Properties

Label 825.4.a.p.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, 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("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
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 \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.59486 q^{2} +3.00000 q^{3} +13.1127 q^{4} -13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.59486 q^{2} +3.00000 q^{3} +13.1127 q^{4} -13.7846 q^{6} -20.6383 q^{7} -23.4921 q^{8} +9.00000 q^{9} +11.0000 q^{11} +39.3381 q^{12} +15.6584 q^{13} +94.8302 q^{14} +3.04132 q^{16} -72.9507 q^{17} -41.3537 q^{18} +61.0513 q^{19} -61.9150 q^{21} -50.5434 q^{22} +13.6605 q^{23} -70.4764 q^{24} -71.9483 q^{26} +27.0000 q^{27} -270.624 q^{28} -31.4663 q^{29} -243.008 q^{31} +173.963 q^{32} +33.0000 q^{33} +335.198 q^{34} +118.014 q^{36} +65.4018 q^{37} -280.522 q^{38} +46.9753 q^{39} -109.087 q^{41} +284.491 q^{42} +121.750 q^{43} +144.240 q^{44} -62.7678 q^{46} +519.530 q^{47} +9.12396 q^{48} +82.9413 q^{49} -218.852 q^{51} +205.324 q^{52} +542.673 q^{53} -124.061 q^{54} +484.839 q^{56} +183.154 q^{57} +144.583 q^{58} +109.478 q^{59} -89.6156 q^{61} +1116.59 q^{62} -185.745 q^{63} -823.664 q^{64} -151.630 q^{66} -488.446 q^{67} -956.581 q^{68} +40.9814 q^{69} +837.423 q^{71} -211.429 q^{72} -351.216 q^{73} -300.512 q^{74} +800.547 q^{76} -227.022 q^{77} -215.845 q^{78} -831.205 q^{79} +81.0000 q^{81} +501.238 q^{82} -1389.13 q^{83} -811.873 q^{84} -559.423 q^{86} -94.3988 q^{87} -258.413 q^{88} +1523.70 q^{89} -323.164 q^{91} +179.125 q^{92} -729.025 q^{93} -2387.17 q^{94} +521.888 q^{96} +426.612 q^{97} -381.103 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 9 q^{3} + 17 q^{4} - 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 9 q^{3} + 17 q^{4} - 3 q^{6} - 6 q^{7} + 3 q^{8} + 27 q^{9} + 33 q^{11} + 51 q^{12} + 20 q^{13} + 144 q^{14} + 25 q^{16} - 32 q^{17} - 9 q^{18} + 116 q^{19} - 18 q^{21} - 11 q^{22} - 240 q^{23} + 9 q^{24} + 302 q^{26} + 81 q^{27} - 160 q^{28} + 238 q^{29} + 92 q^{31} - 197 q^{32} + 99 q^{33} + 354 q^{34} + 153 q^{36} + 90 q^{37} - 324 q^{38} + 60 q^{39} - 46 q^{41} + 432 q^{42} + 134 q^{43} + 187 q^{44} - 240 q^{46} + 220 q^{47} + 75 q^{48} - 457 q^{49} - 96 q^{51} + 1530 q^{52} + 798 q^{53} - 27 q^{54} + 688 q^{56} + 348 q^{57} + 978 q^{58} + 1236 q^{59} + 342 q^{61} + 1792 q^{62} - 54 q^{63} - 1919 q^{64} - 33 q^{66} - 764 q^{67} - 1074 q^{68} - 720 q^{69} + 1816 q^{71} + 27 q^{72} - 100 q^{73} - 1874 q^{74} + 396 q^{76} - 66 q^{77} + 906 q^{78} - 96 q^{79} + 243 q^{81} + 910 q^{82} - 858 q^{83} - 480 q^{84} + 188 q^{86} + 714 q^{87} + 33 q^{88} + 838 q^{89} + 332 q^{91} + 688 q^{92} + 276 q^{93} - 3112 q^{94} - 591 q^{96} + 1322 q^{97} - 1017 q^{98} + 297 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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
\(5\) 0 0
\(6\) −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
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 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
\(15\) 0 0
\(16\) 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 \(-0.379843\pi\)
0.368582 + 0.929595i \(0.379843\pi\)
\(20\) 0 0
\(21\) −61.9150 −0.643379
\(22\) −50.5434 −0.489813
\(23\) 13.6605 0.123844 0.0619218 0.998081i \(-0.480277\pi\)
0.0619218 + 0.998081i \(0.480277\pi\)
\(24\) −70.4764 −0.599414
\(25\) 0 0
\(26\) −71.9483 −0.542701
\(27\) 27.0000 0.192450
\(28\) −270.624 −1.82654
\(29\) −31.4663 −0.201487 −0.100744 0.994912i \(-0.532122\pi\)
−0.100744 + 0.994912i \(0.532122\pi\)
\(30\) 0 0
\(31\) −243.008 −1.40792 −0.703961 0.710239i \(-0.748587\pi\)
−0.703961 + 0.710239i \(0.748587\pi\)
\(32\) 173.963 0.961016
\(33\) 33.0000 0.174078
\(34\) 335.198 1.69076
\(35\) 0 0
\(36\) 118.014 0.546363
\(37\) 65.4018 0.290594 0.145297 0.989388i \(-0.453586\pi\)
0.145297 + 0.989388i \(0.453586\pi\)
\(38\) −280.522 −1.19754
\(39\) 46.9753 0.192874
\(40\) 0 0
\(41\) −109.087 −0.415524 −0.207762 0.978179i \(-0.566618\pi\)
−0.207762 + 0.978179i \(0.566618\pi\)
\(42\) 284.491 1.04519
\(43\) 121.750 0.431783 0.215891 0.976417i \(-0.430734\pi\)
0.215891 + 0.976417i \(0.430734\pi\)
\(44\) 144.240 0.494204
\(45\) 0 0
\(46\) −62.7678 −0.201187
\(47\) 519.530 1.61237 0.806184 0.591665i \(-0.201529\pi\)
0.806184 + 0.591665i \(0.201529\pi\)
\(48\) 9.12396 0.0274361
\(49\) 82.9413 0.241811
\(50\) 0 0
\(51\) −218.852 −0.600891
\(52\) 205.324 0.547565
\(53\) 542.673 1.40645 0.703226 0.710967i \(-0.251742\pi\)
0.703226 + 0.710967i \(0.251742\pi\)
\(54\) −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 \(0.461458\pi\)
\(60\) 0 0
\(61\) −89.6156 −0.188100 −0.0940501 0.995567i \(-0.529981\pi\)
−0.0940501 + 0.995567i \(0.529981\pi\)
\(62\) 1116.59 2.28721
\(63\) −185.745 −0.371455
\(64\) −823.664 −1.60872
\(65\) 0 0
\(66\) −151.630 −0.282794
\(67\) −488.446 −0.890644 −0.445322 0.895371i \(-0.646911\pi\)
−0.445322 + 0.895371i \(0.646911\pi\)
\(68\) −956.581 −1.70592
\(69\) 40.9814 0.0715011
\(70\) 0 0
\(71\) 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
\(75\) 0 0
\(76\) 800.547 1.20828
\(77\) −227.022 −0.335994
\(78\) −215.845 −0.313328
\(79\) −831.205 −1.18377 −0.591885 0.806022i \(-0.701616\pi\)
−0.591885 + 0.806022i \(0.701616\pi\)
\(80\) 0 0
\(81\) 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
\(85\) 0 0
\(86\) −559.423 −0.701443
\(87\) −94.3988 −0.116329
\(88\) −258.413 −0.313034
\(89\) 1523.70 1.81474 0.907369 0.420335i \(-0.138088\pi\)
0.907369 + 0.420335i \(0.138088\pi\)
\(90\) 0 0
\(91\) −323.164 −0.372273
\(92\) 179.125 0.202990
\(93\) −729.025 −0.812864
\(94\) −2387.17 −2.61933
\(95\) 0 0
\(96\) 521.888 0.554843
\(97\) 426.612 0.446555 0.223278 0.974755i \(-0.428324\pi\)
0.223278 + 0.974755i \(0.428324\pi\)
\(98\) −381.103 −0.392829
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 74.1387 0.0730403 0.0365202 0.999333i \(-0.488373\pi\)
0.0365202 + 0.999333i \(0.488373\pi\)
\(102\) 1005.59 0.976163
\(103\) 69.3916 0.0663821 0.0331911 0.999449i \(-0.489433\pi\)
0.0331911 + 0.999449i \(0.489433\pi\)
\(104\) −367.850 −0.346833
\(105\) 0 0
\(106\) −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 \(-0.933734\pi\)
−0.978409 + 0.206680i \(0.933734\pi\)
\(110\) 0 0
\(111\) 196.205 0.167775
\(112\) −62.7678 −0.0529554
\(113\) 1719.76 1.43169 0.715847 0.698257i \(-0.246041\pi\)
0.715847 + 0.698257i \(0.246041\pi\)
\(114\) −841.566 −0.691402
\(115\) 0 0
\(116\) −412.608 −0.330256
\(117\) 140.926 0.111356
\(118\) −503.038 −0.392444
\(119\) 1505.58 1.15980
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 411.771 0.305574
\(123\) −327.260 −0.239903
\(124\) −3186.49 −2.30771
\(125\) 0 0
\(126\) 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
\(130\) 0 0
\(131\) 2004.13 1.33665 0.668327 0.743868i \(-0.267011\pi\)
0.668327 + 0.743868i \(0.267011\pi\)
\(132\) 432.719 0.285329
\(133\) −1260.00 −0.821471
\(134\) 2244.34 1.44687
\(135\) 0 0
\(136\) 1713.77 1.08055
\(137\) 1672.85 1.04322 0.521610 0.853184i \(-0.325331\pi\)
0.521610 + 0.853184i \(0.325331\pi\)
\(138\) −188.303 −0.116155
\(139\) 2540.38 1.55016 0.775080 0.631863i \(-0.217709\pi\)
0.775080 + 0.631863i \(0.217709\pi\)
\(140\) 0 0
\(141\) 1558.59 0.930901
\(142\) −3847.84 −2.27397
\(143\) 172.243 0.100725
\(144\) 27.3719 0.0158402
\(145\) 0 0
\(146\) 1613.79 0.914780
\(147\) 248.824 0.139610
\(148\) 857.594 0.476310
\(149\) 3090.68 1.69932 0.849658 0.527334i \(-0.176808\pi\)
0.849658 + 0.527334i \(0.176808\pi\)
\(150\) 0 0
\(151\) 1358.74 0.732267 0.366134 0.930562i \(-0.380681\pi\)
0.366134 + 0.930562i \(0.380681\pi\)
\(152\) −1434.22 −0.765335
\(153\) −656.557 −0.346925
\(154\) 1043.13 0.545831
\(155\) 0 0
\(156\) 615.973 0.316137
\(157\) 1011.95 0.514411 0.257205 0.966357i \(-0.417198\pi\)
0.257205 + 0.966357i \(0.417198\pi\)
\(158\) 3819.27 1.92307
\(159\) 1628.02 0.812015
\(160\) 0 0
\(161\) −281.929 −0.138007
\(162\) −372.183 −0.180503
\(163\) 2816.37 1.35334 0.676672 0.736285i \(-0.263422\pi\)
0.676672 + 0.736285i \(0.263422\pi\)
\(164\) −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
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) 33.4545 0.0143280
\(177\) 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 \(0.262700\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) 3349.76 1.32052
\(187\) −802.458 −0.313805
\(188\) 6812.44 2.64281
\(189\) −557.235 −0.214460
\(190\) 0 0
\(191\) −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
\(195\) 0 0
\(196\) 1087.58 0.396350
\(197\) 143.991 0.0520756 0.0260378 0.999661i \(-0.491711\pi\)
0.0260378 + 0.999661i \(0.491711\pi\)
\(198\) −454.891 −0.163271
\(199\) 761.249 0.271174 0.135587 0.990765i \(-0.456708\pi\)
0.135587 + 0.990765i \(0.456708\pi\)
\(200\) 0 0
\(201\) −1465.34 −0.514213
\(202\) −340.657 −0.118656
\(203\) 649.411 0.224531
\(204\) −2869.74 −0.984913
\(205\) 0 0
\(206\) −318.844 −0.107840
\(207\) 122.944 0.0412812
\(208\) 47.6223 0.0158751
\(209\) 671.564 0.222263
\(210\) 0 0
\(211\) 3976.58 1.29743 0.648717 0.761029i \(-0.275306\pi\)
0.648717 + 0.761029i \(0.275306\pi\)
\(212\) 7115.91 2.30530
\(213\) 2512.27 0.808159
\(214\) 5245.97 1.67573
\(215\) 0 0
\(216\) −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 \(-0.543536\pi\)
−0.136345 + 0.990661i \(0.543536\pi\)
\(224\) −3590.30 −1.07092
\(225\) 0 0
\(226\) −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 \(0.299793\pi\)
\(230\) 0 0
\(231\) −681.065 −0.193986
\(232\) 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
\(235\) 0 0
\(236\) 1435.56 0.395961
\(237\) −2493.62 −0.683450
\(238\) −6917.93 −1.88413
\(239\) 4024.96 1.08934 0.544672 0.838649i \(-0.316654\pi\)
0.544672 + 0.838649i \(0.316654\pi\)
\(240\) 0 0
\(241\) −2784.27 −0.744194 −0.372097 0.928194i \(-0.621361\pi\)
−0.372097 + 0.928194i \(0.621361\pi\)
\(242\) −555.978 −0.147684
\(243\) 243.000 0.0641500
\(244\) −1175.10 −0.308313
\(245\) 0 0
\(246\) 1503.71 0.389729
\(247\) 955.968 0.246262
\(248\) 5708.78 1.46173
\(249\) −4167.40 −1.06064
\(250\) 0 0
\(251\) −1827.60 −0.459591 −0.229796 0.973239i \(-0.573806\pi\)
−0.229796 + 0.973239i \(0.573806\pi\)
\(252\) −2435.62 −0.608848
\(253\) 150.265 0.0373402
\(254\) −7359.26 −1.81796
\(255\) 0 0
\(256\) −4405.79 −1.07563
\(257\) −585.171 −0.142031 −0.0710155 0.997475i \(-0.522624\pi\)
−0.0710155 + 0.997475i \(0.522624\pi\)
\(258\) −1678.27 −0.404978
\(259\) −1349.78 −0.323828
\(260\) 0 0
\(261\) −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 \(0.491114\pi\)
\(264\) −775.240 −0.180730
\(265\) 0 0
\(266\) 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 \(0.324661\pi\)
\(270\) 0 0
\(271\) −143.439 −0.0321525 −0.0160762 0.999871i \(-0.505117\pi\)
−0.0160762 + 0.999871i \(0.505117\pi\)
\(272\) −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
\(280\) 0 0
\(281\) −2992.81 −0.635360 −0.317680 0.948198i \(-0.602904\pi\)
−0.317680 + 0.948198i \(0.602904\pi\)
\(282\) −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
\(285\) 0 0
\(286\) −791.431 −0.163630
\(287\) 2251.37 0.463046
\(288\) 1565.66 0.320339
\(289\) 408.809 0.0832096
\(290\) 0 0
\(291\) 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
\(295\) 0 0
\(296\) −1536.43 −0.301699
\(297\) 297.000 0.0580259
\(298\) −14201.2 −2.76059
\(299\) 213.901 0.0413720
\(300\) 0 0
\(301\) −2512.71 −0.481164
\(302\) −6243.20 −1.18959
\(303\) 222.416 0.0421699
\(304\) 185.677 0.0350305
\(305\) 0 0
\(306\) 3016.78 0.563588
\(307\) 9572.69 1.77962 0.889808 0.456335i \(-0.150838\pi\)
0.889808 + 0.456335i \(0.150838\pi\)
\(308\) −2976.87 −0.550724
\(309\) 208.175 0.0383257
\(310\) 0 0
\(311\) 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 \(-0.843002\pi\)
−0.880811 + 0.473469i \(0.843002\pi\)
\(314\) −4649.77 −0.835674
\(315\) 0 0
\(316\) −10899.3 −1.94030
\(317\) 4353.75 0.771391 0.385695 0.922626i \(-0.373962\pi\)
0.385695 + 0.922626i \(0.373962\pi\)
\(318\) −7480.52 −1.31914
\(319\) −346.129 −0.0607508
\(320\) 0 0
\(321\) −3425.12 −0.595549
\(322\) 1295.42 0.224196
\(323\) −4453.74 −0.767221
\(324\) 1062.13 0.182121
\(325\) 0 0
\(326\) −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
\(335\) 0 0
\(336\) −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
\(340\) 0 0
\(341\) −2673.09 −0.424504
\(342\) −2524.70 −0.399181
\(343\) 5367.18 0.844899
\(344\) −2860.16 −0.448283
\(345\) 0 0
\(346\) −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 \(-0.410043\pi\)
0.278860 + 0.960332i \(0.410043\pi\)
\(350\) 0 0
\(351\) 422.778 0.0642912
\(352\) 1913.59 0.289757
\(353\) −210.408 −0.0317248 −0.0158624 0.999874i \(-0.505049\pi\)
−0.0158624 + 0.999874i \(0.505049\pi\)
\(354\) −1509.11 −0.226577
\(355\) 0 0
\(356\) 19979.8 2.97451
\(357\) 4516.75 0.669612
\(358\) −14928.9 −2.20396
\(359\) 2499.68 0.367488 0.183744 0.982974i \(-0.441178\pi\)
0.183744 + 0.982974i \(0.441178\pi\)
\(360\) 0 0
\(361\) −3131.74 −0.456588
\(362\) −5378.03 −0.780837
\(363\) 363.000 0.0524864
\(364\) −4237.56 −0.610188
\(365\) 0 0
\(366\) 1235.31 0.176423
\(367\) −5748.70 −0.817656 −0.408828 0.912612i \(-0.634062\pi\)
−0.408828 + 0.912612i \(0.634062\pi\)
\(368\) 41.5458 0.00588512
\(369\) −981.781 −0.138508
\(370\) 0 0
\(371\) −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 \(0.600362\pi\)
\(374\) 3687.18 0.509785
\(375\) 0 0
\(376\) −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 \(0.322623\pi\)
\(380\) 0 0
\(381\) 4804.89 0.646094
\(382\) 12683.4 1.69880
\(383\) 11161.1 1.48904 0.744522 0.667597i \(-0.232677\pi\)
0.744522 + 0.667597i \(0.232677\pi\)
\(384\) 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 \(0.313366\pi\)
\(390\) 0 0
\(391\) −996.540 −0.128893
\(392\) −1948.47 −0.251052
\(393\) 6012.39 0.771717
\(394\) −661.616 −0.0845983
\(395\) 0 0
\(396\) 1298.16 0.164735
\(397\) −6019.74 −0.761013 −0.380507 0.924778i \(-0.624250\pi\)
−0.380507 + 0.924778i \(0.624250\pi\)
\(398\) −3497.83 −0.440529
\(399\) −3779.99 −0.474277
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) −2983.95 −0.364756
\(407\) 719.420 0.0876175
\(408\) 5141.30 0.623854
\(409\) −4733.68 −0.572287 −0.286144 0.958187i \(-0.592373\pi\)
−0.286144 + 0.958187i \(0.592373\pi\)
\(410\) 0 0
\(411\) 5018.55 0.602304
\(412\) 909.911 0.108806
\(413\) −2259.45 −0.269202
\(414\) −564.910 −0.0670624
\(415\) 0 0
\(416\) 2723.98 0.321044
\(417\) 7621.14 0.894985
\(418\) −3085.74 −0.361073
\(419\) −8117.57 −0.946466 −0.473233 0.880937i \(-0.656913\pi\)
−0.473233 + 0.880937i \(0.656913\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −11543.5 −1.31288
\(427\) 1849.52 0.209612
\(428\) −14970.8 −1.69075
\(429\) 516.728 0.0581536
\(430\) 0 0
\(431\) 9335.16 1.04329 0.521646 0.853162i \(-0.325318\pi\)
0.521646 + 0.853162i \(0.325318\pi\)
\(432\) 82.1157 0.00914535
\(433\) 2983.02 0.331074 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(434\) −23044.5 −2.54878
\(435\) 0 0
\(436\) −29200.0 −3.20739
\(437\) 833.988 0.0912931
\(438\) 4841.36 0.528148
\(439\) −5232.32 −0.568850 −0.284425 0.958698i \(-0.591803\pi\)
−0.284425 + 0.958698i \(0.591803\pi\)
\(440\) 0 0
\(441\) 746.472 0.0806038
\(442\) 5248.68 0.564828
\(443\) −7517.71 −0.806269 −0.403135 0.915141i \(-0.632079\pi\)
−0.403135 + 0.915141i \(0.632079\pi\)
\(444\) 2572.78 0.274997
\(445\) 0 0
\(446\) 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 \(0.179850\pi\)
\(450\) 0 0
\(451\) −1199.96 −0.125285
\(452\) 22550.7 2.34667
\(453\) 4076.21 0.422775
\(454\) −9475.29 −0.979510
\(455\) 0 0
\(456\) −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
\(460\) 0 0
\(461\) −14538.0 −1.46877 −0.734385 0.678733i \(-0.762529\pi\)
−0.734385 + 0.678733i \(0.762529\pi\)
\(462\) 3129.40 0.315136
\(463\) 9978.17 1.00157 0.500783 0.865573i \(-0.333045\pi\)
0.500783 + 0.865573i \(0.333045\pi\)
\(464\) −95.6990 −0.00957481
\(465\) 0 0
\(466\) 7732.03 0.768625
\(467\) −15188.2 −1.50498 −0.752489 0.658605i \(-0.771147\pi\)
−0.752489 + 0.658605i \(0.771147\pi\)
\(468\) 1847.92 0.182522
\(469\) 10080.7 0.992503
\(470\) 0 0
\(471\) 3035.85 0.296995
\(472\) −2571.88 −0.250806
\(473\) 1339.25 0.130187
\(474\) 11457.8 1.11028
\(475\) 0 0
\(476\) 19742.3 1.90102
\(477\) 4884.06 0.468817
\(478\) −18494.1 −1.76967
\(479\) 11330.8 1.08083 0.540415 0.841399i \(-0.318267\pi\)
0.540415 + 0.841399i \(0.318267\pi\)
\(480\) 0 0
\(481\) 1024.09 0.0970779
\(482\) 12793.3 1.20896
\(483\) −845.788 −0.0796784
\(484\) 1586.64 0.149008
\(485\) 0 0
\(486\) −1116.55 −0.104213
\(487\) −19086.9 −1.77599 −0.887997 0.459850i \(-0.847903\pi\)
−0.887997 + 0.459850i \(0.847903\pi\)
\(488\) 2105.26 0.195288
\(489\) 8449.11 0.781353
\(490\) 0 0
\(491\) −8112.85 −0.745677 −0.372839 0.927896i \(-0.621616\pi\)
−0.372839 + 0.927896i \(0.621616\pi\)
\(492\) −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 \(0.192765\pi\)
\(500\) 0 0
\(501\) −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
\(505\) 0 0
\(506\) −690.446 −0.0606602
\(507\) −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 \(0.256314\pi\)
\(510\) 0 0
\(511\) 7248.51 0.627505
\(512\) 1100.65 0.0950048
\(513\) 1648.38 0.141867
\(514\) 2688.78 0.230733
\(515\) 0 0
\(516\) 4789.40 0.408608
\(517\) 5714.83 0.486147
\(518\) 6202.07 0.526068
\(519\) 6863.56 0.580495
\(520\) 0 0
\(521\) 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
\(525\) 0 0
\(526\) −1094.02 −0.0906877
\(527\) 17727.6 1.46533
\(528\) 100.364 0.00827228
\(529\) −11980.4 −0.984663
\(530\) 0 0
\(531\) 985.306 0.0805247
\(532\) −16522.0 −1.34646
\(533\) −1708.13 −0.138813
\(534\) −21003.5 −1.70208
\(535\) 0 0
\(536\) 11474.6 0.924680
\(537\) 9747.17 0.783280
\(538\) −21221.2 −1.70058
\(539\) 912.355 0.0729089
\(540\) 0 0
\(541\) 5313.05 0.422229 0.211115 0.977461i \(-0.432291\pi\)
0.211115 + 0.977461i \(0.432291\pi\)
\(542\) 659.084 0.0522326
\(543\) 3511.34 0.277506
\(544\) −12690.7 −1.00020
\(545\) 0 0
\(546\) 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
\(555\) 0 0
\(556\) 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
\(560\) 0 0
\(561\) −2407.37 −0.181175
\(562\) 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
\(565\) 0 0
\(566\) −31515.6 −2.34046
\(567\) −1671.71 −0.123818
\(568\) −19672.9 −1.45327
\(569\) −1348.88 −0.0993814 −0.0496907 0.998765i \(-0.515824\pi\)
−0.0496907 + 0.998765i \(0.515824\pi\)
\(570\) 0 0
\(571\) 3463.51 0.253841 0.126920 0.991913i \(-0.459491\pi\)
0.126920 + 0.991913i \(0.459491\pi\)
\(572\) 2258.57 0.165097
\(573\) −8281.05 −0.603745
\(574\) −10344.7 −0.752231
\(575\) 0 0
\(576\) −7412.97 −0.536239
\(577\) −12052.6 −0.869598 −0.434799 0.900528i \(-0.643181\pi\)
−0.434799 + 0.900528i \(0.643181\pi\)
\(578\) −1878.42 −0.135176
\(579\) 3751.83 0.269293
\(580\) 0 0
\(581\) 28669.4 2.04717
\(582\) −5880.66 −0.418834
\(583\) 5969.41 0.424061
\(584\) 8250.80 0.584624
\(585\) 0 0
\(586\) 18607.8 1.31174
\(587\) 11133.1 0.782813 0.391407 0.920218i \(-0.371989\pi\)
0.391407 + 0.920218i \(0.371989\pi\)
\(588\) 3262.75 0.228833
\(589\) −14836.0 −1.03787
\(590\) 0 0
\(591\) 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 \(0.411352\pi\)
\(594\) −1364.67 −0.0942646
\(595\) 0 0
\(596\) 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 \(0.731229\pi\)
\(600\) 0 0
\(601\) −19946.1 −1.35377 −0.676887 0.736087i \(-0.736671\pi\)
−0.676887 + 0.736087i \(0.736671\pi\)
\(602\) 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
\(610\) 0 0
\(611\) 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
\(615\) 0 0
\(616\) 5333.22 0.348834
\(617\) −20795.5 −1.35688 −0.678440 0.734655i \(-0.737344\pi\)
−0.678440 + 0.734655i \(0.737344\pi\)
\(618\) −956.533 −0.0622612
\(619\) 1677.43 0.108920 0.0544602 0.998516i \(-0.482656\pi\)
0.0544602 + 0.998516i \(0.482656\pi\)
\(620\) 0 0
\(621\) 368.832 0.0238337
\(622\) −24795.8 −1.59842
\(623\) −31446.6 −2.02228
\(624\) 142.867 0.00916547
\(625\) 0 0
\(626\) 44823.0 2.86180
\(627\) 2014.69 0.128324
\(628\) 13269.4 0.843165
\(629\) −4771.11 −0.302443
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 21347.7 1.33096
\(637\) 1298.73 0.0807812
\(638\) 1590.41 0.0986912
\(639\) 7536.81 0.466591
\(640\) 0 0
\(641\) 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 \(-0.813242\pi\)
−0.832763 + 0.553630i \(0.813242\pi\)
\(644\) −3696.85 −0.226206
\(645\) 0 0
\(646\) 20464.3 1.24637
\(647\) −29154.9 −1.77156 −0.885778 0.464110i \(-0.846374\pi\)
−0.885778 + 0.464110i \(0.846374\pi\)
\(648\) −1902.86 −0.115357
\(649\) 1204.26 0.0728374
\(650\) 0 0
\(651\) 15045.9 0.905828
\(652\) 36930.2 2.21825
\(653\) 19141.7 1.14713 0.573564 0.819161i \(-0.305560\pi\)
0.573564 + 0.819161i \(0.305560\pi\)
\(654\) 30696.1 1.83534
\(655\) 0 0
\(656\) −331.768 −0.0197460
\(657\) −3160.94 −0.187702
\(658\) 49267.2 2.91890
\(659\) −24939.6 −1.47422 −0.737110 0.675773i \(-0.763810\pi\)
−0.737110 + 0.675773i \(0.763810\pi\)
\(660\) 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
\(665\) 0 0
\(666\) −2704.61 −0.157359
\(667\) −429.843 −0.0249529
\(668\) −45224.3 −2.61943
\(669\) −2724.25 −0.157438
\(670\) 0 0
\(671\) −985.772 −0.0567143
\(672\) −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
\(675\) 0 0
\(676\) −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
\(680\) 0 0
\(681\) 6186.46 0.348114
\(682\) 12282.5 0.689619
\(683\) 14420.5 0.807887 0.403943 0.914784i \(-0.367639\pi\)
0.403943 + 0.914784i \(0.367639\pi\)
\(684\) 7204.93 0.402759
\(685\) 0 0
\(686\) −24661.4 −1.37256
\(687\) 12232.4 0.679324
\(688\) 370.280 0.0205186
\(689\) 8497.42 0.469849
\(690\) 0 0
\(691\) 30552.4 1.68201 0.841005 0.541027i \(-0.181964\pi\)
0.841005 + 0.541027i \(0.181964\pi\)
\(692\) 29999.9 1.64801
\(693\) −2043.20 −0.111998
\(694\) 27138.7 1.48440
\(695\) 0 0
\(696\) 2217.63 0.120774
\(697\) 7957.96 0.432467
\(698\) −16708.1 −0.906032
\(699\) −5048.27 −0.273166
\(700\) 0 0
\(701\) −9151.47 −0.493076 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(702\) −1942.60 −0.104443
\(703\) 3992.86 0.214216
\(704\) −9060.30 −0.485047
\(705\) 0 0
\(706\) 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 \(0.553036\pi\)
\(710\) 0 0
\(711\) −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 \(0.656739\pi\)
\(720\) 0 0
\(721\) −1432.13 −0.0739740
\(722\) 14389.9 0.741740
\(723\) −8352.81 −0.429660
\(724\) 15347.7 0.787836
\(725\) 0 0
\(726\) −1667.93 −0.0852655
\(727\) 7233.66 0.369026 0.184513 0.982830i \(-0.440929\pi\)
0.184513 + 0.982830i \(0.440929\pi\)
\(728\) 7591.81 0.386499
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −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
\(735\) 0 0
\(736\) 2376.41 0.119016
\(737\) −5372.90 −0.268539
\(738\) 4511.14 0.225010
\(739\) 18490.9 0.920432 0.460216 0.887807i \(-0.347772\pi\)
0.460216 + 0.887807i \(0.347772\pi\)
\(740\) 0 0
\(741\) 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
\(745\) 0 0
\(746\) 20528.8 1.00752
\(747\) −12502.2 −0.612358
\(748\) −10522.4 −0.514354
\(749\) 23562.9 1.14949
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) −7306.86 −0.351518
\(757\) 7014.90 0.336804 0.168402 0.985718i \(-0.446139\pi\)
0.168402 + 0.985718i \(0.446139\pi\)
\(758\) −35858.6 −1.71826
\(759\) 450.795 0.0215584
\(760\) 0 0
\(761\) −30156.9 −1.43651 −0.718256 0.695779i \(-0.755059\pi\)
−0.718256 + 0.695779i \(0.755059\pi\)
\(762\) −22077.8 −1.04960
\(763\) 45958.4 2.18061
\(764\) −36195.7 −1.71402
\(765\) 0 0
\(766\) −51283.5 −2.41899
\(767\) 1714.26 0.0807019
\(768\) −13217.4 −0.621017
\(769\) 11292.2 0.529530 0.264765 0.964313i \(-0.414706\pi\)
0.264765 + 0.964313i \(0.414706\pi\)
\(770\) 0 0
\(771\) −1755.51 −0.0820016
\(772\) 16398.9 0.764519
\(773\) −8524.10 −0.396624 −0.198312 0.980139i \(-0.563546\pi\)
−0.198312 + 0.980139i \(0.563546\pi\)
\(774\) −5034.80 −0.233814
\(775\) 0 0
\(776\) −10022.0 −0.463621
\(777\) −4049.35 −0.186962
\(778\) −39011.4 −1.79772
\(779\) −6659.89 −0.306310
\(780\) 0 0
\(781\) 9211.66 0.422047
\(782\) 4578.96 0.209390
\(783\) −849.589 −0.0387763
\(784\) 252.251 0.0114910
\(785\) 0 0
\(786\) −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
\(790\) 0 0
\(791\) −35493.0 −1.59543
\(792\) −2325.72 −0.104345
\(793\) −1403.24 −0.0628380
\(794\) 27659.9 1.23629
\(795\) 0 0
\(796\) 9982.04 0.444477
\(797\) −37172.3 −1.65208 −0.826041 0.563610i \(-0.809412\pi\)
−0.826041 + 0.563610i \(0.809412\pi\)
\(798\) 17368.5 0.770475
\(799\) −37900.1 −1.67811
\(800\) 0 0
\(801\) 13713.3 0.604913
\(802\) 47781.0 2.10375
\(803\) −3863.37 −0.169783
\(804\) −19214.5 −0.842841
\(805\) 0 0
\(806\) 17484.0 0.764080
\(807\) 13855.4 0.604378
\(808\) −1741.68 −0.0758316
\(809\) 23797.1 1.03419 0.517096 0.855928i \(-0.327013\pi\)
0.517096 + 0.855928i \(0.327013\pi\)
\(810\) 0 0
\(811\) 8988.35 0.389178 0.194589 0.980885i \(-0.437663\pi\)
0.194589 + 0.980885i \(0.437663\pi\)
\(812\) 8515.54 0.368026
\(813\) −430.318 −0.0185633
\(814\) −3305.63 −0.142337
\(815\) 0 0
\(816\) −665.600 −0.0285547
\(817\) 7432.98 0.318295
\(818\) 21750.6 0.929696
\(819\) −2908.48 −0.124091
\(820\) 0 0
\(821\) −25156.8 −1.06940 −0.534702 0.845041i \(-0.679576\pi\)
−0.534702 + 0.845041i \(0.679576\pi\)
\(822\) −23059.5 −0.978459
\(823\) −1318.51 −0.0558447 −0.0279224 0.999610i \(-0.508889\pi\)
−0.0279224 + 0.999610i \(0.508889\pi\)
\(824\) −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 \(0.559604\pi\)
\(830\) 0 0
\(831\) 25807.5 1.07732
\(832\) −12897.3 −0.537419
\(833\) −6050.63 −0.251671
\(834\) −35018.0 −1.45393
\(835\) 0 0
\(836\) 8806.02 0.364309
\(837\) −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 \(0.480372\pi\)
\(840\) 0 0
\(841\) −23398.9 −0.959403
\(842\) 43578.8 1.78364
\(843\) −8978.43 −0.366825
\(844\) 52143.6 2.12661
\(845\) 0 0
\(846\) −21484.5 −0.873111
\(847\) −2497.24 −0.101306
\(848\) 1650.44 0.0668355
\(849\) 20576.7 0.831789
\(850\) 0 0
\(851\) 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
\(855\) 0 0
\(856\) 26821.1 1.07094
\(857\) −26394.1 −1.05205 −0.526024 0.850470i \(-0.676318\pi\)
−0.526024 + 0.850470i \(0.676318\pi\)
\(858\) −2374.29 −0.0944720
\(859\) −29456.2 −1.17000 −0.585002 0.811032i \(-0.698906\pi\)
−0.585002 + 0.811032i \(0.698906\pi\)
\(860\) 0 0
\(861\) 6754.12 0.267340
\(862\) −42893.7 −1.69486
\(863\) 762.616 0.0300808 0.0150404 0.999887i \(-0.495212\pi\)
0.0150404 + 0.999887i \(0.495212\pi\)
\(864\) 4696.99 0.184948
\(865\) 0 0
\(866\) −13706.6 −0.537838
\(867\) 1226.43 0.0480411
\(868\) 65764.0 2.57163
\(869\) −9143.26 −0.356920
\(870\) 0 0
\(871\) −7648.29 −0.297535
\(872\) 52313.3 2.03160
\(873\) 3839.51 0.148852
\(874\) −3832.06 −0.148308
\(875\) 0 0
\(876\) −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 \(-0.543070\pi\)
−0.134896 + 0.990860i \(0.543070\pi\)
\(884\) −14978.6 −0.569891
\(885\) 0 0
\(886\) 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
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) −11907.4 −0.446963
\(893\) 31718.0 1.18858
\(894\) −42603.6 −1.59382
\(895\) 0 0
\(896\) −49385.8 −1.84137
\(897\) 641.704 0.0238862
\(898\) −73843.6 −2.74409
\(899\) 7646.56 0.283679
\(900\) 0 0
\(901\) −39588.4 −1.46380
\(902\) 5513.62 0.203529
\(903\) −7538.14 −0.277800
\(904\) −40400.8 −1.48641
\(905\) 0 0
\(906\) −18729.6 −0.686809
\(907\) −1269.76 −0.0464848 −0.0232424 0.999730i \(-0.507399\pi\)
−0.0232424 + 0.999730i \(0.507399\pi\)
\(908\) 27040.4 0.988289
\(909\) 667.248 0.0243468
\(910\) 0 0
\(911\) −33783.1 −1.22863 −0.614316 0.789060i \(-0.710568\pi\)
−0.614316 + 0.789060i \(0.710568\pi\)
\(912\) 557.030 0.0202249
\(913\) −15280.5 −0.553899
\(914\) 44655.1 1.61604
\(915\) 0 0
\(916\) 53466.6 1.92859
\(917\) −41361.9 −1.48952
\(918\) 9050.35 0.325388
\(919\) 39262.5 1.40930 0.704652 0.709553i \(-0.251103\pi\)
0.704652 + 0.709553i \(0.251103\pi\)
\(920\) 0 0
\(921\) 28718.1 1.02746
\(922\) 66800.1 2.38606
\(923\) 13112.7 0.467618
\(924\) −8930.61 −0.317960
\(925\) 0 0
\(926\) −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 \(0.621984\pi\)
\(930\) 0 0
\(931\) 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
\(940\) 0 0
\(941\) −9702.77 −0.336133 −0.168067 0.985776i \(-0.553752\pi\)
−0.168067 + 0.985776i \(0.553752\pi\)
\(942\) −13949.3 −0.482477
\(943\) −1490.18 −0.0514600
\(944\) 332.959 0.0114798
\(945\) 0 0
\(946\) −6153.65 −0.211493
\(947\) 699.579 0.0240055 0.0120028 0.999928i \(-0.496179\pi\)
0.0120028 + 0.999928i \(0.496179\pi\)
\(948\) −32698.0 −1.12024
\(949\) −5499.49 −0.188115
\(950\) 0 0
\(951\) 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
\(955\) 0 0
\(956\) 52778.1 1.78553
\(957\) −1038.39 −0.0350745
\(958\) −52063.4 −1.75584
\(959\) −34524.9 −1.16253
\(960\) 0 0
\(961\) 29262.0 0.982243
\(962\) −4705.55 −0.157706
\(963\) −10275.3 −0.343841
\(964\) −36509.3 −1.21980
\(965\) 0 0
\(966\) 3886.27 0.129440
\(967\) 32794.8 1.09060 0.545299 0.838242i \(-0.316416\pi\)
0.545299 + 0.838242i \(0.316416\pi\)
\(968\) −2842.55 −0.0943832
\(969\) −13361.2 −0.442955
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −272.550 −0.00893864
\(977\) −22192.5 −0.726716 −0.363358 0.931650i \(-0.618370\pi\)
−0.363358 + 0.931650i \(0.618370\pi\)
\(978\) −38822.4 −1.26933
\(979\) 16760.7 0.547164
\(980\) 0 0
\(981\) −20041.6 −0.652272
\(982\) 37277.4 1.21137
\(983\) −7383.09 −0.239556 −0.119778 0.992801i \(-0.538218\pi\)
−0.119778 + 0.992801i \(0.538218\pi\)
\(984\) 7688.04 0.249071
\(985\) 0 0
\(986\) −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
\(995\) 0 0
\(996\) −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 825.4.a.p.1.1 3
3.2 odd 2 2475.4.a.z.1.3 3
5.2 odd 4 825.4.c.m.199.1 6
5.3 odd 4 825.4.c.m.199.6 6
5.4 even 2 165.4.a.g.1.3 3
15.14 odd 2 495.4.a.i.1.1 3
55.54 odd 2 1815.4.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.g.1.3 3 5.4 even 2
495.4.a.i.1.1 3 15.14 odd 2
825.4.a.p.1.1 3 1.1 even 1 trivial
825.4.c.m.199.1 6 5.2 odd 4
825.4.c.m.199.6 6 5.3 odd 4
1815.4.a.q.1.1 3 55.54 odd 2
2475.4.a.z.1.3 3 3.2 odd 2