Properties

Label 1617.4.a.k.1.2
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
Analytic rank $0$
Dimension $2$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.42443\) 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)
 
\(f(q)\) \(=\) \(q+5.42443 q^{2} +3.00000 q^{3} +21.4244 q^{4} +16.8489 q^{5} +16.2733 q^{6} +72.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.42443 q^{2} +3.00000 q^{3} +21.4244 q^{4} +16.8489 q^{5} +16.2733 q^{6} +72.8199 q^{8} +9.00000 q^{9} +91.3954 q^{10} -11.0000 q^{11} +64.2733 q^{12} -24.8489 q^{13} +50.5466 q^{15} +223.611 q^{16} +15.9420 q^{17} +48.8199 q^{18} -15.1511 q^{19} +360.977 q^{20} -59.6687 q^{22} +17.7557 q^{23} +218.460 q^{24} +158.884 q^{25} -134.791 q^{26} +27.0000 q^{27} -128.547 q^{29} +274.186 q^{30} -219.395 q^{31} +630.402 q^{32} -33.0000 q^{33} +86.4763 q^{34} +192.820 q^{36} +92.0703 q^{37} -82.1863 q^{38} -74.5466 q^{39} +1226.93 q^{40} +459.942 q^{41} +64.9648 q^{43} -235.669 q^{44} +151.640 q^{45} +96.3146 q^{46} -497.408 q^{47} +670.832 q^{48} +861.855 q^{50} +47.8260 q^{51} -532.373 q^{52} -526.919 q^{53} +146.460 q^{54} -185.337 q^{55} -45.4534 q^{57} -697.292 q^{58} +578.443 q^{59} +1082.93 q^{60} +221.569 q^{61} -1190.09 q^{62} +1630.68 q^{64} -418.675 q^{65} -179.006 q^{66} -860.745 q^{67} +341.548 q^{68} +53.2671 q^{69} +580.919 q^{71} +655.379 q^{72} -510.116 q^{73} +499.429 q^{74} +476.652 q^{75} -324.605 q^{76} -404.373 q^{78} +1035.12 q^{79} +3767.59 q^{80} +81.0000 q^{81} +2494.92 q^{82} -606.211 q^{83} +268.605 q^{85} +352.397 q^{86} -385.640 q^{87} -801.018 q^{88} +23.4411 q^{89} +822.559 q^{90} +380.406 q^{92} -658.186 q^{93} -2698.15 q^{94} -255.279 q^{95} +1891.20 q^{96} -719.490 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9} + 104 q^{10} - 22 q^{11} + 99 q^{12} - 30 q^{13} + 42 q^{15} + 201 q^{16} - 106 q^{17} + 9 q^{18} - 50 q^{19} + 328 q^{20} - 11 q^{22} + 134 q^{23} + 171 q^{24} + 42 q^{25} - 112 q^{26} + 54 q^{27} - 198 q^{29} + 312 q^{30} - 360 q^{31} + 857 q^{32} - 66 q^{33} + 626 q^{34} + 297 q^{36} - 328 q^{37} + 72 q^{38} - 90 q^{39} + 1272 q^{40} + 782 q^{41} + 386 q^{43} - 363 q^{44} + 126 q^{45} - 418 q^{46} - 266 q^{47} + 603 q^{48} + 1379 q^{50} - 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 154 q^{55} - 150 q^{57} - 390 q^{58} + 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} + 809 q^{64} - 404 q^{65} - 33 q^{66} - 776 q^{67} - 1070 q^{68} + 402 q^{69} + 630 q^{71} + 513 q^{72} - 1296 q^{73} + 2358 q^{74} + 126 q^{75} - 728 q^{76} - 336 q^{78} + 652 q^{79} + 3832 q^{80} + 162 q^{81} + 1070 q^{82} + 324 q^{83} + 616 q^{85} - 1068 q^{86} - 594 q^{87} - 627 q^{88} + 756 q^{89} + 936 q^{90} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} - 156 q^{95} + 2571 q^{96} + 452 q^{97} - 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.42443 1.91783 0.958913 0.283702i \(-0.0915625\pi\)
0.958913 + 0.283702i \(0.0915625\pi\)
\(3\) 3.00000 0.577350
\(4\) 21.4244 2.67805
\(5\) 16.8489 1.50701 0.753504 0.657444i \(-0.228362\pi\)
0.753504 + 0.657444i \(0.228362\pi\)
\(6\) 16.2733 1.10726
\(7\) 0 0
\(8\) 72.8199 3.21821
\(9\) 9.00000 0.333333
\(10\) 91.3954 2.89018
\(11\) −11.0000 −0.301511
\(12\) 64.2733 1.54617
\(13\) −24.8489 −0.530141 −0.265071 0.964229i \(-0.585395\pi\)
−0.265071 + 0.964229i \(0.585395\pi\)
\(14\) 0 0
\(15\) 50.5466 0.870071
\(16\) 223.611 3.49392
\(17\) 15.9420 0.227441 0.113721 0.993513i \(-0.463723\pi\)
0.113721 + 0.993513i \(0.463723\pi\)
\(18\) 48.8199 0.639275
\(19\) −15.1511 −0.182943 −0.0914713 0.995808i \(-0.529157\pi\)
−0.0914713 + 0.995808i \(0.529157\pi\)
\(20\) 360.977 4.03585
\(21\) 0 0
\(22\) −59.6687 −0.578246
\(23\) 17.7557 0.160971 0.0804853 0.996756i \(-0.474353\pi\)
0.0804853 + 0.996756i \(0.474353\pi\)
\(24\) 218.460 1.85804
\(25\) 158.884 1.27107
\(26\) −134.791 −1.01672
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −128.547 −0.823121 −0.411560 0.911383i \(-0.635016\pi\)
−0.411560 + 0.911383i \(0.635016\pi\)
\(30\) 274.186 1.66864
\(31\) −219.395 −1.27112 −0.635558 0.772053i \(-0.719230\pi\)
−0.635558 + 0.772053i \(0.719230\pi\)
\(32\) 630.402 3.48251
\(33\) −33.0000 −0.174078
\(34\) 86.4763 0.436193
\(35\) 0 0
\(36\) 192.820 0.892685
\(37\) 92.0703 0.409088 0.204544 0.978857i \(-0.434429\pi\)
0.204544 + 0.978857i \(0.434429\pi\)
\(38\) −82.1863 −0.350852
\(39\) −74.5466 −0.306077
\(40\) 1226.93 4.84987
\(41\) 459.942 1.75197 0.875986 0.482336i \(-0.160212\pi\)
0.875986 + 0.482336i \(0.160212\pi\)
\(42\) 0 0
\(43\) 64.9648 0.230396 0.115198 0.993343i \(-0.463250\pi\)
0.115198 + 0.993343i \(0.463250\pi\)
\(44\) −235.669 −0.807464
\(45\) 151.640 0.502336
\(46\) 96.3146 0.308713
\(47\) −497.408 −1.54371 −0.771855 0.635799i \(-0.780671\pi\)
−0.771855 + 0.635799i \(0.780671\pi\)
\(48\) 670.832 2.01721
\(49\) 0 0
\(50\) 861.855 2.43769
\(51\) 47.8260 0.131313
\(52\) −532.373 −1.41975
\(53\) −526.919 −1.36562 −0.682811 0.730596i \(-0.739243\pi\)
−0.682811 + 0.730596i \(0.739243\pi\)
\(54\) 146.460 0.369086
\(55\) −185.337 −0.454380
\(56\) 0 0
\(57\) −45.4534 −0.105622
\(58\) −697.292 −1.57860
\(59\) 578.443 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(60\) 1082.93 2.33010
\(61\) 221.569 0.465067 0.232533 0.972588i \(-0.425299\pi\)
0.232533 + 0.972588i \(0.425299\pi\)
\(62\) −1190.09 −2.43778
\(63\) 0 0
\(64\) 1630.68 3.18493
\(65\) −418.675 −0.798927
\(66\) −179.006 −0.333851
\(67\) −860.745 −1.56950 −0.784752 0.619810i \(-0.787210\pi\)
−0.784752 + 0.619810i \(0.787210\pi\)
\(68\) 341.548 0.609100
\(69\) 53.2671 0.0929364
\(70\) 0 0
\(71\) 580.919 0.971020 0.485510 0.874231i \(-0.338634\pi\)
0.485510 + 0.874231i \(0.338634\pi\)
\(72\) 655.379 1.07274
\(73\) −510.116 −0.817871 −0.408935 0.912563i \(-0.634100\pi\)
−0.408935 + 0.912563i \(0.634100\pi\)
\(74\) 499.429 0.784560
\(75\) 476.652 0.733854
\(76\) −324.605 −0.489930
\(77\) 0 0
\(78\) −404.373 −0.587002
\(79\) 1035.12 1.47418 0.737088 0.675797i \(-0.236200\pi\)
0.737088 + 0.675797i \(0.236200\pi\)
\(80\) 3767.59 5.26536
\(81\) 81.0000 0.111111
\(82\) 2494.92 3.35998
\(83\) −606.211 −0.801690 −0.400845 0.916146i \(-0.631283\pi\)
−0.400845 + 0.916146i \(0.631283\pi\)
\(84\) 0 0
\(85\) 268.605 0.342756
\(86\) 352.397 0.441860
\(87\) −385.640 −0.475229
\(88\) −801.018 −0.970328
\(89\) 23.4411 0.0279186 0.0139593 0.999903i \(-0.495556\pi\)
0.0139593 + 0.999903i \(0.495556\pi\)
\(90\) 822.559 0.963392
\(91\) 0 0
\(92\) 380.406 0.431088
\(93\) −658.186 −0.733879
\(94\) −2698.15 −2.96057
\(95\) −255.279 −0.275696
\(96\) 1891.20 2.01063
\(97\) −719.490 −0.753126 −0.376563 0.926391i \(-0.622894\pi\)
−0.376563 + 0.926391i \(0.622894\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) 3404.00 3.40400
\(101\) −1871.27 −1.84355 −0.921774 0.387727i \(-0.873260\pi\)
−0.921774 + 0.387727i \(0.873260\pi\)
\(102\) 259.429 0.251836
\(103\) 428.745 0.410151 0.205075 0.978746i \(-0.434256\pi\)
0.205075 + 0.978746i \(0.434256\pi\)
\(104\) −1809.49 −1.70611
\(105\) 0 0
\(106\) −2858.24 −2.61902
\(107\) 1148.02 1.03723 0.518616 0.855008i \(-0.326448\pi\)
0.518616 + 0.855008i \(0.326448\pi\)
\(108\) 578.460 0.515392
\(109\) −1828.32 −1.60662 −0.803308 0.595564i \(-0.796929\pi\)
−0.803308 + 0.595564i \(0.796929\pi\)
\(110\) −1005.35 −0.871421
\(111\) 276.211 0.236187
\(112\) 0 0
\(113\) 1126.40 0.937722 0.468861 0.883272i \(-0.344665\pi\)
0.468861 + 0.883272i \(0.344665\pi\)
\(114\) −246.559 −0.202565
\(115\) 299.163 0.242584
\(116\) −2754.04 −2.20436
\(117\) −223.640 −0.176714
\(118\) 3137.72 2.44789
\(119\) 0 0
\(120\) 3680.79 2.80008
\(121\) 121.000 0.0909091
\(122\) 1201.89 0.891916
\(123\) 1379.83 1.01150
\(124\) −4700.42 −3.40412
\(125\) 570.907 0.408508
\(126\) 0 0
\(127\) 661.304 0.462057 0.231029 0.972947i \(-0.425791\pi\)
0.231029 + 0.972947i \(0.425791\pi\)
\(128\) 3802.31 2.62562
\(129\) 194.895 0.133019
\(130\) −2271.07 −1.53220
\(131\) −622.186 −0.414967 −0.207483 0.978239i \(-0.566527\pi\)
−0.207483 + 0.978239i \(0.566527\pi\)
\(132\) −707.006 −0.466189
\(133\) 0 0
\(134\) −4669.05 −3.01003
\(135\) 454.919 0.290024
\(136\) 1160.89 0.731955
\(137\) −1872.84 −1.16794 −0.583969 0.811776i \(-0.698501\pi\)
−0.583969 + 0.811776i \(0.698501\pi\)
\(138\) 288.944 0.178236
\(139\) 954.058 0.582174 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(140\) 0 0
\(141\) −1492.22 −0.891261
\(142\) 3151.15 1.86225
\(143\) 273.337 0.159844
\(144\) 2012.50 1.16464
\(145\) −2165.86 −1.24045
\(146\) −2767.09 −1.56853
\(147\) 0 0
\(148\) 1972.55 1.09556
\(149\) −2047.01 −1.12549 −0.562745 0.826631i \(-0.690255\pi\)
−0.562745 + 0.826631i \(0.690255\pi\)
\(150\) 2585.57 1.40740
\(151\) 475.863 0.256458 0.128229 0.991745i \(-0.459071\pi\)
0.128229 + 0.991745i \(0.459071\pi\)
\(152\) −1103.30 −0.588749
\(153\) 143.478 0.0758138
\(154\) 0 0
\(155\) −3696.56 −1.91558
\(156\) −1597.12 −0.819691
\(157\) 647.466 0.329130 0.164565 0.986366i \(-0.447378\pi\)
0.164565 + 0.986366i \(0.447378\pi\)
\(158\) 5614.92 2.82721
\(159\) −1580.76 −0.788442
\(160\) 10621.5 5.24817
\(161\) 0 0
\(162\) 439.379 0.213092
\(163\) 1093.23 0.525329 0.262665 0.964887i \(-0.415399\pi\)
0.262665 + 0.964887i \(0.415399\pi\)
\(164\) 9853.99 4.69188
\(165\) −556.012 −0.262336
\(166\) −3288.35 −1.53750
\(167\) −1123.25 −0.520479 −0.260240 0.965544i \(-0.583802\pi\)
−0.260240 + 0.965544i \(0.583802\pi\)
\(168\) 0 0
\(169\) −1579.53 −0.718951
\(170\) 1457.03 0.657346
\(171\) −136.360 −0.0609809
\(172\) 1391.83 0.617014
\(173\) −46.0123 −0.0202211 −0.0101106 0.999949i \(-0.503218\pi\)
−0.0101106 + 0.999949i \(0.503218\pi\)
\(174\) −2091.88 −0.911406
\(175\) 0 0
\(176\) −2459.72 −1.05346
\(177\) 1735.33 0.736923
\(178\) 127.155 0.0535430
\(179\) −831.975 −0.347401 −0.173700 0.984799i \(-0.555572\pi\)
−0.173700 + 0.984799i \(0.555572\pi\)
\(180\) 3248.79 1.34528
\(181\) 1810.63 0.743553 0.371776 0.928322i \(-0.378749\pi\)
0.371776 + 0.928322i \(0.378749\pi\)
\(182\) 0 0
\(183\) 664.708 0.268506
\(184\) 1292.97 0.518037
\(185\) 1551.28 0.616499
\(186\) −3570.28 −1.40745
\(187\) −175.362 −0.0685762
\(188\) −10656.7 −4.13414
\(189\) 0 0
\(190\) −1384.75 −0.528737
\(191\) 458.898 0.173847 0.0869233 0.996215i \(-0.472296\pi\)
0.0869233 + 0.996215i \(0.472296\pi\)
\(192\) 4892.05 1.83882
\(193\) 1778.91 0.663465 0.331733 0.943373i \(-0.392367\pi\)
0.331733 + 0.943373i \(0.392367\pi\)
\(194\) −3902.82 −1.44436
\(195\) −1256.02 −0.461260
\(196\) 0 0
\(197\) 5304.53 1.91844 0.959218 0.282666i \(-0.0912188\pi\)
0.959218 + 0.282666i \(0.0912188\pi\)
\(198\) −537.018 −0.192749
\(199\) 5138.40 1.83041 0.915205 0.402989i \(-0.132029\pi\)
0.915205 + 0.402989i \(0.132029\pi\)
\(200\) 11569.9 4.09058
\(201\) −2582.24 −0.906153
\(202\) −10150.6 −3.53560
\(203\) 0 0
\(204\) 1024.65 0.351664
\(205\) 7749.50 2.64024
\(206\) 2325.70 0.786597
\(207\) 159.801 0.0536568
\(208\) −5556.47 −1.85227
\(209\) 166.663 0.0551593
\(210\) 0 0
\(211\) −4262.36 −1.39068 −0.695339 0.718682i \(-0.744746\pi\)
−0.695339 + 0.718682i \(0.744746\pi\)
\(212\) −11288.9 −3.65721
\(213\) 1742.76 0.560619
\(214\) 6227.38 1.98923
\(215\) 1094.58 0.347209
\(216\) 1966.14 0.619345
\(217\) 0 0
\(218\) −9917.58 −3.08121
\(219\) −1530.35 −0.472198
\(220\) −3970.75 −1.21685
\(221\) −396.141 −0.120576
\(222\) 1498.29 0.452966
\(223\) 1377.80 0.413740 0.206870 0.978368i \(-0.433672\pi\)
0.206870 + 0.978368i \(0.433672\pi\)
\(224\) 0 0
\(225\) 1429.96 0.423691
\(226\) 6110.06 1.79839
\(227\) 1227.28 0.358843 0.179422 0.983772i \(-0.442577\pi\)
0.179422 + 0.983772i \(0.442577\pi\)
\(228\) −973.814 −0.282861
\(229\) −3890.28 −1.12261 −0.561304 0.827610i \(-0.689700\pi\)
−0.561304 + 0.827610i \(0.689700\pi\)
\(230\) 1622.79 0.465233
\(231\) 0 0
\(232\) −9360.74 −2.64898
\(233\) −3218.14 −0.904837 −0.452419 0.891806i \(-0.649439\pi\)
−0.452419 + 0.891806i \(0.649439\pi\)
\(234\) −1213.12 −0.338906
\(235\) −8380.75 −2.32638
\(236\) 12392.8 3.41823
\(237\) 3105.35 0.851115
\(238\) 0 0
\(239\) −428.098 −0.115864 −0.0579318 0.998321i \(-0.518451\pi\)
−0.0579318 + 0.998321i \(0.518451\pi\)
\(240\) 11302.8 3.03996
\(241\) −1231.16 −0.329070 −0.164535 0.986371i \(-0.552612\pi\)
−0.164535 + 0.986371i \(0.552612\pi\)
\(242\) 656.356 0.174348
\(243\) 243.000 0.0641500
\(244\) 4747.00 1.24547
\(245\) 0 0
\(246\) 7484.77 1.93988
\(247\) 376.489 0.0969854
\(248\) −15976.3 −4.09072
\(249\) −1818.63 −0.462856
\(250\) 3096.84 0.783446
\(251\) −2838.22 −0.713732 −0.356866 0.934156i \(-0.616155\pi\)
−0.356866 + 0.934156i \(0.616155\pi\)
\(252\) 0 0
\(253\) −195.313 −0.0485344
\(254\) 3587.20 0.886145
\(255\) 805.814 0.197890
\(256\) 7579.90 1.85056
\(257\) −342.007 −0.0830110 −0.0415055 0.999138i \(-0.513215\pi\)
−0.0415055 + 0.999138i \(0.513215\pi\)
\(258\) 1057.19 0.255108
\(259\) 0 0
\(260\) −8969.87 −2.13957
\(261\) −1156.92 −0.274374
\(262\) −3375.01 −0.795834
\(263\) 5895.00 1.38213 0.691067 0.722791i \(-0.257141\pi\)
0.691067 + 0.722791i \(0.257141\pi\)
\(264\) −2403.06 −0.560219
\(265\) −8877.99 −2.05800
\(266\) 0 0
\(267\) 70.3234 0.0161188
\(268\) −18441.0 −4.20322
\(269\) 2496.18 0.565779 0.282890 0.959152i \(-0.408707\pi\)
0.282890 + 0.959152i \(0.408707\pi\)
\(270\) 2467.68 0.556215
\(271\) 2249.68 0.504274 0.252137 0.967692i \(-0.418867\pi\)
0.252137 + 0.967692i \(0.418867\pi\)
\(272\) 3564.80 0.794662
\(273\) 0 0
\(274\) −10159.1 −2.23990
\(275\) −1747.72 −0.383243
\(276\) 1141.22 0.248889
\(277\) 4082.59 0.885556 0.442778 0.896631i \(-0.353993\pi\)
0.442778 + 0.896631i \(0.353993\pi\)
\(278\) 5175.22 1.11651
\(279\) −1974.56 −0.423705
\(280\) 0 0
\(281\) −1033.79 −0.219468 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(282\) −8094.46 −1.70928
\(283\) 7809.14 1.64030 0.820150 0.572148i \(-0.193890\pi\)
0.820150 + 0.572148i \(0.193890\pi\)
\(284\) 12445.9 2.60044
\(285\) −765.838 −0.159173
\(286\) 1482.70 0.306552
\(287\) 0 0
\(288\) 5673.61 1.16084
\(289\) −4658.85 −0.948270
\(290\) −11748.6 −2.37896
\(291\) −2158.47 −0.434817
\(292\) −10928.9 −2.19030
\(293\) 1949.19 0.388645 0.194323 0.980938i \(-0.437749\pi\)
0.194323 + 0.980938i \(0.437749\pi\)
\(294\) 0 0
\(295\) 9746.10 1.92353
\(296\) 6704.55 1.31653
\(297\) −297.000 −0.0580259
\(298\) −11103.9 −2.15849
\(299\) −441.209 −0.0853371
\(300\) 10212.0 1.96530
\(301\) 0 0
\(302\) 2581.28 0.491842
\(303\) −5613.81 −1.06437
\(304\) −3387.96 −0.639187
\(305\) 3733.19 0.700859
\(306\) 778.286 0.145398
\(307\) −2364.09 −0.439497 −0.219748 0.975557i \(-0.570524\pi\)
−0.219748 + 0.975557i \(0.570524\pi\)
\(308\) 0 0
\(309\) 1286.24 0.236801
\(310\) −20051.7 −3.67375
\(311\) 1989.17 0.362686 0.181343 0.983420i \(-0.441956\pi\)
0.181343 + 0.983420i \(0.441956\pi\)
\(312\) −5428.47 −0.985021
\(313\) 3878.67 0.700433 0.350216 0.936669i \(-0.386108\pi\)
0.350216 + 0.936669i \(0.386108\pi\)
\(314\) 3512.13 0.631214
\(315\) 0 0
\(316\) 22176.8 3.94792
\(317\) 2913.73 0.516251 0.258126 0.966111i \(-0.416895\pi\)
0.258126 + 0.966111i \(0.416895\pi\)
\(318\) −8574.71 −1.51209
\(319\) 1414.01 0.248180
\(320\) 27475.1 4.79971
\(321\) 3444.07 0.598846
\(322\) 0 0
\(323\) −241.540 −0.0416087
\(324\) 1735.38 0.297562
\(325\) −3948.09 −0.673847
\(326\) 5930.17 1.00749
\(327\) −5484.95 −0.927580
\(328\) 33492.9 5.63822
\(329\) 0 0
\(330\) −3016.05 −0.503115
\(331\) 8104.46 1.34580 0.672902 0.739731i \(-0.265047\pi\)
0.672902 + 0.739731i \(0.265047\pi\)
\(332\) −12987.7 −2.14697
\(333\) 828.633 0.136363
\(334\) −6093.02 −0.998189
\(335\) −14502.6 −2.36525
\(336\) 0 0
\(337\) 5919.19 0.956792 0.478396 0.878144i \(-0.341218\pi\)
0.478396 + 0.878144i \(0.341218\pi\)
\(338\) −8568.07 −1.37882
\(339\) 3379.19 0.541394
\(340\) 5754.70 0.917919
\(341\) 2413.35 0.383256
\(342\) −739.677 −0.116951
\(343\) 0 0
\(344\) 4730.73 0.741465
\(345\) 897.490 0.140056
\(346\) −249.590 −0.0387805
\(347\) −8540.59 −1.32128 −0.660638 0.750705i \(-0.729714\pi\)
−0.660638 + 0.750705i \(0.729714\pi\)
\(348\) −8262.11 −1.27269
\(349\) −937.337 −0.143767 −0.0718833 0.997413i \(-0.522901\pi\)
−0.0718833 + 0.997413i \(0.522901\pi\)
\(350\) 0 0
\(351\) −670.919 −0.102026
\(352\) −6934.42 −1.05002
\(353\) 211.118 0.0318319 0.0159160 0.999873i \(-0.494934\pi\)
0.0159160 + 0.999873i \(0.494934\pi\)
\(354\) 9413.17 1.41329
\(355\) 9787.82 1.46333
\(356\) 502.213 0.0747675
\(357\) 0 0
\(358\) −4512.99 −0.666254
\(359\) −1376.31 −0.202337 −0.101169 0.994869i \(-0.532258\pi\)
−0.101169 + 0.994869i \(0.532258\pi\)
\(360\) 11042.4 1.61662
\(361\) −6629.44 −0.966532
\(362\) 9821.63 1.42600
\(363\) 363.000 0.0524864
\(364\) 0 0
\(365\) −8594.87 −1.23254
\(366\) 3605.66 0.514948
\(367\) −1030.45 −0.146564 −0.0732821 0.997311i \(-0.523347\pi\)
−0.0732821 + 0.997311i \(0.523347\pi\)
\(368\) 3970.37 0.562418
\(369\) 4139.48 0.583991
\(370\) 8414.81 1.18234
\(371\) 0 0
\(372\) −14101.3 −1.96537
\(373\) 9365.39 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(374\) −951.239 −0.131517
\(375\) 1712.72 0.235852
\(376\) −36221.2 −4.96799
\(377\) 3194.24 0.436370
\(378\) 0 0
\(379\) −7120.23 −0.965017 −0.482509 0.875891i \(-0.660274\pi\)
−0.482509 + 0.875891i \(0.660274\pi\)
\(380\) −5469.22 −0.738329
\(381\) 1983.91 0.266769
\(382\) 2489.26 0.333407
\(383\) 1163.56 0.155235 0.0776176 0.996983i \(-0.475269\pi\)
0.0776176 + 0.996983i \(0.475269\pi\)
\(384\) 11406.9 1.51590
\(385\) 0 0
\(386\) 9649.57 1.27241
\(387\) 584.684 0.0767988
\(388\) −15414.7 −2.01691
\(389\) −10958.9 −1.42838 −0.714188 0.699954i \(-0.753204\pi\)
−0.714188 + 0.699954i \(0.753204\pi\)
\(390\) −6813.22 −0.884617
\(391\) 283.062 0.0366114
\(392\) 0 0
\(393\) −1866.56 −0.239581
\(394\) 28774.0 3.67923
\(395\) 17440.6 2.22159
\(396\) −2121.02 −0.269155
\(397\) 2172.09 0.274595 0.137298 0.990530i \(-0.456158\pi\)
0.137298 + 0.990530i \(0.456158\pi\)
\(398\) 27872.9 3.51041
\(399\) 0 0
\(400\) 35528.2 4.44102
\(401\) 7830.71 0.975180 0.487590 0.873073i \(-0.337876\pi\)
0.487590 + 0.873073i \(0.337876\pi\)
\(402\) −14007.2 −1.73784
\(403\) 5451.73 0.673870
\(404\) −40090.9 −4.93712
\(405\) 1364.76 0.167445
\(406\) 0 0
\(407\) −1012.77 −0.123345
\(408\) 3482.68 0.422594
\(409\) 10731.2 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(410\) 42036.6 5.06351
\(411\) −5618.52 −0.674309
\(412\) 9185.62 1.09841
\(413\) 0 0
\(414\) 866.831 0.102904
\(415\) −10214.0 −1.20815
\(416\) −15664.8 −1.84622
\(417\) 2862.17 0.336118
\(418\) 904.049 0.105786
\(419\) 7315.88 0.852994 0.426497 0.904489i \(-0.359748\pi\)
0.426497 + 0.904489i \(0.359748\pi\)
\(420\) 0 0
\(421\) −12495.7 −1.44657 −0.723284 0.690551i \(-0.757368\pi\)
−0.723284 + 0.690551i \(0.757368\pi\)
\(422\) −23120.9 −2.66708
\(423\) −4476.67 −0.514570
\(424\) −38370.2 −4.39486
\(425\) 2532.93 0.289094
\(426\) 9453.46 1.07517
\(427\) 0 0
\(428\) 24595.8 2.77776
\(429\) 820.012 0.0922857
\(430\) 5937.49 0.665887
\(431\) 6075.01 0.678939 0.339470 0.940617i \(-0.389752\pi\)
0.339470 + 0.940617i \(0.389752\pi\)
\(432\) 6037.49 0.672405
\(433\) −5641.79 −0.626160 −0.313080 0.949727i \(-0.601361\pi\)
−0.313080 + 0.949727i \(0.601361\pi\)
\(434\) 0 0
\(435\) −6497.59 −0.716174
\(436\) −39170.7 −4.30260
\(437\) −269.019 −0.0294484
\(438\) −8301.26 −0.905593
\(439\) −10897.0 −1.18470 −0.592351 0.805680i \(-0.701800\pi\)
−0.592351 + 0.805680i \(0.701800\pi\)
\(440\) −13496.2 −1.46229
\(441\) 0 0
\(442\) −2148.84 −0.231244
\(443\) −7720.83 −0.828054 −0.414027 0.910265i \(-0.635878\pi\)
−0.414027 + 0.910265i \(0.635878\pi\)
\(444\) 5917.66 0.632522
\(445\) 394.956 0.0420735
\(446\) 7473.76 0.793481
\(447\) −6141.04 −0.649802
\(448\) 0 0
\(449\) −7473.86 −0.785553 −0.392776 0.919634i \(-0.628485\pi\)
−0.392776 + 0.919634i \(0.628485\pi\)
\(450\) 7756.70 0.812565
\(451\) −5059.36 −0.528240
\(452\) 24132.4 2.51127
\(453\) 1427.59 0.148066
\(454\) 6657.29 0.688198
\(455\) 0 0
\(456\) −3309.91 −0.339914
\(457\) 11140.5 1.14033 0.570167 0.821529i \(-0.306879\pi\)
0.570167 + 0.821529i \(0.306879\pi\)
\(458\) −21102.6 −2.15296
\(459\) 430.434 0.0437711
\(460\) 6409.41 0.649652
\(461\) −14328.8 −1.44763 −0.723817 0.689992i \(-0.757614\pi\)
−0.723817 + 0.689992i \(0.757614\pi\)
\(462\) 0 0
\(463\) 11760.7 1.18049 0.590246 0.807223i \(-0.299031\pi\)
0.590246 + 0.807223i \(0.299031\pi\)
\(464\) −28744.4 −2.87592
\(465\) −11089.7 −1.10596
\(466\) −17456.5 −1.73532
\(467\) −11854.9 −1.17469 −0.587343 0.809338i \(-0.699826\pi\)
−0.587343 + 0.809338i \(0.699826\pi\)
\(468\) −4791.35 −0.473249
\(469\) 0 0
\(470\) −45460.8 −4.46160
\(471\) 1942.40 0.190023
\(472\) 42122.1 4.10769
\(473\) −714.613 −0.0694671
\(474\) 16844.8 1.63229
\(475\) −2407.27 −0.232533
\(476\) 0 0
\(477\) −4742.27 −0.455207
\(478\) −2322.19 −0.222206
\(479\) 1324.68 0.126359 0.0631796 0.998002i \(-0.479876\pi\)
0.0631796 + 0.998002i \(0.479876\pi\)
\(480\) 31864.6 3.03003
\(481\) −2287.84 −0.216874
\(482\) −6678.34 −0.631100
\(483\) 0 0
\(484\) 2592.36 0.243459
\(485\) −12122.6 −1.13497
\(486\) 1318.14 0.123029
\(487\) 18636.4 1.73408 0.867040 0.498239i \(-0.166020\pi\)
0.867040 + 0.498239i \(0.166020\pi\)
\(488\) 16134.7 1.49668
\(489\) 3279.70 0.303299
\(490\) 0 0
\(491\) 124.552 0.0114480 0.00572398 0.999984i \(-0.498178\pi\)
0.00572398 + 0.999984i \(0.498178\pi\)
\(492\) 29562.0 2.70886
\(493\) −2049.29 −0.187212
\(494\) 2042.24 0.186001
\(495\) −1668.04 −0.151460
\(496\) −49059.2 −4.44117
\(497\) 0 0
\(498\) −9865.04 −0.887677
\(499\) 10230.2 0.917768 0.458884 0.888496i \(-0.348249\pi\)
0.458884 + 0.888496i \(0.348249\pi\)
\(500\) 12231.4 1.09401
\(501\) −3369.76 −0.300499
\(502\) −15395.7 −1.36881
\(503\) −5150.81 −0.456587 −0.228294 0.973592i \(-0.573315\pi\)
−0.228294 + 0.973592i \(0.573315\pi\)
\(504\) 0 0
\(505\) −31528.8 −2.77824
\(506\) −1059.46 −0.0930806
\(507\) −4738.60 −0.415086
\(508\) 14168.1 1.23741
\(509\) 22.7715 0.00198296 0.000991481 1.00000i \(-0.499684\pi\)
0.000991481 1.00000i \(0.499684\pi\)
\(510\) 4371.08 0.379519
\(511\) 0 0
\(512\) 10698.1 0.923429
\(513\) −409.081 −0.0352073
\(514\) −1855.19 −0.159201
\(515\) 7223.87 0.618100
\(516\) 4175.50 0.356233
\(517\) 5471.49 0.465446
\(518\) 0 0
\(519\) −138.037 −0.0116747
\(520\) −30487.8 −2.57112
\(521\) −21521.7 −1.80976 −0.904879 0.425669i \(-0.860039\pi\)
−0.904879 + 0.425669i \(0.860039\pi\)
\(522\) −6275.63 −0.526201
\(523\) −2923.36 −0.244416 −0.122208 0.992504i \(-0.538998\pi\)
−0.122208 + 0.992504i \(0.538998\pi\)
\(524\) −13330.0 −1.11130
\(525\) 0 0
\(526\) 31977.0 2.65069
\(527\) −3497.60 −0.289104
\(528\) −7379.15 −0.608213
\(529\) −11851.7 −0.974088
\(530\) −48158.0 −3.94689
\(531\) 5205.99 0.425462
\(532\) 0 0
\(533\) −11429.0 −0.928792
\(534\) 381.464 0.0309130
\(535\) 19342.9 1.56312
\(536\) −62679.3 −5.05100
\(537\) −2495.93 −0.200572
\(538\) 13540.3 1.08507
\(539\) 0 0
\(540\) 9746.38 0.776699
\(541\) 21272.8 1.69056 0.845278 0.534327i \(-0.179435\pi\)
0.845278 + 0.534327i \(0.179435\pi\)
\(542\) 12203.2 0.967108
\(543\) 5431.89 0.429290
\(544\) 10049.9 0.792067
\(545\) −30805.1 −2.42118
\(546\) 0 0
\(547\) −18730.5 −1.46409 −0.732046 0.681256i \(-0.761434\pi\)
−0.732046 + 0.681256i \(0.761434\pi\)
\(548\) −40124.5 −3.12780
\(549\) 1994.12 0.155022
\(550\) −9480.41 −0.734992
\(551\) 1947.63 0.150584
\(552\) 3878.91 0.299089
\(553\) 0 0
\(554\) 22145.7 1.69834
\(555\) 4653.84 0.355936
\(556\) 20440.1 1.55909
\(557\) −18885.0 −1.43659 −0.718297 0.695736i \(-0.755078\pi\)
−0.718297 + 0.695736i \(0.755078\pi\)
\(558\) −10710.9 −0.812593
\(559\) −1614.30 −0.122143
\(560\) 0 0
\(561\) −526.086 −0.0395925
\(562\) −5607.71 −0.420902
\(563\) −10285.1 −0.769922 −0.384961 0.922933i \(-0.625785\pi\)
−0.384961 + 0.922933i \(0.625785\pi\)
\(564\) −31970.0 −2.38685
\(565\) 18978.5 1.41315
\(566\) 42360.1 3.14581
\(567\) 0 0
\(568\) 42302.5 3.12495
\(569\) 18008.8 1.32683 0.663415 0.748251i \(-0.269106\pi\)
0.663415 + 0.748251i \(0.269106\pi\)
\(570\) −4154.24 −0.305266
\(571\) −7010.79 −0.513822 −0.256911 0.966435i \(-0.582705\pi\)
−0.256911 + 0.966435i \(0.582705\pi\)
\(572\) 5856.10 0.428070
\(573\) 1376.69 0.100370
\(574\) 0 0
\(575\) 2821.10 0.204605
\(576\) 14676.1 1.06164
\(577\) 16398.9 1.18318 0.591589 0.806240i \(-0.298501\pi\)
0.591589 + 0.806240i \(0.298501\pi\)
\(578\) −25271.6 −1.81862
\(579\) 5336.73 0.383052
\(580\) −46402.4 −3.32199
\(581\) 0 0
\(582\) −11708.5 −0.833903
\(583\) 5796.11 0.411750
\(584\) −37146.6 −2.63208
\(585\) −3768.07 −0.266309
\(586\) 10573.3 0.745354
\(587\) −12823.5 −0.901671 −0.450836 0.892607i \(-0.648874\pi\)
−0.450836 + 0.892607i \(0.648874\pi\)
\(588\) 0 0
\(589\) 3324.09 0.232541
\(590\) 52867.0 3.68899
\(591\) 15913.6 1.10761
\(592\) 20587.9 1.42932
\(593\) −16899.5 −1.17029 −0.585144 0.810929i \(-0.698962\pi\)
−0.585144 + 0.810929i \(0.698962\pi\)
\(594\) −1611.06 −0.111284
\(595\) 0 0
\(596\) −43856.1 −3.01412
\(597\) 15415.2 1.05679
\(598\) −2393.31 −0.163662
\(599\) −15074.9 −1.02829 −0.514143 0.857704i \(-0.671890\pi\)
−0.514143 + 0.857704i \(0.671890\pi\)
\(600\) 34709.7 2.36170
\(601\) 11418.8 0.775014 0.387507 0.921867i \(-0.373336\pi\)
0.387507 + 0.921867i \(0.373336\pi\)
\(602\) 0 0
\(603\) −7746.71 −0.523168
\(604\) 10195.1 0.686809
\(605\) 2038.71 0.137001
\(606\) −30451.7 −2.04128
\(607\) 17952.8 1.20046 0.600232 0.799826i \(-0.295075\pi\)
0.600232 + 0.799826i \(0.295075\pi\)
\(608\) −9551.30 −0.637100
\(609\) 0 0
\(610\) 20250.4 1.34412
\(611\) 12360.0 0.818384
\(612\) 3073.94 0.203033
\(613\) 12528.9 0.825507 0.412753 0.910843i \(-0.364567\pi\)
0.412753 + 0.910843i \(0.364567\pi\)
\(614\) −12823.8 −0.842878
\(615\) 23248.5 1.52434
\(616\) 0 0
\(617\) −8586.10 −0.560232 −0.280116 0.959966i \(-0.590373\pi\)
−0.280116 + 0.959966i \(0.590373\pi\)
\(618\) 6977.09 0.454142
\(619\) −18415.4 −1.19576 −0.597882 0.801584i \(-0.703991\pi\)
−0.597882 + 0.801584i \(0.703991\pi\)
\(620\) −79196.7 −5.13003
\(621\) 479.404 0.0309788
\(622\) 10790.1 0.695569
\(623\) 0 0
\(624\) −16669.4 −1.06941
\(625\) −10241.4 −0.655448
\(626\) 21039.6 1.34331
\(627\) 499.988 0.0318462
\(628\) 13871.6 0.881427
\(629\) 1467.79 0.0930436
\(630\) 0 0
\(631\) 2374.38 0.149798 0.0748989 0.997191i \(-0.476137\pi\)
0.0748989 + 0.997191i \(0.476137\pi\)
\(632\) 75377.1 4.74421
\(633\) −12787.1 −0.802909
\(634\) 15805.3 0.990080
\(635\) 11142.2 0.696324
\(636\) −33866.8 −2.11149
\(637\) 0 0
\(638\) 7670.21 0.475966
\(639\) 5228.27 0.323673
\(640\) 64064.6 3.95684
\(641\) 11086.0 0.683104 0.341552 0.939863i \(-0.389047\pi\)
0.341552 + 0.939863i \(0.389047\pi\)
\(642\) 18682.1 1.14848
\(643\) −19934.1 −1.22259 −0.611294 0.791403i \(-0.709351\pi\)
−0.611294 + 0.791403i \(0.709351\pi\)
\(644\) 0 0
\(645\) 3283.75 0.200461
\(646\) −1310.21 −0.0797983
\(647\) 30634.8 1.86148 0.930739 0.365684i \(-0.119165\pi\)
0.930739 + 0.365684i \(0.119165\pi\)
\(648\) 5898.41 0.357579
\(649\) −6362.87 −0.384845
\(650\) −21416.1 −1.29232
\(651\) 0 0
\(652\) 23421.9 1.40686
\(653\) −9818.07 −0.588378 −0.294189 0.955747i \(-0.595049\pi\)
−0.294189 + 0.955747i \(0.595049\pi\)
\(654\) −29752.7 −1.77894
\(655\) −10483.1 −0.625358
\(656\) 102848. 6.12125
\(657\) −4591.04 −0.272624
\(658\) 0 0
\(659\) 16478.5 0.974070 0.487035 0.873383i \(-0.338078\pi\)
0.487035 + 0.873383i \(0.338078\pi\)
\(660\) −11912.2 −0.702551
\(661\) −2958.12 −0.174066 −0.0870328 0.996205i \(-0.527738\pi\)
−0.0870328 + 0.996205i \(0.527738\pi\)
\(662\) 43962.1 2.58102
\(663\) −1188.42 −0.0696146
\(664\) −44144.2 −2.58001
\(665\) 0 0
\(666\) 4494.86 0.261520
\(667\) −2282.44 −0.132498
\(668\) −24065.1 −1.39387
\(669\) 4133.39 0.238873
\(670\) −78668.2 −4.53614
\(671\) −2437.26 −0.140223
\(672\) 0 0
\(673\) −29960.3 −1.71602 −0.858012 0.513630i \(-0.828300\pi\)
−0.858012 + 0.513630i \(0.828300\pi\)
\(674\) 32108.2 1.83496
\(675\) 4289.87 0.244618
\(676\) −33840.6 −1.92539
\(677\) 4514.73 0.256300 0.128150 0.991755i \(-0.459096\pi\)
0.128150 + 0.991755i \(0.459096\pi\)
\(678\) 18330.2 1.03830
\(679\) 0 0
\(680\) 19559.7 1.10306
\(681\) 3681.84 0.207178
\(682\) 13091.0 0.735018
\(683\) −13555.7 −0.759438 −0.379719 0.925102i \(-0.623979\pi\)
−0.379719 + 0.925102i \(0.623979\pi\)
\(684\) −2921.44 −0.163310
\(685\) −31555.2 −1.76009
\(686\) 0 0
\(687\) −11670.8 −0.648137
\(688\) 14526.8 0.804986
\(689\) 13093.3 0.723972
\(690\) 4868.37 0.268603
\(691\) 11471.3 0.631535 0.315768 0.948837i \(-0.397738\pi\)
0.315768 + 0.948837i \(0.397738\pi\)
\(692\) −985.787 −0.0541532
\(693\) 0 0
\(694\) −46327.8 −2.53398
\(695\) 16074.8 0.877340
\(696\) −28082.2 −1.52939
\(697\) 7332.40 0.398471
\(698\) −5084.52 −0.275719
\(699\) −9654.41 −0.522408
\(700\) 0 0
\(701\) 22229.0 1.19769 0.598843 0.800866i \(-0.295627\pi\)
0.598843 + 0.800866i \(0.295627\pi\)
\(702\) −3639.35 −0.195667
\(703\) −1394.97 −0.0748397
\(704\) −17937.5 −0.960292
\(705\) −25142.3 −1.34314
\(706\) 1145.19 0.0610480
\(707\) 0 0
\(708\) 37178.4 1.97352
\(709\) −15081.2 −0.798851 −0.399426 0.916766i \(-0.630790\pi\)
−0.399426 + 0.916766i \(0.630790\pi\)
\(710\) 53093.4 2.80642
\(711\) 9316.06 0.491392
\(712\) 1706.98 0.0898480
\(713\) −3895.52 −0.204612
\(714\) 0 0
\(715\) 4605.42 0.240885
\(716\) −17824.6 −0.930358
\(717\) −1284.30 −0.0668938
\(718\) −7465.71 −0.388047
\(719\) 7399.80 0.383819 0.191910 0.981413i \(-0.438532\pi\)
0.191910 + 0.981413i \(0.438532\pi\)
\(720\) 33908.3 1.75512
\(721\) 0 0
\(722\) −35960.9 −1.85364
\(723\) −3693.48 −0.189989
\(724\) 38791.7 1.99127
\(725\) −20424.0 −1.04625
\(726\) 1969.07 0.100660
\(727\) −1705.77 −0.0870202 −0.0435101 0.999053i \(-0.513854\pi\)
−0.0435101 + 0.999053i \(0.513854\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) −46622.3 −2.36379
\(731\) 1035.67 0.0524017
\(732\) 14241.0 0.719074
\(733\) −37122.6 −1.87061 −0.935303 0.353847i \(-0.884873\pi\)
−0.935303 + 0.353847i \(0.884873\pi\)
\(734\) −5589.60 −0.281084
\(735\) 0 0
\(736\) 11193.2 0.560581
\(737\) 9468.20 0.473223
\(738\) 22454.3 1.11999
\(739\) 34256.3 1.70520 0.852598 0.522568i \(-0.175026\pi\)
0.852598 + 0.522568i \(0.175026\pi\)
\(740\) 33235.3 1.65102
\(741\) 1129.47 0.0559945
\(742\) 0 0
\(743\) 1567.88 0.0774160 0.0387080 0.999251i \(-0.487676\pi\)
0.0387080 + 0.999251i \(0.487676\pi\)
\(744\) −47929.0 −2.36178
\(745\) −34489.8 −1.69612
\(746\) 50801.9 2.49328
\(747\) −5455.90 −0.267230
\(748\) −3757.03 −0.183651
\(749\) 0 0
\(750\) 9290.53 0.452323
\(751\) −955.613 −0.0464325 −0.0232163 0.999730i \(-0.507391\pi\)
−0.0232163 + 0.999730i \(0.507391\pi\)
\(752\) −111226. −5.39360
\(753\) −8514.65 −0.412073
\(754\) 17326.9 0.836881
\(755\) 8017.75 0.386484
\(756\) 0 0
\(757\) −14015.4 −0.672918 −0.336459 0.941698i \(-0.609229\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(758\) −38623.2 −1.85073
\(759\) −585.938 −0.0280214
\(760\) −18589.4 −0.887249
\(761\) 36271.0 1.72776 0.863879 0.503699i \(-0.168028\pi\)
0.863879 + 0.503699i \(0.168028\pi\)
\(762\) 10761.6 0.511616
\(763\) 0 0
\(764\) 9831.63 0.465571
\(765\) 2417.44 0.114252
\(766\) 6311.64 0.297714
\(767\) −14373.6 −0.676665
\(768\) 22739.7 1.06842
\(769\) 18163.6 0.851749 0.425874 0.904782i \(-0.359967\pi\)
0.425874 + 0.904782i \(0.359967\pi\)
\(770\) 0 0
\(771\) −1026.02 −0.0479264
\(772\) 38112.1 1.77680
\(773\) 8345.65 0.388321 0.194160 0.980970i \(-0.437802\pi\)
0.194160 + 0.980970i \(0.437802\pi\)
\(774\) 3171.57 0.147287
\(775\) −34858.4 −1.61568
\(776\) −52393.2 −2.42372
\(777\) 0 0
\(778\) −59445.7 −2.73937
\(779\) −6968.65 −0.320510
\(780\) −26909.6 −1.23528
\(781\) −6390.11 −0.292774
\(782\) 1535.45 0.0702142
\(783\) −3470.76 −0.158410
\(784\) 0 0
\(785\) 10909.1 0.496001
\(786\) −10125.0 −0.459475
\(787\) 22996.2 1.04158 0.520791 0.853684i \(-0.325637\pi\)
0.520791 + 0.853684i \(0.325637\pi\)
\(788\) 113647. 5.13768
\(789\) 17685.0 0.797975
\(790\) 94605.0 4.26063
\(791\) 0 0
\(792\) −7209.17 −0.323443
\(793\) −5505.75 −0.246551
\(794\) 11782.4 0.526626
\(795\) −26634.0 −1.18819
\(796\) 110087. 4.90194
\(797\) 2743.82 0.121946 0.0609730 0.998139i \(-0.480580\pi\)
0.0609730 + 0.998139i \(0.480580\pi\)
\(798\) 0 0
\(799\) −7929.68 −0.351104
\(800\) 100161. 4.42652
\(801\) 210.970 0.00930619
\(802\) 42477.1 1.87022
\(803\) 5611.28 0.246597
\(804\) −55322.9 −2.42673
\(805\) 0 0
\(806\) 29572.5 1.29237
\(807\) 7488.53 0.326653
\(808\) −136266. −5.93293
\(809\) 41241.7 1.79231 0.896156 0.443738i \(-0.146348\pi\)
0.896156 + 0.443738i \(0.146348\pi\)
\(810\) 7403.03 0.321131
\(811\) −12832.9 −0.555641 −0.277820 0.960633i \(-0.589612\pi\)
−0.277820 + 0.960633i \(0.589612\pi\)
\(812\) 0 0
\(813\) 6749.03 0.291142
\(814\) −5493.72 −0.236554
\(815\) 18419.7 0.791675
\(816\) 10694.4 0.458798
\(817\) −984.292 −0.0421493
\(818\) 58210.4 2.48812
\(819\) 0 0
\(820\) 166029. 7.07069
\(821\) 16368.5 0.695817 0.347908 0.937529i \(-0.386892\pi\)
0.347908 + 0.937529i \(0.386892\pi\)
\(822\) −30477.3 −1.29321
\(823\) 3869.53 0.163892 0.0819461 0.996637i \(-0.473886\pi\)
0.0819461 + 0.996637i \(0.473886\pi\)
\(824\) 31221.2 1.31995
\(825\) −5243.17 −0.221265
\(826\) 0 0
\(827\) 7388.69 0.310677 0.155339 0.987861i \(-0.450353\pi\)
0.155339 + 0.987861i \(0.450353\pi\)
\(828\) 3423.65 0.143696
\(829\) −23990.1 −1.00508 −0.502539 0.864554i \(-0.667601\pi\)
−0.502539 + 0.864554i \(0.667601\pi\)
\(830\) −55404.9 −2.31703
\(831\) 12247.8 0.511276
\(832\) −40520.6 −1.68846
\(833\) 0 0
\(834\) 15525.7 0.644616
\(835\) −18925.6 −0.784367
\(836\) 3570.65 0.147720
\(837\) −5923.68 −0.244626
\(838\) 39684.5 1.63589
\(839\) 18228.3 0.750074 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(840\) 0 0
\(841\) −7864.78 −0.322472
\(842\) −67782.3 −2.77427
\(843\) −3101.36 −0.126710
\(844\) −91318.7 −3.72431
\(845\) −26613.3 −1.08346
\(846\) −24283.4 −0.986855
\(847\) 0 0
\(848\) −117825. −4.77137
\(849\) 23427.4 0.947028
\(850\) 13739.7 0.554433
\(851\) 1634.77 0.0658511
\(852\) 37337.6 1.50137
\(853\) 21737.3 0.872534 0.436267 0.899817i \(-0.356300\pi\)
0.436267 + 0.899817i \(0.356300\pi\)
\(854\) 0 0
\(855\) −2297.51 −0.0918987
\(856\) 83599.0 3.33803
\(857\) 18712.2 0.745852 0.372926 0.927861i \(-0.378355\pi\)
0.372926 + 0.927861i \(0.378355\pi\)
\(858\) 4448.10 0.176988
\(859\) −30527.6 −1.21256 −0.606279 0.795252i \(-0.707339\pi\)
−0.606279 + 0.795252i \(0.707339\pi\)
\(860\) 23450.8 0.929845
\(861\) 0 0
\(862\) 32953.4 1.30209
\(863\) 10906.4 0.430196 0.215098 0.976592i \(-0.430993\pi\)
0.215098 + 0.976592i \(0.430993\pi\)
\(864\) 17020.8 0.670209
\(865\) −775.255 −0.0304734
\(866\) −30603.5 −1.20087
\(867\) −13976.6 −0.547484
\(868\) 0 0
\(869\) −11386.3 −0.444481
\(870\) −35245.7 −1.37350
\(871\) 21388.5 0.832058
\(872\) −133138. −5.17043
\(873\) −6475.41 −0.251042
\(874\) −1459.28 −0.0564768
\(875\) 0 0
\(876\) −32786.8 −1.26457
\(877\) 21770.9 0.838256 0.419128 0.907927i \(-0.362336\pi\)
0.419128 + 0.907927i \(0.362336\pi\)
\(878\) −59109.8 −2.27205
\(879\) 5847.58 0.224384
\(880\) −41443.4 −1.58757
\(881\) −47206.9 −1.80527 −0.902634 0.430409i \(-0.858369\pi\)
−0.902634 + 0.430409i \(0.858369\pi\)
\(882\) 0 0
\(883\) −6059.68 −0.230945 −0.115473 0.993311i \(-0.536838\pi\)
−0.115473 + 0.993311i \(0.536838\pi\)
\(884\) −8487.09 −0.322909
\(885\) 29238.3 1.11055
\(886\) −41881.1 −1.58806
\(887\) 37130.2 1.40553 0.702767 0.711420i \(-0.251948\pi\)
0.702767 + 0.711420i \(0.251948\pi\)
\(888\) 20113.6 0.760101
\(889\) 0 0
\(890\) 2142.41 0.0806896
\(891\) −891.000 −0.0335013
\(892\) 29518.5 1.10802
\(893\) 7536.30 0.282410
\(894\) −33311.6 −1.24621
\(895\) −14017.8 −0.523536
\(896\) 0 0
\(897\) −1323.63 −0.0492694
\(898\) −40541.4 −1.50655
\(899\) 28202.5 1.04628
\(900\) 30636.0 1.13467
\(901\) −8400.15 −0.310599
\(902\) −27444.1 −1.01307
\(903\) 0 0
\(904\) 82024.1 3.01779
\(905\) 30507.0 1.12054
\(906\) 7743.85 0.283965
\(907\) 1182.94 0.0433064 0.0216532 0.999766i \(-0.493107\pi\)
0.0216532 + 0.999766i \(0.493107\pi\)
\(908\) 26293.8 0.961001
\(909\) −16841.4 −0.614516
\(910\) 0 0
\(911\) 37676.4 1.37022 0.685112 0.728438i \(-0.259753\pi\)
0.685112 + 0.728438i \(0.259753\pi\)
\(912\) −10163.9 −0.369035
\(913\) 6668.32 0.241719
\(914\) 60431.1 2.18696
\(915\) 11199.6 0.404641
\(916\) −83347.1 −3.00640
\(917\) 0 0
\(918\) 2334.86 0.0839454
\(919\) 8697.82 0.312203 0.156101 0.987741i \(-0.450107\pi\)
0.156101 + 0.987741i \(0.450107\pi\)
\(920\) 21785.0 0.780686
\(921\) −7092.26 −0.253744
\(922\) −77725.7 −2.77631
\(923\) −14435.2 −0.514778
\(924\) 0 0
\(925\) 14628.5 0.519981
\(926\) 63795.3 2.26398
\(927\) 3858.71 0.136717
\(928\) −81036.0 −2.86653
\(929\) 17247.5 0.609119 0.304559 0.952493i \(-0.401491\pi\)
0.304559 + 0.952493i \(0.401491\pi\)
\(930\) −60155.2 −2.12104
\(931\) 0 0
\(932\) −68946.7 −2.42320
\(933\) 5967.51 0.209397
\(934\) −64306.0 −2.25284
\(935\) −2954.65 −0.103345
\(936\) −16285.4 −0.568702
\(937\) 41812.4 1.45779 0.728896 0.684624i \(-0.240034\pi\)
0.728896 + 0.684624i \(0.240034\pi\)
\(938\) 0 0
\(939\) 11636.0 0.404395
\(940\) −179553. −6.23018
\(941\) −37655.9 −1.30451 −0.652257 0.757998i \(-0.726178\pi\)
−0.652257 + 0.757998i \(0.726178\pi\)
\(942\) 10536.4 0.364431
\(943\) 8166.60 0.282016
\(944\) 129346. 4.45959
\(945\) 0 0
\(946\) −3876.37 −0.133226
\(947\) −21244.4 −0.728986 −0.364493 0.931206i \(-0.618758\pi\)
−0.364493 + 0.931206i \(0.618758\pi\)
\(948\) 66530.4 2.27933
\(949\) 12675.8 0.433587
\(950\) −13058.1 −0.445958
\(951\) 8741.20 0.298058
\(952\) 0 0
\(953\) 1324.27 0.0450130 0.0225065 0.999747i \(-0.492835\pi\)
0.0225065 + 0.999747i \(0.492835\pi\)
\(954\) −25724.1 −0.873007
\(955\) 7731.91 0.261988
\(956\) −9171.77 −0.310289
\(957\) 4242.04 0.143287
\(958\) 7185.62 0.242335
\(959\) 0 0
\(960\) 82425.4 2.77111
\(961\) 18343.4 0.615735
\(962\) −12410.2 −0.415927
\(963\) 10332.2 0.345744
\(964\) −26376.9 −0.881268
\(965\) 29972.6 0.999847
\(966\) 0 0
\(967\) 52267.1 1.73815 0.869077 0.494676i \(-0.164713\pi\)
0.869077 + 0.494676i \(0.164713\pi\)
\(968\) 8811.20 0.292565
\(969\) −724.619 −0.0240228
\(970\) −65758.1 −2.17667
\(971\) 52489.8 1.73479 0.867394 0.497622i \(-0.165793\pi\)
0.867394 + 0.497622i \(0.165793\pi\)
\(972\) 5206.14 0.171797
\(973\) 0 0
\(974\) 101092. 3.32566
\(975\) −11844.3 −0.389046
\(976\) 49545.3 1.62490
\(977\) 8324.11 0.272581 0.136291 0.990669i \(-0.456482\pi\)
0.136291 + 0.990669i \(0.456482\pi\)
\(978\) 17790.5 0.581675
\(979\) −257.852 −0.00841777
\(980\) 0 0
\(981\) −16454.9 −0.535539
\(982\) 675.623 0.0219552
\(983\) 44407.1 1.44086 0.720431 0.693527i \(-0.243944\pi\)
0.720431 + 0.693527i \(0.243944\pi\)
\(984\) 100479. 3.25523
\(985\) 89375.3 2.89110
\(986\) −11116.2 −0.359039
\(987\) 0 0
\(988\) 8066.05 0.259732
\(989\) 1153.50 0.0370870
\(990\) −9048.15 −0.290474
\(991\) −45124.7 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(992\) −138307. −4.42667
\(993\) 24313.4 0.777001
\(994\) 0 0
\(995\) 86576.2 2.75844
\(996\) −38963.2 −1.23955
\(997\) −5480.61 −0.174095 −0.0870474 0.996204i \(-0.527743\pi\)
−0.0870474 + 0.996204i \(0.527743\pi\)
\(998\) 55492.9 1.76012
\(999\) 2485.90 0.0787291
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.k.1.2 2
7.6 odd 2 33.4.a.c.1.2 2
21.20 even 2 99.4.a.f.1.1 2
28.27 even 2 528.4.a.p.1.1 2
35.13 even 4 825.4.c.h.199.1 4
35.27 even 4 825.4.c.h.199.4 4
35.34 odd 2 825.4.a.l.1.1 2
56.13 odd 2 2112.4.a.bn.1.2 2
56.27 even 2 2112.4.a.bg.1.2 2
77.76 even 2 363.4.a.i.1.1 2
84.83 odd 2 1584.4.a.bj.1.2 2
105.104 even 2 2475.4.a.p.1.2 2
231.230 odd 2 1089.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 7.6 odd 2
99.4.a.f.1.1 2 21.20 even 2
363.4.a.i.1.1 2 77.76 even 2
528.4.a.p.1.1 2 28.27 even 2
825.4.a.l.1.1 2 35.34 odd 2
825.4.c.h.199.1 4 35.13 even 4
825.4.c.h.199.4 4 35.27 even 4
1089.4.a.u.1.2 2 231.230 odd 2
1584.4.a.bj.1.2 2 84.83 odd 2
1617.4.a.k.1.2 2 1.1 even 1 trivial
2112.4.a.bg.1.2 2 56.27 even 2
2112.4.a.bn.1.2 2 56.13 odd 2
2475.4.a.p.1.2 2 105.104 even 2