Properties

Label 1815.4.a.q.1.1
Level $1815$
Weight $4$
Character 1815.1
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,4,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.12946\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} +13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.59486 q^{2} -3.00000 q^{3} +13.1127 q^{4} +5.00000 q^{5} +13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} -22.9743 q^{10} -39.3381 q^{12} +15.6584 q^{13} +94.8302 q^{14} -15.0000 q^{15} +3.04132 q^{16} -72.9507 q^{17} -41.3537 q^{18} -61.0513 q^{19} +65.5635 q^{20} +61.9150 q^{21} -13.6605 q^{23} +70.4764 q^{24} +25.0000 q^{25} -71.9483 q^{26} -27.0000 q^{27} -270.624 q^{28} +31.4663 q^{29} +68.9228 q^{30} -243.008 q^{31} +173.963 q^{32} +335.198 q^{34} -103.192 q^{35} +118.014 q^{36} -65.4018 q^{37} +280.522 q^{38} -46.9753 q^{39} -117.461 q^{40} +109.087 q^{41} -284.491 q^{42} +121.750 q^{43} +45.0000 q^{45} +62.7678 q^{46} -519.530 q^{47} -9.12396 q^{48} +82.9413 q^{49} -114.871 q^{50} +218.852 q^{51} +205.324 q^{52} -542.673 q^{53} +124.061 q^{54} +484.839 q^{56} +183.154 q^{57} -144.583 q^{58} +109.478 q^{59} -196.691 q^{60} +89.6156 q^{61} +1116.59 q^{62} -185.745 q^{63} -823.664 q^{64} +78.2922 q^{65} +488.446 q^{67} -956.581 q^{68} +40.9814 q^{69} +474.151 q^{70} +837.423 q^{71} -211.429 q^{72} -351.216 q^{73} +300.512 q^{74} -75.0000 q^{75} -800.547 q^{76} +215.845 q^{78} +831.205 q^{79} +15.2066 q^{80} +81.0000 q^{81} -501.238 q^{82} -1389.13 q^{83} +811.873 q^{84} -364.754 q^{85} -559.423 q^{86} -94.3988 q^{87} +1523.70 q^{89} -206.769 q^{90} -323.164 q^{91} -179.125 q^{92} +729.025 q^{93} +2387.17 q^{94} -305.256 q^{95} -521.888 q^{96} -426.612 q^{97} -381.103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 9 q^{3} + 17 q^{4} + 15 q^{5} + 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} - 5 q^{10} - 51 q^{12} + 20 q^{13} + 144 q^{14} - 45 q^{15} + 25 q^{16} - 32 q^{17} - 9 q^{18} - 116 q^{19} + 85 q^{20} + 18 q^{21} + 240 q^{23} - 9 q^{24} + 75 q^{25} + 302 q^{26} - 81 q^{27} - 160 q^{28} - 238 q^{29} + 15 q^{30} + 92 q^{31} - 197 q^{32} + 354 q^{34} - 30 q^{35} + 153 q^{36} - 90 q^{37} + 324 q^{38} - 60 q^{39} + 15 q^{40} + 46 q^{41} - 432 q^{42} + 134 q^{43} + 135 q^{45} + 240 q^{46} - 220 q^{47} - 75 q^{48} - 457 q^{49} - 25 q^{50} + 96 q^{51} + 1530 q^{52} - 798 q^{53} + 27 q^{54} + 688 q^{56} + 348 q^{57} - 978 q^{58} + 1236 q^{59} - 255 q^{60} - 342 q^{61} + 1792 q^{62} - 54 q^{63} - 1919 q^{64} + 100 q^{65} + 764 q^{67} - 1074 q^{68} - 720 q^{69} + 720 q^{70} + 1816 q^{71} + 27 q^{72} - 100 q^{73} + 1874 q^{74} - 225 q^{75} - 396 q^{76} - 906 q^{78} + 96 q^{79} + 125 q^{80} + 243 q^{81} - 910 q^{82} - 858 q^{83} + 480 q^{84} - 160 q^{85} + 188 q^{86} + 714 q^{87} + 838 q^{89} - 45 q^{90} + 332 q^{91} - 688 q^{92} - 276 q^{93} + 3112 q^{94} - 580 q^{95} + 591 q^{96} - 1322 q^{97} - 1017 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.59486 −1.62453 −0.812263 0.583291i \(-0.801765\pi\)
−0.812263 + 0.583291i \(0.801765\pi\)
\(3\) −3.00000 −0.577350
\(4\) 13.1127 1.63909
\(5\) 5.00000 0.447214
\(6\) 13.7846 0.937921
\(7\) −20.6383 −1.11437 −0.557183 0.830390i \(-0.688118\pi\)
−0.557183 + 0.830390i \(0.688118\pi\)
\(8\) −23.4921 −1.03822
\(9\) 9.00000 0.333333
\(10\) −22.9743 −0.726511
\(11\) 0 0
\(12\) −39.3381 −0.946328
\(13\) 15.6584 0.334067 0.167033 0.985951i \(-0.446581\pi\)
0.167033 + 0.985951i \(0.446581\pi\)
\(14\) 94.8302 1.81032
\(15\) −15.0000 −0.258199
\(16\) 3.04132 0.0475206
\(17\) −72.9507 −1.04077 −0.520387 0.853931i \(-0.674212\pi\)
−0.520387 + 0.853931i \(0.674212\pi\)
\(18\) −41.3537 −0.541509
\(19\) −61.0513 −0.737165 −0.368582 0.929595i \(-0.620157\pi\)
−0.368582 + 0.929595i \(0.620157\pi\)
\(20\) 65.5635 0.733022
\(21\) 61.9150 0.643379
\(22\) 0 0
\(23\) −13.6605 −0.123844 −0.0619218 0.998081i \(-0.519723\pi\)
−0.0619218 + 0.998081i \(0.519723\pi\)
\(24\) 70.4764 0.599414
\(25\) 25.0000 0.200000
\(26\) −71.9483 −0.542701
\(27\) −27.0000 −0.192450
\(28\) −270.624 −1.82654
\(29\) 31.4663 0.201487 0.100744 0.994912i \(-0.467878\pi\)
0.100744 + 0.994912i \(0.467878\pi\)
\(30\) 68.9228 0.419451
\(31\) −243.008 −1.40792 −0.703961 0.710239i \(-0.748587\pi\)
−0.703961 + 0.710239i \(0.748587\pi\)
\(32\) 173.963 0.961016
\(33\) 0 0
\(34\) 335.198 1.69076
\(35\) −103.192 −0.498360
\(36\) 118.014 0.546363
\(37\) −65.4018 −0.290594 −0.145297 0.989388i \(-0.546414\pi\)
−0.145297 + 0.989388i \(0.546414\pi\)
\(38\) 280.522 1.19754
\(39\) −46.9753 −0.192874
\(40\) −117.461 −0.464304
\(41\) 109.087 0.415524 0.207762 0.978179i \(-0.433382\pi\)
0.207762 + 0.978179i \(0.433382\pi\)
\(42\) −284.491 −1.04519
\(43\) 121.750 0.431783 0.215891 0.976417i \(-0.430734\pi\)
0.215891 + 0.976417i \(0.430734\pi\)
\(44\) 0 0
\(45\) 45.0000 0.149071
\(46\) 62.7678 0.201187
\(47\) −519.530 −1.61237 −0.806184 0.591665i \(-0.798471\pi\)
−0.806184 + 0.591665i \(0.798471\pi\)
\(48\) −9.12396 −0.0274361
\(49\) 82.9413 0.241811
\(50\) −114.871 −0.324905
\(51\) 218.852 0.600891
\(52\) 205.324 0.547565
\(53\) −542.673 −1.40645 −0.703226 0.710967i \(-0.748258\pi\)
−0.703226 + 0.710967i \(0.748258\pi\)
\(54\) 124.061 0.312640
\(55\) 0 0
\(56\) 484.839 1.15695
\(57\) 183.154 0.425602
\(58\) −144.583 −0.327322
\(59\) 109.478 0.241574 0.120787 0.992678i \(-0.461458\pi\)
0.120787 + 0.992678i \(0.461458\pi\)
\(60\) −196.691 −0.423211
\(61\) 89.6156 0.188100 0.0940501 0.995567i \(-0.470019\pi\)
0.0940501 + 0.995567i \(0.470019\pi\)
\(62\) 1116.59 2.28721
\(63\) −185.745 −0.371455
\(64\) −823.664 −1.60872
\(65\) 78.2922 0.149399
\(66\) 0 0
\(67\) 488.446 0.890644 0.445322 0.895371i \(-0.353089\pi\)
0.445322 + 0.895371i \(0.353089\pi\)
\(68\) −956.581 −1.70592
\(69\) 40.9814 0.0715011
\(70\) 474.151 0.809599
\(71\) 837.423 1.39977 0.699887 0.714254i \(-0.253234\pi\)
0.699887 + 0.714254i \(0.253234\pi\)
\(72\) −211.429 −0.346072
\(73\) −351.216 −0.563105 −0.281553 0.959546i \(-0.590849\pi\)
−0.281553 + 0.959546i \(0.590849\pi\)
\(74\) 300.512 0.472078
\(75\) −75.0000 −0.115470
\(76\) −800.547 −1.20828
\(77\) 0 0
\(78\) 215.845 0.313328
\(79\) 831.205 1.18377 0.591885 0.806022i \(-0.298384\pi\)
0.591885 + 0.806022i \(0.298384\pi\)
\(80\) 15.2066 0.0212519
\(81\) 81.0000 0.111111
\(82\) −501.238 −0.675031
\(83\) −1389.13 −1.83707 −0.918537 0.395335i \(-0.870629\pi\)
−0.918537 + 0.395335i \(0.870629\pi\)
\(84\) 811.873 1.05456
\(85\) −364.754 −0.465448
\(86\) −559.423 −0.701443
\(87\) −94.3988 −0.116329
\(88\) 0 0
\(89\) 1523.70 1.81474 0.907369 0.420335i \(-0.138088\pi\)
0.907369 + 0.420335i \(0.138088\pi\)
\(90\) −206.769 −0.242170
\(91\) −323.164 −0.372273
\(92\) −179.125 −0.202990
\(93\) 729.025 0.812864
\(94\) 2387.17 2.61933
\(95\) −305.256 −0.329670
\(96\) −521.888 −0.554843
\(97\) −426.612 −0.446555 −0.223278 0.974755i \(-0.571676\pi\)
−0.223278 + 0.974755i \(0.571676\pi\)
\(98\) −381.103 −0.392829
\(99\) 0 0
\(100\) 327.818 0.327818
\(101\) −74.1387 −0.0730403 −0.0365202 0.999333i \(-0.511627\pi\)
−0.0365202 + 0.999333i \(0.511627\pi\)
\(102\) −1005.59 −0.976163
\(103\) −69.3916 −0.0663821 −0.0331911 0.999449i \(-0.510567\pi\)
−0.0331911 + 0.999449i \(0.510567\pi\)
\(104\) −367.850 −0.346833
\(105\) 309.575 0.287728
\(106\) 2493.51 2.28482
\(107\) −1141.71 −1.03152 −0.515761 0.856733i \(-0.672491\pi\)
−0.515761 + 0.856733i \(0.672491\pi\)
\(108\) −354.043 −0.315443
\(109\) 2226.85 1.95682 0.978409 0.206680i \(-0.0662659\pi\)
0.978409 + 0.206680i \(0.0662659\pi\)
\(110\) 0 0
\(111\) 196.205 0.167775
\(112\) −62.7678 −0.0529554
\(113\) −1719.76 −1.43169 −0.715847 0.698257i \(-0.753959\pi\)
−0.715847 + 0.698257i \(0.753959\pi\)
\(114\) −841.566 −0.691402
\(115\) −68.3023 −0.0553845
\(116\) 412.608 0.330256
\(117\) 140.926 0.111356
\(118\) −503.038 −0.392444
\(119\) 1505.58 1.15980
\(120\) 352.382 0.268066
\(121\) 0 0
\(122\) −411.771 −0.305574
\(123\) −327.260 −0.239903
\(124\) −3186.49 −2.30771
\(125\) 125.000 0.0894427
\(126\) 853.472 0.603439
\(127\) 1601.63 1.11907 0.559534 0.828807i \(-0.310980\pi\)
0.559534 + 0.828807i \(0.310980\pi\)
\(128\) 2392.91 1.65239
\(129\) −365.249 −0.249290
\(130\) −359.741 −0.242703
\(131\) −2004.13 −1.33665 −0.668327 0.743868i \(-0.732989\pi\)
−0.668327 + 0.743868i \(0.732989\pi\)
\(132\) 0 0
\(133\) 1260.00 0.821471
\(134\) −2244.34 −1.44687
\(135\) −135.000 −0.0860663
\(136\) 1713.77 1.08055
\(137\) −1672.85 −1.04322 −0.521610 0.853184i \(-0.674669\pi\)
−0.521610 + 0.853184i \(0.674669\pi\)
\(138\) −188.303 −0.116155
\(139\) −2540.38 −1.55016 −0.775080 0.631863i \(-0.782291\pi\)
−0.775080 + 0.631863i \(0.782291\pi\)
\(140\) −1353.12 −0.816855
\(141\) 1558.59 0.930901
\(142\) −3847.84 −2.27397
\(143\) 0 0
\(144\) 27.3719 0.0158402
\(145\) 157.331 0.0901079
\(146\) 1613.79 0.914780
\(147\) −248.824 −0.139610
\(148\) −857.594 −0.476310
\(149\) −3090.68 −1.69932 −0.849658 0.527334i \(-0.823192\pi\)
−0.849658 + 0.527334i \(0.823192\pi\)
\(150\) 344.614 0.187584
\(151\) −1358.74 −0.732267 −0.366134 0.930562i \(-0.619319\pi\)
−0.366134 + 0.930562i \(0.619319\pi\)
\(152\) 1434.22 0.765335
\(153\) −656.557 −0.346925
\(154\) 0 0
\(155\) −1215.04 −0.629642
\(156\) −615.973 −0.316137
\(157\) −1011.95 −0.514411 −0.257205 0.966357i \(-0.582802\pi\)
−0.257205 + 0.966357i \(0.582802\pi\)
\(158\) −3819.27 −1.92307
\(159\) 1628.02 0.812015
\(160\) 869.813 0.429780
\(161\) 281.929 0.138007
\(162\) −372.183 −0.180503
\(163\) −2816.37 −1.35334 −0.676672 0.736285i \(-0.736578\pi\)
−0.676672 + 0.736285i \(0.736578\pi\)
\(164\) 1430.42 0.681081
\(165\) 0 0
\(166\) 6382.87 2.98438
\(167\) −3448.89 −1.59810 −0.799052 0.601262i \(-0.794665\pi\)
−0.799052 + 0.601262i \(0.794665\pi\)
\(168\) −1454.52 −0.667966
\(169\) −1951.81 −0.888399
\(170\) 1675.99 0.756133
\(171\) −549.462 −0.245722
\(172\) 1596.47 0.707730
\(173\) 2287.85 1.00545 0.502723 0.864448i \(-0.332332\pi\)
0.502723 + 0.864448i \(0.332332\pi\)
\(174\) 433.749 0.188979
\(175\) −515.959 −0.222873
\(176\) 0 0
\(177\) −328.435 −0.139473
\(178\) −7001.17 −2.94809
\(179\) 3249.06 1.35668 0.678340 0.734748i \(-0.262700\pi\)
0.678340 + 0.734748i \(0.262700\pi\)
\(180\) 590.072 0.244341
\(181\) 1170.45 0.480655 0.240328 0.970692i \(-0.422745\pi\)
0.240328 + 0.970692i \(0.422745\pi\)
\(182\) 1484.89 0.604767
\(183\) −268.847 −0.108600
\(184\) 320.913 0.128576
\(185\) −327.009 −0.129958
\(186\) −3349.76 −1.32052
\(187\) 0 0
\(188\) −6812.44 −2.64281
\(189\) 557.235 0.214460
\(190\) 1402.61 0.535558
\(191\) −2760.35 −1.04572 −0.522859 0.852419i \(-0.675134\pi\)
−0.522859 + 0.852419i \(0.675134\pi\)
\(192\) 2470.99 0.928794
\(193\) 1250.61 0.466430 0.233215 0.972425i \(-0.425075\pi\)
0.233215 + 0.972425i \(0.425075\pi\)
\(194\) 1960.22 0.725441
\(195\) −234.877 −0.0862557
\(196\) 1087.58 0.396350
\(197\) 143.991 0.0520756 0.0260378 0.999661i \(-0.491711\pi\)
0.0260378 + 0.999661i \(0.491711\pi\)
\(198\) 0 0
\(199\) 761.249 0.271174 0.135587 0.990765i \(-0.456708\pi\)
0.135587 + 0.990765i \(0.456708\pi\)
\(200\) −587.303 −0.207643
\(201\) −1465.34 −0.514213
\(202\) 340.657 0.118656
\(203\) −649.411 −0.224531
\(204\) 2869.74 0.984913
\(205\) 545.434 0.185828
\(206\) 318.844 0.107840
\(207\) −122.944 −0.0412812
\(208\) 47.6223 0.0158751
\(209\) 0 0
\(210\) −1422.45 −0.467422
\(211\) −3976.58 −1.29743 −0.648717 0.761029i \(-0.724694\pi\)
−0.648717 + 0.761029i \(0.724694\pi\)
\(212\) −7115.91 −2.30530
\(213\) −2512.27 −0.808159
\(214\) 5245.97 1.67573
\(215\) 608.749 0.193099
\(216\) 634.287 0.199805
\(217\) 5015.29 1.56894
\(218\) −10232.0 −3.17890
\(219\) 1053.65 0.325109
\(220\) 0 0
\(221\) −1142.29 −0.347688
\(222\) −901.535 −0.272555
\(223\) 908.084 0.272690 0.136345 0.990661i \(-0.456464\pi\)
0.136345 + 0.990661i \(0.456464\pi\)
\(224\) −3590.30 −1.07092
\(225\) 225.000 0.0666667
\(226\) 7902.05 2.32583
\(227\) 2062.15 0.602951 0.301475 0.953474i \(-0.402521\pi\)
0.301475 + 0.953474i \(0.402521\pi\)
\(228\) 2401.64 0.697599
\(229\) 4077.47 1.17662 0.588312 0.808634i \(-0.299793\pi\)
0.588312 + 0.808634i \(0.299793\pi\)
\(230\) 313.839 0.0899737
\(231\) 0 0
\(232\) −739.209 −0.209187
\(233\) −1682.76 −0.473138 −0.236569 0.971615i \(-0.576023\pi\)
−0.236569 + 0.971615i \(0.576023\pi\)
\(234\) −647.534 −0.180900
\(235\) −2597.65 −0.721073
\(236\) 1435.56 0.395961
\(237\) −2493.62 −0.683450
\(238\) −6917.93 −1.88413
\(239\) −4024.96 −1.08934 −0.544672 0.838649i \(-0.683346\pi\)
−0.544672 + 0.838649i \(0.683346\pi\)
\(240\) −45.6198 −0.0122698
\(241\) 2784.27 0.744194 0.372097 0.928194i \(-0.378639\pi\)
0.372097 + 0.928194i \(0.378639\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 1175.10 0.308313
\(245\) 414.707 0.108141
\(246\) 1503.71 0.389729
\(247\) −955.968 −0.246262
\(248\) 5708.78 1.46173
\(249\) 4167.40 1.06064
\(250\) −574.357 −0.145302
\(251\) −1827.60 −0.459591 −0.229796 0.973239i \(-0.573806\pi\)
−0.229796 + 0.973239i \(0.573806\pi\)
\(252\) −2435.62 −0.608848
\(253\) 0 0
\(254\) −7359.26 −1.81796
\(255\) 1094.26 0.268727
\(256\) −4405.79 −1.07563
\(257\) 585.171 0.142031 0.0710155 0.997475i \(-0.477376\pi\)
0.0710155 + 0.997475i \(0.477376\pi\)
\(258\) 1678.27 0.404978
\(259\) 1349.78 0.323828
\(260\) 1026.62 0.244878
\(261\) 283.196 0.0671625
\(262\) 9208.69 2.17143
\(263\) 238.098 0.0558241 0.0279120 0.999610i \(-0.491114\pi\)
0.0279120 + 0.999610i \(0.491114\pi\)
\(264\) 0 0
\(265\) −2713.37 −0.628984
\(266\) −5789.51 −1.33450
\(267\) −4571.09 −1.04774
\(268\) 6404.84 1.45984
\(269\) 4618.46 1.04681 0.523406 0.852083i \(-0.324661\pi\)
0.523406 + 0.852083i \(0.324661\pi\)
\(270\) 620.306 0.139817
\(271\) 143.439 0.0321525 0.0160762 0.999871i \(-0.494883\pi\)
0.0160762 + 0.999871i \(0.494883\pi\)
\(272\) −221.867 −0.0494582
\(273\) 969.493 0.214932
\(274\) 7686.51 1.69474
\(275\) 0 0
\(276\) 537.376 0.117197
\(277\) 8602.51 1.86597 0.932987 0.359911i \(-0.117193\pi\)
0.932987 + 0.359911i \(0.117193\pi\)
\(278\) 11672.7 2.51828
\(279\) −2187.07 −0.469307
\(280\) 2424.19 0.517404
\(281\) 2992.81 0.635360 0.317680 0.948198i \(-0.397096\pi\)
0.317680 + 0.948198i \(0.397096\pi\)
\(282\) −7161.50 −1.51227
\(283\) 6858.89 1.44070 0.720351 0.693610i \(-0.243981\pi\)
0.720351 + 0.693610i \(0.243981\pi\)
\(284\) 10980.9 2.29435
\(285\) 915.769 0.190335
\(286\) 0 0
\(287\) −2251.37 −0.463046
\(288\) 1565.66 0.320339
\(289\) 408.809 0.0832096
\(290\) −722.915 −0.146383
\(291\) 1279.84 0.257819
\(292\) −4605.39 −0.922979
\(293\) −4049.70 −0.807461 −0.403731 0.914878i \(-0.632287\pi\)
−0.403731 + 0.914878i \(0.632287\pi\)
\(294\) 1143.31 0.226800
\(295\) 547.392 0.108035
\(296\) 1536.43 0.301699
\(297\) 0 0
\(298\) 14201.2 2.76059
\(299\) −213.901 −0.0413720
\(300\) −983.453 −0.189266
\(301\) −2512.71 −0.481164
\(302\) 6243.20 1.18959
\(303\) 222.416 0.0421699
\(304\) −185.677 −0.0350305
\(305\) 448.078 0.0841209
\(306\) 3016.78 0.563588
\(307\) 9572.69 1.77962 0.889808 0.456335i \(-0.150838\pi\)
0.889808 + 0.456335i \(0.150838\pi\)
\(308\) 0 0
\(309\) 208.175 0.0383257
\(310\) 5582.94 1.02287
\(311\) 5396.42 0.983932 0.491966 0.870614i \(-0.336278\pi\)
0.491966 + 0.870614i \(0.336278\pi\)
\(312\) 1103.55 0.200244
\(313\) 9755.04 1.76162 0.880811 0.473469i \(-0.156998\pi\)
0.880811 + 0.473469i \(0.156998\pi\)
\(314\) 4649.77 0.835674
\(315\) −928.726 −0.166120
\(316\) 10899.3 1.94030
\(317\) −4353.75 −0.771391 −0.385695 0.922626i \(-0.626038\pi\)
−0.385695 + 0.922626i \(0.626038\pi\)
\(318\) −7480.52 −1.31914
\(319\) 0 0
\(320\) −4118.32 −0.719440
\(321\) 3425.12 0.595549
\(322\) −1295.42 −0.224196
\(323\) 4453.74 0.767221
\(324\) 1062.13 0.182121
\(325\) 391.461 0.0668134
\(326\) 12940.8 2.19854
\(327\) −6680.54 −1.12977
\(328\) −2562.68 −0.431404
\(329\) 10722.2 1.79677
\(330\) 0 0
\(331\) 5387.64 0.894656 0.447328 0.894370i \(-0.352376\pi\)
0.447328 + 0.894370i \(0.352376\pi\)
\(332\) −18215.3 −3.01113
\(333\) −588.616 −0.0968648
\(334\) 15847.2 2.59616
\(335\) 2442.23 0.398308
\(336\) 188.303 0.0305738
\(337\) 4500.27 0.727434 0.363717 0.931509i \(-0.381508\pi\)
0.363717 + 0.931509i \(0.381508\pi\)
\(338\) 8968.30 1.44323
\(339\) 5159.28 0.826589
\(340\) −4782.91 −0.762910
\(341\) 0 0
\(342\) 2524.70 0.399181
\(343\) 5367.18 0.844899
\(344\) −2860.16 −0.448283
\(345\) 204.907 0.0319763
\(346\) −10512.4 −1.63337
\(347\) −5906.32 −0.913740 −0.456870 0.889533i \(-0.651030\pi\)
−0.456870 + 0.889533i \(0.651030\pi\)
\(348\) −1237.82 −0.190673
\(349\) −3636.26 −0.557721 −0.278860 0.960332i \(-0.589957\pi\)
−0.278860 + 0.960332i \(0.589957\pi\)
\(350\) 2370.76 0.362063
\(351\) −422.778 −0.0642912
\(352\) 0 0
\(353\) 210.408 0.0317248 0.0158624 0.999874i \(-0.494951\pi\)
0.0158624 + 0.999874i \(0.494951\pi\)
\(354\) 1509.11 0.226577
\(355\) 4187.12 0.625998
\(356\) 19979.8 2.97451
\(357\) −4516.75 −0.669612
\(358\) −14928.9 −2.20396
\(359\) −2499.68 −0.367488 −0.183744 0.982974i \(-0.558822\pi\)
−0.183744 + 0.982974i \(0.558822\pi\)
\(360\) −1057.15 −0.154768
\(361\) −3131.74 −0.456588
\(362\) −5378.03 −0.780837
\(363\) 0 0
\(364\) −4237.56 −0.610188
\(365\) −1756.08 −0.251828
\(366\) 1235.31 0.176423
\(367\) 5748.70 0.817656 0.408828 0.912612i \(-0.365938\pi\)
0.408828 + 0.912612i \(0.365938\pi\)
\(368\) −41.5458 −0.00588512
\(369\) 981.781 0.138508
\(370\) 1502.56 0.211120
\(371\) 11199.9 1.56730
\(372\) 9559.48 1.33236
\(373\) −4467.78 −0.620196 −0.310098 0.950705i \(-0.600362\pi\)
−0.310098 + 0.950705i \(0.600362\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 12204.9 1.67398
\(377\) 492.712 0.0673103
\(378\) −2560.42 −0.348396
\(379\) 7804.08 1.05770 0.528851 0.848715i \(-0.322623\pi\)
0.528851 + 0.848715i \(0.322623\pi\)
\(380\) −4002.74 −0.540358
\(381\) −4804.89 −0.646094
\(382\) 12683.4 1.69880
\(383\) −11161.1 −1.48904 −0.744522 0.667597i \(-0.767323\pi\)
−0.744522 + 0.667597i \(0.767323\pi\)
\(384\) −7178.74 −0.954007
\(385\) 0 0
\(386\) −5746.38 −0.757728
\(387\) 1095.75 0.143928
\(388\) −5594.04 −0.731944
\(389\) 8490.24 1.10661 0.553306 0.832978i \(-0.313366\pi\)
0.553306 + 0.832978i \(0.313366\pi\)
\(390\) 1079.22 0.140125
\(391\) 996.540 0.128893
\(392\) −1948.47 −0.251052
\(393\) 6012.39 0.771717
\(394\) −661.616 −0.0845983
\(395\) 4156.03 0.529398
\(396\) 0 0
\(397\) 6019.74 0.761013 0.380507 0.924778i \(-0.375750\pi\)
0.380507 + 0.924778i \(0.375750\pi\)
\(398\) −3497.83 −0.440529
\(399\) −3779.99 −0.474277
\(400\) 76.0330 0.00950413
\(401\) −10398.8 −1.29499 −0.647495 0.762069i \(-0.724184\pi\)
−0.647495 + 0.762069i \(0.724184\pi\)
\(402\) 6733.01 0.835354
\(403\) −3805.13 −0.470340
\(404\) −972.158 −0.119720
\(405\) 405.000 0.0496904
\(406\) 2983.95 0.364756
\(407\) 0 0
\(408\) −5141.30 −0.623854
\(409\) 4733.68 0.572287 0.286144 0.958187i \(-0.407627\pi\)
0.286144 + 0.958187i \(0.407627\pi\)
\(410\) −2506.19 −0.301883
\(411\) 5018.55 0.602304
\(412\) −909.911 −0.108806
\(413\) −2259.45 −0.269202
\(414\) 564.910 0.0670624
\(415\) −6945.67 −0.821565
\(416\) 2723.98 0.321044
\(417\) 7621.14 0.894985
\(418\) 0 0
\(419\) −8117.57 −0.946466 −0.473233 0.880937i \(-0.656913\pi\)
−0.473233 + 0.880937i \(0.656913\pi\)
\(420\) 4059.37 0.471611
\(421\) −9484.27 −1.09795 −0.548973 0.835840i \(-0.684981\pi\)
−0.548973 + 0.835840i \(0.684981\pi\)
\(422\) 18271.8 2.10772
\(423\) −4675.77 −0.537456
\(424\) 12748.5 1.46020
\(425\) −1823.77 −0.208155
\(426\) 11543.5 1.31288
\(427\) −1849.52 −0.209612
\(428\) −14970.8 −1.69075
\(429\) 0 0
\(430\) −2797.11 −0.313695
\(431\) −9335.16 −1.04329 −0.521646 0.853162i \(-0.674682\pi\)
−0.521646 + 0.853162i \(0.674682\pi\)
\(432\) −82.1157 −0.00914535
\(433\) −2983.02 −0.331074 −0.165537 0.986204i \(-0.552936\pi\)
−0.165537 + 0.986204i \(0.552936\pi\)
\(434\) −23044.5 −2.54878
\(435\) −471.994 −0.0520238
\(436\) 29200.0 3.20739
\(437\) 833.988 0.0912931
\(438\) −4841.36 −0.528148
\(439\) 5232.32 0.568850 0.284425 0.958698i \(-0.408197\pi\)
0.284425 + 0.958698i \(0.408197\pi\)
\(440\) 0 0
\(441\) 746.472 0.0806038
\(442\) 5248.68 0.564828
\(443\) 7517.71 0.806269 0.403135 0.915141i \(-0.367921\pi\)
0.403135 + 0.915141i \(0.367921\pi\)
\(444\) 2572.78 0.274997
\(445\) 7618.49 0.811575
\(446\) −4172.52 −0.442992
\(447\) 9272.03 0.981101
\(448\) 16999.1 1.79270
\(449\) 16070.9 1.68916 0.844581 0.535428i \(-0.179850\pi\)
0.844581 + 0.535428i \(0.179850\pi\)
\(450\) −1033.84 −0.108302
\(451\) 0 0
\(452\) −22550.7 −2.34667
\(453\) 4076.21 0.422775
\(454\) −9475.29 −0.979510
\(455\) −1615.82 −0.166485
\(456\) −4302.67 −0.441867
\(457\) −9718.51 −0.994776 −0.497388 0.867528i \(-0.665707\pi\)
−0.497388 + 0.867528i \(0.665707\pi\)
\(458\) −18735.4 −1.91146
\(459\) 1969.67 0.200297
\(460\) −895.627 −0.0907801
\(461\) 14538.0 1.46877 0.734385 0.678733i \(-0.237471\pi\)
0.734385 + 0.678733i \(0.237471\pi\)
\(462\) 0 0
\(463\) −9978.17 −1.00157 −0.500783 0.865573i \(-0.666955\pi\)
−0.500783 + 0.865573i \(0.666955\pi\)
\(464\) 95.6990 0.00957481
\(465\) 3645.12 0.363524
\(466\) 7732.03 0.768625
\(467\) 15188.2 1.50498 0.752489 0.658605i \(-0.228853\pi\)
0.752489 + 0.658605i \(0.228853\pi\)
\(468\) 1847.92 0.182522
\(469\) −10080.7 −0.992503
\(470\) 11935.8 1.17140
\(471\) 3035.85 0.296995
\(472\) −2571.88 −0.250806
\(473\) 0 0
\(474\) 11457.8 1.11028
\(475\) −1526.28 −0.147433
\(476\) 19742.3 1.90102
\(477\) −4884.06 −0.468817
\(478\) 18494.1 1.76967
\(479\) −11330.8 −1.08083 −0.540415 0.841399i \(-0.681733\pi\)
−0.540415 + 0.841399i \(0.681733\pi\)
\(480\) −2609.44 −0.248133
\(481\) −1024.09 −0.0970779
\(482\) −12793.3 −1.20896
\(483\) −845.788 −0.0796784
\(484\) 0 0
\(485\) −2133.06 −0.199706
\(486\) 1116.55 0.104213
\(487\) 19086.9 1.77599 0.887997 0.459850i \(-0.152097\pi\)
0.887997 + 0.459850i \(0.152097\pi\)
\(488\) −2105.26 −0.195288
\(489\) 8449.11 0.781353
\(490\) −1905.52 −0.175679
\(491\) 8112.85 0.745677 0.372839 0.927896i \(-0.378384\pi\)
0.372839 + 0.927896i \(0.378384\pi\)
\(492\) −4291.27 −0.393222
\(493\) −2295.49 −0.209703
\(494\) 4392.53 0.400060
\(495\) 0 0
\(496\) −739.066 −0.0669053
\(497\) −17283.0 −1.55986
\(498\) −19148.6 −1.72303
\(499\) 18329.1 1.64433 0.822167 0.569246i \(-0.192765\pi\)
0.822167 + 0.569246i \(0.192765\pi\)
\(500\) 1639.09 0.146604
\(501\) 10346.7 0.922666
\(502\) 8397.58 0.746618
\(503\) −7739.57 −0.686064 −0.343032 0.939324i \(-0.611454\pi\)
−0.343032 + 0.939324i \(0.611454\pi\)
\(504\) 4363.55 0.385651
\(505\) −370.693 −0.0326646
\(506\) 0 0
\(507\) 5855.44 0.512918
\(508\) 21001.7 1.83425
\(509\) 15914.9 1.38589 0.692943 0.720993i \(-0.256314\pi\)
0.692943 + 0.720993i \(0.256314\pi\)
\(510\) −5027.97 −0.436554
\(511\) 7248.51 0.627505
\(512\) 1100.65 0.0950048
\(513\) 1648.38 0.141867
\(514\) −2688.78 −0.230733
\(515\) −346.958 −0.0296870
\(516\) −4789.40 −0.408608
\(517\) 0 0
\(518\) −6202.07 −0.526068
\(519\) −6863.56 −0.580495
\(520\) −1839.25 −0.155109
\(521\) 2274.50 0.191262 0.0956312 0.995417i \(-0.469513\pi\)
0.0956312 + 0.995417i \(0.469513\pi\)
\(522\) −1301.25 −0.109107
\(523\) −10971.1 −0.917274 −0.458637 0.888624i \(-0.651662\pi\)
−0.458637 + 0.888624i \(0.651662\pi\)
\(524\) −26279.6 −2.19089
\(525\) 1547.88 0.128676
\(526\) −1094.02 −0.0906877
\(527\) 17727.6 1.46533
\(528\) 0 0
\(529\) −11980.4 −0.984663
\(530\) 12467.5 1.02180
\(531\) 985.306 0.0805247
\(532\) 16522.0 1.34646
\(533\) 1708.13 0.138813
\(534\) 21003.5 1.70208
\(535\) −5708.53 −0.461310
\(536\) −11474.6 −0.924680
\(537\) −9747.17 −0.783280
\(538\) −21221.2 −1.70058
\(539\) 0 0
\(540\) −1770.21 −0.141070
\(541\) −5313.05 −0.422229 −0.211115 0.977461i \(-0.567709\pi\)
−0.211115 + 0.977461i \(0.567709\pi\)
\(542\) −659.084 −0.0522326
\(543\) −3511.34 −0.277506
\(544\) −12690.7 −1.00020
\(545\) 11134.2 0.875115
\(546\) −4454.68 −0.349162
\(547\) −20685.1 −1.61688 −0.808439 0.588581i \(-0.799687\pi\)
−0.808439 + 0.588581i \(0.799687\pi\)
\(548\) −21935.6 −1.70993
\(549\) 806.541 0.0627000
\(550\) 0 0
\(551\) −1921.06 −0.148529
\(552\) −962.739 −0.0742335
\(553\) −17154.7 −1.31915
\(554\) −39527.3 −3.03132
\(555\) 981.027 0.0750311
\(556\) −33311.2 −2.54085
\(557\) 10853.8 0.825659 0.412830 0.910808i \(-0.364540\pi\)
0.412830 + 0.910808i \(0.364540\pi\)
\(558\) 10049.3 0.762402
\(559\) 1906.41 0.144244
\(560\) −313.839 −0.0236824
\(561\) 0 0
\(562\) −13751.5 −1.03216
\(563\) 15381.2 1.15141 0.575704 0.817658i \(-0.304728\pi\)
0.575704 + 0.817658i \(0.304728\pi\)
\(564\) 20437.3 1.52583
\(565\) −8598.80 −0.640273
\(566\) −31515.6 −2.34046
\(567\) −1671.71 −0.123818
\(568\) −19672.9 −1.45327
\(569\) 1348.88 0.0993814 0.0496907 0.998765i \(-0.484176\pi\)
0.0496907 + 0.998765i \(0.484176\pi\)
\(570\) −4207.83 −0.309204
\(571\) −3463.51 −0.253841 −0.126920 0.991913i \(-0.540509\pi\)
−0.126920 + 0.991913i \(0.540509\pi\)
\(572\) 0 0
\(573\) 8281.05 0.603745
\(574\) 10344.7 0.752231
\(575\) −341.511 −0.0247687
\(576\) −7412.97 −0.536239
\(577\) 12052.6 0.869598 0.434799 0.900528i \(-0.356819\pi\)
0.434799 + 0.900528i \(0.356819\pi\)
\(578\) −1878.42 −0.135176
\(579\) −3751.83 −0.269293
\(580\) 2063.04 0.147695
\(581\) 28669.4 2.04717
\(582\) −5880.66 −0.418834
\(583\) 0 0
\(584\) 8250.80 0.584624
\(585\) 704.630 0.0497997
\(586\) 18607.8 1.31174
\(587\) −11133.1 −0.782813 −0.391407 0.920218i \(-0.628011\pi\)
−0.391407 + 0.920218i \(0.628011\pi\)
\(588\) −3262.75 −0.228833
\(589\) 14836.0 1.03787
\(590\) −2515.19 −0.175506
\(591\) −431.972 −0.0300659
\(592\) −198.908 −0.0138092
\(593\) 7939.69 0.549821 0.274911 0.961470i \(-0.411352\pi\)
0.274911 + 0.961470i \(0.411352\pi\)
\(594\) 0 0
\(595\) 7527.91 0.518680
\(596\) −40527.1 −2.78533
\(597\) −2283.75 −0.156562
\(598\) 982.846 0.0672100
\(599\) −19474.7 −1.32840 −0.664202 0.747553i \(-0.731229\pi\)
−0.664202 + 0.747553i \(0.731229\pi\)
\(600\) 1761.91 0.119883
\(601\) 19946.1 1.35377 0.676887 0.736087i \(-0.263329\pi\)
0.676887 + 0.736087i \(0.263329\pi\)
\(602\) 11545.6 0.781664
\(603\) 4396.01 0.296881
\(604\) −17816.7 −1.20025
\(605\) 0 0
\(606\) −1021.97 −0.0685061
\(607\) −1427.44 −0.0954496 −0.0477248 0.998861i \(-0.515197\pi\)
−0.0477248 + 0.998861i \(0.515197\pi\)
\(608\) −10620.6 −0.708427
\(609\) 1948.23 0.129633
\(610\) −2058.85 −0.136657
\(611\) −8135.03 −0.538638
\(612\) −8609.23 −0.568640
\(613\) 8029.40 0.529045 0.264522 0.964380i \(-0.414786\pi\)
0.264522 + 0.964380i \(0.414786\pi\)
\(614\) −43985.1 −2.89103
\(615\) −1636.30 −0.107288
\(616\) 0 0
\(617\) 20795.5 1.35688 0.678440 0.734655i \(-0.262656\pi\)
0.678440 + 0.734655i \(0.262656\pi\)
\(618\) −956.533 −0.0622612
\(619\) 1677.43 0.108920 0.0544602 0.998516i \(-0.482656\pi\)
0.0544602 + 0.998516i \(0.482656\pi\)
\(620\) −15932.5 −1.03204
\(621\) 368.832 0.0238337
\(622\) −24795.8 −1.59842
\(623\) −31446.6 −2.02228
\(624\) −142.867 −0.00916547
\(625\) 625.000 0.0400000
\(626\) −44823.0 −2.86180
\(627\) 0 0
\(628\) −13269.4 −0.843165
\(629\) 4771.11 0.302443
\(630\) 4267.36 0.269866
\(631\) −25225.2 −1.59144 −0.795719 0.605666i \(-0.792907\pi\)
−0.795719 + 0.605666i \(0.792907\pi\)
\(632\) −19526.8 −1.22901
\(633\) 11929.7 0.749074
\(634\) 20004.8 1.25315
\(635\) 8008.15 0.500462
\(636\) 21347.7 1.33096
\(637\) 1298.73 0.0807812
\(638\) 0 0
\(639\) 7536.81 0.466591
\(640\) 11964.6 0.738971
\(641\) 15165.3 0.934468 0.467234 0.884134i \(-0.345251\pi\)
0.467234 + 0.884134i \(0.345251\pi\)
\(642\) −15737.9 −0.967486
\(643\) 27156.1 1.66553 0.832763 0.553630i \(-0.186758\pi\)
0.832763 + 0.553630i \(0.186758\pi\)
\(644\) 3696.85 0.226206
\(645\) −1826.25 −0.111486
\(646\) −20464.3 −1.24637
\(647\) 29154.9 1.77156 0.885778 0.464110i \(-0.153626\pi\)
0.885778 + 0.464110i \(0.153626\pi\)
\(648\) −1902.86 −0.115357
\(649\) 0 0
\(650\) −1798.71 −0.108540
\(651\) −15045.9 −0.905828
\(652\) −36930.2 −2.21825
\(653\) −19141.7 −1.14713 −0.573564 0.819161i \(-0.694440\pi\)
−0.573564 + 0.819161i \(0.694440\pi\)
\(654\) 30696.1 1.83534
\(655\) −10020.7 −0.597770
\(656\) 331.768 0.0197460
\(657\) −3160.94 −0.187702
\(658\) −49267.2 −2.91890
\(659\) 24939.6 1.47422 0.737110 0.675773i \(-0.236190\pi\)
0.737110 + 0.675773i \(0.236190\pi\)
\(660\) 0 0
\(661\) 22617.7 1.33090 0.665452 0.746440i \(-0.268239\pi\)
0.665452 + 0.746440i \(0.268239\pi\)
\(662\) −24755.4 −1.45339
\(663\) 3426.88 0.200738
\(664\) 32633.7 1.90728
\(665\) 6299.99 0.367373
\(666\) 2704.61 0.157359
\(667\) −429.843 −0.0249529
\(668\) −45224.3 −2.61943
\(669\) −2724.25 −0.157438
\(670\) −11221.7 −0.647062
\(671\) 0 0
\(672\) 10770.9 0.618298
\(673\) 13855.8 0.793615 0.396807 0.917902i \(-0.370118\pi\)
0.396807 + 0.917902i \(0.370118\pi\)
\(674\) −20678.1 −1.18174
\(675\) −675.000 −0.0384900
\(676\) −25593.5 −1.45616
\(677\) 24992.8 1.41884 0.709419 0.704787i \(-0.248957\pi\)
0.709419 + 0.704787i \(0.248957\pi\)
\(678\) −23706.1 −1.34282
\(679\) 8804.57 0.497626
\(680\) 8568.84 0.483235
\(681\) −6186.46 −0.348114
\(682\) 0 0
\(683\) −14420.5 −0.807887 −0.403943 0.914784i \(-0.632361\pi\)
−0.403943 + 0.914784i \(0.632361\pi\)
\(684\) −7204.93 −0.402759
\(685\) −8364.25 −0.466543
\(686\) −24661.4 −1.37256
\(687\) −12232.4 −0.679324
\(688\) 370.280 0.0205186
\(689\) −8497.42 −0.469849
\(690\) −941.517 −0.0519463
\(691\) 30552.4 1.68201 0.841005 0.541027i \(-0.181964\pi\)
0.841005 + 0.541027i \(0.181964\pi\)
\(692\) 29999.9 1.64801
\(693\) 0 0
\(694\) 27138.7 1.48440
\(695\) −12701.9 −0.693253
\(696\) 2217.63 0.120774
\(697\) −7957.96 −0.432467
\(698\) 16708.1 0.906032
\(699\) 5048.27 0.273166
\(700\) −6765.61 −0.365309
\(701\) 9151.47 0.493076 0.246538 0.969133i \(-0.420707\pi\)
0.246538 + 0.969133i \(0.420707\pi\)
\(702\) 1942.60 0.104443
\(703\) 3992.86 0.214216
\(704\) 0 0
\(705\) 7792.95 0.416311
\(706\) −966.793 −0.0515378
\(707\) 1530.10 0.0813937
\(708\) −4306.67 −0.228608
\(709\) −6261.96 −0.331697 −0.165848 0.986151i \(-0.553036\pi\)
−0.165848 + 0.986151i \(0.553036\pi\)
\(710\) −19239.2 −1.01695
\(711\) 7480.85 0.394590
\(712\) −35794.9 −1.88409
\(713\) 3319.60 0.174362
\(714\) 20753.8 1.08780
\(715\) 0 0
\(716\) 42603.9 2.22372
\(717\) 12074.9 0.628933
\(718\) 11485.7 0.596995
\(719\) −18228.7 −0.945500 −0.472750 0.881197i \(-0.656739\pi\)
−0.472750 + 0.881197i \(0.656739\pi\)
\(720\) 136.859 0.00708396
\(721\) 1432.13 0.0739740
\(722\) 14389.9 0.741740
\(723\) −8352.81 −0.429660
\(724\) 15347.7 0.787836
\(725\) 786.656 0.0402975
\(726\) 0 0
\(727\) −7233.66 −0.369026 −0.184513 0.982830i \(-0.559071\pi\)
−0.184513 + 0.982830i \(0.559071\pi\)
\(728\) 7591.81 0.386499
\(729\) 729.000 0.0370370
\(730\) 8068.93 0.409102
\(731\) −8881.73 −0.449388
\(732\) −3525.31 −0.178004
\(733\) 13444.8 0.677485 0.338743 0.940879i \(-0.389998\pi\)
0.338743 + 0.940879i \(0.389998\pi\)
\(734\) −26414.4 −1.32830
\(735\) −1244.12 −0.0624355
\(736\) −2376.41 −0.119016
\(737\) 0 0
\(738\) −4511.14 −0.225010
\(739\) −18490.9 −0.920432 −0.460216 0.887807i \(-0.652228\pi\)
−0.460216 + 0.887807i \(0.652228\pi\)
\(740\) −4287.97 −0.213012
\(741\) 2867.90 0.142180
\(742\) −51461.8 −2.54612
\(743\) 25160.9 1.24235 0.621173 0.783674i \(-0.286657\pi\)
0.621173 + 0.783674i \(0.286657\pi\)
\(744\) −17126.3 −0.843927
\(745\) −15453.4 −0.759957
\(746\) 20528.8 1.00752
\(747\) −12502.2 −0.612358
\(748\) 0 0
\(749\) 23562.9 1.14949
\(750\) 1723.07 0.0838902
\(751\) −13419.5 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(752\) −1580.06 −0.0766207
\(753\) 5482.81 0.265345
\(754\) −2263.94 −0.109347
\(755\) −6793.68 −0.327480
\(756\) 7306.86 0.351518
\(757\) −7014.90 −0.336804 −0.168402 0.985718i \(-0.553861\pi\)
−0.168402 + 0.985718i \(0.553861\pi\)
\(758\) −35858.6 −1.71826
\(759\) 0 0
\(760\) 7171.12 0.342268
\(761\) 30156.9 1.43651 0.718256 0.695779i \(-0.244941\pi\)
0.718256 + 0.695779i \(0.244941\pi\)
\(762\) 22077.8 1.04960
\(763\) −45958.4 −2.18061
\(764\) −36195.7 −1.71402
\(765\) −3282.78 −0.155149
\(766\) 51283.5 2.41899
\(767\) 1714.26 0.0807019
\(768\) 13217.4 0.621017
\(769\) −11292.2 −0.529530 −0.264765 0.964313i \(-0.585294\pi\)
−0.264765 + 0.964313i \(0.585294\pi\)
\(770\) 0 0
\(771\) −1755.51 −0.0820016
\(772\) 16398.9 0.764519
\(773\) 8524.10 0.396624 0.198312 0.980139i \(-0.436454\pi\)
0.198312 + 0.980139i \(0.436454\pi\)
\(774\) −5034.80 −0.233814
\(775\) −6075.21 −0.281584
\(776\) 10022.0 0.463621
\(777\) −4049.35 −0.186962
\(778\) −39011.4 −1.79772
\(779\) −6659.89 −0.306310
\(780\) −3079.87 −0.141381
\(781\) 0 0
\(782\) −4578.96 −0.209390
\(783\) −849.589 −0.0387763
\(784\) 252.251 0.0114910
\(785\) −5059.76 −0.230052
\(786\) −27626.1 −1.25368
\(787\) 14983.9 0.678676 0.339338 0.940665i \(-0.389797\pi\)
0.339338 + 0.940665i \(0.389797\pi\)
\(788\) 1888.10 0.0853565
\(789\) −714.293 −0.0322300
\(790\) −19096.3 −0.860022
\(791\) 35493.0 1.59543
\(792\) 0 0
\(793\) 1403.24 0.0628380
\(794\) −27659.9 −1.23629
\(795\) 8140.10 0.363144
\(796\) 9982.04 0.444477
\(797\) 37172.3 1.65208 0.826041 0.563610i \(-0.190588\pi\)
0.826041 + 0.563610i \(0.190588\pi\)
\(798\) 17368.5 0.770475
\(799\) 37900.1 1.67811
\(800\) 4349.06 0.192203
\(801\) 13713.3 0.604913
\(802\) 47781.0 2.10375
\(803\) 0 0
\(804\) −19214.5 −0.842841
\(805\) 1409.65 0.0617186
\(806\) 17484.0 0.764080
\(807\) −13855.4 −0.604378
\(808\) 1741.68 0.0758316
\(809\) −23797.1 −1.03419 −0.517096 0.855928i \(-0.672987\pi\)
−0.517096 + 0.855928i \(0.672987\pi\)
\(810\) −1860.92 −0.0807234
\(811\) −8988.35 −0.389178 −0.194589 0.980885i \(-0.562337\pi\)
−0.194589 + 0.980885i \(0.562337\pi\)
\(812\) −8515.54 −0.368026
\(813\) −430.318 −0.0185633
\(814\) 0 0
\(815\) −14081.8 −0.605234
\(816\) 665.600 0.0285547
\(817\) −7432.98 −0.318295
\(818\) −21750.6 −0.929696
\(819\) −2908.48 −0.124091
\(820\) 7152.11 0.304589
\(821\) 25156.8 1.06940 0.534702 0.845041i \(-0.320424\pi\)
0.534702 + 0.845041i \(0.320424\pi\)
\(822\) −23059.5 −0.978459
\(823\) 1318.51 0.0558447 0.0279224 0.999610i \(-0.491111\pi\)
0.0279224 + 0.999610i \(0.491111\pi\)
\(824\) 1630.16 0.0689189
\(825\) 0 0
\(826\) 10381.9 0.437326
\(827\) −124.982 −0.00525519 −0.00262760 0.999997i \(-0.500836\pi\)
−0.00262760 + 0.999997i \(0.500836\pi\)
\(828\) −1612.13 −0.0676635
\(829\) −8886.80 −0.372318 −0.186159 0.982520i \(-0.559604\pi\)
−0.186159 + 0.982520i \(0.559604\pi\)
\(830\) 31914.3 1.33465
\(831\) −25807.5 −1.07732
\(832\) −12897.3 −0.537419
\(833\) −6050.63 −0.251671
\(834\) −35018.0 −1.45393
\(835\) −17244.5 −0.714694
\(836\) 0 0
\(837\) 6561.22 0.270955
\(838\) 37299.1 1.53756
\(839\) 2995.21 0.123249 0.0616247 0.998099i \(-0.480372\pi\)
0.0616247 + 0.998099i \(0.480372\pi\)
\(840\) −7272.58 −0.298724
\(841\) −23398.9 −0.959403
\(842\) 43578.8 1.78364
\(843\) −8978.43 −0.366825
\(844\) −52143.6 −2.12661
\(845\) −9759.07 −0.397304
\(846\) 21484.5 0.873111
\(847\) 0 0
\(848\) −1650.44 −0.0668355
\(849\) −20576.7 −0.831789
\(850\) 8379.95 0.338153
\(851\) 893.418 0.0359882
\(852\) −32942.7 −1.32464
\(853\) −18130.5 −0.727757 −0.363878 0.931446i \(-0.618548\pi\)
−0.363878 + 0.931446i \(0.618548\pi\)
\(854\) 8498.27 0.340521
\(855\) −2747.31 −0.109890
\(856\) 26821.1 1.07094
\(857\) −26394.1 −1.05205 −0.526024 0.850470i \(-0.676318\pi\)
−0.526024 + 0.850470i \(0.676318\pi\)
\(858\) 0 0
\(859\) −29456.2 −1.17000 −0.585002 0.811032i \(-0.698906\pi\)
−0.585002 + 0.811032i \(0.698906\pi\)
\(860\) 7982.34 0.316506
\(861\) 6754.12 0.267340
\(862\) 42893.7 1.69486
\(863\) −762.616 −0.0300808 −0.0150404 0.999887i \(-0.504788\pi\)
−0.0150404 + 0.999887i \(0.504788\pi\)
\(864\) −4696.99 −0.184948
\(865\) 11439.3 0.449649
\(866\) 13706.6 0.537838
\(867\) −1226.43 −0.0480411
\(868\) 65764.0 2.57163
\(869\) 0 0
\(870\) 2168.74 0.0845141
\(871\) 7648.29 0.297535
\(872\) −52313.3 −2.03160
\(873\) −3839.51 −0.148852
\(874\) −3832.06 −0.148308
\(875\) −2579.79 −0.0996719
\(876\) 13816.2 0.532882
\(877\) −44767.2 −1.72369 −0.861847 0.507168i \(-0.830692\pi\)
−0.861847 + 0.507168i \(0.830692\pi\)
\(878\) −24041.8 −0.924112
\(879\) 12149.1 0.466188
\(880\) 0 0
\(881\) −32057.9 −1.22595 −0.612973 0.790104i \(-0.710027\pi\)
−0.612973 + 0.790104i \(0.710027\pi\)
\(882\) −3429.93 −0.130943
\(883\) 7078.95 0.269791 0.134896 0.990860i \(-0.456930\pi\)
0.134896 + 0.990860i \(0.456930\pi\)
\(884\) −14978.6 −0.569891
\(885\) −1642.18 −0.0623742
\(886\) −34542.8 −1.30981
\(887\) 25148.1 0.951964 0.475982 0.879455i \(-0.342093\pi\)
0.475982 + 0.879455i \(0.342093\pi\)
\(888\) −4609.28 −0.174186
\(889\) −33055.0 −1.24705
\(890\) −35005.9 −1.31843
\(891\) 0 0
\(892\) 11907.4 0.446963
\(893\) 31718.0 1.18858
\(894\) −42603.6 −1.59382
\(895\) 16245.3 0.606726
\(896\) −49385.8 −1.84137
\(897\) 641.704 0.0238862
\(898\) −73843.6 −2.74409
\(899\) −7646.56 −0.283679
\(900\) 2950.36 0.109273
\(901\) 39588.4 1.46380
\(902\) 0 0
\(903\) 7538.14 0.277800
\(904\) 40400.8 1.48641
\(905\) 5852.23 0.214955
\(906\) −18729.6 −0.686809
\(907\) 1269.76 0.0464848 0.0232424 0.999730i \(-0.492601\pi\)
0.0232424 + 0.999730i \(0.492601\pi\)
\(908\) 27040.4 0.988289
\(909\) −667.248 −0.0243468
\(910\) 7424.47 0.270460
\(911\) −33783.1 −1.22863 −0.614316 0.789060i \(-0.710568\pi\)
−0.614316 + 0.789060i \(0.710568\pi\)
\(912\) 557.030 0.0202249
\(913\) 0 0
\(914\) 44655.1 1.61604
\(915\) −1344.23 −0.0485672
\(916\) 53466.6 1.92859
\(917\) 41361.9 1.48952
\(918\) −9050.35 −0.325388
\(919\) −39262.5 −1.40930 −0.704652 0.709553i \(-0.748897\pi\)
−0.704652 + 0.709553i \(0.748897\pi\)
\(920\) 1604.57 0.0575010
\(921\) −28718.1 −1.02746
\(922\) −66800.1 −2.38606
\(923\) 13112.7 0.467618
\(924\) 0 0
\(925\) −1635.04 −0.0581189
\(926\) 45848.3 1.62707
\(927\) −624.525 −0.0221274
\(928\) 5473.95 0.193633
\(929\) −21175.0 −0.747825 −0.373913 0.927464i \(-0.621984\pi\)
−0.373913 + 0.927464i \(0.621984\pi\)
\(930\) −16748.8 −0.590554
\(931\) −5063.68 −0.178255
\(932\) −22065.5 −0.775515
\(933\) −16189.3 −0.568073
\(934\) −69787.5 −2.44488
\(935\) 0 0
\(936\) −3310.65 −0.115611
\(937\) 5135.11 0.179036 0.0895180 0.995985i \(-0.471467\pi\)
0.0895180 + 0.995985i \(0.471467\pi\)
\(938\) 46319.4 1.61235
\(939\) −29265.1 −1.01707
\(940\) −34062.2 −1.18190
\(941\) 9702.77 0.336133 0.168067 0.985776i \(-0.446248\pi\)
0.168067 + 0.985776i \(0.446248\pi\)
\(942\) −13949.3 −0.482477
\(943\) −1490.18 −0.0514600
\(944\) 332.959 0.0114798
\(945\) 2786.18 0.0959094
\(946\) 0 0
\(947\) −699.579 −0.0240055 −0.0120028 0.999928i \(-0.503821\pi\)
−0.0120028 + 0.999928i \(0.503821\pi\)
\(948\) −32698.0 −1.12024
\(949\) −5499.49 −0.188115
\(950\) 7013.05 0.239509
\(951\) 13061.2 0.445363
\(952\) −35369.3 −1.20412
\(953\) −42039.3 −1.42895 −0.714473 0.699663i \(-0.753333\pi\)
−0.714473 + 0.699663i \(0.753333\pi\)
\(954\) 22441.6 0.761606
\(955\) −13801.8 −0.467659
\(956\) −52778.1 −1.78553
\(957\) 0 0
\(958\) 52063.4 1.75584
\(959\) 34524.9 1.16253
\(960\) 12355.0 0.415369
\(961\) 29262.0 0.982243
\(962\) 4705.55 0.157706
\(963\) −10275.3 −0.343841
\(964\) 36509.3 1.21980
\(965\) 6253.05 0.208594
\(966\) 3886.27 0.129440
\(967\) 32794.8 1.09060 0.545299 0.838242i \(-0.316416\pi\)
0.545299 + 0.838242i \(0.316416\pi\)
\(968\) 0 0
\(969\) −13361.2 −0.442955
\(970\) 9801.10 0.324427
\(971\) −3322.53 −0.109810 −0.0549048 0.998492i \(-0.517486\pi\)
−0.0549048 + 0.998492i \(0.517486\pi\)
\(972\) −3186.39 −0.105148
\(973\) 52429.3 1.72745
\(974\) −87701.4 −2.88515
\(975\) −1174.38 −0.0385747
\(976\) 272.550 0.00893864
\(977\) 22192.5 0.726716 0.363358 0.931650i \(-0.381630\pi\)
0.363358 + 0.931650i \(0.381630\pi\)
\(978\) −38822.4 −1.26933
\(979\) 0 0
\(980\) 5437.92 0.177253
\(981\) 20041.6 0.652272
\(982\) −37277.4 −1.21137
\(983\) 7383.09 0.239556 0.119778 0.992801i \(-0.461782\pi\)
0.119778 + 0.992801i \(0.461782\pi\)
\(984\) 7688.04 0.249071
\(985\) 719.953 0.0232889
\(986\) 10547.4 0.340668
\(987\) −32166.7 −1.03736
\(988\) −12535.3 −0.403645
\(989\) −1663.16 −0.0534735
\(990\) 0 0
\(991\) 46260.8 1.48287 0.741434 0.671026i \(-0.234146\pi\)
0.741434 + 0.671026i \(0.234146\pi\)
\(992\) −42274.3 −1.35304
\(993\) −16162.9 −0.516530
\(994\) 79413.1 2.53403
\(995\) 3806.25 0.121273
\(996\) 54645.9 1.73847
\(997\) −41196.8 −1.30864 −0.654320 0.756217i \(-0.727045\pi\)
−0.654320 + 0.756217i \(0.727045\pi\)
\(998\) −84219.5 −2.67127
\(999\) 1765.85 0.0559249
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 1815.4.a.q.1.1 3
11.10 odd 2 165.4.a.g.1.3 3
33.32 even 2 495.4.a.i.1.1 3
55.32 even 4 825.4.c.m.199.6 6
55.43 even 4 825.4.c.m.199.1 6
55.54 odd 2 825.4.a.p.1.1 3
165.164 even 2 2475.4.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 11.10 odd 2
495.4.a.i.1.1 3 33.32 even 2
825.4.a.p.1.1 3 55.54 odd 2
825.4.c.m.199.1 6 55.43 even 4
825.4.c.m.199.6 6 55.32 even 4
1815.4.a.q.1.1 3 1.1 even 1 trivial
2475.4.a.z.1.3 3 165.164 even 2