Properties

Label 289.4.b.f.288.7
Level $289$
Weight $4$
Character 289.288
Analytic conductor $17.052$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,4,Mod(288,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.288");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(17.0515519917\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 288.7
Character \(\chi\) \(=\) 289.288
Dual form 289.4.b.f.288.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.25752 q^{2} -6.51757i q^{3} +2.61146 q^{4} +19.1073i q^{5} +21.2311i q^{6} -22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} +O(q^{10})\) \(q-3.25752 q^{2} -6.51757i q^{3} +2.61146 q^{4} +19.1073i q^{5} +21.2311i q^{6} -22.0995i q^{7} +17.5533 q^{8} -15.4787 q^{9} -62.2426i q^{10} +8.25972i q^{11} -17.0204i q^{12} -0.398269 q^{13} +71.9898i q^{14} +124.533 q^{15} -78.0720 q^{16} +50.4221 q^{18} -103.371 q^{19} +49.8980i q^{20} -144.035 q^{21} -26.9062i q^{22} -138.311i q^{23} -114.405i q^{24} -240.090 q^{25} +1.29737 q^{26} -75.0910i q^{27} -57.7120i q^{28} +222.296i q^{29} -405.670 q^{30} +49.1520i q^{31} +113.895 q^{32} +53.8333 q^{33} +422.263 q^{35} -40.4219 q^{36} +61.3077i q^{37} +336.734 q^{38} +2.59575i q^{39} +335.397i q^{40} +387.391i q^{41} +469.198 q^{42} +20.3847 q^{43} +21.5699i q^{44} -295.756i q^{45} +450.553i q^{46} +44.1322 q^{47} +508.839i q^{48} -145.389 q^{49} +782.100 q^{50} -1.04006 q^{52} -59.6456 q^{53} +244.611i q^{54} -157.821 q^{55} -387.920i q^{56} +673.728i q^{57} -724.134i q^{58} -238.069 q^{59} +325.214 q^{60} +595.816i q^{61} -160.114i q^{62} +342.071i q^{63} +253.561 q^{64} -7.60987i q^{65} -175.363 q^{66} -408.815 q^{67} -901.454 q^{69} -1375.53 q^{70} +1037.38i q^{71} -271.702 q^{72} +22.3789i q^{73} -199.711i q^{74} +1564.80i q^{75} -269.949 q^{76} +182.536 q^{77} -8.45571i q^{78} +682.688i q^{79} -1491.75i q^{80} -907.335 q^{81} -1261.93i q^{82} +312.352 q^{83} -376.142 q^{84} -66.4037 q^{86} +1448.83 q^{87} +144.985i q^{88} -904.392 q^{89} +963.433i q^{90} +8.80157i q^{91} -361.195i q^{92} +320.351 q^{93} -143.762 q^{94} -1975.15i q^{95} -742.317i q^{96} +1006.72i q^{97} +473.610 q^{98} -127.849i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.25752 −1.15171 −0.575854 0.817552i \(-0.695330\pi\)
−0.575854 + 0.817552i \(0.695330\pi\)
\(3\) − 6.51757i − 1.25431i −0.778896 0.627153i \(-0.784220\pi\)
0.778896 0.627153i \(-0.215780\pi\)
\(4\) 2.61146 0.326432
\(5\) 19.1073i 1.70901i 0.519441 + 0.854506i \(0.326140\pi\)
−0.519441 + 0.854506i \(0.673860\pi\)
\(6\) 21.2311i 1.44460i
\(7\) − 22.0995i − 1.19326i −0.802515 0.596631i \(-0.796505\pi\)
0.802515 0.596631i \(-0.203495\pi\)
\(8\) 17.5533 0.775753
\(9\) −15.4787 −0.573284
\(10\) − 62.2426i − 1.96828i
\(11\) 8.25972i 0.226400i 0.993572 + 0.113200i \(0.0361101\pi\)
−0.993572 + 0.113200i \(0.963890\pi\)
\(12\) − 17.0204i − 0.409446i
\(13\) −0.398269 −0.00849693 −0.00424846 0.999991i \(-0.501352\pi\)
−0.00424846 + 0.999991i \(0.501352\pi\)
\(14\) 71.9898i 1.37429i
\(15\) 124.533 2.14362
\(16\) −78.0720 −1.21987
\(17\) 0 0
\(18\) 50.4221 0.660256
\(19\) −103.371 −1.24816 −0.624078 0.781362i \(-0.714525\pi\)
−0.624078 + 0.781362i \(0.714525\pi\)
\(20\) 49.8980i 0.557877i
\(21\) −144.035 −1.49672
\(22\) − 26.9062i − 0.260747i
\(23\) − 138.311i − 1.25391i −0.779056 0.626955i \(-0.784301\pi\)
0.779056 0.626955i \(-0.215699\pi\)
\(24\) − 114.405i − 0.973032i
\(25\) −240.090 −1.92072
\(26\) 1.29737 0.00978598
\(27\) − 75.0910i − 0.535233i
\(28\) − 57.7120i − 0.389520i
\(29\) 222.296i 1.42342i 0.702471 + 0.711712i \(0.252080\pi\)
−0.702471 + 0.711712i \(0.747920\pi\)
\(30\) −405.670 −2.46883
\(31\) 49.1520i 0.284773i 0.989811 + 0.142386i \(0.0454776\pi\)
−0.989811 + 0.142386i \(0.954522\pi\)
\(32\) 113.895 0.629186
\(33\) 53.8333 0.283975
\(34\) 0 0
\(35\) 422.263 2.03930
\(36\) −40.4219 −0.187138
\(37\) 61.3077i 0.272403i 0.990681 + 0.136202i \(0.0434895\pi\)
−0.990681 + 0.136202i \(0.956510\pi\)
\(38\) 336.734 1.43751
\(39\) 2.59575i 0.0106577i
\(40\) 335.397i 1.32577i
\(41\) 387.391i 1.47562i 0.675011 + 0.737808i \(0.264139\pi\)
−0.675011 + 0.737808i \(0.735861\pi\)
\(42\) 469.198 1.72378
\(43\) 20.3847 0.0722939 0.0361470 0.999346i \(-0.488492\pi\)
0.0361470 + 0.999346i \(0.488492\pi\)
\(44\) 21.5699i 0.0739043i
\(45\) − 295.756i − 0.979749i
\(46\) 450.553i 1.44414i
\(47\) 44.1322 0.136965 0.0684824 0.997652i \(-0.478184\pi\)
0.0684824 + 0.997652i \(0.478184\pi\)
\(48\) 508.839i 1.53010i
\(49\) −145.389 −0.423876
\(50\) 782.100 2.21211
\(51\) 0 0
\(52\) −1.04006 −0.00277367
\(53\) −59.6456 −0.154584 −0.0772920 0.997009i \(-0.524627\pi\)
−0.0772920 + 0.997009i \(0.524627\pi\)
\(54\) 244.611i 0.616432i
\(55\) −157.821 −0.386920
\(56\) − 387.920i − 0.925678i
\(57\) 673.728i 1.56557i
\(58\) − 724.134i − 1.63937i
\(59\) −238.069 −0.525321 −0.262660 0.964888i \(-0.584600\pi\)
−0.262660 + 0.964888i \(0.584600\pi\)
\(60\) 325.214 0.699749
\(61\) 595.816i 1.25060i 0.780386 + 0.625298i \(0.215023\pi\)
−0.780386 + 0.625298i \(0.784977\pi\)
\(62\) − 160.114i − 0.327975i
\(63\) 342.071i 0.684078i
\(64\) 253.561 0.495235
\(65\) − 7.60987i − 0.0145214i
\(66\) −175.363 −0.327056
\(67\) −408.815 −0.745442 −0.372721 0.927943i \(-0.621575\pi\)
−0.372721 + 0.927943i \(0.621575\pi\)
\(68\) 0 0
\(69\) −901.454 −1.57279
\(70\) −1375.53 −2.34868
\(71\) 1037.38i 1.73400i 0.498309 + 0.866999i \(0.333954\pi\)
−0.498309 + 0.866999i \(0.666046\pi\)
\(72\) −271.702 −0.444727
\(73\) 22.3789i 0.0358802i 0.999839 + 0.0179401i \(0.00571082\pi\)
−0.999839 + 0.0179401i \(0.994289\pi\)
\(74\) − 199.711i − 0.313729i
\(75\) 1564.80i 2.40917i
\(76\) −269.949 −0.407438
\(77\) 182.536 0.270155
\(78\) − 8.45571i − 0.0122746i
\(79\) 682.688i 0.972259i 0.873887 + 0.486129i \(0.161592\pi\)
−0.873887 + 0.486129i \(0.838408\pi\)
\(80\) − 1491.75i − 2.08478i
\(81\) −907.335 −1.24463
\(82\) − 1261.93i − 1.69948i
\(83\) 312.352 0.413073 0.206536 0.978439i \(-0.433781\pi\)
0.206536 + 0.978439i \(0.433781\pi\)
\(84\) −376.142 −0.488577
\(85\) 0 0
\(86\) −66.4037 −0.0832616
\(87\) 1448.83 1.78541
\(88\) 144.985i 0.175630i
\(89\) −904.392 −1.07714 −0.538570 0.842581i \(-0.681035\pi\)
−0.538570 + 0.842581i \(0.681035\pi\)
\(90\) 963.433i 1.12839i
\(91\) 8.80157i 0.0101391i
\(92\) − 361.195i − 0.409317i
\(93\) 320.351 0.357192
\(94\) −143.762 −0.157744
\(95\) − 1975.15i − 2.13311i
\(96\) − 742.317i − 0.789192i
\(97\) 1006.72i 1.05378i 0.849933 + 0.526891i \(0.176642\pi\)
−0.849933 + 0.526891i \(0.823358\pi\)
\(98\) 473.610 0.488182
\(99\) − 127.849i − 0.129791i
\(100\) −626.986 −0.626986
\(101\) −1124.60 −1.10794 −0.553968 0.832538i \(-0.686887\pi\)
−0.553968 + 0.832538i \(0.686887\pi\)
\(102\) 0 0
\(103\) −396.677 −0.379473 −0.189736 0.981835i \(-0.560763\pi\)
−0.189736 + 0.981835i \(0.560763\pi\)
\(104\) −6.99094 −0.00659152
\(105\) − 2752.13i − 2.55791i
\(106\) 194.297 0.178036
\(107\) − 288.906i − 0.261024i −0.991447 0.130512i \(-0.958338\pi\)
0.991447 0.130512i \(-0.0416621\pi\)
\(108\) − 196.097i − 0.174717i
\(109\) − 1379.15i − 1.21191i −0.795499 0.605955i \(-0.792791\pi\)
0.795499 0.605955i \(-0.207209\pi\)
\(110\) 514.106 0.445619
\(111\) 399.577 0.341677
\(112\) 1725.35i 1.45563i
\(113\) 1364.42i 1.13587i 0.823072 + 0.567937i \(0.192258\pi\)
−0.823072 + 0.567937i \(0.807742\pi\)
\(114\) − 2194.68i − 1.80308i
\(115\) 2642.76 2.14295
\(116\) 580.517i 0.464652i
\(117\) 6.16468 0.00487115
\(118\) 775.515 0.605016
\(119\) 0 0
\(120\) 2185.97 1.66292
\(121\) 1262.78 0.948743
\(122\) − 1940.88i − 1.44032i
\(123\) 2524.84 1.85087
\(124\) 128.358i 0.0929591i
\(125\) − 2199.07i − 1.57353i
\(126\) − 1114.31i − 0.787859i
\(127\) −1003.33 −0.701032 −0.350516 0.936557i \(-0.613994\pi\)
−0.350516 + 0.936557i \(0.613994\pi\)
\(128\) −1737.14 −1.19955
\(129\) − 132.859i − 0.0906787i
\(130\) 24.7893i 0.0167244i
\(131\) − 218.014i − 0.145404i −0.997354 0.0727022i \(-0.976838\pi\)
0.997354 0.0727022i \(-0.0231623\pi\)
\(132\) 140.583 0.0926986
\(133\) 2284.45i 1.48938i
\(134\) 1331.72 0.858532
\(135\) 1434.79 0.914719
\(136\) 0 0
\(137\) −1625.34 −1.01359 −0.506797 0.862066i \(-0.669171\pi\)
−0.506797 + 0.862066i \(0.669171\pi\)
\(138\) 2936.51 1.81139
\(139\) 1034.35i 0.631166i 0.948898 + 0.315583i \(0.102200\pi\)
−0.948898 + 0.315583i \(0.897800\pi\)
\(140\) 1102.72 0.665694
\(141\) − 287.635i − 0.171796i
\(142\) − 3379.28i − 1.99706i
\(143\) − 3.28959i − 0.00192370i
\(144\) 1208.45 0.699334
\(145\) −4247.48 −2.43265
\(146\) − 72.8999i − 0.0413236i
\(147\) 947.585i 0.531670i
\(148\) 160.103i 0.0889213i
\(149\) −650.200 −0.357493 −0.178747 0.983895i \(-0.557204\pi\)
−0.178747 + 0.983895i \(0.557204\pi\)
\(150\) − 5097.39i − 2.77467i
\(151\) −1023.67 −0.551689 −0.275845 0.961202i \(-0.588958\pi\)
−0.275845 + 0.961202i \(0.588958\pi\)
\(152\) −1814.50 −0.968261
\(153\) 0 0
\(154\) −594.615 −0.311139
\(155\) −939.164 −0.486680
\(156\) 6.77869i 0.00347903i
\(157\) 671.612 0.341404 0.170702 0.985323i \(-0.445396\pi\)
0.170702 + 0.985323i \(0.445396\pi\)
\(158\) − 2223.87i − 1.11976i
\(159\) 388.744i 0.193896i
\(160\) 2176.23i 1.07529i
\(161\) −3056.62 −1.49624
\(162\) 2955.66 1.43345
\(163\) 1893.45i 0.909857i 0.890528 + 0.454929i \(0.150335\pi\)
−0.890528 + 0.454929i \(0.849665\pi\)
\(164\) 1011.65i 0.481689i
\(165\) 1028.61i 0.485316i
\(166\) −1017.49 −0.475739
\(167\) 290.246i 0.134491i 0.997736 + 0.0672453i \(0.0214210\pi\)
−0.997736 + 0.0672453i \(0.978579\pi\)
\(168\) −2528.29 −1.16108
\(169\) −2196.84 −0.999928
\(170\) 0 0
\(171\) 1600.05 0.715547
\(172\) 53.2339 0.0235991
\(173\) − 3665.86i − 1.61104i −0.592567 0.805521i \(-0.701886\pi\)
0.592567 0.805521i \(-0.298114\pi\)
\(174\) −4719.59 −2.05627
\(175\) 5305.89i 2.29193i
\(176\) − 644.852i − 0.276179i
\(177\) 1551.63i 0.658913i
\(178\) 2946.08 1.24055
\(179\) 3628.67 1.51519 0.757597 0.652723i \(-0.226373\pi\)
0.757597 + 0.652723i \(0.226373\pi\)
\(180\) − 772.355i − 0.319822i
\(181\) − 3827.78i − 1.57191i −0.618281 0.785957i \(-0.712171\pi\)
0.618281 0.785957i \(-0.287829\pi\)
\(182\) − 28.6713i − 0.0116773i
\(183\) 3883.27 1.56863
\(184\) − 2427.82i − 0.972725i
\(185\) −1171.43 −0.465541
\(186\) −1043.55 −0.411382
\(187\) 0 0
\(188\) 115.249 0.0447097
\(189\) −1659.48 −0.638673
\(190\) 6434.08i 2.45672i
\(191\) −877.247 −0.332332 −0.166166 0.986098i \(-0.553139\pi\)
−0.166166 + 0.986098i \(0.553139\pi\)
\(192\) − 1652.60i − 0.621177i
\(193\) 4769.70i 1.77891i 0.457018 + 0.889457i \(0.348917\pi\)
−0.457018 + 0.889457i \(0.651083\pi\)
\(194\) − 3279.41i − 1.21365i
\(195\) −49.5978 −0.0182142
\(196\) −379.679 −0.138367
\(197\) − 1492.97i − 0.539949i −0.962867 0.269974i \(-0.912985\pi\)
0.962867 0.269974i \(-0.0870152\pi\)
\(198\) 416.473i 0.149482i
\(199\) − 3091.28i − 1.10118i −0.834775 0.550591i \(-0.814402\pi\)
0.834775 0.550591i \(-0.185598\pi\)
\(200\) −4214.38 −1.49001
\(201\) 2664.48i 0.935013i
\(202\) 3663.40 1.27602
\(203\) 4912.63 1.69852
\(204\) 0 0
\(205\) −7402.00 −2.52185
\(206\) 1292.18 0.437042
\(207\) 2140.88i 0.718846i
\(208\) 31.0937 0.0103652
\(209\) − 853.816i − 0.282582i
\(210\) 8965.12i 2.94596i
\(211\) 2776.38i 0.905848i 0.891549 + 0.452924i \(0.149619\pi\)
−0.891549 + 0.452924i \(0.850381\pi\)
\(212\) −155.762 −0.0504612
\(213\) 6761.17 2.17497
\(214\) 941.118i 0.300624i
\(215\) 389.498i 0.123551i
\(216\) − 1318.10i − 0.415209i
\(217\) 1086.24 0.339809
\(218\) 4492.60i 1.39577i
\(219\) 145.856 0.0450048
\(220\) −412.144 −0.126303
\(221\) 0 0
\(222\) −1301.63 −0.393513
\(223\) −4641.22 −1.39372 −0.696859 0.717209i \(-0.745419\pi\)
−0.696859 + 0.717209i \(0.745419\pi\)
\(224\) − 2517.02i − 0.750784i
\(225\) 3716.28 1.10112
\(226\) − 4444.63i − 1.30820i
\(227\) − 1712.30i − 0.500658i −0.968161 0.250329i \(-0.919461\pi\)
0.968161 0.250329i \(-0.0805388\pi\)
\(228\) 1759.41i 0.511052i
\(229\) 6028.18 1.73954 0.869768 0.493461i \(-0.164269\pi\)
0.869768 + 0.493461i \(0.164269\pi\)
\(230\) −8608.86 −2.46805
\(231\) − 1189.69i − 0.338857i
\(232\) 3902.02i 1.10423i
\(233\) 2663.60i 0.748919i 0.927243 + 0.374459i \(0.122172\pi\)
−0.927243 + 0.374459i \(0.877828\pi\)
\(234\) −20.0816 −0.00561015
\(235\) 843.249i 0.234075i
\(236\) −621.707 −0.171482
\(237\) 4449.47 1.21951
\(238\) 0 0
\(239\) 6020.73 1.62949 0.814747 0.579817i \(-0.196876\pi\)
0.814747 + 0.579817i \(0.196876\pi\)
\(240\) −9722.56 −2.61495
\(241\) 6080.97i 1.62535i 0.582716 + 0.812676i \(0.301990\pi\)
−0.582716 + 0.812676i \(0.698010\pi\)
\(242\) −4113.53 −1.09268
\(243\) 3886.16i 1.02591i
\(244\) 1555.95i 0.408235i
\(245\) − 2778.01i − 0.724409i
\(246\) −8224.74 −2.13167
\(247\) 41.1695 0.0106055
\(248\) 862.780i 0.220914i
\(249\) − 2035.77i − 0.518120i
\(250\) 7163.52i 1.81224i
\(251\) −3296.82 −0.829058 −0.414529 0.910036i \(-0.636054\pi\)
−0.414529 + 0.910036i \(0.636054\pi\)
\(252\) 893.305i 0.223305i
\(253\) 1142.41 0.283885
\(254\) 3268.37 0.807384
\(255\) 0 0
\(256\) 3630.28 0.886300
\(257\) 1243.37 0.301786 0.150893 0.988550i \(-0.451785\pi\)
0.150893 + 0.988550i \(0.451785\pi\)
\(258\) 432.790i 0.104435i
\(259\) 1354.87 0.325049
\(260\) − 19.8729i − 0.00474024i
\(261\) − 3440.84i − 0.816026i
\(262\) 710.186i 0.167463i
\(263\) −4523.81 −1.06065 −0.530324 0.847795i \(-0.677930\pi\)
−0.530324 + 0.847795i \(0.677930\pi\)
\(264\) 944.951 0.220294
\(265\) − 1139.67i − 0.264186i
\(266\) − 7441.66i − 1.71533i
\(267\) 5894.44i 1.35106i
\(268\) −1067.60 −0.243337
\(269\) − 2155.56i − 0.488576i −0.969703 0.244288i \(-0.921446\pi\)
0.969703 0.244288i \(-0.0785542\pi\)
\(270\) −4673.86 −1.05349
\(271\) −5433.12 −1.21785 −0.608927 0.793226i \(-0.708400\pi\)
−0.608927 + 0.793226i \(0.708400\pi\)
\(272\) 0 0
\(273\) 57.3648 0.0127175
\(274\) 5294.59 1.16736
\(275\) − 1983.08i − 0.434851i
\(276\) −2354.11 −0.513408
\(277\) − 3349.75i − 0.726595i −0.931673 0.363297i \(-0.881651\pi\)
0.931673 0.363297i \(-0.118349\pi\)
\(278\) − 3369.40i − 0.726919i
\(279\) − 760.807i − 0.163256i
\(280\) 7412.11 1.58199
\(281\) 2668.18 0.566441 0.283221 0.959055i \(-0.408597\pi\)
0.283221 + 0.959055i \(0.408597\pi\)
\(282\) 936.977i 0.197859i
\(283\) − 5452.65i − 1.14532i −0.819792 0.572662i \(-0.805911\pi\)
0.819792 0.572662i \(-0.194089\pi\)
\(284\) 2709.07i 0.566033i
\(285\) −12873.1 −2.67558
\(286\) 10.7159i 0.00221555i
\(287\) 8561.15 1.76080
\(288\) −1762.94 −0.360702
\(289\) 0 0
\(290\) 13836.3 2.80170
\(291\) 6561.36 1.32177
\(292\) 58.4417i 0.0117125i
\(293\) 6341.24 1.26437 0.632183 0.774819i \(-0.282159\pi\)
0.632183 + 0.774819i \(0.282159\pi\)
\(294\) − 3086.78i − 0.612329i
\(295\) − 4548.86i − 0.897780i
\(296\) 1076.15i 0.211318i
\(297\) 620.231 0.121177
\(298\) 2118.04 0.411728
\(299\) 55.0852i 0.0106544i
\(300\) 4086.42i 0.786433i
\(301\) − 450.493i − 0.0862657i
\(302\) 3334.63 0.635385
\(303\) 7329.63i 1.38969i
\(304\) 8070.38 1.52259
\(305\) −11384.5 −2.13729
\(306\) 0 0
\(307\) 820.602 0.152555 0.0762773 0.997087i \(-0.475697\pi\)
0.0762773 + 0.997087i \(0.475697\pi\)
\(308\) 476.685 0.0881872
\(309\) 2585.37i 0.475975i
\(310\) 3059.35 0.560514
\(311\) 7492.46i 1.36610i 0.730369 + 0.683052i \(0.239348\pi\)
−0.730369 + 0.683052i \(0.760652\pi\)
\(312\) 45.5639i 0.00826779i
\(313\) − 1217.76i − 0.219910i −0.993937 0.109955i \(-0.964929\pi\)
0.993937 0.109955i \(-0.0350707\pi\)
\(314\) −2187.79 −0.393198
\(315\) −6536.07 −1.16910
\(316\) 1782.81i 0.317377i
\(317\) 5682.18i 1.00676i 0.864065 + 0.503380i \(0.167910\pi\)
−0.864065 + 0.503380i \(0.832090\pi\)
\(318\) − 1266.34i − 0.223311i
\(319\) −1836.10 −0.322263
\(320\) 4844.87i 0.846363i
\(321\) −1882.96 −0.327404
\(322\) 9957.00 1.72324
\(323\) 0 0
\(324\) −2369.47 −0.406287
\(325\) 95.6206 0.0163202
\(326\) − 6167.97i − 1.04789i
\(327\) −8988.68 −1.52011
\(328\) 6799.98i 1.14471i
\(329\) − 975.301i − 0.163435i
\(330\) − 3350.72i − 0.558943i
\(331\) −7387.94 −1.22682 −0.613411 0.789764i \(-0.710203\pi\)
−0.613411 + 0.789764i \(0.710203\pi\)
\(332\) 815.693 0.134840
\(333\) − 948.962i − 0.156165i
\(334\) − 945.483i − 0.154894i
\(335\) − 7811.36i − 1.27397i
\(336\) 11245.1 1.82581
\(337\) − 5104.29i − 0.825069i −0.910942 0.412534i \(-0.864644\pi\)
0.910942 0.412534i \(-0.135356\pi\)
\(338\) 7156.26 1.15163
\(339\) 8892.69 1.42473
\(340\) 0 0
\(341\) −405.982 −0.0644725
\(342\) −5212.19 −0.824102
\(343\) − 4367.10i − 0.687467i
\(344\) 357.819 0.0560823
\(345\) − 17224.4i − 2.68791i
\(346\) 11941.6i 1.85545i
\(347\) − 8986.15i − 1.39021i −0.718910 0.695103i \(-0.755359\pi\)
0.718910 0.695103i \(-0.244641\pi\)
\(348\) 3783.55 0.582816
\(349\) −1924.65 −0.295198 −0.147599 0.989047i \(-0.547155\pi\)
−0.147599 + 0.989047i \(0.547155\pi\)
\(350\) − 17284.0i − 2.63963i
\(351\) 29.9065i 0.00454783i
\(352\) 940.739i 0.142448i
\(353\) 4574.33 0.689709 0.344854 0.938656i \(-0.387928\pi\)
0.344854 + 0.938656i \(0.387928\pi\)
\(354\) − 5054.47i − 0.758876i
\(355\) −19821.5 −2.96343
\(356\) −2361.78 −0.351613
\(357\) 0 0
\(358\) −11820.5 −1.74506
\(359\) −9358.89 −1.37589 −0.687943 0.725765i \(-0.741486\pi\)
−0.687943 + 0.725765i \(0.741486\pi\)
\(360\) − 5191.50i − 0.760044i
\(361\) 3826.58 0.557892
\(362\) 12469.1i 1.81039i
\(363\) − 8230.23i − 1.19001i
\(364\) 22.9849i 0.00330972i
\(365\) −427.602 −0.0613198
\(366\) −12649.8 −1.80661
\(367\) 1596.51i 0.227077i 0.993534 + 0.113539i \(0.0362186\pi\)
−0.993534 + 0.113539i \(0.963781\pi\)
\(368\) 10798.2i 1.52961i
\(369\) − 5996.29i − 0.845947i
\(370\) 3815.95 0.536167
\(371\) 1318.14i 0.184459i
\(372\) 836.585 0.116599
\(373\) −10884.6 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(374\) 0 0
\(375\) −14332.6 −1.97368
\(376\) 774.666 0.106251
\(377\) − 88.5336i − 0.0120947i
\(378\) 5405.79 0.735565
\(379\) − 1561.47i − 0.211629i −0.994386 0.105814i \(-0.966255\pi\)
0.994386 0.105814i \(-0.0337449\pi\)
\(380\) − 5158.01i − 0.696317i
\(381\) 6539.26i 0.879308i
\(382\) 2857.65 0.382749
\(383\) 4409.48 0.588287 0.294143 0.955761i \(-0.404966\pi\)
0.294143 + 0.955761i \(0.404966\pi\)
\(384\) 11321.9i 1.50461i
\(385\) 3487.78i 0.461697i
\(386\) − 15537.4i − 2.04879i
\(387\) −315.528 −0.0414450
\(388\) 2629.01i 0.343989i
\(389\) 8017.21 1.04496 0.522479 0.852652i \(-0.325007\pi\)
0.522479 + 0.852652i \(0.325007\pi\)
\(390\) 161.566 0.0209775
\(391\) 0 0
\(392\) −2552.06 −0.328823
\(393\) −1420.92 −0.182382
\(394\) 4863.39i 0.621864i
\(395\) −13044.4 −1.66160
\(396\) − 333.874i − 0.0423681i
\(397\) 9437.12i 1.19304i 0.802599 + 0.596519i \(0.203450\pi\)
−0.802599 + 0.596519i \(0.796550\pi\)
\(398\) 10069.9i 1.26824i
\(399\) 14889.1 1.86814
\(400\) 18744.3 2.34304
\(401\) − 4509.28i − 0.561553i −0.959773 0.280777i \(-0.909408\pi\)
0.959773 0.280777i \(-0.0905920\pi\)
\(402\) − 8679.59i − 1.07686i
\(403\) − 19.5757i − 0.00241969i
\(404\) −2936.84 −0.361666
\(405\) − 17336.8i − 2.12709i
\(406\) −16003.0 −1.95620
\(407\) −506.384 −0.0616721
\(408\) 0 0
\(409\) 12446.9 1.50480 0.752398 0.658709i \(-0.228897\pi\)
0.752398 + 0.658709i \(0.228897\pi\)
\(410\) 24112.2 2.90443
\(411\) 10593.3i 1.27136i
\(412\) −1035.91 −0.123872
\(413\) 5261.21i 0.626846i
\(414\) − 6973.95i − 0.827901i
\(415\) 5968.21i 0.705946i
\(416\) −45.3608 −0.00534615
\(417\) 6741.41 0.791675
\(418\) 2781.33i 0.325452i
\(419\) − 1648.22i − 0.192173i −0.995373 0.0960867i \(-0.969367\pi\)
0.995373 0.0960867i \(-0.0306326\pi\)
\(420\) − 7187.07i − 0.834984i
\(421\) 14395.5 1.66649 0.833247 0.552901i \(-0.186479\pi\)
0.833247 + 0.552901i \(0.186479\pi\)
\(422\) − 9044.13i − 1.04327i
\(423\) −683.108 −0.0785197
\(424\) −1046.98 −0.119919
\(425\) 0 0
\(426\) −22024.7 −2.50493
\(427\) 13167.3 1.49229
\(428\) − 754.466i − 0.0852068i
\(429\) −21.4401 −0.00241291
\(430\) − 1268.80i − 0.142295i
\(431\) 5351.92i 0.598127i 0.954233 + 0.299064i \(0.0966743\pi\)
−0.954233 + 0.299064i \(0.903326\pi\)
\(432\) 5862.50i 0.652916i
\(433\) −15578.0 −1.72894 −0.864472 0.502680i \(-0.832347\pi\)
−0.864472 + 0.502680i \(0.832347\pi\)
\(434\) −3538.44 −0.391361
\(435\) 27683.2i 3.05129i
\(436\) − 3601.58i − 0.395607i
\(437\) 14297.4i 1.56507i
\(438\) −475.130 −0.0518324
\(439\) − 16584.8i − 1.80307i −0.432707 0.901535i \(-0.642442\pi\)
0.432707 0.901535i \(-0.357558\pi\)
\(440\) −2770.28 −0.300155
\(441\) 2250.44 0.243001
\(442\) 0 0
\(443\) 5163.31 0.553761 0.276881 0.960904i \(-0.410699\pi\)
0.276881 + 0.960904i \(0.410699\pi\)
\(444\) 1043.48 0.111535
\(445\) − 17280.5i − 1.84084i
\(446\) 15118.9 1.60516
\(447\) 4237.72i 0.448406i
\(448\) − 5603.57i − 0.590946i
\(449\) − 10771.9i − 1.13220i −0.824337 0.566099i \(-0.808452\pi\)
0.824337 0.566099i \(-0.191548\pi\)
\(450\) −12105.9 −1.26817
\(451\) −3199.74 −0.334079
\(452\) 3563.13i 0.370786i
\(453\) 6671.84i 0.691987i
\(454\) 5577.86i 0.576612i
\(455\) −168.175 −0.0173278
\(456\) 11826.1i 1.21450i
\(457\) −5024.59 −0.514312 −0.257156 0.966370i \(-0.582785\pi\)
−0.257156 + 0.966370i \(0.582785\pi\)
\(458\) −19636.9 −2.00344
\(459\) 0 0
\(460\) 6901.47 0.699527
\(461\) 2366.96 0.239133 0.119567 0.992826i \(-0.461849\pi\)
0.119567 + 0.992826i \(0.461849\pi\)
\(462\) 3875.44i 0.390264i
\(463\) −17712.9 −1.77795 −0.888974 0.457958i \(-0.848581\pi\)
−0.888974 + 0.457958i \(0.848581\pi\)
\(464\) − 17355.1i − 1.73640i
\(465\) 6121.06i 0.610446i
\(466\) − 8676.73i − 0.862536i
\(467\) 19592.7 1.94141 0.970707 0.240267i \(-0.0772349\pi\)
0.970707 + 0.240267i \(0.0772349\pi\)
\(468\) 16.0988 0.00159010
\(469\) 9034.61i 0.889509i
\(470\) − 2746.90i − 0.269586i
\(471\) − 4377.27i − 0.428225i
\(472\) −4178.89 −0.407519
\(473\) 168.372i 0.0163673i
\(474\) −14494.2 −1.40452
\(475\) 24818.4 2.39736
\(476\) 0 0
\(477\) 923.234 0.0886205
\(478\) −19612.7 −1.87670
\(479\) − 4060.41i − 0.387317i −0.981069 0.193659i \(-0.937965\pi\)
0.981069 0.193659i \(-0.0620354\pi\)
\(480\) 14183.7 1.34874
\(481\) − 24.4170i − 0.00231459i
\(482\) − 19808.9i − 1.87193i
\(483\) 19921.7i 1.87675i
\(484\) 3297.69 0.309701
\(485\) −19235.7 −1.80093
\(486\) − 12659.2i − 1.18155i
\(487\) 7256.96i 0.675244i 0.941282 + 0.337622i \(0.109623\pi\)
−0.941282 + 0.337622i \(0.890377\pi\)
\(488\) 10458.5i 0.970155i
\(489\) 12340.7 1.14124
\(490\) 9049.42i 0.834308i
\(491\) 769.855 0.0707598 0.0353799 0.999374i \(-0.488736\pi\)
0.0353799 + 0.999374i \(0.488736\pi\)
\(492\) 6593.53 0.604185
\(493\) 0 0
\(494\) −134.111 −0.0122144
\(495\) 2442.86 0.221815
\(496\) − 3837.39i − 0.347387i
\(497\) 22925.5 2.06912
\(498\) 6631.58i 0.596723i
\(499\) − 1432.66i − 0.128526i −0.997933 0.0642630i \(-0.979530\pi\)
0.997933 0.0642630i \(-0.0204697\pi\)
\(500\) − 5742.78i − 0.513650i
\(501\) 1891.70 0.168692
\(502\) 10739.5 0.954833
\(503\) − 16206.3i − 1.43659i −0.695739 0.718295i \(-0.744923\pi\)
0.695739 0.718295i \(-0.255077\pi\)
\(504\) 6004.48i 0.530676i
\(505\) − 21488.0i − 1.89348i
\(506\) −3721.44 −0.326953
\(507\) 14318.1i 1.25422i
\(508\) −2620.15 −0.228839
\(509\) 15009.4 1.30703 0.653517 0.756912i \(-0.273293\pi\)
0.653517 + 0.756912i \(0.273293\pi\)
\(510\) 0 0
\(511\) 494.564 0.0428145
\(512\) 2071.37 0.178794
\(513\) 7762.24i 0.668053i
\(514\) −4050.29 −0.347570
\(515\) − 7579.44i − 0.648524i
\(516\) − 346.955i − 0.0296005i
\(517\) 364.520i 0.0310088i
\(518\) −4413.53 −0.374361
\(519\) −23892.5 −2.02074
\(520\) − 133.578i − 0.0112650i
\(521\) − 19626.5i − 1.65039i −0.564846 0.825196i \(-0.691064\pi\)
0.564846 0.825196i \(-0.308936\pi\)
\(522\) 11208.6i 0.939824i
\(523\) −9267.41 −0.774829 −0.387414 0.921906i \(-0.626632\pi\)
−0.387414 + 0.921906i \(0.626632\pi\)
\(524\) − 569.335i − 0.0474647i
\(525\) 34581.5 2.87478
\(526\) 14736.4 1.22156
\(527\) 0 0
\(528\) −4202.87 −0.346414
\(529\) −6963.04 −0.572289
\(530\) 3712.50i 0.304265i
\(531\) 3684.99 0.301158
\(532\) 5965.76i 0.486181i
\(533\) − 154.286i − 0.0125382i
\(534\) − 19201.3i − 1.55603i
\(535\) 5520.22 0.446094
\(536\) −7176.04 −0.578280
\(537\) − 23650.1i − 1.90052i
\(538\) 7021.79i 0.562697i
\(539\) − 1200.88i − 0.0959655i
\(540\) 3746.90 0.298594
\(541\) − 907.733i − 0.0721377i −0.999349 0.0360689i \(-0.988516\pi\)
0.999349 0.0360689i \(-0.0114836\pi\)
\(542\) 17698.5 1.40261
\(543\) −24947.8 −1.97166
\(544\) 0 0
\(545\) 26351.8 2.07117
\(546\) −186.867 −0.0146468
\(547\) − 11197.7i − 0.875281i −0.899150 0.437640i \(-0.855814\pi\)
0.899150 0.437640i \(-0.144186\pi\)
\(548\) −4244.51 −0.330870
\(549\) − 9222.44i − 0.716947i
\(550\) 6459.92i 0.500822i
\(551\) − 22979.0i − 1.77665i
\(552\) −15823.5 −1.22009
\(553\) 15087.1 1.16016
\(554\) 10911.9i 0.836825i
\(555\) 7634.85i 0.583931i
\(556\) 2701.15i 0.206033i
\(557\) −3207.25 −0.243978 −0.121989 0.992531i \(-0.538927\pi\)
−0.121989 + 0.992531i \(0.538927\pi\)
\(558\) 2478.35i 0.188023i
\(559\) −8.11861 −0.000614276 0
\(560\) −32966.9 −2.48769
\(561\) 0 0
\(562\) −8691.64 −0.652375
\(563\) 6522.79 0.488282 0.244141 0.969740i \(-0.421494\pi\)
0.244141 + 0.969740i \(0.421494\pi\)
\(564\) − 751.146i − 0.0560797i
\(565\) −26070.4 −1.94122
\(566\) 17762.2i 1.31908i
\(567\) 20051.7i 1.48517i
\(568\) 18209.4i 1.34516i
\(569\) 3040.50 0.224015 0.112007 0.993707i \(-0.464272\pi\)
0.112007 + 0.993707i \(0.464272\pi\)
\(570\) 41934.6 3.08148
\(571\) − 17273.2i − 1.26596i −0.774169 0.632979i \(-0.781832\pi\)
0.774169 0.632979i \(-0.218168\pi\)
\(572\) − 8.59064i 0 0.000627959i
\(573\) 5717.51i 0.416846i
\(574\) −27888.2 −2.02793
\(575\) 33207.2i 2.40841i
\(576\) −3924.78 −0.283910
\(577\) −5804.29 −0.418779 −0.209390 0.977832i \(-0.567148\pi\)
−0.209390 + 0.977832i \(0.567148\pi\)
\(578\) 0 0
\(579\) 31086.8 2.23130
\(580\) −11092.1 −0.794096
\(581\) − 6902.83i − 0.492904i
\(582\) −21373.8 −1.52229
\(583\) − 492.656i − 0.0349978i
\(584\) 392.824i 0.0278342i
\(585\) 117.791i 0.00832486i
\(586\) −20656.8 −1.45618
\(587\) −12862.4 −0.904407 −0.452204 0.891915i \(-0.649362\pi\)
−0.452204 + 0.891915i \(0.649362\pi\)
\(588\) 2474.58i 0.173554i
\(589\) − 5080.89i − 0.355441i
\(590\) 14818.0i 1.03398i
\(591\) −9730.55 −0.677261
\(592\) − 4786.41i − 0.332298i
\(593\) −1319.33 −0.0913631 −0.0456816 0.998956i \(-0.514546\pi\)
−0.0456816 + 0.998956i \(0.514546\pi\)
\(594\) −2020.42 −0.139560
\(595\) 0 0
\(596\) −1697.97 −0.116697
\(597\) −20147.6 −1.38122
\(598\) − 179.441i − 0.0122707i
\(599\) −16269.3 −1.10976 −0.554881 0.831930i \(-0.687236\pi\)
−0.554881 + 0.831930i \(0.687236\pi\)
\(600\) 27467.5i 1.86893i
\(601\) 7427.38i 0.504108i 0.967713 + 0.252054i \(0.0811061\pi\)
−0.967713 + 0.252054i \(0.918894\pi\)
\(602\) 1467.49i 0.0993529i
\(603\) 6327.90 0.427350
\(604\) −2673.27 −0.180089
\(605\) 24128.3i 1.62141i
\(606\) − 23876.4i − 1.60052i
\(607\) − 7212.10i − 0.482257i −0.970493 0.241129i \(-0.922482\pi\)
0.970493 0.241129i \(-0.0775176\pi\)
\(608\) −11773.4 −0.785322
\(609\) − 32018.4i − 2.13046i
\(610\) 37085.1 2.46153
\(611\) −17.5765 −0.00116378
\(612\) 0 0
\(613\) −15446.5 −1.01775 −0.508873 0.860842i \(-0.669938\pi\)
−0.508873 + 0.860842i \(0.669938\pi\)
\(614\) −2673.13 −0.175698
\(615\) 48243.1i 3.16317i
\(616\) 3204.11 0.209573
\(617\) − 8746.54i − 0.570701i −0.958423 0.285350i \(-0.907890\pi\)
0.958423 0.285350i \(-0.0921100\pi\)
\(618\) − 8421.89i − 0.548185i
\(619\) − 4740.09i − 0.307787i −0.988087 0.153894i \(-0.950819\pi\)
0.988087 0.153894i \(-0.0491813\pi\)
\(620\) −2452.59 −0.158868
\(621\) −10385.9 −0.671133
\(622\) − 24406.9i − 1.57335i
\(623\) 19986.7i 1.28531i
\(624\) − 202.655i − 0.0130011i
\(625\) 12007.1 0.768453
\(626\) 3966.89i 0.253273i
\(627\) −5564.80 −0.354445
\(628\) 1753.89 0.111445
\(629\) 0 0
\(630\) 21291.4 1.34646
\(631\) 17055.1 1.07600 0.537998 0.842946i \(-0.319181\pi\)
0.537998 + 0.842946i \(0.319181\pi\)
\(632\) 11983.4i 0.754233i
\(633\) 18095.2 1.13621
\(634\) − 18509.8i − 1.15949i
\(635\) − 19170.9i − 1.19807i
\(636\) 1015.19i 0.0632938i
\(637\) 57.9042 0.00360164
\(638\) 5981.14 0.371153
\(639\) − 16057.2i − 0.994074i
\(640\) − 33192.1i − 2.05005i
\(641\) 16050.5i 0.989012i 0.869174 + 0.494506i \(0.164651\pi\)
−0.869174 + 0.494506i \(0.835349\pi\)
\(642\) 6133.80 0.377074
\(643\) 14821.5i 0.909027i 0.890740 + 0.454514i \(0.150187\pi\)
−0.890740 + 0.454514i \(0.849813\pi\)
\(644\) −7982.23 −0.488422
\(645\) 2538.58 0.154971
\(646\) 0 0
\(647\) −30527.0 −1.85493 −0.927465 0.373910i \(-0.878017\pi\)
−0.927465 + 0.373910i \(0.878017\pi\)
\(648\) −15926.7 −0.965526
\(649\) − 1966.38i − 0.118933i
\(650\) −311.486 −0.0187962
\(651\) − 7079.62i − 0.426224i
\(652\) 4944.68i 0.297007i
\(653\) 20173.6i 1.20897i 0.796618 + 0.604483i \(0.206620\pi\)
−0.796618 + 0.604483i \(0.793380\pi\)
\(654\) 29280.8 1.75072
\(655\) 4165.67 0.248498
\(656\) − 30244.3i − 1.80007i
\(657\) − 346.396i − 0.0205696i
\(658\) 3177.07i 0.188229i
\(659\) −21503.0 −1.27107 −0.635537 0.772071i \(-0.719221\pi\)
−0.635537 + 0.772071i \(0.719221\pi\)
\(660\) 2686.17i 0.158423i
\(661\) 1772.12 0.104278 0.0521389 0.998640i \(-0.483396\pi\)
0.0521389 + 0.998640i \(0.483396\pi\)
\(662\) 24066.4 1.41294
\(663\) 0 0
\(664\) 5482.80 0.320443
\(665\) −43649.8 −2.54536
\(666\) 3091.26i 0.179856i
\(667\) 30746.0 1.78485
\(668\) 757.966i 0.0439021i
\(669\) 30249.4i 1.74815i
\(670\) 25445.7i 1.46724i
\(671\) −4921.27 −0.283135
\(672\) −16404.9 −0.941713
\(673\) 33449.4i 1.91587i 0.286983 + 0.957936i \(0.407348\pi\)
−0.286983 + 0.957936i \(0.592652\pi\)
\(674\) 16627.3i 0.950239i
\(675\) 18028.6i 1.02803i
\(676\) −5736.96 −0.326409
\(677\) 805.765i 0.0457431i 0.999738 + 0.0228716i \(0.00728088\pi\)
−0.999738 + 0.0228716i \(0.992719\pi\)
\(678\) −28968.2 −1.64088
\(679\) 22248.0 1.25744
\(680\) 0 0
\(681\) −11160.0 −0.627978
\(682\) 1322.49 0.0742536
\(683\) − 993.802i − 0.0556761i −0.999612 0.0278380i \(-0.991138\pi\)
0.999612 0.0278380i \(-0.00886227\pi\)
\(684\) 4178.46 0.233578
\(685\) − 31056.0i − 1.73224i
\(686\) 14225.9i 0.791762i
\(687\) − 39289.1i − 2.18191i
\(688\) −1591.47 −0.0881895
\(689\) 23.7550 0.00131349
\(690\) 56108.8i 3.09569i
\(691\) 33450.1i 1.84154i 0.390111 + 0.920768i \(0.372437\pi\)
−0.390111 + 0.920768i \(0.627563\pi\)
\(692\) − 9573.25i − 0.525896i
\(693\) −2825.41 −0.154875
\(694\) 29272.6i 1.60111i
\(695\) −19763.6 −1.07867
\(696\) 25431.7 1.38504
\(697\) 0 0
\(698\) 6269.59 0.339982
\(699\) 17360.2 0.939373
\(700\) 13856.1i 0.748159i
\(701\) −18290.5 −0.985481 −0.492740 0.870176i \(-0.664005\pi\)
−0.492740 + 0.870176i \(0.664005\pi\)
\(702\) − 97.4210i − 0.00523778i
\(703\) − 6337.44i − 0.340002i
\(704\) 2094.34i 0.112121i
\(705\) 5495.93 0.293601
\(706\) −14901.0 −0.794343
\(707\) 24853.1i 1.32206i
\(708\) 4052.02i 0.215091i
\(709\) − 6780.00i − 0.359137i −0.983745 0.179569i \(-0.942530\pi\)
0.983745 0.179569i \(-0.0574702\pi\)
\(710\) 64569.0 3.41300
\(711\) − 10567.1i − 0.557380i
\(712\) −15875.1 −0.835595
\(713\) 6798.28 0.357079
\(714\) 0 0
\(715\) 62.8554 0.00328763
\(716\) 9476.13 0.494608
\(717\) − 39240.5i − 2.04388i
\(718\) 30486.8 1.58462
\(719\) 24591.0i 1.27551i 0.770240 + 0.637754i \(0.220136\pi\)
−0.770240 + 0.637754i \(0.779864\pi\)
\(720\) 23090.3i 1.19517i
\(721\) 8766.37i 0.452811i
\(722\) −12465.2 −0.642528
\(723\) 39633.1 2.03869
\(724\) − 9996.08i − 0.513124i
\(725\) − 53371.1i − 2.73400i
\(726\) 26810.2i 1.37055i
\(727\) −9819.79 −0.500957 −0.250479 0.968122i \(-0.580588\pi\)
−0.250479 + 0.968122i \(0.580588\pi\)
\(728\) 154.497i 0.00786542i
\(729\) 830.242 0.0421806
\(730\) 1392.92 0.0706225
\(731\) 0 0
\(732\) 10141.0 0.512052
\(733\) 10751.8 0.541783 0.270892 0.962610i \(-0.412682\pi\)
0.270892 + 0.962610i \(0.412682\pi\)
\(734\) − 5200.68i − 0.261527i
\(735\) −18105.8 −0.908631
\(736\) − 15753.0i − 0.788942i
\(737\) − 3376.69i − 0.168768i
\(738\) 19533.1i 0.974284i
\(739\) 1809.70 0.0900823 0.0450412 0.998985i \(-0.485658\pi\)
0.0450412 + 0.998985i \(0.485658\pi\)
\(740\) −3059.13 −0.151968
\(741\) − 268.325i − 0.0133025i
\(742\) − 4293.87i − 0.212443i
\(743\) 895.790i 0.0442306i 0.999755 + 0.0221153i \(0.00704010\pi\)
−0.999755 + 0.0221153i \(0.992960\pi\)
\(744\) 5623.22 0.277093
\(745\) − 12423.6i − 0.610960i
\(746\) 35456.7 1.74016
\(747\) −4834.79 −0.236808
\(748\) 0 0
\(749\) −6384.69 −0.311470
\(750\) 46688.7 2.27311
\(751\) 11683.1i 0.567672i 0.958873 + 0.283836i \(0.0916071\pi\)
−0.958873 + 0.283836i \(0.908393\pi\)
\(752\) −3445.49 −0.167080
\(753\) 21487.2i 1.03989i
\(754\) 288.400i 0.0139296i
\(755\) − 19559.6i − 0.942844i
\(756\) −4333.66 −0.208484
\(757\) 12621.2 0.605976 0.302988 0.952994i \(-0.402016\pi\)
0.302988 + 0.952994i \(0.402016\pi\)
\(758\) 5086.53i 0.243735i
\(759\) − 7445.75i − 0.356079i
\(760\) − 34670.3i − 1.65477i
\(761\) −5759.46 −0.274350 −0.137175 0.990547i \(-0.543802\pi\)
−0.137175 + 0.990547i \(0.543802\pi\)
\(762\) − 21301.8i − 1.01271i
\(763\) −30478.5 −1.44613
\(764\) −2290.89 −0.108484
\(765\) 0 0
\(766\) −14364.0 −0.677535
\(767\) 94.8156 0.00446361
\(768\) − 23660.6i − 1.11169i
\(769\) 3245.56 0.152195 0.0760975 0.997100i \(-0.475754\pi\)
0.0760975 + 0.997100i \(0.475754\pi\)
\(770\) − 11361.5i − 0.531741i
\(771\) − 8103.72i − 0.378532i
\(772\) 12455.9i 0.580695i
\(773\) 2500.27 0.116337 0.0581684 0.998307i \(-0.481474\pi\)
0.0581684 + 0.998307i \(0.481474\pi\)
\(774\) 1027.84 0.0477325
\(775\) − 11800.9i − 0.546970i
\(776\) 17671.2i 0.817475i
\(777\) − 8830.47i − 0.407711i
\(778\) −26116.2 −1.20349
\(779\) − 40045.0i − 1.84180i
\(780\) −129.523 −0.00594571
\(781\) −8568.44 −0.392577
\(782\) 0 0
\(783\) 16692.4 0.761863
\(784\) 11350.8 0.517075
\(785\) 12832.7i 0.583464i
\(786\) 4628.68 0.210050
\(787\) − 15270.8i − 0.691671i −0.938295 0.345835i \(-0.887596\pi\)
0.938295 0.345835i \(-0.112404\pi\)
\(788\) − 3898.84i − 0.176257i
\(789\) 29484.2i 1.33038i
\(790\) 42492.3 1.91368
\(791\) 30153.0 1.35540
\(792\) − 2244.18i − 0.100686i
\(793\) − 237.295i − 0.0106262i
\(794\) − 30741.7i − 1.37403i
\(795\) −7427.86 −0.331370
\(796\) − 8072.76i − 0.359462i
\(797\) −6885.20 −0.306006 −0.153003 0.988226i \(-0.548894\pi\)
−0.153003 + 0.988226i \(0.548894\pi\)
\(798\) −48501.5 −2.15155
\(799\) 0 0
\(800\) −27345.1 −1.20849
\(801\) 13998.8 0.617507
\(802\) 14689.1i 0.646746i
\(803\) −184.844 −0.00812328
\(804\) 6958.17i 0.305219i
\(805\) − 58403.8i − 2.55710i
\(806\) 63.7684i 0.00278678i
\(807\) −14049.0 −0.612824
\(808\) −19740.4 −0.859485
\(809\) − 13606.4i − 0.591318i −0.955294 0.295659i \(-0.904461\pi\)
0.955294 0.295659i \(-0.0955392\pi\)
\(810\) 56474.9i 2.44978i
\(811\) 12718.7i 0.550694i 0.961345 + 0.275347i \(0.0887927\pi\)
−0.961345 + 0.275347i \(0.911207\pi\)
\(812\) 12829.1 0.554452
\(813\) 35410.7i 1.52756i
\(814\) 1649.56 0.0710283
\(815\) −36178.9 −1.55496
\(816\) 0 0
\(817\) −2107.19 −0.0902341
\(818\) −40546.2 −1.73309
\(819\) − 136.237i − 0.00581257i
\(820\) −19330.0 −0.823212
\(821\) 29119.6i 1.23786i 0.785447 + 0.618929i \(0.212433\pi\)
−0.785447 + 0.618929i \(0.787567\pi\)
\(822\) − 34507.8i − 1.46423i
\(823\) 6284.88i 0.266193i 0.991103 + 0.133097i \(0.0424921\pi\)
−0.991103 + 0.133097i \(0.957508\pi\)
\(824\) −6962.98 −0.294377
\(825\) −12924.8 −0.545437
\(826\) − 17138.5i − 0.721944i
\(827\) − 5383.60i − 0.226368i −0.993574 0.113184i \(-0.963895\pi\)
0.993574 0.113184i \(-0.0361049\pi\)
\(828\) 5590.81i 0.234655i
\(829\) 25382.6 1.06342 0.531709 0.846927i \(-0.321550\pi\)
0.531709 + 0.846927i \(0.321550\pi\)
\(830\) − 19441.6i − 0.813045i
\(831\) −21832.2 −0.911372
\(832\) −100.985 −0.00420798
\(833\) 0 0
\(834\) −21960.3 −0.911779
\(835\) −5545.83 −0.229846
\(836\) − 2229.71i − 0.0922440i
\(837\) 3690.88 0.152420
\(838\) 5369.10i 0.221328i
\(839\) − 2499.44i − 0.102849i −0.998677 0.0514246i \(-0.983624\pi\)
0.998677 0.0514246i \(-0.0163762\pi\)
\(840\) − 48308.9i − 1.98431i
\(841\) −25026.4 −1.02614
\(842\) −46893.7 −1.91932
\(843\) − 17390.0i − 0.710491i
\(844\) 7250.41i 0.295698i
\(845\) − 41975.8i − 1.70889i
\(846\) 2225.24 0.0904318
\(847\) − 27906.8i − 1.13210i
\(848\) 4656.65 0.188573
\(849\) −35538.0 −1.43659
\(850\) 0 0
\(851\) 8479.55 0.341569
\(852\) 17656.5 0.709979
\(853\) − 38293.9i − 1.53711i −0.639781 0.768557i \(-0.720975\pi\)
0.639781 0.768557i \(-0.279025\pi\)
\(854\) −42892.6 −1.71868
\(855\) 30572.6i 1.22288i
\(856\) − 5071.25i − 0.202490i
\(857\) − 426.303i − 0.0169921i −0.999964 0.00849606i \(-0.997296\pi\)
0.999964 0.00849606i \(-0.00270441\pi\)
\(858\) 69.8418 0.00277897
\(859\) −30976.1 −1.23037 −0.615186 0.788382i \(-0.710919\pi\)
−0.615186 + 0.788382i \(0.710919\pi\)
\(860\) 1017.16i 0.0403311i
\(861\) − 55797.9i − 2.20858i
\(862\) − 17434.0i − 0.688868i
\(863\) 38201.4 1.50682 0.753412 0.657549i \(-0.228407\pi\)
0.753412 + 0.657549i \(0.228407\pi\)
\(864\) − 8552.48i − 0.336761i
\(865\) 70044.8 2.75329
\(866\) 50745.8 1.99124
\(867\) 0 0
\(868\) 2836.66 0.110925
\(869\) −5638.81 −0.220119
\(870\) − 90178.8i − 3.51419i
\(871\) 162.818 0.00633397
\(872\) − 24208.6i − 0.940144i
\(873\) − 15582.7i − 0.604116i
\(874\) − 46574.1i − 1.80251i
\(875\) −48598.4 −1.87763
\(876\) 380.898 0.0146910
\(877\) − 13461.8i − 0.518327i −0.965833 0.259163i \(-0.916553\pi\)
0.965833 0.259163i \(-0.0834468\pi\)
\(878\) 54025.2i 2.07661i
\(879\) − 41329.5i − 1.58590i
\(880\) 12321.4 0.471994
\(881\) − 13867.4i − 0.530310i −0.964206 0.265155i \(-0.914577\pi\)
0.964206 0.265155i \(-0.0854231\pi\)
\(882\) −7330.85 −0.279867
\(883\) −330.794 −0.0126072 −0.00630358 0.999980i \(-0.502007\pi\)
−0.00630358 + 0.999980i \(0.502007\pi\)
\(884\) 0 0
\(885\) −29647.5 −1.12609
\(886\) −16819.6 −0.637772
\(887\) 42906.5i 1.62419i 0.583523 + 0.812097i \(0.301674\pi\)
−0.583523 + 0.812097i \(0.698326\pi\)
\(888\) 7013.90 0.265057
\(889\) 22173.1i 0.836515i
\(890\) 56291.7i 2.12012i
\(891\) − 7494.33i − 0.281784i
\(892\) −12120.4 −0.454954
\(893\) −4561.99 −0.170953
\(894\) − 13804.5i − 0.516433i
\(895\) 69334.3i 2.58948i
\(896\) 38389.9i 1.43138i
\(897\) 359.021 0.0133639
\(898\) 35089.7i 1.30396i
\(899\) −10926.3 −0.405353
\(900\) 9704.91 0.359441
\(901\) 0 0
\(902\) 10423.2 0.384762
\(903\) −2936.12 −0.108204
\(904\) 23950.1i 0.881158i
\(905\) 73138.6 2.68642
\(906\) − 21733.7i − 0.796968i
\(907\) − 6908.93i − 0.252930i −0.991971 0.126465i \(-0.959637\pi\)
0.991971 0.126465i \(-0.0403631\pi\)
\(908\) − 4471.60i − 0.163431i
\(909\) 17407.2 0.635162
\(910\) 547.833 0.0199566
\(911\) 33032.6i 1.20134i 0.799497 + 0.600669i \(0.205099\pi\)
−0.799497 + 0.600669i \(0.794901\pi\)
\(912\) − 52599.2i − 1.90980i
\(913\) 2579.94i 0.0935196i
\(914\) 16367.7 0.592337
\(915\) 74198.9i 2.68081i
\(916\) 15742.4 0.567841
\(917\) −4818.01 −0.173506
\(918\) 0 0
\(919\) 8167.05 0.293151 0.146576 0.989199i \(-0.453175\pi\)
0.146576 + 0.989199i \(0.453175\pi\)
\(920\) 46389.2 1.66240
\(921\) − 5348.33i − 0.191350i
\(922\) −7710.44 −0.275412
\(923\) − 413.155i − 0.0147337i
\(924\) − 3106.83i − 0.110614i
\(925\) − 14719.4i − 0.523211i
\(926\) 57700.3 2.04768
\(927\) 6140.03 0.217546
\(928\) 25318.3i 0.895599i
\(929\) 1864.03i 0.0658308i 0.999458 + 0.0329154i \(0.0104792\pi\)
−0.999458 + 0.0329154i \(0.989521\pi\)
\(930\) − 19939.5i − 0.703056i
\(931\) 15029.1 0.529063
\(932\) 6955.87i 0.244471i
\(933\) 48832.6 1.71351
\(934\) −63823.6 −2.23594
\(935\) 0 0
\(936\) 108.210 0.00377881
\(937\) 44071.0 1.53654 0.768270 0.640126i \(-0.221118\pi\)
0.768270 + 0.640126i \(0.221118\pi\)
\(938\) − 29430.5i − 1.02445i
\(939\) −7936.84 −0.275835
\(940\) 2202.11i 0.0764095i
\(941\) − 33030.1i − 1.14426i −0.820162 0.572131i \(-0.806117\pi\)
0.820162 0.572131i \(-0.193883\pi\)
\(942\) 14259.1i 0.493191i
\(943\) 53580.5 1.85029
\(944\) 18586.5 0.640825
\(945\) − 31708.2i − 1.09150i
\(946\) − 548.476i − 0.0188504i
\(947\) − 9340.37i − 0.320508i −0.987076 0.160254i \(-0.948769\pi\)
0.987076 0.160254i \(-0.0512314\pi\)
\(948\) 11619.6 0.398088
\(949\) − 8.91285i 0 0.000304872i
\(950\) −80846.5 −2.76106
\(951\) 37034.0 1.26278
\(952\) 0 0
\(953\) 21211.2 0.720984 0.360492 0.932762i \(-0.382609\pi\)
0.360492 + 0.932762i \(0.382609\pi\)
\(954\) −3007.46 −0.102065
\(955\) − 16761.9i − 0.567959i
\(956\) 15722.9 0.531919
\(957\) 11966.9i 0.404217i
\(958\) 13226.9i 0.446077i
\(959\) 35919.3i 1.20948i
\(960\) 31576.7 1.06160
\(961\) 27375.1 0.918904
\(962\) 79.5389i 0.00266574i
\(963\) 4471.88i 0.149641i
\(964\) 15880.2i 0.530567i
\(965\) −91136.3 −3.04019
\(966\) − 64895.4i − 2.16147i
\(967\) −21688.1 −0.721242 −0.360621 0.932712i \(-0.617435\pi\)
−0.360621 + 0.932712i \(0.617435\pi\)
\(968\) 22165.9 0.735991
\(969\) 0 0
\(970\) 62660.8 2.07414
\(971\) 22478.6 0.742918 0.371459 0.928449i \(-0.378858\pi\)
0.371459 + 0.928449i \(0.378858\pi\)
\(972\) 10148.5i 0.334892i
\(973\) 22858.5 0.753146
\(974\) − 23639.7i − 0.777685i
\(975\) − 623.214i − 0.0204706i
\(976\) − 46516.5i − 1.52557i
\(977\) −21731.7 −0.711626 −0.355813 0.934557i \(-0.615796\pi\)
−0.355813 + 0.934557i \(0.615796\pi\)
\(978\) −40200.2 −1.31438
\(979\) − 7470.03i − 0.243864i
\(980\) − 7254.65i − 0.236471i
\(981\) 21347.3i 0.694769i
\(982\) −2507.82 −0.0814947
\(983\) 34128.8i 1.10737i 0.832728 + 0.553683i \(0.186778\pi\)
−0.832728 + 0.553683i \(0.813222\pi\)
\(984\) 44319.3 1.43582
\(985\) 28526.7 0.922779
\(986\) 0 0
\(987\) −6356.59 −0.204998
\(988\) 107.513 0.00346197
\(989\) − 2819.44i − 0.0906501i
\(990\) −7957.68 −0.255466
\(991\) 39750.8i 1.27419i 0.770784 + 0.637097i \(0.219865\pi\)
−0.770784 + 0.637097i \(0.780135\pi\)
\(992\) 5598.16i 0.179175i
\(993\) 48151.4i 1.53881i
\(994\) −74680.5 −2.38302
\(995\) 59066.2 1.88193
\(996\) − 5316.34i − 0.169131i
\(997\) − 54112.0i − 1.71890i −0.511219 0.859450i \(-0.670806\pi\)
0.511219 0.859450i \(-0.329194\pi\)
\(998\) 4666.91i 0.148024i
\(999\) 4603.66 0.145799
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 289.4.b.f.288.7 24
17.4 even 4 289.4.a.i.1.9 yes 12
17.13 even 4 289.4.a.h.1.9 12
17.16 even 2 inner 289.4.b.f.288.8 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
289.4.a.h.1.9 12 17.13 even 4
289.4.a.i.1.9 yes 12 17.4 even 4
289.4.b.f.288.7 24 1.1 even 1 trivial
289.4.b.f.288.8 24 17.16 even 2 inner