Properties

Label 1521.4.a.bn.1.2
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $18$
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] = [1521,4,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 100 x^{16} + 4066 x^{14} - 86197 x^{12} + 1016064 x^{10} - 6594119 x^{8} + 22251817 x^{6} - 37096332 x^{4} + 27824160 x^{2} - 6889792 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7\cdot 13^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.80856\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.80856 q^{2} +15.1223 q^{4} -1.59043 q^{5} +34.5272 q^{7} -34.2479 q^{8} +O(q^{10})\) \(q-4.80856 q^{2} +15.1223 q^{4} -1.59043 q^{5} +34.5272 q^{7} -34.2479 q^{8} +7.64769 q^{10} -42.6836 q^{11} -166.026 q^{14} +43.7049 q^{16} +45.4494 q^{17} +41.0910 q^{19} -24.0509 q^{20} +205.247 q^{22} -171.733 q^{23} -122.471 q^{25} +522.129 q^{28} -178.040 q^{29} +18.5736 q^{31} +63.8255 q^{32} -218.546 q^{34} -54.9131 q^{35} -331.269 q^{37} -197.588 q^{38} +54.4689 q^{40} +411.934 q^{41} +11.3363 q^{43} -645.473 q^{44} +825.787 q^{46} +494.592 q^{47} +849.125 q^{49} +588.907 q^{50} +217.808 q^{53} +67.8853 q^{55} -1182.48 q^{56} +856.115 q^{58} -775.414 q^{59} +340.831 q^{61} -89.3123 q^{62} -656.547 q^{64} +676.191 q^{67} +687.298 q^{68} +264.053 q^{70} +449.604 q^{71} +645.014 q^{73} +1592.93 q^{74} +621.389 q^{76} -1473.74 q^{77} -778.289 q^{79} -69.5095 q^{80} -1980.81 q^{82} +1056.50 q^{83} -72.2842 q^{85} -54.5114 q^{86} +1461.82 q^{88} -160.228 q^{89} -2596.99 q^{92} -2378.28 q^{94} -65.3523 q^{95} +7.23476 q^{97} -4083.07 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 56 q^{4} + 94 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 56 q^{4} + 94 q^{7} + 156 q^{10} + 88 q^{16} + 448 q^{19} - 256 q^{22} - 16 q^{25} + 1300 q^{28} + 818 q^{31} + 900 q^{34} + 524 q^{37} + 800 q^{40} + 752 q^{43} + 1650 q^{46} + 2948 q^{49} - 464 q^{55} + 1626 q^{58} - 1784 q^{61} - 4274 q^{64} + 2872 q^{67} + 5772 q^{70} + 3078 q^{73} + 5726 q^{76} + 3326 q^{79} - 1038 q^{82} + 2340 q^{85} + 780 q^{88} + 1220 q^{94} + 4630 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.80856 −1.70008 −0.850042 0.526715i \(-0.823423\pi\)
−0.850042 + 0.526715i \(0.823423\pi\)
\(3\) 0 0
\(4\) 15.1223 1.89028
\(5\) −1.59043 −0.142252 −0.0711262 0.997467i \(-0.522659\pi\)
−0.0711262 + 0.997467i \(0.522659\pi\)
\(6\) 0 0
\(7\) 34.5272 1.86429 0.932146 0.362083i \(-0.117934\pi\)
0.932146 + 0.362083i \(0.117934\pi\)
\(8\) −34.2479 −1.51356
\(9\) 0 0
\(10\) 7.64769 0.241841
\(11\) −42.6836 −1.16996 −0.584981 0.811047i \(-0.698898\pi\)
−0.584981 + 0.811047i \(0.698898\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −166.026 −3.16945
\(15\) 0 0
\(16\) 43.7049 0.682888
\(17\) 45.4494 0.648418 0.324209 0.945985i \(-0.394902\pi\)
0.324209 + 0.945985i \(0.394902\pi\)
\(18\) 0 0
\(19\) 41.0910 0.496153 0.248077 0.968740i \(-0.420202\pi\)
0.248077 + 0.968740i \(0.420202\pi\)
\(20\) −24.0509 −0.268897
\(21\) 0 0
\(22\) 205.247 1.98903
\(23\) −171.733 −1.55690 −0.778450 0.627706i \(-0.783994\pi\)
−0.778450 + 0.627706i \(0.783994\pi\)
\(24\) 0 0
\(25\) −122.471 −0.979764
\(26\) 0 0
\(27\) 0 0
\(28\) 522.129 3.52404
\(29\) −178.040 −1.14004 −0.570019 0.821631i \(-0.693064\pi\)
−0.570019 + 0.821631i \(0.693064\pi\)
\(30\) 0 0
\(31\) 18.5736 0.107610 0.0538051 0.998551i \(-0.482865\pi\)
0.0538051 + 0.998551i \(0.482865\pi\)
\(32\) 63.8255 0.352589
\(33\) 0 0
\(34\) −218.546 −1.10236
\(35\) −54.9131 −0.265200
\(36\) 0 0
\(37\) −331.269 −1.47190 −0.735949 0.677037i \(-0.763264\pi\)
−0.735949 + 0.677037i \(0.763264\pi\)
\(38\) −197.588 −0.843502
\(39\) 0 0
\(40\) 54.4689 0.215307
\(41\) 411.934 1.56911 0.784553 0.620062i \(-0.212893\pi\)
0.784553 + 0.620062i \(0.212893\pi\)
\(42\) 0 0
\(43\) 11.3363 0.0402040 0.0201020 0.999798i \(-0.493601\pi\)
0.0201020 + 0.999798i \(0.493601\pi\)
\(44\) −645.473 −2.21156
\(45\) 0 0
\(46\) 825.787 2.64686
\(47\) 494.592 1.53497 0.767486 0.641065i \(-0.221507\pi\)
0.767486 + 0.641065i \(0.221507\pi\)
\(48\) 0 0
\(49\) 849.125 2.47558
\(50\) 588.907 1.66568
\(51\) 0 0
\(52\) 0 0
\(53\) 217.808 0.564495 0.282248 0.959342i \(-0.408920\pi\)
0.282248 + 0.959342i \(0.408920\pi\)
\(54\) 0 0
\(55\) 67.8853 0.166430
\(56\) −1182.48 −2.82171
\(57\) 0 0
\(58\) 856.115 1.93816
\(59\) −775.414 −1.71102 −0.855511 0.517785i \(-0.826757\pi\)
−0.855511 + 0.517785i \(0.826757\pi\)
\(60\) 0 0
\(61\) 340.831 0.715392 0.357696 0.933838i \(-0.383562\pi\)
0.357696 + 0.933838i \(0.383562\pi\)
\(62\) −89.3123 −0.182946
\(63\) 0 0
\(64\) −656.547 −1.28232
\(65\) 0 0
\(66\) 0 0
\(67\) 676.191 1.23298 0.616492 0.787361i \(-0.288553\pi\)
0.616492 + 0.787361i \(0.288553\pi\)
\(68\) 687.298 1.22569
\(69\) 0 0
\(70\) 264.053 0.450862
\(71\) 449.604 0.751524 0.375762 0.926716i \(-0.377381\pi\)
0.375762 + 0.926716i \(0.377381\pi\)
\(72\) 0 0
\(73\) 645.014 1.03415 0.517076 0.855939i \(-0.327020\pi\)
0.517076 + 0.855939i \(0.327020\pi\)
\(74\) 1592.93 2.50235
\(75\) 0 0
\(76\) 621.389 0.937870
\(77\) −1473.74 −2.18115
\(78\) 0 0
\(79\) −778.289 −1.10841 −0.554205 0.832380i \(-0.686978\pi\)
−0.554205 + 0.832380i \(0.686978\pi\)
\(80\) −69.5095 −0.0971425
\(81\) 0 0
\(82\) −1980.81 −2.66761
\(83\) 1056.50 1.39718 0.698591 0.715521i \(-0.253811\pi\)
0.698591 + 0.715521i \(0.253811\pi\)
\(84\) 0 0
\(85\) −72.2842 −0.0922391
\(86\) −54.5114 −0.0683501
\(87\) 0 0
\(88\) 1461.82 1.77080
\(89\) −160.228 −0.190833 −0.0954166 0.995437i \(-0.530418\pi\)
−0.0954166 + 0.995437i \(0.530418\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2596.99 −2.94298
\(93\) 0 0
\(94\) −2378.28 −2.60958
\(95\) −65.3523 −0.0705790
\(96\) 0 0
\(97\) 7.23476 0.00757297 0.00378649 0.999993i \(-0.498795\pi\)
0.00378649 + 0.999993i \(0.498795\pi\)
\(98\) −4083.07 −4.20870
\(99\) 0 0
\(100\) −1852.03 −1.85203
\(101\) −1125.70 −1.10902 −0.554511 0.832176i \(-0.687095\pi\)
−0.554511 + 0.832176i \(0.687095\pi\)
\(102\) 0 0
\(103\) 1204.03 1.15181 0.575907 0.817515i \(-0.304649\pi\)
0.575907 + 0.817515i \(0.304649\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1047.34 −0.959689
\(107\) 1491.74 1.34778 0.673889 0.738833i \(-0.264623\pi\)
0.673889 + 0.738833i \(0.264623\pi\)
\(108\) 0 0
\(109\) 375.430 0.329905 0.164953 0.986301i \(-0.447253\pi\)
0.164953 + 0.986301i \(0.447253\pi\)
\(110\) −326.431 −0.282945
\(111\) 0 0
\(112\) 1509.00 1.27310
\(113\) 810.999 0.675154 0.337577 0.941298i \(-0.390393\pi\)
0.337577 + 0.941298i \(0.390393\pi\)
\(114\) 0 0
\(115\) 273.129 0.221473
\(116\) −2692.36 −2.15500
\(117\) 0 0
\(118\) 3728.62 2.90888
\(119\) 1569.24 1.20884
\(120\) 0 0
\(121\) 490.889 0.368812
\(122\) −1638.91 −1.21623
\(123\) 0 0
\(124\) 280.875 0.203414
\(125\) 393.585 0.281626
\(126\) 0 0
\(127\) −584.796 −0.408600 −0.204300 0.978908i \(-0.565492\pi\)
−0.204300 + 0.978908i \(0.565492\pi\)
\(128\) 2646.45 1.82746
\(129\) 0 0
\(130\) 0 0
\(131\) −1521.40 −1.01470 −0.507350 0.861740i \(-0.669375\pi\)
−0.507350 + 0.861740i \(0.669375\pi\)
\(132\) 0 0
\(133\) 1418.75 0.924975
\(134\) −3251.51 −2.09618
\(135\) 0 0
\(136\) −1556.55 −0.981417
\(137\) 583.044 0.363597 0.181799 0.983336i \(-0.441808\pi\)
0.181799 + 0.983336i \(0.441808\pi\)
\(138\) 0 0
\(139\) −2008.49 −1.22560 −0.612798 0.790240i \(-0.709956\pi\)
−0.612798 + 0.790240i \(0.709956\pi\)
\(140\) −830.410 −0.501303
\(141\) 0 0
\(142\) −2161.95 −1.27765
\(143\) 0 0
\(144\) 0 0
\(145\) 283.160 0.162173
\(146\) −3101.59 −1.75815
\(147\) 0 0
\(148\) −5009.54 −2.78231
\(149\) −1357.74 −0.746515 −0.373257 0.927728i \(-0.621759\pi\)
−0.373257 + 0.927728i \(0.621759\pi\)
\(150\) 0 0
\(151\) 2305.46 1.24249 0.621243 0.783618i \(-0.286628\pi\)
0.621243 + 0.783618i \(0.286628\pi\)
\(152\) −1407.28 −0.750956
\(153\) 0 0
\(154\) 7086.59 3.70814
\(155\) −29.5400 −0.0153078
\(156\) 0 0
\(157\) 1254.03 0.637470 0.318735 0.947844i \(-0.396742\pi\)
0.318735 + 0.947844i \(0.396742\pi\)
\(158\) 3742.45 1.88439
\(159\) 0 0
\(160\) −101.510 −0.0501567
\(161\) −5929.44 −2.90252
\(162\) 0 0
\(163\) −2792.83 −1.34203 −0.671017 0.741442i \(-0.734142\pi\)
−0.671017 + 0.741442i \(0.734142\pi\)
\(164\) 6229.38 2.96605
\(165\) 0 0
\(166\) −5080.25 −2.37533
\(167\) 1218.71 0.564712 0.282356 0.959310i \(-0.408884\pi\)
0.282356 + 0.959310i \(0.408884\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 347.583 0.156814
\(171\) 0 0
\(172\) 171.431 0.0759969
\(173\) −1526.65 −0.670922 −0.335461 0.942054i \(-0.608892\pi\)
−0.335461 + 0.942054i \(0.608892\pi\)
\(174\) 0 0
\(175\) −4228.56 −1.82657
\(176\) −1865.48 −0.798954
\(177\) 0 0
\(178\) 770.468 0.324433
\(179\) −1067.26 −0.445647 −0.222823 0.974859i \(-0.571527\pi\)
−0.222823 + 0.974859i \(0.571527\pi\)
\(180\) 0 0
\(181\) 3579.98 1.47015 0.735076 0.677984i \(-0.237146\pi\)
0.735076 + 0.677984i \(0.237146\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5881.47 2.35646
\(185\) 526.860 0.209381
\(186\) 0 0
\(187\) −1939.94 −0.758625
\(188\) 7479.36 2.90153
\(189\) 0 0
\(190\) 314.251 0.119990
\(191\) 3087.19 1.16953 0.584767 0.811201i \(-0.301186\pi\)
0.584767 + 0.811201i \(0.301186\pi\)
\(192\) 0 0
\(193\) −897.457 −0.334717 −0.167358 0.985896i \(-0.553524\pi\)
−0.167358 + 0.985896i \(0.553524\pi\)
\(194\) −34.7888 −0.0128747
\(195\) 0 0
\(196\) 12840.7 4.67956
\(197\) 23.3043 0.00842825 0.00421412 0.999991i \(-0.498659\pi\)
0.00421412 + 0.999991i \(0.498659\pi\)
\(198\) 0 0
\(199\) 3910.89 1.39314 0.696571 0.717487i \(-0.254708\pi\)
0.696571 + 0.717487i \(0.254708\pi\)
\(200\) 4194.35 1.48293
\(201\) 0 0
\(202\) 5412.99 1.88543
\(203\) −6147.20 −2.12537
\(204\) 0 0
\(205\) −655.153 −0.223209
\(206\) −5789.67 −1.95818
\(207\) 0 0
\(208\) 0 0
\(209\) −1753.91 −0.580481
\(210\) 0 0
\(211\) −191.920 −0.0626177 −0.0313088 0.999510i \(-0.509968\pi\)
−0.0313088 + 0.999510i \(0.509968\pi\)
\(212\) 3293.75 1.06706
\(213\) 0 0
\(214\) −7173.14 −2.29133
\(215\) −18.0296 −0.00571912
\(216\) 0 0
\(217\) 641.294 0.200617
\(218\) −1805.28 −0.560866
\(219\) 0 0
\(220\) 1026.58 0.314600
\(221\) 0 0
\(222\) 0 0
\(223\) −1444.23 −0.433688 −0.216844 0.976206i \(-0.569576\pi\)
−0.216844 + 0.976206i \(0.569576\pi\)
\(224\) 2203.71 0.657329
\(225\) 0 0
\(226\) −3899.74 −1.14782
\(227\) −4307.34 −1.25942 −0.629710 0.776830i \(-0.716826\pi\)
−0.629710 + 0.776830i \(0.716826\pi\)
\(228\) 0 0
\(229\) 1864.12 0.537924 0.268962 0.963151i \(-0.413319\pi\)
0.268962 + 0.963151i \(0.413319\pi\)
\(230\) −1313.36 −0.376522
\(231\) 0 0
\(232\) 6097.48 1.72551
\(233\) 3541.27 0.995691 0.497846 0.867266i \(-0.334125\pi\)
0.497846 + 0.867266i \(0.334125\pi\)
\(234\) 0 0
\(235\) −786.615 −0.218354
\(236\) −11726.0 −3.23432
\(237\) 0 0
\(238\) −7545.79 −2.05513
\(239\) 2043.01 0.552935 0.276467 0.961023i \(-0.410836\pi\)
0.276467 + 0.961023i \(0.410836\pi\)
\(240\) 0 0
\(241\) −228.833 −0.0611636 −0.0305818 0.999532i \(-0.509736\pi\)
−0.0305818 + 0.999532i \(0.509736\pi\)
\(242\) −2360.47 −0.627011
\(243\) 0 0
\(244\) 5154.13 1.35229
\(245\) −1350.48 −0.352158
\(246\) 0 0
\(247\) 0 0
\(248\) −636.106 −0.162874
\(249\) 0 0
\(250\) −1892.58 −0.478788
\(251\) 3409.06 0.857282 0.428641 0.903475i \(-0.358993\pi\)
0.428641 + 0.903475i \(0.358993\pi\)
\(252\) 0 0
\(253\) 7330.16 1.82151
\(254\) 2812.03 0.694654
\(255\) 0 0
\(256\) −7473.22 −1.82452
\(257\) 3963.28 0.961955 0.480978 0.876733i \(-0.340282\pi\)
0.480978 + 0.876733i \(0.340282\pi\)
\(258\) 0 0
\(259\) −11437.8 −2.74405
\(260\) 0 0
\(261\) 0 0
\(262\) 7315.76 1.72507
\(263\) 4890.12 1.14653 0.573266 0.819370i \(-0.305676\pi\)
0.573266 + 0.819370i \(0.305676\pi\)
\(264\) 0 0
\(265\) −346.409 −0.0803008
\(266\) −6822.17 −1.57253
\(267\) 0 0
\(268\) 10225.5 2.33069
\(269\) 3946.47 0.894500 0.447250 0.894409i \(-0.352403\pi\)
0.447250 + 0.894409i \(0.352403\pi\)
\(270\) 0 0
\(271\) 2692.66 0.603570 0.301785 0.953376i \(-0.402418\pi\)
0.301785 + 0.953376i \(0.402418\pi\)
\(272\) 1986.36 0.442797
\(273\) 0 0
\(274\) −2803.60 −0.618146
\(275\) 5227.48 1.14629
\(276\) 0 0
\(277\) −2638.62 −0.572344 −0.286172 0.958178i \(-0.592383\pi\)
−0.286172 + 0.958178i \(0.592383\pi\)
\(278\) 9657.94 2.08361
\(279\) 0 0
\(280\) 1880.66 0.401395
\(281\) −3345.23 −0.710176 −0.355088 0.934833i \(-0.615549\pi\)
−0.355088 + 0.934833i \(0.615549\pi\)
\(282\) 0 0
\(283\) 2799.71 0.588076 0.294038 0.955794i \(-0.405001\pi\)
0.294038 + 0.955794i \(0.405001\pi\)
\(284\) 6799.04 1.42059
\(285\) 0 0
\(286\) 0 0
\(287\) 14222.9 2.92527
\(288\) 0 0
\(289\) −2847.35 −0.579554
\(290\) −1361.59 −0.275708
\(291\) 0 0
\(292\) 9754.07 1.95484
\(293\) 8050.62 1.60520 0.802598 0.596521i \(-0.203451\pi\)
0.802598 + 0.596521i \(0.203451\pi\)
\(294\) 0 0
\(295\) 1233.24 0.243397
\(296\) 11345.2 2.22780
\(297\) 0 0
\(298\) 6528.79 1.26914
\(299\) 0 0
\(300\) 0 0
\(301\) 391.411 0.0749520
\(302\) −11085.9 −2.11233
\(303\) 0 0
\(304\) 1795.87 0.338817
\(305\) −542.068 −0.101766
\(306\) 0 0
\(307\) −5378.68 −0.999926 −0.499963 0.866047i \(-0.666653\pi\)
−0.499963 + 0.866047i \(0.666653\pi\)
\(308\) −22286.3 −4.12299
\(309\) 0 0
\(310\) 142.045 0.0260246
\(311\) −3727.50 −0.679636 −0.339818 0.940491i \(-0.610366\pi\)
−0.339818 + 0.940491i \(0.610366\pi\)
\(312\) 0 0
\(313\) 5336.69 0.963730 0.481865 0.876245i \(-0.339960\pi\)
0.481865 + 0.876245i \(0.339960\pi\)
\(314\) −6030.10 −1.08375
\(315\) 0 0
\(316\) −11769.5 −2.09521
\(317\) −338.871 −0.0600408 −0.0300204 0.999549i \(-0.509557\pi\)
−0.0300204 + 0.999549i \(0.509557\pi\)
\(318\) 0 0
\(319\) 7599.37 1.33380
\(320\) 1044.19 0.182413
\(321\) 0 0
\(322\) 28512.1 4.93452
\(323\) 1867.56 0.321715
\(324\) 0 0
\(325\) 0 0
\(326\) 13429.5 2.28157
\(327\) 0 0
\(328\) −14107.9 −2.37493
\(329\) 17076.9 2.86164
\(330\) 0 0
\(331\) 2076.44 0.344807 0.172404 0.985026i \(-0.444847\pi\)
0.172404 + 0.985026i \(0.444847\pi\)
\(332\) 15976.7 2.64107
\(333\) 0 0
\(334\) −5860.26 −0.960057
\(335\) −1075.44 −0.175395
\(336\) 0 0
\(337\) 2584.64 0.417787 0.208893 0.977938i \(-0.433014\pi\)
0.208893 + 0.977938i \(0.433014\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −1093.10 −0.174358
\(341\) −792.788 −0.125900
\(342\) 0 0
\(343\) 17475.1 2.75092
\(344\) −388.244 −0.0608510
\(345\) 0 0
\(346\) 7341.02 1.14062
\(347\) 4191.32 0.648420 0.324210 0.945985i \(-0.394902\pi\)
0.324210 + 0.945985i \(0.394902\pi\)
\(348\) 0 0
\(349\) −7431.29 −1.13979 −0.569897 0.821716i \(-0.693017\pi\)
−0.569897 + 0.821716i \(0.693017\pi\)
\(350\) 20333.3 3.10532
\(351\) 0 0
\(352\) −2724.30 −0.412516
\(353\) 12249.5 1.84695 0.923474 0.383661i \(-0.125337\pi\)
0.923474 + 0.383661i \(0.125337\pi\)
\(354\) 0 0
\(355\) −715.065 −0.106906
\(356\) −2423.01 −0.360729
\(357\) 0 0
\(358\) 5131.98 0.757636
\(359\) 5485.46 0.806439 0.403219 0.915103i \(-0.367891\pi\)
0.403219 + 0.915103i \(0.367891\pi\)
\(360\) 0 0
\(361\) −5170.53 −0.753832
\(362\) −17214.5 −2.49938
\(363\) 0 0
\(364\) 0 0
\(365\) −1025.85 −0.147111
\(366\) 0 0
\(367\) −8752.18 −1.24485 −0.622425 0.782680i \(-0.713852\pi\)
−0.622425 + 0.782680i \(0.713852\pi\)
\(368\) −7505.54 −1.06319
\(369\) 0 0
\(370\) −2533.44 −0.355966
\(371\) 7520.30 1.05238
\(372\) 0 0
\(373\) 5987.30 0.831128 0.415564 0.909564i \(-0.363584\pi\)
0.415564 + 0.909564i \(0.363584\pi\)
\(374\) 9328.34 1.28973
\(375\) 0 0
\(376\) −16938.7 −2.32327
\(377\) 0 0
\(378\) 0 0
\(379\) 13603.0 1.84364 0.921819 0.387620i \(-0.126703\pi\)
0.921819 + 0.387620i \(0.126703\pi\)
\(380\) −988.276 −0.133414
\(381\) 0 0
\(382\) −14844.9 −1.98831
\(383\) 6648.90 0.887057 0.443528 0.896260i \(-0.353727\pi\)
0.443528 + 0.896260i \(0.353727\pi\)
\(384\) 0 0
\(385\) 2343.89 0.310274
\(386\) 4315.48 0.569047
\(387\) 0 0
\(388\) 109.406 0.0143151
\(389\) 4923.40 0.641713 0.320856 0.947128i \(-0.396029\pi\)
0.320856 + 0.947128i \(0.396029\pi\)
\(390\) 0 0
\(391\) −7805.15 −1.00952
\(392\) −29080.7 −3.74694
\(393\) 0 0
\(394\) −112.060 −0.0143287
\(395\) 1237.82 0.157674
\(396\) 0 0
\(397\) −576.103 −0.0728307 −0.0364153 0.999337i \(-0.511594\pi\)
−0.0364153 + 0.999337i \(0.511594\pi\)
\(398\) −18805.7 −2.36846
\(399\) 0 0
\(400\) −5352.56 −0.669070
\(401\) −9053.52 −1.12746 −0.563730 0.825959i \(-0.690634\pi\)
−0.563730 + 0.825959i \(0.690634\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −17023.1 −2.09637
\(405\) 0 0
\(406\) 29559.2 3.61330
\(407\) 14139.7 1.72207
\(408\) 0 0
\(409\) 15789.9 1.90895 0.954474 0.298293i \(-0.0964171\pi\)
0.954474 + 0.298293i \(0.0964171\pi\)
\(410\) 3150.34 0.379474
\(411\) 0 0
\(412\) 18207.7 2.17726
\(413\) −26772.8 −3.18984
\(414\) 0 0
\(415\) −1680.29 −0.198753
\(416\) 0 0
\(417\) 0 0
\(418\) 8433.78 0.986866
\(419\) 4091.45 0.477041 0.238521 0.971137i \(-0.423338\pi\)
0.238521 + 0.971137i \(0.423338\pi\)
\(420\) 0 0
\(421\) −5540.81 −0.641431 −0.320715 0.947176i \(-0.603923\pi\)
−0.320715 + 0.947176i \(0.603923\pi\)
\(422\) 922.860 0.106455
\(423\) 0 0
\(424\) −7459.46 −0.854395
\(425\) −5566.22 −0.635297
\(426\) 0 0
\(427\) 11767.9 1.33370
\(428\) 22558.5 2.54768
\(429\) 0 0
\(430\) 86.6965 0.00972297
\(431\) −40.7848 −0.00455808 −0.00227904 0.999997i \(-0.500725\pi\)
−0.00227904 + 0.999997i \(0.500725\pi\)
\(432\) 0 0
\(433\) 766.330 0.0850519 0.0425259 0.999095i \(-0.486459\pi\)
0.0425259 + 0.999095i \(0.486459\pi\)
\(434\) −3083.70 −0.341066
\(435\) 0 0
\(436\) 5677.35 0.623614
\(437\) −7056.65 −0.772461
\(438\) 0 0
\(439\) −6054.63 −0.658250 −0.329125 0.944286i \(-0.606754\pi\)
−0.329125 + 0.944286i \(0.606754\pi\)
\(440\) −2324.93 −0.251901
\(441\) 0 0
\(442\) 0 0
\(443\) 7422.44 0.796051 0.398025 0.917374i \(-0.369696\pi\)
0.398025 + 0.917374i \(0.369696\pi\)
\(444\) 0 0
\(445\) 254.832 0.0271465
\(446\) 6944.65 0.737306
\(447\) 0 0
\(448\) −22668.7 −2.39062
\(449\) 1050.84 0.110451 0.0552253 0.998474i \(-0.482412\pi\)
0.0552253 + 0.998474i \(0.482412\pi\)
\(450\) 0 0
\(451\) −17582.8 −1.83579
\(452\) 12264.1 1.27623
\(453\) 0 0
\(454\) 20712.1 2.14112
\(455\) 0 0
\(456\) 0 0
\(457\) −14462.4 −1.48036 −0.740179 0.672409i \(-0.765259\pi\)
−0.740179 + 0.672409i \(0.765259\pi\)
\(458\) −8963.74 −0.914516
\(459\) 0 0
\(460\) 4130.33 0.418647
\(461\) −1400.23 −0.141465 −0.0707323 0.997495i \(-0.522534\pi\)
−0.0707323 + 0.997495i \(0.522534\pi\)
\(462\) 0 0
\(463\) −11922.3 −1.19671 −0.598353 0.801232i \(-0.704178\pi\)
−0.598353 + 0.801232i \(0.704178\pi\)
\(464\) −7781.20 −0.778519
\(465\) 0 0
\(466\) −17028.4 −1.69276
\(467\) −8048.13 −0.797479 −0.398740 0.917064i \(-0.630552\pi\)
−0.398740 + 0.917064i \(0.630552\pi\)
\(468\) 0 0
\(469\) 23347.0 2.29864
\(470\) 3782.49 0.371219
\(471\) 0 0
\(472\) 26556.3 2.58973
\(473\) −483.874 −0.0470371
\(474\) 0 0
\(475\) −5032.43 −0.486113
\(476\) 23730.5 2.28505
\(477\) 0 0
\(478\) −9823.95 −0.940035
\(479\) −15262.5 −1.45587 −0.727933 0.685648i \(-0.759519\pi\)
−0.727933 + 0.685648i \(0.759519\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1100.36 0.103983
\(483\) 0 0
\(484\) 7423.35 0.697159
\(485\) −11.5064 −0.00107727
\(486\) 0 0
\(487\) −5277.64 −0.491073 −0.245537 0.969387i \(-0.578964\pi\)
−0.245537 + 0.969387i \(0.578964\pi\)
\(488\) −11672.7 −1.08279
\(489\) 0 0
\(490\) 6493.84 0.598698
\(491\) 11058.1 1.01638 0.508192 0.861244i \(-0.330314\pi\)
0.508192 + 0.861244i \(0.330314\pi\)
\(492\) 0 0
\(493\) −8091.80 −0.739222
\(494\) 0 0
\(495\) 0 0
\(496\) 811.757 0.0734858
\(497\) 15523.6 1.40106
\(498\) 0 0
\(499\) 112.405 0.0100840 0.00504201 0.999987i \(-0.498395\pi\)
0.00504201 + 0.999987i \(0.498395\pi\)
\(500\) 5951.89 0.532354
\(501\) 0 0
\(502\) −16392.7 −1.45745
\(503\) 10080.5 0.893570 0.446785 0.894641i \(-0.352569\pi\)
0.446785 + 0.894641i \(0.352569\pi\)
\(504\) 0 0
\(505\) 1790.35 0.157761
\(506\) −35247.5 −3.09673
\(507\) 0 0
\(508\) −8843.44 −0.772370
\(509\) 19683.5 1.71406 0.857031 0.515265i \(-0.172306\pi\)
0.857031 + 0.515265i \(0.172306\pi\)
\(510\) 0 0
\(511\) 22270.5 1.92796
\(512\) 14763.9 1.27437
\(513\) 0 0
\(514\) −19057.7 −1.63540
\(515\) −1914.93 −0.163849
\(516\) 0 0
\(517\) −21111.0 −1.79586
\(518\) 54999.2 4.66511
\(519\) 0 0
\(520\) 0 0
\(521\) 14722.3 1.23800 0.618998 0.785393i \(-0.287539\pi\)
0.618998 + 0.785393i \(0.287539\pi\)
\(522\) 0 0
\(523\) −11311.6 −0.945738 −0.472869 0.881133i \(-0.656782\pi\)
−0.472869 + 0.881133i \(0.656782\pi\)
\(524\) −23007.1 −1.91807
\(525\) 0 0
\(526\) −23514.4 −1.94920
\(527\) 844.160 0.0697764
\(528\) 0 0
\(529\) 17325.1 1.42394
\(530\) 1665.73 0.136518
\(531\) 0 0
\(532\) 21454.8 1.74846
\(533\) 0 0
\(534\) 0 0
\(535\) −2372.51 −0.191725
\(536\) −23158.1 −1.86619
\(537\) 0 0
\(538\) −18976.8 −1.52073
\(539\) −36243.7 −2.89634
\(540\) 0 0
\(541\) 12705.1 1.00967 0.504837 0.863215i \(-0.331553\pi\)
0.504837 + 0.863215i \(0.331553\pi\)
\(542\) −12947.8 −1.02612
\(543\) 0 0
\(544\) 2900.83 0.228625
\(545\) −597.095 −0.0469298
\(546\) 0 0
\(547\) −6624.52 −0.517814 −0.258907 0.965902i \(-0.583362\pi\)
−0.258907 + 0.965902i \(0.583362\pi\)
\(548\) 8816.95 0.687302
\(549\) 0 0
\(550\) −25136.7 −1.94878
\(551\) −7315.82 −0.565634
\(552\) 0 0
\(553\) −26872.1 −2.06640
\(554\) 12688.0 0.973032
\(555\) 0 0
\(556\) −30372.9 −2.31672
\(557\) 14637.9 1.11351 0.556757 0.830676i \(-0.312046\pi\)
0.556757 + 0.830676i \(0.312046\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2399.97 −0.181102
\(561\) 0 0
\(562\) 16085.7 1.20736
\(563\) −18465.4 −1.38228 −0.691139 0.722722i \(-0.742891\pi\)
−0.691139 + 0.722722i \(0.742891\pi\)
\(564\) 0 0
\(565\) −1289.84 −0.0960423
\(566\) −13462.6 −0.999778
\(567\) 0 0
\(568\) −15398.0 −1.13747
\(569\) 195.327 0.0143911 0.00719557 0.999974i \(-0.497710\pi\)
0.00719557 + 0.999974i \(0.497710\pi\)
\(570\) 0 0
\(571\) −12479.1 −0.914592 −0.457296 0.889314i \(-0.651182\pi\)
−0.457296 + 0.889314i \(0.651182\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −68391.8 −4.97320
\(575\) 21032.2 1.52540
\(576\) 0 0
\(577\) 5511.41 0.397648 0.198824 0.980035i \(-0.436288\pi\)
0.198824 + 0.980035i \(0.436288\pi\)
\(578\) 13691.7 0.985290
\(579\) 0 0
\(580\) 4282.02 0.306554
\(581\) 36478.0 2.60476
\(582\) 0 0
\(583\) −9296.83 −0.660438
\(584\) −22090.4 −1.56525
\(585\) 0 0
\(586\) −38711.9 −2.72897
\(587\) 11511.5 0.809420 0.404710 0.914445i \(-0.367372\pi\)
0.404710 + 0.914445i \(0.367372\pi\)
\(588\) 0 0
\(589\) 763.207 0.0533912
\(590\) −5930.12 −0.413795
\(591\) 0 0
\(592\) −14478.1 −1.00514
\(593\) −22211.3 −1.53812 −0.769062 0.639174i \(-0.779276\pi\)
−0.769062 + 0.639174i \(0.779276\pi\)
\(594\) 0 0
\(595\) −2495.77 −0.171961
\(596\) −20532.2 −1.41112
\(597\) 0 0
\(598\) 0 0
\(599\) −18477.3 −1.26037 −0.630184 0.776446i \(-0.717021\pi\)
−0.630184 + 0.776446i \(0.717021\pi\)
\(600\) 0 0
\(601\) 1530.40 0.103871 0.0519355 0.998650i \(-0.483461\pi\)
0.0519355 + 0.998650i \(0.483461\pi\)
\(602\) −1882.12 −0.127425
\(603\) 0 0
\(604\) 34863.7 2.34865
\(605\) −780.725 −0.0524644
\(606\) 0 0
\(607\) −18934.8 −1.26613 −0.633063 0.774101i \(-0.718202\pi\)
−0.633063 + 0.774101i \(0.718202\pi\)
\(608\) 2622.65 0.174938
\(609\) 0 0
\(610\) 2606.57 0.173011
\(611\) 0 0
\(612\) 0 0
\(613\) 3888.26 0.256191 0.128096 0.991762i \(-0.459114\pi\)
0.128096 + 0.991762i \(0.459114\pi\)
\(614\) 25863.7 1.69996
\(615\) 0 0
\(616\) 50472.6 3.30130
\(617\) 13927.8 0.908769 0.454385 0.890806i \(-0.349859\pi\)
0.454385 + 0.890806i \(0.349859\pi\)
\(618\) 0 0
\(619\) 29627.9 1.92382 0.961910 0.273368i \(-0.0881376\pi\)
0.961910 + 0.273368i \(0.0881376\pi\)
\(620\) −446.712 −0.0289361
\(621\) 0 0
\(622\) 17923.9 1.15544
\(623\) −5532.23 −0.355769
\(624\) 0 0
\(625\) 14682.8 0.939702
\(626\) −25661.8 −1.63842
\(627\) 0 0
\(628\) 18963.8 1.20500
\(629\) −15056.0 −0.954405
\(630\) 0 0
\(631\) 6066.13 0.382708 0.191354 0.981521i \(-0.438712\pi\)
0.191354 + 0.981521i \(0.438712\pi\)
\(632\) 26654.8 1.67764
\(633\) 0 0
\(634\) 1629.48 0.102074
\(635\) 930.077 0.0581244
\(636\) 0 0
\(637\) 0 0
\(638\) −36542.0 −2.26758
\(639\) 0 0
\(640\) −4208.99 −0.259961
\(641\) −16798.4 −1.03510 −0.517548 0.855654i \(-0.673155\pi\)
−0.517548 + 0.855654i \(0.673155\pi\)
\(642\) 0 0
\(643\) −20373.2 −1.24952 −0.624758 0.780818i \(-0.714802\pi\)
−0.624758 + 0.780818i \(0.714802\pi\)
\(644\) −89666.6 −5.48658
\(645\) 0 0
\(646\) −8980.28 −0.546942
\(647\) −23098.6 −1.40356 −0.701778 0.712396i \(-0.747610\pi\)
−0.701778 + 0.712396i \(0.747610\pi\)
\(648\) 0 0
\(649\) 33097.4 2.00183
\(650\) 0 0
\(651\) 0 0
\(652\) −42234.0 −2.53683
\(653\) −1795.81 −0.107619 −0.0538097 0.998551i \(-0.517136\pi\)
−0.0538097 + 0.998551i \(0.517136\pi\)
\(654\) 0 0
\(655\) 2419.69 0.144344
\(656\) 18003.5 1.07152
\(657\) 0 0
\(658\) −82115.2 −4.86502
\(659\) 14640.3 0.865412 0.432706 0.901535i \(-0.357559\pi\)
0.432706 + 0.901535i \(0.357559\pi\)
\(660\) 0 0
\(661\) −20490.3 −1.20572 −0.602860 0.797847i \(-0.705972\pi\)
−0.602860 + 0.797847i \(0.705972\pi\)
\(662\) −9984.67 −0.586201
\(663\) 0 0
\(664\) −36182.9 −2.11471
\(665\) −2256.43 −0.131580
\(666\) 0 0
\(667\) 30575.2 1.77493
\(668\) 18429.7 1.06746
\(669\) 0 0
\(670\) 5171.30 0.298186
\(671\) −14547.9 −0.836981
\(672\) 0 0
\(673\) 21436.5 1.22781 0.613904 0.789380i \(-0.289598\pi\)
0.613904 + 0.789380i \(0.289598\pi\)
\(674\) −12428.4 −0.710273
\(675\) 0 0
\(676\) 0 0
\(677\) −20158.0 −1.14436 −0.572181 0.820127i \(-0.693903\pi\)
−0.572181 + 0.820127i \(0.693903\pi\)
\(678\) 0 0
\(679\) 249.796 0.0141182
\(680\) 2475.58 0.139609
\(681\) 0 0
\(682\) 3812.17 0.214040
\(683\) −4273.98 −0.239443 −0.119721 0.992808i \(-0.538200\pi\)
−0.119721 + 0.992808i \(0.538200\pi\)
\(684\) 0 0
\(685\) −927.291 −0.0517226
\(686\) −84030.0 −4.67679
\(687\) 0 0
\(688\) 495.452 0.0274548
\(689\) 0 0
\(690\) 0 0
\(691\) 10287.2 0.566346 0.283173 0.959069i \(-0.408613\pi\)
0.283173 + 0.959069i \(0.408613\pi\)
\(692\) −23086.5 −1.26823
\(693\) 0 0
\(694\) −20154.2 −1.10237
\(695\) 3194.36 0.174344
\(696\) 0 0
\(697\) 18722.2 1.01744
\(698\) 35733.8 1.93774
\(699\) 0 0
\(700\) −63945.4 −3.45273
\(701\) −11726.5 −0.631815 −0.315907 0.948790i \(-0.602309\pi\)
−0.315907 + 0.948790i \(0.602309\pi\)
\(702\) 0 0
\(703\) −13612.2 −0.730287
\(704\) 28023.8 1.50027
\(705\) 0 0
\(706\) −58902.3 −3.13997
\(707\) −38867.2 −2.06754
\(708\) 0 0
\(709\) −11833.0 −0.626795 −0.313398 0.949622i \(-0.601467\pi\)
−0.313398 + 0.949622i \(0.601467\pi\)
\(710\) 3438.43 0.181749
\(711\) 0 0
\(712\) 5487.48 0.288837
\(713\) −3189.69 −0.167538
\(714\) 0 0
\(715\) 0 0
\(716\) −16139.4 −0.842398
\(717\) 0 0
\(718\) −26377.2 −1.37101
\(719\) 5964.80 0.309387 0.154694 0.987962i \(-0.450561\pi\)
0.154694 + 0.987962i \(0.450561\pi\)
\(720\) 0 0
\(721\) 41571.9 2.14732
\(722\) 24862.8 1.28158
\(723\) 0 0
\(724\) 54137.4 2.77901
\(725\) 21804.6 1.11697
\(726\) 0 0
\(727\) −25972.8 −1.32500 −0.662501 0.749061i \(-0.730505\pi\)
−0.662501 + 0.749061i \(0.730505\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4932.86 0.250101
\(731\) 515.229 0.0260690
\(732\) 0 0
\(733\) 33025.9 1.66417 0.832087 0.554646i \(-0.187146\pi\)
0.832087 + 0.554646i \(0.187146\pi\)
\(734\) 42085.4 2.11635
\(735\) 0 0
\(736\) −10960.9 −0.548946
\(737\) −28862.3 −1.44255
\(738\) 0 0
\(739\) 7564.70 0.376552 0.188276 0.982116i \(-0.439710\pi\)
0.188276 + 0.982116i \(0.439710\pi\)
\(740\) 7967.32 0.395790
\(741\) 0 0
\(742\) −36161.8 −1.78914
\(743\) 12623.9 0.623319 0.311660 0.950194i \(-0.399115\pi\)
0.311660 + 0.950194i \(0.399115\pi\)
\(744\) 0 0
\(745\) 2159.40 0.106194
\(746\) −28790.3 −1.41299
\(747\) 0 0
\(748\) −29336.4 −1.43402
\(749\) 51505.6 2.51265
\(750\) 0 0
\(751\) 15054.7 0.731497 0.365749 0.930714i \(-0.380813\pi\)
0.365749 + 0.930714i \(0.380813\pi\)
\(752\) 21616.1 1.04822
\(753\) 0 0
\(754\) 0 0
\(755\) −3666.67 −0.176747
\(756\) 0 0
\(757\) 17024.5 0.817391 0.408695 0.912671i \(-0.365984\pi\)
0.408695 + 0.912671i \(0.365984\pi\)
\(758\) −65410.9 −3.13434
\(759\) 0 0
\(760\) 2238.18 0.106825
\(761\) −1037.46 −0.0494191 −0.0247095 0.999695i \(-0.507866\pi\)
−0.0247095 + 0.999695i \(0.507866\pi\)
\(762\) 0 0
\(763\) 12962.5 0.615040
\(764\) 46685.3 2.21075
\(765\) 0 0
\(766\) −31971.6 −1.50807
\(767\) 0 0
\(768\) 0 0
\(769\) 33909.7 1.59014 0.795068 0.606520i \(-0.207435\pi\)
0.795068 + 0.606520i \(0.207435\pi\)
\(770\) −11270.7 −0.527492
\(771\) 0 0
\(772\) −13571.6 −0.632710
\(773\) −13566.7 −0.631253 −0.315626 0.948884i \(-0.602215\pi\)
−0.315626 + 0.948884i \(0.602215\pi\)
\(774\) 0 0
\(775\) −2274.72 −0.105433
\(776\) −247.775 −0.0114621
\(777\) 0 0
\(778\) −23674.5 −1.09097
\(779\) 16926.8 0.778517
\(780\) 0 0
\(781\) −19190.7 −0.879255
\(782\) 37531.5 1.71627
\(783\) 0 0
\(784\) 37110.9 1.69055
\(785\) −1994.46 −0.0906817
\(786\) 0 0
\(787\) 12545.8 0.568246 0.284123 0.958788i \(-0.408298\pi\)
0.284123 + 0.958788i \(0.408298\pi\)
\(788\) 352.414 0.0159318
\(789\) 0 0
\(790\) −5952.11 −0.268059
\(791\) 28001.5 1.25868
\(792\) 0 0
\(793\) 0 0
\(794\) 2770.23 0.123818
\(795\) 0 0
\(796\) 59141.5 2.63344
\(797\) 26658.5 1.18481 0.592403 0.805642i \(-0.298179\pi\)
0.592403 + 0.805642i \(0.298179\pi\)
\(798\) 0 0
\(799\) 22478.9 0.995304
\(800\) −7816.74 −0.345454
\(801\) 0 0
\(802\) 43534.4 1.91677
\(803\) −27531.5 −1.20992
\(804\) 0 0
\(805\) 9430.36 0.412890
\(806\) 0 0
\(807\) 0 0
\(808\) 38552.8 1.67857
\(809\) −3448.15 −0.149852 −0.0749260 0.997189i \(-0.523872\pi\)
−0.0749260 + 0.997189i \(0.523872\pi\)
\(810\) 0 0
\(811\) 33920.4 1.46869 0.734344 0.678778i \(-0.237490\pi\)
0.734344 + 0.678778i \(0.237490\pi\)
\(812\) −92959.7 −4.01754
\(813\) 0 0
\(814\) −67991.8 −2.92766
\(815\) 4441.81 0.190908
\(816\) 0 0
\(817\) 465.820 0.0199473
\(818\) −75926.7 −3.24537
\(819\) 0 0
\(820\) −9907.40 −0.421928
\(821\) 5461.33 0.232158 0.116079 0.993240i \(-0.462967\pi\)
0.116079 + 0.993240i \(0.462967\pi\)
\(822\) 0 0
\(823\) 21155.0 0.896010 0.448005 0.894031i \(-0.352135\pi\)
0.448005 + 0.894031i \(0.352135\pi\)
\(824\) −41235.6 −1.74334
\(825\) 0 0
\(826\) 128739. 5.42300
\(827\) −18974.8 −0.797846 −0.398923 0.916984i \(-0.630616\pi\)
−0.398923 + 0.916984i \(0.630616\pi\)
\(828\) 0 0
\(829\) 9497.90 0.397920 0.198960 0.980008i \(-0.436244\pi\)
0.198960 + 0.980008i \(0.436244\pi\)
\(830\) 8079.79 0.337896
\(831\) 0 0
\(832\) 0 0
\(833\) 38592.3 1.60521
\(834\) 0 0
\(835\) −1938.28 −0.0803316
\(836\) −26523.1 −1.09727
\(837\) 0 0
\(838\) −19674.0 −0.811010
\(839\) 8225.56 0.338472 0.169236 0.985576i \(-0.445870\pi\)
0.169236 + 0.985576i \(0.445870\pi\)
\(840\) 0 0
\(841\) 7309.11 0.299689
\(842\) 26643.3 1.09049
\(843\) 0 0
\(844\) −2902.27 −0.118365
\(845\) 0 0
\(846\) 0 0
\(847\) 16949.0 0.687573
\(848\) 9519.27 0.385487
\(849\) 0 0
\(850\) 26765.5 1.08006
\(851\) 56889.6 2.29160
\(852\) 0 0
\(853\) 19140.4 0.768294 0.384147 0.923272i \(-0.374496\pi\)
0.384147 + 0.923272i \(0.374496\pi\)
\(854\) −56586.8 −2.26740
\(855\) 0 0
\(856\) −51089.0 −2.03994
\(857\) −20602.9 −0.821215 −0.410608 0.911812i \(-0.634683\pi\)
−0.410608 + 0.911812i \(0.634683\pi\)
\(858\) 0 0
\(859\) 5247.92 0.208448 0.104224 0.994554i \(-0.466764\pi\)
0.104224 + 0.994554i \(0.466764\pi\)
\(860\) −272.649 −0.0108107
\(861\) 0 0
\(862\) 196.116 0.00774912
\(863\) 38622.5 1.52343 0.761717 0.647909i \(-0.224356\pi\)
0.761717 + 0.647909i \(0.224356\pi\)
\(864\) 0 0
\(865\) 2428.04 0.0954402
\(866\) −3684.95 −0.144595
\(867\) 0 0
\(868\) 9697.82 0.379223
\(869\) 33220.2 1.29680
\(870\) 0 0
\(871\) 0 0
\(872\) −12857.7 −0.499330
\(873\) 0 0
\(874\) 33932.4 1.31325
\(875\) 13589.4 0.525034
\(876\) 0 0
\(877\) 19963.9 0.768682 0.384341 0.923191i \(-0.374429\pi\)
0.384341 + 0.923191i \(0.374429\pi\)
\(878\) 29114.1 1.11908
\(879\) 0 0
\(880\) 2966.92 0.113653
\(881\) 13175.8 0.503865 0.251933 0.967745i \(-0.418934\pi\)
0.251933 + 0.967745i \(0.418934\pi\)
\(882\) 0 0
\(883\) 14434.0 0.550105 0.275052 0.961429i \(-0.411305\pi\)
0.275052 + 0.961429i \(0.411305\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −35691.2 −1.35335
\(887\) −20962.3 −0.793514 −0.396757 0.917924i \(-0.629864\pi\)
−0.396757 + 0.917924i \(0.629864\pi\)
\(888\) 0 0
\(889\) −20191.3 −0.761750
\(890\) −1225.38 −0.0461513
\(891\) 0 0
\(892\) −21840.0 −0.819794
\(893\) 20323.3 0.761582
\(894\) 0 0
\(895\) 1697.40 0.0633943
\(896\) 91374.3 3.40692
\(897\) 0 0
\(898\) −5053.04 −0.187775
\(899\) −3306.84 −0.122680
\(900\) 0 0
\(901\) 9899.25 0.366029
\(902\) 84548.1 3.12100
\(903\) 0 0
\(904\) −27775.0 −1.02188
\(905\) −5693.71 −0.209133
\(906\) 0 0
\(907\) −14079.6 −0.515441 −0.257721 0.966219i \(-0.582971\pi\)
−0.257721 + 0.966219i \(0.582971\pi\)
\(908\) −65136.8 −2.38066
\(909\) 0 0
\(910\) 0 0
\(911\) −622.987 −0.0226569 −0.0113285 0.999936i \(-0.503606\pi\)
−0.0113285 + 0.999936i \(0.503606\pi\)
\(912\) 0 0
\(913\) −45095.3 −1.63465
\(914\) 69543.5 2.51673
\(915\) 0 0
\(916\) 28189.7 1.01683
\(917\) −52529.8 −1.89170
\(918\) 0 0
\(919\) −30293.2 −1.08735 −0.543677 0.839294i \(-0.682968\pi\)
−0.543677 + 0.839294i \(0.682968\pi\)
\(920\) −9354.08 −0.335212
\(921\) 0 0
\(922\) 6733.09 0.240502
\(923\) 0 0
\(924\) 0 0
\(925\) 40570.7 1.44211
\(926\) 57329.0 2.03450
\(927\) 0 0
\(928\) −11363.5 −0.401965
\(929\) −25045.6 −0.884520 −0.442260 0.896887i \(-0.645823\pi\)
−0.442260 + 0.896887i \(0.645823\pi\)
\(930\) 0 0
\(931\) 34891.4 1.22827
\(932\) 53552.0 1.88214
\(933\) 0 0
\(934\) 38699.9 1.35578
\(935\) 3085.35 0.107916
\(936\) 0 0
\(937\) 4025.83 0.140361 0.0701805 0.997534i \(-0.477642\pi\)
0.0701805 + 0.997534i \(0.477642\pi\)
\(938\) −112265. −3.90788
\(939\) 0 0
\(940\) −11895.4 −0.412750
\(941\) 50575.8 1.75210 0.876049 0.482222i \(-0.160170\pi\)
0.876049 + 0.482222i \(0.160170\pi\)
\(942\) 0 0
\(943\) −70742.5 −2.44294
\(944\) −33889.3 −1.16844
\(945\) 0 0
\(946\) 2326.74 0.0799671
\(947\) −51825.9 −1.77837 −0.889185 0.457547i \(-0.848728\pi\)
−0.889185 + 0.457547i \(0.848728\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 24198.8 0.826433
\(951\) 0 0
\(952\) −53743.1 −1.82965
\(953\) 40037.1 1.36089 0.680446 0.732798i \(-0.261786\pi\)
0.680446 + 0.732798i \(0.261786\pi\)
\(954\) 0 0
\(955\) −4909.96 −0.166369
\(956\) 30895.0 1.04520
\(957\) 0 0
\(958\) 73390.5 2.47509
\(959\) 20130.9 0.677851
\(960\) 0 0
\(961\) −29446.0 −0.988420
\(962\) 0 0
\(963\) 0 0
\(964\) −3460.47 −0.115617
\(965\) 1427.34 0.0476143
\(966\) 0 0
\(967\) 5557.42 0.184813 0.0924067 0.995721i \(-0.470544\pi\)
0.0924067 + 0.995721i \(0.470544\pi\)
\(968\) −16811.9 −0.558218
\(969\) 0 0
\(970\) 55.3292 0.00183146
\(971\) −25730.6 −0.850396 −0.425198 0.905100i \(-0.639796\pi\)
−0.425198 + 0.905100i \(0.639796\pi\)
\(972\) 0 0
\(973\) −69347.4 −2.28487
\(974\) 25377.9 0.834865
\(975\) 0 0
\(976\) 14896.0 0.488533
\(977\) −39868.8 −1.30554 −0.652772 0.757554i \(-0.726394\pi\)
−0.652772 + 0.757554i \(0.726394\pi\)
\(978\) 0 0
\(979\) 6839.12 0.223268
\(980\) −20422.2 −0.665678
\(981\) 0 0
\(982\) −53173.5 −1.72794
\(983\) 27728.9 0.899709 0.449854 0.893102i \(-0.351476\pi\)
0.449854 + 0.893102i \(0.351476\pi\)
\(984\) 0 0
\(985\) −37.0639 −0.00119894
\(986\) 38909.9 1.25674
\(987\) 0 0
\(988\) 0 0
\(989\) −1946.81 −0.0625936
\(990\) 0 0
\(991\) 38088.3 1.22090 0.610452 0.792053i \(-0.290988\pi\)
0.610452 + 0.792053i \(0.290988\pi\)
\(992\) 1185.47 0.0379422
\(993\) 0 0
\(994\) −74646.0 −2.38192
\(995\) −6220.00 −0.198178
\(996\) 0 0
\(997\) −18606.0 −0.591033 −0.295516 0.955338i \(-0.595492\pi\)
−0.295516 + 0.955338i \(0.595492\pi\)
\(998\) −540.505 −0.0171437
\(999\) 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 1521.4.a.bn.1.2 yes 18
3.2 odd 2 inner 1521.4.a.bn.1.17 yes 18
13.12 even 2 1521.4.a.bm.1.17 yes 18
39.38 odd 2 1521.4.a.bm.1.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.4.a.bm.1.2 18 39.38 odd 2
1521.4.a.bm.1.17 yes 18 13.12 even 2
1521.4.a.bn.1.2 yes 18 1.1 even 1 trivial
1521.4.a.bn.1.17 yes 18 3.2 odd 2 inner