Properties

Label 289.4.a.g.1.4
Level $289$
Weight $4$
Character 289.1
Self dual yes
Analytic conductor $17.052$
Analytic rank $1$
Dimension $12$
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(1,289)]
 
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([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 289.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: \(17.0515519917\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 58 x^{10} + 204 x^{9} + 1191 x^{8} - 3456 x^{7} - 10364 x^{6} + 21448 x^{5} + 38476 x^{4} - 32336 x^{3} - 57024 x^{2} - 15776 x + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.651949\) of defining polynomial
Character \(\chi\) \(=\) 289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.49971 q^{2} +2.82539 q^{3} +4.24796 q^{4} +8.71933 q^{5} -9.88806 q^{6} +6.85501 q^{7} +13.1311 q^{8} -19.0171 q^{9} +O(q^{10})\) \(q-3.49971 q^{2} +2.82539 q^{3} +4.24796 q^{4} +8.71933 q^{5} -9.88806 q^{6} +6.85501 q^{7} +13.1311 q^{8} -19.0171 q^{9} -30.5151 q^{10} -61.7928 q^{11} +12.0022 q^{12} -5.37363 q^{13} -23.9905 q^{14} +24.6355 q^{15} -79.9385 q^{16} +66.5545 q^{18} +96.7877 q^{19} +37.0393 q^{20} +19.3681 q^{21} +216.257 q^{22} -116.525 q^{23} +37.1004 q^{24} -48.9733 q^{25} +18.8061 q^{26} -130.017 q^{27} +29.1198 q^{28} +197.375 q^{29} -86.2172 q^{30} -138.070 q^{31} +174.713 q^{32} -174.589 q^{33} +59.7710 q^{35} -80.7840 q^{36} -111.428 q^{37} -338.729 q^{38} -15.1826 q^{39} +114.494 q^{40} +166.487 q^{41} -67.7827 q^{42} +165.887 q^{43} -262.493 q^{44} -165.817 q^{45} +407.804 q^{46} -130.994 q^{47} -225.858 q^{48} -296.009 q^{49} +171.392 q^{50} -22.8269 q^{52} -714.232 q^{53} +455.020 q^{54} -538.792 q^{55} +90.0135 q^{56} +273.463 q^{57} -690.754 q^{58} -846.216 q^{59} +104.651 q^{60} +4.99209 q^{61} +483.205 q^{62} -130.363 q^{63} +28.0634 q^{64} -46.8544 q^{65} +611.011 q^{66} -314.069 q^{67} -329.230 q^{69} -209.181 q^{70} +118.468 q^{71} -249.715 q^{72} -650.548 q^{73} +389.965 q^{74} -138.369 q^{75} +411.150 q^{76} -423.590 q^{77} +53.1347 q^{78} -208.405 q^{79} -697.010 q^{80} +146.115 q^{81} -582.655 q^{82} -742.430 q^{83} +82.2749 q^{84} -580.555 q^{86} +557.662 q^{87} -811.405 q^{88} +215.527 q^{89} +580.310 q^{90} -36.8362 q^{91} -494.994 q^{92} -390.102 q^{93} +458.441 q^{94} +843.923 q^{95} +493.633 q^{96} +705.690 q^{97} +1035.94 q^{98} +1175.12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{2} + 16 q^{4} - 96 q^{8} - 36 q^{9} - 8 q^{13} - 192 q^{15} - 184 q^{16} - 352 q^{19} - 256 q^{21} - 492 q^{25} - 784 q^{26} + 744 q^{30} + 24 q^{32} - 1400 q^{33} - 632 q^{35} - 856 q^{36} - 624 q^{38} - 1664 q^{42} - 1200 q^{43} - 1512 q^{47} - 1052 q^{49} - 2856 q^{50} + 792 q^{52} - 2504 q^{53} - 1424 q^{55} - 3408 q^{59} - 2808 q^{60} + 272 q^{64} + 272 q^{66} - 1080 q^{67} - 344 q^{69} + 2600 q^{70} + 248 q^{72} + 896 q^{76} + 848 q^{77} - 2404 q^{81} - 2960 q^{83} + 4768 q^{84} - 1200 q^{86} - 160 q^{87} - 2144 q^{89} + 3800 q^{93} + 5984 q^{94} + 3464 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.49971 −1.23733 −0.618667 0.785653i \(-0.712327\pi\)
−0.618667 + 0.785653i \(0.712327\pi\)
\(3\) 2.82539 0.543747 0.271874 0.962333i \(-0.412357\pi\)
0.271874 + 0.962333i \(0.412357\pi\)
\(4\) 4.24796 0.530995
\(5\) 8.71933 0.779880 0.389940 0.920840i \(-0.372496\pi\)
0.389940 + 0.920840i \(0.372496\pi\)
\(6\) −9.88806 −0.672797
\(7\) 6.85501 0.370136 0.185068 0.982726i \(-0.440750\pi\)
0.185068 + 0.982726i \(0.440750\pi\)
\(8\) 13.1311 0.580316
\(9\) −19.0171 −0.704339
\(10\) −30.5151 −0.964972
\(11\) −61.7928 −1.69375 −0.846875 0.531793i \(-0.821519\pi\)
−0.846875 + 0.531793i \(0.821519\pi\)
\(12\) 12.0022 0.288727
\(13\) −5.37363 −0.114644 −0.0573221 0.998356i \(-0.518256\pi\)
−0.0573221 + 0.998356i \(0.518256\pi\)
\(14\) −23.9905 −0.457981
\(15\) 24.6355 0.424058
\(16\) −79.9385 −1.24904
\(17\) 0 0
\(18\) 66.5545 0.871502
\(19\) 96.7877 1.16866 0.584332 0.811515i \(-0.301357\pi\)
0.584332 + 0.811515i \(0.301357\pi\)
\(20\) 37.0393 0.414112
\(21\) 19.3681 0.201260
\(22\) 216.257 2.09573
\(23\) −116.525 −1.05640 −0.528200 0.849120i \(-0.677133\pi\)
−0.528200 + 0.849120i \(0.677133\pi\)
\(24\) 37.1004 0.315545
\(25\) −48.9733 −0.391787
\(26\) 18.8061 0.141853
\(27\) −130.017 −0.926730
\(28\) 29.1198 0.196540
\(29\) 197.375 1.26385 0.631924 0.775031i \(-0.282266\pi\)
0.631924 + 0.775031i \(0.282266\pi\)
\(30\) −86.2172 −0.524701
\(31\) −138.070 −0.799939 −0.399970 0.916528i \(-0.630979\pi\)
−0.399970 + 0.916528i \(0.630979\pi\)
\(32\) 174.713 0.965162
\(33\) −174.589 −0.920972
\(34\) 0 0
\(35\) 59.7710 0.288661
\(36\) −80.7840 −0.374000
\(37\) −111.428 −0.495098 −0.247549 0.968875i \(-0.579625\pi\)
−0.247549 + 0.968875i \(0.579625\pi\)
\(38\) −338.729 −1.44603
\(39\) −15.1826 −0.0623375
\(40\) 114.494 0.452577
\(41\) 166.487 0.634167 0.317084 0.948398i \(-0.397296\pi\)
0.317084 + 0.948398i \(0.397296\pi\)
\(42\) −67.7827 −0.249026
\(43\) 165.887 0.588314 0.294157 0.955757i \(-0.404961\pi\)
0.294157 + 0.955757i \(0.404961\pi\)
\(44\) −262.493 −0.899372
\(45\) −165.817 −0.549300
\(46\) 407.804 1.30712
\(47\) −130.994 −0.406541 −0.203271 0.979123i \(-0.565157\pi\)
−0.203271 + 0.979123i \(0.565157\pi\)
\(48\) −225.858 −0.679162
\(49\) −296.009 −0.863000
\(50\) 171.392 0.484771
\(51\) 0 0
\(52\) −22.8269 −0.0608755
\(53\) −714.232 −1.85108 −0.925541 0.378648i \(-0.876389\pi\)
−0.925541 + 0.378648i \(0.876389\pi\)
\(54\) 455.020 1.14667
\(55\) −538.792 −1.32092
\(56\) 90.0135 0.214796
\(57\) 273.463 0.635458
\(58\) −690.754 −1.56380
\(59\) −846.216 −1.86725 −0.933627 0.358247i \(-0.883375\pi\)
−0.933627 + 0.358247i \(0.883375\pi\)
\(60\) 104.651 0.225172
\(61\) 4.99209 0.0104782 0.00523911 0.999986i \(-0.498332\pi\)
0.00523911 + 0.999986i \(0.498332\pi\)
\(62\) 483.205 0.989792
\(63\) −130.363 −0.260701
\(64\) 28.0634 0.0548113
\(65\) −46.8544 −0.0894088
\(66\) 611.011 1.13955
\(67\) −314.069 −0.572681 −0.286341 0.958128i \(-0.592439\pi\)
−0.286341 + 0.958128i \(0.592439\pi\)
\(68\) 0 0
\(69\) −329.230 −0.574415
\(70\) −209.181 −0.357171
\(71\) 118.468 0.198022 0.0990108 0.995086i \(-0.468432\pi\)
0.0990108 + 0.995086i \(0.468432\pi\)
\(72\) −249.715 −0.408739
\(73\) −650.548 −1.04303 −0.521513 0.853243i \(-0.674632\pi\)
−0.521513 + 0.853243i \(0.674632\pi\)
\(74\) 389.965 0.612601
\(75\) −138.369 −0.213033
\(76\) 411.150 0.620554
\(77\) −423.590 −0.626917
\(78\) 53.1347 0.0771323
\(79\) −208.405 −0.296803 −0.148402 0.988927i \(-0.547413\pi\)
−0.148402 + 0.988927i \(0.547413\pi\)
\(80\) −697.010 −0.974101
\(81\) 146.115 0.200432
\(82\) −582.655 −0.784677
\(83\) −742.430 −0.981834 −0.490917 0.871206i \(-0.663338\pi\)
−0.490917 + 0.871206i \(0.663338\pi\)
\(84\) 82.2749 0.106868
\(85\) 0 0
\(86\) −580.555 −0.727940
\(87\) 557.662 0.687214
\(88\) −811.405 −0.982910
\(89\) 215.527 0.256695 0.128348 0.991729i \(-0.459033\pi\)
0.128348 + 0.991729i \(0.459033\pi\)
\(90\) 580.310 0.679667
\(91\) −36.8362 −0.0424339
\(92\) −494.994 −0.560943
\(93\) −390.102 −0.434965
\(94\) 458.441 0.503027
\(95\) 843.923 0.911418
\(96\) 493.633 0.524805
\(97\) 705.690 0.738680 0.369340 0.929294i \(-0.379584\pi\)
0.369340 + 0.929294i \(0.379584\pi\)
\(98\) 1035.94 1.06782
\(99\) 1175.12 1.19297
\(100\) −208.037 −0.208037
\(101\) −546.988 −0.538884 −0.269442 0.963017i \(-0.586839\pi\)
−0.269442 + 0.963017i \(0.586839\pi\)
\(102\) 0 0
\(103\) 1550.96 1.48369 0.741846 0.670570i \(-0.233950\pi\)
0.741846 + 0.670570i \(0.233950\pi\)
\(104\) −70.5614 −0.0665299
\(105\) 168.877 0.156959
\(106\) 2499.60 2.29041
\(107\) −245.779 −0.222059 −0.111029 0.993817i \(-0.535415\pi\)
−0.111029 + 0.993817i \(0.535415\pi\)
\(108\) −552.305 −0.492089
\(109\) 1993.39 1.75167 0.875834 0.482612i \(-0.160312\pi\)
0.875834 + 0.482612i \(0.160312\pi\)
\(110\) 1885.61 1.63442
\(111\) −314.827 −0.269208
\(112\) −547.979 −0.462314
\(113\) −1082.75 −0.901387 −0.450693 0.892679i \(-0.648823\pi\)
−0.450693 + 0.892679i \(0.648823\pi\)
\(114\) −957.042 −0.786273
\(115\) −1016.02 −0.823865
\(116\) 838.440 0.671096
\(117\) 102.191 0.0807484
\(118\) 2961.51 2.31042
\(119\) 0 0
\(120\) 323.491 0.246088
\(121\) 2487.35 1.86879
\(122\) −17.4708 −0.0129651
\(123\) 470.391 0.344827
\(124\) −586.516 −0.424764
\(125\) −1516.93 −1.08543
\(126\) 456.231 0.322574
\(127\) 1016.07 0.709932 0.354966 0.934879i \(-0.384492\pi\)
0.354966 + 0.934879i \(0.384492\pi\)
\(128\) −1495.92 −1.03298
\(129\) 468.695 0.319894
\(130\) 163.977 0.110629
\(131\) −1108.85 −0.739550 −0.369775 0.929121i \(-0.620565\pi\)
−0.369775 + 0.929121i \(0.620565\pi\)
\(132\) −741.647 −0.489031
\(133\) 663.480 0.432564
\(134\) 1099.15 0.708598
\(135\) −1133.66 −0.722738
\(136\) 0 0
\(137\) −1975.27 −1.23182 −0.615909 0.787818i \(-0.711211\pi\)
−0.615909 + 0.787818i \(0.711211\pi\)
\(138\) 1152.21 0.710743
\(139\) −1038.50 −0.633702 −0.316851 0.948475i \(-0.602626\pi\)
−0.316851 + 0.948475i \(0.602626\pi\)
\(140\) 253.905 0.153278
\(141\) −370.110 −0.221056
\(142\) −414.603 −0.245019
\(143\) 332.052 0.194179
\(144\) 1520.20 0.879747
\(145\) 1720.97 0.985650
\(146\) 2276.73 1.29057
\(147\) −836.342 −0.469254
\(148\) −473.340 −0.262894
\(149\) 119.562 0.0657377 0.0328689 0.999460i \(-0.489536\pi\)
0.0328689 + 0.999460i \(0.489536\pi\)
\(150\) 484.251 0.263593
\(151\) −1567.85 −0.844966 −0.422483 0.906371i \(-0.638841\pi\)
−0.422483 + 0.906371i \(0.638841\pi\)
\(152\) 1270.92 0.678194
\(153\) 0 0
\(154\) 1482.44 0.775705
\(155\) −1203.88 −0.623857
\(156\) −64.4951 −0.0331009
\(157\) 2121.62 1.07850 0.539248 0.842147i \(-0.318709\pi\)
0.539248 + 0.842147i \(0.318709\pi\)
\(158\) 729.358 0.367245
\(159\) −2017.99 −1.00652
\(160\) 1523.38 0.752711
\(161\) −798.782 −0.391011
\(162\) −511.359 −0.248001
\(163\) 443.287 0.213012 0.106506 0.994312i \(-0.466034\pi\)
0.106506 + 0.994312i \(0.466034\pi\)
\(164\) 707.229 0.336740
\(165\) −1522.30 −0.718248
\(166\) 2598.29 1.21486
\(167\) 2352.44 1.09005 0.545023 0.838421i \(-0.316521\pi\)
0.545023 + 0.838421i \(0.316521\pi\)
\(168\) 254.324 0.116795
\(169\) −2168.12 −0.986857
\(170\) 0 0
\(171\) −1840.63 −0.823135
\(172\) 704.680 0.312391
\(173\) 1900.27 0.835116 0.417558 0.908650i \(-0.362886\pi\)
0.417558 + 0.908650i \(0.362886\pi\)
\(174\) −1951.65 −0.850313
\(175\) −335.713 −0.145014
\(176\) 4939.63 2.11556
\(177\) −2390.89 −1.01531
\(178\) −754.283 −0.317617
\(179\) 99.7302 0.0416435 0.0208217 0.999783i \(-0.493372\pi\)
0.0208217 + 0.999783i \(0.493372\pi\)
\(180\) −704.382 −0.291675
\(181\) 3878.82 1.59287 0.796437 0.604721i \(-0.206715\pi\)
0.796437 + 0.604721i \(0.206715\pi\)
\(182\) 128.916 0.0525049
\(183\) 14.1046 0.00569750
\(184\) −1530.10 −0.613046
\(185\) −971.575 −0.386117
\(186\) 1365.24 0.538197
\(187\) 0 0
\(188\) −556.457 −0.215871
\(189\) −891.265 −0.343016
\(190\) −2953.48 −1.12773
\(191\) −2326.90 −0.881511 −0.440756 0.897627i \(-0.645289\pi\)
−0.440756 + 0.897627i \(0.645289\pi\)
\(192\) 79.2901 0.0298035
\(193\) 1396.48 0.520835 0.260418 0.965496i \(-0.416140\pi\)
0.260418 + 0.965496i \(0.416140\pi\)
\(194\) −2469.71 −0.913994
\(195\) −132.382 −0.0486158
\(196\) −1257.43 −0.458248
\(197\) −2023.83 −0.731939 −0.365969 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(198\) −4112.59 −1.47611
\(199\) 4143.34 1.47595 0.737973 0.674830i \(-0.235783\pi\)
0.737973 + 0.674830i \(0.235783\pi\)
\(200\) −643.072 −0.227360
\(201\) −887.369 −0.311394
\(202\) 1914.30 0.666780
\(203\) 1353.01 0.467795
\(204\) 0 0
\(205\) 1451.65 0.494575
\(206\) −5427.90 −1.83582
\(207\) 2215.98 0.744063
\(208\) 429.560 0.143195
\(209\) −5980.78 −1.97942
\(210\) −591.019 −0.194211
\(211\) −2350.55 −0.766912 −0.383456 0.923559i \(-0.625266\pi\)
−0.383456 + 0.923559i \(0.625266\pi\)
\(212\) −3034.03 −0.982914
\(213\) 334.718 0.107674
\(214\) 860.153 0.274761
\(215\) 1446.42 0.458814
\(216\) −1707.25 −0.537796
\(217\) −946.471 −0.296086
\(218\) −6976.27 −2.16740
\(219\) −1838.05 −0.567143
\(220\) −2288.77 −0.701402
\(221\) 0 0
\(222\) 1101.80 0.333100
\(223\) 412.388 0.123837 0.0619183 0.998081i \(-0.480278\pi\)
0.0619183 + 0.998081i \(0.480278\pi\)
\(224\) 1197.66 0.357241
\(225\) 931.333 0.275951
\(226\) 3789.31 1.11532
\(227\) 1915.89 0.560186 0.280093 0.959973i \(-0.409635\pi\)
0.280093 + 0.959973i \(0.409635\pi\)
\(228\) 1161.66 0.337425
\(229\) −4863.54 −1.40346 −0.701729 0.712444i \(-0.747588\pi\)
−0.701729 + 0.712444i \(0.747588\pi\)
\(230\) 3555.78 1.01940
\(231\) −1196.81 −0.340884
\(232\) 2591.74 0.733431
\(233\) 555.672 0.156237 0.0781186 0.996944i \(-0.475109\pi\)
0.0781186 + 0.996944i \(0.475109\pi\)
\(234\) −357.639 −0.0999127
\(235\) −1142.18 −0.317053
\(236\) −3594.69 −0.991502
\(237\) −588.828 −0.161386
\(238\) 0 0
\(239\) −3205.56 −0.867575 −0.433787 0.901015i \(-0.642823\pi\)
−0.433787 + 0.901015i \(0.642823\pi\)
\(240\) −1969.33 −0.529665
\(241\) 893.396 0.238791 0.119396 0.992847i \(-0.461904\pi\)
0.119396 + 0.992847i \(0.461904\pi\)
\(242\) −8705.02 −2.31231
\(243\) 3923.28 1.03571
\(244\) 21.2062 0.00556388
\(245\) −2581.00 −0.673036
\(246\) −1646.23 −0.426666
\(247\) −520.101 −0.133981
\(248\) −1813.01 −0.464218
\(249\) −2097.66 −0.533870
\(250\) 5308.81 1.34304
\(251\) −2431.39 −0.611427 −0.305713 0.952124i \(-0.598895\pi\)
−0.305713 + 0.952124i \(0.598895\pi\)
\(252\) −553.775 −0.138431
\(253\) 7200.43 1.78928
\(254\) −3555.94 −0.878422
\(255\) 0 0
\(256\) 5010.77 1.22333
\(257\) −3215.44 −0.780442 −0.390221 0.920721i \(-0.627601\pi\)
−0.390221 + 0.920721i \(0.627601\pi\)
\(258\) −1640.30 −0.395816
\(259\) −763.838 −0.183253
\(260\) −199.035 −0.0474756
\(261\) −3753.50 −0.890177
\(262\) 3880.67 0.915070
\(263\) 2659.16 0.623464 0.311732 0.950170i \(-0.399091\pi\)
0.311732 + 0.950170i \(0.399091\pi\)
\(264\) −2292.54 −0.534455
\(265\) −6227.62 −1.44362
\(266\) −2321.99 −0.535226
\(267\) 608.950 0.139577
\(268\) −1334.15 −0.304091
\(269\) 5712.40 1.29476 0.647381 0.762166i \(-0.275864\pi\)
0.647381 + 0.762166i \(0.275864\pi\)
\(270\) 3967.47 0.894268
\(271\) −250.885 −0.0562369 −0.0281185 0.999605i \(-0.508952\pi\)
−0.0281185 + 0.999605i \(0.508952\pi\)
\(272\) 0 0
\(273\) −104.077 −0.0230733
\(274\) 6912.88 1.52417
\(275\) 3026.20 0.663589
\(276\) −1398.55 −0.305011
\(277\) 3688.27 0.800024 0.400012 0.916510i \(-0.369006\pi\)
0.400012 + 0.916510i \(0.369006\pi\)
\(278\) 3634.46 0.784101
\(279\) 2625.70 0.563428
\(280\) 784.857 0.167515
\(281\) 3532.00 0.749827 0.374914 0.927060i \(-0.377672\pi\)
0.374914 + 0.927060i \(0.377672\pi\)
\(282\) 1295.28 0.273520
\(283\) −788.220 −0.165565 −0.0827824 0.996568i \(-0.526381\pi\)
−0.0827824 + 0.996568i \(0.526381\pi\)
\(284\) 503.246 0.105148
\(285\) 2384.42 0.495581
\(286\) −1162.08 −0.240264
\(287\) 1141.27 0.234728
\(288\) −3322.54 −0.679801
\(289\) 0 0
\(290\) −6022.91 −1.21958
\(291\) 1993.85 0.401656
\(292\) −2763.50 −0.553841
\(293\) −1413.82 −0.281899 −0.140949 0.990017i \(-0.545015\pi\)
−0.140949 + 0.990017i \(0.545015\pi\)
\(294\) 2926.95 0.580624
\(295\) −7378.44 −1.45623
\(296\) −1463.16 −0.287313
\(297\) 8034.09 1.56965
\(298\) −418.433 −0.0813395
\(299\) 626.163 0.121110
\(300\) −587.786 −0.113119
\(301\) 1137.15 0.217756
\(302\) 5487.02 1.04551
\(303\) −1545.46 −0.293017
\(304\) −7737.06 −1.45971
\(305\) 43.5276 0.00817175
\(306\) 0 0
\(307\) 4499.58 0.836498 0.418249 0.908333i \(-0.362644\pi\)
0.418249 + 0.908333i \(0.362644\pi\)
\(308\) −1799.39 −0.332890
\(309\) 4382.06 0.806754
\(310\) 4213.22 0.771919
\(311\) 4716.79 0.860014 0.430007 0.902825i \(-0.358511\pi\)
0.430007 + 0.902825i \(0.358511\pi\)
\(312\) −199.364 −0.0361755
\(313\) 6171.31 1.11445 0.557225 0.830361i \(-0.311866\pi\)
0.557225 + 0.830361i \(0.311866\pi\)
\(314\) −7425.05 −1.33446
\(315\) −1136.67 −0.203315
\(316\) −885.298 −0.157601
\(317\) 1186.87 0.210287 0.105144 0.994457i \(-0.466470\pi\)
0.105144 + 0.994457i \(0.466470\pi\)
\(318\) 7062.37 1.24540
\(319\) −12196.3 −2.14064
\(320\) 244.694 0.0427462
\(321\) −694.421 −0.120744
\(322\) 2795.50 0.483811
\(323\) 0 0
\(324\) 620.690 0.106428
\(325\) 263.164 0.0449161
\(326\) −1551.37 −0.263567
\(327\) 5632.10 0.952465
\(328\) 2186.15 0.368018
\(329\) −897.965 −0.150475
\(330\) 5327.60 0.888712
\(331\) 3070.09 0.509811 0.254905 0.966966i \(-0.417956\pi\)
0.254905 + 0.966966i \(0.417956\pi\)
\(332\) −3153.81 −0.521349
\(333\) 2119.04 0.348716
\(334\) −8232.87 −1.34875
\(335\) −2738.47 −0.446623
\(336\) −1548.26 −0.251382
\(337\) −4437.46 −0.717281 −0.358640 0.933476i \(-0.616760\pi\)
−0.358640 + 0.933476i \(0.616760\pi\)
\(338\) 7587.80 1.22107
\(339\) −3059.20 −0.490127
\(340\) 0 0
\(341\) 8531.74 1.35490
\(342\) 6441.65 1.01849
\(343\) −4380.41 −0.689563
\(344\) 2178.27 0.341408
\(345\) −2870.66 −0.447975
\(346\) −6650.40 −1.03332
\(347\) −8714.21 −1.34814 −0.674068 0.738669i \(-0.735455\pi\)
−0.674068 + 0.738669i \(0.735455\pi\)
\(348\) 2368.92 0.364907
\(349\) −100.929 −0.0154803 −0.00774014 0.999970i \(-0.502464\pi\)
−0.00774014 + 0.999970i \(0.502464\pi\)
\(350\) 1174.90 0.179431
\(351\) 698.660 0.106244
\(352\) −10796.0 −1.63474
\(353\) 8688.88 1.31009 0.655046 0.755589i \(-0.272649\pi\)
0.655046 + 0.755589i \(0.272649\pi\)
\(354\) 8367.43 1.25628
\(355\) 1032.96 0.154433
\(356\) 915.551 0.136304
\(357\) 0 0
\(358\) −349.026 −0.0515269
\(359\) 2411.52 0.354526 0.177263 0.984163i \(-0.443276\pi\)
0.177263 + 0.984163i \(0.443276\pi\)
\(360\) −2177.35 −0.318768
\(361\) 2508.85 0.365775
\(362\) −13574.7 −1.97092
\(363\) 7027.76 1.01615
\(364\) −156.479 −0.0225322
\(365\) −5672.34 −0.813435
\(366\) −49.3620 −0.00704971
\(367\) −2274.20 −0.323467 −0.161734 0.986834i \(-0.551709\pi\)
−0.161734 + 0.986834i \(0.551709\pi\)
\(368\) 9314.86 1.31948
\(369\) −3166.10 −0.446669
\(370\) 3400.23 0.477755
\(371\) −4896.07 −0.685151
\(372\) −1657.14 −0.230964
\(373\) 9493.93 1.31790 0.658951 0.752186i \(-0.271001\pi\)
0.658951 + 0.752186i \(0.271001\pi\)
\(374\) 0 0
\(375\) −4285.93 −0.590198
\(376\) −1720.09 −0.235922
\(377\) −1060.62 −0.144893
\(378\) 3119.17 0.424425
\(379\) −5337.70 −0.723428 −0.361714 0.932289i \(-0.617808\pi\)
−0.361714 + 0.932289i \(0.617808\pi\)
\(380\) 3584.95 0.483958
\(381\) 2870.79 0.386024
\(382\) 8143.47 1.09072
\(383\) −779.426 −0.103986 −0.0519932 0.998647i \(-0.516557\pi\)
−0.0519932 + 0.998647i \(0.516557\pi\)
\(384\) −4226.56 −0.561681
\(385\) −3693.42 −0.488920
\(386\) −4887.29 −0.644447
\(387\) −3154.69 −0.414372
\(388\) 2997.74 0.392235
\(389\) −10826.1 −1.41106 −0.705532 0.708679i \(-0.749292\pi\)
−0.705532 + 0.708679i \(0.749292\pi\)
\(390\) 463.299 0.0601540
\(391\) 0 0
\(392\) −3886.91 −0.500813
\(393\) −3132.95 −0.402128
\(394\) 7082.81 0.905652
\(395\) −1817.16 −0.231471
\(396\) 4991.88 0.633463
\(397\) 6106.87 0.772027 0.386014 0.922493i \(-0.373852\pi\)
0.386014 + 0.922493i \(0.373852\pi\)
\(398\) −14500.5 −1.82624
\(399\) 1874.59 0.235206
\(400\) 3914.86 0.489357
\(401\) 11451.6 1.42609 0.713047 0.701116i \(-0.247314\pi\)
0.713047 + 0.701116i \(0.247314\pi\)
\(402\) 3105.53 0.385298
\(403\) 741.937 0.0917085
\(404\) −2323.58 −0.286145
\(405\) 1274.02 0.156313
\(406\) −4735.12 −0.578818
\(407\) 6885.44 0.838571
\(408\) 0 0
\(409\) −7597.11 −0.918466 −0.459233 0.888316i \(-0.651876\pi\)
−0.459233 + 0.888316i \(0.651876\pi\)
\(410\) −5080.36 −0.611954
\(411\) −5580.93 −0.669798
\(412\) 6588.40 0.787833
\(413\) −5800.82 −0.691137
\(414\) −7755.28 −0.920655
\(415\) −6473.49 −0.765713
\(416\) −938.843 −0.110650
\(417\) −2934.18 −0.344574
\(418\) 20931.0 2.44921
\(419\) 6887.55 0.803052 0.401526 0.915848i \(-0.368480\pi\)
0.401526 + 0.915848i \(0.368480\pi\)
\(420\) 717.381 0.0833444
\(421\) 13586.7 1.57286 0.786431 0.617678i \(-0.211927\pi\)
0.786431 + 0.617678i \(0.211927\pi\)
\(422\) 8226.24 0.948926
\(423\) 2491.13 0.286343
\(424\) −9378.62 −1.07421
\(425\) 0 0
\(426\) −1171.42 −0.133228
\(427\) 34.2208 0.00387836
\(428\) −1044.06 −0.117912
\(429\) 938.177 0.105584
\(430\) −5062.05 −0.567706
\(431\) 16814.5 1.87917 0.939587 0.342311i \(-0.111210\pi\)
0.939587 + 0.342311i \(0.111210\pi\)
\(432\) 10393.3 1.15752
\(433\) −248.824 −0.0276159 −0.0138080 0.999905i \(-0.504395\pi\)
−0.0138080 + 0.999905i \(0.504395\pi\)
\(434\) 3312.37 0.366357
\(435\) 4862.43 0.535944
\(436\) 8467.82 0.930127
\(437\) −11278.2 −1.23458
\(438\) 6432.66 0.701745
\(439\) 9870.05 1.07306 0.536528 0.843882i \(-0.319735\pi\)
0.536528 + 0.843882i \(0.319735\pi\)
\(440\) −7074.91 −0.766552
\(441\) 5629.24 0.607844
\(442\) 0 0
\(443\) −10000.3 −1.07253 −0.536264 0.844051i \(-0.680165\pi\)
−0.536264 + 0.844051i \(0.680165\pi\)
\(444\) −1337.37 −0.142948
\(445\) 1879.25 0.200191
\(446\) −1443.24 −0.153227
\(447\) 337.810 0.0357447
\(448\) 192.375 0.0202876
\(449\) 8580.38 0.901856 0.450928 0.892560i \(-0.351093\pi\)
0.450928 + 0.892560i \(0.351093\pi\)
\(450\) −3259.39 −0.341443
\(451\) −10287.7 −1.07412
\(452\) −4599.48 −0.478632
\(453\) −4429.80 −0.459448
\(454\) −6705.07 −0.693137
\(455\) −321.187 −0.0330934
\(456\) 3590.86 0.368766
\(457\) 8345.86 0.854273 0.427137 0.904187i \(-0.359522\pi\)
0.427137 + 0.904187i \(0.359522\pi\)
\(458\) 17021.0 1.73655
\(459\) 0 0
\(460\) −4316.02 −0.437468
\(461\) −8600.87 −0.868942 −0.434471 0.900686i \(-0.643065\pi\)
−0.434471 + 0.900686i \(0.643065\pi\)
\(462\) 4188.48 0.421788
\(463\) −15888.5 −1.59482 −0.797408 0.603440i \(-0.793796\pi\)
−0.797408 + 0.603440i \(0.793796\pi\)
\(464\) −15777.8 −1.57860
\(465\) −3401.43 −0.339221
\(466\) −1944.69 −0.193318
\(467\) −5941.71 −0.588757 −0.294378 0.955689i \(-0.595113\pi\)
−0.294378 + 0.955689i \(0.595113\pi\)
\(468\) 434.103 0.0428770
\(469\) −2152.95 −0.211970
\(470\) 3997.29 0.392301
\(471\) 5994.41 0.586429
\(472\) −11111.7 −1.08360
\(473\) −10250.6 −0.996456
\(474\) 2060.72 0.199688
\(475\) −4740.02 −0.457867
\(476\) 0 0
\(477\) 13582.7 1.30379
\(478\) 11218.5 1.07348
\(479\) −14915.1 −1.42273 −0.711366 0.702822i \(-0.751923\pi\)
−0.711366 + 0.702822i \(0.751923\pi\)
\(480\) 4304.15 0.409285
\(481\) 598.771 0.0567601
\(482\) −3126.63 −0.295465
\(483\) −2256.87 −0.212611
\(484\) 10566.2 0.992316
\(485\) 6153.15 0.576082
\(486\) −13730.3 −1.28152
\(487\) −8222.85 −0.765119 −0.382559 0.923931i \(-0.624957\pi\)
−0.382559 + 0.923931i \(0.624957\pi\)
\(488\) 65.5514 0.00608068
\(489\) 1252.46 0.115825
\(490\) 9032.74 0.832771
\(491\) −2635.62 −0.242248 −0.121124 0.992637i \(-0.538650\pi\)
−0.121124 + 0.992637i \(0.538650\pi\)
\(492\) 1998.20 0.183101
\(493\) 0 0
\(494\) 1820.20 0.165779
\(495\) 10246.3 0.930376
\(496\) 11037.1 0.999156
\(497\) 812.097 0.0732949
\(498\) 7341.19 0.660575
\(499\) −12763.9 −1.14507 −0.572537 0.819879i \(-0.694041\pi\)
−0.572537 + 0.819879i \(0.694041\pi\)
\(500\) −6443.86 −0.576356
\(501\) 6646.58 0.592709
\(502\) 8509.16 0.756539
\(503\) 6403.16 0.567600 0.283800 0.958884i \(-0.408405\pi\)
0.283800 + 0.958884i \(0.408405\pi\)
\(504\) −1711.80 −0.151289
\(505\) −4769.36 −0.420265
\(506\) −25199.4 −2.21393
\(507\) −6125.81 −0.536601
\(508\) 4316.21 0.376970
\(509\) −7892.36 −0.687274 −0.343637 0.939103i \(-0.611659\pi\)
−0.343637 + 0.939103i \(0.611659\pi\)
\(510\) 0 0
\(511\) −4459.51 −0.386061
\(512\) −5568.89 −0.480688
\(513\) −12584.0 −1.08304
\(514\) 11253.1 0.965667
\(515\) 13523.3 1.15710
\(516\) 1991.00 0.169862
\(517\) 8094.49 0.688579
\(518\) 2673.21 0.226745
\(519\) 5369.02 0.454092
\(520\) −615.248 −0.0518854
\(521\) 8506.79 0.715335 0.357667 0.933849i \(-0.383572\pi\)
0.357667 + 0.933849i \(0.383572\pi\)
\(522\) 13136.2 1.10145
\(523\) 18757.8 1.56830 0.784150 0.620571i \(-0.213099\pi\)
0.784150 + 0.620571i \(0.213099\pi\)
\(524\) −4710.37 −0.392697
\(525\) −948.521 −0.0788511
\(526\) −9306.30 −0.771433
\(527\) 0 0
\(528\) 13956.4 1.15033
\(529\) 1411.14 0.115981
\(530\) 21794.9 1.78624
\(531\) 16092.6 1.31518
\(532\) 2818.44 0.229689
\(533\) −894.638 −0.0727037
\(534\) −2131.15 −0.172704
\(535\) −2143.02 −0.173179
\(536\) −4124.06 −0.332336
\(537\) 281.777 0.0226435
\(538\) −19991.7 −1.60205
\(539\) 18291.2 1.46170
\(540\) −4815.73 −0.383770
\(541\) −13399.5 −1.06486 −0.532429 0.846475i \(-0.678721\pi\)
−0.532429 + 0.846475i \(0.678721\pi\)
\(542\) 878.026 0.0695838
\(543\) 10959.2 0.866121
\(544\) 0 0
\(545\) 17381.0 1.36609
\(546\) 364.239 0.0285494
\(547\) −7133.73 −0.557616 −0.278808 0.960347i \(-0.589939\pi\)
−0.278808 + 0.960347i \(0.589939\pi\)
\(548\) −8390.88 −0.654089
\(549\) −94.9352 −0.00738021
\(550\) −10590.8 −0.821080
\(551\) 19103.4 1.47701
\(552\) −4323.13 −0.333342
\(553\) −1428.62 −0.109857
\(554\) −12907.9 −0.989897
\(555\) −2745.08 −0.209950
\(556\) −4411.52 −0.336493
\(557\) 5175.60 0.393712 0.196856 0.980432i \(-0.436927\pi\)
0.196856 + 0.980432i \(0.436927\pi\)
\(558\) −9189.18 −0.697149
\(559\) −891.413 −0.0674468
\(560\) −4778.01 −0.360550
\(561\) 0 0
\(562\) −12361.0 −0.927787
\(563\) 5569.32 0.416907 0.208454 0.978032i \(-0.433157\pi\)
0.208454 + 0.978032i \(0.433157\pi\)
\(564\) −1572.21 −0.117379
\(565\) −9440.87 −0.702974
\(566\) 2758.54 0.204859
\(567\) 1001.62 0.0741870
\(568\) 1555.61 0.114915
\(569\) −15330.1 −1.12948 −0.564738 0.825270i \(-0.691023\pi\)
−0.564738 + 0.825270i \(0.691023\pi\)
\(570\) −8344.76 −0.613199
\(571\) −22285.0 −1.63327 −0.816636 0.577153i \(-0.804164\pi\)
−0.816636 + 0.577153i \(0.804164\pi\)
\(572\) 1410.54 0.103108
\(573\) −6574.41 −0.479319
\(574\) −3994.11 −0.290437
\(575\) 5706.63 0.413883
\(576\) −533.686 −0.0386057
\(577\) −19321.1 −1.39401 −0.697007 0.717064i \(-0.745485\pi\)
−0.697007 + 0.717064i \(0.745485\pi\)
\(578\) 0 0
\(579\) 3945.62 0.283203
\(580\) 7310.63 0.523375
\(581\) −5089.36 −0.363412
\(582\) −6977.91 −0.496982
\(583\) 44134.4 3.13527
\(584\) −8542.38 −0.605285
\(585\) 891.037 0.0629741
\(586\) 4947.96 0.348803
\(587\) −23571.2 −1.65739 −0.828696 0.559699i \(-0.810917\pi\)
−0.828696 + 0.559699i \(0.810917\pi\)
\(588\) −3552.74 −0.249171
\(589\) −13363.5 −0.934860
\(590\) 25822.4 1.80185
\(591\) −5718.12 −0.397990
\(592\) 8907.37 0.618396
\(593\) 27338.0 1.89315 0.946576 0.322481i \(-0.104517\pi\)
0.946576 + 0.322481i \(0.104517\pi\)
\(594\) −28117.0 −1.94218
\(595\) 0 0
\(596\) 507.895 0.0349064
\(597\) 11706.6 0.802542
\(598\) −2191.39 −0.149854
\(599\) −3930.83 −0.268129 −0.134064 0.990973i \(-0.542803\pi\)
−0.134064 + 0.990973i \(0.542803\pi\)
\(600\) −1816.93 −0.123626
\(601\) 22459.9 1.52439 0.762196 0.647347i \(-0.224121\pi\)
0.762196 + 0.647347i \(0.224121\pi\)
\(602\) −3979.71 −0.269437
\(603\) 5972.70 0.403362
\(604\) −6660.16 −0.448673
\(605\) 21688.1 1.45743
\(606\) 5408.64 0.362560
\(607\) 2227.26 0.148932 0.0744660 0.997224i \(-0.476275\pi\)
0.0744660 + 0.997224i \(0.476275\pi\)
\(608\) 16910.1 1.12795
\(609\) 3822.77 0.254362
\(610\) −152.334 −0.0101112
\(611\) 703.913 0.0466076
\(612\) 0 0
\(613\) 721.642 0.0475479 0.0237739 0.999717i \(-0.492432\pi\)
0.0237739 + 0.999717i \(0.492432\pi\)
\(614\) −15747.2 −1.03503
\(615\) 4101.49 0.268924
\(616\) −5562.19 −0.363810
\(617\) −6831.58 −0.445752 −0.222876 0.974847i \(-0.571545\pi\)
−0.222876 + 0.974847i \(0.571545\pi\)
\(618\) −15335.9 −0.998224
\(619\) −16094.9 −1.04509 −0.522544 0.852612i \(-0.675017\pi\)
−0.522544 + 0.852612i \(0.675017\pi\)
\(620\) −5114.02 −0.331265
\(621\) 15150.2 0.978997
\(622\) −16507.4 −1.06412
\(623\) 1477.44 0.0950120
\(624\) 1213.68 0.0778620
\(625\) −7104.94 −0.454716
\(626\) −21597.8 −1.37895
\(627\) −16898.1 −1.07631
\(628\) 9012.55 0.572675
\(629\) 0 0
\(630\) 3978.03 0.251569
\(631\) −27131.0 −1.71168 −0.855838 0.517245i \(-0.826958\pi\)
−0.855838 + 0.517245i \(0.826958\pi\)
\(632\) −2736.58 −0.172240
\(633\) −6641.23 −0.417007
\(634\) −4153.69 −0.260196
\(635\) 8859.42 0.553662
\(636\) −8572.32 −0.534457
\(637\) 1590.64 0.0989380
\(638\) 42683.7 2.64869
\(639\) −2252.92 −0.139474
\(640\) −13043.4 −0.805603
\(641\) −13901.7 −0.856606 −0.428303 0.903635i \(-0.640888\pi\)
−0.428303 + 0.903635i \(0.640888\pi\)
\(642\) 2430.27 0.149401
\(643\) −8827.31 −0.541392 −0.270696 0.962665i \(-0.587254\pi\)
−0.270696 + 0.962665i \(0.587254\pi\)
\(644\) −3393.19 −0.207625
\(645\) 4086.71 0.249479
\(646\) 0 0
\(647\) −14695.4 −0.892946 −0.446473 0.894797i \(-0.647320\pi\)
−0.446473 + 0.894797i \(0.647320\pi\)
\(648\) 1918.64 0.116314
\(649\) 52290.1 3.16266
\(650\) −920.999 −0.0555762
\(651\) −2674.15 −0.160996
\(652\) 1883.06 0.113108
\(653\) −6942.30 −0.416039 −0.208019 0.978125i \(-0.566702\pi\)
−0.208019 + 0.978125i \(0.566702\pi\)
\(654\) −19710.7 −1.17852
\(655\) −9668.47 −0.576761
\(656\) −13308.7 −0.792100
\(657\) 12371.6 0.734644
\(658\) 3142.61 0.186188
\(659\) −11804.8 −0.697800 −0.348900 0.937160i \(-0.613445\pi\)
−0.348900 + 0.937160i \(0.613445\pi\)
\(660\) −6466.66 −0.381386
\(661\) 3863.35 0.227333 0.113666 0.993519i \(-0.463741\pi\)
0.113666 + 0.993519i \(0.463741\pi\)
\(662\) −10744.4 −0.630806
\(663\) 0 0
\(664\) −9748.88 −0.569774
\(665\) 5785.10 0.337348
\(666\) −7416.01 −0.431479
\(667\) −22999.1 −1.33513
\(668\) 9993.08 0.578808
\(669\) 1165.16 0.0673358
\(670\) 9583.85 0.552622
\(671\) −308.475 −0.0177475
\(672\) 3383.86 0.194249
\(673\) 876.547 0.0502057 0.0251028 0.999685i \(-0.492009\pi\)
0.0251028 + 0.999685i \(0.492009\pi\)
\(674\) 15529.8 0.887516
\(675\) 6367.35 0.363080
\(676\) −9210.10 −0.524016
\(677\) −20341.1 −1.15476 −0.577378 0.816477i \(-0.695924\pi\)
−0.577378 + 0.816477i \(0.695924\pi\)
\(678\) 10706.3 0.606450
\(679\) 4837.51 0.273412
\(680\) 0 0
\(681\) 5413.15 0.304600
\(682\) −29858.6 −1.67646
\(683\) −30034.8 −1.68265 −0.841324 0.540531i \(-0.818223\pi\)
−0.841324 + 0.540531i \(0.818223\pi\)
\(684\) −7818.90 −0.437080
\(685\) −17223.1 −0.960670
\(686\) 15330.2 0.853219
\(687\) −13741.4 −0.763126
\(688\) −13260.7 −0.734827
\(689\) 3838.02 0.212216
\(690\) 10046.5 0.554294
\(691\) 10797.7 0.594450 0.297225 0.954807i \(-0.403939\pi\)
0.297225 + 0.954807i \(0.403939\pi\)
\(692\) 8072.28 0.443442
\(693\) 8055.48 0.441562
\(694\) 30497.2 1.66810
\(695\) −9055.04 −0.494212
\(696\) 7322.68 0.398801
\(697\) 0 0
\(698\) 353.223 0.0191543
\(699\) 1569.99 0.0849536
\(700\) −1426.09 −0.0770018
\(701\) −328.897 −0.0177208 −0.00886039 0.999961i \(-0.502820\pi\)
−0.00886039 + 0.999961i \(0.502820\pi\)
\(702\) −2445.11 −0.131460
\(703\) −10784.8 −0.578603
\(704\) −1734.12 −0.0928366
\(705\) −3227.11 −0.172397
\(706\) −30408.6 −1.62102
\(707\) −3749.60 −0.199460
\(708\) −10156.4 −0.539127
\(709\) 4850.17 0.256914 0.128457 0.991715i \(-0.458998\pi\)
0.128457 + 0.991715i \(0.458998\pi\)
\(710\) −3615.06 −0.191085
\(711\) 3963.28 0.209050
\(712\) 2830.10 0.148964
\(713\) 16088.7 0.845056
\(714\) 0 0
\(715\) 2895.27 0.151436
\(716\) 423.650 0.0221125
\(717\) −9056.97 −0.471741
\(718\) −8439.60 −0.438667
\(719\) −32883.0 −1.70560 −0.852801 0.522236i \(-0.825098\pi\)
−0.852801 + 0.522236i \(0.825098\pi\)
\(720\) 13255.1 0.686097
\(721\) 10631.8 0.549167
\(722\) −8780.24 −0.452586
\(723\) 2524.20 0.129842
\(724\) 16477.1 0.845808
\(725\) −9666.10 −0.495159
\(726\) −24595.1 −1.25731
\(727\) −16213.2 −0.827116 −0.413558 0.910478i \(-0.635714\pi\)
−0.413558 + 0.910478i \(0.635714\pi\)
\(728\) −483.699 −0.0246251
\(729\) 7139.71 0.362735
\(730\) 19851.5 1.00649
\(731\) 0 0
\(732\) 59.9158 0.00302534
\(733\) −24861.1 −1.25275 −0.626374 0.779523i \(-0.715462\pi\)
−0.626374 + 0.779523i \(0.715462\pi\)
\(734\) 7959.05 0.400237
\(735\) −7292.34 −0.365962
\(736\) −20358.5 −1.01960
\(737\) 19407.2 0.969979
\(738\) 11080.4 0.552678
\(739\) 1024.01 0.0509727 0.0254863 0.999675i \(-0.491887\pi\)
0.0254863 + 0.999675i \(0.491887\pi\)
\(740\) −4127.21 −0.205026
\(741\) −1469.49 −0.0728516
\(742\) 17134.8 0.847761
\(743\) 6870.67 0.339247 0.169623 0.985509i \(-0.445745\pi\)
0.169623 + 0.985509i \(0.445745\pi\)
\(744\) −5122.45 −0.252417
\(745\) 1042.50 0.0512676
\(746\) −33226.0 −1.63068
\(747\) 14118.9 0.691544
\(748\) 0 0
\(749\) −1684.81 −0.0821919
\(750\) 14999.5 0.730272
\(751\) −12169.0 −0.591284 −0.295642 0.955299i \(-0.595534\pi\)
−0.295642 + 0.955299i \(0.595534\pi\)
\(752\) 10471.5 0.507786
\(753\) −6869.64 −0.332462
\(754\) 3711.85 0.179281
\(755\) −13670.6 −0.658972
\(756\) −3786.05 −0.182140
\(757\) 28243.4 1.35604 0.678021 0.735043i \(-0.262838\pi\)
0.678021 + 0.735043i \(0.262838\pi\)
\(758\) 18680.4 0.895121
\(759\) 20344.0 0.972914
\(760\) 11081.6 0.528910
\(761\) 30505.2 1.45310 0.726552 0.687112i \(-0.241122\pi\)
0.726552 + 0.687112i \(0.241122\pi\)
\(762\) −10046.9 −0.477640
\(763\) 13664.7 0.648355
\(764\) −9884.58 −0.468078
\(765\) 0 0
\(766\) 2727.76 0.128666
\(767\) 4547.25 0.214070
\(768\) 14157.4 0.665184
\(769\) 20404.5 0.956834 0.478417 0.878133i \(-0.341211\pi\)
0.478417 + 0.878133i \(0.341211\pi\)
\(770\) 12925.9 0.604957
\(771\) −9084.89 −0.424363
\(772\) 5932.21 0.276561
\(773\) 30606.6 1.42412 0.712059 0.702119i \(-0.247763\pi\)
0.712059 + 0.702119i \(0.247763\pi\)
\(774\) 11040.5 0.512716
\(775\) 6761.75 0.313406
\(776\) 9266.46 0.428668
\(777\) −2158.14 −0.0996435
\(778\) 37888.1 1.74596
\(779\) 16113.9 0.741129
\(780\) −562.354 −0.0258147
\(781\) −7320.46 −0.335399
\(782\) 0 0
\(783\) −25662.0 −1.17124
\(784\) 23662.5 1.07792
\(785\) 18499.1 0.841097
\(786\) 10964.4 0.497567
\(787\) 36091.5 1.63472 0.817360 0.576127i \(-0.195437\pi\)
0.817360 + 0.576127i \(0.195437\pi\)
\(788\) −8597.14 −0.388656
\(789\) 7513.19 0.339007
\(790\) 6359.51 0.286407
\(791\) −7422.27 −0.333635
\(792\) 15430.6 0.692302
\(793\) −26.8256 −0.00120127
\(794\) −21372.3 −0.955255
\(795\) −17595.5 −0.784966
\(796\) 17600.7 0.783720
\(797\) −21357.2 −0.949199 −0.474599 0.880202i \(-0.657407\pi\)
−0.474599 + 0.880202i \(0.657407\pi\)
\(798\) −6560.53 −0.291028
\(799\) 0 0
\(800\) −8556.28 −0.378138
\(801\) −4098.72 −0.180800
\(802\) −40077.1 −1.76455
\(803\) 40199.2 1.76662
\(804\) −3769.51 −0.165349
\(805\) −6964.84 −0.304942
\(806\) −2596.56 −0.113474
\(807\) 16139.8 0.704024
\(808\) −7182.52 −0.312723
\(809\) −26168.9 −1.13727 −0.568633 0.822591i \(-0.692527\pi\)
−0.568633 + 0.822591i \(0.692527\pi\)
\(810\) −4458.71 −0.193411
\(811\) −20028.2 −0.867185 −0.433592 0.901109i \(-0.642754\pi\)
−0.433592 + 0.901109i \(0.642754\pi\)
\(812\) 5747.51 0.248397
\(813\) −708.850 −0.0305787
\(814\) −24097.0 −1.03759
\(815\) 3865.16 0.166124
\(816\) 0 0
\(817\) 16055.8 0.687541
\(818\) 26587.7 1.13645
\(819\) 700.520 0.0298879
\(820\) 6166.56 0.262617
\(821\) −2162.08 −0.0919087 −0.0459544 0.998944i \(-0.514633\pi\)
−0.0459544 + 0.998944i \(0.514633\pi\)
\(822\) 19531.6 0.828763
\(823\) −39375.6 −1.66773 −0.833867 0.551965i \(-0.813878\pi\)
−0.833867 + 0.551965i \(0.813878\pi\)
\(824\) 20365.7 0.861011
\(825\) 8550.21 0.360825
\(826\) 20301.2 0.855167
\(827\) −22589.8 −0.949849 −0.474925 0.880026i \(-0.657525\pi\)
−0.474925 + 0.880026i \(0.657525\pi\)
\(828\) 9413.38 0.395094
\(829\) −10977.3 −0.459898 −0.229949 0.973203i \(-0.573856\pi\)
−0.229949 + 0.973203i \(0.573856\pi\)
\(830\) 22655.3 0.947443
\(831\) 10420.8 0.435011
\(832\) −150.802 −0.00628380
\(833\) 0 0
\(834\) 10268.8 0.426353
\(835\) 20511.7 0.850105
\(836\) −25406.1 −1.05106
\(837\) 17951.4 0.741327
\(838\) −24104.4 −0.993643
\(839\) −44593.9 −1.83499 −0.917493 0.397752i \(-0.869790\pi\)
−0.917493 + 0.397752i \(0.869790\pi\)
\(840\) 2217.53 0.0910858
\(841\) 14567.8 0.597310
\(842\) −47549.4 −1.94615
\(843\) 9979.30 0.407717
\(844\) −9985.03 −0.407226
\(845\) −18904.6 −0.769630
\(846\) −8718.23 −0.354301
\(847\) 17050.8 0.691704
\(848\) 57094.6 2.31207
\(849\) −2227.03 −0.0900254
\(850\) 0 0
\(851\) 12984.1 0.523021
\(852\) 1421.87 0.0571742
\(853\) 37075.9 1.48822 0.744112 0.668055i \(-0.232873\pi\)
0.744112 + 0.668055i \(0.232873\pi\)
\(854\) −119.763 −0.00479883
\(855\) −16049.0 −0.641947
\(856\) −3227.33 −0.128864
\(857\) 45571.5 1.81645 0.908223 0.418487i \(-0.137440\pi\)
0.908223 + 0.418487i \(0.137440\pi\)
\(858\) −3283.34 −0.130643
\(859\) 68.5340 0.00272218 0.00136109 0.999999i \(-0.499567\pi\)
0.00136109 + 0.999999i \(0.499567\pi\)
\(860\) 6144.33 0.243628
\(861\) 3224.53 0.127633
\(862\) −58845.7 −2.32516
\(863\) −5064.23 −0.199755 −0.0998774 0.995000i \(-0.531845\pi\)
−0.0998774 + 0.995000i \(0.531845\pi\)
\(864\) −22715.6 −0.894445
\(865\) 16569.1 0.651291
\(866\) 870.810 0.0341701
\(867\) 0 0
\(868\) −4020.57 −0.157220
\(869\) 12878.0 0.502710
\(870\) −17017.1 −0.663142
\(871\) 1687.69 0.0656546
\(872\) 26175.3 1.01652
\(873\) −13420.2 −0.520281
\(874\) 39470.4 1.52758
\(875\) −10398.6 −0.401755
\(876\) −7807.98 −0.301150
\(877\) −3826.86 −0.147348 −0.0736738 0.997282i \(-0.523472\pi\)
−0.0736738 + 0.997282i \(0.523472\pi\)
\(878\) −34542.3 −1.32773
\(879\) −3994.60 −0.153282
\(880\) 43070.2 1.64988
\(881\) 41015.0 1.56848 0.784240 0.620457i \(-0.213053\pi\)
0.784240 + 0.620457i \(0.213053\pi\)
\(882\) −19700.7 −0.752106
\(883\) 7085.32 0.270034 0.135017 0.990843i \(-0.456891\pi\)
0.135017 + 0.990843i \(0.456891\pi\)
\(884\) 0 0
\(885\) −20847.0 −0.791824
\(886\) 34998.2 1.32707
\(887\) 17254.1 0.653141 0.326570 0.945173i \(-0.394107\pi\)
0.326570 + 0.945173i \(0.394107\pi\)
\(888\) −4134.01 −0.156226
\(889\) 6965.14 0.262771
\(890\) −6576.84 −0.247704
\(891\) −9028.85 −0.339481
\(892\) 1751.81 0.0657566
\(893\) −12678.6 −0.475110
\(894\) −1182.24 −0.0442281
\(895\) 869.580 0.0324769
\(896\) −10254.5 −0.382344
\(897\) 1769.16 0.0658534
\(898\) −30028.8 −1.11590
\(899\) −27251.5 −1.01100
\(900\) 3956.26 0.146528
\(901\) 0 0
\(902\) 36003.9 1.32905
\(903\) 3212.91 0.118404
\(904\) −14217.7 −0.523089
\(905\) 33820.7 1.24225
\(906\) 15503.0 0.568491
\(907\) −18106.5 −0.662861 −0.331431 0.943480i \(-0.607531\pi\)
−0.331431 + 0.943480i \(0.607531\pi\)
\(908\) 8138.63 0.297456
\(909\) 10402.1 0.379557
\(910\) 1124.06 0.0409476
\(911\) −19160.7 −0.696840 −0.348420 0.937339i \(-0.613282\pi\)
−0.348420 + 0.937339i \(0.613282\pi\)
\(912\) −21860.2 −0.793712
\(913\) 45876.8 1.66298
\(914\) −29208.1 −1.05702
\(915\) 122.983 0.00444337
\(916\) −20660.1 −0.745229
\(917\) −7601.21 −0.273734
\(918\) 0 0
\(919\) −13214.6 −0.474329 −0.237165 0.971469i \(-0.576218\pi\)
−0.237165 + 0.971469i \(0.576218\pi\)
\(920\) −13341.4 −0.478102
\(921\) 12713.1 0.454843
\(922\) 30100.5 1.07517
\(923\) −636.601 −0.0227020
\(924\) −5084.00 −0.181008
\(925\) 5456.99 0.193973
\(926\) 55605.0 1.97332
\(927\) −29494.8 −1.04502
\(928\) 34483.9 1.21982
\(929\) −8124.36 −0.286923 −0.143462 0.989656i \(-0.545823\pi\)
−0.143462 + 0.989656i \(0.545823\pi\)
\(930\) 11904.0 0.419729
\(931\) −28650.0 −1.00856
\(932\) 2360.47 0.0829612
\(933\) 13326.8 0.467631
\(934\) 20794.2 0.728489
\(935\) 0 0
\(936\) 1341.88 0.0468596
\(937\) −11617.7 −0.405053 −0.202526 0.979277i \(-0.564915\pi\)
−0.202526 + 0.979277i \(0.564915\pi\)
\(938\) 7534.68 0.262277
\(939\) 17436.4 0.605980
\(940\) −4851.93 −0.168354
\(941\) −12810.7 −0.443801 −0.221901 0.975069i \(-0.571226\pi\)
−0.221901 + 0.975069i \(0.571226\pi\)
\(942\) −20978.7 −0.725608
\(943\) −19399.9 −0.669934
\(944\) 67645.3 2.33227
\(945\) −7771.23 −0.267511
\(946\) 35874.1 1.23295
\(947\) 21469.3 0.736703 0.368351 0.929687i \(-0.379922\pi\)
0.368351 + 0.929687i \(0.379922\pi\)
\(948\) −2501.31 −0.0856951
\(949\) 3495.80 0.119577
\(950\) 16588.7 0.566534
\(951\) 3353.37 0.114343
\(952\) 0 0
\(953\) 15897.5 0.540369 0.270184 0.962809i \(-0.412915\pi\)
0.270184 + 0.962809i \(0.412915\pi\)
\(954\) −47535.3 −1.61322
\(955\) −20289.0 −0.687473
\(956\) −13617.1 −0.460678
\(957\) −34459.5 −1.16397
\(958\) 52198.5 1.76039
\(959\) −13540.5 −0.455940
\(960\) 691.356 0.0232432
\(961\) −10727.7 −0.360097
\(962\) −2095.52 −0.0702312
\(963\) 4674.01 0.156405
\(964\) 3795.11 0.126797
\(965\) 12176.4 0.406189
\(966\) 7898.40 0.263071
\(967\) −33191.2 −1.10378 −0.551891 0.833916i \(-0.686094\pi\)
−0.551891 + 0.833916i \(0.686094\pi\)
\(968\) 32661.6 1.08449
\(969\) 0 0
\(970\) −21534.2 −0.712806
\(971\) 31834.5 1.05213 0.526065 0.850445i \(-0.323667\pi\)
0.526065 + 0.850445i \(0.323667\pi\)
\(972\) 16665.9 0.549959
\(973\) −7118.94 −0.234556
\(974\) 28777.6 0.946708
\(975\) 743.543 0.0244230
\(976\) −399.060 −0.0130877
\(977\) −14447.7 −0.473105 −0.236552 0.971619i \(-0.576018\pi\)
−0.236552 + 0.971619i \(0.576018\pi\)
\(978\) −4383.24 −0.143314
\(979\) −13318.0 −0.434777
\(980\) −10964.0 −0.357379
\(981\) −37908.5 −1.23377
\(982\) 9223.90 0.299742
\(983\) 7830.90 0.254086 0.127043 0.991897i \(-0.459451\pi\)
0.127043 + 0.991897i \(0.459451\pi\)
\(984\) 6176.73 0.200109
\(985\) −17646.4 −0.570824
\(986\) 0 0
\(987\) −2537.10 −0.0818206
\(988\) −2209.37 −0.0711430
\(989\) −19330.0 −0.621494
\(990\) −35859.0 −1.15119
\(991\) −14647.7 −0.469526 −0.234763 0.972053i \(-0.575431\pi\)
−0.234763 + 0.972053i \(0.575431\pi\)
\(992\) −24122.6 −0.772071
\(993\) 8674.21 0.277208
\(994\) −2842.10 −0.0906902
\(995\) 36127.1 1.15106
\(996\) −8910.76 −0.283482
\(997\) 49174.6 1.56206 0.781031 0.624492i \(-0.214694\pi\)
0.781031 + 0.624492i \(0.214694\pi\)
\(998\) 44670.1 1.41684
\(999\) 14487.5 0.458822
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.a.g.1.4 12
17.4 even 4 289.4.b.e.288.9 12
17.5 odd 16 17.4.d.a.8.3 12
17.7 odd 16 17.4.d.a.15.3 yes 12
17.13 even 4 289.4.b.e.288.10 12
17.16 even 2 inner 289.4.a.g.1.3 12
51.5 even 16 153.4.l.a.127.1 12
51.41 even 16 153.4.l.a.100.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.d.a.8.3 12 17.5 odd 16
17.4.d.a.15.3 yes 12 17.7 odd 16
153.4.l.a.100.1 12 51.41 even 16
153.4.l.a.127.1 12 51.5 even 16
289.4.a.g.1.3 12 17.16 even 2 inner
289.4.a.g.1.4 12 1.1 even 1 trivial
289.4.b.e.288.9 12 17.4 even 4
289.4.b.e.288.10 12 17.13 even 4