Properties

Label 261.4.a.b.1.2
Level $261$
Weight $4$
Character 261.1
Self dual yes
Analytic conductor $15.399$
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] = [261,4,Mod(1,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, 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("261.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 261.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: \(15.3994985115\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 261.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} -2.17157 q^{4} +10.6569 q^{5} -22.1421 q^{7} -24.5563 q^{8} +O(q^{10})\) \(q+2.41421 q^{2} -2.17157 q^{4} +10.6569 q^{5} -22.1421 q^{7} -24.5563 q^{8} +25.7279 q^{10} -39.3259 q^{11} +23.7696 q^{13} -53.4558 q^{14} -41.9117 q^{16} -4.54416 q^{17} -155.255 q^{19} -23.1421 q^{20} -94.9411 q^{22} +41.8823 q^{23} -11.4315 q^{25} +57.3848 q^{26} +48.0833 q^{28} -29.0000 q^{29} -57.9045 q^{31} +95.2670 q^{32} -10.9706 q^{34} -235.966 q^{35} +235.196 q^{37} -374.818 q^{38} -261.693 q^{40} +175.161 q^{41} -402.831 q^{43} +85.3991 q^{44} +101.113 q^{46} -227.742 q^{47} +147.274 q^{49} -27.5980 q^{50} -51.6173 q^{52} -673.534 q^{53} -419.090 q^{55} +543.730 q^{56} -70.0122 q^{58} +800.725 q^{59} -222.270 q^{61} -139.794 q^{62} +565.288 q^{64} +253.309 q^{65} -524.479 q^{67} +9.86797 q^{68} -569.671 q^{70} +281.917 q^{71} +1229.10 q^{73} +567.813 q^{74} +337.147 q^{76} +870.759 q^{77} +611.247 q^{79} -446.647 q^{80} +422.877 q^{82} -515.490 q^{83} -48.4264 q^{85} -972.519 q^{86} +965.701 q^{88} +358.219 q^{89} -526.309 q^{91} -90.9504 q^{92} -549.818 q^{94} -1654.53 q^{95} +829.415 q^{97} +355.551 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 10 q^{4} + 10 q^{5} - 16 q^{7} - 18 q^{8} + 26 q^{10} + 26 q^{11} - 26 q^{13} - 56 q^{14} + 18 q^{16} - 60 q^{17} - 220 q^{19} - 18 q^{20} - 122 q^{22} - 52 q^{23} - 136 q^{25} + 78 q^{26} - 58 q^{29} - 294 q^{31} + 18 q^{32} + 12 q^{34} - 240 q^{35} + 312 q^{37} - 348 q^{38} - 266 q^{40} - 40 q^{41} - 322 q^{43} - 426 q^{44} + 140 q^{46} + 130 q^{47} - 158 q^{49} + 24 q^{50} + 338 q^{52} - 1002 q^{53} - 462 q^{55} + 584 q^{56} - 58 q^{58} + 900 q^{59} - 948 q^{61} - 42 q^{62} + 118 q^{64} + 286 q^{65} + 320 q^{67} + 444 q^{68} - 568 q^{70} + 660 q^{71} + 648 q^{73} + 536 q^{74} + 844 q^{76} + 1272 q^{77} + 258 q^{79} - 486 q^{80} + 512 q^{82} - 1212 q^{83} - 12 q^{85} - 1006 q^{86} + 1394 q^{88} - 760 q^{89} - 832 q^{91} + 644 q^{92} - 698 q^{94} - 1612 q^{95} + 24 q^{97} + 482 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 0.853553 0.426777 0.904357i \(-0.359649\pi\)
0.426777 + 0.904357i \(0.359649\pi\)
\(3\) 0 0
\(4\) −2.17157 −0.271447
\(5\) 10.6569 0.953178 0.476589 0.879126i \(-0.341873\pi\)
0.476589 + 0.879126i \(0.341873\pi\)
\(6\) 0 0
\(7\) −22.1421 −1.19556 −0.597781 0.801659i \(-0.703951\pi\)
−0.597781 + 0.801659i \(0.703951\pi\)
\(8\) −24.5563 −1.08525
\(9\) 0 0
\(10\) 25.7279 0.813588
\(11\) −39.3259 −1.07793 −0.538964 0.842329i \(-0.681184\pi\)
−0.538964 + 0.842329i \(0.681184\pi\)
\(12\) 0 0
\(13\) 23.7696 0.507114 0.253557 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) −53.4558 −1.02048
\(15\) 0 0
\(16\) −41.9117 −0.654870
\(17\) −4.54416 −0.0648306 −0.0324153 0.999474i \(-0.510320\pi\)
−0.0324153 + 0.999474i \(0.510320\pi\)
\(18\) 0 0
\(19\) −155.255 −1.87463 −0.937313 0.348488i \(-0.886695\pi\)
−0.937313 + 0.348488i \(0.886695\pi\)
\(20\) −23.1421 −0.258737
\(21\) 0 0
\(22\) −94.9411 −0.920069
\(23\) 41.8823 0.379698 0.189849 0.981813i \(-0.439200\pi\)
0.189849 + 0.981813i \(0.439200\pi\)
\(24\) 0 0
\(25\) −11.4315 −0.0914517
\(26\) 57.3848 0.432849
\(27\) 0 0
\(28\) 48.0833 0.324532
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −57.9045 −0.335483 −0.167741 0.985831i \(-0.553647\pi\)
−0.167741 + 0.985831i \(0.553647\pi\)
\(32\) 95.2670 0.526281
\(33\) 0 0
\(34\) −10.9706 −0.0553364
\(35\) −235.966 −1.13958
\(36\) 0 0
\(37\) 235.196 1.04503 0.522513 0.852631i \(-0.324995\pi\)
0.522513 + 0.852631i \(0.324995\pi\)
\(38\) −374.818 −1.60009
\(39\) 0 0
\(40\) −261.693 −1.03443
\(41\) 175.161 0.667210 0.333605 0.942713i \(-0.391735\pi\)
0.333605 + 0.942713i \(0.391735\pi\)
\(42\) 0 0
\(43\) −402.831 −1.42863 −0.714315 0.699824i \(-0.753262\pi\)
−0.714315 + 0.699824i \(0.753262\pi\)
\(44\) 85.3991 0.292600
\(45\) 0 0
\(46\) 101.113 0.324092
\(47\) −227.742 −0.706800 −0.353400 0.935472i \(-0.614975\pi\)
−0.353400 + 0.935472i \(0.614975\pi\)
\(48\) 0 0
\(49\) 147.274 0.429371
\(50\) −27.5980 −0.0780589
\(51\) 0 0
\(52\) −51.6173 −0.137654
\(53\) −673.534 −1.74560 −0.872802 0.488074i \(-0.837699\pi\)
−0.872802 + 0.488074i \(0.837699\pi\)
\(54\) 0 0
\(55\) −419.090 −1.02746
\(56\) 543.730 1.29748
\(57\) 0 0
\(58\) −70.0122 −0.158501
\(59\) 800.725 1.76687 0.883437 0.468551i \(-0.155224\pi\)
0.883437 + 0.468551i \(0.155224\pi\)
\(60\) 0 0
\(61\) −222.270 −0.466537 −0.233268 0.972412i \(-0.574942\pi\)
−0.233268 + 0.972412i \(0.574942\pi\)
\(62\) −139.794 −0.286352
\(63\) 0 0
\(64\) 565.288 1.10408
\(65\) 253.309 0.483370
\(66\) 0 0
\(67\) −524.479 −0.956349 −0.478174 0.878265i \(-0.658701\pi\)
−0.478174 + 0.878265i \(0.658701\pi\)
\(68\) 9.86797 0.0175980
\(69\) 0 0
\(70\) −569.671 −0.972696
\(71\) 281.917 0.471230 0.235615 0.971846i \(-0.424289\pi\)
0.235615 + 0.971846i \(0.424289\pi\)
\(72\) 0 0
\(73\) 1229.10 1.97061 0.985307 0.170790i \(-0.0546320\pi\)
0.985307 + 0.170790i \(0.0546320\pi\)
\(74\) 567.813 0.891986
\(75\) 0 0
\(76\) 337.147 0.508861
\(77\) 870.759 1.28873
\(78\) 0 0
\(79\) 611.247 0.870514 0.435257 0.900306i \(-0.356657\pi\)
0.435257 + 0.900306i \(0.356657\pi\)
\(80\) −446.647 −0.624208
\(81\) 0 0
\(82\) 422.877 0.569500
\(83\) −515.490 −0.681716 −0.340858 0.940115i \(-0.610718\pi\)
−0.340858 + 0.940115i \(0.610718\pi\)
\(84\) 0 0
\(85\) −48.4264 −0.0617951
\(86\) −972.519 −1.21941
\(87\) 0 0
\(88\) 965.701 1.16982
\(89\) 358.219 0.426643 0.213321 0.976982i \(-0.431572\pi\)
0.213321 + 0.976982i \(0.431572\pi\)
\(90\) 0 0
\(91\) −526.309 −0.606287
\(92\) −90.9504 −0.103068
\(93\) 0 0
\(94\) −549.818 −0.603292
\(95\) −1654.53 −1.78685
\(96\) 0 0
\(97\) 829.415 0.868189 0.434095 0.900867i \(-0.357068\pi\)
0.434095 + 0.900867i \(0.357068\pi\)
\(98\) 355.551 0.366491
\(99\) 0 0
\(100\) 24.8242 0.0248242
\(101\) 978.010 0.963521 0.481761 0.876303i \(-0.339997\pi\)
0.481761 + 0.876303i \(0.339997\pi\)
\(102\) 0 0
\(103\) −1217.33 −1.16453 −0.582266 0.812998i \(-0.697834\pi\)
−0.582266 + 0.812998i \(0.697834\pi\)
\(104\) −583.693 −0.550345
\(105\) 0 0
\(106\) −1626.06 −1.48997
\(107\) 707.044 0.638808 0.319404 0.947619i \(-0.396517\pi\)
0.319404 + 0.947619i \(0.396517\pi\)
\(108\) 0 0
\(109\) −1496.14 −1.31471 −0.657357 0.753580i \(-0.728326\pi\)
−0.657357 + 0.753580i \(0.728326\pi\)
\(110\) −1011.77 −0.876989
\(111\) 0 0
\(112\) 928.014 0.782938
\(113\) 1067.23 0.888469 0.444234 0.895911i \(-0.353476\pi\)
0.444234 + 0.895911i \(0.353476\pi\)
\(114\) 0 0
\(115\) 446.333 0.361920
\(116\) 62.9756 0.0504064
\(117\) 0 0
\(118\) 1933.12 1.50812
\(119\) 100.617 0.0775090
\(120\) 0 0
\(121\) 215.527 0.161928
\(122\) −536.607 −0.398214
\(123\) 0 0
\(124\) 125.744 0.0910656
\(125\) −1453.93 −1.04035
\(126\) 0 0
\(127\) −1179.58 −0.824177 −0.412088 0.911144i \(-0.635201\pi\)
−0.412088 + 0.911144i \(0.635201\pi\)
\(128\) 602.591 0.416109
\(129\) 0 0
\(130\) 611.541 0.412582
\(131\) 2357.47 1.57231 0.786156 0.618028i \(-0.212068\pi\)
0.786156 + 0.618028i \(0.212068\pi\)
\(132\) 0 0
\(133\) 3437.67 2.24123
\(134\) −1266.21 −0.816295
\(135\) 0 0
\(136\) 111.588 0.0703572
\(137\) −722.489 −0.450558 −0.225279 0.974294i \(-0.572329\pi\)
−0.225279 + 0.974294i \(0.572329\pi\)
\(138\) 0 0
\(139\) 1398.24 0.853219 0.426610 0.904436i \(-0.359708\pi\)
0.426610 + 0.904436i \(0.359708\pi\)
\(140\) 512.416 0.309336
\(141\) 0 0
\(142\) 680.607 0.402220
\(143\) −934.759 −0.546633
\(144\) 0 0
\(145\) −309.049 −0.177001
\(146\) 2967.30 1.68203
\(147\) 0 0
\(148\) −510.745 −0.283669
\(149\) −2830.63 −1.55634 −0.778168 0.628056i \(-0.783851\pi\)
−0.778168 + 0.628056i \(0.783851\pi\)
\(150\) 0 0
\(151\) 1705.58 0.919194 0.459597 0.888128i \(-0.347994\pi\)
0.459597 + 0.888128i \(0.347994\pi\)
\(152\) 3812.49 2.03443
\(153\) 0 0
\(154\) 2102.20 1.10000
\(155\) −617.080 −0.319775
\(156\) 0 0
\(157\) −2670.84 −1.35768 −0.678841 0.734286i \(-0.737517\pi\)
−0.678841 + 0.734286i \(0.737517\pi\)
\(158\) 1475.68 0.743031
\(159\) 0 0
\(160\) 1015.25 0.501639
\(161\) −927.362 −0.453953
\(162\) 0 0
\(163\) −2151.92 −1.03406 −0.517028 0.855968i \(-0.672962\pi\)
−0.517028 + 0.855968i \(0.672962\pi\)
\(164\) −380.376 −0.181112
\(165\) 0 0
\(166\) −1244.50 −0.581881
\(167\) −999.387 −0.463083 −0.231542 0.972825i \(-0.574377\pi\)
−0.231542 + 0.972825i \(0.574377\pi\)
\(168\) 0 0
\(169\) −1632.01 −0.742835
\(170\) −116.912 −0.0527454
\(171\) 0 0
\(172\) 874.776 0.387797
\(173\) 2534.30 1.11375 0.556877 0.830595i \(-0.311999\pi\)
0.556877 + 0.830595i \(0.311999\pi\)
\(174\) 0 0
\(175\) 253.117 0.109336
\(176\) 1648.21 0.705903
\(177\) 0 0
\(178\) 864.818 0.364162
\(179\) −3550.27 −1.48245 −0.741227 0.671254i \(-0.765756\pi\)
−0.741227 + 0.671254i \(0.765756\pi\)
\(180\) 0 0
\(181\) −3034.68 −1.24622 −0.623110 0.782135i \(-0.714131\pi\)
−0.623110 + 0.782135i \(0.714131\pi\)
\(182\) −1270.62 −0.517499
\(183\) 0 0
\(184\) −1028.48 −0.412066
\(185\) 2506.45 0.996096
\(186\) 0 0
\(187\) 178.703 0.0698827
\(188\) 494.559 0.191859
\(189\) 0 0
\(190\) −3994.38 −1.52517
\(191\) −2224.00 −0.842529 −0.421265 0.906938i \(-0.638414\pi\)
−0.421265 + 0.906938i \(0.638414\pi\)
\(192\) 0 0
\(193\) −632.830 −0.236021 −0.118011 0.993012i \(-0.537652\pi\)
−0.118011 + 0.993012i \(0.537652\pi\)
\(194\) 2002.39 0.741046
\(195\) 0 0
\(196\) −319.817 −0.116551
\(197\) −1369.80 −0.495404 −0.247702 0.968836i \(-0.579675\pi\)
−0.247702 + 0.968836i \(0.579675\pi\)
\(198\) 0 0
\(199\) 1416.38 0.504544 0.252272 0.967656i \(-0.418822\pi\)
0.252272 + 0.967656i \(0.418822\pi\)
\(200\) 280.715 0.0992477
\(201\) 0 0
\(202\) 2361.13 0.822417
\(203\) 642.122 0.222010
\(204\) 0 0
\(205\) 1866.67 0.635970
\(206\) −2938.89 −0.993991
\(207\) 0 0
\(208\) −996.222 −0.332094
\(209\) 6105.54 2.02071
\(210\) 0 0
\(211\) −896.432 −0.292478 −0.146239 0.989249i \(-0.546717\pi\)
−0.146239 + 0.989249i \(0.546717\pi\)
\(212\) 1462.63 0.473838
\(213\) 0 0
\(214\) 1706.95 0.545257
\(215\) −4292.91 −1.36174
\(216\) 0 0
\(217\) 1282.13 0.401091
\(218\) −3611.99 −1.12218
\(219\) 0 0
\(220\) 910.085 0.278900
\(221\) −108.013 −0.0328765
\(222\) 0 0
\(223\) −2268.94 −0.681344 −0.340672 0.940182i \(-0.610655\pi\)
−0.340672 + 0.940182i \(0.610655\pi\)
\(224\) −2109.42 −0.629202
\(225\) 0 0
\(226\) 2576.53 0.758356
\(227\) 2078.09 0.607610 0.303805 0.952734i \(-0.401743\pi\)
0.303805 + 0.952734i \(0.401743\pi\)
\(228\) 0 0
\(229\) −3715.05 −1.07204 −0.536020 0.844205i \(-0.680073\pi\)
−0.536020 + 0.844205i \(0.680073\pi\)
\(230\) 1077.54 0.308918
\(231\) 0 0
\(232\) 712.134 0.201525
\(233\) −2521.35 −0.708923 −0.354461 0.935071i \(-0.615336\pi\)
−0.354461 + 0.935071i \(0.615336\pi\)
\(234\) 0 0
\(235\) −2427.02 −0.673707
\(236\) −1738.83 −0.479612
\(237\) 0 0
\(238\) 242.912 0.0661581
\(239\) −3940.04 −1.06636 −0.533179 0.846002i \(-0.679003\pi\)
−0.533179 + 0.846002i \(0.679003\pi\)
\(240\) 0 0
\(241\) −1973.06 −0.527369 −0.263684 0.964609i \(-0.584938\pi\)
−0.263684 + 0.964609i \(0.584938\pi\)
\(242\) 520.327 0.138214
\(243\) 0 0
\(244\) 482.675 0.126640
\(245\) 1569.48 0.409267
\(246\) 0 0
\(247\) −3690.34 −0.950650
\(248\) 1421.92 0.364082
\(249\) 0 0
\(250\) −3510.10 −0.887992
\(251\) 1236.65 0.310982 0.155491 0.987837i \(-0.450304\pi\)
0.155491 + 0.987837i \(0.450304\pi\)
\(252\) 0 0
\(253\) −1647.06 −0.409287
\(254\) −2847.75 −0.703479
\(255\) 0 0
\(256\) −3067.52 −0.748907
\(257\) −2918.10 −0.708272 −0.354136 0.935194i \(-0.615225\pi\)
−0.354136 + 0.935194i \(0.615225\pi\)
\(258\) 0 0
\(259\) −5207.74 −1.24939
\(260\) −550.078 −0.131209
\(261\) 0 0
\(262\) 5691.43 1.34205
\(263\) 310.789 0.0728673 0.0364336 0.999336i \(-0.488400\pi\)
0.0364336 + 0.999336i \(0.488400\pi\)
\(264\) 0 0
\(265\) −7177.75 −1.66387
\(266\) 8299.28 1.91301
\(267\) 0 0
\(268\) 1138.95 0.259598
\(269\) 1839.98 0.417047 0.208523 0.978017i \(-0.433134\pi\)
0.208523 + 0.978017i \(0.433134\pi\)
\(270\) 0 0
\(271\) 5187.35 1.16277 0.581383 0.813630i \(-0.302512\pi\)
0.581383 + 0.813630i \(0.302512\pi\)
\(272\) 190.453 0.0424556
\(273\) 0 0
\(274\) −1744.24 −0.384575
\(275\) 449.552 0.0985783
\(276\) 0 0
\(277\) 8666.41 1.87983 0.939917 0.341403i \(-0.110902\pi\)
0.939917 + 0.341403i \(0.110902\pi\)
\(278\) 3375.66 0.728268
\(279\) 0 0
\(280\) 5794.45 1.23673
\(281\) −7856.96 −1.66800 −0.833998 0.551767i \(-0.813954\pi\)
−0.833998 + 0.551767i \(0.813954\pi\)
\(282\) 0 0
\(283\) −3054.71 −0.641638 −0.320819 0.947141i \(-0.603958\pi\)
−0.320819 + 0.947141i \(0.603958\pi\)
\(284\) −612.203 −0.127914
\(285\) 0 0
\(286\) −2256.71 −0.466580
\(287\) −3878.45 −0.797692
\(288\) 0 0
\(289\) −4892.35 −0.995797
\(290\) −746.110 −0.151080
\(291\) 0 0
\(292\) −2669.07 −0.534917
\(293\) −2847.83 −0.567822 −0.283911 0.958851i \(-0.591632\pi\)
−0.283911 + 0.958851i \(0.591632\pi\)
\(294\) 0 0
\(295\) 8533.21 1.68414
\(296\) −5775.55 −1.13411
\(297\) 0 0
\(298\) −6833.74 −1.32842
\(299\) 995.522 0.192550
\(300\) 0 0
\(301\) 8919.53 1.70802
\(302\) 4117.64 0.784581
\(303\) 0 0
\(304\) 6506.99 1.22764
\(305\) −2368.70 −0.444693
\(306\) 0 0
\(307\) −2470.79 −0.459334 −0.229667 0.973269i \(-0.573764\pi\)
−0.229667 + 0.973269i \(0.573764\pi\)
\(308\) −1890.92 −0.349822
\(309\) 0 0
\(310\) −1489.76 −0.272945
\(311\) 5653.29 1.03077 0.515384 0.856959i \(-0.327649\pi\)
0.515384 + 0.856959i \(0.327649\pi\)
\(312\) 0 0
\(313\) −8289.72 −1.49701 −0.748503 0.663132i \(-0.769227\pi\)
−0.748503 + 0.663132i \(0.769227\pi\)
\(314\) −6447.97 −1.15885
\(315\) 0 0
\(316\) −1327.37 −0.236298
\(317\) −2577.06 −0.456600 −0.228300 0.973591i \(-0.573317\pi\)
−0.228300 + 0.973591i \(0.573317\pi\)
\(318\) 0 0
\(319\) 1140.45 0.200166
\(320\) 6024.20 1.05238
\(321\) 0 0
\(322\) −2238.85 −0.387473
\(323\) 705.502 0.121533
\(324\) 0 0
\(325\) −271.721 −0.0463765
\(326\) −5195.19 −0.882622
\(327\) 0 0
\(328\) −4301.33 −0.724088
\(329\) 5042.70 0.845024
\(330\) 0 0
\(331\) 7672.12 1.27401 0.637006 0.770859i \(-0.280173\pi\)
0.637006 + 0.770859i \(0.280173\pi\)
\(332\) 1119.42 0.185049
\(333\) 0 0
\(334\) −2412.73 −0.395266
\(335\) −5589.30 −0.911570
\(336\) 0 0
\(337\) 3650.77 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(338\) −3940.02 −0.634049
\(339\) 0 0
\(340\) 105.161 0.0167741
\(341\) 2277.15 0.361626
\(342\) 0 0
\(343\) 4333.79 0.682223
\(344\) 9892.05 1.55042
\(345\) 0 0
\(346\) 6118.35 0.950649
\(347\) 8737.06 1.35167 0.675835 0.737053i \(-0.263783\pi\)
0.675835 + 0.737053i \(0.263783\pi\)
\(348\) 0 0
\(349\) −2143.83 −0.328816 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(350\) 611.078 0.0933243
\(351\) 0 0
\(352\) −3746.46 −0.567293
\(353\) 5111.17 0.770651 0.385326 0.922781i \(-0.374089\pi\)
0.385326 + 0.922781i \(0.374089\pi\)
\(354\) 0 0
\(355\) 3004.35 0.449167
\(356\) −777.900 −0.115811
\(357\) 0 0
\(358\) −8571.10 −1.26535
\(359\) −4520.87 −0.664630 −0.332315 0.943168i \(-0.607830\pi\)
−0.332315 + 0.943168i \(0.607830\pi\)
\(360\) 0 0
\(361\) 17245.1 2.51422
\(362\) −7326.35 −1.06371
\(363\) 0 0
\(364\) 1142.92 0.164575
\(365\) 13098.3 1.87835
\(366\) 0 0
\(367\) −486.272 −0.0691641 −0.0345820 0.999402i \(-0.511010\pi\)
−0.0345820 + 0.999402i \(0.511010\pi\)
\(368\) −1755.36 −0.248653
\(369\) 0 0
\(370\) 6051.10 0.850221
\(371\) 14913.5 2.08698
\(372\) 0 0
\(373\) 10053.1 1.39552 0.697758 0.716333i \(-0.254181\pi\)
0.697758 + 0.716333i \(0.254181\pi\)
\(374\) 431.427 0.0596486
\(375\) 0 0
\(376\) 5592.52 0.767053
\(377\) −689.317 −0.0941688
\(378\) 0 0
\(379\) −11602.8 −1.57255 −0.786273 0.617879i \(-0.787992\pi\)
−0.786273 + 0.617879i \(0.787992\pi\)
\(380\) 3592.93 0.485035
\(381\) 0 0
\(382\) −5369.22 −0.719144
\(383\) 4161.38 0.555187 0.277593 0.960699i \(-0.410463\pi\)
0.277593 + 0.960699i \(0.410463\pi\)
\(384\) 0 0
\(385\) 9279.56 1.22839
\(386\) −1527.79 −0.201457
\(387\) 0 0
\(388\) −1801.14 −0.235667
\(389\) 6843.95 0.892036 0.446018 0.895024i \(-0.352842\pi\)
0.446018 + 0.895024i \(0.352842\pi\)
\(390\) 0 0
\(391\) −190.319 −0.0246160
\(392\) −3616.52 −0.465974
\(393\) 0 0
\(394\) −3307.00 −0.422854
\(395\) 6513.97 0.829755
\(396\) 0 0
\(397\) 5474.73 0.692112 0.346056 0.938214i \(-0.387521\pi\)
0.346056 + 0.938214i \(0.387521\pi\)
\(398\) 3419.43 0.430655
\(399\) 0 0
\(400\) 479.112 0.0598890
\(401\) −890.013 −0.110836 −0.0554178 0.998463i \(-0.517649\pi\)
−0.0554178 + 0.998463i \(0.517649\pi\)
\(402\) 0 0
\(403\) −1376.37 −0.170128
\(404\) −2123.82 −0.261545
\(405\) 0 0
\(406\) 1550.22 0.189498
\(407\) −9249.29 −1.12646
\(408\) 0 0
\(409\) 8107.81 0.980209 0.490104 0.871664i \(-0.336959\pi\)
0.490104 + 0.871664i \(0.336959\pi\)
\(410\) 4506.54 0.542835
\(411\) 0 0
\(412\) 2643.52 0.316109
\(413\) −17729.8 −2.11241
\(414\) 0 0
\(415\) −5493.51 −0.649797
\(416\) 2264.45 0.266885
\(417\) 0 0
\(418\) 14740.1 1.72479
\(419\) −4237.59 −0.494080 −0.247040 0.969005i \(-0.579458\pi\)
−0.247040 + 0.969005i \(0.579458\pi\)
\(420\) 0 0
\(421\) −953.634 −0.110397 −0.0551987 0.998475i \(-0.517579\pi\)
−0.0551987 + 0.998475i \(0.517579\pi\)
\(422\) −2164.18 −0.249646
\(423\) 0 0
\(424\) 16539.5 1.89441
\(425\) 51.9463 0.00592886
\(426\) 0 0
\(427\) 4921.53 0.557774
\(428\) −1535.40 −0.173402
\(429\) 0 0
\(430\) −10364.0 −1.16232
\(431\) 10098.8 1.12864 0.564318 0.825557i \(-0.309139\pi\)
0.564318 + 0.825557i \(0.309139\pi\)
\(432\) 0 0
\(433\) −12833.1 −1.42430 −0.712148 0.702029i \(-0.752278\pi\)
−0.712148 + 0.702029i \(0.752278\pi\)
\(434\) 3095.34 0.342352
\(435\) 0 0
\(436\) 3248.97 0.356875
\(437\) −6502.42 −0.711792
\(438\) 0 0
\(439\) −14434.8 −1.56932 −0.784662 0.619924i \(-0.787164\pi\)
−0.784662 + 0.619924i \(0.787164\pi\)
\(440\) 10291.3 1.11505
\(441\) 0 0
\(442\) −260.765 −0.0280619
\(443\) 7020.65 0.752959 0.376480 0.926425i \(-0.377134\pi\)
0.376480 + 0.926425i \(0.377134\pi\)
\(444\) 0 0
\(445\) 3817.49 0.406666
\(446\) −5477.71 −0.581563
\(447\) 0 0
\(448\) −12516.7 −1.32000
\(449\) −13210.5 −1.38851 −0.694254 0.719730i \(-0.744265\pi\)
−0.694254 + 0.719730i \(0.744265\pi\)
\(450\) 0 0
\(451\) −6888.38 −0.719205
\(452\) −2317.58 −0.241172
\(453\) 0 0
\(454\) 5016.95 0.518628
\(455\) −5608.79 −0.577900
\(456\) 0 0
\(457\) −2811.18 −0.287749 −0.143875 0.989596i \(-0.545956\pi\)
−0.143875 + 0.989596i \(0.545956\pi\)
\(458\) −8968.92 −0.915044
\(459\) 0 0
\(460\) −969.245 −0.0982419
\(461\) 9645.10 0.974441 0.487220 0.873279i \(-0.338011\pi\)
0.487220 + 0.873279i \(0.338011\pi\)
\(462\) 0 0
\(463\) 6923.23 0.694924 0.347462 0.937694i \(-0.387044\pi\)
0.347462 + 0.937694i \(0.387044\pi\)
\(464\) 1215.44 0.121606
\(465\) 0 0
\(466\) −6087.07 −0.605103
\(467\) 124.351 0.0123218 0.00616089 0.999981i \(-0.498039\pi\)
0.00616089 + 0.999981i \(0.498039\pi\)
\(468\) 0 0
\(469\) 11613.1 1.14337
\(470\) −5859.33 −0.575044
\(471\) 0 0
\(472\) −19662.9 −1.91749
\(473\) 15841.7 1.53996
\(474\) 0 0
\(475\) 1774.79 0.171438
\(476\) −218.498 −0.0210396
\(477\) 0 0
\(478\) −9512.09 −0.910194
\(479\) 4461.48 0.425575 0.212788 0.977098i \(-0.431746\pi\)
0.212788 + 0.977098i \(0.431746\pi\)
\(480\) 0 0
\(481\) 5590.50 0.529948
\(482\) −4763.39 −0.450137
\(483\) 0 0
\(484\) −468.032 −0.0439549
\(485\) 8838.96 0.827539
\(486\) 0 0
\(487\) −5660.72 −0.526718 −0.263359 0.964698i \(-0.584830\pi\)
−0.263359 + 0.964698i \(0.584830\pi\)
\(488\) 5458.14 0.506308
\(489\) 0 0
\(490\) 3789.06 0.349331
\(491\) −5587.50 −0.513565 −0.256782 0.966469i \(-0.582662\pi\)
−0.256782 + 0.966469i \(0.582662\pi\)
\(492\) 0 0
\(493\) 131.781 0.0120387
\(494\) −8909.26 −0.811431
\(495\) 0 0
\(496\) 2426.88 0.219698
\(497\) −6242.24 −0.563386
\(498\) 0 0
\(499\) −4210.19 −0.377703 −0.188851 0.982006i \(-0.560476\pi\)
−0.188851 + 0.982006i \(0.560476\pi\)
\(500\) 3157.32 0.282399
\(501\) 0 0
\(502\) 2985.53 0.265440
\(503\) −10796.7 −0.957063 −0.478532 0.878070i \(-0.658831\pi\)
−0.478532 + 0.878070i \(0.658831\pi\)
\(504\) 0 0
\(505\) 10422.5 0.918407
\(506\) −3976.35 −0.349348
\(507\) 0 0
\(508\) 2561.54 0.223720
\(509\) −14983.6 −1.30479 −0.652395 0.757879i \(-0.726236\pi\)
−0.652395 + 0.757879i \(0.726236\pi\)
\(510\) 0 0
\(511\) −27214.8 −2.35599
\(512\) −12226.4 −1.05534
\(513\) 0 0
\(514\) −7044.91 −0.604548
\(515\) −12972.9 −1.11001
\(516\) 0 0
\(517\) 8956.17 0.761880
\(518\) −12572.6 −1.06643
\(519\) 0 0
\(520\) −6220.34 −0.524576
\(521\) 10770.3 0.905671 0.452836 0.891594i \(-0.350412\pi\)
0.452836 + 0.891594i \(0.350412\pi\)
\(522\) 0 0
\(523\) −7538.93 −0.630314 −0.315157 0.949040i \(-0.602057\pi\)
−0.315157 + 0.949040i \(0.602057\pi\)
\(524\) −5119.41 −0.426799
\(525\) 0 0
\(526\) 750.312 0.0621961
\(527\) 263.127 0.0217495
\(528\) 0 0
\(529\) −10412.9 −0.855829
\(530\) −17328.6 −1.42020
\(531\) 0 0
\(532\) −7465.16 −0.608375
\(533\) 4163.51 0.338352
\(534\) 0 0
\(535\) 7534.86 0.608898
\(536\) 12879.3 1.03787
\(537\) 0 0
\(538\) 4442.11 0.355972
\(539\) −5791.69 −0.462831
\(540\) 0 0
\(541\) 595.816 0.0473496 0.0236748 0.999720i \(-0.492463\pi\)
0.0236748 + 0.999720i \(0.492463\pi\)
\(542\) 12523.4 0.992482
\(543\) 0 0
\(544\) −432.908 −0.0341191
\(545\) −15944.1 −1.25316
\(546\) 0 0
\(547\) 12703.4 0.992978 0.496489 0.868043i \(-0.334622\pi\)
0.496489 + 0.868043i \(0.334622\pi\)
\(548\) 1568.94 0.122302
\(549\) 0 0
\(550\) 1085.32 0.0841418
\(551\) 4502.39 0.348109
\(552\) 0 0
\(553\) −13534.3 −1.04075
\(554\) 20922.6 1.60454
\(555\) 0 0
\(556\) −3036.39 −0.231604
\(557\) 14973.0 1.13901 0.569503 0.821990i \(-0.307136\pi\)
0.569503 + 0.821990i \(0.307136\pi\)
\(558\) 0 0
\(559\) −9575.10 −0.724479
\(560\) 9889.71 0.746280
\(561\) 0 0
\(562\) −18968.4 −1.42372
\(563\) 10439.3 0.781466 0.390733 0.920504i \(-0.372222\pi\)
0.390733 + 0.920504i \(0.372222\pi\)
\(564\) 0 0
\(565\) 11373.4 0.846869
\(566\) −7374.71 −0.547672
\(567\) 0 0
\(568\) −6922.85 −0.511402
\(569\) 8859.72 0.652757 0.326379 0.945239i \(-0.394172\pi\)
0.326379 + 0.945239i \(0.394172\pi\)
\(570\) 0 0
\(571\) −5078.17 −0.372180 −0.186090 0.982533i \(-0.559582\pi\)
−0.186090 + 0.982533i \(0.559582\pi\)
\(572\) 2029.90 0.148382
\(573\) 0 0
\(574\) −9363.40 −0.680873
\(575\) −478.775 −0.0347240
\(576\) 0 0
\(577\) 20457.2 1.47599 0.737994 0.674808i \(-0.235773\pi\)
0.737994 + 0.674808i \(0.235773\pi\)
\(578\) −11811.2 −0.849966
\(579\) 0 0
\(580\) 671.122 0.0480462
\(581\) 11414.1 0.815034
\(582\) 0 0
\(583\) 26487.3 1.88164
\(584\) −30182.1 −2.13860
\(585\) 0 0
\(586\) −6875.27 −0.484667
\(587\) −4355.22 −0.306234 −0.153117 0.988208i \(-0.548931\pi\)
−0.153117 + 0.988208i \(0.548931\pi\)
\(588\) 0 0
\(589\) 8989.96 0.628905
\(590\) 20601.0 1.43751
\(591\) 0 0
\(592\) −9857.46 −0.684357
\(593\) −4342.49 −0.300716 −0.150358 0.988632i \(-0.548043\pi\)
−0.150358 + 0.988632i \(0.548043\pi\)
\(594\) 0 0
\(595\) 1072.26 0.0738799
\(596\) 6146.92 0.422462
\(597\) 0 0
\(598\) 2403.40 0.164352
\(599\) 12761.8 0.870506 0.435253 0.900308i \(-0.356659\pi\)
0.435253 + 0.900308i \(0.356659\pi\)
\(600\) 0 0
\(601\) −2804.87 −0.190371 −0.0951857 0.995460i \(-0.530344\pi\)
−0.0951857 + 0.995460i \(0.530344\pi\)
\(602\) 21533.6 1.45788
\(603\) 0 0
\(604\) −3703.80 −0.249512
\(605\) 2296.84 0.154346
\(606\) 0 0
\(607\) 12033.5 0.804654 0.402327 0.915496i \(-0.368202\pi\)
0.402327 + 0.915496i \(0.368202\pi\)
\(608\) −14790.7 −0.986580
\(609\) 0 0
\(610\) −5718.54 −0.379569
\(611\) −5413.33 −0.358429
\(612\) 0 0
\(613\) 7993.67 0.526690 0.263345 0.964702i \(-0.415174\pi\)
0.263345 + 0.964702i \(0.415174\pi\)
\(614\) −5965.02 −0.392066
\(615\) 0 0
\(616\) −21382.7 −1.39859
\(617\) −9922.51 −0.647432 −0.323716 0.946154i \(-0.604932\pi\)
−0.323716 + 0.946154i \(0.604932\pi\)
\(618\) 0 0
\(619\) −15401.4 −1.00005 −0.500027 0.866010i \(-0.666676\pi\)
−0.500027 + 0.866010i \(0.666676\pi\)
\(620\) 1340.03 0.0868018
\(621\) 0 0
\(622\) 13648.3 0.879816
\(623\) −7931.74 −0.510078
\(624\) 0 0
\(625\) −14065.4 −0.900185
\(626\) −20013.2 −1.27777
\(627\) 0 0
\(628\) 5799.92 0.368538
\(629\) −1068.77 −0.0677497
\(630\) 0 0
\(631\) −3776.97 −0.238287 −0.119143 0.992877i \(-0.538015\pi\)
−0.119143 + 0.992877i \(0.538015\pi\)
\(632\) −15010.0 −0.944724
\(633\) 0 0
\(634\) −6221.57 −0.389732
\(635\) −12570.6 −0.785587
\(636\) 0 0
\(637\) 3500.64 0.217740
\(638\) 2753.29 0.170853
\(639\) 0 0
\(640\) 6421.72 0.396626
\(641\) −464.125 −0.0285988 −0.0142994 0.999898i \(-0.504552\pi\)
−0.0142994 + 0.999898i \(0.504552\pi\)
\(642\) 0 0
\(643\) 7607.61 0.466586 0.233293 0.972406i \(-0.425050\pi\)
0.233293 + 0.972406i \(0.425050\pi\)
\(644\) 2013.84 0.123224
\(645\) 0 0
\(646\) 1703.23 0.103735
\(647\) −7776.85 −0.472550 −0.236275 0.971686i \(-0.575927\pi\)
−0.236275 + 0.971686i \(0.575927\pi\)
\(648\) 0 0
\(649\) −31489.2 −1.90456
\(650\) −655.992 −0.0395848
\(651\) 0 0
\(652\) 4673.04 0.280691
\(653\) 24486.1 1.46740 0.733702 0.679471i \(-0.237791\pi\)
0.733702 + 0.679471i \(0.237791\pi\)
\(654\) 0 0
\(655\) 25123.2 1.49869
\(656\) −7341.31 −0.436936
\(657\) 0 0
\(658\) 12174.2 0.721273
\(659\) −910.966 −0.0538486 −0.0269243 0.999637i \(-0.508571\pi\)
−0.0269243 + 0.999637i \(0.508571\pi\)
\(660\) 0 0
\(661\) 5826.64 0.342859 0.171430 0.985196i \(-0.445161\pi\)
0.171430 + 0.985196i \(0.445161\pi\)
\(662\) 18522.1 1.08744
\(663\) 0 0
\(664\) 12658.6 0.739830
\(665\) 36634.8 2.13629
\(666\) 0 0
\(667\) −1214.59 −0.0705081
\(668\) 2170.24 0.125702
\(669\) 0 0
\(670\) −13493.8 −0.778074
\(671\) 8740.97 0.502893
\(672\) 0 0
\(673\) 27236.0 1.55999 0.779994 0.625788i \(-0.215222\pi\)
0.779994 + 0.625788i \(0.215222\pi\)
\(674\) 8813.75 0.503699
\(675\) 0 0
\(676\) 3544.03 0.201640
\(677\) −8989.70 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(678\) 0 0
\(679\) −18365.0 −1.03798
\(680\) 1189.18 0.0670630
\(681\) 0 0
\(682\) 5497.52 0.308667
\(683\) −6934.65 −0.388502 −0.194251 0.980952i \(-0.562228\pi\)
−0.194251 + 0.980952i \(0.562228\pi\)
\(684\) 0 0
\(685\) −7699.46 −0.429462
\(686\) 10462.7 0.582314
\(687\) 0 0
\(688\) 16883.3 0.935567
\(689\) −16009.6 −0.885221
\(690\) 0 0
\(691\) 5234.54 0.288178 0.144089 0.989565i \(-0.453975\pi\)
0.144089 + 0.989565i \(0.453975\pi\)
\(692\) −5503.43 −0.302325
\(693\) 0 0
\(694\) 21093.1 1.15372
\(695\) 14900.9 0.813270
\(696\) 0 0
\(697\) −795.961 −0.0432556
\(698\) −5175.67 −0.280662
\(699\) 0 0
\(700\) −549.662 −0.0296789
\(701\) 33713.7 1.81647 0.908237 0.418457i \(-0.137429\pi\)
0.908237 + 0.418457i \(0.137429\pi\)
\(702\) 0 0
\(703\) −36515.3 −1.95903
\(704\) −22230.5 −1.19012
\(705\) 0 0
\(706\) 12339.4 0.657792
\(707\) −21655.2 −1.15195
\(708\) 0 0
\(709\) −12687.3 −0.672049 −0.336025 0.941853i \(-0.609083\pi\)
−0.336025 + 0.941853i \(0.609083\pi\)
\(710\) 7253.13 0.383388
\(711\) 0 0
\(712\) −8796.56 −0.463013
\(713\) −2425.17 −0.127382
\(714\) 0 0
\(715\) −9961.59 −0.521038
\(716\) 7709.66 0.402407
\(717\) 0 0
\(718\) −10914.3 −0.567297
\(719\) −30604.9 −1.58744 −0.793720 0.608283i \(-0.791859\pi\)
−0.793720 + 0.608283i \(0.791859\pi\)
\(720\) 0 0
\(721\) 26954.2 1.39227
\(722\) 41633.3 2.14602
\(723\) 0 0
\(724\) 6590.02 0.338282
\(725\) 331.512 0.0169821
\(726\) 0 0
\(727\) 3727.96 0.190182 0.0950911 0.995469i \(-0.469686\pi\)
0.0950911 + 0.995469i \(0.469686\pi\)
\(728\) 12924.2 0.657972
\(729\) 0 0
\(730\) 31622.1 1.60327
\(731\) 1830.52 0.0926189
\(732\) 0 0
\(733\) 1107.21 0.0557921 0.0278960 0.999611i \(-0.491119\pi\)
0.0278960 + 0.999611i \(0.491119\pi\)
\(734\) −1173.97 −0.0590352
\(735\) 0 0
\(736\) 3990.00 0.199828
\(737\) 20625.6 1.03087
\(738\) 0 0
\(739\) −29127.7 −1.44991 −0.724953 0.688798i \(-0.758139\pi\)
−0.724953 + 0.688798i \(0.758139\pi\)
\(740\) −5442.94 −0.270387
\(741\) 0 0
\(742\) 36004.3 1.78135
\(743\) −12162.8 −0.600552 −0.300276 0.953852i \(-0.597079\pi\)
−0.300276 + 0.953852i \(0.597079\pi\)
\(744\) 0 0
\(745\) −30165.6 −1.48347
\(746\) 24270.2 1.19115
\(747\) 0 0
\(748\) −388.067 −0.0189694
\(749\) −15655.5 −0.763736
\(750\) 0 0
\(751\) −25067.6 −1.21802 −0.609008 0.793164i \(-0.708432\pi\)
−0.609008 + 0.793164i \(0.708432\pi\)
\(752\) 9545.06 0.462862
\(753\) 0 0
\(754\) −1664.16 −0.0803781
\(755\) 18176.1 0.876156
\(756\) 0 0
\(757\) −12020.6 −0.577140 −0.288570 0.957459i \(-0.593180\pi\)
−0.288570 + 0.957459i \(0.593180\pi\)
\(758\) −28011.6 −1.34225
\(759\) 0 0
\(760\) 40629.2 1.93918
\(761\) −9255.42 −0.440879 −0.220439 0.975401i \(-0.570749\pi\)
−0.220439 + 0.975401i \(0.570749\pi\)
\(762\) 0 0
\(763\) 33127.6 1.57182
\(764\) 4829.58 0.228702
\(765\) 0 0
\(766\) 10046.5 0.473881
\(767\) 19032.9 0.896007
\(768\) 0 0
\(769\) 28690.1 1.34537 0.672687 0.739928i \(-0.265140\pi\)
0.672687 + 0.739928i \(0.265140\pi\)
\(770\) 22402.8 1.04850
\(771\) 0 0
\(772\) 1374.24 0.0640672
\(773\) 8984.36 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(774\) 0 0
\(775\) 661.933 0.0306804
\(776\) −20367.4 −0.942201
\(777\) 0 0
\(778\) 16522.8 0.761400
\(779\) −27194.7 −1.25077
\(780\) 0 0
\(781\) −11086.6 −0.507952
\(782\) −459.472 −0.0210111
\(783\) 0 0
\(784\) −6172.51 −0.281182
\(785\) −28462.7 −1.29411
\(786\) 0 0
\(787\) −5257.13 −0.238115 −0.119058 0.992887i \(-0.537987\pi\)
−0.119058 + 0.992887i \(0.537987\pi\)
\(788\) 2974.63 0.134476
\(789\) 0 0
\(790\) 15726.1 0.708240
\(791\) −23630.9 −1.06222
\(792\) 0 0
\(793\) −5283.26 −0.236588
\(794\) 13217.2 0.590755
\(795\) 0 0
\(796\) −3075.76 −0.136957
\(797\) −7623.92 −0.338837 −0.169419 0.985544i \(-0.554189\pi\)
−0.169419 + 0.985544i \(0.554189\pi\)
\(798\) 0 0
\(799\) 1034.90 0.0458223
\(800\) −1089.04 −0.0481293
\(801\) 0 0
\(802\) −2148.68 −0.0946042
\(803\) −48335.3 −2.12418
\(804\) 0 0
\(805\) −9882.77 −0.432698
\(806\) −3322.84 −0.145213
\(807\) 0 0
\(808\) −24016.4 −1.04566
\(809\) 28238.7 1.22722 0.613609 0.789610i \(-0.289717\pi\)
0.613609 + 0.789610i \(0.289717\pi\)
\(810\) 0 0
\(811\) 29851.2 1.29250 0.646251 0.763125i \(-0.276336\pi\)
0.646251 + 0.763125i \(0.276336\pi\)
\(812\) −1394.41 −0.0602640
\(813\) 0 0
\(814\) −22329.8 −0.961496
\(815\) −22932.7 −0.985640
\(816\) 0 0
\(817\) 62541.4 2.67815
\(818\) 19574.0 0.836661
\(819\) 0 0
\(820\) −4053.61 −0.172632
\(821\) −2043.20 −0.0868552 −0.0434276 0.999057i \(-0.513828\pi\)
−0.0434276 + 0.999057i \(0.513828\pi\)
\(822\) 0 0
\(823\) 123.242 0.00521988 0.00260994 0.999997i \(-0.499169\pi\)
0.00260994 + 0.999997i \(0.499169\pi\)
\(824\) 29893.1 1.26381
\(825\) 0 0
\(826\) −42803.4 −1.80305
\(827\) −34335.2 −1.44371 −0.721857 0.692043i \(-0.756711\pi\)
−0.721857 + 0.692043i \(0.756711\pi\)
\(828\) 0 0
\(829\) −14705.0 −0.616073 −0.308037 0.951375i \(-0.599672\pi\)
−0.308037 + 0.951375i \(0.599672\pi\)
\(830\) −13262.5 −0.554636
\(831\) 0 0
\(832\) 13436.7 0.559894
\(833\) −669.237 −0.0278364
\(834\) 0 0
\(835\) −10650.3 −0.441401
\(836\) −13258.6 −0.548515
\(837\) 0 0
\(838\) −10230.4 −0.421724
\(839\) 22204.3 0.913679 0.456840 0.889549i \(-0.348981\pi\)
0.456840 + 0.889549i \(0.348981\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) −2302.28 −0.0942300
\(843\) 0 0
\(844\) 1946.67 0.0793922
\(845\) −17392.1 −0.708054
\(846\) 0 0
\(847\) −4772.22 −0.193595
\(848\) 28228.9 1.14314
\(849\) 0 0
\(850\) 125.410 0.00506060
\(851\) 9850.54 0.396794
\(852\) 0 0
\(853\) −40095.2 −1.60942 −0.804710 0.593669i \(-0.797679\pi\)
−0.804710 + 0.593669i \(0.797679\pi\)
\(854\) 11881.6 0.476090
\(855\) 0 0
\(856\) −17362.4 −0.693265
\(857\) 12883.7 0.513536 0.256768 0.966473i \(-0.417342\pi\)
0.256768 + 0.966473i \(0.417342\pi\)
\(858\) 0 0
\(859\) −34336.9 −1.36386 −0.681932 0.731416i \(-0.738860\pi\)
−0.681932 + 0.731416i \(0.738860\pi\)
\(860\) 9322.36 0.369639
\(861\) 0 0
\(862\) 24380.7 0.963351
\(863\) −10864.2 −0.428531 −0.214266 0.976775i \(-0.568736\pi\)
−0.214266 + 0.976775i \(0.568736\pi\)
\(864\) 0 0
\(865\) 27007.7 1.06161
\(866\) −30981.9 −1.21571
\(867\) 0 0
\(868\) −2784.24 −0.108875
\(869\) −24037.8 −0.938352
\(870\) 0 0
\(871\) −12466.6 −0.484978
\(872\) 36739.6 1.42679
\(873\) 0 0
\(874\) −15698.2 −0.607552
\(875\) 32193.1 1.24380
\(876\) 0 0
\(877\) 4818.40 0.185525 0.0927626 0.995688i \(-0.470430\pi\)
0.0927626 + 0.995688i \(0.470430\pi\)
\(878\) −34848.6 −1.33950
\(879\) 0 0
\(880\) 17564.8 0.672851
\(881\) 30195.7 1.15473 0.577366 0.816485i \(-0.304080\pi\)
0.577366 + 0.816485i \(0.304080\pi\)
\(882\) 0 0
\(883\) 30734.0 1.17133 0.585663 0.810555i \(-0.300834\pi\)
0.585663 + 0.810555i \(0.300834\pi\)
\(884\) 234.557 0.00892422
\(885\) 0 0
\(886\) 16949.3 0.642691
\(887\) 43409.3 1.64323 0.821614 0.570045i \(-0.193074\pi\)
0.821614 + 0.570045i \(0.193074\pi\)
\(888\) 0 0
\(889\) 26118.3 0.985355
\(890\) 9216.24 0.347111
\(891\) 0 0
\(892\) 4927.17 0.184948
\(893\) 35358.1 1.32499
\(894\) 0 0
\(895\) −37834.7 −1.41304
\(896\) −13342.6 −0.497485
\(897\) 0 0
\(898\) −31892.9 −1.18517
\(899\) 1679.23 0.0622976
\(900\) 0 0
\(901\) 3060.64 0.113169
\(902\) −16630.0 −0.613880
\(903\) 0 0
\(904\) −26207.4 −0.964209
\(905\) −32340.1 −1.18787
\(906\) 0 0
\(907\) −7991.51 −0.292562 −0.146281 0.989243i \(-0.546730\pi\)
−0.146281 + 0.989243i \(0.546730\pi\)
\(908\) −4512.72 −0.164934
\(909\) 0 0
\(910\) −13540.8 −0.493268
\(911\) −24188.5 −0.879694 −0.439847 0.898073i \(-0.644967\pi\)
−0.439847 + 0.898073i \(0.644967\pi\)
\(912\) 0 0
\(913\) 20272.1 0.734840
\(914\) −6786.79 −0.245609
\(915\) 0 0
\(916\) 8067.50 0.291002
\(917\) −52199.4 −1.87980
\(918\) 0 0
\(919\) 27421.3 0.984273 0.492136 0.870518i \(-0.336216\pi\)
0.492136 + 0.870518i \(0.336216\pi\)
\(920\) −10960.3 −0.392773
\(921\) 0 0
\(922\) 23285.3 0.831737
\(923\) 6701.03 0.238968
\(924\) 0 0
\(925\) −2688.63 −0.0955694
\(926\) 16714.1 0.593154
\(927\) 0 0
\(928\) −2762.74 −0.0977279
\(929\) 9284.12 0.327882 0.163941 0.986470i \(-0.447579\pi\)
0.163941 + 0.986470i \(0.447579\pi\)
\(930\) 0 0
\(931\) −22865.0 −0.804910
\(932\) 5475.29 0.192435
\(933\) 0 0
\(934\) 300.210 0.0105173
\(935\) 1904.41 0.0666106
\(936\) 0 0
\(937\) −1891.00 −0.0659297 −0.0329649 0.999457i \(-0.510495\pi\)
−0.0329649 + 0.999457i \(0.510495\pi\)
\(938\) 28036.5 0.975932
\(939\) 0 0
\(940\) 5270.44 0.182875
\(941\) −15163.9 −0.525322 −0.262661 0.964888i \(-0.584600\pi\)
−0.262661 + 0.964888i \(0.584600\pi\)
\(942\) 0 0
\(943\) 7336.16 0.253338
\(944\) −33559.7 −1.15707
\(945\) 0 0
\(946\) 38245.2 1.31444
\(947\) 31251.8 1.07238 0.536192 0.844096i \(-0.319862\pi\)
0.536192 + 0.844096i \(0.319862\pi\)
\(948\) 0 0
\(949\) 29215.1 0.999327
\(950\) 4284.72 0.146331
\(951\) 0 0
\(952\) −2470.79 −0.0841165
\(953\) 12521.4 0.425612 0.212806 0.977094i \(-0.431740\pi\)
0.212806 + 0.977094i \(0.431740\pi\)
\(954\) 0 0
\(955\) −23700.9 −0.803081
\(956\) 8556.07 0.289459
\(957\) 0 0
\(958\) 10771.0 0.363251
\(959\) 15997.5 0.538670
\(960\) 0 0
\(961\) −26438.1 −0.887451
\(962\) 13496.7 0.452339
\(963\) 0 0
\(964\) 4284.64 0.143152
\(965\) −6743.98 −0.224970
\(966\) 0 0
\(967\) −18348.5 −0.610183 −0.305091 0.952323i \(-0.598687\pi\)
−0.305091 + 0.952323i \(0.598687\pi\)
\(968\) −5292.55 −0.175732
\(969\) 0 0
\(970\) 21339.1 0.706349
\(971\) −27188.0 −0.898564 −0.449282 0.893390i \(-0.648320\pi\)
−0.449282 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −30960.1 −1.02008
\(974\) −13666.2 −0.449582
\(975\) 0 0
\(976\) 9315.71 0.305521
\(977\) −4733.67 −0.155009 −0.0775044 0.996992i \(-0.524695\pi\)
−0.0775044 + 0.996992i \(0.524695\pi\)
\(978\) 0 0
\(979\) −14087.3 −0.459890
\(980\) −3408.24 −0.111094
\(981\) 0 0
\(982\) −13489.4 −0.438355
\(983\) 43971.5 1.42673 0.713363 0.700795i \(-0.247171\pi\)
0.713363 + 0.700795i \(0.247171\pi\)
\(984\) 0 0
\(985\) −14597.8 −0.472208
\(986\) 318.146 0.0102757
\(987\) 0 0
\(988\) 8013.84 0.258051
\(989\) −16871.4 −0.542448
\(990\) 0 0
\(991\) −14031.4 −0.449769 −0.224885 0.974385i \(-0.572201\pi\)
−0.224885 + 0.974385i \(0.572201\pi\)
\(992\) −5516.39 −0.176558
\(993\) 0 0
\(994\) −15070.1 −0.480880
\(995\) 15094.1 0.480920
\(996\) 0 0
\(997\) 43177.4 1.37156 0.685779 0.727810i \(-0.259462\pi\)
0.685779 + 0.727810i \(0.259462\pi\)
\(998\) −10164.3 −0.322390
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 261.4.a.b.1.2 2
3.2 odd 2 29.4.a.a.1.1 2
12.11 even 2 464.4.a.f.1.1 2
15.14 odd 2 725.4.a.b.1.2 2
21.20 even 2 1421.4.a.c.1.1 2
24.5 odd 2 1856.4.a.n.1.1 2
24.11 even 2 1856.4.a.h.1.2 2
87.86 odd 2 841.4.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.a.1.1 2 3.2 odd 2
261.4.a.b.1.2 2 1.1 even 1 trivial
464.4.a.f.1.1 2 12.11 even 2
725.4.a.b.1.2 2 15.14 odd 2
841.4.a.a.1.2 2 87.86 odd 2
1421.4.a.c.1.1 2 21.20 even 2
1856.4.a.h.1.2 2 24.11 even 2
1856.4.a.n.1.1 2 24.5 odd 2