Properties

Label 1617.4.a.be.1.4
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.31007\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.31007 q^{2} -3.00000 q^{3} +2.95655 q^{4} -1.76442 q^{5} +9.93020 q^{6} +16.6942 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.31007 q^{2} -3.00000 q^{3} +2.95655 q^{4} -1.76442 q^{5} +9.93020 q^{6} +16.6942 q^{8} +9.00000 q^{9} +5.84035 q^{10} +11.0000 q^{11} -8.86964 q^{12} +75.2178 q^{13} +5.29326 q^{15} -78.9112 q^{16} +81.4617 q^{17} -29.7906 q^{18} -43.8512 q^{19} -5.21659 q^{20} -36.4107 q^{22} -170.889 q^{23} -50.0825 q^{24} -121.887 q^{25} -248.976 q^{26} -27.0000 q^{27} -92.7753 q^{29} -17.5210 q^{30} -141.586 q^{31} +127.648 q^{32} -33.0000 q^{33} -269.644 q^{34} +26.6089 q^{36} +387.424 q^{37} +145.150 q^{38} -225.654 q^{39} -29.4555 q^{40} -215.640 q^{41} +76.8553 q^{43} +32.5220 q^{44} -15.8798 q^{45} +565.654 q^{46} +67.6960 q^{47} +236.734 q^{48} +403.454 q^{50} -244.385 q^{51} +222.385 q^{52} -62.3214 q^{53} +89.3718 q^{54} -19.4086 q^{55} +131.554 q^{57} +307.093 q^{58} +445.069 q^{59} +15.6498 q^{60} -710.347 q^{61} +468.659 q^{62} +208.766 q^{64} -132.716 q^{65} +109.232 q^{66} +87.2411 q^{67} +240.845 q^{68} +512.667 q^{69} +873.751 q^{71} +150.248 q^{72} +648.933 q^{73} -1282.40 q^{74} +365.660 q^{75} -129.648 q^{76} +746.928 q^{78} -1279.70 q^{79} +139.232 q^{80} +81.0000 q^{81} +713.782 q^{82} -1378.74 q^{83} -143.733 q^{85} -254.396 q^{86} +278.326 q^{87} +183.636 q^{88} -271.370 q^{89} +52.5631 q^{90} -505.242 q^{92} +424.758 q^{93} -224.078 q^{94} +77.3719 q^{95} -382.944 q^{96} +1075.26 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20} + 44 q^{22} + 4 q^{23} - 198 q^{24} + 588 q^{25} - 406 q^{26} - 432 q^{27} + 412 q^{29} + 534 q^{30} - 628 q^{31} + 592 q^{32} - 528 q^{33} + 88 q^{34} + 648 q^{36} + 148 q^{37} - 446 q^{38} + 312 q^{39} - 1376 q^{40} - 596 q^{41} - 260 q^{43} + 792 q^{44} - 360 q^{45} - 148 q^{46} - 2220 q^{47} - 660 q^{48} + 82 q^{50} + 540 q^{51} - 1046 q^{52} + 168 q^{53} - 108 q^{54} - 440 q^{55} + 456 q^{57} + 538 q^{58} + 48 q^{59} + 894 q^{60} - 1504 q^{61} - 1276 q^{62} + 630 q^{64} + 1224 q^{65} - 132 q^{66} + 116 q^{67} - 356 q^{68} - 12 q^{69} + 320 q^{71} + 594 q^{72} - 652 q^{73} + 1062 q^{74} - 1764 q^{75} + 594 q^{76} + 1218 q^{78} + 1136 q^{79} - 1970 q^{80} + 1296 q^{81} + 416 q^{82} - 3300 q^{83} - 1148 q^{85} + 1864 q^{86} - 1236 q^{87} + 726 q^{88} - 2416 q^{89} - 1602 q^{90} - 1064 q^{92} + 1884 q^{93} - 914 q^{94} + 1120 q^{95} - 1776 q^{96} - 3616 q^{97} + 1584 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.31007 −1.17029 −0.585143 0.810930i \(-0.698962\pi\)
−0.585143 + 0.810930i \(0.698962\pi\)
\(3\) −3.00000 −0.577350
\(4\) 2.95655 0.369568
\(5\) −1.76442 −0.157814 −0.0789072 0.996882i \(-0.525143\pi\)
−0.0789072 + 0.996882i \(0.525143\pi\)
\(6\) 9.93020 0.675665
\(7\) 0 0
\(8\) 16.6942 0.737785
\(9\) 9.00000 0.333333
\(10\) 5.84035 0.184688
\(11\) 11.0000 0.301511
\(12\) −8.86964 −0.213370
\(13\) 75.2178 1.60474 0.802372 0.596824i \(-0.203571\pi\)
0.802372 + 0.596824i \(0.203571\pi\)
\(14\) 0 0
\(15\) 5.29326 0.0911142
\(16\) −78.9112 −1.23299
\(17\) 81.4617 1.16220 0.581099 0.813833i \(-0.302623\pi\)
0.581099 + 0.813833i \(0.302623\pi\)
\(18\) −29.7906 −0.390095
\(19\) −43.8512 −0.529482 −0.264741 0.964320i \(-0.585286\pi\)
−0.264741 + 0.964320i \(0.585286\pi\)
\(20\) −5.21659 −0.0583232
\(21\) 0 0
\(22\) −36.4107 −0.352854
\(23\) −170.889 −1.54925 −0.774627 0.632419i \(-0.782062\pi\)
−0.774627 + 0.632419i \(0.782062\pi\)
\(24\) −50.0825 −0.425960
\(25\) −121.887 −0.975095
\(26\) −248.976 −1.87801
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −92.7753 −0.594067 −0.297034 0.954867i \(-0.595997\pi\)
−0.297034 + 0.954867i \(0.595997\pi\)
\(30\) −17.5210 −0.106630
\(31\) −141.586 −0.820309 −0.410154 0.912016i \(-0.634525\pi\)
−0.410154 + 0.912016i \(0.634525\pi\)
\(32\) 127.648 0.705163
\(33\) −33.0000 −0.174078
\(34\) −269.644 −1.36010
\(35\) 0 0
\(36\) 26.6089 0.123189
\(37\) 387.424 1.72141 0.860704 0.509105i \(-0.170024\pi\)
0.860704 + 0.509105i \(0.170024\pi\)
\(38\) 145.150 0.619645
\(39\) −225.654 −0.926500
\(40\) −29.4555 −0.116433
\(41\) −215.640 −0.821396 −0.410698 0.911771i \(-0.634715\pi\)
−0.410698 + 0.911771i \(0.634715\pi\)
\(42\) 0 0
\(43\) 76.8553 0.272566 0.136283 0.990670i \(-0.456484\pi\)
0.136283 + 0.990670i \(0.456484\pi\)
\(44\) 32.5220 0.111429
\(45\) −15.8798 −0.0526048
\(46\) 565.654 1.81307
\(47\) 67.6960 0.210095 0.105048 0.994467i \(-0.466500\pi\)
0.105048 + 0.994467i \(0.466500\pi\)
\(48\) 236.734 0.711866
\(49\) 0 0
\(50\) 403.454 1.14114
\(51\) −244.385 −0.670995
\(52\) 222.385 0.593063
\(53\) −62.3214 −0.161519 −0.0807595 0.996734i \(-0.525735\pi\)
−0.0807595 + 0.996734i \(0.525735\pi\)
\(54\) 89.3718 0.225222
\(55\) −19.4086 −0.0475828
\(56\) 0 0
\(57\) 131.554 0.305696
\(58\) 307.093 0.695228
\(59\) 445.069 0.982086 0.491043 0.871135i \(-0.336616\pi\)
0.491043 + 0.871135i \(0.336616\pi\)
\(60\) 15.6498 0.0336729
\(61\) −710.347 −1.49099 −0.745496 0.666510i \(-0.767788\pi\)
−0.745496 + 0.666510i \(0.767788\pi\)
\(62\) 468.659 0.959996
\(63\) 0 0
\(64\) 208.766 0.407746
\(65\) −132.716 −0.253252
\(66\) 109.232 0.203721
\(67\) 87.2411 0.159078 0.0795388 0.996832i \(-0.474655\pi\)
0.0795388 + 0.996832i \(0.474655\pi\)
\(68\) 240.845 0.429511
\(69\) 512.667 0.894462
\(70\) 0 0
\(71\) 873.751 1.46049 0.730247 0.683183i \(-0.239405\pi\)
0.730247 + 0.683183i \(0.239405\pi\)
\(72\) 150.248 0.245928
\(73\) 648.933 1.04044 0.520218 0.854033i \(-0.325851\pi\)
0.520218 + 0.854033i \(0.325851\pi\)
\(74\) −1282.40 −2.01454
\(75\) 365.660 0.562971
\(76\) −129.648 −0.195680
\(77\) 0 0
\(78\) 746.928 1.08427
\(79\) −1279.70 −1.82250 −0.911252 0.411849i \(-0.864883\pi\)
−0.911252 + 0.411849i \(0.864883\pi\)
\(80\) 139.232 0.194583
\(81\) 81.0000 0.111111
\(82\) 713.782 0.961268
\(83\) −1378.74 −1.82333 −0.911664 0.410937i \(-0.865202\pi\)
−0.911664 + 0.410937i \(0.865202\pi\)
\(84\) 0 0
\(85\) −143.733 −0.183412
\(86\) −254.396 −0.318980
\(87\) 278.326 0.342985
\(88\) 183.636 0.222451
\(89\) −271.370 −0.323204 −0.161602 0.986856i \(-0.551666\pi\)
−0.161602 + 0.986856i \(0.551666\pi\)
\(90\) 52.5631 0.0615627
\(91\) 0 0
\(92\) −505.242 −0.572555
\(93\) 424.758 0.473606
\(94\) −224.078 −0.245872
\(95\) 77.3719 0.0835599
\(96\) −382.944 −0.407126
\(97\) 1075.26 1.12553 0.562765 0.826617i \(-0.309737\pi\)
0.562765 + 0.826617i \(0.309737\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) −360.364 −0.360364
\(101\) 313.115 0.308476 0.154238 0.988034i \(-0.450708\pi\)
0.154238 + 0.988034i \(0.450708\pi\)
\(102\) 808.931 0.785256
\(103\) −1321.14 −1.26384 −0.631921 0.775033i \(-0.717733\pi\)
−0.631921 + 0.775033i \(0.717733\pi\)
\(104\) 1255.70 1.18396
\(105\) 0 0
\(106\) 206.288 0.189023
\(107\) 782.734 0.707193 0.353597 0.935398i \(-0.384959\pi\)
0.353597 + 0.935398i \(0.384959\pi\)
\(108\) −79.8268 −0.0711235
\(109\) −2095.05 −1.84101 −0.920503 0.390736i \(-0.872221\pi\)
−0.920503 + 0.390736i \(0.872221\pi\)
\(110\) 64.2438 0.0556855
\(111\) −1162.27 −0.993856
\(112\) 0 0
\(113\) 1930.98 1.60753 0.803766 0.594946i \(-0.202827\pi\)
0.803766 + 0.594946i \(0.202827\pi\)
\(114\) −435.451 −0.357752
\(115\) 301.520 0.244495
\(116\) −274.295 −0.219548
\(117\) 676.961 0.534915
\(118\) −1473.21 −1.14932
\(119\) 0 0
\(120\) 88.3665 0.0672227
\(121\) 121.000 0.0909091
\(122\) 2351.30 1.74489
\(123\) 646.919 0.474233
\(124\) −418.605 −0.303160
\(125\) 435.612 0.311698
\(126\) 0 0
\(127\) 2270.89 1.58669 0.793344 0.608774i \(-0.208338\pi\)
0.793344 + 0.608774i \(0.208338\pi\)
\(128\) −1712.21 −1.18234
\(129\) −230.566 −0.157366
\(130\) 439.298 0.296377
\(131\) −402.008 −0.268119 −0.134060 0.990973i \(-0.542801\pi\)
−0.134060 + 0.990973i \(0.542801\pi\)
\(132\) −97.5661 −0.0643336
\(133\) 0 0
\(134\) −288.774 −0.186166
\(135\) 47.6393 0.0303714
\(136\) 1359.93 0.857452
\(137\) 1848.54 1.15278 0.576392 0.817174i \(-0.304460\pi\)
0.576392 + 0.817174i \(0.304460\pi\)
\(138\) −1696.96 −1.04678
\(139\) 1523.00 0.929346 0.464673 0.885482i \(-0.346172\pi\)
0.464673 + 0.885482i \(0.346172\pi\)
\(140\) 0 0
\(141\) −203.088 −0.121299
\(142\) −2892.17 −1.70920
\(143\) 827.396 0.483849
\(144\) −710.201 −0.410996
\(145\) 163.695 0.0937524
\(146\) −2148.01 −1.21761
\(147\) 0 0
\(148\) 1145.44 0.636178
\(149\) 2337.90 1.28543 0.642713 0.766107i \(-0.277809\pi\)
0.642713 + 0.766107i \(0.277809\pi\)
\(150\) −1210.36 −0.658837
\(151\) −2178.31 −1.17396 −0.586981 0.809600i \(-0.699684\pi\)
−0.586981 + 0.809600i \(0.699684\pi\)
\(152\) −732.059 −0.390644
\(153\) 733.155 0.387399
\(154\) 0 0
\(155\) 249.817 0.129457
\(156\) −667.155 −0.342405
\(157\) −1733.26 −0.881079 −0.440539 0.897733i \(-0.645213\pi\)
−0.440539 + 0.897733i \(0.645213\pi\)
\(158\) 4235.90 2.13285
\(159\) 186.964 0.0932530
\(160\) −225.225 −0.111285
\(161\) 0 0
\(162\) −268.115 −0.130032
\(163\) −2091.37 −1.00496 −0.502482 0.864588i \(-0.667580\pi\)
−0.502482 + 0.864588i \(0.667580\pi\)
\(164\) −637.549 −0.303562
\(165\) 58.2258 0.0274720
\(166\) 4563.72 2.13381
\(167\) −3166.81 −1.46739 −0.733697 0.679476i \(-0.762207\pi\)
−0.733697 + 0.679476i \(0.762207\pi\)
\(168\) 0 0
\(169\) 3460.72 1.57520
\(170\) 475.764 0.214644
\(171\) −394.661 −0.176494
\(172\) 227.226 0.100732
\(173\) 2071.69 0.910448 0.455224 0.890377i \(-0.349559\pi\)
0.455224 + 0.890377i \(0.349559\pi\)
\(174\) −921.278 −0.401390
\(175\) 0 0
\(176\) −868.023 −0.371760
\(177\) −1335.21 −0.567008
\(178\) 898.254 0.378241
\(179\) 240.414 0.100388 0.0501938 0.998739i \(-0.484016\pi\)
0.0501938 + 0.998739i \(0.484016\pi\)
\(180\) −46.9493 −0.0194411
\(181\) −3173.22 −1.30311 −0.651557 0.758600i \(-0.725884\pi\)
−0.651557 + 0.758600i \(0.725884\pi\)
\(182\) 0 0
\(183\) 2131.04 0.860825
\(184\) −2852.85 −1.14302
\(185\) −683.578 −0.271663
\(186\) −1405.98 −0.554254
\(187\) 896.078 0.350416
\(188\) 200.147 0.0776446
\(189\) 0 0
\(190\) −256.106 −0.0977889
\(191\) −4984.15 −1.88817 −0.944085 0.329701i \(-0.893052\pi\)
−0.944085 + 0.329701i \(0.893052\pi\)
\(192\) −626.298 −0.235412
\(193\) 943.449 0.351870 0.175935 0.984402i \(-0.443705\pi\)
0.175935 + 0.984402i \(0.443705\pi\)
\(194\) −3559.20 −1.31719
\(195\) 398.147 0.146215
\(196\) 0 0
\(197\) 1929.50 0.697824 0.348912 0.937155i \(-0.386551\pi\)
0.348912 + 0.937155i \(0.386551\pi\)
\(198\) −327.697 −0.117618
\(199\) −1348.24 −0.480273 −0.240137 0.970739i \(-0.577192\pi\)
−0.240137 + 0.970739i \(0.577192\pi\)
\(200\) −2034.80 −0.719410
\(201\) −261.723 −0.0918435
\(202\) −1036.43 −0.361005
\(203\) 0 0
\(204\) −722.536 −0.247979
\(205\) 380.479 0.129628
\(206\) 4373.06 1.47906
\(207\) −1538.00 −0.516418
\(208\) −5935.53 −1.97863
\(209\) −482.363 −0.159645
\(210\) 0 0
\(211\) −3460.60 −1.12909 −0.564544 0.825403i \(-0.690948\pi\)
−0.564544 + 0.825403i \(0.690948\pi\)
\(212\) −184.256 −0.0596923
\(213\) −2621.25 −0.843217
\(214\) −2590.90 −0.827618
\(215\) −135.605 −0.0430148
\(216\) −450.743 −0.141987
\(217\) 0 0
\(218\) 6934.76 2.15450
\(219\) −1946.80 −0.600696
\(220\) −57.3825 −0.0175851
\(221\) 6127.37 1.86503
\(222\) 3847.20 1.16309
\(223\) 3034.12 0.911119 0.455559 0.890205i \(-0.349439\pi\)
0.455559 + 0.890205i \(0.349439\pi\)
\(224\) 0 0
\(225\) −1096.98 −0.325032
\(226\) −6391.66 −1.88127
\(227\) −101.512 −0.0296809 −0.0148404 0.999890i \(-0.504724\pi\)
−0.0148404 + 0.999890i \(0.504724\pi\)
\(228\) 388.944 0.112976
\(229\) −3232.61 −0.932824 −0.466412 0.884568i \(-0.654454\pi\)
−0.466412 + 0.884568i \(0.654454\pi\)
\(230\) −998.051 −0.286129
\(231\) 0 0
\(232\) −1548.81 −0.438294
\(233\) 4447.25 1.25042 0.625212 0.780455i \(-0.285013\pi\)
0.625212 + 0.780455i \(0.285013\pi\)
\(234\) −2240.79 −0.626003
\(235\) −119.444 −0.0331561
\(236\) 1315.87 0.362948
\(237\) 3839.11 1.05222
\(238\) 0 0
\(239\) 625.565 0.169307 0.0846536 0.996410i \(-0.473022\pi\)
0.0846536 + 0.996410i \(0.473022\pi\)
\(240\) −417.697 −0.112343
\(241\) 3938.19 1.05262 0.526309 0.850293i \(-0.323575\pi\)
0.526309 + 0.850293i \(0.323575\pi\)
\(242\) −400.518 −0.106390
\(243\) −243.000 −0.0641500
\(244\) −2100.17 −0.551024
\(245\) 0 0
\(246\) −2141.34 −0.554988
\(247\) −3298.39 −0.849683
\(248\) −2363.66 −0.605212
\(249\) 4136.21 1.05270
\(250\) −1441.90 −0.364776
\(251\) 2226.20 0.559828 0.279914 0.960025i \(-0.409694\pi\)
0.279914 + 0.960025i \(0.409694\pi\)
\(252\) 0 0
\(253\) −1879.78 −0.467118
\(254\) −7516.81 −1.85688
\(255\) 431.198 0.105893
\(256\) 3997.42 0.975932
\(257\) −4618.81 −1.12106 −0.560532 0.828133i \(-0.689403\pi\)
−0.560532 + 0.828133i \(0.689403\pi\)
\(258\) 763.188 0.184163
\(259\) 0 0
\(260\) −392.381 −0.0935939
\(261\) −834.978 −0.198022
\(262\) 1330.67 0.313776
\(263\) −1267.90 −0.297270 −0.148635 0.988892i \(-0.547488\pi\)
−0.148635 + 0.988892i \(0.547488\pi\)
\(264\) −550.908 −0.128432
\(265\) 109.961 0.0254900
\(266\) 0 0
\(267\) 814.111 0.186602
\(268\) 257.933 0.0587901
\(269\) −1788.22 −0.405315 −0.202658 0.979250i \(-0.564958\pi\)
−0.202658 + 0.979250i \(0.564958\pi\)
\(270\) −157.689 −0.0355432
\(271\) 5882.91 1.31868 0.659338 0.751846i \(-0.270836\pi\)
0.659338 + 0.751846i \(0.270836\pi\)
\(272\) −6428.24 −1.43297
\(273\) 0 0
\(274\) −6118.79 −1.34909
\(275\) −1340.76 −0.294002
\(276\) 1515.73 0.330565
\(277\) 952.123 0.206525 0.103263 0.994654i \(-0.467072\pi\)
0.103263 + 0.994654i \(0.467072\pi\)
\(278\) −5041.23 −1.08760
\(279\) −1274.27 −0.273436
\(280\) 0 0
\(281\) −3721.91 −0.790143 −0.395072 0.918650i \(-0.629280\pi\)
−0.395072 + 0.918650i \(0.629280\pi\)
\(282\) 672.235 0.141954
\(283\) −5593.19 −1.17484 −0.587422 0.809281i \(-0.699857\pi\)
−0.587422 + 0.809281i \(0.699857\pi\)
\(284\) 2583.29 0.539753
\(285\) −232.116 −0.0482433
\(286\) −2738.74 −0.566241
\(287\) 0 0
\(288\) 1148.83 0.235054
\(289\) 1723.00 0.350702
\(290\) −541.840 −0.109717
\(291\) −3225.79 −0.649826
\(292\) 1918.60 0.384513
\(293\) −2114.10 −0.421525 −0.210763 0.977537i \(-0.567595\pi\)
−0.210763 + 0.977537i \(0.567595\pi\)
\(294\) 0 0
\(295\) −785.289 −0.154987
\(296\) 6467.72 1.27003
\(297\) −297.000 −0.0580259
\(298\) −7738.61 −1.50431
\(299\) −12853.9 −2.48616
\(300\) 1081.09 0.208056
\(301\) 0 0
\(302\) 7210.36 1.37387
\(303\) −939.345 −0.178099
\(304\) 3460.35 0.652844
\(305\) 1253.35 0.235300
\(306\) −2426.79 −0.453368
\(307\) −6947.16 −1.29152 −0.645758 0.763542i \(-0.723459\pi\)
−0.645758 + 0.763542i \(0.723459\pi\)
\(308\) 0 0
\(309\) 3963.42 0.729680
\(310\) −826.911 −0.151501
\(311\) −6006.92 −1.09525 −0.547623 0.836725i \(-0.684467\pi\)
−0.547623 + 0.836725i \(0.684467\pi\)
\(312\) −3767.10 −0.683557
\(313\) −6375.86 −1.15139 −0.575695 0.817664i \(-0.695268\pi\)
−0.575695 + 0.817664i \(0.695268\pi\)
\(314\) 5737.21 1.03111
\(315\) 0 0
\(316\) −3783.50 −0.673540
\(317\) 9057.17 1.60474 0.802368 0.596829i \(-0.203573\pi\)
0.802368 + 0.596829i \(0.203573\pi\)
\(318\) −618.864 −0.109133
\(319\) −1020.53 −0.179118
\(320\) −368.350 −0.0643482
\(321\) −2348.20 −0.408298
\(322\) 0 0
\(323\) −3572.19 −0.615362
\(324\) 239.480 0.0410632
\(325\) −9168.06 −1.56478
\(326\) 6922.59 1.17609
\(327\) 6285.16 1.06291
\(328\) −3599.92 −0.606014
\(329\) 0 0
\(330\) −192.731 −0.0321501
\(331\) 4371.80 0.725969 0.362985 0.931795i \(-0.381758\pi\)
0.362985 + 0.931795i \(0.381758\pi\)
\(332\) −4076.31 −0.673844
\(333\) 3486.82 0.573803
\(334\) 10482.3 1.71727
\(335\) −153.930 −0.0251047
\(336\) 0 0
\(337\) −2533.17 −0.409468 −0.204734 0.978818i \(-0.565633\pi\)
−0.204734 + 0.978818i \(0.565633\pi\)
\(338\) −11455.2 −1.84344
\(339\) −5792.93 −0.928109
\(340\) −424.952 −0.0677831
\(341\) −1557.44 −0.247332
\(342\) 1306.35 0.206548
\(343\) 0 0
\(344\) 1283.03 0.201095
\(345\) −904.560 −0.141159
\(346\) −6857.42 −1.06548
\(347\) 4744.90 0.734062 0.367031 0.930209i \(-0.380374\pi\)
0.367031 + 0.930209i \(0.380374\pi\)
\(348\) 822.884 0.126756
\(349\) −10611.4 −1.62755 −0.813773 0.581182i \(-0.802590\pi\)
−0.813773 + 0.581182i \(0.802590\pi\)
\(350\) 0 0
\(351\) −2030.88 −0.308833
\(352\) 1404.13 0.212615
\(353\) 8416.02 1.26895 0.634475 0.772943i \(-0.281216\pi\)
0.634475 + 0.772943i \(0.281216\pi\)
\(354\) 4419.63 0.663561
\(355\) −1541.66 −0.230487
\(356\) −802.319 −0.119446
\(357\) 0 0
\(358\) −795.787 −0.117482
\(359\) 6074.53 0.893040 0.446520 0.894774i \(-0.352663\pi\)
0.446520 + 0.894774i \(0.352663\pi\)
\(360\) −265.100 −0.0388110
\(361\) −4936.07 −0.719649
\(362\) 10503.6 1.52502
\(363\) −363.000 −0.0524864
\(364\) 0 0
\(365\) −1144.99 −0.164196
\(366\) −7053.89 −1.00741
\(367\) −10998.5 −1.56435 −0.782176 0.623057i \(-0.785890\pi\)
−0.782176 + 0.623057i \(0.785890\pi\)
\(368\) 13485.1 1.91021
\(369\) −1940.76 −0.273799
\(370\) 2262.69 0.317923
\(371\) 0 0
\(372\) 1255.82 0.175030
\(373\) −1643.26 −0.228109 −0.114054 0.993475i \(-0.536384\pi\)
−0.114054 + 0.993475i \(0.536384\pi\)
\(374\) −2966.08 −0.410086
\(375\) −1306.84 −0.179959
\(376\) 1130.13 0.155005
\(377\) −6978.36 −0.953326
\(378\) 0 0
\(379\) 816.382 0.110646 0.0553229 0.998469i \(-0.482381\pi\)
0.0553229 + 0.998469i \(0.482381\pi\)
\(380\) 228.754 0.0308811
\(381\) −6812.68 −0.916074
\(382\) 16497.9 2.20970
\(383\) 6523.26 0.870295 0.435147 0.900359i \(-0.356696\pi\)
0.435147 + 0.900359i \(0.356696\pi\)
\(384\) 5136.64 0.682625
\(385\) 0 0
\(386\) −3122.88 −0.411788
\(387\) 691.697 0.0908552
\(388\) 3179.07 0.415961
\(389\) −7891.91 −1.02863 −0.514313 0.857603i \(-0.671953\pi\)
−0.514313 + 0.857603i \(0.671953\pi\)
\(390\) −1317.89 −0.171113
\(391\) −13920.9 −1.80054
\(392\) 0 0
\(393\) 1206.02 0.154799
\(394\) −6386.78 −0.816653
\(395\) 2257.93 0.287618
\(396\) 292.698 0.0371430
\(397\) −9368.49 −1.18436 −0.592180 0.805806i \(-0.701733\pi\)
−0.592180 + 0.805806i \(0.701733\pi\)
\(398\) 4462.77 0.562057
\(399\) 0 0
\(400\) 9618.24 1.20228
\(401\) 4474.52 0.557224 0.278612 0.960404i \(-0.410126\pi\)
0.278612 + 0.960404i \(0.410126\pi\)
\(402\) 866.322 0.107483
\(403\) −10649.8 −1.31639
\(404\) 925.740 0.114003
\(405\) −142.918 −0.0175349
\(406\) 0 0
\(407\) 4261.66 0.519024
\(408\) −4079.80 −0.495050
\(409\) −3463.19 −0.418688 −0.209344 0.977842i \(-0.567133\pi\)
−0.209344 + 0.977842i \(0.567133\pi\)
\(410\) −1259.41 −0.151702
\(411\) −5545.62 −0.665560
\(412\) −3906.01 −0.467076
\(413\) 0 0
\(414\) 5090.89 0.604356
\(415\) 2432.67 0.287747
\(416\) 9601.41 1.13161
\(417\) −4569.00 −0.536558
\(418\) 1596.65 0.186830
\(419\) −4994.05 −0.582280 −0.291140 0.956680i \(-0.594035\pi\)
−0.291140 + 0.956680i \(0.594035\pi\)
\(420\) 0 0
\(421\) −1278.23 −0.147974 −0.0739868 0.997259i \(-0.523572\pi\)
−0.0739868 + 0.997259i \(0.523572\pi\)
\(422\) 11454.8 1.32136
\(423\) 609.264 0.0700318
\(424\) −1040.40 −0.119166
\(425\) −9929.10 −1.13325
\(426\) 8676.52 0.986805
\(427\) 0 0
\(428\) 2314.19 0.261356
\(429\) −2482.19 −0.279350
\(430\) 448.861 0.0503396
\(431\) 7772.18 0.868614 0.434307 0.900765i \(-0.356993\pi\)
0.434307 + 0.900765i \(0.356993\pi\)
\(432\) 2130.60 0.237289
\(433\) −12303.4 −1.36551 −0.682753 0.730649i \(-0.739218\pi\)
−0.682753 + 0.730649i \(0.739218\pi\)
\(434\) 0 0
\(435\) −491.084 −0.0541280
\(436\) −6194.12 −0.680378
\(437\) 7493.69 0.820301
\(438\) 6444.04 0.702986
\(439\) 2520.11 0.273983 0.136991 0.990572i \(-0.456257\pi\)
0.136991 + 0.990572i \(0.456257\pi\)
\(440\) −324.011 −0.0351059
\(441\) 0 0
\(442\) −20282.0 −2.18262
\(443\) 16413.0 1.76029 0.880144 0.474708i \(-0.157446\pi\)
0.880144 + 0.474708i \(0.157446\pi\)
\(444\) −3436.31 −0.367298
\(445\) 478.811 0.0510063
\(446\) −10043.1 −1.06627
\(447\) −7013.70 −0.742141
\(448\) 0 0
\(449\) −2406.33 −0.252922 −0.126461 0.991972i \(-0.540362\pi\)
−0.126461 + 0.991972i \(0.540362\pi\)
\(450\) 3631.08 0.380380
\(451\) −2372.04 −0.247660
\(452\) 5709.03 0.594093
\(453\) 6534.93 0.677788
\(454\) 336.010 0.0347351
\(455\) 0 0
\(456\) 2196.18 0.225538
\(457\) −9462.01 −0.968521 −0.484260 0.874924i \(-0.660911\pi\)
−0.484260 + 0.874924i \(0.660911\pi\)
\(458\) 10700.1 1.09167
\(459\) −2199.46 −0.223665
\(460\) 891.458 0.0903575
\(461\) −7025.94 −0.709827 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(462\) 0 0
\(463\) −312.939 −0.0314114 −0.0157057 0.999877i \(-0.504999\pi\)
−0.0157057 + 0.999877i \(0.504999\pi\)
\(464\) 7321.01 0.732477
\(465\) −749.451 −0.0747418
\(466\) −14720.7 −1.46335
\(467\) −13114.6 −1.29951 −0.649756 0.760143i \(-0.725129\pi\)
−0.649756 + 0.760143i \(0.725129\pi\)
\(468\) 2001.47 0.197688
\(469\) 0 0
\(470\) 395.368 0.0388021
\(471\) 5199.79 0.508691
\(472\) 7430.06 0.724568
\(473\) 845.408 0.0821816
\(474\) −12707.7 −1.23140
\(475\) 5344.88 0.516295
\(476\) 0 0
\(477\) −560.893 −0.0538397
\(478\) −2070.66 −0.198138
\(479\) 10282.1 0.980794 0.490397 0.871499i \(-0.336852\pi\)
0.490397 + 0.871499i \(0.336852\pi\)
\(480\) 675.674 0.0642503
\(481\) 29141.2 2.76242
\(482\) −13035.7 −1.23186
\(483\) 0 0
\(484\) 357.742 0.0335971
\(485\) −1897.22 −0.177625
\(486\) 804.346 0.0750739
\(487\) −15975.4 −1.48648 −0.743239 0.669025i \(-0.766712\pi\)
−0.743239 + 0.669025i \(0.766712\pi\)
\(488\) −11858.6 −1.10003
\(489\) 6274.12 0.580216
\(490\) 0 0
\(491\) −10785.2 −0.991302 −0.495651 0.868522i \(-0.665070\pi\)
−0.495651 + 0.868522i \(0.665070\pi\)
\(492\) 1912.65 0.175262
\(493\) −7557.63 −0.690423
\(494\) 10917.9 0.994372
\(495\) −174.677 −0.0158609
\(496\) 11172.7 1.01143
\(497\) 0 0
\(498\) −13691.2 −1.23196
\(499\) 3397.04 0.304754 0.152377 0.988322i \(-0.451307\pi\)
0.152377 + 0.988322i \(0.451307\pi\)
\(500\) 1287.91 0.115194
\(501\) 9500.42 0.847201
\(502\) −7368.89 −0.655159
\(503\) 14719.8 1.30482 0.652409 0.757867i \(-0.273759\pi\)
0.652409 + 0.757867i \(0.273759\pi\)
\(504\) 0 0
\(505\) −552.466 −0.0486820
\(506\) 6222.20 0.546661
\(507\) −10382.2 −0.909444
\(508\) 6714.01 0.586390
\(509\) −5563.51 −0.484476 −0.242238 0.970217i \(-0.577882\pi\)
−0.242238 + 0.970217i \(0.577882\pi\)
\(510\) −1427.29 −0.123925
\(511\) 0 0
\(512\) 465.991 0.0402228
\(513\) 1183.98 0.101899
\(514\) 15288.6 1.31197
\(515\) 2331.04 0.199453
\(516\) −681.679 −0.0581574
\(517\) 744.656 0.0633461
\(518\) 0 0
\(519\) −6215.06 −0.525647
\(520\) −2215.58 −0.186845
\(521\) −2697.48 −0.226831 −0.113415 0.993548i \(-0.536179\pi\)
−0.113415 + 0.993548i \(0.536179\pi\)
\(522\) 2763.83 0.231743
\(523\) 2249.22 0.188053 0.0940264 0.995570i \(-0.470026\pi\)
0.0940264 + 0.995570i \(0.470026\pi\)
\(524\) −1188.56 −0.0990884
\(525\) 0 0
\(526\) 4196.83 0.347890
\(527\) −11533.8 −0.953361
\(528\) 2604.07 0.214636
\(529\) 17036.1 1.40019
\(530\) −363.979 −0.0298306
\(531\) 4005.62 0.327362
\(532\) 0 0
\(533\) −16219.9 −1.31813
\(534\) −2694.76 −0.218378
\(535\) −1381.07 −0.111605
\(536\) 1456.42 0.117365
\(537\) −721.242 −0.0579589
\(538\) 5919.14 0.474335
\(539\) 0 0
\(540\) 140.848 0.0112243
\(541\) −3866.80 −0.307295 −0.153648 0.988126i \(-0.549102\pi\)
−0.153648 + 0.988126i \(0.549102\pi\)
\(542\) −19472.8 −1.54323
\(543\) 9519.66 0.752353
\(544\) 10398.4 0.819538
\(545\) 3696.55 0.290537
\(546\) 0 0
\(547\) −6935.29 −0.542105 −0.271053 0.962565i \(-0.587372\pi\)
−0.271053 + 0.962565i \(0.587372\pi\)
\(548\) 5465.29 0.426032
\(549\) −6393.12 −0.496998
\(550\) 4437.99 0.344066
\(551\) 4068.31 0.314548
\(552\) 8558.55 0.659921
\(553\) 0 0
\(554\) −3151.59 −0.241694
\(555\) 2050.73 0.156845
\(556\) 4502.82 0.343457
\(557\) −23296.6 −1.77219 −0.886095 0.463504i \(-0.846592\pi\)
−0.886095 + 0.463504i \(0.846592\pi\)
\(558\) 4217.93 0.319999
\(559\) 5780.89 0.437398
\(560\) 0 0
\(561\) −2688.23 −0.202313
\(562\) 12319.8 0.924693
\(563\) −13044.2 −0.976463 −0.488232 0.872714i \(-0.662358\pi\)
−0.488232 + 0.872714i \(0.662358\pi\)
\(564\) −600.440 −0.0448281
\(565\) −3407.05 −0.253692
\(566\) 18513.8 1.37490
\(567\) 0 0
\(568\) 14586.5 1.07753
\(569\) 12764.7 0.940468 0.470234 0.882542i \(-0.344170\pi\)
0.470234 + 0.882542i \(0.344170\pi\)
\(570\) 768.318 0.0564585
\(571\) −18737.3 −1.37326 −0.686630 0.727007i \(-0.740911\pi\)
−0.686630 + 0.727007i \(0.740911\pi\)
\(572\) 2446.24 0.178815
\(573\) 14952.5 1.09014
\(574\) 0 0
\(575\) 20829.1 1.51067
\(576\) 1878.89 0.135915
\(577\) 10057.9 0.725679 0.362839 0.931852i \(-0.381807\pi\)
0.362839 + 0.931852i \(0.381807\pi\)
\(578\) −5703.25 −0.410422
\(579\) −2830.35 −0.203152
\(580\) 483.971 0.0346479
\(581\) 0 0
\(582\) 10677.6 0.760482
\(583\) −685.536 −0.0486998
\(584\) 10833.4 0.767619
\(585\) −1194.44 −0.0844173
\(586\) 6997.80 0.493305
\(587\) −18139.1 −1.27544 −0.637718 0.770270i \(-0.720121\pi\)
−0.637718 + 0.770270i \(0.720121\pi\)
\(588\) 0 0
\(589\) 6208.71 0.434339
\(590\) 2599.36 0.181380
\(591\) −5788.51 −0.402889
\(592\) −30572.1 −2.12248
\(593\) 12472.5 0.863719 0.431860 0.901941i \(-0.357858\pi\)
0.431860 + 0.901941i \(0.357858\pi\)
\(594\) 983.090 0.0679069
\(595\) 0 0
\(596\) 6912.12 0.475053
\(597\) 4044.73 0.277286
\(598\) 42547.3 2.90951
\(599\) 6718.22 0.458262 0.229131 0.973396i \(-0.426412\pi\)
0.229131 + 0.973396i \(0.426412\pi\)
\(600\) 6104.40 0.415352
\(601\) −9752.11 −0.661892 −0.330946 0.943650i \(-0.607368\pi\)
−0.330946 + 0.943650i \(0.607368\pi\)
\(602\) 0 0
\(603\) 785.170 0.0530259
\(604\) −6440.28 −0.433860
\(605\) −213.495 −0.0143468
\(606\) 3109.30 0.208427
\(607\) 17028.9 1.13868 0.569342 0.822101i \(-0.307198\pi\)
0.569342 + 0.822101i \(0.307198\pi\)
\(608\) −5597.52 −0.373371
\(609\) 0 0
\(610\) −4148.67 −0.275368
\(611\) 5091.95 0.337149
\(612\) 2167.61 0.143170
\(613\) 6039.53 0.397935 0.198968 0.980006i \(-0.436241\pi\)
0.198968 + 0.980006i \(0.436241\pi\)
\(614\) 22995.6 1.51144
\(615\) −1141.44 −0.0748409
\(616\) 0 0
\(617\) 14499.5 0.946071 0.473036 0.881043i \(-0.343158\pi\)
0.473036 + 0.881043i \(0.343158\pi\)
\(618\) −13119.2 −0.853933
\(619\) 23675.3 1.53730 0.768650 0.639669i \(-0.220929\pi\)
0.768650 + 0.639669i \(0.220929\pi\)
\(620\) 738.595 0.0478431
\(621\) 4614.00 0.298154
\(622\) 19883.3 1.28175
\(623\) 0 0
\(624\) 17806.6 1.14236
\(625\) 14467.3 0.925904
\(626\) 21104.5 1.34746
\(627\) 1447.09 0.0921709
\(628\) −5124.47 −0.325619
\(629\) 31560.2 2.00062
\(630\) 0 0
\(631\) −5929.96 −0.374117 −0.187059 0.982349i \(-0.559895\pi\)
−0.187059 + 0.982349i \(0.559895\pi\)
\(632\) −21363.6 −1.34462
\(633\) 10381.8 0.651880
\(634\) −29979.9 −1.87800
\(635\) −4006.81 −0.250402
\(636\) 552.769 0.0344634
\(637\) 0 0
\(638\) 3378.02 0.209619
\(639\) 7863.76 0.486832
\(640\) 3021.06 0.186591
\(641\) −15785.3 −0.972670 −0.486335 0.873772i \(-0.661667\pi\)
−0.486335 + 0.873772i \(0.661667\pi\)
\(642\) 7772.70 0.477826
\(643\) 8058.27 0.494226 0.247113 0.968987i \(-0.420518\pi\)
0.247113 + 0.968987i \(0.420518\pi\)
\(644\) 0 0
\(645\) 406.815 0.0248346
\(646\) 11824.2 0.720150
\(647\) −9376.94 −0.569777 −0.284888 0.958561i \(-0.591957\pi\)
−0.284888 + 0.958561i \(0.591957\pi\)
\(648\) 1352.23 0.0819761
\(649\) 4895.76 0.296110
\(650\) 30346.9 1.83124
\(651\) 0 0
\(652\) −6183.25 −0.371403
\(653\) −16317.2 −0.977856 −0.488928 0.872324i \(-0.662612\pi\)
−0.488928 + 0.872324i \(0.662612\pi\)
\(654\) −20804.3 −1.24390
\(655\) 709.311 0.0423131
\(656\) 17016.4 1.01277
\(657\) 5840.40 0.346812
\(658\) 0 0
\(659\) −19633.7 −1.16058 −0.580289 0.814410i \(-0.697061\pi\)
−0.580289 + 0.814410i \(0.697061\pi\)
\(660\) 172.147 0.0101528
\(661\) 2634.58 0.155027 0.0775137 0.996991i \(-0.475302\pi\)
0.0775137 + 0.996991i \(0.475302\pi\)
\(662\) −14471.0 −0.849591
\(663\) −18382.1 −1.07678
\(664\) −23016.9 −1.34522
\(665\) 0 0
\(666\) −11541.6 −0.671513
\(667\) 15854.3 0.920360
\(668\) −9362.82 −0.542303
\(669\) −9102.35 −0.526035
\(670\) 509.518 0.0293797
\(671\) −7813.81 −0.449551
\(672\) 0 0
\(673\) −13585.1 −0.778107 −0.389054 0.921215i \(-0.627198\pi\)
−0.389054 + 0.921215i \(0.627198\pi\)
\(674\) 8384.97 0.479195
\(675\) 3290.94 0.187657
\(676\) 10231.8 0.582146
\(677\) −30035.0 −1.70508 −0.852540 0.522661i \(-0.824939\pi\)
−0.852540 + 0.522661i \(0.824939\pi\)
\(678\) 19175.0 1.08615
\(679\) 0 0
\(680\) −2399.49 −0.135318
\(681\) 304.535 0.0171363
\(682\) 5155.25 0.289450
\(683\) 19594.1 1.09772 0.548862 0.835913i \(-0.315061\pi\)
0.548862 + 0.835913i \(0.315061\pi\)
\(684\) −1166.83 −0.0652266
\(685\) −3261.60 −0.181926
\(686\) 0 0
\(687\) 9697.82 0.538566
\(688\) −6064.74 −0.336070
\(689\) −4687.68 −0.259197
\(690\) 2994.15 0.165196
\(691\) 12211.4 0.672277 0.336139 0.941813i \(-0.390879\pi\)
0.336139 + 0.941813i \(0.390879\pi\)
\(692\) 6125.04 0.336473
\(693\) 0 0
\(694\) −15705.9 −0.859062
\(695\) −2687.21 −0.146664
\(696\) 4646.42 0.253049
\(697\) −17566.4 −0.954624
\(698\) 35124.4 1.90469
\(699\) −13341.7 −0.721932
\(700\) 0 0
\(701\) 10128.9 0.545739 0.272869 0.962051i \(-0.412027\pi\)
0.272869 + 0.962051i \(0.412027\pi\)
\(702\) 6722.36 0.361423
\(703\) −16989.0 −0.911454
\(704\) 2296.42 0.122940
\(705\) 358.333 0.0191427
\(706\) −27857.6 −1.48503
\(707\) 0 0
\(708\) −3947.61 −0.209548
\(709\) 32710.0 1.73265 0.866326 0.499479i \(-0.166475\pi\)
0.866326 + 0.499479i \(0.166475\pi\)
\(710\) 5103.01 0.269736
\(711\) −11517.3 −0.607502
\(712\) −4530.30 −0.238455
\(713\) 24195.5 1.27087
\(714\) 0 0
\(715\) −1459.87 −0.0763583
\(716\) 710.796 0.0371001
\(717\) −1876.69 −0.0977496
\(718\) −20107.1 −1.04511
\(719\) −15769.6 −0.817949 −0.408974 0.912546i \(-0.634113\pi\)
−0.408974 + 0.912546i \(0.634113\pi\)
\(720\) 1253.09 0.0648611
\(721\) 0 0
\(722\) 16338.7 0.842195
\(723\) −11814.6 −0.607730
\(724\) −9381.78 −0.481590
\(725\) 11308.1 0.579272
\(726\) 1201.55 0.0614241
\(727\) −12692.8 −0.647525 −0.323762 0.946138i \(-0.604948\pi\)
−0.323762 + 0.946138i \(0.604948\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 3789.99 0.192156
\(731\) 6260.76 0.316775
\(732\) 6300.52 0.318134
\(733\) −8307.58 −0.418619 −0.209309 0.977849i \(-0.567122\pi\)
−0.209309 + 0.977849i \(0.567122\pi\)
\(734\) 36405.8 1.83074
\(735\) 0 0
\(736\) −21813.7 −1.09248
\(737\) 959.652 0.0479637
\(738\) 6424.03 0.320423
\(739\) 4934.86 0.245645 0.122822 0.992429i \(-0.460805\pi\)
0.122822 + 0.992429i \(0.460805\pi\)
\(740\) −2021.03 −0.100398
\(741\) 9895.18 0.490565
\(742\) 0 0
\(743\) 12725.6 0.628339 0.314169 0.949367i \(-0.398274\pi\)
0.314169 + 0.949367i \(0.398274\pi\)
\(744\) 7090.98 0.349419
\(745\) −4125.04 −0.202859
\(746\) 5439.29 0.266953
\(747\) −12408.6 −0.607776
\(748\) 2649.30 0.129503
\(749\) 0 0
\(750\) 4325.71 0.210604
\(751\) −236.257 −0.0114796 −0.00573978 0.999984i \(-0.501827\pi\)
−0.00573978 + 0.999984i \(0.501827\pi\)
\(752\) −5341.98 −0.259045
\(753\) −6678.61 −0.323217
\(754\) 23098.8 1.11566
\(755\) 3843.45 0.185268
\(756\) 0 0
\(757\) 3821.11 0.183462 0.0917308 0.995784i \(-0.470760\pi\)
0.0917308 + 0.995784i \(0.470760\pi\)
\(758\) −2702.28 −0.129487
\(759\) 5639.34 0.269690
\(760\) 1291.66 0.0616492
\(761\) −10167.2 −0.484311 −0.242156 0.970237i \(-0.577854\pi\)
−0.242156 + 0.970237i \(0.577854\pi\)
\(762\) 22550.4 1.07207
\(763\) 0 0
\(764\) −14735.9 −0.697808
\(765\) −1293.59 −0.0611372
\(766\) −21592.4 −1.01849
\(767\) 33477.1 1.57600
\(768\) −11992.3 −0.563454
\(769\) 13462.4 0.631294 0.315647 0.948877i \(-0.397778\pi\)
0.315647 + 0.948877i \(0.397778\pi\)
\(770\) 0 0
\(771\) 13856.4 0.647247
\(772\) 2789.35 0.130040
\(773\) −998.375 −0.0464541 −0.0232271 0.999730i \(-0.507394\pi\)
−0.0232271 + 0.999730i \(0.507394\pi\)
\(774\) −2289.57 −0.106327
\(775\) 17257.5 0.799879
\(776\) 17950.6 0.830400
\(777\) 0 0
\(778\) 26122.7 1.20379
\(779\) 9456.05 0.434914
\(780\) 1177.14 0.0540365
\(781\) 9611.26 0.440356
\(782\) 46079.1 2.10714
\(783\) 2504.93 0.114328
\(784\) 0 0
\(785\) 3058.20 0.139047
\(786\) −3992.02 −0.181159
\(787\) 11371.7 0.515067 0.257533 0.966269i \(-0.417090\pi\)
0.257533 + 0.966269i \(0.417090\pi\)
\(788\) 5704.66 0.257894
\(789\) 3803.69 0.171629
\(790\) −7473.91 −0.336595
\(791\) 0 0
\(792\) 1652.72 0.0741502
\(793\) −53430.7 −2.39266
\(794\) 31010.3 1.38604
\(795\) −329.883 −0.0147167
\(796\) −3986.14 −0.177494
\(797\) −21559.4 −0.958185 −0.479092 0.877764i \(-0.659034\pi\)
−0.479092 + 0.877764i \(0.659034\pi\)
\(798\) 0 0
\(799\) 5514.63 0.244172
\(800\) −15558.6 −0.687600
\(801\) −2442.33 −0.107735
\(802\) −14811.0 −0.652111
\(803\) 7138.26 0.313703
\(804\) −773.798 −0.0339425
\(805\) 0 0
\(806\) 35251.5 1.54055
\(807\) 5364.67 0.234009
\(808\) 5227.20 0.227589
\(809\) −23243.4 −1.01013 −0.505065 0.863081i \(-0.668532\pi\)
−0.505065 + 0.863081i \(0.668532\pi\)
\(810\) 473.068 0.0205209
\(811\) −1152.29 −0.0498920 −0.0249460 0.999689i \(-0.507941\pi\)
−0.0249460 + 0.999689i \(0.507941\pi\)
\(812\) 0 0
\(813\) −17648.7 −0.761338
\(814\) −14106.4 −0.607406
\(815\) 3690.06 0.158598
\(816\) 19284.7 0.827328
\(817\) −3370.20 −0.144318
\(818\) 11463.4 0.489985
\(819\) 0 0
\(820\) 1124.90 0.0479065
\(821\) 22920.8 0.974351 0.487175 0.873304i \(-0.338027\pi\)
0.487175 + 0.873304i \(0.338027\pi\)
\(822\) 18356.4 0.778895
\(823\) −40562.5 −1.71801 −0.859004 0.511969i \(-0.828916\pi\)
−0.859004 + 0.511969i \(0.828916\pi\)
\(824\) −22055.3 −0.932444
\(825\) 4022.27 0.169742
\(826\) 0 0
\(827\) −21510.0 −0.904444 −0.452222 0.891905i \(-0.649369\pi\)
−0.452222 + 0.891905i \(0.649369\pi\)
\(828\) −4547.18 −0.190852
\(829\) −37411.3 −1.56737 −0.783683 0.621161i \(-0.786661\pi\)
−0.783683 + 0.621161i \(0.786661\pi\)
\(830\) −8052.31 −0.336747
\(831\) −2856.37 −0.119237
\(832\) 15702.9 0.654328
\(833\) 0 0
\(834\) 15123.7 0.627926
\(835\) 5587.58 0.231576
\(836\) −1426.13 −0.0589997
\(837\) 3822.82 0.157869
\(838\) 16530.7 0.681434
\(839\) 30053.3 1.23666 0.618328 0.785920i \(-0.287810\pi\)
0.618328 + 0.785920i \(0.287810\pi\)
\(840\) 0 0
\(841\) −15781.7 −0.647084
\(842\) 4231.01 0.173171
\(843\) 11165.7 0.456190
\(844\) −10231.4 −0.417276
\(845\) −6106.17 −0.248590
\(846\) −2016.71 −0.0819572
\(847\) 0 0
\(848\) 4917.86 0.199151
\(849\) 16779.6 0.678296
\(850\) 32866.0 1.32623
\(851\) −66206.5 −2.66690
\(852\) −7749.86 −0.311626
\(853\) −2559.84 −0.102752 −0.0513760 0.998679i \(-0.516361\pi\)
−0.0513760 + 0.998679i \(0.516361\pi\)
\(854\) 0 0
\(855\) 696.347 0.0278533
\(856\) 13067.1 0.521757
\(857\) −18182.7 −0.724750 −0.362375 0.932032i \(-0.618034\pi\)
−0.362375 + 0.932032i \(0.618034\pi\)
\(858\) 8216.21 0.326919
\(859\) −4229.22 −0.167985 −0.0839924 0.996466i \(-0.526767\pi\)
−0.0839924 + 0.996466i \(0.526767\pi\)
\(860\) −400.922 −0.0158969
\(861\) 0 0
\(862\) −25726.4 −1.01653
\(863\) 16173.7 0.637961 0.318980 0.947761i \(-0.396660\pi\)
0.318980 + 0.947761i \(0.396660\pi\)
\(864\) −3446.50 −0.135709
\(865\) −3655.32 −0.143682
\(866\) 40725.1 1.59803
\(867\) −5169.00 −0.202478
\(868\) 0 0
\(869\) −14076.7 −0.549506
\(870\) 1625.52 0.0633452
\(871\) 6562.09 0.255279
\(872\) −34975.2 −1.35827
\(873\) 9677.37 0.375177
\(874\) −24804.6 −0.959987
\(875\) 0 0
\(876\) −5755.81 −0.221998
\(877\) −18309.9 −0.704994 −0.352497 0.935813i \(-0.614667\pi\)
−0.352497 + 0.935813i \(0.614667\pi\)
\(878\) −8341.74 −0.320638
\(879\) 6342.29 0.243368
\(880\) 1531.56 0.0586691
\(881\) −839.702 −0.0321116 −0.0160558 0.999871i \(-0.505111\pi\)
−0.0160558 + 0.999871i \(0.505111\pi\)
\(882\) 0 0
\(883\) 22004.4 0.838626 0.419313 0.907842i \(-0.362271\pi\)
0.419313 + 0.907842i \(0.362271\pi\)
\(884\) 18115.9 0.689256
\(885\) 2355.87 0.0894820
\(886\) −54328.3 −2.06004
\(887\) −12234.0 −0.463110 −0.231555 0.972822i \(-0.574381\pi\)
−0.231555 + 0.972822i \(0.574381\pi\)
\(888\) −19403.2 −0.733252
\(889\) 0 0
\(890\) −1584.90 −0.0596920
\(891\) 891.000 0.0335013
\(892\) 8970.51 0.336721
\(893\) −2968.55 −0.111242
\(894\) 23215.8 0.868516
\(895\) −424.191 −0.0158426
\(896\) 0 0
\(897\) 38561.7 1.43538
\(898\) 7965.13 0.295991
\(899\) 13135.7 0.487318
\(900\) −3243.28 −0.120121
\(901\) −5076.81 −0.187717
\(902\) 7851.60 0.289833
\(903\) 0 0
\(904\) 32236.1 1.18601
\(905\) 5598.89 0.205650
\(906\) −21631.1 −0.793205
\(907\) −3651.43 −0.133675 −0.0668377 0.997764i \(-0.521291\pi\)
−0.0668377 + 0.997764i \(0.521291\pi\)
\(908\) −300.124 −0.0109691
\(909\) 2818.04 0.102825
\(910\) 0 0
\(911\) 2031.16 0.0738699 0.0369349 0.999318i \(-0.488241\pi\)
0.0369349 + 0.999318i \(0.488241\pi\)
\(912\) −10381.1 −0.376920
\(913\) −15166.1 −0.549754
\(914\) 31319.9 1.13345
\(915\) −3760.05 −0.135851
\(916\) −9557.35 −0.344742
\(917\) 0 0
\(918\) 7280.38 0.261752
\(919\) 31085.1 1.11578 0.557890 0.829915i \(-0.311611\pi\)
0.557890 + 0.829915i \(0.311611\pi\)
\(920\) 5033.62 0.180384
\(921\) 20841.5 0.745657
\(922\) 23256.3 0.830701
\(923\) 65721.6 2.34372
\(924\) 0 0
\(925\) −47221.9 −1.67854
\(926\) 1035.85 0.0367604
\(927\) −11890.3 −0.421281
\(928\) −11842.6 −0.418914
\(929\) −7660.84 −0.270553 −0.135277 0.990808i \(-0.543192\pi\)
−0.135277 + 0.990808i \(0.543192\pi\)
\(930\) 2480.73 0.0874693
\(931\) 0 0
\(932\) 13148.5 0.462117
\(933\) 18020.8 0.632340
\(934\) 43410.3 1.52080
\(935\) −1581.06 −0.0553007
\(936\) 11301.3 0.394652
\(937\) 31354.8 1.09319 0.546594 0.837398i \(-0.315924\pi\)
0.546594 + 0.837398i \(0.315924\pi\)
\(938\) 0 0
\(939\) 19127.6 0.664755
\(940\) −353.142 −0.0122534
\(941\) −54310.4 −1.88148 −0.940738 0.339134i \(-0.889866\pi\)
−0.940738 + 0.339134i \(0.889866\pi\)
\(942\) −17211.6 −0.595314
\(943\) 36850.4 1.27255
\(944\) −35121.0 −1.21090
\(945\) 0 0
\(946\) −2798.36 −0.0961760
\(947\) −17047.6 −0.584977 −0.292488 0.956269i \(-0.594483\pi\)
−0.292488 + 0.956269i \(0.594483\pi\)
\(948\) 11350.5 0.388869
\(949\) 48811.4 1.66963
\(950\) −17691.9 −0.604212
\(951\) −27171.5 −0.926495
\(952\) 0 0
\(953\) −24025.8 −0.816654 −0.408327 0.912836i \(-0.633888\pi\)
−0.408327 + 0.912836i \(0.633888\pi\)
\(954\) 1856.59 0.0630078
\(955\) 8794.13 0.297981
\(956\) 1849.51 0.0625706
\(957\) 3061.59 0.103414
\(958\) −34034.4 −1.14781
\(959\) 0 0
\(960\) 1105.05 0.0371514
\(961\) −9744.44 −0.327093
\(962\) −96459.3 −3.23282
\(963\) 7044.60 0.235731
\(964\) 11643.4 0.389015
\(965\) −1664.64 −0.0555302
\(966\) 0 0
\(967\) −36019.9 −1.19785 −0.598925 0.800805i \(-0.704405\pi\)
−0.598925 + 0.800805i \(0.704405\pi\)
\(968\) 2019.99 0.0670714
\(969\) 10716.6 0.355280
\(970\) 6279.91 0.207872
\(971\) −10860.3 −0.358934 −0.179467 0.983764i \(-0.557437\pi\)
−0.179467 + 0.983764i \(0.557437\pi\)
\(972\) −718.441 −0.0237078
\(973\) 0 0
\(974\) 52879.7 1.73960
\(975\) 27504.2 0.903425
\(976\) 56054.3 1.83838
\(977\) 41310.4 1.35275 0.676374 0.736558i \(-0.263550\pi\)
0.676374 + 0.736558i \(0.263550\pi\)
\(978\) −20767.8 −0.679019
\(979\) −2985.07 −0.0974498
\(980\) 0 0
\(981\) −18855.5 −0.613669
\(982\) 35699.7 1.16011
\(983\) −56605.8 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(984\) 10799.8 0.349882
\(985\) −3404.45 −0.110127
\(986\) 25016.3 0.807992
\(987\) 0 0
\(988\) −9751.85 −0.314016
\(989\) −13133.7 −0.422273
\(990\) 578.194 0.0185618
\(991\) 22226.0 0.712443 0.356222 0.934401i \(-0.384065\pi\)
0.356222 + 0.934401i \(0.384065\pi\)
\(992\) −18073.2 −0.578451
\(993\) −13115.4 −0.419139
\(994\) 0 0
\(995\) 2378.87 0.0757941
\(996\) 12228.9 0.389044
\(997\) 39276.1 1.24763 0.623815 0.781572i \(-0.285582\pi\)
0.623815 + 0.781572i \(0.285582\pi\)
\(998\) −11244.4 −0.356649
\(999\) −10460.4 −0.331285
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 1617.4.a.be.1.4 16
7.6 odd 2 1617.4.a.bf.1.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.4.a.be.1.4 16 1.1 even 1 trivial
1617.4.a.bf.1.4 yes 16 7.6 odd 2