Properties

Label 363.4.a.q.1.3
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $4$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 28x^{2} - 2x + 72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.81838\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.81838 q^{2} +3.00000 q^{3} -4.69349 q^{4} -19.2178 q^{5} +5.45514 q^{6} +14.1043 q^{7} -23.0816 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+1.81838 q^{2} +3.00000 q^{3} -4.69349 q^{4} -19.2178 q^{5} +5.45514 q^{6} +14.1043 q^{7} -23.0816 q^{8} +9.00000 q^{9} -34.9453 q^{10} -14.0805 q^{12} +54.2664 q^{13} +25.6471 q^{14} -57.6534 q^{15} -4.42322 q^{16} +78.2846 q^{17} +16.3654 q^{18} +90.2036 q^{19} +90.1986 q^{20} +42.3130 q^{21} -122.719 q^{23} -69.2448 q^{24} +244.324 q^{25} +98.6770 q^{26} +27.0000 q^{27} -66.1986 q^{28} -123.798 q^{29} -104.836 q^{30} +212.442 q^{31} +176.610 q^{32} +142.351 q^{34} -271.054 q^{35} -42.2414 q^{36} +429.123 q^{37} +164.024 q^{38} +162.799 q^{39} +443.578 q^{40} +224.405 q^{41} +76.9412 q^{42} -408.062 q^{43} -172.960 q^{45} -223.150 q^{46} +224.749 q^{47} -13.2697 q^{48} -144.068 q^{49} +444.274 q^{50} +234.854 q^{51} -254.699 q^{52} +283.806 q^{53} +49.0963 q^{54} -325.551 q^{56} +270.611 q^{57} -225.112 q^{58} -306.981 q^{59} +270.596 q^{60} +111.455 q^{61} +386.301 q^{62} +126.939 q^{63} +356.529 q^{64} -1042.88 q^{65} -153.581 q^{67} -367.428 q^{68} -368.157 q^{69} -492.880 q^{70} -323.300 q^{71} -207.734 q^{72} -194.522 q^{73} +780.309 q^{74} +732.971 q^{75} -423.370 q^{76} +296.031 q^{78} -788.138 q^{79} +85.0045 q^{80} +81.0000 q^{81} +408.054 q^{82} +870.482 q^{83} -198.596 q^{84} -1504.46 q^{85} -742.013 q^{86} -371.394 q^{87} +788.215 q^{89} -314.508 q^{90} +765.392 q^{91} +575.981 q^{92} +637.327 q^{93} +408.679 q^{94} -1733.51 q^{95} +529.829 q^{96} -852.379 q^{97} -261.970 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 12 q^{3} + 25 q^{4} + 14 q^{5} - 3 q^{6} + 20 q^{7} - 75 q^{8} + 36 q^{9} + 3 q^{10} + 75 q^{12} + 32 q^{13} + 62 q^{14} + 42 q^{15} + 289 q^{16} + 92 q^{17} - 9 q^{18} - 34 q^{19} + 391 q^{20} + 60 q^{21} - 26 q^{23} - 225 q^{24} + 334 q^{25} - 181 q^{26} + 108 q^{27} + 692 q^{28} + 174 q^{29} + 9 q^{30} + 422 q^{31} - 1271 q^{32} - 477 q^{34} - 82 q^{35} + 225 q^{36} + 518 q^{37} - 798 q^{38} + 96 q^{39} + 423 q^{40} + 428 q^{41} + 186 q^{42} - 550 q^{43} + 126 q^{45} - 2004 q^{46} + 556 q^{47} + 867 q^{48} + 282 q^{49} + 2074 q^{50} + 276 q^{51} + 467 q^{52} + 882 q^{53} - 27 q^{54} - 2112 q^{56} - 102 q^{57} - 225 q^{58} - 158 q^{59} + 1173 q^{60} + 290 q^{61} + 1142 q^{62} + 180 q^{63} + 3097 q^{64} - 1636 q^{65} + 992 q^{67} - 2033 q^{68} - 78 q^{69} + 948 q^{70} - 42 q^{71} - 675 q^{72} + 1274 q^{73} + 1509 q^{74} + 1002 q^{75} + 632 q^{76} - 543 q^{78} + 362 q^{79} + 1423 q^{80} + 324 q^{81} + 2403 q^{82} + 1500 q^{83} + 2076 q^{84} - 2388 q^{85} - 1272 q^{86} + 522 q^{87} + 1428 q^{89} + 27 q^{90} + 1750 q^{91} + 896 q^{92} + 1266 q^{93} - 1728 q^{94} - 4452 q^{95} - 3813 q^{96} + 1052 q^{97} - 615 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.81838 0.642895 0.321447 0.946927i \(-0.395831\pi\)
0.321447 + 0.946927i \(0.395831\pi\)
\(3\) 3.00000 0.577350
\(4\) −4.69349 −0.586686
\(5\) −19.2178 −1.71889 −0.859446 0.511226i \(-0.829191\pi\)
−0.859446 + 0.511226i \(0.829191\pi\)
\(6\) 5.45514 0.371175
\(7\) 14.1043 0.761563 0.380781 0.924665i \(-0.375655\pi\)
0.380781 + 0.924665i \(0.375655\pi\)
\(8\) −23.0816 −1.02007
\(9\) 9.00000 0.333333
\(10\) −34.9453 −1.10507
\(11\) 0 0
\(12\) −14.0805 −0.338724
\(13\) 54.2664 1.15775 0.578877 0.815415i \(-0.303491\pi\)
0.578877 + 0.815415i \(0.303491\pi\)
\(14\) 25.6471 0.489605
\(15\) −57.6534 −0.992403
\(16\) −4.42322 −0.0691128
\(17\) 78.2846 1.11687 0.558436 0.829548i \(-0.311402\pi\)
0.558436 + 0.829548i \(0.311402\pi\)
\(18\) 16.3654 0.214298
\(19\) 90.2036 1.08916 0.544582 0.838708i \(-0.316688\pi\)
0.544582 + 0.838708i \(0.316688\pi\)
\(20\) 90.1986 1.00845
\(21\) 42.3130 0.439689
\(22\) 0 0
\(23\) −122.719 −1.11255 −0.556276 0.830998i \(-0.687770\pi\)
−0.556276 + 0.830998i \(0.687770\pi\)
\(24\) −69.2448 −0.588939
\(25\) 244.324 1.95459
\(26\) 98.6770 0.744314
\(27\) 27.0000 0.192450
\(28\) −66.1986 −0.446799
\(29\) −123.798 −0.792715 −0.396357 0.918096i \(-0.629726\pi\)
−0.396357 + 0.918096i \(0.629726\pi\)
\(30\) −104.836 −0.638011
\(31\) 212.442 1.23083 0.615416 0.788202i \(-0.288988\pi\)
0.615416 + 0.788202i \(0.288988\pi\)
\(32\) 176.610 0.975640
\(33\) 0 0
\(34\) 142.351 0.718031
\(35\) −271.054 −1.30904
\(36\) −42.2414 −0.195562
\(37\) 429.123 1.90669 0.953343 0.301889i \(-0.0976171\pi\)
0.953343 + 0.301889i \(0.0976171\pi\)
\(38\) 164.024 0.700218
\(39\) 162.799 0.668429
\(40\) 443.578 1.75339
\(41\) 224.405 0.854784 0.427392 0.904066i \(-0.359432\pi\)
0.427392 + 0.904066i \(0.359432\pi\)
\(42\) 76.9412 0.282673
\(43\) −408.062 −1.44718 −0.723592 0.690228i \(-0.757510\pi\)
−0.723592 + 0.690228i \(0.757510\pi\)
\(44\) 0 0
\(45\) −172.960 −0.572964
\(46\) −223.150 −0.715253
\(47\) 224.749 0.697510 0.348755 0.937214i \(-0.386605\pi\)
0.348755 + 0.937214i \(0.386605\pi\)
\(48\) −13.2697 −0.0399023
\(49\) −144.068 −0.420022
\(50\) 444.274 1.25660
\(51\) 234.854 0.644826
\(52\) −254.699 −0.679238
\(53\) 283.806 0.735543 0.367771 0.929916i \(-0.380121\pi\)
0.367771 + 0.929916i \(0.380121\pi\)
\(54\) 49.0963 0.123725
\(55\) 0 0
\(56\) −325.551 −0.776849
\(57\) 270.611 0.628829
\(58\) −225.112 −0.509632
\(59\) −306.981 −0.677381 −0.338690 0.940898i \(-0.609984\pi\)
−0.338690 + 0.940898i \(0.609984\pi\)
\(60\) 270.596 0.582229
\(61\) 111.455 0.233940 0.116970 0.993135i \(-0.462682\pi\)
0.116970 + 0.993135i \(0.462682\pi\)
\(62\) 386.301 0.791295
\(63\) 126.939 0.253854
\(64\) 356.529 0.696347
\(65\) −1042.88 −1.99005
\(66\) 0 0
\(67\) −153.581 −0.280043 −0.140021 0.990148i \(-0.544717\pi\)
−0.140021 + 0.990148i \(0.544717\pi\)
\(68\) −367.428 −0.655253
\(69\) −368.157 −0.642332
\(70\) −492.880 −0.841578
\(71\) −323.300 −0.540403 −0.270202 0.962804i \(-0.587090\pi\)
−0.270202 + 0.962804i \(0.587090\pi\)
\(72\) −207.734 −0.340024
\(73\) −194.522 −0.311877 −0.155939 0.987767i \(-0.549840\pi\)
−0.155939 + 0.987767i \(0.549840\pi\)
\(74\) 780.309 1.22580
\(75\) 732.971 1.12848
\(76\) −423.370 −0.638998
\(77\) 0 0
\(78\) 296.031 0.429730
\(79\) −788.138 −1.12244 −0.561218 0.827668i \(-0.689667\pi\)
−0.561218 + 0.827668i \(0.689667\pi\)
\(80\) 85.0045 0.118797
\(81\) 81.0000 0.111111
\(82\) 408.054 0.549536
\(83\) 870.482 1.15118 0.575589 0.817739i \(-0.304773\pi\)
0.575589 + 0.817739i \(0.304773\pi\)
\(84\) −198.596 −0.257959
\(85\) −1504.46 −1.91978
\(86\) −742.013 −0.930387
\(87\) −371.394 −0.457674
\(88\) 0 0
\(89\) 788.215 0.938771 0.469385 0.882993i \(-0.344475\pi\)
0.469385 + 0.882993i \(0.344475\pi\)
\(90\) −314.508 −0.368356
\(91\) 765.392 0.881702
\(92\) 575.981 0.652719
\(93\) 637.327 0.710621
\(94\) 408.679 0.448426
\(95\) −1733.51 −1.87216
\(96\) 529.829 0.563286
\(97\) −852.379 −0.892226 −0.446113 0.894977i \(-0.647192\pi\)
−0.446113 + 0.894977i \(0.647192\pi\)
\(98\) −261.970 −0.270030
\(99\) 0 0
\(100\) −1146.73 −1.14673
\(101\) 1070.69 1.05483 0.527415 0.849608i \(-0.323161\pi\)
0.527415 + 0.849608i \(0.323161\pi\)
\(102\) 427.054 0.414555
\(103\) 1705.74 1.63176 0.815879 0.578222i \(-0.196253\pi\)
0.815879 + 0.578222i \(0.196253\pi\)
\(104\) −1252.56 −1.18099
\(105\) −813.163 −0.755777
\(106\) 516.068 0.472877
\(107\) 652.066 0.589136 0.294568 0.955630i \(-0.404824\pi\)
0.294568 + 0.955630i \(0.404824\pi\)
\(108\) −126.724 −0.112908
\(109\) 2066.09 1.81555 0.907777 0.419453i \(-0.137778\pi\)
0.907777 + 0.419453i \(0.137778\pi\)
\(110\) 0 0
\(111\) 1287.37 1.10083
\(112\) −62.3866 −0.0526337
\(113\) −181.559 −0.151147 −0.0755736 0.997140i \(-0.524079\pi\)
−0.0755736 + 0.997140i \(0.524079\pi\)
\(114\) 492.073 0.404271
\(115\) 2358.39 1.91236
\(116\) 581.045 0.465075
\(117\) 488.398 0.385918
\(118\) −558.208 −0.435484
\(119\) 1104.15 0.850568
\(120\) 1330.73 1.01232
\(121\) 0 0
\(122\) 202.668 0.150399
\(123\) 673.215 0.493510
\(124\) −997.097 −0.722112
\(125\) −2293.14 −1.64084
\(126\) 230.824 0.163202
\(127\) −755.589 −0.527934 −0.263967 0.964532i \(-0.585031\pi\)
−0.263967 + 0.964532i \(0.585031\pi\)
\(128\) −764.571 −0.527962
\(129\) −1224.19 −0.835532
\(130\) −1896.36 −1.27939
\(131\) 252.052 0.168106 0.0840531 0.996461i \(-0.473213\pi\)
0.0840531 + 0.996461i \(0.473213\pi\)
\(132\) 0 0
\(133\) 1272.26 0.829467
\(134\) −279.268 −0.180038
\(135\) −518.881 −0.330801
\(136\) −1806.94 −1.13929
\(137\) −2149.24 −1.34031 −0.670153 0.742223i \(-0.733771\pi\)
−0.670153 + 0.742223i \(0.733771\pi\)
\(138\) −669.450 −0.412952
\(139\) 607.426 0.370656 0.185328 0.982677i \(-0.440665\pi\)
0.185328 + 0.982677i \(0.440665\pi\)
\(140\) 1272.19 0.767998
\(141\) 674.246 0.402708
\(142\) −587.882 −0.347422
\(143\) 0 0
\(144\) −39.8090 −0.0230376
\(145\) 2379.13 1.36259
\(146\) −353.714 −0.200504
\(147\) −432.203 −0.242500
\(148\) −2014.09 −1.11863
\(149\) −2679.77 −1.47339 −0.736697 0.676224i \(-0.763615\pi\)
−0.736697 + 0.676224i \(0.763615\pi\)
\(150\) 1332.82 0.725496
\(151\) −529.332 −0.285275 −0.142637 0.989775i \(-0.545558\pi\)
−0.142637 + 0.989775i \(0.545558\pi\)
\(152\) −2082.04 −1.11103
\(153\) 704.562 0.372291
\(154\) 0 0
\(155\) −4082.68 −2.11567
\(156\) −764.097 −0.392158
\(157\) 111.990 0.0569283 0.0284642 0.999595i \(-0.490938\pi\)
0.0284642 + 0.999595i \(0.490938\pi\)
\(158\) −1433.13 −0.721608
\(159\) 851.418 0.424666
\(160\) −3394.05 −1.67702
\(161\) −1730.87 −0.847278
\(162\) 147.289 0.0714327
\(163\) 2318.99 1.11434 0.557171 0.830398i \(-0.311887\pi\)
0.557171 + 0.830398i \(0.311887\pi\)
\(164\) −1053.24 −0.501490
\(165\) 0 0
\(166\) 1582.87 0.740087
\(167\) 3361.08 1.55741 0.778707 0.627387i \(-0.215876\pi\)
0.778707 + 0.627387i \(0.215876\pi\)
\(168\) −976.652 −0.448514
\(169\) 747.844 0.340393
\(170\) −2735.68 −1.23422
\(171\) 811.832 0.363055
\(172\) 1915.24 0.849043
\(173\) −872.145 −0.383283 −0.191642 0.981465i \(-0.561381\pi\)
−0.191642 + 0.981465i \(0.561381\pi\)
\(174\) −675.336 −0.294236
\(175\) 3446.03 1.48854
\(176\) 0 0
\(177\) −920.942 −0.391086
\(178\) 1433.27 0.603531
\(179\) 1774.29 0.740874 0.370437 0.928858i \(-0.379208\pi\)
0.370437 + 0.928858i \(0.379208\pi\)
\(180\) 811.787 0.336150
\(181\) −969.085 −0.397964 −0.198982 0.980003i \(-0.563764\pi\)
−0.198982 + 0.980003i \(0.563764\pi\)
\(182\) 1391.77 0.566842
\(183\) 334.365 0.135065
\(184\) 2832.55 1.13488
\(185\) −8246.80 −3.27739
\(186\) 1158.90 0.456855
\(187\) 0 0
\(188\) −1054.86 −0.409220
\(189\) 380.817 0.146563
\(190\) −3152.19 −1.20360
\(191\) 796.117 0.301597 0.150798 0.988565i \(-0.451816\pi\)
0.150798 + 0.988565i \(0.451816\pi\)
\(192\) 1069.59 0.402036
\(193\) −3167.89 −1.18150 −0.590750 0.806855i \(-0.701168\pi\)
−0.590750 + 0.806855i \(0.701168\pi\)
\(194\) −1549.95 −0.573608
\(195\) −3128.64 −1.14896
\(196\) 676.180 0.246421
\(197\) 1309.07 0.473438 0.236719 0.971578i \(-0.423928\pi\)
0.236719 + 0.971578i \(0.423928\pi\)
\(198\) 0 0
\(199\) 2639.16 0.940125 0.470063 0.882633i \(-0.344231\pi\)
0.470063 + 0.882633i \(0.344231\pi\)
\(200\) −5639.38 −1.99382
\(201\) −460.742 −0.161683
\(202\) 1946.93 0.678145
\(203\) −1746.09 −0.603702
\(204\) −1102.28 −0.378311
\(205\) −4312.57 −1.46928
\(206\) 3101.68 1.04905
\(207\) −1104.47 −0.370850
\(208\) −240.032 −0.0800156
\(209\) 0 0
\(210\) −1478.64 −0.485885
\(211\) −1067.34 −0.348239 −0.174120 0.984725i \(-0.555708\pi\)
−0.174120 + 0.984725i \(0.555708\pi\)
\(212\) −1332.04 −0.431533
\(213\) −969.899 −0.312002
\(214\) 1185.70 0.378753
\(215\) 7842.06 2.48755
\(216\) −623.203 −0.196313
\(217\) 2996.36 0.937356
\(218\) 3756.94 1.16721
\(219\) −583.565 −0.180062
\(220\) 0 0
\(221\) 4248.23 1.29306
\(222\) 2340.93 0.707715
\(223\) 2488.35 0.747229 0.373615 0.927584i \(-0.378118\pi\)
0.373615 + 0.927584i \(0.378118\pi\)
\(224\) 2490.96 0.743011
\(225\) 2198.91 0.651530
\(226\) −330.143 −0.0971717
\(227\) −4030.74 −1.17854 −0.589272 0.807935i \(-0.700585\pi\)
−0.589272 + 0.807935i \(0.700585\pi\)
\(228\) −1270.11 −0.368926
\(229\) 2301.18 0.664046 0.332023 0.943271i \(-0.392269\pi\)
0.332023 + 0.943271i \(0.392269\pi\)
\(230\) 4288.45 1.22944
\(231\) 0 0
\(232\) 2857.46 0.808627
\(233\) 1153.47 0.324320 0.162160 0.986765i \(-0.448154\pi\)
0.162160 + 0.986765i \(0.448154\pi\)
\(234\) 888.093 0.248105
\(235\) −4319.18 −1.19894
\(236\) 1440.81 0.397410
\(237\) −2364.41 −0.648038
\(238\) 2007.77 0.546826
\(239\) −633.939 −0.171574 −0.0857869 0.996314i \(-0.527340\pi\)
−0.0857869 + 0.996314i \(0.527340\pi\)
\(240\) 255.014 0.0685877
\(241\) −3185.15 −0.851343 −0.425671 0.904878i \(-0.639962\pi\)
−0.425671 + 0.904878i \(0.639962\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) −523.113 −0.137250
\(245\) 2768.66 0.721973
\(246\) 1224.16 0.317275
\(247\) 4895.02 1.26098
\(248\) −4903.51 −1.25554
\(249\) 2611.45 0.664633
\(250\) −4169.80 −1.05489
\(251\) −2805.97 −0.705622 −0.352811 0.935695i \(-0.614774\pi\)
−0.352811 + 0.935695i \(0.614774\pi\)
\(252\) −595.787 −0.148933
\(253\) 0 0
\(254\) −1373.95 −0.339406
\(255\) −4513.38 −1.10839
\(256\) −4242.52 −1.03577
\(257\) −3930.62 −0.954029 −0.477014 0.878895i \(-0.658281\pi\)
−0.477014 + 0.878895i \(0.658281\pi\)
\(258\) −2226.04 −0.537159
\(259\) 6052.50 1.45206
\(260\) 4894.75 1.16754
\(261\) −1114.18 −0.264238
\(262\) 458.327 0.108075
\(263\) −7363.55 −1.72645 −0.863224 0.504821i \(-0.831559\pi\)
−0.863224 + 0.504821i \(0.831559\pi\)
\(264\) 0 0
\(265\) −5454.13 −1.26432
\(266\) 2313.46 0.533260
\(267\) 2364.64 0.542000
\(268\) 720.829 0.164297
\(269\) −5.56813 −0.00126206 −0.000631032 1.00000i \(-0.500201\pi\)
−0.000631032 1.00000i \(0.500201\pi\)
\(270\) −943.523 −0.212670
\(271\) 2378.32 0.533108 0.266554 0.963820i \(-0.414115\pi\)
0.266554 + 0.963820i \(0.414115\pi\)
\(272\) −346.270 −0.0771901
\(273\) 2296.18 0.509051
\(274\) −3908.14 −0.861676
\(275\) 0 0
\(276\) 1727.94 0.376847
\(277\) 5629.04 1.22100 0.610499 0.792017i \(-0.290969\pi\)
0.610499 + 0.792017i \(0.290969\pi\)
\(278\) 1104.53 0.238293
\(279\) 1911.98 0.410277
\(280\) 6256.37 1.33532
\(281\) 3642.04 0.773189 0.386594 0.922250i \(-0.373651\pi\)
0.386594 + 0.922250i \(0.373651\pi\)
\(282\) 1226.04 0.258899
\(283\) −4526.44 −0.950773 −0.475386 0.879777i \(-0.657692\pi\)
−0.475386 + 0.879777i \(0.657692\pi\)
\(284\) 1517.40 0.317047
\(285\) −5200.54 −1.08089
\(286\) 0 0
\(287\) 3165.08 0.650972
\(288\) 1589.49 0.325213
\(289\) 1215.49 0.247402
\(290\) 4326.16 0.876003
\(291\) −2557.14 −0.515127
\(292\) 912.985 0.182974
\(293\) −3472.04 −0.692282 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(294\) −785.909 −0.155902
\(295\) 5899.49 1.16434
\(296\) −9904.85 −1.94496
\(297\) 0 0
\(298\) −4872.85 −0.947237
\(299\) −6659.52 −1.28806
\(300\) −3440.19 −0.662066
\(301\) −5755.45 −1.10212
\(302\) −962.528 −0.183402
\(303\) 3212.08 0.609007
\(304\) −398.990 −0.0752752
\(305\) −2141.92 −0.402118
\(306\) 1281.16 0.239344
\(307\) 6657.30 1.23763 0.618815 0.785537i \(-0.287613\pi\)
0.618815 + 0.785537i \(0.287613\pi\)
\(308\) 0 0
\(309\) 5117.21 0.942096
\(310\) −7423.86 −1.36015
\(311\) −722.660 −0.131763 −0.0658815 0.997827i \(-0.520986\pi\)
−0.0658815 + 0.997827i \(0.520986\pi\)
\(312\) −3757.67 −0.681846
\(313\) −4727.15 −0.853657 −0.426828 0.904333i \(-0.640369\pi\)
−0.426828 + 0.904333i \(0.640369\pi\)
\(314\) 203.640 0.0365989
\(315\) −2439.49 −0.436348
\(316\) 3699.12 0.658518
\(317\) 11094.3 1.96567 0.982837 0.184474i \(-0.0590583\pi\)
0.982837 + 0.184474i \(0.0590583\pi\)
\(318\) 1548.20 0.273015
\(319\) 0 0
\(320\) −6851.71 −1.19694
\(321\) 1956.20 0.340138
\(322\) −3147.38 −0.544710
\(323\) 7061.56 1.21646
\(324\) −380.173 −0.0651874
\(325\) 13258.6 2.26293
\(326\) 4216.82 0.716404
\(327\) 6198.26 1.04821
\(328\) −5179.62 −0.871942
\(329\) 3169.93 0.531198
\(330\) 0 0
\(331\) −3064.62 −0.508902 −0.254451 0.967086i \(-0.581895\pi\)
−0.254451 + 0.967086i \(0.581895\pi\)
\(332\) −4085.60 −0.675381
\(333\) 3862.11 0.635562
\(334\) 6111.73 1.00125
\(335\) 2951.48 0.481363
\(336\) −187.160 −0.0303881
\(337\) 7186.71 1.16168 0.580838 0.814019i \(-0.302725\pi\)
0.580838 + 0.814019i \(0.302725\pi\)
\(338\) 1359.86 0.218837
\(339\) −544.677 −0.0872648
\(340\) 7061.16 1.12631
\(341\) 0 0
\(342\) 1476.22 0.233406
\(343\) −6869.77 −1.08144
\(344\) 9418.73 1.47623
\(345\) 7075.17 1.10410
\(346\) −1585.89 −0.246411
\(347\) −10910.7 −1.68794 −0.843971 0.536389i \(-0.819788\pi\)
−0.843971 + 0.536389i \(0.819788\pi\)
\(348\) 1743.14 0.268511
\(349\) 7635.29 1.17108 0.585541 0.810643i \(-0.300882\pi\)
0.585541 + 0.810643i \(0.300882\pi\)
\(350\) 6266.19 0.956977
\(351\) 1465.19 0.222810
\(352\) 0 0
\(353\) 5344.39 0.805817 0.402908 0.915240i \(-0.367999\pi\)
0.402908 + 0.915240i \(0.367999\pi\)
\(354\) −1674.62 −0.251427
\(355\) 6213.11 0.928895
\(356\) −3699.48 −0.550764
\(357\) 3312.46 0.491076
\(358\) 3226.33 0.476304
\(359\) 3249.23 0.477682 0.238841 0.971059i \(-0.423233\pi\)
0.238841 + 0.971059i \(0.423233\pi\)
\(360\) 3992.20 0.584465
\(361\) 1277.69 0.186279
\(362\) −1762.17 −0.255849
\(363\) 0 0
\(364\) −3592.36 −0.517283
\(365\) 3738.28 0.536083
\(366\) 608.003 0.0868329
\(367\) 2898.60 0.412277 0.206138 0.978523i \(-0.433910\pi\)
0.206138 + 0.978523i \(0.433910\pi\)
\(368\) 542.813 0.0768915
\(369\) 2019.64 0.284928
\(370\) −14995.8 −2.10702
\(371\) 4002.90 0.560162
\(372\) −2991.29 −0.416912
\(373\) −10883.1 −1.51074 −0.755372 0.655297i \(-0.772544\pi\)
−0.755372 + 0.655297i \(0.772544\pi\)
\(374\) 0 0
\(375\) −6879.42 −0.947338
\(376\) −5187.56 −0.711511
\(377\) −6718.08 −0.917768
\(378\) 692.471 0.0942245
\(379\) 521.221 0.0706420 0.0353210 0.999376i \(-0.488755\pi\)
0.0353210 + 0.999376i \(0.488755\pi\)
\(380\) 8136.23 1.09837
\(381\) −2266.77 −0.304803
\(382\) 1447.64 0.193895
\(383\) −10668.3 −1.42330 −0.711649 0.702536i \(-0.752051\pi\)
−0.711649 + 0.702536i \(0.752051\pi\)
\(384\) −2293.71 −0.304819
\(385\) 0 0
\(386\) −5760.43 −0.759580
\(387\) −3672.56 −0.482395
\(388\) 4000.63 0.523457
\(389\) 2186.58 0.284997 0.142499 0.989795i \(-0.454486\pi\)
0.142499 + 0.989795i \(0.454486\pi\)
\(390\) −5689.07 −0.738659
\(391\) −9607.01 −1.24258
\(392\) 3325.31 0.428453
\(393\) 756.157 0.0970562
\(394\) 2380.39 0.304371
\(395\) 15146.3 1.92935
\(396\) 0 0
\(397\) 2360.44 0.298406 0.149203 0.988807i \(-0.452329\pi\)
0.149203 + 0.988807i \(0.452329\pi\)
\(398\) 4798.99 0.604401
\(399\) 3816.79 0.478893
\(400\) −1080.70 −0.135087
\(401\) 5041.37 0.627815 0.313908 0.949454i \(-0.398362\pi\)
0.313908 + 0.949454i \(0.398362\pi\)
\(402\) −837.804 −0.103945
\(403\) 11528.5 1.42500
\(404\) −5025.29 −0.618855
\(405\) −1556.64 −0.190988
\(406\) −3175.06 −0.388117
\(407\) 0 0
\(408\) −5420.81 −0.657769
\(409\) −574.833 −0.0694955 −0.0347477 0.999396i \(-0.511063\pi\)
−0.0347477 + 0.999396i \(0.511063\pi\)
\(410\) −7841.89 −0.944594
\(411\) −6447.72 −0.773826
\(412\) −8005.85 −0.957330
\(413\) −4329.76 −0.515868
\(414\) −2008.35 −0.238418
\(415\) −16728.7 −1.97875
\(416\) 9583.98 1.12955
\(417\) 1822.28 0.213998
\(418\) 0 0
\(419\) 690.078 0.0804594 0.0402297 0.999190i \(-0.487191\pi\)
0.0402297 + 0.999190i \(0.487191\pi\)
\(420\) 3816.57 0.443404
\(421\) 13982.6 1.61869 0.809345 0.587334i \(-0.199823\pi\)
0.809345 + 0.587334i \(0.199823\pi\)
\(422\) −1940.82 −0.223881
\(423\) 2022.74 0.232503
\(424\) −6550.70 −0.750307
\(425\) 19126.8 2.18303
\(426\) −1763.65 −0.200584
\(427\) 1572.00 0.178160
\(428\) −3060.47 −0.345638
\(429\) 0 0
\(430\) 14259.8 1.59923
\(431\) 11745.4 1.31266 0.656332 0.754472i \(-0.272107\pi\)
0.656332 + 0.754472i \(0.272107\pi\)
\(432\) −119.427 −0.0133008
\(433\) 167.526 0.0185930 0.00929651 0.999957i \(-0.497041\pi\)
0.00929651 + 0.999957i \(0.497041\pi\)
\(434\) 5448.53 0.602621
\(435\) 7137.38 0.786693
\(436\) −9697.17 −1.06516
\(437\) −11069.7 −1.21175
\(438\) −1061.14 −0.115761
\(439\) −3975.68 −0.432229 −0.216115 0.976368i \(-0.569339\pi\)
−0.216115 + 0.976368i \(0.569339\pi\)
\(440\) 0 0
\(441\) −1296.61 −0.140007
\(442\) 7724.90 0.831303
\(443\) −6117.57 −0.656105 −0.328053 0.944659i \(-0.606392\pi\)
−0.328053 + 0.944659i \(0.606392\pi\)
\(444\) −6042.26 −0.645839
\(445\) −15147.8 −1.61365
\(446\) 4524.77 0.480390
\(447\) −8039.32 −0.850664
\(448\) 5028.61 0.530312
\(449\) 12725.6 1.33755 0.668773 0.743467i \(-0.266820\pi\)
0.668773 + 0.743467i \(0.266820\pi\)
\(450\) 3998.46 0.418865
\(451\) 0 0
\(452\) 852.145 0.0886760
\(453\) −1588.00 −0.164703
\(454\) −7329.42 −0.757680
\(455\) −14709.1 −1.51555
\(456\) −6246.13 −0.641451
\(457\) −13086.5 −1.33952 −0.669759 0.742579i \(-0.733603\pi\)
−0.669759 + 0.742579i \(0.733603\pi\)
\(458\) 4184.43 0.426911
\(459\) 2113.69 0.214942
\(460\) −11069.1 −1.12195
\(461\) 7251.82 0.732649 0.366324 0.930487i \(-0.380616\pi\)
0.366324 + 0.930487i \(0.380616\pi\)
\(462\) 0 0
\(463\) −1559.25 −0.156511 −0.0782553 0.996933i \(-0.524935\pi\)
−0.0782553 + 0.996933i \(0.524935\pi\)
\(464\) 547.586 0.0547867
\(465\) −12248.0 −1.22148
\(466\) 2097.45 0.208503
\(467\) 9284.25 0.919966 0.459983 0.887928i \(-0.347856\pi\)
0.459983 + 0.887928i \(0.347856\pi\)
\(468\) −2292.29 −0.226413
\(469\) −2166.15 −0.213270
\(470\) −7853.91 −0.770795
\(471\) 335.969 0.0328676
\(472\) 7085.60 0.690977
\(473\) 0 0
\(474\) −4299.40 −0.416620
\(475\) 22038.9 2.12887
\(476\) −5182.33 −0.499017
\(477\) 2554.25 0.245181
\(478\) −1152.74 −0.110304
\(479\) −10669.5 −1.01775 −0.508873 0.860842i \(-0.669938\pi\)
−0.508873 + 0.860842i \(0.669938\pi\)
\(480\) −10182.2 −0.968228
\(481\) 23287.0 2.20747
\(482\) −5791.82 −0.547324
\(483\) −5192.61 −0.489176
\(484\) 0 0
\(485\) 16380.8 1.53364
\(486\) 441.867 0.0412417
\(487\) −15539.4 −1.44591 −0.722955 0.690895i \(-0.757217\pi\)
−0.722955 + 0.690895i \(0.757217\pi\)
\(488\) −2572.56 −0.238636
\(489\) 6956.98 0.643365
\(490\) 5034.48 0.464152
\(491\) −14350.7 −1.31902 −0.659511 0.751695i \(-0.729237\pi\)
−0.659511 + 0.751695i \(0.729237\pi\)
\(492\) −3159.73 −0.289536
\(493\) −9691.49 −0.885361
\(494\) 8901.02 0.810680
\(495\) 0 0
\(496\) −939.679 −0.0850662
\(497\) −4559.93 −0.411551
\(498\) 4748.60 0.427289
\(499\) 9290.76 0.833490 0.416745 0.909023i \(-0.363171\pi\)
0.416745 + 0.909023i \(0.363171\pi\)
\(500\) 10762.8 0.962657
\(501\) 10083.2 0.899174
\(502\) −5102.32 −0.453640
\(503\) −11744.5 −1.04107 −0.520537 0.853839i \(-0.674268\pi\)
−0.520537 + 0.853839i \(0.674268\pi\)
\(504\) −2929.96 −0.258950
\(505\) −20576.4 −1.81314
\(506\) 0 0
\(507\) 2243.53 0.196526
\(508\) 3546.35 0.309732
\(509\) 2755.59 0.239959 0.119980 0.992776i \(-0.461717\pi\)
0.119980 + 0.992776i \(0.461717\pi\)
\(510\) −8207.04 −0.712576
\(511\) −2743.60 −0.237514
\(512\) −1597.94 −0.137929
\(513\) 2435.50 0.209610
\(514\) −7147.37 −0.613340
\(515\) −32780.5 −2.80482
\(516\) 5745.71 0.490195
\(517\) 0 0
\(518\) 11005.7 0.933523
\(519\) −2616.44 −0.221289
\(520\) 24071.4 2.03000
\(521\) 11366.3 0.955794 0.477897 0.878416i \(-0.341399\pi\)
0.477897 + 0.878416i \(0.341399\pi\)
\(522\) −2026.01 −0.169877
\(523\) 1742.55 0.145691 0.0728455 0.997343i \(-0.476792\pi\)
0.0728455 + 0.997343i \(0.476792\pi\)
\(524\) −1183.01 −0.0986257
\(525\) 10338.1 0.859411
\(526\) −13389.7 −1.10992
\(527\) 16631.0 1.37468
\(528\) 0 0
\(529\) 2892.96 0.237771
\(530\) −9917.68 −0.812824
\(531\) −2762.82 −0.225794
\(532\) −5971.35 −0.486637
\(533\) 12177.6 0.989629
\(534\) 4299.82 0.348449
\(535\) −12531.3 −1.01266
\(536\) 3544.89 0.285664
\(537\) 5322.86 0.427744
\(538\) −10.1250 −0.000811374 0
\(539\) 0 0
\(540\) 2435.36 0.194076
\(541\) −16859.5 −1.33983 −0.669913 0.742439i \(-0.733669\pi\)
−0.669913 + 0.742439i \(0.733669\pi\)
\(542\) 4324.68 0.342733
\(543\) −2907.25 −0.229765
\(544\) 13825.8 1.08966
\(545\) −39705.7 −3.12074
\(546\) 4175.32 0.327266
\(547\) −3129.06 −0.244587 −0.122293 0.992494i \(-0.539025\pi\)
−0.122293 + 0.992494i \(0.539025\pi\)
\(548\) 10087.4 0.786339
\(549\) 1003.10 0.0779801
\(550\) 0 0
\(551\) −11167.0 −0.863397
\(552\) 8497.65 0.655225
\(553\) −11116.2 −0.854805
\(554\) 10235.7 0.784973
\(555\) −24740.4 −1.89220
\(556\) −2850.95 −0.217459
\(557\) −12117.1 −0.921754 −0.460877 0.887464i \(-0.652465\pi\)
−0.460877 + 0.887464i \(0.652465\pi\)
\(558\) 3476.71 0.263765
\(559\) −22144.1 −1.67548
\(560\) 1198.93 0.0904717
\(561\) 0 0
\(562\) 6622.62 0.497079
\(563\) −8619.51 −0.645238 −0.322619 0.946529i \(-0.604563\pi\)
−0.322619 + 0.946529i \(0.604563\pi\)
\(564\) −3164.57 −0.236263
\(565\) 3489.16 0.259806
\(566\) −8230.78 −0.611247
\(567\) 1142.45 0.0846181
\(568\) 7462.28 0.551250
\(569\) 17596.1 1.29642 0.648212 0.761460i \(-0.275517\pi\)
0.648212 + 0.761460i \(0.275517\pi\)
\(570\) −9456.57 −0.694898
\(571\) 16207.2 1.18783 0.593916 0.804527i \(-0.297581\pi\)
0.593916 + 0.804527i \(0.297581\pi\)
\(572\) 0 0
\(573\) 2388.35 0.174127
\(574\) 5755.33 0.418506
\(575\) −29983.2 −2.17458
\(576\) 3208.77 0.232116
\(577\) −16520.8 −1.19197 −0.595987 0.802994i \(-0.703239\pi\)
−0.595987 + 0.802994i \(0.703239\pi\)
\(578\) 2210.22 0.159053
\(579\) −9503.66 −0.682139
\(580\) −11166.4 −0.799414
\(581\) 12277.6 0.876695
\(582\) −4649.85 −0.331173
\(583\) 0 0
\(584\) 4489.87 0.318137
\(585\) −9385.93 −0.663351
\(586\) −6313.49 −0.445065
\(587\) −14514.9 −1.02060 −0.510301 0.859996i \(-0.670466\pi\)
−0.510301 + 0.859996i \(0.670466\pi\)
\(588\) 2028.54 0.142271
\(589\) 19163.1 1.34058
\(590\) 10727.5 0.748551
\(591\) 3927.21 0.273340
\(592\) −1898.10 −0.131776
\(593\) −6639.31 −0.459771 −0.229885 0.973218i \(-0.573835\pi\)
−0.229885 + 0.973218i \(0.573835\pi\)
\(594\) 0 0
\(595\) −21219.4 −1.46203
\(596\) 12577.5 0.864420
\(597\) 7917.47 0.542781
\(598\) −12109.5 −0.828087
\(599\) 2835.72 0.193430 0.0967148 0.995312i \(-0.469167\pi\)
0.0967148 + 0.995312i \(0.469167\pi\)
\(600\) −16918.2 −1.15113
\(601\) −12841.2 −0.871555 −0.435778 0.900054i \(-0.643527\pi\)
−0.435778 + 0.900054i \(0.643527\pi\)
\(602\) −10465.6 −0.708548
\(603\) −1382.23 −0.0933475
\(604\) 2484.42 0.167367
\(605\) 0 0
\(606\) 5840.78 0.391527
\(607\) −15853.7 −1.06010 −0.530050 0.847966i \(-0.677827\pi\)
−0.530050 + 0.847966i \(0.677827\pi\)
\(608\) 15930.8 1.06263
\(609\) −5238.27 −0.348548
\(610\) −3894.83 −0.258520
\(611\) 12196.3 0.807545
\(612\) −3306.85 −0.218418
\(613\) −10069.5 −0.663467 −0.331733 0.943373i \(-0.607633\pi\)
−0.331733 + 0.943373i \(0.607633\pi\)
\(614\) 12105.5 0.795665
\(615\) −12937.7 −0.848290
\(616\) 0 0
\(617\) −4947.89 −0.322843 −0.161422 0.986886i \(-0.551608\pi\)
−0.161422 + 0.986886i \(0.551608\pi\)
\(618\) 9305.03 0.605669
\(619\) −14401.3 −0.935114 −0.467557 0.883963i \(-0.654866\pi\)
−0.467557 + 0.883963i \(0.654866\pi\)
\(620\) 19162.0 1.24123
\(621\) −3313.41 −0.214111
\(622\) −1314.07 −0.0847097
\(623\) 11117.3 0.714933
\(624\) −720.097 −0.0461970
\(625\) 13528.6 0.865833
\(626\) −8595.77 −0.548811
\(627\) 0 0
\(628\) −525.622 −0.0333991
\(629\) 33593.7 2.12952
\(630\) −4435.92 −0.280526
\(631\) −20392.2 −1.28653 −0.643264 0.765645i \(-0.722420\pi\)
−0.643264 + 0.765645i \(0.722420\pi\)
\(632\) 18191.5 1.14497
\(633\) −3202.01 −0.201056
\(634\) 20173.7 1.26372
\(635\) 14520.8 0.907462
\(636\) −3996.12 −0.249146
\(637\) −7818.03 −0.486282
\(638\) 0 0
\(639\) −2909.70 −0.180134
\(640\) 14693.4 0.907511
\(641\) −19353.8 −1.19256 −0.596278 0.802778i \(-0.703354\pi\)
−0.596278 + 0.802778i \(0.703354\pi\)
\(642\) 3557.11 0.218673
\(643\) −4443.84 −0.272547 −0.136274 0.990671i \(-0.543513\pi\)
−0.136274 + 0.990671i \(0.543513\pi\)
\(644\) 8123.83 0.497086
\(645\) 23526.2 1.43619
\(646\) 12840.6 0.782054
\(647\) −5631.62 −0.342198 −0.171099 0.985254i \(-0.554732\pi\)
−0.171099 + 0.985254i \(0.554732\pi\)
\(648\) −1869.61 −0.113341
\(649\) 0 0
\(650\) 24109.1 1.45483
\(651\) 8989.08 0.541183
\(652\) −10884.2 −0.653769
\(653\) 18345.4 1.09940 0.549702 0.835361i \(-0.314741\pi\)
0.549702 + 0.835361i \(0.314741\pi\)
\(654\) 11270.8 0.673889
\(655\) −4843.89 −0.288957
\(656\) −992.592 −0.0590765
\(657\) −1750.69 −0.103959
\(658\) 5764.15 0.341504
\(659\) 8999.50 0.531974 0.265987 0.963977i \(-0.414302\pi\)
0.265987 + 0.963977i \(0.414302\pi\)
\(660\) 0 0
\(661\) −22524.0 −1.32539 −0.662695 0.748890i \(-0.730587\pi\)
−0.662695 + 0.748890i \(0.730587\pi\)
\(662\) −5572.64 −0.327171
\(663\) 12744.7 0.746550
\(664\) −20092.1 −1.17429
\(665\) −24450.1 −1.42576
\(666\) 7022.78 0.408600
\(667\) 15192.4 0.881936
\(668\) −15775.2 −0.913714
\(669\) 7465.05 0.431413
\(670\) 5366.92 0.309466
\(671\) 0 0
\(672\) 7472.89 0.428978
\(673\) −22164.9 −1.26953 −0.634766 0.772704i \(-0.718904\pi\)
−0.634766 + 0.772704i \(0.718904\pi\)
\(674\) 13068.2 0.746836
\(675\) 6596.74 0.376161
\(676\) −3510.00 −0.199704
\(677\) −21982.0 −1.24792 −0.623958 0.781458i \(-0.714476\pi\)
−0.623958 + 0.781458i \(0.714476\pi\)
\(678\) −990.430 −0.0561021
\(679\) −12022.2 −0.679486
\(680\) 34725.3 1.95832
\(681\) −12092.2 −0.680433
\(682\) 0 0
\(683\) −14360.0 −0.804495 −0.402247 0.915531i \(-0.631771\pi\)
−0.402247 + 0.915531i \(0.631771\pi\)
\(684\) −3810.33 −0.212999
\(685\) 41303.6 2.30384
\(686\) −12491.9 −0.695250
\(687\) 6903.55 0.383387
\(688\) 1804.95 0.100019
\(689\) 15401.1 0.851577
\(690\) 12865.4 0.709820
\(691\) 28580.0 1.57342 0.786711 0.617322i \(-0.211782\pi\)
0.786711 + 0.617322i \(0.211782\pi\)
\(692\) 4093.41 0.224867
\(693\) 0 0
\(694\) −19839.8 −1.08517
\(695\) −11673.4 −0.637118
\(696\) 8572.38 0.466861
\(697\) 17567.5 0.954684
\(698\) 13883.9 0.752882
\(699\) 3460.42 0.187246
\(700\) −16173.9 −0.873308
\(701\) −5713.11 −0.307819 −0.153910 0.988085i \(-0.549186\pi\)
−0.153910 + 0.988085i \(0.549186\pi\)
\(702\) 2664.28 0.143243
\(703\) 38708.4 2.07669
\(704\) 0 0
\(705\) −12957.5 −0.692211
\(706\) 9718.14 0.518055
\(707\) 15101.4 0.803320
\(708\) 4322.43 0.229445
\(709\) 17566.5 0.930501 0.465250 0.885179i \(-0.345964\pi\)
0.465250 + 0.885179i \(0.345964\pi\)
\(710\) 11297.8 0.597182
\(711\) −7093.24 −0.374145
\(712\) −18193.3 −0.957614
\(713\) −26070.7 −1.36936
\(714\) 6023.31 0.315710
\(715\) 0 0
\(716\) −8327.60 −0.434661
\(717\) −1901.82 −0.0990582
\(718\) 5908.34 0.307099
\(719\) 3747.65 0.194386 0.0971931 0.995266i \(-0.469014\pi\)
0.0971931 + 0.995266i \(0.469014\pi\)
\(720\) 765.041 0.0395991
\(721\) 24058.3 1.24269
\(722\) 2323.32 0.119758
\(723\) −9555.45 −0.491523
\(724\) 4548.39 0.233480
\(725\) −30246.8 −1.54943
\(726\) 0 0
\(727\) 19566.4 0.998183 0.499092 0.866549i \(-0.333667\pi\)
0.499092 + 0.866549i \(0.333667\pi\)
\(728\) −17666.5 −0.899400
\(729\) 729.000 0.0370370
\(730\) 6797.61 0.344645
\(731\) −31945.0 −1.61632
\(732\) −1569.34 −0.0792411
\(733\) −5341.99 −0.269183 −0.134591 0.990901i \(-0.542972\pi\)
−0.134591 + 0.990901i \(0.542972\pi\)
\(734\) 5270.75 0.265051
\(735\) 8305.98 0.416831
\(736\) −21673.4 −1.08545
\(737\) 0 0
\(738\) 3672.48 0.183179
\(739\) −30927.1 −1.53947 −0.769737 0.638361i \(-0.779613\pi\)
−0.769737 + 0.638361i \(0.779613\pi\)
\(740\) 38706.3 1.92280
\(741\) 14685.1 0.728029
\(742\) 7278.79 0.360125
\(743\) 26069.2 1.28720 0.643598 0.765364i \(-0.277441\pi\)
0.643598 + 0.765364i \(0.277441\pi\)
\(744\) −14710.5 −0.724885
\(745\) 51499.3 2.53260
\(746\) −19789.7 −0.971249
\(747\) 7834.34 0.383726
\(748\) 0 0
\(749\) 9196.96 0.448664
\(750\) −12509.4 −0.609039
\(751\) −5928.45 −0.288059 −0.144029 0.989573i \(-0.546006\pi\)
−0.144029 + 0.989573i \(0.546006\pi\)
\(752\) −994.113 −0.0482069
\(753\) −8417.90 −0.407391
\(754\) −12216.0 −0.590029
\(755\) 10172.6 0.490356
\(756\) −1787.36 −0.0859864
\(757\) 12040.3 0.578089 0.289044 0.957316i \(-0.406663\pi\)
0.289044 + 0.957316i \(0.406663\pi\)
\(758\) 947.779 0.0454154
\(759\) 0 0
\(760\) 40012.3 1.90973
\(761\) −23222.8 −1.10621 −0.553106 0.833111i \(-0.686558\pi\)
−0.553106 + 0.833111i \(0.686558\pi\)
\(762\) −4121.85 −0.195956
\(763\) 29140.8 1.38266
\(764\) −3736.57 −0.176943
\(765\) −13540.1 −0.639927
\(766\) −19399.0 −0.915030
\(767\) −16658.7 −0.784240
\(768\) −12727.6 −0.598003
\(769\) −4256.37 −0.199595 −0.0997975 0.995008i \(-0.531819\pi\)
−0.0997975 + 0.995008i \(0.531819\pi\)
\(770\) 0 0
\(771\) −11791.9 −0.550809
\(772\) 14868.5 0.693170
\(773\) 17107.1 0.795987 0.397994 0.917388i \(-0.369707\pi\)
0.397994 + 0.917388i \(0.369707\pi\)
\(774\) −6678.11 −0.310129
\(775\) 51904.7 2.40577
\(776\) 19674.3 0.910135
\(777\) 18157.5 0.838348
\(778\) 3976.04 0.183223
\(779\) 20242.1 0.931000
\(780\) 14684.3 0.674078
\(781\) 0 0
\(782\) −17469.2 −0.798846
\(783\) −3342.55 −0.152558
\(784\) 637.242 0.0290289
\(785\) −2152.19 −0.0978536
\(786\) 1374.98 0.0623969
\(787\) 21847.6 0.989557 0.494779 0.869019i \(-0.335249\pi\)
0.494779 + 0.869019i \(0.335249\pi\)
\(788\) −6144.10 −0.277760
\(789\) −22090.7 −0.996766
\(790\) 27541.7 1.24037
\(791\) −2560.77 −0.115108
\(792\) 0 0
\(793\) 6048.27 0.270845
\(794\) 4292.18 0.191843
\(795\) −16362.4 −0.729955
\(796\) −12386.9 −0.551559
\(797\) 31883.5 1.41703 0.708513 0.705698i \(-0.249366\pi\)
0.708513 + 0.705698i \(0.249366\pi\)
\(798\) 6940.37 0.307878
\(799\) 17594.4 0.779029
\(800\) 43150.0 1.90698
\(801\) 7093.93 0.312924
\(802\) 9167.13 0.403619
\(803\) 0 0
\(804\) 2162.49 0.0948570
\(805\) 33263.5 1.45638
\(806\) 20963.2 0.916125
\(807\) −16.7044 −0.000728652 0
\(808\) −24713.3 −1.07600
\(809\) 12262.7 0.532921 0.266460 0.963846i \(-0.414146\pi\)
0.266460 + 0.963846i \(0.414146\pi\)
\(810\) −2830.57 −0.122785
\(811\) 21255.2 0.920307 0.460154 0.887839i \(-0.347794\pi\)
0.460154 + 0.887839i \(0.347794\pi\)
\(812\) 8195.26 0.354184
\(813\) 7134.95 0.307790
\(814\) 0 0
\(815\) −44566.0 −1.91543
\(816\) −1038.81 −0.0445657
\(817\) −36808.7 −1.57622
\(818\) −1045.26 −0.0446783
\(819\) 6888.53 0.293901
\(820\) 20241.0 0.862008
\(821\) −34754.2 −1.47738 −0.738690 0.674046i \(-0.764555\pi\)
−0.738690 + 0.674046i \(0.764555\pi\)
\(822\) −11724.4 −0.497489
\(823\) 25847.0 1.09474 0.547368 0.836892i \(-0.315630\pi\)
0.547368 + 0.836892i \(0.315630\pi\)
\(824\) −39371.1 −1.66451
\(825\) 0 0
\(826\) −7873.15 −0.331649
\(827\) 13383.8 0.562758 0.281379 0.959597i \(-0.409208\pi\)
0.281379 + 0.959597i \(0.409208\pi\)
\(828\) 5183.82 0.217573
\(829\) 28625.8 1.19929 0.599647 0.800265i \(-0.295308\pi\)
0.599647 + 0.800265i \(0.295308\pi\)
\(830\) −30419.2 −1.27213
\(831\) 16887.1 0.704944
\(832\) 19347.6 0.806198
\(833\) −11278.3 −0.469111
\(834\) 3313.59 0.137578
\(835\) −64592.6 −2.67703
\(836\) 0 0
\(837\) 5735.95 0.236874
\(838\) 1254.82 0.0517269
\(839\) −43963.0 −1.80903 −0.904513 0.426447i \(-0.859765\pi\)
−0.904513 + 0.426447i \(0.859765\pi\)
\(840\) 18769.1 0.770947
\(841\) −9063.03 −0.371603
\(842\) 25425.6 1.04065
\(843\) 10926.1 0.446401
\(844\) 5009.53 0.204307
\(845\) −14371.9 −0.585099
\(846\) 3678.11 0.149475
\(847\) 0 0
\(848\) −1255.34 −0.0508354
\(849\) −13579.3 −0.548929
\(850\) 34779.8 1.40346
\(851\) −52661.6 −2.12129
\(852\) 4552.21 0.183047
\(853\) −46858.1 −1.88088 −0.940440 0.339959i \(-0.889587\pi\)
−0.940440 + 0.339959i \(0.889587\pi\)
\(854\) 2858.50 0.114538
\(855\) −15601.6 −0.624052
\(856\) −15050.7 −0.600962
\(857\) 13108.4 0.522489 0.261244 0.965273i \(-0.415867\pi\)
0.261244 + 0.965273i \(0.415867\pi\)
\(858\) 0 0
\(859\) −18510.5 −0.735240 −0.367620 0.929976i \(-0.619827\pi\)
−0.367620 + 0.929976i \(0.619827\pi\)
\(860\) −36806.6 −1.45941
\(861\) 9495.25 0.375839
\(862\) 21357.7 0.843905
\(863\) −27702.6 −1.09271 −0.546354 0.837555i \(-0.683984\pi\)
−0.546354 + 0.837555i \(0.683984\pi\)
\(864\) 4768.46 0.187762
\(865\) 16760.7 0.658822
\(866\) 304.626 0.0119534
\(867\) 3646.46 0.142838
\(868\) −14063.4 −0.549934
\(869\) 0 0
\(870\) 12978.5 0.505761
\(871\) −8334.27 −0.324220
\(872\) −47688.6 −1.85200
\(873\) −7671.41 −0.297409
\(874\) −20128.9 −0.779029
\(875\) −32343.2 −1.24960
\(876\) 2738.96 0.105640
\(877\) 15341.4 0.590698 0.295349 0.955389i \(-0.404564\pi\)
0.295349 + 0.955389i \(0.404564\pi\)
\(878\) −7229.29 −0.277878
\(879\) −10416.1 −0.399689
\(880\) 0 0
\(881\) 39390.7 1.50636 0.753182 0.657812i \(-0.228518\pi\)
0.753182 + 0.657812i \(0.228518\pi\)
\(882\) −2357.73 −0.0900100
\(883\) −16427.5 −0.626082 −0.313041 0.949740i \(-0.601348\pi\)
−0.313041 + 0.949740i \(0.601348\pi\)
\(884\) −19939.0 −0.758622
\(885\) 17698.5 0.672235
\(886\) −11124.1 −0.421807
\(887\) 21130.1 0.799865 0.399933 0.916545i \(-0.369034\pi\)
0.399933 + 0.916545i \(0.369034\pi\)
\(888\) −29714.5 −1.12292
\(889\) −10657.1 −0.402055
\(890\) −27544.4 −1.03740
\(891\) 0 0
\(892\) −11679.0 −0.438389
\(893\) 20273.1 0.759703
\(894\) −14618.5 −0.546887
\(895\) −34097.9 −1.27348
\(896\) −10783.8 −0.402077
\(897\) −19978.6 −0.743662
\(898\) 23140.0 0.859901
\(899\) −26300.0 −0.975699
\(900\) −10320.6 −0.382244
\(901\) 22217.7 0.821507
\(902\) 0 0
\(903\) −17266.3 −0.636310
\(904\) 4190.67 0.154181
\(905\) 18623.7 0.684057
\(906\) −2887.58 −0.105887
\(907\) 53222.0 1.94841 0.974205 0.225665i \(-0.0724556\pi\)
0.974205 + 0.225665i \(0.0724556\pi\)
\(908\) 18918.2 0.691436
\(909\) 9636.23 0.351610
\(910\) −26746.8 −0.974340
\(911\) −19976.1 −0.726494 −0.363247 0.931693i \(-0.618332\pi\)
−0.363247 + 0.931693i \(0.618332\pi\)
\(912\) −1196.97 −0.0434601
\(913\) 0 0
\(914\) −23796.2 −0.861169
\(915\) −6425.76 −0.232163
\(916\) −10800.6 −0.389587
\(917\) 3555.03 0.128024
\(918\) 3843.49 0.138185
\(919\) −8732.05 −0.313432 −0.156716 0.987644i \(-0.550091\pi\)
−0.156716 + 0.987644i \(0.550091\pi\)
\(920\) −54435.4 −1.95074
\(921\) 19971.9 0.714546
\(922\) 13186.6 0.471016
\(923\) −17544.3 −0.625654
\(924\) 0 0
\(925\) 104845. 3.72679
\(926\) −2835.31 −0.100620
\(927\) 15351.6 0.543919
\(928\) −21864.0 −0.773404
\(929\) −35095.6 −1.23945 −0.619725 0.784819i \(-0.712756\pi\)
−0.619725 + 0.784819i \(0.712756\pi\)
\(930\) −22271.6 −0.785284
\(931\) −12995.4 −0.457473
\(932\) −5413.81 −0.190274
\(933\) −2167.98 −0.0760734
\(934\) 16882.3 0.591441
\(935\) 0 0
\(936\) −11273.0 −0.393664
\(937\) −18700.7 −0.652002 −0.326001 0.945369i \(-0.605701\pi\)
−0.326001 + 0.945369i \(0.605701\pi\)
\(938\) −3938.89 −0.137110
\(939\) −14181.5 −0.492859
\(940\) 20272.0 0.703404
\(941\) 31357.5 1.08632 0.543160 0.839629i \(-0.317228\pi\)
0.543160 + 0.839629i \(0.317228\pi\)
\(942\) 610.919 0.0211304
\(943\) −27538.7 −0.950991
\(944\) 1357.84 0.0468157
\(945\) −7318.47 −0.251926
\(946\) 0 0
\(947\) −17691.0 −0.607054 −0.303527 0.952823i \(-0.598164\pi\)
−0.303527 + 0.952823i \(0.598164\pi\)
\(948\) 11097.3 0.380195
\(949\) −10556.0 −0.361077
\(950\) 40075.1 1.36864
\(951\) 33282.9 1.13488
\(952\) −25485.6 −0.867641
\(953\) −22457.5 −0.763347 −0.381673 0.924297i \(-0.624652\pi\)
−0.381673 + 0.924297i \(0.624652\pi\)
\(954\) 4644.61 0.157626
\(955\) −15299.6 −0.518412
\(956\) 2975.39 0.100660
\(957\) 0 0
\(958\) −19401.2 −0.654304
\(959\) −30313.6 −1.02073
\(960\) −20555.1 −0.691056
\(961\) 15340.8 0.514947
\(962\) 42344.6 1.41917
\(963\) 5868.59 0.196379
\(964\) 14949.5 0.499471
\(965\) 60879.8 2.03087
\(966\) −9442.15 −0.314489
\(967\) 28238.1 0.939065 0.469532 0.882915i \(-0.344422\pi\)
0.469532 + 0.882915i \(0.344422\pi\)
\(968\) 0 0
\(969\) 21184.7 0.702322
\(970\) 29786.6 0.985970
\(971\) 20726.0 0.684995 0.342497 0.939519i \(-0.388727\pi\)
0.342497 + 0.939519i \(0.388727\pi\)
\(972\) −1140.52 −0.0376359
\(973\) 8567.34 0.282278
\(974\) −28256.6 −0.929568
\(975\) 39775.7 1.30651
\(976\) −492.990 −0.0161683
\(977\) −11209.8 −0.367074 −0.183537 0.983013i \(-0.558755\pi\)
−0.183537 + 0.983013i \(0.558755\pi\)
\(978\) 12650.4 0.413616
\(979\) 0 0
\(980\) −12994.7 −0.423571
\(981\) 18594.8 0.605185
\(982\) −26095.1 −0.847992
\(983\) 53573.8 1.73829 0.869145 0.494557i \(-0.164670\pi\)
0.869145 + 0.494557i \(0.164670\pi\)
\(984\) −15538.9 −0.503416
\(985\) −25157.4 −0.813789
\(986\) −17622.8 −0.569194
\(987\) 9509.80 0.306687
\(988\) −22974.8 −0.739802
\(989\) 50077.0 1.61007
\(990\) 0 0
\(991\) −43260.9 −1.38671 −0.693354 0.720597i \(-0.743868\pi\)
−0.693354 + 0.720597i \(0.743868\pi\)
\(992\) 37519.4 1.20085
\(993\) −9193.85 −0.293815
\(994\) −8291.69 −0.264584
\(995\) −50718.8 −1.61597
\(996\) −12256.8 −0.389931
\(997\) −7938.54 −0.252173 −0.126086 0.992019i \(-0.540242\pi\)
−0.126086 + 0.992019i \(0.540242\pi\)
\(998\) 16894.1 0.535846
\(999\) 11586.3 0.366942
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 363.4.a.q.1.3 4
3.2 odd 2 1089.4.a.bf.1.2 4
11.10 odd 2 363.4.a.s.1.2 yes 4
33.32 even 2 1089.4.a.ba.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
363.4.a.q.1.3 4 1.1 even 1 trivial
363.4.a.s.1.2 yes 4 11.10 odd 2
1089.4.a.ba.1.3 4 33.32 even 2
1089.4.a.bf.1.2 4 3.2 odd 2