Properties

Label 2475.4.a.p.1.2
Level $2475$
Weight $4$
Character 2475.1
Self dual yes
Analytic conductor $146.030$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
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 \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.42443 q^{2} +21.4244 q^{4} +7.69772 q^{7} +72.8199 q^{8} +O(q^{10})\) \(q+5.42443 q^{2} +21.4244 q^{4} +7.69772 q^{7} +72.8199 q^{8} +11.0000 q^{11} -24.8489 q^{13} +41.7557 q^{14} +223.611 q^{16} -15.9420 q^{17} +15.1511 q^{19} +59.6687 q^{22} +17.7557 q^{23} -134.791 q^{26} +164.919 q^{28} +128.547 q^{29} +219.395 q^{31} +630.402 q^{32} -86.4763 q^{34} -92.0703 q^{37} +82.1863 q^{38} +459.942 q^{41} -64.9648 q^{43} +235.669 q^{44} +96.3146 q^{46} +497.408 q^{47} -283.745 q^{49} -532.373 q^{52} -526.919 q^{53} +560.547 q^{56} +697.292 q^{58} +578.443 q^{59} -221.569 q^{61} +1190.09 q^{62} +1630.68 q^{64} +860.745 q^{67} -341.548 q^{68} -580.919 q^{71} -510.116 q^{73} -499.429 q^{74} +324.605 q^{76} +84.6749 q^{77} +1035.12 q^{79} +2494.92 q^{82} +606.211 q^{83} -352.397 q^{86} +801.018 q^{88} +23.4411 q^{89} -191.279 q^{91} +380.406 q^{92} +2698.15 q^{94} -719.490 q^{97} -1539.16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8} + 22 q^{11} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} + 50 q^{19} + 11 q^{22} + 134 q^{23} - 112 q^{26} - 202 q^{28} + 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 328 q^{37} - 72 q^{38} + 782 q^{41} - 386 q^{43} + 363 q^{44} - 418 q^{46} + 266 q^{47} + 378 q^{49} - 592 q^{52} - 522 q^{53} + 1062 q^{56} + 390 q^{58} + 172 q^{59} - 778 q^{61} + 568 q^{62} + 809 q^{64} + 776 q^{67} + 1070 q^{68} - 630 q^{71} - 1296 q^{73} - 2358 q^{74} + 728 q^{76} - 264 q^{77} + 652 q^{79} + 1070 q^{82} - 324 q^{83} + 1068 q^{86} + 627 q^{88} + 756 q^{89} - 28 q^{91} + 1726 q^{92} + 3722 q^{94} + 452 q^{97} - 4467 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.42443 1.91783 0.958913 0.283702i \(-0.0915625\pi\)
0.958913 + 0.283702i \(0.0915625\pi\)
\(3\) 0 0
\(4\) 21.4244 2.67805
\(5\) 0 0
\(6\) 0 0
\(7\) 7.69772 0.415638 0.207819 0.978167i \(-0.433364\pi\)
0.207819 + 0.978167i \(0.433364\pi\)
\(8\) 72.8199 3.21821
\(9\) 0 0
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) −24.8489 −0.530141 −0.265071 0.964229i \(-0.585395\pi\)
−0.265071 + 0.964229i \(0.585395\pi\)
\(14\) 41.7557 0.797120
\(15\) 0 0
\(16\) 223.611 3.49392
\(17\) −15.9420 −0.227441 −0.113721 0.993513i \(-0.536277\pi\)
−0.113721 + 0.993513i \(0.536277\pi\)
\(18\) 0 0
\(19\) 15.1511 0.182943 0.0914713 0.995808i \(-0.470843\pi\)
0.0914713 + 0.995808i \(0.470843\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 59.6687 0.578246
\(23\) 17.7557 0.160971 0.0804853 0.996756i \(-0.474353\pi\)
0.0804853 + 0.996756i \(0.474353\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −134.791 −1.01672
\(27\) 0 0
\(28\) 164.919 1.11310
\(29\) 128.547 0.823121 0.411560 0.911383i \(-0.364984\pi\)
0.411560 + 0.911383i \(0.364984\pi\)
\(30\) 0 0
\(31\) 219.395 1.27112 0.635558 0.772053i \(-0.280770\pi\)
0.635558 + 0.772053i \(0.280770\pi\)
\(32\) 630.402 3.48251
\(33\) 0 0
\(34\) −86.4763 −0.436193
\(35\) 0 0
\(36\) 0 0
\(37\) −92.0703 −0.409088 −0.204544 0.978857i \(-0.565571\pi\)
−0.204544 + 0.978857i \(0.565571\pi\)
\(38\) 82.1863 0.350852
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.536750\pi\)
−0.115198 + 0.993343i \(0.536750\pi\)
\(44\) 235.669 0.807464
\(45\) 0 0
\(46\) 96.3146 0.308713
\(47\) 497.408 1.54371 0.771855 0.635799i \(-0.219329\pi\)
0.771855 + 0.635799i \(0.219329\pi\)
\(48\) 0 0
\(49\) −283.745 −0.827245
\(50\) 0 0
\(51\) 0 0
\(52\) −532.373 −1.41975
\(53\) −526.919 −1.36562 −0.682811 0.730596i \(-0.739243\pi\)
−0.682811 + 0.730596i \(0.739243\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 560.547 1.33761
\(57\) 0 0
\(58\) 697.292 1.57860
\(59\) 578.443 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(60\) 0 0
\(61\) −221.569 −0.465067 −0.232533 0.972588i \(-0.574701\pi\)
−0.232533 + 0.972588i \(0.574701\pi\)
\(62\) 1190.09 2.43778
\(63\) 0 0
\(64\) 1630.68 3.18493
\(65\) 0 0
\(66\) 0 0
\(67\) 860.745 1.56950 0.784752 0.619810i \(-0.212790\pi\)
0.784752 + 0.619810i \(0.212790\pi\)
\(68\) −341.548 −0.609100
\(69\) 0 0
\(70\) 0 0
\(71\) −580.919 −0.971020 −0.485510 0.874231i \(-0.661366\pi\)
−0.485510 + 0.874231i \(0.661366\pi\)
\(72\) 0 0
\(73\) −510.116 −0.817871 −0.408935 0.912563i \(-0.634100\pi\)
−0.408935 + 0.912563i \(0.634100\pi\)
\(74\) −499.429 −0.784560
\(75\) 0 0
\(76\) 324.605 0.489930
\(77\) 84.6749 0.125319
\(78\) 0 0
\(79\) 1035.12 1.47418 0.737088 0.675797i \(-0.236200\pi\)
0.737088 + 0.675797i \(0.236200\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2494.92 3.35998
\(83\) 606.211 0.801690 0.400845 0.916146i \(-0.368717\pi\)
0.400845 + 0.916146i \(0.368717\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −352.397 −0.441860
\(87\) 0 0
\(88\) 801.018 0.970328
\(89\) 23.4411 0.0279186 0.0139593 0.999903i \(-0.495556\pi\)
0.0139593 + 0.999903i \(0.495556\pi\)
\(90\) 0 0
\(91\) −191.279 −0.220347
\(92\) 380.406 0.431088
\(93\) 0 0
\(94\) 2698.15 2.96057
\(95\) 0 0
\(96\) 0 0
\(97\) −719.490 −0.753126 −0.376563 0.926391i \(-0.622894\pi\)
−0.376563 + 0.926391i \(0.622894\pi\)
\(98\) −1539.16 −1.58651
\(99\) 0 0
\(100\) 0 0
\(101\) −1871.27 −1.84355 −0.921774 0.387727i \(-0.873260\pi\)
−0.921774 + 0.387727i \(0.873260\pi\)
\(102\) 0 0
\(103\) 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 \(0.326448\pi\)
\(108\) 0 0
\(109\) −1828.32 −1.60662 −0.803308 0.595564i \(-0.796929\pi\)
−0.803308 + 0.595564i \(0.796929\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1721.29 1.45220
\(113\) 1126.40 0.937722 0.468861 0.883272i \(-0.344665\pi\)
0.468861 + 0.883272i \(0.344665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2754.04 2.20436
\(117\) 0 0
\(118\) 3137.72 2.44789
\(119\) −122.717 −0.0945332
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −1201.89 −0.891916
\(123\) 0 0
\(124\) 4700.42 3.40412
\(125\) 0 0
\(126\) 0 0
\(127\) −661.304 −0.462057 −0.231029 0.972947i \(-0.574209\pi\)
−0.231029 + 0.972947i \(0.574209\pi\)
\(128\) 3802.31 2.62562
\(129\) 0 0
\(130\) 0 0
\(131\) −622.186 −0.414967 −0.207483 0.978239i \(-0.566527\pi\)
−0.207483 + 0.978239i \(0.566527\pi\)
\(132\) 0 0
\(133\) 116.629 0.0760378
\(134\) 4669.05 3.01003
\(135\) 0 0
\(136\) −1160.89 −0.731955
\(137\) −1872.84 −1.16794 −0.583969 0.811776i \(-0.698501\pi\)
−0.583969 + 0.811776i \(0.698501\pi\)
\(138\) 0 0
\(139\) −954.058 −0.582174 −0.291087 0.956697i \(-0.594017\pi\)
−0.291087 + 0.956697i \(0.594017\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3151.15 −1.86225
\(143\) −273.337 −0.159844
\(144\) 0 0
\(145\) 0 0
\(146\) −2767.09 −1.56853
\(147\) 0 0
\(148\) −1972.55 −1.09556
\(149\) 2047.01 1.12549 0.562745 0.826631i \(-0.309745\pi\)
0.562745 + 0.826631i \(0.309745\pi\)
\(150\) 0 0
\(151\) 475.863 0.256458 0.128229 0.991745i \(-0.459071\pi\)
0.128229 + 0.991745i \(0.459071\pi\)
\(152\) 1103.30 0.588749
\(153\) 0 0
\(154\) 459.313 0.240341
\(155\) 0 0
\(156\) 0 0
\(157\) 647.466 0.329130 0.164565 0.986366i \(-0.447378\pi\)
0.164565 + 0.986366i \(0.447378\pi\)
\(158\) 5614.92 2.82721
\(159\) 0 0
\(160\) 0 0
\(161\) 136.678 0.0669054
\(162\) 0 0
\(163\) −1093.23 −0.525329 −0.262665 0.964887i \(-0.584601\pi\)
−0.262665 + 0.964887i \(0.584601\pi\)
\(164\) 9853.99 4.69188
\(165\) 0 0
\(166\) 3288.35 1.53750
\(167\) 1123.25 0.520479 0.260240 0.965544i \(-0.416198\pi\)
0.260240 + 0.965544i \(0.416198\pi\)
\(168\) 0 0
\(169\) −1579.53 −0.718951
\(170\) 0 0
\(171\) 0 0
\(172\) −1391.83 −0.617014
\(173\) 46.0123 0.0202211 0.0101106 0.999949i \(-0.496782\pi\)
0.0101106 + 0.999949i \(0.496782\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2459.72 1.05346
\(177\) 0 0
\(178\) 127.155 0.0535430
\(179\) 831.975 0.347401 0.173700 0.984799i \(-0.444428\pi\)
0.173700 + 0.984799i \(0.444428\pi\)
\(180\) 0 0
\(181\) −1810.63 −0.743553 −0.371776 0.928322i \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) −1037.58 −0.422586
\(183\) 0 0
\(184\) 1292.97 0.518037
\(185\) 0 0
\(186\) 0 0
\(187\) −175.362 −0.0685762
\(188\) 10656.7 4.13414
\(189\) 0 0
\(190\) 0 0
\(191\) −458.898 −0.173847 −0.0869233 0.996215i \(-0.527704\pi\)
−0.0869233 + 0.996215i \(0.527704\pi\)
\(192\) 0 0
\(193\) −1778.91 −0.663465 −0.331733 0.943373i \(-0.607633\pi\)
−0.331733 + 0.943373i \(0.607633\pi\)
\(194\) −3902.82 −1.44436
\(195\) 0 0
\(196\) −6079.08 −2.21541
\(197\) 5304.53 1.91844 0.959218 0.282666i \(-0.0912188\pi\)
0.959218 + 0.282666i \(0.0912188\pi\)
\(198\) 0 0
\(199\) −5138.40 −1.83041 −0.915205 0.402989i \(-0.867971\pi\)
−0.915205 + 0.402989i \(0.867971\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10150.6 −3.53560
\(203\) 989.515 0.342120
\(204\) 0 0
\(205\) 0 0
\(206\) 2325.70 0.786597
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 6227.38 1.98923
\(215\) 0 0
\(216\) 0 0
\(217\) 1688.84 0.528323
\(218\) −9917.58 −3.08121
\(219\) 0 0
\(220\) 0 0
\(221\) 396.141 0.120576
\(222\) 0 0
\(223\) 1377.80 0.413740 0.206870 0.978368i \(-0.433672\pi\)
0.206870 + 0.978368i \(0.433672\pi\)
\(224\) 4852.65 1.44746
\(225\) 0 0
\(226\) 6110.06 1.79839
\(227\) −1227.28 −0.358843 −0.179422 0.983772i \(-0.557423\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(228\) 0 0
\(229\) 3890.28 1.12261 0.561304 0.827610i \(-0.310300\pi\)
0.561304 + 0.827610i \(0.310300\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9360.74 2.64898
\(233\) −3218.14 −0.904837 −0.452419 0.891806i \(-0.649439\pi\)
−0.452419 + 0.891806i \(0.649439\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12392.8 3.41823
\(237\) 0 0
\(238\) −665.670 −0.181298
\(239\) 428.098 0.115864 0.0579318 0.998321i \(-0.481549\pi\)
0.0579318 + 0.998321i \(0.481549\pi\)
\(240\) 0 0
\(241\) 1231.16 0.329070 0.164535 0.986371i \(-0.447388\pi\)
0.164535 + 0.986371i \(0.447388\pi\)
\(242\) 656.356 0.174348
\(243\) 0 0
\(244\) −4747.00 −1.24547
\(245\) 0 0
\(246\) 0 0
\(247\) −376.489 −0.0969854
\(248\) 15976.3 4.09072
\(249\) 0 0
\(250\) 0 0
\(251\) −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
\(255\) 0 0
\(256\) 7579.90 1.85056
\(257\) 342.007 0.0830110 0.0415055 0.999138i \(-0.486785\pi\)
0.0415055 + 0.999138i \(0.486785\pi\)
\(258\) 0 0
\(259\) −708.731 −0.170032
\(260\) 0 0
\(261\) 0 0
\(262\) −3375.01 −0.795834
\(263\) 5895.00 1.38213 0.691067 0.722791i \(-0.257141\pi\)
0.691067 + 0.722791i \(0.257141\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 632.647 0.145827
\(267\) 0 0
\(268\) 18441.0 4.20322
\(269\) 2496.18 0.565779 0.282890 0.959152i \(-0.408707\pi\)
0.282890 + 0.959152i \(0.408707\pi\)
\(270\) 0 0
\(271\) −2249.68 −0.504274 −0.252137 0.967692i \(-0.581133\pi\)
−0.252137 + 0.967692i \(0.581133\pi\)
\(272\) −3564.80 −0.794662
\(273\) 0 0
\(274\) −10159.1 −2.23990
\(275\) 0 0
\(276\) 0 0
\(277\) −4082.59 −0.885556 −0.442778 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(278\) −5175.22 −1.11651
\(279\) 0 0
\(280\) 0 0
\(281\) 1033.79 0.219468 0.109734 0.993961i \(-0.465000\pi\)
0.109734 + 0.993961i \(0.465000\pi\)
\(282\) 0 0
\(283\) 7809.14 1.64030 0.820150 0.572148i \(-0.193890\pi\)
0.820150 + 0.572148i \(0.193890\pi\)
\(284\) −12445.9 −2.60044
\(285\) 0 0
\(286\) −1482.70 −0.306552
\(287\) 3540.50 0.728186
\(288\) 0 0
\(289\) −4658.85 −0.948270
\(290\) 0 0
\(291\) 0 0
\(292\) −10928.9 −2.19030
\(293\) −1949.19 −0.388645 −0.194323 0.980938i \(-0.562251\pi\)
−0.194323 + 0.980938i \(0.562251\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6704.55 −1.31653
\(297\) 0 0
\(298\) 11103.9 2.15849
\(299\) −441.209 −0.0853371
\(300\) 0 0
\(301\) −500.081 −0.0957614
\(302\) 2581.28 0.491842
\(303\) 0 0
\(304\) 3387.96 0.639187
\(305\) 0 0
\(306\) 0 0
\(307\) −2364.09 −0.439497 −0.219748 0.975557i \(-0.570524\pi\)
−0.219748 + 0.975557i \(0.570524\pi\)
\(308\) 1814.11 0.335612
\(309\) 0 0
\(310\) 0 0
\(311\) 1989.17 0.362686 0.181343 0.983420i \(-0.441956\pi\)
0.181343 + 0.983420i \(0.441956\pi\)
\(312\) 0 0
\(313\) 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 \(0.416895\pi\)
\(318\) 0 0
\(319\) 1414.01 0.248180
\(320\) 0 0
\(321\) 0 0
\(322\) 741.402 0.128313
\(323\) −241.540 −0.0416087
\(324\) 0 0
\(325\) 0 0
\(326\) −5930.17 −1.00749
\(327\) 0 0
\(328\) 33492.9 5.63822
\(329\) 3828.90 0.641624
\(330\) 0 0
\(331\) 8104.46 1.34580 0.672902 0.739731i \(-0.265047\pi\)
0.672902 + 0.739731i \(0.265047\pi\)
\(332\) 12987.7 2.14697
\(333\) 0 0
\(334\) 6093.02 0.998189
\(335\) 0 0
\(336\) 0 0
\(337\) −5919.19 −0.956792 −0.478396 0.878144i \(-0.658782\pi\)
−0.478396 + 0.878144i \(0.658782\pi\)
\(338\) −8568.07 −1.37882
\(339\) 0 0
\(340\) 0 0
\(341\) 2413.35 0.383256
\(342\) 0 0
\(343\) −4824.51 −0.759472
\(344\) −4730.73 −0.741465
\(345\) 0 0
\(346\) 249.590 0.0387805
\(347\) −8540.59 −1.32128 −0.660638 0.750705i \(-0.729714\pi\)
−0.660638 + 0.750705i \(0.729714\pi\)
\(348\) 0 0
\(349\) 937.337 0.143767 0.0718833 0.997413i \(-0.477099\pi\)
0.0718833 + 0.997413i \(0.477099\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6934.42 1.05002
\(353\) −211.118 −0.0318319 −0.0159160 0.999873i \(-0.505066\pi\)
−0.0159160 + 0.999873i \(0.505066\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 502.213 0.0747675
\(357\) 0 0
\(358\) 4512.99 0.666254
\(359\) 1376.31 0.202337 0.101169 0.994869i \(-0.467742\pi\)
0.101169 + 0.994869i \(0.467742\pi\)
\(360\) 0 0
\(361\) −6629.44 −0.966532
\(362\) −9821.63 −1.42600
\(363\) 0 0
\(364\) −4098.05 −0.590100
\(365\) 0 0
\(366\) 0 0
\(367\) −1030.45 −0.146564 −0.0732821 0.997311i \(-0.523347\pi\)
−0.0732821 + 0.997311i \(0.523347\pi\)
\(368\) 3970.37 0.562418
\(369\) 0 0
\(370\) 0 0
\(371\) −4056.07 −0.567603
\(372\) 0 0
\(373\) −9365.39 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(374\) −951.239 −0.131517
\(375\) 0 0
\(376\) 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 \(0.660274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2489.26 −0.333407
\(383\) −1163.56 −0.155235 −0.0776176 0.996983i \(-0.524731\pi\)
−0.0776176 + 0.996983i \(0.524731\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9649.57 −1.27241
\(387\) 0 0
\(388\) −15414.7 −2.01691
\(389\) 10958.9 1.42838 0.714188 0.699954i \(-0.246796\pi\)
0.714188 + 0.699954i \(0.246796\pi\)
\(390\) 0 0
\(391\) −283.062 −0.0366114
\(392\) −20662.3 −2.66225
\(393\) 0 0
\(394\) 28774.0 3.67923
\(395\) 0 0
\(396\) 0 0
\(397\) 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
\(400\) 0 0
\(401\) −7830.71 −0.975180 −0.487590 0.873073i \(-0.662124\pi\)
−0.487590 + 0.873073i \(0.662124\pi\)
\(402\) 0 0
\(403\) −5451.73 −0.673870
\(404\) −40090.9 −4.93712
\(405\) 0 0
\(406\) 5367.55 0.656126
\(407\) −1012.77 −0.123345
\(408\) 0 0
\(409\) −10731.2 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9185.62 1.09841
\(413\) 4452.69 0.530515
\(414\) 0 0
\(415\) 0 0
\(416\) −15664.8 −1.84622
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) −38370.2 −4.39486
\(425\) 0 0
\(426\) 0 0
\(427\) −1705.58 −0.193299
\(428\) 24595.8 2.77776
\(429\) 0 0
\(430\) 0 0
\(431\) −6075.01 −0.678939 −0.339470 0.940617i \(-0.610248\pi\)
−0.339470 + 0.940617i \(0.610248\pi\)
\(432\) 0 0
\(433\) −5641.79 −0.626160 −0.313080 0.949727i \(-0.601361\pi\)
−0.313080 + 0.949727i \(0.601361\pi\)
\(434\) 9161.01 1.01323
\(435\) 0 0
\(436\) −39170.7 −4.30260
\(437\) 269.019 0.0294484
\(438\) 0 0
\(439\) 10897.0 1.18470 0.592351 0.805680i \(-0.298200\pi\)
0.592351 + 0.805680i \(0.298200\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2148.84 0.231244
\(443\) −7720.83 −0.828054 −0.414027 0.910265i \(-0.635878\pi\)
−0.414027 + 0.910265i \(0.635878\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7473.76 0.793481
\(447\) 0 0
\(448\) 12552.5 1.32378
\(449\) 7473.86 0.785553 0.392776 0.919634i \(-0.371515\pi\)
0.392776 + 0.919634i \(0.371515\pi\)
\(450\) 0 0
\(451\) 5059.36 0.528240
\(452\) 24132.4 2.51127
\(453\) 0 0
\(454\) −6657.29 −0.688198
\(455\) 0 0
\(456\) 0 0
\(457\) −11140.5 −1.14033 −0.570167 0.821529i \(-0.693121\pi\)
−0.570167 + 0.821529i \(0.693121\pi\)
\(458\) 21102.6 2.15296
\(459\) 0 0
\(460\) 0 0
\(461\) −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 \(-0.700969\pi\)
−0.590246 + 0.807223i \(0.700969\pi\)
\(464\) 28744.4 2.87592
\(465\) 0 0
\(466\) −17456.5 −1.73532
\(467\) 11854.9 1.17469 0.587343 0.809338i \(-0.300174\pi\)
0.587343 + 0.809338i \(0.300174\pi\)
\(468\) 0 0
\(469\) 6625.77 0.652345
\(470\) 0 0
\(471\) 0 0
\(472\) 42122.1 4.10769
\(473\) −714.613 −0.0694671
\(474\) 0 0
\(475\) 0 0
\(476\) −2629.14 −0.253165
\(477\) 0 0
\(478\) 2322.19 0.222206
\(479\) 1324.68 0.126359 0.0631796 0.998002i \(-0.479876\pi\)
0.0631796 + 0.998002i \(0.479876\pi\)
\(480\) 0 0
\(481\) 2287.84 0.216874
\(482\) 6678.34 0.631100
\(483\) 0 0
\(484\) 2592.36 0.243459
\(485\) 0 0
\(486\) 0 0
\(487\) −18636.4 −1.73408 −0.867040 0.498239i \(-0.833980\pi\)
−0.867040 + 0.498239i \(0.833980\pi\)
\(488\) −16134.7 −1.49668
\(489\) 0 0
\(490\) 0 0
\(491\) −124.552 −0.0114480 −0.00572398 0.999984i \(-0.501822\pi\)
−0.00572398 + 0.999984i \(0.501822\pi\)
\(492\) 0 0
\(493\) −2049.29 −0.187212
\(494\) −2042.24 −0.186001
\(495\) 0 0
\(496\) 49059.2 4.44117
\(497\) −4471.75 −0.403592
\(498\) 0 0
\(499\) 10230.2 0.917768 0.458884 0.888496i \(-0.348249\pi\)
0.458884 + 0.888496i \(0.348249\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −15395.7 −1.36881
\(503\) 5150.81 0.456587 0.228294 0.973592i \(-0.426685\pi\)
0.228294 + 0.973592i \(0.426685\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1059.46 0.0930806
\(507\) 0 0
\(508\) −14168.1 −1.23741
\(509\) 22.7715 0.00198296 0.000991481 1.00000i \(-0.499684\pi\)
0.000991481 1.00000i \(0.499684\pi\)
\(510\) 0 0
\(511\) −3926.73 −0.339938
\(512\) 10698.1 0.923429
\(513\) 0 0
\(514\) 1855.19 0.159201
\(515\) 0 0
\(516\) 0 0
\(517\) 5471.49 0.465446
\(518\) −3844.46 −0.326093
\(519\) 0 0
\(520\) 0 0
\(521\) −21521.7 −1.80976 −0.904879 0.425669i \(-0.860039\pi\)
−0.904879 + 0.425669i \(0.860039\pi\)
\(522\) 0 0
\(523\) −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
\(528\) 0 0
\(529\) −11851.7 −0.974088
\(530\) 0 0
\(531\) 0 0
\(532\) 2498.71 0.203633
\(533\) −11429.0 −0.928792
\(534\) 0 0
\(535\) 0 0
\(536\) 62679.3 5.05100
\(537\) 0 0
\(538\) 13540.3 1.08507
\(539\) −3121.20 −0.249424
\(540\) 0 0
\(541\) 21272.8 1.69056 0.845278 0.534327i \(-0.179435\pi\)
0.845278 + 0.534327i \(0.179435\pi\)
\(542\) −12203.2 −0.967108
\(543\) 0 0
\(544\) −10049.9 −0.792067
\(545\) 0 0
\(546\) 0 0
\(547\) 18730.5 1.46409 0.732046 0.681256i \(-0.238566\pi\)
0.732046 + 0.681256i \(0.238566\pi\)
\(548\) −40124.5 −3.12780
\(549\) 0 0
\(550\) 0 0
\(551\) 1947.63 0.150584
\(552\) 0 0
\(553\) 7968.04 0.612723
\(554\) −22145.7 −1.69834
\(555\) 0 0
\(556\) −20440.1 −1.55909
\(557\) −18885.0 −1.43659 −0.718297 0.695736i \(-0.755078\pi\)
−0.718297 + 0.695736i \(0.755078\pi\)
\(558\) 0 0
\(559\) 1614.30 0.122143
\(560\) 0 0
\(561\) 0 0
\(562\) 5607.71 0.420902
\(563\) 10285.1 0.769922 0.384961 0.922933i \(-0.374215\pi\)
0.384961 + 0.922933i \(0.374215\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 42360.1 3.14581
\(567\) 0 0
\(568\) −42302.5 −3.12495
\(569\) −18008.8 −1.32683 −0.663415 0.748251i \(-0.730894\pi\)
−0.663415 + 0.748251i \(0.730894\pi\)
\(570\) 0 0
\(571\) −7010.79 −0.513822 −0.256911 0.966435i \(-0.582705\pi\)
−0.256911 + 0.966435i \(0.582705\pi\)
\(572\) −5856.10 −0.428070
\(573\) 0 0
\(574\) 19205.2 1.39653
\(575\) 0 0
\(576\) 0 0
\(577\) 16398.9 1.18318 0.591589 0.806240i \(-0.298501\pi\)
0.591589 + 0.806240i \(0.298501\pi\)
\(578\) −25271.6 −1.81862
\(579\) 0 0
\(580\) 0 0
\(581\) 4666.44 0.333213
\(582\) 0 0
\(583\) −5796.11 −0.411750
\(584\) −37146.6 −2.63208
\(585\) 0 0
\(586\) −10573.3 −0.745354
\(587\) 12823.5 0.901671 0.450836 0.892607i \(-0.351126\pi\)
0.450836 + 0.892607i \(0.351126\pi\)
\(588\) 0 0
\(589\) 3324.09 0.232541
\(590\) 0 0
\(591\) 0 0
\(592\) −20587.9 −1.42932
\(593\) 16899.5 1.17029 0.585144 0.810929i \(-0.301038\pi\)
0.585144 + 0.810929i \(0.301038\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 43856.1 3.01412
\(597\) 0 0
\(598\) −2393.31 −0.163662
\(599\) 15074.9 1.02829 0.514143 0.857704i \(-0.328110\pi\)
0.514143 + 0.857704i \(0.328110\pi\)
\(600\) 0 0
\(601\) −11418.8 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(602\) −2712.65 −0.183654
\(603\) 0 0
\(604\) 10195.1 0.686809
\(605\) 0 0
\(606\) 0 0
\(607\) 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
\(610\) 0 0
\(611\) −12360.0 −0.818384
\(612\) 0 0
\(613\) −12528.9 −0.825507 −0.412753 0.910843i \(-0.635433\pi\)
−0.412753 + 0.910843i \(0.635433\pi\)
\(614\) −12823.8 −0.842878
\(615\) 0 0
\(616\) 6166.01 0.403305
\(617\) −8586.10 −0.560232 −0.280116 0.959966i \(-0.590373\pi\)
−0.280116 + 0.959966i \(0.590373\pi\)
\(618\) 0 0
\(619\) 18415.4 1.19576 0.597882 0.801584i \(-0.296009\pi\)
0.597882 + 0.801584i \(0.296009\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 10790.1 0.695569
\(623\) 180.443 0.0116040
\(624\) 0 0
\(625\) 0 0
\(626\) 21039.6 1.34331
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 15805.3 0.990080
\(635\) 0 0
\(636\) 0 0
\(637\) 7050.74 0.438557
\(638\) 7670.21 0.475966
\(639\) 0 0
\(640\) 0 0
\(641\) −11086.0 −0.683104 −0.341552 0.939863i \(-0.610953\pi\)
−0.341552 + 0.939863i \(0.610953\pi\)
\(642\) 0 0
\(643\) −19934.1 −1.22259 −0.611294 0.791403i \(-0.709351\pi\)
−0.611294 + 0.791403i \(0.709351\pi\)
\(644\) 2928.26 0.179176
\(645\) 0 0
\(646\) −1310.21 −0.0797983
\(647\) −30634.8 −1.86148 −0.930739 0.365684i \(-0.880835\pi\)
−0.930739 + 0.365684i \(0.880835\pi\)
\(648\) 0 0
\(649\) 6362.87 0.384845
\(650\) 0 0
\(651\) 0 0
\(652\) −23421.9 −1.40686
\(653\) −9818.07 −0.588378 −0.294189 0.955747i \(-0.595049\pi\)
−0.294189 + 0.955747i \(0.595049\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 102848. 6.12125
\(657\) 0 0
\(658\) 20769.6 1.23052
\(659\) −16478.5 −0.974070 −0.487035 0.873383i \(-0.661922\pi\)
−0.487035 + 0.873383i \(0.661922\pi\)
\(660\) 0 0
\(661\) 2958.12 0.174066 0.0870328 0.996205i \(-0.472262\pi\)
0.0870328 + 0.996205i \(0.472262\pi\)
\(662\) 43962.1 2.58102
\(663\) 0 0
\(664\) 44144.2 2.58001
\(665\) 0 0
\(666\) 0 0
\(667\) 2282.44 0.132498
\(668\) 24065.1 1.39387
\(669\) 0 0
\(670\) 0 0
\(671\) −2437.26 −0.140223
\(672\) 0 0
\(673\) 29960.3 1.71602 0.858012 0.513630i \(-0.171700\pi\)
0.858012 + 0.513630i \(0.171700\pi\)
\(674\) −32108.2 −1.83496
\(675\) 0 0
\(676\) −33840.6 −1.92539
\(677\) −4514.73 −0.256300 −0.128150 0.991755i \(-0.540904\pi\)
−0.128150 + 0.991755i \(0.540904\pi\)
\(678\) 0 0
\(679\) −5538.43 −0.313027
\(680\) 0 0
\(681\) 0 0
\(682\) 13091.0 0.735018
\(683\) −13555.7 −0.759438 −0.379719 0.925102i \(-0.623979\pi\)
−0.379719 + 0.925102i \(0.623979\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −26170.2 −1.45653
\(687\) 0 0
\(688\) −14526.8 −0.804986
\(689\) 13093.3 0.723972
\(690\) 0 0
\(691\) −11471.3 −0.631535 −0.315768 0.948837i \(-0.602262\pi\)
−0.315768 + 0.948837i \(0.602262\pi\)
\(692\) 985.787 0.0541532
\(693\) 0 0
\(694\) −46327.8 −2.53398
\(695\) 0 0
\(696\) 0 0
\(697\) −7332.40 −0.398471
\(698\) 5084.52 0.275719
\(699\) 0 0
\(700\) 0 0
\(701\) −22229.0 −1.19769 −0.598843 0.800866i \(-0.704373\pi\)
−0.598843 + 0.800866i \(0.704373\pi\)
\(702\) 0 0
\(703\) −1394.97 −0.0748397
\(704\) 17937.5 0.960292
\(705\) 0 0
\(706\) −1145.19 −0.0610480
\(707\) −14404.5 −0.766248
\(708\) 0 0
\(709\) −15081.2 −0.798851 −0.399426 0.916766i \(-0.630790\pi\)
−0.399426 + 0.916766i \(0.630790\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1706.98 0.0898480
\(713\) 3895.52 0.204612
\(714\) 0 0
\(715\) 0 0
\(716\) 17824.6 0.930358
\(717\) 0 0
\(718\) 7465.71 0.388047
\(719\) 7399.80 0.383819 0.191910 0.981413i \(-0.438532\pi\)
0.191910 + 0.981413i \(0.438532\pi\)
\(720\) 0 0
\(721\) 3300.36 0.170474
\(722\) −35960.9 −1.85364
\(723\) 0 0
\(724\) −38791.7 −1.99127
\(725\) 0 0
\(726\) 0 0
\(727\) −1705.77 −0.0870202 −0.0435101 0.999053i \(-0.513854\pi\)
−0.0435101 + 0.999053i \(0.513854\pi\)
\(728\) −13928.9 −0.709122
\(729\) 0 0
\(730\) 0 0
\(731\) 1035.67 0.0524017
\(732\) 0 0
\(733\) −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
\(738\) 0 0
\(739\) 34256.3 1.70520 0.852598 0.522568i \(-0.175026\pi\)
0.852598 + 0.522568i \(0.175026\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −22001.9 −1.08856
\(743\) 1567.88 0.0774160 0.0387080 0.999251i \(-0.487676\pi\)
0.0387080 + 0.999251i \(0.487676\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −50801.9 −2.49328
\(747\) 0 0
\(748\) −3757.03 −0.183651
\(749\) 8837.17 0.431112
\(750\) 0 0
\(751\) −955.613 −0.0464325 −0.0232163 0.999730i \(-0.507391\pi\)
−0.0232163 + 0.999730i \(0.507391\pi\)
\(752\) 111226. 5.39360
\(753\) 0 0
\(754\) −17326.9 −0.836881
\(755\) 0 0
\(756\) 0 0
\(757\) 14015.4 0.672918 0.336459 0.941698i \(-0.390771\pi\)
0.336459 + 0.941698i \(0.390771\pi\)
\(758\) −38623.2 −1.85073
\(759\) 0 0
\(760\) 0 0
\(761\) 36271.0 1.72776 0.863879 0.503699i \(-0.168028\pi\)
0.863879 + 0.503699i \(0.168028\pi\)
\(762\) 0 0
\(763\) −14073.9 −0.667770
\(764\) −9831.63 −0.465571
\(765\) 0 0
\(766\) −6311.64 −0.297714
\(767\) −14373.6 −0.676665
\(768\) 0 0
\(769\) −18163.6 −0.851749 −0.425874 0.904782i \(-0.640033\pi\)
−0.425874 + 0.904782i \(0.640033\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −38112.1 −1.77680
\(773\) −8345.65 −0.388321 −0.194160 0.980970i \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −52393.2 −2.42372
\(777\) 0 0
\(778\) 59445.7 2.73937
\(779\) 6968.65 0.320510
\(780\) 0 0
\(781\) −6390.11 −0.292774
\(782\) −1535.45 −0.0702142
\(783\) 0 0
\(784\) −63448.5 −2.89033
\(785\) 0 0
\(786\) 0 0
\(787\) 22996.2 1.04158 0.520791 0.853684i \(-0.325637\pi\)
0.520791 + 0.853684i \(0.325637\pi\)
\(788\) 113647. 5.13768
\(789\) 0 0
\(790\) 0 0
\(791\) 8670.69 0.389752
\(792\) 0 0
\(793\) 5505.75 0.246551
\(794\) 11782.4 0.526626
\(795\) 0 0
\(796\) −110087. −4.90194
\(797\) −2743.82 −0.121946 −0.0609730 0.998139i \(-0.519420\pi\)
−0.0609730 + 0.998139i \(0.519420\pi\)
\(798\) 0 0
\(799\) −7929.68 −0.351104
\(800\) 0 0
\(801\) 0 0
\(802\) −42477.1 −1.87022
\(803\) −5611.28 −0.246597
\(804\) 0 0
\(805\) 0 0
\(806\) −29572.5 −1.29237
\(807\) 0 0
\(808\) −136266. −5.93293
\(809\) −41241.7 −1.79231 −0.896156 0.443738i \(-0.853652\pi\)
−0.896156 + 0.443738i \(0.853652\pi\)
\(810\) 0 0
\(811\) 12832.9 0.555641 0.277820 0.960633i \(-0.410388\pi\)
0.277820 + 0.960633i \(0.410388\pi\)
\(812\) 21199.8 0.916215
\(813\) 0 0
\(814\) −5493.72 −0.236554
\(815\) 0 0
\(816\) 0 0
\(817\) −984.292 −0.0421493
\(818\) −58210.4 −2.48812
\(819\) 0 0
\(820\) 0 0
\(821\) −16368.5 −0.695817 −0.347908 0.937529i \(-0.613108\pi\)
−0.347908 + 0.937529i \(0.613108\pi\)
\(822\) 0 0
\(823\) −3869.53 −0.163892 −0.0819461 0.996637i \(-0.526114\pi\)
−0.0819461 + 0.996637i \(0.526114\pi\)
\(824\) 31221.2 1.31995
\(825\) 0 0
\(826\) 24153.3 1.01743
\(827\) 7388.69 0.310677 0.155339 0.987861i \(-0.450353\pi\)
0.155339 + 0.987861i \(0.450353\pi\)
\(828\) 0 0
\(829\) 23990.1 1.00508 0.502539 0.864554i \(-0.332399\pi\)
0.502539 + 0.864554i \(0.332399\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −40520.6 −1.68846
\(833\) 4523.47 0.188150
\(834\) 0 0
\(835\) 0 0
\(836\) 3570.65 0.147720
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) −91318.7 −3.72431
\(845\) 0 0
\(846\) 0 0
\(847\) 931.424 0.0377852
\(848\) −117825. −4.77137
\(849\) 0 0
\(850\) 0 0
\(851\) −1634.77 −0.0658511
\(852\) 0 0
\(853\) 21737.3 0.872534 0.436267 0.899817i \(-0.356300\pi\)
0.436267 + 0.899817i \(0.356300\pi\)
\(854\) −9251.79 −0.370714
\(855\) 0 0
\(856\) 83599.0 3.33803
\(857\) −18712.2 −0.745852 −0.372926 0.927861i \(-0.621645\pi\)
−0.372926 + 0.927861i \(0.621645\pi\)
\(858\) 0 0
\(859\) 30527.6 1.21256 0.606279 0.795252i \(-0.292661\pi\)
0.606279 + 0.795252i \(0.292661\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −32953.4 −1.30209
\(863\) 10906.4 0.430196 0.215098 0.976592i \(-0.430993\pi\)
0.215098 + 0.976592i \(0.430993\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −30603.5 −1.20087
\(867\) 0 0
\(868\) 36182.5 1.41488
\(869\) 11386.3 0.444481
\(870\) 0 0
\(871\) −21388.5 −0.832058
\(872\) −133138. −5.17043
\(873\) 0 0
\(874\) 1459.28 0.0564768
\(875\) 0 0
\(876\) 0 0
\(877\) −21770.9 −0.838256 −0.419128 0.907927i \(-0.637664\pi\)
−0.419128 + 0.907927i \(0.637664\pi\)
\(878\) 59109.8 2.27205
\(879\) 0 0
\(880\) 0 0
\(881\) −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 \(-0.463162\pi\)
0.115473 + 0.993311i \(0.463162\pi\)
\(884\) 8487.09 0.322909
\(885\) 0 0
\(886\) −41881.1 −1.58806
\(887\) −37130.2 −1.40553 −0.702767 0.711420i \(-0.748052\pi\)
−0.702767 + 0.711420i \(0.748052\pi\)
\(888\) 0 0
\(889\) −5090.53 −0.192048
\(890\) 0 0
\(891\) 0 0
\(892\) 29518.5 1.10802
\(893\) 7536.30 0.282410
\(894\) 0 0
\(895\) 0 0
\(896\) 29269.1 1.09131
\(897\) 0 0
\(898\) 40541.4 1.50655
\(899\) 28202.5 1.04628
\(900\) 0 0
\(901\) 8400.15 0.310599
\(902\) 27444.1 1.01307
\(903\) 0 0
\(904\) 82024.1 3.01779
\(905\) 0 0
\(906\) 0 0
\(907\) −1182.94 −0.0433064 −0.0216532 0.999766i \(-0.506893\pi\)
−0.0216532 + 0.999766i \(0.506893\pi\)
\(908\) −26293.8 −0.961001
\(909\) 0 0
\(910\) 0 0
\(911\) −37676.4 −1.37022 −0.685112 0.728438i \(-0.740247\pi\)
−0.685112 + 0.728438i \(0.740247\pi\)
\(912\) 0 0
\(913\) 6668.32 0.241719
\(914\) −60431.1 −2.18696
\(915\) 0 0
\(916\) 83347.1 3.00640
\(917\) −4789.41 −0.172476
\(918\) 0 0
\(919\) 8697.82 0.312203 0.156101 0.987741i \(-0.450107\pi\)
0.156101 + 0.987741i \(0.450107\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −77725.7 −2.77631
\(923\) 14435.2 0.514778
\(924\) 0 0
\(925\) 0 0
\(926\) −63795.3 −2.26398
\(927\) 0 0
\(928\) 81036.0 2.86653
\(929\) 17247.5 0.609119 0.304559 0.952493i \(-0.401491\pi\)
0.304559 + 0.952493i \(0.401491\pi\)
\(930\) 0 0
\(931\) −4299.06 −0.151338
\(932\) −68946.7 −2.42320
\(933\) 0 0
\(934\) 64306.0 2.25284
\(935\) 0 0
\(936\) 0 0
\(937\) 41812.4 1.45779 0.728896 0.684624i \(-0.240034\pi\)
0.728896 + 0.684624i \(0.240034\pi\)
\(938\) 35941.0 1.25108
\(939\) 0 0
\(940\) 0 0
\(941\) −37655.9 −1.30451 −0.652257 0.757998i \(-0.726178\pi\)
−0.652257 + 0.757998i \(0.726178\pi\)
\(942\) 0 0
\(943\) 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 \(0.618758\pi\)
\(948\) 0 0
\(949\) 12675.8 0.433587
\(950\) 0 0
\(951\) 0 0
\(952\) −8936.24 −0.304228
\(953\) 1324.27 0.0450130 0.0225065 0.999747i \(-0.492835\pi\)
0.0225065 + 0.999747i \(0.492835\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 9171.77 0.310289
\(957\) 0 0
\(958\) 7185.62 0.242335
\(959\) −14416.6 −0.485439
\(960\) 0 0
\(961\) 18343.4 0.615735
\(962\) 12410.2 0.415927
\(963\) 0 0
\(964\) 26376.9 0.881268
\(965\) 0 0
\(966\) 0 0
\(967\) −52267.1 −1.73815 −0.869077 0.494676i \(-0.835287\pi\)
−0.869077 + 0.494676i \(0.835287\pi\)
\(968\) 8811.20 0.292565
\(969\) 0 0
\(970\) 0 0
\(971\) 52489.8 1.73479 0.867394 0.497622i \(-0.165793\pi\)
0.867394 + 0.497622i \(0.165793\pi\)
\(972\) 0 0
\(973\) −7344.07 −0.241973
\(974\) −101092. −3.32566
\(975\) 0 0
\(976\) −49545.3 −1.62490
\(977\) 8324.11 0.272581 0.136291 0.990669i \(-0.456482\pi\)
0.136291 + 0.990669i \(0.456482\pi\)
\(978\) 0 0
\(979\) 257.852 0.00841777
\(980\) 0 0
\(981\) 0 0
\(982\) −675.623 −0.0219552
\(983\) −44407.1 −1.44086 −0.720431 0.693527i \(-0.756056\pi\)
−0.720431 + 0.693527i \(0.756056\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −11116.2 −0.359039
\(987\) 0 0
\(988\) −8066.05 −0.259732
\(989\) −1153.50 −0.0370870
\(990\) 0 0
\(991\) −45124.7 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(992\) 138307. 4.42667
\(993\) 0 0
\(994\) −24256.7 −0.774020
\(995\) 0 0
\(996\) 0 0
\(997\) −5480.61 −0.174095 −0.0870474 0.996204i \(-0.527743\pi\)
−0.0870474 + 0.996204i \(0.527743\pi\)
\(998\) 55492.9 1.76012
\(999\) 0 0
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 2475.4.a.p.1.2 2
3.2 odd 2 825.4.a.l.1.1 2
5.4 even 2 99.4.a.f.1.1 2
15.2 even 4 825.4.c.h.199.1 4
15.8 even 4 825.4.c.h.199.4 4
15.14 odd 2 33.4.a.c.1.2 2
20.19 odd 2 1584.4.a.bj.1.2 2
55.54 odd 2 1089.4.a.u.1.2 2
60.59 even 2 528.4.a.p.1.1 2
105.104 even 2 1617.4.a.k.1.2 2
120.29 odd 2 2112.4.a.bn.1.2 2
120.59 even 2 2112.4.a.bg.1.2 2
165.164 even 2 363.4.a.i.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 15.14 odd 2
99.4.a.f.1.1 2 5.4 even 2
363.4.a.i.1.1 2 165.164 even 2
528.4.a.p.1.1 2 60.59 even 2
825.4.a.l.1.1 2 3.2 odd 2
825.4.c.h.199.1 4 15.2 even 4
825.4.c.h.199.4 4 15.8 even 4
1089.4.a.u.1.2 2 55.54 odd 2
1584.4.a.bj.1.2 2 20.19 odd 2
1617.4.a.k.1.2 2 105.104 even 2
2112.4.a.bg.1.2 2 120.59 even 2
2112.4.a.bn.1.2 2 120.29 odd 2
2475.4.a.p.1.2 2 1.1 even 1 trivial