Properties

Label 363.4.a.i.1.1
Level $363$
Weight $4$
Character 363.1
Self dual yes
Analytic conductor $21.418$
Analytic rank $1$
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] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.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: \(21.4176933321\)
Analytic rank: \(1\)
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 1.1
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} +16.2733 q^{6} +7.69772 q^{7} -72.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-5.42443 q^{2} -3.00000 q^{3} +21.4244 q^{4} -16.8489 q^{5} +16.2733 q^{6} +7.69772 q^{7} -72.8199 q^{8} +9.00000 q^{9} +91.3954 q^{10} -64.2733 q^{12} -24.8489 q^{13} -41.7557 q^{14} +50.5466 q^{15} +223.611 q^{16} +15.9420 q^{17} -48.8199 q^{18} -15.1511 q^{19} -360.977 q^{20} -23.0931 q^{21} +17.7557 q^{23} +218.460 q^{24} +158.884 q^{25} +134.791 q^{26} -27.0000 q^{27} +164.919 q^{28} +128.547 q^{29} -274.186 q^{30} +219.395 q^{31} -630.402 q^{32} -86.4763 q^{34} -129.698 q^{35} +192.820 q^{36} +92.0703 q^{37} +82.1863 q^{38} +74.5466 q^{39} +1226.93 q^{40} +459.942 q^{41} +125.267 q^{42} -64.9648 q^{43} -151.640 q^{45} -96.3146 q^{46} +497.408 q^{47} -670.832 q^{48} -283.745 q^{49} -861.855 q^{50} -47.8260 q^{51} -532.373 q^{52} -526.919 q^{53} +146.460 q^{54} -560.547 q^{56} +45.4534 q^{57} -697.292 q^{58} -578.443 q^{59} +1082.93 q^{60} +221.569 q^{61} -1190.09 q^{62} +69.2794 q^{63} +1630.68 q^{64} +418.675 q^{65} -860.745 q^{67} +341.548 q^{68} -53.2671 q^{69} +703.536 q^{70} +580.919 q^{71} -655.379 q^{72} -510.116 q^{73} -499.429 q^{74} -476.652 q^{75} -324.605 q^{76} -404.373 q^{78} -1035.12 q^{79} -3767.59 q^{80} +81.0000 q^{81} -2494.92 q^{82} -606.211 q^{83} -494.757 q^{84} -268.605 q^{85} +352.397 q^{86} -385.640 q^{87} -23.4411 q^{89} +822.559 q^{90} -191.279 q^{91} +380.406 q^{92} -658.186 q^{93} -2698.15 q^{94} +255.279 q^{95} +1891.20 q^{96} +719.490 q^{97} +1539.16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} + 3 q^{6} - 24 q^{7} - 57 q^{8} + 18 q^{9} + 104 q^{10} - 99 q^{12} - 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} - 106 q^{17} - 9 q^{18} - 50 q^{19} - 328 q^{20} + 72 q^{21} + 134 q^{23} + 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} - 202 q^{28} + 198 q^{29} - 312 q^{30} + 360 q^{31} - 857 q^{32} - 626 q^{34} - 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} + 90 q^{39} + 1272 q^{40} + 782 q^{41} + 546 q^{42} - 386 q^{43} - 126 q^{45} + 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} - 1379 q^{50} + 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 1062 q^{56} + 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} - 216 q^{63} + 809 q^{64} + 404 q^{65} - 776 q^{67} - 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} - 513 q^{72} - 1296 q^{73} - 2358 q^{74} - 126 q^{75} - 728 q^{76} - 336 q^{78} - 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} + 324 q^{83} + 606 q^{84} - 616 q^{85} - 1068 q^{86} - 594 q^{87} - 756 q^{89} + 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} + 156 q^{95} + 2571 q^{96} - 452 q^{97} + 4467 q^{98}+O(q^{100}) \) 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 \(-0.908437\pi\)
−0.958913 + 0.283702i \(0.908437\pi\)
\(3\) −3.00000 −0.577350
\(4\) 21.4244 2.67805
\(5\) −16.8489 −1.50701 −0.753504 0.657444i \(-0.771638\pi\)
−0.753504 + 0.657444i \(0.771638\pi\)
\(6\) 16.2733 1.10726
\(7\) 7.69772 0.415638 0.207819 0.978167i \(-0.433364\pi\)
0.207819 + 0.978167i \(0.433364\pi\)
\(8\) −72.8199 −3.21821
\(9\) 9.00000 0.333333
\(10\) 91.3954 2.89018
\(11\) 0 0
\(12\) −64.2733 −1.54617
\(13\) −24.8489 −0.530141 −0.265071 0.964229i \(-0.585395\pi\)
−0.265071 + 0.964229i \(0.585395\pi\)
\(14\) −41.7557 −0.797120
\(15\) 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
\(21\) −23.0931 −0.239968
\(22\) 0 0
\(23\) 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
\(28\) 164.919 1.11310
\(29\) 128.547 0.823121 0.411560 0.911383i \(-0.364984\pi\)
0.411560 + 0.911383i \(0.364984\pi\)
\(30\) −274.186 −1.66864
\(31\) 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
\(35\) −129.698 −0.626369
\(36\) 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 \(0.160212\pi\)
\(42\) 125.267 0.460218
\(43\) −64.9648 −0.230396 −0.115198 0.993343i \(-0.536750\pi\)
−0.115198 + 0.993343i \(0.536750\pi\)
\(44\) 0 0
\(45\) −151.640 −0.502336
\(46\) −96.3146 −0.308713
\(47\) 497.408 1.54371 0.771855 0.635799i \(-0.219329\pi\)
0.771855 + 0.635799i \(0.219329\pi\)
\(48\) −670.832 −2.01721
\(49\) −283.745 −0.827245
\(50\) −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
\(55\) 0 0
\(56\) −560.547 −1.33761
\(57\) 45.4534 0.105622
\(58\) −697.292 −1.57860
\(59\) −578.443 −1.27639 −0.638194 0.769876i \(-0.720318\pi\)
−0.638194 + 0.769876i \(0.720318\pi\)
\(60\) 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
\(63\) 69.2794 0.138546
\(64\) 1630.68 3.18493
\(65\) 418.675 0.798927
\(66\) 0 0
\(67\) −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
\(70\) 703.536 1.20127
\(71\) 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 \(-0.763800\pi\)
−0.737088 + 0.675797i \(0.763800\pi\)
\(80\) −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 \(0.631283\pi\)
\(84\) −494.757 −0.642648
\(85\) −268.605 −0.342756
\(86\) 352.397 0.441860
\(87\) −385.640 −0.475229
\(88\) 0 0
\(89\) −23.4411 −0.0279186 −0.0139593 0.999903i \(-0.504444\pi\)
−0.0139593 + 0.999903i \(0.504444\pi\)
\(90\) 822.559 0.963392
\(91\) −191.279 −0.220347
\(92\) 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 \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(98\) 1539.16 1.58651
\(99\) 0 0
\(100\) 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 \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(104\) 1809.49 1.70611
\(105\) 389.093 0.361634
\(106\) 2858.24 2.61902
\(107\) −1148.02 −1.03723 −0.518616 0.855008i \(-0.673552\pi\)
−0.518616 + 0.855008i \(0.673552\pi\)
\(108\) −578.460 −0.515392
\(109\) 1828.32 1.60662 0.803308 0.595564i \(-0.203071\pi\)
0.803308 + 0.595564i \(0.203071\pi\)
\(110\) 0 0
\(111\) −276.211 −0.236187
\(112\) 1721.29 1.45220
\(113\) 1126.40 0.937722 0.468861 0.883272i \(-0.344665\pi\)
0.468861 + 0.883272i \(0.344665\pi\)
\(114\) −246.559 −0.202565
\(115\) −299.163 −0.242584
\(116\) 2754.04 2.20436
\(117\) −223.640 −0.176714
\(118\) 3137.72 2.44789
\(119\) 122.717 0.0945332
\(120\) −3680.79 −2.80008
\(121\) 0 0
\(122\) −1201.89 −0.891916
\(123\) −1379.83 −1.01150
\(124\) 4700.42 3.40412
\(125\) −570.907 −0.408508
\(126\) −375.801 −0.265707
\(127\) −661.304 −0.462057 −0.231029 0.972947i \(-0.574209\pi\)
−0.231029 + 0.972947i \(0.574209\pi\)
\(128\) −3802.31 −2.62562
\(129\) 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 \(0.566527\pi\)
\(132\) 0 0
\(133\) −116.629 −0.0760378
\(134\) 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 \(0.405983\pi\)
\(140\) −2778.70 −1.67745
\(141\) −1492.22 −0.891261
\(142\) −3151.15 −1.86225
\(143\) 0 0
\(144\) 2012.50 1.16464
\(145\) −2165.86 −1.24045
\(146\) 2767.09 1.56853
\(147\) 851.236 0.477610
\(148\) 1972.55 1.09556
\(149\) 2047.01 1.12549 0.562745 0.826631i \(-0.309745\pi\)
0.562745 + 0.826631i \(0.309745\pi\)
\(150\) 2585.57 1.40740
\(151\) −475.863 −0.256458 −0.128229 0.991745i \(-0.540929\pi\)
−0.128229 + 0.991745i \(0.540929\pi\)
\(152\) 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 \(-0.552622\pi\)
−0.164565 + 0.986366i \(0.552622\pi\)
\(158\) 5614.92 2.82721
\(159\) 1580.76 0.788442
\(160\) 10621.5 5.24817
\(161\) 136.678 0.0669054
\(162\) −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
\(165\) 0 0
\(166\) 3288.35 1.53750
\(167\) −1123.25 −0.520479 −0.260240 0.965544i \(-0.583802\pi\)
−0.260240 + 0.965544i \(0.583802\pi\)
\(168\) 1681.64 0.772270
\(169\) −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
\(175\) 1223.04 0.528305
\(176\) 0 0
\(177\) 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 \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) 1037.58 0.422586
\(183\) −664.708 −0.268506
\(184\) −1292.97 −0.518037
\(185\) −1551.28 −0.616499
\(186\) 3570.28 1.40745
\(187\) 0 0
\(188\) 10656.7 4.13414
\(189\) −207.838 −0.0799895
\(190\) −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 \(-0.607633\pi\)
−0.331733 + 0.943373i \(0.607633\pi\)
\(194\) −3902.82 −1.44436
\(195\) −1256.02 −0.461260
\(196\) −6079.08 −2.21541
\(197\) −5304.53 −1.91844 −0.959218 0.282666i \(-0.908781\pi\)
−0.959218 + 0.282666i \(0.908781\pi\)
\(198\) 0 0
\(199\) −5138.40 −1.83041 −0.915205 0.402989i \(-0.867971\pi\)
−0.915205 + 0.402989i \(0.867971\pi\)
\(200\) −11569.9 −4.09058
\(201\) 2582.24 0.906153
\(202\) 10150.6 3.53560
\(203\) 989.515 0.342120
\(204\) −1024.65 −0.351664
\(205\) −7749.50 −2.64024
\(206\) 2325.70 0.786597
\(207\) 159.801 0.0536568
\(208\) −5556.47 −1.85227
\(209\) 0 0
\(210\) −2110.61 −0.693551
\(211\) 4262.36 1.39068 0.695339 0.718682i \(-0.255254\pi\)
0.695339 + 0.718682i \(0.255254\pi\)
\(212\) −11288.9 −3.65721
\(213\) −1742.76 −0.560619
\(214\) 6227.38 1.98923
\(215\) 1094.58 0.347209
\(216\) 1966.14 0.619345
\(217\) 1688.84 0.528323
\(218\) −9917.58 −3.08121
\(219\) 1530.35 0.472198
\(220\) 0 0
\(221\) −396.141 −0.120576
\(222\) 1498.29 0.452966
\(223\) −1377.80 −0.413740 −0.206870 0.978368i \(-0.566328\pi\)
−0.206870 + 0.978368i \(0.566328\pi\)
\(224\) −4852.65 −1.44746
\(225\) 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 \(-0.310300\pi\)
0.561304 + 0.827610i \(0.310300\pi\)
\(230\) 1622.79 0.465233
\(231\) 0 0
\(232\) −9360.74 −2.64898
\(233\) 3218.14 0.904837 0.452419 0.891806i \(-0.350561\pi\)
0.452419 + 0.891806i \(0.350561\pi\)
\(234\) 1213.12 0.338906
\(235\) −8380.75 −2.32638
\(236\) −12392.8 −3.41823
\(237\) 3105.35 0.851115
\(238\) −665.670 −0.181298
\(239\) 428.098 0.115864 0.0579318 0.998321i \(-0.481549\pi\)
0.0579318 + 0.998321i \(0.481549\pi\)
\(240\) 11302.8 3.03996
\(241\) −1231.16 −0.329070 −0.164535 0.986371i \(-0.552612\pi\)
−0.164535 + 0.986371i \(0.552612\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 4747.00 1.24547
\(245\) 4780.78 1.24667
\(246\) 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 \(-0.383845\pi\)
0.356866 + 0.934156i \(0.383845\pi\)
\(252\) 1484.27 0.371033
\(253\) 0 0
\(254\) 3587.20 0.886145
\(255\) 805.814 0.197890
\(256\) 7579.90 1.85056
\(257\) 342.007 0.0830110 0.0415055 0.999138i \(-0.486785\pi\)
0.0415055 + 0.999138i \(0.486785\pi\)
\(258\) −1057.19 −0.255108
\(259\) 708.731 0.170032
\(260\) 8969.87 2.13957
\(261\) 1156.92 0.274374
\(262\) 3375.01 0.795834
\(263\) −5895.00 −1.38213 −0.691067 0.722791i \(-0.742859\pi\)
−0.691067 + 0.722791i \(0.742859\pi\)
\(264\) 0 0
\(265\) 8877.99 2.05800
\(266\) 632.647 0.145827
\(267\) 70.3234 0.0161188
\(268\) −18441.0 −4.20322
\(269\) −2496.18 −0.565779 −0.282890 0.959152i \(-0.591293\pi\)
−0.282890 + 0.959152i \(0.591293\pi\)
\(270\) −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
\(273\) 573.838 0.127217
\(274\) 10159.1 2.23990
\(275\) 0 0
\(276\) −1141.22 −0.248889
\(277\) −4082.59 −0.885556 −0.442778 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(278\) −5175.22 −1.11651
\(279\) 1974.56 0.423705
\(280\) 9444.57 2.01579
\(281\) 1033.79 0.219468 0.109734 0.993961i \(-0.465000\pi\)
0.109734 + 0.993961i \(0.465000\pi\)
\(282\) 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
\(286\) 0 0
\(287\) 3540.50 0.728186
\(288\) −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 \(0.437749\pi\)
\(294\) −4617.47 −0.915973
\(295\) 9746.10 1.92353
\(296\) −6704.55 −1.31653
\(297\) 0 0
\(298\) −11103.9 −2.15849
\(299\) −441.209 −0.0853371
\(300\) −10212.0 −1.96530
\(301\) −500.081 −0.0957614
\(302\) 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 \(-0.558044\pi\)
−0.181343 + 0.983420i \(0.558044\pi\)
\(312\) −5428.47 −0.985021
\(313\) −3878.67 −0.700433 −0.350216 0.936669i \(-0.613892\pi\)
−0.350216 + 0.936669i \(0.613892\pi\)
\(314\) 3512.13 0.631214
\(315\) −1167.28 −0.208790
\(316\) −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
\(319\) 0 0
\(320\) −27475.1 −4.79971
\(321\) 3444.07 0.598846
\(322\) −741.402 −0.128313
\(323\) −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
\(329\) 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
\(333\) 828.633 0.136363
\(334\) 6093.02 0.998189
\(335\) 14502.6 2.36525
\(336\) −5163.88 −0.838430
\(337\) −5919.19 −0.956792 −0.478396 0.878144i \(-0.658782\pi\)
−0.478396 + 0.878144i \(0.658782\pi\)
\(338\) 8568.07 1.37882
\(339\) −3379.19 −0.541394
\(340\) −5754.70 −0.917919
\(341\) 0 0
\(342\) 739.677 0.116951
\(343\) −4824.51 −0.759472
\(344\) 4730.73 0.741465
\(345\) 897.490 0.140056
\(346\) 249.590 0.0387805
\(347\) 8540.59 1.32128 0.660638 0.750705i \(-0.270286\pi\)
0.660638 + 0.750705i \(0.270286\pi\)
\(348\) −8262.11 −1.27269
\(349\) −937.337 −0.143767 −0.0718833 0.997413i \(-0.522901\pi\)
−0.0718833 + 0.997413i \(0.522901\pi\)
\(350\) −6634.31 −1.01320
\(351\) 670.919 0.102026
\(352\) 0 0
\(353\) −211.118 −0.0318319 −0.0159160 0.999873i \(-0.505066\pi\)
−0.0159160 + 0.999873i \(0.505066\pi\)
\(354\) −9413.17 −1.41329
\(355\) −9787.82 −1.46333
\(356\) −502.213 −0.0747675
\(357\) −368.151 −0.0545788
\(358\) 4512.99 0.666254
\(359\) 1376.31 0.202337 0.101169 0.994869i \(-0.467742\pi\)
0.101169 + 0.994869i \(0.467742\pi\)
\(360\) 11042.4 1.61662
\(361\) −6629.44 −0.966532
\(362\) 9821.63 1.42600
\(363\) 0 0
\(364\) −4098.05 −0.590100
\(365\) 8594.87 1.23254
\(366\) 3605.66 0.514948
\(367\) 1030.45 0.146564 0.0732821 0.997311i \(-0.476653\pi\)
0.0732821 + 0.997311i \(0.476653\pi\)
\(368\) 3970.37 0.562418
\(369\) 4139.48 0.583991
\(370\) 8414.81 1.18234
\(371\) −4056.07 −0.567603
\(372\) −14101.3 −1.96537
\(373\) −9365.39 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(374\) 0 0
\(375\) 1712.72 0.235852
\(376\) −36221.2 −4.96799
\(377\) −3194.24 −0.436370
\(378\) 1127.40 0.153406
\(379\) −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 \(-0.524731\pi\)
−0.0776176 + 0.996983i \(0.524731\pi\)
\(384\) 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
\(392\) 20662.3 2.66225
\(393\) 1866.56 0.239581
\(394\) 28774.0 3.67923
\(395\) 17440.6 2.22159
\(396\) 0 0
\(397\) −2172.09 −0.274595 −0.137298 0.990530i \(-0.543842\pi\)
−0.137298 + 0.990530i \(0.543842\pi\)
\(398\) 27872.9 3.51041
\(399\) 349.888 0.0439005
\(400\) 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
\(406\) −5367.55 −0.656126
\(407\) 0 0
\(408\) 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
\(413\) −4452.69 −0.530515
\(414\) −866.831 −0.102904
\(415\) 10214.0 1.20815
\(416\) 15664.8 1.84622
\(417\) −2862.17 −0.336118
\(418\) 0 0
\(419\) −7315.88 −0.852994 −0.426497 0.904489i \(-0.640252\pi\)
−0.426497 + 0.904489i \(0.640252\pi\)
\(420\) 8336.10 0.968476
\(421\) −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
\(427\) 1705.58 0.193299
\(428\) −24595.8 −2.77776
\(429\) 0 0
\(430\) −5937.49 −0.665887
\(431\) −6075.01 −0.678939 −0.339470 0.940617i \(-0.610248\pi\)
−0.339470 + 0.940617i \(0.610248\pi\)
\(432\) −6037.49 −0.672405
\(433\) 5641.79 0.626160 0.313080 0.949727i \(-0.398639\pi\)
0.313080 + 0.949727i \(0.398639\pi\)
\(434\) −9161.01 −1.01323
\(435\) 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 \(0.701800\pi\)
\(440\) 0 0
\(441\) −2553.71 −0.275748
\(442\) 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
\(448\) 12552.5 1.32378
\(449\) −7473.86 −0.785553 −0.392776 0.919634i \(-0.628485\pi\)
−0.392776 + 0.919634i \(0.628485\pi\)
\(450\) −7756.70 −0.812565
\(451\) 0 0
\(452\) 24132.4 2.51127
\(453\) 1427.59 0.148066
\(454\) −6657.29 −0.688198
\(455\) 3222.84 0.332064
\(456\) −3309.91 −0.339914
\(457\) −11140.5 −1.14033 −0.570167 0.821529i \(-0.693121\pi\)
−0.570167 + 0.821529i \(0.693121\pi\)
\(458\) −21102.6 −2.15296
\(459\) −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 \(-0.300174\pi\)
0.587343 + 0.809338i \(0.300174\pi\)
\(468\) −4791.35 −0.473249
\(469\) −6625.77 −0.652345
\(470\) 45460.8 4.46160
\(471\) 1942.40 0.190023
\(472\) 42122.1 4.10769
\(473\) 0 0
\(474\) −16844.8 −1.63229
\(475\) −2407.27 −0.232533
\(476\) 2629.14 0.253165
\(477\) −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
\(483\) −410.035 −0.0386278
\(484\) 0 0
\(485\) −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
\(490\) −25933.0 −2.39089
\(491\) −124.552 −0.0114480 −0.00572398 0.999984i \(-0.501822\pi\)
−0.00572398 + 0.999984i \(0.501822\pi\)
\(492\) −29562.0 −2.70886
\(493\) 2049.29 0.187212
\(494\) −2042.24 −0.186001
\(495\) 0 0
\(496\) 49059.2 4.44117
\(497\) 4471.75 0.403592
\(498\) −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 \(0.573315\pi\)
\(504\) −5044.92 −0.445870
\(505\) 31528.8 2.77824
\(506\) 0 0
\(507\) 4738.60 0.415086
\(508\) −14168.1 −1.23741
\(509\) −22.7715 −0.00198296 −0.000991481 1.00000i \(-0.500316\pi\)
−0.000991481 1.00000i \(0.500316\pi\)
\(510\) −4371.08 −0.379519
\(511\) −3926.73 −0.339938
\(512\) −10698.1 −0.923429
\(513\) 409.081 0.0352073
\(514\) −1855.19 −0.159201
\(515\) 7223.87 0.618100
\(516\) 4175.50 0.356233
\(517\) 0 0
\(518\) −3844.46 −0.326093
\(519\) 138.037 0.0116747
\(520\) −30487.8 −2.57112
\(521\) 21521.7 1.80976 0.904879 0.425669i \(-0.139961\pi\)
0.904879 + 0.425669i \(0.139961\pi\)
\(522\) −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
\(525\) −3669.13 −0.305017
\(526\) 31977.0 2.65069
\(527\) 3497.60 0.289104
\(528\) 0 0
\(529\) −11851.7 −0.974088
\(530\) −48158.0 −3.94689
\(531\) −5205.99 −0.425462
\(532\) −2498.71 −0.203633
\(533\) −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 \(-0.820565\pi\)
−0.845278 + 0.534327i \(0.820565\pi\)
\(542\) −12203.2 −0.967108
\(543\) 5431.89 0.429290
\(544\) −10049.9 −0.792067
\(545\) −30805.1 −2.42118
\(546\) −3112.75 −0.243980
\(547\) 18730.5 1.46409 0.732046 0.681256i \(-0.238566\pi\)
0.732046 + 0.681256i \(0.238566\pi\)
\(548\) −40124.5 −3.12780
\(549\) 1994.12 0.155022
\(550\) 0 0
\(551\) −1947.63 −0.150584
\(552\) 3878.91 0.299089
\(553\) −7968.04 −0.612723
\(554\) 22145.7 1.69834
\(555\) 4653.84 0.355936
\(556\) 20440.1 1.55909
\(557\) 18885.0 1.43659 0.718297 0.695736i \(-0.244922\pi\)
0.718297 + 0.695736i \(0.244922\pi\)
\(558\) −10710.9 −0.812593
\(559\) 1614.30 0.122143
\(560\) −29001.8 −2.18848
\(561\) 0 0
\(562\) −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
\(567\) 623.515 0.0461820
\(568\) −42302.5 −3.12495
\(569\) −18008.8 −1.32683 −0.663415 0.748251i \(-0.730894\pi\)
−0.663415 + 0.748251i \(0.730894\pi\)
\(570\) 4154.24 0.305266
\(571\) 7010.79 0.513822 0.256911 0.966435i \(-0.417295\pi\)
0.256911 + 0.966435i \(0.417295\pi\)
\(572\) 0 0
\(573\) −1376.69 −0.100370
\(574\) −19205.2 −1.39653
\(575\) 2821.10 0.204605
\(576\) 14676.1 1.06164
\(577\) −16398.9 −1.18318 −0.591589 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(578\) 25271.6 1.81862
\(579\) 5336.73 0.383052
\(580\) −46402.4 −3.32199
\(581\) −4666.44 −0.333213
\(582\) 11708.5 0.833903
\(583\) 0 0
\(584\) 37146.6 2.63208
\(585\) 3768.07 0.266309
\(586\) −10573.3 −0.745354
\(587\) 12823.5 0.901671 0.450836 0.892607i \(-0.351126\pi\)
0.450836 + 0.892607i \(0.351126\pi\)
\(588\) 18237.2 1.27907
\(589\) −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 \(0.698962\pi\)
\(594\) 0 0
\(595\) −2067.64 −0.142462
\(596\) 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 \(0.373336\pi\)
\(602\) 2712.65 0.183654
\(603\) −7746.71 −0.523168
\(604\) −10195.1 −0.686809
\(605\) 0 0
\(606\) −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
\(609\) −2968.54 −0.197523
\(610\) 20250.4 1.34412
\(611\) −12360.0 −0.818384
\(612\) 3073.94 0.203033
\(613\) −12528.9 −0.825507 −0.412753 0.910843i \(-0.635433\pi\)
−0.412753 + 0.910843i \(0.635433\pi\)
\(614\) 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 \(-0.296009\pi\)
0.597882 + 0.801584i \(0.296009\pi\)
\(620\) −79196.7 −5.13003
\(621\) −479.404 −0.0309788
\(622\) 10790.1 0.695569
\(623\) −180.443 −0.0116040
\(624\) 16669.4 1.06941
\(625\) −10241.4 −0.655448
\(626\) 21039.6 1.34331
\(627\) 0 0
\(628\) −13871.6 −0.881427
\(629\) 1467.79 0.0930436
\(630\) 6331.82 0.400422
\(631\) 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
\(637\) 7050.74 0.438557
\(638\) 0 0
\(639\) 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 \(-0.290649\pi\)
0.611294 + 0.791403i \(0.290649\pi\)
\(644\) 2928.26 0.179176
\(645\) −3283.75 −0.200461
\(646\) 1310.21 0.0797983
\(647\) −30634.8 −1.86148 −0.930739 0.365684i \(-0.880835\pi\)
−0.930739 + 0.365684i \(0.880835\pi\)
\(648\) −5898.41 −0.357579
\(649\) 0 0
\(650\) 21416.1 1.29232
\(651\) −5066.53 −0.305028
\(652\) 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
\(658\) −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
\(663\) 1188.42 0.0696146
\(664\) 44144.2 2.58001
\(665\) 1965.07 0.114590
\(666\) −4494.86 −0.261520
\(667\) 2282.44 0.132498
\(668\) −24065.1 −1.39387
\(669\) 4133.39 0.238873
\(670\) −78668.2 −4.53614
\(671\) 0 0
\(672\) 14558.0 0.835692
\(673\) 29960.3 1.71602 0.858012 0.513630i \(-0.171700\pi\)
0.858012 + 0.513630i \(0.171700\pi\)
\(674\) 32108.2 1.83496
\(675\) −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
\(679\) 5538.43 0.313027
\(680\) 19559.7 1.10306
\(681\) −3681.84 −0.207178
\(682\) 0 0
\(683\) −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
\(686\) 26170.2 1.45653
\(687\) −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 \(-0.602262\pi\)
−0.315768 + 0.948837i \(0.602262\pi\)
\(692\) −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
\(700\) 26203.0 1.41483
\(701\) −22229.0 −1.19769 −0.598843 0.800866i \(-0.704373\pi\)
−0.598843 + 0.800866i \(0.704373\pi\)
\(702\) −3639.35 −0.195667
\(703\) −1394.97 −0.0748397
\(704\) 0 0
\(705\) 25142.3 1.34314
\(706\) 1145.19 0.0610480
\(707\) −14404.5 −0.766248
\(708\) 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
\(714\) 1997.01 0.104673
\(715\) 0 0
\(716\) −17824.6 −0.930358
\(717\) −1284.30 −0.0668938
\(718\) −7465.71 −0.388047
\(719\) −7399.80 −0.383819 −0.191910 0.981413i \(-0.561468\pi\)
−0.191910 + 0.981413i \(0.561468\pi\)
\(720\) −33908.3 −1.75512
\(721\) −3300.36 −0.170474
\(722\) 35960.9 1.85364
\(723\) 3693.48 0.189989
\(724\) −38791.7 −1.99127
\(725\) 20424.0 1.04625
\(726\) 0 0
\(727\) 1705.77 0.0870202 0.0435101 0.999053i \(-0.486146\pi\)
0.0435101 + 0.999053i \(0.486146\pi\)
\(728\) 13928.9 0.709122
\(729\) 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
\(735\) −14342.3 −0.719762
\(736\) −11193.2 −0.560581
\(737\) 0 0
\(738\) −22454.3 −1.11999
\(739\) −34256.3 −1.70520 −0.852598 0.522568i \(-0.824974\pi\)
−0.852598 + 0.522568i \(0.824974\pi\)
\(740\) −33235.3 −1.65102
\(741\) −1129.47 −0.0559945
\(742\) 22001.9 1.08856
\(743\) −1567.88 −0.0774160 −0.0387080 0.999251i \(-0.512324\pi\)
−0.0387080 + 0.999251i \(0.512324\pi\)
\(744\) 47929.0 2.36178
\(745\) −34489.8 −1.69612
\(746\) 50801.9 2.49328
\(747\) −5455.90 −0.267230
\(748\) 0 0
\(749\) −8837.17 −0.431112
\(750\) −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
\(756\) −4452.82 −0.214216
\(757\) −14015.4 −0.672918 −0.336459 0.941698i \(-0.609229\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(758\) 38623.2 1.85073
\(759\) 0 0
\(760\) −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
\(763\) 14073.9 0.667770
\(764\) 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 \(-0.562198\pi\)
−0.194160 + 0.980970i \(0.562198\pi\)
\(774\) 3171.57 0.147287
\(775\) 34858.4 1.61568
\(776\) −52393.2 −2.42372
\(777\) −2126.19 −0.0981683
\(778\) 59445.7 2.73937
\(779\) −6968.65 −0.320510
\(780\) −26909.6 −1.23528
\(781\) 0 0
\(782\) −1535.45 −0.0702142
\(783\) −3470.76 −0.158410
\(784\) −63448.5 −2.89033
\(785\) 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
\(791\) 8670.69 0.389752
\(792\) 0 0
\(793\) −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 \(-0.519420\pi\)
−0.0609730 + 0.998139i \(0.519420\pi\)
\(798\) −1897.94 −0.0841934
\(799\) 7929.68 0.351104
\(800\) −100161. −4.42652
\(801\) −210.970 −0.00930619
\(802\) −42477.1 −1.87022
\(803\) 0 0
\(804\) 55322.9 2.42673
\(805\) −2302.88 −0.100827
\(806\) 29572.5 1.29237
\(807\) 7488.53 0.326653
\(808\) 136266. 5.93293
\(809\) −41241.7 −1.79231 −0.896156 0.443738i \(-0.853652\pi\)
−0.896156 + 0.443738i \(0.853652\pi\)
\(810\) 7403.03 0.321131
\(811\) −12832.9 −0.555641 −0.277820 0.960633i \(-0.589612\pi\)
−0.277820 + 0.960633i \(0.589612\pi\)
\(812\) 21199.8 0.916215
\(813\) −6749.03 −0.291142
\(814\) 0 0
\(815\) −18419.7 −0.791675
\(816\) −10694.4 −0.458798
\(817\) 984.292 0.0421493
\(818\) −58210.4 −2.48812
\(819\) −1721.51 −0.0734488
\(820\) −166029. −7.07069
\(821\) −16368.5 −0.695817 −0.347908 0.937529i \(-0.613108\pi\)
−0.347908 + 0.937529i \(0.613108\pi\)
\(822\) −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
\(825\) 0 0
\(826\) 24153.3 1.01743
\(827\) −7388.69 −0.310677 −0.155339 0.987861i \(-0.549647\pi\)
−0.155339 + 0.987861i \(0.549647\pi\)
\(828\) 3423.65 0.143696
\(829\) 23990.1 1.00508 0.502539 0.864554i \(-0.332399\pi\)
0.502539 + 0.864554i \(0.332399\pi\)
\(830\) −55404.9 −2.31703
\(831\) 12247.8 0.511276
\(832\) −40520.6 −1.68846
\(833\) −4523.47 −0.188150
\(834\) 15525.7 0.644616
\(835\) 18925.6 0.784367
\(836\) 0 0
\(837\) −5923.68 −0.244626
\(838\) 39684.5 1.63589
\(839\) −18228.3 −0.750074 −0.375037 0.927010i \(-0.622370\pi\)
−0.375037 + 0.927010i \(0.622370\pi\)
\(840\) −28333.7 −1.16382
\(841\) −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 \(0.356300\pi\)
\(854\) −9251.79 −0.370714
\(855\) 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 \(0.378355\pi\)
\(858\) 0 0
\(859\) 30527.6 1.21256 0.606279 0.795252i \(-0.292661\pi\)
0.606279 + 0.795252i \(0.292661\pi\)
\(860\) 23450.8 0.929845
\(861\) −10621.5 −0.420418
\(862\) 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
\(868\) 36182.5 1.41488
\(869\) 0 0
\(870\) −35245.7 −1.37350
\(871\) 21388.5 0.832058
\(872\) −133138. −5.17043
\(873\) 6475.41 0.251042
\(874\) 1459.28 0.0564768
\(875\) −4394.68 −0.169791
\(876\) 32786.8 1.26457
\(877\) −21770.9 −0.838256 −0.419128 0.907927i \(-0.637664\pi\)
−0.419128 + 0.907927i \(0.637664\pi\)
\(878\) 59109.8 2.27205
\(879\) −5847.58 −0.224384
\(880\) 0 0
\(881\) 47206.9 1.80527 0.902634 0.430409i \(-0.141631\pi\)
0.902634 + 0.430409i \(0.141631\pi\)
\(882\) 13852.4 0.528837
\(883\) −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
\(889\) −5090.53 −0.192048
\(890\) −2142.41 −0.0806896
\(891\) 0 0
\(892\) −29518.5 −1.10802
\(893\) −7536.30 −0.282410
\(894\) 33311.6 1.24621
\(895\) 14017.8 0.523536
\(896\) −29269.1 −1.09131
\(897\) 1323.63 0.0492694
\(898\) 40541.4 1.50655
\(899\) 28202.5 1.04628
\(900\) 30636.0 1.13467
\(901\) −8400.15 −0.310599
\(902\) 0 0
\(903\) 1500.24 0.0552879
\(904\) −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
\(910\) −17482.1 −0.636841
\(911\) 37676.4 1.37022 0.685112 0.728438i \(-0.259753\pi\)
0.685112 + 0.728438i \(0.259753\pi\)
\(912\) 10163.9 0.369035
\(913\) 0 0
\(914\) 60431.1 2.18696
\(915\) 11199.6 0.404641
\(916\) 83347.1 3.00640
\(917\) −4789.41 −0.172476
\(918\) 2334.86 0.0839454
\(919\) −8697.82 −0.312203 −0.156101 0.987741i \(-0.549893\pi\)
−0.156101 + 0.987741i \(0.549893\pi\)
\(920\) 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 \(-0.598509\pi\)
−0.304559 + 0.952493i \(0.598509\pi\)
\(930\) −60155.2 −2.12104
\(931\) 4299.06 0.151338
\(932\) 68946.7 2.42320
\(933\) 5967.51 0.209397
\(934\) −64306.0 −2.25284
\(935\) 0 0
\(936\) 16285.4 0.568702
\(937\) 41812.4 1.45779 0.728896 0.684624i \(-0.240034\pi\)
0.728896 + 0.684624i \(0.240034\pi\)
\(938\) 35941.0 1.25108
\(939\) 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
\(945\) 3501.84 0.120545
\(946\) 0 0
\(947\) −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
\(952\) −8936.24 −0.304228
\(953\) −1324.27 −0.0450130 −0.0225065 0.999747i \(-0.507165\pi\)
−0.0225065 + 0.999747i \(0.507165\pi\)
\(954\) 25724.1 0.873007
\(955\) −7731.91 −0.261988
\(956\) 9171.77 0.310289
\(957\) 0 0
\(958\) −7185.62 −0.242335
\(959\) −14416.6 −0.485439
\(960\) 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
\(966\) 2224.21 0.0740815
\(967\) −52267.1 −1.73815 −0.869077 0.494676i \(-0.835287\pi\)
−0.869077 + 0.494676i \(0.835287\pi\)
\(968\) 0 0
\(969\) 724.619 0.0240228
\(970\) 65758.1 2.17667
\(971\) −52489.8 −1.73479 −0.867394 0.497622i \(-0.834207\pi\)
−0.867394 + 0.497622i \(0.834207\pi\)
\(972\) −5206.14 −0.171797
\(973\) 7344.07 0.241973
\(974\) −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
\(979\) 0 0
\(980\) 102426. 3.33864
\(981\) 16454.9 0.535539
\(982\) 675.623 0.0219552
\(983\) −44407.1 −1.44086 −0.720431 0.693527i \(-0.756056\pi\)
−0.720431 + 0.693527i \(0.756056\pi\)
\(984\) 100479. 3.25523
\(985\) 89375.3 2.89110
\(986\) −11116.2 −0.359039
\(987\) −11486.7 −0.370442
\(988\) 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
\(993\) −24313.4 −0.777001
\(994\) −24256.7 −0.774020
\(995\) 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 363.4.a.i.1.1 2
3.2 odd 2 1089.4.a.u.1.2 2
11.10 odd 2 33.4.a.c.1.2 2
33.32 even 2 99.4.a.f.1.1 2
44.43 even 2 528.4.a.p.1.1 2
55.32 even 4 825.4.c.h.199.4 4
55.43 even 4 825.4.c.h.199.1 4
55.54 odd 2 825.4.a.l.1.1 2
77.76 even 2 1617.4.a.k.1.2 2
88.21 odd 2 2112.4.a.bn.1.2 2
88.43 even 2 2112.4.a.bg.1.2 2
132.131 odd 2 1584.4.a.bj.1.2 2
165.164 even 2 2475.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 11.10 odd 2
99.4.a.f.1.1 2 33.32 even 2
363.4.a.i.1.1 2 1.1 even 1 trivial
528.4.a.p.1.1 2 44.43 even 2
825.4.a.l.1.1 2 55.54 odd 2
825.4.c.h.199.1 4 55.43 even 4
825.4.c.h.199.4 4 55.32 even 4
1089.4.a.u.1.2 2 3.2 odd 2
1584.4.a.bj.1.2 2 132.131 odd 2
1617.4.a.k.1.2 2 77.76 even 2
2112.4.a.bg.1.2 2 88.43 even 2
2112.4.a.bn.1.2 2 88.21 odd 2
2475.4.a.p.1.2 2 165.164 even 2