{91} +1.62092 q^{92} -0.126642 q^{93} +12.4926 q^{94} -0.0361439 q^{96} +13.6598 q^{97} -3.62392 q^{98} +7.39405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 9 q^{4} + 9 q^{5} + 9 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 9 q^{4} + 9 q^{5} + 9 q^{8} + 21 q^{9} + 9 q^{10} + 12 q^{11} + 9 q^{13} + 9 q^{16} + 6 q^{17} + 21 q^{18} + 9 q^{20} + 6 q^{21} + 12 q^{22} + 18 q^{23} + 9 q^{25} + 9 q^{26} - 18 q^{27} - 6 q^{31} + 9 q^{32} + 6 q^{33} + 6 q^{34} + 21 q^{36} + 6 q^{37} + 24 q^{39} + 9 q^{40} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 21 q^{45} + 18 q^{46} - 3 q^{47} + 39 q^{49} + 9 q^{50} - 48 q^{51} + 9 q^{52} - 18 q^{54} + 12 q^{55} + 21 q^{59} + 18 q^{61} - 6 q^{62} - 12 q^{63} + 9 q^{64} + 9 q^{65} + 6 q^{66} + 6 q^{68} - 30 q^{69} + 18 q^{71} + 21 q^{72} - 36 q^{73} + 6 q^{74} + 15 q^{77} + 24 q^{78} - 6 q^{79} + 9 q^{80} + 69 q^{81} - 6 q^{83} + 6 q^{84} + 6 q^{85} + 18 q^{86} - 24 q^{87} + 12 q^{88} - 18 q^{89} + 21 q^{90} - 60 q^{91} + 18 q^{92} - 3 q^{94} + 18 q^{97} + 39 q^{98} + 54 q^{99}+O(q^{100}) \) 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
\(3\) −0.0361439 −0.0208677 −0.0104338 0.999946i \(-0.503321\pi\)
−0.0104338 + 0.999946i \(0.503321\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.0361439 −0.0147557
\(7\) 1.83741 0.694476 0.347238 0.937777i \(-0.387120\pi\)
0.347238 + 0.937777i \(0.387120\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.99869 −0.999565
\(10\) 1.00000 0.316228
\(11\) −2.46576 −0.743453 −0.371727 0.928342i \(-0.621234\pi\)
−0.371727 + 0.928342i \(0.621234\pi\)
\(12\) −0.0361439 −0.0104338
\(13\) −2.39309 −0.663724 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(14\) 1.83741 0.491069
\(15\) −0.0361439 −0.00933232
\(16\) 1.00000 0.250000
\(17\) 6.74198 1.63517 0.817586 0.575807i \(-0.195312\pi\)
0.817586 + 0.575807i \(0.195312\pi\)
\(18\) −2.99869 −0.706799
\(19\) 0 0
\(20\) 1.00000 0.223607
\(21\) −0.0664112 −0.0144921
\(22\) −2.46576 −0.525701
\(23\) 1.62092 0.337985 0.168993 0.985617i \(-0.445949\pi\)
0.168993 + 0.985617i \(0.445949\pi\)
\(24\) −0.0361439 −0.00737785
\(25\) 1.00000 0.200000
\(26\) −2.39309 −0.469324
\(27\) 0.216816 0.0417263
\(28\) 1.83741 0.347238
\(29\) −3.30157 −0.613086 −0.306543 0.951857i \(-0.599172\pi\)
−0.306543 + 0.951857i \(0.599172\pi\)
\(30\) −0.0361439 −0.00659895
\(31\) 3.50384 0.629308 0.314654 0.949206i \(-0.398111\pi\)
0.314654 + 0.949206i \(0.398111\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.0891221 0.0155142
\(34\) 6.74198 1.15624
\(35\) 1.83741 0.310579
\(36\) −2.99869 −0.499782
\(37\) 6.00888 0.987854 0.493927 0.869503i \(-0.335561\pi\)
0.493927 + 0.869503i \(0.335561\pi\)
\(38\) 0 0
\(39\) 0.0864957 0.0138504
\(40\) 1.00000 0.158114
\(41\) 7.98063 1.24637 0.623183 0.782076i \(-0.285839\pi\)
0.623183 + 0.782076i \(0.285839\pi\)
\(42\) −0.0664112 −0.0102475
\(43\) 5.71443 0.871443 0.435721 0.900082i \(-0.356493\pi\)
0.435721 + 0.900082i \(0.356493\pi\)
\(44\) −2.46576 −0.371727
\(45\) −2.99869 −0.447019
\(46\) 1.62092 0.238992
\(47\) 12.4926 1.82224 0.911120 0.412142i \(-0.135219\pi\)
0.911120 + 0.412142i \(0.135219\pi\)
\(48\) −0.0361439 −0.00521692
\(49\) −3.62392 −0.517703
\(50\) 1.00000 0.141421
\(51\) −0.243682 −0.0341223
\(52\) −2.39309 −0.331862
\(53\) 1.92136 0.263919 0.131960 0.991255i \(-0.457873\pi\)
0.131960 + 0.991255i \(0.457873\pi\)
\(54\) 0.216816 0.0295050
\(55\) −2.46576 −0.332482
\(56\) 1.83741 0.245534
\(57\) 0 0
\(58\) −3.30157 −0.433518
\(59\) 4.75396 0.618912 0.309456 0.950914i \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(60\) −0.0361439 −0.00466616
\(61\) 13.6642 1.74952 0.874760 0.484556i \(-0.161019\pi\)
0.874760 + 0.484556i \(0.161019\pi\)
\(62\) 3.50384 0.444988
\(63\) −5.50983 −0.694174
\(64\) 1.00000 0.125000
\(65\) −2.39309 −0.296826
\(66\) 0.0891221 0.0109702
\(67\) 3.71765 0.454183 0.227091 0.973873i \(-0.427078\pi\)
0.227091 + 0.973873i \(0.427078\pi\)
\(68\) 6.74198 0.817586
\(69\) −0.0585864 −0.00705297
\(70\) 1.83741 0.219613
\(71\) −5.83313 −0.692265 −0.346133 0.938186i \(-0.612505\pi\)
−0.346133 + 0.938186i \(0.612505\pi\)
\(72\) −2.99869 −0.353399
\(73\) −0.510408 −0.0597387 −0.0298694 0.999554i \(-0.509509\pi\)
−0.0298694 + 0.999554i \(0.509509\pi\)
\(74\) 6.00888 0.698519
\(75\) −0.0361439 −0.00417354
\(76\) 0 0
\(77\) −4.53061 −0.516311
\(78\) 0.0864957 0.00979371
\(79\) −12.1354 −1.36534 −0.682671 0.730726i \(-0.739182\pi\)
−0.682671 + 0.730726i \(0.739182\pi\)
\(80\) 1.00000 0.111803
\(81\) 8.98824 0.998694
\(82\) 7.98063 0.881314
\(83\) 6.23256 0.684112 0.342056 0.939680i \(-0.388877\pi\)
0.342056 + 0.939680i \(0.388877\pi\)
\(84\) −0.0664112 −0.00724606
\(85\) 6.74198 0.731271
\(86\) 5.71443 0.616203
\(87\) 0.119332 0.0127937
\(88\) −2.46576 −0.262850
\(89\) −17.0020 −1.80220 −0.901102 0.433607i \(-0.857241\pi\)
−0.901102 + 0.433607i \(0.857241\pi\)
\(90\) −2.99869 −0.316090
\(91\) −4.39709 −0.460941
\(92\) 1.62092 0.168993
\(93\) −0.126642 −0.0131322
\(94\) 12.4926 1.28852
\(95\) 0 0
\(96\) −0.0361439 −0.00368892
\(97\) 13.6598 1.38695 0.693473 0.720483i \(-0.256080\pi\)
0.693473 + 0.720483i \(0.256080\pi\)
\(98\) −3.62392 −0.366071
\(99\) 7.39405 0.743130
\(100\) 1.00000 0.100000
\(101\) −10.6583 −1.06054 −0.530270 0.847829i \(-0.677909\pi\)
−0.530270 + 0.847829i \(0.677909\pi\)
\(102\) −0.243682 −0.0241281
\(103\) 9.74930 0.960627 0.480313 0.877097i \(-0.340523\pi\)
0.480313 + 0.877097i \(0.340523\pi\)
\(104\) −2.39309 −0.234662
\(105\) −0.0664112 −0.00648107
\(106\) 1.92136 0.186619
\(107\) −13.2857 −1.28438 −0.642191 0.766545i \(-0.721974\pi\)
−0.642191 + 0.766545i \(0.721974\pi\)
\(108\) 0.216816 0.0208632
\(109\) −1.44392 −0.138303 −0.0691513 0.997606i \(-0.522029\pi\)
−0.0691513 + 0.997606i \(0.522029\pi\)
\(110\) −2.46576 −0.235101
\(111\) −0.217185 −0.0206142
\(112\) 1.83741 0.173619
\(113\) 0.841529 0.0791644 0.0395822 0.999216i \(-0.487397\pi\)
0.0395822 + 0.999216i \(0.487397\pi\)
\(114\) 0 0
\(115\) 1.62092 0.151151
\(116\) −3.30157 −0.306543
\(117\) 7.17615 0.663435
\(118\) 4.75396 0.437637
\(119\) 12.3878 1.13559
\(120\) −0.0361439 −0.00329947
\(121\) −4.92005 −0.447277
\(122\) 13.6642 1.23710
\(123\) −0.288451 −0.0260088
\(124\) 3.50384 0.314654
\(125\) 1.00000 0.0894427
\(126\) −5.50983 −0.490855
\(127\) −12.9858 −1.15230 −0.576152 0.817343i \(-0.695446\pi\)
−0.576152 + 0.817343i \(0.695446\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.206542 −0.0181850
\(130\) −2.39309 −0.209888
\(131\) 8.92899 0.780129 0.390065 0.920787i \(-0.372453\pi\)
0.390065 + 0.920787i \(0.372453\pi\)
\(132\) 0.0891221 0.00775708
\(133\) 0 0
\(134\) 3.71765 0.321156
\(135\) 0.216816 0.0186606
\(136\) 6.74198 0.578120
\(137\) 4.45359 0.380496 0.190248 0.981736i \(-0.439071\pi\)
0.190248 + 0.981736i \(0.439071\pi\)
\(138\) −0.0585864 −0.00498720
\(139\) −1.21147 −0.102755 −0.0513777 0.998679i \(-0.516361\pi\)
−0.0513777 + 0.998679i \(0.516361\pi\)
\(140\) 1.83741 0.155290
\(141\) −0.451533 −0.0380259
\(142\) −5.83313 −0.489505
\(143\) 5.90078 0.493448
\(144\) −2.99869 −0.249891
\(145\) −3.30157 −0.274181
\(146\) −0.510408 −0.0422416
\(147\) 0.130983 0.0108033
\(148\) 6.00888 0.493927
\(149\) 9.28405 0.760579 0.380289 0.924868i \(-0.375824\pi\)
0.380289 + 0.924868i \(0.375824\pi\)
\(150\) −0.0361439 −0.00295114
\(151\) 3.34570 0.272269 0.136135 0.990690i \(-0.456532\pi\)
0.136135 + 0.990690i \(0.456532\pi\)
\(152\) 0 0
\(153\) −20.2171 −1.63446
\(154\) −4.53061 −0.365087
\(155\) 3.50384 0.281435
\(156\) 0.0864957 0.00692520
\(157\) 7.05317 0.562904 0.281452 0.959575i \(-0.409184\pi\)
0.281452 + 0.959575i \(0.409184\pi\)
\(158\) −12.1354 −0.965443
\(159\) −0.0694454 −0.00550738
\(160\) 1.00000 0.0790569
\(161\) 2.97829 0.234723
\(162\) 8.98824 0.706183
\(163\) −15.8559 −1.24193 −0.620964 0.783839i \(-0.713259\pi\)
−0.620964 + 0.783839i \(0.713259\pi\)
\(164\) 7.98063 0.623183
\(165\) 0.0891221 0.00693814
\(166\) 6.23256 0.483740
\(167\) 13.6257 1.05439 0.527196 0.849744i \(-0.323243\pi\)
0.527196 + 0.849744i \(0.323243\pi\)
\(168\) −0.0664112 −0.00512374
\(169\) −7.27311 −0.559470
\(170\) 6.74198 0.517086
\(171\) 0 0
\(172\) 5.71443 0.435721
\(173\) −12.0395 −0.915344 −0.457672 0.889121i \(-0.651317\pi\)
−0.457672 + 0.889121i \(0.651317\pi\)
\(174\) 0.119332 0.00904651
\(175\) 1.83741 0.138895
\(176\) −2.46576 −0.185863
\(177\) −0.171827 −0.0129153
\(178\) −17.0020 −1.27435
\(179\) −16.0730 −1.20135 −0.600675 0.799493i \(-0.705102\pi\)
−0.600675 + 0.799493i \(0.705102\pi\)
\(180\) −2.99869 −0.223509
\(181\) −5.75310 −0.427625 −0.213812 0.976875i \(-0.568588\pi\)
−0.213812 + 0.976875i \(0.568588\pi\)
\(182\) −4.39709 −0.325934
\(183\) −0.493877 −0.0365085
\(184\) 1.62092 0.119496
\(185\) 6.00888 0.441782
\(186\) −0.126642 −0.00928588
\(187\) −16.6241 −1.21567
\(188\) 12.4926 0.911120
\(189\) 0.398381 0.0289779
\(190\) 0 0
\(191\) −20.5460 −1.48666 −0.743329 0.668926i \(-0.766754\pi\)
−0.743329 + 0.668926i \(0.766754\pi\)
\(192\) −0.0361439 −0.00260846
\(193\) 12.5139 0.900774 0.450387 0.892833i \(-0.351286\pi\)
0.450387 + 0.892833i \(0.351286\pi\)
\(194\) 13.6598 0.980719
\(195\) 0.0864957 0.00619408
\(196\) −3.62392 −0.258851
\(197\) 6.02346 0.429154 0.214577 0.976707i \(-0.431163\pi\)
0.214577 + 0.976707i \(0.431163\pi\)
\(198\) 7.39405 0.525472
\(199\) 11.2535 0.797742 0.398871 0.917007i \(-0.369402\pi\)
0.398871 + 0.917007i \(0.369402\pi\)
\(200\) 1.00000 0.0707107
\(201\) −0.134370 −0.00947775
\(202\) −10.6583 −0.749915
\(203\) −6.06634 −0.425774
\(204\) −0.243682 −0.0170611
\(205\) 7.98063 0.557392
\(206\) 9.74930 0.679266
\(207\) −4.86064 −0.337838
\(208\) −2.39309 −0.165931
\(209\) 0 0
\(210\) −0.0664112 −0.00458281
\(211\) −14.0676 −0.968454 −0.484227 0.874942i \(-0.660899\pi\)
−0.484227 + 0.874942i \(0.660899\pi\)
\(212\) 1.92136 0.131960
\(213\) 0.210832 0.0144460
\(214\) −13.2857 −0.908195
\(215\) 5.71443 0.389721
\(216\) 0.216816 0.0147525
\(217\) 6.43799 0.437039
\(218\) −1.44392 −0.0977947
\(219\) 0.0184481 0.00124661
\(220\) −2.46576 −0.166241
\(221\) −16.1342 −1.08530
\(222\) −0.217185 −0.0145765
\(223\) −25.3025 −1.69438 −0.847189 0.531291i \(-0.821707\pi\)
−0.847189 + 0.531291i \(0.821707\pi\)
\(224\) 1.83741 0.122767
\(225\) −2.99869 −0.199913
\(226\) 0.841529 0.0559777
\(227\) −22.0111 −1.46093 −0.730464 0.682951i \(-0.760696\pi\)
−0.730464 + 0.682951i \(0.760696\pi\)
\(228\) 0 0
\(229\) −5.29529 −0.349923 −0.174961 0.984575i \(-0.555980\pi\)
−0.174961 + 0.984575i \(0.555980\pi\)
\(230\) 1.62092 0.106880
\(231\) 0.163754 0.0107742
\(232\) −3.30157 −0.216759
\(233\) 20.5518 1.34639 0.673196 0.739464i \(-0.264921\pi\)
0.673196 + 0.739464i \(0.264921\pi\)
\(234\) 7.17615 0.469119
\(235\) 12.4926 0.814930
\(236\) 4.75396 0.309456
\(237\) 0.438622 0.0284915
\(238\) 12.3878 0.802981
\(239\) 0.887170 0.0573863 0.0286931 0.999588i \(-0.490865\pi\)
0.0286931 + 0.999588i \(0.490865\pi\)
\(240\) −0.0361439 −0.00233308
\(241\) 4.00142 0.257754 0.128877 0.991661i \(-0.458863\pi\)
0.128877 + 0.991661i \(0.458863\pi\)
\(242\) −4.92005 −0.316273
\(243\) −0.975319 −0.0625667
\(244\) 13.6642 0.874760
\(245\) −3.62392 −0.231524
\(246\) −0.288451 −0.0183910
\(247\) 0 0
\(248\) 3.50384 0.222494
\(249\) −0.225269 −0.0142758
\(250\) 1.00000 0.0632456
\(251\) −12.6687 −0.799642 −0.399821 0.916593i \(-0.630928\pi\)
−0.399821 + 0.916593i \(0.630928\pi\)
\(252\) −5.50983 −0.347087
\(253\) −3.99679 −0.251276
\(254\) −12.9858 −0.814801
\(255\) −0.243682 −0.0152599
\(256\) 1.00000 0.0625000
\(257\) 13.0448 0.813712 0.406856 0.913492i \(-0.366625\pi\)
0.406856 + 0.913492i \(0.366625\pi\)
\(258\) −0.206542 −0.0128587
\(259\) 11.0408 0.686041
\(260\) −2.39309 −0.148413
\(261\) 9.90040 0.612819
\(262\) 8.92899 0.551635
\(263\) 28.2516 1.74207 0.871033 0.491224i \(-0.163450\pi\)
0.871033 + 0.491224i \(0.163450\pi\)
\(264\) 0.0891221 0.00548508
\(265\) 1.92136 0.118028
\(266\) 0 0
\(267\) 0.614517 0.0376079
\(268\) 3.71765 0.227091
\(269\) −29.5432 −1.80128 −0.900641 0.434563i \(-0.856903\pi\)
−0.900641 + 0.434563i \(0.856903\pi\)
\(270\) 0.216816 0.0131950
\(271\) 27.0112 1.64081 0.820406 0.571782i \(-0.193748\pi\)
0.820406 + 0.571782i \(0.193748\pi\)
\(272\) 6.74198 0.408793
\(273\) 0.158928 0.00961877
\(274\) 4.45359 0.269051
\(275\) −2.46576 −0.148691
\(276\) −0.0585864 −0.00352648
\(277\) −24.1778 −1.45270 −0.726352 0.687323i \(-0.758786\pi\)
−0.726352 + 0.687323i \(0.758786\pi\)
\(278\) −1.21147 −0.0726591
\(279\) −10.5069 −0.629034
\(280\) 1.83741 0.109806
\(281\) −2.40101 −0.143232 −0.0716162 0.997432i \(-0.522816\pi\)
−0.0716162 + 0.997432i \(0.522816\pi\)
\(282\) −0.451533 −0.0268884
\(283\) 8.15093 0.484523 0.242261 0.970211i \(-0.422111\pi\)
0.242261 + 0.970211i \(0.422111\pi\)
\(284\) −5.83313 −0.346133
\(285\) 0 0
\(286\) 5.90078 0.348920
\(287\) 14.6637 0.865571
\(288\) −2.99869 −0.176700
\(289\) 28.4543 1.67378
\(290\) −3.30157 −0.193875
\(291\) −0.493720 −0.0289424
\(292\) −0.510408 −0.0298694
\(293\) −9.19756 −0.537327 −0.268664 0.963234i \(-0.586582\pi\)
−0.268664 + 0.963234i \(0.586582\pi\)
\(294\) 0.130983 0.00763906
\(295\) 4.75396 0.276786
\(296\) 6.00888 0.349259
\(297\) −0.534616 −0.0310216
\(298\) 9.28405 0.537811
\(299\) −3.87901 −0.224329
\(300\) −0.0361439 −0.00208677
\(301\) 10.4998 0.605196
\(302\) 3.34570 0.192523
\(303\) 0.385232 0.0221310
\(304\) 0 0
\(305\) 13.6642 0.782409
\(306\) −20.2171 −1.15574
\(307\) 28.3472 1.61786 0.808929 0.587907i \(-0.200048\pi\)
0.808929 + 0.587907i \(0.200048\pi\)
\(308\) −4.53061 −0.258155
\(309\) −0.352378 −0.0200461
\(310\) 3.50384 0.199005
\(311\) 17.4409 0.988980 0.494490 0.869183i \(-0.335355\pi\)
0.494490 + 0.869183i \(0.335355\pi\)
\(312\) 0.0864957 0.00489685
\(313\) −13.1983 −0.746012 −0.373006 0.927829i \(-0.621673\pi\)
−0.373006 + 0.927829i \(0.621673\pi\)
\(314\) 7.05317 0.398034
\(315\) −5.50983 −0.310444
\(316\) −12.1354 −0.682671
\(317\) 12.2864 0.690072 0.345036 0.938589i \(-0.387867\pi\)
0.345036 + 0.938589i \(0.387867\pi\)
\(318\) −0.0694454 −0.00389431
\(319\) 8.14087 0.455801
\(320\) 1.00000 0.0559017
\(321\) 0.480199 0.0268021
\(322\) 2.97829 0.165974
\(323\) 0 0
\(324\) 8.98824 0.499347
\(325\) −2.39309 −0.132745
\(326\) −15.8559 −0.878175
\(327\) 0.0521890 0.00288606
\(328\) 7.98063 0.440657
\(329\) 22.9541 1.26550
\(330\) 0.0891221 0.00490601
\(331\) −0.996594 −0.0547778 −0.0273889 0.999625i \(-0.508719\pi\)
−0.0273889 + 0.999625i \(0.508719\pi\)
\(332\) 6.23256 0.342056
\(333\) −18.0188 −0.987424
\(334\) 13.6257 0.745568
\(335\) 3.71765 0.203117
\(336\) −0.0664112 −0.00362303
\(337\) −8.55919 −0.466249 −0.233124 0.972447i \(-0.574895\pi\)
−0.233124 + 0.972447i \(0.574895\pi\)
\(338\) −7.27311 −0.395605
\(339\) −0.0304162 −0.00165198
\(340\) 6.74198 0.365635
\(341\) −8.63961 −0.467861
\(342\) 0 0
\(343\) −19.5205 −1.05401
\(344\) 5.71443 0.308102
\(345\) −0.0585864 −0.00315418
\(346\) −12.0395 −0.647246
\(347\) −6.63773 −0.356332 −0.178166 0.984000i \(-0.557016\pi\)
−0.178166 + 0.984000i \(0.557016\pi\)
\(348\) 0.119332 0.00639685
\(349\) 31.0443 1.66176 0.830881 0.556450i \(-0.187837\pi\)
0.830881 + 0.556450i \(0.187837\pi\)
\(350\) 1.83741 0.0982138
\(351\) −0.518861 −0.0276948
\(352\) −2.46576 −0.131425
\(353\) 17.7629 0.945424 0.472712 0.881217i \(-0.343275\pi\)
0.472712 + 0.881217i \(0.343275\pi\)
\(354\) −0.171827 −0.00913248
\(355\) −5.83313 −0.309590
\(356\) −17.0020 −0.901102
\(357\) −0.447743 −0.0236971
\(358\) −16.0730 −0.849483
\(359\) −5.13615 −0.271076 −0.135538 0.990772i \(-0.543276\pi\)
−0.135538 + 0.990772i \(0.543276\pi\)
\(360\) −2.99869 −0.158045
\(361\) 0 0
\(362\) −5.75310 −0.302376
\(363\) 0.177830 0.00933364
\(364\) −4.39709 −0.230470
\(365\) −0.510408 −0.0267160
\(366\) −0.493877 −0.0258154
\(367\) −5.74589 −0.299933 −0.149966 0.988691i \(-0.547917\pi\)
−0.149966 + 0.988691i \(0.547917\pi\)
\(368\) 1.62092 0.0844963
\(369\) −23.9315 −1.24582
\(370\) 6.00888 0.312387
\(371\) 3.53033 0.183285
\(372\) −0.126642 −0.00656611
\(373\) −37.7854 −1.95645 −0.978227 0.207539i \(-0.933454\pi\)
−0.978227 + 0.207539i \(0.933454\pi\)
\(374\) −16.6241 −0.859611
\(375\) −0.0361439 −0.00186646
\(376\) 12.4926 0.644259
\(377\) 7.90096 0.406920
\(378\) 0.398381 0.0204905
\(379\) 32.2663 1.65741 0.828703 0.559689i \(-0.189079\pi\)
0.828703 + 0.559689i \(0.189079\pi\)
\(380\) 0 0
\(381\) 0.469358 0.0240459
\(382\) −20.5460 −1.05123
\(383\) 21.7799 1.11290 0.556450 0.830881i \(-0.312163\pi\)
0.556450 + 0.830881i \(0.312163\pi\)
\(384\) −0.0361439 −0.00184446
\(385\) −4.53061 −0.230901
\(386\) 12.5139 0.636943
\(387\) −17.1358 −0.871063
\(388\) 13.6598 0.693473
\(389\) 9.68485 0.491041 0.245521 0.969391i \(-0.421041\pi\)
0.245521 + 0.969391i \(0.421041\pi\)
\(390\) 0.0864957 0.00437988
\(391\) 10.9282 0.552663
\(392\) −3.62392 −0.183036
\(393\) −0.322729 −0.0162795
\(394\) 6.02346 0.303458
\(395\) −12.1354 −0.610600
\(396\) 7.39405 0.371565
\(397\) −18.3952 −0.923226 −0.461613 0.887081i \(-0.652729\pi\)
−0.461613 + 0.887081i \(0.652729\pi\)
\(398\) 11.2535 0.564089
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 36.5367 1.82455 0.912277 0.409573i \(-0.134322\pi\)
0.912277 + 0.409573i \(0.134322\pi\)
\(402\) −0.134370 −0.00670178
\(403\) −8.38501 −0.417687
\(404\) −10.6583 −0.530270
\(405\) 8.98824 0.446629
\(406\) −6.06634 −0.301068
\(407\) −14.8164 −0.734424
\(408\) −0.243682 −0.0120640
\(409\) −36.3063 −1.79523 −0.897616 0.440779i \(-0.854702\pi\)
−0.897616 + 0.440779i \(0.854702\pi\)
\(410\) 7.98063 0.394135
\(411\) −0.160970 −0.00794007
\(412\) 9.74930 0.480313
\(413\) 8.73497 0.429820
\(414\) −4.86064 −0.238887
\(415\) 6.23256 0.305944
\(416\) −2.39309 −0.117331
\(417\) 0.0437872 0.00214427
\(418\) 0 0
\(419\) 39.5001 1.92970 0.964852 0.262793i \(-0.0846437\pi\)
0.964852 + 0.262793i \(0.0846437\pi\)
\(420\) −0.0664112 −0.00324054
\(421\) 14.1005 0.687217 0.343609 0.939113i \(-0.388351\pi\)
0.343609 + 0.939113i \(0.388351\pi\)
\(422\) −14.0676 −0.684800
\(423\) −37.4616 −1.82145
\(424\) 1.92136 0.0933095
\(425\) 6.74198 0.327034
\(426\) 0.210832 0.0102149
\(427\) 25.1067 1.21500
\(428\) −13.2857 −0.642191
\(429\) −0.213277 −0.0102971
\(430\) 5.71443 0.275574
\(431\) −21.0431 −1.01361 −0.506806 0.862060i \(-0.669174\pi\)
−0.506806 + 0.862060i \(0.669174\pi\)
\(432\) 0.216816 0.0104316
\(433\) 5.25769 0.252669 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(434\) 6.43799 0.309034
\(435\) 0.119332 0.00572152
\(436\) −1.44392 −0.0691513
\(437\) 0 0
\(438\) 0.0184481 0.000881486 0
\(439\) −36.4838 −1.74128 −0.870639 0.491923i \(-0.836294\pi\)
−0.870639 + 0.491923i \(0.836294\pi\)
\(440\) −2.46576 −0.117550
\(441\) 10.8670 0.517478
\(442\) −16.1342 −0.767425
\(443\) −9.23069 −0.438564 −0.219282 0.975662i \(-0.570371\pi\)
−0.219282 + 0.975662i \(0.570371\pi\)
\(444\) −0.217185 −0.0103071
\(445\) −17.0020 −0.805970
\(446\) −25.3025 −1.19811
\(447\) −0.335562 −0.0158715
\(448\) 1.83741 0.0868095
\(449\) 22.8297 1.07740 0.538699 0.842498i \(-0.318916\pi\)
0.538699 + 0.842498i \(0.318916\pi\)
\(450\) −2.99869 −0.141360
\(451\) −19.6783 −0.926615
\(452\) 0.841529 0.0395822
\(453\) −0.120927 −0.00568163
\(454\) −22.0111 −1.03303
\(455\) −4.39709 −0.206139
\(456\) 0 0
\(457\) −7.82515 −0.366045 −0.183022 0.983109i \(-0.558588\pi\)
−0.183022 + 0.983109i \(0.558588\pi\)
\(458\) −5.29529 −0.247433
\(459\) 1.46177 0.0682296
\(460\) 1.62092 0.0755757
\(461\) −3.93168 −0.183117 −0.0915584 0.995800i \(-0.529185\pi\)
−0.0915584 + 0.995800i \(0.529185\pi\)
\(462\) 0.163754 0.00761852
\(463\) −17.2964 −0.803832 −0.401916 0.915676i \(-0.631656\pi\)
−0.401916 + 0.915676i \(0.631656\pi\)
\(464\) −3.30157 −0.153272
\(465\) −0.126642 −0.00587290
\(466\) 20.5518 0.952044
\(467\) −21.0267 −0.973002 −0.486501 0.873680i \(-0.661727\pi\)
−0.486501 + 0.873680i \(0.661727\pi\)
\(468\) 7.17615 0.331718
\(469\) 6.83084 0.315419
\(470\) 12.4926 0.576243
\(471\) −0.254929 −0.0117465
\(472\) 4.75396 0.218819
\(473\) −14.0904 −0.647877
\(474\) 0.438622 0.0201466
\(475\) 0 0
\(476\) 12.3878 0.567794
\(477\) −5.76157 −0.263804
\(478\) 0.887170 0.0405782
\(479\) 21.1317 0.965531 0.482766 0.875750i \(-0.339632\pi\)
0.482766 + 0.875750i \(0.339632\pi\)
\(480\) −0.0361439 −0.00164974
\(481\) −14.3798 −0.655663
\(482\) 4.00142 0.182260
\(483\) −0.107647 −0.00489812
\(484\) −4.92005 −0.223639
\(485\) 13.6598 0.620261
\(486\) −0.975319 −0.0442414
\(487\) 17.2003 0.779420 0.389710 0.920938i \(-0.372575\pi\)
0.389710 + 0.920938i \(0.372575\pi\)
\(488\) 13.6642 0.618549
\(489\) 0.573093 0.0259162
\(490\) −3.62392 −0.163712
\(491\) −33.1374 −1.49547 −0.747734 0.663998i \(-0.768858\pi\)
−0.747734 + 0.663998i \(0.768858\pi\)
\(492\) −0.288451 −0.0130044
\(493\) −22.2591 −1.00250
\(494\) 0 0
\(495\) 7.39405 0.332338
\(496\) 3.50384 0.157327
\(497\) −10.7179 −0.480762
\(498\) −0.225269 −0.0100945
\(499\) −31.9699 −1.43117 −0.715584 0.698527i \(-0.753839\pi\)
−0.715584 + 0.698527i \(0.753839\pi\)
\(500\) 1.00000 0.0447214
\(501\) −0.492488 −0.0220027
\(502\) −12.6687 −0.565432
\(503\) −25.2415 −1.12546 −0.562732 0.826639i \(-0.690250\pi\)
−0.562732 + 0.826639i \(0.690250\pi\)
\(504\) −5.50983 −0.245427
\(505\) −10.6583 −0.474288
\(506\) −3.99679 −0.177679
\(507\) 0.262879 0.0116749
\(508\) −12.9858 −0.576152
\(509\) 17.0508 0.755763 0.377881 0.925854i \(-0.376653\pi\)
0.377881 + 0.925854i \(0.376653\pi\)
\(510\) −0.243682 −0.0107904
\(511\) −0.937829 −0.0414871
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 13.0448 0.575381
\(515\) 9.74930 0.429605
\(516\) −0.206542 −0.00909250
\(517\) −30.8038 −1.35475
\(518\) 11.0408 0.485104
\(519\) 0.435153 0.0191011
\(520\) −2.39309 −0.104944
\(521\) 6.80513 0.298138 0.149069 0.988827i \(-0.452372\pi\)
0.149069 + 0.988827i \(0.452372\pi\)
\(522\) 9.90040 0.433329
\(523\) −5.22078 −0.228289 −0.114144 0.993464i \(-0.536413\pi\)
−0.114144 + 0.993464i \(0.536413\pi\)
\(524\) 8.92899 0.390065
\(525\) −0.0664112 −0.00289842
\(526\) 28.2516 1.23183
\(527\) 23.6228 1.02903
\(528\) 0.0891221 0.00387854
\(529\) −20.3726 −0.885766
\(530\) 1.92136 0.0834585
\(531\) −14.2557 −0.618643
\(532\) 0 0
\(533\) −19.0984 −0.827243
\(534\) 0.614517 0.0265928
\(535\) −13.2857 −0.574393
\(536\) 3.71765 0.160578
\(537\) 0.580940 0.0250694
\(538\) −29.5432 −1.27370
\(539\) 8.93570 0.384888
\(540\) 0.216816 0.00933029
\(541\) −26.1671 −1.12501 −0.562506 0.826793i \(-0.690163\pi\)
−0.562506 + 0.826793i \(0.690163\pi\)
\(542\) 27.0112 1.16023
\(543\) 0.207940 0.00892355
\(544\) 6.74198 0.289060
\(545\) −1.44392 −0.0618508
\(546\) 0.158928 0.00680150
\(547\) −35.5365 −1.51943 −0.759715 0.650256i \(-0.774662\pi\)
−0.759715 + 0.650256i \(0.774662\pi\)
\(548\) 4.45359 0.190248
\(549\) −40.9747 −1.74876
\(550\) −2.46576 −0.105140
\(551\) 0 0
\(552\) −0.0585864 −0.00249360
\(553\) −22.2978 −0.948197
\(554\) −24.1778 −1.02722
\(555\) −0.217185 −0.00921897
\(556\) −1.21147 −0.0513777
\(557\) 30.6913 1.30043 0.650216 0.759749i \(-0.274678\pi\)
0.650216 + 0.759749i \(0.274678\pi\)
\(558\) −10.5069 −0.444794
\(559\) −13.6752 −0.578398
\(560\) 1.83741 0.0776448
\(561\) 0.600859 0.0253683
\(562\) −2.40101 −0.101281
\(563\) −31.2044 −1.31511 −0.657553 0.753408i \(-0.728409\pi\)
−0.657553 + 0.753408i \(0.728409\pi\)
\(564\) −0.451533 −0.0190130
\(565\) 0.841529 0.0354034
\(566\) 8.15093 0.342609
\(567\) 16.5151 0.693569
\(568\) −5.83313 −0.244753
\(569\) 20.6053 0.863820 0.431910 0.901917i \(-0.357840\pi\)
0.431910 + 0.901917i \(0.357840\pi\)
\(570\) 0 0
\(571\) 4.38785 0.183626 0.0918129 0.995776i \(-0.470734\pi\)
0.0918129 + 0.995776i \(0.470734\pi\)
\(572\) 5.90078 0.246724
\(573\) 0.742614 0.0310231
\(574\) 14.6637 0.612051
\(575\) 1.62092 0.0675970
\(576\) −2.99869 −0.124946
\(577\) 29.8821 1.24401 0.622004 0.783014i \(-0.286319\pi\)
0.622004 + 0.783014i \(0.286319\pi\)
\(578\) 28.4543 1.18354
\(579\) −0.452303 −0.0187971
\(580\) −3.30157 −0.137090
\(581\) 11.4518 0.475100
\(582\) −0.493720 −0.0204653
\(583\) −4.73760 −0.196211
\(584\) −0.510408 −0.0211208
\(585\) 7.17615 0.296697
\(586\) −9.19756 −0.379948
\(587\) 39.3798 1.62538 0.812689 0.582698i \(-0.198003\pi\)
0.812689 + 0.582698i \(0.198003\pi\)
\(588\) 0.130983 0.00540163
\(589\) 0 0
\(590\) 4.75396 0.195717
\(591\) −0.217712 −0.00895546
\(592\) 6.00888 0.246964
\(593\) −40.1118 −1.64719 −0.823597 0.567176i \(-0.808036\pi\)
−0.823597 + 0.567176i \(0.808036\pi\)
\(594\) −0.534616 −0.0219356
\(595\) 12.3878 0.507850
\(596\) 9.28405 0.380289
\(597\) −0.406747 −0.0166470
\(598\) −3.87901 −0.158624
\(599\) −31.4596 −1.28540 −0.642702 0.766116i \(-0.722187\pi\)
−0.642702 + 0.766116i \(0.722187\pi\)
\(600\) −0.0361439 −0.00147557
\(601\) 34.4890 1.40684 0.703418 0.710777i \(-0.251656\pi\)
0.703418 + 0.710777i \(0.251656\pi\)
\(602\) 10.4998 0.427938
\(603\) −11.1481 −0.453985
\(604\) 3.34570 0.136135
\(605\) −4.92005 −0.200028
\(606\) 0.385232 0.0156490
\(607\) −12.9979 −0.527568 −0.263784 0.964582i \(-0.584971\pi\)
−0.263784 + 0.964582i \(0.584971\pi\)
\(608\) 0 0
\(609\) 0.219261 0.00888492
\(610\) 13.6642 0.553247
\(611\) −29.8960 −1.20946
\(612\) −20.2171 −0.817229
\(613\) 8.71181 0.351867 0.175933 0.984402i \(-0.443706\pi\)
0.175933 + 0.984402i \(0.443706\pi\)
\(614\) 28.3472 1.14400
\(615\) −0.288451 −0.0116315
\(616\) −4.53061 −0.182543
\(617\) −24.3566 −0.980559 −0.490279 0.871565i \(-0.663105\pi\)
−0.490279 + 0.871565i \(0.663105\pi\)
\(618\) −0.352378 −0.0141747
\(619\) 1.64497 0.0661168 0.0330584 0.999453i \(-0.489475\pi\)
0.0330584 + 0.999453i \(0.489475\pi\)
\(620\) 3.50384 0.140718
\(621\) 0.351442 0.0141029
\(622\) 17.4409 0.699315
\(623\) −31.2396 −1.25159
\(624\) 0.0864957 0.00346260
\(625\) 1.00000 0.0400000
\(626\) −13.1983 −0.527510
\(627\) 0 0
\(628\) 7.05317 0.281452
\(629\) 40.5118 1.61531
\(630\) −5.50983 −0.219517
\(631\) −43.4987 −1.73166 −0.865829 0.500341i \(-0.833208\pi\)
−0.865829 + 0.500341i \(0.833208\pi\)
\(632\) −12.1354 −0.482721
\(633\) 0.508458 0.0202094
\(634\) 12.2864 0.487955
\(635\) −12.9858 −0.515326
\(636\) −0.0694454 −0.00275369
\(637\) 8.67237 0.343612
\(638\) 8.14087 0.322300
\(639\) 17.4918 0.691964
\(640\) 1.00000 0.0395285
\(641\) 36.8215 1.45436 0.727181 0.686446i \(-0.240830\pi\)
0.727181 + 0.686446i \(0.240830\pi\)
\(642\) 0.480199 0.0189519
\(643\) −14.7356 −0.581117 −0.290558 0.956857i \(-0.593841\pi\)
−0.290558 + 0.956857i \(0.593841\pi\)
\(644\) 2.97829 0.117361
\(645\) −0.206542 −0.00813258
\(646\) 0 0
\(647\) −6.04813 −0.237776 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(648\) 8.98824 0.353092
\(649\) −11.7221 −0.460132
\(650\) −2.39309 −0.0938648
\(651\) −0.232694 −0.00912001
\(652\) −15.8559 −0.620964
\(653\) −2.88867 −0.113043 −0.0565213 0.998401i \(-0.518001\pi\)
−0.0565213 + 0.998401i \(0.518001\pi\)
\(654\) 0.0521890 0.00204075
\(655\) 8.92899 0.348884
\(656\) 7.98063 0.311591
\(657\) 1.53056 0.0597127
\(658\) 22.9541 0.894845
\(659\) −16.3211 −0.635779 −0.317889 0.948128i \(-0.602974\pi\)
−0.317889 + 0.948128i \(0.602974\pi\)
\(660\) 0.0891221 0.00346907
\(661\) −28.5414 −1.11013 −0.555066 0.831806i \(-0.687307\pi\)
−0.555066 + 0.831806i \(0.687307\pi\)
\(662\) −0.996594 −0.0387337
\(663\) 0.583152 0.0226478
\(664\) 6.23256 0.241870
\(665\) 0 0
\(666\) −18.0188 −0.698214
\(667\) −5.35158 −0.207214
\(668\) 13.6257 0.527196
\(669\) 0.914530 0.0353578
\(670\) 3.71765 0.143625
\(671\) −33.6926 −1.30069
\(672\) −0.0664112 −0.00256187
\(673\) −6.23447 −0.240321 −0.120161 0.992754i \(-0.538341\pi\)
−0.120161 + 0.992754i \(0.538341\pi\)
\(674\) −8.55919 −0.329688
\(675\) 0.216816 0.00834526
\(676\) −7.27311 −0.279735
\(677\) 14.6150 0.561700 0.280850 0.959752i \(-0.409384\pi\)
0.280850 + 0.959752i \(0.409384\pi\)
\(678\) −0.0304162 −0.00116813
\(679\) 25.0987 0.963201
\(680\) 6.74198 0.258543
\(681\) 0.795568 0.0304862
\(682\) −8.63961 −0.330828
\(683\) −47.0671 −1.80097 −0.900486 0.434885i \(-0.856789\pi\)
−0.900486 + 0.434885i \(0.856789\pi\)
\(684\) 0 0
\(685\) 4.45359 0.170163
\(686\) −19.5205 −0.745297
\(687\) 0.191392 0.00730208
\(688\) 5.71443 0.217861
\(689\) −4.59799 −0.175169
\(690\) −0.0585864 −0.00223034
\(691\) −27.3484 −1.04038 −0.520191 0.854050i \(-0.674139\pi\)
−0.520191 + 0.854050i \(0.674139\pi\)
\(692\) −12.0395 −0.457672
\(693\) 13.5859 0.516086
\(694\) −6.63773 −0.251965
\(695\) −1.21147 −0.0459536
\(696\) 0.119332 0.00452326
\(697\) 53.8053 2.03802
\(698\) 31.0443 1.17504
\(699\) −0.742822 −0.0280961
\(700\) 1.83741 0.0694476
\(701\) 19.8319 0.749040 0.374520 0.927219i \(-0.377808\pi\)
0.374520 + 0.927219i \(0.377808\pi\)
\(702\) −0.518861 −0.0195831
\(703\) 0 0
\(704\) −2.46576 −0.0929317
\(705\) −0.451533 −0.0170057
\(706\) 17.7629 0.668516
\(707\) −19.5837 −0.736519
\(708\) −0.171827 −0.00645764
\(709\) 32.7102 1.22846 0.614230 0.789127i \(-0.289467\pi\)
0.614230 + 0.789127i \(0.289467\pi\)
\(710\) −5.83313 −0.218913
\(711\) 36.3904 1.36475
\(712\) −17.0020 −0.637175
\(713\) 5.67944 0.212697
\(714\) −0.447743 −0.0167564
\(715\) 5.90078 0.220677
\(716\) −16.0730 −0.600675
\(717\) −0.0320658 −0.00119752
\(718\) −5.13615 −0.191680
\(719\) −3.24776 −0.121121 −0.0605606 0.998165i \(-0.519289\pi\)
−0.0605606 + 0.998165i \(0.519289\pi\)
\(720\) −2.99869 −0.111755
\(721\) 17.9135 0.667132
\(722\) 0 0
\(723\) −0.144627 −0.00537873
\(724\) −5.75310 −0.213812
\(725\) −3.30157 −0.122617
\(726\) 0.177830 0.00659988
\(727\) 19.4096 0.719862 0.359931 0.932979i \(-0.382800\pi\)
0.359931 + 0.932979i \(0.382800\pi\)
\(728\) −4.39709 −0.162967
\(729\) −26.9295 −0.997388
\(730\) −0.510408 −0.0188910
\(731\) 38.5266 1.42496
\(732\) −0.493877 −0.0182542
\(733\) 12.0753 0.446010 0.223005 0.974817i \(-0.428413\pi\)
0.223005 + 0.974817i \(0.428413\pi\)
\(734\) −5.74589 −0.212085
\(735\) 0.130983 0.00483137
\(736\) 1.62092 0.0597479
\(737\) −9.16681 −0.337664
\(738\) −23.9315 −0.880930
\(739\) 3.04613 0.112054 0.0560268 0.998429i \(-0.482157\pi\)
0.0560268 + 0.998429i \(0.482157\pi\)
\(740\) 6.00888 0.220891
\(741\) 0 0
\(742\) 3.53033 0.129602
\(743\) −39.7565 −1.45853 −0.729263 0.684234i \(-0.760137\pi\)
−0.729263 + 0.684234i \(0.760137\pi\)
\(744\) −0.126642 −0.00464294
\(745\) 9.28405 0.340141
\(746\) −37.7854 −1.38342
\(747\) −18.6895 −0.683814
\(748\) −16.6241 −0.607837
\(749\) −24.4114 −0.891972
\(750\) −0.0361439 −0.00131979
\(751\) −35.7867 −1.30588 −0.652938 0.757411i \(-0.726464\pi\)
−0.652938 + 0.757411i \(0.726464\pi\)
\(752\) 12.4926 0.455560
\(753\) 0.457897 0.0166867
\(754\) 7.90096 0.287736
\(755\) 3.34570 0.121763
\(756\) 0.398381 0.0144890
\(757\) −49.9062 −1.81387 −0.906936 0.421267i \(-0.861585\pi\)
−0.906936 + 0.421267i \(0.861585\pi\)
\(758\) 32.2663 1.17196
\(759\) 0.144460 0.00524355
\(760\) 0 0
\(761\) −36.2563 −1.31429 −0.657145 0.753764i \(-0.728236\pi\)
−0.657145 + 0.753764i \(0.728236\pi\)
\(762\) 0.469358 0.0170030
\(763\) −2.65308 −0.0960478
\(764\) −20.5460 −0.743329
\(765\) −20.2171 −0.730952
\(766\) 21.7799 0.786939
\(767\) −11.3767 −0.410787
\(768\) −0.0361439 −0.00130423
\(769\) −5.51540 −0.198891 −0.0994453 0.995043i \(-0.531707\pi\)
−0.0994453 + 0.995043i \(0.531707\pi\)
\(770\) −4.53061 −0.163272
\(771\) −0.471490 −0.0169803
\(772\) 12.5139 0.450387
\(773\) 38.3992 1.38112 0.690562 0.723273i \(-0.257363\pi\)
0.690562 + 0.723273i \(0.257363\pi\)
\(774\) −17.1358 −0.615935
\(775\) 3.50384 0.125862
\(776\) 13.6598 0.490359
\(777\) −0.399057 −0.0143161
\(778\) 9.68485 0.347219
\(779\) 0 0
\(780\) 0.0864957 0.00309704
\(781\) 14.3831 0.514667
\(782\) 10.9282 0.390792
\(783\) −0.715834 −0.0255818
\(784\) −3.62392 −0.129426
\(785\) 7.05317 0.251738
\(786\) −0.322729 −0.0115113
\(787\) −3.12058 −0.111237 −0.0556184 0.998452i \(-0.517713\pi\)
−0.0556184 + 0.998452i \(0.517713\pi\)
\(788\) 6.02346 0.214577
\(789\) −1.02112 −0.0363529
\(790\) −12.1354 −0.431759
\(791\) 1.54624 0.0549778
\(792\) 7.39405 0.262736
\(793\) −32.6997 −1.16120
\(794\) −18.3952 −0.652819
\(795\) −0.0694454 −0.00246298
\(796\) 11.2535 0.398871
\(797\) 0.491495 0.0174096 0.00870482 0.999962i \(-0.497229\pi\)
0.00870482 + 0.999962i \(0.497229\pi\)
\(798\) 0 0
\(799\) 84.2252 2.97967
\(800\) 1.00000 0.0353553
\(801\) 50.9837 1.80142
\(802\) 36.5367 1.29015
\(803\) 1.25854 0.0444129
\(804\) −0.134370 −0.00473887
\(805\) 2.97829 0.104971
\(806\) −8.38501 −0.295349
\(807\) 1.06781 0.0375886
\(808\) −10.6583 −0.374957
\(809\) 33.6552 1.18326 0.591628 0.806211i \(-0.298486\pi\)
0.591628 + 0.806211i \(0.298486\pi\)
\(810\) 8.98824 0.315815
\(811\) 48.1108 1.68940 0.844699 0.535242i \(-0.179780\pi\)
0.844699 + 0.535242i \(0.179780\pi\)
\(812\) −6.06634 −0.212887
\(813\) −0.976289 −0.0342400
\(814\) −14.8164 −0.519316
\(815\) −15.8559 −0.555407
\(816\) −0.243682 −0.00853056
\(817\) 0 0
\(818\) −36.3063 −1.26942
\(819\) 13.1855 0.460740
\(820\) 7.98063 0.278696
\(821\) −1.89435 −0.0661131 −0.0330566 0.999453i \(-0.510524\pi\)
−0.0330566 + 0.999453i \(0.510524\pi\)
\(822\) −0.160970 −0.00561448
\(823\) −26.5567 −0.925707 −0.462853 0.886435i \(-0.653174\pi\)
−0.462853 + 0.886435i \(0.653174\pi\)
\(824\) 9.74930 0.339633
\(825\) 0.0891221 0.00310283
\(826\) 8.73497 0.303929
\(827\) −25.7977 −0.897075 −0.448537 0.893764i \(-0.648055\pi\)
−0.448537 + 0.893764i \(0.648055\pi\)
\(828\) −4.86064 −0.168919
\(829\) 27.9638 0.971221 0.485611 0.874175i \(-0.338597\pi\)
0.485611 + 0.874175i \(0.338597\pi\)
\(830\) 6.23256 0.216335
\(831\) 0.873881 0.0303146
\(832\) −2.39309 −0.0829655
\(833\) −24.4324 −0.846533
\(834\) 0.0437872 0.00151623
\(835\) 13.6257 0.471539
\(836\) 0 0
\(837\) 0.759689 0.0262587
\(838\) 39.5001 1.36451
\(839\) −49.9183 −1.72337 −0.861685 0.507443i \(-0.830591\pi\)
−0.861685 + 0.507443i \(0.830591\pi\)
\(840\) −0.0664112 −0.00229140
\(841\) −18.0996 −0.624125
\(842\) 14.1005 0.485936
\(843\) 0.0867820 0.00298893
\(844\) −14.0676 −0.484227
\(845\) −7.27311 −0.250203
\(846\) −37.4616 −1.28796
\(847\) −9.04015 −0.310623
\(848\) 1.92136 0.0659798
\(849\) −0.294607 −0.0101109
\(850\) 6.74198 0.231248
\(851\) 9.73991 0.333880
\(852\) 0.210832 0.00722299
\(853\) −7.95618 −0.272415 −0.136207 0.990680i \(-0.543491\pi\)
−0.136207 + 0.990680i \(0.543491\pi\)
\(854\) 25.1067 0.859135
\(855\) 0 0
\(856\) −13.2857 −0.454097
\(857\) 25.1550 0.859277 0.429638 0.903001i \(-0.358641\pi\)
0.429638 + 0.903001i \(0.358641\pi\)
\(858\) −0.213277 −0.00728116
\(859\) 3.46880 0.118354 0.0591769 0.998248i \(-0.481152\pi\)
0.0591769 + 0.998248i \(0.481152\pi\)
\(860\) 5.71443 0.194861
\(861\) −0.530004 −0.0180625
\(862\) −21.0431 −0.716732
\(863\) 40.9708 1.39466 0.697331 0.716750i \(-0.254371\pi\)
0.697331 + 0.716750i \(0.254371\pi\)
\(864\) 0.216816 0.00737624
\(865\) −12.0395 −0.409354
\(866\) 5.25769 0.178664
\(867\) −1.02845 −0.0349280
\(868\) 6.43799 0.218520
\(869\) 29.9230 1.01507
\(870\) 0.119332 0.00404572
\(871\) −8.89666 −0.301452
\(872\) −1.44392 −0.0488973
\(873\) −40.9616 −1.38634
\(874\) 0 0
\(875\) 1.83741 0.0621158
\(876\) 0.0184481 0.000623305 0
\(877\) 24.8602 0.839469 0.419735 0.907647i \(-0.362123\pi\)
0.419735 + 0.907647i \(0.362123\pi\)
\(878\) −36.4838 −1.23127
\(879\) 0.332436 0.0112128
\(880\) −2.46576 −0.0831206
\(881\) −15.7411 −0.530332 −0.265166 0.964203i \(-0.585427\pi\)
−0.265166 + 0.964203i \(0.585427\pi\)
\(882\) 10.8670 0.365912
\(883\) −7.75970 −0.261135 −0.130567 0.991439i \(-0.541680\pi\)
−0.130567 + 0.991439i \(0.541680\pi\)
\(884\) −16.1342 −0.542651
\(885\) −0.171827 −0.00577589
\(886\) −9.23069 −0.310111
\(887\) −34.0594 −1.14360 −0.571802 0.820392i \(-0.693755\pi\)
−0.571802 + 0.820392i \(0.693755\pi\)
\(888\) −0.217185 −0.00728824
\(889\) −23.8603 −0.800247
\(890\) −17.0020 −0.569907
\(891\) −22.1628 −0.742482
\(892\) −25.3025 −0.847189
\(893\) 0 0
\(894\) −0.335562 −0.0112229
\(895\) −16.0730 −0.537260
\(896\) 1.83741 0.0613836
\(897\) 0.140203 0.00468123
\(898\) 22.8297 0.761836
\(899\) −11.5682 −0.385820
\(900\) −2.99869 −0.0999565
\(901\) 12.9538 0.431553
\(902\) −19.6783 −0.655216
\(903\) −0.379503 −0.0126291
\(904\) 0.841529 0.0279888
\(905\) −5.75310 −0.191240
\(906\) −0.120927 −0.00401752
\(907\) −4.80756 −0.159633 −0.0798163 0.996810i \(-0.525433\pi\)
−0.0798163 + 0.996810i \(0.525433\pi\)
\(908\) −22.0111 −0.730464
\(909\) 31.9609 1.06008
\(910\) −4.39709 −0.145762
\(911\) −9.14462 −0.302975 −0.151487 0.988459i \(-0.548406\pi\)
−0.151487 + 0.988459i \(0.548406\pi\)
\(912\) 0 0
\(913\) −15.3680 −0.508606
\(914\) −7.82515 −0.258833
\(915\) −0.493877 −0.0163271
\(916\) −5.29529 −0.174961
\(917\) 16.4062 0.541781
\(918\) 1.46177 0.0482456
\(919\) −21.0816 −0.695419 −0.347709 0.937602i \(-0.613040\pi\)
−0.347709 + 0.937602i \(0.613040\pi\)
\(920\) 1.62092 0.0534401
\(921\) −1.02458 −0.0337610
\(922\) −3.93168 −0.129483
\(923\) 13.9592 0.459473
\(924\) 0.163754 0.00538711
\(925\) 6.00888 0.197571
\(926\) −17.2964 −0.568395
\(927\) −29.2352 −0.960209
\(928\) −3.30157 −0.108379
\(929\) 18.8119 0.617200 0.308600 0.951192i \(-0.400140\pi\)
0.308600 + 0.951192i \(0.400140\pi\)
\(930\) −0.126642 −0.00415277
\(931\) 0 0
\(932\) 20.5518 0.673196
\(933\) −0.630381 −0.0206377
\(934\) −21.0267 −0.688016
\(935\) −16.6241 −0.543666
\(936\) 7.17615 0.234560
\(937\) −19.2143 −0.627704 −0.313852 0.949472i \(-0.601620\pi\)
−0.313852 + 0.949472i \(0.601620\pi\)
\(938\) 6.83084 0.223035
\(939\) 0.477038 0.0155676
\(940\) 12.4926 0.407465
\(941\) −15.5583 −0.507185 −0.253593 0.967311i \(-0.581612\pi\)
−0.253593 + 0.967311i \(0.581612\pi\)
\(942\) −0.254929 −0.00830604
\(943\) 12.9360 0.421253
\(944\) 4.75396 0.154728
\(945\) 0.398381 0.0129593
\(946\) −14.0904 −0.458118
\(947\) −22.1758 −0.720618 −0.360309 0.932833i \(-0.617329\pi\)
−0.360309 + 0.932833i \(0.617329\pi\)
\(948\) 0.438622 0.0142458
\(949\) 1.22145 0.0396500
\(950\) 0 0
\(951\) −0.444078 −0.0144002
\(952\) 12.3878 0.401491
\(953\) 4.95575 0.160532 0.0802662 0.996773i \(-0.474423\pi\)
0.0802662 + 0.996773i \(0.474423\pi\)
\(954\) −5.76157 −0.186538
\(955\) −20.5460 −0.664854
\(956\) 0.887170 0.0286931
\(957\) −0.294243 −0.00951152
\(958\) 21.1317 0.682734
\(959\) 8.18308 0.264245
\(960\) −0.0361439 −0.00116654
\(961\) −18.7231 −0.603971
\(962\) −14.3798 −0.463624
\(963\) 39.8399 1.28382
\(964\) 4.00142 0.128877
\(965\) 12.5139 0.402838
\(966\) −0.107647 −0.00346349
\(967\) −17.2293 −0.554056 −0.277028 0.960862i \(-0.589349\pi\)
−0.277028 + 0.960862i \(0.589349\pi\)
\(968\) −4.92005 −0.158136
\(969\) 0 0
\(970\) 13.6598 0.438591
\(971\) 3.80331 0.122054 0.0610270 0.998136i \(-0.480562\pi\)
0.0610270 + 0.998136i \(0.480562\pi\)
\(972\) −0.975319 −0.0312834
\(973\) −2.22597 −0.0713612
\(974\) 17.2003 0.551133
\(975\) 0.0864957 0.00277008
\(976\) 13.6642 0.437380
\(977\) 61.7916 1.97689 0.988445 0.151582i \(-0.0484366\pi\)
0.988445 + 0.151582i \(0.0484366\pi\)
\(978\) 0.573093 0.0183255
\(979\) 41.9227 1.33985
\(980\) −3.62392 −0.115762
\(981\) 4.32988 0.138242
\(982\) −33.1374 −1.05746
\(983\) 11.9175 0.380109 0.190055 0.981774i \(-0.439133\pi\)
0.190055 + 0.981774i \(0.439133\pi\)
\(984\) −0.288451 −0.00919549
\(985\) 6.02346 0.191924
\(986\) −22.2591 −0.708875
\(987\) −0.829652 −0.0264081
\(988\) 0 0
\(989\) 9.26263 0.294535
\(990\) 7.39405 0.234998
\(991\) −16.1454 −0.512874 −0.256437 0.966561i \(-0.582549\pi\)
−0.256437 + 0.966561i \(0.582549\pi\)
\(992\) 3.50384 0.111247
\(993\) 0.0360208 0.00114309
\(994\) −10.7179 −0.339950
\(995\) 11.2535 0.356761
\(996\) −0.225269 −0.00713792
\(997\) −42.5330 −1.34703 −0.673517 0.739172i \(-0.735217\pi\)
−0.673517 + 0.739172i \(0.735217\pi\)
\(998\) −31.9699 −1.01199
\(999\) 1.30282 0.0412195
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 3610.2.a.bj.1.5 9
19.2 odd 18 190.2.k.d.61.2 18
19.10 odd 18 190.2.k.d.81.2 yes 18
19.18 odd 2 3610.2.a.bi.1.5 9
95.2 even 36 950.2.u.g.99.2 36
95.29 odd 18 950.2.l.i.651.2 18
95.48 even 36 950.2.u.g.499.2 36
95.59 odd 18 950.2.l.i.251.2 18
95.67 even 36 950.2.u.g.499.5 36
95.78 even 36 950.2.u.g.99.5 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.k.d.61.2 18 19.2 odd 18
190.2.k.d.81.2 yes 18 19.10 odd 18
950.2.l.i.251.2 18 95.59 odd 18
950.2.l.i.651.2 18 95.29 odd 18
950.2.u.g.99.2 36 95.2 even 36
950.2.u.g.99.5 36 95.78 even 36
950.2.u.g.499.2 36 95.48 even 36
950.2.u.g.499.5 36 95.67 even 36
3610.2.a.bi.1.5 9 19.18 odd 2
3610.2.a.bj.1.5 9 1.1 even 1 trivial